Nuprl Lemma : pv11_p1_commander_state_fun_eq
∀[Cmd:ValueAllType]. ∀[f:pv11_p1_headers_type{i:l}(Cmd)]. ∀[es:EO+(Message(f))]. ∀[e:E]. ∀[accpts:bag(Id)].
∀[b:pv11_p1_Ballot_Num()]. ∀[s:ℤ].
(pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es;e)
= if e ∈b pv11_p1_p2b'base(Cmd;f)
then if first(e)
then pv11_p1_on_p2b() b s loc(e) pv11_p1_p2b'base(Cmd;f)@e accpts
else pv11_p1_on_p2b() b s loc(e) pv11_p1_p2b'base(Cmd;f)@e
pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es;pred(e))
fi
if first(e) then accpts
else pv11_p1_CommanderStateFun(Cmd;accpts;f;b;s;es;pred(e))
fi
∈ bag(Id))
Proof
Definitions occuring in Statement :
pv11_p1_CommanderStateFun: pv11_p1_CommanderStateFun(Cmd;accpts;mf;x;zzs;es;e),
pv11_p1_on_p2b: pv11_p1_on_p2b(),
pv11_p1_p2b'base: pv11_p1_p2b'base(Cmd;mf),
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
pv11_p1_Ballot_Num: pv11_p1_Ballot_Num(),
Message: Message(f),
classfun-res: X@e,
member-eclass: e ∈b X,
event-ordering+: EO+(Info),
es-first: first(e),
es-pred: pred(e),
es-loc: loc(e),
es-E: E,
Id: Id,
vatype: ValueAllType,
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
apply: f a,
int: ℤ,
equal: s = t ∈ T,
bag: bag(T)
Definitions unfolded in proof :
vatype: ValueAllType,
uall: ∀[x:A]. B[x],
pv11_p1_headers_type: pv11_p1_headers_type{i:l}(Cmd),
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_apply: x[s],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
listp: A List+,
name: Name,
pv11_p1_headers: pv11_p1_headers(),
rev_implies: P ⇐ Q,
guard: {T},
or: P ∨ Q,
uimplies: b supposing a,
pv11_p1_headers_fun: pv11_p1_headers_fun(Cmd),
name_eq: name_eq(x;y),
name-deq: NameDeq,
list-deq: list-deq(eq),
list_ind: list_ind,
cons: [a / b],
band: p ∧b q,
ifthenelse: if b then t else f fi ,
atom-deq: AtomDeq,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
nil: [],
it: ⋅,
null: null(as),
pv11_p1_CommanderStateFun: pv11_p1_CommanderStateFun(Cmd;accpts;mf;x;zzs;es;e),
pv11_p1_CommanderState: pv11_p1_CommanderState(Cmd;accpts;mf),
pv11_p1_p2b'base: pv11_p1_p2b'base(Cmd;mf),
encodes-msg-type: hdr encodes T,
bool: 𝔹,
unit: Unit,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
state-class1: state-class1(init;tr;X),
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
not: ¬A
Latex:
\mforall{}[Cmd:ValueAllType]. \mforall{}[f:pv11\_p1\_headers\_type\{i:l\}(Cmd)]. \mforall{}[es:EO+(Message(f))]. \mforall{}[e:E].
\mforall{}[accpts:bag(Id)]. \mforall{}[b:pv11\_p1\_Ballot\_Num()]. \mforall{}[s:\mBbbZ{}].
(pv11\_p1\_CommanderStateFun(Cmd;accpts;f;b;s;es;e)
= if e \mmember{}\msubb{} pv11\_p1\_p2b'base(Cmd;f)
then if first(e)
then pv11\_p1\_on\_p2b() b s loc(e) pv11\_p1\_p2b'base(Cmd;f)@e accpts
else pv11\_p1\_on\_p2b() b s loc(e) pv11\_p1\_p2b'base(Cmd;f)@e
pv11\_p1\_CommanderStateFun(Cmd;accpts;f;b;s;es;pred(e))
fi
if first(e) then accpts
else pv11\_p1\_CommanderStateFun(Cmd;accpts;f;b;s;es;pred(e))
fi )
Date html generated:
2016_05_17-PM-03_42_24
Last ObjectModification:
2015_12_29-PM-11_17_31
Theory : paxos!synod
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