%   ORIGINAL: h4/ratRing/rat__ring__thms_c3
% 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/ratRing/rat__interp__p__def: h4/ratRing/rat__interp__p = h4/ringNorm/interp__p (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/ratRing/rat__ring__thms_c1: !vm p. h4/ratRing/rat__interp__p vm p = h4/ratRing/rat__r__interp__cs vm (h4/ratRing/rat__polynom__simplify p)
% Assm: h4/ratRing/rat__ring__thms_c2: !vm c. h4/ratRing/rat__interp__p vm (h4/ringNorm/Pconst c) = c
% Assm: h4/ringNorm/interp__p__def_c1: !vm r i. h4/ringNorm/interp__p r vm (h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm: h4/bool/TRUTH: T
% Assm: h4/ringNorm/Pvar0: h4/ringNorm/Pvar = h4/ringNorm/_20_40ind__typeringNorm0
% Assm: h4/ratRing/rat__r__interp__cs__def: h4/ratRing/rat__r__interp__cs = h4/ringNorm/r__interp__cs (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ringNorm/polynom__simplify__ok: !r. h4/ring/is__ring r ==> (!vm p. h4/ringNorm/r__interp__cs r vm (h4/ringNorm/polynom__simplify r p) = h4/ringNorm/interp__p r vm p)
% Assm: h4/ratRing/RAT__IS__RING: h4/ring/is__ring (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ratRing/rat__polynom__simplify__def: h4/ratRing/rat__polynom__simplify = h4/ringNorm/polynom__simplify (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ring/ring__accessors_c3: !f1 f0 f a0 a. h4/ring/ring__RM (h4/ring/ring0 a a0 f f0 f1) = f0
% Assm: h4/ring/ring__accessors_c4: !f1 f0 f a0 a. h4/ring/ring__RN (h4/ring/ring0 a a0 f f0 f1) = f1
% Assm: h4/ringNorm/interp__p__def_c2: !vm r p2 p1. h4/ringNorm/interp__p r vm (h4/ringNorm/Pplus p1 p2) = h4/ring/ring__RP r (h4/ringNorm/interp__p r vm p1) (h4/ringNorm/interp__p r vm p2)
% Assm: h4/ringNorm/interp__p__def_c4: !vm r p1. h4/ringNorm/interp__p r vm (h4/ringNorm/Popp p1) = h4/ring/ring__RN r (h4/ringNorm/interp__p r vm p1)
% Assm: h4/ring/ring__accessors_c2: !f1 f0 f a0 a. h4/ring/ring__RP (h4/ring/ring0 a a0 f f0 f1) = f
% Assm: h4/ringNorm/interp__p__def_c3: !vm r p2 p1. h4/ringNorm/interp__p r vm (h4/ringNorm/Pmult p1 p2) = h4/ring/ring__RM r (h4/ringNorm/interp__p r vm p1) (h4/ringNorm/interp__p r vm p2)
% Assm: h4/ringNorm/interp__p__def_c0: !vm r c. h4/ringNorm/interp__p r vm (h4/ringNorm/Pconst c) = c
% Assm: h4/quote/varmap__find__def_c1: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Right__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v2
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/ringNorm/Popp0: h4/ringNorm/Popp = h4/ringNorm/_20_40ind__typeringNorm4
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm3____def: h4/ringNorm/_20_40ind__typeringNorm3 = (\a0 a1. h4/ringNorm/_20_40ind__typeringNorm5 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/ringNorm/_20_40ind__typeringNorm6 a0) (h4/ringNorm/_20_40ind__typeringNorm6 a1)))
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm4____def: h4/ringNorm/_20_40ind__typeringNorm4 = (\a. h4/ringNorm/_20_40ind__typeringNorm5 ((\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) (h4/ringNorm/_20_40ind__typeringNorm6 a)))
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm0____def: h4/ringNorm/_20_40ind__typeringNorm0 = (\a. h4/ringNorm/_20_40ind__typeringNorm5 ((\a0. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 h4/bool/ARB) (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/ringNorm/Pconst0: h4/ringNorm/Pconst = h4/ringNorm/_20_40ind__typeringNorm1
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm2____def: h4/ringNorm/_20_40ind__typeringNorm2 = (\a0 a1. h4/ringNorm/_20_40ind__typeringNorm5 ((\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) (h4/ringNorm/_20_40ind__typeringNorm6 a0) (h4/ringNorm/_20_40ind__typeringNorm6 a1)))
% Assm: h4/ringNorm/hidden____20__40ind____typeringNorm1____def: h4/ringNorm/_20_40ind__typeringNorm1 = (\a. h4/ringNorm/_20_40ind__typeringNorm5 ((\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a0) (\n. h4/ind__type/BOTTOM)) a))
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/ringNorm/Pplus0: h4/ringNorm/Pplus = h4/ringNorm/_20_40ind__typeringNorm2
% Assm: h4/ringNorm/polynom__repfns_c1: !r. (\a0_27. !_27polynom_27. (!a0_270. (?a. a0_270 = (\a0. h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a0 h4/bool/ARB) (\n. h4/ind__type/BOTTOM)) a) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a0) (\n. h4/ind__type/BOTTOM)) a) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27polynom_27 a0 /\ _27polynom_27 a1) \/ (?a0 a1. a0_270 = (\a00 a10. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a00 (h4/ind__type/FCONS a10 (\n. h4/ind__type/BOTTOM)))) a0 a1 /\ _27polynom_27 a0 /\ _27polynom_27 a1) \/ (?a. a0_270 = (\a0. h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (\n. h4/ind__type/BOTTOM))) a /\ _27polynom_27 a) ==> _27polynom_27 a0_270) ==> _27polynom_27 a0_27) r <=> h4/ringNorm/_20_40ind__typeringNorm6 (h4/ringNorm/_20_40ind__typeringNorm5 r) = r
% Assm: h4/ringNorm/Pmult0: h4/ringNorm/Pmult = h4/ringNorm/_20_40ind__typeringNorm3
% Assm: h4/ringNorm/polynom__repfns_c0: !a. h4/ringNorm/_20_40ind__typeringNorm5 (h4/ringNorm/_20_40ind__typeringNorm6 a) = a
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/integerRing/int__ring__thms_c3: !vm i. h4/integerRing/int__interp__p vm (h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm: h4/ringNorm/polynom__normalize__def_c0: !r i. h4/ringNorm/polynom__normalize r (h4/ringNorm/Pvar i) = h4/canonical/Cons__varlist (h4/list/CONS i h4/list/NIL) h4/canonical/Nil__monom
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists P
% Assm: h4/quote/varmap__find__def_c0: !x v2 v1. h4/quote/varmap__find h4/quote/End__idx (h4/quote/Node__vm x v1 v2) = x
% Assm: h4/quote/varmap__find__def_c3: !i. h4/quote/varmap__find i h4/quote/Empty__vm = h4/min/_40 (\x. T)
% Assm: h4/quote/varmap__find__def_c2: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Left__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v1
% Assm: h4/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/ringNorm/polynom__case__def_c0: !f4 f3 f2 f1 f a. h4/ringNorm/polynom__CASE (h4/ringNorm/Pvar a) f f1 f2 f3 f4 = f a
% Assm: h4/ringNorm/polynom__distinct_c0: !a_27 a. ~(h4/ringNorm/Pvar a = h4/ringNorm/Pconst a_27)
% Assm: h4/ringNorm/polynom__distinct_c2: !a1 a0 a. ~(h4/ringNorm/Pvar a = h4/ringNorm/Pmult a0 a1)
% Assm: h4/ringNorm/polynom__distinct_c1: !a1 a0 a. ~(h4/ringNorm/Pvar a = h4/ringNorm/Pplus a0 a1)
% Assm: h4/ringNorm/polynom__nchotomy: !pp. (?i. pp = h4/ringNorm/Pvar i) \/ (?a. pp = h4/ringNorm/Pconst a) \/ (?p p0. pp = h4/ringNorm/Pplus p p0) \/ (?p p0. pp = h4/ringNorm/Pmult p p0) \/ (?p. pp = h4/ringNorm/Popp p)
% Assm: h4/ringNorm/polynom__size__def_c0: !f a. h4/ringNorm/polynom__size f (h4/ringNorm/Pvar a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/quote/index__size a)
% Assm: h4/integerRing/int__ring__thms_c51: !i. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pvar i) = h4/canonical/Cons__varlist (h4/list/CONS i h4/list/NIL) h4/canonical/Nil__monom
% Assm: h4/ringNorm/polynom__case__cong: !f4_27 f4 f3_27 f3 f2_27 f2 f1_27 f1 f_27 f M_27 M. M = M_27 /\ (!a. M_27 = h4/ringNorm/Pvar a ==> f a = f_27 a) /\ (!a. M_27 = h4/ringNorm/Pconst a ==> f1 a = f1_27 a) /\ (!a0 a1. M_27 = h4/ringNorm/Pplus a0 a1 ==> f2 a0 a1 = f2_27 a0 a1) /\ (!a0 a1. M_27 = h4/ringNorm/Pmult a0 a1 ==> f3 a0 a1 = f3_27 a0 a1) /\ (!a. M_27 = h4/ringNorm/Popp a ==> f4 a = f4_27 a) ==> h4/ringNorm/polynom__CASE M f f1 f2 f3 f4 = h4/ringNorm/polynom__CASE M_27 f_27 f1_27 f2_27 f3_27 f4_27
% Assm: h4/ringNorm/polynom__11_c0: !a_27 a. h4/ringNorm/Pvar a = h4/ringNorm/Pvar a_27 <=> a = a_27
% Assm: h4/ringNorm/polynom__Axiom: !f4 f3 f2 f1 f0. ?fn. (!a. fn (h4/ringNorm/Pvar a) = f0 a) /\ (!a. fn (h4/ringNorm/Pconst a) = f1 a) /\ (!a0 a1. fn (h4/ringNorm/Pplus a0 a1) = f2 a0 a1 (fn a0) (fn a1)) /\ (!a0 a1. fn (h4/ringNorm/Pmult a0 a1) = f3 a0 a1 (fn a0) (fn a1)) /\ (!a. fn (h4/ringNorm/Popp a) = f4 a (fn a))
% Assm: h4/ringNorm/polynom__distinct_c3: !a_27 a. ~(h4/ringNorm/Pvar a = h4/ringNorm/Popp a_27)
% Assm: h4/ringNorm/polynom__induction: !P. (!i. P (h4/ringNorm/Pvar i)) /\ (!a. P (h4/ringNorm/Pconst a)) /\ (!p p0. P p /\ P p0 ==> P (h4/ringNorm/Pplus p p0)) /\ (!p p0. P p /\ P p0 ==> P (h4/ringNorm/Pmult p p0)) /\ (!p. P p ==> P (h4/ringNorm/Popp p)) ==> (!p. P p)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/pair/FST0: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% 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/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/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/list/list__Axiom: !f1 f0. ?fn. fn h4/list/NIL = f0 /\ (!a0 a1. fn (h4/list/CONS a0 a1) = f1 a0 a1 (fn a1))
% Assm: h4/ringNorm/datatype__polynom: !polynom. h4/bool/DATATYPE (polynom h4/ringNorm/Pvar h4/ringNorm/Pconst h4/ringNorm/Pplus h4/ringNorm/Pmult h4/ringNorm/Popp)
% Assm: h4/canonical/ivl__aux__def_c1: !x_27 x vm t_27 sr. h4/canonical/ivl__aux sr vm x (h4/list/CONS x_27 t_27) = h4/semi__ring/semi__ring__SRM sr (h4/quote/varmap__find x vm) (h4/canonical/ivl__aux sr vm x_27 t_27)
% Assm: h4/semi__ring/semi__ring__accessors_c3: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm: h4/ringNorm/ivl__aux__def_c1: !x_27 x vm t_27 r. h4/ringNorm/r__ivl__aux r vm x (h4/list/CONS x_27 t_27) = h4/ring/ring__RM r (h4/quote/varmap__find x vm) (h4/ringNorm/r__ivl__aux r vm x_27 t_27)
% Assm: h4/ringNorm/ivl__aux__def_c0: !x vm r. h4/ringNorm/r__ivl__aux r vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm: h4/canonical/ivl__aux__def_c0: !x vm sr. h4/canonical/ivl__aux sr vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm: h4/ring/semi__ring__of__def: !r. h4/ring/semi__ring__of r = h4/semi__ring/semi__ring0 (h4/ring/ring__R0 r) (h4/ring/ring__R1 r) (h4/ring/ring__RP r) (h4/ring/ring__RM r)
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/integerRing/int__interp__p__def: h4/integerRing/int__interp__p = h4/ringNorm/interp__p (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm: h4/integerRing/int__r__ivl__aux__def: h4/integerRing/int__r__ivl__aux = h4/ringNorm/r__ivl__aux (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm: h4/numRing/num__ring__thms_c7: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Right__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v2
% Assm: h4/numRing/num__ring__thms_c3: !vm i. h4/numRing/num__interp__sp vm (h4/canonical/SPvar i) = h4/quote/varmap__find i vm
% Assm: h4/integerRing/int__ring__thms_c9: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Left__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v1
% Assm: h4/integerRing/int__ring__thms_c10: !i. h4/quote/varmap__find i h4/quote/Empty__vm = h4/min/_40 (\x. T)
% Assm: h4/numRing/num__ring__thms_c8: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Left__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v1
% Assm: h4/ringNorm/interp__sp__def_c1: !vm r i. h4/ringNorm/r__interp__sp r vm (h4/canonical/SPvar i) = h4/quote/varmap__find i vm
% Assm: h4/numRing/num__ring__thms_c39: !x_27 x vm t_27. h4/numRing/num__ivl__aux vm x (h4/list/CONS x_27 t_27) = h4/arithmetic/_2A (h4/quote/varmap__find x vm) (h4/numRing/num__ivl__aux vm x_27 t_27)
% Assm: h4/numRing/num__ring__thms_c9: !i. h4/quote/varmap__find i h4/quote/Empty__vm = h4/min/_40 (\x. T)
% Assm: h4/integerRing/int__ring__thms_c40: !x_27 x vm t_27. h4/integerRing/int__r__ivl__aux vm x (h4/list/CONS x_27 t_27) = h4/integer/int__mul (h4/quote/varmap__find x vm) (h4/integerRing/int__r__ivl__aux vm x_27 t_27)
% Assm: h4/integerRing/int__ring__thms_c39: !x vm. h4/integerRing/int__r__ivl__aux vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/ringNorm/r__ivl__aux__def: !r. h4/ringNorm/r__ivl__aux r = h4/canonical/ivl__aux (h4/ring/semi__ring__of r)
% Assm: h4/ring/ring__accessors_c1: !f1 f0 f a0 a. h4/ring/ring__R1 (h4/ring/ring0 a a0 f f0 f1) = a0
% Assm: h4/integerRing/int__r__canonical__sum__prod__def: h4/integerRing/int__r__canonical__sum__prod = h4/ringNorm/r__canonical__sum__prod (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm: h4/integerRing/int__r__canonical__sum__scalar3__def: h4/integerRing/int__r__canonical__sum__scalar3 = h4/ringNorm/r__canonical__sum__scalar3 (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm: h4/integerRing/int__r__canonical__sum__merge__def: h4/integerRing/int__r__canonical__sum__merge = h4/ringNorm/r__canonical__sum__merge (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm: h4/integerRing/int__polynom__normalize__def: h4/integerRing/int__polynom__normalize = h4/ringNorm/polynom__normalize (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm: h4/ringNorm/polynom__normalize__def_c4: !r p. h4/ringNorm/polynom__normalize r (h4/ringNorm/Popp p) = h4/ringNorm/r__canonical__sum__scalar3 r (h4/ring/ring__RN r (h4/ring/ring__R1 r)) h4/list/NIL (h4/ringNorm/polynom__normalize r p)
% Assm: h4/ringNorm/polynom__normalize__def_c3: !r pr pl. h4/ringNorm/polynom__normalize r (h4/ringNorm/Pmult pl pr) = h4/ringNorm/r__canonical__sum__prod r (h4/ringNorm/polynom__normalize r pl) (h4/ringNorm/polynom__normalize r pr)
% Assm: h4/ringNorm/polynom__normalize__def_c2: !r pr pl. h4/ringNorm/polynom__normalize r (h4/ringNorm/Pplus pl pr) = h4/ringNorm/r__canonical__sum__merge r (h4/ringNorm/polynom__normalize r pl) (h4/ringNorm/polynom__normalize r pr)
% Assm: h4/ringNorm/polynom__normalize__def_c1: !r c. h4/ringNorm/polynom__normalize r (h4/ringNorm/Pconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
% Assm: h4/canonical/interp__sp__def_c2: !vm sr p2 p1. h4/canonical/interp__sp sr vm (h4/canonical/SPplus p1 p2) = h4/semi__ring/semi__ring__SRP sr (h4/canonical/interp__sp sr vm p1) (h4/canonical/interp__sp sr vm p2)
% Assm: h4/canonical/interp__sp__def_c3: !vm sr p2 p1. h4/canonical/interp__sp sr vm (h4/canonical/SPmult p1 p2) = h4/semi__ring/semi__ring__SRM sr (h4/canonical/interp__sp sr vm p1) (h4/canonical/interp__sp sr vm p2)
% Assm: h4/canonical/interp__sp__def_c1: !vm sr i. h4/canonical/interp__sp sr vm (h4/canonical/SPvar i) = h4/quote/varmap__find i vm
% Assm: h4/canonical/interp__sp__def_c0: !vm sr c. h4/canonical/interp__sp sr vm (h4/canonical/SPconst c) = c
% Assm: h4/semi__ring/semi__ring__accessors_c2: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% Assm: h4/numRing/num__ivl__aux__def: h4/numRing/num__ivl__aux = h4/canonical/ivl__aux (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/ringNorm/polynom__case__def_c4: !f4 f3 f2 f1 f a. h4/ringNorm/polynom__CASE (h4/ringNorm/Popp a) f f1 f2 f3 f4 = f4 a
% Assm: h4/ringNorm/polynom__case__def_c3: !f4 f3 f2 f1 f a1 a0. h4/ringNorm/polynom__CASE (h4/ringNorm/Pmult a0 a1) f f1 f2 f3 f4 = f3 a0 a1
% Assm: h4/ringNorm/polynom__case__def_c2: !f4 f3 f2 f1 f a1 a0. h4/ringNorm/polynom__CASE (h4/ringNorm/Pplus a0 a1) f f1 f2 f3 f4 = f2 a0 a1
% Assm: h4/ringNorm/polynom__case__def_c1: !f4 f3 f2 f1 f a. h4/ringNorm/polynom__CASE (h4/ringNorm/Pconst a) f f1 f2 f3 f4 = f1 a
% Assm: h4/numRing/num__interp__sp__def: h4/numRing/num__interp__sp = h4/canonical/interp__sp (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm: h4/ringNorm/r__interp__sp__def: !r. h4/ringNorm/r__interp__sp r = h4/canonical/interp__sp (h4/ring/semi__ring__of r)
% Assm: h4/bool/EQ__EXPAND: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm: h4/bool/SELECT__AX: !x P. P x ==> P (h4/min/_40 P)
% Assm: h4/quote/varmap__find__tupled__primitive__def: h4/quote/varmap__find__tupled = h4/relation/WFREC (h4/min/_40 (\R. h4/relation/WF R /\ (!v1 x v2 i1. R (h4/pair/_2C i1 v2) (h4/pair/_2C (h4/quote/Right__idx i1) (h4/quote/Node__vm x v1 v2))) /\ (!v2 x v1 i1. R (h4/pair/_2C i1 v1) (h4/pair/_2C (h4/quote/Left__idx i1) (h4/quote/Node__vm x v1 v2))))) (\varmap__find__tupled a. h4/pair/pair__CASE a (\v3 v4. h4/quote/varmap__CASE v4 (h4/combin/I (h4/min/_40 (\x. T))) (\x v1 v2. h4/quote/index__CASE v3 (\i1_27. h4/combin/I (varmap__find__tupled (h4/pair/_2C i1_27 v1))) (\i1. h4/combin/I (varmap__find__tupled (h4/pair/_2C i1 v2))) (h4/combin/I x))))
% Assm: h4/quote/varmap__find__curried__def: !x1 x. h4/quote/varmap__find x x1 = h4/quote/varmap__find__tupled (h4/pair/_2C x x1)
% Assm: h4/quote/varmap__case__def_c1: !v f a2 a1 a0. h4/quote/varmap__CASE (h4/quote/Node__vm a0 a1 a2) v f = f a0 a1 a2
% Assm: h4/quote/index__case__def_c2: !v f1 f. h4/quote/index__CASE h4/quote/End__idx f f1 v = v
% Assm: h4/quote/index__case__def_c1: !v f1 f a. h4/quote/index__CASE (h4/quote/Right__idx a) f f1 v = f1 a
% Assm: h4/quote/index__case__def_c0: !v f1 f a. h4/quote/index__CASE (h4/quote/Left__idx a) f f1 v = f a
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/quote/index__size__def_c0: !a. h4/quote/index__size (h4/quote/Left__idx a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/quote/index__size a)
% Assm: h4/numeral/numeral__distrib_c27: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm: h4/numeral/numeral__lt_c0: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT1 n) <=> T
% Assm: h4/numeral/numeral__distrib_c21: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Goal: !vm i. h4/ratRing/rat__interp__p vm (h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
%   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_ratRings_ratu_u_interpu_u_pu_u_def]: h4/ratRing/rat__interp__p = h4/ringNorm/interp__p (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c1]: !vm p. happ (happ h4/ratRing/rat__interp__p vm) p = happ (happ h4/ratRing/rat__r__interp__cs vm) (happ h4/ratRing/rat__polynom__simplify p)
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c2]: !vm c. happ (happ h4/ratRing/rat__interp__p vm) (happ h4/ringNorm/Pconst c) = c
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c1]: !vm r i. happ (happ (h4/ringNorm/interp__p r) vm) (happ h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_ringNorms_Pvar0]: h4/ringNorm/Pvar = h4/ringNorm/_20_40ind__typeringNorm0
% Assm [h4s_ratRings_ratu_u_ru_u_interpu_u_csu_u_def]: h4/ratRing/rat__r__interp__cs = h4/ringNorm/r__interp__cs (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ringNorms_polynomu_u_simplifyu_u_ok]: !r. h4/ring/is__ring r ==> (!vm p. happ (happ (h4/ringNorm/r__interp__cs r) vm) (happ (h4/ringNorm/polynom__simplify r) p) = happ (happ (h4/ringNorm/interp__p r) vm) p)
% Assm [h4s_ratRings_RATu_u_ISu_u_RING]: h4/ring/is__ring (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ratRings_ratu_u_polynomu_u_simplifyu_u_def]: h4/ratRing/rat__polynom__simplify = h4/ringNorm/polynom__simplify (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_rings_ringu_u_accessorsu_c3]: !f1 f0 f a0 a. h4/ring/ring__RM (h4/ring/ring0 a a0 f f0 f1) = f0
% Assm [h4s_rings_ringu_u_accessorsu_c4]: !f1 f0 f a0 a. h4/ring/ring__RN (h4/ring/ring0 a a0 f f0 f1) = f1
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c2]: !vm r p2 p1. happ (happ (h4/ringNorm/interp__p r) vm) (happ (happ h4/ringNorm/Pplus p1) p2) = happ (happ (h4/ring/ring__RP r) (happ (happ (h4/ringNorm/interp__p r) vm) p1)) (happ (happ (h4/ringNorm/interp__p r) vm) p2)
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c4]: !vm r p1. happ (happ (h4/ringNorm/interp__p r) vm) (happ h4/ringNorm/Popp p1) = happ (h4/ring/ring__RN r) (happ (happ (h4/ringNorm/interp__p r) vm) p1)
% Assm [h4s_rings_ringu_u_accessorsu_c2]: !f1 f0 f a0 a. h4/ring/ring__RP (h4/ring/ring0 a a0 f f0 f1) = f
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c3]: !vm r p2 p1. happ (happ (h4/ringNorm/interp__p r) vm) (happ (happ h4/ringNorm/Pmult p1) p2) = happ (happ (h4/ring/ring__RM r) (happ (happ (h4/ringNorm/interp__p r) vm) p1)) (happ (happ (h4/ringNorm/interp__p r) vm) p2)
% Assm [h4s_ringNorms_interpu_u_pu_u_defu_c0]: !vm r c. happ (happ (h4/ringNorm/interp__p r) vm) (happ h4/ringNorm/Pconst c) = c
% Assm [h4s_quotes_varmapu_u_findu_u_defu_c1]: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Right__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v2
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_ringNorms_Popp0]: h4/ringNorm/Popp = h4/ringNorm/_20_40ind__typeringNorm4
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm3u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/ringNorm/_20_40ind__typeringNorm3 x) x' = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x') _0))))
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm4u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/ringNorm/_20_40ind__typeringNorm4 x = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x) _0)))
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm0u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/ringNorm/_20_40ind__typeringNorm0 x = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C x h4/bool/ARB) _0))
% Assm [h4s_ringNorms_Pconst0]: h4/ringNorm/Pconst = h4/ringNorm/_20_40ind__typeringNorm1
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm2u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x x'. happ (happ h4/ringNorm/_20_40ind__typeringNorm2 x) x' = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x) (h4/ind__type/FCONS (h4/ringNorm/_20_40ind__typeringNorm6 x') _0))))
% Assm [h4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm1u_u_u_u_def]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!x. happ h4/ringNorm/_20_40ind__typeringNorm1 x = h4/ringNorm/_20_40ind__typeringNorm5 (h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB x) _0))
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_ringNorms_Pplus0]: h4/ringNorm/Pplus = h4/ringNorm/_20_40ind__typeringNorm2
% Assm [h4s_ringNorms_polynomu_u_repfnsu_c1]: !_0. (!n. happ _0 n = h4/ind__type/BOTTOM) ==> (!r. (!_27polynom_27. (!a0_270. (?a. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a h4/bool/ARB) _0) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a) _0) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC h4/num/0)) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27polynom_27 a0 /\ happ _27polynom_27 a1) \/ (?a0 a1. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a0 (h4/ind__type/FCONS a1 _0)) /\ happ _27polynom_27 a0 /\ happ _27polynom_27 a1) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC (h4/num/SUC (h4/num/SUC (h4/num/SUC h4/num/0)))) (h4/pair/_2C h4/bool/ARB h4/bool/ARB) (h4/ind__type/FCONS a _0) /\ happ _27polynom_27 a) ==> happ _27polynom_27 a0_270) ==> happ _27polynom_27 r) <=> h4/ringNorm/_20_40ind__typeringNorm6 (h4/ringNorm/_20_40ind__typeringNorm5 r) = r)
% Assm [h4s_ringNorms_Pmult0]: h4/ringNorm/Pmult = h4/ringNorm/_20_40ind__typeringNorm3
% Assm [h4s_ringNorms_polynomu_u_repfnsu_c0]: !a. h4/ringNorm/_20_40ind__typeringNorm5 (h4/ringNorm/_20_40ind__typeringNorm6 a) = a
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c3]: !vm i. happ (happ h4/integerRing/int__interp__p vm) (happ h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_defu_c0]: !r i. happ (h4/ringNorm/polynom__normalize r) (happ h4/ringNorm/Pvar i) = h4/canonical/Cons__varlist (h4/list/CONS i h4/list/NIL) h4/canonical/Nil__monom
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists P
% Assm [h4s_quotes_varmapu_u_findu_u_defu_c0]: !x v2 v1. h4/quote/varmap__find h4/quote/End__idx (h4/quote/Node__vm x v1 v2) = x
% Assm [h4s_quotes_varmapu_u_findu_u_defu_c3]: !_0. (!x. happ _0 x <=> T) ==> (!i. h4/quote/varmap__find i h4/quote/Empty__vm = h4/min/_40 _0)
% Assm [h4s_quotes_varmapu_u_findu_u_defu_c2]: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Left__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v1
% Assm [h4s_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_ringNorms_polynomu_u_caseu_u_defu_c0]: !f4 f3 f2 f1 f a. h4/ringNorm/polynom__CASE (happ h4/ringNorm/Pvar a) f f1 f2 f3 f4 = happ f a
% Assm [h4s_ringNorms_polynomu_u_distinctu_c0]: !a_27 a. ~(happ h4/ringNorm/Pvar a = happ h4/ringNorm/Pconst a_27)
% Assm [h4s_ringNorms_polynomu_u_distinctu_c2]: !a1 a0 a. ~(happ h4/ringNorm/Pvar a = happ (happ h4/ringNorm/Pmult a0) a1)
% Assm [h4s_ringNorms_polynomu_u_distinctu_c1]: !a1 a0 a. ~(happ h4/ringNorm/Pvar a = happ (happ h4/ringNorm/Pplus a0) a1)
% Assm [h4s_ringNorms_polynomu_u_nchotomy]: !pp. (?i. pp = happ h4/ringNorm/Pvar i) \/ (?a. pp = happ h4/ringNorm/Pconst a) \/ (?p p0. pp = happ (happ h4/ringNorm/Pplus p) p0) \/ (?p p0. pp = happ (happ h4/ringNorm/Pmult p) p0) \/ (?p. pp = happ h4/ringNorm/Popp p)
% Assm [h4s_ringNorms_polynomu_u_sizeu_u_defu_c0]: !f a. h4/ringNorm/polynom__size f (happ h4/ringNorm/Pvar a) = happ (happ h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/quote/index__size a)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c51]: !i. happ h4/integerRing/int__polynom__normalize (happ h4/ringNorm/Pvar i) = h4/canonical/Cons__varlist (h4/list/CONS i h4/list/NIL) h4/canonical/Nil__monom
% Assm [h4s_ringNorms_polynomu_u_caseu_u_cong]: !f4_27 f4 f3_27 f3 f2_27 f2 f1_27 f1 f_27 f M_27 M. M = M_27 /\ (!a. M_27 = happ h4/ringNorm/Pvar a ==> happ f a = happ f_27 a) /\ (!a. M_27 = happ h4/ringNorm/Pconst a ==> happ f1 a = happ f1_27 a) /\ (!a0 a1. M_27 = happ (happ h4/ringNorm/Pplus a0) a1 ==> happ (happ f2 a0) a1 = happ (happ f2_27 a0) a1) /\ (!a0 a1. M_27 = happ (happ h4/ringNorm/Pmult a0) a1 ==> happ (happ f3 a0) a1 = happ (happ f3_27 a0) a1) /\ (!a. M_27 = happ h4/ringNorm/Popp a ==> happ f4 a = happ f4_27 a) ==> h4/ringNorm/polynom__CASE M f f1 f2 f3 f4 = h4/ringNorm/polynom__CASE M_27 f_27 f1_27 f2_27 f3_27 f4_27
% Assm [h4s_ringNorms_polynomu_u_11u_c0]: !a_27 a. happ h4/ringNorm/Pvar a = happ h4/ringNorm/Pvar a_27 <=> a = a_27
% Assm [h4s_ringNorms_polynomu_u_Axiom]: !f4 f3 f2 f1 f0. ?fn. (!a. happ fn (happ h4/ringNorm/Pvar a) = happ f0 a) /\ (!a. happ fn (happ h4/ringNorm/Pconst a) = happ f1 a) /\ (!a0 a1. happ fn (happ (happ h4/ringNorm/Pplus a0) a1) = happ (happ (happ (happ f2 a0) a1) (happ fn a0)) (happ fn a1)) /\ (!a0 a1. happ fn (happ (happ h4/ringNorm/Pmult a0) a1) = happ (happ (happ (happ f3 a0) a1) (happ fn a0)) (happ fn a1)) /\ (!a. happ fn (happ h4/ringNorm/Popp a) = happ (happ f4 a) (happ fn a))
% Assm [h4s_ringNorms_polynomu_u_distinctu_c3]: !a_27 a. ~(happ h4/ringNorm/Pvar a = happ h4/ringNorm/Popp a_27)
% Assm [h4s_ringNorms_polynomu_u_induction]: !P. (!i. happ P (happ h4/ringNorm/Pvar i)) /\ (!a. happ P (happ h4/ringNorm/Pconst a)) /\ (!p p0. happ P p /\ happ P p0 ==> happ P (happ (happ h4/ringNorm/Pplus p) p0)) /\ (!p p0. happ P p /\ happ P p0 ==> happ P (happ (happ h4/ringNorm/Pmult p) p0)) /\ (!p. happ P p ==> happ P (happ h4/ringNorm/Popp p)) ==> (!p. happ P p)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_pairs_FST0]: !y x. h4/pair/FST (h4/pair/_2C x y) = x
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_indu_u_types_FCONS0u_c0]: !f a. happ (h4/ind__type/FCONS a f) h4/num/0 = a
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% 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_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm [h4s_lists_listu_u_Axiom]: !f1 f0. ?fn. happ fn h4/list/NIL = f0 /\ (!a0 a1. happ fn (h4/list/CONS a0 a1) = happ (happ (happ f1 a0) a1) (happ fn a1))
% Assm [h4s_ringNorms_datatypeu_u_polynom]: !polynom. h4/bool/DATATYPE (happ (happ (happ (happ (happ polynom h4/ringNorm/Pvar) h4/ringNorm/Pconst) h4/ringNorm/Pplus) h4/ringNorm/Pmult) h4/ringNorm/Popp)
% Assm [h4s_canonicals_ivlu_u_auxu_u_defu_c1]: !x_27 x vm t_27 sr. happ (happ (happ (h4/canonical/ivl__aux sr) vm) x) (h4/list/CONS x_27 t_27) = happ (happ (h4/semi__ring/semi__ring__SRM sr) (h4/quote/varmap__find x vm)) (happ (happ (happ (h4/canonical/ivl__aux sr) vm) x_27) t_27)
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3]: !f0 f a0 a. h4/semi__ring/semi__ring__SRM (h4/semi__ring/semi__ring0 a a0 f f0) = f0
% Assm [h4s_ringNorms_ivlu_u_auxu_u_defu_c1]: !x_27 x vm t_27 r. happ (happ (happ (h4/ringNorm/r__ivl__aux r) vm) x) (h4/list/CONS x_27 t_27) = happ (happ (h4/ring/ring__RM r) (h4/quote/varmap__find x vm)) (happ (happ (happ (h4/ringNorm/r__ivl__aux r) vm) x_27) t_27)
% Assm [h4s_ringNorms_ivlu_u_auxu_u_defu_c0]: !x vm r. happ (happ (happ (h4/ringNorm/r__ivl__aux r) vm) x) h4/list/NIL = h4/quote/varmap__find x vm
% Assm [h4s_canonicals_ivlu_u_auxu_u_defu_c0]: !x vm sr. happ (happ (happ (h4/canonical/ivl__aux sr) vm) x) h4/list/NIL = h4/quote/varmap__find x vm
% Assm [h4s_rings_semiu_u_ringu_u_ofu_u_def]: !r. h4/ring/semi__ring__of r = h4/semi__ring/semi__ring0 (h4/ring/ring__R0 r) (h4/ring/ring__R1 r) (h4/ring/ring__RP r) (h4/ring/ring__RM r)
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_integerRings_intu_u_interpu_u_pu_u_def]: h4/integerRing/int__interp__p = h4/ringNorm/interp__p (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm [h4s_integerRings_intu_u_ru_u_ivlu_u_auxu_u_def]: h4/integerRing/int__r__ivl__aux = h4/ringNorm/r__ivl__aux (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm [h4s_numRings_numu_u_ringu_u_thmsu_c7]: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Right__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v2
% Assm [h4s_numRings_numu_u_ringu_u_thmsu_c3]: !vm i. happ (happ h4/numRing/num__interp__sp vm) (h4/canonical/SPvar i) = h4/quote/varmap__find i vm
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c9]: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Left__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v1
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c10]: !_0. (!x. happ _0 x <=> T) ==> (!i. h4/quote/varmap__find i h4/quote/Empty__vm = h4/min/_40 _0)
% Assm [h4s_numRings_numu_u_ringu_u_thmsu_c8]: !x v2 v1 i1. h4/quote/varmap__find (h4/quote/Left__idx i1) (h4/quote/Node__vm x v1 v2) = h4/quote/varmap__find i1 v1
% Assm [h4s_ringNorms_interpu_u_spu_u_defu_c1]: !vm r i. happ (happ (h4/ringNorm/r__interp__sp r) vm) (h4/canonical/SPvar i) = h4/quote/varmap__find i vm
% Assm [h4s_numRings_numu_u_ringu_u_thmsu_c39]: !x_27 x vm t_27. happ (happ (happ h4/numRing/num__ivl__aux vm) x) (h4/list/CONS x_27 t_27) = happ (happ h4/arithmetic/_2A (h4/quote/varmap__find x vm)) (happ (happ (happ h4/numRing/num__ivl__aux vm) x_27) t_27)
% Assm [h4s_numRings_numu_u_ringu_u_thmsu_c9]: !_0. (!x. happ _0 x <=> T) ==> (!i. h4/quote/varmap__find i h4/quote/Empty__vm = h4/min/_40 _0)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c40]: !x_27 x vm t_27. happ (happ (happ h4/integerRing/int__r__ivl__aux vm) x) (h4/list/CONS x_27 t_27) = happ (happ h4/integer/int__mul (h4/quote/varmap__find x vm)) (happ (happ (happ h4/integerRing/int__r__ivl__aux vm) x_27) t_27)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c39]: !x vm. happ (happ (happ h4/integerRing/int__r__ivl__aux vm) x) h4/list/NIL = h4/quote/varmap__find x vm
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% 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_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_ringNorms_ru_u_ivlu_u_auxu_u_def]: !r. h4/ringNorm/r__ivl__aux r = h4/canonical/ivl__aux (h4/ring/semi__ring__of r)
% Assm [h4s_rings_ringu_u_accessorsu_c1]: !f1 f0 f a0 a. h4/ring/ring__R1 (h4/ring/ring0 a a0 f f0 f1) = a0
% Assm [h4s_integerRings_intu_u_ru_u_canonicalu_u_sumu_u_produ_u_def]: h4/integerRing/int__r__canonical__sum__prod = h4/ringNorm/r__canonical__sum__prod (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm [h4s_integerRings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3u_u_def]: h4/integerRing/int__r__canonical__sum__scalar3 = h4/ringNorm/r__canonical__sum__scalar3 (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm [h4s_integerRings_intu_u_ru_u_canonicalu_u_sumu_u_mergeu_u_def]: h4/integerRing/int__r__canonical__sum__merge = h4/ringNorm/r__canonical__sum__merge (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm [h4s_integerRings_intu_u_polynomu_u_normalizeu_u_def]: h4/integerRing/int__polynom__normalize = h4/ringNorm/polynom__normalize (h4/ring/ring0 h4/integer/int__0 h4/integer/int__1 h4/integer/int__add h4/integer/int__mul h4/integer/int__neg)
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_defu_c4]: !r p. happ (h4/ringNorm/polynom__normalize r) (happ h4/ringNorm/Popp p) = happ (happ (happ (h4/ringNorm/r__canonical__sum__scalar3 r) (happ (h4/ring/ring__RN r) (h4/ring/ring__R1 r))) h4/list/NIL) (happ (h4/ringNorm/polynom__normalize r) p)
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_defu_c3]: !r pr pl. happ (h4/ringNorm/polynom__normalize r) (happ (happ h4/ringNorm/Pmult pl) pr) = happ (happ (h4/ringNorm/r__canonical__sum__prod r) (happ (h4/ringNorm/polynom__normalize r) pl)) (happ (h4/ringNorm/polynom__normalize r) pr)
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_defu_c2]: !r pr pl. happ (h4/ringNorm/polynom__normalize r) (happ (happ h4/ringNorm/Pplus pl) pr) = happ (happ (h4/ringNorm/r__canonical__sum__merge r) (happ (h4/ringNorm/polynom__normalize r) pl)) (happ (h4/ringNorm/polynom__normalize r) pr)
% Assm [h4s_ringNorms_polynomu_u_normalizeu_u_defu_c1]: !r c. happ (h4/ringNorm/polynom__normalize r) (happ h4/ringNorm/Pconst c) = h4/canonical/Cons__monom c h4/list/NIL h4/canonical/Nil__monom
% Assm [h4s_canonicals_interpu_u_spu_u_defu_c2]: !vm sr p2 p1. happ (happ (h4/canonical/interp__sp sr) vm) (h4/canonical/SPplus p1 p2) = happ (happ (h4/semi__ring/semi__ring__SRP sr) (happ (happ (h4/canonical/interp__sp sr) vm) p1)) (happ (happ (h4/canonical/interp__sp sr) vm) p2)
% Assm [h4s_canonicals_interpu_u_spu_u_defu_c3]: !vm sr p2 p1. happ (happ (h4/canonical/interp__sp sr) vm) (h4/canonical/SPmult p1 p2) = happ (happ (h4/semi__ring/semi__ring__SRM sr) (happ (happ (h4/canonical/interp__sp sr) vm) p1)) (happ (happ (h4/canonical/interp__sp sr) vm) p2)
% Assm [h4s_canonicals_interpu_u_spu_u_defu_c1]: !vm sr i. happ (happ (h4/canonical/interp__sp sr) vm) (h4/canonical/SPvar i) = h4/quote/varmap__find i vm
% Assm [h4s_canonicals_interpu_u_spu_u_defu_c0]: !vm sr c. happ (happ (h4/canonical/interp__sp sr) vm) (h4/canonical/SPconst c) = c
% Assm [h4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2]: !f0 f a0 a. h4/semi__ring/semi__ring__SRP (h4/semi__ring/semi__ring0 a a0 f f0) = f
% Assm [h4s_numRings_numu_u_ivlu_u_auxu_u_def]: h4/numRing/num__ivl__aux = h4/canonical/ivl__aux (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_ringNorms_polynomu_u_caseu_u_defu_c4]: !f4 f3 f2 f1 f a. h4/ringNorm/polynom__CASE (happ h4/ringNorm/Popp a) f f1 f2 f3 f4 = happ f4 a
% Assm [h4s_ringNorms_polynomu_u_caseu_u_defu_c3]: !f4 f3 f2 f1 f a1 a0. h4/ringNorm/polynom__CASE (happ (happ h4/ringNorm/Pmult a0) a1) f f1 f2 f3 f4 = happ (happ f3 a0) a1
% Assm [h4s_ringNorms_polynomu_u_caseu_u_defu_c2]: !f4 f3 f2 f1 f a1 a0. h4/ringNorm/polynom__CASE (happ (happ h4/ringNorm/Pplus a0) a1) f f1 f2 f3 f4 = happ (happ f2 a0) a1
% Assm [h4s_ringNorms_polynomu_u_caseu_u_defu_c1]: !f4 f3 f2 f1 f a. h4/ringNorm/polynom__CASE (happ h4/ringNorm/Pconst a) f f1 f2 f3 f4 = happ f1 a
% Assm [h4s_numRings_numu_u_interpu_u_spu_u_def]: h4/numRing/num__interp__sp = h4/canonical/interp__sp (h4/semi__ring/semi__ring0 h4/num/0 (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/arithmetic/_2B h4/arithmetic/_2A)
% Assm [h4s_ringNorms_ru_u_interpu_u_spu_u_def]: !r. h4/ringNorm/r__interp__sp r = h4/canonical/interp__sp (h4/ring/semi__ring__of r)
% Assm [h4s_bools_EQu_u_EXPAND]: !t2 t1. (t1 <=> t2) <=> t1 /\ t2 \/ ~t1 /\ ~t2
% Assm [h4s_bools_SELECTu_u_AX]: !x P. happ P x ==> happ P (h4/min/_40 P)
% Assm [h4s_quotes_varmapu_u_findu_u_tupledu_u_primitiveu_u_def]: !_9. (!varmap__find__tupled v1 i1_27. happ (happ (happ _9 varmap__find__tupled) v1) i1_27 = h4/combin/I (happ varmap__find__tupled (h4/pair/_2C i1_27 v1))) ==> (!_8. (!v3 v1 varmap__find__tupled x v2. happ (happ (happ (happ (happ _8 v3) v1) varmap__find__tupled) x) v2 = h4/quote/index__CASE v3 (happ (happ _9 varmap__find__tupled) v1) (happ (happ _9 varmap__find__tupled) v2) (h4/combin/I x)) ==> (!_7. (!v3 varmap__find__tupled x v1. happ (happ (happ (happ _7 v3) varmap__find__tupled) x) v1 = happ (happ (happ (happ _8 v3) v1) varmap__find__tupled) x) ==> (!_6. (!v3 varmap__find__tupled x. happ (happ (happ _6 v3) varmap__find__tupled) x = happ (happ (happ _7 v3) varmap__find__tupled) x) ==> (!_5. (!x. happ _5 x <=> T) ==> (!_4. (!v3 varmap__find__tupled v4. happ (happ (happ _4 v3) varmap__find__tupled) v4 = h4/quote/varmap__CASE v4 (h4/combin/I (h4/min/_40 _5)) (happ (happ _6 v3) varmap__find__tupled)) ==> (!_3. (!varmap__find__tupled v3. happ (happ _3 varmap__find__tupled) v3 = happ (happ _4 v3) varmap__find__tupled) ==> (!_2. (!varmap__find__tupled a. happ (happ _2 varmap__find__tupled) a = h4/pair/pair__CASE a (happ _3 varmap__find__tupled)) ==> (!_1. (!varmap__find__tupled. happ _1 varmap__find__tupled = happ _2 varmap__find__tupled) ==> (!_0. (!R. happ _0 R <=> h4/relation/WF R /\ (!v1 x v2 i1. happ (happ R (h4/pair/_2C i1 v2)) (h4/pair/_2C (h4/quote/Right__idx i1) (h4/quote/Node__vm x v1 v2))) /\ (!v2 x v1 i1. happ (happ R (h4/pair/_2C i1 v1)) (h4/pair/_2C (h4/quote/Left__idx i1) (h4/quote/Node__vm x v1 v2)))) ==> h4/quote/varmap__find__tupled = h4/relation/WFREC (h4/min/_40 _0) _1)))))))))
% Assm [h4s_quotes_varmapu_u_findu_u_curriedu_u_def]: !x1 x. h4/quote/varmap__find x x1 = happ h4/quote/varmap__find__tupled (h4/pair/_2C x x1)
% Assm [h4s_quotes_varmapu_u_caseu_u_defu_c1]: !v f a2 a1 a0. h4/quote/varmap__CASE (h4/quote/Node__vm a0 a1 a2) v f = happ (happ (happ f a0) a1) a2
% Assm [h4s_quotes_indexu_u_caseu_u_defu_c2]: !v f1 f. h4/quote/index__CASE h4/quote/End__idx f f1 v = v
% Assm [h4s_quotes_indexu_u_caseu_u_defu_c1]: !v f1 f a. h4/quote/index__CASE (h4/quote/Right__idx a) f f1 v = happ f1 a
% Assm [h4s_quotes_indexu_u_caseu_u_defu_c0]: !v f1 f a. h4/quote/index__CASE (h4/quote/Left__idx a) f f1 v = happ f a
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_quotes_indexu_u_sizeu_u_defu_c0]: !a. h4/quote/index__size (h4/quote/Left__idx a) = happ (happ h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/quote/index__size a)
% Assm [h4s_numerals_numeralu_u_distribu_c27]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/NUMERAL n) h4/num/0 <=> h4/arithmetic/_3C_3D n h4/arithmetic/ZERO
% Assm [h4s_numerals_numeralu_u_ltu_c0]: !n. h4/prim__rec/_3C h4/arithmetic/ZERO (h4/arithmetic/BIT1 n) <=> T
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. h4/prim__rec/_3C h4/num/0 (h4/arithmetic/NUMERAL n) <=> h4/prim__rec/_3C h4/arithmetic/ZERO n
% Goal: !vm i. happ (happ h4/ratRing/rat__interp__p vm) (happ h4/ringNorm/Pvar i) = h4/quote/varmap__find i vm
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_Q1330077,TV_Q1330073]: ![V_f, V_g]: (![V_x]: s(TV_Q1330073,happ(s(t_fun(TV_Q1330077,TV_Q1330073),V_f),s(TV_Q1330077,V_x))) = s(TV_Q1330073,happ(s(t_fun(TV_Q1330077,TV_Q1330073),V_g),s(TV_Q1330077,V_x))) => s(t_fun(TV_Q1330077,TV_Q1330073),V_f) = s(t_fun(TV_Q1330077,TV_Q1330073),V_g))).
fof(ah4s_ratRings_ratu_u_interpu_u_pu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p) = s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
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_ratRings_ratu_u_ringu_u_thmsu_c1, axiom, ![V_vm, V_p]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_interpu_u_cs),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_polynomu_u_simplify),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),V_p)))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c2, axiom, ![V_vm, V_c]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_h4s_ringNorms_polynom(t_h4s_rats_rat)),h4s_ringnorms_pconst),s(t_h4s_rats_rat,V_c))))) = s(t_h4s_rats_rat,V_c)).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c1, axiom, ![TV_u_27a]: ![V_vm, V_r, V_i]: s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_i))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(TV_u_27a),V_vm)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_ringNorms_Pvar0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar) = s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm0)).
fof(ah4s_ratRings_ratu_u_ru_u_interpu_u_csu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_ru_u_interpu_u_cs) = s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ringnorms_ru_u_interpu_u_cs(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ringNorms_polynomu_u_simplifyu_u_ok, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(TV_u_27a),V_r)))) => ![V_vm, V_p]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_cs(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_simplify(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))))) = s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))))).
fof(ah4s_ratRings_RATu_u_ISu_u_RING, axiom, p(s(t_bool,h4s_rings_isu_u_ring(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv))))))).
fof(ah4s_ratRings_ratu_u_polynomu_u_simplifyu_u_def, axiom, s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ratrings_ratu_u_polynomu_u_simplify) = s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),h4s_ringnorms_polynomu_u_simplify(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_rings_ringu_u_accessorsu_c3, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_rings_ringu_u_accessorsu_c4, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,TV_u_27a),V_f1)).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c2, axiom, ![TV_u_27a]: ![V_vm, V_r, V_p2, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))))),s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2)))))).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c4, axiom, ![TV_u_27a]: ![V_vm, V_r, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1)))))).
fof(ah4s_rings_ringu_u_accessorsu_c2, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c3, axiom, ![TV_u_27a]: ![V_vm, V_r, V_p2, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p1))))),s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p2)))))).
fof(ah4s_ringNorms_interpu_u_pu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_r, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_c))))) = s(TV_u_27a,V_c)).
fof(ah4s_quotes_varmapu_u_findu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_v2, V_v1, V_i1]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_v1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c3, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_ringNorms_Popp0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp) = s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm4)).
fof(ah4s_bools_MONOu_u_AND, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) & p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) & p(s(t_bool,V_w)))))).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm3u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_u_20u_40indu_u_typeringnorm3),s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_xi_))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))))))).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm4u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm4),s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))))).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm0u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm0),s(t_h4s_quotes_index,V_x))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_x),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))).
fof(ah4s_ringNorms_Pconst0, axiom, ![TV_u_27a]: s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst) = s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm1)).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm2u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x, V_xi_]: s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_u_20u_40indu_u_typeringnorm2),s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_xi_))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_xi_))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))))))).
fof(ah4s_ringNorms_hiddenu_u_u_u_20u_u_40indu_u_u_u_typeringNorm1u_u_u_u_def, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_x]: s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm1),s(TV_u_27a,V_x))) = s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,V_x))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))).
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_MONOu_u_OR, axiom, ![V_z, V_y, V_x, V_w]: (((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) & (p(s(t_bool,V_z)) => p(s(t_bool,V_w)))) => ((p(s(t_bool,V_x)) | p(s(t_bool,V_z))) => (p(s(t_bool,V_y)) | p(s(t_bool,V_w)))))).
fof(ah4s_ringNorms_Pplus0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus) = s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_u_20u_40indu_u_typeringnorm2)).
fof(ah4s_ringNorms_polynomu_u_repfnsu_c1, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_n]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0),s(t_h4s_nums_num,V_n))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_bottom) => ![V_r]: (![V_uu_27polynomu_27]: (![V_a0u_270]: ((?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_a),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))) | (?[V_a]: s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_indu_u_types_constr(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,V_a))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1)))))) | (?[V_a0, V_a1]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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_suc(s(t_h4s_nums_num,h4s_nums_0))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))))) & (p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0)))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a1)))))) | ?[V_a]: (s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),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_suc(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_nums_0))))))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_bools_arb),s(TV_u_27a,h4s_bools_arb))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),h4s_indu_u_types_fcons(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_uu_0))))) & p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a))))))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_270))))) => p(s(t_bool,happ(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_27polynomu_27),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r))))) <=> s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r))))) = s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_r)))).
fof(ah4s_ringNorms_Pmult0, axiom, ![TV_u_27a]: s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult) = s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_u_20u_40indu_u_typeringnorm3)).
fof(ah4s_ringNorms_polynomu_u_repfnsu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_ringNorms_polynom(TV_u_27a),h4s_ringnorms_u_20u_40indu_u_typeringnorm5(s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),h4s_ringnorms_u_20u_40indu_u_typeringnorm6(s(t_h4s_ringNorms_polynom(TV_u_27a),V_a))))) = s(t_h4s_ringNorms_polynom(TV_u_27a),V_a)).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_bools_MONOu_u_EXISTS, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (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,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) => (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) => ?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c3, axiom, ![V_vm, V_i]: s(t_h4s_integers_int,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_integers_int),t_h4s_integers_int),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_integers_int),t_fun(t_h4s_ringNorms_polynom(t_h4s_integers_int),t_h4s_integers_int)),h4s_integerrings_intu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(t_h4s_integers_int)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_i))))) = s(t_h4s_integers_int,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm)))).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_r, V_i]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_i))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_i),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
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_quotes_varmapu_u_findu_u_defu_c0, axiom, ![TV_u_27a]: ![V_x, V_v2, V_v1]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,h4s_quotes_endu_u_idx),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_v1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))))) = s(TV_u_27a,V_x)).
fof(ah4s_quotes_varmapu_u_findu_u_defu_c3, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_uu_0),s(TV_u_27a,V_x))) = s(t_bool,t) => ![V_i]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_emptyu_u_vm))) = s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_uu_0))))).
fof(ah4s_quotes_varmapu_u_findu_u_defu_c2, axiom, ![TV_u_27a]: ![V_x, V_v2, V_v1, V_i1]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_v1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i1),s(t_h4s_quotes_varmap(TV_u_27a),V_v1)))).
fof(ah4s_bools_DATATYPEu_u_TAGu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(t_bool,h4s_bools_datatype(s(TV_u_27a,V_x))) = s(t_bool,t)).
fof(ah4s_ringNorms_polynomu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_f4, V_f3, V_f2, V_f1, V_f, V_a]: s(TV_u_27b,h4s_ringnorms_polynomu_u_case(s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_a))),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f2),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f3),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_f4))) = s(TV_u_27b,happ(s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_h4s_quotes_index,V_a)))).
fof(ah4s_ringNorms_polynomu_u_distinctu_c0, axiom, ![TV_u_27a]: ![V_au_27, V_a]: ~ (s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_a))) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_au_27))))).
fof(ah4s_ringNorms_polynomu_u_distinctu_c2, axiom, ![TV_u_27a]: ![V_a1, V_a0, V_a]: ~ (s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_a))) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))))).
fof(ah4s_ringNorms_polynomu_u_distinctu_c1, axiom, ![TV_u_27a]: ![V_a1, V_a0, V_a]: ~ (s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_a))) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))))).
fof(ah4s_ringNorms_polynomu_u_nchotomy, axiom, ![TV_u_27a]: ![V_pp]: (?[V_i]: s(t_h4s_ringNorms_polynom(TV_u_27a),V_pp) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_i))) | (?[V_a]: s(t_h4s_ringNorms_polynom(TV_u_27a),V_pp) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_a))) | (?[V_p, V_p0]: s(t_h4s_ringNorms_polynom(TV_u_27a),V_pp) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p0))) | (?[V_p, V_p0]: s(t_h4s_ringNorms_polynom(TV_u_27a),V_pp) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p0))) | ?[V_p]: s(t_h4s_ringNorms_polynom(TV_u_27a),V_pp) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p)))))))).
fof(ah4s_ringNorms_polynomu_u_sizeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(t_h4s_nums_num,h4s_ringnorms_polynomu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_a))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,h4s_quotes_indexu_u_size(s(t_h4s_quotes_index,V_a)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c51, axiom, ![V_i]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),h4s_integerrings_intu_u_polynomu_u_normalize),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(t_h4s_integers_int)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_i))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_i),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_ringNorms_polynomu_u_caseu_u_cong, axiom, ![TV_u_27a,TV_u_27b]: ![V_f4u_27, V_f4, V_f3u_27, V_f3, V_f2u_27, V_f2, V_f1u_27, V_f1, V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_ringNorms_polynom(TV_u_27a),V_M) = s(t_h4s_ringNorms_polynom(TV_u_27a),V_Mu_27) & (![V_a]: (s(t_h4s_ringNorms_polynom(TV_u_27a),V_Mu_27) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_a))) => s(TV_u_27b,happ(s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_h4s_quotes_index,V_a))) = s(TV_u_27b,happ(s(t_fun(t_h4s_quotes_index,TV_u_27b),V_fu_27),s(t_h4s_quotes_index,V_a)))) & (![V_a]: (s(t_h4s_ringNorms_polynom(TV_u_27a),V_Mu_27) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_a))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_a))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1u_27),s(TV_u_27a,V_a)))) & (![V_a0, V_a1]: (s(t_h4s_ringNorms_polynom(TV_u_27a),V_Mu_27) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))) => s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f2),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))) = s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f2u_27),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1)))) & (![V_a0, V_a1]: (s(t_h4s_ringNorms_polynom(TV_u_27a),V_Mu_27) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))) => s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f3),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))) = s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f3u_27),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1)))) & ![V_a]: (s(t_h4s_ringNorms_polynom(TV_u_27a),V_Mu_27) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a))) => s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_f4),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a))) = s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_f4u_27),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a))))))))) => s(TV_u_27b,h4s_ringnorms_polynomu_u_case(s(t_h4s_ringNorms_polynom(TV_u_27a),V_M),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f2),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f3),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_f4))) = s(TV_u_27b,h4s_ringnorms_polynomu_u_case(s(t_h4s_ringNorms_polynom(TV_u_27a),V_Mu_27),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_fu_27),s(t_fun(TV_u_27a,TV_u_27b),V_f1u_27),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f2u_27),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f3u_27),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_f4u_27))))).
fof(ah4s_ringNorms_polynomu_u_11u_c0, axiom, ![TV_u_27a]: ![V_au_27, V_a]: (s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_a))) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_au_27))) <=> s(t_h4s_quotes_index,V_a) = s(t_h4s_quotes_index,V_au_27))).
fof(ah4s_ringNorms_polynomu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f4, V_f3, V_f2, V_f1, V_f0]: ?[V_fn]: (![V_a]: s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_a))))) = s(TV_u_27b,happ(s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f0),s(t_h4s_quotes_index,V_a))) & (![V_a]: s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_a))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_a))) & (![V_a0, V_a1]: s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b)))),V_f2),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))),s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))))),s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))))) & (![V_a0, V_a1]: s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b))),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(TV_u_27b,t_fun(TV_u_27b,TV_u_27b)))),V_f3),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))),s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))))),s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))))) & ![V_a]: s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(TV_u_27b,TV_u_27b)),V_f4),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a))),s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a)))))))))).
fof(ah4s_ringNorms_polynomu_u_distinctu_c3, axiom, ![TV_u_27a]: ![V_au_27, V_a]: ~ (s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_a))) = s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp),s(t_h4s_ringNorms_polynom(TV_u_27a),V_au_27))))).
fof(ah4s_ringNorms_polynomu_u_induction, axiom, ![TV_u_27a]: ![V_P]: ((![V_i]: p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_i)))))) & (![V_a]: p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_a)))))) & (![V_p, V_p0]: ((p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p)))) & p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p0))))) => p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p0))))))) & (![V_p, V_p0]: ((p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p)))) & p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p0))))) => p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p0))))))) & ![V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p)))) => p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))))))))))) => ![V_p]: p(s(t_bool,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_bool),V_P),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p)))))).
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_pairs_FST0, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_x]: s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27a,V_x)).
fof(ah4s_bools_BETAu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_y, V_f]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_y)))).
fof(ah4s_indu_u_types_FCONS0u_c0, axiom, ![TV_u_27a]: ![V_f, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,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_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_pairs_SND0, axiom, ![TV_u_27a,TV_u_27b]: ![V_y, V_x]: s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(TV_u_27b,V_y))))) = s(TV_u_27b,V_y)).
fof(ah4s_lists_listu_u_Axiom, axiom, ![TV_u_27b,TV_u_27a]: ![V_f1, V_f0]: ?[V_fn]: (s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil))) = s(TV_u_27b,V_f0) & ![V_a0, V_a1]: s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),h4s_lists_cons(s(TV_u_27a,V_a0),s(t_h4s_lists_list(TV_u_27a),V_a1))))) = s(TV_u_27b,happ(s(t_fun(TV_u_27b,TV_u_27b),happ(s(t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(TV_u_27a),t_fun(TV_u_27b,TV_u_27b))),V_f1),s(TV_u_27a,V_a0))),s(t_h4s_lists_list(TV_u_27a),V_a1))),s(TV_u_27b,happ(s(t_fun(t_h4s_lists_list(TV_u_27a),TV_u_27b),V_fn),s(t_h4s_lists_list(TV_u_27a),V_a1))))))).
fof(ah4s_ringNorms_datatypeu_u_polynom, axiom, ![TV_u_27a]: ![V_polynom]: p(s(t_bool,h4s_bools_datatype(s(t_bool,happ(s(t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),t_bool),happ(s(t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),t_bool)),happ(s(t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),t_bool))),happ(s(t_fun(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),t_bool)))),happ(s(t_fun(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),t_fun(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),t_fun(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),t_bool))))),V_polynom),s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pvar))),s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst))),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus))),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult))),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp))))))).
fof(ah4s_canonicals_ivlu_u_auxu_u_defu_c1, axiom, ![TV_u_27a]: ![V_xu_27, V_x, V_vm, V_tu_27, V_sr]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_ivlu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_quotes_index,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_xu_27),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_ivlu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_quotes_index,V_xu_27))),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27)))))).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c3, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0)).
fof(ah4s_ringNorms_ivlu_u_auxu_u_defu_c1, axiom, ![TV_u_27a]: ![V_xu_27, V_x, V_vm, V_tu_27, V_r]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_ivlu_u_aux(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_quotes_index,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_xu_27),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))))),s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_ivlu_u_aux(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_quotes_index,V_xu_27))),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27)))))).
fof(ah4s_ringNorms_ivlu_u_auxu_u_defu_c0, axiom, ![TV_u_27a]: ![V_x, V_vm, V_r]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_ivlu_u_aux(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_quotes_index,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_vm)))).
fof(ah4s_canonicals_ivlu_u_auxu_u_defu_c0, axiom, ![TV_u_27a]: ![V_x, V_vm, V_sr]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_ivlu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_quotes_index,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_vm)))).
fof(ah4s_rings_semiu_u_ringu_u_ofu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,h4s_rings_ringu_u_r0(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rm(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
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_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_integerRings_intu_u_interpu_u_pu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_integers_int),t_fun(t_h4s_ringNorms_polynom(t_h4s_integers_int),t_h4s_integers_int)),h4s_integerrings_intu_u_interpu_u_p) = s(t_fun(t_h4s_quotes_varmap(t_h4s_integers_int),t_fun(t_h4s_ringNorms_polynom(t_h4s_integers_int),t_h4s_integers_int)),h4s_ringnorms_interpu_u_p(s(t_h4s_rings_ring(t_h4s_integers_int),h4s_rings_ring0(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_fun(t_h4s_integers_int,t_h4s_integers_int),h4s_integers_intu_u_neg)))))).
fof(ah4s_integerRings_intu_u_ru_u_ivlu_u_auxu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_integers_int),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_ivlu_u_aux) = s(t_fun(t_h4s_quotes_varmap(t_h4s_integers_int),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int))),h4s_ringnorms_ru_u_ivlu_u_aux(s(t_h4s_rings_ring(t_h4s_integers_int),h4s_rings_ring0(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_fun(t_h4s_integers_int,t_h4s_integers_int),h4s_integers_intu_u_neg)))))).
fof(ah4s_numRings_numu_u_ringu_u_thmsu_c7, axiom, ![TV_u_27a]: ![V_x, V_v2, V_v1, V_i1]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_v1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2)))).
fof(ah4s_numRings_numu_u_ringu_u_thmsu_c3, axiom, ![V_vm, V_i]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_canonicals_spolynom(t_h4s_nums_num),t_h4s_nums_num),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_canonicals_spolynom(t_h4s_nums_num),t_h4s_nums_num)),h4s_numrings_numu_u_interpu_u_sp),s(t_h4s_quotes_varmap(t_h4s_nums_num),V_vm))),s(t_h4s_canonicals_spolynom(t_h4s_nums_num),h4s_canonicals_spvar(s(t_h4s_quotes_index,V_i))))) = s(t_h4s_nums_num,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(t_h4s_nums_num),V_vm)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c9, axiom, ![TV_u_27a]: ![V_x, V_v2, V_v1, V_i1]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_v1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i1),s(t_h4s_quotes_varmap(TV_u_27a),V_v1)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c10, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_uu_0),s(TV_u_27a,V_x))) = s(t_bool,t) => ![V_i]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_emptyu_u_vm))) = s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_uu_0))))).
fof(ah4s_numRings_numu_u_ringu_u_thmsu_c8, axiom, ![TV_u_27a]: ![V_x, V_v2, V_v1, V_i1]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_v1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i1),s(t_h4s_quotes_varmap(TV_u_27a),V_v1)))).
fof(ah4s_ringNorms_interpu_u_spu_u_defu_c1, axiom, ![TV_u_27a]: ![V_vm, V_r, V_i]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_sp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_spvar(s(t_h4s_quotes_index,V_i))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(TV_u_27a),V_vm)))).
fof(ah4s_numRings_numu_u_ringu_u_thmsu_c39, axiom, ![V_xu_27, V_x, V_vm, V_tu_27]: s(t_h4s_nums_num,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num)),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num))),h4s_numrings_numu_u_ivlu_u_aux),s(t_h4s_quotes_varmap(t_h4s_nums_num),V_vm))),s(t_h4s_quotes_index,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_xu_27),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a),s(t_h4s_nums_num,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(t_h4s_nums_num),V_vm))))),s(t_h4s_nums_num,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num)),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num))),h4s_numrings_numu_u_ivlu_u_aux),s(t_h4s_quotes_varmap(t_h4s_nums_num),V_vm))),s(t_h4s_quotes_index,V_xu_27))),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27)))))).
fof(ah4s_numRings_numu_u_ringu_u_thmsu_c9, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_uu_0),s(TV_u_27a,V_x))) = s(t_bool,t) => ![V_i]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_emptyu_u_vm))) = s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_uu_0))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c40, axiom, ![V_xu_27, V_x, V_vm, V_tu_27]: s(t_h4s_integers_int,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int)),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_integers_int),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_ivlu_u_aux),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm))),s(t_h4s_quotes_index,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_xu_27),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27))))) = s(t_h4s_integers_int,happ(s(t_fun(t_h4s_integers_int,t_h4s_integers_int),happ(s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_h4s_integers_int,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm))))),s(t_h4s_integers_int,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int)),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_integers_int),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_ivlu_u_aux),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm))),s(t_h4s_quotes_index,V_xu_27))),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c39, axiom, ![V_x, V_vm]: s(t_h4s_integers_int,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int)),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_integers_int),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_ivlu_u_aux),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm))),s(t_h4s_quotes_index,V_x))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(t_h4s_integers_int,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm)))).
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_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_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_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
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_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_c2, axiom, ![V_t]: ((p(s(t_bool,f)) & p(s(t_bool,V_t))) <=> p(s(t_bool,f)))).
fof(ah4s_ringNorms_ru_u_ivlu_u_auxu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_ivlu_u_aux(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_ivlu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_rings_ringu_u_accessorsu_c1, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(TV_u_27a,V_a0)).
fof(ah4s_integerRings_intu_u_ru_u_canonicalu_u_sumu_u_produ_u_def, axiom, s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_rings_ring(t_h4s_integers_int),h4s_rings_ring0(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_fun(t_h4s_integers_int,t_h4s_integers_int),h4s_integers_intu_u_neg)))))).
fof(ah4s_integerRings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3u_u_def, axiom, s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar3) = s(t_fun(t_h4s_integers_int,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_scalar3(s(t_h4s_rings_ring(t_h4s_integers_int),h4s_rings_ring0(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_fun(t_h4s_integers_int,t_h4s_integers_int),h4s_integers_intu_u_neg)))))).
fof(ah4s_integerRings_intu_u_ru_u_canonicalu_u_sumu_u_mergeu_u_def, axiom, s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(t_h4s_integers_int),h4s_rings_ring0(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_fun(t_h4s_integers_int,t_h4s_integers_int),h4s_integers_intu_u_neg)))))).
fof(ah4s_integerRings_intu_u_polynomu_u_normalizeu_u_def, axiom, s(t_fun(t_h4s_ringNorms_polynom(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),h4s_integerrings_intu_u_polynomu_u_normalize) = s(t_fun(t_h4s_ringNorms_polynom(t_h4s_integers_int),t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(t_h4s_integers_int),h4s_rings_ring0(s(t_h4s_integers_int,h4s_integers_intu_u_0),s(t_h4s_integers_int,h4s_integers_intu_u_1),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_add),s(t_fun(t_h4s_integers_int,t_fun(t_h4s_integers_int,t_h4s_integers_int)),h4s_integers_intu_u_mul),s(t_fun(t_h4s_integers_int,t_h4s_integers_int),h4s_integers_intu_u_neg)))))).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_defu_c4, axiom, ![TV_u_27a]: ![V_r, V_p]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_scalar3(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),h4s_rings_ringu_u_rn(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_p)))))).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_defu_c3, axiom, ![TV_u_27a]: ![V_r, V_pr, V_pl]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pl))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pr))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pl))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pr)))))).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_r, V_pr, V_pl]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pl))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pr))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pl))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_pr)))))).
fof(ah4s_ringNorms_polynomu_u_normalizeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_r, V_c]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),h4s_ringnorms_polynomu_u_normalize(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_c))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_canonicals_interpu_u_spu_u_defu_c2, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_p2, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_spplus(s(t_h4s_canonicals_spolynom(TV_u_27a),V_p1),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p1))))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p2)))))).
fof(ah4s_canonicals_interpu_u_spu_u_defu_c3, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_p2, V_p1]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_spmult(s(t_h4s_canonicals_spolynom(TV_u_27a),V_p1),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p2))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srm(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p1))))),s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),V_p2)))))).
fof(ah4s_canonicals_interpu_u_spu_u_defu_c1, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_i]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_spvar(s(t_h4s_quotes_index,V_i))))) = s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(TV_u_27a),V_vm)))).
fof(ah4s_canonicals_interpu_u_spu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),V_sr))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(t_h4s_canonicals_spolynom(TV_u_27a),h4s_canonicals_spconst(s(TV_u_27a,V_c))))) = s(TV_u_27a,V_c)).
fof(ah4s_semiu_u_rings_semiu_u_ringu_u_accessorsu_c2, axiom, ![TV_u_27a]: ![V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_semiu_u_rings_semiu_u_ringu_u_srp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_semiu_u_rings_semiu_u_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_numRings_numu_u_ivlu_u_auxu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num))),h4s_numrings_numu_u_ivlu_u_aux) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_quotes_index,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_h4s_nums_num))),h4s_canonicals_ivlu_u_aux(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_ringNorms_polynomu_u_caseu_u_defu_c4, axiom, ![TV_u_27b,TV_u_27a]: ![V_f4, V_f3, V_f2, V_f1, V_f, V_a]: s(TV_u_27b,h4s_ringnorms_polynomu_u_case(s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_popp),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a))),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f2),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f3),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_f4))) = s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_f4),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a)))).
fof(ah4s_ringNorms_polynomu_u_caseu_u_defu_c3, axiom, ![TV_u_27b,TV_u_27a]: ![V_f4, V_f3, V_f2, V_f1, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_ringnorms_polynomu_u_case(s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pmult),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f2),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f3),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_f4))) = s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f3),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1)))).
fof(ah4s_ringNorms_polynomu_u_caseu_u_defu_c2, axiom, ![TV_u_27b,TV_u_27a]: ![V_f4, V_f3, V_f2, V_f1, V_f, V_a1, V_a0]: s(TV_u_27b,h4s_ringnorms_polynomu_u_case(s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a)),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_ringNorms_polynom(TV_u_27a))),h4s_ringnorms_pplus),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1))),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f2),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f3),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_f4))) = s(TV_u_27b,happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f2),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1)))).
fof(ah4s_ringNorms_polynomu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_f4, V_f3, V_f2, V_f1, V_f, V_a]: s(TV_u_27b,h4s_ringnorms_polynomu_u_case(s(t_h4s_ringNorms_polynom(TV_u_27a),happ(s(t_fun(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_a))),s(t_fun(t_h4s_quotes_index,TV_u_27b),V_f),s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f2),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b)),V_f3),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),TV_u_27b),V_f4))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f1),s(TV_u_27a,V_a)))).
fof(ah4s_numRings_numu_u_interpu_u_spu_u_def, axiom, s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_canonicals_spolynom(t_h4s_nums_num),t_h4s_nums_num)),h4s_numrings_numu_u_interpu_u_sp) = s(t_fun(t_h4s_quotes_varmap(t_h4s_nums_num),t_fun(t_h4s_canonicals_spolynom(t_h4s_nums_num),t_h4s_nums_num)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(t_h4s_nums_num),h4s_semiu_u_rings_semiu_u_ring0(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2a)))))).
fof(ah4s_ringNorms_ru_u_interpu_u_spu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_ringnorms_ru_u_interpu_u_sp(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_canonicals_spolynom(TV_u_27a),TV_u_27a)),h4s_canonicals_interpu_u_sp(s(t_h4s_semiu_u_rings_semiu_u_ring(TV_u_27a),h4s_rings_semiu_u_ringu_u_of(s(t_h4s_rings_ring(TV_u_27a),V_r)))))).
fof(ah4s_bools_EQu_u_EXPAND, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) | (~ (p(s(t_bool,V_t1))) & ~ (p(s(t_bool,V_t2))))))).
fof(ah4s_bools_SELECTu_u_AX, axiom, ![TV_u_27a]: ![V_x, V_P]: (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,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_P)))))))).
fof(ah4s_quotes_varmapu_u_findu_u_tupledu_u_primitiveu_u_def, axiom, ![TV_u_27a]: ![V_uu_9]: (![V_varmapu_u_findu_u_tupled, V_v1, V_i1u_27]: s(TV_u_27a,happ(s(t_fun(t_h4s_quotes_index,TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,TV_u_27a))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(t_h4s_quotes_varmap(TV_u_27a),V_v1))),s(t_h4s_quotes_index,V_i1u_27))) = s(TV_u_27a,h4s_combins_i(s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled),s(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_i1u_27),s(t_h4s_quotes_varmap(TV_u_27a),V_v1))))))) => ![V_uu_8]: (![V_v3, V_v1, V_varmapu_u_findu_u_tupled, V_x, V_v2]: s(TV_u_27a,happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)))),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))))),V_uu_8),s(t_h4s_quotes_index,V_v3))),s(t_h4s_quotes_varmap(TV_u_27a),V_v1))),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(TV_u_27a,V_x))),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))) = s(TV_u_27a,h4s_quotes_indexu_u_case(s(t_h4s_quotes_index,V_v3),s(t_fun(t_h4s_quotes_index,TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,TV_u_27a))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(t_h4s_quotes_varmap(TV_u_27a),V_v1))),s(t_fun(t_h4s_quotes_index,TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_index,TV_u_27a))),V_uu_9),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))),s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))))) => ![V_uu_7]: (![V_v3, V_varmapu_u_findu_u_tupled, V_x, V_v1]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)))),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))))),V_uu_7),s(t_h4s_quotes_index,V_v3))),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(TV_u_27a,V_x))),s(t_h4s_quotes_varmap(TV_u_27a),V_v1))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)))),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))))),V_uu_8),s(t_h4s_quotes_index,V_v3))),s(t_h4s_quotes_varmap(TV_u_27a),V_v1))),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(TV_u_27a,V_x))) => ![V_uu_6]: (![V_v3, V_varmapu_u_findu_u_tupled, V_x]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)))),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))))),V_uu_6),s(t_h4s_quotes_index,V_v3))),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(TV_u_27a,V_x))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)))),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))))),V_uu_7),s(t_h4s_quotes_index,V_v3))),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(TV_u_27a,V_x))) => ![V_uu_5]: (![V_x]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_uu_5),s(TV_u_27a,V_x))) = s(t_bool,t) => ![V_uu_4]: (![V_v3, V_varmapu_u_findu_u_tupled, V_v4]: s(TV_u_27a,happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))),V_uu_4),s(t_h4s_quotes_index,V_v3))),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(t_h4s_quotes_varmap(TV_u_27a),V_v4))) = s(TV_u_27a,h4s_quotes_varmapu_u_case(s(t_h4s_quotes_varmap(TV_u_27a),V_v4),s(TV_u_27a,h4s_combins_i(s(TV_u_27a,h4s_mins_u_40(s(t_fun(TV_u_27a,t_bool),V_uu_5))))),s(t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)))),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))))),V_uu_6),s(t_h4s_quotes_index,V_v3))),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))))) => ![V_uu_3]: (![V_varmapu_u_findu_u_tupled, V_v3]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(t_h4s_quotes_index,V_v3))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_h4s_quotes_index,t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))),V_uu_4),s(t_h4s_quotes_index,V_v3))),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))) => ![V_uu_2]: (![V_varmapu_u_findu_u_tupled, V_a]: s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))),s(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),V_a))) = s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),V_a),s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a)),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27a))),V_uu_3),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))))) => ![V_uu_1]: (![V_varmapu_u_findu_u_tupled]: s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a)),V_uu_1),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))) = s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a)),V_uu_2),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),V_varmapu_u_findu_u_tupled))) => ![V_uu_0]: (![V_R]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_bool)),t_bool),V_uu_0),s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_bool)),V_R)))) <=> (p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_bool)),V_R)))) & (![V_v1, V_x, V_v2, V_i1]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_i1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_v1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2)))))))) & ![V_v2, V_x, V_v1, V_i1]: p(s(t_bool,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_bool)),V_R),s(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_i1),s(t_h4s_quotes_varmap(TV_u_27a),V_v1))))),s(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_i1))),s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_v1),s(t_h4s_quotes_varmap(TV_u_27a),V_v2))))))))))) => s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),h4s_quotes_varmapu_u_findu_u_tupled) = s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),h4s_relations_wfrec(s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_bool)),h4s_mins_u_40(s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),t_bool)),t_bool),V_uu_0))),s(t_fun(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a)),V_uu_1)))))))))))))).
fof(ah4s_quotes_varmapu_u_findu_u_curriedu_u_def, axiom, ![TV_u_27a]: ![V_x1, V_x]: s(TV_u_27a,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_x1))) = s(TV_u_27a,happ(s(t_fun(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),TV_u_27a),h4s_quotes_varmapu_u_findu_u_tupled),s(t_h4s_pairs_prod(t_h4s_quotes_index,t_h4s_quotes_varmap(TV_u_27a)),h4s_pairs_u_2c(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(TV_u_27a),V_x1)))))).
fof(ah4s_quotes_varmapu_u_caseu_u_defu_c1, axiom, ![TV_u_27b,TV_u_27a]: ![V_v, V_f, V_a2, V_a1, V_a0]: s(TV_u_27b,h4s_quotes_varmapu_u_case(s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_nodeu_u_vm(s(TV_u_27a,V_a0),s(t_h4s_quotes_varmap(TV_u_27a),V_a1),s(t_h4s_quotes_varmap(TV_u_27a),V_a2))),s(TV_u_27b,V_v),s(t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27b))),V_f))) = s(TV_u_27b,happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27b),happ(s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(t_h4s_quotes_varmap(TV_u_27a),TV_u_27b))),V_f),s(TV_u_27a,V_a0))),s(t_h4s_quotes_varmap(TV_u_27a),V_a1))),s(t_h4s_quotes_varmap(TV_u_27a),V_a2)))).
fof(ah4s_quotes_indexu_u_caseu_u_defu_c2, axiom, ![TV_u_27a]: ![V_v, V_f1, V_f]: s(TV_u_27a,h4s_quotes_indexu_u_case(s(t_h4s_quotes_index,h4s_quotes_endu_u_idx),s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f),s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f1),s(TV_u_27a,V_v))) = s(TV_u_27a,V_v)).
fof(ah4s_quotes_indexu_u_caseu_u_defu_c1, axiom, ![TV_u_27a]: ![V_v, V_f1, V_f, V_a]: s(TV_u_27a,h4s_quotes_indexu_u_case(s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_a))),s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f),s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f1),s(TV_u_27a,V_v))) = s(TV_u_27a,happ(s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f1),s(t_h4s_quotes_index,V_a)))).
fof(ah4s_quotes_indexu_u_caseu_u_defu_c0, axiom, ![TV_u_27a]: ![V_v, V_f1, V_f, V_a]: s(TV_u_27a,h4s_quotes_indexu_u_case(s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_a))),s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f),s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f1),s(TV_u_27a,V_v))) = s(TV_u_27a,happ(s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f),s(t_h4s_quotes_index,V_a)))).
fof(ah4s_numerals_numeralu_u_lteu_c1, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_bool,f)).
fof(ah4s_quotes_indexu_u_sizeu_u_defu_c0, axiom, ![V_a]: s(t_h4s_nums_num,h4s_quotes_indexu_u_size(s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_a))))) = s(t_h4s_nums_num,happ(s(t_fun(t_h4s_nums_num,t_h4s_nums_num),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_h4s_nums_num)),h4s_arithmetics_u_2b),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_nums_num,h4s_quotes_indexu_u_size(s(t_h4s_quotes_index,V_a)))))).
fof(ah4s_numerals_numeralu_u_distribu_c27, axiom, ![V_n]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_nums_0))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_numerals_numeralu_u_ltu_c0, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))))) = s(t_bool,t)).
fof(ah4s_numerals_numeralu_u_distribu_c21, axiom, ![V_n]: s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,V_n))))) = s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_arithmetics_zero),s(t_h4s_nums_num,V_n)))).
fof(ch4s_ratRings_ratu_u_ringu_u_thmsu_c3, conjecture, ![V_vm, V_i]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_varmap(t_h4s_rats_rat),t_fun(t_h4s_ringNorms_polynom(t_h4s_rats_rat),t_h4s_rats_rat)),h4s_ratrings_ratu_u_interpu_u_p),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm))),s(t_h4s_ringNorms_polynom(t_h4s_rats_rat),happ(s(t_fun(t_h4s_quotes_index,t_h4s_ringNorms_polynom(t_h4s_rats_rat)),h4s_ringnorms_pvar),s(t_h4s_quotes_index,V_i))))) = s(t_h4s_rats_rat,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_i),s(t_h4s_quotes_varmap(t_h4s_rats_rat),V_vm)))).
