WORST_CASE(Omega(n^1),O(n^1))
proof of input_XxItRaJM5m.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 215 ms]
(2) CpxRelTRS
(3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
    (5) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxWeightedTrs
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedTrs
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) CompleteCoflocoProof [FINISHED, 588 ms]
    (20) BOUNDS(1, n^1)
(21) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(22) CdtProblem
    (23) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRelTRS
    (25) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
    (27) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) typed CpxTrs
    (29) OrderProof [LOWER BOUND(ID), 12 ms]
    (30) typed CpxTrs
    (31) RewriteLemmaProof [LOWER BOUND(ID), 419 ms]
    (32) BEST
        (33) proven lower bound
            (34) LowerBoundPropagationProof [FINISHED, 0 ms]
            (35) BOUNDS(n^1, INF)
        (36) typed CpxTrs
            (37) RewriteLemmaProof [LOWER BOUND(ID), 64 ms]
            (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 652 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 561 ms]
            (42) typed CpxTrs
            (43) RewriteLemmaProof [LOWER BOUND(ID), 75 ms]
            (44) typed CpxTrs
            (45) RewriteLemmaProof [LOWER BOUND(ID), 86 ms]
            (46) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   lt0(Nil, Cons(x', xs)) -> True
   lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs)
   g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   lt0(x, Nil) -> False
   g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
   f(x, Cons(x', xs)) -> f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
   number4(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil))

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

   g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y))
   g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs)
   f[Ite][False][Ite](False, Cons(x, xs), y) -> f(xs, Cons(Cons(Nil, Nil), y))
   f[Ite][False][Ite](True, x', Cons(x, xs)) -> f(x', xs)

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, n^1).


The TRS R consists of the following rules:

   lt0(Nil, Cons(x', xs)) -> True
   lt0(Cons(x', xs'), Cons(x, xs)) -> lt0(xs', xs)
   g(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   f(x, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   notEmpty(Cons(x, xs)) -> True
   notEmpty(Nil) -> False
   lt0(x, Nil) -> False
   g(x, Cons(x', xs)) -> g[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
   f(x, Cons(x', xs)) -> f[Ite][False][Ite](lt0(x, Cons(Nil, Nil)), x, Cons(x', xs))
   number4(n) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   goal(x, y) -> Cons(f(x, y), Cons(g(x, y), Nil))

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

   g[Ite][False][Ite](False, Cons(x, xs), y) -> g(xs, Cons(Cons(Nil, Nil), y))
   g[Ite][False][Ite](True, x', Cons(x, xs)) -> g(x', xs)
   f[Ite][False][Ite](False, Cons(x, xs), y) -> f(xs, Cons(Cons(Nil, Nil), y))
   f[Ite][False][Ite](True, x', Cons(x, xs)) -> f(x', xs)

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

(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
   g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
   f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
   f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
   lt0(Nil, Cons(z0, z1)) -> True
   lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
   lt0(z0, Nil) -> False
   g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))
Tuples:
   G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
   G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
   F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
   F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
   LT0(Nil, Cons(z0, z1)) -> c4
   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   LT0(z0, Nil) -> c6
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   NOTEMPTY(Cons(z0, z1)) -> c11
   NOTEMPTY(Nil) -> c12
   NUMBER4(z0) -> c13
   GOAL(z0, z1) -> c14(F(z0, z1))
   GOAL(z0, z1) -> c15(G(z0, z1))
S tuples:
   LT0(Nil, Cons(z0, z1)) -> c4
   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   LT0(z0, Nil) -> c6
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   NOTEMPTY(Cons(z0, z1)) -> c11
   NOTEMPTY(Nil) -> c12
   NUMBER4(z0) -> c13
   GOAL(z0, z1) -> c14(F(z0, z1))
   GOAL(z0, z1) -> c15(G(z0, z1))
K tuples:none
Defined Rule Symbols:   lt0_2, g_2, f_2, notEmpty_1, number4_1, goal_2, g[Ite][False][Ite]_3, f[Ite][False][Ite]_3

Defined Pair Symbols:   G[ITE][FALSE][ITE]_3, F[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2, NOTEMPTY_1, NUMBER4_1, GOAL_2

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4, c5_1, c6, c7, c8_2, c9, c10_2, c11, c12, c13, c14_1, c15_1


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

(5) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   GOAL(z0, z1) -> c15(G(z0, z1))
   GOAL(z0, z1) -> c14(F(z0, z1))
Removed 5 trailing nodes:
   NUMBER4(z0) -> c13
   LT0(z0, Nil) -> c6
   NOTEMPTY(Nil) -> c12
   NOTEMPTY(Cons(z0, z1)) -> c11
   LT0(Nil, Cons(z0, z1)) -> c4

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
   g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
   f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
   f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
   lt0(Nil, Cons(z0, z1)) -> True
   lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
   lt0(z0, Nil) -> False
   g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))
Tuples:
   G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
   G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
   F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
   F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
S tuples:
   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
K tuples:none
Defined Rule Symbols:   lt0_2, g_2, f_2, notEmpty_1, number4_1, goal_2, g[Ite][False][Ite]_3, f[Ite][False][Ite]_3

Defined Pair Symbols:   G[ITE][FALSE][ITE]_3, F[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c5_1, c7, c8_2, c9, c10_2


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
   g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
   f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
   f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
   g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt0(Nil, Cons(z0, z1)) -> True
   lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
   lt0(z0, Nil) -> False
Tuples:
   G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
   G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
   F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
   F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
S tuples:
   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
K tuples:none
Defined Rule Symbols:   lt0_2

Defined Pair Symbols:   G[ITE][FALSE][ITE]_3, F[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c5_1, c7, c8_2, c9, c10_2


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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))

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

   G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
   G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
   F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
   F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
   lt0(Nil, Cons(z0, z1)) -> True
   lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
   lt0(z0, Nil) -> False

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

(11) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3)) [1]
   G(z0, Nil) -> c7 [1]
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) [1]
   F(z0, Nil) -> c9 [1]
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) [1]
   G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) [0]
   G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) [0]
   F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2))) [0]
   F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2)) [0]
   lt0(Nil, Cons(z0, z1)) -> True [0]
   lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) [0]
   lt0(z0, Nil) -> False [0]

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

(13) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(14)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3)) [1]
   G(z0, Nil) -> c7 [1]
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) [1]
   F(z0, Nil) -> c9 [1]
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) [1]
   G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) [0]
   G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) [0]
   F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2))) [0]
   F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2)) [0]
   lt0(Nil, Cons(z0, z1)) -> True [0]
   lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) [0]
   lt0(z0, Nil) -> False [0]

The TRS has the following type information:
LT0 :: Cons:Nil -> Cons:Nil -> c5
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c5 :: c5 -> c5
G :: Cons:Nil -> Cons:Nil -> c7:c8
Nil :: Cons:Nil
c7 :: c7:c8
c8 :: c:c1 -> c5 -> c7:c8
G[ITE][FALSE][ITE] :: False:True -> Cons:Nil -> Cons:Nil -> c:c1
lt0 :: Cons:Nil -> Cons:Nil -> False:True
F :: Cons:Nil -> Cons:Nil -> c9:c10
c9 :: c9:c10
c10 :: c2:c3 -> c5 -> c9:c10
F[ITE][FALSE][ITE] :: False:True -> Cons:Nil -> Cons:Nil -> c2:c3
False :: False:True
c :: c7:c8 -> c:c1
True :: False:True
c1 :: c7:c8 -> c:c1
c2 :: c9:c10 -> c2:c3
c3 :: c9:c10 -> c2:c3

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

(15) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   G[ITE][FALSE][ITE](v0, v1, v2) -> null_G[ITE][FALSE][ITE] [0]
   F[ITE][FALSE][ITE](v0, v1, v2) -> null_F[ITE][FALSE][ITE] [0]
   lt0(v0, v1) -> null_lt0 [0]
   LT0(v0, v1) -> null_LT0 [0]

And the following fresh constants:    null_G[ITE][FALSE][ITE], null_F[ITE][FALSE][ITE], null_lt0, null_LT0

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

(16)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3)) [1]
   G(z0, Nil) -> c7 [1]
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) [1]
   F(z0, Nil) -> c9 [1]
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil))) [1]
   G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2))) [0]
   G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2)) [0]
   F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2))) [0]
   F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2)) [0]
   lt0(Nil, Cons(z0, z1)) -> True [0]
   lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3) [0]
   lt0(z0, Nil) -> False [0]
   G[ITE][FALSE][ITE](v0, v1, v2) -> null_G[ITE][FALSE][ITE] [0]
   F[ITE][FALSE][ITE](v0, v1, v2) -> null_F[ITE][FALSE][ITE] [0]
   lt0(v0, v1) -> null_lt0 [0]
   LT0(v0, v1) -> null_LT0 [0]

The TRS has the following type information:
LT0 :: Cons:Nil -> Cons:Nil -> c5:null_LT0
Cons :: Cons:Nil -> Cons:Nil -> Cons:Nil
c5 :: c5:null_LT0 -> c5:null_LT0
G :: Cons:Nil -> Cons:Nil -> c7:c8
Nil :: Cons:Nil
c7 :: c7:c8
c8 :: c:c1:null_G[ITE][FALSE][ITE] -> c5:null_LT0 -> c7:c8
G[ITE][FALSE][ITE] :: False:True:null_lt0 -> Cons:Nil -> Cons:Nil -> c:c1:null_G[ITE][FALSE][ITE]
lt0 :: Cons:Nil -> Cons:Nil -> False:True:null_lt0
F :: Cons:Nil -> Cons:Nil -> c9:c10
c9 :: c9:c10
c10 :: c2:c3:null_F[ITE][FALSE][ITE] -> c5:null_LT0 -> c9:c10
F[ITE][FALSE][ITE] :: False:True:null_lt0 -> Cons:Nil -> Cons:Nil -> c2:c3:null_F[ITE][FALSE][ITE]
False :: False:True:null_lt0
c :: c7:c8 -> c:c1:null_G[ITE][FALSE][ITE]
True :: False:True:null_lt0
c1 :: c7:c8 -> c:c1:null_G[ITE][FALSE][ITE]
c2 :: c9:c10 -> c2:c3:null_F[ITE][FALSE][ITE]
c3 :: c9:c10 -> c2:c3:null_F[ITE][FALSE][ITE]
null_G[ITE][FALSE][ITE] :: c:c1:null_G[ITE][FALSE][ITE]
null_F[ITE][FALSE][ITE] :: c2:c3:null_F[ITE][FALSE][ITE]
null_lt0 :: False:True:null_lt0
null_LT0 :: c5:null_LT0

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

(17) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   Nil => 0
   c7 => 0
   c9 => 0
   False => 1
   True => 2
   null_G[ITE][FALSE][ITE] => 0
   null_F[ITE][FALSE][ITE] => 0
   null_lt0 => 0
   null_LT0 => 0

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z') -{ 1 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   F(z, z') -{ 1 }-> 1 + F[ITE][FALSE][ITE](lt0(z0, 1 + 0 + 0), z0, 1 + z1 + z2) + LT0(z0, 1 + 0 + 0) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   F[ITE][FALSE][ITE](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   F[ITE][FALSE][ITE](z, z', z'') -{ 0 }-> 1 + F(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   F[ITE][FALSE][ITE](z, z', z'') -{ 0 }-> 1 + F(z1, 1 + (1 + 0 + 0) + z2) :|: z'' = z2, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   G(z, z') -{ 1 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   G(z, z') -{ 1 }-> 1 + G[ITE][FALSE][ITE](lt0(z0, 1 + 0 + 0), z0, 1 + z1 + z2) + LT0(z0, 1 + 0 + 0) :|: z = z0, z1 >= 0, z' = 1 + z1 + z2, z0 >= 0, z2 >= 0
   G[ITE][FALSE][ITE](z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   G[ITE][FALSE][ITE](z, z', z'') -{ 0 }-> 1 + G(z0, z2) :|: z = 2, z1 >= 0, z'' = 1 + z1 + z2, z0 >= 0, z' = z0, z2 >= 0
   G[ITE][FALSE][ITE](z, z', z'') -{ 0 }-> 1 + G(z1, 1 + (1 + 0 + 0) + z2) :|: z'' = z2, z' = 1 + z0 + z1, z1 >= 0, z = 1, z0 >= 0, z2 >= 0
   LT0(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   LT0(z, z') -{ 1 }-> 1 + LT0(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   lt0(z, z') -{ 0 }-> lt0(z1, z3) :|: z1 >= 0, z' = 1 + z2 + z3, z0 >= 0, z = 1 + z0 + z1, z2 >= 0, z3 >= 0
   lt0(z, z') -{ 0 }-> 2 :|: z' = 1 + z0 + z1, z1 >= 0, z0 >= 0, z = 0
   lt0(z, z') -{ 0 }-> 1 :|: z = z0, z0 >= 0, z' = 0
   lt0(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

(19) CompleteCoflocoProof (FINISHED)
Transformed the RNTS (where only complete derivations are relevant) into cost relations for CoFloCo:

eq(start(V1, V, V15),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15),0,[fun1(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15),0,[fun3(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V15),0,[fun2(V1, V, V15, Out)],[V1 >= 0,V >= 0,V15 >= 0]).
eq(start(V1, V, V15),0,[fun4(V1, V, V15, Out)],[V1 >= 0,V >= 0,V15 >= 0]).
eq(start(V1, V, V15),0,[lt0(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[fun(V2, V4, Ret1)],[Out = 1 + Ret1,V2 >= 0,V = 1 + V4 + V5,V3 >= 0,V1 = 1 + V2 + V3,V5 >= 0,V4 >= 0]).
eq(fun1(V1, V, Out),1,[],[Out = 0,V1 = V6,V6 >= 0,V = 0]).
eq(fun1(V1, V, Out),1,[lt0(V9, 1 + 0 + 0, Ret010),fun2(Ret010, V9, 1 + V8 + V7, Ret01),fun(V9, 1 + 0 + 0, Ret11)],[Out = 1 + Ret01 + Ret11,V1 = V9,V8 >= 0,V = 1 + V7 + V8,V9 >= 0,V7 >= 0]).
eq(fun3(V1, V, Out),1,[],[Out = 0,V1 = V10,V10 >= 0,V = 0]).
eq(fun3(V1, V, Out),1,[lt0(V11, 1 + 0 + 0, Ret0101),fun4(Ret0101, V11, 1 + V13 + V12, Ret011),fun(V11, 1 + 0 + 0, Ret12)],[Out = 1 + Ret011 + Ret12,V1 = V11,V13 >= 0,V = 1 + V12 + V13,V11 >= 0,V12 >= 0]).
eq(fun2(V1, V, V15, Out),0,[fun1(V17, 1 + (1 + 0 + 0) + V16, Ret13)],[Out = 1 + Ret13,V15 = V16,V = 1 + V14 + V17,V17 >= 0,V1 = 1,V14 >= 0,V16 >= 0]).
eq(fun2(V1, V, V15, Out),0,[fun1(V19, V20, Ret14)],[Out = 1 + Ret14,V1 = 2,V18 >= 0,V15 = 1 + V18 + V20,V19 >= 0,V = V19,V20 >= 0]).
eq(fun4(V1, V, V15, Out),0,[fun3(V22, 1 + (1 + 0 + 0) + V21, Ret15)],[Out = 1 + Ret15,V15 = V21,V = 1 + V22 + V23,V22 >= 0,V1 = 1,V23 >= 0,V21 >= 0]).
eq(fun4(V1, V, V15, Out),0,[fun3(V26, V24, Ret16)],[Out = 1 + Ret16,V1 = 2,V25 >= 0,V15 = 1 + V24 + V25,V26 >= 0,V = V26,V24 >= 0]).
eq(lt0(V1, V, Out),0,[],[Out = 2,V = 1 + V27 + V28,V28 >= 0,V27 >= 0,V1 = 0]).
eq(lt0(V1, V, Out),0,[lt0(V32, V31, Ret)],[Out = Ret,V32 >= 0,V = 1 + V29 + V31,V30 >= 0,V1 = 1 + V30 + V32,V29 >= 0,V31 >= 0]).
eq(lt0(V1, V, Out),0,[],[Out = 1,V1 = V33,V33 >= 0,V = 0]).
eq(fun2(V1, V, V15, Out),0,[],[Out = 0,V35 >= 0,V15 = V36,V34 >= 0,V1 = V35,V = V34,V36 >= 0]).
eq(fun4(V1, V, V15, Out),0,[],[Out = 0,V39 >= 0,V15 = V37,V38 >= 0,V1 = V39,V = V38,V37 >= 0]).
eq(lt0(V1, V, Out),0,[],[Out = 0,V41 >= 0,V40 >= 0,V1 = V41,V = V40]).
eq(fun(V1, V, Out),0,[],[Out = 0,V42 >= 0,V43 >= 0,V1 = V42,V = V43]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun3(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun2(V1,V,V15,Out),[V1,V,V15],[Out]).
input_output_vars(fun4(V1,V,V15,Out),[V1,V,V15],[Out]).
input_output_vars(lt0(V1,V,Out),[V1,V],[Out]).


CoFloCo proof output:
Preprocessing Cost Relations
=====================================

#### Computed strongly connected components 
0. recursive  : [fun/3]
1. recursive  : [lt0/3]
2. recursive [non_tail] : [fun1/3,fun2/4]
3. recursive [non_tail] : [fun3/3,fun4/4]
4. non_recursive  : [start/3]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into fun/3
1. SCC is partially evaluated into lt0/3
2. SCC is partially evaluated into fun1/3
3. SCC is partially evaluated into fun3/3
4. SCC is partially evaluated into start/3

Control-Flow Refinement of Cost Relations
=====================================

### Specialization of cost equations fun/3 
* CE 19 is refined into CE [24] 
* CE 18 is refined into CE [25] 


### Cost equations --> "Loop" of fun/3 
* CEs [25] --> Loop 17 
* CEs [24] --> Loop 18 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [17]: [V,V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [17]:
  - RF of loop [17:1]:
    V
    V1


### Specialization of cost equations lt0/3 
* CE 23 is refined into CE [26] 
* CE 22 is refined into CE [27] 
* CE 20 is refined into CE [28] 
* CE 21 is refined into CE [29] 


### Cost equations --> "Loop" of lt0/3 
* CEs [29] --> Loop 19 
* CEs [26] --> Loop 20 
* CEs [27] --> Loop 21 
* CEs [28] --> Loop 22 

### Ranking functions of CR lt0(V1,V,Out) 
* RF of phase [19]: [V,V1]

#### Partial ranking functions of CR lt0(V1,V,Out) 
* Partial RF of phase [19]:
  - RF of loop [19:1]:
    V
    V1


### Specialization of cost equations fun1/3 
* CE 14 is refined into CE [30,31,32,33,34] 
* CE 17 is refined into CE [35] 
* CE 16 is refined into CE [36,37] 
* CE 15 is refined into CE [38] 


### Cost equations --> "Loop" of fun1/3 
* CEs [37] --> Loop 23 
* CEs [36] --> Loop 24 
* CEs [38] --> Loop 25 
* CEs [32,34] --> Loop 26 
* CEs [35] --> Loop 27 
* CEs [30,31,33] --> Loop 28 

### Ranking functions of CR fun1(V1,V,Out) 
* RF of phase [23,24]: [V1]
* RF of phase [25]: [V]

#### Partial ranking functions of CR fun1(V1,V,Out) 
* Partial RF of phase [23,24]:
  - RF of loop [23:1,24:1]:
    V1
* Partial RF of phase [25]:
  - RF of loop [25:1]:
    V


### Specialization of cost equations fun3/3 
* CE 10 is refined into CE [39,40,41,42,43] 
* CE 13 is refined into CE [44] 
* CE 12 is refined into CE [45,46] 
* CE 11 is refined into CE [47] 


### Cost equations --> "Loop" of fun3/3 
* CEs [46] --> Loop 29 
* CEs [45] --> Loop 30 
* CEs [47] --> Loop 31 
* CEs [41,43] --> Loop 32 
* CEs [44] --> Loop 33 
* CEs [39,40,42] --> Loop 34 

### Ranking functions of CR fun3(V1,V,Out) 
* RF of phase [29,30]: [V1]
* RF of phase [31]: [V]

#### Partial ranking functions of CR fun3(V1,V,Out) 
* Partial RF of phase [29,30]:
  - RF of loop [29:1,30:1]:
    V1
* Partial RF of phase [31]:
  - RF of loop [31:1]:
    V


### Specialization of cost equations start/3 
* CE 2 is refined into CE [48,49,50,51,52,53] 
* CE 4 is refined into CE [54,55,56,57,58,59] 
* CE 1 is refined into CE [60] 
* CE 3 is refined into CE [61,62,63,64,65] 
* CE 5 is refined into CE [66,67,68,69,70] 
* CE 6 is refined into CE [71,72] 
* CE 7 is refined into CE [73,74,75,76,77,78] 
* CE 8 is refined into CE [79,80,81,82,83,84] 
* CE 9 is refined into CE [85,86,87,88,89] 


### Cost equations --> "Loop" of start/3 
* CEs [74,80,86] --> Loop 35 
* CEs [48,49,50,51,52,53,54,55,56,57,58,59] --> Loop 36 
* CEs [61,62,63,64,65,66,67,68,69,70] --> Loop 37 
* CEs [60,71,72,73,75,76,77,78,79,81,82,83,84,85,87,88,89] --> Loop 38 

### Ranking functions of CR start(V1,V,V15) 

#### Partial ranking functions of CR start(V1,V,V15) 


Computing Bounds
=====================================

#### Cost of chains of fun(V1,V,Out):
* Chain [[17],18]: 1*it(17)+0
  Such that:it(17) =< V

  with precondition: [Out>=1,V1>=Out,V>=Out] 

* Chain [18]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of lt0(V1,V,Out):
* Chain [[19],22]: 0
  with precondition: [Out=2,V1>=1,V>=2] 

* Chain [[19],21]: 0
  with precondition: [Out=1,V1>=1,V>=1] 

* Chain [[19],20]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

* Chain [22]: 0
  with precondition: [V1=0,Out=2,V>=1] 

* Chain [21]: 0
  with precondition: [V=0,Out=1,V1>=0] 

* Chain [20]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun1(V1,V,Out):
* Chain [[25],28]: 1*it(25)+1
  Such that:it(25) =< V

  with precondition: [V1=0,Out>=3,2*V>=Out+1] 

* Chain [[25],27]: 1*it(25)+1
  Such that:it(25) =< V

  with precondition: [V1=0,Out>=2,2*V>=Out] 

* Chain [[23,24],[25],28]: 3*it(23)+1*it(25)+1
  Such that:it(25) =< 2*V1+V
aux(4) =< V1
it(23) =< aux(4)

  with precondition: [V1>=1,V>=1,Out>=5,2*V+7*V1>=Out+1] 

* Chain [[23,24],[25],27]: 3*it(23)+1*it(25)+1
  Such that:it(25) =< 2*V1+V
aux(5) =< V1
it(23) =< aux(5)

  with precondition: [V1>=1,V>=1,Out>=4,2*V+7*V1>=Out] 

* Chain [[23,24],28]: 3*it(23)+1
  Such that:aux(6) =< V1
it(23) =< aux(6)

  with precondition: [V1>=1,V>=1,Out>=3,3*V1+1>=Out] 

* Chain [[23,24],26]: 3*it(23)+3
  Such that:aux(8) =< V1
it(23) =< aux(8)

  with precondition: [V1>=2,V>=1,Out>=4,3*V1>=Out+1] 

* Chain [28]: 1
  with precondition: [Out=1,V1>=0,V>=1] 

* Chain [27]: 1
  with precondition: [V=0,Out=0,V1>=0] 

* Chain [26]: 3
  with precondition: [Out=2,V1>=1,V>=1] 


#### Cost of chains of fun3(V1,V,Out):
* Chain [[31],34]: 1*it(31)+1
  Such that:it(31) =< V

  with precondition: [V1=0,Out>=3,2*V>=Out+1] 

* Chain [[31],33]: 1*it(31)+1
  Such that:it(31) =< V

  with precondition: [V1=0,Out>=2,2*V>=Out] 

* Chain [[29,30],[31],34]: 3*it(29)+1*it(31)+1
  Such that:it(31) =< 2*V1+V
aux(16) =< V1
it(29) =< aux(16)

  with precondition: [V1>=1,V>=1,Out>=5,2*V+7*V1>=Out+1] 

* Chain [[29,30],[31],33]: 3*it(29)+1*it(31)+1
  Such that:it(31) =< 2*V1+V
aux(17) =< V1
it(29) =< aux(17)

  with precondition: [V1>=1,V>=1,Out>=4,2*V+7*V1>=Out] 

* Chain [[29,30],34]: 3*it(29)+1
  Such that:aux(18) =< V1
it(29) =< aux(18)

  with precondition: [V1>=1,V>=1,Out>=3,3*V1+1>=Out] 

* Chain [[29,30],32]: 3*it(29)+3
  Such that:aux(20) =< V1
it(29) =< aux(20)

  with precondition: [V1>=2,V>=1,Out>=4,3*V1>=Out+1] 

* Chain [34]: 1
  with precondition: [Out=1,V1>=0,V>=1] 

* Chain [33]: 1
  with precondition: [V=0,Out=0,V1>=0] 

* Chain [32]: 3
  with precondition: [Out=2,V1>=1,V>=1] 


#### Cost of chains of start(V1,V,V15):
* Chain [38]: 5*s(35)+24*s(39)+4*s(42)+3
  Such that:aux(25) =< V1
aux(26) =< 2*V1+V
aux(27) =< V
s(35) =< aux(27)
s(39) =< aux(25)
s(42) =< aux(26)

  with precondition: [V1>=0,V>=0] 

* Chain [37]: 4*s(53)+24*s(55)+4*s(58)+3
  Such that:aux(28) =< V
aux(29) =< 2*V+V15
aux(30) =< V15+2
s(55) =< aux(28)
s(58) =< aux(29)
s(53) =< aux(30)

  with precondition: [V1=1,V>=1,V15>=0] 

* Chain [36]: 4*s(69)+24*s(71)+4*s(74)+3
  Such that:aux(31) =< V
aux(32) =< 2*V+V15
aux(33) =< V15
s(71) =< aux(31)
s(74) =< aux(32)
s(69) =< aux(33)

  with precondition: [V1=2,V>=0,V15>=1] 

* Chain [35]: 1
  with precondition: [V=0,V1>=0] 


Closed-form bounds of start(V1,V,V15): 
-------------------------------------
* Chain [38] with precondition: [V1>=0,V>=0] 
    - Upper bound: 32*V1+9*V+3 
    - Complexity: n 
* Chain [37] with precondition: [V1=1,V>=1,V15>=0] 
    - Upper bound: 32*V+8*V15+11 
    - Complexity: n 
* Chain [36] with precondition: [V1=2,V>=0,V15>=1] 
    - Upper bound: 32*V+8*V15+3 
    - Complexity: n 
* Chain [35] with precondition: [V=0,V1>=0] 
    - Upper bound: 1 
    - Complexity: constant 

### Maximum cost of start(V1,V,V15): 5*V+2+max([32*V1+4*V,nat(2*V+V15)*4+19*V+max([nat(V15)*4,nat(V15+2)*4])])+1 
Asymptotic class: n 
* Total analysis performed in 523 ms.


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

(20)
BOUNDS(1, n^1)

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
   g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
   f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
   f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
   lt0(Nil, Cons(z0, z1)) -> True
   lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
   lt0(z0, Nil) -> False
   g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))
Tuples:
   G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
   G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
   F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
   F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
   LT0(Nil, Cons(z0, z1)) -> c4
   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   LT0(z0, Nil) -> c6
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   NOTEMPTY(Cons(z0, z1)) -> c11
   NOTEMPTY(Nil) -> c12
   NUMBER4(z0) -> c13
   GOAL(z0, z1) -> c14(F(z0, z1))
   GOAL(z0, z1) -> c15(G(z0, z1))
S tuples:
   LT0(Nil, Cons(z0, z1)) -> c4
   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   LT0(z0, Nil) -> c6
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   NOTEMPTY(Cons(z0, z1)) -> c11
   NOTEMPTY(Nil) -> c12
   NUMBER4(z0) -> c13
   GOAL(z0, z1) -> c14(F(z0, z1))
   GOAL(z0, z1) -> c15(G(z0, z1))
K tuples:none
Defined Rule Symbols:   lt0_2, g_2, f_2, notEmpty_1, number4_1, goal_2, g[Ite][False][Ite]_3, f[Ite][False][Ite]_3

Defined Pair Symbols:   G[ITE][FALSE][ITE]_3, F[ITE][FALSE][ITE]_3, LT0_2, G_2, F_2, NOTEMPTY_1, NUMBER4_1, GOAL_2

Compound Symbols:   c_1, c1_1, c2_1, c3_1, c4, c5_1, c6, c7, c8_2, c9, c10_2, c11, c12, c13, c14_1, c15_1


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

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

(24)
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:

   LT0(Nil, Cons(z0, z1)) -> c4
   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   LT0(z0, Nil) -> c6
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   NOTEMPTY(Cons(z0, z1)) -> c11
   NOTEMPTY(Nil) -> c12
   NUMBER4(z0) -> c13
   GOAL(z0, z1) -> c14(F(z0, z1))
   GOAL(z0, z1) -> c15(G(z0, z1))

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

   G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
   G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
   F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
   F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
   g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
   g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
   f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
   f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
   lt0(Nil, Cons(z0, z1)) -> True
   lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
   lt0(z0, Nil) -> False
   g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

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

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

(26)
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:

   LT0(Nil, Cons(z0, z1)) -> c4
   LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
   LT0(z0, Nil) -> c6
   G(z0, Nil) -> c7
   G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   F(z0, Nil) -> c9
   F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
   NOTEMPTY(Cons(z0, z1)) -> c11
   NOTEMPTY(Nil) -> c12
   NUMBER4(z0) -> c13
   GOAL(z0, z1) -> c14(F(z0, z1))
   GOAL(z0, z1) -> c15(G(z0, z1))

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

   G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
   G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
   F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
   F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
   g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
   g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
   f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
   f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
   lt0(Nil, Cons(z0, z1)) -> True
   lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
   lt0(z0, Nil) -> False
   g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
   notEmpty(Cons(z0, z1)) -> True
   notEmpty(Nil) -> False
   number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
   goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

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

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

(28)
Obligation:
Innermost TRS:
Rules:
LT0(Nil, Cons(z0, z1)) -> c4
LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
LT0(z0, Nil) -> c6
G(z0, Nil) -> c7
G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
F(z0, Nil) -> c9
F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
NOTEMPTY(Cons(z0, z1)) -> c11
NOTEMPTY(Nil) -> c12
NUMBER4(z0) -> c13
GOAL(z0, z1) -> c14(F(z0, z1))
GOAL(z0, z1) -> c15(G(z0, z1))
G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
lt0(Nil, Cons(z0, z1)) -> True
lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
lt0(z0, Nil) -> False
g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

Types:
LT0 :: Nil:Cons -> Nil:Cons -> c4:c5:c6
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
c4 :: c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
c6 :: c4:c5:c6
G :: Nil:Cons -> Nil:Cons -> c7:c8
c7 :: c7:c8
c8 :: c:c1 -> c4:c5:c6 -> c7:c8
G[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c:c1
lt0 :: Nil:Cons -> Nil:Cons -> False:True
F :: Nil:Cons -> Nil:Cons -> c9:c10
c9 :: c9:c10
c10 :: c2:c3 -> c4:c5:c6 -> c9:c10
F[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c2:c3
NOTEMPTY :: Nil:Cons -> c11:c12
c11 :: c11:c12
c12 :: c11:c12
NUMBER4 :: a -> c13
c13 :: c13
GOAL :: Nil:Cons -> Nil:Cons -> c14:c15
c14 :: c9:c10 -> c14:c15
c15 :: c7:c8 -> c14:c15
False :: False:True
c :: c7:c8 -> c:c1
True :: False:True
c1 :: c7:c8 -> c:c1
c2 :: c9:c10 -> c2:c3
c3 :: c9:c10 -> c2:c3
g[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
g :: Nil:Cons -> Nil:Cons -> Nil:Cons
f[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
f :: Nil:Cons -> Nil:Cons -> Nil:Cons
notEmpty :: Nil:Cons -> False:True
number4 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons -> Nil:Cons
hole_c4:c5:c61_16 :: c4:c5:c6
hole_Nil:Cons2_16 :: Nil:Cons
hole_c7:c83_16 :: c7:c8
hole_c:c14_16 :: c:c1
hole_False:True5_16 :: False:True
hole_c9:c106_16 :: c9:c10
hole_c2:c37_16 :: c2:c3
hole_c11:c128_16 :: c11:c12
hole_c139_16 :: c13
hole_a10_16 :: a
hole_c14:c1511_16 :: c14:c15
hole_b12_16 :: b
gen_c4:c5:c613_16 :: Nat -> c4:c5:c6
gen_Nil:Cons14_16 :: Nat -> Nil:Cons

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

(29) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
LT0, G, lt0, F, g, f

They will be analysed ascendingly in the following order:
LT0 < G
LT0 < F
lt0 < G
lt0 < F
lt0 < g
lt0 < f

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

(30)
Obligation:
Innermost TRS:
Rules:
LT0(Nil, Cons(z0, z1)) -> c4
LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
LT0(z0, Nil) -> c6
G(z0, Nil) -> c7
G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
F(z0, Nil) -> c9
F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
NOTEMPTY(Cons(z0, z1)) -> c11
NOTEMPTY(Nil) -> c12
NUMBER4(z0) -> c13
GOAL(z0, z1) -> c14(F(z0, z1))
GOAL(z0, z1) -> c15(G(z0, z1))
G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
lt0(Nil, Cons(z0, z1)) -> True
lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
lt0(z0, Nil) -> False
g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

Types:
LT0 :: Nil:Cons -> Nil:Cons -> c4:c5:c6
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
c4 :: c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
c6 :: c4:c5:c6
G :: Nil:Cons -> Nil:Cons -> c7:c8
c7 :: c7:c8
c8 :: c:c1 -> c4:c5:c6 -> c7:c8
G[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c:c1
lt0 :: Nil:Cons -> Nil:Cons -> False:True
F :: Nil:Cons -> Nil:Cons -> c9:c10
c9 :: c9:c10
c10 :: c2:c3 -> c4:c5:c6 -> c9:c10
F[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c2:c3
NOTEMPTY :: Nil:Cons -> c11:c12
c11 :: c11:c12
c12 :: c11:c12
NUMBER4 :: a -> c13
c13 :: c13
GOAL :: Nil:Cons -> Nil:Cons -> c14:c15
c14 :: c9:c10 -> c14:c15
c15 :: c7:c8 -> c14:c15
False :: False:True
c :: c7:c8 -> c:c1
True :: False:True
c1 :: c7:c8 -> c:c1
c2 :: c9:c10 -> c2:c3
c3 :: c9:c10 -> c2:c3
g[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
g :: Nil:Cons -> Nil:Cons -> Nil:Cons
f[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
f :: Nil:Cons -> Nil:Cons -> Nil:Cons
notEmpty :: Nil:Cons -> False:True
number4 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons -> Nil:Cons
hole_c4:c5:c61_16 :: c4:c5:c6
hole_Nil:Cons2_16 :: Nil:Cons
hole_c7:c83_16 :: c7:c8
hole_c:c14_16 :: c:c1
hole_False:True5_16 :: False:True
hole_c9:c106_16 :: c9:c10
hole_c2:c37_16 :: c2:c3
hole_c11:c128_16 :: c11:c12
hole_c139_16 :: c13
hole_a10_16 :: a
hole_c14:c1511_16 :: c14:c15
hole_b12_16 :: b
gen_c4:c5:c613_16 :: Nat -> c4:c5:c6
gen_Nil:Cons14_16 :: Nat -> Nil:Cons


Generator Equations:
gen_c4:c5:c613_16(0) <=> c4
gen_c4:c5:c613_16(+(x, 1)) <=> c5(gen_c4:c5:c613_16(x))
gen_Nil:Cons14_16(0) <=> Nil
gen_Nil:Cons14_16(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons14_16(x))


The following defined symbols remain to be analysed:
LT0, G, lt0, F, g, f

They will be analysed ascendingly in the following order:
LT0 < G
LT0 < F
lt0 < G
lt0 < F
lt0 < g
lt0 < f

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

(31) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LT0(gen_Nil:Cons14_16(n16_16), gen_Nil:Cons14_16(+(1, n16_16))) -> gen_c4:c5:c613_16(n16_16), rt in Omega(1 + n16_16)

Induction Base:
LT0(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(+(1, 0))) ->_R^Omega(1)
c4

Induction Step:
LT0(gen_Nil:Cons14_16(+(n16_16, 1)), gen_Nil:Cons14_16(+(1, +(n16_16, 1)))) ->_R^Omega(1)
c5(LT0(gen_Nil:Cons14_16(n16_16), gen_Nil:Cons14_16(+(1, n16_16)))) ->_IH
c5(gen_c4:c5:c613_16(c17_16))

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

(32)
Complex Obligation (BEST)

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

(33)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
LT0(Nil, Cons(z0, z1)) -> c4
LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
LT0(z0, Nil) -> c6
G(z0, Nil) -> c7
G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
F(z0, Nil) -> c9
F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
NOTEMPTY(Cons(z0, z1)) -> c11
NOTEMPTY(Nil) -> c12
NUMBER4(z0) -> c13
GOAL(z0, z1) -> c14(F(z0, z1))
GOAL(z0, z1) -> c15(G(z0, z1))
G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
lt0(Nil, Cons(z0, z1)) -> True
lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
lt0(z0, Nil) -> False
g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

Types:
LT0 :: Nil:Cons -> Nil:Cons -> c4:c5:c6
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
c4 :: c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
c6 :: c4:c5:c6
G :: Nil:Cons -> Nil:Cons -> c7:c8
c7 :: c7:c8
c8 :: c:c1 -> c4:c5:c6 -> c7:c8
G[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c:c1
lt0 :: Nil:Cons -> Nil:Cons -> False:True
F :: Nil:Cons -> Nil:Cons -> c9:c10
c9 :: c9:c10
c10 :: c2:c3 -> c4:c5:c6 -> c9:c10
F[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c2:c3
NOTEMPTY :: Nil:Cons -> c11:c12
c11 :: c11:c12
c12 :: c11:c12
NUMBER4 :: a -> c13
c13 :: c13
GOAL :: Nil:Cons -> Nil:Cons -> c14:c15
c14 :: c9:c10 -> c14:c15
c15 :: c7:c8 -> c14:c15
False :: False:True
c :: c7:c8 -> c:c1
True :: False:True
c1 :: c7:c8 -> c:c1
c2 :: c9:c10 -> c2:c3
c3 :: c9:c10 -> c2:c3
g[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
g :: Nil:Cons -> Nil:Cons -> Nil:Cons
f[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
f :: Nil:Cons -> Nil:Cons -> Nil:Cons
notEmpty :: Nil:Cons -> False:True
number4 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons -> Nil:Cons
hole_c4:c5:c61_16 :: c4:c5:c6
hole_Nil:Cons2_16 :: Nil:Cons
hole_c7:c83_16 :: c7:c8
hole_c:c14_16 :: c:c1
hole_False:True5_16 :: False:True
hole_c9:c106_16 :: c9:c10
hole_c2:c37_16 :: c2:c3
hole_c11:c128_16 :: c11:c12
hole_c139_16 :: c13
hole_a10_16 :: a
hole_c14:c1511_16 :: c14:c15
hole_b12_16 :: b
gen_c4:c5:c613_16 :: Nat -> c4:c5:c6
gen_Nil:Cons14_16 :: Nat -> Nil:Cons


Generator Equations:
gen_c4:c5:c613_16(0) <=> c4
gen_c4:c5:c613_16(+(x, 1)) <=> c5(gen_c4:c5:c613_16(x))
gen_Nil:Cons14_16(0) <=> Nil
gen_Nil:Cons14_16(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons14_16(x))


The following defined symbols remain to be analysed:
LT0, G, lt0, F, g, f

They will be analysed ascendingly in the following order:
LT0 < G
LT0 < F
lt0 < G
lt0 < F
lt0 < g
lt0 < f

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

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

(35)
BOUNDS(n^1, INF)

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

(36)
Obligation:
Innermost TRS:
Rules:
LT0(Nil, Cons(z0, z1)) -> c4
LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
LT0(z0, Nil) -> c6
G(z0, Nil) -> c7
G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
F(z0, Nil) -> c9
F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
NOTEMPTY(Cons(z0, z1)) -> c11
NOTEMPTY(Nil) -> c12
NUMBER4(z0) -> c13
GOAL(z0, z1) -> c14(F(z0, z1))
GOAL(z0, z1) -> c15(G(z0, z1))
G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
lt0(Nil, Cons(z0, z1)) -> True
lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
lt0(z0, Nil) -> False
g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

Types:
LT0 :: Nil:Cons -> Nil:Cons -> c4:c5:c6
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
c4 :: c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
c6 :: c4:c5:c6
G :: Nil:Cons -> Nil:Cons -> c7:c8
c7 :: c7:c8
c8 :: c:c1 -> c4:c5:c6 -> c7:c8
G[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c:c1
lt0 :: Nil:Cons -> Nil:Cons -> False:True
F :: Nil:Cons -> Nil:Cons -> c9:c10
c9 :: c9:c10
c10 :: c2:c3 -> c4:c5:c6 -> c9:c10
F[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c2:c3
NOTEMPTY :: Nil:Cons -> c11:c12
c11 :: c11:c12
c12 :: c11:c12
NUMBER4 :: a -> c13
c13 :: c13
GOAL :: Nil:Cons -> Nil:Cons -> c14:c15
c14 :: c9:c10 -> c14:c15
c15 :: c7:c8 -> c14:c15
False :: False:True
c :: c7:c8 -> c:c1
True :: False:True
c1 :: c7:c8 -> c:c1
c2 :: c9:c10 -> c2:c3
c3 :: c9:c10 -> c2:c3
g[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
g :: Nil:Cons -> Nil:Cons -> Nil:Cons
f[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
f :: Nil:Cons -> Nil:Cons -> Nil:Cons
notEmpty :: Nil:Cons -> False:True
number4 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons -> Nil:Cons
hole_c4:c5:c61_16 :: c4:c5:c6
hole_Nil:Cons2_16 :: Nil:Cons
hole_c7:c83_16 :: c7:c8
hole_c:c14_16 :: c:c1
hole_False:True5_16 :: False:True
hole_c9:c106_16 :: c9:c10
hole_c2:c37_16 :: c2:c3
hole_c11:c128_16 :: c11:c12
hole_c139_16 :: c13
hole_a10_16 :: a
hole_c14:c1511_16 :: c14:c15
hole_b12_16 :: b
gen_c4:c5:c613_16 :: Nat -> c4:c5:c6
gen_Nil:Cons14_16 :: Nat -> Nil:Cons


Lemmas:
LT0(gen_Nil:Cons14_16(n16_16), gen_Nil:Cons14_16(+(1, n16_16))) -> gen_c4:c5:c613_16(n16_16), rt in Omega(1 + n16_16)


Generator Equations:
gen_c4:c5:c613_16(0) <=> c4
gen_c4:c5:c613_16(+(x, 1)) <=> c5(gen_c4:c5:c613_16(x))
gen_Nil:Cons14_16(0) <=> Nil
gen_Nil:Cons14_16(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons14_16(x))


The following defined symbols remain to be analysed:
lt0, G, F, g, f

They will be analysed ascendingly in the following order:
lt0 < G
lt0 < F
lt0 < g
lt0 < f

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

(37) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt0(gen_Nil:Cons14_16(n696_16), gen_Nil:Cons14_16(+(1, n696_16))) -> True, rt in Omega(0)

Induction Base:
lt0(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(+(1, 0))) ->_R^Omega(0)
True

Induction Step:
lt0(gen_Nil:Cons14_16(+(n696_16, 1)), gen_Nil:Cons14_16(+(1, +(n696_16, 1)))) ->_R^Omega(0)
lt0(gen_Nil:Cons14_16(n696_16), gen_Nil:Cons14_16(+(1, n696_16))) ->_IH
True

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

(38)
Obligation:
Innermost TRS:
Rules:
LT0(Nil, Cons(z0, z1)) -> c4
LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
LT0(z0, Nil) -> c6
G(z0, Nil) -> c7
G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
F(z0, Nil) -> c9
F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
NOTEMPTY(Cons(z0, z1)) -> c11
NOTEMPTY(Nil) -> c12
NUMBER4(z0) -> c13
GOAL(z0, z1) -> c14(F(z0, z1))
GOAL(z0, z1) -> c15(G(z0, z1))
G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
lt0(Nil, Cons(z0, z1)) -> True
lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
lt0(z0, Nil) -> False
g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

Types:
LT0 :: Nil:Cons -> Nil:Cons -> c4:c5:c6
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
c4 :: c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
c6 :: c4:c5:c6
G :: Nil:Cons -> Nil:Cons -> c7:c8
c7 :: c7:c8
c8 :: c:c1 -> c4:c5:c6 -> c7:c8
G[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c:c1
lt0 :: Nil:Cons -> Nil:Cons -> False:True
F :: Nil:Cons -> Nil:Cons -> c9:c10
c9 :: c9:c10
c10 :: c2:c3 -> c4:c5:c6 -> c9:c10
F[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c2:c3
NOTEMPTY :: Nil:Cons -> c11:c12
c11 :: c11:c12
c12 :: c11:c12
NUMBER4 :: a -> c13
c13 :: c13
GOAL :: Nil:Cons -> Nil:Cons -> c14:c15
c14 :: c9:c10 -> c14:c15
c15 :: c7:c8 -> c14:c15
False :: False:True
c :: c7:c8 -> c:c1
True :: False:True
c1 :: c7:c8 -> c:c1
c2 :: c9:c10 -> c2:c3
c3 :: c9:c10 -> c2:c3
g[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
g :: Nil:Cons -> Nil:Cons -> Nil:Cons
f[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
f :: Nil:Cons -> Nil:Cons -> Nil:Cons
notEmpty :: Nil:Cons -> False:True
number4 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons -> Nil:Cons
hole_c4:c5:c61_16 :: c4:c5:c6
hole_Nil:Cons2_16 :: Nil:Cons
hole_c7:c83_16 :: c7:c8
hole_c:c14_16 :: c:c1
hole_False:True5_16 :: False:True
hole_c9:c106_16 :: c9:c10
hole_c2:c37_16 :: c2:c3
hole_c11:c128_16 :: c11:c12
hole_c139_16 :: c13
hole_a10_16 :: a
hole_c14:c1511_16 :: c14:c15
hole_b12_16 :: b
gen_c4:c5:c613_16 :: Nat -> c4:c5:c6
gen_Nil:Cons14_16 :: Nat -> Nil:Cons


Lemmas:
LT0(gen_Nil:Cons14_16(n16_16), gen_Nil:Cons14_16(+(1, n16_16))) -> gen_c4:c5:c613_16(n16_16), rt in Omega(1 + n16_16)
lt0(gen_Nil:Cons14_16(n696_16), gen_Nil:Cons14_16(+(1, n696_16))) -> True, rt in Omega(0)


Generator Equations:
gen_c4:c5:c613_16(0) <=> c4
gen_c4:c5:c613_16(+(x, 1)) <=> c5(gen_c4:c5:c613_16(x))
gen_Nil:Cons14_16(0) <=> Nil
gen_Nil:Cons14_16(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons14_16(x))


The following defined symbols remain to be analysed:
G, F, g, f
----------------------------------------

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
G(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n1156_16)) -> *15_16, rt in Omega(n1156_16)

Induction Base:
G(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(0))

Induction Step:
G(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(+(n1156_16, 1))) ->_R^Omega(1)
c8(G[ITE][FALSE][ITE](lt0(gen_Nil:Cons14_16(0), Cons(Nil, Nil)), gen_Nil:Cons14_16(0), Cons(Nil, gen_Nil:Cons14_16(n1156_16))), LT0(gen_Nil:Cons14_16(0), Cons(Nil, Nil))) ->_L^Omega(0)
c8(G[ITE][FALSE][ITE](True, gen_Nil:Cons14_16(0), Cons(Nil, gen_Nil:Cons14_16(n1156_16))), LT0(gen_Nil:Cons14_16(0), Cons(Nil, Nil))) ->_R^Omega(0)
c8(c1(G(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n1156_16))), LT0(gen_Nil:Cons14_16(0), Cons(Nil, Nil))) ->_IH
c8(c1(*15_16), LT0(gen_Nil:Cons14_16(0), Cons(Nil, Nil))) ->_L^Omega(1)
c8(c1(*15_16), gen_c4:c5:c613_16(0))

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

(40)
Obligation:
Innermost TRS:
Rules:
LT0(Nil, Cons(z0, z1)) -> c4
LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
LT0(z0, Nil) -> c6
G(z0, Nil) -> c7
G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
F(z0, Nil) -> c9
F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
NOTEMPTY(Cons(z0, z1)) -> c11
NOTEMPTY(Nil) -> c12
NUMBER4(z0) -> c13
GOAL(z0, z1) -> c14(F(z0, z1))
GOAL(z0, z1) -> c15(G(z0, z1))
G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
lt0(Nil, Cons(z0, z1)) -> True
lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
lt0(z0, Nil) -> False
g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

Types:
LT0 :: Nil:Cons -> Nil:Cons -> c4:c5:c6
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
c4 :: c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
c6 :: c4:c5:c6
G :: Nil:Cons -> Nil:Cons -> c7:c8
c7 :: c7:c8
c8 :: c:c1 -> c4:c5:c6 -> c7:c8
G[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c:c1
lt0 :: Nil:Cons -> Nil:Cons -> False:True
F :: Nil:Cons -> Nil:Cons -> c9:c10
c9 :: c9:c10
c10 :: c2:c3 -> c4:c5:c6 -> c9:c10
F[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c2:c3
NOTEMPTY :: Nil:Cons -> c11:c12
c11 :: c11:c12
c12 :: c11:c12
NUMBER4 :: a -> c13
c13 :: c13
GOAL :: Nil:Cons -> Nil:Cons -> c14:c15
c14 :: c9:c10 -> c14:c15
c15 :: c7:c8 -> c14:c15
False :: False:True
c :: c7:c8 -> c:c1
True :: False:True
c1 :: c7:c8 -> c:c1
c2 :: c9:c10 -> c2:c3
c3 :: c9:c10 -> c2:c3
g[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
g :: Nil:Cons -> Nil:Cons -> Nil:Cons
f[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
f :: Nil:Cons -> Nil:Cons -> Nil:Cons
notEmpty :: Nil:Cons -> False:True
number4 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons -> Nil:Cons
hole_c4:c5:c61_16 :: c4:c5:c6
hole_Nil:Cons2_16 :: Nil:Cons
hole_c7:c83_16 :: c7:c8
hole_c:c14_16 :: c:c1
hole_False:True5_16 :: False:True
hole_c9:c106_16 :: c9:c10
hole_c2:c37_16 :: c2:c3
hole_c11:c128_16 :: c11:c12
hole_c139_16 :: c13
hole_a10_16 :: a
hole_c14:c1511_16 :: c14:c15
hole_b12_16 :: b
gen_c4:c5:c613_16 :: Nat -> c4:c5:c6
gen_Nil:Cons14_16 :: Nat -> Nil:Cons


Lemmas:
LT0(gen_Nil:Cons14_16(n16_16), gen_Nil:Cons14_16(+(1, n16_16))) -> gen_c4:c5:c613_16(n16_16), rt in Omega(1 + n16_16)
lt0(gen_Nil:Cons14_16(n696_16), gen_Nil:Cons14_16(+(1, n696_16))) -> True, rt in Omega(0)
G(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n1156_16)) -> *15_16, rt in Omega(n1156_16)


Generator Equations:
gen_c4:c5:c613_16(0) <=> c4
gen_c4:c5:c613_16(+(x, 1)) <=> c5(gen_c4:c5:c613_16(x))
gen_Nil:Cons14_16(0) <=> Nil
gen_Nil:Cons14_16(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons14_16(x))


The following defined symbols remain to be analysed:
F, g, f
----------------------------------------

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
F(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n4129_16)) -> *15_16, rt in Omega(n4129_16)

Induction Base:
F(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(0))

Induction Step:
F(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(+(n4129_16, 1))) ->_R^Omega(1)
c10(F[ITE][FALSE][ITE](lt0(gen_Nil:Cons14_16(0), Cons(Nil, Nil)), gen_Nil:Cons14_16(0), Cons(Nil, gen_Nil:Cons14_16(n4129_16))), LT0(gen_Nil:Cons14_16(0), Cons(Nil, Nil))) ->_L^Omega(0)
c10(F[ITE][FALSE][ITE](True, gen_Nil:Cons14_16(0), Cons(Nil, gen_Nil:Cons14_16(n4129_16))), LT0(gen_Nil:Cons14_16(0), Cons(Nil, Nil))) ->_R^Omega(0)
c10(c3(F(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n4129_16))), LT0(gen_Nil:Cons14_16(0), Cons(Nil, Nil))) ->_IH
c10(c3(*15_16), LT0(gen_Nil:Cons14_16(0), Cons(Nil, Nil))) ->_L^Omega(1)
c10(c3(*15_16), gen_c4:c5:c613_16(0))

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

(42)
Obligation:
Innermost TRS:
Rules:
LT0(Nil, Cons(z0, z1)) -> c4
LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
LT0(z0, Nil) -> c6
G(z0, Nil) -> c7
G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
F(z0, Nil) -> c9
F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
NOTEMPTY(Cons(z0, z1)) -> c11
NOTEMPTY(Nil) -> c12
NUMBER4(z0) -> c13
GOAL(z0, z1) -> c14(F(z0, z1))
GOAL(z0, z1) -> c15(G(z0, z1))
G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
lt0(Nil, Cons(z0, z1)) -> True
lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
lt0(z0, Nil) -> False
g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

Types:
LT0 :: Nil:Cons -> Nil:Cons -> c4:c5:c6
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
c4 :: c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
c6 :: c4:c5:c6
G :: Nil:Cons -> Nil:Cons -> c7:c8
c7 :: c7:c8
c8 :: c:c1 -> c4:c5:c6 -> c7:c8
G[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c:c1
lt0 :: Nil:Cons -> Nil:Cons -> False:True
F :: Nil:Cons -> Nil:Cons -> c9:c10
c9 :: c9:c10
c10 :: c2:c3 -> c4:c5:c6 -> c9:c10
F[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c2:c3
NOTEMPTY :: Nil:Cons -> c11:c12
c11 :: c11:c12
c12 :: c11:c12
NUMBER4 :: a -> c13
c13 :: c13
GOAL :: Nil:Cons -> Nil:Cons -> c14:c15
c14 :: c9:c10 -> c14:c15
c15 :: c7:c8 -> c14:c15
False :: False:True
c :: c7:c8 -> c:c1
True :: False:True
c1 :: c7:c8 -> c:c1
c2 :: c9:c10 -> c2:c3
c3 :: c9:c10 -> c2:c3
g[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
g :: Nil:Cons -> Nil:Cons -> Nil:Cons
f[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
f :: Nil:Cons -> Nil:Cons -> Nil:Cons
notEmpty :: Nil:Cons -> False:True
number4 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons -> Nil:Cons
hole_c4:c5:c61_16 :: c4:c5:c6
hole_Nil:Cons2_16 :: Nil:Cons
hole_c7:c83_16 :: c7:c8
hole_c:c14_16 :: c:c1
hole_False:True5_16 :: False:True
hole_c9:c106_16 :: c9:c10
hole_c2:c37_16 :: c2:c3
hole_c11:c128_16 :: c11:c12
hole_c139_16 :: c13
hole_a10_16 :: a
hole_c14:c1511_16 :: c14:c15
hole_b12_16 :: b
gen_c4:c5:c613_16 :: Nat -> c4:c5:c6
gen_Nil:Cons14_16 :: Nat -> Nil:Cons


Lemmas:
LT0(gen_Nil:Cons14_16(n16_16), gen_Nil:Cons14_16(+(1, n16_16))) -> gen_c4:c5:c613_16(n16_16), rt in Omega(1 + n16_16)
lt0(gen_Nil:Cons14_16(n696_16), gen_Nil:Cons14_16(+(1, n696_16))) -> True, rt in Omega(0)
G(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n1156_16)) -> *15_16, rt in Omega(n1156_16)
F(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n4129_16)) -> *15_16, rt in Omega(n4129_16)


Generator Equations:
gen_c4:c5:c613_16(0) <=> c4
gen_c4:c5:c613_16(+(x, 1)) <=> c5(gen_c4:c5:c613_16(x))
gen_Nil:Cons14_16(0) <=> Nil
gen_Nil:Cons14_16(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons14_16(x))


The following defined symbols remain to be analysed:
g, f
----------------------------------------

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n10234_16)) -> gen_Nil:Cons14_16(4), rt in Omega(0)

Induction Base:
g(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(0)) ->_R^Omega(0)
Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))

Induction Step:
g(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(+(n10234_16, 1))) ->_R^Omega(0)
g[Ite][False][Ite](lt0(gen_Nil:Cons14_16(0), Cons(Nil, Nil)), gen_Nil:Cons14_16(0), Cons(Nil, gen_Nil:Cons14_16(n10234_16))) ->_L^Omega(0)
g[Ite][False][Ite](True, gen_Nil:Cons14_16(0), Cons(Nil, gen_Nil:Cons14_16(n10234_16))) ->_R^Omega(0)
g(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n10234_16)) ->_IH
gen_Nil:Cons14_16(4)

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

(44)
Obligation:
Innermost TRS:
Rules:
LT0(Nil, Cons(z0, z1)) -> c4
LT0(Cons(z0, z1), Cons(z2, z3)) -> c5(LT0(z1, z3))
LT0(z0, Nil) -> c6
G(z0, Nil) -> c7
G(z0, Cons(z1, z2)) -> c8(G[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
F(z0, Nil) -> c9
F(z0, Cons(z1, z2)) -> c10(F[ITE][FALSE][ITE](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2)), LT0(z0, Cons(Nil, Nil)))
NOTEMPTY(Cons(z0, z1)) -> c11
NOTEMPTY(Nil) -> c12
NUMBER4(z0) -> c13
GOAL(z0, z1) -> c14(F(z0, z1))
GOAL(z0, z1) -> c15(G(z0, z1))
G[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c(G(z1, Cons(Cons(Nil, Nil), z2)))
G[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c1(G(z0, z2))
F[ITE][FALSE][ITE](False, Cons(z0, z1), z2) -> c2(F(z1, Cons(Cons(Nil, Nil), z2)))
F[ITE][FALSE][ITE](True, z0, Cons(z1, z2)) -> c3(F(z0, z2))
g[Ite][False][Ite](False, Cons(z0, z1), z2) -> g(z1, Cons(Cons(Nil, Nil), z2))
g[Ite][False][Ite](True, z0, Cons(z1, z2)) -> g(z0, z2)
f[Ite][False][Ite](False, Cons(z0, z1), z2) -> f(z1, Cons(Cons(Nil, Nil), z2))
f[Ite][False][Ite](True, z0, Cons(z1, z2)) -> f(z0, z2)
lt0(Nil, Cons(z0, z1)) -> True
lt0(Cons(z0, z1), Cons(z2, z3)) -> lt0(z1, z3)
lt0(z0, Nil) -> False
g(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
g(z0, Cons(z1, z2)) -> g[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
f(z0, Nil) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
f(z0, Cons(z1, z2)) -> f[Ite][False][Ite](lt0(z0, Cons(Nil, Nil)), z0, Cons(z1, z2))
notEmpty(Cons(z0, z1)) -> True
notEmpty(Nil) -> False
number4(z0) -> Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))
goal(z0, z1) -> Cons(f(z0, z1), Cons(g(z0, z1), Nil))

Types:
LT0 :: Nil:Cons -> Nil:Cons -> c4:c5:c6
Nil :: Nil:Cons
Cons :: Nil:Cons -> Nil:Cons -> Nil:Cons
c4 :: c4:c5:c6
c5 :: c4:c5:c6 -> c4:c5:c6
c6 :: c4:c5:c6
G :: Nil:Cons -> Nil:Cons -> c7:c8
c7 :: c7:c8
c8 :: c:c1 -> c4:c5:c6 -> c7:c8
G[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c:c1
lt0 :: Nil:Cons -> Nil:Cons -> False:True
F :: Nil:Cons -> Nil:Cons -> c9:c10
c9 :: c9:c10
c10 :: c2:c3 -> c4:c5:c6 -> c9:c10
F[ITE][FALSE][ITE] :: False:True -> Nil:Cons -> Nil:Cons -> c2:c3
NOTEMPTY :: Nil:Cons -> c11:c12
c11 :: c11:c12
c12 :: c11:c12
NUMBER4 :: a -> c13
c13 :: c13
GOAL :: Nil:Cons -> Nil:Cons -> c14:c15
c14 :: c9:c10 -> c14:c15
c15 :: c7:c8 -> c14:c15
False :: False:True
c :: c7:c8 -> c:c1
True :: False:True
c1 :: c7:c8 -> c:c1
c2 :: c9:c10 -> c2:c3
c3 :: c9:c10 -> c2:c3
g[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
g :: Nil:Cons -> Nil:Cons -> Nil:Cons
f[Ite][False][Ite] :: False:True -> Nil:Cons -> Nil:Cons -> Nil:Cons
f :: Nil:Cons -> Nil:Cons -> Nil:Cons
notEmpty :: Nil:Cons -> False:True
number4 :: b -> Nil:Cons
goal :: Nil:Cons -> Nil:Cons -> Nil:Cons
hole_c4:c5:c61_16 :: c4:c5:c6
hole_Nil:Cons2_16 :: Nil:Cons
hole_c7:c83_16 :: c7:c8
hole_c:c14_16 :: c:c1
hole_False:True5_16 :: False:True
hole_c9:c106_16 :: c9:c10
hole_c2:c37_16 :: c2:c3
hole_c11:c128_16 :: c11:c12
hole_c139_16 :: c13
hole_a10_16 :: a
hole_c14:c1511_16 :: c14:c15
hole_b12_16 :: b
gen_c4:c5:c613_16 :: Nat -> c4:c5:c6
gen_Nil:Cons14_16 :: Nat -> Nil:Cons


Lemmas:
LT0(gen_Nil:Cons14_16(n16_16), gen_Nil:Cons14_16(+(1, n16_16))) -> gen_c4:c5:c613_16(n16_16), rt in Omega(1 + n16_16)
lt0(gen_Nil:Cons14_16(n696_16), gen_Nil:Cons14_16(+(1, n696_16))) -> True, rt in Omega(0)
G(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n1156_16)) -> *15_16, rt in Omega(n1156_16)
F(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n4129_16)) -> *15_16, rt in Omega(n4129_16)
g(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n10234_16)) -> gen_Nil:Cons14_16(4), rt in Omega(0)


Generator Equations:
gen_c4:c5:c613_16(0) <=> c4
gen_c4:c5:c613_16(+(x, 1)) <=> c5(gen_c4:c5:c613_16(x))
gen_Nil:Cons14_16(0) <=> Nil
gen_Nil:Cons14_16(+(x, 1)) <=> Cons(Nil, gen_Nil:Cons14_16(x))


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

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
f(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n11508_16)) -> gen_Nil:Cons14_16(4), rt in Omega(0)

Induction Base:
f(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(0)) ->_R^Omega(0)
Cons(Nil, Cons(Nil, Cons(Nil, Cons(Nil, Nil))))

Induction Step:
f(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(+(n11508_16, 1))) ->_R^Omega(0)
f[Ite][False][Ite](lt0(gen_Nil:Cons14_16(0), Cons(Nil, Nil)), gen_Nil:Cons14_16(0), Cons(Nil, gen_Nil:Cons14_16(n11508_16))) ->_L^Omega(0)
f[Ite][False][Ite](True, gen_Nil:Cons14_16(0), Cons(Nil, gen_Nil:Cons14_16(n11508_16))) ->_R^Omega(0)
f(gen_Nil:Cons14_16(0), gen_Nil:Cons14_16(n11508_16)) ->_IH
gen_Nil:Cons14_16(4)

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

(46)
BOUNDS(1, INF)