WORST_CASE(Omega(n^1), O(n^2))
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2).

(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) InliningProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 246 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 248 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 72 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 137 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 384 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 137 ms]
    (40) CpxRNTS
    (41) FinalProof [FINISHED, 0 ms]
    (42) BOUNDS(1, n^2)
(43) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(44) CpxTRS
    (45) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) typed CpxTrs
    (47) OrderProof [LOWER BOUND(ID), 0 ms]
    (48) typed CpxTrs
    (49) RewriteLemmaProof [LOWER BOUND(ID), 248 ms]
    (50) BEST
        (51) proven lower bound
            (52) LowerBoundPropagationProof [FINISHED, 0 ms]
            (53) BOUNDS(n^1, INF)
        (54) typed CpxTrs


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

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


The TRS R consists of the following rules:

   p(0) -> 0
   p(s(x)) -> x
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0) -> x
   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y)))))
   if(true, x, y) -> x
   if(false, x, y) -> 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^2).


The TRS R consists of the following rules:

   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y))))) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> 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:

   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y))))) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]

The TRS has the following type information:
p :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
le :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
minus :: 0:s -> 0:s -> 0:s
if :: true:false -> 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:
none

(c) The following functions are completely defined:

   le_2
   p_1
   minus_2
   if_3

Due to the following rules being added:
none

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:

   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(x, 0) -> x [1]
   minus(x, s(y)) -> if(le(x, s(y)), 0, p(minus(x, p(s(y))))) [1]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]

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

   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   le(0, y) -> true [1]
   le(s(x), 0) -> false [1]
   le(s(x), s(y)) -> le(x, y) [1]
   minus(x, 0) -> x [1]
   minus(0, s(y)) -> if(true, 0, p(minus(0, y))) [3]
   minus(s(x'), s(y)) -> if(le(x', y), 0, p(minus(s(x'), y))) [3]
   if(true, x, y) -> x [1]
   if(false, x, y) -> y [1]

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

   if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 3 }-> if(le(x', y), 0, p(minus(1 + x', y))) :|: z = 1 + x', z' = 1 + y, x' >= 0, y >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, y))) :|: z' = 1 + y, y >= 0, z = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0


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

(11) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0

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

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

   if(z, z', z'') -{ 1 }-> x :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   le(z, z') -{ 1 }-> le(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   le(z, z') -{ 1 }-> 1 :|: y >= 0, z = 0, z' = y
   le(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 3 }-> if(le(x', y), 0, p(minus(1 + x', y))) :|: z = 1 + x', z' = 1 + y, x' >= 0, y >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, y))) :|: z' = 1 + y, y >= 0, z = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0


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

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 3 }-> if(le(z - 1, z' - 1), 0, p(minus(1 + (z - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0


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

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

   { le }
   { if }
   { p }
   { minus }

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 3 }-> if(le(z - 1, z' - 1), 0, p(minus(1 + (z - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {le}, {if}, {p}, {minus}

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

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 3 }-> if(le(z - 1, z' - 1), 0, p(minus(1 + (z - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {le}, {if}, {p}, {minus}

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

(19) 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

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 3 }-> if(le(z - 1, z' - 1), 0, p(minus(1 + (z - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(21) 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'

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 3 }-> if(le(z - 1, z' - 1), 0, p(minus(1 + (z - 1), z' - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(37) 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

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

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 1 + 6*z' + z'^2

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

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

   if(z, z', z'') -{ 1 }-> z' :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 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') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 4 + z' }-> if(s', 0, p(minus(1 + (z - 1), z' - 1))) :|: s' >= 0, s' <= 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 3 }-> if(1, 0, p(minus(0, z' - 1))) :|: z' - 1 >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z]
  minus: runtime: O(n^2) [1 + 6*z' + z'^2], size: O(n^1) [z]

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

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

(42)
BOUNDS(1, n^2)

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

(44)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   p(0') -> 0'
   p(s(x)) -> x
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   minus(x, 0') -> x
   minus(x, s(y)) -> if(le(x, s(y)), 0', p(minus(x, p(s(y)))))
   if(true, x, y) -> x
   if(false, x, y) -> y

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

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

(46)
Obligation:
Innermost TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(x, s(y)) -> if(le(x, s(y)), 0', p(minus(x, p(s(y)))))
if(true, x, y) -> x
if(false, x, y) -> y

Types:
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

(47) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
le, minus

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

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

(48)
Obligation:
Innermost TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(x, s(y)) -> if(le(x, s(y)), 0', p(minus(x, p(s(y)))))
if(true, x, y) -> x
if(false, x, y) -> y

Types:
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
le, minus

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

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

(49) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)

Induction Base:
le(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
le(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
true

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

(50)
Complex Obligation (BEST)

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

(51)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(x, s(y)) -> if(le(x, s(y)), 0', p(minus(x, p(s(y)))))
if(true, x, y) -> x
if(false, x, y) -> y

Types:
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
le, minus

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

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

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

(53)
BOUNDS(n^1, INF)

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

(54)
Obligation:
Innermost TRS:
Rules:
p(0') -> 0'
p(s(x)) -> x
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
minus(x, 0') -> x
minus(x, s(y)) -> if(le(x, s(y)), 0', p(minus(x, p(s(y)))))
if(true, x, y) -> x
if(false, x, y) -> y

Types:
p :: 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
le :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
minus :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
le(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> true, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
minus