Nuprl Lemma : section-iota_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[I:fset(ℕ)]. ∀[rho:Gamma(I)]. ∀[a:A(rho)]. ∀[psi:𝔽(I)].
(section-iota(Gamma;A;I;rho;a) ∈ {I,psi ⊢ _:((A)<rho>)iota})
Proof
Definitions occuring in Statement :
section-iota: section-iota(Gamma;A;I;rho;a),
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
subset-iota: iota,
cubical-subset: I,psi,
face-presheaf: 𝔽,
context-map: <rho>,
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
section-iota: section-iota(Gamma;A;I;rho;a),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x]
Lemmas referenced :
csm-ap-term_wf,
cubical-subset_wf,
formal-cube_wf,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
context-map_wf,
subset-iota_wf,
canonical-section_wf,
I_cube_wf,
face-presheaf_wf2,
istype-cubical-type-at,
fset_wf,
nat_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
dependent_functionElimination,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[rho:Gamma(I)]. \mforall{}[a:A(rho)]. \mforall{}[psi:\mBbbF{}(I)].
(section-iota(Gamma;A;I;rho;a) \mmember{} \{I,psi \mvdash{} \_:((A)<rho>)iota\})
Date html generated:
2020_05_20-PM-02_32_58
Last ObjectModification:
2020_04_03-PM-08_43_19
Theory : cubical!type!theory
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