Nuprl Lemma : sub-bag-iff
∀[T:Type]. ∀eq:EqDecider(T). ∀as,bs:bag(T). (sub-bag(T;as;bs) ⇐⇒ ∀x:T. ((#x in as) ≤ (#x in bs)))
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
sub-bag: sub-bag(T;as;bs),
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
le: A ≤ B,
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
sub-bag: sub-bag(T;as;bs),
exists: ∃x:A. B[x],
squash: ↓T,
true: True,
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
false: False,
nat: ℕ,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bag: bag(T),
quotient: x,y:A//B[x; y],
bag-filter: [x∈b|p[x]],
bag-size: #(bs),
bag-map: bag-map(f;bs),
bag-union: bag-union(bbs),
concat: concat(ll),
cons: [a / b],
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
deq: EqDecider(T),
istype: istype(T),
eqof: eqof(d),
cons-bag: x.b,
bnot: ¬bb,
assert: ↑b,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
l_all: (∀x∈L.P[x]),
rev_uimplies: rev_uimplies(P;Q),
sq_or: a ↓∨ b,
sq_stable: SqStable(P),
bag-member: x ↓∈ bs
Lemmas referenced :
le_witness_for_triv,
istype-universe,
sub-bag_wf,
le_wf,
bag-count_wf,
bag_wf,
deq_wf,
equal_wf,
squash_wf,
true_wf,
istype-int,
bag-count-append,
subtype_rel_self,
iff_weakening_equal,
decidable__le,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformnot_wf,
itermAdd_wf,
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_not_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
subtract_wf,
subtract_nat_wf,
subtract-is-int-iff,
itermSubtract_wf,
intformeq_wf,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
false_wf,
repn_wf,
list-subtype-bag,
nat_wf,
bag-union_wf,
bag-map_wf,
bag-to-set_wf,
set_subtype_base,
int_subtype_base,
member-bag-to-set,
count-bag-to-set,
ifthenelse_wf,
lt_int_wf,
bag-member_wf,
list_wf,
permutation_wf,
permutation_weakening,
bag-count-sqequal,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
list-cases,
filter_nil_lemma,
map_nil_lemma,
length_of_nil_lemma,
reduce_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
subtract-1-ge-0,
subtype_base_sq,
spread_cons_lemma,
decidable__equal_int,
filter_cons_lemma,
map_cons_lemma,
length_wf,
filter_wf5,
l_member_wf,
nil_wf,
assert_wf,
bnot_wf,
eqof_wf,
not_wf,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
non_neg_length,
length_wf_nat,
safe-assert-deq,
length_of_cons_lemma,
bool_cases_sqequal,
assert-bnot,
concat-cons,
filter_append_sq,
cons-bag_wf,
append_wf,
subtype_rel_list,
concat_wf,
map_wf,
add-is-int-iff,
filter_is_nil,
list_ind_nil_lemma,
int_seg_wf,
length-repn,
select-repn,
append-nil,
top_wf,
filter_trivial,
l_all_iff,
strong-subtype-deq-subtype,
strong-subtype-set2,
bag-member-cons,
sq_stable__bag-member,
set_wf,
permutation-length,
permutation_functionality_wrt_permutation,
filter_functionality_wrt_permutation,
bag-append_wf,
bag-extensionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
independent_pairFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
hypothesisEquality,
universeIsType,
sqequalRule,
functionIsType,
applyEquality,
because_Cache,
inhabitedIsType,
universeEquality,
hyp_replacement,
applyLambdaEquality,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_functionElimination,
dependent_functionElimination,
unionElimination,
setElimination,
rename,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
dependent_set_memberEquality_alt,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
equalityIsType1,
setEquality,
setIsType,
equalityIsType4,
intEquality,
equalityElimination,
productIsType,
isectIsType,
pertypeElimination,
intWeakElimination,
axiomEquality,
functionIsTypeImplies,
hypothesis_subsumption,
voidEquality,
cumulativity,
addEquality,
axiomSqEquality,
inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}eq:EqDecider(T). \mforall{}as,bs:bag(T). (sub-bag(T;as;bs) \mLeftarrow{}{}\mRightarrow{} \mforall{}x:T. ((\#x in as) \mleq{} (\#x in bs)))
Date html generated:
2019_10_16-AM-11_32_49
Last ObjectModification:
2018_10_11-PM-10_24_35
Theory : bags_2
Home
Index