Nuprl Lemma : cubical-subset_wf
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[psi:I:fset(ℕ) ⟶ alpha:X(I) ⟶ T(alpha) ⟶ ℙ].
X ⊢ {t:T | ∀I,alpha. psi[I;alpha;t]} supposing cubical-type-restriction(X;T;I,a,t.psi[I;a;t])
Proof
Definitions occuring in Statement :
cubical-subset: cubical-subset,
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,
uimplies: b supposing a,
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 :
cubical-type-restriction: cubical-type-restriction,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
so_apply: x[s1;s2;s3],
prop: ℙ,
subtype_rel: A ⊆r B,
cubical-subset: cubical-subset,
cubical-type: {X ⊢ _},
guard: {T},
and: P ∧ Q,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
names-hom_wf,
I_cube_wf,
istype-cubical-type-at,
cube-set-restriction_wf,
cubical-type-ap-morph_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
fset_wf,
nat_wf,
cubical-type_wf,
cubical_set_wf,
cubical-type-at_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type-ap-morph-id,
nh-id_wf,
subtype_rel_self,
iff_weakening_equal,
cubical-type-ap-morph-comp,
nh-comp_wf,
subtype_rel-equal,
cube-set-restriction-id,
cube-set-restriction-comp
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
functionIsType,
because_Cache,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
universeEquality,
dependent_set_memberEquality_alt,
dependent_pairEquality_alt,
lambdaEquality_alt,
setEquality,
functionExtensionality,
setElimination,
rename,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
lambdaFormation_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
setIsType,
productIsType,
equalityIstype
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T:\{X \mvdash{} \_\}]. \mforall{}[psi:I:fset(\mBbbN{}) {}\mrightarrow{} alpha:X(I) {}\mrightarrow{} T(alpha) {}\mrightarrow{} \mBbbP{}].
X \mvdash{} \{t:T | \mforall{}I,alpha. psi[I;alpha;t]\} supposing cubical-type-restriction(X;T;I,a,t.psi[I;a;t])
Date html generated:
2020_05_20-PM-03_13_36
Last ObjectModification:
2020_04_06-PM-05_17_04
Theory : cubical!type!theory
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