KILLED



    


Runtime Complexity (full) proof of /tmp/tmpfEP64E/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 DecreasingLoopProof (⇔, 8587 ms)


↳2 BOUNDS(n^1, INF)


↳3 RenamingProof (⇔, 0 ms)


↳4 CpxRelTRS


    ↳5 TypeInferenceProof (BOTH BOUNDS(ID, ID), 0 ms)


    ↳6 typed CpxTrs


    ↳7 OrderProof (LOWER BOUND(ID), 0 ms)


    ↳8 typed CpxTrs


    ↳9 NoRewriteLemmaProof (LOWER BOUND(ID), 42 ms)


    ↳10 typed CpxTrs


    ↳11 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳12 typed CpxTrs


    ↳13 NoRewriteLemmaProof (LOWER BOUND(ID), 15 ms)


    ↳14 typed CpxTrs


    ↳15 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳16 typed CpxTrs


    ↳17 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳18 typed CpxTrs


    ↳19 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳20 typed CpxTrs


    ↳21 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳22 typed CpxTrs


    ↳23 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳24 typed CpxTrs


    ↳25 NoRewriteLemmaProof (LOWER BOUND(ID), 1 ms)


    ↳26 typed CpxTrs


    ↳27 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳28 typed CpxTrs


    ↳29 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳30 typed CpxTrs


    ↳31 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳32 typed CpxTrs


    ↳33 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳34 typed CpxTrs


    ↳35 NoRewriteLemmaProof (LOWER BOUND(ID), 0 ms)


    ↳36 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) DecreasingLoopProof (EQUIVALENT transformation)
The following loop(s) give(s) rise to the lower bound Ω(n1):
The rewrite sequence
sel(mark(X1), X2) →+ mark(sel(X1, X2))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0].
The pumping substitution is [X1 / mark(X1)].
The result substitution is [ ].





(2) BOUNDS(n^1, INF)




(3) RenamingProof (EQUIVALENT transformation)
Renamed function symbols to avoid clashes with predefined symbol.




(4) Obligation:
Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

active(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




(5) TypeInferenceProof (BOTH BOUNDS(ID, ID) transformation)
Infered types.




(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':ok




(7) 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, top
They 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




(8) 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
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:
sel, active, cons, first, from, s, sel1, quote, cons1, first1, quote1, s1, unquote, fcons, unquote1, proper, top
They 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




(9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol sel.




(10) 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
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, top
They 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




(11) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol cons.




(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':ok
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, top
They 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) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol first.




(14) 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
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, top
They 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




(15) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol from.




(16) 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
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, top
They 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




(17) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol s.




(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':ok
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, top
They 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




(19) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol sel1.




(20) 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
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, top
They 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




(21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol quote.




(22) 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
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, top
They 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




(23) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol cons1.




(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':ok
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, top
They 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




(25) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol first1.




(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':ok
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, top
They 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




(27) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol quote1.




(28) 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
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, top
They 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




(29) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol s1.




(30) 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
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, top
They will be analysed ascendingly in the following order:
unquote < active
fcons < active
unquote1 < active
active < top
unquote < proper
fcons < proper
unquote1 < proper
proper < top




(31) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol unquote.




(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':ok
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, top
They will be analysed ascendingly in the following order:
fcons < active
unquote1 < active
active < top
fcons < proper
unquote1 < proper
proper < top




(33) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol fcons.




(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':ok
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, top
They will be analysed ascendingly in the following order:
unquote1 < active
active < top
unquote1 < proper
proper < top




(35) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)
Could not prove a rewrite lemma for the defined symbol unquote1.




(36) 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
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, top
They will be analysed ascendingly in the following order:
active < top
proper < top