Nuprl Lemma : bag-summation-single-non-zero-no-repeats
∀[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] = f[z] ∈ R) supposing
((bag-no-repeats(T;b) ∧ z ↓∈ b) and
(∀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-no-repeats: bag-no-repeats(T;bs),
bag-summation: Σ(x∈b). f[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,
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,
squash: ↓T,
prop: ℙ,
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
deq: EqDecider(T),
cand: A c∧ B,
uiff: uiff(P;Q),
bag-member: x ↓∈ bs,
or: P ∨ Q,
monoid_p: IsMonoid(T;op;id),
eqof: eqof(d),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
bag-summation-single-non-zero,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
bag-extensionality-no-repeats,
decidable-equal-deq,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
single-bag_wf,
bag-single-no-repeats,
bag-member-single,
bag-member_wf,
bag-summation-single,
bag-summation_wf,
bag-no-repeats_wf,
all_wf,
or_wf,
monoid_p_wf,
comm_wf,
bag_wf,
deq_wf,
bag-filter-no-repeats,
bag-member-filter,
safe-assert-deq,
and_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
productElimination,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
functionExtensionality,
cumulativity,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_functionElimination,
because_Cache,
setElimination,
rename,
setEquality,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality,
productEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
dependent_set_memberEquality
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] = f[z]) supposing
((bag-no-repeats(T;b) \mwedge{} z \mdownarrow{}\mmember{} b) and
(\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:
2017_10_01-AM-09_01_57
Last ObjectModification:
2017_07_26-PM-04_43_16
Theory : bags
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