%   ORIGINAL: 'h4/thm/prim_rec/SIMP_REC_REL_UNIQUE_RESULT_'
% 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: !x f n. 'h4/const/bool/?!' (\y. ?g. 'h4/const/prim_rec/SIMP_REC_REL' g x f ('h4/const/num/SUC' n) /\ y = g n)
%   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: !_0. (!x f n y. happ (happ (happ (happ _0 x) f) n) y <=> (?g. 'h4/const/prim_rec/SIMP_REC_REL' g x f ('h4/const/num/SUC' n) /\ y = happ g n)) ==> (!x f n. 'h4/const/bool/?!' (happ (happ (happ _0 x) f) n))
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_3f86769,V_3f86765]: ![F, G]: (![X]: s(V_3f86765,happ(s(fun(V_3f86769,V_3f86765),F),s(V_3f86769,X))) = s(V_3f86765,happ(s(fun(V_3f86769,V_3f86765),G),s(V_3f86769,X))) => s(fun(V_3f86769,V_3f86765),F) = s(fun(V_3f86769,V_3f86765),G))).
fof('h4/thm/prim_rec/SIMP_REC_REL_UNIQUE_RESULT_', conjecture, ![A]: ![V__0]: (![X, F, N, Y]: (p(s(bool,happ(s(fun(A,bool),happ(s(fun('h4/type/num/num',fun(A,bool)),happ(s(fun(fun(A,A),fun('h4/type/num/num',fun(A,bool))),happ(s(fun(A,fun(fun(A,A),fun('h4/type/num/num',fun(A,bool)))),V__0),s(A,X))),s(fun(A,A),F))),s('h4/type/num/num',N))),s(A,Y)))) <=> ?[G]: (p(s(bool,'h4/const/prim_rec/SIMP_REC_REL'(s(fun('h4/type/num/num',A),G),s(A,X),s(fun(A,A),F),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N)))))) & s(A,Y) = s(A,happ(s(fun('h4/type/num/num',A),G),s('h4/type/num/num',N))))) => ![X, F, N]: p(s(bool,'h4/const/bool/?!'(s(fun(A,bool),happ(s(fun('h4/type/num/num',fun(A,bool)),happ(s(fun(fun(A,A),fun('h4/type/num/num',fun(A,bool))),happ(s(fun(A,fun(fun(A,A),fun('h4/type/num/num',fun(A,bool)))),V__0),s(A,X))),s(fun(A,A),F))),s('h4/type/num/num',N)))))))).
