Nuprl Lemma : cubical-type-restriction_wf
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[psi:I:fset(ℕ) ⟶ alpha:X(I) ⟶ T(alpha) ⟶ ℙ{[i' | j']}].
(cubical-type-restriction(X;T;I,a,t.psi[I;a;t]) ∈ ℙ{[i' | j']})
Proof
Definitions occuring in Statement :
cubical-type-restriction: cubical-type-restriction,
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
I_cube: A(I),
cubical_set: CubicalSet,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-type-restriction: cubical-type-restriction,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_apply: x[s1;s2;s3]
Lemmas referenced :
fset_wf,
nat_wf,
names-hom_wf,
I_cube_wf,
cubical-type-at_wf,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
istype-cubical-type-at,
cubical-type_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
functionEquality,
cumulativity,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
hypothesisEquality,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsType,
universeIsType,
universeEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
instantiate
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T:\{X \mvdash{} \_\}]. \mforall{}[psi:I:fset(\mBbbN{}) {}\mrightarrow{} alpha:X(I) {}\mrightarrow{} T(alpha) {}\mrightarrow{} \mBbbP{}\{[i' | j']\}].
(cubical-type-restriction(X;T;I,a,t.psi[I;a;t]) \mmember{} \mBbbP{}\{[i' | j']\})
Date html generated:
2020_05_20-PM-03_12_59
Last ObjectModification:
2020_04_07-PM-03_14_18
Theory : cubical!type!theory
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