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# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 250 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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 55 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms]
    (14) CdtProblem
    (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) BOUNDS(1, 1)
(17) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(18) TRS for Loop Detection
    (19) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (20) BEST
        (21) proven lower bound
            (22) LowerBoundPropagationProof [FINISHED, 0 ms]
            (23) BOUNDS(n^1, INF)
        (24) TRS for Loop Detection


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

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


The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13

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

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0

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^1).


The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13

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

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0
   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13
Tuples:
   AND'(False, False) -> c14
   AND'(True, False) -> c15
   AND'(False, True) -> c16
   AND'(True, True) -> c17
   AND''(False, False) -> c18
   AND''(True, False) -> c19
   AND''(False, True) -> c20
   AND''(True, True) -> c21
   A'(C(z0, z1), z2) -> c22(A'(z0, z2), A'(z1, C(z0, z1)))
   A'(Z, z0) -> c23
   EQZLIST'(C(z0, z1), C(z2, z3)) -> c24(AND''(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST'(z0, z2), EQZLIST'(z1, z3))
   EQZLIST'(C(z0, z1), Z) -> c25
   EQZLIST'(Z, C(z0, z1)) -> c26
   EQZLIST'(Z, Z) -> c27
   SECOND'(C(z0, z1)) -> c28
   FIRST'(C(z0, z1)) -> c29
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   A''(Z, z0) -> c32
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(AND'(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(AND'(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
   EQZLIST''(C(z0, z1), Z) -> c35
   EQZLIST''(Z, C(z0, z1)) -> c36
   EQZLIST''(Z, Z) -> c37
   SECOND''(C(z0, z1)) -> c38
   FIRST''(C(z0, z1)) -> c39
S tuples:
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   A''(Z, z0) -> c32
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(AND'(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(AND'(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
   EQZLIST''(C(z0, z1), Z) -> c35
   EQZLIST''(Z, C(z0, z1)) -> c36
   EQZLIST''(Z, Z) -> c37
   SECOND''(C(z0, z1)) -> c38
   FIRST''(C(z0, z1)) -> c39
K tuples:none
Defined Rule Symbols:   A_2, EQZLIST_2, SECOND_1, FIRST_1, AND_2, and_2, a_2, eqZList_2, second_1, first_1

Defined Pair Symbols:   AND'_2, AND''_2, A'_2, EQZLIST'_2, SECOND'_1, FIRST'_1, A''_2, EQZLIST''_2, SECOND''_1, FIRST''_1

Compound Symbols:   c14, c15, c16, c17, c18, c19, c20, c21, c22_2, c23, c24_3, c25, c26, c27, c28, c29, c30_1, c31_1, c32, c33_4, c34_4, c35, c36, c37, c38, c39


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

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 20 trailing nodes:
   AND'(True, False) -> c15
   AND''(False, False) -> c18
   SECOND''(C(z0, z1)) -> c38
   EQZLIST'(Z, C(z0, z1)) -> c26
   AND''(False, True) -> c20
   EQZLIST''(Z, C(z0, z1)) -> c36
   AND''(True, True) -> c21
   A''(Z, z0) -> c32
   FIRST''(C(z0, z1)) -> c39
   AND''(True, False) -> c19
   FIRST'(C(z0, z1)) -> c29
   EQZLIST'(C(z0, z1), Z) -> c25
   EQZLIST'(Z, Z) -> c27
   AND'(False, True) -> c16
   EQZLIST''(Z, Z) -> c37
   AND'(True, True) -> c17
   SECOND'(C(z0, z1)) -> c28
   EQZLIST''(C(z0, z1), Z) -> c35
   A'(Z, z0) -> c23
   AND'(False, False) -> c14

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0
   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13
Tuples:
   A'(C(z0, z1), z2) -> c22(A'(z0, z2), A'(z1, C(z0, z1)))
   EQZLIST'(C(z0, z1), C(z2, z3)) -> c24(AND''(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST'(z0, z2), EQZLIST'(z1, z3))
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(AND'(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(AND'(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
S tuples:
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(AND'(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(AND'(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
K tuples:none
Defined Rule Symbols:   A_2, EQZLIST_2, SECOND_1, FIRST_1, AND_2, and_2, a_2, eqZList_2, second_1, first_1

Defined Pair Symbols:   A'_2, EQZLIST'_2, A''_2, EQZLIST''_2

Compound Symbols:   c22_2, c24_3, c30_1, c31_1, c33_4, c34_4


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

(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0
   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13
Tuples:
   A'(C(z0, z1), z2) -> c22(A'(z0, z2), A'(z1, C(z0, z1)))
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST'(C(z0, z1), C(z2, z3)) -> c24(EQZLIST'(z0, z2), EQZLIST'(z1, z3))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
S tuples:
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
K tuples:none
Defined Rule Symbols:   A_2, EQZLIST_2, SECOND_1, FIRST_1, AND_2, and_2, a_2, eqZList_2, second_1, first_1

Defined Pair Symbols:   A'_2, A''_2, EQZLIST'_2, EQZLIST''_2

Compound Symbols:   c22_2, c30_1, c31_1, c24_2, c33_3, c34_3


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0
   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   A'(C(z0, z1), z2) -> c22(A'(z0, z2), A'(z1, C(z0, z1)))
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST'(C(z0, z1), C(z2, z3)) -> c24(EQZLIST'(z0, z2), EQZLIST'(z1, z3))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
S tuples:
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   A'_2, A''_2, EQZLIST'_2, EQZLIST''_2

Compound Symbols:   c22_2, c30_1, c31_1, c24_2, c33_3, c34_3


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

(11) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
We considered the (Usable) Rules:none
And the Tuples:
   A'(C(z0, z1), z2) -> c22(A'(z0, z2), A'(z1, C(z0, z1)))
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST'(C(z0, z1), C(z2, z3)) -> c24(EQZLIST'(z0, z2), EQZLIST'(z1, z3))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A'(x_1, x_2)) = 0
   POL(A''(x_1, x_2)) = [2]x_1
   POL(C(x_1, x_2)) = [1] + x_1 + x_2
   POL(EQZLIST'(x_1, x_2)) = 0
   POL(EQZLIST''(x_1, x_2)) = 0
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c30(x_1)) = x_1
   POL(c31(x_1)) = x_1
   POL(c33(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3

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(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   A'(C(z0, z1), z2) -> c22(A'(z0, z2), A'(z1, C(z0, z1)))
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST'(C(z0, z1), C(z2, z3)) -> c24(EQZLIST'(z0, z2), EQZLIST'(z1, z3))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
S tuples:
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
K tuples:
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
Defined Rule Symbols:none

Defined Pair Symbols:   A'_2, A''_2, EQZLIST'_2, EQZLIST''_2

Compound Symbols:   c22_2, c30_1, c31_1, c24_2, c33_3, c34_3


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

(13) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
We considered the (Usable) Rules:none
And the Tuples:
   A'(C(z0, z1), z2) -> c22(A'(z0, z2), A'(z1, C(z0, z1)))
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST'(C(z0, z1), C(z2, z3)) -> c24(EQZLIST'(z0, z2), EQZLIST'(z1, z3))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(A'(x_1, x_2)) = [1] + x_1
   POL(A''(x_1, x_2)) = x_1
   POL(C(x_1, x_2)) = [1] + x_1 + x_2
   POL(EQZLIST'(x_1, x_2)) = 0
   POL(EQZLIST''(x_1, x_2)) = x_1
   POL(c22(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c30(x_1)) = x_1
   POL(c31(x_1)) = x_1
   POL(c33(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c34(x_1, x_2, x_3)) = x_1 + x_2 + x_3

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   A'(C(z0, z1), z2) -> c22(A'(z0, z2), A'(z1, C(z0, z1)))
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST'(C(z0, z1), C(z2, z3)) -> c24(EQZLIST'(z0, z2), EQZLIST'(z1, z3))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
S tuples:none
K tuples:
   A''(C(z0, z1), z2) -> c30(A''(z0, z2))
   A''(C(z0, z1), z2) -> c31(A''(z1, C(z0, z1)))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c33(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z0, z2))
   EQZLIST''(C(z0, z1), C(z2, z3)) -> c34(EQZLIST'(z0, z2), EQZLIST'(z1, z3), EQZLIST''(z1, z3))
Defined Rule Symbols:none

Defined Pair Symbols:   A'_2, A''_2, EQZLIST'_2, EQZLIST''_2

Compound Symbols:   c22_2, c30_1, c31_1, c24_2, c33_3, c34_3


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

(16)
BOUNDS(1, 1)

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(17) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(18)
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^1).


The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13

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

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0

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

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

The rewrite sequence

A(C(z0, z1), z2) ->^+ c5(A(z1, C(z0, z1)))

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

The pumping substitution is [z1 / C(z0, z1)].

The result substitution is [z2 / C(z0, z1)].




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(20)
Complex Obligation (BEST)

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

(21)
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^1).


The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13

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

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0

Rewrite Strategy: INNERMOST
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(22) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(23)
BOUNDS(n^1, INF)

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

(24)
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^1).


The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   A(Z, z0) -> c6
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   SECOND(C(z0, z1)) -> c12
   FIRST(C(z0, z1)) -> c13

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

   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True
   a(C(z0, z1), z2) -> C(a(z0, z2), a(z1, C(z0, z1)))
   a(Z, z0) -> Z
   eqZList(C(z0, z1), C(z2, z3)) -> and(eqZList(z0, z2), eqZList(z1, z3))
   eqZList(C(z0, z1), Z) -> False
   eqZList(Z, C(z0, z1)) -> False
   eqZList(Z, Z) -> True
   second(C(z0, z1)) -> z1
   first(C(z0, z1)) -> z0

Rewrite Strategy: INNERMOST