Nuprl Lemma : new_23_sig_progress-step8-v2

[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
        ((((coeff flrs) 1) ≤ ||bs||)
        ∧ (∀x∈bs.(loc(fst(snd(x))) loc(e) ∈ Id)
             ∧ e ≤loc fst(snd(x)) 
             ∧ <<<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)));bs)
        ∧ (∀e'∈[e, fst(snd(last(bs)))].(↑new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e';n;0))
               (e' ∈ map(λx.(fst(snd(x)));bs)))
        ∧ b_all(Id;[x∈reps|¬bbag-deq-member(IdDeq;x;faulty)];i.(i ∈ map(λi.(fst(i));bs)))
        ∧ (bs
          mapfilter(λe.<snd(msgval(e)), e, snd(fst(msgval(e)))>;
                      λe.new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e;n;0);
                      [e, fst(snd(last(bs)))])
          ∈ ((Id × E × Cmd) List)))))


Proof




Definitions occuring in Statement :  new_23_sig_vote_with_ballot: new_23_sig_vote_with_ballot(Cmd;notify;propose;f;es;e;n;r) 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) es-info-body: msgval(e) Message: Message(f) classrel: v ∈ X(e) event-ordering+: EO+(Info) es-interval: [e, e'] es-le: e ≤loc e'  es-locl: (e <loc e') es-loc: loc(e) es-E: E id-deq: IdDeq Id: Id deq: EqDecider(T) mapfilter: mapfilter(f;P;L) l_all: (∀x∈L.P[x]) last: last(L) l_member: (x ∈ l) map: map(f;as) length: ||as|| list: List int_upper: {i...} nat: vatype: ValueAllType bnot: ¬bb assert: b uall: [x:A]. B[x] pi1: fst(t) pi2: snd(t) le: A ≤ B 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) l-ordered: l-ordered(T;x,y.R[x; y];L) b_all: b_all(T;b;x.P[x]) bag-member: x ↓∈ bs bag-no-repeats: bag-no-repeats(T;bs) bag-size: #(bs) bag-filter: [x∈b|p[x]] bag: bag(T) set-sig-member: set-sig-member(s) set-sig: set-sig{i:l}(Item)
Lemmas :  new_23_sig_progress-step7 es-info-body_wf new_23_sig_vote_with_ballot-assert-type assert_wf new_23_sig_vote_with_ballot_wf subtype_top subtype_rel_product le-add-cancel zero-add add-associates add_functionality_wrt_le add-commutes add-swap minus-one-mul minus-add condition-implies-le vatype_wf name_wf subtype_rel_dep_function Id_wf length_wf decidable__lt le_wf false_wf int_upper_subtype_nat mul_bounds_1a top_wf subtype_rel_list non_null_iff_length last_wf pi2_wf pi1_wf_top es-interval_wf Message_wf event-ordering+_subtype es-E_wf mapfilter_wf es-causl_wf true_wf squash_wf es-loc_wf equal_wf and_wf l_before_wf es-causl_weakening not_wf es-blocl-iff es-locl_wf es-locl_irreflexivity es-le_weakening es-locl_transitivity2 es-blocl_wf sublist-if-no_repeats es-locl-antireflexive l-ordered-no_repeats l_before-es-interval l_before_mapfilter member-mapfilter l_member_wf l_all_iff subtract_wf decidable__equal_int le-add-cancel2 add-zero less-iff-le sq_stable__le not-le-2 decidable__le list_wf equal-wf-T-base assert_of_null member-es-interval es-le_wf l_all_fwd iff_weakening_equal less_than_wf select_wf es-le-self minus-minus le_antisymmetry_iff le-add-cancel-alt not-equal-2 lelt_wf subtract-is-less l-ordered-inst assert_of_band iff_weakening_uiff iff_transitivity assert-member-eclass int_subtype_base subtype_rel_transitivity es-info-type_wf assert_of_eq_int eq_int_wf eqtt_to_assert bool_wf member-eclass_wf ext-eq_weakening subtype_rel_weakening listp_wf nil_wf cons_wf cons_wf_listp base-noloc-classrel new_23_sig_vote'base_wf classrel_wf le_transitivity length_sublist new_23_sig_vote_with_ballot-assert-classrel es-le-loc valueall-type_wf l_all-mapfilter map_wf member-map es-interval-last last_member last-mapfilter3 bag-member-map list-subtype-bag subtype_rel_self sq_stable__l_member decidable__equal_Id bag-member-sq-list-member bag-member_wf b_all_wf new_23_sig_vote_with_ballot-if-classrel filter_wf5 null_wf3 null-filter null-mapfilter set_wf l-ordered_wf l_all_wf2 null_cons_lemma product_subtype_list length_of_nil_lemma null_nil_lemma list-cases

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
                ((((coeff  *  flrs)  +  1)  \mleq{}  ||bs||)
                \mwedge{}  (\mforall{}x\mmember{}bs.(loc(fst(snd(x)))  =  loc(e))
                          \mwedge{}  e  \mleq{}loc  fst(snd(x)) 
                          \mwedge{}  <<<n,  0>,  snd(snd(x))>,  fst(x)>  \mmember{}  new\_23\_sig\_vote'base(Cmd;notify;propose;f)(
                                                                                                  fst(snd(x))))
                \mwedge{}  l-ordered(Id  \mtimes{}  E  \mtimes{}  Cmd;x,y.(fst(snd(x))  <loc  fst(snd(y)));bs)
                \mwedge{}  (\mforall{}e'\mmember{}[e,  fst(snd(last(bs)))].
                            (\muparrow{}new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e';n;0))
                            {}\mRightarrow{}  (e'  \mmember{}  map(\mlambda{}x.(fst(snd(x)));bs)))
                \mwedge{}  b\_all(Id;[x\mmember{}reps|\mneg{}\msubb{}bag-deq-member(IdDeq;x;faulty)];i.(i  \mmember{}  map(\mlambda{}i.(fst(i));bs)))
                \mwedge{}  (bs
                    =  mapfilter(\mlambda{}e.<snd(msgval(e)),  e,  snd(fst(msgval(e)))>
                                            \mlambda{}e.new\_23\_sig\_vote\_with\_ballot(Cmd;notify;propose;f;es;e;n;0);
                                            [e,  fst(snd(last(bs)))])))))



Date html generated: 2015_07_23-PM-04_05_08
Last ObjectModification: 2015_07_16-AM-09_32_17

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