Nuprl Lemma : cubical-isect-family_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. (cubical-isect-family(X;A;B;I;a) ∈ Type)
Proof
Definitions occuring in Statement :
cubical-isect-family: cubical-isect-family(X;A;B;I;a),
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-isect-family: cubical-isect-family(X;A;B;I;a),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ,
top: Top,
uimplies: b supposing a,
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
fset_wf,
nat_wf,
names-hom_wf,
cubical-type-at_wf,
cube-set-restriction_wf,
cube-context-adjoin_wf,
cc-adjoin-cube_wf,
all_wf,
equal_wf,
cubical-type-ap-morph_wf,
I_cube_wf,
cubical-type_wf,
cubical_set_wf,
nh-comp_wf,
cc-adjoin-cube-restriction,
subtype_rel-equal,
squash_wf,
true_wf,
cube-set-restriction-comp,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
setEquality,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
isectEquality,
because_Cache,
dependent_functionElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
instantiate,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:X(I)].
(cubical-isect-family(X;A;B;I;a) \mmember{} Type)
Date html generated:
2017_10_05-AM-10_02_08
Last ObjectModification:
2017_07_28-AM-11_14_29
Theory : cubical!type!theory
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