/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 13 ms]
(2) CpxTRS
    (3) NonCtorToCtorProof [UPPER BOUND(ID), 1 ms]
    (4) CpxRelTRS
    (5) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (8) CdtProblem
    (9) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CdtProblem
    (13) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 90 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms]
    (18) CdtProblem
    (19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) BOUNDS(1, 1)
(21) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(22) TRS for Loop Detection
    (23) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (24) BEST
        (25) proven lower bound
            (26) LowerBoundPropagationProof [FINISHED, 0 ms]
            (27) BOUNDS(n^1, INF)
        (28) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   D(t) -> s(h)
   D(constant) -> h
   D(b(x, y)) -> b(D(x), D(y))
   D(c(x, y)) -> b(c(y, D(x)), c(x, D(y)))
   D(m(x, y)) -> m(D(x), D(y))
   D(opp(x)) -> opp(D(x))
   D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
   b(h, x) -> x
   b(x, h) -> x
   b(s(x), s(y)) -> s(s(b(x, y)))
   b(b(x, y), z) -> b(x, b(y, z))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of b: b, D
The following defined symbols can occur below the 1th argument of b: b, D

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
D(b(x, y)) -> b(D(x), D(y))

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

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


The TRS R consists of the following rules:

   D(t) -> s(h)
   D(constant) -> h
   D(c(x, y)) -> b(c(y, D(x)), c(x, D(y)))
   D(m(x, y)) -> m(D(x), D(y))
   D(opp(x)) -> opp(D(x))
   D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
   b(h, x) -> x
   b(x, h) -> x
   b(s(x), s(y)) -> s(s(b(x, y)))
   b(b(x, y), z) -> b(x, b(y, z))

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

(3) NonCtorToCtorProof (UPPER BOUND(ID))
transformed non-ctor to ctor-system
----------------------------------------

(4)
Obligation:
The Runtime Complexity (full) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   D(t) -> s(h)
   D(constant) -> h
   D(c(x, y)) -> b(c(y, D(x)), c(x, D(y)))
   D(m(x, y)) -> m(D(x), D(y))
   D(opp(x)) -> opp(D(x))
   D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
   b(h, x) -> x
   b(x, h) -> x
   b(s(x), s(y)) -> s(s(b(x, y)))
   b(c_b(x, y), z) -> b(x, b(y, z))

The (relative) TRS S consists of the following rules:

   b(x0, x1) -> c_b(x0, x1)

Rewrite Strategy: FULL
----------------------------------------

(5) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
D(c(x, []))
D(c([], y))
D(div([], y))
D(div(x, []))
D(ln([]))
D(pow(x, []))
D(pow([], y))


The defined contexts are:
b(x0, [])
b(c(x0, []), c(x2, x3))
b(c(x0, x1), c(x2, []))
b(c(c(x0, pow(x1, m(x2, 1))), []), c(c(pow(x4, x5), ln(x6)), x7))
b(c(c(x0, pow(x1, m(x2, 1))), x3), c(c(pow(x4, x5), ln(x6)), []))


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   D(t) -> s(h)
   D(constant) -> h
   D(c(x, y)) -> b(c(y, D(x)), c(x, D(y)))
   D(m(x, y)) -> m(D(x), D(y))
   D(opp(x)) -> opp(D(x))
   D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
   b(h, x) -> x
   b(x, h) -> x
   b(s(x), s(y)) -> s(s(b(x, y)))
   b(c_b(x, y), z) -> b(x, b(y, z))

The (relative) TRS S consists of the following rules:

   b(x0, x1) -> c_b(x0, x1)

Rewrite Strategy: INNERMOST
----------------------------------------

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   b(z0, z1) -> c_b(z0, z1)
   b(h, z0) -> z0
   b(z0, h) -> z0
   b(s(z0), s(z1)) -> s(s(b(z0, z1)))
   b(c_b(z0, z1), z2) -> b(z0, b(z1, z2))
   D(t) -> s(h)
   D(constant) -> h
   D(c(z0, z1)) -> b(c(z1, D(z0)), c(z0, D(z1)))
   D(m(z0, z1)) -> m(D(z0), D(z1))
   D(opp(z0)) -> opp(D(z0))
   D(div(z0, z1)) -> m(div(D(z0), z1), div(c(z0, D(z1)), pow(z1, 2)))
   D(ln(z0)) -> div(D(z0), z0)
   D(pow(z0, z1)) -> b(c(c(z1, pow(z0, m(z1, 1))), D(z0)), c(c(pow(z0, z1), ln(z0)), D(z1)))
Tuples:
   B(z0, z1) -> c1
   B(h, z0) -> c2
   B(z0, h) -> c3
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(t) -> c6
   D'(constant) -> c7
   D'(c(z0, z1)) -> c8(B(c(z1, D(z0)), c(z0, D(z1))), D'(z0), D'(z1))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(pow(z0, z1)) -> c13(B(c(c(z1, pow(z0, m(z1, 1))), D(z0)), c(c(pow(z0, z1), ln(z0)), D(z1))), D'(z0), D'(z1))
S tuples:
   B(h, z0) -> c2
   B(z0, h) -> c3
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(t) -> c6
   D'(constant) -> c7
   D'(c(z0, z1)) -> c8(B(c(z1, D(z0)), c(z0, D(z1))), D'(z0), D'(z1))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(pow(z0, z1)) -> c13(B(c(c(z1, pow(z0, m(z1, 1))), D(z0)), c(c(pow(z0, z1), ln(z0)), D(z1))), D'(z0), D'(z1))
K tuples:none
Defined Rule Symbols:   D_1, b_2

Defined Pair Symbols:   B_2, D'_1

Compound Symbols:   c1, c2, c3, c4_1, c5_2, c6, c7, c8_3, c9_2, c10_1, c11_2, c12_1, c13_3


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

(9) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   B(z0, h) -> c3
   D'(t) -> c6
   B(z0, z1) -> c1
   D'(constant) -> c7
   B(h, z0) -> c2

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   b(z0, z1) -> c_b(z0, z1)
   b(h, z0) -> z0
   b(z0, h) -> z0
   b(s(z0), s(z1)) -> s(s(b(z0, z1)))
   b(c_b(z0, z1), z2) -> b(z0, b(z1, z2))
   D(t) -> s(h)
   D(constant) -> h
   D(c(z0, z1)) -> b(c(z1, D(z0)), c(z0, D(z1)))
   D(m(z0, z1)) -> m(D(z0), D(z1))
   D(opp(z0)) -> opp(D(z0))
   D(div(z0, z1)) -> m(div(D(z0), z1), div(c(z0, D(z1)), pow(z1, 2)))
   D(ln(z0)) -> div(D(z0), z0)
   D(pow(z0, z1)) -> b(c(c(z1, pow(z0, m(z1, 1))), D(z0)), c(c(pow(z0, z1), ln(z0)), D(z1)))
Tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(c(z0, z1)) -> c8(B(c(z1, D(z0)), c(z0, D(z1))), D'(z0), D'(z1))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(pow(z0, z1)) -> c13(B(c(c(z1, pow(z0, m(z1, 1))), D(z0)), c(c(pow(z0, z1), ln(z0)), D(z1))), D'(z0), D'(z1))
S tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(c(z0, z1)) -> c8(B(c(z1, D(z0)), c(z0, D(z1))), D'(z0), D'(z1))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(pow(z0, z1)) -> c13(B(c(c(z1, pow(z0, m(z1, 1))), D(z0)), c(c(pow(z0, z1), ln(z0)), D(z1))), D'(z0), D'(z1))
K tuples:none
Defined Rule Symbols:   D_1, b_2

Defined Pair Symbols:   B_2, D'_1

Compound Symbols:   c4_1, c5_2, c8_3, c9_2, c10_1, c11_2, c12_1, c13_3


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

(11) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   b(z0, z1) -> c_b(z0, z1)
   b(h, z0) -> z0
   b(z0, h) -> z0
   b(s(z0), s(z1)) -> s(s(b(z0, z1)))
   b(c_b(z0, z1), z2) -> b(z0, b(z1, z2))
   D(t) -> s(h)
   D(constant) -> h
   D(c(z0, z1)) -> b(c(z1, D(z0)), c(z0, D(z1)))
   D(m(z0, z1)) -> m(D(z0), D(z1))
   D(opp(z0)) -> opp(D(z0))
   D(div(z0, z1)) -> m(div(D(z0), z1), div(c(z0, D(z1)), pow(z1, 2)))
   D(ln(z0)) -> div(D(z0), z0)
   D(pow(z0, z1)) -> b(c(c(z1, pow(z0, m(z1, 1))), D(z0)), c(c(pow(z0, z1), ln(z0)), D(z1)))
Tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
S tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
K tuples:none
Defined Rule Symbols:   D_1, b_2

Defined Pair Symbols:   B_2, D'_1

Compound Symbols:   c4_1, c5_2, c9_2, c10_1, c11_2, c12_1, c8_2, c13_2


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

(13) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   D(t) -> s(h)
   D(constant) -> h
   D(c(z0, z1)) -> b(c(z1, D(z0)), c(z0, D(z1)))
   D(m(z0, z1)) -> m(D(z0), D(z1))
   D(opp(z0)) -> opp(D(z0))
   D(div(z0, z1)) -> m(div(D(z0), z1), div(c(z0, D(z1)), pow(z1, 2)))
   D(ln(z0)) -> div(D(z0), z0)
   D(pow(z0, z1)) -> b(c(c(z1, pow(z0, m(z1, 1))), D(z0)), c(c(pow(z0, z1), ln(z0)), D(z1)))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   b(z0, z1) -> c_b(z0, z1)
   b(h, z0) -> z0
   b(z0, h) -> z0
   b(s(z0), s(z1)) -> s(s(b(z0, z1)))
   b(c_b(z0, z1), z2) -> b(z0, b(z1, z2))
Tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
S tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
K tuples:none
Defined Rule Symbols:   b_2

Defined Pair Symbols:   B_2, D'_1

Compound Symbols:   c4_1, c5_2, c9_2, c10_1, c11_2, c12_1, c8_2, c13_2


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

(15) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
We considered the (Usable) Rules:none
And the Tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(B(x_1, x_2)) = x_1
   POL(D'(x_1)) = x_1
   POL(b(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1, x_2)) = [1] + x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(c_b(x_1, x_2)) = [1] + x_1 + x_2
   POL(div(x_1, x_2)) = [1] + x_1 + x_2
   POL(h) = 0
   POL(ln(x_1)) = [1] + x_1
   POL(m(x_1, x_2)) = [1] + x_1 + x_2
   POL(opp(x_1)) = [1] + x_1
   POL(pow(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = x_1

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   b(z0, z1) -> c_b(z0, z1)
   b(h, z0) -> z0
   b(z0, h) -> z0
   b(s(z0), s(z1)) -> s(s(b(z0, z1)))
   b(c_b(z0, z1), z2) -> b(z0, b(z1, z2))
Tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
S tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
K tuples:
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
Defined Rule Symbols:   b_2

Defined Pair Symbols:   B_2, D'_1

Compound Symbols:   c4_1, c5_2, c9_2, c10_1, c11_2, c12_1, c8_2, c13_2


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(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   B(s(z0), s(z1)) -> c4(B(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(B(x_1, x_2)) = [1] + x_1
   POL(D'(x_1)) = 0
   POL(b(x_1, x_2)) = [1] + x_1 + x_2
   POL(c(x_1, x_2)) = [1] + x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(c_b(x_1, x_2)) = [1] + x_1 + x_2
   POL(div(x_1, x_2)) = [1] + x_1 + x_2
   POL(h) = 0
   POL(ln(x_1)) = x_1
   POL(m(x_1, x_2)) = [1] + x_1 + x_2
   POL(opp(x_1)) = x_1
   POL(pow(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1] + x_1

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(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   b(z0, z1) -> c_b(z0, z1)
   b(h, z0) -> z0
   b(z0, h) -> z0
   b(s(z0), s(z1)) -> s(s(b(z0, z1)))
   b(c_b(z0, z1), z2) -> b(z0, b(z1, z2))
Tuples:
   B(s(z0), s(z1)) -> c4(B(z0, z1))
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
S tuples:none
K tuples:
   B(c_b(z0, z1), z2) -> c5(B(z0, b(z1, z2)), B(z1, z2))
   D'(m(z0, z1)) -> c9(D'(z0), D'(z1))
   D'(opp(z0)) -> c10(D'(z0))
   D'(div(z0, z1)) -> c11(D'(z0), D'(z1))
   D'(ln(z0)) -> c12(D'(z0))
   D'(c(z0, z1)) -> c8(D'(z0), D'(z1))
   D'(pow(z0, z1)) -> c13(D'(z0), D'(z1))
   B(s(z0), s(z1)) -> c4(B(z0, z1))
Defined Rule Symbols:   b_2

Defined Pair Symbols:   B_2, D'_1

Compound Symbols:   c4_1, c5_2, c9_2, c10_1, c11_2, c12_1, c8_2, c13_2


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(19) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(20)
BOUNDS(1, 1)

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(21) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(22)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   D(t) -> s(h)
   D(constant) -> h
   D(b(x, y)) -> b(D(x), D(y))
   D(c(x, y)) -> b(c(y, D(x)), c(x, D(y)))
   D(m(x, y)) -> m(D(x), D(y))
   D(opp(x)) -> opp(D(x))
   D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
   b(h, x) -> x
   b(x, h) -> x
   b(s(x), s(y)) -> s(s(b(x, y)))
   b(b(x, y), z) -> b(x, b(y, z))

S is empty.
Rewrite Strategy: FULL
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(23) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

D(div(x, y)) ->^+ m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0].

The pumping substitution is [x / div(x, y)].

The result substitution is [ ].




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(24)
Complex Obligation (BEST)

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(25)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   D(t) -> s(h)
   D(constant) -> h
   D(b(x, y)) -> b(D(x), D(y))
   D(c(x, y)) -> b(c(y, D(x)), c(x, D(y)))
   D(m(x, y)) -> m(D(x), D(y))
   D(opp(x)) -> opp(D(x))
   D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
   b(h, x) -> x
   b(x, h) -> x
   b(s(x), s(y)) -> s(s(b(x, y)))
   b(b(x, y), z) -> b(x, b(y, z))

S is empty.
Rewrite Strategy: FULL
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(26) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(27)
BOUNDS(n^1, INF)

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(28)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   D(t) -> s(h)
   D(constant) -> h
   D(b(x, y)) -> b(D(x), D(y))
   D(c(x, y)) -> b(c(y, D(x)), c(x, D(y)))
   D(m(x, y)) -> m(D(x), D(y))
   D(opp(x)) -> opp(D(x))
   D(div(x, y)) -> m(div(D(x), y), div(c(x, D(y)), pow(y, 2)))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> b(c(c(y, pow(x, m(y, 1))), D(x)), c(c(pow(x, y), ln(x)), D(y)))
   b(h, x) -> x
   b(x, h) -> x
   b(s(x), s(y)) -> s(s(b(x, y)))
   b(b(x, y), z) -> b(x, b(y, z))

S is empty.
Rewrite Strategy: FULL