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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) InliningProof [UPPER BOUND(ID), 132 ms]
    (14) CpxRNTS
    (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (16) CpxRNTS
    (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxRNTS
    (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 216 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 54 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 429 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 169 ms]
    (30) CpxRNTS
    (31) FinalProof [FINISHED, 0 ms]
    (32) BOUNDS(1, n^1)
(33) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(34) TRS for Loop Detection
    (35) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (36) BEST
        (37) proven lower bound
            (38) LowerBoundPropagationProof [FINISHED, 0 ms]
            (39) BOUNDS(n^1, INF)
        (40) 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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

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

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

The duplicating contexts are:
admit([], .(u, .(v, .(w, z))))
admit(x, .([], .(v, .(w, z))))
admit(x, .(u, .([], .(w, z))))


The defined contexts are:
cond(=(sum(x0, x1, x2), w), .(x3, .(x4, .(w, []))))


As the TRS is an overlay system and the defined contexts and the duplicating contexts don't overlap, 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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

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

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

(4)
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:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1]
   cond(true, y) -> y [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1]
   cond(true, y) -> y [1]

The TRS has the following type information:
admit :: carry -> nil:. -> nil:.
nil :: nil:.
. :: w -> nil:. -> nil:.
w :: w
cond :: =:true -> nil:. -> nil:.
= :: sum -> w -> =:true
sum :: carry -> w -> w -> sum
carry :: carry -> w -> w -> carry
true :: =:true

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

(7) 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:
none

(c) The following functions are completely defined:

   admit_2
   cond_2

Due to the following rules being added:

   admit(v0, v1) -> nil [0]
   cond(v0, v1) -> nil [0]

And the following fresh constants:    const, const1

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

(8)
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:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z))))) [1]
   cond(true, y) -> y [1]
   admit(v0, v1) -> nil [0]
   cond(v0, v1) -> nil [0]

The TRS has the following type information:
admit :: carry -> nil:. -> nil:.
nil :: nil:.
. :: w -> nil:. -> nil:.
w :: w
cond :: =:true -> nil:. -> nil:.
= :: sum -> w -> =:true
sum :: carry -> w -> w -> sum
carry :: carry -> w -> w -> carry
true :: =:true
const :: carry
const1 :: sum

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

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

(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:

   admit(x, nil) -> nil [1]
   admit(x, .(u, .(v, .(w, nil)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, nil)))) [2]
   admit(x, .(u, .(v, .(w, .(u', .(v', .(w, z'))))))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, cond(=(sum(carry(x, u, v), u', v'), w), .(u', .(v', .(w, admit(carry(carry(x, u, v), u', v'), z'))))))))) [2]
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, nil)))) [1]
   cond(true, y) -> y [1]
   admit(v0, v1) -> nil [0]
   cond(v0, v1) -> nil [0]

The TRS has the following type information:
admit :: carry -> nil:. -> nil:.
nil :: nil:.
. :: w -> nil:. -> nil:.
w :: w
cond :: =:true -> nil:. -> nil:.
= :: sum -> w -> =:true
sum :: carry -> w -> w -> sum
carry :: carry -> w -> w -> carry
true :: =:true
const :: carry
const1 :: sum

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

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

   nil => 0
   w => 0
   true => 0
   const => 0
   const1 => 0

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + x + u + v) + 0, 1 + u + (1 + v + (1 + 0 + cond(1 + (1 + (1 + x + u + v) + u' + v') + 0, 1 + u' + (1 + v' + (1 + 0 + admit(1 + (1 + x + u + v) + u' + v', z'))))))) :|: v >= 0, x >= 0, z' >= 0, u' >= 0, v' >= 0, z'' = x, z1 = 1 + u + (1 + v + (1 + 0 + (1 + u' + (1 + v' + (1 + 0 + z'))))), u >= 0
   admit(z'', z1) -{ 2 }-> cond(1 + (1 + x + u + v) + 0, 1 + u + (1 + v + (1 + 0 + 0))) :|: v >= 0, x >= 0, z1 = 1 + u + (1 + v + (1 + 0 + 0)), z'' = x, u >= 0
   admit(z'', z1) -{ 1 }-> cond(1 + (1 + x + u + v) + 0, 1 + u + (1 + v + (1 + 0 + 0))) :|: v >= 0, z >= 0, x >= 0, z'' = x, z1 = 1 + u + (1 + v + (1 + 0 + z)), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 0, z'' = x
   admit(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> y :|: z'' = 0, z1 = y, y >= 0
   cond(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0


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

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

   cond(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> y :|: z'' = 0, z1 = y, y >= 0

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + x + u + v) + 0, 1 + u + (1 + v + (1 + 0 + cond(1 + (1 + (1 + x + u + v) + u' + v') + 0, 1 + u' + (1 + v' + (1 + 0 + admit(1 + (1 + x + u + v) + u' + v', z'))))))) :|: v >= 0, x >= 0, z' >= 0, u' >= 0, v' >= 0, z'' = x, z1 = 1 + u + (1 + v + (1 + 0 + (1 + u' + (1 + v' + (1 + 0 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, x >= 0, z'' = x
   admit(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, x >= 0, z1 = 1 + u + (1 + v + (1 + 0 + 0)), z'' = x, u >= 0, 1 + (1 + x + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, x >= 0, z'' = x, z1 = 1 + u + (1 + v + (1 + 0 + z)), u >= 0, 1 + (1 + x + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> y :|: z'' = 0, z1 = y, y >= 0
   cond(z'', z1) -{ 0 }-> 0 :|: z'' = v0, v0 >= 0, z1 = v1, v1 >= 0


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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 0, 1 + u + (1 + v + (1 + 0 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 0, 1 + u' + (1 + v' + (1 + 0 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + (1 + u' + (1 + v' + (1 + 0 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + z)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0


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

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

   { cond }
   { admit }

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 0, 1 + u + (1 + v + (1 + 0 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 0, 1 + u' + (1 + v' + (1 + 0 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + (1 + u' + (1 + v' + (1 + 0 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + z)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0

Function symbols to be analyzed: {cond}, {admit}

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 0, 1 + u + (1 + v + (1 + 0 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 0, 1 + u' + (1 + v' + (1 + 0 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + (1 + u' + (1 + v' + (1 + 0 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + z)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0

Function symbols to be analyzed: {cond}, {admit}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: cond
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z1

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 0, 1 + u + (1 + v + (1 + 0 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 0, 1 + u' + (1 + v' + (1 + 0 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + (1 + u' + (1 + v' + (1 + 0 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + z)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0

Function symbols to be analyzed: {cond}, {admit}
Previous analysis results are:
  cond: runtime: ?, size: O(n^1) [z1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 0, 1 + u + (1 + v + (1 + 0 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 0, 1 + u' + (1 + v' + (1 + 0 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + (1 + u' + (1 + v' + (1 + 0 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + z)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0

Function symbols to be analyzed: {admit}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]

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

(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:

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 0, 1 + u + (1 + v + (1 + 0 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 0, 1 + u' + (1 + v' + (1 + 0 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + (1 + u' + (1 + v' + (1 + 0 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + z)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0

Function symbols to be analyzed: {admit}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: admit
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z1

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

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 0, 1 + u + (1 + v + (1 + 0 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 0, 1 + u' + (1 + v' + (1 + 0 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + (1 + u' + (1 + v' + (1 + 0 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + z)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0

Function symbols to be analyzed: {admit}
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  admit: runtime: ?, size: O(n^1) [z1]

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(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: admit
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 4*z1

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

   admit(z'', z1) -{ 2 }-> cond(1 + (1 + z'' + u + v) + 0, 1 + u + (1 + v + (1 + 0 + cond(1 + (1 + (1 + z'' + u + v) + u' + v') + 0, 1 + u' + (1 + v' + (1 + 0 + admit(1 + (1 + z'' + u + v) + u' + v', z'))))))) :|: v >= 0, z'' >= 0, z' >= 0, u' >= 0, v' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + (1 + u' + (1 + v' + (1 + 0 + z'))))), u >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: z1 = 0, z'' >= 0
   admit(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0
   admit(z'', z1) -{ 2 }-> 0 :|: v >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + 0)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   admit(z'', z1) -{ 1 }-> 0 :|: v >= 0, z >= 0, z'' >= 0, z1 = 1 + u + (1 + v + (1 + 0 + z)), u >= 0, 1 + (1 + z'' + u + v) + 0 = v0, v0 >= 0, 1 + u + (1 + v + (1 + 0 + 0)) = v1, v1 >= 0
   cond(z'', z1) -{ 1 }-> z1 :|: z'' = 0, z1 >= 0
   cond(z'', z1) -{ 0 }-> 0 :|: z'' >= 0, z1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  cond: runtime: O(1) [1], size: O(n^1) [z1]
  admit: runtime: O(n^1) [4 + 4*z1], size: O(n^1) [z1]

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(31) FinalProof (FINISHED)
Computed overall runtime complexity
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(32)
BOUNDS(1, n^1)

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(33) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(34)
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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

S is empty.
Rewrite Strategy: FULL
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(35) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

admit(x, .(u, .(v, .(w, z)))) ->^+ cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))

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

The pumping substitution is [z / .(u, .(v, .(w, z)))].

The result substitution is [x / carry(x, u, v)].




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(36)
Complex Obligation (BEST)

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(37)
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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

S is empty.
Rewrite Strategy: FULL
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(38) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(39)
BOUNDS(n^1, INF)

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(40)
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:

   admit(x, nil) -> nil
   admit(x, .(u, .(v, .(w, z)))) -> cond(=(sum(x, u, v), w), .(u, .(v, .(w, admit(carry(x, u, v), z)))))
   cond(true, y) -> y

S is empty.
Rewrite Strategy: FULL