KILLED
proof of input_8DZDj2Z71a.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 12 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 267 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 8385 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
(23) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(24) CdtProblem
    (25) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CpxRelTRS
        (31) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (32) CpxTRS
        (33) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (34) CpxWeightedTrs
            (35) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (36) CpxTypedWeightedTrs
            (37) CompletionProof [UPPER BOUND(ID), 0 ms]
            (38) CpxTypedWeightedCompleteTrs
                (39) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (40) CpxTypedWeightedCompleteTrs
                (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (42) CpxRNTS
                (43) InliningProof [UPPER BOUND(ID), 278 ms]
                (44) CpxRNTS
                (45) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (46) CpxRNTS
            (47) CompletionProof [UPPER BOUND(ID), 0 ms]
            (48) CpxTypedWeightedCompleteTrs
                (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (50) CpxRNTS
    (51) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
        (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (54) CdtProblem
        (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (56) CdtProblem
        (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CdtProblem
        (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (60) CdtProblem
        (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (62) CdtProblem
        (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CdtProblem
        (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CdtProblem
        (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (68) CdtProblem
        (69) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (70) CdtProblem
(71) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(72) CpxWeightedTrs
    (73) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CpxTypedWeightedTrs
    (75) CompletionProof [UPPER BOUND(ID), 0 ms]
    (76) CpxTypedWeightedCompleteTrs
        (77) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (78) CpxTypedWeightedCompleteTrs
        (79) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (80) CpxRNTS
        (81) InliningProof [UPPER BOUND(ID), 236 ms]
        (82) CpxRNTS
        (83) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (84) CpxRNTS
    (85) CompletionProof [UPPER BOUND(ID), 0 ms]
    (86) CpxTypedWeightedCompleteTrs
        (87) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (88) CpxRNTS


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   a__f(a, b, X) -> a__f(X, X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

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

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

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   a__f(a, b, X) -> a__f(X, X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

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

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   a__f(a, b, X) -> a__f(X, X, mark(X))
   a__c -> a
   a__c -> b
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
   a__f(X1, X2, X3) -> f(X1, X2, X3)
   a__c -> c

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

(5) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   A__C -> c5
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   MARK(c) -> c7(A__C)
   MARK(a) -> c8
   MARK(b) -> c9
S tuples:
   A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   A__C -> c5
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   MARK(c) -> c7(A__C)
   MARK(a) -> c8
   MARK(b) -> c9
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, A__C, MARK_1

Compound Symbols:   c1_2, c2, c3, c4, c5, c6_2, c7_1, c8, c9


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

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

(8)
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__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   A__C -> c5
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   MARK(c) -> c7(A__C)
   MARK(a) -> c8
   MARK(b) -> c9

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

   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b

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

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

(10)
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__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   A__C -> c5
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   MARK(c) -> c7(A__C)
   MARK(a) -> c8
   MARK(b) -> c9

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

   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b

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

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

(12)
Obligation:
Innermost TRS:
Rules:
A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
A__F(z0, z1, z2) -> c2
A__C -> c3
A__C -> c4
A__C -> c5
MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
MARK(c) -> c7(A__C)
MARK(a) -> c8
MARK(b) -> c9
a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
a__f(z0, z1, z2) -> f(z0, z1, z2)
a__c -> a
a__c -> b
a__c -> c
mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
mark(c) -> a__c
mark(a) -> a
mark(b) -> b

Types:
A__F :: a:b:f:c -> a:b:f:c -> a:b:f:c -> c1:c2
a :: a:b:f:c
b :: a:b:f:c
c1 :: c1:c2 -> c6:c7:c8:c9 -> c1:c2
mark :: a:b:f:c -> a:b:f:c
MARK :: a:b:f:c -> c6:c7:c8:c9
c2 :: c1:c2
A__C :: c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c6 :: c1:c2 -> c6:c7:c8:c9 -> c6:c7:c8:c9
c :: a:b:f:c
c7 :: c3:c4:c5 -> c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
hole_c1:c21_10 :: c1:c2
hole_a:b:f:c2_10 :: a:b:f:c
hole_c6:c7:c8:c93_10 :: c6:c7:c8:c9
hole_c3:c4:c54_10 :: c3:c4:c5
gen_c1:c25_10 :: Nat -> c1:c2
gen_a:b:f:c6_10 :: Nat -> a:b:f:c
gen_c6:c7:c8:c97_10 :: Nat -> c6:c7:c8:c9

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
A__F, mark, MARK, a__f

They will be analysed ascendingly in the following order:
mark < A__F
A__F = MARK
mark < MARK
mark = a__f

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

(14)
Obligation:
Innermost TRS:
Rules:
A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
A__F(z0, z1, z2) -> c2
A__C -> c3
A__C -> c4
A__C -> c5
MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
MARK(c) -> c7(A__C)
MARK(a) -> c8
MARK(b) -> c9
a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
a__f(z0, z1, z2) -> f(z0, z1, z2)
a__c -> a
a__c -> b
a__c -> c
mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
mark(c) -> a__c
mark(a) -> a
mark(b) -> b

Types:
A__F :: a:b:f:c -> a:b:f:c -> a:b:f:c -> c1:c2
a :: a:b:f:c
b :: a:b:f:c
c1 :: c1:c2 -> c6:c7:c8:c9 -> c1:c2
mark :: a:b:f:c -> a:b:f:c
MARK :: a:b:f:c -> c6:c7:c8:c9
c2 :: c1:c2
A__C :: c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c6 :: c1:c2 -> c6:c7:c8:c9 -> c6:c7:c8:c9
c :: a:b:f:c
c7 :: c3:c4:c5 -> c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
hole_c1:c21_10 :: c1:c2
hole_a:b:f:c2_10 :: a:b:f:c
hole_c6:c7:c8:c93_10 :: c6:c7:c8:c9
hole_c3:c4:c54_10 :: c3:c4:c5
gen_c1:c25_10 :: Nat -> c1:c2
gen_a:b:f:c6_10 :: Nat -> a:b:f:c
gen_c6:c7:c8:c97_10 :: Nat -> c6:c7:c8:c9


Generator Equations:
gen_c1:c25_10(0) <=> c2
gen_c1:c25_10(+(x, 1)) <=> c1(gen_c1:c25_10(x), c7(c3))
gen_a:b:f:c6_10(0) <=> a
gen_a:b:f:c6_10(+(x, 1)) <=> f(a, a, gen_a:b:f:c6_10(x))
gen_c6:c7:c8:c97_10(0) <=> c7(c3)
gen_c6:c7:c8:c97_10(+(x, 1)) <=> c6(c2, gen_c6:c7:c8:c97_10(x))


The following defined symbols remain to be analysed:
a__f, A__F, mark, MARK

They will be analysed ascendingly in the following order:
mark < A__F
A__F = MARK
mark < MARK
mark = a__f

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mark(gen_a:b:f:c6_10(n20_10)) -> gen_a:b:f:c6_10(n20_10), rt in Omega(0)

Induction Base:
mark(gen_a:b:f:c6_10(0)) ->_R^Omega(0)
a

Induction Step:
mark(gen_a:b:f:c6_10(+(n20_10, 1))) ->_R^Omega(0)
a__f(a, a, mark(gen_a:b:f:c6_10(n20_10))) ->_IH
a__f(a, a, gen_a:b:f:c6_10(c21_10)) ->_R^Omega(0)
f(a, a, gen_a:b:f:c6_10(n20_10))

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

(16)
Obligation:
Innermost TRS:
Rules:
A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
A__F(z0, z1, z2) -> c2
A__C -> c3
A__C -> c4
A__C -> c5
MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
MARK(c) -> c7(A__C)
MARK(a) -> c8
MARK(b) -> c9
a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
a__f(z0, z1, z2) -> f(z0, z1, z2)
a__c -> a
a__c -> b
a__c -> c
mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
mark(c) -> a__c
mark(a) -> a
mark(b) -> b

Types:
A__F :: a:b:f:c -> a:b:f:c -> a:b:f:c -> c1:c2
a :: a:b:f:c
b :: a:b:f:c
c1 :: c1:c2 -> c6:c7:c8:c9 -> c1:c2
mark :: a:b:f:c -> a:b:f:c
MARK :: a:b:f:c -> c6:c7:c8:c9
c2 :: c1:c2
A__C :: c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c6 :: c1:c2 -> c6:c7:c8:c9 -> c6:c7:c8:c9
c :: a:b:f:c
c7 :: c3:c4:c5 -> c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
hole_c1:c21_10 :: c1:c2
hole_a:b:f:c2_10 :: a:b:f:c
hole_c6:c7:c8:c93_10 :: c6:c7:c8:c9
hole_c3:c4:c54_10 :: c3:c4:c5
gen_c1:c25_10 :: Nat -> c1:c2
gen_a:b:f:c6_10 :: Nat -> a:b:f:c
gen_c6:c7:c8:c97_10 :: Nat -> c6:c7:c8:c9


Lemmas:
mark(gen_a:b:f:c6_10(n20_10)) -> gen_a:b:f:c6_10(n20_10), rt in Omega(0)


Generator Equations:
gen_c1:c25_10(0) <=> c2
gen_c1:c25_10(+(x, 1)) <=> c1(gen_c1:c25_10(x), c7(c3))
gen_a:b:f:c6_10(0) <=> a
gen_a:b:f:c6_10(+(x, 1)) <=> f(a, a, gen_a:b:f:c6_10(x))
gen_c6:c7:c8:c97_10(0) <=> c7(c3)
gen_c6:c7:c8:c97_10(+(x, 1)) <=> c6(c2, gen_c6:c7:c8:c97_10(x))


The following defined symbols remain to be analysed:
a__f, A__F, MARK

They will be analysed ascendingly in the following order:
mark < A__F
A__F = MARK
mark < MARK
mark = a__f

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MARK(gen_a:b:f:c6_10(+(1, n432_10))) -> *8_10, rt in Omega(n432_10)

Induction Base:
MARK(gen_a:b:f:c6_10(+(1, 0)))

Induction Step:
MARK(gen_a:b:f:c6_10(+(1, +(n432_10, 1)))) ->_R^Omega(1)
c6(A__F(a, a, mark(gen_a:b:f:c6_10(+(1, n432_10)))), MARK(gen_a:b:f:c6_10(+(1, n432_10)))) ->_L^Omega(0)
c6(A__F(a, a, gen_a:b:f:c6_10(+(1, n432_10))), MARK(gen_a:b:f:c6_10(+(1, n432_10)))) ->_R^Omega(1)
c6(c2, MARK(gen_a:b:f:c6_10(+(1, n432_10)))) ->_IH
c6(c2, *8_10)

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

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
A__F(z0, z1, z2) -> c2
A__C -> c3
A__C -> c4
A__C -> c5
MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
MARK(c) -> c7(A__C)
MARK(a) -> c8
MARK(b) -> c9
a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
a__f(z0, z1, z2) -> f(z0, z1, z2)
a__c -> a
a__c -> b
a__c -> c
mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
mark(c) -> a__c
mark(a) -> a
mark(b) -> b

Types:
A__F :: a:b:f:c -> a:b:f:c -> a:b:f:c -> c1:c2
a :: a:b:f:c
b :: a:b:f:c
c1 :: c1:c2 -> c6:c7:c8:c9 -> c1:c2
mark :: a:b:f:c -> a:b:f:c
MARK :: a:b:f:c -> c6:c7:c8:c9
c2 :: c1:c2
A__C :: c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c6 :: c1:c2 -> c6:c7:c8:c9 -> c6:c7:c8:c9
c :: a:b:f:c
c7 :: c3:c4:c5 -> c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
hole_c1:c21_10 :: c1:c2
hole_a:b:f:c2_10 :: a:b:f:c
hole_c6:c7:c8:c93_10 :: c6:c7:c8:c9
hole_c3:c4:c54_10 :: c3:c4:c5
gen_c1:c25_10 :: Nat -> c1:c2
gen_a:b:f:c6_10 :: Nat -> a:b:f:c
gen_c6:c7:c8:c97_10 :: Nat -> c6:c7:c8:c9


Lemmas:
mark(gen_a:b:f:c6_10(n20_10)) -> gen_a:b:f:c6_10(n20_10), rt in Omega(0)


Generator Equations:
gen_c1:c25_10(0) <=> c2
gen_c1:c25_10(+(x, 1)) <=> c1(gen_c1:c25_10(x), c7(c3))
gen_a:b:f:c6_10(0) <=> a
gen_a:b:f:c6_10(+(x, 1)) <=> f(a, a, gen_a:b:f:c6_10(x))
gen_c6:c7:c8:c97_10(0) <=> c7(c3)
gen_c6:c7:c8:c97_10(+(x, 1)) <=> c6(c2, gen_c6:c7:c8:c97_10(x))


The following defined symbols remain to be analysed:
MARK, A__F

They will be analysed ascendingly in the following order:
A__F = MARK

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
A__F(z0, z1, z2) -> c2
A__C -> c3
A__C -> c4
A__C -> c5
MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
MARK(c) -> c7(A__C)
MARK(a) -> c8
MARK(b) -> c9
a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
a__f(z0, z1, z2) -> f(z0, z1, z2)
a__c -> a
a__c -> b
a__c -> c
mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
mark(c) -> a__c
mark(a) -> a
mark(b) -> b

Types:
A__F :: a:b:f:c -> a:b:f:c -> a:b:f:c -> c1:c2
a :: a:b:f:c
b :: a:b:f:c
c1 :: c1:c2 -> c6:c7:c8:c9 -> c1:c2
mark :: a:b:f:c -> a:b:f:c
MARK :: a:b:f:c -> c6:c7:c8:c9
c2 :: c1:c2
A__C :: c3:c4:c5
c3 :: c3:c4:c5
c4 :: c3:c4:c5
c5 :: c3:c4:c5
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c6 :: c1:c2 -> c6:c7:c8:c9 -> c6:c7:c8:c9
c :: a:b:f:c
c7 :: c3:c4:c5 -> c6:c7:c8:c9
c8 :: c6:c7:c8:c9
c9 :: c6:c7:c8:c9
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
hole_c1:c21_10 :: c1:c2
hole_a:b:f:c2_10 :: a:b:f:c
hole_c6:c7:c8:c93_10 :: c6:c7:c8:c9
hole_c3:c4:c54_10 :: c3:c4:c5
gen_c1:c25_10 :: Nat -> c1:c2
gen_a:b:f:c6_10 :: Nat -> a:b:f:c
gen_c6:c7:c8:c97_10 :: Nat -> c6:c7:c8:c9


Lemmas:
mark(gen_a:b:f:c6_10(n20_10)) -> gen_a:b:f:c6_10(n20_10), rt in Omega(0)
MARK(gen_a:b:f:c6_10(+(1, n432_10))) -> *8_10, rt in Omega(n432_10)


Generator Equations:
gen_c1:c25_10(0) <=> c2
gen_c1:c25_10(+(x, 1)) <=> c1(gen_c1:c25_10(x), c7(c3))
gen_a:b:f:c6_10(0) <=> a
gen_a:b:f:c6_10(+(x, 1)) <=> f(a, a, gen_a:b:f:c6_10(x))
gen_c6:c7:c8:c97_10(0) <=> c7(c3)
gen_c6:c7:c8:c97_10(+(x, 1)) <=> c6(c2, gen_c6:c7:c8:c97_10(x))


The following defined symbols remain to be analysed:
A__F

They will be analysed ascendingly in the following order:
A__F = MARK

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

(23) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   A__C -> c5
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   MARK(c) -> c7(A__C)
   MARK(a) -> c8
   MARK(b) -> c9
S tuples:
   A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   A__C -> c5
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   MARK(c) -> c7(A__C)
   MARK(a) -> c8
   MARK(b) -> c9
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, A__C, MARK_1

Compound Symbols:   c1_2, c2, c3, c4, c5, c6_2, c7_1, c8, c9


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

(25) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   A__F(z0, z1, z2) -> c2
   A__C -> c3
   A__C -> c4
   MARK(a) -> c8
   A__C -> c5
   MARK(c) -> c7(A__C)
   MARK(b) -> c9

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
S tuples:
   A__F(a, b, z0) -> c1(A__F(z0, z0, mark(z0)), MARK(z0))
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, MARK_1

Compound Symbols:   c1_2, c6_2


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

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   A__F(a, b, z0) -> c1(MARK(z0))
S tuples:
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   A__F(a, b, z0) -> c1(MARK(z0))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   MARK_1, A__F_3

Compound Symbols:   c6_2, c1_1


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

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

(30)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   A__F(a, b, z0) -> c1(MARK(z0))

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

   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b

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

(31) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(32)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   A__F(a, b, z0) -> c1(MARK(z0))
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b

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

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

(34)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2)) [1]
   A__F(a, b, z0) -> c1(MARK(z0)) [1]
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0)) [0]
   a__f(z0, z1, z2) -> f(z0, z1, z2) [0]
   a__c -> a [0]
   a__c -> b [0]
   a__c -> c [0]
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2)) [0]
   mark(c) -> a__c [0]
   mark(a) -> a [0]
   mark(b) -> b [0]

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

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

(36)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2)) [1]
   A__F(a, b, z0) -> c1(MARK(z0)) [1]
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0)) [0]
   a__f(z0, z1, z2) -> f(z0, z1, z2) [0]
   a__c -> a [0]
   a__c -> b [0]
   a__c -> c [0]
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2)) [0]
   mark(c) -> a__c [0]
   mark(a) -> a [0]
   mark(b) -> b [0]

The TRS has the following type information:
MARK :: f:a:b:c -> c6
f :: f:a:b:c -> f:a:b:c -> f:a:b:c -> f:a:b:c
c6 :: c1 -> c6 -> c6
A__F :: f:a:b:c -> f:a:b:c -> f:a:b:c -> c1
mark :: f:a:b:c -> f:a:b:c
a :: f:a:b:c
b :: f:a:b:c
c1 :: c6 -> c1
a__f :: f:a:b:c -> f:a:b:c -> f:a:b:c -> f:a:b:c
a__c :: f:a:b:c
c :: f:a:b:c

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

(37) 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:

   MARK_1
   A__F_3

(c) The following functions are completely defined:

   a__f_3
   a__c
   mark_1

Due to the following rules being added:

   a__f(v0, v1, v2) -> null_a__f [0]
   a__c -> null_a__c [0]
   mark(v0) -> null_mark [0]

And the following fresh constants:    null_a__f, null_a__c, null_mark, const, const1

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

(38)
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:

   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2)) [1]
   A__F(a, b, z0) -> c1(MARK(z0)) [1]
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0)) [0]
   a__f(z0, z1, z2) -> f(z0, z1, z2) [0]
   a__c -> a [0]
   a__c -> b [0]
   a__c -> c [0]
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2)) [0]
   mark(c) -> a__c [0]
   mark(a) -> a [0]
   mark(b) -> b [0]
   a__f(v0, v1, v2) -> null_a__f [0]
   a__c -> null_a__c [0]
   mark(v0) -> null_mark [0]

The TRS has the following type information:
MARK :: f:a:b:c:null_a__f:null_a__c:null_mark -> c6
f :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark
c6 :: c1 -> c6 -> c6
A__F :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> c1
mark :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark
a :: f:a:b:c:null_a__f:null_a__c:null_mark
b :: f:a:b:c:null_a__f:null_a__c:null_mark
c1 :: c6 -> c1
a__f :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark
a__c :: f:a:b:c:null_a__f:null_a__c:null_mark
c :: f:a:b:c:null_a__f:null_a__c:null_mark
null_a__f :: f:a:b:c:null_a__f:null_a__c:null_mark
null_a__c :: f:a:b:c:null_a__f:null_a__c:null_mark
null_mark :: f:a:b:c:null_a__f:null_a__c:null_mark
const :: c6
const1 :: c1

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

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

(40)
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:

   MARK(f(z0, z1, f(z0', z1', z2'))) -> c6(A__F(z0, z1, a__f(z0', z1', mark(z2'))), MARK(f(z0', z1', z2'))) [1]
   MARK(f(z0, z1, c)) -> c6(A__F(z0, z1, a__c), MARK(c)) [1]
   MARK(f(z0, z1, a)) -> c6(A__F(z0, z1, a), MARK(a)) [1]
   MARK(f(z0, z1, b)) -> c6(A__F(z0, z1, b), MARK(b)) [1]
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, null_mark), MARK(z2)) [1]
   A__F(a, b, z0) -> c1(MARK(z0)) [1]
   a__f(a, b, f(z0'', z1'', z2'')) -> a__f(f(z0'', z1'', z2''), f(z0'', z1'', z2''), a__f(z0'', z1'', mark(z2''))) [0]
   a__f(a, b, c) -> a__f(c, c, a__c) [0]
   a__f(a, b, a) -> a__f(a, a, a) [0]
   a__f(a, b, b) -> a__f(b, b, b) [0]
   a__f(a, b, z0) -> a__f(z0, z0, null_mark) [0]
   a__f(z0, z1, z2) -> f(z0, z1, z2) [0]
   a__c -> a [0]
   a__c -> b [0]
   a__c -> c [0]
   mark(f(z0, z1, f(z01, z11, z21))) -> a__f(z0, z1, a__f(z01, z11, mark(z21))) [0]
   mark(f(z0, z1, c)) -> a__f(z0, z1, a__c) [0]
   mark(f(z0, z1, a)) -> a__f(z0, z1, a) [0]
   mark(f(z0, z1, b)) -> a__f(z0, z1, b) [0]
   mark(f(z0, z1, z2)) -> a__f(z0, z1, null_mark) [0]
   mark(c) -> a__c [0]
   mark(a) -> a [0]
   mark(b) -> b [0]
   a__f(v0, v1, v2) -> null_a__f [0]
   a__c -> null_a__c [0]
   mark(v0) -> null_mark [0]

The TRS has the following type information:
MARK :: f:a:b:c:null_a__f:null_a__c:null_mark -> c6
f :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark
c6 :: c1 -> c6 -> c6
A__F :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> c1
mark :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark
a :: f:a:b:c:null_a__f:null_a__c:null_mark
b :: f:a:b:c:null_a__f:null_a__c:null_mark
c1 :: c6 -> c1
a__f :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark
a__c :: f:a:b:c:null_a__f:null_a__c:null_mark
c :: f:a:b:c:null_a__f:null_a__c:null_mark
null_a__f :: f:a:b:c:null_a__f:null_a__c:null_mark
null_a__c :: f:a:b:c:null_a__f:null_a__c:null_mark
null_mark :: f:a:b:c:null_a__f:null_a__c:null_mark
const :: c6
const1 :: c1

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

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

   a => 0
   b => 1
   c => 2
   null_a__f => 0
   null_a__c => 0
   null_mark => 0
   const => 0
   const1 => 0

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   A__F(z, z', z'') -{ 1 }-> 1 + MARK(z0) :|: z'' = z0, z0 >= 0, z' = 1, z = 0
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, a__f(z0', z1', mark(z2'))) + MARK(1 + z0' + z1' + z2') :|: z1 >= 0, z = 1 + z0 + z1 + (1 + z0' + z1' + z2'), z0' >= 0, z1' >= 0, z2' >= 0, z0 >= 0
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, a__c) + MARK(2) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 1) + MARK(1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 1
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 0) + MARK(z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 0) + MARK(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 0
   a__c -{ 0 }-> 2 :|: 
   a__c -{ 0 }-> 1 :|: 
   a__c -{ 0 }-> 0 :|: 
   a__f(z, z', z'') -{ 0 }-> a__f(z0, z0, 0) :|: z'' = z0, z0 >= 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(2, 2, a__c) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(1, 1, 1) :|: z' = 1, z = 0, z'' = 1
   a__f(z, z', z'') -{ 0 }-> a__f(0, 0, 0) :|: z'' = 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(1 + z0'' + z1'' + z2'', 1 + z0'' + z1'' + z2'', a__f(z0'', z1'', mark(z2''))) :|: z'' = 1 + z0'' + z1'' + z2'', z' = 1, z0'' >= 0, z = 0, z2'' >= 0, z1'' >= 0
   a__f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   a__f(z, z', z'') -{ 0 }-> 1 + z0 + z1 + z2 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   mark(z) -{ 0 }-> a__f(z0, z1, a__f(z01, z11, mark(z21))) :|: z = 1 + z0 + z1 + (1 + z01 + z11 + z21), z21 >= 0, z1 >= 0, z11 >= 0, z01 >= 0, z0 >= 0
   mark(z) -{ 0 }-> a__f(z0, z1, a__c) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   mark(z) -{ 0 }-> a__f(z0, z1, 1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 1
   mark(z) -{ 0 }-> a__f(z0, z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 0
   mark(z) -{ 0 }-> a__f(z0, z1, 0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0
   mark(z) -{ 0 }-> a__c :|: z = 2
   mark(z) -{ 0 }-> 1 :|: z = 1
   mark(z) -{ 0 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

(43) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   a__c -{ 0 }-> 0 :|: 
   a__c -{ 0 }-> 1 :|: 
   a__c -{ 0 }-> 2 :|: 

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   A__F(z, z', z'') -{ 1 }-> 1 + MARK(z0) :|: z'' = z0, z0 >= 0, z' = 1, z = 0
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, a__f(z0', z1', mark(z2'))) + MARK(1 + z0' + z1' + z2') :|: z1 >= 0, z = 1 + z0 + z1 + (1 + z0' + z1' + z2'), z0' >= 0, z1' >= 0, z2' >= 0, z0 >= 0
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 2) + MARK(2) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 1) + MARK(2) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 1) + MARK(1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 1
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 0) + MARK(z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 0) + MARK(2) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 0) + MARK(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 0
   a__c -{ 0 }-> 2 :|: 
   a__c -{ 0 }-> 1 :|: 
   a__c -{ 0 }-> 0 :|: 
   a__f(z, z', z'') -{ 0 }-> a__f(z0, z0, 0) :|: z'' = z0, z0 >= 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(2, 2, 2) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(2, 2, 1) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(2, 2, 0) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(1, 1, 1) :|: z' = 1, z = 0, z'' = 1
   a__f(z, z', z'') -{ 0 }-> a__f(0, 0, 0) :|: z'' = 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(1 + z0'' + z1'' + z2'', 1 + z0'' + z1'' + z2'', a__f(z0'', z1'', mark(z2''))) :|: z'' = 1 + z0'' + z1'' + z2'', z' = 1, z0'' >= 0, z = 0, z2'' >= 0, z1'' >= 0
   a__f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   a__f(z, z', z'') -{ 0 }-> 1 + z0 + z1 + z2 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   mark(z) -{ 0 }-> a__f(z0, z1, a__f(z01, z11, mark(z21))) :|: z = 1 + z0 + z1 + (1 + z01 + z11 + z21), z21 >= 0, z1 >= 0, z11 >= 0, z01 >= 0, z0 >= 0
   mark(z) -{ 0 }-> a__f(z0, z1, 2) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   mark(z) -{ 0 }-> a__f(z0, z1, 1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 1
   mark(z) -{ 0 }-> a__f(z0, z1, 1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   mark(z) -{ 0 }-> a__f(z0, z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 0
   mark(z) -{ 0 }-> a__f(z0, z1, 0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0
   mark(z) -{ 0 }-> a__f(z0, z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   mark(z) -{ 0 }-> 2 :|: z = 2
   mark(z) -{ 0 }-> 1 :|: z = 1
   mark(z) -{ 0 }-> 1 :|: z = 2
   mark(z) -{ 0 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   mark(z) -{ 0 }-> 0 :|: z = 2


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

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   A__F(z, z', z'') -{ 1 }-> 1 + MARK(z'') :|: z'' >= 0, z' = 1, z = 0
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, a__f(z0', z1', mark(z2'))) + MARK(1 + z0' + z1' + z2') :|: z1 >= 0, z = 1 + z0 + z1 + (1 + z0' + z1' + z2'), z0' >= 0, z1' >= 0, z2' >= 0, z0 >= 0
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 2) + MARK(2) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 1) + MARK(2) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 1) + MARK(1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 1
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 0) + MARK(z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 0) + MARK(2) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, 0) + MARK(0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 0
   a__c -{ 0 }-> 2 :|: 
   a__c -{ 0 }-> 1 :|: 
   a__c -{ 0 }-> 0 :|: 
   a__f(z, z', z'') -{ 0 }-> a__f(z'', z'', 0) :|: z'' >= 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(2, 2, 2) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(2, 2, 1) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(2, 2, 0) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(1, 1, 1) :|: z' = 1, z = 0, z'' = 1
   a__f(z, z', z'') -{ 0 }-> a__f(0, 0, 0) :|: z'' = 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 0 }-> a__f(1 + z0'' + z1'' + z2'', 1 + z0'' + z1'' + z2'', a__f(z0'', z1'', mark(z2''))) :|: z'' = 1 + z0'' + z1'' + z2'', z' = 1, z0'' >= 0, z = 0, z2'' >= 0, z1'' >= 0
   a__f(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   a__f(z, z', z'') -{ 0 }-> 1 + z + z' + z'' :|: z' >= 0, z >= 0, z'' >= 0
   mark(z) -{ 0 }-> a__f(z0, z1, a__f(z01, z11, mark(z21))) :|: z = 1 + z0 + z1 + (1 + z01 + z11 + z21), z21 >= 0, z1 >= 0, z11 >= 0, z01 >= 0, z0 >= 0
   mark(z) -{ 0 }-> a__f(z0, z1, 2) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   mark(z) -{ 0 }-> a__f(z0, z1, 1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 1
   mark(z) -{ 0 }-> a__f(z0, z1, 1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   mark(z) -{ 0 }-> a__f(z0, z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 0
   mark(z) -{ 0 }-> a__f(z0, z1, 0) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0
   mark(z) -{ 0 }-> a__f(z0, z1, 0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1 + 2
   mark(z) -{ 0 }-> 2 :|: z = 2
   mark(z) -{ 0 }-> 1 :|: z = 1
   mark(z) -{ 0 }-> 1 :|: z = 2
   mark(z) -{ 0 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: z >= 0
   mark(z) -{ 0 }-> 0 :|: z = 2


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

(47) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   a__f(v0, v1, v2) -> null_a__f [0]
   a__c -> null_a__c [0]
   mark(v0) -> null_mark [0]
   MARK(v0) -> null_MARK [0]
   A__F(v0, v1, v2) -> null_A__F [0]

And the following fresh constants:    null_a__f, null_a__c, null_mark, null_MARK, null_A__F

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

(48)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2)) [1]
   A__F(a, b, z0) -> c1(MARK(z0)) [1]
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0)) [0]
   a__f(z0, z1, z2) -> f(z0, z1, z2) [0]
   a__c -> a [0]
   a__c -> b [0]
   a__c -> c [0]
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2)) [0]
   mark(c) -> a__c [0]
   mark(a) -> a [0]
   mark(b) -> b [0]
   a__f(v0, v1, v2) -> null_a__f [0]
   a__c -> null_a__c [0]
   mark(v0) -> null_mark [0]
   MARK(v0) -> null_MARK [0]
   A__F(v0, v1, v2) -> null_A__F [0]

The TRS has the following type information:
MARK :: f:a:b:c:null_a__f:null_a__c:null_mark -> c6:null_MARK
f :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark
c6 :: c1:null_A__F -> c6:null_MARK -> c6:null_MARK
A__F :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> c1:null_A__F
mark :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark
a :: f:a:b:c:null_a__f:null_a__c:null_mark
b :: f:a:b:c:null_a__f:null_a__c:null_mark
c1 :: c6:null_MARK -> c1:null_A__F
a__f :: f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark -> f:a:b:c:null_a__f:null_a__c:null_mark
a__c :: f:a:b:c:null_a__f:null_a__c:null_mark
c :: f:a:b:c:null_a__f:null_a__c:null_mark
null_a__f :: f:a:b:c:null_a__f:null_a__c:null_mark
null_a__c :: f:a:b:c:null_a__f:null_a__c:null_mark
null_mark :: f:a:b:c:null_a__f:null_a__c:null_mark
null_MARK :: c6:null_MARK
null_A__F :: c1:null_A__F

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

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

   a => 0
   b => 1
   c => 2
   null_a__f => 0
   null_a__c => 0
   null_mark => 0
   null_MARK => 0
   null_A__F => 0

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   A__F(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   A__F(z, z', z'') -{ 1 }-> 1 + MARK(z0) :|: z'' = z0, z0 >= 0, z' = 1, z = 0
   MARK(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   MARK(z) -{ 1 }-> 1 + A__F(z0, z1, mark(z2)) + MARK(z2) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0
   a__c -{ 0 }-> 2 :|: 
   a__c -{ 0 }-> 1 :|: 
   a__c -{ 0 }-> 0 :|: 
   a__f(z, z', z'') -{ 0 }-> a__f(z0, z0, mark(z0)) :|: z'' = z0, z0 >= 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   a__f(z, z', z'') -{ 0 }-> 1 + z0 + z1 + z2 :|: z'' = z2, z = z0, z1 >= 0, z' = z1, z0 >= 0, z2 >= 0
   mark(z) -{ 0 }-> a__f(z0, z1, mark(z2)) :|: z1 >= 0, z = 1 + z0 + z1 + z2, z0 >= 0, z2 >= 0
   mark(z) -{ 0 }-> a__c :|: z = 2
   mark(z) -{ 0 }-> 1 :|: z = 1
   mark(z) -{ 0 }-> 0 :|: z = 0
   mark(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

(51) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace A__F(a, b, z0) -> c1(MARK(z0)) by 
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
S tuples:
   MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2))
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   MARK_1, A__F_3

Compound Symbols:   c6_2, c1_1


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MARK(f(z0, z1, z2)) -> c6(A__F(z0, z1, mark(z2)), MARK(z2)) by 
   MARK(f(x0, x1, f(z0, z1, z2))) -> c6(A__F(x0, x1, a__f(z0, z1, mark(z2))), MARK(f(z0, z1, z2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c), MARK(c))
   MARK(f(x0, x1, a)) -> c6(A__F(x0, x1, a), MARK(a))
   MARK(f(x0, x1, b)) -> c6(A__F(x0, x1, b), MARK(b))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(z0, z1, z2))) -> c6(A__F(x0, x1, a__f(z0, z1, mark(z2))), MARK(f(z0, z1, z2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c), MARK(c))
   MARK(f(x0, x1, a)) -> c6(A__F(x0, x1, a), MARK(a))
   MARK(f(x0, x1, b)) -> c6(A__F(x0, x1, b), MARK(b))
S tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(z0, z1, z2))) -> c6(A__F(x0, x1, a__f(z0, z1, mark(z2))), MARK(f(z0, z1, z2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c), MARK(c))
   MARK(f(x0, x1, a)) -> c6(A__F(x0, x1, a), MARK(a))
   MARK(f(x0, x1, b)) -> c6(A__F(x0, x1, b), MARK(b))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, MARK_1

Compound Symbols:   c1_1, c6_2


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

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   MARK(f(x0, x1, b)) -> c6(A__F(x0, x1, b), MARK(b))
   MARK(f(x0, x1, a)) -> c6(A__F(x0, x1, a), MARK(a))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(z0, z1, z2))) -> c6(A__F(x0, x1, a__f(z0, z1, mark(z2))), MARK(f(z0, z1, z2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c), MARK(c))
S tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(z0, z1, z2))) -> c6(A__F(x0, x1, a__f(z0, z1, mark(z2))), MARK(f(z0, z1, z2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c), MARK(c))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, MARK_1

Compound Symbols:   c1_1, c6_2


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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(z0, z1, z2))) -> c6(A__F(x0, x1, a__f(z0, z1, mark(z2))), MARK(f(z0, z1, z2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c))
S tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(z0, z1, z2))) -> c6(A__F(x0, x1, a__f(z0, z1, mark(z2))), MARK(f(z0, z1, z2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, MARK_1

Compound Symbols:   c1_1, c6_2, c6_1


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MARK(f(x0, x1, f(z0, z1, z2))) -> c6(A__F(x0, x1, a__f(z0, z1, mark(z2))), MARK(f(z0, z1, z2))) by 
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)), MARK(f(x2, x3, c)))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)), MARK(f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)), MARK(f(x2, x3, b)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c))
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)), MARK(f(x2, x3, c)))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)), MARK(f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)), MARK(f(x2, x3, b)))
S tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c))
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)), MARK(f(x2, x3, c)))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)), MARK(f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)), MARK(f(x2, x3, b)))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, MARK_1

Compound Symbols:   c1_1, c6_1, c6_2


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

(61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c))
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)), MARK(f(x2, x3, c)))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)))
S tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c))
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)), MARK(f(x2, x3, c)))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, MARK_1

Compound Symbols:   c1_1, c6_1, c6_2


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a__c)) by 
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, b))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, c))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)), MARK(f(x2, x3, c)))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, b))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, c))
S tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)), MARK(f(x2, x3, c)))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, b))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, c))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, MARK_1

Compound Symbols:   c1_1, c6_2, c6_1


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

(65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, a))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, b))
   MARK(f(x0, x1, c)) -> c6(A__F(x0, x1, c))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)), MARK(f(x2, x3, c)))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)))
S tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)), MARK(f(x2, x3, c)))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, MARK_1

Compound Symbols:   c1_1, c6_2, c6_1


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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)))
S tuples:
   A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2)))
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   A__F_3, MARK_1

Compound Symbols:   c1_1, c6_2, c6_1


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

(69) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace A__F(a, b, f(y0, y1, y2)) -> c1(MARK(f(y0, y1, y2))) by 
   A__F(a, b, f(z0, z1, f(a, b, y2))) -> c1(MARK(f(z0, z1, f(a, b, y2))))
   A__F(a, b, f(z0, z1, f(y2, y3, y4))) -> c1(MARK(f(z0, z1, f(y2, y3, y4))))
   A__F(a, b, f(z0, z1, f(y2, y3, f(y4, y5, y6)))) -> c1(MARK(f(z0, z1, f(y2, y3, f(y4, y5, y6)))))
   A__F(a, b, f(z0, z1, f(y2, y3, a))) -> c1(MARK(f(z0, z1, f(y2, y3, a))))
   A__F(a, b, f(z0, z1, f(y2, y3, b))) -> c1(MARK(f(z0, z1, f(y2, y3, b))))
   A__F(a, b, f(z0, z1, f(y2, y3, c))) -> c1(MARK(f(z0, z1, f(y2, y3, c))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   a__f(a, b, z0) -> a__f(z0, z0, mark(z0))
   a__f(z0, z1, z2) -> f(z0, z1, z2)
   a__c -> a
   a__c -> b
   a__c -> c
   mark(f(z0, z1, z2)) -> a__f(z0, z1, mark(z2))
   mark(c) -> a__c
   mark(a) -> a
   mark(b) -> b
Tuples:
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)))
   A__F(a, b, f(z0, z1, f(a, b, y2))) -> c1(MARK(f(z0, z1, f(a, b, y2))))
   A__F(a, b, f(z0, z1, f(y2, y3, y4))) -> c1(MARK(f(z0, z1, f(y2, y3, y4))))
   A__F(a, b, f(z0, z1, f(y2, y3, f(y4, y5, y6)))) -> c1(MARK(f(z0, z1, f(y2, y3, f(y4, y5, y6)))))
   A__F(a, b, f(z0, z1, f(y2, y3, a))) -> c1(MARK(f(z0, z1, f(y2, y3, a))))
   A__F(a, b, f(z0, z1, f(y2, y3, b))) -> c1(MARK(f(z0, z1, f(y2, y3, b))))
   A__F(a, b, f(z0, z1, f(y2, y3, c))) -> c1(MARK(f(z0, z1, f(y2, y3, c))))
S tuples:
   MARK(f(x0, x1, f(a, b, x4))) -> c6(A__F(x0, x1, a__f(mark(x4), mark(x4), mark(mark(x4)))), MARK(f(a, b, x4)))
   MARK(f(x0, x1, f(z0, z1, x4))) -> c6(A__F(x0, x1, f(z0, z1, mark(x4))), MARK(f(z0, z1, x4)))
   MARK(f(x0, x1, f(x2, x3, f(z0, z1, z2)))) -> c6(A__F(x0, x1, a__f(x2, x3, a__f(z0, z1, mark(z2)))), MARK(f(x2, x3, f(z0, z1, z2))))
   MARK(f(x0, x1, f(x2, x3, a))) -> c6(A__F(x0, x1, a__f(x2, x3, a)))
   MARK(f(x0, x1, f(x2, x3, b))) -> c6(A__F(x0, x1, a__f(x2, x3, b)))
   MARK(f(x0, x1, f(x2, x3, c))) -> c6(A__F(x0, x1, a__f(x2, x3, a__c)))
   A__F(a, b, f(z0, z1, f(a, b, y2))) -> c1(MARK(f(z0, z1, f(a, b, y2))))
   A__F(a, b, f(z0, z1, f(y2, y3, y4))) -> c1(MARK(f(z0, z1, f(y2, y3, y4))))
   A__F(a, b, f(z0, z1, f(y2, y3, f(y4, y5, y6)))) -> c1(MARK(f(z0, z1, f(y2, y3, f(y4, y5, y6)))))
   A__F(a, b, f(z0, z1, f(y2, y3, a))) -> c1(MARK(f(z0, z1, f(y2, y3, a))))
   A__F(a, b, f(z0, z1, f(y2, y3, b))) -> c1(MARK(f(z0, z1, f(y2, y3, b))))
   A__F(a, b, f(z0, z1, f(y2, y3, c))) -> c1(MARK(f(z0, z1, f(y2, y3, c))))
K tuples:none
Defined Rule Symbols:   a__f_3, a__c, mark_1

Defined Pair Symbols:   MARK_1, A__F_3

Compound Symbols:   c6_2, c6_1, c1_1


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

(71) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(72)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   a__f(a, b, X) -> a__f(X, X, mark(X)) [1]
   a__c -> a [1]
   a__c -> b [1]
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3)) [1]
   mark(c) -> a__c [1]
   mark(a) -> a [1]
   mark(b) -> b [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__c -> c [1]

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

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

(74)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   a__f(a, b, X) -> a__f(X, X, mark(X)) [1]
   a__c -> a [1]
   a__c -> b [1]
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3)) [1]
   mark(c) -> a__c [1]
   mark(a) -> a [1]
   mark(b) -> b [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__c -> c [1]

The TRS has the following type information:
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c :: a:b:f:c

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

(75) 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:
none

(c) The following functions are completely defined:

   mark_1
   a__c
   a__f_3

Due to the following rules being added:
none

And the following fresh constants: none

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

(76)
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__f(a, b, X) -> a__f(X, X, mark(X)) [1]
   a__c -> a [1]
   a__c -> b [1]
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3)) [1]
   mark(c) -> a__c [1]
   mark(a) -> a [1]
   mark(b) -> b [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__c -> c [1]

The TRS has the following type information:
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c :: a:b:f:c

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

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

(78)
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__f(a, b, f(X1', X2', X3')) -> a__f(f(X1', X2', X3'), f(X1', X2', X3'), a__f(X1', X2', mark(X3'))) [2]
   a__f(a, b, c) -> a__f(c, c, a__c) [2]
   a__f(a, b, a) -> a__f(a, a, a) [2]
   a__f(a, b, b) -> a__f(b, b, b) [2]
   a__c -> a [1]
   a__c -> b [1]
   mark(f(X1, X2, f(X1'', X2'', X3''))) -> a__f(X1, X2, a__f(X1'', X2'', mark(X3''))) [2]
   mark(f(X1, X2, c)) -> a__f(X1, X2, a__c) [2]
   mark(f(X1, X2, a)) -> a__f(X1, X2, a) [2]
   mark(f(X1, X2, b)) -> a__f(X1, X2, b) [2]
   mark(c) -> a__c [1]
   mark(a) -> a [1]
   mark(b) -> b [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__c -> c [1]

The TRS has the following type information:
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c :: a:b:f:c

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

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

   a => 0
   b => 1
   c => 2

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

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   a__c -{ 1 }-> 2 :|: 
   a__c -{ 1 }-> 1 :|: 
   a__c -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 2 }-> a__f(2, 2, a__c) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 2 }-> a__f(1, 1, 1) :|: z' = 1, z = 0, z'' = 1
   a__f(z, z', z'') -{ 2 }-> a__f(0, 0, 0) :|: z'' = 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 2 }-> a__f(1 + X1' + X2' + X3', 1 + X1' + X2' + X3', a__f(X1', X2', mark(X3'))) :|: X3' >= 0, X2' >= 0, X1' >= 0, z'' = 1 + X1' + X2' + X3', z' = 1, z = 0
   a__f(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
   mark(z) -{ 2 }-> a__f(X1, X2, a__f(X1'', X2'', mark(X3''))) :|: X1 >= 0, X1'' >= 0, z = 1 + X1 + X2 + (1 + X1'' + X2'' + X3''), X3'' >= 0, X2 >= 0, X2'' >= 0
   mark(z) -{ 2 }-> a__f(X1, X2, a__c) :|: X1 >= 0, z = 1 + X1 + X2 + 2, X2 >= 0
   mark(z) -{ 2 }-> a__f(X1, X2, 1) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 + 1
   mark(z) -{ 2 }-> a__f(X1, X2, 0) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 + 0
   mark(z) -{ 1 }-> a__c :|: z = 2
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 1 }-> 0 :|: z = 0


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

(81) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   a__c -{ 1 }-> 0 :|: 
   a__c -{ 1 }-> 1 :|: 
   a__c -{ 1 }-> 2 :|: 

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

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   a__c -{ 1 }-> 2 :|: 
   a__c -{ 1 }-> 1 :|: 
   a__c -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 3 }-> a__f(2, 2, 2) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 3 }-> a__f(2, 2, 1) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 3 }-> a__f(2, 2, 0) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 2 }-> a__f(1, 1, 1) :|: z' = 1, z = 0, z'' = 1
   a__f(z, z', z'') -{ 2 }-> a__f(0, 0, 0) :|: z'' = 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 2 }-> a__f(1 + X1' + X2' + X3', 1 + X1' + X2' + X3', a__f(X1', X2', mark(X3'))) :|: X3' >= 0, X2' >= 0, X1' >= 0, z'' = 1 + X1' + X2' + X3', z' = 1, z = 0
   a__f(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
   mark(z) -{ 2 }-> a__f(X1, X2, a__f(X1'', X2'', mark(X3''))) :|: X1 >= 0, X1'' >= 0, z = 1 + X1 + X2 + (1 + X1'' + X2'' + X3''), X3'' >= 0, X2 >= 0, X2'' >= 0
   mark(z) -{ 3 }-> a__f(X1, X2, 2) :|: X1 >= 0, z = 1 + X1 + X2 + 2, X2 >= 0
   mark(z) -{ 2 }-> a__f(X1, X2, 1) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 + 1
   mark(z) -{ 3 }-> a__f(X1, X2, 1) :|: X1 >= 0, z = 1 + X1 + X2 + 2, X2 >= 0
   mark(z) -{ 2 }-> a__f(X1, X2, 0) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 + 0
   mark(z) -{ 3 }-> a__f(X1, X2, 0) :|: X1 >= 0, z = 1 + X1 + X2 + 2, X2 >= 0
   mark(z) -{ 2 }-> 2 :|: z = 2
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 2
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 2


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

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

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   a__c -{ 1 }-> 2 :|: 
   a__c -{ 1 }-> 1 :|: 
   a__c -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 3 }-> a__f(2, 2, 2) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 3 }-> a__f(2, 2, 1) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 3 }-> a__f(2, 2, 0) :|: z' = 1, z'' = 2, z = 0
   a__f(z, z', z'') -{ 2 }-> a__f(1, 1, 1) :|: z' = 1, z = 0, z'' = 1
   a__f(z, z', z'') -{ 2 }-> a__f(0, 0, 0) :|: z'' = 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 2 }-> a__f(1 + X1' + X2' + X3', 1 + X1' + X2' + X3', a__f(X1', X2', mark(X3'))) :|: X3' >= 0, X2' >= 0, X1' >= 0, z'' = 1 + X1' + X2' + X3', z' = 1, z = 0
   a__f(z, z', z'') -{ 1 }-> 1 + z + z' + z'' :|: z >= 0, z'' >= 0, z' >= 0
   mark(z) -{ 2 }-> a__f(X1, X2, a__f(X1'', X2'', mark(X3''))) :|: X1 >= 0, X1'' >= 0, z = 1 + X1 + X2 + (1 + X1'' + X2'' + X3''), X3'' >= 0, X2 >= 0, X2'' >= 0
   mark(z) -{ 3 }-> a__f(X1, X2, 2) :|: X1 >= 0, z = 1 + X1 + X2 + 2, X2 >= 0
   mark(z) -{ 2 }-> a__f(X1, X2, 1) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 + 1
   mark(z) -{ 3 }-> a__f(X1, X2, 1) :|: X1 >= 0, z = 1 + X1 + X2 + 2, X2 >= 0
   mark(z) -{ 2 }-> a__f(X1, X2, 0) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2 + 0
   mark(z) -{ 3 }-> a__f(X1, X2, 0) :|: X1 >= 0, z = 1 + X1 + X2 + 2, X2 >= 0
   mark(z) -{ 2 }-> 2 :|: z = 2
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 2 }-> 1 :|: z = 2
   mark(z) -{ 1 }-> 0 :|: z = 0
   mark(z) -{ 2 }-> 0 :|: z = 2


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

(85) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

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

(86)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   a__f(a, b, X) -> a__f(X, X, mark(X)) [1]
   a__c -> a [1]
   a__c -> b [1]
   mark(f(X1, X2, X3)) -> a__f(X1, X2, mark(X3)) [1]
   mark(c) -> a__c [1]
   mark(a) -> a [1]
   mark(b) -> b [1]
   a__f(X1, X2, X3) -> f(X1, X2, X3) [1]
   a__c -> c [1]

The TRS has the following type information:
a__f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
a :: a:b:f:c
b :: a:b:f:c
mark :: a:b:f:c -> a:b:f:c
a__c :: a:b:f:c
f :: a:b:f:c -> a:b:f:c -> a:b:f:c -> a:b:f:c
c :: a:b:f:c

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

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

   a => 0
   b => 1
   c => 2

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

(88)
Obligation:
Complexity RNTS consisting of the following rules:

   a__c -{ 1 }-> 2 :|: 
   a__c -{ 1 }-> 1 :|: 
   a__c -{ 1 }-> 0 :|: 
   a__f(z, z', z'') -{ 1 }-> a__f(X, X, mark(X)) :|: z'' = X, X >= 0, z' = 1, z = 0
   a__f(z, z', z'') -{ 1 }-> 1 + X1 + X2 + X3 :|: X1 >= 0, X3 >= 0, X2 >= 0, z = X1, z' = X2, z'' = X3
   mark(z) -{ 1 }-> a__f(X1, X2, mark(X3)) :|: X1 >= 0, X3 >= 0, z = 1 + X1 + X2 + X3, X2 >= 0
   mark(z) -{ 1 }-> a__c :|: z = 2
   mark(z) -{ 1 }-> 1 :|: z = 1
   mark(z) -{ 1 }-> 0 :|: z = 0

Only complete derivations are relevant for the runtime complexity.