Nuprl Lemma : bag-summation-append
∀[R:Type]. ∀[add:R ⟶ R ⟶ R]. ∀[zero:R].
∀[T:Type]. ∀[f:T ⟶ R]. ∀[b,c:bag(T)]. (Σ(x∈b + c). f[x] = (Σ(x∈b). f[x] add Σ(x∈c). f[x]) ∈ R)
supposing IsMonoid(R;add;zero) ∧ Comm(R;add)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag-append: as + bs,
bag: bag(T),
comm: Comm(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
so_apply: x[s],
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,
bag: bag(T),
quotient: x,y:A//B[x; y],
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
subtype_rel: A ⊆r B,
monoid_p: IsMonoid(T;op;id),
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
empty-bag: {},
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
int_iseg: {i...j},
single-bag: {x},
bag-append: as + bs,
true: True,
ident: Ident(T;op;id),
infix_ap: x f y,
assoc: Assoc(T;op),
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
comm: Comm(T;op)
Lemmas referenced :
list_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
equal_wf,
bag-summation_wf,
bag-append_wf,
infix_ap_wf,
list-subtype-bag,
equal-wf-base,
bag_wf,
monoid_p_wf,
comm_wf,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
le_wf,
length_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
non_neg_length,
decidable__lt,
lelt_wf,
decidable__assert,
null_wf,
list-cases,
bag-summation-empty,
empty_bag_append_lemma,
product_subtype_list,
null_cons_lemma,
last-lemma-sq,
pos_length,
iff_transitivity,
not_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
assert_of_null,
iff_weakening_uiff,
assert_of_bnot,
firstn_wf,
length_firstn,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
length_wf_nat,
single-bag_wf,
last_wf,
squash_wf,
true_wf,
bag-append-assoc,
iff_weakening_equal,
list_accum_append,
bag-subtype-list,
list_accum_cons_lemma,
list_accum_nil_lemma,
subtype_rel_list,
top_wf,
list_accum_wf,
bag-accum_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
pointwiseFunctionalityForEquality,
because_Cache,
sqequalRule,
pertypeElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaFormation,
rename,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
functionExtensionality,
applyEquality,
independent_pairFormation,
productEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
universeEquality,
setElimination,
intWeakElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
computeAll,
unionElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
imageElimination,
promote_hyp,
baseClosed,
impliesFunctionality,
addEquality,
imageMemberEquality
Latex:
\mforall{}[R:Type]. \mforall{}[add:R {}\mrightarrow{} R {}\mrightarrow{} R]. \mforall{}[zero:R].
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} R]. \mforall{}[b,c:bag(T)]. (\mSigma{}(x\mmember{}b + c). f[x] = (\mSigma{}(x\mmember{}b). f[x] add \mSigma{}(x\mmember{}c). f[x]))
supposing IsMonoid(R;add;zero) \mwedge{} Comm(R;add)
Date html generated:
2017_10_01-AM-08_48_37
Last ObjectModification:
2017_07_26-PM-04_32_42
Theory : bags
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