TPTP Problem File: SWV593_5.p
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%------------------------------------------------------------------------------
% File : SWV593_5 : TPTP v8.2.0. Released v6.0.0.
% Domain : Software Verification
% Problem : Fast Fourier Transform line 123
% 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_123 [Bla13]
% Status : Unknown
% Rating : 1.00 v6.4.0
% Syntax : Number of formulae : 157 ( 40 unt; 34 typ; 0 def)
% Number of atoms : 269 ( 62 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 168 ( 22 ~; 6 |; 3 &)
% ( 27 <=>; 110 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 4 avg)
% Maximal term depth : 3 ( 1 avg)
% Number of types : 3 ( 2 usr)
% Number of type conns : 18 ( 12 >; 6 *; 0 +; 0 <<)
% Number of predicates : 19 ( 18 usr; 0 prp; 1-3 aty)
% Number of functors : 14 ( 14 usr; 3 con; 0-4 aty)
% Number of variables : 222 ( 197 !; 1 ?; 222 :)
% ( 24 !>; 0 ?*; 0 @-; 0 @+)
% SPC : TF1_UNK_EQU_NAR
% Comments : This file was generated by Isabelle (most likely Sledgehammer)
% 2011-12-13 16:15:37
%------------------------------------------------------------------------------
%----Should-be-implicit typings (3)
tff(ty_tc_HOL_Obool,type,
bool: $tType ).
tff(ty_tc_RealDef_Oreal,type,
real: $tType ).
tff(ty_tc_fun,type,
fun: ( $tType * $tType ) > $tType ).
%----Explicit typings (31)
tff(sy_cl_Groups_Oone,type,
one:
!>[A: $tType] : $o ).
tff(sy_cl_SEQ_Obanach,type,
banach:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Ozero,type,
zero:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Osgn__if,type,
sgn_if:
!>[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_Orderings_Olinorder,type,
linorder:
!>[A: $tType] : $o ).
tff(sy_cl_Orderings_Opreorder,type,
preorder:
!>[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__semidom,type,
linordered_semidom:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__field,type,
real_normed_field:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__vector,type,
real_normed_vector:
!>[A: $tType] : $o ).
tff(sy_cl_Groups_Oordered__ab__group__add__abs,type,
ordere142940540dd_abs:
!>[A: $tType] : $o ).
tff(sy_cl_RealVector_Oreal__normed__algebra__1,type,
real_n2089651433ebra_1:
!>[A: $tType] : $o ).
tff(sy_c_Groups_Oabs__class_Oabs,type,
abs_abs:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Oone__class_Oone,type,
one_one:
!>[A: $tType] : A ).
tff(sy_c_Groups_Osgn__class_Osgn,type,
sgn_sgn:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Groups_Ozero__class_Ozero,type,
zero_zero:
!>[A: $tType] : A ).
tff(sy_c_Log_Olog,type,
log: ( real * real ) > real ).
tff(sy_c_Log_Opowr,type,
powr: ( real * real ) > real ).
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_Transcendental_Ocos,type,
cos: real > real ).
tff(sy_c_Transcendental_Oexp,type,
exp:
!>[A: $tType] : ( A > A ) ).
tff(sy_c_Transcendental_Osin,type,
sin: real > real ).
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_x,type,
x: real ).
%----Relevant facts (98)
tff(fact_0__C0_C,axiom,
ord_less(real,zero_zero(real),x) ).
tff(fact_1_cos__zero,axiom,
cos(zero_zero(real)) = one_one(real) ).
tff(fact_2_sin__zero,axiom,
sin(zero_zero(real)) = zero_zero(real) ).
tff(fact_3_cos__one__sin__zero,axiom,
! [X: real] :
( ( cos(X) = one_one(real) )
=> ( sin(X) = zero_zero(real) ) ) ).
tff(fact_4_one__neq__zero,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( one_one(A) != zero_zero(A) ) ) ).
tff(fact_5_zero__neq__one,axiom,
! [A: $tType] :
( zero_neq_one(A)
=> ( zero_zero(A) != one_one(A) ) ) ).
tff(fact_6_one__reorient,axiom,
! [A: $tType] :
( one(A)
=> ! [Xa: A] :
( ( one_one(A) = Xa )
<=> ( Xa = one_one(A) ) ) ) ).
tff(fact_7_zero__reorient,axiom,
! [A: $tType] :
( zero(A)
=> ! [Xa: A] :
( ( zero_zero(A) = Xa )
<=> ( Xa = zero_zero(A) ) ) ) ).
tff(fact_8_sin__zero__abs__cos__one,axiom,
! [X: real] :
( ( sin(X) = zero_zero(real) )
=> ( abs_abs(real,cos(X)) = one_one(real) ) ) ).
tff(fact_9_log__one,axiom,
! [A1: real] : log(A1,one_one(real)) = zero_zero(real) ).
tff(fact_10_exp__zero,axiom,
! [A: $tType] :
( ( real_normed_field(A)
& banach(A) )
=> ( exp(A,zero_zero(A)) = one_one(A) ) ) ).
tff(fact_11_exp__less__cancel__iff,axiom,
! [Y1: real,Xa: real] :
( ord_less(real,exp(real,Xa),exp(real,Y1))
<=> ord_less(real,Xa,Y1) ) ).
tff(fact_12_exp__less__mono,axiom,
! [Y: real,X: real] :
( ord_less(real,X,Y)
=> ord_less(real,exp(real,X),exp(real,Y)) ) ).
tff(fact_13_abs__idempotent,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] : abs_abs(A,abs_abs(A,A1)) = abs_abs(A,A1) ) ).
tff(fact_14_abs__exp__cancel,axiom,
! [X: real] : abs_abs(real,exp(real,X)) = exp(real,X) ).
tff(fact_15_abs__eq__0,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A2: A] :
( ( abs_abs(A,A2) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_16_abs__zero,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ( abs_abs(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_17_abs__one,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ( abs_abs(A,one_one(A)) = one_one(A) ) ) ).
tff(fact_18_exp__eq__one__iff,axiom,
! [Xa: real] :
( ( exp(real,Xa) = one_one(real) )
<=> ( Xa = zero_zero(real) ) ) ).
tff(fact_19_zero__less__abs__iff,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A2: A] :
( ord_less(A,zero_zero(A),abs_abs(A,A2))
<=> ( A2 != zero_zero(A) ) ) ) ).
tff(fact_20_exp__less__one__iff,axiom,
! [Xa: real] :
( ord_less(real,exp(real,Xa),one_one(real))
<=> ord_less(real,Xa,zero_zero(real)) ) ).
tff(fact_21_one__less__exp__iff,axiom,
! [Xa: real] :
( ord_less(real,one_one(real),exp(real,Xa))
<=> ord_less(real,zero_zero(real),Xa) ) ).
tff(fact_22_not__exp__less__zero,axiom,
! [X: real] : ~ ord_less(real,exp(real,X),zero_zero(real)) ).
tff(fact_23_abs__not__less__zero,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] : ~ ord_less(A,abs_abs(A,A1),zero_zero(A)) ) ).
tff(fact_24_exp__gt__zero,axiom,
! [X: real] : ord_less(real,zero_zero(real),exp(real,X)) ).
tff(fact_25_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_26_abs__of__pos,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( abs_abs(A,A1) = A1 ) ) ) ).
tff(fact_27_exp__less__cancel,axiom,
! [Y: real,X: real] :
( ord_less(real,exp(real,X),exp(real,Y))
=> ord_less(real,X,Y) ) ).
tff(fact_28_exp__gt__one,axiom,
! [X: real] :
( ord_less(real,zero_zero(real),X)
=> ord_less(real,one_one(real),exp(real,X)) ) ).
tff(fact_29_exp__not__eq__zero,axiom,
! [A: $tType] :
( ( real_normed_field(A)
& banach(A) )
=> ! [X: A] : exp(A,X) != zero_zero(A) ) ).
tff(fact_30_log__less__cancel__iff,axiom,
! [Y1: real,Xa: real,A2: real] :
( ord_less(real,one_one(real),A2)
=> ( ord_less(real,zero_zero(real),Xa)
=> ( ord_less(real,zero_zero(real),Y1)
=> ( ord_less(real,log(A2,Xa),log(A2,Y1))
<=> ord_less(real,Xa,Y1) ) ) ) ) ).
tff(fact_31_log__eq__one,axiom,
! [A1: real] :
( ord_less(real,zero_zero(real),A1)
=> ( ( A1 != one_one(real) )
=> ( log(A1,A1) = one_one(real) ) ) ) ).
tff(fact_32_zero__less__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less(A,zero_zero(A),one_one(A)) ) ).
tff(fact_33_not__one__less__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less(A,one_one(A),zero_zero(A)) ) ).
tff(fact_34_exp__total,axiom,
! [Y: real] :
( ord_less(real,zero_zero(real),Y)
=> ? [X2: real] : exp(real,X2) = Y ) ).
tff(fact_35_powr__log__cancel,axiom,
! [X: real,A1: real] :
( ord_less(real,zero_zero(real),A1)
=> ( ( A1 != one_one(real) )
=> ( ord_less(real,zero_zero(real),X)
=> ( powr(A1,log(A1,X)) = X ) ) ) ) ).
tff(fact_36_log__powr__cancel,axiom,
! [Y: real,A1: real] :
( ord_less(real,zero_zero(real),A1)
=> ( ( A1 != one_one(real) )
=> ( log(A1,powr(A1,Y)) = Y ) ) ) ).
tff(fact_37_log__le__cancel__iff,axiom,
! [Y1: real,Xa: real,A2: real] :
( ord_less(real,one_one(real),A2)
=> ( ord_less(real,zero_zero(real),Xa)
=> ( ord_less(real,zero_zero(real),Y1)
=> ( ord_less_eq(real,log(A2,Xa),log(A2,Y1))
<=> ord_less_eq(real,Xa,Y1) ) ) ) ) ).
tff(fact_38_powr__one__gt__zero__iff,axiom,
! [Xa: real] :
( ( powr(Xa,one_one(real)) = Xa )
<=> ord_less(real,zero_zero(real),Xa) ) ).
tff(fact_39_sgn__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] :
( ord_less(A,zero_zero(A),A1)
=> ( sgn_sgn(A,A1) = one_one(A) ) ) ) ).
tff(fact_40_sgn__1__pos,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( ( sgn_sgn(A,A2) = one_one(A) )
<=> ord_less(A,zero_zero(A),A2) ) ) ).
tff(fact_41_abs__sin__le__one,axiom,
! [X: real] : ord_less_eq(real,abs_abs(real,sin(X)),one_one(real)) ).
tff(fact_42_abs__cos__le__one,axiom,
! [X: real] : ord_less_eq(real,abs_abs(real,cos(X)),one_one(real)) ).
tff(fact_43_sgn__sgn,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A1: A] : sgn_sgn(A,sgn_sgn(A,A1)) = sgn_sgn(A,A1) ) ).
tff(fact_44_exp__inj__iff,axiom,
! [Y1: real,Xa: real] :
( ( exp(real,Xa) = exp(real,Y1) )
<=> ( Xa = Y1 ) ) ).
tff(fact_45_sgn0,axiom,
! [A: $tType] :
( sgn_if(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_46_exp__le__cancel__iff,axiom,
! [Y1: real,Xa: real] :
( ord_less_eq(real,exp(real,Xa),exp(real,Y1))
<=> ord_less_eq(real,Xa,Y1) ) ).
tff(fact_47_powr__one__eq__one,axiom,
! [A1: real] : powr(one_one(real),A1) = one_one(real) ).
tff(fact_48_abs__le__zero__iff,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A2: A] :
( ord_less_eq(A,abs_abs(A,A2),zero_zero(A))
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_49_sgn__less,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( ord_less(A,sgn_sgn(A,A2),zero_zero(A))
<=> ord_less(A,A2,zero_zero(A)) ) ) ).
tff(fact_50_sgn__greater,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( ord_less(A,zero_zero(A),sgn_sgn(A,A2))
<=> ord_less(A,zero_zero(A),A2) ) ) ).
tff(fact_51_powr__zero__eq__one,axiom,
! [X: real] : powr(X,zero_zero(real)) = one_one(real) ).
tff(fact_52_one__le__exp__iff,axiom,
! [Xa: real] :
( ord_less_eq(real,one_one(real),exp(real,Xa))
<=> ord_less_eq(real,zero_zero(real),Xa) ) ).
tff(fact_53_exp__le__one__iff,axiom,
! [Xa: real] :
( ord_less_eq(real,exp(real,Xa),one_one(real))
<=> ord_less_eq(real,Xa,zero_zero(real)) ) ).
tff(fact_54_powr__ge__pzero,axiom,
! [Y: real,X: real] : ord_less_eq(real,zero_zero(real),powr(X,Y)) ).
tff(fact_55_powr__powr__swap,axiom,
! [B1: real,A1: real,X: real] : powr(powr(X,A1),B1) = powr(powr(X,B1),A1) ).
tff(fact_56_powr__mono,axiom,
! [X: real,B1: real,A1: real] :
( ord_less_eq(real,A1,B1)
=> ( ord_less_eq(real,one_one(real),X)
=> ord_less_eq(real,powr(X,A1),powr(X,B1)) ) ) ).
tff(fact_57_powr__mono2,axiom,
! [Y: real,X: real,A1: real] :
( ord_less_eq(real,zero_zero(real),A1)
=> ( ord_less(real,zero_zero(real),X)
=> ( ord_less_eq(real,X,Y)
=> ord_less_eq(real,powr(X,A1),powr(Y,A1)) ) ) ) ).
tff(fact_58_powr__le__cancel__iff,axiom,
! [B2: real,A2: real,Xa: real] :
( ord_less(real,one_one(real),Xa)
=> ( ord_less_eq(real,powr(Xa,A2),powr(Xa,B2))
<=> ord_less_eq(real,A2,B2) ) ) ).
tff(fact_59_ge__one__powr__ge__zero,axiom,
! [A1: real,X: real] :
( ord_less_eq(real,one_one(real),X)
=> ( ord_less_eq(real,zero_zero(real),A1)
=> ord_less_eq(real,one_one(real),powr(X,A1)) ) ) ).
tff(fact_60_exp__ge__zero,axiom,
! [X: real] : ord_less_eq(real,zero_zero(real),exp(real,X)) ).
tff(fact_61_not__exp__le__zero,axiom,
! [X: real] : ~ ord_less_eq(real,exp(real,X),zero_zero(real)) ).
tff(fact_62_abs__le__D1,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [B1: A,A1: A] :
( ord_less_eq(A,abs_abs(A,A1),B1)
=> ord_less_eq(A,A1,B1) ) ) ).
tff(fact_63_abs__ge__self,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] : ord_less_eq(A,A1,abs_abs(A,A1)) ) ).
tff(fact_64_powr__not__zero,axiom,
! [A1: real,X: real] : powr(X,A1) != zero_zero(real) ).
tff(fact_65_sgn__0__0,axiom,
! [A: $tType] :
( linordered_idom(A)
=> ! [A2: A] :
( ( sgn_sgn(A,A2) = zero_zero(A) )
<=> ( A2 = zero_zero(A) ) ) ) ).
tff(fact_66_not__one__le__zero,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ~ ord_less_eq(A,one_one(A),zero_zero(A)) ) ).
tff(fact_67_zero__le__one,axiom,
! [A: $tType] :
( linordered_semidom(A)
=> ord_less_eq(A,zero_zero(A),one_one(A)) ) ).
tff(fact_68_abs__of__nonneg,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] :
( ord_less_eq(A,zero_zero(A),A1)
=> ( abs_abs(A,A1) = A1 ) ) ) ).
tff(fact_69_abs__ge__zero,axiom,
! [A: $tType] :
( ordere142940540dd_abs(A)
=> ! [A1: A] : ord_less_eq(A,zero_zero(A),abs_abs(A,A1)) ) ).
tff(fact_70_powr__less__mono2__neg,axiom,
! [Y: real,X: real,A1: real] :
( ord_less(real,A1,zero_zero(real))
=> ( ord_less(real,zero_zero(real),X)
=> ( ord_less(real,X,Y)
=> ord_less(real,powr(Y,A1),powr(X,A1)) ) ) ) ).
tff(fact_71_powr__less__mono2,axiom,
! [Y: real,X: real,A1: real] :
( ord_less(real,zero_zero(real),A1)
=> ( ord_less(real,zero_zero(real),X)
=> ( ord_less(real,X,Y)
=> ord_less(real,powr(X,A1),powr(Y,A1)) ) ) ) ).
tff(fact_72_powr__gt__zero,axiom,
! [A1: real,X: real] : ord_less(real,zero_zero(real),powr(X,A1)) ).
tff(fact_73_powr__less__cancel,axiom,
! [B1: real,A1: real,X: real] :
( ord_less(real,powr(X,A1),powr(X,B1))
=> ( ord_less(real,one_one(real),X)
=> ord_less(real,A1,B1) ) ) ).
tff(fact_74_powr__less__mono,axiom,
! [X: real,B1: real,A1: real] :
( ord_less(real,A1,B1)
=> ( ord_less(real,one_one(real),X)
=> ord_less(real,powr(X,A1),powr(X,B1)) ) ) ).
tff(fact_75_powr__less__cancel__iff,axiom,
! [B2: real,A2: real,Xa: real] :
( ord_less(real,one_one(real),Xa)
=> ( ord_less(real,powr(Xa,A2),powr(Xa,B2))
<=> ord_less(real,A2,B2) ) ) ).
tff(fact_76_cos__le__one,axiom,
! [X: real] : ord_less_eq(real,cos(X),one_one(real)) ).
tff(fact_77_sin__le__one,axiom,
! [X: real] : ord_less_eq(real,sin(X),one_one(real)) ).
tff(fact_78_sgn__one,axiom,
! [A: $tType] :
( real_n2089651433ebra_1(A)
=> ( sgn_sgn(A,one_one(A)) = one_one(A) ) ) ).
tff(fact_79_sgn__zero,axiom,
! [A: $tType] :
( real_normed_vector(A)
=> ( sgn_sgn(A,zero_zero(A)) = zero_zero(A) ) ) ).
tff(fact_80_real__sgn__pos,axiom,
! [X: real] :
( ord_less(real,zero_zero(real),X)
=> ( sgn_sgn(real,X) = one_one(real) ) ) ).
tff(fact_81_order__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ord_less_eq(A,X,X) ) ).
tff(fact_82_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)
<=> ! [X1: A] : ord_less_eq(B,aa(A,B,F,X1),aa(A,B,G,X1)) ) ) ).
tff(fact_83_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_84_order__eq__iff,axiom,
! [A: $tType] :
( order(A)
=> ! [Y1: A,Xa: A] :
( ( Xa = Y1 )
<=> ( ord_less_eq(A,Xa,Y1)
& ord_less_eq(A,Y1,Xa) ) ) ) ).
tff(fact_85_order__eq__refl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Y: A,X: A] :
( ( X = Y )
=> ord_less_eq(A,X,Y) ) ) ).
tff(fact_86_le__funD,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [Xa: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,Xa),aa(A,B,G,Xa)) ) ) ).
tff(fact_87_order__antisym__conv,axiom,
! [A: $tType] :
( order(A)
=> ! [Xa: A,Y1: A] :
( ord_less_eq(A,Y1,Xa)
=> ( ord_less_eq(A,Xa,Y1)
<=> ( Xa = Y1 ) ) ) ) ).
tff(fact_88_ord__eq__le__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A1: A] :
( ( A1 = B1 )
=> ( ord_less_eq(A,B1,C)
=> ord_less_eq(A,A1,C) ) ) ) ).
tff(fact_89_ord__le__eq__trans,axiom,
! [A: $tType] :
( ord(A)
=> ! [C: A,B1: A,A1: A] :
( ord_less_eq(A,A1,B1)
=> ( ( B1 = C )
=> ord_less_eq(A,A1,C) ) ) ) ).
tff(fact_90_order__antisym,axiom,
! [A: $tType] :
( order(A)
=> ! [Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less_eq(A,Y,X)
=> ( X = Y ) ) ) ) ).
tff(fact_91_order__trans,axiom,
! [A: $tType] :
( preorder(A)
=> ! [Z: A,Y: A,X: A] :
( ord_less_eq(A,X,Y)
=> ( ord_less_eq(A,Y,Z)
=> ord_less_eq(A,X,Z) ) ) ) ).
tff(fact_92_le__funE,axiom,
! [B: $tType,A: $tType] :
( ord(B)
=> ! [Xa: A,G: fun(A,B),F: fun(A,B)] :
( ord_less_eq(fun(A,B),F,G)
=> ord_less_eq(B,aa(A,B,F,Xa),aa(A,B,G,Xa)) ) ) ).
tff(fact_93_linorder__le__cases,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ~ ord_less_eq(A,X,Y)
=> ord_less_eq(A,Y,X) ) ) ).
tff(fact_94_order__less__irrefl,axiom,
! [A: $tType] :
( preorder(A)
=> ! [X: A] : ~ ord_less(A,X,X) ) ).
tff(fact_95_linorder__neq__iff,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,Xa: A] :
( ( Xa != Y1 )
<=> ( ord_less(A,Xa,Y1)
| ord_less(A,Y1,Xa) ) ) ) ).
tff(fact_96_not__less__iff__gr__or__eq,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y1: A,Xa: A] :
( ~ ord_less(A,Xa,Y1)
<=> ( ord_less(A,Y1,Xa)
| ( Xa = Y1 ) ) ) ) ).
tff(fact_97_linorder__less__linear,axiom,
! [A: $tType] :
( linorder(A)
=> ! [Y: A,X: A] :
( ord_less(A,X,Y)
| ( X = Y )
| ord_less(A,Y,X) ) ) ).
%----Arities (22)
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_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___Groups_Oordered__ab__group__add__abs,axiom,
ordere142940540dd_abs(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__vector,axiom,
real_normed_vector(real) ).
tff(arity_RealDef_Oreal___RealVector_Oreal__normed__field,axiom,
real_normed_field(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__semidom,axiom,
linordered_semidom(real) ).
tff(arity_RealDef_Oreal___Rings_Olinordered__idom,axiom,
linordered_idom(real) ).
tff(arity_RealDef_Oreal___Rings_Ozero__neq__one,axiom,
zero_neq_one(real) ).
tff(arity_RealDef_Oreal___Orderings_Opreorder,axiom,
preorder(real) ).
tff(arity_RealDef_Oreal___Orderings_Olinorder,axiom,
linorder(real) ).
tff(arity_RealDef_Oreal___Orderings_Oorder,axiom,
order(real) ).
tff(arity_RealDef_Oreal___Orderings_Oord,axiom,
ord(real) ).
tff(arity_RealDef_Oreal___Groups_Osgn__if,axiom,
sgn_if(real) ).
tff(arity_RealDef_Oreal___Groups_Ozero,axiom,
zero(real) ).
tff(arity_RealDef_Oreal___SEQ_Obanach,axiom,
banach(real) ).
tff(arity_RealDef_Oreal___Groups_Oone,axiom,
one(real) ).
%----Helper facts (2)
tff(help_pp_1_1_U,axiom,
~ pp(fFalse) ).
tff(help_pp_2_1_U,axiom,
pp(fTrue) ).
%----Conjectures (1)
tff(conj_0,conjecture,
( ( sin(x) != zero_zero(real) )
| ( cos(x) != one_one(real) ) ) ).
%------------------------------------------------------------------------------