TPTP Problem File: NUM924_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM924_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 102
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_102 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 156 ( 55 unt; 35 typ; 0 def)
% Number of atoms : 229 ( 63 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 122 ( 14 ~; 0 |; 20 &)
% ( 47 <=>; 41 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 8 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 16 ( 11 >; 5 *; 0 +; 0 <<)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-3 aty)
% Number of functors : 14 ( 14 usr; 6 con; 0-3 aty)
% Number of variables : 180 ( 159 !; 0 ?; 180 :)
% ( 21 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:24:22
%------------------------------------------------------------------------------
%----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 (32)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring,type,
semiring:
!>[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_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_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_Olinordered__ring__strict,type,
linord581940658strict:
!>[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_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_IntPrimes_Ozprime,type,
zprime: int > $o ).
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_TwoSquares__Mirabelle__poiayhyqls_Ois__sum2sq,type,
twoSqu1567020053sum2sq: int > $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_m,type,
m: int ).
tff(sy_v_s____,type,
s: int ).
tff(sy_v_t____,type,
t: int ).
%----Relevant facts (98)
tff(fact_0__096t_A_060_A0_096,axiom,
ord_less(int,t,zero_zero(int)) ).
tff(fact_1_calculation_I1_J,axiom,
ord_less(int,t,one_one(int)) ).
tff(fact_2__096_I4_A_K_Am_A_L_A1_J_A_K_At_A_060_A_I4_A_K_Am_A_L_A1_J_A_K_A0_096,axiom,
ord_less(int,times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t),times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),zero_zero(int))) ).
tff(fact_3_t,axiom,
plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)) = times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t) ).
tff(fact_4_calculation_I2_J,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_5__096_126_A1_A_060_061_At_096,axiom,
~ ord_less_eq(int,one_one(int),t) ).
tff(fact_6_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_7_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_8_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_9_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_10_add__special_I2_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : plus_plus(A,one_one(A),number_number_of(A,W)) = number_number_of(A,plus_plus(int,bit1(pls),W)) ) ).
tff(fact_11_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_12_less__special_I2_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y1: int] :
( ord_less(A,one_one(A),number_number_of(A,Y1))
<=> ord_less(int,bit1(pls),Y1) ) ) ).
tff(fact_13_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_14_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_15_less__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y1: int] :
( ord_less(A,zero_zero(A),number_number_of(A,Y1))
<=> ord_less(int,pls,Y1) ) ) ).
tff(fact_16_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_17_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y1: int,X1: int] :
( ( number_number_of(A,X1) = number_number_of(A,Y1) )
<=> ( X1 = Y1 ) ) ) ).
tff(fact_18_rel__simps_I51_J,axiom,
! [L: int,K: int] :
( ( bit1(K) = bit1(L) )
<=> ( K = L ) ) ).
tff(fact_19_rel__simps_I48_J,axiom,
! [L: int,K: int] :
( ( bit0(K) = bit0(L) )
<=> ( K = L ) ) ).
tff(fact_20_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_21_rel__simps_I46_J,axiom,
! [K3: int] : bit1(K3) != pls ).
tff(fact_22_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_23_rel__simps_I50_J,axiom,
! [L1: int,K3: int] : bit1(K3) != bit0(L1) ).
tff(fact_24_rel__simps_I49_J,axiom,
! [L1: int,K3: int] : bit0(K3) != bit1(L1) ).
tff(fact_25_rel__simps_I44_J,axiom,
! [K: int] :
( ( bit0(K) = pls )
<=> ( K = pls ) ) ).
tff(fact_26_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_27_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_28_rel__simps_I17_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit1(K),bit1(L))
<=> ord_less(int,K,L) ) ).
tff(fact_29_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_30_rel__simps_I14_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit0(K),bit0(L))
<=> ord_less(int,K,L) ) ).
tff(fact_31_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_32_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
tff(fact_33_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_34_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_35_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_36_mult__Bit0,axiom,
! [L1: int,K3: int] : times_times(int,bit0(K3),L1) = bit0(times_times(int,K3,L1)) ).
tff(fact_37_add__Bit0__Bit0,axiom,
! [L1: int,K3: int] : plus_plus(int,bit0(K3),bit0(L1)) = bit0(plus_plus(int,K3,L1)) ).
tff(fact_38_p0,axiom,
ord_less(int,zero_zero(int),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_39_left__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& semiring(B) )
=> ! [V: int,B1: B,A1: B] : times_times(B,plus_plus(B,A1,B1),number_number_of(B,V)) = plus_plus(B,times_times(B,A1,number_number_of(B,V)),times_times(B,B1,number_number_of(B,V))) ) ).
tff(fact_40_right__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& semiring(B) )
=> ! [C: B,B1: B,V: int] : times_times(B,number_number_of(B,V),plus_plus(B,B1,C)) = plus_plus(B,times_times(B,number_number_of(B,V),B1),times_times(B,number_number_of(B,V),C)) ) ).
tff(fact_41_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_42_less__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y1: int,X1: int] :
( ord_less(A,number_number_of(A,X1),number_number_of(A,Y1))
<=> ord_less(int,X1,Y1) ) ) ).
tff(fact_43_le__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y1: int,X1: int] :
( ord_less_eq(A,number_number_of(A,X1),number_number_of(A,Y1))
<=> ord_less_eq(int,X1,Y1) ) ) ).
tff(fact_44_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z3: A,W: int,V: int] : times_times(A,number_number_of(A,V),times_times(A,number_number_of(A,W),Z3)) = times_times(A,number_number_of(A,times_times(int,V,W)),Z3) ) ).
tff(fact_45_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_46_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z3: A,W: int,V: int] : plus_plus(A,number_number_of(A,V),plus_plus(A,number_number_of(A,W),Z3)) = plus_plus(A,number_number_of(A,plus_plus(int,V,W)),Z3) ) ).
tff(fact_47_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_48_rel__simps_I12_J,axiom,
! [K: int] :
( ord_less(int,bit1(K),pls)
<=> ord_less(int,K,pls) ) ).
tff(fact_49_rel__simps_I16_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit1(K),bit0(L))
<=> ord_less(int,K,L) ) ).
tff(fact_50_rel__simps_I10_J,axiom,
! [K: int] :
( ord_less(int,bit0(K),pls)
<=> ord_less(int,K,pls) ) ).
tff(fact_51_rel__simps_I4_J,axiom,
! [K: int] :
( ord_less(int,pls,bit0(K))
<=> ord_less(int,pls,K) ) ).
tff(fact_52_rel__simps_I22_J,axiom,
! [K: int] :
( ord_less_eq(int,pls,bit1(K))
<=> ord_less_eq(int,pls,K) ) ).
tff(fact_53_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_54_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_55_rel__simps_I27_J,axiom,
! [K: int] :
( ord_less_eq(int,bit0(K),pls)
<=> ord_less_eq(int,K,pls) ) ).
tff(fact_56_rel__simps_I21_J,axiom,
! [K: int] :
( ord_less_eq(int,pls,bit0(K))
<=> ord_less_eq(int,pls,K) ) ).
tff(fact_57_t__l__p,axiom,
ord_less(int,t,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_58_add__Bit1__Bit0,axiom,
! [L1: int,K3: int] : plus_plus(int,bit1(K3),bit0(L1)) = bit1(plus_plus(int,K3,L1)) ).
tff(fact_59_add__Bit0__Bit1,axiom,
! [L1: int,K3: int] : plus_plus(int,bit0(K3),bit1(L1)) = bit1(plus_plus(int,K3,L1)) ).
tff(fact_60_less__0__number__of,axiom,
! [V1: int] :
( ord_less(nat,zero_zero(nat),number_number_of(nat,V1))
<=> ord_less(int,pls,V1) ) ).
tff(fact_61_less__nat__number__of,axiom,
! [V3: int,V1: int] :
( ord_less(nat,number_number_of(nat,V1),number_number_of(nat,V3))
<=> ( ( ord_less(int,V1,V3)
=> ord_less(int,pls,V3) )
& ord_less(int,V1,V3) ) ) ).
tff(fact_62_eq__number__of__0,axiom,
! [V1: int] :
( ( number_number_of(nat,V1) = zero_zero(nat) )
<=> ord_less_eq(int,V1,pls) ) ).
tff(fact_63_eq__0__number__of,axiom,
! [V1: int] :
( ( zero_zero(nat) = number_number_of(nat,V1) )
<=> ord_less_eq(int,V1,pls) ) ).
tff(fact_64_numeral__1__eq__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_65_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
tff(fact_66_rel__simps_I29_J,axiom,
! [K: int] :
( ord_less_eq(int,bit1(K),pls)
<=> ord_less(int,K,pls) ) ).
tff(fact_67_rel__simps_I5_J,axiom,
! [K: int] :
( ord_less(int,pls,bit1(K))
<=> ord_less_eq(int,pls,K) ) ).
tff(fact_68_rel__simps_I33_J,axiom,
! [L: int,K: int] :
( ord_less_eq(int,bit1(K),bit0(L))
<=> ord_less(int,K,L) ) ).
tff(fact_69_rel__simps_I15_J,axiom,
! [L: int,K: int] :
( ord_less(int,bit0(K),bit1(L))
<=> ord_less_eq(int,K,L) ) ).
tff(fact_70_mult__Bit1,axiom,
! [L1: int,K3: int] : times_times(int,bit1(K3),L1) = plus_plus(int,bit0(times_times(int,K3,L1)),L1) ).
tff(fact_71_add__nat__number__of,axiom,
! [V2: int,V: int] :
( ( ord_less(int,V,pls)
=> ( plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V2)) = number_number_of(nat,V2) ) )
& ( ~ ord_less(int,V,pls)
=> ( ( ord_less(int,V2,pls)
=> ( plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V2)) = number_number_of(nat,V) ) )
& ( ~ ord_less(int,V2,pls)
=> ( plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V2)) = number_number_of(nat,plus_plus(int,V,V2)) ) ) ) ) ) ).
tff(fact_72_zle__add1__eq__le,axiom,
! [Z: int,W1: int] :
( ord_less(int,W1,plus_plus(int,Z,one_one(int)))
<=> ord_less_eq(int,W1,Z) ) ).
tff(fact_73_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_74_le__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y1: int] :
( ord_less_eq(A,zero_zero(A),number_number_of(A,Y1))
<=> ord_less_eq(int,pls,Y1) ) ) ).
tff(fact_75_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_76_le__special_I2_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y1: int] :
( ord_less_eq(A,one_one(A),number_number_of(A,Y1))
<=> ord_less_eq(int,bit1(pls),Y1) ) ) ).
tff(fact_77__096_B_Bthesis_O_A_I_B_Bt_O_As_A_094_A2_A_L_A1_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_At_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [T: int] : plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)) != times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),T) ).
tff(fact_78_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_79_p,axiom,
zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_80_qf1pt,axiom,
twoSqu1567020053sum2sq(times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),t)) ).
tff(fact_81_number__of__is__id,axiom,
! [K3: int] : number_number_of(int,K3) = K3 ).
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_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_84_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_85_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_86_less__int__def,axiom,
! [W1: int,Z: int] :
( ord_less(int,Z,W1)
<=> ( ord_less_eq(int,Z,W1)
& ( Z != W1 ) ) ) ).
tff(fact_87_semiring__mult__number__of,axiom,
! [A: $tType] :
( number_semiring(A)
=> ! [V2: int,V: int] :
( ord_less_eq(int,pls,V)
=> ( ord_less_eq(int,pls,V2)
=> ( times_times(A,number_number_of(A,V),number_number_of(A,V2)) = number_number_of(A,times_times(int,V,V2)) ) ) ) ) ).
tff(fact_88_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_89_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,plus_plus(A,times_times(A,X1,X1),times_times(A,Y1,Y1)),zero_zero(A))
<=> ( ( X1 = zero_zero(A) )
& ( Y1 = zero_zero(A) ) ) ) ) ).
tff(fact_90_sum__squares__ge__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [Y: A,X: A] : ord_less_eq(A,zero_zero(A),plus_plus(A,times_times(A,X,X),times_times(A,Y,Y))) ) ).
tff(fact_91_le__number__of__eq__not__less,axiom,
! [A: $tType] :
( ( number(A)
& linorder(A) )
=> ! [W1: int,V1: int] :
( ord_less_eq(A,number_number_of(A,V1),number_number_of(A,W1))
<=> ~ ord_less(A,number_number_of(A,W1),number_number_of(A,V1)) ) ) ).
tff(fact_92_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_93_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
tff(fact_94_less__number__of__int__code,axiom,
! [L: int,K: int] :
( ord_less(int,number_number_of(int,K),number_number_of(int,L))
<=> ord_less(int,K,L) ) ).
tff(fact_95_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_96_int__distrib_I1_J,axiom,
! [W: int,Z2: int,Z1: int] : times_times(int,plus_plus(int,Z1,Z2),W) = plus_plus(int,times_times(int,Z1,W),times_times(int,Z2,W)) ).
tff(fact_97_int__distrib_I2_J,axiom,
! [Z2: int,Z1: int,W: int] : times_times(int,W,plus_plus(int,Z1,Z2)) = plus_plus(int,times_times(int,W,Z1),times_times(int,W,Z2)) ).
%----Arities (20)
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_Olinordered__ring__strict,axiom,
linord581940658strict(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___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___Int_Oring__char__0,axiom,
ring_char_0(int) ).
tff(arity_Int_Oint___Int_Onumber__ring,axiom,
number_ring(int) ).
tff(arity_Int_Oint___Rings_Osemiring,axiom,
semiring(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(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_Osemiring,axiom,
semiring(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,
ord_less(int,plus_plus(int,power_power(int,s,number_number_of(nat,bit0(bit1(pls)))),one_one(int)),zero_zero(int)) ).
%------------------------------------------------------------------------------