TPTP Problem File: SWW825_1.p
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%------------------------------------------------------------------------------
% File : SWW825_1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Software Verification
% Problem : Fast Fourier Transform 660068
% 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
% : [BBP11] Blanchette et al. (2011), Extending Sledgehammer with
% Source : [SMTL]
% Names : FFT/smtlib.660068.p [SMTL]
% Status : Unsatisfiable
% Rating : 1.00 v7.0.0
% Syntax : Number of formulae : 70 ( 21 unt; 29 typ; 0 def)
% Number of atoms : 77 ( 47 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 43 ( 7 ~; 0 |; 18 &)
% ( 7 <=>; 11 =>; 0 <=; 0 <~>)
% Maximal formula depth : 10 ( 4 avg)
% Maximal term depth : 6 ( 2 avg)
% Number arithmetic : 136 ( 30 atm; 12 fun; 30 num; 64 var)
% Number of types : 11 ( 9 usr; 2 ari)
% Number of type conns : 14 ( 7 >; 7 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 29 ( 20 usr; 18 con; 0-2 aty)
% Number of variables : 77 ( 60 !; 17 ?; 77 :)
% SPC : TF0_UNS_EQU_ARI
% Comments : This is an SMT-LIB AUFNIRA problem. Translated and contributed
% to SMT-LIB by Andrew Reynolds and Morgan Deters.
% : Translated to TFF using Peter Baumgartner's SMTtoTPTP program.
% : This might align with one of the existing Fast Fourier Transform
% Judgement Day problems in the SWV domain, but I can't pick it.
%------------------------------------------------------------------------------
tff('S2',type,
'S2': $tType ).
tff('S8',type,
'S8': $tType ).
tff('S9',type,
'S9': $tType ).
tff('S4',type,
'S4': $tType ).
tff('S6',type,
'S6': $tType ).
tff('S5',type,
'S5': $tType ).
tff('S1',type,
'S1': $tType ).
tff('S3',type,
'S3': $tType ).
tff('S7',type,
'S7': $tType ).
tff(f7,type,
f7: ( 'S4' * 'S5' ) > $real ).
tff(f11,type,
f11: 'S5' ).
tff(f20,type,
f20: 'S8' ).
tff(f12,type,
f12: ( 'S6' * 'S5' ) > $int ).
tff(f9,type,
f9: 'S5' ).
tff(f6,type,
f6: 'S3' ).
tff(f3,type,
f3: ( 'S2' * $real ) > $real ).
tff(f15,type,
f15: 'S7' ).
tff(f8,type,
f8: 'S4' ).
tff(f10,type,
f10: $real ).
tff(f14,type,
f14: ( 'S7' * $int ) > 'S5' ).
tff(f1,type,
f1: 'S1' ).
tff(f18,type,
f18: 'S9' ).
tff(f19,type,
f19: 'S2' ).
tff(f4,type,
f4: ( 'S3' * $real ) > 'S2' ).
tff(f16,type,
f16: ( 'S8' * $int ) > $int ).
tff(f5,type,
f5: 'S3' ).
tff(f2,type,
f2: 'S1' ).
tff(f17,type,
f17: ( 'S9' * $int ) > 'S8' ).
tff(f13,type,
f13: 'S6' ).
tff(formula_1,axiom,
f1 != f2 ).
tff(formula_2,axiom,
~ $less(0.0,f3(f4(f5,f3(f4(f6,f7(f8,f9)),$product(2.0,f10))),f7(f8,f11))) ).
tff(formula_3,axiom,
$less(0,f12(f13,f9)) ).
tff(formula_4,axiom,
$less(f12(f13,f9),f12(f13,f11)) ).
tff(formula_5,axiom,
$less(0,f12(f13,f9)) ).
tff(formula_6,axiom,
$less(f12(f13,f9),f12(f13,f11)) ).
tff(formula_7,axiom,
! [A__questionmark_v0: 'S5'] : $lesseq(0.0,f7(f8,A__questionmark_v0)) ).
tff(formula_8,axiom,
f3(f4(f5,f10),2.0) != 0.0 ).
tff(formula_9,axiom,
? [A__questionmark_v_0_1: $real] :
( ( A__questionmark_v_0_1 = 2.0 )
& ( f3(f4(f5,f10),A__questionmark_v_0_1) != A__questionmark_v_0_1 ) ) ).
tff(formula_10,axiom,
$less(0.0,f3(f4(f5,f10),2.0)) ).
tff(formula_11,axiom,
$lesseq(0.0,f3(f4(f5,f10),2.0)) ).
tff(formula_12,axiom,
$less($uminus(f3(f4(f5,f10),2.0)),0.0) ).
tff(formula_13,axiom,
? [A__questionmark_v_0_2: $real] :
( ( A__questionmark_v_0_2 = 2.0 )
& $less(f3(f4(f5,f10),A__questionmark_v_0_2),A__questionmark_v_0_2) ) ).
tff(formula_14,axiom,
? [A__questionmark_v_0_3: $real] :
( ( A__questionmark_v_0_3 = 2.0 )
& $lesseq(f3(f4(f5,f10),A__questionmark_v_0_3),A__questionmark_v_0_3) ) ).
tff(formula_15,axiom,
! [A__questionmark_v0: 'S2',A__questionmark_v1: $real,A__questionmark_v2: $real] :
? [A__questionmark_v_0_4: $real,A__questionmark_v_1_5: $real] :
( ( A__questionmark_v_0_4 = f3(A__questionmark_v0,A__questionmark_v1) )
& ( A__questionmark_v_1_5 = f3(A__questionmark_v0,A__questionmark_v2) )
& ? [A__questionmark_v_2_6: 'S2'] :
( ( A__questionmark_v_2_6 = f4(f6,f3(f4(f5,$difference(A__questionmark_v_1_5,A__questionmark_v_0_4)),$difference(A__questionmark_v2,A__questionmark_v1))) )
& ( $difference(A__questionmark_v_0_4,f3(A__questionmark_v_2_6,A__questionmark_v1)) = $difference(A__questionmark_v_1_5,f3(A__questionmark_v_2_6,A__questionmark_v2)) ) ) ) ).
tff(formula_16,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real] :
? [A__questionmark_v_0_7: 'S2'] :
( ( A__questionmark_v_0_7 = f4(f6,A__questionmark_v0) )
& ( f3(f4(f5,f3(A__questionmark_v_0_7,A__questionmark_v1)),f3(A__questionmark_v_0_7,A__questionmark_v0)) = f3(f4(f5,A__questionmark_v1),A__questionmark_v0) ) ) ).
tff(formula_17,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real,A__questionmark_v2: $real] :
? [A__questionmark_v_0_8: 'S2'] :
( ( A__questionmark_v_0_8 = f4(f6,A__questionmark_v0) )
& ( f3(A__questionmark_v_0_8,f3(f4(f5,A__questionmark_v1),A__questionmark_v2)) = f3(f4(f5,f3(A__questionmark_v_0_8,A__questionmark_v1)),A__questionmark_v2) ) ) ).
tff(formula_18,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real,A__questionmark_v2: $real,A__questionmark_v3: $real] : f3(f4(f6,f3(f4(f5,A__questionmark_v0),A__questionmark_v1)),f3(f4(f5,A__questionmark_v2),A__questionmark_v3)) = f3(f4(f5,f3(f4(f6,A__questionmark_v0),A__questionmark_v2)),f3(f4(f6,A__questionmark_v1),A__questionmark_v3)) ).
tff(formula_19,axiom,
! [A__questionmark_v0: 'S5'] : $product(f7(f8,A__questionmark_v0),4.0) = f7(f8,f14(f15,$product(4,f12(f13,A__questionmark_v0)))) ).
tff(formula_20,axiom,
! [A__questionmark_v0: 'S5',A__questionmark_v1: 'S5'] : f7(f8,f14(f15,f16(f17(f18,f12(f13,A__questionmark_v0)),f12(f13,A__questionmark_v1)))) = f3(f4(f6,f7(f8,A__questionmark_v0)),f7(f8,A__questionmark_v1)) ).
tff(formula_21,axiom,
! [A__questionmark_v0: $real] :
? [A__questionmark_v_0_9: $real] :
( ( A__questionmark_v_0_9 = f3(f19,A__questionmark_v0) )
& ( f3(f4(f6,A__questionmark_v_0_9),A__questionmark_v_0_9) = f3(f4(f6,A__questionmark_v0),A__questionmark_v0) ) ) ).
tff(formula_22,axiom,
! [A__questionmark_v0: $int] :
? [A__questionmark_v_0_10: $int] :
( ( A__questionmark_v_0_10 = f16(f20,A__questionmark_v0) )
& ( f16(f17(f18,A__questionmark_v_0_10),A__questionmark_v_0_10) = f16(f17(f18,A__questionmark_v0),A__questionmark_v0) ) ) ).
tff(formula_23,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real] : f3(f19,f3(f4(f6,A__questionmark_v0),A__questionmark_v1)) = f3(f4(f6,f3(f19,A__questionmark_v0)),f3(f19,A__questionmark_v1)) ).
tff(formula_24,axiom,
! [A__questionmark_v0: $int,A__questionmark_v1: $int] : f16(f20,f16(f17(f18,A__questionmark_v0),A__questionmark_v1)) = f16(f17(f18,f16(f20,A__questionmark_v0)),f16(f20,A__questionmark_v1)) ).
tff(formula_25,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real] : f3(f19,f3(f4(f5,A__questionmark_v0),A__questionmark_v1)) = f3(f4(f5,f3(f19,A__questionmark_v0)),f3(f19,A__questionmark_v1)) ).
tff(formula_26,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real,A__questionmark_v2: $real] :
? [A__questionmark_v_0_11: 'S2'] :
( ( A__questionmark_v_0_11 = f4(f6,A__questionmark_v0) )
& ( $less(0.0,A__questionmark_v0)
=> ( $less(A__questionmark_v1,A__questionmark_v2)
=> $less(f3(A__questionmark_v_0_11,A__questionmark_v1),f3(A__questionmark_v_0_11,A__questionmark_v2)) ) ) ) ).
tff(formula_27,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real] :
? [A__questionmark_v_0_12: $real] :
( ( A__questionmark_v_0_12 = 0.0 )
& ( $less(A__questionmark_v_0_12,A__questionmark_v0)
=> ( $less(A__questionmark_v_0_12,A__questionmark_v1)
=> $less(A__questionmark_v_0_12,f3(f4(f6,A__questionmark_v0),A__questionmark_v1)) ) ) ) ).
tff(formula_28,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real,A__questionmark_v2: $real] :
? [A__questionmark_v_0_13: 'S2'] :
( ( A__questionmark_v_0_13 = f4(f6,A__questionmark_v0) )
& ( $less(0.0,A__questionmark_v0)
=> ( $lesseq(f3(A__questionmark_v_0_13,A__questionmark_v1),f3(A__questionmark_v_0_13,A__questionmark_v2))
<=> $lesseq(A__questionmark_v1,A__questionmark_v2) ) ) ) ).
tff(formula_29,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real,A__questionmark_v2: $real] :
( $less(0.0,A__questionmark_v0)
=> ( $lesseq(f3(f4(f6,A__questionmark_v1),A__questionmark_v0),f3(f4(f6,A__questionmark_v2),A__questionmark_v0))
<=> $lesseq(A__questionmark_v1,A__questionmark_v2) ) ) ).
tff(formula_30,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real,A__questionmark_v2: $real] :
( $less(0.0,A__questionmark_v0)
=> ( $less(f3(f4(f6,A__questionmark_v1),A__questionmark_v0),f3(f4(f6,A__questionmark_v2),A__questionmark_v0))
<=> $less(A__questionmark_v1,A__questionmark_v2) ) ) ).
tff(formula_31,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real,A__questionmark_v2: $real] :
? [A__questionmark_v_0_14: 'S2'] :
( ( A__questionmark_v_0_14 = f4(f6,A__questionmark_v0) )
& ( ( A__questionmark_v0 != 0.0 )
=> ( ( f3(A__questionmark_v_0_14,A__questionmark_v1) = f3(A__questionmark_v_0_14,A__questionmark_v2) )
<=> ( A__questionmark_v1 = A__questionmark_v2 ) ) ) ) ).
tff(formula_32,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real,A__questionmark_v2: $real] :
( ( A__questionmark_v0 != 0.0 )
=> ( ( f3(f4(f6,A__questionmark_v1),A__questionmark_v0) = f3(f4(f6,A__questionmark_v2),A__questionmark_v0) )
<=> ( A__questionmark_v1 = A__questionmark_v2 ) ) ) ).
tff(formula_33,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real] :
? [A__questionmark_v_0_15: $real] :
( ( A__questionmark_v_0_15 = 0.0 )
& ( ( $sum(f3(f4(f6,A__questionmark_v0),A__questionmark_v0),f3(f4(f6,A__questionmark_v1),A__questionmark_v1)) = A__questionmark_v_0_15 )
<=> ( ( A__questionmark_v0 = A__questionmark_v_0_15 )
& ( A__questionmark_v1 = A__questionmark_v_0_15 ) ) ) ) ).
tff(formula_34,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real,A__questionmark_v2: $real] : f3(f4(f6,$sum(A__questionmark_v0,A__questionmark_v1)),A__questionmark_v2) = $sum(f3(f4(f6,A__questionmark_v0),A__questionmark_v2),f3(f4(f6,A__questionmark_v1),A__questionmark_v2)) ).
tff(formula_35,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real,A__questionmark_v2: $real] :
? [A__questionmark_v_0_16: 'S2'] :
( ( A__questionmark_v_0_16 = f4(f6,A__questionmark_v0) )
& ( f3(f4(f6,f3(A__questionmark_v_0_16,A__questionmark_v1)),A__questionmark_v2) = f3(A__questionmark_v_0_16,f3(f4(f6,A__questionmark_v1),A__questionmark_v2)) ) ) ).
tff(formula_36,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real] : f3(f4(f6,A__questionmark_v0),A__questionmark_v1) = f3(f4(f6,A__questionmark_v1),A__questionmark_v0) ).
tff(formula_37,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: $real] : $lesseq($uminus(f3(f4(f6,A__questionmark_v0),A__questionmark_v0)),f3(f4(f6,A__questionmark_v1),A__questionmark_v1)) ).
tff(formula_38,axiom,
! [A__questionmark_v0: $real] :
? [A__questionmark_v_0_17: $real] :
( ( A__questionmark_v_0_17 = 0.0 )
& ( ~ $less(A__questionmark_v_0_17,f3(f4(f6,A__questionmark_v0),A__questionmark_v0))
<=> ( A__questionmark_v0 = A__questionmark_v_0_17 ) ) ) ).
tff(formula_39,axiom,
! [A__questionmark_v0: 'S5'] : f14(f15,f12(f13,A__questionmark_v0)) = A__questionmark_v0 ).
tff(formula_40,axiom,
! [A__questionmark_v0: $int] :
( $lesseq(0,A__questionmark_v0)
=> ( f12(f13,f14(f15,A__questionmark_v0)) = A__questionmark_v0 ) ) ).
tff(formula_41,axiom,
! [A__questionmark_v0: $int] :
( $less(A__questionmark_v0,0)
=> ( f12(f13,f14(f15,A__questionmark_v0)) = 0 ) ) ).
%------------------------------------------------------------------------------