Nuprl Lemma : closed-type-to-type_wf
∀[T:{ * ⊢ _}]. ∀[X:j⊢]. X ⊢ closed-type-to-type(T)
Proof
Definitions occuring in Statement :
closed-type-to-type: closed-type-to-type(T),
closed-cubical-type: { * ⊢ _},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
closed-type-to-type: closed-type-to-type(T),
closed-cubical-type: { * ⊢ _},
and: P ∧ Q,
cubical-type: {X ⊢ _},
subtype_rel: A ⊆r B,
cand: A c∧ B,
all: ∀x:A. B[x],
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
I_cube_wf,
fset_wf,
nat_wf,
istype-top,
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_set_wf,
closed-cubical-type_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
lambdaEquality_alt,
applyEquality,
hypothesisEquality,
universeIsType,
instantiate,
extract_by_obid,
isectElimination,
hypothesis,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
Error :memTop,
because_Cache,
functionIsType,
lambdaFormation_alt,
independent_pairFormation,
productIsType,
equalityIstype,
independent_isectElimination,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[T:\{ * \mvdash{} \_\}]. \mforall{}[X:j\mvdash{}]. X \mvdash{} closed-type-to-type(T)
Date html generated:
2020_05_20-PM-01_50_56
Last ObjectModification:
2020_03_20-PM-01_19_40
Theory : cubical!type!theory
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