TPTP Problem File: SWW486_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW486_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 86
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_86 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.25 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 242 ( 62 unt; 55 typ; 0 def)
% Number of atoms : 420 ( 66 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 253 ( 20 ~; 8 |; 10 &)
% ( 31 <=>; 184 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 4 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 18 ( 11 >; 7 *; 0 +; 0 <<)
% Number of predicates : 37 ( 36 usr; 0 prp; 1-3 aty)
% Number of functors : 16 ( 16 usr; 7 con; 0-4 aty)
% Number of variables : 380 ( 336 !; 1 ?; 380 :)
% ( 43 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:53
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_tc_Complex_Ocomplex,type,
complex: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Polynomial_Opoly,type,
poly1: $tType > $tType ).
tff(ty_tc_RealDef_Oreal,type,
real: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (50)
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_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[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_Rings_Olinordered__ring,type,
linordered_ring:
!>[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_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[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_RealVector_Oreal__normed__vector,type,
real_normed_vector:
!>[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_Groups_Oordered__comm__monoid__add,type,
ordere216010020id_add:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__algebra,type,
real_normed_algebra:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add,type,
ordere779506340up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Oab__semigroup__idem__mult,type,
ab_sem1668676832m_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semiring__strict,type,
linord20386208strict:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__div__algebra,type,
real_n1866405975lgebra:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord893533164strict:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_cl_Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,type,
semiri456707255roduct:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Polynomial_OpCons,type,
pCons:
!>[A: $tType] : ( ( A * poly1(A) ) > poly1(A) ) ).
tff(sy_c_Polynomial_Opoly,type,
poly:
!>[A: $tType] : ( poly1(A) > fun(A,A) ) ).
tff(sy_c_RealVector_Onorm__class_Onorm,type,
norm_norm:
!>[A: $tType] : ( A > real ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B1: $tType] : ( ( fun(A,B1) * A ) > B1 ) ).
tff(sy_c_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_c____,type,
c: complex ).
tff(sy_v_cs____,type,
cs: poly1(complex) ).
tff(sy_v_m____,type,
m: real ).
tff(sy_v_r,type,
r: real ).
tff(sy_v_z____,type,
z: complex ).
%----Relevant facts (99)
tff(fact_0__096cmod_A_Ic_A_L_Az_A_K_Apoly_Acs_Az_J_A_060_061_Acmod_Ac_A_L_Acmod_A_Iz_A_K_Apoly_Acs_Az_J_096,axiom,
ord_less_eq(real,norm_norm(complex,plus_plus(complex,c,times_times(complex,z,aa(complex,complex,poly(complex,cs),z)))),plus_plus(real,norm_norm(complex,c),norm_norm(complex,times_times(complex,z,aa(complex,complex,poly(complex,cs),z))))) ).
tff(fact_1_H,axiom,
ord_less_eq(real,norm_norm(complex,z),r) ).
tff(fact_2_th,axiom,
ord_less_eq(real,norm_norm(complex,aa(complex,complex,poly(complex,cs),z)),m) ).
tff(fact_3_m,axiom,
! [Z1: complex] :
( ord_less_eq(real,norm_norm(complex,Z1),r)
=> ord_less_eq(real,norm_norm(complex,aa(complex,complex,poly(complex,cs),Z1)),m) ) ).
tff(fact_4_poly__pCons,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X1: A,P: poly1(A),A1: A] : aa(A,A,poly(A,pCons(A,A1,P)),X1) = plus_plus(A,A1,times_times(A,X1,aa(A,A,poly(A,P),X1))) ) ).
tff(fact_5_poly__add,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X1: A,Q1: poly1(A),P: poly1(A)] : aa(A,A,poly(A,plus_plus(poly1(A),P,Q1)),X1) = plus_plus(A,aa(A,A,poly(A,P),X1),aa(A,A,poly(A,Q1),X1)) ) ).
tff(fact_6_poly__mult,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X1: A,Q1: poly1(A),P: poly1(A)] : aa(A,A,poly(A,times_times(poly1(A),P,Q1)),X1) = times_times(A,aa(A,A,poly(A,P),X1),aa(A,A,poly(A,Q1),X1)) ) ).
tff(fact_7_add__pCons,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Q1: poly1(A),B: A,P: poly1(A),A1: A] : plus_plus(poly1(A),pCons(A,A1,P),pCons(A,B,Q1)) = pCons(A,plus_plus(A,A1,B),plus_plus(poly1(A),P,Q1)) ) ).
tff(fact_8_add__le__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,Ca: A,A2: A] :
( ord_less_eq(A,plus_plus(A,A2,Ca),plus_plus(A,B2,Ca))
<=> ord_less_eq(A,A2,B2) ) ) ).
tff(fact_9_add__le__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,A2: A,Ca: A] :
( ord_less_eq(A,plus_plus(A,Ca,A2),plus_plus(A,Ca,B2))
<=> ord_less_eq(A,A2,B2) ) ) ).
tff(fact_10_norm__triangle__ineq,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [Y1: A,X1: A] : ord_less_eq(real,norm_norm(A,plus_plus(A,X1,Y1)),plus_plus(real,norm_norm(A,X1),norm_norm(A,Y1))) ) ).
tff(fact_11_complex__mod__triangle__sub,axiom,
! [Z: complex,W1: complex] : ord_less_eq(real,norm_norm(complex,W1),plus_plus(real,norm_norm(complex,plus_plus(complex,W1,Z)),norm_norm(complex,Z))) ).
tff(fact_12_norm__mult__ineq,axiom,
! [A: $tType] :
( real_normed_algebra(A)
=> ! [Y1: A,X1: A] : ord_less_eq(real,norm_norm(A,times_times(A,X1,Y1)),times_times(real,norm_norm(A,X1),norm_norm(A,Y1))) ) ).
tff(fact_13__096_B_Bthesis_O_A_I_B_Bm_O_AALL_Az_O_Acmod_Az_A_060_061_Ar_A_N_N_062_Acmod_A_Ipoly_Acs_Az_J_A_060_061_Am_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [M1: real] :
~ ! [Z1: complex] :
( ord_less_eq(real,norm_norm(complex,Z1),r)
=> ord_less_eq(real,norm_norm(complex,aa(complex,complex,poly(complex,cs),Z1)),M1) ) ).
tff(fact_14_pCons__eq__iff,axiom,
! [A: $tType] :
( zero(A)
=> ! [Q: poly1(A),B2: A,Pa: poly1(A),A2: A] :
( ( pCons(A,A2,Pa) = pCons(A,B2,Q) )
<=> ( ( A2 = B2 )
& ( Pa = Q ) ) ) ) ).
tff(fact_15_add__left__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [Ca: A,B2: A,A2: A] :
( ( plus_plus(A,A2,B2) = plus_plus(A,A2,Ca) )
<=> ( B2 = Ca ) ) ) ).
tff(fact_16_add__right__cancel,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [Ca: A,A2: A,B2: A] :
( ( plus_plus(A,B2,A2) = plus_plus(A,Ca,A2) )
<=> ( B2 = Ca ) ) ) ).
tff(fact_17_mult__poly__add__left,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [R: poly1(A),Q1: poly1(A),P: poly1(A)] : times_times(poly1(A),plus_plus(poly1(A),P,Q1),R) = plus_plus(poly1(A),times_times(poly1(A),P,R),times_times(poly1(A),Q1,R)) ) ).
tff(fact_18_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [C: A,B: A,A1: A] : times_times(A,times_times(A,A1,B),C) = times_times(A,A1,times_times(A,B,C)) ) ).
tff(fact_19_add__right__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,A1: A,B: A] :
( ( plus_plus(A,B,A1) = plus_plus(A,C,A1) )
=> ( B = C ) ) ) ).
tff(fact_20_add__imp__eq,axiom,
! [A: $tType] :
( cancel146912293up_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
=> ( B = C ) ) ) ).
tff(fact_21_add__left__imp__eq,axiom,
! [A: $tType] :
( cancel_semigroup_add(A)
=> ! [C: A,B: A,A1: A] :
( ( plus_plus(A,A1,B) = plus_plus(A,A1,C) )
=> ( B = C ) ) ) ).
tff(fact_22_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_add(A)
=> ! [C: A,B: A,A1: A] : plus_plus(A,plus_plus(A,A1,B),C) = plus_plus(A,A1,plus_plus(A,B,C)) ) ).
tff(fact_23_poly__eq__iff,axiom,
! [A: $tType] :
( ( ring_char_0(A)
& idom(A) )
=> ! [Q: poly1(A),Pa: poly1(A)] :
( ( poly(A,Pa) = poly(A,Q) )
<=> ( Pa = Q ) ) ) ).
tff(fact_24_add__le__imp__le__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,A1: A,C: A] :
( ord_less_eq(A,plus_plus(A,C,A1),plus_plus(A,C,B))
=> ord_less_eq(A,A1,B) ) ) ).
tff(fact_25_add__le__imp__le__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,C: A,A1: A] :
( ord_less_eq(A,plus_plus(A,A1,C),plus_plus(A,B,C))
=> ord_less_eq(A,A1,B) ) ) ).
tff(fact_26_add__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [D: A,C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ( ord_less_eq(A,C,D)
=> ord_less_eq(A,plus_plus(A,A1,C),plus_plus(A,B,D)) ) ) ) ).
tff(fact_27_add__left__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ord_less_eq(A,plus_plus(A,C,A1),plus_plus(A,C,B)) ) ) ).
tff(fact_28_add__right__mono,axiom,
! [A: $tType] :
( ordere779506340up_add(A)
=> ! [C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ord_less_eq(A,plus_plus(A,A1,C),plus_plus(A,B,C)) ) ) ).
tff(fact_29_norm__mult,axiom,
! [A: $tType] :
( real_n1866405975lgebra(A)
=> ! [Y1: A,X1: A] : norm_norm(A,times_times(A,X1,Y1)) = times_times(real,norm_norm(A,X1),norm_norm(A,Y1)) ) ).
tff(fact_30_rp,axiom,
ord_less_eq(real,zero_zero(real),r) ).
tff(fact_31_mult__left__idem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [B: A,A1: A] : times_times(A,A1,times_times(A,A1,B)) = times_times(A,A1,B) ) ).
tff(fact_32_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X1: A] : ord_less_eq(A,X1,X1) ) ).
tff(fact_33_pCons,axiom,
? [M1: real] :
( ord_less(real,zero_zero(real),M1)
& ! [Z1: complex] :
( ord_less_eq(real,norm_norm(complex,Z1),r)
=> ord_less_eq(real,norm_norm(complex,aa(complex,complex,poly(complex,cs),Z1)),M1) ) ) ).
tff(fact_34_crossproduct__eq,axiom,
! [A: $tType] :
( semiri456707255roduct(A)
=> ! [Za: A,X: A,Y: A,W: A] :
( ( plus_plus(A,times_times(A,W,Y),times_times(A,X,Za)) = plus_plus(A,times_times(A,W,Za),times_times(A,X,Y)) )
<=> ( ( W = X )
| ( Y = Za ) ) ) ) ).
tff(fact_35_combine__common__factor,axiom,
! [A: $tType] :
( semiring(A)
=> ! [C: A,B: A,E: A,A1: A] : plus_plus(A,times_times(A,A1,E),plus_plus(A,times_times(A,B,E),C)) = plus_plus(A,times_times(A,plus_plus(A,A1,B),E),C) ) ).
tff(fact_36_comm__semiring__1__class_Onormalizing__semiring__rules_I1_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B: A,M: A,A1: A] : plus_plus(A,times_times(A,A1,M),times_times(A,B,M)) = times_times(A,plus_plus(A,A1,B),M) ) ).
tff(fact_37_comm__semiring__1__class_Onormalizing__semiring__rules_I8_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [C: A,B: A,A1: A] : times_times(A,plus_plus(A,A1,B),C) = plus_plus(A,times_times(A,A1,C),times_times(A,B,C)) ) ).
tff(fact_38_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : times_times(A,zero_zero(A),A1) = zero_zero(A) ) ).
tff(fact_39_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : times_times(A,A1,zero_zero(A)) = zero_zero(A) ) ).
tff(fact_40_mult__eq__0__iff,axiom,
! [A: $tType] :
( ring_n68954251visors(A)
=> ! [B2: A,A2: A] :
( ( times_times(A,A2,B2) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ) ) ) ).
tff(fact_41_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( zero_zero(A) = plus_plus(A,A2,A2) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_42_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,Ca: A,A2: A] :
( ord_less(A,plus_plus(A,A2,Ca),plus_plus(A,B2,Ca))
<=> ord_less(A,A2,B2) ) ) ).
tff(fact_43_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B2: A,A2: A,Ca: A] :
( ord_less(A,plus_plus(A,Ca,A2),plus_plus(A,Ca,B2))
<=> ord_less(A,A2,B2) ) ) ).
tff(fact_44_pCons__eq__0__iff,axiom,
! [A: $tType] :
( zero(A)
=> ! [Pa: poly1(A),A2: A] :
( ( pCons(A,A2,Pa) = zero_zero(poly1(A)) )
<=> ( ( A2 = zero_zero(A) )
& ( Pa = zero_zero(poly1(A)) ) ) ) ) ).
tff(fact_45_pCons__0__0,axiom,
! [A: $tType] :
( zero(A)
=> ( pCons(A,zero_zero(A),zero_zero(poly1(A))) = zero_zero(poly1(A)) ) ) ).
tff(fact_46_poly__0,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X1: A] : aa(A,A,poly(A,zero_zero(poly1(A))),X1) = zero_zero(A) ) ).
tff(fact_47_zero__le__double__add__iff__zero__le__single__add,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less_eq(A,zero_zero(A),plus_plus(A,A2,A2))
<=> ord_less_eq(A,zero_zero(A),A2) ) ) ).
tff(fact_48_double__add__le__zero__iff__single__add__le__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less_eq(A,plus_plus(A,A2,A2),zero_zero(A))
<=> ord_less_eq(A,A2,zero_zero(A)) ) ) ).
tff(fact_49_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less(A,zero_zero(A),plus_plus(A,A2,A2))
<=> ord_less(A,zero_zero(A),A2) ) ) ).
tff(fact_50_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ord_less(A,plus_plus(A,A2,A2),zero_zero(A))
<=> ord_less(A,A2,zero_zero(A)) ) ) ).
tff(fact_51_norm__eq__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X: A] :
( ( norm_norm(A,X) = zero_zero(real) )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_52_norm__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ( norm_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).
tff(fact_53_norm__le__zero__iff,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X: A] :
( ord_less_eq(real,norm_norm(A,X),zero_zero(real))
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_54_zero__less__norm__iff,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X: A] :
( ord_less(real,zero_zero(real),norm_norm(A,X))
<=> ( X != zero_zero(A) ) ) ) ).
tff(fact_55_norm__not__less__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X1: A] : ~ ord_less(real,norm_norm(A,X1),zero_zero(real)) ) ).
tff(fact_56_not__square__less__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A1: A] : ~ ord_less(A,times_times(A,A1,A1),zero_zero(A)) ) ).
tff(fact_57_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_58_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,Ca: A,A2: A] :
( ord_less(A,times_times(A,A2,Ca),times_times(A,B2,Ca))
<=> ( ( ord_less(A,zero_zero(A),Ca)
& ord_less(A,A2,B2) )
| ( ord_less(A,Ca,zero_zero(A))
& ord_less(A,B2,A2) ) ) ) ) ).
tff(fact_59_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,A2: A,Ca: A] :
( ord_less(A,times_times(A,Ca,A2),times_times(A,Ca,B2))
<=> ( ( ord_less(A,zero_zero(A),Ca)
& ord_less(A,A2,B2) )
| ( ord_less(A,Ca,zero_zero(A))
& ord_less(A,B2,A2) ) ) ) ) ).
tff(fact_60_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,A2: A,Ca: A] :
( ord_less(A,zero_zero(A),Ca)
=> ( ord_less(A,times_times(A,Ca,A2),times_times(A,Ca,B2))
<=> ord_less(A,A2,B2) ) ) ) ).
tff(fact_61_add__pos__pos,axiom,
! [A: $tType] :
( ordere216010020id_add(A)
=> ! [B: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,zero_zero(A),B)
=> ord_less(A,zero_zero(A),plus_plus(A,A1,B)) ) ) ) ).
tff(fact_62_mult__pos__pos,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,zero_zero(A),B)
=> ord_less(A,zero_zero(A),times_times(A,A1,B)) ) ) ) ).
tff(fact_63_mult__pos__neg,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,B,zero_zero(A))
=> ord_less(A,times_times(A,A1,B),zero_zero(A)) ) ) ) ).
tff(fact_64_mult__pos__neg2,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,B,zero_zero(A))
=> ord_less(A,times_times(A,B,A1),zero_zero(A)) ) ) ) ).
tff(fact_65_pos__add__strict,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [C: A,B: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,B,C)
=> ord_less(A,B,plus_plus(A,A1,C)) ) ) ) ).
tff(fact_66_zero__less__mult__pos,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B: A,A1: A] :
( ord_less(A,zero_zero(A),times_times(A,A1,B))
=> ( ord_less(A,zero_zero(A),A1)
=> ord_less(A,zero_zero(A),B) ) ) ) ).
tff(fact_67_zero__less__mult__pos2,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [A1: A,B: A] :
( ord_less(A,zero_zero(A),times_times(A,B,A1))
=> ( ord_less(A,zero_zero(A),A1)
=> ord_less(A,zero_zero(A),B) ) ) ) ).
tff(fact_68_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,A2: A,Ca: A] :
( ord_less(A,Ca,zero_zero(A))
=> ( ord_less(A,times_times(A,Ca,A2),times_times(A,Ca,B2))
<=> ord_less(A,B2,A2) ) ) ) ).
tff(fact_69_mult__neg__pos,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B: A,A1: A] :
( ord_less(A,A1,zero_zero(A))
=> ( ord_less(A,zero_zero(A),B)
=> ord_less(A,times_times(A,A1,B),zero_zero(A)) ) ) ) ).
tff(fact_70_mult__neg__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B: A,A1: A] :
( ord_less(A,A1,zero_zero(A))
=> ( ord_less(A,B,zero_zero(A))
=> ord_less(A,zero_zero(A),times_times(A,A1,B)) ) ) ) ).
tff(fact_71_add__neg__neg,axiom,
! [A: $tType] :
( ordere216010020id_add(A)
=> ! [B: A,A1: A] :
( ord_less(A,A1,zero_zero(A))
=> ( ord_less(A,B,zero_zero(A))
=> ord_less(A,plus_plus(A,A1,B),zero_zero(A)) ) ) ) ).
tff(fact_72_mult__strict__right__mono,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ord_less(A,zero_zero(A),C)
=> ord_less(A,times_times(A,A1,C),times_times(A,B,C)) ) ) ) ).
tff(fact_73_mult__strict__left__mono,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ord_less(A,zero_zero(A),C)
=> ord_less(A,times_times(A,C,A1),times_times(A,C,B)) ) ) ) ).
tff(fact_74_ext,axiom,
! [B1: $tType,A: $tType,G: fun(A,B1),F: fun(A,B1)] :
( ! [X2: A] : aa(A,B1,F,X2) = aa(A,B1,G,X2)
=> ( F = G ) ) ).
tff(fact_75_comm__mult__strict__left__mono,axiom,
! [A: $tType] :
( linord893533164strict(A)
=> ! [C: A,B: A,A1: A] :
( ord_less(A,A1,B)
=> ( ord_less(A,zero_zero(A),C)
=> ord_less(A,times_times(A,C,A1),times_times(A,C,B)) ) ) ) ).
tff(fact_76_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [C: A,A1: A,B: A] :
( ord_less(A,B,A1)
=> ( ord_less(A,C,zero_zero(A))
=> ord_less(A,times_times(A,A1,C),times_times(A,B,C)) ) ) ) ).
tff(fact_77_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [C: A,A1: A,B: A] :
( ord_less(A,B,A1)
=> ( ord_less(A,C,zero_zero(A))
=> ord_less(A,times_times(A,C,A1),times_times(A,C,B)) ) ) ) ).
tff(fact_78_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ord_less(A,Y1,Z)
=> ord_less(A,X1,Z) ) ) ) ).
tff(fact_79_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ( ord_less_eq(A,Y1,Z)
=> ord_less(A,X1,Z) ) ) ) ).
tff(fact_80_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ( ( A1 != B )
=> ord_less(A,A1,B) ) ) ) ).
tff(fact_81_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
=> ( ord_less(A,X1,Y1)
| ( X1 = Y1 ) ) ) ) ).
tff(fact_82_linorder__antisym__conv2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ~ ord_less(A,X,Y)
<=> ( X = Y ) ) ) ) ).
tff(fact_83_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,X1,Y1)
=> ord_less_eq(A,X1,Y1) ) ) ).
tff(fact_84_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,Y1: A] :
( ord_less_eq(A,Y1,X1)
=> ~ ord_less(A,X1,Y1) ) ) ).
tff(fact_85_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B: A,A1: A] :
( ( A1 != B )
=> ( ord_less_eq(A,A1,B)
=> ord_less(A,A1,B) ) ) ) ).
tff(fact_86_linorder__antisym__conv1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less(A,X,Y)
=> ( ord_less_eq(A,X,Y)
<=> ( X = Y ) ) ) ) ).
tff(fact_87_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X1: A,Y1: A] :
( ~ ord_less_eq(A,Y1,X1)
=> ord_less(A,X1,Y1) ) ) ).
tff(fact_88_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ~ ord_less(A,X1,Y1)
=> ord_less_eq(A,Y1,X1) ) ) ).
tff(fact_89_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
<=> ( ord_less(A,X,Y)
| ( X = Y ) ) ) ) ).
tff(fact_90_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
<=> ( ord_less_eq(A,X,Y)
& ~ ord_less_eq(A,Y,X) ) ) ) ).
tff(fact_91_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
<=> ( ord_less_eq(A,X,Y)
& ( X != Y ) ) ) ) ).
tff(fact_92_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,X1,Y1)
| ord_less(A,Y1,X1) ) ) ).
tff(fact_93_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less_eq(A,X,Y)
<=> ord_less(A,Y,X) ) ) ).
tff(fact_94_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less(A,X,Y)
<=> ord_less_eq(A,Y,X) ) ) ).
tff(fact_95_divisors__zero,axiom,
! [A: $tType] :
( no_zero_divisors(A)
=> ! [B: A,A1: A] :
( ( times_times(A,A1,B) = zero_zero(A) )
=> ( ( A1 = zero_zero(A) )
| ( B = zero_zero(A) ) ) ) ) ).
tff(fact_96_no__zero__divisors,axiom,
! [A: $tType] :
( no_zero_divisors(A)
=> ! [B: A,A1: A] :
( ( A1 != zero_zero(A) )
=> ( ( B != zero_zero(A) )
=> ( times_times(A,A1,B) != zero_zero(A) ) ) ) ) ).
tff(fact_97_comm__semiring__1__class_Onormalizing__semiring__rules_I10_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : times_times(A,A1,zero_zero(A)) = zero_zero(A) ) ).
tff(fact_98_comm__semiring__1__class_Onormalizing__semiring__rules_I9_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [A1: A] : times_times(A,zero_zero(A),A1) = zero_zero(A) ) ).
%----Arities (85)
tff(arity_Polynomial_Opoly___Rings_Olinordered__idom,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linordered_idom(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ocancel__comm__monoid__add,axiom,
! [T_1: $tType] :
( cancel1352612707id_add(T_1)
=> cancel1352612707id_add(poly1(T_1)) ) ).
tff(arity_Complex_Ocomplex___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add(complex) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
tff(arity_RealDef_Oreal___Groups_Ocancel__comm__monoid__add,axiom,
cancel1352612707id_add(real) ).
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_RealDef_Oreal___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(real) ).
tff(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__div__algebra,axiom,
real_n1866405975lgebra(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(real) ).
tff(arity_RealDef_Oreal___Groups_Oordered__ab__semigroup__add,axiom,
ordere779506340up_add(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra,axiom,
real_normed_algebra(real) ).
tff(arity_RealDef_Oreal___Groups_Oordered__comm__monoid__add,axiom,
ordere216010020id_add(real) ).
tff(arity_RealDef_Oreal___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(real) ).
tff(arity_RealDef_Oreal___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(real) ).
tff(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(real) ).
tff(arity_RealDef_Oreal___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(real) ).
tff(arity_RealDef_Oreal___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(real) ).
tff(arity_RealDef_Oreal___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__ring,axiom,
linordered_ring(real) ).
tff(arity_RealDef_Oreal___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(real) ).
tff(arity_RealDef_Oreal___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(real) ).
tff(arity_RealDef_Oreal___Orderings_Opreorder,axiom,
preorder(real) ).
tff(arity_RealDef_Oreal___Orderings_Olinorder,axiom,
linorder(real) ).
tff(arity_RealDef_Oreal___Rings_Omult__zero,axiom,
mult_zero(real) ).
tff(arity_RealDef_Oreal___Orderings_Oorder,axiom,
order(real) ).
tff(arity_RealDef_Oreal___Int_Oring__char__0,axiom,
ring_char_0(real) ).
tff(arity_RealDef_Oreal___Rings_Osemiring,axiom,
semiring(real) ).
tff(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
tff(arity_RealDef_Oreal___Rings_Oidom,axiom,
idom(real) ).
tff(arity_Complex_Ocomplex___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
semiri456707255roduct(complex) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__div__algebra,axiom,
real_n1866405975lgebra(complex) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__algebra,axiom,
real_normed_algebra(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocancel__ab__semigroup__add,axiom,
cancel146912293up_add(complex) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(complex) ).
tff(arity_Complex_Ocomplex___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oab__semigroup__add,axiom,
ab_semigroup_add(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(complex) ).
tff(arity_Complex_Ocomplex___Rings_Omult__zero,axiom,
mult_zero(complex) ).
tff(arity_Complex_Ocomplex___Int_Oring__char__0,axiom,
ring_char_0(complex) ).
tff(arity_Complex_Ocomplex___Rings_Osemiring,axiom,
semiring(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ozero,axiom,
zero(complex) ).
tff(arity_Complex_Ocomplex___Rings_Oidom,axiom,
idom(complex) ).
tff(arity_Polynomial_Opoly___Semiring__Normalization_Ocomm__semiring__1__cancel__crossproduct,axiom,
! [T_1: $tType] :
( idom(T_1)
=> semiri456707255roduct(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> ordere236663937imp_le(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Olinordered__comm__semiring__strict,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linord893533164strict(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Olinordered__semiring__strict,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linord20386208strict(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Oordered__ab__semigroup__add,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> ordere779506340up_add(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Oordered__comm__monoid__add,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> ordere216010020id_add(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Olinordered__ab__group__add,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linord219039673up_add(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ocancel__ab__semigroup__add,axiom,
! [T_1: $tType] :
( cancel1352612707id_add(T_1)
=> cancel146912293up_add(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Olinordered__ring__strict,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linord581940658strict(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Oring__no__zero__divisors,axiom,
! [T_1: $tType] :
( idom(T_1)
=> ring_n68954251visors(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ocancel__semigroup__add,axiom,
! [T_1: $tType] :
( cancel1352612707id_add(T_1)
=> cancel_semigroup_add(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Olinordered__semidom,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linordered_semidom(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Oab__semigroup__mult,axiom,
! [T_1: $tType] :
( comm_semiring_0(T_1)
=> ab_semigroup_mult(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Oab__semigroup__add,axiom,
! [T_1: $tType] :
( comm_monoid_add(T_1)
=> ab_semigroup_add(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Ono__zero__divisors,axiom,
! [T_1: $tType] :
( idom(T_1)
=> no_zero_divisors(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ocomm__monoid__add,axiom,
! [T_1: $tType] :
( comm_monoid_add(T_1)
=> comm_monoid_add(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Olinordered__ring,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linordered_ring(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Ocomm__semiring__1,axiom,
! [T_1: $tType] :
( comm_semiring_1(T_1)
=> comm_semiring_1(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Ocomm__semiring__0,axiom,
! [T_1: $tType] :
( comm_semiring_0(T_1)
=> comm_semiring_0(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Orderings_Opreorder,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> preorder(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Orderings_Olinorder,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linorder(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Omult__zero,axiom,
! [T_1: $tType] :
( comm_semiring_0(T_1)
=> mult_zero(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Orderings_Oorder,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> order(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Int_Oring__char__0,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> ring_char_0(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Osemiring,axiom,
! [T_1: $tType] :
( comm_semiring_0(T_1)
=> semiring(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ozero,axiom,
! [T_1: $tType] :
( zero(T_1)
=> zero(poly1(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Oidom,axiom,
! [T_1: $tType] :
( idom(T_1)
=> idom(poly1(T_1)) ) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
ord_less_eq(real,norm_norm(complex,aa(complex,complex,poly(complex,pCons(complex,c,cs)),z)),plus_plus(real,norm_norm(complex,c),norm_norm(complex,times_times(complex,z,aa(complex,complex,poly(complex,cs),z))))) ).
%------------------------------------------------------------------------------