Proof of Nuprl Lemma : pv11_p1

Step * 1 of Lemma pv11_p1_validity-lemma

1. Cmd : {T:Type| valueall-type(T)}
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

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

  ⇒ pv11_p1_valid-p1b-message(Cmd;es;e;f))
∧ ((header(e) = ``pv11_p1 p2a`` ∈ Name)
  ⇒ has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
  ⇒ pv11_p1_valid-p2a-message(Cmd;es;e;f))
BY
{ (MoveToConcl (-1)
   THEN CausalInd'
   THEN Auto
   THEN Try (UnfoldTopAb 0)
   THEN If (\p. is_term `pv11_p1_valid-proposals` (concl p))
           (Try (UnfoldTopAb 0 THEN RWO "l_all_iff" 0 THEN Auto))
           Id⋅
   THEN Try ((OnSomeHyp UseMessageConstraint THEN SqExRepD THEN SplitOrHyps THEN SqExRepD))⋅) }

1
1. Cmd : {T:Type| valueall-type(T)}
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
      ((e1 < e)
      ⇒ (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e1;f)
         ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e1;f)
         ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.
              (start ≤loc e1  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
         ∧ ((header(e1) = ``pv11_p1 adopted`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-adopted-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p1b`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id
              × pv11_p1_Ballot_Num()
              × pv11_p1_Ballot_Num()
              × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-p1b-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p2a`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
           ⇒ pv11_p1_valid-p2a-message(Cmd;es;e1;f))))
20. p : ℤ × Cmd@i
21. (p ∈ map(λp.(snd(p));snd(pv11_p1_AcceptorStateFun(Cmd;ldrs_uid;f;es;e))))@i
⊢ pv11_p1_valid-proposal(Cmd;es;e;p;f)

2
1. Cmd : {T:Type| valueall-type(T)}
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
      ((e1 < e)
      ⇒ (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e1;f)
         ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e1;f)
         ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.
              (start ≤loc e1  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
         ∧ ((header(e1) = ``pv11_p1 adopted`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-adopted-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p1b`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id
              × pv11_p1_Ballot_Num()
              × pv11_p1_Ballot_Num()
              × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-p1b-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p2a`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
           ⇒ pv11_p1_valid-p2a-message(Cmd;es;e1;f))))
20. pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
21. p : ℤ × Cmd@i
22. (p ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))))@i
⊢ pv11_p1_valid-proposal(Cmd;es;e;p;f)

3
1. Cmd : {T:Type| valueall-type(T)}
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
      ((e1 < e)
      ⇒ (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e1;f)
         ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e1;f)
         ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.
              (start ≤loc e1  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
         ∧ ((header(e1) = ``pv11_p1 adopted`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-adopted-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p1b`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id
              × pv11_p1_Ballot_Num()
              × pv11_p1_Ballot_Num()
              × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-p1b-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p2a`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
           ⇒ pv11_p1_valid-p2a-message(Cmd;es;e1;f))))
20. pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
21. pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
22. b : pv11_p1_Ballot_Num()@i
23. start : E@i
24. start ≤loc e @i
25. p : ℤ × Cmd@i
26. (p ∈ map(λp.(snd(p));snd(pv11_p1_ScoutStateFun(Cmd;accpts;f;b;es.start;e))))@i
⊢ pv11_p1_valid-proposal(Cmd;es;e;p;f)

4
1. Cmd : {T:Type| valueall-type(T)}
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
      ((e1 < e)
      ⇒ (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e1;f)
         ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e1;f)
         ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.
              (start ≤loc e1  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
         ∧ ((header(e1) = ``pv11_p1 adopted`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-adopted-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p1b`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id
              × pv11_p1_Ballot_Num()
              × pv11_p1_Ballot_Num()
              × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-p1b-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p2a`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
           ⇒ pv11_p1_valid-p2a-message(Cmd;es;e1;f))))
20. pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
21. pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
22. ∀b:pv11_p1_Ballot_Num(). ∀start:E.  (start ≤loc e  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f))
23. header(e) = ``pv11_p1 adopted`` ∈ Name@i
24. has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
⊢ pv11_p1_valid-proposals(Cmd;es;e;map(λp.(snd(p));snd(msgval(e)));f)

5
1. Cmd : {T:Type| valueall-type(T)}
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
      ((e1 < e)
      ⇒ (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e1;f)
         ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e1;f)
         ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.
              (start ≤loc e1  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
         ∧ ((header(e1) = ``pv11_p1 adopted`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-adopted-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p1b`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id
              × pv11_p1_Ballot_Num()
              × pv11_p1_Ballot_Num()
              × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-p1b-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p2a`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
           ⇒ pv11_p1_valid-p2a-message(Cmd;es;e1;f))))
20. pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
21. pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
22. ∀b:pv11_p1_Ballot_Num(). ∀start:E.  (start ≤loc e  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f))
23. (header(e) = ``pv11_p1 adopted`` ∈ Name)
⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f)
24. header(e) = ``pv11_p1 p1b`` ∈ Name@i
25. has-es-info-type(es;e;f;Id
× pv11_p1_Ballot_Num()
× pv11_p1_Ballot_Num()
× ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))@i
⊢ pv11_p1_valid-proposals(Cmd;es;e;map(λp.(snd(p));snd(snd(snd(msgval(e)))));f)

6
1. Cmd : {T:Type| valueall-type(T)}
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
      ((e1 < e)
      ⇒ (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e1;f)
         ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e1;f)
         ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.
              (start ≤loc e1  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
         ∧ ((header(e1) = ``pv11_p1 adopted`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-adopted-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p1b`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id
              × pv11_p1_Ballot_Num()
              × pv11_p1_Ballot_Num()
              × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-p1b-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p2a`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
           ⇒ pv11_p1_valid-p2a-message(Cmd;es;e1;f))))
20. pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
21. pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
22. ∀b:pv11_p1_Ballot_Num(). ∀start:E.  (start ≤loc e  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f))
23. (header(e) = ``pv11_p1 adopted`` ∈ Name)
⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f)
24. (header(e) = ``pv11_p1 p1b`` ∈ Name)
⇒

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

⇒ pv11_p1_valid-p1b-message(Cmd;es;e;f)
25. header(e) = ``pv11_p1 p2a`` ∈ Name@i
26. has-es-info-type(es;e;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)@i
27. ∀e:E
      ((header(e) ∈ pv11_p1_headers_no_inputs())
      ⇒ (↓∃e':E
            ∃delay:ℤ

             ((e' < e) ∧ make-msg-interface(delay;loc(e);info(e)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e'))))

28. e' : E
29. delay : ℤ
30. (e' < e)
31. loc(e') ↓∈ ldrs
32. loc(e) ↓∈ accpts
33. delay = 0 ∈ ℤ
34. es-info-auth(es;e) = pv11_p1_init_active()
35. (fst(msgval(e))) = loc(e') ∈ Id
36. header(e') = [propose] ∈ Name
37. has-es-info-type(es;e';f;ℤ × Cmd)
38. ↑(fst(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))))
39. ¬↑(pv11_p1_in_domain(Cmd) (fst(msgval(e'))) (snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')))))
40. (fst(snd(msgval(e)))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()
41. <fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> = msgval(e') ∈ (ℤ × Cmd)
⊢ pv11_p1_valid-proposal(Cmd;es;e;snd(snd(<fst(msgval(e))
                                          , fst(snd(msgval(e)))
                                          , fst(snd(snd(msgval(e))))
                                          , snd(snd(snd(msgval(e))))>));f)

7
1. Cmd : {T:Type| valueall-type(T)}
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. accpts : bag(Id)@i
14. ldrs : bag(Id)@i
15. ldrs_uid : Id ─→ ℤ@i
16. reps : bag(Id)@i
17. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
18. e : E@i
19. ∀e1:E
      ((e1 < e)
      ⇒ (pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e1;f)
         ∧ pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e1;f)
         ∧ (∀b:pv11_p1_Ballot_Num(). ∀start:E.
              (start ≤loc e1  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e1;f)))
         ∧ ((header(e1) = ``pv11_p1 adopted`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-adopted-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p1b`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id
              × pv11_p1_Ballot_Num()
              × pv11_p1_Ballot_Num()
              × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
           ⇒ pv11_p1_valid-p1b-message(Cmd;es;e1;f))
         ∧ ((header(e1) = ``pv11_p1 p2a`` ∈ Name)
           ⇒ has-es-info-type(es;e1;f;Id × pv11_p1_Ballot_Num() × ℤ × Cmd)
           ⇒ pv11_p1_valid-p2a-message(Cmd;es;e1;f))))
20. pv11_p1_valid-AcceptorState(Cmd;ldrs_uid;es;e;f)
21. pv11_p1_valid-LeaderState(Cmd;ldrs_uid;es;e;f)
22. ∀b:pv11_p1_Ballot_Num(). ∀start:E.  (start ≤loc e  ⇒ pv11_p1_valid-ScoutState(Cmd;accpts;b;start;es;e;f))
23. (header(e) = ``pv11_p1 adopted`` ∈ Name)
⇒ has-es-info-type(es;e;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
⇒ pv11_p1_valid-adopted-message(Cmd;es;e;f)
24. (header(e) = ``pv11_p1 p1b`` ∈ Name)
⇒

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

             ((e' < e) ∧ make-msg-interface(delay;loc(e);info(e)) ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e'))))

28. e' : E
29. delay : ℤ
30. (e' < e)
31. loc(e') ↓∈ ldrs
32. loc(e) ↓∈ accpts
33. delay = 0 ∈ ℤ
34. es-info-auth(es;e) = pv11_p1_init_active()
35. (fst(msgval(e))) = loc(e') ∈ Id
36. header(e') = ``pv11_p1 adopted`` ∈ Name
37. has-es-info-type(es;e';f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
38. (fst(msgval(e'))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'))) ∈ pv11_p1_Ballot_Num()

39. (<fst(snd(snd(msgval(e)))), snd(snd(snd(msgval(e))))> ↓∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;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)))), snd(snd(snd(msgval(e))))> ↓∈ snd(msgval(e'))

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

40. (fst(snd(msgval(e)))) = (fst(msgval(e'))) ∈ pv11_p1_Ballot_Num()
⊢ pv11_p1_valid-proposal(Cmd;es;e;snd(snd(<fst(msgval(e))
                                          , fst(snd(msgval(e)))
                                          , fst(snd(snd(msgval(e))))
                                          , snd(snd(snd(msgval(e))))>));f)

Latex:

Latex:
1. Cmd : \{T:Type| valueall-type(T)\} 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. accpts : bag(Id)@i 14. ldrs : bag(Id)@i 15. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 16. reps : bag(Id)@i 17. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 18. e : E@i \mvdash{} pv11\_p1\_valid-AcceptorState(Cmd;ldrs$_{uid}$;es;e;f) \mwedge{} pv11\_p1\_valid-LeaderState(Cmd;ldrs$_{uid}$;es;e;f) \mwedge{} (\mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}start:E. (start \mleq{}loc e {}\mRightarrow{} pv11\_p1\_valid-ScoutState(Cmd;accpts;b;start;es;e;f))) \mwedge{} ((header(e) = ``pv11\_p1 adopted``) {}\mRightarrow{} has-es-info-type(es;e;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) {}\mRightarrow{} pv11\_p1\_valid-adopted-message(Cmd;es;e;f)) \mwedge{} ((header(e) = ``pv11\_p1 p1b``) {}\mRightarrow{} has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) {}\mRightarrow{} pv11\_p1\_valid-p1b-message(Cmd;es;e;f)) \mwedge{} ((header(e) = ``pv11\_p1 p2a``) {}\mRightarrow{} has-es-info-type(es;e;f;Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) {}\mRightarrow{} pv11\_p1\_valid-p2a-message(Cmd;es;e;f)) By Latex: (MoveToConcl (-1) THEN CausalInd' THEN Auto THEN Try (UnfoldTopAb 0) THEN If (\mbackslash{}p. is\_term `pv11\_p1\_valid-proposals` (concl p)) (Try (UnfoldTopAb 0 THEN RWO "l\_all\_iff" 0 THEN Auto)) Id\mcdot{} THEN Try ((OnSomeHyp UseMessageConstraint THEN SqExRepD THEN SplitOrHyps THEN SqExRepD))\mcdot{}) Home Index