Nuprl Lemma : cube-set-map-is
∀[A,B:CubicalSet].
(A ⟶ B ~ {trans:I:(Cname List) ⟶ A(I) ⟶ B(I)|
∀I,J:Cname List. ∀g:name-morph(I;J). ((λs.g(trans I s)) = (λs.(trans J g(s))) ∈ (A(I) ⟶ B(J)))} )
Proof
Definitions occuring in Statement :
cube-set-restriction: f(s),
I-cube: X(I),
cube-set-map: A ⟶ B,
cubical-set: CubicalSet,
name-morph: name-morph(I;J),
coordinate_name: Cname,
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cube-set-restriction: f(s),
I-cube: X(I),
cube-set-map: A ⟶ B,
type-cat: TypeCat,
name-cat: NameCat,
nat-trans: nat-trans(C;D;F;G),
functor-arrow: functor-arrow(F),
compose: f o g,
cat-comp: cat-comp(C),
cat-arrow: cat-arrow(C),
cat-ob: cat-ob(C),
pi1: fst(t),
pi2: snd(t)
Lemmas referenced :
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis,
sqequalAxiom,
lemma_by_obid,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
because_Cache
Latex:
\mforall{}[A,B:CubicalSet].
(A {}\mrightarrow{} B \msim{} \{trans:I:(Cname List) {}\mrightarrow{} A(I) {}\mrightarrow{} B(I)|
\mforall{}I,J:Cname List. \mforall{}g:name-morph(I;J). ((\mlambda{}s.g(trans I s)) = (\mlambda{}s.(trans J g(s))))\} )
Date html generated:
2016_06_16-PM-05_37_40
Last ObjectModification:
2015_12_28-PM-04_37_07
Theory : cubical!sets
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