Nuprl Lemma : context-subset-is-cubical-subset
∀[I:fset(ℕ)]. ∀[phi:{formal-cube(I) ⊢ _:𝔽}]. (formal-cube(I), phi = I,phi(1) ∈ CubicalSet{j})
Proof
Definitions occuring in Statement :
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-subset: I,psi,
formal-cube: formal-cube(I),
cubical_set: CubicalSet,
nh-id: 1,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
formal-cube: formal-cube(I),
names-hom: I ⟶ J,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
cubical-subset: I,psi,
context-subset: Gamma, phi,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
cubical-term-at: u(a),
name-morph-satisfies: (psi f) = 1,
cat-arrow: cat-arrow(C),
all: ∀x:A. B[x],
pi2: snd(t),
cube-cat: CubeCat,
squash: ↓T,
true: True,
bdd-distributive-lattice: BoundedDistributiveLattice,
prop: ℙ,
lattice: Lattice,
rev_uimplies: rev_uimplies(P;Q),
rev_subtype_rel: A ⊇r B,
guard: {T},
cat-comp: cat-comp(C),
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
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-term: {X ⊢ _:A},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2)
Lemmas referenced :
cubical_sets_equal,
context-subset_wf,
formal-cube_wf1,
cubical-subset_wf,
cubical-term-at_wf,
face-type_wf,
nh-id_wf,
names-hom_wf,
istype-cubical-term,
fset_wf,
nat_wf,
I_cube_pair_redex_lemma,
cubical-term-at-morph,
face-type-ap-morph,
face-type-at,
cube_set_restriction_pair_lemma,
nh-id-right,
equal_wf,
lattice-point_wf,
lattice-1_wf,
face_lattice_wf,
bdd-distributive-lattice-subtype-lattice,
lattice_wf,
lattice-structure_wf,
bdd-distributive-lattice_wf,
subtype_rel_functionality_wrt_implies,
subtype_rel_weakening,
ext-eq_inversion,
ext-eq_weakening,
I_cube_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
subtype_rel_self,
nh-comp_wf,
squash_wf,
true_wf,
istype-universe,
iff_weakening_equal,
fl-morph_wf,
fl-morph-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
applyEquality,
independent_isectElimination,
dependent_pairEquality_alt,
functionIsType,
inhabitedIsType,
universeIsType,
functionExtensionality,
dependent_functionElimination,
Error :memTop,
setEquality,
instantiate,
applyLambdaEquality,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
cumulativity,
productEquality,
isectEquality,
dependent_set_memberEquality_alt,
universeEquality,
productElimination,
independent_functionElimination,
equalityIstype
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[phi:\{formal-cube(I) \mvdash{} \_:\mBbbF{}\}]. (formal-cube(I), phi = I,phi(1))
Date html generated:
2020_05_20-PM-02_46_13
Last ObjectModification:
2020_04_20-PM-00_05_02
Theory : cubical!type!theory
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