/export/starexec/sandbox2/solver/bin/starexec_run_standard /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


YES
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


Termination of the given CSR could be proven:

(0) CSR
(1) CSRRRRProof [EQUIVALENT, 112 ms]
(2) CSR
(3) CSRRRRProof [EQUIVALENT, 22 ms]
(4) CSR
(5) CSRRRRProof [EQUIVALENT, 0 ms]
(6) CSR
(7) CSRRRRProof [EQUIVALENT, 0 ms]
(8) CSR
(9) CSRRRRProof [EQUIVALENT, 0 ms]
(10) CSR
(11) CSRRRRProof [EQUIVALENT, 5 ms]
(12) CSR
(13) CSRRRRProof [EQUIVALENT, 17 ms]
(14) CSR
(15) CSRRRRProof [EQUIVALENT, 0 ms]
(16) CSR
(17) CSRRRRProof [EQUIVALENT, 4 ms]
(18) CSR
(19) CSRRRRProof [EQUIVALENT, 0 ms]
(20) CSR
(21) CSRRRRProof [EQUIVALENT, 0 ms]
(22) CSR
(23) CSRRRRProof [EQUIVALENT, 6 ms]
(24) CSR
(25) CSRRRRProof [EQUIVALENT, 0 ms]
(26) CSR
(27) CSRRRRProof [EQUIVALENT, 0 ms]
(28) CSR
(29) RisEmptyProof [EQUIVALENT, 2 ms]
(30) YES


----------------------------------------

(0)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt) -> tt
   U21(tt) -> tt
   U31(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(V) -> U31(isNatList(V))
   isNatIList(zeros) -> tt
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(nil) -> tt
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(nil) -> 0
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U21: {1}
U31: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
nil: empty set

----------------------------------------

(1) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt) -> tt
   U21(tt) -> tt
   U31(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(V) -> U31(isNatList(V))
   isNatIList(zeros) -> tt
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(nil) -> tt
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(nil) -> 0
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U21: {1}
U31: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1)) = 2*x_1
   POL(U21(x_1)) = 2*x_1
   POL(U31(x_1)) = 2*x_1
   POL(U41(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U42(x_1)) = x_1
   POL(U51(x_1, x_2)) = 2*x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2, x_3)) = 2*x_1 + 2*x_2 + 2*x_3
   POL(U62(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 1 + x_1
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 2*x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(tt) = 0
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   isNatIList(V) -> U31(isNatList(V))
   isNatIList(zeros) -> tt




----------------------------------------

(2)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt) -> tt
   U21(tt) -> tt
   U31(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(nil) -> tt
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(nil) -> 0
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U21: {1}
U31: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
nil: empty set

----------------------------------------

(3) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt) -> tt
   U21(tt) -> tt
   U31(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(nil) -> tt
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(nil) -> 0
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U21: {1}
U31: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1)) = 2*x_1
   POL(U21(x_1)) = 2*x_1
   POL(U31(x_1)) = 2 + x_1
   POL(U41(x_1, x_2)) = x_1 + 2*x_2
   POL(U42(x_1)) = x_1
   POL(U51(x_1, x_2)) = 2*x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2, x_3)) = x_1 + x_2 + 2*x_3
   POL(U62(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = x_1
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = x_1
   POL(nil) = 0
   POL(s(x_1)) = x_1
   POL(tt) = 0
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U31(tt) -> tt




----------------------------------------

(4)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt) -> tt
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(nil) -> tt
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(nil) -> 0
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
nil: empty set

----------------------------------------

(5) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt) -> tt
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(nil) -> tt
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(nil) -> 0
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
nil: empty set
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1)) = x_1
   POL(U21(x_1)) = x_1
   POL(U41(x_1, x_2)) = x_1 + x_2
   POL(U42(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2, x_3)) = x_1 + x_2
   POL(U62(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 1
   POL(isNatIList(x_1)) = 1 + x_1
   POL(isNatList(x_1)) = 1
   POL(length(x_1)) = 1 + x_1
   POL(nil) = 1
   POL(s(x_1)) = x_1
   POL(tt) = 1
   POL(zeros) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   length(nil) -> 0




----------------------------------------

(6)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt) -> tt
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(nil) -> tt
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
nil: empty set

----------------------------------------

(7) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt) -> tt
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(nil) -> tt
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
nil: empty set
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(zeros) =  	[[0], [0]]
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0], [0]]
>>>

   <<<
 POL(U11(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[0], [0]]
>>>

   <<<
 POL(U21(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U52(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(isNatList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U62(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(length(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(nil) =  	[[1], [1]]
>>>

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

   isNatList(nil) -> tt




----------------------------------------

(8)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt) -> tt
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}

----------------------------------------

(9) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U11(tt) -> tt
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
U11: {1}
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U11(x_1)) = 1 + x_1
   POL(U21(x_1)) = x_1
   POL(U41(x_1, x_2)) = x_1 + 2*x_2
   POL(U42(x_1)) = 2*x_1
   POL(U51(x_1, x_2)) = x_1 + 2*x_2
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2, x_3)) = 2 + 2*x_1 + 2*x_2 + 2*x_3
   POL(U62(x_1, x_2)) = 2 + x_1 + 2*x_2
   POL(cons(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(isNat(x_1)) = x_1
   POL(isNatIList(x_1)) = x_1
   POL(isNatList(x_1)) = x_1
   POL(length(x_1)) = 2 + 2*x_1
   POL(s(x_1)) = x_1
   POL(tt) = 0
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U11(tt) -> tt
   isNat(length(V1)) -> U11(isNatList(V1))




----------------------------------------

(10)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}

----------------------------------------

(11) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U42(tt) -> tt
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(zeros) =  	[[0], [1]]
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0], [0]]
>>>

   <<<
 POL(U21(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[0], [1]]
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U52(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(isNatList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U62(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(length(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

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

   U42(tt) -> tt




----------------------------------------

(12)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}

----------------------------------------

(13) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   U52(tt) -> tt
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(zeros) =  	[[0], [0]]
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 0]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0], [0]]
>>>

   <<<
 POL(U21(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[0], [1]]
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U52(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(isNatList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[1, 1], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U62(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(length(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

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

   U52(tt) -> tt




----------------------------------------

(14)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}

----------------------------------------

(15) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   U61(tt, L, N) -> U62(isNat(N), L)
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(zeros) =  	[[0], [1]]
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0], [0]]
>>>

   <<<
 POL(U21(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[0], [1]]
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 0], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(U52(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(isNatList(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(U61(x_1, x_2, x_3)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2 	 +  	[[1, 0], [0, 0]] 	* 	x_3
>>>

   <<<
 POL(U62(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1 	 +  	[[1, 1], [0, 0]] 	* 	x_2
>>>

   <<<
 POL(isNat(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(length(x_1)) =  	[[0], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

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

   U61(tt, L, N) -> U62(isNat(N), L)




----------------------------------------

(16)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}

----------------------------------------

(17) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   U62(tt, L) -> s(length(L))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
U62: {1}
isNat: empty set
s: {1}
length: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U21(x_1)) = x_1
   POL(U41(x_1, x_2)) = x_1
   POL(U42(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2, x_3)) = x_1 + x_2
   POL(U62(x_1, x_2)) = 1 + x_1 + x_2
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = x_1
   POL(s(x_1)) = x_1
   POL(tt) = 0
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U62(tt, L) -> s(length(L))




----------------------------------------

(18)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
isNat: empty set
s: {1}
length: {1}

----------------------------------------

(19) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)
   length(cons(N, L)) -> U61(isNatList(L), L, N)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
U61: {1}
isNat: empty set
s: {1}
length: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U21(x_1)) = x_1
   POL(U41(x_1, x_2)) = x_1
   POL(U42(x_1)) = x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(U61(x_1, x_2, x_3)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(length(x_1)) = 1 + x_1
   POL(s(x_1)) = 1 + x_1
   POL(tt) = 0
   POL(zeros) = 0
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   length(cons(N, L)) -> U61(isNatList(L), L, N)




----------------------------------------

(20)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
isNat: empty set
s: {1}

----------------------------------------

(21) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U21(tt) -> tt
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
isNat: empty set
s: {1}
Used ordering:
Matrix interpretation [MATRO] to (N^2, +, *, >=, >) :

   <<<
 POL(zeros) =  	[[0], [0]]
>>>

   <<<
 POL(cons(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 1], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(0) =  	[[0], [0]]
>>>

   <<<
 POL(U21(x_1)) =  	[[1], [0]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(tt) =  	[[0], [0]]
>>>

   <<<
 POL(U41(x_1, x_2)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(U42(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1
>>>

   <<<
 POL(isNatIList(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(U51(x_1, x_2)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 0]] 	* 	x_1 	 +  	[[1, 1], [1, 1]] 	* 	x_2
>>>

   <<<
 POL(U52(x_1)) =  	[[0], [0]] 	 +  	[[1, 0], [0, 1]] 	* 	x_1
>>>

   <<<
 POL(isNatList(x_1)) =  	[[0], [1]] 	 +  	[[1, 0], [1, 1]] 	* 	x_1
>>>

   <<<
 POL(isNat(x_1)) =  	[[0], [1]] 	 +  	[[1, 1], [0, 0]] 	* 	x_1
>>>

   <<<
 POL(s(x_1)) =  	[[1], [1]] 	 +  	[[1, 1], [1, 1]] 	* 	x_1
>>>

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

   U21(tt) -> tt




----------------------------------------

(22)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
isNat: empty set
s: {1}

----------------------------------------

(23) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   zeros -> cons(0, zeros)
   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)

The replacement map contains the following entries:

zeros: empty set
cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
isNat: empty set
s: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 0
   POL(U21(x_1)) = 2*x_1
   POL(U41(x_1, x_2)) = 2*x_1
   POL(U42(x_1)) = 2*x_1
   POL(U51(x_1, x_2)) = x_1
   POL(U52(x_1)) = x_1
   POL(cons(x_1, x_2)) = 2*x_1
   POL(isNat(x_1)) = 0
   POL(isNatIList(x_1)) = 0
   POL(isNatList(x_1)) = 0
   POL(s(x_1)) = 2 + x_1
   POL(tt) = 0
   POL(zeros) = 1
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   zeros -> cons(0, zeros)




----------------------------------------

(24)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)

The replacement map contains the following entries:

cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
isNat: empty set
s: {1}

----------------------------------------

(25) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U41(tt, V2) -> U42(isNatIList(V2))
   U51(tt, V2) -> U52(isNatList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)

The replacement map contains the following entries:

cons: {1}
0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
U51: {1}
U52: {1}
isNatList: empty set
isNat: empty set
s: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 2
   POL(U21(x_1)) = x_1
   POL(U41(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(U42(x_1)) = 2*x_1
   POL(U51(x_1, x_2)) = 2*x_1 + 2*x_2
   POL(U52(x_1)) = x_1
   POL(cons(x_1, x_2)) = 1 + 2*x_1 + 2*x_2
   POL(isNat(x_1)) = x_1
   POL(isNatIList(x_1)) = 2 + x_1
   POL(isNatList(x_1)) = 2 + x_1
   POL(s(x_1)) = x_1
   POL(tt) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U51(tt, V2) -> U52(isNatList(V2))
   isNatIList(cons(V1, V2)) -> U41(isNat(V1), V2)
   isNatList(cons(V1, V2)) -> U51(isNat(V1), V2)




----------------------------------------

(26)
Obligation:
Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U41(tt, V2) -> U42(isNatIList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))

The replacement map contains the following entries:

0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
isNat: empty set
s: {1}

----------------------------------------

(27) CSRRRRProof (EQUIVALENT)
The following CSR is given: Context-sensitive rewrite system:
The TRS R consists of the following rules:

   U41(tt, V2) -> U42(isNatIList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))

The replacement map contains the following entries:

0: empty set
tt: empty set
U21: {1}
U41: {1}
U42: {1}
isNatIList: empty set
isNat: empty set
s: {1}
Used ordering:
Polynomial interpretation [POLO]:

   POL(0) = 2
   POL(U21(x_1)) = 2*x_1
   POL(U41(x_1, x_2)) = 1 + x_1 + 2*x_2
   POL(U42(x_1)) = 2*x_1
   POL(isNat(x_1)) = 1 + 2*x_1
   POL(isNatIList(x_1)) = x_1
   POL(s(x_1)) = 2 + 2*x_1
   POL(tt) = 2
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

   U41(tt, V2) -> U42(isNatIList(V2))
   isNat(0) -> tt
   isNat(s(V1)) -> U21(isNat(V1))




----------------------------------------

(28)
Obligation:
Context-sensitive rewrite system:
R is empty.

----------------------------------------

(29) RisEmptyProof (EQUIVALENT)
The CSR R is empty. Hence, termination is trivially proven.
----------------------------------------

(30)
YES