{"unit":"calls to Claude","notProved":"Not proved. Every division Claude proposed ({calls} calls) was tried, and the automated provers could not complete any of them. Select the goal, or the red badge, to read each attempt and its subgoals.","notes":{"apply NNPP; intro":"Proof by contradiction: assume the statement is false and show that this assumption is impossible."},"who":{"model":["Claude","Proposed by Claude (claude-sonnet-5-5, with no further training), then checked by Rocq."],"auto":["Automation","Rocq's standard automation (firstorder, intuition, tauto), tried on every goal before Claude is asked."],"hammer":["CoqHammer","CoqHammer: the external provers Vampire and E found a proof, rebuilt as a Rocq tactic."]},"runLabels":{"claude1":"Claude · run 1 of 3","claude2":"Claude · run 2 of 3","claude3":"Claude states subgoals only · run 3 of 3"},"stages":["The search starts from the statement to be proved, exactly as posed to Rocq.","If the automation fails, Claude proposes up to three complete proofs, or up to two divisions of the goal into two to four subgoals.","Rocq checks every proposal. A division is accepted only if the goal follows from its subgoals; each subgoal then becomes a new goal.","Rocq's automation tries every goal, and every subgoal, before Claude is asked.","When every subgoal is proved, the lemmas are assembled into one file and compiled by Rocq from scratch."],"tasks":[{"key":"claude1/pr001_relational_exhaustive","id":"pr001_relational_exhaustive","run":"claude1","title":"Dalīl al-kalima, premise 1","english":"The four possibilities are the only ones.","stmt":"forall x, (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b) \\/ ((forall y, ~ DependsOn x y) \\/ exists y, clos_trans Thing DependsOn x y /\\ forall z, ~ DependsOn y z)","proved":true,"seconds":13.7,"expansions":1,"calls":35,"root":0,"goals":["forall x, (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b) \\/ ((forall y, ~ DependsOn x y) \\/ exists y, clos_trans Thing DependsOn x y /\\ forall z, ~ DependsOn y z)","x : Thing\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\nH : exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\n============================\n(forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\n============================\nexists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nH1 : exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\n============================\n(forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\n============================\n~ ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\n============================\nFalse","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\n============================\nexists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\n============================\n(x = x \\/ clos_trans Thing DependsOn x x) /\\ (forall a : Thing, a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\n============================\nx = x \\/ clos_trans Thing DependsOn x x","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\n============================\nforall a : Thing, a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\n============================\nexists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\n============================\n~ ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\nFalse","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\nexists a0 : Thing, (a0 = x \\/ clos_trans Thing DependsOn x a0) /\\ (forall b : Thing, ~ DependsOn a0 b)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\n(a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\nforall b : Thing, ~ DependsOn a b","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHb : DependsOn a b\n============================\nFalse","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHb : DependsOn a b\n============================\nexists b0 : Thing, (b0 = x \\/ clos_trans Thing DependsOn x b0) /\\ DependsOn a b0","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHb : DependsOn a b\n============================\n(b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHb : DependsOn a b\n============================\nb = x \\/ clos_trans Thing DependsOn x b","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x \\/ clos_trans Thing DependsOn x a\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHb : DependsOn a b\n============================\nclos_trans Thing DependsOn x b","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : a = x\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHb : DependsOn a b\n============================\nclos_trans Thing DependsOn x b","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\na : Thing\nHa : clos_trans Thing DependsOn x a\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHb : DependsOn a b\n============================\nclos_trans Thing DependsOn x b","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nN : ~ exists a : Thing, (a = x \\/ clos_trans Thing DependsOn x a) /\\ (forall b : Thing, ~ DependsOn a b)\nN2 : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn x b\nb : Thing\nHb : DependsOn x b\n============================\nclos_trans Thing DependsOn x b","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\na : Thing\nHa : a = x\nHb : forall b : Thing, ~ DependsOn a b\n============================\n(forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\na : Thing\nHa : clos_trans Thing DependsOn x a\nHb : forall b : Thing, ~ DependsOn a b\n============================\n(forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\nH : ~ exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)\nHb : forall b : Thing, ~ DependsOn x b\n============================\n(forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intro x.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intro x.\ndestruct (classic (exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b)) as [H|H].\n- right; right; left; exact H.\n- right; right; right.\n  assert (H1: exists a, (a = x \\/ clos_trans Thing DependsOn x a) /\\ forall b, ~ DependsOn a b).\n  { apply NNPP. intro N. apply H.\n    exists (fun a => a = x \\/ clos_trans Thing DependsOn x a).\n    split.\n    - left; reflexivity.\n    - intros a Ha. apply NNPP. intro N2. apply N. exists a. split; [exact Ha|].\n      intros b Hb. apply N2. exists b. split; [|exact Hb].\n      cbn. right.\n      destruct Ha as [Ha|Ha].\n      + subst. apply t_step; exact Hb.\n      + eapply t_trans; [exact Ha| apply t_step; exact Hb]. }\n  destruct H1 as [a [[Ha|Ha] Hb]].\n  + subst. left; exact Hb.\n  + right; exists a; split; assumption."},{"g":1,"t":"destruct (classic (exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b)) as [H|H].","w":"model","kind":"step","ok":true,"c":[2,3],"d":"Tactic 2 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"right; right; left; exact H.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 3 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"right; right; right.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 4 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"assert (H1: exists a, (a = x \\/ clos_trans Thing DependsOn x a) /\\ forall b, ~ DependsOn a b).","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 5 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"apply NNPP.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 6 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"intro N.","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 7 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"apply H.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 8 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"exists (fun a => a = x \\/ clos_trans Thing DependsOn x a).","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 9 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"split.","w":"model","kind":"step","ok":true,"c":[11,12],"d":"Tactic 10 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"left; reflexivity.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 11 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"intros a Ha.","w":"model","kind":"step","ok":true,"c":[13],"d":"Tactic 12 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"apply NNPP.","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 13 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"intro N2.","w":"model","kind":"step","ok":true,"c":[15],"d":"Tactic 14 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"apply N.","w":"model","kind":"step","ok":true,"c":[16],"d":"Tactic 15 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"exists a.","w":"model","kind":"step","ok":true,"c":[17],"d":"Tactic 16 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"split; [exact Ha|].","w":"model","kind":"step","ok":true,"c":[18],"d":"Tactic 17 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"intros b Hb.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 18 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"apply N2.","w":"model","kind":"step","ok":true,"c":[20],"d":"Tactic 19 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"exists b.","w":"model","kind":"step","ok":true,"c":[21],"d":"Tactic 20 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":21,"t":"split; [|exact Hb].","w":"model","kind":"step","ok":true,"c":[22],"d":"Tactic 21 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"cbn.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 22 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"right.","w":"model","kind":"step","ok":true,"c":[23],"d":"Tactic 23 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"destruct Ha as [Ha|Ha].","w":"model","kind":"step","ok":true,"c":[24,25],"d":"Tactic 24 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"subst.","w":"model","kind":"step","ok":true,"c":[26],"d":"Tactic 25 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":26,"t":"apply t_step; exact Hb.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 26 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"eapply t_trans; [exact Ha| apply t_step; exact Hb].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 27 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"destruct H1 as [a [[Ha|Ha] Hb]].","w":"model","kind":"step","ok":true,"c":[27,28],"d":"Tactic 28 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"subst.","w":"model","kind":"step","ok":true,"c":[29],"d":"Tactic 29 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":29,"t":"left; exact Hb.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 30 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":28,"t":"right; exists a; split; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 31 of 31 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,31,32,33,34],"proofFile":"files/claude/run1/pr001_relational_exhaustive.v"},{"key":"claude1/pr002_no_self_creation","id":"pr002_no_self_creation","run":"claude1","title":"Dalīl al-kalima, premise 2","english":"Nothing gives existence to itself.","stmt":"MP001 -> MP002 -> forall x, ~ Gives x x","proved":true,"seconds":0.4,"expansions":0,"calls":1,"root":0,"goals":["MP001 -> MP002 -> forall x, ~ Gives x x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run1/pr002_no_self_creation.v"},{"key":"claude1/pr003_no_circle","id":"pr003_no_circle","run":"claude1","title":"Dalīl al-kalima, premise 3","english":"No chain of existence-giving returns to where it started.","stmt":"MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing DependsOn x x","proved":true,"seconds":96.9,"expansions":10,"calls":61,"root":0,"goals":["MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing DependsOn x x","MP002 -> MP003 -> forall x y, clos_trans Thing DependsOn x y -> Through x y","forall x, clos_trans Thing DependsOn x x -> exists y, Gives y x /\\ (y = x \\/ clos_trans Thing DependsOn y x)","forall x y, clos_trans Thing DependsOn x y -> clos_trans_1n Thing DependsOn x y","forall x y, clos_trans_1n Thing DependsOn x y -> exists z, Gives z x /\\ (z = y \\/ clos_trans Thing DependsOn z y)","forall x y, clos_trans_1n Thing DependsOn x y -> Gives y x \\/ (exists z, Gives z x /\\ clos_trans_1n Thing DependsOn z y)","forall x y, clos_trans_1n Thing DependsOn x y -> clos_trans Thing DependsOn x y","A2 : MP002\nA3 : MP003\nx : Thing\ny : Thing\nH : clos_trans Thing DependsOn x y\n============================\nThrough x y","A2 : MP002\nA3 : MP003\nx : Thing\ny : Thing\nH : DependsOn x y\n============================\nThrough x y","A2 : MP002\nA3 : MP003\nx : Thing\ny : Thing\nz : Thing\nH1 : clos_trans Thing DependsOn x y\nH1' : clos_trans Thing DependsOn y z\nIH1 : Through x y\nIH2 : Through y z\n============================\nThrough x z","A2 : forall x y : Thing, Gives y x -> Through x y\nA3 : MP003\nx : Thing\ny : Thing\nH : DependsOn x y\n============================\nThrough x y","A2 : forall x y : Thing, Gives y x -> Through x y\nA3 : MP003\nx : Thing\ny : Thing\nH : DependsOn x y\n============================\nGives y x","A2 : MP002\nA3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nx : Thing\ny : Thing\nz : Thing\nH1 : clos_trans Thing DependsOn x y\nH1' : clos_trans Thing DependsOn y z\nIH1 : Through x y\nIH2 : Through y z\n============================\nThrough x z","x : Thing\ny : Thing\nH : clos_trans Thing DependsOn x y\n============================\nclos_trans_1n Thing DependsOn x y","x : Thing\ny : Thing\nH : clos_trans Thing DependsOn x y\n============================\nclos_trans Thing DependsOn x y","x : Thing\ny : Thing\nH : clos_trans_1n Thing DependsOn x y\n============================\nGives y x \\/ exists z : Thing, Gives z x /\\ clos_trans_1n Thing DependsOn z y","x : Thing\ny0 : Thing\nH : DependsOn x y0\n============================\nGives y0 x \\/ exists z : Thing, Gives z x /\\ clos_trans_1n Thing DependsOn z y0","x : Thing\ny0 : Thing\nz : Thing\nH : DependsOn x y0\nH0 : clos_trans_1n Thing DependsOn y0 z\n============================\nGives z x \\/ exists z0 : Thing, Gives z0 x /\\ clos_trans_1n Thing DependsOn z0 z","x : Thing\ny0 : Thing\nH : DependsOn x y0\n============================\nGives y0 x","x : Thing\ny0 : Thing\nz : Thing\nH : DependsOn x y0\nH0 : clos_trans_1n Thing DependsOn y0 z\n============================\nexists z0 : Thing, Gives z0 x /\\ clos_trans_1n Thing DependsOn z0 z","x : Thing\ny0 : Thing\nz : Thing\nH : DependsOn x y0\nH0 : clos_trans_1n Thing DependsOn y0 z\n============================\nGives y0 x /\\ clos_trans_1n Thing DependsOn y0 z","x : Thing\ny : Thing\nH : clos_trans_1n Thing DependsOn x y\n============================\nclos_trans Thing DependsOn x y","x : Thing\ny : Thing\nH : DependsOn x y\n============================\nclos_trans Thing DependsOn x y","x : Thing\ny : Thing\nz : Thing\nH1 : DependsOn x y\nH2 : clos_trans_1n Thing DependsOn y z\nIH : clos_trans Thing DependsOn y z\n============================\nclos_trans Thing DependsOn x z","x : Thing\ny : Thing\nH : DependsOn x y\n============================\nDependsOn x y","x : Thing\ny : Thing\nz : Thing\nH1 : DependsOn x y\nH2 : clos_trans_1n Thing DependsOn y z\nIH : clos_trans Thing DependsOn y z\n============================\nclos_trans Thing DependsOn x ?y","x : Thing\ny : Thing\nz : Thing\nH1 : DependsOn x y\nH2 : clos_trans_1n Thing DependsOn y z\nIH : clos_trans Thing DependsOn y z\n============================\nDependsOn x ?y","x : Thing\ny : Thing\nz : Thing\nH1 : DependsOn x y\nH2 : clos_trans_1n Thing DependsOn y z\nIH : clos_trans Thing DependsOn y z\n============================\nclos_trans Thing DependsOn y z"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 x H.\nassert (T: forall a b, clos_trans Thing DependsOn a b -> Through a b).\n{ intros a b Hab. induction Hab as [a b Hab | a b c _ IH1 _ IH2].\n  - apply H2. exact Hab.\n  - eapply H3; eauto. }\napply clos_trans_t1n in H.\ninversion H as [x0 y Hxy | x0 y z Hxy Hyz].\n- exact (H1 _ _ Hxy (H2 _ _ Hxy)).\n- apply clos_t1n_trans in Hyz.\n  exact (H1 x y Hxy (T _ _ Hyz)).","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: In environment Thing : Type"},{"g":0,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y, clos_trans Thing DependsOn x y -> Through x y","forall x, clos_trans Thing DependsOn x x -> exists y, Gives y x /\\ (y = x \\/ clos_trans Thing DependsOn y x)"],"combine":"intros H1 H2 H3 x H. destruct (sub2 x H) as [y [Hg [E|Hc]]]. - subst. exact (H1 _ _ Hg (H2 _ _ Hg)). - exact (H1 x y Hg (sub1 H2 H3 y x Hc)).","ok":true,"c":[1,2],"assembled":true},{"g":1,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"intros A2 A3 x y H.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 1 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros A2 A3 x y H. induction H as [x y H | x y z H1 IH1 H1' IH2].\n- unfold MP002 in A2. apply A2. exact H.\n- unfold MP003 in A3. exact (A3 x y z IH1 IH2)."},{"g":7,"t":"induction H as [x y H | x y z H1 IH1 H1' IH2].","w":"model","kind":"step","ok":true,"c":[8,9],"d":"Tactic 2 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"unfold MP002 in A2.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 3 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"apply A2.","w":"model","kind":"step","ok":true,"c":[11],"d":"Tactic 4 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"exact H.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"unfold MP003 in A3.","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 6 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"exact (A3 x y z IH1 IH2).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"intros x H. apply clos_trans_t1n_iff in H. inversion H; subst. - exists x0. split. + exact H0. + left. reflexivity. - exists y. split. + exact H0. + right. apply clos_trans_t1n_iff. exact H1.","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: The variable x0 was not found in the current environment."},{"g":2,"t":"intros x H. apply clos_trans_t1n_iff in H. remember x as z in |- * at 2. inversion H; subst. - exists y. split. + exact H0. + left. reflexivity. - exists y. split. + exact H0. + right. apply clos_trans_t1n_iff. exact H1.","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: The variable y was not found in the current environment."},{"g":2,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans Thing DependsOn x y -> clos_trans_1n Thing DependsOn x y","forall x y, clos_trans_1n Thing DependsOn x y -> exists z, Gives z x /\\ (z = y \\/ clos_trans Thing DependsOn z y)"],"combine":"intros x H. destruct (sub2 x x (sub1 x x H)) as [z [Hz Hr]]. exists z. split; [exact Hz | exact Hr].","ok":true,"c":[3,4],"assembled":true},{"g":3,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"intros x y H.","w":"model","kind":"step","ok":true,"c":[13],"d":"Tactic 1 of 3 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros x y H. apply clos_trans_t1n_iff. exact H."},{"g":13,"t":"apply clos_trans_t1n_iff.","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 2 of 3 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"exact H.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 3 of 3 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"intros x y H. destruct H as [y1 Hxy | y1 z1 Hxy Hyz].\n- exists y1; split; [exact Hxy | left; reflexivity].\n- exists y1; split; [exact Hxy | right]. clear Hxy. induction Hyz as [a h | a b h hs IH].\n  + apply t_step; exact h.\n  + eapply t_trans; [apply t_step; exact h | exact IH].","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: In environment Thing : Type"},{"g":4,"t":"divide into 1 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans_1n Thing DependsOn x y -> clos_trans Thing DependsOn x y"],"combine":"intros x y H. destruct H as [x y Hd | x y' y Hd Hc].\n- exists y. split; [exact Hd | left; reflexivity].\n- exists y'. split; [exact Hd | right; apply sub1; exact Hc].","ok":false,"err":"Rocq could not derive the goal from these subgoals with the proposed combining proof: Error: x is already used."},{"g":4,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans_1n Thing DependsOn x y -> Gives y x \\/ (exists z, Gives z x /\\ clos_trans_1n Thing DependsOn z y)","forall x y, clos_trans_1n Thing DependsOn x y -> clos_trans Thing DependsOn x y"],"combine":"intros x y H. destruct (sub1 x y H) as [Hg | [z [Hz Hc]]].\n- exists y. split; [exact Hg | left; reflexivity].\n- exists z. split; [exact Hz | right; apply sub2; exact Hc].","ok":true,"c":[5,6],"assembled":true},{"g":5,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":5,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":5,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":5,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":5,"t":"intros x y H.","w":"model","kind":"step","ok":true,"c":[15],"d":"Tactic 1 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros x y H. destruct H as [y0 H | y0 z H H0].\n- left. exact H.\n- right. exists y0. split; assumption."},{"g":15,"t":"destruct H as [y0 H | y0 z H H0].","w":"model","kind":"step","ok":true,"c":[16,17],"d":"Tactic 2 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"left.","w":"model","kind":"step","ok":true,"c":[18],"d":"Tactic 3 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"exact H.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 4 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"right.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 5 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"exists y0.","w":"model","kind":"step","ok":true,"c":[20],"d":"Tactic 6 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"split; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":6,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":6,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":6,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":6,"t":"intros x y H.","w":"model","kind":"step","ok":true,"c":[21],"d":"Tactic 1 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros x y H. induction H as [x y H | x y z H1 H2 IH].\n- apply t_step. exact H.\n- eapply t_trans.\n  + apply t_step. exact H1.\n  + exact IH."},{"g":21,"t":"induction H as [x y H | x y z H1 H2 IH].","w":"model","kind":"step","ok":true,"c":[22,23],"d":"Tactic 2 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"apply t_step.","w":"model","kind":"step","ok":true,"c":[24],"d":"Tactic 3 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"exact H.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 4 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"eapply t_trans.","w":"model","kind":"step","ok":true,"c":[25],"d":"Tactic 5 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"apply t_step.","w":"model","kind":"step","ok":true,"c":[26],"d":"Tactic 6 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":26,"t":"exact H1.","w":"model","kind":"step","ok":true,"c":[27],"d":"Tactic 7 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"exact IH.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 8 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[5,10,11,12,13,14,15,16,23,28,29,30,37,42,43,44,45,46,47,48,53,54,55,56,57,58,59,60],"proofFile":"files/claude/run1/pr003_no_circle.v"},{"key":"claude1/pr004_no_two_step_giving","id":"pr004_no_two_step_giving","run":"claude1","title":"Dalīl al-kalima, premise 4 (key step)","english":"What receives existence does not give existence to anything.","stmt":"OP001 -> forall x y z, Gives y x -> ~ Gives z y","proved":true,"seconds":0.4,"expansions":0,"calls":1,"root":0,"goals":["OP001 -> forall x y z, Gives y x -> ~ Gives z y"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run1/pr004_no_two_step_giving.v"},{"key":"claude1/kalima_first_cause","id":"kalima_first_cause","run":"claude1","title":"Dalīl al-kalima, the first cause","english":"If the universe has its existence through another, it has a first cause that exists of itself.","stmt":"OP001 -> ThroughAnother Universe -> exists b, Gives b Universe /\\ OfItself b /\\ (forall y, ~ Gives y b)","proved":true,"seconds":7.3,"expansions":1,"calls":17,"root":0,"goals":["OP001 -> ThroughAnother Universe -> exists b, Gives b Universe /\\ OfItself b /\\ (forall y, ~ Gives y b)","H : OP001\nb : Thing\nHb : Gives b Universe\n============================\nexists b0 : Thing, Gives b0 Universe /\\ OfItself b0 /\\ (forall y : Thing, ~ Gives y b0)","H : OP001\nb : Thing\nHb : Gives b Universe\n============================\nGives b Universe /\\ OfItself b /\\ (forall y : Thing, ~ Gives y b)","H : OP001\nb : Thing\nHb : Gives b Universe\nHi : OfItself b\n============================\nGives b Universe /\\ OfItself b /\\ (forall y : Thing, ~ Gives y b)","H : OP001\nb : Thing\nHb : Gives b Universe\nHi : OfItself b\n============================\nGives b Universe","H : OP001\nb : Thing\nHb : Gives b Universe\nHi : OfItself b\n============================\nOfItself b /\\ (forall y : Thing, ~ Gives y b)","H : OP001\nb : Thing\nHb : Gives b Universe\nHi : OfItself b\n============================\nOfItself b","H : OP001\nb : Thing\nHb : Gives b Universe\nHi : OfItself b\n============================\nforall y : Thing, ~ Gives y b","H : OP001\nb : Thing\nHb : Gives b Universe\nHi : OfItself b\ny : Thing\nHy : Gives y b\n============================\nFalse","H : OP001\nb : Thing\nHb : Gives b Universe\nN : ~ ThroughAnother b\ny : Thing\nHy : Gives y b\n============================\nFalse","H : OP001\nb : Thing\nHb : Gives b Universe\nN : ~ ThroughAnother b\ny : Thing\nHy : Gives y b\n============================\nThroughAnother b","H : OP001\nb : Thing\nHb : Gives b Universe\nN : ~ ThroughAnother b\ny : Thing\nHy : Gives y b\n============================\nexists y0 : Thing, Gives y0 b","H : OP001\nb : Thing\nHb : Gives b Universe\nN : ~ ThroughAnother b\ny : Thing\nHy : Gives y b\n============================\nGives y b"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H [b Hb].","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H [b Hb]. exists b. pose proof (H Universe b Hb) as Hi. split. exact Hb. split. exact Hi. intros y Hy. destruct Hi as [_ N]. apply N. unfold ThroughAnother. exists y. exact Hy."},{"g":1,"t":"exists b.","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"pose proof (H Universe b Hb) as Hi.","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 3 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"split.","w":"model","kind":"step","ok":true,"c":[4,5],"d":"Tactic 4 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"exact Hb.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"split.","w":"model","kind":"step","ok":true,"c":[6,7],"d":"Tactic 6 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"exact Hi.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"intros y Hy.","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 8 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"destruct Hi as [_ N].","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 9 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"apply N.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 10 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"unfold ThroughAnother.","w":"model","kind":"step","ok":true,"c":[11],"d":"Tactic 11 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"exists y.","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 12 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"exact Hy.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 13 of 13 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16],"proofFile":"files/claude/run1/kalima_first_cause.v"},{"key":"claude1/pr011_something_caused","id":"pr011_something_caused","run":"claude1","title":"The universe is caused","english":"Something in the universe has its existence through another.","stmt":"MP008 -> OP006 -> exists x, InUniverse x /\\ Possible x /\\ ThroughAnother x","proved":true,"seconds":8.6,"expansions":1,"calls":16,"root":0,"goals":["MP008 -> OP006 -> exists x, InUniverse x /\\ Possible x /\\ ThroughAnother x","H8 : MP008\nH6 : OP006\n============================\nexists x : Thing, InUniverse x /\\ Possible x /\\ ThroughAnother x","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\n============================\nexists x0 : Thing, InUniverse x0 /\\ Possible x0 /\\ ThroughAnother x0","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\n============================\nInUniverse x /\\ Possible x /\\ ThroughAnother x","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\n============================\nInUniverse x","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\n============================\nPossible x /\\ ThroughAnother x","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\n============================\nPossible x","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\n============================\nThroughAnother x","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\nT : ThroughAnother x\n============================\nThroughAnother x","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\nN : ~ ThroughAnother x\n============================\nThroughAnother x","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\nN : ~ ThroughAnother x\n============================\nFalse","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\nN : ~ ThroughAnother x\n============================\nEss x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H8 H6.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H8 H6. destruct H6 as [x [Hu [He Hn]]]. exists x. split. exact Hu. split. split; assumption. destruct (classic (ThroughAnother x)) as [T|N]. exact T. exfalso. apply Hn. exact (H8 x He N)."},{"g":1,"t":"destruct H6 as [x [Hu [He Hn]]].","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"exists x.","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 3 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"split.","w":"model","kind":"step","ok":true,"c":[4,5],"d":"Tactic 4 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"exact Hu.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"split.","w":"model","kind":"step","ok":true,"c":[6,7],"d":"Tactic 6 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"split; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"destruct (classic (ThroughAnother x)) as [T|N].","w":"model","kind":"step","ok":true,"c":[8,9],"d":"Tactic 8 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"exact T.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 9 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"exfalso.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 10 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"apply Hn.","w":"model","kind":"step","ok":true,"c":[11],"d":"Tactic 11 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"exact (H8 x He N).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 12 of 12 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15],"proofFile":"files/claude/run1/pr011_something_caused.v"},{"key":"claude1/kalima_unconditional","id":"kalima_unconditional","run":"claude1","title":"Dalīl al-kalima, the conclusion without the “if”","english":"Something in the universe has a giver of existence that is necessary and depends on nothing.","stmt":"MP001 -> MP002 -> MP003 -> OP001 -> MP008 -> OP006 -> exists x b, InUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","proved":true,"seconds":11.2,"expansions":1,"calls":15,"root":0,"goals":["MP001 -> MP002 -> MP003 -> OP001 -> MP008 -> OP006 -> exists x b, InUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nOP1 : OP001\nH8 : MP008\nH6 : OP006\n============================\nexists x b : Thing, InUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nOP1 : OP001\nH8 : MP008\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHness : ~ Ess x\n============================\nexists x0 b : Thing, InUniverse x0 /\\ Gives b x0 /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nOP1 : OP001\nH8 : MP008\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHness : ~ Ess x\nHT : ThroughAnother x\n============================\nexists x0 b : Thing, InUniverse x0 /\\ Gives b x0 /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nOP1 : OP001\nH8 : MP008\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHness : ~ Ess x\nb : Thing\nHb : Gives b x\n============================\nexists x0 b0 : Thing, InUniverse x0 /\\ Gives b0 x0 /\\ Exists b0 /\\ Necessary b0 /\\ Independent b0","H1 : MP001\nH2 : MP002\nH3 : MP003\nOP1 : OP001\nH8 : MP008\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHness : ~ Ess x\nb : Thing\nHb : Gives b x\nHbe : Exists b\nHnb : ~ ThroughAnother b\n============================\nexists x0 b0 : Thing, InUniverse x0 /\\ Gives b0 x0 /\\ Exists b0 /\\ Necessary b0 /\\ Independent b0","H1 : MP001\nH2 : MP002\nH3 : MP003\nOP1 : OP001\nH8 : MP008\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHness : ~ Ess x\nb : Thing\nHb : Gives b x\nHbe : Exists b\nHnb : ~ ThroughAnother b\n============================\nInUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nOP1 : OP001\nH8 : MP008\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHness : ~ Ess x\nb : Thing\nHb : Gives b x\nHbe : Exists b\nHnb : ~ ThroughAnother b\n============================\nIndependent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nOP1 : OP001\nH8 : MP008\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHness : ~ Ess x\nb : Thing\nHb : Gives b x\nHbe : Exists b\nHnb : ~ ThroughAnother b\ny : Thing\nHy : DependsOn b y\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nOP1 : OP001\nH8 : MP008\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHness : ~ Ess x\nb : Thing\nHb : Gives b x\nHbe : Exists b\nHnb : ~ ThroughAnother b\ny : Thing\nHy : DependsOn b y\n============================\nThroughAnother b","H1 : MP001\nH2 : MP002\nH3 : MP003\nOP1 : OP001\nH8 : MP008\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHness : ~ Ess x\nb : Thing\nHb : Gives b x\nHbe : Exists b\nHnb : ~ ThroughAnother b\ny : Thing\nHy : DependsOn b y\n============================\nGives y b"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 OP1 H8 H6.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 OP1 H8 H6.\ndestruct H6 as [x [Hu [Hex Hness]]].\nassert (HT : ThroughAnother x) by (apply NNPP; intro n; apply Hness; apply H8; assumption).\ndestruct HT as [b Hb].\ndestruct (OP1 x b Hb) as [Hbe Hnb].\nexists x, b.\nrefine (conj Hu (conj Hb (conj Hbe (conj (conj Hbe (H8 b Hbe Hnb)) _)))).\nintros y Hy.\napply Hnb.\nexists y.\nexact Hy."},{"g":1,"t":"destruct H6 as [x [Hu [Hex Hness]]].","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"assert (HT : ThroughAnother x) by (apply NNPP; intro n; apply Hness; apply H8; assumption).","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 3 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"destruct HT as [b Hb].","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 4 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"destruct (OP1 x b Hb) as [Hbe Hnb].","w":"model","kind":"step","ok":true,"c":[5],"d":"Tactic 5 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"exists x, b.","w":"model","kind":"step","ok":true,"c":[6],"d":"Tactic 6 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"refine (conj Hu (conj Hb (conj Hbe (conj (conj Hbe (H8 b Hbe Hnb)) _)))).","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 7 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"intros y Hy.","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 8 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"apply Hnb.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 9 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"exists y.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 10 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"exact Hy.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 11 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14],"proofFile":"files/claude/run1/kalima_unconditional.v"},{"key":"claude1/imkan_s1_division","id":"imkan_s1_division","run":"claude1","title":"Dalīl al-imkān, step 1","english":"Every existent is necessary or possible in itself, and not both.","stmt":"forall x, Exists x -> (Necessary x \\/ Possible x) /\\ ~ (Necessary x /\\ Possible x)","proved":true,"seconds":0.9,"expansions":0,"calls":4,"root":0,"goals":["forall x, Exists x -> (Necessary x \\/ Possible x) /\\ ~ (Necessary x /\\ Possible x)"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":true,"c":[]}],"proof":[3],"proofFile":"files/claude/run1/imkan_s1_division.v"},{"key":"claude1/imkan_s2_necessary_cannot_fail","id":"imkan_s2_necessary_cannot_fail","run":"claude1","title":"Dalīl al-imkān, step 2","english":"The non-existence of what is necessary in itself is not possible.","stmt":"MP007 -> forall x, Necessary x -> ~ NonExistencePossible x","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["MP007 -> forall x, Necessary x -> ~ NonExistencePossible x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run1/imkan_s2_necessary_cannot_fail.v"},{"key":"claude1/step3_possible_needs_cause","id":"step3_possible_needs_cause","run":"claude1","title":"Dalīl al-imkān, step 3","english":"A possible existent needs a cause.","stmt":"MP008 -> forall x, Possible x -> ThroughAnother x","proved":true,"seconds":5.1,"expansions":1,"calls":9,"root":0,"goals":["MP008 -> forall x, Possible x -> ThroughAnother x","H : MP008\nx : Thing\nHe : Exists x\nHn : ~ Ess x\n============================\nThroughAnother x","H : MP008\nx : Thing\nHe : Exists x\nHn : ~ Ess x\n============================\n~ ~ ThroughAnother x","H : MP008\nx : Thing\nHe : Exists x\nHn : ~ Ess x\nHc : ~ ThroughAnother x\n============================\nFalse","H : MP008\nx : Thing\nHe : Exists x\nHn : ~ Ess x\nHc : ~ ThroughAnother x\n============================\nEss x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H x [He Hn].","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H x [He Hn]. apply NNPP. intro Hc. apply Hn. apply H; assumption."},{"g":1,"t":"apply NNPP.","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intro Hc.","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 3 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"apply Hn.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 4 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"apply H; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8],"proofFile":"files/claude/run1/step3_possible_needs_cause.v"},{"key":"claude1/imkan_s4_no_self_causation","id":"imkan_s4_no_self_causation","run":"claude1","title":"Dalīl al-imkān, step 4","english":"Nothing gives existence to itself.","stmt":"MP001 -> MP002 -> forall A, ~ Gives A A","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["MP001 -> MP002 -> forall A, ~ Gives A A"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run1/imkan_s4_no_self_causation.v"},{"key":"claude1/imkan_s5_none_outside","id":"imkan_s5_none_outside","run":"claude1","title":"Dalīl al-imkān, step 5","english":"No possible existent lies outside the aggregate of possibles.","stmt":"forall A, AggregateOfPossibles A -> forall p, p <> A -> Possible p -> MemberOf p A","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["forall A, AggregateOfPossibles A -> forall p, p <> A -> Possible p -> MemberOf p A"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run1/imkan_s5_none_outside.v"},{"key":"claude1/imkan_s6_aggregate_possible_caused","id":"imkan_s6_aggregate_possible_caused","run":"claude1","title":"Dalīl al-imkān, step 6","english":"The aggregate is itself possible, and so has its existence through another.","stmt":"MP008 -> MP010 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> Possible A /\\ ThroughAnother A","proved":true,"seconds":221.2,"expansions":1,"calls":24,"root":0,"goals":["MP008 -> MP010 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> Possible A /\\ ThroughAnother A","============================\n(forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x) -> (forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A) -> forall A : Thing, (forall p : Thing, p <> A -> MemberOf p A <-> Possible p) /\\ ~ MemberOf A A -> Exists A -> (exists p : Thing, p <> A /\\ Possible p) -> Possible A /\\ ThroughAnother A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\n============================\nPossible A /\\ ThroughAnother A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\n============================\n~ Ess A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nhn : ~ Ess A\n============================\nPossible A /\\ ThroughAnother A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\n============================\nexists p0 : Thing, MemberOf p0 A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\n============================\nforall p0 : Thing, MemberOf p0 A -> ~ Ess p0","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\n============================\nMemberOf p A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\n============================\nPossible p","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nq : Thing\nHq : MemberOf q A\n============================\n~ Ess q","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nq : Thing\nHq : MemberOf q A\nHq' : q <> A\n============================\n~ Ess q","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nq : Thing\nHq : Possible q\nHq' : q <> A\n============================\n~ Ess q","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nq : Thing\nn : ~ Ess q\nHq' : q <> A\n============================\n~ Ess q","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nhn : ~ Ess A\n============================\nPossible A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nhn : ~ Ess A\n============================\nThroughAnother A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nhn : ~ Ess A\n============================\nExists A /\\ ~ Ess A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nhn : ~ Ess A\nh : ThroughAnother A\n============================\nThroughAnother A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nhn : ~ Ess A\nh : ~ ThroughAnother A\n============================\nThroughAnother A","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nhn : ~ Ess A\nh : ~ ThroughAnother A\n============================\nFalse","H8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH10 : forall A : Thing, (exists p : Thing, MemberOf p A) -> (forall p : Thing, MemberOf p A -> ~ Ess p) -> ~ Ess A\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHp : p <> A\nHpos : Possible p\nhn : ~ Ess A\nh : ~ ThroughAnother A\n============================\nEss A"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP008, MP010, AggregateOfPossibles.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"unfold MP008, MP010, AggregateOfPossibles. intros H8 H10 A [HA HnA] HE [p [Hp Hpos]].\nassert (hn: ~ Ess A).\n{ apply H10.\n  - exists p. apply (HA p Hp). exact Hpos.\n  - intros q Hq.\n    assert (Hq' : q <> A) by (intro e; subst; apply HnA; exact Hq).\n    apply (HA q Hq') in Hq.\n    destruct Hq as [_ n]. exact n. }\nsplit.\n- unfold Possible. split; assumption.\n- destruct (classic (ThroughAnother A)) as [h|h].\n  + exact h.\n  + exfalso. apply hn. apply H8; assumption."},{"g":1,"t":"intros H8 H10 A [HA HnA] HE [p [Hp Hpos]].","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"assert (hn: ~ Ess A).","w":"model","kind":"step","ok":true,"c":[3,4],"d":"Tactic 3 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"apply H10.","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 4 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"exists p.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 5 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"apply (HA p Hp).","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 6 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"exact Hpos.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"intros q Hq.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 8 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"assert (Hq' : q <> A) by (intro e; subst; apply HnA; exact Hq).","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 9 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"apply (HA q Hq') in Hq.","w":"model","kind":"step","ok":true,"c":[11],"d":"Tactic 10 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"destruct Hq as [_ n].","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 11 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"exact n.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 12 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"split.","w":"model","kind":"step","ok":true,"c":[13,14],"d":"Tactic 13 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"unfold Possible.","w":"model","kind":"step","ok":true,"c":[15],"d":"Tactic 14 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"split; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 15 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"destruct (classic (ThroughAnother A)) as [h|h].","w":"model","kind":"step","ok":true,"c":[16,17],"d":"Tactic 16 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"exact h.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 17 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"exfalso.","w":"model","kind":"step","ok":true,"c":[18],"d":"Tactic 18 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"apply hn.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 19 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"apply H8; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 20 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],"proofFile":"files/claude/run1/imkan_s6_aggregate_possible_caused.v"},{"key":"claude1/imkan_s7_cause_not_member","id":"imkan_s7_cause_not_member","run":"claude1","title":"Dalīl al-imkān, step 7","english":"What gives existence to the aggregate is not one of its members.","stmt":"MP001 -> MP002 -> MP003 -> MP011 -> forall c A, Gives c A -> ~ MemberOf c A","proved":true,"seconds":55.4,"expansions":4,"calls":34,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP011 -> forall c A, Gives c A -> ~ MemberOf c A","MP001 -> MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> exists x, Gives a x /\\ (x = b \\/ Through b x)","MP001 -> MP002 -> forall a x, Gives a x -> (x = a \\/ Through a x) -> False","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nH : clos_trans Thing Gives a b\n============================\nexists x : Thing, Gives a x /\\ (x = b \\/ Through b x)","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nHab : Gives a b\n============================\nexists x : Thing, Gives a x /\\ (x = b \\/ Through b x)","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nIH1 : exists x : Thing, Gives a x /\\ (x = b \\/ Through b x)\nIH2 : exists x : Thing, Gives b x /\\ (x = c \\/ Through c x)\n============================\nexists x : Thing, Gives a x /\\ (x = c \\/ Through c x)","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : x = b \\/ Through b x\nIH2 : exists x : Thing, Gives b x /\\ (x = c \\/ Through c x)\n============================\nexists x0 : Thing, Gives a x0 /\\ (x0 = c \\/ Through c x0)","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : x = b \\/ Through b x\ny : Thing\nHby : Gives b y\nHy : y = c \\/ Through c y\n============================\nexists x0 : Thing, Gives a x0 /\\ (x0 = c \\/ Through c x0)","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : x = b \\/ Through b x\ny : Thing\nHby : Gives b y\nHy : y = c \\/ Through c y\n============================\nThrough c b","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : x = b \\/ Through b x\ny : Thing\nHby : Gives b y\nHy : y = c \\/ Through c y\nTcb : Through c b\n============================\nexists x0 : Thing, Gives a x0 /\\ (x0 = c \\/ Through c x0)","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : x = b \\/ Through b x\ny : Thing\nHby : Gives b y\nHy : y = c\n============================\nThrough c b","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : x = b \\/ Through b x\ny : Thing\nHby : Gives b y\nHy : Through c y\n============================\nThrough c b","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : x = b \\/ Through b x\nHby : Gives b c\n============================\nThrough c b","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : x = b \\/ Through b x\ny : Thing\nHby : Gives b y\nHy : y = c \\/ Through c y\nTcb : Through c b\n============================\nx = c \\/ Through c x","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : x = b \\/ Through b x\ny : Thing\nHby : Gives b y\nHy : y = c \\/ Through c y\nTcb : Through c b\n============================\nThrough c x","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : x = b\ny : Thing\nHby : Gives b y\nHy : y = c \\/ Through c y\nTcb : Through c b\n============================\nThrough c x","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nx : Thing\nHax : Gives a x\nHx : Through b x\ny : Thing\nHby : Gives b y\nHy : y = c \\/ Through c y\nTcb : Through c b\n============================\nThrough c x","H1 : MP001\nH2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nc : Thing\nHab : clos_trans Thing Gives a b\nHbc : clos_trans Thing Gives b c\nHax : Gives a b\ny : Thing\nHby : Gives b y\nHy : y = c \\/ Through c y\nTcb : Through c b\n============================\nThrough c b","H1 : MP001\nH2 : MP002\na : Thing\nx : Thing\nG : Gives a x\nE : x = a\n============================\nFalse","H1 : MP001\nH2 : MP002\na : Thing\nx : Thing\nG : Gives a x\nT : Through a x\n============================\nFalse","H1 : MP001\nH2 : MP002\na : Thing\nG : Gives a a\n============================\nFalse"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H11 c A hg hm.\nassert (L: forall y x, clos_trans Thing Gives y x -> Through x y).\n{ intros y x h. induction h as [a b hab | a b d h1 IH1 h2 IH2].\n  - apply H2; exact hab.\n  - exact (H3 _ _ _ IH2 IH1). }\nhave_h: assert (h := H11 c A c hg hm).\napply (proj1 (clos_trans_t1n_iff Thing Gives c c)) in h.\ninversion h; subst.\n- exact (H1 _ _ H (H2 _ _ H)).\n- rename y into z.\n  match goal with H5 : clos_trans_1n Thing Gives z c |- _ =>\n    apply (proj2 (clos_trans_t1n_iff Thing Gives z c)) in H5;\n    exact (H1 _ _ H (L _ _ H5)) end.","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: Syntax error: [ltac_use_default] expected after [tactic] (in [tactic_command])."},{"g":0,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> exists x, Gives a x /\\ (x = b \\/ Through b x)","MP001 -> MP002 -> forall a x, Gives a x -> (x = a \\/ Through a x) -> False"],"combine":"intros h1 h2 h3 h11 c A hg hm. pose proof (h11 c A c hg hm) as H. destruct (sub1 h1 h2 h3 c c H) as [x [hx hor]]. exact (sub2 h1 h2 c x hx hor).","ok":true,"c":[1,2],"assembled":true},{"g":1,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"intros H1 H2 H3 a b H.","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 1 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 a b H. induction H as [a b Hab | a b c Hab IH1 Hbc IH2].\n- exists b; split; [exact Hab | left; reflexivity].\n- destruct IH1 as [x [Hax Hx]]. destruct IH2 as [y [Hby Hy]].\n  assert (Tcb : Through c b).\n  { destruct Hy as [Hy|Hy].\n    - subst y. apply H2; exact Hby.\n    - apply H3 with y; [exact Hy | apply H2; exact Hby]. }\n  exists x; split; [exact Hax|]. right.\n  destruct Hx as [Hx|Hx].\n  + subst x. exact Tcb.\n  + apply H3 with b; assumption."},{"g":3,"t":"induction H as [a b Hab | a b c Hab IH1 Hbc IH2].","w":"model","kind":"step","ok":true,"c":[4,5],"d":"Tactic 2 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"exists b; split; [exact Hab | left; reflexivity].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 3 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"destruct IH1 as [x [Hax Hx]].","w":"model","kind":"step","ok":true,"c":[6],"d":"Tactic 4 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"destruct IH2 as [y [Hby Hy]].","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 5 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"assert (Tcb : Through c b).","w":"model","kind":"step","ok":true,"c":[8,9],"d":"Tactic 6 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"destruct Hy as [Hy|Hy].","w":"model","kind":"step","ok":true,"c":[10,11],"d":"Tactic 7 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"subst y.","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 8 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"apply H2; exact Hby.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 9 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"apply H3 with y; [exact Hy | apply H2; exact Hby].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 10 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"exists x; split; [exact Hax|].","w":"model","kind":"step","ok":true,"c":[13],"d":"Tactic 11 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"right.","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 12 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"destruct Hx as [Hx|Hx].","w":"model","kind":"step","ok":true,"c":[15,16],"d":"Tactic 13 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"subst x.","w":"model","kind":"step","ok":true,"c":[17],"d":"Tactic 14 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"exact Tcb.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 15 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"apply H3 with b; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 16 of 16 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"intros H1 H2 a x G [E|T].","w":"model","kind":"step","ok":true,"c":[18,19],"d":"Tactic 1 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 a x G [E|T].\n- subst. exact (H1 _ _ G (H2 _ _ G)).\n- exact (H1 _ _ G T)."},{"g":18,"t":"subst.","w":"model","kind":"step","ok":true,"c":[20],"d":"Tactic 2 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"exact (H1 _ _ G (H2 _ _ G)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 3 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"exact (H1 _ _ G T).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 4 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[5,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,30,31,32,33],"proofFile":"files/claude/run1/imkan_s7_cause_not_member.v"},{"key":"claude1/imkan_s8_no_internal_cause","id":"imkan_s8_no_internal_cause","run":"claude1","title":"Dalīl al-imkān, step 8","english":"Neither a member, nor a group of members, nor the aggregate gives the aggregate existence.","stmt":"MP001 -> MP002 -> MP003 -> MP010 -> MP011 -> forall A, AggregateOfPossibles A -> (forall x, ~ Gives x x) /\\ ~ Gives A A /\\ (forall c, MemberOf c A -> ~ Gives c A) /\\ (forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> ~ Gives B A)","proved":true,"seconds":213.7,"expansions":4,"calls":49,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP010 -> MP011 -> forall A, AggregateOfPossibles A -> (forall x, ~ Gives x x) /\\ ~ Gives A A /\\ (forall c, MemberOf c A -> ~ Gives c A) /\\ (forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> ~ Gives B A)","MP001 -> MP002 -> forall x, ~ Gives x x","MP001 -> MP002 -> MP003 -> MP011 -> forall A c, MemberOf c A -> ~ Gives c A","MP010 -> forall A, AggregateOfPossibles A -> forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> MemberOf B A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\n============================\nforall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\na : Thing\nb : Thing\nH : clos_trans Thing Gives a b\n============================\nexists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\na : Thing\nb : Thing\nHab : Gives a b\n============================\nexists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\na : Thing\ny : Thing\nb : Thing\nHay : clos_trans Thing Gives a y\nHyb : clos_trans Thing Gives y b\nIH1 : exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m y\nIH2 : exists m : Thing, (y = m \\/ clos_trans Thing Gives y m) /\\ Gives m b\n============================\nexists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\na : Thing\nb : Thing\nHab : Gives a b\n============================\n(a = a \\/ clos_trans Thing Gives a a) /\\ Gives a b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\na : Thing\ny : Thing\nb : Thing\nHay : clos_trans Thing Gives a y\nHyb : clos_trans Thing Gives y b\nIH1 : exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m y\nm : Thing\nE : y = m\nHg' : Gives m b\n============================\nexists m0 : Thing, (a = m0 \\/ clos_trans Thing Gives a m0) /\\ Gives m0 b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\na : Thing\ny : Thing\nb : Thing\nHay : clos_trans Thing Gives a y\nHyb : clos_trans Thing Gives y b\nIH1 : exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m y\nm : Thing\nHm' : clos_trans Thing Gives y m\nHg' : Gives m b\n============================\nexists m0 : Thing, (a = m0 \\/ clos_trans Thing Gives a m0) /\\ Gives m0 b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\na : Thing\ny : Thing\nb : Thing\nHay : clos_trans Thing Gives a y\nHyb : clos_trans Thing Gives y b\nIH1 : exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m y\nm : Thing\nE : y = m\nHg' : Gives m b\n============================\n(a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\na : Thing\ny : Thing\nb : Thing\nHay : clos_trans Thing Gives a y\nHyb : clos_trans Thing Gives y b\nIH1 : exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m y\nm : Thing\nHm' : clos_trans Thing Gives y m\nHg' : Gives m b\n============================\n(a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\n============================\nforall a b : Thing, clos_trans Thing Gives a b -> Through b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\nT : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\na : Thing\nb : Thing\nH : clos_trans Thing Gives a b\n============================\nThrough b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\na : Thing\nb : Thing\nHab : Gives a b\n============================\nThrough b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\na : Thing\ny : Thing\nb : Thing\nHay : clos_trans Thing Gives a y\nHyb : clos_trans Thing Gives y b\nIH1 : Through y a\nIH2 : Through b y\n============================\nThrough b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\nT : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nHc : clos_trans Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\nT : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nHc : clos_trans Thing Gives c c\nm : Thing\nE : c = m\nHg' : Gives m c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\nT : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nHc : clos_trans Thing Gives c c\nm : Thing\nHm' : clos_trans Thing Gives c m\nHg' : Gives m c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\nT : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nHc : clos_trans Thing Gives c c\nm : Thing\nE : c = m\nHg' : Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\nT : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nHc : clos_trans Thing Gives c c\nm : Thing\nHm' : clos_trans Thing Gives c m\nHg' : Gives m c\n============================\nThrough m c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHm : MemberOf c A\nHg : Gives c A\nK : forall a b : Thing, clos_trans Thing Gives a b -> exists m : Thing, (a = m \\/ clos_trans Thing Gives a m) /\\ Gives m b\nT : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nHc : clos_trans Thing Gives c c\nm : Thing\nHm' : clos_trans Thing Gives c m\nHg' : Gives m c\n============================\nclos_trans Thing Gives c m","H10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nB : Thing\nHBA : B <> A\nHEB : Exists B\nHex : exists p : Thing, MemberOf p B\nHsub : forall p : Thing, MemberOf p B -> MemberOf p A\n============================\nMemberOf B A","H10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nB : Thing\nHBA : B <> A\nHEB : Exists B\nHex : exists p : Thing, MemberOf p B\nHsub : forall p : Thing, MemberOf p B -> MemberOf p A\n============================\nPossible B","H10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nB : Thing\nHBA : B <> A\nHEB : Exists B\nHex : exists p : Thing, MemberOf p B\nHsub : forall p : Thing, MemberOf p B -> MemberOf p A\n============================\nExists B","H10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nB : Thing\nHBA : B <> A\nHEB : Exists B\nHex : exists p : Thing, MemberOf p B\nHsub : forall p : Thing, MemberOf p B -> MemberOf p A\n============================\n~ Ess B","H10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nB : Thing\nHBA : B <> A\nHEB : Exists B\nHex : exists p : Thing, MemberOf p B\nHsub : forall p : Thing, MemberOf p B -> MemberOf p A\n============================\nexists p : Thing, MemberOf p B","H10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nB : Thing\nHBA : B <> A\nHEB : Exists B\nHex : exists p : Thing, MemberOf p B\nHsub : forall p : Thing, MemberOf p B -> MemberOf p A\n============================\nforall p : Thing, MemberOf p B -> ~ Ess p","H10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nB : Thing\nHBA : B <> A\nHEB : Exists B\nHex : exists p : Thing, MemberOf p B\nHsub : forall p : Thing, MemberOf p B -> MemberOf p A\np : Thing\nHp : MemberOf p B\n============================\n~ Ess p","H10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nB : Thing\nHBA : B <> A\nHEB : Exists B\nHex : exists p : Thing, MemberOf p B\nHsub : forall p : Thing, MemberOf p B -> MemberOf p A\np : Thing\nHp : MemberOf p B\nH1 : MemberOf p A\n============================\n~ Ess p","H10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nB : Thing\nHBA : B <> A\nHEB : Exists B\nHex : exists p : Thing, MemberOf p B\nHsub : forall p : Thing, MemberOf p B -> MemberOf p A\np : Thing\nHp : MemberOf p B\nH1 : MemberOf p A\nH2 : p <> A\n============================\n~ Ess p","H10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nB : Thing\nHBA : B <> A\nHEB : Exists B\nHex : exists p : Thing, MemberOf p B\nHsub : forall p : Thing, MemberOf p B -> MemberOf p A\np : Thing\nHp : MemberOf p B\nH1 : Possible p\nH2 : p <> A\n============================\n~ Ess p"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H10 H11 A [HA HAA].\nassert (L: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b h. induction h as [x y h|x y z h1 IH1 h2 IH2].\n  - apply H2; exact h.\n  - exact (H3 _ _ _ IH2 IH1). }\nassert (NC: forall c, ~ clos_trans Thing Gives c c).\n{ intros c h. apply clos_trans_t1n in h. inversion h as [x y Hg|x y z Hg Hc]; subst.\n  - exact (H1 _ _ Hg (H2 _ _ Hg)).\n  - apply clos_trans_t1n_iff in Hc. exact (H1 _ _ Hg (L _ _ Hc)). }\nassert (NG: forall x, ~ Gives x x).\n{ intros x h. exact (H1 x x h (H2 x x h)). }\nassert (P3: forall c, MemberOf c A -> ~ Gives c A).\n{ intros c hm hg. apply (NC c). exact (H11 c A c hg hm). }\nrepeat split.\n- exact NG.\n- exact (NG A).\n- exact P3.\n- intros B hne hex hne2 hsub hg.\n  apply (P3 B); [ | exact hg ].\n  apply (proj2 (HA B hne)).\n  unfold Possible. split; [exact hex|].\n  apply H10; [exact hne2|].\n  intros p hp he.\n  assert (hneq: p <> A).\n  { intro e. subst. exact (HAA (hsub _ hp)). }\n  destruct ((proj1 (HA p hneq)) (hsub p hp)) as [_ hn].\n  exact (hn he).","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: The variable Hg was not found in the current environment."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> forall x, ~ Gives x x","MP001 -> MP002 -> MP003 -> MP011 -> forall A c, MemberOf c A -> ~ Gives c A","MP010 -> forall A, AggregateOfPossibles A -> forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> MemberOf B A"],"combine":"intros H1 H2 H3 H10 H11 A HA. refine (conj (sub1 H1 H2) (conj _ (conj _ _))).\n- intro h. exact (sub1 H1 H2 A h).\n- intros c hc. exact (sub2 H1 H2 H3 H11 A c hc).\n- intros B hne hex hm hs hg. exact (sub2 H1 H2 H3 H11 A B (sub3 H10 A HA B hne hex hm hs) hg).","ok":true,"c":[1,2,3],"assembled":true},{"g":1,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":2,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"intros H1 H2 H3 H11 A c Hm Hg.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 1 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 H11 A c Hm Hg.\nassert (K: forall a b, clos_trans Thing Gives a b -> exists m, (a=m \\/ clos_trans Thing Gives a m) /\\ Gives m b).\n{ intros a b H. induction H as [a b Hab | a y b Hay IH1 Hyb IH2].\n  - exists a. split; [left; reflexivity | exact Hab].\n  - destruct IH2 as [m [[E|Hm'] Hg']].\n    + exists m. split; [right; rewrite <- E; exact Hay | exact Hg'].\n    + exists m. split; [right; eapply t_trans; eauto | exact Hg']. }\nassert (T: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b H. induction H as [a b Hab | a y b Hay IH1 Hyb IH2].\n  - apply H2; exact Hab.\n  - eapply H3; eauto. }\nassert (Hc := H11 c A c Hg Hm).\ndestruct (K c c Hc) as [m [[E|Hm'] Hg']].\n- rewrite <- E in Hg'. exact (imkan_s8_no_internal_cause_L7 H1 H2 c Hg').\n- apply (H1 c m Hg'). apply T. exact Hm'."},{"g":4,"t":"assert (K: forall a b, clos_trans Thing Gives a b -> exists m, (a=m \\/ clos_trans Thing Gives a m) /\\ Gives m b).","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 2 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"intros a b H.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 3 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"induction H as [a b Hab | a y b Hay IH1 Hyb IH2].","w":"model","kind":"step","ok":true,"c":[8,9],"d":"Tactic 4 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"exists a.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 5 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"split; [left; reflexivity | exact Hab].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"destruct IH2 as [m [[E|Hm'] Hg']].","w":"model","kind":"step","ok":true,"c":[11,12],"d":"Tactic 7 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"exists m.","w":"model","kind":"step","ok":true,"c":[13],"d":"Tactic 8 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"split; [right; rewrite <- E; exact Hay | exact Hg'].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 9 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"exists m.","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 10 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"split; [right; eapply t_trans; eauto | exact Hg'].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 11 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"assert (T: forall a b, clos_trans Thing Gives a b -> Through b a).","w":"model","kind":"step","ok":true,"c":[15,16],"d":"Tactic 12 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"intros a b H.","w":"model","kind":"step","ok":true,"c":[17],"d":"Tactic 13 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"induction H as [a b Hab | a y b Hay IH1 Hyb IH2].","w":"model","kind":"step","ok":true,"c":[18,19],"d":"Tactic 14 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"apply H2; exact Hab.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 15 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"eapply H3; eauto.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 16 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"assert (Hc := H11 c A c Hg Hm).","w":"model","kind":"step","ok":true,"c":[20],"d":"Tactic 17 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"destruct (K c c Hc) as [m [[E|Hm'] Hg']].","w":"model","kind":"step","ok":true,"c":[21,22],"d":"Tactic 18 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":21,"t":"rewrite <- E in Hg'.","w":"model","kind":"step","ok":true,"c":[23],"d":"Tactic 19 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"exact (imkan_s8_no_internal_cause_L7 H1 H2 c Hg').","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 20 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"apply (H1 c m Hg').","w":"model","kind":"step","ok":true,"c":[24],"d":"Tactic 21 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"apply T.","w":"model","kind":"step","ok":true,"c":[25],"d":"Tactic 22 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"exact Hm'.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 23 of 23 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"intros H10 A [HA HnA] B HBA HEB Hex Hsub.","w":"model","kind":"step","ok":true,"c":[26],"d":"Tactic 1 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H10 A [HA HnA] B HBA HEB Hex Hsub.\napply (proj2 (HA B HBA)).\nsplit.\n- exact HEB.\n- apply H10.\n  + exact Hex.\n  + intros p Hp.\n    assert (H1 : MemberOf p A) by (apply Hsub; exact Hp).\n    assert (H2 : p <> A) by (intro e; subst; exact (HnA H1)).\n    apply (proj1 (HA p H2)) in H1.\n    destruct H1 as [_ H3]; exact H3."},{"g":26,"t":"apply (proj2 (HA B HBA)).","w":"model","kind":"step","ok":true,"c":[27],"d":"Tactic 2 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"split.","w":"model","kind":"step","ok":true,"c":[28,29],"d":"Tactic 3 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":28,"t":"exact HEB.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 4 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":29,"t":"apply H10.","w":"model","kind":"step","ok":true,"c":[30,31],"d":"Tactic 5 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":30,"t":"exact Hex.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":31,"t":"intros p Hp.","w":"model","kind":"step","ok":true,"c":[32],"d":"Tactic 7 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":32,"t":"assert (H1 : MemberOf p A) by (apply Hsub; exact Hp).","w":"model","kind":"step","ok":true,"c":[33],"d":"Tactic 8 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":33,"t":"assert (H2 : p <> A) by (intro e; subst; exact (HnA H1)).","w":"model","kind":"step","ok":true,"c":[34],"d":"Tactic 9 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":34,"t":"apply (proj1 (HA p H2)) in H1.","w":"model","kind":"step","ok":true,"c":[35],"d":"Tactic 10 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":35,"t":"destruct H1 as [_ H3]; exact H3.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 11 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[5,6,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,38,39,40,41,42,43,44,45,46,47,48],"proofFile":"files/claude/run1/imkan_s8_no_internal_cause.v"},{"key":"claude1/imkan_s9_cause_outside","id":"imkan_s9_cause_outside","run":"claude1","title":"Dalīl al-imkān, step 9","english":"The cause of the aggregate lies outside it and is not possible.","stmt":"MP001 -> MP002 -> MP003 -> MP011 -> forall A c, AggregateOfPossibles A -> Gives c A -> ~ MemberOf c A /\\ c <> A /\\ ~ Possible c","proved":true,"seconds":65.5,"expansions":1,"calls":39,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP011 -> forall A c, AggregateOfPossibles A -> Gives c A -> ~ MemberOf c A /\\ c <> A /\\ ~ Possible c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\n============================\n~ MemberOf c A /\\ c <> A /\\ ~ Possible c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\n============================\nforall a b : Thing, clos_trans Thing Gives a b -> Through b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\n============================\n~ MemberOf c A /\\ c <> A /\\ ~ Possible c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\na : Thing\nb : Thing\nH : clos_trans Thing Gives a b\n============================\nThrough b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nx : Thing\ny : Thing\nHg : Gives x y\n============================\nThrough y x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nx : Thing\ny : Thing\nz : Thing\nHxy : clos_trans Thing Gives x y\nHyz : clos_trans Thing Gives y z\nIH1 : Through y x\nIH2 : Through z y\n============================\nThrough z x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\n============================\nforall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\n============================\n~ MemberOf c A /\\ c <> A /\\ ~ Possible c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\na : Thing\nb : Thing\nH : clos_trans Thing Gives a b\n============================\nexists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nx : Thing\ny : Thing\nHg : Gives x y\n============================\nexists y0 : Thing, Gives y0 y /\\ (y0 = x \\/ clos_trans Thing Gives x y0)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nx : Thing\ny : Thing\nz : Thing\nHxy : clos_trans Thing Gives x y\nHyz : clos_trans Thing Gives y z\nIH1 : exists y0 : Thing, Gives y0 y /\\ (y0 = x \\/ clos_trans Thing Gives x y0)\nIH2 : exists y0 : Thing, Gives y0 z /\\ (y0 = y \\/ clos_trans Thing Gives y y0)\n============================\nexists y0 : Thing, Gives y0 z /\\ (y0 = x \\/ clos_trans Thing Gives x y0)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nx : Thing\ny : Thing\nHg : Gives x y\n============================\nGives x y /\\ (x = x \\/ clos_trans Thing Gives x x)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nx : Thing\ny : Thing\nz : Thing\nHxy : clos_trans Thing Gives x y\nHyz : clos_trans Thing Gives y z\nIH1 : exists y0 : Thing, Gives y0 y /\\ (y0 = x \\/ clos_trans Thing Gives x y0)\nw : Thing\nHw : Gives w z\nHw1 : w = y\n============================\nexists y0 : Thing, Gives y0 z /\\ (y0 = x \\/ clos_trans Thing Gives x y0)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nx : Thing\ny : Thing\nz : Thing\nHxy : clos_trans Thing Gives x y\nHyz : clos_trans Thing Gives y z\nIH1 : exists y0 : Thing, Gives y0 y /\\ (y0 = x \\/ clos_trans Thing Gives x y0)\nw : Thing\nHw : Gives w z\nHw1 : clos_trans Thing Gives y w\n============================\nexists y0 : Thing, Gives y0 z /\\ (y0 = x \\/ clos_trans Thing Gives x y0)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nx : Thing\ny : Thing\nz : Thing\nHxy : clos_trans Thing Gives x y\nHyz : clos_trans Thing Gives y z\nIH1 : exists y0 : Thing, Gives y0 y /\\ (y0 = x \\/ clos_trans Thing Gives x y0)\nw : Thing\nHw : Gives w z\nHw1 : w = y\n============================\nGives w z /\\ (w = x \\/ clos_trans Thing Gives x w)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nx : Thing\ny : Thing\nz : Thing\nHxy : clos_trans Thing Gives x y\nHyz : clos_trans Thing Gives y z\nIH1 : exists y0 : Thing, Gives y0 y /\\ (y0 = x \\/ clos_trans Thing Gives x y0)\nw : Thing\nHw : Gives w z\nHw1 : w = y\n============================\nclos_trans Thing Gives x w","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nx : Thing\ny : Thing\nz : Thing\nHxy : clos_trans Thing Gives x y\nHyz : clos_trans Thing Gives y z\nIH1 : exists y0 : Thing, Gives y0 y /\\ (y0 = x \\/ clos_trans Thing Gives x y0)\nHw : Gives y z\n============================\nclos_trans Thing Gives x y","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nx : Thing\ny : Thing\nz : Thing\nHxy : clos_trans Thing Gives x y\nHyz : clos_trans Thing Gives y z\nIH1 : exists y0 : Thing, Gives y0 y /\\ (y0 = x \\/ clos_trans Thing Gives x y0)\nw : Thing\nHw : Gives w z\nHw1 : clos_trans Thing Gives y w\n============================\nGives w z /\\ (w = x \\/ clos_trans Thing Gives x w)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nx : Thing\ny : Thing\nz : Thing\nHxy : clos_trans Thing Gives x y\nHyz : clos_trans Thing Gives y z\nIH1 : exists y0 : Thing, Gives y0 y /\\ (y0 = x \\/ clos_trans Thing Gives x y0)\nw : Thing\nHw : Gives w z\nHw1 : clos_trans Thing Gives y w\n============================\nclos_trans Thing Gives x w","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\n============================\nc <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\n============================\n~ MemberOf c A /\\ c <> A /\\ ~ Possible c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\ne : c = A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nHA : AggregateOfPossibles A\nHG : Gives A A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\n============================\n~ MemberOf c A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\nHnm : ~ MemberOf c A\n============================\n~ MemberOf c A /\\ c <> A /\\ ~ Possible c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\nHm : MemberOf c A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\nHm : MemberOf c A\nHc : clos_trans Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\nHm : MemberOf c A\nHc : clos_trans Thing Gives c c\ny : Thing\nHy : Gives y c\ne : y = c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\nHm : MemberOf c A\nHc : clos_trans Thing Gives c c\ny : Thing\nHy : Gives y c\nHy2 : clos_trans Thing Gives c y\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\nHm : MemberOf c A\nHc : clos_trans Thing Gives c c\nHy : Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\nHnm : ~ MemberOf c A\n============================\n~ Possible c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\nHnm : ~ MemberOf c A\nHp : Possible c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA : AggregateOfPossibles A\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\nHnm : ~ MemberOf c A\nHp : Possible c\n============================\nMemberOf c A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH11 : MP011\nA : Thing\nc : Thing\nHA1 : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHG : Gives c A\nL1 : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nL2 : forall a b : Thing, clos_trans Thing Gives a b -> exists y : Thing, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)\nHne : c <> A\nHnm : ~ MemberOf c A\nHp : Possible c\n============================\nMemberOf c A"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H11 A c HA HG.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 H11 A c HA HG.\nassert (L1: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b H. induction H as [x y Hg | x y z Hxy IH1 Hyz IH2].\n  - apply H2; exact Hg.\n  - eapply H3; [exact IH2 | exact IH1]. }\nassert (L2: forall a b, clos_trans Thing Gives a b -> exists y, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)).\n{ intros a b H. induction H as [x y Hg | x y z Hxy IH1 Hyz IH2].\n  - exists x. split; [exact Hg | left; reflexivity].\n  - destruct IH2 as [w [Hw [Hw1 | Hw1]]].\n    + exists w. split; [exact Hw | right]. subst w. exact Hxy.\n    + exists w. split; [exact Hw | right]. eapply t_trans; [exact Hxy | exact Hw1]. }\nassert (Hne: c <> A).\n{ intro e. subst. exact (H1 _ _ HG (H2 _ _ HG)). }\nassert (Hnm: ~ MemberOf c A).\n{ intro Hm. pose proof (H11 _ _ _ HG Hm) as Hc.\n  destruct (L2 _ _ Hc) as [y [Hy [e | Hy2]]].\n  - subst y. exact (H1 _ _ Hy (H2 _ _ Hy)).\n  - exact (H1 _ _ Hy (L1 _ _ Hy2)). }\nrepeat split; auto.\nintro Hp. apply Hnm. destruct HA as [HA1 _]. apply HA1; auto."},{"g":1,"t":"assert (L1: forall a b, clos_trans Thing Gives a b -> Through b a).","w":"model","kind":"step","ok":true,"c":[2,3],"d":"Tactic 2 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros a b H.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 3 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"induction H as [x y Hg | x y z Hxy IH1 Hyz IH2].","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 4 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"apply H2; exact Hg.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"eapply H3; [exact IH2 | exact IH1].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"assert (L2: forall a b, clos_trans Thing Gives a b -> exists y, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)).","w":"model","kind":"step","ok":true,"c":[7,8],"d":"Tactic 7 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"intros a b H.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 8 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"induction H as [x y Hg | x y z Hxy IH1 Hyz IH2].","w":"model","kind":"step","ok":true,"c":[10,11],"d":"Tactic 9 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"exists x.","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 10 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"split; [exact Hg | left; reflexivity].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 11 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"destruct IH2 as [w [Hw [Hw1 | Hw1]]].","w":"model","kind":"step","ok":true,"c":[13,14],"d":"Tactic 12 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"exists w.","w":"model","kind":"step","ok":true,"c":[15],"d":"Tactic 13 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"split; [exact Hw | right].","w":"model","kind":"step","ok":true,"c":[16],"d":"Tactic 14 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"subst w.","w":"model","kind":"step","ok":true,"c":[17],"d":"Tactic 15 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"exact Hxy.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 16 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"exists w.","w":"model","kind":"step","ok":true,"c":[18],"d":"Tactic 17 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"split; [exact Hw | right].","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 18 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"eapply t_trans; [exact Hxy | exact Hw1].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 19 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"assert (Hne: c <> A).","w":"model","kind":"step","ok":true,"c":[20,21],"d":"Tactic 20 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"intro e.","w":"model","kind":"step","ok":true,"c":[22],"d":"Tactic 21 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"subst.","w":"model","kind":"step","ok":true,"c":[23],"d":"Tactic 22 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"exact (H1 _ _ HG (H2 _ _ HG)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 23 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":21,"t":"assert (Hnm: ~ MemberOf c A).","w":"model","kind":"step","ok":true,"c":[24,25],"d":"Tactic 24 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"intro Hm.","w":"model","kind":"step","ok":true,"c":[26],"d":"Tactic 25 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":26,"t":"pose proof (H11 _ _ _ HG Hm) as Hc.","w":"model","kind":"step","ok":true,"c":[27],"d":"Tactic 26 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"destruct (L2 _ _ Hc) as [y [Hy [e | Hy2]]].","w":"model","kind":"step","ok":true,"c":[28,29],"d":"Tactic 27 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":28,"t":"subst y.","w":"model","kind":"step","ok":true,"c":[30],"d":"Tactic 28 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":30,"t":"exact (H1 _ _ Hy (H2 _ _ Hy)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 29 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":29,"t":"exact (H1 _ _ Hy (L1 _ _ Hy2)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 30 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"repeat split; auto.","w":"model","kind":"step","ok":true,"c":[31],"d":"Tactic 31 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":31,"t":"intro Hp.","w":"model","kind":"step","ok":true,"c":[32],"d":"Tactic 32 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":32,"t":"apply Hnm.","w":"model","kind":"step","ok":true,"c":[33],"d":"Tactic 33 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":33,"t":"destruct HA as [HA1 _].","w":"model","kind":"step","ok":true,"c":[34],"d":"Tactic 34 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":34,"t":"apply HA1; auto.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 35 of 35 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38],"proofFile":"files/claude/run1/imkan_s9_cause_outside.v"},{"key":"claude1/imkan_s10_cause_necessary","id":"imkan_s10_cause_necessary","run":"claude1","title":"Dalīl al-imkān, step 10","english":"That cause is necessary in itself.","stmt":"MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Necessary c","proved":true,"seconds":67.9,"expansions":5,"calls":33,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Necessary c","MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","MP001 -> MP002 -> MP003 -> forall c, ~ clos_trans Thing Gives c c","MP001 -> MP002 -> forall c A, Gives c A -> c <> A","============================\n(forall x y : Thing, Gives y x -> Through x y) -> (forall x y z : Thing, Through x y -> Through y z -> Through x z) -> forall x y : Thing, clos_trans Thing Gives x y -> Through y x","H2 : forall x y : Thing, Gives y x -> Through x y\nH3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nx : Thing\ny : Thing\nH : clos_trans Thing Gives x y\n============================\nThrough y x","H2 : forall x y : Thing, Gives y x -> Through x y\nH3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\na : Thing\nb : Thing\nHab : Gives a b\n============================\nThrough b a","H2 : forall x y : Thing, Gives y x -> Through x y\nH3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\na : Thing\nb : Thing\nc : Thing\nH1 : clos_trans Thing Gives a b\nH4 : clos_trans Thing Gives b c\nIH1 : Through b a\nIH2 : Through c b\n============================\nThrough c a","H2 : forall x y : Thing, Gives y x -> Through x y\nH3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\na : Thing\nb : Thing\nHab : Gives a b\n============================\nGives a b","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_1n Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_1n Thing Gives c c\nH0 : Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_1n Thing Gives c c\ny : Thing\nH0 : Gives c y\nH4 : clos_trans_1n Thing Gives y c\n============================\nFalse","============================\n(forall x y : Thing, Gives y x -> ~ Through y x) -> (forall x y : Thing, Gives y x -> Through x y) -> forall c A : Thing, Gives c A -> c <> A","H1 : forall x y : Thing, Gives y x -> ~ Through y x\nH2 : forall x y : Thing, Gives y x -> Through x y\nc : Thing\nA : Thing\nH : Gives c A\nE : c = A\n============================\nFalse","H1 : forall x y : Thing, Gives y x -> ~ Through y x\nH2 : forall x y : Thing, Gives y x -> Through x y\nA : Thing\nH : Gives A A\n============================\nFalse"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H11 H12 A c [HA _] HG.\nassert (L: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b H. induction H as [a b H|a b d _ IH1 _ IH2].\n - apply H2; exact H.\n - eapply H3; [exact IH2|exact IH1]. }\nassert (Hne: c <> A).\n{ intro e. subst. exact (H1 _ _ HG (H2 _ _ HG)). }\nsplit.\n- exact (H12 _ _ HG).\n- destruct (classic (Ess c)) as [E|NE]; [exact E|].\n  exfalso.\n  assert (Pc: Possible c) by (split; [exact (H12 _ _ HG)|exact NE]).\n  pose proof (proj2 (HA c Hne) Pc) as M.\n  pose proof (H11 c A c HG M) as C.\n  apply clos_trans_t1n in C.\n  inversion C as [? H|? y ? H H0]; subst.\n  + exact (H1 _ _ H (H2 _ _ H)).\n  + apply clos_t1n_trans in H0. exact (H1 _ _ H (L _ _ H0)).","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: H0 is already used."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","MP001 -> MP002 -> MP003 -> forall c, ~ clos_trans Thing Gives c c","MP001 -> MP002 -> forall c A, Gives c A -> c <> A"],"combine":"intros h1 h2 h3 h11 h12 A c hAg hg. destruct hAg as [hA hnn]. unfold Necessary. split. - exact (h12 _ _ hg). - destruct (classic (Ess c)) as [e|ne]. + exact e. + exfalso. assert (hp : Possible c) by exact (conj (h12 _ _ hg) ne). pose proof (proj2 (hA c (sub3 h1 h2 c A hg)) hp) as hm. exact (sub2 h1 h2 h3 c (h11 c A c hg hm)).","ok":true,"c":[1,2,3],"assembled":true},{"g":1,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP002, MP003.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 1 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"unfold MP002, MP003. intros H2 H3 x y H. induction H as [a b Hab | a b c H1 IH1 H4 IH2].\n- apply H2. exact Hab.\n- apply (H3 c b a); assumption."},{"g":4,"t":"intros H2 H3 x y H.","w":"model","kind":"step","ok":true,"c":[5],"d":"Tactic 2 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"induction H as [a b Hab | a b c H1 IH1 H4 IH2].","w":"model","kind":"step","ok":true,"c":[6,7],"d":"Tactic 3 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"apply H2.","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 4 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"exact Hab.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"apply (H3 c b a); assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"intros H1 H2 H3 c H.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 1 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 c H.\napply clos_trans_t1n in H.\ninversion H; subst.\n- eapply H1; [eassumption | eapply H2; eassumption].\n- eapply H1; [eassumption | eapply imkan_s10_cause_necessary_L11; try eassumption; apply clos_t1n_trans; eassumption]."},{"g":9,"t":"apply clos_trans_t1n in H.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 2 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"inversion H; subst.","w":"model","kind":"step","ok":true,"c":[11,12],"d":"Tactic 3 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"eapply H1; [eassumption | eapply H2; eassumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 4 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"eapply H1; [eassumption | eapply imkan_s10_cause_necessary_L11; try eassumption; apply clos_t1n_trans; eassumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002.","w":"model","kind":"step","ok":true,"c":[13],"d":"Tactic 1 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"unfold MP001, MP002. intros H1 H2 c A H E. subst. exact (H1 _ _ H (H2 _ _ H))."},{"g":13,"t":"intros H1 H2 c A H E.","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 2 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"subst.","w":"model","kind":"step","ok":true,"c":[15],"d":"Tactic 3 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"exact (H1 _ _ H (H2 _ _ H)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 4 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[5,10,11,12,13,14,15,20,21,22,23,24,29,30,31,32],"proofFile":"files/claude/run1/imkan_s10_cause_necessary.v"},{"key":"claude1/imkan_s11_necessary_cause","id":"imkan_s11_necessary_cause","run":"claude1","title":"Dalīl al-imkān, step 11","english":"The aggregate has a cause that is necessary in itself.","stmt":"MP001 -> MP002 -> MP003 -> MP008 -> MP010 -> MP011 -> MP012 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> exists c, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","proved":true,"seconds":124.4,"expansions":1,"calls":42,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP008 -> MP010 -> MP011 -> MP012 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> exists c, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\n============================\nexists c : Thing, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\n============================\nforall a b : Thing, clos_trans Thing Gives a b -> Through b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\n============================\nexists c : Thing, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\na : Thing\nb : Thing\nH : clos_trans Thing Gives a b\n============================\nThrough b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\na : Thing\nb : Thing\nH : Gives a b\n============================\nThrough b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\na : Thing\nb : Thing\nc : Thing\nHa : clos_trans Thing Gives a b\nHb : clos_trans Thing Gives b c\nIHa : Through b a\nIHb : Through c b\n============================\nThrough c a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\n============================\nforall c : Thing, ~ clos_trans Thing Gives c c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\n============================\nexists c : Thing, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nc : Thing\nH : clos_trans Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nc : Thing\nH : clos_trans_1n Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nc : Thing\nH : clos_trans_1n Thing Gives c c\nH0 : Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nc : Thing\nH : clos_trans_1n Thing Gives c c\ny : Thing\nH0 : Gives c y\nH4 : clos_trans_1n Thing Gives y c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nc : Thing\nH : clos_trans_1n Thing Gives c c\ny : Thing\nH0 : Gives c y\nH4 : clos_trans Thing Gives y c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\n============================\nexists c : Thing, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\n============================\n~ Ess A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\n============================\nexists c : Thing, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\n============================\nexists p : Thing, MemberOf p A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\n============================\nforall p : Thing, MemberOf p A -> ~ Ess p","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\np : Thing\nHp : MemberOf p A\n============================\n~ Ess p","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\np : Thing\nHp : MemberOf p A\npn : p <> A\n============================\n~ Ess p","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\np : Thing\nHp : MemberOf p A\npn : p <> A\nPp : Possible p\n============================\n~ Ess p","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\n============================\nexists c0 : Thing, Gives c0 A /\\ Necessary c0 /\\ ~ MemberOf c0 A /\\ c0 <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nN : ~ ThroughAnother A\n============================\nexists c : Thing, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\n============================\nGives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\n============================\nGives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\nEx : Exists c\n============================\nGives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\nEx : Exists c\n============================\nEss c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\nEx : Exists c\nEs : Ess c\n============================\nGives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\nEx : Exists c\nNE : ~ Ess c\n============================\nEss c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\nEx : Exists c\nNE : ~ Ess c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\nEx : Exists c\nNE : ~ Ess c\nPc : Possible c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\nEx : Exists c\nNE : ~ Ess c\nPc : Possible c\nMc : MemberOf c A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\nEx : Exists c\nEs : Ess c\n============================\n~ MemberOf c A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\nEx : Exists c\nEs : Ess c\nNM : ~ MemberOf c A\n============================\nGives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nc : Thing\nHc : Gives c A\nne : c <> A\nEx : Exists c\nEs : Ess c\nMc : MemberOf c A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nN : ~ ThroughAnother A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnM : ~ MemberOf A A\nHE : Exists A\nq : Thing\nHq : q <> A\nHqp : Possible q\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nNC : forall c : Thing, ~ clos_trans Thing Gives c c\nHqm : MemberOf q A\nNEss : ~ Ess A\nN : ~ ThroughAnother A\n============================\nEss A"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H8 H10 H11 H12 A [HA HnM] HE [q [Hq Hqp]].","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 H8 H10 H11 H12 A [HA HnM] HE [q [Hq Hqp]].\nassert (L: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b H. induction H as [a b H|a b c Ha IHa Hb IHb].\n  - apply H2; exact H.\n  - exact (H3 c b a IHb IHa). }\nassert (NC: forall c, ~ clos_trans Thing Gives c c).\n{ intros c H. apply clos_trans_t1n in H. inversion H; subst.\n  - eapply H1; [eassumption | eapply H2; eassumption].\n  - match goal with Hx: clos_trans_1n _ _ _ _ |- _ => apply clos_t1n_trans in Hx end.\n    eapply H1; [eassumption | eapply L; eassumption]. }\nassert (Hqm: MemberOf q A) by (apply (proj2 (HA q Hq)); exact Hqp).\nassert (NEss: ~ Ess A).\n{ apply H10.\n  - exists q; exact Hqm.\n  - intros p Hp. assert (pn: p <> A) by (intro e; subst; exact (HnM Hp)).\n    pose proof (proj1 (HA p pn) Hp) as Pp. exact (proj2 Pp). }\ndestruct (classic (ThroughAnother A)) as [[c Hc]|N].\n- exists c.\n  assert (ne: c <> A) by (intro e; subst; eapply H1; [exact Hc | eapply H2; exact Hc]).\n  assert (Ex: Exists c) by (eapply H12; exact Hc).\n  assert (Es: Ess c).\n  { destruct (classic (Ess c)) as [E|NE]; [exact E|].\n    exfalso.\n    assert (Pc: Possible c) by (split; assumption).\n    pose proof (proj2 (HA c ne) Pc) as Mc.\n    exact (NC c (H11 c A c Hc Mc)). }\n  assert (NM: ~ MemberOf c A).\n  { intro Mc. exact (NC c (H11 c A c Hc Mc)). }\n  repeat split; try assumption.\n- exfalso. apply NEss. apply H8; assumption."},{"g":1,"t":"assert (L: forall a b, clos_trans Thing Gives a b -> Through b a).","w":"model","kind":"step","ok":true,"c":[2,3],"d":"Tactic 2 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros a b H.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 3 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"induction H as [a b H|a b c Ha IHa Hb IHb].","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 4 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"apply H2; exact H.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"exact (H3 c b a IHb IHa).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"assert (NC: forall c, ~ clos_trans Thing Gives c c).","w":"model","kind":"step","ok":true,"c":[7,8],"d":"Tactic 7 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"intros c H.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 8 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"apply clos_trans_t1n in H.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 9 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"inversion H; subst.","w":"model","kind":"step","ok":true,"c":[11,12],"d":"Tactic 10 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"eapply H1; [eassumption | eapply H2; eassumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 11 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"match goal with Hx: clos_trans_1n _ _ _ _ |- _ => apply clos_t1n_trans in Hx end.","w":"model","kind":"step","ok":true,"c":[13],"d":"Tactic 12 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"eapply H1; [eassumption | eapply L; eassumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 13 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"assert (Hqm: MemberOf q A) by (apply (proj2 (HA q Hq)); exact Hqp).","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 14 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"assert (NEss: ~ Ess A).","w":"model","kind":"step","ok":true,"c":[15,16],"d":"Tactic 15 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"apply H10.","w":"model","kind":"step","ok":true,"c":[17,18],"d":"Tactic 16 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"exists q; exact Hqm.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 17 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"intros p Hp.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 18 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"assert (pn: p <> A) by (intro e; subst; exact (HnM Hp)).","w":"model","kind":"step","ok":true,"c":[20],"d":"Tactic 19 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"pose proof (proj1 (HA p pn) Hp) as Pp.","w":"model","kind":"step","ok":true,"c":[21],"d":"Tactic 20 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":21,"t":"exact (proj2 Pp).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 21 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"destruct (classic (ThroughAnother A)) as [[c Hc]|N].","w":"model","kind":"step","ok":true,"c":[22,23],"d":"Tactic 22 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"exists c.","w":"model","kind":"step","ok":true,"c":[24],"d":"Tactic 23 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"assert (ne: c <> A) by (intro e; subst; eapply H1; [exact Hc | eapply H2; exact Hc]).","w":"model","kind":"step","ok":true,"c":[25],"d":"Tactic 24 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"assert (Ex: Exists c) by (eapply H12; exact Hc).","w":"model","kind":"step","ok":true,"c":[26],"d":"Tactic 25 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":26,"t":"assert (Es: Ess c).","w":"model","kind":"step","ok":true,"c":[27,28],"d":"Tactic 26 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"destruct (classic (Ess c)) as [E|NE]; [exact E|].","w":"model","kind":"step","ok":true,"c":[29],"d":"Tactic 27 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":29,"t":"exfalso.","w":"model","kind":"step","ok":true,"c":[30],"d":"Tactic 28 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":30,"t":"assert (Pc: Possible c) by (split; assumption).","w":"model","kind":"step","ok":true,"c":[31],"d":"Tactic 29 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":31,"t":"pose proof (proj2 (HA c ne) Pc) as Mc.","w":"model","kind":"step","ok":true,"c":[32],"d":"Tactic 30 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":32,"t":"exact (NC c (H11 c A c Hc Mc)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 31 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":28,"t":"assert (NM: ~ MemberOf c A).","w":"model","kind":"step","ok":true,"c":[33,34],"d":"Tactic 32 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":33,"t":"intro Mc.","w":"model","kind":"step","ok":true,"c":[35],"d":"Tactic 33 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":35,"t":"exact (NC c (H11 c A c Hc Mc)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 34 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":34,"t":"repeat split; try assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 35 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"exfalso.","w":"model","kind":"step","ok":true,"c":[36],"d":"Tactic 36 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":36,"t":"apply NEss.","w":"model","kind":"step","ok":true,"c":[37],"d":"Tactic 37 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":37,"t":"apply H8; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 38 of 38 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41],"proofFile":"files/claude/run1/imkan_s11_necessary_cause.v"},{"key":"claude1/imkan_s12_necessary_exists","id":"imkan_s12_necessary_exists","run":"claude1","title":"Dalīl al-imkān, step 12 (aggregate route)","english":"A Necessary Existent exists.","stmt":"MP001 -> MP002 -> MP003 -> MP008 -> MP009 -> MP010 -> MP011 -> MP012 -> SP001 -> exists n, Necessary n","proved":true,"seconds":138.5,"expansions":9,"calls":75,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP008 -> MP009 -> MP010 -> MP011 -> MP012 -> SP001 -> exists n, Necessary n","MP001 -> MP002 -> MP003 -> forall c, ~ clos_trans Thing Gives c c","MP001 -> MP002 -> MP008 -> MP009 -> MP012 -> SP001 -> ~ (exists n, Necessary n) -> exists c A, Gives c A /\\ MemberOf c A","MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> Through b a","forall c, clos_trans Thing Gives c c -> exists y, Gives y c /\\ (y = c \\/ clos_trans Thing Gives c y)","forall a b : Thing, Gives a b -> exists y, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)","forall a m b y : Thing, clos_trans Thing Gives a m -> Gives y b -> (y = m \\/ clos_trans Thing Gives m y) -> (y = a \\/ clos_trans Thing Gives a y)","============================\n(forall x y : Thing, Gives y x -> Through x y) -> (forall x y z : Thing, Through x y -> Through y z -> Through x z) -> forall a b : Thing, clos_trans Thing Gives a b -> Through b a","A2 : forall x y : Thing, Gives y x -> Through x y\nA3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\na : Thing\nb : Thing\nH : clos_trans Thing Gives a b\n============================\nThrough b a","A2 : forall x y : Thing, Gives y x -> Through x y\nA3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nx : Thing\ny : Thing\nHg : Gives x y\n============================\nThrough y x","A2 : forall x y : Thing, Gives y x -> Through x y\nA3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nx : Thing\ny : Thing\nz : Thing\nH1 : clos_trans Thing Gives x y\nH2 : clos_trans Thing Gives y z\nIH1 : Through y x\nIH2 : Through z y\n============================\nThrough z x","A2 : forall x y : Thing, Gives y x -> Through x y\nA3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nx : Thing\ny : Thing\nHg : Gives x y\n============================\nGives x y","A2 : forall x y : Thing, Gives y x -> Through x y\nA3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nx : Thing\ny : Thing\nz : Thing\nH1 : clos_trans Thing Gives x y\nH2 : clos_trans Thing Gives y z\nIH1 : Through y x\nIH2 : Through z y\n============================\nThrough z y","A2 : forall x y : Thing, Gives y x -> Through x y\nA3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nx : Thing\ny : Thing\nz : Thing\nH1 : clos_trans Thing Gives x y\nH2 : clos_trans Thing Gives y z\nIH1 : Through y x\nIH2 : Through z y\n============================\nThrough y x","a : Thing\nm : Thing\nb : Thing\ny : Thing\nH1 : clos_trans Thing Gives a m\nH2 : Gives y b\nH3 : y = m \\/ clos_trans Thing Gives m y\n============================\ny = a \\/ clos_trans Thing Gives a y","a : Thing\nm : Thing\nb : Thing\ny : Thing\nH1 : clos_trans Thing Gives a m\nH2 : Gives y b\nE : y = m\n============================\ny = a \\/ clos_trans Thing Gives a y","a : Thing\nm : Thing\nb : Thing\ny : Thing\nH1 : clos_trans Thing Gives a m\nH2 : Gives y b\nH3 : clos_trans Thing Gives m y\n============================\ny = a \\/ clos_trans Thing Gives a y","a : Thing\nm : Thing\nb : Thing\ny : Thing\nH1 : clos_trans Thing Gives a m\nH2 : Gives y b\nE : y = m\n============================\nclos_trans Thing Gives a y","a : Thing\nm : Thing\nb : Thing\nH1 : clos_trans Thing Gives a m\nH2 : Gives m b\n============================\nclos_trans Thing Gives a m","a : Thing\nm : Thing\nb : Thing\ny : Thing\nH1 : clos_trans Thing Gives a m\nH2 : Gives y b\nH3 : clos_trans Thing Gives m y\n============================\nclos_trans Thing Gives a y","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nS1 : SP001\nHN : ~ exists n : Thing, Necessary n\n============================\nexists c A : Thing, Gives c A /\\ MemberOf c A","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\n============================\nexists c A : Thing, Gives c A /\\ MemberOf c A","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\n============================\nforall y : Thing, Exists y -> Possible y","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\n============================\nexists c A : Thing, Gives c A /\\ MemberOf c A","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\ny : Thing\nHy : Exists y\n============================\nPossible y","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\ny : Thing\nHy : Exists y\n============================\n~ Ess y","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\ny : Thing\nHy : Exists y\nHe : Ess y\n============================\nFalse","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\ny : Thing\nHy : Exists y\nHe : Ess y\n============================\nexists n : Thing, Necessary n","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\ny : Thing\nHy : Exists y\nHe : Ess y\n============================\nNecessary y","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\n============================\nforall y : Thing, Exists y -> ThroughAnother y","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\n============================\nexists c A : Thing, Gives c A /\\ MemberOf c A","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\ny : Thing\nHy : Exists y\n============================\nThroughAnother y","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\ny : Thing\nHy : Exists y\n============================\n~ ~ ThroughAnother y","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\ny : Thing\nHy : Exists y\nHn : ~ ThroughAnother y\n============================\nFalse","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\ny : Thing\nHy : Exists y\nHn : ~ ThroughAnother y\n============================\nexists n : Thing, Necessary n","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\ny : Thing\nHy : Exists y\nHn : ~ ThroughAnother y\n============================\nNecessary y","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHAA : AggregateOfPossibles A\nHEA : Exists A\n============================\nexists c A0 : Thing, Gives c A0 /\\ MemberOf c A0","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHEA : Exists A\n============================\nexists c A0 : Thing, Gives c A0 /\\ MemberOf c A0","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHEA : Exists A\nc : Thing\nHc : Gives c A\n============================\nexists c0 A0 : Thing, Gives c0 A0 /\\ MemberOf c0 A0","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHEA : Exists A\nc : Thing\nHc : Gives c A\n============================\nGives c A /\\ MemberOf c A","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHEA : Exists A\nc : Thing\nHc : Gives c A\n============================\nMemberOf c A","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHEA : Exists A\nc : Thing\nHc : Gives c A\n============================\nc <> A","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHEA : Exists A\nc : Thing\nHc : Gives c A\nHne : c <> A\n============================\nMemberOf c A","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHEA : Exists A\nc : Thing\nHc : Gives c A\ne : c = A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHEA : Exists A\nHc : Gives A A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHEA : Exists A\nc : Thing\nHc : Gives c A\nHne : c <> A\n============================\nPossible c","H1 : MP001\nH2 : MP002\nH8 : MP008\nH9 : MP009\nH12 : MP012\nx : Thing\nHx : Exists x\nHN : ~ exists n : Thing, Necessary n\nHP : forall y : Thing, Exists y -> Possible y\nHT : forall y : Thing, Exists y -> ThroughAnother y\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHEA : Exists A\nc : Thing\nHc : Gives c A\nHne : c <> A\n============================\nExists c"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H8 H9 H10 H11 H12 S.\nunfold MP001, MP002, MP003, MP008, MP009, MP010, MP011, MP012, SP001 in *.\napply NNPP; intro Hn.\nassert (HP: forall x, Exists x -> ~ Ess x).\n{ intros x hx he; apply Hn; exists x; unfold Necessary; split; assumption. }\nassert (HT: forall x, Exists x -> ThroughAnother x).\n{ intros x hx; apply NNPP; intro h; apply (HP x hx); apply H8; assumption. }\nassert (HC: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b h; induction h as [a b hg | a b d h1 IH1 h2 IH2].\n  - apply H2; exact hg.\n  - apply (H3 d b a IH2 IH1). }\ndestruct S as [s hs].\nassert (HPos: exists p, Possible p).\n{ exists s; unfold Possible; split; [exact hs | apply HP; exact hs]. }\ndestruct (H9 HPos) as [A [[HA1 HA2] HAe]].\ndestruct (HT A HAe) as [c hc].\nassert (hce: Exists c) by exact (H12 c A hc).\nassert (hcA: c <> A).\n{ intro e; subst c; exact (H1 _ _ hc (H2 _ _ hc)). }\nassert (hm: MemberOf c A).\n{ apply (proj2 (HA1 c hcA)); unfold Possible; split; [exact hce | apply HP; exact hce]. }\npose proof (H11 c A c hc hm) as cyc.\napply clos_trans_tn1 in cyc.\ndestruct cyc as [hg | y hg h].\n- exact (H1 c c hg (H2 c c hg)).\n- exact (H1 c y hg (HC c y (clos_n1_trans _ _ _ _ h))).","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: In environment Thing : Type"},{"g":0,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> MP003 -> forall c, ~ clos_trans Thing Gives c c","MP001 -> MP002 -> MP008 -> MP009 -> MP012 -> SP001 -> ~ (exists n, Necessary n) -> exists c A, Gives c A /\\ MemberOf c A"],"combine":"intros H1 H2 H3 H8 H9 H10 H11 H12 S1. apply NNPP. intro HN. destruct (sub2 H1 H2 H8 H9 H12 S1 HN) as [c [A [G M]]]. exact (sub1 H1 H2 H3 c (H11 c A c G M)).","ok":true,"c":[1,2],"assembled":true},{"g":1,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"intros H1 H2 H3 c H.\nassert (key: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b h. induction h as [x y hg | x y z h1 IH1 h2 IH2].\n  - apply H2. exact hg.\n  - eapply H3; [exact IH2 | exact IH1]. }\napply clos_trans_t1n_iff in H.\ninversion H as [x y hg | x y z hg hr]; subst.\n- exact (H1 _ _ hg (H2 _ _ hg)).\n- apply clos_trans_t1n_iff in hr. exact (H1 _ _ hg (key _ _ hr)).","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: The variable hg was not found in the current environment."},{"g":1,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> Through b a","forall c, clos_trans Thing Gives c c -> exists y, Gives y c /\\ (y = c \\/ clos_trans Thing Gives c y)"],"combine":"intros H1 H2 H3 c Hc. destruct (sub2 c Hc) as [y [Hg [->|Hy]]].\n- exact (H1 _ _ Hg (H2 _ _ Hg)).\n- exact (H1 _ _ Hg (sub1 H2 H3 c y Hy)).","ok":true,"c":[3,4],"assembled":true},{"g":3,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP002, MP003.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 1 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"unfold MP002, MP003. intros A2 A3 a b H. induction H as [x y Hg | x y z H1 IH1 H2 IH2].\n- apply A2. exact Hg.\n- apply A3 with y.\n  + exact IH2.\n  + exact IH1."},{"g":7,"t":"intros A2 A3 a b H.","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 2 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"induction H as [x y Hg | x y z H1 IH1 H2 IH2].","w":"model","kind":"step","ok":true,"c":[9,10],"d":"Tactic 3 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"apply A2.","w":"model","kind":"step","ok":true,"c":[11],"d":"Tactic 4 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"exact Hg.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"apply A3 with y.","w":"model","kind":"step","ok":true,"c":[12,13],"d":"Tactic 6 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"exact IH2.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"exact IH1.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 8 of 8 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"intros c H. apply clos_trans_tn1 in H. inversion H; subst.\n- exists c; split; [assumption | left; reflexivity].\n- exists y; split; [assumption | right; apply clos_trans_n1_trans; assumption].","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: The variable clos_trans_n1_trans was not found in the current environment."},{"g":4,"t":"intros c H. apply clos_trans_tn1 in H. destruct H as [y H0 | y z H0 H1].\n- exists c; split; [exact H0 | left; reflexivity].\n- exists y; split; [exact H0 | right; apply clos_trans_n1_trans; exact H1].","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: In environment Thing : Type"},{"g":4,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall a b : Thing, Gives a b -> exists y, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)","forall a m b y : Thing, clos_trans Thing Gives a m -> Gives y b -> (y = m \\/ clos_trans Thing Gives m y) -> (y = a \\/ clos_trans Thing Gives a y)"],"combine":"intros c H. cut (forall a b, clos_trans Thing Gives a b -> exists y, Gives y b /\\ (y = a \\/ clos_trans Thing Gives a y)).\n- intro G. exact (G c c H).\n- intros a b Hab. induction Hab as [a b Hg | a m b H1 IH1 H2 IH2].\n  + exact (sub1 a b Hg).\n  + destruct IH2 as [y [Hy Hor]]. exists y. split; [exact Hy | exact (sub2 a m b y H1 Hy Hor)].","ok":true,"c":[5,6],"assembled":true},{"g":5,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":6,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":6,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":6,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":6,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":6,"t":"intros a m b y H1 H2 H3.","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 1 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros a m b y H1 H2 H3. destruct H3 as [E|H3].\n- right. subst. exact H1.\n- right. eapply t_trans; eauto."},{"g":14,"t":"destruct H3 as [E|H3].","w":"model","kind":"step","ok":true,"c":[15,16],"d":"Tactic 2 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"right.","w":"model","kind":"step","ok":true,"c":[17],"d":"Tactic 3 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"subst.","w":"model","kind":"step","ok":true,"c":[18],"d":"Tactic 4 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"exact H1.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"right.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 6 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"eapply t_trans; eauto.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"intros H1 H2 H8 H9 H12 S1 HN.","w":"model","kind":"step","ok":true,"c":[20],"d":"Tactic 1 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H8 H9 H12 S1 HN.\ndestruct S1 as [x Hx].\nassert (HP: forall y, Exists y -> Possible y).\n{ intros y Hy. split; [exact Hy|]. intro He. apply HN. exists y. split; assumption. }\nassert (HT: forall y, Exists y -> ThroughAnother y).\n{ intros y Hy. apply NNPP. intro Hn. apply HN. exists y. split; [exact Hy | apply H8; assumption]. }\ndestruct (H9 (ex_intro _ x (HP x Hx))) as [A [HAA HEA]].\ndestruct HAA as [HA _].\ndestruct (HT A HEA) as [c Hc].\nexists c, A. split; [exact Hc|].\nassert (Hne: c <> A).\n{ intro e. subst c. exact (H1 _ _ Hc (H2 _ _ Hc)). }\napply (proj2 (HA c Hne)). apply HP. exact (H12 _ _ Hc)."},{"g":20,"t":"destruct S1 as [x Hx].","w":"model","kind":"step","ok":true,"c":[21],"d":"Tactic 2 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":21,"t":"assert (HP: forall y, Exists y -> Possible y).","w":"model","kind":"step","ok":true,"c":[22,23],"d":"Tactic 3 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"intros y Hy.","w":"model","kind":"step","ok":true,"c":[24],"d":"Tactic 4 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"split; [exact Hy|].","w":"model","kind":"step","ok":true,"c":[25],"d":"Tactic 5 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"intro He.","w":"model","kind":"step","ok":true,"c":[26],"d":"Tactic 6 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":26,"t":"apply HN.","w":"model","kind":"step","ok":true,"c":[27],"d":"Tactic 7 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"exists y.","w":"model","kind":"step","ok":true,"c":[28],"d":"Tactic 8 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":28,"t":"split; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 9 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"assert (HT: forall y, Exists y -> ThroughAnother y).","w":"model","kind":"step","ok":true,"c":[29,30],"d":"Tactic 10 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":29,"t":"intros y Hy.","w":"model","kind":"step","ok":true,"c":[31],"d":"Tactic 11 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":31,"t":"apply NNPP.","w":"model","kind":"step","ok":true,"c":[32],"d":"Tactic 12 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":32,"t":"intro Hn.","w":"model","kind":"step","ok":true,"c":[33],"d":"Tactic 13 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":33,"t":"apply HN.","w":"model","kind":"step","ok":true,"c":[34],"d":"Tactic 14 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":34,"t":"exists y.","w":"model","kind":"step","ok":true,"c":[35],"d":"Tactic 15 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":35,"t":"split; [exact Hy | apply H8; assumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 16 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":30,"t":"destruct (H9 (ex_intro _ x (HP x Hx))) as [A [HAA HEA]].","w":"model","kind":"step","ok":true,"c":[36],"d":"Tactic 17 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":36,"t":"destruct HAA as [HA _].","w":"model","kind":"step","ok":true,"c":[37],"d":"Tactic 18 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":37,"t":"destruct (HT A HEA) as [c Hc].","w":"model","kind":"step","ok":true,"c":[38],"d":"Tactic 19 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":38,"t":"exists c, A.","w":"model","kind":"step","ok":true,"c":[39],"d":"Tactic 20 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":39,"t":"split; [exact Hc|].","w":"model","kind":"step","ok":true,"c":[40],"d":"Tactic 21 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":40,"t":"assert (Hne: c <> A).","w":"model","kind":"step","ok":true,"c":[41,42],"d":"Tactic 22 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":41,"t":"intro e.","w":"model","kind":"step","ok":true,"c":[43],"d":"Tactic 23 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":43,"t":"subst c.","w":"model","kind":"step","ok":true,"c":[44],"d":"Tactic 24 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":44,"t":"exact (H1 _ _ Hc (H2 _ _ Hc)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 25 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":42,"t":"apply (proj2 (HA c Hne)).","w":"model","kind":"step","ok":true,"c":[45],"d":"Tactic 26 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":45,"t":"apply HP.","w":"model","kind":"step","ok":true,"c":[46],"d":"Tactic 27 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":46,"t":"exact (H12 _ _ Hc).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 28 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[5,11,16,17,18,19,20,21,22,23,30,31,36,37,38,39,40,41,42,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74],"proofFile":"files/claude/run1/imkan_s12_necessary_exists.v"},{"key":"claude1/imkan_b_necessary_exists","id":"imkan_b_necessary_exists","run":"claude1","title":"Dalīl al-imkān, conclusion (shorter route)","english":"A Necessary Existent exists.","stmt":"MP008 -> OP001 -> SP001 -> exists n, Necessary n","proved":true,"seconds":8.3,"expansions":1,"calls":14,"root":0,"goals":["MP008 -> OP001 -> SP001 -> exists n, Necessary n","H8 : MP008\nH1 : OP001\nHs : SP001\n============================\nexists n : Thing, Necessary n","H8 : MP008\nH1 : OP001\nx : Thing\nHx : Exists x\n============================\nexists n : Thing, Necessary n","H8 : MP008\nH1 : OP001\nx : Thing\nHx : Exists x\ny : Thing\nHy : Gives y x\n============================\nexists n : Thing, Necessary n","H8 : MP008\nH1 : OP001\nx : Thing\nHx : Exists x\nHn : ~ ThroughAnother x\n============================\nexists n : Thing, Necessary n","H8 : MP008\nH1 : OP001\nx : Thing\nHx : Exists x\ny : Thing\nHy : Gives y x\nH2 : OfItself y\n============================\nexists n : Thing, Necessary n","H8 : MP008\nH1 : OP001\nx : Thing\nHx : Exists x\ny : Thing\nHy : Gives y x\nH2 : Exists y /\\ ~ ThroughAnother y\n============================\nexists n : Thing, Necessary n","H8 : MP008\nH1 : OP001\nx : Thing\nHx : Exists x\ny : Thing\nHy : Gives y x\nHe : Exists y\nHn2 : ~ ThroughAnother y\n============================\nexists n : Thing, Necessary n","H8 : MP008\nH1 : OP001\nx : Thing\nHx : Exists x\ny : Thing\nHy : Gives y x\nHe : Exists y\nHn2 : ~ ThroughAnother y\n============================\nNecessary y","H8 : MP008\nH1 : OP001\nx : Thing\nHx : Exists x\nHn : ~ ThroughAnother x\n============================\nNecessary x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H8 H1 Hs.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H8 H1 Hs. destruct Hs as [x Hx].\ndestruct (classic (ThroughAnother x)) as [[y Hy]|Hn].\n- pose proof (H1 x y Hy) as H2. unfold OfItself in H2. destruct H2 as [He Hn2].\n  exists y. split; [exact He | apply H8; assumption].\n- exists x. split; [exact Hx | apply H8; assumption]."},{"g":1,"t":"destruct Hs as [x Hx].","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"destruct (classic (ThroughAnother x)) as [[y Hy]|Hn].","w":"model","kind":"step","ok":true,"c":[3,4],"d":"Tactic 3 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"pose proof (H1 x y Hy) as H2.","w":"model","kind":"step","ok":true,"c":[5],"d":"Tactic 4 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"unfold OfItself in H2.","w":"model","kind":"step","ok":true,"c":[6],"d":"Tactic 5 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"destruct H2 as [He Hn2].","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 6 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"exists y.","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 7 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"split; [exact He | apply H8; assumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 8 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"exists x.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 9 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"split; [exact Hx | apply H8; assumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 10 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13],"proofFile":"files/claude/run1/imkan_b_necessary_exists.v"},{"key":"claude2/pr001_relational_exhaustive","id":"pr001_relational_exhaustive","run":"claude2","title":"Dalīl al-kalima, premise 1","english":"The four possibilities are the only ones.","stmt":"forall x, (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b) \\/ ((forall y, ~ DependsOn x y) \\/ exists y, clos_trans Thing DependsOn x y /\\ forall z, ~ DependsOn y z)","proved":true,"seconds":17.0,"expansions":1,"calls":38,"root":0,"goals":["forall x, (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b) \\/ ((forall y, ~ DependsOn x y) \\/ exists y, clos_trans Thing DependsOn x y /\\ forall z, ~ DependsOn y z)","x : Thing\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\nH : forall a : Thing, a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\nH : ~ (forall a : Thing, a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\nH : forall a : Thing, a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\nexists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)","x : Thing\nH : forall a : Thing, a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\n(x = x \\/ clos_trans Thing DependsOn x x) /\\ (forall a : Thing, a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)","x : Thing\nH : forall a : Thing, a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\nx = x \\/ clos_trans Thing DependsOn x x","x : Thing\nH : forall a : Thing, a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\nforall a : Thing, a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b","x : Thing\nH : exists n : Thing, ~ (n = x \\/ clos_trans Thing DependsOn x n -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn n b)\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a : Thing, S a -> exists b : Thing, S b /\\ DependsOn a b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a0 : Thing, S a0 -> exists b : Thing, S b /\\ DependsOn a0 b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a0 : Thing, S a0 -> exists b : Thing, S b /\\ DependsOn a0 b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a0 : Thing, S a0 -> exists b : Thing, S b /\\ DependsOn a0 b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\n============================\nforall b : Thing, ~ DependsOn a b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nHno : forall b : Thing, ~ DependsOn a b\n============================\n(exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y : Thing, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ (forall a0 : Thing, S a0 -> exists b : Thing, S b /\\ DependsOn a0 b)) \\/ (forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\nFalse","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\nexists b0 : Thing, (b0 = x \\/ clos_trans Thing DependsOn x b0) /\\ DependsOn a b0","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\n(b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\nb = x \\/ clos_trans Thing DependsOn x b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\nDependsOn a b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\nclos_trans Thing DependsOn x b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nE : a = x\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\nclos_trans Thing DependsOn x b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nHc : clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\nclos_trans Thing DependsOn x b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nE : a = x\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn x b\n============================\nclos_trans Thing DependsOn x b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nE : a = x\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn x b\n============================\nDependsOn x b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nHc : clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\nclos_trans Thing DependsOn x ?y","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nHc : clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\nclos_trans Thing DependsOn a b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nHc : clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nb : Thing\nHab : DependsOn a b\n============================\nDependsOn a b","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nH1 : a = x \\/ clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nHno : forall b : Thing, ~ DependsOn a b\n============================\n(forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nE : a = x\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nHno : forall b : Thing, ~ DependsOn a b\n============================\n(forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nHc : clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nHno : forall b : Thing, ~ DependsOn a b\n============================\n(forall y : Thing, ~ DependsOn x y) \\/ exists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nE : a = x\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nHno : forall b : Thing, ~ DependsOn a b\n============================\nforall y : Thing, ~ DependsOn x y","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nE : a = x\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nHno : forall b : Thing, ~ DependsOn a b\n============================\nforall y : Thing, ~ DependsOn a y","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nHc : clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nHno : forall b : Thing, ~ DependsOn a b\n============================\nexists y : Thing, clos_trans Thing DependsOn x y /\\ (forall z : Thing, ~ DependsOn y z)","x : Thing\na : Thing\nHa : ~ (a = x \\/ clos_trans Thing DependsOn x a -> exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)\nHc : clos_trans Thing DependsOn x a\nHn : ~ exists b : Thing, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b\nHno : forall b : Thing, ~ DependsOn a b\n============================\nclos_trans Thing DependsOn x a /\\ (forall z : Thing, ~ DependsOn a z)"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intro x.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intro x.\ndestruct (classic (forall a, (a = x \\/ clos_trans Thing DependsOn x a) -> exists b, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)) as [H|H].\n- right; right; left. exists (fun a => a = x \\/ clos_trans Thing DependsOn x a). split.\n  + left; reflexivity.\n  + exact H.\n- apply not_all_ex_not in H. destruct H as [a Ha].\n  pose proof (not_imply_elim _ _ Ha) as H1.\n  pose proof (not_imply_elim2 _ _ Ha) as Hn.\n  assert (Hno: forall b, ~ DependsOn a b).\n  { intros b Hab. apply Hn. exists b. split.\n    - right. destruct H1 as [E|Hc].\n      + rewrite E in Hab. apply t_step. exact Hab.\n      + eapply t_trans. exact Hc. apply t_step. exact Hab.\n    - exact Hab. }\n  right; right; right.\n  destruct H1 as [E|Hc].\n  + left. rewrite <- E. exact Hno.\n  + right. exists a. split; assumption."},{"g":1,"t":"destruct (classic (forall a, (a = x \\/ clos_trans Thing DependsOn x a) -> exists b, (b = x \\/ clos_trans Thing DependsOn x b) /\\ DependsOn a b)) as [H|H].","w":"model","kind":"step","ok":true,"c":[2,3],"d":"Tactic 2 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"right; right; left.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 3 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"exists (fun a => a = x \\/ clos_trans Thing DependsOn x a).","w":"model","kind":"step","ok":true,"c":[5],"d":"Tactic 4 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"split.","w":"model","kind":"step","ok":true,"c":[6,7],"d":"Tactic 5 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"left; reflexivity.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"exact H.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"apply not_all_ex_not in H.","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 8 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"destruct H as [a Ha].","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 9 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"pose proof (not_imply_elim _ _ Ha) as H1.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 10 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"pose proof (not_imply_elim2 _ _ Ha) as Hn.","w":"model","kind":"step","ok":true,"c":[11],"d":"Tactic 11 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"assert (Hno: forall b, ~ DependsOn a b).","w":"model","kind":"step","ok":true,"c":[12,13],"d":"Tactic 12 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"intros b Hab.","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 13 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"apply Hn.","w":"model","kind":"step","ok":true,"c":[15],"d":"Tactic 14 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"exists b.","w":"model","kind":"step","ok":true,"c":[16],"d":"Tactic 15 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"split.","w":"model","kind":"step","ok":true,"c":[17,18],"d":"Tactic 16 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"right.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 17 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"destruct H1 as [E|Hc].","w":"model","kind":"step","ok":true,"c":[20,21],"d":"Tactic 18 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"rewrite E in Hab.","w":"model","kind":"step","ok":true,"c":[22],"d":"Tactic 19 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"apply t_step.","w":"model","kind":"step","ok":true,"c":[23],"d":"Tactic 20 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"exact Hab.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 21 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":21,"t":"eapply t_trans.","w":"model","kind":"step","ok":true,"c":[24],"d":"Tactic 22 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"exact Hc.","w":"model","kind":"step","ok":true,"c":[25,18,13],"d":"Tactic 23 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"apply t_step.","w":"model","kind":"step","ok":true,"c":[26],"d":"Tactic 24 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":26,"t":"exact Hab.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 25 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"exact Hab.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 26 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"right; right; right.","w":"model","kind":"step","ok":true,"c":[27],"d":"Tactic 27 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"destruct H1 as [E|Hc].","w":"model","kind":"step","ok":true,"c":[28,29],"d":"Tactic 28 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":28,"t":"left.","w":"model","kind":"step","ok":true,"c":[30],"d":"Tactic 29 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":30,"t":"rewrite <- E.","w":"model","kind":"step","ok":true,"c":[31],"d":"Tactic 30 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":31,"t":"exact Hno.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 31 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":29,"t":"right.","w":"model","kind":"step","ok":true,"c":[32],"d":"Tactic 32 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":32,"t":"exists a.","w":"model","kind":"step","ok":true,"c":[33],"d":"Tactic 33 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":33,"t":"split; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 34 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],"proofFile":"files/claude/run2/pr001_relational_exhaustive.v"},{"key":"claude2/pr002_no_self_creation","id":"pr002_no_self_creation","run":"claude2","title":"Dalīl al-kalima, premise 2","english":"Nothing gives existence to itself.","stmt":"MP001 -> MP002 -> forall x, ~ Gives x x","proved":true,"seconds":0.5,"expansions":0,"calls":1,"root":0,"goals":["MP001 -> MP002 -> forall x, ~ Gives x x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run2/pr002_no_self_creation.v"},{"key":"claude2/pr003_no_circle","id":"pr003_no_circle","run":"claude2","title":"Dalīl al-kalima, premise 3","english":"No chain of existence-giving returns to where it started.","stmt":"MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing DependsOn x x","proved":true,"seconds":18.9,"expansions":1,"calls":14,"root":0,"goals":["MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing DependsOn x x","H1 : MP001\nH2 : MP002\nH3 : MP003\nx : Thing\nH : clos_trans Thing DependsOn x x\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nx : Thing\nH : clos_trans Thing DependsOn x x\n============================\nforall a b : Thing, clos_trans Thing DependsOn a b -> Through a b","H1 : MP001\nH2 : MP002\nH3 : MP003\nx : Thing\nH : clos_trans Thing DependsOn x x\nK : forall a b : Thing, clos_trans Thing DependsOn a b -> Through a b\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nx : Thing\nH : clos_trans Thing DependsOn x x\na : Thing\nb : Thing\nHab : clos_trans Thing DependsOn a b\n============================\nThrough a b","H1 : MP001\nH2 : MP002\nH3 : MP003\nx : Thing\nH : clos_trans Thing DependsOn x x\na : Thing\nb : Thing\nHab : DependsOn a b\n============================\nThrough a b","H1 : MP001\nH2 : MP002\nH3 : MP003\nx : Thing\nH : clos_trans Thing DependsOn x x\na : Thing\nb : Thing\nc : Thing\nIH1 : Through a b\nIH2 : Through b c\n============================\nThrough a c","H1 : MP001\nH2 : MP002\nH3 : MP003\nx : Thing\nH : clos_trans_1n Thing DependsOn x x\nK : forall a b : Thing, clos_trans Thing DependsOn a b -> Through a b\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nx : Thing\nH : clos_trans_1n Thing DependsOn x x\nK : forall a b : Thing, clos_trans Thing DependsOn a b -> Through a b\nH0 : DependsOn x x\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nx : Thing\nH : clos_trans_1n Thing DependsOn x x\nK : forall a b : Thing, clos_trans Thing DependsOn a b -> Through a b\ny : Thing\nH0 : DependsOn x y\nH4 : clos_trans_1n Thing DependsOn y x\n============================\nFalse"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 x H.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 x H.\nassert (K: forall a b, clos_trans Thing DependsOn a b -> Through a b).\n{ intros a b Hab. induction Hab as [a b Hab|a b c _ IH1 _ IH2].\n  - apply H2; exact Hab.\n  - eapply H3; eauto. }\napply clos_trans_t1n_iff in H.\ninversion H; subst.\n- match goal with G: DependsOn x x |- _ => exact (H1 _ _ G (H2 _ _ G)) end.\n- match goal with G: DependsOn x ?y, G2: clos_trans_1n Thing DependsOn ?y x |- _ => exact (H1 _ _ G (K _ _ (proj2 (clos_trans_t1n_iff _ _ _ _) G2))) end."},{"g":1,"t":"assert (K: forall a b, clos_trans Thing DependsOn a b -> Through a b).","w":"model","kind":"step","ok":true,"c":[2,3],"d":"Tactic 2 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros a b Hab.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 3 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"induction Hab as [a b Hab|a b c _ IH1 _ IH2].","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 4 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"apply H2; exact Hab.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"eapply H3; eauto.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"apply clos_trans_t1n_iff in H.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 7 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"inversion H; subst.","w":"model","kind":"step","ok":true,"c":[8,9],"d":"Tactic 8 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"match goal with G: DependsOn x x |- _ => exact (H1 _ _ G (H2 _ _ G)) end.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 9 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"match goal with G: DependsOn x ?y, G2: clos_trans_1n Thing DependsOn ?y x |- _ => exact (H1 _ _ G (K _ _ (proj2 (clos_trans_t1n_iff _ _ _ _) G2))) end.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 10 of 10 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13],"proofFile":"files/claude/run2/pr003_no_circle.v"},{"key":"claude2/pr004_no_two_step_giving","id":"pr004_no_two_step_giving","run":"claude2","title":"Dalīl al-kalima, premise 4 (key step)","english":"What receives existence does not give existence to anything.","stmt":"OP001 -> forall x y z, Gives y x -> ~ Gives z y","proved":true,"seconds":0.5,"expansions":0,"calls":1,"root":0,"goals":["OP001 -> forall x y z, Gives y x -> ~ Gives z y"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run2/pr004_no_two_step_giving.v"},{"key":"claude2/kalima_first_cause","id":"kalima_first_cause","run":"claude2","title":"Dalīl al-kalima, the first cause","english":"If the universe has its existence through another, it has a first cause that exists of itself.","stmt":"OP001 -> ThroughAnother Universe -> exists b, Gives b Universe /\\ OfItself b /\\ (forall y, ~ Gives y b)","proved":true,"seconds":7.2,"expansions":1,"calls":15,"root":0,"goals":["OP001 -> ThroughAnother Universe -> exists b, Gives b Universe /\\ OfItself b /\\ (forall y, ~ Gives y b)","H : OP001\ny : Thing\nHy : Gives y Universe\n============================\nexists b : Thing, Gives b Universe /\\ OfItself b /\\ (forall y0 : Thing, ~ Gives y0 b)","H : OP001\ny : Thing\nHy : Gives y Universe\n============================\nGives y Universe /\\ OfItself y /\\ (forall y0 : Thing, ~ Gives y0 y)","H : forall x y : Thing, Gives y x -> OfItself y\ny : Thing\nHy : Gives y Universe\n============================\nGives y Universe /\\ OfItself y /\\ (forall y0 : Thing, ~ Gives y0 y)","H : forall x y : Thing, Gives y x -> OfItself y\ny : Thing\nHy : Gives y Universe\nHo : OfItself y\n============================\nGives y Universe /\\ OfItself y /\\ (forall y0 : Thing, ~ Gives y0 y)","H : forall x y : Thing, Gives y x -> OfItself y\ny : Thing\nHy : Gives y Universe\nHo : OfItself y\n============================\nOfItself y /\\ (forall y0 : Thing, ~ Gives y0 y)","H : forall x y : Thing, Gives y x -> OfItself y\ny : Thing\nHy : Gives y Universe\nHo : OfItself y\n============================\nforall y0 : Thing, ~ Gives y0 y","H : forall x y : Thing, Gives y x -> OfItself y\ny : Thing\nHy : Gives y Universe\nHo : OfItself y\nz : Thing\nHz : Gives z y\n============================\nFalse","H : forall x y : Thing, Gives y x -> OfItself y\ny : Thing\nHy : Gives y Universe\nHn : ~ ThroughAnother y\nz : Thing\nHz : Gives z y\n============================\nFalse","H : forall x y : Thing, Gives y x -> OfItself y\ny : Thing\nHy : Gives y Universe\nHn : ~ ThroughAnother y\nz : Thing\nHz : Gives z y\n============================\nThroughAnother y","H : forall x y : Thing, Gives y x -> OfItself y\ny : Thing\nHy : Gives y Universe\nHn : ~ ThroughAnother y\nz : Thing\nHz : Gives z y\n============================\nGives z y"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H [y Hy].","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H [y Hy]. exists y. unfold OP001 in H. pose proof (H _ _ Hy) as Ho. split; [exact Hy|]. split; [exact Ho|]. intros z Hz. destruct Ho as [_ Hn]. apply Hn. exists z. exact Hz."},{"g":1,"t":"exists y.","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"unfold OP001 in H.","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 3 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"pose proof (H _ _ Hy) as Ho.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 4 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"split; [exact Hy|].","w":"model","kind":"step","ok":true,"c":[5],"d":"Tactic 5 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"split; [exact Ho|].","w":"model","kind":"step","ok":true,"c":[6],"d":"Tactic 6 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"intros z Hz.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 7 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"destruct Ho as [_ Hn].","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 8 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"apply Hn.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 9 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"exists z.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 10 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"exact Hz.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 11 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14],"proofFile":"files/claude/run2/kalima_first_cause.v"},{"key":"claude2/pr011_something_caused","id":"pr011_something_caused","run":"claude2","title":"The universe is caused","english":"Something in the universe has its existence through another.","stmt":"MP008 -> OP006 -> exists x, InUniverse x /\\ Possible x /\\ ThroughAnother x","proved":true,"seconds":7.1,"expansions":1,"calls":10,"root":0,"goals":["MP008 -> OP006 -> exists x, InUniverse x /\\ Possible x /\\ ThroughAnother x","H8 : MP008\nH6 : OP006\n============================\nexists x : Thing, InUniverse x /\\ Possible x /\\ ThroughAnother x","H8 : MP008\nx : Thing\nHu : InUniverse x\nHp : Possible x\n============================\nexists x0 : Thing, InUniverse x0 /\\ Possible x0 /\\ ThroughAnother x0","H8 : MP008\nx : Thing\nHu : InUniverse x\nHp : Exists x /\\ ~ Ess x\n============================\nexists x0 : Thing, InUniverse x0 /\\ Possible x0 /\\ ThroughAnother x0","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\n============================\nexists x0 : Thing, InUniverse x0 /\\ Possible x0 /\\ ThroughAnother x0","H8 : MP008\nx : Thing\nHu : InUniverse x\nHe : Exists x\nHn : ~ Ess x\n============================\nInUniverse x /\\ Possible x /\\ ThroughAnother x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H8 H6.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H8 H6. destruct H6 as [x [Hu Hp]]. unfold Possible in Hp. destruct Hp as [He Hn]. exists x. split; [exact Hu | split; [unfold Possible; split; assumption | destruct (classic (ThroughAnother x)) as [H|H]; [exact H | exfalso; apply Hn; apply H8; assumption]]]."},{"g":1,"t":"destruct H6 as [x [Hu Hp]].","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"unfold Possible in Hp.","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 3 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"destruct Hp as [He Hn].","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 4 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"exists x.","w":"model","kind":"step","ok":true,"c":[5],"d":"Tactic 5 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"split; [exact Hu | split; [unfold Possible; split; assumption | destruct (classic (ThroughAnother x)) as [H|H]; [exact H | exfalso; apply Hn; apply H8; assumption]]].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9],"proofFile":"files/claude/run2/pr011_something_caused.v"},{"key":"claude2/kalima_unconditional","id":"kalima_unconditional","run":"claude2","title":"Dalīl al-kalima, the conclusion without the “if”","english":"Something in the universe has a giver of existence that is necessary and depends on nothing.","stmt":"MP001 -> MP002 -> MP003 -> OP001 -> MP008 -> OP006 -> exists x b, InUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","proved":true,"seconds":11.7,"expansions":1,"calls":24,"root":0,"goals":["MP001 -> MP002 -> MP003 -> OP001 -> MP008 -> OP006 -> exists x b, InUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : OP001\nH8 : MP008\nH6 : OP006\n============================\nexists x b : Thing, InUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : MP008\nH6 : OP006\n============================\nexists x b : Thing, InUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nH6 : OP006\n============================\nexists x b : Thing, InUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\n============================\nexists x0 b : Thing, InUniverse x0 /\\ Gives b x0 /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\n============================\nThroughAnother x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nHT : ThroughAnother x\n============================\nexists x0 b : Thing, InUniverse x0 /\\ Gives b x0 /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\n============================\n~ ~ ThroughAnother x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nn : ~ ThroughAnother x\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nn : ~ ThroughAnother x\n============================\nEss x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nb : Thing\nHb : Gives b x\n============================\nexists x0 b0 : Thing, InUniverse x0 /\\ Gives b0 x0 /\\ Exists b0 /\\ Necessary b0 /\\ Independent b0","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nb : Thing\nHb : Gives b x\nHeb : Exists b\nHnb : ~ ThroughAnother b\n============================\nexists x0 b0 : Thing, InUniverse x0 /\\ Gives b0 x0 /\\ Exists b0 /\\ Necessary b0 /\\ Independent b0","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nb : Thing\nHb : Gives b x\nHeb : Exists b\nHnb : ~ ThroughAnother b\n============================\nInUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nb : Thing\nHb : Gives b x\nHeb : Exists b\nHnb : ~ ThroughAnother b\n============================\nNecessary b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nb : Thing\nHb : Gives b x\nHeb : Exists b\nHnb : ~ ThroughAnother b\n============================\nIndependent b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nb : Thing\nHb : Gives b x\nHeb : Exists b\nHnb : ~ ThroughAnother b\n============================\nExists b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nb : Thing\nHb : Gives b x\nHeb : Exists b\nHnb : ~ ThroughAnother b\n============================\nEss b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nb : Thing\nHb : Gives b x\nHeb : Exists b\nHnb : ~ ThroughAnother b\ny : Thing\nHy : DependsOn b y\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nb : Thing\nHb : Gives b x\nHeb : Exists b\nHnb : ~ ThroughAnother b\ny : Thing\nHy : DependsOn b y\n============================\nThroughAnother b","H1 : MP001\nH2 : MP002\nH3 : MP003\nH4 : forall x y : Thing, Gives y x -> OfItself y\nH8 : forall x : Thing, Exists x -> ~ ThroughAnother x -> Ess x\nx : Thing\nHu : InUniverse x\nHex : Exists x\nHne : ~ Ess x\nb : Thing\nHb : Gives b x\nHeb : Exists b\nHnb : ~ ThroughAnother b\ny : Thing\nHy : DependsOn b y\n============================\nGives y b"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H4 H8 H6.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 H4 H8 H6.\nunfold OP001 in H4. unfold MP008 in H8.\ndestruct H6 as [x [Hu [Hex Hne]]].\nassert (HT : ThroughAnother x).\n{ apply NNPP. intro n. apply Hne. exact (H8 x Hex n). }\ndestruct HT as [b Hb].\ndestruct (H4 x b Hb) as [Heb Hnb].\nexists x, b.\nrefine (conj Hu (conj Hb (conj Heb (conj _ _)))).\n- split. exact Heb. exact (H8 b Heb Hnb).\n- intros y Hy. apply Hnb. exists y. exact Hy."},{"g":1,"t":"unfold OP001 in H4.","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"unfold MP008 in H8.","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 3 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"destruct H6 as [x [Hu [Hex Hne]]].","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 4 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"assert (HT : ThroughAnother x).","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 5 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"apply NNPP.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 6 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"intro n.","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 7 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"apply Hne.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 8 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"exact (H8 x Hex n).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 9 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"destruct HT as [b Hb].","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 10 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"destruct (H4 x b Hb) as [Heb Hnb].","w":"model","kind":"step","ok":true,"c":[11],"d":"Tactic 11 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"exists x, b.","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 12 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"refine (conj Hu (conj Hb (conj Heb (conj _ _)))).","w":"model","kind":"step","ok":true,"c":[13,14],"d":"Tactic 13 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"split.","w":"model","kind":"step","ok":true,"c":[15,16],"d":"Tactic 14 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"exact Heb.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 15 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"exact (H8 b Heb Hnb).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 16 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"intros y Hy.","w":"model","kind":"step","ok":true,"c":[17],"d":"Tactic 17 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"apply Hnb.","w":"model","kind":"step","ok":true,"c":[18],"d":"Tactic 18 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"exists y.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 19 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"exact Hy.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 20 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],"proofFile":"files/claude/run2/kalima_unconditional.v"},{"key":"claude2/imkan_s1_division","id":"imkan_s1_division","run":"claude2","title":"Dalīl al-imkān, step 1","english":"Every existent is necessary or possible in itself, and not both.","stmt":"forall x, Exists x -> (Necessary x \\/ Possible x) /\\ ~ (Necessary x /\\ Possible x)","proved":true,"seconds":1.0,"expansions":0,"calls":4,"root":0,"goals":["forall x, Exists x -> (Necessary x \\/ Possible x) /\\ ~ (Necessary x /\\ Possible x)"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":true,"c":[]}],"proof":[3],"proofFile":"files/claude/run2/imkan_s1_division.v"},{"key":"claude2/imkan_s2_necessary_cannot_fail","id":"imkan_s2_necessary_cannot_fail","run":"claude2","title":"Dalīl al-imkān, step 2","english":"The non-existence of what is necessary in itself is not possible.","stmt":"MP007 -> forall x, Necessary x -> ~ NonExistencePossible x","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["MP007 -> forall x, Necessary x -> ~ NonExistencePossible x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run2/imkan_s2_necessary_cannot_fail.v"},{"key":"claude2/step3_possible_needs_cause","id":"step3_possible_needs_cause","run":"claude2","title":"Dalīl al-imkān, step 3","english":"A possible existent needs a cause.","stmt":"MP008 -> forall x, Possible x -> ThroughAnother x","proved":true,"seconds":5.3,"expansions":1,"calls":9,"root":0,"goals":["MP008 -> forall x, Possible x -> ThroughAnother x","H : MP008\nx : Thing\nHe : Exists x\nHn : ~ Ess x\n============================\nThroughAnother x","H : MP008\nx : Thing\nHe : Exists x\nHn : ~ Ess x\n============================\n~ ~ ThroughAnother x","H : MP008\nx : Thing\nHe : Exists x\nHn : ~ Ess x\nHc : ~ ThroughAnother x\n============================\nFalse","H : MP008\nx : Thing\nHe : Exists x\nHn : ~ Ess x\nHc : ~ ThroughAnother x\n============================\nEss x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H x [He Hn].","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H x [He Hn]. apply Classical_Prop.NNPP. intro Hc. apply Hn. apply H; assumption."},{"g":1,"t":"apply Classical_Prop.NNPP.","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intro Hc.","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 3 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"apply Hn.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 4 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"apply H; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8],"proofFile":"files/claude/run2/step3_possible_needs_cause.v"},{"key":"claude2/imkan_s4_no_self_causation","id":"imkan_s4_no_self_causation","run":"claude2","title":"Dalīl al-imkān, step 4","english":"Nothing gives existence to itself.","stmt":"MP001 -> MP002 -> forall A, ~ Gives A A","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["MP001 -> MP002 -> forall A, ~ Gives A A"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run2/imkan_s4_no_self_causation.v"},{"key":"claude2/imkan_s5_none_outside","id":"imkan_s5_none_outside","run":"claude2","title":"Dalīl al-imkān, step 5","english":"No possible existent lies outside the aggregate of possibles.","stmt":"forall A, AggregateOfPossibles A -> forall p, p <> A -> Possible p -> MemberOf p A","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["forall A, AggregateOfPossibles A -> forall p, p <> A -> Possible p -> MemberOf p A"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run2/imkan_s5_none_outside.v"},{"key":"claude2/imkan_s6_aggregate_possible_caused","id":"imkan_s6_aggregate_possible_caused","run":"claude2","title":"Dalīl al-imkān, step 6","english":"The aggregate is itself possible, and so has its existence through another.","stmt":"MP008 -> MP010 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> Possible A /\\ ThroughAnother A","proved":true,"seconds":250.7,"expansions":1,"calls":24,"root":0,"goals":["MP008 -> MP010 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> Possible A /\\ ThroughAnother A","H8 : MP008\nH10 : MP010\nA : Thing\nHAgg : AggregateOfPossibles A\nHE : Exists A\nHex : exists p : Thing, p <> A /\\ Possible p\n============================\nPossible A /\\ ThroughAnother A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\nHex : exists p : Thing, p <> A /\\ Possible p\n============================\nPossible A /\\ ThroughAnother A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\n============================\nPossible A /\\ ThroughAnother A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\n============================\nPossible A /\\ ThroughAnother A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\n============================\n~ Ess A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\n============================\nPossible A /\\ ThroughAnother A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\n============================\nexists p0 : Thing, MemberOf p0 A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\n============================\nforall p0 : Thing, MemberOf p0 A -> ~ Ess p0","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nq : Thing\nHq : MemberOf q A\n============================\n~ Ess q","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nq : Thing\nHq : MemberOf q A\nE : q = A\n============================\n~ Ess q","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nq : Thing\nHq : MemberOf q A\nHqa : q <> A\n============================\n~ Ess q","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nHq : MemberOf A A\n============================\n~ Ess A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nHq : MemberOf A A\n============================\nFalse","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nq : Thing\nHq : MemberOf q A\nHqa : q <> A\nHpq : Possible q\n============================\n~ Ess q","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\n============================\nPossible A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\n============================\nThroughAnother A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\n============================\n~ ~ ThroughAnother A","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nH : ~ ThroughAnother A\n============================\nFalse","H8 : MP008\nH10 : MP010\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHE : Exists A\np : Thing\nHne : p <> A\nHp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nH : ~ ThroughAnother A\n============================\nEss A"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H8 H10 A HAgg HE Hex.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H8 H10 A HAgg HE Hex. destruct HAgg as [HA HnA]. destruct Hex as [p [Hne Hp]].\nassert (Hmem: MemberOf p A) by (apply (proj2 (HA p Hne)); exact Hp).\nassert (HnE: ~ Ess A).\n{ apply H10.\n  - exists p; exact Hmem.\n  - intros q Hq. destruct (classic (q = A)) as [E|Hqa].\n    + subst q. exfalso. exact (HnA Hq).\n    + pose proof (proj1 (HA q Hqa) Hq) as Hpq. exact (proj2 Hpq). }\nsplit.\n- split; assumption.\n- apply NNPP. intro H. apply HnE. apply H8; assumption."},{"g":1,"t":"destruct HAgg as [HA HnA].","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"destruct Hex as [p [Hne Hp]].","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 3 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"assert (Hmem: MemberOf p A) by (apply (proj2 (HA p Hne)); exact Hp).","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 4 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"assert (HnE: ~ Ess A).","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 5 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"apply H10.","w":"model","kind":"step","ok":true,"c":[7,8],"d":"Tactic 6 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"exists p; exact Hmem.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"intros q Hq.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 8 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"destruct (classic (q = A)) as [E|Hqa].","w":"model","kind":"step","ok":true,"c":[10,11],"d":"Tactic 9 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"subst q.","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 10 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"exfalso.","w":"model","kind":"step","ok":true,"c":[13],"d":"Tactic 11 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"exact (HnA Hq).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 12 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"pose proof (proj1 (HA q Hqa) Hq) as Hpq.","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 13 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"exact (proj2 Hpq).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 14 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"split.","w":"model","kind":"step","ok":true,"c":[15,16],"d":"Tactic 15 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"split; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 16 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"apply NNPP.","w":"model","kind":"step","ok":true,"c":[17],"d":"Tactic 17 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"intro H.","w":"model","kind":"step","ok":true,"c":[18],"d":"Tactic 18 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"apply HnE.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 19 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"apply H8; assumption.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 20 of 20 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23],"proofFile":"files/claude/run2/imkan_s6_aggregate_possible_caused.v"},{"key":"claude2/imkan_s7_cause_not_member","id":"imkan_s7_cause_not_member","run":"claude2","title":"Dalīl al-imkān, step 7","english":"What gives existence to the aggregate is not one of its members.","stmt":"MP001 -> MP002 -> MP003 -> MP011 -> forall c A, Gives c A -> ~ MemberOf c A","proved":true,"seconds":80.8,"expansions":6,"calls":33,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP011 -> forall c A, Gives c A -> ~ MemberOf c A","MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> Through b a","MP001 -> MP002 -> MP003 -> forall a, ~ clos_trans Thing Gives a a","forall a : Thing, clos_trans Thing Gives a a -> exists c : Thing, Gives a c /\\ clos_trans Thing Gives c a","H2 : MP002\nH3 : MP003\na : Thing\nb : Thing\nH : clos_trans Thing Gives a b\n============================\nThrough b a","H2 : MP002\nH3 : MP003\nx : Thing\ny : Thing\nHxy : Gives x y\n============================\nThrough y x","H2 : MP002\nH3 : MP003\nx : Thing\ny : Thing\nz : Thing\nH1 : clos_trans Thing Gives x y\nH4 : clos_trans Thing Gives y z\nIH1 : Through y x\nIH2 : Through z y\n============================\nThrough z x","H2 : MP002\nH3 : MP003\nx : Thing\ny : Thing\nHxy : Gives x y\n============================\nGives x y","a : Thing\nH : clos_trans Thing Gives a a\n============================\nexists c : Thing, Gives a c /\\ clos_trans Thing Gives c a","a : Thing\nH : clos_trans_1n Thing Gives a a\n============================\nexists c : Thing, Gives a c /\\ clos_trans Thing Gives c a","a : Thing\nH : clos_trans_1n Thing Gives a a\nH0 : Gives a a\n============================\nexists c : Thing, Gives a c /\\ clos_trans Thing Gives c a","a : Thing\nH : clos_trans_1n Thing Gives a a\ny : Thing\nH0 : Gives a y\nH1 : clos_trans_1n Thing Gives y a\n============================\nexists c : Thing, Gives a c /\\ clos_trans Thing Gives c a","a : Thing\nH : clos_trans_1n Thing Gives a a\nH0 : Gives a a\n============================\nGives a a /\\ clos_trans Thing Gives a a","a : Thing\nH : clos_trans_1n Thing Gives a a\ny : Thing\nH0 : Gives a y\nH1 : clos_trans_1n Thing Gives y a\n============================\nGives a ?c /\\ clos_trans Thing Gives ?c a"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H11 c A G M.\nassert (L: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b H. induction H as [x y Hg | x y z _ IH1 _ IH2].\n  - exact (H2 y x Hg).\n  - exact (H3 z y x IH2 IH1). }\nassert (T := H11 c A c G M).\napply clos_trans_tn1 in T.\ninversion T as [? ? Hg | ? m ? Hg Hr]; subst.\n- exact (H1 c c Hg (H2 c c Hg)).\n- apply clos_tn1_trans in Hr.\n  exact (H1 c m Hg (L c m Hr)).","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: The variable Hg was not found in the current environment."},{"g":0,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> Through b a","MP001 -> MP002 -> MP003 -> forall a, ~ clos_trans Thing Gives a a"],"combine":"intros H1 H2 H3 H11 c A hg hm. exact (sub2 H1 H2 H3 c (H11 c A c hg hm)).","ok":true,"c":[1,2],"assembled":true},{"g":1,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"intros H2 H3 a b H.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 1 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H2 H3 a b H. induction H as [x y Hxy | x y z H1 IH1 H4 IH2].\n- apply H2. exact Hxy.\n- exact (H3 _ _ _ IH2 IH1)."},{"g":4,"t":"induction H as [x y Hxy | x y z H1 IH1 H4 IH2].","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 2 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"apply H2.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 3 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"exact Hxy.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 4 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"exact (H3 _ _ _ IH2 IH1).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 5 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"intros H1 H2 H3 a H.\napply clos_trans_t1n in H.\ninversion H as [y Hg | y z Hg Hc]; subst.\n- exact (H1 a a Hg (H2 a a Hg)).\n- apply clos_t1n_trans in Hc.\n  exact (H1 a y Hg (imkan_s7_cause_not_member_L11 H2 H3 y a Hc)).","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: In environment Thing : Type"},{"g":2,"t":"intros H1 H2 H3 a H.\napply clos_trans_t1n in H.\ninversion H as [y Hg | y z Hg Hc]; subst.\n- apply (H1 a a Hg). apply H2. exact Hg.\n- apply (H1 a y Hg). apply (imkan_s7_cause_not_member_L11 H2 H3 y a). apply clos_t1n_trans. exact Hc.","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: In environment Thing : Type"},{"g":2,"t":"divide into 1 subgoals","w":"model","kind":"division","subs":["forall a : Thing, clos_trans Thing Gives a a -> exists c : Thing, Gives a c /\\ clos_trans Thing Gives c a"],"combine":"intros H1 H2 H3 a Hc. destruct (sub1 a Hc) as [c [Hg Hct]]. apply (H1 c a Hg). exact (imkan_s7_cause_not_member_L11 H2 H3 c a Hct).","ok":true,"c":[3],"assembled":true},{"g":3,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"intros a H.","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 1 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros a H. apply clos_trans_t1n in H. inversion H; subst.\n- exists a. split; [assumption | apply t_step; assumption].\n- eexists. split; [eassumption | apply clos_t1n_trans; eassumption]."},{"g":8,"t":"apply clos_trans_t1n in H.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 2 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"inversion H; subst.","w":"model","kind":"step","ok":true,"c":[10,11],"d":"Tactic 3 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"exists a.","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 4 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"split; [assumption | apply t_step; assumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"eexists.","w":"model","kind":"step","ok":true,"c":[13],"d":"Tactic 6 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"split; [eassumption | apply clos_t1n_trans; eassumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[5,10,11,12,13,14,21,26,27,28,29,30,31,32],"proofFile":"files/claude/run2/imkan_s7_cause_not_member.v"},{"key":"claude2/imkan_s8_no_internal_cause","id":"imkan_s8_no_internal_cause","run":"claude2","title":"Dalīl al-imkān, step 8","english":"Neither a member, nor a group of members, nor the aggregate gives the aggregate existence.","stmt":"MP001 -> MP002 -> MP003 -> MP010 -> MP011 -> forall A, AggregateOfPossibles A -> (forall x, ~ Gives x x) /\\ ~ Gives A A /\\ (forall c, MemberOf c A -> ~ Gives c A) /\\ (forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> ~ Gives B A)","proved":true,"seconds":182.0,"expansions":1,"calls":32,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP010 -> MP011 -> forall A, AggregateOfPossibles A -> (forall x, ~ Gives x x) /\\ ~ Gives A A /\\ (forall c, MemberOf c A -> ~ Gives c A) /\\ (forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> ~ Gives B A)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\n============================\n(forall x : Thing, ~ Gives x x) /\\ ~ Gives A A /\\ (forall c : Thing, MemberOf c A -> ~ Gives c A) /\\ (forall B : Thing, B <> A -> Exists B -> (exists p : Thing, MemberOf p B) -> (forall p : Thing, MemberOf p B -> MemberOf p A) -> ~ Gives B A)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\n============================\nforall x : Thing, ~ Gives x x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\n============================\n(forall x : Thing, ~ Gives x x) /\\ ~ Gives A A /\\ (forall c : Thing, MemberOf c A -> ~ Gives c A) /\\ (forall B : Thing, B <> A -> Exists B -> (exists p : Thing, MemberOf p B) -> (forall p : Thing, MemberOf p B -> MemberOf p A) -> ~ Gives B A)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\nx : Thing\nh : Gives x x\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\n============================\nforall a b : Thing, clos_trans Thing Gives a b -> Through b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\n============================\n(forall x : Thing, ~ Gives x x) /\\ ~ Gives A A /\\ (forall c : Thing, MemberOf c A -> ~ Gives c A) /\\ (forall B : Thing, B <> A -> Exists B -> (exists p : Thing, MemberOf p B) -> (forall p : Thing, MemberOf p B -> MemberOf p A) -> ~ Gives B A)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\na : Thing\nb : Thing\nh : clos_trans Thing Gives a b\n============================\nThrough b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\na : Thing\nb : Thing\nh : Gives a b\n============================\nThrough b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\na : Thing\nb : Thing\nc : Thing\nh1 : clos_trans Thing Gives a b\nh2 : clos_trans Thing Gives b c\nIH1 : Through b a\nIH2 : Through c b\n============================\nThrough c a","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\n============================\nforall c : Thing, MemberOf c A -> ~ Gives c A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\n============================\n(forall x : Thing, ~ Gives x x) /\\ ~ Gives A A /\\ (forall c : Thing, MemberOf c A -> ~ Gives c A) /\\ (forall B : Thing, B <> A -> Exists B -> (exists p : Thing, MemberOf p B) -> (forall p : Thing, MemberOf p B -> MemberOf p A) -> ~ Gives B A)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nc : Thing\nhc : MemberOf c A\nhg : Gives c A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nc : Thing\nhc : MemberOf c A\nhg : Gives c A\nh : clos_trans Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nc : Thing\nhc : MemberOf c A\nhg : Gives c A\nh : clos_trans_1n Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\n============================\nforall x : Thing, ~ Gives x x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\n============================\n~ Gives A A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\n============================\nforall c : Thing, MemberOf c A -> ~ Gives c A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\n============================\nforall B : Thing, B <> A -> Exists B -> (exists p : Thing, MemberOf p B) -> (forall p : Thing, MemberOf p B -> MemberOf p A) -> ~ Gives B A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\nB : Thing\nhne : B <> A\nhE : Exists B\nhex : exists p : Thing, MemberOf p B\nhsub : forall p : Thing, MemberOf p B -> MemberOf p A\nhg : Gives B A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\nB : Thing\nhne : B <> A\nhE : Exists B\nhex : exists p : Thing, MemberOf p B\nhsub : forall p : Thing, MemberOf p B -> MemberOf p A\nhg : Gives B A\n============================\nPossible B","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\nB : Thing\nhne : B <> A\nhE : Exists B\nhex : exists p : Thing, MemberOf p B\nhsub : forall p : Thing, MemberOf p B -> MemberOf p A\nhg : Gives B A\nhP : Possible B\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\nB : Thing\nhne : B <> A\nhE : Exists B\nhex : exists p : Thing, MemberOf p B\nhsub : forall p : Thing, MemberOf p B -> MemberOf p A\nhg : Gives B A\n============================\n~ Ess B","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\nB : Thing\nhne : B <> A\nhE : Exists B\nhex : exists p : Thing, MemberOf p B\nhsub : forall p : Thing, MemberOf p B -> MemberOf p A\nhg : Gives B A\n============================\nforall p : Thing, MemberOf p B -> ~ Ess p","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\nB : Thing\nhne : B <> A\nhE : Exists B\nhex : exists p : Thing, MemberOf p B\nhsub : forall p : Thing, MemberOf p B -> MemberOf p A\nhg : Gives B A\np : Thing\nhp : MemberOf p B\n============================\n~ Ess p","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\nB : Thing\nhne : B <> A\nhE : Exists B\nhex : exists p : Thing, MemberOf p B\nhsub : forall p : Thing, MemberOf p B -> MemberOf p A\nhg : Gives B A\np : Thing\nhp : MemberOf p B\nhpA : MemberOf p A\n============================\n~ Ess p","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\nB : Thing\nhne : B <> A\nhE : Exists B\nhex : exists p : Thing, MemberOf p B\nhsub : forall p : Thing, MemberOf p B -> MemberOf p A\nhg : Gives B A\np : Thing\nhp : MemberOf p B\nhpA : MemberOf p A\npne : p <> A\n============================\n~ Ess p","H1 : MP001\nH2 : MP002\nH3 : MP003\nH10 : MP010\nH11 : MP011\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHAA : ~ MemberOf A A\ng : forall x : Thing, ~ Gives x x\nct : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\nm : forall c : Thing, MemberOf c A -> ~ Gives c A\nB : Thing\nhne : B <> A\nhE : Exists B\nhex : exists p : Thing, MemberOf p B\nhsub : forall p : Thing, MemberOf p B -> MemberOf p A\nhg : Gives B A\np : Thing\nhp : MemberOf p B\nhpA : MemberOf p A\npne : p <> A\nne : ~ Ess p\n============================\n~ Ess p"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H10 H11 A [HA HAA].","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 H10 H11 A [HA HAA].\nassert (g: forall x, ~ Gives x x).\n{ intros x h. exact (H1 x x h (H2 x x h)). }\nassert (ct: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b h. induction h as [a b h|a b c h1 IH1 h2 IH2].\n  - exact (H2 b a h).\n  - exact (H3 c b a IH2 IH1). }\nassert (m: forall c, MemberOf c A -> ~ Gives c A).\n{ intros c hc hg.\n  pose proof (H11 c A c hg hc) as h.\n  apply clos_trans_t1n in h.\n  inversion h; subst;\n  [ exact (g _ H) | exact (H1 _ _ H (ct _ _ (clos_t1n_trans _ _ _ _ H0))) ]. }\nrepeat split.\n- exact g.\n- exact (g A).\n- exact m.\n- intros B hne hE hex hsub hg.\n  assert (hP: Possible B).\n  { split; [exact hE|].\n    apply (H10 B hex).\n    intros p hp.\n    assert (hpA := hsub p hp).\n    assert (pne: p <> A) by (intro e; subst; exact (HAA hpA)).\n    pose proof (proj1 (HA p pne) hpA) as [_ ne].\n    exact ne. }\n  exact (m B (proj2 (HA B hne) hP) hg)."},{"g":1,"t":"assert (g: forall x, ~ Gives x x).","w":"model","kind":"step","ok":true,"c":[2,3],"d":"Tactic 2 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros x h.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 3 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"exact (H1 x x h (H2 x x h)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 4 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"assert (ct: forall a b, clos_trans Thing Gives a b -> Through b a).","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 5 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"intros a b h.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 6 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"induction h as [a b h|a b c h1 IH1 h2 IH2].","w":"model","kind":"step","ok":true,"c":[8,9],"d":"Tactic 7 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"exact (H2 b a h).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 8 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"exact (H3 c b a IH2 IH1).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 9 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"assert (m: forall c, MemberOf c A -> ~ Gives c A).","w":"model","kind":"step","ok":true,"c":[10,11],"d":"Tactic 10 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"intros c hc hg.","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 11 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"pose proof (H11 c A c hg hc) as h.","w":"model","kind":"step","ok":true,"c":[13],"d":"Tactic 12 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"apply clos_trans_t1n in h.","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 13 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"inversion h; subst;\n  [ exact (g _ H) | exact (H1 _ _ H (ct _ _ (clos_t1n_trans _ _ _ _ H0))) ].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 14 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"repeat split.","w":"model","kind":"step","ok":true,"c":[15,16,17,18],"d":"Tactic 15 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"exact g.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 16 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"exact (g A).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 17 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"exact m.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 18 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"intros B hne hE hex hsub hg.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 19 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"assert (hP: Possible B).","w":"model","kind":"step","ok":true,"c":[20,21],"d":"Tactic 20 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"split; [exact hE|].","w":"model","kind":"step","ok":true,"c":[22],"d":"Tactic 21 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"apply (H10 B hex).","w":"model","kind":"step","ok":true,"c":[23],"d":"Tactic 22 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"intros p hp.","w":"model","kind":"step","ok":true,"c":[24],"d":"Tactic 23 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"assert (hpA := hsub p hp).","w":"model","kind":"step","ok":true,"c":[25],"d":"Tactic 24 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"assert (pne: p <> A) by (intro e; subst; exact (HAA hpA)).","w":"model","kind":"step","ok":true,"c":[26],"d":"Tactic 25 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":26,"t":"pose proof (proj1 (HA p pne) hpA) as [_ ne].","w":"model","kind":"step","ok":true,"c":[27],"d":"Tactic 26 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"exact ne.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 27 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":21,"t":"exact (m B (proj2 (HA B hne) hP) hg).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 28 of 28 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31],"proofFile":"files/claude/run2/imkan_s8_no_internal_cause.v"},{"key":"claude2/imkan_s9_cause_outside","id":"imkan_s9_cause_outside","run":"claude2","title":"Dalīl al-imkān, step 9","english":"The cause of the aggregate lies outside it and is not possible.","stmt":"MP001 -> MP002 -> MP003 -> MP011 -> forall A c, AggregateOfPossibles A -> Gives c A -> ~ MemberOf c A /\\ c <> A /\\ ~ Possible c","proved":true,"seconds":99.9,"expansions":4,"calls":33,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP011 -> forall A c, AggregateOfPossibles A -> Gives c A -> ~ MemberOf c A /\\ c <> A /\\ ~ Possible c","MP001 -> MP002 -> forall A c, Gives c A -> c <> A","MP001 -> MP002 -> MP003 -> forall c, ~ clos_trans Thing Gives c c","H1 : MP001\nH2 : MP002\nA : Thing\nc : Thing\nH : Gives c A\nHeq : c = A\n============================\nFalse","H1 : MP001\nH2 : MP002\nA : Thing\nH : Gives A A\n============================\nFalse","H1 : MP001\nH2 : MP002\nA : Thing\nH : Gives A A\n============================\nThrough A A","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans Thing Gives c c\n============================\nforall a b : Thing, clos_trans Thing Gives a b -> Through b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans Thing Gives c c\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans Thing Gives c c\na : Thing\nb : Thing\nh : clos_trans Thing Gives a b\n============================\nThrough b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans Thing Gives c c\na : Thing\nb : Thing\ng : Gives a b\n============================\nThrough b a","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans Thing Gives c c\na : Thing\nb : Thing\nd : Thing\nh1 : clos_trans Thing Gives a b\nh2 : clos_trans Thing Gives b d\nIH1 : Through b a\nIH2 : Through d b\n============================\nThrough d a","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_n1 Thing Gives c c\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_n1 Thing Gives c c\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\ny : Thing\ng : Gives c c\nH0 : y = c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_n1 Thing Gives c c\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\ny : Thing\nz : Thing\ng : Gives y c\nh' : clos_trans_n1 Thing Gives c y\nH0 : z = c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_n1 Thing Gives c c\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\ng : Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_n1 Thing Gives c c\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\ny : Thing\ng : Gives y c\nh' : clos_trans_n1 Thing Gives c y\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_n1 Thing Gives c c\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\ny : Thing\ng : Gives y c\nh' : clos_trans_n1 Thing Gives c y\n============================\nThrough y c","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_n1 Thing Gives c c\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\ny : Thing\ng : Gives y c\nh' : clos_trans_n1 Thing Gives c y\n============================\nclos_trans Thing Gives c y","H1 : MP001\nH2 : MP002\nH3 : MP003\nc : Thing\nH : clos_trans_n1 Thing Gives c c\nL : forall a b : Thing, clos_trans Thing Gives a b -> Through b a\ny : Thing\ng : Gives y c\nh' : clos_trans_n1 Thing Gives c y\n============================\nclos_trans_n1 Thing Gives c y"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros M1 M2 M3 M11 A c [HA _] HG.\nassert (L: forall x y, clos_trans Thing Gives x y -> Through y x).\n{ intros x y H. induction H as [x y H | x y z H1 IH1 H2 IH2].\n  - apply M2; exact H.\n  - exact (M3 _ _ _ IH2 IH1). }\nassert (HM: ~ MemberOf c A).\n{ intro HM. pose proof (M11 c A c HG HM) as C.\n  apply clos_trans_t1n_iff in C.\n  inversion C as [a b Hs | a z b Hs Hr]; subst.\n  - exact (M1 _ _ Hs (M2 _ _ Hs)).\n  - apply (M1 _ _ Hs). apply L. apply clos_trans_t1n_iff. exact Hr. }\nassert (HN: c <> A).\n{ intro e. subst c. exact (M1 _ _ HG (M2 _ _ HG)). }\nrepeat split.\n- exact HM.\n- exact HN.\n- intro P. apply HM. apply (proj2 (HA c HN)). exact P.","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: The variable Hs was not found in the current environment."},{"g":0,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> forall A c, Gives c A -> c <> A","MP001 -> MP002 -> MP003 -> forall c, ~ clos_trans Thing Gives c c"],"combine":"intros h1 h2 h3 h11 A c [hA _] hg. assert (hne : c <> A) by exact (sub1 h1 h2 A c hg). assert (hnm : ~ MemberOf c A). { intro hm. apply (sub2 h1 h2 h3 c). exact (h11 c A c hg hm). } refine (conj hnm (conj hne _)). intro hp. apply hnm. apply (proj2 (hA c hne)). exact hp.","ok":true,"c":[1,2],"assembled":true},{"g":1,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"intros H1 H2 A c H Heq.","w":"model","kind":"step","ok":true,"c":[3],"d":"Tactic 1 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 A c H Heq. subst c. apply (H1 A A H). apply (H2 A A H)."},{"g":3,"t":"subst c.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 2 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"apply (H1 A A H).","w":"model","kind":"step","ok":true,"c":[5],"d":"Tactic 3 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"apply (H2 A A H).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 4 of 4 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"intros H1 H2 H3 c H.","w":"model","kind":"step","ok":true,"c":[6],"d":"Tactic 1 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 c H.\nassert (L: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b h. induction h as [a b g | a b d h1 IH1 h2 IH2].\n  - apply H2; exact g.\n  - exact (H3 _ _ _ IH2 IH1). }\napply clos_trans_tn1 in H.\ninversion H as [y g | y z g h'].\n- subst. exact (H1 _ _ g (H2 _ _ g)).\n- subst. apply (H1 _ _ g). apply L. apply clos_trans_tn1_iff. exact h'."},{"g":6,"t":"assert (L: forall a b, clos_trans Thing Gives a b -> Through b a).","w":"model","kind":"step","ok":true,"c":[7,8],"d":"Tactic 2 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"intros a b h.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 3 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"induction h as [a b g | a b d h1 IH1 h2 IH2].","w":"model","kind":"step","ok":true,"c":[10,11],"d":"Tactic 4 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"apply H2; exact g.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"exact (H3 _ _ _ IH2 IH1).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"apply clos_trans_tn1 in H.","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 7 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"inversion H as [y g | y z g h'].","w":"model","kind":"step","ok":true,"c":[13,14],"d":"Tactic 8 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"subst.","w":"model","kind":"step","ok":true,"c":[15],"d":"Tactic 9 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"exact (H1 _ _ g (H2 _ _ g)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 10 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"subst.","w":"model","kind":"step","ok":true,"c":[16],"d":"Tactic 11 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"apply (H1 _ _ g).","w":"model","kind":"step","ok":true,"c":[17],"d":"Tactic 12 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"apply L.","w":"model","kind":"step","ok":true,"c":[18],"d":"Tactic 13 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"apply clos_trans_tn1_iff.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 14 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"exact h'.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 15 of 15 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[5,10,11,12,13,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32],"proofFile":"files/claude/run2/imkan_s9_cause_outside.v"},{"key":"claude2/imkan_s10_cause_necessary","id":"imkan_s10_cause_necessary","run":"claude2","title":"Dalīl al-imkān, step 10","english":"That cause is necessary in itself.","stmt":"MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Necessary c","proved":true,"seconds":112.1,"expansions":8,"calls":55,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Necessary c","MP001 -> MP002 -> MP003 -> forall c, ~ clos_trans Thing Gives c c","MP001 -> MP002 -> forall A c, Gives c A -> c <> A","MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> c <> A -> ~ Ess c -> clos_trans Thing Gives c c","MP002 -> MP003 -> forall c d, clos_trans Thing Gives c d -> Through d c","forall c, clos_trans Thing Gives c c -> exists y, Gives y c /\\ (y = c \\/ clos_trans Thing Gives c y)","============================\n(forall x y : Thing, Gives y x -> Through x y) -> (forall x y z : Thing, Through x y -> Through y z -> Through x z) -> forall c d : Thing, clos_trans Thing Gives c d -> Through d c","H2 : forall x y : Thing, Gives y x -> Through x y\nH3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nc : Thing\nd : Thing\nH : clos_trans Thing Gives c d\n============================\nThrough d c","H2 : forall x y : Thing, Gives y x -> Through x y\nH3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nx : Thing\ny : Thing\nHxy : Gives x y\n============================\nThrough y x","H2 : forall x y : Thing, Gives y x -> Through x y\nH3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nx : Thing\ny : Thing\nz : Thing\nHxy : clos_trans Thing Gives x y\nHyz : clos_trans Thing Gives y z\nIH1 : Through y x\nIH2 : Through z y\n============================\nThrough z x","H2 : forall x y : Thing, Gives y x -> Through x y\nH3 : forall x y z : Thing, Through x y -> Through y z -> Through x z\nx : Thing\ny : Thing\nHxy : Gives x y\n============================\nGives x y","c : Thing\nH : clos_trans Thing Gives c c\n============================\nexists y : Thing, Gives y c /\\ (y = c \\/ clos_trans Thing Gives c y)","c : Thing\nH : clos_trans_n1 Thing Gives c c\n============================\nexists y : Thing, Gives y c /\\ (y = c \\/ clos_trans Thing Gives c y)","c : Thing\nH : clos_trans_n1 Thing Gives c c\nH0 : Gives c c\n============================\nexists y : Thing, Gives y c /\\ (y = c \\/ clos_trans Thing Gives c y)","c : Thing\nH : clos_trans_n1 Thing Gives c c\ny : Thing\nH0 : Gives y c\nH1 : clos_trans_n1 Thing Gives c y\n============================\nexists y0 : Thing, Gives y0 c /\\ (y0 = c \\/ clos_trans Thing Gives c y0)","c : Thing\nH : clos_trans_n1 Thing Gives c c\nH0 : Gives c c\n============================\nGives c c /\\ (c = c \\/ clos_trans Thing Gives c c)","c : Thing\nH : clos_trans_n1 Thing Gives c c\ny : Thing\nH0 : Gives y c\nH1 : clos_trans_n1 Thing Gives c y\n============================\nGives ?y c /\\ (?y = c \\/ clos_trans Thing Gives c ?y)","H1 : MP001\nH2 : MP002\nA : Thing\nc : Thing\nH : Gives c A\nE : c = A\n============================\nFalse","H1 : MP001\nH2 : MP002\nA : Thing\nH : Gives A A\n============================\nFalse","H11 : MP011\nH12 : MP012\nA : Thing\nc : Thing\nHAg : AggregateOfPossibles A\nHg : Gives c A\nHne : c <> A\nHn : ~ Ess c\n============================\nclos_trans Thing Gives c c","H11 : MP011\nH12 : MP012\nA : Thing\nc : Thing\nHAg : (forall p : Thing, p <> A -> MemberOf p A <-> Possible p) /\\ ~ MemberOf A A\nHg : Gives c A\nHne : c <> A\nHn : ~ Ess c\n============================\nclos_trans Thing Gives c c","H11 : MP011\nH12 : MP012\nA : Thing\nc : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHg : Gives c A\nHne : c <> A\nHn : ~ Ess c\n============================\nclos_trans Thing Gives c c","H11 : forall c A p : Thing, Gives c A -> MemberOf p A -> clos_trans Thing Gives c p\nH12 : MP012\nA : Thing\nc : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHg : Gives c A\nHne : c <> A\nHn : ~ Ess c\n============================\nclos_trans Thing Gives c c","H11 : forall c A p : Thing, Gives c A -> MemberOf p A -> clos_trans Thing Gives c p\nH12 : forall y x : Thing, Gives y x -> Exists y\nA : Thing\nc : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHg : Gives c A\nHne : c <> A\nHn : ~ Ess c\n============================\nclos_trans Thing Gives c c","H11 : forall c A p : Thing, Gives c A -> MemberOf p A -> clos_trans Thing Gives c p\nH12 : forall y x : Thing, Gives y x -> Exists y\nA : Thing\nc : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHg : Gives c A\nHne : c <> A\nHn : ~ Ess c\n============================\nMemberOf c A","H11 : forall c A p : Thing, Gives c A -> MemberOf p A -> clos_trans Thing Gives c p\nH12 : forall y x : Thing, Gives y x -> Exists y\nA : Thing\nc : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHg : Gives c A\nHne : c <> A\nHn : ~ Ess c\n============================\nPossible c","H11 : forall c A p : Thing, Gives c A -> MemberOf p A -> clos_trans Thing Gives c p\nH12 : forall y x : Thing, Gives y x -> Exists y\nA : Thing\nc : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHg : Gives c A\nHne : c <> A\nHn : ~ Ess c\n============================\nExists c /\\ ~ Ess c","H11 : forall c A p : Thing, Gives c A -> MemberOf p A -> clos_trans Thing Gives c p\nH12 : forall y x : Thing, Gives y x -> Exists y\nA : Thing\nc : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHg : Gives c A\nHne : c <> A\nHn : ~ Ess c\n============================\nExists c","H11 : forall c A p : Thing, Gives c A -> MemberOf p A -> clos_trans Thing Gives c p\nH12 : forall y x : Thing, Gives y x -> Exists y\nA : Thing\nc : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHg : Gives c A\nHne : c <> A\nHn : ~ Ess c\n============================\n~ Ess c"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H11 H12 A c [HA _] G.\nassert (key: forall z, clos_trans_n1 Thing Gives c z -> Through z c).\n{ intros z H. induction H as [y G0 | y z G0 H IH].\n  - apply H2. exact G0.\n  - apply (H3 z y c).\n    + apply H2. exact G0.\n    + exact IH. }\nassert (Hne: c <> A).\n{ intro e. subst. apply (H1 A A G). apply H2. exact G. }\nsplit.\n- exact (H12 c A G).\n- destruct (classic (Ess c)) as [E|NE].\n  + exact E.\n  + exfalso.\n    assert (Hmem: MemberOf c A).\n    { apply (proj2 (HA c Hne)). split.\n      - exact (H12 c A G).\n      - exact NE. }\n    pose proof (H11 c A c G Hmem) as ct.\n    apply clos_trans_clos_trans_n1 in ct.\n    assert (aux: forall z, clos_trans_n1 Thing Gives c z -> z = c -> False).\n    { intros z H. destruct H as [y G0 | y z0 G0 H].\n      - intro e. subst. apply (H1 _ _ G0). apply H2. exact G0.\n      - intro e. subst. exact (H1 c y G0 (key y H)). }\n    exact (aux c ct eq_refl).","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: The variable clos_trans_clos_trans_n1 was not found in the current environment."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> MP003 -> forall c, ~ clos_trans Thing Gives c c","MP001 -> MP002 -> forall A c, Gives c A -> c <> A","MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> c <> A -> ~ Ess c -> clos_trans Thing Gives c c"],"combine":"intros H1 H2 H3 H11 H12 A c HA HG. unfold Necessary. split. - exact (H12 _ _ HG). - apply NNPP. intro HE. apply (sub1 H1 H2 H3 c). exact (sub3 H11 H12 A c HA HG (sub2 H1 H2 A c HG) HE).","ok":true,"c":[1,2,3],"assembled":true},{"g":1,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":1,"t":"intros H1 H2 H3 c H.\nassert (L: forall a b, clos_trans Thing Gives a b -> Through b a).\n{ intros a b h. induction h as [a b g | a b d h1 IH1 h2 IH2].\n - apply H2; exact g.\n - exact (H3 _ _ _ IH2 IH1). }\napply clos_trans_t1n in H.\ninversion H; subst.\n- match goal with g: Gives _ _ |- _ => exact (H1 _ _ g (H2 _ _ g)) end.\n- match goal with g: Gives _ _, h: clos_trans_1n _ _ _ _ |- _ => apply clos_trans_1n_trans in h; exact (H1 _ _ g (L _ _ h)) end.","w":"model","kind":"proof","ok":false,"c":[],"err":"Error: No matching clauses for match."},{"g":1,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall c d, clos_trans Thing Gives c d -> Through d c","forall c, clos_trans Thing Gives c c -> exists y, Gives y c /\\ (y = c \\/ clos_trans Thing Gives c y)"],"combine":"intros H1 H2 H3 c Hc. destruct (sub2 c Hc) as [y [Hg [He|Hy]]]. - subst. exact (H1 _ _ Hg (H2 _ _ Hg)). - exact (H1 _ _ Hg (sub1 H2 H3 _ _ Hy)).","ok":true,"c":[4,5],"assembled":true},{"g":4,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP002, MP003.","w":"model","kind":"step","ok":true,"c":[6],"d":"Tactic 1 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"unfold MP002, MP003. intros H2 H3 c d H. induction H as [x y Hxy | x y z Hxy IH1 Hyz IH2].\n- apply H2. exact Hxy.\n- exact (H3 z y x IH2 IH1)."},{"g":6,"t":"intros H2 H3 c d H.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 2 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"induction H as [x y Hxy | x y z Hxy IH1 Hyz IH2].","w":"model","kind":"step","ok":true,"c":[8,9],"d":"Tactic 3 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"apply H2.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 4 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"exact Hxy.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"exact (H3 z y x IH2 IH1).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 6 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":5,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":5,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":5,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":5,"t":"intros c H.","w":"model","kind":"step","ok":true,"c":[11],"d":"Tactic 1 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros c H. apply clos_trans_tn1 in H. inversion H; subst.\n- exists c. split; auto.\n- eexists. split; [eassumption | right; apply clos_trans_tn1_iff; eassumption]."},{"g":11,"t":"apply clos_trans_tn1 in H.","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 2 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"inversion H; subst.","w":"model","kind":"step","ok":true,"c":[13,14],"d":"Tactic 3 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"exists c.","w":"model","kind":"step","ok":true,"c":[15],"d":"Tactic 4 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"split; auto.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"eexists.","w":"model","kind":"step","ok":true,"c":[16],"d":"Tactic 6 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"split; [eassumption | right; apply clos_trans_tn1_iff; eassumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 7 of 7 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"intros H1 H2 A c H E.","w":"model","kind":"step","ok":true,"c":[17],"d":"Tactic 1 of 3 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 A c H E. subst. exact (H1 _ _ H (H2 _ _ H))."},{"g":17,"t":"subst.","w":"model","kind":"step","ok":true,"c":[18],"d":"Tactic 2 of 3 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"exact (H1 _ _ H (H2 _ _ H)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 3 of 3 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":3,"t":"intros H11 H12 A c HAg Hg Hne Hn.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 1 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H11 H12 A c HAg Hg Hne Hn.\nunfold AggregateOfPossibles in HAg.\ndestruct HAg as [HA _].\nunfold MP011 in H11.\nunfold MP012 in H12.\napply (H11 c A c Hg).\napply (proj2 (HA c Hne)).\nunfold Possible.\nsplit.\n- exact (H12 c A Hg).\n- exact Hn."},{"g":19,"t":"unfold AggregateOfPossibles in HAg.","w":"model","kind":"step","ok":true,"c":[20],"d":"Tactic 2 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"destruct HAg as [HA _].","w":"model","kind":"step","ok":true,"c":[21],"d":"Tactic 3 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":21,"t":"unfold MP011 in H11.","w":"model","kind":"step","ok":true,"c":[22],"d":"Tactic 4 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"unfold MP012 in H12.","w":"model","kind":"step","ok":true,"c":[23],"d":"Tactic 5 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"apply (H11 c A c Hg).","w":"model","kind":"step","ok":true,"c":[24],"d":"Tactic 6 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"apply (proj2 (HA c Hne)).","w":"model","kind":"step","ok":true,"c":[25],"d":"Tactic 7 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"unfold Possible.","w":"model","kind":"step","ok":true,"c":[26],"d":"Tactic 8 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":26,"t":"split.","w":"model","kind":"step","ok":true,"c":[27,28],"d":"Tactic 9 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"exact (H12 c A Hg).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 10 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":28,"t":"exact Hn.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 11 of 11 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[5,11,16,17,18,19,20,21,26,27,28,29,30,31,32,37,38,39,44,45,46,47,48,49,50,51,52,53,54],"proofFile":"files/claude/run2/imkan_s10_cause_necessary.v"},{"key":"claude2/imkan_s11_necessary_cause","id":"imkan_s11_necessary_cause","run":"claude2","title":"Dalīl al-imkān, step 11","english":"The aggregate has a cause that is necessary in itself.","stmt":"MP001 -> MP002 -> MP003 -> MP008 -> MP010 -> MP011 -> MP012 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> exists c, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","proved":true,"seconds":208.8,"expansions":1,"calls":46,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP008 -> MP010 -> MP011 -> MP012 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> exists c, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\n============================\nexists c : Thing, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\n============================\nexists c : Thing, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\n============================\n~ Ess A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\n============================\nexists c : Thing, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\n============================\nexists p0 : Thing, MemberOf p0 A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\n============================\nforall p0 : Thing, MemberOf p0 A -> ~ Ess p0","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nq : Thing\nHq : MemberOf q A\n============================\n~ Ess q","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nq : Thing\nHq : MemberOf q A\nHq2 : q <> A\n============================\n~ Ess q","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nq : Thing\nHq : MemberOf q A\nHq2 : q <> A\nHq3 : Possible q\n============================\n~ Ess q","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nq : Thing\nHq : MemberOf q A\nHq2 : q <> A\nH : ~ Ess q\n============================\n~ Ess q","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nHT : ThroughAnother A\n============================\nexists c : Thing, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\n============================\nexists c0 : Thing, Gives c0 A /\\ Necessary c0 /\\ ~ MemberOf c0 A /\\ c0 <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\n============================\nforall x y : Thing, clos_trans Thing Gives x y -> Through y x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\n============================\nexists c0 : Thing, Gives c0 A /\\ Necessary c0 /\\ ~ MemberOf c0 A /\\ c0 <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nx : Thing\ny : Thing\nh : clos_trans Thing Gives x y\n============================\nThrough y x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nx : Thing\ny : Thing\nHg : Gives x y\n============================\nThrough y x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nx : Thing\ny : Thing\nz : Thing\nh1 : clos_trans Thing Gives x y\nh2 : clos_trans Thing Gives y z\nIH1 : Through y x\nIH2 : Through z y\n============================\nThrough z x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\n============================\nforall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\n============================\nexists c0 : Thing, Gives c0 A /\\ Necessary c0 /\\ ~ MemberOf c0 A /\\ c0 <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nx : Thing\ny : Thing\nh : clos_trans Thing Gives x y\n============================\nexists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nx : Thing\ny : Thing\nHg : Gives x y\n============================\nexists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nx : Thing\ny : Thing\nz : Thing\nh1 : clos_trans Thing Gives x y\nh2 : clos_trans Thing Gives y z\nIH1 : exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nIH2 : exists w : Thing, Gives w z /\\ (w = y \\/ clos_trans Thing Gives y w)\n============================\nexists w : Thing, Gives w z /\\ (w = x \\/ clos_trans Thing Gives x w)","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nx : Thing\ny : Thing\nz : Thing\nh1 : clos_trans Thing Gives x y\nh2 : clos_trans Thing Gives y z\nIH1 : exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nw : Thing\nHw : Gives w z\nE : w = y\n============================\nw = x \\/ clos_trans Thing Gives x w","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nx : Thing\ny : Thing\nz : Thing\nh1 : clos_trans Thing Gives x y\nh2 : clos_trans Thing Gives y z\nIH1 : exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nw : Thing\nHw : Gives w z\nC : clos_trans Thing Gives y w\n============================\nw = x \\/ clos_trans Thing Gives x w","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\n============================\n~ MemberOf c A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf c A\n============================\nexists c0 : Thing, Gives c0 A /\\ Necessary c0 /\\ ~ MemberOf c0 A /\\ c0 <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHm : MemberOf c A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHm : MemberOf c A\nHcc : clos_trans Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHm : MemberOf c A\nHcc : clos_trans Thing Gives c c\nw : Thing\nHw : Gives w c\nE : w = c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHm : MemberOf c A\nHcc : clos_trans Thing Gives c c\nw : Thing\nHw : Gives w c\nC : clos_trans Thing Gives c w\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHm : MemberOf c A\nHcc : clos_trans Thing Gives c c\nHw : Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf c A\n============================\nc <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf c A\nHne : c <> A\n============================\nexists c0 : Thing, Gives c0 A /\\ Necessary c0 /\\ ~ MemberOf c0 A /\\ c0 <> A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf c A\nE : c = A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nHc : Gives A A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf A A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf c A\nHne : c <> A\n============================\nNecessary c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf c A\nHne : c <> A\n============================\nExists c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf c A\nHne : c <> A\n============================\nEss c","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf c A\nHne : c <> A\nHn : ~ Ess c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf c A\nHne : c <> A\nHn : ~ Ess c\n============================\nMemberOf c A","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH10 : MP010\nH11 : MP011\nH12 : MP012\nA : Thing\nHA : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHnA : ~ MemberOf A A\nHEx : Exists A\np : Thing\nHp : p <> A\nHpp : Possible p\nHmem : MemberOf p A\nHnE : ~ Ess A\nc : Thing\nHc : Gives c A\nHth : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nHcl : forall x y : Thing, clos_trans Thing Gives x y -> exists w : Thing, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)\nHnm : ~ MemberOf c A\nHne : c <> A\nHn : ~ Ess c\n============================\nPossible c"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H8 H10 H11 H12 A [HA HnA] HEx [p [Hp Hpp]].","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 H8 H10 H11 H12 A [HA HnA] HEx [p [Hp Hpp]].\nassert (Hmem: MemberOf p A) by (apply (proj2 (HA p Hp)); exact Hpp).\nassert (HnE: ~Ess A).\n{ apply H10. exists p; exact Hmem. intros q Hq. assert (Hq2: q <> A) by (intro E; subst; exact (HnA Hq)). pose proof (proj1 (HA q Hq2) Hq) as Hq3. destruct Hq3 as [_ H]. exact H. }\nassert (HT: ThroughAnother A) by (apply NNPP; intro HN; apply HnE; apply H8; auto).\ndestruct HT as [c Hc].\nassert (Hth: forall x y, clos_trans Thing Gives x y -> Through y x).\n{ intros x y h. induction h as [x y Hg | x y z h1 IH1 h2 IH2]. exact (H2 y x Hg). exact (H3 z y x IH2 IH1). }\nassert (Hcl: forall x y, clos_trans Thing Gives x y -> exists w, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)).\n{ intros x y h. induction h as [x y Hg | x y z h1 IH1 h2 IH2].\n  - exists x; auto.\n  - destruct IH2 as [w [Hw [E|C]]]; exists w; split; auto.\n    + right; subst; exact h1.\n    + right; eapply t_trans; eauto. }\nassert (Hnm: ~MemberOf c A).\n{ intro Hm. pose proof (H11 c A c Hc Hm) as Hcc. destruct (Hcl c c Hcc) as [w [Hw [E|C]]].\n  - subst. exact (H1 _ _ Hw (H2 _ _ Hw)).\n  - exact (H1 c w Hw (Hth c w C)). }\nassert (Hne: c <> A).\n{ intro E. subst. exact (H1 _ _ Hc (H2 _ _ Hc)). }\nrefine (ex_intro _ c (conj Hc (conj _ (conj Hnm Hne)))).\nsplit.\n- exact (H12 c A Hc).\n- apply NNPP; intro Hn. apply Hnm. apply (proj2 (HA c Hne)). split; [exact (H12 c A Hc) | exact Hn]."},{"g":1,"t":"assert (Hmem: MemberOf p A) by (apply (proj2 (HA p Hp)); exact Hpp).","w":"model","kind":"step","ok":true,"c":[2],"d":"Tactic 2 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"assert (HnE: ~Ess A).","w":"model","kind":"step","ok":true,"c":[3,4],"d":"Tactic 3 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"apply H10.","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 4 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"exists p; exact Hmem.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"intros q Hq.","w":"model","kind":"step","ok":true,"c":[7],"d":"Tactic 6 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"assert (Hq2: q <> A) by (intro E; subst; exact (HnA Hq)).","w":"model","kind":"step","ok":true,"c":[8],"d":"Tactic 7 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"pose proof (proj1 (HA q Hq2) Hq) as Hq3.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 8 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"destruct Hq3 as [_ H].","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 9 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"exact H.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 10 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"assert (HT: ThroughAnother A) by (apply NNPP; intro HN; apply HnE; apply H8; auto).","w":"model","kind":"step","ok":true,"c":[11],"d":"Tactic 11 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"destruct HT as [c Hc].","w":"model","kind":"step","ok":true,"c":[12],"d":"Tactic 12 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"assert (Hth: forall x y, clos_trans Thing Gives x y -> Through y x).","w":"model","kind":"step","ok":true,"c":[13,14],"d":"Tactic 13 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"intros x y h.","w":"model","kind":"step","ok":true,"c":[15],"d":"Tactic 14 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"induction h as [x y Hg | x y z h1 IH1 h2 IH2].","w":"model","kind":"step","ok":true,"c":[16,17],"d":"Tactic 15 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"exact (H2 y x Hg).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 16 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"exact (H3 z y x IH2 IH1).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 17 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"assert (Hcl: forall x y, clos_trans Thing Gives x y -> exists w, Gives w y /\\ (w = x \\/ clos_trans Thing Gives x w)).","w":"model","kind":"step","ok":true,"c":[18,19],"d":"Tactic 18 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"intros x y h.","w":"model","kind":"step","ok":true,"c":[20],"d":"Tactic 19 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"induction h as [x y Hg | x y z h1 IH1 h2 IH2].","w":"model","kind":"step","ok":true,"c":[21,22],"d":"Tactic 20 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":21,"t":"exists x; auto.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 21 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"destruct IH2 as [w [Hw [E|C]]]; exists w; split; auto.","w":"model","kind":"step","ok":true,"c":[23,24],"d":"Tactic 22 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"right; subst; exact h1.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 23 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"right; eapply t_trans; eauto.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 24 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"assert (Hnm: ~MemberOf c A).","w":"model","kind":"step","ok":true,"c":[25,26],"d":"Tactic 25 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"intro Hm.","w":"model","kind":"step","ok":true,"c":[27],"d":"Tactic 26 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"pose proof (H11 c A c Hc Hm) as Hcc.","w":"model","kind":"step","ok":true,"c":[28],"d":"Tactic 27 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":28,"t":"destruct (Hcl c c Hcc) as [w [Hw [E|C]]].","w":"model","kind":"step","ok":true,"c":[29,30],"d":"Tactic 28 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":29,"t":"subst.","w":"model","kind":"step","ok":true,"c":[31],"d":"Tactic 29 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":31,"t":"exact (H1 _ _ Hw (H2 _ _ Hw)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 30 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":30,"t":"exact (H1 c w Hw (Hth c w C)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 31 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":26,"t":"assert (Hne: c <> A).","w":"model","kind":"step","ok":true,"c":[32,33],"d":"Tactic 32 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":32,"t":"intro E.","w":"model","kind":"step","ok":true,"c":[34],"d":"Tactic 33 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":34,"t":"subst.","w":"model","kind":"step","ok":true,"c":[35],"d":"Tactic 34 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":35,"t":"exact (H1 _ _ Hc (H2 _ _ Hc)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 35 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":33,"t":"refine (ex_intro _ c (conj Hc (conj _ (conj Hnm Hne)))).","w":"model","kind":"step","ok":true,"c":[36],"d":"Tactic 36 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":36,"t":"split.","w":"model","kind":"step","ok":true,"c":[37,38],"d":"Tactic 37 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":37,"t":"exact (H12 c A Hc).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 38 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":38,"t":"apply NNPP; intro Hn.","w":"model","kind":"step","ok":true,"c":[39],"d":"Tactic 39 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":39,"t":"apply Hnm.","w":"model","kind":"step","ok":true,"c":[40],"d":"Tactic 40 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":40,"t":"apply (proj2 (HA c Hne)).","w":"model","kind":"step","ok":true,"c":[41],"d":"Tactic 41 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":41,"t":"split; [exact (H12 c A Hc) | exact Hn].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 42 of 42 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45],"proofFile":"files/claude/run2/imkan_s11_necessary_cause.v"},{"key":"claude2/imkan_s12_necessary_exists","id":"imkan_s12_necessary_exists","run":"claude2","title":"Dalīl al-imkān, step 12 (aggregate route)","english":"A Necessary Existent exists.","stmt":"MP001 -> MP002 -> MP003 -> MP008 -> MP009 -> MP010 -> MP011 -> MP012 -> SP001 -> exists n, Necessary n","proved":true,"seconds":49.4,"expansions":1,"calls":38,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP008 -> MP009 -> MP010 -> MP011 -> MP012 -> SP001 -> exists n, Necessary n","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\n============================\nexists n : Thing, Necessary n","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\n============================\nforall x y : Thing, clos_trans Thing Gives x y -> Through y x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\n============================\nexists n : Thing, Necessary n","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nx : Thing\ny : Thing\nh : clos_trans Thing Gives x y\n============================\nThrough y x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nx : Thing\ny : Thing\nH : Gives x y\n============================\nThrough y x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nx : Thing\ny : Thing\nz : Thing\nh1 : clos_trans Thing Gives x y\nh2 : clos_trans Thing Gives y z\nIHh1 : Through y x\nIHh2 : Through z y\n============================\nThrough z x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\n============================\nforall c : Thing, clos_trans Thing Gives c c -> False","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\n============================\nexists n : Thing, Necessary n","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nc : Thing\nh : clos_trans Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nc : Thing\nh : clos_trans_1n Thing Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nc : Thing\nh : clos_trans_1n Thing Gives c c\nH : Gives c c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\nc : Thing\nh : clos_trans_1n Thing Gives c c\ny : Thing\nH : Gives c y\nH0 : clos_trans_1n Thing Gives y c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\n============================\n~ ~ exists n : Thing, Necessary n","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\n============================\nforall x : Thing, Exists x -> ~ Ess x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nx : Thing\nex : Exists x\nes : Ess x\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nx : Thing\nex : Exists x\nes : Ess x\n============================\nexists n : Thing, Necessary n","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nx : Thing\nex : Exists x\nes : Ess x\n============================\nNecessary x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\n============================\nforall x : Thing, Exists x -> ThroughAnother x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nthr : forall x : Thing, Exists x -> ThroughAnother x\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nx : Thing\nex : Exists x\n============================\nThroughAnother x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nx : Thing\nex : Exists x\n============================\n~ ~ ThroughAnother x","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nS : SP001\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nx : Thing\nex : Exists x\nnt : ~ ThroughAnother x\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nx0 : Thing\ne0 : Exists x0\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nthr : forall x : Thing, Exists x -> ThroughAnother x\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nx0 : Thing\ne0 : Exists x0\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nthr : forall x : Thing, Exists x -> ThroughAnother x\nA : Thing\nHM : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHA : ~ MemberOf A A\neA : Exists A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nx0 : Thing\ne0 : Exists x0\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nthr : forall x : Thing, Exists x -> ThroughAnother x\nA : Thing\nHM : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHA : ~ MemberOf A A\neA : Exists A\nc : Thing\ngc : Gives c A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nx0 : Thing\ne0 : Exists x0\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nthr : forall x : Thing, Exists x -> ThroughAnother x\nA : Thing\nHM : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHA : ~ MemberOf A A\neA : Exists A\nc : Thing\ngc : Gives c A\nec : Exists c\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nx0 : Thing\ne0 : Exists x0\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nthr : forall x : Thing, Exists x -> ThroughAnother x\nA : Thing\nHM : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHA : ~ MemberOf A A\neA : Exists A\nc : Thing\ngc : Gives c A\nec : Exists c\ne : c = A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nx0 : Thing\ne0 : Exists x0\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nthr : forall x : Thing, Exists x -> ThroughAnother x\nA : Thing\nHM : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHA : ~ MemberOf A A\neA : Exists A\nc : Thing\ngc : Gives c A\nec : Exists c\nne : c <> A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nx0 : Thing\ne0 : Exists x0\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nthr : forall x : Thing, Exists x -> ThroughAnother x\nA : Thing\nHM : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHA : ~ MemberOf A A\neA : Exists A\nec : Exists A\ngc : Gives A A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nx0 : Thing\ne0 : Exists x0\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nthr : forall x : Thing, Exists x -> ThroughAnother x\nA : Thing\nHM : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHA : ~ MemberOf A A\neA : Exists A\nc : Thing\ngc : Gives c A\nec : Exists c\nne : c <> A\nmem : MemberOf c A\n============================\nFalse","H1 : MP001\nH2 : MP002\nH3 : MP003\nH8 : MP008\nH9 : MP009\nH10 : MP010\nH11 : MP011\nH12 : MP012\nx0 : Thing\ne0 : Exists x0\nkey : forall x y : Thing, clos_trans Thing Gives x y -> Through y x\ncyc : forall c : Thing, clos_trans Thing Gives c c -> False\nN : ~ exists n : Thing, Necessary n\nallnon : forall x : Thing, Exists x -> ~ Ess x\nthr : forall x : Thing, Exists x -> ThroughAnother x\nA : Thing\nHM : forall p : Thing, p <> A -> MemberOf p A <-> Possible p\nHA : ~ MemberOf A A\neA : Exists A\nc : Thing\ngc : Gives c A\nec : Exists c\nne : c <> A\nmem : MemberOf c A\n============================\nclos_trans Thing Gives c c"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"intros H1 H2 H3 H8 H9 H10 H11 H12 S.","w":"model","kind":"step","ok":true,"c":[1],"d":"Tactic 1 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H1 H2 H3 H8 H9 H10 H11 H12 S.\nassert (key: forall x y, clos_trans Thing Gives x y -> Through y x).\n{ intros x y h. induction h.\n  - apply H2; exact H.\n  - eapply H3; eauto. }\nassert (cyc: forall c, clos_trans Thing Gives c c -> False).\n{ intros c h. apply clos_trans_t1n_iff in h. inversion h; subst.\n  - eapply H1; [eassumption | apply H2; eassumption].\n  - eapply H1; [eassumption | apply key; apply clos_trans_t1n_iff; eassumption]. }\napply NNPP. intro N.\nassert (allnon: forall x, Exists x -> ~ Ess x).\n{ intros x ex es. apply N. exists x. split; auto. }\nassert (thr: forall x, Exists x -> ThroughAnother x).\n{ intros x ex. apply NNPP. intro nt. exact (allnon x ex (H8 x ex nt)). }\ndestruct S as [x0 e0].\ndestruct (H9 (ex_intro _ x0 (conj e0 (allnon x0 e0)) : exists p, Possible p)) as [A [[HM HA] eA]].\ndestruct (thr A eA) as [c gc].\nassert (ec : Exists c) by exact (H12 c A gc).\ndestruct (classic (c = A)) as [e|ne].\n- subst. eapply H1; [exact gc | apply H2; exact gc].\n- assert (mem : MemberOf c A) by (apply (HM c ne); exact (conj ec (allnon c ec))).\n  apply (cyc c). exact (H11 c A c gc mem)."},{"g":1,"t":"assert (key: forall x y, clos_trans Thing Gives x y -> Through y x).","w":"model","kind":"step","ok":true,"c":[2,3],"d":"Tactic 2 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":2,"t":"intros x y h.","w":"model","kind":"step","ok":true,"c":[4],"d":"Tactic 3 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":4,"t":"induction h.","w":"model","kind":"step","ok":true,"c":[5,6],"d":"Tactic 4 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":5,"t":"apply H2; exact H.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 5 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"eapply H3; eauto.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 6 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":3,"t":"assert (cyc: forall c, clos_trans Thing Gives c c -> False).","w":"model","kind":"step","ok":true,"c":[7,8],"d":"Tactic 7 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":7,"t":"intros c h.","w":"model","kind":"step","ok":true,"c":[9],"d":"Tactic 8 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":9,"t":"apply clos_trans_t1n_iff in h.","w":"model","kind":"step","ok":true,"c":[10],"d":"Tactic 9 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":10,"t":"inversion h; subst.","w":"model","kind":"step","ok":true,"c":[11,12],"d":"Tactic 10 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":11,"t":"eapply H1; [eassumption | apply H2; eassumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 11 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":12,"t":"eapply H1; [eassumption | apply key; apply clos_trans_t1n_iff; eassumption].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 12 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":8,"t":"apply NNPP.","w":"model","kind":"step","ok":true,"c":[13],"d":"Tactic 13 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":13,"t":"intro N.","w":"model","kind":"step","ok":true,"c":[14],"d":"Tactic 14 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":14,"t":"assert (allnon: forall x, Exists x -> ~ Ess x).","w":"model","kind":"step","ok":true,"c":[15,16],"d":"Tactic 15 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":15,"t":"intros x ex es.","w":"model","kind":"step","ok":true,"c":[17],"d":"Tactic 16 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":17,"t":"apply N.","w":"model","kind":"step","ok":true,"c":[18],"d":"Tactic 17 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":18,"t":"exists x.","w":"model","kind":"step","ok":true,"c":[19],"d":"Tactic 18 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":19,"t":"split; auto.","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 19 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":16,"t":"assert (thr: forall x, Exists x -> ThroughAnother x).","w":"model","kind":"step","ok":true,"c":[20,21],"d":"Tactic 20 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":20,"t":"intros x ex.","w":"model","kind":"step","ok":true,"c":[22],"d":"Tactic 21 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":22,"t":"apply NNPP.","w":"model","kind":"step","ok":true,"c":[23],"d":"Tactic 22 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":23,"t":"intro nt.","w":"model","kind":"step","ok":true,"c":[24],"d":"Tactic 23 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":24,"t":"exact (allnon x ex (H8 x ex nt)).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 24 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":21,"t":"destruct S as [x0 e0].","w":"model","kind":"step","ok":true,"c":[25],"d":"Tactic 25 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":25,"t":"destruct (H9 (ex_intro _ x0 (conj e0 (allnon x0 e0)) : exists p, Possible p)) as [A [[HM HA] eA]].","w":"model","kind":"step","ok":true,"c":[26],"d":"Tactic 26 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":26,"t":"destruct (thr A eA) as [c gc].","w":"model","kind":"step","ok":true,"c":[27],"d":"Tactic 27 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":27,"t":"assert (ec : Exists c) by exact (H12 c A gc).","w":"model","kind":"step","ok":true,"c":[28],"d":"Tactic 28 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":28,"t":"destruct (classic (c = A)) as [e|ne].","w":"model","kind":"step","ok":true,"c":[29,30],"d":"Tactic 29 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":29,"t":"subst.","w":"model","kind":"step","ok":true,"c":[31],"d":"Tactic 30 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":31,"t":"eapply H1; [exact gc | apply H2; exact gc].","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 31 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":30,"t":"assert (mem : MemberOf c A) by (apply (HM c ne); exact (conj ec (allnon c ec))).","w":"model","kind":"step","ok":true,"c":[32],"d":"Tactic 32 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":32,"t":"apply (cyc c).","w":"model","kind":"step","ok":true,"c":[33],"d":"Tactic 33 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":33,"t":"exact (H11 c A c gc mem).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 34 of 34 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37],"proofFile":"files/claude/run2/imkan_s12_necessary_exists.v"},{"key":"claude2/imkan_b_necessary_exists","id":"imkan_b_necessary_exists","run":"claude2","title":"Dalīl al-imkān, conclusion (shorter route)","english":"A Necessary Existent exists.","stmt":"MP008 -> OP001 -> SP001 -> exists n, Necessary n","proved":true,"seconds":34.8,"expansions":5,"calls":19,"root":0,"goals":["MP008 -> OP001 -> SP001 -> exists n, Necessary n","MP008 -> forall x, OfItself x -> Necessary x","OP001 -> forall x, Exists x -> exists y, OfItself y","forall x, Exists x -> ~ ThroughAnother x -> exists y, OfItself y","OP001 -> forall x, ThroughAnother x -> exists y, OfItself y","H : OP001\nx : Thing\ny : Thing\nHy : Gives y x\n============================\nexists y0 : Thing, OfItself y0","H : OP001\nx : Thing\ny : Thing\nHy : Gives y x\n============================\nOfItself y"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":0,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP008 -> forall x, OfItself x -> Necessary x","OP001 -> forall x, Exists x -> exists y, OfItself y"],"combine":"intros H8 H1 Hs. destruct Hs as [x Hx]. destruct (sub2 H1 x Hx) as [y Hy]. exists y. exact (sub1 H8 y Hy).","ok":true,"c":[1,2],"assembled":true},{"g":1,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":2,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":2,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x, Exists x -> ~ ThroughAnother x -> exists y, OfItself y","OP001 -> forall x, ThroughAnother x -> exists y, OfItself y"],"combine":"intros H x Hx. destruct (classic (ThroughAnother x)) as [T|N]. - exact (sub2 H x T). - exact (sub1 x Hx N).","ok":true,"c":[3,4],"assembled":true},{"g":3,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":4,"t":"intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; tauto.","w":"auto","ok":false,"err":"The automation could not prove it."},{"g":4,"t":"intros H x [y Hy].","w":"model","kind":"step","ok":true,"c":[5],"d":"Tactic 1 of 3 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time.","full":"intros H x [y Hy]. exists y. exact (H x y Hy)."},{"g":5,"t":"exists y.","w":"model","kind":"step","ok":true,"c":[6],"d":"Tactic 2 of 3 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."},{"g":6,"t":"exact (H x y Hy).","w":"model","kind":"step","ok":true,"c":[],"d":"Tactic 3 of 3 in a complete proof that Claude wrote. Rocq checked the whole proof; it is replayed here one tactic at a time."}],"proof":[4,5,10,11,16,17,18],"proofFile":"files/claude/run2/imkan_b_necessary_exists.v"},{"key":"claude3/pr001_relational_exhaustive","id":"pr001_relational_exhaustive","run":"claude3","title":"Dalīl al-kalima, premise 1","english":"The four possibilities are the only ones.","stmt":"forall x, (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b) \\/ ((forall y, ~ DependsOn x y) \\/ exists y, clos_trans Thing DependsOn x y /\\ forall z, ~ DependsOn y z)","proved":false,"seconds":659.6,"expansions":4,"calls":12,"root":0,"goals":["forall x, (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ DependsOn y y) \\/ (exists y, (x = y \\/ clos_trans Thing DependsOn x y) /\\ clos_trans Thing DependsOn y y /\\ ~ DependsOn y y) \\/ (exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b) \\/ ((forall y, ~ DependsOn x y) \\/ exists y, clos_trans Thing DependsOn x y /\\ forall z, ~ DependsOn y z)","forall x : Thing, (exists a, (x = a \\/ clos_trans Thing DependsOn x a) /\\ forall z, ~ DependsOn a z) \\/ (forall a, (x = a \\/ clos_trans Thing DependsOn x a) -> exists b, DependsOn a b)","forall x : Thing, (forall a, (x = a \\/ clos_trans Thing DependsOn x a) -> exists b, DependsOn a b) -> exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b","forall x : Thing, (exists a, (x = a \\/ clos_trans Thing DependsOn x a) /\\ forall z, ~ DependsOn a z) -> (forall y, ~ DependsOn x y) \\/ exists y, clos_trans Thing DependsOn x y /\\ forall z, ~ DependsOn y z","forall P : Thing -> Prop, (exists a, P a /\\ forall z, ~ DependsOn a z) \\/ (forall a, P a -> exists b, DependsOn a b)","forall a : Thing, (forall z, ~ DependsOn a z) \\/ (exists b, DependsOn a b)","forall P : Thing -> Prop, (exists a, P a /\\ forall z, ~ DependsOn a z) \\/ ~ (exists a, P a /\\ forall z, ~ DependsOn a z)"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x : Thing, (forall a, (x = a \\/ clos_trans Thing DependsOn x a) -> exists b, DependsOn a b) -> exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b","forall x : Thing, (exists a, (x = a \\/ clos_trans Thing DependsOn x a) /\\ forall z, ~ DependsOn a z) -> (forall y, ~ DependsOn x y) \\/ exists y, clos_trans Thing DependsOn x y /\\ forall z, ~ DependsOn y z"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x : Thing, (exists a, (x = a \\/ clos_trans Thing DependsOn x a) /\\ forall z, ~ DependsOn a z) \\/ (forall a, (x = a \\/ clos_trans Thing DependsOn x a) -> exists b, DependsOn a b)","forall x : Thing, (forall a, (x = a \\/ clos_trans Thing DependsOn x a) -> exists b, DependsOn a b) -> exists S : Thing -> Prop, S x /\\ forall a, S a -> exists b, S b /\\ DependsOn a b","forall x : Thing, (exists a, (x = a \\/ clos_trans Thing DependsOn x a) /\\ forall z, ~ DependsOn a z) -> (forall y, ~ DependsOn x y) \\/ exists y, clos_trans Thing DependsOn x y /\\ forall z, ~ DependsOn y z"],"ok":true,"c":[1,2,3],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":1,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":1,"t":"divide into 1 subgoals","w":"model","kind":"division","subs":["forall P : Thing -> Prop, (exists a, P a /\\ forall z, ~ DependsOn a z) \\/ (forall a, P a -> exists b, DependsOn a b)"],"ok":true,"c":[4],"combine":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":4,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":4,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall a : Thing, (forall z, ~ DependsOn a z) \\/ (exists b, DependsOn a b)","forall P : Thing -> Prop, (exists a, P a /\\ forall z, ~ DependsOn a z) \\/ ~ (exists a, P a /\\ forall z, ~ DependsOn a z)"],"ok":true,"c":[5,6],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":5,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","w":"auto","ok":true,"c":[]},{"g":6,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","w":"auto","ok":true,"c":[]},{"g":2,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":2,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x a b : Thing, (x = a \\/ clos_trans Thing DependsOn x a) -> DependsOn a b -> (x = b \\/ clos_trans Thing DependsOn x b)","forall x : Thing, (x = x \\/ clos_trans Thing DependsOn x x)"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":2,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x a : Thing, (x = a \\/ clos_trans Thing DependsOn x a) <-> clos_refl_trans Thing DependsOn x a","forall x a b : Thing, clos_refl_trans Thing DependsOn x a -> DependsOn a b -> clos_refl_trans Thing DependsOn x b","forall x : Thing, clos_refl_trans Thing DependsOn x x"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."}],"proof":[],"proofFile":null},{"key":"claude3/pr002_no_self_creation","id":"pr002_no_self_creation","run":"claude3","title":"Dalīl al-kalima, premise 2","english":"Nothing gives existence to itself.","stmt":"MP001 -> MP002 -> forall x, ~ Gives x x","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["MP001 -> MP002 -> forall x, ~ Gives x x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run3_subgoals_only/pr002_no_self_creation.v"},{"key":"claude3/pr003_no_circle","id":"pr003_no_circle","run":"claude3","title":"Dalīl al-kalima, premise 3","english":"No chain of existence-giving returns to where it started.","stmt":"MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing DependsOn x x","proved":true,"seconds":874.8,"expansions":5,"calls":22,"root":0,"goals":["MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing DependsOn x x","MP002 -> MP003 -> forall x y, clos_trans Thing DependsOn x y -> Through x y","forall x, clos_trans Thing DependsOn x x -> exists y, Gives y x /\\ (y = x \\/ clos_trans Thing DependsOn y x)","forall x z, clos_trans Thing DependsOn x z -> clos_trans_1n Thing DependsOn x z","forall x z, clos_trans_1n Thing DependsOn x z -> exists y, DependsOn x y /\\ (y = z \\/ clos_trans Thing DependsOn y z)","forall x y, DependsOn x y -> clos_trans_1n Thing DependsOn x y","forall x y z, clos_trans_1n Thing DependsOn x y -> clos_trans_1n Thing DependsOn y z -> clos_trans_1n Thing DependsOn x z","forall x y, clos_trans_1n Thing DependsOn x y -> clos_trans Thing DependsOn x y","forall x y, clos_trans Thing DependsOn x y -> clos_trans_1n Thing DependsOn x y","forall x y z, clos_trans Thing DependsOn x y -> clos_trans Thing DependsOn y z -> clos_trans Thing DependsOn x z","forall y z, clos_trans_1n Thing DependsOn y z -> forall x, clos_trans_1n Thing DependsOn x y -> clos_trans_1n Thing DependsOn x z","forall x y z, DependsOn x y -> clos_trans_1n Thing DependsOn y z -> clos_trans_1n Thing DependsOn x z","forall x y z, clos_trans_1n Thing DependsOn x y -> clos_trans_1n Thing DependsOn y z -> (exists w, DependsOn x w /\\ clos_trans_1n Thing DependsOn w y) \\/ x = y","forall x z, clos_trans Thing DependsOn x z -> clos_trans_n1 Thing DependsOn x z","forall x y z, clos_trans_1n Thing DependsOn x y -> DependsOn y z -> clos_trans_1n Thing DependsOn x z","forall x z, clos_trans_n1 Thing DependsOn x z -> clos_trans_1n Thing DependsOn x z","forall x z, clos_trans_1n Thing DependsOn x z -> (DependsOn x z \\/ exists y, DependsOn x y /\\ clos_trans_1n Thing DependsOn y z)","forall y z, clos_trans_1n Thing DependsOn y z -> clos_trans Thing DependsOn y z"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y, clos_trans Thing DependsOn x y -> Through x y","forall x, clos_trans Thing DependsOn x x -> exists y, Gives y x /\\ (y = x \\/ clos_trans Thing DependsOn y x)"],"ok":true,"c":[1,2],"combine":"intros; qauto unfold: MP001.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":1,"t":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":2,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":2,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x z, clos_trans Thing DependsOn x z -> clos_trans_1n Thing DependsOn x z","forall x z, clos_trans_1n Thing DependsOn x z -> exists y, DependsOn x y /\\ (y = z \\/ clos_trans Thing DependsOn y z)"],"ok":true,"c":[3,4],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":3,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":3,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x y, DependsOn x y -> clos_trans_1n Thing DependsOn x y","forall x y z, clos_trans_1n Thing DependsOn x y -> clos_trans_1n Thing DependsOn y z -> clos_trans_1n Thing DependsOn x z"],"ok":true,"c":[5,6],"combine":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":5,"t":"intros; sauto q: on.","w":"hammer","ok":true,"c":[]},{"g":6,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":6,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans_1n Thing DependsOn x y -> clos_trans Thing DependsOn x y","forall x y, clos_trans Thing DependsOn x y -> clos_trans_1n Thing DependsOn x y","forall x y z, clos_trans Thing DependsOn x y -> clos_trans Thing DependsOn y z -> clos_trans Thing DependsOn x z"],"ok":true,"c":[7,8,9],"combine":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":7,"t":"intros; srun (eauto) use: clos_t1n_trans unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":8,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":6,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall y z, clos_trans_1n Thing DependsOn y z -> forall x, clos_trans_1n Thing DependsOn x y -> clos_trans_1n Thing DependsOn x z","forall x y z, DependsOn x y -> clos_trans_1n Thing DependsOn y z -> clos_trans_1n Thing DependsOn x z","forall x y z, clos_trans_1n Thing DependsOn x y -> clos_trans_1n Thing DependsOn y z -> (exists w, DependsOn x w /\\ clos_trans_1n Thing DependsOn w y) \\/ x = y"],"ok":true,"c":[10,11,12],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":10,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":3,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x z, clos_trans Thing DependsOn x z -> clos_trans_n1 Thing DependsOn x z","forall x y z, clos_trans_1n Thing DependsOn x y -> DependsOn y z -> clos_trans_1n Thing DependsOn x z","forall x z, clos_trans_n1 Thing DependsOn x z -> clos_trans_1n Thing DependsOn x z"],"ok":true,"c":[13,14,15],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":13,"t":"intros; sfirstorder use: clos_trans_tn1 unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":14,"t":"intros; hauto lq: on use: t_trans, pr003_no_circle_L3, pr003_no_circle_L2, clos_trans_t1n unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":15,"t":"intros; srun (eauto) use: clos_tn1_trans, clos_trans_t1n unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":4,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":4,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x z, clos_trans_1n Thing DependsOn x z -> (DependsOn x z \\/ exists y, DependsOn x y /\\ clos_trans_1n Thing DependsOn y z)","forall y z, clos_trans_1n Thing DependsOn y z -> clos_trans Thing DependsOn y z"],"ok":true,"c":[16,17],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":16,"t":"intros; match goal with H : clos_trans_1n _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":17,"t":"intros; srun (eauto) use: pr003_no_circle_L3.","w":"hammer","ok":true,"c":[]}],"proof":[1,2,4,14,15,16,17,19,20,21],"proofFile":"files/claude/run3_subgoals_only/pr003_no_circle.v"},{"key":"claude3/pr004_no_two_step_giving","id":"pr004_no_two_step_giving","run":"claude3","title":"Dalīl al-kalima, premise 4 (key step)","english":"What receives existence does not give existence to anything.","stmt":"OP001 -> forall x y z, Gives y x -> ~ Gives z y","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["OP001 -> forall x y z, Gives y x -> ~ Gives z y"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run3_subgoals_only/pr004_no_two_step_giving.v"},{"key":"claude3/kalima_first_cause","id":"kalima_first_cause","run":"claude3","title":"Dalīl al-kalima, the first cause","english":"If the universe has its existence through another, it has a first cause that exists of itself.","stmt":"OP001 -> ThroughAnother Universe -> exists b, Gives b Universe /\\ OfItself b /\\ (forall y, ~ Gives y b)","proved":true,"seconds":2.4,"expansions":0,"calls":1,"root":0,"goals":["OP001 -> ThroughAnother Universe -> exists b, Gives b Universe /\\ OfItself b /\\ (forall y, ~ Gives y b)"],"events":[{"g":0,"t":"intros; sfirstorder.","w":"hammer","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run3_subgoals_only/kalima_first_cause.v"},{"key":"claude3/pr011_something_caused","id":"pr011_something_caused","run":"claude3","title":"The universe is caused","english":"Something in the universe has its existence through another.","stmt":"MP008 -> OP006 -> exists x, InUniverse x /\\ Possible x /\\ ThroughAnother x","proved":true,"seconds":0.8,"expansions":0,"calls":1,"root":0,"goals":["MP008 -> OP006 -> exists x, InUniverse x /\\ Possible x /\\ ThroughAnother x"],"events":[{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run3_subgoals_only/pr011_something_caused.v"},{"key":"claude3/kalima_unconditional","id":"kalima_unconditional","run":"claude3","title":"Dalīl al-kalima, the conclusion without the “if”","english":"Something in the universe has a giver of existence that is necessary and depends on nothing.","stmt":"MP001 -> MP002 -> MP003 -> OP001 -> MP008 -> OP006 -> exists x b, InUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b","proved":false,"seconds":340.1,"expansions":1,"calls":3,"root":0,"goals":["MP001 -> MP002 -> MP003 -> OP001 -> MP008 -> OP006 -> exists x b, InUniverse x /\\ Gives b x /\\ Exists b /\\ Necessary b /\\ Independent b"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP008 -> forall x, Possible x -> ThroughAnother x","MP008 -> OP001 -> forall x b, Gives b x -> Exists b /\\ Necessary b /\\ Independent b","OP006 -> exists x, InUniverse x /\\ Possible x"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP008 -> OP006 -> exists x, InUniverse x /\\ ThroughAnother x","OP001 -> forall x b, Gives b x -> OfItself b","MP008 -> forall b, OfItself b -> Necessary b /\\ Independent b"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."}],"proof":[],"proofFile":null},{"key":"claude3/imkan_s1_division","id":"imkan_s1_division","run":"claude3","title":"Dalīl al-imkān, step 1","english":"Every existent is necessary or possible in itself, and not both.","stmt":"forall x, Exists x -> (Necessary x \\/ Possible x) /\\ ~ (Necessary x /\\ Possible x)","proved":true,"seconds":1.1,"expansions":0,"calls":1,"root":0,"goals":["forall x, Exists x -> (Necessary x \\/ Possible x) /\\ ~ (Necessary x /\\ Possible x)"],"events":[{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run3_subgoals_only/imkan_s1_division.v"},{"key":"claude3/imkan_s2_necessary_cannot_fail","id":"imkan_s2_necessary_cannot_fail","run":"claude3","title":"Dalīl al-imkān, step 2","english":"The non-existence of what is necessary in itself is not possible.","stmt":"MP007 -> forall x, Necessary x -> ~ NonExistencePossible x","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["MP007 -> forall x, Necessary x -> ~ NonExistencePossible x"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run3_subgoals_only/imkan_s2_necessary_cannot_fail.v"},{"key":"claude3/step3_possible_needs_cause","id":"step3_possible_needs_cause","run":"claude3","title":"Dalīl al-imkān, step 3","english":"A possible existent needs a cause.","stmt":"MP008 -> forall x, Possible x -> ThroughAnother x","proved":true,"seconds":0.8,"expansions":0,"calls":1,"root":0,"goals":["MP008 -> forall x, Possible x -> ThroughAnother x"],"events":[{"g":0,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run3_subgoals_only/step3_possible_needs_cause.v"},{"key":"claude3/imkan_s4_no_self_causation","id":"imkan_s4_no_self_causation","run":"claude3","title":"Dalīl al-imkān, step 4","english":"Nothing gives existence to itself.","stmt":"MP001 -> MP002 -> forall A, ~ Gives A A","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["MP001 -> MP002 -> forall A, ~ Gives A A"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run3_subgoals_only/imkan_s4_no_self_causation.v"},{"key":"claude3/imkan_s5_none_outside","id":"imkan_s5_none_outside","run":"claude3","title":"Dalīl al-imkān, step 5","english":"No possible existent lies outside the aggregate of possibles.","stmt":"forall A, AggregateOfPossibles A -> forall p, p <> A -> Possible p -> MemberOf p A","proved":true,"seconds":0.2,"expansions":0,"calls":1,"root":0,"goals":["forall A, AggregateOfPossibles A -> forall p, p <> A -> Possible p -> MemberOf p A"],"events":[{"g":0,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[0],"proofFile":"files/claude/run3_subgoals_only/imkan_s5_none_outside.v"},{"key":"claude3/imkan_s6_aggregate_possible_caused","id":"imkan_s6_aggregate_possible_caused","run":"claude3","title":"Dalīl al-imkān, step 6","english":"The aggregate is itself possible, and so has its existence through another.","stmt":"MP008 -> MP010 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> Possible A /\\ ThroughAnother A","proved":true,"seconds":676.5,"expansions":1,"calls":6,"root":0,"goals":["MP008 -> MP010 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> Possible A /\\ ThroughAnother A","MP010 -> forall A, AggregateOfPossibles A -> (exists p, p <> A /\\ Possible p) -> ~ Ess A","forall A, Exists A -> ~ Ess A -> Possible A","MP008 -> forall A, Exists A -> ~ Ess A -> ThroughAnother A"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall A, AggregateOfPossibles A -> (exists p, p <> A /\\ Possible p) -> exists p, MemberOf p A","forall A, AggregateOfPossibles A -> forall p, MemberOf p A -> ~ Ess p","MP008 -> forall A, Exists A -> ~ Ess A -> ThroughAnother A"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP010 -> forall A, AggregateOfPossibles A -> (exists p, p <> A /\\ Possible p) -> ~ Ess A","forall A, Exists A -> ~ Ess A -> Possible A","MP008 -> forall A, Exists A -> ~ Ess A -> ThroughAnother A"],"ok":true,"c":[1,2,3],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":1,"t":"intros; hauto lq: on unfold: MP010, AggregateOfPossibles, Possible.","w":"hammer","ok":true,"c":[]},{"g":2,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":3,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[2,3,4,5],"proofFile":"files/claude/run3_subgoals_only/imkan_s6_aggregate_possible_caused.v"},{"key":"claude3/imkan_s7_cause_not_member","id":"imkan_s7_cause_not_member","run":"claude3","title":"Dalīl al-imkān, step 7","english":"What gives existence to the aggregate is not one of its members.","stmt":"MP001 -> MP002 -> MP003 -> MP011 -> forall c A, Gives c A -> ~ MemberOf c A","proved":true,"seconds":592.8,"expansions":5,"calls":17,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP011 -> forall c A, Gives c A -> ~ MemberOf c A","MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> Through b a","MP001 -> MP002 -> MP003 -> forall a, ~ clos_trans Thing Gives a a","MP011 -> (forall a, ~ clos_trans Thing Gives a a) -> forall c A, Gives c A -> ~ MemberOf c A","forall a : Thing, clos_trans Thing Gives a a -> exists z, Gives a z /\\ clos_trans Thing Gives z a","MP001 -> MP002 -> MP003 -> forall a z : Thing, Gives a z -> clos_trans Thing Gives z a -> False","forall x y : Thing, clos_trans Thing Gives x y -> exists z, Gives x z /\\ (z = y \\/ clos_trans Thing Gives z y)","forall x y : Thing, clos_trans Thing Gives x y -> clos_trans_1n Thing Gives x y","forall x y : Thing, clos_trans_1n Thing Gives x y -> Gives x y \\/ (exists z, Gives x z /\\ clos_trans_1n Thing Gives z y)","forall x y : Thing, clos_trans_1n Thing Gives x y -> clos_trans Thing Gives x y","MP002 -> MP003 -> forall a z : Thing, clos_trans Thing Gives z a -> Through a z","MP001 -> forall a z : Thing, Gives a z -> ~ Through a z"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> Through b a","MP001 -> MP002 -> MP003 -> forall a, ~ clos_trans Thing Gives a a","MP011 -> (forall a, ~ clos_trans Thing Gives a a) -> forall c A, Gives c A -> ~ MemberOf c A"],"ok":true,"c":[1,2,3],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":1,"t":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":2,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":2,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall a : Thing, clos_trans Thing Gives a a -> exists z, Gives a z /\\ clos_trans Thing Gives z a","MP001 -> MP002 -> MP003 -> forall a z : Thing, Gives a z -> clos_trans Thing Gives z a -> False"],"ok":true,"c":[4,5],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":4,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":4,"t":"divide into 1 subgoals","w":"model","kind":"division","subs":["forall x y : Thing, clos_trans Thing Gives x y -> exists z, Gives x z /\\ (z = y \\/ clos_trans Thing Gives z y)"],"ok":true,"c":[6],"combine":"intros; hauto.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":6,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":6,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x y : Thing, clos_trans Thing Gives x y -> clos_trans_1n Thing Gives x y","forall x y : Thing, clos_trans_1n Thing Gives x y -> Gives x y \\/ (exists z, Gives x z /\\ clos_trans_1n Thing Gives z y)","forall x y : Thing, clos_trans_1n Thing Gives x y -> clos_trans Thing Gives x y"],"ok":true,"c":[7,8,9],"combine":"intros; hauto.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":7,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; srun (eauto) use: clos_trans_t1n unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":8,"t":"intros; match goal with H : clos_trans_1n _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":9,"t":"intros; srun (eauto) use: clos_t1n_trans unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":5,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":5,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall a z : Thing, clos_trans Thing Gives z a -> Through a z","MP001 -> forall a z : Thing, Gives a z -> ~ Through a z"],"ok":true,"c":[10,11],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":10,"t":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":11,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":3,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[1,2,4,6,8,9,10,11,13,14,15,16],"proofFile":"files/claude/run3_subgoals_only/imkan_s7_cause_not_member.v"},{"key":"claude3/imkan_s8_no_internal_cause","id":"imkan_s8_no_internal_cause","run":"claude3","title":"Dalīl al-imkān, step 8","english":"Neither a member, nor a group of members, nor the aggregate gives the aggregate existence.","stmt":"MP001 -> MP002 -> MP003 -> MP010 -> MP011 -> forall A, AggregateOfPossibles A -> (forall x, ~ Gives x x) /\\ ~ Gives A A /\\ (forall c, MemberOf c A -> ~ Gives c A) /\\ (forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> ~ Gives B A)","proved":true,"seconds":1389.1,"expansions":8,"calls":30,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP010 -> MP011 -> forall A, AggregateOfPossibles A -> (forall x, ~ Gives x x) /\\ ~ Gives A A /\\ (forall c, MemberOf c A -> ~ Gives c A) /\\ (forall B, B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> ~ Gives B A)","MP001 -> MP002 -> forall x, ~ Gives x x","MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing Gives x x","MP010 -> forall A B, AggregateOfPossibles A -> B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> MemberOf B A","MP001 -> MP002 -> MP003 -> MP011 -> forall A c, MemberOf c A -> Gives c A -> False","MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","forall x, clos_trans Thing Gives x x -> exists y, Gives x y /\\ (y = x \\/ clos_trans Thing Gives y x)","MP001 -> MP002 -> MP003 -> forall x y, Gives x y -> (y = x \\/ clos_trans Thing Gives y x) -> False","MP001 -> MP002 -> forall x y, Gives x y -> Through x y -> False","forall x y, clos_trans Thing Gives x y -> clos_trans_1n Thing Gives x y","forall x y, clos_trans_1n Thing Gives x y -> exists z, Gives x z /\\ (z = y \\/ clos_trans Thing Gives z y)","forall x y, clos_trans_1n Thing Gives x y -> Gives x y \\/ (exists z, Gives x z /\\ clos_trans_1n Thing Gives z y)","forall x y, clos_trans_1n Thing Gives x y -> clos_trans Thing Gives x y","MP011 -> forall A c, MemberOf c A -> Gives c A -> clos_trans Thing Gives c c","MP001 -> MP002 -> MP003 -> forall x, clos_trans Thing Gives x x -> False","MP001 -> MP002 -> forall x, Gives x x -> False","MP001 -> MP002 -> MP003 -> forall x y, Gives x y -> clos_trans Thing Gives y x -> False","forall x y, Gives x y -> clos_trans Thing Gives y x -> clos_trans Thing Gives x x"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 4 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> forall x, ~ Gives x x","MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing Gives x x","MP010 -> forall A B, AggregateOfPossibles A -> B <> A -> Exists B -> (exists p, MemberOf p B) -> (forall p, MemberOf p B -> MemberOf p A) -> MemberOf B A","MP001 -> MP002 -> MP003 -> MP011 -> forall A c, MemberOf c A -> Gives c A -> False"],"ok":true,"c":[1,2,3,4],"combine":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; intuition eauto.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":1,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":2,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":2,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","forall x, clos_trans Thing Gives x x -> exists y, Gives x y /\\ (y = x \\/ clos_trans Thing Gives y x)","MP001 -> MP002 -> MP003 -> forall x y, Gives x y -> (y = x \\/ clos_trans Thing Gives y x) -> False"],"ok":true,"c":[5,6,7],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":5,"t":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":6,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":6,"t":"divide into 1 subgoals","w":"model","kind":"division","subs":["forall x z, clos_trans Thing Gives x z -> Gives x z / (exists y, Gives x y /\\ clos_trans Thing Gives y z)"],"ok":false,"err":"Rejected before checking: a subgoal is not a well-formed Rocq statement."},{"g":6,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x z, clos_trans Thing Gives x z -> clos_trans_1n Thing Gives x z","forall x z, clos_trans_1n Thing Gives x z -> Gives x z / (exists y, Gives x y /\\ clos_trans Thing Gives y z)"],"ok":false,"err":"Rejected before checking: a subgoal is not a well-formed Rocq statement."},{"g":2,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","MP001 -> MP002 -> forall x y, Gives x y -> Through x y -> False","forall x, clos_trans Thing Gives x x -> exists y, Gives x y /\\ (y = x \\/ clos_trans Thing Gives y x)"],"ok":true,"c":[5,8,6],"combine":"intros; hauto.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":8,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":6,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":6,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans Thing Gives x y -> clos_trans_1n Thing Gives x y","forall x y, clos_trans_1n Thing Gives x y -> exists z, Gives x z /\\ (z = y \\/ clos_trans Thing Gives z y)"],"ok":true,"c":[9,10],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":9,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; srun (eauto) use: clos_trans_t1n unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":10,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":10,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans_1n Thing Gives x y -> Gives x y \\/ (exists z, Gives x z /\\ clos_trans_1n Thing Gives z y)","forall x y, clos_trans_1n Thing Gives x y -> clos_trans Thing Gives x y"],"ok":true,"c":[11,12],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":11,"t":"intros; match goal with H : clos_trans_1n _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":12,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; srun (eauto) use: clos_t1n_trans unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":3,"t":"intros; hauto lq: on unfold: MP010, AggregateOfPossibles, Possible.","w":"hammer","ok":true,"c":[]},{"g":4,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":4,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP011 -> forall A c, MemberOf c A -> Gives c A -> clos_trans Thing Gives c c","MP001 -> MP002 -> MP003 -> forall x, clos_trans Thing Gives x x -> False"],"ok":true,"c":[13,14],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":13,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":14,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":14,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> MP003 -> forall x y, Gives x y -> clos_trans Thing Gives y x -> False","MP001 -> MP002 -> forall x, Gives x x -> False"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":14,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y, Gives x y -> (y = x \\/ clos_trans Thing Gives y x) -> Through x y","MP001 -> MP002 -> forall x y, Gives x y -> Through x y -> False"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":4,"t":"divide into 4 subgoals","w":"model","kind":"division","subs":["MP011 -> forall A c, MemberOf c A -> Gives c A -> clos_trans Thing Gives c c","MP001 -> MP002 -> forall x, Gives x x -> False","MP001 -> MP002 -> MP003 -> forall x y, Gives x y -> clos_trans Thing Gives y x -> False","forall x, clos_trans Thing Gives x x -> exists y, Gives x y /\\ (y = x \\/ clos_trans Thing Gives y x)"],"ok":true,"c":[13,15,16,6],"combine":"intros; sauto.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":15,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":16,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":16,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing Gives x x","forall x y, Gives x y -> clos_trans Thing Gives y x -> clos_trans Thing Gives x x"],"ok":true,"c":[2,17],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":17,"t":"intros; sauto lq: on.","w":"hammer","ok":true,"c":[]}],"proof":[1,2,9,5,10,12,13,15,16,17,18,25,21,26,28,29],"proofFile":"files/claude/run3_subgoals_only/imkan_s8_no_internal_cause.v"},{"key":"claude3/imkan_s9_cause_outside","id":"imkan_s9_cause_outside","run":"claude3","title":"Dalīl al-imkān, step 9","english":"The cause of the aggregate lies outside it and is not possible.","stmt":"MP001 -> MP002 -> MP003 -> MP011 -> forall A c, AggregateOfPossibles A -> Gives c A -> ~ MemberOf c A /\\ c <> A /\\ ~ Possible c","proved":true,"seconds":1543.5,"expansions":7,"calls":32,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP011 -> forall A c, AggregateOfPossibles A -> Gives c A -> ~ MemberOf c A /\\ c <> A /\\ ~ Possible c","MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> Through b a","MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing Gives x x","MP001 -> MP002 -> forall A, ~ Gives A A","forall A c, AggregateOfPossibles A -> c <> A -> ~ MemberOf c A -> ~ Possible c","forall x, clos_trans Thing Gives x x -> exists z, Gives x z /\\ clos_refl_trans Thing Gives z x","MP002 -> MP003 -> forall a b, clos_refl_trans Thing Gives a b -> a = b \\/ Through b a","MP001 -> MP002 -> forall x z, Gives x z -> ~ (x = z \\/ Through x z)","forall a b, clos_trans Thing Gives a b -> clos_trans_1n Thing Gives a b","forall a b, clos_trans_1n Thing Gives a b -> exists z, Gives a z /\\ clos_refl_trans Thing Gives z b","forall a b, clos_trans_1n Thing Gives a b -> (Gives a b \\/ exists z, Gives a z /\\ clos_trans_1n Thing Gives z b)","forall z b, clos_trans_1n Thing Gives z b -> clos_refl_trans Thing Gives z b","forall a b, Gives a b -> clos_refl_trans Thing Gives b b","forall a b, clos_trans_1n Thing Gives a b -> exists z, Gives a z /\\ clos_refl_trans_1n Thing Gives z b","forall z b, clos_refl_trans_1n Thing Gives z b -> clos_refl_trans Thing Gives z b","forall a b, clos_trans Thing Gives a b -> clos_refl_trans Thing Gives a b","forall a b c, (exists z, Gives a z /\\ clos_refl_trans Thing Gives z b) -> clos_refl_trans Thing Gives b c -> exists z, Gives a z /\\ clos_refl_trans Thing Gives z c","forall a b, Gives a b -> exists z, Gives a z /\\ clos_refl_trans Thing Gives z b","forall a b, Gives a b -> clos_refl_trans Thing Gives a b","forall a b c, Gives a b -> clos_refl_trans Thing Gives b c -> clos_refl_trans Thing Gives a c","forall a b, clos_trans_1n Thing Gives a b -> clos_refl_trans Thing Gives a b","forall a b, clos_refl_trans Thing Gives a b -> a = b \\/ clos_trans Thing Gives a b","forall a b, clos_refl_trans Thing Gives a b -> forall c, Gives b c -> clos_trans Thing Gives a c \\/ a = c \\/ False","forall a b, clos_refl_trans Thing Gives a b -> a = b \\/ (exists z, clos_refl_trans Thing Gives a z /\\ Gives z b)","forall a b, (exists z, clos_refl_trans Thing Gives a z /\\ Gives z b) -> clos_trans Thing Gives a b"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 4 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> Through b a","MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing Gives x x","MP001 -> MP002 -> forall A, ~ Gives A A","forall A c, AggregateOfPossibles A -> c <> A -> ~ MemberOf c A -> ~ Possible c"],"ok":true,"c":[1,2,3,4],"combine":"intros; qauto unfold: MP011, MP001, AggregateOfPossibles, MP002.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":1,"t":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":2,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":2,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x, clos_trans Thing Gives x x -> exists z, Gives x z /\\ clos_refl_trans Thing Gives z x","MP002 -> MP003 -> forall a b, clos_refl_trans Thing Gives a b -> a = b \\/ Through b a","MP001 -> MP002 -> forall x z, Gives x z -> ~ (x = z \\/ Through x z)"],"ok":true,"c":[5,6,7],"combine":"intros; hcrush use: imkan_s9_cause_outside_L1 unfold: MP001.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":5,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":5,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall a b, clos_trans Thing Gives a b -> clos_trans_1n Thing Gives a b","forall a b, clos_trans_1n Thing Gives a b -> exists z, Gives a z /\\ clos_refl_trans Thing Gives z b"],"ok":true,"c":[8,9],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":8,"t":"intros; srun (eauto) use: clos_trans_t1n unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":9,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":9,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall a b, clos_trans_1n Thing Gives a b -> (Gives a b \\/ exists z, Gives a z /\\ clos_trans_1n Thing Gives z b)","forall z b, clos_trans_1n Thing Gives z b -> clos_refl_trans Thing Gives z b","forall a b, Gives a b -> clos_refl_trans Thing Gives b b"],"ok":true,"c":[10,11,12],"combine":"intros; match goal with H : clos_trans_1n _ _ _ _ |- _ => induction H end; firstorder eauto.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":10,"t":"intros; match goal with H : clos_trans_1n _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":11,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":9,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall a b, clos_trans_1n Thing Gives a b -> exists z, Gives a z /\\ clos_refl_trans_1n Thing Gives z b","forall z b, clos_refl_trans_1n Thing Gives z b -> clos_refl_trans Thing Gives z b"],"ok":true,"c":[13,14],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":13,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":5,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall a b, clos_trans Thing Gives a b -> clos_refl_trans Thing Gives a b","forall a b c, (exists z, Gives a z /\\ clos_refl_trans Thing Gives z b) -> clos_refl_trans Thing Gives b c -> exists z, Gives a z /\\ clos_refl_trans Thing Gives z c","forall a b, Gives a b -> exists z, Gives a z /\\ clos_refl_trans Thing Gives z b"],"ok":true,"c":[15,16,17],"combine":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; qauto use: imkan_s9_cause_outside_L2, imkan_s9_cause_outside_L3, clos_t1n_trans unfold: relation.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":15,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":15,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall a b, Gives a b -> clos_refl_trans Thing Gives a b","forall a b c, Gives a b -> clos_refl_trans Thing Gives b c -> clos_refl_trans Thing Gives a c","forall a b, clos_trans_1n Thing Gives a b -> clos_refl_trans Thing Gives a b"],"ok":true,"c":[18,19,20],"combine":"intros; srun (eauto) use: imkan_s9_cause_outside_L2.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":18,"t":"intros; sauto lq: on.","w":"hammer","ok":true,"c":[]},{"g":19,"t":"intros; sauto lq: on.","w":"hammer","ok":true,"c":[]},{"g":20,"t":"intros; srun (eauto) use: imkan_s9_cause_outside_L4, clos_t1n_trans, clos_rt_t, imkan_s9_cause_outside_L3, clos_t_clos_rt unfold: relation, inclusion.","w":"hammer","ok":true,"c":[]},{"g":16,"t":"intros; sauto lq: on.","w":"hammer","ok":true,"c":[]},{"g":17,"t":"intros; sfirstorder.","w":"hammer","ok":true,"c":[]},{"g":6,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":6,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall a b, clos_refl_trans Thing Gives a b -> a = b \\/ clos_trans Thing Gives a b","MP002 -> MP003 -> forall a b, clos_trans Thing Gives a b -> Through b a"],"ok":true,"c":[21,1],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":21,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":21,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall a b, clos_refl_trans Thing Gives a b -> forall c, Gives b c -> clos_trans Thing Gives a c \\/ a = c \\/ False","forall a b, clos_refl_trans Thing Gives a b -> a = b \\/ (exists z, clos_refl_trans Thing Gives a z /\\ Gives z b)","forall a b, (exists z, clos_refl_trans Thing Gives a z /\\ Gives z b) -> clos_trans Thing Gives a b"],"ok":true,"c":[22,23,24],"combine":"intros; hauto.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":22,"t":"intros; srun (eauto) use: clos_rt_t, t_step unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":23,"t":"intros; qauto use: clos_rt_rtn1, clos_rt_rtn1_iff unfold: relation inv: clos_refl_trans_n1, clos_refl_trans.","w":"hammer","ok":true,"c":[]},{"g":24,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; hauto lq: on use: clos_rt_t, imkan_s9_cause_outside_L11, t_step unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":7,"t":"intros; hauto lq: on unfold: MP001, MP002.","w":"hammer","ok":true,"c":[]},{"g":3,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":4,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[1,2,4,14,16,17,18,19,20,21,23,25,26,27,28,29,30,31],"proofFile":"files/claude/run3_subgoals_only/imkan_s9_cause_outside.v"},{"key":"claude3/imkan_s10_cause_necessary","id":"imkan_s10_cause_necessary","run":"claude3","title":"Dalīl al-imkān, step 10","english":"That cause is necessary in itself.","stmt":"MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Necessary c","proved":true,"seconds":1809.3,"expansions":10,"calls":43,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Necessary c","MP001 -> MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> ~ Gives y x","MP001 -> MP002 -> forall A c, Gives c A -> c <> A","MP011 -> MP001 -> MP002 -> MP003 -> forall A c, AggregateOfPossibles A -> Gives c A -> Exists c -> Ess c","MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","MP001 -> forall x y, Gives y x -> Through y x -> False","MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing Gives x x","MP011 -> MP001 -> MP002 -> MP003 -> forall A c, Gives c A -> MemberOf c A -> False","forall A c, AggregateOfPossibles A -> c <> A -> ~ MemberOf c A -> Exists c -> Ess c","forall x, clos_trans Thing Gives x x -> exists z, clos_trans Thing Gives x z /\\ Gives z x","MP001 -> MP002 -> MP003 -> forall x z, clos_trans Thing Gives x z -> Gives z x -> False","forall x y, clos_trans Thing Gives x y -> Gives x y \\/ (exists z, clos_trans Thing Gives x z /\\ Gives z y)","forall x, clos_trans Thing Gives x x -> Gives x x -> exists z, clos_trans Thing Gives x z /\\ Gives z x","forall x y, clos_trans Thing Gives x y -> exists z, (x = z \\/ clos_trans Thing Gives x z) /\\ Gives z y","forall x z, clos_trans Thing Gives x x -> (x = z \\/ clos_trans Thing Gives x z) -> clos_trans Thing Gives x z","MP002 -> MP003 -> forall x z, clos_trans Thing Gives x z -> Gives z x -> Through z x /\\ Through x z","MP001 -> forall x z, Gives z x -> Through z x -> False","forall x z, (x = z \\/ clos_trans Thing Gives x z) -> forall y, Gives z y -> clos_trans Thing Gives x y -> clos_trans Thing Gives x z \\/ x = z","forall x y, clos_trans Thing Gives x y -> (Gives x y) \\/ (exists z, clos_trans Thing Gives x z /\\ Gives z y)","forall x, Gives x x -> exists z, clos_trans Thing Gives x z /\\ Gives z x","forall x, clos_trans Thing Gives x x -> Gives x x \\/ (exists z, clos_trans Thing Gives x z /\\ Gives z x)","MP001 -> MP002 -> MP003 -> forall x y, Gives x y -> clos_trans Thing Gives y x -> False","MP011 -> MP001 -> MP002 -> MP003 -> forall A c, Gives c A -> ~ MemberOf c A","MP002 -> MP003 -> forall x y, clos_trans Thing Gives y x -> Through x y","MP001 -> forall x y, Gives x y -> Through x y -> False","forall x : Thing, clos_trans Thing Gives x x -> exists y : Thing, Gives x y /\\ clos_trans Thing Gives y x","MP001 -> MP002 -> MP003 -> forall x : Thing, clos_trans Thing Gives x x -> False","MP011 -> forall A c : Thing, Gives c A -> MemberOf c A -> clos_trans Thing Gives c c","forall x y : Thing, clos_trans Thing Gives x y -> exists z : Thing, Gives x z /\\ (z = y \\/ clos_trans Thing Gives z y)","forall x z : Thing, (z = x \\/ clos_trans Thing Gives z x) -> clos_trans Thing Gives z x \\/ z = x","forall x z : Thing, Gives x z -> z = x -> clos_trans Thing Gives z x -> exists y : Thing, Gives x y /\\ clos_trans Thing Gives y x","forall x y : Thing, clos_trans Thing Gives x y -> (Gives x y \\/ exists z : Thing, Gives x z /\\ clos_trans Thing Gives z y)","forall x : Thing, Gives x x -> exists y : Thing, Gives x y /\\ clos_trans Thing Gives y x","forall x z : Thing, Gives x z -> clos_trans Thing Gives z x -> exists y : Thing, Gives x y /\\ clos_trans Thing Gives y x","MP001 -> MP002 -> MP003 -> forall x y : Thing, Gives x y -> clos_trans Thing Gives y x -> False","MP002 -> MP003 -> forall x y : Thing, clos_trans Thing Gives y x -> Through x y","MP001 -> forall x y : Thing, Gives x y -> Through x y -> False"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 4 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing Gives x x","MP001 -> MP002 -> forall A c, Gives c A -> c <> A","MP011 -> forall A c, AggregateOfPossibles A -> Gives c A -> c <> A -> Exists c -> ~ Ess c -> clos_trans Thing Gives c c"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> ~ Gives y x","MP001 -> MP002 -> forall A c, Gives c A -> c <> A","MP011 -> MP001 -> MP002 -> MP003 -> forall A c, AggregateOfPossibles A -> Gives c A -> Exists c -> Ess c"],"ok":true,"c":[1,2,3],"combine":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":1,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":1,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","MP001 -> forall x y, Gives y x -> Through y x -> False"],"ok":true,"c":[4,5],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":4,"t":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":5,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":2,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; hauto lq: on drew: off.","w":"hammer","ok":true,"c":[]},{"g":3,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":3,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing Gives x x","MP011 -> MP001 -> MP002 -> MP003 -> forall A c, Gives c A -> MemberOf c A -> False","forall A c, AggregateOfPossibles A -> c <> A -> ~ MemberOf c A -> Exists c -> Ess c"],"ok":true,"c":[6,7,8],"combine":"intros; sauto lq: on.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal."},{"g":6,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":6,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x, clos_trans Thing Gives x x -> exists z, clos_trans Thing Gives x z /\\ Gives z x","MP001 -> MP002 -> MP003 -> forall x z, clos_trans Thing Gives x z -> Gives z x -> False"],"ok":true,"c":[9,10],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":9,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":9,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans Thing Gives x y -> Gives x y \\/ (exists z, clos_trans Thing Gives x z /\\ Gives z y)","forall x, clos_trans Thing Gives x x -> Gives x x -> exists z, clos_trans Thing Gives x z /\\ Gives z x"],"ok":true,"c":[11,12],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":11,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":9,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans Thing Gives x y -> exists z, (x = z \\/ clos_trans Thing Gives x z) /\\ Gives z y","forall x z, clos_trans Thing Gives x x -> (x = z \\/ clos_trans Thing Gives x z) -> clos_trans Thing Gives x z"],"ok":true,"c":[13,14],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":13,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":6,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x, clos_trans Thing Gives x x -> exists z, clos_trans Thing Gives x z /\\ Gives z x","MP002 -> MP003 -> forall x z, clos_trans Thing Gives x z -> Gives z x -> Through z x /\\ Through x z","MP001 -> forall x z, Gives z x -> Through z x -> False"],"ok":true,"c":[9,15,16],"combine":"intros; sfirstorder.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal."},{"g":9,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":9,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans Thing Gives x y -> exists z, (x = z \\/ clos_trans Thing Gives x z) /\\ Gives z y","forall x z, (x = z \\/ clos_trans Thing Gives x z) -> forall y, Gives z y -> clos_trans Thing Gives x y -> clos_trans Thing Gives x z \\/ x = z"],"ok":true,"c":[13,17],"combine":"intros; hauto.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal."},{"g":13,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":9,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans Thing Gives x y -> (Gives x y) \\/ (exists z, clos_trans Thing Gives x z /\\ Gives z y)","forall x, Gives x x -> exists z, clos_trans Thing Gives x z /\\ Gives z x","forall x, clos_trans Thing Gives x x -> Gives x x \\/ (exists z, clos_trans Thing Gives x z /\\ Gives z x)"],"ok":true,"c":[18,19,20],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":18,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":3,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> MP003 -> forall x y, Gives x y -> clos_trans Thing Gives y x -> False","MP011 -> MP001 -> MP002 -> MP003 -> forall A c, Gives c A -> ~ MemberOf c A","forall A c, AggregateOfPossibles A -> c <> A -> ~ MemberOf c A -> Exists c -> Ess c"],"ok":true,"c":[21,22,8],"combine":"intros; sauto lq: on drew: off.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":21,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":21,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y, clos_trans Thing Gives y x -> Through x y","MP001 -> forall x y, Gives x y -> Through x y -> False"],"ok":true,"c":[23,24],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":23,"t":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":24,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":22,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":22,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x : Thing, clos_trans Thing Gives x x -> exists y : Thing, Gives x y /\\ clos_trans Thing Gives y x","MP001 -> MP002 -> MP003 -> forall x : Thing, clos_trans Thing Gives x x -> False","MP011 -> forall A c : Thing, Gives c A -> MemberOf c A -> clos_trans Thing Gives c c"],"ok":true,"c":[25,26,27],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":25,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":25,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x y : Thing, clos_trans Thing Gives x y -> exists z : Thing, Gives x z /\\ (z = y \\/ clos_trans Thing Gives z y)","forall x z : Thing, (z = x \\/ clos_trans Thing Gives z x) -> clos_trans Thing Gives z x \\/ z = x","forall x z : Thing, Gives x z -> z = x -> clos_trans Thing Gives z x -> exists y : Thing, Gives x y /\\ clos_trans Thing Gives y x"],"ok":true,"c":[28,29,30],"combine":"intros; hauto.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal."},{"g":28,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":25,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x y : Thing, clos_trans Thing Gives x y -> (Gives x y \\/ exists z : Thing, Gives x z /\\ clos_trans Thing Gives z y)","forall x : Thing, Gives x x -> exists y : Thing, Gives x y /\\ clos_trans Thing Gives y x","forall x z : Thing, Gives x z -> clos_trans Thing Gives z x -> exists y : Thing, Gives x y /\\ clos_trans Thing Gives y x"],"ok":true,"c":[31,32,33],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal."},{"g":31,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":22,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> MP003 -> forall x y : Thing, Gives x y -> clos_trans Thing Gives y x -> False","forall x : Thing, clos_trans Thing Gives x x -> exists y : Thing, Gives x y /\\ clos_trans Thing Gives y x","MP011 -> forall A c : Thing, Gives c A -> MemberOf c A -> clos_trans Thing Gives c c"],"ok":true,"c":[34,25,27],"combine":"intros; sauto lq: on drew: off.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":34,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":34,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y : Thing, clos_trans Thing Gives y x -> Through x y","MP001 -> forall x y : Thing, Gives x y -> Through x y -> False"],"ok":true,"c":[35,36],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":35,"t":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":36,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":25,"t":"intros; hauto lq: on use: clos_trans_t1n, clos_t1n_trans unfold: relation inv: clos_trans_1n.","w":"hammer","ok":true,"c":[]},{"g":27,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":8,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[2,4,5,6,7,23,25,26,27,35,37,38,39,40,41,42],"proofFile":"files/claude/run3_subgoals_only/imkan_s10_cause_necessary.v"},{"key":"claude3/imkan_s11_necessary_cause","id":"imkan_s11_necessary_cause","run":"claude3","title":"Dalīl al-imkān, step 11","english":"The aggregate has a cause that is necessary in itself.","stmt":"MP001 -> MP002 -> MP003 -> MP008 -> MP010 -> MP011 -> MP012 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> exists c, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A","proved":false,"seconds":934.5,"expansions":1,"calls":3,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP008 -> MP010 -> MP011 -> MP012 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> exists c, Gives c A /\\ Necessary c /\\ ~ MemberOf c A /\\ c <> A"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 4 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","MP001 -> MP002 -> MP003 -> forall c, ~ clos_trans Thing Gives c c","MP008 -> MP010 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> ThroughAnother A","MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Necessary c /\\ ~ MemberOf c A /\\ c <> A"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":0,"t":"divide into 4 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> forall c, ~ Gives c c","MP001 -> MP002 -> MP003 -> MP011 -> forall A c, AggregateOfPossibles A -> Gives c A -> ~ MemberOf c A","MP008 -> MP010 -> forall A, AggregateOfPossibles A -> Exists A -> (exists p, p <> A /\\ Possible p) -> exists c, Gives c A","MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Exists c /\\ Ess c"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."}],"proof":[],"proofFile":null},{"key":"claude3/imkan_s12_necessary_exists","id":"imkan_s12_necessary_exists","run":"claude3","title":"Dalīl al-imkān, step 12 (aggregate route)","english":"A Necessary Existent exists.","stmt":"MP001 -> MP002 -> MP003 -> MP008 -> MP009 -> MP010 -> MP011 -> MP012 -> SP001 -> exists n, Necessary n","proved":true,"seconds":1371.3,"expansions":8,"calls":35,"root":0,"goals":["MP001 -> MP002 -> MP003 -> MP008 -> MP009 -> MP010 -> MP011 -> MP012 -> SP001 -> exists n, Necessary n","MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing Gives x x","MP001 -> MP002 -> MP003 -> MP008 -> MP009 -> MP010 -> MP011 -> MP012 -> (exists p, Possible p) -> exists n, Necessary n","MP008 -> SP001 -> ~ (exists p, Possible p) -> exists n, Necessary n","forall x y, clos_trans Thing Gives x y -> exists z, Gives x z /\\ (z = y \\/ clos_trans Thing Gives z y)","MP001 -> MP002 -> forall x z, Gives x z -> (z = x \\/ clos_trans Thing Gives z x) -> False","forall x y, clos_trans Thing Gives x y -> clos_trans_1n Thing Gives x y","forall x y, clos_trans_1n Thing Gives x y -> exists z, Gives x z /\\ (z = y \\/ clos_trans_1n Thing Gives z y)","forall x y, clos_trans_1n Thing Gives x y -> clos_trans Thing Gives x y","MP001 -> MP002 -> forall x, Gives x x -> False","MP001 -> MP002 -> forall x z, Gives x z -> clos_trans Thing Gives z x -> Through x z","MP001 -> forall x z, Gives x z -> Through x z -> False","forall x, clos_trans Thing Gives x x -> exists z, Gives x z /\\ (z = x \\/ clos_trans Thing Gives z x)","MP001 -> MP002 -> forall x, ~ Gives x x","MP001 -> MP002 -> MP003 -> forall x z, Gives x z -> clos_trans Thing Gives z x -> False","MP002 -> MP003 -> forall x z, clos_trans Thing Gives z x -> Through x z","MP001 -> forall x z, Gives x z -> ~ Through x z","MP008 -> forall A, Exists A -> ~ Ess A -> exists c, Gives c A","MP009 -> MP010 -> (exists p, Possible p) -> exists A, AggregateOfPossibles A /\\ Exists A /\\ ~ Ess A","MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Necessary c","MP001 -> MP002 -> forall A c, Gives c A -> c <> A","MP001 -> MP002 -> MP003 -> MP011 -> forall A c, Gives c A -> ~ MemberOf c A","forall A c, AggregateOfPossibles A -> c <> A -> ~ MemberOf c A -> ~ Possible c","forall c, Exists c -> ~ Possible c -> Ess c"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 4 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x y, clos_trans Thing Gives x y -> Through y x","MP001 -> MP002 -> MP003 -> forall x, ~ clos_trans Thing Gives x x","MP001 -> MP002 -> MP003 -> MP008 -> MP009 -> MP010 -> MP011 -> MP012 -> (exists p, Possible p) -> exists n, Necessary n","MP008 -> SP001 -> ~ (exists p, Possible p) -> exists n, Necessary n"],"ok":true,"c":[1,2,3,4],"combine":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":1,"t":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":2,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":2,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans Thing Gives x y -> exists z, Gives x z /\\ (z = y \\/ clos_trans Thing Gives z y)","MP001 -> MP002 -> forall x z, Gives x z -> (z = x \\/ clos_trans Thing Gives z x) -> False"],"ok":true,"c":[5,6],"combine":"intros; sauto lq: on drew: off.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal."},{"g":5,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":5,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x y, clos_trans Thing Gives x y -> clos_trans_1n Thing Gives x y","forall x y, clos_trans_1n Thing Gives x y -> exists z, Gives x z /\\ (z = y \\/ clos_trans_1n Thing Gives z y)","forall x y, clos_trans_1n Thing Gives x y -> clos_trans Thing Gives x y"],"ok":true,"c":[7,8,9],"combine":"intros; hauto.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":7,"t":"intros; srun (eauto) use: clos_trans_t1n unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":8,"t":"intros; match goal with H : clos_trans_1n _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":9,"t":"intros; srun (eauto) use: clos_t1n_trans unfold: relation.","w":"hammer","ok":true,"c":[]},{"g":6,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":6,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> forall x, Gives x x -> False","MP001 -> MP002 -> forall x z, Gives x z -> clos_trans Thing Gives z x -> Through x z","MP001 -> forall x z, Gives x z -> Through x z -> False"],"ok":true,"c":[10,11,12],"combine":"intros; hauto.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal."},{"g":10,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":11,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":11,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP002 -> forall x z, Gives x z -> Through z x","MP002 -> forall x z, clos_trans Thing Gives z x -> exists y, Gives z y /\\ (y = x \\/ clos_trans Thing Gives y x)","MP003 -> MP002 -> forall x z, clos_trans Thing Gives z x -> Through x z"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":11,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x z, clos_trans Thing Gives z x -> clos_trans_1n Thing Gives z x","MP002 -> forall x z, Gives z x -> Through x z","MP003 -> forall x y z, Through x y -> Through y z -> Through x z"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":6,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> forall x z, Gives x z -> z = x -> False","MP001 -> MP002 -> forall x z, Gives x z -> clos_trans Thing Gives z x -> exists y, Gives z y /\\ Through x y","MP002 -> forall x z, Gives x z -> Through z x"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":2,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["forall x, clos_trans Thing Gives x x -> exists z, Gives x z /\\ (z = x \\/ clos_trans Thing Gives z x)","MP001 -> MP002 -> forall x, ~ Gives x x","MP001 -> MP002 -> MP003 -> forall x z, Gives x z -> clos_trans Thing Gives z x -> False"],"ok":true,"c":[13,14,15],"combine":"intros; hauto.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":13,"t":"intros; srun (eauto) use: imkan_s12_necessary_exists_L5.","w":"hammer","ok":true,"c":[]},{"g":14,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":15,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":15,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP002 -> MP003 -> forall x z, clos_trans Thing Gives z x -> Through x z","MP001 -> forall x z, Gives x z -> ~ Through x z"],"ok":true,"c":[16,17],"combine":"intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":16,"t":"intros; match goal with H : clos_trans _ _ _ _ |- _ => induction H end; firstorder eauto.","w":"auto","ok":true,"c":[]},{"g":17,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":3,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":3,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP008 -> forall A, Exists A -> ~ Ess A -> exists c, Gives c A","MP009 -> MP010 -> (exists p, Possible p) -> exists A, AggregateOfPossibles A /\\ Exists A /\\ ~ Ess A","MP001 -> MP002 -> MP003 -> MP011 -> MP012 -> forall A c, AggregateOfPossibles A -> Gives c A -> Necessary c"],"ok":true,"c":[18,19,20],"combine":"intros; hauto lq: on.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":18,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","w":"auto","ok":true,"c":[]},{"g":19,"t":"intros; hauto lq: on unfold: MP010, MP009, AggregateOfPossibles, Possible.","w":"hammer","ok":true,"c":[]},{"g":20,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":20,"t":"divide into 4 subgoals","w":"model","kind":"division","subs":["MP001 -> MP002 -> forall A c, Gives c A -> c <> A","MP001 -> MP002 -> MP003 -> MP011 -> forall A c, Gives c A -> ~ MemberOf c A","forall A c, AggregateOfPossibles A -> c <> A -> ~ MemberOf c A -> ~ Possible c","forall c, Exists c -> ~ Possible c -> Ess c"],"ok":true,"c":[21,22,23,24],"combine":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; firstorder.","d":"Claude stated these subgoals only. automation derived the goal from them (shown below); each subgoal then became a new goal.","assembled":true},{"g":21,"t":"intros; hfcrush use: imkan_s12_necessary_exists_L8.","w":"hammer","ok":true,"c":[]},{"g":22,"t":"intros; sfirstorder use: imkan_s12_necessary_exists_L12 unfold: MP011.","w":"hammer","ok":true,"c":[]},{"g":23,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":24,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","w":"auto","ok":true,"c":[]},{"g":4,"t":"unfold MP001, MP002, MP003, MP007, MP008, MP009, MP010, MP011, MP012, OP001, OP006, SP001, OfItself, ThroughAnother, Necessary, Possible, InUniverse, DependsOn, Independent, NonExistencePossible, AggregateOfPossibles in *; intros; apply NNPP; intro; firstorder.","w":"auto","ok":true,"c":[]}],"proof":[1,2,17,18,19,21,22,23,25,26,27,29,30,31,32,33,34],"proofFile":"files/claude/run3_subgoals_only/imkan_s12_necessary_exists.v"},{"key":"claude3/imkan_b_necessary_exists","id":"imkan_b_necessary_exists","run":"claude3","title":"Dalīl al-imkān, conclusion (shorter route)","english":"A Necessary Existent exists.","stmt":"MP008 -> OP001 -> SP001 -> exists n, Necessary n","proved":false,"seconds":536.3,"expansions":2,"calls":8,"root":0,"goals":["MP008 -> OP001 -> SP001 -> exists n, Necessary n","MP008 -> forall x, Exists x -> ~ ThroughAnother x -> Necessary x","OP001 -> forall x, ThroughAnother x -> exists y, Exists y /\\ ~ ThroughAnother y","forall x, Exists x -> (exists n, Necessary n) \\/ ThroughAnother x"],"events":[{"g":0,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP008 -> forall x, OfItself x -> Necessary x","OP001 -> forall x, ThroughAnother x -> exists y, OfItself y","SP001 -> exists x, Exists x"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":0,"t":"divide into 3 subgoals","w":"model","kind":"division","subs":["MP008 -> forall x, Exists x -> ~ ThroughAnother x -> Necessary x","OP001 -> forall x, ThroughAnother x -> exists y, Exists y /\\ ~ ThroughAnother y","forall x, Exists x -> (exists n, Necessary n) \\/ ThroughAnother x"],"ok":true,"c":[1,2,3],"combine":"intros; hauto lq: on drew: off.","d":"Claude stated these subgoals only. CoqHammer derived the goal from them (shown below); each subgoal then became a new goal."},{"g":1,"t":"intros; firstorder.","w":"auto","ok":true,"c":[]},{"g":2,"t":"intros; sfirstorder.","w":"hammer","ok":true,"c":[]},{"g":3,"t":"automated provers","w":"auto","ok":false,"err":"Rocq’s automation, the induction tactic and CoqHammer could not prove this goal."},{"g":3,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["forall x, ThroughAnother x \\/ ~ ThroughAnother x","MP008 -> forall x, Exists x -> ~ ThroughAnother x -> exists n, Necessary n"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."},{"g":3,"t":"divide into 2 subgoals","w":"model","kind":"division","subs":["MP008 -> forall y, Exists y -> ~ ThroughAnother y -> Necessary y","forall x, Exists x -> (exists y, Exists y /\\ ~ ThroughAnother y) \\/ ThroughAnother x"],"ok":false,"err":"The automated provers could not derive the goal from these subgoals."}],"proof":[],"proofFile":null}]}