MAYBE
proof of input_kmNEsfCB7R.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 3 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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxTRS
    (25) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (26) CpxTRS
    (27) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CpxWeightedTrs
        (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (30) CpxTypedWeightedTrs
        (31) CompletionProof [UPPER BOUND(ID), 0 ms]
        (32) CpxTypedWeightedCompleteTrs
            (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (34) CpxRNTS
        (35) CompletionProof [UPPER BOUND(ID), 0 ms]
        (36) CpxTypedWeightedCompleteTrs
            (37) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (38) CpxTypedWeightedCompleteTrs
            (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
            (40) CpxRNTS
            (41) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
            (42) CpxRNTS
            (43) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
            (44) CpxRNTS
            (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
            (46) CpxRNTS
            (47) IntTrsBoundProof [UPPER BOUND(ID), 208 ms]
            (48) CpxRNTS
            (49) IntTrsBoundProof [UPPER BOUND(ID), 63 ms]
            (50) CpxRNTS
(51) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(52) CpxWeightedTrs
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CpxTypedWeightedTrs
    (55) CompletionProof [UPPER BOUND(ID), 0 ms]
    (56) CpxTypedWeightedCompleteTrs
        (57) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (58) CpxTypedWeightedCompleteTrs
        (59) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (60) CpxRNTS
        (61) InliningProof [UPPER BOUND(ID), 39 ms]
        (62) CpxRNTS
        (63) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (64) CpxRNTS
        (65) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (66) CpxRNTS
        (67) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (68) CpxRNTS
        (69) IntTrsBoundProof [UPPER BOUND(ID), 93 ms]
        (70) CpxRNTS
        (71) IntTrsBoundProof [UPPER BOUND(ID), 17 ms]
        (72) CpxRNTS
        (73) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (74) CpxRNTS
        (75) IntTrsBoundProof [UPPER BOUND(ID), 125 ms]
        (76) CpxRNTS
        (77) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
        (78) CpxRNTS
        (79) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (80) CpxRNTS
        (81) IntTrsBoundProof [UPPER BOUND(ID), 221 ms]
        (82) CpxRNTS
        (83) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
        (84) CpxRNTS
    (85) CompletionProof [UPPER BOUND(ID), 0 ms]
    (86) CpxTypedWeightedCompleteTrs
        (87) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (88) 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:

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

(4)
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


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

(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:

   F(z0) -> c(F(z0))
   H(z0) -> c1
   ACTIVATE(n__h(z0)) -> c2(H(z0))
   ACTIVATE(z0) -> c3

The (relative) TRS S consists of the following rules:

   f(z0) -> g(n__h(f(z0)))
   h(z0) -> n__h(z0)
   activate(n__h(z0)) -> h(z0)
   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:

   F(z0) -> c(F(z0))
   H(z0) -> c1
   ACTIVATE(n__h(z0)) -> c2(H(z0))
   ACTIVATE(z0) -> c3

The (relative) TRS S consists of the following rules:

   f(z0) -> g(n__h(f(z0)))
   h(z0) -> n__h(z0)
   activate(n__h(z0)) -> h(z0)
   activate(z0) -> z0

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

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

(10)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(z0))
H(z0) -> c1
ACTIVATE(n__h(z0)) -> c2(H(z0))
ACTIVATE(z0) -> c3
f(z0) -> g(n__h(f(z0)))
h(z0) -> n__h(z0)
activate(n__h(z0)) -> h(z0)
activate(z0) -> z0

Types:
F :: a -> c
c :: c -> c
H :: g -> c1
c1 :: c1
ACTIVATE :: n__h -> c2:c3
n__h :: g -> n__h
c2 :: c1 -> c2:c3
c3 :: c2:c3
f :: b -> g
g :: n__h -> g
h :: g -> n__h
activate :: n__h -> n__h
hole_c1_4 :: c
hole_a2_4 :: a
hole_c13_4 :: c1
hole_g4_4 :: g
hole_c2:c35_4 :: c2:c3
hole_n__h6_4 :: n__h
hole_b7_4 :: b
gen_c8_4 :: Nat -> c

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
F, f
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
F(z0) -> c(F(z0))
H(z0) -> c1
ACTIVATE(n__h(z0)) -> c2(H(z0))
ACTIVATE(z0) -> c3
f(z0) -> g(n__h(f(z0)))
h(z0) -> n__h(z0)
activate(n__h(z0)) -> h(z0)
activate(z0) -> z0

Types:
F :: a -> c
c :: c -> c
H :: g -> c1
c1 :: c1
ACTIVATE :: n__h -> c2:c3
n__h :: g -> n__h
c2 :: c1 -> c2:c3
c3 :: c2:c3
f :: b -> g
g :: n__h -> g
h :: g -> n__h
activate :: n__h -> n__h
hole_c1_4 :: c
hole_a2_4 :: a
hole_c13_4 :: c1
hole_g4_4 :: g
hole_c2:c35_4 :: c2:c3
hole_n__h6_4 :: n__h
hole_b7_4 :: b
gen_c8_4 :: Nat -> c


Generator Equations:
gen_c8_4(0) <=> hole_c1_4
gen_c8_4(+(x, 1)) <=> c(gen_c8_4(x))


The following defined symbols remain to be analysed:
F, f
----------------------------------------

(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:

   F(z0) -> c(F(z0))
   H(z0) -> c1
   ACTIVATE(n__h(z0)) -> c2(H(z0))
   ACTIVATE(z0) -> c3

The (relative) TRS S consists of the following rules:

   f(z0) -> g(n__h(f(z0)))
   h(z0) -> n__h(z0)
   activate(n__h(z0)) -> h(z0)
   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:

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

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

(18)
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


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

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

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

(20)
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


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

(21) 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

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

(22)
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


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

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

(24)
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:

   F(z0) -> c(F(z0))

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

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

(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:

   F(z0) -> c(F(z0))

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

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

(28)
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(z0) -> c(F(z0)) [1]

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

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

(30)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   F(z0) -> c(F(z0)) [1]

The TRS has the following type information:
F :: a -> c
c :: c -> c

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

(31) 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

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

(32)
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(z0) -> c(F(z0)) [1]

The TRS has the following type information:
F :: a -> c
c :: c -> c
const :: c
const1 :: a

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

(33) 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

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

(35) 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

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const, const1

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

(36)
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(z0) -> c(F(z0)) [1]

The TRS has the following type information:
F :: a -> c
c :: c -> c
const :: c
const1 :: a

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

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

(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:

   F(z0) -> c(F(z0)) [1]

The TRS has the following type information:
F :: a -> c
c :: c -> c
const :: c
const1 :: a

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

(39) 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

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

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

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


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

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

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

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


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

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

   { F }

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

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

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

Function symbols to be analyzed: {F}

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

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

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

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

Function symbols to be analyzed: {F}

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

(47) 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

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

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

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

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

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

(49) 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: ?

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

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

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

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

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

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

(52)
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
----------------------------------------

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

(54)
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
----------------------------------------

(55) 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

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

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

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

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

(59) 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

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

(60)
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


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

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

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

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(62)
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


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(63) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(64)
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


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

   { activate }
   { h }
   { f }

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(66)
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}

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

(68)
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}

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(69) 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

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

(70)
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]

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(71) 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

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

(72)
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]

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

(74)
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]

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(75) 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

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

(76)
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]

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(77) 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

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

(78)
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]

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

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

(80)
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]

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

(81) 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

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

(82)
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]

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(83) 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: ?

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

(84)
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]

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(85) 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

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(86)
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
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(87) 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

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

(88)
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.