WORST_CASE(Omega(n^1),O(n^1))
proof of input_mzt0szb2Kx.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) InliningProof [UPPER BOUND(ID), 1058 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), 3843 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 1642 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 342 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 145 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
    (40) CpxRNTS
    (41) FinalProof [FINISHED, 0 ms]
    (42) BOUNDS(1, n^1)
(43) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(44) CdtProblem
    (45) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CpxRelTRS
    (47) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CpxRelTRS
    (49) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) typed CpxTrs
    (51) OrderProof [LOWER BOUND(ID), 0 ms]
    (52) typed CpxTrs
    (53) RewriteLemmaProof [LOWER BOUND(ID), 348 ms]
    (54) typed CpxTrs
    (55) RewriteLemmaProof [LOWER BOUND(ID), 146 ms]
    (56) BEST
        (57) proven lower bound
            (58) LowerBoundPropagationProof [FINISHED, 0 ms]
            (59) BOUNDS(n^1, INF)
        (60) typed CpxTrs


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

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


The TRS R consists of the following rules:

   cond1(true, x, y) -> cond2(gr(x, 0), x, y)
   cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y)
   cond2(false, x, y) -> cond3(gr(y, 0), x, y)
   cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y))
   cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y)
   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: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

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


The TRS R consists of the following rules:

   cond1(true, x, y) -> cond2(gr(x, 0), x, y) [1]
   cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1]
   cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1]
   cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1]
   cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) [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) -> cond2(gr(x, 0), x, y) [1]
   cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1]
   cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1]
   cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1]
   cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) [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 -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 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 -> 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_3
   cond2_3
   cond3_3

(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) -> cond2(gr(x, 0), x, y) [1]
   cond2(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), p(x), y) [1]
   cond2(false, x, y) -> cond3(gr(y, 0), x, y) [1]
   cond3(true, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, p(y)) [1]
   cond3(false, x, y) -> cond1(or(gr(x, 0), gr(y, 0)), x, y) [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 -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 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 -> 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) -> cond2(false, 0, y) [2]
   cond1(true, s(x'), y) -> cond2(true, s(x'), y) [2]
   cond2(true, 0, 0) -> cond1(or(false, false), 0, 0) [4]
   cond2(true, 0, s(x1)) -> cond1(or(false, true), 0, s(x1)) [4]
   cond2(true, s(x''), 0) -> cond1(or(true, false), x'', 0) [4]
   cond2(true, s(x''), s(x2)) -> cond1(or(true, true), x'', s(x2)) [4]
   cond2(false, x, 0) -> cond3(false, x, 0) [2]
   cond2(false, x, s(x3)) -> cond3(true, x, s(x3)) [2]
   cond3(true, 0, 0) -> cond1(or(false, false), 0, 0) [4]
   cond3(true, 0, s(x5)) -> cond1(or(false, true), 0, x5) [4]
   cond3(true, s(x4), 0) -> cond1(or(true, false), s(x4), 0) [4]
   cond3(true, s(x4), s(x6)) -> cond1(or(true, true), s(x4), x6) [4]
   cond3(false, 0, 0) -> cond1(or(false, false), 0, 0) [3]
   cond3(false, 0, s(x8)) -> cond1(or(false, true), 0, s(x8)) [3]
   cond3(false, s(x7), 0) -> cond1(or(true, false), s(x7), 0) [3]
   cond3(false, s(x7), s(x9)) -> cond1(or(true, true), s(x7), s(x9)) [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 -> cond1:cond2:cond3
true :: true:false
cond2 :: true:false -> 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 -> 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', z'') -{ 2 }-> cond2(1, 1 + x', y) :|: z' = 1 + x', z'' = y, z = 1, x' >= 0, y >= 0
   cond1(z, z', z'') -{ 2 }-> cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
   cond2(z, z', z'') -{ 2 }-> cond3(1, x, 1 + x3) :|: z' = x, z'' = 1 + x3, x >= 0, z = 0, x3 >= 0
   cond2(z, z', z'') -{ 2 }-> cond3(0, x, 0) :|: z'' = 0, z' = x, x >= 0, z = 0
   cond2(z, z', z'') -{ 4 }-> cond1(or(1, 1), x'', 1 + x2) :|: z' = 1 + x'', z = 1, z'' = 1 + x2, x'' >= 0, x2 >= 0
   cond2(z, z', z'') -{ 4 }-> cond1(or(1, 0), x'', 0) :|: z'' = 0, z' = 1 + x'', z = 1, x'' >= 0
   cond2(z, z', z'') -{ 4 }-> cond1(or(0, 1), 0, 1 + x1) :|: x1 >= 0, z = 1, z'' = 1 + x1, z' = 0
   cond2(z, z', z'') -{ 4 }-> cond1(or(0, 0), 0, 0) :|: z'' = 0, z = 1, z' = 0
   cond3(z, z', z'') -{ 4 }-> cond1(or(1, 1), 1 + x4, x6) :|: x4 >= 0, z = 1, z' = 1 + x4, x6 >= 0, z'' = 1 + x6
   cond3(z, z', z'') -{ 3 }-> cond1(or(1, 1), 1 + x7, 1 + x9) :|: z' = 1 + x7, x7 >= 0, z = 0, x9 >= 0, z'' = 1 + x9
   cond3(z, z', z'') -{ 4 }-> cond1(or(1, 0), 1 + x4, 0) :|: z'' = 0, x4 >= 0, z = 1, z' = 1 + x4
   cond3(z, z', z'') -{ 3 }-> cond1(or(1, 0), 1 + x7, 0) :|: z' = 1 + x7, z'' = 0, x7 >= 0, z = 0
   cond3(z, z', z'') -{ 4 }-> cond1(or(0, 1), 0, x5) :|: x5 >= 0, z = 1, z'' = 1 + x5, z' = 0
   cond3(z, z', z'') -{ 3 }-> cond1(or(0, 1), 0, 1 + x8) :|: x8 >= 0, z'' = 1 + x8, z = 0, z' = 0
   cond3(z, z', z'') -{ 4 }-> cond1(or(0, 0), 0, 0) :|: z'' = 0, z = 1, z' = 0
   cond3(z, z', z'') -{ 3 }-> cond1(or(0, 0), 0, 0) :|: z'' = 0, z = 0, z' = 0
   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z' = x, z = 1, x >= 0
   or(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1, z = x
   or(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0


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

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

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

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

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

   cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + x', y) :|: z' = 1 + x', z'' = y, z = 1, x' >= 0, y >= 0
   cond1(z, z', z'') -{ 2 }-> cond2(0, 0, y) :|: z'' = y, z = 1, y >= 0, z' = 0
   cond2(z, z', z'') -{ 2 }-> cond3(1, x, 1 + x3) :|: z' = x, z'' = 1 + x3, x >= 0, z = 0, x3 >= 0
   cond2(z, z', z'') -{ 2 }-> cond3(0, x, 0) :|: z'' = 0, z' = x, x >= 0, z = 0
   cond2(z, z', z'') -{ 5 }-> cond1(1, x'', 0) :|: z'' = 0, z' = 1 + x'', z = 1, x'' >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 5 }-> cond1(1, x'', 1 + x2) :|: z' = 1 + x'', z = 1, z'' = 1 + x2, x'' >= 0, x2 >= 0, 1 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 5 }-> cond1(1, 0, 1 + x1) :|: x1 >= 0, z = 1, z'' = 1 + x1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 5 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 5 }-> cond1(1, 0, x5) :|: x5 >= 0, z = 1, z'' = 1 + x5, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 1 + x8) :|: x8 >= 0, z'' = 1 + x8, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 5 }-> cond1(1, 1 + x4, x6) :|: x4 >= 0, z = 1, z' = 1 + x4, x6 >= 0, z'' = 1 + x6, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 5 }-> cond1(1, 1 + x4, 0) :|: z'' = 0, x4 >= 0, z = 1, z' = 1 + x4, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + x7, 0) :|: z' = 1 + x7, z'' = 0, x7 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + x7, 1 + x9) :|: z' = 1 + x7, x7 >= 0, z = 0, x9 >= 0, z'' = 1 + x9, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 5 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 0, z' = 0, 0 = 0
   gr(z, z') -{ 1 }-> gr(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   gr(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + x, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z' = x, z = 1, x >= 0
   or(z, z') -{ 1 }-> 1 :|: x >= 0, z' = 1, z = x
   or(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0


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

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

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

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

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

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

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

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

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

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

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

(19) 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

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

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

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

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: cond2
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 48 + 7*z' + 9*z''

Computed RUNTIME bound using CoFloCo for: cond1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 50 + 7*z' + 9*z''

Computed RUNTIME bound using CoFloCo for: cond3
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 55 + 7*z' + 9*z''

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

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

   cond1(z, z', z'') -{ 2 }-> cond2(1, 1 + (z' - 1), z'') :|: z = 1, z' - 1 >= 0, z'' >= 0
   cond1(z, z', z'') -{ 2 }-> cond2(0, 0, z'') :|: z = 1, z'' >= 0, z' = 0
   cond2(z, z', z'') -{ 2 }-> cond3(1, z', 1 + (z'' - 1)) :|: z' >= 0, z = 0, z'' - 1 >= 0
   cond2(z, z', z'') -{ 2 }-> cond3(0, z', 0) :|: z'' = 0, z' >= 0, z = 0
   cond2(z, z', z'') -{ 5 }-> cond1(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 5 }-> cond1(1, z' - 1, 0) :|: z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 5 }-> cond1(1, z' - 1, 1 + (z'' - 1)) :|: z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 5 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 5 }-> cond1(1, 0, z'' - 1) :|: z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 4 }-> cond1(1, 0, 1 + (z'' - 1)) :|: z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 5 }-> cond1(1, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 5 }-> cond1(1, 1 + (z' - 1), z'' - 1) :|: z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 4 }-> cond1(1, 1 + (z' - 1), 1 + (z'' - 1)) :|: z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 5 }-> cond1(0, 0, 0) :|: z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 4 }-> cond1(0, 0, 0) :|: z'' = 0, z = 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 - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   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}
Previous analysis results are:
  cond2: runtime: O(n^1) [48 + 7*z' + 9*z''], size: O(1) [0]
  cond1: runtime: O(n^1) [50 + 7*z' + 9*z''], size: O(1) [0]
  cond3: runtime: O(n^1) [55 + 7*z' + 9*z''], size: O(1) [0]

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

(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', z'') -{ 50 + 9*z'' }-> s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
   cond1(z, z', z'') -{ 50 + 7*z' + 9*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
   cond2(z, z', z'') -{ 57 + 7*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
   cond2(z, z', z'') -{ 57 + 7*z' + 9*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   cond2(z, z', z'') -{ 55 }-> s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond2(z, z', z'') -{ 55 + 9*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 48 + 7*z' }-> s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 48 + 7*z' + 9*z'' }-> s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 }-> s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 54 + 9*z'' }-> s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 54 + 7*z' }-> s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 + 7*z' + 9*z'' }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 55 }-> s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 46 + 9*z'' }-> s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 55 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 46 + 7*z' + 9*z'' }-> s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   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}
Previous analysis results are:
  cond2: runtime: O(n^1) [48 + 7*z' + 9*z''], size: O(1) [0]
  cond1: runtime: O(n^1) [50 + 7*z' + 9*z''], size: O(1) [0]
  cond3: runtime: O(n^1) [55 + 7*z' + 9*z''], size: O(1) [0]

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

(25) 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

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

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

   cond1(z, z', z'') -{ 50 + 9*z'' }-> s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
   cond1(z, z', z'') -{ 50 + 7*z' + 9*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
   cond2(z, z', z'') -{ 57 + 7*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
   cond2(z, z', z'') -{ 57 + 7*z' + 9*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   cond2(z, z', z'') -{ 55 }-> s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond2(z, z', z'') -{ 55 + 9*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 48 + 7*z' }-> s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 48 + 7*z' + 9*z'' }-> s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 }-> s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 54 + 9*z'' }-> s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 54 + 7*z' }-> s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 + 7*z' + 9*z'' }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 55 }-> s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 46 + 9*z'' }-> s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 55 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 46 + 7*z' + 9*z'' }-> s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   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}
Previous analysis results are:
  cond2: runtime: O(n^1) [48 + 7*z' + 9*z''], size: O(1) [0]
  cond1: runtime: O(n^1) [50 + 7*z' + 9*z''], size: O(1) [0]
  cond3: runtime: O(n^1) [55 + 7*z' + 9*z''], size: O(1) [0]
  p: runtime: ?, size: O(n^1) [z]

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

(27) 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

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

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

   cond1(z, z', z'') -{ 50 + 9*z'' }-> s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
   cond1(z, z', z'') -{ 50 + 7*z' + 9*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
   cond2(z, z', z'') -{ 57 + 7*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
   cond2(z, z', z'') -{ 57 + 7*z' + 9*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   cond2(z, z', z'') -{ 55 }-> s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond2(z, z', z'') -{ 55 + 9*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 48 + 7*z' }-> s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 48 + 7*z' + 9*z'' }-> s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 }-> s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 54 + 9*z'' }-> s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 54 + 7*z' }-> s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 + 7*z' + 9*z'' }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 55 }-> s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 46 + 9*z'' }-> s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 55 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 46 + 7*z' + 9*z'' }-> s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   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}
Previous analysis results are:
  cond2: runtime: O(n^1) [48 + 7*z' + 9*z''], size: O(1) [0]
  cond1: runtime: O(n^1) [50 + 7*z' + 9*z''], size: O(1) [0]
  cond3: runtime: O(n^1) [55 + 7*z' + 9*z''], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(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', z'') -{ 50 + 9*z'' }-> s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
   cond1(z, z', z'') -{ 50 + 7*z' + 9*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
   cond2(z, z', z'') -{ 57 + 7*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
   cond2(z, z', z'') -{ 57 + 7*z' + 9*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   cond2(z, z', z'') -{ 55 }-> s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond2(z, z', z'') -{ 55 + 9*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 48 + 7*z' }-> s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 48 + 7*z' + 9*z'' }-> s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 }-> s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 54 + 9*z'' }-> s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 54 + 7*z' }-> s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 + 7*z' + 9*z'' }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 55 }-> s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 46 + 9*z'' }-> s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 55 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 46 + 7*z' + 9*z'' }-> s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   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}
Previous analysis results are:
  cond2: runtime: O(n^1) [48 + 7*z' + 9*z''], size: O(1) [0]
  cond1: runtime: O(n^1) [50 + 7*z' + 9*z''], size: O(1) [0]
  cond3: runtime: O(n^1) [55 + 7*z' + 9*z''], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]

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

(31) 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

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

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

   cond1(z, z', z'') -{ 50 + 9*z'' }-> s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
   cond1(z, z', z'') -{ 50 + 7*z' + 9*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
   cond2(z, z', z'') -{ 57 + 7*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
   cond2(z, z', z'') -{ 57 + 7*z' + 9*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   cond2(z, z', z'') -{ 55 }-> s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond2(z, z', z'') -{ 55 + 9*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 48 + 7*z' }-> s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 48 + 7*z' + 9*z'' }-> s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 }-> s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 54 + 9*z'' }-> s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 54 + 7*z' }-> s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 + 7*z' + 9*z'' }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 55 }-> s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 46 + 9*z'' }-> s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 55 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 46 + 7*z' + 9*z'' }-> s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   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}
Previous analysis results are:
  cond2: runtime: O(n^1) [48 + 7*z' + 9*z''], size: O(1) [0]
  cond1: runtime: O(n^1) [50 + 7*z' + 9*z''], size: O(1) [0]
  cond3: runtime: O(n^1) [55 + 7*z' + 9*z''], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: ?, size: O(1) [1]

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

(33) 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'

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

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

   cond1(z, z', z'') -{ 50 + 9*z'' }-> s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
   cond1(z, z', z'') -{ 50 + 7*z' + 9*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
   cond2(z, z', z'') -{ 57 + 7*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
   cond2(z, z', z'') -{ 57 + 7*z' + 9*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   cond2(z, z', z'') -{ 55 }-> s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond2(z, z', z'') -{ 55 + 9*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 48 + 7*z' }-> s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 48 + 7*z' + 9*z'' }-> s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 }-> s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 54 + 9*z'' }-> s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 54 + 7*z' }-> s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 + 7*z' + 9*z'' }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 55 }-> s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 46 + 9*z'' }-> s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 55 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 46 + 7*z' + 9*z'' }-> s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   gr(z, z') -{ 1 }-> gr(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   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}
Previous analysis results are:
  cond2: runtime: O(n^1) [48 + 7*z' + 9*z''], size: O(1) [0]
  cond1: runtime: O(n^1) [50 + 7*z' + 9*z''], size: O(1) [0]
  cond3: runtime: O(n^1) [55 + 7*z' + 9*z''], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: O(n^1) [2 + z'], 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', z'') -{ 50 + 9*z'' }-> s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
   cond1(z, z', z'') -{ 50 + 7*z' + 9*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
   cond2(z, z', z'') -{ 57 + 7*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
   cond2(z, z', z'') -{ 57 + 7*z' + 9*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   cond2(z, z', z'') -{ 55 }-> s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond2(z, z', z'') -{ 55 + 9*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 48 + 7*z' }-> s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 48 + 7*z' + 9*z'' }-> s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 }-> s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 54 + 9*z'' }-> s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 54 + 7*z' }-> s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 + 7*z' + 9*z'' }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 55 }-> s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 46 + 9*z'' }-> s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 55 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 46 + 7*z' + 9*z'' }-> s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   gr(z, z') -{ 2 + z' }-> s14 :|: s14 >= 0, s14 <= 1, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   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}
Previous analysis results are:
  cond2: runtime: O(n^1) [48 + 7*z' + 9*z''], size: O(1) [0]
  cond1: runtime: O(n^1) [50 + 7*z' + 9*z''], size: O(1) [0]
  cond3: runtime: O(n^1) [55 + 7*z' + 9*z''], size: O(1) [0]
  p: runtime: O(1) [1], size: O(n^1) [z]
  gr: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(37) 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

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

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

   cond1(z, z', z'') -{ 50 + 9*z'' }-> s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
   cond1(z, z', z'') -{ 50 + 7*z' + 9*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
   cond2(z, z', z'') -{ 57 + 7*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
   cond2(z, z', z'') -{ 57 + 7*z' + 9*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   cond2(z, z', z'') -{ 55 }-> s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond2(z, z', z'') -{ 55 + 9*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 48 + 7*z' }-> s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 48 + 7*z' + 9*z'' }-> s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 }-> s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 54 + 9*z'' }-> s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 54 + 7*z' }-> s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 + 7*z' + 9*z'' }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 55 }-> s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 46 + 9*z'' }-> s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 55 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 46 + 7*z' + 9*z'' }-> s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   gr(z, z') -{ 2 + z' }-> s14 :|: s14 >= 0, s14 <= 1, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   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}
Previous analysis results are:
  cond2: runtime: O(n^1) [48 + 7*z' + 9*z''], size: O(1) [0]
  cond1: runtime: O(n^1) [50 + 7*z' + 9*z''], size: O(1) [0]
  cond3: runtime: O(n^1) [55 + 7*z' + 9*z''], size: O(1) [0]
  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]

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

(39) 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

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

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

   cond1(z, z', z'') -{ 50 + 9*z'' }-> s :|: s >= 0, s <= 0, z = 1, z'' >= 0, z' = 0
   cond1(z, z', z'') -{ 50 + 7*z' + 9*z'' }-> s' :|: s' >= 0, s' <= 0, z = 1, z' - 1 >= 0, z'' >= 0
   cond2(z, z', z'') -{ 57 + 7*z' }-> s'' :|: s'' >= 0, s'' <= 0, z'' = 0, z' >= 0, z = 0
   cond2(z, z', z'') -{ 57 + 7*z' + 9*z'' }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z = 0, z'' - 1 >= 0
   cond2(z, z', z'') -{ 55 }-> s2 :|: s2 >= 0, s2 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond2(z, z', z'') -{ 55 + 9*z'' }-> s3 :|: s3 >= 0, s3 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond2(z, z', z'') -{ 48 + 7*z' }-> s4 :|: s4 >= 0, s4 <= 0, z'' = 0, z = 1, z' - 1 >= 0, 0 = x, 1 = 1, x >= 0
   cond2(z, z', z'') -{ 48 + 7*z' + 9*z'' }-> s5 :|: s5 >= 0, s5 <= 0, z = 1, z' - 1 >= 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 }-> s10 :|: s10 >= 0, s10 <= 0, z'' = 0, z = 0, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 54 + 9*z'' }-> s11 :|: s11 >= 0, s11 <= 0, z'' - 1 >= 0, z = 0, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 54 + 7*z' }-> s12 :|: s12 >= 0, s12 <= 0, z'' = 0, z' - 1 >= 0, z = 0, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 54 + 7*z' + 9*z'' }-> s13 :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z = 0, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 55 }-> s6 :|: s6 >= 0, s6 <= 0, z'' = 0, z = 1, z' = 0, 0 = 0
   cond3(z, z', z'') -{ 46 + 9*z'' }-> s7 :|: s7 >= 0, s7 <= 0, z'' - 1 >= 0, z = 1, z' = 0, x >= 0, 1 = 1, 0 = x
   cond3(z, z', z'') -{ 55 + 7*z' }-> s8 :|: s8 >= 0, s8 <= 0, z'' = 0, z' - 1 >= 0, z = 1, 0 = x, 1 = 1, x >= 0
   cond3(z, z', z'') -{ 46 + 7*z' + 9*z'' }-> s9 :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z = 1, z'' - 1 >= 0, 1 = x, 1 = 1, x >= 0
   gr(z, z') -{ 2 + z' }-> s14 :|: s14 >= 0, s14 <= 1, z - 1 >= 0, z' - 1 >= 0
   gr(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   gr(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   or(z, z') -{ 1 }-> 1 :|: z = 1, z' >= 0
   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:
  cond2: runtime: O(n^1) [48 + 7*z' + 9*z''], size: O(1) [0]
  cond1: runtime: O(n^1) [50 + 7*z' + 9*z''], size: O(1) [0]
  cond3: runtime: O(n^1) [55 + 7*z' + 9*z''], size: O(1) [0]
  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]

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

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

(42)
BOUNDS(1, n^1)

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

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond1(true, z0, z1) -> cond2(gr(z0, 0), z0, z1)
   cond2(true, z0, z1) -> cond1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1)
   cond2(false, z0, z1) -> cond3(gr(z1, 0), z0, z1)
   cond3(true, z0, z1) -> cond1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1))
   cond3(false, z0, z1) -> cond1(or(gr(z0, 0), gr(z1, 0)), z0, z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   or(false, false) -> false
   or(true, z0) -> true
   or(z0, true) -> true
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, 0), z0, z1), GR(z0, 0))
   COND2(true, z0, z1) -> c1(COND1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1), OR(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND2(true, z0, z1) -> c2(COND1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1), OR(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   COND2(true, z0, z1) -> c3(COND1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1), P(z0))
   COND2(false, z0, z1) -> c4(COND3(gr(z1, 0), z0, z1), GR(z1, 0))
   COND3(true, z0, z1) -> c5(COND1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1)), OR(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND3(true, z0, z1) -> c6(COND1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1)), OR(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   COND3(true, z0, z1) -> c7(COND1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1)), P(z1))
   COND3(false, z0, z1) -> c8(COND1(or(gr(z0, 0), gr(z1, 0)), z0, z1), OR(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND3(false, z0, z1) -> c9(COND1(or(gr(z0, 0), gr(z1, 0)), z0, z1), OR(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   GR(0, z0) -> c10
   GR(s(z0), 0) -> c11
   GR(s(z0), s(z1)) -> c12(GR(z0, z1))
   OR(false, false) -> c13
   OR(true, z0) -> c14
   OR(z0, true) -> c15
   P(0) -> c16
   P(s(z0)) -> c17
S tuples:
   COND1(true, z0, z1) -> c(COND2(gr(z0, 0), z0, z1), GR(z0, 0))
   COND2(true, z0, z1) -> c1(COND1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1), OR(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND2(true, z0, z1) -> c2(COND1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1), OR(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   COND2(true, z0, z1) -> c3(COND1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1), P(z0))
   COND2(false, z0, z1) -> c4(COND3(gr(z1, 0), z0, z1), GR(z1, 0))
   COND3(true, z0, z1) -> c5(COND1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1)), OR(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND3(true, z0, z1) -> c6(COND1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1)), OR(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   COND3(true, z0, z1) -> c7(COND1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1)), P(z1))
   COND3(false, z0, z1) -> c8(COND1(or(gr(z0, 0), gr(z1, 0)), z0, z1), OR(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND3(false, z0, z1) -> c9(COND1(or(gr(z0, 0), gr(z1, 0)), z0, z1), OR(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   GR(0, z0) -> c10
   GR(s(z0), 0) -> c11
   GR(s(z0), s(z1)) -> c12(GR(z0, z1))
   OR(false, false) -> c13
   OR(true, z0) -> c14
   OR(z0, true) -> c15
   P(0) -> c16
   P(s(z0)) -> c17
K tuples:none
Defined Rule Symbols:   cond1_3, cond2_3, cond3_3, gr_2, or_2, p_1

Defined Pair Symbols:   COND1_3, COND2_3, COND3_3, GR_2, OR_2, P_1

Compound Symbols:   c_2, c1_3, c2_3, c3_2, c4_2, c5_3, c6_3, c7_2, c8_3, c9_3, c10, c11, c12_1, c13, c14, c15, c16, c17


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

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

(46)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   COND1(true, z0, z1) -> c(COND2(gr(z0, 0), z0, z1), GR(z0, 0))
   COND2(true, z0, z1) -> c1(COND1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1), OR(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND2(true, z0, z1) -> c2(COND1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1), OR(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   COND2(true, z0, z1) -> c3(COND1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1), P(z0))
   COND2(false, z0, z1) -> c4(COND3(gr(z1, 0), z0, z1), GR(z1, 0))
   COND3(true, z0, z1) -> c5(COND1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1)), OR(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND3(true, z0, z1) -> c6(COND1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1)), OR(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   COND3(true, z0, z1) -> c7(COND1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1)), P(z1))
   COND3(false, z0, z1) -> c8(COND1(or(gr(z0, 0), gr(z1, 0)), z0, z1), OR(gr(z0, 0), gr(z1, 0)), GR(z0, 0))
   COND3(false, z0, z1) -> c9(COND1(or(gr(z0, 0), gr(z1, 0)), z0, z1), OR(gr(z0, 0), gr(z1, 0)), GR(z1, 0))
   GR(0, z0) -> c10
   GR(s(z0), 0) -> c11
   GR(s(z0), s(z1)) -> c12(GR(z0, z1))
   OR(false, false) -> c13
   OR(true, z0) -> c14
   OR(z0, true) -> c15
   P(0) -> c16
   P(s(z0)) -> c17

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

   cond1(true, z0, z1) -> cond2(gr(z0, 0), z0, z1)
   cond2(true, z0, z1) -> cond1(or(gr(z0, 0), gr(z1, 0)), p(z0), z1)
   cond2(false, z0, z1) -> cond3(gr(z1, 0), z0, z1)
   cond3(true, z0, z1) -> cond1(or(gr(z0, 0), gr(z1, 0)), z0, p(z1))
   cond3(false, z0, z1) -> cond1(or(gr(z0, 0), gr(z1, 0)), z0, z1)
   gr(0, z0) -> false
   gr(s(z0), 0) -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   or(false, false) -> false
   or(true, z0) -> true
   or(z0, true) -> true
   p(0) -> 0
   p(s(z0)) -> z0

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

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

(48)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   COND1(true, z0, z1) -> c(COND2(gr(z0, 0'), z0, z1), GR(z0, 0'))
   COND2(true, z0, z1) -> c1(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
   COND2(true, z0, z1) -> c2(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
   COND2(true, z0, z1) -> c3(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), P(z0))
   COND2(false, z0, z1) -> c4(COND3(gr(z1, 0'), z0, z1), GR(z1, 0'))
   COND3(true, z0, z1) -> c5(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
   COND3(true, z0, z1) -> c6(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
   COND3(true, z0, z1) -> c7(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), P(z1))
   COND3(false, z0, z1) -> c8(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
   COND3(false, z0, z1) -> c9(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
   GR(0', z0) -> c10
   GR(s(z0), 0') -> c11
   GR(s(z0), s(z1)) -> c12(GR(z0, z1))
   OR(false, false) -> c13
   OR(true, z0) -> c14
   OR(z0, true) -> c15
   P(0') -> c16
   P(s(z0)) -> c17

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

   cond1(true, z0, z1) -> cond2(gr(z0, 0'), z0, z1)
   cond2(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1)
   cond2(false, z0, z1) -> cond3(gr(z1, 0'), z0, z1)
   cond3(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1))
   cond3(false, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, z1)
   gr(0', z0) -> false
   gr(s(z0), 0') -> true
   gr(s(z0), s(z1)) -> gr(z0, z1)
   or(false, false) -> false
   or(true, z0) -> true
   or(z0, true) -> true
   p(0') -> 0'
   p(s(z0)) -> z0

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

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

(50)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0, z1) -> c(COND2(gr(z0, 0'), z0, z1), GR(z0, 0'))
COND2(true, z0, z1) -> c1(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND2(true, z0, z1) -> c2(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND2(true, z0, z1) -> c3(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), P(z0))
COND2(false, z0, z1) -> c4(COND3(gr(z1, 0'), z0, z1), GR(z1, 0'))
COND3(true, z0, z1) -> c5(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND3(true, z0, z1) -> c6(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND3(true, z0, z1) -> c7(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), P(z1))
COND3(false, z0, z1) -> c8(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND3(false, z0, z1) -> c9(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
GR(0', z0) -> c10
GR(s(z0), 0') -> c11
GR(s(z0), s(z1)) -> c12(GR(z0, z1))
OR(false, false) -> c13
OR(true, z0) -> c14
OR(z0, true) -> c15
P(0') -> c16
P(s(z0)) -> c17
cond1(true, z0, z1) -> cond2(gr(z0, 0'), z0, z1)
cond2(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1)
cond2(false, z0, z1) -> cond3(gr(z1, 0'), z0, z1)
cond3(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1))
cond3(false, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
or(false, false) -> false
or(true, z0) -> true
or(z0, true) -> true
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s -> 0':s -> c
true :: true:false
c :: c1:c2:c3:c4 -> c10:c11:c12 -> c
COND2 :: true:false -> 0':s -> 0':s -> c1:c2:c3:c4
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
GR :: 0':s -> 0':s -> c10:c11:c12
c1 :: c -> c13:c14:c15 -> c10:c11:c12 -> c1:c2:c3:c4
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
OR :: true:false -> true:false -> c13:c14:c15
c2 :: c -> c13:c14:c15 -> c10:c11:c12 -> c1:c2:c3:c4
c3 :: c -> c16:c17 -> c1:c2:c3:c4
P :: 0':s -> c16:c17
false :: true:false
c4 :: c5:c6:c7:c8:c9 -> c10:c11:c12 -> c1:c2:c3:c4
COND3 :: true:false -> 0':s -> 0':s -> c5:c6:c7:c8:c9
c5 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c6 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c7 :: c -> c16:c17 -> c5:c6:c7:c8:c9
c8 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c9 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c10 :: c10:c11:c12
s :: 0':s -> 0':s
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15
c16 :: c16:c17
c17 :: c16:c17
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
hole_c1_18 :: c
hole_true:false2_18 :: true:false
hole_0':s3_18 :: 0':s
hole_c1:c2:c3:c44_18 :: c1:c2:c3:c4
hole_c10:c11:c125_18 :: c10:c11:c12
hole_c13:c14:c156_18 :: c13:c14:c15
hole_c16:c177_18 :: c16:c17
hole_c5:c6:c7:c8:c98_18 :: c5:c6:c7:c8:c9
hole_cond1:cond2:cond39_18 :: cond1:cond2:cond3
gen_0':s10_18 :: Nat -> 0':s
gen_c10:c11:c1211_18 :: Nat -> c10:c11:c12

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

(51) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
COND1, COND2, gr, GR, COND3, cond1, cond2, cond3

They will be analysed ascendingly in the following order:
COND1 = COND2
gr < COND1
GR < COND1
COND1 = COND3
gr < COND2
GR < COND2
COND2 = COND3
gr < COND3
gr < cond1
gr < cond2
gr < cond3
GR < COND3
cond1 = cond2
cond1 = cond3
cond2 = cond3

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

(52)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0, z1) -> c(COND2(gr(z0, 0'), z0, z1), GR(z0, 0'))
COND2(true, z0, z1) -> c1(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND2(true, z0, z1) -> c2(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND2(true, z0, z1) -> c3(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), P(z0))
COND2(false, z0, z1) -> c4(COND3(gr(z1, 0'), z0, z1), GR(z1, 0'))
COND3(true, z0, z1) -> c5(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND3(true, z0, z1) -> c6(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND3(true, z0, z1) -> c7(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), P(z1))
COND3(false, z0, z1) -> c8(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND3(false, z0, z1) -> c9(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
GR(0', z0) -> c10
GR(s(z0), 0') -> c11
GR(s(z0), s(z1)) -> c12(GR(z0, z1))
OR(false, false) -> c13
OR(true, z0) -> c14
OR(z0, true) -> c15
P(0') -> c16
P(s(z0)) -> c17
cond1(true, z0, z1) -> cond2(gr(z0, 0'), z0, z1)
cond2(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1)
cond2(false, z0, z1) -> cond3(gr(z1, 0'), z0, z1)
cond3(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1))
cond3(false, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
or(false, false) -> false
or(true, z0) -> true
or(z0, true) -> true
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s -> 0':s -> c
true :: true:false
c :: c1:c2:c3:c4 -> c10:c11:c12 -> c
COND2 :: true:false -> 0':s -> 0':s -> c1:c2:c3:c4
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
GR :: 0':s -> 0':s -> c10:c11:c12
c1 :: c -> c13:c14:c15 -> c10:c11:c12 -> c1:c2:c3:c4
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
OR :: true:false -> true:false -> c13:c14:c15
c2 :: c -> c13:c14:c15 -> c10:c11:c12 -> c1:c2:c3:c4
c3 :: c -> c16:c17 -> c1:c2:c3:c4
P :: 0':s -> c16:c17
false :: true:false
c4 :: c5:c6:c7:c8:c9 -> c10:c11:c12 -> c1:c2:c3:c4
COND3 :: true:false -> 0':s -> 0':s -> c5:c6:c7:c8:c9
c5 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c6 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c7 :: c -> c16:c17 -> c5:c6:c7:c8:c9
c8 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c9 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c10 :: c10:c11:c12
s :: 0':s -> 0':s
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15
c16 :: c16:c17
c17 :: c16:c17
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
hole_c1_18 :: c
hole_true:false2_18 :: true:false
hole_0':s3_18 :: 0':s
hole_c1:c2:c3:c44_18 :: c1:c2:c3:c4
hole_c10:c11:c125_18 :: c10:c11:c12
hole_c13:c14:c156_18 :: c13:c14:c15
hole_c16:c177_18 :: c16:c17
hole_c5:c6:c7:c8:c98_18 :: c5:c6:c7:c8:c9
hole_cond1:cond2:cond39_18 :: cond1:cond2:cond3
gen_0':s10_18 :: Nat -> 0':s
gen_c10:c11:c1211_18 :: Nat -> c10:c11:c12


Generator Equations:
gen_0':s10_18(0) <=> 0'
gen_0':s10_18(+(x, 1)) <=> s(gen_0':s10_18(x))
gen_c10:c11:c1211_18(0) <=> c10
gen_c10:c11:c1211_18(+(x, 1)) <=> c12(gen_c10:c11:c1211_18(x))


The following defined symbols remain to be analysed:
gr, COND1, COND2, GR, COND3, cond1, cond2, cond3

They will be analysed ascendingly in the following order:
COND1 = COND2
gr < COND1
GR < COND1
COND1 = COND3
gr < COND2
GR < COND2
COND2 = COND3
gr < COND3
gr < cond1
gr < cond2
gr < cond3
GR < COND3
cond1 = cond2
cond1 = cond3
cond2 = cond3

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
gr(gen_0':s10_18(n13_18), gen_0':s10_18(n13_18)) -> false, rt in Omega(0)

Induction Base:
gr(gen_0':s10_18(0), gen_0':s10_18(0)) ->_R^Omega(0)
false

Induction Step:
gr(gen_0':s10_18(+(n13_18, 1)), gen_0':s10_18(+(n13_18, 1))) ->_R^Omega(0)
gr(gen_0':s10_18(n13_18), gen_0':s10_18(n13_18)) ->_IH
false

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

(54)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0, z1) -> c(COND2(gr(z0, 0'), z0, z1), GR(z0, 0'))
COND2(true, z0, z1) -> c1(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND2(true, z0, z1) -> c2(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND2(true, z0, z1) -> c3(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), P(z0))
COND2(false, z0, z1) -> c4(COND3(gr(z1, 0'), z0, z1), GR(z1, 0'))
COND3(true, z0, z1) -> c5(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND3(true, z0, z1) -> c6(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND3(true, z0, z1) -> c7(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), P(z1))
COND3(false, z0, z1) -> c8(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND3(false, z0, z1) -> c9(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
GR(0', z0) -> c10
GR(s(z0), 0') -> c11
GR(s(z0), s(z1)) -> c12(GR(z0, z1))
OR(false, false) -> c13
OR(true, z0) -> c14
OR(z0, true) -> c15
P(0') -> c16
P(s(z0)) -> c17
cond1(true, z0, z1) -> cond2(gr(z0, 0'), z0, z1)
cond2(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1)
cond2(false, z0, z1) -> cond3(gr(z1, 0'), z0, z1)
cond3(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1))
cond3(false, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
or(false, false) -> false
or(true, z0) -> true
or(z0, true) -> true
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s -> 0':s -> c
true :: true:false
c :: c1:c2:c3:c4 -> c10:c11:c12 -> c
COND2 :: true:false -> 0':s -> 0':s -> c1:c2:c3:c4
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
GR :: 0':s -> 0':s -> c10:c11:c12
c1 :: c -> c13:c14:c15 -> c10:c11:c12 -> c1:c2:c3:c4
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
OR :: true:false -> true:false -> c13:c14:c15
c2 :: c -> c13:c14:c15 -> c10:c11:c12 -> c1:c2:c3:c4
c3 :: c -> c16:c17 -> c1:c2:c3:c4
P :: 0':s -> c16:c17
false :: true:false
c4 :: c5:c6:c7:c8:c9 -> c10:c11:c12 -> c1:c2:c3:c4
COND3 :: true:false -> 0':s -> 0':s -> c5:c6:c7:c8:c9
c5 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c6 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c7 :: c -> c16:c17 -> c5:c6:c7:c8:c9
c8 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c9 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c10 :: c10:c11:c12
s :: 0':s -> 0':s
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15
c16 :: c16:c17
c17 :: c16:c17
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
hole_c1_18 :: c
hole_true:false2_18 :: true:false
hole_0':s3_18 :: 0':s
hole_c1:c2:c3:c44_18 :: c1:c2:c3:c4
hole_c10:c11:c125_18 :: c10:c11:c12
hole_c13:c14:c156_18 :: c13:c14:c15
hole_c16:c177_18 :: c16:c17
hole_c5:c6:c7:c8:c98_18 :: c5:c6:c7:c8:c9
hole_cond1:cond2:cond39_18 :: cond1:cond2:cond3
gen_0':s10_18 :: Nat -> 0':s
gen_c10:c11:c1211_18 :: Nat -> c10:c11:c12


Lemmas:
gr(gen_0':s10_18(n13_18), gen_0':s10_18(n13_18)) -> false, rt in Omega(0)


Generator Equations:
gen_0':s10_18(0) <=> 0'
gen_0':s10_18(+(x, 1)) <=> s(gen_0':s10_18(x))
gen_c10:c11:c1211_18(0) <=> c10
gen_c10:c11:c1211_18(+(x, 1)) <=> c12(gen_c10:c11:c1211_18(x))


The following defined symbols remain to be analysed:
GR, COND1, COND2, COND3, cond1, cond2, cond3

They will be analysed ascendingly in the following order:
COND1 = COND2
GR < COND1
COND1 = COND3
GR < COND2
COND2 = COND3
GR < COND3
cond1 = cond2
cond1 = cond3
cond2 = cond3

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

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GR(gen_0':s10_18(n402_18), gen_0':s10_18(n402_18)) -> gen_c10:c11:c1211_18(n402_18), rt in Omega(1 + n402_18)

Induction Base:
GR(gen_0':s10_18(0), gen_0':s10_18(0)) ->_R^Omega(1)
c10

Induction Step:
GR(gen_0':s10_18(+(n402_18, 1)), gen_0':s10_18(+(n402_18, 1))) ->_R^Omega(1)
c12(GR(gen_0':s10_18(n402_18), gen_0':s10_18(n402_18))) ->_IH
c12(gen_c10:c11:c1211_18(c403_18))

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

(56)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
COND1(true, z0, z1) -> c(COND2(gr(z0, 0'), z0, z1), GR(z0, 0'))
COND2(true, z0, z1) -> c1(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND2(true, z0, z1) -> c2(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND2(true, z0, z1) -> c3(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), P(z0))
COND2(false, z0, z1) -> c4(COND3(gr(z1, 0'), z0, z1), GR(z1, 0'))
COND3(true, z0, z1) -> c5(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND3(true, z0, z1) -> c6(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND3(true, z0, z1) -> c7(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), P(z1))
COND3(false, z0, z1) -> c8(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND3(false, z0, z1) -> c9(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
GR(0', z0) -> c10
GR(s(z0), 0') -> c11
GR(s(z0), s(z1)) -> c12(GR(z0, z1))
OR(false, false) -> c13
OR(true, z0) -> c14
OR(z0, true) -> c15
P(0') -> c16
P(s(z0)) -> c17
cond1(true, z0, z1) -> cond2(gr(z0, 0'), z0, z1)
cond2(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1)
cond2(false, z0, z1) -> cond3(gr(z1, 0'), z0, z1)
cond3(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1))
cond3(false, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
or(false, false) -> false
or(true, z0) -> true
or(z0, true) -> true
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s -> 0':s -> c
true :: true:false
c :: c1:c2:c3:c4 -> c10:c11:c12 -> c
COND2 :: true:false -> 0':s -> 0':s -> c1:c2:c3:c4
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
GR :: 0':s -> 0':s -> c10:c11:c12
c1 :: c -> c13:c14:c15 -> c10:c11:c12 -> c1:c2:c3:c4
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
OR :: true:false -> true:false -> c13:c14:c15
c2 :: c -> c13:c14:c15 -> c10:c11:c12 -> c1:c2:c3:c4
c3 :: c -> c16:c17 -> c1:c2:c3:c4
P :: 0':s -> c16:c17
false :: true:false
c4 :: c5:c6:c7:c8:c9 -> c10:c11:c12 -> c1:c2:c3:c4
COND3 :: true:false -> 0':s -> 0':s -> c5:c6:c7:c8:c9
c5 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c6 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c7 :: c -> c16:c17 -> c5:c6:c7:c8:c9
c8 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c9 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c10 :: c10:c11:c12
s :: 0':s -> 0':s
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15
c16 :: c16:c17
c17 :: c16:c17
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
hole_c1_18 :: c
hole_true:false2_18 :: true:false
hole_0':s3_18 :: 0':s
hole_c1:c2:c3:c44_18 :: c1:c2:c3:c4
hole_c10:c11:c125_18 :: c10:c11:c12
hole_c13:c14:c156_18 :: c13:c14:c15
hole_c16:c177_18 :: c16:c17
hole_c5:c6:c7:c8:c98_18 :: c5:c6:c7:c8:c9
hole_cond1:cond2:cond39_18 :: cond1:cond2:cond3
gen_0':s10_18 :: Nat -> 0':s
gen_c10:c11:c1211_18 :: Nat -> c10:c11:c12


Lemmas:
gr(gen_0':s10_18(n13_18), gen_0':s10_18(n13_18)) -> false, rt in Omega(0)


Generator Equations:
gen_0':s10_18(0) <=> 0'
gen_0':s10_18(+(x, 1)) <=> s(gen_0':s10_18(x))
gen_c10:c11:c1211_18(0) <=> c10
gen_c10:c11:c1211_18(+(x, 1)) <=> c12(gen_c10:c11:c1211_18(x))


The following defined symbols remain to be analysed:
GR, COND1, COND2, COND3, cond1, cond2, cond3

They will be analysed ascendingly in the following order:
COND1 = COND2
GR < COND1
COND1 = COND3
GR < COND2
COND2 = COND3
GR < COND3
cond1 = cond2
cond1 = cond3
cond2 = cond3

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

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

(59)
BOUNDS(n^1, INF)

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

(60)
Obligation:
Innermost TRS:
Rules:
COND1(true, z0, z1) -> c(COND2(gr(z0, 0'), z0, z1), GR(z0, 0'))
COND2(true, z0, z1) -> c1(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND2(true, z0, z1) -> c2(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND2(true, z0, z1) -> c3(COND1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1), P(z0))
COND2(false, z0, z1) -> c4(COND3(gr(z1, 0'), z0, z1), GR(z1, 0'))
COND3(true, z0, z1) -> c5(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND3(true, z0, z1) -> c6(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
COND3(true, z0, z1) -> c7(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1)), P(z1))
COND3(false, z0, z1) -> c8(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z0, 0'))
COND3(false, z0, z1) -> c9(COND1(or(gr(z0, 0'), gr(z1, 0')), z0, z1), OR(gr(z0, 0'), gr(z1, 0')), GR(z1, 0'))
GR(0', z0) -> c10
GR(s(z0), 0') -> c11
GR(s(z0), s(z1)) -> c12(GR(z0, z1))
OR(false, false) -> c13
OR(true, z0) -> c14
OR(z0, true) -> c15
P(0') -> c16
P(s(z0)) -> c17
cond1(true, z0, z1) -> cond2(gr(z0, 0'), z0, z1)
cond2(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), p(z0), z1)
cond2(false, z0, z1) -> cond3(gr(z1, 0'), z0, z1)
cond3(true, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, p(z1))
cond3(false, z0, z1) -> cond1(or(gr(z0, 0'), gr(z1, 0')), z0, z1)
gr(0', z0) -> false
gr(s(z0), 0') -> true
gr(s(z0), s(z1)) -> gr(z0, z1)
or(false, false) -> false
or(true, z0) -> true
or(z0, true) -> true
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND1 :: true:false -> 0':s -> 0':s -> c
true :: true:false
c :: c1:c2:c3:c4 -> c10:c11:c12 -> c
COND2 :: true:false -> 0':s -> 0':s -> c1:c2:c3:c4
gr :: 0':s -> 0':s -> true:false
0' :: 0':s
GR :: 0':s -> 0':s -> c10:c11:c12
c1 :: c -> c13:c14:c15 -> c10:c11:c12 -> c1:c2:c3:c4
or :: true:false -> true:false -> true:false
p :: 0':s -> 0':s
OR :: true:false -> true:false -> c13:c14:c15
c2 :: c -> c13:c14:c15 -> c10:c11:c12 -> c1:c2:c3:c4
c3 :: c -> c16:c17 -> c1:c2:c3:c4
P :: 0':s -> c16:c17
false :: true:false
c4 :: c5:c6:c7:c8:c9 -> c10:c11:c12 -> c1:c2:c3:c4
COND3 :: true:false -> 0':s -> 0':s -> c5:c6:c7:c8:c9
c5 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c6 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c7 :: c -> c16:c17 -> c5:c6:c7:c8:c9
c8 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c9 :: c -> c13:c14:c15 -> c10:c11:c12 -> c5:c6:c7:c8:c9
c10 :: c10:c11:c12
s :: 0':s -> 0':s
c11 :: c10:c11:c12
c12 :: c10:c11:c12 -> c10:c11:c12
c13 :: c13:c14:c15
c14 :: c13:c14:c15
c15 :: c13:c14:c15
c16 :: c16:c17
c17 :: c16:c17
cond1 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
cond2 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
cond3 :: true:false -> 0':s -> 0':s -> cond1:cond2:cond3
hole_c1_18 :: c
hole_true:false2_18 :: true:false
hole_0':s3_18 :: 0':s
hole_c1:c2:c3:c44_18 :: c1:c2:c3:c4
hole_c10:c11:c125_18 :: c10:c11:c12
hole_c13:c14:c156_18 :: c13:c14:c15
hole_c16:c177_18 :: c16:c17
hole_c5:c6:c7:c8:c98_18 :: c5:c6:c7:c8:c9
hole_cond1:cond2:cond39_18 :: cond1:cond2:cond3
gen_0':s10_18 :: Nat -> 0':s
gen_c10:c11:c1211_18 :: Nat -> c10:c11:c12


Lemmas:
gr(gen_0':s10_18(n13_18), gen_0':s10_18(n13_18)) -> false, rt in Omega(0)
GR(gen_0':s10_18(n402_18), gen_0':s10_18(n402_18)) -> gen_c10:c11:c1211_18(n402_18), rt in Omega(1 + n402_18)


Generator Equations:
gen_0':s10_18(0) <=> 0'
gen_0':s10_18(+(x, 1)) <=> s(gen_0':s10_18(x))
gen_c10:c11:c1211_18(0) <=> c10
gen_c10:c11:c1211_18(+(x, 1)) <=> c12(gen_c10:c11:c1211_18(x))


The following defined symbols remain to be analysed:
cond2, COND1, COND2, COND3, cond1, cond3

They will be analysed ascendingly in the following order:
COND1 = COND2
COND1 = COND3
COND2 = COND3
cond1 = cond2
cond1 = cond3
cond2 = cond3