MAYBE
proof of input_heOVRPZ99w.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 27 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), 117 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 158 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 262 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
        (38) CpxRNTS
    (39) CompletionProof [UPPER BOUND(ID), 0 ms]
    (40) CpxTypedWeightedCompleteTrs
        (41) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
(43) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(44) CdtProblem
    (45) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem


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

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


The TRS R consists of the following rules:

   f(X) -> g(n__h(f(X)))
   h(X) -> n__h(X)
   activate(n__h(X)) -> h(X)
   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:

   f(X) -> g(n__h(f(X)))
   h(X) -> n__h(X)
   activate(n__h(X)) -> h(X)
   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:

   f(X) -> g(n__h(f(X)))
   h(X) -> n__h(X)
   activate(n__h(X)) -> h(X)
   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:

   f(X) -> g(n__h(f(X))) [1]
   h(X) -> n__h(X) [1]
   activate(n__h(X)) -> h(X) [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:

   f(X) -> g(n__h(f(X))) [1]
   h(X) -> n__h(X) [1]
   activate(n__h(X)) -> h(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
f :: a -> g
g :: n__h -> g
n__h :: g -> n__h
h :: g -> n__h
activate :: n__h -> n__h

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:

   f_1
   h_1
   activate_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

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

(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:

   f(X) -> g(n__h(f(X))) [1]
   h(X) -> n__h(X) [1]
   activate(n__h(X)) -> h(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
f :: a -> g
g :: n__h -> g
n__h :: g -> n__h
h :: g -> n__h
activate :: n__h -> n__h
const :: g
const1 :: a
const2 :: n__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:

   f(X) -> g(n__h(f(X))) [1]
   h(X) -> n__h(X) [1]
   activate(n__h(X)) -> h(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
f :: a -> g
g :: n__h -> g
n__h :: g -> n__h
h :: g -> n__h
activate :: n__h -> n__h
const :: g
const1 :: a
const2 :: n__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:

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

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

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

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


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

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

   h(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) -{ 2 }-> 1 + X' :|: z = 1 + X, X >= 0, X' >= 0, X = X'
   f(z) -{ 1 }-> 1 + (1 + f(X)) :|: X >= 0, z = X
   h(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) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0
   h(z) -{ 1 }-> 1 + z :|: z >= 0


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

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

   { activate }
   { h }
   { f }

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

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

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

Function symbols to be analyzed: {activate}, {h}, {f}

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

(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 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0
   h(z) -{ 1 }-> 1 + z :|: z >= 0

Function symbols to be analyzed: {activate}, {h}, {f}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

Function symbols to be analyzed: {h}, {f}
Previous analysis results are:
  activate: runtime: O(1) [3], size: O(n^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) -{ 2 }-> 1 + X' :|: z - 1 >= 0, X' >= 0, z - 1 = X'
   f(z) -{ 1 }-> 1 + (1 + f(z)) :|: z >= 0
   h(z) -{ 1 }-> 1 + z :|: z >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

And the following fresh constants:    const, const1, const2

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

(40)
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:

   f(X) -> g(n__h(f(X))) [1]
   h(X) -> n__h(X) [1]
   activate(n__h(X)) -> h(X) [1]
   activate(X) -> X [1]

The TRS has the following type information:
f :: a -> g
g :: n__h -> g
n__h :: g -> n__h
h :: g -> n__h
activate :: n__h -> n__h
const :: g
const1 :: a
const2 :: n__h

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:

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

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> g(n__h(f(z0)))
   h(z0) -> n__h(z0)
   activate(n__h(z0)) -> h(z0)
   activate(z0) -> z0
Tuples:
   F(z0) -> c(F(z0))
   H(z0) -> c1
   ACTIVATE(n__h(z0)) -> c2(H(z0))
   ACTIVATE(z0) -> c3
S tuples:
   F(z0) -> c(F(z0))
   H(z0) -> c1
   ACTIVATE(n__h(z0)) -> c2(H(z0))
   ACTIVATE(z0) -> c3
K tuples:none
Defined Rule Symbols:   f_1, h_1, activate_1

Defined Pair Symbols:   F_1, H_1, ACTIVATE_1

Compound Symbols:   c_1, c1, c2_1, c3


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

(45) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   H(z0) -> c1
   ACTIVATE(n__h(z0)) -> c2(H(z0))
   ACTIVATE(z0) -> c3

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(z0) -> g(n__h(f(z0)))
   h(z0) -> n__h(z0)
   activate(n__h(z0)) -> h(z0)
   activate(z0) -> z0
Tuples:
   F(z0) -> c(F(z0))
S tuples:
   F(z0) -> c(F(z0))
K tuples:none
Defined Rule Symbols:   f_1, h_1, activate_1

Defined Pair Symbols:   F_1

Compound Symbols:   c_1


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

(47) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(z0) -> g(n__h(f(z0)))
   h(z0) -> n__h(z0)
   activate(n__h(z0)) -> h(z0)
   activate(z0) -> z0

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   F(z0) -> c(F(z0))
S tuples:
   F(z0) -> c(F(z0))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   F_1

Compound Symbols:   c_1