Nuprl Lemma : context-subset-type-iota
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ({Gamma, phi ⊢ _:(A)iota} = {Gamma, phi ⊢ _:A} ∈ 𝕌{[i' | j']})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
subset-iota: iota,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
implies: P ⇒ Q,
true: True
Lemmas referenced :
context-subset-ap-iota,
cubical-term_wf,
squash_wf,
true_wf,
cubical-type_wf,
cubical_set_wf,
context-subset_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
equal_functionality_wrt_subtype_rel2,
face-type_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
instantiate,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
sqequalRule,
cumulativity,
because_Cache,
independent_isectElimination,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}].
(\{Gamma, phi \mvdash{} \_:(A)iota\} = \{Gamma, phi \mvdash{} \_:A\})
Date html generated:
2020_05_20-PM-04_08_50
Last ObjectModification:
2020_04_10-AM-03_50_58
Theory : cubical!type!theory
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