Nuprl Lemma : bag-member-cons
∀[T:Type]. ∀[x,u:T]. ∀[v:bag(T)]. uiff(x ↓∈ u.v;(x = u ∈ T) ↓∨ x ↓∈ v)
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
cons-bag: x.b,
bag: bag(T),
sq_or: a ↓∨ b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
sq_or: a ↓∨ b,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
squash: ↓T,
prop: ℙ,
uall: ∀[x:A]. B[x],
bag-member: x ↓∈ bs,
exists: ∃x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
cons-bag: x.b,
append: as @ bs,
all: ∀x:A. B[x],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
bag-append: as + bs,
single-bag: {x},
or: P ∨ Q,
sq_stable: SqStable(P),
cand: A c∧ B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
bag-member_wf,
cons-bag_wf,
squash_wf,
or_wf,
equal_wf,
bag_wf,
bag_to_squash_list,
true_wf,
iff_weakening_equal,
list_ind_cons_lemma,
list_ind_nil_lemma,
bag-member-append,
cons_wf,
nil_wf,
list-subtype-bag,
bag-member-single,
sq_stable__bag-member,
cons_member,
l_member_wf,
and_wf,
exists_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
imageElimination,
hypothesis,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
cumulativity,
because_Cache,
universeEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
natural_numberEquality,
hyp_replacement,
applyLambdaEquality,
applyEquality,
lambdaEquality,
independent_isectElimination,
independent_functionElimination,
dependent_functionElimination,
voidElimination,
voidEquality,
unionElimination,
inlFormation,
inrFormation,
dependent_pairFormation,
productEquality,
rename,
addLevel,
existsFunctionality,
dependent_set_memberEquality,
setElimination
Latex:
\mforall{}[T:Type]. \mforall{}[x,u:T]. \mforall{}[v:bag(T)]. uiff(x \mdownarrow{}\mmember{} u.v;(x = u) \mdownarrow{}\mvee{} x \mdownarrow{}\mmember{} v)
Date html generated:
2017_10_01-AM-08_54_02
Last ObjectModification:
2017_07_26-PM-04_35_50
Theory : bags
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