Nuprl Lemma : bag-summation-zero
∀[T,R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)].
Σ(x∈b). zero = zero ∈ R supposing IsMonoid(R;add;zero) ∧ Comm(R;add)
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],
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,
bag: bag(T),
quotient: x,y:A//B[x; y],
all: ∀x:A. B[x],
implies: P ⇒ Q,
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
squash: ↓T,
monoid_p: IsMonoid(T;op;id),
assoc: Assoc(T;op),
ident: Ident(T;op;id),
infix_ap: x f y,
true: True
Lemmas referenced :
monoid_p_wf,
comm_wf,
bag_wf,
list_wf,
permutation_wf,
equal_wf,
equal-wf-base,
list_induction,
list_accum_wf,
top_wf,
subtype_rel_list,
list_accum_nil_lemma,
list_accum_cons_lemma,
squash_wf,
true_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis,
productEquality,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
lambdaFormation,
rename,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality,
independent_isectElimination,
voidElimination,
voidEquality,
addLevel,
hyp_replacement,
imageElimination,
imageMemberEquality,
baseClosed,
natural_numberEquality,
levelHypothesis
Latex:
\mforall{}[T,R:Type]. \mforall{}[add:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R]. \mforall{}[b:bag(T)].
\mSigma{}(x\mmember{}b). zero = zero supposing IsMonoid(R;add;zero) \mwedge{} Comm(R;add)
Date html generated:
2017_10_01-AM-08_50_47
Last ObjectModification:
2017_07_26-PM-04_32_56
Theory : bags
Home
Index