/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 3 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), 161 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 303 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 156 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 2662 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 1773 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^2)
(35) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CpxTRS
    (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) typed CpxTrs
    (39) OrderProof [LOWER BOUND(ID), 0 ms]
    (40) typed CpxTrs
    (41) RewriteLemmaProof [LOWER BOUND(ID), 280 ms]
    (42) BEST
        (43) proven lower bound
            (44) LowerBoundPropagationProof [FINISHED, 0 ms]
            (45) BOUNDS(n^1, INF)
        (46) typed CpxTrs


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


The TRS R consists of the following rules:

   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
   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: INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

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


The TRS R consists of the following rules:

   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z) [1]
   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z) [1]
   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z) [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:

   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z) [1]
   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z) [1]
   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z) [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:
cond1 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2
gr :: 0:s -> 0:s -> true:false
p :: 0:s -> 0:s
false :: true:false
0 :: 0:s
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:

   cond1_4
   cond2_4

(c) The following functions are completely defined:

   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:

   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z) [1]
   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z) [1]
   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z) [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:
cond1 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2
gr :: 0:s -> 0:s -> true:false
p :: 0:s -> 0:s
false :: true:false
0 :: 0:s
s :: 0:s -> 0:s
const :: cond1:cond2

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:

   cond1(true, x, 0, z) -> cond2(false, x, 0, z) [2]
   cond1(true, x, s(x'), 0) -> cond2(true, x, s(x'), 0) [2]
   cond1(true, x, s(x''), s(y')) -> cond2(gr(x'', y'), x, s(x''), s(y')) [2]
   cond2(true, x, 0, z) -> cond2(false, x, 0, z) [3]
   cond2(true, x, s(x1), 0) -> cond2(true, x, x1, 0) [3]
   cond2(true, x, s(x2), s(y'')) -> cond2(gr(x2, y''), x, x2, s(y'')) [3]
   cond2(false, 0, y, z) -> cond1(false, 0, y, z) [3]
   cond2(false, s(x3), y, 0) -> cond1(true, x3, y, 0) [3]
   cond2(false, s(x4), y, s(y1)) -> cond1(gr(x4, y1), x4, y, s(y1)) [3]
   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:
cond1 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2
gr :: 0:s -> 0:s -> true:false
p :: 0:s -> 0:s
false :: true:false
0 :: 0:s
s :: 0:s -> 0:s
const :: cond1:cond2

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
   false => 0
   0 => 0
   const => 0

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(gr(x'', y'), x, 1 + x'', 1 + y') :|: z1 = 1 + x'', x >= 0, z'' = x, y' >= 0, z2 = 1 + y', z' = 1, x'' >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, x, 1 + x', 0) :|: z2 = 0, x >= 0, x' >= 0, z'' = x, z' = 1, z1 = 1 + x'
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, x, 0, z) :|: z1 = 0, z >= 0, z2 = z, x >= 0, z'' = x, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(gr(x2, y''), x, x2, 1 + y'') :|: z2 = 1 + y'', x >= 0, z'' = x, y'' >= 0, z' = 1, x2 >= 0, z1 = 1 + x2
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, x, x1, 0) :|: x1 >= 0, z2 = 0, x >= 0, z'' = x, z' = 1, z1 = 1 + x1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, x, 0, z) :|: z1 = 0, z >= 0, z2 = z, x >= 0, z'' = x, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(gr(x4, y1), x4, y, 1 + y1) :|: y1 >= 0, z2 = 1 + y1, z1 = y, x4 >= 0, z'' = 1 + x4, y >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, x3, y, 0) :|: z1 = y, z'' = 1 + x3, z2 = 0, y >= 0, x3 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, y, z) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 0
   gr(z', z'') -{ 1 }-> gr(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   gr(z', z'') -{ 1 }-> 0 :|: x >= 0, z'' = x, z' = 0
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   p(z') -{ 1 }-> 0 :|: z' = 0


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

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   gr(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> 0 :|: z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0


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

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

   { p }
   { gr }
   { cond2, cond1 }

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}

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

(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:

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}

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

(17) 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'

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}
Previous analysis results are:
  p: runtime: ?, size: O(n^1) [z']

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

(19) 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

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}
Previous analysis results are:
  p: runtime: O(1) [1], size: O(n^1) [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:

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}
Previous analysis results are:
  p: runtime: O(1) [1], size: O(n^1) [z']

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

(23) 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

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 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}, {cond2,cond1}
Previous analysis results are:
  p: runtime: O(1) [1], size: O(n^1) [z']
  gr: runtime: ?, size: O(1) [1]

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

(25) 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: 2 + z''

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(gr(z1 - 1, z2 - 1), z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(gr(z1 - 1, z2 - 1), z'', z1 - 1, 1 + (z2 - 1)) :|: z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(gr(z'' - 1, z2 - 1), z'' - 1, z1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 1 }-> gr(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 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: {cond2,cond1}
Previous analysis results are:
  p: runtime: O(1) [1], size: O(n^1) [z']
  gr: runtime: O(n^1) [2 + z''], 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:

   cond1(z', z'', z1, z2) -{ 3 + z2 }-> cond2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 + z2 }-> cond2(s', z'', z1 - 1, 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 + z2 }-> cond1(s'', z'' - 1, z1, 1 + (z2 - 1)) :|: s'' >= 0, s'' <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 2 + z'' }-> s1 :|: s1 >= 0, s1 <= 1, z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 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: {cond2,cond1}
Previous analysis results are:
  p: runtime: O(1) [1], size: O(n^1) [z']
  gr: runtime: O(n^1) [2 + z''], size: O(1) [1]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   cond1(z', z'', z1, z2) -{ 3 + z2 }-> cond2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 + z2 }-> cond2(s', z'', z1 - 1, 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 + z2 }-> cond1(s'', z'' - 1, z1, 1 + (z2 - 1)) :|: s'' >= 0, s'' <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 2 + z'' }-> s1 :|: s1 >= 0, s1 <= 1, z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 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: {cond2,cond1}
Previous analysis results are:
  p: runtime: O(1) [1], size: O(n^1) [z']
  gr: runtime: O(n^1) [2 + z''], size: O(1) [1]
  cond2: runtime: ?, size: O(1) [0]
  cond1: runtime: ?, size: O(1) [0]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 21 + 7*z'' + 2*z''*z2 + 11*z1 + 3*z1*z2 + 3*z2

Computed RUNTIME bound using KoAT for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 70 + 21*z'' + 4*z''*z2 + 22*z1 + 3*z1*z2 + 7*z2

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

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

   cond1(z', z'', z1, z2) -{ 3 + z2 }-> cond2(s, z'', 1 + (z1 - 1), 1 + (z2 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, z'', 1 + (z1 - 1), 0) :|: z2 = 0, z'' >= 0, z1 - 1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 + z2 }-> cond2(s', z'', z1 - 1, 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, z'' >= 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(1, z'', z1 - 1, 0) :|: z1 - 1 >= 0, z2 = 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 3 }-> cond2(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 + z2 }-> cond1(s'', z'' - 1, z1, 1 + (z2 - 1)) :|: s'' >= 0, s'' <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(1, z'' - 1, z1, 0) :|: z2 = 0, z1 >= 0, z'' - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 3 }-> cond1(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 0
   gr(z', z'') -{ 2 + z'' }-> s1 :|: s1 >= 0, s1 <= 1, z' - 1 >= 0, z'' - 1 >= 0
   gr(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 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:
  p: runtime: O(1) [1], size: O(n^1) [z']
  gr: runtime: O(n^1) [2 + z''], size: O(1) [1]
  cond2: runtime: O(n^2) [21 + 7*z'' + 2*z''*z2 + 11*z1 + 3*z1*z2 + 3*z2], size: O(1) [0]
  cond1: runtime: O(n^2) [70 + 21*z'' + 4*z''*z2 + 22*z1 + 3*z1*z2 + 7*z2], size: O(1) [0]

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

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

(34)
BOUNDS(1, n^2)

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

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

(36)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
   cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
   cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
   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: INNERMOST
----------------------------------------

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

(38)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
p :: 0':s -> 0':s
false :: true:false
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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

(39) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond1, cond2, gr

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
gr < cond2

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

(40)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
p :: 0':s -> 0':s
false :: true:false
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
gr < cond2

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)

Induction Base:
gr(gen_0':s4_0(0), gen_0':s4_0(0)) ->_R^Omega(1)
false

Induction Step:
gr(gen_0':s4_0(+(n6_0, 1)), gen_0':s4_0(+(n6_0, 1))) ->_R^Omega(1)
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) ->_IH
false

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

(42)
Complex Obligation (BEST)

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

(43)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
p :: 0':s -> 0':s
false :: true:false
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
gr, cond1, cond2

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
gr < cond2

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

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

(45)
BOUNDS(n^1, INF)

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

(46)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y, z) -> cond2(gr(y, z), x, y, z)
cond2(true, x, y, z) -> cond2(gr(y, z), x, p(y), z)
cond2(false, x, y, z) -> cond1(gr(x, z), p(x), y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2
gr :: 0':s -> 0':s -> true:false
p :: 0':s -> 0':s
false :: true:false
0' :: 0':s
s :: 0':s -> 0':s
hole_cond1:cond21_0 :: cond1:cond2
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
gr(gen_0':s4_0(n6_0), gen_0':s4_0(n6_0)) -> false, rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
cond2, cond1

They will be analysed ascendingly in the following order:
cond1 = cond2