WORST_CASE(Omega(n^1),O(n^1))
proof of input_bHl8mVFGiw.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 21 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), 215 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) 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:

   decrease(Cons(x, xs)) -> decrease(xs)
   decrease(Nil) -> number42(Nil)
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(x) -> decrease(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:

   decrease(Cons(x, xs)) -> decrease(xs)
   decrease(Nil) -> number42(Nil)
   number42(x) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(x) -> decrease(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 2. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
decrease0(0) -> 1
number420(0) -> 2
goal0(0) -> 3
decrease1(0) -> 1
Nil1() -> 4
number421(4) -> 1
Nil1() -> 5
Nil1() -> 8
Cons1(5, 8) -> 7
Cons1(5, 7) -> 6
Cons1(5, 6) -> 6
Cons1(5, 6) -> 2
decrease1(0) -> 3
number421(4) -> 3
Nil2() -> 9
Nil2() -> 12
Cons2(9, 12) -> 11
Cons2(9, 11) -> 10
Cons2(9, 10) -> 10
Cons2(9, 10) -> 1
Cons2(9, 10) -> 3

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

(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:
   decrease(Cons(z0, z1)) -> decrease(z1)
   decrease(Nil) -> number42(Nil)
   number42(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(z0) -> decrease(z0)
Tuples:
   DECREASE(Cons(z0, z1)) -> c(DECREASE(z1))
   DECREASE(Nil) -> c1(NUMBER42(Nil))
   NUMBER42(z0) -> c2
   GOAL(z0) -> c3(DECREASE(z0))
S tuples:
   DECREASE(Cons(z0, z1)) -> c(DECREASE(z1))
   DECREASE(Nil) -> c1(NUMBER42(Nil))
   NUMBER42(z0) -> c2
   GOAL(z0) -> c3(DECREASE(z0))
K tuples:none
Defined Rule Symbols:   decrease_1, number42_1, goal_1

Defined Pair Symbols:   DECREASE_1, NUMBER42_1, GOAL_1

Compound Symbols:   c_1, c1_1, c2, c3_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:

   DECREASE(Cons(z0, z1)) -> c(DECREASE(z1))
   DECREASE(Nil) -> c1(NUMBER42(Nil))
   NUMBER42(z0) -> c2
   GOAL(z0) -> c3(DECREASE(z0))

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

   decrease(Cons(z0, z1)) -> decrease(z1)
   decrease(Nil) -> number42(Nil)
   number42(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(z0) -> decrease(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:

   DECREASE(Cons(z0, z1)) -> c(DECREASE(z1))
   DECREASE(Nil) -> c1(NUMBER42(Nil))
   NUMBER42(z0) -> c2
   GOAL(z0) -> c3(DECREASE(z0))

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

   decrease(Cons(z0, z1)) -> decrease(z1)
   decrease(Nil) -> number42(Nil)
   number42(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
   goal(z0) -> decrease(z0)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
DECREASE(Cons(z0, z1)) -> c(DECREASE(z1))
DECREASE(Nil) -> c1(NUMBER42(Nil))
NUMBER42(z0) -> c2
GOAL(z0) -> c3(DECREASE(z0))
decrease(Cons(z0, z1)) -> decrease(z1)
decrease(Nil) -> number42(Nil)
number42(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(z0) -> decrease(z0)

Types:
DECREASE :: Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c2 -> c:c1
NUMBER42 :: Cons:Nil -> c2
c2 :: c2
GOAL :: Cons:Nil -> c3
c3 :: c:c1 -> c3
decrease :: Cons:Nil -> Cons:Nil
number42 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_4 :: c:c1
hole_Cons:Nil2_4 :: Cons:Nil
hole_c23_4 :: c2
hole_c34_4 :: c3
gen_c:c15_4 :: Nat -> c:c1
gen_Cons:Nil6_4 :: Nat -> Cons:Nil

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
DECREASE, decrease
----------------------------------------

(14)
Obligation:
Innermost TRS:
Rules:
DECREASE(Cons(z0, z1)) -> c(DECREASE(z1))
DECREASE(Nil) -> c1(NUMBER42(Nil))
NUMBER42(z0) -> c2
GOAL(z0) -> c3(DECREASE(z0))
decrease(Cons(z0, z1)) -> decrease(z1)
decrease(Nil) -> number42(Nil)
number42(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(z0) -> decrease(z0)

Types:
DECREASE :: Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c2 -> c:c1
NUMBER42 :: Cons:Nil -> c2
c2 :: c2
GOAL :: Cons:Nil -> c3
c3 :: c:c1 -> c3
decrease :: Cons:Nil -> Cons:Nil
number42 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_4 :: c:c1
hole_Cons:Nil2_4 :: Cons:Nil
hole_c23_4 :: c2
hole_c34_4 :: c3
gen_c:c15_4 :: Nat -> c:c1
gen_Cons:Nil6_4 :: Nat -> Cons:Nil


Generator Equations:
gen_c:c15_4(0) <=> c1(c2)
gen_c:c15_4(+(x, 1)) <=> c(gen_c:c15_4(x))
gen_Cons:Nil6_4(0) <=> Nil
gen_Cons:Nil6_4(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_4(x))


The following defined symbols remain to be analysed:
DECREASE, decrease
----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
DECREASE(gen_Cons:Nil6_4(n8_4)) -> gen_c:c15_4(n8_4), rt in Omega(1 + n8_4)

Induction Base:
DECREASE(gen_Cons:Nil6_4(0)) ->_R^Omega(1)
c1(NUMBER42(Nil)) ->_R^Omega(1)
c1(c2)

Induction Step:
DECREASE(gen_Cons:Nil6_4(+(n8_4, 1))) ->_R^Omega(1)
c(DECREASE(gen_Cons:Nil6_4(n8_4))) ->_IH
c(gen_c:c15_4(c9_4))

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

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
DECREASE(Cons(z0, z1)) -> c(DECREASE(z1))
DECREASE(Nil) -> c1(NUMBER42(Nil))
NUMBER42(z0) -> c2
GOAL(z0) -> c3(DECREASE(z0))
decrease(Cons(z0, z1)) -> decrease(z1)
decrease(Nil) -> number42(Nil)
number42(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(z0) -> decrease(z0)

Types:
DECREASE :: Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c2 -> c:c1
NUMBER42 :: Cons:Nil -> c2
c2 :: c2
GOAL :: Cons:Nil -> c3
c3 :: c:c1 -> c3
decrease :: Cons:Nil -> Cons:Nil
number42 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_4 :: c:c1
hole_Cons:Nil2_4 :: Cons:Nil
hole_c23_4 :: c2
hole_c34_4 :: c3
gen_c:c15_4 :: Nat -> c:c1
gen_Cons:Nil6_4 :: Nat -> Cons:Nil


Generator Equations:
gen_c:c15_4(0) <=> c1(c2)
gen_c:c15_4(+(x, 1)) <=> c(gen_c:c15_4(x))
gen_Cons:Nil6_4(0) <=> Nil
gen_Cons:Nil6_4(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_4(x))


The following defined symbols remain to be analysed:
DECREASE, decrease
----------------------------------------

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
DECREASE(Cons(z0, z1)) -> c(DECREASE(z1))
DECREASE(Nil) -> c1(NUMBER42(Nil))
NUMBER42(z0) -> c2
GOAL(z0) -> c3(DECREASE(z0))
decrease(Cons(z0, z1)) -> decrease(z1)
decrease(Nil) -> number42(Nil)
number42(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))))))))))))))))))))))))))))))))))))))))
goal(z0) -> decrease(z0)

Types:
DECREASE :: Cons:Nil -> c:c1
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c :: c:c1 -> c:c1
Nil :: Cons:Nil
c1 :: c2 -> c:c1
NUMBER42 :: Cons:Nil -> c2
c2 :: c2
GOAL :: Cons:Nil -> c3
c3 :: c:c1 -> c3
decrease :: Cons:Nil -> Cons:Nil
number42 :: Cons:Nil -> Cons:Nil
goal :: Cons:Nil -> Cons:Nil
hole_c:c11_4 :: c:c1
hole_Cons:Nil2_4 :: Cons:Nil
hole_c23_4 :: c2
hole_c34_4 :: c3
gen_c:c15_4 :: Nat -> c:c1
gen_Cons:Nil6_4 :: Nat -> Cons:Nil


Lemmas:
DECREASE(gen_Cons:Nil6_4(n8_4)) -> gen_c:c15_4(n8_4), rt in Omega(1 + n8_4)


Generator Equations:
gen_c:c15_4(0) <=> c1(c2)
gen_c:c15_4(+(x, 1)) <=> c(gen_c:c15_4(x))
gen_Cons:Nil6_4(0) <=> Nil
gen_Cons:Nil6_4(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil6_4(x))


The following defined symbols remain to be analysed:
decrease