Nuprl Lemma : bag-sum-count
∀[A,B:Type]. ∀[f:B ⟶ bag(A)]. ∀[eq:EqDecider(A)]. ∀[bb:bag(B)]. ∀[z:A].
(bag-sum(bb;b.(#z in f[b])) ~ bag-sum([b∈bb|1 ≤z (#z in f[b])];b.(#z in f[b])))
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-sum: bag-sum(ba;x.f[x]),
bag-filter: [x∈b|p[x]],
bag: bag(T),
deq: EqDecider(T),
le_int: i ≤z j,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
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],
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
bag-filter: [x∈b|p[x]],
bag-sum: bag-sum(ba;x.f[x]),
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
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,
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},
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bnot: ¬bb
Lemmas referenced :
subtype_base_sq,
int_subtype_base,
list_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
equal_wf,
bag-sum_wf,
bag-count_wf,
assert_wf,
le_int_wf,
bag-filter_wf,
list-subtype-bag,
equal-wf-base,
bag_wf,
deq_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_wf3,
subtype_rel_list,
top_wf,
list-cases,
list_accum_nil_lemma,
filter_nil_lemma,
product_subtype_list,
null_cons_lemma,
last-lemma-sq,
pos_length,
iff_transitivity,
not_wf,
equal-wf-T-base,
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,
cons_wf,
last_wf,
nil_wf,
filter_wf5,
l_member_wf,
list_accum_cons_lemma,
filter_cons_lemma,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
list_accum_append,
filter_append,
set_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
hypothesis,
pointwiseFunctionalityForEquality,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
hypothesisEquality,
lambdaFormation,
because_Cache,
rename,
lambdaEquality,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
natural_numberEquality,
setElimination,
productEquality,
sqequalAxiom,
isect_memberEquality,
functionEquality,
universeEquality,
intWeakElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
axiomEquality,
unionElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
imageElimination,
promote_hyp,
baseClosed,
impliesFunctionality,
addEquality,
equalityElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:B {}\mrightarrow{} bag(A)]. \mforall{}[eq:EqDecider(A)]. \mforall{}[bb:bag(B)]. \mforall{}[z:A].
(bag-sum(bb;b.(\#z in f[b])) \msim{} bag-sum([b\mmember{}bb|1 \mleq{}z (\#z in f[b])];b.(\#z in f[b])))
Date html generated:
2018_05_21-PM-09_46_19
Last ObjectModification:
2017_07_26-PM-06_29_59
Theory : bags_2
Home
Index