/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RcToIrcProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTRS
    (5) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
    (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (14) CpxRNTS
    (15) InliningProof [UPPER BOUND(ID), 53 ms]
    (16) CpxRNTS
    (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 165 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 138 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 52 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 931 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 238 ms]
    (38) CpxRNTS
    (39) FinalProof [FINISHED, 0 ms]
    (40) BOUNDS(1, n^1)
(41) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(42) CpxTRS
    (43) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) typed CpxTrs
    (45) OrderProof [LOWER BOUND(ID), 0 ms]
    (46) typed CpxTrs
    (47) RewriteLemmaProof [LOWER BOUND(ID), 295 ms]
    (48) BEST
        (49) proven lower bound
            (50) LowerBoundPropagationProof [FINISHED, 0 ms]
            (51) BOUNDS(n^1, INF)
        (52) typed CpxTrs


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

(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:

   circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t))
   circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t))
   circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t))
   circ(circ(s, t), u) -> circ(s, circ(t, u))
   circ(s, id) -> s
   circ(id, s) -> s
   circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u)
   subst(a, id) -> a
   msubst(a, id) -> a
   msubst(msubst(a, s), t) -> msubst(a, circ(s, t))

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

(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The TRS does not nest defined symbols.
Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
circ(circ(s, t), u) -> circ(s, circ(t, u))
circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u)
msubst(msubst(a, s), t) -> msubst(a, circ(s, t))

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

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


The TRS R consists of the following rules:

   circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t))
   circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t))
   circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t))
   circ(s, id) -> s
   circ(id, s) -> s
   subst(a, id) -> a
   msubst(a, id) -> a

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

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

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

(4)
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:

   circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t))
   circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t))
   circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t))
   circ(s, id) -> s
   circ(id, s) -> s
   subst(a, id) -> a
   msubst(a, id) -> a

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

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

(6)
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:

   circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) [1]
   circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) [1]
   circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) [1]
   circ(s, id) -> s [1]
   circ(id, s) -> s [1]
   subst(a, id) -> a [1]
   msubst(a, id) -> a [1]

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) [1]
   circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) [1]
   circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) [1]
   circ(s, id) -> s [1]
   circ(id, s) -> s [1]
   subst(a, id) -> a [1]
   msubst(a, id) -> a [1]

The TRS has the following type information:
circ :: cons:id -> cons:id -> cons:id
cons :: lift -> cons:id -> cons:id
msubst :: lift -> cons:id -> lift
lift :: lift
id :: cons:id
subst :: subst -> cons:id -> subst

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

(9) 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:

   circ_2
   subst_2
   msubst_2

(c) The following functions are completely defined:
none

Due to the following rules being added:
none

And the following fresh constants:    const

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

(10)
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:

   circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) [1]
   circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) [1]
   circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) [1]
   circ(s, id) -> s [1]
   circ(id, s) -> s [1]
   subst(a, id) -> a [1]
   msubst(a, id) -> a [1]

The TRS has the following type information:
circ :: cons:id -> cons:id -> cons:id
cons :: lift -> cons:id -> cons:id
msubst :: lift -> cons:id -> lift
lift :: lift
id :: cons:id
subst :: subst -> cons:id -> subst
const :: subst

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

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

(12)
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:

   circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t)) [1]
   circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t)) [1]
   circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t)) [1]
   circ(s, id) -> s [1]
   circ(id, s) -> s [1]
   subst(a, id) -> a [1]
   msubst(a, id) -> a [1]

The TRS has the following type information:
circ :: cons:id -> cons:id -> cons:id
cons :: lift -> cons:id -> cons:id
msubst :: lift -> cons:id -> lift
lift :: lift
id :: cons:id
subst :: subst -> cons:id -> subst
const :: subst

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

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

   lift => 0
   id => 0
   const => 0

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

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   circ(z, z') -{ 1 }-> s :|: z = s, s >= 0, z' = 0
   circ(z, z') -{ 1 }-> s :|: s >= 0, z = 0, z' = s
   circ(z, z') -{ 1 }-> 1 + a + circ(s, t) :|: z = 1 + 0 + s, a >= 0, z' = 1 + a + t, t >= 0, s >= 0
   circ(z, z') -{ 1 }-> 1 + msubst(a, t) + circ(s, t) :|: z' = t, a >= 0, z = 1 + a + s, t >= 0, s >= 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(s, t) :|: z = 1 + 0 + s, z' = 1 + 0 + t, t >= 0, s >= 0
   msubst(z, z') -{ 1 }-> a :|: z = a, a >= 0, z' = 0
   subst(z, z') -{ 1 }-> a :|: z = a, a >= 0, z' = 0


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

(15) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   msubst(z, z') -{ 1 }-> a :|: z = a, a >= 0, z' = 0

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

   circ(z, z') -{ 1 }-> s :|: z = s, s >= 0, z' = 0
   circ(z, z') -{ 1 }-> s :|: s >= 0, z = 0, z' = s
   circ(z, z') -{ 1 }-> 1 + a + circ(s, t) :|: z = 1 + 0 + s, a >= 0, z' = 1 + a + t, t >= 0, s >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, t) :|: z' = t, a >= 0, z = 1 + a + s, t >= 0, s >= 0, a = a', a' >= 0, t = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(s, t) :|: z = 1 + 0 + s, z' = 1 + 0 + t, t >= 0, s >= 0
   msubst(z, z') -{ 1 }-> a :|: z = a, a >= 0, z' = 0
   subst(z, z') -{ 1 }-> a :|: z = a, a >= 0, z' = 0


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

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0


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

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

   { subst }
   { msubst }
   { circ }

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0

Function symbols to be analyzed: {subst}, {msubst}, {circ}

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

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0

Function symbols to be analyzed: {subst}, {msubst}, {circ}

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0

Function symbols to be analyzed: {subst}, {msubst}, {circ}
Previous analysis results are:
  subst: runtime: ?, size: O(n^1) [z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0

Function symbols to be analyzed: {msubst}, {circ}
Previous analysis results are:
  subst: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0

Function symbols to be analyzed: {msubst}, {circ}
Previous analysis results are:
  subst: runtime: O(1) [1], size: O(n^1) [z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0

Function symbols to be analyzed: {msubst}, {circ}
Previous analysis results are:
  subst: runtime: O(1) [1], size: O(n^1) [z]
  msubst: runtime: ?, size: O(n^1) [z]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0

Function symbols to be analyzed: {circ}
Previous analysis results are:
  subst: runtime: O(1) [1], size: O(n^1) [z]
  msubst: runtime: O(1) [1], size: O(n^1) [z]

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

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0

Function symbols to be analyzed: {circ}
Previous analysis results are:
  subst: runtime: O(1) [1], size: O(n^1) [z]
  msubst: runtime: O(1) [1], size: O(n^1) [z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0

Function symbols to be analyzed: {circ}
Previous analysis results are:
  subst: runtime: O(1) [1], size: O(n^1) [z]
  msubst: runtime: O(1) [1], size: O(n^1) [z]
  circ: runtime: ?, size: O(n^1) [z + z']

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   circ(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   circ(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   circ(z, z') -{ 1 }-> 1 + a + circ(z - 1, t) :|: a >= 0, z' = 1 + a + t, t >= 0, z - 1 >= 0
   circ(z, z') -{ 2 }-> 1 + a' + circ(s, z') :|: a >= 0, z = 1 + a + s, z' >= 0, s >= 0, a = a', a' >= 0, z' = 0
   circ(z, z') -{ 1 }-> 1 + 0 + circ(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   msubst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   subst(z, z') -{ 1 }-> z :|: z >= 0, z' = 0

Function symbols to be analyzed: 
Previous analysis results are:
  subst: runtime: O(1) [1], size: O(n^1) [z]
  msubst: runtime: O(1) [1], size: O(n^1) [z]
  circ: runtime: O(n^1) [2 + 4*z], size: O(n^1) [z + z']

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(39) FinalProof (FINISHED)
Computed overall runtime complexity
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(40)
BOUNDS(1, n^1)

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(41) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
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(42)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t))
   circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t))
   circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t))
   circ(circ(s, t), u) -> circ(s, circ(t, u))
   circ(s, id) -> s
   circ(id, s) -> s
   circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u)
   subst(a, id) -> a
   msubst(a, id) -> a
   msubst(msubst(a, s), t) -> msubst(a, circ(s, t))

S is empty.
Rewrite Strategy: FULL
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(43) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
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(44)
Obligation:
TRS:
Rules:
circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t))
circ(circ(s, t), u) -> circ(s, circ(t, u))
circ(s, id) -> s
circ(id, s) -> s
circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u)
subst(a, id) -> a
msubst(a, id) -> a
msubst(msubst(a, s), t) -> msubst(a, circ(s, t))

Types:
circ :: cons:id -> cons:id -> cons:id
cons :: lift -> cons:id -> cons:id
msubst :: lift -> cons:id -> lift
lift :: lift
id :: cons:id
subst :: subst -> cons:id -> subst
hole_cons:id1_0 :: cons:id
hole_lift2_0 :: lift
hole_subst3_0 :: subst
gen_cons:id4_0 :: Nat -> cons:id

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(45) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
circ, msubst

They will be analysed ascendingly in the following order:
circ = msubst

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(46)
Obligation:
TRS:
Rules:
circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t))
circ(circ(s, t), u) -> circ(s, circ(t, u))
circ(s, id) -> s
circ(id, s) -> s
circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u)
subst(a, id) -> a
msubst(a, id) -> a
msubst(msubst(a, s), t) -> msubst(a, circ(s, t))

Types:
circ :: cons:id -> cons:id -> cons:id
cons :: lift -> cons:id -> cons:id
msubst :: lift -> cons:id -> lift
lift :: lift
id :: cons:id
subst :: subst -> cons:id -> subst
hole_cons:id1_0 :: cons:id
hole_lift2_0 :: lift
hole_subst3_0 :: subst
gen_cons:id4_0 :: Nat -> cons:id


Generator Equations:
gen_cons:id4_0(0) <=> id
gen_cons:id4_0(+(x, 1)) <=> cons(lift, gen_cons:id4_0(x))


The following defined symbols remain to be analysed:
msubst, circ

They will be analysed ascendingly in the following order:
circ = msubst

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(47) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
circ(gen_cons:id4_0(n16_0), gen_cons:id4_0(n16_0)) -> gen_cons:id4_0(n16_0), rt in Omega(1 + n16_0)

Induction Base:
circ(gen_cons:id4_0(0), gen_cons:id4_0(0)) ->_R^Omega(1)
gen_cons:id4_0(0)

Induction Step:
circ(gen_cons:id4_0(+(n16_0, 1)), gen_cons:id4_0(+(n16_0, 1))) ->_R^Omega(1)
cons(lift, circ(gen_cons:id4_0(n16_0), gen_cons:id4_0(n16_0))) ->_IH
cons(lift, gen_cons:id4_0(c17_0))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(48)
Complex Obligation (BEST)

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(49)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t))
circ(circ(s, t), u) -> circ(s, circ(t, u))
circ(s, id) -> s
circ(id, s) -> s
circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u)
subst(a, id) -> a
msubst(a, id) -> a
msubst(msubst(a, s), t) -> msubst(a, circ(s, t))

Types:
circ :: cons:id -> cons:id -> cons:id
cons :: lift -> cons:id -> cons:id
msubst :: lift -> cons:id -> lift
lift :: lift
id :: cons:id
subst :: subst -> cons:id -> subst
hole_cons:id1_0 :: cons:id
hole_lift2_0 :: lift
hole_subst3_0 :: subst
gen_cons:id4_0 :: Nat -> cons:id


Generator Equations:
gen_cons:id4_0(0) <=> id
gen_cons:id4_0(+(x, 1)) <=> cons(lift, gen_cons:id4_0(x))


The following defined symbols remain to be analysed:
circ

They will be analysed ascendingly in the following order:
circ = msubst

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(50) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(51)
BOUNDS(n^1, INF)

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(52)
Obligation:
TRS:
Rules:
circ(cons(a, s), t) -> cons(msubst(a, t), circ(s, t))
circ(cons(lift, s), cons(a, t)) -> cons(a, circ(s, t))
circ(cons(lift, s), cons(lift, t)) -> cons(lift, circ(s, t))
circ(circ(s, t), u) -> circ(s, circ(t, u))
circ(s, id) -> s
circ(id, s) -> s
circ(cons(lift, s), circ(cons(lift, t), u)) -> circ(cons(lift, circ(s, t)), u)
subst(a, id) -> a
msubst(a, id) -> a
msubst(msubst(a, s), t) -> msubst(a, circ(s, t))

Types:
circ :: cons:id -> cons:id -> cons:id
cons :: lift -> cons:id -> cons:id
msubst :: lift -> cons:id -> lift
lift :: lift
id :: cons:id
subst :: subst -> cons:id -> subst
hole_cons:id1_0 :: cons:id
hole_lift2_0 :: lift
hole_subst3_0 :: subst
gen_cons:id4_0 :: Nat -> cons:id


Lemmas:
circ(gen_cons:id4_0(n16_0), gen_cons:id4_0(n16_0)) -> gen_cons:id4_0(n16_0), rt in Omega(1 + n16_0)


Generator Equations:
gen_cons:id4_0(0) <=> id
gen_cons:id4_0(+(x, 1)) <=> cons(lift, gen_cons:id4_0(x))


The following defined symbols remain to be analysed:
msubst

They will be analysed ascendingly in the following order:
circ = msubst