357.16/291.45 WORST_CASE(Omega(n^1), O(n^2)) 357.16/291.46 proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml 357.16/291.46 # AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty 357.16/291.46 357.16/291.46 357.16/291.46 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 357.16/291.46 357.16/291.46 (0) CpxTRS 357.16/291.46 (1) CpxTrsToCdtProof [UPPER BOUND(ID), 15 ms] 357.16/291.46 (2) CdtProblem 357.16/291.46 (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms] 357.16/291.46 (4) CdtProblem 357.16/291.46 (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms] 357.16/291.46 (6) CdtProblem 357.16/291.46 (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 116 ms] 357.16/291.46 (8) CdtProblem 357.16/291.46 (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 30 ms] 357.16/291.46 (10) CdtProblem 357.16/291.46 (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms] 357.16/291.46 (12) CdtProblem 357.16/291.46 (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 171 ms] 357.16/291.46 (14) CdtProblem 357.16/291.46 (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 180 ms] 357.16/291.46 (16) CdtProblem 357.16/291.46 (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms] 357.16/291.46 (18) BOUNDS(1, 1) 357.16/291.46 (19) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms] 357.16/291.46 (20) TRS for Loop Detection 357.16/291.46 (21) DecreasingLoopProof [LOWER BOUND(ID), 0 ms] 357.16/291.46 (22) BEST 357.16/291.46 (23) proven lower bound 357.16/291.46 (24) LowerBoundPropagationProof [FINISHED, 0 ms] 357.16/291.46 (25) BOUNDS(n^1, INF) 357.16/291.46 (26) TRS for Loop Detection 357.16/291.46 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (0) 357.16/291.46 Obligation: 357.16/291.46 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 357.16/291.46 357.16/291.46 357.16/291.46 The TRS R consists of the following rules: 357.16/291.46 357.16/291.46 din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) 357.16/291.46 u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) 357.16/291.46 u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) 357.16/291.46 din(der(times(X, Y))) -> u31(din(der(X)), X, Y) 357.16/291.46 u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) 357.16/291.46 u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) 357.16/291.46 din(der(der(X))) -> u41(din(der(X)), X) 357.16/291.46 u41(dout(DX), X) -> u42(din(der(DX)), X, DX) 357.16/291.46 u42(dout(DDX), X, DX) -> dout(DDX) 357.16/291.46 357.16/291.46 S is empty. 357.16/291.46 Rewrite Strategy: INNERMOST 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (1) CpxTrsToCdtProof (UPPER BOUND(ID)) 357.16/291.46 Converted Cpx (relative) TRS to CDT 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (2) 357.16/291.46 Obligation: 357.16/291.46 Complexity Dependency Tuples Problem 357.16/291.46 357.16/291.46 Rules: 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2))) 357.16/291.46 U22(dout(z0), z1, z2, z3) -> c4 357.16/291.46 U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2))) 357.16/291.46 U32(dout(z0), z1, z2, z3) -> c6 357.16/291.46 U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0))) 357.16/291.46 U42(dout(z0), z1, z2) -> c8 357.16/291.46 S tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2))) 357.16/291.46 U22(dout(z0), z1, z2, z3) -> c4 357.16/291.46 U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2))) 357.16/291.46 U32(dout(z0), z1, z2, z3) -> c6 357.16/291.46 U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0))) 357.16/291.46 U42(dout(z0), z1, z2) -> c8 357.16/291.46 K tuples:none 357.16/291.46 Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 357.16/291.46 357.16/291.46 Defined Pair Symbols: DIN_1, U21_3, U22_4, U31_3, U32_4, U41_2, U42_3 357.16/291.46 357.16/291.46 Compound Symbols: c_2, c1_2, c2_2, c3_2, c4, c5_2, c6, c7_2, c8 357.16/291.46 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID)) 357.16/291.46 Removed 3 trailing nodes: 357.16/291.46 U32(dout(z0), z1, z2, z3) -> c6 357.16/291.46 U42(dout(z0), z1, z2) -> c8 357.16/291.46 U22(dout(z0), z1, z2, z3) -> c4 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (4) 357.16/291.46 Obligation: 357.16/291.46 Complexity Dependency Tuples Problem 357.16/291.46 357.16/291.46 Rules: 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0))) 357.16/291.46 S tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0))) 357.16/291.46 K tuples:none 357.16/291.46 Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 357.16/291.46 357.16/291.46 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 357.16/291.46 357.16/291.46 Compound Symbols: c_2, c1_2, c2_2, c3_2, c5_2, c7_2 357.16/291.46 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID)) 357.16/291.46 Removed 3 trailing tuple parts 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (6) 357.16/291.46 Obligation: 357.16/291.46 Complexity Dependency Tuples Problem 357.16/291.46 357.16/291.46 Rules: 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 S tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 K tuples:none 357.16/291.46 Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 357.16/291.46 357.16/291.46 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 357.16/291.46 357.16/291.46 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 357.16/291.46 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) 357.16/291.46 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 We considered the (Usable) Rules: 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 And the Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 The order we found is given by the following interpretation: 357.16/291.46 357.16/291.46 Polynomial interpretation : 357.16/291.46 357.16/291.46 POL(DIN(x_1)) = x_1 357.16/291.46 POL(U21(x_1, x_2, x_3)) = x_1 357.16/291.46 POL(U31(x_1, x_2, x_3)) = 0 357.16/291.46 POL(U41(x_1, x_2)) = 0 357.16/291.46 POL(c(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c1(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c2(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c3(x_1)) = x_1 357.16/291.46 POL(c5(x_1)) = x_1 357.16/291.46 POL(c7(x_1)) = x_1 357.16/291.46 POL(der(x_1)) = 0 357.16/291.46 POL(din(x_1)) = 0 357.16/291.46 POL(dout(x_1)) = [1] 357.16/291.46 POL(plus(x_1, x_2)) = [1] 357.16/291.46 POL(times(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(u21(x_1, x_2, x_3)) = 0 357.16/291.46 POL(u22(x_1, x_2, x_3, x_4)) = x_1 357.16/291.46 POL(u31(x_1, x_2, x_3)) = 0 357.16/291.46 POL(u32(x_1, x_2, x_3, x_4)) = x_1 357.16/291.46 POL(u41(x_1, x_2)) = x_1 357.16/291.46 POL(u42(x_1, x_2, x_3)) = [1] 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (8) 357.16/291.46 Obligation: 357.16/291.46 Complexity Dependency Tuples Problem 357.16/291.46 357.16/291.46 Rules: 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 S tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 K tuples: 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 357.16/291.46 357.16/291.46 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 357.16/291.46 357.16/291.46 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 357.16/291.46 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) 357.16/291.46 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 We considered the (Usable) Rules: 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 And the Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 The order we found is given by the following interpretation: 357.16/291.46 357.16/291.46 Polynomial interpretation : 357.16/291.46 357.16/291.46 POL(DIN(x_1)) = x_1 357.16/291.46 POL(U21(x_1, x_2, x_3)) = 0 357.16/291.46 POL(U31(x_1, x_2, x_3)) = x_1 357.16/291.46 POL(U41(x_1, x_2)) = 0 357.16/291.46 POL(c(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c1(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c2(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c3(x_1)) = x_1 357.16/291.46 POL(c5(x_1)) = x_1 357.16/291.46 POL(c7(x_1)) = x_1 357.16/291.46 POL(der(x_1)) = 0 357.16/291.46 POL(din(x_1)) = 0 357.16/291.46 POL(dout(x_1)) = [1] 357.16/291.46 POL(plus(x_1, x_2)) = [1] 357.16/291.46 POL(times(x_1, x_2)) = x_2 357.16/291.46 POL(u21(x_1, x_2, x_3)) = 0 357.16/291.46 POL(u22(x_1, x_2, x_3, x_4)) = x_1 357.16/291.46 POL(u31(x_1, x_2, x_3)) = x_1 357.16/291.46 POL(u32(x_1, x_2, x_3, x_4)) = [1] 357.16/291.46 POL(u41(x_1, x_2)) = x_1 357.16/291.46 POL(u42(x_1, x_2, x_3)) = [1] 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (10) 357.16/291.46 Obligation: 357.16/291.46 Complexity Dependency Tuples Problem 357.16/291.46 357.16/291.46 Rules: 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 S tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 K tuples: 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 357.16/291.46 357.16/291.46 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 357.16/291.46 357.16/291.46 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 357.16/291.46 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1))) 357.16/291.46 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 We considered the (Usable) Rules: 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 And the Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 The order we found is given by the following interpretation: 357.16/291.46 357.16/291.46 Polynomial interpretation : 357.16/291.46 357.16/291.46 POL(DIN(x_1)) = x_1 357.16/291.46 POL(U21(x_1, x_2, x_3)) = 0 357.16/291.46 POL(U31(x_1, x_2, x_3)) = x_1 357.16/291.46 POL(U41(x_1, x_2)) = x_1 357.16/291.46 POL(c(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c1(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c2(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c3(x_1)) = x_1 357.16/291.46 POL(c5(x_1)) = x_1 357.16/291.46 POL(c7(x_1)) = x_1 357.16/291.46 POL(der(x_1)) = 0 357.16/291.46 POL(din(x_1)) = 0 357.16/291.46 POL(dout(x_1)) = [1] 357.16/291.46 POL(plus(x_1, x_2)) = [1] 357.16/291.46 POL(times(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(u21(x_1, x_2, x_3)) = 0 357.16/291.46 POL(u22(x_1, x_2, x_3, x_4)) = x_1 357.16/291.46 POL(u31(x_1, x_2, x_3)) = x_1 357.16/291.46 POL(u32(x_1, x_2, x_3, x_4)) = [1] 357.16/291.46 POL(u41(x_1, x_2)) = 0 357.16/291.46 POL(u42(x_1, x_2, x_3)) = x_1 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (12) 357.16/291.46 Obligation: 357.16/291.46 Complexity Dependency Tuples Problem 357.16/291.46 357.16/291.46 Rules: 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 S tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 K tuples: 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 357.16/291.46 357.16/291.46 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 357.16/291.46 357.16/291.46 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 357.16/291.46 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 357.16/291.46 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 We considered the (Usable) Rules: 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 And the Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 The order we found is given by the following interpretation: 357.16/291.46 357.16/291.46 Polynomial interpretation : 357.16/291.46 357.16/291.46 POL(DIN(x_1)) = [2]x_1 + x_1^2 357.16/291.46 POL(U21(x_1, x_2, x_3)) = x_3^2 + x_2*x_3 + x_1*x_3 357.16/291.46 POL(U31(x_1, x_2, x_3)) = [1] + [2]x_2 + [2]x_3 + x_3^2 + [2]x_2*x_3 357.16/291.46 POL(U41(x_1, x_2)) = x_1^2 357.16/291.46 POL(c(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c1(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c2(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c3(x_1)) = x_1 357.16/291.46 POL(c5(x_1)) = x_1 357.16/291.46 POL(c7(x_1)) = x_1 357.16/291.46 POL(der(x_1)) = x_1 357.16/291.46 POL(din(x_1)) = 0 357.16/291.46 POL(dout(x_1)) = [2] + x_1 357.16/291.46 POL(plus(x_1, x_2)) = [2] + x_1 + x_2 357.16/291.46 POL(times(x_1, x_2)) = [1] + x_1 + x_2 357.16/291.46 POL(u21(x_1, x_2, x_3)) = [2]x_1 + x_1^2 357.16/291.46 POL(u22(x_1, x_2, x_3, x_4)) = x_1 + x_4^2 + [2]x_1*x_4 + [2]x_1^2 357.16/291.46 POL(u31(x_1, x_2, x_3)) = x_1 + x_1*x_3 + x_1^2 + [2]x_1*x_2 357.16/291.46 POL(u32(x_1, x_2, x_3, x_4)) = x_2 + x_3 + [2]x_4 + x_4^2 + [2]x_2*x_4 + [2]x_1^2 357.16/291.46 POL(u41(x_1, x_2)) = 0 357.16/291.46 POL(u42(x_1, x_2, x_3)) = [2]x_1 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (14) 357.16/291.46 Obligation: 357.16/291.46 Complexity Dependency Tuples Problem 357.16/291.46 357.16/291.46 Rules: 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 S tuples: 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 K tuples: 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 357.16/291.46 357.16/291.46 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 357.16/291.46 357.16/291.46 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 357.16/291.46 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2))) 357.16/291.46 Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S. 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 We considered the (Usable) Rules: 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 And the Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 The order we found is given by the following interpretation: 357.16/291.46 357.16/291.46 Polynomial interpretation : 357.16/291.46 357.16/291.46 POL(DIN(x_1)) = [1] + [2]x_1 + [2]x_1^2 357.16/291.46 POL(U21(x_1, x_2, x_3)) = [1] + [2]x_3 + [2]x_3^2 + x_2*x_3 + [2]x_1*x_3 + x_1^2 357.16/291.46 POL(U31(x_1, x_2, x_3)) = [2] + [2]x_1 + [2]x_2 + [2]x_3 + [2]x_3^2 + [2]x_2*x_3 + [2]x_1*x_3 357.16/291.46 POL(U41(x_1, x_2)) = [2]x_1 + [2]x_1^2 357.16/291.46 POL(c(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c1(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c2(x_1, x_2)) = x_1 + x_2 357.16/291.46 POL(c3(x_1)) = x_1 357.16/291.46 POL(c5(x_1)) = x_1 357.16/291.46 POL(c7(x_1)) = x_1 357.16/291.46 POL(der(x_1)) = [1] + x_1 357.16/291.46 POL(din(x_1)) = 0 357.16/291.46 POL(dout(x_1)) = [2] + x_1 357.16/291.46 POL(plus(x_1, x_2)) = [2] + x_1 + x_2 357.16/291.46 POL(times(x_1, x_2)) = [2] + x_1 + x_2 357.16/291.46 POL(u21(x_1, x_2, x_3)) = x_1 + x_1*x_3 357.16/291.46 POL(u22(x_1, x_2, x_3, x_4)) = [2]x_1*x_4 + [2]x_1^2 357.16/291.46 POL(u31(x_1, x_2, x_3)) = 0 357.16/291.46 POL(u32(x_1, x_2, x_3, x_4)) = [2]x_1*x_4 + [2]x_1^2 + [2]x_1*x_2 + x_1*x_3 357.16/291.46 POL(u41(x_1, x_2)) = x_1 + x_1^2 357.16/291.46 POL(u42(x_1, x_2, x_3)) = [2]x_1 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (16) 357.16/291.46 Obligation: 357.16/291.46 Complexity Dependency Tuples Problem 357.16/291.46 357.16/291.46 Rules: 357.16/291.46 din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1) 357.16/291.46 din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1) 357.16/291.46 din(der(der(z0))) -> u41(din(der(z0)), z0) 357.16/291.46 u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0) 357.16/291.46 u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0)) 357.16/291.46 u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0) 357.16/291.46 u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3))) 357.16/291.46 u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0) 357.16/291.46 u42(dout(z0), z1, z2) -> dout(z0) 357.16/291.46 Tuples: 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 S tuples:none 357.16/291.46 K tuples: 357.16/291.46 U21(dout(z0), z1, z2) -> c3(DIN(der(z2))) 357.16/291.46 U31(dout(z0), z1, z2) -> c5(DIN(der(z2))) 357.16/291.46 U41(dout(z0), z1) -> c7(DIN(der(z0))) 357.16/291.46 DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0))) 357.16/291.46 DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0))) 357.16/291.46 Defined Rule Symbols: din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3 357.16/291.46 357.16/291.46 Defined Pair Symbols: DIN_1, U21_3, U31_3, U41_2 357.16/291.46 357.16/291.46 Compound Symbols: c_2, c1_2, c2_2, c3_1, c5_1, c7_1 357.16/291.46 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (17) SIsEmptyProof (BOTH BOUNDS(ID, ID)) 357.16/291.46 The set S is empty 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (18) 357.16/291.46 BOUNDS(1, 1) 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (19) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID)) 357.16/291.46 Transformed a relative TRS into a decreasing-loop problem. 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (20) 357.16/291.46 Obligation: 357.16/291.46 Analyzing the following TRS for decreasing loops: 357.16/291.46 357.16/291.46 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 357.16/291.46 357.16/291.46 357.16/291.46 The TRS R consists of the following rules: 357.16/291.46 357.16/291.46 din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) 357.16/291.46 u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) 357.16/291.46 u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) 357.16/291.46 din(der(times(X, Y))) -> u31(din(der(X)), X, Y) 357.16/291.46 u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) 357.16/291.46 u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) 357.16/291.46 din(der(der(X))) -> u41(din(der(X)), X) 357.16/291.46 u41(dout(DX), X) -> u42(din(der(DX)), X, DX) 357.16/291.46 u42(dout(DDX), X, DX) -> dout(DDX) 357.16/291.46 357.16/291.46 S is empty. 357.16/291.46 Rewrite Strategy: INNERMOST 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (21) DecreasingLoopProof (LOWER BOUND(ID)) 357.16/291.46 The following loop(s) give(s) rise to the lower bound Omega(n^1): 357.16/291.46 357.16/291.46 The rewrite sequence 357.16/291.46 357.16/291.46 din(der(plus(X, Y))) ->^+ u21(din(der(X)), X, Y) 357.16/291.46 357.16/291.46 gives rise to a decreasing loop by considering the right hand sides subterm at position [0]. 357.16/291.46 357.16/291.46 The pumping substitution is [X / plus(X, Y)]. 357.16/291.46 357.16/291.46 The result substitution is [ ]. 357.16/291.46 357.16/291.46 357.16/291.46 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (22) 357.16/291.46 Complex Obligation (BEST) 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (23) 357.16/291.46 Obligation: 357.16/291.46 Proved the lower bound n^1 for the following obligation: 357.16/291.46 357.16/291.46 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 357.16/291.46 357.16/291.46 357.16/291.46 The TRS R consists of the following rules: 357.16/291.46 357.16/291.46 din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) 357.16/291.46 u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) 357.16/291.46 u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) 357.16/291.46 din(der(times(X, Y))) -> u31(din(der(X)), X, Y) 357.16/291.46 u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) 357.16/291.46 u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) 357.16/291.46 din(der(der(X))) -> u41(din(der(X)), X) 357.16/291.46 u41(dout(DX), X) -> u42(din(der(DX)), X, DX) 357.16/291.46 u42(dout(DDX), X, DX) -> dout(DDX) 357.16/291.46 357.16/291.46 S is empty. 357.16/291.46 Rewrite Strategy: INNERMOST 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (24) LowerBoundPropagationProof (FINISHED) 357.16/291.46 Propagated lower bound. 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (25) 357.16/291.46 BOUNDS(n^1, INF) 357.16/291.46 357.16/291.46 ---------------------------------------- 357.16/291.46 357.16/291.46 (26) 357.16/291.46 Obligation: 357.16/291.46 Analyzing the following TRS for decreasing loops: 357.16/291.46 357.16/291.46 The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2). 357.16/291.46 357.16/291.46 357.16/291.46 The TRS R consists of the following rules: 357.16/291.46 357.16/291.46 din(der(plus(X, Y))) -> u21(din(der(X)), X, Y) 357.16/291.46 u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX) 357.16/291.46 u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY)) 357.16/291.46 din(der(times(X, Y))) -> u31(din(der(X)), X, Y) 357.16/291.46 u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX) 357.16/291.46 u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX))) 357.16/291.46 din(der(der(X))) -> u41(din(der(X)), X) 357.16/291.46 u41(dout(DX), X) -> u42(din(der(DX)), X, DX) 357.16/291.46 u42(dout(DDX), X, DX) -> dout(DDX) 357.16/291.46 357.16/291.46 S is empty. 357.16/291.46 Rewrite Strategy: INNERMOST 357.54/291.61 EOF