TPTP Problem File: SWW482_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SWW482_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fundamental Theorem of Algebra line 43
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fta_43 [Bla13]
% Status : Theorem
% Rating : 0.33 v7.4.0, 0.50 v7.1.0, 1.00 v6.4.0
% Syntax : Number of formulae : 188 ( 74 unt; 42 typ; 0 def)
% Number of atoms : 268 ( 133 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 142 ( 20 ~; 2 |; 23 &)
% ( 19 <=>; 78 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 3 avg)
% Maximal term depth : 6 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 17 ( 12 >; 5 *; 0 +; 0 <<)
% Number of predicates : 23 ( 22 usr; 0 prp; 1-2 aty)
% Number of functors : 16 ( 16 usr; 5 con; 0-4 aty)
% Number of variables : 252 ( 223 !; 0 ?; 252 :)
% ( 29 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:14:13
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
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_RealDef_Oreal,type,
real: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (37)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Ofield,type,
field:
!>[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_Fields_Odivision__ring,type,
division_ring:
!>[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_Fields_Ofield__inverse__zero,type,
field_inverse_zero:
!>[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_Fields_Odivision__ring__inverse__zero,type,
divisi14063676e_zero:
!>[A: $tType] : $o ).
tff(sy_cl_Fields_Olinordered__field__inverse__zero,type,
linord1117847801e_zero:
!>[A: $tType] : $o ).
tff(sy_c_Fields_Oinverse__class_Odivide,type,
inverse_divide:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( 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_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_Oiszero,type,
iszero:
!>[A: $tType] : ( A > $o ) ).
tff(sy_c_Int_Onumber__class_Onumber__of,type,
number_number_of:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_NthRoot_Oroot,type,
root: nat > fun(real,real) ).
tff(sy_c_NthRoot_Osqrt,type,
sqrt: fun(real,real) ).
tff(sy_c_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( ( A * nat ) > A ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B2: $tType] : ( ( fun(A,B2) * A ) > B2 ) ).
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_y____,type,
y: real ).
%----Relevant facts (98)
tff(fact_0_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_1_rel__simps_I38_J,axiom,
! [L1: int] :
( ( pls = bit0(L1) )
<=> ( pls = L1 ) ) ).
tff(fact_2_rel__simps_I44_J,axiom,
! [K1: int] :
( ( bit0(K1) = pls )
<=> ( K1 = pls ) ) ).
tff(fact_3_rel__simps_I49_J,axiom,
! [L: int,K: int] : bit0(K) != bit1(L) ).
tff(fact_4_rel__simps_I50_J,axiom,
! [L: int,K: int] : bit1(K) != bit0(L) ).
tff(fact_5_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_6_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_7_sq4,axiom,
! [X: real] : inverse_divide(real,power_power(real,X,number_number_of(nat,bit0(bit1(pls)))),number_number_of(real,bit0(bit0(bit1(pls))))) = power_power(real,inverse_divide(real,X,number_number_of(real,bit0(bit1(pls)))),number_number_of(nat,bit0(bit1(pls)))) ).
tff(fact_8_real__sqrt__eq__iff,axiom,
! [Ya: real,Xa: real] :
( ( aa(real,real,sqrt,Xa) = aa(real,real,sqrt,Ya) )
<=> ( Xa = Ya ) ) ).
tff(fact_9_rel__simps_I48_J,axiom,
! [L1: int,K1: int] :
( ( bit0(K1) = bit0(L1) )
<=> ( K1 = L1 ) ) ).
tff(fact_10_rel__simps_I51_J,axiom,
! [L1: int,K1: int] :
( ( bit1(K1) = bit1(L1) )
<=> ( K1 = L1 ) ) ).
tff(fact_11_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Ya: int,Xa: int] :
( ( number_number_of(A,Xa) = number_number_of(A,Ya) )
<=> ( Xa = Ya ) ) ) ).
tff(fact_12_real__sqrt__divide,axiom,
! [Y: real,X: real] : aa(real,real,sqrt,inverse_divide(real,X,Y)) = inverse_divide(real,aa(real,real,sqrt,X),aa(real,real,sqrt,Y)) ).
tff(fact_13_divide__Numeral1,axiom,
! [A: $tType] :
( ( field(A)
& number_ring(A) )
=> ! [X: A] : inverse_divide(A,X,number_number_of(A,bit1(pls))) = X ) ).
tff(fact_14_divide__numeral__1,axiom,
! [A: $tType] :
( ( field(A)
& number_ring(A) )
=> ! [A1: A] : inverse_divide(A,A1,number_number_of(A,bit1(pls))) = A1 ) ).
tff(fact_15_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [Xa: A,W1: int] :
( ( number_number_of(A,W1) = Xa )
<=> ( Xa = number_number_of(A,W1) ) ) ) ).
tff(fact_16_real__sqrt__power,axiom,
! [K: nat,X: real] : aa(real,real,sqrt,power_power(real,X,K)) = power_power(real,aa(real,real,sqrt,X),K) ).
tff(fact_17_real__sqrt__abs,axiom,
! [X: real] : aa(real,real,sqrt,power_power(real,X,number_number_of(nat,bit0(bit1(pls))))) = abs_abs(real,X) ).
tff(fact_18_power__divide,axiom,
! [A: $tType] :
( field_inverse_zero(A)
=> ! [N: nat,B: A,A1: A] : power_power(A,inverse_divide(A,A1,B),N) = inverse_divide(A,power_power(A,A1,N),power_power(A,B,N)) ) ).
tff(fact_19_four__x__squared,axiom,
! [X: real] : times_times(real,number_number_of(real,bit0(bit0(bit1(pls)))),power_power(real,X,number_number_of(nat,bit0(bit1(pls))))) = power_power(real,times_times(real,number_number_of(real,bit0(bit1(pls))),X),number_number_of(nat,bit0(bit1(pls)))) ).
tff(fact_20_not__iszero__Numeral1,axiom,
! [A: $tType] :
( number_ring(A)
=> ~ iszero(A,number_number_of(A,bit1(pls))) ) ).
tff(fact_21_power2__eq__square__number__of,axiom,
! [B2: $tType] :
( ( monoid_mult(B2)
& number(B2) )
=> ! [W: int] : power_power(B2,number_number_of(B2,W),number_number_of(nat,bit0(bit1(pls)))) = times_times(B2,number_number_of(B2,W),number_number_of(B2,W)) ) ).
tff(fact_22_iszero__number__of__Bit0,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [W1: int] :
( iszero(A,number_number_of(A,bit0(W1)))
<=> iszero(A,number_number_of(A,W1)) ) ) ).
tff(fact_23_sqrt__def,axiom,
sqrt = root(number_number_of(nat,bit0(bit1(pls)))) ).
tff(fact_24_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_25_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_26_y0,axiom,
y != zero_zero(real) ).
tff(fact_27_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_28_mult__Bit0,axiom,
! [L: int,K: int] : times_times(int,bit0(K),L) = bit0(times_times(int,K,L)) ).
tff(fact_29_real__sqrt__zero,axiom,
aa(real,real,sqrt,zero_zero(real)) = zero_zero(real) ).
tff(fact_30_real__sqrt__eq__0__iff,axiom,
! [Xa: real] :
( ( aa(real,real,sqrt,Xa) = zero_zero(real) )
<=> ( Xa = zero_zero(real) ) ) ).
tff(fact_31_real__root__zero,axiom,
! [N: nat] : aa(real,real,root(N),zero_zero(real)) = zero_zero(real) ).
tff(fact_32_arith__simps_I32_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : times_times(A,number_number_of(A,V),number_number_of(A,W)) = number_number_of(A,times_times(int,V,W)) ) ).
tff(fact_33_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V: int] : times_times(A,number_number_of(A,V),times_times(A,number_number_of(A,W),Z)) = times_times(A,number_number_of(A,times_times(int,V,W)),Z) ) ).
tff(fact_34_power__eq__0__iff,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [N1: nat,A2: A] :
( ( power_power(A,A2,N1) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
& ( N1 != zero_zero(nat) ) ) ) ) ).
tff(fact_35_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_36_real__divide__square__eq,axiom,
! [A1: real,R: real] : inverse_divide(real,times_times(real,R,A1),times_times(real,R,R)) = inverse_divide(real,A1,R) ).
tff(fact_37_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_38_real__sqrt__abs2,axiom,
! [X: real] : aa(real,real,sqrt,times_times(real,X,X)) = abs_abs(real,X) ).
tff(fact_39_divide__eq__eq__number__of1,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [A2: A,W1: int,B1: A] :
( ( inverse_divide(A,B1,number_number_of(A,W1)) = A2 )
<=> ( ( ( number_number_of(A,W1) != zero_zero(A) )
=> ( B1 = times_times(A,A2,number_number_of(A,W1)) ) )
& ( ( number_number_of(A,W1) = zero_zero(A) )
=> ( A2 = zero_zero(A) ) ) ) ) ) ).
tff(fact_40_eq__divide__eq__number__of1,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [W1: int,B1: A,A2: A] :
( ( A2 = inverse_divide(A,B1,number_number_of(A,W1)) )
<=> ( ( ( number_number_of(A,W1) != zero_zero(A) )
=> ( times_times(A,A2,number_number_of(A,W1)) = B1 ) )
& ( ( number_number_of(A,W1) = zero_zero(A) )
=> ( A2 = zero_zero(A) ) ) ) ) ) ).
tff(fact_41_abs__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : abs_abs(A,power_power(A,A1,number_number_of(nat,bit0(bit1(pls))))) = power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_42_power2__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : power_power(A,abs_abs(A,A1),number_number_of(nat,bit0(bit1(pls)))) = power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_43_iszero__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> iszero(A,zero_zero(A)) ) ).
tff(fact_44_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_45_iszero__def,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [Za: A] :
( iszero(A,Za)
<=> ( Za = zero_zero(A) ) ) ) ).
tff(fact_46_number__of__mult,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,times_times(int,V,W)) = times_times(A,number_number_of(A,V),number_number_of(A,W)) ) ).
tff(fact_47_power__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat,A1: A] : abs_abs(A,power_power(A,A1,N)) = power_power(A,abs_abs(A,A1),N) ) ).
tff(fact_48_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_49_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_50_power__mult,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat,M: nat,A1: A] : power_power(A,A1,times_times(nat,M,N)) = power_power(A,power_power(A,A1,M),N) ) ).
tff(fact_51_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_52_divide__eq__eq__number__of,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [W1: int,C1: A,B1: A] :
( ( inverse_divide(A,B1,C1) = number_number_of(A,W1) )
<=> ( ( ( C1 != zero_zero(A) )
=> ( B1 = times_times(A,number_number_of(A,W1),C1) ) )
& ( ( C1 = zero_zero(A) )
=> ( number_number_of(A,W1) = zero_zero(A) ) ) ) ) ) ).
tff(fact_53_eq__divide__eq__number__of,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [C1: A,B1: A,W1: int] :
( ( number_number_of(A,W1) = inverse_divide(A,B1,C1) )
<=> ( ( ( C1 != zero_zero(A) )
=> ( times_times(A,number_number_of(A,W1),C1) = B1 ) )
& ( ( C1 = zero_zero(A) )
=> ( number_number_of(A,W1) = zero_zero(A) ) ) ) ) ) ).
tff(fact_54_real__sqrt__mult,axiom,
! [Y: real,X: real] : aa(real,real,sqrt,times_times(real,X,Y)) = times_times(real,aa(real,real,sqrt,X),aa(real,real,sqrt,Y)) ).
tff(fact_55_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_56_semiring__norm_I112_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( zero_zero(A) = number_number_of(A,pls) ) ) ).
tff(fact_57_nonzero__power__divide,axiom,
! [A: $tType] :
( field(A)
=> ! [N: nat,A1: A,B: A] :
( ( B != zero_zero(A) )
=> ( power_power(A,inverse_divide(A,A1,B),N) = inverse_divide(A,power_power(A,A1,N),power_power(A,B,N)) ) ) ) ).
tff(fact_58_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
tff(fact_59_semiring__numeral__0__eq__0,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_60_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_61_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_62_mult__numeral__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : times_times(A,number_number_of(A,bit1(pls)),A1) = A1 ) ).
tff(fact_63_divide__Numeral0,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number_ring(A) )
=> ! [X: A] : inverse_divide(A,X,number_number_of(A,pls)) = zero_zero(A) ) ).
tff(fact_64_iszero__number__of__Bit1,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [W: int] : ~ iszero(A,number_number_of(A,bit1(W))) ) ).
tff(fact_65_iszero__Numeral0,axiom,
! [A: $tType] :
( number_ring(A)
=> iszero(A,number_number_of(A,pls)) ) ).
tff(fact_66_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_67_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_68_power__eq__0__iff__number__of,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [W1: int,A2: A] :
( ( power_power(A,A2,number_number_of(nat,W1)) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
& ( number_number_of(nat,W1) != zero_zero(nat) ) ) ) ) ).
tff(fact_69_eq__divide__2__times__iff,axiom,
! [Za: real,Ya: real,Xa: real] :
( ( Xa = inverse_divide(real,Ya,times_times(real,number_number_of(real,bit0(bit1(pls))),Za)) )
<=> ( times_times(real,number_number_of(real,bit0(bit1(pls))),Xa) = inverse_divide(real,Ya,Za) ) ) ).
tff(fact_70_comm__semiring__1__class_Onormalizing__semiring__rules_I36_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [N: nat,X: A] : power_power(A,X,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N)) = times_times(A,power_power(A,X,N),power_power(A,X,N)) ) ).
tff(fact_71_comm__semiring__1__class_Onormalizing__semiring__rules_I29_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X: A] : times_times(A,X,X) = power_power(A,X,number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_72_abs__divide,axiom,
! [A: $tType] :
( linord1117847801e_zero(A)
=> ! [B: A,A1: A] : abs_abs(A,inverse_divide(A,A1,B)) = inverse_divide(A,abs_abs(A,A1),abs_abs(A,B)) ) ).
tff(fact_73_ext,axiom,
! [B2: $tType,A: $tType,G: fun(A,B2),F: fun(A,B2)] :
( ! [X1: A] : aa(A,B2,F,X1) = aa(A,B2,G,X1)
=> ( F = G ) ) ).
tff(fact_74_abs__mult__self,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : times_times(A,abs_abs(A,A1),abs_abs(A,A1)) = times_times(A,A1,A1) ) ).
tff(fact_75_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : times_times(A,zero_zero(A),A1) = zero_zero(A) ) ).
tff(fact_76_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : times_times(A,A1,zero_zero(A)) = zero_zero(A) ) ).
tff(fact_77_mult__eq__0__iff,axiom,
! [A: $tType] :
( ring_n68954251visors(A)
=> ! [B1: A,A2: A] :
( ( times_times(A,A2,B1) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
| ( B1 = zero_zero(A) ) ) ) ) ).
tff(fact_78_divide__zero__left,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [A1: A] : inverse_divide(A,zero_zero(A),A1) = zero_zero(A) ) ).
tff(fact_79_divide__zero,axiom,
! [A: $tType] :
( divisi14063676e_zero(A)
=> ! [A1: A] : inverse_divide(A,A1,zero_zero(A)) = zero_zero(A) ) ).
tff(fact_80_times__divide__eq__right,axiom,
! [A: $tType] :
( division_ring(A)
=> ! [C: A,B: A,A1: A] : times_times(A,A1,inverse_divide(A,B,C)) = inverse_divide(A,times_times(A,A1,B),C) ) ).
tff(fact_81_number__of__is__id,axiom,
! [K: int] : number_number_of(int,K) = K ).
tff(fact_82_times__numeral__code_I5_J,axiom,
! [W: int,V: int] : times_times(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,times_times(int,V,W)) ).
tff(fact_83_zpower__zpower,axiom,
! [Z: nat,Y: nat,X: int] : power_power(int,power_power(int,X,Y),Z) = power_power(int,X,times_times(nat,Y,Z)) ).
tff(fact_84_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_85_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_86_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_87_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_88_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_89_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_90_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_91_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_92_comm__semiring__1__class_Onormalizing__semiring__rules_I31_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Q: nat,P: nat,X: A] : power_power(A,power_power(A,X,P),Q) = power_power(A,X,times_times(nat,P,Q)) ) ).
tff(fact_93_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_94_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_95_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_96_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_97_comm__semiring__1__class_Onormalizing__semiring__rules_I30_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [Q: nat,Y: A,X: A] : power_power(A,times_times(A,X,Y),Q) = times_times(A,power_power(A,X,Q),power_power(A,Y,Q)) ) ).
%----Arities (45)
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_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___Int_Onumber,axiom,
number(int) ).
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___Int_Onumber,axiom,
number(nat) ).
tff(arity_RealDef_Oreal___Fields_Olinordered__field__inverse__zero,axiom,
linord1117847801e_zero(real) ).
tff(arity_RealDef_Oreal___Fields_Odivision__ring__inverse__zero,axiom,
divisi14063676e_zero(real) ).
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___Fields_Ofield__inverse__zero,axiom,
field_inverse_zero(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___Fields_Odivision__ring,axiom,
division_ring(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___Fields_Ofield,axiom,
field(real) ).
tff(arity_RealDef_Oreal___Power_Opower,axiom,
power(real) ).
tff(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
%----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,
aa(real,real,sqrt,number_number_of(real,bit0(bit0(bit1(pls))))) = aa(real,real,sqrt,power_power(real,number_number_of(real,bit0(bit1(pls))),number_number_of(nat,bit0(bit1(pls))))) ).
%------------------------------------------------------------------------------