%   ORIGINAL: 'h4/thm/option/SOME_APPLY_PERMUTE_'
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Goal: !x f. 'h4/const/option/OPTION_APPLY' f ('h4/const/option/SOME' x) = 'h4/const/option/OPTION_APPLY' ('h4/const/option/SOME' (\f0. f0 x)) f
%   PROCESSED
% Assm ['HL_TRUTH']: T
% Assm ['HL_FALSITY']: ~F
% Assm ['HL_BOOL_CASES']: !t. (t <=> T) \/ (t <=> F)
% Assm ['HL_EXT']: !f g. (!x. happ f x = happ g x) ==> f = g
% Goal: !_0. (!x f0. happ (happ _0 x) f0 = happ f0 x) ==> (!x f. 'h4/const/option/OPTION_APPLY' f ('h4/const/option/SOME' x) = 'h4/const/option/OPTION_APPLY' ('h4/const/option/SOME' (happ _0 x)) f)
fof('HL_TRUTH', axiom, p(s(bool,'T'))).
fof('HL_FALSITY', axiom, ~ (p(s(bool,'F')))).
fof('HL_BOOL_CASES', axiom, ![T]: (s(bool,T) = s(bool,'T') | s(bool,T) = s(bool,'F'))).
fof('HL_EXT', axiom, ![V_3f100657,V_3f100653]: ![F, G]: (![X]: s(V_3f100653,happ(s(fun(V_3f100657,V_3f100653),F),s(V_3f100657,X))) = s(V_3f100653,happ(s(fun(V_3f100657,V_3f100653),G),s(V_3f100657,X))) => s(fun(V_3f100657,V_3f100653),F) = s(fun(V_3f100657,V_3f100653),G))).
fof('h4/thm/option/SOME_APPLY_PERMUTE_', conjecture, ![A,B]: ![V__0]: (![X, F0]: s(B,happ(s(fun(fun(A,B),B),happ(s(fun(A,fun(fun(A,B),B)),V__0),s(A,X))),s(fun(A,B),F0))) = s(B,happ(s(fun(A,B),F0),s(A,X))) => ![X, F]: s('h4/type/option/option'(B),'h4/const/option/OPTION_APPLY'(s('h4/type/option/option'(fun(A,B)),F),s('h4/type/option/option'(A),'h4/const/option/SOME'(s(A,X))))) = s('h4/type/option/option'(B),'h4/const/option/OPTION_APPLY'(s('h4/type/option/option'(fun(fun(A,B),B)),'h4/const/option/SOME'(s(fun(fun(A,B),B),happ(s(fun(A,fun(fun(A,B),B)),V__0),s(A,X))))),s('h4/type/option/option'(fun(A,B)),F))))).
