/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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 (innermost) 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) 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), 308 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), 78 ms]
        (14) typed CpxTrs


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

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


The TRS R consists of the following rules:

   eq(0, 0) -> true
   eq(0, s(x)) -> false
   eq(s(x), 0) -> false
   eq(s(x), s(y)) -> eq(x, y)
   or(true, y) -> true
   or(false, y) -> y
   union(empty, h) -> h
   union(edge(x, y, i), h) -> edge(x, y, union(i, h))
   reach(x, y, empty, h) -> false
   reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
   if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
   if_reach_2(true, x, y, edge(u, v, i), h) -> true
   if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
   if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))

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

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

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


The TRS R consists of the following rules:

   eq(0', 0') -> true
   eq(0', s(x)) -> false
   eq(s(x), 0') -> false
   eq(s(x), s(y)) -> eq(x, y)
   or(true, y) -> true
   or(false, y) -> y
   union(empty, h) -> h
   union(edge(x, y, i), h) -> edge(x, y, union(i, h))
   reach(x, y, empty, h) -> false
   reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
   if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
   if_reach_2(true, x, y, edge(u, v, i), h) -> true
   if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
   if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))

S is empty.
Rewrite Strategy: INNERMOST
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(3) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(4)
Obligation:
Innermost TRS:
Rules:
eq(0', 0') -> true
eq(0', s(x)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) -> true
if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
or :: true:false -> true:false -> true:false
union :: empty:edge -> empty:edge -> empty:edge
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
reach :: 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if_reach_1 :: true:false -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if_reach_2 :: true:false -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat -> 0':s
gen_empty:edge5_0 :: Nat -> empty:edge

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

(5) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
eq, union, reach

They will be analysed ascendingly in the following order:
eq < reach
union < reach

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

(6)
Obligation:
Innermost TRS:
Rules:
eq(0', 0') -> true
eq(0', s(x)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) -> true
if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
or :: true:false -> true:false -> true:false
union :: empty:edge -> empty:edge -> empty:edge
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
reach :: 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if_reach_1 :: true:false -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if_reach_2 :: true:false -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat -> 0':s
gen_empty:edge5_0 :: Nat -> empty:edge


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_empty:edge5_0(0) <=> empty
gen_empty:edge5_0(+(x, 1)) <=> edge(0', 0', gen_empty:edge5_0(x))


The following defined symbols remain to be analysed:
eq, union, reach

They will be analysed ascendingly in the following order:
eq < reach
union < reach

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(7) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)

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

Induction Step:
eq(gen_0':s4_0(+(n7_0, 1)), gen_0':s4_0(+(n7_0, 1))) ->_R^Omega(1)
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) ->_IH
true

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:

Innermost TRS:
Rules:
eq(0', 0') -> true
eq(0', s(x)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) -> true
if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
or :: true:false -> true:false -> true:false
union :: empty:edge -> empty:edge -> empty:edge
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
reach :: 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if_reach_1 :: true:false -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if_reach_2 :: true:false -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat -> 0':s
gen_empty:edge5_0 :: Nat -> empty:edge


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_empty:edge5_0(0) <=> empty
gen_empty:edge5_0(+(x, 1)) <=> edge(0', 0', gen_empty:edge5_0(x))


The following defined symbols remain to be analysed:
eq, union, reach

They will be analysed ascendingly in the following order:
eq < reach
union < reach

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
Innermost TRS:
Rules:
eq(0', 0') -> true
eq(0', s(x)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) -> true
if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
or :: true:false -> true:false -> true:false
union :: empty:edge -> empty:edge -> empty:edge
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
reach :: 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if_reach_1 :: true:false -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if_reach_2 :: true:false -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat -> 0':s
gen_empty:edge5_0 :: Nat -> empty:edge


Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_empty:edge5_0(0) <=> empty
gen_empty:edge5_0(+(x, 1)) <=> edge(0', 0', gen_empty:edge5_0(x))


The following defined symbols remain to be analysed:
union, reach

They will be analysed ascendingly in the following order:
union < reach

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
union(gen_empty:edge5_0(n494_0), gen_empty:edge5_0(b)) -> gen_empty:edge5_0(+(n494_0, b)), rt in Omega(1 + n494_0)

Induction Base:
union(gen_empty:edge5_0(0), gen_empty:edge5_0(b)) ->_R^Omega(1)
gen_empty:edge5_0(b)

Induction Step:
union(gen_empty:edge5_0(+(n494_0, 1)), gen_empty:edge5_0(b)) ->_R^Omega(1)
edge(0', 0', union(gen_empty:edge5_0(n494_0), gen_empty:edge5_0(b))) ->_IH
edge(0', 0', gen_empty:edge5_0(+(b, c495_0)))

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

(14)
Obligation:
Innermost TRS:
Rules:
eq(0', 0') -> true
eq(0', s(x)) -> false
eq(s(x), 0') -> false
eq(s(x), s(y)) -> eq(x, y)
or(true, y) -> true
or(false, y) -> y
union(empty, h) -> h
union(edge(x, y, i), h) -> edge(x, y, union(i, h))
reach(x, y, empty, h) -> false
reach(x, y, edge(u, v, i), h) -> if_reach_1(eq(x, u), x, y, edge(u, v, i), h)
if_reach_1(true, x, y, edge(u, v, i), h) -> if_reach_2(eq(y, v), x, y, edge(u, v, i), h)
if_reach_2(true, x, y, edge(u, v, i), h) -> true
if_reach_2(false, x, y, edge(u, v, i), h) -> or(reach(x, y, i, h), reach(v, y, union(i, h), empty))
if_reach_1(false, x, y, edge(u, v, i), h) -> reach(x, y, i, edge(u, v, h))

Types:
eq :: 0':s -> 0':s -> true:false
0' :: 0':s
true :: true:false
s :: 0':s -> 0':s
false :: true:false
or :: true:false -> true:false -> true:false
union :: empty:edge -> empty:edge -> empty:edge
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
reach :: 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if_reach_1 :: true:false -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if_reach_2 :: true:false -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
hole_true:false1_0 :: true:false
hole_0':s2_0 :: 0':s
hole_empty:edge3_0 :: empty:edge
gen_0':s4_0 :: Nat -> 0':s
gen_empty:edge5_0 :: Nat -> empty:edge


Lemmas:
eq(gen_0':s4_0(n7_0), gen_0':s4_0(n7_0)) -> true, rt in Omega(1 + n7_0)
union(gen_empty:edge5_0(n494_0), gen_empty:edge5_0(b)) -> gen_empty:edge5_0(+(n494_0, b)), rt in Omega(1 + n494_0)


Generator Equations:
gen_0':s4_0(0) <=> 0'
gen_0':s4_0(+(x, 1)) <=> s(gen_0':s4_0(x))
gen_empty:edge5_0(0) <=> empty
gen_empty:edge5_0(+(x, 1)) <=> edge(0', 0', gen_empty:edge5_0(x))


The following defined symbols remain to be analysed:
reach