Nuprl Lemma : bag-mapfilter-fast-eq
∀[A,B:Type]. ∀[bs:bag(A)]. ∀[P:A ⟶ 𝔹]. ∀[f:{x:A| ↑P[x]} ⟶ B].
(bag-mapfilter-fast(f;P;bs) = bag-mapfilter(f;P;bs) ∈ bag(B))
Proof
Definitions occuring in Statement :
bag-mapfilter-fast: bag-mapfilter-fast(f;P;bs),
bag-mapfilter: bag-mapfilter(f;P;bs),
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
set: {x:A| B[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-mapfilter: bag-mapfilter(f;P;bs),
so_apply: x[s],
prop: ℙ,
so_lambda: λ2x.t[x],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
bag-mapfilter-fast: bag-mapfilter-fast(f;P;bs),
exists: ∃x:A. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
empty-bag: {},
bag-accum: bag-accum(v,x.f[v; x];init;bs),
cons-bag: x.b,
bnot: ¬bb,
assert: ↑b,
label: ...$L... t,
bag-append: as + bs,
single-bag: {x},
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
bag-map-as-accum,
assert_wf,
bag-filter_wf,
equal_wf,
squash_wf,
true_wf,
bag_wf,
bag-mapfilter-fast_wf,
iff_weakening_equal,
bag-filter-as-accum,
bool_wf,
set_wf,
empty-bag_wf,
cons-bag_wf,
cons-bag-as-append,
bag-append_wf,
bag-append-comm,
single-bag_wf,
bag-accum_wf,
all_wf,
bag-append-assoc,
bag_to_squash_list,
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_wf,
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,
eqtt_to_assert,
list_accum_nil_lemma,
nil_wf,
list-subtype-bag,
bag-append-assoc-comm,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
subtype_rel_list,
top_wf,
list_accum_append,
bag-accum-single,
list_accum_wf,
ifthenelse_wf,
append_wf,
bag-subtype-list,
list_ind_cons_lemma,
list_ind_nil_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
sqequalRule,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
functionEquality,
isect_memberEquality,
axiomEquality,
setElimination,
rename,
dependent_set_memberEquality,
lambdaFormation,
voidElimination,
voidEquality,
dependent_functionElimination,
promote_hyp,
intWeakElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
unionElimination,
hypothesis_subsumption,
applyLambdaEquality,
addEquality,
instantiate,
hyp_replacement,
equalityElimination
Latex:
\mforall{}[A,B:Type]. \mforall{}[bs:bag(A)]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:A| \muparrow{}P[x]\} {}\mrightarrow{} B].
(bag-mapfilter-fast(f;P;bs) = bag-mapfilter(f;P;bs))
Date html generated:
2017_10_01-AM-08_58_35
Last ObjectModification:
2017_07_26-PM-04_40_28
Theory : bags
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