%   ORIGINAL: 'h4/thm/numeral/enumeral_mult_e5'
% 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. 'h4/const/arithmetic/*' ('h4/const/arithmetic/BIT2' x) ('h4/const/arithmetic/BIT2' y) = 'h4/const/bool/LET' (\m. 'h4/const/bool/LET' (\n. 'h4/const/bool/COND' ('h4/const/arithmetic/ODD' m) ('h4/const/numeral/texp_help' ('h4/const/arithmetic/DIV2' m) ('h4/const/prim_rec/PRE' ('h4/const/arithmetic/BIT2' y))) ('h4/const/bool/COND' ('h4/const/arithmetic/ODD' n) ('h4/const/numeral/texp_help' ('h4/const/arithmetic/DIV2' n) ('h4/const/prim_rec/PRE' ('h4/const/arithmetic/BIT2' x))) ('h4/const/numeral/internal_mult' ('h4/const/arithmetic/BIT2' x) ('h4/const/arithmetic/BIT2' y)))) ('h4/const/numeral/exactlog' ('h4/const/arithmetic/BIT2' y))) ('h4/const/numeral/exactlog' ('h4/const/arithmetic/BIT2' x))
%   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. (!m x y n. happ (happ (happ (happ _1 m) x) y) n = 'h4/const/bool/COND' ('h4/const/arithmetic/ODD' m) ('h4/const/numeral/texp_help' ('h4/const/arithmetic/DIV2' m) ('h4/const/prim_rec/PRE' ('h4/const/arithmetic/BIT2' y))) ('h4/const/bool/COND' ('h4/const/arithmetic/ODD' n) ('h4/const/numeral/texp_help' ('h4/const/arithmetic/DIV2' n) ('h4/const/prim_rec/PRE' ('h4/const/arithmetic/BIT2' x))) ('h4/const/numeral/internal_mult' ('h4/const/arithmetic/BIT2' x) ('h4/const/arithmetic/BIT2' y)))) ==> (!_0. (!x y m. happ (happ (happ _0 x) y) m = 'h4/const/bool/LET' (happ (happ (happ _1 m) x) y) ('h4/const/numeral/exactlog' ('h4/const/arithmetic/BIT2' y))) ==> (!y x. 'h4/const/arithmetic/*' ('h4/const/arithmetic/BIT2' x) ('h4/const/arithmetic/BIT2' y) = 'h4/const/bool/LET' (happ (happ _0 x) y) ('h4/const/numeral/exactlog' ('h4/const/arithmetic/BIT2' x))))
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_3f45859,V_3f45855]: ![F, G]: (![X]: s(V_3f45855,happ(s(fun(V_3f45859,V_3f45855),F),s(V_3f45859,X))) = s(V_3f45855,happ(s(fun(V_3f45859,V_3f45855),G),s(V_3f45859,X))) => s(fun(V_3f45859,V_3f45855),F) = s(fun(V_3f45859,V_3f45855),G))).
fof('h4/thm/numeral/enumeral_mult_e5', conjecture, ![V__1]: (![M, X, Y, N]: s('h4/type/num/num',happ(s(fun('h4/type/num/num','h4/type/num/num'),happ(s(fun('h4/type/num/num',fun('h4/type/num/num','h4/type/num/num')),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num','h4/type/num/num'))),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num','h4/type/num/num')))),V__1),s('h4/type/num/num',M))),s('h4/type/num/num',X))),s('h4/type/num/num',Y))),s('h4/type/num/num',N))) = s('h4/type/num/num','h4/const/bool/COND'(s(bool,'h4/const/arithmetic/ODD'(s('h4/type/num/num',M))),s('h4/type/num/num','h4/const/numeral/texp_help'(s('h4/type/num/num','h4/const/arithmetic/DIV2'(s('h4/type/num/num',M))),s('h4/type/num/num','h4/const/prim_rec/PRE'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',Y))))))),s('h4/type/num/num','h4/const/bool/COND'(s(bool,'h4/const/arithmetic/ODD'(s('h4/type/num/num',N))),s('h4/type/num/num','h4/const/numeral/texp_help'(s('h4/type/num/num','h4/const/arithmetic/DIV2'(s('h4/type/num/num',N))),s('h4/type/num/num','h4/const/prim_rec/PRE'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',X))))))),s('h4/type/num/num','h4/const/numeral/internal_mult'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',X))),s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',Y))))))))) => ![V__0]: (![X, Y, M]: s('h4/type/num/num',happ(s(fun('h4/type/num/num','h4/type/num/num'),happ(s(fun('h4/type/num/num',fun('h4/type/num/num','h4/type/num/num')),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num','h4/type/num/num'))),V__0),s('h4/type/num/num',X))),s('h4/type/num/num',Y))),s('h4/type/num/num',M))) = s('h4/type/num/num','h4/const/bool/LET'(s(fun('h4/type/num/num','h4/type/num/num'),happ(s(fun('h4/type/num/num',fun('h4/type/num/num','h4/type/num/num')),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num','h4/type/num/num'))),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num','h4/type/num/num')))),V__1),s('h4/type/num/num',M))),s('h4/type/num/num',X))),s('h4/type/num/num',Y))),s('h4/type/num/num','h4/const/numeral/exactlog'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',Y))))))) => ![Y, X]: s('h4/type/num/num','h4/const/arithmetic/*'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',X))),s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',Y))))) = s('h4/type/num/num','h4/const/bool/LET'(s(fun('h4/type/num/num','h4/type/num/num'),happ(s(fun('h4/type/num/num',fun('h4/type/num/num','h4/type/num/num')),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num','h4/type/num/num'))),V__0),s('h4/type/num/num',X))),s('h4/type/num/num',Y))),s('h4/type/num/num','h4/const/numeral/exactlog'(s('h4/type/num/num','h4/const/arithmetic/BIT2'(s('h4/type/num/num',X)))))))))).
