Nuprl Lemma : collect_filter-wf2
∀[A:Type]. ∀[P:{L:A List| 0 < ||L||} ─→ 𝔹].
(collect_filter() ∈ {s:ℤ × {L:A List| 0 < ||L|| ⇒ (¬↑P[L])} × ({L:A List| 0 < ||L|| ∧ (↑P[L])} + Top)|
(↑isl(snd(snd(s)))) ⇒ (1 ≤ (fst(s)))} ─→ bag(ℕ × {L:A List| 0 < ||L|| ∧ (↑P[L])} ))
Proof
Definitions occuring in Statement :
collect_filter: collect_filter(),
length: ||as||,
list: T List,
nat: ℕ,
assert: ↑b,
isl: isl(x),
bool: 𝔹,
less_than: a < b,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ─→ B[x],
product: x:A × B[x],
union: left + right,
natural_number: $n,
int: ℤ,
universe: Type,
bag: bag(T)
Lemmas :
single-bag_wf,
nat_wf,
less_than_wf,
length_wf,
assert_wf,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
le_wf,
empty-bag_wf,
not_wf,
top_wf,
isl_wf,
list_wf,
bool_wf
\mforall{}[A:Type]. \mforall{}[P:\{L:A List| 0 < ||L||\} {}\mrightarrow{} \mBbbB{}].
(collect\_filter() \mmember{} \{s:\mBbbZ{}
\mtimes{} \{L:A List| 0 < ||L|| {}\mRightarrow{} (\mneg{}\muparrow{}P[L])\}
\mtimes{} (\{L:A List| 0 < ||L|| \mwedge{} (\muparrow{}P[L])\} + Top)|
(\muparrow{}isl(snd(snd(s)))) {}\mRightarrow{} (1 \mleq{} (fst(s)))\} {}\mrightarrow{} bag(\mBbbN{} \mtimes{} \{L:A List| 0 < ||L|| \mwedge{} (\muparrow{}P\000C[L])\} ))
Date html generated:
2015_07_17-AM-08_59_19
Last ObjectModification:
2015_01_27-PM-01_02_44
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