Nuprl Lemma : csm-id-adjoin-ap-type
∀Gamma,Delta:j⊢. ∀A:{Gamma ⊢ _}. ∀B:{Gamma.A ⊢ _}. ∀sigma:Delta j⟶ Gamma. ∀u:{Delta ⊢ _:(A)sigma}.
(((B)(sigma o p;q))[u] = (B)(sigma;u) ∈ {Delta ⊢ _})
Proof
Definitions occuring in Statement :
csm-id-adjoin: [u],
csm-adjoin: (s;u),
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
csm-comp: G o F,
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
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,
csm-ap-type: (AF)s,
pscm-ap-type: (AF)s,
csm-ap: (s)x,
pscm-ap: (s)x,
csm-adjoin: (s;u),
pscm-adjoin: (s;u),
csm-comp: G o F,
pscm-comp: G o F,
cc-fst: p,
psc-fst: p,
cc-snd: q,
psc-snd: q,
csm-id-adjoin: [u],
pscm-id-adjoin: [u],
csm-id: 1(X),
pscm-id: 1(X)
Lemmas referenced :
pscm-id-adjoin-ap-type,
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,
dependent_functionElimination,
thin,
hypothesis,
sqequalRule,
isectElimination,
Error :memTop
Latex:
\mforall{}Gamma,Delta:j\mvdash{}. \mforall{}A:\{Gamma \mvdash{} \_\}. \mforall{}B:\{Gamma.A \mvdash{} \_\}. \mforall{}sigma:Delta j{}\mrightarrow{} Gamma. \mforall{}u:\{Delta \mvdash{} \_:(A)sigma\}.
(((B)(sigma o p;q))[u] = (B)(sigma;u))
Date html generated:
2020_05_20-PM-01_57_45
Last ObjectModification:
2020_04_03-PM-08_31_45
Theory : cubical!type!theory
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