### Nuprl Lemma : es-bound-list

`∀es:EO`
`  ∀[T:Type]`
`    ∀i:Id`
`      ∀[P:T ─→ ℙ]. ∀[Q:E ─→ T ─→ ℙ].`
`        ((∀x:T. Dec(P[x]))`
`        `` (∀L:T List`
`              (∀x∈L.P[x] `` (∃e:E. Q[e;x]))`
`              `` ∃e'@i.True supposing ¬(∃x∈L. P[x])`
`              `` ∃e'@i.(∀x∈L.P[x] `` (∃e:E. (e ≤loc e'  ∧ Q[e;x]))) `
`              supposing (∀x∈L.∀e:E. (Q[e;x] `` (loc(e) = i ∈ Id)))))`

Proof

Definitions occuring in Statement :  existse-at: `∃e@i.P[e]` es-le: `e ≤loc e' ` es-loc: `loc(e)` es-E: `E` event_ordering: `EO` Id: `Id` l_exists: `(∃x∈L. P[x])` l_all: `(∀x∈L.P[x])` list: `T List` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2]` so_apply: `x[s]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` and: `P ∧ Q` true: `True` function: `x:A ─→ B[x]` universe: `Type` equal: `s = t ∈ T`
Lemmas :  list_induction isect_wf l_all_wf2 l_member_wf all_wf es-E_wf es-loc_wf exists_wf not_wf l_exists_wf existse-at_wf true_wf es-le_wf select_wf nil_wf sq_stable__le int_seg_wf length_wf false_wf l_exists_nil l_all_nil l_exists_wf_nil cons_wf list_wf decidable_wf Id_wf event_ordering_wf l_all_cons l_all_iff cons_member es-le-total and_wf equal_wf es-le-self es-le_transitivity or_wf l_exists_cons
\mforall{}es:EO
\mforall{}[T:Type]
\mforall{}i:Id
\mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[Q:E  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
((\mforall{}x:T.  Dec(P[x]))
{}\mRightarrow{}  (\mforall{}L:T  List
(\mforall{}x\mmember{}L.P[x]  {}\mRightarrow{}  (\mexists{}e:E.  Q[e;x]))
{}\mRightarrow{}  \mexists{}e'@i.True  supposing  \mneg{}(\mexists{}x\mmember{}L.  P[x])
{}\mRightarrow{}  \mexists{}e'@i.(\mforall{}x\mmember{}L.P[x]  {}\mRightarrow{}  (\mexists{}e:E.  (e  \mleq{}loc  e'    \mwedge{}  Q[e;x])))
supposing  (\mforall{}x\mmember{}L.\mforall{}e:E.  (Q[e;x]  {}\mRightarrow{}  (loc(e)  =  i)))))

Date html generated: 2015_07_17-AM-08_50_51
Last ObjectModification: 2015_01_27-PM-01_20_46

Home Index