Nuprl Lemma : Kanfiller_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _(Kan)}]. ∀[I:Cname List]. ∀[alpha:X(I)]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[bx:A-open-box(X;Kan-type(A);I;alpha;J;x;i)].
(filler(x;i;bx) ∈ {cube:Kan-type(A)(alpha)| fills-A-open-box(X;Kan-type(A);I;alpha;bx;cube)} )
Proof
Definitions occuring in Statement :
Kanfiller: filler(x;i;bx),
Kan-type: Kan-type(Ak),
Kan-cubical-type: {X ⊢ _(Kan)},
fills-A-open-box: fills-A-open-box(X;A;I;alpha;bx;cube),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
cubical-type-at: A(a),
I-cube: X(I),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n
Definitions unfolded in proof :
member: t ∈ T,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
Kan-cubical-type: {X ⊢ _(Kan)},
Kan-type: Kan-type(Ak),
pi1: fst(t),
Kanfiller: filler(x;i;bx),
pi2: snd(t),
sq_stable: SqStable(P),
implies: P ⇒ Q,
and: P ∧ Q,
squash: ↓T,
Kan-A-filler: Kan-A-filler(X;A;filler),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
prop: ℙ
Lemmas referenced :
decidable__equal-coordinate_name,
sq_stable__l_subset,
fills-A-open-box_wf,
sq_stable_Kan-A-filler,
cubical-set_wf,
Kan-cubical-type_wf,
I-cube_wf,
list_wf,
int_seg_wf,
coordinate_name_wf,
nameset_wf,
subtype_rel_list,
Kan-type_wf,
A-open-box_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_set_memberEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
hypothesis,
applyEquality,
independent_isectElimination,
lambdaEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
natural_numberEquality,
isect_memberFormation,
introduction,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
productElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaFormation
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_(Kan)\}]. \mforall{}[I:Cname List]. \mforall{}[alpha:X(I)]. \mforall{}[J:nameset(I) List].
\mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2]. \mforall{}[bx:A-open-box(X;Kan-type(A);I;alpha;J;x;i)].
(filler(x;i;bx) \mmember{} \{cube:Kan-type(A)(alpha)| fills-A-open-box(X;Kan-type(A);I;alpha;bx;cube)\} )
Date html generated:
2016_06_16-PM-06_44_25
Last ObjectModification:
2016_01_18-PM-04_52_14
Theory : cubical!sets
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