Nuprl Lemma : mk-extended-eo_wf
∀[Info,E:Type]. ∀[dom:E ⟶ 𝔹]. ∀[l:E ⟶ Id]. ∀[R:E ⟶ E ⟶ ℙ]. ∀[locless:E ⟶ E ⟶ 𝔹]. ∀[pred:E ⟶ E]. ∀[rank:E ⟶ ℕ].
∀[info:E ⟶ Info].
mk-extended-eo(type: E;
domain: dom;
loc: l;
info: info;
causal: R;
local: locless;
pred: pred;
rank: rank) ∈ EO+(Info)
supposing (∀x,y:E. ((↓x R y) ⇒ rank x < rank y))
∧ (∀e:E. ((l (pred e)) = (l e) ∈ Id))
∧ (∀e:E. (¬↓e R (pred e)))
∧ (∀e,x:E. ((↓x R e) ⇒ ((l x) = (l e) ∈ Id) ⇒ ((↓(pred e) R e) ∧ (¬↓(pred e) R x))))
∧ (∀x,y,z:E. ((↓x R y) ⇒ (↓y R z) ⇒ (↓x R z)))
∧ (∀e1,e2:E.
(↓e1 R e2 ⇐⇒ ↑(e1 locless e2)) ∧ ((¬↓e1 R e2) ⇒ (¬↓e2 R e1) ⇒ (e1 = e2 ∈ E)) supposing (l e1) = (l e2) ∈ Id)
Proof
Definitions occuring in Statement :
mk-extended-eo: mk-extended-eo,
event-ordering+: EO+(Info),
Id: Id,
nat: ℕ,
assert: ↑b,
bool: 𝔹,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
squash: ↓T,
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
event-ordering+: EO+(Info),
mk-extended-eo: mk-extended-eo,
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
record-select: r.x,
mk-eo: mk-eo(E;dom;l;R;locless;pred;rank),
mk-eo-record: mk-eo-record(E;dom;l;R;locless;pred;rank),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
prop: ℙ,
event_ordering: EO,
eo_record: eo_record{i:l}(),
record+: record+,
subtype_rel: A ⊆r B,
guard: {T},
es-base-E: es-base-E(es),
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
infix_ap: x f y,
so_apply: x[s],
eo_axioms: eo_axioms(r)
Latex:
\mforall{}[Info,E:Type]. \mforall{}[dom:E {}\mrightarrow{} \mBbbB{}]. \mforall{}[l:E {}\mrightarrow{} Id]. \mforall{}[R:E {}\mrightarrow{} E {}\mrightarrow{} \mBbbP{}]. \mforall{}[locless:E {}\mrightarrow{} E {}\mrightarrow{} \mBbbB{}].
\mforall{}[pred:E {}\mrightarrow{} E]. \mforall{}[rank:E {}\mrightarrow{} \mBbbN{}]. \mforall{}[info:E {}\mrightarrow{} Info].
mk-extended-eo(type: E;
domain: dom;
loc: l;
info: info;
causal: R;
local: locless;
pred: pred;
rank: rank) \mmember{} EO+(Info)
supposing (\mforall{}x,y:E. ((\mdownarrow{}x R y) {}\mRightarrow{} rank x < rank y))
\mwedge{} (\mforall{}e:E. ((l (pred e)) = (l e)))
\mwedge{} (\mforall{}e:E. (\mneg{}\mdownarrow{}e R (pred e)))
\mwedge{} (\mforall{}e,x:E. ((\mdownarrow{}x R e) {}\mRightarrow{} ((l x) = (l e)) {}\mRightarrow{} ((\mdownarrow{}(pred e) R e) \mwedge{} (\mneg{}\mdownarrow{}(pred e) R x))))
\mwedge{} (\mforall{}x,y,z:E. ((\mdownarrow{}x R y) {}\mRightarrow{} (\mdownarrow{}y R z) {}\mRightarrow{} (\mdownarrow{}x R z)))
\mwedge{} (\mforall{}e1,e2:E.
(\mdownarrow{}e1 R e2 \mLeftarrow{}{}\mRightarrow{} \muparrow{}(e1 locless e2)) \mwedge{} ((\mneg{}\mdownarrow{}e1 R e2) {}\mRightarrow{} (\mneg{}\mdownarrow{}e2 R e1) {}\mRightarrow{} (e1 = e2))
supposing (l e1) = (l e2))
Date html generated:
2016_05_16-PM-01_02_25
Last ObjectModification:
2015_12_29-PM-01_48_46
Theory : event-ordering
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