(0) Obligation:
Q restricted rewrite system:
The TRS R consists of the following rules:
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))
div(div(x, y), z) → div(x, times(y, z))
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:
PLUS(s(x), y) → PLUS(x, y)
TIMES(s(x), y) → PLUS(y, times(x, y))
TIMES(s(x), y) → TIMES(x, y)
DIV(x, y) → QUOT(x, y, y)
QUOT(s(x), s(y), z) → QUOT(x, y, z)
QUOT(x, 0, s(z)) → DIV(x, s(z))
DIV(div(x, y), z) → DIV(x, times(y, z))
DIV(div(x, y), z) → TIMES(y, z)
The TRS R consists of the following rules:
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))
div(div(x, y), z) → div(x, times(y, z))
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 2 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:
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))
div(div(x, y), z) → div(x, times(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(6) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
PLUS(s(x), y) → PLUS(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
PLUS(
x0,
x1,
x2) =
PLUS(
x1)
Tags:
PLUS has argument tags [2,3,2] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
PLUS(
x1,
x2) =
x2
s(
x1) =
s(
x1)
Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial
Status:
s1: [1]
The following usable rules [FROCOS05] were oriented:
none
(7) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))
div(div(x, y), z) → div(x, times(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(8) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(9) TRUE
(10) 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:
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))
div(div(x, y), z) → div(x, times(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(11) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
TIMES(s(x), y) → TIMES(x, y)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
TIMES(
x0,
x1,
x2) =
TIMES(
x1)
Tags:
TIMES has argument tags [2,3,2] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
TIMES(
x1,
x2) =
x2
s(
x1) =
s(
x1)
Lexicographic path order with status [LPO].
Quasi-Precedence:
trivial
Status:
s1: [1]
The following usable rules [FROCOS05] were oriented:
none
(12) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))
div(div(x, y), z) → div(x, times(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(13) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(14) TRUE
(15) Obligation:
Q DP problem:
The TRS P consists of the following rules:
DIV(x, y) → QUOT(x, y, y)
QUOT(s(x), s(y), z) → QUOT(x, y, z)
QUOT(x, 0, s(z)) → DIV(x, s(z))
DIV(div(x, y), z) → DIV(x, times(y, z))
The TRS R consists of the following rules:
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))
div(div(x, y), z) → div(x, times(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(16) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
QUOT(s(x), s(y), z) → QUOT(x, y, z)
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
DIV(
x0,
x1,
x2) =
DIV(
x0,
x1)
QUOT(
x0,
x1,
x2,
x3) =
QUOT(
x1)
Tags:
DIV has argument tags [1,0,0] and root tag 1
QUOT has argument tags [6,1,0,2] and root tag 1
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
DIV(
x1,
x2) =
x1
QUOT(
x1,
x2,
x3) =
QUOT(
x1,
x2,
x3)
s(
x1) =
s(
x1)
0 =
0
div(
x1,
x2) =
x1
times(
x1,
x2) =
x1
plus(
x1,
x2) =
plus
Lexicographic path order with status [LPO].
Quasi-Precedence:
QUOT3 > s1
0 > s1
plus > s1
Status:
QUOT3: [3,1,2]
s1: [1]
0: []
plus: []
The following usable rules [FROCOS05] were oriented:
none
(17) Obligation:
Q DP problem:
The TRS P consists of the following rules:
DIV(x, y) → QUOT(x, y, y)
QUOT(x, 0, s(z)) → DIV(x, s(z))
DIV(div(x, y), z) → DIV(x, times(y, z))
The TRS R consists of the following rules:
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))
div(div(x, y), z) → div(x, times(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(18) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
DIV(div(x, y), z) → DIV(x, times(y, z))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
DIV(
x0,
x1,
x2) =
DIV(
x0,
x1)
QUOT(
x0,
x1,
x2,
x3) =
QUOT(
x0)
Tags:
DIV has argument tags [0,1,0] and root tag 0
QUOT has argument tags [1,4,7,6] and root tag 0
Comparison: MAX
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
DIV(
x1,
x2) =
x1
QUOT(
x1,
x2,
x3) =
x1
0 =
0
s(
x1) =
s
div(
x1,
x2) =
div(
x1)
times(
x1,
x2) =
x1
plus(
x1,
x2) =
plus(
x1)
Lexicographic path order with status [LPO].
Quasi-Precedence:
plus1 > s
Status:
0: []
s: []
div1: [1]
plus1: [1]
The following usable rules [FROCOS05] were oriented:
none
(19) Obligation:
Q DP problem:
The TRS P consists of the following rules:
DIV(x, y) → QUOT(x, y, y)
QUOT(x, 0, s(z)) → DIV(x, s(z))
The TRS R consists of the following rules:
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))
div(div(x, y), z) → div(x, times(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(20) QDPOrderProof (EQUIVALENT transformation)
We use the reduction pair processor [LPAR04].
The following pairs can be oriented strictly and are deleted.
DIV(x, y) → QUOT(x, y, y)
QUOT(x, 0, s(z)) → DIV(x, s(z))
The remaining pairs can at least be oriented weakly.
Used ordering: SCNP Order with the following components:
Level mapping:
Top level AFS:
DIV(
x0,
x1,
x2) =
DIV(
x0,
x1,
x2)
QUOT(
x0,
x1,
x2,
x3) =
QUOT(
x0,
x1,
x2,
x3)
Tags:
DIV has argument tags [2,1,2] and root tag 1
QUOT has argument tags [2,1,2,4] and root tag 0
Comparison: DMS
Underlying order for the size change arcs and the rules of R:
Combined order from the following AFS and order.
DIV(
x1,
x2) =
DIV
QUOT(
x1,
x2,
x3) =
QUOT
0 =
0
s(
x1) =
s
Lexicographic path order with status [LPO].
Quasi-Precedence:
[DIV, QUOT, 0] > s
Status:
DIV: []
QUOT: []
0: []
s: []
The following usable rules [FROCOS05] were oriented:
none
(21) Obligation:
Q DP problem:
P is empty.
The TRS R consists of the following rules:
plus(x, 0) → x
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(0), y) → y
times(s(x), y) → plus(y, times(x, y))
div(0, y) → 0
div(x, y) → quot(x, y, y)
quot(0, s(y), z) → 0
quot(s(x), s(y), z) → quot(x, y, z)
quot(x, 0, s(z)) → s(div(x, s(z)))
div(div(x, y), z) → div(x, times(y, z))
Q is empty.
We have to consider all minimal (P,Q,R)-chains.
(22) PisEmptyProof (EQUIVALENT transformation)
The TRS P is empty. Hence, there is no (P,Q,R) chain.
(23) TRUE