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Step * of Lemma map-path_wf

No Annotations
∀[G:j⊢]. ∀[S,T:{G ⊢ _}]. ∀[f:{G ⊢ _:(S ⟶ T)}]. ∀[a,b:{G ⊢ _:S}]. ∀[pth:{G ⊢ _:(Path_S a b)}].
  (map-path(G;f;pth) ∈ {G ⊢ _:(Path_T app(f; a) app(f; b))})
BY
{ ((Auto
    THEN (Assert {G.𝕀 ⊢ _:((S ⟶ T))p} = {G.𝕀 ⊢ _:((S)p ⟶ (T)p)} ∈ 𝕌{[i' | j']} BY
                (EqCDA THEN Auto))
    THEN (Assert (f)p ∈ {G.𝕀 ⊢ _:((S)p ⟶ (T)p)} BY
                ((Assert (f)p ∈ {G.𝕀 ⊢ _:((S ⟶ T))p} BY Auto) THEN InferEqualTypeUp THEN Eq)))
   THEN Unfold `map-path` 0
   THEN (InstLemma `term-to-path_wf` [⌜G⌝;⌜T⌝;⌜app((f)p; path-eta(pth))⌝]⋅ THENA Auto)
   THEN InferEqualType
   THEN Auto
   THEN EqCDA
   THEN (RWO "csm-cubical-app" 0 THENA Auto)
   THEN Symmetry
   THEN EqCDA
   THEN RWW  "path-eta-0 path-eta-1" 0
   THEN Auto
   THEN Symmetry
   THEN Fold `cubical-path-app` 0
   THEN Auto) }

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Latex:

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Latex:
No  Annotations
\mforall{}[G:j\mvdash{}].  \mforall{}[S,T:\{G  \mvdash{}  \_\}].  \mforall{}[f:\{G  \mvdash{}  \_:(S  {}\mrightarrow{}  T)\}].  \mforall{}[a,b:\{G  \mvdash{}  \_:S\}].  \mforall{}[pth:\{G  \mvdash{}  \_:(Path\_S  a  b)\}].
    (map-path(G;f;pth)  \mmember{}  \{G  \mvdash{}  \_:(Path\_T  app(f;  a)  app(f;  b))\})


By

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Latex:
((Auto
    THEN  (Assert  \{G.\mBbbI{}  \mvdash{}  \_:((S  {}\mrightarrow{}  T))p\}  =  \{G.\mBbbI{}  \mvdash{}  \_:((S)p  {}\mrightarrow{}  (T)p)\}  BY
                            (EqCDA  THEN  Auto))
    THEN  (Assert  (f)p  \mmember{}  \{G.\mBbbI{}  \mvdash{}  \_:((S)p  {}\mrightarrow{}  (T)p)\}  BY
                            ((Assert  (f)p  \mmember{}  \{G.\mBbbI{}  \mvdash{}  \_:((S  {}\mrightarrow{}  T))p\}  BY  Auto)  THEN  InferEqualTypeUp  THEN  Eq)))
  THEN  Unfold  `map-path`  0
  THEN  (InstLemma  `term-to-path\_wf`  [\mkleeneopen{}G\mkleeneclose{};\mkleeneopen{}T\mkleeneclose{};\mkleeneopen{}app((f)p;  path-eta(pth))\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  InferEqualType
  THEN  Auto
  THEN  EqCDA
  THEN  (RWO  "csm-cubical-app"  0  THENA  Auto)
  THEN  Symmetry
  THEN  EqCDA
  THEN  RWW    "path-eta-0  path-eta-1"  0
  THEN  Auto
  THEN  Symmetry
  THEN  Fold  `cubical-path-app`  0
  THEN  Auto)

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