Step
*
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. (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)))↓
⊢ 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)) ≤ 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.(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
{ (HasValueD (-1)
THEN (Assert ⌜(case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt)↓⌝⋅
THENA Auto
)
THEN HasValueD (-1)
THEN (Assert ⌜(if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff)↓⌝⋅ THENA Auto)
THEN HasValueD (-1)
THEN (Assert ⌜(case M n s of inl() => 0 | inr() => n + 1)↓⌝⋅ THENA Auto)
THEN HasValueD (-1)) }
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. (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. case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff of inl() => ff | inr() => tt ∈ Top + Top
12. (case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff of inl() => ff | inr() => tt)↓
13. if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff ∈ Top + Top
14. (if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff)↓
15. n ∈ ℤ
16. case M n s of inl() => 0 | inr() => n + 1 ∈ ℤ
17. (case M n s of inl() => 0 | inr() => n + 1)↓
18. M n s ∈ Top + Top
⊢ 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)) ≤ 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.(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. (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)))\mdownarrow{}
\mvdash{} 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)) \mleq{} 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.(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:
(HasValueD (-1)
THEN (Assert \mkleeneopen{}(case if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff
of inl() =>
ff
| inr() =>
tt)\mdownarrow{}\mkleeneclose{}\mcdot{}
THENA Auto
)
THEN HasValueD (-1)
THEN (Assert \mkleeneopen{}(if (n) < (case M n s of inl() => 0 | inr() => n + 1) then tt else ff)\mdownarrow{}\mkleeneclose{}\mcdot{}
THENA Auto
)
THEN HasValueD (-1)
THEN (Assert \mkleeneopen{}(case M n s of inl() => 0 | inr() => n + 1)\mdownarrow{}\mkleeneclose{}\mcdot{} THENA Auto)
THEN HasValueD (-1))
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