Nuprl Lemma : filter-interface-predecessors-lower-bound
∀[Info:Type]. ∀[es:EO+(Info)]. ∀[T:Type]. ∀[X:EClass(T)]. ∀[P:E(X) ─→ 𝔹]. ∀[n:ℕ]. ∀[f:ℕn ─→ {e:E(X)| ↑P[e]} ].
∀[k:ℕn]. n ≤ ||filter(λe.P[e];≤(X)(f k))|| supposing ∀i:ℕn. f i ≤loc f k supposing Inj(ℕn;{e:E(X)| ↑P[e]} ;f)
Proof
Definitions occuring in Statement :
es-interface-predecessors: ≤(X)(e),
es-E-interface: E(X),
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
filter: filter(P;l),
length: ||as||,
inject: Inj(A;B;f),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x],
natural_number: $n,
universe: Type
Lemmas :
less_than_wf,
all_wf,
int_seg_wf,
es-le_wf,
event-ordering+_subtype,
es-E-interface_wf,
es-interface-subtype_rel2,
es-E_wf,
event-ordering+_wf,
top_wf,
assert_wf,
inject_wf,
nat_wf,
bool_wf,
eclass_wf,
member_filter,
member-interface-predecessors,
set_wf,
sq_stable__assert,
equal_wf,
filter_type,
es-interface-predecessors_wf,
subtype_rel_list,
Id_wf,
es-loc_wf,
list_wf,
l_member_wf,
decidable__lt,
length_wf,
false_wf,
less-iff-le,
add_functionality_wrt_le,
add-swap,
add-commutes,
le-add-cancel,
lelt_wf,
select_wf,
sq_stable__le,
exists_wf,
zero-le-nat,
int_seg_subtype-nat,
non_neg_length,
length_wf_nat,
pigeon-hole,
es-E-interface-property,
iff_weakening_equal,
in-eclass_wf,
squash_wf,
le_wf
Latex:
\mforall{}[Info:Type]. \mforall{}[es:EO+(Info)]. \mforall{}[T:Type]. \mforall{}[X:EClass(T)]. \mforall{}[P:E(X) {}\mrightarrow{} \mBbbB{}]. \mforall{}[n:\mBbbN{}].
\mforall{}[f:\mBbbN{}n {}\mrightarrow{} \{e:E(X)| \muparrow{}P[e]\} ].
\mforall{}[k:\mBbbN{}n]. n \mleq{} ||filter(\mlambda{}e.P[e];\mleq{}(X)(f k))|| supposing \mforall{}i:\mBbbN{}n. f i \mleq{}loc f k
supposing Inj(\mBbbN{}n;\{e:E(X)| \muparrow{}P[e]\} ;f)
Date html generated:
2015_07_21-PM-03_40_09
Last ObjectModification:
2015_02_04-PM-06_14_39
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