Nuprl Lemma : face-term-implies-subtype
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[phi,psi:{Gamma ⊢ _:𝔽}].
{Gamma, phi ⊢ _:A} ⊆r {Gamma, psi ⊢ _:A} supposing Gamma ⊢ (psi ⇒ phi)
Proof
Definitions occuring in Statement :
face-term-implies: Gamma ⊢ (phi ⇒ psi),
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
prop: ℙ
Lemmas referenced :
face-term-implies_wf,
subset-cubical-term,
context-subset_wf,
thin-context-subset,
face-term-implies-subset
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
axiomEquality,
hypothesis,
universeIsType,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
because_Cache,
independent_isectElimination
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[phi,psi:\{Gamma \mvdash{} \_:\mBbbF{}\}].
\{Gamma, phi \mvdash{} \_:A\} \msubseteq{}r \{Gamma, psi \mvdash{} \_:A\} supposing Gamma \mvdash{} (psi {}\mRightarrow{} phi)
Date html generated:
2020_05_20-PM-02_55_47
Last ObjectModification:
2020_04_06-AM-10_31_11
Theory : cubical!type!theory
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