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Step * 1 of Lemma csm-ap-term-cube+


1. [Gamma] : CubicalSet{j}
2. [A] : {Gamma ⊢ _}
3. [I] : fset(ℕ)
4. [i] : {i:ℕ| ¬i ∈ I} 
5. [rho] : Gamma(I+i)
6. [phi] : 𝔽(I)
7. [u] : {I+i,s(phi) ⊢ _:(A)<rho> o iota}
8. canonical-section(Gamma;𝔽;I+i;rho;s(phi)) ∈ {formal-cube(I+i) ⊢ _:𝔽}
9. canonical-section(();𝔽;I;⋅;phi) ∈ {formal-cube(I) ⊢ _:𝔽}
10. formal-cube(I+i), canonical-section(Gamma;𝔽;I+i;rho;s(phi)) = I+i,s(phi) ∈ CubicalSet{j}
⊢ (u)cube+(I;i) ∈ {formal-cube(I), canonical-section(();𝔽;I;⋅;phi).𝕀 ⊢ _:(A)<rho> o cube+(I;i)}
BY
{ ((InstLemma `context-subset-map` [formal-cube(I+i);⌜canonical-section(Gamma;𝔽;I+i;rho;s(phi))⌝;⌜formal-cube(I).𝕀⌝;
    ⌜cube+(I;i)⌝]⋅
    THENA Auto
    )
   THEN DoSubsume
   ) }

1
1. Gamma : CubicalSet{j}
2. A : {Gamma ⊢ _}
3. I : fset(ℕ)
4. i : {i:ℕ| ¬i ∈ I} 
5. rho : Gamma(I+i)
6. phi : 𝔽(I)
7. u : {I+i,s(phi) ⊢ _:(A)<rho> o iota}
8. canonical-section(Gamma;𝔽;I+i;rho;s(phi)) ∈ {formal-cube(I+i) ⊢ _:𝔽}
9. canonical-section(();𝔽;I;⋅;phi) ∈ {formal-cube(I) ⊢ _:𝔽}
10. formal-cube(I+i), canonical-section(Gamma;𝔽;I+i;rho;s(phi)) = I+i,s(phi) ∈ CubicalSet{j}
11. cube+(I;i)
    ∈ formal-cube(I).𝕀, (canonical-section(Gamma;𝔽;I+i;rho;s(phi)))cube+(I;i) j⟶ formal-cube(I+i), ...
⊢ (u)cube+(I;i) ∈ {formal-cube(I).𝕀, (canonical-section(Gamma;𝔽;I+i;rho;s(phi)))cube+(I;i) ⊢ _
                   :((A)<rho> o iota)cube+(I;i)}

2
1. Gamma : CubicalSet{j}
2. A : {Gamma ⊢ _}
3. I : fset(ℕ)
4. i : {i:ℕ| ¬i ∈ I} 
5. rho : Gamma(I+i)
6. phi : 𝔽(I)
7. u : {I+i,s(phi) ⊢ _:(A)<rho> o iota}
8. canonical-section(Gamma;𝔽;I+i;rho;s(phi)) ∈ {formal-cube(I+i) ⊢ _:𝔽}
9. canonical-section(();𝔽;I;⋅;phi) ∈ {formal-cube(I) ⊢ _:𝔽}
10. formal-cube(I+i), canonical-section(Gamma;𝔽;I+i;rho;s(phi)) = I+i,s(phi) ∈ CubicalSet{j}
11. cube+(I;i)
    ∈ formal-cube(I).𝕀, (canonical-section(Gamma;𝔽;I+i;rho;s(phi)))cube+(I;i) j⟶ formal-cube(I+i), ...
12. (u)cube+(I;i)
= (u)cube+(I;i)
∈ {formal-cube(I).𝕀, (canonical-section(Gamma;𝔽;I+i;rho;s(phi)))cube+(I;i) ⊢ _:((A)<rho> o iota)cube+(I;i)}
⊢ {formal-cube(I).𝕀, (canonical-section(Gamma;𝔽;I+i;rho;s(phi)))cube+(I;i) ⊢ _:((A)<rho> o iota)cube+(I;i)}
    ⊆r {formal-cube(I), canonical-section(();𝔽;I;⋅;phi).𝕀 ⊢ _:(A)<rho> o cube+(I;i)}

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Latex:

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Latex:

1.  [Gamma]  :  CubicalSet\{j\}
2.  [A]  :  \{Gamma  \mvdash{}  \_\}
3.  [I]  :  fset(\mBbbN{})
4.  [i]  :  \{i:\mBbbN{}|  \mneg{}i  \mmember{}  I\} 
5.  [rho]  :  Gamma(I+i)
6.  [phi]  :  \mBbbF{}(I)
7.  [u]  :  \{I+i,s(phi)  \mvdash{}  \_:(A)<rho>  o  iota\}
8.  canonical-section(Gamma;\mBbbF{};I+i;rho;s(phi))  \mmember{}  \{formal-cube(I+i)  \mvdash{}  \_:\mBbbF{}\}
9.  canonical-section(();\mBbbF{};I;\mcdot{};phi)  \mmember{}  \{formal-cube(I)  \mvdash{}  \_:\mBbbF{}\}
10.  formal-cube(I+i),  canonical-section(Gamma;\mBbbF{};I+i;rho;s(phi))  =  I+i,s(phi)
\mvdash{}  (u)cube+(I;i)  \mmember{}  \{formal-cube(I),  canonical-section(();\mBbbF{};I;\mcdot{};phi).\mBbbI{}  \mvdash{}  \_:(A)<rho>  o  cube+(I;i)\}


By

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Latex:
((InstLemma  `context-subset-map`  [formal-cube(I+i);\mkleeneopen{}canonical-section(Gamma;\mBbbF{};I+i;rho;s(phi))\mkleeneclose{};
    \mkleeneopen{}formal-cube(I).\mBbbI{}\mkleeneclose{};\mkleeneopen{}cube+(I;i)\mkleeneclose{}]\mcdot{}
    THENA  Auto
    )
  THEN  DoSubsume
  )

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