TPTP Problem File: SWC468_1.p
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%------------------------------------------------------------------------------
% File : SWC468_1 : TPTP v9.1.0. Released v9.0.0.
% Domain : Software Creation
% Problem : Prove equivalence of small and fast program for sequence A171226
% Version : Especial.
% English : Terms: 19 919 90109 9001009 900010009 90000100009 9000001000009
% 900000010000009 90000000100000009 9000000001000000009
% 900000000010000000009 90000000000100000000009
% 9000000000001000000000009 900000000000010000000000009
% 90000000000000100000000000009
% Small: loop(loop((x+x)+x,2,1+(x*x))+x,1,loop2(x*y,y,x,1,
% 2+(2*(2+2))))
% Fast : loop((1+((2*(2*(x+x)))+x))*x,1,loop2(x*y,y,x,1,
% 2+(2*(2+2))))+(1+(2*(2+2)))
% Refs : [GB+23] Gauthier et al. (2023), A Mathematical Benchmark for I
% : [Git23] https://github.com/ai4reason/oeis-atp-benchmark
% Source : [Git23]
% Names : A171226 [Git23]
% Status : Unknown
% Rating : 1.00 v9.0.0
% Syntax : Number of formulae : 67 ( 26 unt; 33 typ; 0 def)
% Number of atoms : 56 ( 41 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 31 ( 9 ~; 0 |; 8 &)
% ( 0 <=>; 14 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 7 ( 2 avg)
% Number arithmetic : 138 ( 15 atm; 34 fun; 47 num; 42 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 41 ( 26 >; 15 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 39 ( 33 usr; 10 con; 0-3 aty)
% Number of variables : 42 (; 41 !; 1 ?; 42 :)
% SPC : TF0_UNK_EQU_ARI
% Comments : Not in an "aind_*" subset, i.e., unlikely to require induction.
%------------------------------------------------------------------------------
tff(v0,type,
v0: $int > $int ).
tff(v2,type,
v2: ( $int * $int * $int ) > $int ).
tff(i2,type,
i2: $int ).
tff(v4,type,
v4: ( $int * $int * $int ) > $int ).
tff(u1,type,
u1: ( $int * $int ) > $int ).
tff(u4,type,
u4: ( $int * $int * $int ) > $int ).
tff(v3,type,
v3: $int > $int ).
tff(j2,type,
j2: $int ).
tff(i4,type,
i4: $int ).
tff(u0,type,
u0: ( $int * $int ) > $int ).
tff(w4,type,
w4: $int > $int ).
tff(u2,type,
u2: ( $int * $int * $int ) > $int ).
tff(g4,type,
g4: ( $int * $int ) > $int ).
tff(g2,type,
g2: ( $int * $int ) > $int ).
tff(v1,type,
v1: $int > $int ).
tff(u3,type,
u3: ( $int * $int ) > $int ).
tff(h4,type,
h4: $int > $int ).
tff(h3,type,
h3: $int > $int ).
tff(w2,type,
w2: $int > $int ).
tff(h1,type,
h1: $int > $int ).
tff(h2,type,
h2: $int > $int ).
tff(g3,type,
g3: $int ).
tff(f3,type,
f3: $int > $int ).
tff(j4,type,
j4: $int ).
tff(g1,type,
g1: $int ).
tff(f2,type,
f2: ( $int * $int ) > $int ).
tff(fast,type,
fast: $int > $int ).
tff(g0,type,
g0: $int ).
tff(h0,type,
h0: $int > $int ).
tff(small,type,
small: $int > $int ).
tff(f0,type,
f0: $int > $int ).
tff(f4,type,
f4: ( $int * $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) = (1 + (x * x)))
tff(formula_3,axiom,
! [X: $int] : ( h1(X) = $sum(1,$product(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 (f0(x) = (v1(x) + x))
tff(formula_6,axiom,
! [X: $int] : ( f0(X) = $sum(v1(X),X) ) ).
%----(g0 = 1)
tff(formula_7,axiom,
g0 = 1 ).
%----∀ x:Int y:Int (f2(x, y) = (x * y))
tff(formula_8,axiom,
! [X: $int,Y: $int] : ( f2(X,Y) = $product(X,Y) ) ).
%----∀ x:Int y:Int (g2(x, y) = y)
tff(formula_9,axiom,
! [X: $int,Y: $int] : ( g2(X,Y) = Y ) ).
%----∀ x:Int (h2(x) = x)
tff(formula_10,axiom,
! [X: $int] : ( h2(X) = X ) ).
%----(i2 = 1)
tff(formula_11,axiom,
i2 = 1 ).
%----(j2 = (2 + (2 * (2 + 2))))
tff(formula_12,axiom,
j2 = $sum(2,$product(2,$sum(2,2))) ).
%----∀ 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_13,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), v2((x - 1), y, z))))
tff(formula_14,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),v2($difference(X,1),Y,Z)) ) ) ) ).
%----∀ x:Int (w2(x) = u2(h2(x), i2, j2))
tff(formula_15,axiom,
! [X: $int] : ( w2(X) = u2(h2(X),i2,j2) ) ).
%----∀ x:Int (h0(x) = w2(x))
tff(formula_16,axiom,
! [X: $int] : ( h0(X) = w2(X) ) ).
%----∀ x:Int y:Int (u0(x, y) = (if (x ≤ 0) y else f0(u0((x - 1), y))))
tff(formula_17,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
=> ( u0(X,Y) = Y ) )
& ( ~ $lesseq(X,0)
=> ( u0(X,Y) = f0(u0($difference(X,1),Y)) ) ) ) ).
%----∀ x:Int (v0(x) = u0(g0, h0(x)))
tff(formula_18,axiom,
! [X: $int] : ( v0(X) = u0(g0,h0(X)) ) ).
%----∀ x:Int (small(x) = v0(x))
tff(formula_19,axiom,
! [X: $int] : ( small(X) = v0(X) ) ).
%----∀ x:Int (f3(x) = ((1 + ((2 * (2 * (x + x))) + x)) * x))
tff(formula_20,axiom,
! [X: $int] : ( f3(X) = $product($sum(1,$sum($product(2,$product(2,$sum(X,X))),X)),X) ) ).
%----(g3 = 1)
tff(formula_21,axiom,
g3 = 1 ).
%----∀ x:Int y:Int (f4(x, y) = (x * y))
tff(formula_22,axiom,
! [X: $int,Y: $int] : ( f4(X,Y) = $product(X,Y) ) ).
%----∀ x:Int y:Int (g4(x, y) = y)
tff(formula_23,axiom,
! [X: $int,Y: $int] : ( g4(X,Y) = Y ) ).
%----∀ x:Int (h4(x) = x)
tff(formula_24,axiom,
! [X: $int] : ( h4(X) = X ) ).
%----(i4 = 1)
tff(formula_25,axiom,
i4 = 1 ).
%----(j4 = (2 + (2 * (2 + 2))))
tff(formula_26,axiom,
j4 = $sum(2,$product(2,$sum(2,2))) ).
%----∀ x:Int y:Int z:Int (u4(x, y, z) = (if (x ≤ 0) y else f4(u4((x - 1), y,
%----z), v4((x - 1), y, z))))
tff(formula_27,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $lesseq(X,0)
=> ( u4(X,Y,Z) = Y ) )
& ( ~ $lesseq(X,0)
=> ( u4(X,Y,Z) = f4(u4($difference(X,1),Y,Z),v4($difference(X,1),Y,Z)) ) ) ) ).
%----∀ x:Int y:Int z:Int (v4(x, y, z) = (if (x ≤ 0) z else g4(u4((x - 1), y,
%----z), v4((x - 1), y, z))))
tff(formula_28,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $lesseq(X,0)
=> ( v4(X,Y,Z) = Z ) )
& ( ~ $lesseq(X,0)
=> ( v4(X,Y,Z) = g4(u4($difference(X,1),Y,Z),v4($difference(X,1),Y,Z)) ) ) ) ).
%----∀ x:Int (w4(x) = u4(h4(x), i4, j4))
tff(formula_29,axiom,
! [X: $int] : ( w4(X) = u4(h4(X),i4,j4) ) ).
%----∀ x:Int (h3(x) = w4(x))
tff(formula_30,axiom,
! [X: $int] : ( h3(X) = w4(X) ) ).
%----∀ x:Int y:Int (u3(x, y) = (if (x ≤ 0) y else f3(u3((x - 1), y))))
tff(formula_31,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
=> ( u3(X,Y) = Y ) )
& ( ~ $lesseq(X,0)
=> ( u3(X,Y) = f3(u3($difference(X,1),Y)) ) ) ) ).
%----∀ x:Int (v3(x) = u3(g3, h3(x)))
tff(formula_32,axiom,
! [X: $int] : ( v3(X) = u3(g3,h3(X)) ) ).
%----∀ x:Int (fast(x) = (v3(x) + (1 + (2 * (2 + 2)))))
tff(formula_33,axiom,
! [X: $int] : ( fast(X) = $sum(v3(X),$sum(1,$product(2,$sum(2,2)))) ) ).
%----∃ c:Int ((c ≥ 0) ∧ ¬(small(c) = fast(c)))
tff(conjecture_1,conjecture,
~ ? [C: $int] :
( $greatereq(C,0)
& ( small(C) != fast(C) ) ) ).
%------------------------------------------------------------------------------