%   ORIGINAL: h4/float/IS__VALID__FINITE
% 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
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/float/SIGN: !a. h4/ieee/sign a = h4/pair/FST a
% Assm: h4/float/EXPONENT: !a. h4/ieee/exponent a = h4/pair/FST (h4/pair/SND a)
% Assm: h4/float/FRACTION: !a. h4/ieee/fraction a = h4/pair/SND (h4/pair/SND a)
% Assm: h4/float/IS__VALID: !a X. h4/ieee/is__valid X a <=> h4/prim__rec/_3C (h4/ieee/sign a) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) /\ h4/prim__rec/_3C (h4/ieee/exponent a) (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/ieee/expwidth X)) /\ h4/prim__rec/_3C (h4/ieee/fraction a) (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/ieee/fracwidth X))
% Assm: h4/float/FINITE__R3: !p n m. h4/pred__set/FINITE (h4/pred__set/GSPEC (\a. h4/pair/_2C a (h4/prim__rec/_3C (h4/pair/FST a) m /\ h4/prim__rec/_3C (h4/pair/FST (h4/pair/SND a)) n /\ h4/prim__rec/_3C (h4/pair/SND (h4/pair/SND a)) p)))
% Goal: !X. h4/pred__set/FINITE (h4/pred__set/GSPEC (\a. h4/pair/_2C a (h4/ieee/is__valid X a)))
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_floats_SIGN]: !a. h4/ieee/sign a = h4/pair/FST a
% Assm [h4s_floats_EXPONENT]: !a. h4/ieee/exponent a = h4/pair/FST (h4/pair/SND a)
% Assm [h4s_floats_FRACTION]: !a. h4/ieee/fraction a = h4/pair/SND (h4/pair/SND a)
% Assm [h4s_floats_ISu_u_VALID]: !a X. h4/ieee/is__valid X a <=> h4/prim__rec/_3C (h4/ieee/sign a) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) /\ h4/prim__rec/_3C (h4/ieee/exponent a) (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/ieee/expwidth X)) /\ h4/prim__rec/_3C (h4/ieee/fraction a) (h4/arithmetic/EXP (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) (h4/ieee/fracwidth X))
% Assm [h4s_floats_FINITEu_u_R3]: !_0. (!m n p a. ?v. (v <=> h4/prim__rec/_3C (h4/pair/FST a) m /\ h4/prim__rec/_3C (h4/pair/FST (h4/pair/SND a)) n /\ h4/prim__rec/_3C (h4/pair/SND (h4/pair/SND a)) p) /\ happ (happ (happ (happ _0 m) n) p) a = h4/pair/_2C a v) ==> (!p n m. h4/pred__set/FINITE (h4/pred__set/GSPEC (happ (happ (happ _0 m) n) p)))
% Goal: !_0. (!X a. happ (happ _0 X) a = h4/pair/_2C a (h4/ieee/is__valid X a)) ==> (!X. h4/pred__set/FINITE (h4/pred__set/GSPEC (happ _0 X)))
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q98438,TV_Q98434]: ![V_f, V_g]: (![V_x]: s(TV_Q98434,happ(s(t_fun(TV_Q98438,TV_Q98434),V_f),s(TV_Q98438,V_x))) = s(TV_Q98434,happ(s(t_fun(TV_Q98438,TV_Q98434),V_g),s(TV_Q98438,V_x))) => s(t_fun(TV_Q98438,TV_Q98434),V_f) = s(t_fun(TV_Q98438,TV_Q98434),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_floats_SIGN, axiom, ![V_a]: s(t_h4s_nums_num,h4s_ieees_sign(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))) = s(t_h4s_nums_num,h4s_pairs_fst(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a)))).
fof(ah4s_floats_EXPONENT, axiom, ![V_a]: s(t_h4s_nums_num,h4s_ieees_exponent(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))) = s(t_h4s_nums_num,h4s_pairs_fst(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_snd(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a)))))).
fof(ah4s_floats_FRACTION, axiom, ![V_a]: s(t_h4s_nums_num,h4s_ieees_fraction(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))) = s(t_h4s_nums_num,h4s_pairs_snd(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_snd(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a)))))).
fof(ah4s_floats_ISu_u_VALID, axiom, ![V_a, V_X]: (p(s(t_bool,h4s_ieees_isu_u_valid(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_X),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a)))) <=> (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_ieees_sign(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))) & (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_ieees_exponent(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_ieees_expwidth(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_X)))))))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_ieees_fraction(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))),s(t_h4s_nums_num,h4s_arithmetics_exp(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_ieees_fracwidth(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_X)))))))))))).
fof(ah4s_floats_FINITEu_u_R3, axiom, ![V_uu_0]: (![V_m, V_n, V_p, V_a]: ?[V_v]: ((p(s(t_bool,V_v)) <=> (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_pairs_fst(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))),s(t_h4s_nums_num,V_m)))) & (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_pairs_fst(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_snd(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))))),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_pairs_snd(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),h4s_pairs_snd(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))))),s(t_h4s_nums_num,V_p))))))) & s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool))))),V_uu_0),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))) = s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a),s(t_bool,V_v)))) => ![V_p, V_n, V_m]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool)),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool))))),V_uu_0),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))))))))).
fof(ch4s_floats_ISu_u_VALIDu_u_FINITE, conjecture, ![V_uu_0]: (![V_X, V_a]: s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool))),V_uu_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_X))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))) = s(t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),h4s_pairs_u_2c(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a),s(t_bool,h4s_ieees_isu_u_valid(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_X),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),V_a))))) => ![V_X]: p(s(t_bool,h4s_predu_u_sets_finite(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool),h4s_predu_u_sets_gspec(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool)),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_h4s_pairs_prod(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num)),t_bool))),V_uu_0),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_nums_num),V_X)))))))))).
