Proof of Nuprl Lemma : pv11

Step * of Lemma pv11_p1-decision

∀[Cmd:{T:Type| valueall-type(T)} ]. ∀[accpts,ldrs:bag(Id)]. ∀[ldrs_uid:Id ⟶ ℤ]. ∀[reps:bag(Id)].

∀[mf:pv11_p1_headers_type{i:l}(Cmd)]. ∀[es:EO+(Message(mf))]. ∀[e:E]. ∀[d:ℤ]. ∀[i:Id]. ∀[auth:𝔹]. ∀[k:ℤ]. ∀[v:Cmd].

  (<d, i, mk-msg(auth;[decision];<k, v>)> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;mf)(e)
  ⇐⇒ loc(e) ↓∈ ldrs
      ∧ (↓∃e':{e':E| e' ≤loc e }
           ((header(e) = ``pv11_p1 p2b`` ∈ Name)
           ∧ has-es-info-type(es;e;mf;Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num())
           ∧ (d = 0 ∈ ℤ)
           ∧ ((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e))))) ∈ pv11_p1_Ballot_Num())

           ∧ #(pv11_p1_CommanderStateFun(Cmd;accpts;mf;fst(snd(msgval(e)));fst(snd(snd(msgval(e))));es.e';e)) < ...

           ∧ i ↓∈ reps
           ∧ auth = pv11_p1_init_active()
           ∧ (k = (fst(snd(snd(msgval(e))))) ∈ ℤ)
           ∧ ((((header(e') = [propose] ∈ Name) ∧ has-es-info-type(es;e';mf;ℤ × Cmd))
             ∧ ((↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e')))))
               ∧ (¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e')))
                     (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e')))))))

             ∧ ((fst(snd(msgval(e)))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e'))) ∈ pv11_p1_Ballot_Num())

∧ (<fst(snd(snd(msgval(e)))), v> = msgval(e') ∈ (ℤ × Cmd)))
∨ (((header(e') = ``pv11_p1 adopted`` ∈ Name)

                ∧ has-es-info-type(es;e';mf;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)))

               ∧ ((fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e'))) ∈ pv11_p1_Ballot_Num())

               ∧ ((<fst(snd(snd(msgval(e)))), v> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;mf;es;e')))

                 ∧ (¬(∃p2:Cmd. (<fst(snd(snd(msgval(e)))), p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e')))))))

                 ∨ (∃v2:pv11_p1_Ballot_Num()
                     (<v2, fst(snd(snd(msgval(e)))), v> ↓∈ snd(msgval(e'))
                     ∧ (¬(∃z5:pv11_p1_Ballot_Num()

                           ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, fst(snd(snd(msgval(e)))), z8> ∈ snd(msgval(e')))))))))

∧ ((fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num())))

           ∧ (no ((pv11_p1_commander_output(Cmd;accpts;reps;mf) <fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), v> o pv1\000C1_p1_p2b'base(Cmd;mf)) o

                 pv11_p1_CommanderState(Cmd;accpts;mf) (fst(snd(msgval(e)))) (fst(snd(snd(msgval(e)))))) between e' and

e))))
BY
{ ProveILF_instance "decision""pv11_p1" }

Latex:

Latex:
\mforall{}[Cmd:\{T:Type| valueall-type(T)\} ]. \mforall{}[accpts,ldrs:bag(Id)]. \mforall{}[ldrs$_{uid}$:Id {}\mrightarrow{}\000C \mBbbZ{}]. \mforall{}[reps:bag(Id)]. \mforall{}[mf:pv11\_p1\_headers\_type\{i:l\}(Cmd)]. \mforall{}[es:EO+(Message(mf))]. \mforall{}[e:E]. \mforall{}[d:\mBbbZ{}]. \mforall{}[i:Id]. \mforall{}[auth:\mBbbB{}]. \mforall{}[k:\mBbbZ{}]. \mforall{}[v:Cmd]. (<d, i, mk-msg(auth;[decision];<k, v>)> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid}\000C$;reps;mf)(e) \mLeftarrow{}{}\mRightarrow{} loc(e) \mdownarrow{}\mmember{} ldrs \mwedge{} (\mdownarrow{}\mexists{}e':\{e':E| e' \mleq{}loc e \} ((header(e) = ``pv11\_p1 p2b``) \mwedge{} has-es-info-type(es;e;mf;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num()) \mwedge{} (d = 0) \mwedge{} ((fst(snd(msgval(e)))) = (snd(snd(snd(msgval(e)))))) \mwedge{} \#(pv11\_p1\_CommanderStateFun(Cmd;accpts;mf;fst(snd(msgval(e)));fst(...);es.e';e)) < ... \mwedge{} i \mdownarrow{}\mmember{} reps \mwedge{} auth = pv11\_p1\_init\_active() \mwedge{} (k = (fst(snd(snd(msgval(e)))))) \mwedge{} ((((header(e') = [propose]) \mwedge{} has-es-info-type(es;e';mf;\mBbbZ{} \mtimes{} Cmd)) \mwedge{} ((\muparrow{}(fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;e'))))) \mwedge{} (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) (fst(msgval(e'))) (snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;e'))\000C))))) \mwedge{} ((fst(snd(msgval(e)))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}\mbackslash{}\000Cff24;mf;es;e')))) \mwedge{} (<fst(snd(snd(msgval(e)))), v> = msgval(e'))) \mvee{} (((header(e') = ``pv11\_p1 adopted``) \mwedge{} has-es-info-type(es;e';mf;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))) \mwedge{} ((fst(msgval(e'))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}\mbackslash{}ff\000C24;mf;es;e')))) \mwedge{} ((<fst(snd(snd(msgval(e)))) , v > \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;mf;es;e'\000C))) \mwedge{} (\mneg{}(\mexists{}p2:Cmd (<fst(snd(snd(msgval(e)))), p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid\mbackslash{}ff7\000Cd$) (snd(msgval(e'))))))) \mvee{} (\mexists{}v2:pv11\_p1\_Ballot\_Num() (<v2, fst(snd(snd(msgval(e)))), v> \mdownarrow{}\mmember{} snd(msgval(e')) \mwedge{} (\mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num() \mexists{}z8:Cmd ((\muparrow{}(v2 < z5)) \mwedge{} (<z5, fst(snd(snd(msgval(e)))), z8> \mmember{} snd(msgval(e'))))))))) \mwedge{} ((fst(snd(msgval(e)))) = (fst(msgval(e')))))) \mwedge{} (no ((pv11\_p1\_commander\_output(Cmd;accpts;reps;mf) <fst(snd(msgval(e))), fst(snd(snd(msgval(e)))), v> o pv11\_p1\_p2b'base(Cmd;mf)) o pv11\_p1\_CommanderState(Cmd;accpts;mf) (fst(snd(msgval(e)))) (fst(snd(snd(msgval(e)))))) between e' and e)))) By Latex: ProveILF\_instance "decision""pv11\_p1" Home Index