WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


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

(0) CpxTRS
(1) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
    (3) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
    (4) CdtProblem
    (5) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CdtProblem
    (7) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CdtProblem
    (9) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms]
    (10) CdtProblem
    (11) SIsEmptyProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) BOUNDS(1, 1)
(13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(14) TRS for Loop Detection
    (15) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (16) BEST
        (17) proven lower bound
            (18) LowerBoundPropagationProof [FINISHED, 0 ms]
            (19) BOUNDS(n^1, INF)
        (20) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (full) 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, l1)
   ifappend(l1, l2, nil) -> l2
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

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

(1) RcToIrcProof (BOTH BOUNDS(ID, ID))
Converted rc-obligation to irc-obligation.

As the TRS does not nest defined symbols, we have rc = irc.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(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, l1)
   ifappend(l1, l2, nil) -> l2
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

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

(3) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(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, z0)
   ifappend(z0, z1, nil) -> z1
   ifappend(z0, z1, cons(z2, z3)) -> cons(z2, append(z3, 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, z0))
   IFAPPEND(z0, z1, nil) -> c5
   IFAPPEND(z0, z1, cons(z2, z3)) -> c6(APPEND(z3, z1))
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, z0))
   IFAPPEND(z0, z1, nil) -> c5
   IFAPPEND(z0, z1, cons(z2, z3)) -> c6(APPEND(z3, z1))
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_1, c5, c6_1


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

(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   IS_EMPTY(cons(z0, z1)) -> c1
   IFAPPEND(z0, z1, nil) -> c5
   TL(cons(z0, z1)) -> c3
   HD(cons(z0, z1)) -> c2
   IS_EMPTY(nil) -> c

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

(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, z0)
   ifappend(z0, z1, nil) -> z1
   ifappend(z0, z1, cons(z2, z3)) -> cons(z2, append(z3, z1))
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, z0))
   IFAPPEND(z0, z1, cons(z2, z3)) -> c6(APPEND(z3, z1))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, z0))
   IFAPPEND(z0, z1, cons(z2, z3)) -> c6(APPEND(z3, 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, c6_1


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

(7) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   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, z0)
   ifappend(z0, z1, nil) -> z1
   ifappend(z0, z1, cons(z2, z3)) -> cons(z2, append(z3, z1))

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

(8)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, z0))
   IFAPPEND(z0, z1, cons(z2, z3)) -> c6(APPEND(z3, z1))
S tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, z0))
   IFAPPEND(z0, z1, cons(z2, z3)) -> c6(APPEND(z3, z1))
K tuples:none
Defined Rule Symbols:none

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_1, c6_1


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

(9) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, z0))
   IFAPPEND(z0, z1, cons(z2, z3)) -> c6(APPEND(z3, z1))
We considered the (Usable) Rules:none
And the Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, z0))
   IFAPPEND(z0, z1, cons(z2, z3)) -> c6(APPEND(z3, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(APPEND(x_1, x_2)) = [3] + [2]x_1
   POL(IFAPPEND(x_1, x_2, x_3)) = [2]x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(cons(x_1, x_2)) = [3] + x_2

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

(10)
Obligation:
Complexity Dependency Tuples Problem

Rules:none
Tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, z0))
   IFAPPEND(z0, z1, cons(z2, z3)) -> c6(APPEND(z3, z1))
S tuples:none
K tuples:
   APPEND(z0, z1) -> c4(IFAPPEND(z0, z1, z0))
   IFAPPEND(z0, z1, cons(z2, z3)) -> c6(APPEND(z3, z1))
Defined Rule Symbols:none

Defined Pair Symbols:   APPEND_2, IFAPPEND_3

Compound Symbols:   c4_1, c6_1


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

(11) SIsEmptyProof (BOTH BOUNDS(ID, ID))
The set S is empty
----------------------------------------

(12)
BOUNDS(1, 1)

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

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) 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, l1)
   ifappend(l1, l2, nil) -> l2
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

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

(15) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

append(cons(x3_0, l4_0), l2) ->^+ cons(x3_0, append(l4_0, l2))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [l4_0 / cons(x3_0, l4_0)].

The result substitution is [ ].




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

(16)
Complex Obligation (BEST)

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

(17)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) 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, l1)
   ifappend(l1, l2, nil) -> l2
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

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

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

(19)
BOUNDS(n^1, INF)

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

(20)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) 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, l1)
   ifappend(l1, l2, nil) -> l2
   ifappend(l1, l2, cons(x, l)) -> cons(x, append(l, l2))

S is empty.
Rewrite Strategy: FULL