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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 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), 222 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 108 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 151 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 360 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 167 ms]
    (32) CpxRNTS
    (33) FinalProof [FINISHED, 0 ms]
    (34) BOUNDS(1, n^2)
(35) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(36) CpxTRS
    (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) typed CpxTrs
    (39) OrderProof [LOWER BOUND(ID), 0 ms]
    (40) typed CpxTrs
    (41) RewriteLemmaProof [LOWER BOUND(ID), 255 ms]
    (42) BEST
        (43) proven lower bound
            (44) LowerBoundPropagationProof [FINISHED, 0 ms]
            (45) BOUNDS(n^1, INF)
        (46) typed CpxTrs


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

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


The TRS R consists of the following rules:

   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: INNERMOST
----------------------------------------

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

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


The TRS R consists of the following rules:

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

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


The TRS R consists of the following rules:

   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: INNERMOST
----------------------------------------

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

(38)
Obligation:
Innermost TRS:
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

Types:
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
hole_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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

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

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

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

(40)
Obligation:
Innermost TRS:
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

Types:
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
hole_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


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


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

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

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
odd(gen_0':s4_0(*(2, n6_0))) -> false, rt in Omega(1 + n6_0)

Induction Base:
odd(gen_0':s4_0(*(2, 0))) ->_R^Omega(1)
false

Induction Step:
odd(gen_0':s4_0(*(2, +(n6_0, 1)))) ->_R^Omega(1)
odd(gen_0':s4_0(*(2, n6_0))) ->_IH
false

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

(42)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
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

Types:
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
hole_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


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


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

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

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

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

(45)
BOUNDS(n^1, INF)

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

(46)
Obligation:
Innermost TRS:
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

Types:
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
hole_cond1_0 :: cond
hole_true:false2_0 :: true:false
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
odd(gen_0':s4_0(*(2, n6_0))) -> false, rt in Omega(1 + n6_0)


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


The following defined symbols remain to be analysed:
cond