Nuprl Lemma : cubical-subset-is-context-subset
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. (I,psi = formal-cube(I), λJ,f. f(psi) ∈ CubicalSet{j})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
cubical-subset: I,psi,
face-presheaf: 𝔽,
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
cubical-term: {X ⊢ _:A},
subtype_rel: A ⊆r B,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
formal-cube: formal-cube(I),
names-hom: I ⟶ J,
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,
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
all: ∀x:A. B[x],
cubical-type-ap-morph: (u a f),
pi2: snd(t),
context-subset: Gamma, phi,
cubical-subset: I,psi,
cube-cat: CubeCat,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
cubical-term-at: u(a),
name-morph-satisfies: (psi f) = 1,
squash: ↓T,
prop: ℙ,
guard: {T},
fl-morph: <f>,
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
cube-set-restriction: f(s),
bdd-distributive-lattice: BoundedDistributiveLattice,
true: True,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2)
Lemmas referenced :
formal-cube_wf,
cubical_sets_equal,
I_cube_wf,
face-presheaf_wf2,
fset_wf,
nat_wf,
cubical-subset_wf,
context-subset_wf,
cube-set-restriction_wf,
subtype_rel_self,
names-hom_wf,
cubical-type-at_wf_face-type,
cube_set_restriction_pair_lemma,
cubical_type_at_pair_lemma,
cube-set-restriction-comp,
istype-cubical-type-at,
face-type_wf,
cubical-type-ap-morph_wf,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
I_cube_pair_redex_lemma,
face_lattice-point_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
subtype_rel_universe1,
lattice-1_wf,
face_lattice_wf,
name-morph-satisfies_wf,
lattice-point_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,
nh-comp_wf,
fl-morph-comp2,
iff_weakening_equal,
fl-morph_wf,
fl-morph-1,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
independent_isectElimination,
universeIsType,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
applyEquality,
sqequalRule,
Error :memTop,
inhabitedIsType,
lambdaFormation_alt,
dependent_functionElimination,
functionIsType,
because_Cache,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_pairEquality_alt,
functionExtensionality,
setEquality,
imageElimination,
universeEquality,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productEquality,
cumulativity,
isectEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[psi:\mBbbF{}(I)]. (I,psi = formal-cube(I), \mlambda{}J,f. f(psi))
Date html generated:
2020_05_20-PM-02_45_45
Last ObjectModification:
2020_04_05-PM-02_50_39
Theory : cubical!type!theory
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