Nuprl Lemma : bag-count-combine
∀[T1,T2:Type]. ∀[f:T1 ⟶ bag(T2)]. ∀[eq:EqDecider(T2)]. ∀[z:T2]. ∀[bs:bag(T1)].
((#z in ⋃x∈bs.f[x]) ~ bag-sum(bs;x.(#z in f[x])))
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-sum: bag-sum(ba;x.f[x]),
bag-combine: ⋃x∈bs.f[x],
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
prop: ℙ,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
guard: {T},
bag-sum: bag-sum(ba;x.f[x]),
bag-combine: ⋃x∈bs.f[x],
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
concat: concat(ll),
bag-count: (#x in bs),
count: count(P;L),
reduce: reduce(f;k;as),
list_ind: list_ind,
nil: [],
it: ⋅,
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
int_iseg: {i...j},
cand: A c∧ B,
bag-append: as + bs
Lemmas referenced :
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
list_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
equal_wf,
bag-count_wf,
bag-combine_wf,
bag-sum_wf_nat,
list-subtype-bag,
equal-wf-base,
bag_wf,
deq_wf,
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,
length_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
non_neg_length,
decidable__lt,
lelt_wf,
decidable__assert,
null_wf3,
subtype_rel_list,
top_wf,
list-cases,
map_nil_lemma,
list_accum_nil_lemma,
reduce_nil_lemma,
product_subtype_list,
null_cons_lemma,
last-lemma-sq,
pos_length,
iff_transitivity,
not_wf,
equal-wf-T-base,
assert_wf,
bnot_wf,
assert_of_null,
iff_weakening_uiff,
assert_of_bnot,
firstn_wf,
length_firstn,
itermAdd_wf,
int_term_value_add_lemma,
length_wf_nat,
map_cons_lemma,
list_accum_cons_lemma,
concat-single,
last_wf,
bag-subtype-list,
map_append_sq,
list_accum_append,
concat_append,
bag-union_wf,
bag-map_wf,
add-commutes,
bag-count-append,
bag-append_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
natural_numberEquality,
hypothesisEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
because_Cache,
rename,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
applyEquality,
functionExtensionality,
productEquality,
sqequalAxiom,
isect_memberEquality,
functionEquality,
universeEquality,
setElimination,
intWeakElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
axiomEquality,
unionElimination,
hypothesis_subsumption,
dependent_set_memberEquality,
imageElimination,
promote_hyp,
baseClosed,
impliesFunctionality,
addEquality
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[f:T1 {}\mrightarrow{} bag(T2)]. \mforall{}[eq:EqDecider(T2)]. \mforall{}[z:T2]. \mforall{}[bs:bag(T1)].
((\#z in \mcup{}x\mmember{}bs.f[x]) \msim{} bag-sum(bs;x.(\#z in f[x])))
Date html generated:
2018_05_21-PM-09_46_16
Last ObjectModification:
2017_07_26-PM-06_29_58
Theory : bags_2
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