Nuprl Lemma : cubical-type-subtype-cubical-subset
∀[I:fset(ℕ)]. ∀[psi:𝔽(I)]. ({formal-cube(I) ⊢ _} ⊆r {I,psi ⊢ _})
Proof
Definitions occuring in Statement :
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-type: {X ⊢ _},
and: P ∧ Q,
formal-cube: formal-cube(I),
all: ∀x:A. B[x],
top: Top,
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
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,
so_apply: x[s]
Lemmas referenced :
cubical-type_wf,
formal-cube_wf,
I_cube_wf,
face-presheaf_wf,
fset_wf,
nat_wf,
I_cube_pair_redex_lemma,
cube_set_restriction_pair_lemma,
cubical-subset-I_cube,
cubical-subset_wf,
names-hom_wf,
cube-set-restriction_wf,
cubical-subset-restriction,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
set_wf,
name-morph-satisfies_wf,
subtype_rel_self,
fset-all_wf,
fset-contains-none_wf,
face-lattice-constraints_wf,
all_wf,
nh-id_wf,
subtype_rel-equal,
cube-set-restriction-id,
nh-comp_wf,
cube-set-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
productElimination,
dependent_set_memberEquality,
sqequalRule,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_pairEquality,
functionExtensionality,
applyEquality,
because_Cache,
functionEquality,
independent_pairFormation,
lambdaFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
independent_functionElimination,
setEquality,
productEquality,
instantiate
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[psi:\mBbbF{}(I)]. (\{formal-cube(I) \mvdash{} \_\} \msubseteq{}r \{I,psi \mvdash{} \_\})
Date html generated:
2017_10_05-AM-06_42_13
Last ObjectModification:
2017_07_28-AM-10_32_53
Theory : cubical!type!theory
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