WORST_CASE(Omega(n^1),O(n^2))
proof of input_xxQgTi8Jy7.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# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxWeightedTrs
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedTrs
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
    (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRNTS
    (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 118 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 290 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 179 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
    (42) CpxRNTS
    (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 1902 ms]
    (46) CpxRNTS
    (47) IntTrsBoundProof [UPPER BOUND(ID), 987 ms]
    (48) CpxRNTS
    (49) FinalProof [FINISHED, 0 ms]
    (50) BOUNDS(1, n^2)
(51) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(52) CdtProblem
    (53) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CpxRelTRS
    (55) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CpxRelTRS
    (57) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) typed CpxTrs
    (59) OrderProof [LOWER BOUND(ID), 10 ms]
    (60) typed CpxTrs
    (61) RewriteLemmaProof [LOWER BOUND(ID), 254 ms]
    (62) typed CpxTrs
    (63) RewriteLemmaProof [LOWER BOUND(ID), 110 ms]
    (64) BEST
        (65) proven lower bound
            (66) LowerBoundPropagationProof [FINISHED, 0 ms]
            (67) BOUNDS(n^1, INF)
        (68) typed CpxTrs


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

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


The TRS R consists of the following rules:

   cond(true, x) -> cond(odd(x), p(x))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(x))) -> odd(x)
   p(0) -> 0
   p(s(x)) -> x

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

(1) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond(true, z0) -> cond(odd(z0), p(z0))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
   ODD(0) -> c2
   ODD(s(0)) -> c3
   ODD(s(s(z0))) -> c4(ODD(z0))
   P(0) -> c5
   P(s(z0)) -> c6
S tuples:
   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
   ODD(0) -> c2
   ODD(s(0)) -> c3
   ODD(s(s(z0))) -> c4(ODD(z0))
   P(0) -> c5
   P(s(z0)) -> c6
K tuples:none
Defined Rule Symbols:   cond_2, odd_1, p_1

Defined Pair Symbols:   COND_2, ODD_1, P_1

Compound Symbols:   c_2, c1_2, c2, c3, c4_1, c5, c6


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   ODD(0) -> c2
   P(s(z0)) -> c6
   P(0) -> c5
   ODD(s(0)) -> c3

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond(true, z0) -> cond(odd(z0), p(z0))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
   ODD(s(s(z0))) -> c4(ODD(z0))
S tuples:
   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
   ODD(s(s(z0))) -> c4(ODD(z0))
K tuples:none
Defined Rule Symbols:   cond_2, odd_1, p_1

Defined Pair Symbols:   COND_2, ODD_1

Compound Symbols:   c_2, c1_2, c4_1


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond(true, z0) -> cond(odd(z0), p(z0))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   ODD(s(s(z0))) -> c4(ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)))
S tuples:
   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   ODD(s(s(z0))) -> c4(ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)))
K tuples:none
Defined Rule Symbols:   cond_2, odd_1, p_1

Defined Pair Symbols:   COND_2, ODD_1

Compound Symbols:   c_2, c4_1, c1_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   cond(true, z0) -> cond(odd(z0), p(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   ODD(s(s(z0))) -> c4(ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)))
S tuples:
   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   ODD(s(s(z0))) -> c4(ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)))
K tuples:none
Defined Rule Symbols:   odd_1, p_1

Defined Pair Symbols:   COND_2, ODD_1

Compound Symbols:   c_2, c4_1, c1_1


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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   ODD(s(s(z0))) -> c4(ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)))

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

   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0) -> 0
   p(s(z0)) -> z0

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

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

(12)
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:

   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0)) [1]
   ODD(s(s(z0))) -> c4(ODD(z0)) [1]
   COND(true, z0) -> c1(COND(odd(z0), p(z0))) [1]
   odd(0) -> false [0]
   odd(s(0)) -> true [0]
   odd(s(s(z0))) -> odd(z0) [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0)) [1]
   ODD(s(s(z0))) -> c4(ODD(z0)) [1]
   COND(true, z0) -> c1(COND(odd(z0), p(z0))) [1]
   odd(0) -> false [0]
   odd(s(0)) -> true [0]
   odd(s(s(z0))) -> odd(z0) [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]

The TRS has the following type information:
COND :: true:false -> s:0 -> c:c1
true :: true:false
c :: c:c1 -> c4 -> c:c1
odd :: s:0 -> true:false
p :: s:0 -> s:0
ODD :: s:0 -> c4
s :: s:0 -> s:0
c4 :: c4 -> c4
c1 :: c:c1 -> c:c1
0 :: s:0
false :: true:false

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

(15) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   COND_2
   ODD_1

(c) The following functions are completely defined:

   odd_1
   p_1

Due to the following rules being added:

   odd(v0) -> null_odd [0]
   p(v0) -> 0 [0]

And the following fresh constants:    null_odd, const, const1

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

(16)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0)) [1]
   ODD(s(s(z0))) -> c4(ODD(z0)) [1]
   COND(true, z0) -> c1(COND(odd(z0), p(z0))) [1]
   odd(0) -> false [0]
   odd(s(0)) -> true [0]
   odd(s(s(z0))) -> odd(z0) [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   odd(v0) -> null_odd [0]
   p(v0) -> 0 [0]

The TRS has the following type information:
COND :: true:false:null_odd -> s:0 -> c:c1
true :: true:false:null_odd
c :: c:c1 -> c4 -> c:c1
odd :: s:0 -> true:false:null_odd
p :: s:0 -> s:0
ODD :: s:0 -> c4
s :: s:0 -> s:0
c4 :: c4 -> c4
c1 :: c:c1 -> c:c1
0 :: s:0
false :: true:false:null_odd
null_odd :: true:false:null_odd
const :: c:c1
const1 :: c4

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

(17) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(18)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   COND(true, 0) -> c(COND(false, 0), ODD(0)) [1]
   COND(true, 0) -> c(COND(false, 0), ODD(0)) [1]
   COND(true, s(0)) -> c(COND(true, 0), ODD(s(0))) [1]
   COND(true, s(0)) -> c(COND(true, 0), ODD(s(0))) [1]
   COND(true, s(s(z0'))) -> c(COND(odd(z0'), s(z0')), ODD(s(s(z0')))) [1]
   COND(true, s(s(z0'))) -> c(COND(odd(z0'), 0), ODD(s(s(z0')))) [1]
   COND(true, 0) -> c(COND(null_odd, 0), ODD(0)) [1]
   COND(true, s(z0'')) -> c(COND(null_odd, z0''), ODD(s(z0''))) [1]
   COND(true, z0) -> c(COND(null_odd, 0), ODD(z0)) [1]
   ODD(s(s(z0))) -> c4(ODD(z0)) [1]
   COND(true, 0) -> c1(COND(false, 0)) [1]
   COND(true, 0) -> c1(COND(false, 0)) [1]
   COND(true, s(0)) -> c1(COND(true, 0)) [1]
   COND(true, s(0)) -> c1(COND(true, 0)) [1]
   COND(true, s(s(z01))) -> c1(COND(odd(z01), s(z01))) [1]
   COND(true, s(s(z01))) -> c1(COND(odd(z01), 0)) [1]
   COND(true, 0) -> c1(COND(null_odd, 0)) [1]
   COND(true, s(z02)) -> c1(COND(null_odd, z02)) [1]
   COND(true, z0) -> c1(COND(null_odd, 0)) [1]
   odd(0) -> false [0]
   odd(s(0)) -> true [0]
   odd(s(s(z0))) -> odd(z0) [0]
   p(0) -> 0 [0]
   p(s(z0)) -> z0 [0]
   odd(v0) -> null_odd [0]
   p(v0) -> 0 [0]

The TRS has the following type information:
COND :: true:false:null_odd -> s:0 -> c:c1
true :: true:false:null_odd
c :: c:c1 -> c4 -> c:c1
odd :: s:0 -> true:false:null_odd
p :: s:0 -> s:0
ODD :: s:0 -> c4
s :: s:0 -> s:0
c4 :: c4 -> c4
c1 :: c:c1 -> c:c1
0 :: s:0
false :: true:false:null_odd
null_odd :: true:false:null_odd
const :: c:c1
const1 :: c4

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

(19) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   true => 2
   0 => 0
   false => 1
   null_odd => 0
   const => 0
   const1 => 0

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

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

   COND(z, z') -{ 1 }-> 1 + COND(odd(z01), 0) :|: z = 2, z01 >= 0, z' = 1 + (1 + z01)
   COND(z, z') -{ 1 }-> 1 + COND(odd(z01), 1 + z01) :|: z = 2, z01 >= 0, z' = 1 + (1 + z01)
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z02) :|: z = 2, z02 >= 0, z' = 1 + z02
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z0 >= 0, z' = z0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z0'), 0) + ODD(1 + (1 + z0')) :|: z = 2, z' = 1 + (1 + z0'), z0' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z0'), 1 + z0') + ODD(1 + (1 + z0')) :|: z = 2, z' = 1 + (1 + z0'), z0' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) + ODD(1 + 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z0'') + ODD(1 + z0'') :|: z = 2, z' = 1 + z0'', z0'' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(z0) :|: z = 2, z0 >= 0, z' = z0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(0) :|: z = 2, z' = 0
   ODD(z) -{ 1 }-> 1 + ODD(z0) :|: z0 >= 0, z = 1 + (1 + z0)
   odd(z) -{ 0 }-> odd(z0) :|: z0 >= 0, z = 1 + (1 + z0)
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   p(z) -{ 0 }-> z0 :|: z = 1 + z0, z0 >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) + ODD(1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) + ODD(1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) + ODD(1 + 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(z') :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) + ODD(1 + (z' - 1)) :|: z = 2, z' - 1 >= 0
   ODD(z) -{ 1 }-> 1 + ODD(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0


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

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

   { ODD }
   { odd }
   { p }
   { COND }

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

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

   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) + ODD(1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) + ODD(1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) + ODD(1 + 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(z') :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) + ODD(1 + (z' - 1)) :|: z = 2, z' - 1 >= 0
   ODD(z) -{ 1 }-> 1 + ODD(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {ODD}, {odd}, {p}, {COND}

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

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

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

   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) + ODD(1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) + ODD(1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) + ODD(1 + 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(z') :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) + ODD(1 + (z' - 1)) :|: z = 2, z' - 1 >= 0
   ODD(z) -{ 1 }-> 1 + ODD(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {ODD}, {odd}, {p}, {COND}

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) + ODD(1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) + ODD(1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) + ODD(1 + 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(z') :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) + ODD(1 + (z' - 1)) :|: z = 2, z' - 1 >= 0
   ODD(z) -{ 1 }-> 1 + ODD(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) + ODD(1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) + ODD(1 + (1 + (z' - 2))) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) + ODD(1 + 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(z') :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + ODD(0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) + ODD(1 + (z' - 1)) :|: z = 2, z' - 1 >= 0
   ODD(z) -{ 1 }-> 1 + ODD(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {odd}, {p}, {COND}
Previous analysis results are:
  ODD: runtime: O(n^1) [z], size: O(1) [0]

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

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

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

   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(odd(z' - 2), 0) + s1 :|: s1 >= 0, s1 <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) + s'' :|: s'' >= 0, s'' <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 2 }-> 1 + COND(2, 0) + s' :|: s' >= 0, s' <= 0, z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + s :|: s >= 0, s <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + s2 :|: s2 >= 0, s2 <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, 0) + s4 :|: s4 >= 0, s4 <= 0, z = 2, z' >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, z' - 1) + s3 :|: s3 >= 0, s3 <= 0, z = 2, z' - 1 >= 0
   ODD(z) -{ -1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 2 >= 0
   odd(z) -{ 0 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {odd}, {p}, {COND}
Previous analysis results are:
  ODD: runtime: O(n^1) [z], size: O(1) [0]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: odd
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(odd(z' - 2), 0) + s1 :|: s1 >= 0, s1 <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) + s'' :|: s'' >= 0, s'' <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 2 }-> 1 + COND(2, 0) + s' :|: s' >= 0, s' <= 0, z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + s :|: s >= 0, s <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + s2 :|: s2 >= 0, s2 <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, 0) + s4 :|: s4 >= 0, s4 <= 0, z = 2, z' >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, z' - 1) + s3 :|: s3 >= 0, s3 <= 0, z = 2, z' - 1 >= 0
   ODD(z) -{ -1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 2 >= 0
   odd(z) -{ 0 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {odd}, {p}, {COND}
Previous analysis results are:
  ODD: runtime: O(n^1) [z], size: O(1) [0]
  odd: runtime: ?, size: O(1) [2]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 0) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) :|: z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(odd(z' - 2), 0) + s1 :|: s1 >= 0, s1 <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(odd(z' - 2), 1 + (z' - 2)) + s'' :|: s'' >= 0, s'' <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 2 }-> 1 + COND(2, 0) + s' :|: s' >= 0, s' <= 0, z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + s :|: s >= 0, s <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + s2 :|: s2 >= 0, s2 <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, 0) + s4 :|: s4 >= 0, s4 <= 0, z = 2, z' >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, z' - 1) + s3 :|: s3 >= 0, s3 <= 0, z = 2, z' - 1 >= 0
   ODD(z) -{ -1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 2 >= 0
   odd(z) -{ 0 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {COND}
Previous analysis results are:
  ODD: runtime: O(n^1) [z], size: O(1) [0]
  odd: runtime: O(1) [0], size: O(1) [2]

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

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

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

   COND(z, z') -{ 1 }-> 1 + COND(s8, 1 + (z' - 2)) :|: s8 >= 0, s8 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(s9, 0) :|: s9 >= 0, s9 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s6, 1 + (z' - 2)) + s'' :|: s6 >= 0, s6 <= 2, s'' >= 0, s'' <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s7, 0) + s1 :|: s7 >= 0, s7 <= 2, s1 >= 0, s1 <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 2 }-> 1 + COND(2, 0) + s' :|: s' >= 0, s' <= 0, z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + s :|: s >= 0, s <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + s2 :|: s2 >= 0, s2 <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, 0) + s4 :|: s4 >= 0, s4 <= 0, z = 2, z' >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, z' - 1) + s3 :|: s3 >= 0, s3 <= 0, z = 2, z' - 1 >= 0
   ODD(z) -{ -1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 2 >= 0
   odd(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2, z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {COND}
Previous analysis results are:
  ODD: runtime: O(n^1) [z], size: O(1) [0]
  odd: runtime: O(1) [0], size: O(1) [2]

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

(39) 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

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

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

   COND(z, z') -{ 1 }-> 1 + COND(s8, 1 + (z' - 2)) :|: s8 >= 0, s8 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(s9, 0) :|: s9 >= 0, s9 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s6, 1 + (z' - 2)) + s'' :|: s6 >= 0, s6 <= 2, s'' >= 0, s'' <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s7, 0) + s1 :|: s7 >= 0, s7 <= 2, s1 >= 0, s1 <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 2 }-> 1 + COND(2, 0) + s' :|: s' >= 0, s' <= 0, z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + s :|: s >= 0, s <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + s2 :|: s2 >= 0, s2 <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, 0) + s4 :|: s4 >= 0, s4 <= 0, z = 2, z' >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, z' - 1) + s3 :|: s3 >= 0, s3 <= 0, z = 2, z' - 1 >= 0
   ODD(z) -{ -1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 2 >= 0
   odd(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2, z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {p}, {COND}
Previous analysis results are:
  ODD: runtime: O(n^1) [z], size: O(1) [0]
  odd: runtime: O(1) [0], size: O(1) [2]
  p: runtime: ?, size: O(n^1) [z]

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

(41) 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: 0

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

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

   COND(z, z') -{ 1 }-> 1 + COND(s8, 1 + (z' - 2)) :|: s8 >= 0, s8 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(s9, 0) :|: s9 >= 0, s9 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s6, 1 + (z' - 2)) + s'' :|: s6 >= 0, s6 <= 2, s'' >= 0, s'' <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s7, 0) + s1 :|: s7 >= 0, s7 <= 2, s1 >= 0, s1 <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 2 }-> 1 + COND(2, 0) + s' :|: s' >= 0, s' <= 0, z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + s :|: s >= 0, s <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + s2 :|: s2 >= 0, s2 <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, 0) + s4 :|: s4 >= 0, s4 <= 0, z = 2, z' >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, z' - 1) + s3 :|: s3 >= 0, s3 <= 0, z = 2, z' - 1 >= 0
   ODD(z) -{ -1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 2 >= 0
   odd(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2, z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {COND}
Previous analysis results are:
  ODD: runtime: O(n^1) [z], size: O(1) [0]
  odd: runtime: O(1) [0], size: O(1) [2]
  p: runtime: O(1) [0], size: O(n^1) [z]

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

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

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

   COND(z, z') -{ 1 }-> 1 + COND(s8, 1 + (z' - 2)) :|: s8 >= 0, s8 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(s9, 0) :|: s9 >= 0, s9 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s6, 1 + (z' - 2)) + s'' :|: s6 >= 0, s6 <= 2, s'' >= 0, s'' <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s7, 0) + s1 :|: s7 >= 0, s7 <= 2, s1 >= 0, s1 <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 2 }-> 1 + COND(2, 0) + s' :|: s' >= 0, s' <= 0, z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + s :|: s >= 0, s <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + s2 :|: s2 >= 0, s2 <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, 0) + s4 :|: s4 >= 0, s4 <= 0, z = 2, z' >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, z' - 1) + s3 :|: s3 >= 0, s3 <= 0, z = 2, z' - 1 >= 0
   ODD(z) -{ -1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 2 >= 0
   odd(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2, z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {COND}
Previous analysis results are:
  ODD: runtime: O(n^1) [z], size: O(1) [0]
  odd: runtime: O(1) [0], size: O(1) [2]
  p: runtime: O(1) [0], size: O(n^1) [z]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   COND(z, z') -{ 1 }-> 1 + COND(s8, 1 + (z' - 2)) :|: s8 >= 0, s8 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(s9, 0) :|: s9 >= 0, s9 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s6, 1 + (z' - 2)) + s'' :|: s6 >= 0, s6 <= 2, s'' >= 0, s'' <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s7, 0) + s1 :|: s7 >= 0, s7 <= 2, s1 >= 0, s1 <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 2 }-> 1 + COND(2, 0) + s' :|: s' >= 0, s' <= 0, z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + s :|: s >= 0, s <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + s2 :|: s2 >= 0, s2 <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, 0) + s4 :|: s4 >= 0, s4 <= 0, z = 2, z' >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, z' - 1) + s3 :|: s3 >= 0, s3 <= 0, z = 2, z' - 1 >= 0
   ODD(z) -{ -1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 2 >= 0
   odd(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2, z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {COND}
Previous analysis results are:
  ODD: runtime: O(n^1) [z], size: O(1) [0]
  odd: runtime: O(1) [0], size: O(1) [2]
  p: runtime: O(1) [0], size: O(n^1) [z]
  COND: runtime: ?, size: O(1) [0]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: COND
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + 3*z' + z'^2

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   COND(z, z') -{ 1 }-> 1 + COND(s8, 1 + (z' - 2)) :|: s8 >= 0, s8 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(s9, 0) :|: s9 >= 0, s9 <= 2, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 }-> 1 + COND(2, 0) :|: z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) :|: z = 2, z' >= 0
   COND(z, z') -{ 1 }-> 1 + COND(0, z' - 1) :|: z = 2, z' - 1 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s6, 1 + (z' - 2)) + s'' :|: s6 >= 0, s6 <= 2, s'' >= 0, s'' <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(s7, 0) + s1 :|: s7 >= 0, s7 <= 2, s1 >= 0, s1 <= 0, z = 2, z' - 2 >= 0
   COND(z, z') -{ 2 }-> 1 + COND(2, 0) + s' :|: s' >= 0, s' <= 0, z = 2, z' = 1 + 0
   COND(z, z') -{ 1 }-> 1 + COND(1, 0) + s :|: s >= 0, s <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 }-> 1 + COND(0, 0) + s2 :|: s2 >= 0, s2 <= 0, z = 2, z' = 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, 0) + s4 :|: s4 >= 0, s4 <= 0, z = 2, z' >= 0
   COND(z, z') -{ 1 + z' }-> 1 + COND(0, z' - 1) + s3 :|: s3 >= 0, s3 <= 0, z = 2, z' - 1 >= 0
   ODD(z) -{ -1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= 0, z - 2 >= 0
   odd(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2, z - 2 >= 0
   odd(z) -{ 0 }-> 2 :|: z = 1 + 0
   odd(z) -{ 0 }-> 1 :|: z = 0
   odd(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 0 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  ODD: runtime: O(n^1) [z], size: O(1) [0]
  odd: runtime: O(1) [0], size: O(1) [2]
  p: runtime: O(1) [0], size: O(n^1) [z]
  COND: runtime: O(n^2) [4 + 3*z' + z'^2], size: O(1) [0]

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

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

(50)
BOUNDS(1, n^2)

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

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   cond(true, z0) -> cond(odd(z0), p(z0))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0) -> 0
   p(s(z0)) -> z0
Tuples:
   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
   ODD(0) -> c2
   ODD(s(0)) -> c3
   ODD(s(s(z0))) -> c4(ODD(z0))
   P(0) -> c5
   P(s(z0)) -> c6
S tuples:
   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
   ODD(0) -> c2
   ODD(s(0)) -> c3
   ODD(s(s(z0))) -> c4(ODD(z0))
   P(0) -> c5
   P(s(z0)) -> c6
K tuples:none
Defined Rule Symbols:   cond_2, odd_1, p_1

Defined Pair Symbols:   COND_2, ODD_1, P_1

Compound Symbols:   c_2, c1_2, c2, c3, c4_1, c5, c6


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

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

(54)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
   ODD(0) -> c2
   ODD(s(0)) -> c3
   ODD(s(s(z0))) -> c4(ODD(z0))
   P(0) -> c5
   P(s(z0)) -> c6

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

   cond(true, z0) -> cond(odd(z0), p(z0))
   odd(0) -> false
   odd(s(0)) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0) -> 0
   p(s(z0)) -> z0

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

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

(56)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
   COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
   ODD(0') -> c2
   ODD(s(0')) -> c3
   ODD(s(s(z0))) -> c4(ODD(z0))
   P(0') -> c5
   P(s(z0)) -> c6

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

   cond(true, z0) -> cond(odd(z0), p(z0))
   odd(0') -> false
   odd(s(0')) -> true
   odd(s(s(z0))) -> odd(z0)
   p(0') -> 0'
   p(s(z0)) -> z0

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

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

(58)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
ODD(0') -> c2
ODD(s(0')) -> c3
ODD(s(s(z0))) -> c4(ODD(z0))
P(0') -> c5
P(s(z0)) -> c6
cond(true, z0) -> cond(odd(z0), p(z0))
odd(0') -> false
odd(s(0')) -> true
odd(s(s(z0))) -> odd(z0)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> c:c1
true :: true:false
c :: c:c1 -> c2:c3:c4 -> c:c1
odd :: 0':s -> true:false
p :: 0':s -> 0':s
ODD :: 0':s -> c2:c3:c4
c1 :: c:c1 -> c5:c6 -> c:c1
P :: 0':s -> c5:c6
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6
cond :: true:false -> 0':s -> cond
false :: true:false
hole_c:c11_7 :: c:c1
hole_true:false2_7 :: true:false
hole_0':s3_7 :: 0':s
hole_c2:c3:c44_7 :: c2:c3:c4
hole_c5:c65_7 :: c5:c6
hole_cond6_7 :: cond
gen_c:c17_7 :: Nat -> c:c1
gen_0':s8_7 :: Nat -> 0':s
gen_c2:c3:c49_7 :: Nat -> c2:c3:c4

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

(59) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
COND, odd, ODD, cond

They will be analysed ascendingly in the following order:
odd < COND
ODD < COND
odd < cond

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

(60)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
ODD(0') -> c2
ODD(s(0')) -> c3
ODD(s(s(z0))) -> c4(ODD(z0))
P(0') -> c5
P(s(z0)) -> c6
cond(true, z0) -> cond(odd(z0), p(z0))
odd(0') -> false
odd(s(0')) -> true
odd(s(s(z0))) -> odd(z0)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> c:c1
true :: true:false
c :: c:c1 -> c2:c3:c4 -> c:c1
odd :: 0':s -> true:false
p :: 0':s -> 0':s
ODD :: 0':s -> c2:c3:c4
c1 :: c:c1 -> c5:c6 -> c:c1
P :: 0':s -> c5:c6
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6
cond :: true:false -> 0':s -> cond
false :: true:false
hole_c:c11_7 :: c:c1
hole_true:false2_7 :: true:false
hole_0':s3_7 :: 0':s
hole_c2:c3:c44_7 :: c2:c3:c4
hole_c5:c65_7 :: c5:c6
hole_cond6_7 :: cond
gen_c:c17_7 :: Nat -> c:c1
gen_0':s8_7 :: Nat -> 0':s
gen_c2:c3:c49_7 :: Nat -> c2:c3:c4


Generator Equations:
gen_c:c17_7(0) <=> hole_c:c11_7
gen_c:c17_7(+(x, 1)) <=> c(gen_c:c17_7(x), c2)
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c2:c3:c49_7(0) <=> c2
gen_c2:c3:c49_7(+(x, 1)) <=> c4(gen_c2:c3:c49_7(x))


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

They will be analysed ascendingly in the following order:
odd < COND
ODD < COND
odd < cond

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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
odd(gen_0':s8_7(*(2, n11_7))) -> false, rt in Omega(0)

Induction Base:
odd(gen_0':s8_7(*(2, 0))) ->_R^Omega(0)
false

Induction Step:
odd(gen_0':s8_7(*(2, +(n11_7, 1)))) ->_R^Omega(0)
odd(gen_0':s8_7(*(2, n11_7))) ->_IH
false

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

(62)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
ODD(0') -> c2
ODD(s(0')) -> c3
ODD(s(s(z0))) -> c4(ODD(z0))
P(0') -> c5
P(s(z0)) -> c6
cond(true, z0) -> cond(odd(z0), p(z0))
odd(0') -> false
odd(s(0')) -> true
odd(s(s(z0))) -> odd(z0)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> c:c1
true :: true:false
c :: c:c1 -> c2:c3:c4 -> c:c1
odd :: 0':s -> true:false
p :: 0':s -> 0':s
ODD :: 0':s -> c2:c3:c4
c1 :: c:c1 -> c5:c6 -> c:c1
P :: 0':s -> c5:c6
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6
cond :: true:false -> 0':s -> cond
false :: true:false
hole_c:c11_7 :: c:c1
hole_true:false2_7 :: true:false
hole_0':s3_7 :: 0':s
hole_c2:c3:c44_7 :: c2:c3:c4
hole_c5:c65_7 :: c5:c6
hole_cond6_7 :: cond
gen_c:c17_7 :: Nat -> c:c1
gen_0':s8_7 :: Nat -> 0':s
gen_c2:c3:c49_7 :: Nat -> c2:c3:c4


Lemmas:
odd(gen_0':s8_7(*(2, n11_7))) -> false, rt in Omega(0)


Generator Equations:
gen_c:c17_7(0) <=> hole_c:c11_7
gen_c:c17_7(+(x, 1)) <=> c(gen_c:c17_7(x), c2)
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c2:c3:c49_7(0) <=> c2
gen_c2:c3:c49_7(+(x, 1)) <=> c4(gen_c2:c3:c49_7(x))


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

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

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

(63) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ODD(gen_0':s8_7(*(2, n171_7))) -> gen_c2:c3:c49_7(n171_7), rt in Omega(1 + n171_7)

Induction Base:
ODD(gen_0':s8_7(*(2, 0))) ->_R^Omega(1)
c2

Induction Step:
ODD(gen_0':s8_7(*(2, +(n171_7, 1)))) ->_R^Omega(1)
c4(ODD(gen_0':s8_7(*(2, n171_7)))) ->_IH
c4(gen_c2:c3:c49_7(c172_7))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(64)
Complex Obligation (BEST)

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(65)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
ODD(0') -> c2
ODD(s(0')) -> c3
ODD(s(s(z0))) -> c4(ODD(z0))
P(0') -> c5
P(s(z0)) -> c6
cond(true, z0) -> cond(odd(z0), p(z0))
odd(0') -> false
odd(s(0')) -> true
odd(s(s(z0))) -> odd(z0)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> c:c1
true :: true:false
c :: c:c1 -> c2:c3:c4 -> c:c1
odd :: 0':s -> true:false
p :: 0':s -> 0':s
ODD :: 0':s -> c2:c3:c4
c1 :: c:c1 -> c5:c6 -> c:c1
P :: 0':s -> c5:c6
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6
cond :: true:false -> 0':s -> cond
false :: true:false
hole_c:c11_7 :: c:c1
hole_true:false2_7 :: true:false
hole_0':s3_7 :: 0':s
hole_c2:c3:c44_7 :: c2:c3:c4
hole_c5:c65_7 :: c5:c6
hole_cond6_7 :: cond
gen_c:c17_7 :: Nat -> c:c1
gen_0':s8_7 :: Nat -> 0':s
gen_c2:c3:c49_7 :: Nat -> c2:c3:c4


Lemmas:
odd(gen_0':s8_7(*(2, n11_7))) -> false, rt in Omega(0)


Generator Equations:
gen_c:c17_7(0) <=> hole_c:c11_7
gen_c:c17_7(+(x, 1)) <=> c(gen_c:c17_7(x), c2)
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c2:c3:c49_7(0) <=> c2
gen_c2:c3:c49_7(+(x, 1)) <=> c4(gen_c2:c3:c49_7(x))


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

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

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(66) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(67)
BOUNDS(n^1, INF)

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(68)
Obligation:
Innermost TRS:
Rules:
COND(true, z0) -> c(COND(odd(z0), p(z0)), ODD(z0))
COND(true, z0) -> c1(COND(odd(z0), p(z0)), P(z0))
ODD(0') -> c2
ODD(s(0')) -> c3
ODD(s(s(z0))) -> c4(ODD(z0))
P(0') -> c5
P(s(z0)) -> c6
cond(true, z0) -> cond(odd(z0), p(z0))
odd(0') -> false
odd(s(0')) -> true
odd(s(s(z0))) -> odd(z0)
p(0') -> 0'
p(s(z0)) -> z0

Types:
COND :: true:false -> 0':s -> c:c1
true :: true:false
c :: c:c1 -> c2:c3:c4 -> c:c1
odd :: 0':s -> true:false
p :: 0':s -> 0':s
ODD :: 0':s -> c2:c3:c4
c1 :: c:c1 -> c5:c6 -> c:c1
P :: 0':s -> c5:c6
0' :: 0':s
c2 :: c2:c3:c4
s :: 0':s -> 0':s
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
c5 :: c5:c6
c6 :: c5:c6
cond :: true:false -> 0':s -> cond
false :: true:false
hole_c:c11_7 :: c:c1
hole_true:false2_7 :: true:false
hole_0':s3_7 :: 0':s
hole_c2:c3:c44_7 :: c2:c3:c4
hole_c5:c65_7 :: c5:c6
hole_cond6_7 :: cond
gen_c:c17_7 :: Nat -> c:c1
gen_0':s8_7 :: Nat -> 0':s
gen_c2:c3:c49_7 :: Nat -> c2:c3:c4


Lemmas:
odd(gen_0':s8_7(*(2, n11_7))) -> false, rt in Omega(0)
ODD(gen_0':s8_7(*(2, n171_7))) -> gen_c2:c3:c49_7(n171_7), rt in Omega(1 + n171_7)


Generator Equations:
gen_c:c17_7(0) <=> hole_c:c11_7
gen_c:c17_7(+(x, 1)) <=> c(gen_c:c17_7(x), c2)
gen_0':s8_7(0) <=> 0'
gen_0':s8_7(+(x, 1)) <=> s(gen_0':s8_7(x))
gen_c2:c3:c49_7(0) <=> c2
gen_c2:c3:c49_7(+(x, 1)) <=> c4(gen_c2:c3:c49_7(x))


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