Nuprl Lemma : bag-count-bag-lub
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs:bag(T)]. ∀[x:T]. ((#x in bag-lub(eq;as;bs)) = imax((#x in as);(#x in bs)) ∈ ℤ)
Proof
Definitions occuring in Statement :
bag-lub: bag-lub(eq;b1;b2),
bag-count: (#x in bs),
bag: bag(T),
deq: EqDecider(T),
imax: imax(a;b),
uall: ∀[x:A]. B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bag-lub: bag-lub(eq;b1;b2),
nat: ℕ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
deq: EqDecider(T),
cand: A c∧ B,
uiff: uiff(P;Q),
bag-member: x ↓∈ bs,
squash: ↓T,
eqof: eqof(d),
rev_uimplies: rev_uimplies(P;Q),
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
assert: ↑b,
bool: 𝔹,
unit: Unit,
it: ⋅,
sq_or: a ↓∨ b,
sq_stable: SqStable(P),
single-bag: {x},
bag-sum: bag-sum(ba;x.f[x]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
empty-bag: {}
Lemmas referenced :
bag_wf,
deq_wf,
bag-rep_wf,
imax_wf,
bag-count_wf,
imax_nat,
nat_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_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_eq_lemma,
int_formula_prop_wf,
equal_wf,
le_wf,
list-subtype-bag,
bag-to-set_wf,
bag-append_wf,
decidable__bag-member,
decidable-equal-deq,
bag-count-combine,
bag-sum-count,
bag-sum_wf,
bag-extensionality-no-repeats,
bag-filter_wf,
subtype_rel_bag,
assert_wf,
single-bag_wf,
bag-single-no-repeats,
bag-member-single,
bag-member_wf,
bag-no-repeats-filter,
no-repeats-bag-to-set,
bag-member-filter,
safe-assert-deq,
squash_wf,
true_wf,
bag-filter-equal,
le_int_wf,
assert_of_le_int,
assert_functionality_wrt_uiff,
ifthenelse_wf,
bag-count-rep,
bnot_wf,
eqof_wf,
not_wf,
bool_cases,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
iff_weakening_equal,
bool_cases_sqequal,
member-bag-to-set,
bag-member-append,
sq_stable__le,
imax_ub,
bag-member-count,
or_wf,
list_accum_cons_lemma,
list_accum_nil_lemma,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
assert-bnot,
add_functionality_wrt_eq,
bag-filter-is-empty,
false_wf,
and_wf,
imax-idempotent,
bag-count-is-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
hypothesisEquality,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
axiomEquality,
because_Cache,
extract_by_obid,
cumulativity,
universeEquality,
lambdaEquality,
dependent_set_memberEquality,
applyEquality,
setElimination,
rename,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
hyp_replacement,
setEquality,
productElimination,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
impliesFunctionality,
promote_hyp,
equalityElimination,
addLevel,
orFunctionality,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs:bag(T)]. \mforall{}[x:T].
((\#x in bag-lub(eq;as;bs)) = imax((\#x in as);(\#x in bs)))
Date html generated:
2018_05_21-PM-09_52_28
Last ObjectModification:
2017_07_26-PM-06_31_50
Theory : bags_2
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