Nuprl Lemma : cubical-isect_wf
∀[X:⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. X ⊢ ⋂A B
Proof
Definitions occuring in Statement :
cubical-isect: ⋂A B,
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-isect: ⋂A B,
cubical-type: {X ⊢ _},
cubical-isect-family: cubical-isect-family(X;A;B;I;a),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
squash: ↓T,
all: ∀x:A. B[x],
true: True,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
top: Top,
cand: A c∧ B
Lemmas referenced :
cubical-type_wf,
cube-context-adjoin_wf,
cubical_set_wf,
cubical-isect-family_wf,
I_cube_wf,
fset_wf,
nat_wf,
names-hom_wf,
cube-set-restriction_wf,
nh-comp_wf,
subtype_rel-equal,
cubical-type-at_wf,
cube-set-restriction-comp,
cc-adjoin-cube_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
istype-cubical-type-at,
subtype_rel_self,
iff_weakening_equal,
cubical-type-ap-morph_wf,
nh-comp-assoc,
cc-adjoin-cube-restriction,
istype-void,
nh-id-right,
cube-set-restriction-id,
nh-id_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
dependent_pairEquality_alt,
lambdaEquality_alt,
setElimination,
rename,
functionIsType,
because_Cache,
applyEquality,
independent_isectElimination,
imageElimination,
dependent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
productElimination,
independent_functionElimination,
lambdaFormation_alt,
hyp_replacement,
independent_pairFormation,
productIsType,
equalityIstype,
applyLambdaEquality,
voidElimination,
functionExtensionality,
isectEquality,
equalityIsType1
Latex:
\mforall{}[X:\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. X \mvdash{} \mcap{}A B
Date html generated:
2019_11_05-PM-00_23_27
Last ObjectModification:
2018_12_16-PM-04_53_27
Theory : cubical!type!theory
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