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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) 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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxWeightedTrs
    (13) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxTypedWeightedTrs
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
    (17) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
    (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (22) CpxRNTS
    (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 191 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (30) CpxRNTS
    (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 759 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 273 ms]
    (36) CpxRNTS
    (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 147 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
    (42) CpxRNTS
    (43) FinalProof [FINISHED, 0 ms]
    (44) BOUNDS(1, n^1)
(45) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(46) CdtProblem
    (47) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CpxRelTRS
    (49) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CpxRelTRS
    (51) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) typed CpxTrs
    (53) OrderProof [LOWER BOUND(ID), 21 ms]
    (54) typed CpxTrs
    (55) RewriteLemmaProof [LOWER BOUND(ID), 755 ms]
    (56) BEST
        (57) proven lower bound
            (58) LowerBoundPropagationProof [FINISHED, 0 ms]
            (59) BOUNDS(n^1, INF)
        (60) typed CpxTrs
            (61) RewriteLemmaProof [LOWER BOUND(ID), 79 ms]
            (62) BOUNDS(1, INF)


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

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


The TRS R consists of the following rules:

   is_empty(nil) -> true
   is_empty(cons(x, l)) -> false
   hd(cons(x, l)) -> x
   tl(cons(x, l)) -> l
   append(l1, l2) -> ifappend(l1, l2, is_empty(l1))
   ifappend(l1, l2, true) -> l2
   ifappend(l1, l2, false) -> cons(hd(l1), append(tl(l1), l2))

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:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
S tuples:
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   IS_EMPTY_1, HD_1, TL_1, APPEND_2, IFAPPEND_3

Compound Symbols:   c, c1, c2, c3, c4_2, c5, c6_1, c7_2


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

(3) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   IS_EMPTY(cons(z0, z1)) -> c1
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   IFAPPEND(z0, z1, true) -> c5
   IS_EMPTY(nil) -> c

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

(4)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_2, c7_2


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

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

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_1, c7_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   hd(cons(z0, z1)) -> z0
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   tl(cons(z0, z1)) -> z1
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))
K tuples:none
Defined Rule Symbols:   is_empty_1, tl_1

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_1, c7_1


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

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

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


The TRS R consists of the following rules:

   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1))

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

   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   tl(cons(z0, z1)) -> z1

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

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

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


The TRS R consists of the following rules:

   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0))) [1]
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1)) [1]
   is_empty(nil) -> true [0]
   is_empty(cons(z0, z1)) -> false [0]
   tl(cons(z0, z1)) -> z1 [0]

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

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

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

   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0))) [1]
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1)) [1]
   is_empty(nil) -> true [0]
   is_empty(cons(z0, z1)) -> false [0]
   tl(cons(z0, z1)) -> z1 [0]

The TRS has the following type information:
APPEND :: nil:cons -> a -> c4
c4 :: c7 -> c4
IFAPPEND :: nil:cons -> a -> false:true -> c7
is_empty :: nil:cons -> false:true
false :: false:true
c7 :: c4 -> c7
tl :: nil:cons -> nil:cons
nil :: nil:cons
true :: false:true
cons :: b -> nil:cons -> nil:cons

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

(15) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   APPEND_2
   IFAPPEND_3

(c) The following functions are completely defined:

   is_empty_1
   tl_1

Due to the following rules being added:

   is_empty(v0) -> null_is_empty [0]
   tl(v0) -> nil [0]

And the following fresh constants:    null_is_empty, const, const1, const2, const3

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

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

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

   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0))) [1]
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1)) [1]
   is_empty(nil) -> true [0]
   is_empty(cons(z0, z1)) -> false [0]
   tl(cons(z0, z1)) -> z1 [0]
   is_empty(v0) -> null_is_empty [0]
   tl(v0) -> nil [0]

The TRS has the following type information:
APPEND :: nil:cons -> a -> c4
c4 :: c7 -> c4
IFAPPEND :: nil:cons -> a -> false:true:null_is_empty -> c7
is_empty :: nil:cons -> false:true:null_is_empty
false :: false:true:null_is_empty
c7 :: c4 -> c7
tl :: nil:cons -> nil:cons
nil :: nil:cons
true :: false:true:null_is_empty
cons :: b -> nil:cons -> nil:cons
null_is_empty :: false:true:null_is_empty
const :: c4
const1 :: a
const2 :: c7
const3 :: b

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

(17) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(18)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   APPEND(nil, z1) -> c4(IFAPPEND(nil, z1, true)) [1]
   APPEND(cons(z0', z1'), z1) -> c4(IFAPPEND(cons(z0', z1'), z1, false)) [1]
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, null_is_empty)) [1]
   IFAPPEND(cons(z0'', z1''), z1, false) -> c7(APPEND(z1'', z1)) [1]
   IFAPPEND(z0, z1, false) -> c7(APPEND(nil, z1)) [1]
   is_empty(nil) -> true [0]
   is_empty(cons(z0, z1)) -> false [0]
   tl(cons(z0, z1)) -> z1 [0]
   is_empty(v0) -> null_is_empty [0]
   tl(v0) -> nil [0]

The TRS has the following type information:
APPEND :: nil:cons -> a -> c4
c4 :: c7 -> c4
IFAPPEND :: nil:cons -> a -> false:true:null_is_empty -> c7
is_empty :: nil:cons -> false:true:null_is_empty
false :: false:true:null_is_empty
c7 :: c4 -> c7
tl :: nil:cons -> nil:cons
nil :: nil:cons
true :: false:true:null_is_empty
cons :: b -> nil:cons -> nil:cons
null_is_empty :: false:true:null_is_empty
const :: c4
const1 :: a
const2 :: c7
const3 :: b

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
   true => 2
   null_is_empty => 0
   const => 0
   const1 => 0
   const2 => 0
   const3 => 0

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

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

   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(z0, z1, 0) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(0, z1, 2) :|: z1 >= 0, z' = z1, z = 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(1 + z0' + z1', z1, 1) :|: z1 >= 0, z0' >= 0, z1' >= 0, z' = z1, z = 1 + z0' + z1'
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(z1'', z1) :|: z = 1 + z0'' + z1'', z1 >= 0, z' = z1, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(0, z1) :|: z = z0, z1 >= 0, z' = z1, z0 >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

(21) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(z, z', 0) :|: z' >= 0, z >= 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(0, z', 2) :|: z' >= 0, z = 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(1 + z0' + z1', z', 1) :|: z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(z1'', z') :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(0, z') :|: z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0


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

(23) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { is_empty }
   { IFAPPEND, APPEND }
   { tl }

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

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(z, z', 0) :|: z' >= 0, z >= 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(0, z', 2) :|: z' >= 0, z = 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(1 + z0' + z1', z', 1) :|: z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(z1'', z') :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(0, z') :|: z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {is_empty}, {IFAPPEND,APPEND}, {tl}

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

(25) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(z, z', 0) :|: z' >= 0, z >= 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(0, z', 2) :|: z' >= 0, z = 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(1 + z0' + z1', z', 1) :|: z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(z1'', z') :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(0, z') :|: z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {is_empty}, {IFAPPEND,APPEND}, {tl}

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: is_empty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(z, z', 0) :|: z' >= 0, z >= 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(0, z', 2) :|: z' >= 0, z = 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(1 + z0' + z1', z', 1) :|: z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(z1'', z') :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(0, z') :|: z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {is_empty}, {IFAPPEND,APPEND}, {tl}
Previous analysis results are:
  is_empty: runtime: ?, size: O(1) [2]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: is_empty
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(z, z', 0) :|: z' >= 0, z >= 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(0, z', 2) :|: z' >= 0, z = 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(1 + z0' + z1', z', 1) :|: z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(z1'', z') :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(0, z') :|: z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFAPPEND,APPEND}, {tl}
Previous analysis results are:
  is_empty: runtime: O(1) [0], size: O(1) [2]

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

(31) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(z, z', 0) :|: z' >= 0, z >= 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(0, z', 2) :|: z' >= 0, z = 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(1 + z0' + z1', z', 1) :|: z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(z1'', z') :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(0, z') :|: z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFAPPEND,APPEND}, {tl}
Previous analysis results are:
  is_empty: runtime: O(1) [0], size: O(1) [2]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: IFAPPEND
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

Computed SIZE bound using CoFloCo for: APPEND
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(z, z', 0) :|: z' >= 0, z >= 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(0, z', 2) :|: z' >= 0, z = 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(1 + z0' + z1', z', 1) :|: z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(z1'', z') :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(0, z') :|: z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {IFAPPEND,APPEND}, {tl}
Previous analysis results are:
  is_empty: runtime: O(1) [0], size: O(1) [2]
  IFAPPEND: runtime: ?, size: O(1) [0]
  APPEND: runtime: ?, size: O(1) [1]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: IFAPPEND
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 2*z

Computed RUNTIME bound using CoFloCo for: APPEND
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 2*z

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(z, z', 0) :|: z' >= 0, z >= 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(0, z', 2) :|: z' >= 0, z = 0
   APPEND(z, z') -{ 1 }-> 1 + IFAPPEND(1 + z0' + z1', z', 1) :|: z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(z1'', z') :|: z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 1 }-> 1 + APPEND(0, z') :|: z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tl}
Previous analysis results are:
  is_empty: runtime: O(1) [0], size: O(1) [2]
  IFAPPEND: runtime: O(n^1) [4 + 2*z], size: O(1) [0]
  APPEND: runtime: O(n^1) [5 + 2*z], size: O(1) [1]

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

(37) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 5 }-> 1 + s :|: s >= 0, s <= 0, z' >= 0, z = 0
   APPEND(z, z') -{ 7 + 2*z0' + 2*z1' }-> 1 + s' :|: s' >= 0, s' <= 0, z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   APPEND(z, z') -{ 5 + 2*z }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' >= 0, z >= 0
   IFAPPEND(z, z', z'') -{ 6 + 2*z1'' }-> 1 + s1 :|: s1 >= 0, s1 <= 1, z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 6 }-> 1 + s2 :|: s2 >= 0, s2 <= 1, z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tl}
Previous analysis results are:
  is_empty: runtime: O(1) [0], size: O(1) [2]
  IFAPPEND: runtime: O(n^1) [4 + 2*z], size: O(1) [0]
  APPEND: runtime: O(n^1) [5 + 2*z], size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: tl
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

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

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 5 }-> 1 + s :|: s >= 0, s <= 0, z' >= 0, z = 0
   APPEND(z, z') -{ 7 + 2*z0' + 2*z1' }-> 1 + s' :|: s' >= 0, s' <= 0, z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   APPEND(z, z') -{ 5 + 2*z }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' >= 0, z >= 0
   IFAPPEND(z, z', z'') -{ 6 + 2*z1'' }-> 1 + s1 :|: s1 >= 0, s1 <= 1, z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 6 }-> 1 + s2 :|: s2 >= 0, s2 <= 1, z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {tl}
Previous analysis results are:
  is_empty: runtime: O(1) [0], size: O(1) [2]
  IFAPPEND: runtime: O(n^1) [4 + 2*z], size: O(1) [0]
  APPEND: runtime: O(n^1) [5 + 2*z], size: O(1) [1]
  tl: runtime: ?, size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: tl
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   APPEND(z, z') -{ 5 }-> 1 + s :|: s >= 0, s <= 0, z' >= 0, z = 0
   APPEND(z, z') -{ 7 + 2*z0' + 2*z1' }-> 1 + s' :|: s' >= 0, s' <= 0, z' >= 0, z0' >= 0, z1' >= 0, z = 1 + z0' + z1'
   APPEND(z, z') -{ 5 + 2*z }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z' >= 0, z >= 0
   IFAPPEND(z, z', z'') -{ 6 + 2*z1'' }-> 1 + s1 :|: s1 >= 0, s1 <= 1, z = 1 + z0'' + z1'', z' >= 0, z0'' >= 0, z'' = 1, z1'' >= 0
   IFAPPEND(z, z', z'') -{ 6 }-> 1 + s2 :|: s2 >= 0, s2 <= 1, z' >= 0, z >= 0, z'' = 1
   is_empty(z) -{ 0 }-> 2 :|: z = 0
   is_empty(z) -{ 0 }-> 1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   is_empty(z) -{ 0 }-> 0 :|: z >= 0
   tl(z) -{ 0 }-> z1 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   tl(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  is_empty: runtime: O(1) [0], size: O(1) [2]
  IFAPPEND: runtime: O(n^1) [4 + 2*z], size: O(1) [0]
  APPEND: runtime: O(n^1) [5 + 2*z], size: O(1) [1]
  tl: runtime: O(1) [0], size: O(n^1) [z]

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

(43) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(44)
BOUNDS(1, n^1)

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

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))
Tuples:
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
S tuples:
   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
K tuples:none
Defined Rule Symbols:   is_empty_1, hd_1, tl_1, append_2, ifappend_3

Defined Pair Symbols:   IS_EMPTY_1, HD_1, TL_1, APPEND_2, IFAPPEND_3

Compound Symbols:   c, c1, c2, c3, c4_2, c5, c6_1, c7_2


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

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

(48)
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:

   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))

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

   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

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

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

(50)
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:

   IS_EMPTY(nil) -> c
   IS_EMPTY(cons(z0, z1)) -> c1
   HD(cons(z0, z1)) -> c2
   TL(cons(z0, z1)) -> c3
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
   IFAPPEND(z0, z1, true) -> c5
   IFAPPEND(z0, z1, false) -> c6(HD(z0))
   IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))

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

   is_empty(nil) -> true
   is_empty(cons(z0, z1)) -> false
   hd(cons(z0, z1)) -> z0
   tl(cons(z0, z1)) -> z1
   append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
   ifappend(z0, z1, true) -> z1
   ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

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

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

(52)
Obligation:
Innermost TRS:
Rules:
IS_EMPTY(nil) -> c
IS_EMPTY(cons(z0, z1)) -> c1
HD(cons(z0, z1)) -> c2
TL(cons(z0, z1)) -> c3
APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
IFAPPEND(z0, z1, true) -> c5
IFAPPEND(z0, z1, false) -> c6(HD(z0))
IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
is_empty(nil) -> true
is_empty(cons(z0, z1)) -> false
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1
append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
ifappend(z0, z1, true) -> z1
ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

Types:
IS_EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: hd -> nil:cons -> nil:cons
c1 :: c:c1
HD :: nil:cons -> c2
c2 :: c2
TL :: nil:cons -> c3
c3 :: c3
APPEND :: nil:cons -> a -> c4
c4 :: c5:c6:c7 -> c:c1 -> c4
IFAPPEND :: nil:cons -> a -> true:false -> c5:c6:c7
is_empty :: nil:cons -> true:false
true :: true:false
c5 :: c5:c6:c7
false :: true:false
c6 :: c2 -> c5:c6:c7
c7 :: c4 -> c3 -> c5:c6:c7
tl :: nil:cons -> nil:cons
hd :: nil:cons -> hd
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_c:c11_8 :: c:c1
hole_nil:cons2_8 :: nil:cons
hole_hd3_8 :: hd
hole_c24_8 :: c2
hole_c35_8 :: c3
hole_c46_8 :: c4
hole_a7_8 :: a
hole_c5:c6:c78_8 :: c5:c6:c7
hole_true:false9_8 :: true:false
gen_nil:cons10_8 :: Nat -> nil:cons

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

(53) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
APPEND, append
----------------------------------------

(54)
Obligation:
Innermost TRS:
Rules:
IS_EMPTY(nil) -> c
IS_EMPTY(cons(z0, z1)) -> c1
HD(cons(z0, z1)) -> c2
TL(cons(z0, z1)) -> c3
APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
IFAPPEND(z0, z1, true) -> c5
IFAPPEND(z0, z1, false) -> c6(HD(z0))
IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
is_empty(nil) -> true
is_empty(cons(z0, z1)) -> false
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1
append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
ifappend(z0, z1, true) -> z1
ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

Types:
IS_EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: hd -> nil:cons -> nil:cons
c1 :: c:c1
HD :: nil:cons -> c2
c2 :: c2
TL :: nil:cons -> c3
c3 :: c3
APPEND :: nil:cons -> a -> c4
c4 :: c5:c6:c7 -> c:c1 -> c4
IFAPPEND :: nil:cons -> a -> true:false -> c5:c6:c7
is_empty :: nil:cons -> true:false
true :: true:false
c5 :: c5:c6:c7
false :: true:false
c6 :: c2 -> c5:c6:c7
c7 :: c4 -> c3 -> c5:c6:c7
tl :: nil:cons -> nil:cons
hd :: nil:cons -> hd
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_c:c11_8 :: c:c1
hole_nil:cons2_8 :: nil:cons
hole_hd3_8 :: hd
hole_c24_8 :: c2
hole_c35_8 :: c3
hole_c46_8 :: c4
hole_a7_8 :: a
hole_c5:c6:c78_8 :: c5:c6:c7
hole_true:false9_8 :: true:false
gen_nil:cons10_8 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons10_8(0) <=> nil
gen_nil:cons10_8(+(x, 1)) <=> cons(hole_hd3_8, gen_nil:cons10_8(x))


The following defined symbols remain to be analysed:
APPEND, append
----------------------------------------

(55) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
APPEND(gen_nil:cons10_8(n12_8), hole_a7_8) -> *11_8, rt in Omega(n12_8)

Induction Base:
APPEND(gen_nil:cons10_8(0), hole_a7_8)

Induction Step:
APPEND(gen_nil:cons10_8(+(n12_8, 1)), hole_a7_8) ->_R^Omega(1)
c4(IFAPPEND(gen_nil:cons10_8(+(n12_8, 1)), hole_a7_8, is_empty(gen_nil:cons10_8(+(n12_8, 1)))), IS_EMPTY(gen_nil:cons10_8(+(n12_8, 1)))) ->_R^Omega(0)
c4(IFAPPEND(gen_nil:cons10_8(+(1, n12_8)), hole_a7_8, false), IS_EMPTY(gen_nil:cons10_8(+(1, n12_8)))) ->_R^Omega(1)
c4(c7(APPEND(tl(gen_nil:cons10_8(+(1, n12_8))), hole_a7_8), TL(gen_nil:cons10_8(+(1, n12_8)))), IS_EMPTY(gen_nil:cons10_8(+(1, n12_8)))) ->_R^Omega(0)
c4(c7(APPEND(gen_nil:cons10_8(n12_8), hole_a7_8), TL(gen_nil:cons10_8(+(1, n12_8)))), IS_EMPTY(gen_nil:cons10_8(+(1, n12_8)))) ->_IH
c4(c7(*11_8, TL(gen_nil:cons10_8(+(1, n12_8)))), IS_EMPTY(gen_nil:cons10_8(+(1, n12_8)))) ->_R^Omega(1)
c4(c7(*11_8, c3), IS_EMPTY(gen_nil:cons10_8(+(1, n12_8)))) ->_R^Omega(1)
c4(c7(*11_8, c3), c1)

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

(56)
Complex Obligation (BEST)

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

(57)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
IS_EMPTY(nil) -> c
IS_EMPTY(cons(z0, z1)) -> c1
HD(cons(z0, z1)) -> c2
TL(cons(z0, z1)) -> c3
APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
IFAPPEND(z0, z1, true) -> c5
IFAPPEND(z0, z1, false) -> c6(HD(z0))
IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
is_empty(nil) -> true
is_empty(cons(z0, z1)) -> false
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1
append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
ifappend(z0, z1, true) -> z1
ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

Types:
IS_EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: hd -> nil:cons -> nil:cons
c1 :: c:c1
HD :: nil:cons -> c2
c2 :: c2
TL :: nil:cons -> c3
c3 :: c3
APPEND :: nil:cons -> a -> c4
c4 :: c5:c6:c7 -> c:c1 -> c4
IFAPPEND :: nil:cons -> a -> true:false -> c5:c6:c7
is_empty :: nil:cons -> true:false
true :: true:false
c5 :: c5:c6:c7
false :: true:false
c6 :: c2 -> c5:c6:c7
c7 :: c4 -> c3 -> c5:c6:c7
tl :: nil:cons -> nil:cons
hd :: nil:cons -> hd
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_c:c11_8 :: c:c1
hole_nil:cons2_8 :: nil:cons
hole_hd3_8 :: hd
hole_c24_8 :: c2
hole_c35_8 :: c3
hole_c46_8 :: c4
hole_a7_8 :: a
hole_c5:c6:c78_8 :: c5:c6:c7
hole_true:false9_8 :: true:false
gen_nil:cons10_8 :: Nat -> nil:cons


Generator Equations:
gen_nil:cons10_8(0) <=> nil
gen_nil:cons10_8(+(x, 1)) <=> cons(hole_hd3_8, gen_nil:cons10_8(x))


The following defined symbols remain to be analysed:
APPEND, append
----------------------------------------

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

(59)
BOUNDS(n^1, INF)

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

(60)
Obligation:
Innermost TRS:
Rules:
IS_EMPTY(nil) -> c
IS_EMPTY(cons(z0, z1)) -> c1
HD(cons(z0, z1)) -> c2
TL(cons(z0, z1)) -> c3
APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, is_empty(z0)), IS_EMPTY(z0))
IFAPPEND(z0, z1, true) -> c5
IFAPPEND(z0, z1, false) -> c6(HD(z0))
IFAPPEND(z0, z1, false) -> c7(APPEND(tl(z0), z1), TL(z0))
is_empty(nil) -> true
is_empty(cons(z0, z1)) -> false
hd(cons(z0, z1)) -> z0
tl(cons(z0, z1)) -> z1
append(z0, z1) -> ifappend(z0, z1, is_empty(z0))
ifappend(z0, z1, true) -> z1
ifappend(z0, z1, false) -> cons(hd(z0), append(tl(z0), z1))

Types:
IS_EMPTY :: nil:cons -> c:c1
nil :: nil:cons
c :: c:c1
cons :: hd -> nil:cons -> nil:cons
c1 :: c:c1
HD :: nil:cons -> c2
c2 :: c2
TL :: nil:cons -> c3
c3 :: c3
APPEND :: nil:cons -> a -> c4
c4 :: c5:c6:c7 -> c:c1 -> c4
IFAPPEND :: nil:cons -> a -> true:false -> c5:c6:c7
is_empty :: nil:cons -> true:false
true :: true:false
c5 :: c5:c6:c7
false :: true:false
c6 :: c2 -> c5:c6:c7
c7 :: c4 -> c3 -> c5:c6:c7
tl :: nil:cons -> nil:cons
hd :: nil:cons -> hd
append :: nil:cons -> nil:cons -> nil:cons
ifappend :: nil:cons -> nil:cons -> true:false -> nil:cons
hole_c:c11_8 :: c:c1
hole_nil:cons2_8 :: nil:cons
hole_hd3_8 :: hd
hole_c24_8 :: c2
hole_c35_8 :: c3
hole_c46_8 :: c4
hole_a7_8 :: a
hole_c5:c6:c78_8 :: c5:c6:c7
hole_true:false9_8 :: true:false
gen_nil:cons10_8 :: Nat -> nil:cons


Lemmas:
APPEND(gen_nil:cons10_8(n12_8), hole_a7_8) -> *11_8, rt in Omega(n12_8)


Generator Equations:
gen_nil:cons10_8(0) <=> nil
gen_nil:cons10_8(+(x, 1)) <=> cons(hole_hd3_8, gen_nil:cons10_8(x))


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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
append(gen_nil:cons10_8(n1460_8), gen_nil:cons10_8(b)) -> gen_nil:cons10_8(+(n1460_8, b)), rt in Omega(0)

Induction Base:
append(gen_nil:cons10_8(0), gen_nil:cons10_8(b)) ->_R^Omega(0)
ifappend(gen_nil:cons10_8(0), gen_nil:cons10_8(b), is_empty(gen_nil:cons10_8(0))) ->_R^Omega(0)
ifappend(gen_nil:cons10_8(0), gen_nil:cons10_8(b), true) ->_R^Omega(0)
gen_nil:cons10_8(b)

Induction Step:
append(gen_nil:cons10_8(+(n1460_8, 1)), gen_nil:cons10_8(b)) ->_R^Omega(0)
ifappend(gen_nil:cons10_8(+(n1460_8, 1)), gen_nil:cons10_8(b), is_empty(gen_nil:cons10_8(+(n1460_8, 1)))) ->_R^Omega(0)
ifappend(gen_nil:cons10_8(+(1, n1460_8)), gen_nil:cons10_8(b), false) ->_R^Omega(0)
cons(hd(gen_nil:cons10_8(+(1, n1460_8))), append(tl(gen_nil:cons10_8(+(1, n1460_8))), gen_nil:cons10_8(b))) ->_R^Omega(0)
cons(hole_hd3_8, append(tl(gen_nil:cons10_8(+(1, n1460_8))), gen_nil:cons10_8(b))) ->_R^Omega(0)
cons(hole_hd3_8, append(gen_nil:cons10_8(n1460_8), gen_nil:cons10_8(b))) ->_IH
cons(hole_hd3_8, gen_nil:cons10_8(+(b, c1461_8)))

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

(62)
BOUNDS(1, INF)