Nuprl Lemma : subset-constrained-cubical-term
∀[X,Y:j⊢].
∀[A:{X ⊢ _}]. ∀[phi:{X ⊢ _:𝔽}]. ∀[t:{X, phi ⊢ _:A}]. ({X ⊢ _:A[phi |⟶ t]} ⊆r {Y ⊢ _:A[phi |⟶ t]})
supposing sub_cubical_set{j:l}(Y; X)
Proof
Definitions occuring in Statement :
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
sub_cubical_set: Y ⊆ 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,
guard: {T},
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q
Lemmas referenced :
subset-cubical-term,
cubical-term_wf,
context-subset_wf,
thin-context-subset,
cubical-type-cumulativity2,
face-type_wf,
cubical-type_wf,
sub_cubical_set_wf,
cubical_set_wf,
sub_cubical_set_functionality2,
constrained-cubical-term_wf,
cubical_set_cumulativity-i-j,
subset-cubical-type,
sub_cubical_set_transitivity,
context-subset-is-subset,
subset-cubical-term2
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
sqequalRule,
axiomEquality,
universeIsType,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
lambdaEquality_alt,
setElimination,
rename,
dependent_set_memberEquality_alt,
equalityIstype,
independent_pairFormation,
lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[X,Y:j\mvdash{}].
\mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[phi:\{X \mvdash{} \_:\mBbbF{}\}]. \mforall{}[t:\{X, phi \mvdash{} \_:A\}].
(\{X \mvdash{} \_:A[phi |{}\mrightarrow{} t]\} \msubseteq{}r \{Y \mvdash{} \_:A[phi |{}\mrightarrow{} t]\})
supposing sub\_cubical\_set\{j:l\}(Y; X)
Date html generated:
2020_05_20-PM-02_58_52
Last ObjectModification:
2020_04_06-PM-00_06_58
Theory : cubical!type!theory
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