Nuprl Lemma : cubical-fun-comp_wf
∀[X:j⊢]. ∀[A,B,C:{X ⊢ _}]. ∀[g:{X ⊢ _:(A ⟶ B)}]. ∀[f:{X ⊢ _:(B ⟶ C)}]. ((f o g) ∈ {X ⊢ _:(A ⟶ C)})
Proof
Definitions occuring in Statement :
cubical-fun-comp: (f o g),
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
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],
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-fun-comp: (f o g),
presheaf-fun-comp: (f o g),
cubical-lam: cubical-lam(X;b),
presheaf-lam: presheaf-lam(X;b),
cubical-lambda: (λb),
presheaf-lambda: (λb),
cubical-app: app(w; u),
presheaf-app: app(w; u),
csm-ap-term: (t)s,
pscm-ap-term: (t)s,
csm-ap: (s)x,
pscm-ap: (s)x,
cc-fst: p,
psc-fst: p,
cc-snd: q,
psc-snd: q,
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u)
Lemmas referenced :
presheaf-fun-comp_wf,
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:j\mvdash{}]. \mforall{}[A,B,C:\{X \mvdash{} \_\}]. \mforall{}[g:\{X \mvdash{} \_:(A {}\mrightarrow{} B)\}]. \mforall{}[f:\{X \mvdash{} \_:(B {}\mrightarrow{} C)\}].
((f o g) \mmember{} \{X \mvdash{} \_:(A {}\mrightarrow{} C)\})
Date html generated:
2020_05_20-PM-02_26_08
Last ObjectModification:
2020_04_03-PM-08_36_18
Theory : cubical!type!theory
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