Nuprl Lemma : csm-constrained-cubical-term
∀[Gamma,K:j⊢]. ∀[s:K j⟶ Gamma]. ∀[A:{Gamma ⊢ _}]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[t:{Gamma, phi ⊢ _:A}].
∀[v:{Gamma ⊢ _:A[phi |⟶ t]}].
((v)s ∈ {K ⊢ _:(A)s[(phi)s |⟶ (t)s]})
Proof
Definitions occuring in Statement :
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
cubical-term_wf,
face-type_wf,
cubical-type_wf,
cube_set_map_wf,
cubical_set_wf,
thin-context-subset,
csm-ap-term_wf,
context-subset_wf,
csm-face-type,
csm-ap-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-context-subset-subtype2,
context-subset-map,
constrained-cubical-term_wf,
context-subset-term-subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
universeIsType,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
inhabitedIsType,
dependent_set_memberEquality_alt,
setElimination,
rename,
equalityIstype,
sqequalRule,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
applyEquality,
applyLambdaEquality
Latex:
\mforall{}[Gamma,K:j\mvdash{}]. \mforall{}[s:K j{}\mrightarrow{} Gamma]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[t:\{Gamma, phi \mvdash{} \_:A\}].
\mforall{}[v:\{Gamma \mvdash{} \_:A[phi |{}\mrightarrow{} t]\}].
((v)s \mmember{} \{K \mvdash{} \_:(A)s[(phi)s |{}\mrightarrow{} (t)s]\})
Date html generated:
2020_05_20-PM-02_59_26
Last ObjectModification:
2020_04_06-AM-11_57_13
Theory : cubical!type!theory
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