Nuprl Lemma : bag-extensionality2
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[as,bs:bag(T)].
uiff(as = bs ∈ bag(T);(∀x:T. (x ↓∈ as ⇒ ((#x in as) = (#x in bs) ∈ ℤ))) ∧ (∀x:T. (x ↓∈ bs ⇒ x ↓∈ as)))
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: 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,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
nat: ℕ,
ge: i ≥ j ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
bag-extensionality,
equal_wf,
squash_wf,
true_wf,
bag-count_wf,
iff_weakening_equal,
bag-member_wf,
and_wf,
bag_wf,
decidable__lt,
bag-member-count,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
decidable__equal_int,
nat_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
all_wf,
nat_wf,
deq_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
productElimination,
independent_isectElimination,
lambdaFormation,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
intEquality,
dependent_functionElimination,
cumulativity,
because_Cache,
sqequalRule,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
addLevel,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
rename,
levelHypothesis,
independent_pairEquality,
axiomEquality,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productEquality,
functionEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs:bag(T)].
uiff(as = bs;(\mforall{}x:T. (x \mdownarrow{}\mmember{} as {}\mRightarrow{} ((\#x in as) = (\#x in bs)))) \mwedge{} (\mforall{}x:T. (x \mdownarrow{}\mmember{} bs {}\mRightarrow{} x \mdownarrow{}\mmember{} as)))
Date html generated:
2018_05_21-PM-09_47_05
Last ObjectModification:
2017_07_26-PM-06_30_07
Theory : bags_2
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