Nuprl Lemma : cubical-term-at-comp
∀Gamma:j⊢. ∀T:{Gamma ⊢ _}. ∀t:{Gamma ⊢ _:T}. ∀I:fset(ℕ). ∀rho:Gamma(I). ∀J:fset(ℕ). ∀f:J ⟶ I. ∀K:fset(ℕ). ∀g:K ⟶ J.
(t(f ⋅ g(rho)) = t(g(f(rho))) ∈ T(g(f(rho))))
Proof
Definitions occuring in Statement :
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
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-cat: CubeCat,
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-term-at: u(a),
presheaf-term-at: u(a)
Lemmas referenced :
presheaf-term-at-comp,
cube-cat_wf,
cubical-type-sq-presheaf-type,
cubical-term-sq-presheaf-term,
cat_ob_pair_lemma,
cat_arrow_triple_lemma,
cat_comp_tuple_lemma
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:j\mvdash{}. \mforall{}T:\{Gamma \mvdash{} \_\}. \mforall{}t:\{Gamma \mvdash{} \_:T\}. \mforall{}I:fset(\mBbbN{}). \mforall{}rho:Gamma(I). \mforall{}J:fset(\mBbbN{}). \mforall{}f:J {}\mrightarrow{} I.
\mforall{}K:fset(\mBbbN{}). \mforall{}g:K {}\mrightarrow{} J.
(t(f \mcdot{} g(rho)) = t(g(f(rho))))
Date html generated:
2020_05_20-PM-02_33_07
Last ObjectModification:
2020_04_03-PM-08_43_28
Theory : cubical!type!theory
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