(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) → s(plus(x, 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, 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, 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) → s(plus(x, 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, 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) → s(plus(x, 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, y)
The graph contains the following edges 1 > 1, 2 >= 2
(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, 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) → s(plus(x, 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, 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) → s(plus(x, 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, 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) → s(plus(x, 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' = [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:
intermediate steps: Equivalent (Remove Unused) - Instantiate mu - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming) - Equivalent (Simplify mu) - Instantiate mu - Equivalent (Remove Unused) - Instantiate mu - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
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
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
intermediate steps: Equivalent (Remove Unused)
F(true, s(s(zl2)), s(zl3))[zl2 / s(zl2), zl3 / s(zl3), zr2 / s(zr2), zr3 / s(zr3), zt1 / plus(s(s(0)), zt1)]n[zl2 / y1, zl3 / 0, x0 / y1, zr2 / y1, zr3 / 0, zt1 / times(y1, s(s(0)))] → F(true, plus(s(s(0)), plus(s(s(0)), zt1)), s(s(zr3)))[zl2 / s(zl2), zl3 / s(zl3), zr2 / s(zr2), zr3 / s(zr3), zt1 / plus(s(s(0)), zt1)]n[zl2 / y1, zl3 / 0, x0 / y1, zr2 / y1, zr3 / 0, zt1 / times(y1, s(s(0)))]
by Narrowing at position: [1]
intermediate steps: Equivalent (Add Unused) - Instantiate mu - Equivalent (Add Unused) - Equivalent (Domain Renaming) - Equivalent (Domain Renaming) - Equivalent (Remove Unused)
F(true, s(s(zl2)), s(zl3))[zl2 / s(zl2), zl3 / s(zl3), zr2 / s(zr2), zr3 / s(zr3)]n[zl2 / x0, zl3 / 0, zr2 / x0, zr3 / 0] → F(true, times(s(s(zr2)), s(s(0))), s(s(zr3)))[zl2 / s(zl2), zl3 / s(zl3), zr2 / s(zr2), zr3 / s(zr3)]n[zl2 / x0, zl3 / 0, zr2 / x0, zr3 / 0]
by Narrowing at position: [1]
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming) - Equivalent (Remove Unused)
F(true, s(s(zl2)), s(zl3))[zl2 / s(zl2), zl3 / s(zl3), zr2 / s(zr2), zr3 / s(zr3)]n[zl2 / y0, zl3 / 0, zr2 / y0, zr3 / 0] → F(true, double(s(s(zr2))), s(s(zr3)))[zl2 / s(zl2), zl3 / s(zl3), zr2 / s(zr2), zr3 / s(zr3)]n[zl2 / y0, zl3 / 0, zr2 / y0, zr3 / 0]
by Narrowing at position: [0]
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Instantiation - Equivalent (Simplify mu) - Instantiation - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
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 OriginalRule from TRS P
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming)
gt(s(x), s(y))[x / s(x), y / s(y)]n[ ] → gt(x, y)[ ]n[ ]
by PatternCreation I
gt(s(x), s(y))[ ]n[ ] → gt(x, y)[ ]n[ ]
by OriginalRule from TRS R
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Instantiation
gt(s(x), 0)[ ]n[ ] → true[ ]n[ ]
by OriginalRule from TRS R
intermediate steps: Equivalent (Add Unused) - Instantiate mu - Equivalent (Add Unused) - Instantiate Sigma - Instantiation - Instantiation
double(x)[ ]n[ ] → times(x, s(s(0)))[ ]n[ ]
by OriginalRule from TRS R
intermediate steps: Equivalent (Add Unused) - Equivalent (Add Unused) - Instantiation - Equivalent (Domain Renaming) - Expand Sigma - Equivalent (Domain Renaming) - Instantiation - Equivalent (Domain Renaming) - Equivalent (Remove Unused)
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
times(s(x), y)[ ]n[ ] → plus(y, times(x, y))[ ]n[ ]
by OriginalRule from TRS R
(16) NO