%   ORIGINAL: 'h4/thm/list/MEM_ZIP_'
% 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/bool/IN' p ('h4/const/list/LIST_TO_SET' ('h4/const/list/ZIP' ('h4/const/pair/,' l1 l2))) <=> (?n. 'h4/const/prim_rec/<' n ('h4/const/list/LENGTH' l1) /\ p = 'h4/const/pair/,' ('h4/const/list/EL' n l1) ('h4/const/list/EL' n l2)))
%   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: !l1 l2 p. 'h4/const/list/LENGTH' l1 = 'h4/const/list/LENGTH' l2 ==> ('h4/const/bool/IN' p ('h4/const/list/LIST_TO_SET' ('h4/const/list/ZIP' ('h4/const/pair/,' l1 l2))) <=> (?n. 'h4/const/prim_rec/<' n ('h4/const/list/LENGTH' l1) /\ p = 'h4/const/pair/,' ('h4/const/list/EL' n l1) ('h4/const/list/EL' n l2)))
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_3f6389,V_3f6385]: ![F, G]: (![X]: s(V_3f6385,happ(s(fun(V_3f6389,V_3f6385),F),s(V_3f6389,X))) = s(V_3f6385,happ(s(fun(V_3f6389,V_3f6385),G),s(V_3f6389,X))) => s(fun(V_3f6389,V_3f6385),F) = s(fun(V_3f6389,V_3f6385),G))).
fof('h4/thm/list/MEM_ZIP_', conjecture, ![A,B]: ![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))) => (p(s(bool,'h4/const/bool/IN'(s('h4/type/pair/prod'(A,B),P),s(fun('h4/type/pair/prod'(A,B),bool),'h4/const/list/LIST_TO_SET'(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)))))))))) <=> ?[N]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/list/LENGTH'(s('h4/type/list/list'(A),L1)))))) & s('h4/type/pair/prod'(A,B),P) = s('h4/type/pair/prod'(A,B),'h4/const/pair/,'(s(A,'h4/const/list/EL'(s('h4/type/num/num',N),s('h4/type/list/list'(A),L1))),s(B,'h4/const/list/EL'(s('h4/type/num/num',N),s('h4/type/list/list'(B),L2))))))))).
