Nuprl Lemma : bag-combine-size
∀[A,B:Type]. ∀[f:A ⟶ bag(B)]. ∀[ba:bag(A)]. (#(⋃a∈ba.f[a]) = bag-sum(ba;a.#(f[a])) ∈ ℕ)
Proof
Definitions occuring in Statement :
bag-sum: bag-sum(ba;x.f[x]),
bag-combine: ⋃x∈bs.f[x],
bag-size: #(bs),
bag: bag(T),
nat: ℕ,
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,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
prop: ℙ,
bag-combine: ⋃x∈bs.f[x],
bag-size: #(bs),
bag-sum: bag-sum(ba;x.f[x]),
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
concat: concat(ll),
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
or: P ∨ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
le: A ≤ B,
less_than': less_than'(a;b),
cons: [a / b],
colength: colength(L),
nil: [],
it: ⋅,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
nat_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
permutation_wf,
list_wf,
bag_wf,
istype-universe,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
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,
list_accum_nil_lemma,
map_nil_lemma,
reduce_nil_lemma,
length_of_nil_lemma,
decidable__equal_int,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
istype-false,
istype-le,
product_subtype_list,
colength-cons-not-zero,
istype-nat,
colength_wf_list,
subtract-1-ge-0,
spread_cons_lemma,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
list_accum_cons_lemma,
map_cons_lemma,
reduce_cons_lemma,
length-append,
bag-size_wf,
zero-le-nat,
list_accum_wf,
add_nat_wf,
add-swap,
add-commutes,
equal_wf,
iff_weakening_equal,
trivial-equal,
quotient-member-eq,
permutation-equiv,
bag-combine_wf,
list-subtype-bag,
equal-wf-base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
thin,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
rename,
instantiate,
isectElimination,
cumulativity,
independent_isectElimination,
intEquality,
lambdaEquality_alt,
closedConclusion,
natural_numberEquality,
hypothesisEquality,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
equalityIsType1,
productIsType,
equalityIsType4,
because_Cache,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
functionIsType,
universeEquality,
setElimination,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
independent_pairFormation,
functionIsTypeImplies,
unionElimination,
dependent_set_memberEquality_alt,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
imageElimination,
baseApply,
baseClosed,
applyEquality,
addEquality,
imageMemberEquality,
hyp_replacement
Latex:
\mforall{}[A,B:Type]. \mforall{}[f:A {}\mrightarrow{} bag(B)]. \mforall{}[ba:bag(A)]. (\#(\mcup{}a\mmember{}ba.f[a]) = bag-sum(ba;a.\#(f[a])))
Date html generated:
2019_10_15-AM-11_00_28
Last ObjectModification:
2018_10_18-PM-11_34_06
Theory : bags
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