TPTP Problem File: SWC441_1.p
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%------------------------------------------------------------------------------
% File : SWC441_1 : TPTP v9.0.0. Released v9.0.0.
% Domain : Software Creation
% Problem : Prove equivalence of small and fast program for sequence A25967
% Version : Especial.
% English : Terms: 1 21 289 3321 34705 343257 3282913 30740457 283960369
% 2599944633 23667405697 214622461833 1941330570193 17530479823449
% 158125883415841 1425244488222249 12839878208747377
% 115635005126816505
% Small: loop(loop2(((2*((2*(x+x))+(y*y)))-y)+x,y+y,y,1,2)+
% (2*((x+x)+x)),x,1)
% Fast : loop((loop(((loop2((2*(x+y))+x,y+y,x,1,2+2) div 2)+1)*
% loop(x+x,x,1),1,y)+(2*(2*(x+x))))+x,x,1)
% Refs : [GB+23] Gauthier et al. (2023), A Mathematical Benchmark for I
% : [Git23] https://github.com/ai4reason/oeis-atp-benchmark
% Source : [Git23]
% Names : A25967 [Git23]
% Status : Unknown
% Rating : 1.00 v9.0.0
% Syntax : Number of formulae : 78 ( 30 unt; 39 typ; 0 def)
% Number of atoms : 64 ( 47 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 35 ( 10 ~; 0 |; 9 &)
% ( 0 <=>; 16 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 7 ( 2 avg)
% Number arithmetic : 153 ( 17 atm; 37 fun; 46 num; 53 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 54 ( 31 >; 23 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 45 ( 39 usr; 11 con; 0-3 aty)
% Number of variables : 53 (; 52 !; 1 ?; 53 :)
% SPC : TF0_UNK_EQU_ARI
% Comments : In the "aind_sem" subset, i.e., very likely to require induction.
%------------------------------------------------------------------------------
tff(v0,type,
v0: $int > $int ).
tff(v3,type,
v3: ( $int * $int ) > $int ).
tff(u2,type,
u2: ( $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(v4,type,
v4: ( $int * $int * $int ) > $int ).
tff(g5,type,
g5: $int > $int ).
tff(u4,type,
u4: ( $int * $int * $int ) > $int ).
tff(i4,type,
i4: $int ).
tff(f5,type,
f5: $int > $int ).
tff(u0,type,
u0: ( $int * $int ) > $int ).
tff(w4,type,
w4: $int > $int ).
tff(g4,type,
g4: ( $int * $int ) > $int ).
tff(w1,type,
w1: ( $int * $int ) > $int ).
tff(i1,type,
i1: $int ).
tff(v1,type,
v1: ( $int * $int * $int ) > $int ).
tff(h5,type,
h5: $int ).
tff(u3,type,
u3: ( $int * $int ) > $int ).
tff(h4,type,
h4: $int > $int ).
tff(v2,type,
v2: $int > $int ).
tff(g0,type,
g0: $int > $int ).
tff(u1,type,
u1: ( $int * $int * $int ) > $int ).
tff(g1,type,
g1: ( $int * $int ) > $int ).
tff(g3,type,
g3: $int ).
tff(f0,type,
f0: ( $int * $int ) > $int ).
tff(f3,type,
f3: $int > $int ).
tff(j4,type,
j4: $int ).
tff(h0,type,
h0: $int ).
tff(u5,type,
u5: ( $int * $int ) > $int ).
tff(h1,type,
h1: ( $int * $int ) > $int ).
tff(f2,type,
f2: ( $int * $int ) > $int ).
tff(j1,type,
j1: $int ).
tff(fast,type,
fast: $int > $int ).
tff(f1,type,
f1: ( $int * $int ) > $int ).
tff(h3,type,
h3: ( $int * $int ) > $int ).
tff(small,type,
small: $int > $int ).
tff(f4,type,
f4: ( $int * $int ) > $int ).
tff(v5,type,
v5: $int > $int ).
%----∀ x:Int y:Int (f1(x, y) = (((2 * ((2 * (x + x)) + (y * y))) - y) + x))
tff(formula_1,axiom,
! [X: $int,Y: $int] : ( f1(X,Y) = $sum($difference($product(2,$sum($product(2,$sum(X,X)),$product(Y,Y))),Y),X) ) ).
%----∀ x:Int y:Int (g1(x, y) = (y + y))
tff(formula_2,axiom,
! [X: $int,Y: $int] : ( g1(X,Y) = $sum(Y,Y) ) ).
%----∀ x:Int y:Int (h1(x, y) = y)
tff(formula_3,axiom,
! [X: $int,Y: $int] : ( h1(X,Y) = Y ) ).
%----(i1 = 1)
tff(formula_4,axiom,
i1 = 1 ).
%----(j1 = 2)
tff(formula_5,axiom,
j1 = 2 ).
%----∀ x:Int y:Int z:Int (u1(x, y, z) = (if (x ≤ 0) y else f1(u1((x - 1), y,
%----z), v1((x - 1), y, z))))
tff(formula_6,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $lesseq(X,0)
=> ( u1(X,Y,Z) = Y ) )
& ( ~ $lesseq(X,0)
=> ( u1(X,Y,Z) = f1(u1($difference(X,1),Y,Z),v1($difference(X,1),Y,Z)) ) ) ) ).
%----∀ x:Int y:Int z:Int (v1(x, y, z) = (if (x ≤ 0) z else g1(u1((x - 1), y,
%----z), v1((x - 1), y, z))))
tff(formula_7,axiom,
! [X: $int,Y: $int,Z: $int] :
( ( $lesseq(X,0)
=> ( v1(X,Y,Z) = Z ) )
& ( ~ $lesseq(X,0)
=> ( v1(X,Y,Z) = g1(u1($difference(X,1),Y,Z),v1($difference(X,1),Y,Z)) ) ) ) ).
%----∀ x:Int y:Int (w1(x, y) = u1(h1(x, y), i1, j1))
tff(formula_8,axiom,
! [X: $int,Y: $int] : ( w1(X,Y) = u1(h1(X,Y),i1,j1) ) ).
%----∀ x:Int y:Int (f0(x, y) = (w1(x, y) + (2 * ((x + x) + x))))
tff(formula_9,axiom,
! [X: $int,Y: $int] : ( f0(X,Y) = $sum(w1(X,Y),$product(2,$sum($sum(X,X),X))) ) ).
%----∀ x:Int (g0(x) = x)
tff(formula_10,axiom,
! [X: $int] : ( g0(X) = X ) ).
%----(h0 = 1)
tff(formula_11,axiom,
h0 = 1 ).
%----∀ x:Int y:Int (u0(x, y) = (if (x ≤ 0) y else f0(u0((x - 1), y), x)))
tff(formula_12,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) ) ) ) ).
%----∀ x:Int (v0(x) = u0(g0(x), h0))
tff(formula_13,axiom,
! [X: $int] : ( v0(X) = u0(g0(X),h0) ) ).
%----∀ x:Int (small(x) = v0(x))
tff(formula_14,axiom,
! [X: $int] : ( small(X) = v0(X) ) ).
%----∀ x:Int y:Int (f4(x, y) = ((2 * (x + y)) + x))
tff(formula_15,axiom,
! [X: $int,Y: $int] : ( f4(X,Y) = $sum($product(2,$sum(X,Y)),X) ) ).
%----∀ x:Int y:Int (g4(x, y) = (y + y))
tff(formula_16,axiom,
! [X: $int,Y: $int] : ( g4(X,Y) = $sum(Y,Y) ) ).
%----∀ x:Int (h4(x) = x)
tff(formula_17,axiom,
! [X: $int] : ( h4(X) = X ) ).
%----(i4 = 1)
tff(formula_18,axiom,
i4 = 1 ).
%----(j4 = (2 + 2))
tff(formula_19,axiom,
j4 = $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_20,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_21,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_22,axiom,
! [X: $int] : ( w4(X) = u4(h4(X),i4,j4) ) ).
%----∀ x:Int (f5(x) = (x + x))
tff(formula_23,axiom,
! [X: $int] : ( f5(X) = $sum(X,X) ) ).
%----∀ x:Int (g5(x) = x)
tff(formula_24,axiom,
! [X: $int] : ( g5(X) = X ) ).
%----(h5 = 1)
tff(formula_25,axiom,
h5 = 1 ).
%----∀ x:Int y:Int (u5(x, y) = (if (x ≤ 0) y else f5(u5((x - 1), y))))
tff(formula_26,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
=> ( u5(X,Y) = Y ) )
& ( ~ $lesseq(X,0)
=> ( u5(X,Y) = f5(u5($difference(X,1),Y)) ) ) ) ).
%----∀ x:Int (v5(x) = u5(g5(x), h5))
tff(formula_27,axiom,
! [X: $int] : ( v5(X) = u5(g5(X),h5) ) ).
%----∀ x:Int (f3(x) = (((w4(x) div 2) + 1) * v5(x)))
tff(formula_28,axiom,
! [X: $int] : ( f3(X) = $product($sum('div:(Int*Int)>Int'(w4(X),2),1),v5(X)) ) ).
%----(g3 = 1)
tff(formula_29,axiom,
g3 = 1 ).
%----∀ x:Int y:Int (h3(x, y) = y)
tff(formula_30,axiom,
! [X: $int,Y: $int] : ( h3(X,Y) = Y ) ).
%----∀ 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 y:Int (v3(x, y) = u3(g3, h3(x, y)))
tff(formula_32,axiom,
! [X: $int,Y: $int] : ( v3(X,Y) = u3(g3,h3(X,Y)) ) ).
%----∀ x:Int y:Int (f2(x, y) = ((v3(x, y) + (2 * (2 * (x + x)))) + x))
tff(formula_33,axiom,
! [X: $int,Y: $int] : ( f2(X,Y) = $sum($sum(v3(X,Y),$product(2,$product(2,$sum(X,X)))),X) ) ).
%----∀ x:Int (g2(x) = x)
tff(formula_34,axiom,
! [X: $int] : ( g2(X) = X ) ).
%----(h2 = 1)
tff(formula_35,axiom,
h2 = 1 ).
%----∀ x:Int y:Int (u2(x, y) = (if (x ≤ 0) y else f2(u2((x - 1), y), x)))
tff(formula_36,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) ) ) ) ).
%----∀ x:Int (v2(x) = u2(g2(x), h2))
tff(formula_37,axiom,
! [X: $int] : ( v2(X) = u2(g2(X),h2) ) ).
%----∀ x:Int (fast(x) = v2(x))
tff(formula_38,axiom,
! [X: $int] : ( fast(X) = v2(X) ) ).
%----∃ c:Int ((c ≥ 0) ∧ ¬(small(c) = fast(c)))
tff(conjecture_1,conjecture,
~ ? [C: $int] :
( $greatereq(C,0)
& ( small(C) != fast(C) ) ) ).
%------------------------------------------------------------------------------