WORST_CASE(Omega(n^1), O(n^3))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 251 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 99 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 724 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 250 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 330 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 14 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^3)
(35) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(36) TRS for Loop Detection
    (37) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (38) BEST
        (39) proven lower bound
            (40) LowerBoundPropagationProof [FINISHED, 0 ms]
            (41) BOUNDS(n^1, INF)
        (42) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   minus(0, Y) -> 0
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
   ifMinus(true, s(X), Y) -> 0
   ifMinus(false, s(X), Y) -> s(minus(X, Y))
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

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

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

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   le(0, Y) -> true [1]
   le(s(X), 0) -> false [1]
   le(s(X), s(Y)) -> le(X, Y) [1]
   minus(0, Y) -> 0 [1]
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y) [1]
   ifMinus(true, s(X), Y) -> 0 [1]
   ifMinus(false, s(X), Y) -> s(minus(X, Y)) [1]
   quot(0, s(Y)) -> 0 [1]
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   le(0, Y) -> true [1]
   le(s(X), 0) -> false [1]
   le(s(X), s(Y)) -> le(X, Y) [1]
   minus(0, Y) -> 0 [1]
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y) [1]
   ifMinus(true, s(X), Y) -> 0 [1]
   ifMinus(false, s(X), Y) -> s(minus(X, Y)) [1]
   quot(0, s(Y)) -> 0 [1]
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) [1]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
ifMinus :: true:false -> 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s

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

(5) 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:

   quot_2

(c) The following functions are completely defined:

   minus_2
   le_2
   ifMinus_3

Due to the following rules being added:

   ifMinus(v0, v1, v2) -> 0 [0]

And the following fresh constants: none

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

(6)
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:

   le(0, Y) -> true [1]
   le(s(X), 0) -> false [1]
   le(s(X), s(Y)) -> le(X, Y) [1]
   minus(0, Y) -> 0 [1]
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y) [1]
   ifMinus(true, s(X), Y) -> 0 [1]
   ifMinus(false, s(X), Y) -> s(minus(X, Y)) [1]
   quot(0, s(Y)) -> 0 [1]
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y))) [1]
   ifMinus(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
ifMinus :: true:false -> 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s

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

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

(8)
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:

   le(0, Y) -> true [1]
   le(s(X), 0) -> false [1]
   le(s(X), s(Y)) -> le(X, Y) [1]
   minus(0, Y) -> 0 [1]
   minus(s(X), 0) -> ifMinus(false, s(X), 0) [2]
   minus(s(X), s(Y')) -> ifMinus(le(X, Y'), s(X), s(Y')) [2]
   ifMinus(true, s(X), Y) -> 0 [1]
   ifMinus(false, s(X), Y) -> s(minus(X, Y)) [1]
   quot(0, s(Y)) -> 0 [1]
   quot(s(0), s(Y)) -> s(quot(0, s(Y))) [2]
   quot(s(s(X')), s(Y)) -> s(quot(ifMinus(le(s(X'), Y), s(X'), Y), s(Y))) [2]
   ifMinus(v0, v1, v2) -> 0 [0]

The TRS has the following type information:
le :: 0:s -> 0:s -> true:false
0 :: 0:s
true :: true:false
s :: 0:s -> 0:s
false :: true:false
minus :: 0:s -> 0:s -> 0:s
ifMinus :: true:false -> 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s

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

(9) 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
   true => 1
   false => 0

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: Y >= 0, z = 1, z'' = Y, z' = 1 + X, X >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   ifMinus(z, z', z'') -{ 1 }-> 1 + minus(X, Y) :|: Y >= 0, z'' = Y, z' = 1 + X, X >= 0, z = 0
   le(z, z') -{ 1 }-> le(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   le(z, z') -{ 1 }-> 1 :|: z' = Y, Y >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z = 1 + X, X >= 0, z' = 0
   minus(z, z') -{ 2 }-> ifMinus(le(X, Y'), 1 + X, 1 + Y') :|: z = 1 + X, Y' >= 0, X >= 0, z' = 1 + Y'
   minus(z, z') -{ 2 }-> ifMinus(0, 1 + X, 0) :|: z = 1 + X, X >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(ifMinus(le(1 + X', Y), 1 + X', Y), 1 + Y) :|: Y >= 0, z' = 1 + Y, X' >= 0, z = 1 + (1 + X')
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + Y) :|: Y >= 0, z = 1 + 0, z' = 1 + Y


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

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> ifMinus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(ifMinus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0


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

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

   { le }
   { ifMinus, minus }
   { quot }

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

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

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> ifMinus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(ifMinus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {le}, {ifMinus,minus}, {quot}

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

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

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

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> ifMinus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(ifMinus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {le}, {ifMinus,minus}, {quot}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> ifMinus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(ifMinus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {le}, {ifMinus,minus}, {quot}
Previous analysis results are:
  le: runtime: ?, size: O(1) [1]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 1 }-> le(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 2 }-> ifMinus(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 2 }-> 1 + quot(ifMinus(le(1 + (z - 2), z' - 1), 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {ifMinus,minus}, {quot}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

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

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

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 3 + z' }-> ifMinus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(ifMinus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {ifMinus,minus}, {quot}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 3 + z' }-> ifMinus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(ifMinus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: ifMinus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 4*z' + z'*z'' + z''

Computed RUNTIME bound using KoAT for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 18 + 8*z + z*z' + 2*z'

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

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

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 1 }-> 1 + minus(z' - 1, z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 3 + z' }-> ifMinus(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> ifMinus(0, 1 + (z - 1), 0) :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 3 + z' }-> 1 + quot(ifMinus(s'', 1 + (z - 2), z' - 1), 1 + (z' - 1)) :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {quot}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  ifMinus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  minus: runtime: O(n^2) [18 + 8*z + z*z' + 2*z'], size: O(n^1) [z]

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(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(28)
Obligation:
Complexity RNTS consisting of the following rules:

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 11 + 8*z' + z'*z'' + z'' }-> 1 + s3 :|: s3 >= 0, s3 <= z' - 1, z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 5 + 3*z + z*z' + z' }-> 1 + quot(s4, 1 + (z' - 1)) :|: s4 >= 0, s4 <= 1 + (z - 2), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {quot}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  ifMinus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  minus: runtime: O(n^2) [18 + 8*z + z*z' + 2*z'], size: O(n^1) [z]

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(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 11 + 8*z' + z'*z'' + z'' }-> 1 + s3 :|: s3 >= 0, s3 <= z' - 1, z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 5 + 3*z + z*z' + z' }-> 1 + quot(s4, 1 + (z' - 1)) :|: s4 >= 0, s4 <= 1 + (z - 2), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: {quot}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  ifMinus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  minus: runtime: O(n^2) [18 + 8*z + z*z' + 2*z'], size: O(n^1) [z]
  quot: runtime: ?, size: O(n^1) [z]

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(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 1 + 7*z + z*z' + 3*z^2 + z^2*z'

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

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

   ifMinus(z, z', z'') -{ 1 }-> 0 :|: z'' >= 0, z = 1, z' - 1 >= 0
   ifMinus(z, z', z'') -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0
   ifMinus(z, z', z'') -{ 11 + 8*z' + z'*z'' + z'' }-> 1 + s3 :|: s3 >= 0, s3 <= z' - 1, z'' >= 0, z' - 1 >= 0, z = 0
   le(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z' - 1 >= 0, z - 1 >= 0
   le(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   le(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   minus(z, z') -{ 8 + 4*z }-> s1 :|: s1 >= 0, s1 <= 1 + (z - 1), z - 1 >= 0, z' = 0
   minus(z, z') -{ 9 + 4*z + z*z' + 2*z' }-> s2 :|: s2 >= 0, s2 <= 1 + (z - 1), s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   quot(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   quot(z, z') -{ 5 + 3*z + z*z' + z' }-> 1 + quot(s4, 1 + (z' - 1)) :|: s4 >= 0, s4 <= 1 + (z - 2), s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 2 >= 0
   quot(z, z') -{ 2 }-> 1 + quot(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0

Function symbols to be analyzed: 
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  ifMinus: runtime: O(n^2) [6 + 4*z' + z'*z'' + z''], size: O(n^1) [z']
  minus: runtime: O(n^2) [18 + 8*z + z*z' + 2*z'], size: O(n^1) [z]
  quot: runtime: O(n^3) [1 + 7*z + z*z' + 3*z^2 + z^2*z'], size: O(n^1) [z]

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(33) FinalProof (FINISHED)
Computed overall runtime complexity
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(34)
BOUNDS(1, n^3)

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(35) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(36)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   minus(0, Y) -> 0
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
   ifMinus(true, s(X), Y) -> 0
   ifMinus(false, s(X), Y) -> s(minus(X, Y))
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

S is empty.
Rewrite Strategy: INNERMOST
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(37) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

le(s(X), s(Y)) ->^+ le(X, Y)

gives rise to a decreasing loop by considering the right hand sides subterm at position [].

The pumping substitution is [X / s(X), Y / s(Y)].

The result substitution is [ ].




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(38)
Complex Obligation (BEST)

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(39)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   minus(0, Y) -> 0
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
   ifMinus(true, s(X), Y) -> 0
   ifMinus(false, s(X), Y) -> s(minus(X, Y))
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

S is empty.
Rewrite Strategy: INNERMOST
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(40) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(41)
BOUNDS(n^1, INF)

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(42)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^3).


The TRS R consists of the following rules:

   le(0, Y) -> true
   le(s(X), 0) -> false
   le(s(X), s(Y)) -> le(X, Y)
   minus(0, Y) -> 0
   minus(s(X), Y) -> ifMinus(le(s(X), Y), s(X), Y)
   ifMinus(true, s(X), Y) -> 0
   ifMinus(false, s(X), Y) -> s(minus(X, Y))
   quot(0, s(Y)) -> 0
   quot(s(X), s(Y)) -> s(quot(minus(X, Y), s(Y)))

S is empty.
Rewrite Strategy: INNERMOST