WORST_CASE(Omega(n^1),O(n^1))
proof of input_TecPEO8FB0.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^1).

(0) CpxTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsMatchBoundsTAProof [FINISHED, 27 ms]
    (4) BOUNDS(1, n^1)
(5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 193 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 1581 ms]
    (18) BEST
        (19) proven lower bound
            (20) LowerBoundPropagationProof [FINISHED, 0 ms]
            (21) BOUNDS(n^1, INF)
        (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 1462 ms]
            (24) typed CpxTrs


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

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


The TRS R consists of the following rules:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

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

(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

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


The TRS R consists of the following rules:

   odd(Cons(x, xs)) -> even(xs)
   odd(Nil) -> False
   even(Cons(x, xs)) -> odd(xs)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   even(Nil) -> True
   evenodd(x) -> even(x)

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

(3) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 1. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
False0() -> 0
True0() -> 0
odd0(0) -> 1
even0(0) -> 2
notEmpty0(0) -> 3
evenodd0(0) -> 4
even1(0) -> 1
False1() -> 1
odd1(0) -> 2
True1() -> 3
False1() -> 3
True1() -> 2
even1(0) -> 4
even1(0) -> 2
False1() -> 2
odd1(0) -> 1
odd1(0) -> 4
True1() -> 1
True1() -> 4
False1() -> 4

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

(4)
BOUNDS(1, n^1)

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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   odd(Cons(z0, z1)) -> even(z1)
   odd(Nil) -> False
   even(Cons(z0, z1)) -> odd(z1)
   even(Nil) -> True
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   evenodd(z0) -> even(z0)
Tuples:
   ODD(Cons(z0, z1)) -> c(EVEN(z1))
   ODD(Nil) -> c1
   EVEN(Cons(z0, z1)) -> c2(ODD(z1))
   EVEN(Nil) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   EVENODD(z0) -> c6(EVEN(z0))
S tuples:
   ODD(Cons(z0, z1)) -> c(EVEN(z1))
   ODD(Nil) -> c1
   EVEN(Cons(z0, z1)) -> c2(ODD(z1))
   EVEN(Nil) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   EVENODD(z0) -> c6(EVEN(z0))
K tuples:none
Defined Rule Symbols:   odd_1, even_1, notEmpty_1, evenodd_1

Defined Pair Symbols:   ODD_1, EVEN_1, NOTEMPTY_1, EVENODD_1

Compound Symbols:   c_1, c1, c2_1, c3, c4, c5, c6_1


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

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

(8)
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:

   ODD(Cons(z0, z1)) -> c(EVEN(z1))
   ODD(Nil) -> c1
   EVEN(Cons(z0, z1)) -> c2(ODD(z1))
   EVEN(Nil) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   EVENODD(z0) -> c6(EVEN(z0))

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

   odd(Cons(z0, z1)) -> even(z1)
   odd(Nil) -> False
   even(Cons(z0, z1)) -> odd(z1)
   even(Nil) -> True
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   evenodd(z0) -> even(z0)

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

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

(10)
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:

   ODD(Cons(z0, z1)) -> c(EVEN(z1))
   ODD(Nil) -> c1
   EVEN(Cons(z0, z1)) -> c2(ODD(z1))
   EVEN(Nil) -> c3
   NOTEMPTY(Cons(z0, z1)) -> c4
   NOTEMPTY(Nil) -> c5
   EVENODD(z0) -> c6(EVEN(z0))

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

   odd(Cons(z0, z1)) -> even(z1)
   odd(Nil) -> False
   even(Cons(z0, z1)) -> odd(z1)
   even(Nil) -> True
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   evenodd(z0) -> even(z0)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
ODD(Cons(z0, z1)) -> c(EVEN(z1))
ODD(Nil) -> c1
EVEN(Cons(z0, z1)) -> c2(ODD(z1))
EVEN(Nil) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
EVENODD(z0) -> c6(EVEN(z0))
odd(Cons(z0, z1)) -> even(z1)
odd(Nil) -> False
even(Cons(z0, z1)) -> odd(z1)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
evenodd(z0) -> even(z0)

Types:
ODD :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1
EVEN :: Cons:Nil -> c2:c3
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
EVENODD :: Cons:Nil -> c6
c6 :: c2:c3 -> c6
odd :: Cons:Nil -> False:True
even :: Cons:Nil -> False:True
False :: False:True
True :: False:True
notEmpty :: Cons:Nil -> False:True
evenodd :: Cons:Nil -> False:True
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_False:True7_7 :: False:True
gen_Cons:Nil8_7 :: Nat -> Cons:Nil

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
ODD, EVEN, odd, even

They will be analysed ascendingly in the following order:
ODD = EVEN
odd = even

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

(14)
Obligation:
Innermost TRS:
Rules:
ODD(Cons(z0, z1)) -> c(EVEN(z1))
ODD(Nil) -> c1
EVEN(Cons(z0, z1)) -> c2(ODD(z1))
EVEN(Nil) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
EVENODD(z0) -> c6(EVEN(z0))
odd(Cons(z0, z1)) -> even(z1)
odd(Nil) -> False
even(Cons(z0, z1)) -> odd(z1)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
evenodd(z0) -> even(z0)

Types:
ODD :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1
EVEN :: Cons:Nil -> c2:c3
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
EVENODD :: Cons:Nil -> c6
c6 :: c2:c3 -> c6
odd :: Cons:Nil -> False:True
even :: Cons:Nil -> False:True
False :: False:True
True :: False:True
notEmpty :: Cons:Nil -> False:True
evenodd :: Cons:Nil -> False:True
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_False:True7_7 :: False:True
gen_Cons:Nil8_7 :: Nat -> Cons:Nil


Generator Equations:
gen_Cons:Nil8_7(0) <=> Nil
gen_Cons:Nil8_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil8_7(x))


The following defined symbols remain to be analysed:
even, ODD, EVEN, odd

They will be analysed ascendingly in the following order:
ODD = EVEN
odd = even

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
even(gen_Cons:Nil8_7(*(2, n10_7))) -> True, rt in Omega(0)

Induction Base:
even(gen_Cons:Nil8_7(*(2, 0))) ->_R^Omega(0)
True

Induction Step:
even(gen_Cons:Nil8_7(*(2, +(n10_7, 1)))) ->_R^Omega(0)
odd(gen_Cons:Nil8_7(+(1, *(2, n10_7)))) ->_R^Omega(0)
even(gen_Cons:Nil8_7(*(2, n10_7))) ->_IH
True

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

(16)
Obligation:
Innermost TRS:
Rules:
ODD(Cons(z0, z1)) -> c(EVEN(z1))
ODD(Nil) -> c1
EVEN(Cons(z0, z1)) -> c2(ODD(z1))
EVEN(Nil) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
EVENODD(z0) -> c6(EVEN(z0))
odd(Cons(z0, z1)) -> even(z1)
odd(Nil) -> False
even(Cons(z0, z1)) -> odd(z1)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
evenodd(z0) -> even(z0)

Types:
ODD :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1
EVEN :: Cons:Nil -> c2:c3
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
EVENODD :: Cons:Nil -> c6
c6 :: c2:c3 -> c6
odd :: Cons:Nil -> False:True
even :: Cons:Nil -> False:True
False :: False:True
True :: False:True
notEmpty :: Cons:Nil -> False:True
evenodd :: Cons:Nil -> False:True
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_False:True7_7 :: False:True
gen_Cons:Nil8_7 :: Nat -> Cons:Nil


Lemmas:
even(gen_Cons:Nil8_7(*(2, n10_7))) -> True, rt in Omega(0)


Generator Equations:
gen_Cons:Nil8_7(0) <=> Nil
gen_Cons:Nil8_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil8_7(x))


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

They will be analysed ascendingly in the following order:
ODD = EVEN
odd = even

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
EVEN(gen_Cons:Nil8_7(+(1, *(2, n132_7)))) -> *9_7, rt in Omega(n132_7)

Induction Base:
EVEN(gen_Cons:Nil8_7(+(1, *(2, 0))))

Induction Step:
EVEN(gen_Cons:Nil8_7(+(1, *(2, +(n132_7, 1))))) ->_R^Omega(1)
c2(ODD(gen_Cons:Nil8_7(+(2, *(2, n132_7))))) ->_R^Omega(1)
c2(c(EVEN(gen_Cons:Nil8_7(+(1, *(2, n132_7)))))) ->_IH
c2(c(*9_7))

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

(18)
Complex Obligation (BEST)

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

(19)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
ODD(Cons(z0, z1)) -> c(EVEN(z1))
ODD(Nil) -> c1
EVEN(Cons(z0, z1)) -> c2(ODD(z1))
EVEN(Nil) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
EVENODD(z0) -> c6(EVEN(z0))
odd(Cons(z0, z1)) -> even(z1)
odd(Nil) -> False
even(Cons(z0, z1)) -> odd(z1)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
evenodd(z0) -> even(z0)

Types:
ODD :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1
EVEN :: Cons:Nil -> c2:c3
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
EVENODD :: Cons:Nil -> c6
c6 :: c2:c3 -> c6
odd :: Cons:Nil -> False:True
even :: Cons:Nil -> False:True
False :: False:True
True :: False:True
notEmpty :: Cons:Nil -> False:True
evenodd :: Cons:Nil -> False:True
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_False:True7_7 :: False:True
gen_Cons:Nil8_7 :: Nat -> Cons:Nil


Lemmas:
even(gen_Cons:Nil8_7(*(2, n10_7))) -> True, rt in Omega(0)


Generator Equations:
gen_Cons:Nil8_7(0) <=> Nil
gen_Cons:Nil8_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil8_7(x))


The following defined symbols remain to be analysed:
EVEN, ODD

They will be analysed ascendingly in the following order:
ODD = EVEN

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

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

(21)
BOUNDS(n^1, INF)

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

(22)
Obligation:
Innermost TRS:
Rules:
ODD(Cons(z0, z1)) -> c(EVEN(z1))
ODD(Nil) -> c1
EVEN(Cons(z0, z1)) -> c2(ODD(z1))
EVEN(Nil) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
EVENODD(z0) -> c6(EVEN(z0))
odd(Cons(z0, z1)) -> even(z1)
odd(Nil) -> False
even(Cons(z0, z1)) -> odd(z1)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
evenodd(z0) -> even(z0)

Types:
ODD :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1
EVEN :: Cons:Nil -> c2:c3
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
EVENODD :: Cons:Nil -> c6
c6 :: c2:c3 -> c6
odd :: Cons:Nil -> False:True
even :: Cons:Nil -> False:True
False :: False:True
True :: False:True
notEmpty :: Cons:Nil -> False:True
evenodd :: Cons:Nil -> False:True
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_False:True7_7 :: False:True
gen_Cons:Nil8_7 :: Nat -> Cons:Nil


Lemmas:
even(gen_Cons:Nil8_7(*(2, n10_7))) -> True, rt in Omega(0)
EVEN(gen_Cons:Nil8_7(+(1, *(2, n132_7)))) -> *9_7, rt in Omega(n132_7)


Generator Equations:
gen_Cons:Nil8_7(0) <=> Nil
gen_Cons:Nil8_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil8_7(x))


The following defined symbols remain to be analysed:
ODD

They will be analysed ascendingly in the following order:
ODD = EVEN

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ODD(gen_Cons:Nil8_7(+(1, *(2, n428_7)))) -> *9_7, rt in Omega(n428_7)

Induction Base:
ODD(gen_Cons:Nil8_7(+(1, *(2, 0))))

Induction Step:
ODD(gen_Cons:Nil8_7(+(1, *(2, +(n428_7, 1))))) ->_R^Omega(1)
c(EVEN(gen_Cons:Nil8_7(+(2, *(2, n428_7))))) ->_R^Omega(1)
c(c2(ODD(gen_Cons:Nil8_7(+(1, *(2, n428_7)))))) ->_IH
c(c2(*9_7))

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

(24)
Obligation:
Innermost TRS:
Rules:
ODD(Cons(z0, z1)) -> c(EVEN(z1))
ODD(Nil) -> c1
EVEN(Cons(z0, z1)) -> c2(ODD(z1))
EVEN(Nil) -> c3
NOTEMPTY(Cons(z0, z1)) -> c4
NOTEMPTY(Nil) -> c5
EVENODD(z0) -> c6(EVEN(z0))
odd(Cons(z0, z1)) -> even(z1)
odd(Nil) -> False
even(Cons(z0, z1)) -> odd(z1)
even(Nil) -> True
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
evenodd(z0) -> even(z0)

Types:
ODD :: Cons:Nil -> c:c1
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c2:c3 -> c:c1
EVEN :: Cons:Nil -> c2:c3
Nil :: Cons:Nil
c1 :: c:c1
c2 :: c:c1 -> c2:c3
c3 :: c2:c3
NOTEMPTY :: Cons:Nil -> c4:c5
c4 :: c4:c5
c5 :: c4:c5
EVENODD :: Cons:Nil -> c6
c6 :: c2:c3 -> c6
odd :: Cons:Nil -> False:True
even :: Cons:Nil -> False:True
False :: False:True
True :: False:True
notEmpty :: Cons:Nil -> False:True
evenodd :: Cons:Nil -> False:True
hole_c:c11_7 :: c:c1
hole_Cons:Nil2_7 :: Cons:Nil
hole_a3_7 :: a
hole_c2:c34_7 :: c2:c3
hole_c4:c55_7 :: c4:c5
hole_c66_7 :: c6
hole_False:True7_7 :: False:True
gen_Cons:Nil8_7 :: Nat -> Cons:Nil


Lemmas:
even(gen_Cons:Nil8_7(*(2, n10_7))) -> True, rt in Omega(0)
EVEN(gen_Cons:Nil8_7(+(1, *(2, n132_7)))) -> *9_7, rt in Omega(n132_7)
ODD(gen_Cons:Nil8_7(+(1, *(2, n428_7)))) -> *9_7, rt in Omega(n428_7)


Generator Equations:
gen_Cons:Nil8_7(0) <=> Nil
gen_Cons:Nil8_7(+(x, 1)) <=> Cons(hole_a3_7, gen_Cons:Nil8_7(x))


The following defined symbols remain to be analysed:
EVEN

They will be analysed ascendingly in the following order:
ODD = EVEN