%   ORIGINAL: 'h4/thm/real/sum_compute_1'
% 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: !n m f. 'h4/const/real/sum' ('h4/const/pair/,' n ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' m))) f = 'h4/const/realax/real_add' ('h4/const/real/sum' ('h4/const/pair/,' n ('h4/const/arithmetic/-' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' m)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))) f) (f ('h4/const/arithmetic/+' n ('h4/const/arithmetic/-' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' m)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))))
%   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: !n m f. 'h4/const/real/sum' ('h4/const/pair/,' n ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' m))) f = 'h4/const/realax/real_add' ('h4/const/real/sum' ('h4/const/pair/,' n ('h4/const/arithmetic/-' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' m)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))) f) (happ f ('h4/const/arithmetic/+' n ('h4/const/arithmetic/-' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' m)) ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')))))
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_3f94044,V_3f94040]: ![F, G]: (![X]: s(V_3f94040,happ(s(fun(V_3f94044,V_3f94040),F),s(V_3f94044,X))) = s(V_3f94040,happ(s(fun(V_3f94044,V_3f94040),G),s(V_3f94044,X))) => s(fun(V_3f94044,V_3f94040),F) = s(fun(V_3f94044,V_3f94040),G))).
fof('h4/thm/real/sum_compute_1', conjecture, ![N, M, F]: s('h4/type/realax/real','h4/const/real/sum'(s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num',M))))))),s(fun('h4/type/num/num','h4/type/realax/real'),F))) = s('h4/type/realax/real','h4/const/realax/real_add'(s('h4/type/realax/real','h4/const/real/sum'(s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/arithmetic/-'(s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num',M))))),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num','h4/const/arithmetic/ZERO'))))))))),s(fun('h4/type/num/num','h4/type/realax/real'),F))),s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/num/num','h4/const/arithmetic/+'(s('h4/type/num/num',N),s('h4/type/num/num','h4/const/arithmetic/-'(s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num',M))))),s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num','h4/const/arithmetic/ZERO')))))))))))))).
