Nuprl Lemma : csm-ap-cubical-lambda
∀[X,Delta:CubicalSet]. ∀[A:{X ⊢ _}]. ∀[B:{X.A ⊢ _}]. ∀[w:{X ⊢ _:ΠA B}]. ∀[u:{X ⊢ _:A}]. ∀[s:Delta ⟶ X].
((app(w; u))s = app((w)s; (u)s) ∈ {Delta ⊢ _:((B)[u])s})
Proof
Definitions occuring in Statement :
cubical-app: app(w; u),
cubical-pi: ΠA B,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:AF},
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
Lemmas referenced :
csm-ap-cubical-app,
cube-set-map_wf,
cubical-term_wf,
cubical-pi_wf,
cubical-type_wf,
cube-context-adjoin_wf,
cubical-set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache
Latex:
\mforall{}[X,Delta:CubicalSet]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[B:\{X.A \mvdash{} \_\}]. \mforall{}[w:\{X \mvdash{} \_:\mPi{}A B\}]. \mforall{}[u:\{X \mvdash{} \_:A\}].
\mforall{}[s:Delta {}\mrightarrow{} X].
((app(w; u))s = app((w)s; (u)s))
Date html generated:
2016_06_16-PM-05_47_20
Last ObjectModification:
2015_12_28-PM-04_32_24
Theory : cubical!sets
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