%   ORIGINAL: h4/ringNorm/polynom__nchotomy
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/bool/TRUTH: T
% Assm: h4/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/bool/DATATYPE__TAG__THM: !x. h4/bool/DATATYPE x <=> T
% Assm: h4/ringNorm/Pvar0: h4/ringNorm/Pvar = h4/ringNorm/_20_40ind__typeringNorm0
% Assm: h4/ringNorm/Pplus0: h4/ringNorm/Pplus = h4/ringNorm/_20_40ind__typeringNorm2
% Assm: h4/ringNorm/Pmult0: h4/ringNorm/Pmult = h4/ringNorm/_20_40ind__typeringNorm3
% Assm: h4/ringNorm/Popp0: h4/ringNorm/Popp = h4/ringNorm/_20_40ind__typeringNorm4
% Assm: h4/ringNorm/Pconst0: h4/ringNorm/Pconst = h4/ringNorm/_20_40ind__typeringNorm1
% 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____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____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/polynom__repfns_c0: !a. h4/ringNorm/_20_40ind__typeringNorm5 (h4/ringNorm/_20_40ind__typeringNorm6 a) = a
% 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/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/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/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/MONO__EXISTS: !Q P. (!x. P x ==> Q x) ==> (?x. P x) ==> (?x. Q x)
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/bool/MONO__AND: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm: h4/bool/IMP__CLAUSES_c3: !t. t ==> t <=> T
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/bool/MONO__OR: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm: h4/bool/JRH__INDUCT__UTIL: !t P. (!x. x = t ==> P x) ==> $exists 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/pair/SND0: !y x. h4/pair/SND (h4/pair/_2C x y) = y
% Assm: h4/ind__type/FCONS0_c0: !f a. h4/ind__type/FCONS a f h4/num/0 = a
% Assm: h4/ind__type/CONSTR__REC: !Fn. ?f. !c i r. f (h4/ind__type/CONSTR c i r) = Fn c i r (\n. f (r n))
% Assm: h4/ind__type/FCONS0_c1: !n f a. h4/ind__type/FCONS a f (h4/num/SUC n) = f n
% Assm: h4/bool/BETA__THM: !y f. (\x. f x) y = f y
% Assm: h4/ringNorm/polynom__distinct_c5: !a1 a0 a. ~(h4/ringNorm/Pconst a = h4/ringNorm/Pmult a0 a1)
% Assm: h4/ringNorm/polynom__distinct_c4: !a1 a0 a. ~(h4/ringNorm/Pconst a = h4/ringNorm/Pplus a0 a1)
% Assm: h4/ringNorm/polynom__distinct_c6: !a_27 a. ~(h4/ringNorm/Pconst a = h4/ringNorm/Popp a_27)
% Assm: h4/ringNorm/polynom__distinct_c8: !a1 a0 a. ~(h4/ringNorm/Pplus a0 a1 = h4/ringNorm/Popp a)
% Assm: h4/ringNorm/polynom__distinct_c7: !a1_27 a1 a0_27 a0. ~(h4/ringNorm/Pplus a0 a1 = h4/ringNorm/Pmult a0_27 a1_27)
% Assm: h4/ringNorm/polynom__distinct_c9: !a1 a0 a. ~(h4/ringNorm/Pmult a0 a1 = h4/ringNorm/Popp a)
% Assm: h4/ringNorm/polynom__distinct_c1: !a1 a0 a. ~(h4/ringNorm/Pvar a = h4/ringNorm/Pplus a0 a1)
% Assm: h4/ringNorm/polynom__distinct_c2: !a1 a0 a. ~(h4/ringNorm/Pvar a = h4/ringNorm/Pmult a0 a1)
% Assm: h4/ringNorm/polynom__distinct_c3: !a_27 a. ~(h4/ringNorm/Pvar a = h4/ringNorm/Popp a_27)
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/ringNorm/polynom__distinct_c0: !a_27 a. ~(h4/ringNorm/Pvar a = h4/ringNorm/Pconst a_27)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> 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/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> 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/polynom__11_c4: !a_27 a. h4/ringNorm/Popp a = h4/ringNorm/Popp a_27 <=> a = a_27
% Assm: h4/ringNorm/polynom__11_c2: !a1_27 a1 a0_27 a0. h4/ringNorm/Pplus a0 a1 = h4/ringNorm/Pplus a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/ringNorm/polynom__11_c3: !a1_27 a1 a0_27 a0. h4/ringNorm/Pmult a0 a1 = h4/ringNorm/Pmult a0_27 a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm: h4/ringNorm/polynom__size__def_c3: !f a1 a0. h4/ringNorm/polynom__size f (h4/ringNorm/Pmult a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/ringNorm/polynom__size f a0) (h4/ringNorm/polynom__size f a1))
% Assm: h4/ringNorm/polynom__size__def_c2: !f a1 a0. h4/ringNorm/polynom__size f (h4/ringNorm/Pplus a0 a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/ringNorm/polynom__size f a0) (h4/ringNorm/polynom__size f a1))
% Assm: h4/ringNorm/polynom__size__def_c4: !f a. h4/ringNorm/polynom__size f (h4/ringNorm/Popp a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/ringNorm/polynom__size f a)
% Assm: h4/ringNorm/polynom__size__def_c1: !f a. h4/ringNorm/polynom__size f (h4/ringNorm/Pconst a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (f a)
% Assm: h4/ringNorm/polynom__11_c1: !a_27 a. h4/ringNorm/Pconst a = h4/ringNorm/Pconst a_27 <=> a = a_27
% 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/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/ringNorm/polynom__11_c0: !a_27 a. h4/ringNorm/Pvar a = h4/ringNorm/Pvar a_27 <=> a = a_27
% Assm: h4/ringNorm/polynom__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\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) rep
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% 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__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/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/ringNorm/interp__m__def_c1: !x vm t r c. h4/ringNorm/r__interp__m r vm c (h4/list/CONS x t) = h4/ring/ring__RM r c (h4/ringNorm/r__ivl__aux r vm x t)
% Assm: h4/quote/index__case__cong: !v_27 v f1_27 f1 f_27 f M_27 M. M = M_27 /\ (!a. M_27 = h4/quote/Left__idx a ==> f a = f_27 a) /\ (!a. M_27 = h4/quote/Right__idx a ==> f1 a = f1_27 a) /\ (M_27 = h4/quote/End__idx ==> v = v_27) ==> h4/quote/index__CASE M f f1 v = h4/quote/index__CASE M_27 f_27 f1_27 v_27
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/ringNorm/r__interp__m__def: !r. h4/ringNorm/r__interp__m r = h4/canonical/interp__m (h4/ring/semi__ring__of r)
% 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/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/canonical/interp__m__def_c0: !vm sr c. h4/canonical/interp__m sr vm c h4/list/NIL = c
% Assm: h4/canonical/interp__m__def_c1: !x vm t sr c. h4/canonical/interp__m sr vm c (h4/list/CONS x t) = h4/semi__ring/semi__ring__SRM sr c (h4/canonical/ivl__aux sr vm x t)
% 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/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_c0: !v f1 f a. h4/quote/index__CASE (h4/quote/Left__idx a) f f1 v = f a
% 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/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__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/varmap__case__def_c0: !v f. h4/quote/varmap__CASE h4/quote/Empty__vm v f = v
% Assm: h4/quote/index__size__def_c1: !a. h4/quote/index__size (h4/quote/Right__idx a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/quote/index__size a)
% 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__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
% 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__distrib_c1: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm: h4/numeral/numeral__lte_c1: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/NOT__LESS: !n m. ~h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D n m
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/ADD__CLAUSES_c0: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm: h4/arithmetic/ADD__SYM: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/LESS__EQ: !n m. h4/prim__rec/_3C m n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm: h4/pair/pair__case__thm: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = f x y
% Assm: h4/pair/UNCURRY__DEF: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = f x y
% Assm: h4/relation/WFREC__COROLLARY: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. f x = M (h4/relation/RESTRICT f R x) x)
% Assm: h4/prim__rec/WF__LESS: h4/relation/WF h4/prim__rec/_3C
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/relation/WF__inv__image: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm: h4/pair/PAIR: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm: h4/relation/RESTRICT__LEMMA: !z y f R. R y z ==> h4/relation/RESTRICT f R z y = f y
% Assm: h4/relation/inv__image__def: !f R. h4/relation/inv__image R f = (\x y. R (f x) (f y))
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/combin/I__THM: !x. h4/combin/I x = x
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/bool/IMP__F__EQ__F: !t. t ==> F <=> t <=> F
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Goal: !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)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_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_bools_DATATYPEu_u_TAGu_u_THM]: !x. h4/bool/DATATYPE x <=> T
% Assm [h4s_ringNorms_Pvar0]: h4/ringNorm/Pvar = h4/ringNorm/_20_40ind__typeringNorm0
% Assm [h4s_ringNorms_Pplus0]: h4/ringNorm/Pplus = h4/ringNorm/_20_40ind__typeringNorm2
% Assm [h4s_ringNorms_Pmult0]: h4/ringNorm/Pmult = h4/ringNorm/_20_40ind__typeringNorm3
% Assm [h4s_ringNorms_Popp0]: h4/ringNorm/Popp = h4/ringNorm/_20_40ind__typeringNorm4
% 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_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_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_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_polynomu_u_repfnsu_c0]: !a. h4/ringNorm/_20_40ind__typeringNorm5 (h4/ringNorm/_20_40ind__typeringNorm6 a) = a
% 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_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_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_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_MONOu_u_EXISTS]: !Q P. (!x. happ P x ==> happ Q x) ==> (?x. happ P x) ==> (?x. happ Q x)
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_bools_MONOu_u_AND]: !z y x w. (x ==> y) /\ (z ==> w) ==> x /\ z ==> y /\ w
% Assm [h4s_bools_IMPu_u_CLAUSESu_c3]: !t. t ==> t <=> T
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_bools_MONOu_u_OR]: !z y x w. (x ==> y) /\ (z ==> w) ==> x \/ z ==> y \/ w
% Assm [h4s_bools_JRHu_u_INDUCTu_u_UTIL]: !t P. (!x. x = t ==> happ P x) ==> $exists 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_pairs_SND0]: !y x. h4/pair/SND (h4/pair/_2C x y) = 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_CONSTRu_u_REC]: !_0. (!f r n. happ (happ (happ _0 f) r) n = happ f (happ r n)) ==> (!Fn. ?f. !c i r. happ f (h4/ind__type/CONSTR c i r) = happ (happ (happ (happ Fn c) i) r) (happ (happ _0 f) r))
% Assm [h4s_indu_u_types_FCONS0u_c1]: !n f a. happ (h4/ind__type/FCONS a f) (h4/num/SUC n) = happ f n
% Assm [h4s_bools_BETAu_u_THM]: !y f. happ f y = happ f y
% Assm [h4s_ringNorms_polynomu_u_distinctu_c5]: !a1 a0 a. ~(happ h4/ringNorm/Pconst a = happ (happ h4/ringNorm/Pmult a0) a1)
% Assm [h4s_ringNorms_polynomu_u_distinctu_c4]: !a1 a0 a. ~(happ h4/ringNorm/Pconst a = happ (happ h4/ringNorm/Pplus a0) a1)
% Assm [h4s_ringNorms_polynomu_u_distinctu_c6]: !a_27 a. ~(happ h4/ringNorm/Pconst a = happ h4/ringNorm/Popp a_27)
% Assm [h4s_ringNorms_polynomu_u_distinctu_c8]: !a1 a0 a. ~(happ (happ h4/ringNorm/Pplus a0) a1 = happ h4/ringNorm/Popp a)
% Assm [h4s_ringNorms_polynomu_u_distinctu_c7]: !a1_27 a1 a0_27 a0. ~(happ (happ h4/ringNorm/Pplus a0) a1 = happ (happ h4/ringNorm/Pmult a0_27) a1_27)
% Assm [h4s_ringNorms_polynomu_u_distinctu_c9]: !a1 a0 a. ~(happ (happ h4/ringNorm/Pmult a0) a1 = happ h4/ringNorm/Popp a)
% 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_distinctu_c2]: !a1 a0 a. ~(happ h4/ringNorm/Pvar a = happ (happ h4/ringNorm/Pmult a0) a1)
% Assm [h4s_ringNorms_polynomu_u_distinctu_c3]: !a_27 a. ~(happ h4/ringNorm/Pvar a = happ h4/ringNorm/Popp a_27)
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_ringNorms_polynomu_u_distinctu_c0]: !a_27 a. ~(happ h4/ringNorm/Pvar a = happ h4/ringNorm/Pconst a_27)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> 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_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> 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_polynomu_u_11u_c4]: !a_27 a. happ h4/ringNorm/Popp a = happ h4/ringNorm/Popp a_27 <=> a = a_27
% Assm [h4s_ringNorms_polynomu_u_11u_c2]: !a1_27 a1 a0_27 a0. happ (happ h4/ringNorm/Pplus a0) a1 = happ (happ h4/ringNorm/Pplus a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_ringNorms_polynomu_u_11u_c3]: !a1_27 a1 a0_27 a0. happ (happ h4/ringNorm/Pmult a0) a1 = happ (happ h4/ringNorm/Pmult a0_27) a1_27 <=> a0 = a0_27 /\ a1 = a1_27
% Assm [h4s_ringNorms_polynomu_u_sizeu_u_defu_c3]: !f a1 a0. h4/ringNorm/polynom__size f (happ (happ h4/ringNorm/Pmult a0) a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/ringNorm/polynom__size f a0) (h4/ringNorm/polynom__size f a1))
% Assm [h4s_ringNorms_polynomu_u_sizeu_u_defu_c2]: !f a1 a0. h4/ringNorm/polynom__size f (happ (happ h4/ringNorm/Pplus a0) a1) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/arithmetic/_2B (h4/ringNorm/polynom__size f a0) (h4/ringNorm/polynom__size f a1))
% Assm [h4s_ringNorms_polynomu_u_sizeu_u_defu_c4]: !f a. h4/ringNorm/polynom__size f (happ h4/ringNorm/Popp a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/ringNorm/polynom__size f a)
% Assm [h4s_ringNorms_polynomu_u_sizeu_u_defu_c1]: !f a. h4/ringNorm/polynom__size f (happ h4/ringNorm/Pconst a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (happ f a)
% Assm [h4s_ringNorms_polynomu_u_11u_c1]: !a_27 a. happ h4/ringNorm/Pconst a = happ h4/ringNorm/Pconst a_27 <=> a = a_27
% Assm [h4s_ringNorms_polynomu_u_sizeu_u_defu_c0]: !f a. h4/ringNorm/polynom__size f (happ h4/ringNorm/Pvar a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/quote/index__size a)
% 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_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_TYu_u_DEF]: !_1. (!n. happ _1 n = h4/ind__type/BOTTOM) ==> (!_0. (!a0_27. happ _0 a0_27 <=> (!_27polynom_27. (!a0_270. (?a. a0_270 = h4/ind__type/CONSTR h4/num/0 (h4/pair/_2C a h4/bool/ARB) _1) \/ (?a. a0_270 = h4/ind__type/CONSTR (h4/num/SUC h4/num/0) (h4/pair/_2C h4/bool/ARB a) _1) \/ (?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 _1)) /\ 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 _1)) /\ 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 _1) /\ happ _27polynom_27 a) ==> happ _27polynom_27 a0_270) ==> happ _27polynom_27 a0_27)) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep))
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% 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_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_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_ringNorms_interpu_u_mu_u_defu_c1]: !x vm t r c. happ (happ (happ (h4/ringNorm/r__interp__m r) vm) c) (h4/list/CONS x t) = happ (happ (h4/ring/ring__RM r) c) (happ (happ (happ (h4/ringNorm/r__ivl__aux r) vm) x) t)
% Assm [h4s_quotes_indexu_u_caseu_u_cong]: !v_27 v f1_27 f1 f_27 f M_27 M. M = M_27 /\ (!a. M_27 = h4/quote/Left__idx a ==> happ f a = happ f_27 a) /\ (!a. M_27 = h4/quote/Right__idx a ==> happ f1 a = happ f1_27 a) /\ (M_27 = h4/quote/End__idx ==> v = v_27) ==> h4/quote/index__CASE M f f1 v = h4/quote/index__CASE M_27 f_27 f1_27 v_27
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_ringNorms_ru_u_interpu_u_mu_u_def]: !r. h4/ringNorm/r__interp__m r = h4/canonical/interp__m (h4/ring/semi__ring__of r)
% 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_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_canonicals_interpu_u_mu_u_defu_c0]: !vm sr c. happ (happ (happ (h4/canonical/interp__m sr) vm) c) h4/list/NIL = c
% Assm [h4s_canonicals_interpu_u_mu_u_defu_c1]: !x vm t sr c. happ (happ (happ (h4/canonical/interp__m sr) vm) c) (h4/list/CONS x t) = happ (happ (h4/semi__ring/semi__ring__SRM sr) c) (happ (happ (happ (h4/canonical/ivl__aux sr) vm) x) t)
% 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_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_c0]: !v f1 f a. h4/quote/index__CASE (h4/quote/Left__idx a) f f1 v = happ f a
% 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_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_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_varmapu_u_caseu_u_defu_c0]: !v f. h4/quote/varmap__CASE h4/quote/Empty__vm v f = v
% Assm [h4s_quotes_indexu_u_sizeu_u_defu_c1]: !a. h4/quote/index__size (h4/quote/Right__idx a) = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) (h4/quote/index__size a)
% Assm [h4s_quotes_indexu_u_sizeu_u_defu_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 [h4s_numerals_numeralu_u_ltu_c0]: !n. happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) (h4/arithmetic/BIT1 n) <=> T
% Assm [h4s_numerals_numeralu_u_distribu_c21]: !n. happ (happ h4/prim__rec/_3C h4/num/0) (h4/arithmetic/NUMERAL n) <=> happ (happ h4/prim__rec/_3C h4/arithmetic/ZERO) n
% 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_distribu_c1]: !n. h4/arithmetic/_2B n h4/num/0 = n
% Assm [h4s_numerals_numeralu_u_lteu_c1]: !n. h4/arithmetic/_3C_3D (h4/arithmetic/BIT1 n) h4/arithmetic/ZERO <=> F
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_NOTu_u_LESS]: !n m. ~happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D n m
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_ADDu_u_CLAUSESu_c0]: !m. h4/arithmetic/_2B h4/num/0 m = m
% Assm [h4s_arithmetics_ADDu_u_SYM]: !n m. h4/arithmetic/_2B m n = h4/arithmetic/_2B n m
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_LESSu_u_EQ]: !n m. happ (happ h4/prim__rec/_3C m) n <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n
% Assm [h4s_pairs_pairu_u_caseu_u_thm]: !y x f. h4/pair/pair__CASE (h4/pair/_2C x y) f = happ (happ f x) y
% Assm [h4s_pairs_UNCURRYu_u_DEF]: !y x f. h4/pair/UNCURRY f (h4/pair/_2C x y) = happ (happ f x) y
% Assm [h4s_relations_WFRECu_u_COROLLARY]: !f R M. f = h4/relation/WFREC R M ==> h4/relation/WF R ==> (!x. happ f x = happ (happ M (h4/relation/RESTRICT f R x)) x)
% Assm [h4s_primu_u_recs_WFu_u_LESS]: h4/relation/WF h4/prim__rec/_3C
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_relations_WFu_u_invu_u_image]: !f R. h4/relation/WF R ==> h4/relation/WF (h4/relation/inv__image R f)
% Assm [h4s_pairs_PAIR]: !x. h4/pair/_2C (h4/pair/FST x) (h4/pair/SND x) = x
% Assm [h4s_relations_RESTRICTu_u_LEMMA]: !z y f R. happ (happ R y) z ==> happ (h4/relation/RESTRICT f R z) y = happ f y
% Assm [h4s_relations_invu_u_imageu_u_def]: !f R x x'. happ (happ (h4/relation/inv__image R f) x) x' <=> happ (happ R (happ f x)) (happ f x')
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_combins_Iu_u_THM]: !x. h4/combin/I x = x
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_bools_IMPu_u_Fu_u_EQu_u_F]: !t. t ==> F <=> t <=> F
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Goal: !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)
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_Q1384104,TV_Q1384100]: ![V_f, V_g]: (![V_x]: s(TV_Q1384100,happ(s(t_fun(TV_Q1384104,TV_Q1384100),V_f),s(TV_Q1384104,V_x))) = s(TV_Q1384100,happ(s(t_fun(TV_Q1384104,TV_Q1384100),V_g),s(TV_Q1384104,V_x))) => s(t_fun(TV_Q1384104,TV_Q1384100),V_f) = s(t_fun(TV_Q1384104,TV_Q1384100),V_g))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_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_CONSTRu_u_REC, axiom, ![TV_u_27b,TV_u_27a]: ![V_uu_0]: (![V_f, V_r, V_n]: s(TV_u_27b,happ(s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_h4s_nums_num,V_n))) = s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r),s(t_h4s_nums_num,V_n))))) => ![V_Fn]: ?[V_f]: ![V_c, V_i, V_r]: s(TV_u_27b,happ(s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f),s(t_h4s_indu_u_types_recspace(TV_u_27a),h4s_indu_u_types_constr(s(t_h4s_nums_num,V_c),s(TV_u_27a,V_i),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))) = s(TV_u_27b,happ(s(t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b))),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_fun(t_h4s_nums_num,TV_u_27b),TV_u_27b)))),V_Fn),s(t_h4s_nums_num,V_c))),s(TV_u_27a,V_i))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))),s(t_fun(t_h4s_nums_num,TV_u_27b),happ(s(t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b)),happ(s(t_fun(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),t_fun(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),t_fun(t_h4s_nums_num,TV_u_27b))),V_uu_0),s(t_fun(t_h4s_indu_u_types_recspace(TV_u_27a),TV_u_27b),V_f))),s(t_fun(t_h4s_nums_num,t_h4s_indu_u_types_recspace(TV_u_27a)),V_r))))))).
fof(ah4s_indu_u_types_FCONS0u_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_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_ringNorms_polynomu_u_distinctu_c5, axiom, ![TV_u_27a]: ![V_a1, V_a0, 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_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_c4, axiom, ![TV_u_27a]: ![V_a1, V_a0, 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_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_distinctu_c6, axiom, ![TV_u_27a]: ![V_au_27, 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_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_distinctu_c8, axiom, ![TV_u_27a]: ![V_a1, V_a0, 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))) = 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))))).
fof(ah4s_ringNorms_polynomu_u_distinctu_c7, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: ~ (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_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_a0u_27))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1u_27))))).
fof(ah4s_ringNorms_polynomu_u_distinctu_c9, axiom, ![TV_u_27a]: ![V_a1, V_a0, 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))) = 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))))).
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_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_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_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_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_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_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_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_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,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_polynomu_u_11u_c4, axiom, ![TV_u_27a]: ![V_au_27, 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_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))) <=> s(t_h4s_ringNorms_polynom(TV_u_27a),V_a) = s(t_h4s_ringNorms_polynom(TV_u_27a),V_au_27))).
fof(ah4s_ringNorms_polynomu_u_11u_c2, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (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_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_a0u_27))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1u_27))) <=> (s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0) = s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0u_27) & s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1) = s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1u_27)))).
fof(ah4s_ringNorms_polynomu_u_11u_c3, axiom, ![TV_u_27a]: ![V_a1u_27, V_a1, V_a0u_27, V_a0]: (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_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_a0u_27))),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1u_27))) <=> (s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0) = s(t_h4s_ringNorms_polynom(TV_u_27a),V_a0u_27) & s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1) = s(t_h4s_ringNorms_polynom(TV_u_27a),V_a1u_27)))).
fof(ah4s_ringNorms_polynomu_u_sizeu_u_defu_c3, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: 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_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_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_arithmetics_u_2b(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),V_a0))),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),V_a1)))))))).
fof(ah4s_ringNorms_polynomu_u_sizeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_f, V_a1, V_a0]: 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_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_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_arithmetics_u_2b(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),V_a0))),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),V_a1)))))))).
fof(ah4s_ringNorms_polynomu_u_sizeu_u_defu_c4, 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_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_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_ringnorms_polynomu_u_size(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(t_h4s_ringNorms_polynom(TV_u_27a),V_a)))))).
fof(ah4s_ringNorms_polynomu_u_sizeu_u_defu_c1, 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(TV_u_27a,t_h4s_ringNorms_polynom(TV_u_27a)),h4s_ringnorms_pconst),s(TV_u_27a,V_a))))) = s(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,happ(s(t_fun(TV_u_27a,t_h4s_nums_num),V_f),s(TV_u_27a,V_a)))))).
fof(ah4s_ringNorms_polynomu_u_11u_c1, axiom, ![TV_u_27a]: ![V_au_27, 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_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))) <=> s(TV_u_27a,V_a) = s(TV_u_27a,V_au_27))).
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,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_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_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_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_1]: (![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_1),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_uu_0]: (![V_a0u_27]: (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_0),s(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),V_a0u_27)))) <=> ![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_1))) | (?[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_1))) | (?[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_1))))))) & (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_1))))))) & (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_1))))) & 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_a0u_27)))))) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a)),t_bool),V_uu_0),s(t_fun(t_h4s_ringNorms_polynom(TV_u_27a),t_h4s_indu_u_types_recspace(t_h4s_pairs_prod(t_h4s_quotes_index,TV_u_27a))),V_rep))))))).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
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_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_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_ringNorms_interpu_u_mu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_vm, V_t, V_r, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(TV_u_27a,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(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_interpu_u_m(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_quotes_varmap(TV_u_27a),V_vm))),s(TV_u_27a,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_t))))) = 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,V_c))),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),V_t)))))).
fof(ah4s_quotes_indexu_u_caseu_u_cong, axiom, ![TV_u_27a]: ![V_vu_27, V_v, V_f1u_27, V_f1, V_fu_27, V_f, V_Mu_27, V_M]: ((s(t_h4s_quotes_index,V_M) = s(t_h4s_quotes_index,V_Mu_27) & (![V_a]: (s(t_h4s_quotes_index,V_Mu_27) = s(t_h4s_quotes_index,h4s_quotes_leftu_u_idx(s(t_h4s_quotes_index,V_a))) => s(TV_u_27a,happ(s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f),s(t_h4s_quotes_index,V_a))) = s(TV_u_27a,happ(s(t_fun(t_h4s_quotes_index,TV_u_27a),V_fu_27),s(t_h4s_quotes_index,V_a)))) & (![V_a]: (s(t_h4s_quotes_index,V_Mu_27) = s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_a))) => s(TV_u_27a,happ(s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f1),s(t_h4s_quotes_index,V_a))) = s(TV_u_27a,happ(s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f1u_27),s(t_h4s_quotes_index,V_a)))) & (s(t_h4s_quotes_index,V_Mu_27) = s(t_h4s_quotes_index,h4s_quotes_endu_u_idx) => s(TV_u_27a,V_v) = s(TV_u_27a,V_vu_27))))) => s(TV_u_27a,h4s_quotes_indexu_u_case(s(t_h4s_quotes_index,V_M),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,h4s_quotes_indexu_u_case(s(t_h4s_quotes_index,V_Mu_27),s(t_fun(t_h4s_quotes_index,TV_u_27a),V_fu_27),s(t_fun(t_h4s_quotes_index,TV_u_27a),V_f1u_27),s(TV_u_27a,V_vu_27))))).
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_ringNorms_ru_u_interpu_u_mu_u_def, axiom, ![TV_u_27a]: ![V_r]: s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_ringnorms_ru_u_interpu_u_m(s(t_h4s_rings_ring(TV_u_27a),V_r))) = s(t_fun(t_h4s_quotes_varmap(TV_u_27a),t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_interpu_u_m(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_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_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_canonicals_interpu_u_mu_u_defu_c0, axiom, ![TV_u_27a]: ![V_vm, V_sr, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(TV_u_27a,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(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_interpu_u_m(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(TV_u_27a,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(TV_u_27a,V_c)).
fof(ah4s_canonicals_interpu_u_mu_u_defu_c1, axiom, ![TV_u_27a]: ![V_x, V_vm, V_t, V_sr, V_c]: s(TV_u_27a,happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a),happ(s(t_fun(TV_u_27a,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(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),TV_u_27a))),h4s_canonicals_interpu_u_m(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(TV_u_27a,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_t))))) = 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,V_c))),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),V_t)))))).
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_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_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_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_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_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_varmapu_u_caseu_u_defu_c0, axiom, ![TV_u_27a,TV_u_27b]: ![V_v, V_f]: s(TV_u_27b,h4s_quotes_varmapu_u_case(s(t_h4s_quotes_varmap(TV_u_27a),h4s_quotes_emptyu_u_vm),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,V_v)).
fof(ah4s_quotes_indexu_u_sizeu_u_defu_c1, axiom, ![V_a]: s(t_h4s_nums_num,h4s_quotes_indexu_u_size(s(t_h4s_quotes_index,h4s_quotes_rightu_u_idx(s(t_h4s_quotes_index,V_a))))) = s(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_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,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_ltu_c0, axiom, ![V_n]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,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,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,h4s_arithmetics_zero))),s(t_h4s_nums_num,V_n)))).
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_distribu_c1, axiom, ![V_n]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,h4s_nums_0))) = s(t_h4s_nums_num,V_n)).
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_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(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,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_NOTu_u_LESS, axiom, ![V_n, V_m]: (~ (p(s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_ADDu_u_CLAUSESu_c0, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_ADDu_u_SYM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(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,V_n)))).
fof(ah4s_arithmetics_LESSu_u_EQ, axiom, ![V_n, V_m]: s(t_bool,happ(s(t_fun(t_h4s_nums_num,t_bool),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c),s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_pairs_pairu_u_caseu_u_thm, axiom, ![TV_u_27a,TV_u_27b,TV_u_27c]: ![V_y, V_x, V_f]: s(TV_u_27a,h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(TV_u_27b,TV_u_27c),h4s_pairs_u_2c(s(TV_u_27b,V_x),s(TV_u_27c,V_y))),s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f))) = s(TV_u_27a,happ(s(t_fun(TV_u_27c,TV_u_27a),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27c,TV_u_27a)),V_f),s(TV_u_27b,V_x))),s(TV_u_27c,V_y)))).
fof(ah4s_pairs_UNCURRYu_u_DEF, axiom, ![TV_u_27c,TV_u_27a,TV_u_27b]: ![V_y, V_x, V_f]: s(TV_u_27c,h4s_pairs_uncurry(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),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_27c,happ(s(t_fun(TV_u_27b,TV_u_27c),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27b,TV_u_27c)),V_f),s(TV_u_27a,V_x))),s(TV_u_27b,V_y)))).
fof(ah4s_relations_WFRECu_u_COROLLARY, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_M]: (s(t_fun(TV_u_27a,TV_u_27b),V_f) = s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_wfrec(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M))) => (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R)))) => ![V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),happ(s(t_fun(t_fun(TV_u_27a,TV_u_27b),t_fun(TV_u_27a,TV_u_27b)),V_M),s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_x))))),s(TV_u_27a,V_x)))))).
fof(ah4s_primu_u_recs_WFu_u_LESS, axiom, p(s(t_bool,h4s_relations_wf(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_nums_num,t_bool)),h4s_primu_u_recs_u_3c))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_relations_WFu_u_invu_u_image, axiom, ![TV_u_27a,TV_u_27b]: ![V_f, V_R]: (p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R)))) => p(s(t_bool,h4s_relations_wf(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f)))))))).
fof(ah4s_pairs_PAIR, axiom, ![TV_u_27a,TV_u_27b]: ![V_x]: s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),h4s_pairs_u_2c(s(TV_u_27a,h4s_pairs_fst(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))),s(TV_u_27b,h4s_pairs_snd(s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x))))) = s(t_h4s_pairs_prod(TV_u_27a,TV_u_27b),V_x)).
fof(ah4s_relations_RESTRICTu_u_LEMMA, axiom, ![TV_u_27b,TV_u_27a]: ![V_z, V_y, V_f, V_R]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_y))),s(TV_u_27a,V_z)))) => s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),h4s_relations_restrict(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),V_R),s(TV_u_27a,V_z))),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_relations_invu_u_imageu_u_def, axiom, ![TV_u_27b,TV_u_27a]: ![V_f, V_R, V_x, V_xi_]: s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,t_bool)),h4s_relations_invu_u_image(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(t_fun(TV_u_27a,TV_u_27b),V_f))),s(TV_u_27a,V_x))),s(TV_u_27a,V_xi_))) = s(t_bool,happ(s(t_fun(TV_u_27b,t_bool),happ(s(t_fun(TV_u_27b,t_fun(TV_u_27b,t_bool)),V_R),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_x))))),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_f),s(TV_u_27a,V_xi_)))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_combins_Iu_u_THM, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_combins_i(s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_bools_IMPu_u_Fu_u_EQu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ch4s_ringNorms_polynomu_u_nchotomy, conjecture, ![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)))))))).
