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

0 QTRS
↳1 QTRSRRRProof (⇔)
↳2 QTRS
↳3 QTRSRRRProof (⇔)
↳4 QTRS
↳5 QTRSRRRProof (⇔)
↳6 QTRS
↳7 QTRSRRRProof (⇔)
↳8 QTRS
↳9 QTRSRRRProof (⇔)
↳10 QTRS
↳11 QTRSRRRProof (⇔)
↳12 QTRS
↳13 QTRSRRRProof (⇔)
↳14 QTRS
↳15 QTRSRRRProof (⇔)
↳16 QTRS
  ↳17 RisEmptyProof (⇔)
    ↳18 TRUE
  ↳19 RisEmptyProof (⇔)
    ↳20 TRUE

(0) Obligation:

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

+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 0) → 2
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 0) → 4
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 8) → c(1, 3)
+(5, 9) → c(1, 4)
+(6, 0) → 6
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 6) → c(1, 3)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 0) → 8
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 1) → c(1, 0)
+(9, 2) → c(1, 1)
+(9, 3) → c(1, 2)
+(9, 4) → c(1, 3)
+(9, 5) → c(1, 4)
+(9, 6) → c(1, 5)
+(9, 7) → c(1, 6)
+(9, 8) → c(1, 7)
+(9, 9) → c(1, 8)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))
c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)

Q is empty.

(1) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = x1 + x2   
POL(0) = 0   
POL(1) = 4   
POL(2) = 3   
POL(3) = 6   
POL(4) = 5   
POL(5) = 6   
POL(6) = 5   
POL(7) = 7   
POL(8) = 6   
POL(9) = 7   
POL(c(x1, x2)) = 2 + x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(1, 1) → 2
+(1, 2) → 3
+(1, 3) → 4
+(1, 4) → 5
+(1, 5) → 6
+(1, 6) → 7
+(1, 7) → 8
+(1, 8) → 9
+(1, 9) → c(1, 0)
+(2, 1) → 3
+(2, 2) → 4
+(2, 3) → 5
+(2, 4) → 6
+(2, 5) → 7
+(2, 6) → 8
+(2, 7) → 9
+(2, 8) → c(1, 0)
+(3, 1) → 4
+(3, 2) → 5
+(3, 3) → 6
+(3, 4) → 7
+(3, 5) → 8
+(3, 6) → 9
+(3, 7) → c(1, 0)
+(3, 8) → c(1, 1)
+(3, 9) → c(1, 2)
+(4, 1) → 5
+(4, 2) → 6
+(4, 3) → 7
+(4, 4) → 8
+(4, 5) → 9
+(4, 6) → c(1, 0)
+(4, 7) → c(1, 1)
+(4, 8) → c(1, 2)
+(5, 1) → 6
+(5, 2) → 7
+(5, 3) → 8
+(5, 4) → 9
+(5, 5) → c(1, 0)
+(5, 6) → c(1, 1)
+(5, 7) → c(1, 2)
+(5, 9) → c(1, 4)
+(6, 1) → 7
+(6, 2) → 8
+(6, 3) → 9
+(6, 4) → c(1, 0)
+(6, 5) → c(1, 1)
+(6, 6) → c(1, 2)
+(7, 1) → 8
+(7, 2) → 9
+(7, 3) → c(1, 0)
+(7, 4) → c(1, 1)
+(7, 5) → c(1, 2)
+(7, 7) → c(1, 4)
+(7, 8) → c(1, 5)
+(7, 9) → c(1, 6)
+(8, 1) → 9
+(8, 2) → c(1, 0)
+(8, 3) → c(1, 1)
+(8, 4) → c(1, 2)
+(8, 7) → c(1, 5)
+(8, 8) → c(1, 6)
+(9, 1) → c(1, 0)
+(9, 3) → c(1, 2)
+(9, 5) → c(1, 4)
+(9, 7) → c(1, 6)
+(9, 9) → c(1, 8)
c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)


(2) Obligation:

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

+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(2, 0) → 2
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(4, 0) → 4
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(5, 8) → c(1, 3)
+(6, 0) → 6
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(7, 6) → c(1, 3)
+(8, 0) → 8
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 2) → c(1, 1)
+(9, 4) → c(1, 3)
+(9, 6) → c(1, 5)
+(9, 8) → c(1, 7)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))

Q is empty.

(3) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = x1 + x2   
POL(0) = 0   
POL(1) = 0   
POL(2) = 0   
POL(3) = 0   
POL(4) = 0   
POL(5) = 0   
POL(6) = 0   
POL(7) = 1   
POL(8) = 1   
POL(9) = 0   
POL(c(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(5, 8) → c(1, 3)
+(6, 7) → c(1, 3)
+(6, 8) → c(1, 4)
+(7, 6) → c(1, 3)
+(8, 5) → c(1, 3)
+(8, 6) → c(1, 4)


(4) Obligation:

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

+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(2, 0) → 2
+(2, 9) → c(1, 1)
+(3, 0) → 3
+(4, 0) → 4
+(4, 9) → c(1, 3)
+(5, 0) → 5
+(6, 0) → 6
+(6, 9) → c(1, 5)
+(7, 0) → 7
+(8, 0) → 8
+(8, 9) → c(1, 7)
+(9, 0) → 9
+(9, 2) → c(1, 1)
+(9, 4) → c(1, 3)
+(9, 6) → c(1, 5)
+(9, 8) → c(1, 7)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))

Q is empty.

(5) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = x1 + x2   
POL(0) = 1   
POL(1) = 0   
POL(2) = 0   
POL(3) = 0   
POL(4) = 0   
POL(5) = 0   
POL(6) = 0   
POL(7) = 0   
POL(8) = 0   
POL(9) = 0   
POL(c(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(0, 0) → 0
+(0, 1) → 1
+(0, 2) → 2
+(0, 3) → 3
+(0, 4) → 4
+(0, 5) → 5
+(0, 6) → 6
+(0, 7) → 7
+(0, 8) → 8
+(0, 9) → 9
+(1, 0) → 1
+(2, 0) → 2
+(3, 0) → 3
+(4, 0) → 4
+(5, 0) → 5
+(6, 0) → 6
+(7, 0) → 7
+(8, 0) → 8
+(9, 0) → 9


(6) Obligation:

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

+(2, 9) → c(1, 1)
+(4, 9) → c(1, 3)
+(6, 9) → c(1, 5)
+(8, 9) → c(1, 7)
+(9, 2) → c(1, 1)
+(9, 4) → c(1, 3)
+(9, 6) → c(1, 5)
+(9, 8) → c(1, 7)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))

Q is empty.

(7) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = x1 + x2   
POL(1) = 0   
POL(2) = 1   
POL(3) = 0   
POL(4) = 0   
POL(5) = 0   
POL(6) = 0   
POL(7) = 0   
POL(8) = 0   
POL(9) = 0   
POL(c(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(2, 9) → c(1, 1)
+(9, 2) → c(1, 1)


(8) Obligation:

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

+(4, 9) → c(1, 3)
+(6, 9) → c(1, 5)
+(8, 9) → c(1, 7)
+(9, 4) → c(1, 3)
+(9, 6) → c(1, 5)
+(9, 8) → c(1, 7)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))

Q is empty.

(9) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = x1 + x2   
POL(1) = 0   
POL(3) = 0   
POL(4) = 1   
POL(5) = 0   
POL(6) = 0   
POL(7) = 0   
POL(8) = 0   
POL(9) = 0   
POL(c(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(4, 9) → c(1, 3)
+(9, 4) → c(1, 3)


(10) Obligation:

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

+(6, 9) → c(1, 5)
+(8, 9) → c(1, 7)
+(9, 6) → c(1, 5)
+(9, 8) → c(1, 7)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))

Q is empty.

(11) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = x1 + x2   
POL(1) = 0   
POL(5) = 0   
POL(6) = 1   
POL(7) = 0   
POL(8) = 0   
POL(9) = 0   
POL(c(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(6, 9) → c(1, 5)
+(9, 6) → c(1, 5)


(12) Obligation:

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

+(8, 9) → c(1, 7)
+(9, 8) → c(1, 7)
+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))

Q is empty.

(13) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = 1 + x1 + x2   
POL(1) = 0   
POL(7) = 0   
POL(8) = 0   
POL(9) = 0   
POL(c(x1, x2)) = x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(8, 9) → c(1, 7)
+(9, 8) → c(1, 7)


(14) Obligation:

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

+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))

Q is empty.

(15) QTRSRRRProof (EQUIVALENT transformation)

Used ordering:
Polynomial interpretation [POLO]:

POL(+(x1, x2)) = 2 + 2·x1 + 2·x2   
POL(c(x1, x2)) = 1 + x1 + x2   
With this ordering the following rules can be removed by the rule removal processor [LPAR04] because they are oriented strictly:

+(x, c(y, z)) → c(y, +(x, z))
+(c(x, y), z) → c(x, +(y, z))


(16) Obligation:

Q restricted rewrite system:
R is empty.
Q is empty.

(17) RisEmptyProof (EQUIVALENT transformation)

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

(18) TRUE

(19) RisEmptyProof (EQUIVALENT transformation)

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

(20) TRUE