Nuprl Lemma : bag-size-filter-member-bound
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs:bag(T)].
#([x∈as|bag-deq-member(eq;x;bs)]) ≤ #(bs) supposing bag-no-repeats(T;as)
Proof
Definitions occuring in Statement :
bag-deq-member: bag-deq-member(eq;x;b),
bag-no-repeats: bag-no-repeats(T;bs),
bag-size: #(bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
squash: ↓T,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
sq_stable: SqStable(P),
implies: P ⇒ Q,
exists: ∃x:A. B[x],
bag-size: #(bs),
bag-deq-member: bag-deq-member(eq;x;b),
bag-filter: [x∈b|p[x]],
uiff: uiff(P;Q),
and: P ∧ Q,
all: ∀x:A. B[x],
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
or: P ∨ Q,
cons: [a / b],
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
guard: {T},
sq_type: SQType(T),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
bool: 𝔹,
unit: Unit,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda3,
so_apply: x[s1;s2;s3]
Lemmas referenced :
bag_to_squash_list,
sq_stable__le,
bag-size_wf,
assert_wf,
bag-deq-member_wf,
bag-filter_wf,
list-subtype-bag,
bag-no-repeats_wf,
bag-no-repeats-list,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
le_witness_for_triv,
list-cases,
list_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-void,
istype-le,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
le_wf,
istype-nat,
bag_wf,
deq_wf,
istype-universe,
no_repeats_wf,
nil_wf,
true_wf,
length_wf,
non_neg_length,
istype-true,
cons_wf,
not_wf,
l_member_wf,
filter_wf5,
deq-member_wf,
eqtt_to_assert,
length_of_cons_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
sqequal-tt-to-assert,
assert-deq-member,
l_member_decomp,
length-append,
length-singleton,
sqequal-ff-to-assert,
implies_functionality_wrt_iff,
iff_weakening_uiff,
no_repeats_nil_uiff,
filter-nil,
length-nil,
iff_weakening_equal,
no_repeats_cons,
filter-cons,
filter-sq,
list_ind_cons_lemma,
list_ind_nil_lemma,
append_wf,
istype-assert,
member_append,
cons_member,
add-is-int-iff,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
setEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
universeIsType,
applyEquality,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
productElimination,
promote_hyp,
because_Cache,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality,
lambdaFormation_alt,
intWeakElimination,
natural_numberEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
independent_pairFormation,
voidElimination,
functionIsTypeImplies,
unionElimination,
hypothesis_subsumption,
equalityIstype,
dependent_set_memberEquality_alt,
instantiate,
baseApply,
closedConclusion,
baseClosed,
intEquality,
sqequalBase,
imageMemberEquality,
universeEquality,
productEquality,
setIsType,
equalityElimination,
addEquality,
cumulativity,
productIsType,
functionIsType,
inlFormation_alt,
unionIsType,
inrFormation_alt,
pointwiseFunctionality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs:bag(T)].
\#([x\mmember{}as|bag-deq-member(eq;x;bs)]) \mleq{} \#(bs) supposing bag-no-repeats(T;as)
Date html generated:
2020_05_20-AM-09_04_23
Last ObjectModification:
2020_01_27-PM-05_19_58
Theory : bags_2
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