%   ORIGINAL: 'h4/thm/list/every_zip_fst_'
% 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: !l1 l2 P. 'h4/const/list/LENGTH' l1 = 'h4/const/list/LENGTH' l2 ==> ('h4/const/list/EVERY' (\x. P ('h4/const/pair/FST' x)) ('h4/const/list/ZIP' ('h4/const/pair/,' l1 l2)) <=> 'h4/const/list/EVERY' P l1)
%   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. (!P x. happ (happ _0 P) x <=> happ P ('h4/const/pair/FST' x)) ==> (!l1 l2 P. 'h4/const/list/LENGTH' l1 = 'h4/const/list/LENGTH' l2 ==> ('h4/const/list/EVERY' (happ _0 P) ('h4/const/list/ZIP' ('h4/const/pair/,' l1 l2)) <=> 'h4/const/list/EVERY' P l1))
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_3f12544,V_3f12540]: ![F, G]: (![X]: s(V_3f12540,happ(s(fun(V_3f12544,V_3f12540),F),s(V_3f12544,X))) = s(V_3f12540,happ(s(fun(V_3f12544,V_3f12540),G),s(V_3f12544,X))) => s(fun(V_3f12544,V_3f12540),F) = s(fun(V_3f12544,V_3f12540),G))).
fof('h4/thm/list/every_zip_fst_', conjecture, ![B,A]: ![V__0]: (![P, X]: s(bool,happ(s(fun('h4/type/pair/prod'(A,B),bool),happ(s(fun(fun(A,bool),fun('h4/type/pair/prod'(A,B),bool)),V__0),s(fun(A,bool),P))),s('h4/type/pair/prod'(A,B),X))) = s(bool,happ(s(fun(A,bool),P),s(A,'h4/const/pair/FST'(s('h4/type/pair/prod'(A,B),X))))) => ![L1, L2, P]: (s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(A),L1))) = s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(B),L2))) => s(bool,'h4/const/list/EVERY'(s(fun('h4/type/pair/prod'(A,B),bool),happ(s(fun(fun(A,bool),fun('h4/type/pair/prod'(A,B),bool)),V__0),s(fun(A,bool),P))),s('h4/type/list/list'('h4/type/pair/prod'(A,B)),'h4/const/list/ZIP'(s('h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)),'h4/const/pair/,'(s('h4/type/list/list'(A),L1),s('h4/type/list/list'(B),L2))))))) = s(bool,'h4/const/list/EVERY'(s(fun(A,bool),P),s('h4/type/list/list'(A),L1)))))).
