Termination w.r.t. Q proof of Transformed_CSR_04_PEANO_nokinds

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

0 QTRS

↳1 QTRSToCSRProof (⇔, 0 ms)

↳2 CSR

↳3 CSRRRRProof (⇔, 51 ms)

↳4 CSR

↳5 CSRRRRProof (⇔, 0 ms)

↳6 CSR

↳7 CSRRRRProof (⇔, 0 ms)

↳8 CSR

↳9 CSRRRRProof (⇔, 0 ms)

↳10 CSR

↳11 CSRRRRProof (⇔, 0 ms)

↳12 CSR

↳13 CSRRRRProof (⇔, 3 ms)

↳14 CSR

↳15 CSRRRRProof (⇔, 0 ms)

↳16 CSR

↳17 RisEmptyProof (⇔, 0 ms)

↳18 YES

(0) Obligation:

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

active(U11(tt, N)) → mark(N)
active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) → mark(isNat(V1))
active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2) → mark(U11(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(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(U11(tt, N)) → mark(N)
active(U21(tt, M, N)) → mark(s(plus(N, M)))
active(and(tt, X)) → mark(X)
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(and(isNat(V1), isNat(V2)))
active(isNat(s(V1))) → mark(isNat(V1))
active(plus(N, 0)) → mark(U11(isNat(N), N))
active(plus(N, s(M))) → mark(U21(and(isNat(M), isNat(N)), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U21(X1, X2, X3)) → U21(active(X1), X2, X3)
active(s(X)) → s(active(X))
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(and(X1, X2)) → and(active(X1), X2)
U11(mark(X1), X2) → mark(U11(X1, X2))
U21(mark(X1), X2, X3) → mark(U21(X1, X2, X3))
s(mark(X)) → mark(s(X))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
and(mark(X1), X2) → mark(and(X1, X2))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U21(X1, X2, X3)) → U21(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(0) → ok(0)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U21(ok(X1), ok(X2), ok(X3)) → ok(U21(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
isNat(ok(X)) → ok(isNat(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:

U11: {1}
tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set
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:

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set

(3) CSRRRRProof (EQUIVALENT transformation)

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

U11(tt, N) → N
U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(U21(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(and(x₁, x₂)) = x₁ + 2·x₂
POL(isNat(x₁)) = 0
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U11(tt, N) → N

(4) Obligation:

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

U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set

(5) CSRRRRProof (EQUIVALENT transformation)

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

U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, 0) → U11(isNat(N), N)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(U11(x₁, x₂)) = 1 + x₁ + x₂
POL(U21(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(and(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = 0
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

plus(N, 0) → U11(isNat(N), N)

(6) Obligation:

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

U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

The replacement map contains the following entries:

tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set

(7) CSRRRRProof (EQUIVALENT transformation)

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

U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)
plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

The replacement map contains the following entries:

tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U21(x₁, x₂, x₃)) = 2 + x₁ + 2·x₂ + x₃
POL(and(x₁, x₂)) = 2·x₁ + x₂
POL(isNat(x₁)) = 0
POL(plus(x₁, x₂)) = x₁ + 2·x₂
POL(s(x₁)) = 2 + x₁
POL(tt) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

plus(N, s(M)) → U21(and(isNat(M), isNat(N)), M, N)

(8) Obligation:

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

U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)

The replacement map contains the following entries:

tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set

(9) CSRRRRProof (EQUIVALENT transformation)

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

U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X
isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)

The replacement map contains the following entries:

tt: empty set
U21: {1}
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(U21(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(and(x₁, x₂)) = x₁ + x₂
POL(isNat(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 1

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

U21(tt, M, N) → s(plus(N, M))
and(tt, X) → X

(10) Obligation:

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

isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)

The replacement map contains the following entries:

tt: empty set
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set

(11) CSRRRRProof (EQUIVALENT transformation)

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

isNat(0) → tt
isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)

The replacement map contains the following entries:

tt: empty set
s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(and(x₁, x₂)) = x₁
POL(isNat(x₁)) = 1 + x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

isNat(0) → tt

(12) Obligation:

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

isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)

The replacement map contains the following entries:

s: {1}
plus: {1, 2}
and: {1}
isNat: empty set

(13) CSRRRRProof (EQUIVALENT transformation)

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

isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))
isNat(s(V1)) → isNat(V1)

The replacement map contains the following entries:

s: {1}
plus: {1, 2}
and: {1}
isNat: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(and(x₁, x₂)) = x₁
POL(isNat(x₁)) = x₁
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = x₁

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

isNat(plus(V1, V2)) → and(isNat(V1), isNat(V2))

(14) Obligation:

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

isNat(s(V1)) → isNat(V1)

The replacement map contains the following entries:

s: {1}
isNat: empty set

(15) CSRRRRProof (EQUIVALENT transformation)

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

isNat(s(V1)) → isNat(V1)

The replacement map contains the following entries:

s: {1}
isNat: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(isNat(x₁)) = x₁
POL(s(x₁)) = 1 + x₁

With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

isNat(s(V1)) → isNat(V1)

(16) Obligation:

Context-sensitive rewrite system:
R is empty.

(17) RisEmptyProof (EQUIVALENT transformation)

The CSR R is empty. Hence, termination is trivially proven.

(0) Obligation:

(1) QTRSToCSRProof (EQUIVALENT transformation)

(2) Obligation:

(3) CSRRRRProof (EQUIVALENT transformation)

(4) Obligation:

(5) CSRRRRProof (EQUIVALENT transformation)

(6) Obligation:

(7) CSRRRRProof (EQUIVALENT transformation)

(8) Obligation:

(9) CSRRRRProof (EQUIVALENT transformation)

(10) Obligation:

(11) CSRRRRProof (EQUIVALENT transformation)

(12) Obligation:

(13) CSRRRRProof (EQUIVALENT transformation)

(14) Obligation:

(15) CSRRRRProof (EQUIVALENT transformation)

(16) Obligation:

(17) RisEmptyProof (EQUIVALENT transformation)

(18) YES