Step
*
1
1
3
of Lemma
new_23_sig_progress-step6
1. Cmd : {T:Type| valueall-type(T)}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : {2...}
6. flrs : ℕ
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig{i:l}(ℤ)
10. f : new_23_sig_headers_type{i:l}(Cmd;notify;propose)
11. (f propose) = (ℤ × Cmd) ∈ Type
12. (f notify) = (ℤ × Cmd) ∈ Type
13. (f ``new_23_sig decided``) = (ℤ × Cmd) ∈ Type
14. (f ``new_23_sig retry``) = (ℤ × ℤ × Cmd) ∈ Type
15. (f ``new_23_sig vote``) = (ℤ × ℤ × Cmd × Id) ∈ Type
16. f ∈ Name ─→ Type
17. es : EO+(Message(f))
18. e : E
19. n : ℤ
20. c : Cmd
21. faulty : bag(Id)
22. msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)@i
23. bag-no-repeats(Id;reps)@i
24. #(reps) = ((coeff * flrs) + flrs + 1) ∈ ℤ@i
25. loc(e) ↓∈ reps@i
26. ¬loc(e) ↓∈ faulty@i
27. <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)@i
28. ¬↑(set-sig-member(slots) n new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
29. bs : (Id × E × Cmd) List
30. bag-map(λi.(fst(i));bs) = [x∈reps|¬bbag-deq-member(IdDeq;x;faulty)] ∈ bag(Id)
31. (∀x∈bs.(loc(fst(snd(x))) = loc(e) ∈ Id)
∧ <<<n, 0>, snd(snd(x))>, fst(x)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(fst(snd(x))))
32. L' : (Id × E × Cmd) List
33. l-ordered(Id × E × Cmd;x,y.↑((λy.fst(snd(x)) ≤loc fst(snd(y))) y);L')
34. permutation(Id × E × Cmd;bs;L')
35. bag-map(λi.(fst(i));L') = [x∈reps|¬bbag-deq-member(IdDeq;x;faulty)] ∈ bag(Id)
36. (∀x∈L'.(loc(fst(snd(x))) = loc(e) ∈ Id)
∧ <<<n, 0>, snd(snd(x))>, fst(x)> ∈ new_23_sig_vote'base(Cmd;notify;propose;f)(fst(snd(x))))
⊢ l-ordered(Id × E × Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));L')
BY
{ (AllReduce
THEN All(\i.((RWO "l-ordered-is-sorted-by" i THENA Auto) THEN Unfold `sorted-by` i))⋅
THEN ParallelOp -4
THEN ParallelLast
THEN AllReduce
THEN AllPushDown
THEN D (-1)
THEN Auto
THEN Assert ⌈False⌉⋅
THEN Auto
THEN (Using [`x',⌈L'[j]⌉] (FLemma `l_all_fwd` [-5])⋅ THENA Auto)
THEN (Using [`x',⌈L'[i]⌉] (FLemma `l_all_fwd` [-6])⋅ THENA Auto)
THEN RepD
THEN (RWO "-5" (-3) THENA Auto)
THEN MaUseClassRel (-3)
THEN MaUseClassRel (-1)
THEN (RevHypSubst (-5) (-2) THENA Auto)
THEN SimpEqPairs
THEN (Subst ⌈fst(L'[i]) ~ bag-map(λi.(fst(i));L')[i]⌉ (-2)⋅
THENA (Auto THEN Unfold `bag-map` 0 THEN RWO "select-map" 0 THEN Reduce 0 THEN Auto)
)
THEN (Subst ⌈fst(L'[j]) ~ bag-map(λi.(fst(i));L')[j]⌉ (-2)⋅
THENA (Auto THEN Unfold `bag-map` 0 THEN RWO "select-map" 0 THEN Reduce 0 THEN Auto)
)
THEN (Assert ⌈no_repeats(Id;bag-map(λi.(fst(i));L'))⌉⋅
THENM (UnfoldTopAb (-1) THEN InstHyp [⌈i⌉;⌈j⌉] (-1)⋅ THEN Auto THEN RepUR ``bag-map`` 0 THEN RW ListC 0 THEN Auto)
)
THEN (Assert ⌈bag-no-repeats(Id;bag-map(λi.(fst(i));L'))⌉⋅
THENM (D (-1)
THEN All(Unfold `bag-map`)
THEN (Unhide THENA Auto)
THEN ExRepD
THEN Using [`bs',⌈L⌉] (BLemma `no_repeats-bag`)⋅
THEN Auto)
)
THEN (HypSubst (-16) 0 THENA Auto)
THEN BLemma `bag-no-repeats-filter`
THEN Auto) }
Latex:
Latex:
1. Cmd : \{T:Type| valueall-type(T)\}
2. eq : EqDecider(Cmd)
3. reps : bag(Id)
4. clients : bag(Id)
5. coeff : \{2...\}
6. flrs : \mBbbN{}
7. propose : Atom List
8. notify : Atom List
9. slots : set-sig\{i:l\}(\mBbbZ{})
10. f : new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose)
11. (f propose) = (\mBbbZ{} \mtimes{} Cmd)
12. (f notify) = (\mBbbZ{} \mtimes{} Cmd)
13. (f ``new\_23\_sig decided``) = (\mBbbZ{} \mtimes{} Cmd)
14. (f ``new\_23\_sig retry``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd)
15. (f ``new\_23\_sig vote``) = (\mBbbZ{} \mtimes{} \mBbbZ{} \mtimes{} Cmd \mtimes{} Id)
16. f \mmember{} Name {}\mrightarrow{} Type
17. es : EO+(Message(f))
18. e : E
19. n : \mBbbZ{}
20. c : Cmd
21. faulty : bag(Id)
22. msgs-interface-delivered-with-omissions(f;es;new\_23\_sig\_main();faulty;flrs;reps)@i
23. bag-no-repeats(Id;reps)@i
24. \#(reps) = ((coeff * flrs) + flrs + 1)@i
25. loc(e) \mdownarrow{}\mmember{} reps@i
26. \mneg{}loc(e) \mdownarrow{}\mmember{} faulty@i
27. <n, c> \mmember{} new\_23\_sig\_Proposal(Cmd;notify;propose;f)(e)@i
28. \mneg{}\muparrow{}(set-sig-member(slots) n new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e))@i
29. bs : (Id \mtimes{} E \mtimes{} Cmd) List
30. bag-map(\mlambda{}i.(fst(i));bs) = [x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)]
31. (\mforall{}x\mmember{}bs.(loc(fst(snd(x))) = loc(e))
\mwedge{} <<<n, 0>, snd(snd(x))>, fst(x)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(fst(snd(x))))
32. L' : (Id \mtimes{} E \mtimes{} Cmd) List
33. l-ordered(Id \mtimes{} E \mtimes{} Cmd;x,y.\muparrow{}((\mlambda{}y.fst(snd(x)) \mleq{}loc fst(snd(y))) y);L')
34. permutation(Id \mtimes{} E \mtimes{} Cmd;bs;L')
35. bag-map(\mlambda{}i.(fst(i));L') = [x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)]
36. (\mforall{}x\mmember{}L'.(loc(fst(snd(x))) = loc(e))
\mwedge{} <<<n, 0>, snd(snd(x))>, fst(x)> \mmember{} new\_23\_sig\_vote'base(Cmd;notify;propose;f)(fst(snd(x))))
\mvdash{} l-ordered(Id \mtimes{} E \mtimes{} Cmd;x,y.(fst(snd(x)) <loc fst(snd(y)));L')
By
Latex:
(AllReduce
THEN All(\mbackslash{}i.((RWO "l-ordered-is-sorted-by" i THENA Auto) THEN Unfold `sorted-by` i))\mcdot{}
THEN ParallelOp -4
THEN ParallelLast
THEN AllReduce
THEN AllPushDown
THEN D (-1)
THEN Auto
THEN Assert \mkleeneopen{}False\mkleeneclose{}\mcdot{}
THEN Auto
THEN (Using [`x',\mkleeneopen{}L'[j]\mkleeneclose{}] (FLemma `l\_all\_fwd` [-5])\mcdot{} THENA Auto)
THEN (Using [`x',\mkleeneopen{}L'[i]\mkleeneclose{}] (FLemma `l\_all\_fwd` [-6])\mcdot{} THENA Auto)
THEN RepD
THEN (RWO "-5" (-3) THENA Auto)
THEN MaUseClassRel (-3)
THEN MaUseClassRel (-1)
THEN (RevHypSubst (-5) (-2) THENA Auto)
THEN SimpEqPairs
THEN (Subst \mkleeneopen{}fst(L'[i]) \msim{} bag-map(\mlambda{}i.(fst(i));L')[i]\mkleeneclose{} (-2)\mcdot{}
THENA (Auto THEN Unfold `bag-map` 0 THEN RWO "select-map" 0 THEN Reduce 0 THEN Auto)
)
THEN (Subst \mkleeneopen{}fst(L'[j]) \msim{} bag-map(\mlambda{}i.(fst(i));L')[j]\mkleeneclose{} (-2)\mcdot{}
THENA (Auto THEN Unfold `bag-map` 0 THEN RWO "select-map" 0 THEN Reduce 0 THEN Auto)
)
THEN (Assert \mkleeneopen{}no\_repeats(Id;bag-map(\mlambda{}i.(fst(i));L'))\mkleeneclose{}\mcdot{}
THENM (UnfoldTopAb (-1)
THEN InstHyp [\mkleeneopen{}i\mkleeneclose{};\mkleeneopen{}j\mkleeneclose{}] (-1)\mcdot{}
THEN Auto
THEN RepUR ``bag-map`` 0
THEN RW ListC 0
THEN Auto)
)
THEN (Assert \mkleeneopen{}bag-no-repeats(Id;bag-map(\mlambda{}i.(fst(i));L'))\mkleeneclose{}\mcdot{}
THENM (D (-1)
THEN All(Unfold `bag-map`)
THEN (Unhide THENA Auto)
THEN ExRepD
THEN Using [`bs',\mkleeneopen{}L\mkleeneclose{}] (BLemma `no\_repeats-bag`)\mcdot{}
THEN Auto)
)
THEN (HypSubst (-16) 0 THENA Auto)
THEN BLemma `bag-no-repeats-filter`
THEN Auto)
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