Termination w.r.t. Q proof of Transformed_CSR_04_PEANO_complete_noand

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

0 QTRS

↳1 QTRSToCSRProof (⇔, 0 ms)

↳2 CSR

↳3 CSRRRRProof (⇔, 59 ms)

↳4 CSR

↳5 CSRRRRProof (⇔, 0 ms)

↳6 CSR

↳7 CSRRRRProof (⇔, 8 ms)

↳8 CSR

↳9 CSRRRRProof (⇔, 0 ms)

↳10 CSR

↳11 CSRRRRProof (⇔, 0 ms)

↳12 CSR

↳13 CSRRRRProof (⇔, 0 ms)

↳14 CSR

↳15 CSRRRRProof (⇔, 0 ms)

↳16 CSR

↳17 CSRRRRProof (⇔, 0 ms)

↳18 CSR

↳19 CSRRRRProof (⇔, 1 ms)

↳20 CSR

↳21 CSRRRRProof (⇔, 0 ms)

↳22 CSR

↳23 RisEmptyProof (⇔, 0 ms)

↳24 YES

(0) Obligation:

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

active(U11(tt, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U13(X1, X2, X3)) → U13(active(X1), X2, X3)
active(U14(X1, X2, X3)) → U14(active(X1), X2, X3)
active(U15(X1, X2)) → U15(active(X1), X2)
active(U16(X)) → U16(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X)) → U41(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2, X3)) → U62(active(X1), X2, X3)
active(U63(X1, X2, X3)) → U63(active(X1), X2, X3)
active(U64(X1, X2, X3)) → U64(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, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U13(mark(X1), X2, X3) → mark(U13(X1, X2, X3))
U14(mark(X1), X2, X3) → mark(U14(X1, X2, X3))
U15(mark(X1), X2) → mark(U15(X1, X2))
U16(mark(X)) → mark(U16(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X)) → mark(U41(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2, X3) → mark(U62(X1, X2, X3))
U63(mark(X1), X2, X3) → mark(U63(X1, X2, X3))
U64(mark(X1), X2, X3) → mark(U64(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, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U13(X1, X2, X3)) → U13(proper(X1), proper(X2), proper(X3))
proper(U14(X1, X2, X3)) → U14(proper(X1), proper(X2), proper(X3))
proper(U15(X1, X2)) → U15(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U16(X)) → U16(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X)) → U41(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2, X3)) → U62(proper(X1), proper(X2), proper(X3))
proper(U63(X1, X2, X3)) → U63(proper(X1), proper(X2), proper(X3))
proper(U64(X1, X2, X3)) → U64(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(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U13(ok(X1), ok(X2), ok(X3)) → ok(U13(X1, X2, X3))
U14(ok(X1), ok(X2), ok(X3)) → ok(U14(X1, X2, X3))
U15(ok(X1), ok(X2)) → ok(U15(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U16(ok(X)) → ok(U16(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X)) → ok(U41(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2), ok(X3)) → ok(U62(X1, X2, X3))
U63(ok(X1), ok(X2), ok(X3)) → ok(U63(X1, X2, X3))
U64(ok(X1), ok(X2), ok(X3)) → ok(U64(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, V1, V2)) → mark(U12(isNatKind(V1), V1, V2))
active(U12(tt, V1, V2)) → mark(U13(isNatKind(V2), V1, V2))
active(U13(tt, V1, V2)) → mark(U14(isNatKind(V2), V1, V2))
active(U14(tt, V1, V2)) → mark(U15(isNat(V1), V2))
active(U15(tt, V2)) → mark(U16(isNat(V2)))
active(U16(tt)) → mark(tt)
active(U21(tt, V1)) → mark(U22(isNatKind(V1), V1))
active(U22(tt, V1)) → mark(U23(isNat(V1)))
active(U23(tt)) → mark(tt)
active(U31(tt, V2)) → mark(U32(isNatKind(V2)))
active(U32(tt)) → mark(tt)
active(U41(tt)) → mark(tt)
active(U51(tt, N)) → mark(U52(isNatKind(N), N))
active(U52(tt, N)) → mark(N)
active(U61(tt, M, N)) → mark(U62(isNatKind(M), M, N))
active(U62(tt, M, N)) → mark(U63(isNat(N), M, N))
active(U63(tt, M, N)) → mark(U64(isNatKind(N), M, N))
active(U64(tt, M, N)) → mark(s(plus(N, M)))
active(isNat(0)) → mark(tt)
active(isNat(plus(V1, V2))) → mark(U11(isNatKind(V1), V1, V2))
active(isNat(s(V1))) → mark(U21(isNatKind(V1), V1))
active(isNatKind(0)) → mark(tt)
active(isNatKind(plus(V1, V2))) → mark(U31(isNatKind(V1), V2))
active(isNatKind(s(V1))) → mark(U41(isNatKind(V1)))
active(plus(N, 0)) → mark(U51(isNat(N), N))
active(plus(N, s(M))) → mark(U61(isNat(M), M, N))
active(U11(X1, X2, X3)) → U11(active(X1), X2, X3)
active(U12(X1, X2, X3)) → U12(active(X1), X2, X3)
active(U13(X1, X2, X3)) → U13(active(X1), X2, X3)
active(U14(X1, X2, X3)) → U14(active(X1), X2, X3)
active(U15(X1, X2)) → U15(active(X1), X2)
active(U16(X)) → U16(active(X))
active(U21(X1, X2)) → U21(active(X1), X2)
active(U22(X1, X2)) → U22(active(X1), X2)
active(U23(X)) → U23(active(X))
active(U31(X1, X2)) → U31(active(X1), X2)
active(U32(X)) → U32(active(X))
active(U41(X)) → U41(active(X))
active(U51(X1, X2)) → U51(active(X1), X2)
active(U52(X1, X2)) → U52(active(X1), X2)
active(U61(X1, X2, X3)) → U61(active(X1), X2, X3)
active(U62(X1, X2, X3)) → U62(active(X1), X2, X3)
active(U63(X1, X2, X3)) → U63(active(X1), X2, X3)
active(U64(X1, X2, X3)) → U64(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, X3) → mark(U11(X1, X2, X3))
U12(mark(X1), X2, X3) → mark(U12(X1, X2, X3))
U13(mark(X1), X2, X3) → mark(U13(X1, X2, X3))
U14(mark(X1), X2, X3) → mark(U14(X1, X2, X3))
U15(mark(X1), X2) → mark(U15(X1, X2))
U16(mark(X)) → mark(U16(X))
U21(mark(X1), X2) → mark(U21(X1, X2))
U22(mark(X1), X2) → mark(U22(X1, X2))
U23(mark(X)) → mark(U23(X))
U31(mark(X1), X2) → mark(U31(X1, X2))
U32(mark(X)) → mark(U32(X))
U41(mark(X)) → mark(U41(X))
U51(mark(X1), X2) → mark(U51(X1, X2))
U52(mark(X1), X2) → mark(U52(X1, X2))
U61(mark(X1), X2, X3) → mark(U61(X1, X2, X3))
U62(mark(X1), X2, X3) → mark(U62(X1, X2, X3))
U63(mark(X1), X2, X3) → mark(U63(X1, X2, X3))
U64(mark(X1), X2, X3) → mark(U64(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, X3)) → U11(proper(X1), proper(X2), proper(X3))
proper(tt) → ok(tt)
proper(U12(X1, X2, X3)) → U12(proper(X1), proper(X2), proper(X3))
proper(isNatKind(X)) → isNatKind(proper(X))
proper(U13(X1, X2, X3)) → U13(proper(X1), proper(X2), proper(X3))
proper(U14(X1, X2, X3)) → U14(proper(X1), proper(X2), proper(X3))
proper(U15(X1, X2)) → U15(proper(X1), proper(X2))
proper(isNat(X)) → isNat(proper(X))
proper(U16(X)) → U16(proper(X))
proper(U21(X1, X2)) → U21(proper(X1), proper(X2))
proper(U22(X1, X2)) → U22(proper(X1), proper(X2))
proper(U23(X)) → U23(proper(X))
proper(U31(X1, X2)) → U31(proper(X1), proper(X2))
proper(U32(X)) → U32(proper(X))
proper(U41(X)) → U41(proper(X))
proper(U51(X1, X2)) → U51(proper(X1), proper(X2))
proper(U52(X1, X2)) → U52(proper(X1), proper(X2))
proper(U61(X1, X2, X3)) → U61(proper(X1), proper(X2), proper(X3))
proper(U62(X1, X2, X3)) → U62(proper(X1), proper(X2), proper(X3))
proper(U63(X1, X2, X3)) → U63(proper(X1), proper(X2), proper(X3))
proper(U64(X1, X2, X3)) → U64(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(X3)) → ok(U11(X1, X2, X3))
U12(ok(X1), ok(X2), ok(X3)) → ok(U12(X1, X2, X3))
isNatKind(ok(X)) → ok(isNatKind(X))
U13(ok(X1), ok(X2), ok(X3)) → ok(U13(X1, X2, X3))
U14(ok(X1), ok(X2), ok(X3)) → ok(U14(X1, X2, X3))
U15(ok(X1), ok(X2)) → ok(U15(X1, X2))
isNat(ok(X)) → ok(isNat(X))
U16(ok(X)) → ok(U16(X))
U21(ok(X1), ok(X2)) → ok(U21(X1, X2))
U22(ok(X1), ok(X2)) → ok(U22(X1, X2))
U23(ok(X)) → ok(U23(X))
U31(ok(X1), ok(X2)) → ok(U31(X1, X2))
U32(ok(X)) → ok(U32(X))
U41(ok(X)) → ok(U41(X))
U51(ok(X1), ok(X2)) → ok(U51(X1, X2))
U52(ok(X1), ok(X2)) → ok(U52(X1, X2))
U61(ok(X1), ok(X2), ok(X3)) → ok(U61(X1, X2, X3))
U62(ok(X1), ok(X2), ok(X3)) → ok(U62(X1, X2, X3))
U63(ok(X1), ok(X2), ok(X3)) → ok(U63(X1, X2, X3))
U64(ok(X1), ok(X2), ok(X3)) → ok(U64(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}
isNatKind: empty set
U13: {1}
U14: {1}
U15: {1}
isNat: empty set
U16: {1}
U21: {1}
U22: {1}
U23: {1}
U31: {1}
U32: {1}
U41: {1}
U51: {1}
U52: {1}
U61: {1}
U62: {1}
U63: {1}
U64: {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, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)

The replacement map contains the following entries:

(3) CSRRRRProof (EQUIVALENT transformation)

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))
plus(N, 0) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)

The replacement map contains the following entries:

POL(0) = 1
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂, x₃)) = x₁
POL(U13(x₁, x₂, x₃)) = x₁
POL(U14(x₁, x₂, x₃)) = x₁
POL(U15(x₁, x₂)) = x₁
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = 2·x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U51(x₁, x₂)) = x₁ + x₂
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃
POL(U62(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃
POL(U64(x₁, x₂, x₃)) = 1 + x₁ + 2·x₂ + 2·x₃
POL(isNat(x₁)) = 0
POL(isNatKind(x₁)) = 0
POL(plus(x₁, x₂)) = 2·x₁ + 2·x₂
POL(s(x₁)) = 1 + 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) → U51(isNat(N), N)
plus(N, s(M)) → U61(isNat(M), M, N)

(4) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

(5) CSRRRRProof (EQUIVALENT transformation)

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
U64(tt, M, N) → s(plus(N, M))
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

POL(0) = 0
POL(U11(x₁, x₂, x₃)) = x₁
POL(U12(x₁, x₂, x₃)) = x₁
POL(U13(x₁, x₂, x₃)) = x₁
POL(U14(x₁, x₂, x₃)) = x₁
POL(U15(x₁, x₂)) = x₁
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = x₁
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U51(x₁, x₂)) = 1 + x₁ + x₂
POL(U52(x₁, x₂)) = x₁ + x₂
POL(U61(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U62(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U64(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(isNat(x₁)) = 1
POL(isNatKind(x₁)) = 1
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:

U51(tt, N) → U52(isNatKind(N), N)
U52(tt, N) → N
U61(tt, M, N) → U62(isNatKind(M), M, N)
U64(tt, M, N) → s(plus(N, M))

(6) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U13: {1}
U14: {1}
U15: {1}
isNat: empty set
U16: {1}
U21: {1}
U22: {1}
U23: {1}
U31: {1}
U32: {1}
U41: {1}
U62: {1}
U63: {1}
U64: {1}
s: {1}
plus: {1, 2}
0: empty set

(7) CSRRRRProof (EQUIVALENT transformation)

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U21(tt, V1) → U22(isNatKind(V1), V1)
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U62(tt, M, N) → U63(isNat(N), M, N)
U63(tt, M, N) → U64(isNatKind(N), M, N)
isNat(0) → tt
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

POL(0) = 1
POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U12(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U13(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U14(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U15(x₁, x₂)) = x₁ + x₂
POL(U16(x₁)) = x₁
POL(U21(x₁, x₂)) = 1 + x₁ + x₂
POL(U22(x₁, x₂)) = x₁ + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U62(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₁ + x₂
POL(U64(x₁, x₂, x₃)) = x₁
POL(isNat(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = 1
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 1

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

U13(tt, V1, V2) → U14(isNatKind(V2), V1, V2)
U21(tt, V1) → U22(isNatKind(V1), V1)
U63(tt, M, N) → U64(isNatKind(N), M, N)
isNat(0) → tt

(8) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U62(tt, M, N) → U63(isNat(N), M, N)
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U13: {1}
U14: {1}
U15: {1}
isNat: empty set
U16: {1}
U21: {1}
U22: {1}
U23: {1}
U31: {1}
U32: {1}
U41: {1}
U62: {1}
U63: {1}
s: {1}
plus: {1, 2}
0: empty set

(9) CSRRRRProof (EQUIVALENT transformation)

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U16(tt) → tt
U22(tt, V1) → U23(isNat(V1))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U62(tt, M, N) → U63(isNat(N), M, N)
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNat(s(V1)) → U21(isNatKind(V1), V1)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

POL(0) = 0
POL(U11(x₁, x₂, x₃)) = 1 + x₁
POL(U12(x₁, x₂, x₃)) = 1 + x₁
POL(U13(x₁, x₂, x₃)) = x₁
POL(U14(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U15(x₁, x₂)) = 1 + x₁ + x₂
POL(U16(x₁)) = 1 + x₁
POL(U21(x₁, x₂)) = x₁
POL(U22(x₁, x₂)) = 1 + x₁ + x₂
POL(U23(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U62(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₁
POL(isNat(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = 1
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 1

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

U12(tt, V1, V2) → U13(isNatKind(V2), V1, V2)
U16(tt) → tt
U22(tt, V1) → U23(isNat(V1))
isNat(s(V1)) → U21(isNatKind(V1), V1)

(10) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U62(tt, M, N) → U63(isNat(N), M, N)
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U14: {1}
U15: {1}
isNat: empty set
U16: {1}
U23: {1}
U31: {1}
U32: {1}
U41: {1}
U62: {1}
U63: {1}
s: {1}
plus: {1, 2}
0: empty set

(11) CSRRRRProof (EQUIVALENT transformation)

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U14(tt, V1, V2) → U15(isNat(V1), V2)
U15(tt, V2) → U16(isNat(V2))
U23(tt) → tt
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U62(tt, M, N) → U63(isNat(N), M, N)
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U14: {1}
U15: {1}
isNat: empty set
U16: {1}
U23: {1}
U31: {1}
U32: {1}
U41: {1}
U62: {1}
U63: {1}
s: {1}
plus: {1, 2}
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂, x₃)) = x₁ + x₃
POL(U12(x₁, x₂, x₃)) = x₁ + x₃
POL(U14(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U15(x₁, x₂)) = x₁ + x₂
POL(U16(x₁)) = x₁
POL(U23(x₁)) = 1 + x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U62(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = 1 + x₁
POL(isNat(x₁)) = x₁
POL(isNatKind(x₁)) = 0
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:

U14(tt, V1, V2) → U15(isNat(V1), V2)
U23(tt) → tt

(12) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U15(tt, V2) → U16(isNat(V2))
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U62(tt, M, N) → U63(isNat(N), M, N)
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U15: {1}
isNat: empty set
U16: {1}
U31: {1}
U32: {1}
U41: {1}
U62: {1}
U63: {1}
s: {1}
plus: {1, 2}
0: empty set

(13) CSRRRRProof (EQUIVALENT transformation)

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U15(tt, V2) → U16(isNat(V2))
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
U62(tt, M, N) → U63(isNat(N), M, N)
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U15: {1}
isNat: empty set
U16: {1}
U31: {1}
U32: {1}
U41: {1}
U62: {1}
U63: {1}
s: {1}
plus: {1, 2}
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U12(x₁, x₂, x₃)) = x₁ + x₃
POL(U15(x₁, x₂)) = 1 + x₁ + x₂
POL(U16(x₁)) = x₁
POL(U31(x₁, x₂)) = x₁
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(U62(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U63(x₁, x₂, x₃)) = x₁
POL(isNat(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = 1
POL(plus(x₁, x₂)) = 1 + 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:

U15(tt, V2) → U16(isNat(V2))
U62(tt, M, N) → U63(isNat(N), M, N)
isNat(plus(V1, V2)) → U11(isNatKind(V1), V1, V2)

(14) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U31: {1}
U32: {1}
U41: {1}
s: {1}
plus: {1, 2}
0: empty set

(15) CSRRRRProof (EQUIVALENT transformation)

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U31(tt, V2) → U32(isNatKind(V2))
U32(tt) → tt
U41(tt) → tt
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)
isNatKind(s(V1)) → U41(isNatKind(V1))

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U31: {1}
U32: {1}
U41: {1}
s: {1}
plus: {1, 2}
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U11(x₁, x₂, x₃)) = x₁ + x₂ + x₃
POL(U12(x₁, x₂, x₃)) = x₁
POL(U31(x₁, x₂)) = 1 + x₁ + x₂
POL(U32(x₁)) = x₁
POL(U41(x₁)) = x₁
POL(isNatKind(x₁)) = x₁
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(s(x₁)) = 1 + x₁
POL(tt) = 0

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

U31(tt, V2) → U32(isNatKind(V2))
isNatKind(s(V1)) → U41(isNatKind(V1))

(16) Obligation:

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U32(tt) → tt
U41(tt) → tt
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U31: {1}
U32: {1}
U41: {1}
plus: {1, 2}
0: empty set

(17) CSRRRRProof (EQUIVALENT transformation)

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U32(tt) → tt
U41(tt) → tt
isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)

The replacement map contains the following entries:

U11: {1}
tt: empty set
U12: {1}
isNatKind: empty set
U31: {1}
U32: {1}
U41: {1}
plus: {1, 2}
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(U11(x₁, x₂, x₃)) = 1 + x₁ + x₂ + x₃
POL(U12(x₁, x₂, x₃)) = x₁
POL(U31(x₁, x₂)) = x₁ + x₂
POL(U32(x₁)) = 1 + x₁
POL(U41(x₁)) = 1 + x₁
POL(isNatKind(x₁)) = x₁
POL(plus(x₁, x₂)) = x₁ + x₂
POL(tt) = 1

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

U11(tt, V1, V2) → U12(isNatKind(V1), V1, V2)
U32(tt) → tt
U41(tt) → tt

(18) Obligation:

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

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)

The replacement map contains the following entries:

tt: empty set
isNatKind: empty set
U31: {1}
plus: {1, 2}
0: empty set

(19) CSRRRRProof (EQUIVALENT transformation)

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

isNatKind(0) → tt
isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)

The replacement map contains the following entries:

tt: empty set
isNatKind: empty set
U31: {1}
plus: {1, 2}
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 0
POL(U31(x₁, x₂)) = x₁ + x₂
POL(isNatKind(x₁)) = x₁
POL(plus(x₁, x₂)) = 1 + x₁ + x₂
POL(tt) = 0

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

isNatKind(plus(V1, V2)) → U31(isNatKind(V1), V2)

(20) Obligation:

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

isNatKind(0) → tt

The replacement map contains the following entries:

tt: empty set
isNatKind: empty set
0: empty set

(21) CSRRRRProof (EQUIVALENT transformation)

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

isNatKind(0) → tt

The replacement map contains the following entries:

tt: empty set
isNatKind: empty set
0: empty set
Used ordering:
Polynomial interpretation [POLO]:

POL(0) = 1
POL(isNatKind(x₁)) = 1 + x₁
POL(tt) = 1

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

isNatKind(0) → tt

(22) Obligation:

Context-sensitive rewrite system:
R is empty.

(23) 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) CSRRRRProof (EQUIVALENT transformation)

(18) Obligation:

(19) CSRRRRProof (EQUIVALENT transformation)

(20) Obligation:

(21) CSRRRRProof (EQUIVALENT transformation)

(22) Obligation:

(23) RisEmptyProof (EQUIVALENT transformation)

(24) YES