Nuprl Lemma : cubical-pi-implies-sigma
∀G:j⊢. ∀A:{G ⊢ _}. ∀B:{G.A ⊢ _}. ({G ⊢ _:A} ⇒ {G ⊢ _:ΠA B} ⇒ {G ⊢ _:Σ A B})
Proof
Definitions occuring in Statement :
cubical-sigma: Σ A B,
cubical-pi: ΠA B,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
all: ∀x:A. B[x],
implies: P ⇒ Q
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
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,
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u),
cubical-sigma: Σ A B,
presheaf-sigma: Σ A B
Lemmas referenced :
presheaf-pi-implies-sigma,
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,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
isectElimination,
Error :memTop
Latex:
\mforall{}G:j\mvdash{}. \mforall{}A:\{G \mvdash{} \_\}. \mforall{}B:\{G.A \mvdash{} \_\}. (\{G \mvdash{} \_:A\} {}\mRightarrow{} \{G \mvdash{} \_:\mPi{}A B\} {}\mRightarrow{} \{G \mvdash{} \_:\mSigma{} A B\})
Date html generated:
2020_05_20-PM-02_34_50
Last ObjectModification:
2020_04_03-PM-08_45_14
Theory : cubical!type!theory
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