TPTP Problem File: NUM994_5.p
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%------------------------------------------------------------------------------
% File : NUM994_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 155
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_155 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 161 ( 51 unt; 34 typ; 0 def)
% Number of atoms : 290 ( 72 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 172 ( 9 ~; 11 |; 17 &)
% ( 36 <=>; 99 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 5 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 9 ( 7 >; 2 *; 0 +; 0 <<)
% Number of predicates : 21 ( 20 usr; 0 prp; 1-3 aty)
% Number of functors : 11 ( 11 usr; 5 con; 0-3 aty)
% Number of variables : 222 ( 199 !; 0 ?; 222 :)
% ( 23 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:25:36
%------------------------------------------------------------------------------
%----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 (31)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[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_Oordered__ring,type,
ordered_ring:
!>[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_Olinordered__ring,type,
linordered_ring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Ono__zero__divisors,type,
no_zero_divisors:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__ring__abs,type,
ordered_ring_abs:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__semiring,type,
ordered_semiring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__comm__semiring,type,
ordere1490568538miring:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__no__zero__divisors,type,
ring_n68954251visors:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__ring__strict,type,
linord581940658strict:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oordered__cancel__semiring,type,
ordere453448008miring:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere142940540dd_abs:
!>[A: $tType] : $o ).
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_Onumber__class_Onumber__of,type,
number_number_of:
!>[A: $tType] : ( int > A ) ).
tff(sy_c_Orderings_Oord__class_Oless__eq,type,
ord_less_eq:
!>[A: $tType] : ( ( A * A ) > $o ) ).
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_v____,type,
v: int ).
tff(sy_v_w____,type,
w: int ).
%----Relevant facts (97)
tff(fact_0_le__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int] :
( ord_less_eq(A,zero_zero(A),number_number_of(A,Ya))
<=> ord_less_eq(int,pls,Ya) ) ) ).
tff(fact_1_le__special_I3_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Xa: int] :
( ord_less_eq(A,number_number_of(A,Xa),zero_zero(A))
<=> ord_less_eq(int,Xa,pls) ) ) ).
tff(fact_2_rel__simps_I21_J,axiom,
! [K: int] :
( ord_less_eq(int,pls,bit0(K))
<=> ord_less_eq(int,pls,K) ) ).
tff(fact_3_rel__simps_I27_J,axiom,
! [K: int] :
( ord_less_eq(int,bit0(K),pls)
<=> ord_less_eq(int,K,pls) ) ).
tff(fact_4_rel__simps_I32_J,axiom,
! [L: int,K: int] :
( ord_less_eq(int,bit0(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
tff(fact_5_rel__simps_I22_J,axiom,
! [K: int] :
( ord_less_eq(int,pls,bit1(K))
<=> ord_less_eq(int,pls,K) ) ).
tff(fact_6_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_7_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_8_le__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int,Xa: int] :
( ord_less_eq(A,number_number_of(A,Xa),number_number_of(A,Ya))
<=> ord_less_eq(int,Xa,Ya) ) ) ).
tff(fact_9_abs__le__zero__iff,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A2: A] :
( ord_less_eq(A,abs_abs(A,A2),zero_zero(A))
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_10_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_11_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_12_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_13_rel__simps_I51_J,axiom,
! [L: int,K: int] :
( ( bit1(K) = bit1(L) )
<=> ( K = L ) ) ).
tff(fact_14_rel__simps_I48_J,axiom,
! [L: int,K: int] :
( ( bit0(K) = bit0(L) )
<=> ( K = L ) ) ).
tff(fact_15_abs__idempotent,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] : abs_abs(A,abs_abs(A,A1)) = abs_abs(A,A1) ) ).
tff(fact_16_abs__eq__0,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A2: A] :
( ( abs_abs(A,A2) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_17_abs__zero,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ( abs_abs(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_18_rel__simps_I46_J,axiom,
! [K3: int] : bit1(K3) != pls ).
tff(fact_19_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_20_rel__simps_I50_J,axiom,
! [L1: int,K3: int] : bit1(K3) != bit0(L1) ).
tff(fact_21_rel__simps_I49_J,axiom,
! [L1: int,K3: int] : bit0(K3) != bit1(L1) ).
tff(fact_22_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_23_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_24_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_25_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_26_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
tff(fact_27_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_28_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_29_mult__Bit0,axiom,
! [L1: int,K3: int] : times_times(int,bit0(K3),L1) = bit0(times_times(int,K3,L1)) ).
tff(fact_30_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xa: A] :
( ( zero_zero(A) = Xa )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_31_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_32_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_33_number__of__is__id,axiom,
! [K3: int] : number_number_of(int,K3) = K3 ).
tff(fact_34_abs__le__D1,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [B: A,A1: A] :
( ord_less_eq(A,abs_abs(A,A1),B)
=> ord_less_eq(A,A1,B) ) ) ).
tff(fact_35_abs__ge__self,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] : ord_less_eq(A,A1,abs_abs(A,A1)) ) ).
tff(fact_36_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_37_less__eq__int__code_I16_J,axiom,
! [K2: int,K1: int] :
( ord_less_eq(int,bit1(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
tff(fact_38_less__eq__int__code_I13_J,axiom,
! [K2: int,K1: int] :
( ord_less_eq(int,bit0(K1),bit0(K2))
<=> ord_less_eq(int,K1,K2) ) ).
tff(fact_39_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq(int,zero_zero(int),zero_zero(int)) ).
tff(fact_40_less__eq__number__of__int__code,axiom,
! [L: int,K: int] :
( ord_less_eq(int,number_number_of(int,K),number_number_of(int,L))
<=> ord_less_eq(int,K,L) ) ).
tff(fact_41_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_42_semiring__numeral__0__eq__0,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_43_abs__of__nonneg,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] :
( ord_less_eq(A,zero_zero(A),A1)
=> ( abs_abs(A,A1) = A1 ) ) ) ).
tff(fact_44_abs__ge__zero,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] : ord_less_eq(A,zero_zero(A),abs_abs(A,A1)) ) ).
tff(fact_45_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_46_less__eq__int__code_I14_J,axiom,
! [K2: int,K1: int] :
( ord_less_eq(int,bit0(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
tff(fact_47_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_48_Nat__Transfer_Otransfer__nat__int__function__closures_I2_J,axiom,
! [Y: int,X: int] :
( ord_less_eq(int,zero_zero(int),X)
=> ( ord_less_eq(int,zero_zero(int),Y)
=> ord_less_eq(int,zero_zero(int),times_times(int,X,Y)) ) ) ).
tff(fact_49_mult__numeral__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : times_times(A,number_number_of(A,bit1(pls)),A1) = A1 ) ).
tff(fact_50_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_51_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_52_le__nat__number__of,axiom,
! [V2: int,Va: int] :
( ord_less_eq(nat,number_number_of(nat,Va),number_number_of(nat,V2))
<=> ( ~ ord_less_eq(int,Va,V2)
=> ord_less_eq(int,Va,pls) ) ) ).
tff(fact_53_eq__number__of__0,axiom,
! [Va: int] :
( ( number_number_of(nat,Va) = zero_zero(nat) )
<=> ord_less_eq(int,Va,pls) ) ).
tff(fact_54_eq__0__number__of,axiom,
! [Va: int] :
( ( zero_zero(nat) = number_number_of(nat,Va) )
<=> ord_less_eq(int,Va,pls) ) ).
tff(fact_55_number__of1,axiom,
! [N: int] :
( ord_less_eq(int,zero_zero(int),number_number_of(int,N))
=> ( ord_less_eq(int,zero_zero(int),number_number_of(int,bit0(N)))
& ord_less_eq(int,zero_zero(int),number_number_of(int,bit1(N))) ) ) ).
tff(fact_56_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_57_semiring__mult__number__of,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [V1: int,V: int] :
( ord_less_eq(int,pls,V)
=> ( ord_less_eq(int,pls,V1)
=> ( times_times(A,number_number_of(A,V),number_number_of(A,V1)) = number_number_of(A,times_times(int,V,V1)) ) ) ) ) ).
tff(fact_58_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_59_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : times_times(A,A1,zero_zero(A)) = zero_zero(A) ) ).
tff(fact_60_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : times_times(A,zero_zero(A),A1) = zero_zero(A) ) ).
tff(fact_61_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_62_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
tff(fact_63_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_64_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_65_abs__mult,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [B: A,A1: A] : abs_abs(A,times_times(A,A1,B)) = times_times(A,abs_abs(A,A1),abs_abs(A,B)) ) ).
tff(fact_66_conj__le__cong,axiom,
! [P1: bool,P: bool,Xa: int] :
( ( ord_less_eq(int,zero_zero(int),Xa)
=> ( pp(P)
<=> pp(P1) ) )
=> ( ( ord_less_eq(int,zero_zero(int),Xa)
& pp(P) )
<=> ( ord_less_eq(int,zero_zero(int),Xa)
& pp(P1) ) ) ) ).
tff(fact_67_imp__le__cong,axiom,
! [P1: bool,P: bool,Xa: int] :
( ( ord_less_eq(int,zero_zero(int),Xa)
=> ( pp(P)
<=> pp(P1) ) )
=> ( ( ord_less_eq(int,zero_zero(int),Xa)
=> pp(P) )
<=> ( ord_less_eq(int,zero_zero(int),Xa)
=> pp(P1) ) ) ) ).
tff(fact_68_zero__le__square,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A1: A] : ord_less_eq(A,zero_zero(A),times_times(A,A1,A1)) ) ).
tff(fact_69_zero__le__mult__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B1: A,A2: A] :
( ord_less_eq(A,zero_zero(A),times_times(A,A2,B1))
<=> ( ( ord_less_eq(A,zero_zero(A),A2)
& ord_less_eq(A,zero_zero(A),B1) )
| ( ord_less_eq(A,A2,zero_zero(A))
& ord_less_eq(A,B1,zero_zero(A)) ) ) ) ) ).
tff(fact_70_mult__le__0__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B1: A,A2: A] :
( ord_less_eq(A,times_times(A,A2,B1),zero_zero(A))
<=> ( ( ord_less_eq(A,zero_zero(A),A2)
& ord_less_eq(A,B1,zero_zero(A)) )
| ( ord_less_eq(A,A2,zero_zero(A))
& ord_less_eq(A,zero_zero(A),B1) ) ) ) ) ).
tff(fact_71_mult__nonneg__nonneg,axiom,
! [A: $tType] :
( ordere453448008miring(A)
=> ! [B: A,A1: A] :
( ord_less_eq(A,zero_zero(A),A1)
=> ( ord_less_eq(A,zero_zero(A),B)
=> ord_less_eq(A,zero_zero(A),times_times(A,A1,B)) ) ) ) ).
tff(fact_72_mult__nonneg__nonpos,axiom,
! [A: $tType] :
( ordere453448008miring(A)
=> ! [B: A,A1: A] :
( ord_less_eq(A,zero_zero(A),A1)
=> ( ord_less_eq(A,B,zero_zero(A))
=> ord_less_eq(A,times_times(A,A1,B),zero_zero(A)) ) ) ) ).
tff(fact_73_mult__nonneg__nonpos2,axiom,
! [A: $tType] :
( ordere453448008miring(A)
=> ! [B: A,A1: A] :
( ord_less_eq(A,zero_zero(A),A1)
=> ( ord_less_eq(A,B,zero_zero(A))
=> ord_less_eq(A,times_times(A,B,A1),zero_zero(A)) ) ) ) ).
tff(fact_74_mult__nonpos__nonneg,axiom,
! [A: $tType] :
( ordere453448008miring(A)
=> ! [B: A,A1: A] :
( ord_less_eq(A,A1,zero_zero(A))
=> ( ord_less_eq(A,zero_zero(A),B)
=> ord_less_eq(A,times_times(A,A1,B),zero_zero(A)) ) ) ) ).
tff(fact_75_mult__nonpos__nonpos,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [B: A,A1: A] :
( ord_less_eq(A,A1,zero_zero(A))
=> ( ord_less_eq(A,B,zero_zero(A))
=> ord_less_eq(A,zero_zero(A),times_times(A,A1,B)) ) ) ) ).
tff(fact_76_mult__right__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ( ord_less_eq(A,zero_zero(A),C)
=> ord_less_eq(A,times_times(A,A1,C),times_times(A,B,C)) ) ) ) ).
tff(fact_77_mult__left__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ( ord_less_eq(A,zero_zero(A),C)
=> ord_less_eq(A,times_times(A,C,A1),times_times(A,C,B)) ) ) ) ).
tff(fact_78_comm__mult__left__mono,axiom,
! [A: $tType] :
( ordere1490568538miring(A)
=> ! [C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ( ord_less_eq(A,zero_zero(A),C)
=> ord_less_eq(A,times_times(A,C,A1),times_times(A,C,B)) ) ) ) ).
tff(fact_79_mult__right__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [C: A,A1: A,B: A] :
( ord_less_eq(A,B,A1)
=> ( ord_less_eq(A,C,zero_zero(A))
=> ord_less_eq(A,times_times(A,A1,C),times_times(A,B,C)) ) ) ) ).
tff(fact_80_mult__left__mono__neg,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [C: A,A1: A,B: A] :
( ord_less_eq(A,B,A1)
=> ( ord_less_eq(A,C,zero_zero(A))
=> ord_less_eq(A,times_times(A,C,A1),times_times(A,C,B)) ) ) ) ).
tff(fact_81_mult__mono_H,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [D: A,C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ( ord_less_eq(A,C,D)
=> ( ord_less_eq(A,zero_zero(A),A1)
=> ( ord_less_eq(A,zero_zero(A),C)
=> ord_less_eq(A,times_times(A,A1,C),times_times(A,B,D)) ) ) ) ) ) ).
tff(fact_82_mult__mono,axiom,
! [A: $tType] :
( ordered_semiring(A)
=> ! [D: A,C: A,B: A,A1: A] :
( ord_less_eq(A,A1,B)
=> ( ord_less_eq(A,C,D)
=> ( ord_less_eq(A,zero_zero(A),B)
=> ( ord_less_eq(A,zero_zero(A),C)
=> ord_less_eq(A,times_times(A,A1,C),times_times(A,B,D)) ) ) ) ) ) ).
tff(fact_83_split__mult__pos__le,axiom,
! [A: $tType] :
( ordered_ring(A)
=> ! [B: A,A1: A] :
( ( ( ord_less_eq(A,zero_zero(A),A1)
& ord_less_eq(A,zero_zero(A),B) )
| ( ord_less_eq(A,A1,zero_zero(A))
& ord_less_eq(A,B,zero_zero(A)) ) )
=> ord_less_eq(A,zero_zero(A),times_times(A,A1,B)) ) ) ).
tff(fact_84_split__mult__neg__le,axiom,
! [A: $tType] :
( ordere453448008miring(A)
=> ! [B: A,A1: A] :
( ( ( ord_less_eq(A,zero_zero(A),A1)
& ord_less_eq(A,B,zero_zero(A)) )
| ( ord_less_eq(A,A1,zero_zero(A))
& ord_less_eq(A,zero_zero(A),B) ) )
=> ord_less_eq(A,times_times(A,A1,B),zero_zero(A)) ) ) ).
tff(fact_85_semiring__norm_I112_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( zero_zero(A) = number_number_of(A,pls) ) ) ).
tff(fact_86_abs__mult__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,zero_zero(A),X)
=> ( times_times(A,abs_abs(A,Y),X) = abs_abs(A,times_times(A,Y,X)) ) ) ) ).
tff(fact_87_abs__eq__mult,axiom,
! [A: $tType] :
( ordered_ring_abs(A)
=> ! [B: A,A1: A] :
( ( ( ord_less_eq(A,zero_zero(A),A1)
| ord_less_eq(A,A1,zero_zero(A)) )
& ( ord_less_eq(A,zero_zero(A),B)
| ord_less_eq(A,B,zero_zero(A)) ) )
=> ( abs_abs(A,times_times(A,A1,B)) = times_times(A,abs_abs(A,A1),abs_abs(A,B)) ) ) ) ).
tff(fact_88_number__of2,axiom,
ord_less_eq(int,zero_zero(int),number_number_of(int,pls)) ).
tff(fact_89_le0,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_90_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_91_le__0__eq,axiom,
! [Na: nat] :
( ord_less_eq(nat,Na,zero_zero(nat))
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_92_mult__cancel2,axiom,
! [Na: nat,K: nat,Ma: nat] :
( ( times_times(nat,Ma,K) = times_times(nat,Na,K) )
<=> ( ( Ma = Na )
| ( K = zero_zero(nat) ) ) ) ).
tff(fact_93_mult__cancel1,axiom,
! [Na: nat,Ma: nat,K: nat] :
( ( times_times(nat,K,Ma) = times_times(nat,K,Na) )
<=> ( ( Ma = Na )
| ( K = zero_zero(nat) ) ) ) ).
tff(fact_94_mult__is__0,axiom,
! [Na: nat,Ma: nat] :
( ( times_times(nat,Ma,Na) = zero_zero(nat) )
<=> ( ( Ma = zero_zero(nat) )
| ( Na = zero_zero(nat) ) ) ) ).
tff(fact_95_mult__0__right,axiom,
! [M: nat] : times_times(nat,M,zero_zero(nat)) = zero_zero(nat) ).
tff(fact_96_mult__0,axiom,
! [N: nat] : times_times(nat,zero_zero(nat),N) = zero_zero(nat) ).
%----Arities (27)
tff(arity_Int_Oint___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs(int) ).
tff(arity_Int_Oint___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(int) ).
tff(arity_Int_Oint___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(int) ).
tff(arity_Int_Oint___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(int) ).
tff(arity_Int_Oint___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(int) ).
tff(arity_Int_Oint___Rings_Oordered__semiring,axiom,
ordered_semiring(int) ).
tff(arity_Int_Oint___Rings_Oordered__ring__abs,axiom,
ordered_ring_abs(int) ).
tff(arity_Int_Oint___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(int) ).
tff(arity_Int_Oint___Rings_Olinordered__ring,axiom,
linordered_ring(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_Oordered__ring,axiom,
ordered_ring(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___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Rings_Oordered__cancel__semiring,axiom,
ordere453448008miring(nat) ).
tff(arity_Nat_Onat___Rings_Oordered__comm__semiring,axiom,
ordere1490568538miring(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(nat) ).
tff(arity_Nat_Onat___Rings_Oordered__semiring,axiom,
ordered_semiring(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_Omult__zero,axiom,
mult_zero(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_eq(int,zero_zero(int),times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,v)))
& ord_less_eq(int,zero_zero(int),times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,w))) ) ).
%------------------------------------------------------------------------------