Nuprl Lemma : cubical-pi-family_wf
∀[X:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[I:Cname List]. ∀[a:X(I)]. (cubical-pi-family(X;A;B;I;a) ∈ Type)
Proof
Definitions occuring in Statement :
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
I-cube: X(I),
cubical-set: CubicalSet,
coordinate_name: Cname,
list: T List,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_apply: x[s]
Lemmas referenced :
list_wf,
coordinate_name_wf,
name-morph_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,
subtype_rel-equal,
cc-adjoin-cube-restriction,
squash_wf,
true_wf,
name-comp_wf,
cube-set-restriction-comp,
iff_weakening_equal,
I-cube_wf,
cubical-type_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
setEquality,
functionEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
because_Cache,
dependent_functionElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
instantiate,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
axiomEquality
Latex:
\mforall{}[X:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[I:Cname List]. \mforall{}[a:X(I)].
(cubical-pi-family(X;A;B;I;a) \mmember{} Type)
Date html generated:
2017_10_05-AM-10_13_48
Last ObjectModification:
2017_07_28-AM-11_19_07
Theory : cubical!sets
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