Nuprl Lemma : sigma-elim-unelim
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. (SigmaUnElim o SigmaElim = 1(X.Σ A B) ∈ X.Σ A B ij⟶ X.Σ A B)
Proof
Definitions occuring in Statement :
sigma-unelim-csm: SigmaUnElim,
sigma-elim-csm: SigmaElim,
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
cubical-type: {X ⊢ _},
csm-id: 1(X),
csm-comp: G o F,
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),
cube_set_map: A ⟶ B,
cubical-sigma: Σ A B,
presheaf-sigma: Σ A B,
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u),
csm-comp: G o F,
pscm-comp: G o F,
sigma-unelim-csm: SigmaUnElim,
sigma-unelim-pscm: SigmaUnElim,
sigma-elim-csm: SigmaElim,
sigma-elim-pscm: SigmaElim,
csm-id: 1(X),
pscm-id: 1(X)
Lemmas referenced :
ps-sigma-elim-unelim,
cube-cat_wf,
cubical-type-sq-presheaf-type
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isectElimination,
thin,
hypothesis,
sqequalRule,
Error :memTop
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. (SigmaUnElim o SigmaElim = 1(X.\mSigma{} A B))
Date html generated:
2020_05_20-PM-02_28_03
Last ObjectModification:
2020_04_04-AM-09_26_12
Theory : cubical!type!theory
Home
Index