(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), double(x), s(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(x, s(s(0)))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(x, s(s(0)))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
The TRS R 2 is
f(true, x, y) → f(gt(x, y), double(x), s(y))
The signature Sigma is {
f}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), double(x), s(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(x, s(s(0)))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
The set Q consists of the following terms:
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
(3) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(4) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y) → F(gt(x, y), double(x), s(y))
F(true, x, y) → GT(x, y)
F(true, x, y) → DOUBLE(x)
GT(s(x), s(y)) → GT(x, y)
DOUBLE(x) → TIMES(x, s(s(0)))
TIMES(s(x), y) → PLUS(y, times(x, y))
TIMES(s(x), y) → TIMES(x, y)
PLUS(s(x), y) → PLUS(x, s(y))
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), double(x), s(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(x, s(s(0)))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
The set Q consists of the following terms:
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 4 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PLUS(s(x), y) → PLUS(x, s(y))
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), double(x), s(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(x, s(s(0)))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
The set Q consists of the following terms:
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(8) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(9) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PLUS(s(x), y) → PLUS(x, s(y))
R is empty.
The set Q consists of the following terms:
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(10) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
PLUS(s(x), y) → PLUS(x, s(y))
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(12) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- PLUS(s(x), y) → PLUS(x, s(y))
The graph contains the following edges 1 > 1
(13) YES
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TIMES(s(x), y) → TIMES(x, y)
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), double(x), s(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(x, s(s(0)))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
The set Q consists of the following terms:
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(15) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(16) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TIMES(s(x), y) → TIMES(x, y)
R is empty.
The set Q consists of the following terms:
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(17) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
TIMES(s(x), y) → TIMES(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(19) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- TIMES(s(x), y) → TIMES(x, y)
The graph contains the following edges 1 > 1, 2 >= 2
(20) YES
(21) Obligation:
Q DP problem:
The TRS P consists of the following rules:
GT(s(x), s(y)) → GT(x, y)
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), double(x), s(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(x, s(s(0)))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
The set Q consists of the following terms:
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(22) UsableRulesProof (EQUIVALENT transformation)
As all Q-normal forms are R-normal forms we are in the innermost case. Hence, by the usable rules processor [LPAR04] we can delete all non-usable rules [FROCOS05] from R.
(23) Obligation:
Q DP problem:
The TRS P consists of the following rules:
GT(s(x), s(y)) → GT(x, y)
R is empty.
The set Q consists of the following terms:
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(24) QReductionProof (EQUIVALENT transformation)
We deleted the following terms from Q as each root-symbol of these terms does neither occur in P nor in R.[THIEMANN].
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
(25) Obligation:
Q DP problem:
The TRS P consists of the following rules:
GT(s(x), s(y)) → GT(x, y)
R is empty.
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(26) QDPSizeChangeProof (EQUIVALENT transformation)
By using the subterm criterion [SUBTERM_CRITERION] together with the size-change analysis [AAECC05] we have proven that there are no infinite chains for this DP problem.
From the DPs we obtained the following set of size-change graphs:
- GT(s(x), s(y)) → GT(x, y)
The graph contains the following edges 1 > 1, 2 > 2
(27) YES
(28) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y) → F(gt(x, y), double(x), s(y))
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), double(x), s(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(x, s(s(0)))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
The set Q consists of the following terms:
f(true, x0, x1)
gt(s(x0), 0)
gt(0, x0)
gt(s(x0), s(x1))
double(x0)
times(0, x0)
times(s(x0), x1)
plus(0, x0)
plus(s(x0), x1)
We have to consider all minimal (P,Q,R)-chains.
(29) MNOCProof (EQUIVALENT transformation)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
(30) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(true, x, y) → F(gt(x, y), double(x), s(y))
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), double(x), s(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(x, s(s(0)))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
Q is empty.
We have to consider all (P,Q,R)-chains.
(31) NonLoopProof (COMPLETE transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 1,
σ' = [
zr3 /
s(
zr3)], and μ' = [
zr3 /
times(
s(
zr3),
s(
s(
0)))] on the rule
F(
true,
s(
s(
s(
zr3))),
s(
zr2))[
zr3 /
s(
zr3),
zr2 /
s(
zr2)]
n[
zr2 /
0] →
F(
true,
s(
s(
s(
s(
zr3)))),
s(
s(
zr2)))[
zr2 /
s(
zr2),
zr3 /
s(
s(
zr3))]
n[
zr2 /
0,
zr3 /
times(
s(
zr3),
s(
s(
0)))]
This rule is correct for the QDP as the following derivation shows:
F(
true,
s(
s(
s(
zr3))),
s(
zr2))[
zr3 /
s(
zr3),
zr2 /
s(
zr2)]
n[
zr2 /
0] →
F(
true,
s(
s(
s(
s(
zr3)))),
s(
s(
zr2)))[
zr2 /
s(
zr2),
zr3 /
s(
s(
zr3))]
n[
zr2 /
0,
zr3 /
times(
s(
zr3),
s(
s(
0)))]
by Equivalence by Irrelevant Pattern Substitutions σ: [
zr2 /
s(
zr2),
zr3 /
s(
s(
zr3))] µ: [
zr2 /
0,
zr3 /
times(
s(
zr3),
s(
s(
0)))]
intermediate steps: Equiv IPS (lhs) - Instantiate mu - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Equiv Sµ (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs)
F(
true,
s(
s(
zl2)),
s(
zl3))[
zl2 /
s(
zl2),
zl3 /
s(
zl3)]
n[
zl2 /
y1,
zl3 /
0] →
F(
true,
s(
s(
s(
s(
zt1)))),
s(
s(
zr3)))[
zr3 /
s(
zr3),
zt1 /
s(
s(
zt1))]
n[
zr3 /
0,
zt1 /
times(
y1,
s(
s(
0)))]
by Rewrite sigma at the term of variable:
zt1 with the rewrite sequence
: [([],plus(s(x), y) -> plus(x, s(y))), ([],plus(s(x), y) -> plus(x, s(y))), ([],plus(0, y) -> y)]
F(true, s(s(zl2)), s(zl3))[zl2 / s(zl2), zl3 / s(zl3)]n[zl2 / y1, zl3 / 0] → F(true, s(s(s(s(zt1)))), s(s(zr3)))[zr3 / s(zr3), zt1 / plus(s(s(0)), zt1)]n[zr3 / 0, zt1 / times(y1, s(s(0)))]
by Rewrite t with the rewrite sequence : [([1],plus(s(x), y) -> plus(x, s(y))), ([1],plus(s(x), y) -> plus(x, s(y))), ([1],plus(0, y) -> y), ([1,0,0],plus(s(x), y) -> plus(x, s(y))), ([1,0,0],plus(s(x), y) -> plus(x, s(y))), ([1,0,0],plus(0, y) -> y)]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs)
F(true, s(s(zl2)), s(zl3))[zr2 / s(zr2), zr3 / s(zr3), zt1 / plus(s(s(0)), zt1), zl2 / s(zl2), zl3 / s(zl3)]n[zr2 / y1, zr3 / 0, zt1 / times(y1, s(s(0))), zl2 / y1, zl3 / 0, x0 / y1] → F(true, plus(s(s(0)), plus(s(s(0)), zt1)), s(s(zr3)))[zr2 / s(zr2), zr3 / s(zr3), zt1 / plus(s(s(0)), zt1), zl2 / s(zl2), zl3 / s(zl3)]n[zr2 / y1, zr3 / 0, zt1 / times(y1, s(s(0))), zl2 / y1, zl3 / 0, x0 / y1]
by Narrowing at position: [1]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv DR (rhs) - Equiv DR (lhs) - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
F(true, s(s(zl2)), s(zl3))[zr2 / s(zr2), zr3 / s(zr3), zl2 / s(zl2), zl3 / s(zl3)]n[zr2 / x0, zr3 / 0, zl2 / x0, zl3 / 0] → F(true, times(s(s(zr2)), s(s(0))), s(s(zr3)))[zr2 / s(zr2), zr3 / s(zr3), zl2 / s(zl2), zl3 / s(zl3)]n[zr2 / x0, zr3 / 0, zl2 / x0, zl3 / 0]
by Narrowing at position: [1]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
F(true, s(s(zl2)), s(zl3))[zr2 / s(zr2), zr3 / s(zr3), zl2 / s(zl2), zl3 / s(zl3)]n[zr2 / y0, zr3 / 0, zl2 / y0, zl3 / 0] → F(true, double(s(s(zr2))), s(s(zr3)))[zr2 / s(zr2), zr3 / s(zr3), zl2 / s(zl2), zl3 / s(zl3)]n[zr2 / y0, zr3 / 0, zl2 / y0, zl3 / 0]
by Narrowing at position: [0]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation - Equiv Sµ (rhs) - Equiv Sµ (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs)
F(true, s(zs2), s(zs3))[zs2 / s(zs2), zs3 / s(zs3)]n[zs2 / y1, zs3 / y0] → F(gt(y1, y0), double(s(zs2)), s(s(zs3)))[zs2 / s(zs2), zs3 / s(zs3)]n[zs2 / y1, zs3 / y0]
by Narrowing at position: [0]
intermediate steps: Instantiate mu - Instantiate Sigma - Instantiation - Instantiation - Instantiation
F(true, x, y)[ ]n[ ] → F(gt(x, y), double(x), s(y))[ ]n[ ]
by Rule from TRS P
intermediate steps: Equiv IPS (rhs) - Equiv IPS (rhs) - Instantiation - Equiv DR (lhs) - Instantiation - Equiv DR (lhs)
gt(s(x), s(y))[x / s(x), y / s(y)]n[ ] → gt(x, y)[ ]n[ ]
by PatternCreation I with delta: [ ], theta: [ ], sigma: [x / s(x), y / s(y)]
gt(s(x), s(y))[ ]n[ ] → gt(x, y)[ ]n[ ]
by Rule from TRS R
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation
gt(s(x), 0)[ ]n[ ] → true[ ]n[ ]
by Rule from TRS R
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate Sigma - Instantiation - Instantiation
double(x)[ ]n[ ] → times(x, s(s(0)))[ ]n[ ]
by Rule from TRS R
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation - Equiv DR (lhs) - Expand Sigma - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs)
times(s(x), y)[x / s(x)]n[ ] → plus(y, z)[x / s(x), z / plus(y, z)]n[z / times(x, y)]
by PatternCreation II with pi: [1], sigma: [x / s(x)]
times(s(x), y)[ ]n[ ] → plus(y, times(x, y))[ ]n[ ]
by Rule from TRS R
(32) NO