Nuprl Lemma : csm-ap-cubical-app
∀[X,Delta:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[u:{X ⊢ _:A}]. ∀[s:Delta j⟶ X].
((app(w; u))s = app((w)s; (u)s) ∈ {Delta ⊢ _:((B)[u])s})
Proof
Definitions occuring in Statement :
cubical-app: app(w; u),
cubical-pi: ΠA B,
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 ⊢ _},
cube_set_map: A ⟶ B,
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),
cubical-pi: ΠA B,
presheaf-pi: ΠA B,
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
presheaf-pi-family: presheaf-pi-family(C; X; A; B; I; a),
cube-cat: CubeCat,
all: ∀x:A. B[x],
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u),
cube_set_map: A ⟶ B,
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),
csm-ap-term: (t)s,
pscm-ap-term: (t)s,
cubical-app: app(w; u),
presheaf-app: app(w; u)
Lemmas referenced :
pscm-ap-presheaf-app,
cube-cat_wf,
cubical-type-sq-presheaf-type,
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,
Error :memTop,
dependent_functionElimination
Latex:
\mforall{}[X,Delta:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:\mPi{}A B\}]. \mforall{}[u:\{X \mvdash{} \_:A\}]. \mforall{}[s:Delta j{}\mrightarrow{} X].
((app(w; u))s = app((w)s; (u)s))
Date html generated:
2020_05_20-PM-02_29_23
Last ObjectModification:
2020_04_03-PM-08_39_41
Theory : cubical!type!theory
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