Nuprl Lemma : bag-summation-cons
∀[R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R].
∀[T:Type]. ∀[f:T ⟶ R]. ∀[b:bag(T)]. ∀[a:T]. (Σ(x∈a.b). f[x] = (f[a] add Σ(x∈b). f[x]) ∈ R)
supposing IsMonoid(R;add;zero) ∧ Comm(R;add)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
cons-bag: x.b,
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],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
monoid_p: IsMonoid(T;op;id)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
single-bag: {x},
bag-append: as + bs,
cons-bag: x.b,
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
prop: ℙ,
cand: A c∧ B,
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
squash: ↓T,
infix_ap: x f y,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
monoid_p: IsMonoid(T;op;id)
Lemmas referenced :
list_ind_cons_lemma,
list_ind_nil_lemma,
bag_wf,
monoid_p_wf,
comm_wf,
single-bag_wf,
bag-summation_wf,
infix_ap_wf,
equal_wf,
squash_wf,
true_wf,
bag-summation-append,
bag-summation-single,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
hypothesisEquality,
isectElimination,
axiomEquality,
because_Cache,
cumulativity,
functionEquality,
universeEquality,
productEquality,
functionExtensionality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
lambdaEquality,
independent_isectElimination,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination
Latex:
\mforall{}[R:Type]. \mforall{}[add:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R].
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} R]. \mforall{}[b:bag(T)]. \mforall{}[a:T]. (\mSigma{}(x\mmember{}a.b). f[x] = (f[a] add \mSigma{}(x\mmember{}b). f[x]))
supposing IsMonoid(R;add;zero) \mwedge{} Comm(R;add)
Date html generated:
2017_10_01-AM-08_48_41
Last ObjectModification:
2017_07_26-PM-04_32_45
Theory : bags
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