Nuprl Lemma : csm-cubical-lam
∀[X:j⊢]. ∀[A,B:{X ⊢ _}]. ∀[b:{X.A ⊢ _:(B)p}]. ∀[H:j⊢]. ∀[s:H j⟶ X].
((cubical-lam(X;b))s = cubical-lam(H;(b)s+) ∈ {H ⊢ _:((A ⟶ B))s})
Proof
Definitions occuring in Statement :
cubical-lam: cubical-lam(X;b),
cubical-fun: (A ⟶ B),
csm+: tau+,
cc-fst: p,
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,
csm-ap-type: (AF)s,
pscm-ap-type: (AF)s,
csm-ap: (s)x,
pscm-ap: (s)x,
cc-fst: p,
psc-fst: p,
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),
cube_set_map: A ⟶ B,
cubical-fun: (A ⟶ B),
presheaf-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
presheaf-fun-family: presheaf-fun-family(C; X; A; B; I; a),
cube-cat: CubeCat,
all: ∀x:A. B[x],
csm-ap-term: (t)s,
pscm-ap-term: (t)s,
cubical-lam: cubical-lam(X;b),
presheaf-lam: presheaf-lam(X;b),
cubical-lambda: (λb),
presheaf-lambda: (λb),
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u),
csm+: tau+,
pscm+: tau+,
csm-adjoin: (s;u),
pscm-adjoin: (s;u),
csm-comp: G o F,
pscm-comp: G o F,
cc-snd: q,
psc-snd: q
Lemmas referenced :
pscm-presheaf-lam,
cube-cat_wf,
cubical-type-sq-presheaf-type,
cubical-term-sq-presheaf-term,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
cat_comp_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,B:\{X \mvdash{} \_\}]. \mforall{}[b:\{X.A \mvdash{} \_:(B)p\}]. \mforall{}[H:j\mvdash{}]. \mforall{}[s:H j{}\mrightarrow{} X].
((cubical-lam(X;b))s = cubical-lam(H;(b)s+))
Date html generated:
2020_05_20-PM-02_25_07
Last ObjectModification:
2020_04_03-PM-08_35_19
Theory : cubical!type!theory
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