TPTP Problem File: NUM970_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : NUM970_5 : TPTP v9.0.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 103
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_103 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 175 ( 66 unt; 36 typ; 0 def)
% Number of atoms : 284 ( 67 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 161 ( 16 ~; 6 |; 21 &)
% ( 45 <=>; 73 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 16 ( 10 >; 6 *; 0 +; 0 <<)
% Number of predicates : 22 ( 21 usr; 0 prp; 1-3 aty)
% Number of functors : 12 ( 12 usr; 4 con; 0-4 aty)
% Number of variables : 217 ( 191 !; 0 ?; 217 :)
% ( 26 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:24:24
%------------------------------------------------------------------------------
%----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_Orderings_Oord,type,
ord:
!>[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_Orderings_Oorder,type,
order:
!>[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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[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_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
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_Power_Opower__class_Opower,type,
power_power:
!>[A: $tType] : ( ( A * nat ) > A ) ).
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_s____,type,
s: int ).
%----Relevant facts (97)
tff(fact_0_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_1_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_2_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_3_le__special_I3_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [X: int] :
( ord_less_eq(A,number_number_of(A,X),zero_zero(A))
<=> ord_less_eq(int,X,pls) ) ) ).
tff(fact_4_rel__simps_I21_J,axiom,
! [K1: int] :
( ord_less_eq(int,pls,bit0(K1))
<=> ord_less_eq(int,pls,K1) ) ).
tff(fact_5_rel__simps_I27_J,axiom,
! [K1: int] :
( ord_less_eq(int,bit0(K1),pls)
<=> ord_less_eq(int,K1,pls) ) ).
tff(fact_6_rel__simps_I32_J,axiom,
! [L1: int,K1: int] :
( ord_less_eq(int,bit0(K1),bit1(L1))
<=> ord_less_eq(int,K1,L1) ) ).
tff(fact_7_rel__simps_I22_J,axiom,
! [K1: int] :
( ord_less_eq(int,pls,bit1(K1))
<=> ord_less_eq(int,pls,K1) ) ).
tff(fact_8_power__eq__0__iff__number__of,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [W1: int,A1: A] :
( ( power_power(A,A1,number_number_of(nat,W1)) = zero_zero(A) )
<=> ( ( A1 = zero_zero(A) )
& ( number_number_of(nat,W1) != zero_zero(nat) ) ) ) ) ).
tff(fact_9_le__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int,X: int] :
( ord_less_eq(A,number_number_of(A,X),number_number_of(A,Y))
<=> ord_less_eq(int,X,Y) ) ) ).
tff(fact_10_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_11_zero__le__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] : ord_less_eq(A,zero_zero(A),power_power(A,A2,number_number_of(nat,bit0(bit1(pls))))) ) ).
tff(fact_12_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y: int,X: int] :
( ( number_number_of(A,X) = number_number_of(A,Y) )
<=> ( X = Y ) ) ) ).
tff(fact_13_rel__simps_I51_J,axiom,
! [L1: int,K1: int] :
( ( bit1(K1) = bit1(L1) )
<=> ( K1 = L1 ) ) ).
tff(fact_14_rel__simps_I48_J,axiom,
! [L1: int,K1: int] :
( ( bit0(K1) = bit0(L1) )
<=> ( K1 = L1 ) ) ).
tff(fact_15_rel__simps_I46_J,axiom,
! [K: int] : ( bit1(K) != pls ) ).
tff(fact_16_rel__simps_I39_J,axiom,
! [L: int] : ( pls != bit1(L) ) ).
tff(fact_17_rel__simps_I50_J,axiom,
! [L: int,K: int] : ( bit1(K) != bit0(L) ) ).
tff(fact_18_rel__simps_I49_J,axiom,
! [L: int,K: int] : ( bit0(K) != bit1(L) ) ).
tff(fact_19_rel__simps_I44_J,axiom,
! [K1: int] :
( ( bit0(K1) = pls )
<=> ( K1 = pls ) ) ).
tff(fact_20_rel__simps_I38_J,axiom,
! [L1: int] :
( ( pls = bit0(L1) )
<=> ( pls = L1 ) ) ).
tff(fact_21_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_22_rel__simps_I34_J,axiom,
! [L1: int,K1: int] :
( ord_less_eq(int,bit1(K1),bit1(L1))
<=> ord_less_eq(int,K1,L1) ) ).
tff(fact_23_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
tff(fact_24_rel__simps_I31_J,axiom,
! [L1: int,K1: int] :
( ord_less_eq(int,bit0(K1),bit0(L1))
<=> ord_less_eq(int,K1,L1) ) ).
tff(fact_25_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_26_le__nat__number__of,axiom,
! [V1: int,V: int] :
( ord_less_eq(nat,number_number_of(nat,V),number_number_of(nat,V1))
<=> ( ~ ord_less_eq(int,V,V1)
=> ord_less_eq(int,V,pls) ) ) ).
tff(fact_27_eq__number__of__0,axiom,
! [V: int] :
( ( number_number_of(nat,V) = zero_zero(nat) )
<=> ord_less_eq(int,V,pls) ) ).
tff(fact_28_eq__0__number__of,axiom,
! [V: int] :
( ( zero_zero(nat) = number_number_of(nat,V) )
<=> ord_less_eq(int,V,pls) ) ).
tff(fact_29_less__eq__number__of__int__code,axiom,
! [L1: int,K1: int] :
( ord_less_eq(int,number_number_of(int,K1),number_number_of(int,L1))
<=> ord_less_eq(int,K1,L1) ) ).
tff(fact_30_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_31_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
tff(fact_32_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [X: A,W1: int] :
( ( number_number_of(A,W1) = X )
<=> ( X = number_number_of(A,W1) ) ) ) ).
tff(fact_33_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_34_less__eq__int__code_I16_J,axiom,
! [K2: int,K11: int] :
( ord_less_eq(int,bit1(K11),bit1(K2))
<=> ord_less_eq(int,K11,K2) ) ).
tff(fact_35_less__eq__int__code_I13_J,axiom,
! [K2: int,K11: int] :
( ord_less_eq(int,bit0(K11),bit0(K2))
<=> ord_less_eq(int,K11,K2) ) ).
tff(fact_36_semiring__numeral__0__eq__0,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_37_semiring__norm_I112_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( zero_zero(A) = number_number_of(A,pls) ) ) ).
tff(fact_38_less__eq__int__code_I14_J,axiom,
! [K2: int,K11: int] :
( ord_less_eq(int,bit0(K11),bit1(K2))
<=> ord_less_eq(int,K11,K2) ) ).
tff(fact_39_power2__eq__imp__eq,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Y1: A,X2: A] :
( ( power_power(A,X2,number_number_of(nat,bit0(bit1(pls)))) = power_power(A,Y1,number_number_of(nat,bit0(bit1(pls)))) )
=> ( ord_less_eq(A,zero_zero(A),X2)
=> ( ord_less_eq(A,zero_zero(A),Y1)
=> ( X2 = Y1 ) ) ) ) ) ).
tff(fact_40_power2__le__imp__le,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Y1: A,X2: A] :
( ord_less_eq(A,power_power(A,X2,number_number_of(nat,bit0(bit1(pls)))),power_power(A,Y1,number_number_of(nat,bit0(bit1(pls)))))
=> ( ord_less_eq(A,zero_zero(A),Y1)
=> ord_less_eq(A,X2,Y1) ) ) ) ).
tff(fact_41_power__eq__0__iff,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [N: nat,A1: A] :
( ( power_power(A,A1,N) = zero_zero(A) )
<=> ( ( A1 = zero_zero(A) )
& ( N != zero_zero(nat) ) ) ) ) ).
tff(fact_42_power2__ge__self,axiom,
! [X2: int] : ord_less_eq(int,X2,power_power(int,X2,number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_43_quartic__square__square,axiom,
! [X2: int] : ( power_power(int,power_power(int,X2,number_number_of(nat,bit0(bit1(pls)))),number_number_of(nat,bit0(bit1(pls)))) = power_power(int,X2,number_number_of(nat,bit0(bit0(bit1(pls))))) ) ).
tff(fact_44_Nat__Transfer_Otransfer__nat__int__function__closures_I7_J,axiom,
ord_less_eq(int,zero_zero(int),number_number_of(int,bit0(bit1(pls)))) ).
tff(fact_45_number__of1,axiom,
! [N1: int] :
( ord_less_eq(int,zero_zero(int),number_number_of(int,N1))
=> ( ord_less_eq(int,zero_zero(int),number_number_of(int,bit0(N1)))
& ord_less_eq(int,zero_zero(int),number_number_of(int,bit1(N1))) ) ) ).
tff(fact_46_Nat__Transfer_Otransfer__nat__int__function__closures_I8_J,axiom,
ord_less_eq(int,zero_zero(int),number_number_of(int,bit1(bit1(pls)))) ).
tff(fact_47_Nat__Transfer_Otransfer__nat__int__function__closures_I4_J,axiom,
! [N1: nat,X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
=> ord_less_eq(int,zero_zero(int),power_power(int,X2,N1)) ) ).
tff(fact_48_number__of2,axiom,
ord_less_eq(int,zero_zero(int),number_number_of(int,pls)) ).
tff(fact_49_power__mono,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N1: nat,B1: A,A2: A] :
( ord_less_eq(A,A2,B1)
=> ( ord_less_eq(A,zero_zero(A),A2)
=> ord_less_eq(A,power_power(A,A2,N1),power_power(A,B1,N1)) ) ) ) ).
tff(fact_50_number__of__is__id,axiom,
! [K: int] : ( number_number_of(int,K) = K ) ).
tff(fact_51_field__power__not__zero,axiom,
! [A: $tType] :
( ring_11004092258visors(A)
=> ! [N1: nat,A2: A] :
( ( A2 != zero_zero(A) )
=> ( power_power(A,A2,N1) != zero_zero(A) ) ) ) ).
tff(fact_52_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq(int,zero_zero(int),zero_zero(int)) ).
tff(fact_53_conj__le__cong,axiom,
! [P1: bool,P: bool,X: int] :
( ( ord_less_eq(int,zero_zero(int),X)
=> ( pp(P)
<=> pp(P1) ) )
=> ( ( ord_less_eq(int,zero_zero(int),X)
& pp(P) )
<=> ( ord_less_eq(int,zero_zero(int),X)
& pp(P1) ) ) ) ).
tff(fact_54_imp__le__cong,axiom,
! [P1: bool,P: bool,X: int] :
( ( ord_less_eq(int,zero_zero(int),X)
=> ( pp(P)
<=> pp(P1) ) )
=> ( ( ord_less_eq(int,zero_zero(int),X)
=> pp(P) )
<=> ( ord_less_eq(int,zero_zero(int),X)
=> pp(P1) ) ) ) ).
tff(fact_55_zero__le__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N1: nat,A2: A] :
( ord_less_eq(A,zero_zero(A),A2)
=> ord_less_eq(A,zero_zero(A),power_power(A,A2,N1)) ) ) ).
tff(fact_56_le__0__eq,axiom,
! [N: nat] :
( ord_less_eq(nat,N,zero_zero(nat))
<=> ( N = zero_zero(nat) ) ) ).
tff(fact_57_less__eq__nat_Osimps_I1_J,axiom,
! [N1: nat] : ord_less_eq(nat,zero_zero(nat),N1) ).
tff(fact_58_le0,axiom,
! [N1: nat] : ord_less_eq(nat,zero_zero(nat),N1) ).
tff(fact_59_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X2: A] : ord_less_eq(A,X2,X2) ) ).
tff(fact_60_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ~ ord_less_eq(A,X2,Y1)
=> ord_less_eq(A,Y1,X2) ) ) ).
tff(fact_61_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X: 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,X),aa(A,B,G,X)) ) ) ).
tff(fact_62_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
=> ( ord_less_eq(A,Y1,Z)
=> ord_less_eq(A,X2,Z) ) ) ) ).
tff(fact_63_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
=> ( ord_less_eq(A,Y1,X2)
=> ( X2 = Y1 ) ) ) ) ).
tff(fact_64_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A2: A] :
( ord_less_eq(A,A2,B1)
=> ( ( B1 = C )
=> ord_less_eq(A,A2,C) ) ) ) ).
tff(fact_65_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A2: A] :
( ( A2 = B1 )
=> ( ord_less_eq(A,B1,C)
=> ord_less_eq(A,A2,C) ) ) ) ).
tff(fact_66_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [X: A,Y: A] :
( ord_less_eq(A,Y,X)
=> ( ord_less_eq(A,X,Y)
<=> ( X = Y ) ) ) ) ).
tff(fact_67_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [X: 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,X),aa(A,B,G,X)) ) ) ).
tff(fact_68_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y1: A,X2: A] :
( ( X2 = Y1 )
=> ord_less_eq(A,X2,Y1) ) ) ).
tff(fact_69_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ( X = Y )
<=> ( ord_less_eq(A,X,Y)
& ord_less_eq(A,Y,X) ) ) ) ).
tff(fact_70_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,X2: A] :
( ord_less_eq(A,X2,Y1)
| ord_less_eq(A,Y1,X2) ) ) ).
tff(fact_71_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)
<=> ! [X1: A] : ord_less_eq(B,aa(A,B,F,X1),aa(A,B,G,X1)) ) ) ).
tff(fact_72_le__antisym,axiom,
! [N1: nat,M: nat] :
( ord_less_eq(nat,M,N1)
=> ( ord_less_eq(nat,N1,M)
=> ( M = N1 ) ) ) ).
tff(fact_73_le__trans,axiom,
! [K: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ( ord_less_eq(nat,J,K)
=> ord_less_eq(nat,I,K) ) ) ).
tff(fact_74_eq__imp__le,axiom,
! [N1: nat,M: nat] :
( ( M = N1 )
=> ord_less_eq(nat,M,N1) ) ).
tff(fact_75_nat__le__linear,axiom,
! [N1: nat,M: nat] :
( ord_less_eq(nat,M,N1)
| ord_less_eq(nat,N1,M) ) ).
tff(fact_76_le__refl,axiom,
! [N1: nat] : ord_less_eq(nat,N1,N1) ).
tff(fact_77_power2__eq__square__number__of,axiom,
! [B: $tType] :
( ( monoid_mult(B)
& number(B) )
=> ! [W: int] : ( power_power(B,number_number_of(B,W),number_number_of(nat,bit0(bit1(pls)))) = times_times(B,number_number_of(B,W),number_number_of(B,W)) ) ) ).
tff(fact_78_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_79_mult__cancel2,axiom,
! [N: nat,K1: nat,Ma: nat] :
( ( times_times(nat,Ma,K1) = times_times(nat,N,K1) )
<=> ( ( Ma = N )
| ( K1 = zero_zero(nat) ) ) ) ).
tff(fact_80_mult__cancel1,axiom,
! [N: nat,Ma: nat,K1: nat] :
( ( times_times(nat,K1,Ma) = times_times(nat,K1,N) )
<=> ( ( Ma = N )
| ( K1 = zero_zero(nat) ) ) ) ).
tff(fact_81_mult__is__0,axiom,
! [N: nat,Ma: nat] :
( ( times_times(nat,Ma,N) = zero_zero(nat) )
<=> ( ( Ma = zero_zero(nat) )
| ( N = zero_zero(nat) ) ) ) ).
tff(fact_82_mult__0__right,axiom,
! [M: nat] : ( times_times(nat,M,zero_zero(nat)) = zero_zero(nat) ) ).
tff(fact_83_mult__0,axiom,
! [N1: nat] : ( times_times(nat,zero_zero(nat),N1) = zero_zero(nat) ) ).
tff(fact_84_less__zeroE,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_85_less__nat__zero__code,axiom,
! [N1: nat] : ~ ord_less(nat,N1,zero_zero(nat)) ).
tff(fact_86_neq0__conv,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_87_mult__less__cancel2,axiom,
! [N: nat,K1: nat,Ma: nat] :
( ord_less(nat,times_times(nat,Ma,K1),times_times(nat,N,K1))
<=> ( ord_less(nat,zero_zero(nat),K1)
& ord_less(nat,Ma,N) ) ) ).
tff(fact_88_mult__less__cancel1,axiom,
! [N: nat,Ma: nat,K1: nat] :
( ord_less(nat,times_times(nat,K1,Ma),times_times(nat,K1,N))
<=> ( ord_less(nat,zero_zero(nat),K1)
& ord_less(nat,Ma,N) ) ) ).
tff(fact_89_nat__0__less__mult__iff,axiom,
! [N: nat,Ma: nat] :
( ord_less(nat,zero_zero(nat),times_times(nat,Ma,N))
<=> ( ord_less(nat,zero_zero(nat),Ma)
& ord_less(nat,zero_zero(nat),N) ) ) ).
tff(fact_90_nat__zero__less__power__iff,axiom,
! [N: nat,X: nat] :
( ord_less(nat,zero_zero(nat),power_power(nat,X,N))
<=> ( ord_less(nat,zero_zero(nat),X)
| ( N = zero_zero(nat) ) ) ) ).
tff(fact_91_rel__simps_I17_J,axiom,
! [L1: int,K1: int] :
( ord_less(int,bit1(K1),bit1(L1))
<=> ord_less(int,K1,L1) ) ).
tff(fact_92_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_93_mult__Pls,axiom,
! [W: int] : ( times_times(int,pls,W) = pls ) ).
tff(fact_94_rel__simps_I14_J,axiom,
! [L1: int,K1: int] :
( ord_less(int,bit0(K1),bit0(L1))
<=> ord_less(int,K1,L1) ) ).
tff(fact_95_mult__Bit0,axiom,
! [L: int,K: int] : ( times_times(int,bit0(K),L) = bit0(times_times(int,K,L)) ) ).
tff(fact_96_less__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int,X: int] :
( ord_less(A,number_number_of(A,X),number_number_of(A,Y))
<=> ord_less(int,X,Y) ) ) ).
%----Arities (39)
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_Int_Oint___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(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___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(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___Orderings_Oorder,axiom,
order(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___Orderings_Oord,axiom,
ord(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___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___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(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___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(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_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) ).
%----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_eq(int,zero_zero(int),power_power(int,s,number_number_of(nat,bit0(bit1(pls))))) ).
%------------------------------------------------------------------------------