WORST_CASE(Omega(n^1),O(n^2))
proof of input_3LJmaNnLO8.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^2).

(0) CpxTRS
(1) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(2) CdtProblem
    (3) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CdtProblem
    (5) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (8) CdtProblem
    (9) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CdtProblem
    (11) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRelTRS
    (13) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxWeightedTrs
    (15) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxTypedWeightedTrs
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
    (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) CompleteCoflocoProof [FINISHED, 2930 ms]
    (22) BOUNDS(1, n^2)
(23) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(24) CdtProblem
    (25) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CpxRelTRS
    (27) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CpxRelTRS
    (29) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) typed CpxTrs
    (31) OrderProof [LOWER BOUND(ID), 16 ms]
    (32) typed CpxTrs
    (33) RewriteLemmaProof [LOWER BOUND(ID), 356 ms]
    (34) BEST
        (35) proven lower bound
            (36) LowerBoundPropagationProof [FINISHED, 0 ms]
            (37) BOUNDS(n^1, INF)
        (38) typed CpxTrs
            (39) RewriteLemmaProof [LOWER BOUND(ID), 16 ms]
            (40) typed CpxTrs
            (41) RewriteLemmaProof [LOWER BOUND(ID), 13 ms]
            (42) typed CpxTrs
            (43) RewriteLemmaProof [LOWER BOUND(ID), 77 ms]
            (44) typed CpxTrs
            (45) RewriteLemmaProof [LOWER BOUND(ID), 50 ms]
            (46) typed CpxTrs
            (47) RewriteLemmaProof [LOWER BOUND(ID), 39 ms]
            (48) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   rev(x) -> if(x, eq(0, length(x)), nil, 0, length(x))
   if(x, true, z, c, l) -> z
   if(x, false, z, c, l) -> help(s(c), l, x, z)
   help(c, l, cons(x, y), z) -> if(append(y, cons(x, nil)), ge(c, l), cons(x, z), c, l)
   append(nil, y) -> y
   append(cons(x, y), z) -> cons(x, append(y, z))
   length(nil) -> 0
   length(cons(x, y)) -> s(length(y))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

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

(2)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   rev(z0) -> if(z0, eq(0, length(z0)), nil, 0, length(z0))
   if(z0, true, z1, z2, z3) -> z1
   if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
   help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   GE(z0, 0) -> c
   GE(0, s(z0)) -> c1
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   REV(z0) -> c3(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   REV(z0) -> c4(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   IF(z0, true, z1, z2, z3) -> c5
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(nil) -> c11
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
S tuples:
   GE(z0, 0) -> c
   GE(0, s(z0)) -> c1
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   REV(z0) -> c3(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   REV(z0) -> c4(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   IF(z0, true, z1, z2, z3) -> c5
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(nil) -> c11
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
K tuples:none
Defined Rule Symbols:   ge_2, rev_1, if_5, help_4, append_2, length_1

Defined Pair Symbols:   GE_2, REV_1, IF_5, HELP_4, APPEND_2, LENGTH_1

Compound Symbols:   c, c1, c2_1, c3_2, c4_2, c5, c6_1, c7_2, c8_2, c9, c10_1, c11, c12_1


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   LENGTH(nil) -> c11
   APPEND(nil, z0) -> c9
   GE(z0, 0) -> c
   GE(0, s(z0)) -> c1
   IF(z0, true, z1, z2, z3) -> c5

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   rev(z0) -> if(z0, eq(0, length(z0)), nil, 0, length(z0))
   if(z0, true, z1, z2, z3) -> z1
   if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
   help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   REV(z0) -> c3(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   REV(z0) -> c4(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
S tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   REV(z0) -> c3(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   REV(z0) -> c4(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
K tuples:none
Defined Rule Symbols:   ge_2, rev_1, if_5, help_4, append_2, length_1

Defined Pair Symbols:   GE_2, REV_1, IF_5, HELP_4, APPEND_2, LENGTH_1

Compound Symbols:   c2_1, c3_2, c4_2, c6_1, c7_2, c8_2, c10_1, c12_1


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

(5) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   rev(z0) -> if(z0, eq(0, length(z0)), nil, 0, length(z0))
   if(z0, true, z1, z2, z3) -> z1
   if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
   help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
   REV(z0) -> c3(LENGTH(z0))
   REV(z0) -> c4(LENGTH(z0))
S tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
   REV(z0) -> c3(LENGTH(z0))
   REV(z0) -> c4(LENGTH(z0))
K tuples:none
Defined Rule Symbols:   ge_2, rev_1, if_5, help_4, append_2, length_1

Defined Pair Symbols:   GE_2, IF_5, HELP_4, APPEND_2, LENGTH_1, REV_1

Compound Symbols:   c2_1, c6_1, c7_2, c8_2, c10_1, c12_1, c3_1, c4_1


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

(7) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   REV(z0) -> c3(LENGTH(z0))
   REV(z0) -> c4(LENGTH(z0))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   rev(z0) -> if(z0, eq(0, length(z0)), nil, 0, length(z0))
   if(z0, true, z1, z2, z3) -> z1
   if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
   help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
S tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
K tuples:none
Defined Rule Symbols:   ge_2, rev_1, if_5, help_4, append_2, length_1

Defined Pair Symbols:   GE_2, IF_5, HELP_4, APPEND_2, LENGTH_1

Compound Symbols:   c2_1, c6_1, c7_2, c8_2, c10_1, c12_1


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

(9) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   rev(z0) -> if(z0, eq(0, length(z0)), nil, 0, length(z0))
   if(z0, true, z1, z2, z3) -> z1
   if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
   help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
S tuples:
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
K tuples:none
Defined Rule Symbols:   append_2, ge_2

Defined Pair Symbols:   GE_2, IF_5, HELP_4, APPEND_2, LENGTH_1

Compound Symbols:   c2_1, c6_1, c7_2, c8_2, c10_1, c12_1


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

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

(12)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))

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

   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)

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

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

(14)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   GE(s(z0), s(z1)) -> c2(GE(z0, z1)) [1]
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1)) [1]
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil))) [1]
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1)) [1]
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2)) [1]
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1)) [1]
   append(nil, z0) -> z0 [0]
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) [0]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]

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

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

(16)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   GE(s(z0), s(z1)) -> c2(GE(z0, z1)) [1]
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1)) [1]
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil))) [1]
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1)) [1]
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2)) [1]
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1)) [1]
   append(nil, z0) -> z0 [0]
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) [0]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]

The TRS has the following type information:
GE :: s:0 -> s:0 -> c2
s :: s:0 -> s:0
c2 :: c2 -> c2
IF :: cons:nil -> false:true -> cons:nil -> s:0 -> s:0 -> c6
false :: false:true
c6 :: c7:c8 -> c6
HELP :: s:0 -> s:0 -> cons:nil -> cons:nil -> c7:c8
cons :: a -> cons:nil -> cons:nil
c7 :: c6 -> c10 -> c7:c8
append :: cons:nil -> cons:nil -> cons:nil
nil :: cons:nil
ge :: s:0 -> s:0 -> false:true
APPEND :: cons:nil -> cons:nil -> c10
c8 :: c6 -> c2 -> c7:c8
c10 :: c10 -> c10
LENGTH :: cons:nil -> c12
c12 :: c12 -> c12
0 :: s:0
true :: false:true

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

(17) 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:

   append(v0, v1) -> null_append [0]
   ge(v0, v1) -> null_ge [0]
   GE(v0, v1) -> null_GE [0]
   IF(v0, v1, v2, v3, v4) -> null_IF [0]
   HELP(v0, v1, v2, v3) -> null_HELP [0]
   APPEND(v0, v1) -> null_APPEND [0]
   LENGTH(v0) -> null_LENGTH [0]

And the following fresh constants:    null_append, null_ge, null_GE, null_IF, null_HELP, null_APPEND, null_LENGTH, const

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

(18)
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:

   GE(s(z0), s(z1)) -> c2(GE(z0, z1)) [1]
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1)) [1]
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil))) [1]
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1)) [1]
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2)) [1]
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1)) [1]
   append(nil, z0) -> z0 [0]
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2)) [0]
   ge(z0, 0) -> true [0]
   ge(0, s(z0)) -> false [0]
   ge(s(z0), s(z1)) -> ge(z0, z1) [0]
   append(v0, v1) -> null_append [0]
   ge(v0, v1) -> null_ge [0]
   GE(v0, v1) -> null_GE [0]
   IF(v0, v1, v2, v3, v4) -> null_IF [0]
   HELP(v0, v1, v2, v3) -> null_HELP [0]
   APPEND(v0, v1) -> null_APPEND [0]
   LENGTH(v0) -> null_LENGTH [0]

The TRS has the following type information:
GE :: s:0 -> s:0 -> c2:null_GE
s :: s:0 -> s:0
c2 :: c2:null_GE -> c2:null_GE
IF :: cons:nil:null_append -> false:true:null_ge -> cons:nil:null_append -> s:0 -> s:0 -> c6:null_IF
false :: false:true:null_ge
c6 :: c7:c8:null_HELP -> c6:null_IF
HELP :: s:0 -> s:0 -> cons:nil:null_append -> cons:nil:null_append -> c7:c8:null_HELP
cons :: a -> cons:nil:null_append -> cons:nil:null_append
c7 :: c6:null_IF -> c10:null_APPEND -> c7:c8:null_HELP
append :: cons:nil:null_append -> cons:nil:null_append -> cons:nil:null_append
nil :: cons:nil:null_append
ge :: s:0 -> s:0 -> false:true:null_ge
APPEND :: cons:nil:null_append -> cons:nil:null_append -> c10:null_APPEND
c8 :: c6:null_IF -> c2:null_GE -> c7:c8:null_HELP
c10 :: c10:null_APPEND -> c10:null_APPEND
LENGTH :: cons:nil:null_append -> c12:null_LENGTH
c12 :: c12:null_LENGTH -> c12:null_LENGTH
0 :: s:0
true :: false:true:null_ge
null_append :: cons:nil:null_append
null_ge :: false:true:null_ge
null_GE :: c2:null_GE
null_IF :: c6:null_IF
null_HELP :: c7:c8:null_HELP
null_APPEND :: c10:null_APPEND
null_LENGTH :: c12:null_LENGTH
const :: a

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

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

   false => 1
   nil => 0
   0 => 0
   true => 2
   null_append => 0
   null_ge => 0
   null_GE => 0
   null_IF => 0
   null_HELP => 0
   null_APPEND => 0
   null_LENGTH => 0
   const => 0

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   APPEND(z, z') -{ 1 }-> 1 + APPEND(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   GE(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   GE(z, z') -{ 1 }-> 1 + GE(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   HELP(z, z', z'', z5) -{ 0 }-> 0 :|: z5 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   HELP(z, z', z'', z5) -{ 1 }-> 1 + IF(append(z3, 1 + z2 + 0), ge(z0, z1), 1 + z2 + z4, z0, z1) + GE(z0, z1) :|: z = z0, z1 >= 0, z5 = z4, z' = z1, z0 >= 0, z4 >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   HELP(z, z', z'', z5) -{ 1 }-> 1 + IF(append(z3, 1 + z2 + 0), ge(z0, z1), 1 + z2 + z4, z0, z1) + APPEND(z3, 1 + z2 + 0) :|: z = z0, z1 >= 0, z5 = z4, z' = z1, z0 >= 0, z4 >= 0, z'' = 1 + z2 + z3, z2 >= 0, z3 >= 0
   IF(z, z', z'', z5, z6) -{ 0 }-> 0 :|: z5 = v3, v0 >= 0, v4 >= 0, z6 = v4, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   IF(z, z', z'', z5, z6) -{ 1 }-> 1 + HELP(1 + z2, z3, z0, z1) :|: z = z0, z1 >= 0, z6 = z3, z0 >= 0, z' = 1, z5 = z2, z2 >= 0, z3 >= 0, z'' = z1
   LENGTH(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   LENGTH(z) -{ 1 }-> 1 + LENGTH(z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   append(z, z') -{ 0 }-> z0 :|: z0 >= 0, z = 0, z' = z0
   append(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   append(z, z') -{ 0 }-> 1 + z0 + append(z1, z2) :|: z1 >= 0, z' = z2, z0 >= 0, z = 1 + z0 + z1, z2 >= 0
   ge(z, z') -{ 0 }-> ge(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   ge(z, z') -{ 0 }-> 2 :|: z = z0, z0 >= 0, z' = 0
   ge(z, z') -{ 0 }-> 1 :|: z0 >= 0, z' = 1 + z0, z = 0
   ge(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

eq(start(V1, V, V6, V5, V4),0,[fun(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6, V5, V4),0,[fun1(V1, V, V6, V5, V4, Out)],[V1 >= 0,V >= 0,V6 >= 0,V5 >= 0,V4 >= 0]).
eq(start(V1, V, V6, V5, V4),0,[fun2(V1, V, V6, V5, Out)],[V1 >= 0,V >= 0,V6 >= 0,V5 >= 0]).
eq(start(V1, V, V6, V5, V4),0,[fun3(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6, V5, V4),0,[fun4(V1, Out)],[V1 >= 0]).
eq(start(V1, V, V6, V5, V4),0,[append(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(start(V1, V, V6, V5, V4),0,[ge(V1, V, Out)],[V1 >= 0,V >= 0]).
eq(fun(V1, V, Out),1,[fun(V3, V2, Ret1)],[Out = 1 + Ret1,V2 >= 0,V1 = 1 + V3,V3 >= 0,V = 1 + V2]).
eq(fun1(V1, V, V6, V5, V4, Out),1,[fun2(1 + V10, V9, V8, V7, Ret11)],[Out = 1 + Ret11,V1 = V8,V7 >= 0,V4 = V9,V8 >= 0,V = 1,V5 = V10,V10 >= 0,V9 >= 0,V6 = V7]).
eq(fun2(V1, V, V6, V5, Out),1,[append(V15, 1 + V11 + 0, Ret010),ge(V13, V12, Ret011),fun1(Ret010, Ret011, 1 + V11 + V14, V13, V12, Ret01),fun3(V15, 1 + V11 + 0, Ret12)],[Out = 1 + Ret01 + Ret12,V1 = V13,V12 >= 0,V5 = V14,V = V12,V13 >= 0,V14 >= 0,V6 = 1 + V11 + V15,V11 >= 0,V15 >= 0]).
eq(fun2(V1, V, V6, V5, Out),1,[append(V17, 1 + V18 + 0, Ret0101),ge(V16, V19, Ret0111),fun1(Ret0101, Ret0111, 1 + V18 + V20, V16, V19, Ret012),fun(V16, V19, Ret13)],[Out = 1 + Ret012 + Ret13,V1 = V16,V19 >= 0,V5 = V20,V = V19,V16 >= 0,V20 >= 0,V6 = 1 + V17 + V18,V18 >= 0,V17 >= 0]).
eq(fun3(V1, V, Out),1,[fun3(V21, V23, Ret14)],[Out = 1 + Ret14,V21 >= 0,V = V23,V22 >= 0,V1 = 1 + V21 + V22,V23 >= 0]).
eq(fun4(V1, Out),1,[fun4(V24, Ret15)],[Out = 1 + Ret15,V24 >= 0,V25 >= 0,V1 = 1 + V24 + V25]).
eq(append(V1, V, Out),0,[],[Out = V26,V26 >= 0,V1 = 0,V = V26]).
eq(append(V1, V, Out),0,[append(V27, V28, Ret16)],[Out = 1 + Ret16 + V29,V27 >= 0,V = V28,V29 >= 0,V1 = 1 + V27 + V29,V28 >= 0]).
eq(ge(V1, V, Out),0,[],[Out = 2,V1 = V30,V30 >= 0,V = 0]).
eq(ge(V1, V, Out),0,[],[Out = 1,V31 >= 0,V = 1 + V31,V1 = 0]).
eq(ge(V1, V, Out),0,[ge(V33, V32, Ret)],[Out = Ret,V32 >= 0,V1 = 1 + V33,V33 >= 0,V = 1 + V32]).
eq(append(V1, V, Out),0,[],[Out = 0,V35 >= 0,V34 >= 0,V1 = V35,V = V34]).
eq(ge(V1, V, Out),0,[],[Out = 0,V37 >= 0,V36 >= 0,V1 = V37,V = V36]).
eq(fun(V1, V, Out),0,[],[Out = 0,V39 >= 0,V38 >= 0,V1 = V39,V = V38]).
eq(fun1(V1, V, V6, V5, V4, Out),0,[],[Out = 0,V5 = V43,V40 >= 0,V42 >= 0,V4 = V42,V6 = V44,V41 >= 0,V1 = V40,V = V41,V44 >= 0,V43 >= 0]).
eq(fun2(V1, V, V6, V5, Out),0,[],[Out = 0,V5 = V48,V46 >= 0,V6 = V47,V45 >= 0,V1 = V46,V = V45,V47 >= 0,V48 >= 0]).
eq(fun3(V1, V, Out),0,[],[Out = 0,V50 >= 0,V49 >= 0,V1 = V50,V = V49]).
eq(fun4(V1, Out),0,[],[Out = 0,V51 >= 0,V1 = V51]).
input_output_vars(fun(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun1(V1,V,V6,V5,V4,Out),[V1,V,V6,V5,V4],[Out]).
input_output_vars(fun2(V1,V,V6,V5,Out),[V1,V,V6,V5],[Out]).
input_output_vars(fun3(V1,V,Out),[V1,V],[Out]).
input_output_vars(fun4(V1,Out),[V1],[Out]).
input_output_vars(append(V1,V,Out),[V1,V],[Out]).
input_output_vars(ge(V1,V,Out),[V1,V],[Out]).


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

#### Computed strongly connected components 
0. recursive  : [append/3]
1. recursive  : [fun/3]
2. recursive  : [fun3/3]
3. recursive  : [ge/3]
4. recursive [non_tail] : [fun1/6,fun2/5]
5. recursive  : [fun4/2]
6. non_recursive  : [start/5]

#### Obtained direct recursion through partial evaluation 
0. SCC is partially evaluated into append/3
1. SCC is partially evaluated into fun/3
2. SCC is partially evaluated into fun3/3
3. SCC is partially evaluated into ge/3
4. SCC is partially evaluated into fun2/5
5. SCC is partially evaluated into fun4/2
6. SCC is partially evaluated into start/5

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

### Specialization of cost equations append/3 
* 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 append/3 
* CEs [29] --> Loop 18 
* CEs [27] --> Loop 19 
* CEs [28] --> Loop 20 

### Ranking functions of CR append(V1,V,Out) 
* RF of phase [18]: [V1]

#### Partial ranking functions of CR append(V1,V,Out) 
* Partial RF of phase [18]:
  - RF of loop [18:1]:
    V1


### Specialization of cost equations fun/3 
* CE 15 is refined into CE [30] 
* CE 14 is refined into CE [31] 


### Cost equations --> "Loop" of fun/3 
* CEs [31] --> Loop 21 
* CEs [30] --> Loop 22 

### Ranking functions of CR fun(V1,V,Out) 
* RF of phase [21]: [V,V1]

#### Partial ranking functions of CR fun(V1,V,Out) 
* Partial RF of phase [21]:
  - RF of loop [21:1]:
    V
    V1


### Specialization of cost equations fun3/3 
* CE 17 is refined into CE [32] 
* CE 16 is refined into CE [33] 


### Cost equations --> "Loop" of fun3/3 
* CEs [33] --> Loop 23 
* CEs [32] --> Loop 24 

### Ranking functions of CR fun3(V1,V,Out) 
* RF of phase [23]: [V1]

#### Partial ranking functions of CR fun3(V1,V,Out) 
* Partial RF of phase [23]:
  - RF of loop [23:1]:
    V1


### Specialization of cost equations ge/3 
* CE 26 is refined into CE [34] 
* CE 23 is refined into CE [35] 
* CE 24 is refined into CE [36] 
* CE 25 is refined into CE [37] 


### Cost equations --> "Loop" of ge/3 
* CEs [37] --> Loop 25 
* CEs [34] --> Loop 26 
* CEs [35] --> Loop 27 
* CEs [36] --> Loop 28 

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

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


### Specialization of cost equations fun2/5 
* CE 9 is refined into CE [38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69] 
* CE 10 is refined into CE [70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104] 
* CE 13 is refined into CE [105] 
* CE 11 is refined into CE [106,107,108,109,110,111,112,113,114,115,116,117] 
* CE 12 is refined into CE [118,119,120,121,122,123,124,125,126,127,128,129,130,131] 


### Cost equations --> "Loop" of fun2/5 
* CEs [117] --> Loop 29 
* CEs [131] --> Loop 30 
* CEs [116,130] --> Loop 31 
* CEs [114] --> Loop 32 
* CEs [127] --> Loop 33 
* CEs [108] --> Loop 34 
* CEs [113,126] --> Loop 35 
* CEs [107,119] --> Loop 36 
* CEs [111] --> Loop 37 
* CEs [123] --> Loop 38 
* CEs [110,122] --> Loop 39 
* CEs [129] --> Loop 40 
* CEs [115,128] --> Loop 41 
* CEs [125] --> Loop 42 
* CEs [112,124] --> Loop 43 
* CEs [106,118] --> Loop 44 
* CEs [121] --> Loop 45 
* CEs [109,120] --> Loop 46 
* CEs [45,53] --> Loop 47 
* CEs [41,43,49,51,57,59,61,65,67,69] --> Loop 48 
* CEs [105] --> Loop 49 
* CEs [78,88,98] --> Loop 50 
* CEs [39,47,55,63,71,77,87,97] --> Loop 51 
* CEs [76,80,82,84,86,90,92,94,96,100,102,104] --> Loop 52 
* CEs [38,40,42,44,46,48,50,52,54,56,58,60,62,64,66,68,70,72,73,74,75,79,81,83,85,89,91,93,95,99,101,103] --> Loop 53 

### Ranking functions of CR fun2(V1,V,V6,V5,Out) 
* RF of phase [29,30,31,32,33,34,35,36]: [-V1+V]

#### Partial ranking functions of CR fun2(V1,V,V6,V5,Out) 
* Partial RF of phase [29,30,31,32,33,34,35,36]:
  - RF of loop [29:1,30:1,31:1]:
    V6-1
  - RF of loop [29:1,30:1,31:1,32:1,33:1,34:1,35:1,36:1]:
    -V1+V


### Specialization of cost equations fun4/2 
* CE 19 is refined into CE [132] 
* CE 18 is refined into CE [133] 


### Cost equations --> "Loop" of fun4/2 
* CEs [133] --> Loop 54 
* CEs [132] --> Loop 55 

### Ranking functions of CR fun4(V1,Out) 
* RF of phase [54]: [V1]

#### Partial ranking functions of CR fun4(V1,Out) 
* Partial RF of phase [54]:
  - RF of loop [54:1]:
    V1


### Specialization of cost equations start/5 
* CE 1 is refined into CE [134] 
* CE 2 is refined into CE [135,136,137,138,139] 
* CE 3 is refined into CE [140,141] 
* CE 4 is refined into CE [142,143,144,145,146,147,148,149,150,151,152,153,154] 
* CE 5 is refined into CE [155,156] 
* CE 6 is refined into CE [157,158] 
* CE 7 is refined into CE [159,160,161,162] 
* CE 8 is refined into CE [163,164,165,166,167] 


### Cost equations --> "Loop" of start/5 
* CEs [135,136,137,138,139] --> Loop 56 
* CEs [164] --> Loop 57 
* CEs [134,140,141,142,143,144,145,146,147,148,149,150,151,152,153,154,155,156,157,158,159,160,161,162,163,165,166,167] --> Loop 58 

### Ranking functions of CR start(V1,V,V6,V5,V4) 

#### Partial ranking functions of CR start(V1,V,V6,V5,V4) 


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

#### Cost of chains of append(V1,V,Out):
* Chain [[18],20]: 0
  with precondition: [V+V1=Out,V1>=1,V>=0] 

* Chain [[18],19]: 0
  with precondition: [V>=0,Out>=1,V1>=Out] 

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

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


#### Cost of chains of fun(V1,V,Out):
* Chain [[21],22]: 1*it(21)+0
  Such that:it(21) =< Out

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

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


#### Cost of chains of fun3(V1,V,Out):
* Chain [[23],24]: 1*it(23)+0
  Such that:it(23) =< V1

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

* Chain [24]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of ge(V1,V,Out):
* Chain [[25],28]: 0
  with precondition: [Out=1,V1>=1,V>=V1+1] 

* Chain [[25],27]: 0
  with precondition: [Out=2,V>=1,V1>=V] 

* Chain [[25],26]: 0
  with precondition: [Out=0,V1>=1,V>=1] 

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

* Chain [27]: 0
  with precondition: [V=0,Out=2,V1>=0] 

* Chain [26]: 0
  with precondition: [Out=0,V1>=0,V>=0] 


#### Cost of chains of fun2(V1,V,V6,V5,Out):
* Chain [[29,30,31,32,33,34,35,36],53]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1
  Such that:aux(1) =< V
aux(10) =< -V1+V
aux(11) =< V6
it(29) =< aux(10)
it(32) =< aux(10)
it(29) =< aux(11)
aux(3) =< aux(1)
aux(4) =< aux(11)
s(12) =< it(29)*aux(11)
s(11) =< it(29)*aux(1)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1>=1,V6>=1,V5>=0,Out>=3,V>=V1+1] 

* Chain [[29,30,31,32,33,34,35,36],52]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+12*s(16)+1
  Such that:aux(1) =< V
aux(13) =< -V1+V
aux(14) =< V6
s(16) =< aux(14)
it(29) =< aux(13)
it(32) =< aux(13)
it(29) =< aux(14)
aux(3) =< aux(1)
aux(4) =< aux(14)
s(12) =< it(29)*aux(14)
s(11) =< it(29)*aux(1)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1>=1,V6>=2,V5>=0,Out>=4,V>=V1+1] 

* Chain [[29,30,31,32,33,34,35,36],49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+0
  Such that:aux(1) =< V
aux(15) =< -V1+V
aux(16) =< V6
it(29) =< aux(15)
it(32) =< aux(15)
it(29) =< aux(16)
aux(3) =< aux(1)
aux(4) =< aux(16)
s(12) =< it(29)*aux(16)
s(11) =< it(29)*aux(1)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1>=1,V6>=1,V5>=0,Out>=2,V>=V1+1] 

* Chain [[29,30,31,32,33,34,35,36],48]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+10*s(28)+1
  Such that:aux(19) =< -V1+V
aux(20) =< V
aux(21) =< V6
s(28) =< aux(20)
it(29) =< aux(19)
it(32) =< aux(19)
it(29) =< aux(21)
aux(3) =< aux(20)
aux(4) =< aux(21)
s(12) =< it(29)*aux(21)
s(11) =< it(29)*aux(20)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1>=1,V6>=1,V5>=0,Out>=4,V>=V1+1] 

* Chain [[29,30,31,32,33,34,35,36],47]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+2*s(38)+1
  Such that:aux(23) =< -V1+V
aux(24) =< V
aux(25) =< V6
s(38) =< aux(24)
it(29) =< aux(23)
it(32) =< aux(23)
it(29) =< aux(25)
aux(3) =< aux(24)
aux(4) =< aux(25)
s(12) =< it(29)*aux(25)
s(11) =< it(29)*aux(24)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1>=1,V6>=1,V5>=0,V>=V1+1,Out+2*V1>=2*V+2] 

* Chain [[29,30,31,32,33,34,35,36],39,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+2
  Such that:aux(1) =< V
aux(26) =< -V1+V
aux(27) =< V6
it(29) =< aux(26)
it(32) =< aux(26)
it(29) =< aux(27)
aux(3) =< aux(1)
aux(4) =< aux(27)
s(12) =< it(29)*aux(27)
s(11) =< it(29)*aux(1)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1>=1,V6>=1,V5>=0,Out>=4,V>=V1+2] 

* Chain [[29,30,31,32,33,34,35,36],38,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(40)+2
  Such that:aux(1) =< V
aux(28) =< -V1+V
aux(29) =< V6
s(40) =< aux(29)
it(29) =< aux(28)
it(32) =< aux(28)
it(29) =< aux(29)
aux(3) =< aux(1)
aux(4) =< aux(29)
s(12) =< it(29)*aux(29)
s(11) =< it(29)*aux(1)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1>=1,V6>=2,V5>=0,Out>=5,V>=V1+2] 

* Chain [[29,30,31,32,33,34,35,36],37,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(41)+2
  Such that:aux(30) =< -V1+V
aux(31) =< V
aux(32) =< V6
s(41) =< aux(31)
it(29) =< aux(30)
it(32) =< aux(30)
it(29) =< aux(32)
aux(3) =< aux(31)
aux(4) =< aux(32)
s(12) =< it(29)*aux(32)
s(11) =< it(29)*aux(31)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1>=1,V6>=1,V5>=0,Out>=5,V>=V1+2] 

* Chain [53]: 1
  with precondition: [Out=1,V1>=0,V>=0,V6>=1,V5>=0] 

* Chain [52]: 12*s(16)+1
  Such that:aux(12) =< V6
s(16) =< aux(12)

  with precondition: [V1>=0,V>=0,V5>=0,Out>=2,V6>=Out] 

* Chain [51]: 1
  with precondition: [V=0,Out=1,V1>=0,V6>=1,V5>=0] 

* Chain [50]: 3*s(42)+1
  Such that:aux(33) =< V6
s(42) =< aux(33)

  with precondition: [V=0,V1>=0,V5>=0,Out>=2,V6>=Out] 

* Chain [49]: 0
  with precondition: [Out=0,V1>=0,V>=0,V6>=0,V5>=0] 

* Chain [48]: 6*s(28)+4*s(29)+1
  Such that:aux(17) =< V1
aux(18) =< V
s(29) =< aux(17)
s(28) =< aux(18)

  with precondition: [V6>=1,V5>=0,Out>=2,V1+1>=Out,V+1>=Out] 

* Chain [47]: 2*s(38)+1
  Such that:aux(22) =< V
s(38) =< aux(22)

  with precondition: [V6>=1,V5>=0,Out>=2,V1>=V,V+1>=Out] 

* Chain [46,49]: 2
  with precondition: [V1=0,Out=2,V>=1,V6>=1,V5>=0] 

* Chain [45,49]: 1*s(45)+2
  Such that:s(45) =< V6

  with precondition: [V1=0,V>=1,V5>=0,Out>=3,V6+1>=Out] 

* Chain [44,[29,30,31,32,33,34,35,36],53]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+3
  Such that:aux(11) =< V6
aux(34) =< V
it(29) =< aux(34)
it(32) =< aux(34)
it(29) =< aux(11)
aux(3) =< aux(34)
aux(4) =< aux(11)
s(12) =< it(29)*aux(11)
s(11) =< it(29)*aux(34)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=1,V5>=0,Out>=5] 

* Chain [44,[29,30,31,32,33,34,35,36],52]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+12*s(16)+3
  Such that:aux(14) =< V6
aux(35) =< V
s(16) =< aux(14)
it(29) =< aux(35)
it(32) =< aux(35)
it(29) =< aux(14)
aux(3) =< aux(35)
aux(4) =< aux(14)
s(12) =< it(29)*aux(14)
s(11) =< it(29)*aux(35)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=6] 

* Chain [44,[29,30,31,32,33,34,35,36],49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+2
  Such that:aux(16) =< V6
aux(36) =< V
it(29) =< aux(36)
it(32) =< aux(36)
it(29) =< aux(16)
aux(3) =< aux(36)
aux(4) =< aux(16)
s(12) =< it(29)*aux(16)
s(11) =< it(29)*aux(36)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=1,V5>=0,Out>=4] 

* Chain [44,[29,30,31,32,33,34,35,36],48]: 6*it(29)+20*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+3
  Such that:aux(21) =< V6
aux(37) =< V
it(32) =< aux(37)
it(29) =< aux(37)
it(29) =< aux(21)
aux(3) =< aux(37)
aux(4) =< aux(21)
s(12) =< it(29)*aux(21)
s(11) =< it(29)*aux(37)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=1,V5>=0,Out>=6] 

* Chain [44,[29,30,31,32,33,34,35,36],47]: 6*it(29)+12*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+3
  Such that:aux(25) =< V6
aux(38) =< V
it(32) =< aux(38)
it(29) =< aux(38)
it(29) =< aux(25)
aux(3) =< aux(38)
aux(4) =< aux(25)
s(12) =< it(29)*aux(25)
s(11) =< it(29)*aux(38)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=1,V5>=0,Out>=2*V+2] 

* Chain [44,[29,30,31,32,33,34,35,36],39,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+4
  Such that:aux(27) =< V6
aux(39) =< V
it(29) =< aux(39)
it(32) =< aux(39)
it(29) =< aux(27)
aux(3) =< aux(39)
aux(4) =< aux(27)
s(12) =< it(29)*aux(27)
s(11) =< it(29)*aux(39)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=1,V5>=0,Out>=6] 

* Chain [44,[29,30,31,32,33,34,35,36],38,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(40)+4
  Such that:aux(29) =< V6
aux(40) =< V
s(40) =< aux(29)
it(29) =< aux(40)
it(32) =< aux(40)
it(29) =< aux(29)
aux(3) =< aux(40)
aux(4) =< aux(29)
s(12) =< it(29)*aux(29)
s(11) =< it(29)*aux(40)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=7] 

* Chain [44,[29,30,31,32,33,34,35,36],37,49]: 6*it(29)+11*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+4
  Such that:aux(32) =< V6
aux(41) =< V
it(32) =< aux(41)
it(29) =< aux(41)
it(29) =< aux(32)
aux(3) =< aux(41)
aux(4) =< aux(32)
s(12) =< it(29)*aux(32)
s(11) =< it(29)*aux(41)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=1,V5>=0,Out>=7] 

* Chain [44,53]: 3
  with precondition: [V1=0,Out=3,V>=1,V6>=1,V5>=0] 

* Chain [44,52]: 12*s(16)+3
  Such that:aux(12) =< V6
s(16) =< aux(12)

  with precondition: [V1=0,V>=1,V5>=0,Out>=4,V6+2>=Out] 

* Chain [44,49]: 2
  with precondition: [V1=0,Out=2,V>=1,V6>=1,V5>=0] 

* Chain [44,48]: 6*s(28)+4*s(29)+3
  Such that:aux(17) =< 1
aux(18) =< V
s(29) =< aux(17)
s(28) =< aux(18)

  with precondition: [V1=0,Out=4,V>=1,V6>=1,V5>=0] 

* Chain [44,47]: 2*s(38)+3
  Such that:aux(22) =< 1
s(38) =< aux(22)

  with precondition: [V1=0,V=1,Out=4,V6>=1,V5>=0] 

* Chain [44,39,49]: 4
  with precondition: [V1=0,Out=4,V>=2,V6>=1,V5>=0] 

* Chain [44,38,49]: 1*s(40)+4
  Such that:s(40) =< V6

  with precondition: [V1=0,V>=2,V5>=0,Out>=5,V6+3>=Out] 

* Chain [44,37,49]: 1*s(41)+4
  Such that:s(41) =< 1

  with precondition: [V1=0,Out=5,V>=2,V6>=1,V5>=0] 

* Chain [43,[29,30,31,32,33,34,35,36],53]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+3
  Such that:aux(11) =< V6
aux(42) =< V
it(29) =< aux(42)
it(32) =< aux(42)
it(29) =< aux(11)
aux(3) =< aux(42)
aux(4) =< aux(11)
s(12) =< it(29)*aux(11)
s(11) =< it(29)*aux(42)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=5] 

* Chain [43,[29,30,31,32,33,34,35,36],52]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+12*s(16)+3
  Such that:aux(14) =< V6
aux(43) =< V
s(16) =< aux(14)
it(29) =< aux(43)
it(32) =< aux(43)
it(29) =< aux(14)
aux(3) =< aux(43)
aux(4) =< aux(14)
s(12) =< it(29)*aux(14)
s(11) =< it(29)*aux(43)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=6] 

* Chain [43,[29,30,31,32,33,34,35,36],49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+2
  Such that:aux(16) =< V6
aux(44) =< V
it(29) =< aux(44)
it(32) =< aux(44)
it(29) =< aux(16)
aux(3) =< aux(44)
aux(4) =< aux(16)
s(12) =< it(29)*aux(16)
s(11) =< it(29)*aux(44)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=4] 

* Chain [43,[29,30,31,32,33,34,35,36],48]: 6*it(29)+20*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+3
  Such that:aux(21) =< V6
aux(45) =< V
it(32) =< aux(45)
it(29) =< aux(45)
it(29) =< aux(21)
aux(3) =< aux(45)
aux(4) =< aux(21)
s(12) =< it(29)*aux(21)
s(11) =< it(29)*aux(45)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=6] 

* Chain [43,[29,30,31,32,33,34,35,36],47]: 6*it(29)+12*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+3
  Such that:aux(25) =< V6
aux(46) =< V
it(32) =< aux(46)
it(29) =< aux(46)
it(29) =< aux(25)
aux(3) =< aux(46)
aux(4) =< aux(25)
s(12) =< it(29)*aux(25)
s(11) =< it(29)*aux(46)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=2*V+2] 

* Chain [43,[29,30,31,32,33,34,35,36],39,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+4
  Such that:aux(27) =< V6
aux(47) =< V
it(29) =< aux(47)
it(32) =< aux(47)
it(29) =< aux(27)
aux(3) =< aux(47)
aux(4) =< aux(27)
s(12) =< it(29)*aux(27)
s(11) =< it(29)*aux(47)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=6] 

* Chain [43,[29,30,31,32,33,34,35,36],38,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(40)+4
  Such that:aux(29) =< V6
aux(48) =< V
s(40) =< aux(29)
it(29) =< aux(48)
it(32) =< aux(48)
it(29) =< aux(29)
aux(3) =< aux(48)
aux(4) =< aux(29)
s(12) =< it(29)*aux(29)
s(11) =< it(29)*aux(48)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=7] 

* Chain [43,[29,30,31,32,33,34,35,36],37,49]: 6*it(29)+11*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+4
  Such that:aux(32) =< V6
aux(49) =< V
it(32) =< aux(49)
it(29) =< aux(49)
it(29) =< aux(32)
aux(3) =< aux(49)
aux(4) =< aux(32)
s(12) =< it(29)*aux(32)
s(11) =< it(29)*aux(49)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=7] 

* Chain [43,53]: 3
  with precondition: [V1=0,Out=3,V>=1,V6>=2,V5>=0] 

* Chain [43,52]: 12*s(16)+3
  Such that:aux(12) =< V6
s(16) =< aux(12)

  with precondition: [V1=0,V>=1,V5>=0,Out>=4,V6+2>=Out] 

* Chain [43,49]: 2
  with precondition: [V1=0,Out=2,V>=1,V6>=2,V5>=0] 

* Chain [43,48]: 6*s(28)+4*s(29)+3
  Such that:aux(17) =< 1
aux(18) =< V
s(29) =< aux(17)
s(28) =< aux(18)

  with precondition: [V1=0,Out=4,V>=1,V6>=2,V5>=0] 

* Chain [43,47]: 2*s(38)+3
  Such that:aux(22) =< 1
s(38) =< aux(22)

  with precondition: [V1=0,V=1,Out=4,V6>=2,V5>=0] 

* Chain [43,39,49]: 4
  with precondition: [V1=0,Out=4,V>=2,V6>=2,V5>=0] 

* Chain [43,38,49]: 1*s(40)+4
  Such that:s(40) =< V6

  with precondition: [V1=0,V>=2,V5>=0,Out>=5,V6+3>=Out] 

* Chain [43,37,49]: 1*s(41)+4
  Such that:s(41) =< 1

  with precondition: [V1=0,Out=5,V>=2,V6>=2,V5>=0] 

* Chain [42,[29,30,31,32,33,34,35,36],53]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(46)+3
  Such that:aux(50) =< V
aux(51) =< V6
s(46) =< aux(51)
it(29) =< aux(50)
it(32) =< aux(50)
it(29) =< aux(51)
aux(3) =< aux(50)
aux(4) =< aux(51)
s(12) =< it(29)*aux(51)
s(11) =< it(29)*aux(50)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=6] 

* Chain [42,[29,30,31,32,33,34,35,36],52]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+13*s(16)+3
  Such that:aux(52) =< V
aux(53) =< V6
s(16) =< aux(53)
it(29) =< aux(52)
it(32) =< aux(52)
it(29) =< aux(53)
aux(3) =< aux(52)
aux(4) =< aux(53)
s(12) =< it(29)*aux(53)
s(11) =< it(29)*aux(52)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=7] 

* Chain [42,[29,30,31,32,33,34,35,36],49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(46)+2
  Such that:aux(54) =< V
aux(55) =< V6
s(46) =< aux(55)
it(29) =< aux(54)
it(32) =< aux(54)
it(29) =< aux(55)
aux(3) =< aux(54)
aux(4) =< aux(55)
s(12) =< it(29)*aux(55)
s(11) =< it(29)*aux(54)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=5] 

* Chain [42,[29,30,31,32,33,34,35,36],48]: 6*it(29)+20*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(46)+3
  Such that:aux(56) =< V
aux(57) =< V6
s(46) =< aux(57)
it(32) =< aux(56)
it(29) =< aux(56)
it(29) =< aux(57)
aux(3) =< aux(56)
aux(4) =< aux(57)
s(12) =< it(29)*aux(57)
s(11) =< it(29)*aux(56)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=7] 

* Chain [42,[29,30,31,32,33,34,35,36],47]: 6*it(29)+12*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(46)+3
  Such that:aux(58) =< V
aux(59) =< V6
s(46) =< aux(59)
it(32) =< aux(58)
it(29) =< aux(58)
it(29) =< aux(59)
aux(3) =< aux(58)
aux(4) =< aux(59)
s(12) =< it(29)*aux(59)
s(11) =< it(29)*aux(58)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=2*V+3] 

* Chain [42,[29,30,31,32,33,34,35,36],39,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(46)+4
  Such that:aux(60) =< V
aux(61) =< V6
s(46) =< aux(61)
it(29) =< aux(60)
it(32) =< aux(60)
it(29) =< aux(61)
aux(3) =< aux(60)
aux(4) =< aux(61)
s(12) =< it(29)*aux(61)
s(11) =< it(29)*aux(60)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=7] 

* Chain [42,[29,30,31,32,33,34,35,36],38,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+2*s(40)+4
  Such that:aux(62) =< V
aux(63) =< V6
s(40) =< aux(63)
it(29) =< aux(62)
it(32) =< aux(62)
it(29) =< aux(63)
aux(3) =< aux(62)
aux(4) =< aux(63)
s(12) =< it(29)*aux(63)
s(11) =< it(29)*aux(62)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=8] 

* Chain [42,[29,30,31,32,33,34,35,36],37,49]: 6*it(29)+11*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(46)+4
  Such that:aux(64) =< V
aux(65) =< V6
s(46) =< aux(65)
it(32) =< aux(64)
it(29) =< aux(64)
it(29) =< aux(65)
aux(3) =< aux(64)
aux(4) =< aux(65)
s(12) =< it(29)*aux(65)
s(11) =< it(29)*aux(64)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=8] 

* Chain [42,53]: 1*s(46)+3
  Such that:s(46) =< V6

  with precondition: [V1=0,V>=1,V5>=0,Out>=4,V6+2>=Out] 

* Chain [42,52]: 13*s(16)+3
  Such that:aux(66) =< V6
s(16) =< aux(66)

  with precondition: [V1=0,V>=1,V5>=0,Out>=5,2*V6+1>=Out] 

* Chain [42,49]: 1*s(46)+2
  Such that:s(46) =< V6

  with precondition: [V1=0,V>=1,V5>=0,Out>=3,V6+1>=Out] 

* Chain [42,48]: 6*s(28)+4*s(29)+1*s(46)+3
  Such that:aux(17) =< 1
aux(18) =< V
s(46) =< V6
s(29) =< aux(17)
s(28) =< aux(18)

  with precondition: [V1=0,V>=1,V5>=0,Out>=5,V6+3>=Out] 

* Chain [42,47]: 2*s(38)+1*s(46)+3
  Such that:aux(22) =< 1
s(46) =< V6
s(38) =< aux(22)

  with precondition: [V1=0,V=1,V5>=0,Out>=5,V6+3>=Out] 

* Chain [42,39,49]: 1*s(46)+4
  Such that:s(46) =< V6

  with precondition: [V1=0,V>=2,V5>=0,Out>=5,V6+3>=Out] 

* Chain [42,38,49]: 2*s(40)+4
  Such that:aux(67) =< V6
s(40) =< aux(67)

  with precondition: [V1=0,V>=2,V5>=0,Out>=6,2*V6+2>=Out] 

* Chain [42,37,49]: 1*s(41)+1*s(46)+4
  Such that:s(41) =< 1
s(46) =< V6

  with precondition: [V1=0,V>=2,V5>=0,Out>=6,V6+4>=Out] 

* Chain [41,[29,30,31,32,33,34,35,36],53]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+3
  Such that:aux(11) =< V6
aux(68) =< V
it(29) =< aux(68)
it(32) =< aux(68)
it(29) =< aux(11)
aux(3) =< aux(68)
aux(4) =< aux(11)
s(12) =< it(29)*aux(11)
s(11) =< it(29)*aux(68)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=5] 

* Chain [41,[29,30,31,32,33,34,35,36],52]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+12*s(16)+3
  Such that:aux(14) =< V6
aux(69) =< V
s(16) =< aux(14)
it(29) =< aux(69)
it(32) =< aux(69)
it(29) =< aux(14)
aux(3) =< aux(69)
aux(4) =< aux(14)
s(12) =< it(29)*aux(14)
s(11) =< it(29)*aux(69)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=3,V5>=0,Out>=6] 

* Chain [41,[29,30,31,32,33,34,35,36],49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+2
  Such that:aux(16) =< V6
aux(70) =< V
it(29) =< aux(70)
it(32) =< aux(70)
it(29) =< aux(16)
aux(3) =< aux(70)
aux(4) =< aux(16)
s(12) =< it(29)*aux(16)
s(11) =< it(29)*aux(70)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=4] 

* Chain [41,[29,30,31,32,33,34,35,36],48]: 6*it(29)+20*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+3
  Such that:aux(21) =< V6
aux(71) =< V
it(32) =< aux(71)
it(29) =< aux(71)
it(29) =< aux(21)
aux(3) =< aux(71)
aux(4) =< aux(21)
s(12) =< it(29)*aux(21)
s(11) =< it(29)*aux(71)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=6] 

* Chain [41,[29,30,31,32,33,34,35,36],47]: 6*it(29)+12*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+3
  Such that:aux(25) =< V6
aux(72) =< V
it(32) =< aux(72)
it(29) =< aux(72)
it(29) =< aux(25)
aux(3) =< aux(72)
aux(4) =< aux(25)
s(12) =< it(29)*aux(25)
s(11) =< it(29)*aux(72)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=2*V+2] 

* Chain [41,[29,30,31,32,33,34,35,36],39,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+4
  Such that:aux(27) =< V6
aux(73) =< V
it(29) =< aux(73)
it(32) =< aux(73)
it(29) =< aux(27)
aux(3) =< aux(73)
aux(4) =< aux(27)
s(12) =< it(29)*aux(27)
s(11) =< it(29)*aux(73)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=6] 

* Chain [41,[29,30,31,32,33,34,35,36],38,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(40)+4
  Such that:aux(29) =< V6
aux(74) =< V
s(40) =< aux(29)
it(29) =< aux(74)
it(32) =< aux(74)
it(29) =< aux(29)
aux(3) =< aux(74)
aux(4) =< aux(29)
s(12) =< it(29)*aux(29)
s(11) =< it(29)*aux(74)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=3,V5>=0,Out>=7] 

* Chain [41,[29,30,31,32,33,34,35,36],37,49]: 6*it(29)+11*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+4
  Such that:aux(32) =< V6
aux(75) =< V
it(32) =< aux(75)
it(29) =< aux(75)
it(29) =< aux(32)
aux(3) =< aux(75)
aux(4) =< aux(32)
s(12) =< it(29)*aux(32)
s(11) =< it(29)*aux(75)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=7] 

* Chain [41,53]: 3
  with precondition: [V1=0,Out=3,V>=1,V6>=2,V5>=0] 

* Chain [41,52]: 12*s(16)+3
  Such that:aux(12) =< V6
s(16) =< aux(12)

  with precondition: [V1=0,V>=1,V5>=0,Out>=4,V6+1>=Out] 

* Chain [41,49]: 2
  with precondition: [V1=0,Out=2,V>=1,V6>=2,V5>=0] 

* Chain [41,48]: 6*s(28)+4*s(29)+3
  Such that:aux(17) =< 1
aux(18) =< V
s(29) =< aux(17)
s(28) =< aux(18)

  with precondition: [V1=0,Out=4,V>=1,V6>=2,V5>=0] 

* Chain [41,47]: 2*s(38)+3
  Such that:aux(22) =< 1
s(38) =< aux(22)

  with precondition: [V1=0,V=1,Out=4,V6>=2,V5>=0] 

* Chain [41,39,49]: 4
  with precondition: [V1=0,Out=4,V>=2,V6>=2,V5>=0] 

* Chain [41,38,49]: 1*s(40)+4
  Such that:s(40) =< V6

  with precondition: [V1=0,V>=2,V5>=0,Out>=5,V6+2>=Out] 

* Chain [41,37,49]: 1*s(41)+4
  Such that:s(41) =< 1

  with precondition: [V1=0,Out=5,V>=2,V6>=2,V5>=0] 

* Chain [40,[29,30,31,32,33,34,35,36],53]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(47)+3
  Such that:aux(76) =< V
aux(77) =< V6
s(47) =< aux(77)
it(29) =< aux(76)
it(32) =< aux(76)
it(29) =< aux(77)
aux(3) =< aux(76)
aux(4) =< aux(77)
s(12) =< it(29)*aux(77)
s(11) =< it(29)*aux(76)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=6] 

* Chain [40,[29,30,31,32,33,34,35,36],52]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+13*s(16)+3
  Such that:aux(78) =< V
aux(79) =< V6
s(16) =< aux(79)
it(29) =< aux(78)
it(32) =< aux(78)
it(29) =< aux(79)
aux(3) =< aux(78)
aux(4) =< aux(79)
s(12) =< it(29)*aux(79)
s(11) =< it(29)*aux(78)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=3,V5>=0,Out>=7] 

* Chain [40,[29,30,31,32,33,34,35,36],49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(47)+2
  Such that:aux(80) =< V
aux(81) =< V6
s(47) =< aux(81)
it(29) =< aux(80)
it(32) =< aux(80)
it(29) =< aux(81)
aux(3) =< aux(80)
aux(4) =< aux(81)
s(12) =< it(29)*aux(81)
s(11) =< it(29)*aux(80)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=5] 

* Chain [40,[29,30,31,32,33,34,35,36],48]: 6*it(29)+20*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(47)+3
  Such that:aux(82) =< V
aux(83) =< V6
s(47) =< aux(83)
it(32) =< aux(82)
it(29) =< aux(82)
it(29) =< aux(83)
aux(3) =< aux(82)
aux(4) =< aux(83)
s(12) =< it(29)*aux(83)
s(11) =< it(29)*aux(82)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=7] 

* Chain [40,[29,30,31,32,33,34,35,36],47]: 6*it(29)+12*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(47)+3
  Such that:aux(84) =< V
aux(85) =< V6
s(47) =< aux(85)
it(32) =< aux(84)
it(29) =< aux(84)
it(29) =< aux(85)
aux(3) =< aux(84)
aux(4) =< aux(85)
s(12) =< it(29)*aux(85)
s(11) =< it(29)*aux(84)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=2,V6>=2,V5>=0,Out>=2*V+3] 

* Chain [40,[29,30,31,32,33,34,35,36],39,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(47)+4
  Such that:aux(86) =< V
aux(87) =< V6
s(47) =< aux(87)
it(29) =< aux(86)
it(32) =< aux(86)
it(29) =< aux(87)
aux(3) =< aux(86)
aux(4) =< aux(87)
s(12) =< it(29)*aux(87)
s(11) =< it(29)*aux(86)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=7] 

* Chain [40,[29,30,31,32,33,34,35,36],38,49]: 6*it(29)+10*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+2*s(40)+4
  Such that:aux(88) =< V
aux(89) =< V6
s(40) =< aux(89)
it(29) =< aux(88)
it(32) =< aux(88)
it(29) =< aux(89)
aux(3) =< aux(88)
aux(4) =< aux(89)
s(12) =< it(29)*aux(89)
s(11) =< it(29)*aux(88)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=3,V5>=0,Out>=8] 

* Chain [40,[29,30,31,32,33,34,35,36],37,49]: 6*it(29)+11*it(32)+1*s(11)+1*s(12)+2*s(13)+1*s(14)+1*s(47)+4
  Such that:aux(90) =< V
aux(91) =< V6
s(47) =< aux(91)
it(32) =< aux(90)
it(29) =< aux(90)
it(29) =< aux(91)
aux(3) =< aux(90)
aux(4) =< aux(91)
s(12) =< it(29)*aux(91)
s(11) =< it(29)*aux(90)
s(13) =< it(32)*aux(3)
s(14) =< it(32)*aux(4)

  with precondition: [V1=0,V>=3,V6>=2,V5>=0,Out>=8] 

* Chain [40,53]: 1*s(47)+3
  Such that:s(47) =< V6

  with precondition: [V1=0,V>=1,V5>=0,Out>=4,V6+2>=Out] 

* Chain [40,52]: 13*s(16)+3
  Such that:aux(92) =< V6
s(16) =< aux(92)

  with precondition: [V1=0,V>=1,V6>=3,V5>=0,Out>=5,2*V6>=Out] 

* Chain [40,49]: 1*s(47)+2
  Such that:s(47) =< V6

  with precondition: [V1=0,V>=1,V5>=0,Out>=3,V6+1>=Out] 

* Chain [40,48]: 6*s(28)+4*s(29)+1*s(47)+3
  Such that:aux(17) =< 1
aux(18) =< V
s(47) =< V6
s(29) =< aux(17)
s(28) =< aux(18)

  with precondition: [V1=0,V>=1,V5>=0,Out>=5,V6+3>=Out] 

* Chain [40,47]: 2*s(38)+1*s(47)+3
  Such that:aux(22) =< 1
s(47) =< V6
s(38) =< aux(22)

  with precondition: [V1=0,V=1,V5>=0,Out>=5,V6+3>=Out] 

* Chain [40,39,49]: 1*s(47)+4
  Such that:s(47) =< V6

  with precondition: [V1=0,V>=2,V5>=0,Out>=5,V6+3>=Out] 

* Chain [40,38,49]: 2*s(40)+4
  Such that:aux(93) =< V6
s(40) =< aux(93)

  with precondition: [V1=0,V>=2,V6>=3,V5>=0,Out>=6,2*V6+1>=Out] 

* Chain [40,37,49]: 1*s(41)+1*s(47)+4
  Such that:s(41) =< 1
s(47) =< V6

  with precondition: [V1=0,V>=2,V5>=0,Out>=6,V6+4>=Out] 

* Chain [39,49]: 2
  with precondition: [Out=2,V1>=1,V6>=1,V5>=0,V>=V1+1] 

* Chain [38,49]: 1*s(40)+2
  Such that:s(40) =< V6

  with precondition: [V1>=1,V5>=0,Out>=3,V>=V1+1,V6+1>=Out] 

* Chain [37,49]: 1*s(41)+2
  Such that:s(41) =< V1

  with precondition: [V6>=1,V5>=0,Out>=3,V>=V1+1,V1+2>=Out] 


#### Cost of chains of fun4(V1,Out):
* Chain [[54],55]: 1*it(54)+0
  Such that:it(54) =< V1

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

* Chain [55]: 0
  with precondition: [Out=0,V1>=0] 


#### Cost of chains of start(V1,V,V6,V5,V4):
* Chain [58]: 517*s(638)+35*s(641)+192*s(646)+240*s(655)+40*s(658)+40*s(659)+80*s(660)+40*s(661)+7*s(670)+48*s(676)+80*s(677)+8*s(680)+8*s(681)+16*s(682)+8*s(683)+4
  Such that:s(671) =< -V1+V
aux(109) =< 1
aux(110) =< V1
aux(111) =< V
aux(112) =< V6
s(670) =< aux(110)
s(638) =< aux(111)
s(641) =< aux(109)
s(646) =< aux(112)
s(655) =< aux(111)
s(655) =< aux(112)
s(656) =< aux(111)
s(657) =< aux(112)
s(658) =< s(655)*aux(112)
s(659) =< s(655)*aux(111)
s(660) =< s(638)*s(656)
s(661) =< s(638)*s(657)
s(676) =< s(671)
s(677) =< s(671)
s(676) =< aux(112)
s(680) =< s(676)*aux(112)
s(681) =< s(676)*aux(111)
s(682) =< s(677)*s(656)
s(683) =< s(677)*s(657)

  with precondition: [V1>=0] 

* Chain [57]: 0
  with precondition: [V=0,V1>=0] 

* Chain [56]: 29*s(691)+5*s(692)+21*s(697)+48*s(698)+80*s(699)+8*s(702)+8*s(703)+16*s(704)+8*s(705)+3
  Such that:s(693) =< -V5+V4
aux(113) =< V1
aux(114) =< V5+1
aux(115) =< V4
s(692) =< aux(114)
s(691) =< aux(113)
s(697) =< aux(115)
s(698) =< s(693)
s(699) =< s(693)
s(698) =< aux(113)
s(700) =< aux(115)
s(701) =< aux(113)
s(702) =< s(698)*aux(113)
s(703) =< s(698)*aux(115)
s(704) =< s(699)*s(700)
s(705) =< s(699)*s(701)

  with precondition: [V=1,V1>=0,V6>=0,V5>=0,V4>=0] 


Closed-form bounds of start(V1,V,V6,V5,V4): 
-------------------------------------
* Chain [58] with precondition: [V1>=0] 
    - Upper bound: 7*V1+39+nat(V)*757+nat(V)*120*nat(V)+nat(V)*80*nat(V6)+nat(V)*24*nat(-V1+V)+nat(V6)*192+nat(V6)*16*nat(-V1+V)+nat(-V1+V)*128 
    - Complexity: n^2 
* Chain [57] with precondition: [V=0,V1>=0] 
    - Upper bound: 0 
    - Complexity: constant 
* Chain [56] with precondition: [V=1,V1>=0,V6>=0,V5>=0,V4>=0] 
    - Upper bound: 29*V1+3+16*V1*nat(-V5+V4)+21*V4+24*V4*nat(-V5+V4)+(5*V5+5)+nat(-V5+V4)*128 
    - Complexity: n^2 

### Maximum cost of start(V1,V,V6,V5,V4): 7*V1+3+max([16*V1*nat(-V5+V4)+22*V1+nat(V4)*21+nat(V4)*24*nat(-V5+V4)+nat(V5+1)*5+nat(-V5+V4)*128,nat(V)*757+36+nat(V)*120*nat(V)+nat(V)*80*nat(V6)+nat(V)*24*nat(-V1+V)+nat(V6)*192+nat(V6)*16*nat(-V1+V)+nat(-V1+V)*128]) 
Asymptotic class: n^2 
* Total analysis performed in 3037 ms.


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

(22)
BOUNDS(1, n^2)

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

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   rev(z0) -> if(z0, eq(0, length(z0)), nil, 0, length(z0))
   if(z0, true, z1, z2, z3) -> z1
   if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
   help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))
Tuples:
   GE(z0, 0) -> c
   GE(0, s(z0)) -> c1
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   REV(z0) -> c3(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   REV(z0) -> c4(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   IF(z0, true, z1, z2, z3) -> c5
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(nil) -> c11
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
S tuples:
   GE(z0, 0) -> c
   GE(0, s(z0)) -> c1
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   REV(z0) -> c3(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   REV(z0) -> c4(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   IF(z0, true, z1, z2, z3) -> c5
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(nil) -> c11
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
K tuples:none
Defined Rule Symbols:   ge_2, rev_1, if_5, help_4, append_2, length_1

Defined Pair Symbols:   GE_2, REV_1, IF_5, HELP_4, APPEND_2, LENGTH_1

Compound Symbols:   c, c1, c2_1, c3_2, c4_2, c5, c6_1, c7_2, c8_2, c9, c10_1, c11, c12_1


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

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

(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:

   GE(z0, 0) -> c
   GE(0, s(z0)) -> c1
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   REV(z0) -> c3(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   REV(z0) -> c4(IF(z0, eq(0, length(z0)), nil, 0, length(z0)), LENGTH(z0))
   IF(z0, true, z1, z2, z3) -> c5
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(nil) -> c11
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))

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

   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   rev(z0) -> if(z0, eq(0, length(z0)), nil, 0, length(z0))
   if(z0, true, z1, z2, z3) -> z1
   if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
   help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
   length(nil) -> 0
   length(cons(z0, z1)) -> s(length(z1))

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

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

(28)
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:

   GE(z0, 0') -> c
   GE(0', s(z0)) -> c1
   GE(s(z0), s(z1)) -> c2(GE(z0, z1))
   REV(z0) -> c3(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
   REV(z0) -> c4(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
   IF(z0, true, z1, z2, z3) -> c5
   IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
   HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
   HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
   APPEND(nil, z0) -> c9
   APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
   LENGTH(nil) -> c11
   LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))

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

   ge(z0, 0') -> true
   ge(0', s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   rev(z0) -> if(z0, eq(0', length(z0)), nil, 0', length(z0))
   if(z0, true, z1, z2, z3) -> z1
   if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
   help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
   append(nil, z0) -> z0
   append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
   length(nil) -> 0'
   length(cons(z0, z1)) -> s(length(z1))

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

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

(30)
Obligation:
Innermost TRS:
Rules:
GE(z0, 0') -> c
GE(0', s(z0)) -> c1
GE(s(z0), s(z1)) -> c2(GE(z0, z1))
REV(z0) -> c3(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
REV(z0) -> c4(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
IF(z0, true, z1, z2, z3) -> c5
IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
LENGTH(nil) -> c11
LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
rev(z0) -> if(z0, eq(0', length(z0)), nil, 0', length(z0))
if(z0, true, z1, z2, z3) -> z1
if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
GE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4
c3 :: c5:c6 -> c11:c12 -> c3:c4
IF :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> c5:c6
eq :: 0':s -> 0':s -> eq:true:false
length :: nil:cons -> 0':s
nil :: nil:cons
LENGTH :: nil:cons -> c11:c12
c4 :: c5:c6 -> c11:c12 -> c3:c4
true :: eq:true:false
c5 :: c5:c6
false :: eq:true:false
c6 :: c7:c8 -> c5:c6
HELP :: 0':s -> 0':s -> nil:cons -> nil:cons -> c7:c8
cons :: a -> nil:cons -> nil:cons
c7 :: c5:c6 -> c9:c10 -> c7:c8
append :: nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> eq:true:false
APPEND :: nil:cons -> nil:cons -> c9:c10
c8 :: c5:c6 -> c:c1:c2 -> c7:c8
c9 :: c9:c10
c10 :: c9:c10 -> c9:c10
c11 :: c11:c12
c12 :: c11:c12 -> c11:c12
rev :: nil:cons -> nil:cons
if :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_nil:cons4_13 :: nil:cons
hole_c5:c65_13 :: c5:c6
hole_c11:c126_13 :: c11:c12
hole_eq:true:false7_13 :: eq:true:false
hole_c7:c88_13 :: c7:c8
hole_a9_13 :: a
hole_c9:c1010_13 :: c9:c10
gen_c:c1:c211_13 :: Nat -> c:c1:c2
gen_0':s12_13 :: Nat -> 0':s
gen_nil:cons13_13 :: Nat -> nil:cons
gen_c11:c1214_13 :: Nat -> c11:c12
gen_c9:c1015_13 :: Nat -> c9:c10

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

(31) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
GE, length, LENGTH, HELP, append, ge, APPEND, help

They will be analysed ascendingly in the following order:
GE < HELP
append < HELP
ge < HELP
APPEND < HELP
append < help
ge < help

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

(32)
Obligation:
Innermost TRS:
Rules:
GE(z0, 0') -> c
GE(0', s(z0)) -> c1
GE(s(z0), s(z1)) -> c2(GE(z0, z1))
REV(z0) -> c3(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
REV(z0) -> c4(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
IF(z0, true, z1, z2, z3) -> c5
IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
LENGTH(nil) -> c11
LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
rev(z0) -> if(z0, eq(0', length(z0)), nil, 0', length(z0))
if(z0, true, z1, z2, z3) -> z1
if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
GE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4
c3 :: c5:c6 -> c11:c12 -> c3:c4
IF :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> c5:c6
eq :: 0':s -> 0':s -> eq:true:false
length :: nil:cons -> 0':s
nil :: nil:cons
LENGTH :: nil:cons -> c11:c12
c4 :: c5:c6 -> c11:c12 -> c3:c4
true :: eq:true:false
c5 :: c5:c6
false :: eq:true:false
c6 :: c7:c8 -> c5:c6
HELP :: 0':s -> 0':s -> nil:cons -> nil:cons -> c7:c8
cons :: a -> nil:cons -> nil:cons
c7 :: c5:c6 -> c9:c10 -> c7:c8
append :: nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> eq:true:false
APPEND :: nil:cons -> nil:cons -> c9:c10
c8 :: c5:c6 -> c:c1:c2 -> c7:c8
c9 :: c9:c10
c10 :: c9:c10 -> c9:c10
c11 :: c11:c12
c12 :: c11:c12 -> c11:c12
rev :: nil:cons -> nil:cons
if :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_nil:cons4_13 :: nil:cons
hole_c5:c65_13 :: c5:c6
hole_c11:c126_13 :: c11:c12
hole_eq:true:false7_13 :: eq:true:false
hole_c7:c88_13 :: c7:c8
hole_a9_13 :: a
hole_c9:c1010_13 :: c9:c10
gen_c:c1:c211_13 :: Nat -> c:c1:c2
gen_0':s12_13 :: Nat -> 0':s
gen_nil:cons13_13 :: Nat -> nil:cons
gen_c11:c1214_13 :: Nat -> c11:c12
gen_c9:c1015_13 :: Nat -> c9:c10


Generator Equations:
gen_c:c1:c211_13(0) <=> c
gen_c:c1:c211_13(+(x, 1)) <=> c2(gen_c:c1:c211_13(x))
gen_0':s12_13(0) <=> 0'
gen_0':s12_13(+(x, 1)) <=> s(gen_0':s12_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(hole_a9_13, gen_nil:cons13_13(x))
gen_c11:c1214_13(0) <=> c11
gen_c11:c1214_13(+(x, 1)) <=> c12(gen_c11:c1214_13(x))
gen_c9:c1015_13(0) <=> c9
gen_c9:c1015_13(+(x, 1)) <=> c10(gen_c9:c1015_13(x))


The following defined symbols remain to be analysed:
GE, length, LENGTH, HELP, append, ge, APPEND, help

They will be analysed ascendingly in the following order:
GE < HELP
append < HELP
ge < HELP
APPEND < HELP
append < help
ge < help

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

(33) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
GE(gen_0':s12_13(n17_13), gen_0':s12_13(n17_13)) -> gen_c:c1:c211_13(n17_13), rt in Omega(1 + n17_13)

Induction Base:
GE(gen_0':s12_13(0), gen_0':s12_13(0)) ->_R^Omega(1)
c

Induction Step:
GE(gen_0':s12_13(+(n17_13, 1)), gen_0':s12_13(+(n17_13, 1))) ->_R^Omega(1)
c2(GE(gen_0':s12_13(n17_13), gen_0':s12_13(n17_13))) ->_IH
c2(gen_c:c1:c211_13(c18_13))

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

(34)
Complex Obligation (BEST)

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

(35)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
GE(z0, 0') -> c
GE(0', s(z0)) -> c1
GE(s(z0), s(z1)) -> c2(GE(z0, z1))
REV(z0) -> c3(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
REV(z0) -> c4(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
IF(z0, true, z1, z2, z3) -> c5
IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
LENGTH(nil) -> c11
LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
rev(z0) -> if(z0, eq(0', length(z0)), nil, 0', length(z0))
if(z0, true, z1, z2, z3) -> z1
if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
GE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4
c3 :: c5:c6 -> c11:c12 -> c3:c4
IF :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> c5:c6
eq :: 0':s -> 0':s -> eq:true:false
length :: nil:cons -> 0':s
nil :: nil:cons
LENGTH :: nil:cons -> c11:c12
c4 :: c5:c6 -> c11:c12 -> c3:c4
true :: eq:true:false
c5 :: c5:c6
false :: eq:true:false
c6 :: c7:c8 -> c5:c6
HELP :: 0':s -> 0':s -> nil:cons -> nil:cons -> c7:c8
cons :: a -> nil:cons -> nil:cons
c7 :: c5:c6 -> c9:c10 -> c7:c8
append :: nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> eq:true:false
APPEND :: nil:cons -> nil:cons -> c9:c10
c8 :: c5:c6 -> c:c1:c2 -> c7:c8
c9 :: c9:c10
c10 :: c9:c10 -> c9:c10
c11 :: c11:c12
c12 :: c11:c12 -> c11:c12
rev :: nil:cons -> nil:cons
if :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_nil:cons4_13 :: nil:cons
hole_c5:c65_13 :: c5:c6
hole_c11:c126_13 :: c11:c12
hole_eq:true:false7_13 :: eq:true:false
hole_c7:c88_13 :: c7:c8
hole_a9_13 :: a
hole_c9:c1010_13 :: c9:c10
gen_c:c1:c211_13 :: Nat -> c:c1:c2
gen_0':s12_13 :: Nat -> 0':s
gen_nil:cons13_13 :: Nat -> nil:cons
gen_c11:c1214_13 :: Nat -> c11:c12
gen_c9:c1015_13 :: Nat -> c9:c10


Generator Equations:
gen_c:c1:c211_13(0) <=> c
gen_c:c1:c211_13(+(x, 1)) <=> c2(gen_c:c1:c211_13(x))
gen_0':s12_13(0) <=> 0'
gen_0':s12_13(+(x, 1)) <=> s(gen_0':s12_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(hole_a9_13, gen_nil:cons13_13(x))
gen_c11:c1214_13(0) <=> c11
gen_c11:c1214_13(+(x, 1)) <=> c12(gen_c11:c1214_13(x))
gen_c9:c1015_13(0) <=> c9
gen_c9:c1015_13(+(x, 1)) <=> c10(gen_c9:c1015_13(x))


The following defined symbols remain to be analysed:
GE, length, LENGTH, HELP, append, ge, APPEND, help

They will be analysed ascendingly in the following order:
GE < HELP
append < HELP
ge < HELP
APPEND < HELP
append < help
ge < help

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

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

(37)
BOUNDS(n^1, INF)

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

(38)
Obligation:
Innermost TRS:
Rules:
GE(z0, 0') -> c
GE(0', s(z0)) -> c1
GE(s(z0), s(z1)) -> c2(GE(z0, z1))
REV(z0) -> c3(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
REV(z0) -> c4(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
IF(z0, true, z1, z2, z3) -> c5
IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
LENGTH(nil) -> c11
LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
rev(z0) -> if(z0, eq(0', length(z0)), nil, 0', length(z0))
if(z0, true, z1, z2, z3) -> z1
if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
GE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4
c3 :: c5:c6 -> c11:c12 -> c3:c4
IF :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> c5:c6
eq :: 0':s -> 0':s -> eq:true:false
length :: nil:cons -> 0':s
nil :: nil:cons
LENGTH :: nil:cons -> c11:c12
c4 :: c5:c6 -> c11:c12 -> c3:c4
true :: eq:true:false
c5 :: c5:c6
false :: eq:true:false
c6 :: c7:c8 -> c5:c6
HELP :: 0':s -> 0':s -> nil:cons -> nil:cons -> c7:c8
cons :: a -> nil:cons -> nil:cons
c7 :: c5:c6 -> c9:c10 -> c7:c8
append :: nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> eq:true:false
APPEND :: nil:cons -> nil:cons -> c9:c10
c8 :: c5:c6 -> c:c1:c2 -> c7:c8
c9 :: c9:c10
c10 :: c9:c10 -> c9:c10
c11 :: c11:c12
c12 :: c11:c12 -> c11:c12
rev :: nil:cons -> nil:cons
if :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_nil:cons4_13 :: nil:cons
hole_c5:c65_13 :: c5:c6
hole_c11:c126_13 :: c11:c12
hole_eq:true:false7_13 :: eq:true:false
hole_c7:c88_13 :: c7:c8
hole_a9_13 :: a
hole_c9:c1010_13 :: c9:c10
gen_c:c1:c211_13 :: Nat -> c:c1:c2
gen_0':s12_13 :: Nat -> 0':s
gen_nil:cons13_13 :: Nat -> nil:cons
gen_c11:c1214_13 :: Nat -> c11:c12
gen_c9:c1015_13 :: Nat -> c9:c10


Lemmas:
GE(gen_0':s12_13(n17_13), gen_0':s12_13(n17_13)) -> gen_c:c1:c211_13(n17_13), rt in Omega(1 + n17_13)


Generator Equations:
gen_c:c1:c211_13(0) <=> c
gen_c:c1:c211_13(+(x, 1)) <=> c2(gen_c:c1:c211_13(x))
gen_0':s12_13(0) <=> 0'
gen_0':s12_13(+(x, 1)) <=> s(gen_0':s12_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(hole_a9_13, gen_nil:cons13_13(x))
gen_c11:c1214_13(0) <=> c11
gen_c11:c1214_13(+(x, 1)) <=> c12(gen_c11:c1214_13(x))
gen_c9:c1015_13(0) <=> c9
gen_c9:c1015_13(+(x, 1)) <=> c10(gen_c9:c1015_13(x))


The following defined symbols remain to be analysed:
length, LENGTH, HELP, append, ge, APPEND, help

They will be analysed ascendingly in the following order:
append < HELP
ge < HELP
APPEND < HELP
append < help
ge < help

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

(39) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
length(gen_nil:cons13_13(n612_13)) -> gen_0':s12_13(n612_13), rt in Omega(0)

Induction Base:
length(gen_nil:cons13_13(0)) ->_R^Omega(0)
0'

Induction Step:
length(gen_nil:cons13_13(+(n612_13, 1))) ->_R^Omega(0)
s(length(gen_nil:cons13_13(n612_13))) ->_IH
s(gen_0':s12_13(c613_13))

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

(40)
Obligation:
Innermost TRS:
Rules:
GE(z0, 0') -> c
GE(0', s(z0)) -> c1
GE(s(z0), s(z1)) -> c2(GE(z0, z1))
REV(z0) -> c3(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
REV(z0) -> c4(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
IF(z0, true, z1, z2, z3) -> c5
IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
LENGTH(nil) -> c11
LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
rev(z0) -> if(z0, eq(0', length(z0)), nil, 0', length(z0))
if(z0, true, z1, z2, z3) -> z1
if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
GE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4
c3 :: c5:c6 -> c11:c12 -> c3:c4
IF :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> c5:c6
eq :: 0':s -> 0':s -> eq:true:false
length :: nil:cons -> 0':s
nil :: nil:cons
LENGTH :: nil:cons -> c11:c12
c4 :: c5:c6 -> c11:c12 -> c3:c4
true :: eq:true:false
c5 :: c5:c6
false :: eq:true:false
c6 :: c7:c8 -> c5:c6
HELP :: 0':s -> 0':s -> nil:cons -> nil:cons -> c7:c8
cons :: a -> nil:cons -> nil:cons
c7 :: c5:c6 -> c9:c10 -> c7:c8
append :: nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> eq:true:false
APPEND :: nil:cons -> nil:cons -> c9:c10
c8 :: c5:c6 -> c:c1:c2 -> c7:c8
c9 :: c9:c10
c10 :: c9:c10 -> c9:c10
c11 :: c11:c12
c12 :: c11:c12 -> c11:c12
rev :: nil:cons -> nil:cons
if :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_nil:cons4_13 :: nil:cons
hole_c5:c65_13 :: c5:c6
hole_c11:c126_13 :: c11:c12
hole_eq:true:false7_13 :: eq:true:false
hole_c7:c88_13 :: c7:c8
hole_a9_13 :: a
hole_c9:c1010_13 :: c9:c10
gen_c:c1:c211_13 :: Nat -> c:c1:c2
gen_0':s12_13 :: Nat -> 0':s
gen_nil:cons13_13 :: Nat -> nil:cons
gen_c11:c1214_13 :: Nat -> c11:c12
gen_c9:c1015_13 :: Nat -> c9:c10


Lemmas:
GE(gen_0':s12_13(n17_13), gen_0':s12_13(n17_13)) -> gen_c:c1:c211_13(n17_13), rt in Omega(1 + n17_13)
length(gen_nil:cons13_13(n612_13)) -> gen_0':s12_13(n612_13), rt in Omega(0)


Generator Equations:
gen_c:c1:c211_13(0) <=> c
gen_c:c1:c211_13(+(x, 1)) <=> c2(gen_c:c1:c211_13(x))
gen_0':s12_13(0) <=> 0'
gen_0':s12_13(+(x, 1)) <=> s(gen_0':s12_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(hole_a9_13, gen_nil:cons13_13(x))
gen_c11:c1214_13(0) <=> c11
gen_c11:c1214_13(+(x, 1)) <=> c12(gen_c11:c1214_13(x))
gen_c9:c1015_13(0) <=> c9
gen_c9:c1015_13(+(x, 1)) <=> c10(gen_c9:c1015_13(x))


The following defined symbols remain to be analysed:
LENGTH, HELP, append, ge, APPEND, help

They will be analysed ascendingly in the following order:
append < HELP
ge < HELP
APPEND < HELP
append < help
ge < help

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

(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LENGTH(gen_nil:cons13_13(n962_13)) -> gen_c11:c1214_13(n962_13), rt in Omega(1 + n962_13)

Induction Base:
LENGTH(gen_nil:cons13_13(0)) ->_R^Omega(1)
c11

Induction Step:
LENGTH(gen_nil:cons13_13(+(n962_13, 1))) ->_R^Omega(1)
c12(LENGTH(gen_nil:cons13_13(n962_13))) ->_IH
c12(gen_c11:c1214_13(c963_13))

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:
GE(z0, 0') -> c
GE(0', s(z0)) -> c1
GE(s(z0), s(z1)) -> c2(GE(z0, z1))
REV(z0) -> c3(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
REV(z0) -> c4(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
IF(z0, true, z1, z2, z3) -> c5
IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
LENGTH(nil) -> c11
LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
rev(z0) -> if(z0, eq(0', length(z0)), nil, 0', length(z0))
if(z0, true, z1, z2, z3) -> z1
if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
GE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4
c3 :: c5:c6 -> c11:c12 -> c3:c4
IF :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> c5:c6
eq :: 0':s -> 0':s -> eq:true:false
length :: nil:cons -> 0':s
nil :: nil:cons
LENGTH :: nil:cons -> c11:c12
c4 :: c5:c6 -> c11:c12 -> c3:c4
true :: eq:true:false
c5 :: c5:c6
false :: eq:true:false
c6 :: c7:c8 -> c5:c6
HELP :: 0':s -> 0':s -> nil:cons -> nil:cons -> c7:c8
cons :: a -> nil:cons -> nil:cons
c7 :: c5:c6 -> c9:c10 -> c7:c8
append :: nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> eq:true:false
APPEND :: nil:cons -> nil:cons -> c9:c10
c8 :: c5:c6 -> c:c1:c2 -> c7:c8
c9 :: c9:c10
c10 :: c9:c10 -> c9:c10
c11 :: c11:c12
c12 :: c11:c12 -> c11:c12
rev :: nil:cons -> nil:cons
if :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_nil:cons4_13 :: nil:cons
hole_c5:c65_13 :: c5:c6
hole_c11:c126_13 :: c11:c12
hole_eq:true:false7_13 :: eq:true:false
hole_c7:c88_13 :: c7:c8
hole_a9_13 :: a
hole_c9:c1010_13 :: c9:c10
gen_c:c1:c211_13 :: Nat -> c:c1:c2
gen_0':s12_13 :: Nat -> 0':s
gen_nil:cons13_13 :: Nat -> nil:cons
gen_c11:c1214_13 :: Nat -> c11:c12
gen_c9:c1015_13 :: Nat -> c9:c10


Lemmas:
GE(gen_0':s12_13(n17_13), gen_0':s12_13(n17_13)) -> gen_c:c1:c211_13(n17_13), rt in Omega(1 + n17_13)
length(gen_nil:cons13_13(n612_13)) -> gen_0':s12_13(n612_13), rt in Omega(0)
LENGTH(gen_nil:cons13_13(n962_13)) -> gen_c11:c1214_13(n962_13), rt in Omega(1 + n962_13)


Generator Equations:
gen_c:c1:c211_13(0) <=> c
gen_c:c1:c211_13(+(x, 1)) <=> c2(gen_c:c1:c211_13(x))
gen_0':s12_13(0) <=> 0'
gen_0':s12_13(+(x, 1)) <=> s(gen_0':s12_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(hole_a9_13, gen_nil:cons13_13(x))
gen_c11:c1214_13(0) <=> c11
gen_c11:c1214_13(+(x, 1)) <=> c12(gen_c11:c1214_13(x))
gen_c9:c1015_13(0) <=> c9
gen_c9:c1015_13(+(x, 1)) <=> c10(gen_c9:c1015_13(x))


The following defined symbols remain to be analysed:
append, HELP, ge, APPEND, help

They will be analysed ascendingly in the following order:
append < HELP
ge < HELP
APPEND < HELP
append < help
ge < help

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

(43) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
append(gen_nil:cons13_13(n1384_13), gen_nil:cons13_13(b)) -> gen_nil:cons13_13(+(n1384_13, b)), rt in Omega(0)

Induction Base:
append(gen_nil:cons13_13(0), gen_nil:cons13_13(b)) ->_R^Omega(0)
gen_nil:cons13_13(b)

Induction Step:
append(gen_nil:cons13_13(+(n1384_13, 1)), gen_nil:cons13_13(b)) ->_R^Omega(0)
cons(hole_a9_13, append(gen_nil:cons13_13(n1384_13), gen_nil:cons13_13(b))) ->_IH
cons(hole_a9_13, gen_nil:cons13_13(+(b, c1385_13)))

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:
GE(z0, 0') -> c
GE(0', s(z0)) -> c1
GE(s(z0), s(z1)) -> c2(GE(z0, z1))
REV(z0) -> c3(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
REV(z0) -> c4(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
IF(z0, true, z1, z2, z3) -> c5
IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
LENGTH(nil) -> c11
LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
rev(z0) -> if(z0, eq(0', length(z0)), nil, 0', length(z0))
if(z0, true, z1, z2, z3) -> z1
if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
GE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4
c3 :: c5:c6 -> c11:c12 -> c3:c4
IF :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> c5:c6
eq :: 0':s -> 0':s -> eq:true:false
length :: nil:cons -> 0':s
nil :: nil:cons
LENGTH :: nil:cons -> c11:c12
c4 :: c5:c6 -> c11:c12 -> c3:c4
true :: eq:true:false
c5 :: c5:c6
false :: eq:true:false
c6 :: c7:c8 -> c5:c6
HELP :: 0':s -> 0':s -> nil:cons -> nil:cons -> c7:c8
cons :: a -> nil:cons -> nil:cons
c7 :: c5:c6 -> c9:c10 -> c7:c8
append :: nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> eq:true:false
APPEND :: nil:cons -> nil:cons -> c9:c10
c8 :: c5:c6 -> c:c1:c2 -> c7:c8
c9 :: c9:c10
c10 :: c9:c10 -> c9:c10
c11 :: c11:c12
c12 :: c11:c12 -> c11:c12
rev :: nil:cons -> nil:cons
if :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_nil:cons4_13 :: nil:cons
hole_c5:c65_13 :: c5:c6
hole_c11:c126_13 :: c11:c12
hole_eq:true:false7_13 :: eq:true:false
hole_c7:c88_13 :: c7:c8
hole_a9_13 :: a
hole_c9:c1010_13 :: c9:c10
gen_c:c1:c211_13 :: Nat -> c:c1:c2
gen_0':s12_13 :: Nat -> 0':s
gen_nil:cons13_13 :: Nat -> nil:cons
gen_c11:c1214_13 :: Nat -> c11:c12
gen_c9:c1015_13 :: Nat -> c9:c10


Lemmas:
GE(gen_0':s12_13(n17_13), gen_0':s12_13(n17_13)) -> gen_c:c1:c211_13(n17_13), rt in Omega(1 + n17_13)
length(gen_nil:cons13_13(n612_13)) -> gen_0':s12_13(n612_13), rt in Omega(0)
LENGTH(gen_nil:cons13_13(n962_13)) -> gen_c11:c1214_13(n962_13), rt in Omega(1 + n962_13)
append(gen_nil:cons13_13(n1384_13), gen_nil:cons13_13(b)) -> gen_nil:cons13_13(+(n1384_13, b)), rt in Omega(0)


Generator Equations:
gen_c:c1:c211_13(0) <=> c
gen_c:c1:c211_13(+(x, 1)) <=> c2(gen_c:c1:c211_13(x))
gen_0':s12_13(0) <=> 0'
gen_0':s12_13(+(x, 1)) <=> s(gen_0':s12_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(hole_a9_13, gen_nil:cons13_13(x))
gen_c11:c1214_13(0) <=> c11
gen_c11:c1214_13(+(x, 1)) <=> c12(gen_c11:c1214_13(x))
gen_c9:c1015_13(0) <=> c9
gen_c9:c1015_13(+(x, 1)) <=> c10(gen_c9:c1015_13(x))


The following defined symbols remain to be analysed:
ge, HELP, APPEND, help

They will be analysed ascendingly in the following order:
ge < HELP
APPEND < HELP
ge < help

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

(45) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s12_13(n2871_13), gen_0':s12_13(n2871_13)) -> true, rt in Omega(0)

Induction Base:
ge(gen_0':s12_13(0), gen_0':s12_13(0)) ->_R^Omega(0)
true

Induction Step:
ge(gen_0':s12_13(+(n2871_13, 1)), gen_0':s12_13(+(n2871_13, 1))) ->_R^Omega(0)
ge(gen_0':s12_13(n2871_13), gen_0':s12_13(n2871_13)) ->_IH
true

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

(46)
Obligation:
Innermost TRS:
Rules:
GE(z0, 0') -> c
GE(0', s(z0)) -> c1
GE(s(z0), s(z1)) -> c2(GE(z0, z1))
REV(z0) -> c3(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
REV(z0) -> c4(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
IF(z0, true, z1, z2, z3) -> c5
IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
LENGTH(nil) -> c11
LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
rev(z0) -> if(z0, eq(0', length(z0)), nil, 0', length(z0))
if(z0, true, z1, z2, z3) -> z1
if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
GE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4
c3 :: c5:c6 -> c11:c12 -> c3:c4
IF :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> c5:c6
eq :: 0':s -> 0':s -> eq:true:false
length :: nil:cons -> 0':s
nil :: nil:cons
LENGTH :: nil:cons -> c11:c12
c4 :: c5:c6 -> c11:c12 -> c3:c4
true :: eq:true:false
c5 :: c5:c6
false :: eq:true:false
c6 :: c7:c8 -> c5:c6
HELP :: 0':s -> 0':s -> nil:cons -> nil:cons -> c7:c8
cons :: a -> nil:cons -> nil:cons
c7 :: c5:c6 -> c9:c10 -> c7:c8
append :: nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> eq:true:false
APPEND :: nil:cons -> nil:cons -> c9:c10
c8 :: c5:c6 -> c:c1:c2 -> c7:c8
c9 :: c9:c10
c10 :: c9:c10 -> c9:c10
c11 :: c11:c12
c12 :: c11:c12 -> c11:c12
rev :: nil:cons -> nil:cons
if :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_nil:cons4_13 :: nil:cons
hole_c5:c65_13 :: c5:c6
hole_c11:c126_13 :: c11:c12
hole_eq:true:false7_13 :: eq:true:false
hole_c7:c88_13 :: c7:c8
hole_a9_13 :: a
hole_c9:c1010_13 :: c9:c10
gen_c:c1:c211_13 :: Nat -> c:c1:c2
gen_0':s12_13 :: Nat -> 0':s
gen_nil:cons13_13 :: Nat -> nil:cons
gen_c11:c1214_13 :: Nat -> c11:c12
gen_c9:c1015_13 :: Nat -> c9:c10


Lemmas:
GE(gen_0':s12_13(n17_13), gen_0':s12_13(n17_13)) -> gen_c:c1:c211_13(n17_13), rt in Omega(1 + n17_13)
length(gen_nil:cons13_13(n612_13)) -> gen_0':s12_13(n612_13), rt in Omega(0)
LENGTH(gen_nil:cons13_13(n962_13)) -> gen_c11:c1214_13(n962_13), rt in Omega(1 + n962_13)
append(gen_nil:cons13_13(n1384_13), gen_nil:cons13_13(b)) -> gen_nil:cons13_13(+(n1384_13, b)), rt in Omega(0)
ge(gen_0':s12_13(n2871_13), gen_0':s12_13(n2871_13)) -> true, rt in Omega(0)


Generator Equations:
gen_c:c1:c211_13(0) <=> c
gen_c:c1:c211_13(+(x, 1)) <=> c2(gen_c:c1:c211_13(x))
gen_0':s12_13(0) <=> 0'
gen_0':s12_13(+(x, 1)) <=> s(gen_0':s12_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(hole_a9_13, gen_nil:cons13_13(x))
gen_c11:c1214_13(0) <=> c11
gen_c11:c1214_13(+(x, 1)) <=> c12(gen_c11:c1214_13(x))
gen_c9:c1015_13(0) <=> c9
gen_c9:c1015_13(+(x, 1)) <=> c10(gen_c9:c1015_13(x))


The following defined symbols remain to be analysed:
APPEND, HELP, help

They will be analysed ascendingly in the following order:
APPEND < HELP

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

(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
APPEND(gen_nil:cons13_13(n3253_13), gen_nil:cons13_13(b)) -> gen_c9:c1015_13(n3253_13), rt in Omega(1 + n3253_13)

Induction Base:
APPEND(gen_nil:cons13_13(0), gen_nil:cons13_13(b)) ->_R^Omega(1)
c9

Induction Step:
APPEND(gen_nil:cons13_13(+(n3253_13, 1)), gen_nil:cons13_13(b)) ->_R^Omega(1)
c10(APPEND(gen_nil:cons13_13(n3253_13), gen_nil:cons13_13(b))) ->_IH
c10(gen_c9:c1015_13(c3254_13))

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

(48)
Obligation:
Innermost TRS:
Rules:
GE(z0, 0') -> c
GE(0', s(z0)) -> c1
GE(s(z0), s(z1)) -> c2(GE(z0, z1))
REV(z0) -> c3(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
REV(z0) -> c4(IF(z0, eq(0', length(z0)), nil, 0', length(z0)), LENGTH(z0))
IF(z0, true, z1, z2, z3) -> c5
IF(z0, false, z1, z2, z3) -> c6(HELP(s(z2), z3, z0, z1))
HELP(z0, z1, cons(z2, z3), z4) -> c7(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), APPEND(z3, cons(z2, nil)))
HELP(z0, z1, cons(z2, z3), z4) -> c8(IF(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1), GE(z0, z1))
APPEND(nil, z0) -> c9
APPEND(cons(z0, z1), z2) -> c10(APPEND(z1, z2))
LENGTH(nil) -> c11
LENGTH(cons(z0, z1)) -> c12(LENGTH(z1))
ge(z0, 0') -> true
ge(0', s(z0)) -> false
ge(s(z0), s(z1)) -> ge(z0, z1)
rev(z0) -> if(z0, eq(0', length(z0)), nil, 0', length(z0))
if(z0, true, z1, z2, z3) -> z1
if(z0, false, z1, z2, z3) -> help(s(z2), z3, z0, z1)
help(z0, z1, cons(z2, z3), z4) -> if(append(z3, cons(z2, nil)), ge(z0, z1), cons(z2, z4), z0, z1)
append(nil, z0) -> z0
append(cons(z0, z1), z2) -> cons(z0, append(z1, z2))
length(nil) -> 0'
length(cons(z0, z1)) -> s(length(z1))

Types:
GE :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
s :: 0':s -> 0':s
c1 :: c:c1:c2
c2 :: c:c1:c2 -> c:c1:c2
REV :: nil:cons -> c3:c4
c3 :: c5:c6 -> c11:c12 -> c3:c4
IF :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> c5:c6
eq :: 0':s -> 0':s -> eq:true:false
length :: nil:cons -> 0':s
nil :: nil:cons
LENGTH :: nil:cons -> c11:c12
c4 :: c5:c6 -> c11:c12 -> c3:c4
true :: eq:true:false
c5 :: c5:c6
false :: eq:true:false
c6 :: c7:c8 -> c5:c6
HELP :: 0':s -> 0':s -> nil:cons -> nil:cons -> c7:c8
cons :: a -> nil:cons -> nil:cons
c7 :: c5:c6 -> c9:c10 -> c7:c8
append :: nil:cons -> nil:cons -> nil:cons
ge :: 0':s -> 0':s -> eq:true:false
APPEND :: nil:cons -> nil:cons -> c9:c10
c8 :: c5:c6 -> c:c1:c2 -> c7:c8
c9 :: c9:c10
c10 :: c9:c10 -> c9:c10
c11 :: c11:c12
c12 :: c11:c12 -> c11:c12
rev :: nil:cons -> nil:cons
if :: nil:cons -> eq:true:false -> nil:cons -> 0':s -> 0':s -> nil:cons
help :: 0':s -> 0':s -> nil:cons -> nil:cons -> nil:cons
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_nil:cons4_13 :: nil:cons
hole_c5:c65_13 :: c5:c6
hole_c11:c126_13 :: c11:c12
hole_eq:true:false7_13 :: eq:true:false
hole_c7:c88_13 :: c7:c8
hole_a9_13 :: a
hole_c9:c1010_13 :: c9:c10
gen_c:c1:c211_13 :: Nat -> c:c1:c2
gen_0':s12_13 :: Nat -> 0':s
gen_nil:cons13_13 :: Nat -> nil:cons
gen_c11:c1214_13 :: Nat -> c11:c12
gen_c9:c1015_13 :: Nat -> c9:c10


Lemmas:
GE(gen_0':s12_13(n17_13), gen_0':s12_13(n17_13)) -> gen_c:c1:c211_13(n17_13), rt in Omega(1 + n17_13)
length(gen_nil:cons13_13(n612_13)) -> gen_0':s12_13(n612_13), rt in Omega(0)
LENGTH(gen_nil:cons13_13(n962_13)) -> gen_c11:c1214_13(n962_13), rt in Omega(1 + n962_13)
append(gen_nil:cons13_13(n1384_13), gen_nil:cons13_13(b)) -> gen_nil:cons13_13(+(n1384_13, b)), rt in Omega(0)
ge(gen_0':s12_13(n2871_13), gen_0':s12_13(n2871_13)) -> true, rt in Omega(0)
APPEND(gen_nil:cons13_13(n3253_13), gen_nil:cons13_13(b)) -> gen_c9:c1015_13(n3253_13), rt in Omega(1 + n3253_13)


Generator Equations:
gen_c:c1:c211_13(0) <=> c
gen_c:c1:c211_13(+(x, 1)) <=> c2(gen_c:c1:c211_13(x))
gen_0':s12_13(0) <=> 0'
gen_0':s12_13(+(x, 1)) <=> s(gen_0':s12_13(x))
gen_nil:cons13_13(0) <=> nil
gen_nil:cons13_13(+(x, 1)) <=> cons(hole_a9_13, gen_nil:cons13_13(x))
gen_c11:c1214_13(0) <=> c11
gen_c11:c1214_13(+(x, 1)) <=> c12(gen_c11:c1214_13(x))
gen_c9:c1015_13(0) <=> c9
gen_c9:c1015_13(+(x, 1)) <=> c10(gen_c9:c1015_13(x))


The following defined symbols remain to be analysed:
HELP, help