%   ORIGINAL: 'h4/thm/real/SUM_PERMUTE_0_'
% 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 p. (!y. 'h4/const/prim_rec/<' y n ==> 'h4/const/bool/?!' (\x. 'h4/const/prim_rec/<' x n /\ p x = y)) ==> (!f. 'h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' n) (\n0. f (p n0)) = 'h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' n) f)
%   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. (!f p n0. happ (happ (happ _1 f) p) n0 = happ f (happ p n0)) ==> (!_0. (!n p y x. happ (happ (happ (happ _0 n) p) y) x <=> 'h4/const/prim_rec/<' x n /\ happ p x = y) ==> (!n p. (!y. 'h4/const/prim_rec/<' y n ==> 'h4/const/bool/?!' (happ (happ (happ _0 n) p) y)) ==> (!f. 'h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' n) (happ (happ _1 f) p) = 'h4/const/real/sum' ('h4/const/pair/,' 'h4/const/num/0' n) f)))
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_3f94669,V_3f94665]: ![F, G]: (![X]: s(V_3f94665,happ(s(fun(V_3f94669,V_3f94665),F),s(V_3f94669,X))) = s(V_3f94665,happ(s(fun(V_3f94669,V_3f94665),G),s(V_3f94669,X))) => s(fun(V_3f94669,V_3f94665),F) = s(fun(V_3f94669,V_3f94665),G))).
fof('h4/thm/real/SUM_PERMUTE_0_', conjecture, ![V__1]: (![F, P, N0]: s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),happ(s(fun(fun('h4/type/num/num','h4/type/num/num'),fun('h4/type/num/num','h4/type/realax/real')),happ(s(fun(fun('h4/type/num/num','h4/type/realax/real'),fun(fun('h4/type/num/num','h4/type/num/num'),fun('h4/type/num/num','h4/type/realax/real'))),V__1),s(fun('h4/type/num/num','h4/type/realax/real'),F))),s(fun('h4/type/num/num','h4/type/num/num'),P))),s('h4/type/num/num',N0))) = s('h4/type/realax/real',happ(s(fun('h4/type/num/num','h4/type/realax/real'),F),s('h4/type/num/num',happ(s(fun('h4/type/num/num','h4/type/num/num'),P),s('h4/type/num/num',N0))))) => ![V__0]: (![N, P, Y, X]: (p(s(bool,happ(s(fun('h4/type/num/num',bool),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',bool)),happ(s(fun(fun('h4/type/num/num','h4/type/num/num'),fun('h4/type/num/num',fun('h4/type/num/num',bool))),happ(s(fun('h4/type/num/num',fun(fun('h4/type/num/num','h4/type/num/num'),fun('h4/type/num/num',fun('h4/type/num/num',bool)))),V__0),s('h4/type/num/num',N))),s(fun('h4/type/num/num','h4/type/num/num'),P))),s('h4/type/num/num',Y))),s('h4/type/num/num',X)))) <=> (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',X),s('h4/type/num/num',N)))) & s('h4/type/num/num',happ(s(fun('h4/type/num/num','h4/type/num/num'),P),s('h4/type/num/num',X))) = s('h4/type/num/num',Y))) => ![N, P]: (![Y]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',Y),s('h4/type/num/num',N)))) => p(s(bool,'h4/const/bool/?!'(s(fun('h4/type/num/num',bool),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',bool)),happ(s(fun(fun('h4/type/num/num','h4/type/num/num'),fun('h4/type/num/num',fun('h4/type/num/num',bool))),happ(s(fun('h4/type/num/num',fun(fun('h4/type/num/num','h4/type/num/num'),fun('h4/type/num/num',fun('h4/type/num/num',bool)))),V__0),s('h4/type/num/num',N))),s(fun('h4/type/num/num','h4/type/num/num'),P))),s('h4/type/num/num',Y))))))) => ![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','h4/const/num/0'),s('h4/type/num/num',N))),s(fun('h4/type/num/num','h4/type/realax/real'),happ(s(fun(fun('h4/type/num/num','h4/type/num/num'),fun('h4/type/num/num','h4/type/realax/real')),happ(s(fun(fun('h4/type/num/num','h4/type/realax/real'),fun(fun('h4/type/num/num','h4/type/num/num'),fun('h4/type/num/num','h4/type/realax/real'))),V__1),s(fun('h4/type/num/num','h4/type/realax/real'),F))),s(fun('h4/type/num/num','h4/type/num/num'),P))))) = 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','h4/const/num/0'),s('h4/type/num/num',N))),s(fun('h4/type/num/num','h4/type/realax/real'),F))))))).
