Nuprl Lemma : Kanfiller-uniform
∀[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)].
∀K:Cname List. ∀f:name-morph(I;K).
((∀i:nameset(I). ((i ∈ J) ⇒ (↑isname(f i))))
⇒ (↑isname(f x))
⇒ ((filler(x;i;bx) alpha f)
= filler(f x;i;A-open-box-image(X;Kan-type(A);I;K;f;alpha;bx))
∈ Kan-type(A)(f(alpha))))
Proof
Definitions occuring in Statement :
Kanfiller: filler(x;i;bx),
Kan-type: Kan-type(Ak),
Kan-cubical-type: {X ⊢ _(Kan)},
A-open-box-image: A-open-box-image(X;A;I;K;f;alpha;bx),
A-open-box: A-open-box(X;A;I;alpha;J;x;i),
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
name-morph: name-morph(I;J),
isname: isname(z),
nameset: nameset(L),
coordinate_name: Cname,
l_member: (x ∈ l),
map: map(f;as),
list: T List,
int_seg: {i..j-},
assert: ↑b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
Kan-cubical-type: {X ⊢ _(Kan)},
Kan-type: Kan-type(Ak),
pi1: fst(t),
Kanfiller: filler(x;i;bx),
pi2: snd(t),
and: P ∧ Q,
uniform-Kan-A-filler: uniform-Kan-A-filler(X;A;filler),
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
name-morph: name-morph(I;J),
so_lambda: λ2x.t[x],
prop: ℙ,
nameset: nameset(L),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
so_apply: x[s]
Lemmas referenced :
assert_wf,
isname_wf,
all_wf,
nameset_wf,
l_member_wf,
coordinate_name_wf,
subtype_rel_list,
name-morph_wf,
list_wf,
A-open-box_wf,
Kan-type_wf,
int_seg_wf,
I-cube_wf,
Kan-cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
setElimination,
thin,
rename,
cut,
productElimination,
sqequalRule,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
independent_functionElimination,
lemma_by_obid,
isectElimination,
applyEquality,
lambdaEquality,
functionEquality,
independent_isectElimination,
because_Cache,
natural_numberEquality,
isect_memberFormation,
introduction,
lambdaFormation,
axiomEquality,
isect_memberEquality
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)].
\mforall{}K:Cname List. \mforall{}f:name-morph(I;K).
((\mforall{}i:nameset(I). ((i \mmember{} J) {}\mRightarrow{} (\muparrow{}isname(f i))))
{}\mRightarrow{} (\muparrow{}isname(f x))
{}\mRightarrow{} ((filler(x;i;bx) alpha f) = filler(f x;i;A-open-box-image(X;Kan-type(A);I;K;f;alpha;bx))))
Date html generated:
2016_06_16-PM-06_44_30
Last ObjectModification:
2015_12_28-PM-04_25_32
Theory : cubical!sets
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