(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), plus2(x), plus1(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
plus1(x) → plus(s(0), x)
plus2(x) → plus(s(s(0)), x)
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), plus2(x), plus1(y))
F(true, x, y) → GT(x, y)
F(true, x, y) → PLUS2(x)
F(true, x, y) → PLUS1(y)
GT(s(x), s(y)) → GT(x, y)
PLUS1(x) → PLUS(s(0), x)
PLUS2(x) → PLUS(s(s(0)), x)
PLUS(s(x), y) → PLUS(x, y)
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), plus2(x), plus1(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
plus1(x) → plus(s(0), x)
plus2(x) → plus(s(s(0)), x)
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 3 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, y)
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), plus2(x), plus1(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
plus1(x) → plus(s(0), x)
plus2(x) → plus(s(s(0)), x)
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:
GT(s(x), s(y)) → GT(x, y)
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), plus2(x), plus1(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
plus1(x) → plus(s(0), x)
plus2(x) → plus(s(s(0)), x)
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:
- GT(s(x), s(y)) → GT(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:
F(true, x, y) → F(gt(x, y), plus2(x), plus1(y))
The TRS R consists of the following rules:
f(true, x, y) → f(gt(x, y), plus2(x), plus1(y))
gt(s(x), 0) → true
gt(0, y) → false
gt(s(x), s(y)) → gt(x, y)
plus1(x) → plus(s(0), x)
plus2(x) → plus(s(s(0)), x)
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) 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' = [zr0 / s(zr0)] on the rule
F(true, s(s(s(zr0))), s(zr1))[zr0 / s(zr0), zr1 / s(zr1)]n[zr1 / 0] → F(true, s(s(s(s(zr0)))), s(s(zr1)))[zr0 / s(zr0), zr1 / s(zr1)]n[zr1 / 0, zr0 / s(zr0)]
This rule is correct for the QDP as the following derivation shows:
intermediate steps: 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 / y0, zl3 / 0] → F(true, s(s(s(s(zr2)))), s(s(zr3)))[zr2 / s(zr2), zr3 / s(zr3)]n[zr2 / y0, zr3 / 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)]n[zl2 / y0, zl3 / 0, zr2 / y0, zr3 / 0] → F(true, plus2(s(s(zr2))), plus1(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), plus2(s(zs2)), plus1(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), plus2(x), plus1(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
(13) NO