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
Home
Index