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


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


WORST_CASE(Omega(n^1), ?)
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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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^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), 267 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), 11 ms]
        (14) typed CpxTrs
        (15) RewriteLemmaProof [LOWER BOUND(ID), 10 ms]
        (16) typed CpxTrs
        (17) RewriteLemmaProof [LOWER BOUND(ID), 1183 ms]
        (18) 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:

   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
   and(true, y) -> y
   and(false, y) -> false
   size(empty) -> 0
   size(edge(x, y, i)) -> s(size(i))
   le(0, y) -> true
   le(s(x), 0) -> false
   le(s(x), s(y)) -> le(x, y)
   reachable(x, y, i) -> reach(x, y, 0, i, i)
   reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
   if1(true, x, y, c, i, j) -> true
   if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
   if2(false, x, y, c, i, j) -> false
   if2(true, x, y, c, empty, j) -> false
   if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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:

   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
   and(true, y) -> y
   and(false, y) -> false
   size(empty) -> 0'
   size(edge(x, y, i)) -> s(size(i))
   le(0', y) -> true
   le(s(x), 0') -> false
   le(s(x), s(y)) -> le(x, y)
   reachable(x, y, i) -> reach(x, y, 0', i, i)
   reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
   if1(true, x, y, c, i, j) -> true
   if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
   if2(false, x, y, c, i, j) -> false
   if2(true, x, y, c, empty, j) -> false
   if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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

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

(4)
Obligation:
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
and(true, y) -> y
and(false, y) -> false
size(empty) -> 0'
size(edge(x, y, i)) -> s(size(i))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
reachable(x, y, i) -> reach(x, y, 0', i, i)
reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) -> true
if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) -> false
if2(true, x, y, c, empty, j) -> false
if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0':s
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
le :: 0':s -> 0':s -> true:false
reachable :: 0':s -> 0':s -> empty:edge -> true:false
reach :: 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0':s -> 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, size, le, reach, if2

They will be analysed ascendingly in the following order:
eq < reach
eq < if2
size < reach
le < reach
reach = if2

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

(6)
Obligation:
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
and(true, y) -> y
and(false, y) -> false
size(empty) -> 0'
size(edge(x, y, i)) -> s(size(i))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
reachable(x, y, i) -> reach(x, y, 0', i, i)
reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) -> true
if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) -> false
if2(true, x, y, c, empty, j) -> false
if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0':s
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
le :: 0':s -> 0':s -> true:false
reachable :: 0':s -> 0':s -> empty:edge -> true:false
reach :: 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0':s -> 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, size, le, reach, if2

They will be analysed ascendingly in the following order:
eq < reach
eq < if2
size < reach
le < reach
reach = if2

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

(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:

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
and(true, y) -> y
and(false, y) -> false
size(empty) -> 0'
size(edge(x, y, i)) -> s(size(i))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
reachable(x, y, i) -> reach(x, y, 0', i, i)
reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) -> true
if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) -> false
if2(true, x, y, c, empty, j) -> false
if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0':s
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
le :: 0':s -> 0':s -> true:false
reachable :: 0':s -> 0':s -> empty:edge -> true:false
reach :: 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0':s -> 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, size, le, reach, if2

They will be analysed ascendingly in the following order:
eq < reach
eq < if2
size < reach
le < reach
reach = if2

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

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

(11)
BOUNDS(n^1, INF)

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

(12)
Obligation:
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
and(true, y) -> y
and(false, y) -> false
size(empty) -> 0'
size(edge(x, y, i)) -> s(size(i))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
reachable(x, y, i) -> reach(x, y, 0', i, i)
reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) -> true
if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) -> false
if2(true, x, y, c, empty, j) -> false
if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0':s
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
le :: 0':s -> 0':s -> true:false
reachable :: 0':s -> 0':s -> empty:edge -> true:false
reach :: 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0':s -> 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:
size, le, reach, if2

They will be analysed ascendingly in the following order:
size < reach
le < reach
reach = if2

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
size(gen_empty:edge5_0(n530_0)) -> gen_0':s4_0(n530_0), rt in Omega(1 + n530_0)

Induction Base:
size(gen_empty:edge5_0(0)) ->_R^Omega(1)
0'

Induction Step:
size(gen_empty:edge5_0(+(n530_0, 1))) ->_R^Omega(1)
s(size(gen_empty:edge5_0(n530_0))) ->_IH
s(gen_0':s4_0(c531_0))

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

(14)
Obligation:
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
and(true, y) -> y
and(false, y) -> false
size(empty) -> 0'
size(edge(x, y, i)) -> s(size(i))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
reachable(x, y, i) -> reach(x, y, 0', i, i)
reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) -> true
if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) -> false
if2(true, x, y, c, empty, j) -> false
if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0':s
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
le :: 0':s -> 0':s -> true:false
reachable :: 0':s -> 0':s -> empty:edge -> true:false
reach :: 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0':s -> 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)
size(gen_empty:edge5_0(n530_0)) -> gen_0':s4_0(n530_0), rt in Omega(1 + n530_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:
le, reach, if2

They will be analysed ascendingly in the following order:
le < reach
reach = if2

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
le(gen_0':s4_0(n804_0), gen_0':s4_0(n804_0)) -> true, rt in Omega(1 + n804_0)

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

Induction Step:
le(gen_0':s4_0(+(n804_0, 1)), gen_0':s4_0(+(n804_0, 1))) ->_R^Omega(1)
le(gen_0':s4_0(n804_0), gen_0':s4_0(n804_0)) ->_IH
true

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

(16)
Obligation:
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
and(true, y) -> y
and(false, y) -> false
size(empty) -> 0'
size(edge(x, y, i)) -> s(size(i))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
reachable(x, y, i) -> reach(x, y, 0', i, i)
reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) -> true
if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) -> false
if2(true, x, y, c, empty, j) -> false
if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0':s
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
le :: 0':s -> 0':s -> true:false
reachable :: 0':s -> 0':s -> empty:edge -> true:false
reach :: 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0':s -> 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)
size(gen_empty:edge5_0(n530_0)) -> gen_0':s4_0(n530_0), rt in Omega(1 + n530_0)
le(gen_0':s4_0(n804_0), gen_0':s4_0(n804_0)) -> true, rt in Omega(1 + n804_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:
if2, reach

They will be analysed ascendingly in the following order:
reach = if2

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

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
if2(true, gen_0':s4_0(0), gen_0':s4_0(0), gen_0':s4_0(c), gen_empty:edge5_0(n1127_0), gen_empty:edge5_0(e)) -> *6_0, rt in Omega(n1127_0)

Induction Base:
if2(true, gen_0':s4_0(0), gen_0':s4_0(0), gen_0':s4_0(c), gen_empty:edge5_0(0), gen_empty:edge5_0(e))

Induction Step:
if2(true, gen_0':s4_0(0), gen_0':s4_0(0), gen_0':s4_0(c), gen_empty:edge5_0(+(n1127_0, 1)), gen_empty:edge5_0(e)) ->_R^Omega(1)
or(if2(true, gen_0':s4_0(0), gen_0':s4_0(0), gen_0':s4_0(c), gen_empty:edge5_0(n1127_0), gen_empty:edge5_0(e)), and(eq(gen_0':s4_0(0), 0'), reach(0', gen_0':s4_0(0), s(gen_0':s4_0(c)), gen_empty:edge5_0(e), gen_empty:edge5_0(e)))) ->_IH
or(*6_0, and(eq(gen_0':s4_0(0), 0'), reach(0', gen_0':s4_0(0), s(gen_0':s4_0(c)), gen_empty:edge5_0(e), gen_empty:edge5_0(e)))) ->_L^Omega(1)
or(*6_0, and(true, reach(0', gen_0':s4_0(0), s(gen_0':s4_0(c)), gen_empty:edge5_0(e), gen_empty:edge5_0(e)))) ->_R^Omega(1)
or(*6_0, and(true, if1(eq(0', gen_0':s4_0(0)), 0', gen_0':s4_0(0), s(gen_0':s4_0(c)), gen_empty:edge5_0(e), gen_empty:edge5_0(e)))) ->_L^Omega(1)
or(*6_0, and(true, if1(true, 0', gen_0':s4_0(0), s(gen_0':s4_0(c)), gen_empty:edge5_0(e), gen_empty:edge5_0(e)))) ->_R^Omega(1)
or(*6_0, and(true, true)) ->_R^Omega(1)
or(*6_0, true)

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

(18)
Obligation:
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
and(true, y) -> y
and(false, y) -> false
size(empty) -> 0'
size(edge(x, y, i)) -> s(size(i))
le(0', y) -> true
le(s(x), 0') -> false
le(s(x), s(y)) -> le(x, y)
reachable(x, y, i) -> reach(x, y, 0', i, i)
reach(x, y, c, i, j) -> if1(eq(x, y), x, y, c, i, j)
if1(true, x, y, c, i, j) -> true
if1(false, x, y, c, i, j) -> if2(le(c, size(j)), x, y, c, i, j)
if2(false, x, y, c, i, j) -> false
if2(true, x, y, c, empty, j) -> false
if2(true, x, y, c, edge(u, v, i), j) -> or(if2(true, x, y, c, i, j), and(eq(x, u), reach(v, y, s(c), j, j)))

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
and :: true:false -> true:false -> true:false
size :: empty:edge -> 0':s
empty :: empty:edge
edge :: 0':s -> 0':s -> empty:edge -> empty:edge
le :: 0':s -> 0':s -> true:false
reachable :: 0':s -> 0':s -> empty:edge -> true:false
reach :: 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if1 :: true:false -> 0':s -> 0':s -> 0':s -> empty:edge -> empty:edge -> true:false
if2 :: true:false -> 0':s -> 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)
size(gen_empty:edge5_0(n530_0)) -> gen_0':s4_0(n530_0), rt in Omega(1 + n530_0)
le(gen_0':s4_0(n804_0), gen_0':s4_0(n804_0)) -> true, rt in Omega(1 + n804_0)
if2(true, gen_0':s4_0(0), gen_0':s4_0(0), gen_0':s4_0(c), gen_empty:edge5_0(n1127_0), gen_empty:edge5_0(e)) -> *6_0, rt in Omega(n1127_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

They will be analysed ascendingly in the following order:
reach = if2