Nuprl Lemma : csm-context-subset-subtype3
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma ⊢ _}]. ∀[X:j⊢]. (Gamma.A ij⟶ X ⊆r Gamma, phi.A ij⟶ X)
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
cube_set_map_subtype3,
cube-context-adjoin_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
context-subset_wf,
thin-context-subset,
sub_cubical_set_self,
sub_cubical_set_functionality,
context-subset-is-subset,
cubical-type_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
axiomEquality,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
universeIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[X:j\mvdash{}].
(Gamma.A ij{}\mrightarrow{} X \msubseteq{}r Gamma, phi.A ij{}\mrightarrow{} X)
Date html generated:
2020_05_20-PM-02_59_14
Last ObjectModification:
2020_04_06-AM-11_30_46
Theory : cubical!type!theory
Home
Index