%   ORIGINAL: 'h4/thm/list/list_Axiom_old_'
% 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/bool/?!' (\fn1. fn1 'h4/const/list/NIL' = x /\ (!h t. fn1 ('h4/const/list/CONS' h t) = f (fn1 t) h t))
%   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 f fn1. happ (happ (happ _0 x) f) fn1 <=> happ fn1 'h4/const/list/NIL' = x /\ (!h t. happ fn1 ('h4/const/list/CONS' h t) = happ (happ (happ f (happ fn1 t)) h) t)) ==> (!x f. 'h4/const/bool/?!' (happ (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_3f2513,V_3f2509]: ![F, G]: (![X]: s(V_3f2509,happ(s(fun(V_3f2513,V_3f2509),F),s(V_3f2513,X))) = s(V_3f2509,happ(s(fun(V_3f2513,V_3f2509),G),s(V_3f2513,X))) => s(fun(V_3f2513,V_3f2509),F) = s(fun(V_3f2513,V_3f2509),G))).
fof('h4/thm/list/list_Axiom_old_', conjecture, ![A,B]: ![V__0]: (![X, F, Fn1]: (p(s(bool,happ(s(fun(fun('h4/type/list/list'(A),B),bool),happ(s(fun(fun(B,fun(A,fun('h4/type/list/list'(A),B))),fun(fun('h4/type/list/list'(A),B),bool)),happ(s(fun(B,fun(fun(B,fun(A,fun('h4/type/list/list'(A),B))),fun(fun('h4/type/list/list'(A),B),bool))),V__0),s(B,X))),s(fun(B,fun(A,fun('h4/type/list/list'(A),B))),F))),s(fun('h4/type/list/list'(A),B),Fn1)))) <=> (s(B,happ(s(fun('h4/type/list/list'(A),B),Fn1),s('h4/type/list/list'(A),'h4/const/list/NIL'))) = s(B,X) & ![H, T]: s(B,happ(s(fun('h4/type/list/list'(A),B),Fn1),s('h4/type/list/list'(A),'h4/const/list/CONS'(s(A,H),s('h4/type/list/list'(A),T))))) = s(B,happ(s(fun('h4/type/list/list'(A),B),happ(s(fun(A,fun('h4/type/list/list'(A),B)),happ(s(fun(B,fun(A,fun('h4/type/list/list'(A),B))),F),s(B,happ(s(fun('h4/type/list/list'(A),B),Fn1),s('h4/type/list/list'(A),T))))),s(A,H))),s('h4/type/list/list'(A),T))))) => ![X, F]: p(s(bool,'h4/const/bool/?!'(s(fun(fun('h4/type/list/list'(A),B),bool),happ(s(fun(fun(B,fun(A,fun('h4/type/list/list'(A),B))),fun(fun('h4/type/list/list'(A),B),bool)),happ(s(fun(B,fun(fun(B,fun(A,fun('h4/type/list/list'(A),B))),fun(fun('h4/type/list/list'(A),B),bool))),V__0),s(B,X))),s(fun(B,fun(A,fun('h4/type/list/list'(A),B))),F)))))))).
