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of Lemma
coW-equiv-iff3
1. [A] : 𝕌'
2. B : A ⟶ Type
3. w : coW(A;a.B[a])
4. w' : coW(A;a.B[a])
5. e : coW-equiv(a.B[a];w;w')
6. p : ℕ ⟶ copath(a.B[a];w')
7. [%2] : ∀n:ℕ. ((∀i:ℕn. (copath-length(p i) = i ∈ ℤ)) ⇒ (∀i:ℕn - 1. copathAgree(a.B[a];w';p i;p (i + 1))))
8. n : ℕ
9. q : copath(a.B[a];w)
10. ∀i:ℕn + 2. (copath-length(p i) = i ∈ ℤ)
11. copath-length(q) = n ∈ ℤ
12. ∀z:coW(A;a.B[a]). (coWmem(a.B[a];z;copath-at(w;q)) ⇐⇒ coWmem(a.B[a];z;copath-at(w';p n)))
13. copathAgree(a.B[a];w';p n;p (n + 1))
14. coWmem(a.B[a];copath-at(w';p (n + 1));copath-at(w;q)) ⇒ coWmem(a.B[a];copath-at(w';p (n + 1));copath-at(w';p n))
15. (fst(copath-last(w';p (n + 1)))) = copath-at(w';p n) ∈ coW(A;a.B[a])
16. copath-at(w';p (n + 1)) = coW-item(copath-at(w';p n);snd(copath-last(w';p (n + 1)))) ∈ coW(A;a.B[a])
⊢ coWmem(a.B[a];copath-at(w';p (n + 1));copath-at(w';p n))
BY
{ (D 0 With ⌜snd(copath-last(w';p (n + 1)))⌝ THEN Auto) }
1
1. A : 𝕌'
2. B : A ⟶ Type
3. w : coW(A;a.B[a])
4. w' : coW(A;a.B[a])
5. e : coW-equiv(a.B[a];w;w')
6. p : ℕ ⟶ copath(a.B[a];w')
7. ∀n:ℕ. ((∀i:ℕn. (copath-length(p i) = i ∈ ℤ)) ⇒ (∀i:ℕn - 1. copathAgree(a.B[a];w';p i;p (i + 1))))
8. n : ℕ
9. q : copath(a.B[a];w)
10. ∀i:ℕn + 2. (copath-length(p i) = i ∈ ℤ)
11. copath-length(q) = n ∈ ℤ
12. ∀z:coW(A;a.B[a]). (coWmem(a.B[a];z;copath-at(w;q)) ⇐⇒ coWmem(a.B[a];z;copath-at(w';p n)))
13. copathAgree(a.B[a];w';p n;p (n + 1))
14. coWmem(a.B[a];copath-at(w';p (n + 1));copath-at(w;q)) ⇒ coWmem(a.B[a];copath-at(w';p (n + 1));copath-at(w';p n))
15. (fst(copath-last(w';p (n + 1)))) = copath-at(w';p n) ∈ coW(A;a.B[a])
16. copath-at(w';p (n + 1)) = coW-item(copath-at(w';p n);snd(copath-last(w';p (n + 1)))) ∈ coW(A;a.B[a])
⊢ 0 < copath-length(p (n + 1))
2
1. [A] : 𝕌'
2. B : A ⟶ Type
3. w : coW(A;a.B[a])
4. w' : coW(A;a.B[a])
5. e : coW-equiv(a.B[a];w;w')
6. p : ℕ ⟶ copath(a.B[a];w')
7. [%2] : ∀n:ℕ. ((∀i:ℕn. (copath-length(p i) = i ∈ ℤ)) ⇒ (∀i:ℕn - 1. copathAgree(a.B[a];w';p i;p (i + 1))))
8. n : ℕ
9. q : copath(a.B[a];w)
10. ∀i:ℕn + 2. (copath-length(p i) = i ∈ ℤ)
11. copath-length(q) = n ∈ ℤ
12. ∀z:coW(A;a.B[a]). (coWmem(a.B[a];z;copath-at(w;q)) ⇐⇒ coWmem(a.B[a];z;copath-at(w';p n)))
13. copathAgree(a.B[a];w';p n;p (n + 1))
14. coWmem(a.B[a];copath-at(w';p (n + 1));copath-at(w;q)) ⇒ coWmem(a.B[a];copath-at(w';p (n + 1));copath-at(w';p n))
15. (fst(copath-last(w';p (n + 1)))) = copath-at(w';p n) ∈ coW(A;a.B[a])
16. copath-at(w';p (n + 1)) = coW-item(copath-at(w';p n);snd(copath-last(w';p (n + 1)))) ∈ coW(A;a.B[a])
⊢ coW-equiv(a.B[a];copath-at(w';p (n + 1));coW-item(copath-at(w';p n);snd(copath-last(w';p (n + 1)))))
Latex:
Latex:
1. [A] : \mBbbU{}'
2. B : A {}\mrightarrow{} Type
3. w : coW(A;a.B[a])
4. w' : coW(A;a.B[a])
5. e : coW-equiv(a.B[a];w;w')
6. p : \mBbbN{} {}\mrightarrow{} copath(a.B[a];w')
7. [\%2] : \mforall{}n:\mBbbN{}
((\mforall{}i:\mBbbN{}n. (copath-length(p i) = i)) {}\mRightarrow{} (\mforall{}i:\mBbbN{}n - 1. copathAgree(a.B[a];w';p i;p (i + 1))))
8. n : \mBbbN{}
9. q : copath(a.B[a];w)
10. \mforall{}i:\mBbbN{}n + 2. (copath-length(p i) = i)
11. copath-length(q) = n
12. \mforall{}z:coW(A;a.B[a]). (coWmem(a.B[a];z;copath-at(w;q)) \mLeftarrow{}{}\mRightarrow{} coWmem(a.B[a];z;copath-at(w';p n)))
13. copathAgree(a.B[a];w';p n;p (n + 1))
14. coWmem(a.B[a];copath-at(w';p (n + 1));copath-at(w;q))
{}\mRightarrow{} coWmem(a.B[a];copath-at(w';p (n + 1));copath-at(w';p n))
15. (fst(copath-last(w';p (n + 1)))) = copath-at(w';p n)
16. copath-at(w';p (n + 1)) = coW-item(copath-at(w';p n);snd(copath-last(w';p (n + 1))))
\mvdash{} coWmem(a.B[a];copath-at(w';p (n + 1));copath-at(w';p n))
By
Latex:
(D 0 With \mkleeneopen{}snd(copath-last(w';p (n + 1)))\mkleeneclose{} THEN Auto)
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