Proof of Nuprl Lemma : unit-cube-cubical-type

Step * of Lemma unit-cube-cubical-type

∀[I:Cname List]
({unit-cube(I) ⊢ _} ~ {AF:A:L:(Cname List) ⟶ name-morph(I;L) ⟶ Type × (L:(Cname List)

                                                                          ⟶ J:(Cname List)

                                                                          ⟶ f:name-morph(L;J)

                                                                          ⟶ a:name-morph(I;L)

                                                                          ⟶ (A L a)

                                                                          ⟶ (A J (a o f)))|

let A,F = AF

                         in (∀K:Cname List. ∀a:name-morph(I;K). ∀u:A K a.  ((F K K 1 a u) = u ∈ (A K a)))

                            ∧ (∀L,J,K:Cname List. ∀f:name-morph(L;J). ∀g:name-morph(J;K). ∀a:name-morph(I;L). ∀u:A L a.

                                 ((F L K (f o g) a u) = (F J K g (a o f) (F L J f a u)) ∈ (A K (a o (f o g)))))} )

{ ((D 0 THENA Auto) THEN RepUR ``cubical-type unit-cube I-cube functor-ob cube-set-restriction`` 0 THEN EqCD) }

Latex:

Latex:
\mforall{}[I:Cname List] (\{unit-cube(I) \mvdash{} \_\} \msim{} \{AF:A:L:(Cname List) {}\mrightarrow{} name-morph(I;L) {}\mrightarrow{} Type \mtimes{} (L:(Cname List) {}\mrightarrow{} J:(Cname List) {}\mrightarrow{} f:name-morph(L;J) {}\mrightarrow{} a:name-morph(I;L) {}\mrightarrow{} (A L a) {}\mrightarrow{} (A J (a o f)))| let A,F = AF in (\mforall{}K:Cname List. \mforall{}a:name-morph(I;K). \mforall{}u:A K a. ((F K K 1 a u) = u)) \mwedge{} (\mforall{}L,J,K:Cname List. \mforall{}f:name-morph(L;J). \mforall{}g:name-morph(J;K). \mforall{}a:name-morph(I;L). \mforall{}u:A L a. ((F L K (f o g) a u) = (F J K g (a o f) (F L J f a u))))\} ) By Latex: ((D 0 THENA Auto) THEN RepUR ``cubical-type unit-cube I-cube functor-ob cube-set-restriction`` 0 THEN EqCD) Home Index