WORST_CASE(Omega(n^1),O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 64 ms]
    (8) CdtProblem
    (9) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) BOUNDS(1, 1)
(11) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxTRS
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) typed CpxTrs
    (15) OrderProof [LOWER BOUND(ID), 0 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 1585 ms]
    (18) proven lower bound
    (19) LowerBoundPropagationProof [FINISHED, 0 ms]
    (20) BOUNDS(n^1, INF)


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

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


The TRS R consists of the following rules:

   D(t) -> 1
   D(constant) -> 0
   D(+(x, y)) -> +(D(x), D(y))
   D(*(x, y)) -> +(*(y, D(x)), *(x, D(y)))
   D(-(x, y)) -> -(D(x), D(y))
   D(minus(x)) -> minus(D(x))
   D(div(x, y)) -> -(div(D(x), y), div(*(x, D(y)), pow(y, 2)))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> +(*(*(y, pow(x, -(y, 1))), D(x)), *(*(pow(x, y), ln(x)), D(y)))

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   D(t) -> 1
   D(constant) -> 0
   D(+(z0, z1)) -> +(D(z0), D(z1))
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1)))
   D(-(z0, z1)) -> -(D(z0), D(z1))
   D(minus(z0)) -> minus(D(z0))
   D(div(z0, z1)) -> -(div(D(z0), z1), div(*(z0, D(z1)), pow(z1, 2)))
   D(ln(z0)) -> div(D(z0), z0)
   D(pow(z0, z1)) -> +(*(*(z1, pow(z0, -(z1, 1))), D(z0)), *(*(pow(z0, z1), ln(z0)), D(z1)))
Tuples:
   D'(t) -> c
   D'(constant) -> c1
   D'(+(z0, z1)) -> c2(D'(z0), D'(z1))
   D'(*(z0, z1)) -> c3(D'(z0), D'(z1))
   D'(-(z0, z1)) -> c4(D'(z0), D'(z1))
   D'(minus(z0)) -> c5(D'(z0))
   D'(div(z0, z1)) -> c6(D'(z0), D'(z1))
   D'(ln(z0)) -> c7(D'(z0))
   D'(pow(z0, z1)) -> c8(D'(z0), D'(z1))
S tuples:
   D'(t) -> c
   D'(constant) -> c1
   D'(+(z0, z1)) -> c2(D'(z0), D'(z1))
   D'(*(z0, z1)) -> c3(D'(z0), D'(z1))
   D'(-(z0, z1)) -> c4(D'(z0), D'(z1))
   D'(minus(z0)) -> c5(D'(z0))
   D'(div(z0, z1)) -> c6(D'(z0), D'(z1))
   D'(ln(z0)) -> c7(D'(z0))
   D'(pow(z0, z1)) -> c8(D'(z0), D'(z1))
K tuples:none
Defined Rule Symbols:   D_1

Defined Pair Symbols:   D'_1

Compound Symbols:   c, c1, c2_2, c3_2, c4_2, c5_1, c6_2, c7_1, c8_2


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   D'(t) -> c
   D'(constant) -> c1

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   D(t) -> 1
   D(constant) -> 0
   D(+(z0, z1)) -> +(D(z0), D(z1))
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1)))
   D(-(z0, z1)) -> -(D(z0), D(z1))
   D(minus(z0)) -> minus(D(z0))
   D(div(z0, z1)) -> -(div(D(z0), z1), div(*(z0, D(z1)), pow(z1, 2)))
   D(ln(z0)) -> div(D(z0), z0)
   D(pow(z0, z1)) -> +(*(*(z1, pow(z0, -(z1, 1))), D(z0)), *(*(pow(z0, z1), ln(z0)), D(z1)))
Tuples:
   D'(+(z0, z1)) -> c2(D'(z0), D'(z1))
   D'(*(z0, z1)) -> c3(D'(z0), D'(z1))
   D'(-(z0, z1)) -> c4(D'(z0), D'(z1))
   D'(minus(z0)) -> c5(D'(z0))
   D'(div(z0, z1)) -> c6(D'(z0), D'(z1))
   D'(ln(z0)) -> c7(D'(z0))
   D'(pow(z0, z1)) -> c8(D'(z0), D'(z1))
S tuples:
   D'(+(z0, z1)) -> c2(D'(z0), D'(z1))
   D'(*(z0, z1)) -> c3(D'(z0), D'(z1))
   D'(-(z0, z1)) -> c4(D'(z0), D'(z1))
   D'(minus(z0)) -> c5(D'(z0))
   D'(div(z0, z1)) -> c6(D'(z0), D'(z1))
   D'(ln(z0)) -> c7(D'(z0))
   D'(pow(z0, z1)) -> c8(D'(z0), D'(z1))
K tuples:none
Defined Rule Symbols:   D_1

Defined Pair Symbols:   D'_1

Compound Symbols:   c2_2, c3_2, c4_2, c5_1, c6_2, c7_1, c8_2


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   D(t) -> 1
   D(constant) -> 0
   D(+(z0, z1)) -> +(D(z0), D(z1))
   D(*(z0, z1)) -> +(*(z1, D(z0)), *(z0, D(z1)))
   D(-(z0, z1)) -> -(D(z0), D(z1))
   D(minus(z0)) -> minus(D(z0))
   D(div(z0, z1)) -> -(div(D(z0), z1), div(*(z0, D(z1)), pow(z1, 2)))
   D(ln(z0)) -> div(D(z0), z0)
   D(pow(z0, z1)) -> +(*(*(z1, pow(z0, -(z1, 1))), D(z0)), *(*(pow(z0, z1), ln(z0)), D(z1)))

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   D'(+(z0, z1)) -> c2(D'(z0), D'(z1))
   D'(*(z0, z1)) -> c3(D'(z0), D'(z1))
   D'(-(z0, z1)) -> c4(D'(z0), D'(z1))
   D'(minus(z0)) -> c5(D'(z0))
   D'(div(z0, z1)) -> c6(D'(z0), D'(z1))
   D'(ln(z0)) -> c7(D'(z0))
   D'(pow(z0, z1)) -> c8(D'(z0), D'(z1))
S tuples:
   D'(+(z0, z1)) -> c2(D'(z0), D'(z1))
   D'(*(z0, z1)) -> c3(D'(z0), D'(z1))
   D'(-(z0, z1)) -> c4(D'(z0), D'(z1))
   D'(minus(z0)) -> c5(D'(z0))
   D'(div(z0, z1)) -> c6(D'(z0), D'(z1))
   D'(ln(z0)) -> c7(D'(z0))
   D'(pow(z0, z1)) -> c8(D'(z0), D'(z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   D'_1

Compound Symbols:   c2_2, c3_2, c4_2, c5_1, c6_2, c7_1, c8_2


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   D'(+(z0, z1)) -> c2(D'(z0), D'(z1))
   D'(*(z0, z1)) -> c3(D'(z0), D'(z1))
   D'(-(z0, z1)) -> c4(D'(z0), D'(z1))
   D'(minus(z0)) -> c5(D'(z0))
   D'(div(z0, z1)) -> c6(D'(z0), D'(z1))
   D'(ln(z0)) -> c7(D'(z0))
   D'(pow(z0, z1)) -> c8(D'(z0), D'(z1))
We considered the (Usable) Rules:none
And the Tuples:
   D'(+(z0, z1)) -> c2(D'(z0), D'(z1))
   D'(*(z0, z1)) -> c3(D'(z0), D'(z1))
   D'(-(z0, z1)) -> c4(D'(z0), D'(z1))
   D'(minus(z0)) -> c5(D'(z0))
   D'(div(z0, z1)) -> c6(D'(z0), D'(z1))
   D'(ln(z0)) -> c7(D'(z0))
   D'(pow(z0, z1)) -> c8(D'(z0), D'(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = [1] + x_1 + x_2
   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(-(x_1, x_2)) = [1] + x_1 + x_2
   POL(D'(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(div(x_1, x_2)) = [1] + x_1 + x_2
   POL(ln(x_1)) = [1] + x_1
   POL(minus(x_1)) = [1] + x_1
   POL(pow(x_1, x_2)) = [1] + x_1 + x_2

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   D'(+(z0, z1)) -> c2(D'(z0), D'(z1))
   D'(*(z0, z1)) -> c3(D'(z0), D'(z1))
   D'(-(z0, z1)) -> c4(D'(z0), D'(z1))
   D'(minus(z0)) -> c5(D'(z0))
   D'(div(z0, z1)) -> c6(D'(z0), D'(z1))
   D'(ln(z0)) -> c7(D'(z0))
   D'(pow(z0, z1)) -> c8(D'(z0), D'(z1))
S tuples:none
K tuples:
   D'(+(z0, z1)) -> c2(D'(z0), D'(z1))
   D'(*(z0, z1)) -> c3(D'(z0), D'(z1))
   D'(-(z0, z1)) -> c4(D'(z0), D'(z1))
   D'(minus(z0)) -> c5(D'(z0))
   D'(div(z0, z1)) -> c6(D'(z0), D'(z1))
   D'(ln(z0)) -> c7(D'(z0))
   D'(pow(z0, z1)) -> c8(D'(z0), D'(z1))
Defined Rule Symbols:none

Defined Pair Symbols:   D'_1

Compound Symbols:   c2_2, c3_2, c4_2, c5_1, c6_2, c7_1, c8_2


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

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

(10)
BOUNDS(1, 1)

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

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

(12)
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:

   D(t) -> 1'
   D(constant) -> 0'
   D(+'(x, y)) -> +'(D(x), D(y))
   D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y)))
   D(-(x, y)) -> -(D(x), D(y))
   D(minus(x)) -> minus(D(x))
   D(div(x, y)) -> -(div(D(x), y), div(*'(x, D(y)), pow(y, 2')))
   D(ln(x)) -> div(D(x), x)
   D(pow(x, y)) -> +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y)))

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

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

(14)
Obligation:
Innermost TRS:
Rules:
D(t) -> 1'
D(constant) -> 0'
D(+'(x, y)) -> +'(D(x), D(y))
D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) -> -(D(x), D(y))
D(minus(x)) -> minus(D(x))
D(div(x, y)) -> -(div(D(x), y), div(*'(x, D(y)), pow(y, 2')))
D(ln(x)) -> div(D(x), x)
D(pow(x, y)) -> +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y)))

Types:
D :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
t :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
1' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
constant :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
0' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
+' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
*' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
- :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
minus :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
div :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
pow :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
2' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
ln :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
hole_t:1':constant:0':+':*':-:minus:div:2':pow:ln1_0 :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0 :: Nat -> t:1':constant:0':+':*':-:minus:div:2':pow:ln

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

(15) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
D
----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
D(t) -> 1'
D(constant) -> 0'
D(+'(x, y)) -> +'(D(x), D(y))
D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) -> -(D(x), D(y))
D(minus(x)) -> minus(D(x))
D(div(x, y)) -> -(div(D(x), y), div(*'(x, D(y)), pow(y, 2')))
D(ln(x)) -> div(D(x), x)
D(pow(x, y)) -> +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y)))

Types:
D :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
t :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
1' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
constant :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
0' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
+' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
*' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
- :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
minus :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
div :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
pow :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
2' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
ln :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
hole_t:1':constant:0':+':*':-:minus:div:2':pow:ln1_0 :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0 :: Nat -> t:1':constant:0':+':*':-:minus:div:2':pow:ln


Generator Equations:
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(0) <=> t
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(+(x, 1)) <=> +'(t, gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(x))


The following defined symbols remain to be analysed:
D
----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0)) -> *3_0, rt in Omega(n4_0)

Induction Base:
D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(0))

Induction Step:
D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(+(n4_0, 1))) ->_R^Omega(1)
+'(D(t), D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0))) ->_R^Omega(1)
+'(1', D(gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(n4_0))) ->_IH
+'(1', *3_0)

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

(18)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
D(t) -> 1'
D(constant) -> 0'
D(+'(x, y)) -> +'(D(x), D(y))
D(*'(x, y)) -> +'(*'(y, D(x)), *'(x, D(y)))
D(-(x, y)) -> -(D(x), D(y))
D(minus(x)) -> minus(D(x))
D(div(x, y)) -> -(div(D(x), y), div(*'(x, D(y)), pow(y, 2')))
D(ln(x)) -> div(D(x), x)
D(pow(x, y)) -> +'(*'(*'(y, pow(x, -(y, 1'))), D(x)), *'(*'(pow(x, y), ln(x)), D(y)))

Types:
D :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
t :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
1' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
constant :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
0' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
+' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
*' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
- :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
minus :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
div :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
pow :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
2' :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
ln :: t:1':constant:0':+':*':-:minus:div:2':pow:ln -> t:1':constant:0':+':*':-:minus:div:2':pow:ln
hole_t:1':constant:0':+':*':-:minus:div:2':pow:ln1_0 :: t:1':constant:0':+':*':-:minus:div:2':pow:ln
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0 :: Nat -> t:1':constant:0':+':*':-:minus:div:2':pow:ln


Generator Equations:
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(0) <=> t
gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(+(x, 1)) <=> +'(t, gen_t:1':constant:0':+':*':-:minus:div:2':pow:ln2_0(x))


The following defined symbols remain to be analysed:
D
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(19) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(20)
BOUNDS(n^1, INF)