WORST_CASE(Omega(n^1),O(n^1))
proof of input_GRLmV7haAo.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 112 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) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTRS
    (13) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxWeightedTrs
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxTypedWeightedTrs
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
    (19) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxTypedWeightedCompleteTrs
    (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRNTS
    (25) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 295 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 76 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 213 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 135 ms]
    (38) CpxRNTS
    (39) FinalProof [FINISHED, 0 ms]
    (40) BOUNDS(1, n^1)
(41) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 3 ms]
(42) CdtProblem
    (43) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CpxRelTRS
    (45) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CpxRelTRS
    (47) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) typed CpxTrs
    (49) OrderProof [LOWER BOUND(ID), 19 ms]
    (50) typed CpxTrs
    (51) RewriteLemmaProof [LOWER BOUND(ID), 307 ms]
    (52) BEST
        (53) proven lower bound
            (54) LowerBoundPropagationProof [FINISHED, 0 ms]
            (55) BOUNDS(n^1, INF)
        (56) typed CpxTrs
            (57) RewriteLemmaProof [LOWER BOUND(ID), 82 ms]
            (58) typed CpxTrs
            (59) RewriteLemmaProof [LOWER BOUND(ID), 393 ms]
            (60) typed CpxTrs
            (61) RewriteLemmaProof [LOWER BOUND(ID), 32 ms]
            (62) BOUNDS(1, INF)


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

(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

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

(2)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST 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(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
S tuples:
   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
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_1, c5_1, c6, c7_2, c8_2, c9, c10, c11, c12, c13


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

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   EQZLIST(Z, C(z0, z1)) -> c10
   EQZLIST(Z, Z) -> c11
   AND(True, False) -> c1
   SECOND(C(z0, z1)) -> c12
   A(Z, z0) -> c6
   FIRST(C(z0, z1)) -> c13
   EQZLIST(C(z0, z1), Z) -> c9
   AND(False, False) -> c
   AND(True, True) -> c3
   AND(False, True) -> c2

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

(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(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   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))
S tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   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))
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_1, c5_1, c7_2, c8_2


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

(7) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 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(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(EQZLIST(z1, z3))
S tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(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_1, c5_1, c7_1, c8_1


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

(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(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(EQZLIST(z1, z3))
S tuples:
   A(C(z0, z1), z2) -> c4(A(z0, z2))
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(EQZLIST(z1, z3))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   A_2, EQZLIST_2

Compound Symbols:   c4_1, c5_1, c7_1, c8_1


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

(11) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(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)))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(EQZLIST(z0, z2))
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(EQZLIST(z1, z3))

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

(13) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2)) [1]
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1))) [1]
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(EQZLIST(z0, z2)) [1]
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(EQZLIST(z1, z3)) [1]

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

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

(16)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2)) [1]
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1))) [1]
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(EQZLIST(z0, z2)) [1]
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(EQZLIST(z1, z3)) [1]

The TRS has the following type information:
A :: C -> C -> c4:c5
C :: C -> C -> C
c4 :: c4:c5 -> c4:c5
c5 :: c4:c5 -> c4:c5
EQZLIST :: C -> C -> c7:c8
c7 :: c7:c8 -> c7:c8
c8 :: c7:c8 -> c7:c8

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

(17) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   A_2
   EQZLIST_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1, const2

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

(18)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2)) [1]
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1))) [1]
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(EQZLIST(z0, z2)) [1]
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(EQZLIST(z1, z3)) [1]

The TRS has the following type information:
A :: C -> C -> c4:c5
C :: C -> C -> C
c4 :: c4:c5 -> c4:c5
c5 :: c4:c5 -> c4:c5
EQZLIST :: C -> C -> c7:c8
c7 :: c7:c8 -> c7:c8
c8 :: c7:c8 -> c7:c8
const :: c4:c5
const1 :: C
const2 :: c7:c8

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

(19) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(20)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   A(C(z0, z1), z2) -> c4(A(z0, z2)) [1]
   A(C(z0, z1), z2) -> c5(A(z1, C(z0, z1))) [1]
   EQZLIST(C(z0, z1), C(z2, z3)) -> c7(EQZLIST(z0, z2)) [1]
   EQZLIST(C(z0, z1), C(z2, z3)) -> c8(EQZLIST(z1, z3)) [1]

The TRS has the following type information:
A :: C -> C -> c4:c5
C :: C -> C -> C
c4 :: c4:c5 -> c4:c5
c5 :: c4:c5 -> c4:c5
EQZLIST :: C -> C -> c7:c8
c7 :: c7:c8 -> c7:c8
c8 :: c7:c8 -> c7:c8
const :: c4:c5
const1 :: C
const2 :: c7:c8

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

(21) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   const => 0
   const1 => 0
   const2 => 0

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

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   A(z, z') -{ 1 }-> 1 + A(z0, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   A(z, z') -{ 1 }-> 1 + A(z1, 1 + z0 + z1) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0


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

(23) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   A(z, z') -{ 1 }-> 1 + A(z0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   A(z, z') -{ 1 }-> 1 + A(z1, 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0


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

(25) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { EQZLIST }
   { A }

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

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   A(z, z') -{ 1 }-> 1 + A(z0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   A(z, z') -{ 1 }-> 1 + A(z1, 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {EQZLIST}, {A}

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

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   A(z, z') -{ 1 }-> 1 + A(z0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   A(z, z') -{ 1 }-> 1 + A(z1, 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {EQZLIST}, {A}

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: EQZLIST
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   A(z, z') -{ 1 }-> 1 + A(z0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   A(z, z') -{ 1 }-> 1 + A(z1, 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {EQZLIST}, {A}
Previous analysis results are:
  EQZLIST: runtime: ?, size: O(1) [0]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: EQZLIST
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z'

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   A(z, z') -{ 1 }-> 1 + A(z0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   A(z, z') -{ 1 }-> 1 + A(z1, 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z0, z2) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   EQZLIST(z, z') -{ 1 }-> 1 + EQZLIST(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {A}
Previous analysis results are:
  EQZLIST: runtime: O(n^1) [z'], size: O(1) [0]

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

(33) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   A(z, z') -{ 1 }-> 1 + A(z0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   A(z, z') -{ 1 }-> 1 + A(z1, 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   EQZLIST(z, z') -{ 1 + z2 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   EQZLIST(z, z') -{ 1 + z3 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {A}
Previous analysis results are:
  EQZLIST: runtime: O(n^1) [z'], size: O(1) [0]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: A
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   A(z, z') -{ 1 }-> 1 + A(z0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   A(z, z') -{ 1 }-> 1 + A(z1, 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   EQZLIST(z, z') -{ 1 + z2 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   EQZLIST(z, z') -{ 1 + z3 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: {A}
Previous analysis results are:
  EQZLIST: runtime: O(n^1) [z'], size: O(1) [0]
  A: runtime: ?, size: O(1) [0]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: A
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   A(z, z') -{ 1 }-> 1 + A(z0, z') :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   A(z, z') -{ 1 }-> 1 + A(z1, 1 + z0 + z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1, z' >= 0
   EQZLIST(z, z') -{ 1 + z2 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   EQZLIST(z, z') -{ 1 + z3 }-> 1 + s' :|: s' >= 0, s' <= 0, z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  EQZLIST: runtime: O(n^1) [z'], size: O(1) [0]
  A: runtime: O(n^1) [z], size: O(1) [0]

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

(39) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(40)
BOUNDS(1, n^1)

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

(41) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(42)
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(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
S tuples:
   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
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_1, c5_1, c6, c7_2, c8_2, c9, c10, c11, c12, c13


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

(43) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(44)
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(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
----------------------------------------

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

(46)
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(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
----------------------------------------

(47) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(48)
Obligation:
Innermost TRS:
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
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

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11

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

(49) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
A, EQZLIST, eqZList, a

They will be analysed ascendingly in the following order:
eqZList < EQZLIST

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

(50)
Obligation:
Innermost TRS:
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
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

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11


Generator Equations:
gen_c4:c5:c68_14(0) <=> c6
gen_c4:c5:c68_14(+(x, 1)) <=> c4(gen_c4:c5:c68_14(x))
gen_C:Z9_14(0) <=> Z
gen_C:Z9_14(+(x, 1)) <=> C(gen_C:Z9_14(x), Z)
gen_c7:c8:c9:c10:c1110_14(0) <=> c9
gen_c7:c8:c9:c10:c1110_14(+(x, 1)) <=> c7(c, gen_c7:c8:c9:c10:c1110_14(x))


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

They will be analysed ascendingly in the following order:
eqZList < EQZLIST

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
A(gen_C:Z9_14(n12_14), gen_C:Z9_14(b)) -> gen_c4:c5:c68_14(n12_14), rt in Omega(1 + n12_14)

Induction Base:
A(gen_C:Z9_14(0), gen_C:Z9_14(b)) ->_R^Omega(1)
c6

Induction Step:
A(gen_C:Z9_14(+(n12_14, 1)), gen_C:Z9_14(b)) ->_R^Omega(1)
c4(A(gen_C:Z9_14(n12_14), gen_C:Z9_14(b))) ->_IH
c4(gen_c4:c5:c68_14(c13_14))

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

(52)
Complex Obligation (BEST)

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

(53)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
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
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

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11


Generator Equations:
gen_c4:c5:c68_14(0) <=> c6
gen_c4:c5:c68_14(+(x, 1)) <=> c4(gen_c4:c5:c68_14(x))
gen_C:Z9_14(0) <=> Z
gen_C:Z9_14(+(x, 1)) <=> C(gen_C:Z9_14(x), Z)
gen_c7:c8:c9:c10:c1110_14(0) <=> c9
gen_c7:c8:c9:c10:c1110_14(+(x, 1)) <=> c7(c, gen_c7:c8:c9:c10:c1110_14(x))


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

They will be analysed ascendingly in the following order:
eqZList < EQZLIST

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

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

(55)
BOUNDS(n^1, INF)

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

(56)
Obligation:
Innermost TRS:
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
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

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11


Lemmas:
A(gen_C:Z9_14(n12_14), gen_C:Z9_14(b)) -> gen_c4:c5:c68_14(n12_14), rt in Omega(1 + n12_14)


Generator Equations:
gen_c4:c5:c68_14(0) <=> c6
gen_c4:c5:c68_14(+(x, 1)) <=> c4(gen_c4:c5:c68_14(x))
gen_C:Z9_14(0) <=> Z
gen_C:Z9_14(+(x, 1)) <=> C(gen_C:Z9_14(x), Z)
gen_c7:c8:c9:c10:c1110_14(0) <=> c9
gen_c7:c8:c9:c10:c1110_14(+(x, 1)) <=> c7(c, gen_c7:c8:c9:c10:c1110_14(x))


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

They will be analysed ascendingly in the following order:
eqZList < EQZLIST

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eqZList(gen_C:Z9_14(+(1, n675_14)), gen_C:Z9_14(n675_14)) -> False, rt in Omega(0)

Induction Base:
eqZList(gen_C:Z9_14(+(1, 0)), gen_C:Z9_14(0)) ->_R^Omega(0)
False

Induction Step:
eqZList(gen_C:Z9_14(+(1, +(n675_14, 1))), gen_C:Z9_14(+(n675_14, 1))) ->_R^Omega(0)
and(eqZList(gen_C:Z9_14(+(1, n675_14)), gen_C:Z9_14(n675_14)), eqZList(Z, Z)) ->_IH
and(False, eqZList(Z, Z)) ->_R^Omega(0)
and(False, True) ->_R^Omega(0)
False

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(58)
Obligation:
Innermost TRS:
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
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

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11


Lemmas:
A(gen_C:Z9_14(n12_14), gen_C:Z9_14(b)) -> gen_c4:c5:c68_14(n12_14), rt in Omega(1 + n12_14)
eqZList(gen_C:Z9_14(+(1, n675_14)), gen_C:Z9_14(n675_14)) -> False, rt in Omega(0)


Generator Equations:
gen_c4:c5:c68_14(0) <=> c6
gen_c4:c5:c68_14(+(x, 1)) <=> c4(gen_c4:c5:c68_14(x))
gen_C:Z9_14(0) <=> Z
gen_C:Z9_14(+(x, 1)) <=> C(gen_C:Z9_14(x), Z)
gen_c7:c8:c9:c10:c1110_14(0) <=> c9
gen_c7:c8:c9:c10:c1110_14(+(x, 1)) <=> c7(c, gen_c7:c8:c9:c10:c1110_14(x))


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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14))) -> *11_14, rt in Omega(n2887_14)

Induction Base:
EQZLIST(gen_C:Z9_14(+(2, 0)), gen_C:Z9_14(+(1, 0)))

Induction Step:
EQZLIST(gen_C:Z9_14(+(2, +(n2887_14, 1))), gen_C:Z9_14(+(1, +(n2887_14, 1)))) ->_R^Omega(1)
c7(AND(eqZList(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14))), eqZList(Z, Z)), EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14)))) ->_L^Omega(0)
c7(AND(False, eqZList(Z, Z)), EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14)))) ->_R^Omega(0)
c7(AND(False, True), EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14)))) ->_R^Omega(0)
c7(c2, EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14)))) ->_IH
c7(c2, *11_14)

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

(60)
Obligation:
Innermost TRS:
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
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

Types:
A :: C:Z -> C:Z -> c4:c5:c6
C :: C:Z -> C:Z -> C:Z
c4 :: c4:c5:c6 -> c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
Z :: C:Z
c6 :: c4:c5:c6
EQZLIST :: C:Z -> C:Z -> c7:c8:c9:c10:c11
c7 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
AND :: False:True -> False:True -> c:c1:c2:c3
eqZList :: C:Z -> C:Z -> False:True
c8 :: c:c1:c2:c3 -> c7:c8:c9:c10:c11 -> c7:c8:c9:c10:c11
c9 :: c7:c8:c9:c10:c11
c10 :: c7:c8:c9:c10:c11
c11 :: c7:c8:c9:c10:c11
SECOND :: C:Z -> c12
c12 :: c12
FIRST :: C:Z -> c13
c13 :: c13
False :: False:True
c :: c:c1:c2:c3
True :: False:True
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
and :: False:True -> False:True -> False:True
a :: C:Z -> C:Z -> C:Z
second :: C:Z -> C:Z
first :: C:Z -> C:Z
hole_c4:c5:c61_14 :: c4:c5:c6
hole_C:Z2_14 :: C:Z
hole_c7:c8:c9:c10:c113_14 :: c7:c8:c9:c10:c11
hole_c:c1:c2:c34_14 :: c:c1:c2:c3
hole_False:True5_14 :: False:True
hole_c126_14 :: c12
hole_c137_14 :: c13
gen_c4:c5:c68_14 :: Nat -> c4:c5:c6
gen_C:Z9_14 :: Nat -> C:Z
gen_c7:c8:c9:c10:c1110_14 :: Nat -> c7:c8:c9:c10:c11


Lemmas:
A(gen_C:Z9_14(n12_14), gen_C:Z9_14(b)) -> gen_c4:c5:c68_14(n12_14), rt in Omega(1 + n12_14)
eqZList(gen_C:Z9_14(+(1, n675_14)), gen_C:Z9_14(n675_14)) -> False, rt in Omega(0)
EQZLIST(gen_C:Z9_14(+(2, n2887_14)), gen_C:Z9_14(+(1, n2887_14))) -> *11_14, rt in Omega(n2887_14)


Generator Equations:
gen_c4:c5:c68_14(0) <=> c6
gen_c4:c5:c68_14(+(x, 1)) <=> c4(gen_c4:c5:c68_14(x))
gen_C:Z9_14(0) <=> Z
gen_C:Z9_14(+(x, 1)) <=> C(gen_C:Z9_14(x), Z)
gen_c7:c8:c9:c10:c1110_14(0) <=> c9
gen_c7:c8:c9:c10:c1110_14(+(x, 1)) <=> c7(c, gen_c7:c8:c9:c10:c1110_14(x))


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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
a(gen_C:Z9_14(n5491_14), gen_C:Z9_14(b)) -> gen_C:Z9_14(n5491_14), rt in Omega(0)

Induction Base:
a(gen_C:Z9_14(0), gen_C:Z9_14(b)) ->_R^Omega(0)
Z

Induction Step:
a(gen_C:Z9_14(+(n5491_14, 1)), gen_C:Z9_14(b)) ->_R^Omega(0)
C(a(gen_C:Z9_14(n5491_14), gen_C:Z9_14(b)), a(Z, C(gen_C:Z9_14(n5491_14), Z))) ->_IH
C(gen_C:Z9_14(c5492_14), a(Z, C(gen_C:Z9_14(n5491_14), Z))) ->_R^Omega(0)
C(gen_C:Z9_14(n5491_14), Z)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(62)
BOUNDS(1, INF)