Nuprl Lemma : csm-ap-cubical-fst
∀[X,Delta:j⊢]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[p:{X ⊢ _:Σ A B}]. ∀[s:Delta j⟶ X]. ((p.1)s = (p)s.1 ∈ {Delta ⊢ _:(A)s})
Proof
Definitions occuring in Statement :
cubical-fst: p.1,
cubical-sigma: Σ A B,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
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),
cubical-sigma: Σ A B,
presheaf-sigma: Σ A B,
cc-adjoin-cube: (v;u),
psc-adjoin-set: (v;u),
cube_set_map: A ⟶ B,
csm-ap-type: (AF)s,
pscm-ap-type: (AF)s,
csm-ap: (s)x,
pscm-ap: (s)x,
csm-ap-term: (t)s,
pscm-ap-term: (t)s,
cubical-fst: p.1,
presheaf-fst: p.1
Lemmas referenced :
pscm-ap-presheaf-fst,
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,Delta:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[p:\{X \mvdash{} \_:\mSigma{} A B\}]. \mforall{}[s:Delta j{}\mrightarrow{} X].
((p.1)s = (p)s.1)
Date html generated:
2020_05_20-PM-02_27_10
Last ObjectModification:
2020_04_03-PM-08_37_30
Theory : cubical!type!theory
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