%   ORIGINAL: h4/ratRing/rat__ring__thms_c24
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/ratRing/rat__r__canonical__sum__scalar__def: h4/ratRing/rat__r__canonical__sum__scalar = h4/ringNorm/r__canonical__sum__scalar (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/ring/ring__accessors_c2: !f1 f0 f a0 a. h4/ring/ring__RP (h4/ring/ring0 a a0 f f0 f1) = f
% Assm: h4/ring/ring__accessors_c1: !f1 f0 f a0 a. h4/ring/ring__R1 (h4/ring/ring0 a a0 f f0 f1) = a0
% Assm: h4/ringNorm/canonical__sum__merge__def_c1: !t2 t1 r l2 l1 c1. h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__monom c1 l1 t1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/ringNorm/r__canonical__sum__merge r t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/ring/ring__RP r c1 (h4/ring/ring__R1 r)) l1 (h4/ringNorm/r__canonical__sum__merge r t1 t2)) (h4/canonical/Cons__varlist l2 (h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm: h4/ratRing/rat__r__canonical__sum__merge__def: h4/ratRing/rat__r__canonical__sum__merge = h4/ringNorm/r__canonical__sum__merge (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ringNorm/canonical__sum__merge__def_c6: !v8 v7 r. h4/ringNorm/r__canonical__sum__merge r h4/canonical/Nil__monom (h4/canonical/Cons__varlist v7 v8) = h4/canonical/Cons__varlist v7 v8
% Assm: h4/ringNorm/canonical__sum__merge__def_c4: !s1 r. h4/ringNorm/r__canonical__sum__merge r s1 h4/canonical/Nil__monom = s1
% Assm: h4/ringNorm/canonical__sum__merge__def_c0: !t2 t1 r l2 l1 c2 c1. h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__monom c1 l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/ringNorm/r__canonical__sum__merge r t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/ring/ring__RP r c1 c2) l1 (h4/ringNorm/r__canonical__sum__merge r t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm: h4/ringNorm/canonical__sum__merge__def_c3: !t2 t1 r l2 l1. h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__varlist l1 t1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/ringNorm/r__canonical__sum__merge r t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/ring/ring__RP r (h4/ring/ring__R1 r) (h4/ring/ring__R1 r)) l1 (h4/ringNorm/r__canonical__sum__merge r t1 t2)) (h4/canonical/Cons__varlist l2 (h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__varlist l1 t1) t2))
% Assm: h4/ringNorm/canonical__sum__merge__def_c5: !v6 v5 v4 r. h4/ringNorm/r__canonical__sum__merge r h4/canonical/Nil__monom (h4/canonical/Cons__monom v4 v5 v6) = h4/canonical/Cons__monom v4 v5 v6
% Assm: h4/ringNorm/canonical__sum__merge__def_c2: !t2 t1 r l2 l1 c2. h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__varlist l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/ringNorm/r__canonical__sum__merge r t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/ring/ring__RP r (h4/ring/ring__R1 r) c2) l1 (h4/ringNorm/r__canonical__sum__merge r t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/ringNorm/r__canonical__sum__merge r (h4/canonical/Cons__varlist l1 t1) t2))
% Assm: h4/ratRing/rat__ring__thms_c21: !t2 l2 l1 c2. h4/ratRing/rat__r__varlist__insert l1 (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (h4/ratRing/rat__r__varlist__insert l1 t2))
% Assm: h4/ratRing/rat__ring__thms_c16: !v6 v5 v4. h4/ratRing/rat__r__canonical__sum__merge h4/canonical/Nil__monom (h4/canonical/Cons__monom v4 v5 v6) = h4/canonical/Cons__monom v4 v5 v6
% Assm: h4/ratRing/rat__ring__thms_c18: !t2 l2 l1 c2 c1. h4/ratRing/rat__r__monom__insert c1 l1 (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (h4/rat/rat__add c1 c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (h4/ratRing/rat__r__monom__insert c1 l1 t2))
% Assm: h4/ratRing/rat__ring__thms_c12: !t2 t1 l2 l1 c1. h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/ratRing/rat__r__canonical__sum__merge t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/rat/rat__add c1 (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 (h4/ratRing/rat__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__varlist l2 (h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm: h4/ratRing/rat__ring__thms_c11: !t2 t1 l2 l1 c2 c1. h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/ratRing/rat__r__canonical__sum__merge t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/rat/rat__add c1 c2) l1 (h4/ratRing/rat__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1) t2))
% Assm: h4/ratRing/rat__ring__thms_c13: !t2 t1 l2 l1 c2. h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/ratRing/rat__r__canonical__sum__merge t1 (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) c2) l1 (h4/ratRing/rat__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__monom c2 l2 (h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) t2))
% Assm: h4/ringNorm/monom__insert__def_c0: !t2 r l2 l1 c2 c1. h4/ringNorm/r__monom__insert r c1 l1 (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (h4/ring/ring__RP r c1 c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (h4/ringNorm/r__monom__insert r c1 l1 t2))
% Assm: h4/ratRing/rat__r__monom__insert__def: h4/ratRing/rat__r__monom__insert = h4/ringNorm/r__monom__insert (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ringNorm/monom__insert__def_c1: !t2 r l2 l1 c1. h4/ringNorm/r__monom__insert r c1 l1 (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (h4/ring/ring__RP r c1 (h4/ring/ring__R1 r)) l1 t2) (h4/canonical/Cons__varlist l2 (h4/ringNorm/r__monom__insert r c1 l1 t2))
% Assm: h4/ringNorm/monom__insert__def_c2: !r l1 c1. h4/ringNorm/r__monom__insert r c1 l1 h4/canonical/Nil__monom = h4/canonical/Cons__monom c1 l1 h4/canonical/Nil__monom
% Assm: h4/ratRing/rat__r__varlist__insert__def: h4/ratRing/rat__r__varlist__insert = h4/ringNorm/r__varlist__insert (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm: h4/ringNorm/varlist__insert__def_c1: !t2 r l2 l1. h4/ringNorm/r__varlist__insert r l1 (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (h4/ring/ring__RP r (h4/ring/ring__R1 r) (h4/ring/ring__R1 r)) l1 t2) (h4/canonical/Cons__varlist l2 (h4/ringNorm/r__varlist__insert r l1 t2))
% Assm: h4/ringNorm/varlist__insert__def_c0: !t2 r l2 l1 c2. h4/ringNorm/r__varlist__insert r l1 (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (h4/ring/ring__RP r (h4/ring/ring__R1 r) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (h4/ringNorm/r__varlist__insert r l1 t2))
% Assm: h4/ringNorm/varlist__insert__def_c2: !r l1. h4/ringNorm/r__varlist__insert r l1 h4/canonical/Nil__monom = h4/canonical/Cons__varlist l1 h4/canonical/Nil__monom
% Assm: h4/ratRing/rat__ring__thms_c19: !t2 l2 l1 c1. h4/ratRing/rat__r__monom__insert c1 l1 (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (h4/rat/rat__add c1 (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 t2) (h4/canonical/Cons__varlist l2 (h4/ratRing/rat__r__monom__insert c1 l1 t2))
% Assm: h4/ratRing/rat__ring__thms_c22: !t2 l2 l1. h4/ratRing/rat__r__varlist__insert l1 (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 t2) (h4/canonical/Cons__varlist l2 (h4/ratRing/rat__r__varlist__insert l1 t2))
% Assm: h4/ratRing/rat__ring__thms_c14: !t2 t1 l2 l1. h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/ratRing/rat__r__canonical__sum__merge t1 (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 (h4/ratRing/rat__r__canonical__sum__merge t1 t2)) (h4/canonical/Cons__varlist l2 (h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1) t2))
% Assm: h4/ratRing/rat__ring__thms_c17: !v8 v7. h4/ratRing/rat__r__canonical__sum__merge h4/canonical/Nil__monom (h4/canonical/Cons__varlist v7 v8) = h4/canonical/Cons__varlist v7 v8
% Assm: h4/ratRing/rat__ring__thms_c15: !s1. h4/ratRing/rat__r__canonical__sum__merge s1 h4/canonical/Nil__monom = s1
% Assm: h4/rat/RAT__LES__LMUL__POS: !r3 r2 r1. h4/rat/rat__les (h4/rat/rat__of__num h4/num/0) r3 ==> (h4/rat/rat__les (h4/rat/rat__mul r3 r1) (h4/rat/rat__mul r3 r2) <=> h4/rat/rat__les r1 r2)
% Assm: h4/rat/RAT__MUL__SIGN__CASES_c1: !q p. h4/rat/rat__les (h4/rat/rat__mul p q) (h4/rat/rat__of__num h4/num/0) <=> h4/rat/rat__les (h4/rat/rat__of__num h4/num/0) p /\ h4/rat/rat__les q (h4/rat/rat__of__num h4/num/0) \/ h4/rat/rat__les p (h4/rat/rat__of__num h4/num/0) /\ h4/rat/rat__les (h4/rat/rat__of__num h4/num/0) q
% Assm: h4/rat/RAT__LDIV__LES__POS: !r3 r2 r1. h4/rat/rat__les (h4/rat/rat__of__num h4/num/0) r2 ==> (h4/rat/rat__les (h4/rat/rat__div r1 r2) r3 <=> h4/rat/rat__les r1 (h4/rat/rat__mul r2 r3))
% Assm: h4/rat/RAT__LES__LMUL__NEG: !r3 r2 r1. h4/rat/rat__les r3 (h4/rat/rat__of__num h4/num/0) ==> (h4/rat/rat__les (h4/rat/rat__mul r3 r2) (h4/rat/rat__mul r3 r1) <=> h4/rat/rat__les r1 r2)
% Assm: h4/rat/RAT__RDIV__EQ: !r3 r2 r1. ~(r3 = h4/rat/rat__of__num h4/num/0) ==> (r1 = h4/rat/rat__div r2 r3 <=> h4/rat/rat__mul r1 r3 = r2)
% Assm: h4/rat/RAT__RDISTRIB: !c b a. h4/rat/rat__mul (h4/rat/rat__add a b) c = h4/rat/rat__add (h4/rat/rat__mul a c) (h4/rat/rat__mul b c)
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/TRUTH: T
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% 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__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/rat/RAT__MUL__COMM: !b a. h4/rat/rat__mul a b = h4/rat/rat__mul b a
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/NOT__FORALL__THM: !P. ~(!x. P x) <=> (?x. ~P x)
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/rat/RAT__LES__RMUL__NEG: !r3 r2 r1. h4/rat/rat__les r3 (h4/rat/rat__of__num h4/num/0) ==> (h4/rat/rat__les (h4/rat/rat__mul r2 r3) (h4/rat/rat__mul r1 r3) <=> h4/rat/rat__les r1 r2)
% Assm: h4/rat/RAT__LES__RMUL__POS: !r3 r2 r1. h4/rat/rat__les (h4/rat/rat__of__num h4/num/0) r3 ==> (h4/rat/rat__les (h4/rat/rat__mul r1 r3) (h4/rat/rat__mul r2 r3) <=> h4/rat/rat__les r1 r2)
% Assm: h4/numeral/numeral__distrib_c18: !n. h4/num/0 = h4/arithmetic/NUMERAL n <=> n = h4/arithmetic/ZERO
% Assm: h4/integer/INT__NEG__MUL2: !y x. h4/integer/int__mul (h4/integer/int__neg x) (h4/integer/int__neg y) = h4/integer/int__mul x y
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/rat/RAT__SGN__TOTAL: !r1. h4/rat/rat__sgn r1 = h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) \/ h4/rat/rat__sgn r1 = h4/integer/int__of__num h4/num/0 \/ h4/rat/rat__sgn r1 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/rat/RAT__SGN__MUL: !r2 r1. h4/rat/rat__sgn (h4/rat/rat__mul r1 r2) = h4/integer/int__mul (h4/rat/rat__sgn r1) (h4/rat/rat__sgn r2)
% Assm: h4/rat/RAT__SGN__AINV: !r1. h4/integer/int__neg (h4/rat/rat__sgn (h4/rat/rat__ainv r1)) = h4/rat/rat__sgn r1
% Assm: h4/rat/RAT__SUB__LID: !r1. h4/rat/rat__sub (h4/rat/rat__of__num h4/num/0) r1 = h4/rat/rat__ainv r1
% Assm: h4/integer/INT__MUL__RZERO: !x. h4/integer/int__mul x (h4/integer/int__of__num h4/num/0) = h4/integer/int__of__num h4/num/0
% Assm: h4/rat/RAT__SUB__RID: !r1. h4/rat/rat__sub r1 (h4/rat/rat__of__num h4/num/0) = r1
% Assm: h4/integer/INT__MUL__EQ__1: !y x. h4/integer/int__mul x y = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) <=> x = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) /\ y = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) \/ x = h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ y = h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm: h4/integer/INT__EQ__NEG: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/AND__CLAUSES_c2: !t. F /\ t <=> F
% Assm: h4/integer/INT__EQ__CALCULATE_c2: !n m. h4/integer/int__of__num n = h4/integer/int__neg (h4/integer/int__of__num m) <=> n = h4/num/0 /\ m = h4/num/0
% Assm: h4/integer/INT__EQ__CALCULATE_c1: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% Assm: h4/integer/INT__INJ: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm: h4/integer/INT__MUL__LZERO: !x. h4/integer/int__mul (h4/integer/int__of__num h4/num/0) x = h4/integer/int__of__num h4/num/0
% Assm: h4/integer/INT__MUL__RID: !x. h4/integer/int__mul x (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm: h4/numeral/numeral__distrib_c17: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/numeral/numeral__eq_c1: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm: h4/bool/OR__CLAUSES_c1: !t. t \/ T <=> T
% Assm: h4/rat/rat__les__def: !r2 r1. h4/rat/rat__les r1 r2 <=> h4/rat/rat__sgn (h4/rat/rat__sub r2 r1) = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/integer/INT__MUL__LID: !x. h4/integer/int__mul (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x = x
% Assm: h4/bool/OR__CLAUSES_c4: !t. t \/ t <=> t
% Assm: h4/rat/RAT__LDIV__EQ: !r3 r2 r1. ~(r2 = h4/rat/rat__of__num h4/num/0) ==> (h4/rat/rat__div r1 r2 = r3 <=> r1 = h4/rat/rat__mul r2 r3)
% Assm: h4/rat/RAT__LES__REF: !r1. ~h4/rat/rat__les r1 r1
% Assm: h4/rat/RAT__DIV__MULMINV: !r2 r1. h4/rat/rat__div r1 r2 = h4/rat/rat__mul r1 (h4/rat/rat__minv r2)
% Assm: h4/rat/RAT__MUL__RINV: !a. ~(a = h4/rat/rat__of__num h4/num/0) ==> h4/rat/rat__mul a (h4/rat/rat__minv a) = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm: h4/rat/RAT__MUL__RID: !a. h4/rat/rat__mul a (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = a
% Assm: h4/rat/RAT__MUL__ASSOC: !c b a. h4/rat/rat__mul a (h4/rat/rat__mul b c) = h4/rat/rat__mul (h4/rat/rat__mul a b) c
% Assm: h4/frac/frac__mul__def: !f2 f1. h4/frac/frac__mul f1 f2 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__nmr f2)) (h4/integer/int__mul (h4/frac/frac__dnm f1) (h4/frac/frac__dnm f2)))
% Assm: h4/frac/NMR: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__nmr (h4/frac/abs__frac (h4/pair/_2C a b)) = a
% Assm: h4/frac/frac__add__def: !f2 f1. h4/frac/frac__add f1 f2 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__add (h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2)) (h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1))) (h4/integer/int__mul (h4/frac/frac__dnm f1) (h4/frac/frac__dnm f2)))
% Assm: h4/frac/DNM: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__dnm (h4/frac/abs__frac (h4/pair/_2C a b)) = b
% Assm: h4/intExtension/INT__MUL__POS__SIGN: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) a ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__mul a b)
% Assm: h4/integerRing/int__ring__thms_c56: !x. h4/integerRing/int__polynom__simplify x = h4/integerRing/int__r__canonical__sum__simplify (h4/integerRing/int__polynom__normalize x)
% Assm: h4/rat/rat__equiv__def: !f2 f1. h4/rat/rat__equiv f1 f2 <=> h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2) = h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1)
% Assm: h4/numRing/num__rewrites_c75: !ALT__ZERO. ALT__ZERO = ALT__ZERO <=> T
% Assm: h4/integerRing/int__ring__thms_c54: !pr pl. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pmult pl pr) = h4/integerRing/int__r__canonical__sum__prod (h4/integerRing/int__polynom__normalize pl) (h4/integerRing/int__polynom__normalize pr)
% Assm: h4/integerRing/int__ring__thms_c53: !pr pl. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pplus pl pr) = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__polynom__normalize pl) (h4/integerRing/int__polynom__normalize pr)
% Assm: h4/integerRing/int__ring__thms_c46: !vm t l a. h4/integerRing/int__r__ics__aux vm a (h4/canonical/Cons__varlist l t) = h4/integer/int__add a (h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__vl vm l) t)
% Assm: h4/integerRing/int__ring__thms_c45: !vm a. h4/integerRing/int__r__ics__aux vm a h4/canonical/Nil__monom = a
% Assm: h4/integerRing/int__ring__thms_c40: !x_27 x vm t_27. h4/integerRing/int__r__ivl__aux vm x (h4/list/CONS x_27 t_27) = h4/integer/int__mul (h4/quote/varmap__find x vm) (h4/integerRing/int__r__ivl__aux vm x_27 t_27)
% Assm: h4/frac/FRAC__DNMPOS: !f. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/frac/frac__dnm f)
% Assm: h4/integerRing/int__ring__thms_c49: !vm t l. h4/integerRing/int__r__interp__cs vm (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__vl vm l) t
% Assm: h4/integerRing/int__ring__thms_c42: !x vm t. h4/integerRing/int__r__interp__vl vm (h4/list/CONS x t) = h4/integerRing/int__r__ivl__aux vm x t
% Assm: h4/integerRing/int__ring__thms_c39: !x vm. h4/integerRing/int__r__ivl__aux vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm: h4/integerRing/int__ring__thms_c35: !s2. h4/integerRing/int__r__canonical__sum__prod h4/canonical/Nil__monom s2 = h4/canonical/Nil__monom
% Assm: h4/integerRing/int__ring__thms_c37: !t l. h4/integerRing/int__r__canonical__sum__simplify (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__varlist l (h4/integerRing/int__r__canonical__sum__simplify t)
% Assm: h4/integerRing/int__ring__thms_c34: !t1 s2 l1. h4/integerRing/int__r__canonical__sum__prod (h4/canonical/Cons__varlist l1 t1) s2 = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__r__canonical__sum__scalar2 l1 s2) (h4/integerRing/int__r__canonical__sum__prod t1 s2)
% Assm: h4/integerRing/int__ring__thms_c28: !t l0 l. h4/integerRing/int__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar2 l0 t)
% Goal: !t l c0 c. h4/ratRing/rat__r__canonical__sum__scalar c0 (h4/canonical/Cons__monom c l t) = h4/canonical/Cons__monom (h4/rat/rat__mul c0 c) l (h4/ratRing/rat__r__canonical__sum__scalar c0 t)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_scalaru_u_def]: h4/ratRing/rat__r__canonical__sum__scalar = h4/ringNorm/r__canonical__sum__scalar (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_rings_ringu_u_accessorsu_c2]: !f1 f0 f a0 a. h4/ring/ring__RP (h4/ring/ring0 a a0 f f0 f1) = f
% Assm [h4s_rings_ringu_u_accessorsu_c1]: !f1 f0 f a0 a. h4/ring/ring__R1 (h4/ring/ring0 a a0 f f0 f1) = a0
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c1]: !t2 t1 r l2 l1 c1. happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__monom c1 l1 t1)) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) c1) (h4/ring/ring__R1 r)) l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) t2)) (h4/canonical/Cons__varlist l2 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__monom c1 l1 t1)) t2))
% Assm [h4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_mergeu_u_def]: h4/ratRing/rat__r__canonical__sum__merge = h4/ringNorm/r__canonical__sum__merge (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c6]: !v8 v7 r. happ (happ (h4/ringNorm/r__canonical__sum__merge r) h4/canonical/Nil__monom) (h4/canonical/Cons__varlist v7 v8) = h4/canonical/Cons__varlist v7 v8
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c4]: !s1 r. happ (happ (h4/ringNorm/r__canonical__sum__merge r) s1) h4/canonical/Nil__monom = s1
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c0]: !t2 t1 r l2 l1 c2 c1. happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__monom c1 l1 t1)) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) c1) c2) l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) t2)) (h4/canonical/Cons__monom c2 l2 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__monom c1 l1 t1)) t2))
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c3]: !t2 t1 r l2 l1. happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__varlist l1 t1)) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R1 r)) (h4/ring/ring__R1 r)) l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) t2)) (h4/canonical/Cons__varlist l2 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__varlist l1 t1)) t2))
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c5]: !v6 v5 v4 r. happ (happ (h4/ringNorm/r__canonical__sum__merge r) h4/canonical/Nil__monom) (h4/canonical/Cons__monom v4 v5 v6) = h4/canonical/Cons__monom v4 v5 v6
% Assm [h4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c2]: !t2 t1 r l2 l1 c2. happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__varlist l1 t1)) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R1 r)) c2) l1 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) t1) t2)) (h4/canonical/Cons__monom c2 l2 (happ (happ (h4/ringNorm/r__canonical__sum__merge r) (h4/canonical/Cons__varlist l1 t1)) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c21]: !t2 l2 l1 c2. happ (happ h4/ratRing/rat__r__varlist__insert l1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (happ (happ h4/ratRing/rat__r__varlist__insert l1) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c16]: !v6 v5 v4. happ (happ h4/ratRing/rat__r__canonical__sum__merge h4/canonical/Nil__monom) (h4/canonical/Cons__monom v4 v5 v6) = h4/canonical/Cons__monom v4 v5 v6
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c18]: !t2 l2 l1 c2 c1. happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add c1) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c12]: !t2 t1 l2 l1 c1. happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1)) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add c1) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) t2)) (h4/canonical/Cons__varlist l2 (happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1)) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c11]: !t2 t1 l2 l1 c2 c1. happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1)) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add c1) c2) l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) t2)) (h4/canonical/Cons__monom c2 l2 (happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__monom c1 l1 t1)) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c13]: !t2 t1 l2 l1 c2. happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1)) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (h4/canonical/Cons__monom c2 l2 t2))) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) c2) l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) t2)) (h4/canonical/Cons__monom c2 l2 (happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1)) t2))
% Assm [h4s_ringNorms_monomu_u_insertu_u_defu_c0]: !t2 r l2 l1 c2 c1. happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) c1) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) t2))
% Assm [h4s_ratRings_ratu_u_ru_u_monomu_u_insertu_u_def]: h4/ratRing/rat__r__monom__insert = h4/ringNorm/r__monom__insert (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ringNorms_monomu_u_insertu_u_defu_c1]: !t2 r l2 l1 c1. happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) c1) (h4/ring/ring__R1 r)) l1 t2) (h4/canonical/Cons__varlist l2 (happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) t2))
% Assm [h4s_ringNorms_monomu_u_insertu_u_defu_c2]: !r l1 c1. happ (happ (happ (h4/ringNorm/r__monom__insert r) c1) l1) h4/canonical/Nil__monom = h4/canonical/Cons__monom c1 l1 h4/canonical/Nil__monom
% Assm [h4s_ratRings_ratu_u_ru_u_varlistu_u_insertu_u_def]: h4/ratRing/rat__r__varlist__insert = h4/ringNorm/r__varlist__insert (h4/ring/ring0 (h4/rat/rat__of__num h4/num/0) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) h4/rat/rat__add h4/rat/rat__mul h4/rat/rat__ainv)
% Assm [h4s_ringNorms_varlistu_u_insertu_u_defu_c1]: !t2 r l2 l1. happ (happ (h4/ringNorm/r__varlist__insert r) l1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R1 r)) (h4/ring/ring__R1 r)) l1 t2) (h4/canonical/Cons__varlist l2 (happ (happ (h4/ringNorm/r__varlist__insert r) l1) t2))
% Assm [h4s_ringNorms_varlistu_u_insertu_u_defu_c0]: !t2 r l2 l1 c2. happ (happ (h4/ringNorm/r__varlist__insert r) l1) (h4/canonical/Cons__monom c2 l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/canonical/Cons__monom c2 l2 t2)) (h4/canonical/Cons__monom (happ (happ (h4/ring/ring__RP r) (h4/ring/ring__R1 r)) c2) l1 t2) (h4/canonical/Cons__monom c2 l2 (happ (happ (h4/ringNorm/r__varlist__insert r) l1) t2))
% Assm [h4s_ringNorms_varlistu_u_insertu_u_defu_c2]: !r l1. happ (happ (h4/ringNorm/r__varlist__insert r) l1) h4/canonical/Nil__monom = h4/canonical/Cons__varlist l1 h4/canonical/Nil__monom
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c19]: !t2 l2 l1 c1. happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__monom c1 l1 (h4/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add c1) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 t2) (h4/canonical/Cons__varlist l2 (happ (happ (happ h4/ratRing/rat__r__monom__insert c1) l1) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c22]: !t2 l2 l1. happ (happ h4/ratRing/rat__r__varlist__insert l1) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (h4/canonical/Cons__varlist l2 t2)) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 t2) (h4/canonical/Cons__varlist l2 (happ (happ h4/ratRing/rat__r__varlist__insert l1) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c14]: !t2 t1 l2 l1. happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1)) (h4/canonical/Cons__varlist l2 t2) = h4/prelim/compare (h4/prelim/list__compare h4/quote/index__compare l1 l2) (h4/canonical/Cons__varlist l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) (h4/canonical/Cons__varlist l2 t2))) (h4/canonical/Cons__monom (happ (happ h4/rat/rat__add (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))) l1 (happ (happ h4/ratRing/rat__r__canonical__sum__merge t1) t2)) (h4/canonical/Cons__varlist l2 (happ (happ h4/ratRing/rat__r__canonical__sum__merge (h4/canonical/Cons__varlist l1 t1)) t2))
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c17]: !v8 v7. happ (happ h4/ratRing/rat__r__canonical__sum__merge h4/canonical/Nil__monom) (h4/canonical/Cons__varlist v7 v8) = h4/canonical/Cons__varlist v7 v8
% Assm [h4s_ratRings_ratu_u_ringu_u_thmsu_c15]: !s1. happ (happ h4/ratRing/rat__r__canonical__sum__merge s1) h4/canonical/Nil__monom = s1
% Assm [h4s_rats_RATu_u_LESu_u_LMULu_u_POS]: !r3 r2 r1. h4/rat/rat__les (h4/rat/rat__of__num h4/num/0) r3 ==> (h4/rat/rat__les (happ (happ h4/rat/rat__mul r3) r1) (happ (happ h4/rat/rat__mul r3) r2) <=> h4/rat/rat__les r1 r2)
% Assm [h4s_rats_RATu_u_MULu_u_SIGNu_u_CASESu_c1]: !q p. h4/rat/rat__les (happ (happ h4/rat/rat__mul p) q) (h4/rat/rat__of__num h4/num/0) <=> h4/rat/rat__les (h4/rat/rat__of__num h4/num/0) p /\ h4/rat/rat__les q (h4/rat/rat__of__num h4/num/0) \/ h4/rat/rat__les p (h4/rat/rat__of__num h4/num/0) /\ h4/rat/rat__les (h4/rat/rat__of__num h4/num/0) q
% Assm [h4s_rats_RATu_u_LDIVu_u_LESu_u_POS]: !r3 r2 r1. h4/rat/rat__les (h4/rat/rat__of__num h4/num/0) r2 ==> (h4/rat/rat__les (h4/rat/rat__div r1 r2) r3 <=> h4/rat/rat__les r1 (happ (happ h4/rat/rat__mul r2) r3))
% Assm [h4s_rats_RATu_u_LESu_u_LMULu_u_NEG]: !r3 r2 r1. h4/rat/rat__les r3 (h4/rat/rat__of__num h4/num/0) ==> (h4/rat/rat__les (happ (happ h4/rat/rat__mul r3) r2) (happ (happ h4/rat/rat__mul r3) r1) <=> h4/rat/rat__les r1 r2)
% Assm [h4s_rats_RATu_u_RDIVu_u_EQ]: !r3 r2 r1. ~(r3 = h4/rat/rat__of__num h4/num/0) ==> (r1 = h4/rat/rat__div r2 r3 <=> happ (happ h4/rat/rat__mul r1) r3 = r2)
% Assm [h4s_rats_RATu_u_RDISTRIB]: !c b a. happ (happ h4/rat/rat__mul (happ (happ h4/rat/rat__add a) b)) c = happ (happ h4/rat/rat__add (happ (happ h4/rat/rat__mul a) c)) (happ (happ h4/rat/rat__mul b) c)
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% 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_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% 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_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_rats_RATu_u_MULu_u_COMM]: !b a. happ (happ h4/rat/rat__mul a) b = happ (happ h4/rat/rat__mul b) a
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_NOTu_u_FORALLu_u_THM]: !P. ~(!x. happ P x) <=> (?x. ~happ P x)
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_rats_RATu_u_LESu_u_RMULu_u_NEG]: !r3 r2 r1. h4/rat/rat__les r3 (h4/rat/rat__of__num h4/num/0) ==> (h4/rat/rat__les (happ (happ h4/rat/rat__mul r2) r3) (happ (happ h4/rat/rat__mul r1) r3) <=> h4/rat/rat__les r1 r2)
% Assm [h4s_rats_RATu_u_LESu_u_RMULu_u_POS]: !r3 r2 r1. h4/rat/rat__les (h4/rat/rat__of__num h4/num/0) r3 ==> (h4/rat/rat__les (happ (happ h4/rat/rat__mul r1) r3) (happ (happ h4/rat/rat__mul r2) r3) <=> h4/rat/rat__les r1 r2)
% Assm [h4s_numerals_numeralu_u_distribu_c18]: !n. h4/num/0 = h4/arithmetic/NUMERAL n <=> n = h4/arithmetic/ZERO
% Assm [h4s_integers_INTu_u_NEGu_u_MUL2]: !y x. h4/integer/int__mul (h4/integer/int__neg x) (h4/integer/int__neg y) = h4/integer/int__mul x y
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_rats_RATu_u_SGNu_u_TOTAL]: !r1. h4/rat/rat__sgn r1 = h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) \/ h4/rat/rat__sgn r1 = h4/integer/int__of__num h4/num/0 \/ h4/rat/rat__sgn r1 = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_rats_RATu_u_SGNu_u_MUL]: !r2 r1. h4/rat/rat__sgn (happ (happ h4/rat/rat__mul r1) r2) = h4/integer/int__mul (h4/rat/rat__sgn r1) (h4/rat/rat__sgn r2)
% Assm [h4s_rats_RATu_u_SGNu_u_AINV]: !r1. h4/integer/int__neg (h4/rat/rat__sgn (happ h4/rat/rat__ainv r1)) = h4/rat/rat__sgn r1
% Assm [h4s_rats_RATu_u_SUBu_u_LID]: !r1. h4/rat/rat__sub (h4/rat/rat__of__num h4/num/0) r1 = happ h4/rat/rat__ainv r1
% Assm [h4s_integers_INTu_u_MULu_u_RZERO]: !x. h4/integer/int__mul x (h4/integer/int__of__num h4/num/0) = h4/integer/int__of__num h4/num/0
% Assm [h4s_rats_RATu_u_SUBu_u_RID]: !r1. h4/rat/rat__sub r1 (h4/rat/rat__of__num h4/num/0) = r1
% Assm [h4s_integers_INTu_u_MULu_u_EQu_u_1]: !y x. h4/integer/int__mul x y = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) <=> x = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) /\ y = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) \/ x = h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) /\ y = h4/integer/int__neg (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)))
% Assm [h4s_integers_INTu_u_EQu_u_NEG]: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c2]: !t. F /\ t <=> F
% Assm [h4s_integers_INTu_u_EQu_u_CALCULATEu_c2]: !n m. h4/integer/int__of__num n = h4/integer/int__neg (h4/integer/int__of__num m) <=> n = h4/num/0 /\ m = h4/num/0
% Assm [h4s_integers_INTu_u_EQu_u_CALCULATEu_c1]: !y x. h4/integer/int__neg x = h4/integer/int__neg y <=> x = y
% Assm [h4s_integers_INTu_u_INJ]: !n m. h4/integer/int__of__num m = h4/integer/int__of__num n <=> m = n
% Assm [h4s_integers_INTu_u_MULu_u_LZERO]: !x. h4/integer/int__mul (h4/integer/int__of__num h4/num/0) x = h4/integer/int__of__num h4/num/0
% Assm [h4s_integers_INTu_u_MULu_u_RID]: !x. h4/integer/int__mul x (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = x
% Assm [h4s_numerals_numeralu_u_distribu_c17]: !n. h4/arithmetic/NUMERAL n = h4/num/0 <=> n = h4/arithmetic/ZERO
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_numerals_numeralu_u_equ_c1]: !n. h4/arithmetic/BIT1 n = h4/arithmetic/ZERO <=> F
% Assm [h4s_bools_ORu_u_CLAUSESu_c1]: !t. t \/ T <=> T
% Assm [h4s_rats_ratu_u_lesu_u_def]: !r2 r1. h4/rat/rat__les r1 r2 <=> h4/rat/rat__sgn (h4/rat/rat__sub r2 r1) = h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_integers_INTu_u_MULu_u_LID]: !x. h4/integer/int__mul (h4/integer/int__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) x = x
% Assm [h4s_bools_ORu_u_CLAUSESu_c4]: !t. t \/ t <=> t
% Assm [h4s_rats_RATu_u_LDIVu_u_EQ]: !r3 r2 r1. ~(r2 = h4/rat/rat__of__num h4/num/0) ==> (h4/rat/rat__div r1 r2 = r3 <=> r1 = happ (happ h4/rat/rat__mul r2) r3)
% Assm [h4s_rats_RATu_u_LESu_u_REF]: !r1. ~h4/rat/rat__les r1 r1
% Assm [h4s_rats_RATu_u_DIVu_u_MULMINV]: !r2 r1. h4/rat/rat__div r1 r2 = happ (happ h4/rat/rat__mul r1) (h4/rat/rat__minv r2)
% Assm [h4s_rats_RATu_u_MULu_u_RINV]: !a. ~(a = h4/rat/rat__of__num h4/num/0) ==> happ (happ h4/rat/rat__mul a) (h4/rat/rat__minv a) = h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Assm [h4s_rats_RATu_u_MULu_u_RID]: !a. happ (happ h4/rat/rat__mul a) (h4/rat/rat__of__num (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) = a
% Assm [h4s_rats_RATu_u_MULu_u_ASSOC]: !c b a. happ (happ h4/rat/rat__mul a) (happ (happ h4/rat/rat__mul b) c) = happ (happ h4/rat/rat__mul (happ (happ h4/rat/rat__mul a) b)) c
% Assm [h4s_fracs_fracu_u_mulu_u_def]: !f2 f1. h4/frac/frac__mul f1 f2 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__nmr f2)) (h4/integer/int__mul (h4/frac/frac__dnm f1) (h4/frac/frac__dnm f2)))
% Assm [h4s_fracs_NMR]: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__nmr (h4/frac/abs__frac (h4/pair/_2C a b)) = a
% Assm [h4s_fracs_fracu_u_addu_u_def]: !f2 f1. h4/frac/frac__add f1 f2 = h4/frac/abs__frac (h4/pair/_2C (h4/integer/int__add (h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2)) (h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1))) (h4/integer/int__mul (h4/frac/frac__dnm f1) (h4/frac/frac__dnm f2)))
% Assm [h4s_fracs_DNM]: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/frac/frac__dnm (h4/frac/abs__frac (h4/pair/_2C a b)) = b
% Assm [h4s_intExtensions_INTu_u_MULu_u_POSu_u_SIGN]: !b a. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) a ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) b ==> h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/integer/int__mul a b)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c56]: !x. h4/integerRing/int__polynom__simplify x = h4/integerRing/int__r__canonical__sum__simplify (h4/integerRing/int__polynom__normalize x)
% Assm [h4s_rats_ratu_u_equivu_u_def]: !f2 f1. h4/rat/rat__equiv f1 f2 <=> h4/integer/int__mul (h4/frac/frac__nmr f1) (h4/frac/frac__dnm f2) = h4/integer/int__mul (h4/frac/frac__nmr f2) (h4/frac/frac__dnm f1)
% Assm [h4s_numRings_numu_u_rewritesu_c75]: !ALT__ZERO. ALT__ZERO = ALT__ZERO <=> T
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c54]: !pr pl. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pmult pl pr) = h4/integerRing/int__r__canonical__sum__prod (h4/integerRing/int__polynom__normalize pl) (h4/integerRing/int__polynom__normalize pr)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c53]: !pr pl. h4/integerRing/int__polynom__normalize (h4/ringNorm/Pplus pl pr) = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__polynom__normalize pl) (h4/integerRing/int__polynom__normalize pr)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c46]: !vm t l a. h4/integerRing/int__r__ics__aux vm a (h4/canonical/Cons__varlist l t) = h4/integer/int__add a (h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__vl vm l) t)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c45]: !vm a. h4/integerRing/int__r__ics__aux vm a h4/canonical/Nil__monom = a
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c40]: !x_27 x vm t_27. h4/integerRing/int__r__ivl__aux vm x (h4/list/CONS x_27 t_27) = h4/integer/int__mul (h4/quote/varmap__find x vm) (h4/integerRing/int__r__ivl__aux vm x_27 t_27)
% Assm [h4s_fracs_FRACu_u_DNMPOS]: !f. h4/integer/int__lt (h4/integer/int__of__num h4/num/0) (h4/frac/frac__dnm f)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c49]: !vm t l. h4/integerRing/int__r__interp__cs vm (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__ics__aux vm (h4/integerRing/int__r__interp__vl vm l) t
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c42]: !x vm t. h4/integerRing/int__r__interp__vl vm (h4/list/CONS x t) = h4/integerRing/int__r__ivl__aux vm x t
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c39]: !x vm. h4/integerRing/int__r__ivl__aux vm x h4/list/NIL = h4/quote/varmap__find x vm
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c35]: !s2. h4/integerRing/int__r__canonical__sum__prod h4/canonical/Nil__monom s2 = h4/canonical/Nil__monom
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c37]: !t l. h4/integerRing/int__r__canonical__sum__simplify (h4/canonical/Cons__varlist l t) = h4/canonical/Cons__varlist l (h4/integerRing/int__r__canonical__sum__simplify t)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c34]: !t1 s2 l1. h4/integerRing/int__r__canonical__sum__prod (h4/canonical/Cons__varlist l1 t1) s2 = h4/integerRing/int__r__canonical__sum__merge (h4/integerRing/int__r__canonical__sum__scalar2 l1 s2) (h4/integerRing/int__r__canonical__sum__prod t1 s2)
% Assm [h4s_integerRings_intu_u_ringu_u_thmsu_c28]: !t l0 l. h4/integerRing/int__r__canonical__sum__scalar2 l0 (h4/canonical/Cons__varlist l t) = h4/integerRing/int__r__varlist__insert (h4/prelim/list__merge h4/quote/index__lt l0 l) (h4/integerRing/int__r__canonical__sum__scalar2 l0 t)
% Goal: !t l c0 c. happ (happ h4/ratRing/rat__r__canonical__sum__scalar c0) (h4/canonical/Cons__monom c l t) = h4/canonical/Cons__monom (happ (happ h4/rat/rat__mul c0) c) l (happ (happ h4/ratRing/rat__r__canonical__sum__scalar c0) t)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t0))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1330602,TV_Q1330598]: ![V_f, V_g]: (![V_x]: s(TV_Q1330598,happ(s(t_fun(TV_Q1330602,TV_Q1330598),V_f),s(TV_Q1330602,V_x))) = s(TV_Q1330598,happ(s(t_fun(TV_Q1330602,TV_Q1330598),V_g),s(TV_Q1330602,V_x))) => s(t_fun(TV_Q1330602,TV_Q1330598),V_f) = s(t_fun(TV_Q1330602,TV_Q1330598),V_g))).
fof(ah4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_scalaru_u_def, axiom, s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_scalar) = s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_scalar(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_rings_ringu_u_accessorsu_c2, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f)).
fof(ah4s_rings_ringu_u_accessorsu_c1, axiom, ![TV_u_27a]: ![V_f1, V_f0, V_f, V_a0, V_a]: s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),h4s_rings_ring0(s(TV_u_27a,V_a),s(TV_u_27a,V_a0),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),V_f0),s(t_fun(TV_u_27a,TV_u_27a),V_f1))))) = s(TV_u_27a,V_a0)).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_r, V_l2, V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ru_u_canonicalu_u_sumu_u_mergeu_u_def, axiom, s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge) = s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c6, axiom, ![TV_u_27a]: ![V_v8, V_v7, V_r]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_v7),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_v8))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_v7),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_v8)))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c4, axiom, ![TV_u_27a]: ![V_s1, V_r]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_s1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_s1)).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_r, V_l2, V_l1, V_c2, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(TV_u_27a,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c3, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_r, V_l2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c5, axiom, ![TV_u_27a]: ![V_v6, V_v5, V_v4, V_r]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_v4),s(t_h4s_lists_list(t_h4s_quotes_index),V_v5),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_v6))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_v4),s(t_h4s_lists_list(t_h4s_quotes_index),V_v5),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_v6)))).
fof(ah4s_ringNorms_canonicalu_u_sumu_u_mergeu_u_defu_c2, axiom, ![TV_u_27a]: ![V_t2, V_t1, V_r, V_l2, V_l1, V_c2]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c21, axiom, ![V_t2, V_l2, V_l1, V_c2]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_varlistu_u_insert),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_rats_rat,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_varlistu_u_insert),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c16, axiom, ![V_v6, V_v5, V_v4]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_nilu_u_monom))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_v4),s(t_h4s_lists_list(t_h4s_quotes_index),V_v5),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_v6))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_v4),s(t_h4s_lists_list(t_h4s_quotes_index),V_v5),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_v6)))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c18, axiom, ![V_t2, V_l2, V_l1, V_c2, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ratrings_ratu_u_ru_u_monomu_u_insert),s(t_h4s_rats_rat,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,V_c1))),s(t_h4s_rats_rat,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ratrings_ratu_u_ru_u_monomu_u_insert),s(t_h4s_rats_rat,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c12, axiom, ![V_t2, V_t1, V_l2, V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,V_c1))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c11, axiom, ![V_t2, V_t1, V_l2, V_l1, V_c2, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,V_c1))),s(t_h4s_rats_rat,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c13, axiom, ![V_t2, V_t1, V_l2, V_l1, V_c2]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_rats_rat,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ringNorms_monomu_u_insertu_u_defu_c0, axiom, ![TV_u_27a]: ![V_t2, V_r, V_l2, V_l1, V_c2, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(TV_u_27a,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ru_u_monomu_u_insertu_u_def, axiom, s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ratrings_ratu_u_ru_u_monomu_u_insert) = s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ringNorms_monomu_u_insertu_u_defu_c1, axiom, ![TV_u_27a]: ![V_t2, V_r, V_l2, V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_monomu_u_insertu_u_defu_c2, axiom, ![TV_u_27a]: ![V_r, V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),happ(s(t_fun(TV_u_27a,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)))),h4s_ringnorms_ru_u_monomu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_ratRings_ratu_u_ru_u_varlistu_u_insertu_u_def, axiom, s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_varlistu_u_insert) = s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(t_h4s_rats_rat),h4s_rings_ring0(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv)))))).
fof(ah4s_ringNorms_varlistu_u_insertu_u_defu_c1, axiom, ![TV_u_27a]: ![V_t2, V_r, V_l2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_varlistu_u_insertu_u_defu_c0, axiom, ![TV_u_27a]: ![V_t2, V_r, V_l2, V_l1, V_c2]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(TV_u_27a,t_fun(TV_u_27a,TV_u_27a)),h4s_rings_ringu_u_rp(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(TV_u_27a,h4s_rings_ringu_u_r1(s(t_h4s_rings_ring(TV_u_27a),V_r))))),s(TV_u_27a,V_c2))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_monom(s(TV_u_27a,V_c2),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),V_t2)))))))).
fof(ah4s_ringNorms_varlistu_u_insertu_u_defu_c2, axiom, ![TV_u_27a]: ![V_r, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),t_h4s_canonicals_canonicalu_u_sum(TV_u_27a))),h4s_ringnorms_ru_u_varlistu_u_insert(s(t_h4s_rings_ring(TV_u_27a),V_r))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(TV_u_27a),h4s_canonicals_nilu_u_monom)))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c19, axiom, ![V_t2, V_l2, V_l1, V_c1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ratrings_ratu_u_ru_u_monomu_u_insert),s(t_h4s_rats_rat,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,V_c1))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)))),h4s_ratrings_ratu_u_ru_u_monomu_u_insert),s(t_h4s_rats_rat,V_c1))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c22, axiom, ![V_t2, V_l2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_varlistu_u_insert),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_lists_list(t_h4s_quotes_index),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_varlistu_u_insert),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c14, axiom, ![V_t2, V_t1, V_l2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_prelims_compare(s(t_h4s_prelims_ordering,h4s_prelims_listu_u_compare(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_h4s_prelims_ordering)),h4s_quotes_indexu_u_compare),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_lists_list(t_h4s_quotes_index),V_l2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l2),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t1))))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t2)))))))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c17, axiom, ![V_v8, V_v7]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_nilu_u_monom))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_v7),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_v8))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_v7),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_v8)))).
fof(ah4s_ratRings_ratu_u_ringu_u_thmsu_c15, axiom, ![V_s1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_merge),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_s1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_nilu_u_monom))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_s1)).
fof(ah4s_rats_RATu_u_LESu_u_LMULu_u_POS, axiom, ![V_r3, V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,V_r3)))) => s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r3))),s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r3))),s(t_h4s_rats_rat,V_r2))))) = s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))))).
fof(ah4s_rats_RATu_u_MULu_u_SIGNu_u_CASESu_c1, axiom, ![V_q, V_p]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_p))),s(t_h4s_rats_rat,V_q))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) <=> ((p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,V_p)))) & p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_q),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))))) | (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_p),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) & p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,V_q)))))))).
fof(ah4s_rats_RATu_u_LDIVu_u_LESu_u_POS, axiom, ![V_r3, V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,V_r2)))) => s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,h4s_rats_ratu_u_div(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))),s(t_h4s_rats_rat,V_r3))) = s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r2))),s(t_h4s_rats_rat,V_r3))))))).
fof(ah4s_rats_RATu_u_LESu_u_LMULu_u_NEG, axiom, ![V_r3, V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r3),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) => s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r3))),s(t_h4s_rats_rat,V_r2))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r3))),s(t_h4s_rats_rat,V_r1))))) = s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))))).
fof(ah4s_rats_RATu_u_RDIVu_u_EQ, axiom, ![V_r3, V_r2, V_r1]: (~ (s(t_h4s_rats_rat,V_r3) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => (s(t_h4s_rats_rat,V_r1) = s(t_h4s_rats_rat,h4s_rats_ratu_u_div(s(t_h4s_rats_rat,V_r2),s(t_h4s_rats_rat,V_r3))) <=> s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,V_r3))) = s(t_h4s_rats_rat,V_r2)))).
fof(ah4s_rats_RATu_u_RDISTRIB, axiom, ![V_c, V_b, V_a]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,V_b))))),s(t_h4s_rats_rat,V_c))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_add),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,V_c))))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_b))),s(t_h4s_rats_rat,V_c)))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t0))).
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_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
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_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_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_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_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_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
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_rats_RATu_u_MULu_u_COMM, axiom, ![V_b, V_a]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,V_b))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_b))),s(t_h4s_rats_rat,V_a)))).
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_sats_dcu_u_imp, 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_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_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_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,t0)))).
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_bools_NOTu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![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))))) <=> ?[V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t0)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t0)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t0))) <=> p(s(t_bool,f)))).
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_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t0) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_rats_RATu_u_LESu_u_RMULu_u_NEG, axiom, ![V_r3, V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r3),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))))) => s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r2))),s(t_h4s_rats_rat,V_r3))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,V_r3))))) = s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))))).
fof(ah4s_rats_RATu_u_LESu_u_RMULu_u_POS, axiom, ![V_r3, V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,V_r3)))) => s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,V_r3))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r2))),s(t_h4s_rats_rat,V_r3))))) = s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))))).
fof(ah4s_numerals_numeralu_u_distribu_c18, axiom, ![V_n]: (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_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_integers_INTu_u_NEGu_u_MUL2, axiom, ![V_y, V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x))),s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y)))).
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_rats_RATu_u_SGNu_u_TOTAL, axiom, ![V_r1]: (s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) | (s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))) | s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))))).
fof(ah4s_rats_RATu_u_SGNu_u_MUL, axiom, ![V_r2, V_r1]: s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,V_r2))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1))),s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r2)))))).
fof(ah4s_rats_RATu_u_SGNu_u_AINV, axiom, ![V_r1]: s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv),s(t_h4s_rats_rat,V_r1))))))) = s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,V_r1)))).
fof(ah4s_rats_RATu_u_SUBu_u_LID, axiom, ![V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_rats_rat,V_r1))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),h4s_rats_ratu_u_ainv),s(t_h4s_rats_rat,V_r1)))).
fof(ah4s_integers_INTu_u_MULu_u_RZERO, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_rats_RATu_u_SUBu_u_RID, axiom, ![V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))))) = s(t_h4s_rats_rat,V_r1)).
fof(ah4s_integers_INTu_u_MULu_u_EQu_u_1, axiom, ![V_y, V_x]: (s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,V_y))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) <=> ((s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))) & s(t_h4s_integers_int,V_y) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))) | (s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) & s(t_h4s_integers_int,V_y) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))))))).
fof(ah4s_integers_INTu_u_EQu_u_NEG, axiom, ![V_y, V_x]: (s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y))) <=> s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,V_y))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t0))) <=> 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_integers_INTu_u_EQu_u_CALCULATEu_c2, axiom, ![V_n, V_m]: (s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))))) <=> (s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0) & s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_integers_INTu_u_EQu_u_CALCULATEu_c1, axiom, ![V_y, V_x]: (s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,h4s_integers_intu_u_neg(s(t_h4s_integers_int,V_y))) <=> s(t_h4s_integers_int,V_x) = s(t_h4s_integers_int,V_y))).
fof(ah4s_integers_INTu_u_INJ, axiom, ![V_n, V_m]: (s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_m))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,V_n))) <=> s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n))).
fof(ah4s_integers_INTu_u_MULu_u_LZERO, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_integers_INTu_u_MULu_u_RID, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_x),s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) = s(t_h4s_integers_int,V_x)).
fof(ah4s_numerals_numeralu_u_distribu_c17, axiom, ![V_n]: (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_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_arithmetics_zero))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_numerals_numeralu_u_equ_c1, axiom, ![V_n]: (s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_zero) <=> p(s(t_bool,f)))).
fof(ah4s_bools_ORu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,t0))) <=> p(s(t_bool,t0)))).
fof(ah4s_rats_ratu_u_lesu_u_def, axiom, ![V_r2, V_r1]: (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2)))) <=> s(t_h4s_integers_int,h4s_rats_ratu_u_sgn(s(t_h4s_rats_rat,h4s_rats_ratu_u_sub(s(t_h4s_rats_rat,V_r2),s(t_h4s_rats_rat,V_r1))))) = s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
fof(ah4s_integers_INTu_u_MULu_u_LID, axiom, ![V_x]: s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_integers_int,V_x))) = s(t_h4s_integers_int,V_x)).
fof(ah4s_bools_ORu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_rats_RATu_u_LDIVu_u_EQ, axiom, ![V_r3, V_r2, V_r1]: (~ (s(t_h4s_rats_rat,V_r2) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => (s(t_h4s_rats_rat,h4s_rats_ratu_u_div(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,V_r3) <=> s(t_h4s_rats_rat,V_r1) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r2))),s(t_h4s_rats_rat,V_r3)))))).
fof(ah4s_rats_RATu_u_LESu_u_REF, axiom, ![V_r1]: ~ (p(s(t_bool,h4s_rats_ratu_u_les(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r1)))))).
fof(ah4s_rats_RATu_u_DIVu_u_MULMINV, axiom, ![V_r2, V_r1]: s(t_h4s_rats_rat,h4s_rats_ratu_u_div(s(t_h4s_rats_rat,V_r1),s(t_h4s_rats_rat,V_r2))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_r1))),s(t_h4s_rats_rat,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,V_r2)))))).
fof(ah4s_rats_RATu_u_MULu_u_RINV, axiom, ![V_a]: (~ (s(t_h4s_rats_rat,V_a) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0)))) => s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,h4s_rats_ratu_u_minv(s(t_h4s_rats_rat,V_a))))) = s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))).
fof(ah4s_rats_RATu_u_MULu_u_RID, axiom, ![V_a]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,h4s_rats_ratu_u_ofu_u_num(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))))) = s(t_h4s_rats_rat,V_a)).
fof(ah4s_rats_RATu_u_MULu_u_ASSOC, axiom, ![V_c, V_b, V_a]: s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_b))),s(t_h4s_rats_rat,V_c))))) = s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_a))),s(t_h4s_rats_rat,V_b))))),s(t_h4s_rats_rat,V_c)))).
fof(ah4s_fracs_fracu_u_mulu_u_def, axiom, ![V_f2, V_f1]: s(t_h4s_fracs_frac,h4s_fracs_fracu_u_mul(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2))) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f2))))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2)))))))))).
fof(ah4s_fracs_NMR, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b)))) => s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,V_b))))))) = s(t_h4s_integers_int,V_a))).
fof(ah4s_fracs_fracu_u_addu_u_def, axiom, ![V_f2, V_f1]: s(t_h4s_fracs_frac,h4s_fracs_fracu_u_add(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2))) = s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2))))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f2))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))))))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2)))))))))).
fof(ah4s_fracs_DNM, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b)))) => s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,h4s_fracs_absu_u_frac(s(t_h4s_pairs_prod(t_h4s_integers_int,t_h4s_integers_int),h4s_pairs_u_2c(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,V_b))))))) = s(t_h4s_integers_int,V_b))).
fof(ah4s_intExtensions_INTu_u_MULu_u_POSu_u_SIGN, axiom, ![V_b, V_a]: (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_a)))) => (p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,V_b)))) => p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,V_b))))))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c56, axiom, ![V_x]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_simplify(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_x))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_x)))))).
fof(ah4s_rats_ratu_u_equivu_u_def, axiom, ![V_f2, V_f1]: (p(s(t_bool,h4s_rats_ratu_u_equiv(s(t_h4s_fracs_frac,V_f1),s(t_h4s_fracs_frac,V_f2)))) <=> s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f1))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f2))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_fracs_fracu_u_nmr(s(t_h4s_fracs_frac,V_f2))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f1))))))).
fof(ah4s_numRings_numu_u_rewritesu_c75, axiom, ![TV_u_27a]: ![V_ALTu_u_ZERO]: (s(TV_u_27a,V_ALTu_u_ZERO) = s(TV_u_27a,V_ALTu_u_ZERO) <=> p(s(t_bool,t0)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c54, axiom, ![V_pr, V_pl]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pmult(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pl),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pr))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pl))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pr)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c53, axiom, ![V_pr, V_pl]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),h4s_ringnorms_pplus(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pl),s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pr))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pl))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_polynomu_u_normalize(s(t_h4s_ringNorms_polynom(t_h4s_integers_int),V_pr)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c46, axiom, ![V_vm, V_t, V_l, V_a]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_icsu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_integers_int,V_a),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))) = s(t_h4s_integers_int,h4s_integers_intu_u_add(s(t_h4s_integers_int,V_a),s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_icsu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_vl(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c45, axiom, ![V_vm, V_a]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_icsu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_integers_int,V_a),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom))) = s(t_h4s_integers_int,V_a)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c40, axiom, ![V_xu_27, V_x, V_vm, V_tu_27]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_ivlu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_cons(s(t_h4s_quotes_index,V_xu_27),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27))))) = s(t_h4s_integers_int,h4s_integers_intu_u_mul(s(t_h4s_integers_int,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm))),s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_ivlu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_quotes_index,V_xu_27),s(t_h4s_lists_list(t_h4s_quotes_index),V_tu_27)))))).
fof(ah4s_fracs_FRACu_u_DNMPOS, axiom, ![V_f]: p(s(t_bool,h4s_integers_intu_u_lt(s(t_h4s_integers_int,h4s_integers_intu_u_ofu_u_num(s(t_h4s_nums_num,h4s_nums_0))),s(t_h4s_integers_int,h4s_fracs_fracu_u_dnm(s(t_h4s_fracs_frac,V_f))))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c49, axiom, ![V_vm, V_t, V_l]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_cs(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))) = s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_icsu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_vl(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c42, axiom, ![V_x, V_vm, V_t]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_interpu_u_vl(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),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(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_ivlu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),V_t)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c39, axiom, ![V_x, V_vm]: s(t_h4s_integers_int,h4s_integerrings_intu_u_ru_u_ivlu_u_aux(s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm),s(t_h4s_quotes_index,V_x),s(t_h4s_lists_list(t_h4s_quotes_index),h4s_lists_nil))) = s(t_h4s_integers_int,h4s_quotes_varmapu_u_find(s(t_h4s_quotes_index,V_x),s(t_h4s_quotes_varmap(t_h4s_integers_int),V_vm)))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c35, axiom, ![V_s2]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_nilu_u_monom)).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c37, axiom, ![V_t, V_l]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_simplify(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c34, axiom, ![V_t1, V_s2, V_l1]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_merge(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_lists_list(t_h4s_quotes_index),V_l1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_prod(s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t1),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_s2)))))).
fof(ah4s_integerRings_intu_u_ringu_u_thmsu_c28, axiom, ![V_t, V_l0, V_l]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_canonicals_consu_u_varlist(s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_varlistu_u_insert(s(t_h4s_lists_list(t_h4s_quotes_index),h4s_prelims_listu_u_merge(s(t_fun(t_h4s_quotes_index,t_fun(t_h4s_quotes_index,t_bool)),h4s_quotes_indexu_u_lt),s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_lists_list(t_h4s_quotes_index),V_l))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),h4s_integerrings_intu_u_ru_u_canonicalu_u_sumu_u_scalar2(s(t_h4s_lists_list(t_h4s_quotes_index),V_l0),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_integers_int),V_t)))))).
fof(ch4s_ratRings_ratu_u_ringu_u_thmsu_c24, conjecture, ![V_t, V_l, V_c0, V_c]: s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_scalar),s(t_h4s_rats_rat,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,V_c),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t))))) = s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),h4s_canonicals_consu_u_monom(s(t_h4s_rats_rat,happ(s(t_fun(t_h4s_rats_rat,t_h4s_rats_rat),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_rats_rat,t_h4s_rats_rat)),h4s_rats_ratu_u_mul),s(t_h4s_rats_rat,V_c0))),s(t_h4s_rats_rat,V_c))),s(t_h4s_lists_list(t_h4s_quotes_index),V_l),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),happ(s(t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat)),happ(s(t_fun(t_h4s_rats_rat,t_fun(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat))),h4s_ratrings_ratu_u_ru_u_canonicalu_u_sumu_u_scalar),s(t_h4s_rats_rat,V_c0))),s(t_h4s_canonicals_canonicalu_u_sum(t_h4s_rats_rat),V_t)))))).
