Nuprl Lemma : context-subset-type-subtype
ā[G:jā¢]. ā[phi:{G ā¢ _:š½}]. ({G ā¢j _} ār {G, phi ā¢j _})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi
,
face-type: š½
,
cubical-term: {X ā¢ _:A}
,
cubical-type: {X ā¢ _}
,
cubical_set: CubicalSet
,
subtype_rel: A ār B
,
uall: ā[x:A]. B[x]
Definitions unfolded in proof :
uall: ā[x:A]. B[x]
,
subtype_rel: A ār B
,
member: t ā T
,
cubical-type: {X ā¢ _}
,
and: P ā§ Q
,
all: āx:A. B[x]
,
uimplies: b supposing a
,
squash: āT
,
prop: ā
,
true: True
,
guard: {T}
,
iff: P
āā Q
,
rev_implies: P
ā Q
,
implies: P
ā Q
,
so_lambda: Ī»2x.t[x]
,
so_apply: x[s]
,
cube-set-restriction: f(s)
,
pi2: snd(t)
,
context-subset: Gamma, phi
Lemmas referenced :
fset_wf,
nat_wf,
I_cube_wf,
context-subset_wf,
nh-id_wf,
subtype_rel-equal,
cube-set-restriction_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction-id,
subtype_rel_self,
iff_weakening_equal,
names-hom_wf,
nh-comp_wf,
cube-set-restriction-comp,
cubical-type_wf,
cubical-term_wf,
face-type_wf,
cubical_set_wf,
subtype_rel_dep_function,
subset-I_cube,
context-subset-is-subset,
context-subset-restriction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaEquality_alt,
rename,
sqequalHypSubstitution,
setElimination,
thin,
cut,
productElimination,
dependent_set_memberEquality_alt,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
hypothesisEquality,
applyEquality,
equalityIstype,
because_Cache,
instantiate,
independent_isectElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
dependent_pairEquality_alt,
functionExtensionality,
lambdaFormation_alt,
cumulativity,
functionEquality,
inhabitedIsType,
Error :memTop,
independent_pairFormation,
promote_hyp
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}]. (\{G \mvdash{}j \_\} \msubseteq{}r \{G, phi \mvdash{}j \_\})
Date html generated:
2020_05_20-PM-02_57_53
Last ObjectModification:
2020_04_04-PM-05_13_00
Theory : cubical!type!theory
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