Nuprl Lemma : cubical-sigma-equal
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w,y:{X ⊢ _:Σ A B}].
(w = y ∈ {X ⊢ _:Σ A B}) supposing ((w.2 = y.2 ∈ {X ⊢ _:(B)[w.1]}) and (w.1 = y.1 ∈ {X ⊢ _:A}))
Proof
Definitions occuring in Statement :
cubical-snd: p.2,
cubical-fst: p.1,
cubical-sigma: Σ A B,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uimplies: b supposing a,
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),
cubical-sigma: Σ A B,
presheaf-sigma: Σ A B,
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u),
cubical-fst: p.1,
presheaf-fst: p.1,
csm-ap-type: (AF)s,
pscm-ap-type: (AF)s,
csm-ap: (s)x,
pscm-ap: (s)x,
csm-id-adjoin: [u],
pscm-id-adjoin: [u],
csm-adjoin: (s;u),
pscm-adjoin: (s;u),
csm-id: 1(X),
pscm-id: 1(X),
cubical-snd: p.2,
presheaf-snd: p.2
Lemmas referenced :
presheaf-sigma-equal,
cube-cat_wf,
cubical-type-sq-presheaf-type,
cubical-term-sq-presheaf-term
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{} \_\}]. \mforall{}[w,y:\{X \mvdash{} \_:\mSigma{} A B\}].
(w = y) supposing ((w.2 = y.2) and (w.1 = y.1))
Date html generated:
2020_05_20-PM-02_30_18
Last ObjectModification:
2020_04_03-PM-08_40_35
Theory : cubical!type!theory
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