WORST_CASE(?,O(n^1))
proof of input_ga0Bee3FSC.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 225 ms]
(2) CpxRelTRS
(3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
(5) CdtLeafRemovalProof [ComplexityIfPolyImplication, 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) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CpxRNTS
(23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CpxRNTS
(25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 188 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 72 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 158 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
(36) CpxRNTS
(37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
(42) CpxRNTS
(43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 219 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 15 ms]
(48) CpxRNTS
(49) FinalProof [FINISHED, 0 ms]
(50) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   map(Cons(x, xs)) -> Cons(f(x), map(xs))
   map(Nil) -> Nil
   goal(xs) -> map(xs)
   f(x) -> *(x, x)
   +Full(S(x), y) -> +Full(x, S(y))
   +Full(0, y) -> y

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

   *(x, S(S(y))) -> +(x, *(x, S(y)))
   *(x, S(0)) -> x
   *(x, 0) -> 0
   *(0, y) -> 0

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   map(Cons(x, xs)) -> Cons(f(x), map(xs))
   map(Nil) -> Nil
   goal(xs) -> map(xs)
   f(x) -> *(x, x)
   +Full(S(x), y) -> +Full(x, S(y))
   +Full(0, y) -> y

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

   *(x, S(S(y))) -> +(x, *(x, S(y)))
   *(x, S(0)) -> x
   *(x, 0) -> 0
   *(0, y) -> 0

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:
   *(z0, S(S(z1))) -> +(z0, *(z0, S(z1)))
   *(z0, S(0)) -> z0
   *(z0, 0) -> 0
   *(0, z0) -> 0
   map(Cons(z0, z1)) -> Cons(f(z0), map(z1))
   map(Nil) -> Nil
   goal(z0) -> map(z0)
   f(z0) -> *(z0, z0)
   +Full(S(z0), z1) -> +Full(z0, S(z1))
   +Full(0, z0) -> z0
Tuples:
   *'(z0, S(S(z1))) -> c(*'(z0, S(z1)))
   *'(z0, S(0)) -> c1
   *'(z0, 0) -> c2
   *'(0, z0) -> c3
   MAP(Cons(z0, z1)) -> c4(F(z0))
   MAP(Cons(z0, z1)) -> c5(MAP(z1))
   MAP(Nil) -> c6
   GOAL(z0) -> c7(MAP(z0))
   F(z0) -> c8(*'(z0, z0))
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1)))
   +FULL(0, z0) -> c10
S tuples:
   MAP(Cons(z0, z1)) -> c4(F(z0))
   MAP(Cons(z0, z1)) -> c5(MAP(z1))
   MAP(Nil) -> c6
   GOAL(z0) -> c7(MAP(z0))
   F(z0) -> c8(*'(z0, z0))
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1)))
   +FULL(0, z0) -> c10
K tuples:none
Defined Rule Symbols:   map_1, goal_1, f_1, +Full_2, *_2

Defined Pair Symbols:   *'_2, MAP_1, GOAL_1, F_1, +FULL_2

Compound Symbols:   c_1, c1, c2, c3, c4_1, c5_1, c6, c7_1, c8_1, c9_1, c10


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

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GOAL(z0) -> c7(MAP(z0))
Removed 5 trailing nodes:
   *'(0, z0) -> c3
   *'(z0, S(0)) -> c1
   *'(z0, 0) -> c2
   +FULL(0, z0) -> c10
   MAP(Nil) -> c6

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, S(S(z1))) -> +(z0, *(z0, S(z1)))
   *(z0, S(0)) -> z0
   *(z0, 0) -> 0
   *(0, z0) -> 0
   map(Cons(z0, z1)) -> Cons(f(z0), map(z1))
   map(Nil) -> Nil
   goal(z0) -> map(z0)
   f(z0) -> *(z0, z0)
   +Full(S(z0), z1) -> +Full(z0, S(z1))
   +Full(0, z0) -> z0
Tuples:
   *'(z0, S(S(z1))) -> c(*'(z0, S(z1)))
   MAP(Cons(z0, z1)) -> c4(F(z0))
   MAP(Cons(z0, z1)) -> c5(MAP(z1))
   F(z0) -> c8(*'(z0, z0))
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1)))
S tuples:
   MAP(Cons(z0, z1)) -> c4(F(z0))
   MAP(Cons(z0, z1)) -> c5(MAP(z1))
   F(z0) -> c8(*'(z0, z0))
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1)))
K tuples:none
Defined Rule Symbols:   map_1, goal_1, f_1, +Full_2, *_2

Defined Pair Symbols:   *'_2, MAP_1, F_1, +FULL_2

Compound Symbols:   c_1, c4_1, c5_1, c8_1, c9_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   *(z0, S(S(z1))) -> +(z0, *(z0, S(z1)))
   *(z0, S(0)) -> z0
   *(z0, 0) -> 0
   *(0, z0) -> 0
   map(Cons(z0, z1)) -> Cons(f(z0), map(z1))
   map(Nil) -> Nil
   goal(z0) -> map(z0)
   f(z0) -> *(z0, z0)
   +Full(S(z0), z1) -> +Full(z0, S(z1))
   +Full(0, z0) -> z0

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   *'(z0, S(S(z1))) -> c(*'(z0, S(z1)))
   MAP(Cons(z0, z1)) -> c4(F(z0))
   MAP(Cons(z0, z1)) -> c5(MAP(z1))
   F(z0) -> c8(*'(z0, z0))
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1)))
S tuples:
   MAP(Cons(z0, z1)) -> c4(F(z0))
   MAP(Cons(z0, z1)) -> c5(MAP(z1))
   F(z0) -> c8(*'(z0, z0))
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1)))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   *'_2, MAP_1, F_1, +FULL_2

Compound Symbols:   c_1, c4_1, c5_1, c8_1, c9_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:

   MAP(Cons(z0, z1)) -> c4(F(z0))
   MAP(Cons(z0, z1)) -> c5(MAP(z1))
   F(z0) -> c8(*'(z0, z0))
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1)))

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

   *'(z0, S(S(z1))) -> c(*'(z0, S(z1)))

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:

   MAP(Cons(z0, z1)) -> c4(F(z0)) [1]
   MAP(Cons(z0, z1)) -> c5(MAP(z1)) [1]
   F(z0) -> c8(*'(z0, z0)) [1]
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1))) [1]
   *'(z0, S(S(z1))) -> c(*'(z0, S(z1))) [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:

   MAP(Cons(z0, z1)) -> c4(F(z0)) [1]
   MAP(Cons(z0, z1)) -> c5(MAP(z1)) [1]
   F(z0) -> c8(*'(z0, z0)) [1]
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1))) [1]
   *'(z0, S(S(z1))) -> c(*'(z0, S(z1))) [0]

The TRS has the following type information:
MAP :: Cons -> c4:c5
Cons :: S -> Cons -> Cons
c4 :: c8 -> c4:c5
F :: S -> c8
c5 :: c4:c5 -> c4:c5
c8 :: c -> c8
*' :: S -> S -> c
+FULL :: S -> S -> c9
S :: S -> S
c9 :: c9 -> c9
c :: c -> c

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:

   MAP_1
   F_1
   +FULL_2

(c) The following functions are completely defined:

   *'_2

Due to the following rules being added:

   *'(v0, v1) -> const4 [0]

And the following fresh constants:    const4, const, const1, const2, const3, const5

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

(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:

   MAP(Cons(z0, z1)) -> c4(F(z0)) [1]
   MAP(Cons(z0, z1)) -> c5(MAP(z1)) [1]
   F(z0) -> c8(*'(z0, z0)) [1]
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1))) [1]
   *'(z0, S(S(z1))) -> c(*'(z0, S(z1))) [0]
   *'(v0, v1) -> const4 [0]

The TRS has the following type information:
MAP :: Cons -> c4:c5
Cons :: S -> Cons -> Cons
c4 :: c8 -> c4:c5
F :: S -> c8
c5 :: c4:c5 -> c4:c5
c8 :: c:const4 -> c8
*' :: S -> S -> c:const4
+FULL :: S -> S -> c9
S :: S -> S
c9 :: c9 -> c9
c :: c:const4 -> c:const4
const4 :: c:const4
const :: c4:c5
const1 :: Cons
const2 :: S
const3 :: c8
const5 :: c9

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:

   MAP(Cons(z0, z1)) -> c4(F(z0)) [1]
   MAP(Cons(z0, z1)) -> c5(MAP(z1)) [1]
   F(z0) -> c8(*'(z0, z0)) [1]
   +FULL(S(z0), z1) -> c9(+FULL(z0, S(z1))) [1]
   *'(z0, S(S(z1))) -> c(*'(z0, S(z1))) [0]
   *'(v0, v1) -> const4 [0]

The TRS has the following type information:
MAP :: Cons -> c4:c5
Cons :: S -> Cons -> Cons
c4 :: c8 -> c4:c5
F :: S -> c8
c5 :: c4:c5 -> c4:c5
c8 :: c:const4 -> c8
*' :: S -> S -> c:const4
+FULL :: S -> S -> c9
S :: S -> S
c9 :: c9 -> c9
c :: c:const4 -> c:const4
const4 :: c:const4
const :: c4:c5
const1 :: Cons
const2 :: S
const3 :: c8
const5 :: c9

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:

   const4 => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0
   const5 => 0

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

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

   *'(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   *'(z, z') -{ 0 }-> 1 + *'(z0, 1 + z1) :|: z' = 1 + (1 + z1), z = z0, z1 >= 0, z0 >= 0
   +FULL(z, z') -{ 1 }-> 1 + +FULL(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   F(z) -{ 1 }-> 1 + *'(z0, z0) :|: z = z0, z0 >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1


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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + *'(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ 1 }-> 1 + +FULL(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + *'(z, z) :|: z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1


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

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

   { *' }
   { +FULL }
   { F }
   { MAP }

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + *'(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ 1 }-> 1 + +FULL(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + *'(z, z) :|: z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {*'}, {+FULL}, {F}, {MAP}

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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + *'(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ 1 }-> 1 + +FULL(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + *'(z, z) :|: z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {*'}, {+FULL}, {F}, {MAP}

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + *'(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ 1 }-> 1 + +FULL(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + *'(z, z) :|: z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {*'}, {+FULL}, {F}, {MAP}
Previous analysis results are:
  *': runtime: ?, size: O(n^1) [z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + *'(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ 1 }-> 1 + +FULL(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + *'(z, z) :|: z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {+FULL}, {F}, {MAP}
Previous analysis results are:
  *': runtime: O(1) [0], size: O(n^1) [z']

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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + s' :|: s' >= 0, s' <= 1 + (z' - 2), z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ 1 }-> 1 + +FULL(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + s :|: s >= 0, s <= z, z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {+FULL}, {F}, {MAP}
Previous analysis results are:
  *': runtime: O(1) [0], size: O(n^1) [z']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + s' :|: s' >= 0, s' <= 1 + (z' - 2), z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ 1 }-> 1 + +FULL(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + s :|: s >= 0, s <= z, z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {+FULL}, {F}, {MAP}
Previous analysis results are:
  *': runtime: O(1) [0], size: O(n^1) [z']
  +FULL: runtime: ?, size: O(1) [0]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + s' :|: s' >= 0, s' <= 1 + (z' - 2), z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ 1 }-> 1 + +FULL(z - 1, 1 + z') :|: z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + s :|: s >= 0, s <= z, z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {F}, {MAP}
Previous analysis results are:
  *': runtime: O(1) [0], size: O(n^1) [z']
  +FULL: runtime: O(n^1) [z], size: O(1) [0]

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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + s' :|: s' >= 0, s' <= 1 + (z' - 2), z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ z }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + s :|: s >= 0, s <= z, z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {F}, {MAP}
Previous analysis results are:
  *': runtime: O(1) [0], size: O(n^1) [z']
  +FULL: runtime: O(n^1) [z], size: O(1) [0]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + s' :|: s' >= 0, s' <= 1 + (z' - 2), z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ z }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + s :|: s >= 0, s <= z, z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {F}, {MAP}
Previous analysis results are:
  *': runtime: O(1) [0], size: O(n^1) [z']
  +FULL: runtime: O(n^1) [z], size: O(1) [0]
  F: runtime: ?, size: O(n^1) [1 + z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + s' :|: s' >= 0, s' <= 1 + (z' - 2), z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ z }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + s :|: s >= 0, s <= z, z >= 0
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + F(z0) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {MAP}
Previous analysis results are:
  *': runtime: O(1) [0], size: O(n^1) [z']
  +FULL: runtime: O(n^1) [z], size: O(1) [0]
  F: runtime: O(1) [1], size: O(n^1) [1 + z]

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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + s' :|: s' >= 0, s' <= 1 + (z' - 2), z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ z }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + s :|: s >= 0, s <= z, z >= 0
   MAP(z) -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {MAP}
Previous analysis results are:
  *': runtime: O(1) [0], size: O(n^1) [z']
  +FULL: runtime: O(n^1) [z], size: O(1) [0]
  F: runtime: O(1) [1], size: O(n^1) [1 + z]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + s' :|: s' >= 0, s' <= 1 + (z' - 2), z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ z }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + s :|: s >= 0, s <= z, z >= 0
   MAP(z) -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: {MAP}
Previous analysis results are:
  *': runtime: O(1) [0], size: O(n^1) [z']
  +FULL: runtime: O(n^1) [z], size: O(1) [0]
  F: runtime: O(1) [1], size: O(n^1) [1 + z]
  MAP: runtime: ?, size: O(n^1) [1 + z]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: MAP
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z

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

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

   *'(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   *'(z, z') -{ 0 }-> 1 + s' :|: s' >= 0, s' <= 1 + (z' - 2), z' - 2 >= 0, z >= 0
   +FULL(z, z') -{ z }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' >= 0, z - 1 >= 0
   F(z) -{ 1 }-> 1 + s :|: s >= 0, s <= z, z >= 0
   MAP(z) -{ 2 }-> 1 + s1 :|: s1 >= 0, s1 <= z0 + 1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   MAP(z) -{ 1 }-> 1 + MAP(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1

Function symbols to be analyzed: 
Previous analysis results are:
  *': runtime: O(1) [0], size: O(n^1) [z']
  +FULL: runtime: O(n^1) [z], size: O(1) [0]
  F: runtime: O(1) [1], size: O(n^1) [1 + z]
  MAP: runtime: O(n^1) [2 + z], size: O(n^1) [1 + z]

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

(50)
BOUNDS(1, n^1)