Nuprl Lemma : free-vs-bag-add
∀[G:Type]. ∀[K:CRng]. ∀[S:Type]. ∀[f:S ⟶ bag(|K| × G)]. ∀[bs:bag(S)].
(Σ{f[b] | b ∈ bs} = ⋃b∈bs.f[b] ∈ Point(free-vs(K;G)))
Proof
Definitions occuring in Statement :
free-vs: free-vs(K;S),
vs-bag-add: Σ{f[b] | b ∈ bs},
vs-point: Point(vs),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|,
bag-combine: ⋃x∈bs.f[x],
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
free-vs: free-vs(K;S),
vs-bag-add: Σ{f[b] | b ∈ bs},
mk-vs: mk-vs,
vs-add: x + y,
vs-0: 0,
all: ∀x:A. B[x],
member: t ∈ T,
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
formal-sum-add: x + y,
bag-summation: Σ(x∈b). f[x],
basic-formal-sum: basic-formal-sum(K;S),
subtype_rel: A ⊆r B,
vs-point: Point(vs),
formal-sum: formal-sum(K;S),
crng: CRng,
rng: Rng,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
rec_select_update_lemma,
istype-void,
subtype_quotient,
basic-formal-sum_wf,
bfs-equiv_wf,
bfs-equiv-rel,
bag_wf,
rng_car_wf,
crng_wf,
istype-universe,
bag-combine_wf,
equal_wf,
squash_wf,
true_wf,
bag-combine-as-accum,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
applyEquality,
isectElimination,
setElimination,
rename,
hypothesisEquality,
lambdaEquality_alt,
because_Cache,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
universeIsType,
functionIsType,
productEquality,
instantiate,
universeEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[G:Type]. \mforall{}[K:CRng]. \mforall{}[S:Type]. \mforall{}[f:S {}\mrightarrow{} bag(|K| \mtimes{} G)]. \mforall{}[bs:bag(S)].
(\mSigma{}\{f[b] | b \mmember{} bs\} = \mcup{}b\mmember{}bs.f[b])
Date html generated:
2019_10_31-AM-06_31_23
Last ObjectModification:
2019_08_01-PM-01_39_59
Theory : linear!algebra
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