(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(tt, x) → f(swap(x, 0), s(x))
swap(s(x), y) → swap(x, s(y))
swap(0, s(y)) → decr(s(y))
decr(s(y)) → decr(y)
decr(0) → tt
Q is empty.
(1) AAECC Innermost (EQUIVALENT transformation)
We have applied [NOC,AAECCNOC] to switch to innermost. The TRS R 1 is
swap(s(x), y) → swap(x, s(y))
swap(0, s(y)) → decr(s(y))
decr(s(y)) → decr(y)
decr(0) → tt
The TRS R 2 is
f(tt, x) → f(swap(x, 0), s(x))
The signature Sigma is {
f}
(2) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(tt, x) → f(swap(x, 0), s(x))
swap(s(x), y) → swap(x, s(y))
swap(0, s(y)) → decr(s(y))
decr(s(y)) → decr(y)
decr(0) → tt
The set Q consists of the following terms:
f(tt, x0)
swap(s(x0), x1)
swap(0, s(x0))
decr(s(x0))
decr(0)
(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(tt, x) → F(swap(x, 0), s(x))
F(tt, x) → SWAP(x, 0)
SWAP(s(x), y) → SWAP(x, s(y))
SWAP(0, s(y)) → DECR(s(y))
DECR(s(y)) → DECR(y)
The TRS R consists of the following rules:
f(tt, x) → f(swap(x, 0), s(x))
swap(s(x), y) → swap(x, s(y))
swap(0, s(y)) → decr(s(y))
decr(s(y)) → decr(y)
decr(0) → tt
The set Q consists of the following terms:
f(tt, x0)
swap(s(x0), x1)
swap(0, s(x0))
decr(s(x0))
decr(0)
We have to consider all minimal (P,Q,R)-chains.
(5) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 3 SCCs with 2 less nodes.
(6) Complex Obligation (AND)
(7) Obligation:
Q DP problem:
The TRS P consists of the following rules:
DECR(s(y)) → DECR(y)
The TRS R consists of the following rules:
f(tt, x) → f(swap(x, 0), s(x))
swap(s(x), y) → swap(x, s(y))
swap(0, s(y)) → decr(s(y))
decr(s(y)) → decr(y)
decr(0) → tt
The set Q consists of the following terms:
f(tt, x0)
swap(s(x0), x1)
swap(0, s(x0))
decr(s(x0))
decr(0)
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:
DECR(s(y)) → DECR(y)
R is empty.
The set Q consists of the following terms:
f(tt, x0)
swap(s(x0), x1)
swap(0, s(x0))
decr(s(x0))
decr(0)
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(tt, x0)
swap(s(x0), x1)
swap(0, s(x0))
decr(s(x0))
decr(0)
(11) Obligation:
Q DP problem:
The TRS P consists of the following rules:
DECR(s(y)) → DECR(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:
- DECR(s(y)) → DECR(y)
The graph contains the following edges 1 > 1
(13) YES
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SWAP(s(x), y) → SWAP(x, s(y))
The TRS R consists of the following rules:
f(tt, x) → f(swap(x, 0), s(x))
swap(s(x), y) → swap(x, s(y))
swap(0, s(y)) → decr(s(y))
decr(s(y)) → decr(y)
decr(0) → tt
The set Q consists of the following terms:
f(tt, x0)
swap(s(x0), x1)
swap(0, s(x0))
decr(s(x0))
decr(0)
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:
SWAP(s(x), y) → SWAP(x, s(y))
R is empty.
The set Q consists of the following terms:
f(tt, x0)
swap(s(x0), x1)
swap(0, s(x0))
decr(s(x0))
decr(0)
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(tt, x0)
swap(s(x0), x1)
swap(0, s(x0))
decr(s(x0))
decr(0)
(18) Obligation:
Q DP problem:
The TRS P consists of the following rules:
SWAP(s(x), y) → SWAP(x, s(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:
- SWAP(s(x), y) → SWAP(x, s(y))
The graph contains the following edges 1 > 1
(20) YES
(21) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(tt, x) → F(swap(x, 0), s(x))
The TRS R consists of the following rules:
f(tt, x) → f(swap(x, 0), s(x))
swap(s(x), y) → swap(x, s(y))
swap(0, s(y)) → decr(s(y))
decr(s(y)) → decr(y)
decr(0) → tt
The set Q consists of the following terms:
f(tt, x0)
swap(s(x0), x1)
swap(0, s(x0))
decr(s(x0))
decr(0)
We have to consider all minimal (P,Q,R)-chains.
(22) MNOCProof (EQUIVALENT transformation)
We use the modular non-overlap check [FROCOS05] to decrease Q to the empty set.
(23) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(tt, x) → F(swap(x, 0), s(x))
The TRS R consists of the following rules:
f(tt, x) → f(swap(x, 0), s(x))
swap(s(x), y) → swap(x, s(y))
swap(0, s(y)) → decr(s(y))
decr(s(y)) → decr(y)
decr(0) → tt
Q is empty.
We have to consider all (P,Q,R)-chains.
(24) NonLoopProof (COMPLETE transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 1, b = 1,
σ' = [ ], and μ' = [ ] on the rule
F(tt, s(zr0))[zr0 / s(zr0)]n[zr0 / 0] → F(tt, s(s(zr0)))[zr0 / s(zr0)]n[zr0 / 0]
This rule is correct for the QDP as the following derivation shows:
F(tt, s(zr0))[zr0 / s(zr0)]n[zr0 / 0] → F(tt, s(s(zr0)))[zr0 / s(zr0)]n[zr0 / 0]
by Equivalence by Domain Renaming of the lhs with [zl0 / zr0]
intermediate steps: Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
F(tt, s(zl1))[zr1 / s(zr1), zl1 / s(zl1)]n[zr1 / 0, zl1 / 0] → F(tt, s(s(zr1)))[zr1 / s(zr1), zl1 / s(zl1)]n[zr1 / 0, zl1 / 0]
by Narrowing at position: [0]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv DR (rhs) - Equiv DR (lhs) - Equiv DR (rhs) - Equiv DR (lhs) - Equiv Sµ (rhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
F(tt, s(zl1))[zr2 / s(zr2), zr3 / s(zr3), zl1 / s(zl1)]n[zr2 / 0, zr3 / 0, y0 / 0, zl1 / 0] → F(decr(y0), s(s(zr3)))[zr2 / s(zr2), zr3 / s(zr3), zl1 / s(zl1)]n[zr2 / 0, zr3 / 0, y0 / 0, zl1 / 0]
by Narrowing at position: [0]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - 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(tt, s(zl1))[zr2 / s(zr2), zr3 / s(zr3), zl1 / s(zl1)]n[zr2 / 0, zr3 / 0, zl1 / 0] → F(decr(s(zr2)), s(s(zr3)))[zr2 / s(zr2), zr3 / s(zr3), zl1 / s(zl1)]n[zr2 / 0, zr3 / 0, zl1 / 0]
by Narrowing at position: [0]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Equiv IPS (lhs)
F(tt, s(zs1))[zt1 / s(zt1), zs1 / s(zs1)]n[zt1 / 0, zs1 / y1] → F(swap(y1, s(zt1)), s(s(zs1)))[zt1 / s(zt1), zs1 / s(zs1)]n[zt1 / 0, zs1 / y1]
by Narrowing at position: [0]
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate Sigma - Instantiation - Instantiation
F(tt, x)[ ]n[ ] → F(swap(x, 0), s(x))[ ]n[ ]
by Rule from TRS P
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiation - Instantiation - Equiv DR (rhs) - Equiv DR (lhs) - Instantiation - Equiv DR (rhs) - Equiv DR (lhs)
swap(s(x), y)[x / s(x)]n[ ] → swap(x, s(y))[y / s(y)]n[ ]
by PatternCreation I with delta: [ ], theta: [y / s(y)], sigma: [x / s(x)]
swap(s(x), y)[ ]n[ ] → swap(x, s(y))[ ]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
swap(0, s(y))[ ]n[ ] → decr(s(y))[ ]n[ ]
by Rule from TRS R
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Instantiate mu - Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv DR (lhs) - Equiv DR (lhs) - Instantiation - Equiv DR (lhs)
decr(s(y))[y / s(y)]n[ ] → decr(y)[ ]n[ ]
by PatternCreation I with delta: [ ], theta: [ ], sigma: [y / s(y)]
decr(s(y))[ ]n[ ] → decr(y)[ ]n[ ]
by Rule from TRS R
intermediate steps: Equiv IPS (rhs) - Equiv IPS (lhs) - Equiv IPS (rhs) - Equiv IPS (lhs)
decr(0)[ ]n[ ] → tt[ ]n[ ]
by Rule from TRS R
(25) NO