Nuprl Lemma : bag-filter-union
∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[bbs:bag(bag(T))].
([x∈bag-union(bbs)|p[x]] = bag-union(bag-map(λb.[x∈b|p[x]];bbs)) ∈ bag({x:T| ↑p[x]} ))
Proof
Definitions occuring in Statement :
bag-union: bag-union(bbs),
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag: bag(T),
so_apply: x[s],
prop: ℙ,
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
bag-union: bag-union(bbs),
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
concat: concat(ll),
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},
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
bag_wf,
assert_wf,
list_wf,
permutation_wf,
equal_wf,
equal-wf-base,
bool_wf,
subtype_rel_list,
top_wf,
bag-subtype-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-cases,
map_nil_lemma,
reduce_nil_lemma,
filter_nil_lemma,
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,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
map_cons_lemma,
reduce_cons_lemma,
filter_append_sq,
bag-filter_wf,
squash_wf,
true_wf,
bag-union_wf,
quotient-member-eq,
permutation-equiv
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
because_Cache,
rename,
dependent_functionElimination,
independent_functionElimination,
productEquality,
isect_memberEquality,
axiomEquality,
functionEquality,
independent_isectElimination,
setElimination,
intWeakElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
sqequalAxiom,
unionElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[bbs:bag(bag(T))].
([x\mmember{}bag-union(bbs)|p[x]] = bag-union(bag-map(\mlambda{}b.[x\mmember{}b|p[x]];bbs)))
Date html generated:
2017_10_01-AM-08_46_38
Last ObjectModification:
2017_07_26-PM-04_31_22
Theory : bags
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