MAYBE
proof of input_17QnAd4NBw.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 112 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 164 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 47 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 1191 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 19 ms]
        (32) CpxRNTS
    (33) CompletionProof [UPPER BOUND(ID), 0 ms]
    (34) CpxTypedWeightedCompleteTrs
        (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 5 ms]
        (36) CpxRNTS
(37) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(38) CdtProblem
    (39) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) 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:

   c -> f(n__g(n__c))
   f(n__g(X)) -> g(activate(X))
   g(X) -> n__g(X)
   c -> n__c
   activate(n__g(X)) -> g(X)
   activate(n__c) -> c
   activate(X) -> X

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:

   c -> f(n__g(n__c))
   f(n__g(X)) -> g(activate(X))
   g(X) -> n__g(X)
   c -> n__c
   activate(n__g(X)) -> g(X)
   activate(n__c) -> c
   activate(X) -> X

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:

   c -> f(n__g(n__c))
   f(n__g(X)) -> g(activate(X))
   g(X) -> n__g(X)
   c -> n__c
   activate(n__g(X)) -> g(X)
   activate(n__c) -> c
   activate(X) -> X

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:

   c -> f(n__g(n__c)) [1]
   f(n__g(X)) -> g(activate(X)) [1]
   g(X) -> n__g(X) [1]
   c -> n__c [1]
   activate(n__g(X)) -> g(X) [1]
   activate(n__c) -> c [1]
   activate(X) -> X [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:

   c -> f(n__g(n__c)) [1]
   f(n__g(X)) -> g(activate(X)) [1]
   g(X) -> n__g(X) [1]
   c -> n__c [1]
   activate(n__g(X)) -> g(X) [1]
   activate(n__c) -> c [1]
   activate(X) -> X [1]

The TRS has the following type information:
c :: n__c:n__g
f :: n__c:n__g -> n__c:n__g
n__g :: n__c:n__g -> n__c:n__g
n__c :: n__c:n__g
g :: n__c:n__g -> n__c:n__g
activate :: n__c:n__g -> n__c:n__g

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:

   activate_1
   c
   g_1
   f_1

Due to the following rules being added:

   f(v0) -> n__c [0]

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:

   c -> f(n__g(n__c)) [1]
   f(n__g(X)) -> g(activate(X)) [1]
   g(X) -> n__g(X) [1]
   c -> n__c [1]
   activate(n__g(X)) -> g(X) [1]
   activate(n__c) -> c [1]
   activate(X) -> X [1]
   f(v0) -> n__c [0]

The TRS has the following type information:
c :: n__c:n__g
f :: n__c:n__g -> n__c:n__g
n__g :: n__c:n__g -> n__c:n__g
n__c :: n__c:n__g
g :: n__c:n__g -> n__c:n__g
activate :: n__c:n__g -> n__c:n__g

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:

   c -> f(n__g(n__c)) [1]
   f(n__g(n__g(X'))) -> g(g(X')) [2]
   f(n__g(n__c)) -> g(c) [2]
   f(n__g(X)) -> g(X) [2]
   g(X) -> n__g(X) [1]
   c -> n__c [1]
   activate(n__g(X)) -> g(X) [1]
   activate(n__c) -> c [1]
   activate(X) -> X [1]
   f(v0) -> n__c [0]

The TRS has the following type information:
c :: n__c:n__g
f :: n__c:n__g -> n__c:n__g
n__g :: n__c:n__g -> n__c:n__g
n__c :: n__c:n__g
g :: n__c:n__g -> n__c:n__g
activate :: n__c:n__g -> n__c:n__g

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:

   n__c => 0

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> g(X) :|: z = 1 + X, X >= 0
   activate(z) -{ 1 }-> c :|: z = 0
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> g(X) :|: z = 1 + X, X >= 0
   f(z) -{ 2 }-> g(g(X')) :|: X' >= 0, z = 1 + (1 + X')
   f(z) -{ 2 }-> g(c) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X


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

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

   g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> c :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> g(c) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z) -{ 3 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
   f(z) -{ 4 }-> 1 + X'' :|: X' >= 0, z = 1 + (1 + X'), X >= 0, X' = X, X'' >= 0, 1 + X = X''
   g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X


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

(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:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> c :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> g(c) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X''
   g(z) -{ 1 }-> 1 + z :|: z >= 0


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

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

   { g }
   { c, f }
   { activate }

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> c :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> g(c) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X''
   g(z) -{ 1 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {c,f}, {activate}

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> c :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> g(c) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X''
   g(z) -{ 1 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {c,f}, {activate}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> c :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> g(c) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X''
   g(z) -{ 1 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {g}, {c,f}, {activate}
Previous analysis results are:
  g: runtime: ?, size: O(n^1) [1 + z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> c :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> g(c) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X''
   g(z) -{ 1 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {c,f}, {activate}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [1 + z]

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> c :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> g(c) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X''
   g(z) -{ 1 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {c,f}, {activate}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [1 + z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: c
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> c :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> g(c) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X''
   g(z) -{ 1 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {c,f}, {activate}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [1 + z]
  c: runtime: ?, size: INF
  f: runtime: ?, size: INF

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: c
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> c :|: z = 0
   activate(z) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 2 }-> g(c) :|: z = 1 + 0
   f(z) -{ 0 }-> 0 :|: z >= 0
   f(z) -{ 3 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 4 }-> 1 + X'' :|: z - 2 >= 0, X >= 0, z - 2 = X, X'' >= 0, 1 + X = X''
   g(z) -{ 1 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {c,f}, {activate}
Previous analysis results are:
  g: runtime: O(1) [1], size: O(n^1) [1 + z]
  c: runtime: INF, size: INF
  f: runtime: ?, size: INF

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

(33) 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:

   f(v0) -> null_f [0]

And the following fresh constants:    null_f

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

(34)
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:

   c -> f(n__g(n__c)) [1]
   f(n__g(X)) -> g(activate(X)) [1]
   g(X) -> n__g(X) [1]
   c -> n__c [1]
   activate(n__g(X)) -> g(X) [1]
   activate(n__c) -> c [1]
   activate(X) -> X [1]
   f(v0) -> null_f [0]

The TRS has the following type information:
c :: n__c:n__g:null_f
f :: n__c:n__g:null_f -> n__c:n__g:null_f
n__g :: n__c:n__g:null_f -> n__c:n__g:null_f
n__c :: n__c:n__g:null_f
g :: n__c:n__g:null_f -> n__c:n__g:null_f
activate :: n__c:n__g:null_f -> n__c:n__g:null_f
null_f :: n__c:n__g:null_f

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

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

   n__c => 0
   null_f => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> g(X) :|: z = 1 + X, X >= 0
   activate(z) -{ 1 }-> c :|: z = 0
   c -{ 1 }-> f(1 + 0) :|: 
   c -{ 1 }-> 0 :|: 
   f(z) -{ 1 }-> g(activate(X)) :|: z = 1 + X, X >= 0
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + X :|: X >= 0, z = X

Only complete derivations are relevant for the runtime complexity.

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

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   c -> f(n__g(n__c))
   c -> n__c
   f(n__g(z0)) -> g(activate(z0))
   g(z0) -> n__g(z0)
   activate(n__g(z0)) -> g(z0)
   activate(n__c) -> c
   activate(z0) -> z0
Tuples:
   C -> c1(F(n__g(n__c)))
   C -> c2
   F(n__g(z0)) -> c3(G(activate(z0)), ACTIVATE(z0))
   G(z0) -> c4
   ACTIVATE(n__g(z0)) -> c5(G(z0))
   ACTIVATE(n__c) -> c6(C)
   ACTIVATE(z0) -> c7
S tuples:
   C -> c1(F(n__g(n__c)))
   C -> c2
   F(n__g(z0)) -> c3(G(activate(z0)), ACTIVATE(z0))
   G(z0) -> c4
   ACTIVATE(n__g(z0)) -> c5(G(z0))
   ACTIVATE(n__c) -> c6(C)
   ACTIVATE(z0) -> c7
K tuples:none
Defined Rule Symbols:   c, f_1, g_1, activate_1

Defined Pair Symbols:   C, F_1, G_1, ACTIVATE_1

Compound Symbols:   c1_1, c2, c3_2, c4, c5_1, c6_1, c7


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

(39) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   C -> c2
   ACTIVATE(z0) -> c7
   G(z0) -> c4
   ACTIVATE(n__g(z0)) -> c5(G(z0))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   c -> f(n__g(n__c))
   c -> n__c
   f(n__g(z0)) -> g(activate(z0))
   g(z0) -> n__g(z0)
   activate(n__g(z0)) -> g(z0)
   activate(n__c) -> c
   activate(z0) -> z0
Tuples:
   C -> c1(F(n__g(n__c)))
   F(n__g(z0)) -> c3(G(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__c) -> c6(C)
S tuples:
   C -> c1(F(n__g(n__c)))
   F(n__g(z0)) -> c3(G(activate(z0)), ACTIVATE(z0))
   ACTIVATE(n__c) -> c6(C)
K tuples:none
Defined Rule Symbols:   c, f_1, g_1, activate_1

Defined Pair Symbols:   C, F_1, ACTIVATE_1

Compound Symbols:   c1_1, c3_2, c6_1


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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   c -> f(n__g(n__c))
   c -> n__c
   f(n__g(z0)) -> g(activate(z0))
   g(z0) -> n__g(z0)
   activate(n__g(z0)) -> g(z0)
   activate(n__c) -> c
   activate(z0) -> z0
Tuples:
   C -> c1(F(n__g(n__c)))
   ACTIVATE(n__c) -> c6(C)
   F(n__g(z0)) -> c3(ACTIVATE(z0))
S tuples:
   C -> c1(F(n__g(n__c)))
   ACTIVATE(n__c) -> c6(C)
   F(n__g(z0)) -> c3(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:   c, f_1, g_1, activate_1

Defined Pair Symbols:   C, ACTIVATE_1, F_1

Compound Symbols:   c1_1, c6_1, c3_1


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

(43) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   c -> f(n__g(n__c))
   c -> n__c
   f(n__g(z0)) -> g(activate(z0))
   g(z0) -> n__g(z0)
   activate(n__g(z0)) -> g(z0)
   activate(n__c) -> c
   activate(z0) -> z0

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   C -> c1(F(n__g(n__c)))
   ACTIVATE(n__c) -> c6(C)
   F(n__g(z0)) -> c3(ACTIVATE(z0))
S tuples:
   C -> c1(F(n__g(n__c)))
   ACTIVATE(n__c) -> c6(C)
   F(n__g(z0)) -> c3(ACTIVATE(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   C, ACTIVATE_1, F_1

Compound Symbols:   c1_1, c6_1, c3_1


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

(45) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(n__g(z0)) -> c3(ACTIVATE(z0)) by 
   F(n__g(n__c)) -> c3(ACTIVATE(n__c))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   C -> c1(F(n__g(n__c)))
   ACTIVATE(n__c) -> c6(C)
   F(n__g(n__c)) -> c3(ACTIVATE(n__c))
S tuples:
   C -> c1(F(n__g(n__c)))
   ACTIVATE(n__c) -> c6(C)
   F(n__g(n__c)) -> c3(ACTIVATE(n__c))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   C, ACTIVATE_1, F_1

Compound Symbols:   c1_1, c6_1, c3_1