Proof of Nuprl Lemma : pv11_p1_bnum_p2a

Step * 1 of Lemma pv11_p1_bnum_p2a_loc

1. Cmd : {T:Type| valueall-type(T)} @i'
2. f : pv11_p1_headers_type{i:l}(Cmd)@i'
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))@i'
13. e : E@i
14. ldrs_uid : Id ─→ ℤ@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. reps : bag(Id)@i
18. b : pv11_p1_Ballot_Num()@i
19. i : Id@i
20. l : Id@i
21. s : ℤ@i
22. c : Cmd@i
23. pv11_p1_p2a'send(Cmd;f) i <l, b, s, c> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e)@i
⊢ ∃n:ℤ. (↑(pv11_p1_eq_bnums() b (pv11_p1_mk_bnum() n loc(e))))
BY
{ (RepUR ``pv11_p1_p2a'send mk-msg-interface make-msg-interface`` (-1)
   THEN (RWO "make-Msg-as-mk-msg" (-1) THENA Trivial)
   THEN RWO "pv11_p1-p2a" (-1)
   THEN Auto
   THEN (Assert ⌈↓∃n:ℤ. (↑(pv11_p1_eq_bnums() b (pv11_p1_mk_bnum() n loc(e))))⌉⋅
   THENM (DVar `b'
          THEN Try (Complete ((Assert ⌈False⌉⋅ THEN Auto THEN SquashExRepD THEN Auto)))
          THEN DVar `x'
          THEN (InstConcl [⌈x1⌉]⋅ THENA Auto)
          THEN (D (-1) THEN Auto)
          THEN ExRepD
          THEN (RW assert_pushdownC (-1) THENA Auto)
          THEN All (RepUR ``pv11_p1_mk_bnum pv11_p1_Ballot_Num``)
          THEN Repeat ((EqCD THEN Auto)))
   )) }

1
.....assertion.....
1. Cmd : {T:Type| valueall-type(T)} @i'
2. f : pv11_p1_headers_type{i:l}(Cmd)@i'
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

       ∧ (¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))))))

     ∧ (b = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))) ∈ pv11_p1_Ballot_Num())
     ∧ (<s, c> = msgval(e) ∈ (ℤ × Cmd)))
     ∨ (((header(e) = ``pv11_p1 adopted`` ∈ Name)

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

       ∧ ((fst(msgval(e))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))) ∈ pv11_p1_Ballot_Num())
       ∧ ((<s, c> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))
         ∧ (¬(∃p2:Cmd. (<s, p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e)))))))
         ∨ (∃v2:pv11_p1_Ballot_Num()
             (<v2, s, c> ↓∈ snd(msgval(e))

             ∧ (¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2  < z5)) ∧ (<z5, s, z8> ∈ snd(msgval(e)))))))))

∧ (b = (fst(msgval(e))) ∈ pv11_p1_Ballot_Num()))
⊢ ↓∃n:ℤ. (↑(pv11_p1_eq_bnums() b (pv11_p1_mk_bnum() n loc(e))))

Latex:

Latex:
1. Cmd : \{T:Type| valueall-type(T)\} @i' 2. f : pv11\_p1\_headers\_type\{i:l\}(Cmd)@i' 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))@i' 13. e : E@i 14. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 15. accpts : bag(Id)@i 16. ldrs : bag(Id)@i 17. reps : bag(Id)@i 18. b : pv11\_p1\_Ballot\_Num()@i 19. i : Id@i 20. l : Id@i 21. s : \mBbbZ{}@i 22. c : Cmd@i 23. pv11\_p1\_p2a'send(Cmd;f) i <l, b, s, c> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$_{uid\mbackslash{}ff7\000Cd$;reps;f)(e)@i \mvdash{} \mexists{}n:\mBbbZ{}. (\muparrow{}(pv11\_p1\_eq\_bnums() b (pv11\_p1\_mk\_bnum() n loc(e)))) By Latex: (RepUR ``pv11\_p1\_p2a'send mk-msg-interface make-msg-interface`` (-1) THEN (RWO "make-Msg-as-mk-msg" (-1) THENA Trivial) THEN RWO "pv11\_p1-p2a" (-1) THEN Auto THEN (Assert \mkleeneopen{}\mdownarrow{}\mexists{}n:\mBbbZ{}. (\muparrow{}(pv11\_p1\_eq\_bnums() b (pv11\_p1\_mk\_bnum() n loc(e))))\mkleeneclose{}\mcdot{} THENM (DVar `b' THEN Try (Complete ((Assert \mkleeneopen{}False\mkleeneclose{}\mcdot{} THEN Auto THEN SquashExRepD THEN Auto))) THEN DVar `x' THEN (InstConcl [\mkleeneopen{}x1\mkleeneclose{}]\mcdot{} THENA Auto) THEN (D (-1) THEN Auto) THEN ExRepD THEN (RW assert\_pushdownC (-1) THENA Auto) THEN All (RepUR ``pv11\_p1\_mk\_bnum pv11\_p1\_Ballot\_Num``) THEN Repeat ((EqCD THEN Auto))) )) Home Index