%   ORIGINAL: h4/binary__ieee/float__mul__nan
% 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/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/OR__CLAUSES_c4: !t. t \/ t <=> t
% Assm: h4/bool/IMP__CLAUSES_c0: !t. T ==> t <=> t
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/binary__ieee/float__value__repfns_c0: !a. h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/binary__ieee/_20_40ind__typebinary__ieee7 a) = a
% Assm: h4/binary__ieee/float__value__repfns_c1: !r. (\a0. !_27float__value_27. (!a00. (?a. a00 = (\a1. h4/ind__type/CONSTR h4/num/0 a1 (\n. h4/ind__type/BOTTOM)) a) \/ a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM) \/ a00 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (\n. h4/ind__type/BOTTOM) ==> _27float__value_27 a00) ==> _27float__value_27 a0) r <=> h4/binary__ieee/_20_40ind__typebinary__ieee7 (h4/binary__ieee/_20_40ind__typebinary__ieee6 r) = r
% Assm: h4/binary__ieee/hidden____20__40ind____typebinary____ieee3____def: h4/binary__ieee/_20_40ind__typebinary__ieee3 = (\a. h4/binary__ieee/_20_40ind__typebinary__ieee6 ((\a0. h4/ind__type/CONSTR h4/num/0 a0 (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/binary__ieee/hidden____20__40ind____typebinary____ieee4____def: h4/binary__ieee/_20_40ind__typebinary__ieee4 = h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/binary__ieee/hidden____20__40ind____typebinary____ieee5____def: h4/binary__ieee/_20_40ind__typebinary__ieee5 = h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB (\n. h4/ind__type/BOTTOM))
% Assm: h4/binary__ieee/Float0: h4/binary__ieee/Float = h4/binary__ieee/_20_40ind__typebinary__ieee3
% Assm: h4/binary__ieee/Infinity0: h4/binary__ieee/Infinity = h4/binary__ieee/_20_40ind__typebinary__ieee4
% Assm: h4/binary__ieee/NaN0: h4/binary__ieee/NaN = h4/binary__ieee/_20_40ind__typebinary__ieee5
% Assm: h4/binary__ieee/float__value__case__def_c0: !v1 v f a. h4/binary__ieee/float__value__CASE (h4/binary__ieee/Float a) f v v1 = f a
% Assm: h4/binary__ieee/float__value__case__def_c1: !v1 v f. h4/binary__ieee/float__value__CASE h4/binary__ieee/Infinity f v v1 = v
% Assm: h4/binary__ieee/float__value__case__def_c2: !v1 v f. h4/binary__ieee/float__value__CASE h4/binary__ieee/NaN f v v1 = v1
% Assm: h4/binary__ieee/float__mul__def: !y x mode. h4/binary__ieee/float__mul mode x y = h4/pair/pair__CASE (h4/pair/_2C (h4/binary__ieee/float__value0 x) (h4/binary__ieee/float__value0 y)) (\v v1. h4/binary__ieee/float__value__CASE v (\r_27. h4/binary__ieee/float__value__CASE v1 (\r2. h4/binary__ieee/float__round mode (~(h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y)) (h4/realax/real__mul r_27 r2)) (h4/bool/COND (r_27 = h4/real/real__of__num h4/num/0) (h4/binary__ieee/float__some__nan h4/bool/the__value) (h4/bool/COND (h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) (h4/binary__ieee/float__plus__infinity h4/bool/the__value) (h4/binary__ieee/float__minus__infinity h4/bool/the__value))) (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__value__CASE v1 (\r. h4/bool/COND (r = h4/real/real__of__num h4/num/0) (h4/binary__ieee/float__some__nan h4/bool/the__value) (h4/bool/COND (h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) (h4/binary__ieee/float__plus__infinity h4/bool/the__value) (h4/binary__ieee/float__minus__infinity h4/bool/the__value))) (h4/bool/COND (h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) (h4/binary__ieee/float__plus__infinity h4/bool/the__value) (h4/binary__ieee/float__minus__infinity h4/bool/the__value)) (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__some__nan h4/bool/the__value))
% Goal: !y x mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/NaN \/ h4/binary__ieee/float__value0 y = h4/binary__ieee/NaN ==> h4/binary__ieee/float__mul mode x y = h4/binary__ieee/float__some__nan h4/bool/the__value
%   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_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_ORu_u_CLAUSESu_c4]: !t. t \/ t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c0]: !t. T ==> t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_indu_u_types_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = happ f n
% Assm [h4s_binaryu_u_ieees_floatu_u_valueu_u_repfnsu_c0]: !a. h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/binary__ieee/_20_40ind__typebinary__ieee7 a) = a
% Assm [h4s_binaryu_u_ieees_floatu_u_valueu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27float__value_27. (!a00. (?a. a00 = h4/ind__type/CONSTR h4/num/0 a _0) \/ a00 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0 \/ a00 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB _0 ==> happ _27float__value_27 a00) ==> happ _27float__value_27 r) <=> h4/binary__ieee/_20_40ind__typebinary__ieee7 (h4/binary__ieee/_20_40ind__typebinary__ieee6 r) = r)
% Assm [h4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee3u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/binary__ieee/_20_40ind__typebinary__ieee3 x = h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/ind__type/CONSTR h4/num/0 x _0))
% Assm [h4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee4u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/binary__ieee/_20_40ind__typebinary__ieee4 = h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) h4/bool/ARB _0)
% Assm [h4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee5u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> h4/binary__ieee/_20_40ind__typebinary__ieee5 = h4/binary__ieee/_20_40ind__typebinary__ieee6 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) h4/bool/ARB _0)
% Assm [h4s_binaryu_u_ieees_Float0]: h4/binary__ieee/Float = h4/binary__ieee/_20_40ind__typebinary__ieee3
% Assm [h4s_binaryu_u_ieees_Infinity0]: h4/binary__ieee/Infinity = h4/binary__ieee/_20_40ind__typebinary__ieee4
% Assm [h4s_binaryu_u_ieees_NaN0]: h4/binary__ieee/NaN = h4/binary__ieee/_20_40ind__typebinary__ieee5
% Assm [h4s_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c0]: !v1 v f a. h4/binary__ieee/float__value__CASE (happ h4/binary__ieee/Float a) f v v1 = happ f a
% Assm [h4s_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c1]: !v1 v f. h4/binary__ieee/float__value__CASE h4/binary__ieee/Infinity f v v1 = v
% Assm [h4s_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c2]: !v1 v f. h4/binary__ieee/float__value__CASE h4/binary__ieee/NaN f v v1 = v1
% Assm [h4s_binaryu_u_ieees_floatu_u_mulu_u_def]: !_4. (!x y r. ?v. (v <=> h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) /\ (?v'. (v' <=> r = h4/real/real__of__num h4/num/0) /\ happ (happ (happ _4 x) y) r = h4/bool/COND v' (h4/binary__ieee/float__some__nan h4/bool/the__value) (h4/bool/COND v (h4/binary__ieee/float__plus__infinity h4/bool/the__value) (h4/binary__ieee/float__minus__infinity h4/bool/the__value)))) ==> (!_3. (!mode x y r_27 r2. ?v. (v <=> ~(h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y)) /\ happ (happ (happ (happ (happ _3 mode) x) y) r_27) r2 = h4/binary__ieee/float__round mode v (h4/realax/real__mul r_27 r2)) ==> (!_2. (!v1 mode x y r_27. ?v. (v <=> h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) /\ (?v'. (v' <=> r_27 = h4/real/real__of__num h4/num/0) /\ happ (happ (happ (happ (happ _2 v1) mode) x) y) r_27 = h4/binary__ieee/float__value__CASE v1 (happ (happ (happ (happ _3 mode) x) y) r_27) (h4/bool/COND v' (h4/binary__ieee/float__some__nan h4/bool/the__value) (h4/bool/COND v (h4/binary__ieee/float__plus__infinity h4/bool/the__value) (h4/binary__ieee/float__minus__infinity h4/bool/the__value))) (h4/binary__ieee/float__some__nan h4/bool/the__value))) ==> (!_1. (!v mode x y v1. ?v. (v <=> h4/binary__ieee/float__Sign x = h4/binary__ieee/float__Sign y) /\ happ (happ (happ (happ (happ _1 v) mode) x) y) v1 = h4/binary__ieee/float__value__CASE v (happ (happ (happ (happ _2 v1) mode) x) y) (h4/binary__ieee/float__value__CASE v1 (happ (happ _4 x) y) (h4/bool/COND v (h4/binary__ieee/float__plus__infinity h4/bool/the__value) (h4/binary__ieee/float__minus__infinity h4/bool/the__value)) (h4/binary__ieee/float__some__nan h4/bool/the__value)) (h4/binary__ieee/float__some__nan h4/bool/the__value)) ==> (!_0. (!mode x y v. happ (happ (happ (happ _0 mode) x) y) v = happ (happ (happ (happ _1 v) mode) x) y) ==> (!y x mode. h4/binary__ieee/float__mul mode x y = h4/pair/pair__CASE (h4/pair/_2C (h4/binary__ieee/float__value0 x) (h4/binary__ieee/float__value0 y)) (happ (happ (happ _0 mode) x) y))))))
% Goal: !y x mode. h4/binary__ieee/float__value0 x = h4/binary__ieee/NaN \/ h4/binary__ieee/float__value0 y = h4/binary__ieee/NaN ==> h4/binary__ieee/float__mul mode x y = h4/binary__ieee/float__some__nan h4/bool/the__value
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_Q109675,TV_Q109671]: ![V_f, V_g]: (![V_x]: s(TV_Q109671,happ(s(t_fun(TV_Q109675,TV_Q109671),V_f),s(TV_Q109675,V_x))) = s(TV_Q109671,happ(s(t_fun(TV_Q109675,TV_Q109671),V_g),s(TV_Q109675,V_x))) => s(t_fun(TV_Q109675,TV_Q109671),V_f) = s(t_fun(TV_Q109675,TV_Q109671),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_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_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_JRHu_u_INDUCTu_u_UTIL, axiom, ![TV_u_27a]: ![V_t, V_P]: (![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_t) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) => p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_P)))))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_indu_u_types_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_a)).
fof(ah4s_indu_u_types_FCONS0u_c1, axiom, ![TV_u_27a]: ![V_n, V_f, V_a]: s(TV_u_27a,h4s_indu_u_types_fcons(s(TV_u_27a,V_a),s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_f),s(t_h4s_nums_num,V_n)))).
fof(ah4s_binaryu_u_ieees_floatu_u_valueu_u_repfnsu_c0, axiom, ![V_a]: s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee6(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee7(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_a))))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,V_a)).
fof(ah4s_binaryu_u_ieees_floatu_u_valueu_u_repfnsu_c1, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27floatu_u_valueu_27]: (![V_a00]: ((?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_realaxs_real,V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))) | (s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))) | s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),t_bool),V_uu_27floatu_u_valueu_27),s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_a00))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),t_bool),V_uu_27floatu_u_valueu_27),s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee7(s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee6(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),V_r)))).
fof(ah4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee3u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee3),s(t_h4s_realaxs_real,V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee6(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_realaxs_real,V_x),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee4u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee4) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee6(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_binaryu_u_ieees_hiddenu_u_u_u_20u_u_40indu_u_u_u_typebinaryu_u_u_u_ieee5u_u_u_u_def, axiom, ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_bottom) => s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee5) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee6(s(t_h4s_indu_u_types_recspace(t_h4s_realaxs_real),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_realaxs_real,h4s_bools_arb),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_realaxs_real)),V_uu_0))))))).
fof(ah4s_binaryu_u_ieees_Float0, axiom, s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float) = s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee3)).
fof(ah4s_binaryu_u_ieees_Infinity0, axiom, s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_infinity) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee4)).
fof(ah4s_binaryu_u_ieees_NaN0, axiom, s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_nan) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_u_20u_40indu_u_typebinaryu_u_ieee5)).
fof(ah4s_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f, V_a]: s(TV_u_27a,h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_binaryu_u_ieees_float),s(t_h4s_realaxs_real,V_a))),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(TV_u_27a,V_v),s(TV_u_27a,V_v1))) = s(TV_u_27a,happ(s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(t_h4s_realaxs_real,V_a)))).
fof(ah4s_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f]: s(TV_u_27a,h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_infinity),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(TV_u_27a,V_v),s(TV_u_27a,V_v1))) = s(TV_u_27a,V_v)).
fof(ah4s_binaryu_u_ieees_floatu_u_valueu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_v1, V_v, V_f]: s(TV_u_27a,h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_nan),s(t_fun(t_h4s_realaxs_real,TV_u_27a),V_f),s(TV_u_27a,V_v),s(TV_u_27a,V_v1))) = s(TV_u_27a,V_v1)).
fof(ah4s_binaryu_u_ieees_floatu_u_mulu_u_def, axiom, ![TV_u_27t,TV_u_27w]: ![V_uu_4]: (![V_x, V_y, V_r]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y)))) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(t_h4s_realaxs_real,V_r) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),V_uu_4),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_realaxs_real,V_r))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,V_vi_),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))))))) => ![V_uu_3]: (![V_mode, V_x, V_y, V_ru_27, V_r2]: ?[V_v]: ((p(s(t_bool,V_v)) <=> ~ (s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))))) & s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_3),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_realaxs_real,V_ru_27))),s(t_h4s_realaxs_real,V_r2))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_round(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_bool,V_v),s(t_h4s_realaxs_real,h4s_realaxs_realu_u_mul(s(t_h4s_realaxs_real,V_ru_27),s(t_h4s_realaxs_real,V_r2)))))) => ![V_uu_2]: (![V_v1, V_mode, V_x, V_y, V_ru_27]: ?[V_v]: ((p(s(t_bool,V_v)) <=> s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y)))) & ?[V_vi_]: ((p(s(t_bool,V_vi_)) <=> s(t_h4s_realaxs_real,V_ru_27) = s(t_h4s_realaxs_real,h4s_reals_realu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) & s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_2),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_realaxs_real,V_ru_27))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1),s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_3),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_realaxs_real,V_ru_27))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,V_vi_),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,V_v),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))))) => ![V_uu_1]: (![V_v, V_mode, V_x, V_y, V_v1]: ?[V_v0]: ((p(s(t_bool,V_v0)) <=> s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_fcps_cart(t_bool,t_h4s_ones_one),h4s_binaryu_u_ieees_floatu_u_sign(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y)))) & s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_1),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v),s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_2),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_valueu_u_case(s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v1),s(t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_realaxs_real,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),V_uu_4),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_bools_cond(s(t_bool,V_v0),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_plusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_minusu_u_infinity(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value)))))) => ![V_uu_0]: (![V_mode, V_x, V_y, V_v]: s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_0),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v))) = s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_1),s(t_h4s_binaryu_u_ieees_floatu_u_value,V_v))),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) => ![V_y, V_x, V_mode]: s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_mul(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_floatu_u_value),h4s_pairs_u_2c(s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))))),s(t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))),happ(s(t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w))))),happ(s(t_fun(t_h4s_binaryu_u_ieees_rounding,t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_fun(t_h4s_binaryu_u_ieees_floatu_u_value,t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w)))))),V_uu_0),s(t_h4s_binaryu_u_ieees_rounding,V_mode))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))))))))))).
fof(ch4s_binaryu_u_ieees_floatu_u_mulu_u_nan, conjecture, ![TV_u_27t,TV_u_27w]: ![V_y, V_x, V_mode]: ((s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_nan) | s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_floatu_u_value0(s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) = s(t_h4s_binaryu_u_ieees_floatu_u_value,h4s_binaryu_u_ieees_nan)) => s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_mul(s(t_h4s_binaryu_u_ieees_rounding,V_mode),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_x),s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),V_y))) = s(t_h4s_binaryu_u_ieees_float(TV_u_27t,TV_u_27w),h4s_binaryu_u_ieees_floatu_u_someu_u_nan(s(t_h4s_bools_itself(t_h4s_pairs_prod(TV_u_27t,TV_u_27w)),h4s_bools_theu_u_value))))).
