(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
while(true, s(s(s(i)))) → while(gt(s(s(s(i))), s(0)), f(s(s(s(i)))))
f(i) → if(neq(i, s(s(0))), i)
gt(s(x), s(y)) → gt(x, y)
gt(s(x), 0) → true
gt(0, 0) → false
gt(0, s(y)) → false
if(true, i) → plus(i, s(0))
if(false, i) → i
neq(s(x), s(y)) → neq(x, y)
neq(0, 0) → false
neq(0, s(y)) → true
neq(s(x), 0) → true
plus(s(x), y) → plus(x, s(y))
plus(0, y) → 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:
WHILE(true, s(s(s(i)))) → WHILE(gt(s(s(s(i))), s(0)), f(s(s(s(i)))))
WHILE(true, s(s(s(i)))) → GT(s(s(s(i))), s(0))
WHILE(true, s(s(s(i)))) → F(s(s(s(i))))
F(i) → IF(neq(i, s(s(0))), i)
F(i) → NEQ(i, s(s(0)))
GT(s(x), s(y)) → GT(x, y)
IF(true, i) → PLUS(i, s(0))
NEQ(s(x), s(y)) → NEQ(x, y)
PLUS(s(x), y) → PLUS(x, s(y))
The TRS R consists of the following rules:
while(true, s(s(s(i)))) → while(gt(s(s(s(i))), s(0)), f(s(s(s(i)))))
f(i) → if(neq(i, s(s(0))), i)
gt(s(x), s(y)) → gt(x, y)
gt(s(x), 0) → true
gt(0, 0) → false
gt(0, s(y)) → false
if(true, i) → plus(i, s(0))
if(false, i) → i
neq(s(x), s(y)) → neq(x, y)
neq(0, 0) → false
neq(0, s(y)) → true
neq(s(x), 0) → true
plus(s(x), y) → plus(x, s(y))
plus(0, y) → 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 5 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:
while(true, s(s(s(i)))) → while(gt(s(s(s(i))), s(0)), f(s(s(s(i)))))
f(i) → if(neq(i, s(s(0))), i)
gt(s(x), s(y)) → gt(x, y)
gt(s(x), 0) → true
gt(0, 0) → false
gt(0, s(y)) → false
if(true, i) → plus(i, s(0))
if(false, i) → i
neq(s(x), s(y)) → neq(x, y)
neq(0, 0) → false
neq(0, s(y)) → true
neq(s(x), 0) → true
plus(s(x), y) → plus(x, s(y))
plus(0, y) → 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:
NEQ(s(x), s(y)) → NEQ(x, y)
The TRS R consists of the following rules:
while(true, s(s(s(i)))) → while(gt(s(s(s(i))), s(0)), f(s(s(s(i)))))
f(i) → if(neq(i, s(s(0))), i)
gt(s(x), s(y)) → gt(x, y)
gt(s(x), 0) → true
gt(0, 0) → false
gt(0, s(y)) → false
if(true, i) → plus(i, s(0))
if(false, i) → i
neq(s(x), s(y)) → neq(x, y)
neq(0, 0) → false
neq(0, s(y)) → true
neq(s(x), 0) → true
plus(s(x), y) → plus(x, s(y))
plus(0, y) → 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:
- NEQ(s(x), s(y)) → NEQ(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:
while(true, s(s(s(i)))) → while(gt(s(s(s(i))), s(0)), f(s(s(s(i)))))
f(i) → if(neq(i, s(s(0))), i)
gt(s(x), s(y)) → gt(x, y)
gt(s(x), 0) → true
gt(0, 0) → false
gt(0, s(y)) → false
if(true, i) → plus(i, s(0))
if(false, i) → i
neq(s(x), s(y)) → neq(x, y)
neq(0, 0) → false
neq(0, s(y)) → true
neq(s(x), 0) → true
plus(s(x), y) → plus(x, s(y))
plus(0, y) → 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:
WHILE(true, s(s(s(i)))) → WHILE(gt(s(s(s(i))), s(0)), f(s(s(s(i)))))
The TRS R consists of the following rules:
while(true, s(s(s(i)))) → while(gt(s(s(s(i))), s(0)), f(s(s(s(i)))))
f(i) → if(neq(i, s(s(0))), i)
gt(s(x), s(y)) → gt(x, y)
gt(s(x), 0) → true
gt(0, 0) → false
gt(0, s(y)) → false
if(true, i) → plus(i, s(0))
if(false, i) → i
neq(s(x), s(y)) → neq(x, y)
neq(0, 0) → false
neq(0, s(y)) → true
neq(s(x), 0) → true
plus(s(x), y) → plus(x, s(y))
plus(0, y) → 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 = 1, b = 1,
σ' = [ ], and μ' = [ ] on the rule
WHILE(true, s(s(s(s(zr0)))))[zr0 / s(zr0)]n[zr0 / 0] → WHILE(true, s(s(s(s(s(zr0))))))[zr0 / 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)
WHILE(true, s(s(s(s(zl1)))))[zl1 / s(zl1), zr1 / s(zr1)]n[zl1 / 0, zr1 / 0] → WHILE(true, s(s(s(s(s(zr1))))))[zl1 / s(zl1), zr1 / 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)
WHILE(true, s(s(s(s(zs1)))))[zs1 / s(zs1), zt1 / s(zt1)]n[zs1 / y1, zt1 / 0] → WHILE(true, plus(y1, s(s(s(s(s(zt1)))))))[zs1 / s(zs1), zt1 / s(zt1)]n[zs1 / y1, zt1 / 0]
by Narrowing at position: [1]
intermediate steps: Equivalent (Add Unused) - Instantiate mu - Equivalent (Add Unused) - Instantiate Sigma - Instantiation - Instantiation
WHILE(true, s(s(s(i))))[ ]n[ ] → WHILE(true, plus(i, s(s(s(s(0))))))[ ]n[ ]
by Rewrite t
WHILE(true, s(s(s(i))))[ ]n[ ] → WHILE(gt(s(s(s(i))), s(0)), f(s(s(s(i)))))[ ]n[ ]
by OriginalRule from TRS P
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - 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