Step
*
2
of Lemma
univ-trans-equiv_path
1. G : CubicalSet{j}
2. A : {G ⊢ _:c𝕌}
3. B : {G ⊢ _:c𝕌}
4. f : {G ⊢ _:Equiv(decode(A);decode(B))}
5. equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f) ∈ G.𝕀 +⊢ Compositon(equiv-path1(G;decode(A);decode(B);f))
6. 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))
∈ G.𝕀 ⊢ CompOp(equiv-path1(G;decode(A);decode(B);f))
7. decode(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))))
= equiv-path1(G;decode(A);decode(B);f)
∈ {G.𝕀 ⊢ _}
8. compOp(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))))
= 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))
∈ G.𝕀 ⊢ CompOp(equiv-path1(G;decode(A);decode(B);f))
9. transprt-fun(G;equiv-path1(G;decode(A);decode(B);f);equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))
= trans-equiv-path(G;A;B;f)
∈ {G ⊢ _:(decode(A) ⟶ decode(B))}
⊢ transprt-fun(G;decode(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))));cop-to-cfun(
compOp(encode(...;
...))))
= trans-equiv-path(G;A;B;f)
∈ {G ⊢ _:(decode(A) ⟶ decode(B))}
BY
{ (NthHypEqGen (-1) THEN EqCDA THEN Thin (-1) THEN Symmetry THEN InferEqualTypeGen) }
1
1. G : CubicalSet{j}
2. A : {G ⊢ _:c𝕌}
3. B : {G ⊢ _:c𝕌}
4. f : {G ⊢ _:Equiv(decode(A);decode(B))}
5. equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f) ∈ G.𝕀 +⊢ Compositon(equiv-path1(G;decode(A);decode(B);f))
6. 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))
∈ G.𝕀 ⊢ CompOp(equiv-path1(G;decode(A);decode(B);f))
7. decode(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))))
= equiv-path1(G;decode(A);decode(B);f)
∈ {G.𝕀 ⊢ _}
8. compOp(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))))
= 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))
∈ G.𝕀 ⊢ CompOp(equiv-path1(G;decode(A);decode(B);f))
⊢ {G ⊢ _:((equiv-path1(G;decode(A);decode(B);f))[0(𝕀)] ⟶ (equiv-path1(G;decode(A);decode(B);f))[1(𝕀)])}
= {G ⊢ _:(decode(A) ⟶ decode(B))}
∈ 𝕌{[i | j']}
2
1. G : CubicalSet{j}
2. A : {G ⊢ _:c𝕌}
3. B : {G ⊢ _:c𝕌}
4. f : {G ⊢ _:Equiv(decode(A);decode(B))}
5. equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f) ∈ G.𝕀 +⊢ Compositon(equiv-path1(G;decode(A);decode(B);f))
6. 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))
∈ G.𝕀 ⊢ CompOp(equiv-path1(G;decode(A);decode(B);f))
7. decode(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))))
= equiv-path1(G;decode(A);decode(B);f)
∈ {G.𝕀 ⊢ _}
8. compOp(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))))
= 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))
∈ G.𝕀 ⊢ CompOp(equiv-path1(G;decode(A);decode(B);f))
9. {G ⊢ _:((equiv-path1(G;decode(A);decode(B);f))[0(𝕀)] ⟶ (equiv-path1(G;decode(A);decode(B);f))[1(𝕀)])}
= {G ⊢ _:(decode(A) ⟶ decode(B))}
∈ 𝕌{[i | j']}
⊢ transprt-fun(G;equiv-path1(G;decode(A);decode(B);f);equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f))
= transprt-fun(G;decode(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))));cop-to-cfun(
compOp(encode(...;
...))))
∈ {G ⊢ _:((equiv-path1(G;decode(A);decode(B);f))[0(𝕀)] ⟶ (equiv-path1(G;decode(A);decode(B);f))[1(𝕀)])}
Latex:
Latex:
1. G : CubicalSet\{j\}
2. A : \{G \mvdash{} \_:c\mBbbU{}\}
3. B : \{G \mvdash{} \_:c\mBbbU{}\}
4. f : \{G \mvdash{} \_:Equiv(decode(A);decode(B))\}
5. equiv-path2(G;decode(A);decode(B);CompFun(A);CompFun(B);f)
\mmember{} G.\mBbbI{} +\mvdash{} Compositon(equiv-path1(G;decode(A);decode(B);f))
6. 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))
\mmember{} G.\mBbbI{} \mvdash{} CompOp(equiv-path1(G;decode(A);decode(B);f))
7. decode(encode(equiv-path1(G;decode(A);decode(B);f);
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))))
= equiv-path1(G;decode(A);decode(B);f)
8. compOp(encode(equiv-path1(G;decode(A);decode(B);f);
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))))
= 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))
9. transprt-fun(G;equiv-path1(G;decode(A);decode(B);f);equiv-path2(G;decode(A);decode(B);...;...;f))
= trans-equiv-path(G;A;B;f)
\mvdash{} transprt-fun(G;decode(...);cop-to-cfun(
compOp(encode(equiv-path1(G;decode(A);decode(B);f);
...))))
= trans-equiv-path(G;A;B;f)
By
Latex:
(NthHypEqGen (-1) THEN EqCDA THEN Thin (-1) THEN Symmetry THEN InferEqualTypeGen)
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