neuper@42289
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(* Title: partial fraction decomposition
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neuper@42289
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Author: Jan Rocnik
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(c) due to copyright terms
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*)
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"--------------------------------------------------------";
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"table of contents --------------------------------------";
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"--------------------------------------------------------";
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neuper@42305
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"----------- why helpless here ? ------------------------";
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neuper@42310
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"----------- why not nxt = Model_Problem here ? ---------";
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neuper@42359
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"----------- fun factors_from_solution ------------------";
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neuper@42359
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"----------- Logic.unvarify_global ----------------------";
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neuper@42359
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"----------- eval_drop_questionmarks --------------------";
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neuper@42376
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"----------- = me for met_partial_fraction --------------";
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neuper@42386
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"----------- progr.vers.2: check erls for multiply_ansatz";
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neuper@42386
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"----------- progr.vers.2: improve program --------------";
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"--------------------------------------------------------";
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"--------------------------------------------------------";
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"--------------------------------------------------------";
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"----------- why helpless here ? ------------------------";
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"----------- why helpless here ? ------------------------";
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"----------- why helpless here ? ------------------------";
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val fmz = ["filterExpression (X z = 3 / (z - 1/4 + -1/8 * (1/(z::real))))",
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"stepResponse (x[n::real]::bool)"];
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val (dI,pI,mI) = ("Isac", ["Inverse", "Z_Transform", "SignalProcessing"],
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["SignalProcessing","Z_Transform","Inverse"]);
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val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI,pI,mI))];
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val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Add_Given";
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val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Add_Find";
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val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Specify_Theory";
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val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Specify_Problem";
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val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Specify_Method";
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val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Apply_Method";
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neuper@42315
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val (p,_,f,nxt,_,pt) = me nxt p [] pt; "nxt = Rewrite (ruleZY, Inverse_Z_Transform.ruleZY) --> X z = 3 / (z - 1 / 4 + -1 / 8 * (1 / z))";
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val (p''',_,f,nxt''',_,pt''') = me nxt p [] pt; "nxt = Rewrite_Set norm_Rational --> X' z = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))";
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"~~~~~ fun me, args:"; val ((_,tac), (p:pos'), _, (pt:ptree)) = (nxt, p, [], pt);
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val ("ok", (_, _, ptp)) = locatetac tac (pt,p)
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val (pt, p) = ptp;
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"~~~~~ fun step, args:"; val (((ip as (_,p_)):pos'), ((ptp as (pt,p), tacis):calcstate)) =
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(p, ((pt, e_pos'),[]));
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val pIopt = get_pblID (pt,ip);
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ip = ([],Res); "false";
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tacis; " = []";
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pIopt; (* = SOME ["Inverse", "Z_Transform", "SignalProcessing"]*)
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member op = [Pbl,Met] p_ andalso is_none (get_obj g_env pt (fst p)); "false";
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(*nxt_solve_ (pt,ip); "WAS isalist2list applied to NON-list 'no_meth'"
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THIS MEANS: replace no_meth by [no_meth] in Script.*)
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(*WAS val ("helpless",_) = step p ((pt, e_pos'),[]) *)
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(*WAS val (p,_,f,nxt,_,pt) = me nxt p [] pt; "Empty_Tac instead SubProblem";*)
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neuper@42310
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"----------- why not nxt = Model_Problem here ? ---------";
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neuper@42310
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"----------- why not nxt = Model_Problem here ? ---------";
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neuper@42310
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"----------- why not nxt = Model_Problem here ? ---------";
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neuper@42313
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val (p,_,f,nxt,_,pt) = me nxt''' p''' [] pt''';
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"~~~~~ fun me, args:"; val (((_,tac):tac'_), (p:pos'), _, (pt:ptree)) = (nxt, p, c, pt);
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val ("ok", (_, _, ptp)) = locatetac tac (pt,p);
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val (pt, p) = ptp;
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"~~~~~ fun step, args:"; val (((ip as (_,p_)):pos'), ((ptp as (pt,p), tacis):calcstate)) =
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(p, ((pt, e_pos'),[]));
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val pIopt = get_pblID (pt,ip);
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ip = ([],Res); " = false";
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tacis; " = []";
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pIopt (* = SOME ["Inverse", "Z_Transform", "SignalProcessing"]*);
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neuper@42313
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member op = [Pbl,Met] p_ andalso is_none (get_obj g_env pt (fst p)); " = false";
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neuper@42313
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(* ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ leads into
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nxt_solve_, which is definitely WRONG (should be nxt_specify_ for FIND_ADD).
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neuper@42313
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This ERROR seems to be introduced together with ctxt, concerns Apply_Method without init_form.
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See TODO.txt
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*)
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neuper@42310
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"----------- fun factors_from_solution ------------------";
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"----------- fun factors_from_solution ------------------";
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"----------- fun factors_from_solution ------------------";
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val ctxt = ProofContext.init_global @{theory Isac};
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val SOME t = parseNEW ctxt "factors_from_solution [(z::real) = 1 / 2, z = -1 / 4]";
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val SOME (_, t') = eval_factors_from_solution "" 0 t thy;
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if term2str t' =
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"factors_from_solution [z = 1 / 2, z = -1 / 4] = (z - 1 / 2) * (z - -1 / 4)"
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then () else error "factors_from_solution broken";
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"----------- Logic.unvarify_global ----------------------";
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"----------- Logic.unvarify_global ----------------------";
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"----------- Logic.unvarify_global ----------------------";
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val A_var = parseNEW ctxt "A::real" |> the |> Logic.varify_global
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val B_var = parseNEW ctxt "B::real" |> the |> Logic.varify_global
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val denom1 = parseNEW ctxt "(z + -1 * (1 / 2))::real" |> the;
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val denom2 = parseNEW ctxt "(z + -1 * (-1 / 4))::real" |> the;
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val test = HOLogic.mk_binop "Groups.plus_class.plus"
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(HOLogic.mk_binop "Rings.inverse_class.divide" (A_var, denom1),
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HOLogic.mk_binop "Rings.inverse_class.divide" (B_var, denom2));
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if term2str test = "?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))" then ()
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else error "HOLogic.mk_binop broken ?";
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(* Logic.unvarify_global test
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---> exception TERM raised (line 539 of "logic.ML"): Illegal fixed variable: "z"
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thus we need another fun var2free in termC.sml*)
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"----------- eval_drop_questionmarks --------------------";
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"----------- eval_drop_questionmarks --------------------";
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"----------- eval_drop_questionmarks --------------------";
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if term2str test = "?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))"
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then () else error "test setup (ctxt!) probably changed";
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val t = Const ("Partial_Fractions.drop_questionmarks", HOLogic.realT --> HOLogic.realT) $ test;
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val SOME (_, t') = eval_drop_questionmarks "drop_questionmarks" 0 t @{theory Isac};
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if term2str t' = "drop_questionmarks (?A / (z + -1 * (1 / 2)) + ?B / (z + -1 * (-1 / 4))) =" ^
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"\nA / (z + -1 * (1 / 2)) + B / (z + -1 * (-1 / 4))"
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then () else error "eval_drop_questionmarks broken";
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"----------- = me for met_partial_fraction --------------";
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neuper@42376
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"----------- = me for met_partial_fraction --------------";
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"----------- = me for met_partial_fraction --------------";
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val fmz =
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["functionTerm (3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))))",
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"solveFor z", "decomposedFunction p_p"];
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val (dI',pI',mI') =
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("Partial_Fractions",
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["partial_fraction", "rational", "simplification"],
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["simplification","of_rationals","to_partial_fraction"]);
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val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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val (p,_,f,nxt,_,pt) = me nxt p [] pt;
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show_pt pt;
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if f2str f = "4 / (z - 1 / 2) + -4 / (z - -1 / 4)" then()
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else error "= me .. met_partial_fraction broken";
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neuper@42376
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223 |
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neuper@42386
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224 |
"----------- progr.vers.2: check erls for multiply_ansatz";
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neuper@42386
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225 |
"----------- progr.vers.2: check erls for multiply_ansatz";
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neuper@42386
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226 |
"----------- progr.vers.2: check erls for multiply_ansatz";
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neuper@42386
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(*test for outcommented 3 lines in script: is norm_Rational strong enough?*)
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val t = str2term "(3 / ((-1 + -2 * z + 8 * z ^^^ 2) *3/24)) = (3 / ((z - 1 / 2) * (z - -1 / 4)))";
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val SOME (t', _) = rewrite_set_ @{theory Isac} true ansatz_rls t;
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term2str t' = "3 / ((-1 + -2 * z + 8 * z ^^^ 2) * 3 / 24) =\n?A / (z - 1 / 2) + ?B / (z - -1 / 4)";
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231 |
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neuper@42386
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val SOME (t'', _) = rewrite_set_ @{theory Isac} true multiply_ansatz t'; (*true*)
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term2str t'' = "(z - 1 / 2) * (z - -1 / 4) * 3 / ((-1 + -2 * z + 8 * z ^^^ 2) * 3 / 24) =\n" ^
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neuper@42386
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"?A * (z - -1 / 4) + ?B * (z - 1 / 2)"; (*true*)
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neuper@42386
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235 |
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neuper@42386
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val SOME (t''', _) = rewrite_set_ @{theory Isac} true norm_Rational t'';
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neuper@42386
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if term2str t''' = "3 = ?A * (1 + 4 * z) / 4 + ?B * (-1 + 2 * z) / 2" then ()
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neuper@42386
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else error "ansatz_rls - multiply_ansatz - norm_Rational broken";
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neuper@42386
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239 |
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neuper@42386
|
240 |
(*test for outcommented 3 lines in script: empower erls for x = a*b ==> ...*)
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neuper@42386
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val xxx = append_rls "multiply_ansatz_erls" norm_Rational
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[Calc ("HOL.eq",eval_equal "#equal_")];
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243 |
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neuper@42386
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val multiply_ansatz = prep_rls(
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neuper@42386
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Rls {id = "multiply_ansatz", preconds = [], rew_ord = ("dummy_ord",dummy_ord),
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neuper@42386
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erls = xxx,
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neuper@42386
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srls = Erls, calc = [],
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neuper@42386
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rules =
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neuper@42386
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[Thm ("multiply_2nd_order",num_str @{thm multiply_2nd_order})
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neuper@42386
|
250 |
],
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neuper@42386
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scr = EmptyScr}:rls);
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neuper@42386
|
252 |
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neuper@42386
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253 |
rewrite_set_ thy true xxx @{term "a+b = a+(b::real)"}; (*SOME ok*)
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neuper@42386
|
254 |
rewrite_set_ thy true multiply_ansatz @{term "a+b = a+(b::real)"}; (*why NONE?: GOON*)
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neuper@42386
|
255 |
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neuper@42386
|
256 |
"----------- progr.vers.2: improve program --------------";
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neuper@42386
|
257 |
"----------- progr.vers.2: improve program --------------";
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neuper@42386
|
258 |
"----------- progr.vers.2: improve program --------------";
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neuper@42386
|
259 |
(*WN120318 stopped due to much effort with the test above*)
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neuper@42386
|
260 |
"Script PartFracScript (f_f::real) (zzz::real) = " ^(*f_f: 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)), zzz: z*)
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neuper@42386
|
261 |
" (let f_f = Take f_f; " ^
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neuper@42386
|
262 |
" (num_orig::real) = get_numerator f_f; " ^(*num_orig: 3*)
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neuper@42386
|
263 |
" f_f = (Rewrite_Set norm_Rational False) f_f; " ^(*f_f: 24 / (-1 + -2 * z + 8 * z ^^^ 2)*)
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neuper@42386
|
264 |
" (numer::real) = get_numerator f_f; " ^(*numer: 24*)
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neuper@42386
|
265 |
" (denom::real) = get_denominator f_f; " ^(*denom: -1 + -2 * z + 8 * z ^^^ 2*)
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neuper@42386
|
266 |
" (equ::bool) = (denom = (0::real)); " ^(*equ: -1 + -2 * z + 8 * z ^^^ 2 = 0*)
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neuper@42386
|
267 |
" (L_L::bool list) = (SubProblem (PolyEq', " ^
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neuper@42386
|
268 |
" [abcFormula, degree_2, polynomial, univariate, equation], " ^
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neuper@42386
|
269 |
" [no_met]) [BOOL equ, REAL zzz]); " ^(*L_L: [z = 1 / 2, z = -1 / 4]*)
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neuper@42386
|
270 |
" (facs::real) = factors_from_solution L_L; " ^(*facs: (z - 1 / 2) * (z - -1 / 4)*)
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neuper@42386
|
271 |
" (eql::real) = Take (num_orig / facs); " ^(*eql: 3 / ((z - 1 / 2) * (z - -1 / 4)) *)
|
neuper@42386
|
272 |
" (eqr::real) = (Try (Rewrite_Set ansatz_rls False)) eql; " ^(*eqr: ?A / (z - 1 / 2) + ?B / (z - -1 / 4)*)
|
neuper@42386
|
273 |
" (eq::bool) = Take (eql = eqr); " ^(*eq: 3 / ((z - 1 / 2) * (z - -1 / 4)) = ?A / (z - 1 / 2) + ?B / (z - -1 / 4)*)
|
neuper@42386
|
274 |
(*this has been tested below by rewrite_set_
|
neuper@42386
|
275 |
" (eeeee::bool) = Take ((num_orig / denom * denom / numer) = (num_orig / facs));" ^(**)
|
neuper@42386
|
276 |
" (eeeee::bool) = (Rewrite_Set ansatz_rls False) eeeee;" ^(**)
|
neuper@42386
|
277 |
" eq = (Try (Rewrite_Set multiply_ansatz False)) eeeee; " ^(*eq: 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)*)
|
neuper@42386
|
278 |
NEXT try to outcomment the very next line..*)
|
neuper@42386
|
279 |
" eq = (Try (Rewrite_Set equival_trans False)) eeeee; " ^(*eq: 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)*)
|
neuper@42386
|
280 |
" eq = drop_questionmarks eq; " ^(*eq: 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
|
neuper@42386
|
281 |
" (z1::real) = (rhs (NTH 1 L_L)); " ^(*z1: 1 / 2*)
|
neuper@42386
|
282 |
" (z2::real) = (rhs (NTH 2 L_L)); " ^(*z2: -1 / 4*)
|
neuper@42386
|
283 |
" (eq_a::bool) = Take eq; " ^(*eq_a: 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
|
neuper@42386
|
284 |
" eq_a = (Substitute [zzz = z1]) eq; " ^(*eq_a: 3 = A * (1 / 2 - -1 / 4) + B * (1 / 2 - 1 / 2)*)
|
neuper@42386
|
285 |
" eq_a = (Rewrite_Set norm_Rational False) eq_a; " ^(*eq_a: 3 = 3 * A / 4*)
|
neuper@42386
|
286 |
" (sol_a::bool list) = " ^
|
neuper@42386
|
287 |
" (SubProblem (Isac', [univariate,equation], [no_met]) " ^
|
neuper@42386
|
288 |
" [BOOL eq_a, REAL (A::real)]); " ^(*sol_a: [A = 4]*)
|
neuper@42386
|
289 |
" (a::real) = (rhs (NTH 1 sol_a)); " ^(*a: 4*)
|
neuper@42386
|
290 |
" (eq_b::bool) = Take eq; " ^(*eq_b: 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
|
neuper@42386
|
291 |
" eq_b = (Substitute [zzz = z2]) eq_b; " ^(*eq_b: *)
|
neuper@42386
|
292 |
" eq_b = (Rewrite_Set norm_Rational False) eq_b; " ^(*eq_b: *)
|
neuper@42386
|
293 |
" (sol_b::bool list) = " ^
|
neuper@42386
|
294 |
" (SubProblem (Isac', [univariate,equation], [no_met]) " ^
|
neuper@42386
|
295 |
" [BOOL eq_b, REAL (B::real)]); " ^(*sol_b: [B = -4]*)
|
neuper@42386
|
296 |
" (b::real) = (rhs (NTH 1 sol_b)); " ^(*b: -4*)
|
neuper@42386
|
297 |
" eqr = drop_questionmarks eqr; " ^(*eqr: A / (z - 1 / 2) + B / (z - -1 / 4)*)
|
neuper@42386
|
298 |
" (pbz::real) = Take eqr; " ^(*pbz: A / (z - 1 / 2) + B / (z - -1 / 4)*)
|
neuper@42386
|
299 |
" pbz = ((Substitute [A = a, B = b]) pbz) " ^(*pbz: 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
|
neuper@42386
|
300 |
" in pbz)"
|
neuper@42386
|
301 |
|