Nuprl Lemma : bag-member-count
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)]. uiff(x ↓∈ bs;1 ≤ (#x in bs))
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
deq: EqDecider(T),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
bag-member: x ↓∈ bs,
squash: ↓T,
nat: ℕ,
sq_stable: SqStable(P),
exists: ∃x:A. B[x],
bag-filter: [x∈b|p[x]],
bag-size: #(bs),
all: ∀x:A. B[x],
or: P ∨ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
l_member: (x ∈ l),
int_seg: {i..j-},
lelt: i ≤ j < k,
eqof: eqof(d),
ge: i ≥ j ,
cand: A c∧ B,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
rev_uimplies: rev_uimplies(P;Q),
cons: [a / b],
bfalse: ff,
guard: {T},
less_than: a < b,
bag: bag(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
quotient: x,y:A//B[x; y],
less_than': less_than'(a;b),
sq_type: SQType(T)
Lemmas referenced :
bag-count-sqequal,
less_than'_wf,
bag-size_wf,
assert_wf,
bag-filter_wf,
bag-member_wf,
le_wf,
bag-count_wf,
nat_wf,
bag_wf,
deq_wf,
sq_stable__le,
filter_is_empty,
filter_wf5,
l_member_wf,
list_wf,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
lelt_wf,
length_wf,
safe-assert-deq,
select_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
non_neg_length,
itermAdd_wf,
int_term_value_add_lemma,
uiff_wf,
null_wf3,
subtype_rel_list,
top_wf,
uall_wf,
int_seg_wf,
not_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
equal_wf,
permutation_wf,
permutation_weakening,
quotient-member-eq,
permutation-equiv,
list-subtype-bag,
equal-wf-base,
member_wf,
squash_wf,
true_wf,
decidable__exists_int_seg,
decidable__assert,
int_seg_subtype_nat,
false_wf,
less_than_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalTransitivity,
computationStep,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
independent_pairFormation,
isect_memberFormation,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
voidElimination,
setEquality,
cumulativity,
applyEquality,
setElimination,
rename,
natural_numberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
isect_memberEquality,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
lambdaFormation,
unionElimination,
independent_isectElimination,
dependent_set_memberEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll,
promote_hyp,
hypothesis_subsumption,
functionEquality,
instantiate,
pointwiseFunctionality,
pertypeElimination,
productEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[bs:bag(T)]. uiff(x \mdownarrow{}\mmember{} bs;1 \mleq{} (\#x in bs))
Date html generated:
2018_05_21-PM-09_46_00
Last ObjectModification:
2017_07_26-PM-06_29_54
Theory : bags_2
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