%   ORIGINAL: 'h4/thm/arithmetic/NRC_SUC_RECURSE_LEFT_'
% 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 n R. 'h4/const/arithmetic/NRC' R ('h4/const/num/SUC' n) x y <=> (?z. 'h4/const/arithmetic/NRC' R n x z /\ R z y)
%   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: !y x n R. 'h4/const/arithmetic/NRC' R ('h4/const/num/SUC' n) x y <=> (?z. 'h4/const/arithmetic/NRC' R n x z /\ happ (happ R z) y)
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_3f85217,V_3f85213]: ![F, G]: (![X]: s(V_3f85213,happ(s(fun(V_3f85217,V_3f85213),F),s(V_3f85217,X))) = s(V_3f85213,happ(s(fun(V_3f85217,V_3f85213),G),s(V_3f85217,X))) => s(fun(V_3f85217,V_3f85213),F) = s(fun(V_3f85217,V_3f85213),G))).
fof('h4/thm/arithmetic/NRC_SUC_RECURSE_LEFT_', conjecture, ![A]: ![Y, X, N, R]: (p(s(bool,'h4/const/arithmetic/NRC'(s(fun(A,fun(A,bool)),R),s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',N))),s(A,X),s(A,Y)))) <=> ?[Z]: (p(s(bool,'h4/const/arithmetic/NRC'(s(fun(A,fun(A,bool)),R),s('h4/type/num/num',N),s(A,X),s(A,Z)))) & p(s(bool,happ(s(fun(A,bool),happ(s(fun(A,fun(A,bool)),R),s(A,Z))),s(A,Y))))))).
