TPTP Problem File: SWC444_1.p
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%------------------------------------------------------------------------------
% File : SWC444_1 : TPTP v9.0.0. Released v9.0.0.
% Domain : Software Creation
% Problem : Prove equivalence of small and fast program for sequence A41145
% Version : Especial.
% English : Terms: 1 18 325 5868 105949 1912950 34539049 623615832
% 11259624025 203296848282 3670602893101 66274148924100
% 1196605283526901 21605169252408318 390089651826876625
% 7043218902136187568
% Small: loop2((2*loop((x+x)+x,2,x))+y,x,x,1,0)
% Fast : loop2(((x/y)*x)+y,x,x-1,if x<=0 then 1 else
% (2+(2*(2*(2+2)))),1)
% Refs : [GB+23] Gauthier et al. (2023), A Mathematical Benchmark for I
% : [Git23] https://github.com/ai4reason/oeis-atp-benchmark
% Source : [Git23]
% Names : A41145 [Git23]
% Status : Unknown
% Rating : 1.00 v9.0.0
% Syntax : Number of formulae : 48 ( 17 unt; 24 typ; 0 def)
% Number of atoms : 43 ( 30 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 27 ( 8 ~; 0 |; 7 &)
% ( 0 <=>; 12 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 5 ( 1 avg)
% Number arithmetic : 94 ( 13 atm; 18 fun; 32 num; 31 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 32 ( 20 >; 12 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 30 ( 24 usr; 7 con; 0-3 aty)
% Number of variables : 31 (; 30 !; 1 ?; 31 :)
% SPC : TF0_UNK_EQU_ARI
% Comments : In the "aind_sem" subset, i.e., very likely to require induction.
%------------------------------------------------------------------------------
tff(v2,type,
v2: ( $int * $int * $int ) > $int ).
tff(v0,type,
v0: ( $int * $int * $int ) > $int ).
tff(div,type,
'div:(Int*Int)>Int': ( $int * $int ) > $int ).
tff(g2,type,
g2: $int > $int ).
tff(i2,type,
i2: $int > $int ).
tff(u1,type,
u1: ( $int * $int ) > $int ).
tff(j2,type,
j2: $int ).
tff(w0,type,
w0: $int > $int ).
tff(u2,type,
u2: ( $int * $int * $int ) > $int ).
tff(v1,type,
v1: $int > $int ).
tff(g0,type,
g0: $int > $int ).
tff(i0,type,
i0: $int ).
tff(w2,type,
w2: $int > $int ).
tff(h1,type,
h1: $int > $int ).
tff(h2,type,
h2: $int > $int ).
tff(u0,type,
u0: ( $int * $int * $int ) > $int ).
tff(f0,type,
f0: ( $int * $int ) > $int ).
tff(g1,type,
g1: $int ).
tff(f2,type,
f2: ( $int * $int ) > $int ).
tff(j0,type,
j0: $int ).
tff(fast,type,
fast: $int > $int ).
tff(h0,type,
h0: $int > $int ).
tff(small,type,
small: $int > $int ).
tff(f1,type,
f1: $int > $int ).
%----∀ x:Int (f1(x) = ((x + x) + x))
tff(formula_1,axiom,
! [X: $int] : ( f1(X) = $sum($sum(X,X),X) ) ).
%----(g1 = 2)
tff(formula_2,axiom,
g1 = 2 ).
%----∀ x:Int (h1(x) = x)
tff(formula_3,axiom,
! [X: $int] : ( h1(X) = X ) ).
%----∀ x:Int y:Int (u1(x, y) = (if (x ≤ 0) y else f1(u1((x - 1), y))))
tff(formula_4,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
=> ( u1(X,Y) = Y ) )
& ( ~ $lesseq(X,0)
=> ( u1(X,Y) = f1(u1($difference(X,1),Y)) ) ) ) ).
%----∀ x:Int (v1(x) = u1(g1, h1(x)))
tff(formula_5,axiom,
! [X: $int] : ( v1(X) = u1(g1,h1(X)) ) ).
%----∀ x:Int y:Int (f0(x, y) = ((2 * v1(x)) + y))
tff(formula_6,axiom,
! [X: $int,Y: $int] : ( f0(X,Y) = $sum($product(2,v1(X)),Y) ) ).
%----∀ x:Int (g0(x) = x)
tff(formula_7,axiom,
! [X: $int] : ( g0(X) = X ) ).
%----∀ x:Int (h0(x) = x)
tff(formula_8,axiom,
! [X: $int] : ( h0(X) = X ) ).
%----(i0 = 1)
tff(formula_9,axiom,
i0 = 1 ).
%----(j0 = 0)
tff(formula_10,axiom,
j0 = 0 ).
%----∀ x:Int y:Int z:Int (u0(x, y, z) = (if (x ≤ 0) y else f0(u0((x - 1), y,
%----z), v0((x - 1), y, z))))
tff(formula_11,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $lesseq(X,0)
=> ( u0(X,Y,Z) = Y ) )
& ( ~ $lesseq(X,0)
=> ( u0(X,Y,Z) = f0(u0($difference(X,1),Y,Z),v0($difference(X,1),Y,Z)) ) ) ) ).
%----∀ x:Int y:Int z:Int (v0(x, y, z) = (if (x ≤ 0) z else g0(u0((x - 1), y,
%----z))))
tff(formula_12,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $lesseq(X,0)
=> ( v0(X,Y,Z) = Z ) )
& ( ~ $lesseq(X,0)
=> ( v0(X,Y,Z) = g0(u0($difference(X,1),Y,Z)) ) ) ) ).
%----∀ x:Int (w0(x) = u0(h0(x), i0, j0))
tff(formula_13,axiom,
! [X: $int] : ( w0(X) = u0(h0(X),i0,j0) ) ).
%----∀ x:Int (small(x) = w0(x))
tff(formula_14,axiom,
! [X: $int] : ( small(X) = w0(X) ) ).
%----∀ x:Int y:Int (f2(x, y) = (((x div y) * x) + y))
tff(formula_15,axiom,
! [X: $int,Y: $int] : ( f2(X,Y) = $sum($product('div:(Int*Int)>Int'(X,Y),X),Y) ) ).
%----∀ x:Int (g2(x) = x)
tff(formula_16,axiom,
! [X: $int] : ( g2(X) = X ) ).
%----∀ x:Int (h2(x) = (x - 1))
tff(formula_17,axiom,
! [X: $int] : ( h2(X) = $difference(X,1) ) ).
%----∀ x:Int (i2(x) = (if (x ≤ 0) 1 else (2 + (2 * (2 * (2 + 2))))))
tff(formula_18,axiom,
! [X: $int] :
( ( $lesseq(X,0)
=> ( i2(X) = 1 ) )
& ( ~ $lesseq(X,0)
=> ( i2(X) = $sum(2,$product(2,$product(2,$sum(2,2)))) ) ) ) ).
%----(j2 = 1)
tff(formula_19,axiom,
j2 = 1 ).
%----∀ x:Int y:Int z:Int (u2(x, y, z) = (if (x ≤ 0) y else f2(u2((x - 1), y,
%----z), v2((x - 1), y, z))))
tff(formula_20,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $lesseq(X,0)
=> ( u2(X,Y,Z) = Y ) )
& ( ~ $lesseq(X,0)
=> ( u2(X,Y,Z) = f2(u2($difference(X,1),Y,Z),v2($difference(X,1),Y,Z)) ) ) ) ).
%----∀ x:Int y:Int z:Int (v2(x, y, z) = (if (x ≤ 0) z else g2(u2((x - 1), y,
%----z))))
tff(formula_21,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $lesseq(X,0)
=> ( v2(X,Y,Z) = Z ) )
& ( ~ $lesseq(X,0)
=> ( v2(X,Y,Z) = g2(u2($difference(X,1),Y,Z)) ) ) ) ).
%----∀ x:Int (w2(x) = u2(h2(x), i2(x), j2))
tff(formula_22,axiom,
! [X: $int] : ( w2(X) = u2(h2(X),i2(X),j2) ) ).
%----∀ x:Int (fast(x) = w2(x))
tff(formula_23,axiom,
! [X: $int] : ( fast(X) = w2(X) ) ).
%----∃ c:Int ((c ≥ 0) ∧ ¬(small(c) = fast(c)))
tff(conjecture_1,conjecture,
~ ? [C: $int] :
( $greatereq(C,0)
& ( small(C) != fast(C) ) ) ).
%------------------------------------------------------------------------------