Nuprl Lemma : cubical-subst_wf
∀[G:j⊢]. ∀[A:{G ⊢' _}]. ∀[a,b:{G ⊢ _:A}]. ∀[p:{G ⊢ _:(Path_A a b)}]. ∀[f:{G ⊢ _:(A ⟶ c𝕌)}].
∀[x:{G ⊢ _:decode(app(f; a))}].
(cubical-subst(G;f;p;x) ∈ {G ⊢ _:decode(app(f; b))})
Proof
Definitions occuring in Statement :
cubical-subst: cubical-subst(G;f;pth;x),
universe-decode: decode(t),
cubical-universe: c𝕌,
path-type: (Path_A a b),
cubical-app: app(w; u),
cubical-fun: (A ⟶ B),
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-subst: cubical-subst(G;f;pth;x)
Lemmas referenced :
cubical-app_wf_fun,
cubical-universe_wf,
map-path_wf,
universe-decode_wf,
path-trans_wf,
istype-cubical-term,
cubical-fun_wf,
path-type_wf,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
hypothesisEquality,
hypothesis,
sqequalRule,
universeIsType
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{}' \_\}]. \mforall{}[a,b:\{G \mvdash{} \_:A\}]. \mforall{}[p:\{G \mvdash{} \_:(Path\_A a b)\}]. \mforall{}[f:\{G \mvdash{} \_:(A {}\mrightarrow{} c\mBbbU{})\}].
\mforall{}[x:\{G \mvdash{} \_:decode(app(f; a))\}].
(cubical-subst(G;f;p;x) \mmember{} \{G \mvdash{} \_:decode(app(f; b))\})
Date html generated:
2020_05_20-PM-07_33_18
Last ObjectModification:
2020_04_30-AM-00_10_54
Theory : cubical!type!theory
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