Nuprl Lemma : bag-intersection
∀[A:Type]. ∀[as,as',bs,bs':bag(A)].
(↓∃x:A. (x ↓∈ as ∧ x ↓∈ bs)) supposing (#(as') < #(as) and #(bs') < #(bs) and ((as + as') = (bs + bs') ∈ bag(A)))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-size: #(bs),
bag-append: as + bs,
bag: bag(T),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
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,
squash: ↓T,
exists: ∃x:A. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
bag-size: #(bs),
bag-append: as + bs,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
cand: A c∧ B,
permutation: permutation(T;L1;L2),
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
implies: P ⇒ Q,
not: ¬A,
less_than: a < b,
guard: {T},
sq_type: SQType(T),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
inject: Inj(A;B;f),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
equipollent: A ~ B,
biject: Bij(A;B;f),
less_than': less_than'(a;b)
Lemmas referenced :
bag_to_squash_list,
less_than_wf,
bag-size_wf,
equal_wf,
bag_wf,
bag-append_wf,
list-subtype-bag,
nat_wf,
permutation-length,
decidable__exists_int_seg,
length_wf,
int_seg_wf,
length-append,
non_neg_length,
length_append,
subtype_rel_list,
top_wf,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermAdd_wf,
intformle_wf,
itermConstant_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lelt_wf,
member_wf,
list_wf,
append_wf,
permutation_wf,
squash_wf,
exists_wf,
bag-member_wf,
select_wf,
int_seg_properties,
decidable__le,
list-member-bag-member,
bag-member-select,
subtype_base_sq,
int_subtype_base,
length_wf_nat,
permute_list_select,
nat_properties,
iff_weakening_equal,
select_member,
l_member_wf,
true_wf,
select_append_front,
not_over_exists,
all_wf,
le_wf,
inject_wf,
decidable__equal_int,
le_weakening2,
injection_le,
ifthenelse_wf,
lt_int_wf,
subtract_wf,
equipollent_functionality_wrt_equipollent,
equipollent-int_seg,
equipollent_weakening_ext-eq,
ext-eq_weakening,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
itermSubtract_wf,
int_term_value_subtract_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
compose_wf,
injection-composition,
int_seg_subtype,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
equalitySymmetry,
hyp_replacement,
applyLambdaEquality,
cumulativity,
applyEquality,
because_Cache,
sqequalRule,
equalityTransitivity,
rename,
independent_isectElimination,
lambdaEquality,
setElimination,
pertypeElimination,
instantiate,
dependent_functionElimination,
natural_numberEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
independent_functionElimination,
lambdaFormation,
imageMemberEquality,
baseClosed,
productEquality,
universeEquality,
addEquality,
functionEquality,
equalityElimination
Latex:
\mforall{}[A:Type]. \mforall{}[as,as',bs,bs':bag(A)].
(\mdownarrow{}\mexists{}x:A. (x \mdownarrow{}\mmember{} as \mwedge{} x \mdownarrow{}\mmember{} bs)) supposing
(\#(as') < \#(as) and
\#(bs') < \#(bs) and
((as + as') = (bs + bs')))
Date html generated:
2017_10_01-AM-08_59_39
Last ObjectModification:
2017_07_26-PM-04_41_42
Theory : bags
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