Nuprl Lemma : discrete-cubical-term-is-constant
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)]. ∀[t:{formal-cube(I) ⊢ _:discr(T)}]. (t = discr(t(1)) ∈ {formal-cube(I) ⊢ _:discr(T)})
Proof
Definitions occuring in Statement :
discrete-cubical-term: discr(t),
discrete-cubical-type: discr(T),
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
formal-cube: formal-cube(I),
nh-id: 1,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cube-cat: CubeCat,
all: ∀x:A. B[x],
top: Top,
discrete-cubical-type: discr(T),
discrete-presheaf-type: discr(T),
formal-cube: formal-cube(I),
Yoneda: Yoneda(I),
discrete-cubical-term: discr(t),
discrete-presheaf-term: discr(t),
cubical-term-at: u(a),
presheaf-term-at: u(a)
Lemmas referenced :
discrete-presheaf-term-is-constant,
cube-cat_wf,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma,
cubical-term-sq-presheaf-term,
cat_id_tuple_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
sqequalRule,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality
Latex:
\mforall{}[T:Type]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[t:\{formal-cube(I) \mvdash{} \_:discr(T)\}]. (t = discr(t(1)))
Date html generated:
2018_05_23-AM-09_14_26
Last ObjectModification:
2018_05_20-PM-06_13_24
Theory : cubical!type!theory
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