%   ORIGINAL: 'h4/thm/pair/LET2_RAND_'
% 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 M N. P ('h4/const/bool/LET' ('h4/const/pair/UNCURRY' (\x y. N x y)) M) = 'h4/const/bool/LET' ('h4/const/pair/UNCURRY' (\x y. P (N x y))) M
%   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: !_3. (!P N x y. happ (happ (happ (happ _3 P) N) x) y = happ P (happ (happ N x) y)) ==> (!_2. (!P N x. happ (happ (happ _2 P) N) x = happ (happ (happ _3 P) N) x) ==> (!_1. (!N x y. happ (happ (happ _1 N) x) y = happ (happ N x) y) ==> (!_0. (!N x. happ (happ _0 N) x = happ (happ _1 N) x) ==> (!P M N. happ P ('h4/const/bool/LET' ('h4/const/pair/UNCURRY' (happ _0 N)) M) = 'h4/const/bool/LET' ('h4/const/pair/UNCURRY' (happ (happ _2 P) N)) M))))
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_3f61461,V_3f61457]: ![F, G]: (![X]: s(V_3f61457,happ(s(fun(V_3f61461,V_3f61457),F),s(V_3f61461,X))) = s(V_3f61457,happ(s(fun(V_3f61461,V_3f61457),G),s(V_3f61461,X))) => s(fun(V_3f61461,V_3f61457),F) = s(fun(V_3f61461,V_3f61457),G))).
fof('h4/thm/pair/LET2_RAND_', conjecture, ![D,C,A,B]: ![V__3]: (![P, N, X, Y]: s(D,happ(s(fun(B,D),happ(s(fun(A,fun(B,D)),happ(s(fun(fun(A,fun(B,C)),fun(A,fun(B,D))),happ(s(fun(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D)))),V__3),s(fun(C,D),P))),s(fun(A,fun(B,C)),N))),s(A,X))),s(B,Y))) = s(D,happ(s(fun(C,D),P),s(C,happ(s(fun(B,C),happ(s(fun(A,fun(B,C)),N),s(A,X))),s(B,Y))))) => ![V__2]: (![P, N, X]: s(fun(B,D),happ(s(fun(A,fun(B,D)),happ(s(fun(fun(A,fun(B,C)),fun(A,fun(B,D))),happ(s(fun(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D)))),V__2),s(fun(C,D),P))),s(fun(A,fun(B,C)),N))),s(A,X))) = s(fun(B,D),happ(s(fun(A,fun(B,D)),happ(s(fun(fun(A,fun(B,C)),fun(A,fun(B,D))),happ(s(fun(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D)))),V__3),s(fun(C,D),P))),s(fun(A,fun(B,C)),N))),s(A,X))) => ![V__1]: (![N, X, Y]: s(C,happ(s(fun(B,C),happ(s(fun(A,fun(B,C)),happ(s(fun(fun(A,fun(B,C)),fun(A,fun(B,C))),V__1),s(fun(A,fun(B,C)),N))),s(A,X))),s(B,Y))) = s(C,happ(s(fun(B,C),happ(s(fun(A,fun(B,C)),N),s(A,X))),s(B,Y))) => ![V__0]: (![N, X]: s(fun(B,C),happ(s(fun(A,fun(B,C)),happ(s(fun(fun(A,fun(B,C)),fun(A,fun(B,C))),V__0),s(fun(A,fun(B,C)),N))),s(A,X))) = s(fun(B,C),happ(s(fun(A,fun(B,C)),happ(s(fun(fun(A,fun(B,C)),fun(A,fun(B,C))),V__1),s(fun(A,fun(B,C)),N))),s(A,X))) => ![P, M, N]: s(D,happ(s(fun(C,D),P),s(C,'h4/const/bool/LET'(s(fun('h4/type/pair/prod'(A,B),C),'h4/const/pair/UNCURRY'(s(fun(A,fun(B,C)),happ(s(fun(fun(A,fun(B,C)),fun(A,fun(B,C))),V__0),s(fun(A,fun(B,C)),N))))),s('h4/type/pair/prod'(A,B),M))))) = s(D,'h4/const/bool/LET'(s(fun('h4/type/pair/prod'(A,B),D),'h4/const/pair/UNCURRY'(s(fun(A,fun(B,D)),happ(s(fun(fun(A,fun(B,C)),fun(A,fun(B,D))),happ(s(fun(fun(C,D),fun(fun(A,fun(B,C)),fun(A,fun(B,D)))),V__2),s(fun(C,D),P))),s(fun(A,fun(B,C)),N))))),s('h4/type/pair/prod'(A,B),M)))))))).
