TPTP Problem File: SWV573_5.p
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%------------------------------------------------------------------------------
% File : SWV573_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fast Fourier Transform line 27
% 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_27 [Bla13]
% Status : Theorem
% Rating : 0.00 v6.4.0
% Syntax : Number of formulae : 183 ( 75 unt; 38 typ; 0 def)
% Number of atoms : 261 ( 85 equ)
% Maximal formula atoms : 7 ( 1 avg)
% Number of connectives : 141 ( 25 ~; 7 |; 8 &)
% ( 32 <=>; 69 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of types : 6 ( 5 usr)
% Number of type conns : 28 ( 17 >; 11 *; 0 +; 0 <<)
% Number of predicates : 15 ( 14 usr; 0 prp; 1-3 aty)
% Number of functors : 19 ( 19 usr; 4 con; 0-4 aty)
% Number of variables : 224 ( 198 !; 2 ?; 224 :)
% ( 24 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_THM_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:13:47
%------------------------------------------------------------------------------
%----Should-be-implicit typings (6)
tff(ty_tc_Complex_Ocomplex,type,
complex1: $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_RealDef_Oreal,type,
real: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (32)
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oord,type,
ord:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Oorder,type,
order:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Osemiring__1,type,
semiring_1:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[A: $tType] : $o ).
tff(sy_cl_Nat_Osemiring__char__0,type,
semiring_char_0:
!>[A: $tType] : $o ).
tff(sy_cl_Rings_Olinordered__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__algebra__1,type,
real_algebra_1:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__vector,type,
real_normed_vector:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__algebra__1,type,
real_n2089651433ebra_1:
!>[A: $tType] : $o ).
tff(sy_c_Complex_OIm,type,
im: complex1 > real ).
tff(sy_c_Complex_ORe,type,
re: complex1 > real ).
tff(sy_c_Complex_Ocomplex_OComplex,type,
complex: ( real * real ) > complex1 ).
tff(sy_c_Complex_Ocomplex_Ocomplex__case,type,
complex_case:
!>[T1: $tType] : ( ( fun(real,fun(real,T1)) * complex1 ) > T1 ) ).
tff(sy_c_Complex_Ocomplex_Ocomplex__rec,type,
complex_rec:
!>[T1: $tType] : ( ( fun(real,fun(real,T1)) * complex1 ) > T1 ) ).
tff(sy_c_Complex_Ocomplex_Ocomplex__size,type,
complex_size: complex1 > nat ).
tff(sy_c_Complex_Oii,type,
ii: complex1 ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_If,type,
if:
!>[A: $tType] : ( ( bool * A * A ) > A ) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat,type,
semiring_1_of_nat:
!>[A: $tType] : ( nat > A ) ).
tff(sy_c_Nat_Osemiring__1__class_Oof__nat__aux,type,
semiri532925092at_aux:
!>[A: $tType] : ( ( fun(A,A) * nat * A ) > A ) ).
tff(sy_c_Nat_Osize__class_Osize,type,
size_size:
!>[A: $tType] : ( A > 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_RealVector_Onorm__class_Onorm,type,
norm_norm:
!>[A: $tType] : ( A > real ) ).
tff(sy_c_RealVector_Oof__real,type,
of_real:
!>[A: $tType] : ( real > A ) ).
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_n,type,
n: nat ).
%----Relevant facts (99)
tff(fact_0_of__nat__0,axiom,
! [A: $tType] :
( semiring_1(A)
=> ( semiring_1_of_nat(A,zero_zero(nat)) = zero_zero(A) ) ) ).
tff(fact_1_complex_Oinject,axiom,
! [Real23: real,Real13: real,Real22: real,Real12: real] :
( ( complex(Real12,Real22) = complex(Real13,Real23) )
<=> ( ( Real12 = Real13 )
& ( Real22 = Real23 ) ) ) ).
tff(fact_2_of__nat__eq__iff,axiom,
! [A: $tType] :
( semiring_char_0(A)
=> ! [Na: nat,M1: nat] :
( ( semiring_1_of_nat(A,M1) = semiring_1_of_nat(A,Na) )
<=> ( M1 = Na ) ) ) ).
tff(fact_3_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [X1: A] :
( ( zero_zero(A) = X1 )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_4_complex__Re__of__nat,axiom,
! [N: nat] : re(semiring_1_of_nat(complex1,N)) = semiring_1_of_nat(real,N) ).
tff(fact_5_complex_Orecs,axiom,
! [A: $tType,Real22: real,Real12: real,F1: fun(real,fun(real,A))] : complex_rec(A,F1,complex(Real12,Real22)) = aa(real,A,aa(real,fun(real,A),F1,Real12),Real22) ).
tff(fact_6_complex_Osimps_I2_J,axiom,
! [A: $tType,Real22: real,Real12: real,F1: fun(real,fun(real,A))] : complex_case(A,F1,complex(Real12,Real22)) = aa(real,A,aa(real,fun(real,A),F1,Real12),Real22) ).
tff(fact_7_complex__Im__of__nat,axiom,
! [N: nat] : im(semiring_1_of_nat(complex1,N)) = zero_zero(real) ).
tff(fact_8_of__real__of__nat__eq,axiom,
! [A: $tType] :
( real_algebra_1(A)
=> ! [N: nat] : of_real(A,semiring_1_of_nat(real,N)) = semiring_1_of_nat(A,N) ) ).
tff(fact_9_norm__of__nat,axiom,
! [A: $tType] :
( real_n2089651433ebra_1(A)
=> ! [N: nat] : norm_norm(A,semiring_1_of_nat(A,N)) = semiring_1_of_nat(real,N) ) ).
tff(fact_10_complex_Oexhaust,axiom,
! [Y: complex1] :
~ ! [Real11: real,Real21: real] : Y != complex(Real11,Real21) ).
tff(fact_11_of__real__eq__iff,axiom,
! [A: $tType] :
( real_algebra_1(A)
=> ! [Y1: real,X1: real] :
( ( of_real(A,X1) = of_real(A,Y1) )
<=> ( X1 = Y1 ) ) ) ).
tff(fact_12_complex__Re__zero,axiom,
re(zero_zero(complex1)) = zero_zero(real) ).
tff(fact_13_complex__Im__zero,axiom,
im(zero_zero(complex1)) = zero_zero(real) ).
tff(fact_14_Im__complex__of__real,axiom,
! [Z: real] : im(of_real(complex1,Z)) = zero_zero(real) ).
tff(fact_15_complex__eq__cancel__iff2,axiom,
! [Xa: real,Y1: real,X1: real] :
( ( complex(X1,Y1) = of_real(complex1,Xa) )
<=> ( ( X1 = Xa )
& ( Y1 = zero_zero(real) ) ) ) ).
tff(fact_16_Complex__eq__0,axiom,
! [B2: real,A2: real] :
( ( complex(A2,B2) = zero_zero(complex1) )
<=> ( ( A2 = zero_zero(real) )
& ( B2 = zero_zero(real) ) ) ) ).
tff(fact_17_Im,axiom,
! [Y: real,X: real] : im(complex(X,Y)) = Y ).
tff(fact_18_norm__eq__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X1: A] :
( ( norm_norm(A,X1) = zero_zero(real) )
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_19_norm__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ( norm_norm(A,zero_zero(A)) = zero_zero(real) ) ) ).
tff(fact_20_of__real__eq__0__iff,axiom,
! [A: $tType] :
( real_algebra_1(A)
=> ! [X1: real] :
( ( of_real(A,X1) = zero_zero(A) )
<=> ( X1 = zero_zero(real) ) ) ) ).
tff(fact_21_of__real__0,axiom,
! [A: $tType] :
( real_algebra_1(A)
=> ( of_real(A,zero_zero(real)) = zero_zero(A) ) ) ).
tff(fact_22_Re__complex__of__real,axiom,
! [Z: real] : re(of_real(complex1,Z)) = Z ).
tff(fact_23_complex__eq__iff,axiom,
! [Y1: complex1,X1: complex1] :
( ( X1 = Y1 )
<=> ( ( re(X1) = re(Y1) )
& ( im(X1) = im(Y1) ) ) ) ).
tff(fact_24_complex__surj,axiom,
! [Z: complex1] : complex(re(Z),im(Z)) = Z ).
tff(fact_25_complex__eqI,axiom,
! [Y: complex1,X: complex1] :
( ( re(X) = re(Y) )
=> ( ( im(X) = im(Y) )
=> ( X = Y ) ) ) ).
tff(fact_26_Re,axiom,
! [Y: real,X: real] : re(complex(X,Y)) = X ).
tff(fact_27_complex__of__real__def,axiom,
! [R: real] : of_real(complex1,R) = complex(R,zero_zero(real)) ).
tff(fact_28_complex__zero__def,axiom,
zero_zero(complex1) = complex(zero_zero(real),zero_zero(real)) ).
tff(fact_29_complex_Osize_I1_J,axiom,
! [Real2: real,Real1: real] : complex_size(complex(Real1,Real2)) = zero_zero(nat) ).
tff(fact_30_complex_Osize_I2_J,axiom,
! [Real2: real,Real1: real] : size_size(complex1,complex(Real1,Real2)) = zero_zero(nat) ).
tff(fact_31_complex__Re__i,axiom,
re(ii) = zero_zero(real) ).
tff(fact_32_norm__le__zero__iff,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X1: A] :
( ord_less_eq(real,norm_norm(A,X1),zero_zero(real))
<=> ( X1 = zero_zero(A) ) ) ) ).
tff(fact_33_zero__less__norm__iff,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X1: A] :
( ord_less(real,zero_zero(real),norm_norm(A,X1))
<=> ( X1 != zero_zero(A) ) ) ) ).
tff(fact_34_transfer__int__nat__numerals_I1_J,axiom,
zero_zero(int) = semiring_1_of_nat(int,zero_zero(nat)) ).
tff(fact_35_int__0,axiom,
semiring_1_of_nat(int,zero_zero(nat)) = zero_zero(int) ).
tff(fact_36_int__eq__0__conv,axiom,
! [Na: nat] :
( ( semiring_1_of_nat(int,Na) = zero_zero(int) )
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_37_of__nat__aux_Osimps_I1_J,axiom,
! [A: $tType] :
( semiring_1(A)
=> ! [I1: A,Inc: fun(A,A)] : semiri532925092at_aux(A,Inc,zero_zero(nat),I1) = I1 ) ).
tff(fact_38_of__nat__le__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat,M1: nat] :
( ord_less_eq(A,semiring_1_of_nat(A,M1),semiring_1_of_nat(A,Na))
<=> ord_less_eq(nat,M1,Na) ) ) ).
tff(fact_39_of__nat__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat,M1: nat] :
( ord_less(A,semiring_1_of_nat(A,M1),semiring_1_of_nat(A,Na))
<=> ord_less(nat,M1,Na) ) ) ).
tff(fact_40_of__nat__0__less__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [Na: nat] :
( ord_less(A,zero_zero(A),semiring_1_of_nat(A,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ) ).
tff(fact_41_complex__i__not__zero,axiom,
ii != zero_zero(complex1) ).
tff(fact_42_int__int__eq,axiom,
! [Na: nat,M1: nat] :
( ( semiring_1_of_nat(int,M1) = semiring_1_of_nat(int,Na) )
<=> ( M1 = Na ) ) ).
tff(fact_43_Nat__Transfer_Otransfer__int__nat__relations_I1_J,axiom,
! [Y1: nat,X1: nat] :
( ( semiring_1_of_nat(int,X1) = semiring_1_of_nat(int,Y1) )
<=> ( X1 = Y1 ) ) ).
tff(fact_44_int__if__cong,axiom,
! [Y1: nat,X1: nat,P1: bool] :
( ( pp(P1)
=> ( semiring_1_of_nat(int,X1) = semiring_1_of_nat(int,if(nat,P1,X1,Y1)) ) )
& ( ~ pp(P1)
=> ( semiring_1_of_nat(int,Y1) = semiring_1_of_nat(int,if(nat,P1,X1,Y1)) ) ) ) ).
tff(fact_45_less__imp__of__nat__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N)) ) ) ).
tff(fact_46_of__nat__less__imp__less,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat,M: nat] :
( ord_less(A,semiring_1_of_nat(A,M),semiring_1_of_nat(A,N))
=> ord_less(nat,M,N) ) ) ).
tff(fact_47_complex__Re__le__cmod,axiom,
! [X: complex1] : ord_less_eq(real,re(X),norm_norm(complex1,X)) ).
tff(fact_48_of__nat__less__0__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M: nat] : ~ ord_less(A,semiring_1_of_nat(A,M),zero_zero(A)) ) ).
tff(fact_49_zero__le__imp__of__nat,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [M: nat] : ord_less_eq(A,zero_zero(A),semiring_1_of_nat(A,M)) ) ).
tff(fact_50_of__nat__0__le__iff,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ! [N: nat] : ord_less_eq(A,zero_zero(A),semiring_1_of_nat(A,N)) ) ).
tff(fact_51_norm__not__less__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X: A] : ~ ord_less(real,norm_norm(A,X),zero_zero(real)) ) ).
tff(fact_52_norm__ge__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ! [X: A] : ord_less_eq(real,zero_zero(real),norm_norm(A,X)) ) ).
tff(fact_53_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ord_less_eq(A,X,X) ) ).
tff(fact_54_less__eq__real__def,axiom,
! [Y1: real,X1: real] :
( ord_less_eq(real,X1,Y1)
<=> ( ord_less(real,X1,Y1)
| ( X1 = Y1 ) ) ) ).
tff(fact_55_order__le__less__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less(A,Y,Z)
=> ord_less(A,X,Z) ) ) ) ).
tff(fact_56_order__less__le__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( ord_less(A,X,Y)
=> ( ord_less_eq(A,Y,Z)
=> ord_less(A,X,Z) ) ) ) ).
tff(fact_57_order__le__neq__trans,axiom,
! [A: $tType] :
( order(A)
=> ! [B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ( A1 != B1 )
=> ord_less(A,A1,B1) ) ) ) ).
tff(fact_58_neq0__conv,axiom,
! [Na: nat] :
( ( Na != zero_zero(nat) )
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_59_less__nat__zero__code,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_60_less__zeroE,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_61_le0,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_62_less__eq__nat_Osimps_I1_J,axiom,
! [N: nat] : ord_less_eq(nat,zero_zero(nat),N) ).
tff(fact_63_le__0__eq,axiom,
! [Na: nat] :
( ord_less_eq(nat,Na,zero_zero(nat))
<=> ( Na = zero_zero(nat) ) ) ).
tff(fact_64_less__not__refl,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_65_le__refl,axiom,
! [N: nat] : ord_less_eq(nat,N,N) ).
tff(fact_66_nat__neq__iff,axiom,
! [Na: nat,M1: nat] :
( ( M1 != Na )
<=> ( ord_less(nat,M1,Na)
| ord_less(nat,Na,M1) ) ) ).
tff(fact_67_nat__le__linear,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
| ord_less_eq(nat,N,M) ) ).
tff(fact_68_nat__less__le,axiom,
! [Na: nat,M1: nat] :
( ord_less(nat,M1,Na)
<=> ( ord_less_eq(nat,M1,Na)
& ( M1 != Na ) ) ) ).
tff(fact_69_le__eq__less__or__eq,axiom,
! [Na: nat,M1: nat] :
( ord_less_eq(nat,M1,Na)
<=> ( ord_less(nat,M1,Na)
| ( M1 = Na ) ) ) ).
tff(fact_70_zless__int,axiom,
! [Na: nat,M1: nat] :
( ord_less(int,semiring_1_of_nat(int,M1),semiring_1_of_nat(int,Na))
<=> ord_less(nat,M1,Na) ) ).
tff(fact_71_Nat__Transfer_Otransfer__int__nat__relations_I2_J,axiom,
! [Y1: nat,X1: nat] :
( ord_less(int,semiring_1_of_nat(int,X1),semiring_1_of_nat(int,Y1))
<=> ord_less(nat,X1,Y1) ) ).
tff(fact_72_zle__int,axiom,
! [Na: nat,M1: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,M1),semiring_1_of_nat(int,Na))
<=> ord_less_eq(nat,M1,Na) ) ).
tff(fact_73_Nat__Transfer_Otransfer__int__nat__relations_I3_J,axiom,
! [Y1: nat,X1: nat] :
( ord_less_eq(int,semiring_1_of_nat(int,X1),semiring_1_of_nat(int,Y1))
<=> ord_less_eq(nat,X1,Y1) ) ).
tff(fact_74_linorder__neqE__nat,axiom,
! [Y: nat,X: nat] :
( ( X != Y )
=> ( ~ ord_less(nat,X,Y)
=> ord_less(nat,Y,X) ) ) ).
tff(fact_75_less__irrefl__nat,axiom,
! [N: nat] : ~ ord_less(nat,N,N) ).
tff(fact_76_less__not__refl2,axiom,
! [M: nat,N: nat] :
( ord_less(nat,N,M)
=> ( M != N ) ) ).
tff(fact_77_less__not__refl3,axiom,
! [T: nat,S: nat] :
( ord_less(nat,S,T)
=> ( S != T ) ) ).
tff(fact_78_eq__imp__le,axiom,
! [N: nat,M: nat] :
( ( M = N )
=> ord_less_eq(nat,M,N) ) ).
tff(fact_79_less__imp__le__nat,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ord_less_eq(nat,M,N) ) ).
tff(fact_80_le__neq__implies__less,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
=> ( ( M != N )
=> ord_less(nat,M,N) ) ) ).
tff(fact_81_le__trans,axiom,
! [K: nat,J: nat,I: nat] :
( ord_less_eq(nat,I,J)
=> ( ord_less_eq(nat,J,K)
=> ord_less_eq(nat,I,K) ) ) ).
tff(fact_82_le__antisym,axiom,
! [N: nat,M: nat] :
( ord_less_eq(nat,M,N)
=> ( ord_less_eq(nat,N,M)
=> ( M = N ) ) ) ).
tff(fact_83_less__or__eq__imp__le,axiom,
! [N: nat,M: nat] :
( ( ord_less(nat,M,N)
| ( M = N ) )
=> ord_less_eq(nat,M,N) ) ).
tff(fact_84_nat__less__cases,axiom,
! [P1: fun(nat,fun(nat,bool)),Na: nat,M1: nat] :
( ( ord_less(nat,M1,Na)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) )
=> ( ( ( M1 = Na )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) )
=> ( ( ord_less(nat,Na,M1)
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) )
=> pp(aa(nat,bool,aa(nat,fun(nat,bool),P1,Na),M1)) ) ) ) ).
tff(fact_85_not__less0,axiom,
! [N: nat] : ~ ord_less(nat,N,zero_zero(nat)) ).
tff(fact_86_gr__implies__not0,axiom,
! [N: nat,M: nat] :
( ord_less(nat,M,N)
=> ( N != zero_zero(nat) ) ) ).
tff(fact_87_gr0I,axiom,
! [N: nat] :
( ( N != zero_zero(nat) )
=> ord_less(nat,zero_zero(nat),N) ) ).
tff(fact_88_Nat__Transfer_Otransfer__nat__int__function__closures_I5_J,axiom,
ord_less_eq(int,zero_zero(int),zero_zero(int)) ).
tff(fact_89_zero__less__int__conv,axiom,
! [Na: nat] :
( ord_less(int,zero_zero(int),semiring_1_of_nat(int,Na))
<=> ord_less(nat,zero_zero(nat),Na) ) ).
tff(fact_90_int__less__0__conv,axiom,
! [K: nat] : ~ ord_less(int,semiring_1_of_nat(int,K),zero_zero(int)) ).
tff(fact_91_transfer__int__nat__quantifiers_I1_J,axiom,
! [P1: fun(int,bool)] :
( ! [X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
=> pp(aa(int,bool,P1,X2)) )
<=> ! [X2: nat] : pp(aa(int,bool,P1,semiring_1_of_nat(int,X2))) ) ).
tff(fact_92_transfer__int__nat__quantifiers_I2_J,axiom,
! [P1: fun(int,bool)] :
( ? [X2: int] :
( ord_less_eq(int,zero_zero(int),X2)
& pp(aa(int,bool,P1,X2)) )
<=> ? [X2: nat] : pp(aa(int,bool,P1,semiring_1_of_nat(int,X2))) ) ).
tff(fact_93_Nat__Transfer_Otransfer__nat__int__function__closures_I9_J,axiom,
! [Z: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,Z)) ).
tff(fact_94_zero__zle__int,axiom,
! [N: nat] : ord_less_eq(int,zero_zero(int),semiring_1_of_nat(int,N)) ).
tff(fact_95_le__fun__def,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
<=> ! [X2: A] : ord_less_eq(B,aa(A,B,F,X2),aa(A,B,G,X2)) ) ) ).
tff(fact_96_linorder__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
| ord_less_eq(A,Y,X) ) ) ).
tff(fact_97_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,X1: A] :
( ( X1 = Y1 )
<=> ( ord_less_eq(A,X1,Y1)
& ord_less_eq(A,Y1,X1) ) ) ) ).
tff(fact_98_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ( X = Y )
=> ord_less_eq(A,X,Y) ) ) ).
%----Arities (40)
tff(arity_fun___Orderings_Opreorder,axiom,
! [T_1: $tType,T_2: $tType] :
( preorder(T_2)
=> preorder(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oorder,axiom,
! [T_1: $tType,T_2: $tType] :
( order(T_2)
=> order(fun(T_1,T_2)) ) ).
tff(arity_fun___Orderings_Oord,axiom,
! [T_1: $tType,T_2: $tType] :
( ord(T_2)
=> ord(fun(T_1,T_2)) ) ).
tff(arity_Int_Oint___Rings_Olinordered__semidom,axiom,
linordered_semidom(int) ).
tff(arity_Int_Oint___Nat_Osemiring__char__0,axiom,
semiring_char_0(int) ).
tff(arity_Int_Oint___Orderings_Opreorder,axiom,
preorder(int) ).
tff(arity_Int_Oint___Orderings_Olinorder,axiom,
linorder(int) ).
tff(arity_Int_Oint___Rings_Osemiring__1,axiom,
semiring_1(int) ).
tff(arity_Int_Oint___Orderings_Oorder,axiom,
order(int) ).
tff(arity_Int_Oint___Orderings_Oord,axiom,
ord(int) ).
tff(arity_Int_Oint___Groups_Ozero,axiom,
zero(int) ).
tff(arity_Nat_Onat___Rings_Olinordered__semidom,axiom,
linordered_semidom(nat) ).
tff(arity_Nat_Onat___Nat_Osemiring__char__0,axiom,
semiring_char_0(nat) ).
tff(arity_Nat_Onat___Orderings_Opreorder,axiom,
preorder(nat) ).
tff(arity_Nat_Onat___Orderings_Olinorder,axiom,
linorder(nat) ).
tff(arity_Nat_Onat___Rings_Osemiring__1,axiom,
semiring_1(nat) ).
tff(arity_Nat_Onat___Orderings_Oorder,axiom,
order(nat) ).
tff(arity_Nat_Onat___Orderings_Oord,axiom,
ord(nat) ).
tff(arity_Nat_Onat___Groups_Ozero,axiom,
zero(nat) ).
tff(arity_HOL_Obool___Orderings_Opreorder,axiom,
preorder(bool) ).
tff(arity_HOL_Obool___Orderings_Olinorder,axiom,
linorder(bool) ).
tff(arity_HOL_Obool___Orderings_Oorder,axiom,
order(bool) ).
tff(arity_HOL_Obool___Orderings_Oord,axiom,
ord(bool) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__algebra__1,axiom,
real_n2089651433ebra_1(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__algebra__1,axiom,
real_algebra_1(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom(real) ).
tff(arity_RealDef_Oreal___Nat_Osemiring__char__0,axiom,
semiring_char_0(real) ).
tff(arity_RealDef_Oreal___Orderings_Opreorder,axiom,
preorder(real) ).
tff(arity_RealDef_Oreal___Orderings_Olinorder,axiom,
linorder(real) ).
tff(arity_RealDef_Oreal___Rings_Osemiring__1,axiom,
semiring_1(real) ).
tff(arity_RealDef_Oreal___Orderings_Oorder,axiom,
order(real) ).
tff(arity_RealDef_Oreal___Orderings_Oord,axiom,
ord(real) ).
tff(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__algebra__1,axiom,
real_n2089651433ebra_1(complex1) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(complex1) ).
tff(arity_Complex_Ocomplex___RealVector_Oreal__algebra__1,axiom,
real_algebra_1(complex1) ).
tff(arity_Complex_Ocomplex___Nat_Osemiring__char__0,axiom,
semiring_char_0(complex1) ).
tff(arity_Complex_Ocomplex___Rings_Osemiring__1,axiom,
semiring_1(complex1) ).
tff(arity_Complex_Ocomplex___Groups_Ozero,axiom,
zero(complex1) ).
%----Helper facts (5)
tff(help_If_1_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fTrue,X,Y) = X ).
tff(help_If_2_1_T,axiom,
! [A: $tType,Y: A,X: A] : if(A,fFalse,X,Y) = Y ).
tff(help_If_3_1_T,axiom,
! [P: bool] :
( ( P = fTrue )
| ( P = fFalse ) ) ).
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
semiring_1_of_nat(complex1,n) = complex(semiring_1_of_nat(real,n),zero_zero(real)) ).
%------------------------------------------------------------------------------