WORST_CASE(Omega(n^1),O(n^1))
proof of input_siqZKC8ReW.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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) InliningProof [UPPER BOUND(ID), 423 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), 148 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 49 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 542 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 161 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 130 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 314 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 167 ms]
    (40) CpxRNTS
    (41) FinalProof [FINISHED, 0 ms]
    (42) BOUNDS(1, n^1)
(43) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(44) CdtProblem
    (45) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CpxRelTRS
    (47) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CpxRelTRS
    (49) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) typed CpxTrs
    (51) OrderProof [LOWER BOUND(ID), 5 ms]
    (52) typed CpxTrs
    (53) RewriteLemmaProof [LOWER BOUND(ID), 285 ms]
    (54) typed CpxTrs
    (55) RewriteLemmaProof [LOWER BOUND(ID), 172 ms]
    (56) BEST
        (57) proven lower bound
            (58) LowerBoundPropagationProof [FINISHED, 0 ms]
            (59) BOUNDS(n^1, INF)
        (60) typed CpxTrs


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

(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:

   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y))
   and(true, true) -> true
   and(x, false) -> false
   and(false, x) -> false
   gr(0, 0) -> false
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   p(0) -> 0
   p(s(x)) -> x

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:

   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y)) [1]
   and(true, true) -> true [1]
   and(x, false) -> false [1]
   and(false, x) -> false [1]
   gr(0, 0) -> false [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [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:

   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y)) [1]
   and(true, true) -> true [1]
   and(x, false) -> false [1]
   and(false, x) -> false [1]
   gr(0, 0) -> false [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond :: true:false -> 0:s -> 0:s -> cond
true :: true:false
and :: true:false -> true:false -> true:false
gr :: 0:s -> 0:s -> true:false
0 :: 0:s
p :: 0:s -> 0:s
false :: true:false
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:

   cond_3

(c) The following functions are completely defined:

   and_2
   gr_2
   p_1

Due to the following rules being added:
none

And the following fresh constants:    const

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

(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:

   cond(true, x, y) -> cond(and(gr(x, 0), gr(y, 0)), p(x), p(y)) [1]
   and(true, true) -> true [1]
   and(x, false) -> false [1]
   and(false, x) -> false [1]
   gr(0, 0) -> false [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond :: true:false -> 0:s -> 0:s -> cond
true :: true:false
and :: true:false -> true:false -> true:false
gr :: 0:s -> 0:s -> true:false
0 :: 0:s
p :: 0:s -> 0:s
false :: true:false
s :: 0:s -> 0:s
const :: cond

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:

   cond(true, 0, 0) -> cond(and(false, false), 0, 0) [5]
   cond(true, 0, 0) -> cond(and(false, false), 0, 0) [5]
   cond(true, 0, s(x'')) -> cond(and(false, true), 0, x'') [5]
   cond(true, 0, 0) -> cond(and(false, false), 0, 0) [5]
   cond(true, 0, 0) -> cond(and(false, false), 0, 0) [5]
   cond(true, 0, s(x1)) -> cond(and(false, true), 0, x1) [5]
   cond(true, s(x'), 0) -> cond(and(true, false), x', 0) [5]
   cond(true, s(x'), 0) -> cond(and(true, false), x', 0) [5]
   cond(true, s(x'), s(x2)) -> cond(and(true, true), x', x2) [5]
   and(true, true) -> true [1]
   and(x, false) -> false [1]
   and(false, x) -> false [1]
   gr(0, 0) -> false [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

The TRS has the following type information:
cond :: true:false -> 0:s -> 0:s -> cond
true :: true:false
and :: true:false -> true:false -> true:false
gr :: 0:s -> 0:s -> true:false
0 :: 0:s
p :: 0:s -> 0:s
false :: true:false
s :: 0:s -> 0:s
const :: cond

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:

   true => 1
   0 => 0
   false => 0
   const => 0

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

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

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   cond(z, z', z'') -{ 5 }-> cond(and(1, 1), x', x2) :|: z' = 1 + x', z = 1, x' >= 0, z'' = 1 + x2, x2 >= 0
   cond(z, z', z'') -{ 5 }-> cond(and(1, 0), x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0
   cond(z, z', z'') -{ 5 }-> cond(and(0, 1), 0, x'') :|: z = 1, z'' = 1 + x'', x'' >= 0, z' = 0
   cond(z, z', z'') -{ 5 }-> cond(and(0, 1), 0, x1) :|: x1 >= 0, z = 1, z'' = 1 + x1, z' = 0
   cond(z, z', z'') -{ 5 }-> cond(and(0, 0), 0, 0) :|: z'' = 0, z = 1, z' = 0
   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 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:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0

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

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

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> cond(1, x', x2) :|: z' = 1 + x', z = 1, x' >= 0, z'' = 1 + x2, x2 >= 0, 1 = 1
   cond(z, z', z'') -{ 6 }-> cond(0, x', 0) :|: z'' = 0, z' = 1 + x', z = 1, x' >= 0, x >= 0, 1 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, x'') :|: z = 1, z'' = 1 + x'', x'' >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, x1) :|: x1 >= 0, z = 1, z'' = 1 + x1, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 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:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 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:

   { and }
   { cond }
   { p }
   { gr }

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

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

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {and}, {cond}, {p}, {gr}

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

(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:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {and}, {cond}, {p}, {gr}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: and
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:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {and}, {cond}, {p}, {gr}
Previous analysis results are:
  and: runtime: ?, size: O(1) [1]

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond}, {p}, {gr}
Previous analysis results are:
  and: runtime: O(1) [1], 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:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond}, {p}, {gr}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [1]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: cond
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {cond}, {p}, {gr}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [1]
  cond: runtime: ?, size: O(1) [0]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: cond
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 24*z + 24*z'

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

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

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> cond(1, z' - 1, z'' - 1) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   cond(z, z', z'') -{ 6 }-> cond(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, 0, z'' - 1) :|: z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ 6 }-> cond(0, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {gr}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [1]
  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]

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

(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:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {gr}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [1]
  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]

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

(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:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {gr}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [1]
  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
  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:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [1]
  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
  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:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [1]
  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: gr
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:

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {gr}
Previous analysis results are:
  and: runtime: O(1) [1], size: O(1) [1]
  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: ?, size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   and(z, z') -{ 1 }-> 1 :|: z = 1, z' = 1
   and(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   and(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   cond(z, z', z'') -{ 6 }-> s :|: s >= 0, s <= 0, z'' = 0, z = 1, z' = 0, x >= 0, 0 = x, 0 = 0
   cond(z, z', z'') -{ 6 }-> s' :|: s' >= 0, s' <= 0, z = 1, z'' - 1 >= 0, z' = 0, 1 = x, x >= 0, 0 = 0
   cond(z, z', z'') -{ -18 + 24*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z = 1, z' - 1 >= 0, x >= 0, 1 = x, 0 = 0
   cond(z, z', z'') -{ 6 + 24*z' }-> s1 :|: s1 >= 0, s1 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = 1
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 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:
  and: runtime: O(1) [1], size: O(1) [1]
  cond: runtime: O(n^1) [24*z + 24*z'], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: O(n^1) [3 + z'], size: O(1) [1]

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

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

(42)
BOUNDS(1, n^1)

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

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond(true, z0, z1) -> cond(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1))
   and(true, true) -> true
   and(z0, false) -> false
   and(false, z0) -> false
   gr(0, 0) -> false
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND(true, z0, z1) -> c(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND(true, z0, z1) -> c1(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), AND(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   COND(true, z0, z1) -> c2(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), P(z0))
   COND(true, z0, z1) -> c3(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), P(z1))
   AND(true, true) -> c4
   AND(z0, false) -> c5
   AND(false, z0) -> c6
   GR(0, 0) -> c7
   GR(0, z0) -> c8
   GR(s(z0), 0) -> c9
   GR(s(z0), s(z1)) -> c10(GR(z0, z1))
   P(0) -> c11
   P(s(z0)) -> c12
S tuples:
   COND(true, z0, z1) -> c(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND(true, z0, z1) -> c1(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), AND(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   COND(true, z0, z1) -> c2(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), P(z0))
   COND(true, z0, z1) -> c3(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), P(z1))
   AND(true, true) -> c4
   AND(z0, false) -> c5
   AND(false, z0) -> c6
   GR(0, 0) -> c7
   GR(0, z0) -> c8
   GR(s(z0), 0) -> c9
   GR(s(z0), s(z1)) -> c10(GR(z0, z1))
   P(0) -> c11
   P(s(z0)) -> c12
K tuples:none
Defined Rule Symbols:   cond_3, and_2, gr_2, p_1

Defined Pair Symbols:   COND_3, AND_2, GR_2, P_1

Compound Symbols:   c_3, c1_3, c2_2, c3_2, c4, c5, c6, c7, c8, c9, c10_1, c11, c12


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

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

(46)
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:

   COND(true, z0, z1) -> c(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), AND(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND(true, z0, z1) -> c1(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), AND(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   COND(true, z0, z1) -> c2(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), P(z0))
   COND(true, z0, z1) -> c3(COND(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1)), P(z1))
   AND(true, true) -> c4
   AND(z0, false) -> c5
   AND(false, z0) -> c6
   GR(0, 0) -> c7
   GR(0, z0) -> c8
   GR(s(z0), 0) -> c9
   GR(s(z0), s(z1)) -> c10(GR(z0, z1))
   P(0) -> c11
   P(s(z0)) -> c12

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

   cond(true, z0, z1) -> cond(and(gr(z0, 0), gr(z1, 0)), p(z0), p(z1))
   and(true, true) -> true
   and(z0, false) -> false
   and(false, z0) -> false
   gr(0, 0) -> false
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   p(0) -> 0
   p(s(z0)) -> z0

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

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

(48)
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:

   COND(true, z0, z1) -> c(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
   COND(true, z0, z1) -> c1(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
   COND(true, z0, z1) -> c2(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z0))
   COND(true, z0, z1) -> c3(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z1))
   AND(true, true) -> c4
   AND(z0, false) -> c5
   AND(false, z0) -> c6
   GR(0', 0') -> c7
   GR(0', z0) -> c8
   GR(s(z0), 0') -> c9
   GR(s(z0), s(z1)) -> c10(GR(z0, z1))
   P(0') -> c11
   P(s(z0)) -> c12

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

   cond(true, z0, z1) -> cond(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1))
   and(true, true) -> true
   and(z0, false) -> false
   and(false, z0) -> false
   gr(0', 0') -> false
   gr(0', z0) -> false
   gr(s(z0), 0') -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   p(0') -> 0'
   p(s(z0)) -> z0

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

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

(50)
Obligation:
Innermost TRS:
Rules:
COND(true, z0, z1) -> c(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND(true, z0, z1) -> c1(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND(true, z0, z1) -> c2(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z0))
COND(true, z0, z1) -> c3(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z1))
AND(true, true) -> c4
AND(z0, false) -> c5
AND(false, z0) -> c6
GR(0', 0') -> c7
GR(0', z0) -> c8
GR(s(z0), 0') -> c9
GR(s(z0), s(z1)) -> c10(GR(z0, z1))
P(0') -> c11
P(s(z0)) -> c12
cond(true, z0, z1) -> cond(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1))
and(true, true) -> true
and(z0, false) -> false
and(false, z0) -> false
gr(0', 0') -> false
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> 0':s -> c:c1:c2:c3
true :: true:false
c :: c:c1:c2:c3 -> c4:c5:c6 -> c7:c8:c9:c10 -> c:c1:c2:c3
and :: true:false -> true:false -> true:false
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
p :: 0':s -> 0':s
AND :: true:false -> true:false -> c4:c5:c6
GR :: 0':s -> 0':s -> c7:c8:c9:c10
c1 :: c:c1:c2:c3 -> c4:c5:c6 -> c7:c8:c9:c10 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c11:c12 -> c:c1:c2:c3
P :: 0':s -> c11:c12
c3 :: c:c1:c2:c3 -> c11:c12 -> c:c1:c2:c3
c4 :: c4:c5:c6
false :: true:false
c5 :: c4:c5:c6
c6 :: c4:c5:c6
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
s :: 0':s -> 0':s
c9 :: c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c7:c8:c9:c10
c11 :: c11:c12
c12 :: c11:c12
cond :: true:false -> 0':s -> 0':s -> cond
hole_c:c1:c2:c31_13 :: c:c1:c2:c3
hole_true:false2_13 :: true:false
hole_0':s3_13 :: 0':s
hole_c4:c5:c64_13 :: c4:c5:c6
hole_c7:c8:c9:c105_13 :: c7:c8:c9:c10
hole_c11:c126_13 :: c11:c12
hole_cond7_13 :: cond
gen_c:c1:c2:c38_13 :: Nat -> c:c1:c2:c3
gen_0':s9_13 :: Nat -> 0':s
gen_c7:c8:c9:c1010_13 :: Nat -> c7:c8:c9:c10

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

(51) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
COND, gr, GR, cond

They will be analysed ascendingly in the following order:
gr < COND
GR < COND
gr < cond

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

(52)
Obligation:
Innermost TRS:
Rules:
COND(true, z0, z1) -> c(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND(true, z0, z1) -> c1(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND(true, z0, z1) -> c2(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z0))
COND(true, z0, z1) -> c3(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z1))
AND(true, true) -> c4
AND(z0, false) -> c5
AND(false, z0) -> c6
GR(0', 0') -> c7
GR(0', z0) -> c8
GR(s(z0), 0') -> c9
GR(s(z0), s(z1)) -> c10(GR(z0, z1))
P(0') -> c11
P(s(z0)) -> c12
cond(true, z0, z1) -> cond(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1))
and(true, true) -> true
and(z0, false) -> false
and(false, z0) -> false
gr(0', 0') -> false
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> 0':s -> c:c1:c2:c3
true :: true:false
c :: c:c1:c2:c3 -> c4:c5:c6 -> c7:c8:c9:c10 -> c:c1:c2:c3
and :: true:false -> true:false -> true:false
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
p :: 0':s -> 0':s
AND :: true:false -> true:false -> c4:c5:c6
GR :: 0':s -> 0':s -> c7:c8:c9:c10
c1 :: c:c1:c2:c3 -> c4:c5:c6 -> c7:c8:c9:c10 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c11:c12 -> c:c1:c2:c3
P :: 0':s -> c11:c12
c3 :: c:c1:c2:c3 -> c11:c12 -> c:c1:c2:c3
c4 :: c4:c5:c6
false :: true:false
c5 :: c4:c5:c6
c6 :: c4:c5:c6
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
s :: 0':s -> 0':s
c9 :: c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c7:c8:c9:c10
c11 :: c11:c12
c12 :: c11:c12
cond :: true:false -> 0':s -> 0':s -> cond
hole_c:c1:c2:c31_13 :: c:c1:c2:c3
hole_true:false2_13 :: true:false
hole_0':s3_13 :: 0':s
hole_c4:c5:c64_13 :: c4:c5:c6
hole_c7:c8:c9:c105_13 :: c7:c8:c9:c10
hole_c11:c126_13 :: c11:c12
hole_cond7_13 :: cond
gen_c:c1:c2:c38_13 :: Nat -> c:c1:c2:c3
gen_0':s9_13 :: Nat -> 0':s
gen_c7:c8:c9:c1010_13 :: Nat -> c7:c8:c9:c10


Generator Equations:
gen_c:c1:c2:c38_13(0) <=> hole_c:c1:c2:c31_13
gen_c:c1:c2:c38_13(+(x, 1)) <=> c(gen_c:c1:c2:c38_13(x), c4, c7)
gen_0':s9_13(0) <=> 0'
gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x))
gen_c7:c8:c9:c1010_13(0) <=> c7
gen_c7:c8:c9:c1010_13(+(x, 1)) <=> c10(gen_c7:c8:c9:c1010_13(x))


The following defined symbols remain to be analysed:
gr, COND, GR, cond

They will be analysed ascendingly in the following order:
gr < COND
GR < COND
gr < cond

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_0':s9_13(n12_13), gen_0':s9_13(n12_13)) -> false, rt in Omega(0)

Induction Base:
gr(gen_0':s9_13(0), gen_0':s9_13(0)) ->_R^Omega(0)
false

Induction Step:
gr(gen_0':s9_13(+(n12_13, 1)), gen_0':s9_13(+(n12_13, 1))) ->_R^Omega(0)
gr(gen_0':s9_13(n12_13), gen_0':s9_13(n12_13)) ->_IH
false

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

(54)
Obligation:
Innermost TRS:
Rules:
COND(true, z0, z1) -> c(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND(true, z0, z1) -> c1(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND(true, z0, z1) -> c2(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z0))
COND(true, z0, z1) -> c3(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z1))
AND(true, true) -> c4
AND(z0, false) -> c5
AND(false, z0) -> c6
GR(0', 0') -> c7
GR(0', z0) -> c8
GR(s(z0), 0') -> c9
GR(s(z0), s(z1)) -> c10(GR(z0, z1))
P(0') -> c11
P(s(z0)) -> c12
cond(true, z0, z1) -> cond(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1))
and(true, true) -> true
and(z0, false) -> false
and(false, z0) -> false
gr(0', 0') -> false
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> 0':s -> c:c1:c2:c3
true :: true:false
c :: c:c1:c2:c3 -> c4:c5:c6 -> c7:c8:c9:c10 -> c:c1:c2:c3
and :: true:false -> true:false -> true:false
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
p :: 0':s -> 0':s
AND :: true:false -> true:false -> c4:c5:c6
GR :: 0':s -> 0':s -> c7:c8:c9:c10
c1 :: c:c1:c2:c3 -> c4:c5:c6 -> c7:c8:c9:c10 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c11:c12 -> c:c1:c2:c3
P :: 0':s -> c11:c12
c3 :: c:c1:c2:c3 -> c11:c12 -> c:c1:c2:c3
c4 :: c4:c5:c6
false :: true:false
c5 :: c4:c5:c6
c6 :: c4:c5:c6
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
s :: 0':s -> 0':s
c9 :: c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c7:c8:c9:c10
c11 :: c11:c12
c12 :: c11:c12
cond :: true:false -> 0':s -> 0':s -> cond
hole_c:c1:c2:c31_13 :: c:c1:c2:c3
hole_true:false2_13 :: true:false
hole_0':s3_13 :: 0':s
hole_c4:c5:c64_13 :: c4:c5:c6
hole_c7:c8:c9:c105_13 :: c7:c8:c9:c10
hole_c11:c126_13 :: c11:c12
hole_cond7_13 :: cond
gen_c:c1:c2:c38_13 :: Nat -> c:c1:c2:c3
gen_0':s9_13 :: Nat -> 0':s
gen_c7:c8:c9:c1010_13 :: Nat -> c7:c8:c9:c10


Lemmas:
gr(gen_0':s9_13(n12_13), gen_0':s9_13(n12_13)) -> false, rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c38_13(0) <=> hole_c:c1:c2:c31_13
gen_c:c1:c2:c38_13(+(x, 1)) <=> c(gen_c:c1:c2:c38_13(x), c4, c7)
gen_0':s9_13(0) <=> 0'
gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x))
gen_c7:c8:c9:c1010_13(0) <=> c7
gen_c7:c8:c9:c1010_13(+(x, 1)) <=> c10(gen_c7:c8:c9:c1010_13(x))


The following defined symbols remain to be analysed:
GR, COND, cond

They will be analysed ascendingly in the following order:
GR < COND

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

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GR(gen_0':s9_13(n432_13), gen_0':s9_13(n432_13)) -> gen_c7:c8:c9:c1010_13(n432_13), rt in Omega(1 + n432_13)

Induction Base:
GR(gen_0':s9_13(0), gen_0':s9_13(0)) ->_R^Omega(1)
c7

Induction Step:
GR(gen_0':s9_13(+(n432_13, 1)), gen_0':s9_13(+(n432_13, 1))) ->_R^Omega(1)
c10(GR(gen_0':s9_13(n432_13), gen_0':s9_13(n432_13))) ->_IH
c10(gen_c7:c8:c9:c1010_13(c433_13))

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

(56)
Complex Obligation (BEST)

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

(57)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
COND(true, z0, z1) -> c(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND(true, z0, z1) -> c1(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND(true, z0, z1) -> c2(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z0))
COND(true, z0, z1) -> c3(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z1))
AND(true, true) -> c4
AND(z0, false) -> c5
AND(false, z0) -> c6
GR(0', 0') -> c7
GR(0', z0) -> c8
GR(s(z0), 0') -> c9
GR(s(z0), s(z1)) -> c10(GR(z0, z1))
P(0') -> c11
P(s(z0)) -> c12
cond(true, z0, z1) -> cond(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1))
and(true, true) -> true
and(z0, false) -> false
and(false, z0) -> false
gr(0', 0') -> false
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> 0':s -> c:c1:c2:c3
true :: true:false
c :: c:c1:c2:c3 -> c4:c5:c6 -> c7:c8:c9:c10 -> c:c1:c2:c3
and :: true:false -> true:false -> true:false
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
p :: 0':s -> 0':s
AND :: true:false -> true:false -> c4:c5:c6
GR :: 0':s -> 0':s -> c7:c8:c9:c10
c1 :: c:c1:c2:c3 -> c4:c5:c6 -> c7:c8:c9:c10 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c11:c12 -> c:c1:c2:c3
P :: 0':s -> c11:c12
c3 :: c:c1:c2:c3 -> c11:c12 -> c:c1:c2:c3
c4 :: c4:c5:c6
false :: true:false
c5 :: c4:c5:c6
c6 :: c4:c5:c6
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
s :: 0':s -> 0':s
c9 :: c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c7:c8:c9:c10
c11 :: c11:c12
c12 :: c11:c12
cond :: true:false -> 0':s -> 0':s -> cond
hole_c:c1:c2:c31_13 :: c:c1:c2:c3
hole_true:false2_13 :: true:false
hole_0':s3_13 :: 0':s
hole_c4:c5:c64_13 :: c4:c5:c6
hole_c7:c8:c9:c105_13 :: c7:c8:c9:c10
hole_c11:c126_13 :: c11:c12
hole_cond7_13 :: cond
gen_c:c1:c2:c38_13 :: Nat -> c:c1:c2:c3
gen_0':s9_13 :: Nat -> 0':s
gen_c7:c8:c9:c1010_13 :: Nat -> c7:c8:c9:c10


Lemmas:
gr(gen_0':s9_13(n12_13), gen_0':s9_13(n12_13)) -> false, rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c38_13(0) <=> hole_c:c1:c2:c31_13
gen_c:c1:c2:c38_13(+(x, 1)) <=> c(gen_c:c1:c2:c38_13(x), c4, c7)
gen_0':s9_13(0) <=> 0'
gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x))
gen_c7:c8:c9:c1010_13(0) <=> c7
gen_c7:c8:c9:c1010_13(+(x, 1)) <=> c10(gen_c7:c8:c9:c1010_13(x))


The following defined symbols remain to be analysed:
GR, COND, cond

They will be analysed ascendingly in the following order:
GR < COND

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

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

(59)
BOUNDS(n^1, INF)

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

(60)
Obligation:
Innermost TRS:
Rules:
COND(true, z0, z1) -> c(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND(true, z0, z1) -> c1(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), AND(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND(true, z0, z1) -> c2(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z0))
COND(true, z0, z1) -> c3(COND(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1)), P(z1))
AND(true, true) -> c4
AND(z0, false) -> c5
AND(false, z0) -> c6
GR(0', 0') -> c7
GR(0', z0) -> c8
GR(s(z0), 0') -> c9
GR(s(z0), s(z1)) -> c10(GR(z0, z1))
P(0') -> c11
P(s(z0)) -> c12
cond(true, z0, z1) -> cond(and(gr(z0, 0'), gr(z1, 0')), p(z0), p(z1))
and(true, true) -> true
and(z0, false) -> false
and(false, z0) -> false
gr(0', 0') -> false
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> 0':s -> c:c1:c2:c3
true :: true:false
c :: c:c1:c2:c3 -> c4:c5:c6 -> c7:c8:c9:c10 -> c:c1:c2:c3
and :: true:false -> true:false -> true:false
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
p :: 0':s -> 0':s
AND :: true:false -> true:false -> c4:c5:c6
GR :: 0':s -> 0':s -> c7:c8:c9:c10
c1 :: c:c1:c2:c3 -> c4:c5:c6 -> c7:c8:c9:c10 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c11:c12 -> c:c1:c2:c3
P :: 0':s -> c11:c12
c3 :: c:c1:c2:c3 -> c11:c12 -> c:c1:c2:c3
c4 :: c4:c5:c6
false :: true:false
c5 :: c4:c5:c6
c6 :: c4:c5:c6
c7 :: c7:c8:c9:c10
c8 :: c7:c8:c9:c10
s :: 0':s -> 0':s
c9 :: c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c7:c8:c9:c10
c11 :: c11:c12
c12 :: c11:c12
cond :: true:false -> 0':s -> 0':s -> cond
hole_c:c1:c2:c31_13 :: c:c1:c2:c3
hole_true:false2_13 :: true:false
hole_0':s3_13 :: 0':s
hole_c4:c5:c64_13 :: c4:c5:c6
hole_c7:c8:c9:c105_13 :: c7:c8:c9:c10
hole_c11:c126_13 :: c11:c12
hole_cond7_13 :: cond
gen_c:c1:c2:c38_13 :: Nat -> c:c1:c2:c3
gen_0':s9_13 :: Nat -> 0':s
gen_c7:c8:c9:c1010_13 :: Nat -> c7:c8:c9:c10


Lemmas:
gr(gen_0':s9_13(n12_13), gen_0':s9_13(n12_13)) -> false, rt in Omega(0)
GR(gen_0':s9_13(n432_13), gen_0':s9_13(n432_13)) -> gen_c7:c8:c9:c1010_13(n432_13), rt in Omega(1 + n432_13)


Generator Equations:
gen_c:c1:c2:c38_13(0) <=> hole_c:c1:c2:c31_13
gen_c:c1:c2:c38_13(+(x, 1)) <=> c(gen_c:c1:c2:c38_13(x), c4, c7)
gen_0':s9_13(0) <=> 0'
gen_0':s9_13(+(x, 1)) <=> s(gen_0':s9_13(x))
gen_c7:c8:c9:c1010_13(0) <=> c7
gen_c7:c8:c9:c1010_13(+(x, 1)) <=> c10(gen_c7:c8:c9:c1010_13(x))


The following defined symbols remain to be analysed:
COND, cond