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Step * 1 1 1 1 1 of Lemma coW-equiv-implies

.....subterm..... T:t
2:n
1. A : 𝕌'
2. B : A ⟶ Type
3. w : coW(A;a.B[a])
4. w' : coW(A;a.B[a])
5. z : coW(A;a.B[a])
6. b : coW-dom(a.B[a];w)
7. coW-equiv(a.B[a];z;coW-item(w;b))
8. q1 : copath(a.B[a];w)
9. q2 : copath(a.B[a];w')
10. win2(coW-game(a.B[a];w;w')@<q1, q2>)
11. copath-cons(b;()) = q1 ∈ copath(a.B[a];w)
12. copath-length(q2) = 1 ∈ ℤ
13. sg-normalize(coW-game(a.B[a];coW-item(w;b);coW-item(w';copath-hd(q2)))) ≅
    coW-game(a.B[a];w;w')@<copath-cons(b;()), copath-cons(copath-hd(q2);())>
⊢ copath-cons(copath-hd(q2);()) = q2 ∈ copath(a.B[a];w')
BY
{ (InstLemma `copath-eta` [⌜A⌝;⌜B⌝;⌜w'⌝;⌜q2⌝]⋅ THENA Auto) }

1
1. A : 𝕌'
2. B : A ⟶ Type
3. w : coW(A;a.B[a])
4. w' : coW(A;a.B[a])
5. z : coW(A;a.B[a])
6. b : coW-dom(a.B[a];w)
7. coW-equiv(a.B[a];z;coW-item(w;b))
8. q1 : copath(a.B[a];w)
9. q2 : copath(a.B[a];w')
10. win2(coW-game(a.B[a];w;w')@<q1, q2>)
11. copath-cons(b;()) = q1 ∈ copath(a.B[a];w)
12. copath-length(q2) = 1 ∈ ℤ
13. sg-normalize(coW-game(a.B[a];coW-item(w;b);coW-item(w';copath-hd(q2)))) ≅
    coW-game(a.B[a];w;w')@<copath-cons(b;()), copath-cons(copath-hd(q2);())>
14. copath-cons(copath-hd(q2);copath-tl(q2)) = q2 ∈ copath(a.B[a];w')
⊢ copath-cons(copath-hd(q2);()) = q2 ∈ copath(a.B[a];w')

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Latex:

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Latex:
.....subterm.....  T:t
2:n
1.  A  :  \mBbbU{}'
2.  B  :  A  {}\mrightarrow{}  Type
3.  w  :  coW(A;a.B[a])
4.  w'  :  coW(A;a.B[a])
5.  z  :  coW(A;a.B[a])
6.  b  :  coW-dom(a.B[a];w)
7.  coW-equiv(a.B[a];z;coW-item(w;b))
8.  q1  :  copath(a.B[a];w)
9.  q2  :  copath(a.B[a];w')
10.  win2(coW-game(a.B[a];w;w')@<q1,  q2>)
11.  copath-cons(b;())  =  q1
12.  copath-length(q2)  =  1
13.  sg-normalize(coW-game(a.B[a];coW-item(w;b);coW-item(w';copath-hd(q2))))  \mcong{}
        coW-game(a.B[a];w;w')@<copath-cons(b;()),  copath-cons(copath-hd(q2);())>
\mvdash{}  copath-cons(copath-hd(q2);())  =  q2


By

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Latex:
(InstLemma  `copath-eta`  [\mkleeneopen{}A\mkleeneclose{};\mkleeneopen{}B\mkleeneclose{};\mkleeneopen{}w'\mkleeneclose{};\mkleeneopen{}q2\mkleeneclose{}]\mcdot{}  THENA  Auto)

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