(0) Obligation:
Runtime Complexity TRS:
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)
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
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)
+(z0, c(z1, z2)) → c(z1, +(z0, z2))
+(c(z0, z1), z2) → c(z0, +(z1, z2))
c(0, z0) → z0
c(z0, c(z1, z2)) → c(+(z0, z1), z2)
Tuples:
+'(1, 9) → c20(C(1, 0))
+'(2, 8) → c29(C(1, 0))
+'(2, 9) → c30(C(1, 1))
+'(3, 7) → c38(C(1, 0))
+'(3, 8) → c39(C(1, 1))
+'(3, 9) → c40(C(1, 2))
+'(4, 6) → c47(C(1, 0))
+'(4, 7) → c48(C(1, 1))
+'(4, 8) → c49(C(1, 2))
+'(4, 9) → c50(C(1, 3))
+'(5, 5) → c56(C(1, 0))
+'(5, 6) → c57(C(1, 1))
+'(5, 7) → c58(C(1, 2))
+'(5, 8) → c59(C(1, 3))
+'(5, 9) → c60(C(1, 4))
+'(6, 4) → c65(C(1, 0))
+'(6, 5) → c66(C(1, 1))
+'(6, 6) → c67(C(1, 2))
+'(6, 7) → c68(C(1, 3))
+'(6, 8) → c69(C(1, 4))
+'(6, 9) → c70(C(1, 5))
+'(7, 3) → c74(C(1, 0))
+'(7, 4) → c75(C(1, 1))
+'(7, 5) → c76(C(1, 2))
+'(7, 6) → c77(C(1, 3))
+'(7, 7) → c78(C(1, 4))
+'(7, 8) → c79(C(1, 5))
+'(7, 9) → c80(C(1, 6))
+'(8, 2) → c83(C(1, 0))
+'(8, 3) → c84(C(1, 1))
+'(8, 4) → c85(C(1, 2))
+'(8, 5) → c86(C(1, 3))
+'(8, 6) → c87(C(1, 4))
+'(8, 7) → c88(C(1, 5))
+'(8, 8) → c89(C(1, 6))
+'(8, 9) → c90(C(1, 7))
+'(9, 1) → c92(C(1, 0))
+'(9, 2) → c93(C(1, 1))
+'(9, 3) → c94(C(1, 2))
+'(9, 4) → c95(C(1, 3))
+'(9, 5) → c96(C(1, 4))
+'(9, 6) → c97(C(1, 5))
+'(9, 7) → c98(C(1, 6))
+'(9, 8) → c99(C(1, 7))
+'(9, 9) → c100(C(1, 8))
+'(z0, c(z1, z2)) → c101(C(z1, +(z0, z2)), +'(z0, z2))
+'(c(z0, z1), z2) → c102(C(z0, +(z1, z2)), +'(z1, z2))
C(z0, c(z1, z2)) → c104(C(+(z0, z1), z2), +'(z0, z1))
S tuples:
+'(1, 9) → c20(C(1, 0))
+'(2, 8) → c29(C(1, 0))
+'(2, 9) → c30(C(1, 1))
+'(3, 7) → c38(C(1, 0))
+'(3, 8) → c39(C(1, 1))
+'(3, 9) → c40(C(1, 2))
+'(4, 6) → c47(C(1, 0))
+'(4, 7) → c48(C(1, 1))
+'(4, 8) → c49(C(1, 2))
+'(4, 9) → c50(C(1, 3))
+'(5, 5) → c56(C(1, 0))
+'(5, 6) → c57(C(1, 1))
+'(5, 7) → c58(C(1, 2))
+'(5, 8) → c59(C(1, 3))
+'(5, 9) → c60(C(1, 4))
+'(6, 4) → c65(C(1, 0))
+'(6, 5) → c66(C(1, 1))
+'(6, 6) → c67(C(1, 2))
+'(6, 7) → c68(C(1, 3))
+'(6, 8) → c69(C(1, 4))
+'(6, 9) → c70(C(1, 5))
+'(7, 3) → c74(C(1, 0))
+'(7, 4) → c75(C(1, 1))
+'(7, 5) → c76(C(1, 2))
+'(7, 6) → c77(C(1, 3))
+'(7, 7) → c78(C(1, 4))
+'(7, 8) → c79(C(1, 5))
+'(7, 9) → c80(C(1, 6))
+'(8, 2) → c83(C(1, 0))
+'(8, 3) → c84(C(1, 1))
+'(8, 4) → c85(C(1, 2))
+'(8, 5) → c86(C(1, 3))
+'(8, 6) → c87(C(1, 4))
+'(8, 7) → c88(C(1, 5))
+'(8, 8) → c89(C(1, 6))
+'(8, 9) → c90(C(1, 7))
+'(9, 1) → c92(C(1, 0))
+'(9, 2) → c93(C(1, 1))
+'(9, 3) → c94(C(1, 2))
+'(9, 4) → c95(C(1, 3))
+'(9, 5) → c96(C(1, 4))
+'(9, 6) → c97(C(1, 5))
+'(9, 7) → c98(C(1, 6))
+'(9, 8) → c99(C(1, 7))
+'(9, 9) → c100(C(1, 8))
+'(z0, c(z1, z2)) → c101(C(z1, +(z0, z2)), +'(z0, z2))
+'(c(z0, z1), z2) → c102(C(z0, +(z1, z2)), +'(z1, z2))
C(z0, c(z1, z2)) → c104(C(+(z0, z1), z2), +'(z0, z1))
K tuples:none
Defined Rule Symbols:
+, c
Defined Pair Symbols:
+', C
Compound Symbols:
c20, c29, c30, c38, c39, c40, c47, c48, c49, c50, c56, c57, c58, c59, c60, c65, c66, c67, c68, c69, c70, c74, c75, c76, c77, c78, c79, c80, c83, c84, c85, c86, c87, c88, c89, c90, c92, c93, c94, c95, c96, c97, c98, c99, c100, c101, c102, c104
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
+'(z0, c(z1, z2)) → c101(C(z1, +(z0, z2)), +'(z0, z2))
+'(c(z0, z1), z2) → c102(C(z0, +(z1, z2)), +'(z1, z2))
C(z0, c(z1, z2)) → c104(C(+(z0, z1), z2), +'(z0, z1))
(4) Obligation:
Complexity Dependency Tuples Problem
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)
+(z0, c(z1, z2)) → c(z1, +(z0, z2))
+(c(z0, z1), z2) → c(z0, +(z1, z2))
c(0, z0) → z0
c(z0, c(z1, z2)) → c(+(z0, z1), z2)
Tuples:
+'(1, 9) → c20(C(1, 0))
+'(2, 8) → c29(C(1, 0))
+'(2, 9) → c30(C(1, 1))
+'(3, 7) → c38(C(1, 0))
+'(3, 8) → c39(C(1, 1))
+'(3, 9) → c40(C(1, 2))
+'(4, 6) → c47(C(1, 0))
+'(4, 7) → c48(C(1, 1))
+'(4, 8) → c49(C(1, 2))
+'(4, 9) → c50(C(1, 3))
+'(5, 5) → c56(C(1, 0))
+'(5, 6) → c57(C(1, 1))
+'(5, 7) → c58(C(1, 2))
+'(5, 8) → c59(C(1, 3))
+'(5, 9) → c60(C(1, 4))
+'(6, 4) → c65(C(1, 0))
+'(6, 5) → c66(C(1, 1))
+'(6, 6) → c67(C(1, 2))
+'(6, 7) → c68(C(1, 3))
+'(6, 8) → c69(C(1, 4))
+'(6, 9) → c70(C(1, 5))
+'(7, 3) → c74(C(1, 0))
+'(7, 4) → c75(C(1, 1))
+'(7, 5) → c76(C(1, 2))
+'(7, 6) → c77(C(1, 3))
+'(7, 7) → c78(C(1, 4))
+'(7, 8) → c79(C(1, 5))
+'(7, 9) → c80(C(1, 6))
+'(8, 2) → c83(C(1, 0))
+'(8, 3) → c84(C(1, 1))
+'(8, 4) → c85(C(1, 2))
+'(8, 5) → c86(C(1, 3))
+'(8, 6) → c87(C(1, 4))
+'(8, 7) → c88(C(1, 5))
+'(8, 8) → c89(C(1, 6))
+'(8, 9) → c90(C(1, 7))
+'(9, 1) → c92(C(1, 0))
+'(9, 2) → c93(C(1, 1))
+'(9, 3) → c94(C(1, 2))
+'(9, 4) → c95(C(1, 3))
+'(9, 5) → c96(C(1, 4))
+'(9, 6) → c97(C(1, 5))
+'(9, 7) → c98(C(1, 6))
+'(9, 8) → c99(C(1, 7))
+'(9, 9) → c100(C(1, 8))
S tuples:
+'(1, 9) → c20(C(1, 0))
+'(2, 8) → c29(C(1, 0))
+'(2, 9) → c30(C(1, 1))
+'(3, 7) → c38(C(1, 0))
+'(3, 8) → c39(C(1, 1))
+'(3, 9) → c40(C(1, 2))
+'(4, 6) → c47(C(1, 0))
+'(4, 7) → c48(C(1, 1))
+'(4, 8) → c49(C(1, 2))
+'(4, 9) → c50(C(1, 3))
+'(5, 5) → c56(C(1, 0))
+'(5, 6) → c57(C(1, 1))
+'(5, 7) → c58(C(1, 2))
+'(5, 8) → c59(C(1, 3))
+'(5, 9) → c60(C(1, 4))
+'(6, 4) → c65(C(1, 0))
+'(6, 5) → c66(C(1, 1))
+'(6, 6) → c67(C(1, 2))
+'(6, 7) → c68(C(1, 3))
+'(6, 8) → c69(C(1, 4))
+'(6, 9) → c70(C(1, 5))
+'(7, 3) → c74(C(1, 0))
+'(7, 4) → c75(C(1, 1))
+'(7, 5) → c76(C(1, 2))
+'(7, 6) → c77(C(1, 3))
+'(7, 7) → c78(C(1, 4))
+'(7, 8) → c79(C(1, 5))
+'(7, 9) → c80(C(1, 6))
+'(8, 2) → c83(C(1, 0))
+'(8, 3) → c84(C(1, 1))
+'(8, 4) → c85(C(1, 2))
+'(8, 5) → c86(C(1, 3))
+'(8, 6) → c87(C(1, 4))
+'(8, 7) → c88(C(1, 5))
+'(8, 8) → c89(C(1, 6))
+'(8, 9) → c90(C(1, 7))
+'(9, 1) → c92(C(1, 0))
+'(9, 2) → c93(C(1, 1))
+'(9, 3) → c94(C(1, 2))
+'(9, 4) → c95(C(1, 3))
+'(9, 5) → c96(C(1, 4))
+'(9, 6) → c97(C(1, 5))
+'(9, 7) → c98(C(1, 6))
+'(9, 8) → c99(C(1, 7))
+'(9, 9) → c100(C(1, 8))
K tuples:none
Defined Rule Symbols:
+, c
Defined Pair Symbols:
+'
Compound Symbols:
c20, c29, c30, c38, c39, c40, c47, c48, c49, c50, c56, c57, c58, c59, c60, c65, c66, c67, c68, c69, c70, c74, c75, c76, c77, c78, c79, c80, c83, c84, c85, c86, c87, c88, c89, c90, c92, c93, c94, c95, c96, c97, c98, c99, c100
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 45 trailing nodes:
+'(6, 4) → c65(C(1, 0))
+'(3, 7) → c38(C(1, 0))
+'(5, 9) → c60(C(1, 4))
+'(8, 6) → c87(C(1, 4))
+'(7, 3) → c74(C(1, 0))
+'(7, 6) → c77(C(1, 3))
+'(5, 7) → c58(C(1, 2))
+'(9, 3) → c94(C(1, 2))
+'(9, 7) → c98(C(1, 6))
+'(8, 4) → c85(C(1, 2))
+'(8, 2) → c83(C(1, 0))
+'(6, 9) → c70(C(1, 5))
+'(7, 5) → c76(C(1, 2))
+'(2, 8) → c29(C(1, 0))
+'(7, 9) → c80(C(1, 6))
+'(8, 7) → c88(C(1, 5))
+'(6, 5) → c66(C(1, 1))
+'(1, 9) → c20(C(1, 0))
+'(6, 8) → c69(C(1, 4))
+'(9, 6) → c97(C(1, 5))
+'(4, 6) → c47(C(1, 0))
+'(9, 2) → c93(C(1, 1))
+'(3, 9) → c40(C(1, 2))
+'(9, 4) → c95(C(1, 3))
+'(9, 8) → c99(C(1, 7))
+'(8, 3) → c84(C(1, 1))
+'(6, 6) → c67(C(1, 2))
+'(9, 9) → c100(C(1, 8))
+'(3, 8) → c39(C(1, 1))
+'(8, 5) → c86(C(1, 3))
+'(5, 8) → c59(C(1, 3))
+'(8, 9) → c90(C(1, 7))
+'(7, 4) → c75(C(1, 1))
+'(7, 8) → c79(C(1, 5))
+'(4, 9) → c50(C(1, 3))
+'(8, 8) → c89(C(1, 6))
+'(5, 6) → c57(C(1, 1))
+'(9, 5) → c96(C(1, 4))
+'(5, 5) → c56(C(1, 0))
+'(6, 7) → c68(C(1, 3))
+'(9, 1) → c92(C(1, 0))
+'(4, 7) → c48(C(1, 1))
+'(7, 7) → c78(C(1, 4))
+'(2, 9) → c30(C(1, 1))
+'(4, 8) → c49(C(1, 2))
(6) Obligation:
Complexity Dependency Tuples Problem
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)
+(z0, c(z1, z2)) → c(z1, +(z0, z2))
+(c(z0, z1), z2) → c(z0, +(z1, z2))
c(0, z0) → z0
c(z0, c(z1, z2)) → c(+(z0, z1), z2)
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
+, c
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (EQUIVALENT transformation)
The set S is empty
(8) BOUNDS(O(1), O(1))