/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^1).

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 126 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)), 31 ms]
    (12) CdtProblem
    (13) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 6 ms]
    (14) CdtProblem
    (15) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) BOUNDS(1, 1)
(17) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxRelTRS
    (19) SlicingProof [LOWER BOUND(ID), 0 ms]
    (20) CpxRelTRS
    (21) TypeInferenceProof [BOTH BOUNDS(ID, ID), 5 ms]
    (22) typed CpxTrs
    (23) OrderProof [LOWER BOUND(ID), 0 ms]
    (24) typed CpxTrs
    (25) RewriteLemmaProof [LOWER BOUND(ID), 208 ms]
    (26) BEST
        (27) proven lower bound
            (28) LowerBoundPropagationProof [FINISHED, 0 ms]
            (29) BOUNDS(n^1, INF)
        (30) typed CpxTrs
            (31) RewriteLemmaProof [LOWER BOUND(ID), 91 ms]
            (32) BOUNDS(1, INF)


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

(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(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2)))
   a(Z, y) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

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(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2)))
   a(Z, y) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   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
Tuples:
   AND(False, False) -> c
   AND(True, False) -> c1
   AND(False, True) -> c2
   AND(True, True) -> c3
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   A(Z, z0) -> c5
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c7
   EQZLIST(Z, C(z0, z1)) -> c8
   EQZLIST(Z, Z) -> c9
   SECOND(C(z0, z1)) -> c10
   FIRST(C(z0, z1)) -> c11
S tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   A(Z, z0) -> c5
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2), EQZLIST(z1, z3))
   EQZLIST(C(z0, z1), Z) -> c7
   EQZLIST(Z, C(z0, z1)) -> c8
   EQZLIST(Z, Z) -> c9
   SECOND(C(z0, z1)) -> c10
   FIRST(C(z0, z1)) -> c11
K tuples:none
Defined Rule Symbols:   a_2, eqZList_2, second_1, first_1, and_2

Defined Pair Symbols:   AND_2, A_2, EQZLIST_2, SECOND_1, FIRST_1

Compound Symbols:   c, c1, c2, c3, c4_2, c5, c6_3, c7, c8, c9, c10, c11


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

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   EQZLIST(C(z0, z1), Z) -> c7
   AND(True, False) -> c1
   A(Z, z0) -> c5
   AND(True, True) -> c3
   SECOND(C(z0, z1)) -> c10
   AND(False, True) -> c2
   FIRST(C(z0, z1)) -> c11
   EQZLIST(Z, Z) -> c9
   EQZLIST(Z, C(z0, z1)) -> c8
   AND(False, False) -> c

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   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
Tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2), EQZLIST(z1, z3))
S tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(AND(eqZList(z0, z2), eqZList(z1, z3)), EQZLIST(z0, z2), EQZLIST(z1, z3))
K tuples:none
Defined Rule Symbols:   a_2, eqZList_2, second_1, first_1, and_2

Defined Pair Symbols:   A_2, EQZLIST_2

Compound Symbols:   c4_2, c6_3


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

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   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
Tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(EQZLIST(z0, z2), EQZLIST(z1, z3))
S tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(EQZLIST(z0, z2), EQZLIST(z1, z3))
K tuples:none
Defined Rule Symbols:   a_2, eqZList_2, second_1, first_1, and_2

Defined Pair Symbols:   A_2, EQZLIST_2

Compound Symbols:   c4_2, c6_2


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   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

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(EQZLIST(z0, z2), EQZLIST(z1, z3))
S tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(EQZLIST(z0, z2), EQZLIST(z1, z3))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   A_2, EQZLIST_2

Compound Symbols:   c4_2, c6_2


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

(11) 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)) -> c6(EQZLIST(z0, z2), EQZLIST(z1, z3))
We considered the (Usable) Rules:none
And the Tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(EQZLIST(z0, z2), 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(C(x_1, x_2)) = [1] + x_1 + x_2
   POL(EQZLIST(x_1, x_2)) = [1] + x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2

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

(12)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(EQZLIST(z0, z2), EQZLIST(z1, z3))
S tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
K tuples:
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(EQZLIST(z0, z2), EQZLIST(z1, z3))
Defined Rule Symbols:none

Defined Pair Symbols:   A_2, EQZLIST_2

Compound Symbols:   c4_2, c6_2


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

(13) 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) -> c4(A(z0, z2), A(z1, C(z0, z1)))
We considered the (Usable) Rules:none
And the Tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(EQZLIST(z0, z2), EQZLIST(z1, z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   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)) = [1] + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2

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

(14)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(EQZLIST(z0, z2), EQZLIST(z1, z3))
S tuples:none
K tuples:
   EQZLIST(C(z0, z1), C(z2, z3)) -> c6(EQZLIST(z0, z2), EQZLIST(z1, z3))
   A(C(z0, z1), z2) -> c4(A(z0, z2), A(z1, C(z0, z1)))
Defined Rule Symbols:none

Defined Pair Symbols:   A_2, EQZLIST_2

Compound Symbols:   c4_2, c6_2


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

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

(16)
BOUNDS(1, 1)

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

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

(18)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a(C(x1, x2), y) -> C(a(x1, y), a(x2, C(x1, x2)))
   a(Z, y) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

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

(19) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
a/1

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

(20)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   a(C(x1, x2)) -> C(a(x1), a(x2))
   a(Z) -> Z
   eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
   eqZList(C(x1, x2), Z) -> False
   eqZList(Z, C(y1, y2)) -> False
   eqZList(Z, Z) -> True
   second(C(x1, x2)) -> x2
   first(C(x1, x2)) -> x1

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

   and(False, False) -> False
   and(True, False) -> False
   and(False, True) -> False
   and(True, True) -> True

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

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

(22)
Obligation:
Innermost TRS:
Rules:
a(C(x1, x2)) -> C(a(x1), a(x2))
a(Z) -> Z
eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) -> False
eqZList(Z, C(y1, y2)) -> False
eqZList(Z, Z) -> True
second(C(x1, x2)) -> x2
first(C(x1, x2)) -> x1
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True

Types:
a :: C:Z -> C:Z
C :: C:Z -> C:Z -> C:Z
Z :: C:Z
eqZList :: C:Z -> C:Z -> False:True
and :: False:True -> False:True -> False:True
False :: False:True
True :: False:True
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_C:Z1_0 :: C:Z
hole_False:True2_0 :: False:True
gen_C:Z3_0 :: Nat -> C:Z

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

(23) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
a, eqZList
----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
a(C(x1, x2)) -> C(a(x1), a(x2))
a(Z) -> Z
eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) -> False
eqZList(Z, C(y1, y2)) -> False
eqZList(Z, Z) -> True
second(C(x1, x2)) -> x2
first(C(x1, x2)) -> x1
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True

Types:
a :: C:Z -> C:Z
C :: C:Z -> C:Z -> C:Z
Z :: C:Z
eqZList :: C:Z -> C:Z -> False:True
and :: False:True -> False:True -> False:True
False :: False:True
True :: False:True
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_C:Z1_0 :: C:Z
hole_False:True2_0 :: False:True
gen_C:Z3_0 :: Nat -> C:Z


Generator Equations:
gen_C:Z3_0(0) <=> Z
gen_C:Z3_0(+(x, 1)) <=> C(Z, gen_C:Z3_0(x))


The following defined symbols remain to be analysed:
a, eqZList
----------------------------------------

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a(gen_C:Z3_0(n5_0)) -> gen_C:Z3_0(n5_0), rt in Omega(1 + n5_0)

Induction Base:
a(gen_C:Z3_0(0)) ->_R^Omega(1)
Z

Induction Step:
a(gen_C:Z3_0(+(n5_0, 1))) ->_R^Omega(1)
C(a(Z), a(gen_C:Z3_0(n5_0))) ->_R^Omega(1)
C(Z, a(gen_C:Z3_0(n5_0))) ->_IH
C(Z, gen_C:Z3_0(c6_0))

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

(26)
Complex Obligation (BEST)

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

(27)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
a(C(x1, x2)) -> C(a(x1), a(x2))
a(Z) -> Z
eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) -> False
eqZList(Z, C(y1, y2)) -> False
eqZList(Z, Z) -> True
second(C(x1, x2)) -> x2
first(C(x1, x2)) -> x1
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True

Types:
a :: C:Z -> C:Z
C :: C:Z -> C:Z -> C:Z
Z :: C:Z
eqZList :: C:Z -> C:Z -> False:True
and :: False:True -> False:True -> False:True
False :: False:True
True :: False:True
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_C:Z1_0 :: C:Z
hole_False:True2_0 :: False:True
gen_C:Z3_0 :: Nat -> C:Z


Generator Equations:
gen_C:Z3_0(0) <=> Z
gen_C:Z3_0(+(x, 1)) <=> C(Z, gen_C:Z3_0(x))


The following defined symbols remain to be analysed:
a, eqZList
----------------------------------------

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

(29)
BOUNDS(n^1, INF)

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

(30)
Obligation:
Innermost TRS:
Rules:
a(C(x1, x2)) -> C(a(x1), a(x2))
a(Z) -> Z
eqZList(C(x1, x2), C(y1, y2)) -> and(eqZList(x1, y1), eqZList(x2, y2))
eqZList(C(x1, x2), Z) -> False
eqZList(Z, C(y1, y2)) -> False
eqZList(Z, Z) -> True
second(C(x1, x2)) -> x2
first(C(x1, x2)) -> x1
and(False, False) -> False
and(True, False) -> False
and(False, True) -> False
and(True, True) -> True

Types:
a :: C:Z -> C:Z
C :: C:Z -> C:Z -> C:Z
Z :: C:Z
eqZList :: C:Z -> C:Z -> False:True
and :: False:True -> False:True -> False:True
False :: False:True
True :: False:True
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_C:Z1_0 :: C:Z
hole_False:True2_0 :: False:True
gen_C:Z3_0 :: Nat -> C:Z


Lemmas:
a(gen_C:Z3_0(n5_0)) -> gen_C:Z3_0(n5_0), rt in Omega(1 + n5_0)


Generator Equations:
gen_C:Z3_0(0) <=> Z
gen_C:Z3_0(+(x, 1)) <=> C(Z, gen_C:Z3_0(x))


The following defined symbols remain to be analysed:
eqZList
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(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eqZList(gen_C:Z3_0(+(1, n148_0)), gen_C:Z3_0(n148_0)) -> False, rt in Omega(1 + n148_0)

Induction Base:
eqZList(gen_C:Z3_0(+(1, 0)), gen_C:Z3_0(0)) ->_R^Omega(1)
False

Induction Step:
eqZList(gen_C:Z3_0(+(1, +(n148_0, 1))), gen_C:Z3_0(+(n148_0, 1))) ->_R^Omega(1)
and(eqZList(Z, Z), eqZList(gen_C:Z3_0(+(1, n148_0)), gen_C:Z3_0(n148_0))) ->_R^Omega(1)
and(True, eqZList(gen_C:Z3_0(+(1, n148_0)), gen_C:Z3_0(n148_0))) ->_IH
and(True, False) ->_R^Omega(0)
False

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(32)
BOUNDS(1, INF)