Proof of Nuprl Lemma : pv11_p1_acc_state_fun

Step * 1 of Lemma pv11_p1_acc_state_fun_eq2

1. Cmd : {T:Type| valueall-type(T)}
2. f : pv11_p1_headers_type{i:l}(Cmd)
3. (f [decision]) = (ℤ × Cmd) ∈ Type
4. (f [propose]) = (ℤ × Cmd) ∈ Type

5. (f ``pv11_p1 adopted``) = (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)) ∈ Type

6. (f ``pv11_p1 preempted``) = pv11_p1_Ballot_Num() ∈ Type
7. (f ``pv11_p1 p2b``) = (Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num()) ∈ Type
8. (f ``pv11_p1 p2a``) = (Id × pv11_p1_Ballot_Num() × ℤ × Cmd) ∈ Type
9. (f ``pv11_p1 p1b``)
= (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
∈ Type
10. (f ``pv11_p1 p1a``) = (Id × pv11_p1_Ballot_Num()) ∈ Type
11. f ∈ Name ─→ Type
12. es : EO+(Message(f))
13. e : E
14. ldrs_uid : Id ─→ ℤ
⊢ pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e)
= if e ∈_b pv11_p1_p1a'base(Cmd;f)
    then if first(e)
         then pv11_p1_on_p1a(Cmd;ldrs_uid) loc(e) pv11_p1_p1a'base(Cmd;f)@e pv11_p1_init_acceptor(Cmd)
         else pv11_p1_on_p1a(Cmd;ldrs_uid) loc(e) pv11_p1_p1a'base(Cmd;f)@e
              pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
         fi
  if e ∈_b pv11_p1_p2a'base(Cmd;f)
    then if first(e)
         then pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e) pv11_p1_p2a'base(Cmd;f)@e pv11_p1_init_acceptor(Cmd)
         else pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e) pv11_p1_p2a'base(Cmd;f)@e
              pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
         fi
  if first(e) then pv11_p1_init_acceptor(Cmd)
  else pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
  fi
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
BY
{ (UnfoldAtAddr [2] 0
   THEN RepUR ``pv11_p1_AcceptorState`` 0
   THEN (RWO "state-class2-fun-eq" 0 THENA Auto)
   THEN ProveFunctional
   THEN Auto) }

1
1. Cmd : {T:Type| valueall-type(T)}
2. f : pv11_p1_headers_type{i:l}(Cmd)
3. (f [decision]) = (ℤ × Cmd) ∈ Type
4. (f [propose]) = (ℤ × Cmd) ∈ Type

5. (f ``pv11_p1 adopted``) = (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)) ∈ Type

              state-class2(λloc.pv11_p1_init_acceptor(Cmd);pv11_p1_on_p1a(Cmd;ldrs_uid);pv11_p1_p1a'base(Cmd;f);

                           pv11_p1_on_p2a(Cmd;ldrs_uid);pv11_p1_p2a'base(Cmd;f))(pred(e))
         fi
if e ∈_b pv11_p1_p2a'base(Cmd;f)
  then if first(e)
       then pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e) pv11_p1_p2a'base(Cmd;f)@e pv11_p1_init_acceptor(Cmd)
       else pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e) pv11_p1_p2a'base(Cmd;f)@e

            state-class2(λloc.pv11_p1_init_acceptor(Cmd);pv11_p1_on_p1a(Cmd;ldrs_uid);pv11_p1_p1a'base(Cmd;f);

                         pv11_p1_on_p2a(Cmd;ldrs_uid);pv11_p1_p2a'base(Cmd;f))(pred(e))
       fi
if first(e) then pv11_p1_init_acceptor(Cmd)
else state-class2(λloc.pv11_p1_init_acceptor(Cmd);pv11_p1_on_p1a(Cmd;ldrs_uid);pv11_p1_p1a'base(Cmd;f);
                  pv11_p1_on_p2a(Cmd;ldrs_uid);pv11_p1_p2a'base(Cmd;f))(pred(e))
fi
= if e ∈_b pv11_p1_p1a'base(Cmd;f)
    then if first(e)
         then pv11_p1_on_p1a(Cmd;ldrs_uid) loc(e) pv11_p1_p1a'base(Cmd;f)@e pv11_p1_init_acceptor(Cmd)
         else pv11_p1_on_p1a(Cmd;ldrs_uid) loc(e) pv11_p1_p1a'base(Cmd;f)@e
              pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
         fi
  if e ∈_b pv11_p1_p2a'base(Cmd;f)
    then if first(e)
         then pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e) pv11_p1_p2a'base(Cmd;f)@e pv11_p1_init_acceptor(Cmd)
         else pv11_p1_on_p2a(Cmd;ldrs_uid) loc(e) pv11_p1_p2a'base(Cmd;f)@e
              pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
         fi
  if first(e) then pv11_p1_init_acceptor(Cmd)
  else pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;pred(e))
  fi
∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))

Latex:

Latex:
1. Cmd : \{T:Type| valueall-type(T)\} 2. f : pv11\_p1\_headers\_type\{i:l\}(Cmd) 3. (f [decision]) = (\mBbbZ{} \mtimes{} Cmd) 4. (f [propose]) = (\mBbbZ{} \mtimes{} Cmd) 5. (f ``pv11\_p1 adopted``) = (pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 6. (f ``pv11\_p1 preempted``) = pv11\_p1\_Ballot\_Num() 7. (f ``pv11\_p1 p2b``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num()) 8. (f ``pv11\_p1 p2a``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) 9. (f ``pv11\_p1 p1b``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 10. (f ``pv11\_p1 p1a``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num()) 11. f \mmember{} Name {}\mrightarrow{} Type 12. es : EO+(Message(f)) 13. e : E 14. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{} \mvdash{} pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;e) = if e \mmember{}\msubb{} pv11\_p1\_p1a'base(Cmd;f) then if first(e) then pv11\_p1\_on\_p1a(Cmd;ldrs$_{uid}$) loc(e) pv11\_p1\_p1a'base(Cmd;f)@e pv11\_p1\_init\_acceptor(Cmd) else pv11\_p1\_on\_p1a(Cmd;ldrs$_{uid}$) loc(e) pv11\_p1\_p1a'base(Cmd;f)@e pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e)) fi if e \mmember{}\msubb{} pv11\_p1\_p2a'base(Cmd;f) then if first(e) then pv11\_p1\_on\_p2a(Cmd;ldrs$_{uid}$) loc(e) pv11\_p1\_p2a'base(Cmd;f)@e pv11\_p1\_init\_acceptor(Cmd) else pv11\_p1\_on\_p2a(Cmd;ldrs$_{uid}$) loc(e) pv11\_p1\_p2a'base(Cmd;f)@e pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e)) fi if first(e) then pv11\_p1\_init\_acceptor(Cmd) else pv11\_p1\_AcceptorStateFun(Cmd;ldrs$_{uid}$;f;es;pred(e)) fi By Latex: (UnfoldAtAddr [2] 0 THEN RepUR ``pv11\_p1\_AcceptorState`` 0 THEN (RWO "state-class2-fun-eq" 0 THENA Auto) THEN ProveFunctional THEN Auto) Home Index