TPTP Problem File: SWC461_1.p
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%------------------------------------------------------------------------------
% File : SWC461_1 : TPTP v9.0.0. Released v9.0.0.
% Domain : Software Creation
% Problem : Prove equivalence of small and fast program for sequence A140656
% Version : Especial.
% English : Terms: 1 3 40322 6402373705728003
% 263130836933693530167218012160000004
% 30414093201713378043612608166064768844377641568960512000000000005
% 612344583768860868615240703852746727407780917846973289838230149...
% ...63978384987221689274204160000000000000006
% Small: loop(x*y2*(x*x)1)+x
% Fast : loop(loop(2*((x-(x/y))+x),x,1)*loop(x*x,1,loop(x*yx1)),
% 1,x*x)+x
% Refs : [GB+23] Gauthier et al. (2023), A Mathematical Benchmark for I
% : [Git23] https://github.com/ai4reason/oeis-atp-benchmark
% Source : [Git23]
% Names : A140656 [Git23]
% Status : Unknown
% Rating : 1.00 v9.0.0
% Syntax : Number of formulae : 56 ( 22 unt; 28 typ; 0 def)
% Number of atoms : 44 ( 33 equ)
% Maximal formula atoms : 4 ( 1 avg)
% Number of connectives : 23 ( 7 ~; 0 |; 6 &)
% ( 0 <=>; 10 =>; 0 <=; 0 <~>)
% Maximal formula depth : 6 ( 3 avg)
% Maximal term depth : 5 ( 2 avg)
% Number arithmetic : 82 ( 11 atm; 17 fun; 23 num; 31 var)
% Number of types : 1 ( 0 usr; 1 ari)
% Number of type conns : 32 ( 23 >; 9 *; 0 +; 0 <<)
% Number of predicates : 3 ( 0 usr; 0 prp; 2-2 aty)
% Number of functors : 34 ( 28 usr; 8 con; 0-2 aty)
% Number of variables : 31 (; 30 !; 1 ?; 31 :)
% SPC : TF0_UNK_EQU_ARI
% Comments : In the "aind_syn" subset, i.e., likely to require induction.
%------------------------------------------------------------------------------
tff(v0,type,
v0: $int > $int ).
tff(u2,type,
u2: ( $int * $int ) > $int ).
tff(g4,type,
g4: $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(v3,type,
v3: $int > $int ).
tff(u0,type,
u0: ( $int * $int ) > $int ).
tff(v1,type,
v1: $int > $int ).
tff(u3,type,
u3: ( $int * $int ) > $int ).
tff(v2,type,
v2: $int > $int ).
tff(g0,type,
g0: $int > $int ).
tff(h3,type,
h3: $int > $int ).
tff(h1,type,
h1: $int > $int ).
tff(h4,type,
h4: $int ).
tff(g3,type,
g3: $int ).
tff(u4,type,
u4: ( $int * $int ) > $int ).
tff(f0,type,
f0: ( $int * $int ) > $int ).
tff(f3,type,
f3: $int > $int ).
tff(h0,type,
h0: $int ).
tff(g1,type,
g1: $int ).
tff(f2,type,
f2: ( $int * $int ) > $int ).
tff(fast,type,
fast: $int > $int ).
tff(small,type,
small: $int > $int ).
tff(v4,type,
v4: $int > $int ).
tff(f4,type,
f4: ( $int * $int ) > $int ).
tff(f1,type,
f1: $int > $int ).
%----∀ x:Int y:Int (f0(x, y) = (x * y))
tff(formula_1,axiom,
! [X: $int,Y: $int] : ( f0(X,Y) = $product(X,Y) ) ).
%----∀ x:Int (g0(x) = (2 * (x * x)))
tff(formula_2,axiom,
! [X: $int] : ( g0(X) = $product(2,$product(X,X)) ) ).
%----(h0 = 1)
tff(formula_3,axiom,
h0 = 1 ).
%----∀ x:Int y:Int (u0(x, y) = (if (x ≤ 0) y else f0(u0((x - 1), y), x)))
tff(formula_4,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_5,axiom,
! [X: $int] : ( v0(X) = u0(g0(X),h0) ) ).
%----∀ x:Int (small(x) = (v0(x) + x))
tff(formula_6,axiom,
! [X: $int] : ( small(X) = $sum(v0(X),X) ) ).
%----∀ x:Int y:Int (f2(x, y) = (2 * ((x - (x div y)) + x)))
tff(formula_7,axiom,
! [X: $int,Y: $int] : ( f2(X,Y) = $product(2,$sum($difference(X,'div:(Int*Int)>Int'(X,Y)),X)) ) ).
%----∀ x:Int (g2(x) = x)
tff(formula_8,axiom,
! [X: $int] : ( g2(X) = X ) ).
%----(h2 = 1)
tff(formula_9,axiom,
h2 = 1 ).
%----∀ x:Int y:Int (u2(x, y) = (if (x ≤ 0) y else f2(u2((x - 1), y), x)))
tff(formula_10,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_11,axiom,
! [X: $int] : ( v2(X) = u2(g2(X),h2) ) ).
%----∀ x:Int (f3(x) = (x * x))
tff(formula_12,axiom,
! [X: $int] : ( f3(X) = $product(X,X) ) ).
%----(g3 = 1)
tff(formula_13,axiom,
g3 = 1 ).
%----∀ x:Int y:Int (f4(x, y) = (x * y))
tff(formula_14,axiom,
! [X: $int,Y: $int] : ( f4(X,Y) = $product(X,Y) ) ).
%----∀ x:Int (g4(x) = x)
tff(formula_15,axiom,
! [X: $int] : ( g4(X) = X ) ).
%----(h4 = 1)
tff(formula_16,axiom,
h4 = 1 ).
%----∀ x:Int y:Int (u4(x, y) = (if (x ≤ 0) y else f4(u4((x - 1), y), x)))
tff(formula_17,axiom,
! [X: $int,Y: $int] :
( ( $lesseq(X,0)
=> ( u4(X,Y) = Y ) )
& ( ~ $lesseq(X,0)
=> ( u4(X,Y) = f4(u4($difference(X,1),Y),X) ) ) ) ).
%----∀ x:Int (v4(x) = u4(g4(x), h4))
tff(formula_18,axiom,
! [X: $int] : ( v4(X) = u4(g4(X),h4) ) ).
%----∀ x:Int (h3(x) = v4(x))
tff(formula_19,axiom,
! [X: $int] : ( h3(X) = v4(X) ) ).
%----∀ x:Int y:Int (u3(x, y) = (if (x ≤ 0) y else f3(u3((x - 1), y))))
tff(formula_20,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_21,axiom,
! [X: $int] : ( v3(X) = u3(g3,h3(X)) ) ).
%----∀ x:Int (f1(x) = (v2(x) * v3(x)))
tff(formula_22,axiom,
! [X: $int] : ( f1(X) = $product(v2(X),v3(X)) ) ).
%----(g1 = 1)
tff(formula_23,axiom,
g1 = 1 ).
%----∀ x:Int (h1(x) = (x * x))
tff(formula_24,axiom,
! [X: $int] : ( h1(X) = $product(X,X) ) ).
%----∀ x:Int y:Int (u1(x, y) = (if (x ≤ 0) y else f1(u1((x - 1), y))))
tff(formula_25,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_26,axiom,
! [X: $int] : ( v1(X) = u1(g1,h1(X)) ) ).
%----∀ x:Int (fast(x) = (v1(x) + x))
tff(formula_27,axiom,
! [X: $int] : ( fast(X) = $sum(v1(X),X) ) ).
%----∃ c:Int ((c ≥ 0) ∧ ¬(small(c) = fast(c)))
tff(conjecture_1,conjecture,
~ ? [C: $int] :
( $greatereq(C,0)
& ( small(C) != fast(C) ) ) ).
%------------------------------------------------------------------------------