TPTP Problem File: SWW835_1.p
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%------------------------------------------------------------------------------
% File : SWW835_1 : TPTP v8.2.0. Released v7.0.0.
% Domain : Software Verification
% Problem : Fast Fourier Transform 640466
% 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/z3.640466.p [SMTL]
% Status : Unsatisfiable
% Rating : 1.00 v7.0.0
% Syntax : Number of formulae : 58 ( 23 unt; 19 typ; 0 def)
% Number of atoms : 65 ( 40 equ)
% Maximal formula atoms : 5 ( 1 avg)
% Number of connectives : 31 ( 5 ~; 1 |; 8 &)
% ( 8 <=>; 9 =>; 0 <=; 0 <~>)
% Maximal formula depth : 8 ( 3 avg)
% Maximal term depth : 9 ( 2 avg)
% Number arithmetic : 141 ( 24 atm; 49 fun; 50 num; 18 var)
% Number of types : 8 ( 6 usr; 2 ari)
% Number of type conns : 8 ( 4 >; 4 *; 0 +; 0 <<)
% Number of predicates : 4 ( 0 usr; 1 prp; 0-2 aty)
% Number of functors : 23 ( 13 usr; 14 con; 0-2 aty)
% Number of variables : 47 ( 40 !; 7 ?; 47 :)
% 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('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(f12,type,
f12: ( 'S6' * 'S4' ) > $int ).
tff(f11,type,
f11: 'S5' ).
tff(f4,type,
f4: 'S2' ).
tff(f10,type,
f10: ( 'S5' * $int ) > 'S4' ).
tff(f9,type,
f9: 'S2' ).
tff(f7,type,
f7: 'S3' ).
tff(f3,type,
f3: ( 'S2' * $real ) > $real ).
tff(f8,type,
f8: 'S4' ).
tff(f5,type,
f5: $real ).
tff(f1,type,
f1: 'S1' ).
tff(f2,type,
f2: 'S1' ).
tff(f6,type,
f6: ( 'S3' * 'S4' ) > $real ).
tff(f13,type,
f13: 'S6' ).
tff(formula_1,axiom,
f1 != f2 ).
tff(formula_2,axiom,
$true ).
tff(formula_3,axiom,
f3(f4,$quotient($product(2.0,f5),f6(f7,f8))) = 0.0 ).
tff(formula_4,axiom,
f3(f9,$quotient($product(2.0,f5),f6(f7,f8))) = 0.0 ).
tff(formula_5,axiom,
! [A__questionmark_v0: $real] : f3(f9,$difference(A__questionmark_v0,f5)) = $uminus(f3(f9,A__questionmark_v0)) ).
tff(formula_6,axiom,
! [A__questionmark_v0: $real] :
? [A__questionmark_v_0_1: $real] :
( ( A__questionmark_v_0_1 = 0.0 )
& ( $less(A__questionmark_v_0_1,A__questionmark_v0)
=> ( $less(A__questionmark_v0,$product(2.0,f5))
=> ( ( f3(f9,A__questionmark_v0) != A__questionmark_v_0_1 )
| ( f3(f4,A__questionmark_v0) != 1.0 ) ) ) ) ) ).
tff(formula_7,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: 'S4'] :
? [A__questionmark_v_0_2: $real] :
( ( A__questionmark_v_0_2 = 2.0 )
& ( f3(f4,$sum(A__questionmark_v0,$quotient($product(f6(f7,f10(f11,$sum(f12(f13,A__questionmark_v1),1))),f5),A__questionmark_v_0_2))) = $uminus(f3(f9,$sum(A__questionmark_v0,$quotient($product(f6(f7,A__questionmark_v1),f5),A__questionmark_v_0_2)))) ) ) ).
tff(formula_8,axiom,
! [A__questionmark_v0: $real,A__questionmark_v1: 'S4'] :
? [A__questionmark_v_0_3: $real] :
( ( A__questionmark_v_0_3 = 2.0 )
& ( f3(f9,$sum(A__questionmark_v0,$quotient($product(f6(f7,f10(f11,$sum(f12(f13,A__questionmark_v1),1))),f5),A__questionmark_v_0_3))) = f3(f4,$sum(A__questionmark_v0,$quotient($product(f6(f7,A__questionmark_v1),f5),A__questionmark_v_0_3))) ) ) ).
tff(formula_9,axiom,
! [A__questionmark_v0: $real] : f3(f4,A__questionmark_v0) = f3(f9,$difference($quotient(f5,2.0),A__questionmark_v0)) ).
tff(formula_10,axiom,
! [A__questionmark_v0: $real] : f3(f9,A__questionmark_v0) = f3(f4,$difference($quotient(f5,2.0),A__questionmark_v0)) ).
tff(formula_11,axiom,
! [A__questionmark_v0: $real] : $uminus(f3(f9,A__questionmark_v0)) = f3(f4,$sum(A__questionmark_v0,$quotient(f5,2.0))) ).
tff(formula_12,axiom,
! [A__questionmark_v0: 'S4'] : f3(f9,$product(f5,f6(f7,A__questionmark_v0))) = 0.0 ).
tff(formula_13,axiom,
! [A__questionmark_v0: 'S4'] : f3(f9,$product(f6(f7,A__questionmark_v0),f5)) = 0.0 ).
tff(formula_14,axiom,
! [A__questionmark_v0: 'S4'] : f3(f9,$product($product(2.0,f6(f7,A__questionmark_v0)),f5)) = 0.0 ).
tff(formula_15,axiom,
! [A__questionmark_v0: 'S4'] : f3(f4,$product($product(2.0,f6(f7,A__questionmark_v0)),f5)) = 1.0 ).
tff(formula_16,axiom,
! [A__questionmark_v0: 'S4'] : $lesseq(0.0,f6(f7,A__questionmark_v0)) ).
tff(formula_17,axiom,
f3(f9,f5) = 0.0 ).
tff(formula_18,axiom,
! [A__questionmark_v0: $real] : f3(f9,$sum(f5,A__questionmark_v0)) = $uminus(f3(f9,A__questionmark_v0)) ).
tff(formula_19,axiom,
! [A__questionmark_v0: 'S4',A__questionmark_v1: 'S4'] :
? [A__questionmark_v_0_4: $int,A__questionmark_v_1_5: $int] :
( ( A__questionmark_v_0_4 = f12(f13,A__questionmark_v1) )
& ( A__questionmark_v_1_5 = f12(f13,A__questionmark_v0) )
& ( $lesseq(A__questionmark_v_1_5,A__questionmark_v_0_4)
=> ( f6(f7,f10(f11,$difference(A__questionmark_v_0_4,A__questionmark_v_1_5))) = $difference(f6(f7,A__questionmark_v1),f6(f7,A__questionmark_v0)) ) ) ) ).
tff(formula_20,axiom,
! [A__questionmark_v0: 'S4',A__questionmark_v1: 'S4'] :
( $lesseq(f6(f7,A__questionmark_v0),f6(f7,A__questionmark_v1))
<=> $lesseq(f12(f13,A__questionmark_v0),f12(f13,A__questionmark_v1)) ) ).
tff(formula_21,axiom,
! [A__questionmark_v0: 'S4',A__questionmark_v1: 'S4'] :
( $less(f6(f7,A__questionmark_v0),f6(f7,A__questionmark_v1))
<=> $less(f12(f13,A__questionmark_v0),f12(f13,A__questionmark_v1)) ) ).
tff(formula_22,axiom,
! [A__questionmark_v0: 'S4',A__questionmark_v1: 'S4'] :
( ( f6(f7,A__questionmark_v0) = f6(f7,A__questionmark_v1) )
<=> ( A__questionmark_v0 = A__questionmark_v1 ) ) ).
tff(formula_23,axiom,
! [A__questionmark_v0: 'S4'] :
( $lesseq(f6(f7,A__questionmark_v0),0.0)
<=> ( A__questionmark_v0 = f10(f11,0) ) ) ).
tff(formula_24,axiom,
! [A__questionmark_v0: 'S4'] :
( ( f6(f7,A__questionmark_v0) = 0.0 )
<=> ( A__questionmark_v0 = f10(f11,0) ) ) ).
tff(formula_25,axiom,
! [A__questionmark_v0: 'S4',A__questionmark_v1: 'S4'] :
( $less(f12(f13,A__questionmark_v0),f12(f13,A__questionmark_v1))
<=> $lesseq($sum(f6(f7,A__questionmark_v0),1.0),f6(f7,A__questionmark_v1)) ) ).
tff(formula_26,axiom,
! [A__questionmark_v0: 'S4',A__questionmark_v1: 'S4'] :
( $lesseq(f12(f13,A__questionmark_v0),f12(f13,A__questionmark_v1))
<=> $less(f6(f7,A__questionmark_v0),$sum(f6(f7,A__questionmark_v1),1.0)) ) ).
tff(formula_27,axiom,
! [A__questionmark_v0: 'S4'] :
( $less(0.0,f6(f7,A__questionmark_v0))
<=> $less(0,f12(f13,A__questionmark_v0)) ) ).
tff(formula_28,axiom,
! [A__questionmark_v0: 'S4',A__questionmark_v1: 'S4'] : f6(f7,f10(f11,$sum(f12(f13,A__questionmark_v0),f12(f13,A__questionmark_v1)))) = $sum(f6(f7,A__questionmark_v0),f6(f7,A__questionmark_v1)) ).
tff(formula_29,axiom,
! [A__questionmark_v0: 'S4'] : f6(f7,f10(f11,$sum(f12(f13,A__questionmark_v0),1))) = $sum(f6(f7,A__questionmark_v0),1.0) ).
tff(formula_30,axiom,
! [A__questionmark_v0: 'S4'] :
? [A__questionmark_v_0_6: $real] :
( ( A__questionmark_v_0_6 = f6(f7,A__questionmark_v0) )
& ( $less(A__questionmark_v_0_6,0.0)
=> ( $uminus(A__questionmark_v_0_6) = A__questionmark_v_0_6 ) )
& ( ~ $less(A__questionmark_v_0_6,0.0)
=> ( A__questionmark_v_0_6 = A__questionmark_v_0_6 ) ) ) ).
tff(formula_31,axiom,
f6(f7,f10(f11,$sum(0,1))) = 1.0 ).
tff(formula_32,axiom,
f6(f7,f10(f11,0)) = 0.0 ).
tff(formula_33,axiom,
f6(f7,f10(f11,1)) = 1.0 ).
tff(formula_34,axiom,
! [A__questionmark_v0: 'S4'] : $less(0.0,f6(f7,f10(f11,$sum(f12(f13,A__questionmark_v0),1)))) ).
tff(formula_35,axiom,
! [A__questionmark_v0: 'S4'] : ~ $less(f6(f7,A__questionmark_v0),0.0) ).
tff(formula_36,axiom,
! [A__questionmark_v0: $real] :
? [A__questionmark_v_0_7: $real] :
( ( A__questionmark_v_0_7 = 2.0 )
& ( $less(0.0,A__questionmark_v0)
=> ( $less(A__questionmark_v0,A__questionmark_v_0_7)
=> $less(f3(f4,$product(A__questionmark_v_0_7,A__questionmark_v0)),1.0) ) ) ) ).
tff(formula_37,axiom,
! [A__questionmark_v0: 'S4'] : f10(f11,f12(f13,A__questionmark_v0)) = A__questionmark_v0 ).
tff(formula_38,axiom,
! [A__questionmark_v0: $int] :
( $lesseq(0,A__questionmark_v0)
=> ( f12(f13,f10(f11,A__questionmark_v0)) = A__questionmark_v0 ) ) ).
tff(formula_39,axiom,
! [A__questionmark_v0: $int] :
( $less(A__questionmark_v0,0)
=> ( f12(f13,f10(f11,A__questionmark_v0)) = 0 ) ) ).
%------------------------------------------------------------------------------