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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) SlicingProof [LOWER BOUND(ID), 0 ms]
(4) CpxTRS
(5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) typed CpxTrs
(7) OrderProof [LOWER BOUND(ID), 0 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 270 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   lt(0, s(X)) -> true
   lt(s(X), 0) -> false
   lt(s(X), s(Y)) -> lt(X, Y)
   append(nil, Y) -> Y
   append(add(N, X), Y) -> add(N, append(X, Y))
   split(N, nil) -> pair(nil, nil)
   split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y)
   f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z)
   f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z))
   f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z)
   qsort(nil) -> nil
   qsort(add(N, X)) -> f_3(split(N, X), N, X)
   f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z)))

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

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

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


The TRS R consists of the following rules:

   lt(0', s(X)) -> true
   lt(s(X), 0') -> false
   lt(s(X), s(Y)) -> lt(X, Y)
   append(nil, Y) -> Y
   append(add(N, X), Y) -> add(N, append(X, Y))
   split(N, nil) -> pair(nil, nil)
   split(N, add(M, Y)) -> f_1(split(N, Y), N, M, Y)
   f_1(pair(X, Z), N, M, Y) -> f_2(lt(N, M), N, M, Y, X, Z)
   f_2(true, N, M, Y, X, Z) -> pair(X, add(M, Z))
   f_2(false, N, M, Y, X, Z) -> pair(add(M, X), Z)
   qsort(nil) -> nil
   qsort(add(N, X)) -> f_3(split(N, X), N, X)
   f_3(pair(Y, Z), N, X) -> append(qsort(Y), add(X, qsort(Z)))

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
f_1/3
f_2/1
f_2/3
f_3/1

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

(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   lt(0', s(X)) -> true
   lt(s(X), 0') -> false
   lt(s(X), s(Y)) -> lt(X, Y)
   append(nil, Y) -> Y
   append(add(N, X), Y) -> add(N, append(X, Y))
   split(N, nil) -> pair(nil, nil)
   split(N, add(M, Y)) -> f_1(split(N, Y), N, M)
   f_1(pair(X, Z), N, M) -> f_2(lt(N, M), M, X, Z)
   f_2(true, M, X, Z) -> pair(X, add(M, Z))
   f_2(false, M, X, Z) -> pair(add(M, X), Z)
   qsort(nil) -> nil
   qsort(add(N, X)) -> f_3(split(N, X), X)
   f_3(pair(Y, Z), X) -> append(qsort(Y), add(X, qsort(Z)))

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

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

(6)
Obligation:
TRS:
Rules:
lt(0', s(X)) -> true
lt(s(X), 0') -> false
lt(s(X), s(Y)) -> lt(X, Y)
append(nil, Y) -> Y
append(add(N, X), Y) -> add(N, append(X, Y))
split(N, nil) -> pair(nil, nil)
split(N, add(M, Y)) -> f_1(split(N, Y), N, M)
f_1(pair(X, Z), N, M) -> f_2(lt(N, M), M, X, Z)
f_2(true, M, X, Z) -> pair(X, add(M, Z))
f_2(false, M, X, Z) -> pair(add(M, X), Z)
qsort(nil) -> nil
qsort(add(N, X)) -> f_3(split(N, X), X)
f_3(pair(Y, Z), X) -> append(qsort(Y), add(X, qsort(Z)))

Types:
lt :: 0':s:nil:add -> 0':s:nil:add -> true:false
0' :: 0':s:nil:add
s :: 0':s:nil:add -> 0':s:nil:add
true :: true:false
false :: true:false
append :: 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add
nil :: 0':s:nil:add
add :: 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add
split :: 0':s:nil:add -> 0':s:nil:add -> pair
pair :: 0':s:nil:add -> 0':s:nil:add -> pair
f_1 :: pair -> 0':s:nil:add -> 0':s:nil:add -> pair
f_2 :: true:false -> 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add -> pair
qsort :: 0':s:nil:add -> 0':s:nil:add
f_3 :: pair -> 0':s:nil:add -> 0':s:nil:add
hole_true:false1_0 :: true:false
hole_0':s:nil:add2_0 :: 0':s:nil:add
hole_pair3_0 :: pair
gen_0':s:nil:add4_0 :: Nat -> 0':s:nil:add

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
lt, append, split, qsort

They will be analysed ascendingly in the following order:
append < qsort
split < qsort

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

(8)
Obligation:
TRS:
Rules:
lt(0', s(X)) -> true
lt(s(X), 0') -> false
lt(s(X), s(Y)) -> lt(X, Y)
append(nil, Y) -> Y
append(add(N, X), Y) -> add(N, append(X, Y))
split(N, nil) -> pair(nil, nil)
split(N, add(M, Y)) -> f_1(split(N, Y), N, M)
f_1(pair(X, Z), N, M) -> f_2(lt(N, M), M, X, Z)
f_2(true, M, X, Z) -> pair(X, add(M, Z))
f_2(false, M, X, Z) -> pair(add(M, X), Z)
qsort(nil) -> nil
qsort(add(N, X)) -> f_3(split(N, X), X)
f_3(pair(Y, Z), X) -> append(qsort(Y), add(X, qsort(Z)))

Types:
lt :: 0':s:nil:add -> 0':s:nil:add -> true:false
0' :: 0':s:nil:add
s :: 0':s:nil:add -> 0':s:nil:add
true :: true:false
false :: true:false
append :: 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add
nil :: 0':s:nil:add
add :: 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add
split :: 0':s:nil:add -> 0':s:nil:add -> pair
pair :: 0':s:nil:add -> 0':s:nil:add -> pair
f_1 :: pair -> 0':s:nil:add -> 0':s:nil:add -> pair
f_2 :: true:false -> 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add -> pair
qsort :: 0':s:nil:add -> 0':s:nil:add
f_3 :: pair -> 0':s:nil:add -> 0':s:nil:add
hole_true:false1_0 :: true:false
hole_0':s:nil:add2_0 :: 0':s:nil:add
hole_pair3_0 :: pair
gen_0':s:nil:add4_0 :: Nat -> 0':s:nil:add


Generator Equations:
gen_0':s:nil:add4_0(0) <=> 0'
gen_0':s:nil:add4_0(+(x, 1)) <=> s(gen_0':s:nil:add4_0(x))


The following defined symbols remain to be analysed:
lt, append, split, qsort

They will be analysed ascendingly in the following order:
append < qsort
split < qsort

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s:nil:add4_0(n6_0), gen_0':s:nil:add4_0(+(1, n6_0))) -> true, rt in Omega(1 + n6_0)

Induction Base:
lt(gen_0':s:nil:add4_0(0), gen_0':s:nil:add4_0(+(1, 0))) ->_R^Omega(1)
true

Induction Step:
lt(gen_0':s:nil:add4_0(+(n6_0, 1)), gen_0':s:nil:add4_0(+(1, +(n6_0, 1)))) ->_R^Omega(1)
lt(gen_0':s:nil:add4_0(n6_0), gen_0':s:nil:add4_0(+(1, n6_0))) ->_IH
true

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(10)
Complex Obligation (BEST)

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

(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
lt(0', s(X)) -> true
lt(s(X), 0') -> false
lt(s(X), s(Y)) -> lt(X, Y)
append(nil, Y) -> Y
append(add(N, X), Y) -> add(N, append(X, Y))
split(N, nil) -> pair(nil, nil)
split(N, add(M, Y)) -> f_1(split(N, Y), N, M)
f_1(pair(X, Z), N, M) -> f_2(lt(N, M), M, X, Z)
f_2(true, M, X, Z) -> pair(X, add(M, Z))
f_2(false, M, X, Z) -> pair(add(M, X), Z)
qsort(nil) -> nil
qsort(add(N, X)) -> f_3(split(N, X), X)
f_3(pair(Y, Z), X) -> append(qsort(Y), add(X, qsort(Z)))

Types:
lt :: 0':s:nil:add -> 0':s:nil:add -> true:false
0' :: 0':s:nil:add
s :: 0':s:nil:add -> 0':s:nil:add
true :: true:false
false :: true:false
append :: 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add
nil :: 0':s:nil:add
add :: 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add
split :: 0':s:nil:add -> 0':s:nil:add -> pair
pair :: 0':s:nil:add -> 0':s:nil:add -> pair
f_1 :: pair -> 0':s:nil:add -> 0':s:nil:add -> pair
f_2 :: true:false -> 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add -> pair
qsort :: 0':s:nil:add -> 0':s:nil:add
f_3 :: pair -> 0':s:nil:add -> 0':s:nil:add
hole_true:false1_0 :: true:false
hole_0':s:nil:add2_0 :: 0':s:nil:add
hole_pair3_0 :: pair
gen_0':s:nil:add4_0 :: Nat -> 0':s:nil:add


Generator Equations:
gen_0':s:nil:add4_0(0) <=> 0'
gen_0':s:nil:add4_0(+(x, 1)) <=> s(gen_0':s:nil:add4_0(x))


The following defined symbols remain to be analysed:
lt, append, split, qsort

They will be analysed ascendingly in the following order:
append < qsort
split < qsort

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
lt(0', s(X)) -> true
lt(s(X), 0') -> false
lt(s(X), s(Y)) -> lt(X, Y)
append(nil, Y) -> Y
append(add(N, X), Y) -> add(N, append(X, Y))
split(N, nil) -> pair(nil, nil)
split(N, add(M, Y)) -> f_1(split(N, Y), N, M)
f_1(pair(X, Z), N, M) -> f_2(lt(N, M), M, X, Z)
f_2(true, M, X, Z) -> pair(X, add(M, Z))
f_2(false, M, X, Z) -> pair(add(M, X), Z)
qsort(nil) -> nil
qsort(add(N, X)) -> f_3(split(N, X), X)
f_3(pair(Y, Z), X) -> append(qsort(Y), add(X, qsort(Z)))

Types:
lt :: 0':s:nil:add -> 0':s:nil:add -> true:false
0' :: 0':s:nil:add
s :: 0':s:nil:add -> 0':s:nil:add
true :: true:false
false :: true:false
append :: 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add
nil :: 0':s:nil:add
add :: 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add
split :: 0':s:nil:add -> 0':s:nil:add -> pair
pair :: 0':s:nil:add -> 0':s:nil:add -> pair
f_1 :: pair -> 0':s:nil:add -> 0':s:nil:add -> pair
f_2 :: true:false -> 0':s:nil:add -> 0':s:nil:add -> 0':s:nil:add -> pair
qsort :: 0':s:nil:add -> 0':s:nil:add
f_3 :: pair -> 0':s:nil:add -> 0':s:nil:add
hole_true:false1_0 :: true:false
hole_0':s:nil:add2_0 :: 0':s:nil:add
hole_pair3_0 :: pair
gen_0':s:nil:add4_0 :: Nat -> 0':s:nil:add


Lemmas:
lt(gen_0':s:nil:add4_0(n6_0), gen_0':s:nil:add4_0(+(1, n6_0))) -> true, rt in Omega(1 + n6_0)


Generator Equations:
gen_0':s:nil:add4_0(0) <=> 0'
gen_0':s:nil:add4_0(+(x, 1)) <=> s(gen_0':s:nil:add4_0(x))


The following defined symbols remain to be analysed:
append, split, qsort

They will be analysed ascendingly in the following order:
append < qsort
split < qsort