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


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WORST_CASE(Omega(n^2), O(n^2))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [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), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 935 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 383 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 419 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 104 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 636 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 206 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), 262 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), 52 ms]
            (52) BEST
                (53) proven lower bound
                    (54) LowerBoundPropagationProof [FINISHED, 0 ms]
                    (55) BOUNDS(n^2, INF)
                (56) typed CpxTrs
                    (57) RewriteLemmaProof [LOWER BOUND(ID), 10 ms]
                    (58) typed CpxTrs


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

(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:

   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(y, z))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 1th argument of plus: plus, times

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
div(div(x, y), z) -> div(x, times(y, z))

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

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


The TRS R consists of the following rules:

   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))

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

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

The duplicating contexts are:
times(s(x), [])
div(x, [])


The defined contexts are:
plus(x0, [])


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

(4)
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:

   plus(x, 0) -> x
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(0), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0, y) -> 0
   div(x, y) -> quot(x, y, y)
   quot(0, s(y), z) -> 0
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0, s(z)) -> s(div(x, s(z)))

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

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

(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:

   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [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:

   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s -> 0: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:

   div_2
   quot_3

(c) The following functions are completely defined:

   times_2
   plus_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(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:

   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s -> 0:s

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:

   plus(x, 0) -> x [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> y [1]
   times(s(0), y) -> plus(y, 0) [2]
   times(s(s(0)), y) -> plus(y, y) [2]
   times(s(s(x')), y) -> plus(y, plus(y, times(x', y))) [2]
   div(0, y) -> 0 [1]
   div(x, y) -> quot(x, y, y) [1]
   quot(0, s(y), z) -> 0 [1]
   quot(s(x), s(y), z) -> quot(x, y, z) [1]
   quot(x, 0, s(z)) -> s(div(x, s(z))) [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s -> 0:s

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

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

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

   div(z', z'') -{ 1 }-> quot(x, y, y) :|: z' = x, z'' = y, x >= 0, y >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   quot(z', z'', z1) -{ 1 }-> quot(x, y, z) :|: z' = 1 + x, z1 = z, z >= 0, x >= 0, y >= 0, z'' = 1 + y
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 = z, z >= 0, y >= 0, z'' = 1 + y, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(x, 1 + z) :|: z'' = 0, z >= 0, z' = x, x >= 0, z1 = 1 + z
   times(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(y, y) :|: z' = 1 + (1 + 0), z'' = y, y >= 0
   times(z', z'') -{ 2 }-> plus(y, plus(y, times(x', y))) :|: z' = 1 + (1 + x'), z'' = y, x' >= 0, y >= 0
   times(z', z'') -{ 2 }-> plus(y, 0) :|: z'' = y, y >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0


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

(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:

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0


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

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

   { div, quot }
   { plus }
   { times }

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {div,quot}, {plus}, {times}

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

(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:

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {div,quot}, {plus}, {times}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed RUNTIME bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 3*z' + z''

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', z'') :|: z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 1 }-> quot(z' - 1, z'' - 1, z1) :|: z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 1 }-> 1 + div(z', 1 + (z1 - 1)) :|: z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

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

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

(27) 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''

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

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

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

(29) 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'

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', z'') :|: z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

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

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

(33) 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: 2*z'*z'' + 4*z''

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: {times}
Previous analysis results are:
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: ?, size: O(n^2) [2*z'*z'' + 4*z'']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 8 + 4*z' + 2*z'*z'' + 2*z''

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

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

   div(z', z'') -{ 6 + 3*z' + z'' }-> s :|: s >= 0, s <= z' + 1, z' >= 0, z'' >= 0
   div(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 2 + 3*z' + z'' }-> s' :|: s' >= 0, s' <= z' - 1 + 1, z1 >= 0, z' - 1 >= 0, z'' - 1 >= 0
   quot(z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z'' - 1 >= 0, z' = 0
   quot(z', z'', z1) -{ 4 + 3*z' }-> 1 + s'' :|: s'' >= 0, s'' <= z', z'' = 0, z1 - 1 >= 0, z' >= 0
   times(z', z'') -{ 3 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 3 + z'' }-> s3 :|: s3 >= 0, s3 <= z'' + z'', z' = 1 + (1 + 0), z'' >= 0
   times(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  div: runtime: O(n^1) [3 + 3*z'], size: O(n^1) [z']
  quot: runtime: O(n^1) [5 + 3*z' + z''], size: O(n^1) [1 + z']
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: O(n^2) [8 + 4*z' + 2*z'*z'' + 2*z''], size: O(n^2) [2*z'*z'' + 4*z'']

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

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

(38)
BOUNDS(1, n^2)

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

(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:

   plus(x, 0') -> x
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0', y) -> 0'
   times(s(0'), y) -> y
   times(s(x), y) -> plus(y, times(x, y))
   div(0', y) -> 0'
   div(x, y) -> quot(x, y, y)
   quot(0', s(y), z) -> 0'
   quot(s(x), s(y), z) -> quot(x, y, z)
   quot(x, 0', s(z)) -> s(div(x, s(z)))
   div(div(x, y), z) -> div(x, times(y, z))

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

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

(42)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s

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

(43) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
plus, times, div, quot

They will be analysed ascendingly in the following order:
plus < times
times < div
div = quot

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

(44)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


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


The following defined symbols remain to be analysed:
plus, times, div, quot

They will be analysed ascendingly in the following order:
plus < times
times < div
div = quot

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

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

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

Induction Step:
plus(gen_0':s2_0(+(n4_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
s(plus(gen_0':s2_0(n4_0), gen_0':s2_0(b))) ->_IH
s(gen_0':s2_0(+(b, 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:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


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


The following defined symbols remain to be analysed:
plus, times, div, quot

They will be analysed ascendingly in the following order:
plus < times
times < div
div = quot

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

(48) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(49)
BOUNDS(n^1, INF)

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

(50)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


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


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


The following defined symbols remain to be analysed:
times, div, quot

They will be analysed ascendingly in the following order:
times < div
div = quot

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n577_0, b)), rt in Omega(1 + b*n577_0 + n577_0)

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

Induction Step:
times(gen_0':s2_0(+(n577_0, 1)), gen_0':s2_0(b)) ->_R^Omega(1)
plus(gen_0':s2_0(b), times(gen_0':s2_0(n577_0), gen_0':s2_0(b))) ->_IH
plus(gen_0':s2_0(b), gen_0':s2_0(*(c578_0, b))) ->_L^Omega(1 + b)
gen_0':s2_0(+(b, *(n577_0, b)))

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

(52)
Complex Obligation (BEST)

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

(53)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


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


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


The following defined symbols remain to be analysed:
times, div, quot

They will be analysed ascendingly in the following order:
times < div
div = quot

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

(54) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(55)
BOUNDS(n^2, INF)

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

(56)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n577_0, b)), rt in Omega(1 + b*n577_0 + n577_0)


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


The following defined symbols remain to be analysed:
quot, div

They will be analysed ascendingly in the following order:
div = quot

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) -> gen_0':s2_0(0), rt in Omega(1 + n1370_0)

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

Induction Step:
quot(gen_0':s2_0(+(n1370_0, 1)), gen_0':s2_0(+(1, +(n1370_0, 1))), gen_0':s2_0(c)) ->_R^Omega(1)
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) ->_IH
gen_0':s2_0(0)

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

(58)
Obligation:
TRS:
Rules:
plus(x, 0') -> x
plus(0', y) -> y
plus(s(x), y) -> s(plus(x, y))
times(0', y) -> 0'
times(s(0'), y) -> y
times(s(x), y) -> plus(y, times(x, y))
div(0', y) -> 0'
div(x, y) -> quot(x, y, y)
quot(0', s(y), z) -> 0'
quot(s(x), s(y), z) -> quot(x, y, z)
quot(x, 0', s(z)) -> s(div(x, s(z)))
div(div(x, y), z) -> div(x, times(y, z))

Types:
plus :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
times :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
gen_0':s2_0 :: Nat -> 0':s


Lemmas:
plus(gen_0':s2_0(n4_0), gen_0':s2_0(b)) -> gen_0':s2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
times(gen_0':s2_0(n577_0), gen_0':s2_0(b)) -> gen_0':s2_0(*(n577_0, b)), rt in Omega(1 + b*n577_0 + n577_0)
quot(gen_0':s2_0(n1370_0), gen_0':s2_0(+(1, n1370_0)), gen_0':s2_0(c)) -> gen_0':s2_0(0), rt in Omega(1 + n1370_0)


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


The following defined symbols remain to be analysed:
div

They will be analysed ascendingly in the following order:
div = quot