Termination w.r.t. Q proof of /tmp/tmpRBiC9N/PEANO_nokinds_noand

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 RisEmptyProof (⇔)

↳12 TRUE

(0) Obligation:

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

active(U11(tt, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
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, V2)) → mark(U12(isNat(V2)))
active(U12(tt)) → mark(tt)
active(U21(tt)) → mark(tt)
active(U31(tt, N)) → mark(N)
active(U41(tt, M, N)) → mark(U42(isNat(N), M, N))
active(U42(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNat(V1), V2))
active(isNat(s(V1))) → mark(U21(isNat(V1)))
active(plus(N, 0)) → mark(U31(isNat(N), N))
active(plus(N, s(M))) → mark(U41(isNat(M), M, N))
active(U11(X1, X2)) → U11(active(X1), X2)
active(U12(X)) → U12(active(X))
active(U21(X)) → U21(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U41(X1, X2, X3)) → U41(active(X1), X2, X3)
active(U42(X1, X2, X3)) → U42(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))
U11(mark(X1), X2) → mark(U11(X1, X2))
U12(mark(X)) → mark(U12(X))
U21(mark(X)) → mark(U21(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U41(mark(X1), X2, X3) → mark(U41(X1, X2, X3))
U42(mark(X1), X2, X3) → mark(U42(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))
proper(U11(X1, X2)) → U11(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(U12(X)) → U12(proper(X))
proper(isNat(X)) → isNat(proper(X))
proper(U21(X)) → U21(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U41(X1, X2, X3)) → U41(proper(X1), proper(X2), proper(X3))
proper(U42(X1, X2, X3)) → U42(proper(X1), proper(X2), proper(X3))
proper(s(X)) → s(proper(X))
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
U11(ok(X1), ok(X2)) → ok(U11(X1, X2))
U12(ok(X)) → ok(U12(X))
isNat(ok(X)) → ok(isNat(X))
U21(ok(X)) → ok(U21(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U41(ok(X1), ok(X2), ok(X3)) → ok(U41(X1, X2, X3))
U42(ok(X1), ok(X2), ok(X3)) → ok(U42(X1, X2, X3))
s(ok(X)) → ok(s(X))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
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
U12: {1}
isNat: empty set
U21: {1}
U31: {1}
U41: {1}
U42: {1}
s: {1}
plus: {1, 2}
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, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U31(tt, N) → N
U41(tt, M, N) → U42(isNat(N), M, N)
U42(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
plus(N, 0) → U31(isNat(N), N)
plus(N, s(M)) → U41(isNat(M), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U31: {1}
U41: {1}
U42: {1}
s: {1}
plus: {1, 2}
0: empty set

(3) PoloCSRProof (EQUIVALENT transformation)

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

U31(tt, N) → N
plus(N, 0) → U31(isNat(N), N)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁)) = 2·x₁
POL(U31(x₁, x₂)) = 1 + x₁ + x₂
POL(U41(x₁, x₂, x₃)) = 1 + 2·x₁ + x₂ + x₃
POL(U42(x₁, x₂, x₃)) = 1 + 2·x₁ + x₂ + x₃
POL(isNat(x₁)) = 0
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = x₁
POL(tt) = 0

(4) Obligation:

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

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U41(tt, M, N) → U42(isNat(N), M, N)
U42(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
plus(N, s(M)) → U41(isNat(M), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U41: {1}
U42: {1}
s: {1}
plus: {1, 2}
0: empty set

(5) PoloCSRProof (EQUIVALENT transformation)

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

U42(tt, M, N) → s(plus(N, M))

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(U11(x₁, x₂)) = x₁
POL(U12(x₁)) = x₁
POL(U21(x₁)) = x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(U42(x₁, x₂, x₃)) = 2·x₁ + 2·x₂ + 2·x₃
POL(isNat(x₁)) = 2
POL(plus(x₁, x₂)) = 2·x₁ + 2·x₂
POL(s(x₁)) = 2 + x₁
POL(tt) = 2

(6) Obligation:

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

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U41(tt, M, N) → U42(isNat(N), M, N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
isNat(s(V1)) → U21(isNat(V1))
plus(N, s(M)) → U41(isNat(M), M, N)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U41: {1}
U42: {1}
s: {1}
plus: {1, 2}
0: empty set

(7) PoloCSRProof (EQUIVALENT transformation)

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

isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNat(V1), V2)
plus(N, s(M)) → U41(isNat(M), M, N)

Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(U11(x₁, x₂)) = 2·x₁ + 2·x₂
POL(U12(x₁)) = x₁
POL(U21(x₁)) = 2·x₁
POL(U41(x₁, x₂, x₃)) = 2·x₁ + x₂ + x₃
POL(U42(x₁, x₂, x₃)) = x₁
POL(isNat(x₁)) = x₁
POL(plus(x₁, x₂)) = 1 + 2·x₁ + 2·x₂
POL(s(x₁)) = 2·x₁
POL(tt) = 0

(8) Obligation:

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

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U41(tt, M, N) → U42(isNat(N), M, N)
isNat(s(V1)) → U21(isNat(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNat: empty set
U21: {1}
U41: {1}
U42: {1}
s: {1}

(9) PoloCSRProof (EQUIVALENT transformation)

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

U11(tt, V2) → U12(isNat(V2))
U12(tt) → tt
U21(tt) → tt
U41(tt, M, N) → U42(isNat(N), M, N)
isNat(s(V1)) → U21(isNat(V1))

Used ordering:
Polynomial interpretation [POLO]:

POL(U11(x₁, x₂)) = 2 + x₁ + 2·x₂
POL(U12(x₁)) = 2 + x₁
POL(U21(x₁)) = 1 + x₁
POL(U41(x₁, x₂, x₃)) = x₁ + 2·x₂ + 2·x₃
POL(U42(x₁, x₂, x₃)) = x₁ + 2·x₂
POL(isNat(x₁)) = 1 + 2·x₁
POL(s(x₁)) = 2 + 2·x₁
POL(tt) = 2

(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) RisEmptyProof (EQUIVALENT transformation)

(12) TRUE