Nuprl Lemma : run-event-cases
∀[M:Type ⟶ Type]
∀S0:System(P.M[P]). ∀r:pRunType(P.M[P]). ∀e1,e2:runEvents(r).
(((run-event-local-pred(r;e2) = run-event-local-pred(r;e1) ∈ (runEvents(r)?))
∧ (run-event-interval(r;e1;e2) = [e2] ∈ (runEvents(r) List)))
∨ (∃e:runEvents(r)
(run-event-step(e) < run-event-step(e2)
∧ (run-event-step(e1) ≤ run-event-step(e))
∧ ((run-event-loc(e1) = run-event-loc(e) ∈ Id) ∧ (run-event-local-pred(r;e2) = (inl e) ∈ (runEvents(r)?)))
∧ (run-event-interval(r;e1;e2) = (run-event-interval(r;e1;e) @ [e2]) ∈ (runEvents(r) List))))) supposing
((run-event-step(e1) ≤ run-event-step(e2)) and
(run-event-loc(e1) = run-event-loc(e2) ∈ Id))
Proof
Definitions occuring in Statement :
run-event-local-pred: run-event-local-pred(r;e),
run-event-interval: run-event-interval(r;e1;e2),
run-event-step: run-event-step(e),
run-event-loc: run-event-loc(e),
runEvents: runEvents(r),
pRunType: pRunType(T.M[T]),
System: System(P.M[P]),
Id: Id,
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
and: P ∧ Q,
unit: Unit,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
runEvents: runEvents(r),
run-event-step: run-event-step(e),
pi1: fst(t),
run-event-loc: run-event-loc(e),
pi2: snd(t),
run-event-interval: run-event-interval(r;e1;e2),
run-event-local-pred: run-event-local-pred(r;e),
let: let,
run-event-history: run-event-history(r;e),
sq_stable: SqStable(P),
sq_type: SQType(T),
guard: {T},
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
true: True,
squash: ↓T,
Id: Id,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
cand: A c∧ B,
less_than': less_than'(a;b),
is-run-event: is-run-event(r;t;x),
int_seg: {i..j-},
lelt: i ≤ j < k,
from-upto: [n, m),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
has-value: (a)↓,
bfalse: ff,
bnot: ¬bb,
mapfilter: mapfilter(f;P;L),
exposed-bfalse: exposed-bfalse,
band: p ∧b q,
isl: isl(x),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
last: last(L),
subtract: n - m,
select: L[n]
Latex:
\mforall{}[M:Type {}\mrightarrow{} Type]
\mforall{}S0:System(P.M[P]). \mforall{}r:pRunType(P.M[P]). \mforall{}e1,e2:runEvents(r).
(((run-event-local-pred(r;e2) = run-event-local-pred(r;e1))
\mwedge{} (run-event-interval(r;e1;e2) = [e2]))
\mvee{} (\mexists{}e:runEvents(r)
(run-event-step(e) < run-event-step(e2)
\mwedge{} (run-event-step(e1) \mleq{} run-event-step(e))
\mwedge{} ((run-event-loc(e1) = run-event-loc(e)) \mwedge{} (run-event-local-pred(r;e2) = (inl e)))
\mwedge{} (run-event-interval(r;e1;e2) = (run-event-interval(r;e1;e) @ [e2]))))) supposing
((run-event-step(e1) \mleq{} run-event-step(e2)) and
(run-event-loc(e1) = run-event-loc(e2)))
Date html generated:
2016_05_17-AM-10_45_24
Last ObjectModification:
2016_01_18-AM-00_26_38
Theory : process-model
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