(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
c(0, x) → x
c(x, c(y, z)) → c(+(x, y), z)
+(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))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(x, c(y, z)) → c(*(x, y), *(x, z))
*(c(x, y), z) → c(*(x, z), *(y, z))
Rewrite Strategy: INNERMOST
(1) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted CpxTRS to CDT
(2) Obligation:
Complexity Dependency Tuples Problem
Rules:
c(0, z0) → z0
c(z0, c(z1, z2)) → c(+(z0, z1), z2)
+(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))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(z0, c(z1, z2)) → c(*(z0, z1), *(z0, z2))
*(c(z0, z1), z2) → c(*(z0, z2), *(z1, z2))
Tuples:
C(z0, c(z1, z2)) → c2(C(+(z0, z1), z2), +'(z0, z1))
+'(1, 9) → c22(C(1, 0))
+'(2, 8) → c31(C(1, 0))
+'(2, 9) → c32(C(1, 1))
+'(3, 7) → c40(C(1, 0))
+'(3, 8) → c41(C(1, 1))
+'(3, 9) → c42(C(1, 2))
+'(4, 6) → c49(C(1, 0))
+'(4, 7) → c50(C(1, 1))
+'(4, 8) → c51(C(1, 2))
+'(4, 9) → c52(C(1, 3))
+'(5, 5) → c58(C(1, 0))
+'(5, 6) → c59(C(1, 1))
+'(5, 7) → c60(C(1, 2))
+'(5, 8) → c61(C(1, 3))
+'(5, 9) → c62(C(1, 4))
+'(6, 4) → c67(C(1, 0))
+'(6, 5) → c68(C(1, 1))
+'(6, 6) → c69(C(1, 2))
+'(6, 7) → c70(C(1, 3))
+'(6, 8) → c71(C(1, 4))
+'(6, 9) → c72(C(1, 5))
+'(7, 3) → c76(C(1, 0))
+'(7, 4) → c77(C(1, 1))
+'(7, 5) → c78(C(1, 2))
+'(7, 6) → c79(C(1, 3))
+'(7, 7) → c80(C(1, 4))
+'(7, 8) → c81(C(1, 5))
+'(7, 9) → c82(C(1, 6))
+'(8, 2) → c85(C(1, 0))
+'(8, 3) → c86(C(1, 1))
+'(8, 4) → c87(C(1, 2))
+'(8, 5) → c88(C(1, 3))
+'(8, 6) → c89(C(1, 4))
+'(8, 7) → c90(C(1, 5))
+'(8, 8) → c91(C(1, 6))
+'(8, 9) → c92(C(1, 7))
+'(9, 1) → c94(C(1, 0))
+'(9, 2) → c95(C(1, 1))
+'(9, 3) → c96(C(1, 2))
+'(9, 4) → c97(C(1, 3))
+'(9, 5) → c98(C(1, 4))
+'(9, 6) → c99(C(1, 5))
+'(9, 7) → c100(C(1, 6))
+'(9, 8) → c101(C(1, 7))
+'(9, 9) → c102(C(1, 8))
+'(z0, c(z1, z2)) → c103(C(z1, +(z0, z2)), +'(z0, z2))
+'(c(z0, z1), z2) → c104(C(z0, +(z1, z2)), +'(z1, z2))
*'(2, 5) → c130(C(1, 0))
*'(2, 6) → c131(C(1, 2))
*'(2, 7) → c132(C(1, 4))
*'(2, 8) → c133(C(1, 6))
*'(2, 9) → c134(C(1, 8))
*'(3, 4) → c139(C(1, 2))
*'(3, 5) → c140(C(1, 5))
*'(3, 6) → c141(C(1, 8))
*'(3, 7) → c142(C(2, 1))
*'(3, 8) → c143(C(2, 4))
*'(3, 9) → c144(C(2, 7))
*'(4, 3) → c148(C(1, 2))
*'(4, 4) → c149(C(1, 6))
*'(4, 5) → c150(C(2, 0))
*'(4, 6) → c151(C(2, 4))
*'(4, 7) → c152(C(2, 8))
*'(4, 8) → c153(C(3, 2))
*'(4, 9) → c154(C(3, 6))
*'(5, 2) → c157(C(1, 0))
*'(5, 3) → c158(C(1, 5))
*'(5, 4) → c159(C(2, 0))
*'(5, 5) → c160(C(2, 5))
*'(5, 6) → c161(C(3, 0))
*'(5, 7) → c162(C(3, 5))
*'(5, 8) → c163(C(4, 0))
*'(5, 9) → c164(C(4, 5))
*'(6, 2) → c167(C(1, 2))
*'(6, 3) → c168(C(1, 8))
*'(6, 4) → c169(C(2, 4))
*'(6, 5) → c170(C(3, 0))
*'(6, 6) → c171(C(3, 6))
*'(6, 7) → c172(C(4, 2))
*'(6, 8) → c173(C(4, 8))
*'(6, 9) → c174(C(5, 4))
*'(7, 2) → c177(C(1, 4))
*'(7, 3) → c178(C(2, 1))
*'(7, 4) → c179(C(2, 8))
*'(7, 5) → c180(C(3, 5))
*'(7, 6) → c181(C(4, 2))
*'(7, 7) → c182(C(4, 9))
*'(7, 8) → c183(C(5, 6))
*'(7, 9) → c184(C(6, 3))
*'(8, 2) → c187(C(1, 8))
*'(8, 3) → c188(C(2, 4))
*'(8, 4) → c189(C(3, 2))
*'(8, 5) → c190(C(4, 0))
*'(8, 6) → c191(C(4, 8))
*'(8, 7) → c192(C(5, 6))
*'(8, 8) → c193(C(6, 4))
*'(8, 9) → c194(C(7, 2))
*'(9, 2) → c197(C(1, 8))
*'(9, 3) → c198(C(2, 7))
*'(9, 4) → c199(C(3, 6))
*'(9, 5) → c200(C(4, 5))
*'(9, 6) → c201(C(5, 4))
*'(9, 7) → c202(C(6, 3))
*'(9, 8) → c203(C(7, 2))
*'(9, 9) → c204(C(8, 1))
*'(z0, c(z1, z2)) → c205(C(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(c(z0, z1), z2) → c206(C(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
S tuples:
C(z0, c(z1, z2)) → c2(C(+(z0, z1), z2), +'(z0, z1))
+'(1, 9) → c22(C(1, 0))
+'(2, 8) → c31(C(1, 0))
+'(2, 9) → c32(C(1, 1))
+'(3, 7) → c40(C(1, 0))
+'(3, 8) → c41(C(1, 1))
+'(3, 9) → c42(C(1, 2))
+'(4, 6) → c49(C(1, 0))
+'(4, 7) → c50(C(1, 1))
+'(4, 8) → c51(C(1, 2))
+'(4, 9) → c52(C(1, 3))
+'(5, 5) → c58(C(1, 0))
+'(5, 6) → c59(C(1, 1))
+'(5, 7) → c60(C(1, 2))
+'(5, 8) → c61(C(1, 3))
+'(5, 9) → c62(C(1, 4))
+'(6, 4) → c67(C(1, 0))
+'(6, 5) → c68(C(1, 1))
+'(6, 6) → c69(C(1, 2))
+'(6, 7) → c70(C(1, 3))
+'(6, 8) → c71(C(1, 4))
+'(6, 9) → c72(C(1, 5))
+'(7, 3) → c76(C(1, 0))
+'(7, 4) → c77(C(1, 1))
+'(7, 5) → c78(C(1, 2))
+'(7, 6) → c79(C(1, 3))
+'(7, 7) → c80(C(1, 4))
+'(7, 8) → c81(C(1, 5))
+'(7, 9) → c82(C(1, 6))
+'(8, 2) → c85(C(1, 0))
+'(8, 3) → c86(C(1, 1))
+'(8, 4) → c87(C(1, 2))
+'(8, 5) → c88(C(1, 3))
+'(8, 6) → c89(C(1, 4))
+'(8, 7) → c90(C(1, 5))
+'(8, 8) → c91(C(1, 6))
+'(8, 9) → c92(C(1, 7))
+'(9, 1) → c94(C(1, 0))
+'(9, 2) → c95(C(1, 1))
+'(9, 3) → c96(C(1, 2))
+'(9, 4) → c97(C(1, 3))
+'(9, 5) → c98(C(1, 4))
+'(9, 6) → c99(C(1, 5))
+'(9, 7) → c100(C(1, 6))
+'(9, 8) → c101(C(1, 7))
+'(9, 9) → c102(C(1, 8))
+'(z0, c(z1, z2)) → c103(C(z1, +(z0, z2)), +'(z0, z2))
+'(c(z0, z1), z2) → c104(C(z0, +(z1, z2)), +'(z1, z2))
*'(2, 5) → c130(C(1, 0))
*'(2, 6) → c131(C(1, 2))
*'(2, 7) → c132(C(1, 4))
*'(2, 8) → c133(C(1, 6))
*'(2, 9) → c134(C(1, 8))
*'(3, 4) → c139(C(1, 2))
*'(3, 5) → c140(C(1, 5))
*'(3, 6) → c141(C(1, 8))
*'(3, 7) → c142(C(2, 1))
*'(3, 8) → c143(C(2, 4))
*'(3, 9) → c144(C(2, 7))
*'(4, 3) → c148(C(1, 2))
*'(4, 4) → c149(C(1, 6))
*'(4, 5) → c150(C(2, 0))
*'(4, 6) → c151(C(2, 4))
*'(4, 7) → c152(C(2, 8))
*'(4, 8) → c153(C(3, 2))
*'(4, 9) → c154(C(3, 6))
*'(5, 2) → c157(C(1, 0))
*'(5, 3) → c158(C(1, 5))
*'(5, 4) → c159(C(2, 0))
*'(5, 5) → c160(C(2, 5))
*'(5, 6) → c161(C(3, 0))
*'(5, 7) → c162(C(3, 5))
*'(5, 8) → c163(C(4, 0))
*'(5, 9) → c164(C(4, 5))
*'(6, 2) → c167(C(1, 2))
*'(6, 3) → c168(C(1, 8))
*'(6, 4) → c169(C(2, 4))
*'(6, 5) → c170(C(3, 0))
*'(6, 6) → c171(C(3, 6))
*'(6, 7) → c172(C(4, 2))
*'(6, 8) → c173(C(4, 8))
*'(6, 9) → c174(C(5, 4))
*'(7, 2) → c177(C(1, 4))
*'(7, 3) → c178(C(2, 1))
*'(7, 4) → c179(C(2, 8))
*'(7, 5) → c180(C(3, 5))
*'(7, 6) → c181(C(4, 2))
*'(7, 7) → c182(C(4, 9))
*'(7, 8) → c183(C(5, 6))
*'(7, 9) → c184(C(6, 3))
*'(8, 2) → c187(C(1, 8))
*'(8, 3) → c188(C(2, 4))
*'(8, 4) → c189(C(3, 2))
*'(8, 5) → c190(C(4, 0))
*'(8, 6) → c191(C(4, 8))
*'(8, 7) → c192(C(5, 6))
*'(8, 8) → c193(C(6, 4))
*'(8, 9) → c194(C(7, 2))
*'(9, 2) → c197(C(1, 8))
*'(9, 3) → c198(C(2, 7))
*'(9, 4) → c199(C(3, 6))
*'(9, 5) → c200(C(4, 5))
*'(9, 6) → c201(C(5, 4))
*'(9, 7) → c202(C(6, 3))
*'(9, 8) → c203(C(7, 2))
*'(9, 9) → c204(C(8, 1))
*'(z0, c(z1, z2)) → c205(C(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(c(z0, z1), z2) → c206(C(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
K tuples:none
Defined Rule Symbols:
c, +, *
Defined Pair Symbols:
C, +', *'
Compound Symbols:
c2, c22, c31, c32, c40, c41, c42, c49, c50, c51, c52, c58, c59, c60, c61, c62, c67, c68, c69, c70, c71, c72, c76, c77, c78, c79, c80, c81, c82, c85, c86, c87, c88, c89, c90, c91, c92, c94, c95, c96, c97, c98, c99, c100, c101, c102, c103, c104, c130, c131, c132, c133, c134, c139, c140, c141, c142, c143, c144, c148, c149, c150, c151, c152, c153, c154, c157, c158, c159, c160, c161, c162, c163, c164, c167, c168, c169, c170, c171, c172, c173, c174, c177, c178, c179, c180, c181, c182, c183, c184, c187, c188, c189, c190, c191, c192, c193, c194, c197, c198, c199, c200, c201, c202, c203, c204, c205, c206
(3) CdtUnreachableProof (EQUIVALENT transformation)
The following tuples could be removed as they are not reachable from basic start terms:
C(z0, c(z1, z2)) → c2(C(+(z0, z1), z2), +'(z0, z1))
+'(z0, c(z1, z2)) → c103(C(z1, +(z0, z2)), +'(z0, z2))
+'(c(z0, z1), z2) → c104(C(z0, +(z1, z2)), +'(z1, z2))
*'(z0, c(z1, z2)) → c205(C(*(z0, z1), *(z0, z2)), *'(z0, z1), *'(z0, z2))
*'(c(z0, z1), z2) → c206(C(*(z0, z2), *(z1, z2)), *'(z0, z2), *'(z1, z2))
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
c(0, z0) → z0
c(z0, c(z1, z2)) → c(+(z0, z1), z2)
+(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))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(z0, c(z1, z2)) → c(*(z0, z1), *(z0, z2))
*(c(z0, z1), z2) → c(*(z0, z2), *(z1, z2))
Tuples:
+'(1, 9) → c22(C(1, 0))
+'(2, 8) → c31(C(1, 0))
+'(2, 9) → c32(C(1, 1))
+'(3, 7) → c40(C(1, 0))
+'(3, 8) → c41(C(1, 1))
+'(3, 9) → c42(C(1, 2))
+'(4, 6) → c49(C(1, 0))
+'(4, 7) → c50(C(1, 1))
+'(4, 8) → c51(C(1, 2))
+'(4, 9) → c52(C(1, 3))
+'(5, 5) → c58(C(1, 0))
+'(5, 6) → c59(C(1, 1))
+'(5, 7) → c60(C(1, 2))
+'(5, 8) → c61(C(1, 3))
+'(5, 9) → c62(C(1, 4))
+'(6, 4) → c67(C(1, 0))
+'(6, 5) → c68(C(1, 1))
+'(6, 6) → c69(C(1, 2))
+'(6, 7) → c70(C(1, 3))
+'(6, 8) → c71(C(1, 4))
+'(6, 9) → c72(C(1, 5))
+'(7, 3) → c76(C(1, 0))
+'(7, 4) → c77(C(1, 1))
+'(7, 5) → c78(C(1, 2))
+'(7, 6) → c79(C(1, 3))
+'(7, 7) → c80(C(1, 4))
+'(7, 8) → c81(C(1, 5))
+'(7, 9) → c82(C(1, 6))
+'(8, 2) → c85(C(1, 0))
+'(8, 3) → c86(C(1, 1))
+'(8, 4) → c87(C(1, 2))
+'(8, 5) → c88(C(1, 3))
+'(8, 6) → c89(C(1, 4))
+'(8, 7) → c90(C(1, 5))
+'(8, 8) → c91(C(1, 6))
+'(8, 9) → c92(C(1, 7))
+'(9, 1) → c94(C(1, 0))
+'(9, 2) → c95(C(1, 1))
+'(9, 3) → c96(C(1, 2))
+'(9, 4) → c97(C(1, 3))
+'(9, 5) → c98(C(1, 4))
+'(9, 6) → c99(C(1, 5))
+'(9, 7) → c100(C(1, 6))
+'(9, 8) → c101(C(1, 7))
+'(9, 9) → c102(C(1, 8))
*'(2, 5) → c130(C(1, 0))
*'(2, 6) → c131(C(1, 2))
*'(2, 7) → c132(C(1, 4))
*'(2, 8) → c133(C(1, 6))
*'(2, 9) → c134(C(1, 8))
*'(3, 4) → c139(C(1, 2))
*'(3, 5) → c140(C(1, 5))
*'(3, 6) → c141(C(1, 8))
*'(3, 7) → c142(C(2, 1))
*'(3, 8) → c143(C(2, 4))
*'(3, 9) → c144(C(2, 7))
*'(4, 3) → c148(C(1, 2))
*'(4, 4) → c149(C(1, 6))
*'(4, 5) → c150(C(2, 0))
*'(4, 6) → c151(C(2, 4))
*'(4, 7) → c152(C(2, 8))
*'(4, 8) → c153(C(3, 2))
*'(4, 9) → c154(C(3, 6))
*'(5, 2) → c157(C(1, 0))
*'(5, 3) → c158(C(1, 5))
*'(5, 4) → c159(C(2, 0))
*'(5, 5) → c160(C(2, 5))
*'(5, 6) → c161(C(3, 0))
*'(5, 7) → c162(C(3, 5))
*'(5, 8) → c163(C(4, 0))
*'(5, 9) → c164(C(4, 5))
*'(6, 2) → c167(C(1, 2))
*'(6, 3) → c168(C(1, 8))
*'(6, 4) → c169(C(2, 4))
*'(6, 5) → c170(C(3, 0))
*'(6, 6) → c171(C(3, 6))
*'(6, 7) → c172(C(4, 2))
*'(6, 8) → c173(C(4, 8))
*'(6, 9) → c174(C(5, 4))
*'(7, 2) → c177(C(1, 4))
*'(7, 3) → c178(C(2, 1))
*'(7, 4) → c179(C(2, 8))
*'(7, 5) → c180(C(3, 5))
*'(7, 6) → c181(C(4, 2))
*'(7, 7) → c182(C(4, 9))
*'(7, 8) → c183(C(5, 6))
*'(7, 9) → c184(C(6, 3))
*'(8, 2) → c187(C(1, 8))
*'(8, 3) → c188(C(2, 4))
*'(8, 4) → c189(C(3, 2))
*'(8, 5) → c190(C(4, 0))
*'(8, 6) → c191(C(4, 8))
*'(8, 7) → c192(C(5, 6))
*'(8, 8) → c193(C(6, 4))
*'(8, 9) → c194(C(7, 2))
*'(9, 2) → c197(C(1, 8))
*'(9, 3) → c198(C(2, 7))
*'(9, 4) → c199(C(3, 6))
*'(9, 5) → c200(C(4, 5))
*'(9, 6) → c201(C(5, 4))
*'(9, 7) → c202(C(6, 3))
*'(9, 8) → c203(C(7, 2))
*'(9, 9) → c204(C(8, 1))
S tuples:
+'(1, 9) → c22(C(1, 0))
+'(2, 8) → c31(C(1, 0))
+'(2, 9) → c32(C(1, 1))
+'(3, 7) → c40(C(1, 0))
+'(3, 8) → c41(C(1, 1))
+'(3, 9) → c42(C(1, 2))
+'(4, 6) → c49(C(1, 0))
+'(4, 7) → c50(C(1, 1))
+'(4, 8) → c51(C(1, 2))
+'(4, 9) → c52(C(1, 3))
+'(5, 5) → c58(C(1, 0))
+'(5, 6) → c59(C(1, 1))
+'(5, 7) → c60(C(1, 2))
+'(5, 8) → c61(C(1, 3))
+'(5, 9) → c62(C(1, 4))
+'(6, 4) → c67(C(1, 0))
+'(6, 5) → c68(C(1, 1))
+'(6, 6) → c69(C(1, 2))
+'(6, 7) → c70(C(1, 3))
+'(6, 8) → c71(C(1, 4))
+'(6, 9) → c72(C(1, 5))
+'(7, 3) → c76(C(1, 0))
+'(7, 4) → c77(C(1, 1))
+'(7, 5) → c78(C(1, 2))
+'(7, 6) → c79(C(1, 3))
+'(7, 7) → c80(C(1, 4))
+'(7, 8) → c81(C(1, 5))
+'(7, 9) → c82(C(1, 6))
+'(8, 2) → c85(C(1, 0))
+'(8, 3) → c86(C(1, 1))
+'(8, 4) → c87(C(1, 2))
+'(8, 5) → c88(C(1, 3))
+'(8, 6) → c89(C(1, 4))
+'(8, 7) → c90(C(1, 5))
+'(8, 8) → c91(C(1, 6))
+'(8, 9) → c92(C(1, 7))
+'(9, 1) → c94(C(1, 0))
+'(9, 2) → c95(C(1, 1))
+'(9, 3) → c96(C(1, 2))
+'(9, 4) → c97(C(1, 3))
+'(9, 5) → c98(C(1, 4))
+'(9, 6) → c99(C(1, 5))
+'(9, 7) → c100(C(1, 6))
+'(9, 8) → c101(C(1, 7))
+'(9, 9) → c102(C(1, 8))
*'(2, 5) → c130(C(1, 0))
*'(2, 6) → c131(C(1, 2))
*'(2, 7) → c132(C(1, 4))
*'(2, 8) → c133(C(1, 6))
*'(2, 9) → c134(C(1, 8))
*'(3, 4) → c139(C(1, 2))
*'(3, 5) → c140(C(1, 5))
*'(3, 6) → c141(C(1, 8))
*'(3, 7) → c142(C(2, 1))
*'(3, 8) → c143(C(2, 4))
*'(3, 9) → c144(C(2, 7))
*'(4, 3) → c148(C(1, 2))
*'(4, 4) → c149(C(1, 6))
*'(4, 5) → c150(C(2, 0))
*'(4, 6) → c151(C(2, 4))
*'(4, 7) → c152(C(2, 8))
*'(4, 8) → c153(C(3, 2))
*'(4, 9) → c154(C(3, 6))
*'(5, 2) → c157(C(1, 0))
*'(5, 3) → c158(C(1, 5))
*'(5, 4) → c159(C(2, 0))
*'(5, 5) → c160(C(2, 5))
*'(5, 6) → c161(C(3, 0))
*'(5, 7) → c162(C(3, 5))
*'(5, 8) → c163(C(4, 0))
*'(5, 9) → c164(C(4, 5))
*'(6, 2) → c167(C(1, 2))
*'(6, 3) → c168(C(1, 8))
*'(6, 4) → c169(C(2, 4))
*'(6, 5) → c170(C(3, 0))
*'(6, 6) → c171(C(3, 6))
*'(6, 7) → c172(C(4, 2))
*'(6, 8) → c173(C(4, 8))
*'(6, 9) → c174(C(5, 4))
*'(7, 2) → c177(C(1, 4))
*'(7, 3) → c178(C(2, 1))
*'(7, 4) → c179(C(2, 8))
*'(7, 5) → c180(C(3, 5))
*'(7, 6) → c181(C(4, 2))
*'(7, 7) → c182(C(4, 9))
*'(7, 8) → c183(C(5, 6))
*'(7, 9) → c184(C(6, 3))
*'(8, 2) → c187(C(1, 8))
*'(8, 3) → c188(C(2, 4))
*'(8, 4) → c189(C(3, 2))
*'(8, 5) → c190(C(4, 0))
*'(8, 6) → c191(C(4, 8))
*'(8, 7) → c192(C(5, 6))
*'(8, 8) → c193(C(6, 4))
*'(8, 9) → c194(C(7, 2))
*'(9, 2) → c197(C(1, 8))
*'(9, 3) → c198(C(2, 7))
*'(9, 4) → c199(C(3, 6))
*'(9, 5) → c200(C(4, 5))
*'(9, 6) → c201(C(5, 4))
*'(9, 7) → c202(C(6, 3))
*'(9, 8) → c203(C(7, 2))
*'(9, 9) → c204(C(8, 1))
K tuples:none
Defined Rule Symbols:
c, +, *
Defined Pair Symbols:
+', *'
Compound Symbols:
c22, c31, c32, c40, c41, c42, c49, c50, c51, c52, c58, c59, c60, c61, c62, c67, c68, c69, c70, c71, c72, c76, c77, c78, c79, c80, c81, c82, c85, c86, c87, c88, c89, c90, c91, c92, c94, c95, c96, c97, c98, c99, c100, c101, c102, c130, c131, c132, c133, c134, c139, c140, c141, c142, c143, c144, c148, c149, c150, c151, c152, c153, c154, c157, c158, c159, c160, c161, c162, c163, c164, c167, c168, c169, c170, c171, c172, c173, c174, c177, c178, c179, c180, c181, c182, c183, c184, c187, c188, c189, c190, c191, c192, c193, c194, c197, c198, c199, c200, c201, c202, c203, c204
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 103 trailing nodes:
*'(8, 7) → c192(C(5, 6))
*'(8, 8) → c193(C(6, 4))
+'(5, 7) → c60(C(1, 2))
+'(9, 5) → c98(C(1, 4))
+'(8, 2) → c85(C(1, 0))
+'(7, 8) → c81(C(1, 5))
+'(7, 7) → c80(C(1, 4))
+'(8, 5) → c88(C(1, 3))
+'(5, 9) → c62(C(1, 4))
*'(4, 5) → c150(C(2, 0))
+'(6, 6) → c69(C(1, 2))
*'(7, 2) → c177(C(1, 4))
*'(7, 3) → c178(C(2, 1))
*'(2, 9) → c134(C(1, 8))
*'(7, 8) → c183(C(5, 6))
+'(6, 4) → c67(C(1, 0))
+'(2, 9) → c32(C(1, 1))
*'(5, 6) → c161(C(3, 0))
*'(5, 8) → c163(C(4, 0))
*'(6, 9) → c174(C(5, 4))
+'(6, 9) → c72(C(1, 5))
*'(5, 2) → c157(C(1, 0))
*'(9, 7) → c202(C(6, 3))
+'(8, 6) → c89(C(1, 4))
*'(6, 4) → c169(C(2, 4))
+'(9, 3) → c96(C(1, 2))
+'(6, 5) → c68(C(1, 1))
*'(6, 7) → c172(C(4, 2))
*'(3, 6) → c141(C(1, 8))
+'(3, 9) → c42(C(1, 2))
*'(2, 6) → c131(C(1, 2))
*'(4, 6) → c151(C(2, 4))
*'(6, 8) → c173(C(4, 8))
+'(2, 8) → c31(C(1, 0))
*'(6, 5) → c170(C(3, 0))
*'(5, 3) → c158(C(1, 5))
+'(6, 8) → c71(C(1, 4))
*'(8, 2) → c187(C(1, 8))
+'(5, 5) → c58(C(1, 0))
*'(4, 9) → c154(C(3, 6))
+'(6, 7) → c70(C(1, 3))
*'(9, 4) → c199(C(3, 6))
*'(9, 5) → c200(C(4, 5))
+'(3, 7) → c40(C(1, 0))
*'(3, 8) → c143(C(2, 4))
*'(5, 5) → c160(C(2, 5))
*'(2, 7) → c132(C(1, 4))
+'(8, 3) → c86(C(1, 1))
+'(8, 7) → c90(C(1, 5))
*'(7, 6) → c181(C(4, 2))
+'(4, 7) → c50(C(1, 1))
+'(4, 9) → c52(C(1, 3))
+'(8, 8) → c91(C(1, 6))
*'(4, 8) → c153(C(3, 2))
*'(6, 2) → c167(C(1, 2))
+'(1, 9) → c22(C(1, 0))
*'(6, 6) → c171(C(3, 6))
+'(5, 8) → c61(C(1, 3))
*'(8, 4) → c189(C(3, 2))
*'(2, 5) → c130(C(1, 0))
+'(9, 1) → c94(C(1, 0))
*'(8, 9) → c194(C(7, 2))
*'(7, 7) → c182(C(4, 9))
+'(9, 4) → c97(C(1, 3))
+'(3, 8) → c41(C(1, 1))
*'(5, 7) → c162(C(3, 5))
+'(8, 9) → c92(C(1, 7))
*'(9, 3) → c198(C(2, 7))
*'(3, 5) → c140(C(1, 5))
*'(4, 3) → c148(C(1, 2))
+'(9, 8) → c101(C(1, 7))
+'(9, 6) → c99(C(1, 5))
+'(7, 9) → c82(C(1, 6))
*'(2, 8) → c133(C(1, 6))
+'(5, 6) → c59(C(1, 1))
+'(7, 6) → c79(C(1, 3))
+'(4, 8) → c51(C(1, 2))
*'(7, 5) → c180(C(3, 5))
*'(3, 7) → c142(C(2, 1))
*'(8, 6) → c191(C(4, 8))
*'(9, 9) → c204(C(8, 1))
*'(8, 3) → c188(C(2, 4))
+'(9, 9) → c102(C(1, 8))
*'(3, 4) → c139(C(1, 2))
*'(7, 9) → c184(C(6, 3))
+'(8, 4) → c87(C(1, 2))
+'(7, 4) → c77(C(1, 1))
+'(7, 3) → c76(C(1, 0))
*'(9, 2) → c197(C(1, 8))
*'(6, 3) → c168(C(1, 8))
+'(9, 2) → c95(C(1, 1))
*'(5, 4) → c159(C(2, 0))
*'(3, 9) → c144(C(2, 7))
*'(7, 4) → c179(C(2, 8))
+'(9, 7) → c100(C(1, 6))
*'(9, 6) → c201(C(5, 4))
*'(4, 4) → c149(C(1, 6))
*'(4, 7) → c152(C(2, 8))
*'(9, 8) → c203(C(7, 2))
*'(5, 9) → c164(C(4, 5))
+'(7, 5) → c78(C(1, 2))
*'(8, 5) → c190(C(4, 0))
+'(4, 6) → c49(C(1, 0))
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
c(0, z0) → z0
c(z0, c(z1, z2)) → c(+(z0, z1), z2)
+(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))
*(0, 0) → 0
*(0, 1) → 0
*(0, 2) → 0
*(0, 3) → 0
*(0, 4) → 0
*(0, 5) → 0
*(0, 6) → 0
*(0, 7) → 0
*(0, 8) → 0
*(0, 9) → 0
*(1, 0) → 0
*(1, 1) → 1
*(1, 2) → 2
*(1, 3) → 3
*(1, 4) → 4
*(1, 5) → 5
*(1, 6) → 6
*(1, 7) → 7
*(1, 8) → 8
*(1, 9) → 9
*(2, 0) → 0
*(2, 1) → 2
*(2, 2) → 4
*(2, 3) → 6
*(2, 4) → 8
*(2, 5) → c(1, 0)
*(2, 6) → c(1, 2)
*(2, 7) → c(1, 4)
*(2, 8) → c(1, 6)
*(2, 9) → c(1, 8)
*(3, 0) → 0
*(3, 1) → 3
*(3, 2) → 6
*(3, 3) → 9
*(3, 4) → c(1, 2)
*(3, 5) → c(1, 5)
*(3, 6) → c(1, 8)
*(3, 7) → c(2, 1)
*(3, 8) → c(2, 4)
*(3, 9) → c(2, 7)
*(4, 0) → 0
*(4, 1) → 4
*(4, 2) → 8
*(4, 3) → c(1, 2)
*(4, 4) → c(1, 6)
*(4, 5) → c(2, 0)
*(4, 6) → c(2, 4)
*(4, 7) → c(2, 8)
*(4, 8) → c(3, 2)
*(4, 9) → c(3, 6)
*(5, 0) → 0
*(5, 1) → 5
*(5, 2) → c(1, 0)
*(5, 3) → c(1, 5)
*(5, 4) → c(2, 0)
*(5, 5) → c(2, 5)
*(5, 6) → c(3, 0)
*(5, 7) → c(3, 5)
*(5, 8) → c(4, 0)
*(5, 9) → c(4, 5)
*(6, 0) → 0
*(6, 1) → 6
*(6, 2) → c(1, 2)
*(6, 3) → c(1, 8)
*(6, 4) → c(2, 4)
*(6, 5) → c(3, 0)
*(6, 6) → c(3, 6)
*(6, 7) → c(4, 2)
*(6, 8) → c(4, 8)
*(6, 9) → c(5, 4)
*(7, 0) → 0
*(7, 1) → 7
*(7, 2) → c(1, 4)
*(7, 3) → c(2, 1)
*(7, 4) → c(2, 8)
*(7, 5) → c(3, 5)
*(7, 6) → c(4, 2)
*(7, 7) → c(4, 9)
*(7, 8) → c(5, 6)
*(7, 9) → c(6, 3)
*(8, 0) → 0
*(8, 1) → 8
*(8, 2) → c(1, 8)
*(8, 3) → c(2, 4)
*(8, 4) → c(3, 2)
*(8, 5) → c(4, 0)
*(8, 6) → c(4, 8)
*(8, 7) → c(5, 6)
*(8, 8) → c(6, 4)
*(8, 9) → c(7, 2)
*(9, 0) → 0
*(9, 1) → 9
*(9, 2) → c(1, 8)
*(9, 3) → c(2, 7)
*(9, 4) → c(3, 6)
*(9, 5) → c(4, 5)
*(9, 6) → c(5, 4)
*(9, 7) → c(6, 3)
*(9, 8) → c(7, 2)
*(9, 9) → c(8, 1)
*(z0, c(z1, z2)) → c(*(z0, z1), *(z0, z2))
*(c(z0, z1), z2) → c(*(z0, z2), *(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))