TPTP Problem File: SWC433_1.p
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%------------------------------------------------------------------------------
% File : SWC433_1 : TPTP v9.1.0. Released v9.0.0.
% Domain : Software Creation
% Problem : Prove equivalence of small and fast program for sequence A10904
% Version : Especial.
% English : Terms: 4 14 49 172 604 2121 7448 26154 91841 322504 1132488
% 3976785 13964668 49037590 172197809 604680724 2123364868
% 7456295833 26183134320 91943310482
% Small: loop2((((((y*y)/(x+y))+x)+x)+x)+y,x+y,1+x,1,1)
% Fast : loop2((((((y*y)/(x+y))+x)+x)+x)+y,x+y,x,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 : A10904 [Git23]
% Status : Unknown
% Rating : 1.00 v9.0.0
% Syntax : Number of formulae : 38 ( 14 unt; 19 typ; 0 def)
% Number of atoms : 32 ( 23 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 : 7 ( 2 avg)
% Number arithmetic : 83 ( 9 atm; 24 fun; 23 num; 27 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 28 ( 15 >; 13 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 25 ( 19 usr; 7 con; 0-3 aty)
% Number of variables : 27 (; 26 !; 1 ?; 27 :)
% 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(div,type,
'div:(Int*Int)>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(g1,type,
g1: ( $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(small,type,
small: $int > $int ).
%----∀ x:Int y:Int (f0(x, y) = ((((((y * y) div (x + y)) + x) + x) + x) + y))
tff(formula_1,axiom,
! [X: $int,Y: $int] : ( f0(X,Y) = $sum($sum($sum($sum('div:(Int*Int)>Int'($product(Y,Y),$sum(X,Y)),X),X),X),Y) ) ).
%----∀ x:Int y:Int (g0(x, y) = (x + y))
tff(formula_2,axiom,
! [X: $int,Y: $int] : ( g0(X,Y) = $sum(X,Y) ) ).
%----∀ x:Int (h0(x) = (1 + x))
tff(formula_3,axiom,
! [X: $int] : ( h0(X) = $sum(1,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) = ((((((y * y) div (x + y)) + x) + x) + x) + y))
tff(formula_10,axiom,
! [X: $int,Y: $int] : ( f1(X,Y) = $sum($sum($sum($sum('div:(Int*Int)>Int'($product(Y,Y),$sum(X,Y)),X),X),X),Y) ) ).
%----∀ x:Int y:Int (g1(x, y) = (x + y))
tff(formula_11,axiom,
! [X: $int,Y: $int] : ( g1(X,Y) = $sum(X,Y) ) ).
%----∀ x:Int (h1(x) = x)
tff(formula_12,axiom,
! [X: $int] : ( h1(X) = X ) ).
%----(i1 = (2 + 2))
tff(formula_13,axiom,
i1 = $sum(2,2) ).
%----(j1 = 2)
tff(formula_14,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_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), v1((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),v1($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) = w1(x))
tff(formula_18,axiom,
! [X: $int] : ( 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) ) ) ).
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