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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 CpxRelTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 160 ms]
(2) CpxRelTRS
(3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (12) CdtProblem
    (13) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CdtProblem
    (15) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CdtProblem
    (17) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 79 ms]
    (18) CdtProblem
    (19) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 40 ms]
    (20) CdtProblem
    (21) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms]
    (22) CdtProblem
    (23) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 180 ms]
    (24) CdtProblem
    (25) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) BOUNDS(1, 1)
(27) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(28) TRS for Loop Detection
    (29) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (30) BEST
        (31) proven lower bound
            (32) LowerBoundPropagationProof [FINISHED, 0 ms]
            (33) BOUNDS(n^1, INF)
        (34) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   bsort(S(x'), Cons(x, xs)) -> bsort(x', bubble(x, xs))
   len(Cons(x, xs)) -> +(S(0), len(xs))
   bubble(x', Cons(x, xs)) -> bubble[Ite][False][Ite](<(x', x), x', Cons(x, xs))
   len(Nil) -> 0
   bubble(x, Nil) -> Cons(x, Nil)
   bsort(0, xs) -> xs
   bubblesort(xs) -> bsort(len(xs), xs)

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

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   bubble[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, bubble(x', xs))
   bubble[Ite][False][Ite](True, x', Cons(x, xs)) -> Cons(x', bubble(x, xs))

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

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   bsort(S(x'), Cons(x, xs)) -> bsort(x', bubble(x, xs))
   len(Cons(x, xs)) -> +(S(0), len(xs))
   bubble(x', Cons(x, xs)) -> bubble[Ite][False][Ite](<(x', x), x', Cons(x, xs))
   len(Nil) -> 0
   bubble(x, Nil) -> Cons(x, Nil)
   bsort(0, xs) -> xs
   bubblesort(xs) -> bsort(len(xs), xs)

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

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   bubble[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, bubble(x', xs))
   bubble[Ite][False][Ite](True, x', Cons(x, xs)) -> Cons(x', bubble(x, xs))

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   bsort(S(z0), Cons(z1, z2)) -> bsort(z0, bubble(z1, z2))
   bsort(0, z0) -> z0
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubblesort(z0) -> bsort(len(z0), z0)
Tuples:
   +'(z0, S(0)) -> c
   +'(S(0), z0) -> c1
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   <'(0, S(z0)) -> c3
   <'(z0, 0) -> c4
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BSORT(0, z0) -> c8
   LEN(Cons(z0, z1)) -> c9(+'(S(0), len(z1)), LEN(z1))
   LEN(Nil) -> c10
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   BUBBLESORT(z0) -> c13(BSORT(len(z0), z0), LEN(z0))
S tuples:
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BSORT(0, z0) -> c8
   LEN(Cons(z0, z1)) -> c9(+'(S(0), len(z1)), LEN(z1))
   LEN(Nil) -> c10
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   BUBBLESORT(z0) -> c13(BSORT(len(z0), z0), LEN(z0))
K tuples:none
Defined Rule Symbols:   bsort_2, len_1, bubble_2, bubblesort_1, +_2, <_2, bubble[Ite][False][Ite]_3

Defined Pair Symbols:   +'_2, <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, LEN_1, BUBBLE_2, BUBBLESORT_1

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6_1, c7_2, c8, c9_2, c10, c11_2, c12, c13_2


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

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   <'(0, S(z0)) -> c3
   +'(S(0), z0) -> c1
   LEN(Nil) -> c10
   <'(z0, 0) -> c4
   +'(z0, S(0)) -> c
   BSORT(0, z0) -> c8

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   bsort(S(z0), Cons(z1, z2)) -> bsort(z0, bubble(z1, z2))
   bsort(0, z0) -> z0
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubblesort(z0) -> bsort(len(z0), z0)
Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   LEN(Cons(z0, z1)) -> c9(+'(S(0), len(z1)), LEN(z1))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   BUBBLESORT(z0) -> c13(BSORT(len(z0), z0), LEN(z0))
S tuples:
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   LEN(Cons(z0, z1)) -> c9(+'(S(0), len(z1)), LEN(z1))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   BUBBLESORT(z0) -> c13(BSORT(len(z0), z0), LEN(z0))
K tuples:none
Defined Rule Symbols:   bsort_2, len_1, bubble_2, bubblesort_1, +_2, <_2, bubble[Ite][False][Ite]_3

Defined Pair Symbols:   <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, LEN_1, BUBBLE_2, BUBBLESORT_1

Compound Symbols:   c2_1, c5_1, c6_1, c7_2, c9_2, c11_2, c12, c13_2


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

(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   bsort(S(z0), Cons(z1, z2)) -> bsort(z0, bubble(z1, z2))
   bsort(0, z0) -> z0
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubblesort(z0) -> bsort(len(z0), z0)
Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   BUBBLESORT(z0) -> c13(BSORT(len(z0), z0), LEN(z0))
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
S tuples:
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   BUBBLESORT(z0) -> c13(BSORT(len(z0), z0), LEN(z0))
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
K tuples:none
Defined Rule Symbols:   bsort_2, len_1, bubble_2, bubblesort_1, +_2, <_2, bubble[Ite][False][Ite]_3

Defined Pair Symbols:   <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, BUBBLE_2, BUBBLESORT_1, LEN_1

Compound Symbols:   c2_1, c5_1, c6_1, c7_2, c11_2, c12, c13_2, c9_1


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

(9) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   bsort(S(z0), Cons(z1, z2)) -> bsort(z0, bubble(z1, z2))
   bsort(0, z0) -> z0
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubblesort(z0) -> bsort(len(z0), z0)
Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
   BUBBLESORT(z0) -> c(LEN(z0))
S tuples:
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
   BUBBLESORT(z0) -> c(LEN(z0))
K tuples:none
Defined Rule Symbols:   bsort_2, len_1, bubble_2, bubblesort_1, +_2, <_2, bubble[Ite][False][Ite]_3

Defined Pair Symbols:   <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, BUBBLE_2, LEN_1, BUBBLESORT_1

Compound Symbols:   c2_1, c5_1, c6_1, c7_2, c11_2, c12, c9_1, c_1


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

(11) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   BUBBLESORT(z0) -> c(LEN(z0))

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   bsort(S(z0), Cons(z1, z2)) -> bsort(z0, bubble(z1, z2))
   bsort(0, z0) -> z0
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubblesort(z0) -> bsort(len(z0), z0)
Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
S tuples:
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
K tuples:none
Defined Rule Symbols:   bsort_2, len_1, bubble_2, bubblesort_1, +_2, <_2, bubble[Ite][False][Ite]_3

Defined Pair Symbols:   <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, BUBBLE_2, LEN_1, BUBBLESORT_1

Compound Symbols:   c2_1, c5_1, c6_1, c7_2, c11_2, c12, c9_1, c_1


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

(13) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   bsort(S(z0), Cons(z1, z2)) -> bsort(z0, bubble(z1, z2))
   bsort(0, z0) -> z0
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubblesort(z0) -> bsort(len(z0), z0)
Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
S tuples:
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
K tuples:
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
Defined Rule Symbols:   bsort_2, len_1, bubble_2, bubblesort_1, +_2, <_2, bubble[Ite][False][Ite]_3

Defined Pair Symbols:   <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, BUBBLE_2, LEN_1, BUBBLESORT_1

Compound Symbols:   c2_1, c5_1, c6_1, c7_2, c11_2, c12, c9_1, c_1


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

(15) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   bsort(S(z0), Cons(z1, z2)) -> bsort(z0, bubble(z1, z2))
   bsort(0, z0) -> z0
   bubblesort(z0) -> bsort(len(z0), z0)

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

(16)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
S tuples:
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
K tuples:
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
Defined Rule Symbols:   bubble_2, bubble[Ite][False][Ite]_3, <_2, len_1, +_2

Defined Pair Symbols:   <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, BUBBLE_2, LEN_1, BUBBLESORT_1

Compound Symbols:   c2_1, c5_1, c6_1, c7_2, c11_2, c12, c9_1, c_1


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

(17) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
We considered the (Usable) Rules:none
And the Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(0) = 0
   POL(<(x_1, x_2)) = [1] + x_2
   POL(<'(x_1, x_2)) = 0
   POL(BSORT(x_1, x_2)) = [1]
   POL(BUBBLE(x_1, x_2)) = 0
   POL(BUBBLESORT(x_1)) = [1]
   POL(BUBBLE[ITE][FALSE][ITE](x_1, x_2, x_3)) = 0
   POL(Cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(False) = [1]
   POL(LEN(x_1)) = x_1
   POL(Nil) = [1]
   POL(S(x_1)) = 0
   POL(True) = [1]
   POL(bubble(x_1, x_2)) = [1] + x_1 + x_2
   POL(bubble[Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(c(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12) = 0
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(len(x_1)) = [1] + x_1

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

(18)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
S tuples:
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
K tuples:
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
Defined Rule Symbols:   bubble_2, bubble[Ite][False][Ite]_3, <_2, len_1, +_2

Defined Pair Symbols:   <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, BUBBLE_2, LEN_1, BUBBLESORT_1

Compound Symbols:   c2_1, c5_1, c6_1, c7_2, c11_2, c12, c9_1, c_1


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

(19) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BUBBLE(z0, Nil) -> c12
We considered the (Usable) Rules:
   <(z0, 0) -> False
   len(Nil) -> 0
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   <(0, S(z0)) -> True
   <(S(z0), S(z1)) -> <(z0, z1)
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
And the Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = x_1 + x_2
   POL(0) = 0
   POL(<(x_1, x_2)) = [1]
   POL(<'(x_1, x_2)) = 0
   POL(BSORT(x_1, x_2)) = [3] + [2]x_1
   POL(BUBBLE(x_1, x_2)) = [2]
   POL(BUBBLESORT(x_1)) = [3] + [2]x_1
   POL(BUBBLE[ITE][FALSE][ITE](x_1, x_2, x_3)) = [1] + x_1
   POL(Cons(x_1, x_2)) = [2] + x_2
   POL(False) = [1]
   POL(LEN(x_1)) = [3]x_1
   POL(Nil) = 0
   POL(S(x_1)) = [1] + x_1
   POL(True) = [1]
   POL(bubble(x_1, x_2)) = [3] + [2]x_1 + [2]x_2
   POL(bubble[Ite][False][Ite](x_1, x_2, x_3)) = [3] + [3]x_2
   POL(c(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12) = 0
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(len(x_1)) = x_1

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

(20)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
S tuples:
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
K tuples:
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLE(z0, Nil) -> c12
Defined Rule Symbols:   bubble_2, bubble[Ite][False][Ite]_3, <_2, len_1, +_2

Defined Pair Symbols:   <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, BUBBLE_2, LEN_1, BUBBLESORT_1

Compound Symbols:   c2_1, c5_1, c6_1, c7_2, c11_2, c12, c9_1, c_1


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

(21) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
We considered the (Usable) Rules:
   <(z0, 0) -> False
   len(Nil) -> 0
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   <(0, S(z0)) -> True
   <(S(z0), S(z1)) -> <(z0, z1)
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
And the Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = x_1 + x_2
   POL(0) = 0
   POL(<(x_1, x_2)) = [1]
   POL(<'(x_1, x_2)) = 0
   POL(BSORT(x_1, x_2)) = [3] + [3]x_1
   POL(BUBBLE(x_1, x_2)) = [3]
   POL(BUBBLESORT(x_1)) = [3] + [3]x_1
   POL(BUBBLE[ITE][FALSE][ITE](x_1, x_2, x_3)) = [1] + [2]x_1
   POL(Cons(x_1, x_2)) = [3] + x_2
   POL(False) = [1]
   POL(LEN(x_1)) = [3]x_1
   POL(Nil) = 0
   POL(S(x_1)) = [3] + x_1
   POL(True) = [1]
   POL(bubble(x_1, x_2)) = [3] + [2]x_1 + [2]x_2
   POL(bubble[Ite][False][Ite](x_1, x_2, x_3)) = [3] + [3]x_2
   POL(c(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12) = 0
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(len(x_1)) = x_1

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
S tuples:
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
K tuples:
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLE(z0, Nil) -> c12
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
Defined Rule Symbols:   bubble_2, bubble[Ite][False][Ite]_3, <_2, len_1, +_2

Defined Pair Symbols:   <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, BUBBLE_2, LEN_1, BUBBLESORT_1

Compound Symbols:   c2_1, c5_1, c6_1, c7_2, c11_2, c12, c9_1, c_1


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

(23) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
We considered the (Usable) Rules:
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   len(Nil) -> 0
   bubble(z0, Nil) -> Cons(z0, Nil)
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
And the Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = x_1 + x_2
   POL(0) = 0
   POL(<(x_1, x_2)) = 0
   POL(<'(x_1, x_2)) = 0
   POL(BSORT(x_1, x_2)) = [2]x_1 + [2]x_1*x_2
   POL(BUBBLE(x_1, x_2)) = [2] + [2]x_2
   POL(BUBBLESORT(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(BUBBLE[ITE][FALSE][ITE](x_1, x_2, x_3)) = [2]x_3
   POL(Cons(x_1, x_2)) = [1] + x_2
   POL(False) = 0
   POL(LEN(x_1)) = [2]x_1 + x_1^2
   POL(Nil) = 0
   POL(S(x_1)) = [1] + x_1
   POL(True) = 0
   POL(bubble(x_1, x_2)) = [1] + x_2
   POL(bubble[Ite][False][Ite](x_1, x_2, x_3)) = [1] + x_3
   POL(c(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c12) = 0
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(len(x_1)) = x_1

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   bubble(z0, Cons(z1, z2)) -> bubble[Ite][False][Ite](<(z0, z1), z0, Cons(z1, z2))
   bubble(z0, Nil) -> Cons(z0, Nil)
   bubble[Ite][False][Ite](False, z0, Cons(z1, z2)) -> Cons(z1, bubble(z0, z2))
   bubble[Ite][False][Ite](True, z0, Cons(z1, z2)) -> Cons(z0, bubble(z1, z2))
   <(S(z0), S(z1)) -> <(z0, z1)
   <(0, S(z0)) -> True
   <(z0, 0) -> False
   len(Cons(z0, z1)) -> +(S(0), len(z1))
   len(Nil) -> 0
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
Tuples:
   <'(S(z0), S(z1)) -> c2(<'(z0, z1))
   BUBBLE[ITE][FALSE][ITE](False, z0, Cons(z1, z2)) -> c5(BUBBLE(z0, z2))
   BUBBLE[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c6(BUBBLE(z1, z2))
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
   BUBBLE(z0, Nil) -> c12
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
S tuples:none
K tuples:
   BUBBLESORT(z0) -> c(BSORT(len(z0), z0))
   LEN(Cons(z0, z1)) -> c9(LEN(z1))
   BUBBLE(z0, Nil) -> c12
   BSORT(S(z0), Cons(z1, z2)) -> c7(BSORT(z0, bubble(z1, z2)), BUBBLE(z1, z2))
   BUBBLE(z0, Cons(z1, z2)) -> c11(BUBBLE[ITE][FALSE][ITE](<(z0, z1), z0, Cons(z1, z2)), <'(z0, z1))
Defined Rule Symbols:   bubble_2, bubble[Ite][False][Ite]_3, <_2, len_1, +_2

Defined Pair Symbols:   <'_2, BUBBLE[ITE][FALSE][ITE]_3, BSORT_2, BUBBLE_2, LEN_1, BUBBLESORT_1

Compound Symbols:   c2_1, c5_1, c6_1, c7_2, c11_2, c12, c9_1, c_1


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(25) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
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(26)
BOUNDS(1, 1)

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(27) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(28)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   bsort(S(x'), Cons(x, xs)) -> bsort(x', bubble(x, xs))
   len(Cons(x, xs)) -> +(S(0), len(xs))
   bubble(x', Cons(x, xs)) -> bubble[Ite][False][Ite](<(x', x), x', Cons(x, xs))
   len(Nil) -> 0
   bubble(x, Nil) -> Cons(x, Nil)
   bsort(0, xs) -> xs
   bubblesort(xs) -> bsort(len(xs), xs)

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

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   bubble[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, bubble(x', xs))
   bubble[Ite][False][Ite](True, x', Cons(x, xs)) -> Cons(x', bubble(x, xs))

Rewrite Strategy: INNERMOST
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(29) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

len(Cons(x, xs)) ->^+ +(S(0), len(xs))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [xs / Cons(x, xs)].

The result substitution is [ ].




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(30)
Complex Obligation (BEST)

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(31)
Obligation:
Proved the lower bound n^1 for the following obligation:

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


The TRS R consists of the following rules:

   bsort(S(x'), Cons(x, xs)) -> bsort(x', bubble(x, xs))
   len(Cons(x, xs)) -> +(S(0), len(xs))
   bubble(x', Cons(x, xs)) -> bubble[Ite][False][Ite](<(x', x), x', Cons(x, xs))
   len(Nil) -> 0
   bubble(x, Nil) -> Cons(x, Nil)
   bsort(0, xs) -> xs
   bubblesort(xs) -> bsort(len(xs), xs)

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

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   bubble[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, bubble(x', xs))
   bubble[Ite][False][Ite](True, x', Cons(x, xs)) -> Cons(x', bubble(x, xs))

Rewrite Strategy: INNERMOST
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(32) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(33)
BOUNDS(n^1, INF)

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(34)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   bsort(S(x'), Cons(x, xs)) -> bsort(x', bubble(x, xs))
   len(Cons(x, xs)) -> +(S(0), len(xs))
   bubble(x', Cons(x, xs)) -> bubble[Ite][False][Ite](<(x', x), x', Cons(x, xs))
   len(Nil) -> 0
   bubble(x, Nil) -> Cons(x, Nil)
   bsort(0, xs) -> xs
   bubblesort(xs) -> bsort(len(xs), xs)

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

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)
   <(S(x), S(y)) -> <(x, y)
   <(0, S(y)) -> True
   <(x, 0) -> False
   bubble[Ite][False][Ite](False, x', Cons(x, xs)) -> Cons(x, bubble(x', xs))
   bubble[Ite][False][Ite](True, x', Cons(x, xs)) -> Cons(x', bubble(x, xs))

Rewrite Strategy: INNERMOST