TPTP Problem File: SWW487_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW487_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 100
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_100 [Bla13]
% Status : Theorem
% Rating : 0.67 v7.4.0, 0.75 v7.1.0, 1.00 v6.4.0
% Syntax : Number of formulae : 207 ( 25 unt; 51 typ; 0 def)
% Number of atoms : 331 ( 149 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 190 ( 15 ~; 8 |; 8 &)
% ( 16 <=>; 143 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 5 avg)
% Maximal term depth : 12 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 29 ( 17 >; 12 *; 0 +; 0 <<)
% Number of predicates : 24 ( 23 usr; 0 prp; 1-2 aty)
% Number of functors : 25 ( 25 usr; 3 con; 0-5 aty)
% Number of variables : 454 ( 403 !; 0 ?; 454 :)
% ( 51 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:06
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_t_a,type,
a: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_Polynomial_Opoly,type,
poly: $tType > $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (46)
tff(sy_cl_Groups_Ocancel__comm__monoid__add,type,
cancel1352612707id_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring,type,
comm_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ono__zero__divisors,type,
no_zero_divisors:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__no__zero__divisors,type,
ring_n68954251visors:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__ab__semigroup__add,type,
cancel146912293up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Oab__semigroup__idem__mult,type,
ab_sem1668676832m_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $o ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : fun(fun(B,C),fun(fun(A,B),fun(A,C))) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : fun(A,fun(B,A)) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( fun(A,fun(B,C)) > fun(fun(A,B),fun(A,C)) ) ).
tff(sy_c_Fundamental__Theorem__Algebra__Mirabelle__jmqnahvvas_Ooffset__poly,type,
fundam296178794t_poly:
!>[A: $tType] : ( ( poly(A) * A ) > poly(A) ) ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > fun(A,A) ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Polynomial_OAbs__poly,type,
abs_poly:
!>[A: $tType] : ( fun(nat,A) > poly(A) ) ).
tff(sy_c_Polynomial_Ocoeff,type,
coeff:
!>[A: $tType] : ( poly(A) > fun(nat,A) ) ).
tff(sy_c_Polynomial_Oorder,type,
order:
!>[A: $tType] : ( ( A * poly(A) ) > nat ) ).
tff(sy_c_Polynomial_OpCons,type,
pCons:
!>[A: $tType] : fun(A,fun(poly(A),poly(A))) ).
tff(sy_c_Polynomial_Opcompose,type,
pcompose:
!>[A: $tType] : ( ( poly(A) * poly(A) ) > poly(A) ) ).
tff(sy_c_Polynomial_Opoly,type,
poly1:
!>[A: $tType] : ( poly(A) > fun(A,A) ) ).
tff(sy_c_Polynomial_Opoly__rec,type,
poly_rec:
!>[B: $tType,A: $tType] : ( ( B * fun(A,fun(poly(A),fun(B,B))) * poly(A) ) > B ) ).
tff(sy_c_Polynomial_Osmult,type,
smult:
!>[A: $tType] : fun(A,fun(poly(A),poly(A))) ).
tff(sy_c_Polynomial_Osynthetic__div,type,
synthetic_div:
!>[A: $tType] : ( ( poly(A) * A ) > poly(A) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_fequal,type,
fequal:
!>[A: $tType] : ( ( A * A ) > bool ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_h,type,
h: a ).
%----Relevant facts (99)
tff(fact_0_offset__poly__def,axiom,
! [B: $tType] :
( comm_semiring_0(B)
=> ! [Ha: B,P2: poly(B)] : fundam296178794t_poly(B,P2,Ha) = poly_rec(poly(B),B,zero_zero(poly(B)),aa(fun(B,fun(poly(B),poly(B))),fun(B,fun(poly(B),fun(poly(B),poly(B)))),aa(fun(fun(poly(B),poly(B)),fun(poly(B),fun(poly(B),poly(B)))),fun(fun(B,fun(poly(B),poly(B))),fun(B,fun(poly(B),fun(poly(B),poly(B))))),combb(fun(poly(B),poly(B)),fun(poly(B),fun(poly(B),poly(B))),B),combk(fun(poly(B),poly(B)),poly(B))),aa(fun(B,fun(poly(B),poly(B))),fun(B,fun(poly(B),poly(B))),aa(fun(fun(poly(B),poly(B)),fun(poly(B),poly(B))),fun(fun(B,fun(poly(B),poly(B))),fun(B,fun(poly(B),poly(B)))),combb(fun(poly(B),poly(B)),fun(poly(B),poly(B)),B),combs(poly(B),poly(B),poly(B),aa(fun(poly(B),poly(B)),fun(poly(B),fun(poly(B),poly(B))),aa(fun(poly(B),fun(poly(B),poly(B))),fun(fun(poly(B),poly(B)),fun(poly(B),fun(poly(B),poly(B)))),combb(poly(B),fun(poly(B),poly(B)),poly(B)),plus_plus(poly(B))),aa(B,fun(poly(B),poly(B)),smult(B),Ha)))),pCons(B))),P2) ) ).
tff(fact_1_add__pCons,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Q2: poly(A),B1: A,P1: poly(A),A1: A] : aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),A1),P1)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),B1),Q2)) = aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),P1),Q2)) ) ).
tff(fact_2_smult__0__left,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [P1: poly(A)] : aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),zero_zero(A)),P1) = zero_zero(poly(A)) ) ).
tff(fact_3_smult__eq__0__iff,axiom,
! [B: $tType] :
( idom(B)
=> ! [P2: poly(B),A2: B] :
( ( aa(poly(B),poly(B),aa(B,fun(poly(B),poly(B)),smult(B),A2),P2) = zero_zero(poly(B)) )
<=> ( ( A2 = zero_zero(B) )
| ( P2 = zero_zero(poly(B)) ) ) ) ) ).
tff(fact_4_pCons__0__0,axiom,
! [A: $tType] :
( zero(A)
=> ( aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),zero_zero(A)),zero_zero(poly(A))) = zero_zero(poly(A)) ) ) ).
tff(fact_5_pCons__eq__0__iff,axiom,
! [B: $tType] :
( zero(B)
=> ! [P2: poly(B),A2: B] :
( ( aa(poly(B),poly(B),aa(B,fun(poly(B),poly(B)),pCons(B),A2),P2) = zero_zero(poly(B)) )
<=> ( ( A2 = zero_zero(B) )
& ( P2 = zero_zero(poly(B)) ) ) ) ) ).
tff(fact_6_smult__0__right,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [A1: A] : aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),zero_zero(poly(A))) = zero_zero(poly(A)) ) ).
tff(fact_7_double__zero__sym,axiom,
! [B: $tType] :
( linord219039673up_add(B)
=> ! [A2: B] :
( ( zero_zero(B) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),A2) )
<=> ( A2 = zero_zero(B) ) ) ) ).
tff(fact_8_double__eq__0__iff,axiom,
! [B: $tType] :
( linord219039673up_add(B)
=> ! [A2: B] :
( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),A2) = zero_zero(B) )
<=> ( A2 = zero_zero(B) ) ) ) ).
tff(fact_9_poly__rec__pCons,axiom,
! [B: $tType,C: $tType] :
( zero(C)
=> ! [P2: poly(C),A2: C,Z1: B,F: fun(C,fun(poly(C),fun(B,B)))] :
( ( aa(B,B,aa(poly(C),fun(B,B),aa(C,fun(poly(C),fun(B,B)),F,zero_zero(C)),zero_zero(poly(C))),Z1) = Z1 )
=> ( poly_rec(B,C,Z1,F,aa(poly(C),poly(C),aa(C,fun(poly(C),poly(C)),pCons(C),A2),P2)) = aa(B,B,aa(poly(C),fun(B,B),aa(C,fun(poly(C),fun(B,B)),F,A2),P2),poly_rec(B,C,Z1,F,P2)) ) ) ) ).
tff(fact_10_poly__rec_Osimps,axiom,
! [B: $tType,C: $tType] :
( zero(C)
=> ! [P2: poly(C),A2: C,F: fun(C,fun(poly(C),fun(B,B))),Z1: B] : poly_rec(B,C,Z1,F,aa(poly(C),poly(C),aa(C,fun(poly(C),poly(C)),pCons(C),A2),P2)) = aa(B,B,aa(poly(C),fun(B,B),aa(C,fun(poly(C),fun(B,B)),F,A2),P2),if(B,fequal(poly(C),P2,zero_zero(poly(C))),Z1,poly_rec(B,C,Z1,F,P2))) ) ).
tff(fact_11_poly__rec__0,axiom,
! [C: $tType,B: $tType] :
( zero(C)
=> ! [Z1: B,F: fun(C,fun(poly(C),fun(B,B)))] :
( ( aa(B,B,aa(poly(C),fun(B,B),aa(C,fun(poly(C),fun(B,B)),F,zero_zero(C)),zero_zero(poly(C))),Z1) = Z1 )
=> ( poly_rec(B,C,Z1,F,zero_zero(poly(C))) = Z1 ) ) ) ).
tff(fact_12_add__right__cancel,axiom,
! [B: $tType] :
( cancel_semigroup_add(B)
=> ! [C2: B,A2: B,B2: B] :
( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),C2),A2) )
<=> ( B2 = C2 ) ) ) ).
tff(fact_13_add__left__cancel,axiom,
! [B: $tType] :
( cancel_semigroup_add(B)
=> ! [C2: B,B2: B,A2: B] :
( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),B2) = aa(B,B,aa(B,fun(B,B),plus_plus(B),A2),C2) )
<=> ( B2 = C2 ) ) ) ).
tff(fact_14_pCons__eq__iff,axiom,
! [B: $tType] :
( zero(B)
=> ! [Q1: poly(B),B2: B,P2: poly(B),A2: B] :
( ( aa(poly(B),poly(B),aa(B,fun(poly(B),poly(B)),pCons(B),A2),P2) = aa(poly(B),poly(B),aa(B,fun(poly(B),poly(B)),pCons(B),B2),Q1) )
<=> ( ( A2 = B2 )
& ( P2 = Q1 ) ) ) ) ).
tff(fact_15_zero__reorient,axiom,
! [B: $tType] :
( zero(B)
=> ! [X2: B] :
( ( zero_zero(B) = X2 )
<=> ( X2 = zero_zero(B) ) ) ) ).
tff(fact_16_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,A1: A,B1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),A1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),A1) )
=> ( B1 = C1 ) ) ) ).
tff(fact_17_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C1: A,B1: A,A1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C1) )
=> ( B1 = C1 ) ) ) ).
tff(fact_18_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C1: A,B1: A,A1: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C1) )
=> ( B1 = C1 ) ) ) ).
tff(fact_19_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C1: A,B1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),C1)) ) ).
tff(fact_20_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),zero_zero(A)) = A1 ) ).
tff(fact_21_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),zero_zero(A)) = A1 ) ).
tff(fact_22_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A1) = A1 ) ).
tff(fact_23_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A1) = A1 ) ).
tff(fact_24_add__poly__code_I2_J,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [P1: poly(A)] : aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),P1),zero_zero(poly(A))) = P1 ) ).
tff(fact_25_add__poly__code_I1_J,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Q2: poly(A)] : aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),zero_zero(poly(A))),Q2) = Q2 ) ).
tff(fact_26_smult__add__right,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [Q2: poly(A),P1: poly(A),A1: A] : aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),P1),Q2)) = aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),P1)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),Q2)) ) ).
tff(fact_27_smult__add__left,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [P1: poly(A),B1: A,A1: A] : aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),P1) = aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),P1)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),B1),P1)) ) ).
tff(fact_28_synthetic__div__unique__lemma,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [A1: A,P1: poly(A),C1: A] :
( ( aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),C1),P1) = aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),A1),P1) )
=> ( P1 = zero_zero(poly(A)) ) ) ) ).
tff(fact_29_pCons__induct,axiom,
! [B: $tType] :
( zero(B)
=> ! [P2: poly(B),P3: fun(poly(B),bool)] :
( pp(aa(poly(B),bool,P3,zero_zero(poly(B))))
=> ( ! [A3: B,P4: poly(B)] :
( pp(aa(poly(B),bool,P3,P4))
=> pp(aa(poly(B),bool,P3,aa(poly(B),poly(B),aa(B,fun(poly(B),poly(B)),pCons(B),A3),P4))) )
=> pp(aa(poly(B),bool,P3,P2)) ) ) ) ).
tff(fact_30_times__poly__def,axiom,
! [B: $tType] :
( comm_semiring_0(B)
=> ! [Q1: poly(B),P2: poly(B)] : aa(poly(B),poly(B),times_times(poly(B),P2),Q1) = poly_rec(poly(B),B,zero_zero(poly(B)),aa(fun(B,fun(poly(B),poly(B))),fun(B,fun(poly(B),fun(poly(B),poly(B)))),aa(fun(fun(poly(B),poly(B)),fun(poly(B),fun(poly(B),poly(B)))),fun(fun(B,fun(poly(B),poly(B))),fun(B,fun(poly(B),fun(poly(B),poly(B))))),combb(fun(poly(B),poly(B)),fun(poly(B),fun(poly(B),poly(B))),B),combk(fun(poly(B),poly(B)),poly(B))),combc(B,fun(poly(B),poly(B)),fun(poly(B),poly(B)),aa(fun(B,fun(poly(B),poly(B))),fun(B,fun(fun(poly(B),poly(B)),fun(poly(B),poly(B)))),aa(fun(fun(poly(B),poly(B)),fun(fun(poly(B),poly(B)),fun(poly(B),poly(B)))),fun(fun(B,fun(poly(B),poly(B))),fun(B,fun(fun(poly(B),poly(B)),fun(poly(B),poly(B))))),combb(fun(poly(B),poly(B)),fun(fun(poly(B),poly(B)),fun(poly(B),poly(B))),B),combb(poly(B),poly(B),poly(B))),aa(fun(B,poly(B)),fun(B,fun(poly(B),poly(B))),aa(fun(poly(B),fun(poly(B),poly(B))),fun(fun(B,poly(B)),fun(B,fun(poly(B),poly(B)))),combb(poly(B),fun(poly(B),poly(B)),B),plus_plus(poly(B))),combc(B,poly(B),poly(B),smult(B),Q1))),aa(B,fun(poly(B),poly(B)),pCons(B),zero_zero(B)))),P2) ) ).
tff(fact_31_add__0__iff,axiom,
! [B: $tType] :
( semiri456707255roduct(B)
=> ! [A2: B,B2: B] :
( ( B2 = aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),A2) )
<=> ( A2 = zero_zero(B) ) ) ) ).
tff(fact_32_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),zero_zero(A)) = A1 ) ).
tff(fact_33_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),zero_zero(A)),A1) = A1 ) ).
tff(fact_34_mult__pCons__left,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [Q2: poly(A),P1: poly(A),A1: A] : aa(poly(A),poly(A),times_times(poly(A),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),A1),P1)),Q2) = aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),Q2)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),zero_zero(A)),aa(poly(A),poly(A),times_times(poly(A),P1),Q2))) ) ).
tff(fact_35_mult__pCons__right,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [Q2: poly(A),A1: A,P1: poly(A)] : aa(poly(A),poly(A),times_times(poly(A),P1),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),A1),Q2)) = aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),P1)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),zero_zero(A)),aa(poly(A),poly(A),times_times(poly(A),P1),Q2))) ) ).
tff(fact_36_pCons__cases,axiom,
! [A: $tType] :
( zero(A)
=> ! [P1: poly(A)] :
~ ! [A3: A,Q3: poly(A)] : P1 != aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),A3),Q3) ) ).
tff(fact_37_pcompose__0,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [Q2: poly(A)] : pcompose(A,zero_zero(poly(A)),Q2) = zero_zero(poly(A)) ) ).
tff(fact_38_smult__smult,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [P1: poly(A),B1: A,A1: A] : aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),B1),P1)) = aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),aa(A,A,times_times(A,A1),B1)),P1) ) ).
tff(fact_39_mult__smult__right,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [Q2: poly(A),A1: A,P1: poly(A)] : aa(poly(A),poly(A),times_times(poly(A),P1),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),Q2)) = aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),aa(poly(A),poly(A),times_times(poly(A),P1),Q2)) ) ).
tff(fact_40_mult__smult__left,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [Q2: poly(A),P1: poly(A),A1: A] : aa(poly(A),poly(A),times_times(poly(A),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),P1)),Q2) = aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),aa(poly(A),poly(A),times_times(poly(A),P1),Q2)) ) ).
tff(fact_41_smult__pCons,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [P1: poly(A),B1: A,A1: A] : aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),B1),P1)) = aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),aa(A,A,times_times(A,A1),B1)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),P1)) ) ).
tff(fact_42_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [C1: A,B1: A,A1: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,A1),B1)),C1) = aa(A,A,times_times(A,A1),aa(A,A,times_times(A,B1),C1)) ) ).
tff(fact_43_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : aa(A,A,times_times(A,A1),zero_zero(A)) = zero_zero(A) ) ).
tff(fact_44_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : aa(A,A,times_times(A,zero_zero(A)),A1) = zero_zero(A) ) ).
tff(fact_45_crossproduct__eq,axiom,
! [B: $tType] :
( semiri456707255roduct(B)
=> ! [Z1: B,X2: B,Y1: B,W: B] :
( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,times_times(B,W),Y1)),aa(B,B,times_times(B,X2),Z1)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,times_times(B,W),Z1)),aa(B,B,times_times(B,X2),Y1)) )
<=> ( ( W = X2 )
| ( Y1 = Z1 ) ) ) ) ).
tff(fact_46_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B1: A,M: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,A1),M)),aa(A,A,times_times(A,B1),M)) = aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),M) ) ).
tff(fact_47_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,B1: A,A1: A] : aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,A1),C1)),aa(A,A,times_times(A,B1),C1)) ) ).
tff(fact_48_crossproduct__noteq,axiom,
! [B: $tType] :
( semiri456707255roduct(B)
=> ! [D1: B,C2: B,B2: B,A2: B] :
( ( ( A2 != B2 )
& ( C2 != D1 ) )
<=> ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,times_times(B,A2),C2)),aa(B,B,times_times(B,B2),D1)) != aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,times_times(B,A2),D1)),aa(B,B,times_times(B,B2),C2)) ) ) ) ).
tff(fact_49_comm__semiring__1__class_Onormalizing__semiring__rules_I34_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Z: A,Y: A,X: A] : aa(A,A,times_times(A,X),aa(A,A,aa(A,fun(A,A),plus_plus(A),Y),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,X),Y)),aa(A,A,times_times(A,X),Z)) ) ).
tff(fact_50_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),aa(A,A,times_times(A,Rx),Ry)) = aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Rx)),aa(A,A,times_times(A,Ly),Ry)) ) ).
tff(fact_51_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),aa(A,A,times_times(A,Rx),Ry)) = aa(A,A,times_times(A,Rx),aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),Ry)) ) ).
tff(fact_52_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),aa(A,A,times_times(A,Rx),Ry)) = aa(A,A,times_times(A,Lx),aa(A,A,times_times(A,Ly),aa(A,A,times_times(A,Rx),Ry))) ) ).
tff(fact_53_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),Rx) = aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Rx)),Ly) ) ).
tff(fact_54_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Ly)),Rx) = aa(A,A,times_times(A,Lx),aa(A,A,times_times(A,Ly),Rx)) ) ).
tff(fact_55_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : aa(A,A,times_times(A,Lx),aa(A,A,times_times(A,Rx),Ry)) = aa(A,A,times_times(A,aa(A,A,times_times(A,Lx),Rx)),Ry) ) ).
tff(fact_56_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : aa(A,A,times_times(A,Lx),aa(A,A,times_times(A,Rx),Ry)) = aa(A,A,times_times(A,Rx),aa(A,A,times_times(A,Lx),Ry)) ) ).
tff(fact_57_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B1: A,A1: A] : aa(A,A,times_times(A,A1),B1) = aa(A,A,times_times(A,B1),A1) ) ).
tff(fact_58_mult__poly__0__left,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [Q2: poly(A)] : aa(poly(A),poly(A),times_times(poly(A),zero_zero(poly(A))),Q2) = zero_zero(poly(A)) ) ).
tff(fact_59_mult__poly__0__right,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [P1: poly(A)] : aa(poly(A),poly(A),times_times(poly(A),P1),zero_zero(poly(A))) = zero_zero(poly(A)) ) ).
tff(fact_60_add__scale__eq__noteq,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [D: A,C1: A,B1: A,A1: A,R1: A] :
( ( R1 != zero_zero(A) )
=> ( ( ( A1 = B1 )
& ( C1 != D ) )
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,times_times(A,R1),C1)) != aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),aa(A,A,times_times(A,R1),D)) ) ) ) ) ).
tff(fact_61_mult__poly__add__left,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [R1: poly(A),Q2: poly(A),P1: poly(A)] : aa(poly(A),poly(A),times_times(poly(A),aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),P1),Q2)),R1) = aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),aa(poly(A),poly(A),times_times(poly(A),P1),R1)),aa(poly(A),poly(A),times_times(poly(A),Q2),R1)) ) ).
tff(fact_62_pcompose__pCons,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [Q2: poly(A),P1: poly(A),A1: A] : pcompose(A,aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),A1),P1),Q2) = aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),A1),zero_zero(poly(A)))),aa(poly(A),poly(A),times_times(poly(A),Q2),pcompose(A,P1,Q2))) ) ).
tff(fact_63_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),A1) ) ).
tff(fact_64_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),D)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),D)) ) ).
tff(fact_65_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),D)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C1)),D) ) ).
tff(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,B1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),C1)) ) ).
tff(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C1: A,B1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C1)),B1) ) ).
tff(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C1: A,B1: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),C1),D)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),C1)),aa(A,A,aa(A,fun(A,A),plus_plus(A),B1),D)) ) ).
tff(fact_69_pcompose__def,axiom,
! [B: $tType] :
( comm_semiring_0(B)
=> ! [Q1: poly(B),P2: poly(B)] : pcompose(B,P2,Q1) = poly_rec(poly(B),B,zero_zero(poly(B)),aa(fun(B,fun(poly(B),poly(B))),fun(B,fun(poly(B),fun(poly(B),poly(B)))),aa(fun(fun(poly(B),poly(B)),fun(poly(B),fun(poly(B),poly(B)))),fun(fun(B,fun(poly(B),poly(B))),fun(B,fun(poly(B),fun(poly(B),poly(B))))),combb(fun(poly(B),poly(B)),fun(poly(B),fun(poly(B),poly(B))),B),combk(fun(poly(B),poly(B)),poly(B))),combc(B,fun(poly(B),poly(B)),fun(poly(B),poly(B)),aa(fun(B,fun(poly(B),poly(B))),fun(B,fun(fun(poly(B),poly(B)),fun(poly(B),poly(B)))),aa(fun(fun(poly(B),poly(B)),fun(fun(poly(B),poly(B)),fun(poly(B),poly(B)))),fun(fun(B,fun(poly(B),poly(B))),fun(B,fun(fun(poly(B),poly(B)),fun(poly(B),poly(B))))),combb(fun(poly(B),poly(B)),fun(fun(poly(B),poly(B)),fun(poly(B),poly(B))),B),combb(poly(B),poly(B),poly(B))),aa(fun(B,poly(B)),fun(B,fun(poly(B),poly(B))),aa(fun(poly(B),fun(poly(B),poly(B))),fun(fun(B,poly(B)),fun(B,fun(poly(B),poly(B)))),combb(poly(B),fun(poly(B),poly(B)),B),plus_plus(poly(B))),combc(B,poly(B),poly(B),pCons(B),zero_zero(poly(B))))),times_times(poly(B),Q1))),P2) ) ).
tff(fact_70_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : aa(A,A,times_times(A,zero_zero(A)),A1) = zero_zero(A) ) ).
tff(fact_71_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : aa(A,A,times_times(A,A1),zero_zero(A)) = zero_zero(A) ) ).
tff(fact_72_mult__eq__0__iff,axiom,
! [B: $tType] :
( ring_n68954251visors(B)
=> ! [B2: B,A2: B] :
( ( aa(B,B,times_times(B,A2),B2) = zero_zero(B) )
<=> ( ( A2 = zero_zero(B) )
| ( B2 = zero_zero(B) ) ) ) ) ).
tff(fact_73_sum__squares__eq__zero__iff,axiom,
! [B: $tType] :
( linord581940658strict(B)
=> ! [Y1: B,X2: B] :
( ( aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,times_times(B,X2),X2)),aa(B,B,times_times(B,Y1),Y1)) = zero_zero(B) )
<=> ( ( X2 = zero_zero(B) )
& ( Y1 = zero_zero(B) ) ) ) ) ).
tff(fact_74_ext,axiom,
! [C: $tType,B: $tType,G: fun(B,C),F: fun(B,C)] :
( ! [X1: B] : aa(B,C,F,X1) = aa(B,C,G,X1)
=> ( F = G ) ) ).
tff(fact_75_mult__left__idem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [B1: A,A1: A] : aa(A,A,times_times(A,A1),aa(A,A,times_times(A,A1),B1)) = aa(A,A,times_times(A,A1),B1) ) ).
tff(fact_76_times_Oidem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [A1: A] : aa(A,A,times_times(A,A1),A1) = A1 ) ).
tff(fact_77_mult__idem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [X: A] : aa(A,A,times_times(A,X),X) = X ) ).
tff(fact_78_no__zero__divisors,axiom,
! [A: $tType] :
( no_zero_divisors(A)
=> ! [B1: A,A1: A] :
( ( A1 != zero_zero(A) )
=> ( ( B1 != zero_zero(A) )
=> ( aa(A,A,times_times(A,A1),B1) != zero_zero(A) ) ) ) ) ).
tff(fact_79_divisors__zero,axiom,
! [A: $tType] :
( no_zero_divisors(A)
=> ! [B1: A,A1: A] :
( ( aa(A,A,times_times(A,A1),B1) = zero_zero(A) )
=> ( ( A1 = zero_zero(A) )
| ( B1 = zero_zero(A) ) ) ) ) ).
tff(fact_80_comm__semiring__class_Odistrib,axiom,
! [A: $tType] :
( comm_semiring(A)
=> ! [C1: A,B1: A,A1: A] : aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),C1) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,A1),C1)),aa(A,A,times_times(A,B1),C1)) ) ).
tff(fact_81_combine__common__factor,axiom,
! [A: $tType] :
( semiring(A)
=> ! [C1: A,B1: A,E: A,A1: A] : aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,A1),E)),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,B1),E)),C1)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),B1)),E)),C1) ) ).
tff(fact_82_poly__pCons,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X: A,P1: poly(A),A1: A] : aa(A,A,poly1(A,aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),A1),P1)),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),A1),aa(A,A,times_times(A,X),aa(A,A,poly1(A,P1),X))) ) ).
tff(fact_83_zero__poly__def,axiom,
! [B: $tType] :
( zero(B)
=> ( zero_zero(poly(B)) = abs_poly(B,aa(B,fun(nat,B),combk(B,nat),zero_zero(B))) ) ) ).
tff(fact_84_poly__0,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X: A] : aa(A,A,poly1(A,zero_zero(poly(A))),X) = zero_zero(A) ) ).
tff(fact_85_poly__smult,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X: A,P1: poly(A),A1: A] : aa(A,A,poly1(A,aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),P1)),X) = aa(A,A,times_times(A,A1),aa(A,A,poly1(A,P1),X)) ) ).
tff(fact_86_poly__add,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X: A,Q2: poly(A),P1: poly(A)] : aa(A,A,poly1(A,aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),P1),Q2)),X) = aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,poly1(A,P1),X)),aa(A,A,poly1(A,Q2),X)) ) ).
tff(fact_87_poly__mult,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X: A,Q2: poly(A),P1: poly(A)] : aa(A,A,poly1(A,aa(poly(A),poly(A),times_times(poly(A),P1),Q2)),X) = aa(A,A,times_times(A,aa(A,A,poly1(A,P1),X)),aa(A,A,poly1(A,Q2),X)) ) ).
tff(fact_88_poly__zero,axiom,
! [B: $tType] :
( ( ring_char_0(B)
& idom(B) )
=> ! [P2: poly(B)] :
( ( poly1(B,P2) = poly1(B,zero_zero(poly(B))) )
<=> ( P2 = zero_zero(poly(B)) ) ) ) ).
tff(fact_89_poly__pcompose,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X: A,Q2: poly(A),P1: poly(A)] : aa(A,A,poly1(A,pcompose(A,P1,Q2)),X) = aa(A,A,poly1(A,P1),aa(A,A,poly1(A,Q2),X)) ) ).
tff(fact_90_poly__eq__iff,axiom,
! [B: $tType] :
( ( ring_char_0(B)
& idom(B) )
=> ! [Q1: poly(B),P2: poly(B)] :
( ( poly1(B,P2) = poly1(B,Q1) )
<=> ( P2 = Q1 ) ) ) ).
tff(fact_91_synthetic__div__unique,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [R1: A,Q2: poly(A),C1: A,P1: poly(A)] :
( ( aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),P1),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),C1),Q2)) = aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),R1),Q2) )
=> ( ( R1 = aa(A,A,poly1(A,P1),C1) )
& ( Q2 = synthetic_div(A,P1,C1) ) ) ) ) ).
tff(fact_92_synthetic__div__correct,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [C1: A,P1: poly(A)] : aa(poly(A),poly(A),aa(poly(A),fun(poly(A),poly(A)),plus_plus(poly(A)),P1),aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),C1),synthetic_div(A,P1,C1))) = aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),aa(A,A,poly1(A,P1),C1)),synthetic_div(A,P1,C1)) ) ).
tff(fact_93_synthetic__div__0,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [C1: A] : synthetic_div(A,zero_zero(poly(A)),C1) = zero_zero(poly(A)) ) ).
tff(fact_94_synthetic__div__pCons,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [C1: A,P1: poly(A),A1: A] : synthetic_div(A,aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),A1),P1),C1) = aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),pCons(A),aa(A,A,poly1(A,P1),C1)),synthetic_div(A,P1,C1)) ) ).
tff(fact_95_order__root,axiom,
! [B: $tType] :
( idom(B)
=> ! [A2: B,P2: poly(B)] :
( ( aa(B,B,poly1(B,P2),A2) = zero_zero(B) )
<=> ( ( P2 = zero_zero(poly(B)) )
| ( order(B,A2,P2) != zero_zero(nat) ) ) ) ) ).
tff(fact_96_plus__poly__def,axiom,
! [B: $tType] :
( comm_monoid_add(B)
=> ! [Q1: poly(B),P2: poly(B)] : aa(poly(B),poly(B),aa(poly(B),fun(poly(B),poly(B)),plus_plus(poly(B)),P2),Q1) = abs_poly(B,aa(fun(nat,B),fun(nat,B),combs(nat,B,B,aa(fun(nat,B),fun(nat,fun(B,B)),aa(fun(B,fun(B,B)),fun(fun(nat,B),fun(nat,fun(B,B))),combb(B,fun(B,B),nat),plus_plus(B)),coeff(B,P2))),coeff(B,Q1))) ) ).
tff(fact_97_coeff__0,axiom,
! [A: $tType] :
( zero(A)
=> ! [N: nat] : aa(nat,A,coeff(A,zero_zero(poly(A))),N) = zero_zero(A) ) ).
tff(fact_98_coeff__smult,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [N: nat,P1: poly(A),A1: A] : aa(nat,A,coeff(A,aa(poly(A),poly(A),aa(A,fun(poly(A),poly(A)),smult(A),A1),P1)),N) = aa(A,A,times_times(A,A1),aa(nat,A,coeff(A,P1),N)) ) ).
%----Class relationships (8)
tff(clar_Rings_Ocomm__semiring__0__Groups_Oab__semigroup__mult,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ab_semigroup_mult(A) ) ).
tff(clar_Rings_Ocomm__semiring__0__Groups_Oab__semigroup__add,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ab_semigroup_add(A) ) ).
tff(clar_Rings_Ocomm__semiring__0__Groups_Ocomm__monoid__add,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> comm_monoid_add(A) ) ).
tff(clar_Rings_Ocomm__semiring__0__Rings_Ocomm__semiring,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> comm_semiring(A) ) ).
tff(clar_Rings_Ocomm__semiring__0__Groups_Omonoid__add,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> monoid_add(A) ) ).
tff(clar_Rings_Ocomm__semiring__0__Rings_Omult__zero,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> mult_zero(A) ) ).
tff(clar_Rings_Ocomm__semiring__0__Rings_Osemiring,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> semiring(A) ) ).
tff(clar_Rings_Ocomm__semiring__0__Groups_Ozero,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> zero(A) ) ).
%----Arities (36)
tff(arity_Polynomial_Opoly___Rings_Olinordered__idom,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linordered_idom(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ocancel__comm__monoid__add,axiom,
! [T_1: $tType] :
( cancel1352612707id_add(T_1)
=> cancel1352612707id_add(poly(T_1)) ) ).
tff(arity_Nat_Onat___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add(nat) ).
tff(arity_Nat_Onat___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(nat) ).
tff(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(nat) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring,axiom,
comm_semiring(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__add,axiom,
monoid_add(nat) ).
tff(arity_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring,axiom,
semiring(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Polynomial_Opoly___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
! [T_1: $tType] :
( idom(T_1)
=> semiri456707255roduct(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Olinordered__ab__group__add,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linord219039673up_add(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ocancel__ab__semigroup__add,axiom,
! [T_1: $tType] :
( cancel1352612707id_add(T_1)
=> cancel146912293up_add(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Olinordered__ring__strict,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linord581940658strict(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Oring__no__zero__divisors,axiom,
! [T_1: $tType] :
( idom(T_1)
=> ring_n68954251visors(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ocancel__semigroup__add,axiom,
! [T_1: $tType] :
( cancel1352612707id_add(T_1)
=> cancel_semigroup_add(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Oab__semigroup__mult,axiom,
! [T_1: $tType] :
( comm_semiring_0(T_1)
=> ab_semigroup_mult(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Oab__semigroup__add,axiom,
! [T_1: $tType] :
( comm_monoid_add(T_1)
=> ab_semigroup_add(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Ono__zero__divisors,axiom,
! [T_1: $tType] :
( idom(T_1)
=> no_zero_divisors(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ocomm__monoid__add,axiom,
! [T_1: $tType] :
( comm_monoid_add(T_1)
=> comm_monoid_add(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Ocomm__semiring__1,axiom,
! [T_1: $tType] :
( comm_semiring_1(T_1)
=> comm_semiring_1(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Ocomm__semiring__0,axiom,
! [T_1: $tType] :
( comm_semiring_0(T_1)
=> comm_semiring_0(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Ocomm__semiring,axiom,
! [T_1: $tType] :
( comm_semiring_0(T_1)
=> comm_semiring(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Omonoid__add,axiom,
! [T_1: $tType] :
( comm_monoid_add(T_1)
=> monoid_add(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Omult__zero,axiom,
! [T_1: $tType] :
( comm_semiring_0(T_1)
=> mult_zero(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Int_Oring__char__0,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> ring_char_0(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Osemiring,axiom,
! [T_1: $tType] :
( comm_semiring_0(T_1)
=> semiring(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ozero,axiom,
! [T_1: $tType] :
( zero(T_1)
=> zero(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Oidom,axiom,
! [T_1: $tType] :
( idom(T_1)
=> idom(poly(T_1)) ) ).
%----Helper facts (11)
tff(help_If_1_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fTrue,X,Y) = X ).
tff(help_If_2_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fFalse,X,Y) = Y ).
tff(help_If_3_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,aa(fun(A,B),fun(A,C),aa(fun(B,C),fun(fun(A,B),fun(A,C)),combb(B,C,A),P),Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,aa(A,fun(B,A),combk(A,B),P),Q) = P ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,aa(fun(A,B),fun(A,C),combs(A,B,C,P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
tff(help_fequal_1_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ~ pp(fequal(A,X,Y))
| ( X = Y ) ) ).
tff(help_fequal_2_1_T,axiom,
! [A: $tType,Y: A,X: A] :
( ( X != Y )
| pp(fequal(A,X,Y)) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
poly_rec(poly(a),a,zero_zero(poly(a)),aa(fun(a,fun(poly(a),poly(a))),fun(a,fun(poly(a),fun(poly(a),poly(a)))),aa(fun(fun(poly(a),poly(a)),fun(poly(a),fun(poly(a),poly(a)))),fun(fun(a,fun(poly(a),poly(a))),fun(a,fun(poly(a),fun(poly(a),poly(a))))),combb(fun(poly(a),poly(a)),fun(poly(a),fun(poly(a),poly(a))),a),combk(fun(poly(a),poly(a)),poly(a))),aa(fun(a,fun(poly(a),poly(a))),fun(a,fun(poly(a),poly(a))),aa(fun(fun(poly(a),poly(a)),fun(poly(a),poly(a))),fun(fun(a,fun(poly(a),poly(a))),fun(a,fun(poly(a),poly(a)))),combb(fun(poly(a),poly(a)),fun(poly(a),poly(a)),a),combs(poly(a),poly(a),poly(a),aa(fun(poly(a),poly(a)),fun(poly(a),fun(poly(a),poly(a))),aa(fun(poly(a),fun(poly(a),poly(a))),fun(fun(poly(a),poly(a)),fun(poly(a),fun(poly(a),poly(a)))),combb(poly(a),fun(poly(a),poly(a)),poly(a)),plus_plus(poly(a))),aa(a,fun(poly(a),poly(a)),smult(a),h)))),pCons(a))),zero_zero(poly(a))) = zero_zero(poly(a)) ).
%----Type variables (1)
tff(tfree_0,hypothesis,
comm_semiring_0(a) ).
%------------------------------------------------------------------------------