Nuprl Lemma : cubical-subset-term
∀[X:j⊢]. ∀[T:{X ⊢ _}]. ∀[psi:I:fset(ℕ) ⟶ alpha:X(I) ⟶ T(alpha) ⟶ ℙ].
∀[t:{X ⊢ _:T}]
t ∈ {X ⊢ _:{t:T | ∀I,alpha. psi[I;alpha;t]}} supposing ∀I:fset(ℕ). ∀alpha:X(I). psi[I;alpha;t(alpha)]
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-term-at: u(a),
cubical-term: {X ⊢ _:A},
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],
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
cubical-term: {X ⊢ _:A},
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: so_lambda3,
prop: ℙ,
so_apply: x[s1;s2;s3],
cubical-term-at: u(a),
cubical-subset: cubical-subset,
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
names-hom_wf,
I_cube_wf,
istype-cubical-type-at,
cube-set-restriction_wf,
cubical-subset_wf,
cubical-type-ap-morph_wf,
cubical-term-at_wf,
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type-restriction_wf,
fset_wf,
nat_wf,
cubical-type_wf,
cubical_set_wf,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type-at_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality_alt,
sqequalRule,
functionIsType,
because_Cache,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
equalityIstype,
instantiate,
applyEquality,
lambdaEquality_alt,
cumulativity,
inhabitedIsType,
independent_isectElimination,
universeEquality,
functionExtensionality,
dependent_functionElimination,
Error :memTop,
lambdaFormation_alt,
equalitySymmetry,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[T:\{X \mvdash{} \_\}]. \mforall{}[psi:I:fset(\mBbbN{}) {}\mrightarrow{} alpha:X(I) {}\mrightarrow{} T(alpha) {}\mrightarrow{} \mBbbP{}].
\mforall{}[t:\{X \mvdash{} \_:T\}]
t \mmember{} \{X \mvdash{} \_:\{t:T | \mforall{}I,alpha. psi[I;alpha;t]\}\}
supposing \mforall{}I:fset(\mBbbN{}). \mforall{}alpha:X(I). psi[I;alpha;t(alpha)]
supposing cubical-type-restriction(X;T;I,a,t.psi[I;a;t])
Date html generated:
2020_05_20-PM-03_13_49
Last ObjectModification:
2020_04_06-PM-05_17_59
Theory : cubical!type!theory
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