Nuprl Lemma : cubical-sigma-fun_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[B,B':{G.A ⊢ _}]. ∀[f:{G.A ⊢ _:(B ⟶ B')}].
(cubical-sigma-fun(G;A;B;f) ∈ {G ⊢ _:(Σ A B ⟶ Σ A B')})
Proof
Definitions occuring in Statement :
cubical-sigma-fun: cubical-sigma-fun(G;A;B;f),
cubical-sigma: Σ A B,
cubical-fun: (A ⟶ B),
cube-context-adjoin: X.A,
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,
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-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-sigma: Σ A B,
presheaf-sigma: Σ A B,
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u),
cubical-sigma-fun: cubical-sigma-fun(G;A;B;f),
presheaf-sigma-fun: presheaf-sigma-fun(G;A;B;f),
cubical-lam: cubical-lam(X;b),
presheaf-lam: presheaf-lam(X;b),
cubical-lambda: (λb),
presheaf-lambda: (λb),
cubical-pair: cubical-pair(u;v),
presheaf-pair: presheaf-pair(u;v),
cubical-fst: p.1,
presheaf-fst: p.1,
cc-snd: q,
psc-snd: q,
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,
cubical-snd: p.2,
presheaf-snd: p.2
Lemmas referenced :
presheaf-sigma-fun_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{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[B,B':\{G.A \mvdash{} \_\}]. \mforall{}[f:\{G.A \mvdash{} \_:(B {}\mrightarrow{} B')\}].
(cubical-sigma-fun(G;A;B;f) \mmember{} \{G \mvdash{} \_:(\mSigma{} A B {}\mrightarrow{} \mSigma{} A B')\})
Date html generated:
2020_05_20-PM-02_31_03
Last ObjectModification:
2020_04_03-PM-08_41_20
Theory : cubical!type!theory
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