Nuprl Lemma : new_23_sig_progress-step9-v2
∀[Cmd:ValueAllType]. ∀[eq:EqDecider(Cmd)]. ∀[reps,clients:bag(Id)]. ∀[coeff:{2...}]. ∀[flrs:ℕ].
∀[propose,notify:Atom List]. ∀[slots:set-sig{i:l}(ℤ)]. ∀[f:new_23_sig_headers_type{i:l}(Cmd;notify;propose)].
∀[es:EO+(Message(f))]. ∀[e:E]. ∀[n:ℤ]. ∀[c:Cmd]. ∀[faulty:bag(Id)].
  (msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)
  
⇒ bag-no-repeats(Id;reps)
  
⇒ (#(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ)
  
⇒ loc(e) ↓∈ reps
  
⇒ (¬loc(e) ↓∈ faulty)
  
⇒ <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)
  
⇒ (¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e)))
  
⇒ (↓∃e':E
        (((coeff * flrs) + 1)
        = ||mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>
                      λe'.new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es.e;e';n;0);
                      [e, e'])||
        ∈ ℤ)))
Proof
Definitions occuring in Statement : 
new_23_sig_vote_with_ballot_first: new_23_sig_vote_with_ballot_first(Cmd;notify;propose;f;es;e;n;r)
, 
new_23_sig_main: new_23_sig_main()
, 
new_23_sig_ReplicaStateFun: new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e)
, 
new_23_sig_Proposal: new_23_sig_Proposal(Cmd;notify;propose;f)
, 
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose)
, 
set-sig-member: set-sig-member(s)
, 
set-sig: set-sig{i:l}(Item)
, 
msgs-interface-delivered-with-omissions: msgs-interface-delivered-with-omissions(f;es;X;faulty;failures;ids)
, 
es-info-body: msgval(e)
, 
Message: Message(f)
, 
classrel: v ∈ X(e)
, 
eo-forward: eo.e
, 
event-ordering+: EO+(Info)
, 
es-interval: [e, e']
, 
es-loc: loc(e)
, 
es-E: E
, 
Id: Id
, 
mapfilter: mapfilter(f;P;L)
, 
length: ||as||
, 
list: T List
, 
deq: EqDecider(T)
, 
int_upper: {i...}
, 
nat: ℕ
, 
vatype: ValueAllType
, 
assert: ↑b
, 
uall: ∀[x:A]. B[x]
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
squash: ↓T
, 
implies: P 
⇒ Q
, 
apply: f a
, 
lambda: λx.A[x]
, 
pair: <a, b>
, 
product: x:A × B[x]
, 
multiply: n * m
, 
add: n + m
, 
natural_number: $n
, 
int: ℤ
, 
atom: Atom
, 
equal: s = t ∈ T
, 
bag-member: x ↓∈ bs
, 
bag-no-repeats: bag-no-repeats(T;bs)
, 
bag-size: #(bs)
, 
bag: bag(T)
Definitions unfolded in proof : 
vatype: ValueAllType
, 
uall: ∀[x:A]. B[x]
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose)
, 
name: Name
, 
subtype_rel: A ⊆r B
, 
prop: ℙ
, 
and: P ∧ Q
, 
listp: A List+
, 
sq_stable: SqStable(P)
, 
l_all: (∀x∈L.P[x])
, 
all: ∀x:A. B[x]
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
iff: P 
⇐⇒ Q
, 
new_23_sig_headers: new_23_sig_headers(notify;propose)
, 
rev_implies: P 
⇐ Q
, 
guard: {T}
, 
or: P ∨ Q
, 
uimplies: b supposing a
, 
new_23_sig_headers_fun: new_23_sig_headers_fun(Cmd;notify;propose)
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
cand: A c∧ B
, 
new_23_sig_headers_no_rep: new_23_sig_headers_no_rep(notify;propose)
, 
squash: ↓T
, 
no_repeats: no_repeats(T;l)
, 
nat: ℕ
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
false: False
, 
not: ¬A
, 
length: ||as||
, 
list_ind: list_ind, 
cons: [a / b]
, 
nil: []
, 
less_than: a < b
, 
true: True
, 
sq_type: SQType(T)
, 
int_upper: {i...}
, 
ge: i ≥ j 
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
top: Top
, 
select: L[n]
, 
subtract: n - m
, 
bfalse: ff
, 
bnot: ¬bb
, 
assert: ↑b
, 
name_eq: name_eq(x;y)
, 
name-deq: NameDeq
, 
list-deq: list-deq(eq)
, 
band: p ∧b q
, 
atom-deq: AtomDeq
, 
eq_atom: x =a y
, 
null: null(as)
, 
msgs-interface-delivered-with-omissions: msgs-interface-delivered-with-omissions(f;es;X;faulty;failures;ids)
, 
decidable: Dec(P)
, 
has-es-info-type: has-es-info-type(es;e;f;T)
, 
pi2: snd(t)
, 
bag-remove-repeats: bag-remove-repeats(eq;bs)
, 
so_apply: x[s1;s2]
, 
irrefl: Irrefl(T;x,y.E[x; y])
, 
st_anti_sym: StAntiSym(T;x,y.R[x; y])
, 
sorted-by: sorted-by(R;L)
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
new_23_sig_vote_with_ballot_first_alt: new_23_sig_vote_with_ballot_first_alt(Cmd;notify;propose;f;es;e;e';n;r;a)
, 
Id: Id
, 
mapfilter: mapfilter(f;P;L)
, 
bag-size: #(bs)
, 
rev_uimplies: rev_uimplies(P;Q)
Latex:
\mforall{}[Cmd:ValueAllType].  \mforall{}[eq:EqDecider(Cmd)].  \mforall{}[reps,clients:bag(Id)].  \mforall{}[coeff:\{2...\}].  \mforall{}[flrs:\mBbbN{}].
\mforall{}[propose,notify:Atom  List].  \mforall{}[slots:set-sig\{i:l\}(\mBbbZ{})].
\mforall{}[f:new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose)].  \mforall{}[es:EO+(Message(f))].  \mforall{}[e:E].  \mforall{}[n:\mBbbZ{}].
\mforall{}[c:Cmd].  \mforall{}[faulty:bag(Id)].
    (msgs-interface-delivered-with-omissions(f;es;new\_23\_sig\_main();faulty;flrs;reps)
    {}\mRightarrow{}  bag-no-repeats(Id;reps)
    {}\mRightarrow{}  (\#(reps)  =  ((coeff  *  flrs)  +  flrs  +  1))
    {}\mRightarrow{}  loc(e)  \mdownarrow{}\mmember{}  reps
    {}\mRightarrow{}  (\mneg{}loc(e)  \mdownarrow{}\mmember{}  faulty)
    {}\mRightarrow{}  <n,  c>  \mmember{}  new\_23\_sig\_Proposal(Cmd;notify;propose;f)(e)
    {}\mRightarrow{}  (\mneg{}\muparrow{}(set-sig-member(slots)  n  new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e)))
    {}\mRightarrow{}  (\mdownarrow{}\mexists{}e':E
                (((coeff  *  flrs)  +  1)
                =  ||mapfilter(\mlambda{}e.<snd(msgval(e)),  e,  snd(fst(msgval(e)))>
                                            \mlambda{}e'.new\_23\_sig\_vote\_with\_ballot\_first(Cmd;notify;propose;f;es.e;e';n;0);
                                            [e,  e'])||)))
Date html generated:
2016_05_17-PM-02_39_13
Last ObjectModification:
2016_01_18-AM-09_31_55
Theory : 2!3!consensus!with!signatures
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