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


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WORST_CASE(Omega(n^1), ?)
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, 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), 5 ms]
(8) typed CpxTrs
(9) RewriteLemmaProof [LOWER BOUND(ID), 417 ms]
(10) BEST
    (11) proven lower bound
        (12) LowerBoundPropagationProof [FINISHED, 0 ms]
        (13) BOUNDS(n^1, INF)
    (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 160 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 148 ms]
        (18) BOUNDS(1, INF)


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

(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:

   g(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldB(t, 0) -> t
   foldB(t, s(n)) -> f(foldB(t, n), B)
   foldC(t, 0) -> t
   foldC(t, s(n)) -> f(foldC(t, n), C)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, s(c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0, c), b))
   f''(triple(a, b, c)) -> foldC(triple(a, b, 0), c)

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:

   g(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldB(t, 0') -> t
   foldB(t, s(n)) -> f(foldB(t, n), B)
   foldC(t, 0') -> t
   foldC(t, s(n)) -> f(foldC(t, n), C)
   f(t, x) -> f'(t, g(x))
   f'(triple(a, b, c), C) -> triple(a, b, s(c))
   f'(triple(a, b, c), B) -> f(triple(a, b, c), A)
   f'(triple(a, b, c), A) -> f''(foldB(triple(s(a), 0', c), b))
   f''(triple(a, b, c)) -> foldC(triple(a, b, 0'), c)

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

(3) SlicingProof (LOWER BOUND(ID))
Sliced the following arguments:
triple/0

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

(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:

   g(A) -> A
   g(B) -> A
   g(B) -> B
   g(C) -> A
   g(C) -> B
   g(C) -> C
   foldB(t, 0') -> t
   foldB(t, s(n)) -> f(foldB(t, n), B)
   foldC(t, 0') -> t
   foldC(t, s(n)) -> f(foldC(t, n), C)
   f(t, x) -> f'(t, g(x))
   f'(triple(b, c), C) -> triple(b, s(c))
   f'(triple(b, c), B) -> f(triple(b, c), A)
   f'(triple(b, c), A) -> f''(foldB(triple(0', c), b))
   f''(triple(b, c)) -> foldC(triple(b, 0'), c)

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

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

(6)
Obligation:
TRS:
Rules:
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0') -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0') -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(b, c), C) -> triple(b, s(c))
f'(triple(b, c), B) -> f(triple(b, c), A)
f'(triple(b, c), A) -> f''(foldB(triple(0', c), b))
f''(triple(b, c)) -> foldC(triple(b, 0'), c)

Types:
g :: A:B:C -> A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldB :: triple -> 0':s -> triple
0' :: 0':s
s :: 0':s -> 0':s
f :: triple -> A:B:C -> triple
foldC :: triple -> 0':s -> triple
f' :: triple -> A:B:C -> triple
triple :: 0':s -> 0':s -> triple
f'' :: triple -> triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s

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

(7) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
foldB, f, foldC, f', f''

They will be analysed ascendingly in the following order:
foldB = f
foldB = foldC
foldB = f'
foldB = f''
f = foldC
f = f'
f = f''
foldC = f'
foldC = f''
f' = f''

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

(8)
Obligation:
TRS:
Rules:
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0') -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0') -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(b, c), C) -> triple(b, s(c))
f'(triple(b, c), B) -> f(triple(b, c), A)
f'(triple(b, c), A) -> f''(foldB(triple(0', c), b))
f''(triple(b, c)) -> foldC(triple(b, 0'), c)

Types:
g :: A:B:C -> A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldB :: triple -> 0':s -> triple
0' :: 0':s
s :: 0':s -> 0':s
f :: triple -> A:B:C -> triple
foldC :: triple -> 0':s -> triple
f' :: triple -> A:B:C -> triple
triple :: 0':s -> 0':s -> triple
f'' :: triple -> triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
f, foldB, foldC, f', f''

They will be analysed ascendingly in the following order:
foldB = f
foldB = foldC
foldB = f'
foldB = f''
f = foldC
f = f'
f = f''
foldC = f'
foldC = f''
f' = f''

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

(9) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
foldC(triple(0', 0'), gen_0':s4_0(n143_0)) -> triple(gen_0':s4_0(0), gen_0':s4_0(0)), rt in Omega(1 + n143_0)

Induction Base:
foldC(triple(0', 0'), gen_0':s4_0(0)) ->_R^Omega(1)
triple(0', 0')

Induction Step:
foldC(triple(0', 0'), gen_0':s4_0(+(n143_0, 1))) ->_R^Omega(1)
f(foldC(triple(0', 0'), gen_0':s4_0(n143_0)), C) ->_IH
f(triple(gen_0':s4_0(0), gen_0':s4_0(0)), C) ->_R^Omega(1)
f'(triple(gen_0':s4_0(0), gen_0':s4_0(0)), g(C)) ->_R^Omega(1)
f'(triple(gen_0':s4_0(0), gen_0':s4_0(0)), A) ->_R^Omega(1)
f''(foldB(triple(0', gen_0':s4_0(0)), gen_0':s4_0(0))) ->_R^Omega(1)
f''(triple(0', gen_0':s4_0(0))) ->_R^Omega(1)
foldC(triple(0', 0'), gen_0':s4_0(0)) ->_R^Omega(1)
triple(0', 0')

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:
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0') -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0') -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(b, c), C) -> triple(b, s(c))
f'(triple(b, c), B) -> f(triple(b, c), A)
f'(triple(b, c), A) -> f''(foldB(triple(0', c), b))
f''(triple(b, c)) -> foldC(triple(b, 0'), c)

Types:
g :: A:B:C -> A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldB :: triple -> 0':s -> triple
0' :: 0':s
s :: 0':s -> 0':s
f :: triple -> A:B:C -> triple
foldC :: triple -> 0':s -> triple
f' :: triple -> A:B:C -> triple
triple :: 0':s -> 0':s -> triple
f'' :: triple -> triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
foldC, foldB

They will be analysed ascendingly in the following order:
foldB = f
foldB = foldC
foldB = f'
foldB = f''
f = foldC
f = f'
f = f''
foldC = f'
foldC = f''
f' = f''

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

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

(13)
BOUNDS(n^1, INF)

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

(14)
Obligation:
TRS:
Rules:
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0') -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0') -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(b, c), C) -> triple(b, s(c))
f'(triple(b, c), B) -> f(triple(b, c), A)
f'(triple(b, c), A) -> f''(foldB(triple(0', c), b))
f''(triple(b, c)) -> foldC(triple(b, 0'), c)

Types:
g :: A:B:C -> A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldB :: triple -> 0':s -> triple
0' :: 0':s
s :: 0':s -> 0':s
f :: triple -> A:B:C -> triple
foldC :: triple -> 0':s -> triple
f' :: triple -> A:B:C -> triple
triple :: 0':s -> 0':s -> triple
f'' :: triple -> triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
foldC(triple(0', 0'), gen_0':s4_0(n143_0)) -> triple(gen_0':s4_0(0), gen_0':s4_0(0)), rt in Omega(1 + n143_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
foldB, f, f', f''

They will be analysed ascendingly in the following order:
foldB = f
foldB = foldC
foldB = f'
foldB = f''
f = foldC
f = f'
f = f''
foldC = f'
foldC = f''
f' = f''

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
foldB(triple(0', 0'), gen_0':s4_0(n2037_0)) -> triple(gen_0':s4_0(0), gen_0':s4_0(0)), rt in Omega(1 + n2037_0)

Induction Base:
foldB(triple(0', 0'), gen_0':s4_0(0)) ->_R^Omega(1)
triple(0', 0')

Induction Step:
foldB(triple(0', 0'), gen_0':s4_0(+(n2037_0, 1))) ->_R^Omega(1)
f(foldB(triple(0', 0'), gen_0':s4_0(n2037_0)), B) ->_IH
f(triple(gen_0':s4_0(0), gen_0':s4_0(0)), B) ->_R^Omega(1)
f'(triple(gen_0':s4_0(0), gen_0':s4_0(0)), g(B)) ->_R^Omega(1)
f'(triple(gen_0':s4_0(0), gen_0':s4_0(0)), A) ->_R^Omega(1)
f''(foldB(triple(0', gen_0':s4_0(0)), gen_0':s4_0(0))) ->_R^Omega(1)
f''(triple(0', gen_0':s4_0(0))) ->_R^Omega(1)
foldC(triple(0', 0'), gen_0':s4_0(0)) ->_L^Omega(1)
triple(gen_0':s4_0(0), gen_0':s4_0(0))

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

(16)
Obligation:
TRS:
Rules:
g(A) -> A
g(B) -> A
g(B) -> B
g(C) -> A
g(C) -> B
g(C) -> C
foldB(t, 0') -> t
foldB(t, s(n)) -> f(foldB(t, n), B)
foldC(t, 0') -> t
foldC(t, s(n)) -> f(foldC(t, n), C)
f(t, x) -> f'(t, g(x))
f'(triple(b, c), C) -> triple(b, s(c))
f'(triple(b, c), B) -> f(triple(b, c), A)
f'(triple(b, c), A) -> f''(foldB(triple(0', c), b))
f''(triple(b, c)) -> foldC(triple(b, 0'), c)

Types:
g :: A:B:C -> A:B:C
A :: A:B:C
B :: A:B:C
C :: A:B:C
foldB :: triple -> 0':s -> triple
0' :: 0':s
s :: 0':s -> 0':s
f :: triple -> A:B:C -> triple
foldC :: triple -> 0':s -> triple
f' :: triple -> A:B:C -> triple
triple :: 0':s -> 0':s -> triple
f'' :: triple -> triple
hole_A:B:C1_0 :: A:B:C
hole_triple2_0 :: triple
hole_0':s3_0 :: 0':s
gen_0':s4_0 :: Nat -> 0':s


Lemmas:
foldC(triple(0', 0'), gen_0':s4_0(n143_0)) -> triple(gen_0':s4_0(0), gen_0':s4_0(0)), rt in Omega(1 + n143_0)
foldB(triple(0', 0'), gen_0':s4_0(n2037_0)) -> triple(gen_0':s4_0(0), gen_0':s4_0(0)), rt in Omega(1 + n2037_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))


The following defined symbols remain to be analysed:
f, foldC, f', f''

They will be analysed ascendingly in the following order:
foldB = f
foldB = foldC
foldB = f'
foldB = f''
f = foldC
f = f'
f = f''
foldC = f'
foldC = f''
f' = f''

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
foldC(triple(0', 0'), gen_0':s4_0(n4048_0)) -> triple(gen_0':s4_0(0), gen_0':s4_0(0)), rt in Omega(1 + n4048_0)

Induction Base:
foldC(triple(0', 0'), gen_0':s4_0(0)) ->_R^Omega(1)
triple(0', 0')

Induction Step:
foldC(triple(0', 0'), gen_0':s4_0(+(n4048_0, 1))) ->_R^Omega(1)
f(foldC(triple(0', 0'), gen_0':s4_0(n4048_0)), C) ->_IH
f(triple(gen_0':s4_0(0), gen_0':s4_0(0)), C) ->_R^Omega(1)
f'(triple(gen_0':s4_0(0), gen_0':s4_0(0)), g(C)) ->_R^Omega(1)
f'(triple(gen_0':s4_0(0), gen_0':s4_0(0)), A) ->_R^Omega(1)
f''(foldB(triple(0', gen_0':s4_0(0)), gen_0':s4_0(0))) ->_L^Omega(1)
f''(triple(gen_0':s4_0(0), gen_0':s4_0(0))) ->_R^Omega(1)
foldC(triple(gen_0':s4_0(0), 0'), gen_0':s4_0(0)) ->_R^Omega(1)
triple(gen_0':s4_0(0), 0')

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

(18)
BOUNDS(1, INF)