YES
Termination Proof using AProVETerm Rewriting System R:
[V1, V2, N, M, X]
U11(tt, V1, V2) -> U12(isNat(V1), V2)
U12(tt, V2) -> U13(isNat(V2))
U13(tt) -> tt
U21(tt, V1) -> U22(isNat(V1))
U22(tt) -> tt
U31(tt, V1, V2) -> U32(isNat(V1), V2)
U32(tt, V2) -> U33(isNat(V2))
U33(tt) -> tt
U41(tt, N) -> N
U51(tt, M, N) -> s(plus(N, M))
U61(tt) -> 0
U71(tt, M, N) -> plus(x(N, M), N)
and(tt, X) -> X
isNat(0) -> tt
isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) -> U21(isNatKind(V1), V1)
isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) -> tt
isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) -> isNatKind(V1)
isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
Termination of R to be shown.
TRS
↳CSR to TRS Transformation
Trivial-Transformation successful.
Replacement map:
U71(1, 2, 3)
and(1, 2)
plus(1, 2)
U31(1, 2, 3)
x(1, 2)
U12(1, 2)
U11(1, 2, 3)
U22(1)
U33(1)
isNat(1)
U51(1, 2, 3)
U41(1, 2)
U32(1, 2)
U21(1, 2)
U13(1)
U61(1)
s(1)
isNatKind(1)
Old CSR:
U11(tt, V1, V2) -> U12(isNat(V1), V2)
U12(tt, V2) -> U13(isNat(V2))
U13(tt) -> tt
U21(tt, V1) -> U22(isNat(V1))
U22(tt) -> tt
U31(tt, V1, V2) -> U32(isNat(V1), V2)
U32(tt, V2) -> U33(isNat(V2))
U33(tt) -> tt
U41(tt, N) -> N
U51(tt, M, N) -> s(plus(N, M))
U61(tt) -> 0
U71(tt, M, N) -> plus(x(N, M), N)
and(tt, X) -> X
isNat(0) -> tt
isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) -> U21(isNatKind(V1), V1)
isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNatKind(0) -> tt
isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) -> isNatKind(V1)
isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
new TRS:
U11(tt, V1, V2) -> U12(isNat(V1), V2)
U12(tt, V2) -> U13(isNat(V2))
isNat(0) -> tt
isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) -> U21(isNatKind(V1), V1)
isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U13(tt) -> tt
U21(tt, V1) -> U22(isNat(V1))
U22(tt) -> tt
U31(tt, V1, V2) -> U32(isNat(V1), V2)
U32(tt, V2) -> U33(isNat(V2))
U33(tt) -> tt
U41(tt, N) -> N
U51(tt, M, N) -> s(plus(N, M))
plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U61(tt) -> 0
U71(tt, M, N) -> plus(x(N, M), N)
x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
and(tt, X) -> X
isNatKind(0) -> tt
isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) -> isNatKind(V1)
isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
TRS
↳CSRtoTRS
→TRS1
↳Dependency Pair Analysis
R contains the following Dependency Pairs:
U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
U11'(tt, V1, V2) -> ISNAT(V1)
U12'(tt, V2) -> U13'(isNat(V2))
U12'(tt, V2) -> ISNAT(V2)
ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
ISNAT(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
ISNAT(plus(V1, V2)) -> ISNATKIND(V1)
ISNAT(plus(V1, V2)) -> ISNATKIND(V2)
ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
ISNAT(s(V1)) -> ISNATKIND(V1)
ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
ISNAT(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
ISNAT(x(V1, V2)) -> ISNATKIND(V1)
ISNAT(x(V1, V2)) -> ISNATKIND(V2)
U21'(tt, V1) -> U22'(isNat(V1))
U21'(tt, V1) -> ISNAT(V1)
U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
U31'(tt, V1, V2) -> ISNAT(V1)
U32'(tt, V2) -> U33'(isNat(V2))
U32'(tt, V2) -> ISNAT(V2)
U51'(tt, M, N) -> PLUS(N, M)
PLUS(N, 0) -> U41'(and(isNat(N), isNatKind(N)), N)
PLUS(N, 0) -> AND(isNat(N), isNatKind(N))
PLUS(N, 0) -> ISNAT(N)
PLUS(N, 0) -> ISNATKIND(N)
PLUS(N, s(M)) -> U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
PLUS(N, s(M)) -> AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
PLUS(N, s(M)) -> AND(isNat(M), isNatKind(M))
PLUS(N, s(M)) -> ISNAT(M)
PLUS(N, s(M)) -> ISNATKIND(M)
PLUS(N, s(M)) -> AND(isNat(N), isNatKind(N))
PLUS(N, s(M)) -> ISNAT(N)
PLUS(N, s(M)) -> ISNATKIND(N)
U71'(tt, M, N) -> PLUS(x(N, M), N)
U71'(tt, M, N) -> X(N, M)
X(N, 0) -> U61'(and(isNat(N), isNatKind(N)))
X(N, 0) -> AND(isNat(N), isNatKind(N))
X(N, 0) -> ISNAT(N)
X(N, 0) -> ISNATKIND(N)
X(N, s(M)) -> U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
X(N, s(M)) -> AND(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N)))
X(N, s(M)) -> AND(isNat(M), isNatKind(M))
X(N, s(M)) -> ISNAT(M)
X(N, s(M)) -> ISNATKIND(M)
X(N, s(M)) -> AND(isNat(N), isNatKind(N))
X(N, s(M)) -> ISNAT(N)
X(N, s(M)) -> ISNATKIND(N)
ISNATKIND(plus(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
ISNATKIND(plus(V1, V2)) -> ISNATKIND(V2)
ISNATKIND(s(V1)) -> ISNATKIND(V1)
ISNATKIND(x(V1, V2)) -> AND(isNatKind(V1), isNatKind(V2))
ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
ISNATKIND(x(V1, V2)) -> ISNATKIND(V2)
Furthermore, R contains four SCCs.
TRS
↳CSRtoTRS
→TRS1
↳DPs
→DP Problem 1
↳Size-Change Principle
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
Dependency Pairs:
ISNATKIND(x(V1, V2)) -> ISNATKIND(V2)
ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
ISNATKIND(s(V1)) -> ISNATKIND(V1)
ISNATKIND(plus(V1, V2)) -> ISNATKIND(V2)
ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
Rules:
U11(tt, V1, V2) -> U12(isNat(V1), V2)
U12(tt, V2) -> U13(isNat(V2))
isNat(0) -> tt
isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) -> U21(isNatKind(V1), V1)
isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U13(tt) -> tt
U21(tt, V1) -> U22(isNat(V1))
U22(tt) -> tt
U31(tt, V1, V2) -> U32(isNat(V1), V2)
U32(tt, V2) -> U33(isNat(V2))
U33(tt) -> tt
U41(tt, N) -> N
U51(tt, M, N) -> s(plus(N, M))
plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U61(tt) -> 0
U71(tt, M, N) -> plus(x(N, M), N)
x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
and(tt, X) -> X
isNatKind(0) -> tt
isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) -> isNatKind(V1)
isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
We number the DPs as follows:
- ISNATKIND(x(V1, V2)) -> ISNATKIND(V2)
- ISNATKIND(x(V1, V2)) -> ISNATKIND(V1)
- ISNATKIND(s(V1)) -> ISNATKIND(V1)
- ISNATKIND(plus(V1, V2)) -> ISNATKIND(V2)
- ISNATKIND(plus(V1, V2)) -> ISNATKIND(V1)
and get the following Size-Change Graph(s): {5, 4, 3, 2, 1} | , | {5, 4, 3, 2, 1} |
---|
1 | > | 1 |
|
which lead(s) to this/these maximal multigraph(s): {5, 4, 3, 2, 1} | , | {5, 4, 3, 2, 1} |
---|
1 | > | 1 |
|
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
plus(x1, x2) -> plus(x1, x2)
x(x1, x2) -> x(x1, x2)
s(x1) -> s(x1)
We obtain no new DP problems.
TRS
↳CSRtoTRS
→TRS1
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳Size-Change Principle
→DP Problem 3
↳SCP
→DP Problem 4
↳SCP
Dependency Pairs:
U31'(tt, V1, V2) -> ISNAT(V1)
U32'(tt, V2) -> ISNAT(V2)
U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U21'(tt, V1) -> ISNAT(V1)
ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
U11'(tt, V1, V2) -> ISNAT(V1)
ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U12'(tt, V2) -> ISNAT(V2)
U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
Rules:
U11(tt, V1, V2) -> U12(isNat(V1), V2)
U12(tt, V2) -> U13(isNat(V2))
isNat(0) -> tt
isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) -> U21(isNatKind(V1), V1)
isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U13(tt) -> tt
U21(tt, V1) -> U22(isNat(V1))
U22(tt) -> tt
U31(tt, V1, V2) -> U32(isNat(V1), V2)
U32(tt, V2) -> U33(isNat(V2))
U33(tt) -> tt
U41(tt, N) -> N
U51(tt, M, N) -> s(plus(N, M))
plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U61(tt) -> 0
U71(tt, M, N) -> plus(x(N, M), N)
x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
and(tt, X) -> X
isNatKind(0) -> tt
isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) -> isNatKind(V1)
isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
We number the DPs as follows:
- U31'(tt, V1, V2) -> ISNAT(V1)
- U32'(tt, V2) -> ISNAT(V2)
- U31'(tt, V1, V2) -> U32'(isNat(V1), V2)
- ISNAT(x(V1, V2)) -> U31'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
- U21'(tt, V1) -> ISNAT(V1)
- ISNAT(s(V1)) -> U21'(isNatKind(V1), V1)
- U11'(tt, V1, V2) -> ISNAT(V1)
- ISNAT(plus(V1, V2)) -> U11'(and(isNatKind(V1), isNatKind(V2)), V1, V2)
- U12'(tt, V2) -> ISNAT(V2)
- U11'(tt, V1, V2) -> U12'(isNat(V1), V2)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
plus(x1, x2) -> plus(x1, x2)
x(x1, x2) -> x(x1, x2)
s(x1) -> s(x1)
We obtain no new DP problems.
TRS
↳CSRtoTRS
→TRS1
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳Size-Change Principle
→DP Problem 4
↳SCP
Dependency Pairs:
PLUS(N, s(M)) -> U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U51'(tt, M, N) -> PLUS(N, M)
Rules:
U11(tt, V1, V2) -> U12(isNat(V1), V2)
U12(tt, V2) -> U13(isNat(V2))
isNat(0) -> tt
isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) -> U21(isNatKind(V1), V1)
isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U13(tt) -> tt
U21(tt, V1) -> U22(isNat(V1))
U22(tt) -> tt
U31(tt, V1, V2) -> U32(isNat(V1), V2)
U32(tt, V2) -> U33(isNat(V2))
U33(tt) -> tt
U41(tt, N) -> N
U51(tt, M, N) -> s(plus(N, M))
plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U61(tt) -> 0
U71(tt, M, N) -> plus(x(N, M), N)
x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
and(tt, X) -> X
isNatKind(0) -> tt
isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) -> isNatKind(V1)
isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
We number the DPs as follows:
- PLUS(N, s(M)) -> U51'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
- U51'(tt, M, N) -> PLUS(N, M)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
s(x1) -> s(x1)
We obtain no new DP problems.
TRS
↳CSRtoTRS
→TRS1
↳DPs
→DP Problem 1
↳SCP
→DP Problem 2
↳SCP
→DP Problem 3
↳SCP
→DP Problem 4
↳Size-Change Principle
Dependency Pairs:
X(N, s(M)) -> U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U71'(tt, M, N) -> X(N, M)
Rules:
U11(tt, V1, V2) -> U12(isNat(V1), V2)
U12(tt, V2) -> U13(isNat(V2))
isNat(0) -> tt
isNat(plus(V1, V2)) -> U11(and(isNatKind(V1), isNatKind(V2)), V1, V2)
isNat(s(V1)) -> U21(isNatKind(V1), V1)
isNat(x(V1, V2)) -> U31(and(isNatKind(V1), isNatKind(V2)), V1, V2)
U13(tt) -> tt
U21(tt, V1) -> U22(isNat(V1))
U22(tt) -> tt
U31(tt, V1, V2) -> U32(isNat(V1), V2)
U32(tt, V2) -> U33(isNat(V2))
U33(tt) -> tt
U41(tt, N) -> N
U51(tt, M, N) -> s(plus(N, M))
plus(N, 0) -> U41(and(isNat(N), isNatKind(N)), N)
plus(N, s(M)) -> U51(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
U61(tt) -> 0
U71(tt, M, N) -> plus(x(N, M), N)
x(N, 0) -> U61(and(isNat(N), isNatKind(N)))
x(N, s(M)) -> U71(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
and(tt, X) -> X
isNatKind(0) -> tt
isNatKind(plus(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
isNatKind(s(V1)) -> isNatKind(V1)
isNatKind(x(V1, V2)) -> and(isNatKind(V1), isNatKind(V2))
We number the DPs as follows:
- X(N, s(M)) -> U71'(and(and(isNat(M), isNatKind(M)), and(isNat(N), isNatKind(N))), M, N)
- U71'(tt, M, N) -> X(N, M)
and get the following Size-Change Graph(s):
which lead(s) to this/these maximal multigraph(s):
DP: empty set
Oriented Rules: none
We used the order Homeomorphic Embedding Order with Non-Strict Precedence.
trivial
with Argument Filtering System:
s(x1) -> s(x1)
We obtain no new DP problems.
Termination of R successfully shown.
Duration:
0:35 minutes