%   ORIGINAL: 'h4/thm/combin/UPDATE_APPLY_ID_'
% 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: !f a b. f a = b <=> 'h4/const/combin/UPDATE' a b f = 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: !f a b. happ f a = b <=> 'h4/const/combin/UPDATE' a b f = 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_3f38756,V_3f38752]: ![F, G]: (![X]: s(V_3f38752,happ(s(fun(V_3f38756,V_3f38752),F),s(V_3f38756,X))) = s(V_3f38752,happ(s(fun(V_3f38756,V_3f38752),G),s(V_3f38756,X))) => s(fun(V_3f38756,V_3f38752),F) = s(fun(V_3f38756,V_3f38752),G))).
fof('h4/thm/combin/UPDATE_APPLY_ID_', conjecture, ![A,B]: ![F, A0, B0]: (s(B,happ(s(fun(A,B),F),s(A,A0))) = s(B,B0) <=> s(fun(A,B),'h4/const/combin/UPDATE'(s(A,A0),s(B,B0),s(fun(A,B),F))) = s(fun(A,B),F))).
