Nuprl Lemma : csm-subtype-cubical-subset
∀[Gamma:j⊢]. ∀[I:fset(ℕ)]. ∀[psi:𝔽(I)].  (formal-cube(I) j⟶ Gamma ⊆r I,psi j⟶ Gamma)
Proof
Definitions occuring in Statement : 
cubical-subset: I,psi
, 
face-presheaf: 𝔽
, 
cube_set_map: A ⟶ B
, 
formal-cube: formal-cube(I)
, 
I_cube: A(I)
, 
cubical_set: CubicalSet
, 
fset: fset(T)
, 
nat: ℕ
, 
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
, 
cube_set_map: A ⟶ B
, 
psc_map: A ⟶ B
, 
nat-trans: nat-trans(C;D;F;G)
, 
formal-cube: formal-cube(I)
, 
type-cat: TypeCat
, 
cube-cat: CubeCat
, 
all: ∀x:A. B[x]
, 
cubical-subset: I,psi
, 
rep-sub-sheaf: rep-sub-sheaf(C;X;P)
, 
compose: f o g
, 
functor-arrow: arrow(F)
, 
functor-ob: ob(F)
, 
op-cat: op-cat(C)
, 
cat-arrow: cat-arrow(C)
, 
cat-ob: cat-ob(C)
, 
spreadn: spread4, 
pi1: fst(t)
, 
pi2: snd(t)
, 
I_cube: A(I)
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
face-presheaf: 𝔽
, 
lattice-point: Point(l)
, 
record-select: r.x
, 
face_lattice: face_lattice(I)
, 
face-lattice: face-lattice(T;eq)
, 
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x])
, 
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P)
, 
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice, 
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o)
, 
record-update: r[x := v]
, 
ifthenelse: if b then t else f fi 
, 
eq_atom: x =a y
, 
bfalse: ff
, 
btrue: tt
, 
bdd-distributive-lattice: BoundedDistributiveLattice
, 
prop: ℙ
, 
and: P ∧ Q
, 
uimplies: b supposing a
, 
istype: istype(T)
, 
cubical_set: CubicalSet
, 
ps_context: __⊢
, 
cat-functor: Functor(C1;C2)
Lemmas referenced : 
cat_arrow_triple_lemma, 
ob_pair_lemma, 
cat_comp_tuple_lemma, 
arrow_pair_lemma, 
subtype_rel_dep_function, 
names-hom_wf, 
I_cube_wf, 
name-morph-satisfies_wf, 
subtype_rel_self, 
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, 
equal_wf, 
lattice-meet_wf, 
lattice-join_wf, 
fset_wf, 
nat_wf, 
cat-ob_wf, 
op-cat_wf, 
cube-cat_wf, 
cat-arrow_wf, 
type-cat_wf, 
functor-ob_wf, 
cubical-subset_wf, 
cat-comp_wf, 
small-category-cumulativity-2, 
cat-functor_wf, 
functor-arrow_wf, 
cube_set_map_wf, 
formal-cube_wf1, 
face-presheaf_wf2, 
cubical_set_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
lambdaEquality_alt, 
sqequalHypSubstitution, 
setElimination, 
thin, 
rename, 
cut, 
dependent_set_memberEquality_alt, 
sqequalRule, 
introduction, 
extract_by_obid, 
dependent_functionElimination, 
Error :memTop, 
hypothesis, 
functionExtensionality, 
applyEquality, 
hypothesisEquality, 
instantiate, 
isectElimination, 
cumulativity, 
universeIsType, 
setEquality, 
productEquality, 
isectEquality, 
because_Cache, 
independent_isectElimination, 
setIsType, 
lambdaFormation_alt, 
functionIsType, 
equalityIstype, 
applyLambdaEquality
Latex:
\mforall{}[Gamma:j\mvdash{}].  \mforall{}[I:fset(\mBbbN{})].  \mforall{}[psi:\mBbbF{}(I)].    (formal-cube(I)  j{}\mrightarrow{}  Gamma  \msubseteq{}r  I,psi  j{}\mrightarrow{}  Gamma)
Date html generated:
2020_05_20-PM-04_20_08
Last ObjectModification:
2020_04_21-AM-00_50_09
Theory : cubical!type!theory
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