Nuprl Lemma : bag-extensionality1
∀[T:Type]. ∀[eq:T ⟶ T ⟶ 𝔹].
∀[as,bs:bag(T)]. uiff(as = bs ∈ bag(T);∀x:T. (#([y∈as|eq x y]) = #([y∈bs|eq x y]) ∈ ℤ))
supposing ∀[x,y:T]. (↑(eq x y) ⇐⇒ x = y ∈ T)
Proof
Definitions occuring in Statement :
bag-size: #(bs),
bag-filter: [x∈b|p[x]],
bag: bag(T),
assert: ↑b,
bool: 𝔹,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
all: ∀x:A. B[x],
squash: ↓T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
true: True,
subtype_rel: A ⊆r B,
bag: bag(T),
quotient: x,y:A//B[x; y],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
bag-filter: [x∈b|p[x]],
bag-size: #(bs)
Lemmas referenced :
bag-size_wf,
set_wf,
assert_wf,
bag-filter_wf,
equal_wf,
bag_wf,
all_wf,
uall_wf,
iff_wf,
bool_wf,
list_wf,
permutation_wf,
permutation_weakening,
quotient-member-eq,
permutation-equiv,
equal-wf-base,
list-subtype-bag,
squash_wf,
true_wf,
permutation-iff-count1,
filter_functionality,
eta_conv,
length_wf,
filter_wf5,
subtype_rel_dep_function,
l_member_wf,
subtype_rel_self,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
because_Cache,
hypothesis,
sqequalRule,
functionExtensionality,
hypothesisEquality,
cumulativity,
imageMemberEquality,
baseClosed,
natural_numberEquality,
dependent_functionElimination,
axiomEquality,
intEquality,
setEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
promote_hyp,
independent_isectElimination,
pointwiseFunctionality,
pertypeElimination,
independent_functionElimination,
productEquality,
universeEquality,
hyp_replacement,
setElimination,
rename
Latex:
\mforall{}[T:Type]. \mforall{}[eq:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}].
\mforall{}[as,bs:bag(T)]. uiff(as = bs;\mforall{}x:T. (\#([y\mmember{}as|eq x y]) = \#([y\mmember{}bs|eq x y])))
supposing \mforall{}[x,y:T]. (\muparrow{}(eq x y) \mLeftarrow{}{}\mRightarrow{} x = y)
Date html generated:
2017_10_01-AM-08_59_27
Last ObjectModification:
2017_07_26-PM-04_41_37
Theory : bags
Home
Index