Proof of Nuprl Lemma : pv11_p1_bnum_p2a

Step * 1 1 of Lemma pv11_p1_bnum_p2a_loc

.....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))))
BY
{ (SquashExRepD THEN DProdsAndUnions) }

1
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. loc(e) ↓∈ ldrs
24. i ↓∈ accpts
25. 0 = 0 ∈ ℤ
26. ff = pv11_p1_init_active()
27. l = loc(e) ∈ Id
28. header(e) = [propose] ∈ Name
29. has-es-info-type(es;e;f;ℤ × Cmd)
30. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))))
31. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))))
32. b = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))) ∈ pv11_p1_Ballot_Num()
33. <s, c> = msgval(e) ∈ (ℤ × Cmd)
⊢ ↓∃n:ℤ. (↑(pv11_p1_eq_bnums() b (pv11_p1_mk_bnum() n loc(e))))

2
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. loc(e) ↓∈ ldrs
24. i ↓∈ accpts
25. 0 = 0 ∈ ℤ
26. ff = pv11_p1_init_active()
27. l = loc(e) ∈ Id
28. header(e) = ``pv11_p1 adopted`` ∈ Name
29. has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
30. (fst(msgval(e))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))) ∈ pv11_p1_Ballot_Num()
31. <s, c> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e)))
32. ¬(∃p2:Cmd. (<s, p2> ∈ pv11_p1_pmax(Cmd;ldrs_uid) (snd(msgval(e)))))
33. b = (fst(msgval(e))) ∈ pv11_p1_Ballot_Num()
⊢ ↓∃n:ℤ. (↑(pv11_p1_eq_bnums() b (pv11_p1_mk_bnum() n loc(e))))

3
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. loc(e) ↓∈ ldrs
24. i ↓∈ accpts
25. 0 = 0 ∈ ℤ
26. ff = pv11_p1_init_active()
27. l = loc(e) ∈ Id
28. header(e) = ``pv11_p1 adopted`` ∈ Name
29. has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
30. (fst(msgval(e))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))) ∈ pv11_p1_Ballot_Num()
31. v2 : pv11_p1_Ballot_Num()
32. <v2, s, c> ↓∈ snd(msgval(e))
33. ¬(∃z5:pv11_p1_Ballot_Num(). ∃z8:Cmd. ((↑(v2 < z5)) ∧ (<z5, s, z8> ∈ snd(msgval(e)))))
34. b = (fst(msgval(e))) ∈ pv11_p1_Ballot_Num()
⊢ ↓∃n:ℤ. (↑(pv11_p1_eq_bnums() b (pv11_p1_mk_bnum() n loc(e))))

Latex:

Latex:
.....assertion..... 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. loc(e) \mdownarrow{}\mmember{} ldrs 24. i \mdownarrow{}\mmember{} accpts 25. 0 = 0 26. ff = pv11\_p1\_init\_active() 27. l = loc(e) 28. \mdownarrow{}(((header(e) = [propose]) \mwedge{} has-es-info-type(es;e;f;\mBbbZ{} \mtimes{} Cmd)) \mwedge{} ((\muparrow{}(fst(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))))) \mwedge{} (\mneg{}\muparrow{}(pv11\_p1\_in\_domain(Cmd) (fst(msgval(e))) (snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))))))) \mwedge{} (b = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))) \mwedge{} (<s, c> = msgval(e))) \mvee{} (((header(e) = ``pv11\_p1 adopted``) \mwedge{} has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List))) \mwedge{} ((fst(msgval(e))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e\000C)))) \mwedge{} ((<s, c> \mdownarrow{}\mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e))) \mwedge{} (\mneg{}(\mexists{}p2:Cmd. (<s, p2> \mmember{} pv11\_p1\_pmax(Cmd;ldrs$_{uid}$) (snd(msgval(e))\000C))))) \mvee{} (\mexists{}v2:pv11\_p1\_Ballot\_Num() (<v2, s, c> \mdownarrow{}\mmember{} snd(msgval(e)) \mwedge{} (\mneg{}(\mexists{}z5:pv11\_p1\_Ballot\_Num() \mexists{}z8:Cmd. ((\muparrow{}(v2 < z5)) \mwedge{} (<z5, s, z8> \mmember{} snd(msgval(e))))))))) \mwedge{} (b = (fst(msgval(e))))) \mvdash{} \mdownarrow{}\mexists{}n:\mBbbZ{}. (\muparrow{}(pv11\_p1\_eq\_bnums() b (pv11\_p1\_mk\_bnum() n loc(e)))) By Latex: (SquashExRepD THEN DProdsAndUnions) Home Index