TPTP Problem File: NUM993_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM993_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 154
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_154 [Bla13]
% Status : Theorem
% Rating : 0.33 v7.4.0, 0.75 v7.1.0, 0.67 v6.4.0
% Syntax : Number of formulae : 142 ( 47 unt; 29 typ; 0 def)
% Number of atoms : 211 ( 52 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 107 ( 9 ~; 0 |; 17 &)
% ( 50 <=>; 31 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 15 ( 10 >; 5 *; 0 +; 0 <<)
% Number of predicates : 12 ( 11 usr; 0 prp; 1-3 aty)
% Number of functors : 15 ( 15 usr; 6 con; 0-3 aty)
% Number of variables : 183 ( 165 !; 1 ?; 183 :)
% ( 17 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:25:34
%------------------------------------------------------------------------------
%----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 (26)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring,type,
ring:
!>[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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Olinordered__ab__group__add,type,
linord219039673up_add:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[A: $tType] : ( ( A * A ) > A ) ).
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_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_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > 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_fFalse,type,
fFalse: bool ).
tff(sy_c_fTrue,type,
fTrue: bool ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_m1____,type,
m1: int ).
tff(sy_v_n____,type,
n: nat ).
tff(sy_v_t____,type,
t: int ).
%----Relevant facts (98)
tff(fact_0_t1,axiom,
ord_less(int,one_one(int),t) ).
tff(fact_1__096_126_Am1_A_060_A1_A_L_Aint_An_096,axiom,
~ ord_less(int,m1,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ).
tff(fact_2__096_I1_A_L_Aint_An_J_A_K_A_I2_A_K_A_I1_A_L_Aint_An_J_A_N_A4_A_K_Am1_J_A_060_A_I1_A_L_Aint_An_J_A_K_A0_096,axiom,
ord_less(int,times_times(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),minus_minus(int,times_times(int,number_number_of(int,bit0(bit1(pls))),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m1))),times_times(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),zero_zero(int))) ).
tff(fact_3_n1pos,axiom,
ord_less(int,zero_zero(int),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ).
tff(fact_4__0962_A_K_A_I1_A_L_Aint_An_J_A_N_A4_A_K_Am1_A_060_A0_096,axiom,
ord_less(int,minus_minus(int,times_times(int,number_number_of(int,bit0(bit1(pls))),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m1)),zero_zero(int)) ).
tff(fact_5_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_6_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_7_less__special_I2_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int] :
( ord_less(A,one_one(A),number_number_of(A,Ya))
<=> ord_less(int,bit1(pls),Ya) ) ) ).
tff(fact_8_less__special_I4_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Xa: int] :
( ord_less(A,number_number_of(A,Xa),one_one(A))
<=> ord_less(int,Xa,bit1(pls)) ) ) ).
tff(fact_9_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_10_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_11_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_12__0964_A_K_A_I1_A_L_Aint_An_J_A_N_A4_A_K_Am1_A_060_061_A0_096,axiom,
ord_less_eq(int,minus_minus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m1)),zero_zero(int)) ).
tff(fact_13_mult__Bit1,axiom,
! [L1: int,K: int] : times_times(int,bit1(K),L1) = plus_plus(int,bit0(times_times(int,K,L1)),L1) ).
tff(fact_14_numeral__1__eq__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_15_diff__bin__simps_I3_J,axiom,
! [L1: int] : minus_minus(int,pls,bit0(L1)) = bit0(minus_minus(int,pls,L1)) ).
tff(fact_16_diff__bin__simps_I9_J,axiom,
! [L1: int,K: int] : minus_minus(int,bit1(K),bit0(L1)) = bit1(minus_minus(int,K,L1)) ).
tff(fact_17_n0,axiom,
ord_less(nat,zero_zero(nat),n) ).
tff(fact_18_tpos,axiom,
ord_less_eq(int,one_one(int),t) ).
tff(fact_19_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_20_rel__simps_I51_J,axiom,
! [L: int,K3: int] :
( ( bit1(K3) = bit1(L) )
<=> ( K3 = L ) ) ).
tff(fact_21_rel__simps_I48_J,axiom,
! [L: int,K3: int] :
( ( bit0(K3) = bit0(L) )
<=> ( K3 = L ) ) ).
tff(fact_22_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_23_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_24_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_25_rel__simps_I50_J,axiom,
! [L1: int,K: int] : bit1(K) != bit0(L1) ).
tff(fact_26_rel__simps_I49_J,axiom,
! [L1: int,K: int] : bit0(K) != bit1(L1) ).
tff(fact_27_rel__simps_I44_J,axiom,
! [K3: int] :
( ( bit0(K3) = pls )
<=> ( K3 = pls ) ) ).
tff(fact_28_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_29_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_30_rel__simps_I17_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit1(K3),bit1(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_31_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_32_rel__simps_I14_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit0(K3),bit0(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_33_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_34_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
tff(fact_35_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_36_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_37_mult__Bit0,axiom,
! [L1: int,K: int] : times_times(int,bit0(K),L1) = bit0(times_times(int,K,L1)) ).
tff(fact_38_add__Bit0__Bit0,axiom,
! [L1: int,K: int] : plus_plus(int,bit0(K),bit0(L1)) = bit0(plus_plus(int,K,L1)) ).
tff(fact_39_diff__bin__simps_I7_J,axiom,
! [L1: int,K: int] : minus_minus(int,bit0(K),bit0(L1)) = bit0(minus_minus(int,K,L1)) ).
tff(fact_40__0961_A_L_Aint_An_A_N_Am1_A_060_061_A0_096,axiom,
ord_less_eq(int,minus_minus(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),m1),zero_zero(int)) ).
tff(fact_41_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_42_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_43_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_44_left__diff__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& ring(B) )
=> ! [V: int,B1: B,A1: B] : times_times(B,minus_minus(B,A1,B1),number_number_of(B,V)) = minus_minus(B,times_times(B,A1,number_number_of(B,V)),times_times(B,B1,number_number_of(B,V))) ) ).
tff(fact_45_right__diff__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& ring(B) )
=> ! [C: B,B1: B,V: int] : times_times(B,number_number_of(B,V),minus_minus(B,B1,C)) = minus_minus(B,times_times(B,number_number_of(B,V),B1),times_times(B,number_number_of(B,V),C)) ) ).
tff(fact_46_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_47_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_48_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_49_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_50_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_51_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_52_rel__simps_I12_J,axiom,
! [K3: int] :
( ord_less(int,bit1(K3),pls)
<=> ord_less(int,K3,pls) ) ).
tff(fact_53_rel__simps_I16_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit1(K3),bit0(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_54_rel__simps_I10_J,axiom,
! [K3: int] :
( ord_less(int,bit0(K3),pls)
<=> ord_less(int,K3,pls) ) ).
tff(fact_55_rel__simps_I4_J,axiom,
! [K3: int] :
( ord_less(int,pls,bit0(K3))
<=> ord_less(int,pls,K3) ) ).
tff(fact_56_rel__simps_I22_J,axiom,
! [K3: int] :
( ord_less_eq(int,pls,bit1(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_57_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_58_rel__simps_I27_J,axiom,
! [K3: int] :
( ord_less_eq(int,bit0(K3),pls)
<=> ord_less_eq(int,K3,pls) ) ).
tff(fact_59_rel__simps_I21_J,axiom,
! [K3: int] :
( ord_less_eq(int,pls,bit0(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_60_add__Bit1__Bit0,axiom,
! [L1: int,K: int] : plus_plus(int,bit1(K),bit0(L1)) = bit1(plus_plus(int,K,L1)) ).
tff(fact_61_add__Bit0__Bit1,axiom,
! [L1: int,K: int] : plus_plus(int,bit0(K),bit1(L1)) = bit1(plus_plus(int,K,L1)) ).
tff(fact_62_diff__bin__simps_I10_J,axiom,
! [L1: int,K: int] : minus_minus(int,bit1(K),bit1(L1)) = bit0(minus_minus(int,K,L1)) ).
tff(fact_63_rel__simps_I29_J,axiom,
! [K3: int] :
( ord_less_eq(int,bit1(K3),pls)
<=> ord_less(int,K3,pls) ) ).
tff(fact_64_rel__simps_I5_J,axiom,
! [K3: int] :
( ord_less(int,pls,bit1(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_65_rel__simps_I33_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit1(K3),bit0(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_66_rel__simps_I15_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit0(K3),bit1(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_67_zle__add1__eq__le,axiom,
! [Z1: int,Wa: int] :
( ord_less(int,Wa,plus_plus(int,Z1,one_one(int)))
<=> ord_less_eq(int,Wa,Z1) ) ).
tff(fact_68_zle__diff1__eq,axiom,
! [Z1: int,Wa: int] :
( ord_less_eq(int,Wa,minus_minus(int,Z1,one_one(int)))
<=> ord_less(int,Wa,Z1) ) ).
tff(fact_69_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_70_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_71_le__special_I4_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Xa: int] :
( ord_less_eq(A,number_number_of(A,Xa),one_one(A))
<=> ord_less_eq(int,Xa,bit1(pls)) ) ) ).
tff(fact_72_le__special_I2_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Ya: int] :
( ord_less_eq(A,one_one(A),number_number_of(A,Ya))
<=> ord_less_eq(int,bit1(pls),Ya) ) ) ).
tff(fact_73_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_74_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_75_less__int__def,axiom,
! [Wa: int,Z1: int] :
( ord_less(int,Z1,Wa)
<=> ( ord_less_eq(int,Z1,Wa)
& ( Z1 != Wa ) ) ) ).
tff(fact_76_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_77_int__le__0__conv,axiom,
! [Na: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,Na),zero_zero(int))
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_78_zero__less__int__conv,axiom,
! [Na: nat] :
( ord_less(int,zero_zero(int),semiring_1_of_nat(int,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_79_zmult__zless__mono2__lemma,axiom,
! [K: nat,J: int,I: int] :
( ord_less(int,I,J)
=> ( ord_less(nat,zero_zero(nat),K)
=> ord_less(int,times_times(int,semiring_1_of_nat(int,K),I),times_times(int,semiring_1_of_nat(int,K),J)) ) ) ).
tff(fact_80_le__number__of__eq__not__less,axiom,
! [A: $tType] :
( ( number(A)
& linorder(A) )
=> ! [Wa: int,Va: int] :
( ord_less_eq(A,number_number_of(A,Va),number_number_of(A,Wa))
<=> ~ ord_less(A,number_number_of(A,Wa),number_number_of(A,Va)) ) ) ).
tff(fact_81_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_82_zero__zle__int,axiom,
! [N1: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,N1)) ).
tff(fact_83_zadd__int__left,axiom,
! [Z: int,N1: nat,M: nat] : plus_plus(int,semiring_1_of_nat(int,M),plus_plus(int,semiring_1_of_nat(int,N1),Z)) = plus_plus(int,semiring_1_of_nat(int,plus_plus(nat,M,N1)),Z) ).
tff(fact_84_zadd__int,axiom,
! [N1: nat,M: nat] : plus_plus(int,semiring_1_of_nat(int,M),semiring_1_of_nat(int,N1)) = semiring_1_of_nat(int,plus_plus(nat,M,N1)) ).
tff(fact_85_zless__int,axiom,
! [Na: nat,Ma: nat] :
( ord_less(int,semiring_1_of_nat(int,Ma),semiring_1_of_nat(int,Na))
<=> ord_less(nat,Ma,Na) ) ).
tff(fact_86_zle__iff__zadd,axiom,
! [Z1: int,Wa: int] :
( ord_less_eq(int,Wa,Z1)
<=> ? [N: nat] : Z1 = plus_plus(int,Wa,semiring_1_of_nat(int,N)) ) ).
tff(fact_87_int__1,axiom,
semiring_1_of_nat(int,one_one(nat)) = one_one(int) ).
tff(fact_88_int__eq__0__conv,axiom,
! [Na: nat] :
( ( semiring_1_of_nat(int,Na) = zero_zero(int) )
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_89_int__0,axiom,
semiring_1_of_nat(int,zero_zero(nat)) = zero_zero(int) ).
tff(fact_90_less__eq__int__code_I15_J,axiom,
! [K2: int,K1: int] :
( ord_less_eq(int,bit1(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
tff(fact_91_less__int__code_I14_J,axiom,
! [K2: int,K1: int] :
( ord_less(int,bit0(K1),bit1(K2))
<=> ord_less_eq(int,K1,K2) ) ).
tff(fact_92_int__one__le__iff__zero__less,axiom,
! [Z1: int] :
( ord_less_eq(int,one_one(int),Z1)
<=> ord_less(int,zero_zero(int),Z1) ) ).
tff(fact_93_zless__imp__add1__zle,axiom,
! [Z: int,W: int] :
( ord_less(int,W,Z)
=> ord_less_eq(int,plus_plus(int,W,one_one(int)),Z) ) ).
tff(fact_94_add1__zle__eq,axiom,
! [Z1: int,Wa: int] :
( ord_less_eq(int,plus_plus(int,Wa,one_one(int)),Z1)
<=> ord_less(int,Wa,Z1) ) ).
tff(fact_95_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_96_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_97_number__of__is__id,axiom,
! [K: int] : number_number_of(int,K) = K ).
%----Arities (12)
tff(arity_Int_Oint___Groups_Olinordered__ab__group__add,axiom,
linord219039673up_add(int) ).
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(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___Rings_Oring,axiom,
ring(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(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,times_times(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),minus_minus(int,times_times(int,number_number_of(int,bit0(bit1(pls))),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m1))),zero_zero(int)) ).
%------------------------------------------------------------------------------