Nuprl Lemma : subset-cubical-type-general
∀[X,Y:j⊢]. {X ⊢j _} ⊆r {Y ⊢j _} supposing sub_cubical_set{j:l}(Y; X)
Proof
Definitions occuring in Statement :
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],
uimplies: b supposing a,
subtype_rel: A ⊆r B,
member: t ∈ T,
cubical-type: {X ⊢ _},
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
prop: ℙ
Lemmas referenced :
subset-I_cube,
subset-restriction,
subtype_rel_dep_function,
I_cube_wf,
istype-universe,
fset_wf,
nat_wf,
names-hom_wf,
cube-set-restriction_wf,
subtype_rel-equal,
equal_wf,
iff_weakening_equal,
nh-id_wf,
squash_wf,
true_wf,
cube-set-restriction-id,
subtype_rel_self,
nh-comp_wf,
cube-set-restriction-comp,
cubical-type_wf,
sub_cubical_set_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
because_Cache,
promote_hyp,
setElimination,
rename,
productElimination,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
functionExtensionality,
applyEquality,
instantiate,
sqequalRule,
universeEquality,
universeIsType,
dependent_functionElimination,
lambdaFormation_alt,
cumulativity,
functionEquality,
inhabitedIsType,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionIsType,
independent_pairFormation,
productIsType,
equalityIstype
Latex:
\mforall{}[X,Y:j\mvdash{}]. \{X \mvdash{}j \_\} \msubseteq{}r \{Y \mvdash{}j \_\} supposing sub\_cubical\_set\{j:l\}(Y; X)
Date html generated:
2020_05_20-PM-02_33_16
Last ObjectModification:
2020_04_03-PM-08_44_09
Theory : cubical!type!theory
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