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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) InliningProof [UPPER BOUND(ID), 943 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), 192 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 241 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 96 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 119 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 4124 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 1807 ms]
    (40) CpxRNTS
    (41) FinalProof [FINISHED, 0 ms]
    (42) BOUNDS(1, n^2)
(43) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(44) CpxTRS
    (45) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) typed CpxTrs
    (47) OrderProof [LOWER BOUND(ID), 0 ms]
    (48) typed CpxTrs
    (49) RewriteLemmaProof [LOWER BOUND(ID), 287 ms]
    (50) BEST
        (51) proven lower bound
            (52) LowerBoundPropagationProof [FINISHED, 0 ms]
            (53) BOUNDS(n^1, INF)
        (54) typed CpxTrs


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

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


The TRS R consists of the following rules:

   cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z)
   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z)
   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
   gr(0, x) -> false
   gr(s(x), 0) -> true
   gr(s(x), s(y)) -> gr(x, y)
   or(false, false) -> false
   or(true, x) -> true
   or(x, true) -> true
   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(x, 0), x, y, z) [1]
   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) [1]
   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) [1]
   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) [1]
   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   or(false, false) -> false [1]
   or(true, x) -> true [1]
   or(x, true) -> true [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(x, 0), x, y, z) [1]
   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) [1]
   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) [1]
   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) [1]
   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   or(false, false) -> false [1]
   or(true, x) -> true [1]
   or(x, true) -> true [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:cond3
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2:cond3
gr :: 0:s -> 0:s -> true:false
0 :: 0:s
or :: true:false -> true:false -> true:false
p :: 0:s -> 0:s
false :: true:false
cond3 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2:cond3
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
   cond3_4

(c) The following functions are completely defined:

   or_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:

   cond1(true, x, y, z) -> cond2(gr(x, 0), x, y, z) [1]
   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z) [1]
   cond2(false, x, y, z) -> cond3(gr(y, 0), x, y, z) [1]
   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z) [1]
   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z) [1]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   or(false, false) -> false [1]
   or(true, x) -> true [1]
   or(x, true) -> true [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:cond3
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2:cond3
gr :: 0:s -> 0:s -> true:false
0 :: 0:s
or :: true:false -> true:false -> true:false
p :: 0:s -> 0:s
false :: true:false
cond3 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2:cond3
s :: 0:s -> 0:s
const :: cond1:cond2:cond3

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, 0, y, z) -> cond2(false, 0, y, z) [2]
   cond1(true, s(x'), y, z) -> cond2(true, s(x'), y, z) [2]
   cond2(true, 0, 0, z) -> cond1(or(false, false), 0, 0, z) [4]
   cond2(true, 0, s(x2), 0) -> cond1(or(false, true), 0, s(x2), 0) [4]
   cond2(true, 0, s(x3), s(y'')) -> cond1(or(false, gr(x3, y'')), 0, s(x3), s(y'')) [4]
   cond2(true, s(x''), 0, 0) -> cond1(or(true, false), x'', 0, 0) [4]
   cond2(true, s(x''), s(x4), 0) -> cond1(or(true, true), x'', s(x4), 0) [4]
   cond2(true, s(x1), 0, s(y')) -> cond1(or(gr(x1, y'), false), x1, 0, s(y')) [4]
   cond2(true, s(x1), s(x5), s(y')) -> cond1(or(gr(x1, y'), gr(x5, y')), x1, s(x5), s(y')) [4]
   cond2(false, x, 0, z) -> cond3(false, x, 0, z) [2]
   cond2(false, x, s(x6), z) -> cond3(true, x, s(x6), z) [2]
   cond3(true, 0, 0, z) -> cond1(or(false, false), 0, 0, z) [4]
   cond3(true, 0, s(x9), 0) -> cond1(or(false, true), 0, x9, 0) [4]
   cond3(true, 0, s(x10), s(y2)) -> cond1(or(false, gr(x10, y2)), 0, x10, s(y2)) [4]
   cond3(true, s(x7), 0, 0) -> cond1(or(true, false), s(x7), 0, 0) [4]
   cond3(true, s(x7), s(x11), 0) -> cond1(or(true, true), s(x7), x11, 0) [4]
   cond3(true, s(x8), 0, s(y1)) -> cond1(or(gr(x8, y1), false), s(x8), 0, s(y1)) [4]
   cond3(true, s(x8), s(x12), s(y1)) -> cond1(or(gr(x8, y1), gr(x12, y1)), s(x8), x12, s(y1)) [4]
   cond3(false, 0, 0, z) -> cond1(or(false, false), 0, 0, z) [3]
   cond3(false, 0, s(x15), 0) -> cond1(or(false, true), 0, s(x15), 0) [3]
   cond3(false, 0, s(x16), s(y4)) -> cond1(or(false, gr(x16, y4)), 0, s(x16), s(y4)) [3]
   cond3(false, s(x13), 0, 0) -> cond1(or(true, false), s(x13), 0, 0) [3]
   cond3(false, s(x13), s(x17), 0) -> cond1(or(true, true), s(x13), s(x17), 0) [3]
   cond3(false, s(x14), 0, s(y3)) -> cond1(or(gr(x14, y3), false), s(x14), 0, s(y3)) [3]
   cond3(false, s(x14), s(x18), s(y3)) -> cond1(or(gr(x14, y3), gr(x18, y3)), s(x14), s(x18), s(y3)) [3]
   gr(0, x) -> false [1]
   gr(s(x), 0) -> true [1]
   gr(s(x), s(y)) -> gr(x, y) [1]
   or(false, false) -> false [1]
   or(true, x) -> true [1]
   or(x, true) -> true [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:cond3
true :: true:false
cond2 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2:cond3
gr :: 0:s -> 0:s -> true:false
0 :: 0:s
or :: true:false -> true:false -> true:false
p :: 0:s -> 0:s
false :: true:false
cond3 :: true:false -> 0:s -> 0:s -> 0:s -> cond1:cond2:cond3
s :: 0:s -> 0:s
const :: cond1:cond2:cond3

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:

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + x', y, z) :|: z1 = y, z >= 0, z'' = 1 + x', z2 = z, x' >= 0, y >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, y, z) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, x, 1 + x6, z) :|: z >= 0, z2 = z, x >= 0, x6 >= 0, z'' = x, z1 = 1 + x6, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, x, 0, z) :|: z1 = 0, z >= 0, z2 = z, x >= 0, z'' = x, z' = 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(x1, y'), gr(x5, y')), x1, 1 + x5, 1 + y') :|: x1 >= 0, x5 >= 0, z'' = 1 + x1, y' >= 0, z2 = 1 + y', z' = 1, z1 = 1 + x5
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(x1, y'), 0), x1, 0, 1 + y') :|: z1 = 0, x1 >= 0, z'' = 1 + x1, y' >= 0, z2 = 1 + y', z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(1, 1), x'', 1 + x4, 0) :|: x4 >= 0, z2 = 0, z1 = 1 + x4, z' = 1, z'' = 1 + x'', x'' >= 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(1, 0), x'', 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' = 1 + x'', x'' >= 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(x3, y'')), 0, 1 + x3, 1 + y'') :|: z'' = 0, z2 = 1 + y'', y'' >= 0, z' = 1, z1 = 1 + x3, x3 >= 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, 1), 0, 1 + x2, 0) :|: z'' = 0, z2 = 0, z' = 1, x2 >= 0, z1 = 1 + x2
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, 0), 0, 0, z) :|: z'' = 0, z1 = 0, z >= 0, z2 = z, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(x14, y3), gr(x18, y3)), 1 + x14, 1 + x18, 1 + y3) :|: z1 = 1 + x18, z2 = 1 + y3, y3 >= 0, x14 >= 0, x18 >= 0, z'' = 1 + x14, z' = 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(x14, y3), 0), 1 + x14, 0, 1 + y3) :|: z1 = 0, z2 = 1 + y3, y3 >= 0, x14 >= 0, z'' = 1 + x14, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(x8, y1), gr(x12, y1)), 1 + x8, x12, 1 + y1) :|: y1 >= 0, z2 = 1 + y1, x8 >= 0, z1 = 1 + x12, x12 >= 0, z'' = 1 + x8, z' = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(x8, y1), 0), 1 + x8, 0, 1 + y1) :|: y1 >= 0, z2 = 1 + y1, z1 = 0, x8 >= 0, z'' = 1 + x8, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(1, 1), 1 + x13, 1 + x17, 0) :|: x13 >= 0, x17 >= 0, z1 = 1 + x17, z2 = 0, z'' = 1 + x13, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(1, 1), 1 + x7, x11, 0) :|: z1 = 1 + x11, x7 >= 0, z2 = 0, z'' = 1 + x7, x11 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(1, 0), 1 + x13, 0, 0) :|: x13 >= 0, z1 = 0, z2 = 0, z'' = 1 + x13, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(1, 0), 1 + x7, 0, 0) :|: z1 = 0, x7 >= 0, z2 = 0, z'' = 1 + x7, z' = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(x10, y2)), 0, x10, 1 + y2) :|: z'' = 0, z2 = 1 + y2, z' = 1, z1 = 1 + x10, x10 >= 0, y2 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, gr(x16, y4)), 0, 1 + x16, 1 + y4) :|: z'' = 0, z1 = 1 + x16, z2 = 1 + y4, y4 >= 0, x16 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, 1), 0, x9, 0) :|: z'' = 0, z2 = 0, z1 = 1 + x9, z' = 1, x9 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, 1), 0, 1 + x15, 0) :|: z'' = 0, z1 = 1 + x15, z2 = 0, x15 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, 0), 0, 0, z) :|: z'' = 0, z1 = 0, z >= 0, z2 = z, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, 0), 0, 0, z) :|: z'' = 0, z1 = 0, z >= 0, z2 = z, 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
   or(z', z'') -{ 1 }-> 1 :|: x >= 0, z'' = x, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1
   or(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   p(z') -{ 1 }-> 0 :|: z' = 0


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

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

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

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + x', y, z) :|: z1 = y, z >= 0, z'' = 1 + x', z2 = z, x' >= 0, y >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, y, z) :|: z'' = 0, z1 = y, z >= 0, z2 = z, y >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, x, 1 + x6, z) :|: z >= 0, z2 = z, x >= 0, x6 >= 0, z'' = x, z1 = 1 + x6, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, x, 0, z) :|: z1 = 0, z >= 0, z2 = z, x >= 0, z'' = x, z' = 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(x1, y'), gr(x5, y')), x1, 1 + x5, 1 + y') :|: x1 >= 0, x5 >= 0, z'' = 1 + x1, y' >= 0, z2 = 1 + y', z' = 1, z1 = 1 + x5
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(x1, y'), 0), x1, 0, 1 + y') :|: z1 = 0, x1 >= 0, z'' = 1 + x1, y' >= 0, z2 = 1 + y', z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(x3, y'')), 0, 1 + x3, 1 + y'') :|: z'' = 0, z2 = 1 + y'', y'' >= 0, z' = 1, z1 = 1 + x3, x3 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, x'', 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' = 1 + x'', x'' >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, x'', 1 + x4, 0) :|: x4 >= 0, z2 = 0, z1 = 1 + x4, z' = 1, z'' = 1 + x'', x'' >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + x2, 0) :|: z'' = 0, z2 = 0, z' = 1, x2 >= 0, z1 = 1 + x2, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z) :|: z'' = 0, z1 = 0, z >= 0, z2 = z, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(x14, y3), gr(x18, y3)), 1 + x14, 1 + x18, 1 + y3) :|: z1 = 1 + x18, z2 = 1 + y3, y3 >= 0, x14 >= 0, x18 >= 0, z'' = 1 + x14, z' = 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(x14, y3), 0), 1 + x14, 0, 1 + y3) :|: z1 = 0, z2 = 1 + y3, y3 >= 0, x14 >= 0, z'' = 1 + x14, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(x8, y1), gr(x12, y1)), 1 + x8, x12, 1 + y1) :|: y1 >= 0, z2 = 1 + y1, x8 >= 0, z1 = 1 + x12, x12 >= 0, z'' = 1 + x8, z' = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(x8, y1), 0), 1 + x8, 0, 1 + y1) :|: y1 >= 0, z2 = 1 + y1, z1 = 0, x8 >= 0, z'' = 1 + x8, z' = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(x10, y2)), 0, x10, 1 + y2) :|: z'' = 0, z2 = 1 + y2, z' = 1, z1 = 1 + x10, x10 >= 0, y2 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, gr(x16, y4)), 0, 1 + x16, 1 + y4) :|: z'' = 0, z1 = 1 + x16, z2 = 1 + y4, y4 >= 0, x16 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, x9, 0) :|: z'' = 0, z2 = 0, z1 = 1 + x9, z' = 1, x9 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + x15, 0) :|: z'' = 0, z1 = 1 + x15, z2 = 0, x15 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + x13, 0, 0) :|: x13 >= 0, z1 = 0, z2 = 0, z'' = 1 + x13, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + x13, 1 + x17, 0) :|: x13 >= 0, x17 >= 0, z1 = 1 + x17, z2 = 0, z'' = 1 + x13, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + x7, x11, 0) :|: z1 = 1 + x11, x7 >= 0, z2 = 0, z'' = 1 + x7, x11 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + x7, 0, 0) :|: z1 = 0, x7 >= 0, z2 = 0, z'' = 1 + x7, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z) :|: z'' = 0, z1 = 0, z >= 0, z2 = z, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z) :|: z'' = 0, z1 = 0, z >= 0, z2 = z, z' = 0, 0 = 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
   or(z', z'') -{ 1 }-> 1 :|: x >= 0, z'' = x, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1
   or(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = 0
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   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:

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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:

   { p }
   { gr }
   { or }
   { cond1, cond2, cond3 }

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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}, {or}, {cond1,cond2,cond3}

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

(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:

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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}, {or}, {cond1,cond2,cond3}

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(19) 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'

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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}, {or}, {cond1,cond2,cond3}
Previous analysis results are:
  p: runtime: ?, size: O(n^1) [z']

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

(21) 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

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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}, {or}, {cond1,cond2,cond3}
Previous analysis results are:
  p: runtime: O(1) [1], size: O(n^1) [z']

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

(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:

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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}, {or}, {cond1,cond2,cond3}
Previous analysis results are:
  p: runtime: O(1) [1], size: O(n^1) [z']

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

(25) 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

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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}, {or}, {cond1,cond2,cond3}
Previous analysis results are:
  p: runtime: O(1) [1], size: O(n^1) [z']
  gr: runtime: ?, size: O(1) [1]

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

(27) 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''

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), z'' - 1, 0, 1 + (z2 - 1)) :|: z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), gr(z1 - 1, z2 - 1)), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(gr(z'' - 1, z2 - 1), 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, z1 - 1, 1 + (z2 - 1)) :|: z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 3 }-> cond1(or(0, gr(z1 - 1, z2 - 1)), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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: {or}, {cond1,cond2,cond3}
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) 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:

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(s'', 0), z'' - 1, 0, 1 + (z2 - 1)) :|: s'' >= 0, s'' <= 1, z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 6 + 2*z2 }-> cond1(or(s1, s2), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(0, s'), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(s4, 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 1, z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 6 + 2*z2 }-> cond1(or(s5, s6), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 4 + z2 }-> cond1(or(s8, 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s8 >= 0, s8 <= 1, z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 + 2*z2 }-> cond1(or(s9, s10), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(0, s3), 0, z1 - 1, 1 + (z2 - 1)) :|: s3 >= 0, s3 <= 1, z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 4 + z2 }-> cond1(or(0, s7), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s7 >= 0, s7 <= 1, z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 0
   gr(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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: {or}, {cond1,cond2,cond3}
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]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: or
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:

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(s'', 0), z'' - 1, 0, 1 + (z2 - 1)) :|: s'' >= 0, s'' <= 1, z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 6 + 2*z2 }-> cond1(or(s1, s2), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(0, s'), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(s4, 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 1, z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 6 + 2*z2 }-> cond1(or(s5, s6), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 4 + z2 }-> cond1(or(s8, 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s8 >= 0, s8 <= 1, z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 + 2*z2 }-> cond1(or(s9, s10), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(0, s3), 0, z1 - 1, 1 + (z2 - 1)) :|: s3 >= 0, s3 <= 1, z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 4 + z2 }-> cond1(or(0, s7), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s7 >= 0, s7 <= 1, z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 0
   gr(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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: {or}, {cond1,cond2,cond3}
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]
  or: runtime: ?, size: O(1) [1]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: or
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:

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(s'', 0), z'' - 1, 0, 1 + (z2 - 1)) :|: s'' >= 0, s'' <= 1, z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 6 + 2*z2 }-> cond1(or(s1, s2), z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(0, s'), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s' >= 0, s' <= 1, z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(s4, 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s4 >= 0, s4 <= 1, z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 6 + 2*z2 }-> cond1(or(s5, s6), 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 4 + z2 }-> cond1(or(s8, 0), 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s8 >= 0, s8 <= 1, z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 + 2*z2 }-> cond1(or(s9, s10), 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(or(0, s3), 0, z1 - 1, 1 + (z2 - 1)) :|: s3 >= 0, s3 <= 1, z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 4 + z2 }-> cond1(or(0, s7), 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s7 >= 0, s7 <= 1, z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 0
   gr(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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: {cond1,cond2,cond3}
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]
  or: runtime: O(1) [1], size: O(1) [1]

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

(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:

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s11, 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s11 >= 0, s11 <= 1, s' >= 0, s' <= 1, z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s12, z'' - 1, 0, 1 + (z2 - 1)) :|: s12 >= 0, s12 <= 1, s'' >= 0, s'' <= 1, z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 7 + 2*z2 }-> cond1(s13, z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s13 >= 0, s13 <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s14, 0, z1 - 1, 1 + (z2 - 1)) :|: s14 >= 0, s14 <= 1, s3 >= 0, s3 <= 1, z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s15, 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s15 >= 0, s15 <= 1, s4 >= 0, s4 <= 1, z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 7 + 2*z2 }-> cond1(s16, 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: s16 >= 0, s16 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(s17, 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s17 >= 0, s17 <= 1, s7 >= 0, s7 <= 1, z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(s18, 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s18 >= 0, s18 <= 1, s8 >= 0, s8 <= 1, z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 6 + 2*z2 }-> cond1(s19, 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: s19 >= 0, s19 <= 1, s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 0
   gr(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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: {cond1,cond2,cond3}
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]
  or: runtime: O(1) [1], size: O(1) [1]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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: cond3
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s11, 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s11 >= 0, s11 <= 1, s' >= 0, s' <= 1, z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s12, z'' - 1, 0, 1 + (z2 - 1)) :|: s12 >= 0, s12 <= 1, s'' >= 0, s'' <= 1, z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 7 + 2*z2 }-> cond1(s13, z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s13 >= 0, s13 <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s14, 0, z1 - 1, 1 + (z2 - 1)) :|: s14 >= 0, s14 <= 1, s3 >= 0, s3 <= 1, z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s15, 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s15 >= 0, s15 <= 1, s4 >= 0, s4 <= 1, z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 7 + 2*z2 }-> cond1(s16, 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: s16 >= 0, s16 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(s17, 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s17 >= 0, s17 <= 1, s7 >= 0, s7 <= 1, z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(s18, 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s18 >= 0, s18 <= 1, s8 >= 0, s8 <= 1, z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 6 + 2*z2 }-> cond1(s19, 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: s19 >= 0, s19 <= 1, s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 0
   gr(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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: {cond1,cond2,cond3}
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]
  or: runtime: O(1) [1], size: O(1) [1]
  cond1: runtime: ?, size: O(1) [0]
  cond2: runtime: ?, size: O(1) [0]
  cond3: runtime: ?, size: O(1) [0]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 14 + 9*z'' + 2*z''*z2 + 10*z1 + z1*z2 + 2*z2

Computed RUNTIME bound using KoAT for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 199 + 72*z'' + 8*z''*z2 + 80*z1 + 4*z1*z2 + 18*z2

Computed RUNTIME bound using KoAT for: cond3
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 227 + 72*z'' + 8*z''*z2 + 80*z1 + 4*z1*z2 + 22*z2

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

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

   cond1(z', z'', z1, z2) -{ 2 }-> cond2(1, 1 + (z'' - 1), z1, z2) :|: z2 >= 0, z'' - 1 >= 0, z1 >= 0, z' = 1
   cond1(z', z'', z1, z2) -{ 2 }-> cond2(0, 0, z1, z2) :|: z'' = 0, z2 >= 0, z1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(1, z'', 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 2 }-> cond3(0, z'', 0, z2) :|: z1 = 0, z2 >= 0, z'' >= 0, z' = 0
   cond2(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s11, 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s11 >= 0, s11 <= 1, s' >= 0, s' <= 1, z'' = 0, z2 - 1 >= 0, z' = 1, z1 - 1 >= 0
   cond2(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s12, z'' - 1, 0, 1 + (z2 - 1)) :|: s12 >= 0, s12 <= 1, s'' >= 0, s'' <= 1, z1 = 0, z'' - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 7 + 2*z2 }-> cond1(s13, z'' - 1, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s13 >= 0, s13 <= 1, s1 >= 0, s1 <= 1, s2 >= 0, s2 <= 1, z'' - 1 >= 0, z1 - 1 >= 0, z2 - 1 >= 0, z' = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 0, 0) :|: z1 = 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 0 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(1, z'' - 1, 1 + (z1 - 1), 0) :|: z1 - 1 >= 0, z2 = 0, z' = 1, z'' - 1 >= 0, x >= 0, 1 = x, 1 = 1
   cond2(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s14, 0, z1 - 1, 1 + (z2 - 1)) :|: s14 >= 0, s14 <= 1, s3 >= 0, s3 <= 1, z'' = 0, z' = 1, z1 - 1 >= 0, z2 - 1 >= 0
   cond3(z', z'', z1, z2) -{ 6 + z2 }-> cond1(s15, 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s15 >= 0, s15 <= 1, s4 >= 0, s4 <= 1, z2 - 1 >= 0, z1 = 0, z'' - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 7 + 2*z2 }-> cond1(s16, 1 + (z'' - 1), z1 - 1, 1 + (z2 - 1)) :|: s16 >= 0, s16 <= 1, s5 >= 0, s5 <= 1, s6 >= 0, s6 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 1
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(s17, 0, 1 + (z1 - 1), 1 + (z2 - 1)) :|: s17 >= 0, s17 <= 1, s7 >= 0, s7 <= 1, z'' = 0, z2 - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 + z2 }-> cond1(s18, 1 + (z'' - 1), 0, 1 + (z2 - 1)) :|: s18 >= 0, s18 <= 1, s8 >= 0, s8 <= 1, z1 = 0, z2 - 1 >= 0, z'' - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 6 + 2*z2 }-> cond1(s19, 1 + (z'' - 1), 1 + (z1 - 1), 1 + (z2 - 1)) :|: s19 >= 0, s19 <= 1, s9 >= 0, s9 <= 1, s10 >= 0, s10 <= 1, z2 - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 0, z1 - 1, 0) :|: z'' = 0, z2 = 0, z' = 1, z1 - 1 >= 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 0, 1 + (z1 - 1), 0) :|: z'' = 0, z2 = 0, z1 - 1 >= 0, z' = 0, 0 = x, x >= 0, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z1 = 0, z'' - 1 >= 0, z2 = 0, z' = 1, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 0, 0) :|: z'' - 1 >= 0, z1 = 0, z2 = 0, z' = 0, x >= 0, 0 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(1, 1 + (z'' - 1), z1 - 1, 0) :|: z'' - 1 >= 0, z2 = 0, z1 - 1 >= 0, z' = 1, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(1, 1 + (z'' - 1), 1 + (z1 - 1), 0) :|: z'' - 1 >= 0, z1 - 1 >= 0, z2 = 0, z' = 0, x >= 0, 1 = x, 1 = 1
   cond3(z', z'', z1, z2) -{ 5 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 1, 0 = 0
   cond3(z', z'', z1, z2) -{ 4 }-> cond1(0, 0, 0, z2) :|: z'' = 0, z1 = 0, z2 >= 0, z' = 0, 0 = 0
   gr(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 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
   or(z', z'') -{ 1 }-> 1 :|: z'' >= 0, z' = 1
   or(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1
   or(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]
  or: runtime: O(1) [1], size: O(1) [1]
  cond1: runtime: O(n^2) [14 + 9*z'' + 2*z''*z2 + 10*z1 + z1*z2 + 2*z2], size: O(1) [0]
  cond2: runtime: O(n^2) [199 + 72*z'' + 8*z''*z2 + 80*z1 + 4*z1*z2 + 18*z2], size: O(1) [0]
  cond3: runtime: O(n^2) [227 + 72*z'' + 8*z''*z2 + 80*z1 + 4*z1*z2 + 22*z2], size: O(1) [0]

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

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

(42)
BOUNDS(1, n^2)

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

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

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


The TRS R consists of the following rules:

   cond1(true, x, y, z) -> cond2(gr(x, 0'), x, y, z)
   cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
   cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
   cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
   cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
   gr(0', x) -> false
   gr(s(x), 0') -> true
   gr(s(x), s(y)) -> gr(x, y)
   or(false, false) -> false
   or(true, x) -> true
   or(x, true) -> true
   p(0') -> 0'
   p(s(x)) -> x

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

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

(46)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y, z) -> cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
or(false, false) -> false
or(true, x) -> true
or(x, true) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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

(47) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
cond1, cond2, gr, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
cond2 = cond3
gr < cond3

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

(48)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y, z) -> cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
or(false, false) -> false
or(true, x) -> true
or(x, true) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
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, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
cond2 = cond3
gr < cond3

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

(49) 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).
----------------------------------------

(50)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
cond1(true, x, y, z) -> cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
or(false, false) -> false
or(true, x) -> true
or(x, true) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
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, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
gr < cond1
cond1 = cond3
gr < cond2
cond2 = cond3
gr < cond3

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

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

(53)
BOUNDS(n^1, INF)

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

(54)
Obligation:
Innermost TRS:
Rules:
cond1(true, x, y, z) -> cond2(gr(x, 0'), x, y, z)
cond2(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), p(x), y, z)
cond2(false, x, y, z) -> cond3(gr(y, 0'), x, y, z)
cond3(true, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, p(y), z)
cond3(false, x, y, z) -> cond1(or(gr(x, z), gr(y, z)), x, y, z)
gr(0', x) -> false
gr(s(x), 0') -> true
gr(s(x), s(y)) -> gr(x, y)
or(false, false) -> false
or(true, x) -> true
or(x, true) -> true
p(0') -> 0'
p(s(x)) -> x

Types:
cond1 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
false :: true:false
cond3 :: true:false -> 0':s -> 0':s -> 0':s -> cond1:cond2:cond3
s :: 0':s -> 0':s
hole_cond1:cond2:cond31_0 :: cond1:cond2:cond3
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, cond3

They will be analysed ascendingly in the following order:
cond1 = cond2
cond1 = cond3
cond2 = cond3