TPTP Problem File: SWW491_5.p
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%------------------------------------------------------------------------------
% File : SWW491_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 132
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_132 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 183 ( 46 unt; 42 typ; 0 def)
% Number of atoms : 286 ( 160 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 166 ( 21 ~; 9 |; 12 &)
% ( 20 <=>; 104 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 32 ( 19 >; 13 *; 0 +; 0 <<)
% Number of predicates : 16 ( 15 usr; 0 prp; 1-2 aty)
% Number of functors : 24 ( 24 usr; 5 con; 0-5 aty)
% Number of variables : 314 ( 283 !; 1 ?; 314 :)
% ( 30 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:45
%------------------------------------------------------------------------------
%----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 (37)
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_Int_Oring__char__0,type,
ring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__add,type,
monoid_add:
!>[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_Groups_Oab__semigroup__add,type,
ab_semigroup_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocancel__semigroup__add,type,
cancel_semigroup_add:
!>[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_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $o ).
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] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_HOL_Obool_Obool__size,type,
bool_size: bool > nat ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Nat_OSuc,type,
suc: nat > nat ).
tff(sy_c_Nat_Onat_Onat__size,type,
nat_size: nat > nat ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > nat ) ).
tff(sy_c_Polynomial_Odegree,type,
degree:
!>[A: $tType] : ( poly(A) > nat ) ).
tff(sy_c_Polynomial_Oorder,type,
order:
!>[A: $tType] : ( ( A * poly(A) ) > nat ) ).
tff(sy_c_Polynomial_OpCons,type,
pCons:
!>[A: $tType] : ( ( 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] : ( ( A * 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_a,type,
a1: a ).
tff(sy_v_h,type,
h: a ).
tff(sy_v_pa,type,
pa: poly(a) ).
%----Relevant facts (98)
tff(fact_0_offset__poly__0,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [H: A] : fundam296178794t_poly(A,zero_zero(poly(A)),H) = zero_zero(poly(A)) ) ).
tff(fact_1_offset__poly__single,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [H: A,A2: A] : fundam296178794t_poly(A,pCons(A,A2,zero_zero(poly(A))),H) = pCons(A,A2,zero_zero(poly(A))) ) ).
tff(fact_2_offset__poly__eq__0__iff,axiom,
! [B: $tType] :
( comm_semiring_0(B)
=> ! [Ha: B,Pa: poly(B)] :
( ( fundam296178794t_poly(B,Pa,Ha) = zero_zero(poly(B)) )
<=> ( Pa = zero_zero(poly(B)) ) ) ) ).
tff(fact_3_offset__poly__pCons,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [H: A,P1: poly(A),A2: A] : fundam296178794t_poly(A,pCons(A,A2,P1),H) = plus_plus(poly(A),smult(A,H,fundam296178794t_poly(A,P1,H)),pCons(A,A2,fundam296178794t_poly(A,P1,H))) ) ).
tff(fact_4_offset__poly__eq__0__lemma,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [A2: A,P1: poly(A),C: A] :
( ( plus_plus(poly(A),smult(A,C,P1),pCons(A,A2,P1)) = zero_zero(poly(A)) )
=> ( P1 = zero_zero(poly(A)) ) ) ) ).
tff(fact_5_add__pCons,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Q1: poly(A),B2: A,P1: poly(A),A2: A] : plus_plus(poly(A),pCons(A,A2,P1),pCons(A,B2,Q1)) = pCons(A,plus_plus(A,A2,B2),plus_plus(poly(A),P1,Q1)) ) ).
tff(fact_6_degree__smult__eq,axiom,
! [A: $tType] :
( idom(A)
=> ! [P1: poly(A),A2: A] :
( ( ( A2 = zero_zero(A) )
=> ( degree(A,smult(A,A2,P1)) = zero_zero(nat) ) )
& ( ( A2 != zero_zero(A) )
=> ( degree(A,smult(A,A2,P1)) = degree(A,P1) ) ) ) ) ).
tff(fact_7_smult__0__left,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [P1: poly(A)] : smult(A,zero_zero(A),P1) = zero_zero(poly(A)) ) ).
tff(fact_8_smult__eq__0__iff,axiom,
! [B: $tType] :
( idom(B)
=> ! [Pa: poly(B),A1: B] :
( ( smult(B,A1,Pa) = zero_zero(poly(B)) )
<=> ( ( A1 = zero_zero(B) )
| ( Pa = zero_zero(poly(B)) ) ) ) ) ).
tff(fact_9_pCons__0__0,axiom,
! [A: $tType] :
( zero(A)
=> ( pCons(A,zero_zero(A),zero_zero(poly(A))) = zero_zero(poly(A)) ) ) ).
tff(fact_10_pCons__eq__0__iff,axiom,
! [B: $tType] :
( zero(B)
=> ! [Pa: poly(B),A1: B] :
( ( pCons(B,A1,Pa) = zero_zero(poly(B)) )
<=> ( ( A1 = zero_zero(B) )
& ( Pa = zero_zero(poly(B)) ) ) ) ) ).
tff(fact_11_smult__0__right,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [A2: A] : smult(A,A2,zero_zero(poly(A))) = zero_zero(poly(A)) ) ).
tff(fact_12_double__zero__sym,axiom,
! [B: $tType] :
( linord219039673up_add(B)
=> ! [A1: B] :
( ( zero_zero(B) = plus_plus(B,A1,A1) )
<=> ( A1 = zero_zero(B) ) ) ) ).
tff(fact_13_double__eq__0__iff,axiom,
! [B: $tType] :
( linord219039673up_add(B)
=> ! [A1: B] :
( ( plus_plus(B,A1,A1) = zero_zero(B) )
<=> ( A1 = zero_zero(B) ) ) ) ).
tff(fact_14_degree__pCons__eq,axiom,
! [A: $tType] :
( zero(A)
=> ! [A2: A,P1: poly(A)] :
( ( P1 != zero_zero(poly(A)) )
=> ( degree(A,pCons(A,A2,P1)) = suc(degree(A,P1)) ) ) ) ).
tff(fact_15_synthetic__div__unique__lemma,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [A2: A,P1: poly(A),C: A] :
( ( smult(A,C,P1) = pCons(A,A2,P1) )
=> ( P1 = zero_zero(poly(A)) ) ) ) ).
tff(fact_16_smult__add__left,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [P1: poly(A),B2: A,A2: A] : smult(A,plus_plus(A,A2,B2),P1) = plus_plus(poly(A),smult(A,A2,P1),smult(A,B2,P1)) ) ).
tff(fact_17_add__right__cancel,axiom,
! [B: $tType] :
( cancel_semigroup_add(B)
=> ! [C2: B,A1: B,B1: B] :
( ( plus_plus(B,B1,A1) = plus_plus(B,C2,A1) )
<=> ( B1 = C2 ) ) ) ).
tff(fact_18_add__left__cancel,axiom,
! [B: $tType] :
( cancel_semigroup_add(B)
=> ! [C2: B,B1: B,A1: B] :
( ( plus_plus(B,A1,B1) = plus_plus(B,A1,C2) )
<=> ( B1 = C2 ) ) ) ).
tff(fact_19_pCons__eq__iff,axiom,
! [B: $tType] :
( zero(B)
=> ! [Q: poly(B),B1: B,Pa: poly(B),A1: B] :
( ( pCons(B,A1,Pa) = pCons(B,B1,Q) )
<=> ( ( A1 = B1 )
& ( Pa = Q ) ) ) ) ).
tff(fact_20_degree__0,axiom,
! [A: $tType] :
( zero(A)
=> ( degree(A,zero_zero(poly(A))) = zero_zero(nat) ) ) ).
tff(fact_21_degree__pCons__eq__if,axiom,
! [A: $tType] :
( zero(A)
=> ! [A2: A,P1: poly(A)] :
( ( ( P1 = zero_zero(poly(A)) )
=> ( degree(A,pCons(A,A2,P1)) = zero_zero(nat) ) )
& ( ( P1 != zero_zero(poly(A)) )
=> ( degree(A,pCons(A,A2,P1)) = suc(degree(A,P1)) ) ) ) ) ).
tff(fact_22_degree__pCons__0,axiom,
! [A: $tType] :
( zero(A)
=> ! [A2: A] : degree(A,pCons(A,A2,zero_zero(poly(A)))) = zero_zero(nat) ) ).
tff(fact_23_zero__reorient,axiom,
! [B: $tType] :
( zero(B)
=> ! [X2: B] :
( ( zero_zero(B) = X2 )
<=> ( X2 = zero_zero(B) ) ) ) ).
tff(fact_24_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A2: A,B2: A] :
( ( plus_plus(A,B2,A2) = plus_plus(A,C,A2) )
=> ( B2 = C ) ) ) ).
tff(fact_25_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B2: A,A2: A] :
( ( plus_plus(A,A2,B2) = plus_plus(A,A2,C) )
=> ( B2 = C ) ) ) ).
tff(fact_26_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B2: A,A2: A] :
( ( plus_plus(A,A2,B2) = plus_plus(A,A2,C) )
=> ( B2 = C ) ) ) ).
tff(fact_27_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C: A,B2: A,A2: A] : plus_plus(A,plus_plus(A,A2,B2),C) = plus_plus(A,A2,plus_plus(A,B2,C)) ) ).
tff(fact_28_add_Ocomm__neutral,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A2: A] : plus_plus(A,A2,zero_zero(A)) = A2 ) ).
tff(fact_29_add__0__right,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A2: A] : plus_plus(A,A2,zero_zero(A)) = A2 ) ).
tff(fact_30_add__0,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [A2: A] : plus_plus(A,zero_zero(A),A2) = A2 ) ).
tff(fact_31_add__0__left,axiom,
! [A: $tType] :
( monoid_add(A)
=> ! [A2: A] : plus_plus(A,zero_zero(A),A2) = A2 ) ).
tff(fact_32_add__poly__code_I2_J,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [P1: poly(A)] : plus_plus(poly(A),P1,zero_zero(poly(A))) = P1 ) ).
tff(fact_33_add__poly__code_I1_J,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Q1: poly(A)] : plus_plus(poly(A),zero_zero(poly(A)),Q1) = Q1 ) ).
tff(fact_34_smult__add__right,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [Q1: poly(A),P1: poly(A),A2: A] : smult(A,A2,plus_plus(poly(A),P1,Q1)) = plus_plus(poly(A),smult(A,A2,P1),smult(A,A2,Q1)) ) ).
tff(fact_35_pCons__induct,axiom,
! [B: $tType] :
( zero(B)
=> ! [Pa: poly(B),P2: fun(poly(B),bool)] :
( pp(aa(poly(B),bool,P2,zero_zero(poly(B))))
=> ( ! [A3: B,P3: poly(B)] :
( pp(aa(poly(B),bool,P2,P3))
=> pp(aa(poly(B),bool,P2,pCons(B,A3,P3))) )
=> pp(aa(poly(B),bool,P2,Pa)) ) ) ) ).
tff(fact_36_add__Suc,axiom,
! [N: nat,M1: nat] : plus_plus(nat,suc(M1),N) = suc(plus_plus(nat,M1,N)) ).
tff(fact_37_add__Suc__right,axiom,
! [N: nat,M1: nat] : plus_plus(nat,M1,suc(N)) = suc(plus_plus(nat,M1,N)) ).
tff(fact_38_nat_Oinject,axiom,
! [Nat5: nat,Nat4: nat] :
( ( suc(Nat4) = suc(Nat5) )
<=> ( Nat4 = Nat5 ) ) ).
tff(fact_39_add__is__0,axiom,
! [N2: nat,M2: nat] :
( ( plus_plus(nat,M2,N2) = zero_zero(nat) )
<=> ( ( M2 = zero_zero(nat) )
& ( N2 = zero_zero(nat) ) ) ) ).
tff(fact_40_one__is__add,axiom,
! [N2: nat,M2: nat] :
( ( suc(zero_zero(nat)) = plus_plus(nat,M2,N2) )
<=> ( ( ( M2 = suc(zero_zero(nat)) )
& ( N2 = zero_zero(nat) ) )
| ( ( M2 = zero_zero(nat) )
& ( N2 = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_41_add__is__1,axiom,
! [N2: nat,M2: nat] :
( ( plus_plus(nat,M2,N2) = suc(zero_zero(nat)) )
<=> ( ( ( M2 = suc(zero_zero(nat)) )
& ( N2 = zero_zero(nat) ) )
| ( ( M2 = zero_zero(nat) )
& ( N2 = suc(zero_zero(nat)) ) ) ) ) ).
tff(fact_42_nat__add__left__cancel,axiom,
! [N2: nat,M2: nat,K: nat] :
( ( plus_plus(nat,K,M2) = plus_plus(nat,K,N2) )
<=> ( M2 = N2 ) ) ).
tff(fact_43_nat__add__right__cancel,axiom,
! [N2: nat,K: nat,M2: nat] :
( ( plus_plus(nat,M2,K) = plus_plus(nat,N2,K) )
<=> ( M2 = N2 ) ) ).
tff(fact_44_nat__add__commute,axiom,
! [N: nat,M1: nat] : plus_plus(nat,M1,N) = plus_plus(nat,N,M1) ).
tff(fact_45_nat__add__left__commute,axiom,
! [Z1: nat,Y: nat,X: nat] : plus_plus(nat,X,plus_plus(nat,Y,Z1)) = plus_plus(nat,Y,plus_plus(nat,X,Z1)) ).
tff(fact_46_nat__add__assoc,axiom,
! [K1: nat,N: nat,M1: nat] : plus_plus(nat,plus_plus(nat,M1,N),K1) = plus_plus(nat,M1,plus_plus(nat,N,K1)) ).
tff(fact_47_add__eq__self__zero,axiom,
! [N: nat,M1: nat] :
( ( plus_plus(nat,M1,N) = M1 )
=> ( N = zero_zero(nat) ) ) ).
tff(fact_48_Nat_Oadd__0__right,axiom,
! [M1: nat] : plus_plus(nat,M1,zero_zero(nat)) = M1 ).
tff(fact_49_plus__nat_Oadd__0,axiom,
! [N: nat] : plus_plus(nat,zero_zero(nat),N) = N ).
tff(fact_50_n__not__Suc__n,axiom,
! [N: nat] : N != suc(N) ).
tff(fact_51_Suc__n__not__n,axiom,
! [N: nat] : suc(N) != N ).
tff(fact_52_Suc__inject,axiom,
! [Y: nat,X: nat] :
( ( suc(X) = suc(Y) )
=> ( X = Y ) ) ).
tff(fact_53_add__Suc__shift,axiom,
! [N: nat,M1: nat] : plus_plus(nat,suc(M1),N) = plus_plus(nat,M1,suc(N)) ).
tff(fact_54_Zero__not__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_55_nat_Osimps_I2_J,axiom,
! [Nat3: nat] : zero_zero(nat) != suc(Nat3) ).
tff(fact_56_Suc__not__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_57_nat_Osimps_I3_J,axiom,
! [Nat2: nat] : suc(Nat2) != zero_zero(nat) ).
tff(fact_58_Zero__neq__Suc,axiom,
! [M1: nat] : zero_zero(nat) != suc(M1) ).
tff(fact_59_Suc__neq__Zero,axiom,
! [M1: nat] : suc(M1) != zero_zero(nat) ).
tff(fact_60_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ? [M: nat] : N = suc(M) ) ).
tff(fact_61_nat__induct,axiom,
! [N2: nat,P2: fun(nat,bool)] :
( pp(aa(nat,bool,P2,zero_zero(nat)))
=> ( ! [N1: nat] :
( pp(aa(nat,bool,P2,N1))
=> pp(aa(nat,bool,P2,suc(N1))) )
=> pp(aa(nat,bool,P2,N2)) ) ) ).
tff(fact_62_zero__induct,axiom,
! [K: nat,P2: fun(nat,bool)] :
( pp(aa(nat,bool,P2,K))
=> ( ! [N1: nat] :
( pp(aa(nat,bool,P2,suc(N1)))
=> pp(aa(nat,bool,P2,N1)) )
=> pp(aa(nat,bool,P2,zero_zero(nat))) ) ) ).
tff(fact_63_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero(nat) )
=> ~ ! [Nat1: nat] : Y != suc(Nat1) ) ).
tff(fact_64_add__0__iff,axiom,
! [B: $tType] :
( semiri456707255roduct(B)
=> ! [A1: B,B1: B] :
( ( B1 = plus_plus(B,B1,A1) )
<=> ( A1 = zero_zero(B) ) ) ) ).
tff(fact_65_comm__semiring__1__class_Onormalizing__semiring__rules_I6_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A2: A] : plus_plus(A,A2,zero_zero(A)) = A2 ) ).
tff(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I5_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A2: A] : plus_plus(A,zero_zero(A),A2) = A2 ) ).
tff(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I24_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,A2: A] : plus_plus(A,A2,C) = plus_plus(A,C,A2) ) ).
tff(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I22_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,A2: A] : plus_plus(A,A2,plus_plus(A,C,D)) = plus_plus(A,C,plus_plus(A,A2,D)) ) ).
tff(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I25_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,A2: A] : plus_plus(A,A2,plus_plus(A,C,D)) = plus_plus(A,plus_plus(A,A2,C),D) ) ).
tff(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I21_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B2: A,A2: A] : plus_plus(A,plus_plus(A,A2,B2),C) = plus_plus(A,A2,plus_plus(A,B2,C)) ) ).
tff(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I23_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B2: A,A2: A] : plus_plus(A,plus_plus(A,A2,B2),C) = plus_plus(A,plus_plus(A,A2,C),B2) ) ).
tff(fact_72_comm__semiring__1__class_Onormalizing__semiring__rules_I20_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [D: A,C: A,B2: A,A2: A] : plus_plus(A,plus_plus(A,A2,B2),plus_plus(A,C,D)) = plus_plus(A,plus_plus(A,A2,C),plus_plus(A,B2,D)) ) ).
tff(fact_73_ext,axiom,
! [C1: $tType,B: $tType,G: fun(B,C1),F: fun(B,C1)] :
( ! [X1: B] : aa(B,C1,F,X1) = aa(B,C1,G,X1)
=> ( F = G ) ) ).
tff(fact_74_nat_Osize_I2_J,axiom,
! [Nat: nat] : nat_size(suc(Nat)) = plus_plus(nat,nat_size(Nat),suc(zero_zero(nat))) ).
tff(fact_75_poly__rec__pCons,axiom,
! [B: $tType,C1: $tType] :
( zero(C1)
=> ! [Pa: poly(C1),A1: C1,Z: B,F: fun(C1,fun(poly(C1),fun(B,B)))] :
( ( aa(B,B,aa(poly(C1),fun(B,B),aa(C1,fun(poly(C1),fun(B,B)),F,zero_zero(C1)),zero_zero(poly(C1))),Z) = Z )
=> ( poly_rec(B,C1,Z,F,pCons(C1,A1,Pa)) = aa(B,B,aa(poly(C1),fun(B,B),aa(C1,fun(poly(C1),fun(B,B)),F,A1),Pa),poly_rec(B,C1,Z,F,Pa)) ) ) ) ).
tff(fact_76_synthetic__div__eq__0__iff,axiom,
! [B: $tType] :
( comm_semiring_0(B)
=> ! [C2: B,Pa: poly(B)] :
( ( synthetic_div(B,Pa,C2) = zero_zero(poly(B)) )
<=> ( degree(B,Pa) = zero_zero(nat) ) ) ) ).
tff(fact_77_synthetic__div__0,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [C: A] : synthetic_div(A,zero_zero(poly(A)),C) = zero_zero(poly(A)) ) ).
tff(fact_78_nat_Osize_I1_J,axiom,
nat_size(zero_zero(nat)) = zero_zero(nat) ).
tff(fact_79_poly__rec__0,axiom,
! [C1: $tType,B: $tType] :
( zero(C1)
=> ! [Z: B,F: fun(C1,fun(poly(C1),fun(B,B)))] :
( ( aa(B,B,aa(poly(C1),fun(B,B),aa(C1,fun(poly(C1),fun(B,B)),F,zero_zero(C1)),zero_zero(poly(C1))),Z) = Z )
=> ( poly_rec(B,C1,Z,F,zero_zero(poly(C1))) = Z ) ) ) ).
tff(fact_80_poly__rec_Osimps,axiom,
! [B: $tType,C1: $tType] :
( zero(C1)
=> ! [Pa: poly(C1),A1: C1,F: fun(C1,fun(poly(C1),fun(B,B))),Z: B] : poly_rec(B,C1,Z,F,pCons(C1,A1,Pa)) = aa(B,B,aa(poly(C1),fun(B,B),aa(C1,fun(poly(C1),fun(B,B)),F,A1),Pa),if(B,fequal(poly(C1),Pa,zero_zero(poly(C1))),Z,poly_rec(B,C1,Z,F,Pa))) ) ).
tff(fact_81_bool_Osize_I1_J,axiom,
bool_size(fTrue) = zero_zero(nat) ).
tff(fact_82_bool_Osize_I2_J,axiom,
bool_size(fFalse) = zero_zero(nat) ).
tff(fact_83_nat_Osize_I4_J,axiom,
! [Nat: nat] : size_size(nat,suc(Nat)) = plus_plus(nat,size_size(nat,Nat),suc(zero_zero(nat))) ).
tff(fact_84_bool_Osize_I3_J,axiom,
size_size(bool,fTrue) = zero_zero(nat) ).
tff(fact_85_nat_Osize_I3_J,axiom,
size_size(nat,zero_zero(nat)) = zero_zero(nat) ).
tff(fact_86_bool_Osize_I4_J,axiom,
size_size(bool,fFalse) = zero_zero(nat) ).
tff(fact_87_nat__size,axiom,
! [N: nat] : size_size(nat,N) = N ).
tff(fact_88_size__bool,axiom,
! [B1: bool] : size_size(bool,B1) = zero_zero(nat) ).
tff(fact_89_synthetic__div__unique,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [R: A,Q1: poly(A),C: A,P1: poly(A)] :
( ( plus_plus(poly(A),P1,smult(A,C,Q1)) = pCons(A,R,Q1) )
=> ( ( R = aa(A,A,poly1(A,P1),C) )
& ( Q1 = synthetic_div(A,P1,C) ) ) ) ) ).
tff(fact_90_synthetic__div__correct,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [C: A,P1: poly(A)] : plus_plus(poly(A),P1,smult(A,C,synthetic_div(A,P1,C))) = pCons(A,aa(A,A,poly1(A,P1),C),synthetic_div(A,P1,C)) ) ).
tff(fact_91_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_92_poly__add,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X: A,Q1: poly(A),P1: poly(A)] : aa(A,A,poly1(A,plus_plus(poly(A),P1,Q1)),X) = plus_plus(A,aa(A,A,poly1(A,P1),X),aa(A,A,poly1(A,Q1),X)) ) ).
tff(fact_93_synthetic__div__pCons,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [C: A,P1: poly(A),A2: A] : synthetic_div(A,pCons(A,A2,P1),C) = pCons(A,aa(A,A,poly1(A,P1),C),synthetic_div(A,P1,C)) ) ).
tff(fact_94_poly__zero,axiom,
! [B: $tType] :
( ( ring_char_0(B)
& idom(B) )
=> ! [Pa: poly(B)] :
( ( poly1(B,Pa) = poly1(B,zero_zero(poly(B))) )
<=> ( Pa = zero_zero(poly(B)) ) ) ) ).
tff(fact_95_poly__eq__iff,axiom,
! [B: $tType] :
( ( ring_char_0(B)
& idom(B) )
=> ! [Q: poly(B),Pa: poly(B)] :
( ( poly1(B,Pa) = poly1(B,Q) )
<=> ( Pa = Q ) ) ) ).
tff(fact_96_poly__offset__poly,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X: A,H: A,P1: poly(A)] : aa(A,A,poly1(A,fundam296178794t_poly(A,P1,H)),X) = aa(A,A,poly1(A,P1),plus_plus(A,H,X)) ) ).
tff(fact_97_order__root,axiom,
! [B: $tType] :
( idom(B)
=> ! [A1: B,Pa: poly(B)] :
( ( aa(B,B,poly1(B,Pa),A1) = zero_zero(B) )
<=> ( ( Pa = zero_zero(poly(B)) )
| ( order(B,A1,Pa) != zero_zero(nat) ) ) ) ) ).
%----Class relationships (4)
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__Groups_Omonoid__add,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> monoid_add(A) ) ).
tff(clar_Rings_Ocomm__semiring__0__Groups_Ozero,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> zero(A) ) ).
%----Arities (24)
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__add,axiom,
ab_semigroup_add(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___Groups_Omonoid__add,axiom,
monoid_add(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___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__add,axiom,
! [T_1: $tType] :
( comm_monoid_add(T_1)
=> ab_semigroup_add(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___Groups_Omonoid__add,axiom,
! [T_1: $tType] :
( comm_monoid_add(T_1)
=> monoid_add(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___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_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
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 (3)
tff(conj_0,hypothesis,
degree(a,fundam296178794t_poly(a,pa,h)) = degree(a,pa) ).
tff(conj_1,hypothesis,
pa != zero_zero(poly(a)) ).
tff(conj_2,conjecture,
degree(a,plus_plus(poly(a),smult(a,h,fundam296178794t_poly(a,pa,h)),pCons(a,a1,fundam296178794t_poly(a,pa,h)))) = suc(degree(a,pa)) ).
%----Type variables (1)
tff(tfree_0,hypothesis,
comm_semiring_0(a) ).
%------------------------------------------------------------------------------