(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