/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^1))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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), 11 ms]
(2) CpxTRS
    (3) NonCtorToCtorProof [UPPER BOUND(ID), 0 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)), 63 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 17 ms]
    (18) CdtProblem
    (19) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) BOUNDS(1, 1)
(21) RenamingProof [BOTH BOUNDS(ID, 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), 553 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 (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:
   D'(constant) -> c7
   B(z0, h) -> c3
   B(z0, z1) -> c1
   D'(t) -> c6
   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


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

(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

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

(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


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

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

(20)
BOUNDS(1, 1)

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

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

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


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

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

(24)
Obligation:
TRS:
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))

Types:
D :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
t :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
s :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
h :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
constant :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
b :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
c :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
m :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
opp :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
div :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
pow :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
2' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
ln :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
1' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
hole_t:h:s:constant:c:m:opp:div:2':pow:ln:1'1_0 :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0 :: Nat -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'

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

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

They will be analysed ascendingly in the following order:
b < D

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

(26)
Obligation:
TRS:
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))

Types:
D :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
t :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
s :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
h :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
constant :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
b :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
c :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
m :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
opp :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
div :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
pow :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
2' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
ln :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
1' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
hole_t:h:s:constant:c:m:opp:div:2':pow:ln:1'1_0 :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0 :: Nat -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'


Generator Equations:
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(0) <=> t
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(x, 1)) <=> s(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(x))


The following defined symbols remain to be analysed:
b, D

They will be analysed ascendingly in the following order:
b < D

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)

Induction Base:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, 0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, 0)))

Induction Step:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, +(n4_0, 1))), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, +(n4_0, 1)))) ->_R^Omega(1)
s(s(b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0))))) ->_IH
s(s(*3_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(28)
Complex Obligation (BEST)

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

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

TRS:
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))

Types:
D :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
t :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
s :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
h :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
constant :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
b :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
c :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
m :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
opp :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
div :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
pow :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
2' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
ln :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
1' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
hole_t:h:s:constant:c:m:opp:div:2':pow:ln:1'1_0 :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0 :: Nat -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'


Generator Equations:
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(0) <=> t
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(x, 1)) <=> s(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(x))


The following defined symbols remain to be analysed:
b, D

They will be analysed ascendingly in the following order:
b < D

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

(30) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(31)
BOUNDS(n^1, INF)

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

(32)
Obligation:
TRS:
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))

Types:
D :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
t :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
s :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
h :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
constant :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
b :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
c :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
m :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
opp :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
div :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
pow :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
2' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
ln :: t:h:s:constant:c:m:opp:div:2':pow:ln:1' -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'
1' :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
hole_t:h:s:constant:c:m:opp:div:2':pow:ln:1'1_0 :: t:h:s:constant:c:m:opp:div:2':pow:ln:1'
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0 :: Nat -> t:h:s:constant:c:m:opp:div:2':pow:ln:1'


Lemmas:
b(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0)), gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(1, n4_0))) -> *3_0, rt in Omega(n4_0)


Generator Equations:
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(0) <=> t
gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(+(x, 1)) <=> s(gen_t:h:s:constant:c:m:opp:div:2':pow:ln:1'2_0(x))


The following defined symbols remain to be analysed:
D