es:ES, Cmd:Type, In:AbsInterface(Cmd), isupdate:(Cmd), Sys, Out:AbsInterface(Top).
(E(In) r E(Sys))
 (f:sys-antecedent(es;Sys).
 (e:E(In). f(e) = e  E)
  (chain:(E(Sys)(Id List)).
  chain-consistent(f;chain)
   (x, y:E(Sys).
   x is f*(y)
    (loc(x) = loc(y)  Id)
    (b:E(Sys). (x <loc b)  ((loc(f(b)) = loc(b)  Id))  (y <loc b))))) 

latex

Up	abstract chain replication
Definitions	s = t, t  T, ES, Type, AbsInterface(A), , x:AB(x), sys-antecedent(es;Sys), E(X), <a, b>, E, x:A. B(x), type List, chain-consistent(f;chain), x:A  B(x), b, left + right, {x:A| B(x)} , x:A. B(x), Id, P  Q, a < b, A, P  Q, P & Q, hd(l), L1  L2, e loc e' , adjacent(T;L;x;y), (x  l), no_repeats(T;l), Top, f(a), e c e', a:A fp B(a), strong-subtype(A;B), , !Void(), False, y is f*(x), y=f*(x) via L, (e <loc e'), let x,y = A in B(x;y), t.1, loc(e), e < e', (e < e'), Atom$n, X(e), e(e1,e2].P(e), @e(xv), (last change to x before e), A c B, pred(e), IdLnk, Knd, a = b, P  Q, P  Q, locl(a), x.A(x), {T}, x. t(x), x,y. t(x;y), EqDecider(T), Unit, EOrderAxioms(E; pred?; info), EState(T), kindcase(k; a.f(a); l,t.g(l;t) ), Msg(M), , , val-axiom(E;V;M;info;pred?;init;Trans;Choose;Send;val;time), r  s, constant_function(f;A;B), loc(e), kind(e), SWellFounded(R(x;y)), pred!(e;e'), pred(e), first(e), s ~ t, SQType(T), e  X, Dec(P), case b of inl(x) => s(x) | inr(y) => t(y), if b then t else f fi , T, True, , A  B, Atom, {i..j}, x,y:A//B(x;y), b | a, a ~ b, |p|, a  b, a <p b, |g|, a < b, x f y, |r|, xL. P(x), (xL.P(x)), , r < s, q-rel(r;x), Outcome, dstype(TypeNames; d; a), l_disjoint(T;l1;l2), MaName, e<e'.P(e), ee'.P(e), e<e'. P(e), ee'.P(e), e[e1,e2).P(e), e[e1,e2).P(e), e[e1,e2].P(e), e[e1,e2].P(e), e(e1,e2].P(e), f**(e), x <<= y, x << y, x before y  l, increasing(f;k), ||as||, #$n, last(L), kind(e), first(e), source(l), destination(l), es-init(es;e), isrcv(e), es-first-from(es;e;l;tg), isrcv(k), lastchange(x;e), ff, inr x , tt, inl x , SqStable(P), a =!x:T. Q(x), InvFuns(A;B;f;g), Inj(A;B;f), IsEqFun(T;eq), Refl(T;x,y.E(x;y)), Sym(T;x,y.E(x;y)), Trans(T;x,y.E(x;y)), AntiSym(T;x,y.R(x;y)), Connex(T;x,y.R(x;y)), CoPrime(a,b), Ident(T;op;id), Assoc(T;op), Comm(T;op), Inverse(T;op;id;inv), BiLinear(T;pl;tm), IsBilinear(A;B;C;+a;+b;+c;f), IsAction(A;x;e;S;f), Dist1op2opLR(A;1op;2op), fun_thru_1op(A;B;opa;opb;f), FunThru2op(A;B;opa;opb;f), Cancel(T;S;op), monot(T;x,y.R(x;y);f), IsMonoid(T;op;id), IsGroup(T;op;id;inv), IsMonHom{M1,M2}(f), a  b, IsIntegDom(r), IsPrimeIdeal(R;P), f  g, f(x)?z
Lemmas	es-locl transitivity2, es-locl irreflexivity, es-le-not-locl, sq stable from decidable, decidable assert, btrue wf, bfalse wf, es-isrcv-loc, es-le-loc, true wf, squash wf, es-loc-pred, es-first wf, adjacent wf, es-le wf, and functionality wrt iff, es-locl-iff, sublist wf, decidable es-locl, chain-order-total2, chain-order-antisymmetric, Id sq, decidable equal Id, chain-path-ordered, fun-connected-step, decidable wf, es-is-interface wf, guard wf, loc wf, pred! wf, strongwellfounded wf, es-causle wf, es-interface wf, top wf, constant function wf, bool wf, qle wf, cless wf, val-axiom wf, rationals wf, nat wf, Msg wf, kindcase wf, Knd wf, EState wf, EOrderAxioms wf, IdLnk wf, unit wf, deq wf, event system wf, chain-consistent wf, fun-connected-induction, all functionality wrt iff, implies functionality wrt iff, not functionality wrt iff, iff wf, rev implies wf, assert-eq-id, eq id wf, false wf, not wf, Id wf, es-loc wf, es-causl wf, es-locl wf, fun-path wf, fun-connected wf, es-E-interface-subtype rel, assert wf, es-E wf, member wf, es-E-interface wf, sys-antecedent wf, subtype rel wf