(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(p(0)) → mark(0)
active(p(s(X))) → mark(X)
active(leq(0, Y)) → mark(true)
active(leq(s(X), 0)) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0, s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: FULL
(1) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
p(mark(X)) →+ mark(p(X))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X / mark(X)].
The result substitution is [ ].
(2) BOUNDS(n^1, INF)
(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.
(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: FULL
(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.
(6) Obligation:
TRS:
Rules:
active(p(0')) → mark(0')
active(p(s(X))) → mark(X)
active(leq(0', Y)) → mark(true)
active(leq(s(X), 0')) → mark(false)
active(leq(s(X), s(Y))) → mark(leq(X, Y))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(diff(X, Y)) → mark(if(leq(X, Y), 0', s(diff(p(X), Y))))
active(p(X)) → p(active(X))
active(s(X)) → s(active(X))
active(leq(X1, X2)) → leq(active(X1), X2)
active(leq(X1, X2)) → leq(X1, active(X2))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(diff(X1, X2)) → diff(active(X1), X2)
active(diff(X1, X2)) → diff(X1, active(X2))
p(mark(X)) → mark(p(X))
s(mark(X)) → mark(s(X))
leq(mark(X1), X2) → mark(leq(X1, X2))
leq(X1, mark(X2)) → mark(leq(X1, X2))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
diff(mark(X1), X2) → mark(diff(X1, X2))
diff(X1, mark(X2)) → mark(diff(X1, X2))
proper(p(X)) → p(proper(X))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(leq(X1, X2)) → leq(proper(X1), proper(X2))
proper(true) → ok(true)
proper(false) → ok(false)
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(diff(X1, X2)) → diff(proper(X1), proper(X2))
p(ok(X)) → ok(p(X))
s(ok(X)) → ok(s(X))
leq(ok(X1), ok(X2)) → ok(leq(X1, X2))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
diff(ok(X1), ok(X2)) → ok(diff(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
leq,
if,
s,
diff,
p,
proper,
topThey will be analysed ascendingly in the following order:
leq < active
if < active
s < active
diff < active
p < active
active < top
leq < proper
if < proper
s < proper
diff < proper
p < proper
proper < top
(8) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
The following defined symbols remain to be analysed:
leq, active, if, s, diff, p, proper, top
They will be analysed ascendingly in the following order:
leq < active
if < active
s < active
diff < active
p < active
active < top
leq < proper
if < proper
s < proper
diff < proper
p < proper
proper < top
(9) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
leq(
gen_0':mark:true:false:ok3_0(
+(
1,
n5_0)),
gen_0':mark:true:false:ok3_0(
b)) →
*4_0, rt ∈ Ω(n5
0)
Induction Base:
leq(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b))
Induction Step:
leq(gen_0':mark:true:false:ok3_0(+(1, +(n5_0, 1))), gen_0':mark:true:false:ok3_0(b)) →RΩ(1)
mark(leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(10) Complex Obligation (BEST)
(11) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
The following defined symbols remain to be analysed:
if, active, s, diff, p, proper, top
They will be analysed ascendingly in the following order:
if < active
s < active
diff < active
p < active
active < top
if < proper
s < proper
diff < proper
p < proper
proper < top
(12) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
if(
gen_0':mark:true:false:ok3_0(
+(
1,
n1185_0)),
gen_0':mark:true:false:ok3_0(
b),
gen_0':mark:true:false:ok3_0(
c)) →
*4_0, rt ∈ Ω(n1185
0)
Induction Base:
if(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c))
Induction Step:
if(gen_0':mark:true:false:ok3_0(+(1, +(n1185_0, 1))), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) →RΩ(1)
mark(if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(13) Complex Obligation (BEST)
(14) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
The following defined symbols remain to be analysed:
s, active, diff, p, proper, top
They will be analysed ascendingly in the following order:
s < active
diff < active
p < active
active < top
s < proper
diff < proper
p < proper
proper < top
(15) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(
gen_0':mark:true:false:ok3_0(
+(
1,
n3331_0))) →
*4_0, rt ∈ Ω(n3331
0)
Induction Base:
s(gen_0':mark:true:false:ok3_0(+(1, 0)))
Induction Step:
s(gen_0':mark:true:false:ok3_0(+(1, +(n3331_0, 1)))) →RΩ(1)
mark(s(gen_0':mark:true:false:ok3_0(+(1, n3331_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(16) Complex Obligation (BEST)
(17) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
s(gen_0':mark:true:false:ok3_0(+(1, n3331_0))) → *4_0, rt ∈ Ω(n33310)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
The following defined symbols remain to be analysed:
diff, active, p, proper, top
They will be analysed ascendingly in the following order:
diff < active
p < active
active < top
diff < proper
p < proper
proper < top
(18) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
diff(
gen_0':mark:true:false:ok3_0(
+(
1,
n4058_0)),
gen_0':mark:true:false:ok3_0(
b)) →
*4_0, rt ∈ Ω(n4058
0)
Induction Base:
diff(gen_0':mark:true:false:ok3_0(+(1, 0)), gen_0':mark:true:false:ok3_0(b))
Induction Step:
diff(gen_0':mark:true:false:ok3_0(+(1, +(n4058_0, 1))), gen_0':mark:true:false:ok3_0(b)) →RΩ(1)
mark(diff(gen_0':mark:true:false:ok3_0(+(1, n4058_0)), gen_0':mark:true:false:ok3_0(b))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(19) Complex Obligation (BEST)
(20) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
s(gen_0':mark:true:false:ok3_0(+(1, n3331_0))) → *4_0, rt ∈ Ω(n33310)
diff(gen_0':mark:true:false:ok3_0(+(1, n4058_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n40580)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
The following defined symbols remain to be analysed:
p, active, proper, top
They will be analysed ascendingly in the following order:
p < active
active < top
p < proper
proper < top
(21) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
p(
gen_0':mark:true:false:ok3_0(
+(
1,
n6150_0))) →
*4_0, rt ∈ Ω(n6150
0)
Induction Base:
p(gen_0':mark:true:false:ok3_0(+(1, 0)))
Induction Step:
p(gen_0':mark:true:false:ok3_0(+(1, +(n6150_0, 1)))) →RΩ(1)
mark(p(gen_0':mark:true:false:ok3_0(+(1, n6150_0)))) →IH
mark(*4_0)
We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(22) Complex Obligation (BEST)
(23) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
s(gen_0':mark:true:false:ok3_0(+(1, n3331_0))) → *4_0, rt ∈ Ω(n33310)
diff(gen_0':mark:true:false:ok3_0(+(1, n4058_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n40580)
p(gen_0':mark:true:false:ok3_0(+(1, n6150_0))) → *4_0, rt ∈ Ω(n61500)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
The following defined symbols remain to be analysed:
active, proper, top
They will be analysed ascendingly in the following order:
active < top
proper < top
(24) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(25) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
s(gen_0':mark:true:false:ok3_0(+(1, n3331_0))) → *4_0, rt ∈ Ω(n33310)
diff(gen_0':mark:true:false:ok3_0(+(1, n4058_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n40580)
p(gen_0':mark:true:false:ok3_0(+(1, n6150_0))) → *4_0, rt ∈ Ω(n61500)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
The following defined symbols remain to be analysed:
proper, top
They will be analysed ascendingly in the following order:
proper < top
(26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(27) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
s(gen_0':mark:true:false:ok3_0(+(1, n3331_0))) → *4_0, rt ∈ Ω(n33310)
diff(gen_0':mark:true:false:ok3_0(+(1, n4058_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n40580)
p(gen_0':mark:true:false:ok3_0(+(1, n6150_0))) → *4_0, rt ∈ Ω(n61500)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
The following defined symbols remain to be analysed:
top
(28) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(29) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
s(gen_0':mark:true:false:ok3_0(+(1, n3331_0))) → *4_0, rt ∈ Ω(n33310)
diff(gen_0':mark:true:false:ok3_0(+(1, n4058_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n40580)
p(gen_0':mark:true:false:ok3_0(+(1, n6150_0))) → *4_0, rt ∈ Ω(n61500)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
No more defined symbols left to analyse.
(30) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(31) BOUNDS(n^1, INF)
(32) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
s(gen_0':mark:true:false:ok3_0(+(1, n3331_0))) → *4_0, rt ∈ Ω(n33310)
diff(gen_0':mark:true:false:ok3_0(+(1, n4058_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n40580)
p(gen_0':mark:true:false:ok3_0(+(1, n6150_0))) → *4_0, rt ∈ Ω(n61500)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
No more defined symbols left to analyse.
(33) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(34) BOUNDS(n^1, INF)
(35) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
s(gen_0':mark:true:false:ok3_0(+(1, n3331_0))) → *4_0, rt ∈ Ω(n33310)
diff(gen_0':mark:true:false:ok3_0(+(1, n4058_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n40580)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
No more defined symbols left to analyse.
(36) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(37) BOUNDS(n^1, INF)
(38) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
s(gen_0':mark:true:false:ok3_0(+(1, n3331_0))) → *4_0, rt ∈ Ω(n33310)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
No more defined symbols left to analyse.
(39) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(40) BOUNDS(n^1, INF)
(41) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
if(gen_0':mark:true:false:ok3_0(+(1, n1185_0)), gen_0':mark:true:false:ok3_0(b), gen_0':mark:true:false:ok3_0(c)) → *4_0, rt ∈ Ω(n11850)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
No more defined symbols left to analyse.
(42) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(43) BOUNDS(n^1, INF)
(44) Obligation:
TRS:
Rules:
active(
p(
0')) →
mark(
0')
active(
p(
s(
X))) →
mark(
X)
active(
leq(
0',
Y)) →
mark(
true)
active(
leq(
s(
X),
0')) →
mark(
false)
active(
leq(
s(
X),
s(
Y))) →
mark(
leq(
X,
Y))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
diff(
X,
Y)) →
mark(
if(
leq(
X,
Y),
0',
s(
diff(
p(
X),
Y))))
active(
p(
X)) →
p(
active(
X))
active(
s(
X)) →
s(
active(
X))
active(
leq(
X1,
X2)) →
leq(
active(
X1),
X2)
active(
leq(
X1,
X2)) →
leq(
X1,
active(
X2))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
diff(
X1,
X2)) →
diff(
active(
X1),
X2)
active(
diff(
X1,
X2)) →
diff(
X1,
active(
X2))
p(
mark(
X)) →
mark(
p(
X))
s(
mark(
X)) →
mark(
s(
X))
leq(
mark(
X1),
X2) →
mark(
leq(
X1,
X2))
leq(
X1,
mark(
X2)) →
mark(
leq(
X1,
X2))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
diff(
mark(
X1),
X2) →
mark(
diff(
X1,
X2))
diff(
X1,
mark(
X2)) →
mark(
diff(
X1,
X2))
proper(
p(
X)) →
p(
proper(
X))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
leq(
X1,
X2)) →
leq(
proper(
X1),
proper(
X2))
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
diff(
X1,
X2)) →
diff(
proper(
X1),
proper(
X2))
p(
ok(
X)) →
ok(
p(
X))
s(
ok(
X)) →
ok(
s(
X))
leq(
ok(
X1),
ok(
X2)) →
ok(
leq(
X1,
X2))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
diff(
ok(
X1),
ok(
X2)) →
ok(
diff(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: 0':mark:true:false:ok → 0':mark:true:false:ok
p :: 0':mark:true:false:ok → 0':mark:true:false:ok
0' :: 0':mark:true:false:ok
mark :: 0':mark:true:false:ok → 0':mark:true:false:ok
s :: 0':mark:true:false:ok → 0':mark:true:false:ok
leq :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
true :: 0':mark:true:false:ok
false :: 0':mark:true:false:ok
if :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
diff :: 0':mark:true:false:ok → 0':mark:true:false:ok → 0':mark:true:false:ok
proper :: 0':mark:true:false:ok → 0':mark:true:false:ok
ok :: 0':mark:true:false:ok → 0':mark:true:false:ok
top :: 0':mark:true:false:ok → top
hole_0':mark:true:false:ok1_0 :: 0':mark:true:false:ok
hole_top2_0 :: top
gen_0':mark:true:false:ok3_0 :: Nat → 0':mark:true:false:ok
Lemmas:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
Generator Equations:
gen_0':mark:true:false:ok3_0(0) ⇔ 0'
gen_0':mark:true:false:ok3_0(+(x, 1)) ⇔ mark(gen_0':mark:true:false:ok3_0(x))
No more defined symbols left to analyse.
(45) LowerBoundsProof (EQUIVALENT transformation)
The lowerbound Ω(n1) was proven with the following lemma:
leq(gen_0':mark:true:false:ok3_0(+(1, n5_0)), gen_0':mark:true:false:ok3_0(b)) → *4_0, rt ∈ Ω(n50)
(46) BOUNDS(n^1, INF)