TPTP Problem File: SWV609_5.p
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%------------------------------------------------------------------------------
% File : SWV609_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fast Fourier Transform line 165
% 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_165 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 174 ( 58 unt; 40 typ; 0 def)
% Number of atoms : 287 ( 98 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 200 ( 47 ~; 9 |; 12 &)
% ( 22 <=>; 110 =>; 0 <=; 0 <~>)
% Maximal formula depth : 11 ( 4 avg)
% Maximal term depth : 8 ( 1 avg)
% Number of types : 5 ( 4 usr)
% Number of type conns : 19 ( 13 >; 6 *; 0 +; 0 <<)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-3 aty)
% Number of functors : 18 ( 18 usr; 5 con; 0-5 aty)
% Number of variables : 266 ( 226 !; 7 ?; 266 :)
% ( 33 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:16:14
%------------------------------------------------------------------------------
%----Should-be-implicit typings (5)
tff(ty_tc_Complex_Ocomplex,type,
complex: $tType ).
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_Nat_Onat,type,
nat: $tType ).
tff(ty_tc_String_Ochar,type,
char: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (35)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Power_Opower,type,
power:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[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_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_Ocomm__semiring__1,type,
comm_semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__idom,type,
linordered_idom:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ocomm__monoid__add,type,
comm_monoid_add:
!>[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_Rings_Oring__1__no__zero__divisors,type,
ring_11004092258visors:
!>[A: $tType] : $o ).
tff(sy_c_Big__Operators_Ocomm__monoid__add__class_Osetsum,type,
big_co1399186613setsum:
!>[B: $tType,A: $tType] : fun(fun(B,A),fun(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)) > fun(B,fun(A,C)) ) ).
tff(sy_c_COMBK,type,
combk:
!>[A: $tType,B: $tType] : ( A > fun(B,A) ) ).
tff(sy_c_FFT__Mirabelle__uutasbvzez_Oroot,type,
fFT_Mirabelle_root: nat > complex ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_HOL_Obool_Obool__size,type,
bool_size: bool > nat ).
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_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_String_Ochar_Ochar__size,type,
char_size: char > nat ).
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_member,type,
member:
!>[A: $tType] : ( ( A * fun(A,bool) ) > $o ) ).
tff(sy_c_pp,type,
pp: bool > $o ).
tff(sy_v_k,type,
k: nat ).
tff(sy_v_n,type,
n: nat ).
%----Relevant facts (97)
tff(fact_0_assms_I1_J,axiom,
ord_less(nat,zero_zero(nat),k) ).
tff(fact_1_assms_I2_J,axiom,
ord_less(nat,k,n) ).
tff(fact_2_root__nonzero,axiom,
! [N: nat] : fFT_Mirabelle_root(N) != zero_zero(complex) ).
tff(fact_3_power__eq__0__iff,axiom,
! [A: $tType] :
( ( power(A)
& mult_zero(A)
& no_zero_divisors(A)
& zero_neq_one(A) )
=> ! [Na: nat,A3: A] :
( ( aa(nat,A,power_power(A,A3),Na) = zero_zero(A) )
<=> ( ( A3 = zero_zero(A) )
& ( Na != zero_zero(nat) ) ) ) ) ).
tff(fact_4_setsum__0,axiom,
! [B: $tType,A: $tType] :
( comm_monoid_add(A)
=> ! [A1: fun(B,bool)] : aa(fun(B,bool),A,aa(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),combk(A,B,zero_zero(A))),A1) = zero_zero(A) ) ).
tff(fact_5_field__power__not__zero,axiom,
! [A: $tType] :
( ring_11004092258visors(A)
=> ! [N: nat,A2: A] :
( ( A2 != zero_zero(A) )
=> ( aa(nat,A,power_power(A,A2),N) != zero_zero(A) ) ) ) ).
tff(fact_6_setsum__commute,axiom,
! [B: $tType,A: $tType,C: $tType] :
( comm_monoid_add(A)
=> ! [A1: fun(B,bool),B2: fun(C,bool),F: fun(B,fun(C,A))] : aa(fun(B,bool),A,aa(fun(B,A),fun(fun(B,bool),A),big_co1399186613setsum(B,A),aa(fun(C,bool),fun(B,A),combc(B,fun(C,bool),A,combb(fun(C,A),fun(fun(C,bool),A),B,big_co1399186613setsum(C,A),F)),B2)),A1) = aa(fun(C,bool),A,aa(fun(C,A),fun(fun(C,bool),A),big_co1399186613setsum(C,A),aa(fun(B,bool),fun(C,A),combc(C,fun(B,bool),A,combb(fun(B,A),fun(fun(B,bool),A),C,big_co1399186613setsum(B,A),combc(B,C,A,F))),A1)),B2) ) ).
tff(fact_7_setsum__0_H,axiom,
! [A: $tType,B: $tType] :
( comm_monoid_add(B)
=> ! [F: fun(A,B),A1: fun(A,bool)] :
( ! [X3: A] :
( member(A,X3,A1)
=> ( aa(A,B,F,X3) = zero_zero(B) ) )
=> ( aa(fun(A,bool),B,aa(fun(A,B),fun(fun(A,bool),B),big_co1399186613setsum(A,B),F),A1) = zero_zero(B) ) ) ) ).
tff(fact_8_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X2: A] :
( ( zero_zero(A) = X2 )
<=> ( X2 = zero_zero(A) ) ) ) ).
tff(fact_9_root__unity,axiom,
! [N: nat] : aa(nat,complex,power_power(complex,fFT_Mirabelle_root(N)),N) = one_one(complex) ).
tff(fact_10_setsum__shift__lb__Suc0__0__upt,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Ka: nat,F: fun(nat,A)] :
( ( aa(nat,A,F,zero_zero(nat)) = zero_zero(A) )
=> ( aa(fun(nat,bool),A,aa(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),F),ord_atLeastLessThan(nat,aa(nat,nat,suc,zero_zero(nat)),Ka)) = aa(fun(nat,bool),A,aa(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),F),ord_atLeastLessThan(nat,zero_zero(nat),Ka)) ) ) ) ).
tff(fact_11_bool_Osize_I1_J,axiom,
bool_size(fTrue) = zero_zero(nat) ).
tff(fact_12_bool_Osize_I2_J,axiom,
bool_size(fFalse) = zero_zero(nat) ).
tff(fact_13_char__size,axiom,
! [C2: char] : char_size(C2) = zero_zero(nat) ).
tff(fact_14_power__one,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [N: nat] : aa(nat,A,power_power(A,one_one(A)),N) = one_one(A) ) ).
tff(fact_15_power__Suc__0,axiom,
! [N: nat] : aa(nat,nat,power_power(nat,aa(nat,nat,suc,zero_zero(nat))),N) = aa(nat,nat,suc,zero_zero(nat)) ).
tff(fact_16_nat__power__eq__Suc__0__iff,axiom,
! [M2: nat,X2: nat] :
( ( aa(nat,nat,power_power(nat,X2),M2) = aa(nat,nat,suc,zero_zero(nat)) )
<=> ( ( M2 = zero_zero(nat) )
| ( X2 = aa(nat,nat,suc,zero_zero(nat)) ) ) ) ).
tff(fact_17_nat__zero__less__power__iff,axiom,
! [Na: nat,X2: nat] :
( ord_less(nat,zero_zero(nat),aa(nat,nat,power_power(nat,X2),Na))
<=> ( ord_less(nat,zero_zero(nat),X2)
| ( Na = zero_zero(nat) ) ) ) ).
tff(fact_18_power__0__Suc,axiom,
! [A: $tType] :
( ( power(A)
& semiring_0(A) )
=> ! [N: nat] : aa(nat,A,power_power(A,zero_zero(A)),aa(nat,nat,suc,N)) = zero_zero(A) ) ).
tff(fact_19_power__0,axiom,
! [A: $tType] :
( power(A)
=> ! [A2: A] : aa(nat,A,power_power(A,A2),zero_zero(nat)) = one_one(A) ) ).
tff(fact_20_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [X2: A] :
( ( one_one(A) = X2 )
<=> ( X2 = one_one(A) ) ) ) ).
tff(fact_21_power__gt1,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A2: A] :
( ord_less(A,one_one(A),A2)
=> ord_less(A,one_one(A),aa(nat,A,power_power(A,A2),aa(nat,nat,suc,N))) ) ) ).
tff(fact_22_power__inject__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat,M2: nat,A3: A] :
( ord_less(A,one_one(A),A3)
=> ( ( aa(nat,A,power_power(A,A3),M2) = aa(nat,A,power_power(A,A3),Na) )
<=> ( M2 = Na ) ) ) ) ).
tff(fact_23_power__strict__increasing__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Y1: nat,X2: nat,B1: A] :
( ord_less(A,one_one(A),B1)
=> ( ord_less(A,aa(nat,A,power_power(A,B1),X2),aa(nat,A,power_power(A,B1),Y1))
<=> ord_less(nat,X2,Y1) ) ) ) ).
tff(fact_24_power__less__imp__less__exp,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M3: nat,A2: A] :
( ord_less(A,one_one(A),A2)
=> ( ord_less(A,aa(nat,A,power_power(A,A2),M3),aa(nat,A,power_power(A,A2),N))
=> ord_less(nat,M3,N) ) ) ) ).
tff(fact_25_nat__power__less__imp__less,axiom,
! [N: nat,M3: nat,I: nat] :
( ord_less(nat,zero_zero(nat),I)
=> ( ord_less(nat,aa(nat,nat,power_power(nat,I),M3),aa(nat,nat,power_power(nat,I),N))
=> ord_less(nat,M3,N) ) ) ).
tff(fact_26_power__strict__increasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A2: A,N3: nat,N: nat] :
( ord_less(nat,N,N3)
=> ( ord_less(A,one_one(A),A2)
=> ord_less(A,aa(nat,A,power_power(A,A2),N),aa(nat,A,power_power(A,A2),N3)) ) ) ) ).
tff(fact_27_power__Suc__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A2: A] :
( ord_less(A,zero_zero(A),A2)
=> ( ord_less(A,A2,one_one(A))
=> ord_less(A,aa(nat,A,power_power(A,A2),aa(nat,nat,suc,N)),one_one(A)) ) ) ) ).
tff(fact_28_power__strict__decreasing,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [A2: A,N3: nat,N: nat] :
( ord_less(nat,N,N3)
=> ( ord_less(A,zero_zero(A),A2)
=> ( ord_less(A,A2,one_one(A))
=> ord_less(A,aa(nat,A,power_power(A,A2),N3),aa(nat,A,power_power(A,A2),N)) ) ) ) ) ).
tff(fact_29_one__less__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A2: A] :
( ord_less(A,one_one(A),A2)
=> ( ord_less(nat,zero_zero(nat),N)
=> ord_less(A,one_one(A),aa(nat,A,power_power(A,A2),N)) ) ) ) ).
tff(fact_30_atLeastLessThan__inj_I2_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [D: A,C1: A,B1: A,A3: A] :
( ( ord_atLeastLessThan(A,A3,B1) = ord_atLeastLessThan(A,C1,D) )
=> ( ord_less(A,A3,B1)
=> ( ord_less(A,C1,D)
=> ( B1 = D ) ) ) ) ) ).
tff(fact_31_atLeastLessThan__inj_I1_J,axiom,
! [A: $tType] :
( linorder(A)
=> ! [D: A,C1: A,B1: A,A3: A] :
( ( ord_atLeastLessThan(A,A3,B1) = ord_atLeastLessThan(A,C1,D) )
=> ( ord_less(A,A3,B1)
=> ( ord_less(A,C1,D)
=> ( A3 = C1 ) ) ) ) ) ).
tff(fact_32_atLeastLessThan__eq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [D: A,C1: A,B1: A,A3: A] :
( ord_less(A,A3,B1)
=> ( ord_less(A,C1,D)
=> ( ( ord_atLeastLessThan(A,A3,B1) = ord_atLeastLessThan(A,C1,D) )
<=> ( ( A3 = C1 )
& ( B1 = D ) ) ) ) ) ) ).
tff(fact_33_zero__less__power,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,A2: A] :
( ord_less(A,zero_zero(A),A2)
=> ord_less(A,zero_zero(A),aa(nat,A,power_power(A,A2),N)) ) ) ).
tff(fact_34_setsum__shift__bounds__Suc__ivl,axiom,
! [A: $tType] :
( comm_monoid_add(A)
=> ! [Na: nat,M2: nat,F: fun(nat,A)] : aa(fun(nat,bool),A,aa(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),F),ord_atLeastLessThan(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,Na))) = aa(fun(nat,bool),A,aa(fun(nat,A),fun(fun(nat,bool),A),big_co1399186613setsum(nat,A),combb(nat,A,nat,F,suc)),ord_atLeastLessThan(nat,M2,Na)) ) ).
tff(fact_35_power__0__left,axiom,
! [A: $tType] :
( ( power(A)
& semiring_0(A) )
=> ! [N: nat] :
( ( ( N = zero_zero(nat) )
=> ( aa(nat,A,power_power(A,zero_zero(A)),N) = one_one(A) ) )
& ( ( N != zero_zero(nat) )
=> ( aa(nat,A,power_power(A,zero_zero(A)),N) = zero_zero(A) ) ) ) ) ).
tff(fact_36_zero__less__Suc,axiom,
! [N: nat] : ord_less(nat,zero_zero(nat),aa(nat,nat,suc,N)) ).
tff(fact_37_less__Suc0,axiom,
! [Na: nat] :
( ord_less(nat,Na,aa(nat,nat,suc,zero_zero(nat)))
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_38_lessI,axiom,
! [N: nat] : ord_less(nat,N,aa(nat,nat,suc,N)) ).
tff(fact_39_Suc__less__eq,axiom,
! [Na: nat,M2: nat] :
( ord_less(nat,aa(nat,nat,suc,M2),aa(nat,nat,suc,Na))
<=> ord_less(nat,M2,Na) ) ).
tff(fact_40_Suc__mono,axiom,
! [N: nat,M3: nat] :
( ord_less(nat,M3,N)
=> ord_less(nat,aa(nat,nat,suc,M3),aa(nat,nat,suc,N)) ) ).
tff(fact_41_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_42_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_43_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_44_nat_Oinject,axiom,
! [Nat4: nat,Nat3: nat] :
( ( aa(nat,nat,suc,Nat3) = aa(nat,nat,suc,Nat4) )
<=> ( Nat3 = Nat4 ) ) ).
tff(fact_45_One__nat__def,axiom,
one_one(nat) = aa(nat,nat,suc,zero_zero(nat)) ).
tff(fact_46_power__one__right,axiom,
! [A: $tType] :
( monoid_mult(A)
=> ! [A2: A] : aa(nat,A,power_power(A,A2),one_one(nat)) = A2 ) ).
tff(fact_47_Suc__inject,axiom,
! [Y: nat,X: nat] :
( ( aa(nat,nat,suc,X) = aa(nat,nat,suc,Y) )
=> ( X = Y ) ) ).
tff(fact_48_Suc__n__not__n,axiom,
! [N: nat] : aa(nat,nat,suc,N) != N ).
tff(fact_49_n__not__Suc__n,axiom,
! [N: nat] : N != aa(nat,nat,suc,N) ).
tff(fact_50_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),Na: nat,M2: nat] :
( ( ord_less(nat,M2,Na)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) )
=> ( ( ( M2 = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) )
=> ( ( ord_less(nat,Na,M2)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M2)) ) ) ) ).
tff(fact_51_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_52_less__not__refl2,axiom,
! [M3: nat,N: nat] :
( ord_less(nat,N,M3)
=> ( M3 != N ) ) ).
tff(fact_53_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_54_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_55_nat__neq__iff,axiom,
! [Na: nat,M2: nat] :
( ( M2 != Na )
<=> ( ord_less(nat,M2,Na)
| ord_less(nat,Na,M2) ) ) ).
tff(fact_56_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_57_Zero__not__Suc,axiom,
! [M3: nat] : zero_zero(nat) != aa(nat,nat,suc,M3) ).
tff(fact_58_nat_Osimps_I2_J,axiom,
! [Nat2: nat] : zero_zero(nat) != aa(nat,nat,suc,Nat2) ).
tff(fact_59_Suc__not__Zero,axiom,
! [M3: nat] : aa(nat,nat,suc,M3) != zero_zero(nat) ).
tff(fact_60_nat_Osimps_I3_J,axiom,
! [Nat1: nat] : aa(nat,nat,suc,Nat1) != zero_zero(nat) ).
tff(fact_61_Zero__neq__Suc,axiom,
! [M3: nat] : zero_zero(nat) != aa(nat,nat,suc,M3) ).
tff(fact_62_Suc__neq__Zero,axiom,
! [M3: nat] : aa(nat,nat,suc,M3) != zero_zero(nat) ).
tff(fact_63_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_64_gr__implies__not0,axiom,
! [N: nat,M3: nat] :
( ord_less(nat,M3,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_65_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_66_Suc__less__SucD,axiom,
! [N: nat,M3: nat] :
( ord_less(nat,aa(nat,nat,suc,M3),aa(nat,nat,suc,N))
=> ord_less(nat,M3,N) ) ).
tff(fact_67_Suc__lessD,axiom,
! [N: nat,M3: nat] :
( ord_less(nat,aa(nat,nat,suc,M3),N)
=> ord_less(nat,M3,N) ) ).
tff(fact_68_less__SucE,axiom,
! [N: nat,M3: nat] :
( ord_less(nat,M3,aa(nat,nat,suc,N))
=> ( ~ ord_less(nat,M3,N)
=> ( M3 = N ) ) ) ).
tff(fact_69_less__trans__Suc,axiom,
! [K: nat,J2: nat,I: nat] :
( ord_less(nat,I,J2)
=> ( ord_less(nat,J2,K)
=> ord_less(nat,aa(nat,nat,suc,I),K) ) ) ).
tff(fact_70_Suc__lessI,axiom,
! [N: nat,M3: nat] :
( ord_less(nat,M3,N)
=> ( ( aa(nat,nat,suc,M3) != N )
=> ord_less(nat,aa(nat,nat,suc,M3),N) ) ) ).
tff(fact_71_less__SucI,axiom,
! [N: nat,M3: nat] :
( ord_less(nat,M3,N)
=> ord_less(nat,M3,aa(nat,nat,suc,N)) ) ).
tff(fact_72_less__antisym,axiom,
! [M3: nat,N: nat] :
( ~ ord_less(nat,N,M3)
=> ( ord_less(nat,N,aa(nat,nat,suc,M3))
=> ( M3 = N ) ) ) ).
tff(fact_73_not__less__less__Suc__eq,axiom,
! [M2: nat,Na: nat] :
( ~ ord_less(nat,Na,M2)
=> ( ord_less(nat,Na,aa(nat,nat,suc,M2))
<=> ( Na = M2 ) ) ) ).
tff(fact_74_less__Suc__eq,axiom,
! [Na: nat,M2: nat] :
( ord_less(nat,M2,aa(nat,nat,suc,Na))
<=> ( ord_less(nat,M2,Na)
| ( M2 = Na ) ) ) ).
tff(fact_75_not__less__eq,axiom,
! [Na: nat,M2: nat] :
( ~ ord_less(nat,M2,Na)
<=> ord_less(nat,Na,aa(nat,nat,suc,M2)) ) ).
tff(fact_76_less__Suc__eq__0__disj,axiom,
! [Na: nat,M2: nat] :
( ord_less(nat,M2,aa(nat,nat,suc,Na))
<=> ( ( M2 = zero_zero(nat) )
| ? [J1: nat] :
( ( M2 = aa(nat,nat,suc,J1) )
& ord_less(nat,J1,Na) ) ) ) ).
tff(fact_77_gr0__conv__Suc,axiom,
! [Na: nat] :
( ord_less(nat,zero_zero(nat),Na)
<=> ? [M1: nat] : Na = aa(nat,nat,suc,M1) ) ).
tff(fact_78_gr0__implies__Suc,axiom,
! [N: nat] :
( ord_less(nat,zero_zero(nat),N)
=> ? [M: nat] : N = aa(nat,nat,suc,M) ) ).
tff(fact_79_setsum__SucD,axiom,
! [A: $tType,Na: nat,A1: fun(A,bool),F: fun(A,nat)] :
( ( aa(fun(A,bool),nat,aa(fun(A,nat),fun(fun(A,bool),nat),big_co1399186613setsum(A,nat),F),A1) = aa(nat,nat,suc,Na) )
=> ? [X3: A] :
( member(A,X3,A1)
& ord_less(nat,zero_zero(nat),aa(A,nat,F,X3)) ) ) ).
tff(fact_80_lift__Suc__mono__less,axiom,
! [A: $tType] :
( order(A)
=> ! [N2: nat,Na: nat,F: fun(nat,A)] :
( ! [N1: nat] : ord_less(A,aa(nat,A,F,N1),aa(nat,A,F,aa(nat,nat,suc,N1)))
=> ( ord_less(nat,Na,N2)
=> ord_less(A,aa(nat,A,F,Na),aa(nat,A,F,N2)) ) ) ) ).
tff(fact_81_lift__Suc__mono__less__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [M2: nat,Na: nat,F: fun(nat,A)] :
( ! [N1: nat] : ord_less(A,aa(nat,A,F,N1),aa(nat,A,F,aa(nat,nat,suc,N1)))
=> ( ord_less(A,aa(nat,A,F,Na),aa(nat,A,F,M2))
<=> ord_less(nat,Na,M2) ) ) ) ).
tff(fact_82_zero__less__power__nat__eq,axiom,
! [Na: nat,X2: nat] :
( ord_less(nat,zero_zero(nat),aa(nat,nat,power_power(nat,X2),Na))
<=> ( ( Na = zero_zero(nat) )
| ord_less(nat,zero_zero(nat),X2) ) ) ).
tff(fact_83_all__nat__less__eq,axiom,
! [P1: fun(nat,bool),Na: nat] :
( ! [M1: nat] :
( ord_less(nat,M1,Na)
=> pp(aa(nat,bool,P1,M1)) )
<=> ! [X1: nat] :
( member(nat,X1,ord_atLeastLessThan(nat,zero_zero(nat),Na))
=> pp(aa(nat,bool,P1,X1)) ) ) ).
tff(fact_84_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_85_ex__nat__less__eq,axiom,
! [P1: fun(nat,bool),Na: nat] :
( ? [M1: nat] :
( ord_less(nat,M1,Na)
& pp(aa(nat,bool,P1,M1)) )
<=> ? [X1: nat] :
( member(nat,X1,ord_atLeastLessThan(nat,zero_zero(nat),Na))
& pp(aa(nat,bool,P1,X1)) ) ) ).
tff(fact_86_comm__semiring__1__class_Onormalizing__semiring__rules_I32_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X: A] : aa(nat,A,power_power(A,X),zero_zero(nat)) = one_one(A) ) ).
tff(fact_87_zero__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less(A,zero_zero(A),one_one(A)) ) ).
tff(fact_88_linorder__neqE__linordered__idom,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [Y: A,X: A] :
( ( X != Y )
=> ( ~ ord_less(A,X,Y)
=> ord_less(A,Y,X) ) ) ) ).
tff(fact_89_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_90_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_91_comm__semiring__1__class_Onormalizing__semiring__rules_I33_J,axiom,
! [A: $tType] :
( comm_semiring_1(A)
=> ! [X: A] : aa(nat,A,power_power(A,X),one_one(nat)) = X ) ).
tff(fact_92_not__one__less__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less(A,one_one(A),zero_zero(A)) ) ).
tff(fact_93_lessE,axiom,
! [K: nat,I: nat] :
( ord_less(nat,I,K)
=> ( ( K != aa(nat,nat,suc,I) )
=> ~ ! [J: nat] :
( ord_less(nat,I,J)
=> ( K != aa(nat,nat,suc,J) ) ) ) ) ).
tff(fact_94_Suc__lessE,axiom,
! [K: nat,I: nat] :
( ord_less(nat,aa(nat,nat,suc,I),K)
=> ~ ! [J: nat] :
( ord_less(nat,I,J)
=> ( K != aa(nat,nat,suc,J) ) ) ) ).
tff(fact_95_not0__implies__Suc,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ? [M: nat] : N = aa(nat,nat,suc,M) ) ).
tff(fact_96_nat_Oexhaust,axiom,
! [Y: nat] :
( ( Y != zero_zero(nat) )
=> ~ ! [Nat: nat] : Y != aa(nat,nat,suc,Nat) ) ).
%----Arities (27)
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
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___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(nat) ).
tff(arity_Nat_Onat___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(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__0,axiom,
semiring_0(nat) ).
tff(arity_Nat_Onat___Rings_Omult__zero,axiom,
mult_zero(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Power_Opower,axiom,
power(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_Nat_Onat___Groups_Oone,axiom,
one(nat) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_Complex_Ocomplex___Rings_Oring__1__no__zero__divisors,axiom,
ring_11004092258visors(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ono__zero__divisors,axiom,
no_zero_divisors(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ocomm__monoid__add,axiom,
comm_monoid_add(complex) ).
tff(arity_Complex_Ocomplex___Rings_Ocomm__semiring__1,axiom,
comm_semiring_1(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__0,axiom,
semiring_0(complex) ).
tff(arity_Complex_Ocomplex___Rings_Omult__zero,axiom,
mult_zero(complex) ).
tff(arity_Complex_Ocomplex___Power_Opower,axiom,
power(complex) ).
tff(arity_Complex_Ocomplex___Groups_Ozero,axiom,
zero(complex) ).
tff(arity_Complex_Ocomplex___Groups_Oone,axiom,
one(complex) ).
%----Helper facts (9)
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,aa(B,fun(A,C),combc(A,B,C,P),Q),R) = aa(B,C,aa(A,fun(B,C),P,R),Q) ).
tff(help_COMBK_1_1_U,axiom,
! [B: $tType,A: $tType,Q: B,P: A] : aa(B,A,combk(A,B,P),Q) = P ).
tff(help_fTrue_1_1_U,axiom,
pp(fTrue) ).
tff(help_fTrue_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_fFalse_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_fFalse_1_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
%----Conjectures (1)
tff(conj_0,conjecture,
aa(fun(nat,bool),complex,aa(fun(nat,complex),fun(fun(nat,bool),complex),big_co1399186613setsum(nat,complex),power_power(complex,aa(nat,complex,power_power(complex,fFT_Mirabelle_root(n)),k))),ord_atLeastLessThan(nat,zero_zero(nat),n)) = zero_zero(complex) ).
%------------------------------------------------------------------------------