/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 8 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 138 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 2 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 255 ms]
    (14) CdtProblem
    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 202 ms]
    (16) CdtProblem
    (17) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) BOUNDS(1, 1)
(19) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(20) CpxTRS
    (21) SlicingProof [LOWER BOUND(ID), 0 ms]
    (22) CpxTRS
    (23) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) typed CpxTrs
    (25) OrderProof [LOWER BOUND(ID), 0 ms]
    (26) typed CpxTrs
    (27) RewriteLemmaProof [LOWER BOUND(ID), 470 ms]
    (28) BEST
        (29) proven lower bound
            (30) LowerBoundPropagationProof [FINISHED, 0 ms]
            (31) BOUNDS(n^1, INF)
        (32) typed CpxTrs


----------------------------------------

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   din(der(plus(X, Y))) -> u21(din(der(X)), X, Y)
   u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX)
   u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY))
   din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
   u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
   u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
   din(der(der(X))) -> u41(din(der(X)), X)
   u41(dout(DX), X) -> u42(din(der(DX)), X, DX)
   u42(dout(DDX), X, DX) -> dout(DDX)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   u42(dout(z0), z1, z2) -> dout(z0)
Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2)))
   U22(dout(z0), z1, z2, z3) -> c4
   U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2)))
   U32(dout(z0), z1, z2, z3) -> c6
   U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0)))
   U42(dout(z0), z1, z2) -> c8
S tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2)))
   U22(dout(z0), z1, z2, z3) -> c4
   U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2)))
   U32(dout(z0), z1, z2, z3) -> c6
   U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0)))
   U42(dout(z0), z1, z2) -> c8
K tuples:none
Defined Rule Symbols:   din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3

Defined Pair Symbols:   DIN_1, U21_3, U22_4, U31_3, U32_4, U41_2, U42_3

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c4, c5_2, c6, c7_2, c8


----------------------------------------

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   U42(dout(z0), z1, z2) -> c8
   U32(dout(z0), z1, z2, z3) -> c6
   U22(dout(z0), z1, z2, z3) -> c4

----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   u42(dout(z0), z1, z2) -> dout(z0)
Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2)))
   U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0)))
S tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(U22(din(der(z2)), z1, z2, z0), DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(U32(din(der(z2)), z1, z2, z0), DIN(der(z2)))
   U41(dout(z0), z1) -> c7(U42(din(der(z0)), z1, z0), DIN(der(z0)))
K tuples:none
Defined Rule Symbols:   din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3

Defined Pair Symbols:   DIN_1, U21_3, U31_3, U41_2

Compound Symbols:   c_2, c1_2, c2_2, c3_2, c5_2, c7_2


----------------------------------------

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   u42(dout(z0), z1, z2) -> dout(z0)
Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
S tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
K tuples:none
Defined Rule Symbols:   din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3

Defined Pair Symbols:   DIN_1, U21_3, U31_3, U41_2

Compound Symbols:   c_2, c1_2, c2_2, c3_1, c5_1, c7_1


----------------------------------------

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
We considered the (Usable) Rules:
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   u42(dout(z0), z1, z2) -> dout(z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
And the Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(DIN(x_1)) = [3]x_1
   POL(U21(x_1, x_2, x_3)) = 0
   POL(U31(x_1, x_2, x_3)) = [2]x_1
   POL(U41(x_1, x_2)) = 0
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(der(x_1)) = 0
   POL(din(x_1)) = 0
   POL(dout(x_1)) = [1] + x_1
   POL(plus(x_1, x_2)) = x_2
   POL(times(x_1, x_2)) = [1]
   POL(u21(x_1, x_2, x_3)) = 0
   POL(u22(x_1, x_2, x_3, x_4)) = x_1
   POL(u31(x_1, x_2, x_3)) = x_1
   POL(u32(x_1, x_2, x_3, x_4)) = [2]x_1 + x_4
   POL(u41(x_1, x_2)) = 0
   POL(u42(x_1, x_2, x_3)) = x_1

----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   u42(dout(z0), z1, z2) -> dout(z0)
Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
S tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
K tuples:
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
Defined Rule Symbols:   din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3

Defined Pair Symbols:   DIN_1, U21_3, U31_3, U41_2

Compound Symbols:   c_2, c1_2, c2_2, c3_1, c5_1, c7_1


----------------------------------------

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
We considered the (Usable) Rules:
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   u42(dout(z0), z1, z2) -> dout(z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
And the Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(DIN(x_1)) = x_1
   POL(U21(x_1, x_2, x_3)) = x_1
   POL(U31(x_1, x_2, x_3)) = 0
   POL(U41(x_1, x_2)) = 0
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(der(x_1)) = 0
   POL(din(x_1)) = 0
   POL(dout(x_1)) = [1]
   POL(plus(x_1, x_2)) = [1]
   POL(times(x_1, x_2)) = x_2
   POL(u21(x_1, x_2, x_3)) = 0
   POL(u22(x_1, x_2, x_3, x_4)) = x_1
   POL(u31(x_1, x_2, x_3)) = x_1
   POL(u32(x_1, x_2, x_3, x_4)) = [1]
   POL(u41(x_1, x_2)) = x_1
   POL(u42(x_1, x_2, x_3)) = [1]

----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   u42(dout(z0), z1, z2) -> dout(z0)
Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
S tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
K tuples:
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
Defined Rule Symbols:   din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3

Defined Pair Symbols:   DIN_1, U21_3, U31_3, U41_2

Compound Symbols:   c_2, c1_2, c2_2, c3_1, c5_1, c7_1


----------------------------------------

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
We considered the (Usable) Rules:
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   u42(dout(z0), z1, z2) -> dout(z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
And the Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(DIN(x_1)) = x_1
   POL(U21(x_1, x_2, x_3)) = 0
   POL(U31(x_1, x_2, x_3)) = x_1
   POL(U41(x_1, x_2)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(der(x_1)) = 0
   POL(din(x_1)) = 0
   POL(dout(x_1)) = [1]
   POL(plus(x_1, x_2)) = [1]
   POL(times(x_1, x_2)) = x_1 + x_2
   POL(u21(x_1, x_2, x_3)) = 0
   POL(u22(x_1, x_2, x_3, x_4)) = x_1
   POL(u31(x_1, x_2, x_3)) = x_1
   POL(u32(x_1, x_2, x_3, x_4)) = [1]
   POL(u41(x_1, x_2)) = 0
   POL(u42(x_1, x_2, x_3)) = x_1

----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   u42(dout(z0), z1, z2) -> dout(z0)
Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
S tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
K tuples:
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
Defined Rule Symbols:   din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3

Defined Pair Symbols:   DIN_1, U21_3, U31_3, U41_2

Compound Symbols:   c_2, c1_2, c2_2, c3_1, c5_1, c7_1


----------------------------------------

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
We considered the (Usable) Rules:
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   u42(dout(z0), z1, z2) -> dout(z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
And the Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(DIN(x_1)) = [2]x_1 + x_1^2
   POL(U21(x_1, x_2, x_3)) = x_3^2 + x_2*x_3 + x_1*x_3
   POL(U31(x_1, x_2, x_3)) = [1] + [2]x_2 + [2]x_3 + x_3^2 + [2]x_2*x_3
   POL(U41(x_1, x_2)) = x_1^2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(der(x_1)) = x_1
   POL(din(x_1)) = 0
   POL(dout(x_1)) = [2] + x_1
   POL(plus(x_1, x_2)) = [2] + x_1 + x_2
   POL(times(x_1, x_2)) = [1] + x_1 + x_2
   POL(u21(x_1, x_2, x_3)) = [2]x_1 + x_1^2
   POL(u22(x_1, x_2, x_3, x_4)) = x_1 + x_4^2 + [2]x_1*x_4 + [2]x_1^2
   POL(u31(x_1, x_2, x_3)) = x_1 + x_1*x_3 + x_1^2 + [2]x_1*x_2
   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
   POL(u41(x_1, x_2)) = 0
   POL(u42(x_1, x_2, x_3)) = [2]x_1

----------------------------------------

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   u42(dout(z0), z1, z2) -> dout(z0)
Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
S tuples:
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
K tuples:
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
Defined Rule Symbols:   din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3

Defined Pair Symbols:   DIN_1, U21_3, U31_3, U41_2

Compound Symbols:   c_2, c1_2, c2_2, c3_1, c5_1, c7_1


----------------------------------------

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
We considered the (Usable) Rules:
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   u42(dout(z0), z1, z2) -> dout(z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
And the Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(DIN(x_1)) = [1] + [2]x_1 + [2]x_1^2
   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
   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
   POL(U41(x_1, x_2)) = [2]x_1 + [2]x_1^2
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(der(x_1)) = [1] + x_1
   POL(din(x_1)) = 0
   POL(dout(x_1)) = [2] + x_1
   POL(plus(x_1, x_2)) = [2] + x_1 + x_2
   POL(times(x_1, x_2)) = [2] + x_1 + x_2
   POL(u21(x_1, x_2, x_3)) = x_1 + x_1*x_3
   POL(u22(x_1, x_2, x_3, x_4)) = [2]x_1*x_4 + [2]x_1^2
   POL(u31(x_1, x_2, x_3)) = 0
   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
   POL(u41(x_1, x_2)) = x_1 + x_1^2
   POL(u42(x_1, x_2, x_3)) = [2]x_1

----------------------------------------

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   din(der(plus(z0, z1))) -> u21(din(der(z0)), z0, z1)
   din(der(times(z0, z1))) -> u31(din(der(z0)), z0, z1)
   din(der(der(z0))) -> u41(din(der(z0)), z0)
   u21(dout(z0), z1, z2) -> u22(din(der(z2)), z1, z2, z0)
   u22(dout(z0), z1, z2, z3) -> dout(plus(z3, z0))
   u31(dout(z0), z1, z2) -> u32(din(der(z2)), z1, z2, z0)
   u32(dout(z0), z1, z2, z3) -> dout(plus(times(z1, z0), times(z2, z3)))
   u41(dout(z0), z1) -> u42(din(der(z0)), z1, z0)
   u42(dout(z0), z1, z2) -> dout(z0)
Tuples:
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
S tuples:none
K tuples:
   U31(dout(z0), z1, z2) -> c5(DIN(der(z2)))
   U21(dout(z0), z1, z2) -> c3(DIN(der(z2)))
   U41(dout(z0), z1) -> c7(DIN(der(z0)))
   DIN(der(plus(z0, z1))) -> c(U21(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(times(z0, z1))) -> c1(U31(din(der(z0)), z0, z1), DIN(der(z0)))
   DIN(der(der(z0))) -> c2(U41(din(der(z0)), z0), DIN(der(z0)))
Defined Rule Symbols:   din_1, u21_3, u22_4, u31_3, u32_4, u41_2, u42_3

Defined Pair Symbols:   DIN_1, U21_3, U31_3, U41_2

Compound Symbols:   c_2, c1_2, c2_2, c3_1, c5_1, c7_1


----------------------------------------

(17) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(18)
BOUNDS(1, 1)

----------------------------------------

(19) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(20)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   din(der(plus(X, Y))) -> u21(din(der(X)), X, Y)
   u21(dout(DX), X, Y) -> u22(din(der(Y)), X, Y, DX)
   u22(dout(DY), X, Y, DX) -> dout(plus(DX, DY))
   din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
   u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
   u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
   din(der(der(X))) -> u41(din(der(X)), X)
   u41(dout(DX), X) -> u42(din(der(DX)), X, DX)
   u42(dout(DDX), X, DX) -> dout(DDX)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(21) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
u21/1
u22/1
u22/2
u41/1
u42/1
u42/2

----------------------------------------

(22)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   din(der(plus(X, Y))) -> u21(din(der(X)), Y)
   u21(dout(DX), Y) -> u22(din(der(Y)), DX)
   u22(dout(DY), DX) -> dout(plus(DX, DY))
   din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
   u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
   u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
   din(der(der(X))) -> u41(din(der(X)))
   u41(dout(DX)) -> u42(din(der(DX)))
   u42(dout(DDX)) -> dout(DDX)

S is empty.
Rewrite Strategy: INNERMOST
----------------------------------------

(23) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
din(der(plus(X, Y))) -> u21(din(der(X)), Y)
u21(dout(DX), Y) -> u22(din(der(Y)), DX)
u22(dout(DY), DX) -> dout(plus(DX, DY))
din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) -> u41(din(der(X)))
u41(dout(DX)) -> u42(din(der(DX)))
u42(dout(DDX)) -> dout(DDX)

Types:
din :: plus:der:times -> dout
der :: plus:der:times -> plus:der:times
plus :: plus:der:times -> plus:der:times -> plus:der:times
u21 :: dout -> plus:der:times -> dout
dout :: plus:der:times -> dout
u22 :: dout -> plus:der:times -> dout
times :: plus:der:times -> plus:der:times -> plus:der:times
u31 :: dout -> plus:der:times -> plus:der:times -> dout
u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
u41 :: dout -> dout
u42 :: dout -> dout
hole_dout1_0 :: dout
hole_plus:der:times2_0 :: plus:der:times
gen_plus:der:times3_0 :: Nat -> plus:der:times

----------------------------------------

(25) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
din, u41

They will be analysed ascendingly in the following order:
din = u41

----------------------------------------

(26)
Obligation:
Innermost TRS:
Rules:
din(der(plus(X, Y))) -> u21(din(der(X)), Y)
u21(dout(DX), Y) -> u22(din(der(Y)), DX)
u22(dout(DY), DX) -> dout(plus(DX, DY))
din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) -> u41(din(der(X)))
u41(dout(DX)) -> u42(din(der(DX)))
u42(dout(DDX)) -> dout(DDX)

Types:
din :: plus:der:times -> dout
der :: plus:der:times -> plus:der:times
plus :: plus:der:times -> plus:der:times -> plus:der:times
u21 :: dout -> plus:der:times -> dout
dout :: plus:der:times -> dout
u22 :: dout -> plus:der:times -> dout
times :: plus:der:times -> plus:der:times -> plus:der:times
u31 :: dout -> plus:der:times -> plus:der:times -> dout
u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
u41 :: dout -> dout
u42 :: dout -> dout
hole_dout1_0 :: dout
hole_plus:der:times2_0 :: plus:der:times
gen_plus:der:times3_0 :: Nat -> plus:der:times


Generator Equations:
gen_plus:der:times3_0(0) <=> hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) <=> der(gen_plus:der:times3_0(x))


The following defined symbols remain to be analysed:
u41, din

They will be analysed ascendingly in the following order:
din = u41

----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
din(gen_plus:der:times3_0(+(2, n29_0))) -> *4_0, rt in Omega(n29_0)

Induction Base:
din(gen_plus:der:times3_0(+(2, 0)))

Induction Step:
din(gen_plus:der:times3_0(+(2, +(n29_0, 1)))) ->_R^Omega(1)
u41(din(der(gen_plus:der:times3_0(+(1, n29_0))))) ->_IH
u41(*4_0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(28)
Complex Obligation (BEST)

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(29)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
din(der(plus(X, Y))) -> u21(din(der(X)), Y)
u21(dout(DX), Y) -> u22(din(der(Y)), DX)
u22(dout(DY), DX) -> dout(plus(DX, DY))
din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) -> u41(din(der(X)))
u41(dout(DX)) -> u42(din(der(DX)))
u42(dout(DDX)) -> dout(DDX)

Types:
din :: plus:der:times -> dout
der :: plus:der:times -> plus:der:times
plus :: plus:der:times -> plus:der:times -> plus:der:times
u21 :: dout -> plus:der:times -> dout
dout :: plus:der:times -> dout
u22 :: dout -> plus:der:times -> dout
times :: plus:der:times -> plus:der:times -> plus:der:times
u31 :: dout -> plus:der:times -> plus:der:times -> dout
u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
u41 :: dout -> dout
u42 :: dout -> dout
hole_dout1_0 :: dout
hole_plus:der:times2_0 :: plus:der:times
gen_plus:der:times3_0 :: Nat -> plus:der:times


Generator Equations:
gen_plus:der:times3_0(0) <=> hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) <=> der(gen_plus:der:times3_0(x))


The following defined symbols remain to be analysed:
din

They will be analysed ascendingly in the following order:
din = u41

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(30) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(31)
BOUNDS(n^1, INF)

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(32)
Obligation:
Innermost TRS:
Rules:
din(der(plus(X, Y))) -> u21(din(der(X)), Y)
u21(dout(DX), Y) -> u22(din(der(Y)), DX)
u22(dout(DY), DX) -> dout(plus(DX, DY))
din(der(times(X, Y))) -> u31(din(der(X)), X, Y)
u31(dout(DX), X, Y) -> u32(din(der(Y)), X, Y, DX)
u32(dout(DY), X, Y, DX) -> dout(plus(times(X, DY), times(Y, DX)))
din(der(der(X))) -> u41(din(der(X)))
u41(dout(DX)) -> u42(din(der(DX)))
u42(dout(DDX)) -> dout(DDX)

Types:
din :: plus:der:times -> dout
der :: plus:der:times -> plus:der:times
plus :: plus:der:times -> plus:der:times -> plus:der:times
u21 :: dout -> plus:der:times -> dout
dout :: plus:der:times -> dout
u22 :: dout -> plus:der:times -> dout
times :: plus:der:times -> plus:der:times -> plus:der:times
u31 :: dout -> plus:der:times -> plus:der:times -> dout
u32 :: dout -> plus:der:times -> plus:der:times -> plus:der:times -> dout
u41 :: dout -> dout
u42 :: dout -> dout
hole_dout1_0 :: dout
hole_plus:der:times2_0 :: plus:der:times
gen_plus:der:times3_0 :: Nat -> plus:der:times


Lemmas:
din(gen_plus:der:times3_0(+(2, n29_0))) -> *4_0, rt in Omega(n29_0)


Generator Equations:
gen_plus:der:times3_0(0) <=> hole_plus:der:times2_0
gen_plus:der:times3_0(+(x, 1)) <=> der(gen_plus:der:times3_0(x))


The following defined symbols remain to be analysed:
u41

They will be analysed ascendingly in the following order:
din = u41