TPTP Problem File: NUN018_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : NUN018_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 206
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_206 [Bla13]
% Status : ContradictoryAxioms
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 163 ( 60 unt; 36 typ; 0 def)
% Number of atoms : 244 ( 81 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 148 ( 31 ~; 5 |; 15 &)
% ( 45 <=>; 52 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 14 ( 9 >; 5 *; 0 +; 0 <<)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-3 aty)
% Number of functors : 15 ( 15 usr; 7 con; 0-4 aty)
% Number of variables : 190 ( 167 !; 0 ?; 190 :)
% ( 23 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_CAX_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:26:40
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
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_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (32)
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_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_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_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__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__semigroup__add__imp__le,type,
ordere236663937imp_le:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( ( A * A ) > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_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_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( ( A * nat ) > A ) ).
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_n____,type,
n: nat ).
tff(sy_v_tn____,type,
tn: nat ).
tff(sy_v_v____,type,
v: int ).
tff(sy_v_w____,type,
w: int ).
%----Relevant facts (97)
tff(fact_0__096w_A_126_061_A0_096,axiom,
w != zero_zero(int) ).
tff(fact_1_zero__less__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ord_less(A,zero_zero(A),power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))))
<=> ( A1 != zero_zero(A) ) ) ) ).
tff(fact_2_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_3_zero__eq__power2,axiom,
! [A: $tType] :
( ring_11004092258visors(A)
=> ! [A1: A] :
( ( power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_4_less__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int] :
( ord_less(A,zero_zero(A),number_number_of(A,Ya))
<=> ord_less(int,pls,Ya) ) ) ).
tff(fact_5_less__special_I3_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Xa: int] :
( ord_less(A,number_number_of(A,Xa),zero_zero(A))
<=> ord_less(int,Xa,pls) ) ) ).
tff(fact_6_rel__simps_I4_J,axiom,
! [K: int] :
( ord_less(int,pls,bit0(K))
<=> ord_less(int,pls,K) ) ).
tff(fact_7_rel__simps_I10_J,axiom,
! [K: int] :
( ord_less(int,bit0(K),pls)
<=> ord_less(int,K,pls) ) ).
tff(fact_8_rel__simps_I16_J,axiom,
! [L1: int,K: int] :
( ord_less(int,bit1(K),bit0(L1))
<=> ord_less(int,K,L1) ) ).
tff(fact_9_rel__simps_I12_J,axiom,
! [K: int] :
( ord_less(int,bit1(K),pls)
<=> ord_less(int,K,pls) ) ).
tff(fact_10_power__eq__0__iff__number__of,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [Wa: int,A1: A] :
( ( power_power(A,A1,number_number_of(nat,Wa)) = zero_zero(A) )
<=> ( ( A1 = zero_zero(A) )
& ( number_number_of(nat,Wa) != zero_zero(nat) ) ) ) ) ).
tff(fact_11_less__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int,Xa: int] :
( ord_less(A,number_number_of(A,Xa),number_number_of(A,Ya))
<=> ord_less(int,Xa,Ya) ) ) ).
tff(fact_12_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_13_tn0,axiom,
ord_less(nat,zero_zero(nat),tn) ).
tff(fact_14_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_15_rel__simps_I51_J,axiom,
! [L1: int,K: int] :
( ( bit1(K) = bit1(L1) )
<=> ( K = L1 ) ) ).
tff(fact_16_rel__simps_I48_J,axiom,
! [L1: int,K: int] :
( ( bit0(K) = bit0(L1) )
<=> ( K = L1 ) ) ).
tff(fact_17_n0,axiom,
ord_less(nat,zero_zero(nat),n) ).
tff(fact_18_rel__simps_I46_J,axiom,
! [K1: int] : bit1(K1) != pls ).
tff(fact_19_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_20_rel__simps_I50_J,axiom,
! [L: int,K1: int] : bit1(K1) != bit0(L) ).
tff(fact_21_rel__simps_I49_J,axiom,
! [L: int,K1: int] : bit0(K1) != bit1(L) ).
tff(fact_22_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_23_rel__simps_I38_J,axiom,
! [L1: int] :
( ( pls = bit0(L1) )
<=> ( pls = L1 ) ) ).
tff(fact_24_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_25_rel__simps_I17_J,axiom,
! [L1: int,K: int] :
( ord_less(int,bit1(K),bit1(L1))
<=> ord_less(int,K,L1) ) ).
tff(fact_26_zero__less__power__nat__eq__number__of,axiom,
! [Wa: int,Xa: nat] :
( ord_less(nat,zero_zero(nat),power_power(nat,Xa,number_number_of(nat,Wa)))
<=> ( ( number_number_of(nat,Wa) = zero_zero(nat) )
| ord_less(nat,zero_zero(nat),Xa) ) ) ).
tff(fact_27_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_28_rel__simps_I14_J,axiom,
! [L1: int,K: int] :
( ord_less(int,bit0(K),bit0(L1))
<=> ord_less(int,K,L1) ) ).
tff(fact_29_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_30_less__nat__number__of,axiom,
! [V1: int,Va: int] :
( ord_less(nat,number_number_of(nat,Va),number_number_of(nat,V1))
<=> ( ( ord_less(int,Va,V1)
=> ord_less(int,pls,V1) )
& ord_less(int,Va,V1) ) ) ).
tff(fact_31_less__0__number__of,axiom,
! [Va: int] :
( ord_less(nat,zero_zero(nat),number_number_of(nat,Va))
<=> ord_less(int,pls,Va) ) ).
tff(fact_32_zero__less__power__nat__eq,axiom,
! [Na: nat,Xa: nat] :
( ord_less(nat,zero_zero(nat),power_power(nat,Xa,Na))
<=> ( ( Na = zero_zero(nat) )
| ord_less(nat,zero_zero(nat),Xa) ) ) ).
tff(fact_33_less__number__of__int__code,axiom,
! [L1: int,K: int] :
( ord_less(int,number_number_of(int,K),number_number_of(int,L1))
<=> ord_less(int,K,L1) ) ).
tff(fact_34_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_35_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
tff(fact_36_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [Xa: A,Wa: int] :
( ( number_number_of(A,Wa) = Xa )
<=> ( Xa = number_number_of(A,Wa) ) ) ) ).
tff(fact_37_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_38_less__int__code_I16_J,axiom,
! [K2: int,K11: int] :
( ord_less(int,bit1(K11),bit1(K2))
<=> ord_less(int,K11,K2) ) ).
tff(fact_39_less__int__code_I13_J,axiom,
! [K2: int,K11: int] :
( ord_less(int,bit0(K11),bit0(K2))
<=> ord_less(int,K11,K2) ) ).
tff(fact_40_semiring__numeral__0__eq__0,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_41_semiring__norm_I112_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( zero_zero(A) = number_number_of(A,pls) ) ) ).
tff(fact_42_less__int__code_I15_J,axiom,
! [K2: int,K11: int] :
( ord_less(int,bit1(K11),bit0(K2))
<=> ord_less(int,K11,K2) ) ).
tff(fact_43_bin__less__0__simps_I4_J,axiom,
! [Wa: int] :
( ord_less(int,bit1(Wa),zero_zero(int))
<=> ord_less(int,Wa,zero_zero(int)) ) ).
tff(fact_44_bin__less__0__simps_I1_J,axiom,
~ ord_less(int,pls,zero_zero(int)) ).
tff(fact_45_bin__less__0__simps_I3_J,axiom,
! [Wa: int] :
( ord_less(int,bit0(Wa),zero_zero(int))
<=> ord_less(int,Wa,zero_zero(int)) ) ).
tff(fact_46_power2__less__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] : ~ ord_less(A,power_power(A,A2,number_number_of(nat,bit0(bit1(pls)))),zero_zero(A)) ) ).
tff(fact_47_power__eq__0__iff,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [Na: nat,A1: A] :
( ( power_power(A,A1,Na) = zero_zero(A) )
<=> ( ( A1 = zero_zero(A) )
& ( Na != zero_zero(nat) ) ) ) ) ).
tff(fact_48_quartic__square__square,axiom,
! [X: int] : power_power(int,power_power(int,X,number_number_of(nat,bit0(bit1(pls)))),number_number_of(nat,bit0(bit1(pls)))) = power_power(int,X,number_number_of(nat,bit0(bit0(bit1(pls))))) ).
tff(fact_49_pos2,axiom,
ord_less(nat,zero_zero(nat),number_number_of(nat,bit0(bit1(pls)))) ).
tff(fact_50_calculation_I1_J,axiom,
plus_plus(int,power_power(int,v,number_number_of(nat,bit0(bit1(pls)))),power_power(int,w,number_number_of(nat,bit0(bit1(pls))))) = zero_zero(int) ).
tff(fact_51_zero__less__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A2: A] :
( ord_less(A,zero_zero(A),A2)
=> ord_less(A,zero_zero(A),power_power(A,A2,N)) ) ) ).
tff(fact_52_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_53_nat__zero__less__power__iff,axiom,
! [Na: nat,Xa: nat] :
( ord_less(nat,zero_zero(nat),power_power(nat,Xa,Na))
<=> ( ord_less(nat,zero_zero(nat),Xa)
| ( Na = zero_zero(nat) ) ) ) ).
tff(fact_54_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_55_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_56_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( plus_plus(A,A1,A1) = zero_zero(A) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_57_add__Bit0__Bit0,axiom,
! [L: int,K1: int] : plus_plus(int,bit0(K1),bit0(L)) = bit0(plus_plus(int,K1,L)) ).
tff(fact_58_add__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : plus_plus(A,number_number_of(A,V),number_number_of(A,W)) = number_number_of(A,plus_plus(int,V,W)) ) ).
tff(fact_59_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V: int] : plus_plus(A,number_number_of(A,V),plus_plus(A,number_number_of(A,W),Z)) = plus_plus(A,number_number_of(A,plus_plus(int,V,W)),Z) ) ).
tff(fact_60_add__Bit0__Bit1,axiom,
! [L: int,K1: int] : plus_plus(int,bit0(K1),bit1(L)) = bit1(plus_plus(int,K1,L)) ).
tff(fact_61_add__Bit1__Bit0,axiom,
! [L: int,K1: int] : plus_plus(int,bit1(K1),bit0(L)) = bit1(plus_plus(int,K1,L)) ).
tff(fact_62_calculation_I2_J,axiom,
( ( v != zero_zero(int) )
=> ord_less(int,zero_zero(int),plus_plus(int,power_power(int,v,number_number_of(nat,bit0(bit1(pls)))),power_power(int,w,number_number_of(nat,bit0(bit1(pls)))))) ) ).
tff(fact_63_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_64_number__of__is__id,axiom,
! [K1: int] : number_number_of(int,K1) = K1 ).
tff(fact_65_number__of__add,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,plus_plus(int,V,W)) = plus_plus(A,number_number_of(A,V),number_number_of(A,W)) ) ).
tff(fact_66_nat__neq__iff,axiom,
! [Na: nat,Ma: nat] :
( ( Ma != Na )
<=> ( ord_less(nat,Ma,Na)
| ord_less(nat,Na,Ma) ) ) ).
tff(fact_67_plus__numeral__code_I9_J,axiom,
! [W: int,V: int] : plus_plus(int,number_number_of(int,V),number_number_of(int,W)) = number_number_of(int,plus_plus(int,V,W)) ).
tff(fact_68_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_69_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_70_less__not__refl2,axiom,
! [M: nat,N: nat] :
( ord_less(nat,N,M)
=> ( M != N ) ) ).
tff(fact_71_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_72_nat__less__cases,axiom,
! [P: fun(nat,fun(nat,bool)),Na: nat,Ma: nat] :
( ( ord_less(nat,Ma,Na)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) )
=> ( ( ( Ma = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) )
=> ( ( ord_less(nat,Na,Ma)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P,Na),Ma)) ) ) ) ).
tff(fact_73_add__Pls,axiom,
! [K1: int] : plus_plus(int,pls,K1) = K1 ).
tff(fact_74_add__Pls__right,axiom,
! [K1: int] : plus_plus(int,K1,pls) = K1 ).
tff(fact_75_Bit0__def,axiom,
! [K1: int] : bit0(K1) = plus_plus(int,K1,K1) ).
tff(fact_76_even__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ord_less(A,plus_plus(A,A1,A1),zero_zero(A))
<=> ord_less(A,A1,zero_zero(A)) ) ) ).
tff(fact_77_add__numeral__0__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A2: A] : plus_plus(A,A2,number_number_of(A,pls)) = A2 ) ).
tff(fact_78_add__numeral__0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A2: A] : plus_plus(A,number_number_of(A,pls),A2) = A2 ) ).
tff(fact_79_number__of__Bit0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : number_number_of(A,bit0(W)) = plus_plus(A,plus_plus(A,zero_zero(A),number_number_of(A,W)),number_number_of(A,W)) ) ).
tff(fact_80_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_81_gr__implies__not0,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_82_nat__power__less__imp__less,axiom,
! [N: nat,M: nat,I: nat] :
( ord_less(nat,zero_zero(nat),I)
=> ( ord_less(nat,power_power(nat,I,M),power_power(nat,I,N))
=> ord_less(nat,M,N) ) ) ).
tff(fact_83_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_84_sum__power2__eq__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Ya: A,Xa: A] :
( ( plus_plus(A,power_power(A,Xa,number_number_of(nat,bit0(bit1(pls)))),power_power(A,Ya,number_number_of(nat,bit0(bit1(pls))))) = zero_zero(A) )
<=> ( ( Xa = zero_zero(A) )
& ( Ya = zero_zero(A) ) ) ) ) ).
tff(fact_85_field__power__not__zero,axiom,
! [A: $tType] :
( ring_11004092258visors(A)
=> ! [N: nat,A2: A] :
( ( A2 != zero_zero(A) )
=> ( power_power(A,A2,N) != zero_zero(A) ) ) ) ).
tff(fact_86_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] : ~ ord_less(A,plus_plus(A,power_power(A,X,number_number_of(nat,bit0(bit1(pls)))),power_power(A,Y,number_number_of(nat,bit0(bit1(pls))))),zero_zero(A)) ) ).
tff(fact_87_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Ya: A,Xa: A] :
( ord_less(A,zero_zero(A),plus_plus(A,power_power(A,Xa,number_number_of(nat,bit0(bit1(pls)))),power_power(A,Ya,number_number_of(nat,bit0(bit1(pls))))))
<=> ( ( Xa != zero_zero(A) )
| ( Ya != zero_zero(A) ) ) ) ) ).
tff(fact_88_zero__less__double__add__iff__zero__less__single__add,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ord_less(A,zero_zero(A),plus_plus(A,A1,A1))
<=> ord_less(A,zero_zero(A),A1) ) ) ).
tff(fact_89_double__add__less__zero__iff__single__add__less__zero,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ord_less(A,plus_plus(A,A1,A1),zero_zero(A))
<=> ord_less(A,A1,zero_zero(A)) ) ) ).
tff(fact_90_add__less__cancel__right,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,C: A,A1: A] :
( ord_less(A,plus_plus(A,A1,C),plus_plus(A,B,C))
<=> ord_less(A,A1,B) ) ) ).
tff(fact_91_add__left__cancel,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_92_add__right__cancel,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_93_add__is__0,axiom,
! [Na: nat,Ma: nat] :
( ( plus_plus(nat,Ma,Na) = zero_zero(nat) )
<=> ( ( Ma = zero_zero(nat) )
& ( Na = zero_zero(nat) ) ) ) ).
tff(fact_94_nat__add__left__cancel__less,axiom,
! [Na: nat,Ma: nat,K: nat] :
( ord_less(nat,plus_plus(nat,K,Ma),plus_plus(nat,K,Na))
<=> ord_less(nat,Ma,Na) ) ).
tff(fact_95_double__zero__sym,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A1: A] :
( ( zero_zero(A) = plus_plus(A,A1,A1) )
<=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_96_add__less__cancel__left,axiom,
! [A: $tType] :
( ordere236663937imp_le(A)
=> ! [B: A,A1: A,C: A] :
( ord_less(A,plus_plus(A,C,A1),plus_plus(A,C,B))
<=> ord_less(A,A1,B) ) ) ).
%----Arities (27)
tff(arity_Int_Oint___Groups_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(int) ).
tff(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(int) ).
tff(arity_Int_Oint___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(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___Int_Onumber__semiring,axiom,
number_semiring(int) ).
tff(arity_Int_Oint___Rings_Ozero__neq__one,axiom,
zero_neq_one(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_Oordered__ab__semigroup__add__imp__le,axiom,
ordere236663937imp_le(nat) ).
tff(arity_Nat_Onat___Groups_Ocancel__semigroup__add,axiom,
cancel_semigroup_add(nat) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(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___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) ).
%----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(int,zero_zero(int),power_power(int,w,number_number_of(nat,bit0(bit1(pls))))) ).
%------------------------------------------------------------------------------