### Nuprl Lemma : es-bound-list2

`∀es:EO`
`  ∀[T:Type]`
`    ∀i:Id`
`      ∀[P:T ─→ ℙ]`
`        ∀L:T List`
`          ∀[Q:E ─→ {x:T| (x ∈ L)}  ─→ ℙ]`
`            ((∀x:T. Dec(P[x]))`
`            `` (∀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])` l_member: `(x ∈ l)` 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` set: `{x:A| B[x]} ` function: `x:A ─→ B[x]` universe: `Type` equal: `s = t ∈ T`
Lemmas :  select_wf l_member_wf list-subtype sq_stable__le es-E_wf int_seg_wf length_wf list-set-type es-bound-list subtype_rel_dep_function subtype_rel_self set_wf isect_wf not_wf l_exists_wf existse-at_wf es-loc_wf true_wf l_all_wf2 exists_wf all_wf decidable_wf list_wf Id_wf event_ordering_wf
\mforall{}es:EO
\mforall{}[T:Type]
\mforall{}i:Id
\mforall{}[P:T  {}\mrightarrow{}  \mBbbP{}]
\mforall{}L:T  List
\mforall{}[Q:E  {}\mrightarrow{}  \{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \mBbbP{}]
((\mforall{}x:T.  Dec(P[x]))
{}\mRightarrow{}  (\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_54
Last ObjectModification: 2015_01_27-PM-01_19_35

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