%   ORIGINAL: 'h4/thm/real/sum_ind_'
% 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. (!n f. P ('h4/const/pair/,' n 'h4/const/num/0') f) /\ (!n m f. P ('h4/const/pair/,' n m) f ==> P ('h4/const/pair/,' n ('h4/const/num/SUC' m)) f) ==> (!v v1 v2. P ('h4/const/pair/,' v v1) v2)
%   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: !P. (!n f. happ (happ P ('h4/const/pair/,' n 'h4/const/num/0')) f) /\ (!n m f. happ (happ P ('h4/const/pair/,' n m)) f ==> happ (happ P ('h4/const/pair/,' n ('h4/const/num/SUC' m))) f) ==> (!v v1 v2. happ (happ P ('h4/const/pair/,' v v1)) v2)
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_3f93944,V_3f93940]: ![F, G]: (![X]: s(V_3f93940,happ(s(fun(V_3f93944,V_3f93940),F),s(V_3f93944,X))) = s(V_3f93940,happ(s(fun(V_3f93944,V_3f93940),G),s(V_3f93944,X))) => s(fun(V_3f93944,V_3f93940),F) = s(fun(V_3f93944,V_3f93940),G))).
fof('h4/thm/real/sum_ind_', conjecture, ![P]: ((![N, F]: p(s(bool,happ(s(fun(fun('h4/type/num/num','h4/type/realax/real'),bool),happ(s(fun('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),fun(fun('h4/type/num/num','h4/type/realax/real'),bool)),P),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/num/0'))))),s(fun('h4/type/num/num','h4/type/realax/real'),F)))) & ![N, M, F]: (p(s(bool,happ(s(fun(fun('h4/type/num/num','h4/type/realax/real'),bool),happ(s(fun('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),fun(fun('h4/type/num/num','h4/type/realax/real'),bool)),P),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',M))))),s(fun('h4/type/num/num','h4/type/realax/real'),F)))) => p(s(bool,happ(s(fun(fun('h4/type/num/num','h4/type/realax/real'),bool),happ(s(fun('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),fun(fun('h4/type/num/num','h4/type/realax/real'),bool)),P),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/num/SUC'(s('h4/type/num/num',M))))))),s(fun('h4/type/num/num','h4/type/realax/real'),F)))))) => ![V, V1, V2]: p(s(bool,happ(s(fun(fun('h4/type/num/num','h4/type/realax/real'),bool),happ(s(fun('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),fun(fun('h4/type/num/num','h4/type/realax/real'),bool)),P),s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num',V),s('h4/type/num/num',V1))))),s(fun('h4/type/num/num','h4/type/realax/real'),V2)))))).
