%   ORIGINAL: 'h4/thm/list/EVERY2_THM_e3'
% 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: !P xxs y ys. 'h4/const/list/LIST_REL' P xxs ('h4/const/list/CONS' y ys) <=> (?x xs. xxs = 'h4/const/list/CONS' x xs /\ P x y /\ 'h4/const/list/LIST_REL' P xs ys)
%   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: !P xxs y ys. 'h4/const/list/LIST_REL' P xxs ('h4/const/list/CONS' y ys) <=> (?x xs. xxs = 'h4/const/list/CONS' x xs /\ happ (happ P x) y /\ 'h4/const/list/LIST_REL' P xs ys)
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_3f11869,V_3f11865]: ![F, G0]: (![X]: s(V_3f11865,happ(s(fun(V_3f11869,V_3f11865),F),s(V_3f11869,X))) = s(V_3f11865,happ(s(fun(V_3f11869,V_3f11865),G0),s(V_3f11869,X))) => s(fun(V_3f11869,V_3f11865),F) = s(fun(V_3f11869,V_3f11865),G0))).
fof('h4/thm/list/EVERY2_THM_e3', conjecture, ![G,H]: ![P, Xxs, Y, Ys]: (p(s(bool,'h4/const/list/LIST_REL'(s(fun(G,fun(H,bool)),P),s('h4/type/list/list'(G),Xxs),s('h4/type/list/list'(H),'h4/const/list/CONS'(s(H,Y),s('h4/type/list/list'(H),Ys)))))) <=> ?[X, Xs]: (s('h4/type/list/list'(G),Xxs) = s('h4/type/list/list'(G),'h4/const/list/CONS'(s(G,X),s('h4/type/list/list'(G),Xs))) & (p(s(bool,happ(s(fun(H,bool),happ(s(fun(G,fun(H,bool)),P),s(G,X))),s(H,Y)))) & p(s(bool,'h4/const/list/LIST_REL'(s(fun(G,fun(H,bool)),P),s('h4/type/list/list'(G),Xs),s('h4/type/list/list'(H),Ys)))))))).
