Nuprl Lemma : equal-count-bag-to-set
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[bs:bag(T)]. ∀[x,y:T].
uiff((#x in bag-to-set(eq;bs)) = (#y in bag-to-set(eq;bs)) ∈ ℤ;x ↓∈ bs ⇐⇒ y ↓∈ bs)
Proof
Definitions occuring in Statement :
bag-to-set: bag-to-set(eq;bs),
bag-count: (#x in bs),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs,
squash: ↓T,
subtype_rel: A ⊆r B,
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q),
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
less_than: a < b,
le: A ≤ B
Lemmas referenced :
bag-member-count,
member-bag-to-set,
bag-member_wf,
equal_wf,
bag-count_wf,
bag-to-set_wf,
nat_wf,
iff_wf,
bag_wf,
deq_wf,
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,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
count-bag-to-set,
intformless_wf,
int_formula_prop_less_lemma
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
lambdaFormation,
cumulativity,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
intEquality,
applyEquality,
setElimination,
rename,
because_Cache,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
universeEquality,
independent_isectElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
computeAll,
equalityElimination,
promote_hyp,
instantiate,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[bs:bag(T)]. \mforall{}[x,y:T].
uiff((\#x in bag-to-set(eq;bs)) = (\#y in bag-to-set(eq;bs));x \mdownarrow{}\mmember{} bs \mLeftarrow{}{}\mRightarrow{} y \mdownarrow{}\mmember{} bs)
Date html generated:
2018_05_21-PM-09_47_39
Last ObjectModification:
2017_07_26-PM-06_30_21
Theory : bags_2
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