WORST_CASE(Omega(n^1),O(n^1))
proof of input_N9yYQe5BVO.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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, 109 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), 9 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 0 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 282 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), 25 ms]
            (22) 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:

   dec(Cons(Nil, Nil)) -> Nil
   dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs))
   dec(Cons(Cons(x, xs), Nil)) -> dec(Nil)
   dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs))
   isNilNil(Cons(Nil, Nil)) -> True
   isNilNil(Cons(Nil, Cons(x, xs))) -> False
   isNilNil(Cons(Cons(x, xs), Nil)) -> False
   isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False
   nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs)))
   number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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) -> nestdec(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:

   dec(Cons(Nil, Nil)) -> Nil
   dec(Cons(Nil, Cons(x, xs))) -> dec(Cons(x, xs))
   dec(Cons(Cons(x, xs), Nil)) -> dec(Nil)
   dec(Cons(Cons(x', xs'), Cons(x, xs))) -> dec(Cons(x, xs))
   isNilNil(Cons(Nil, Nil)) -> True
   isNilNil(Cons(Nil, Cons(x, xs))) -> False
   isNilNil(Cons(Cons(x, xs), Nil)) -> False
   isNilNil(Cons(Cons(x', xs'), Cons(x, xs))) -> False
   nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nestdec(Cons(x, xs)) -> nestdec(dec(Cons(x, xs)))
   number17(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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) -> nestdec(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, 4, 5]
transitions: 
Cons0(0, 0) -> 0
Nil0() -> 0
True0() -> 0
False0() -> 0
dec0(0) -> 1
isNilNil0(0) -> 2
nestdec0(0) -> 3
number170(0) -> 4
goal0(0) -> 5
Nil1() -> 1
Cons1(0, 0) -> 6
dec1(6) -> 1
Nil1() -> 7
dec1(7) -> 1
True1() -> 2
False1() -> 2
Nil1() -> 8
Nil1() -> 11
Cons1(8, 11) -> 10
Cons1(8, 10) -> 9
Cons1(8, 9) -> 9
Cons1(8, 9) -> 3
dec1(6) -> 12
nestdec1(12) -> 3
Cons1(8, 9) -> 4
nestdec1(0) -> 5
Nil1() -> 12
dec1(7) -> 12
Cons1(8, 9) -> 5
nestdec1(12) -> 5
Nil2() -> 13
Nil2() -> 16
Cons2(13, 16) -> 15
Cons2(13, 15) -> 14
Cons2(13, 14) -> 14
Cons2(13, 14) -> 3
Cons2(13, 14) -> 5

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

(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:
   dec(Cons(Nil, Nil)) -> Nil
   dec(Cons(Nil, Cons(z0, z1))) -> dec(Cons(z0, z1))
   dec(Cons(Cons(z0, z1), Nil)) -> dec(Nil)
   dec(Cons(Cons(z0, z1), Cons(z2, z3))) -> dec(Cons(z2, z3))
   isNilNil(Cons(Nil, Nil)) -> True
   isNilNil(Cons(Nil, Cons(z0, z1))) -> False
   isNilNil(Cons(Cons(z0, z1), Nil)) -> False
   isNilNil(Cons(Cons(z0, z1), Cons(z2, z3))) -> False
   nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nestdec(Cons(z0, z1)) -> nestdec(dec(Cons(z0, z1)))
   number17(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, Nil)))))))))))))))))
   goal(z0) -> nestdec(z0)
Tuples:
   DEC(Cons(Nil, Nil)) -> c
   DEC(Cons(Nil, Cons(z0, z1))) -> c1(DEC(Cons(z0, z1)))
   DEC(Cons(Cons(z0, z1), Nil)) -> c2(DEC(Nil))
   DEC(Cons(Cons(z0, z1), Cons(z2, z3))) -> c3(DEC(Cons(z2, z3)))
   ISNILNIL(Cons(Nil, Nil)) -> c4
   ISNILNIL(Cons(Nil, Cons(z0, z1))) -> c5
   ISNILNIL(Cons(Cons(z0, z1), Nil)) -> c6
   ISNILNIL(Cons(Cons(z0, z1), Cons(z2, z3))) -> c7
   NESTDEC(Nil) -> c8
   NESTDEC(Cons(z0, z1)) -> c9(NESTDEC(dec(Cons(z0, z1))), DEC(Cons(z0, z1)))
   NUMBER17(z0) -> c10
   GOAL(z0) -> c11(NESTDEC(z0))
S tuples:
   DEC(Cons(Nil, Nil)) -> c
   DEC(Cons(Nil, Cons(z0, z1))) -> c1(DEC(Cons(z0, z1)))
   DEC(Cons(Cons(z0, z1), Nil)) -> c2(DEC(Nil))
   DEC(Cons(Cons(z0, z1), Cons(z2, z3))) -> c3(DEC(Cons(z2, z3)))
   ISNILNIL(Cons(Nil, Nil)) -> c4
   ISNILNIL(Cons(Nil, Cons(z0, z1))) -> c5
   ISNILNIL(Cons(Cons(z0, z1), Nil)) -> c6
   ISNILNIL(Cons(Cons(z0, z1), Cons(z2, z3))) -> c7
   NESTDEC(Nil) -> c8
   NESTDEC(Cons(z0, z1)) -> c9(NESTDEC(dec(Cons(z0, z1))), DEC(Cons(z0, z1)))
   NUMBER17(z0) -> c10
   GOAL(z0) -> c11(NESTDEC(z0))
K tuples:none
Defined Rule Symbols:   dec_1, isNilNil_1, nestdec_1, number17_1, goal_1

Defined Pair Symbols:   DEC_1, ISNILNIL_1, NESTDEC_1, NUMBER17_1, GOAL_1

Compound Symbols:   c, c1_1, c2_1, c3_1, c4, c5, c6, c7, c8, c9_2, c10, c11_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:

   DEC(Cons(Nil, Nil)) -> c
   DEC(Cons(Nil, Cons(z0, z1))) -> c1(DEC(Cons(z0, z1)))
   DEC(Cons(Cons(z0, z1), Nil)) -> c2(DEC(Nil))
   DEC(Cons(Cons(z0, z1), Cons(z2, z3))) -> c3(DEC(Cons(z2, z3)))
   ISNILNIL(Cons(Nil, Nil)) -> c4
   ISNILNIL(Cons(Nil, Cons(z0, z1))) -> c5
   ISNILNIL(Cons(Cons(z0, z1), Nil)) -> c6
   ISNILNIL(Cons(Cons(z0, z1), Cons(z2, z3))) -> c7
   NESTDEC(Nil) -> c8
   NESTDEC(Cons(z0, z1)) -> c9(NESTDEC(dec(Cons(z0, z1))), DEC(Cons(z0, z1)))
   NUMBER17(z0) -> c10
   GOAL(z0) -> c11(NESTDEC(z0))

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

   dec(Cons(Nil, Nil)) -> Nil
   dec(Cons(Nil, Cons(z0, z1))) -> dec(Cons(z0, z1))
   dec(Cons(Cons(z0, z1), Nil)) -> dec(Nil)
   dec(Cons(Cons(z0, z1), Cons(z2, z3))) -> dec(Cons(z2, z3))
   isNilNil(Cons(Nil, Nil)) -> True
   isNilNil(Cons(Nil, Cons(z0, z1))) -> False
   isNilNil(Cons(Cons(z0, z1), Nil)) -> False
   isNilNil(Cons(Cons(z0, z1), Cons(z2, z3))) -> False
   nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nestdec(Cons(z0, z1)) -> nestdec(dec(Cons(z0, z1)))
   number17(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, Nil)))))))))))))))))
   goal(z0) -> nestdec(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:

   DEC(Cons(Nil, Nil)) -> c
   DEC(Cons(Nil, Cons(z0, z1))) -> c1(DEC(Cons(z0, z1)))
   DEC(Cons(Cons(z0, z1), Nil)) -> c2(DEC(Nil))
   DEC(Cons(Cons(z0, z1), Cons(z2, z3))) -> c3(DEC(Cons(z2, z3)))
   ISNILNIL(Cons(Nil, Nil)) -> c4
   ISNILNIL(Cons(Nil, Cons(z0, z1))) -> c5
   ISNILNIL(Cons(Cons(z0, z1), Nil)) -> c6
   ISNILNIL(Cons(Cons(z0, z1), Cons(z2, z3))) -> c7
   NESTDEC(Nil) -> c8
   NESTDEC(Cons(z0, z1)) -> c9(NESTDEC(dec(Cons(z0, z1))), DEC(Cons(z0, z1)))
   NUMBER17(z0) -> c10
   GOAL(z0) -> c11(NESTDEC(z0))

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

   dec(Cons(Nil, Nil)) -> Nil
   dec(Cons(Nil, Cons(z0, z1))) -> dec(Cons(z0, z1))
   dec(Cons(Cons(z0, z1), Nil)) -> dec(Nil)
   dec(Cons(Cons(z0, z1), Cons(z2, z3))) -> dec(Cons(z2, z3))
   isNilNil(Cons(Nil, Nil)) -> True
   isNilNil(Cons(Nil, Cons(z0, z1))) -> False
   isNilNil(Cons(Cons(z0, z1), Nil)) -> False
   isNilNil(Cons(Cons(z0, z1), Cons(z2, z3))) -> False
   nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
   nestdec(Cons(z0, z1)) -> nestdec(dec(Cons(z0, z1)))
   number17(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, Nil)))))))))))))))))
   goal(z0) -> nestdec(z0)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
DEC(Cons(Nil, Nil)) -> c
DEC(Cons(Nil, Cons(z0, z1))) -> c1(DEC(Cons(z0, z1)))
DEC(Cons(Cons(z0, z1), Nil)) -> c2(DEC(Nil))
DEC(Cons(Cons(z0, z1), Cons(z2, z3))) -> c3(DEC(Cons(z2, z3)))
ISNILNIL(Cons(Nil, Nil)) -> c4
ISNILNIL(Cons(Nil, Cons(z0, z1))) -> c5
ISNILNIL(Cons(Cons(z0, z1), Nil)) -> c6
ISNILNIL(Cons(Cons(z0, z1), Cons(z2, z3))) -> c7
NESTDEC(Nil) -> c8
NESTDEC(Cons(z0, z1)) -> c9(NESTDEC(dec(Cons(z0, z1))), DEC(Cons(z0, z1)))
NUMBER17(z0) -> c10
GOAL(z0) -> c11(NESTDEC(z0))
dec(Cons(Nil, Nil)) -> Nil
dec(Cons(Nil, Cons(z0, z1))) -> dec(Cons(z0, z1))
dec(Cons(Cons(z0, z1), Nil)) -> dec(Nil)
dec(Cons(Cons(z0, z1), Cons(z2, z3))) -> dec(Cons(z2, z3))
isNilNil(Cons(Nil, Nil)) -> True
isNilNil(Cons(Nil, Cons(z0, z1))) -> False
isNilNil(Cons(Cons(z0, z1), Nil)) -> False
isNilNil(Cons(Cons(z0, z1), Cons(z2, z3))) -> False
nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
nestdec(Cons(z0, z1)) -> nestdec(dec(Cons(z0, z1)))
number17(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, Nil)))))))))))))))))
goal(z0) -> nestdec(z0)

Types:
DEC :: Nil:Cons -> c:c1:c2:c3
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
c :: c:c1:c2:c3
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ISNILNIL :: Nil:Cons -> c4:c5:c6:c7
c4 :: c4:c5:c6:c7
c5 :: c4:c5:c6:c7
c6 :: c4:c5:c6:c7
c7 :: c4:c5:c6:c7
NESTDEC :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c:c1:c2:c3 -> c8:c9
dec :: Nil:Cons -> Nil:Cons
NUMBER17 :: a -> c10
c10 :: c10
GOAL :: Nil:Cons -> c11
c11 :: c8:c9 -> c11
isNilNil :: Nil:Cons -> True:False
True :: True:False
False :: True:False
nestdec :: Nil:Cons -> Nil:Cons
number17 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons
hole_c:c1:c2:c31_12 :: c:c1:c2:c3
hole_Nil:Cons2_12 :: Nil:Cons
hole_c4:c5:c6:c73_12 :: c4:c5:c6:c7
hole_c8:c94_12 :: c8:c9
hole_c105_12 :: c10
hole_a6_12 :: a
hole_c117_12 :: c11
hole_True:False8_12 :: True:False
hole_b9_12 :: b
gen_c:c1:c2:c310_12 :: Nat -> c:c1:c2:c3
gen_Nil:Cons11_12 :: Nat -> Nil:Cons
gen_c8:c912_12 :: Nat -> c8:c9

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
DEC, NESTDEC, dec, nestdec

They will be analysed ascendingly in the following order:
DEC < NESTDEC
dec < NESTDEC
dec < nestdec

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

(14)
Obligation:
Innermost TRS:
Rules:
DEC(Cons(Nil, Nil)) -> c
DEC(Cons(Nil, Cons(z0, z1))) -> c1(DEC(Cons(z0, z1)))
DEC(Cons(Cons(z0, z1), Nil)) -> c2(DEC(Nil))
DEC(Cons(Cons(z0, z1), Cons(z2, z3))) -> c3(DEC(Cons(z2, z3)))
ISNILNIL(Cons(Nil, Nil)) -> c4
ISNILNIL(Cons(Nil, Cons(z0, z1))) -> c5
ISNILNIL(Cons(Cons(z0, z1), Nil)) -> c6
ISNILNIL(Cons(Cons(z0, z1), Cons(z2, z3))) -> c7
NESTDEC(Nil) -> c8
NESTDEC(Cons(z0, z1)) -> c9(NESTDEC(dec(Cons(z0, z1))), DEC(Cons(z0, z1)))
NUMBER17(z0) -> c10
GOAL(z0) -> c11(NESTDEC(z0))
dec(Cons(Nil, Nil)) -> Nil
dec(Cons(Nil, Cons(z0, z1))) -> dec(Cons(z0, z1))
dec(Cons(Cons(z0, z1), Nil)) -> dec(Nil)
dec(Cons(Cons(z0, z1), Cons(z2, z3))) -> dec(Cons(z2, z3))
isNilNil(Cons(Nil, Nil)) -> True
isNilNil(Cons(Nil, Cons(z0, z1))) -> False
isNilNil(Cons(Cons(z0, z1), Nil)) -> False
isNilNil(Cons(Cons(z0, z1), Cons(z2, z3))) -> False
nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
nestdec(Cons(z0, z1)) -> nestdec(dec(Cons(z0, z1)))
number17(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, Nil)))))))))))))))))
goal(z0) -> nestdec(z0)

Types:
DEC :: Nil:Cons -> c:c1:c2:c3
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
c :: c:c1:c2:c3
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ISNILNIL :: Nil:Cons -> c4:c5:c6:c7
c4 :: c4:c5:c6:c7
c5 :: c4:c5:c6:c7
c6 :: c4:c5:c6:c7
c7 :: c4:c5:c6:c7
NESTDEC :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c:c1:c2:c3 -> c8:c9
dec :: Nil:Cons -> Nil:Cons
NUMBER17 :: a -> c10
c10 :: c10
GOAL :: Nil:Cons -> c11
c11 :: c8:c9 -> c11
isNilNil :: Nil:Cons -> True:False
True :: True:False
False :: True:False
nestdec :: Nil:Cons -> Nil:Cons
number17 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons
hole_c:c1:c2:c31_12 :: c:c1:c2:c3
hole_Nil:Cons2_12 :: Nil:Cons
hole_c4:c5:c6:c73_12 :: c4:c5:c6:c7
hole_c8:c94_12 :: c8:c9
hole_c105_12 :: c10
hole_a6_12 :: a
hole_c117_12 :: c11
hole_True:False8_12 :: True:False
hole_b9_12 :: b
gen_c:c1:c2:c310_12 :: Nat -> c:c1:c2:c3
gen_Nil:Cons11_12 :: Nat -> Nil:Cons
gen_c8:c912_12 :: Nat -> c8:c9


Generator Equations:
gen_c:c1:c2:c310_12(0) <=> c
gen_c:c1:c2:c310_12(+(x, 1)) <=> c1(gen_c:c1:c2:c310_12(x))
gen_Nil:Cons11_12(0) <=> Nil
gen_Nil:Cons11_12(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons11_12(x))
gen_c8:c912_12(0) <=> c8
gen_c8:c912_12(+(x, 1)) <=> c9(gen_c8:c912_12(x), c)


The following defined symbols remain to be analysed:
DEC, NESTDEC, dec, nestdec

They will be analysed ascendingly in the following order:
DEC < NESTDEC
dec < NESTDEC
dec < nestdec

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
DEC(gen_Nil:Cons11_12(+(1, n14_12))) -> gen_c:c1:c2:c310_12(n14_12), rt in Omega(1 + n14_12)

Induction Base:
DEC(gen_Nil:Cons11_12(+(1, 0))) ->_R^Omega(1)
c

Induction Step:
DEC(gen_Nil:Cons11_12(+(1, +(n14_12, 1)))) ->_R^Omega(1)
c1(DEC(Cons(Nil, gen_Nil:Cons11_12(n14_12)))) ->_IH
c1(gen_c:c1:c2:c310_12(c15_12))

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:
DEC(Cons(Nil, Nil)) -> c
DEC(Cons(Nil, Cons(z0, z1))) -> c1(DEC(Cons(z0, z1)))
DEC(Cons(Cons(z0, z1), Nil)) -> c2(DEC(Nil))
DEC(Cons(Cons(z0, z1), Cons(z2, z3))) -> c3(DEC(Cons(z2, z3)))
ISNILNIL(Cons(Nil, Nil)) -> c4
ISNILNIL(Cons(Nil, Cons(z0, z1))) -> c5
ISNILNIL(Cons(Cons(z0, z1), Nil)) -> c6
ISNILNIL(Cons(Cons(z0, z1), Cons(z2, z3))) -> c7
NESTDEC(Nil) -> c8
NESTDEC(Cons(z0, z1)) -> c9(NESTDEC(dec(Cons(z0, z1))), DEC(Cons(z0, z1)))
NUMBER17(z0) -> c10
GOAL(z0) -> c11(NESTDEC(z0))
dec(Cons(Nil, Nil)) -> Nil
dec(Cons(Nil, Cons(z0, z1))) -> dec(Cons(z0, z1))
dec(Cons(Cons(z0, z1), Nil)) -> dec(Nil)
dec(Cons(Cons(z0, z1), Cons(z2, z3))) -> dec(Cons(z2, z3))
isNilNil(Cons(Nil, Nil)) -> True
isNilNil(Cons(Nil, Cons(z0, z1))) -> False
isNilNil(Cons(Cons(z0, z1), Nil)) -> False
isNilNil(Cons(Cons(z0, z1), Cons(z2, z3))) -> False
nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
nestdec(Cons(z0, z1)) -> nestdec(dec(Cons(z0, z1)))
number17(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, Nil)))))))))))))))))
goal(z0) -> nestdec(z0)

Types:
DEC :: Nil:Cons -> c:c1:c2:c3
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
c :: c:c1:c2:c3
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ISNILNIL :: Nil:Cons -> c4:c5:c6:c7
c4 :: c4:c5:c6:c7
c5 :: c4:c5:c6:c7
c6 :: c4:c5:c6:c7
c7 :: c4:c5:c6:c7
NESTDEC :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c:c1:c2:c3 -> c8:c9
dec :: Nil:Cons -> Nil:Cons
NUMBER17 :: a -> c10
c10 :: c10
GOAL :: Nil:Cons -> c11
c11 :: c8:c9 -> c11
isNilNil :: Nil:Cons -> True:False
True :: True:False
False :: True:False
nestdec :: Nil:Cons -> Nil:Cons
number17 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons
hole_c:c1:c2:c31_12 :: c:c1:c2:c3
hole_Nil:Cons2_12 :: Nil:Cons
hole_c4:c5:c6:c73_12 :: c4:c5:c6:c7
hole_c8:c94_12 :: c8:c9
hole_c105_12 :: c10
hole_a6_12 :: a
hole_c117_12 :: c11
hole_True:False8_12 :: True:False
hole_b9_12 :: b
gen_c:c1:c2:c310_12 :: Nat -> c:c1:c2:c3
gen_Nil:Cons11_12 :: Nat -> Nil:Cons
gen_c8:c912_12 :: Nat -> c8:c9


Generator Equations:
gen_c:c1:c2:c310_12(0) <=> c
gen_c:c1:c2:c310_12(+(x, 1)) <=> c1(gen_c:c1:c2:c310_12(x))
gen_Nil:Cons11_12(0) <=> Nil
gen_Nil:Cons11_12(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons11_12(x))
gen_c8:c912_12(0) <=> c8
gen_c8:c912_12(+(x, 1)) <=> c9(gen_c8:c912_12(x), c)


The following defined symbols remain to be analysed:
DEC, NESTDEC, dec, nestdec

They will be analysed ascendingly in the following order:
DEC < NESTDEC
dec < NESTDEC
dec < nestdec

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

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Innermost TRS:
Rules:
DEC(Cons(Nil, Nil)) -> c
DEC(Cons(Nil, Cons(z0, z1))) -> c1(DEC(Cons(z0, z1)))
DEC(Cons(Cons(z0, z1), Nil)) -> c2(DEC(Nil))
DEC(Cons(Cons(z0, z1), Cons(z2, z3))) -> c3(DEC(Cons(z2, z3)))
ISNILNIL(Cons(Nil, Nil)) -> c4
ISNILNIL(Cons(Nil, Cons(z0, z1))) -> c5
ISNILNIL(Cons(Cons(z0, z1), Nil)) -> c6
ISNILNIL(Cons(Cons(z0, z1), Cons(z2, z3))) -> c7
NESTDEC(Nil) -> c8
NESTDEC(Cons(z0, z1)) -> c9(NESTDEC(dec(Cons(z0, z1))), DEC(Cons(z0, z1)))
NUMBER17(z0) -> c10
GOAL(z0) -> c11(NESTDEC(z0))
dec(Cons(Nil, Nil)) -> Nil
dec(Cons(Nil, Cons(z0, z1))) -> dec(Cons(z0, z1))
dec(Cons(Cons(z0, z1), Nil)) -> dec(Nil)
dec(Cons(Cons(z0, z1), Cons(z2, z3))) -> dec(Cons(z2, z3))
isNilNil(Cons(Nil, Nil)) -> True
isNilNil(Cons(Nil, Cons(z0, z1))) -> False
isNilNil(Cons(Cons(z0, z1), Nil)) -> False
isNilNil(Cons(Cons(z0, z1), Cons(z2, z3))) -> False
nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
nestdec(Cons(z0, z1)) -> nestdec(dec(Cons(z0, z1)))
number17(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, Nil)))))))))))))))))
goal(z0) -> nestdec(z0)

Types:
DEC :: Nil:Cons -> c:c1:c2:c3
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
c :: c:c1:c2:c3
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ISNILNIL :: Nil:Cons -> c4:c5:c6:c7
c4 :: c4:c5:c6:c7
c5 :: c4:c5:c6:c7
c6 :: c4:c5:c6:c7
c7 :: c4:c5:c6:c7
NESTDEC :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c:c1:c2:c3 -> c8:c9
dec :: Nil:Cons -> Nil:Cons
NUMBER17 :: a -> c10
c10 :: c10
GOAL :: Nil:Cons -> c11
c11 :: c8:c9 -> c11
isNilNil :: Nil:Cons -> True:False
True :: True:False
False :: True:False
nestdec :: Nil:Cons -> Nil:Cons
number17 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons
hole_c:c1:c2:c31_12 :: c:c1:c2:c3
hole_Nil:Cons2_12 :: Nil:Cons
hole_c4:c5:c6:c73_12 :: c4:c5:c6:c7
hole_c8:c94_12 :: c8:c9
hole_c105_12 :: c10
hole_a6_12 :: a
hole_c117_12 :: c11
hole_True:False8_12 :: True:False
hole_b9_12 :: b
gen_c:c1:c2:c310_12 :: Nat -> c:c1:c2:c3
gen_Nil:Cons11_12 :: Nat -> Nil:Cons
gen_c8:c912_12 :: Nat -> c8:c9


Lemmas:
DEC(gen_Nil:Cons11_12(+(1, n14_12))) -> gen_c:c1:c2:c310_12(n14_12), rt in Omega(1 + n14_12)


Generator Equations:
gen_c:c1:c2:c310_12(0) <=> c
gen_c:c1:c2:c310_12(+(x, 1)) <=> c1(gen_c:c1:c2:c310_12(x))
gen_Nil:Cons11_12(0) <=> Nil
gen_Nil:Cons11_12(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons11_12(x))
gen_c8:c912_12(0) <=> c8
gen_c8:c912_12(+(x, 1)) <=> c9(gen_c8:c912_12(x), c)


The following defined symbols remain to be analysed:
dec, NESTDEC, nestdec

They will be analysed ascendingly in the following order:
dec < NESTDEC
dec < nestdec

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
dec(gen_Nil:Cons11_12(+(1, n887_12))) -> gen_Nil:Cons11_12(0), rt in Omega(0)

Induction Base:
dec(gen_Nil:Cons11_12(+(1, 0))) ->_R^Omega(0)
Nil

Induction Step:
dec(gen_Nil:Cons11_12(+(1, +(n887_12, 1)))) ->_R^Omega(0)
dec(Cons(Nil, gen_Nil:Cons11_12(n887_12))) ->_IH
gen_Nil:Cons11_12(0)

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

(22)
Obligation:
Innermost TRS:
Rules:
DEC(Cons(Nil, Nil)) -> c
DEC(Cons(Nil, Cons(z0, z1))) -> c1(DEC(Cons(z0, z1)))
DEC(Cons(Cons(z0, z1), Nil)) -> c2(DEC(Nil))
DEC(Cons(Cons(z0, z1), Cons(z2, z3))) -> c3(DEC(Cons(z2, z3)))
ISNILNIL(Cons(Nil, Nil)) -> c4
ISNILNIL(Cons(Nil, Cons(z0, z1))) -> c5
ISNILNIL(Cons(Cons(z0, z1), Nil)) -> c6
ISNILNIL(Cons(Cons(z0, z1), Cons(z2, z3))) -> c7
NESTDEC(Nil) -> c8
NESTDEC(Cons(z0, z1)) -> c9(NESTDEC(dec(Cons(z0, z1))), DEC(Cons(z0, z1)))
NUMBER17(z0) -> c10
GOAL(z0) -> c11(NESTDEC(z0))
dec(Cons(Nil, Nil)) -> Nil
dec(Cons(Nil, Cons(z0, z1))) -> dec(Cons(z0, z1))
dec(Cons(Cons(z0, z1), Nil)) -> dec(Nil)
dec(Cons(Cons(z0, z1), Cons(z2, z3))) -> dec(Cons(z2, z3))
isNilNil(Cons(Nil, Nil)) -> True
isNilNil(Cons(Nil, Cons(z0, z1))) -> False
isNilNil(Cons(Cons(z0, z1), Nil)) -> False
isNilNil(Cons(Cons(z0, z1), Cons(z2, z3))) -> False
nestdec(Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, 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)))))))))))))))))
nestdec(Cons(z0, z1)) -> nestdec(dec(Cons(z0, z1)))
number17(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, Nil)))))))))))))))))
goal(z0) -> nestdec(z0)

Types:
DEC :: Nil:Cons -> c:c1:c2:c3
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
Nil :: Nil:Cons
c :: c:c1:c2:c3
c1 :: c:c1:c2:c3 -> c:c1:c2:c3
c2 :: c:c1:c2:c3 -> c:c1:c2:c3
c3 :: c:c1:c2:c3 -> c:c1:c2:c3
ISNILNIL :: Nil:Cons -> c4:c5:c6:c7
c4 :: c4:c5:c6:c7
c5 :: c4:c5:c6:c7
c6 :: c4:c5:c6:c7
c7 :: c4:c5:c6:c7
NESTDEC :: Nil:Cons -> c8:c9
c8 :: c8:c9
c9 :: c8:c9 -> c:c1:c2:c3 -> c8:c9
dec :: Nil:Cons -> Nil:Cons
NUMBER17 :: a -> c10
c10 :: c10
GOAL :: Nil:Cons -> c11
c11 :: c8:c9 -> c11
isNilNil :: Nil:Cons -> True:False
True :: True:False
False :: True:False
nestdec :: Nil:Cons -> Nil:Cons
number17 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons
hole_c:c1:c2:c31_12 :: c:c1:c2:c3
hole_Nil:Cons2_12 :: Nil:Cons
hole_c4:c5:c6:c73_12 :: c4:c5:c6:c7
hole_c8:c94_12 :: c8:c9
hole_c105_12 :: c10
hole_a6_12 :: a
hole_c117_12 :: c11
hole_True:False8_12 :: True:False
hole_b9_12 :: b
gen_c:c1:c2:c310_12 :: Nat -> c:c1:c2:c3
gen_Nil:Cons11_12 :: Nat -> Nil:Cons
gen_c8:c912_12 :: Nat -> c8:c9


Lemmas:
DEC(gen_Nil:Cons11_12(+(1, n14_12))) -> gen_c:c1:c2:c310_12(n14_12), rt in Omega(1 + n14_12)
dec(gen_Nil:Cons11_12(+(1, n887_12))) -> gen_Nil:Cons11_12(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c2:c310_12(0) <=> c
gen_c:c1:c2:c310_12(+(x, 1)) <=> c1(gen_c:c1:c2:c310_12(x))
gen_Nil:Cons11_12(0) <=> Nil
gen_Nil:Cons11_12(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons11_12(x))
gen_c8:c912_12(0) <=> c8
gen_c8:c912_12(+(x, 1)) <=> c9(gen_c8:c912_12(x), c)


The following defined symbols remain to be analysed:
NESTDEC, nestdec