Nuprl Lemma : cubical-beta
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[b:{X.A ⊢ _:B}]. ∀[u:{X ⊢ _:A}]. (app((λb); u) = (b)[u] ∈ {X ⊢ _:(B)[u]})
Proof
Definitions occuring in Statement :
cubical-app: app(w; u),
cubical-lambda: (λb),
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
cube-context-adjoin: X.A,
psc-adjoin: X.A,
I_cube: A(I),
I_set: A(I),
cubical-type-at: A(a),
presheaf-type-at: A(a),
cube-set-restriction: f(s),
psc-restriction: f(s),
cubical-type-ap-morph: (u a f),
presheaf-type-ap-morph: (u a f),
csm-ap-type: (AF)s,
pscm-ap-type: (AF)s,
csm-ap: (s)x,
pscm-ap: (s)x,
csm-id-adjoin: [u],
pscm-id-adjoin: [u],
csm-adjoin: (s;u),
pscm-adjoin: (s;u),
csm-id: 1(X),
pscm-id: 1(X),
cubical-app: app(w; u),
presheaf-app: app(w; u),
cubical-lambda: (λb),
presheaf-lambda: (λb),
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u),
cube-cat: CubeCat,
all: ∀x:A. B[x],
csm-ap-term: (t)s,
pscm-ap-term: (t)s
Lemmas referenced :
presheaf-beta,
cube-cat_wf,
cubical-type-sq-presheaf-type,
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,
Error :memTop,
dependent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[b:\{X.A \mvdash{} \_:B\}]. \mforall{}[u:\{X \mvdash{} \_:A\}]. (app((\mlambda{}b); u) = (b)[u])
Date html generated:
2020_05_20-PM-02_30_45
Last ObjectModification:
2020_04_03-PM-08_41_01
Theory : cubical!type!theory
Home
Index