Nuprl Lemma : cubical-type-cumulativity-i-j
∀[X:j⊢]. ({X ⊢j _} ⊆r cubical-type{[j | i]:l}(X))
Proof
Definitions occuring in Statement :
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
member: t ∈ T,
cubical-type: {X ⊢ _},
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
subtype_rel_dep_function,
I_cube_wf,
fset_wf,
nat_wf,
names-hom_wf,
cube-set-restriction_wf,
nh-id_wf,
subtype_rel-equal,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction-id,
subtype_rel_self,
iff_weakening_equal,
nh-comp_wf,
cube-set-restriction-comp,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
productElimination,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
functionExtensionality,
applyEquality,
hypothesisEquality,
hypothesis,
instantiate,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
sqequalRule,
universeEquality,
universeIsType,
because_Cache,
independent_isectElimination,
lambdaFormation_alt,
functionIsType,
inhabitedIsType,
productIsType,
equalityIstype,
imageElimination,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. (\{X \mvdash{}j \_\} \msubseteq{}r cubical-type\{[j | i]:l\}(X))
Date html generated:
2020_05_20-PM-01_47_23
Last ObjectModification:
2020_04_03-PM-07_58_46
Theory : cubical!type!theory
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