Nuprl Lemma : cubical-term-subtype-cubical-subset
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. ∀[T:{formal-cube(I) ⊢ _}]. ({formal-cube(I) ⊢ _:T} ⊆r {I,psi ⊢ _:T})
Proof
Definitions occuring in Statement :
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical-subset: I,psi,
face-presheaf: 𝔽,
formal-cube: formal-cube(I),
I_cube: A(I),
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,
cubical-term: {X ⊢ _:A},
top: Top,
formal-cube: formal-cube(I),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
names-hom: I ⟶ J,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
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,
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
implies: P ⇒ Q
Lemmas referenced :
cubical-term_wf,
formal-cube_wf,
cubical-type_wf,
I_cube_wf,
face-presheaf_wf,
fset_wf,
nat_wf,
cubical-subset-I_cube,
I_cube_pair_redex_lemma,
subtype_rel_dep_function,
names-hom_wf,
cubical-type-at_wf,
subtype_rel_self,
names_wf,
lattice-point_wf,
dM_wf,
name-morph-satisfies_wf,
fset-all_wf,
fset-contains-none_wf,
face-lattice-constraints_wf,
assert_wf,
fset-antichain_wf,
union-deq_wf,
names-deq_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
set_wf,
cube_set_restriction_pair_lemma,
cubical-subset-restriction,
cubical-type-subtype-cubical-subset,
cubical-subset_wf,
cubical_set_wf,
all_wf,
cube-set-restriction_wf,
cubical-type-ap-morph_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
dependent_set_memberEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_functionElimination,
functionExtensionality,
applyEquality,
functionEquality,
because_Cache,
setEquality,
productEquality,
lambdaFormation,
unionEquality,
independent_isectElimination,
instantiate,
cumulativity,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[psi:\mBbbF{}(I)]. \mforall{}[T:\{formal-cube(I) \mvdash{} \_\}]. (\{formal-cube(I) \mvdash{} \_:T\} \msubseteq{}r \{I,psi \mvdash{} \_:T\})
Date html generated:
2017_10_05-AM-06_42_28
Last ObjectModification:
2017_07_28-AM-10_33_06
Theory : cubical!type!theory
Home
Index