Nuprl Lemma : joint-embedding-preserves-causal-invariant
∀[Info:Type]. ∀[R:Id ⟶ Id ⟶ Info List+ ⟶ Info List+ ⟶ ℙ]. ∀[P:Id ⟶ Info List+ ⟶ ℙ].
∀eo1,eo2,eo:EO+(Info). ∀f:E ⟶ E. ∀g:E ⟶ E.
(es-joint-embedding(Info;eo1;eo2;eo;f;g)
⇒ (causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) eo1)
⇒ (causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) eo2)
⇒ (causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) eo))
Proof
Definitions occuring in Statement :
causal-invariant: causal-invariant(i,L.P[i; L];a,b,L1,L2.R[a; b; L1; L2]),
es-joint-embedding: es-joint-embedding(Info;eo1;eo2;eo;f;g),
event-ordering+: EO+(Info),
es-E: E,
Id: Id,
listp: A 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],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
causal-invariant: causal-invariant(i,L.P[i; L];a,b,L1,L2.R[a; b; L1; L2]),
es-joint-embedding: es-joint-embedding(Info;eo1;eo2;eo;f;g),
and: P ∧ Q,
or: P ∨ Q,
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
es-embedding: (f embeds eo1 into eo2),
cand: A c∧ B,
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
guard: {T},
squash: ↓T,
true: True
Latex:
\mforall{}[Info:Type]. \mforall{}[R:Id {}\mrightarrow{} Id {}\mrightarrow{} Info List\msupplus{} {}\mrightarrow{} Info List\msupplus{} {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:Id {}\mrightarrow{} Info List\msupplus{} {}\mrightarrow{} \mBbbP{}].
\mforall{}eo1,eo2,eo:EO+(Info). \mforall{}f:E {}\mrightarrow{} E. \mforall{}g:E {}\mrightarrow{} E.
(es-joint-embedding(Info;eo1;eo2;eo;f;g)
{}\mRightarrow{} (causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) eo1)
{}\mRightarrow{} (causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) eo2)
{}\mRightarrow{} (causal-invariant(i,L.P[i;L];a,b,L1,L2.R[a;b;L1;L2]) eo))
Date html generated:
2016_05_16-PM-01_25_14
Last ObjectModification:
2016_01_17-PM-07_53_08
Theory : event-ordering
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