TPTP Problem File: SWV649_5.p
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%------------------------------------------------------------------------------
% File : SWV649_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fast Fourier Transform line 322
% 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_322 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 199 ( 72 unt; 52 typ; 0 def)
% Number of atoms : 355 ( 96 equ)
% Maximal formula atoms : 10 ( 1 avg)
% Number of connectives : 242 ( 34 ~; 6 |; 49 &)
% ( 36 <=>; 117 =>; 0 <=; 0 <~>)
% Maximal formula depth : 12 ( 4 avg)
% Maximal term depth : 14 ( 2 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 27 ( 17 >; 10 *; 0 +; 0 <<)
% Number of predicates : 25 ( 24 usr; 0 prp; 1-3 aty)
% Number of functors : 24 ( 24 usr; 7 con; 0-5 aty)
% Number of variables : 277 ( 236 !; 0 ?; 277 :)
% ( 41 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:19:19
%------------------------------------------------------------------------------
%----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_Power_Opower,type,
power:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Oring__1,type,
ring_1:
!>[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__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_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_Ono__zero__divisors,type,
no_zero_divisors:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[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_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_OIDFT,type,
fFT_Mirabelle_IDFT: ( nat * fun(nat,complex) * nat ) > complex ).
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_Oplus__class_Oplus,type,
plus_plus:
!>[A: $tType] : ( A > fun(A,A) ) ).
tff(sy_c_Groups_Otimes__class_Otimes,type,
times_times:
!>[A: $tType] : fun(A,fun(A,A)) ).
tff(sy_c_Groups_Ouminus__class_Ouminus,type,
uminus_uminus:
!>[A: $tType] : ( 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_ibound,axiom,
ord_less_eq(nat,m,i) ).
tff(fact_2_root__cancel1,axiom,
! [J: nat,I: nat,M: nat] : aa(nat,complex,power_power(complex,fFT_Mirabelle_root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))),M))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),I),aa(nat,nat,aa(nat,fun(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,aa(nat,fun(nat,nat),times_times(nat),I),J)) ).
tff(fact_3_ring__1__class_Opower__minus__even,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [N: nat,A1: A] : aa(nat,A,power_power(A,uminus_uminus(A,A1)),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))),N)) = aa(nat,A,power_power(A,A1),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))),N)) ) ).
tff(fact_4_power2__eq__square__number__of,axiom,
! [B: $tType] :
( ( monoid_mult(B)
& number(B) )
=> ! [W: int] : aa(nat,B,power_power(B,number_number_of(B,W)),number_number_of(nat,bit0(bit1(pls)))) = aa(B,B,aa(B,fun(B,B),times_times(B),number_number_of(B,W)),number_number_of(B,W)) ) ).
tff(fact_5_add__2__eq__Suc_H,axiom,
! [N: nat] : aa(nat,nat,plus_plus(nat,N),number_number_of(nat,bit0(bit1(pls)))) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).
tff(fact_6_add__2__eq__Suc,axiom,
! [N: nat] : aa(nat,nat,plus_plus(nat,number_number_of(nat,bit0(bit1(pls)))),N) = aa(nat,nat,suc,aa(nat,nat,suc,N)) ).
tff(fact_7_power2__minus,axiom,
! [A: $tType] :
( ring_1(A)
=> ! [A1: A] : aa(nat,A,power_power(A,uminus_uminus(A,A1)),number_number_of(nat,bit0(bit1(pls)))) = aa(nat,A,power_power(A,A1),number_number_of(nat,bit0(bit1(pls)))) ) ).
tff(fact_8_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_9_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_10_IDFT__def,axiom,
! [Aa: fun(nat,complex),N1: nat,X2: nat] : fFT_Mirabelle_IDFT(N1,Aa,X2) = big_co1399186613setsum(nat,complex,combs(nat,complex,complex,combb(complex,fun(complex,complex),nat,inverse_divide(complex),Aa),combb(nat,complex,nat,power_power(complex,fFT_Mirabelle_root(N1)),aa(nat,fun(nat,nat),times_times(nat),X2))),ord_atLeastLessThan(nat,zero_zero(nat),N1)) ).
tff(fact_11_power__number__of__even__number__of,axiom,
! [B: $tType] :
( ( monoid_mult(B)
& number(B) )
=> ! [W: int,V: int] : aa(nat,B,power_power(B,number_number_of(B,V)),number_number_of(nat,bit0(W))) = aa(B,B,aa(B,fun(B,B),times_times(B),aa(nat,B,power_power(B,number_number_of(B,V)),number_number_of(nat,W))),aa(nat,B,power_power(B,number_number_of(B,V)),number_number_of(nat,W))) ) ).
tff(fact_12_power__eq__0__iff__number__of,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [W1: int,Aa: A] :
( ( aa(nat,A,power_power(A,Aa),number_number_of(nat,W1)) = zero_zero(A) )
<=> ( ( Aa = zero_zero(A) )
& ( number_number_of(nat,W1) != zero_zero(nat) ) ) ) ) ).
tff(fact_13_eq__divide__eq__number__of1,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [W1: int,B1: A,Aa: A] :
( ( Aa = aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),number_number_of(A,W1)) )
<=> ( ( ( number_number_of(A,W1) != zero_zero(A) )
=> ( aa(A,A,aa(A,fun(A,A),times_times(A),Aa),number_number_of(A,W1)) = B1 ) )
& ( ( number_number_of(A,W1) = zero_zero(A) )
=> ( Aa = zero_zero(A) ) ) ) ) ) ).
tff(fact_14_divide__eq__eq__number__of1,axiom,
! [A: $tType] :
( ( field_inverse_zero(A)
& number(A) )
=> ! [Aa: A,W1: int,B1: A] :
( ( aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),number_number_of(A,W1)) = Aa )
<=> ( ( ( number_number_of(A,W1) != zero_zero(A) )
=> ( B1 = aa(A,A,aa(A,fun(A,A),times_times(A),Aa),number_number_of(A,W1)) ) )
& ( ( number_number_of(A,W1) = zero_zero(A) )
=> ( Aa = zero_zero(A) ) ) ) ) ) ).
tff(fact_15_eq__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& ring_char_0(A) )
=> ! [Y: int,X: int] :
( ( number_number_of(A,X) = number_number_of(A,Y) )
<=> ( X = Y ) ) ) ).
tff(fact_16_rel__simps_I51_J,axiom,
! [L: int,K1: int] :
( ( bit1(K1) = bit1(L) )
<=> ( K1 = L ) ) ).
tff(fact_17_minus__Pls,axiom,
uminus_uminus(int,pls) = pls ).
tff(fact_18_mult__Pls,axiom,
! [W: int] : aa(int,int,aa(int,fun(int,int),times_times(int),pls),W) = pls ).
tff(fact_19_minus__Bit0,axiom,
! [K: int] : uminus_uminus(int,bit0(K)) = bit0(uminus_uminus(int,K)) ).
tff(fact_20_mult__Bit0,axiom,
! [L1: int,K: int] : aa(int,int,aa(int,fun(int,int),times_times(int),bit0(K)),L1) = bit0(aa(int,int,aa(int,fun(int,int),times_times(int),K),L1)) ).
tff(fact_21_rel__simps_I48_J,axiom,
! [L: int,K1: int] :
( ( bit0(K1) = bit0(L) )
<=> ( K1 = L ) ) ).
tff(fact_22_double__eq__0__iff,axiom,
! [A: $tType] :
( linord219039673up_add(A)
=> ! [Aa: A] :
( ( aa(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) )
=> ! [Y: int,X: int] :
( ord_less_eq(A,number_number_of(A,X),number_number_of(A,Y))
<=> ord_less_eq(int,X,Y) ) ) ).
tff(fact_24_less__number__of,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int,X: int] :
( ord_less(A,number_number_of(A,X),number_number_of(A,Y))
<=> ord_less(int,X,Y) ) ) ).
tff(fact_25_mult__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V: int] : aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,V)),aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,W)),Z)) = aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,aa(int,int,aa(int,fun(int,int),times_times(int),V),W))),Z) ) ).
tff(fact_26_arith__simps_I32_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,V)),number_number_of(A,W)) = number_number_of(A,aa(int,int,aa(int,fun(int,int),times_times(int),V),W)) ) ).
tff(fact_27_add__number__of__left,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int,V: int] : aa(A,A,plus_plus(A,number_number_of(A,V)),aa(A,A,plus_plus(A,number_number_of(A,W)),Z)) = aa(A,A,plus_plus(A,number_number_of(A,aa(int,int,plus_plus(int,V),W))),Z) ) ).
tff(fact_28_add__number__of__eq,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : aa(A,A,plus_plus(A,number_number_of(A,V)),number_number_of(A,W)) = number_number_of(A,aa(int,int,plus_plus(int,V),W)) ) ).
tff(fact_29_arith__simps_I30_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : uminus_uminus(A,number_number_of(A,W)) = number_number_of(A,uminus_uminus(int,W)) ) ).
tff(fact_30_rel__simps_I46_J,axiom,
! [K: int] : bit1(K) != pls ).
tff(fact_31_rel__simps_I39_J,axiom,
! [L1: int] : pls != bit1(L1) ).
tff(fact_32_rel__simps_I50_J,axiom,
! [L1: int,K: int] : bit1(K) != bit0(L1) ).
tff(fact_33_rel__simps_I49_J,axiom,
! [L1: int,K: int] : bit0(K) != bit1(L1) ).
tff(fact_34_rel__simps_I44_J,axiom,
! [K1: int] :
( ( bit0(K1) = pls )
<=> ( K1 = pls ) ) ).
tff(fact_35_rel__simps_I38_J,axiom,
! [L: int] :
( ( pls = bit0(L) )
<=> ( pls = L ) ) ).
tff(fact_36_Bit0__Pls,axiom,
bit0(pls) = pls ).
tff(fact_37_left__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& semiring(B) )
=> ! [V: int,B2: B,A1: B] : aa(B,B,aa(B,fun(B,B),times_times(B),aa(B,B,plus_plus(B,A1),B2)),number_number_of(B,V)) = aa(B,B,plus_plus(B,aa(B,B,aa(B,fun(B,B),times_times(B),A1),number_number_of(B,V))),aa(B,B,aa(B,fun(B,B),times_times(B),B2),number_number_of(B,V))) ) ).
tff(fact_38_right__distrib__number__of,axiom,
! [B: $tType] :
( ( number(B)
& semiring(B) )
=> ! [C2: B,B2: B,V: int] : aa(B,B,aa(B,fun(B,B),times_times(B),number_number_of(B,V)),aa(B,B,plus_plus(B,B2),C2)) = aa(B,B,plus_plus(B,aa(B,B,aa(B,fun(B,B),times_times(B),number_number_of(B,V)),B2)),aa(B,B,aa(B,fun(B,B),times_times(B),number_number_of(B,V)),C2)) ) ).
tff(fact_39_number__of__Pls,axiom,
! [A: $tType] :
( number_ring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_40_minus__number__of__mult,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [Z: A,W: int] : aa(A,A,aa(A,fun(A,A),times_times(A),uminus_uminus(A,number_number_of(A,W))),Z) = aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,uminus_uminus(int,W))),Z) ) ).
tff(fact_41_nat__number__of__Pls,axiom,
number_number_of(nat,pls) = zero_zero(nat) ).
tff(fact_42_zero__less__power__nat__eq__number__of,axiom,
! [W1: int,X: nat] :
( ord_less(nat,zero_zero(nat),aa(nat,nat,power_power(nat,X),number_number_of(nat,W1)))
<=> ( ( number_number_of(nat,W1) = zero_zero(nat) )
| ord_less(nat,zero_zero(nat),X) ) ) ).
tff(fact_43_le__special_I3_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [X: int] :
( ord_less_eq(A,number_number_of(A,X),zero_zero(A))
<=> ord_less_eq(int,X,pls) ) ) ).
tff(fact_44_le__special_I1_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [Y: int] :
( ord_less_eq(A,zero_zero(A),number_number_of(A,Y))
<=> ord_less_eq(int,pls,Y) ) ) ).
tff(fact_45_less__special_I3_J,axiom,
! [A: $tType] :
( ( number_ring(A)
& linordered_idom(A) )
=> ! [X: int] :
( ord_less(A,number_number_of(A,X),zero_zero(A))
<=> ord_less(int,X,pls) ) ) ).
tff(fact_46_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_47_divide__less__eq__number__of1,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [Aa: A,W1: int,B1: A] :
( ord_less(A,aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),number_number_of(A,W1)),Aa)
<=> ( ( ord_less(A,zero_zero(A),number_number_of(A,W1))
=> ord_less(A,B1,aa(A,A,aa(A,fun(A,A),times_times(A),Aa),number_number_of(A,W1))) )
& ( ~ ord_less(A,zero_zero(A),number_number_of(A,W1))
=> ( ( ord_less(A,number_number_of(A,W1),zero_zero(A))
=> ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa),number_number_of(A,W1)),B1) )
& ( ~ ord_less(A,number_number_of(A,W1),zero_zero(A))
=> ord_less(A,zero_zero(A),Aa) ) ) ) ) ) ) ).
tff(fact_48_less__divide__eq__number__of1,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [W1: 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,W1)))
<=> ( ( ord_less(A,zero_zero(A),number_number_of(A,W1))
=> ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa),number_number_of(A,W1)),B1) )
& ( ~ ord_less(A,zero_zero(A),number_number_of(A,W1))
=> ( ( ord_less(A,number_number_of(A,W1),zero_zero(A))
=> ord_less(A,B1,aa(A,A,aa(A,fun(A,A),times_times(A),Aa),number_number_of(A,W1))) )
& ( ~ ord_less(A,number_number_of(A,W1),zero_zero(A))
=> ord_less(A,Aa,zero_zero(A)) ) ) ) ) ) ) ).
tff(fact_49_divide__le__eq__number__of1,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [Aa: A,W1: int,B1: A] :
( ord_less_eq(A,aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),number_number_of(A,W1)),Aa)
<=> ( ( ord_less(A,zero_zero(A),number_number_of(A,W1))
=> ord_less_eq(A,B1,aa(A,A,aa(A,fun(A,A),times_times(A),Aa),number_number_of(A,W1))) )
& ( ~ ord_less(A,zero_zero(A),number_number_of(A,W1))
=> ( ( ord_less(A,number_number_of(A,W1),zero_zero(A))
=> ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa),number_number_of(A,W1)),B1) )
& ( ~ ord_less(A,number_number_of(A,W1),zero_zero(A))
=> ord_less_eq(A,zero_zero(A),Aa) ) ) ) ) ) ) ).
tff(fact_50_le__divide__eq__number__of1,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [W1: 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,W1)))
<=> ( ( ord_less(A,zero_zero(A),number_number_of(A,W1))
=> ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),Aa),number_number_of(A,W1)),B1) )
& ( ~ ord_less(A,zero_zero(A),number_number_of(A,W1))
=> ( ( ord_less(A,number_number_of(A,W1),zero_zero(A))
=> ord_less_eq(A,B1,aa(A,A,aa(A,fun(A,A),times_times(A),Aa),number_number_of(A,W1))) )
& ( ~ ord_less(A,number_number_of(A,W1),zero_zero(A))
=> ord_less_eq(A,Aa,zero_zero(A)) ) ) ) ) ) ) ).
tff(fact_51_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_52_zero__is__num__zero,axiom,
zero_zero(int) = number_number_of(int,pls) ).
tff(fact_53_zero__less__power__nat__eq,axiom,
! [N1: nat,X: nat] :
( ord_less(nat,zero_zero(nat),aa(nat,nat,power_power(nat,X),N1))
<=> ( ( N1 = zero_zero(nat) )
| ord_less(nat,zero_zero(nat),X) ) ) ).
tff(fact_54_le__number__of__eq__not__less,axiom,
! [A: $tType] :
( ( number(A)
& linorder(A) )
=> ! [W1: int,V1: int] :
( ord_less_eq(A,number_number_of(A,V1),number_number_of(A,W1))
<=> ~ ord_less(A,number_number_of(A,W1),number_number_of(A,V1)) ) ) ).
tff(fact_55_zpower__zpower,axiom,
! [Z: nat,Y1: nat,X1: int] : aa(nat,int,power_power(int,aa(nat,int,power_power(int,X1),Y1)),Z) = aa(nat,int,power_power(int,X1),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),Y1),Z)) ).
tff(fact_56_Pls__def,axiom,
pls = zero_zero(int) ).
tff(fact_57_zpower__number__of__even,axiom,
! [W: int,Z: int] : aa(nat,int,power_power(int,Z),number_number_of(nat,bit0(W))) = aa(int,int,aa(int,fun(int,int),times_times(int),aa(nat,int,power_power(int,Z),number_number_of(nat,W))),aa(nat,int,power_power(int,Z),number_number_of(nat,W))) ).
tff(fact_58_even__less__0__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Aa: A] :
( ord_less(A,aa(A,A,plus_plus(A,Aa),Aa),zero_zero(A))
<=> ord_less(A,Aa,zero_zero(A)) ) ) ).
tff(fact_59_nat__lt__two__imp__zero__or__one,axiom,
! [X1: nat] :
( ord_less(nat,X1,aa(nat,nat,suc,aa(nat,nat,suc,zero_zero(nat))))
=> ( ( X1 = zero_zero(nat) )
| ( X1 = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
tff(fact_60_divide__le__eq__number__of,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [W1: 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,W1))
<=> ( ( ord_less(A,zero_zero(A),C1)
=> ord_less_eq(A,B1,aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,W1)),C1)) )
& ( ~ ord_less(A,zero_zero(A),C1)
=> ( ( ord_less(A,C1,zero_zero(A))
=> ord_less_eq(A,aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,W1)),C1),B1) )
& ( ~ ord_less(A,C1,zero_zero(A))
=> ord_less_eq(A,zero_zero(A),number_number_of(A,W1)) ) ) ) ) ) ) ).
tff(fact_61_le__divide__eq__number__of,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [C1: A,B1: A,W1: int] :
( ord_less_eq(A,number_number_of(A,W1),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,aa(A,fun(A,A),times_times(A),number_number_of(A,W1)),C1),B1) )
& ( ~ ord_less(A,zero_zero(A),C1)
=> ( ( ord_less(A,C1,zero_zero(A))
=> ord_less_eq(A,B1,aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,W1)),C1)) )
& ( ~ ord_less(A,C1,zero_zero(A))
=> ord_less_eq(A,number_number_of(A,W1),zero_zero(A)) ) ) ) ) ) ) ).
tff(fact_62_root__nonzero,axiom,
! [N: nat] : fFT_Mirabelle_root(N) != zero_zero(complex) ).
tff(fact_63_sum__squares__gt__zero__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [Y: A,X: A] :
( ord_less(A,zero_zero(A),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)))
<=> ( ( X != zero_zero(A) )
| ( Y != zero_zero(A) ) ) ) ) ).
tff(fact_64_not__sum__squares__lt__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [Y1: A,X1: A] : ~ ord_less(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X1),X1)),aa(A,A,aa(A,fun(A,A),times_times(A),Y1),Y1)),zero_zero(A)) ) ).
tff(fact_65_sum__squares__le__zero__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)),zero_zero(A))
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
tff(fact_66_sum__squares__ge__zero,axiom,
! [A: $tType] :
( linordered_ring(A)
=> ! [Y1: A,X1: A] : ord_less_eq(A,zero_zero(A),aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X1),X1)),aa(A,A,aa(A,fun(A,A),times_times(A),Y1),Y1))) ) ).
tff(fact_67_power2__less__imp__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Y1: A,X1: A] :
( ord_less(A,aa(nat,A,power_power(A,X1),number_number_of(nat,bit0(bit1(pls)))),aa(nat,A,power_power(A,Y1),number_number_of(nat,bit0(bit1(pls)))))
=> ( ord_less_eq(A,zero_zero(A),Y1)
=> ord_less(A,X1,Y1) ) ) ) ).
tff(fact_68_divide__less__eq__number__of,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [W1: int,C1: A,B1: A] :
( ord_less(A,aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),C1),number_number_of(A,W1))
<=> ( ( ord_less(A,zero_zero(A),C1)
=> ord_less(A,B1,aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,W1)),C1)) )
& ( ~ ord_less(A,zero_zero(A),C1)
=> ( ( ord_less(A,C1,zero_zero(A))
=> ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,W1)),C1),B1) )
& ( ~ ord_less(A,C1,zero_zero(A))
=> ord_less(A,zero_zero(A),number_number_of(A,W1)) ) ) ) ) ) ) ).
tff(fact_69_less__divide__eq__number__of,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number(A) )
=> ! [C1: A,B1: A,W1: int] :
( ord_less(A,number_number_of(A,W1),aa(A,A,aa(A,fun(A,A),inverse_divide(A),B1),C1))
<=> ( ( ord_less(A,zero_zero(A),C1)
=> ord_less(A,aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,W1)),C1),B1) )
& ( ~ ord_less(A,zero_zero(A),C1)
=> ( ( ord_less(A,C1,zero_zero(A))
=> ord_less(A,B1,aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,W1)),C1)) )
& ( ~ ord_less(A,C1,zero_zero(A))
=> ord_less(A,number_number_of(A,W1),zero_zero(A)) ) ) ) ) ) ) ).
tff(fact_70_root__summation,axiom,
! [N1: nat,K1: nat] :
( ord_less(nat,zero_zero(nat),K1)
=> ( ord_less(nat,K1,N1)
=> ( big_co1399186613setsum(nat,complex,power_power(complex,aa(nat,complex,power_power(complex,fFT_Mirabelle_root(N1)),K1)),ord_atLeastLessThan(nat,zero_zero(nat),N1)) = zero_zero(complex) ) ) ) ).
tff(fact_71_number__of__reorient,axiom,
! [A: $tType] :
( number(A)
=> ! [X: A,W1: int] :
( ( number_number_of(A,W1) = X )
<=> ( X = number_number_of(A,W1) ) ) ) ).
tff(fact_72_half__gt__zero,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number_ring(A) )
=> ! [R2: A] :
( ord_less(A,zero_zero(A),R2)
=> ord_less(A,zero_zero(A),aa(A,A,aa(A,fun(A,A),inverse_divide(A),R2),number_number_of(A,bit0(bit1(pls))))) ) ) ).
tff(fact_73_half__gt__zero__iff,axiom,
! [A: $tType] :
( ( linord1117847801e_zero(A)
& number_ring(A) )
=> ! [R1: A] :
( ord_less(A,zero_zero(A),aa(A,A,aa(A,fun(A,A),inverse_divide(A),R1),number_number_of(A,bit0(bit1(pls)))))
<=> ord_less(A,zero_zero(A),R1) ) ) ).
tff(fact_74_power2__less__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : ~ ord_less(A,aa(nat,A,power_power(A,A1),number_number_of(nat,bit0(bit1(pls)))),zero_zero(A)) ) ).
tff(fact_75_power2__eq__imp__eq,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Y1: A,X1: A] :
( ( aa(nat,A,power_power(A,X1),number_number_of(nat,bit0(bit1(pls)))) = aa(nat,A,power_power(A,Y1),number_number_of(nat,bit0(bit1(pls)))) )
=> ( ord_less_eq(A,zero_zero(A),X1)
=> ( ord_less_eq(A,zero_zero(A),Y1)
=> ( X1 = Y1 ) ) ) ) ) ).
tff(fact_76_power2__le__imp__le,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Y1: A,X1: A] :
( ord_less_eq(A,aa(nat,A,power_power(A,X1),number_number_of(nat,bit0(bit1(pls)))),aa(nat,A,power_power(A,Y1),number_number_of(nat,bit0(bit1(pls)))))
=> ( ord_less_eq(A,zero_zero(A),Y1)
=> ord_less_eq(A,X1,Y1) ) ) ) ).
tff(fact_77_zero__le__power2,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : ord_less_eq(A,zero_zero(A),aa(nat,A,power_power(A,A1),number_number_of(nat,bit0(bit1(pls))))) ) ).
tff(fact_78_less__2__cases,axiom,
! [N: nat] :
( ord_less(nat,N,number_number_of(nat,bit0(bit1(pls))))
=> ( ( N = zero_zero(nat) )
| ( N = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
tff(fact_79_root__cancel,axiom,
! [K: nat,N: nat,D: nat] :
( ord_less(nat,zero_zero(nat),D)
=> ( aa(nat,complex,power_power(complex,fFT_Mirabelle_root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D),N))),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),D),K)) = aa(nat,complex,power_power(complex,fFT_Mirabelle_root(N)),K) ) ) ).
tff(fact_80_sum__power2__gt__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ord_less(A,zero_zero(A),aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),number_number_of(nat,bit0(bit1(pls))))),aa(nat,A,power_power(A,Y),number_number_of(nat,bit0(bit1(pls))))))
<=> ( ( X != zero_zero(A) )
| ( Y != zero_zero(A) ) ) ) ) ).
tff(fact_81_not__sum__power2__lt__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y1: A,X1: A] : ~ ord_less(A,aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X1),number_number_of(nat,bit0(bit1(pls))))),aa(nat,A,power_power(A,Y1),number_number_of(nat,bit0(bit1(pls))))),zero_zero(A)) ) ).
tff(fact_82_sum__power2__le__zero__iff,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X),number_number_of(nat,bit0(bit1(pls))))),aa(nat,A,power_power(A,Y),number_number_of(nat,bit0(bit1(pls))))),zero_zero(A))
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
tff(fact_83_sum__power2__ge__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y1: A,X1: A] : ord_less_eq(A,zero_zero(A),aa(A,A,plus_plus(A,aa(nat,A,power_power(A,X1),number_number_of(nat,bit0(bit1(pls))))),aa(nat,A,power_power(A,Y1),number_number_of(nat,bit0(bit1(pls)))))) ) ).
tff(fact_84_even__power__le__0__imp__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [K: nat,A1: A] :
( ord_less_eq(A,aa(nat,A,power_power(A,A1),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))),K)),zero_zero(A))
=> ( A1 = zero_zero(A) ) ) ) ).
tff(fact_85_zero__le__even__power_H,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat,A1: A] : ord_less_eq(A,zero_zero(A),aa(nat,A,power_power(A,A1),aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))),N))) ) ).
tff(fact_86_odd__power__less__zero,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat,A1: A] :
( ord_less(A,A1,zero_zero(A))
=> ord_less(A,aa(nat,A,power_power(A,A1),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))),N))),zero_zero(A)) ) ) ).
tff(fact_87_odd__0__le__power__imp__0__le,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [N: nat,A1: A] :
( ord_less_eq(A,zero_zero(A),aa(nat,A,power_power(A,A1),aa(nat,nat,suc,aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))),N))))
=> ord_less_eq(A,zero_zero(A),A1) ) ) ).
tff(fact_88_number__of__mult,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,aa(int,int,aa(int,fun(int,int),times_times(int),V),W)) = aa(A,A,aa(A,fun(A,A),times_times(A),number_number_of(A,V)),number_number_of(A,W)) ) ).
tff(fact_89_number__of__add,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int,V: int] : number_number_of(A,aa(int,int,plus_plus(int,V),W)) = aa(A,A,plus_plus(A,number_number_of(A,V)),number_number_of(A,W)) ) ).
tff(fact_90_number__of__minus,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [W: int] : number_number_of(A,uminus_uminus(int,W)) = uminus_uminus(A,number_number_of(A,W)) ) ).
tff(fact_91_sum__squares__eq__zero__iff,axiom,
! [A: $tType] :
( linord581940658strict(A)
=> ! [Y: A,X: A] :
( ( aa(A,A,plus_plus(A,aa(A,A,aa(A,fun(A,A),times_times(A),X),X)),aa(A,A,aa(A,fun(A,A),times_times(A),Y),Y)) = zero_zero(A) )
<=> ( ( X = zero_zero(A) )
& ( Y = zero_zero(A) ) ) ) ) ).
tff(fact_92_semiring__numeral__0__eq__0,axiom,
! [A: $tType] :
( number_semiring(A)
=> ( number_number_of(A,pls) = zero_zero(A) ) ) ).
tff(fact_93_semiring__norm_I112_J,axiom,
! [A: $tType] :
( number_ring(A)
=> ( zero_zero(A) = number_number_of(A,pls) ) ) ).
tff(fact_94_add__numeral__0,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : aa(A,A,plus_plus(A,number_number_of(A,pls)),A1) = A1 ) ).
tff(fact_95_add__numeral__0__right,axiom,
! [A: $tType] :
( number_ring(A)
=> ! [A1: A] : aa(A,A,plus_plus(A,A1),number_number_of(A,pls)) = A1 ) ).
tff(fact_96_semiring__norm_I113_J,axiom,
zero_zero(nat) = number_number_of(nat,pls) ).
%----Arities (45)
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__semidom,axiom,
linordered_semidom(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___Int_Onumber__semiring,axiom,
number_semiring(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_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___Rings_Oring__1,axiom,
ring_1(int) ).
tff(arity_Int_Oint___Power_Opower,axiom,
power(int) ).
tff(arity_Int_Oint___Int_Onumber,axiom,
number(int) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(nat) ).
tff(arity_Nat_Onat___Int_Onumber__semiring,axiom,
number_semiring(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_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___Int_Onumber__semiring,axiom,
number_semiring(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_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___Rings_Oring__1,axiom,
ring_1(complex) ).
tff(arity_Complex_Ocomplex___Power_Opower,axiom,
power(complex) ).
tff(arity_Complex_Ocomplex___Int_Onumber,axiom,
number(complex) ).
%----Helper facts (4)
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_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,
big_co1399186613setsum(nat,complex,combs(nat,complex,complex,combb(complex,fun(complex,complex),nat,times_times(complex),combb(nat,complex,nat,power_power(complex,fFT_Mirabelle_root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))),m))),combb(nat,nat,nat,plus_plus(nat,i),combb(nat,nat,nat,aa(nat,fun(nat,nat),times_times(nat),i),aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))))))),combb(nat,complex,nat,a,combb(nat,nat,nat,suc,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls))))))),ord_atLeastLessThan(nat,zero_zero(nat),m)) = uminus_uminus(complex,aa(complex,complex,aa(complex,fun(complex,complex),inverse_divide(complex),aa(complex,complex,aa(complex,fun(complex,complex),times_times(complex),aa(nat,complex,power_power(complex,fFT_Mirabelle_root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))),m))),i)),big_co1399186613setsum(nat,complex,combs(nat,complex,complex,combb(complex,fun(complex,complex),nat,times_times(complex),power_power(complex,aa(nat,complex,power_power(complex,fFT_Mirabelle_root(m)),i))),combb(nat,complex,nat,a,combb(nat,nat,nat,suc,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls))))))),ord_atLeastLessThan(nat,zero_zero(nat),m)))),aa(nat,complex,power_power(complex,fFT_Mirabelle_root(aa(nat,nat,aa(nat,fun(nat,nat),times_times(nat),number_number_of(nat,bit0(bit1(pls)))),m))),m))) ).
%------------------------------------------------------------------------------