TPTP Problem File: NUM971_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : NUM971_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 104
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_104 [Bla13]
% Status : Theorem
% Rating : 0.00 v7.5.0, 0.33 v7.4.0, 0.25 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 157 ( 53 unt; 32 typ; 0 def)
% Number of atoms : 242 ( 65 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 132 ( 15 ~; 2 |; 19 &)
% ( 53 <=>; 43 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 12 ( 8 >; 4 *; 0 +; 0 <<)
% Number of predicates : 19 ( 17 usr; 1 prp; 0-3 aty)
% Number of functors : 12 ( 12 usr; 5 con; 0-3 aty)
% Number of variables : 169 ( 148 !; 0 ?; 169 :)
% ( 21 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:24:26
%------------------------------------------------------------------------------
%----Should-be-implicit typings (3)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
%----Explicit typings (29)
tff(sy_cl_Int_Onumber,type,
number:
!>[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__0,type,
semiring_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[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_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_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
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_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * 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_s____,type,
s: int ).
tff(sy_v_t____,type,
t: int ).
%----Relevant facts (98)
tff(fact_0__096_126_A1_A_060_061_At_096,axiom,
~ ord_less_eq(int,one_one(int),t) ).
tff(fact_1__0960_A_060_061_As_A_094_A2_096,axiom,
ord_less_eq(int,zero_zero(int),power_power(int,s,number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_2__096t_A_060_A1_096,axiom,
ord_less(int,t,one_one(int)) ).
tff(fact_3__096t_A_061_A0_A_061_061_062_As_A_094_A2_A_L_A1_A_061_A0_096,axiom,
( ( t = zero_zero(int) )
=> ( plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)) = zero_zero(int) ) ) ).
tff(fact_4__096t_A_060_A0_A_061_061_062_As_A_094_A2_A_L_A1_A_060_A0_096,axiom,
( ord_less(int,t,zero_zero(int))
=> ord_less(int,plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)),zero_zero(int)) ) ).
tff(fact_5_one__power2,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( power_power(A,one_one(A),number_number_of(nat,bit0(bit1(pls)))) = one_one(A) ) ) ).
tff(fact_6_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_7_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_8_le__special_I2_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int] :
( ord_less_eq(A,one_one(A),number_number_of(A,Y))
<=> ord_less_eq(int,bit1(pls),Y) ) ) ).
tff(fact_9_le__special_I4_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [X1: int] :
( ord_less_eq(A,number_number_of(A,X1),one_one(A))
<=> ord_less_eq(int,X1,bit1(pls)) ) ) ).
tff(fact_10_le__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int] :
( ord_less_eq(A,zero_zero(A),number_number_of(A,Y))
<=> ord_less_eq(int,pls,Y) ) ) ).
tff(fact_11_le__special_I3_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [X1: int] :
( ord_less_eq(A,number_number_of(A,X1),zero_zero(A))
<=> ord_less_eq(int,X1,pls) ) ) ).
tff(fact_12_power__0__left__number__of,axiom,
! [A: $tType] :
( ( power(A)
& semiring_0(A) )
=> ! [W1: int] :
( ( ( number_number_of(nat,W1) = zero_zero(nat) )
=> ( power_power(A,zero_zero(A),number_number_of(nat,W1)) = one_one(A) ) )
& ( ( number_number_of(nat,W1) != zero_zero(nat) )
=> ( power_power(A,zero_zero(A),number_number_of(nat,W1)) = zero_zero(A) ) ) ) ) ).
tff(fact_13_numeral__1__eq__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_14_rel__simps_I21_J,axiom,
! [K3: int] :
( ord_less_eq(int,pls,bit0(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_15_rel__simps_I27_J,axiom,
! [K3: int] :
( ord_less_eq(int,bit0(K3),pls)
<=> ord_less_eq(int,K3,pls) ) ).
tff(fact_16_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y: int,X1: int] :
( ( number_number_of(A,X1) = number_number_of(A,Y) )
<=> ( X1 = Y ) ) ) ).
tff(fact_17_rel__simps_I51_J,axiom,
! [L: int,K3: int] :
( ( bit1(K3) = bit1(L) )
<=> ( K3 = L ) ) ).
tff(fact_18_rel__simps_I48_J,axiom,
! [L: int,K3: int] :
( ( bit0(K3) = bit0(L) )
<=> ( K3 = L ) ) ).
tff(fact_19_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [A2: A] :
( ( plus_plus(A,A2,A2) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_20_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_21_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_22_rel__simps_I50_J,axiom,
! [L1: int,K: int] : bit1(K) != bit0(L1) ).
tff(fact_23_rel__simps_I49_J,axiom,
! [L1: int,K: int] : bit0(K) != bit1(L1) ).
tff(fact_24_rel__simps_I44_J,axiom,
! [K3: int] :
( ( bit0(K3) = pls )
<=> ( K3 = pls ) ) ).
tff(fact_25_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_26_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_27_rel__simps_I17_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit1(K3),bit1(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_28_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_29_rel__simps_I14_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit0(K3),bit0(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_30_rel__simps_I34_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit1(K3),bit1(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_31_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
tff(fact_32_rel__simps_I31_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit0(K3),bit0(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_33_add__Bit0__Bit0,axiom,
! [L1: int,K: int] : plus_plus(int,bit0(K),bit0(L1)) = bit0(plus_plus(int,K,L1)) ).
tff(fact_34_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_35_less__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int,X1: int] :
( ord_less(A,number_number_of(A,X1),number_number_of(A,Y))
<=> ord_less(int,X1,Y) ) ) ).
tff(fact_36_le__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int,X1: int] :
( ord_less_eq(A,number_number_of(A,X1),number_number_of(A,Y))
<=> ord_less_eq(int,X1,Y) ) ) ).
tff(fact_37_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W1: int,V: int] : plus_plus(A,number_number_of(A,V),plus_plus(A,number_number_of(A,W1),Z)) = plus_plus(A,number_number_of(A,plus_plus(int,V,W1)),Z) ) ).
tff(fact_38_add__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : plus_plus(A,number_number_of(A,V),number_number_of(A,W1)) = number_number_of(A,plus_plus(int,V,W1)) ) ).
tff(fact_39_rel__simps_I12_J,axiom,
! [K3: int] :
( ord_less(int,bit1(K3),pls)
<=> ord_less(int,K3,pls) ) ).
tff(fact_40_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_41_rel__simps_I16_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit1(K3),bit0(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_42_rel__simps_I10_J,axiom,
! [K3: int] :
( ord_less(int,bit0(K3),pls)
<=> ord_less(int,K3,pls) ) ).
tff(fact_43_rel__simps_I4_J,axiom,
! [K3: int] :
( ord_less(int,pls,bit0(K3))
<=> ord_less(int,pls,K3) ) ).
tff(fact_44_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_45_rel__simps_I22_J,axiom,
! [K3: int] :
( ord_less_eq(int,pls,bit1(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_46_rel__simps_I32_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit0(K3),bit1(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_47_add__Bit1__Bit0,axiom,
! [L1: int,K: int] : plus_plus(int,bit1(K),bit0(L1)) = bit1(plus_plus(int,K,L1)) ).
tff(fact_48_add__Bit0__Bit1,axiom,
! [L1: int,K: int] : plus_plus(int,bit0(K),bit1(L1)) = bit1(plus_plus(int,K,L1)) ).
tff(fact_49_le__nat__number__of,axiom,
! [V3: int,V2: int] :
( ord_less_eq(nat,number_number_of(nat,V2),number_number_of(nat,V3))
<=> ( ~ ord_less_eq(int,V2,V3)
=> ord_less_eq(int,V2,pls) ) ) ).
tff(fact_50_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_51_rel__simps_I29_J,axiom,
! [K3: int] :
( ord_less_eq(int,bit1(K3),pls)
<=> ord_less(int,K3,pls) ) ).
tff(fact_52_rel__simps_I5_J,axiom,
! [K3: int] :
( ord_less(int,pls,bit1(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_53_eq__number__of__0,axiom,
! [V2: int] :
( ( number_number_of(nat,V2) = zero_zero(nat) )
<=> ord_less_eq(int,V2,pls) ) ).
tff(fact_54_eq__0__number__of,axiom,
! [V2: int] :
( ( zero_zero(nat) = number_number_of(nat,V2) )
<=> ord_less_eq(int,V2,pls) ) ).
tff(fact_55_rel__simps_I33_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit1(K3),bit0(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_56_rel__simps_I15_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit0(K3),bit1(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_57_zle__add1__eq__le,axiom,
! [Z1: int,W: int] :
( ord_less(int,W,plus_plus(int,Z1,one_one(int)))
<=> ord_less_eq(int,W,Z1) ) ).
tff(fact_58_less__special_I3_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [X1: int] :
( ord_less(A,number_number_of(A,X1),zero_zero(A))
<=> ord_less(int,X1,pls) ) ) ).
tff(fact_59_less__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int] :
( ord_less(A,zero_zero(A),number_number_of(A,Y))
<=> ord_less(int,pls,Y) ) ) ).
tff(fact_60_one__add__one__is__two,axiom,
! [A: $tType] :
( number_ring(A)
=> ( plus_plus(A,one_one(A),one_one(A)) = number_number_of(A,bit0(bit1(pls))) ) ) ).
tff(fact_61_less__special_I4_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [X1: int] :
( ord_less(A,number_number_of(A,X1),one_one(A))
<=> ord_less(int,X1,bit1(pls)) ) ) ).
tff(fact_62_less__special_I2_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int] :
( ord_less(A,one_one(A),number_number_of(A,Y))
<=> ord_less(int,bit1(pls),Y) ) ) ).
tff(fact_63_add__special_I3_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [V: int] : plus_plus(A,number_number_of(A,V),one_one(A)) = number_number_of(A,plus_plus(int,V,bit1(pls))) ) ).
tff(fact_64_add__special_I2_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int] : plus_plus(A,one_one(A),number_number_of(A,W1)) = number_number_of(A,plus_plus(int,bit1(pls),W1)) ) ).
tff(fact_65_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_66_plus__numeral__code_I9_J,axiom,
! [W1: int,V: int] : plus_plus(int,number_number_of(int,V),number_number_of(int,W1)) = number_number_of(int,plus_plus(int,V,W1)) ).
tff(fact_67_less__number__of__int__code,axiom,
! [L: int,K3: int] :
( ord_less(int,number_number_of(int,K3),number_number_of(int,L))
<=> ord_less(int,K3,L) ) ).
tff(fact_68_less__eq__number__of__int__code,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,number_number_of(int,K3),number_number_of(int,L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_69_even__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( ord_less(A,plus_plus(A,A2,A2),zero_zero(A))
<=> ord_less(A,A2,zero_zero(A)) ) ) ).
tff(fact_70_number__of__add,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : number_number_of(A,plus_plus(int,V,W1)) = plus_plus(A,number_number_of(A,V),number_number_of(A,W1)) ) ).
tff(fact_71_zless__add1__eq,axiom,
! [Z1: int,W: int] :
( ord_less(int,W,plus_plus(int,Z1,one_one(int)))
<=> ( ord_less(int,W,Z1)
| ( W = Z1 ) ) ) ).
tff(fact_72_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_73_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
tff(fact_74_add__Pls__right,axiom,
! [K: int] : plus_plus(int,K,pls) = K ).
tff(fact_75_add__Pls,axiom,
! [K: int] : plus_plus(int,pls,K) = K ).
tff(fact_76_Bit0__def,axiom,
! [K: int] : bit0(K) = plus_plus(int,K,K) ).
tff(fact_77_less__int__code_I16_J,axiom,
! [K2: int,K1: int] :
( ord_less(int,bit1(K1),bit1(K2))
<=> ord_less(int,K1,K2) ) ).
tff(fact_78_less__int__code_I13_J,axiom,
! [K2: int,K1: int] :
( ord_less(int,bit0(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
tff(fact_79_less__int__def,axiom,
! [W: int,Z1: int] :
( ord_less(int,Z1,W)
<=> ( ord_less_eq(int,Z1,W)
& ( Z1 != W ) ) ) ).
tff(fact_80_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
tff(fact_81_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
tff(fact_82_odd__less__0__iff,axiom,
! [Z1: int] :
( ord_less(int,plus_plus(int,plus_plus(int,one_one(int),Z1),Z1),zero_zero(int))
<=> ord_less(int,Z1,zero_zero(int)) ) ).
tff(fact_83_zless__imp__add1__zle,axiom,
! [Z: int,W1: int] :
( ord_less(int,W1,Z)
=> ord_less_eq(int,plus_plus(int,W1,one_one(int)),Z) ) ).
tff(fact_84_add1__zle__eq,axiom,
! [Z1: int,W: int] :
( ord_less_eq(int,plus_plus(int,W,one_one(int)),Z1)
<=> ord_less(int,W,Z1) ) ).
tff(fact_85_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [X1: A,W: int] :
( ( number_number_of(A,W) = X1 )
<=> ( X1 = number_number_of(A,W) ) ) ) ).
tff(fact_86_le__number__of__eq__not__less,axiom,
! [A: $tType] :
( ( number(A)
& linorder(A) )
=> ! [W: int,V2: int] :
( ord_less_eq(A,number_number_of(A,V2),number_number_of(A,W))
<=> ~ ord_less(A,number_number_of(A,W),number_number_of(A,V2)) ) ) ).
tff(fact_87_add__numeral__0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : plus_plus(A,number_number_of(A,pls),A1) = A1 ) ).
tff(fact_88_add__numeral__0__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : plus_plus(A,A1,number_number_of(A,pls)) = A1 ) ).
tff(fact_89_less__int__code_I15_J,axiom,
! [K2: int,K1: int] :
( ord_less(int,bit1(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
tff(fact_90_bin__less__0__simps_I4_J,axiom,
! [W: int] :
( ord_less(int,bit1(W),zero_zero(int))
<=> ord_less(int,W,zero_zero(int)) ) ).
tff(fact_91_bin__less__0__simps_I1_J,axiom,
~ ord_less(int,pls,zero_zero(int)) ).
tff(fact_92_bin__less__0__simps_I3_J,axiom,
! [W: int] :
( ord_less(int,bit0(W),zero_zero(int))
<=> ord_less(int,W,zero_zero(int)) ) ).
tff(fact_93_Bit1__def,axiom,
! [K: int] : bit1(K) = plus_plus(int,plus_plus(int,one_one(int),K),K) ).
tff(fact_94_odd__nonzero,axiom,
! [Z: int] : plus_plus(int,plus_plus(int,one_one(int),Z),Z) != zero_zero(int) ).
tff(fact_95_semiring__add__number__of,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [V1: int,V: int] :
( ord_less_eq(int,pls,V)
=> ( ord_less_eq(int,pls,V1)
=> ( plus_plus(A,number_number_of(A,V),number_number_of(A,V1)) = number_number_of(A,plus_plus(int,V,V1)) ) ) ) ) ).
tff(fact_96_le__imp__0__less,axiom,
! [Z: int] :
( ord_less_eq(int,zero_zero(int),Z)
=> ord_less(int,zero_zero(int),plus_plus(int,one_one(int),Z)) ) ).
tff(fact_97_int__pos__lt__two__imp__zero__or__one,axiom,
! [X: int] :
( ord_less_eq(int,zero_zero(int),X)
=> ( ord_less(int,X,number_number_of(int,bit0(bit1(pls))))
=> ( ( X = zero_zero(int) )
| ( X = one_one(int) ) ) ) ) ).
%----Arities (24)
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___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___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Rings_Osemiring__0,axiom,
semiring_0(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___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___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__0,axiom,
semiring_0(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_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
%----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,
$false ).
%------------------------------------------------------------------------------