Nuprl Lemma : count-bag-remove-repeats
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. ∀[x:T].
((#x in bag-remove-repeats(eq;bs)) ~ if 0 <z (#x in bs) then 1 else 0 fi )
Proof
Definitions occuring in Statement :
bag-remove-repeats: bag-remove-repeats(eq;bs)
,
bag-count: (#x in bs)
,
bag: bag(T)
,
deq: EqDecider(T)
,
ifthenelse: if b then t else f fi
,
lt_int: i <z j
,
uall: ∀[x:A]. B[x]
,
natural_number: $n
,
universe: Type
,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
nat: ℕ
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
bag: bag(T)
,
quotient: x,y:A//B[x; y]
,
and: P ∧ Q
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
squash: ↓T
,
so_lambda: λ2x y.t[x; y]
,
so_apply: x[s1;s2]
,
true: True
,
subtype_rel: A ⊆r B
,
sq_type: SQType(T)
,
guard: {T}
,
prop: ℙ
,
bag-filter: [x∈b|p[x]]
,
bag-size: #(bs)
,
bag-remove-repeats: bag-remove-repeats(eq;bs)
,
iff: P
⇐⇒ Q
,
deq: EqDecider(T)
,
istype: istype(T)
,
rev_implies: P
⇐ Q
,
ge: i ≥ j
,
decidable: Dec(P)
,
or: P ∨ Q
,
false: False
,
le: A ≤ B
,
not: ¬A
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
exists: ∃x:A. B[x]
,
top: Top
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
ifthenelse: if b then t else f fi
,
bfalse: ff
,
bnot: ¬bb
,
assert: ↑b
,
less_than': less_than'(a;b)
,
cons: [a / b]
,
sq_stable: SqStable(P)
,
subtract: n - m
Lemmas referenced :
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
bag-count_wf,
istype-universe,
bag-remove-repeats_wf,
quotient-member-eq,
list_wf,
permutation_wf,
permutation-equiv,
bag_wf,
deq_wf,
list-subtype-bag,
deq-member-length-filter2,
l_member-iff-length-filter,
list-to-set_wf,
member-list-to-set,
length_wf,
filter_wf5,
subtype_rel_dep_function,
bool_wf,
l_member_wf,
bag-count-sqequal,
non_neg_length,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
filter_functionality,
eta_conv,
deq-member_wf,
eqtt_to_assert,
assert-deq-member,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
list-to-set-property,
no-repeats-iff-count,
decidable__equal_nat,
length_wf_nat,
istype-false,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
instantiate,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesis,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality_alt,
closedConclusion,
natural_numberEquality,
hypothesisEquality,
pointwiseFunctionalityForEquality,
pertypeElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
lambdaFormation_alt,
rename,
applyEquality,
imageElimination,
because_Cache,
universeIsType,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
imageMemberEquality,
baseClosed,
equalityIsType1,
productIsType,
equalityIsType4,
axiomSqEquality,
isect_memberEquality_alt,
setElimination,
setEquality,
setIsType,
independent_pairFormation,
promote_hyp,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
dependent_set_memberEquality_alt,
equalityElimination,
hypothesis_subsumption,
addEquality,
minusEquality,
functionIsType
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)]. \mforall{}[x:T].
((\#x in bag-remove-repeats(eq;bs)) \msim{} if 0 <z (\#x in bs) then 1 else 0 fi )
Date html generated:
2019_10_16-AM-11_30_39
Last ObjectModification:
2018_10_11-PM-11_28_59
Theory : bags_2
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