%   ORIGINAL: 'h4/thm/arithmetic/DIV_P_'
% 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 p q. 'h4/const/prim_rec/<' 'h4/const/num/0' q ==> (P ('h4/const/arithmetic/DIV' p q) <=> (?k r. p = 'h4/const/arithmetic/+' ('h4/const/arithmetic/*' k q) r /\ 'h4/const/prim_rec/<' r q /\ P k))
%   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 p q. 'h4/const/prim_rec/<' 'h4/const/num/0' q ==> (happ P ('h4/const/arithmetic/DIV' p q) <=> (?k r. p = 'h4/const/arithmetic/+' ('h4/const/arithmetic/*' k q) r /\ 'h4/const/prim_rec/<' r q /\ happ P k))
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_3f82516,V_3f82512]: ![F, G]: (![X]: s(V_3f82512,happ(s(fun(V_3f82516,V_3f82512),F),s(V_3f82516,X))) = s(V_3f82512,happ(s(fun(V_3f82516,V_3f82512),G),s(V_3f82516,X))) => s(fun(V_3f82516,V_3f82512),F) = s(fun(V_3f82516,V_3f82512),G))).
fof('h4/thm/arithmetic/DIV_P_', conjecture, ![P, P0, Q]: (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num','h4/const/num/0'),s('h4/type/num/num',Q)))) => (p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num','h4/const/arithmetic/DIV'(s('h4/type/num/num',P0),s('h4/type/num/num',Q)))))) <=> ?[K, R]: (s('h4/type/num/num',P0) = s('h4/type/num/num','h4/const/arithmetic/+'(s('h4/type/num/num','h4/const/arithmetic/*'(s('h4/type/num/num',K),s('h4/type/num/num',Q))),s('h4/type/num/num',R))) & (p(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',R),s('h4/type/num/num',Q)))) & p(s(bool,happ(s(fun('h4/type/num/num',bool),P),s('h4/type/num/num',K))))))))).
