%   ORIGINAL: 'h4/thm/arithmetic/MOD_ELIM_'
% 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 x n. 'h4/const/prim_rec/<' 'h4/const/num/0' n /\ P x /\ (!y. P ('h4/const/arithmetic/+' y n) ==> P y) ==> P ('h4/const/arithmetic/MOD' x 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: !P x n. 'h4/const/prim_rec/<' 'h4/const/num/0' n /\ happ P x /\ (!y. happ P ('h4/const/arithmetic/+' y n) ==> happ P y) ==> happ P ('h4/const/arithmetic/MOD' x 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_3f85467,V_3f85463]: ![F, G]: (![X]: s(V_3f85463,happ(s(fun(V_3f85467,V_3f85463),F),s(V_3f85467,X))) = s(V_3f85463,happ(s(fun(V_3f85467,V_3f85463),G),s(V_3f85467,X))) => s(fun(V_3f85467,V_3f85463),F) = s(fun(V_3f85467,V_3f85463),G))).
fof('h4/thm/arithmetic/MOD_ELIM_', conjecture, ![P, X, N]: ((p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num','h4/const/num/0'),s('h4/type/num/num',N)))) & (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',X)))) & ![Y]: (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num','h4/const/arithmetic/+'(s('h4/type/num/num',Y),s('h4/type/num/num',N)))))) => p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',Y))))))) => p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num','h4/const/arithmetic/MOD'(s('h4/type/num/num',X),s('h4/type/num/num',N)))))))).
