%   ORIGINAL: 'h4/thm/arithmetic/MOD_SUC_IFF_'
% 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. 'h4/const/prim_rec/<' 'h4/const/num/0' y ==> ('h4/const/arithmetic/MOD' ('h4/const/num/SUC' x) y = 'h4/const/num/SUC' ('h4/const/arithmetic/MOD' x y) <=> ~('h4/const/num/SUC' x = 'h4/const/arithmetic/*' ('h4/const/num/SUC' ('h4/const/arithmetic/DIV' x y)) 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. 'h4/const/prim_rec/<' 'h4/const/num/0' y ==> ('h4/const/arithmetic/MOD' ('h4/const/num/SUC' x) y = 'h4/const/num/SUC' ('h4/const/arithmetic/MOD' x y) <=> ~('h4/const/num/SUC' x = 'h4/const/arithmetic/*' ('h4/const/num/SUC' ('h4/const/arithmetic/DIV' x y)) 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_3f83016,V_3f83012]: ![F, G]: (![X]: s(V_3f83012,happ(s(fun(V_3f83016,V_3f83012),F),s(V_3f83016,X))) = s(V_3f83012,happ(s(fun(V_3f83016,V_3f83012),G),s(V_3f83016,X))) => s(fun(V_3f83016,V_3f83012),F) = s(fun(V_3f83016,V_3f83012),G))).
fof('h4/thm/arithmetic/MOD_SUC_IFF_', conjecture, ![Y, X]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num','h4/const/num/0'),s('h4/type/num/num',Y)))) => (s('h4/type/num/num','h4/const/arithmetic/MOD'(s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',X))),s('h4/type/num/num',Y))) = s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num','h4/const/arithmetic/MOD'(s('h4/type/num/num',X),s('h4/type/num/num',Y))))) <=> ~ (s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num',X))) = s('h4/type/num/num','h4/const/arithmetic/*'(s('h4/type/num/num','h4/const/num/SUC'(s('h4/type/num/num','h4/const/arithmetic/DIV'(s('h4/type/num/num',X),s('h4/type/num/num',Y))))),s('h4/type/num/num',Y))))))).
