Nuprl Lemma : map-concat-filter-lemma1
∀[A,B:Type]. ∀[L2:(tg:Id × (A ─→ B ─→ (Top List))) List]. ∀[L:(Top × Id × Top) List]. ∀[tg:Id]. ∀[a:A]. ∀[b:B].
{(filter(λms.fst(snd(ms)) = tg;L) = [] ∈ ((Top × Id × Top) List)) supposing
((¬(tg ∈ map(λp.(fst(p));L2))) and
(map(λx.(snd(x));L) = concat(map(λtgf.map(λx.<fst(tgf), x>;(snd(tgf)) a b);L2)) ∈ ((tg:Id × Top) List)))}
Proof
Definitions occuring in Statement :
eq_id: a = b,
Id: Id,
l_member: (x ∈ l),
concat: concat(ll),
filter: filter(P;l),
map: map(f;as),
nil: [],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
guard: {T},
pi1: fst(t),
pi2: snd(t),
not: ¬A,
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Lemmas :
nil-iff-no-member,
filter_wf5,
l_member_wf,
eq_id_wf,
not_wf,
Id_wf,
map_wf,
list_wf,
top_wf,
equal_wf,
concat_wf,
member_filter,
assert-eq-id,
member_map,
member-concat,
and_wf,
pi1_wf_top
\mforall{}[A,B:Type]. \mforall{}[L2:(tg:Id \mtimes{} (A {}\mrightarrow{} B {}\mrightarrow{} (Top List))) List]. \mforall{}[L:(Top \mtimes{} Id \mtimes{} Top) List]. \mforall{}[tg:Id].
\mforall{}[a:A]. \mforall{}[b:B].
\{(filter(\mlambda{}ms.fst(snd(ms)) = tg;L) = []) supposing
((\mneg{}(tg \mmember{} map(\mlambda{}p.(fst(p));L2))) and
(map(\mlambda{}x.(snd(x));L) = concat(map(\mlambda{}tgf.map(\mlambda{}x.<fst(tgf), x>(snd(tgf)) a b);L2))))\}
Date html generated:
2015_07_17-AM-09_13_45
Last ObjectModification:
2015_01_28-AM-07_56_33
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