Step
*
1
2
1
1
of Lemma
coW-equiv-iff2
1. [A] : 𝕌'
2. B : A ⟶ Type
3. n : ℤ
4. [%1] : 0 < n
5. ∀w,w':coW(A;a.B[a]).
(coW-equiv(a.B[a];w;w')
⇒ (∀p1:coPath(a.B[a];w';n - 1)
∃q:copath(a.B[a];w)
((copath-length(q) = (n - 1) ∈ ℤ) ∧ coW-equiv(a.B[a];coPath-at(n - 1;w';p1);copath-at(w;q)))))
6. w : coW(A;a.B[a])
7. w' : coW(A;a.B[a])
8. coW-equiv(a.B[a];w;w')
9. t : coW-dom(a.B[a];w')
10. p2 : coPath(a.B[a];coW-item(w';t);n - 1)
11. b : coW-dom(a.B[a];w)
12. coW-equiv(a.B[a];coW-item(w';t);coW-item(w;b))
13. q : copath(a.B[a];coW-item(w;b))
14. copath-length(q) = (n - 1) ∈ ℤ
15. coW-equiv(a.B[a];coPath-at(n - 1;coW-item(w';t);p2);copath-at(coW-item(w;b);q))
⊢ ∃q:copath(a.B[a];w). ((copath-length(q) = n ∈ ℤ) ∧ coW-equiv(a.B[a];coPath-at(n;w';<t, p2>);copath-at(w;q)))
BY
{ (D 0 With ⌜copath-cons(b;q)⌝ THEN Auto) }
1
1. A : 𝕌'
2. B : A ⟶ Type
3. n : ℤ
4. 0 < n
5. ∀w,w':coW(A;a.B[a]).
(coW-equiv(a.B[a];w;w')
⇒ (∀p1:coPath(a.B[a];w';n - 1)
∃q:copath(a.B[a];w)
((copath-length(q) = (n - 1) ∈ ℤ) ∧ coW-equiv(a.B[a];coPath-at(n - 1;w';p1);copath-at(w;q)))))
6. w : coW(A;a.B[a])
7. w' : coW(A;a.B[a])
8. coW-equiv(a.B[a];w;w')
9. t : coW-dom(a.B[a];w')
10. p2 : coPath(a.B[a];coW-item(w';t);n - 1)
11. b : coW-dom(a.B[a];w)
12. coW-equiv(a.B[a];coW-item(w';t);coW-item(w;b))
13. q : copath(a.B[a];coW-item(w;b))
14. copath-length(q) = (n - 1) ∈ ℤ
15. coW-equiv(a.B[a];coPath-at(n - 1;coW-item(w';t);p2);copath-at(coW-item(w;b);q))
⊢ copath-length(copath-cons(b;q)) = n ∈ ℤ
2
1. [A] : 𝕌'
2. B : A ⟶ Type
3. n : ℤ
4. [%1] : 0 < n
5. ∀w,w':coW(A;a.B[a]).
(coW-equiv(a.B[a];w;w')
⇒ (∀p1:coPath(a.B[a];w';n - 1)
∃q:copath(a.B[a];w)
((copath-length(q) = (n - 1) ∈ ℤ) ∧ coW-equiv(a.B[a];coPath-at(n - 1;w';p1);copath-at(w;q)))))
6. w : coW(A;a.B[a])
7. w' : coW(A;a.B[a])
8. coW-equiv(a.B[a];w;w')
9. t : coW-dom(a.B[a];w')
10. p2 : coPath(a.B[a];coW-item(w';t);n - 1)
11. b : coW-dom(a.B[a];w)
12. coW-equiv(a.B[a];coW-item(w';t);coW-item(w;b))
13. q : copath(a.B[a];coW-item(w;b))
14. copath-length(q) = (n - 1) ∈ ℤ
15. coW-equiv(a.B[a];coPath-at(n - 1;coW-item(w';t);p2);copath-at(coW-item(w;b);q))
16. copath-length(copath-cons(b;q)) = n ∈ ℤ
⊢ coW-equiv(a.B[a];coPath-at(n;w';<t, p2>);copath-at(w;copath-cons(b;q)))
3
1. A : 𝕌'
2. B : A ⟶ Type
3. n : ℤ
4. 0 < n
5. ∀w,w':coW(A;a.B[a]).
(coW-equiv(a.B[a];w;w')
⇒ (∀p1:coPath(a.B[a];w';n - 1)
∃q:copath(a.B[a];w)
((copath-length(q) = (n - 1) ∈ ℤ) ∧ coW-equiv(a.B[a];coPath-at(n - 1;w';p1);copath-at(w;q)))))
6. w : coW(A;a.B[a])
7. w' : coW(A;a.B[a])
8. coW-equiv(a.B[a];w;w')
9. t : coW-dom(a.B[a];w')
10. p2 : coPath(a.B[a];coW-item(w';t);n - 1)
11. b : coW-dom(a.B[a];w)
12. coW-equiv(a.B[a];coW-item(w';t);coW-item(w;b))
13. q : copath(a.B[a];coW-item(w;b))
14. copath-length(q) = (n - 1) ∈ ℤ
15. coW-equiv(a.B[a];coPath-at(n - 1;coW-item(w';t);p2);copath-at(coW-item(w;b);q))
16. q1 : copath(a.B[a];w)
17. copath-length(q1) = n ∈ ℤ
⊢ <t, p2> ∈ coPath(a.B[a];w';n)
Latex:
Latex:
1. [A] : \mBbbU{}'
2. B : A {}\mrightarrow{} Type
3. n : \mBbbZ{}
4. [\%1] : 0 < n
5. \mforall{}w,w':coW(A;a.B[a]).
(coW-equiv(a.B[a];w;w')
{}\mRightarrow{} (\mforall{}p1:coPath(a.B[a];w';n - 1)
\mexists{}q:copath(a.B[a];w)
((copath-length(q) = (n - 1))
\mwedge{} coW-equiv(a.B[a];coPath-at(n - 1;w';p1);copath-at(w;q)))))
6. w : coW(A;a.B[a])
7. w' : coW(A;a.B[a])
8. coW-equiv(a.B[a];w;w')
9. t : coW-dom(a.B[a];w')
10. p2 : coPath(a.B[a];coW-item(w';t);n - 1)
11. b : coW-dom(a.B[a];w)
12. coW-equiv(a.B[a];coW-item(w';t);coW-item(w;b))
13. q : copath(a.B[a];coW-item(w;b))
14. copath-length(q) = (n - 1)
15. coW-equiv(a.B[a];coPath-at(n - 1;coW-item(w';t);p2);copath-at(coW-item(w;b);q))
\mvdash{} \mexists{}q:copath(a.B[a];w). ((copath-length(q) = n) \mwedge{} coW-equiv(a.B[a];coPath-at(n;w';<t, p2>);copath-at(\000Cw;q)))
By
Latex:
(D 0 With \mkleeneopen{}copath-cons(b;q)\mkleeneclose{} THEN Auto)
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