Nuprl Lemma : get_face-dimension
∀[X:CubicalSet]. ∀[I,J:Cname List]. ∀[x:nameset(I)]. ∀[i:ℕ2]. ∀[box:open_box(X;I;J;x;i)]. ∀[y:nameset(J)]. ∀[c:ℕ2].
(dimension(get_face(y;c;box)) ~ y)
Proof
Definitions occuring in Statement :
get_face: get_face(y;c;box),
open_box: open_box(X;I;J;x;i),
face-dimension: dimension(f),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
natural_number: $n,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
I-face: I-face(X;I),
face-dimension: dimension(f),
pi1: fst(t),
face-name: face-name(f),
pi2: snd(t),
top: Top,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
get_face_wf,
nameset_wf,
open_box_wf,
int_seg_wf,
list_wf,
coordinate_name_wf,
cubical-set_wf,
pi1_wf_top,
istype-void,
subtype_base_sq,
nameset_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
equalityIsType1,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
axiomSqEquality,
sqequalRule,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType,
natural_numberEquality,
setElimination,
rename,
productElimination,
applyLambdaEquality,
independent_pairEquality,
voidElimination,
instantiate,
cumulativity,
independent_isectElimination
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[I,J:Cname List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2]. \mforall{}[box:open\_box(X;I;J;x;i)].
\mforall{}[y:nameset(J)]. \mforall{}[c:\mBbbN{}2].
(dimension(get\_face(y;c;box)) \msim{} y)
Date html generated:
2019_11_05-PM-00_28_17
Last ObjectModification:
2018_11_08-PM-00_51_52
Theory : cubical!sets
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