TPTP Problem File: SWC463_1.p
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%------------------------------------------------------------------------------
% File : SWC463_1 : TPTP v9.0.0. Released v9.0.0.
% Domain : Software Creation
% Problem : Prove equivalence of small and fast program for sequence A163073
% Version : Especial.
% English : Terms: 1 5 29 177 1097 6829 42565 265401 1654993 10320533
% 64359341 401348865 2502838169 15607867837 97331722837
% 606967236489 3785088940705 23604071924261 147196597046333
% 917927985203793
% Small: loop2((2*((x+x)+x))-y,(y-x)+y,x,1,1)
% Fast : if x<=0 then 1 else loop2((2*(2*((2*(x-y))-y)))+y,x,x-1,
% 1+(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 : A163073 [Git23]
% Status : Unknown
% Rating : 1.00 v9.0.0
% Syntax : Number of formulae : 37 ( 13 unt; 18 typ; 0 def)
% Number of atoms : 35 ( 24 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 23 ( 7 ~; 0 |; 6 &)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 7 ( 3 avg)
% Maximal term depth : 7 ( 2 avg)
% Number arithmetic : 89 ( 11 atm; 22 fun; 30 num; 26 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 25 ( 14 >; 11 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 24 ( 18 usr; 7 con; 0-3 aty)
% Number of variables : 26 (; 25 !; 1 ?; 26 :)
% SPC : TF0_UNK_EQU_ARI
% Comments : In the "aind_sem" subset, i.e., very likely to require induction.
%------------------------------------------------------------------------------
tff(v0,type,
v0: ( $int * $int * $int ) > $int ).
tff(w0,type,
w0: $int > $int ).
tff(i1,type,
i1: $int ).
tff(v1,type,
v1: ( $int * $int * $int ) > $int ).
tff(w1,type,
w1: $int > $int ).
tff(u1,type,
u1: ( $int * $int * $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(j0,type,
j0: $int ).
tff(g0,type,
g0: ( $int * $int ) > $int ).
tff(j1,type,
j1: $int ).
tff(fast,type,
fast: $int > $int ).
tff(f1,type,
f1: ( $int * $int ) > $int ).
tff(h0,type,
h0: $int > $int ).
tff(g1,type,
g1: $int > $int ).
tff(small,type,
small: $int > $int ).
%----∀ x:Int y:Int (f0(x, y) = ((2 * ((x + x) + x)) - y))
tff(formula_1,axiom,
! [X: $int,Y: $int] : ( f0(X,Y) = $difference($product(2,$sum($sum(X,X),X)),Y) ) ).
%----∀ x:Int y:Int (g0(x, y) = ((y - x) + y))
tff(formula_2,axiom,
! [X: $int,Y: $int] : ( g0(X,Y) = $sum($difference(Y,X),Y) ) ).
%----∀ x:Int (h0(x) = x)
tff(formula_3,axiom,
! [X: $int] : ( h0(X) = X ) ).
%----(i0 = 1)
tff(formula_4,axiom,
i0 = 1 ).
%----(j0 = 1)
tff(formula_5,axiom,
j0 = 1 ).
%----∀ 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))
tff(formula_9,axiom,
! [X: $int] : ( small(X) = w0(X) ) ).
%----∀ x:Int y:Int (f1(x, y) = ((2 * (2 * ((2 * (x - y)) - y))) + y))
tff(formula_10,axiom,
! [X: $int,Y: $int] : ( f1(X,Y) = $sum($product(2,$product(2,$difference($product(2,$difference(X,Y)),Y))),Y) ) ).
%----∀ x:Int (g1(x) = x)
tff(formula_11,axiom,
! [X: $int] : ( g1(X) = X ) ).
%----∀ x:Int (h1(x) = (x - 1))
tff(formula_12,axiom,
! [X: $int] : ( h1(X) = $difference(X,1) ) ).
%----(i1 = (1 + (2 + 2)))
tff(formula_13,axiom,
i1 = $sum(1,$sum(2,2)) ).
%----(j1 = 1)
tff(formula_14,axiom,
j1 = 1 ).
%----∀ 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_15,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))))
tff(formula_16,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)) ) ) ) ).
%----∀ x:Int (w1(x) = u1(h1(x), i1, j1))
tff(formula_17,axiom,
! [X: $int] : ( w1(X) = u1(h1(X),i1,j1) ) ).
%----∀ x:Int (fast(x) = (if (x ≤ 0) 1 else w1(x)))
tff(formula_18,axiom,
! [X: $int] :
( ( $lesseq(X,0)
=> ( fast(X) = 1 ) )
& ( ~ $lesseq(X,0)
=> ( fast(X) = w1(X) ) ) ) ).
%----∃ c:Int ((c ≥ 0) ∧ ¬(small(c) = fast(c)))
tff(conjecture_1,conjecture,
~ ? [C: $int] :
( $greatereq(C,0)
& ( small(C) != fast(C) ) ) ).
%------------------------------------------------------------------------------