Nuprl Lemma : composition-type-lemma5
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[u:{Gamma, phi.𝕀 ⊢ _:A}].
({Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]} ∈ 𝕌{[i | j']})
Proof
Definitions occuring in Statement :
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
csm-id-adjoin: [u],
csm-id: 1(X),
guard: {T}
Lemmas referenced :
constrained-cubical-term_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
csm-ap-term_wf,
context-subset_wf,
subset-cubical-type,
sub_cubical_set_functionality,
context-subset-is-subset,
context-subset-adjoin-subtype,
cubical-term_wf,
cubical-type_wf,
face-type_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,
hypothesis,
applyEquality,
sqequalRule,
because_Cache,
independent_isectElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[u:\{Gamma, phi.\mBbbI{} \mvdash{} \_:A\}].
(\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})][phi |{}\mrightarrow{} (u)[0(\mBbbI{})]]\} \mmember{} \mBbbU{}\{[i | j']\})
Date html generated:
2020_05_20-PM-04_09_42
Last ObjectModification:
2020_04_13-PM-00_33_26
Theory : cubical!type!theory
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