WORST_CASE(Omega(n^1),O(n^2))
proof of input_20w4EZG02g.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 202 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 101 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 98 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 888 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 429 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^2)
(35) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CdtProblem
    (37) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CpxRelTRS
    (39) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CpxRelTRS
    (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) typed CpxTrs
    (43) OrderProof [LOWER BOUND(ID), 9 ms]
    (44) typed CpxTrs
    (45) RewriteLemmaProof [LOWER BOUND(ID), 366 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), 37 ms]
            (52) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-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, y) -> if(le(x, y), x, y)
   if(true, x, y) -> 0
   if(false, x, y) -> s(minus(p(x), y))

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

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

   minus_2
   if_3

(c) The following functions are completely defined:

   p_1
   le_2

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, y) -> if(le(x, y), x, y) [1]
   if(true, x, y) -> 0 [1]
   if(false, x, y) -> s(minus(p(x), 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(0, y) -> if(true, 0, y) [2]
   minus(s(x'), 0) -> if(false, s(x'), 0) [2]
   minus(s(x''), s(y')) -> if(le(x'', y'), s(x''), s(y')) [2]
   if(true, x, y) -> 0 [1]
   if(false, 0, y) -> s(minus(0, y)) [2]
   if(false, s(x1), y) -> s(minus(x1, y)) [2]

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 }-> 0 :|: z' = x, z'' = y, z = 1, x >= 0, y >= 0
   if(z, z', z'') -{ 2 }-> 1 + minus(x1, y) :|: x1 >= 0, z'' = y, y >= 0, z' = 1 + x1, z = 0
   if(z, z', z'') -{ 2 }-> 1 + minus(0, y) :|: z'' = y, y >= 0, z = 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') -{ 2 }-> if(le(x'', y'), 1 + x'', 1 + y') :|: z = 1 + x'', y' >= 0, z' = 1 + y', x'' >= 0
   minus(z, z') -{ 2 }-> if(1, 0, y) :|: y >= 0, z = 0, z' = y
   minus(z, z') -{ 2 }-> if(0, 1 + x', 0) :|: z = 1 + x', x' >= 0, z' = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0


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

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

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

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


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

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

   { le }
   { p }
   { if, minus }

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

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

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

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

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

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

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

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 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') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: 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}, {p}, {if,minus}
Previous analysis results are:
  le: runtime: ?, size: O(1) [1]

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 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') -{ 2 }-> if(le(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: 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}, {if,minus}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

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

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

   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 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') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: 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}, {if,minus}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 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') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: 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}, {if,minus}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  p: runtime: ?, size: O(n^1) [z]

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

(25) 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

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

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

   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 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') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: 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,minus}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 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') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: 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,minus}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(29) 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: 1 + z'

Computed SIZE bound using CoFloCo for: minus
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:

   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 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') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: 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,minus}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z'], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]
  if: runtime: ?, size: O(n^1) [1 + z']
  minus: runtime: ?, size: O(n^1) [1 + z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 10 + 5*z' + z'*z'' + z''

Computed RUNTIME bound using KoAT for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 37 + 10*z + z*z' + 3*z'

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

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

   if(z, z', z'') -{ 1 }-> 0 :|: z = 1, z' >= 0, z'' >= 0
   if(z, z', z'') -{ 2 }-> 1 + minus(0, z'') :|: z'' >= 0, z = 0, z' = 0
   if(z, z', z'') -{ 2 }-> 1 + minus(z' - 1, z'') :|: z' - 1 >= 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') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 2 }-> if(1, 0, z') :|: z' >= 0, z = 0
   minus(z, z') -{ 2 }-> if(0, 1 + (z - 1), 0) :|: 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]
  p: runtime: O(1) [1], size: O(n^1) [z]
  if: runtime: O(n^2) [10 + 5*z' + z'*z'' + z''], size: O(n^1) [1 + z']
  minus: runtime: O(n^2) [37 + 10*z + z*z' + 3*z'], size: O(n^1) [1 + z]

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

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

(34)
BOUNDS(1, n^2)

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

(35) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> 0
   p(s(z0)) -> z0
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, z1) -> if(le(z0, z1), z0, z1)
   if(true, z0, z1) -> 0
   if(false, z0, z1) -> s(minus(p(z0), z1))
Tuples:
   P(0) -> c
   P(s(z0)) -> c1
   LE(0, z0) -> c2
   LE(s(z0), 0) -> c3
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, z1) -> c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7(MINUS(p(z0), z1), P(z0))
S tuples:
   P(0) -> c
   P(s(z0)) -> c1
   LE(0, z0) -> c2
   LE(s(z0), 0) -> c3
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, z1) -> c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7(MINUS(p(z0), z1), P(z0))
K tuples:none
Defined Rule Symbols:   p_1, le_2, minus_2, if_3

Defined Pair Symbols:   P_1, LE_2, MINUS_2, IF_3

Compound Symbols:   c, c1, c2, c3, c4_1, c5_2, c6, c7_2


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

(37) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(38)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   P(0) -> c
   P(s(z0)) -> c1
   LE(0, z0) -> c2
   LE(s(z0), 0) -> c3
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, z1) -> c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7(MINUS(p(z0), z1), P(z0))

The (relative) TRS S consists of the following rules:

   p(0) -> 0
   p(s(z0)) -> z0
   le(0, z0) -> true
   le(s(z0), 0) -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, z1) -> if(le(z0, z1), z0, z1)
   if(true, z0, z1) -> 0
   if(false, z0, z1) -> s(minus(p(z0), z1))

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

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

(40)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   P(0') -> c
   P(s(z0)) -> c1
   LE(0', z0) -> c2
   LE(s(z0), 0') -> c3
   LE(s(z0), s(z1)) -> c4(LE(z0, z1))
   MINUS(z0, z1) -> c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
   IF(true, z0, z1) -> c6
   IF(false, z0, z1) -> c7(MINUS(p(z0), z1), P(z0))

The (relative) TRS S consists of the following rules:

   p(0') -> 0'
   p(s(z0)) -> z0
   le(0', z0) -> true
   le(s(z0), 0') -> false
   le(s(z0), s(z1)) -> le(z0, z1)
   minus(z0, z1) -> if(le(z0, z1), z0, z1)
   if(true, z0, z1) -> 0'
   if(false, z0, z1) -> s(minus(p(z0), z1))

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

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

(42)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LE(0', z0) -> c2
LE(s(z0), 0') -> c3
LE(s(z0), s(z1)) -> c4(LE(z0, z1))
MINUS(z0, z1) -> c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7(MINUS(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, z1) -> if(le(z0, z1), z0, z1)
if(true, z0, z1) -> 0'
if(false, z0, z1) -> s(minus(p(z0), z1))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LE :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
MINUS :: 0':s -> 0':s -> c5
c5 :: c6:c7 -> c2:c3:c4 -> c5
IF :: true:false -> 0':s -> 0':s -> c6:c7
le :: 0':s -> 0':s -> true:false
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c5 -> c:c1 -> c6:c7
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_8 :: c:c1
hole_0':s2_8 :: 0':s
hole_c2:c3:c43_8 :: c2:c3:c4
hole_c54_8 :: c5
hole_c6:c75_8 :: c6:c7
hole_true:false6_8 :: true:false
gen_0':s7_8 :: Nat -> 0':s
gen_c2:c3:c48_8 :: Nat -> c2:c3:c4

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

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

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

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

(44)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LE(0', z0) -> c2
LE(s(z0), 0') -> c3
LE(s(z0), s(z1)) -> c4(LE(z0, z1))
MINUS(z0, z1) -> c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7(MINUS(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, z1) -> if(le(z0, z1), z0, z1)
if(true, z0, z1) -> 0'
if(false, z0, z1) -> s(minus(p(z0), z1))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LE :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
MINUS :: 0':s -> 0':s -> c5
c5 :: c6:c7 -> c2:c3:c4 -> c5
IF :: true:false -> 0':s -> 0':s -> c6:c7
le :: 0':s -> 0':s -> true:false
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c5 -> c:c1 -> c6:c7
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_8 :: c:c1
hole_0':s2_8 :: 0':s
hole_c2:c3:c43_8 :: c2:c3:c4
hole_c54_8 :: c5
hole_c6:c75_8 :: c6:c7
hole_true:false6_8 :: true:false
gen_0':s7_8 :: Nat -> 0':s
gen_c2:c3:c48_8 :: Nat -> c2:c3:c4


Generator Equations:
gen_0':s7_8(0) <=> 0'
gen_0':s7_8(+(x, 1)) <=> s(gen_0':s7_8(x))
gen_c2:c3:c48_8(0) <=> c2
gen_c2:c3:c48_8(+(x, 1)) <=> c4(gen_c2:c3:c48_8(x))


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

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

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

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LE(gen_0':s7_8(n10_8), gen_0':s7_8(n10_8)) -> gen_c2:c3:c48_8(n10_8), rt in Omega(1 + n10_8)

Induction Base:
LE(gen_0':s7_8(0), gen_0':s7_8(0)) ->_R^Omega(1)
c2

Induction Step:
LE(gen_0':s7_8(+(n10_8, 1)), gen_0':s7_8(+(n10_8, 1))) ->_R^Omega(1)
c4(LE(gen_0':s7_8(n10_8), gen_0':s7_8(n10_8))) ->_IH
c4(gen_c2:c3:c48_8(c11_8))

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:

Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LE(0', z0) -> c2
LE(s(z0), 0') -> c3
LE(s(z0), s(z1)) -> c4(LE(z0, z1))
MINUS(z0, z1) -> c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7(MINUS(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, z1) -> if(le(z0, z1), z0, z1)
if(true, z0, z1) -> 0'
if(false, z0, z1) -> s(minus(p(z0), z1))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LE :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
MINUS :: 0':s -> 0':s -> c5
c5 :: c6:c7 -> c2:c3:c4 -> c5
IF :: true:false -> 0':s -> 0':s -> c6:c7
le :: 0':s -> 0':s -> true:false
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c5 -> c:c1 -> c6:c7
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_8 :: c:c1
hole_0':s2_8 :: 0':s
hole_c2:c3:c43_8 :: c2:c3:c4
hole_c54_8 :: c5
hole_c6:c75_8 :: c6:c7
hole_true:false6_8 :: true:false
gen_0':s7_8 :: Nat -> 0':s
gen_c2:c3:c48_8 :: Nat -> c2:c3:c4


Generator Equations:
gen_0':s7_8(0) <=> 0'
gen_0':s7_8(+(x, 1)) <=> s(gen_0':s7_8(x))
gen_c2:c3:c48_8(0) <=> c2
gen_c2:c3:c48_8(+(x, 1)) <=> c4(gen_c2:c3:c48_8(x))


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

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

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

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

(49)
BOUNDS(n^1, INF)

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

(50)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LE(0', z0) -> c2
LE(s(z0), 0') -> c3
LE(s(z0), s(z1)) -> c4(LE(z0, z1))
MINUS(z0, z1) -> c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7(MINUS(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, z1) -> if(le(z0, z1), z0, z1)
if(true, z0, z1) -> 0'
if(false, z0, z1) -> s(minus(p(z0), z1))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LE :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
MINUS :: 0':s -> 0':s -> c5
c5 :: c6:c7 -> c2:c3:c4 -> c5
IF :: true:false -> 0':s -> 0':s -> c6:c7
le :: 0':s -> 0':s -> true:false
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c5 -> c:c1 -> c6:c7
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_8 :: c:c1
hole_0':s2_8 :: 0':s
hole_c2:c3:c43_8 :: c2:c3:c4
hole_c54_8 :: c5
hole_c6:c75_8 :: c6:c7
hole_true:false6_8 :: true:false
gen_0':s7_8 :: Nat -> 0':s
gen_c2:c3:c48_8 :: Nat -> c2:c3:c4


Lemmas:
LE(gen_0':s7_8(n10_8), gen_0':s7_8(n10_8)) -> gen_c2:c3:c48_8(n10_8), rt in Omega(1 + n10_8)


Generator Equations:
gen_0':s7_8(0) <=> 0'
gen_0':s7_8(+(x, 1)) <=> s(gen_0':s7_8(x))
gen_c2:c3:c48_8(0) <=> c2
gen_c2:c3:c48_8(+(x, 1)) <=> c4(gen_c2:c3:c48_8(x))


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

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

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

(51) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s7_8(n572_8), gen_0':s7_8(n572_8)) -> true, rt in Omega(0)

Induction Base:
le(gen_0':s7_8(0), gen_0':s7_8(0)) ->_R^Omega(0)
true

Induction Step:
le(gen_0':s7_8(+(n572_8, 1)), gen_0':s7_8(+(n572_8, 1))) ->_R^Omega(0)
le(gen_0':s7_8(n572_8), gen_0':s7_8(n572_8)) ->_IH
true

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(52)
Obligation:
Innermost TRS:
Rules:
P(0') -> c
P(s(z0)) -> c1
LE(0', z0) -> c2
LE(s(z0), 0') -> c3
LE(s(z0), s(z1)) -> c4(LE(z0, z1))
MINUS(z0, z1) -> c5(IF(le(z0, z1), z0, z1), LE(z0, z1))
IF(true, z0, z1) -> c6
IF(false, z0, z1) -> c7(MINUS(p(z0), z1), P(z0))
p(0') -> 0'
p(s(z0)) -> z0
le(0', z0) -> true
le(s(z0), 0') -> false
le(s(z0), s(z1)) -> le(z0, z1)
minus(z0, z1) -> if(le(z0, z1), z0, z1)
if(true, z0, z1) -> 0'
if(false, z0, z1) -> s(minus(p(z0), z1))

Types:
P :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
LE :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
MINUS :: 0':s -> 0':s -> c5
c5 :: c6:c7 -> c2:c3:c4 -> c5
IF :: true:false -> 0':s -> 0':s -> c6:c7
le :: 0':s -> 0':s -> true:false
true :: true:false
c6 :: c6:c7
false :: true:false
c7 :: c5 -> c:c1 -> c6:c7
p :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s
hole_c:c11_8 :: c:c1
hole_0':s2_8 :: 0':s
hole_c2:c3:c43_8 :: c2:c3:c4
hole_c54_8 :: c5
hole_c6:c75_8 :: c6:c7
hole_true:false6_8 :: true:false
gen_0':s7_8 :: Nat -> 0':s
gen_c2:c3:c48_8 :: Nat -> c2:c3:c4


Lemmas:
LE(gen_0':s7_8(n10_8), gen_0':s7_8(n10_8)) -> gen_c2:c3:c48_8(n10_8), rt in Omega(1 + n10_8)
le(gen_0':s7_8(n572_8), gen_0':s7_8(n572_8)) -> true, rt in Omega(0)


Generator Equations:
gen_0':s7_8(0) <=> 0'
gen_0':s7_8(+(x, 1)) <=> s(gen_0':s7_8(x))
gen_c2:c3:c48_8(0) <=> c2
gen_c2:c3:c48_8(+(x, 1)) <=> c4(gen_c2:c3:c48_8(x))


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