Nuprl Lemma : pRun-System-invariant
∀[M:Type ⟶ Type]
∀[Q:ℕ ⟶ System(P.M[P]) ⟶ ℙ]
∀nat2msg:ℕ ⟶ pMsg(P.M[P]). ∀loc2msg:Id ⟶ pMsg(P.M[P]). ∀S0:System(P.M[P]).
(Q[0;S0]
⇒ (∀t:ℕ. ∀S:System(P.M[P]).
(Q[t;S]
⇒ (∀env:pEnvType(P.M[P])
let n,m,nm = env (t + 1) pRun(S0;env;nat2msg;loc2msg) in
Q[t + 1;snd(do-chosen-command(nat2msg;loc2msg;t + 1;S;n;m;nm))])))
⇒ {∀env:pEnvType(P.M[P]). ∀t:ℕ. Q[t;snd((pRun(S0;env;nat2msg;loc2msg) t))]})
supposing Continuous+(P.M[P])
Proof
Definitions occuring in Statement :
pRun: pRun(S0;env;nat2msg;loc2msg),
pEnvType: pEnvType(T.M[T]),
do-chosen-command: do-chosen-command(nat2msg;loc2msg;t;S;n;m;nm),
System: System(P.M[P]),
pMsg: pMsg(P.M[P]),
Id: Id,
strong-type-continuous: Continuous+(T.F[T]),
nat: ℕ,
spreadn: spread3,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s1;s2],
so_apply: x[s],
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
strong-type-continuous: Continuous+(T.F[T]),
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
pRun: pRun(S0;env;nat2msg;loc2msg),
ycomb: Y,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
pi2: snd(t),
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
label: ...$L... t,
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
int_upper: {i...},
fulpRunType: fulpRunType(T.M[T]),
pEnvType: pEnvType(T.M[T]),
nat_plus: ℕ+,
subtract: n - m,
true: True,
pRunType: pRunType(T.M[T]),
spreadn: spread3
Latex:
\mforall{}[M:Type {}\mrightarrow{} Type]
\mforall{}[Q:\mBbbN{} {}\mrightarrow{} System(P.M[P]) {}\mrightarrow{} \mBbbP{}]
\mforall{}nat2msg:\mBbbN{} {}\mrightarrow{} pMsg(P.M[P]). \mforall{}loc2msg:Id {}\mrightarrow{} pMsg(P.M[P]). \mforall{}S0:System(P.M[P]).
(Q[0;S0]
{}\mRightarrow{} (\mforall{}t:\mBbbN{}. \mforall{}S:System(P.M[P]).
(Q[t;S]
{}\mRightarrow{} (\mforall{}env:pEnvType(P.M[P])
let n,m,nm = env (t + 1) pRun(S0;env;nat2msg;loc2msg) in
Q[t + 1;snd(do-chosen-command(nat2msg;loc2msg;t + 1;S;n;m;nm))])))
{}\mRightarrow{} \{\mforall{}env:pEnvType(P.M[P]). \mforall{}t:\mBbbN{}. Q[t;snd((pRun(S0;env;nat2msg;loc2msg) t))]\})
supposing Continuous+(P.M[P])
Date html generated:
2016_05_17-AM-10_41_43
Last ObjectModification:
2016_01_18-AM-00_18_08
Theory : process-model
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