WORST_CASE(Omega(n^1),O(n^1))
proof of input_4MB1FDdGiX.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^1).

(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), 1294 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 494 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 177 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 42 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 147 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 129 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^1)
(47) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
    (49) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CpxRelTRS
    (51) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CpxRelTRS
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) typed CpxTrs
    (55) OrderProof [LOWER BOUND(ID), 11 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 1612 ms]
    (58) BEST
        (59) proven lower bound
            (60) LowerBoundPropagationProof [FINISHED, 0 ms]
            (61) BOUNDS(n^1, INF)
        (62) typed CpxTrs
            (63) RewriteLemmaProof [LOWER BOUND(ID), 21 ms]
            (64) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   nonZero(0) -> false
   nonZero(s(x)) -> true
   p(0) -> 0
   p(s(x)) -> x
   id_inc(x) -> x
   id_inc(x) -> s(x)
   random(x) -> rand(x, 0)
   rand(x, y) -> if(nonZero(x), x, y)
   if(false, x, y) -> y
   if(true, x, y) -> rand(p(x), id_inc(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^1).


The TRS R consists of the following rules:

   nonZero(0) -> false [1]
   nonZero(s(x)) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   random(x) -> rand(x, 0) [1]
   rand(x, y) -> if(nonZero(x), x, y) [1]
   if(false, x, y) -> y [1]
   if(true, x, y) -> rand(p(x), id_inc(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:

   nonZero(0) -> false [1]
   nonZero(s(x)) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   random(x) -> rand(x, 0) [1]
   rand(x, y) -> if(nonZero(x), x, y) [1]
   if(false, x, y) -> y [1]
   if(true, x, y) -> rand(p(x), id_inc(y)) [1]

The TRS has the following type information:
nonZero :: 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
p :: 0:s -> 0:s
id_inc :: 0:s -> 0:s
random :: 0:s -> 0:s
rand :: 0:s -> 0:s -> 0:s
if :: false:true -> 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:

   random_1
   rand_2
   if_3

(c) The following functions are completely defined:

   p_1
   id_inc_1
   nonZero_1

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:

   nonZero(0) -> false [1]
   nonZero(s(x)) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   random(x) -> rand(x, 0) [1]
   rand(x, y) -> if(nonZero(x), x, y) [1]
   if(false, x, y) -> y [1]
   if(true, x, y) -> rand(p(x), id_inc(y)) [1]

The TRS has the following type information:
nonZero :: 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
p :: 0:s -> 0:s
id_inc :: 0:s -> 0:s
random :: 0:s -> 0:s
rand :: 0:s -> 0:s -> 0:s
if :: false:true -> 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:

   nonZero(0) -> false [1]
   nonZero(s(x)) -> true [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]
   id_inc(x) -> x [1]
   id_inc(x) -> s(x) [1]
   random(x) -> rand(x, 0) [1]
   rand(0, y) -> if(false, 0, y) [2]
   rand(s(x'), y) -> if(true, s(x'), y) [2]
   if(false, x, y) -> y [1]
   if(true, 0, y) -> rand(0, y) [3]
   if(true, 0, y) -> rand(0, s(y)) [3]
   if(true, s(x''), y) -> rand(x'', y) [3]
   if(true, s(x''), y) -> rand(x'', s(y)) [3]

The TRS has the following type information:
nonZero :: 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
p :: 0:s -> 0:s
id_inc :: 0:s -> 0:s
random :: 0:s -> 0:s
rand :: 0:s -> 0:s -> 0:s
if :: false:true -> 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
   false => 0
   true => 1

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

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

   id_inc(z) -{ 1 }-> x :|: x >= 0, z = x
   id_inc(z) -{ 1 }-> 1 + x :|: x >= 0, z = x
   if(z, z', z'') -{ 1 }-> y :|: z' = x, z'' = y, x >= 0, y >= 0, z = 0
   if(z, z', z'') -{ 3 }-> rand(x'', y) :|: z' = 1 + x'', z'' = y, z = 1, y >= 0, x'' >= 0
   if(z, z', z'') -{ 3 }-> rand(x'', 1 + y) :|: z' = 1 + x'', z'' = y, z = 1, y >= 0, x'' >= 0
   if(z, z', z'') -{ 3 }-> rand(0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(0, 1 + y) :|: z'' = y, z = 1, y >= 0, z' = 0
   nonZero(z) -{ 1 }-> 1 :|: x >= 0, z = 1 + x
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0
   rand(z, z') -{ 2 }-> if(1, 1 + x', y) :|: z = 1 + x', x' >= 0, y >= 0, z' = y
   rand(z, z') -{ 2 }-> if(0, 0, y) :|: y >= 0, z = 0, z' = y
   random(z) -{ 1 }-> rand(x, 0) :|: x >= 0, z = x


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

(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:

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 3 }-> rand(0, z'') :|: z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 3 }-> rand(z' - 1, 1 + z'') :|: z = 1, z'' >= 0, z' - 1 >= 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 2 }-> if(1, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   rand(z, z') -{ 2 }-> if(0, 0, z') :|: z' >= 0, z = 0
   random(z) -{ 1 }-> rand(z, 0) :|: z >= 0


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

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

   { if, rand }
   { nonZero }
   { id_inc }
   { p }
   { random }

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

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

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 3 }-> rand(0, z'') :|: z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 3 }-> rand(z' - 1, 1 + z'') :|: z = 1, z'' >= 0, z' - 1 >= 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 2 }-> if(1, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   rand(z, z') -{ 2 }-> if(0, 0, z') :|: z' >= 0, z = 0
   random(z) -{ 1 }-> rand(z, 0) :|: z >= 0

Function symbols to be analyzed: {if,rand}, {nonZero}, {id_inc}, {p}, {random}

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

(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:

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 3 }-> rand(0, z'') :|: z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 3 }-> rand(z' - 1, 1 + z'') :|: z = 1, z'' >= 0, z' - 1 >= 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 2 }-> if(1, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   rand(z, z') -{ 2 }-> if(0, 0, z') :|: z' >= 0, z = 0
   random(z) -{ 1 }-> rand(z, 0) :|: z >= 0

Function symbols to be analyzed: {if,rand}, {nonZero}, {id_inc}, {p}, {random}

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

(17) 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' + z''

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

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

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

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 3 }-> rand(0, z'') :|: z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 3 }-> rand(z' - 1, 1 + z'') :|: z = 1, z'' >= 0, z' - 1 >= 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 2 }-> if(1, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   rand(z, z') -{ 2 }-> if(0, 0, z') :|: z' >= 0, z = 0
   random(z) -{ 1 }-> rand(z, 0) :|: z >= 0

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

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 11 + 5*z'

Computed RUNTIME bound using CoFloCo for: rand
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 13 + 5*z

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

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

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   if(z, z', z'') -{ 3 }-> rand(0, z'') :|: z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(0, 1 + z'') :|: z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 3 }-> rand(z' - 1, z'') :|: z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 3 }-> rand(z' - 1, 1 + z'') :|: z = 1, z'' >= 0, z' - 1 >= 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 2 }-> if(1, 1 + (z - 1), z') :|: z - 1 >= 0, z' >= 0
   rand(z, z') -{ 2 }-> if(0, 0, z') :|: z' >= 0, z = 0
   random(z) -{ 1 }-> rand(z, 0) :|: z >= 0

Function symbols to be analyzed: {nonZero}, {id_inc}, {p}, {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']

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

(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:

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {nonZero}, {id_inc}, {p}, {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {nonZero}, {id_inc}, {p}, {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: ?, size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: nonZero
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:

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {id_inc}, {p}, {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: O(1) [1], size: O(1) [1]

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

(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:

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {id_inc}, {p}, {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: O(1) [1], size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: id_inc
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:

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {id_inc}, {p}, {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: O(1) [1], size: O(1) [1]
  id_inc: runtime: ?, size: O(n^1) [1 + z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {p}, {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: O(1) [1], size: O(1) [1]
  id_inc: runtime: O(1) [1], size: O(n^1) [1 + z]

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

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

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

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {p}, {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: O(1) [1], size: O(1) [1]
  id_inc: runtime: O(1) [1], size: O(n^1) [1 + z]

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

(35) 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

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

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

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {p}, {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: O(1) [1], size: O(1) [1]
  id_inc: runtime: O(1) [1], size: O(n^1) [1 + z]
  p: runtime: ?, size: O(n^1) [z]

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

(37) 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

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

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

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: O(1) [1], size: O(1) [1]
  id_inc: runtime: O(1) [1], size: O(n^1) [1 + z]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: O(1) [1], size: O(1) [1]
  id_inc: runtime: O(1) [1], size: O(n^1) [1 + z]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: {random}
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: O(1) [1], size: O(1) [1]
  id_inc: runtime: O(1) [1], size: O(n^1) [1 + z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  random: runtime: ?, size: O(n^1) [1 + z]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: random
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 14 + 5*z

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   id_inc(z) -{ 1 }-> z :|: z >= 0
   id_inc(z) -{ 1 }-> 1 + z :|: z >= 0
   if(z, z', z'') -{ 16 }-> s1 :|: s1 >= 0, s1 <= 0 + z'' + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 16 }-> s2 :|: s2 >= 0, s2 <= 0 + (1 + z'') + 1, z = 1, z'' >= 0, z' = 0
   if(z, z', z'') -{ 11 + 5*z' }-> s3 :|: s3 >= 0, s3 <= z' - 1 + z'' + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 11 + 5*z' }-> s4 :|: s4 >= 0, s4 <= z' - 1 + (1 + z'') + 1, z = 1, z'' >= 0, z' - 1 >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z' >= 0, z'' >= 0, z = 0
   nonZero(z) -{ 1 }-> 1 :|: z - 1 >= 0
   nonZero(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   rand(z, z') -{ 13 }-> s' :|: s' >= 0, s' <= z' + 1 + 0, z' >= 0, z = 0
   rand(z, z') -{ 13 + 5*z }-> s'' :|: s'' >= 0, s'' <= z' + 1 + (1 + (z - 1)), z - 1 >= 0, z' >= 0
   random(z) -{ 14 + 5*z }-> s :|: s >= 0, s <= z + 0 + 1, z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  if: runtime: O(n^1) [11 + 5*z'], size: O(n^1) [1 + z' + z'']
  rand: runtime: O(n^1) [13 + 5*z], size: O(n^1) [1 + z + z']
  nonZero: runtime: O(1) [1], size: O(1) [1]
  id_inc: runtime: O(1) [1], size: O(n^1) [1 + z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  random: runtime: O(n^1) [14 + 5*z], size: O(n^1) [1 + z]

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

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

(46)
BOUNDS(1, n^1)

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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   nonZero(0) -> false
   nonZero(s(z0)) -> true
   p(0) -> 0
   p(s(z0)) -> z0
   id_inc(z0) -> z0
   id_inc(z0) -> s(z0)
   random(z0) -> rand(z0, 0)
   rand(z0, z1) -> if(nonZero(z0), z0, z1)
   if(false, z0, z1) -> z1
   if(true, z0, z1) -> rand(p(z0), id_inc(z1))
Tuples:
   NONZERO(0) -> c
   NONZERO(s(z0)) -> c1
   P(0) -> c2
   P(s(z0)) -> c3
   ID_INC(z0) -> c4
   ID_INC(z0) -> c5
   RANDOM(z0) -> c6(RAND(z0, 0))
   RAND(z0, z1) -> c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
   IF(false, z0, z1) -> c8
   IF(true, z0, z1) -> c9(RAND(p(z0), id_inc(z1)), P(z0))
   IF(true, z0, z1) -> c10(RAND(p(z0), id_inc(z1)), ID_INC(z1))
S tuples:
   NONZERO(0) -> c
   NONZERO(s(z0)) -> c1
   P(0) -> c2
   P(s(z0)) -> c3
   ID_INC(z0) -> c4
   ID_INC(z0) -> c5
   RANDOM(z0) -> c6(RAND(z0, 0))
   RAND(z0, z1) -> c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
   IF(false, z0, z1) -> c8
   IF(true, z0, z1) -> c9(RAND(p(z0), id_inc(z1)), P(z0))
   IF(true, z0, z1) -> c10(RAND(p(z0), id_inc(z1)), ID_INC(z1))
K tuples:none
Defined Rule Symbols:   nonZero_1, p_1, id_inc_1, random_1, rand_2, if_3

Defined Pair Symbols:   NONZERO_1, P_1, ID_INC_1, RANDOM_1, RAND_2, IF_3

Compound Symbols:   c, c1, c2, c3, c4, c5, c6_1, c7_2, c8, c9_2, c10_2


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

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

(50)
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:

   NONZERO(0) -> c
   NONZERO(s(z0)) -> c1
   P(0) -> c2
   P(s(z0)) -> c3
   ID_INC(z0) -> c4
   ID_INC(z0) -> c5
   RANDOM(z0) -> c6(RAND(z0, 0))
   RAND(z0, z1) -> c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
   IF(false, z0, z1) -> c8
   IF(true, z0, z1) -> c9(RAND(p(z0), id_inc(z1)), P(z0))
   IF(true, z0, z1) -> c10(RAND(p(z0), id_inc(z1)), ID_INC(z1))

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

   nonZero(0) -> false
   nonZero(s(z0)) -> true
   p(0) -> 0
   p(s(z0)) -> z0
   id_inc(z0) -> z0
   id_inc(z0) -> s(z0)
   random(z0) -> rand(z0, 0)
   rand(z0, z1) -> if(nonZero(z0), z0, z1)
   if(false, z0, z1) -> z1
   if(true, z0, z1) -> rand(p(z0), id_inc(z1))

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

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

(52)
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:

   NONZERO(0') -> c
   NONZERO(s(z0)) -> c1
   P(0') -> c2
   P(s(z0)) -> c3
   ID_INC(z0) -> c4
   ID_INC(z0) -> c5
   RANDOM(z0) -> c6(RAND(z0, 0'))
   RAND(z0, z1) -> c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
   IF(false, z0, z1) -> c8
   IF(true, z0, z1) -> c9(RAND(p(z0), id_inc(z1)), P(z0))
   IF(true, z0, z1) -> c10(RAND(p(z0), id_inc(z1)), ID_INC(z1))

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

   nonZero(0') -> false
   nonZero(s(z0)) -> true
   p(0') -> 0'
   p(s(z0)) -> z0
   id_inc(z0) -> z0
   id_inc(z0) -> s(z0)
   random(z0) -> rand(z0, 0')
   rand(z0, z1) -> if(nonZero(z0), z0, z1)
   if(false, z0, z1) -> z1
   if(true, z0, z1) -> rand(p(z0), id_inc(z1))

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

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

(54)
Obligation:
Innermost TRS:
Rules:
NONZERO(0') -> c
NONZERO(s(z0)) -> c1
P(0') -> c2
P(s(z0)) -> c3
ID_INC(z0) -> c4
ID_INC(z0) -> c5
RANDOM(z0) -> c6(RAND(z0, 0'))
RAND(z0, z1) -> c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
IF(false, z0, z1) -> c8
IF(true, z0, z1) -> c9(RAND(p(z0), id_inc(z1)), P(z0))
IF(true, z0, z1) -> c10(RAND(p(z0), id_inc(z1)), ID_INC(z1))
nonZero(0') -> false
nonZero(s(z0)) -> true
p(0') -> 0'
p(s(z0)) -> z0
id_inc(z0) -> z0
id_inc(z0) -> s(z0)
random(z0) -> rand(z0, 0')
rand(z0, z1) -> if(nonZero(z0), z0, z1)
if(false, z0, z1) -> z1
if(true, z0, z1) -> rand(p(z0), id_inc(z1))

Types:
NONZERO :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
P :: 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
ID_INC :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
RANDOM :: 0':s -> c6
c6 :: c7 -> c6
RAND :: 0':s -> 0':s -> c7
c7 :: c8:c9:c10 -> c:c1 -> c7
IF :: false:true -> 0':s -> 0':s -> c8:c9:c10
nonZero :: 0':s -> false:true
false :: false:true
c8 :: c8:c9:c10
true :: false:true
c9 :: c7 -> c2:c3 -> c8:c9:c10
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
c10 :: c7 -> c4:c5 -> c8:c9:c10
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c33_11 :: c2:c3
hole_c4:c54_11 :: c4:c5
hole_c65_11 :: c6
hole_c76_11 :: c7
hole_c8:c9:c107_11 :: c8:c9:c10
hole_false:true8_11 :: false:true
gen_0':s9_11 :: Nat -> 0':s

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

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
RAND, rand
----------------------------------------

(56)
Obligation:
Innermost TRS:
Rules:
NONZERO(0') -> c
NONZERO(s(z0)) -> c1
P(0') -> c2
P(s(z0)) -> c3
ID_INC(z0) -> c4
ID_INC(z0) -> c5
RANDOM(z0) -> c6(RAND(z0, 0'))
RAND(z0, z1) -> c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
IF(false, z0, z1) -> c8
IF(true, z0, z1) -> c9(RAND(p(z0), id_inc(z1)), P(z0))
IF(true, z0, z1) -> c10(RAND(p(z0), id_inc(z1)), ID_INC(z1))
nonZero(0') -> false
nonZero(s(z0)) -> true
p(0') -> 0'
p(s(z0)) -> z0
id_inc(z0) -> z0
id_inc(z0) -> s(z0)
random(z0) -> rand(z0, 0')
rand(z0, z1) -> if(nonZero(z0), z0, z1)
if(false, z0, z1) -> z1
if(true, z0, z1) -> rand(p(z0), id_inc(z1))

Types:
NONZERO :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
P :: 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
ID_INC :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
RANDOM :: 0':s -> c6
c6 :: c7 -> c6
RAND :: 0':s -> 0':s -> c7
c7 :: c8:c9:c10 -> c:c1 -> c7
IF :: false:true -> 0':s -> 0':s -> c8:c9:c10
nonZero :: 0':s -> false:true
false :: false:true
c8 :: c8:c9:c10
true :: false:true
c9 :: c7 -> c2:c3 -> c8:c9:c10
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
c10 :: c7 -> c4:c5 -> c8:c9:c10
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c33_11 :: c2:c3
hole_c4:c54_11 :: c4:c5
hole_c65_11 :: c6
hole_c76_11 :: c7
hole_c8:c9:c107_11 :: c8:c9:c10
hole_false:true8_11 :: false:true
gen_0':s9_11 :: Nat -> 0':s


Generator Equations:
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))


The following defined symbols remain to be analysed:
RAND, rand
----------------------------------------

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
RAND(gen_0':s9_11(n11_11), gen_0':s9_11(b)) -> *10_11, rt in Omega(n11_11)

Induction Base:
RAND(gen_0':s9_11(0), gen_0':s9_11(b))

Induction Step:
RAND(gen_0':s9_11(+(n11_11, 1)), gen_0':s9_11(b)) ->_R^Omega(1)
c7(IF(nonZero(gen_0':s9_11(+(n11_11, 1))), gen_0':s9_11(+(n11_11, 1)), gen_0':s9_11(b)), NONZERO(gen_0':s9_11(+(n11_11, 1)))) ->_R^Omega(0)
c7(IF(true, gen_0':s9_11(+(1, n11_11)), gen_0':s9_11(b)), NONZERO(gen_0':s9_11(+(1, n11_11)))) ->_R^Omega(1)
c7(c9(RAND(p(gen_0':s9_11(+(1, n11_11))), id_inc(gen_0':s9_11(b))), P(gen_0':s9_11(+(1, n11_11)))), NONZERO(gen_0':s9_11(+(1, n11_11)))) ->_R^Omega(0)
c7(c9(RAND(gen_0':s9_11(n11_11), id_inc(gen_0':s9_11(b))), P(gen_0':s9_11(+(1, n11_11)))), NONZERO(gen_0':s9_11(+(1, n11_11)))) ->_R^Omega(0)
c7(c9(RAND(gen_0':s9_11(n11_11), gen_0':s9_11(b)), P(gen_0':s9_11(+(1, n11_11)))), NONZERO(gen_0':s9_11(+(1, n11_11)))) ->_IH
c7(c9(*10_11, P(gen_0':s9_11(+(1, n11_11)))), NONZERO(gen_0':s9_11(+(1, n11_11)))) ->_R^Omega(1)
c7(c9(*10_11, c3), NONZERO(gen_0':s9_11(+(1, n11_11)))) ->_R^Omega(1)
c7(c9(*10_11, c3), c1)

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

(58)
Complex Obligation (BEST)

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

(59)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
NONZERO(0') -> c
NONZERO(s(z0)) -> c1
P(0') -> c2
P(s(z0)) -> c3
ID_INC(z0) -> c4
ID_INC(z0) -> c5
RANDOM(z0) -> c6(RAND(z0, 0'))
RAND(z0, z1) -> c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
IF(false, z0, z1) -> c8
IF(true, z0, z1) -> c9(RAND(p(z0), id_inc(z1)), P(z0))
IF(true, z0, z1) -> c10(RAND(p(z0), id_inc(z1)), ID_INC(z1))
nonZero(0') -> false
nonZero(s(z0)) -> true
p(0') -> 0'
p(s(z0)) -> z0
id_inc(z0) -> z0
id_inc(z0) -> s(z0)
random(z0) -> rand(z0, 0')
rand(z0, z1) -> if(nonZero(z0), z0, z1)
if(false, z0, z1) -> z1
if(true, z0, z1) -> rand(p(z0), id_inc(z1))

Types:
NONZERO :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
P :: 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
ID_INC :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
RANDOM :: 0':s -> c6
c6 :: c7 -> c6
RAND :: 0':s -> 0':s -> c7
c7 :: c8:c9:c10 -> c:c1 -> c7
IF :: false:true -> 0':s -> 0':s -> c8:c9:c10
nonZero :: 0':s -> false:true
false :: false:true
c8 :: c8:c9:c10
true :: false:true
c9 :: c7 -> c2:c3 -> c8:c9:c10
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
c10 :: c7 -> c4:c5 -> c8:c9:c10
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c33_11 :: c2:c3
hole_c4:c54_11 :: c4:c5
hole_c65_11 :: c6
hole_c76_11 :: c7
hole_c8:c9:c107_11 :: c8:c9:c10
hole_false:true8_11 :: false:true
gen_0':s9_11 :: Nat -> 0':s


Generator Equations:
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))


The following defined symbols remain to be analysed:
RAND, rand
----------------------------------------

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

(61)
BOUNDS(n^1, INF)

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

(62)
Obligation:
Innermost TRS:
Rules:
NONZERO(0') -> c
NONZERO(s(z0)) -> c1
P(0') -> c2
P(s(z0)) -> c3
ID_INC(z0) -> c4
ID_INC(z0) -> c5
RANDOM(z0) -> c6(RAND(z0, 0'))
RAND(z0, z1) -> c7(IF(nonZero(z0), z0, z1), NONZERO(z0))
IF(false, z0, z1) -> c8
IF(true, z0, z1) -> c9(RAND(p(z0), id_inc(z1)), P(z0))
IF(true, z0, z1) -> c10(RAND(p(z0), id_inc(z1)), ID_INC(z1))
nonZero(0') -> false
nonZero(s(z0)) -> true
p(0') -> 0'
p(s(z0)) -> z0
id_inc(z0) -> z0
id_inc(z0) -> s(z0)
random(z0) -> rand(z0, 0')
rand(z0, z1) -> if(nonZero(z0), z0, z1)
if(false, z0, z1) -> z1
if(true, z0, z1) -> rand(p(z0), id_inc(z1))

Types:
NONZERO :: 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1
P :: 0':s -> c2:c3
c2 :: c2:c3
c3 :: c2:c3
ID_INC :: 0':s -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
RANDOM :: 0':s -> c6
c6 :: c7 -> c6
RAND :: 0':s -> 0':s -> c7
c7 :: c8:c9:c10 -> c:c1 -> c7
IF :: false:true -> 0':s -> 0':s -> c8:c9:c10
nonZero :: 0':s -> false:true
false :: false:true
c8 :: c8:c9:c10
true :: false:true
c9 :: c7 -> c2:c3 -> c8:c9:c10
p :: 0':s -> 0':s
id_inc :: 0':s -> 0':s
c10 :: c7 -> c4:c5 -> c8:c9:c10
random :: 0':s -> 0':s
rand :: 0':s -> 0':s -> 0':s
if :: false:true -> 0':s -> 0':s -> 0':s
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c33_11 :: c2:c3
hole_c4:c54_11 :: c4:c5
hole_c65_11 :: c6
hole_c76_11 :: c7
hole_c8:c9:c107_11 :: c8:c9:c10
hole_false:true8_11 :: false:true
gen_0':s9_11 :: Nat -> 0':s


Lemmas:
RAND(gen_0':s9_11(n11_11), gen_0':s9_11(b)) -> *10_11, rt in Omega(n11_11)


Generator Equations:
gen_0':s9_11(0) <=> 0'
gen_0':s9_11(+(x, 1)) <=> s(gen_0':s9_11(x))


The following defined symbols remain to be analysed:
rand
----------------------------------------

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
rand(gen_0':s9_11(n6259_11), gen_0':s9_11(b)) -> gen_0':s9_11(b), rt in Omega(0)

Induction Base:
rand(gen_0':s9_11(0), gen_0':s9_11(b)) ->_R^Omega(0)
if(nonZero(gen_0':s9_11(0)), gen_0':s9_11(0), gen_0':s9_11(b)) ->_R^Omega(0)
if(false, gen_0':s9_11(0), gen_0':s9_11(b)) ->_R^Omega(0)
gen_0':s9_11(b)

Induction Step:
rand(gen_0':s9_11(+(n6259_11, 1)), gen_0':s9_11(b)) ->_R^Omega(0)
if(nonZero(gen_0':s9_11(+(n6259_11, 1))), gen_0':s9_11(+(n6259_11, 1)), gen_0':s9_11(b)) ->_R^Omega(0)
if(true, gen_0':s9_11(+(1, n6259_11)), gen_0':s9_11(b)) ->_R^Omega(0)
rand(p(gen_0':s9_11(+(1, n6259_11))), id_inc(gen_0':s9_11(b))) ->_R^Omega(0)
rand(gen_0':s9_11(n6259_11), id_inc(gen_0':s9_11(b))) ->_R^Omega(0)
rand(gen_0':s9_11(n6259_11), gen_0':s9_11(b)) ->_IH
gen_0':s9_11(b)

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

(64)
BOUNDS(1, INF)