% Leo-III cross-check.
% Dalil al-imkan, proof (a), in one problem: the conclusion assembled from the steps, each supplied
% exactly as proved in Rocq (theorem names in the comments), with SP-001.
% 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 > $o).
% agg is "the aggregate of all possibles", defined in PR-009 (AggregateOfPossibles); the proved steps
% below already carry its content, so the definition is left abstract here.
thf(step1, axiom, ![X:$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(sp001, axiom, ?[X:$i]: (ex @ X)).                                                % SP-001
thf(conclusion, conjecture, ?[N:$i]: (nec @ N)).                                     % a Necessary Existent exists