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Step * 1 of Lemma pcw-path-copathAgree


1. [A] : 𝕌'
2. [B] : A ⟶ Type
3. w : coW(A;a.B[a])
4. path : Path
5. StepAgree(path 0;⋅;w)
6. i : ℕ
7. (pcw-path-coPath(i;path) ∈ copath(a.B[a];w))
∧ ((copath-length(pcw-path-coPath(i;path)) = i ∈ ℤ)
  ⇒ (copath-at(w;pcw-path-coPath(i;path)) = (fst(snd((path i)))) ∈ coW(A;a.B[a])))
8. (pcw-path-coPath(i + 1;path) ∈ copath(a.B[a];w))
∧ ((copath-length(pcw-path-coPath(i + 1;path)) = (i + 1) ∈ ℤ)
  ⇒ (copath-at(w;pcw-path-coPath(i + 1;path)) = (fst(snd((path (i + 1))))) ∈ coW(A;a.B[a])))
⊢ (copath-length(pcw-path-coPath(i + 1;path)) = (i + 1) ∈ ℤ)
⇒ (copath-length(pcw-path-coPath(i;path)) = i ∈ ℤ)
⇒ copathAgree(a.B[a];w;pcw-path-coPath(i;path);pcw-path-coPath(i + 1;path))
BY
{ (SplitAndHyps
   THEN ParallelLast
   THEN ParallelOp -4
   THEN (Assert ⌜↓copathAgree(a.B[a];w;pcw-path-coPath(i;path);pcw-path-coPath(i + 1;path))⌝⋅
   THENM (D -1 THEN Unhide THEN Auto)
   )) }

1
.....assertion..... 
1. [A] : 𝕌'
2. [B] : A ⟶ Type
3. w : coW(A;a.B[a])
4. path : Path
5. StepAgree(path 0;⋅;w)
6. i : ℕ
7. pcw-path-coPath(i;path) ∈ copath(a.B[a];w)
8. pcw-path-coPath(i + 1;path) ∈ copath(a.B[a];w)
9. copath-length(pcw-path-coPath(i + 1;path)) = (i + 1) ∈ ℤ
10. copath-at(w;pcw-path-coPath(i + 1;path)) = (fst(snd((path (i + 1))))) ∈ coW(A;a.B[a])
11. copath-length(pcw-path-coPath(i;path)) = i ∈ ℤ
12. copath-at(w;pcw-path-coPath(i;path)) = (fst(snd((path i)))) ∈ coW(A;a.B[a])
⊢ ↓copathAgree(a.B[a];w;pcw-path-coPath(i;path);pcw-path-coPath(i + 1;path))

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Latex:

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Latex:

1.  [A]  :  \mBbbU{}'
2.  [B]  :  A  {}\mrightarrow{}  Type
3.  w  :  coW(A;a.B[a])
4.  path  :  Path
5.  StepAgree(path  0;\mcdot{};w)
6.  i  :  \mBbbN{}
7.  (pcw-path-coPath(i;path)  \mmember{}  copath(a.B[a];w))
\mwedge{}  ((copath-length(pcw-path-coPath(i;path))  =  i)
    {}\mRightarrow{}  (copath-at(w;pcw-path-coPath(i;path))  =  (fst(snd((path  i))))))
8.  (pcw-path-coPath(i  +  1;path)  \mmember{}  copath(a.B[a];w))
\mwedge{}  ((copath-length(pcw-path-coPath(i  +  1;path))  =  (i  +  1))
    {}\mRightarrow{}  (copath-at(w;pcw-path-coPath(i  +  1;path))  =  (fst(snd((path  (i  +  1)))))))
\mvdash{}  (copath-length(pcw-path-coPath(i  +  1;path))  =  (i  +  1))
{}\mRightarrow{}  (copath-length(pcw-path-coPath(i;path))  =  i)
{}\mRightarrow{}  copathAgree(a.B[a];w;pcw-path-coPath(i;path);pcw-path-coPath(i  +  1;path))


By

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Latex:
(SplitAndHyps
  THEN  ParallelLast
  THEN  ParallelOp  -4
  THEN  (Assert  \mkleeneopen{}\mdownarrow{}copathAgree(a.B[a];w;pcw-path-coPath(i;path);pcw-path-coPath(i  +  1;path))\mkleeneclose{}\mcdot{}
  THENM  (D  -1  THEN  Unhide  THEN  Auto)
  ))

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