%   ORIGINAL: 'h4/thm/list/EVERY2_THM_1'
% 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 yys x xs. 'h4/const/list/LIST_REL' P ('h4/const/list/CONS' x xs) yys <=> (?y ys. yys = 'h4/const/list/CONS' y ys /\ 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 yys x xs. 'h4/const/list/LIST_REL' P ('h4/const/list/CONS' x xs) yys <=> (?y ys. yys = 'h4/const/list/CONS' y ys /\ 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_3f11819,V_3f11815]: ![F, G]: (![X]: s(V_3f11815,happ(s(fun(V_3f11819,V_3f11815),F),s(V_3f11819,X))) = s(V_3f11815,happ(s(fun(V_3f11819,V_3f11815),G),s(V_3f11819,X))) => s(fun(V_3f11819,V_3f11815),F) = s(fun(V_3f11819,V_3f11815),G))).
fof('h4/thm/list/EVERY2_THM_1', conjecture, ![C,D]: ![P, Yys, X, Xs]: (p(s(bool,'h4/const/list/LIST_REL'(s(fun(C,fun(D,bool)),P),s('h4/type/list/list'(C),'h4/const/list/CONS'(s(C,X),s('h4/type/list/list'(C),Xs))),s('h4/type/list/list'(D),Yys)))) <=> ?[Y, Ys]: (s('h4/type/list/list'(D),Yys) = s('h4/type/list/list'(D),'h4/const/list/CONS'(s(D,Y),s('h4/type/list/list'(D),Ys))) & (p(s(bool,happ(s(fun(D,bool),happ(s(fun(C,fun(D,bool)),P),s(C,X))),s(D,Y)))) & p(s(bool,'h4/const/list/LIST_REL'(s(fun(C,fun(D,bool)),P),s('h4/type/list/list'(C),Xs),s('h4/type/list/list'(D),Ys)))))))).
