Nuprl Lemma : pRun-intransit-invariant
∀[M:Type ⟶ Type]
∀n2m:ℕ ⟶ pMsg(P.M[P]). ∀l2m:Id ⟶ pMsg(P.M[P]). ∀Cs0:component(P.M[P]) List. ∀G0:LabeledDAG(pInTransit(P.M[P])).
∀env:pEnvType(P.M[P]). ∀t:ℕ.
let r = pRun(<Cs0, G0>;env;n2m;l2m) in
let info,Cs,G = r t in
∀x∈G.let ev = fst(x) in
((fst(ev)) ≤ t) ∨ (∃m:ℕlg-size(G0). (ev = (fst(lg-label(G0;m))) ∈ (ℤ × Id)))
supposing Continuous+(P.M[P])
Proof
Definitions occuring in Statement :
pRun: pRun(S0;env;nat2msg;loc2msg),
pEnvType: pEnvType(T.M[T]),
pInTransit: pInTransit(P.M[P]),
component: component(P.M[P]),
pMsg: pMsg(P.M[P]),
lg-all: ∀x∈G.P[x],
ldag: LabeledDAG(T),
lg-label: lg-label(g;x),
lg-size: lg-size(g),
Id: Id,
list: T List,
strong-type-continuous: Continuous+(T.F[T]),
int_seg: {i..j-},
nat: ℕ,
let: let,
spreadn: spread3,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi1: fst(t),
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
pair: <a, b>,
product: x:A × B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
let: let,
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],
pInTransit: pInTransit(P.M[P]),
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
ldag: LabeledDAG(T),
nat: ℕ,
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
System: System(P.M[P]),
so_apply: x[s1;s2],
prop: ℙ,
guard: {T},
lg-all: ∀x∈G.P[x],
or: P ∨ Q,
exists: ∃x:A. B[x],
pEnvType: pEnvType(T.M[T]),
nat_plus: ℕ+,
le: A ≤ B,
decidable: Dec(P),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
pRunType: pRunType(T.M[T]),
spreadn: spread3,
do-chosen-command: do-chosen-command(nat2msg;loc2msg;t;S;n;m;nm),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
pi2: snd(t),
lg-is-source: lg-is-source(g;i),
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
list_accum: list_accum,
nil: [],
deliver-msg-to-comp: deliver-msg-to-comp(t;m;x;S;C),
component: component(P.M[P]),
cand: A c∧ B,
pExt: pExt(P.M[P]),
add-cause: add-cause(ev;ext),
deliver-msg: deliver-msg(t;m;x;Cs;L),
create-component: create-component(t;P;x;Cs;L),
nequal: a ≠ b ∈ T ,
fulpRunType: fulpRunType(T.M[T])
Latex:
\mforall{}[M:Type {}\mrightarrow{} Type]
\mforall{}n2m:\mBbbN{} {}\mrightarrow{} pMsg(P.M[P]). \mforall{}l2m:Id {}\mrightarrow{} pMsg(P.M[P]). \mforall{}Cs0:component(P.M[P]) List.
\mforall{}G0:LabeledDAG(pInTransit(P.M[P])). \mforall{}env:pEnvType(P.M[P]). \mforall{}t:\mBbbN{}.
let r = pRun(<Cs0, G0>env;n2m;l2m) in
let info,Cs,G = r t in
\mforall{}x\mmember{}G.let ev = fst(x) in
((fst(ev)) \mleq{} t) \mvee{} (\mexists{}m:\mBbbN{}lg-size(G0). (ev = (fst(lg-label(G0;m)))))
supposing Continuous+(P.M[P])
Date html generated:
2016_05_17-AM-10_48_14
Last ObjectModification:
2016_01_18-AM-00_18_02
Theory : process-model
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