WORST_CASE(Omega(n^1),?)
proof of input_JK5EPmfCaW.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 193 ms]
(2) CpxRelTRS
(3) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CdtProblem
(5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CpxRelTRS
(7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxRelTRS
(9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(10) typed CpxTrs
(11) OrderProof [LOWER BOUND(ID), 17 ms]
(12) typed CpxTrs
(13) RewriteLemmaProof [LOWER BOUND(ID), 3008 ms]
(14) typed CpxTrs
(15) RewriteLemmaProof [LOWER BOUND(ID), 3597 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), 71.1 s]
        (22) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

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

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

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


The TRS R consists of the following rules:

   subsets(Cons(x, xs)) -> subsets[Ite][True][Let](Cons(x, xs), subsets(xs))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(x', Cons(x, xs), rest) -> Cons(Cons(x', x), mapconsapp(x', xs, rest))
   mapconsapp(x, Nil, rest) -> rest
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   goal(xs) -> subsets(xs)

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

   subsets[Ite][True][Let](Cons(x, xs), subs) -> mapconsapp(x, subs, subs)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> subsets(z0)
Tuples:
   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   SUBSETS(Nil) -> c2
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   GOAL(z0) -> c7(SUBSETS(z0))
S tuples:
   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   SUBSETS(Nil) -> c2
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   GOAL(z0) -> c7(SUBSETS(z0))
K tuples:none
Defined Rule Symbols:   subsets_1, mapconsapp_3, notEmpty_1, goal_1, subsets[Ite][True][Let]_2

Defined Pair Symbols:   SUBSETS[ITE][TRUE][LET]_2, SUBSETS_1, MAPCONSAPP_3, NOTEMPTY_1, GOAL_1

Compound Symbols:   c_1, c1_2, c2, c3_1, c4, c5, c6, c7_1


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

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

(6)
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:

   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   SUBSETS(Nil) -> c2
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   GOAL(z0) -> c7(SUBSETS(z0))

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

   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> subsets(z0)

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

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

(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:

   SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
   SUBSETS(Nil) -> c2
   MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
   MAPCONSAPP(z0, Nil, z1) -> c4
   NOTEMPTY(Cons(z0, z1)) -> c5
   NOTEMPTY(Nil) -> c6
   GOAL(z0) -> c7(SUBSETS(z0))

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

   SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
   subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
   subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
   subsets(Nil) -> Cons(Nil, Nil)
   mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
   mapconsapp(z0, Nil, z1) -> z1
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   goal(z0) -> subsets(z0)

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

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

(10)
Obligation:
Innermost TRS:
Rules:
SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
SUBSETS(Nil) -> c2
MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
MAPCONSAPP(z0, Nil, z1) -> c4
NOTEMPTY(Cons(z0, z1)) -> c5
NOTEMPTY(Nil) -> c6
GOAL(z0) -> c7(SUBSETS(z0))
SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
subsets(Nil) -> Cons(Nil, Nil)
mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
mapconsapp(z0, Nil, z1) -> z1
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> subsets(z0)

Types:
SUBSETS :: Cons:Nil -> c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c1 :: c -> c1:c2 -> c1:c2
SUBSETS[ITE][TRUE][LET] :: Cons:Nil -> Cons:Nil -> c
subsets :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c2 :: c1:c2
MAPCONSAPP :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c3:c4
c3 :: c3:c4 -> c3:c4
c4 :: c3:c4
NOTEMPTY :: Cons:Nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: Cons:Nil -> c7
c7 :: c1:c2 -> c7
c :: c3:c4 -> c
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c1:c21_8 :: c1:c2
hole_Cons:Nil2_8 :: Cons:Nil
hole_c3_8 :: c
hole_c3:c44_8 :: c3:c4
hole_c5:c65_8 :: c5:c6
hole_c76_8 :: c7
hole_True:False7_8 :: True:False
gen_c1:c28_8 :: Nat -> c1:c2
gen_Cons:Nil9_8 :: Nat -> Cons:Nil
gen_c3:c410_8 :: Nat -> c3:c4

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
SUBSETS, subsets, MAPCONSAPP, mapconsapp

They will be analysed ascendingly in the following order:
subsets < SUBSETS

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

(12)
Obligation:
Innermost TRS:
Rules:
SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
SUBSETS(Nil) -> c2
MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
MAPCONSAPP(z0, Nil, z1) -> c4
NOTEMPTY(Cons(z0, z1)) -> c5
NOTEMPTY(Nil) -> c6
GOAL(z0) -> c7(SUBSETS(z0))
SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
subsets(Nil) -> Cons(Nil, Nil)
mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
mapconsapp(z0, Nil, z1) -> z1
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> subsets(z0)

Types:
SUBSETS :: Cons:Nil -> c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c1 :: c -> c1:c2 -> c1:c2
SUBSETS[ITE][TRUE][LET] :: Cons:Nil -> Cons:Nil -> c
subsets :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c2 :: c1:c2
MAPCONSAPP :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c3:c4
c3 :: c3:c4 -> c3:c4
c4 :: c3:c4
NOTEMPTY :: Cons:Nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: Cons:Nil -> c7
c7 :: c1:c2 -> c7
c :: c3:c4 -> c
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c1:c21_8 :: c1:c2
hole_Cons:Nil2_8 :: Cons:Nil
hole_c3_8 :: c
hole_c3:c44_8 :: c3:c4
hole_c5:c65_8 :: c5:c6
hole_c76_8 :: c7
hole_True:False7_8 :: True:False
gen_c1:c28_8 :: Nat -> c1:c2
gen_Cons:Nil9_8 :: Nat -> Cons:Nil
gen_c3:c410_8 :: Nat -> c3:c4


Generator Equations:
gen_c1:c28_8(0) <=> c2
gen_c1:c28_8(+(x, 1)) <=> c1(c(c4), gen_c1:c28_8(x))
gen_Cons:Nil9_8(0) <=> Nil
gen_Cons:Nil9_8(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil9_8(x))
gen_c3:c410_8(0) <=> c4
gen_c3:c410_8(+(x, 1)) <=> c3(gen_c3:c410_8(x))


The following defined symbols remain to be analysed:
subsets, SUBSETS, MAPCONSAPP, mapconsapp

They will be analysed ascendingly in the following order:
subsets < SUBSETS

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
subsets(gen_Cons:Nil9_8(+(1, n12_8))) -> *11_8, rt in Omega(0)

Induction Base:
subsets(gen_Cons:Nil9_8(+(1, 0)))

Induction Step:
subsets(gen_Cons:Nil9_8(+(1, +(n12_8, 1)))) ->_R^Omega(0)
subsets[Ite][True][Let](Cons(Nil, gen_Cons:Nil9_8(+(1, n12_8))), subsets(gen_Cons:Nil9_8(+(1, n12_8)))) ->_IH
subsets[Ite][True][Let](Cons(Nil, gen_Cons:Nil9_8(+(1, n12_8))), *11_8)

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

(14)
Obligation:
Innermost TRS:
Rules:
SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
SUBSETS(Nil) -> c2
MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
MAPCONSAPP(z0, Nil, z1) -> c4
NOTEMPTY(Cons(z0, z1)) -> c5
NOTEMPTY(Nil) -> c6
GOAL(z0) -> c7(SUBSETS(z0))
SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
subsets(Nil) -> Cons(Nil, Nil)
mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
mapconsapp(z0, Nil, z1) -> z1
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> subsets(z0)

Types:
SUBSETS :: Cons:Nil -> c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c1 :: c -> c1:c2 -> c1:c2
SUBSETS[ITE][TRUE][LET] :: Cons:Nil -> Cons:Nil -> c
subsets :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c2 :: c1:c2
MAPCONSAPP :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c3:c4
c3 :: c3:c4 -> c3:c4
c4 :: c3:c4
NOTEMPTY :: Cons:Nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: Cons:Nil -> c7
c7 :: c1:c2 -> c7
c :: c3:c4 -> c
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c1:c21_8 :: c1:c2
hole_Cons:Nil2_8 :: Cons:Nil
hole_c3_8 :: c
hole_c3:c44_8 :: c3:c4
hole_c5:c65_8 :: c5:c6
hole_c76_8 :: c7
hole_True:False7_8 :: True:False
gen_c1:c28_8 :: Nat -> c1:c2
gen_Cons:Nil9_8 :: Nat -> Cons:Nil
gen_c3:c410_8 :: Nat -> c3:c4


Lemmas:
subsets(gen_Cons:Nil9_8(+(1, n12_8))) -> *11_8, rt in Omega(0)


Generator Equations:
gen_c1:c28_8(0) <=> c2
gen_c1:c28_8(+(x, 1)) <=> c1(c(c4), gen_c1:c28_8(x))
gen_Cons:Nil9_8(0) <=> Nil
gen_Cons:Nil9_8(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil9_8(x))
gen_c3:c410_8(0) <=> c4
gen_c3:c410_8(+(x, 1)) <=> c3(gen_c3:c410_8(x))


The following defined symbols remain to be analysed:
SUBSETS, MAPCONSAPP, mapconsapp
----------------------------------------

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MAPCONSAPP(gen_Cons:Nil9_8(a), gen_Cons:Nil9_8(n118997_8), gen_Cons:Nil9_8(c)) -> gen_c3:c410_8(n118997_8), rt in Omega(1 + n118997_8)

Induction Base:
MAPCONSAPP(gen_Cons:Nil9_8(a), gen_Cons:Nil9_8(0), gen_Cons:Nil9_8(c)) ->_R^Omega(1)
c4

Induction Step:
MAPCONSAPP(gen_Cons:Nil9_8(a), gen_Cons:Nil9_8(+(n118997_8, 1)), gen_Cons:Nil9_8(c)) ->_R^Omega(1)
c3(MAPCONSAPP(gen_Cons:Nil9_8(a), gen_Cons:Nil9_8(n118997_8), gen_Cons:Nil9_8(c))) ->_IH
c3(gen_c3:c410_8(c118998_8))

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:
SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
SUBSETS(Nil) -> c2
MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
MAPCONSAPP(z0, Nil, z1) -> c4
NOTEMPTY(Cons(z0, z1)) -> c5
NOTEMPTY(Nil) -> c6
GOAL(z0) -> c7(SUBSETS(z0))
SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
subsets(Nil) -> Cons(Nil, Nil)
mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
mapconsapp(z0, Nil, z1) -> z1
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> subsets(z0)

Types:
SUBSETS :: Cons:Nil -> c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c1 :: c -> c1:c2 -> c1:c2
SUBSETS[ITE][TRUE][LET] :: Cons:Nil -> Cons:Nil -> c
subsets :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c2 :: c1:c2
MAPCONSAPP :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c3:c4
c3 :: c3:c4 -> c3:c4
c4 :: c3:c4
NOTEMPTY :: Cons:Nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: Cons:Nil -> c7
c7 :: c1:c2 -> c7
c :: c3:c4 -> c
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c1:c21_8 :: c1:c2
hole_Cons:Nil2_8 :: Cons:Nil
hole_c3_8 :: c
hole_c3:c44_8 :: c3:c4
hole_c5:c65_8 :: c5:c6
hole_c76_8 :: c7
hole_True:False7_8 :: True:False
gen_c1:c28_8 :: Nat -> c1:c2
gen_Cons:Nil9_8 :: Nat -> Cons:Nil
gen_c3:c410_8 :: Nat -> c3:c4


Lemmas:
subsets(gen_Cons:Nil9_8(+(1, n12_8))) -> *11_8, rt in Omega(0)


Generator Equations:
gen_c1:c28_8(0) <=> c2
gen_c1:c28_8(+(x, 1)) <=> c1(c(c4), gen_c1:c28_8(x))
gen_Cons:Nil9_8(0) <=> Nil
gen_Cons:Nil9_8(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil9_8(x))
gen_c3:c410_8(0) <=> c4
gen_c3:c410_8(+(x, 1)) <=> c3(gen_c3:c410_8(x))


The following defined symbols remain to be analysed:
MAPCONSAPP, mapconsapp
----------------------------------------

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
SUBSETS(Cons(z0, z1)) -> c1(SUBSETS[ITE][TRUE][LET](Cons(z0, z1), subsets(z1)), SUBSETS(z1))
SUBSETS(Nil) -> c2
MAPCONSAPP(z0, Cons(z1, z2), z3) -> c3(MAPCONSAPP(z0, z2, z3))
MAPCONSAPP(z0, Nil, z1) -> c4
NOTEMPTY(Cons(z0, z1)) -> c5
NOTEMPTY(Nil) -> c6
GOAL(z0) -> c7(SUBSETS(z0))
SUBSETS[ITE][TRUE][LET](Cons(z0, z1), z2) -> c(MAPCONSAPP(z0, z2, z2))
subsets[Ite][True][Let](Cons(z0, z1), z2) -> mapconsapp(z0, z2, z2)
subsets(Cons(z0, z1)) -> subsets[Ite][True][Let](Cons(z0, z1), subsets(z1))
subsets(Nil) -> Cons(Nil, Nil)
mapconsapp(z0, Cons(z1, z2), z3) -> Cons(Cons(z0, z1), mapconsapp(z0, z2, z3))
mapconsapp(z0, Nil, z1) -> z1
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
goal(z0) -> subsets(z0)

Types:
SUBSETS :: Cons:Nil -> c1:c2
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c1 :: c -> c1:c2 -> c1:c2
SUBSETS[ITE][TRUE][LET] :: Cons:Nil -> Cons:Nil -> c
subsets :: Cons:Nil -> Cons:Nil
Nil :: Cons:Nil
c2 :: c1:c2
MAPCONSAPP :: Cons:Nil -> Cons:Nil -> Cons:Nil -> c3:c4
c3 :: c3:c4 -> c3:c4
c4 :: c3:c4
NOTEMPTY :: Cons:Nil -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
GOAL :: Cons:Nil -> c7
c7 :: c1:c2 -> c7
c :: c3:c4 -> c
subsets[Ite][True][Let] :: Cons:Nil -> Cons:Nil -> Cons:Nil
mapconsapp :: Cons:Nil -> Cons:Nil -> Cons:Nil -> Cons:Nil
notEmpty :: Cons:Nil -> True:False
True :: True:False
False :: True:False
goal :: Cons:Nil -> Cons:Nil
hole_c1:c21_8 :: c1:c2
hole_Cons:Nil2_8 :: Cons:Nil
hole_c3_8 :: c
hole_c3:c44_8 :: c3:c4
hole_c5:c65_8 :: c5:c6
hole_c76_8 :: c7
hole_True:False7_8 :: True:False
gen_c1:c28_8 :: Nat -> c1:c2
gen_Cons:Nil9_8 :: Nat -> Cons:Nil
gen_c3:c410_8 :: Nat -> c3:c4


Lemmas:
subsets(gen_Cons:Nil9_8(+(1, n12_8))) -> *11_8, rt in Omega(0)
MAPCONSAPP(gen_Cons:Nil9_8(a), gen_Cons:Nil9_8(n118997_8), gen_Cons:Nil9_8(c)) -> gen_c3:c410_8(n118997_8), rt in Omega(1 + n118997_8)


Generator Equations:
gen_c1:c28_8(0) <=> c2
gen_c1:c28_8(+(x, 1)) <=> c1(c(c4), gen_c1:c28_8(x))
gen_Cons:Nil9_8(0) <=> Nil
gen_Cons:Nil9_8(+(x, 1)) <=> Cons(Nil, gen_Cons:Nil9_8(x))
gen_c3:c410_8(0) <=> c4
gen_c3:c410_8(+(x, 1)) <=> c3(gen_c3:c410_8(x))


The following defined symbols remain to be analysed:
mapconsapp
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(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mapconsapp(gen_Cons:Nil9_8(a), gen_Cons:Nil9_8(+(1, n119855_8)), gen_Cons:Nil9_8(c)) -> *11_8, rt in Omega(0)

Induction Base:
mapconsapp(gen_Cons:Nil9_8(a), gen_Cons:Nil9_8(+(1, 0)), gen_Cons:Nil9_8(c))

Induction Step:
mapconsapp(gen_Cons:Nil9_8(a), gen_Cons:Nil9_8(+(1, +(n119855_8, 1))), gen_Cons:Nil9_8(c)) ->_R^Omega(0)
Cons(Cons(gen_Cons:Nil9_8(a), Nil), mapconsapp(gen_Cons:Nil9_8(a), gen_Cons:Nil9_8(+(1, n119855_8)), gen_Cons:Nil9_8(c))) ->_IH
Cons(Cons(gen_Cons:Nil9_8(a), Nil), *11_8)

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)