TPTP Problem File: SWW485_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW485_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 78
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_78 [Bla13]
% Status : Theorem
% Rating : 0.33 v7.5.0, 0.00 v7.4.0, 0.25 v7.1.0, 0.33 v6.4.0
% Syntax : Number of formulae : 171 ( 37 unt; 30 typ; 0 def)
% Number of atoms : 384 ( 57 equ)
% Maximal formula atoms : 7 ( 2 avg)
% Number of connectives : 293 ( 50 ~; 14 |; 17 &)
% ( 27 <=>; 185 =>; 0 <=; 0 <~>)
% Maximal formula depth : 14 ( 5 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 14 ( 9 >; 5 *; 0 +; 0 <<)
% Number of predicates : 16 ( 15 usr; 1 prp; 0-3 aty)
% Number of functors : 11 ( 11 usr; 4 con; 0-4 aty)
% Number of variables : 333 ( 311 !; 3 ?; 333 :)
% ( 19 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:45
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
tff(ty_tc_Complex_Ocomplex,type,
complex: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $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 (24)
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_Orderings_Oord,type,
ord:
!>[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_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_Orderings_Odense__linorder,type,
dense_linorder:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__vector,type,
real_normed_vector:
!>[A: $tType] : $o ).
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_Oorder,type,
order1:
!>[A: $tType] : ( ( A * poly1(A) ) > nat ) ).
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,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
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_cs____,type,
cs: poly1(complex) ).
tff(sy_v_r,type,
r: real ).
tff(sy_v_thesis____,type,
thesis: $o ).
%----Relevant facts (99)
tff(fact_0_pCons,axiom,
? [M3: real] :
( ord_less(real,zero_zero(real),M3)
& ! [Z2: complex] :
( ord_less_eq(real,norm_norm(complex,Z2),r)
=> ord_less_eq(real,norm_norm(complex,aa(complex,complex,poly(complex,cs),Z2)),M3) ) ) ).
tff(fact_1_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ord_less_eq(A,X,X) ) ).
tff(fact_2_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_3_norm__ge__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X: A] : ord_less_eq(real,zero_zero(real),norm_norm(A,X)) ) ).
tff(fact_4_norm__le__zero__iff,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X2: A] :
( ord_less_eq(real,norm_norm(A,X2),zero_zero(real))
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_5_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
<=> ! [X3: A] : ord_less_eq(B,aa(A,B,F,X3),aa(A,B,G,X3)) ) ) ).
tff(fact_6_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
| ord_less_eq(A,Y,X) ) ) ).
tff(fact_7_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ( X2 = Y2 )
<=> ( ord_less_eq(A,X2,Y2)
& ord_less_eq(A,Y2,X2) ) ) ) ).
tff(fact_8_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ( X = Y )
=> ord_less_eq(A,X,Y) ) ) ).
tff(fact_9_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X2: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,X2),aa(A,B,G,X2)) ) ) ).
tff(fact_10_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [X2: A,Y2: A] :
( ord_less_eq(A,Y2,X2)
=> ( ord_less_eq(A,X2,Y2)
<=> ( X2 = Y2 ) ) ) ) ).
tff(fact_11_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B2: A,A2: A] :
( ( A2 = B2 )
=> ( ord_less_eq(A,B2,C)
=> ord_less_eq(A,A2,C) ) ) ) ).
tff(fact_12_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B2: A,A2: A] :
( ord_less_eq(A,A2,B2)
=> ( ( B2 = C )
=> ord_less_eq(A,A2,C) ) ) ) ).
tff(fact_13_poly__0,axiom,
! [A: $tType] :
( comm_semiring_0(A)
=> ! [X: A] : aa(A,A,poly(A,zero_zero(poly1(A))),X) = zero_zero(A) ) ).
tff(fact_14_norm__eq__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X2: A] :
( ( norm_norm(A,X2) = zero_zero(real) )
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_15_norm__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ( norm_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).
tff(fact_16_zero__less__norm__iff,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X2: A] :
( ord_less(real,zero_zero(real),norm_norm(A,X2))
<=> ( X2 != zero_zero(A) ) ) ) ).
tff(fact_17_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ ord_less(A,X,X) ) ).
tff(fact_18_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ( X2 != Y2 )
<=> ( ord_less(A,X2,Y2)
| ord_less(A,Y2,X2) ) ) ) ).
tff(fact_19_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ ord_less(A,X2,Y2)
<=> ( ord_less(A,Y2,X2)
| ( X2 = Y2 ) ) ) ) ).
tff(fact_20_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
| ( X = Y )
| ord_less(A,Y,X) ) ) ).
tff(fact_21_linorder__antisym__conv3,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X2: A,Y2: A] :
( ~ ord_less(A,Y2,X2)
=> ( ~ ord_less(A,X2,Y2)
<=> ( X2 = Y2 ) ) ) ) ).
tff(fact_22_linorder__neqE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_23_less__imp__neq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( X != Y ) ) ) ).
tff(fact_24_order__less__not__sym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_25_order__less__imp__not__less,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_26_order__less__imp__not__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( X != Y ) ) ) ).
tff(fact_27_order__less__imp__not__eq2,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ( Y != X ) ) ) ).
tff(fact_28_order__less__imp__triv,axiom,
! [A: $tType] :
( preorder(A)
=> ! [P: bool,Y2: A,X2: A] :
( ord_less(A,X2,Y2)
=> ( ord_less(A,Y2,X2)
=> pp(P) ) ) ) ).
tff(fact_29_order__less__asym_H,axiom,
! [A: $tType] :
( preorder(A)
=> ! [B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ~ ord_less(A,B2,A2) ) ) ).
tff(fact_30_ord__eq__less__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B2: A,A2: A] :
( ( A2 = B2 )
=> ( ord_less(A,B2,C)
=> ord_less(A,A2,C) ) ) ) ).
tff(fact_31_ord__less__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B2: A,A2: A] :
( ord_less(A,A2,B2)
=> ( ( B2 = C )
=> ord_less(A,A2,C) ) ) ) ).
tff(fact_32_order__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y: A,X: A] :
( ord_less(A,X,Y)
=> ( ord_less(A,Y,Z1)
=> ord_less(A,X,Z1) ) ) ) ).
tff(fact_33_order__less__asym,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ~ ord_less(A,Y,X) ) ) ).
tff(fact_34_linorder__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less(A,X,Y)
=> ( ( X != Y )
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_35_poly__zero,axiom,
! [A: $tType] :
( ( ring_char_0(A)
& idom(A) )
=> ! [Pa: poly1(A)] :
( ( poly(A,Pa) = poly(A,zero_zero(poly1(A))) )
<=> ( Pa = zero_zero(poly1(A)) ) ) ) ).
tff(fact_36_norm__not__less__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X: A] : ~ ord_less(real,norm_norm(A,X),zero_zero(real)) ) ).
tff(fact_37_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less(A,Y,Z1)
=> ord_less(A,X,Z1) ) ) ) ).
tff(fact_38_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y: A,X: A] :
( ord_less(A,X,Y)
=> ( ord_less_eq(A,Y,Z1)
=> ord_less(A,X,Z1) ) ) ) ).
tff(fact_39_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A2: A] :
( ord_less_eq(A,A2,B2)
=> ( ( A2 != B2 )
=> ord_less(A,A2,B2) ) ) ) ).
tff(fact_40_order__le__imp__less__or__eq,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less(A,X,Y)
| ( X = Y ) ) ) ) ).
tff(fact_41_linorder__antisym__conv2,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ord_less_eq(A,X2,Y2)
=> ( ~ ord_less(A,X2,Y2)
<=> ( X2 = Y2 ) ) ) ) ).
tff(fact_42_order__less__imp__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
=> ord_less_eq(A,X,Y) ) ) ).
tff(fact_43_leD,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ord_less_eq(A,Y,X)
=> ~ ord_less(A,X,Y) ) ) ).
tff(fact_44_order__neq__le__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B2: A,A2: A] :
( ( A2 != B2 )
=> ( ord_less_eq(A,A2,B2)
=> ord_less(A,A2,B2) ) ) ) ).
tff(fact_45_linorder__antisym__conv1,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ ord_less(A,X2,Y2)
=> ( ord_less_eq(A,X2,Y2)
<=> ( X2 = Y2 ) ) ) ) ).
tff(fact_46_not__leE,axiom,
! [A: $tType] :
( linorder(A)
=> ! [X: A,Y: A] :
( ~ ord_less_eq(A,Y,X)
=> ord_less(A,X,Y) ) ) ).
tff(fact_47_leI,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less(A,X,Y)
=> ord_less_eq(A,Y,X) ) ) ).
tff(fact_48_order__le__less,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ord_less_eq(A,X2,Y2)
<=> ( ord_less(A,X2,Y2)
| ( X2 = Y2 ) ) ) ) ).
tff(fact_49_less__le__not__le,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
<=> ( ord_less_eq(A,X2,Y2)
& ~ ord_less_eq(A,Y2,X2) ) ) ) ).
tff(fact_50_order__less__le,axiom,
! [A: $tType] :
( order(A)
=> ! [Y2: A,X2: A] :
( ord_less(A,X2,Y2)
<=> ( ord_less_eq(A,X2,Y2)
& ( X2 != Y2 ) ) ) ) ).
tff(fact_51_linorder__le__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
| ord_less(A,Y,X) ) ) ).
tff(fact_52_linorder__not__le,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ ord_less_eq(A,X2,Y2)
<=> ord_less(A,Y2,X2) ) ) ).
tff(fact_53_linorder__not__less,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y2: A,X2: A] :
( ~ ord_less(A,X2,Y2)
<=> ord_less_eq(A,Y2,X2) ) ) ).
tff(fact_54_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less_eq(A,X,Y)
=> ord_less_eq(A,Y,X) ) ) ).
tff(fact_55_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X2: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,X2),aa(A,B,G,X2)) ) ) ).
tff(fact_56_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z1: A,Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less_eq(A,Y,Z1)
=> ord_less_eq(A,X,Z1) ) ) ) ).
tff(fact_57_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less_eq(A,Y,X)
=> ( X = Y ) ) ) ) ).
tff(fact_58_le__funI,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : ord_less_eq(B,aa(A,B,F,X1),aa(A,B,G,X1))
=> ord_less_eq(fun(A,B),F,G) ) ) ).
tff(fact_59_order__le__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [Ca: B,B1: B,F: fun(B,A),A1: A] :
( ord_less_eq(A,A1,aa(B,A,F,B1))
=> ( ord_less(B,B1,Ca)
=> ( ! [X1: B,Y1: B] :
( ord_less(B,X1,Y1)
=> ord_less(A,aa(B,A,F,X1),aa(B,A,F,Y1)) )
=> ord_less(A,A1,aa(B,A,F,Ca)) ) ) ) ) ).
tff(fact_60_order__less__le__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [Ca: B,B1: B,F: fun(B,A),A1: A] :
( ord_less(A,A1,aa(B,A,F,B1))
=> ( ord_less_eq(B,B1,Ca)
=> ( ! [X1: B,Y1: B] :
( ord_less_eq(B,X1,Y1)
=> ord_less_eq(A,aa(B,A,F,X1),aa(B,A,F,Y1)) )
=> ord_less(A,A1,aa(B,A,F,Ca)) ) ) ) ) ).
tff(fact_61_order__le__less__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [Ca: B,F: fun(A,B),B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ord_less(B,aa(A,B,F,B1),Ca)
=> ( ! [X1: A,Y1: A] :
( ord_less_eq(A,X1,Y1)
=> ord_less_eq(B,aa(A,B,F,X1),aa(A,B,F,Y1)) )
=> ord_less(B,aa(A,B,F,A1),Ca) ) ) ) ) ).
tff(fact_62_order__less__le__subst2,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [Ca: B,F: fun(A,B),B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ord_less_eq(B,aa(A,B,F,B1),Ca)
=> ( ! [X1: A,Y1: A] :
( ord_less(A,X1,Y1)
=> ord_less(B,aa(A,B,F,X1),aa(A,B,F,Y1)) )
=> ord_less(B,aa(A,B,F,A1),Ca) ) ) ) ) ).
tff(fact_63_dense__le__bounded,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z1: A,Y: A,X: A] :
( ord_less(A,X,Y)
=> ( ! [W: A] :
( ord_less(A,X,W)
=> ( ord_less(A,W,Y)
=> ord_less_eq(A,W,Z1) ) )
=> ord_less_eq(A,Y,Z1) ) ) ) ).
tff(fact_64_dense__le,axiom,
! [A: $tType] :
( dense_linorder(A)
=> ! [Z1: A,Y: A] :
( ! [X1: A] :
( ord_less(A,X1,Y)
=> ord_less_eq(A,X1,Z1) )
=> ord_less_eq(A,Y,Z1) ) ) ).
tff(fact_65_less__eq__real__def,axiom,
! [Y2: real,X2: real] :
( ord_less_eq(real,X2,Y2)
<=> ( ord_less(real,X2,Y2)
| ( X2 = Y2 ) ) ) ).
tff(fact_66_less__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ord_less(fun(A,B),F,G)
<=> ( ord_less_eq(fun(A,B),F,G)
& ~ ord_less_eq(fun(A,B),G,F) ) ) ) ).
tff(fact_67_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X2: A] :
( ( zero_zero(A) = X2 )
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_68_real__lbound__gt__zero,axiom,
! [D2: real,D1: real] :
( ord_less(real,zero_zero(real),D1)
=> ( ord_less(real,zero_zero(real),D2)
=> ? [E: real] :
( ord_less(real,zero_zero(real),E)
& ord_less(real,E,D1)
& ord_less(real,E,D2) ) ) ) ).
tff(fact_69_order__root,axiom,
! [A: $tType] :
( idom(A)
=> ! [A1: A,Pa: poly1(A)] :
( ( aa(A,A,poly(A,Pa),A1) = zero_zero(A) )
<=> ( ( Pa = zero_zero(poly1(A)) )
| ( order1(A,A1,Pa) != zero_zero(nat) ) ) ) ) ).
tff(fact_70_order__less__subst1,axiom,
! [A: $tType,B: $tType] :
( ( order(B)
& order(A) )
=> ! [Ca: B,B1: B,F: fun(B,A),A1: A] :
( ord_less(A,A1,aa(B,A,F,B1))
=> ( ord_less(B,B1,Ca)
=> ( ! [X1: B,Y1: B] :
( ord_less(B,X1,Y1)
=> ord_less(A,aa(B,A,F,X1),aa(B,A,F,Y1)) )
=> ord_less(A,A1,aa(B,A,F,Ca)) ) ) ) ) ).
tff(fact_71_le0,axiom,
! [N1: nat] : ord_less_eq(nat,zero_zero(nat),N1) ).
tff(fact_72_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_73_less__zeroE,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_74_ext,axiom,
! [B: $tType,A: $tType,G: fun(A,B),F: fun(A,B)] :
( ! [X1: A] : aa(A,B,F,X1) = aa(A,B,G,X1)
=> ( F = G ) ) ).
tff(fact_75_less__nat__zero__code,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_76_le__0__eq,axiom,
! [N: nat] :
( ord_less_eq(nat,N,zero_zero(nat))
<=> ( N = zero_zero(nat) ) ) ).
tff(fact_77_less__eq__nat_Osimps_I1_J,axiom,
! [N1: nat] : ord_less_eq(nat,zero_zero(nat),N1) ).
tff(fact_78_less__not__refl,axiom,
! [N1: nat] : ~ ord_less(nat,N1,N1) ).
tff(fact_79_nat__neq__iff,axiom,
! [N: nat,M2: nat] :
( ( M2 != N )
<=> ( ord_less(nat,M2,N)
| ord_less(nat,N,M2) ) ) ).
tff(fact_80_nat__less__le,axiom,
! [N: nat,M2: nat] :
( ord_less(nat,M2,N)
<=> ( ord_less_eq(nat,M2,N)
& ( M2 != N ) ) ) ).
tff(fact_81_le__eq__less__or__eq,axiom,
! [N: nat,M2: nat] :
( ord_less_eq(nat,M2,N)
<=> ( ord_less(nat,M2,N)
| ( M2 = N ) ) ) ).
tff(fact_82_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_83_less__irrefl__nat,axiom,
! [N1: nat] : ~ ord_less(nat,N1,N1) ).
tff(fact_84_less__not__refl2,axiom,
! [M1: nat,N1: nat] :
( ord_less(nat,N1,M1)
=> ( M1 != N1 ) ) ).
tff(fact_85_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_86_less__imp__le__nat,axiom,
! [N1: nat,M1: nat] :
( ord_less(nat,M1,N1)
=> ord_less_eq(nat,M1,N1) ) ).
tff(fact_87_le__neq__implies__less,axiom,
! [N1: nat,M1: nat] :
( ord_less_eq(nat,M1,N1)
=> ( ( M1 != N1 )
=> ord_less(nat,M1,N1) ) ) ).
tff(fact_88_less__or__eq__imp__le,axiom,
! [N1: nat,M1: nat] :
( ( ord_less(nat,M1,N1)
| ( M1 = N1 ) )
=> ord_less_eq(nat,M1,N1) ) ).
tff(fact_89_nat__less__cases,axiom,
! [P: fun(nat,fun(nat,bool)),N: nat,M2: nat] :
( ( ord_less(nat,M2,N)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N),M2)) )
=> ( ( ( M2 = N )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N),M2)) )
=> ( ( ord_less(nat,N,M2)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N),M2)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,N),M2)) ) ) ) ).
tff(fact_90_le__refl,axiom,
! [N1: nat] : ord_less_eq(nat,N1,N1) ).
tff(fact_91_nat__le__linear,axiom,
! [N1: nat,M1: nat] :
( ord_less_eq(nat,M1,N1)
| ord_less_eq(nat,N1,M1) ) ).
tff(fact_92_eq__imp__le,axiom,
! [N1: nat,M1: nat] :
( ( M1 = N1 )
=> ord_less_eq(nat,M1,N1) ) ).
tff(fact_93_le__trans,axiom,
! [K1: nat,J: nat,I1: nat] :
( ord_less_eq(nat,I1,J)
=> ( ord_less_eq(nat,J,K1)
=> ord_less_eq(nat,I1,K1) ) ) ).
tff(fact_94_le__antisym,axiom,
! [N1: nat,M1: nat] :
( ord_less_eq(nat,M1,N1)
=> ( ord_less_eq(nat,N1,M1)
=> ( M1 = N1 ) ) ) ).
tff(fact_95_gr0I,axiom,
! [N1: nat] :
( ( N1 != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N1) ) ).
tff(fact_96_gr__implies__not0,axiom,
! [N1: nat,M1: nat] :
( ord_less(nat,M1,N1)
=> ( N1 != zero_zero(nat) ) ) ).
tff(fact_97_not__less0,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_98_ex__least__nat__le,axiom,
! [N: nat,P: fun(nat,bool)] :
( ~ pp(aa(nat,bool,P,zero_zero(nat)))
=> ( pp(aa(nat,bool,P,N))
=> ? [K: nat] :
( ord_less_eq(nat,K,N)
& ! [I: nat] :
( ord_less(nat,I,K)
=> ~ pp(aa(nat,bool,P,I)) )
& pp(aa(nat,bool,P,K)) ) ) ) ).
%----Arities (38)
tff(arity_Polynomial_Opoly___Rings_Olinordered__idom,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linordered_idom(poly1(T_1)) ) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(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_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(fun(T_1,T_2)) ) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(nat) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
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_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(real) ).
tff(arity_RealDef_Oreal___Orderings_Odense__linorder,axiom,
dense_linorder(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___Orderings_Oorder,axiom,
order(real) ).
tff(arity_RealDef_Oreal___Int_Oring__char__0,axiom,
ring_char_0(real) ).
tff(arity_RealDef_Oreal___Orderings_Oord,axiom,
ord(real) ).
tff(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
tff(arity_RealDef_Oreal___Rings_Oidom,axiom,
idom(real) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(complex) ).
tff(arity_Complex_Ocomplex___Int_Oring__char__0,axiom,
ring_char_0(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ozero,axiom,
zero(complex) ).
tff(arity_Complex_Ocomplex___Rings_Oidom,axiom,
idom(complex) ).
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___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___Orderings_Oord,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> ord(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 (2)
tff(conj_0,hypothesis,
! [M: real] :
( ! [Z: complex] :
( ord_less_eq(real,norm_norm(complex,Z),r)
=> ord_less_eq(real,norm_norm(complex,aa(complex,complex,poly(complex,cs),Z)),M) )
=> thesis ) ).
tff(conj_1,conjecture,
thesis ).
%------------------------------------------------------------------------------