KILLEDRuntime Complexity (full) proof of /tmp/tmpWZK1Qj/Ex6_15_AEL02_C.xml
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).0 CpxTRS↳1 RenamingProof (⇔, 0 ms)↳2 CpxRelTRS↳3 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)↳4 typed CpxTrs↳5 OrderProof (LOWER BOUND(ID), 6 ms)↳6 typed CpxTrs↳7 RewriteLemmaProof (LOWER BOUND(ID), 487 ms)↳8 BEST↳9 typed CpxTrs↳10 RewriteLemmaProof (LOWER BOUND(ID), 164 ms)↳11 BEST↳12 typed CpxTrs↳13 RewriteLemmaProof (LOWER BOUND(ID), 72 ms)↳14 BEST↳15 typed CpxTrs↳16 RewriteLemmaProof (LOWER BOUND(ID), 102 ms)↳17 BEST↳18 typed CpxTrs↳19 RewriteLemmaProof (LOWER BOUND(ID), 90 ms)↳20 BEST↳21 typed CpxTrs↳22 RewriteLemmaProof (LOWER BOUND(ID), 77 ms)↳23 BEST↳24 typed CpxTrs↳25 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳26 typed CpxTrs↳27 RewriteLemmaProof (LOWER BOUND(ID), 109 ms)↳28 BEST↳29 typed CpxTrs↳30 RewriteLemmaProof (LOWER BOUND(ID), 55 ms)↳31 BEST↳32 typed CpxTrs↳33 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳34 typed CpxTrs↳35 RewriteLemmaProof (LOWER BOUND(ID), 4 ms)↳36 BEST↳37 typed CpxTrs↳38 RewriteLemmaProof (LOWER BOUND(ID), 14 ms)↳39 BEST↳40 typed CpxTrs↳41 RewriteLemmaProof (LOWER BOUND(ID), 128 ms)↳42 BEST↳43 typed CpxTrs↳44 RewriteLemmaProof (LOWER BOUND(ID), 0 ms)↳45 BEST↳46 typed CpxTrs↳47 typed CpxTrs↳48 typed CpxTrs↳49 typed CpxTrs↳50 typed CpxTrs↳51 typed CpxTrs↳52 typed CpxTrs↳53 typed CpxTrs↳54 typed CpxTrs↳55 typed CpxTrs↳56 typed CpxTrs↳57 typed CpxTrs↳58 typed CpxTrs(0) Obligation:
Runtime Complexity TRS:
The TRS R consists of the following rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0, cons(X, Z))) → mark(X)
active(first(0, Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0, cons(X, Z))) → mark(quote(X))
active(first1(0, Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0)) → mark(01)
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01)) → mark(0)
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0) → ok(0)
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01) → ok(01)
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Rewrite Strategy: FULL(1) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.(2) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
S is empty.
Rewrite Strategy: FULL(3) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.(4) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':ok(5) OrderProof (LOWER BOUND(ID) transformation)
Heuristically decided to analyse the following defined symbols:
active, sel, cons, first, from, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
sel < active
cons < active
first < active
from < active
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
sel < proper
cons < proper
first < proper
from < proper
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(6) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okGenerator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
sel, active, cons, first, from, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
sel < active
cons < active
first < active
from < active
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
sel < proper
cons < proper
first < proper
from < proper
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(7) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)Induction Base:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))Induction Step:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n5_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →RΩ(1)
mark(sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(8) Complex Obligation (BEST)
(9) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
cons, active, first, from, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
cons < active
first < active
from < active
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
cons < proper
first < proper
from < proper
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(10) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)Induction Base:
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))Induction Step:
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n1821_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →RΩ(1)
mark(cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(11) Complex Obligation (BEST)
(12) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
first, active, from, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
first < active
from < active
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
first < proper
from < proper
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(13) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)Induction Base:
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))Induction Step:
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n3742_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →RΩ(1)
mark(first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(14) Complex Obligation (BEST)
(15) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
from, active, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
from < active
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
from < proper
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(16) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)Induction Base:
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)))Induction Step:
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n6166_0, 1)))) →RΩ(1)
mark(from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(17) Complex Obligation (BEST)
(18) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
s, active, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
s < active
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
s < proper
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(19) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)Induction Base:
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)))Induction Step:
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n7153_0, 1)))) →RΩ(1)
mark(s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(20) Complex Obligation (BEST)
(21) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
sel1, active, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
sel1 < active
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
sel1 < proper
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(22) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)Induction Base:
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))Induction Step:
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n8241_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →RΩ(1)
mark(sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(23) Complex Obligation (BEST)
(24) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
quote, active, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
quote < active
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
quote < proper
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol quote.(26) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
cons1, active, first1, quote1, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
cons1 < active
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
cons1 < proper
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(27) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)Induction Base:
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))Induction Step:
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n11411_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →RΩ(1)
mark(cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(28) Complex Obligation (BEST)
(29) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
first1, active, quote1, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
first1 < active
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
first1 < proper
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(30) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)Induction Base:
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))Induction Step:
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n14855_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →RΩ(1)
mark(first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(31) Complex Obligation (BEST)
(32) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
quote1, active, s1, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
quote1 < active
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
quote1 < proper
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(33) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol quote1.(34) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
s1, active, unquote, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
s1 < active
unquote < active
fcons < active
unquote1 < active
active < top
s1 < proper
unquote < proper
fcons < proper
unquote1 < proper
proper < top(35) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n186430)Induction Base:
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)))Induction Step:
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n18643_0, 1)))) →RΩ(1)
mark(s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(36) Complex Obligation (BEST)
(37) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n186430)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
unquote, active, fcons, unquote1, proper, topThey will be analysed ascendingly in the following order:
unquote < active
fcons < active
unquote1 < active
active < top
unquote < proper
fcons < proper
unquote1 < proper
proper < top(38) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n202820)Induction Base:
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)))Induction Step:
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n20282_0, 1)))) →RΩ(1)
mark(unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(39) Complex Obligation (BEST)
(40) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n186430)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n202820)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
fcons, active, unquote1, proper, topThey will be analysed ascendingly in the following order:
fcons < active
unquote1 < active
active < top
fcons < proper
unquote1 < proper
proper < top(41) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n220220)Induction Base:
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)), gen_mark:0':nil:nil1:01':ok3_0(b))Induction Step:
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n22022_0, 1))), gen_mark:0':nil:nil1:01':ok3_0(b)) →RΩ(1)
mark(fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(42) Complex Obligation (BEST)
(43) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n186430)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n202820)
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n220220)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
unquote1, active, proper, topThey will be analysed ascendingly in the following order:
unquote1 < active
active < top
unquote1 < proper
proper < top(44) RewriteLemmaProof (LOWER BOUND(ID) transformation)
Proved the following rewrite lemma:
unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n26486_0))) → *4_0, rt ∈ Ω(n264860)Induction Base:
unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, 0)))Induction Step:
unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, +(n26486_0, 1)))) →RΩ(1)
mark(unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n26486_0)))) →IH
mark(*4_0)We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).
(45) Complex Obligation (BEST)
(46) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n186430)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n202820)
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n220220)
unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n26486_0))) → *4_0, rt ∈ Ω(n264860)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))The following defined symbols remain to be analysed:
active, proper, topThey will be analysed ascendingly in the following order:
active < top
proper < top(47) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n186430)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n202820)
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n220220)
unquote1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n26486_0))) → *4_0, rt ∈ Ω(n264860)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(48) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n186430)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n202820)
fcons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n22022_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n220220)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(49) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n186430)
unquote(gen_mark:0':nil:nil1:01':ok3_0(+(1, n20282_0))) → *4_0, rt ∈ Ω(n202820)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(50) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)
s1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n18643_0))) → *4_0, rt ∈ Ω(n186430)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(51) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)
first1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n14855_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n148550)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(52) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)
cons1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n11411_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n114110)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(53) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)
sel1(gen_mark:0':nil:nil1:01':ok3_0(+(1, n8241_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n82410)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(54) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)
s(gen_mark:0':nil:nil1:01':ok3_0(+(1, n7153_0))) → *4_0, rt ∈ Ω(n71530)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(55) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)
from(gen_mark:0':nil:nil1:01':ok3_0(+(1, n6166_0))) → *4_0, rt ∈ Ω(n61660)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(56) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)
first(gen_mark:0':nil:nil1:01':ok3_0(+(1, n3742_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n37420)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(57) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)
cons(gen_mark:0':nil:nil1:01':ok3_0(+(1, n1821_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n18210)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.
(58) Obligation:
TRS:
Rules:
active(sel(s(X), cons(Y, Z))) → mark(sel(X, Z))
active(sel(0', cons(X, Z))) → mark(X)
active(first(0', Z)) → mark(nil)
active(first(s(X), cons(Y, Z))) → mark(cons(Y, first(X, Z)))
active(from(X)) → mark(cons(X, from(s(X))))
active(sel1(s(X), cons(Y, Z))) → mark(sel1(X, Z))
active(sel1(0', cons(X, Z))) → mark(quote(X))
active(first1(0', Z)) → mark(nil1)
active(first1(s(X), cons(Y, Z))) → mark(cons1(quote(Y), first1(X, Z)))
active(quote(0')) → mark(01')
active(quote1(cons(X, Z))) → mark(cons1(quote(X), quote1(Z)))
active(quote1(nil)) → mark(nil1)
active(quote(s(X))) → mark(s1(quote(X)))
active(quote(sel(X, Z))) → mark(sel1(X, Z))
active(quote1(first(X, Z))) → mark(first1(X, Z))
active(unquote(01')) → mark(0')
active(unquote(s1(X))) → mark(s(unquote(X)))
active(unquote1(nil1)) → mark(nil)
active(unquote1(cons1(X, Z))) → mark(fcons(unquote(X), unquote1(Z)))
active(fcons(X, Z)) → mark(cons(X, Z))
active(sel(X1, X2)) → sel(active(X1), X2)
active(sel(X1, X2)) → sel(X1, active(X2))
active(s(X)) → s(active(X))
active(cons(X1, X2)) → cons(active(X1), X2)
active(first(X1, X2)) → first(active(X1), X2)
active(first(X1, X2)) → first(X1, active(X2))
active(from(X)) → from(active(X))
active(sel1(X1, X2)) → sel1(active(X1), X2)
active(sel1(X1, X2)) → sel1(X1, active(X2))
active(first1(X1, X2)) → first1(active(X1), X2)
active(first1(X1, X2)) → first1(X1, active(X2))
active(cons1(X1, X2)) → cons1(active(X1), X2)
active(cons1(X1, X2)) → cons1(X1, active(X2))
active(s1(X)) → s1(active(X))
active(unquote(X)) → unquote(active(X))
active(unquote1(X)) → unquote1(active(X))
active(fcons(X1, X2)) → fcons(active(X1), X2)
active(fcons(X1, X2)) → fcons(X1, active(X2))
sel(mark(X1), X2) → mark(sel(X1, X2))
sel(X1, mark(X2)) → mark(sel(X1, X2))
s(mark(X)) → mark(s(X))
cons(mark(X1), X2) → mark(cons(X1, X2))
first(mark(X1), X2) → mark(first(X1, X2))
first(X1, mark(X2)) → mark(first(X1, X2))
from(mark(X)) → mark(from(X))
sel1(mark(X1), X2) → mark(sel1(X1, X2))
sel1(X1, mark(X2)) → mark(sel1(X1, X2))
first1(mark(X1), X2) → mark(first1(X1, X2))
first1(X1, mark(X2)) → mark(first1(X1, X2))
cons1(mark(X1), X2) → mark(cons1(X1, X2))
cons1(X1, mark(X2)) → mark(cons1(X1, X2))
s1(mark(X)) → mark(s1(X))
unquote(mark(X)) → mark(unquote(X))
unquote1(mark(X)) → mark(unquote1(X))
fcons(mark(X1), X2) → mark(fcons(X1, X2))
fcons(X1, mark(X2)) → mark(fcons(X1, X2))
proper(sel(X1, X2)) → sel(proper(X1), proper(X2))
proper(s(X)) → s(proper(X))
proper(cons(X1, X2)) → cons(proper(X1), proper(X2))
proper(0') → ok(0')
proper(first(X1, X2)) → first(proper(X1), proper(X2))
proper(nil) → ok(nil)
proper(from(X)) → from(proper(X))
proper(sel1(X1, X2)) → sel1(proper(X1), proper(X2))
proper(quote(X)) → quote(proper(X))
proper(first1(X1, X2)) → first1(proper(X1), proper(X2))
proper(nil1) → ok(nil1)
proper(cons1(X1, X2)) → cons1(proper(X1), proper(X2))
proper(01') → ok(01')
proper(quote1(X)) → quote1(proper(X))
proper(s1(X)) → s1(proper(X))
proper(unquote(X)) → unquote(proper(X))
proper(unquote1(X)) → unquote1(proper(X))
proper(fcons(X1, X2)) → fcons(proper(X1), proper(X2))
sel(ok(X1), ok(X2)) → ok(sel(X1, X2))
s(ok(X)) → ok(s(X))
cons(ok(X1), ok(X2)) → ok(cons(X1, X2))
first(ok(X1), ok(X2)) → ok(first(X1, X2))
from(ok(X)) → ok(from(X))
sel1(ok(X1), ok(X2)) → ok(sel1(X1, X2))
quote(ok(X)) → ok(quote(X))
first1(ok(X1), ok(X2)) → ok(first1(X1, X2))
cons1(ok(X1), ok(X2)) → ok(cons1(X1, X2))
quote1(ok(X)) → ok(quote1(X))
s1(ok(X)) → ok(s1(X))
unquote(ok(X)) → ok(unquote(X))
unquote1(ok(X)) → ok(unquote1(X))
fcons(ok(X1), ok(X2)) → ok(fcons(X1, X2))
top(mark(X)) → top(proper(X))
top(ok(X)) → top(active(X))
Types:
active :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
cons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
mark :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
0' :: mark:0':nil:nil1:01':ok
first :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil :: mark:0':nil:nil1:01':ok
from :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
sel1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
quote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
first1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
nil1 :: mark:0':nil:nil1:01':ok
cons1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
01' :: mark:0':nil:nil1:01':ok
quote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
s1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
unquote1 :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
fcons :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
proper :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
ok :: mark:0':nil:nil1:01':ok → mark:0':nil:nil1:01':ok
top :: mark:0':nil:nil1:01':ok → top
hole_mark:0':nil:nil1:01':ok1_0 :: mark:0':nil:nil1:01':ok
hole_top2_0 :: top
gen_mark:0':nil:nil1:01':ok3_0 :: Nat → mark:0':nil:nil1:01':okLemmas:
sel(gen_mark:0':nil:nil1:01':ok3_0(+(1, n5_0)), gen_mark:0':nil:nil1:01':ok3_0(b)) → *4_0, rt ∈ Ω(n50)Generator Equations:
gen_mark:0':nil:nil1:01':ok3_0(0) ⇔ 0'
gen_mark:0':nil:nil1:01':ok3_0(+(x, 1)) ⇔ mark(gen_mark:0':nil:nil1:01':ok3_0(x))No more defined symbols left to analyse.