Nuprl Lemma : es-first-at-exists-cases
∀es:EO. ∀e:E.
∀[P:{e':E| loc(e') = loc(e) ∈ Id} ⟶ ℙ]
((∀e':{e':E| loc(e') = loc(e) ∈ Id} . Dec(P[e']))
⇒ ((∀e'≤e.¬P[e'] ∧ (¬∃e'≤e.e' is first@ loc(e) s.t. e'.P[e']))
∨ ((¬∀e'≤e.¬P[e']) ∧ ∃e'≤e.e' is first@ loc(e) s.t. e'.P[e'])))
Proof
Definitions occuring in Statement :
es-first-at: e is first@ i s.t. e.P[e],
alle-le: ∀e≤e'.P[e],
existse-le: ∃e≤e'.P[e],
es-loc: loc(e),
es-E: E,
event_ordering: EO,
Id: Id,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
alle-at: ∀e@i.P[e],
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
false: False,
guard: {T},
uimplies: b supposing a,
existse-le: ∃e≤e'.P[e],
exists: ∃x:A. B[x],
es-first-at: e is first@ i s.t. e.P[e],
alle-le: ∀e≤e'.P[e]
Latex:
\mforall{}es:EO. \mforall{}e:E.
\mforall{}[P:\{e':E| loc(e') = loc(e)\} {}\mrightarrow{} \mBbbP{}]
((\mforall{}e':\{e':E| loc(e') = loc(e)\} . Dec(P[e']))
{}\mRightarrow{} ((\mforall{}e'\mleq{}e.\mneg{}P[e'] \mwedge{} (\mneg{}\mexists{}e'\mleq{}e.e' is first@ loc(e) s.t. e'.P[e']))
\mvee{} ((\mneg{}\mforall{}e'\mleq{}e.\mneg{}P[e']) \mwedge{} \mexists{}e'\mleq{}e.e' is first@ loc(e) s.t. e'.P[e'])))
Date html generated:
2016_05_16-AM-09_49_12
Last ObjectModification:
2015_12_28-PM-09_36_04
Theory : new!event-ordering
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