Nuprl Lemma : face-forall-type-subtype
∀[H:j⊢]. ∀[phi:{H.𝕀 ⊢ _:𝔽}]. ({H.𝕀, phi ⊢ _} ⊆r {H.𝕀, ((∀ phi))p ⊢ _})
Proof
Definitions occuring in Statement :
face-forall: (∀ phi),
context-subset: Gamma, phi,
face-type: 𝔽,
interval-type: 𝕀,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
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,
uimplies: b supposing a
Lemmas referenced :
subset-cubical-type,
context-subset_wf,
cube-context-adjoin_wf,
cubical_set_cumulativity-i-j,
interval-type_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf,
face-forall_wf,
face-term-implies-subset,
face-forall-implies,
cubical-term_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,
hypothesis,
sqequalRule,
because_Cache,
Error :memTop,
independent_isectElimination,
axiomEquality,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[phi:\{H.\mBbbI{} \mvdash{} \_:\mBbbF{}\}]. (\{H.\mBbbI{}, phi \mvdash{} \_\} \msubseteq{}r \{H.\mBbbI{}, ((\mforall{} phi))p \mvdash{} \_\})
Date html generated:
2020_05_20-PM-03_03_53
Last ObjectModification:
2020_04_04-PM-05_19_54
Theory : cubical!type!theory
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