Step
*
2
2
2
1
2
1
2
1
of Lemma
howard-bar-rec-equal-spector
1. j : ℤ
2. 0 < j
3. ∀n:ℕ. ∀s,ind,base,mon,M:Base.
(λspector-bar-rec,n,s. case case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case M n s of inl(x) => let k,p = x in base n s (mon n k s p) | inr(x) => ⊥
| inr() =>
ind n s (λt.(spector-bar-rec (n + 1) s.t@n))^j - 1
⊥
n
s ≤ fix((λhoward-bar-rec,n,s. case M n s
of inl(x) =>
let k,p = x
in base n s (mon n k s p)
| inr(x) =>
ind n s (λt.(howard-bar-rec (n + 1) s.t@n))))
n
s)
4. n : ℕ
5. s : Base
6. ind : Base
7. base : Base
8. mon : Base
9. M : Base
10. is-exception(case case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case M n s of inl(x) => let k,p = x in base n s (mon n k s p) | inr(x) => ⊥
| inr() =>
ind n s
(λt.(λspector-bar-rec,n,s. case case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case M n s of inl(x) => let k,p = x in base n s (mon n k s p) | inr(x) => ⊥
| inr() =>
ind n s (λt.(spector-bar-rec (n + 1) s.t@n))^j - 1
⊥
(n + 1)
s.t@n)))
11. is-exception(case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt)
12. if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff ∈ Top + Top
13. (if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff)↓
14. n ∈ ℤ
15. case M n s of inl() => 0 | inr() => n + 1 ∈ ℤ
16. (case M n s of inl() => 0 | inr() => n + 1)↓
17. M n s ∈ Top + Top
18. y : Top
19. (M n s) = (inr y ) ∈ (Top + Top)
20. t : Base
⊢ λspector-bar-rec,n,s. case case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case M n s of inl(x) => let k,p = x in base n s (mon n k s p) | inr(x) => ⊥
| inr() =>
ind n s (λt.(spector-bar-rec (n + 1) s.t@n))^j - 1
⊥
(n + 1)
s.t@n ≤ fix((λhoward-bar-rec,n,s. case M n s
of inl(x) =>
let k,p = x
in base n s (mon n k s p)
| inr(x) =>
ind n s (λt.(howard-bar-rec (n + 1) s.t@n))))
(n + 1)
s.t@n
BY
{ (BHyp 3 THEN Auto) }
Latex:
Latex:
1. j : \mBbbZ{}
2. 0 < j
3. \mforall{}n:\mBbbN{}. \mforall{}s,ind,base,mon,M:Base.
(\mlambda{}spector-bar-rec,n,s. case case if (n) < (case M n s of inl() => 0 | inr() => n + 1)
then tt
else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case M n s
of inl(x) =>
let k,p = x
in base n s (mon n k s p)
| 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{}howard-bar-rec,n,s. case M n s
of inl(x) =>
let k,p = x
in base n s (mon n k s p)
| inr(x) =>
ind n s (\mlambda{}t.(howard-bar-rec (n + 1) s.t@n))))
n
s)
4. n : \mBbbN{}
5. s : Base
6. ind : Base
7. base : Base
8. mon : Base
9. M : Base
10. is-exception(case case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case M n s of inl(x) => let k,p = x in base n s (mon n k s p) | inr(x) => \mbot{}
| inr() =>
ind n s
(\mlambda{}t.(\mlambda{}spector-bar-rec,n,s. case case if (n) < (case M n s of inl() => 0 | inr() => n + 1)
then tt
else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case M n s
of inl(x) =>
let k,p = x
in base n s (mon n k s p)
| 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)))
11. is-exception(case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt)
12. if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff \mmember{} Top + Top
13. (if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff)\mdownarrow{}
14. n \mmember{} \mBbbZ{}
15. case M n s of inl() => 0 | inr() => n + 1 \mmember{} \mBbbZ{}
16. (case M n s of inl() => 0 | inr() => n + 1)\mdownarrow{}
17. M n s \mmember{} Top + Top
18. y : Top
19. (M n s) = (inr y )
20. t : Base
\mvdash{} \mlambda{}spector-bar-rec,n,s. case case if (n) < (case M n s of inl() => 0 | inr() => n + 1)
then tt
else ff
of inl() =>
ff
| inr() =>
tt
of inl() =>
case M n s of inl(x) => let k,p = x in base n s (mon n k s p) | 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{} fix((\mlambda{}howard-bar-rec,n,s. case M n s
of inl(x) =>
let k,p = x
in base n s (mon n k s p)
| inr(x) =>
ind n s (\mlambda{}t.(howard-bar-rec (n + 1) s.t@n))))
(n + 1)
s.t@n
By
Latex:
(BHyp 3 THEN Auto)
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