TPTP Problem File: NUN006_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : NUN006_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 177
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_177 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 150 ( 64 unt; 37 typ; 0 def)
% Number of atoms : 179 ( 88 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 79 ( 13 ~; 0 |; 13 &)
% ( 24 <=>; 29 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 9 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 19 ( 13 >; 6 *; 0 +; 0 <<)
% Number of predicates : 13 ( 12 usr; 0 prp; 1-3 aty)
% Number of functors : 22 ( 22 usr; 12 con; 0-3 aty)
% Number of variables : 188 ( 169 !; 1 ?; 188 :)
% ( 18 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:26:08
%------------------------------------------------------------------------------
%----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 (34)
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_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Omonoid__mult,type,
monoid_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
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_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_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
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_Rings_Odvd__class_Odvd,type,
dvd_dvd:
!>[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_m,type,
m: int ).
tff(sy_v_m1____,type,
m1: int ).
tff(sy_v_n____,type,
n: nat ).
tff(sy_v_r____,type,
r: int ).
tff(sy_v_sa____,type,
sa: int ).
tff(sy_v_v____,type,
v: int ).
tff(sy_v_w____,type,
w: int ).
tff(sy_v_x____,type,
x: int ).
tff(sy_v_y____,type,
y: int ).
%----Relevant facts (98)
tff(fact_0__096_Ix_A_094_A2_A_L_Ay_A_094_A2_J_A_K_A_Iv_A_094_A2_A_L_Aw_A_094_A2_J_A_061_Ix_A_K_Av_A_L_Ay_A_K_Aw_J_A_094_A2_A_L_A_Ix_A_K_Aw_A_N_Ay_A_K_Av_J_A_094_A2_096,axiom,
times_times(int,plus_plus(int,power_power(int,x,number_number_of(nat,bit0(bit1(pls)))),power_power(int,y,number_number_of(nat,bit0(bit1(pls))))),plus_plus(int,power_power(int,v,number_number_of(nat,bit0(bit1(pls)))),power_power(int,w,number_number_of(nat,bit0(bit1(pls)))))) = plus_plus(int,power_power(int,plus_plus(int,times_times(int,x,v),times_times(int,y,w)),number_number_of(nat,bit0(bit1(pls)))),power_power(int,minus_minus(int,times_times(int,x,w),times_times(int,y,v)),number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_1_m1,axiom,
plus_plus(int,power_power(int,v,number_number_of(nat,bit0(bit1(pls)))),power_power(int,w,number_number_of(nat,bit0(bit1(pls))))) = times_times(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),m1) ).
tff(fact_2_rv,axiom,
( ( v = minus_minus(int,x,times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)))) )
& ord_less_eq(int,times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,v)),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ) ).
tff(fact_3_sw,axiom,
( ( w = minus_minus(int,y,times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)))) )
& ord_less_eq(int,times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,w)),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ) ).
tff(fact_4__096_B_Bthesis_O_A_I_B_Bm1_O_Av_A_094_A2_A_L_Aw_A_094_A2_A_061_A_I1_A_L_Aint_An_J_A_K_Am1_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [M1: int] : plus_plus(int,power_power(int,v,number_number_of(nat,bit0(bit1(pls)))),power_power(int,w,number_number_of(nat,bit0(bit1(pls))))) != times_times(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),M1) ).
tff(fact_5_power2__eq__square__number__of,axiom,
! [B1: $tType] :
( ( monoid_mult(B1)
& number(B1) )
=> ! [W: int] : power_power(B1,number_number_of(B1,W),number_number_of(nat,bit0(bit1(pls)))) = times_times(B1,number_number_of(B1,W),number_number_of(B1,W)) ) ).
tff(fact_6_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_7_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_8_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_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_xy,axiom,
plus_plus(int,power_power(int,x,number_number_of(nat,bit0(bit1(pls)))),power_power(int,y,number_number_of(nat,bit0(bit1(pls))))) = times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ).
tff(fact_11_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_12_numeral__1__eq__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_13__0961_A_L_Aint_An_Advd_Av_A_094_A2_A_L_Aw_A_094_A2_096,axiom,
dvd_dvd(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),plus_plus(int,power_power(int,v,number_number_of(nat,bit0(bit1(pls)))),power_power(int,w,number_number_of(nat,bit0(bit1(pls)))))) ).
tff(fact_14_power2__diff,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Y: A,X: A] : power_power(A,minus_minus(A,X,Y),number_number_of(nat,bit0(bit1(pls)))) = minus_minus(A,plus_plus(A,power_power(A,X,number_number_of(nat,bit0(bit1(pls)))),power_power(A,Y,number_number_of(nat,bit0(bit1(pls))))),times_times(A,times_times(A,number_number_of(A,bit0(bit1(pls))),X),Y)) ) ).
tff(fact_15_zspecial__product,axiom,
! [B: int,A1: int] : times_times(int,plus_plus(int,A1,B),minus_minus(int,A1,B)) = minus_minus(int,power_power(int,A1,number_number_of(nat,bit0(bit1(pls)))),power_power(int,B,number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_16__096_B_Bthesis_O_A_I_B_Br_Av_O_Av_A_061_Ax_A_N_Ar_A_K_A_I1_A_L_Aint_An_J_A_G_A2_A_K_Aabs_Av_A_060_061_A1_A_L_Aint_An_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [R: int,V1: int] :
~ ( ( V1 = minus_minus(int,x,times_times(int,R,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)))) )
& ord_less_eq(int,times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,V1)),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ) ).
tff(fact_17_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_18_rel__simps_I51_J,axiom,
! [L: int,K3: int] :
( ( bit1(K3) = bit1(L) )
<=> ( K3 = L ) ) ).
tff(fact_19_rel__simps_I48_J,axiom,
! [L: int,K3: int] :
( ( bit0(K3) = bit0(L) )
<=> ( K3 = L ) ) ).
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_I34_J,axiom,
! [L: int,K3: int] :
( ord_less_eq(int,bit1(K3),bit1(L))
<=> ord_less_eq(int,K3,L) ) ).
tff(fact_28_rel__simps_I19_J,axiom,
ord_less_eq(int,pls,pls) ).
tff(fact_29_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_30_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_31_mult__Bit0,axiom,
! [L1: int,K: int] : times_times(int,bit0(K),L1) = bit0(times_times(int,K,L1)) ).
tff(fact_32_add__Bit0__Bit0,axiom,
! [L1: int,K: int] : plus_plus(int,bit0(K),bit0(L1)) = bit0(plus_plus(int,K,L1)) ).
tff(fact_33_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_34_left__distrib__number__of,axiom,
! [B1: $tType] :
( ( number(B1)
& semiring(B1) )
=> ! [V: int,B: B1,A1: B1] : times_times(B1,plus_plus(B1,A1,B),number_number_of(B1,V)) = plus_plus(B1,times_times(B1,A1,number_number_of(B1,V)),times_times(B1,B,number_number_of(B1,V))) ) ).
tff(fact_35_right__distrib__number__of,axiom,
! [B1: $tType] :
( ( number(B1)
& semiring(B1) )
=> ! [C1: B1,B: B1,V: int] : times_times(B1,number_number_of(B1,V),plus_plus(B1,B,C1)) = plus_plus(B1,times_times(B1,number_number_of(B1,V),B),times_times(B1,number_number_of(B1,V),C1)) ) ).
tff(fact_36_left__diff__distrib__number__of,axiom,
! [B1: $tType] :
( ( number(B1)
& ring(B1) )
=> ! [V: int,B: B1,A1: B1] : times_times(B1,minus_minus(B1,A1,B),number_number_of(B1,V)) = minus_minus(B1,times_times(B1,A1,number_number_of(B1,V)),times_times(B1,B,number_number_of(B1,V))) ) ).
tff(fact_37_right__diff__distrib__number__of,axiom,
! [B1: $tType] :
( ( number(B1)
& ring(B1) )
=> ! [C1: B1,B: B1,V: int] : times_times(B1,number_number_of(B1,V),minus_minus(B1,B,C1)) = minus_minus(B1,times_times(B1,number_number_of(B1,V),B),times_times(B1,number_number_of(B1,V),C1)) ) ).
tff(fact_38_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_39_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_40_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_41_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_42_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_43_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_44_rel__simps_I22_J,axiom,
! [K3: int] :
( ord_less_eq(int,pls,bit1(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_45_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_46_rel__simps_I27_J,axiom,
! [K3: int] :
( ord_less_eq(int,bit0(K3),pls)
<=> ord_less_eq(int,K3,pls) ) ).
tff(fact_47_rel__simps_I21_J,axiom,
! [K3: int] :
( ord_less_eq(int,pls,bit0(K3))
<=> ord_less_eq(int,pls,K3) ) ).
tff(fact_48_add__Bit1__Bit0,axiom,
! [L1: int,K: int] : plus_plus(int,bit1(K),bit0(L1)) = bit1(plus_plus(int,K,L1)) ).
tff(fact_49_add__Bit0__Bit1,axiom,
! [L1: int,K: int] : plus_plus(int,bit0(K),bit1(L1)) = bit1(plus_plus(int,K,L1)) ).
tff(fact_50_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_51_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_52_diff__bin__simps_I3_J,axiom,
! [L1: int] : minus_minus(int,pls,bit0(L1)) = bit0(minus_minus(int,pls,L1)) ).
tff(fact_53_zdvd1__eq,axiom,
! [Xa: int] :
( dvd_dvd(int,Xa,one_one(int))
<=> ( abs_abs(int,Xa) = one_one(int) ) ) ).
tff(fact_54_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
tff(fact_55__096_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_K_A_Iv_A_094_A2_A_L_Aw_A_094_A2_J_A_061_Ix_A_094_A2_A_L_Ay_A_094_A2_J_A_K_A_Iv_A_094_A2_A_L_Aw_A_094_A2_J_096,axiom,
times_times(int,times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),plus_plus(int,power_power(int,v,number_number_of(nat,bit0(bit1(pls)))),power_power(int,w,number_number_of(nat,bit0(bit1(pls)))))) = times_times(int,plus_plus(int,power_power(int,x,number_number_of(nat,bit0(bit1(pls)))),power_power(int,y,number_number_of(nat,bit0(bit1(pls))))),plus_plus(int,power_power(int,v,number_number_of(nat,bit0(bit1(pls)))),power_power(int,w,number_number_of(nat,bit0(bit1(pls)))))) ).
tff(fact_56_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_57_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_58_power2__abs,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : power_power(A,abs_abs(A,A1),number_number_of(nat,bit0(bit1(pls)))) = power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_59_abs__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : abs_abs(A,power_power(A,A1,number_number_of(nat,bit0(bit1(pls))))) = power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_60_calculation,axiom,
times_times(int,times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),plus_plus(int,power_power(int,v,number_number_of(nat,bit0(bit1(pls)))),power_power(int,w,number_number_of(nat,bit0(bit1(pls)))))) = plus_plus(int,power_power(int,plus_plus(int,times_times(int,x,v),times_times(int,y,w)),number_number_of(nat,bit0(bit1(pls)))),power_power(int,minus_minus(int,times_times(int,x,w),times_times(int,y,v)),number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_61__096_B_Bthesis_O_A_I_B_Bx_Ay_O_Ax_A_094_A2_A_L_Ay_A_094_A2_A_061_A_I4_A_K_Am_A_L_A1_J_A_K_A_I1_A_L_Aint_An_J_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [X1: int,Y1: int] : plus_plus(int,power_power(int,X1,number_number_of(nat,bit0(bit1(pls)))),power_power(int,Y1,number_number_of(nat,bit0(bit1(pls))))) != times_times(int,plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ).
tff(fact_62__096_B_Bthesis_O_A_I_B_Bs_Aw_O_Aw_A_061_Ay_A_N_As_A_K_A_I1_A_L_Aint_An_J_A_G_A2_A_K_Aabs_Aw_A_060_061_A1_A_L_Aint_An_A_061_061_062_Athesis_J_A_061_061_062_Athesis_096,axiom,
~ ! [S: int,W1: int] :
~ ( ( W1 = minus_minus(int,y,times_times(int,S,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)))) )
& ord_less_eq(int,times_times(int,number_number_of(int,bit0(bit1(pls))),abs_abs(int,W1)),plus_plus(int,one_one(int),semiring_1_of_nat(int,n))) ) ).
tff(fact_63_p,axiom,
zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_64_number__of__is__id,axiom,
! [K: int] : number_number_of(int,K) = K ).
tff(fact_65_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_66_zdvd__antisym__abs,axiom,
! [B: int,A1: int] :
( dvd_dvd(int,A1,B)
=> ( dvd_dvd(int,B,A1)
=> ( abs_abs(int,A1) = abs_abs(int,B) ) ) ) ).
tff(fact_67_zdvd__zdiffD,axiom,
! [N1: int,M: int,K: int] :
( dvd_dvd(int,K,minus_minus(int,M,N1))
=> ( dvd_dvd(int,K,N1)
=> dvd_dvd(int,K,M) ) ) ).
tff(fact_68_zpower__zpower,axiom,
! [Z3: nat,Y: nat,X: int] : power_power(int,power_power(int,X,Y),Z3) = power_power(int,X,times_times(nat,Y,Z3)) ).
tff(fact_69_zdvd__period,axiom,
! [C: int,Ta: int,Xa: int,D: int,A2: int] :
( dvd_dvd(int,A2,D)
=> ( dvd_dvd(int,A2,plus_plus(int,Xa,Ta))
<=> dvd_dvd(int,A2,plus_plus(int,plus_plus(int,Xa,times_times(int,C,D)),Ta)) ) ) ).
tff(fact_70_zdvd__reduce,axiom,
! [Ma: int,Na: int,K3: int] :
( dvd_dvd(int,K3,plus_plus(int,Na,times_times(int,K3,Ma)))
<=> dvd_dvd(int,K3,Na) ) ).
tff(fact_71_zpower__int,axiom,
! [N1: nat,M: nat] : power_power(int,semiring_1_of_nat(int,M),N1) = semiring_1_of_nat(int,power_power(nat,M,N1)) ).
tff(fact_72_int__power,axiom,
! [N1: nat,M: nat] : semiring_1_of_nat(int,power_power(nat,M,N1)) = power_power(int,semiring_1_of_nat(int,M),N1) ).
tff(fact_73_zadd__int__left,axiom,
! [Z3: int,N1: nat,M: nat] : plus_plus(int,semiring_1_of_nat(int,M),plus_plus(int,semiring_1_of_nat(int,N1),Z3)) = plus_plus(int,semiring_1_of_nat(int,plus_plus(nat,M,N1)),Z3) ).
tff(fact_74_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_75_zmult__int,axiom,
! [N1: nat,M: nat] : times_times(int,semiring_1_of_nat(int,M),semiring_1_of_nat(int,N1)) = semiring_1_of_nat(int,times_times(nat,M,N1)) ).
tff(fact_76_int__mult,axiom,
! [N1: nat,M: nat] : semiring_1_of_nat(int,times_times(nat,M,N1)) = times_times(int,semiring_1_of_nat(int,M),semiring_1_of_nat(int,N1)) ).
tff(fact_77_int__1,axiom,
semiring_1_of_nat(int,one_one(nat)) = one_one(int) ).
tff(fact_78_nat__mult__2,axiom,
! [Z3: nat] : times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z3) = plus_plus(nat,Z3,Z3) ).
tff(fact_79_nat__mult__2__right,axiom,
! [Z3: nat] : times_times(nat,Z3,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,Z3,Z3) ).
tff(fact_80_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_81_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_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_abs__int__eq,axiom,
! [M: nat] : abs_abs(int,semiring_1_of_nat(int,M)) = semiring_1_of_nat(int,M) ).
tff(fact_84_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_85_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
tff(fact_86_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_87_abs__zmult__eq__1,axiom,
! [N1: int,M: int] :
( ( abs_abs(int,times_times(int,M,N1)) = one_one(int) )
=> ( abs_abs(int,M) = one_one(int) ) ) ).
tff(fact_88_int__int__eq,axiom,
! [Na: nat,Ma: nat] :
( ( semiring_1_of_nat(int,Ma) = semiring_1_of_nat(int,Na) )
<=> ( Ma = Na ) ) ).
tff(fact_89_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_90_power__even__eq,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N1: nat,A1: A] : power_power(A,A1,times_times(nat,number_number_of(nat,bit0(bit1(pls))),N1)) = power_power(A,power_power(A,A1,N1),number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_91_zle__iff__zadd,axiom,
! [Z: int,Wa: int] :
( ord_less_eq(int,Wa,Z)
<=> ? [N: nat] : Z = plus_plus(int,Wa,semiring_1_of_nat(int,N)) ) ).
tff(fact_92_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
tff(fact_93_add__Pls__right,axiom,
! [K: int] : plus_plus(int,K,pls) = K ).
tff(fact_94_add__Pls,axiom,
! [K: int] : plus_plus(int,pls,K) = K ).
tff(fact_95_Bit0__def,axiom,
! [K: int] : bit0(K) = plus_plus(int,K,K) ).
tff(fact_96_diff__bin__simps_I1_J,axiom,
! [K: int] : minus_minus(int,K,pls) = K ).
tff(fact_97_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)) ).
%----Arities (12)
tff(arity_Int_Oint___Rings_Olinordered__idom,axiom,
linordered_idom(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___Rings_Oring,axiom,
ring(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
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) ).
%----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,
plus_plus(int,power_power(int,plus_plus(int,times_times(int,x,v),times_times(int,y,w)),number_number_of(nat,bit0(bit1(pls)))),power_power(int,minus_minus(int,times_times(int,x,w),times_times(int,y,v)),number_number_of(nat,bit0(bit1(pls))))) = plus_plus(int,power_power(int,plus_plus(int,times_times(int,plus_plus(int,times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),v),v),times_times(int,plus_plus(int,times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),w),w)),number_number_of(nat,bit0(bit1(pls)))),power_power(int,minus_minus(int,times_times(int,plus_plus(int,times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),v),w),times_times(int,plus_plus(int,times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),w),v)),number_number_of(nat,bit0(bit1(pls))))) ).
%------------------------------------------------------------------------------