/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^2), ?)
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^2, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) typed CpxTrs
(5) OrderProof [LOWER BOUND(ID), 0 ms]
(6) typed CpxTrs
(7) RewriteLemmaProof [LOWER BOUND(ID), 250 ms]
(8) BEST
    (9) proven lower bound
        (10) LowerBoundPropagationProof [FINISHED, 0 ms]
        (11) BOUNDS(n^1, INF)
    (12) typed CpxTrs
        (13) RewriteLemmaProof [LOWER BOUND(ID), 76 ms]
        (14) BEST
            (15) proven lower bound
                (16) LowerBoundPropagationProof [FINISHED, 0 ms]
                (17) BOUNDS(n^2, INF)
            (18) typed CpxTrs
                (19) RewriteLemmaProof [LOWER BOUND(ID), 46 ms]
                (20) typed CpxTrs
                (21) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
                (22) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^2, INF).


The TRS R consists of the following rules:

   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(*(x, y), x)
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   fact(x) -> iffact(x, ge(x, s(s(0))))
   iffact(x, true) -> *(x, fact(-(x, s(0))))
   iffact(x, false) -> s(0)

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^2, INF).


The TRS R consists of the following rules:

   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   *'(x, 0') -> 0'
   *'(x, s(y)) -> +'(*'(x, y), x)
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   fact(x) -> iffact(x, ge(x, s(s(0'))))
   iffact(x, true) -> *'(x, fact(-(x, s(0'))))
   iffact(x, false) -> s(0')

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

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

(4)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0'))))
iffact(x, true) -> *'(x, fact(-(x, s(0'))))
iffact(x, false) -> s(0')

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
- :: 0':s -> 0':s -> 0':s
fact :: 0':s -> 0':s
iffact :: 0':s -> true:false -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
+', *', ge, -, fact

They will be analysed ascendingly in the following order:
+' < *'
*' < fact
ge < fact
- < fact

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

(6)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0'))))
iffact(x, true) -> *'(x, fact(-(x, s(0'))))
iffact(x, false) -> s(0')

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
- :: 0':s -> 0':s -> 0':s
fact :: 0':s -> 0':s
iffact :: 0':s -> true:false -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
+', *', ge, -, fact

They will be analysed ascendingly in the following order:
+' < *'
*' < fact
ge < fact
- < fact

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

(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0)

Induction Base:
+'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(a)

Induction Step:
+'(gen_0':s3_0(a), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
s(+'(gen_0':s3_0(a), gen_0':s3_0(n5_0))) ->_IH
s(gen_0':s3_0(+(a, c6_0)))

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

(8)
Complex Obligation (BEST)

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

(9)
Obligation:
Proved the lower bound n^1 for the following obligation:

TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0'))))
iffact(x, true) -> *'(x, fact(-(x, s(0'))))
iffact(x, false) -> s(0')

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
- :: 0':s -> 0':s -> 0':s
fact :: 0':s -> 0':s
iffact :: 0':s -> true:false -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
+', *', ge, -, fact

They will be analysed ascendingly in the following order:
+' < *'
*' < fact
ge < fact
- < fact

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0'))))
iffact(x, true) -> *'(x, fact(-(x, s(0'))))
iffact(x, false) -> s(0')

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
- :: 0':s -> 0':s -> 0':s
fact :: 0':s -> 0':s
iffact :: 0':s -> true:false -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
*', ge, -, fact

They will be analysed ascendingly in the following order:
*' < fact
ge < fact
- < fact

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) -> gen_0':s3_0(*(n548_0, a)), rt in Omega(1 + a*n548_0 + n548_0)

Induction Base:
*'(gen_0':s3_0(a), gen_0':s3_0(0)) ->_R^Omega(1)
0'

Induction Step:
*'(gen_0':s3_0(a), gen_0':s3_0(+(n548_0, 1))) ->_R^Omega(1)
+'(*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)), gen_0':s3_0(a)) ->_IH
+'(gen_0':s3_0(*(c549_0, a)), gen_0':s3_0(a)) ->_L^Omega(1 + a)
gen_0':s3_0(+(a, *(n548_0, a)))

We have rt in Omega(n^2) and sz in O(n). Thus, we have irc_R in Omega(n^2).
----------------------------------------

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^2 for the following obligation:

TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0'))))
iffact(x, true) -> *'(x, fact(-(x, s(0'))))
iffact(x, false) -> s(0')

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
- :: 0':s -> 0':s -> 0':s
fact :: 0':s -> 0':s
iffact :: 0':s -> true:false -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
*', ge, -, fact

They will be analysed ascendingly in the following order:
*' < fact
ge < fact
- < fact

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

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

(17)
BOUNDS(n^2, INF)

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

(18)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0'))))
iffact(x, true) -> *'(x, fact(-(x, s(0'))))
iffact(x, false) -> s(0')

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
- :: 0':s -> 0':s -> 0':s
fact :: 0':s -> 0':s
iffact :: 0':s -> true:false -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0)
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) -> gen_0':s3_0(*(n548_0, a)), rt in Omega(1 + a*n548_0 + n548_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
ge, -, fact

They will be analysed ascendingly in the following order:
ge < fact
- < fact

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) -> true, rt in Omega(1 + n1208_0)

Induction Base:
ge(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
true

Induction Step:
ge(gen_0':s3_0(+(n1208_0, 1)), gen_0':s3_0(+(n1208_0, 1))) ->_R^Omega(1)
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) ->_IH
true

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

(20)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0'))))
iffact(x, true) -> *'(x, fact(-(x, s(0'))))
iffact(x, false) -> s(0')

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
- :: 0':s -> 0':s -> 0':s
fact :: 0':s -> 0':s
iffact :: 0':s -> true:false -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0)
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) -> gen_0':s3_0(*(n548_0, a)), rt in Omega(1 + a*n548_0 + n548_0)
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) -> true, rt in Omega(1 + n1208_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
-, fact

They will be analysed ascendingly in the following order:
- < fact

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s3_0(n1478_0), gen_0':s3_0(n1478_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1478_0)

Induction Base:
-(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
-(gen_0':s3_0(+(n1478_0, 1)), gen_0':s3_0(+(n1478_0, 1))) ->_R^Omega(1)
-(gen_0':s3_0(n1478_0), gen_0':s3_0(n1478_0)) ->_IH
gen_0':s3_0(0)

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

(22)
Obligation:
TRS:
Rules:
+'(x, 0') -> x
+'(x, s(y)) -> s(+'(x, y))
*'(x, 0') -> 0'
*'(x, s(y)) -> +'(*'(x, y), x)
ge(x, 0') -> true
ge(0', s(y)) -> false
ge(s(x), s(y)) -> ge(x, y)
-(x, 0') -> x
-(s(x), s(y)) -> -(x, y)
fact(x) -> iffact(x, ge(x, s(s(0'))))
iffact(x, true) -> *'(x, fact(-(x, s(0'))))
iffact(x, false) -> s(0')

Types:
+' :: 0':s -> 0':s -> 0':s
0' :: 0':s
s :: 0':s -> 0':s
*' :: 0':s -> 0':s -> 0':s
ge :: 0':s -> 0':s -> true:false
true :: true:false
false :: true:false
- :: 0':s -> 0':s -> 0':s
fact :: 0':s -> 0':s
iffact :: 0':s -> true:false -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
+'(gen_0':s3_0(a), gen_0':s3_0(n5_0)) -> gen_0':s3_0(+(n5_0, a)), rt in Omega(1 + n5_0)
*'(gen_0':s3_0(a), gen_0':s3_0(n548_0)) -> gen_0':s3_0(*(n548_0, a)), rt in Omega(1 + a*n548_0 + n548_0)
ge(gen_0':s3_0(n1208_0), gen_0':s3_0(n1208_0)) -> true, rt in Omega(1 + n1208_0)
-(gen_0':s3_0(n1478_0), gen_0':s3_0(n1478_0)) -> gen_0':s3_0(0), rt in Omega(1 + n1478_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
fact