---

Step * 2 1 1 2 1 1 of Lemma twkl!-implies-dfan

.....wf..... 
1. T : Type
2. ∃k:ℕ. T ~ ℕk
3. WKL!(T)
4. s : (T List) ⟶ ℙ
5. ∀t:T List. Dec(s t)
6. ∀f:ℕ ⟶ T. ∃n:ℕ. (s map(f;upto(n)))
7. ¬(s [])
8. ∀a,b:ℕ.  ((a ≤ b) ⇒ tree-big(T;upwd-closure(T;s);a) ⇒ tree-big(T;upwd-closure(T;s);b))
9. ∀n:{n:ℕ| tree-big(T;upwd-closure(T;s);n)} 
     ∃k:ℕn. ((¬tree-big(T;upwd-closure(T;s);k)) ∧ tree-big(T;upwd-closure(T;s);k + 1))
10. ∀n:{n:ℕ| ¬tree-big(T;upwd-closure(T;s);n)} . ∃as:T List. ((||as|| = n ∈ ℤ) ∧ (¬(upwd-closure(T;s) as)))
11. nb : n:{n:ℕ| tree-big(T;upwd-closure(T;s);n)}  ⟶ ℕn
12. ∀n:{n:ℕ| tree-big(T;upwd-closure(T;s);n)} 
      ((¬tree-big(T;upwd-closure(T;s);nb n)) ∧ tree-big(T;upwd-closure(T;s);(nb n) + 1))
13. u : n:{n:ℕ| ¬tree-big(T;upwd-closure(T;s);n)}  ⟶ (T List)
14. ∀n:{n:ℕ| ¬tree-big(T;upwd-closure(T;s);n)} . ((||u n|| = n ∈ ℤ) ∧ (¬(upwd-closure(T;s) (u n))))
⊢ u o nb ∈ {n:ℕ| tree-big(T;upwd-closure(T;s);n)}  ⟶ (T List)
BY
{ (Auto THEN ExtWith [`x'] [⌜n:{n:ℕ| tree-big(T;upwd-closure(T;s);n)}  ⟶ ℕn⌝]⋅ THEN Auto) }

---

Latex:

---

Latex:
.....wf..... 
1.  T  :  Type
2.  \mexists{}k:\mBbbN{}.  T  \msim{}  \mBbbN{}k
3.  WKL!(T)
4.  s  :  (T  List)  {}\mrightarrow{}  \mBbbP{}
5.  \mforall{}t:T  List.  Dec(s  t)
6.  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  T.  \mexists{}n:\mBbbN{}.  (s  map(f;upto(n)))
7.  \mneg{}(s  [])
8.  \mforall{}a,b:\mBbbN{}.    ((a  \mleq{}  b)  {}\mRightarrow{}  tree-big(T;upwd-closure(T;s);a)  {}\mRightarrow{}  tree-big(T;upwd-closure(T;s);b))
9.  \mforall{}n:\{n:\mBbbN{}|  tree-big(T;upwd-closure(T;s);n)\} 
          \mexists{}k:\mBbbN{}n.  ((\mneg{}tree-big(T;upwd-closure(T;s);k))  \mwedge{}  tree-big(T;upwd-closure(T;s);k  +  1))
10.  \mforall{}n:\{n:\mBbbN{}|  \mneg{}tree-big(T;upwd-closure(T;s);n)\} 
            \mexists{}as:T  List.  ((||as||  =  n)  \mwedge{}  (\mneg{}(upwd-closure(T;s)  as)))
11.  nb  :  n:\{n:\mBbbN{}|  tree-big(T;upwd-closure(T;s);n)\}    {}\mrightarrow{}  \mBbbN{}n
12.  \mforall{}n:\{n:\mBbbN{}|  tree-big(T;upwd-closure(T;s);n)\} 
            ((\mneg{}tree-big(T;upwd-closure(T;s);nb  n))  \mwedge{}  tree-big(T;upwd-closure(T;s);(nb  n)  +  1))
13.  u  :  n:\{n:\mBbbN{}|  \mneg{}tree-big(T;upwd-closure(T;s);n)\}    {}\mrightarrow{}  (T  List)
14.  \mforall{}n:\{n:\mBbbN{}|  \mneg{}tree-big(T;upwd-closure(T;s);n)\}  .  ((||u  n||  =  n)  \mwedge{}  (\mneg{}(upwd-closure(T;s)  (u  n))))
\mvdash{}  u  o  nb  \mmember{}  \{n:\mBbbN{}|  tree-big(T;upwd-closure(T;s);n)\}    {}\mrightarrow{}  (T  List)


By

---

Latex:
(Auto  THEN  ExtWith  [`x']  [\mkleeneopen{}n:\{n:\mBbbN{}|  tree-big(T;upwd-closure(T;s);n)\}    {}\mrightarrow{}  \mBbbN{}n\mkleeneclose{}]\mcdot{}  THEN  Auto)

---

Home Index