KILLED
proof of input_RM3HbLY4MK.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) InliningProof [UPPER BOUND(ID), 265 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
    (19) CompletionProof [UPPER BOUND(ID), 0 ms]
    (20) CpxTypedWeightedCompleteTrs
        (21) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
(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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem


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

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

(6)
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
----------------------------------------

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

(8)
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
----------------------------------------

(9) 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

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

(10)
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
----------------------------------------

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

(12)
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
----------------------------------------

(13) 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

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

(14)
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


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

(15) 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 :|: 

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

(16)
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


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

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

(18)
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


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

(19) 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

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

(20)
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
----------------------------------------

(21) 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

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

(22)
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.

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

(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:
   MARK(b) -> c9
   MARK(c) -> c7(A__C)
   MARK(a) -> c8
   A__C -> c3
   A__F(z0, z1, z2) -> c2
   A__C -> c4
   A__C -> c5

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

(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) 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)))

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

(30)
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


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

(31) 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))

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

(32)
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


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

(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   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))

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

(34)
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


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

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

(36)
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


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

(37) 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)))

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

(38)
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


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

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

(40)
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


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

(41) 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))

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

(42)
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


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

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

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

(44)
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


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

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

(46)
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


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

(47) 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))))

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

(48)
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