Nuprl Lemma : bag-count-filter
∀[T:Type]. ∀[p:T ⟶ 𝔹]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)]. ((#x in [t∈bs|p[t]]) ≤ (#x in bs))
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
deq: EqDecider(T),
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
exists: ∃x:A. B[x],
bag-filter: [x∈b|p[x]],
bag-size: #(bs),
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
deq: EqDecider(T),
bfalse: ff,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
less_than: a < b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
eqof: eqof(d)
Lemmas referenced :
sq_stable__le,
bag_wf,
deq_wf,
bool_wf,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
bag-count_wf,
nat_wf,
equal_wf,
bag_to_squash_list,
le_wf,
list-subtype-bag,
bag-count-sqequal,
filter-filter,
length-filter-le,
eqtt_to_assert,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
int_seg_wf,
band_wf,
eqof_wf,
l_member_wf,
l_all_wf2,
l_all_functionality,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
safe-assert-deq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
imageElimination,
cumulativity,
functionEquality,
universeEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
setEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
productElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
productEquality,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
addLevel,
impliesFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[p:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[bs:bag(T)].
((\#x in [t\mmember{}bs|p[t]]) \mleq{} (\#x in bs))
Date html generated:
2018_05_21-PM-09_46_09
Last ObjectModification:
2017_07_26-PM-06_29_57
Theory : bags_2
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