/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, n^2).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 3 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 411 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 98 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 596 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 379 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 509 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (36) CpxRNTS
    (37) FinalProof [FINISHED, 0 ms]
    (38) BOUNDS(1, n^2)
(39) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(40) CpxTRS
    (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) typed CpxTrs
    (43) OrderProof [LOWER BOUND(ID), 0 ms]
    (44) typed CpxTrs
    (45) RewriteLemmaProof [LOWER BOUND(ID), 290 ms]
    (46) BEST
        (47) proven lower bound
            (48) LowerBoundPropagationProof [FINISHED, 0 ms]
            (49) BOUNDS(n^1, INF)
        (50) typed CpxTrs
            (51) RewriteLemmaProof [LOWER BOUND(ID), 739 ms]
            (52) proven lower bound
            (53) LowerBoundPropagationProof [FINISHED, 0 ms]
            (54) BOUNDS(n^2, INF)


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

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


The TRS R consists of the following rules:

   f(0) -> 1
   f(s(x)) -> g(x, s(x))
   g(0, y) -> y
   g(s(x), y) -> g(x, +(y, s(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   g(s(x), y) -> g(x, s(+(y, x)))

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

The duplicating contexts are:
f(s([]))
g(s([]), y)


The defined contexts are:
g(x0, [])
g(x0, s([]))
+([], s(x1))
+([], x1)


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, we have rc = irc.
----------------------------------------

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


The TRS R consists of the following rules:

   f(0) -> 1
   f(s(x)) -> g(x, s(x))
   g(0, y) -> y
   g(s(x), y) -> g(x, +(y, s(x)))
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   g(s(x), y) -> g(x, s(+(y, x)))

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

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

(4)
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:

   f(0) -> 1 [1]
   f(s(x)) -> g(x, s(x)) [1]
   g(0, y) -> y [1]
   g(s(x), y) -> g(x, +(y, s(x))) [1]
   +(x, 0) -> x [1]
   +(x, s(y)) -> s(+(x, y)) [1]
   g(s(x), y) -> g(x, s(+(y, x))) [1]

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

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

+ => plus

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

(6)
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:

   f(0) -> 1 [1]
   f(s(x)) -> g(x, s(x)) [1]
   g(0, y) -> y [1]
   g(s(x), y) -> g(x, plus(y, s(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   g(s(x), y) -> g(x, s(plus(y, x))) [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(0) -> 1 [1]
   f(s(x)) -> g(x, s(x)) [1]
   g(0, y) -> y [1]
   g(s(x), y) -> g(x, plus(y, s(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   g(s(x), y) -> g(x, s(plus(y, x))) [1]

The TRS has the following type information:
f :: 0:1:s -> 0:1:s
0 :: 0:1:s
1 :: 0:1:s
s :: 0:1:s -> 0:1:s
g :: 0:1:s -> 0:1:s -> 0:1:s
plus :: 0:1:s -> 0:1:s -> 0:1:s

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

(9) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   f_1
   g_2

(c) The following functions are completely defined:

   plus_2

Due to the following rules being added:

   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_plus

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(0) -> 1 [1]
   f(s(x)) -> g(x, s(x)) [1]
   g(0, y) -> y [1]
   g(s(x), y) -> g(x, plus(y, s(x))) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   g(s(x), y) -> g(x, s(plus(y, x))) [1]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
f :: 0:1:s:null_plus -> 0:1:s:null_plus
0 :: 0:1:s:null_plus
1 :: 0:1:s:null_plus
s :: 0:1:s:null_plus -> 0:1:s:null_plus
g :: 0:1:s:null_plus -> 0:1:s:null_plus -> 0:1:s:null_plus
plus :: 0:1:s:null_plus -> 0:1:s:null_plus -> 0:1:s:null_plus
null_plus :: 0:1:s:null_plus

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

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

(12)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(0) -> 1 [1]
   f(s(x)) -> g(x, s(x)) [1]
   g(0, y) -> y [1]
   g(s(x), y) -> g(x, s(plus(y, x))) [2]
   g(s(x), y) -> g(x, null_plus) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   g(s(0), y) -> g(0, s(y)) [2]
   g(s(s(y')), y) -> g(s(y'), s(s(plus(y, y')))) [2]
   g(s(x), y) -> g(x, s(null_plus)) [1]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
f :: 0:1:s:null_plus -> 0:1:s:null_plus
0 :: 0:1:s:null_plus
1 :: 0:1:s:null_plus
s :: 0:1:s:null_plus -> 0:1:s:null_plus
g :: 0:1:s:null_plus -> 0:1:s:null_plus -> 0:1:s:null_plus
plus :: 0:1:s:null_plus -> 0:1:s:null_plus -> 0:1:s:null_plus
null_plus :: 0:1:s:null_plus

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

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

   0 => 0
   1 => 1
   null_plus => 0

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

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

   f(z) -{ 1 }-> g(x, 1 + x) :|: x >= 0, z = 1 + x
   f(z) -{ 1 }-> 1 :|: z = 0
   g(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   g(z, z') -{ 1 }-> g(x, 0) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   g(z, z') -{ 2 }-> g(x, 1 + plus(y, x)) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   g(z, z') -{ 1 }-> g(x, 1 + 0) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   g(z, z') -{ 2 }-> g(0, 1 + y) :|: z = 1 + 0, y >= 0, z' = y
   g(z, z') -{ 2 }-> g(1 + y', 1 + (1 + plus(y, y'))) :|: y >= 0, z = 1 + (1 + y'), y' >= 0, z' = y
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x


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

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

   { plus }
   { g }
   { f }

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

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

Function symbols to be analyzed: {plus}, {g}, {f}

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

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

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

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

Function symbols to be analyzed: {plus}, {g}, {f}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(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:

   f(z) -{ 1 }-> g(z - 1, 1 + (z - 1)) :|: z - 1 >= 0
   f(z) -{ 1 }-> 1 :|: z = 0
   g(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   g(z, z') -{ 2 }-> g(0, 1 + z') :|: z = 1 + 0, z' >= 0
   g(z, z') -{ 1 }-> g(z - 1, 0) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 2 + z }-> g(z - 1, 1 + s) :|: s >= 0, s <= z' + (z - 1), z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> g(z - 1, 1 + 0) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 + z }-> g(1 + (z - 2), 1 + (1 + s'')) :|: s'' >= 0, s'' <= z' + (z - 2), z' >= 0, z - 2 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 1 }-> g(z - 1, 1 + (z - 1)) :|: z - 1 >= 0
   f(z) -{ 1 }-> 1 :|: z = 0
   g(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   g(z, z') -{ 2 }-> g(0, 1 + z') :|: z = 1 + 0, z' >= 0
   g(z, z') -{ 1 }-> g(z - 1, 0) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 2 + z }-> g(z - 1, 1 + s) :|: s >= 0, s <= z' + (z - 1), z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> g(z - 1, 1 + 0) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 + z }-> g(1 + (z - 2), 1 + (1 + s'')) :|: s'' >= 0, s'' <= z' + (z - 2), z' >= 0, z - 2 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {g}, {f}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  g: runtime: ?, size: O(n^2) [z^2 + z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 5 + 3*z + z^2

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

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

   f(z) -{ 1 }-> g(z - 1, 1 + (z - 1)) :|: z - 1 >= 0
   f(z) -{ 1 }-> 1 :|: z = 0
   g(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   g(z, z') -{ 2 }-> g(0, 1 + z') :|: z = 1 + 0, z' >= 0
   g(z, z') -{ 1 }-> g(z - 1, 0) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 2 + z }-> g(z - 1, 1 + s) :|: s >= 0, s <= z' + (z - 1), z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> g(z - 1, 1 + 0) :|: z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 + z }-> g(1 + (z - 2), 1 + (1 + s'')) :|: s'' >= 0, s'' <= z' + (z - 2), z' >= 0, z - 2 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  g: runtime: O(n^2) [5 + 3*z + z^2], size: O(n^2) [z^2 + 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:

   f(z) -{ 4 + z + z^2 }-> s1 :|: s1 >= 0, s1 <= (z - 1) * (z - 1) + (1 + (z - 1)), z - 1 >= 0
   f(z) -{ 1 }-> 1 :|: z = 0
   g(z, z') -{ 5 + 2*z + z^2 }-> s2 :|: s2 >= 0, s2 <= (z - 1) * (z - 1) + (1 + s), s >= 0, s <= z' + (z - 1), z - 1 >= 0, z' >= 0
   g(z, z') -{ 4 + z + z^2 }-> s3 :|: s3 >= 0, s3 <= (z - 1) * (z - 1) + 0, z - 1 >= 0, z' >= 0
   g(z, z') -{ 7 }-> s4 :|: s4 >= 0, s4 <= 0 * 0 + (1 + z'), z = 1 + 0, z' >= 0
   g(z, z') -{ 4 + 2*z + z^2 }-> s5 :|: s5 >= 0, s5 <= (1 + (z - 2)) * (1 + (z - 2)) + (1 + (1 + s'')), s'' >= 0, s'' <= z' + (z - 2), z' >= 0, z - 2 >= 0
   g(z, z') -{ 4 + z + z^2 }-> s6 :|: s6 >= 0, s6 <= (z - 1) * (z - 1) + (1 + 0), z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  g: runtime: O(n^2) [5 + 3*z + z^2], size: O(n^2) [z^2 + z']

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(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 4 + z + z^2 }-> s1 :|: s1 >= 0, s1 <= (z - 1) * (z - 1) + (1 + (z - 1)), z - 1 >= 0
   f(z) -{ 1 }-> 1 :|: z = 0
   g(z, z') -{ 5 + 2*z + z^2 }-> s2 :|: s2 >= 0, s2 <= (z - 1) * (z - 1) + (1 + s), s >= 0, s <= z' + (z - 1), z - 1 >= 0, z' >= 0
   g(z, z') -{ 4 + z + z^2 }-> s3 :|: s3 >= 0, s3 <= (z - 1) * (z - 1) + 0, z - 1 >= 0, z' >= 0
   g(z, z') -{ 7 }-> s4 :|: s4 >= 0, s4 <= 0 * 0 + (1 + z'), z = 1 + 0, z' >= 0
   g(z, z') -{ 4 + 2*z + z^2 }-> s5 :|: s5 >= 0, s5 <= (1 + (z - 2)) * (1 + (z - 2)) + (1 + (1 + s'')), s'' >= 0, s'' <= z' + (z - 2), z' >= 0, z - 2 >= 0
   g(z, z') -{ 4 + z + z^2 }-> s6 :|: s6 >= 0, s6 <= (z - 1) * (z - 1) + (1 + 0), z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  g: runtime: O(n^2) [5 + 3*z + z^2], size: O(n^2) [z^2 + z']
  f: runtime: ?, size: O(n^2) [2 + z + z^2]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 5 + z + z^2

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

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

   f(z) -{ 4 + z + z^2 }-> s1 :|: s1 >= 0, s1 <= (z - 1) * (z - 1) + (1 + (z - 1)), z - 1 >= 0
   f(z) -{ 1 }-> 1 :|: z = 0
   g(z, z') -{ 5 + 2*z + z^2 }-> s2 :|: s2 >= 0, s2 <= (z - 1) * (z - 1) + (1 + s), s >= 0, s <= z' + (z - 1), z - 1 >= 0, z' >= 0
   g(z, z') -{ 4 + z + z^2 }-> s3 :|: s3 >= 0, s3 <= (z - 1) * (z - 1) + 0, z - 1 >= 0, z' >= 0
   g(z, z') -{ 7 }-> s4 :|: s4 >= 0, s4 <= 0 * 0 + (1 + z'), z = 1 + 0, z' >= 0
   g(z, z') -{ 4 + 2*z + z^2 }-> s5 :|: s5 >= 0, s5 <= (1 + (z - 2)) * (1 + (z - 2)) + (1 + (1 + s'')), s'' >= 0, s'' <= z' + (z - 2), z' >= 0, z - 2 >= 0
   g(z, z') -{ 4 + z + z^2 }-> s6 :|: s6 >= 0, s6 <= (z - 1) * (z - 1) + (1 + 0), z - 1 >= 0, z' >= 0
   g(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 + z' }-> 1 + s' :|: s' >= 0, s' <= z + (z' - 1), z >= 0, z' - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  g: runtime: O(n^2) [5 + 3*z + z^2], size: O(n^2) [z^2 + z']
  f: runtime: O(n^2) [5 + z + z^2], size: O(n^2) [2 + z + z^2]

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

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

(38)
BOUNDS(1, n^2)

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(39) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(40)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   f(0') -> 1'
   f(s(x)) -> g(x, s(x))
   g(0', y) -> y
   g(s(x), y) -> g(x, +'(y, s(x)))
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   g(s(x), y) -> g(x, s(+'(y, x)))

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

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

(42)
Obligation:
TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))

Types:
f :: 0':1':s -> 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s -> 0':1':s
g :: 0':1':s -> 0':1':s -> 0':1':s
+' :: 0':1':s -> 0':1':s -> 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat -> 0':1':s

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

(43) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
g, +'

They will be analysed ascendingly in the following order:
+' < g

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

(44)
Obligation:
TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))

Types:
f :: 0':1':s -> 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s -> 0':1':s
g :: 0':1':s -> 0':1':s -> 0':1':s
+' :: 0':1':s -> 0':1':s -> 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat -> 0':1':s


Generator Equations:
gen_0':1':s2_0(0) <=> 0'
gen_0':1':s2_0(+(x, 1)) <=> s(gen_0':1':s2_0(x))


The following defined symbols remain to be analysed:
+', g

They will be analysed ascendingly in the following order:
+' < g

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(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) -> gen_0':1':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)

Induction Base:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(0)) ->_R^Omega(1)
gen_0':1':s2_0(a)

Induction Step:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(+(n4_0, 1))) ->_R^Omega(1)
s(+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0))) ->_IH
s(gen_0':1':s2_0(+(a, c5_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(46)
Complex Obligation (BEST)

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

(47)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))

Types:
f :: 0':1':s -> 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s -> 0':1':s
g :: 0':1':s -> 0':1':s -> 0':1':s
+' :: 0':1':s -> 0':1':s -> 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat -> 0':1':s


Generator Equations:
gen_0':1':s2_0(0) <=> 0'
gen_0':1':s2_0(+(x, 1)) <=> s(gen_0':1':s2_0(x))


The following defined symbols remain to be analysed:
+', g

They will be analysed ascendingly in the following order:
+' < g

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(48) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(49)
BOUNDS(n^1, INF)

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

(50)
Obligation:
TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))

Types:
f :: 0':1':s -> 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s -> 0':1':s
g :: 0':1':s -> 0':1':s -> 0':1':s
+' :: 0':1':s -> 0':1':s -> 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat -> 0':1':s


Lemmas:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) -> gen_0':1':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':1':s2_0(0) <=> 0'
gen_0':1':s2_0(+(x, 1)) <=> s(gen_0':1':s2_0(x))


The following defined symbols remain to be analysed:
g
----------------------------------------

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':1':s2_0(n457_0), gen_0':1':s2_0(b)) -> *3_0, rt in Omega(n457_0 + n457_0^2)

Induction Base:
g(gen_0':1':s2_0(0), gen_0':1':s2_0(b))

Induction Step:
g(gen_0':1':s2_0(+(n457_0, 1)), gen_0':1':s2_0(b)) ->_R^Omega(1)
g(gen_0':1':s2_0(n457_0), +'(gen_0':1':s2_0(b), s(gen_0':1':s2_0(n457_0)))) ->_L^Omega(2 + n457_0)
g(gen_0':1':s2_0(n457_0), gen_0':1':s2_0(+(+(n457_0, 1), b))) ->_IH
*3_0

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(52)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
f(0') -> 1'
f(s(x)) -> g(x, s(x))
g(0', y) -> y
g(s(x), y) -> g(x, +'(y, s(x)))
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
g(s(x), y) -> g(x, s(+'(y, x)))

Types:
f :: 0':1':s -> 0':1':s
0' :: 0':1':s
1' :: 0':1':s
s :: 0':1':s -> 0':1':s
g :: 0':1':s -> 0':1':s -> 0':1':s
+' :: 0':1':s -> 0':1':s -> 0':1':s
hole_0':1':s1_0 :: 0':1':s
gen_0':1':s2_0 :: Nat -> 0':1':s


Lemmas:
+'(gen_0':1':s2_0(a), gen_0':1':s2_0(n4_0)) -> gen_0':1':s2_0(+(n4_0, a)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':1':s2_0(0) <=> 0'
gen_0':1':s2_0(+(x, 1)) <=> s(gen_0':1':s2_0(x))


The following defined symbols remain to be analysed:
g
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(53) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(54)
BOUNDS(n^2, INF)