Nuprl Lemma : new_23_sig_progress-step5

[Cmd:ValueAllType]. ∀[eq:EqDecider(Cmd)]. ∀[reps,clients:bag(Id)]. ∀[coeff:{2...}]. ∀[flrs:ℕ].
[propose,notify:Atom List]. ∀[slots:set-sig{i:l}(ℤ)]. ∀[f:new_23_sig_headers_type{i:l}(Cmd;notify;propose)].
[es:EO+(Message(f))]. ∀[e:E]. ∀[n:ℤ]. ∀[c:Cmd]. ∀[faulty:bag(Id)].
  (msgs-interface-delivered-with-omissions(f;es;new_23_sig_main();faulty;flrs;reps)
   bag-no-repeats(Id;reps)
   (#(reps) ((coeff flrs) flrs 1) ∈ ℤ)
   loc(e) ↓∈ reps
   loc(e) ↓∈ faulty)
   <n, c> ∈ new_23_sig_Proposal(Cmd;notify;propose;f)(e)
   (¬↑(set-sig-member(slots) new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e)))
   (↓∃bs:(Id × E × Cmd) List
        ((bag-map(λi.(fst(i));bs) [x∈reps|¬bbag-deq-member(IdDeq;x;faulty)] ∈ bag(Id))
        ∧ (∀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)))))))


Proof




Definitions occuring in Statement :  new_23_sig_main: new_23_sig_main() new_23_sig_ReplicaStateFun: new_23_sig_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e) new_23_sig_Proposal: new_23_sig_Proposal(Cmd;notify;propose;f) new_23_sig_vote'base: new_23_sig_vote'base(Cmd;notify;propose;f) new_23_sig_headers_type: new_23_sig_headers_type{i:l}(Cmd;notify;propose) msgs-interface-delivered-with-omissions: msgs-interface-delivered-with-omissions(f;es;X;faulty;failures;ids) Message: Message(f) classrel: v ∈ X(e) event-ordering+: EO+(Info) es-loc: loc(e) es-E: E id-deq: IdDeq Id: Id deq: EqDecider(T) l_all: (∀x∈L.P[x]) list: List int_upper: {i...} nat: vatype: ValueAllType bnot: ¬bb assert: b uall: [x:A]. B[x] pi1: fst(t) pi2: snd(t) exists: x:A. B[x] not: ¬A squash: T implies:  Q and: P ∧ Q apply: a lambda: λx.A[x] pair: <a, b> product: x:A × B[x] multiply: m add: m natural_number: $n int: atom: Atom equal: t ∈ T bag-deq-member: bag-deq-member(eq;x;b) bag-member: x ↓∈ bs bag-no-repeats: bag-no-repeats(T;bs) bag-size: #(bs) bag-filter: [x∈b|p[x]] bag-map: bag-map(f;bs) bag: bag(T) set-sig-member: set-sig-member(s) set-sig: set-sig{i:l}(Item)
Lemmas :  sq_stable__and equal_wf vatype_wf cons_wf_listp cons_wf nil_wf listp_wf sq_stable__equal squash_wf int_seg_wf length_wf name_wf new_23_sig_headers_wf l_all_iff l_member_wf new_23_sig_headers_fun_wf cons_member equal-wf-base list_subtype_base atom_subtype_base iff_weakening_equal name_eq_wf bool_wf eqtt_to_assert assert-name_eq sq_stable__no_repeats true_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot not_wf assert_wf set-sig-member_wf new_23_sig_ReplicaStateFun_wf classrel_wf new_23_sig_Proposal_wf bag-member_wf es-loc_wf bag-size_wf nat_wf bag-no-repeats_wf msgs-interface-delivered-with-omissions_wf new_23_sig_main_wf es-E_wf event-ordering+_subtype event-ordering+_wf Message_wf subtype_rel_dep_function new_23_sig_headers_type_wf set-sig_wf list_wf int_upper_wf bag_wf Id_wf deq_wf set_wf valueall-type_wf new_23_sig_progress-step4 b_all-squash-exists-list bag-filter_wf bnot_wf bag-deq-member_wf id-deq_wf subtype_rel_bag exists_wf new_23_sig_vote'base_wf l_all-squash-exists-list map_wf bag-map_wf list-subtype-bag subtype_rel_self l_all_wf2 map-map map_wf_bag member-map and_wf pi2_wf pi1_wf_top subtype_rel_product top_wf subtype_top sq_stable__classrel eclass_wf

Latex:
\mforall{}[Cmd:ValueAllType].  \mforall{}[eq:EqDecider(Cmd)].  \mforall{}[reps,clients:bag(Id)].  \mforall{}[coeff:\{2...\}].  \mforall{}[flrs:\mBbbN{}].
\mforall{}[propose,notify:Atom  List].  \mforall{}[slots:set-sig\{i:l\}(\mBbbZ{})].
\mforall{}[f:new\_23\_sig\_headers\_type\{i:l\}(Cmd;notify;propose)].  \mforall{}[es:EO+(Message(f))].  \mforall{}[e:E].  \mforall{}[n:\mBbbZ{}].
\mforall{}[c:Cmd].  \mforall{}[faulty:bag(Id)].
    (msgs-interface-delivered-with-omissions(f;es;new\_23\_sig\_main();faulty;flrs;reps)
    {}\mRightarrow{}  bag-no-repeats(Id;reps)
    {}\mRightarrow{}  (\#(reps)  =  ((coeff  *  flrs)  +  flrs  +  1))
    {}\mRightarrow{}  loc(e)  \mdownarrow{}\mmember{}  reps
    {}\mRightarrow{}  (\mneg{}loc(e)  \mdownarrow{}\mmember{}  faulty)
    {}\mRightarrow{}  <n,  c>  \mmember{}  new\_23\_sig\_Proposal(Cmd;notify;propose;f)(e)
    {}\mRightarrow{}  (\mneg{}\muparrow{}(set-sig-member(slots)  n  new\_23\_sig\_ReplicaStateFun(Cmd;notify;propose;slots;f;es;e)))
    {}\mRightarrow{}  (\mdownarrow{}\mexists{}bs:(Id  \mtimes{}  E  \mtimes{}  Cmd)  List
                ((bag-map(\mlambda{}i.(fst(i));bs)  =  [x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)])
                \mwedge{}  (\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)))))))



Date html generated: 2015_07_23-PM-04_03_53
Last ObjectModification: 2015_02_04-PM-03_53_15

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