WORST_CASE(?,O(n^1))
proof of input_NQzc6Il7hl.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^1).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CdtProblem
(9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxRelTRS
(11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(12) CpxWeightedTrs
(13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(14) CpxTypedWeightedTrs
(15) CompletionProof [UPPER BOUND(ID), 0 ms]
(16) CpxTypedWeightedCompleteTrs
(17) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(18) CpxTypedWeightedCompleteTrs
(19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(20) CpxRNTS
(21) InliningProof [UPPER BOUND(ID), 143 ms]
(22) CpxRNTS
(23) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CpxRNTS
(25) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(26) CpxRNTS
(27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 244 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 1 ms]
(32) CpxRNTS
(33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 22 ms]
(38) CpxRNTS
(39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 438 ms]
(42) CpxRNTS
(43) IntTrsBoundProof [UPPER BOUND(ID), 249 ms]
(44) CpxRNTS
(45) FinalProof [FINISHED, 0 ms]
(46) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   from(X) -> cons(X, n__from(s(X)))
   after(0, XS) -> XS
   after(s(N), cons(X, XS)) -> after(N, activate(XS))
   from(X) -> n__from(X)
   activate(n__from(X)) -> from(X)
   activate(X) -> X

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

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   FROM(z0) -> c
   FROM(z0) -> c1
   AFTER(0, z0) -> c2
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__from(z0)) -> c4(FROM(z0))
   ACTIVATE(z0) -> c5
S tuples:
   FROM(z0) -> c
   FROM(z0) -> c1
   AFTER(0, z0) -> c2
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__from(z0)) -> c4(FROM(z0))
   ACTIVATE(z0) -> c5
K tuples:none
Defined Rule Symbols:   from_1, after_2, activate_1

Defined Pair Symbols:   FROM_1, AFTER_2, ACTIVATE_1

Compound Symbols:   c, c1, c2, c3_2, c4_1, c5


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   FROM(z0) -> c
   ACTIVATE(z0) -> c5
   AFTER(0, z0) -> c2
   ACTIVATE(n__from(z0)) -> c4(FROM(z0))
   FROM(z0) -> c1

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
S tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)), ACTIVATE(z2))
K tuples:none
Defined Rule Symbols:   from_1, after_2, activate_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   c3_2


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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
S tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
K tuples:none
Defined Rule Symbols:   from_1, after_2, activate_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   c3_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   after(0, z0) -> z0
   after(s(z0), cons(z1, z2)) -> after(z0, activate(z2))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)
Tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
S tuples:
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))
K tuples:none
Defined Rule Symbols:   activate_1, from_1

Defined Pair Symbols:   AFTER_2

Compound Symbols:   c3_1


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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2)))

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

   activate(n__from(z0)) -> from(z0)
   activate(z0) -> z0
   from(z0) -> cons(z0, n__from(s(z0)))
   from(z0) -> n__from(z0)

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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2))) [1]
   activate(n__from(z0)) -> from(z0) [0]
   activate(z0) -> z0 [0]
   from(z0) -> cons(z0, n__from(s(z0))) [0]
   from(z0) -> n__from(z0) [0]

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2))) [1]
   activate(n__from(z0)) -> from(z0) [0]
   activate(z0) -> z0 [0]
   from(z0) -> cons(z0, n__from(s(z0))) [0]
   from(z0) -> n__from(z0) [0]

The TRS has the following type information:
AFTER :: s -> cons:n__from -> c3
s :: s -> s
cons :: s -> cons:n__from -> cons:n__from
c3 :: c3 -> c3
activate :: cons:n__from -> cons:n__from
n__from :: s -> cons:n__from
from :: s -> cons:n__from

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

(15) 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:

   AFTER_2

(c) The following functions are completely defined:

   activate_1
   from_1

Due to the following rules being added:

   activate(v0) -> const2 [0]
   from(v0) -> const2 [0]

And the following fresh constants:    const2, const, const1

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

(16)
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:

   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, activate(z2))) [1]
   activate(n__from(z0)) -> from(z0) [0]
   activate(z0) -> z0 [0]
   from(z0) -> cons(z0, n__from(s(z0))) [0]
   from(z0) -> n__from(z0) [0]
   activate(v0) -> const2 [0]
   from(v0) -> const2 [0]

The TRS has the following type information:
AFTER :: s -> cons:n__from:const2 -> c3
s :: s -> s
cons :: s -> cons:n__from:const2 -> cons:n__from:const2
c3 :: c3 -> c3
activate :: cons:n__from:const2 -> cons:n__from:const2
n__from :: s -> cons:n__from:const2
from :: s -> cons:n__from:const2
const2 :: cons:n__from:const2
const :: c3
const1 :: s

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

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

(18)
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:

   AFTER(s(z0), cons(z1, n__from(z0'))) -> c3(AFTER(z0, from(z0'))) [1]
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, z2)) [1]
   AFTER(s(z0), cons(z1, z2)) -> c3(AFTER(z0, const2)) [1]
   activate(n__from(z0)) -> from(z0) [0]
   activate(z0) -> z0 [0]
   from(z0) -> cons(z0, n__from(s(z0))) [0]
   from(z0) -> n__from(z0) [0]
   activate(v0) -> const2 [0]
   from(v0) -> const2 [0]

The TRS has the following type information:
AFTER :: s -> cons:n__from:const2 -> c3
s :: s -> s
cons :: s -> cons:n__from:const2 -> cons:n__from:const2
c3 :: c3 -> c3
activate :: cons:n__from:const2 -> cons:n__from:const2
n__from :: s -> cons:n__from:const2
from :: s -> cons:n__from:const2
const2 :: cons:n__from:const2
const :: c3
const1 :: s

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

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

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

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

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

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z0, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z0, from(z0')) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z = 1 + z0, z0 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z0, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> from(z0) :|: z = 1 + z0, z0 >= 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   from(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   from(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0
   from(z) -{ 0 }-> 1 + z0 + (1 + (1 + z0)) :|: z = z0, z0 >= 0


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

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

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

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

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

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z0, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z0, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z = 1 + z0, z0 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z0, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z = 1 + z0, z0 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z0, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z = 1 + z0, z0 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z0, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z = 1 + z0, z0 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z0 :|: z = z0, z0 >= 0
   activate(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   activate(z) -{ 0 }-> 0 :|: z = 1 + z0, z0 >= 0, v0 >= 0, z0 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z = 1 + z0, z0 >= 0, z0 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z = 1 + z0, z0 >= 0, z0 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   from(z) -{ 0 }-> 1 + z0 :|: z = z0, z0 >= 0
   from(z) -{ 0 }-> 1 + z0 + (1 + (1 + z0)) :|: z = z0, z0 >= 0


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

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

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

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0


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

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

   { activate }
   { from }
   { AFTER }

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

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

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {activate}, {from}, {AFTER}

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

(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:

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {activate}, {from}, {AFTER}

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

(29) 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 + 2*z

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

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

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

(31) 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: 0

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

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

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {from}, {AFTER}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + 2*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:

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {from}, {AFTER}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + 2*z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: from
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 3 + 2*z

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

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

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {AFTER}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + 2*z]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*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:

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: {AFTER}
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + 2*z]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: AFTER
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:

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

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

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, z2) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z1 >= 0, z' = 1 + z1 + z2, z - 1 >= 0, z2 >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 0) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, v0 >= 0, z0' = v0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'') :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   AFTER(z, z') -{ 1 }-> 1 + AFTER(z - 1, 1 + z0'' + (1 + (1 + z0''))) :|: z' = 1 + z1 + (1 + z0'), z1 >= 0, z0' >= 0, z - 1 >= 0, z0' = z0'', z0'' >= 0
   activate(z) -{ 0 }-> z :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z >= 0
   activate(z) -{ 0 }-> 0 :|: z - 1 >= 0, v0 >= 0, z - 1 = v0
   activate(z) -{ 0 }-> 1 + z0' :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   activate(z) -{ 0 }-> 1 + z0' + (1 + (1 + z0')) :|: z - 1 >= 0, z - 1 = z0', z0' >= 0
   from(z) -{ 0 }-> 0 :|: z >= 0
   from(z) -{ 0 }-> 1 + z :|: z >= 0
   from(z) -{ 0 }-> 1 + z + (1 + (1 + z)) :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  activate: runtime: O(1) [0], size: O(n^1) [1 + 2*z]
  from: runtime: O(1) [0], size: O(n^1) [3 + 2*z]
  AFTER: runtime: O(n^1) [z], size: O(1) [0]

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

(45) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(46)
BOUNDS(1, n^1)