Termination w.r.t. Q proof of /tmp/tmpWaouLW/thiemann32.trs

Termination w.r.t. Q of the given QTRS could not be shown:

0 QTRS

↳1 Overlay + Local Confluence (⇔)

↳2 QTRS

↳3 DependencyPairsProof (⇔)

↳4 QDP

↳5 DependencyGraphProof (⇔)

↳6 AND

  ↳7 QDP

    ↳8 QDPOrderProof (⇔)

    ↳9 QDP

    ↳10 PisEmptyProof (⇔)

    ↳11 TRUE

  ↳12 QDP

    ↳13 QDPOrderProof (⇔)

    ↳14 QDP

    ↳15 PisEmptyProof (⇔)

    ↳16 TRUE

  ↳17 QDP

    ↳18 QDPOrderProof (⇔)

    ↳19 QDP

    ↳20 PisEmptyProof (⇔)

    ↳21 TRUE

  ↳22 QDP

(0) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

Q is empty.

(1) Overlay + Local Confluence (EQUIVALENT transformation)

The TRS is overlay and locally confluent. By [NOC] we can switch to innermost.

(2) Obligation:

Q restricted rewrite system:
The TRS R consists of the following rules:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

The set Q consists of the following terms:

table
gen(x0)
if1(false, x0)
if1(true, x0)
if2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
10

(3) DependencyPairsProof (EQUIVALENT transformation)

Using Dependency Pairs [AG00,LPAR04] we result in the following initial DP problem.

(4) Obligation:

Q DP problem:
The TRS P consists of the following rules:

TABLE → GEN(s(0))
GEN(x) → IF1(le(x, 10), x)
GEN(x) → LE(x, 10)
GEN(x) → 10¹
IF1(true, x) → IF2(x, x)
IF2(x, y) → IF3(le(y, 10), x, y)
IF2(x, y) → LE(y, 10)
IF2(x, y) → 10¹
IF3(true, x, y) → TIMES(x, y)
IF3(true, x, y) → IF2(x, s(y))
IF3(false, x, y) → GEN(s(x))
LE(s(x), s(y)) → LE(x, y)
PLUS(s(x), y) → PLUS(x, y)
TIMES(s(x), y) → PLUS(y, times(x, y))
TIMES(s(x), y) → TIMES(x, y)

The TRS R consists of the following rules:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

The set Q consists of the following terms:

table
gen(x0)
if1(false, x0)
if1(true, x0)
if2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
10

We have to consider all minimal (P,Q,R)-chains.

(5) DependencyGraphProof (EQUIVALENT transformation)

The approximation of the Dependency Graph [LPAR04,FROCOS05,EDGSTAR] contains 4 SCCs with 7 less nodes.

(6) Complex Obligation (AND)

(7) 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:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

The set Q consists of the following terms:

table
gen(x0)
if1(false, x0)
if1(true, x0)
if2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
10

We have to consider all minimal (P,Q,R)-chains.

(8) 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: Combined order from the following AFS and order.
PLUS(x1, x2) = PLUS(x1, x2)
s(x1) = s(x1)
table = table
gen(x1) = gen
0 = 0
if1(x1, x2) = x1
le(x1, x2) = le
10 = 10
false = false
nil = nil
true = true
if2(x1, x2) = if2
if3(x1, x2, x3) = x1
cons(x1, x2) = cons
entry(x1, x2, x3) = entry(x3)
times(x1, x2) = times(x1, x2)
plus(x1, x2) = plus(x1, x2)

Lexicographic path order with status [LPO].
Quasi-Precedence:

[table, gen, le, false, true, if2, cons] > [PLUS₂, s₁, 0, nil]
10 > [PLUS₂, s₁, 0, nil]
entry₁ > [PLUS₂, s₁, 0, nil]
times₂ > plus₂ > [PLUS₂, s₁, 0, nil]

Status:

PLUS₂: [1,2]
s₁: [1]
table: []
gen: []
0: []
le: []
10: []
false: []
nil: []
true: []
if2: []
cons: []
entry₁: [1]
times₂: [2,1]
plus₂: [1,2]

The following usable rules [FROCOS05] were oriented:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

(9) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

The set Q consists of the following terms:

table
gen(x0)
if1(false, x0)
if1(true, x0)
if2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
10

We have to consider all minimal (P,Q,R)-chains.

(10) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(11) TRUE

(12) 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:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

The set Q consists of the following terms:

table
gen(x0)
if1(false, x0)
if1(true, x0)
if2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
10

We have to consider all minimal (P,Q,R)-chains.

(13) 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: Combined order from the following AFS and order.
TIMES(x1, x2) = TIMES(x1, x2)
s(x1) = s(x1)
table = table
gen(x1) = gen
0 = 0
if1(x1, x2) = x1
le(x1, x2) = le
10 = 10
false = false
nil = nil
true = true
if2(x1, x2) = if2
if3(x1, x2, x3) = x1
cons(x1, x2) = cons
entry(x1, x2, x3) = entry(x3)
times(x1, x2) = times(x1, x2)
plus(x1, x2) = plus(x1, x2)

Lexicographic path order with status [LPO].
Quasi-Precedence:

[table, gen, le, false, true, if2, cons] > [TIMES₂, s₁, 0, nil]
10 > [TIMES₂, s₁, 0, nil]
entry₁ > [TIMES₂, s₁, 0, nil]
times₂ > plus₂ > [TIMES₂, s₁, 0, nil]

Status:

TIMES₂: [1,2]
s₁: [1]
table: []
gen: []
0: []
le: []
10: []
false: []
nil: []
true: []
if2: []
cons: []
entry₁: [1]
times₂: [2,1]
plus₂: [1,2]

The following usable rules [FROCOS05] were oriented:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

(14) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

The set Q consists of the following terms:

table
gen(x0)
if1(false, x0)
if1(true, x0)
if2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
10

We have to consider all minimal (P,Q,R)-chains.

(15) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(16) TRUE

(17) Obligation:

Q DP problem:
The TRS P consists of the following rules:

LE(s(x), s(y)) → LE(x, y)

The TRS R consists of the following rules:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

The set Q consists of the following terms:

table
gen(x0)
if1(false, x0)
if1(true, x0)
if2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
10

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.

LE(s(x), s(y)) → LE(x, y)

The remaining pairs can at least be oriented weakly.
Used ordering: Combined order from the following AFS and order.
LE(x1, x2) = LE(x1)
s(x1) = s(x1)
table = table
gen(x1) = gen
0 = 0
if1(x1, x2) = x1
le(x1, x2) = le
10 = 10
false = false
nil = nil
true = true
if2(x1, x2) = if2
if3(x1, x2, x3) = if3
cons(x1, x2) = cons
entry(x1, x2, x3) = entry(x1, x2, x3)
times(x1, x2) = times(x1, x2)
plus(x1, x2) = plus(x1, x2)

Lexicographic path order with status [LPO].
Quasi-Precedence:

LE₁ > [s₁, 0]
table > [gen, le, false, true, if2, if3, cons] > 10 > [s₁, 0]
table > [gen, le, false, true, if2, if3, cons] > nil > [s₁, 0]
entry₃ > [s₁, 0]
times₂ > plus₂ > [s₁, 0]

Status:

LE₁: [1]
s₁: [1]
table: []
gen: []
0: []
le: []
10: []
false: []
nil: []
true: []
if2: []
if3: []
cons: []
entry₃: [3,2,1]
times₂: [2,1]
plus₂: [2,1]

The following usable rules [FROCOS05] were oriented:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

(19) Obligation:

Q DP problem:
P is empty.
The TRS R consists of the following rules:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

The set Q consists of the following terms:

table
gen(x0)
if1(false, x0)
if1(true, x0)
if2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
10

We have to consider all minimal (P,Q,R)-chains.

(20) PisEmptyProof (EQUIVALENT transformation)

The TRS P is empty. Hence, there is no (P,Q,R) chain.

(21) TRUE

(22) Obligation:

Q DP problem:
The TRS P consists of the following rules:

IF1(true, x) → IF2(x, x)
IF2(x, y) → IF3(le(y, 10), x, y)
IF3(true, x, y) → IF2(x, s(y))
IF3(false, x, y) → GEN(s(x))
GEN(x) → IF1(le(x, 10), x)

The TRS R consists of the following rules:

table → gen(s(0))
gen(x) → if1(le(x, 10), x)
if1(false, x) → nil
if1(true, x) → if2(x, x)
if2(x, y) → if3(le(y, 10), x, y)
if3(true, x, y) → cons(entry(x, y, times(x, y)), if2(x, s(y)))
if3(false, x, y) → gen(s(x))
le(0, y) → true
le(s(x), 0) → false
le(s(x), s(y)) → le(x, y)
plus(0, y) → y
plus(s(x), y) → s(plus(x, y))
times(0, y) → 0
times(s(x), y) → plus(y, times(x, y))
10 → s(s(s(s(s(s(s(s(s(s(0))))))))))

The set Q consists of the following terms:

table
gen(x0)
if1(false, x0)
if1(true, x0)
if2(x0, x1)
if3(true, x0, x1)
if3(false, x0, x1)
le(0, x0)
le(s(x0), 0)
le(s(x0), s(x1))
plus(0, x0)
plus(s(x0), x1)
times(0, x0)
times(s(x0), x1)
10

We have to consider all minimal (P,Q,R)-chains.