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


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty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), 200 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), 219 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 66 ms]
    (24) CpxRNTS
    (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 575 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 830 ms]
    (30) CpxRNTS
    (31) FinalProof [FINISHED, 0 ms]
    (32) BOUNDS(1, n^1)
(33) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(34) CpxTRS
    (35) SlicingProof [LOWER BOUND(ID), 0 ms]
    (36) CpxTRS
    (37) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) typed CpxTrs
    (39) OrderProof [LOWER BOUND(ID), 0 ms]
    (40) typed CpxTrs
    (41) RewriteLemmaProof [LOWER BOUND(ID), 204 ms]
    (42) proven lower bound
    (43) LowerBoundPropagationProof [FINISHED, 0 ms]
    (44) BOUNDS(n^1, INF)


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

(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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

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

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

The duplicating contexts are:
mem([], union(y, z))


The defined contexts are:
or([], x1)
or(x0, [])


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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> =(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

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:

   or(true, y) -> true [1]
   or(x, true) -> true [1]
   or(false, false) -> false [1]
   mem(x, nil) -> false [1]
   mem(x, set(y)) -> =(x, y) [1]
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) [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:

   or(true, y) -> true [1]
   or(x, true) -> true [1]
   or(false, false) -> false [1]
   mem(x, nil) -> false [1]
   mem(x, set(y)) -> =(x, y) [1]
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) [1]

The TRS has the following type information:
or :: true:false:= -> true:false:= -> true:false:=
true :: true:false:=
false :: true:false:=
mem :: a -> nil:set:union -> true:false:=
nil :: nil:set:union
set :: b -> nil:set:union
= :: a -> b -> true:false:=
union :: nil:set:union -> nil:set:union -> nil:set:union

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:

   mem_2
   or_2

Due to the following rules being added:

   or(v0, v1) -> null_or [0]

And the following fresh constants:    null_or, 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:

   or(true, y) -> true [1]
   or(x, true) -> true [1]
   or(false, false) -> false [1]
   mem(x, nil) -> false [1]
   mem(x, set(y)) -> =(x, y) [1]
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z)) [1]
   or(v0, v1) -> null_or [0]

The TRS has the following type information:
or :: true:false:=:null_or -> true:false:=:null_or -> true:false:=:null_or
true :: true:false:=:null_or
false :: true:false:=:null_or
mem :: a -> nil:set:union -> true:false:=:null_or
nil :: nil:set:union
set :: b -> nil:set:union
= :: a -> b -> true:false:=:null_or
union :: nil:set:union -> nil:set:union -> nil:set:union
null_or :: true:false:=:null_or
const :: a
const1 :: b

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:

   or(true, y) -> true [1]
   or(x, true) -> true [1]
   or(false, false) -> false [1]
   mem(x, nil) -> false [1]
   mem(x, set(y)) -> =(x, y) [1]
   mem(x, union(nil, nil)) -> or(false, false) [3]
   mem(x, union(nil, set(y1))) -> or(false, =(x, y1)) [3]
   mem(x, union(nil, union(y2, z''))) -> or(false, or(mem(x, y2), mem(x, z''))) [3]
   mem(x, union(set(y'), nil)) -> or(=(x, y'), false) [3]
   mem(x, union(set(y'), set(y3))) -> or(=(x, y'), =(x, y3)) [3]
   mem(x, union(set(y'), union(y4, z1))) -> or(=(x, y'), or(mem(x, y4), mem(x, z1))) [3]
   mem(x, union(union(y'', z'), nil)) -> or(or(mem(x, y''), mem(x, z')), false) [3]
   mem(x, union(union(y'', z'), set(y5))) -> or(or(mem(x, y''), mem(x, z')), =(x, y5)) [3]
   mem(x, union(union(y'', z'), union(y6, z2))) -> or(or(mem(x, y''), mem(x, z')), or(mem(x, y6), mem(x, z2))) [3]
   or(v0, v1) -> null_or [0]

The TRS has the following type information:
or :: true:false:=:null_or -> true:false:=:null_or -> true:false:=:null_or
true :: true:false:=:null_or
false :: true:false:=:null_or
mem :: a -> nil:set:union -> true:false:=:null_or
nil :: nil:set:union
set :: b -> nil:set:union
= :: a -> b -> true:false:=:null_or
union :: nil:set:union -> nil:set:union -> nil:set:union
null_or :: true:false:=:null_or
const :: a
const1 :: b

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:

   true => 1
   false => 0
   nil => 0
   null_or => 0
   const => 0
   const1 => 0

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

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

   mem(z, z3) -{ 3 }-> or(or(mem(x, y''), mem(x, z')), or(mem(x, y6), mem(x, z2))) :|: x >= 0, z' >= 0, y6 >= 0, y'' >= 0, z = x, z3 = 1 + (1 + y'' + z') + (1 + y6 + z2), z2 >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(x, y''), mem(x, z')), 0) :|: z3 = 1 + (1 + y'' + z') + 0, x >= 0, z' >= 0, y'' >= 0, z = x
   mem(z, z3) -{ 3 }-> or(or(mem(x, y''), mem(x, z')), 1 + x + y5) :|: y5 >= 0, z3 = 1 + (1 + y'' + z') + (1 + y5), x >= 0, z' >= 0, y'' >= 0, z = x
   mem(z, z3) -{ 3 }-> or(0, or(mem(x, y2), mem(x, z''))) :|: z'' >= 0, x >= 0, z3 = 1 + 0 + (1 + y2 + z''), y2 >= 0, z = x
   mem(z, z3) -{ 3 }-> or(0, 0) :|: x >= 0, z3 = 1 + 0 + 0, z = x
   mem(z, z3) -{ 3 }-> or(0, 1 + x + y1) :|: y1 >= 0, z3 = 1 + 0 + (1 + y1), x >= 0, z = x
   mem(z, z3) -{ 3 }-> or(1 + x + y', or(mem(x, y4), mem(x, z1))) :|: z1 >= 0, z3 = 1 + (1 + y') + (1 + y4 + z1), x >= 0, y' >= 0, y4 >= 0, z = x
   mem(z, z3) -{ 3 }-> or(1 + x + y', 0) :|: z3 = 1 + (1 + y') + 0, x >= 0, y' >= 0, z = x
   mem(z, z3) -{ 3 }-> or(1 + x + y', 1 + x + y3) :|: x >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), z = x
   mem(z, z3) -{ 1 }-> 0 :|: z3 = 0, x >= 0, z = x
   mem(z, z3) -{ 1 }-> 1 + x + y :|: z3 = 1 + y, x >= 0, y >= 0, z = x
   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 = y, y >= 0
   or(z, z3) -{ 1 }-> 1 :|: x >= 0, z3 = 1, z = x
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0
   or(z, z3) -{ 0 }-> 0 :|: z3 = v1, v0 >= 0, v1 >= 0, z = v0


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

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

   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 = y, y >= 0
   or(z, z3) -{ 1 }-> 1 :|: x >= 0, z3 = 1, z = x
   or(z, z3) -{ 0 }-> 0 :|: z3 = v1, v0 >= 0, v1 >= 0, z = v0
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0

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

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

   mem(z, z3) -{ 3 }-> or(or(mem(x, y''), mem(x, z')), or(mem(x, y6), mem(x, z2))) :|: x >= 0, z' >= 0, y6 >= 0, y'' >= 0, z = x, z3 = 1 + (1 + y'' + z') + (1 + y6 + z2), z2 >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(x, y''), mem(x, z')), 0) :|: z3 = 1 + (1 + y'' + z') + 0, x >= 0, z' >= 0, y'' >= 0, z = x
   mem(z, z3) -{ 3 }-> or(or(mem(x, y''), mem(x, z')), 1 + x + y5) :|: y5 >= 0, z3 = 1 + (1 + y'' + z') + (1 + y5), x >= 0, z' >= 0, y'' >= 0, z = x
   mem(z, z3) -{ 3 }-> or(0, or(mem(x, y2), mem(x, z''))) :|: z'' >= 0, x >= 0, z3 = 1 + 0 + (1 + y2 + z''), y2 >= 0, z = x
   mem(z, z3) -{ 3 }-> or(1 + x + y', or(mem(x, y4), mem(x, z1))) :|: z1 >= 0, z3 = 1 + (1 + y') + (1 + y4 + z1), x >= 0, y' >= 0, y4 >= 0, z = x
   mem(z, z3) -{ 4 }-> 1 :|: y1 >= 0, z3 = 1 + 0 + (1 + y1), x >= 0, z = x, x' >= 0, 1 + x + y1 = 1, 0 = x'
   mem(z, z3) -{ 4 }-> 1 :|: z3 = 1 + (1 + y') + 0, x >= 0, y' >= 0, z = x, 1 + x + y' = 1, 0 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: x >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), z = x, 1 + x + y' = 1, 1 + x + y3 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: x >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), z = x, x' >= 0, 1 + x + y3 = 1, 1 + x + y' = x'
   mem(z, z3) -{ 1 }-> 0 :|: z3 = 0, x >= 0, z = x
   mem(z, z3) -{ 3 }-> 0 :|: x >= 0, z3 = 1 + 0 + 0, z = x, 0 = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 4 }-> 0 :|: x >= 0, z3 = 1 + 0 + 0, z = x, 0 = 0
   mem(z, z3) -{ 3 }-> 0 :|: y1 >= 0, z3 = 1 + 0 + (1 + y1), x >= 0, z = x, 1 + x + y1 = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 3 }-> 0 :|: z3 = 1 + (1 + y') + 0, x >= 0, y' >= 0, z = x, 0 = v1, v0 >= 0, v1 >= 0, 1 + x + y' = v0
   mem(z, z3) -{ 3 }-> 0 :|: x >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), z = x, 1 + x + y3 = v1, v0 >= 0, v1 >= 0, 1 + x + y' = v0
   mem(z, z3) -{ 1 }-> 1 + x + y :|: z3 = 1 + y, x >= 0, y >= 0, z = x
   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 = y, y >= 0
   or(z, z3) -{ 1 }-> 1 :|: x >= 0, z3 = 1, z = x
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0
   or(z, z3) -{ 0 }-> 0 :|: z3 = v1, v0 >= 0, v1 >= 0, z = v0


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

(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:

   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), or(mem(z, y6), mem(z, z2))) :|: z >= 0, z' >= 0, y6 >= 0, y'' >= 0, z3 = 1 + (1 + y'' + z') + (1 + y6 + z2), z2 >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 0) :|: z3 = 1 + (1 + y'' + z') + 0, z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 1 + z + y5) :|: y5 >= 0, z3 = 1 + (1 + y'' + z') + (1 + y5), z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(0, or(mem(z, y2), mem(z, z''))) :|: z'' >= 0, z >= 0, z3 = 1 + 0 + (1 + y2 + z''), y2 >= 0
   mem(z, z3) -{ 3 }-> or(1 + z + y', or(mem(z, y4), mem(z, z1))) :|: z1 >= 0, z3 = 1 + (1 + y') + (1 + y4 + z1), z >= 0, y' >= 0, y4 >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z3 - 2 >= 0, z >= 0, x' >= 0, 1 + z + (z3 - 2) = 1, 0 = x'
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, z3 - 2 >= 0, 1 + z + (z3 - 2) = 1, 0 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y' = 1, 1 + z + y3 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), x' >= 0, 1 + z + y3 = 1, 1 + z + y' = x'
   mem(z, z3) -{ 1 }-> 0 :|: z3 = 0, z >= 0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 4 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = 0
   mem(z, z3) -{ 3 }-> 0 :|: z3 - 2 >= 0, z >= 0, 1 + z + (z3 - 2) = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 - 2 >= 0, 0 = v1, v0 >= 0, v1 >= 0, 1 + z + (z3 - 2) = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y3 = v1, v0 >= 0, v1 >= 0, 1 + z + y' = v0
   mem(z, z3) -{ 1 }-> 1 + z + (z3 - 1) :|: z >= 0, z3 - 1 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z >= 0, z3 = 1
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0
   or(z, z3) -{ 0 }-> 0 :|: z >= 0, z3 >= 0


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

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

   { or }
   { mem }

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

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

   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), or(mem(z, y6), mem(z, z2))) :|: z >= 0, z' >= 0, y6 >= 0, y'' >= 0, z3 = 1 + (1 + y'' + z') + (1 + y6 + z2), z2 >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 0) :|: z3 = 1 + (1 + y'' + z') + 0, z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 1 + z + y5) :|: y5 >= 0, z3 = 1 + (1 + y'' + z') + (1 + y5), z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(0, or(mem(z, y2), mem(z, z''))) :|: z'' >= 0, z >= 0, z3 = 1 + 0 + (1 + y2 + z''), y2 >= 0
   mem(z, z3) -{ 3 }-> or(1 + z + y', or(mem(z, y4), mem(z, z1))) :|: z1 >= 0, z3 = 1 + (1 + y') + (1 + y4 + z1), z >= 0, y' >= 0, y4 >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z3 - 2 >= 0, z >= 0, x' >= 0, 1 + z + (z3 - 2) = 1, 0 = x'
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, z3 - 2 >= 0, 1 + z + (z3 - 2) = 1, 0 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y' = 1, 1 + z + y3 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), x' >= 0, 1 + z + y3 = 1, 1 + z + y' = x'
   mem(z, z3) -{ 1 }-> 0 :|: z3 = 0, z >= 0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 4 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = 0
   mem(z, z3) -{ 3 }-> 0 :|: z3 - 2 >= 0, z >= 0, 1 + z + (z3 - 2) = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 - 2 >= 0, 0 = v1, v0 >= 0, v1 >= 0, 1 + z + (z3 - 2) = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y3 = v1, v0 >= 0, v1 >= 0, 1 + z + y' = v0
   mem(z, z3) -{ 1 }-> 1 + z + (z3 - 1) :|: z >= 0, z3 - 1 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z >= 0, z3 = 1
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0
   or(z, z3) -{ 0 }-> 0 :|: z >= 0, z3 >= 0

Function symbols to be analyzed: {or}, {mem}

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

(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:

   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), or(mem(z, y6), mem(z, z2))) :|: z >= 0, z' >= 0, y6 >= 0, y'' >= 0, z3 = 1 + (1 + y'' + z') + (1 + y6 + z2), z2 >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 0) :|: z3 = 1 + (1 + y'' + z') + 0, z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 1 + z + y5) :|: y5 >= 0, z3 = 1 + (1 + y'' + z') + (1 + y5), z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(0, or(mem(z, y2), mem(z, z''))) :|: z'' >= 0, z >= 0, z3 = 1 + 0 + (1 + y2 + z''), y2 >= 0
   mem(z, z3) -{ 3 }-> or(1 + z + y', or(mem(z, y4), mem(z, z1))) :|: z1 >= 0, z3 = 1 + (1 + y') + (1 + y4 + z1), z >= 0, y' >= 0, y4 >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z3 - 2 >= 0, z >= 0, x' >= 0, 1 + z + (z3 - 2) = 1, 0 = x'
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, z3 - 2 >= 0, 1 + z + (z3 - 2) = 1, 0 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y' = 1, 1 + z + y3 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), x' >= 0, 1 + z + y3 = 1, 1 + z + y' = x'
   mem(z, z3) -{ 1 }-> 0 :|: z3 = 0, z >= 0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 4 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = 0
   mem(z, z3) -{ 3 }-> 0 :|: z3 - 2 >= 0, z >= 0, 1 + z + (z3 - 2) = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 - 2 >= 0, 0 = v1, v0 >= 0, v1 >= 0, 1 + z + (z3 - 2) = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y3 = v1, v0 >= 0, v1 >= 0, 1 + z + y' = v0
   mem(z, z3) -{ 1 }-> 1 + z + (z3 - 1) :|: z >= 0, z3 - 1 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z >= 0, z3 = 1
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0
   or(z, z3) -{ 0 }-> 0 :|: z >= 0, z3 >= 0

Function symbols to be analyzed: {or}, {mem}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), or(mem(z, y6), mem(z, z2))) :|: z >= 0, z' >= 0, y6 >= 0, y'' >= 0, z3 = 1 + (1 + y'' + z') + (1 + y6 + z2), z2 >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 0) :|: z3 = 1 + (1 + y'' + z') + 0, z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 1 + z + y5) :|: y5 >= 0, z3 = 1 + (1 + y'' + z') + (1 + y5), z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(0, or(mem(z, y2), mem(z, z''))) :|: z'' >= 0, z >= 0, z3 = 1 + 0 + (1 + y2 + z''), y2 >= 0
   mem(z, z3) -{ 3 }-> or(1 + z + y', or(mem(z, y4), mem(z, z1))) :|: z1 >= 0, z3 = 1 + (1 + y') + (1 + y4 + z1), z >= 0, y' >= 0, y4 >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z3 - 2 >= 0, z >= 0, x' >= 0, 1 + z + (z3 - 2) = 1, 0 = x'
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, z3 - 2 >= 0, 1 + z + (z3 - 2) = 1, 0 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y' = 1, 1 + z + y3 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), x' >= 0, 1 + z + y3 = 1, 1 + z + y' = x'
   mem(z, z3) -{ 1 }-> 0 :|: z3 = 0, z >= 0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 4 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = 0
   mem(z, z3) -{ 3 }-> 0 :|: z3 - 2 >= 0, z >= 0, 1 + z + (z3 - 2) = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 - 2 >= 0, 0 = v1, v0 >= 0, v1 >= 0, 1 + z + (z3 - 2) = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y3 = v1, v0 >= 0, v1 >= 0, 1 + z + y' = v0
   mem(z, z3) -{ 1 }-> 1 + z + (z3 - 1) :|: z >= 0, z3 - 1 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z >= 0, z3 = 1
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0
   or(z, z3) -{ 0 }-> 0 :|: z >= 0, z3 >= 0

Function symbols to be analyzed: {or}, {mem}
Previous analysis results are:
  or: runtime: ?, size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: or
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:

   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), or(mem(z, y6), mem(z, z2))) :|: z >= 0, z' >= 0, y6 >= 0, y'' >= 0, z3 = 1 + (1 + y'' + z') + (1 + y6 + z2), z2 >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 0) :|: z3 = 1 + (1 + y'' + z') + 0, z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 1 + z + y5) :|: y5 >= 0, z3 = 1 + (1 + y'' + z') + (1 + y5), z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(0, or(mem(z, y2), mem(z, z''))) :|: z'' >= 0, z >= 0, z3 = 1 + 0 + (1 + y2 + z''), y2 >= 0
   mem(z, z3) -{ 3 }-> or(1 + z + y', or(mem(z, y4), mem(z, z1))) :|: z1 >= 0, z3 = 1 + (1 + y') + (1 + y4 + z1), z >= 0, y' >= 0, y4 >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z3 - 2 >= 0, z >= 0, x' >= 0, 1 + z + (z3 - 2) = 1, 0 = x'
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, z3 - 2 >= 0, 1 + z + (z3 - 2) = 1, 0 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y' = 1, 1 + z + y3 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), x' >= 0, 1 + z + y3 = 1, 1 + z + y' = x'
   mem(z, z3) -{ 1 }-> 0 :|: z3 = 0, z >= 0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 4 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = 0
   mem(z, z3) -{ 3 }-> 0 :|: z3 - 2 >= 0, z >= 0, 1 + z + (z3 - 2) = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 - 2 >= 0, 0 = v1, v0 >= 0, v1 >= 0, 1 + z + (z3 - 2) = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y3 = v1, v0 >= 0, v1 >= 0, 1 + z + y' = v0
   mem(z, z3) -{ 1 }-> 1 + z + (z3 - 1) :|: z >= 0, z3 - 1 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z >= 0, z3 = 1
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0
   or(z, z3) -{ 0 }-> 0 :|: z >= 0, z3 >= 0

Function symbols to be analyzed: {mem}
Previous analysis results are:
  or: runtime: O(1) [1], size: O(1) [1]

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

(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:

   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), or(mem(z, y6), mem(z, z2))) :|: z >= 0, z' >= 0, y6 >= 0, y'' >= 0, z3 = 1 + (1 + y'' + z') + (1 + y6 + z2), z2 >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 0) :|: z3 = 1 + (1 + y'' + z') + 0, z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 1 + z + y5) :|: y5 >= 0, z3 = 1 + (1 + y'' + z') + (1 + y5), z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(0, or(mem(z, y2), mem(z, z''))) :|: z'' >= 0, z >= 0, z3 = 1 + 0 + (1 + y2 + z''), y2 >= 0
   mem(z, z3) -{ 3 }-> or(1 + z + y', or(mem(z, y4), mem(z, z1))) :|: z1 >= 0, z3 = 1 + (1 + y') + (1 + y4 + z1), z >= 0, y' >= 0, y4 >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z3 - 2 >= 0, z >= 0, x' >= 0, 1 + z + (z3 - 2) = 1, 0 = x'
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, z3 - 2 >= 0, 1 + z + (z3 - 2) = 1, 0 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y' = 1, 1 + z + y3 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), x' >= 0, 1 + z + y3 = 1, 1 + z + y' = x'
   mem(z, z3) -{ 1 }-> 0 :|: z3 = 0, z >= 0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 4 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = 0
   mem(z, z3) -{ 3 }-> 0 :|: z3 - 2 >= 0, z >= 0, 1 + z + (z3 - 2) = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 - 2 >= 0, 0 = v1, v0 >= 0, v1 >= 0, 1 + z + (z3 - 2) = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y3 = v1, v0 >= 0, v1 >= 0, 1 + z + y' = v0
   mem(z, z3) -{ 1 }-> 1 + z + (z3 - 1) :|: z >= 0, z3 - 1 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z >= 0, z3 = 1
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0
   or(z, z3) -{ 0 }-> 0 :|: z >= 0, z3 >= 0

Function symbols to be analyzed: {mem}
Previous analysis results are:
  or: runtime: O(1) [1], size: O(1) [1]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: mem
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z3

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

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

   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), or(mem(z, y6), mem(z, z2))) :|: z >= 0, z' >= 0, y6 >= 0, y'' >= 0, z3 = 1 + (1 + y'' + z') + (1 + y6 + z2), z2 >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 0) :|: z3 = 1 + (1 + y'' + z') + 0, z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 1 + z + y5) :|: y5 >= 0, z3 = 1 + (1 + y'' + z') + (1 + y5), z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(0, or(mem(z, y2), mem(z, z''))) :|: z'' >= 0, z >= 0, z3 = 1 + 0 + (1 + y2 + z''), y2 >= 0
   mem(z, z3) -{ 3 }-> or(1 + z + y', or(mem(z, y4), mem(z, z1))) :|: z1 >= 0, z3 = 1 + (1 + y') + (1 + y4 + z1), z >= 0, y' >= 0, y4 >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z3 - 2 >= 0, z >= 0, x' >= 0, 1 + z + (z3 - 2) = 1, 0 = x'
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, z3 - 2 >= 0, 1 + z + (z3 - 2) = 1, 0 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y' = 1, 1 + z + y3 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), x' >= 0, 1 + z + y3 = 1, 1 + z + y' = x'
   mem(z, z3) -{ 1 }-> 0 :|: z3 = 0, z >= 0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 4 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = 0
   mem(z, z3) -{ 3 }-> 0 :|: z3 - 2 >= 0, z >= 0, 1 + z + (z3 - 2) = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 - 2 >= 0, 0 = v1, v0 >= 0, v1 >= 0, 1 + z + (z3 - 2) = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y3 = v1, v0 >= 0, v1 >= 0, 1 + z + y' = v0
   mem(z, z3) -{ 1 }-> 1 + z + (z3 - 1) :|: z >= 0, z3 - 1 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z >= 0, z3 = 1
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0
   or(z, z3) -{ 0 }-> 0 :|: z >= 0, z3 >= 0

Function symbols to be analyzed: {mem}
Previous analysis results are:
  or: runtime: O(1) [1], size: O(1) [1]
  mem: runtime: ?, size: O(n^1) [z + z3]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: mem
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 10*z3

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

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

   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), or(mem(z, y6), mem(z, z2))) :|: z >= 0, z' >= 0, y6 >= 0, y'' >= 0, z3 = 1 + (1 + y'' + z') + (1 + y6 + z2), z2 >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 0) :|: z3 = 1 + (1 + y'' + z') + 0, z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(or(mem(z, y''), mem(z, z')), 1 + z + y5) :|: y5 >= 0, z3 = 1 + (1 + y'' + z') + (1 + y5), z >= 0, z' >= 0, y'' >= 0
   mem(z, z3) -{ 3 }-> or(0, or(mem(z, y2), mem(z, z''))) :|: z'' >= 0, z >= 0, z3 = 1 + 0 + (1 + y2 + z''), y2 >= 0
   mem(z, z3) -{ 3 }-> or(1 + z + y', or(mem(z, y4), mem(z, z1))) :|: z1 >= 0, z3 = 1 + (1 + y') + (1 + y4 + z1), z >= 0, y' >= 0, y4 >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z3 - 2 >= 0, z >= 0, x' >= 0, 1 + z + (z3 - 2) = 1, 0 = x'
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, z3 - 2 >= 0, 1 + z + (z3 - 2) = 1, 0 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y' = 1, 1 + z + y3 = y, y >= 0
   mem(z, z3) -{ 4 }-> 1 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), x' >= 0, 1 + z + y3 = 1, 1 + z + y' = x'
   mem(z, z3) -{ 1 }-> 0 :|: z3 = 0, z >= 0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 4 }-> 0 :|: z >= 0, z3 = 1 + 0 + 0, 0 = 0
   mem(z, z3) -{ 3 }-> 0 :|: z3 - 2 >= 0, z >= 0, 1 + z + (z3 - 2) = v1, v0 >= 0, v1 >= 0, 0 = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, z3 - 2 >= 0, 0 = v1, v0 >= 0, v1 >= 0, 1 + z + (z3 - 2) = v0
   mem(z, z3) -{ 3 }-> 0 :|: z >= 0, y' >= 0, y3 >= 0, z3 = 1 + (1 + y') + (1 + y3), 1 + z + y3 = v1, v0 >= 0, v1 >= 0, 1 + z + y' = v0
   mem(z, z3) -{ 1 }-> 1 + z + (z3 - 1) :|: z >= 0, z3 - 1 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z = 1, z3 >= 0
   or(z, z3) -{ 1 }-> 1 :|: z >= 0, z3 = 1
   or(z, z3) -{ 1 }-> 0 :|: z3 = 0, z = 0
   or(z, z3) -{ 0 }-> 0 :|: z >= 0, z3 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  or: runtime: O(1) [1], size: O(1) [1]
  mem: runtime: O(n^1) [4 + 10*z3], size: O(n^1) [z + z3]

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

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

(32)
BOUNDS(1, n^1)

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

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

(34)
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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(x, nil) -> false
   mem(x, set(y)) -> ='(x, y)
   mem(x, union(y, z)) -> or(mem(x, y), mem(x, z))

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

(35) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
mem/0
set/0
='/0
='/1

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

(36)
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:

   or(true, y) -> true
   or(x, true) -> true
   or(false, false) -> false
   mem(nil) -> false
   mem(set) -> ='
   mem(union(y, z)) -> or(mem(y), mem(z))

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

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

(38)
Obligation:
TRS:
Rules:
or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(nil) -> false
mem(set) -> ='
mem(union(y, z)) -> or(mem(y), mem(z))

Types:
or :: true:false:=' -> true:false:=' -> true:false:='
true :: true:false:='
false :: true:false:='
mem :: nil:set:union -> true:false:='
nil :: nil:set:union
set :: nil:set:union
=' :: true:false:='
union :: nil:set:union -> nil:set:union -> nil:set:union
hole_true:false:='1_0 :: true:false:='
hole_nil:set:union2_0 :: nil:set:union
gen_nil:set:union3_0 :: Nat -> nil:set:union

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

(39) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
mem
----------------------------------------

(40)
Obligation:
TRS:
Rules:
or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(nil) -> false
mem(set) -> ='
mem(union(y, z)) -> or(mem(y), mem(z))

Types:
or :: true:false:=' -> true:false:=' -> true:false:='
true :: true:false:='
false :: true:false:='
mem :: nil:set:union -> true:false:='
nil :: nil:set:union
set :: nil:set:union
=' :: true:false:='
union :: nil:set:union -> nil:set:union -> nil:set:union
hole_true:false:='1_0 :: true:false:='
hole_nil:set:union2_0 :: nil:set:union
gen_nil:set:union3_0 :: Nat -> nil:set:union


Generator Equations:
gen_nil:set:union3_0(0) <=> nil
gen_nil:set:union3_0(+(x, 1)) <=> union(nil, gen_nil:set:union3_0(x))


The following defined symbols remain to be analysed:
mem
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(41) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
mem(gen_nil:set:union3_0(n5_0)) -> false, rt in Omega(1 + n5_0)

Induction Base:
mem(gen_nil:set:union3_0(0)) ->_R^Omega(1)
false

Induction Step:
mem(gen_nil:set:union3_0(+(n5_0, 1))) ->_R^Omega(1)
or(mem(nil), mem(gen_nil:set:union3_0(n5_0))) ->_R^Omega(1)
or(false, mem(gen_nil:set:union3_0(n5_0))) ->_IH
or(false, false) ->_R^Omega(1)
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
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(42)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
or(true, y) -> true
or(x, true) -> true
or(false, false) -> false
mem(nil) -> false
mem(set) -> ='
mem(union(y, z)) -> or(mem(y), mem(z))

Types:
or :: true:false:=' -> true:false:=' -> true:false:='
true :: true:false:='
false :: true:false:='
mem :: nil:set:union -> true:false:='
nil :: nil:set:union
set :: nil:set:union
=' :: true:false:='
union :: nil:set:union -> nil:set:union -> nil:set:union
hole_true:false:='1_0 :: true:false:='
hole_nil:set:union2_0 :: nil:set:union
gen_nil:set:union3_0 :: Nat -> nil:set:union


Generator Equations:
gen_nil:set:union3_0(0) <=> nil
gen_nil:set:union3_0(+(x, 1)) <=> union(nil, gen_nil:set:union3_0(x))


The following defined symbols remain to be analysed:
mem
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(43) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(44)
BOUNDS(n^1, INF)