Nuprl Lemma : bag-filter-as-accum
∀[A:Type]. ∀[p:A ⟶ 𝔹]. ∀[bs:bag(A)].
([x∈bs|p[x]] = bag-accum(b,x.if p[x] then x.b else b fi ;{};bs) ∈ bag({x:A| ↑p[x]} ))
Proof
Definitions occuring in Statement :
bag-accum: bag-accum(v,x.f[v; x];init;bs),
bag-filter: [x∈b|p[x]],
cons-bag: x.b,
empty-bag: {},
bag: bag(T),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
bag-accum: bag-accum(v,x.f[v; x];init;bs),
list_accum: list_accum,
nil: [],
it: ⋅,
empty-bag: {},
bag-filter: [x∈b|p[x]],
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
so_lambda: λ2x.t[x],
so_apply: x[s],
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
sq_type: SQType(T),
less_than: a < b,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
cons-bag: x.b,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bnot: ¬bb,
assert: ↑b,
bag-append: as + bs
Lemmas referenced :
bag_to_squash_list,
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,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list_wf,
list-cases,
bag-filter_wf,
nil_wf,
list-subtype-bag,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
bag-accum_wf,
bool_wf,
eqtt_to_assert,
cons-bag_wf,
assert_wf,
bag_wf,
filter_cons_lemma,
uiff_transitivity,
bnot_wf,
not_wf,
eqff_to_assert,
assert_of_bnot,
empty-bag_wf,
cons-bag-as-append,
iff_weakening_equal,
set_wf,
bag-append_wf,
squash_wf,
true_wf,
bag-append-comm,
single-bag_wf,
bag-append-assoc-comm,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
list_accum_append,
subtype_rel_list,
top_wf,
bag-accum-single,
bag-append-assoc,
all_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
rename,
lambdaFormation,
setElimination,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
cumulativity,
applyEquality,
unionElimination,
functionExtensionality,
hypothesis_subsumption,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
hyp_replacement,
equalityElimination,
setEquality,
functionEquality,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[A:Type]. \mforall{}[p:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[bs:bag(A)].
([x\mmember{}bs|p[x]] = bag-accum(b,x.if p[x] then x.b else b fi ;\{\};bs))
Date html generated:
2017_10_01-AM-08_48_21
Last ObjectModification:
2017_07_26-PM-04_32_30
Theory : bags
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