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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