Step
*
2
2
2
2
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. is-exception(if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff)
13. n ∈ ℤ
14. is-exception(case M n s of inl() => 0 | inr() => n + 1)
15. M n s ∈ Top + Top
16. y : Top
17. (M n s) = (inr y ) ∈ (Top + Top)
⊢ 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)) ≤ ind n s
(λt.(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
{ SqLeCD }
1
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. is-exception(if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff)
13. n ∈ ℤ
14. is-exception(case M n s of inl() => 0 | inr() => n + 1)
15. M n s ∈ Top + Top
16. y : Top
17. (M n s) = (inr y ) ∈ (Top + Top)
⊢ ind n s ≤ ind n s
2
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. is-exception(if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff)
13. n ∈ ℤ
14. is-exception(case M n s of inl() => 0 | inr() => n + 1)
15. M n s ∈ Top + Top
16. y : Top
17. (M n s) = (inr y ) ∈ (Top + Top)
⊢ 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) ≤ λt.(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)
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. is-exception(if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff)
13. n \mmember{} \mBbbZ{}
14. is-exception(case M n s of inl() => 0 | inr() => n + 1)
15. M n s \mmember{} Top + Top
16. y : Top
17. (M n s) = (inr y )
\mvdash{} 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)) \mleq{} ind n s
(\mlambda{}t.(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:
SqLeCD
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