/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^3), O(n^3))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3, n^3).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 1 ms]
    (4) CdtProblem
    (5) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 88 ms]
    (6) CdtProblem
    (7) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 80 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 522 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTRS
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) typed CpxTrs
    (17) OrderProof [LOWER BOUND(ID), 0 ms]
    (18) typed CpxTrs
    (19) RewriteLemmaProof [LOWER BOUND(ID), 291 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 1580 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 69 ms]
            (28) proven lower bound
            (29) LowerBoundPropagationProof [FINISHED, 0 ms]
            (30) BOUNDS(n^3, INF)


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, n^3).


The TRS R consists of the following rules:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   +(p(x), y) -> p(+(x, y))
   minus(0) -> 0
   minus(s(x)) -> p(minus(x))
   minus(p(x)) -> s(minus(x))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   *(p(x), y) -> +(*(x, y), minus(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:
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   +(p(z0), z1) -> p(+(z0, z1))
   minus(0) -> 0
   minus(s(z0)) -> p(minus(z0))
   minus(p(z0)) -> s(minus(z0))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   *(p(z0), z1) -> +(*(z0, z1), minus(z1))
Tuples:
   +'(0, z0) -> c
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(0) -> c3
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(0, z0) -> c6
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
S tuples:
   +'(0, z0) -> c
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(0) -> c3
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(0, z0) -> c6
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
K tuples:none
Defined Rule Symbols:   +_2, minus_1, *_2

Defined Pair Symbols:   +'_2, MINUS_1, *'_2

Compound Symbols:   c, c1_1, c2_1, c3, c4_1, c5_1, c6, c7_2, c8_3


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   *'(0, z0) -> c6
   MINUS(0) -> c3
   +'(0, z0) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   +(p(z0), z1) -> p(+(z0, z1))
   minus(0) -> 0
   minus(s(z0)) -> p(minus(z0))
   minus(p(z0)) -> s(minus(z0))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   *(p(z0), z1) -> +(*(z0, z1), minus(z1))
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
K tuples:none
Defined Rule Symbols:   +_2, minus_1, *_2

Defined Pair Symbols:   +'_2, MINUS_1, *'_2

Compound Symbols:   c1_1, c2_1, c4_1, c5_1, c7_2, c8_3


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

(5) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
We considered the (Usable) Rules:none
And the Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(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)) = x_1
   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(+'(x_1, x_2)) = 0
   POL(0) = [1]
   POL(MINUS(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(minus(x_1)) = [1] + x_1
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   +(p(z0), z1) -> p(+(z0, z1))
   minus(0) -> 0
   minus(s(z0)) -> p(minus(z0))
   minus(p(z0)) -> s(minus(z0))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   *(p(z0), z1) -> +(*(z0, z1), minus(z1))
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
K tuples:
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
Defined Rule Symbols:   +_2, minus_1, *_2

Defined Pair Symbols:   +'_2, MINUS_1, *'_2

Compound Symbols:   c1_1, c2_1, c4_1, c5_1, c7_2, c8_3


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

(7) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
We considered the (Usable) Rules:none
And the Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = x_2^2
   POL(*'(x_1, x_2)) = x_1*x_2
   POL(+(x_1, x_2)) = [1]
   POL(+'(x_1, x_2)) = 0
   POL(0) = 0
   POL(MINUS(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(minus(x_1)) = 0
   POL(p(x_1)) = [2] + x_1
   POL(s(x_1)) = [1] + x_1

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   +(p(z0), z1) -> p(+(z0, z1))
   minus(0) -> 0
   minus(s(z0)) -> p(minus(z0))
   minus(p(z0)) -> s(minus(z0))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   *(p(z0), z1) -> +(*(z0, z1), minus(z1))
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
Defined Rule Symbols:   +_2, minus_1, *_2

Defined Pair Symbols:   +'_2, MINUS_1, *'_2

Compound Symbols:   c1_1, c2_1, c4_1, c5_1, c7_2, c8_3


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
We considered the (Usable) Rules:
   +(s(z0), z1) -> s(+(z0, z1))
   *(p(z0), z1) -> +(*(z0, z1), minus(z1))
   minus(0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   minus(p(z0)) -> s(minus(z0))
   *(0, z0) -> 0
   minus(s(z0)) -> p(minus(z0))
   +(p(z0), z1) -> p(+(z0, z1))
   +(0, z0) -> z0
And the Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = x_1*x_2
   POL(*'(x_1, x_2)) = x_1^2*x_2
   POL(+(x_1, x_2)) = x_1 + x_2
   POL(+'(x_1, x_2)) = x_1
   POL(0) = 0
   POL(MINUS(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(minus(x_1)) = x_1
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   +(p(z0), z1) -> p(+(z0, z1))
   minus(0) -> 0
   minus(s(z0)) -> p(minus(z0))
   minus(p(z0)) -> s(minus(z0))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   *(p(z0), z1) -> +(*(z0, z1), minus(z1))
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
S tuples:none
K tuples:
   *'(s(z0), z1) -> c7(+'(*(z0, z1), z1), *'(z0, z1))
   *'(p(z0), z1) -> c8(+'(*(z0, z1), minus(z1)), *'(z0, z1), MINUS(z1))
   MINUS(s(z0)) -> c4(MINUS(z0))
   MINUS(p(z0)) -> c5(MINUS(z0))
   +'(s(z0), z1) -> c1(+'(z0, z1))
   +'(p(z0), z1) -> c2(+'(z0, z1))
Defined Rule Symbols:   +_2, minus_1, *_2

Defined Pair Symbols:   +'_2, MINUS_1, *'_2

Compound Symbols:   c1_1, c2_1, c4_1, c5_1, c7_2, c8_3


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

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

(12)
BOUNDS(1, 1)

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

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^3, INF).


The TRS R consists of the following rules:

   +'(0', y) -> y
   +'(s(x), y) -> s(+'(x, y))
   +'(p(x), y) -> p(+'(x, y))
   minus(0') -> 0'
   minus(s(x)) -> p(minus(x))
   minus(p(x)) -> s(minus(x))
   *'(0', y) -> 0'
   *'(s(x), y) -> +'(*'(x, y), y)
   *'(p(x), y) -> +'(*'(x, y), minus(y))

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

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

(16)
Obligation:
Innermost TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p

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

(17) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+', minus, *'

They will be analysed ascendingly in the following order:
+' < *'
minus < *'

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

(18)
Obligation:
Innermost TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p


Generator Equations:
gen_0':s:p2_0(0) <=> 0'
gen_0':s:p2_0(+(x, 1)) <=> s(gen_0':s:p2_0(x))


The following defined symbols remain to be analysed:
+', minus, *'

They will be analysed ascendingly in the following order:
+' < *'
minus < *'

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) -> gen_0':s:p2_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
+'(gen_0':s:p2_0(0), gen_0':s:p2_0(b)) ->_R^Omega(1)
gen_0':s:p2_0(b)

Induction Step:
+'(gen_0':s:p2_0(+(n4_0, 1)), gen_0':s:p2_0(b)) ->_R^Omega(1)
s(+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b))) ->_IH
s(gen_0':s:p2_0(+(b, c5_0)))

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

(20)
Complex Obligation (BEST)

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

(21)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p


Generator Equations:
gen_0':s:p2_0(0) <=> 0'
gen_0':s:p2_0(+(x, 1)) <=> s(gen_0':s:p2_0(x))


The following defined symbols remain to be analysed:
+', minus, *'

They will be analysed ascendingly in the following order:
+' < *'
minus < *'

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

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

(23)
BOUNDS(n^1, INF)

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

(24)
Obligation:
Innermost TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p


Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) -> gen_0':s:p2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:p2_0(0) <=> 0'
gen_0':s:p2_0(+(x, 1)) <=> s(gen_0':s:p2_0(x))


The following defined symbols remain to be analysed:
minus, *'

They will be analysed ascendingly in the following order:
minus < *'

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(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s:p2_0(+(1, n667_0))) -> *3_0, rt in Omega(n667_0)

Induction Base:
minus(gen_0':s:p2_0(+(1, 0)))

Induction Step:
minus(gen_0':s:p2_0(+(1, +(n667_0, 1)))) ->_R^Omega(1)
p(minus(gen_0':s:p2_0(+(1, n667_0)))) ->_IH
p(*3_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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(26)
Obligation:
Innermost TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p


Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) -> gen_0':s:p2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
minus(gen_0':s:p2_0(+(1, n667_0))) -> *3_0, rt in Omega(n667_0)


Generator Equations:
gen_0':s:p2_0(0) <=> 0'
gen_0':s:p2_0(+(x, 1)) <=> s(gen_0':s:p2_0(x))


The following defined symbols remain to be analysed:
*'
----------------------------------------

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_0':s:p2_0(n1827_0), gen_0':s:p2_0(b)) -> gen_0':s:p2_0(*(n1827_0, b)), rt in Omega(1 + b*n1827_0^2 + n1827_0)

Induction Base:
*'(gen_0':s:p2_0(0), gen_0':s:p2_0(b)) ->_R^Omega(1)
0'

Induction Step:
*'(gen_0':s:p2_0(+(n1827_0, 1)), gen_0':s:p2_0(b)) ->_R^Omega(1)
+'(*'(gen_0':s:p2_0(n1827_0), gen_0':s:p2_0(b)), gen_0':s:p2_0(b)) ->_IH
+'(gen_0':s:p2_0(*(c1828_0, b)), gen_0':s:p2_0(b)) ->_L^Omega(1 + b*n1827_0)
gen_0':s:p2_0(+(*(n1827_0, b), b))

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
----------------------------------------

(28)
Obligation:
Proved the lower bound n^3 for the following obligation:

Innermost TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p


Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) -> gen_0':s:p2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
minus(gen_0':s:p2_0(+(1, n667_0))) -> *3_0, rt in Omega(n667_0)


Generator Equations:
gen_0':s:p2_0(0) <=> 0'
gen_0':s:p2_0(+(x, 1)) <=> s(gen_0':s:p2_0(x))


The following defined symbols remain to be analysed:
*'
----------------------------------------

(29) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(30)
BOUNDS(n^3, INF)