Nuprl Lemma : bag-combine-as-accum
∀[A,B:Type]. ∀[f:A ⟶ bag(B)]. ∀[bs:bag(A)].  (⋃b∈bs.f[b] = bag-accum(c,b.f[b] + c;{};bs) ∈ bag(B))
Proof
Definitions occuring in Statement : 
bag-accum: bag-accum(v,x.f[v; x];init;bs)
, 
bag-combine: ⋃x∈bs.f[x]
, 
bag-append: as + bs
, 
empty-bag: {}
, 
bag: bag(T)
, 
uall: ∀[x:A]. B[x]
, 
so_apply: x[s]
, 
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
, 
not: ¬A
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
, 
and: P ∧ Q
, 
prop: ℙ
, 
or: P ∨ Q
, 
cons: [a / b]
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
colength: colength(L)
, 
nil: []
, 
it: ⋅
, 
guard: {T}
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
sq_type: SQType(T)
, 
less_than: a < b
, 
so_lambda: λ2x y.t[x; y]
, 
so_apply: x[s1;s2]
, 
decidable: Dec(P)
, 
subtype_rel: A ⊆r B
, 
bag-combine: ⋃x∈bs.f[x]
, 
bag-union: bag-union(bbs)
, 
concat: concat(ll)
, 
reduce: reduce(f;k;as)
, 
list_ind: list_ind, 
bag-map: bag-map(f;bs)
, 
map: map(f;as)
, 
empty-bag: {}
, 
bag-accum: bag-accum(v,x.f[v; x];init;bs)
, 
list_accum: list_accum, 
cons-bag: x.b
, 
true: True
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
bag-append: as + bs
Lemmas referenced : 
bag_to_squash_list, 
nat_properties, 
full-omega-unsat, 
intformand_wf, 
intformle_wf, 
itermConstant_wf, 
itermVar_wf, 
intformless_wf, 
istype-int, 
int_formula_prop_and_lemma, 
istype-void, 
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, 
istype-less_than, 
list-cases, 
product_subtype_list, 
colength-cons-not-zero, 
colength_wf_list, 
istype-le, 
list_wf, 
subtract-1-ge-0, 
subtype_base_sq, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
set_subtype_base, 
int_subtype_base, 
spread_cons_lemma, 
decidable__equal_int, 
subtract_wf, 
intformnot_wf, 
itermSubtract_wf, 
itermAdd_wf, 
int_formula_prop_not_lemma, 
int_term_value_subtract_lemma, 
int_term_value_add_lemma, 
decidable__le, 
le_wf, 
istype-nat, 
equal_wf, 
bag_wf, 
bag-combine_wf, 
bag-accum_wf, 
empty-bag_wf, 
bag-append_wf, 
bag-append-assoc-comm, 
istype-universe, 
bag-combine-cons-left, 
cons-bag_wf, 
list-subtype-bag, 
squash_wf, 
true_wf, 
subtype_rel_self, 
iff_weakening_equal, 
cons-bag-as-append, 
bag-append-comm, 
single-bag_wf, 
list_accum_append, 
subtype_rel_list, 
top_wf, 
bag-accum-single
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation_alt, 
introduction, 
cut, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesisEquality, 
imageElimination, 
productElimination, 
promote_hyp, 
hypothesis, 
rename, 
lambdaFormation_alt, 
setElimination, 
intWeakElimination, 
natural_numberEquality, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
dependent_pairFormation_alt, 
lambdaEquality_alt, 
int_eqEquality, 
dependent_functionElimination, 
isect_memberEquality_alt, 
voidElimination, 
sqequalRule, 
independent_pairFormation, 
universeIsType, 
axiomEquality, 
functionIsTypeImplies, 
inhabitedIsType, 
unionElimination, 
hypothesis_subsumption, 
equalityIstype, 
because_Cache, 
dependent_set_memberEquality_alt, 
instantiate, 
equalityTransitivity, 
equalitySymmetry, 
applyLambdaEquality, 
baseApply, 
closedConclusion, 
baseClosed, 
applyEquality, 
intEquality, 
sqequalBase, 
hyp_replacement, 
isectIsTypeImplies, 
functionIsType, 
universeEquality, 
imageMemberEquality, 
functionEquality
Latex:
\mforall{}[A,B:Type].  \mforall{}[f:A  {}\mrightarrow{}  bag(B)].  \mforall{}[bs:bag(A)].    (\mcup{}b\mmember{}bs.f[b]  =  bag-accum(c,b.f[b]  +  c;\{\};bs))
Date html generated:
2019_10_15-AM-11_00_34
Last ObjectModification:
2019_08_01-PM-01_36_17
Theory : bags
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