Nuprl Lemma : adjacent-run-states
∀[M:Type ⟶ Type]
∀n2m:ℕ ⟶ pMsg(P.M[P]). ∀l2m:Id ⟶ pMsg(P.M[P]). ∀S:System(P.M[P]). ∀env:pEnvType(P.M[P]). ∀x:Id. ∀n,m:ℕ+.
(run-event-state-when(pRun(S;env;n2m;l2m);<n, x>) ⊆ run-event-state-when(pRun(S;env;n2m;l2m);<m, x>)) supposing
((∀a:runEvents(pRun(S;env;n2m;l2m))
¬((n ≤ run-event-step(a)) ∧ run-event-step(a) < m) supposing run-event-loc(a) = x ∈ Id) and
(n ≤ m))
supposing Continuous+(P.M[P])
Proof
Definitions occuring in Statement :
run-event-step: run-event-step(e),
run-event-loc: run-event-loc(e),
run-event-state-when: run-event-state-when(r;e),
runEvents: runEvents(r),
pRun: pRun(S0;env;nat2msg;loc2msg),
pEnvType: pEnvType(T.M[T]),
System: System(P.M[P]),
pMsg: pMsg(P.M[P]),
Process: Process(P.M[P]),
Id: Id,
l_contains: A ⊆ B,
strong-type-continuous: Continuous+(T.F[T]),
nat_plus: ℕ+,
nat: ℕ,
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],
not: ¬A,
and: P ∧ Q,
function: x:A ⟶ B[x],
pair: <a, b>,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
strong-type-continuous: Continuous+(T.F[T]),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
nat: ℕ,
ge: i ≥ j ,
cand: A c∧ B,
le: A ≤ B,
component: component(P.M[P]),
pi1: fst(t),
deliver-msg: deliver-msg(t;m;x;Cs;L),
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
pi2: snd(t),
System: System(P.M[P]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
list_accum: list_accum,
nil: [],
it: ⋅,
mapfilter: mapfilter(f;P;L),
deliver-msg-to-comp: deliver-msg-to-comp(t;m;x;S;C),
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
l_contains: A ⊆ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
run-event-state-when: run-event-state-when(r;e),
pRun: pRun(S0;env;nat2msg;loc2msg),
ycomb: Y,
runEvents: runEvents(r),
run-event-loc: run-event-loc(e),
run-event-step: run-event-step(e),
is-run-event: is-run-event(r;t;x),
exposed-bfalse: exposed-bfalse,
spreadn: spread3,
isl: isl(x),
outl: outl(x),
band: p ∧b q,
nequal: a ≠ b ∈ T ,
fulpRunType: fulpRunType(T.M[T]),
pEnvType: pEnvType(T.M[T]),
pRunType: pRunType(T.M[T]),
less_than': less_than'(a;b),
do-chosen-command: do-chosen-command(nat2msg;loc2msg;t;S;n;m;nm),
ldag: LabeledDAG(T),
int_seg: {i..j-},
lelt: i ≤ j < k,
lg-is-source: lg-is-source(g;i),
pInTransit: pInTransit(P.M[P]),
let: let,
create-component: create-component(t;P;x;Cs;L),
subtract: n - m
Latex:
\mforall{}[M:Type {}\mrightarrow{} Type]
\mforall{}n2m:\mBbbN{} {}\mrightarrow{} pMsg(P.M[P]). \mforall{}l2m:Id {}\mrightarrow{} pMsg(P.M[P]). \mforall{}S:System(P.M[P]). \mforall{}env:pEnvType(P.M[P]). \mforall{}x:Id.
\mforall{}n,m:\mBbbN{}\msupplus{}.
(run-event-state-when(pRun(S;env;n2m;l2m);<n, x>) \msubseteq{} run-event-state-when(pRun(S;env;n2m;l2m);<m,\000C x>)) supposing
((\mforall{}a:runEvents(pRun(S;env;n2m;l2m))
\mneg{}((n \mleq{} run-event-step(a)) \mwedge{} run-event-step(a) < m) supposing run-event-loc(a) = x) and
(n \mleq{} m))
supposing Continuous+(P.M[P])
Date html generated:
2016_05_17-AM-10_47_07
Last ObjectModification:
2016_01_18-AM-00_22_11
Theory : process-model
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