TPTP Problem File: SWW499_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : SWW499_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 209
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_209 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.00 v6.4.0
% Syntax : Number of formulae : 236 ( 105 unt; 44 typ; 0 def)
% Number of atoms : 310 ( 112 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 138 ( 20 ~; 8 |; 10 &)
% ( 27 <=>; 73 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 16 ( 12 >; 4 *; 0 +; 0 <<)
% Number of predicates : 26 ( 25 usr; 0 prp; 1-3 aty)
% Number of functors : 14 ( 14 usr; 5 con; 0-3 aty)
% Number of variables : 251 ( 220 !; 2 ?; 251 :)
% ( 29 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:17:21
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
tff(ty_tc_Complex_Ocomplex,type,
complex: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_Polynomial_Opoly,type,
poly: $tType > $tType ).
tff(ty_tc_RealDef_Oreal,type,
real: $tType ).
%----Explicit typings (38)
tff(sy_cl_Rings_Ocomm__ring__1,type,
comm_ring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__0,type,
comm_semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oidom,type,
idom:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber__ring,type,
number_ring:
!>[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_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[A: $tType] : $o ).
tff(sy_cl_Int_Onumber__semiring,type,
number_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ono__zero__divisors,type,
no_zero_divisors:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__mult,type,
comm_monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__no__zero__divisors,type,
ring_n68954251visors:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
tff(sy_cl_Lattices_Oab__semigroup__idem__mult,type,
ab_sem1668676832m_mult:
!>[A: $tType] : $o ).
tff(sy_c_Fundamental__Theorem__Algebra__Mirabelle__jmqnahvvas_Ocsqrt,type,
fundam1563812824_csqrt: complex > complex ).
tff(sy_c_Fundamental__Theorem__Algebra__Mirabelle__jmqnahvvas_Opsize,type,
fundam1280195782_psize:
!>[A: $tType] : ( poly(A) > nat ) ).
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_Int_OBit0,type,
bit0: int > int ).
tff(sy_c_Int_OBit1,type,
bit1: int > int ).
tff(sy_c_Int_OPls,type,
pls: int ).
tff(sy_c_Int_Onumber__class_Onumber__of,type,
number_number_of:
!>[A: $tType] : ( int > 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_Parity_Oeven__odd__class_Oeven,type,
even_odd_even:
!>[A: $tType] : ( A > $o ) ).
tff(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( ( A * nat ) > A ) ).
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_n,type,
n: nat ).
tff(sy_v_na____,type,
na: nat ).
%----Relevant facts (97)
tff(fact_0_e,axiom,
even_odd_even(nat,na) ).
tff(fact_1_n,axiom,
na != zero_zero(nat) ).
tff(fact_2_assms_I2_J,axiom,
n != zero_zero(nat) ).
tff(fact_3_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_4_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_5_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_6_rel__simps_I49_J,axiom,
! [L1: int,K1: int] : bit0(K1) != bit1(L1) ).
tff(fact_7_rel__simps_I50_J,axiom,
! [L1: int,K1: int] : bit1(K1) != bit0(L1) ).
tff(fact_8_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_9_rel__simps_I46_J,axiom,
! [K1: int] : bit1(K1) != pls ).
tff(fact_10_arith__simps_I32_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : times_times(A,number_number_of(A,V),number_number_of(A,W1)) = number_number_of(A,times_times(int,V,W1)) ) ).
tff(fact_11_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W1: int,V: int] : times_times(A,number_number_of(A,V),times_times(A,number_number_of(A,W1),Z)) = times_times(A,number_number_of(A,times_times(int,V,W1)),Z) ) ).
tff(fact_12_mult__numeral__1__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : times_times(A,A1,number_number_of(A,bit1(pls))) = A1 ) ).
tff(fact_13_mult__numeral__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : times_times(A,number_number_of(A,bit1(pls)),A1) = A1 ) ).
tff(fact_14_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y1: int,X: int] :
( ( number_number_of(A,X) = number_number_of(A,Y1) )
<=> ( X = Y1 ) ) ) ).
tff(fact_15_rel__simps_I51_J,axiom,
! [L: int,K: int] :
( ( bit1(K) = bit1(L) )
<=> ( K = L ) ) ).
tff(fact_16_rel__simps_I48_J,axiom,
! [L: int,K: int] :
( ( bit0(K) = bit0(L) )
<=> ( K = L ) ) ).
tff(fact_17_mult__Pls,axiom,
! [W1: int] : times_times(int,pls,W1) = pls ).
tff(fact_18_mult__Bit0,axiom,
! [L1: int,K1: int] : times_times(int,bit0(K1),L1) = bit0(times_times(int,K1,L1)) ).
tff(fact_19_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_20_times__numeral__code_I5_J,axiom,
! [W1: int,V: int] : times_times(int,number_number_of(int,V),number_number_of(int,W1)) = number_number_of(int,times_times(int,V,W1)) ).
tff(fact_21_semiring__numeral__0__eq__0,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_22_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [X: A,W: int] :
( ( number_number_of(A,W) = X )
<=> ( X = number_number_of(A,W) ) ) ) ).
tff(fact_23_number__of__mult,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : number_number_of(A,times_times(int,V,W1)) = times_times(A,number_number_of(A,V),number_number_of(A,W1)) ) ).
tff(fact_24_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_25_even__mult__two__ex,axiom,
! [Nb: nat] :
( even_odd_even(nat,Nb)
<=> ? [M3: nat] : Nb = times_times(nat,number_number_of(nat,bit0(bit1(pls))),M3) ) ).
tff(fact_26_even__product__nat,axiom,
! [Y1: nat,X: nat] :
( even_odd_even(nat,times_times(nat,X,Y1))
<=> ( even_odd_even(nat,X)
| even_odd_even(nat,Y1) ) ) ).
tff(fact_27_mult__cancel2,axiom,
! [Nb: nat,K: nat,M2: nat] :
( ( times_times(nat,M2,K) = times_times(nat,Nb,K) )
<=> ( ( M2 = Nb )
| ( K = zero_zero(nat) ) ) ) ).
tff(fact_28_mult__cancel1,axiom,
! [Nb: nat,M2: nat,K: nat] :
( ( times_times(nat,K,M2) = times_times(nat,K,Nb) )
<=> ( ( M2 = Nb )
| ( K = zero_zero(nat) ) ) ) ).
tff(fact_29_mult__is__0,axiom,
! [Nb: nat,M2: nat] :
( ( times_times(nat,M2,Nb) = zero_zero(nat) )
<=> ( ( M2 = zero_zero(nat) )
| ( Nb = zero_zero(nat) ) ) ) ).
tff(fact_30_mult__0__right,axiom,
! [M1: nat] : times_times(nat,M1,zero_zero(nat)) = zero_zero(nat) ).
tff(fact_31_mult__0,axiom,
! [N: nat] : times_times(nat,zero_zero(nat),N) = zero_zero(nat) ).
tff(fact_32_mult__eq__0__iff,axiom,
! [A: $tType] :
( ring_n68954251visors(A)
=> ! [Ba: A,A2: A] :
( ( times_times(A,A2,Ba) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
| ( Ba = zero_zero(A) ) ) ) ) ).
tff(fact_33_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : times_times(A,zero_zero(A),A1) = zero_zero(A) ) ).
tff(fact_34_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : times_times(A,A1,zero_zero(A)) = zero_zero(A) ) ).
tff(fact_35_number__of__is__id,axiom,
! [K1: int] : number_number_of(int,K1) = K1 ).
tff(fact_36_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_37_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_38_nat__mult__commute,axiom,
! [N: nat,M1: nat] : times_times(nat,M1,N) = times_times(nat,N,M1) ).
tff(fact_39_nat__mult__assoc,axiom,
! [K1: nat,N: nat,M1: nat] : times_times(nat,times_times(nat,M1,N),K1) = times_times(nat,M1,times_times(nat,N,K1)) ).
tff(fact_40_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_41_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_42_even__zero__nat,axiom,
even_odd_even(nat,zero_zero(nat)) ).
tff(fact_43_semiring__norm_I112_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( zero_zero(A) = number_number_of(A,pls) ) ) ).
tff(fact_44_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
tff(fact_45_psize__eq__0__iff,axiom,
! [A: $tType] :
( zero(A)
=> ! [P1: poly(A)] :
( ( fundam1280195782_psize(A,P1) = zero_zero(nat) )
<=> ( P1 = zero_zero(poly(A)) ) ) ) ).
tff(fact_46_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_47_nat__mult__eq__cancel__disj,axiom,
! [Nb: nat,M2: nat,K: nat] :
( ( times_times(nat,K,M2) = times_times(nat,K,Nb) )
<=> ( ( K = zero_zero(nat) )
| ( M2 = Nb ) ) ) ).
tff(fact_48_power2__eq__square__number__of,axiom,
! [B1: $tType] :
( ( monoid_mult(B1)
& number(B1) )
=> ! [W1: int] : power_power(B1,number_number_of(B1,W1),number_number_of(nat,bit0(bit1(pls)))) = times_times(B1,number_number_of(B1,W1),number_number_of(B1,W1)) ) ).
tff(fact_49_power__eq__0__iff__number__of,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [W: int,A2: A] :
( ( power_power(A,A2,number_number_of(nat,W)) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
& ( number_number_of(nat,W) != zero_zero(nat) ) ) ) ) ).
tff(fact_50_zero__eq__power2,axiom,
! [A: $tType] :
( ring_11004092258visors(A)
=> ! [A2: A] :
( ( power_power(A,A2,number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_51_zero__power2,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( power_power(A,zero_zero(A),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) ) ) ).
tff(fact_52_zpower__zpower,axiom,
! [Z: nat,Y: nat,X1: int] : power_power(int,power_power(int,X1,Y),Z) = power_power(int,X1,times_times(nat,Y,Z)) ).
tff(fact_53_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X: A] :
( ( zero_zero(A) = X )
<=> ( X = zero_zero(A) ) ) ) ).
tff(fact_54_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_55_mult__idem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [X1: A] : times_times(A,X1,X1) = X1 ) ).
tff(fact_56_times_Oidem,axiom,
! [A: $tType] :
( ab_sem1668676832m_mult(A)
=> ! [A1: A] : times_times(A,A1,A1) = A1 ) ).
tff(fact_57_power3__eq__cube,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : power_power(A,A1,number_number_of(nat,bit1(bit1(pls)))) = times_times(A,times_times(A,A1,A1),A1) ) ).
tff(fact_58_power2__eq__square,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A1: A] : power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) = times_times(A,A1,A1) ) ).
tff(fact_59_power__even__eq,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,A1: A] : power_power(A,A1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = power_power(A,power_power(A,A1,N),number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_60_power__eq__0__iff,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [Nb: nat,A2: A] :
( ( power_power(A,A2,Nb) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
& ( Nb != zero_zero(nat) ) ) ) ) ).
tff(fact_61_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat,X1: A] : power_power(A,X1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = times_times(A,power_power(A,X1,N),power_power(A,X1,N)) ) ).
tff(fact_62_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X1: A] : times_times(A,X1,X1) = power_power(A,X1,number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_63_csqrt,axiom,
! [Z: complex] : power_power(complex,fundam1563812824_csqrt(Z),number_number_of(nat,bit0(bit1(pls)))) = Z ).
tff(fact_64_comm__semiring__1__class_Onormalizing__semiring__rules_I7_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [B: A,A1: A] : times_times(A,A1,B) = times_times(A,B,A1) ) ).
tff(fact_65_comm__semiring__1__class_Onormalizing__semiring__rules_I19_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : times_times(A,Lx,times_times(A,Rx,Ry)) = times_times(A,Rx,times_times(A,Lx,Ry)) ) ).
tff(fact_66_comm__semiring__1__class_Onormalizing__semiring__rules_I18_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Lx: A] : times_times(A,Lx,times_times(A,Rx,Ry)) = times_times(A,times_times(A,Lx,Rx),Ry) ) ).
tff(fact_67_comm__semiring__1__class_Onormalizing__semiring__rules_I17_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),Rx) = times_times(A,Lx,times_times(A,Ly,Rx)) ) ).
tff(fact_68_comm__semiring__1__class_Onormalizing__semiring__rules_I16_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),Rx) = times_times(A,times_times(A,Lx,Rx),Ly) ) ).
tff(fact_69_comm__semiring__1__class_Onormalizing__semiring__rules_I14_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,Lx,times_times(A,Ly,times_times(A,Rx,Ry))) ) ).
tff(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I15_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,Rx,times_times(A,times_times(A,Lx,Ly),Ry)) ) ).
tff(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I13_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Ry: A,Rx: A,Ly: A,Lx: A] : times_times(A,times_times(A,Lx,Ly),times_times(A,Rx,Ry)) = times_times(A,times_times(A,Lx,Rx),times_times(A,Ly,Ry)) ) ).
tff(fact_72_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) ) ).
tff(fact_73_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_74_field__power__not__zero,axiom,
! [A: $tType] :
( ring_11004092258visors(A)
=> ! [N: nat,A1: A] :
( ( A1 != zero_zero(A) )
=> ( power_power(A,A1,N) != zero_zero(A) ) ) ) ).
tff(fact_75_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Q: nat,Y: A,X1: A] : power_power(A,times_times(A,X1,Y),Q) = times_times(A,power_power(A,X1,Q),power_power(A,Y,Q)) ) ).
tff(fact_76_power__mult__distrib,axiom,
! [A: $tType] :
( comm_monoid_mult(A)
=> ! [N: nat,B: A,A1: A] : power_power(A,times_times(A,A1,B),N) = times_times(A,power_power(A,A1,N),power_power(A,B,N)) ) ).
tff(fact_77_power__commutes,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,A1: A] : times_times(A,power_power(A,A1,N),A1) = times_times(A,A1,power_power(A,A1,N)) ) ).
tff(fact_78_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Q: nat,P: nat,X1: A] : power_power(A,power_power(A,X1,P),Q) = power_power(A,X1,times_times(nat,P,Q)) ) ).
tff(fact_79_power__mult,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,M1: nat,A1: A] : power_power(A,A1,times_times(nat,M1,N)) = power_power(A,power_power(A,A1,M1),N) ) ).
tff(fact_80_four__x__squared,axiom,
! [X1: real] : times_times(real,number_number_of(real,bit0(bit0(bit1(pls)))),power_power(real,X1,number_number_of(nat,bit0(bit1(pls))))) = power_power(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X1),number_number_of(nat,bit0(bit1(pls)))) ).
tff(fact_81_zero__less__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( ord_less(A,zero_zero(A),power_power(A,A2,number_number_of(nat,bit0(bit1(pls)))))
<=> ( A2 != zero_zero(A) ) ) ) ).
tff(fact_82_zero__le__even__power_H,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat,A1: A] : ord_less_eq(A,zero_zero(A),power_power(A,A1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N))) ) ).
tff(fact_83_nat__zero__less__power__iff,axiom,
! [Nb: nat,X: nat] :
( ord_less(nat,zero_zero(nat),power_power(nat,X,Nb))
<=> ( ord_less(nat,zero_zero(nat),X)
| ( Nb = zero_zero(nat) ) ) ) ).
tff(fact_84_le__0__eq,axiom,
! [Nb: nat] :
( ord_less_eq(nat,Nb,zero_zero(nat))
<=> ( Nb = zero_zero(nat) ) ) ).
tff(fact_85_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_86_le0,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_87_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_88_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_89_neq0__conv,axiom,
! [Nb: nat] :
( ( Nb != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),Nb) ) ).
tff(fact_90_rel__simps_I34_J,axiom,
! [L: int,K: int] :
( ord_less_eq(int,bit1(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
tff(fact_91_rel__simps_I17_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit1(K),bit1(L))
<=> ord_less(int,K,L) ) ).
tff(fact_92_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
tff(fact_93_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_94_rel__simps_I31_J,axiom,
! [L: int,K: int] :
( ord_less_eq(int,bit0(K),bit0(L))
<=> ord_less_eq(int,K,L) ) ).
tff(fact_95_rel__simps_I14_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit0(K),bit0(L))
<=> ord_less(int,K,L) ) ).
tff(fact_96_not__real__square__gt__zero,axiom,
! [X: real] :
( ~ ord_less(real,zero_zero(real),times_times(real,X,X))
<=> ( X = zero_zero(real) ) ) ).
%----Arities (92)
tff(arity_Polynomial_Opoly___Rings_Oidom,axiom,
! [T_1: $tType] :
( idom(T_1)
=> idom(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__ring__1,axiom,
! [T_1: $tType] :
( comm_ring_1(T_1)
=> comm_ring_1(poly(T_1)) ) ).
tff(arity_Complex_Ocomplex___Rings_Oidom,axiom,
idom(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ocomm__ring__1,axiom,
comm_ring_1(complex) ).
tff(arity_RealDef_Oreal___Rings_Oidom,axiom,
idom(real) ).
tff(arity_RealDef_Oreal___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(real) ).
tff(arity_RealDef_Oreal___Rings_Ocomm__ring__1,axiom,
comm_ring_1(real) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(nat) ).
tff(arity_Int_Oint___Rings_Oidom,axiom,
idom(int) ).
tff(arity_Int_Oint___Rings_Ocomm__semiring__0,axiom,
comm_semiring_0(int) ).
tff(arity_Int_Oint___Rings_Ocomm__ring__1,axiom,
comm_ring_1(int) ).
tff(arity_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(int) ).
tff(arity_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
tff(arity_Int_Oint___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(int) ).
tff(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(int) ).
tff(arity_Int_Oint___Int_Onumber__semiring,axiom,
number_semiring(int) ).
tff(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(int) ).
tff(arity_Int_Oint___Groups_Omonoid__mult,axiom,
monoid_mult(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Rings_Omult__zero,axiom,
mult_zero(int) ).
tff(arity_Int_Oint___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
tff(arity_Int_Oint___Power_Opower,axiom,
power(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(nat) ).
tff(arity_Nat_Onat___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(nat) ).
tff(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(nat) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(nat) ).
tff(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(nat) ).
tff(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(nat) ).
tff(arity_Nat_Onat___Groups_Omonoid__mult,axiom,
monoid_mult(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero(nat) ).
tff(arity_Nat_Onat___Power_Opower,axiom,
power(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
tff(arity_RealDef_Oreal___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(real) ).
tff(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(real) ).
tff(arity_RealDef_Oreal___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(real) ).
tff(arity_RealDef_Oreal___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
tff(arity_RealDef_Oreal___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__semiring,axiom,
number_semiring(real) ).
tff(arity_RealDef_Oreal___Rings_Ozero__neq__one,axiom,
zero_neq_one(real) ).
tff(arity_RealDef_Oreal___Groups_Omonoid__mult,axiom,
monoid_mult(real) ).
tff(arity_RealDef_Oreal___Rings_Osemiring__1,axiom,
semiring_1(real) ).
tff(arity_RealDef_Oreal___Rings_Omult__zero,axiom,
mult_zero(real) ).
tff(arity_RealDef_Oreal___Int_Oring__char__0,axiom,
ring_char_0(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__ring,axiom,
number_ring(real) ).
tff(arity_RealDef_Oreal___Power_Opower,axiom,
power(real) ).
tff(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
tff(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
tff(arity_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(complex) ).
tff(arity_Complex_Ocomplex___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocomm__monoid__mult,axiom,
comm_monoid_mult(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber__semiring,axiom,
number_semiring(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ozero__neq__one,axiom,
zero_neq_one(complex) ).
tff(arity_Complex_Ocomplex___Groups_Omonoid__mult,axiom,
monoid_mult(complex) ).
tff(arity_Complex_Ocomplex___Rings_Osemiring__1,axiom,
semiring_1(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___Int_Onumber__ring,axiom,
number_ring(complex) ).
tff(arity_Complex_Ocomplex___Power_Opower,axiom,
power(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ozero,axiom,
zero(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber,axiom,
number(complex) ).
tff(arity_Polynomial_Opoly___Rings_Oring__1__no__zero__divisors,axiom,
! [T_1: $tType] :
( idom(T_1)
=> ring_11004092258visors(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_Oab__semigroup__mult,axiom,
! [T_1: $tType] :
( comm_semiring_0(T_1)
=> ab_semigroup_mult(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ocomm__monoid__mult,axiom,
! [T_1: $tType] :
( comm_semiring_1(T_1)
=> comm_monoid_mult(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___Rings_Olinordered__idom,axiom,
! [T_1: $tType] :
( linordered_idom(T_1)
=> linordered_idom(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___Int_Onumber__semiring,axiom,
! [T_1: $tType] :
( comm_ring_1(T_1)
=> number_semiring(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Ozero__neq__one,axiom,
! [T_1: $tType] :
( comm_semiring_1(T_1)
=> zero_neq_one(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Omonoid__mult,axiom,
! [T_1: $tType] :
( comm_semiring_1(T_1)
=> monoid_mult(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Rings_Osemiring__1,axiom,
! [T_1: $tType] :
( comm_semiring_1(T_1)
=> semiring_1(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___Int_Onumber__ring,axiom,
! [T_1: $tType] :
( comm_ring_1(T_1)
=> number_ring(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Power_Opower,axiom,
! [T_1: $tType] :
( comm_semiring_1(T_1)
=> power(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Groups_Ozero,axiom,
! [T_1: $tType] :
( zero(T_1)
=> zero(poly(T_1)) ) ).
tff(arity_Polynomial_Opoly___Int_Onumber,axiom,
! [T_1: $tType] :
( comm_ring_1(T_1)
=> number(poly(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,
? [M: nat] : na = times_times(nat,number_number_of(nat,bit0(bit1(pls))),M) ).
%------------------------------------------------------------------------------