(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(f(X)) → mark(if(X, c, f(true)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(f(X)) → f(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(if(X1, X2, X3)) → if(X1, active(X2), X3)
f(mark(X)) → mark(f(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(X1, mark(X2), X3) → mark(if(X1, X2, X3))
proper(f(X)) → f(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
f(ok(X)) → ok(f(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
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
f(mark(X)) →+ mark(f(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(f(X)) → mark(if(X, c, f(true)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(f(X)) → f(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(if(X1, X2, X3)) → if(X1, active(X2), X3)
f(mark(X)) → mark(f(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(X1, mark(X2), X3) → mark(if(X1, X2, X3))
proper(f(X)) → f(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
f(ok(X)) → ok(f(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
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(f(X)) → mark(if(X, c, f(true)))
active(if(true, X, Y)) → mark(X)
active(if(false, X, Y)) → mark(Y)
active(f(X)) → f(active(X))
active(if(X1, X2, X3)) → if(active(X1), X2, X3)
active(if(X1, X2, X3)) → if(X1, active(X2), X3)
f(mark(X)) → mark(f(X))
if(mark(X1), X2, X3) → mark(if(X1, X2, X3))
if(X1, mark(X2), X3) → mark(if(X1, X2, X3))
proper(f(X)) → f(proper(X))
proper(if(X1, X2, X3)) → if(proper(X1), proper(X2), proper(X3))
proper(c) → ok(c)
proper(true) → ok(true)
proper(false) → ok(false)
f(ok(X)) → ok(f(X))
if(ok(X1), ok(X2), ok(X3)) → ok(if(X1, X2, X3))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: c:true:mark:false:ok → c:true:mark:false:ok
f :: c:true:mark:false:ok → c:true:mark:false:ok
mark :: c:true:mark:false:ok → c:true:mark:false:ok
if :: c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok
c :: c:true:mark:false:ok
true :: c:true:mark:false:ok
false :: c:true:mark:false:ok
proper :: c:true:mark:false:ok → c:true:mark:false:ok
ok :: c:true:mark:false:ok → c:true:mark:false:ok
top :: c:true:mark:false:ok → top
hole_c:true:mark:false:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
if,
f,
proper,
topThey will be analysed ascendingly in the following order:
if < active
f < active
active < top
if < proper
f < proper
proper < top
(8) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
if(
X,
c,
f(
true)))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
f(
X)) →
f(
active(
X))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
if(
X1,
X2,
X3)) →
if(
X1,
active(
X2),
X3)
f(
mark(
X)) →
mark(
f(
X))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
if(
X1,
mark(
X2),
X3) →
mark(
if(
X1,
X2,
X3))
proper(
f(
X)) →
f(
proper(
X))
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
c) →
ok(
c)
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
f(
ok(
X)) →
ok(
f(
X))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: c:true:mark:false:ok → c:true:mark:false:ok
f :: c:true:mark:false:ok → c:true:mark:false:ok
mark :: c:true:mark:false:ok → c:true:mark:false:ok
if :: c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok
c :: c:true:mark:false:ok
true :: c:true:mark:false:ok
false :: c:true:mark:false:ok
proper :: c:true:mark:false:ok → c:true:mark:false:ok
ok :: c:true:mark:false:ok → c:true:mark:false:ok
top :: c:true:mark:false:ok → top
hole_c:true:mark:false:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok
Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(x))
The following defined symbols remain to be analysed:
if, active, f, proper, top
They will be analysed ascendingly in the following order:
if < active
f < active
active < top
if < proper
f < proper
proper < top
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol if.
(10) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
if(
X,
c,
f(
true)))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
f(
X)) →
f(
active(
X))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
if(
X1,
X2,
X3)) →
if(
X1,
active(
X2),
X3)
f(
mark(
X)) →
mark(
f(
X))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
if(
X1,
mark(
X2),
X3) →
mark(
if(
X1,
X2,
X3))
proper(
f(
X)) →
f(
proper(
X))
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
c) →
ok(
c)
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
f(
ok(
X)) →
ok(
f(
X))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: c:true:mark:false:ok → c:true:mark:false:ok
f :: c:true:mark:false:ok → c:true:mark:false:ok
mark :: c:true:mark:false:ok → c:true:mark:false:ok
if :: c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok
c :: c:true:mark:false:ok
true :: c:true:mark:false:ok
false :: c:true:mark:false:ok
proper :: c:true:mark:false:ok → c:true:mark:false:ok
ok :: c:true:mark:false:ok → c:true:mark:false:ok
top :: c:true:mark:false:ok → top
hole_c:true:mark:false:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok
Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(x))
The following defined symbols remain to be analysed:
f, active, proper, top
They will be analysed ascendingly in the following order:
f < active
active < top
f < proper
proper < top
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol f.
(12) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
if(
X,
c,
f(
true)))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
f(
X)) →
f(
active(
X))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
if(
X1,
X2,
X3)) →
if(
X1,
active(
X2),
X3)
f(
mark(
X)) →
mark(
f(
X))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
if(
X1,
mark(
X2),
X3) →
mark(
if(
X1,
X2,
X3))
proper(
f(
X)) →
f(
proper(
X))
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
c) →
ok(
c)
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
f(
ok(
X)) →
ok(
f(
X))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: c:true:mark:false:ok → c:true:mark:false:ok
f :: c:true:mark:false:ok → c:true:mark:false:ok
mark :: c:true:mark:false:ok → c:true:mark:false:ok
if :: c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok
c :: c:true:mark:false:ok
true :: c:true:mark:false:ok
false :: c:true:mark:false:ok
proper :: c:true:mark:false:ok → c:true:mark:false:ok
ok :: c:true:mark:false:ok → c:true:mark:false:ok
top :: c:true:mark:false:ok → top
hole_c:true:mark:false:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok
Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark: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
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(14) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
if(
X,
c,
f(
true)))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
f(
X)) →
f(
active(
X))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
if(
X1,
X2,
X3)) →
if(
X1,
active(
X2),
X3)
f(
mark(
X)) →
mark(
f(
X))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
if(
X1,
mark(
X2),
X3) →
mark(
if(
X1,
X2,
X3))
proper(
f(
X)) →
f(
proper(
X))
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
c) →
ok(
c)
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
f(
ok(
X)) →
ok(
f(
X))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: c:true:mark:false:ok → c:true:mark:false:ok
f :: c:true:mark:false:ok → c:true:mark:false:ok
mark :: c:true:mark:false:ok → c:true:mark:false:ok
if :: c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok
c :: c:true:mark:false:ok
true :: c:true:mark:false:ok
false :: c:true:mark:false:ok
proper :: c:true:mark:false:ok → c:true:mark:false:ok
ok :: c:true:mark:false:ok → c:true:mark:false:ok
top :: c:true:mark:false:ok → top
hole_c:true:mark:false:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok
Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark: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
(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(16) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
if(
X,
c,
f(
true)))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
f(
X)) →
f(
active(
X))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
if(
X1,
X2,
X3)) →
if(
X1,
active(
X2),
X3)
f(
mark(
X)) →
mark(
f(
X))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
if(
X1,
mark(
X2),
X3) →
mark(
if(
X1,
X2,
X3))
proper(
f(
X)) →
f(
proper(
X))
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
c) →
ok(
c)
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
f(
ok(
X)) →
ok(
f(
X))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: c:true:mark:false:ok → c:true:mark:false:ok
f :: c:true:mark:false:ok → c:true:mark:false:ok
mark :: c:true:mark:false:ok → c:true:mark:false:ok
if :: c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok
c :: c:true:mark:false:ok
true :: c:true:mark:false:ok
false :: c:true:mark:false:ok
proper :: c:true:mark:false:ok → c:true:mark:false:ok
ok :: c:true:mark:false:ok → c:true:mark:false:ok
top :: c:true:mark:false:ok → top
hole_c:true:mark:false:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok
Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(x))
The following defined symbols remain to be analysed:
top
(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(18) Obligation:
TRS:
Rules:
active(
f(
X)) →
mark(
if(
X,
c,
f(
true)))
active(
if(
true,
X,
Y)) →
mark(
X)
active(
if(
false,
X,
Y)) →
mark(
Y)
active(
f(
X)) →
f(
active(
X))
active(
if(
X1,
X2,
X3)) →
if(
active(
X1),
X2,
X3)
active(
if(
X1,
X2,
X3)) →
if(
X1,
active(
X2),
X3)
f(
mark(
X)) →
mark(
f(
X))
if(
mark(
X1),
X2,
X3) →
mark(
if(
X1,
X2,
X3))
if(
X1,
mark(
X2),
X3) →
mark(
if(
X1,
X2,
X3))
proper(
f(
X)) →
f(
proper(
X))
proper(
if(
X1,
X2,
X3)) →
if(
proper(
X1),
proper(
X2),
proper(
X3))
proper(
c) →
ok(
c)
proper(
true) →
ok(
true)
proper(
false) →
ok(
false)
f(
ok(
X)) →
ok(
f(
X))
if(
ok(
X1),
ok(
X2),
ok(
X3)) →
ok(
if(
X1,
X2,
X3))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: c:true:mark:false:ok → c:true:mark:false:ok
f :: c:true:mark:false:ok → c:true:mark:false:ok
mark :: c:true:mark:false:ok → c:true:mark:false:ok
if :: c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok → c:true:mark:false:ok
c :: c:true:mark:false:ok
true :: c:true:mark:false:ok
false :: c:true:mark:false:ok
proper :: c:true:mark:false:ok → c:true:mark:false:ok
ok :: c:true:mark:false:ok → c:true:mark:false:ok
top :: c:true:mark:false:ok → top
hole_c:true:mark:false:ok1_0 :: c:true:mark:false:ok
hole_top2_0 :: top
gen_c:true:mark:false:ok3_0 :: Nat → c:true:mark:false:ok
Generator Equations:
gen_c:true:mark:false:ok3_0(0) ⇔ c
gen_c:true:mark:false:ok3_0(+(x, 1)) ⇔ mark(gen_c:true:mark:false:ok3_0(x))
No more defined symbols left to analyse.