(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(true, x) → f(gt(x, 0), double(x))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(s(s(0)), x)
times(0, y) → 0
times(s(x), y) → plus(times(x, y), y)
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
Q is empty.
(1) DependencyPairsProof (EQUIVALENT transformation)
Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.
(2) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(true, x) → F(gt(x, 0), double(x))
F(true, x) → GT(x, 0)
F(true, x) → DOUBLE(x)
GT(s(x), s(y)) → GT(x, y)
DOUBLE(x) → TIMES(s(s(0)), x)
TIMES(s(x), y) → PLUS(times(x, y), 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) → f(gt(x, 0), double(x))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(s(s(0)), x)
times(0, y) → 0
times(s(x), y) → plus(times(x, y), y)
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(3) DependencyGraphProof (EQUIVALENT transformation)
The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 4 less nodes.
(4) Complex Obligation (AND)
(5) 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) → f(gt(x, 0), double(x))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(s(s(0)), x)
times(0, y) → 0
times(s(x), y) → plus(times(x, y), y)
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) 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
(7) YES
(8) 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) → f(gt(x, 0), double(x))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(s(s(0)), x)
times(0, y) → 0
times(s(x), y) → plus(times(x, y), y)
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(9) 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
(10) YES
(11) 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) → f(gt(x, 0), double(x))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(s(s(0)), x)
times(0, y) → 0
times(s(x), y) → plus(times(x, y), y)
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
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:
- GT(s(x), s(y)) → GT(x, y)
The graph contains the following edges 1 > 1, 2 > 2
(13) YES
(14) Obligation:
Q DP problem:
The TRS P consists of the following rules:
F(true, x) → F(gt(x, 0), double(x))
The TRS R consists of the following rules:
f(true, x) → f(gt(x, 0), double(x))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
double(x) → times(s(s(0)), x)
times(0, y) → 0
times(s(x), y) → plus(times(x, y), y)
plus(0, y) → y
plus(s(x), y) → plus(x, s(y))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(15) NonLoopProof (EQUIVALENT transformation)
By Theorem 8 [NONLOOP] we deduce infiniteness of the QDP.
We apply the theorem with m = 2, b = 2,
σ' = [ ], and μ' = [ ] on the rule
F(true, s(s(zr0)))[zr0 / s(zr0)]n[zr0 / 0] → F(true, s(s(s(s(zr0)))))[zr0 / s(s(zr0))]n[zr0 / 0]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: Equivalent (Domain Renaming) - Equivalent (Domain Renaming) - Equivalent (Remove Unused)
F(true, s(s(zl1)))[zl1 / s(zl1), zr1 / s(s(zr1))]n[zl1 / 0, zr1 / 0] → F(true, s(s(s(s(zr1)))))[zl1 / s(zl1), zr1 / s(s(zr1))]n[zl1 / 0, zr1 / 0]
by Narrowing at position: [1]
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Instantiation - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming) - Equivalent (Remove Unused)
F(true, s(s(zs1)))[zs1 / s(zs1), zt1 / s(s(zt1))]n[zs1 / y1, zt1 / y1] → F(true, plus(y1, s(s(s(s(zt1))))))[zs1 / s(zs1), zt1 / s(s(zt1))]n[zs1 / y1, zt1 / y1]
by Narrowing at position: [1]
intermediate steps: Equivalent (Add Unused) - Instantiate mu - Equivalent (Add Unused) - Instantiate Sigma - Instantiation - Instantiation
F(true, s(y0))[ ]n[ ] → F(true, plus(y0, s(s(y0))))[ ]n[ ]
by Rewrite t
F(true, s(y0))[ ]n[ ] → F(true, double(s(y0)))[ ]n[ ]
by Narrowing at position: [0]
intermediate steps: Instantiation - Instantiation
F(true, x)[ ]n[ ] → F(gt(x, 0), double(x))[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Instantiation
gt(s(x), 0)[ ]n[ ] → true[ ]n[ ]
by OriginalRule from TRS R
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Equivalent (Domain Renaming) - Equivalent (Remove Unused) - Equivalent (Simplify mu) - Instantiate mu - Instantiate Sigma - Equivalent (Simplify mu) - Equivalent (Remove Unused) - Instantiate mu - Equivalent (Simplify mu) - Instantiation - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
plus(s(x), y)[x / s(x)]n[ ] → plus(x, s(y))[y / s(y)]n[ ]
by PatternCreation I
plus(s(x), y)[ ]n[ ] → plus(x, s(y))[ ]n[ ]
by OriginalRule from TRS R
intermediate steps: Equivalent (Add Unused) - Instantiate mu - Equivalent (Add Unused) - Instantiate Sigma - Instantiation - Instantiation
plus(0, y)[ ]n[ ] → y[ ]n[ ]
by OriginalRule from TRS R
(16) NO