Nuprl Lemma : bag-summation-constant
∀[T:Type]. ∀[r:Rng]. ∀[b:bag(T)]. ∀a:|r|. (Σ(x∈b). a = (#(b) ⋅r a) ∈ |r|)
Proof
Definitions occuring in Statement :
bag-summation: Σ(x∈b). f[x],
bag-size: #(bs),
bag: bag(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
universe: Type,
equal: s = t ∈ T,
rng_nat_op: n ⋅r e,
rng: Rng,
rng_zero: 0,
rng_plus: +r,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
squash: ↓T,
exists: ∃x:A. B[x],
bag-size: #(bs),
rng_nat_op: n ⋅r e,
bag-summation: Σ(x∈b). f[x],
mon_nat_op: n ⋅ e,
bag-accum: bag-accum(v,x.f[v; x];init;bs),
add_grp_of_rng: r↓+gp,
grp_op: *,
pi2: snd(t),
pi1: fst(t),
grp_id: e,
nat_op: n x(op;id) e,
prop: ℙ,
rng: Rng,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
infix_ap: x f y,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
itop: Π(op,id) lb ≤ i < ub. E[i],
ycomb: Y,
lt_int: i <z j,
subtract: n - m,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
bag_to_squash_list,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
rng_zero_wf,
itop_wf,
rng_plus_wf,
length_wf,
int_seg_wf,
iff_weakening_equal,
bag-summation_wf,
rng_all_properties,
rng_plus_comm2,
rng_nat_op_wf,
bag-size_wf,
bag_wf,
rng_wf,
rng_plus_comm,
rng_plus_zero,
list_induction,
all_wf,
list_accum_wf,
top_wf,
subtype_rel_list,
list_wf,
list_accum_nil_lemma,
length_of_nil_lemma,
list_accum_cons_lemma,
length_of_cons_lemma,
add-subtract-cancel,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
infix_ap_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
non_neg_length,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
intformless_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
rng_plus_assoc,
rng_plus_ac_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
rename,
sqequalRule,
applyEquality,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
setElimination,
dependent_functionElimination,
natural_numberEquality,
cumulativity,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
independent_pairFormation,
axiomEquality,
isect_memberEquality,
voidElimination,
voidEquality,
addEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
instantiate,
int_eqEquality,
intEquality,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[r:Rng]. \mforall{}[b:bag(T)]. \mforall{}a:|r|. (\mSigma{}(x\mmember{}b). a = (\#(b) \mcdot{}r a))
Date html generated:
2017_10_01-AM-08_50_56
Last ObjectModification:
2017_07_26-PM-04_33_01
Theory : bags
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