% Leo-III cross-check.
% Proof (a), part 2b -- a possible p exists, and the aggregate A is the only possible thing:
% then A is possible, has a giver (step 3), the giver is not A (step 4) and not possible,
% it exists (MP-012), so it is necessary (step 1).
% Vocabulary.  Nothing is assumed about these symbols.
thf(ex_type, type, ex: $i > $o).                     % (ex @ X): X exists
thf(ess_type, type, ess: $i > $o).                   % (ess @ X): X's existence is from its nature
thf(gives_type, type, gives: $i > $i > $o).          % (gives @ Y @ X): Y gives existence to X
thf(nec_type, type, nec: $i > $o).
thf(poss_type, type, poss: $i > $o).
thf(ta_type, type, through_another: $i > $o).
thf(nec_def, definition, nec = (^[X:$i]: ((ex @ X) & (ess @ X)))).            % necessary in itself
thf(poss_def, definition, poss = (^[X:$i]: ((ex @ X) & ~ (ess @ X)))).        % possible in itself
thf(ta_def, definition, through_another = (^[X:$i]: ?[Y:$i]: (gives @ Y @ X))). % something gives X existence
thf(mem_type, type, mem: $i > $i > $o).              % (mem @ P @ A): P is a member of collection A
thf(agg_type, type, agg: $i > $o).
thf(agg_def, definition, agg = (^[A:$i]: ((![P:$i]: ((P != A) => ((mem @ P @ A) <=> (poss @ P)))) & ~ (mem @ A @ A)))).
thf(step1, axiom, ![X:$i]: ((ex @ X) => ((nec @ X) | (poss @ X)))).                    % PR-006 division_exhaustive
thf(step3, axiom, ![X:$i]: ((poss @ X) => (through_another @ X))).                       % PR-008 possible_requires_murajjih (MP-008)
thf(step4, axiom, ![X:$i]: ~ (gives @ X @ X)).                                           % PR-002 no_self_creation
thf(mp009, axiom, (?[P:$i]: (poss @ P)) => (?[A:$i]: ((agg @ A) & (ex @ A)))).          % MP-009
thf(step6, axiom, ![A:$i]: (((agg @ A) & (ex @ A) & (?[P:$i]: ((P != A) & (poss @ P)))) => (poss @ A))).  % PR-009 aggregate_is_possible (MP-010)
thf(step11, axiom, ![A:$i, C:$i]: (((agg @ A) & (gives @ C @ A)) => (nec @ C))).        % PR-010 cause_is_necessary (MP-011, MP-012)
thf(mp012, axiom, ![Y:$i, X:$i]: ((gives @ Y @ X) => (ex @ Y))).                        % MP-012
thf(a_type, type, a: $i).
thf(p_type, type, p: $i).
thf(case_agg, axiom, (agg @ a) & (ex @ a)).
thf(case_possible, axiom, poss @ p).
thf(case_only, axiom, ~ (?[Q:$i]: ((Q != a) & (poss @ Q)))).
thf(conclusion, conjecture, ?[N:$i]: (nec @ N)).