%   ORIGINAL: 'h4/thm/arithmetic/DIVMOD_THM_'
% 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: !n m a. 'h4/const/arithmetic/DIVMOD' ('h4/const/pair/,' a ('h4/const/pair/,' m n)) = 'h4/const/bool/COND' (n = 'h4/const/num/0') ('h4/const/pair/,' 'h4/const/num/0' 'h4/const/num/0') ('h4/const/bool/COND' ('h4/const/prim_rec/<' m n) ('h4/const/pair/,' a m) ('h4/const/bool/LET' (\q. 'h4/const/arithmetic/DIVMOD' ('h4/const/pair/,' ('h4/const/arithmetic/+' a q) ('h4/const/pair/,' ('h4/const/arithmetic/-' m ('h4/const/arithmetic/*' n q)) n))) ('h4/const/arithmetic/findq' ('h4/const/pair/,' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')) ('h4/const/pair/,' m 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. (!a m n q. happ (happ (happ (happ _0 a) m) n) q = 'h4/const/arithmetic/DIVMOD' ('h4/const/pair/,' ('h4/const/arithmetic/+' a q) ('h4/const/pair/,' ('h4/const/arithmetic/-' m ('h4/const/arithmetic/*' n q)) n))) ==> (!n m a. ?v. (v <=> n = 'h4/const/num/0') /\ 'h4/const/arithmetic/DIVMOD' ('h4/const/pair/,' a ('h4/const/pair/,' m n)) = 'h4/const/bool/COND' v ('h4/const/pair/,' 'h4/const/num/0' 'h4/const/num/0') ('h4/const/bool/COND' ('h4/const/prim_rec/<' m n) ('h4/const/pair/,' a m) ('h4/const/bool/LET' (happ (happ (happ _0 a) m) n) ('h4/const/arithmetic/findq' ('h4/const/pair/,' ('h4/const/arithmetic/NUMERAL' ('h4/const/arithmetic/BIT1' 'h4/const/arithmetic/ZERO')) ('h4/const/pair/,' m 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_3f85743,V_3f85739]: ![F, G]: (![X]: s(V_3f85739,happ(s(fun(V_3f85743,V_3f85739),F),s(V_3f85743,X))) = s(V_3f85739,happ(s(fun(V_3f85743,V_3f85739),G),s(V_3f85743,X))) => s(fun(V_3f85743,V_3f85739),F) = s(fun(V_3f85743,V_3f85739),G))).
fof('h4/thm/arithmetic/DIVMOD_THM_', conjecture, ![V__0]: (![A, M, N, Q]: s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),happ(s(fun('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num')),happ(s(fun('h4/type/num/num',fun('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'))),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num')))),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'))))),V__0),s('h4/type/num/num',A))),s('h4/type/num/num',M))),s('h4/type/num/num',N))),s('h4/type/num/num',Q))) = s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/arithmetic/DIVMOD'(s('h4/type/pair/prod'('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num')),'h4/const/pair/,'(s('h4/type/num/num','h4/const/arithmetic/+'(s('h4/type/num/num',A),s('h4/type/num/num',Q))),s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num','h4/const/arithmetic/-'(s('h4/type/num/num',M),s('h4/type/num/num','h4/const/arithmetic/*'(s('h4/type/num/num',N),s('h4/type/num/num',Q))))),s('h4/type/num/num',N))))))) => ![N, M, A]: ?[V]: ((p(s(bool,V)) <=> s('h4/type/num/num',N) = s('h4/type/num/num','h4/const/num/0')) & s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/arithmetic/DIVMOD'(s('h4/type/pair/prod'('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num')),'h4/const/pair/,'(s('h4/type/num/num',A),s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num',M),s('h4/type/num/num',N))))))) = s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/bool/COND'(s(bool,V),s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num','h4/const/num/0'),s('h4/type/num/num','h4/const/num/0'))),s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/bool/COND'(s(bool,'h4/const/prim_rec/<'(s('h4/type/num/num',M),s('h4/type/num/num',N))),s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num',A),s('h4/type/num/num',M))),s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/bool/LET'(s(fun('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num')),happ(s(fun('h4/type/num/num',fun('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'))),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num')))),happ(s(fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num',fun('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'))))),V__0),s('h4/type/num/num',A))),s('h4/type/num/num',M))),s('h4/type/num/num',N))),s('h4/type/num/num','h4/const/arithmetic/findq'(s('h4/type/pair/prod'('h4/type/num/num','h4/type/pair/prod'('h4/type/num/num','h4/type/num/num')),'h4/const/pair/,'(s('h4/type/num/num','h4/const/arithmetic/NUMERAL'(s('h4/type/num/num','h4/const/arithmetic/BIT1'(s('h4/type/num/num','h4/const/arithmetic/ZERO'))))),s('h4/type/pair/prod'('h4/type/num/num','h4/type/num/num'),'h4/const/pair/,'(s('h4/type/num/num',M),s('h4/type/num/num',N)))))))))))))))).
