The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).

0 CpxRelTRS
↳1 RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID), 0 ms)
↳2 CpxWeightedTrs
↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)
↳4 CpxTypedWeightedTrs
↳5 CompletionProof (UPPER BOUND(ID), 0 ms)
↳6 CpxTypedWeightedCompleteTrs
↳7 NarrowingProof (BOTH BOUNDS(ID, ID), 108 ms)
↳8 CpxTypedWeightedCompleteTrs
↳9 CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID), 12 ms)
↳10 CpxRNTS
↳11 InliningProof (UPPER BOUND(ID), 958 ms)
↳12 CpxRNTS
↳13 SimplificationProof (BOTH BOUNDS(ID, ID), 14 ms)
↳14 CpxRNTS
↳15 CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID), 0 ms)
↳16 CpxRNTS
↳17 IntTrsBoundProof (UPPER BOUND(ID), 1287 ms)
↳18 CpxRNTS
↳19 IntTrsBoundProof (UPPER BOUND(ID), 265 ms)
↳20 CpxRNTS
↳21 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳22 CpxRNTS
↳23 IntTrsBoundProof (UPPER BOUND(ID), 242 ms)
↳24 CpxRNTS
↳25 IntTrsBoundProof (UPPER BOUND(ID), 69 ms)
↳26 CpxRNTS
↳27 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳28 CpxRNTS
↳29 IntTrsBoundProof (UPPER BOUND(ID), 349 ms)
↳30 CpxRNTS
↳31 IntTrsBoundProof (UPPER BOUND(ID), 84 ms)
↳32 CpxRNTS
↳33 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳34 CpxRNTS
↳35 IntTrsBoundProof (UPPER BOUND(ID), 288 ms)
↳36 CpxRNTS
↳37 IntTrsBoundProof (UPPER BOUND(ID), 18 ms)
↳38 CpxRNTS
↳39 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳40 CpxRNTS
↳41 IntTrsBoundProof (UPPER BOUND(ID), 166 ms)
↳42 CpxRNTS
↳43 IntTrsBoundProof (UPPER BOUND(ID), 4 ms)
↳44 CpxRNTS
↳45 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳46 CpxRNTS
↳47 IntTrsBoundProof (UPPER BOUND(ID), 354 ms)
↳48 CpxRNTS
↳49 IntTrsBoundProof (UPPER BOUND(ID), 6 ms)
↳50 CpxRNTS
↳51 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳52 CpxRNTS
↳53 IntTrsBoundProof (UPPER BOUND(ID), 832 ms)
↳54 CpxRNTS
↳55 IntTrsBoundProof (UPPER BOUND(ID), 16 ms)
↳56 CpxRNTS
↳57 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳58 CpxRNTS
↳59 IntTrsBoundProof (UPPER BOUND(ID), 773 ms)
↳60 CpxRNTS
↳61 IntTrsBoundProof (UPPER BOUND(ID), 349 ms)
↳62 CpxRNTS
↳63 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳64 CpxRNTS
↳65 IntTrsBoundProof (UPPER BOUND(ID), 103 ms)
↳66 CpxRNTS
↳67 IntTrsBoundProof (UPPER BOUND(ID), 4 ms)
↳68 CpxRNTS
↳69 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳70 CpxRNTS
↳71 IntTrsBoundProof (UPPER BOUND(ID), 853 ms)
↳72 CpxRNTS
↳73 IntTrsBoundProof (UPPER BOUND(ID), 322 ms)
↳74 CpxRNTS
↳75 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳76 CpxRNTS
↳77 IntTrsBoundProof (UPPER BOUND(ID), 121 ms)
↳78 CpxRNTS
↳79 IntTrsBoundProof (UPPER BOUND(ID), 27 ms)
↳80 CpxRNTS
↳81 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳82 CpxRNTS
↳83 IntTrsBoundProof (UPPER BOUND(ID), 1883 ms)
↳84 CpxRNTS
↳85 IntTrsBoundProof (UPPER BOUND(ID), 636 ms)
↳86 CpxRNTS
↳87 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳88 CpxRNTS
↳89 IntTrsBoundProof (UPPER BOUND(ID), 305 ms)
↳90 CpxRNTS
↳91 IntTrsBoundProof (UPPER BOUND(ID), 98 ms)
↳92 CpxRNTS
↳93 ResultPropagationProof (UPPER BOUND(ID), 0 ms)
↳94 CpxRNTS
↳95 IntTrsBoundProof (UPPER BOUND(ID), 212 ms)
↳96 CpxRNTS
↳97 IntTrsBoundProof (UPPER BOUND(ID), 0 ms)
↳98 CpxRNTS
↳99 FinalProof (⇔, 0 ms)
↳100 BOUNDS(1, n^2)

(0) Obligation:

The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

#equal(@x, @y) → #eq(@x, @y)
#less(@x, @y) → #cklt(#compare(@x, @y))
and(@x, @y) → #and(@x, @y)
insert(@x, @l) → insert#1(@l, @x)
insert#1(::(@y, @ys), @x) → insert#2(leq(@x, @y), @x, @y, @ys)
insert#1(nil, @x) → ::(@x, nil)
insert#2(#false, @x, @y, @ys) → ::(@y, insert(@x, @ys))
insert#2(#true, @x, @y, @ys) → ::(@x, ::(@y, @ys))
isortlist(@l) → isortlist#1(@l)
isortlist#1(::(@x, @xs)) → insert(@x, isortlist(@xs))
isortlist#1(nil) → nil
leq(@l1, @l2) → leq#1(@l1, @l2)
leq#1(::(@x, @xs), @l2) → leq#2(@l2, @x, @xs)
leq#1(nil, @l2) → #true
leq#2(::(@y, @ys), @x, @xs) → or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))
leq#2(nil, @x, @xs) → #false
or(@x, @y) → #or(@x, @y)

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

#and(#false, #false) → #false
#and(#false, #true) → #false
#and(#true, #false) → #false
#and(#true, #true) → #true
#cklt(#EQ) → #false
#cklt(#GT) → #false
#cklt(#LT) → #true
#compare(#0, #0) → #EQ
#compare(#0, #neg(@y)) → #GT
#compare(#0, #pos(@y)) → #LT
#compare(#0, #s(@y)) → #LT
#compare(#neg(@x), #0) → #LT
#compare(#neg(@x), #neg(@y)) → #compare(@y, @x)
#compare(#neg(@x), #pos(@y)) → #LT
#compare(#pos(@x), #0) → #GT
#compare(#pos(@x), #neg(@y)) → #GT
#compare(#pos(@x), #pos(@y)) → #compare(@x, @y)
#compare(#s(@x), #0) → #GT
#compare(#s(@x), #s(@y)) → #compare(@x, @y)
#eq(#0, #0) → #true
#eq(#0, #neg(@y)) → #false
#eq(#0, #pos(@y)) → #false
#eq(#0, #s(@y)) → #false
#eq(#neg(@x), #0) → #false
#eq(#neg(@x), #neg(@y)) → #eq(@x, @y)
#eq(#neg(@x), #pos(@y)) → #false
#eq(#pos(@x), #0) → #false
#eq(#pos(@x), #neg(@y)) → #false
#eq(#pos(@x), #pos(@y)) → #eq(@x, @y)
#eq(#s(@x), #0) → #false
#eq(#s(@x), #s(@y)) → #eq(@x, @y)
#eq(::(@x_1, @x_2), ::(@y_1, @y_2)) → #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))
#eq(::(@x_1, @x_2), nil) → #false
#eq(nil, ::(@y_1, @y_2)) → #false
#eq(nil, nil) → #true
#or(#false, #false) → #false
#or(#false, #true) → #true
#or(#true, #false) → #true
#or(#true, #true) → #true

Rewrite Strategy: INNERMOST

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID) transformation)

Transformed relative TRS to weighted TRS

(2) Obligation:

The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

#equal(@x, @y) → #eq(@x, @y) [1]
#less(@x, @y) → #cklt(#compare(@x, @y)) [1]
and(@x, @y) → #and(@x, @y) [1]
insert(@x, @l) → insert#1(@l, @x) [1]
insert#1(::(@y, @ys), @x) → insert#2(leq(@x, @y), @x, @y, @ys) [1]
insert#1(nil, @x) → ::(@x, nil) [1]
insert#2(#false, @x, @y, @ys) → ::(@y, insert(@x, @ys)) [1]
insert#2(#true, @x, @y, @ys) → ::(@x, ::(@y, @ys)) [1]
isortlist(@l) → isortlist#1(@l) [1]
isortlist#1(::(@x, @xs)) → insert(@x, isortlist(@xs)) [1]
isortlist#1(nil) → nil [1]
leq(@l1, @l2) → leq#1(@l1, @l2) [1]
leq#1(::(@x, @xs), @l2) → leq#2(@l2, @x, @xs) [1]
leq#1(nil, @l2) → #true [1]
leq#2(::(@y, @ys), @x, @xs) → or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) [1]
leq#2(nil, @x, @xs) → #false [1]
or(@x, @y) → #or(@x, @y) [1]
#and(#false, #false) → #false [0]
#and(#false, #true) → #false [0]
#and(#true, #false) → #false [0]
#and(#true, #true) → #true [0]
#cklt(#EQ) → #false [0]
#cklt(#GT) → #false [0]
#cklt(#LT) → #true [0]
#compare(#0, #0) → #EQ [0]
#compare(#0, #neg(@y)) → #GT [0]
#compare(#0, #pos(@y)) → #LT [0]
#compare(#0, #s(@y)) → #LT [0]
#compare(#neg(@x), #0) → #LT [0]
#compare(#neg(@x), #neg(@y)) → #compare(@y, @x) [0]
#compare(#neg(@x), #pos(@y)) → #LT [0]
#compare(#pos(@x), #0) → #GT [0]
#compare(#pos(@x), #neg(@y)) → #GT [0]
#compare(#pos(@x), #pos(@y)) → #compare(@x, @y) [0]
#compare(#s(@x), #0) → #GT [0]
#compare(#s(@x), #s(@y)) → #compare(@x, @y) [0]
#eq(#0, #0) → #true [0]
#eq(#0, #neg(@y)) → #false [0]
#eq(#0, #pos(@y)) → #false [0]
#eq(#0, #s(@y)) → #false [0]
#eq(#neg(@x), #0) → #false [0]
#eq(#neg(@x), #neg(@y)) → #eq(@x, @y) [0]
#eq(#neg(@x), #pos(@y)) → #false [0]
#eq(#pos(@x), #0) → #false [0]
#eq(#pos(@x), #neg(@y)) → #false [0]
#eq(#pos(@x), #pos(@y)) → #eq(@x, @y) [0]
#eq(#s(@x), #0) → #false [0]
#eq(#s(@x), #s(@y)) → #eq(@x, @y) [0]
#eq(::(@x_1, @x_2), ::(@y_1, @y_2)) → #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
#eq(::(@x_1, @x_2), nil) → #false [0]
#eq(nil, ::(@y_1, @y_2)) → #false [0]
#eq(nil, nil) → #true [0]
#or(#false, #false) → #false [0]
#or(#false, #true) → #true [0]
#or(#true, #false) → #true [0]
#or(#true, #true) → #true [0]

Rewrite Strategy: INNERMOST

(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)

Infered types.

(4) Obligation:

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

#equal(@x, @y) → #eq(@x, @y) [1]
#less(@x, @y) → #cklt(#compare(@x, @y)) [1]
and(@x, @y) → #and(@x, @y) [1]
insert(@x, @l) → insert#1(@l, @x) [1]
insert#1(::(@y, @ys), @x) → insert#2(leq(@x, @y), @x, @y, @ys) [1]
insert#1(nil, @x) → ::(@x, nil) [1]
insert#2(#false, @x, @y, @ys) → ::(@y, insert(@x, @ys)) [1]
insert#2(#true, @x, @y, @ys) → ::(@x, ::(@y, @ys)) [1]
isortlist(@l) → isortlist#1(@l) [1]
isortlist#1(::(@x, @xs)) → insert(@x, isortlist(@xs)) [1]
isortlist#1(nil) → nil [1]
leq(@l1, @l2) → leq#1(@l1, @l2) [1]
leq#1(::(@x, @xs), @l2) → leq#2(@l2, @x, @xs) [1]
leq#1(nil, @l2) → #true [1]
leq#2(::(@y, @ys), @x, @xs) → or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) [1]
leq#2(nil, @x, @xs) → #false [1]
or(@x, @y) → #or(@x, @y) [1]
#and(#false, #false) → #false [0]
#and(#false, #true) → #false [0]
#and(#true, #false) → #false [0]
#and(#true, #true) → #true [0]
#cklt(#EQ) → #false [0]
#cklt(#GT) → #false [0]
#cklt(#LT) → #true [0]
#compare(#0, #0) → #EQ [0]
#compare(#0, #neg(@y)) → #GT [0]
#compare(#0, #pos(@y)) → #LT [0]
#compare(#0, #s(@y)) → #LT [0]
#compare(#neg(@x), #0) → #LT [0]
#compare(#neg(@x), #neg(@y)) → #compare(@y, @x) [0]
#compare(#neg(@x), #pos(@y)) → #LT [0]
#compare(#pos(@x), #0) → #GT [0]
#compare(#pos(@x), #neg(@y)) → #GT [0]
#compare(#pos(@x), #pos(@y)) → #compare(@x, @y) [0]
#compare(#s(@x), #0) → #GT [0]
#compare(#s(@x), #s(@y)) → #compare(@x, @y) [0]
#eq(#0, #0) → #true [0]
#eq(#0, #neg(@y)) → #false [0]
#eq(#0, #pos(@y)) → #false [0]
#eq(#0, #s(@y)) → #false [0]
#eq(#neg(@x), #0) → #false [0]
#eq(#neg(@x), #neg(@y)) → #eq(@x, @y) [0]
#eq(#neg(@x), #pos(@y)) → #false [0]
#eq(#pos(@x), #0) → #false [0]
#eq(#pos(@x), #neg(@y)) → #false [0]
#eq(#pos(@x), #pos(@y)) → #eq(@x, @y) [0]
#eq(#s(@x), #0) → #false [0]
#eq(#s(@x), #s(@y)) → #eq(@x, @y) [0]
#eq(::(@x_1, @x_2), ::(@y_1, @y_2)) → #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
#eq(::(@x_1, @x_2), nil) → #false [0]
#eq(nil, ::(@y_1, @y_2)) → #false [0]
#eq(nil, nil) → #true [0]
#or(#false, #false) → #false [0]
#or(#false, #true) → #true [0]
#or(#true, #false) → #true [0]
#or(#true, #true) → #true [0]

The TRS has the following type information:
#equal :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → #false:#true
#eq :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → #false:#true
#less :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → #false:#true
#cklt :: #EQ:#GT:#LT → #false:#true
#compare :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → #EQ:#GT:#LT
and :: #false:#true → #false:#true → #false:#true
#and :: #false:#true → #false:#true → #false:#true
insert :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s
insert#1 :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s
:: :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s
insert#2 :: #false:#true → :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s
leq :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → #false:#true
nil :: :::nil:#0:#neg:#pos:#s
#false :: #false:#true
#true :: #false:#true
isortlist :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s
isortlist#1 :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s
leq#1 :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → #false:#true
leq#2 :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s → #false:#true
or :: #false:#true → #false:#true → #false:#true
#or :: #false:#true → #false:#true → #false:#true
#EQ :: #EQ:#GT:#LT
#GT :: #EQ:#GT:#LT
#LT :: #EQ:#GT:#LT
#0 :: :::nil:#0:#neg:#pos:#s
#neg :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s
#pos :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s
#s :: :::nil:#0:#neg:#pos:#s → :::nil:#0:#neg:#pos:#s

Rewrite Strategy: INNERMOST

(5) CompletionProof (UPPER BOUND(ID) transformation)

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:


leq
isortlist
#less
and
#equal
leq#1
isortlist#1
leq#2
insert
or
insert#1
insert#2
#and
#cklt
#compare
#eq
#or

Due to the following rules being added:

#and(v0, v1) → null_#and [0]
#cklt(v0) → null_#cklt [0]
#compare(v0, v1) → null_#compare [0]
#eq(v0, v1) → null_#eq [0]
#or(v0, v1) → null_#or [0]
leq#1(v0, v1) → null_leq#1 [0]
isortlist#1(v0) → null_isortlist#1 [0]
leq#2(v0, v1, v2) → null_leq#2 [0]
insert#1(v0, v1) → null_insert#1 [0]
insert#2(v0, v1, v2, v3) → null_insert#2 [0]

And the following fresh constants:

null_#and, null_#cklt, null_#compare, null_#eq, null_#or, null_leq#1, null_isortlist#1, null_leq#2, null_insert#1, null_insert#2

(6) 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:

#equal(@x, @y) → #eq(@x, @y) [1]
#less(@x, @y) → #cklt(#compare(@x, @y)) [1]
and(@x, @y) → #and(@x, @y) [1]
insert(@x, @l) → insert#1(@l, @x) [1]
insert#1(::(@y, @ys), @x) → insert#2(leq(@x, @y), @x, @y, @ys) [1]
insert#1(nil, @x) → ::(@x, nil) [1]
insert#2(#false, @x, @y, @ys) → ::(@y, insert(@x, @ys)) [1]
insert#2(#true, @x, @y, @ys) → ::(@x, ::(@y, @ys)) [1]
isortlist(@l) → isortlist#1(@l) [1]
isortlist#1(::(@x, @xs)) → insert(@x, isortlist(@xs)) [1]
isortlist#1(nil) → nil [1]
leq(@l1, @l2) → leq#1(@l1, @l2) [1]
leq#1(::(@x, @xs), @l2) → leq#2(@l2, @x, @xs) [1]
leq#1(nil, @l2) → #true [1]
leq#2(::(@y, @ys), @x, @xs) → or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) [1]
leq#2(nil, @x, @xs) → #false [1]
or(@x, @y) → #or(@x, @y) [1]
#and(#false, #false) → #false [0]
#and(#false, #true) → #false [0]
#and(#true, #false) → #false [0]
#and(#true, #true) → #true [0]
#cklt(#EQ) → #false [0]
#cklt(#GT) → #false [0]
#cklt(#LT) → #true [0]
#compare(#0, #0) → #EQ [0]
#compare(#0, #neg(@y)) → #GT [0]
#compare(#0, #pos(@y)) → #LT [0]
#compare(#0, #s(@y)) → #LT [0]
#compare(#neg(@x), #0) → #LT [0]
#compare(#neg(@x), #neg(@y)) → #compare(@y, @x) [0]
#compare(#neg(@x), #pos(@y)) → #LT [0]
#compare(#pos(@x), #0) → #GT [0]
#compare(#pos(@x), #neg(@y)) → #GT [0]
#compare(#pos(@x), #pos(@y)) → #compare(@x, @y) [0]
#compare(#s(@x), #0) → #GT [0]
#compare(#s(@x), #s(@y)) → #compare(@x, @y) [0]
#eq(#0, #0) → #true [0]
#eq(#0, #neg(@y)) → #false [0]
#eq(#0, #pos(@y)) → #false [0]
#eq(#0, #s(@y)) → #false [0]
#eq(#neg(@x), #0) → #false [0]
#eq(#neg(@x), #neg(@y)) → #eq(@x, @y) [0]
#eq(#neg(@x), #pos(@y)) → #false [0]
#eq(#pos(@x), #0) → #false [0]
#eq(#pos(@x), #neg(@y)) → #false [0]
#eq(#pos(@x), #pos(@y)) → #eq(@x, @y) [0]
#eq(#s(@x), #0) → #false [0]
#eq(#s(@x), #s(@y)) → #eq(@x, @y) [0]
#eq(::(@x_1, @x_2), ::(@y_1, @y_2)) → #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
#eq(::(@x_1, @x_2), nil) → #false [0]
#eq(nil, ::(@y_1, @y_2)) → #false [0]
#eq(nil, nil) → #true [0]
#or(#false, #false) → #false [0]
#or(#false, #true) → #true [0]
#or(#true, #false) → #true [0]
#or(#true, #true) → #true [0]
#and(v0, v1) → null_#and [0]
#cklt(v0) → null_#cklt [0]
#compare(v0, v1) → null_#compare [0]
#eq(v0, v1) → null_#eq [0]
#or(v0, v1) → null_#or [0]
leq#1(v0, v1) → null_leq#1 [0]
isortlist#1(v0) → null_isortlist#1 [0]
leq#2(v0, v1, v2) → null_leq#2 [0]
insert#1(v0, v1) → null_insert#1 [0]
insert#2(v0, v1, v2, v3) → null_insert#2 [0]

The TRS has the following type information:
#equal :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#eq :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#less :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#cklt :: #EQ:#GT:#LT:null_#compare → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#compare :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #EQ:#GT:#LT:null_#compare
and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
insert :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
insert#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
:: :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
insert#2 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
leq :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
nil :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
#false :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#true :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
isortlist :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
isortlist#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
leq#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
leq#2 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#EQ :: #EQ:#GT:#LT:null_#compare
#GT :: #EQ:#GT:#LT:null_#compare
#LT :: #EQ:#GT:#LT:null_#compare
#0 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
#neg :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
#pos :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
#s :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
null_#and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_#cklt :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_#compare :: #EQ:#GT:#LT:null_#compare
null_#eq :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_#or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_leq#1 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_isortlist#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
null_leq#2 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_insert#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
null_insert#2 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2

Rewrite Strategy: INNERMOST

(7) NarrowingProof (BOTH BOUNDS(ID, ID) transformation)

Narrowed the inner basic terms of all right-hand sides by a single narrowing step.

(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:

#equal(@x, @y) → #eq(@x, @y) [1]
#less(#0, #0) → #cklt(#EQ) [1]
#less(#0, #neg(@y')) → #cklt(#GT) [1]
#less(#0, #pos(@y'')) → #cklt(#LT) [1]
#less(#0, #s(@y1)) → #cklt(#LT) [1]
#less(#neg(@x'), #0) → #cklt(#LT) [1]
#less(#neg(@x''), #neg(@y2)) → #cklt(#compare(@y2, @x'')) [1]
#less(#neg(@x1), #pos(@y3)) → #cklt(#LT) [1]
#less(#pos(@x2), #0) → #cklt(#GT) [1]
#less(#pos(@x3), #neg(@y4)) → #cklt(#GT) [1]
#less(#pos(@x4), #pos(@y5)) → #cklt(#compare(@x4, @y5)) [1]
#less(#s(@x5), #0) → #cklt(#GT) [1]
#less(#s(@x6), #s(@y6)) → #cklt(#compare(@x6, @y6)) [1]
#less(@x, @y) → #cklt(null_#compare) [1]
and(@x, @y) → #and(@x, @y) [1]
insert(@x, @l) → insert#1(@l, @x) [1]
insert#1(::(@y, @ys), @x) → insert#2(leq#1(@x, @y), @x, @y, @ys) [2]
insert#1(nil, @x) → ::(@x, nil) [1]
insert#2(#false, @x, @y, @ys) → ::(@y, insert(@x, @ys)) [1]
insert#2(#true, @x, @y, @ys) → ::(@x, ::(@y, @ys)) [1]
isortlist(@l) → isortlist#1(@l) [1]
isortlist#1(::(@x, @xs)) → insert(@x, isortlist#1(@xs)) [2]
isortlist#1(nil) → nil [1]
leq(@l1, @l2) → leq#1(@l1, @l2) [1]
leq#1(::(@x, @xs), @l2) → leq#2(@l2, @x, @xs) [1]
leq#1(nil, @l2) → #true [1]
leq#2(::(@y, @ys), @x, @xs) → or(#cklt(#compare(@x, @y)), and(#eq(@x, @y), leq#1(@xs, @ys))) [4]
leq#2(nil, @x, @xs) → #false [1]
or(@x, @y) → #or(@x, @y) [1]
#and(#false, #false) → #false [0]
#and(#false, #true) → #false [0]
#and(#true, #false) → #false [0]
#and(#true, #true) → #true [0]
#cklt(#EQ) → #false [0]
#cklt(#GT) → #false [0]
#cklt(#LT) → #true [0]
#compare(#0, #0) → #EQ [0]
#compare(#0, #neg(@y)) → #GT [0]
#compare(#0, #pos(@y)) → #LT [0]
#compare(#0, #s(@y)) → #LT [0]
#compare(#neg(@x), #0) → #LT [0]
#compare(#neg(@x), #neg(@y)) → #compare(@y, @x) [0]
#compare(#neg(@x), #pos(@y)) → #LT [0]
#compare(#pos(@x), #0) → #GT [0]
#compare(#pos(@x), #neg(@y)) → #GT [0]
#compare(#pos(@x), #pos(@y)) → #compare(@x, @y) [0]
#compare(#s(@x), #0) → #GT [0]
#compare(#s(@x), #s(@y)) → #compare(@x, @y) [0]
#eq(#0, #0) → #true [0]
#eq(#0, #neg(@y)) → #false [0]
#eq(#0, #pos(@y)) → #false [0]
#eq(#0, #s(@y)) → #false [0]
#eq(#neg(@x), #0) → #false [0]
#eq(#neg(@x), #neg(@y)) → #eq(@x, @y) [0]
#eq(#neg(@x), #pos(@y)) → #false [0]
#eq(#pos(@x), #0) → #false [0]
#eq(#pos(@x), #neg(@y)) → #false [0]
#eq(#pos(@x), #pos(@y)) → #eq(@x, @y) [0]
#eq(#s(@x), #0) → #false [0]
#eq(#s(@x), #s(@y)) → #eq(@x, @y) [0]
#eq(::(@x_1, @x_2), ::(@y_1, @y_2)) → #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) [0]
#eq(::(@x_1, @x_2), nil) → #false [0]
#eq(nil, ::(@y_1, @y_2)) → #false [0]
#eq(nil, nil) → #true [0]
#or(#false, #false) → #false [0]
#or(#false, #true) → #true [0]
#or(#true, #false) → #true [0]
#or(#true, #true) → #true [0]
#and(v0, v1) → null_#and [0]
#cklt(v0) → null_#cklt [0]
#compare(v0, v1) → null_#compare [0]
#eq(v0, v1) → null_#eq [0]
#or(v0, v1) → null_#or [0]
leq#1(v0, v1) → null_leq#1 [0]
isortlist#1(v0) → null_isortlist#1 [0]
leq#2(v0, v1, v2) → null_leq#2 [0]
insert#1(v0, v1) → null_insert#1 [0]
insert#2(v0, v1, v2, v3) → null_insert#2 [0]

The TRS has the following type information:
#equal :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#eq :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#less :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#cklt :: #EQ:#GT:#LT:null_#compare → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#compare :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #EQ:#GT:#LT:null_#compare
and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
insert :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
insert#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
:: :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
insert#2 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
leq :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
nil :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
#false :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#true :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
isortlist :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
isortlist#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
leq#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
leq#2 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2 → #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
#EQ :: #EQ:#GT:#LT:null_#compare
#GT :: #EQ:#GT:#LT:null_#compare
#LT :: #EQ:#GT:#LT:null_#compare
#0 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
#neg :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
#pos :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
#s :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2 → :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
null_#and :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_#cklt :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_#compare :: #EQ:#GT:#LT:null_#compare
null_#eq :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_#or :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_leq#1 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_isortlist#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
null_leq#2 :: #false:#true:null_#and:null_#cklt:null_#eq:null_#or:null_leq#1:null_leq#2
null_insert#1 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2
null_insert#2 :: :::nil:#0:#neg:#pos:#s:null_isortlist#1:null_insert#1:null_insert#2

Rewrite Strategy: INNERMOST

(9) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID) transformation)

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

nil => 1
#false => 1
#true => 2
#EQ => 1
#GT => 2
#LT => 3
#0 => 0
null_#and => 0
null_#cklt => 0
null_#compare => 0
null_#eq => 0
null_#or => 0
null_leq#1 => 0
null_isortlist#1 => 0
null_leq#2 => 0
null_insert#1 => 0
null_insert#2 => 0

(10) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' = 1 + @y, @y >= 0
#compare(z, z') -{ 0 }→ 3 :|: @x >= 0, z = 1 + @x, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' = 1 + @y, @y >= 0
#compare(z, z') -{ 0 }→ 2 :|: @x >= 0, z = 1 + @x, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
#compare(z, z') -{ 0 }→ #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#compare(z, z') -{ 0 }→ #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' = 1 + @y, @y >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x >= 0, z = 1 + @x, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
#eq(z, z') -{ 0 }→ #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0
#less(z, z') -{ 1 }→ #cklt(3) :|: z' = 1 + @y'', @y'' >= 0, z = 0
#less(z, z') -{ 1 }→ #cklt(3) :|: z' = 1 + @y1, @y1 >= 0, z = 0
#less(z, z') -{ 1 }→ #cklt(3) :|: z = 1 + @x', @x' >= 0, z' = 0
#less(z, z') -{ 1 }→ #cklt(3) :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1
#less(z, z') -{ 1 }→ #cklt(2) :|: @y' >= 0, z' = 1 + @y', z = 0
#less(z, z') -{ 1 }→ #cklt(2) :|: @x2 >= 0, z = 1 + @x2, z' = 0
#less(z, z') -{ 1 }→ #cklt(2) :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0
#less(z, z') -{ 1 }→ #cklt(2) :|: z = 1 + @x5, @x5 >= 0, z' = 0
#less(z, z') -{ 1 }→ #cklt(1) :|: z = 0, z' = 0
#less(z, z') -{ 1 }→ #cklt(0) :|: z = @x, @x >= 0, z' = @y, @y >= 0
#less(z, z') -{ 1 }→ #cklt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0
#less(z, z') -{ 1 }→ #cklt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0
#less(z, z') -{ 1 }→ #cklt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
and(z, z') -{ 1 }→ #and(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0
insert(z, z') -{ 1 }→ insert#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l
insert#1(z, z') -{ 2 }→ insert#2(leq#1(@x, @y), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
insert#1(z, z') -{ 1 }→ 1 + @x + 1 :|: @x >= 0, z = 1, z' = @x
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + @x + (1 + @y + @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
insert#2(z, z', z'', z1) -{ 1 }→ 1 + @y + insert(@x, @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
isortlist(z) -{ 1 }→ isortlist#1(@l) :|: z = @l, @l >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
leq(z, z') -{ 1 }→ leq#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1
leq#1(z, z') -{ 1 }→ leq#2(@l2, @x, @xs) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z' = @l2, z = 1, @l2 >= 0
leq#1(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
leq#2(z, z', z'') -{ 4 }→ or(#cklt(#compare(@x, @y)), and(#eq(@x, @y), leq#1(@xs, @ys))) :|: z = 1 + @y + @ys, @x >= 0, @xs >= 0, @y >= 0, z' = @x, z'' = @xs, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: @x >= 0, z = 1, @xs >= 0, z' = @x, z'' = @xs
leq#2(z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
or(z, z') -{ 1 }→ #or(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0

(11) InliningProof (UPPER BOUND(ID) transformation)

Inlined the following terminating rules on right-hand sides where appropriate:

#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
and(z, z') -{ 1 }→ #and(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0
or(z, z') -{ 1 }→ #or(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0

(12) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' = 1 + @y, @y >= 0
#compare(z, z') -{ 0 }→ 3 :|: @x >= 0, z = 1 + @x, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' = 1 + @y, @y >= 0
#compare(z, z') -{ 0 }→ 2 :|: @x >= 0, z = 1 + @x, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
#compare(z, z') -{ 0 }→ #compare(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#compare(z, z') -{ 0 }→ #compare(@y, @x) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' = 1 + @y, @y >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x >= 0, z = 1 + @x, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
#eq(z, z') -{ 0 }→ #eq(@x, @y) :|: @x >= 0, z = 1 + @x, z' = 1 + @y, @y >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(@x, @y) :|: z = @x, @x >= 0, z' = @y, @y >= 0
#less(z, z') -{ 1 }→ 2 :|: z' = 1 + @y'', @y'' >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' = 1 + @y1, @y1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z = 1 + @x', @x' >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: @y' >= 0, z' = 1 + @y', z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: @x2 >= 0, z = 1 + @x2, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z = 1 + @x5, @x5 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: @y' >= 0, z' = 1 + @y', z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' = 1 + @y'', @y'' >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' = 1 + @y1, @y1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z = 1 + @x', @x' >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: @y3 >= 0, @x1 >= 0, z' = 1 + @y3, z = 1 + @x1, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: @x2 >= 0, z = 1 + @x2, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: @x3 >= 0, z' = 1 + @y4, z = 1 + @x3, @y4 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z = 1 + @x5, @x5 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(#compare(@x4, @y5)) :|: z' = 1 + @y5, @y5 >= 0, z = 1 + @x4, @x4 >= 0
#less(z, z') -{ 1 }→ #cklt(#compare(@x6, @y6)) :|: z = 1 + @x6, z' = 1 + @y6, @x6 >= 0, @y6 >= 0
#less(z, z') -{ 1 }→ #cklt(#compare(@y2, @x'')) :|: z = 1 + @x'', z' = 1 + @y2, @y2 >= 0, @x'' >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
and(z, z') -{ 1 }→ 2 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 2, @y = 2
and(z, z') -{ 1 }→ 1 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 1, @y = 1
and(z, z') -{ 1 }→ 1 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @y = 2, @x = 1
and(z, z') -{ 1 }→ 1 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 2, @y = 1
and(z, z') -{ 1 }→ 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, v1 >= 0, @x = v0, @y = v1
insert(z, z') -{ 1 }→ insert#1(@l, @x) :|: z = @x, @l >= 0, @x >= 0, z' = @l
insert#1(z, z') -{ 2 }→ insert#2(leq#1(@x, @y), @x, @y, @ys) :|: z = 1 + @y + @ys, @x >= 0, @y >= 0, z' = @x, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
insert#1(z, z') -{ 1 }→ 1 + @x + 1 :|: @x >= 0, z = 1, z' = @x
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + @x + (1 + @y + @ys) :|: z = 2, @x >= 0, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
insert#2(z, z', z'', z1) -{ 1 }→ 1 + @y + insert(@x, @ys) :|: @x >= 0, z = 1, z1 = @ys, @y >= 0, z' = @x, @ys >= 0, z'' = @y
isortlist(z) -{ 1 }→ isortlist#1(@l) :|: z = @l, @l >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: v0 >= 0, z = v0
leq(z, z') -{ 1 }→ leq#1(@l1, @l2) :|: @l1 >= 0, z' = @l2, @l2 >= 0, z = @l1
leq#1(z, z') -{ 1 }→ leq#2(@l2, @x, @xs) :|: z' = @l2, @x >= 0, z = 1 + @x + @xs, @l2 >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z' = @l2, z = 1, @l2 >= 0
leq#1(z, z') -{ 0 }→ 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
leq#2(z, z', z'') -{ 4 }→ or(#cklt(#compare(@x, @y)), and(#eq(@x, @y), leq#1(@xs, @ys))) :|: z = 1 + @y + @ys, @x >= 0, @xs >= 0, @y >= 0, z' = @x, z'' = @xs, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: @x >= 0, z = 1, @xs >= 0, z' = @x, z'' = @xs
leq#2(z, z', z'') -{ 0 }→ 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
or(z, z') -{ 1 }→ 2 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @y = 2, @x = 1
or(z, z') -{ 1 }→ 2 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 2, @y = 1
or(z, z') -{ 1 }→ 2 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 2, @y = 2
or(z, z') -{ 1 }→ 1 :|: z = @x, @x >= 0, z' = @y, @y >= 0, @x = 1, @y = 1
or(z, z') -{ 1 }→ 0 :|: z = @x, @x >= 0, z' = @y, @y >= 0, v0 >= 0, v1 >= 0, @x = v0, @y = v1

(13) SimplificationProof (BOTH BOUNDS(ID, ID) transformation)

Simplified the RNTS by moving equalities from the constraints into the right-hand sides.

(14) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#compare(z, z') -{ 0 }→ #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(#compare(z', @y)), and(#eq(z', @y), leq#1(z'', @ys))) :|: z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

(15) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID) transformation)

Found the following analysis order by SCC decomposition:

{ #compare }
{ #or }
{ #and }
{ and }
{ #cklt }
{ or }
{ #less }
{ #eq }
{ #equal }
{ leq#1, leq#2 }
{ leq }
{ insert#2, insert, insert#1 }
{ isortlist#1 }
{ isortlist }

(16) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#compare(z, z') -{ 0 }→ #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(#compare(z', @y)), and(#eq(z', @y), leq#1(z'', @ys))) :|: z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#compare}, {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}

(17) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: #compare
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 3

(18) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#compare(z, z') -{ 0 }→ #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(#compare(z', @y)), and(#eq(z', @y), leq#1(z'', @ys))) :|: z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#compare}, {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: ?, size: O(1) [3]

(19) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(20) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#compare(z, z') -{ 0 }→ #compare(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ #compare(z' - 1, z - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(#compare(z - 1, z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(#compare(z' - 1, z - 1)) :|: z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(#compare(z', @y)), and(#eq(z', @y), leq#1(z'', @ys))) :|: z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]

(21) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(22) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]

(23) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(24) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#or}, {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: ?, size: O(1) [2]

(25) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(26) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]

(27) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(28) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]

(29) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(30) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#and}, {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: ?, size: O(1) [2]

(31) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(32) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]

(33) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(34) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]

(35) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(36) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {and}, {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: ?, size: O(1) [2]

(37) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(38) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]

(39) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(40) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]

(41) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(42) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#cklt}, {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: ?, size: O(1) [2]

(43) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(44) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#less(z, z') -{ 1 }→ #cklt(s'') :|: s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ #cklt(s1) :|: s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(#cklt(s2), and(#eq(z', @y), leq#1(z'', @ys))) :|: s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]

(45) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(46) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]

(47) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(48) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {or}, {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: ?, size: O(1) [2]

(49) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(50) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]

(51) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(52) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]

(53) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(54) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#less}, {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: ?, size: O(1) [2]

(55) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(56) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]

(57) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(58) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]

(59) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(60) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#eq}, {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: ?, size: O(1) [2]

(61) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(62) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ #eq(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#equal(z, z') -{ 1 }→ #eq(z, z') :|: z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(#eq(z', @y), leq#1(z'', @ys))) :|: s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]

(63) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(64) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]

(65) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(66) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {#equal}, {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: ?, size: O(1) [2]

(67) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(68) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]

(69) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(70) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]

(71) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

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

(72) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {leq#1,leq#2}, {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: ?, size: O(1) [2]
leq#2: runtime: ?, size: O(1) [2]

(73) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using PUBS for: leq#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 2 + 7·z'

Computed RUNTIME bound using CoFloCo for: leq#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 1 + 7·z

(74) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 2 }→ insert#2(leq#1(z', @y), z', @y, @ys) :|: z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 1 }→ leq#1(z, z') :|: z >= 0, z' >= 0
leq#1(z, z') -{ 1 }→ leq#2(z', @x, @xs) :|: @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 4 }→ or(s5, and(s11, leq#1(z'', @ys))) :|: s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]

(75) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(76) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 4 + 7·@y }→ insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]

(77) IntTrsBoundProof (UPPER BOUND(ID) transformation)


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

(78) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 4 + 7·@y }→ insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {leq}, {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: ?, size: O(1) [2]

(79) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: leq
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 3 + 7·z'

(80) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 4 + 7·@y }→ insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: O(n1) [3 + 7·z'], size: O(1) [2]

(81) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(82) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 4 + 7·@y }→ insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: O(n1) [3 + 7·z'], size: O(1) [2]

(83) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using PUBS for: insert#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 2 + z' + z'' + z1

Computed SIZE bound using CoFloCo for: insert
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 1 + z + z'

Computed SIZE bound using CoFloCo for: insert#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 1 + z + z'

(84) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 4 + 7·@y }→ insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {insert#2,insert,insert#1}, {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: O(n1) [3 + 7·z'], size: O(1) [2]
insert#2: runtime: ?, size: O(n1) [2 + z' + z'' + z1]
insert: runtime: ?, size: O(n1) [1 + z + z']
insert#1: runtime: ?, size: O(n1) [1 + z + z']

(85) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using CoFloCo for: insert#2
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 12 + 9·z1

Computed RUNTIME bound using CoFloCo for: insert
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 8 + 9·z'

Computed RUNTIME bound using CoFloCo for: insert#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: 7 + 9·z

(86) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 1 }→ insert#1(z', z) :|: z' >= 0, z >= 0
insert#1(z, z') -{ 4 + 7·@y }→ insert#2(s12, z', @y, @ys) :|: s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z'' + insert(z', z1) :|: z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: O(n1) [3 + 7·z'], size: O(1) [2]
insert#2: runtime: O(n1) [12 + 9·z1], size: O(n1) [2 + z' + z'' + z1]
insert: runtime: O(n1) [8 + 9·z'], size: O(n1) [1 + z + z']
insert#1: runtime: O(n1) [7 + 9·z], size: O(n1) [1 + z + z']

(87) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(88) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 8 + 9·z' }→ s18 :|: s18 >= 0, s18 <= 1 * z' + 1 * z + 1, z' >= 0, z >= 0
insert#1(z, z') -{ 16 + 7·@y + 9·@ys }→ s19 :|: s19 >= 0, s19 <= 1 * @y + 1 * @ys + 1 * z' + 2, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 9 + 9·z1 }→ 1 + z'' + s20 :|: s20 >= 0, s20 <= 1 * z' + 1 * z1 + 1, z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: O(n1) [3 + 7·z'], size: O(1) [2]
insert#2: runtime: O(n1) [12 + 9·z1], size: O(n1) [2 + z' + z'' + z1]
insert: runtime: O(n1) [8 + 9·z'], size: O(n1) [1 + z + z']
insert#1: runtime: O(n1) [7 + 9·z], size: O(n1) [1 + z + z']

(89) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using KoAT for: isortlist#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: z

(90) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 8 + 9·z' }→ s18 :|: s18 >= 0, s18 <= 1 * z' + 1 * z + 1, z' >= 0, z >= 0
insert#1(z, z') -{ 16 + 7·@y + 9·@ys }→ s19 :|: s19 >= 0, s19 <= 1 * @y + 1 * @ys + 1 * z' + 2, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 9 + 9·z1 }→ 1 + z'' + s20 :|: s20 >= 0, s20 <= 1 * z' + 1 * z1 + 1, z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {isortlist#1}, {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: O(n1) [3 + 7·z'], size: O(1) [2]
insert#2: runtime: O(n1) [12 + 9·z1], size: O(n1) [2 + z' + z'' + z1]
insert: runtime: O(n1) [8 + 9·z'], size: O(n1) [1 + z + z']
insert#1: runtime: O(n1) [7 + 9·z], size: O(n1) [1 + z + z']
isortlist#1: runtime: ?, size: O(n1) [z]

(91) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using KoAT for: isortlist#1
after applying outer abstraction to obtain an ITS,
resulting in: O(n2) with polynomial bound: 1 + 10·z + 9·z2

(92) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 8 + 9·z' }→ s18 :|: s18 >= 0, s18 <= 1 * z' + 1 * z + 1, z' >= 0, z >= 0
insert#1(z, z') -{ 16 + 7·@y + 9·@ys }→ s19 :|: s19 >= 0, s19 <= 1 * @y + 1 * @ys + 1 * z' + 2, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 9 + 9·z1 }→ 1 + z'' + s20 :|: s20 >= 0, s20 <= 1 * z' + 1 * z1 + 1, z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 1 }→ isortlist#1(z) :|: z >= 0
isortlist#1(z) -{ 2 }→ insert(@x, isortlist#1(@xs)) :|: @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: O(n1) [3 + 7·z'], size: O(1) [2]
insert#2: runtime: O(n1) [12 + 9·z1], size: O(n1) [2 + z' + z'' + z1]
insert: runtime: O(n1) [8 + 9·z'], size: O(n1) [1 + z + z']
insert#1: runtime: O(n1) [7 + 9·z], size: O(n1) [1 + z + z']
isortlist#1: runtime: O(n2) [1 + 10·z + 9·z2], size: O(n1) [z]

(93) ResultPropagationProof (UPPER BOUND(ID) transformation)

Applied inner abstraction using the recently inferred runtime/size bounds where possible.

(94) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 8 + 9·z' }→ s18 :|: s18 >= 0, s18 <= 1 * z' + 1 * z + 1, z' >= 0, z >= 0
insert#1(z, z') -{ 16 + 7·@y + 9·@ys }→ s19 :|: s19 >= 0, s19 <= 1 * @y + 1 * @ys + 1 * z' + 2, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 9 + 9·z1 }→ 1 + z'' + s20 :|: s20 >= 0, s20 <= 1 * z' + 1 * z1 + 1, z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 2 + 10·z + 9·z2 }→ s21 :|: s21 >= 0, s21 <= 1 * z, z >= 0
isortlist#1(z) -{ 11 + 10·@xs + 9·@xs2 + 9·s22 }→ s23 :|: s22 >= 0, s22 <= 1 * @xs, s23 >= 0, s23 <= 1 * @x + 1 * s22 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: O(n1) [3 + 7·z'], size: O(1) [2]
insert#2: runtime: O(n1) [12 + 9·z1], size: O(n1) [2 + z' + z'' + z1]
insert: runtime: O(n1) [8 + 9·z'], size: O(n1) [1 + z + z']
insert#1: runtime: O(n1) [7 + 9·z], size: O(n1) [1 + z + z']
isortlist#1: runtime: O(n2) [1 + 10·z + 9·z2], size: O(n1) [z]

(95) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed SIZE bound using CoFloCo for: isortlist
after applying outer abstraction to obtain an ITS,
resulting in: O(n1) with polynomial bound: z

(96) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 8 + 9·z' }→ s18 :|: s18 >= 0, s18 <= 1 * z' + 1 * z + 1, z' >= 0, z >= 0
insert#1(z, z') -{ 16 + 7·@y + 9·@ys }→ s19 :|: s19 >= 0, s19 <= 1 * @y + 1 * @ys + 1 * z' + 2, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 9 + 9·z1 }→ 1 + z'' + s20 :|: s20 >= 0, s20 <= 1 * z' + 1 * z1 + 1, z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 2 + 10·z + 9·z2 }→ s21 :|: s21 >= 0, s21 <= 1 * z, z >= 0
isortlist#1(z) -{ 11 + 10·@xs + 9·@xs2 + 9·s22 }→ s23 :|: s22 >= 0, s22 <= 1 * @xs, s23 >= 0, s23 <= 1 * @x + 1 * s22 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {isortlist}
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: O(n1) [3 + 7·z'], size: O(1) [2]
insert#2: runtime: O(n1) [12 + 9·z1], size: O(n1) [2 + z' + z'' + z1]
insert: runtime: O(n1) [8 + 9·z'], size: O(n1) [1 + z + z']
insert#1: runtime: O(n1) [7 + 9·z], size: O(n1) [1 + z + z']
isortlist#1: runtime: O(n2) [1 + 10·z + 9·z2], size: O(n1) [z]
isortlist: runtime: ?, size: O(n1) [z]

(97) IntTrsBoundProof (UPPER BOUND(ID) transformation)


Computed RUNTIME bound using KoAT for: isortlist
after applying outer abstraction to obtain an ITS,
resulting in: O(n2) with polynomial bound: 2 + 10·z + 9·z2

(98) Obligation:

Complexity RNTS consisting of the following rules:

#and(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#and(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#and(z, z') -{ 0 }→ 1 :|: z' = 2, z = 1
#and(z, z') -{ 0 }→ 1 :|: z = 2, z' = 1
#and(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#cklt(z) -{ 0 }→ 2 :|: z = 3
#cklt(z) -{ 0 }→ 1 :|: z = 1
#cklt(z) -{ 0 }→ 1 :|: z = 2
#cklt(z) -{ 0 }→ 0 :|: z >= 0
#compare(z, z') -{ 0 }→ s :|: s >= 0, s <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ s' :|: s' >= 0, s' <= 3, z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 3 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z = 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' = 0
#compare(z, z') -{ 0 }→ 2 :|: z - 1 >= 0, z' - 1 >= 0
#compare(z, z') -{ 0 }→ 1 :|: z = 0, z' = 0
#compare(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#eq(z, z') -{ 0 }→ s10 :|: s8 >= 0, s8 <= 2, s9 >= 0, s9 <= 2, s10 >= 0, s10 <= 2, @x_1 >= 0, z = 1 + @x_1 + @x_2, @y_1 >= 0, @x_2 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ s7 :|: s7 >= 0, s7 <= 2, z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 2 :|: z = 0, z' = 0
#eq(z, z') -{ 0 }→ 2 :|: z = 1, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' = 0
#eq(z, z') -{ 0 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0
#eq(z, z') -{ 0 }→ 1 :|: @x_1 >= 0, z = 1 + @x_1 + @x_2, @x_2 >= 0, z' = 1
#eq(z, z') -{ 0 }→ 1 :|: z = 1, @y_1 >= 0, @y_2 >= 0, z' = 1 + @y_1 + @y_2
#eq(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
#equal(z, z') -{ 1 }→ s6 :|: s6 >= 0, s6 <= 2, z >= 0, z' >= 0
#less(z, z') -{ 1 }→ s3 :|: s3 >= 0, s3 <= 2, s'' >= 0, s'' <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ s4 :|: s4 >= 0, s4 <= 2, s1 >= 0, s1 <= 3, z' - 1 >= 0, z - 1 >= 0
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z - 1 >= 0, z' = 0, 3 = 3
#less(z, z') -{ 1 }→ 2 :|: z' - 1 >= 0, z - 1 >= 0, 3 = 3
#less(z, z') -{ 1 }→ 1 :|: z = 0, z' = 0, 1 = 1
#less(z, z') -{ 1 }→ 1 :|: z' - 1 >= 0, z = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' = 0, 2 = 2
#less(z, z') -{ 1 }→ 1 :|: z - 1 >= 0, z' - 1 >= 0, 2 = 2
#less(z, z') -{ 1 }→ 0 :|: z = 0, z' = 0, v0 >= 0, 1 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z' - 1 >= 0, z - 1 >= 0, v0 >= 0, 3 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' = 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 2 = v0
#less(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0, v0 >= 0, 0 = v0
#or(z, z') -{ 0 }→ 2 :|: z' = 2, z = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 1
#or(z, z') -{ 0 }→ 2 :|: z = 2, z' = 2
#or(z, z') -{ 0 }→ 1 :|: z = 1, z' = 1
#or(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
and(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z' = 2, z = 1
and(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 2, z' = 1
and(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0
insert(z, z') -{ 8 + 9·z' }→ s18 :|: s18 >= 0, s18 <= 1 * z' + 1 * z + 1, z' >= 0, z >= 0
insert#1(z, z') -{ 16 + 7·@y + 9·@ys }→ s19 :|: s19 >= 0, s19 <= 1 * @y + 1 * @ys + 1 * z' + 2, s12 >= 0, s12 <= 2, z = 1 + @y + @ys, z' >= 0, @y >= 0, @ys >= 0
insert#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
insert#1(z, z') -{ 1 }→ 1 + z' + 1 :|: z' >= 0, z = 1
insert#2(z, z', z'', z1) -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 1 }→ 1 + z' + (1 + z'' + z1) :|: z = 2, z' >= 0, z'' >= 0, z1 >= 0
insert#2(z, z', z'', z1) -{ 9 + 9·z1 }→ 1 + z'' + s20 :|: s20 >= 0, s20 <= 1 * z' + 1 * z1 + 1, z' >= 0, z = 1, z'' >= 0, z1 >= 0
isortlist(z) -{ 2 + 10·z + 9·z2 }→ s21 :|: s21 >= 0, s21 <= 1 * z, z >= 0
isortlist#1(z) -{ 11 + 10·@xs + 9·@xs2 + 9·s22 }→ s23 :|: s22 >= 0, s22 <= 1 * @xs, s23 >= 0, s23 <= 1 * @x + 1 * s22 + 1, @x >= 0, z = 1 + @x + @xs, @xs >= 0
isortlist#1(z) -{ 1 }→ 1 :|: z = 1
isortlist#1(z) -{ 0 }→ 0 :|: z >= 0
leq(z, z') -{ 3 + 7·z' }→ s13 :|: s13 >= 0, s13 <= 2, z >= 0, z' >= 0
leq#1(z, z') -{ 2 + 7·z' }→ s14 :|: s14 >= 0, s14 <= 2, @x >= 0, z = 1 + @x + @xs, z' >= 0, @xs >= 0
leq#1(z, z') -{ 1 }→ 2 :|: z = 1, z' >= 0
leq#1(z, z') -{ 0 }→ 0 :|: z >= 0, z' >= 0
leq#2(z, z', z'') -{ 8 + 7·@ys }→ s17 :|: s15 >= 0, s15 <= 2, s16 >= 0, s16 <= 2, s17 >= 0, s17 <= 2, s11 >= 0, s11 <= 2, s5 >= 0, s5 <= 2, s2 >= 0, s2 <= 3, z = 1 + @y + @ys, z' >= 0, z'' >= 0, @y >= 0, @ys >= 0
leq#2(z, z', z'') -{ 1 }→ 1 :|: z' >= 0, z = 1, z'' >= 0
leq#2(z, z', z'') -{ 0 }→ 0 :|: z >= 0, z' >= 0, z'' >= 0
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z' = 2, z = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 1
or(z, z') -{ 1 }→ 2 :|: z >= 0, z' >= 0, z = 2, z' = 2
or(z, z') -{ 1 }→ 1 :|: z >= 0, z' >= 0, z = 1, z' = 1
or(z, z') -{ 1 }→ 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed:
Previous analysis results are:
#compare: runtime: O(1) [0], size: O(1) [3]
#or: runtime: O(1) [0], size: O(1) [2]
#and: runtime: O(1) [0], size: O(1) [2]
and: runtime: O(1) [1], size: O(1) [2]
#cklt: runtime: O(1) [0], size: O(1) [2]
or: runtime: O(1) [1], size: O(1) [2]
#less: runtime: O(1) [1], size: O(1) [2]
#eq: runtime: O(1) [0], size: O(1) [2]
#equal: runtime: O(1) [1], size: O(1) [2]
leq#1: runtime: O(n1) [2 + 7·z'], size: O(1) [2]
leq#2: runtime: O(n1) [1 + 7·z], size: O(1) [2]
leq: runtime: O(n1) [3 + 7·z'], size: O(1) [2]
insert#2: runtime: O(n1) [12 + 9·z1], size: O(n1) [2 + z' + z'' + z1]
insert: runtime: O(n1) [8 + 9·z'], size: O(n1) [1 + z + z']
insert#1: runtime: O(n1) [7 + 9·z], size: O(n1) [1 + z + z']
isortlist#1: runtime: O(n2) [1 + 10·z + 9·z2], size: O(n1) [z]
isortlist: runtime: O(n2) [2 + 10·z + 9·z2], size: O(n1) [z]

(99) FinalProof (EQUIVALENT transformation)

Computed overall runtime complexity

(100) BOUNDS(1, n^2)