(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(and(tt, X)) → mark(X)
active(plus(N, 0)) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(x(N, 0)) → mark(0)
active(x(N, s(M))) → mark(plus(x(N, M), N))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
s(mark(X)) → mark(s(X))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0) → ok(0)
proper(s(X)) → s(proper(X))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(X))
x(ok(X1), ok(X2)) → ok(x(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
and(mark(X1), X2) →+ mark(and(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
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(and(tt, X)) → mark(X)
active(plus(N, 0')) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(x(N, 0')) → mark(0')
active(x(N, s(M))) → mark(plus(x(N, M), N))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
s(mark(X)) → mark(s(X))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(X))
x(ok(X1), ok(X2)) → ok(x(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(and(tt, X)) → mark(X)
active(plus(N, 0')) → mark(N)
active(plus(N, s(M))) → mark(s(plus(N, M)))
active(x(N, 0')) → mark(0')
active(x(N, s(M))) → mark(plus(x(N, M), N))
active(and(X1, X2)) → and(active(X1), X2)
active(plus(X1, X2)) → plus(active(X1), X2)
active(plus(X1, X2)) → plus(X1, active(X2))
active(s(X)) → s(active(X))
active(x(X1, X2)) → x(active(X1), X2)
active(x(X1, X2)) → x(X1, active(X2))
and(mark(X1), X2) → mark(and(X1, X2))
plus(mark(X1), X2) → mark(plus(X1, X2))
plus(X1, mark(X2)) → mark(plus(X1, X2))
s(mark(X)) → mark(s(X))
x(mark(X1), X2) → mark(x(X1, X2))
x(X1, mark(X2)) → mark(x(X1, X2))
proper(and(X1, X2)) → and(proper(X1), proper(X2))
proper(tt) → ok(tt)
proper(plus(X1, X2)) → plus(proper(X1), proper(X2))
proper(0') → ok(0')
proper(s(X)) → s(proper(X))
proper(x(X1, X2)) → x(proper(X1), proper(X2))
and(ok(X1), ok(X2)) → ok(and(X1, X2))
plus(ok(X1), ok(X2)) → ok(plus(X1, X2))
s(ok(X)) → ok(s(X))
x(ok(X1), ok(X2)) → ok(x(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
(7) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active,
s,
plus,
x,
and,
proper,
topThey will be analysed ascendingly in the following order:
s < active
plus < active
x < active
and < active
active < top
s < proper
plus < proper
x < proper
and < proper
proper < top
(8) Obligation:
TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
x(
N,
0')) →
mark(
0')
active(
x(
N,
s(
M))) →
mark(
plus(
x(
N,
M),
N))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
active(
x(
X1,
X2)) →
x(
active(
X1),
X2)
active(
x(
X1,
X2)) →
x(
X1,
active(
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
x(
mark(
X1),
X2) →
mark(
x(
X1,
X2))
x(
X1,
mark(
X2)) →
mark(
x(
X1,
X2))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
x(
X1,
X2)) →
x(
proper(
X1),
proper(
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
x(
ok(
X1),
ok(
X2)) →
ok(
x(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
s, active, plus, x, and, proper, top
They will be analysed ascendingly in the following order:
s < active
plus < active
x < active
and < active
active < top
s < proper
plus < proper
x < proper
and < proper
proper < top
(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol s.
(10) Obligation:
TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
x(
N,
0')) →
mark(
0')
active(
x(
N,
s(
M))) →
mark(
plus(
x(
N,
M),
N))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
active(
x(
X1,
X2)) →
x(
active(
X1),
X2)
active(
x(
X1,
X2)) →
x(
X1,
active(
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
x(
mark(
X1),
X2) →
mark(
x(
X1,
X2))
x(
X1,
mark(
X2)) →
mark(
x(
X1,
X2))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
x(
X1,
X2)) →
x(
proper(
X1),
proper(
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
x(
ok(
X1),
ok(
X2)) →
ok(
x(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
plus, active, x, and, proper, top
They will be analysed ascendingly in the following order:
plus < active
x < active
and < active
active < top
plus < proper
x < proper
and < proper
proper < top
(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol plus.
(12) Obligation:
TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
x(
N,
0')) →
mark(
0')
active(
x(
N,
s(
M))) →
mark(
plus(
x(
N,
M),
N))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
active(
x(
X1,
X2)) →
x(
active(
X1),
X2)
active(
x(
X1,
X2)) →
x(
X1,
active(
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
x(
mark(
X1),
X2) →
mark(
x(
X1,
X2))
x(
X1,
mark(
X2)) →
mark(
x(
X1,
X2))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
x(
X1,
X2)) →
x(
proper(
X1),
proper(
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
x(
ok(
X1),
ok(
X2)) →
ok(
x(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
x, active, and, proper, top
They will be analysed ascendingly in the following order:
x < active
and < active
active < top
x < proper
and < proper
proper < top
(13) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol x.
(14) Obligation:
TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
x(
N,
0')) →
mark(
0')
active(
x(
N,
s(
M))) →
mark(
plus(
x(
N,
M),
N))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
active(
x(
X1,
X2)) →
x(
active(
X1),
X2)
active(
x(
X1,
X2)) →
x(
X1,
active(
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
x(
mark(
X1),
X2) →
mark(
x(
X1,
X2))
x(
X1,
mark(
X2)) →
mark(
x(
X1,
X2))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
x(
X1,
X2)) →
x(
proper(
X1),
proper(
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
x(
ok(
X1),
ok(
X2)) →
ok(
x(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
and, active, proper, top
They will be analysed ascendingly in the following order:
and < active
active < top
and < proper
proper < top
(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol and.
(16) Obligation:
TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
x(
N,
0')) →
mark(
0')
active(
x(
N,
s(
M))) →
mark(
plus(
x(
N,
M),
N))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
active(
x(
X1,
X2)) →
x(
active(
X1),
X2)
active(
x(
X1,
X2)) →
x(
X1,
active(
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
x(
mark(
X1),
X2) →
mark(
x(
X1,
X2))
x(
X1,
mark(
X2)) →
mark(
x(
X1,
X2))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
x(
X1,
X2)) →
x(
proper(
X1),
proper(
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
x(
ok(
X1),
ok(
X2)) →
ok(
x(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':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
(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol active.
(18) Obligation:
TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
x(
N,
0')) →
mark(
0')
active(
x(
N,
s(
M))) →
mark(
plus(
x(
N,
M),
N))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
active(
x(
X1,
X2)) →
x(
active(
X1),
X2)
active(
x(
X1,
X2)) →
x(
X1,
active(
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
x(
mark(
X1),
X2) →
mark(
x(
X1,
X2))
x(
X1,
mark(
X2)) →
mark(
x(
X1,
X2))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
x(
X1,
X2)) →
x(
proper(
X1),
proper(
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
x(
ok(
X1),
ok(
X2)) →
ok(
x(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
proper, top
They will be analysed ascendingly in the following order:
proper < top
(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol proper.
(20) Obligation:
TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
x(
N,
0')) →
mark(
0')
active(
x(
N,
s(
M))) →
mark(
plus(
x(
N,
M),
N))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
active(
x(
X1,
X2)) →
x(
active(
X1),
X2)
active(
x(
X1,
X2)) →
x(
X1,
active(
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
x(
mark(
X1),
X2) →
mark(
x(
X1,
X2))
x(
X1,
mark(
X2)) →
mark(
x(
X1,
X2))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
x(
X1,
X2)) →
x(
proper(
X1),
proper(
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
x(
ok(
X1),
ok(
X2)) →
ok(
x(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
The following defined symbols remain to be analysed:
top
(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol top.
(22) Obligation:
TRS:
Rules:
active(
and(
tt,
X)) →
mark(
X)
active(
plus(
N,
0')) →
mark(
N)
active(
plus(
N,
s(
M))) →
mark(
s(
plus(
N,
M)))
active(
x(
N,
0')) →
mark(
0')
active(
x(
N,
s(
M))) →
mark(
plus(
x(
N,
M),
N))
active(
and(
X1,
X2)) →
and(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
active(
X1),
X2)
active(
plus(
X1,
X2)) →
plus(
X1,
active(
X2))
active(
s(
X)) →
s(
active(
X))
active(
x(
X1,
X2)) →
x(
active(
X1),
X2)
active(
x(
X1,
X2)) →
x(
X1,
active(
X2))
and(
mark(
X1),
X2) →
mark(
and(
X1,
X2))
plus(
mark(
X1),
X2) →
mark(
plus(
X1,
X2))
plus(
X1,
mark(
X2)) →
mark(
plus(
X1,
X2))
s(
mark(
X)) →
mark(
s(
X))
x(
mark(
X1),
X2) →
mark(
x(
X1,
X2))
x(
X1,
mark(
X2)) →
mark(
x(
X1,
X2))
proper(
and(
X1,
X2)) →
and(
proper(
X1),
proper(
X2))
proper(
tt) →
ok(
tt)
proper(
plus(
X1,
X2)) →
plus(
proper(
X1),
proper(
X2))
proper(
0') →
ok(
0')
proper(
s(
X)) →
s(
proper(
X))
proper(
x(
X1,
X2)) →
x(
proper(
X1),
proper(
X2))
and(
ok(
X1),
ok(
X2)) →
ok(
and(
X1,
X2))
plus(
ok(
X1),
ok(
X2)) →
ok(
plus(
X1,
X2))
s(
ok(
X)) →
ok(
s(
X))
x(
ok(
X1),
ok(
X2)) →
ok(
x(
X1,
X2))
top(
mark(
X)) →
top(
proper(
X))
top(
ok(
X)) →
top(
active(
X))
Types:
active :: tt:mark:0':ok → tt:mark:0':ok
and :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
tt :: tt:mark:0':ok
mark :: tt:mark:0':ok → tt:mark:0':ok
plus :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
0' :: tt:mark:0':ok
s :: tt:mark:0':ok → tt:mark:0':ok
x :: tt:mark:0':ok → tt:mark:0':ok → tt:mark:0':ok
proper :: tt:mark:0':ok → tt:mark:0':ok
ok :: tt:mark:0':ok → tt:mark:0':ok
top :: tt:mark:0':ok → top
hole_tt:mark:0':ok1_0 :: tt:mark:0':ok
hole_top2_0 :: top
gen_tt:mark:0':ok3_0 :: Nat → tt:mark:0':ok
Generator Equations:
gen_tt:mark:0':ok3_0(0) ⇔ tt
gen_tt:mark:0':ok3_0(+(x, 1)) ⇔ mark(gen_tt:mark:0':ok3_0(x))
No more defined symbols left to analyse.