Nuprl Lemma : cubical-term-at-morph1
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[u:{X ⊢ _:A}]. ∀[I:fset(ℕ)]. ∀[a:X(I)]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I].
((u(a) a f) = u(f(a)) ∈ A(f(a)))
Proof
Definitions occuring in Statement :
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
names-hom: I ⟶ J,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-type: {X ⊢ _},
cubical-term: {X ⊢ _:A},
all: ∀x:A. B[x],
cubical-term-at: u(a),
and: P ∧ Q,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
names-hom_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity-i-j,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
productElimination,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
Error :memTop,
hypothesis,
universeIsType,
isectElimination,
hypothesisEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
instantiate,
applyEquality
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[u:\{X \mvdash{} \_:A\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[a:X(I)]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I].
((u(a) a f) = u(f(a)))
Date html generated:
2020_05_20-PM-01_52_05
Last ObjectModification:
2020_03_28-PM-00_52_56
Theory : cubical!type!theory
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