Nuprl Lemma : map-concat-filter-lemma2
∀[C:Id ─→ Type]. ∀[A,B:Type]. ∀[L2:(tg:Id × (A ─→ B ─→ (C[tg] List))) List]. ∀[L:(l:IdLnk × t:Id × C[t]) List].
∀[tg:Id]. ∀[a:A]. ∀[b:B].
{(||filter(λms.fst(snd(ms)) = tg;L)|| = 0 ∈ ℤ) 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 × C[tg]) List)))}
Proof
Definitions occuring in Statement :
IdLnk: IdLnk,
eq_id: a = b,
Id: Id,
l_member: (x ∈ l),
concat: concat(ll),
filter: filter(P;l),
map: map(f;as),
length: ||as||,
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
guard: {T},
so_apply: x[s],
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],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Lemmas :
map-concat-filter-lemma1,
subtype_rel_list,
list_wf,
top_wf,
subtype_rel_product,
subtype_rel_dep_function,
subtype_top,
not_wf,
l_member_wf,
Id_wf,
map_wf,
equal_wf,
IdLnk_wf,
concat_wf,
length_zero
\mforall{}[C:Id {}\mrightarrow{} Type]. \mforall{}[A,B:Type]. \mforall{}[L2:(tg:Id \mtimes{} (A {}\mrightarrow{} B {}\mrightarrow{} (C[tg] List))) List]. \mforall{}[L:(l:IdLnk
\mtimes{} t:Id
\mtimes{} C[t]) List].
\mforall{}[tg:Id]. \mforall{}[a:A]. \mforall{}[b:B].
\{(||filter(\mlambda{}ms.fst(snd(ms)) = tg;L)|| = 0) 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_47
Last ObjectModification:
2015_01_28-AM-07_56_12
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