Nuprl Lemma : bag-summation-equal
∀[T,R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[f,g:T ⟶ R].
Σ(x∈b). f[x] = Σ(x∈b). g[x] ∈ R supposing (∀x:T. (x ↓∈ b ⇒ (f[x] = g[x] ∈ R))) ∧ IsMonoid(R;add;zero) ∧ Comm(R;add)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
comm: Comm(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
prop: ℙ,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
so_apply: x[s],
all: ∀x:A. B[x],
squash: ↓T,
cand: A c∧ B,
sq_stable: SqStable(P),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
monoid_p: IsMonoid(T;op;id)
Lemmas referenced :
all_wf,
bag-member_wf,
equal_wf,
monoid_p_wf,
comm_wf,
bag_wf,
bag-subtype,
bag-summation_wf,
squash_wf,
assoc_wf,
set_wf,
true_wf,
sq_stable__bag-member,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
productEquality,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
setEquality,
lambdaFormation,
independent_functionElimination,
imageElimination,
independent_isectElimination,
independent_pairFormation,
universeEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
natural_numberEquality
Latex:
\mforall{}[T,R:Type]. \mforall{}[add:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R]. \mforall{}[b:bag(T)]. \mforall{}[f,g:T {}\mrightarrow{} R].
\mSigma{}(x\mmember{}b). f[x] = \mSigma{}(x\mmember{}b). g[x]
supposing (\mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (f[x] = g[x]))) \mwedge{} IsMonoid(R;add;zero) \mwedge{} Comm(R;add)
Date html generated:
2017_10_01-AM-09_01_30
Last ObjectModification:
2017_07_26-PM-04_42_54
Theory : bags
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