Nuprl Lemma : bag-partitions-cons
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[x:X]. ∀[b:bag(X)].
(bag-partitions(eq;x.b)
= (bag-map(λp.<x.fst(p), snd(p)>;[p∈bag-partitions(eq;b)|((#x in snd(p)) =z 0)])
+ bag-map(λp.<fst(p), x.snd(p)>;bag-partitions(eq;b)))
∈ bag(bag(X) × bag(X)))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
bag-partitions: bag-partitions(eq;bs),
bag-count: (#x in bs),
bag-filter: [x∈b|p[x]],
bag-append: as + bs,
bag-map: bag-map(f;bs),
cons-bag: x.b,
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
uiff: uiff(P;Q),
bag-member: x ↓∈ bs,
squash: ↓T,
not: ¬A,
false: False,
inject: Inj(A;B;f),
pi2: snd(t),
pi1: fst(t),
true: True,
exists: ∃x:A. B[x],
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
iff: P ⇐⇒ Q,
single-bag: {x},
deq: EqDecider(T),
or: P ∨ Q,
sq_type: SQType(T),
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
btrue: tt,
rev_implies: P ⇐ Q,
bfalse: ff,
decidable: Dec(P),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
sq_or: a ↓∨ b,
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
bag-extensionality-no-repeats,
bag_wf,
decidable__equal_product,
decidable__equal_bag,
decidable-equal-deq,
bag-partitions_wf,
cons-bag_wf,
bag-append_wf,
bag-map_wf,
assert_wf,
eq_int_wf,
bag-count_wf,
pi2_wf,
nat_wf,
pi1_wf_top,
subtype_rel_product,
top_wf,
bag-filter_wf,
no-repeats-bag-partitions,
bag-member_wf,
deq_wf,
valueall-type_wf,
bag-no-repeats-append,
subtype_rel_bag,
bag-map-no-repeats,
equal_wf,
bag-append-cancel,
single-bag_wf,
squash_wf,
true_wf,
cons-bag-as-append,
bag-filter-no-repeats,
bag-member-map,
not_wf,
exists_wf,
assert_functionality_wrt_uiff,
assert_of_eq_int,
bag-count-append,
subtype_rel_self,
iff_weakening_equal,
cons_wf,
nil_wf,
list-subtype-bag,
ifthenelse_wf,
add_functionality_wrt_eq,
bag-count-single,
bnot_wf,
eqof_wf,
member_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
le_weakening2,
decidable__lt,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
bag-member-partitions,
bag-member-append,
bag-member-filter,
or_wf,
sq_or_wf,
decidable__equal_int,
bag-drop-property,
bag-drop_wf,
bag-append-assoc,
bag-member-count,
int_subtype_base,
add-zero,
add-commutes,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
bool_cases_sqequal,
assert-bnot,
bag-append-comm,
cons_bag_append_lemma,
bag-append-assoc-comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
hypothesisEquality,
hypothesis,
independent_functionElimination,
lambdaFormation,
sqequalRule,
lambdaEquality,
because_Cache,
dependent_functionElimination,
independent_isectElimination,
setEquality,
applyEquality,
setElimination,
rename,
natural_numberEquality,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
independent_pairFormation,
imageElimination,
imageMemberEquality,
baseClosed,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
applyLambdaEquality,
hyp_replacement,
addLevel,
impliesFunctionality,
intEquality,
instantiate,
unionElimination,
cumulativity,
promote_hyp,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
orFunctionality,
existsFunctionality,
andLevelFunctionality,
inlFormation,
equalityElimination,
addEquality,
inrFormation
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[x:X]. \mforall{}[b:bag(X)].
(bag-partitions(eq;x.b)
= (bag-map(\mlambda{}p.<x.fst(p), snd(p)>[p\mmember{}bag-partitions(eq;b)|((\#x in snd(p)) =\msubz{} 0)])
+ bag-map(\mlambda{}p.<fst(p), x.snd(p)>bag-partitions(eq;b))))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_50_07
Last ObjectModification:
2018_05_19-PM-04_21_18
Theory : bags_2
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