WORST_CASE(Omega(n^1),O(n^2))
proof of input_MccRPH4FDi.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(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) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 231 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 100 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 142 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 43 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 326 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^2)
(35) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CdtProblem
    (37) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CpxRelTRS
    (39) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CpxRelTRS
    (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) typed CpxTrs
    (43) OrderProof [LOWER BOUND(ID), 0 ms]
    (44) typed CpxTrs
    (45) RewriteLemmaProof [LOWER BOUND(ID), 211 ms]
    (46) typed CpxTrs
    (47) RewriteLemmaProof [LOWER BOUND(ID), 86 ms]
    (48) BEST
        (49) proven lower bound
            (50) LowerBoundPropagationProof [FINISHED, 0 ms]
            (51) BOUNDS(n^1, INF)
        (52) 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) 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^2).


The TRS R consists of the following rules:

   cond(true, x) -> cond(odd(x), p(x)) [1]
   odd(0) -> false [1]
   odd(s(0)) -> true [1]
   odd(s(s(x))) -> odd(x) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   cond(true, x) -> cond(odd(x), p(x)) [1]
   odd(0) -> false [1]
   odd(s(0)) -> true [1]
   odd(s(s(x))) -> odd(x) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

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

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

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

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

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

   cond_2

(c) The following functions are completely defined:

   odd_1
   p_1

Due to the following rules being added:
none

And the following fresh constants:    const

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

(6)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   cond(true, x) -> cond(odd(x), p(x)) [1]
   odd(0) -> false [1]
   odd(s(0)) -> true [1]
   odd(s(s(x))) -> odd(x) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

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

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   cond(true, 0) -> cond(false, 0) [3]
   cond(true, s(0)) -> cond(true, 0) [3]
   cond(true, s(s(x'))) -> cond(odd(x'), s(x')) [3]
   odd(0) -> false [1]
   odd(s(0)) -> true [1]
   odd(s(s(x))) -> odd(x) [1]
   p(0) -> 0 [1]
   p(s(x)) -> x [1]

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

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

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

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

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

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

   cond(z, z') -{ 3 }-> cond(odd(x'), 1 + x') :|: z' = 1 + (1 + x'), z = 1, x' >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 }-> odd(x) :|: x >= 0, z = 1 + (1 + x)
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 1 }-> 0 :|: z = 0


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

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

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

   cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0


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

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

   { odd }
   { p }
   { cond }

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

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

   cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {odd}, {p}, {cond}

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

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

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

   cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {odd}, {p}, {cond}

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

(17) 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: 1

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

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

   cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cond(z, z') -{ 3 }-> cond(odd(z' - 2), 1 + (z' - 2)) :|: z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 }-> odd(z - 2) :|: z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

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

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

   cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 + z }-> s' :|: s' >= 0, s' <= 1, z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(23) 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

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

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

   cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 + z }-> s' :|: s' >= 0, s' <= 1, z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(25) 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

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

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

   cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 + z }-> s' :|: s' >= 0, s' <= 1, z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

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

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

   cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 + z }-> s' :|: s' >= 0, s' <= 1, z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(29) 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

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

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

   cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 + z }-> s' :|: s' >= 0, s' <= 1, z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> 0 :|: z = 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(31) 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: 9 + 4*z' + z'^2

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

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

   cond(z, z') -{ 3 + z' }-> cond(s, 1 + (z' - 2)) :|: s >= 0, s <= 1, z = 1, z' - 2 >= 0
   cond(z, z') -{ 3 }-> cond(1, 0) :|: z = 1, z' = 1 + 0
   cond(z, z') -{ 3 }-> cond(0, 0) :|: z = 1, z' = 0
   odd(z) -{ 1 + z }-> s' :|: s' >= 0, s' <= 1, z - 2 >= 0
   odd(z) -{ 1 }-> 1 :|: z = 1 + 0
   odd(z) -{ 1 }-> 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:
  odd: runtime: O(n^1) [2 + z], size: O(1) [1]
  p: runtime: O(1) [1], size: O(n^1) [z]
  cond: runtime: O(n^2) [9 + 4*z' + z'^2], size: O(1) [0]

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

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

(34)
BOUNDS(1, n^2)

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

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

(36)
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


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

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

(38)
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
----------------------------------------

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

(40)
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
----------------------------------------

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

(42)
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

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

(43) 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

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

(44)
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

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

(45) 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).
----------------------------------------

(46)
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

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

(47) 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).
----------------------------------------

(48)
Complex Obligation (BEST)

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

(49)
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

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

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

(51)
BOUNDS(n^1, INF)

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

(52)
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