Termination w.r.t. Q proof of /tmp/tmpW0nXUV/Ex5_DLMMU04

Termination w.r.t. Q of the given QTRS could be proven:

0 QTRS

↳1 QTRSToCSRProof (⇔)

↳2 CSR

↳3 PoloCSRProof (⇔)

↳4 CSR

↳5 PoloCSRProof (⇔)

↳6 CSR

↳7 PoloCSRProof (⇔)

↳8 CSR

↳9 PoloCSRProof (⇔)

↳10 CSR

↳11 PoloCSRProof (⇔)

↳12 CSR

↳13 RisEmptyProof (⇔)

↳14 TRUE

(0) Obligation:

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.

(1) QTRSToCSRProof (EQUIVALENT transformation)

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

active(pairNs) → mark(cons(0, incr(oddNs)))
active(oddNs) → mark(incr(pairNs))
active(incr(cons(X, XS))) → mark(cons(s(X), incr(XS)))
active(take(0, XS)) → mark(nil)
active(take(s(N), cons(X, XS))) → mark(cons(X, take(N, XS)))
active(zip(nil, XS)) → mark(nil)
active(zip(X, nil)) → mark(nil)
active(zip(cons(X, XS), cons(Y, YS))) → mark(cons(pair(X, Y), zip(XS, YS)))
active(tail(cons(X, XS))) → mark(XS)
active(repItems(nil)) → mark(nil)
active(repItems(cons(X, XS))) → mark(cons(X, cons(X, repItems(XS))))
active(cons(X1, X2)) → cons(active(X1), X2)
active(incr(X)) → incr(active(X))
active(s(X)) → s(active(X))
active(take(X1, X2)) → take(active(X1), X2)
active(take(X1, X2)) → take(X1, active(X2))
active(zip(X1, X2)) → zip(active(X1), X2)
active(zip(X1, X2)) → zip(X1, active(X2))
active(pair(X1, X2)) → pair(active(X1), X2)
active(pair(X1, X2)) → pair(X1, active(X2))
active(tail(X)) → tail(active(X))
active(repItems(X)) → repItems(active(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
incr(mark(X)) → mark(incr(X))
s(mark(X)) → mark(s(X))
take(mark(X1), X2) → mark(take(X1, X2))
take(X1, mark(X2)) → mark(take(X1, X2))
zip(mark(X1), X2) → mark(zip(X1, X2))
zip(X1, mark(X2)) → mark(zip(X1, X2))
pair(mark(X1), X2) → mark(pair(X1, X2))
pair(X1, mark(X2)) → mark(pair(X1, X2))
tail(mark(X)) → mark(tail(X))
repItems(mark(X)) → mark(repItems(X))
proper(pairNs) → ok(pairNs)
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(incr(X)) → incr(proper(X))
proper(oddNs) → ok(oddNs)
proper(s(X)) → s(proper(X))
proper(take(X1, X2)) → take(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(zip(X1, X2)) → zip(proper(X1), proper(X2))
proper(pair(X1, X2)) → pair(proper(X1), proper(X2))
proper(tail(X)) → tail(proper(X))
proper(repItems(X)) → repItems(proper(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
incr(ok(X)) → ok(incr(X))
s(ok(X)) → ok(s(X))
take(ok(X1), ok(X2)) → ok(take(X1, X2))
zip(ok(X1), ok(X2)) → ok(zip(X1, X2))
pair(ok(X1), ok(X2)) → ok(pair(X1, X2))
tail(ok(X)) → ok(tail(X))
repItems(ok(X)) → ok(repItems(X))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))

Q is empty.
Special symbols used for the transformation (see [GM04]):
top: top, active: active, mark: mark, ok: ok, proper: proper
The replacement map contains the following entries:

pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}
nil: empty set
zip: {1, 2}
pair: {1, 2}
tail: {1}
repItems: {1}
The QTRS contained all rules created by the complete Giesl-Middeldorp transformation. Therefore, the inverse transformation is complete (and sound).

(2) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

pairNs → cons(0, incr(oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), incr(XS))
take(0, XS) → nil
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
zip(nil, XS) → nil
zip(X, nil) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), zip(XS, YS))
tail(cons(X, XS)) → XS
repItems(nil) → nil
repItems(cons(X, XS)) → cons(X, cons(X, repItems(XS)))

The replacement map contains the following entries:

pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}
nil: empty set
zip: {1, 2}
pair: {1, 2}
tail: {1}
repItems: {1}

(3) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

take(0, XS) → nil
zip(X, nil) → nil
tail(cons(X, XS)) → XS
repItems(nil) → nil

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(cons(x₁, x₂)) = x₁ + x₂
POL(incr(x₁)) = 2·x₁
POL(nil) = 1
POL(oddNs) = 0
POL(pair(x₁, x₂)) = x₁ + x₂
POL(pairNs) = 0
POL(repItems(x₁)) = 2·x₁
POL(s(x₁)) = x₁
POL(tail(x₁)) = 2 + 2·x₁
POL(take(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(zip(x₁, x₂)) = x₁ + 2·x₂

(4) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

pairNs → cons(0, incr(oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), incr(XS))
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
zip(nil, XS) → nil
zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), zip(XS, YS))
repItems(cons(X, XS)) → cons(X, cons(X, repItems(XS)))

The replacement map contains the following entries:

pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}
nil: empty set
zip: {1, 2}
pair: {1, 2}
repItems: {1}

(5) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

zip(cons(X, XS), cons(Y, YS)) → cons(pair(X, Y), zip(XS, YS))
repItems(cons(X, XS)) → cons(X, cons(X, repItems(XS)))

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(cons(x₁, x₂)) = 1 + 2·x₁
POL(incr(x₁)) = x₁
POL(nil) = 0
POL(oddNs) = 1
POL(pair(x₁, x₂)) = x₁ + x₂
POL(pairNs) = 1
POL(repItems(x₁)) = 2 + 2·x₁
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + x₂
POL(zip(x₁, x₂)) = x₁ + x₂

(6) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

pairNs → cons(0, incr(oddNs))
oddNs → incr(pairNs)
incr(cons(X, XS)) → cons(s(X), incr(XS))
take(s(N), cons(X, XS)) → cons(X, take(N, XS))
zip(nil, XS) → nil

The replacement map contains the following entries:

pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}
nil: empty set
zip: {1, 2}

(7) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

oddNs → incr(pairNs)
zip(nil, XS) → nil

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(cons(x₁, x₂)) = x₁
POL(incr(x₁)) = 2·x₁
POL(nil) = 2
POL(oddNs) = 2
POL(pairNs) = 0
POL(s(x₁)) = x₁
POL(take(x₁, x₂)) = x₁ + 2·x₂
POL(zip(x₁, x₂)) = 2 + x₁ + x₂

(8) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

pairNs → cons(0, incr(oddNs))
incr(cons(X, XS)) → cons(s(X), incr(XS))
take(s(N), cons(X, XS)) → cons(X, take(N, XS))

The replacement map contains the following entries:

pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}

(9) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

incr(cons(X, XS)) → cons(s(X), incr(XS))

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(cons(x₁, x₂)) = 2 + x₁
POL(incr(x₁)) = 2·x₁
POL(oddNs) = 0
POL(pairNs) = 2
POL(s(x₁)) = 2·x₁
POL(take(x₁, x₂)) = 2·x₁ + x₂

(10) Obligation:

Context-sensitive rewrite system:
The TRS R consists of the following rules:

pairNs → cons(0, incr(oddNs))
take(s(N), cons(X, XS)) → cons(X, take(N, XS))

The replacement map contains the following entries:

pairNs: empty set
cons: {1}
0: empty set
incr: {1}
oddNs: empty set
s: {1}
take: {1, 2}

(11) PoloCSRProof (EQUIVALENT transformation)

The following rules can be removed because they are oriented strictly by a µ-monotonic polynomial ordering:

pairNs → cons(0, incr(oddNs))
take(s(N), cons(X, XS)) → cons(X, take(N, XS))

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(cons(x₁, x₂)) = 1 + x₁ + x₂
POL(incr(x₁)) = x₁
POL(oddNs) = 0
POL(pairNs) = 2
POL(s(x₁)) = 2 + 2·x₁
POL(take(x₁, x₂)) = 2 + 2·x₁ + 2·x₂

(0) Obligation:

(1) QTRSToCSRProof (EQUIVALENT transformation)

(2) Obligation:

(3) PoloCSRProof (EQUIVALENT transformation)

(4) Obligation:

(5) PoloCSRProof (EQUIVALENT transformation)

(6) Obligation:

(7) PoloCSRProof (EQUIVALENT transformation)

(8) Obligation:

(9) PoloCSRProof (EQUIVALENT transformation)

(10) Obligation:

(11) PoloCSRProof (EQUIVALENT transformation)

(12) Obligation:

(13) RisEmptyProof (EQUIVALENT transformation)

(14) TRUE