TPTP Problem File: NUM985_5.p
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- Solve Problem
%------------------------------------------------------------------------------
% File : NUM985_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Number Theory
% Problem : Sum of two squares line 141
% Version : Especial.
% English :
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : s2s_141 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 161 ( 63 unt; 42 typ; 0 def)
% Number of atoms : 213 ( 86 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 117 ( 23 ~; 3 |; 17 &)
% ( 28 <=>; 46 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 12 ( 2 avg)
% Number of types : 4 ( 3 usr)
% Number of type conns : 17 ( 12 >; 5 *; 0 +; 0 <<)
% Number of predicates : 18 ( 17 usr; 0 prp; 1-3 aty)
% Number of functors : 22 ( 22 usr; 12 con; 0-3 aty)
% Number of variables : 186 ( 164 !; 0 ?; 186 :)
% ( 22 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:25:14
%------------------------------------------------------------------------------
%----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 (39)
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_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_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_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,type,
ord_less:
!>[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_n____,type,
n: nat ).
tff(sy_v_r____,type,
r: int ).
tff(sy_v_sa____,type,
sa: int ).
tff(sy_v_t____,type,
t: 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_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_1__096_Ix_A_N_Ar_A_K_A_I1_A_L_Aint_An_J_J_A_094_A2_A_L_A_Iy_A_N_As_A_K_A_I1_A_L_Aint_An_J_J_A_094_A2_A_061x_A_094_A2_A_L_Ay_A_094_A2_A_N_A2_A_K_Ax_A_K_A_Ir_A_K_A_I1_A_L_Aint_An_J_J_A_N_A2_A_K_Ay_A_K_A_Is_A_K_A_I1_A_L_Aint_An_J_J_A_L_Ir_A_K_A_I1_A_L_Aint_An_J_J_A_094_A2_A_L_Is_A_K_A_I1_A_L_Aint_An_J_J_A_094_A2_096,axiom,
plus_plus(int,power_power(int,minus_minus(int,x,times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)))),number_number_of(nat,bit0(bit1(pls)))),power_power(int,minus_minus(int,y,times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)))),number_number_of(nat,bit0(bit1(pls))))) = plus_plus(int,plus_plus(int,minus_minus(int,minus_minus(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))))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),x),times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),y),times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))))),power_power(int,times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),number_number_of(nat,bit0(bit1(pls))))),power_power(int,times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_2__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_3_p,axiom,
zprime(plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))) ).
tff(fact_4_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_5_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_6_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_7_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_8_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_9_calculation,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))))) = plus_plus(int,plus_plus(int,minus_minus(int,minus_minus(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))))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),x),times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),y),times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))))),power_power(int,times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),number_number_of(nat,bit0(bit1(pls))))),power_power(int,times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_10__096v_A_094_A2_A_L_Aw_A_094_A2_A_061_A_Ix_A_N_Ar_A_K_A_I1_A_L_Aint_An_J_J_A_094_A2_A_L_A_Iy_A_N_As_A_K_A_I1_A_L_Aint_An_J_J_A_094_A2_096,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))))) = plus_plus(int,power_power(int,minus_minus(int,x,times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)))),number_number_of(nat,bit0(bit1(pls)))),power_power(int,minus_minus(int,y,times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)))),number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_11_nQ1,axiom,
~ twoSqu1567020053sum2sq(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,zero_zero(nat))))) ).
tff(fact_12_smaller_I2_J,axiom,
~ ( ord_less(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))
=> ~ twoSqu1567020053sum2sq(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_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_zdiff__power2,axiom,
! [B1: int,A1: int] : power_power(int,minus_minus(int,A1,B1),number_number_of(nat,bit0(bit1(pls)))) = plus_plus(int,minus_minus(int,power_power(int,A1,number_number_of(nat,bit0(bit1(pls)))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),A1),B1)),power_power(int,B1,number_number_of(nat,bit0(bit1(pls))))) ).
tff(fact_16_t1,axiom,
ord_less(int,one_one(int),t) ).
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_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,
! [K: int] : bit1(K) != pls ).
tff(fact_22_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_23_rel__simps_I50_J,axiom,
! [L1: int,K: int] : bit1(K) != bit0(L1) ).
tff(fact_24_rel__simps_I49_J,axiom,
! [L1: int,K: int] : bit0(K) != bit1(L1) ).
tff(fact_25_rel__simps_I44_J,axiom,
! [K3: int] :
( ( bit0(K3) = pls )
<=> ( K3 = 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,K3: int] :
( ord_less(int,bit1(K3),bit1(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_29_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_30_rel__simps_I14_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit0(K3),bit0(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_31_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_32_mult__Bit0,axiom,
! [L1: int,K: int] : times_times(int,bit0(K),L1) = bit0(times_times(int,K,L1)) ).
tff(fact_33_add__Bit0__Bit0,axiom,
! [L1: int,K: int] : plus_plus(int,bit0(K),bit0(L1)) = bit0(plus_plus(int,K,L1)) ).
tff(fact_34_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_35_n0,axiom,
ord_less(nat,zero_zero(nat),n) ).
tff(fact_36_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_37_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_38_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_39_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_40_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_41_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_42_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_43_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_44_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_45_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_46_rel__simps_I12_J,axiom,
! [K3: int] :
( ord_less(int,bit1(K3),pls)
<=> ord_less(int,K3,pls) ) ).
tff(fact_47_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_48_rel__simps_I16_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit1(K3),bit0(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_49_rel__simps_I10_J,axiom,
! [K3: int] :
( ord_less(int,bit0(K3),pls)
<=> ord_less(int,K3,pls) ) ).
tff(fact_50_rel__simps_I4_J,axiom,
! [K3: int] :
( ord_less(int,pls,bit0(K3))
<=> ord_less(int,pls,K3) ) ).
tff(fact_51_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_52_add__Bit1__Bit0,axiom,
! [L1: int,K: int] : plus_plus(int,bit1(K),bit0(L1)) = bit1(plus_plus(int,K,L1)) ).
tff(fact_53_add__Bit0__Bit1,axiom,
! [L1: int,K: int] : plus_plus(int,bit0(K),bit1(L1)) = bit1(plus_plus(int,K,L1)) ).
tff(fact_54_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_55_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_56_diff__bin__simps_I3_J,axiom,
! [L1: int] : minus_minus(int,pls,bit0(L1)) = bit0(minus_minus(int,pls,L1)) ).
tff(fact_57_IH,axiom,
( ord_less(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int)))
& twoSqu1567020053sum2sq(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_58_nat__1__add__1,axiom,
plus_plus(nat,one_one(nat),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
tff(fact_59_add__nat__number__of,axiom,
! [V1: int,V: int] :
( ( ord_less(int,V,pls)
=> ( plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V1)) = number_number_of(nat,V1) ) )
& ( ~ ord_less(int,V,pls)
=> ( ( ord_less(int,V1,pls)
=> ( plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V1)) = number_number_of(nat,V) ) )
& ( ~ ord_less(int,V1,pls)
=> ( plus_plus(nat,number_number_of(nat,V),number_number_of(nat,V1)) = number_number_of(nat,plus_plus(int,V,V1)) ) ) ) ) ) ).
tff(fact_60_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_61_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_62_mult__nat__number__of,axiom,
! [V1: int,V: int] :
( ( ord_less(int,V,pls)
=> ( times_times(nat,number_number_of(nat,V),number_number_of(nat,V1)) = zero_zero(nat) ) )
& ( ~ ord_less(int,V,pls)
=> ( times_times(nat,number_number_of(nat,V),number_number_of(nat,V1)) = number_number_of(nat,times_times(int,V,V1)) ) ) ) ).
tff(fact_63_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_64_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_65_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_66_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_67_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_68_even__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( ord_less(A,plus_plus(A,A2,A2),zero_zero(A))
<=> ord_less(A,A2,zero_zero(A)) ) ) ).
tff(fact_69_less__int__code_I16_J,axiom,
! [K2: int,K1: int] :
( ord_less(int,bit1(K1),bit1(K2))
<=> ord_less(int,K1,K2) ) ).
tff(fact_70_less__int__code_I13_J,axiom,
! [K2: int,K1: int] :
( ord_less(int,bit0(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
tff(fact_71_less__number__of__int__code,axiom,
! [L: int,K3: int] :
( ord_less(int,number_number_of(int,K3),number_number_of(int,L))
<=> ord_less(int,K3,L) ) ).
tff(fact_72_zpower__zpower,axiom,
! [Z: nat,Y: nat,X: int] : power_power(int,power_power(int,X,Y),Z) = power_power(int,X,times_times(nat,Y,Z)) ).
tff(fact_73_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [Ya: A,Xa: A] :
( ord_less(A,zero_zero(A),plus_plus(A,times_times(A,Xa,Xa),times_times(A,Ya,Ya)))
<=> ( ( Xa != zero_zero(A) )
| ( Ya != zero_zero(A) ) ) ) ) ).
tff(fact_74_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [Y: A,X: A] : ~ ord_less(A,plus_plus(A,times_times(A,X,X),times_times(A,Y,Y)),zero_zero(A)) ) ).
tff(fact_75_nat__number__of__mult__left,axiom,
! [K: nat,V1: int,V: int] :
( ( ord_less(int,V,pls)
=> ( times_times(nat,number_number_of(nat,V),times_times(nat,number_number_of(nat,V1),K)) = zero_zero(nat) ) )
& ( ~ ord_less(int,V,pls)
=> ( times_times(nat,number_number_of(nat,V),times_times(nat,number_number_of(nat,V1),K)) = times_times(nat,number_number_of(nat,times_times(int,V,V1)),K) ) ) ) ).
tff(fact_76_less__int__code_I15_J,axiom,
! [K2: int,K1: int] :
( ord_less(int,bit1(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
tff(fact_77_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [Ya: A,Xa: A] :
( ( plus_plus(A,times_times(A,Xa,Xa),times_times(A,Ya,Ya)) = zero_zero(A) )
<=> ( ( Xa = zero_zero(A) )
& ( Ya = zero_zero(A) ) ) ) ) ).
tff(fact_78_semiring__numeral__0__eq__0,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_79_semiring__norm_I112_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( zero_zero(A) = number_number_of(A,pls) ) ) ).
tff(fact_80_zless__add1__eq,axiom,
! [Z1: int,Wa: int] :
( ord_less(int,Wa,plus_plus(int,Z1,one_one(int)))
<=> ( ord_less(int,Wa,Z1)
| ( Wa = Z1 ) ) ) ).
tff(fact_81_zpower__int,axiom,
! [N: nat,M: nat] : power_power(int,semiring_1_of_nat(int,M),N) = semiring_1_of_nat(int,power_power(nat,M,N)) ).
tff(fact_82_int__power,axiom,
! [N: nat,M: nat] : semiring_1_of_nat(int,power_power(nat,M,N)) = power_power(int,semiring_1_of_nat(int,M),N) ).
tff(fact_83_zadd__int__left,axiom,
! [Z: int,N: nat,M: nat] : plus_plus(int,semiring_1_of_nat(int,M),plus_plus(int,semiring_1_of_nat(int,N),Z)) = plus_plus(int,semiring_1_of_nat(int,plus_plus(nat,M,N)),Z) ).
tff(fact_84_zadd__int,axiom,
! [N: nat,M: nat] : plus_plus(int,semiring_1_of_nat(int,M),semiring_1_of_nat(int,N)) = semiring_1_of_nat(int,plus_plus(nat,M,N)) ).
tff(fact_85_zmult__int,axiom,
! [N: nat,M: nat] : times_times(int,semiring_1_of_nat(int,M),semiring_1_of_nat(int,N)) = semiring_1_of_nat(int,times_times(nat,M,N)) ).
tff(fact_86_int__mult,axiom,
! [N: nat,M: nat] : semiring_1_of_nat(int,times_times(nat,M,N)) = times_times(int,semiring_1_of_nat(int,M),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_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
tff(fact_89_nat__mult__2,axiom,
! [Z: nat] : times_times(nat,number_number_of(nat,bit0(bit1(pls))),Z) = plus_plus(nat,Z,Z) ).
tff(fact_90_nat__mult__2__right,axiom,
! [Z: nat] : times_times(nat,Z,number_number_of(nat,bit0(bit1(pls)))) = plus_plus(nat,Z,Z) ).
tff(fact_91_power2__less__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : ~ ord_less(A,power_power(A,A1,number_number_of(nat,bit0(bit1(pls)))),zero_zero(A)) ) ).
tff(fact_92_number__of__Bit0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : number_number_of(A,bit0(W)) = plus_plus(A,plus_plus(A,zero_zero(A),number_number_of(A,W)),number_number_of(A,W)) ) ).
tff(fact_93_Numeral1__eq1__nat,axiom,
one_one(nat) = number_number_of(nat,bit1(pls)) ).
tff(fact_94_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Ya: A,Xa: A] :
( ord_less(A,zero_zero(A),plus_plus(A,power_power(A,Xa,number_number_of(nat,bit0(bit1(pls)))),power_power(A,Ya,number_number_of(nat,bit0(bit1(pls))))))
<=> ( ( Xa != zero_zero(A) )
| ( Ya != zero_zero(A) ) ) ) ) ).
tff(fact_95_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] : ~ ord_less(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))))),zero_zero(A)) ) ).
tff(fact_96_is__mult__sum2sq,axiom,
! [Y: int,X: int] :
( twoSqu1567020053sum2sq(X)
=> ( twoSqu1567020053sum2sq(Y)
=> twoSqu1567020053sum2sq(times_times(int,X,Y)) ) ) ).
tff(fact_97_zprime__2,axiom,
zprime(number_number_of(int,bit0(bit1(pls)))) ).
%----Arities (18)
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___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___Int_Onumber__semiring,axiom,
number_semiring(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) ).
%----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,plus_plus(int,minus_minus(int,minus_minus(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))))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),x),times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))))),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),y),times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))))),power_power(int,times_times(int,r,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),number_number_of(nat,bit0(bit1(pls))))),power_power(int,times_times(int,sa,plus_plus(int,one_one(int),semiring_1_of_nat(int,n))),number_number_of(nat,bit0(bit1(pls))))) = plus_plus(int,plus_plus(int,minus_minus(int,minus_minus(int,times_times(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),plus_plus(int,times_times(int,number_number_of(int,bit0(bit0(bit1(pls)))),m),one_one(int))),times_times(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),x),r))),times_times(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),times_times(int,times_times(int,number_number_of(int,bit0(bit1(pls))),y),sa))),times_times(int,power_power(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),number_number_of(nat,bit0(bit1(pls)))),power_power(int,r,number_number_of(nat,bit0(bit1(pls)))))),times_times(int,power_power(int,plus_plus(int,one_one(int),semiring_1_of_nat(int,n)),number_number_of(nat,bit0(bit1(pls)))),power_power(int,sa,number_number_of(nat,bit0(bit1(pls)))))) ).
%------------------------------------------------------------------------------