Nuprl Lemma : bag-summation-partition
∀[A:Type]
∀[R,T:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R]. ∀[b:bag(T)]. ∀[case:T ⟶ A ⟶ 𝔹]. ∀[f:T ⟶ R]. ∀[c:bag(A)].
Σ(x∈b). f[x] = Σ(z∈c). Σ(x∈[x∈b|case[x;z]]). f[x] ∈ R
supposing (IsMonoid(R;add;zero) ∧ Comm(R;add))
∧ (∀x:{x:T| x ↓∈ b} . (∃z:{A| (z ↓∈ c ∧ (↑case[x;z]))}))
∧ bag-no-repeats(A;c)
∧ (∀z1,z2:A. ∀x:T. ((↑case[x;z1]) ⇒ (↑case[x;z2]) ⇒ (z1 = z2 ∈ A)))
supposing ∀x,y:A. Dec(x = y ∈ A)
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-filter: [x∈b|p[x]],
bag: bag(T),
comm: Comm(T;op),
assert: ↑b,
bool: 𝔹,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
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,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
sq_exists: ∃x:{A| B[x]},
cand: A c∧ B,
monoid_p: IsMonoid(T;op;id),
bag-filter: [x∈b|p[x]],
top: Top,
empty-bag: {},
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cons-bag: x.b,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
sq_or: a ↓∨ b,
or: P ∨ Q,
single-bag: {x},
bag-append: as + bs,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
infix_ap: x f y,
bag-member: x ↓∈ bs,
sq_type: SQType(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt
Lemmas referenced :
bag_to_squash_list,
all_wf,
bag-member_wf,
sq_exists_wf,
assert_wf,
list_induction,
list-subtype-bag,
equal_wf,
bag-summation_wf,
bag-filter_wf,
list_wf,
nil_wf,
cons_wf,
monoid_p_wf,
comm_wf,
bag-no-repeats_wf,
bag_wf,
bool_wf,
decidable_wf,
filter_nil_lemma,
bag-summation-empty,
squash_wf,
true_wf,
bag-summation-zero,
iff_weakening_equal,
bag-summation-cons,
cons-bag_wf,
set_wf,
bag-member-cons,
infix_ap_wf,
list_ind_cons_lemma,
list_ind_nil_lemma,
bag-summation-add,
single-bag_wf,
bag-append_wf,
assoc_wf,
bag-filter-append,
bag-summation-append,
ifthenelse_wf,
bag-summation-filter,
bag-summation-single,
bag-extensionality-no-repeats,
subtype_rel_bag,
bag-filter-no-repeats,
bag-single-no-repeats,
bag-member-single,
bag-member-filter,
and_wf,
assert_elim,
subtype_base_sq,
bool_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
hypothesisEquality,
imageElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
setEquality,
cumulativity,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
setElimination,
rename,
functionEquality,
because_Cache,
independent_isectElimination,
lambdaFormation,
independent_pairFormation,
independent_functionElimination,
voidEquality,
voidElimination,
dependent_functionElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
natural_numberEquality,
universeEquality,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality,
inrFormation,
inlFormation,
instantiate
Latex:
\mforall{}[A:Type]
\mforall{}[R,T:Type]. \mforall{}[add:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R]. \mforall{}[b:bag(T)]. \mforall{}[case:T {}\mrightarrow{} A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:T {}\mrightarrow{} R].
\mforall{}[c:bag(A)].
\mSigma{}(x\mmember{}b). f[x] = \mSigma{}(z\mmember{}c). \mSigma{}(x\mmember{}[x\mmember{}b|case[x;z]]). f[x]
supposing (IsMonoid(R;add;zero) \mwedge{} Comm(R;add))
\mwedge{} (\mforall{}x:\{x:T| x \mdownarrow{}\mmember{} b\} . (\mexists{}z:\{A| (z \mdownarrow{}\mmember{} c \mwedge{} (\muparrow{}case[x;z]))\}))
\mwedge{} bag-no-repeats(A;c)
\mwedge{} (\mforall{}z1,z2:A. \mforall{}x:T. ((\muparrow{}case[x;z1]) {}\mRightarrow{} (\muparrow{}case[x;z2]) {}\mRightarrow{} (z1 = z2)))
supposing \mforall{}x,y:A. Dec(x = y)
Date html generated:
2017_10_01-AM-09_02_30
Last ObjectModification:
2017_07_26-PM-04_43_33
Theory : bags
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