Nuprl Lemma : bag-summation-linear
∀[T,R:Type]. ∀[add,mul:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[f,g:T ⟶ R].
∀a:R. (Σ(x∈b). a mul (f[x] add g[x]) = (a mul (Σ(x∈b). f[x] add Σ(x∈b). g[x])) ∈ R)
supposing (∃minus:R ⟶ R. IsGroup(R;add;zero;minus)) ∧ Comm(R;add) ∧ BiLinear(R;add;mul)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
comm: Comm(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
group_p: IsGroup(T;op;id;inv),
bilinear: BiLinear(T;pl;tm)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q,
exists: ∃x:A. B[x],
bag: bag(T),
quotient: x,y:A//B[x; y],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
subtype_rel: A ⊆r B,
group_p: IsGroup(T;op;id;inv),
monoid_p: IsMonoid(T;op;id),
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
top: Top,
infix_ap: x f y,
bilinear: BiLinear(T;pl;tm),
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assoc: Assoc(T;op),
comm: Comm(T;op),
ident: Ident(T;op;id),
inverse: Inverse(T;op;id;inv)
Lemmas referenced :
list_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
equal_wf,
bag-summation_wf,
infix_ap_wf,
list-subtype-bag,
equal-wf-base,
exists_wf,
group_p_wf,
comm_wf,
bilinear_wf,
bag_wf,
list_induction,
all_wf,
list_accum_wf,
list_accum_nil_lemma,
list_accum_cons_lemma,
squash_wf,
true_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
pointwiseFunctionalityForEquality,
because_Cache,
sqequalRule,
pertypeElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
rename,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
functionExtensionality,
applyEquality,
independent_pairFormation,
productEquality,
axiomEquality,
functionEquality,
isect_memberEquality,
universeEquality,
voidElimination,
voidEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T,R:Type]. \mforall{}[add,mul:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R]. \mforall{}[b:bag(T)]. \mforall{}[f,g:T {}\mrightarrow{} R].
\mforall{}a:R. (\mSigma{}(x\mmember{}b). a mul (f[x] add g[x]) = (a mul (\mSigma{}(x\mmember{}b). f[x] add \mSigma{}(x\mmember{}b). g[x])))
supposing (\mexists{}minus:R {}\mrightarrow{} R. IsGroup(R;add;zero;minus)) \mwedge{} Comm(R;add) \mwedge{} BiLinear(R;add;mul)
Date html generated:
2017_10_01-AM-08_48_51
Last ObjectModification:
2017_07_26-PM-04_32_47
Theory : bags
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