MAYBE
proof of input_otvooEpeMC.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 187 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), 90 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 24 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 20 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 33 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 168 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 65 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 449 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 137 ms]
        (44) CpxRNTS
    (45) CompletionProof [UPPER BOUND(ID), 0 ms]
    (46) CpxTypedWeightedCompleteTrs
        (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
(49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(50) CdtProblem
    (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 16 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) 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:

   g(X) -> h(activate(X))
   c -> d
   h(n__d) -> g(n__c)
   d -> n__d
   c -> n__c
   activate(n__d) -> d
   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:

   g(X) -> h(activate(X))
   c -> d
   h(n__d) -> g(n__c)
   d -> n__d
   c -> n__c
   activate(n__d) -> d
   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:

   g(X) -> h(activate(X))
   c -> d
   h(n__d) -> g(n__c)
   d -> n__d
   c -> n__c
   activate(n__d) -> d
   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:

   g(X) -> h(activate(X)) [1]
   c -> d [1]
   h(n__d) -> g(n__c) [1]
   d -> n__d [1]
   c -> n__c [1]
   activate(n__d) -> d [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:

   g(X) -> h(activate(X)) [1]
   c -> d [1]
   h(n__d) -> g(n__c) [1]
   d -> n__d [1]
   c -> n__c [1]
   activate(n__d) -> d [1]
   activate(n__c) -> c [1]
   activate(X) -> X [1]

The TRS has the following type information:
g :: n__d:n__c -> g:h
h :: n__d:n__c -> g:h
activate :: n__d:n__c -> n__d:n__c
c :: n__d:n__c
d :: n__d:n__c
n__d :: n__d:n__c
n__c :: n__d:n__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:

   g_1
   h_1

(c) The following functions are completely defined:

   activate_1
   d
   c

Due to the following rules being added:
none

And the following fresh constants:    const

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

(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:

   g(X) -> h(activate(X)) [1]
   c -> d [1]
   h(n__d) -> g(n__c) [1]
   d -> n__d [1]
   c -> n__c [1]
   activate(n__d) -> d [1]
   activate(n__c) -> c [1]
   activate(X) -> X [1]

The TRS has the following type information:
g :: n__d:n__c -> g:h
h :: n__d:n__c -> g:h
activate :: n__d:n__c -> n__d:n__c
c :: n__d:n__c
d :: n__d:n__c
n__d :: n__d:n__c
n__c :: n__d:n__c
const :: g:h

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:

   g(n__d) -> h(d) [2]
   g(n__c) -> h(c) [2]
   g(X) -> h(X) [2]
   c -> d [1]
   h(n__d) -> g(n__c) [1]
   d -> n__d [1]
   c -> n__c [1]
   activate(n__d) -> d [1]
   activate(n__c) -> c [1]
   activate(X) -> X [1]

The TRS has the following type information:
g :: n__d:n__c -> g:h
h :: n__d:n__c -> g:h
activate :: n__d:n__c -> n__d:n__c
c :: n__d:n__c
d :: n__d:n__c
n__d :: n__d:n__c
n__c :: n__d:n__c
const :: g:h

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__d => 1
   n__c => 0
   const => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> d :|: z = 1
   activate(z) -{ 1 }-> c :|: z = 0
   c -{ 1 }-> d :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 2 }-> h(X) :|: X >= 0, z = X
   g(z) -{ 2 }-> h(d) :|: z = 1
   g(z) -{ 2 }-> h(c) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1


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

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

   d -{ 1 }-> 1 :|: 
   c -{ 1 }-> d :|: 
   c -{ 1 }-> 0 :|: 

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 2 }-> d :|: z = 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 2 }-> h(X) :|: X >= 0, z = X
   g(z) -{ 3 }-> h(d) :|: z = 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1


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

(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) -{ 2 }-> d :|: z = 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(d) :|: z = 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1


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

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

   { d }
   { c }
   { activate }
   { h, g }

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> d :|: z = 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(d) :|: z = 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

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

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

(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) -{ 2 }-> d :|: z = 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(d) :|: z = 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> d :|: z = 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(d) :|: z = 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {d}, {c}, {activate}, {h,g}
Previous analysis results are:
  d: runtime: ?, size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: d
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) -{ 2 }-> d :|: z = 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(d) :|: z = 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {c}, {activate}, {h,g}
Previous analysis results are:
  d: runtime: O(1) [1], size: O(1) [1]

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

(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) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {c}, {activate}, {h,g}
Previous analysis results are:
  d: runtime: O(1) [1], size: O(1) [1]

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

(29) 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: 1

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

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

   activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {c}, {activate}, {h,g}
Previous analysis results are:
  d: runtime: O(1) [1], size: O(1) [1]
  c: runtime: ?, size: O(1) [1]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {activate}, {h,g}
Previous analysis results are:
  d: runtime: O(1) [1], size: O(1) [1]
  c: runtime: O(1) [2], size: O(1) [1]

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

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

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

   activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {activate}, {h,g}
Previous analysis results are:
  d: runtime: O(1) [1], size: O(1) [1]
  c: runtime: O(1) [2], size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {activate}, {h,g}
Previous analysis results are:
  d: runtime: O(1) [1], size: O(1) [1]
  c: runtime: O(1) [2], size: O(1) [1]
  activate: runtime: ?, size: O(n^1) [1 + z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: activate
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

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

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

   activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {h,g}
Previous analysis results are:
  d: runtime: O(1) [1], size: O(1) [1]
  c: runtime: O(1) [2], size: O(1) [1]
  activate: runtime: O(1) [3], size: O(n^1) [1 + z]

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

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {h,g}
Previous analysis results are:
  d: runtime: O(1) [1], size: O(1) [1]
  c: runtime: O(1) [2], size: O(1) [1]
  activate: runtime: O(1) [3], size: O(n^1) [1 + z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {h,g}
Previous analysis results are:
  d: runtime: O(1) [1], size: O(1) [1]
  c: runtime: O(1) [2], size: O(1) [1]
  activate: runtime: O(1) [3], size: O(n^1) [1 + z]
  h: runtime: ?, size: O(1) [0]
  g: runtime: ?, size: O(1) [0]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 3 }-> s' :|: s' >= 0, s' <= 1, z = 0
   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 2 }-> 1 :|: z = 1
   activate(z) -{ 2 }-> 0 :|: z = 0
   c -{ 2 }-> 1 :|: 
   c -{ 1 }-> 0 :|: 
   d -{ 1 }-> 1 :|: 
   g(z) -{ 4 }-> h(s) :|: s >= 0, s <= 1, z = 0
   g(z) -{ 2 }-> h(z) :|: z >= 0
   g(z) -{ 3 }-> h(1) :|: z = 1
   g(z) -{ 3 }-> h(0) :|: z = 0
   h(z) -{ 1 }-> g(0) :|: z = 1

Function symbols to be analyzed: {h,g}
Previous analysis results are:
  d: runtime: O(1) [1], size: O(1) [1]
  c: runtime: O(1) [2], size: O(1) [1]
  activate: runtime: O(1) [3], size: O(n^1) [1 + z]
  h: runtime: INF, size: O(1) [0]
  g: runtime: ?, size: O(1) [0]

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

(45) 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:

   h(v0) -> null_h [0]

And the following fresh constants:    null_h

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

(46)
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:

   g(X) -> h(activate(X)) [1]
   c -> d [1]
   h(n__d) -> g(n__c) [1]
   d -> n__d [1]
   c -> n__c [1]
   activate(n__d) -> d [1]
   activate(n__c) -> c [1]
   activate(X) -> X [1]
   h(v0) -> null_h [0]

The TRS has the following type information:
g :: n__d:n__c -> null_h
h :: n__d:n__c -> null_h
activate :: n__d:n__c -> n__d:n__c
c :: n__d:n__c
d :: n__d:n__c
n__d :: n__d:n__c
n__c :: n__d:n__c
null_h :: null_h

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

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

   n__d => 1
   n__c => 0
   null_h => 0

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> h(activate(z0))
   c -> d
   c -> n__c
   h(n__d) -> g(n__c)
   d -> n__d
   activate(n__d) -> d
   activate(n__c) -> c
   activate(z0) -> z0
Tuples:
   G(z0) -> c1(H(activate(z0)), ACTIVATE(z0))
   C -> c2(D)
   C -> c3
   H(n__d) -> c4(G(n__c))
   D -> c5
   ACTIVATE(n__d) -> c6(D)
   ACTIVATE(n__c) -> c7(C)
   ACTIVATE(z0) -> c8
S tuples:
   G(z0) -> c1(H(activate(z0)), ACTIVATE(z0))
   C -> c2(D)
   C -> c3
   H(n__d) -> c4(G(n__c))
   D -> c5
   ACTIVATE(n__d) -> c6(D)
   ACTIVATE(n__c) -> c7(C)
   ACTIVATE(z0) -> c8
K tuples:none
Defined Rule Symbols:   g_1, c, h_1, d, activate_1

Defined Pair Symbols:   G_1, C, H_1, D, ACTIVATE_1

Compound Symbols:   c1_2, c2_1, c3, c4_1, c5, c6_1, c7_1, c8


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

(51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   ACTIVATE(n__d) -> c6(D)
   D -> c5
   ACTIVATE(z0) -> c8
   ACTIVATE(n__c) -> c7(C)
   C -> c2(D)
   C -> c3

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> h(activate(z0))
   c -> d
   c -> n__c
   h(n__d) -> g(n__c)
   d -> n__d
   activate(n__d) -> d
   activate(n__c) -> c
   activate(z0) -> z0
Tuples:
   G(z0) -> c1(H(activate(z0)), ACTIVATE(z0))
   H(n__d) -> c4(G(n__c))
S tuples:
   G(z0) -> c1(H(activate(z0)), ACTIVATE(z0))
   H(n__d) -> c4(G(n__c))
K tuples:none
Defined Rule Symbols:   g_1, c, h_1, d, activate_1

Defined Pair Symbols:   G_1, H_1

Compound Symbols:   c1_2, c4_1


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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(z0) -> h(activate(z0))
   c -> d
   c -> n__c
   h(n__d) -> g(n__c)
   d -> n__d
   activate(n__d) -> d
   activate(n__c) -> c
   activate(z0) -> z0
Tuples:
   H(n__d) -> c4(G(n__c))
   G(z0) -> c1(H(activate(z0)))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(z0) -> c1(H(activate(z0)))
K tuples:none
Defined Rule Symbols:   g_1, c, h_1, d, activate_1

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   g(z0) -> h(activate(z0))
   h(n__d) -> g(n__c)

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__d) -> d
   activate(n__c) -> c
   activate(z0) -> z0
   d -> n__d
   c -> d
   c -> n__c
Tuples:
   H(n__d) -> c4(G(n__c))
   G(z0) -> c1(H(activate(z0)))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(z0) -> c1(H(activate(z0)))
K tuples:none
Defined Rule Symbols:   activate_1, d, c

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(z0) -> c1(H(activate(z0))) by 
   G(n__d) -> c1(H(d))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__d) -> d
   activate(n__c) -> c
   activate(z0) -> z0
   d -> n__d
   c -> d
   c -> n__c
Tuples:
   H(n__d) -> c4(G(n__c))
   G(n__d) -> c1(H(d))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__d) -> c1(H(d))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))
K tuples:none
Defined Rule Symbols:   activate_1, d, c

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(59) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   G(n__d) -> c1(H(d))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__d) -> d
   activate(n__c) -> c
   activate(z0) -> z0
   d -> n__d
   c -> d
   c -> n__c
Tuples:
   H(n__d) -> c4(G(n__c))
   G(n__d) -> c1(H(d))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))
K tuples:
   G(n__d) -> c1(H(d))
Defined Rule Symbols:   activate_1, d, c

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(61) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   activate(n__d) -> d
   activate(n__c) -> c
   activate(z0) -> z0

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   d -> n__d
   c -> d
   c -> n__c
Tuples:
   H(n__d) -> c4(G(n__c))
   G(n__d) -> c1(H(d))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))
K tuples:
   G(n__d) -> c1(H(d))
Defined Rule Symbols:   d, c

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(z0) -> c1(H(z0))
We considered the (Usable) Rules:
   c -> d
   d -> n__d
   c -> n__c
And the Tuples:
   H(n__d) -> c4(G(n__c))
   G(n__d) -> c1(H(d))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(G(x_1)) = [2] + [2]x_1
   POL(H(x_1)) = x_1
   POL(c) = [2]
   POL(c1(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(d) = [2]
   POL(n__c) = 0
   POL(n__d) = [2]

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   d -> n__d
   c -> d
   c -> n__c
Tuples:
   H(n__d) -> c4(G(n__c))
   G(n__d) -> c1(H(d))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(c))
K tuples:
   G(n__d) -> c1(H(d))
   G(z0) -> c1(H(z0))
Defined Rule Symbols:   d, c

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(n__d) -> c1(H(d)) by 
   G(n__d) -> c1(H(n__d))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   d -> n__d
   c -> d
   c -> n__c
Tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))
   G(n__d) -> c1(H(n__d))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(c))
K tuples:
   G(n__d) -> c1(H(d))
   G(z0) -> c1(H(z0))
Defined Rule Symbols:   d, c

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(67) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   G(n__d) -> c1(H(n__d))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   d -> n__d
   c -> d
   c -> n__c
Tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(c))
   G(z0) -> c1(H(z0))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(c))
K tuples:
   G(z0) -> c1(H(z0))
Defined Rule Symbols:   d, c

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(n__c) -> c1(H(c)) by 
   G(n__c) -> c1(H(d))
   G(n__c) -> c1(H(n__c))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   d -> n__d
   c -> d
   c -> n__c
Tuples:
   H(n__d) -> c4(G(n__c))
   G(z0) -> c1(H(z0))
   G(n__c) -> c1(H(d))
   G(n__c) -> c1(H(n__c))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(d))
   G(n__c) -> c1(H(n__c))
K tuples:
   G(z0) -> c1(H(z0))
Defined Rule Symbols:   d, c

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(71) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   G(n__c) -> c1(H(n__c))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   d -> n__d
   c -> d
   c -> n__c
Tuples:
   H(n__d) -> c4(G(n__c))
   G(z0) -> c1(H(z0))
   G(n__c) -> c1(H(d))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(d))
K tuples:
   G(z0) -> c1(H(z0))
Defined Rule Symbols:   d, c

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(73) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   c -> d
   c -> n__c

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   d -> n__d
Tuples:
   H(n__d) -> c4(G(n__c))
   G(z0) -> c1(H(z0))
   G(n__c) -> c1(H(d))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(d))
K tuples:
   G(z0) -> c1(H(z0))
Defined Rule Symbols:   d

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(n__c) -> c1(H(d)) by 
   G(n__c) -> c1(H(n__d))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   d -> n__d
Tuples:
   H(n__d) -> c4(G(n__c))
   G(z0) -> c1(H(z0))
   G(n__c) -> c1(H(n__d))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(n__d))
K tuples:
   G(z0) -> c1(H(z0))
Defined Rule Symbols:   d

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(77) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   d -> n__d

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   H(n__d) -> c4(G(n__c))
   G(z0) -> c1(H(z0))
   G(n__c) -> c1(H(n__d))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(n__d))
K tuples:
   G(z0) -> c1(H(z0))
Defined Rule Symbols:none

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(79) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace G(z0) -> c1(H(z0)) by 
   G(n__c) -> c1(H(n__c))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(n__d))
   G(n__c) -> c1(H(n__c))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(n__d))
K tuples:
   G(n__c) -> c1(H(n__c))
Defined Rule Symbols:none

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1


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

(81) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   G(n__c) -> c1(H(n__c))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(n__d))
S tuples:
   H(n__d) -> c4(G(n__c))
   G(n__c) -> c1(H(n__d))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   H_1, G_1

Compound Symbols:   c4_1, c1_1