TPTP Problem File: SWC460_1.p
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%------------------------------------------------------------------------------
% File : SWC460_1 : TPTP v9.0.0. Released v9.0.0.
% Domain : Software Creation
% Problem : Prove equivalence of small and fast program for sequence A115243
% Version : Especial.
% English : Terms: 0 2 12 50 204 818 3276 13106 52428 209714 838860 3355442
% 13421772 53687090 214748364 858993458 3435973836 13743895346
% 54975581388 219902325554
% Small: loop2((y*y)-x,y+y,x,1,2)-1
% Fast : (loop(x*x,1,loop(x+x,x,2))-2)/(1+(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 : A115243 [Git23]
% Status : Unknown
% Rating : 1.00 v9.0.0
% Syntax : Number of formulae : 42 ( 16 unt; 21 typ; 0 def)
% Number of atoms : 34 ( 25 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 19 ( 6 ~; 0 |; 5 &)
% ( 0 <=>; 8 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 4 ( 1 avg)
% Number arithmetic : 73 ( 9 atm; 15 fun; 24 num; 25 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 26 ( 17 >; 9 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 27 ( 21 usr; 7 con; 0-3 aty)
% Number of variables : 25 (; 24 !; 1 ?; 25 :)
% SPC : TF0_UNK_EQU_ARI
% Comments : In the "aind_sem" subset, i.e., very likely to require induction.
%------------------------------------------------------------------------------
tff(u2,type,
u2: ( $int * $int ) > $int ).
tff(f2,type,
f2: $int > $int ).
tff(v0,type,
v0: ( $int * $int * $int ) > $int ).
tff(div,type,
'div:(Int*Int)>Int': ( $int * $int ) > $int ).
tff(h2,type,
h2: $int ).
tff(g2,type,
g2: $int > $int ).
tff(u1,type,
u1: ( $int * $int ) > $int ).
tff(w0,type,
w0: $int > $int ).
tff(v1,type,
v1: $int > $int ).
tff(v2,type,
v2: $int > $int ).
tff(i0,type,
i0: $int ).
tff(h1,type,
h1: $int > $int ).
tff(u0,type,
u0: ( $int * $int * $int ) > $int ).
tff(f0,type,
f0: ( $int * $int ) > $int ).
tff(g1,type,
g1: $int ).
tff(j0,type,
j0: $int ).
tff(g0,type,
g0: ( $int * $int ) > $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 y:Int (f0(x, y) = ((y * y) - x))
tff(formula_1,axiom,
! [X: $int,Y: $int] : ( f0(X,Y) = $difference($product(Y,Y),X) ) ).
%----∀ x:Int y:Int (g0(x, y) = (y + y))
tff(formula_2,axiom,
! [X: $int,Y: $int] : ( g0(X,Y) = $sum(Y,Y) ) ).
%----∀ x:Int (h0(x) = x)
tff(formula_3,axiom,
! [X: $int] : ( h0(X) = X ) ).
%----(i0 = 1)
tff(formula_4,axiom,
i0 = 1 ).
%----(j0 = 2)
tff(formula_5,axiom,
j0 = 2 ).
%----∀ 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_6,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), v0((x - 1), y, z))))
tff(formula_7,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),v0($difference(X,1),Y,Z)) ) ) ) ).
%----∀ x:Int (w0(x) = u0(h0(x), i0, j0))
tff(formula_8,axiom,
! [X: $int] : ( w0(X) = u0(h0(X),i0,j0) ) ).
%----∀ x:Int (small(x) = (w0(x) - 1))
tff(formula_9,axiom,
! [X: $int] : ( small(X) = $difference(w0(X),1) ) ).
%----∀ x:Int (f1(x) = (x * x))
tff(formula_10,axiom,
! [X: $int] : ( f1(X) = $product(X,X) ) ).
%----(g1 = 1)
tff(formula_11,axiom,
g1 = 1 ).
%----∀ x:Int (f2(x) = (x + x))
tff(formula_12,axiom,
! [X: $int] : ( f2(X) = $sum(X,X) ) ).
%----∀ x:Int (g2(x) = x)
tff(formula_13,axiom,
! [X: $int] : ( g2(X) = X ) ).
%----(h2 = 2)
tff(formula_14,axiom,
h2 = 2 ).
%----∀ x:Int y:Int (u2(x, y) = (if (x ≤ 0) y else f2(u2((x - 1), y))))
tff(formula_15,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
=> ( u2(X,Y) = Y ) )
& ( ~ $lesseq(X,0)
=> ( u2(X,Y) = f2(u2($difference(X,1),Y)) ) ) ) ).
%----∀ x:Int (v2(x) = u2(g2(x), h2))
tff(formula_16,axiom,
! [X: $int] : ( v2(X) = u2(g2(X),h2) ) ).
%----∀ x:Int (h1(x) = v2(x))
tff(formula_17,axiom,
! [X: $int] : ( h1(X) = v2(X) ) ).
%----∀ x:Int y:Int (u1(x, y) = (if (x ≤ 0) y else f1(u1((x - 1), y))))
tff(formula_18,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_19,axiom,
! [X: $int] : ( v1(X) = u1(g1,h1(X)) ) ).
%----∀ x:Int (fast(x) = ((v1(x) - 2) div (1 + (2 + 2))))
tff(formula_20,axiom,
! [X: $int] : ( fast(X) = 'div:(Int*Int)>Int'($difference(v1(X),2),$sum(1,$sum(2,2))) ) ).
%----∃ c:Int ((c ≥ 0) ∧ ¬(small(c) = fast(c)))
tff(conjecture_1,conjecture,
~ ? [C: $int] :
( $greatereq(C,0)
& ( small(C) != fast(C) ) ) ).
%------------------------------------------------------------------------------