Nuprl Lemma : global-order-compat-invariant
∀[Info:Type]. ∀[P:Id ─→ Info List+ ─→ ℙ]. ∀[R:Id ─→ Id ─→ Info List+ ─→ Info List+ ─→ ℙ].
∀L1,L2:(Id × Info) List.
(L1 || L2
⇒ (causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) global-eo(L1))
⇒ (causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) global-eo(L2))
⇒ (∃L:(Id × Info) List
((causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) global-eo(L))
∧ (∀L':(Id × Info) List. (L1 || L' ⇒ L2 || L' ⇒ L || L'))
∧ (∃f:E ─→ E. es-local-embedding(Info;global-eo(L1);global-eo(L);f))
∧ (∃g:E ─→ E. es-local-embedding(Info;global-eo(L2);global-eo(L);g)))))
Proof
Definitions occuring in Statement :
global-order-compat: L1 || L2,
global-eo: global-eo(L),
causal-invariant: causal-invariant(i,L.P[i; L];a,b,L1,L2.R[a; b; L1; L2]),
es-local-embedding: es-local-embedding(Info;eo1;eo2;f),
es-E: E,
Id: Id,
listp: A List+,
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3;s4],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
product: x:A × B[x],
universe: Type
Lemmas :
causal-invariant_wf,
global-order-compat-joint-embedding,
weak-joint-embedding-preserves-causal-invariant,
listp_wf,
global-eo_wf,
es-local-embedding_wf,
all_wf,
list_wf,
global-order-compat_wf,
exists_wf,
es-E_wf,
event-ordering+_subtype,
Id_wf
Latex:
\mforall{}[Info:Type]. \mforall{}[P:Id {}\mrightarrow{} Info List\msupplus{} {}\mrightarrow{} \mBbbP{}]. \mforall{}[R:Id {}\mrightarrow{} Id {}\mrightarrow{} Info List\msupplus{} {}\mrightarrow{} Info List\msupplus{} {}\mrightarrow{} \mBbbP{}].
\mforall{}L1,L2:(Id \mtimes{} Info) List.
(L1 || L2
{}\mRightarrow{} (causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) global-eo(L1))
{}\mRightarrow{} (causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) global-eo(L2))
{}\mRightarrow{} (\mexists{}L:(Id \mtimes{} Info) List
((causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) global-eo(L))
\mwedge{} (\mforall{}L':(Id \mtimes{} Info) List. (L1 || L' {}\mRightarrow{} L2 || L' {}\mRightarrow{} L || L'))
\mwedge{} (\mexists{}f:E {}\mrightarrow{} E. es-local-embedding(Info;global-eo(L1);global-eo(L);f))
\mwedge{} (\mexists{}g:E {}\mrightarrow{} E. es-local-embedding(Info;global-eo(L2);global-eo(L);g)))))
Date html generated:
2015_07_21-PM-04_40_32
Last ObjectModification:
2015_01_27-PM-05_03_22
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