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Step * 1 of Lemma term-to-pathtype-eta


1. X : CubicalSet{j}
2. A : {X ⊢ _}
3. pth : {X ⊢ _:(𝕀 ⟶ A)}
⊢ (λapp((pth)p; q)) = pth ∈ {X ⊢ _:(𝕀 ⟶ A)}
BY
{ ((InstLemma `cubical-fun-as-cubical-pi` [⌜X⌝;⌜𝕀⌝;⌜A⌝]⋅ THENA Auto)
   THEN (InstLemma `cubical-eta` [⌜X⌝;⌜𝕀⌝;⌜(A)p⌝;⌜pth⌝]⋅ THENA Auto)
   THEN InferEqualType
   THEN EqCDA
   THEN Auto) }

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Latex:

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Latex:

1.  X  :  CubicalSet\{j\}
2.  A  :  \{X  \mvdash{}  \_\}
3.  pth  :  \{X  \mvdash{}  \_:(\mBbbI{}  {}\mrightarrow{}  A)\}
\mvdash{}  (\mlambda{}app((pth)p;  q))  =  pth


By

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Latex:
((InstLemma  `cubical-fun-as-cubical-pi`  [\mkleeneopen{}X\mkleeneclose{};\mkleeneopen{}\mBbbI{}\mkleeneclose{};\mkleeneopen{}A\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (InstLemma  `cubical-eta`  [\mkleeneopen{}X\mkleeneclose{};\mkleeneopen{}\mBbbI{}\mkleeneclose{};\mkleeneopen{}(A)p\mkleeneclose{};\mkleeneopen{}pth\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  InferEqualType
  THEN  EqCDA
  THEN  Auto)

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