TPTP Problem File: SWV597_5.p
View Solutions
- Solve Problem
%------------------------------------------------------------------------------
% File : SWV597_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fast Fourier Transform line 133
% Version : Especial.
% English : Formalization of a functional implementation of the FFT algorithm
% over the complex numbers, and its inverse. Both are shown
% equivalent to the usual definitions of these operations through
% Vandermonde matrices. They are also shown to be inverse to each
% other, more precisely, that composition of the inverse and the
% transformation yield the identity up to a scalar.
% Refs : [BN10] Boehme & Nipkow (2010), Sledgehammer: Judgement Day
% : [Bla13] Blanchette (2011), Email to Geoff Sutcliffe
% Source : [Bla13]
% Names : fft_133 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 173 ( 74 unt; 34 typ; 0 def)
% Number of atoms : 271 ( 67 equ)
% Maximal formula atoms : 6 ( 1 avg)
% Number of connectives : 151 ( 19 ~; 4 |; 10 &)
% ( 34 <=>; 84 =>; 0 <=; 0 <~>)
% Maximal formula depth : 9 ( 3 avg)
% Maximal term depth : 6 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 11 ( 8 >; 3 *; 0 +; 0 <<)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-3 aty)
% Number of functors : 12 ( 12 usr; 5 con; 0-3 aty)
% Number of variables : 213 ( 192 !; 0 ?; 213 :)
% ( 21 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:48
%------------------------------------------------------------------------------
%----Should-be-implicit typings (4)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Int_Oint,type,
int: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_RealDef_Oreal,type,
real: $tType ).
%----Explicit typings (30)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring,type,
ring:
!>[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_Groups_Ogroup__add,type,
group_add:
!>[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_Groups_Oab__semigroup__mult,type,
ab_semigroup_mult:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add,type,
ordered_ab_group_add:
!>[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_Olinordered__semiring__strict,type,
linord20386208strict:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__comm__semiring__strict,type,
linord893533164strict:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Ominus__class_Ominus,type,
minus_minus:
!>[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_Orderings_Oord__class_Oless,type,
ord_less:
!>[A: $tType] : ( ( A * A ) > $o ) ).
tff(sy_c_Transcendental_Opi,type,
pi: real ).
tff(sy_c_Transcendental_Osin,type,
sin: real > real ).
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_x,type,
x: real ).
%----Relevant facts (97)
tff(fact_0_pi1,axiom,
ord_less(real,pi,x) ).
tff(fact_1__C0_C,axiom,
ord_less(real,zero_zero(real),x) ).
tff(fact_2__096sin_A_Ix_A_N_Api_J_A_126_061_A0_096,axiom,
sin(minus_minus(real,x,pi)) != zero_zero(real) ).
tff(fact_3_calculation_I1_J,axiom,
( ord_less(real,zero_zero(real),x)
=> ( ord_less(real,x,pi)
=> ( sin(x) != zero_zero(real) ) ) ) ).
tff(fact_4_pi,axiom,
ord_less(real,x,times_times(real,number_number_of(real,bit0(bit1(pls))),pi)) ).
tff(fact_5__0960_A_060_Ax_A_N_Api_096,axiom,
ord_less(real,zero_zero(real),minus_minus(real,x,pi)) ).
tff(fact_6_sin__zero,axiom,
sin(zero_zero(real)) = zero_zero(real) ).
tff(fact_7__096x_A_N_Api_A_060_Api_096,axiom,
ord_less(real,minus_minus(real,x,pi),pi) ).
tff(fact_8_sin__pi,axiom,
sin(pi) = zero_zero(real) ).
tff(fact_9_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xa: A] :
( ( zero_zero(A) = Xa )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_10_sin__gt__zero__pi,axiom,
! [X: real] :
( ord_less(real,zero_zero(real),X)
=> ( ord_less(real,X,pi)
=> ord_less(real,zero_zero(real),sin(X)) ) ) ).
tff(fact_11_mult__zero__left,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : times_times(A,zero_zero(A),A1) = zero_zero(A) ) ).
tff(fact_12_mult__zero__right,axiom,
! [A: $tType] :
( mult_zero(A)
=> ! [A1: A] : times_times(A,A1,zero_zero(A)) = zero_zero(A) ) ).
tff(fact_13_mult__eq__0__iff,axiom,
! [A: $tType] :
( ring_n68954251visors(A)
=> ! [B2: A,A2: A] :
( ( times_times(A,A2,B2) = zero_zero(A) )
<=> ( ( A2 = zero_zero(A) )
| ( B2 = zero_zero(A) ) ) ) ) ).
tff(fact_14_diff__self,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A1: A] : minus_minus(A,A1,A1) = zero_zero(A) ) ).
tff(fact_15_sin__two__pi,axiom,
sin(times_times(real,number_number_of(real,bit0(bit1(pls))),pi)) = zero_zero(real) ).
tff(fact_16_pi__less__4,axiom,
ord_less(real,pi,number_number_of(real,bit0(bit0(bit1(pls))))) ).
tff(fact_17_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
! [A: $tType] :
( ab_semigroup_mult(A)
=> ! [C: A,B1: A,A1: A] : times_times(A,times_times(A,A1,B1),C) = times_times(A,A1,times_times(A,B1,C)) ) ).
tff(fact_18_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y1: A,X: A] :
( ( X != Y1 )
=> ( ~ ord_less(A,X,Y1)
=> ord_less(A,Y1,X) ) ) ) ).
tff(fact_19_diff__eq__diff__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [D: A,C1: A,B2: A,A2: A] :
( ( minus_minus(A,A2,B2) = minus_minus(A,C1,D) )
=> ( ( A2 = B2 )
<=> ( C1 = D ) ) ) ) ).
tff(fact_20_diff__eq__diff__less,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [D: A,C1: A,B2: A,A2: A] :
( ( minus_minus(A,A2,B2) = minus_minus(A,C1,D) )
=> ( ord_less(A,A2,B2)
<=> ord_less(A,C1,D) ) ) ) ).
tff(fact_21_less__iff__diff__less__0,axiom,
! [A: $tType] :
( ordered_ab_group_add(A)
=> ! [B2: A,A2: A] :
( ord_less(A,A2,B2)
<=> ord_less(A,minus_minus(A,A2,B2),zero_zero(A)) ) ) ).
tff(fact_22_mult__strict__left__mono__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [C: A,A1: A,B1: A] :
( ord_less(A,B1,A1)
=> ( ord_less(A,C,zero_zero(A))
=> ord_less(A,times_times(A,C,A1),times_times(A,C,B1)) ) ) ) ).
tff(fact_23_mult__strict__right__mono__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [C: A,A1: A,B1: A] :
( ord_less(A,B1,A1)
=> ( ord_less(A,C,zero_zero(A))
=> ord_less(A,times_times(A,A1,C),times_times(A,B1,C)) ) ) ) ).
tff(fact_24_comm__mult__strict__left__mono,axiom,
! [A: $tType] :
( linord893533164strict(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ord_less(A,zero_zero(A),C)
=> ord_less(A,times_times(A,C,A1),times_times(A,C,B1)) ) ) ) ).
tff(fact_25_mult__strict__left__mono,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ord_less(A,zero_zero(A),C)
=> ord_less(A,times_times(A,C,A1),times_times(A,C,B1)) ) ) ) ).
tff(fact_26_mult__strict__right__mono,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less(A,A1,B1)
=> ( ord_less(A,zero_zero(A),C)
=> ord_less(A,times_times(A,A1,C),times_times(A,B1,C)) ) ) ) ).
tff(fact_27_mult__neg__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B1: A,A1: A] :
( ord_less(A,A1,zero_zero(A))
=> ( ord_less(A,B1,zero_zero(A))
=> ord_less(A,zero_zero(A),times_times(A,A1,B1)) ) ) ) ).
tff(fact_28_mult__neg__pos,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B1: A,A1: A] :
( ord_less(A,A1,zero_zero(A))
=> ( ord_less(A,zero_zero(A),B1)
=> ord_less(A,times_times(A,A1,B1),zero_zero(A)) ) ) ) ).
tff(fact_29_mult__less__cancel__left__neg,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,A2: A,C1: A] :
( ord_less(A,C1,zero_zero(A))
=> ( ord_less(A,times_times(A,C1,A2),times_times(A,C1,B2))
<=> ord_less(A,B2,A2) ) ) ) ).
tff(fact_30_zero__less__mult__pos2,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [A1: A,B1: A] :
( ord_less(A,zero_zero(A),times_times(A,B1,A1))
=> ( ord_less(A,zero_zero(A),A1)
=> ord_less(A,zero_zero(A),B1) ) ) ) ).
tff(fact_31_zero__less__mult__pos,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B1: A,A1: A] :
( ord_less(A,zero_zero(A),times_times(A,A1,B1))
=> ( ord_less(A,zero_zero(A),A1)
=> ord_less(A,zero_zero(A),B1) ) ) ) ).
tff(fact_32_mult__pos__neg2,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B1: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,B1,zero_zero(A))
=> ord_less(A,times_times(A,B1,A1),zero_zero(A)) ) ) ) ).
tff(fact_33_mult__pos__neg,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B1: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,B1,zero_zero(A))
=> ord_less(A,times_times(A,A1,B1),zero_zero(A)) ) ) ) ).
tff(fact_34_mult__pos__pos,axiom,
! [A: $tType] :
( linord20386208strict(A)
=> ! [B1: A,A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( ord_less(A,zero_zero(A),B1)
=> ord_less(A,zero_zero(A),times_times(A,A1,B1)) ) ) ) ).
tff(fact_35_mult__less__cancel__left__pos,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,A2: A,C1: A] :
( ord_less(A,zero_zero(A),C1)
=> ( ord_less(A,times_times(A,C1,A2),times_times(A,C1,B2))
<=> ord_less(A,A2,B2) ) ) ) ).
tff(fact_36_mult__less__cancel__left__disj,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,A2: A,C1: A] :
( ord_less(A,times_times(A,C1,A2),times_times(A,C1,B2))
<=> ( ( ord_less(A,zero_zero(A),C1)
& ord_less(A,A2,B2) )
| ( ord_less(A,C1,zero_zero(A))
& ord_less(A,B2,A2) ) ) ) ) ).
tff(fact_37_mult__less__cancel__right__disj,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [B2: A,C1: A,A2: A] :
( ord_less(A,times_times(A,A2,C1),times_times(A,B2,C1))
<=> ( ( ord_less(A,zero_zero(A),C1)
& ord_less(A,A2,B2) )
| ( ord_less(A,C1,zero_zero(A))
& ord_less(A,B2,A2) ) ) ) ) ).
tff(fact_38_not__square__less__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [A1: A] : ~ ord_less(A,times_times(A,A1,A1),zero_zero(A)) ) ).
tff(fact_39_pi__gt__zero,axiom,
ord_less(real,zero_zero(real),pi) ).
tff(fact_40_pi__not__less__zero,axiom,
~ ord_less(real,pi,zero_zero(real)) ).
tff(fact_41_sin__gt__zero,axiom,
! [X: real] :
( ord_less(real,zero_zero(real),X)
=> ( ord_less(real,X,number_number_of(real,bit0(bit1(pls))))
=> ord_less(real,zero_zero(real),sin(X)) ) ) ).
tff(fact_42_right__minus__eq,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A2: A] :
( ( minus_minus(A,A2,B2) = zero_zero(A) )
<=> ( A2 = B2 ) ) ) ).
tff(fact_43_eq__iff__diff__eq__0,axiom,
! [A: $tType] :
( group_add(A)
=> ! [B2: A,A2: A] :
( ( A2 = B2 )
<=> ( minus_minus(A,A2,B2) = zero_zero(A) ) ) ) ).
tff(fact_44_diff__0__right,axiom,
! [A: $tType] :
( group_add(A)
=> ! [A1: A] : minus_minus(A,A1,zero_zero(A)) = A1 ) ).
tff(fact_45_divisors__zero,axiom,
! [A: $tType] :
( no_zero_divisors(A)
=> ! [B1: A,A1: A] :
( ( times_times(A,A1,B1) = zero_zero(A) )
=> ( ( A1 = zero_zero(A) )
| ( B1 = zero_zero(A) ) ) ) ) ).
tff(fact_46_no__zero__divisors,axiom,
! [A: $tType] :
( no_zero_divisors(A)
=> ! [B1: A,A1: A] :
( ( A1 != zero_zero(A) )
=> ( ( B1 != zero_zero(A) )
=> ( times_times(A,A1,B1) != zero_zero(A) ) ) ) ) ).
tff(fact_47_pi__neq__zero,axiom,
pi != zero_zero(real) ).
tff(fact_48_less__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int] :
( ord_less(A,zero_zero(A),number_number_of(A,Y))
<=> ord_less(int,pls,Y) ) ) ).
tff(fact_49_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_50_not__real__square__gt__zero,axiom,
! [Xa: real] :
( ~ ord_less(real,zero_zero(real),times_times(real,Xa,Xa))
<=> ( Xa = zero_zero(real) ) ) ).
tff(fact_51_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_52_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_53_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_54_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_55_diff__bin__simps_I3_J,axiom,
! [L1: int] : minus_minus(int,pls,bit0(L1)) = bit0(minus_minus(int,pls,L1)) ).
tff(fact_56_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y: int,Xa: int] :
( ( number_number_of(A,Xa) = number_number_of(A,Y) )
<=> ( Xa = Y ) ) ) ).
tff(fact_57_rel__simps_I51_J,axiom,
! [L: int,K3: int] :
( ( bit1(K3) = bit1(L) )
<=> ( K3 = L ) ) ).
tff(fact_58_mult__Pls,axiom,
! [W: int] : times_times(int,pls,W) = pls ).
tff(fact_59_rel__simps_I48_J,axiom,
! [L: int,K3: int] :
( ( bit0(K3) = bit0(L) )
<=> ( K3 = L ) ) ).
tff(fact_60_mult__Bit0,axiom,
! [L1: int,K: int] : times_times(int,bit0(K),L1) = bit0(times_times(int,K,L1)) ).
tff(fact_61_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_62_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_63_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_64_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_65_rel__simps_I50_J,axiom,
! [L1: int,K: int] : bit1(K) != bit0(L1) ).
tff(fact_66_rel__simps_I49_J,axiom,
! [L1: int,K: int] : bit0(K) != bit1(L1) ).
tff(fact_67_rel__simps_I44_J,axiom,
! [K3: int] :
( ( bit0(K3) = pls )
<=> ( K3 = pls ) ) ).
tff(fact_68_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_69_rel__simps_I17_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit1(K3),bit1(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_70_rel__simps_I2_J,axiom,
~ ord_less(int,pls,pls) ).
tff(fact_71_rel__simps_I14_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit0(K3),bit0(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_72_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_73_less__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int,Xa: int] :
( ord_less(A,number_number_of(A,Xa),number_number_of(A,Y))
<=> ord_less(int,Xa,Y) ) ) ).
tff(fact_74_rel__simps_I12_J,axiom,
! [K3: int] :
( ord_less(int,bit1(K3),pls)
<=> ord_less(int,K3,pls) ) ).
tff(fact_75_rel__simps_I16_J,axiom,
! [L: int,K3: int] :
( ord_less(int,bit1(K3),bit0(L))
<=> ord_less(int,K3,L) ) ).
tff(fact_76_rel__simps_I10_J,axiom,
! [K3: int] :
( ord_less(int,bit0(K3),pls)
<=> ord_less(int,K3,pls) ) ).
tff(fact_77_rel__simps_I4_J,axiom,
! [K3: int] :
( ord_less(int,pls,bit0(K3))
<=> ord_less(int,pls,K3) ) ).
tff(fact_78_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_79_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_80_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_81_bin__less__0__simps_I1_J,axiom,
~ ord_less(int,pls,zero_zero(int)) ).
tff(fact_82_less__bin__lemma,axiom,
! [L: int,K3: int] :
( ord_less(int,K3,L)
<=> ord_less(int,minus_minus(int,K3,L),zero_zero(int)) ) ).
tff(fact_83_int__distrib_I4_J,axiom,
! [Z2: int,Z1: int,W: int] : times_times(int,W,minus_minus(int,Z1,Z2)) = minus_minus(int,times_times(int,W,Z1),times_times(int,W,Z2)) ).
tff(fact_84_bin__less__0__simps_I3_J,axiom,
! [W1: int] :
( ord_less(int,bit0(W1),zero_zero(int))
<=> ord_less(int,W1,zero_zero(int)) ) ).
tff(fact_85_bin__less__0__simps_I4_J,axiom,
! [W1: int] :
( ord_less(int,bit1(W1),zero_zero(int))
<=> ord_less(int,W1,zero_zero(int)) ) ).
tff(fact_86_less__int__code_I13_J,axiom,
! [K2: int,K1: int] :
( ord_less(int,bit0(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
tff(fact_87_less__int__code_I16_J,axiom,
! [K2: int,K1: int] :
( ord_less(int,bit1(K1),bit1(K2))
<=> ord_less(int,K1,K2) ) ).
tff(fact_88_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_89_int__distrib_I3_J,axiom,
! [W: int,Z2: int,Z1: int] : times_times(int,minus_minus(int,Z1,Z2),W) = minus_minus(int,times_times(int,Z1,W),times_times(int,Z2,W)) ).
tff(fact_90_zmult__zless__mono2,axiom,
! [K: int,J: int,I: int] :
( ord_less(int,I,J)
=> ( ord_less(int,zero_zero(int),K)
=> ord_less(int,times_times(int,K,I),times_times(int,K,J)) ) ) ).
tff(fact_91_pos2,axiom,
ord_less(nat,zero_zero(nat),number_number_of(nat,bit0(bit1(pls)))) ).
tff(fact_92_less__int__code_I15_J,axiom,
! [K2: int,K1: int] :
( ord_less(int,bit1(K1),bit0(K2))
<=> ord_less(int,K1,K2) ) ).
tff(fact_93_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [Xa: A,W1: int] :
( ( number_number_of(A,W1) = Xa )
<=> ( Xa = number_number_of(A,W1) ) ) ) ).
tff(fact_94_diff__bin__simps_I1_J,axiom,
! [K: int] : minus_minus(int,K,pls) = K ).
tff(fact_95_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_96_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)) ) ).
%----Arities (39)
tff(arity_Int_Oint___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(int) ).
tff(arity_Int_Oint___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(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___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(int) ).
tff(arity_Int_Oint___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(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___Groups_Ogroup__add,axiom,
group_add(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___Rings_Oring,axiom,
ring(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(nat) ).
tff(arity_Nat_Onat___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(nat) ).
tff(arity_Nat_Onat___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(nat) ).
tff(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(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) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__comm__semiring__strict,axiom,
linord893533164strict(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__semiring__strict,axiom,
linord20386208strict(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__ring__strict,axiom,
linord581940658strict(real) ).
tff(arity_RealDef_Oreal___Rings_Oring__no__zero__divisors,axiom,
ring_n68954251visors(real) ).
tff(arity_RealDef_Oreal___Groups_Oordered__ab__group__add,axiom,
ordered_ab_group_add(real) ).
tff(arity_RealDef_Oreal___Groups_Oab__semigroup__mult,axiom,
ab_semigroup_mult(real) ).
tff(arity_RealDef_Oreal___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__ring,axiom,
linordered_ring(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
tff(arity_RealDef_Oreal___Groups_Ogroup__add,axiom,
group_add(real) ).
tff(arity_RealDef_Oreal___Rings_Omult__zero,axiom,
mult_zero(real) ).
tff(arity_RealDef_Oreal___Int_Oring__char__0,axiom,
ring_char_0(real) ).
tff(arity_RealDef_Oreal___Int_Onumber__ring,axiom,
number_ring(real) ).
tff(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
tff(arity_RealDef_Oreal___Rings_Oring,axiom,
ring(real) ).
tff(arity_RealDef_Oreal___Int_Onumber,axiom,
number(real) ).
%----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,
sin(x) != zero_zero(real) ).
%------------------------------------------------------------------------------