Nuprl Lemma : bag-member-map
∀[T,U:Type]. ∀x:U. ∀f:T ⟶ U. ∀bs:bag(T). uiff(x ↓∈ bag-map(f;bs);↓∃v:T. (v ↓∈ bs ∧ (x = (f v) ∈ U)))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-map: bag-map(f;bs),
bag: bag(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
squash: ↓T,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
bag-member: x ↓∈ bs,
so_lambda: λ2x.t[x],
so_apply: x[s],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
bag-map: bag-map(f;bs),
top: Top,
empty-bag: {},
false: False,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
single-bag: {x},
bag-append: as + bs,
iff: P ⇐⇒ Q,
sq_or: a ↓∨ b,
or: P ∨ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
cons-bag: x.b,
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
sq_stable: SqStable(P)
Lemmas referenced :
bag-member_wf,
bag-map_wf,
squash_wf,
exists_wf,
equal_wf,
bag_wf,
bag_to_squash_list,
list_induction,
list-subtype-bag,
list_wf,
map_nil_lemma,
empty-bag_wf,
bag-member-empty-iff,
list_ind_cons_lemma,
list_ind_nil_lemma,
bag-map-append,
single-bag_wf,
top_wf,
bag-member-append,
map_cons_lemma,
cons_wf,
bag-member-single,
bag-member-cons,
sq_stable__bag-member,
map_wf,
member_map,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
imageElimination,
hypothesis,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
thin,
baseClosed,
extract_by_obid,
isectElimination,
cumulativity,
functionExtensionality,
applyEquality,
lambdaEquality,
productEquality,
functionEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
universeEquality,
hyp_replacement,
applyLambdaEquality,
independent_functionElimination,
independent_isectElimination,
rename,
voidElimination,
voidEquality,
unionElimination,
dependent_pairFormation,
inlFormation,
inrFormation
Latex:
\mforall{}[T,U:Type]. \mforall{}x:U. \mforall{}f:T {}\mrightarrow{} U. \mforall{}bs:bag(T). uiff(x \mdownarrow{}\mmember{} bag-map(f;bs);\mdownarrow{}\mexists{}v:T. (v \mdownarrow{}\mmember{} bs \mwedge{} (x = (f v))))
Date html generated:
2017_10_01-AM-08_54_08
Last ObjectModification:
2017_07_26-PM-04_35_53
Theory : bags
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