Nuprl Lemma : context-subset-type-subtype
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ({G ⊢j _} ⊆r {G, phi ⊢j _})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
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,
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,
so_lambda: λ2x.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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