Nuprl Lemma : bag-map-union
∀[T,S:Type]. ∀[f:T ⟶ bag(S)]. ∀[bbs:bag(bag(T))].
  (bag-map(f;bag-union(bbs)) = bag-union(bag-map(λb.bag-map(f;b);bbs)) ∈ bag(bag(S)))
Proof
Definitions occuring in Statement : 
bag-union: bag-union(bbs)
, 
bag-map: bag-map(f;bs)
, 
bag: bag(T)
, 
uall: ∀[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)
, 
quotient: x,y:A//B[x; y]
, 
and: P ∧ Q
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
prop: ℙ
, 
nat: ℕ
, 
false: False
, 
ge: i ≥ j 
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
top: Top
, 
subtype_rel: A ⊆r B
, 
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]
, 
so_apply: x[s]
, 
sq_type: SQType(T)
, 
less_than: a < b
, 
squash: ↓T
, 
less_than': less_than'(a;b)
, 
bag-union: bag-union(bbs)
, 
bag-map: bag-map(f;bs)
, 
concat: concat(ll)
, 
true: True
Lemmas referenced : 
bag_wf, 
list_wf, 
permutation_wf, 
equal_wf, 
equal-wf-base, 
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, 
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_nil_lemma, 
reduce_nil_lemma, 
map_cons_lemma, 
reduce_cons_lemma, 
map_append_sq, 
bag-map_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, 
cumulativity, 
hypothesisEquality, 
hypothesis, 
sqequalRule, 
pertypeElimination, 
productElimination, 
equalityTransitivity, 
equalitySymmetry, 
lambdaFormation, 
because_Cache, 
rename, 
dependent_functionElimination, 
independent_functionElimination, 
productEquality, 
isect_memberEquality, 
axiomEquality, 
functionEquality, 
universeEquality, 
setElimination, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
voidElimination, 
voidEquality, 
independent_pairFormation, 
computeAll, 
sqequalAxiom, 
applyEquality, 
unionElimination, 
promote_hyp, 
hypothesis_subsumption, 
applyLambdaEquality, 
dependent_set_memberEquality, 
addEquality, 
baseClosed, 
instantiate, 
imageElimination, 
imageMemberEquality
Latex:
\mforall{}[T,S:Type].  \mforall{}[f:T  {}\mrightarrow{}  bag(S)].  \mforall{}[bbs:bag(bag(T))].
    (bag-map(f;bag-union(bbs))  =  bag-union(bag-map(\mlambda{}b.bag-map(f;b);bbs)))
Date html generated:
2017_10_01-AM-08_46_42
Last ObjectModification:
2017_07_26-PM-04_31_25
Theory : bags
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