/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^3), O(n^3))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^3, n^3).

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 11 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), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 140 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 78 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 301 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 82 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 1235 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 90 ms]
    (36) CpxRNTS
    (37) FinalProof [FINISHED, 0 ms]
    (38) BOUNDS(1, n^3)
(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), 254 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), 1617 ms]
            (52) typed CpxTrs
            (53) RewriteLemmaProof [LOWER BOUND(ID), 73 ms]
            (54) proven lower bound
            (55) LowerBoundPropagationProof [FINISHED, 0 ms]
            (56) BOUNDS(n^3, INF)


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

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


The TRS R consists of the following rules:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   +(p(x), y) -> p(+(x, y))
   minus(0) -> 0
   minus(s(x)) -> p(minus(x))
   minus(p(x)) -> s(minus(x))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   *(p(x), y) -> +(*(x, y), minus(y))

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

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

The duplicating contexts are:
*(s(x), [])
*(p(x), [])


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


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^3).


The TRS R consists of the following rules:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   +(p(x), y) -> p(+(x, y))
   minus(0) -> 0
   minus(s(x)) -> p(minus(x))
   minus(p(x)) -> s(minus(x))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   *(p(x), y) -> +(*(x, y), minus(y))

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^3).


The TRS R consists of the following rules:

   +(0, y) -> y [1]
   +(s(x), y) -> s(+(x, y)) [1]
   +(p(x), y) -> p(+(x, y)) [1]
   minus(0) -> 0 [1]
   minus(s(x)) -> p(minus(x)) [1]
   minus(p(x)) -> s(minus(x)) [1]
   *(0, y) -> 0 [1]
   *(s(x), y) -> +(*(x, y), y) [1]
   *(p(x), y) -> +(*(x, y), minus(y)) [1]

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

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

+ => plus
* => times

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

(6)
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:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(p(x), y) -> p(plus(x, y)) [1]
   minus(0) -> 0 [1]
   minus(s(x)) -> p(minus(x)) [1]
   minus(p(x)) -> s(minus(x)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(times(x, y), y) [1]
   times(p(x), y) -> plus(times(x, y), minus(y)) [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(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(p(x), y) -> p(plus(x, y)) [1]
   minus(0) -> 0 [1]
   minus(s(x)) -> p(minus(x)) [1]
   minus(p(x)) -> s(minus(x)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(times(x, y), y) [1]
   times(p(x), y) -> plus(times(x, y), minus(y)) [1]

The TRS has the following type information:
plus :: 0:s:p -> 0:s:p -> 0:s:p
0 :: 0:s:p
s :: 0:s:p -> 0:s:p
p :: 0:s:p -> 0:s:p
minus :: 0:s:p -> 0:s:p
times :: 0:s:p -> 0:s:p -> 0:s:p

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:
none

(c) The following functions are completely defined:

   times_2
   minus_1
   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(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(p(x), y) -> p(plus(x, y)) [1]
   minus(0) -> 0 [1]
   minus(s(x)) -> p(minus(x)) [1]
   minus(p(x)) -> s(minus(x)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(times(x, y), y) [1]
   times(p(x), y) -> plus(times(x, y), minus(y)) [1]

The TRS has the following type information:
plus :: 0:s:p -> 0:s:p -> 0:s:p
0 :: 0:s:p
s :: 0:s:p -> 0:s:p
p :: 0:s:p -> 0:s:p
minus :: 0:s:p -> 0:s:p
times :: 0:s:p -> 0:s:p -> 0:s:p

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(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(p(x), y) -> p(plus(x, y)) [1]
   minus(0) -> 0 [1]
   minus(s(x)) -> p(minus(x)) [1]
   minus(p(x)) -> s(minus(x)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> plus(0, y) [2]
   times(s(s(x')), y) -> plus(plus(times(x', y), y), y) [2]
   times(s(p(x'')), y) -> plus(plus(times(x'', y), minus(y)), y) [2]
   times(p(0), 0) -> plus(0, 0) [3]
   times(p(0), s(x3)) -> plus(0, p(minus(x3))) [3]
   times(p(0), p(x4)) -> plus(0, s(minus(x4))) [3]
   times(p(s(x1)), 0) -> plus(plus(times(x1, 0), 0), 0) [3]
   times(p(s(x1)), s(x5)) -> plus(plus(times(x1, s(x5)), s(x5)), p(minus(x5))) [3]
   times(p(s(x1)), p(x6)) -> plus(plus(times(x1, p(x6)), p(x6)), s(minus(x6))) [3]
   times(p(p(x2)), 0) -> plus(plus(times(x2, 0), minus(0)), 0) [3]
   times(p(p(x2)), s(x7)) -> plus(plus(times(x2, s(x7)), minus(s(x7))), p(minus(x7))) [3]
   times(p(p(x2)), p(x8)) -> plus(plus(times(x2, p(x8)), minus(p(x8))), s(minus(x8))) [3]

The TRS has the following type information:
plus :: 0:s:p -> 0:s:p -> 0:s:p
0 :: 0:s:p
s :: 0:s:p -> 0:s:p
p :: 0:s:p -> 0:s:p
minus :: 0:s:p -> 0:s:p
times :: 0:s:p -> 0:s:p -> 0:s:p

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:

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(x) :|: x >= 0, z = 1 + x
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   times(z, z') -{ 2 }-> plus(plus(times(x', y), y), y) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y
   times(z, z') -{ 2 }-> plus(plus(times(x'', y), minus(y)), y) :|: y >= 0, x'' >= 0, z = 1 + (1 + x''), z' = y
   times(z, z') -{ 3 }-> plus(plus(times(x1, 0), 0), 0) :|: z = 1 + (1 + x1), x1 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(x1, 1 + x5), 1 + x5), 1 + minus(x5)) :|: z = 1 + (1 + x1), x1 >= 0, x5 >= 0, z' = 1 + x5
   times(z, z') -{ 3 }-> plus(plus(times(x1, 1 + x6), 1 + x6), 1 + minus(x6)) :|: z' = 1 + x6, z = 1 + (1 + x1), x1 >= 0, x6 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(x2, 0), minus(0)), 0) :|: z = 1 + (1 + x2), x2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(x2, 1 + x7), minus(1 + x7)), 1 + minus(x7)) :|: z' = 1 + x7, z = 1 + (1 + x2), x7 >= 0, x2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(x2, 1 + x8), minus(1 + x8)), 1 + minus(x8)) :|: z = 1 + (1 + x2), z' = 1 + x8, x8 >= 0, x2 >= 0
   times(z, z') -{ 2 }-> plus(0, y) :|: z = 1 + 0, y >= 0, z' = y
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(x3)) :|: z' = 1 + x3, z = 1 + 0, x3 >= 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(x4)) :|: x4 >= 0, z = 1 + 0, z' = 1 + x4
   times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y


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

(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:

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 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:

   { minus }
   { plus }
   { times }

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

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus}, {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:

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {minus}, {plus}, {times}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: minus
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:

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 + z' }-> plus(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 + z' }-> plus(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 + z' }-> plus(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + z', z = 1 + 0, z' >= 0
   times(z, z') -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z = 1 + 0, z' = 0
   times(z, z') -{ 4 + z' }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + s''), s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + z', z = 1 + 0, z' >= 0
   times(z, z') -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z = 1 + 0, z' = 0
   times(z, z') -{ 4 + z' }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + s''), s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  times: runtime: ?, size: O(n^2) [2*z*z' + 3*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^3) with polynomial bound: 12 + 31*z + z*z' + 16*z^2*z' + z'

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

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + z', z = 1 + 0, z' >= 0
   times(z, z') -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z = 1 + 0, z' = 0
   times(z, z') -{ 4 + z' }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + s''), s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  times: runtime: O(n^3) [12 + 31*z + z*z' + 16*z^2*z' + z'], size: O(n^2) [2*z*z' + 3*z']

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

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

(38)
BOUNDS(1, n^3)

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

(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^3, INF).


The TRS R consists of the following rules:

   +'(0', y) -> y
   +'(s(x), y) -> s(+'(x, y))
   +'(p(x), y) -> p(+'(x, y))
   minus(0') -> 0'
   minus(s(x)) -> p(minus(x))
   minus(p(x)) -> s(minus(x))
   *'(0', y) -> 0'
   *'(s(x), y) -> +'(*'(x, y), y)
   *'(p(x), y) -> +'(*'(x, y), minus(y))

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

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

(42)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p

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

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

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

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

(44)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p


Generator Equations:
gen_0':s:p2_0(0) <=> 0'
gen_0':s:p2_0(+(x, 1)) <=> s(gen_0':s:p2_0(x))


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

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

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

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) -> gen_0':s:p2_0(+(n4_0, b)), rt in Omega(1 + n4_0)

Induction Base:
+'(gen_0':s:p2_0(0), gen_0':s:p2_0(b)) ->_R^Omega(1)
gen_0':s:p2_0(b)

Induction Step:
+'(gen_0':s:p2_0(+(n4_0, 1)), gen_0':s:p2_0(b)) ->_R^Omega(1)
s(+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b))) ->_IH
s(gen_0':s:p2_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:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p


Generator Equations:
gen_0':s:p2_0(0) <=> 0'
gen_0':s:p2_0(+(x, 1)) <=> s(gen_0':s:p2_0(x))


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

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

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

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

(49)
BOUNDS(n^1, INF)

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

(50)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p


Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) -> gen_0':s:p2_0(+(n4_0, b)), rt in Omega(1 + n4_0)


Generator Equations:
gen_0':s:p2_0(0) <=> 0'
gen_0':s:p2_0(+(x, 1)) <=> s(gen_0':s:p2_0(x))


The following defined symbols remain to be analysed:
minus, *'

They will be analysed ascendingly in the following order:
minus < *'

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s:p2_0(+(1, n595_0))) -> *3_0, rt in Omega(n595_0)

Induction Base:
minus(gen_0':s:p2_0(+(1, 0)))

Induction Step:
minus(gen_0':s:p2_0(+(1, +(n595_0, 1)))) ->_R^Omega(1)
p(minus(gen_0':s:p2_0(+(1, n595_0)))) ->_IH
p(*3_0)

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

(52)
Obligation:
TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p


Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) -> gen_0':s:p2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
minus(gen_0':s:p2_0(+(1, n595_0))) -> *3_0, rt in Omega(n595_0)


Generator Equations:
gen_0':s:p2_0(0) <=> 0'
gen_0':s:p2_0(+(x, 1)) <=> s(gen_0':s:p2_0(x))


The following defined symbols remain to be analysed:
*'
----------------------------------------

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_0':s:p2_0(n1725_0), gen_0':s:p2_0(b)) -> gen_0':s:p2_0(*(n1725_0, b)), rt in Omega(1 + b*n1725_0^2 + n1725_0)

Induction Base:
*'(gen_0':s:p2_0(0), gen_0':s:p2_0(b)) ->_R^Omega(1)
0'

Induction Step:
*'(gen_0':s:p2_0(+(n1725_0, 1)), gen_0':s:p2_0(b)) ->_R^Omega(1)
+'(*'(gen_0':s:p2_0(n1725_0), gen_0':s:p2_0(b)), gen_0':s:p2_0(b)) ->_IH
+'(gen_0':s:p2_0(*(c1726_0, b)), gen_0':s:p2_0(b)) ->_L^Omega(1 + b*n1725_0)
gen_0':s:p2_0(+(*(n1725_0, b), b))

We have rt in Omega(n^3) and sz in O(n). Thus, we have irc_R in Omega(n^3).
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(54)
Obligation:
Proved the lower bound n^3 for the following obligation:

TRS:
Rules:
+'(0', y) -> y
+'(s(x), y) -> s(+'(x, y))
+'(p(x), y) -> p(+'(x, y))
minus(0') -> 0'
minus(s(x)) -> p(minus(x))
minus(p(x)) -> s(minus(x))
*'(0', y) -> 0'
*'(s(x), y) -> +'(*'(x, y), y)
*'(p(x), y) -> +'(*'(x, y), minus(y))

Types:
+' :: 0':s:p -> 0':s:p -> 0':s:p
0' :: 0':s:p
s :: 0':s:p -> 0':s:p
p :: 0':s:p -> 0':s:p
minus :: 0':s:p -> 0':s:p
*' :: 0':s:p -> 0':s:p -> 0':s:p
hole_0':s:p1_0 :: 0':s:p
gen_0':s:p2_0 :: Nat -> 0':s:p


Lemmas:
+'(gen_0':s:p2_0(n4_0), gen_0':s:p2_0(b)) -> gen_0':s:p2_0(+(n4_0, b)), rt in Omega(1 + n4_0)
minus(gen_0':s:p2_0(+(1, n595_0))) -> *3_0, rt in Omega(n595_0)


Generator Equations:
gen_0':s:p2_0(0) <=> 0'
gen_0':s:p2_0(+(x, 1)) <=> s(gen_0':s:p2_0(x))


The following defined symbols remain to be analysed:
*'
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(55) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(56)
BOUNDS(n^3, INF)