Nuprl Lemma : bag-member-parts
∀[T:Type]
∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. ∀[L:bag(T) List+].
uiff(L ↓∈ bag-parts(eq;bs);(bag-union(L) = bs ∈ bag(T)) ∧ (∀x∈L.¬(x = {} ∈ bag(T))))
supposing valueall-type(T)
Proof
Definitions occuring in Statement :
bag-parts: bag-parts(eq;bs),
bag-member: x ↓∈ bs,
bag-union: bag-union(bbs),
empty-bag: {},
bag: bag(T),
l_all: (∀x∈L.P[x]),
listp: A List+,
deq: EqDecider(T),
valueall-type: valueall-type(T),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
uiff: uiff(P;Q),
l_all: (∀x∈L.P[x]),
listp: A List+,
int_seg: {i..j-},
lelt: i ≤ j < k,
bag-member: x ↓∈ bs,
squash: ↓T,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
less_than: a < b,
bag-parts: bag-parts(eq;bs),
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-union: bag-union(bbs),
concat: concat(ll),
empty-bag: {},
bag-append: as + bs,
true: True,
cons-bag: x.b,
le: A ≤ B,
less_than': less_than'(a;b),
length: ||as||,
list_ind: list_ind,
nil: [],
cons: [a / b],
rev_uimplies: rev_uimplies(P;Q),
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
full-omega-unsat,
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,
bag_wf,
select_wf,
int_seg_properties,
length_wf,
int_seg_wf,
bag-member_wf,
listp_wf,
bag-parts_wf,
equal_wf,
bag-union_wf,
subtype_rel_set,
list_wf,
list-subtype-bag,
l_all_wf2,
not_wf,
l_member_wf,
le_wf,
bag-size_wf,
nat_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
subtype_base_sq,
set_subtype_base,
lelt_wf,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
subtype_rel_self,
itermAdd_wf,
int_term_value_add_lemma,
deq_wf,
valueall-type_wf,
valueall-type-has-valueall,
bag-valueall-type,
product-valueall-type,
bag-partitions_wf,
evalall-reduce,
squash_wf,
exists_wf,
bag-null_wf,
bool_wf,
eqtt_to_assert,
assert-bag-null,
empty-bag_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
not_functionality_wrt_uiff,
assert_wf,
single-bag_wf,
cons_wf_listp,
nil_wf,
set-valueall-type,
list-valueall-type,
bag-map_wf,
bag-combine_wf,
pi1_wf_top,
pi2_wf,
iff_transitivity,
iff_weakening_uiff,
bag-member-combine,
exists-pair,
bag-member-partitions,
bool_cases,
uiff_transitivity,
bnot_wf,
assert_of_bnot,
bag-member-empty-iff,
bag-member-single,
reduce_cons_lemma,
reduce_nil_lemma,
true_wf,
bag-append_wf,
iff_weakening_equal,
bag-member-map,
bag-union-cons,
bag-size-append,
bag-size-zero,
add-is-int-iff,
false_wf,
l_all_iff,
member_singleton,
l_all_cons,
cons_wf,
and_wf,
listp_properties,
list-cases,
product_subtype_list,
bag-append-is-empty,
length_of_nil_lemma,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
productElimination,
independent_pairEquality,
axiomEquality,
cumulativity,
because_Cache,
baseClosed,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
productEquality,
applyEquality,
setEquality,
unionElimination,
instantiate,
applyLambdaEquality,
dependent_set_memberEquality,
hypothesis_subsumption,
addEquality,
universeIsType,
universeEquality,
callbyvalueReduce,
equalityElimination,
promote_hyp,
productIsType,
hyp_replacement,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]
\mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)]. \mforall{}[L:bag(T) List\msupplus{}].
uiff(L \mdownarrow{}\mmember{} bag-parts(eq;bs);(bag-union(L) = bs) \mwedge{} (\mforall{}x\mmember{}L.\mneg{}(x = \{\})))
supposing valueall-type(T)
Date html generated:
2019_10_16-AM-11_32_10
Last ObjectModification:
2018_09_26-PM-11_46_08
Theory : bags_2
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