Proof of Nuprl Lemma : pv11_p1_A4

Step * 1 2 4 1 1 1 1 4 1 1 2 1 1 of Lemma pv11_p1_A4_C1

1. Cmd : Type@i'
2. valueall-type(Cmd)@i'
3. f : pv11_p1_headers_type{i:l}(Cmd)@i'
4. es : EO+(Message(f))@i'
5. (f [decision]) = (ℤ × Cmd) ∈ Type
6. (f [propose]) = (ℤ × Cmd) ∈ Type

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

8. (f ``pv11_p1 preempted``) = pv11_p1_Ballot_Num() ∈ Type
9. (f ``pv11_p1 p2b``) = (Id × pv11_p1_Ballot_Num() × ℤ × pv11_p1_Ballot_Num()) ∈ Type
10. (f ``pv11_p1 p2a``) = (Id × pv11_p1_Ballot_Num() × ℤ × Cmd) ∈ Type
11. (f ``pv11_p1 p1b``)
= (Id × pv11_p1_Ballot_Num() × pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
∈ Type
12. (f ``pv11_p1 p1a``) = (Id × pv11_p1_Ballot_Num()) ∈ Type
13. f ∈ Name ⟶ Type
14. e1 : E@i
15. accpts : bag(Id)@i
16. ldrs : bag(Id)@i
17. ldrs_uid : Id ⟶ ℤ@i
18. reps : bag(Id)@i
19. Inj(Id;ℤ;ldrs_uid)@i
20. pv11_p1_message-constraint{paxos-v11-part1.esh:o}(Cmd; accpts; ldrs; ldrs_uid; reps; f; es)@i
21. e2 : E@i
22. ∀e3:E
((e3 < e2)
⇒

 (∀e1:E. ∀bnum1,bnum2:pv11_p1_Ballot_Num(). ∀accepted1,accepted2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.

          ∀i1,i2,l:Id. ∀b:pv11_p1_Ballot_Num(). ∀s:ℤ. ∀p1,p2:Cmd.
            (((<bnum1, accepted1> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e1)
            ∧ <bnum2, accepted2> ∈ pv11_p1_AcceptorState(Cmd;ldrs_uid;f)(e3)
            ∧ (<b, s, p1> ∈ accepted1)
            ∧ (<b, s, p2> ∈ accepted2))

            ∨ (pv11_p1_p2a'send(Cmd;f) i1 <l, b, s, p1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)

              ∧ pv11_p1_p2a'send(Cmd;f) i2 <l, b, s, p2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e3)))

            ⇒ (p1 = p2 ∈ Cmd))))
23. bnum1 : pv11_p1_Ballot_Num()@i
24. bnum2 : pv11_p1_Ballot_Num()@i
25. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. accepted2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
27. i1 : Id@i
28. i2 : Id@i
29. l : Id@i
30. b : pv11_p1_Ballot_Num()@i
31. s : ℤ@i
32. p1 : Cmd@i
33. p2 : Cmd@i
34. loc(e1) ↓∈ ldrs
35. i1 ↓∈ accpts
36. 0 = 0 ∈ ℤ
37. ff = pv11_p1_init_active()
38. l = loc(e1) ∈ Id
39. header(e1) = ``pv11_p1 adopted`` ∈ Name
40. has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
41. (fst(msgval(e1))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))) ∈ pv11_p1_Ballot_Num()
42. b = (fst(msgval(e1))) ∈ pv11_p1_Ballot_Num()
43. loc(e2) ↓∈ ldrs
44. i2 ↓∈ accpts
45. 0 = 0 ∈ ℤ
46. ff = pv11_p1_init_active()
47. l = loc(e2) ∈ Id
48. header(e2) = ``pv11_p1 adopted`` ∈ Name
49. has-es-info-type(es;e2;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
50. (fst(msgval(e1))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2))) ∈ pv11_p1_Ballot_Num()
51. b = (fst(msgval(e1))) ∈ pv11_p1_Ballot_Num()
52. loc(e1) = loc(e2) ∈ Id
53. (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))
= (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2)))
∈ pv11_p1_Ballot_Num()
54. <b, snd(msgval(e1))> ∈ pv11_p1_adopted'base(Cmd;f)(e1)
55. <b, snd(msgval(e1))> ∈ pv11_p1_adopted'base(Cmd;f)(e2)
56. (snd(msgval(e1))) = (snd(msgval(e1))) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
57. msgval(e1) = msgval(e2) ∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
58. (<s, p1> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))))
59. ¬(∃p2:Cmd
       ∃b:pv11_p1_Ballot_Num()
        ((<b, s, p2> ∈ snd(msgval(e1)))
        ∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.
             ((<b', s, c'> ∈ snd(msgval(e1))) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))
60. (<s, p2> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2))))
61. ¬(∃p2:Cmd
       ∃b:pv11_p1_Ballot_Num()
        ((<b, s, p2> ∈ snd(msgval(e1)))
        ∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.
             ((<b', s, c'> ∈ snd(msgval(e1))) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))
62. e3 : E
63. (e3 <loc e1)
64. ∀e'':E. ((e3 <loc e'') ⇒ e'' ≤loc e1  ⇒ (<s, p1> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'')))))
65. p3 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
66. b2 : pv11_p1_Ballot_Num()
67. <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e3)), p3> ∈ pv11_p1_adopted'base(Cmd;f)(e3)
68. (<b2, s, p1> ∈ p3)
69. ∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.  ((<b', s, c'> ∈ p3) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b2)))
70. e' : E
71. (e' <loc e2)
72. ∀e'':E. ((e' <loc e'') ⇒ e'' ≤loc e2  ⇒ (<s, p2> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'')))))
73. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
74. b1 : pv11_p1_Ballot_Num()
75. <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')), pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e')
76. (<b1, s, p2> ∈ pvals)
77. ∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.  ((<b', s, c'> ∈ pvals) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b1)))
78. (e3 <loc e')
79. (e' <loc e1)
80. e : E
81. ¬↑first(e)
82. e' = pred(e) ∈ E
83. ¬↑first(e)
84. e' = pred(e) ∈ E
⊢ (<s, p1> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))))
BY
{ (Assert e ≤loc e1  BY
         ((HypSubst' (-1) (-6) THENA Auto) THEN FLemma `es-pred-locl-implies-le` [-1] THEN Auto)) }

1
1. Cmd : Type@i'
2. valueall-type(Cmd)@i'
3. f : pv11_p1_headers_type{i:l}(Cmd)@i'
4. es : EO+(Message(f))@i'
5. (f [decision]) = (ℤ × Cmd) ∈ Type
6. (f [propose]) = (ℤ × Cmd) ∈ Type

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

 (∀e1:E. ∀bnum1,bnum2:pv11_p1_Ballot_Num(). ∀accepted1,accepted2:(pv11_p1_Ballot_Num() × ℤ × Cmd) List.

            ∨ (pv11_p1_p2a'send(Cmd;f) i1 <l, b, s, p1> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e1)

              ∧ pv11_p1_p2a'send(Cmd;f) i2 <l, b, s, p2> ∈ pv11_p1_main(Cmd;accpts;ldrs;ldrs_uid;reps;f)(e3)))

            ⇒ (p1 = p2 ∈ Cmd))))
23. bnum1 : pv11_p1_Ballot_Num()@i
24. bnum2 : pv11_p1_Ballot_Num()@i
25. accepted1 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
26. accepted2 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List@i
27. i1 : Id@i
28. i2 : Id@i
29. l : Id@i
30. b : pv11_p1_Ballot_Num()@i
31. s : ℤ@i
32. p1 : Cmd@i
33. p2 : Cmd@i
34. loc(e1) ↓∈ ldrs
35. i1 ↓∈ accpts
36. 0 = 0 ∈ ℤ
37. ff = pv11_p1_init_active()
38. l = loc(e1) ∈ Id
39. header(e1) = ``pv11_p1 adopted`` ∈ Name
40. has-es-info-type(es;e1;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
41. (fst(msgval(e1))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))) ∈ pv11_p1_Ballot_Num()
42. b = (fst(msgval(e1))) ∈ pv11_p1_Ballot_Num()
43. loc(e2) ↓∈ ldrs
44. i2 ↓∈ accpts
45. 0 = 0 ∈ ℤ
46. ff = pv11_p1_init_active()
47. l = loc(e2) ∈ Id
48. header(e2) = ``pv11_p1 adopted`` ∈ Name
49. has-es-info-type(es;e2;f;pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
50. (fst(msgval(e1))) = (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2))) ∈ pv11_p1_Ballot_Num()
51. b = (fst(msgval(e1))) ∈ pv11_p1_Ballot_Num()
52. loc(e1) = loc(e2) ∈ Id
53. (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1)))
= (fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2)))
∈ pv11_p1_Ballot_Num()
54. <b, snd(msgval(e1))> ∈ pv11_p1_adopted'base(Cmd;f)(e1)
55. <b, snd(msgval(e1))> ∈ pv11_p1_adopted'base(Cmd;f)(e2)
56. (snd(msgval(e1))) = (snd(msgval(e1))) ∈ ((pv11_p1_Ballot_Num() × ℤ × Cmd) List)
57. msgval(e1) = msgval(e2) ∈ (pv11_p1_Ballot_Num() × ((pv11_p1_Ballot_Num() × ℤ × Cmd) List))
58. (<s, p1> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e1))))
59. ¬(∃p2:Cmd
       ∃b:pv11_p1_Ballot_Num()
        ((<b, s, p2> ∈ snd(msgval(e1)))
        ∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.
             ((<b', s, c'> ∈ snd(msgval(e1))) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))
60. (<s, p2> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e2))))
61. ¬(∃p2:Cmd
       ∃b:pv11_p1_Ballot_Num()
        ((<b, s, p2> ∈ snd(msgval(e1)))
        ∧ (∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.
             ((<b', s, c'> ∈ snd(msgval(e1))) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b))))))
62. e3 : E
63. (e3 <loc e1)
64. ∀e'':E. ((e3 <loc e'') ⇒ e'' ≤loc e1  ⇒ (<s, p1> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'')))))
65. p3 : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
66. b2 : pv11_p1_Ballot_Num()
67. <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e3)), p3> ∈ pv11_p1_adopted'base(Cmd;f)(e3)
68. (<b2, s, p1> ∈ p3)
69. ∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.  ((<b', s, c'> ∈ p3) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b2)))
70. e' : E
71. (e' <loc e2)
72. ∀e'':E. ((e' <loc e'') ⇒ e'' ≤loc e2  ⇒ (<s, p2> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e'')))))
73. pvals : (pv11_p1_Ballot_Num() × ℤ × Cmd) List
74. b1 : pv11_p1_Ballot_Num()
75. <fst(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e')), pvals> ∈ pv11_p1_adopted'base(Cmd;f)(e')
76. (<b1, s, p2> ∈ pvals)
77. ∀b':pv11_p1_Ballot_Num(). ∀c':Cmd.  ((<b', s, c'> ∈ pvals) ⇒ (↑(pv11_p1_leq_bnum(ldrs_uid) b' b1)))
78. (e3 <loc e')
79. (e' <loc e1)
80. e : E
81. ¬↑first(e)
82. e' = pred(e) ∈ E
83. ¬↑first(e)
84. e' = pred(e) ∈ E
85. e ≤loc e1
⊢ (<s, p1> ∈ snd(snd(pv11_p1_LeaderStateFun(Cmd;ldrs_uid;f;es;e))))

Latex:

Latex:
1. Cmd : Type@i' 2. valueall-type(Cmd)@i' 3. f : pv11\_p1\_headers\_type\{i:l\}(Cmd)@i' 4. es : EO+(Message(f))@i' 5. (f [decision]) = (\mBbbZ{} \mtimes{} Cmd) 6. (f [propose]) = (\mBbbZ{} \mtimes{} Cmd) 7. (f ``pv11\_p1 adopted``) = (pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 8. (f ``pv11\_p1 preempted``) = pv11\_p1\_Ballot\_Num() 9. (f ``pv11\_p1 p2b``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} pv11\_p1\_Ballot\_Num()) 10. (f ``pv11\_p1 p2a``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) 11. (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)) 12. (f ``pv11\_p1 p1a``) = (Id \mtimes{} pv11\_p1\_Ballot\_Num()) 13. f \mmember{} Name {}\mrightarrow{} Type 14. e1 : E@i 15. accpts : bag(Id)@i 16. ldrs : bag(Id)@i 17. ldrs$_{uid}$ : Id {}\mrightarrow{} \mBbbZ{}@i 18. reps : bag(Id)@i 19. Inj(Id;\mBbbZ{};ldrs$_{uid}$)@i 20. pv11\_p1\_message-constraint\{paxos-v11-part1.esh:o\}(Cmd; accpts; ldrs; ldrs$_{uid}\000C$; reps; f; es)@i 21. e2 : E@i 22. \mforall{}e3:E ((e3 < e2) {}\mRightarrow{} (\mforall{}e1:E. \mforall{}bnum1,bnum2:pv11\_p1\_Ballot\_Num(). \mforall{}accepted1,accepted2:(pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List. \mforall{}i1,i2,l:Id. \mforall{}b:pv11\_p1\_Ballot\_Num(). \mforall{}s:\mBbbZ{}. \mforall{}p1,p2:Cmd. (((<bnum1, accepted1> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e1\000C) \mwedge{} <bnum2, accepted2> \mmember{} pv11\_p1\_AcceptorState(Cmd;ldrs$_{uid}$;f)(e3) \mwedge{} (<b, s, p1> \mmember{} accepted1) \mwedge{} (<b, s, p2> \mmember{} accepted2)) \mvee{} (pv11\_p1\_p2a'send(Cmd;f) i1 <l, b, s, p1> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$\mbackslash{}ff5\000Cf{uid}$;reps;f)(e1) \mwedge{} pv11\_p1\_p2a'send(Cmd;f) i2 <l, b, s, p2> \mmember{} pv11\_p1\_main(Cmd;accpts;ldrs;ldrs$\mbackslash{}ff\000C5f{uid}$;reps;f)(e3))) {}\mRightarrow{} (p1 = p2)))) 23. bnum1 : pv11\_p1\_Ballot\_Num()@i 24. bnum2 : pv11\_p1\_Ballot\_Num()@i 25. accepted1 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 26. accepted2 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List@i 27. i1 : Id@i 28. i2 : Id@i 29. l : Id@i 30. b : pv11\_p1\_Ballot\_Num()@i 31. s : \mBbbZ{}@i 32. p1 : Cmd@i 33. p2 : Cmd@i 34. loc(e1) \mdownarrow{}\mmember{} ldrs 35. i1 \mdownarrow{}\mmember{} accpts 36. 0 = 0 37. ff = pv11\_p1\_init\_active() 38. l = loc(e1) 39. header(e1) = ``pv11\_p1 adopted`` 40. has-es-info-type(es;e1;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 41. (fst(msgval(e1))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1))) 42. b = (fst(msgval(e1))) 43. loc(e2) \mdownarrow{}\mmember{} ldrs 44. i2 \mdownarrow{}\mmember{} accpts 45. 0 = 0 46. ff = pv11\_p1\_init\_active() 47. l = loc(e2) 48. header(e2) = ``pv11\_p1 adopted`` 49. has-es-info-type(es;e2;f;pv11\_p1\_Ballot\_Num() \mtimes{} ((pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List)) 50. (fst(msgval(e1))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e2))) 51. b = (fst(msgval(e1))) 52. loc(e1) = loc(e2) 53. (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1))) = (fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e2))) 54. <b, snd(msgval(e1))> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e1) 55. <b, snd(msgval(e1))> \mmember{} pv11\_p1\_adopted'base(Cmd;f)(e2) 56. (snd(msgval(e1))) = (snd(msgval(e1))) 57. msgval(e1) = msgval(e2) 58. (<s, p1> \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e1)))) 59. \mneg{}(\mexists{}p2:Cmd \mexists{}b:pv11\_p1\_Ballot\_Num() ((<b, s, p2> \mmember{} snd(msgval(e1))) \mwedge{} (\mforall{}b':pv11\_p1\_Ballot\_Num(). \mforall{}c':Cmd. ((<b', s, c'> \mmember{} snd(msgval(e1))) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}\mbackslash{}ff2\000C4) b' b)))))) 60. (<s, p2> \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e2)))) 61. \mneg{}(\mexists{}p2:Cmd \mexists{}b:pv11\_p1\_Ballot\_Num() ((<b, s, p2> \mmember{} snd(msgval(e1))) \mwedge{} (\mforall{}b':pv11\_p1\_Ballot\_Num(). \mforall{}c':Cmd. ((<b', s, c'> \mmember{} snd(msgval(e1))) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}\mbackslash{}ff2\000C4) b' b)))))) 62. e3 : E 63. (e3 <loc e1) 64. \mforall{}e'':E ((e3 <loc e'') {}\mRightarrow{} e'' \mleq{}loc e1 {}\mRightarrow{} (<s, p1> \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e''))))\000C) 65. p3 : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List 66. b2 : pv11\_p1\_Ballot\_Num() 67. <fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e3)), p3> \mmember{} pv11\_p1\_adopte\000Cd'base(Cmd;f)(e3) 68. (<b2, s, p1> \mmember{} p3) 69. \mforall{}b':pv11\_p1\_Ballot\_Num(). \mforall{}c':Cmd. ((<b', s, c'> \mmember{} p3) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_\mbackslash{}ff\000C7buid}$) b' b2))) 70. e' : E 71. (e' <loc e2) 72. \mforall{}e'':E ((e' <loc e'') {}\mRightarrow{} e'' \mleq{}loc e2 {}\mRightarrow{} (<s, p2> \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e''))))\000C) 73. pvals : (pv11\_p1\_Ballot\_Num() \mtimes{} \mBbbZ{} \mtimes{} Cmd) List 74. b1 : pv11\_p1\_Ballot\_Num() 75. <fst(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e')), pvals> \mmember{} pv11\_p1\_ado\000Cpted'base(Cmd;f)(e') 76. (<b1, s, p2> \mmember{} pvals) 77. \mforall{}b':pv11\_p1\_Ballot\_Num(). \mforall{}c':Cmd. ((<b', s, c'> \mmember{} pvals) {}\mRightarrow{} (\muparrow{}(pv11\_p1\_leq\_bnum(ldrs$_{uid}$) b' b1))) 78. (e3 <loc e') 79. (e' <loc e1) 80. e : E 81. \mneg{}\muparrow{}first(e) 82. e' = pred(e) 83. \mneg{}\muparrow{}first(e) 84. e' = pred(e) \mvdash{} (<s, p1> \mmember{} snd(snd(pv11\_p1\_LeaderStateFun(Cmd;ldrs$_{uid}$;f;es;e)))) By Latex: (Assert e \mleq{}loc e1 BY ((HypSubst' (-1) (-6) THENA Auto) THEN FLemma `es-pred-locl-implies-le` [-1] THEN Auto)) Home Index