WORST_CASE(?,O(n^2))
proof of input_izDxwugT2s.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
(3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
(7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
(9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) CpxWeightedTrs
(11) CpxWeightedTrsRenamingProof [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), 252 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 63 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 188 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), 147 ms]
(40) CpxRNTS
(41) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
(42) CpxRNTS
(43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(44) CpxRNTS
(45) IntTrsBoundProof [UPPER BOUND(ID), 1002 ms]
(46) CpxRNTS
(47) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
(48) CpxRNTS
(49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(50) CpxRNTS
(51) IntTrsBoundProof [UPPER BOUND(ID), 283 ms]
(52) CpxRNTS
(53) IntTrsBoundProof [UPPER BOUND(ID), 206 ms]
(54) CpxRNTS
(55) FinalProof [FINISHED, 0 ms]
(56) BOUNDS(1, n^2)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   exp(x, 0) -> s(0)
   exp(x, s(y)) -> *(x, exp(x, y))
   *(0, y) -> 0
   *(s(x), y) -> +(y, *(x, y))
   -(0, y) -> 0
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)

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:
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> *(z0, exp(z0, z1))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(z1, *(z0, z1))
   -(0, z0) -> 0
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   EXP(z0, 0) -> c
   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
   *'(0, z0) -> c2
   *'(s(z0), z1) -> c3(*'(z0, z1))
   -'(0, z0) -> c4
   -'(z0, 0) -> c5
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
S tuples:
   EXP(z0, 0) -> c
   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
   *'(0, z0) -> c2
   *'(s(z0), z1) -> c3(*'(z0, z1))
   -'(0, z0) -> c4
   -'(z0, 0) -> c5
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
K tuples:none
Defined Rule Symbols:   exp_2, *_2, -_2

Defined Pair Symbols:   EXP_2, *'_2, -'_2

Compound Symbols:   c, c1_2, c2, c3_1, c4, c5, c6_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   -'(z0, 0) -> c5
   EXP(z0, 0) -> c
   -'(0, z0) -> c4
   *'(0, z0) -> c2

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> *(z0, exp(z0, z1))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(z1, *(z0, z1))
   -(0, z0) -> 0
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
   *'(s(z0), z1) -> c3(*'(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
S tuples:
   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
   *'(s(z0), z1) -> c3(*'(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
K tuples:none
Defined Rule Symbols:   exp_2, *_2, -_2

Defined Pair Symbols:   EXP_2, *'_2, -'_2

Compound Symbols:   c1_2, c3_1, c6_1


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

(5) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   -(0, z0) -> 0
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> *(z0, exp(z0, z1))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(z1, *(z0, z1))
Tuples:
   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
   *'(s(z0), z1) -> c3(*'(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
S tuples:
   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
   *'(s(z0), z1) -> c3(*'(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))
K tuples:none
Defined Rule Symbols:   exp_2, *_2

Defined Pair Symbols:   EXP_2, *'_2, -'_2

Compound Symbols:   c1_2, c3_1, c6_1


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

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

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1))
   *'(s(z0), z1) -> c3(*'(z0, z1))
   -'(s(z0), s(z1)) -> c6(-'(z0, z1))

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

   exp(z0, 0) -> s(0)
   exp(z0, s(z1)) -> *(z0, exp(z0, z1))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(z1, *(z0, z1))

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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) [1]
   *'(s(z0), z1) -> c3(*'(z0, z1)) [1]
   -'(s(z0), s(z1)) -> c6(-'(z0, z1)) [1]
   exp(z0, 0) -> s(0) [0]
   exp(z0, s(z1)) -> *(z0, exp(z0, z1)) [0]
   *(0, z0) -> 0 [0]
   *(s(z0), z1) -> +(z1, *(z0, z1)) [0]

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

(11) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

* => times

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) [1]
   *'(s(z0), z1) -> c3(*'(z0, z1)) [1]
   -'(s(z0), s(z1)) -> c6(-'(z0, z1)) [1]
   exp(z0, 0) -> s(0) [0]
   exp(z0, s(z1)) -> times(z0, exp(z0, z1)) [0]
   times(0, z0) -> 0 [0]
   times(s(z0), z1) -> +(z1, times(z0, 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:

   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) [1]
   *'(s(z0), z1) -> c3(*'(z0, z1)) [1]
   -'(s(z0), s(z1)) -> c6(-'(z0, z1)) [1]
   exp(z0, 0) -> s(0) [0]
   exp(z0, s(z1)) -> times(z0, exp(z0, z1)) [0]
   times(0, z0) -> 0 [0]
   times(s(z0), z1) -> +(z1, times(z0, z1)) [0]

The TRS has the following type information:
EXP :: s:0:+ -> s:0:+ -> c1
s :: s:0:+ -> s:0:+
c1 :: c3 -> c1 -> c1
*' :: s:0:+ -> s:0:+ -> c3
exp :: s:0:+ -> s:0:+ -> s:0:+
c3 :: c3 -> c3
-' :: s:0:+ -> s:0:+ -> c6
c6 :: c6 -> c6
0 :: s:0:+
times :: s:0:+ -> s:0:+ -> s:0:+
+ :: s:0:+ -> s:0:+ -> s:0:+

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:

   EXP_2
   *'_2
   -'_2

(c) The following functions are completely defined:

   exp_2
   times_2

Due to the following rules being added:

   exp(v0, v1) -> 0 [0]
   times(v0, v1) -> 0 [0]

And the following fresh constants:    const, const1, const2

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

(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:

   EXP(z0, s(z1)) -> c1(*'(z0, exp(z0, z1)), EXP(z0, z1)) [1]
   *'(s(z0), z1) -> c3(*'(z0, z1)) [1]
   -'(s(z0), s(z1)) -> c6(-'(z0, z1)) [1]
   exp(z0, 0) -> s(0) [0]
   exp(z0, s(z1)) -> times(z0, exp(z0, z1)) [0]
   times(0, z0) -> 0 [0]
   times(s(z0), z1) -> +(z1, times(z0, z1)) [0]
   exp(v0, v1) -> 0 [0]
   times(v0, v1) -> 0 [0]

The TRS has the following type information:
EXP :: s:0:+ -> s:0:+ -> c1
s :: s:0:+ -> s:0:+
c1 :: c3 -> c1 -> c1
*' :: s:0:+ -> s:0:+ -> c3
exp :: s:0:+ -> s:0:+ -> s:0:+
c3 :: c3 -> c3
-' :: s:0:+ -> s:0:+ -> c6
c6 :: c6 -> c6
0 :: s:0:+
times :: s:0:+ -> s:0:+ -> s:0:+
+ :: s:0:+ -> s:0:+ -> s:0:+
const :: c1
const1 :: c3
const2 :: c6

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:

   EXP(z0, s(0)) -> c1(*'(z0, s(0)), EXP(z0, 0)) [1]
   EXP(z0, s(s(z1'))) -> c1(*'(z0, times(z0, exp(z0, z1'))), EXP(z0, s(z1'))) [1]
   EXP(z0, s(z1)) -> c1(*'(z0, 0), EXP(z0, z1)) [1]
   *'(s(z0), z1) -> c3(*'(z0, z1)) [1]
   -'(s(z0), s(z1)) -> c6(-'(z0, z1)) [1]
   exp(z0, 0) -> s(0) [0]
   exp(z0, s(0)) -> times(z0, s(0)) [0]
   exp(z0, s(s(z1''))) -> times(z0, times(z0, exp(z0, z1''))) [0]
   exp(z0, s(z1)) -> times(z0, 0) [0]
   times(0, z0) -> 0 [0]
   times(s(z0), z1) -> +(z1, times(z0, z1)) [0]
   exp(v0, v1) -> 0 [0]
   times(v0, v1) -> 0 [0]

The TRS has the following type information:
EXP :: s:0:+ -> s:0:+ -> c1
s :: s:0:+ -> s:0:+
c1 :: c3 -> c1 -> c1
*' :: s:0:+ -> s:0:+ -> c3
exp :: s:0:+ -> s:0:+ -> s:0:+
c3 :: c3 -> c3
-' :: s:0:+ -> s:0:+ -> c6
c6 :: c6 -> c6
0 :: s:0:+
times :: s:0:+ -> s:0:+ -> s:0:+
+ :: s:0:+ -> s:0:+ -> s:0:+
const :: c1
const1 :: c3
const2 :: c6

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:

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

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

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

   *'(z, z') -{ 1 }-> 1 + *'(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   EXP(z, z') -{ 1 }-> 1 + *'(z0, times(z0, exp(z0, z1'))) + EXP(z0, 1 + z1') :|: z' = 1 + (1 + z1'), z = z0, z1' >= 0, z0 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z0, 0) + EXP(z0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   EXP(z, z') -{ 1 }-> 1 + *'(z0, 1 + 0) + EXP(z0, 0) :|: z = z0, z' = 1 + 0, z0 >= 0
   exp(z, z') -{ 0 }-> times(z0, times(z0, exp(z0, z1''))) :|: z = z0, z' = 1 + (1 + z1''), z0 >= 0, z1'' >= 0
   exp(z, z') -{ 0 }-> times(z0, 0) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   exp(z, z') -{ 0 }-> times(z0, 1 + 0) :|: z = z0, z' = 1 + 0, z0 >= 0
   exp(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   exp(z, z') -{ 0 }-> 1 + 0 :|: z = z0, z0 >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   times(z, z') -{ 0 }-> 1 + z1 + times(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0


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

(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') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> times(z, 0) :|: z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, 1 + 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + times(z - 1, z') :|: z' >= 0, z - 1 >= 0


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

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

   { times }
   { -' }
   { *' }
   { exp }
   { EXP }

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

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

   *'(z, z') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> times(z, 0) :|: z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, 1 + 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + times(z - 1, z') :|: z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {times}, {-'}, {*'}, {exp}, {EXP}

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

(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') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> times(z, 0) :|: z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, 1 + 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + times(z - 1, z') :|: z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {times}, {-'}, {*'}, {exp}, {EXP}

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   *'(z, z') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> times(z, 0) :|: z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, 1 + 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + times(z - 1, z') :|: z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {times}, {-'}, {*'}, {exp}, {EXP}
Previous analysis results are:
  times: runtime: ?, size: O(n^2) [z + z*z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: times
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') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> times(z, 0) :|: z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, 1 + 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + times(z - 1, z') :|: z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {-'}, {*'}, {exp}, {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*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') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {-'}, {*'}, {exp}, {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: -'
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') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {-'}, {*'}, {exp}, {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': runtime: ?, size: O(1) [0]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: -'
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') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {*'}, {exp}, {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': 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') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {*'}, {exp}, {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': runtime: O(n^1) [z'], size: O(1) [0]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   *'(z, z') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {*'}, {exp}, {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': runtime: O(n^1) [z'], size: O(1) [0]
  *': runtime: ?, size: O(1) [0]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   *'(z, z') -{ 1 }-> 1 + *'(z - 1, z') :|: z' >= 0, z - 1 >= 0
   -'(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 0) + EXP(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, 1 + 0) + EXP(z, 0) :|: z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {exp}, {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': runtime: O(n^1) [z'], size: O(1) [0]
  *': runtime: O(n^1) [z], size: O(1) [0]

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

(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') -{ z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0
   -'(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s2 + EXP(z, 0) :|: s2 >= 0, s2 <= 0, z' = 1 + 0, z >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s3 + EXP(z, z' - 1) :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {exp}, {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': runtime: O(n^1) [z'], size: O(1) [0]
  *': runtime: O(n^1) [z], size: O(1) [0]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   *'(z, z') -{ z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0
   -'(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s2 + EXP(z, 0) :|: s2 >= 0, s2 <= 0, z' = 1 + 0, z >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s3 + EXP(z, z' - 1) :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {exp}, {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': runtime: O(n^1) [z'], size: O(1) [0]
  *': runtime: O(n^1) [z], size: O(1) [0]
  exp: runtime: ?, size: INF

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: exp
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:

   *'(z, z') -{ z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0
   -'(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s2 + EXP(z, 0) :|: s2 >= 0, s2 <= 0, z' = 1 + 0, z >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s3 + EXP(z, z' - 1) :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 }-> 1 + *'(z, times(z, exp(z, z' - 2))) + EXP(z, 1 + (z' - 2)) :|: z' - 2 >= 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': runtime: O(n^1) [z'], size: O(1) [0]
  *': runtime: O(n^1) [z], size: O(1) [0]
  exp: runtime: O(1) [0], size: INF

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

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

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

   *'(z, z') -{ z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0
   -'(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s2 + EXP(z, 0) :|: s2 >= 0, s2 <= 0, z' = 1 + 0, z >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s3 + EXP(z, z' - 1) :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s7 + EXP(z, 1 + (z' - 2)) :|: s5 >= 0, s5 <= inf, s6 >= 0, s6 <= s5 * z + z, s7 >= 0, s7 <= 0, z' - 2 >= 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= inf', s9 >= 0, s9 <= s8 * z + z, s10 >= 0, s10 <= s9 * z + z, z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': runtime: O(n^1) [z'], size: O(1) [0]
  *': runtime: O(n^1) [z], size: O(1) [0]
  exp: runtime: O(1) [0], size: INF

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(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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(52)
Obligation:
Complexity RNTS consisting of the following rules:

   *'(z, z') -{ z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0
   -'(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s2 + EXP(z, 0) :|: s2 >= 0, s2 <= 0, z' = 1 + 0, z >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s3 + EXP(z, z' - 1) :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s7 + EXP(z, 1 + (z' - 2)) :|: s5 >= 0, s5 <= inf, s6 >= 0, s6 <= s5 * z + z, s7 >= 0, s7 <= 0, z' - 2 >= 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= inf', s9 >= 0, s9 <= s8 * z + z, s10 >= 0, s10 <= s9 * z + z, z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: {EXP}
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': runtime: O(n^1) [z'], size: O(1) [0]
  *': runtime: O(n^1) [z], size: O(1) [0]
  exp: runtime: O(1) [0], size: INF
  EXP: runtime: ?, size: O(1) [0]

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(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: EXP
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 3*z*z' + 3*z'

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(54)
Obligation:
Complexity RNTS consisting of the following rules:

   *'(z, z') -{ z }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' >= 0, z - 1 >= 0
   -'(z, z') -{ z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s2 + EXP(z, 0) :|: s2 >= 0, s2 <= 0, z' = 1 + 0, z >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s3 + EXP(z, z' - 1) :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z >= 0
   EXP(z, z') -{ 1 + z }-> 1 + s7 + EXP(z, 1 + (z' - 2)) :|: s5 >= 0, s5 <= inf, s6 >= 0, s6 <= s5 * z + z, s7 >= 0, s7 <= 0, z' - 2 >= 0, z >= 0
   exp(z, z') -{ 0 }-> s :|: s >= 0, s <= (1 + 0) * z + z, z' = 1 + 0, z >= 0
   exp(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0 * z + z, z' - 1 >= 0, z >= 0
   exp(z, z') -{ 0 }-> s10 :|: s8 >= 0, s8 <= inf', s9 >= 0, s9 <= s8 * z + z, s10 >= 0, s10 <= s9 * z + z, z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 0 }-> 1 + 0 :|: z >= 0, z' = 0
   times(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 0 }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z' >= 0, z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  times: runtime: O(1) [0], size: O(n^2) [z + z*z']
  -': runtime: O(n^1) [z'], size: O(1) [0]
  *': runtime: O(n^1) [z], size: O(1) [0]
  exp: runtime: O(1) [0], size: INF
  EXP: runtime: O(n^2) [3*z*z' + 3*z'], size: O(1) [0]

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(55) FinalProof (FINISHED)
Computed overall runtime complexity
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(56)
BOUNDS(1, n^2)