Nuprl Lemma : bag-extensionality-no-repeats
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ (∀[as,bs:bag(T)].
uiff(as = bs ∈ bag(T);∀x:T. uiff(x ↓∈ as;x ↓∈ bs)) supposing bag-no-repeats(T;as) ∧ bag-no-repeats(T;bs)))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-no-repeats: bag-no-repeats(T;bs),
bag: bag(T),
decidable: Dec(P),
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bag-member: x ↓∈ bs,
exists: ∃x:A. B[x],
bag-no-repeats: bag-no-repeats(T;bs),
bag: bag(T),
quotient: x,y:A//B[x; y],
cand: A c∧ B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
respects-equality: respects-equality(S;T)
Lemmas referenced :
bag-member_wf,
squash_wf,
true_wf,
bag_wf,
istype-universe,
subtype_rel_self,
iff_weakening_equal,
bag-no-repeats_wf,
decidable_wf,
equal_wf,
bag_to_squash_list,
uiff_wf,
list-subtype-bag,
quotient-member-eq,
list_wf,
permutation_wf,
permutation-equiv,
permutation-when-no_repeats,
l_member_wf,
no_repeats_functionality_wrt_permutation,
permutation_inversion,
respects-equality-list-bag,
respects-equality-trivial,
bag-member-list-member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
applyEquality,
lambdaEquality_alt,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
because_Cache,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
equalityIstype,
functionIsType,
productIsType,
isectIsType,
axiomEquality,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
functionEquality,
rename,
pertypeElimination,
dependent_pairFormation_alt
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}[as,bs:bag(T)].
uiff(as = bs;\mforall{}x:T. uiff(x \mdownarrow{}\mmember{} as;x \mdownarrow{}\mmember{} bs))
supposing bag-no-repeats(T;as) \mwedge{} bag-no-repeats(T;bs)))
Date html generated:
2020_05_20-AM-08_02_16
Last ObjectModification:
2020_01_04-PM-10_19_37
Theory : bags
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