TPTP Problem File: SWV665_5.p
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%------------------------------------------------------------------------------
% File : SWV665_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fast Fourier Transform line 376
% 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_376 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 198 ( 89 unt; 52 typ; 0 def)
% Number of atoms : 313 ( 105 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 196 ( 29 ~; 4 |; 48 &)
% ( 33 <=>; 82 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 3 avg)
% Maximal term depth : 13 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 27 ( 17 >; 10 *; 0 +; 0 <<)
% Number of predicates : 24 ( 23 usr; 0 prp; 1-3 aty)
% Number of functors : 25 ( 25 usr; 7 con; 0-5 aty)
% Number of variables : 252 ( 208 !; 0 ?; 252 :)
% ( 44 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:21:12
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_tc_Complex_Ocomplex,type,
complex: $tType ).
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_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (47)
tff(sy_cl_Int_Onumber,type,
number:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring,type,
ring:
!>[A: $tType] : $o ).
tff(sy_cl_Power_Opower,type,
power:
!>[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_Omult__zero,type,
mult_zero:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__0,type,
semiring_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_Rings_Ozero__neq__one,type,
zero_neq_one:
!>[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_Fields_Ofield__inverse__zero,type,
field_inverse_zero:
!>[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_cl_Fields_Olinordered__field__inverse__zero,type,
linord1117847801e_zero:
!>[A: $tType] : $o ).
tff(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum,type,
big_co1399186613setsum:
!>[B: $tType,A: $tType] : ( ( fun(B,A) * fun(B,bool) ) > A ) ).
tff(sy_c_COMBB,type,
combb:
!>[B: $tType,C: $tType,A: $tType] : ( ( fun(B,C) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_COMBC,type,
combc:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * B ) > fun(A,C) ) ).
tff(sy_c_COMBS,type,
combs:
!>[A: $tType,B: $tType,C: $tType] : ( ( fun(A,fun(B,C)) * fun(A,B) ) > fun(A,C) ) ).
tff(sy_c_FFT__Mirabelle__uutasbvzez_Oroot,type,
fFT_Mirabelle_root: nat > complex ).
tff(sy_c_Fields_Oinverse__class_Odivide,type,
inverse_divide:
!>[A: $tType] : fun(A,fun(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] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : ( A > fun(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_OSuc,type,
suc: fun(nat,nat) ).
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 > fun(nat,A) ) ).
tff(sy_c_SetInterval_Oord__class_OatLeastLessThan,type,
ord_atLeastLessThan:
!>[A: $tType] : ( ( A * A ) > fun(A,bool) ) ).
tff(sy_c_aa,type,
aa:
!>[A: $tType,B: $tType] : ( ( fun(A,B) * A ) > B ) ).
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_a,type,
a: fun(nat,complex) ).
tff(sy_v_i,type,
i: nat ).
tff(sy_v_m,type,
m: nat ).
%----Relevant facts (97)
tff(fact_0_mbound,axiom,
ord_less(nat,zero_zero(nat),m) ).
tff(fact_1_root0,axiom,
fFT_Mirabelle_root(zero_zero(nat)) = one_one(complex) ).
tff(fact_2_ibound,axiom,
ord_less_eq(nat,m,i) ).
tff(fact_3_root__unity,axiom,
! [N: nat] : aa(nat,complex,power_power(complex,fFT_Mirabelle_root(N)),N) = one_one(complex) ).
tff(fact_4_root__cancel1,axiom,
! [J: nat,I: nat,M: nat] : aa(nat,complex,power_power(complex,fFT_Mirabelle_root(aa(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),M))),aa(nat,nat,times_times(nat,I),aa(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),J))) = aa(nat,complex,power_power(complex,fFT_Mirabelle_root(M)),aa(nat,nat,times_times(nat,I),J)) ).
tff(fact_5_calculation,axiom,
big_co1399186613setsum(nat,complex,combs(nat,complex,complex,combb(complex,fun(complex,complex),nat,inverse_divide(complex),a),combb(nat,complex,nat,power_power(complex,fFT_Mirabelle_root(aa(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),m))),times_times(nat,i))),ord_atLeastLessThan(nat,zero_zero(nat),aa(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),m))) = aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),big_co1399186613setsum(nat,complex,combs(nat,complex,complex,combb(complex,fun(complex,complex),nat,inverse_divide(complex),combb(nat,complex,nat,a,times_times(nat,number_number_of(nat,bit0(bit1(pls)))))),combb(nat,complex,nat,power_power(complex,fFT_Mirabelle_root(aa(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),m))),combb(nat,nat,nat,times_times(nat,i),times_times(nat,number_number_of(nat,bit0(bit1(pls))))))),ord_atLeastLessThan(nat,zero_zero(nat),m))),big_co1399186613setsum(nat,complex,combs(nat,complex,complex,combb(complex,fun(complex,complex),nat,inverse_divide(complex),combb(nat,complex,nat,a,combc(nat,nat,nat,combb(nat,fun(nat,nat),nat,plus_plus(nat),times_times(nat,number_number_of(nat,bit0(bit1(pls))))),one_one(nat)))),combb(nat,complex,nat,power_power(complex,fFT_Mirabelle_root(aa(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),m))),combb(nat,nat,nat,times_times(nat,i),combc(nat,nat,nat,combb(nat,fun(nat,nat),nat,plus_plus(nat),times_times(nat,number_number_of(nat,bit0(bit1(pls))))),one_one(nat))))),ord_atLeastLessThan(nat,zero_zero(nat),m))) ).
tff(fact_6_power2__eq__square__number__of,axiom,
! [B: $tType] :
( ( monoid_mult(B)
& number(B) )
=> ! [W1: int] : aa(nat,B,power_power(B,number_number_of(B,W1)),number_number_of(nat,bit0(bit1(pls)))) = aa(B,B,times_times(B,number_number_of(B,W1)),number_number_of(B,W1)) ) ).
tff(fact_7_add__2__eq__Suc_H,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),number_number_of(nat,bit0(bit1(pls)))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).
tff(fact_8_add__2__eq__Suc,axiom,
! [N: nat] : aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),number_number_of(nat,bit0(bit1(pls)))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).
tff(fact_9_one__power2,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( aa(nat,A,power_power(A,one_one(A)),number_number_of(nat,bit0(bit1(pls)))) = one_one(A) ) ) ).
tff(fact_10_zero__power2,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( aa(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) ) ) ).
tff(fact_11_zero__eq__power2,axiom,
! [A: $tType] :
( ring_11004092258visors(A)
=> ! [Aa: A] :
( ( aa(nat,A,power_power(A,Aa),number_number_of(nat,bit0(bit1(pls)))) = zero_zero(A) )
<=> ( Aa = zero_zero(A) ) ) ) ).
tff(fact_12_one__add__one__is__two,axiom,
! [A: $tType] :
( number_ring(A)
=> ( aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),one_one(A)) = number_number_of(A,bit0(bit1(pls))) ) ) ).
tff(fact_13_power__number__of__even__number__of,axiom,
! [B: $tType] :
( ( monoid_mult(B)
& number(B) )
=> ! [W1: int,V: int] : aa(nat,B,power_power(B,number_number_of(B,V)),number_number_of(nat,bit0(W1))) = aa(B,B,times_times(B,aa(nat,B,power_power(B,number_number_of(B,V)),number_number_of(nat,W1))),aa(nat,B,power_power(B,number_number_of(B,V)),number_number_of(nat,W1))) ) ).
tff(fact_14_power__0__left__number__of,axiom,
! [A: $tType] :
( ( power(A)
& semiring_0(A) )
=> ! [W1: int] :
( ( ( number_number_of(nat,W1) = zero_zero(nat) )
=> ( aa(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,W1)) = one_one(A) ) )
& ( ( number_number_of(nat,W1) != zero_zero(nat) )
=> ( aa(nat,A,power_power(A,zero_zero(A)),number_number_of(nat,W1)) = zero_zero(A) ) ) ) ) ).
tff(fact_15_add__special_I2_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int] : aa(A,A,aa(A,fun(A,A),plus_plus(A),one_one(A)),number_number_of(A,W1)) = number_number_of(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),bit1(pls)),W1)) ) ).
tff(fact_16_add__special_I3_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [V: int] : aa(A,A,aa(A,fun(A,A),plus_plus(A),number_number_of(A,V)),one_one(A)) = number_number_of(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),V),bit1(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,
! [L1: int,K1: int] :
( ( bit1(K1) = bit1(L1) )
<=> ( K1 = L1 ) ) ).
tff(fact_19_mult__Pls,axiom,
! [W1: int] : aa(int,int,times_times(int,pls),W1) = pls ).
tff(fact_20_mult__Bit0,axiom,
! [L: int,K: int] : aa(int,int,times_times(int,bit0(K)),L) = bit0(aa(int,int,times_times(int,K),L)) ).
tff(fact_21_rel__simps_I48_J,axiom,
! [L1: int,K1: int] :
( ( bit0(K1) = bit0(L1) )
<=> ( K1 = L1 ) ) ).
tff(fact_22_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [Aa: A] :
( ( aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa),Aa) = zero_zero(A) )
<=> ( Aa = zero_zero(A) ) ) ) ).
tff(fact_23_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_24_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_25_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W1: int,V: int] : aa(A,A,times_times(A,number_number_of(A,V)),aa(A,A,times_times(A,number_number_of(A,W1)),Z)) = aa(A,A,times_times(A,number_number_of(A,aa(int,int,times_times(int,V),W1))),Z) ) ).
tff(fact_26_arith__simps_I32_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : aa(A,A,times_times(A,number_number_of(A,V)),number_number_of(A,W1)) = number_number_of(A,aa(int,int,times_times(int,V),W1)) ) ).
tff(fact_27_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_28_rel__simps_I39_J,axiom,
! [L: int] : pls != bit1(L) ).
tff(fact_29_rel__simps_I50_J,axiom,
! [L: int,K: int] : bit1(K) != bit0(L) ).
tff(fact_30_rel__simps_I49_J,axiom,
! [L: int,K: int] : bit0(K) != bit1(L) ).
tff(fact_31_rel__simps_I44_J,axiom,
! [K1: int] :
( ( bit0(K1) = pls )
<=> ( K1 = pls ) ) ).
tff(fact_32_rel__simps_I38_J,axiom,
! [L1: int] :
( ( pls = bit0(L1) )
<=> ( pls = L1 ) ) ).
tff(fact_33_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_34_add__Bit0__Bit0,axiom,
! [L: int,K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),bit0(K)),bit0(L)) = bit0(aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).
tff(fact_35_left__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& semiring(B) )
=> ! [V: int,B2: B,A1: B] : aa(B,B,times_times(B,aa(B,B,aa(B,fun(B,B),plus_plus(B),A1),B2)),number_number_of(B,V)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,times_times(B,A1),number_number_of(B,V))),aa(B,B,times_times(B,B2),number_number_of(B,V))) ) ).
tff(fact_36_right__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& semiring(B) )
=> ! [C2: B,B2: B,V: int] : aa(B,B,times_times(B,number_number_of(B,V)),aa(B,B,aa(B,fun(B,B),plus_plus(B),B2),C2)) = aa(B,B,aa(B,fun(B,B),plus_plus(B),aa(B,B,times_times(B,number_number_of(B,V)),B2)),aa(B,B,times_times(B,number_number_of(B,V)),C2)) ) ).
tff(fact_37_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_38_left__diff__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& ring(B) )
=> ! [V: int,B2: B,A1: B] : aa(B,B,times_times(B,minus_minus(B,A1,B2)),number_number_of(B,V)) = minus_minus(B,aa(B,B,times_times(B,A1),number_number_of(B,V)),aa(B,B,times_times(B,B2),number_number_of(B,V))) ) ).
tff(fact_39_right__diff__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& ring(B) )
=> ! [C2: B,B2: B,V: int] : aa(B,B,times_times(B,number_number_of(B,V)),minus_minus(B,B2,C2)) = minus_minus(B,aa(B,B,times_times(B,number_number_of(B,V)),B2),aa(B,B,times_times(B,number_number_of(B,V)),C2)) ) ).
tff(fact_40_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W1: int,V: int] : aa(A,A,aa(A,fun(A,A),plus_plus(A),number_number_of(A,V)),aa(A,A,aa(A,fun(A,A),plus_plus(A),number_number_of(A,W1)),Z)) = aa(A,A,aa(A,fun(A,A),plus_plus(A),number_number_of(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),V),W1))),Z) ) ).
tff(fact_41_add__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W1: int,V: int] : aa(A,A,aa(A,fun(A,A),plus_plus(A),number_number_of(A,V)),number_number_of(A,W1)) = number_number_of(A,aa(int,int,aa(int,fun(int,int),plus_plus(int),V),W1)) ) ).
tff(fact_42_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_43_zero__less__power__nat__eq__number__of,axiom,
! [W: int,X1: nat] :
( ord_less(nat,zero_zero(nat),aa(nat,nat,power_power(nat,X1),number_number_of(nat,W)))
<=> ( ( number_number_of(nat,W) = zero_zero(nat) )
| ord_less(nat,zero_zero(nat),X1) ) ) ).
tff(fact_44_mult__Bit1,axiom,
! [L: int,K: int] : aa(int,int,times_times(int,bit1(K)),L) = aa(int,int,aa(int,fun(int,int),plus_plus(int),bit0(aa(int,int,times_times(int,K),L))),L) ).
tff(fact_45_add__Bit1__Bit0,axiom,
! [L: int,K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),bit1(K)),bit0(L)) = bit1(aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).
tff(fact_46_add__Bit0__Bit1,axiom,
! [L: int,K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),bit0(K)),bit1(L)) = bit1(aa(int,int,aa(int,fun(int,int),plus_plus(int),K),L)) ).
tff(fact_47_root1,axiom,
fFT_Mirabelle_root(one_one(nat)) = one_one(complex) ).
tff(fact_48_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_49_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_50_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_51_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_52_divide__eq__eq__number__of1,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [Aa: A,W: int,B1: A] :
( ( aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),number_number_of(A,W)) = Aa )
<=> ( ( ( number_number_of(A,W) != zero_zero(A) )
=> ( B1 = aa(A,A,times_times(A,Aa),number_number_of(A,W)) ) )
& ( ( number_number_of(A,W) = zero_zero(A) )
=> ( Aa = zero_zero(A) ) ) ) ) ) ).
tff(fact_53_eq__divide__eq__number__of1,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [W: int,B1: A,Aa: A] :
( ( Aa = aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),number_number_of(A,W)) )
<=> ( ( ( number_number_of(A,W) != zero_zero(A) )
=> ( aa(A,A,times_times(A,Aa),number_number_of(A,W)) = B1 ) )
& ( ( number_number_of(A,W) = zero_zero(A) )
=> ( Aa = zero_zero(A) ) ) ) ) ) ).
tff(fact_54_numeral__1__eq__1,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,bit1(pls)) = one_one(A) ) ) ).
tff(fact_55_power__eq__0__iff__number__of,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [W: int,Aa: A] :
( ( aa(nat,A,power_power(A,Aa),number_number_of(nat,W)) = zero_zero(A) )
<=> ( ( Aa = zero_zero(A) )
& ( number_number_of(nat,W) != zero_zero(nat) ) ) ) ) ).
tff(fact_56_nat__numeral__1__eq__1,axiom,
number_number_of(nat,bit1(pls)) = one_one(nat) ).
tff(fact_57_divide__less__eq__number__of1,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [Aa: A,W: int,B1: A] :
( ord_less(A,aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),number_number_of(A,W)),Aa)
<=> ( ( ord_less(A,zero_zero(A),number_number_of(A,W))
=> ord_less(A,B1,aa(A,A,times_times(A,Aa),number_number_of(A,W))) )
& ( ~ ord_less(A,zero_zero(A),number_number_of(A,W))
=> ( ( ord_less(A,number_number_of(A,W),zero_zero(A))
=> ord_less(A,aa(A,A,times_times(A,Aa),number_number_of(A,W)),B1) )
& ( ~ ord_less(A,number_number_of(A,W),zero_zero(A))
=> ord_less(A,zero_zero(A),Aa) ) ) ) ) ) ) ).
tff(fact_58_less__divide__eq__number__of1,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [W: int,B1: A,Aa: A] :
( ord_less(A,Aa,aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),number_number_of(A,W)))
<=> ( ( ord_less(A,zero_zero(A),number_number_of(A,W))
=> ord_less(A,aa(A,A,times_times(A,Aa),number_number_of(A,W)),B1) )
& ( ~ ord_less(A,zero_zero(A),number_number_of(A,W))
=> ( ( ord_less(A,number_number_of(A,W),zero_zero(A))
=> ord_less(A,B1,aa(A,A,times_times(A,Aa),number_number_of(A,W))) )
& ( ~ ord_less(A,number_number_of(A,W),zero_zero(A))
=> ord_less(A,Aa,zero_zero(A)) ) ) ) ) ) ) ).
tff(fact_59_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_60_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_61_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_62_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_63_divide__le__eq__number__of1,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [Aa: A,W: int,B1: A] :
( ord_less_eq(A,aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),number_number_of(A,W)),Aa)
<=> ( ( ord_less(A,zero_zero(A),number_number_of(A,W))
=> ord_less_eq(A,B1,aa(A,A,times_times(A,Aa),number_number_of(A,W))) )
& ( ~ ord_less(A,zero_zero(A),number_number_of(A,W))
=> ( ( ord_less(A,number_number_of(A,W),zero_zero(A))
=> ord_less_eq(A,aa(A,A,times_times(A,Aa),number_number_of(A,W)),B1) )
& ( ~ ord_less(A,number_number_of(A,W),zero_zero(A))
=> ord_less_eq(A,zero_zero(A),Aa) ) ) ) ) ) ) ).
tff(fact_64_le__divide__eq__number__of1,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [W: int,B1: A,Aa: A] :
( ord_less_eq(A,Aa,aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),number_number_of(A,W)))
<=> ( ( ord_less(A,zero_zero(A),number_number_of(A,W))
=> ord_less_eq(A,aa(A,A,times_times(A,Aa),number_number_of(A,W)),B1) )
& ( ~ ord_less(A,zero_zero(A),number_number_of(A,W))
=> ( ( ord_less(A,number_number_of(A,W),zero_zero(A))
=> ord_less_eq(A,B1,aa(A,A,times_times(A,Aa),number_number_of(A,W))) )
& ( ~ ord_less(A,number_number_of(A,W),zero_zero(A))
=> ord_less_eq(A,Aa,zero_zero(A)) ) ) ) ) ) ) ).
tff(fact_65_nat__1__add__1,axiom,
aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),one_one(nat)) = number_number_of(nat,bit0(bit1(pls))) ).
tff(fact_66_zero__less__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ord_less(A,zero_zero(A),aa(nat,A,power_power(A,Aa),number_number_of(nat,bit0(bit1(pls)))))
<=> ( Aa != zero_zero(A) ) ) ) ).
tff(fact_67_odd__nonzero,axiom,
! [Z: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),Z)),Z) != zero_zero(int) ).
tff(fact_68_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_69_Bit1__def,axiom,
! [K: int] : bit1(K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,aa(int,fun(int,int),plus_plus(int),one_one(int)),K)),K) ).
tff(fact_70_zero__less__power__nat__eq,axiom,
! [N1: nat,X1: nat] :
( ord_less(nat,zero_zero(nat),aa(nat,nat,power_power(nat,X1),N1))
<=> ( ( N1 = zero_zero(nat) )
| ord_less(nat,zero_zero(nat),X1) ) ) ).
tff(fact_71_int__distrib_I2_J,axiom,
! [Z2: int,Z1: int,W1: int] : aa(int,int,times_times(int,W1),aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z2)) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,times_times(int,W1),Z1)),aa(int,int,times_times(int,W1),Z2)) ).
tff(fact_72_plus__numeral__code_I9_J,axiom,
! [W1: int,V: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),number_number_of(int,V)),number_number_of(int,W1)) = number_number_of(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),V),W1)) ).
tff(fact_73_le__number__of__eq__not__less,axiom,
! [A: $tType] :
( ( number(A)
& linorder(A) )
=> ! [W: int,V1: int] :
( ord_less_eq(A,number_number_of(A,V1),number_number_of(A,W))
<=> ~ ord_less(A,number_number_of(A,W),number_number_of(A,V1)) ) ) ).
tff(fact_74_int__distrib_I1_J,axiom,
! [W1: int,Z2: int,Z1: int] : aa(int,int,times_times(int,aa(int,int,aa(int,fun(int,int),plus_plus(int),Z1),Z2)),W1) = aa(int,int,aa(int,fun(int,int),plus_plus(int),aa(int,int,times_times(int,Z1),W1)),aa(int,int,times_times(int,Z2),W1)) ).
tff(fact_75_add__Pls__right,axiom,
! [K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),K),pls) = K ).
tff(fact_76_add__Pls,axiom,
! [K: int] : aa(int,int,aa(int,fun(int,int),plus_plus(int),pls),K) = K ).
tff(fact_77_Bit0__def,axiom,
! [K: int] : bit0(K) = aa(int,int,aa(int,fun(int,int),plus_plus(int),K),K) ).
tff(fact_78_zpower__zpower,axiom,
! [Z: nat,Y: nat,X: int] : aa(nat,int,power_power(int,aa(nat,int,power_power(int,X),Y)),Z) = aa(nat,int,power_power(int,X),aa(nat,nat,times_times(nat,Y),Z)) ).
tff(fact_79_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_80_one__is__num__one,axiom,
one_one(int) = number_number_of(int,bit1(pls)) ).
tff(fact_81_zpower__number__of__even,axiom,
! [W1: int,Z: int] : aa(nat,int,power_power(int,Z),number_number_of(nat,bit0(W1))) = aa(int,int,times_times(int,aa(nat,int,power_power(int,Z),number_number_of(nat,W1))),aa(nat,int,power_power(int,Z),number_number_of(nat,W1))) ).
tff(fact_82_power__eq__if,axiom,
! [P1: nat,M: nat] :
( ( ( M = zero_zero(nat) )
=> ( aa(nat,nat,power_power(nat,P1),M) = one_one(nat) ) )
& ( ( M != zero_zero(nat) )
=> ( aa(nat,nat,power_power(nat,P1),M) = aa(nat,nat,times_times(nat,P1),aa(nat,nat,power_power(nat,P1),minus_minus(nat,M,one_one(nat)))) ) ) ) ).
tff(fact_83_Suc__pred_H,axiom,
! [N: nat] :
( ord_less(nat,zero_zero(nat),N)
=> ( N = aa(nat,nat,suc,minus_minus(nat,N,one_one(nat))) ) ) ).
tff(fact_84_even__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),Aa),Aa),zero_zero(A))
<=> ord_less(A,Aa,zero_zero(A)) ) ) ).
tff(fact_85_nat__lt__two__imp__zero__or__one,axiom,
! [X: nat] :
( ord_less(nat,X,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))
=> ( ( X = zero_zero(nat) )
| ( X = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
tff(fact_86_Suc__eq__plus1,axiom,
! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),N),one_one(nat)) ).
tff(fact_87_Suc__eq__plus1__left,axiom,
! [N: nat] : aa(nat,nat,suc,N) = aa(nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),one_one(nat)),N) ).
tff(fact_88_diff__Suc__eq__diff__pred,axiom,
! [N: nat,M: nat] : minus_minus(nat,M,aa(nat,nat,suc,N)) = minus_minus(nat,minus_minus(nat,M,one_one(nat)),N) ).
tff(fact_89_divide__le__eq__number__of,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [W: int,C1: A,B1: A] :
( ord_less_eq(A,aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),C1),number_number_of(A,W))
<=> ( ( ord_less(A,zero_zero(A),C1)
=> ord_less_eq(A,B1,aa(A,A,times_times(A,number_number_of(A,W)),C1)) )
& ( ~ ord_less(A,zero_zero(A),C1)
=> ( ( ord_less(A,C1,zero_zero(A))
=> ord_less_eq(A,aa(A,A,times_times(A,number_number_of(A,W)),C1),B1) )
& ( ~ ord_less(A,C1,zero_zero(A))
=> ord_less_eq(A,zero_zero(A),number_number_of(A,W)) ) ) ) ) ) ) ).
tff(fact_90_le__divide__eq__number__of,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [C1: A,B1: A,W: int] :
( ord_less_eq(A,number_number_of(A,W),aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),C1))
<=> ( ( ord_less(A,zero_zero(A),C1)
=> ord_less_eq(A,aa(A,A,times_times(A,number_number_of(A,W)),C1),B1) )
& ( ~ ord_less(A,zero_zero(A),C1)
=> ( ( ord_less(A,C1,zero_zero(A))
=> ord_less_eq(A,B1,aa(A,A,times_times(A,number_number_of(A,W)),C1)) )
& ( ~ ord_less(A,C1,zero_zero(A))
=> ord_less_eq(A,number_number_of(A,W),zero_zero(A)) ) ) ) ) ) ) ).
tff(fact_91_root__nonzero,axiom,
! [N: nat] : fFT_Mirabelle_root(N) != zero_zero(complex) ).
tff(fact_92_expand__Suc,axiom,
! [V: int] :
( ord_less(nat,zero_zero(nat),number_number_of(nat,V))
=> ( number_number_of(nat,V) = aa(nat,nat,suc,minus_minus(nat,number_number_of(nat,V),one_one(nat))) ) ) ).
tff(fact_93_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,X1),X1)),aa(A,A,times_times(A,Y1),Y1)))
<=> ( ( X1 != zero_zero(A) )
| ( Y1 != zero_zero(A) ) ) ) ) ).
tff(fact_94_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [Y: A,X: A] : ~ ord_less(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,X),X)),aa(A,A,times_times(A,Y),Y)),zero_zero(A)) ) ).
tff(fact_95_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,X1),X1)),aa(A,A,times_times(A,Y1),Y1)),zero_zero(A))
<=> ( ( X1 = zero_zero(A) )
& ( Y1 = zero_zero(A) ) ) ) ) ).
tff(fact_96_sum__squares__ge__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [Y: A,X: A] : ord_less_eq(A,zero_zero(A),aa(A,A,aa(A,fun(A,A),plus_plus(A),aa(A,A,times_times(A,X),X)),aa(A,A,times_times(A,Y),Y))) ) ).
%----Arities (43)
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_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___Rings_Ozero__neq__one,axiom,
zero_neq_one(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___Rings_Osemiring__0,axiom,
semiring_0(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___Rings_Osemiring,axiom,
semiring(int) ).
tff(arity_Int_Oint___Power_Opower,axiom,
power(int) ).
tff(arity_Int_Oint___Rings_Oring,axiom,
ring(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(nat) ).
tff(arity_Nat_Onat___Rings_Ozero__neq__one,axiom,
zero_neq_one(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__0,axiom,
semiring_0(nat) ).
tff(arity_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring,axiom,
semiring(nat) ).
tff(arity_Nat_Onat___Power_Opower,axiom,
power(nat) ).
tff(arity_Nat_Onat___Int_Onumber,axiom,
number(nat) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(complex) ).
tff(arity_Complex_Ocomplex___Fields_Ofield__inverse__zero,axiom,
field_inverse_zero(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ozero__neq__one,axiom,
zero_neq_one(complex) ).
tff(arity_Complex_Ocomplex___Groups_Omonoid__mult,axiom,
monoid_mult(complex) ).
tff(arity_Complex_Ocomplex___Rings_Osemiring__1,axiom,
semiring_1(complex) ).
tff(arity_Complex_Ocomplex___Rings_Osemiring__0,axiom,
semiring_0(complex) ).
tff(arity_Complex_Ocomplex___Rings_Omult__zero,axiom,
mult_zero(complex) ).
tff(arity_Complex_Ocomplex___Int_Oring__char__0,axiom,
ring_char_0(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber__ring,axiom,
number_ring(complex) ).
tff(arity_Complex_Ocomplex___Rings_Osemiring,axiom,
semiring(complex) ).
tff(arity_Complex_Ocomplex___Power_Opower,axiom,
power(complex) ).
tff(arity_Complex_Ocomplex___Rings_Oring,axiom,
ring(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber,axiom,
number(complex) ).
%----Helper facts (5)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
tff(help_COMBB_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(B,C)] : aa(A,C,combb(B,C,A,P,Q),R) = aa(B,C,P,aa(A,B,Q,R)) ).
tff(help_COMBC_1_1_U,axiom,
! [A: $tType,C: $tType,B: $tType,R: A,Q: B,P: fun(A,fun(B,C))] : aa(A,C,combc(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBS_1_1_U,axiom,
! [C: $tType,B: $tType,A: $tType,R: A,Q: fun(A,B),P: fun(A,fun(B,C))] : aa(A,C,combs(A,B,C,P,Q),R) = aa(B,C,aa(A,fun(B,C),P,R),aa(A,B,Q,R)) ).
%----Conjectures (1)
tff(conj_0,conjecture,
aa(complex,complex,aa(complex,fun(complex,complex),plus_plus(complex),big_co1399186613setsum(nat,complex,combs(nat,complex,complex,combb(complex,fun(complex,complex),nat,inverse_divide(complex),combb(nat,complex,nat,a,times_times(nat,number_number_of(nat,bit0(bit1(pls)))))),combb(nat,complex,nat,power_power(complex,fFT_Mirabelle_root(m)),times_times(nat,i))),ord_atLeastLessThan(nat,zero_zero(nat),m))),big_co1399186613setsum(nat,complex,combs(nat,complex,complex,combb(complex,fun(complex,complex),nat,inverse_divide(complex),combb(nat,complex,nat,a,combb(nat,nat,nat,suc,times_times(nat,number_number_of(nat,bit0(bit1(pls))))))),combb(nat,complex,nat,power_power(complex,fFT_Mirabelle_root(aa(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),m))),combb(nat,nat,nat,aa(nat,fun(nat,nat),plus_plus(nat),i),combb(nat,nat,nat,times_times(nat,i),times_times(nat,number_number_of(nat,bit0(bit1(pls)))))))),ord_atLeastLessThan(nat,zero_zero(nat),m))) = minus_minus(complex,big_co1399186613setsum(nat,complex,combs(nat,complex,complex,combb(complex,fun(complex,complex),nat,inverse_divide(complex),combb(nat,complex,nat,a,times_times(nat,number_number_of(nat,bit0(bit1(pls)))))),combb(nat,complex,nat,power_power(complex,fFT_Mirabelle_root(m)),times_times(nat,minus_minus(nat,i,m)))),ord_atLeastLessThan(nat,zero_zero(nat),m)),aa(complex,complex,times_times(complex,aa(nat,complex,power_power(complex,aa(complex,complex,aa(complex,fun(complex,complex),inverse_divide(complex),one_one(complex)),fFT_Mirabelle_root(aa(nat,nat,times_times(nat,number_number_of(nat,bit0(bit1(pls)))),m)))),minus_minus(nat,i,m))),big_co1399186613setsum(nat,complex,combs(nat,complex,complex,combb(complex,fun(complex,complex),nat,inverse_divide(complex),combb(nat,complex,nat,a,combb(nat,nat,nat,suc,times_times(nat,number_number_of(nat,bit0(bit1(pls))))))),combb(nat,complex,nat,power_power(complex,fFT_Mirabelle_root(m)),times_times(nat,minus_minus(nat,i,m)))),ord_atLeastLessThan(nat,zero_zero(nat),m)))) ).
%------------------------------------------------------------------------------