%   ORIGINAL: 'h4/thm/list/UNZIP_THM_e1'
% 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: !y x t. 'h4/const/list/UNZIP' ('h4/const/list/CONS' ('h4/const/pair/,' x y) t) = 'h4/const/bool/LET' ('h4/const/pair/UNCURRY' (\L1 L2. 'h4/const/pair/,' ('h4/const/list/CONS' x L1) ('h4/const/list/CONS' y L2))) ('h4/const/list/UNZIP' 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: !_1. (!x L1 y L2. happ (happ (happ (happ _1 x) L1) y) L2 = 'h4/const/pair/,' ('h4/const/list/CONS' x L1) ('h4/const/list/CONS' y L2)) ==> (!_0. (!x y L1. happ (happ (happ _0 x) y) L1 = happ (happ (happ _1 x) L1) y) ==> (!y x t. 'h4/const/list/UNZIP' ('h4/const/list/CONS' ('h4/const/pair/,' x y) t) = 'h4/const/bool/LET' ('h4/const/pair/UNCURRY' (happ (happ _0 x) y)) ('h4/const/list/UNZIP' t)))
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_3f6189,V_3f6185]: ![F, G]: (![X]: s(V_3f6185,happ(s(fun(V_3f6189,V_3f6185),F),s(V_3f6189,X))) = s(V_3f6185,happ(s(fun(V_3f6189,V_3f6185),G),s(V_3f6189,X))) => s(fun(V_3f6189,V_3f6185),F) = s(fun(V_3f6189,V_3f6185),G))).
fof('h4/thm/list/UNZIP_THM_e1', conjecture, ![A,B]: ![V__1]: (![X, L1, Y, L2]: s('h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)),happ(s(fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B))),happ(s(fun(B,fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)))),happ(s(fun('h4/type/list/list'(A),fun(B,fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B))))),happ(s(fun(A,fun('h4/type/list/list'(A),fun(B,fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)))))),V__1),s(A,X))),s('h4/type/list/list'(A),L1))),s(B,Y))),s('h4/type/list/list'(B),L2))) = s('h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)),'h4/const/pair/,'(s('h4/type/list/list'(A),'h4/const/list/CONS'(s(A,X),s('h4/type/list/list'(A),L1))),s('h4/type/list/list'(B),'h4/const/list/CONS'(s(B,Y),s('h4/type/list/list'(B),L2))))) => ![V__0]: (![X, Y, L1]: s(fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B))),happ(s(fun('h4/type/list/list'(A),fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)))),happ(s(fun(B,fun('h4/type/list/list'(A),fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B))))),happ(s(fun(A,fun(B,fun('h4/type/list/list'(A),fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)))))),V__0),s(A,X))),s(B,Y))),s('h4/type/list/list'(A),L1))) = s(fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B))),happ(s(fun(B,fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)))),happ(s(fun('h4/type/list/list'(A),fun(B,fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B))))),happ(s(fun(A,fun('h4/type/list/list'(A),fun(B,fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)))))),V__1),s(A,X))),s('h4/type/list/list'(A),L1))),s(B,Y))) => ![Y, X, T]: s('h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)),'h4/const/list/UNZIP'(s('h4/type/list/list'('h4/type/pair/prod'(A,B)),'h4/const/list/CONS'(s('h4/type/pair/prod'(A,B),'h4/const/pair/,'(s(A,X),s(B,Y))),s('h4/type/list/list'('h4/type/pair/prod'(A,B)),T))))) = s('h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)),'h4/const/bool/LET'(s(fun('h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B))),'h4/const/pair/UNCURRY'(s(fun('h4/type/list/list'(A),fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)))),happ(s(fun(B,fun('h4/type/list/list'(A),fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B))))),happ(s(fun(A,fun(B,fun('h4/type/list/list'(A),fun('h4/type/list/list'(B),'h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)))))),V__0),s(A,X))),s(B,Y))))),s('h4/type/pair/prod'('h4/type/list/list'(A),'h4/type/list/list'(B)),'h4/const/list/UNZIP'(s('h4/type/list/list'('h4/type/pair/prod'(A,B)),T)))))))).
