Nuprl Lemma : bag-map-filter
∀[T,A:Type]. ∀[f:T ⟶ A]. ∀[P:T ⟶ 𝔹]. ∀[Q:A ⟶ 𝔹].
∀[L:bag(T)]. (bag-map(f;[x∈L|P[x]]) = [x∈bag-map(f;L)|Q[x]] ∈ bag(A)) supposing ∀x:T. Q[f x] = P[x]
Proof
Definitions occuring in Statement :
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
bag: bag(T),
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-map: bag-map(f;bs),
bag-filter: [x∈b|p[x]],
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
bag_wf,
list_wf,
permutation_wf,
equal_wf,
equal-wf-base,
all_wf,
bool_wf,
map-filter,
map_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,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
filter_nil_lemma,
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,
filter_cons_lemma,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
quotient-member-eq,
permutation-equiv,
bag-filter_wf,
bag-map_wf,
list-subtype-bag,
subtype_rel_bag,
assert_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
because_Cache,
rename,
dependent_functionElimination,
independent_functionElimination,
productEquality,
isect_memberEquality,
axiomEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
independent_isectElimination,
setElimination,
intWeakElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
sqequalAxiom,
unionElimination,
promote_hyp,
hypothesis_subsumption,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
equalityElimination,
setEquality,
hyp_replacement
Latex:
\mforall{}[T,A:Type]. \mforall{}[f:T {}\mrightarrow{} A]. \mforall{}[P:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[Q:A {}\mrightarrow{} \mBbbB{}].
\mforall{}[L:bag(T)]. (bag-map(f;[x\mmember{}L|P[x]]) = [x\mmember{}bag-map(f;L)|Q[x]]) supposing \mforall{}x:T. Q[f x] = P[x]
Date html generated:
2017_10_01-AM-08_46_01
Last ObjectModification:
2017_07_26-PM-04_31_04
Theory : bags
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