Nuprl Lemma : respects-equality-context-subset-term
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[phi:{X ⊢ _:𝔽}]. respects-equality({X ⊢ _:A};{X, phi ⊢ _:A})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
respects-equality: respects-equality(S;T),
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-term: {X ⊢ _:A},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
guard: {T},
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
respects-equality: respects-equality(S;T)
Lemmas referenced :
respects-equality-set,
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
fset_wf,
nat_wf,
I_cube_wf,
context-subset_wf,
cubical-type-at_wf,
thin-context-subset,
all_wf,
names-hom_wf,
equal_wf,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
istype-cubical-type-at,
subset-I_cube,
context-subset-is-subset,
subtype-respects-equality,
subtype_rel_set,
subtype_rel_dep_function,
face-type_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,
applyEquality,
hypothesis,
functionEquality,
cumulativity,
lambdaEquality_alt,
universeIsType,
universeEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
functionIsType,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
lambdaFormation_alt,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[phi:\{X \mvdash{} \_:\mBbbF{}\}]. respects-equality(\{X \mvdash{} \_:A\};\{X, phi \mvdash{} \_:A\})
Date html generated:
2020_05_20-PM-02_55_36
Last ObjectModification:
2020_04_06-AM-10_24_40
Theory : cubical!type!theory
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