Step
*
2
2
2
2
3
of Lemma
decidable-bar-rec-equal-spector
1. j : ℤ
2. 0 < j
3. ∀n:ℕ. ∀s,ind,base,dec:Base.
(λspector-bar-rec,n,s. case case if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case dec n s of inl(r) => base n s r | inr(x) => ⊥
| inr() =>
ind n s (λt.(spector-bar-rec (n + 1) s.t@n))^j - 1
⊥
n
s ≤ fix((λdecidable-bar-rec,n,s. case dec n s
of inl(r) =>
base n s r
| inr(x) =>
ind n s (λt.(decidable-bar-rec (n + 1) s.t@n))))
n
s)
4. n : ℕ
5. s : Base
6. ind : Base
7. base : Base
8. dec : Base
9. is-exception(case case if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case dec n s of inl(r) => base n s r | inr(x) => ⊥
| inr() =>
ind n s
(λt.(λspector-bar-rec,n,s. case case if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case dec n s of inl(r) => base n s r | inr(x) => ⊥
| inr() =>
ind n s (λt.(spector-bar-rec (n + 1) s.t@n))^j - 1
⊥
(n + 1)
s.t@n)))
10. is-exception(case if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt)
11. is-exception(if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff)
12. is-exception(n)
⊢ case case if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff of inl() => ff | inr() => tt
of inl() =>
case dec n s of inl(r) => base n s r | inr(x) => ⊥
| inr() =>
ind n s
(λt.(λspector-bar-rec,n,s. case case if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case dec n s of inl(r) => base n s r | inr(x) => ⊥
| inr() =>
ind n s (λt.(spector-bar-rec (n + 1) s.t@n))^j - 1
⊥
(n + 1)
s.t@n)) ≤ case dec n s
of inl(r) =>
base n s r
| inr(x) =>
ind n s
(λt.(fix((λdecidable-bar-rec,n,s. case dec n s
of inl(r) =>
base n s r
| inr(x) =>
ind n s (λt.(decidable-bar-rec (n + 1) s.t@n))))
(n + 1)
s.t@n))
BY
{ (ExceptionSqequal (-1)
THEN (Assert ⌜(n)↓⌝⋅ THENA Auto)
THEN (RWO "-2" (-1) THENA Auto)
THEN FLemma `not-exception-has-value` [-1]
THEN Auto) }
Latex:
Latex:
1. j : \mBbbZ{}
2. 0 < j
3. \mforall{}n:\mBbbN{}. \mforall{}s,ind,base,dec:Base.
(\mlambda{}spector-bar-rec,n,s. case case if (n) < (case dec n s of inl() => 0 | inr() => n + 1)
then tt
else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case dec n s of inl(r) => base n s r | inr(x) => \mbot{}
| inr() =>
ind n s (\mlambda{}t.(spector-bar-rec (n + 1) s.t@n))\^{}j - 1
\mbot{}
n
s \mleq{} fix((\mlambda{}decidable-bar-rec,n,s. case dec n s
of inl(r) =>
base n s r
| inr(x) =>
ind n s (\mlambda{}t.(decidable-bar-rec (n + 1) s.t@n))))
n
s)
4. n : \mBbbN{}
5. s : Base
6. ind : Base
7. base : Base
8. dec : Base
9. is-exception(case case if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case dec n s of inl(r) => base n s r | inr(x) => \mbot{}
| inr() =>
ind n s
(\mlambda{}t.(\mlambda{}spector-bar-rec,n,s. case case if (n) < (case dec n s of inl() => 0 | inr() => n + 1)
then tt
else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case dec n s of inl(r) => base n s r | inr(x) => \mbot{}
| inr() =>
ind n s (\mlambda{}t.(spector-bar-rec (n + 1) s.t@n))\^{}j - 1
\mbot{}
(n + 1)
s.t@n)))
10. is-exception(case if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt)
11. is-exception(if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff)
12. is-exception(n)
\mvdash{} case case if (n) < (case dec n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case dec n s of inl(r) => base n s r | inr(x) => \mbot{}
| inr() =>
ind n s
(\mlambda{}t.(\mlambda{}spector-bar-rec,n,s. case case if (n) < (case dec n s of inl() => 0 | inr() => n + 1)
then tt
else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case dec n s of inl(r) => base n s r | inr(x) => \mbot{}
| inr() =>
ind n s (\mlambda{}t.(spector-bar-rec (n + 1) s.t@n))\^{}j - 1
\mbot{}
(n + 1)
s.t@n)) \mleq{} case dec n s
of inl(r) =>
base n s r
| inr(x) =>
ind n s
(\mlambda{}t.(fix((\mlambda{}decidable-bar-rec,n,s. case dec n s
of inl(r) =>
base n s r
| inr(x) =>
ind n s (\mlambda{}t.(decidable-bar-rec (n + 1) s.t@n))))
(n + 1)
s.t@n))
By
Latex:
(ExceptionSqequal (-1)
THEN (Assert \mkleeneopen{}(n)\mdownarrow{}\mkleeneclose{}\mcdot{} THENA Auto)
THEN (RWO "-2" (-1) THENA Auto)
THEN FLemma `not-exception-has-value` [-1]
THEN Auto)
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