Nuprl Lemma : bag-member-combine
∀[A,B:Type]. ∀[bs:bag(A)]. ∀[f:A ⟶ bag(B)]. ∀[b:B]. uiff(b ↓∈ ⋃x∈bs.f[x];↓∃x:A. (x ↓∈ bs ∧ b ↓∈ f[x]))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-combine: ⋃x∈bs.f[x],
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
so_apply: x[s],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
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),
squash: ↓T,
bag-member: x ↓∈ bs,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
empty-bag: {},
bag-combine: ⋃x∈bs.f[x],
bag-size: #(bs),
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
concat: concat(ll),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_stable: SqStable(P),
length: ||as||,
list_ind: list_ind,
bag-append: as + bs,
append: as @ bs,
single-bag: {x},
cons: [a / b],
nil: [],
it: ⋅,
true: True,
sq_or: a ↓∨ b,
cand: A c∧ B
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
bag-cases,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
bag-size_wf,
bag_wf,
istype-universe,
length_of_nil_lemma,
map_nil_lemma,
reduce_nil_lemma,
squash_wf,
false_wf,
bag-member_wf,
iff_weakening_uiff,
empty-bag_wf,
bag-member-empty-iff,
le_wf,
uiff_wf,
bag-combine_wf,
sq_stable__all,
sq_stable__uiff,
sq_stable__bag-member,
sq_stable__squash,
equal-wf-base,
subtract_nat_wf,
add-is-int-iff,
bag-combine-single-left,
true_wf,
bag-combine-append-left,
single-bag_wf,
iff_weakening_equal,
sq_or_wf,
bag-member-append,
bag-append_wf,
bag-member-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
productElimination,
independent_pairEquality,
imageElimination,
imageMemberEquality,
baseClosed,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
unionElimination,
applyEquality,
instantiate,
because_Cache,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
addEquality,
functionIsType,
universeEquality,
productEquality,
promote_hyp,
hyp_replacement,
functionEquality,
intEquality,
equalityIstype,
pointwiseFunctionality,
baseApply,
closedConclusion,
inlFormation_alt,
unionEquality,
inrFormation_alt
Latex:
\mforall{}[A,B:Type]. \mforall{}[bs:bag(A)]. \mforall{}[f:A {}\mrightarrow{} bag(B)]. \mforall{}[b:B].
uiff(b \mdownarrow{}\mmember{} \mcup{}x\mmember{}bs.f[x];\mdownarrow{}\mexists{}x:A. (x \mdownarrow{}\mmember{} bs \mwedge{} b \mdownarrow{}\mmember{} f[x]))
Date html generated:
2019_10_15-AM-11_01_38
Last ObjectModification:
2019_06_25-PM-03_26_03
Theory : bags
Home
Index