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Step * of Lemma csm-equiv_path

No Annotations
∀[G:j⊢]. ∀[A,B:{G ⊢ _:c𝕌}]. ∀[f:{G ⊢ _:Equiv(decode(A);decode(B))}]. ∀[H:j⊢]. ∀[s:H j⟶ G].
  ((equiv_path(G;A;B;f))s+ = equiv_path(H;(A)s;(B)s;(f)s) ∈ {H.𝕀 ⊢ _:c𝕌})
BY
{ (Auto
   THEN Unfold `equiv_path` 0
   THEN RepUR
   ``let`` 0⋅
   THEN ((InstLemma `equiv-path2_wf` [⌜G⌝]⋅ THENA Auto)
         THEN (InstHyp [⌜decode(A)⌝;⌜decode(B)⌝;⌜CompFun(A)⌝;⌜CompFun(B)⌝;⌜f⌝] (-1)⋅ THENA Auto)
         )
   THEN (InstLemma `csm-universe-encode` [⌜G.𝕀⌝]⋅ THENA Auto)
   THEN (InstHyp [⌜equiv-path1(G;decode(A);decode(B);f)⌝;
         ⌜cfun-to-cop(G.𝕀;equiv-path1(G;decode(A);decode(B);f)
              ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))⌝;⌜H.𝕀⌝;⌜s+⌝] (-1)⋅
         THENA Try (Complete (Auto))
         )) }

1
1. G : CubicalSet{j}
2. A : {G ⊢ _:c𝕌}
3. B : {G ⊢ _:c𝕌}
4. f : {G ⊢ _:Equiv(decode(A);decode(B))}
5. H : CubicalSet{j}
6. s : H j⟶ G
7. ∀[A,B:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[cB:G +⊢ Compositon(B)]. ∀[f:{G ⊢ _:Equiv(A;B)}].
     (equiv-path2(G;A;B;cA;cB;f) ∈ G.𝕀 +⊢ Compositon(equiv-path1(G;A;B;f)))
8. equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f) ∈ G.𝕀 +⊢ Compositon(equiv-path1(G;decode(A);decode(B);f))
9. ∀[T:{G.𝕀 ⊢ _}]. ∀[cT:G.𝕀 ⊢ CompOp(T)]. ∀[H:j⊢]. ∀[s:H j⟶ G.𝕀].  ((encode(T;cT))s = encode((T)s;(cT)s) ∈ {H ⊢ _:c𝕌})
10. (encode(equiv-path1(G;decode(A);decode(B);f);cfun-to-cop(G.𝕀;equiv-path1(G;decode(A);decode(B);f)
                                                     ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))))s+
= encode((equiv-path1(G;decode(A);decode(B);f))s+;(cfun-to-cop(G.𝕀;equiv-path1(G;decode(A);decode(B);f)
                                                       ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f)))s+)
∈ {H.𝕀 ⊢ _:c𝕌}
⊢ (encode(equiv-path1(G;decode(A);decode(B);f);cfun-to-cop(G.𝕀;equiv-path1(G;decode(A);decode(B);f)
                                                   ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))))s+
= encode(equiv-path1(H;decode((A)s);decode((B)s);(f)s);
         cfun-to-cop(H.𝕀;equiv-path1(H;decode((A)s);decode((B)s);(f)s)
             ;equiv-path2(H;decode((A)s);decode((B)s);CompFun((A)s);CompFun((B)s);(f)s)))
∈ {H.𝕀 ⊢ _:c𝕌}

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Latex:

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Latex:
No  Annotations
\mforall{}[G:j\mvdash{}].  \mforall{}[A,B:\{G  \mvdash{}  \_:c\mBbbU{}\}].  \mforall{}[f:\{G  \mvdash{}  \_:Equiv(decode(A);decode(B))\}].  \mforall{}[H:j\mvdash{}].  \mforall{}[s:H  j{}\mrightarrow{}  G].
    ((equiv\_path(G;A;B;f))s+  =  equiv\_path(H;(A)s;(B)s;(f)s))


By

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Latex:
(Auto
  THEN  Unfold  `equiv\_path`  0
  THEN  RepUR
  ``let``  0\mcdot{}
  THEN  ((InstLemma  `equiv-path2\_wf`  [\mkleeneopen{}G\mkleeneclose{}]\mcdot{}  THENA  Auto)
              THEN  (InstHyp  [\mkleeneopen{}decode(A)\mkleeneclose{};\mkleeneopen{}decode(B)\mkleeneclose{};\mkleeneopen{}CompFun(A)\mkleeneclose{};\mkleeneopen{}CompFun(B)\mkleeneclose{};\mkleeneopen{}f\mkleeneclose{}]  (-1)\mcdot{}  THENA  Auto)
              )
  THEN  (InstLemma  `csm-universe-encode`  [\mkleeneopen{}G.\mBbbI{}\mkleeneclose{}]\mcdot{}  THENA  Auto)
  THEN  (InstHyp  [\mkleeneopen{}equiv-path1(G;decode(A);decode(B);f)\mkleeneclose{};
              \mkleeneopen{}cfun-to-cop(G.\mBbbI{};equiv-path1(G;decode(A);decode(B);f)
                        ;equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))\mkleeneclose{};\mkleeneopen{}H.\mBbbI{}\mkleeneclose{};\mkleeneopen{}s+\mkleeneclose{}]  (-1)\mcdot{}
              THENA  Try  (Complete  (Auto))
              ))

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