Nuprl Lemma : bag-summation-single-non-zero
∀[T,R:Type]. ∀[eq:EqDecider(T)]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[f:T ⟶ R].
∀z:T. Σ(x∈b). f[x] = Σ(x∈[x∈b|eq x z]). f[x] ∈ R supposing ∀x:T. (x ↓∈ b ⇒ ((x = z ∈ T) ∨ (f[x] = zero ∈ R)))
supposing IsMonoid(R;add;zero) ∧ Comm(R;add)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-summation: Σ(x∈b). f[x],
bag-filter: [x∈b|p[x]],
bag: bag(T),
deq: EqDecider(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,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
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,
all: ∀x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
deq: EqDecider(T),
so_apply: x[s],
cand: A c∧ B,
prop: ℙ,
implies: P ⇒ Q,
or: P ∨ Q,
monoid_p: IsMonoid(T;op;id),
uiff: uiff(P;Q),
not: ¬A,
false: False,
eqof: eqof(d),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
ident: Ident(T;op;id)
Lemmas referenced :
bag-summation-split,
equal_wf,
infix_ap_wf,
bag-summation_wf,
assert_wf,
bag-filter_wf,
bag-member_wf,
istype-universe,
monoid_p_wf,
comm_wf,
bag_wf,
deq_wf,
bag-summation-is-zero,
bnot_wf,
bag-member-filter-set,
eqof_wf,
not_wf,
istype-void,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
safe-assert-deq
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
promote_hyp,
lambdaFormation_alt,
productElimination,
sqequalRule,
lambdaEquality_alt,
applyEquality,
setElimination,
rename,
because_Cache,
inhabitedIsType,
independent_isectElimination,
independent_pairFormation,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
equalityTransitivity,
setEquality,
setIsType,
universeIsType,
functionIsType,
unionIsType,
equalityIsType1,
dependent_functionElimination,
isect_memberEquality_alt,
axiomEquality,
functionIsTypeImplies,
productIsType,
universeEquality,
independent_functionElimination,
unionElimination,
voidElimination
Latex:
\mforall{}[T,R:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[add:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R]. \mforall{}[b:bag(T)]. \mforall{}[f:T {}\mrightarrow{} R].
\mforall{}z:T
\mSigma{}(x\mmember{}b). f[x] = \mSigma{}(x\mmember{}[x\mmember{}b|eq x z]). f[x] supposing \mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} ((x = z) \mvee{} (f[x] = zero)))
supposing IsMonoid(R;add;zero) \mwedge{} Comm(R;add)
Date html generated:
2019_10_15-AM-11_03_29
Last ObjectModification:
2018_10_09-PM-00_13_42
Theory : bags
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