WORST_CASE(Omega(n^1),O(n^1))
proof of input_A4UKT7w2Vg.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 0 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), 7 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 285 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
            (22) BOUNDS(1, INF)


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

(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:

   list(Cons(x, xs)) -> list(xs)
   list(Nil) -> True
   list(Nil) -> isEmpty[Match](Nil)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(x) -> list(x)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
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(1) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(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:

   list(Cons(x, xs)) -> list(xs)
   list(Nil) -> True
   list(Nil) -> isEmpty[Match](Nil)
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(x) -> list(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]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
True0() -> 0
isEmpty[Match]0(0) -> 0
False0() -> 0
list0(0) -> 1
notEmpty0(0) -> 2
goal0(0) -> 3
list1(0) -> 1
True1() -> 1
Nil1() -> 4
isEmpty[Match]1(4) -> 1
True1() -> 2
False1() -> 2
list1(0) -> 3
True1() -> 3
isEmpty[Match]1(4) -> 3

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

(4)
BOUNDS(1, n^1)

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(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
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(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   list(Cons(z0, z1)) -> list(z1)
   list(Nil) -> True
   list(Nil) -> isEmpty[Match](Nil)
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> list(z0)
Tuples:
   LIST(Cons(z0, z1)) -> c(LIST(z1))
   LIST(Nil) -> c1
   LIST(Nil) -> c2
   NOTEMPTY(Cons(z0, z1)) -> c3
   NOTEMPTY(Nil) -> c4
   GOAL(z0) -> c5(LIST(z0))
S tuples:
   LIST(Cons(z0, z1)) -> c(LIST(z1))
   LIST(Nil) -> c1
   LIST(Nil) -> c2
   NOTEMPTY(Cons(z0, z1)) -> c3
   NOTEMPTY(Nil) -> c4
   GOAL(z0) -> c5(LIST(z0))
K tuples:none
Defined Rule Symbols:   list_1, notEmpty_1, goal_1

Defined Pair Symbols:   LIST_1, NOTEMPTY_1, GOAL_1

Compound Symbols:   c_1, c1, c2, c3, c4, c5_1


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(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:

   LIST(Cons(z0, z1)) -> c(LIST(z1))
   LIST(Nil) -> c1
   LIST(Nil) -> c2
   NOTEMPTY(Cons(z0, z1)) -> c3
   NOTEMPTY(Nil) -> c4
   GOAL(z0) -> c5(LIST(z0))

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

   list(Cons(z0, z1)) -> list(z1)
   list(Nil) -> True
   list(Nil) -> isEmpty[Match](Nil)
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> list(z0)

Rewrite Strategy: INNERMOST
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(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:

   LIST(Cons(z0, z1)) -> c(LIST(z1))
   LIST(Nil) -> c1
   LIST(Nil) -> c2
   NOTEMPTY(Cons(z0, z1)) -> c3
   NOTEMPTY(Nil) -> c4
   GOAL(z0) -> c5(LIST(z0))

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

   list(Cons(z0, z1)) -> list(z1)
   list(Nil) -> True
   list(Nil) -> isEmpty[Match](Nil)
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> list(z0)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
LIST(Cons(z0, z1)) -> c(LIST(z1))
LIST(Nil) -> c1
LIST(Nil) -> c2
NOTEMPTY(Cons(z0, z1)) -> c3
NOTEMPTY(Nil) -> c4
GOAL(z0) -> c5(LIST(z0))
list(Cons(z0, z1)) -> list(z1)
list(Nil) -> True
list(Nil) -> isEmpty[Match](Nil)
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> list(z0)

Types:
LIST :: Cons:Nil -> c:c1:c2
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c1 :: c:c1:c2
c2 :: c:c1:c2
NOTEMPTY :: Cons:Nil -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
GOAL :: Cons:Nil -> c5
c5 :: c:c1:c2 -> c5
list :: Cons:Nil -> True:isEmpty[Match]:False
True :: True:isEmpty[Match]:False
isEmpty[Match] :: Cons:Nil -> True:isEmpty[Match]:False
notEmpty :: Cons:Nil -> True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil -> True:isEmpty[Match]:False
hole_c:c1:c21_6 :: c:c1:c2
hole_Cons:Nil2_6 :: Cons:Nil
hole_a3_6 :: a
hole_c3:c44_6 :: c3:c4
hole_c55_6 :: c5
hole_True:isEmpty[Match]:False6_6 :: True:isEmpty[Match]:False
gen_c:c1:c27_6 :: Nat -> c:c1:c2
gen_Cons:Nil8_6 :: Nat -> Cons:Nil

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

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

(14)
Obligation:
Innermost TRS:
Rules:
LIST(Cons(z0, z1)) -> c(LIST(z1))
LIST(Nil) -> c1
LIST(Nil) -> c2
NOTEMPTY(Cons(z0, z1)) -> c3
NOTEMPTY(Nil) -> c4
GOAL(z0) -> c5(LIST(z0))
list(Cons(z0, z1)) -> list(z1)
list(Nil) -> True
list(Nil) -> isEmpty[Match](Nil)
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> list(z0)

Types:
LIST :: Cons:Nil -> c:c1:c2
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c1 :: c:c1:c2
c2 :: c:c1:c2
NOTEMPTY :: Cons:Nil -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
GOAL :: Cons:Nil -> c5
c5 :: c:c1:c2 -> c5
list :: Cons:Nil -> True:isEmpty[Match]:False
True :: True:isEmpty[Match]:False
isEmpty[Match] :: Cons:Nil -> True:isEmpty[Match]:False
notEmpty :: Cons:Nil -> True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil -> True:isEmpty[Match]:False
hole_c:c1:c21_6 :: c:c1:c2
hole_Cons:Nil2_6 :: Cons:Nil
hole_a3_6 :: a
hole_c3:c44_6 :: c3:c4
hole_c55_6 :: c5
hole_True:isEmpty[Match]:False6_6 :: True:isEmpty[Match]:False
gen_c:c1:c27_6 :: Nat -> c:c1:c2
gen_Cons:Nil8_6 :: Nat -> Cons:Nil


Generator Equations:
gen_c:c1:c27_6(0) <=> c1
gen_c:c1:c27_6(+(x, 1)) <=> c(gen_c:c1:c27_6(x))
gen_Cons:Nil8_6(0) <=> Nil
gen_Cons:Nil8_6(+(x, 1)) <=> Cons(hole_a3_6, gen_Cons:Nil8_6(x))


The following defined symbols remain to be analysed:
LIST, list
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(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LIST(gen_Cons:Nil8_6(n10_6)) -> gen_c:c1:c27_6(n10_6), rt in Omega(1 + n10_6)

Induction Base:
LIST(gen_Cons:Nil8_6(0)) ->_R^Omega(1)
c1

Induction Step:
LIST(gen_Cons:Nil8_6(+(n10_6, 1))) ->_R^Omega(1)
c(LIST(gen_Cons:Nil8_6(n10_6))) ->_IH
c(gen_c:c1:c27_6(c11_6))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(16)
Complex Obligation (BEST)

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

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

Innermost TRS:
Rules:
LIST(Cons(z0, z1)) -> c(LIST(z1))
LIST(Nil) -> c1
LIST(Nil) -> c2
NOTEMPTY(Cons(z0, z1)) -> c3
NOTEMPTY(Nil) -> c4
GOAL(z0) -> c5(LIST(z0))
list(Cons(z0, z1)) -> list(z1)
list(Nil) -> True
list(Nil) -> isEmpty[Match](Nil)
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> list(z0)

Types:
LIST :: Cons:Nil -> c:c1:c2
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c1 :: c:c1:c2
c2 :: c:c1:c2
NOTEMPTY :: Cons:Nil -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
GOAL :: Cons:Nil -> c5
c5 :: c:c1:c2 -> c5
list :: Cons:Nil -> True:isEmpty[Match]:False
True :: True:isEmpty[Match]:False
isEmpty[Match] :: Cons:Nil -> True:isEmpty[Match]:False
notEmpty :: Cons:Nil -> True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil -> True:isEmpty[Match]:False
hole_c:c1:c21_6 :: c:c1:c2
hole_Cons:Nil2_6 :: Cons:Nil
hole_a3_6 :: a
hole_c3:c44_6 :: c3:c4
hole_c55_6 :: c5
hole_True:isEmpty[Match]:False6_6 :: True:isEmpty[Match]:False
gen_c:c1:c27_6 :: Nat -> c:c1:c2
gen_Cons:Nil8_6 :: Nat -> Cons:Nil


Generator Equations:
gen_c:c1:c27_6(0) <=> c1
gen_c:c1:c27_6(+(x, 1)) <=> c(gen_c:c1:c27_6(x))
gen_Cons:Nil8_6(0) <=> Nil
gen_Cons:Nil8_6(+(x, 1)) <=> Cons(hole_a3_6, gen_Cons:Nil8_6(x))


The following defined symbols remain to be analysed:
LIST, list
----------------------------------------

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
LIST(Cons(z0, z1)) -> c(LIST(z1))
LIST(Nil) -> c1
LIST(Nil) -> c2
NOTEMPTY(Cons(z0, z1)) -> c3
NOTEMPTY(Nil) -> c4
GOAL(z0) -> c5(LIST(z0))
list(Cons(z0, z1)) -> list(z1)
list(Nil) -> True
list(Nil) -> isEmpty[Match](Nil)
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> list(z0)

Types:
LIST :: Cons:Nil -> c:c1:c2
Cons :: a -> Cons:Nil -> Cons:Nil
c :: c:c1:c2 -> c:c1:c2
Nil :: Cons:Nil
c1 :: c:c1:c2
c2 :: c:c1:c2
NOTEMPTY :: Cons:Nil -> c3:c4
c3 :: c3:c4
c4 :: c3:c4
GOAL :: Cons:Nil -> c5
c5 :: c:c1:c2 -> c5
list :: Cons:Nil -> True:isEmpty[Match]:False
True :: True:isEmpty[Match]:False
isEmpty[Match] :: Cons:Nil -> True:isEmpty[Match]:False
notEmpty :: Cons:Nil -> True:isEmpty[Match]:False
False :: True:isEmpty[Match]:False
goal :: Cons:Nil -> True:isEmpty[Match]:False
hole_c:c1:c21_6 :: c:c1:c2
hole_Cons:Nil2_6 :: Cons:Nil
hole_a3_6 :: a
hole_c3:c44_6 :: c3:c4
hole_c55_6 :: c5
hole_True:isEmpty[Match]:False6_6 :: True:isEmpty[Match]:False
gen_c:c1:c27_6 :: Nat -> c:c1:c2
gen_Cons:Nil8_6 :: Nat -> Cons:Nil


Lemmas:
LIST(gen_Cons:Nil8_6(n10_6)) -> gen_c:c1:c27_6(n10_6), rt in Omega(1 + n10_6)


Generator Equations:
gen_c:c1:c27_6(0) <=> c1
gen_c:c1:c27_6(+(x, 1)) <=> c(gen_c:c1:c27_6(x))
gen_Cons:Nil8_6(0) <=> Nil
gen_Cons:Nil8_6(+(x, 1)) <=> Cons(hole_a3_6, gen_Cons:Nil8_6(x))


The following defined symbols remain to be analysed:
list
----------------------------------------

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
list(gen_Cons:Nil8_6(n218_6)) -> True, rt in Omega(0)

Induction Base:
list(gen_Cons:Nil8_6(0)) ->_R^Omega(0)
True

Induction Step:
list(gen_Cons:Nil8_6(+(n218_6, 1))) ->_R^Omega(0)
list(gen_Cons:Nil8_6(n218_6)) ->_IH
True

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
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(22)
BOUNDS(1, INF)