Nuprl Lemma : discrete-cubical-term-is-constant-on-irr-face
Not every term of a discrete cubical type is constant, but when the
context is the formal-cube(I) -- Yoneda(I) -- then it is.⋅
∀[T:Type]. ∀[I:fset(ℕ)]. ∀[as,bs:fset(names(I))].
∀[t:{I,irr_face(I;as;bs) ⊢ _:discr(T)}]. (t = discr(t(irr-face-morph(I;as;bs))) ∈ {I,irr_face(I;as;bs) ⊢ _:discr(T)})
supposing ↑fset-disjoint(NamesDeq;as;bs)
Proof
Definitions occuring in Statement :
discrete-cubical-term: discr(t),
discrete-cubical-type: discr(T),
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-subset: I,psi,
irr-face-morph: irr-face-morph(I;as;bs),
irr_face: irr_face(I;as;bs),
names-deq: NamesDeq,
names: names(I),
fset-disjoint: fset-disjoint(eq;as;bs),
fset: fset(T),
nat: ℕ,
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
ext-eq: A ≡ B,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
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,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
cubical-term-at: u(a),
discrete-cubical-term: discr(t),
implies: P ⇒ Q
Lemmas referenced :
discrete-cubical-term-is-map,
irr-face-morph_wf,
irr-face-morph-satisfies,
name-morph-satisfies_wf,
irr_face_wf,
discrete-map-is-constant2,
subtype_rel_self,
I_cube_wf,
face-presheaf_wf2,
istype-cubical-term,
cubical-subset_wf,
discrete-cubical-type_wf,
istype-assert,
fset-disjoint_wf,
names_wf,
names-deq_wf,
fset_wf,
nat_wf,
istype-universe,
irr-face-morph-property,
names-hom_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
productElimination,
dependent_set_memberEquality_alt,
hypothesis,
independent_isectElimination,
universeIsType,
instantiate,
cumulativity,
applyEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeEquality,
lambdaFormation_alt,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[as,bs:fset(names(I))].
\mforall{}[t:\{I,irr\_face(I;as;bs) \mvdash{} \_:discr(T)\}]. (t = discr(t(irr-face-morph(I;as;bs))))
supposing \muparrow{}fset-disjoint(NamesDeq;as;bs)
Date html generated:
2020_05_20-PM-02_32_14
Last ObjectModification:
2020_04_20-PM-01_45_57
Theory : cubical!type!theory
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