Nuprl Lemma : bag-size-partition
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. (#(bs) = Σ(x∈bag-remove-repeats(eq;bs)). (#x in bs) ∈ ℤ)
Proof
Definitions occuring in Statement :
bag-remove-repeats: bag-remove-repeats(eq;bs),
bag-count: (#x in bs),
bag-summation: Σ(x∈b). f[x],
bag-size: #(bs),
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
deq: EqDecider(T),
so_apply: x[s1;s2],
and: P ∧ Q,
cand: A c∧ B,
monoid_p: IsMonoid(T;op;id),
assoc: Assoc(T;op),
infix_ap: x f y,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
ident: Ident(T;op;id),
comm: Comm(T;op),
sq_exists: ∃x:A [B[x]],
uiff: uiff(P;Q),
eqof: eqof(d),
rev_uimplies: rev_uimplies(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
nat: ℕ
Lemmas referenced :
bag_wf,
deq_wf,
decidable-equal-deq,
bag-remove-repeats_wf,
decidable__equal_int,
full-omega-unsat,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
itermConstant_wf,
int_term_value_constant_lemma,
member-bag-remove-repeats,
safe-assert-deq,
bag-member_wf,
assert_wf,
set_wf,
bag-remove-repeats-no-repeats,
bag-summation-partition,
iff_weakening_equal,
bag-size-as-summation,
true_wf,
squash_wf,
equal_wf,
satisfiable-full-omega-tt,
comm_wf,
assoc_wf,
bag-summation_wf,
bag-filter_wf,
nat_wf,
bag-count_wf,
bag-count-sqequal,
iff_wf,
eqof_wf,
iff_imp_equal_bool,
bag-size_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
universeEquality,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
intEquality,
lambdaEquality,
addEquality,
natural_numberEquality,
setElimination,
rename,
independent_pairFormation,
unionElimination,
independent_isectElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
productElimination,
independent_pairEquality,
dependent_set_memberFormation,
productEquality,
applyEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
imageMemberEquality,
imageElimination,
hyp_replacement,
computeAll,
functionEquality,
functionExtensionality,
cumulativity,
setEquality,
impliesLevelFunctionality,
andLevelFunctionality,
levelHypothesis,
impliesFunctionality,
addLevel
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)]. (\#(bs) = \mSigma{}(x\mmember{}bag-remove-repeats(eq;bs)). (\#x in bs))
Date html generated:
2019_10_16-AM-11_31_40
Last ObjectModification:
2018_08_22-AM-09_39_28
Theory : bags_2
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