MAYBE
proof of input_HBl6y2VDWN.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
    (5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
        (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (10) typed CpxTrs
        (11) OrderProof [LOWER BOUND(ID), 8 ms]
        (12) typed CpxTrs
    (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (14) TRS for Loop Detection
(15) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxTRS
(17) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(18) CdtProblem
    (19) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CdtProblem
    (21) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxTRS
        (27) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (28) CpxTRS
        (29) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxWeightedTrs
            (31) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (32) CpxTypedWeightedTrs
            (33) CompletionProof [UPPER BOUND(ID), 0 ms]
            (34) CpxTypedWeightedCompleteTrs
                (35) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (36) CpxRNTS
            (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) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (44) CpxRNTS
                (45) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (46) CpxRNTS
                (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (48) CpxRNTS
                (49) IntTrsBoundProof [UPPER BOUND(ID), 785 ms]
                (50) CpxRNTS
                (51) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
                (52) CpxRNTS
    (53) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms]
    (54) CdtProblem
(55) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(56) CpxWeightedTrs
    (57) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CpxTypedWeightedTrs
    (59) CompletionProof [UPPER BOUND(ID), 0 ms]
    (60) CpxTypedWeightedCompleteTrs
        (61) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (62) CpxRNTS
    (63) CompletionProof [UPPER BOUND(ID), 0 ms]
    (64) CpxTypedWeightedCompleteTrs
        (65) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CpxTypedWeightedCompleteTrs
        (67) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (68) CpxRNTS
        (69) InliningProof [UPPER BOUND(ID), 114 ms]
        (70) CpxRNTS
        (71) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (72) CpxRNTS
        (73) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (74) CpxRNTS
        (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (76) CpxRNTS
        (77) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
        (78) CpxRNTS
        (79) IntTrsBoundProof [UPPER BOUND(ID), 24 ms]
        (80) CpxRNTS
        (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (82) CpxRNTS
        (83) IntTrsBoundProof [UPPER BOUND(ID), 1356 ms]
        (84) CpxRNTS
        (85) IntTrsBoundProof [UPPER BOUND(ID), 28 ms]
        (86) 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:

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

(4)
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


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

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

(6)
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:

   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

The (relative) TRS S consists of the following 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

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

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

(8)
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:

   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

The (relative) TRS S consists of the following 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

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

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

(10)
Obligation:
Innermost TRS:
Rules:
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
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

Types:
C :: c1:c2
c1 :: c3 -> c1:c2
F :: n__c:n__g -> c3
n__g :: n__c:n__g -> n__c:n__g
n__c :: n__c:n__g
c2 :: c1:c2
c3 :: c4 -> c5:c6:c7 -> c3
G :: n__c:n__g -> c4
activate :: n__c:n__g -> n__c:n__g
ACTIVATE :: n__c:n__g -> c5:c6:c7
c4 :: c4
c5 :: c4 -> c5:c6:c7
c6 :: c1:c2 -> c5:c6:c7
c7 :: c5:c6:c7
c :: n__c:n__g
f :: n__c:n__g -> n__c:n__g
g :: n__c:n__g -> n__c:n__g
hole_c1:c21_8 :: c1:c2
hole_c32_8 :: c3
hole_n__c:n__g3_8 :: n__c:n__g
hole_c44_8 :: c4
hole_c5:c6:c75_8 :: c5:c6:c7
gen_n__c:n__g6_8 :: Nat -> n__c:n__g

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
C, activate, c

They will be analysed ascendingly in the following order:
activate < C
activate = c

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

(12)
Obligation:
Innermost TRS:
Rules:
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
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

Types:
C :: c1:c2
c1 :: c3 -> c1:c2
F :: n__c:n__g -> c3
n__g :: n__c:n__g -> n__c:n__g
n__c :: n__c:n__g
c2 :: c1:c2
c3 :: c4 -> c5:c6:c7 -> c3
G :: n__c:n__g -> c4
activate :: n__c:n__g -> n__c:n__g
ACTIVATE :: n__c:n__g -> c5:c6:c7
c4 :: c4
c5 :: c4 -> c5:c6:c7
c6 :: c1:c2 -> c5:c6:c7
c7 :: c5:c6:c7
c :: n__c:n__g
f :: n__c:n__g -> n__c:n__g
g :: n__c:n__g -> n__c:n__g
hole_c1:c21_8 :: c1:c2
hole_c32_8 :: c3
hole_n__c:n__g3_8 :: n__c:n__g
hole_c44_8 :: c4
hole_c5:c6:c75_8 :: c5:c6:c7
gen_n__c:n__g6_8 :: Nat -> n__c:n__g


Generator Equations:
gen_n__c:n__g6_8(0) <=> n__c
gen_n__c:n__g6_8(+(x, 1)) <=> n__g(gen_n__c:n__g6_8(x))


The following defined symbols remain to be analysed:
c, C, activate

They will be analysed ascendingly in the following order:
activate < C
activate = c

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

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   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

The (relative) TRS S consists of the following 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

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

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

(16)
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
----------------------------------------

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

(18)
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


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

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

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

(20)
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


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

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

(22)
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


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

(23) 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

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

(24)
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


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

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

(26)
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:

   C -> c1(F(n__g(n__c)))
   ACTIVATE(n__c) -> c6(C)
   F(n__g(z0)) -> c3(ACTIVATE(z0))

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

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

(28)
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:

   C -> c1(F(n__g(n__c)))
   ACTIVATE(n__c) -> c6(C)
   F(n__g(z0)) -> c3(ACTIVATE(z0))

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

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

(30)
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 -> c1(F(n__g(n__c))) [1]
   ACTIVATE(n__c) -> c6(C) [1]
   F(n__g(z0)) -> c3(ACTIVATE(z0)) [1]

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

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

(32)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   C -> c1(F(n__g(n__c))) [1]
   ACTIVATE(n__c) -> c6(C) [1]
   F(n__g(z0)) -> c3(ACTIVATE(z0)) [1]

The TRS has the following type information:
C :: c1
c1 :: c3 -> c1
F :: n__g -> c3
n__g :: n__c -> n__g
n__c :: n__c
ACTIVATE :: n__c -> c6
c6 :: c1 -> c6
c3 :: c6 -> c3

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

(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, const, const1, const2

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

(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 -> c1(F(n__g(n__c))) [1]
   ACTIVATE(n__c) -> c6(C) [1]
   F(n__g(z0)) -> c3(ACTIVATE(z0)) [1]
   F(v0) -> null_F [0]

The TRS has the following type information:
C :: c1
c1 :: c3:null_F -> c1
F :: n__g -> c3:null_F
n__g :: n__c -> n__g
n__c :: n__c
ACTIVATE :: n__c -> c6
c6 :: c1 -> c6
c3 :: c6 -> c3:null_F
null_F :: c3:null_F
const :: c1
const1 :: n__g
const2 :: c6

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

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

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

   ACTIVATE(z) -{ 1 }-> 1 + C :|: z = 0
   C -{ 1 }-> 1 + F(1 + 0) :|: 
   F(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   F(z) -{ 1 }-> 1 + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

(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:

   C
   ACTIVATE_1
   F_1

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1, const2, const3

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

(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:

   C -> c1(F(n__g(n__c))) [1]
   ACTIVATE(n__c) -> c6(C) [1]
   F(n__g(z0)) -> c3(ACTIVATE(z0)) [1]

The TRS has the following type information:
C :: c1
c1 :: c3 -> c1
F :: n__g -> c3
n__g :: n__c -> n__g
n__c :: n__c
ACTIVATE :: n__c -> c6
c6 :: c1 -> c6
c3 :: c6 -> c3
const :: c1
const1 :: c3
const2 :: n__g
const3 :: c6

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:

   C -> c1(F(n__g(n__c))) [1]
   ACTIVATE(n__c) -> c6(C) [1]
   F(n__g(z0)) -> c3(ACTIVATE(z0)) [1]

The TRS has the following type information:
C :: c1
c1 :: c3 -> c1
F :: n__g -> c3
n__g :: n__c -> n__g
n__c :: n__c
ACTIVATE :: n__c -> c6
c6 :: c1 -> c6
c3 :: c6 -> c3
const :: c1
const1 :: c3
const2 :: n__g
const3 :: c6

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:

   n__c => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0

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

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

   ACTIVATE(z) -{ 1 }-> 1 + C :|: z = 0
   C -{ 1 }-> 1 + F(1 + 0) :|: 
   F(z) -{ 1 }-> 1 + ACTIVATE(z0) :|: z = 1 + z0, z0 >= 0


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

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

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

   ACTIVATE(z) -{ 1 }-> 1 + C :|: z = 0
   C -{ 1 }-> 1 + F(1 + 0) :|: 
   F(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0


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

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

   { C, F, ACTIVATE }

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

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

   ACTIVATE(z) -{ 1 }-> 1 + C :|: z = 0
   C -{ 1 }-> 1 + F(1 + 0) :|: 
   F(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {C,F,ACTIVATE}

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

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   ACTIVATE(z) -{ 1 }-> 1 + C :|: z = 0
   C -{ 1 }-> 1 + F(1 + 0) :|: 
   F(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {C,F,ACTIVATE}

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: F
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

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

   ACTIVATE(z) -{ 1 }-> 1 + C :|: z = 0
   C -{ 1 }-> 1 + F(1 + 0) :|: 
   F(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {C,F,ACTIVATE}
Previous analysis results are:
  C: runtime: ?, size: O(1) [0]
  F: runtime: ?, size: O(1) [2]
  ACTIVATE: runtime: ?, size: O(1) [1]

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

(51) 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: ?

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   ACTIVATE(z) -{ 1 }-> 1 + C :|: z = 0
   C -{ 1 }-> 1 + F(1 + 0) :|: 
   F(z) -{ 1 }-> 1 + ACTIVATE(z - 1) :|: z - 1 >= 0

Function symbols to be analyzed: {C,F,ACTIVATE}
Previous analysis results are:
  C: runtime: INF, size: O(1) [0]
  F: runtime: ?, size: O(1) [2]
  ACTIVATE: runtime: ?, size: O(1) [1]

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

(53) 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))

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

(54)
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


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

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

(56)
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
----------------------------------------

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

(58)
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
----------------------------------------

(59) 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

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

(60)
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
----------------------------------------

(61) 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

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

(62)
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.

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

(63) 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

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

(64)
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
----------------------------------------

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

(66)
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
----------------------------------------

(67) 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

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

(68)
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


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

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

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

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

(70)
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


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

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

(72)
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


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

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

   { g }
   { c, f }
   { activate }

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

(74)
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}

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

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

(76)
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}

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

(77) 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

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

(78)
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]

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

(79) 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

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

(80)
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]

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

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

(82)
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]

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

(83) 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: ?

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

(84)
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

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

(85) 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: ?

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

(86)
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