Nuprl Lemma : subtype-context-subset-0
∀[X,Y:j⊢]. ({X ⊢ _} ⊆r {Y, 0(𝔽) ⊢ _})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-0: 0(𝔽),
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],
member: t ∈ T,
subtype_rel: A ⊆r B,
cubical-type: {X ⊢ _},
face-0: 0(𝔽),
context-subset: Gamma, phi,
all: ∀x:A. B[x],
cubical-term-at: u(a),
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
false: False
Lemmas referenced :
I_cube_pair_redex_lemma,
cube_set_restriction_pair_lemma,
cubical-type_wf,
cubical_set_wf,
I_cube_wf,
equal_wf,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
lattice-0_wf,
lattice-1_wf,
fset_wf,
nat_wf,
face-lattice-0-not-1,
names-hom_wf,
nh-id_wf,
subtype_rel-equal,
cube-set-restriction_wf,
cube-set-restriction-id
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaEquality_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
universeIsType,
isectElimination,
hypothesisEquality,
axiomEquality,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
instantiate,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
functionExtensionality,
setEquality,
cumulativity,
applyEquality,
productEquality,
isectEquality,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
functionIsType,
setIsType,
equalityIstype,
independent_pairFormation,
lambdaFormation_alt,
productIsType
Latex:
\mforall{}[X,Y:j\mvdash{}]. (\{X \mvdash{} \_\} \msubseteq{}r \{Y, 0(\mBbbF{}) \mvdash{} \_\})
Date html generated:
2020_05_20-PM-03_01_18
Last ObjectModification:
2020_04_04-PM-05_16_16
Theory : cubical!type!theory
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