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(* Partial_Fractions
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author: Jan Rocnik, isac team
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Copyright (c) isac team 2011
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Use is subject to license terms.
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*)
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(* Partial Fraction Decomposition *)
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theory Partial_Fractions imports RootRatEq begin
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ML\<open>
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(*
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signature PARTFRAC =
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sig
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val ansatz_rls : rls
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val ansatz_rls_ : theory -> term -> (term * term list) option
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end
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*)
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\<close>
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subsection \<open>eval_ functions\<close>
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consts
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factors_from_solution :: "bool list => real"
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AA :: real
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BB :: real
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text \<open>these might be used for variants of fac_from_sol\<close>
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ML \<open>
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fun mk_minus_1 T = Free("-1", T); (*TODO DELETE WITH numbers_to_string*)
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fun flip_sign t = (*TODO improve for use in factors_from_solution: -(-1) etc*)
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let val minus_1 = t |> type_of |> mk_minus_1
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in HOLogic.mk_binop "Groups.times_class.times" (minus_1, t) end;
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\<close>
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text \<open>from solutions (e.g. [z = 1, z = -2]) make linear factors (e.g. (z - 1)*(z - -2))\<close>
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ML \<open>
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fun fac_from_sol s =
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let val (lhs, rhs) = HOLogic.dest_eq s
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in HOLogic.mk_binop "Groups.minus_class.minus" (lhs, rhs) end;
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fun mk_prod prod [] =
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if prod = TermC.empty then raise ERROR "mk_prod called with []" else prod
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| mk_prod prod (t :: []) =
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if prod = TermC.empty then t else HOLogic.mk_binop "Groups.times_class.times" (prod, t)
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| mk_prod prod (t1 :: t2 :: ts) =
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if prod = TermC.empty
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then
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let val p = HOLogic.mk_binop "Groups.times_class.times" (t1, t2)
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in mk_prod p ts end
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else
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let val p = HOLogic.mk_binop "Groups.times_class.times" (prod, t1)
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in mk_prod p (t2 :: ts) end
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fun factors_from_solution sol =
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let val ts = HOLogic.dest_list sol
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in mk_prod TermC.empty (map fac_from_sol ts) end;
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(*("factors_from_solution", ("Partial_Fractions.factors_from_solution",
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eval_factors_from_solution ""))*)
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fun eval_factors_from_solution (thmid:string) _
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(t as Const ("Partial_Fractions.factors_from_solution", _) $ sol) thy =
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((let val prod = factors_from_solution sol
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in SOME (TermC.mk_thmid thmid (UnparseC.term_in_thy thy prod) "",
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HOLogic.Trueprop $ (TermC.mk_equality (t, prod)))
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end)
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handle _ => NONE)
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| eval_factors_from_solution _ _ _ _ = NONE;
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\<close>
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subsection \<open>'ansatz' for partial fractions\<close>
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axiomatization where
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ansatz_2nd_order: "n / (a*b) = AA/a + BB/b" and
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ansatz_3rd_order: "n / (a*b*c) = AA/a + BB/b + C/c" and
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ansatz_4th_order: "n / (a*b*c*d) = AA/a + BB/b + C/c + D/d" and
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(*version 1*)
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equival_trans_2nd_order: "(n/(a*b) = AA/a + BB/b) = (n = AA*b + BB*a)" and
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equival_trans_3rd_order: "(n/(a*b*c) = AA/a + BB/b + C/c) = (n = AA*b*c + BB*a*c + C*a*b)" and
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equival_trans_4th_order: "(n/(a*b*c*d) = AA/a + BB/b + C/c + D/d) =
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(n = AA*b*c*d + BB*a*c*d + C*a*b*d + D*a*b*c)" and
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(*version 2: not yet used, see partial_fractions.sml*)
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multiply_2nd_order: "(n/x = AA/a + BB/b) = (a*b*n/x = AA*b + BB*a)" and
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multiply_3rd_order: "(n/x = AA/a + BB/b + C/c) = (a*b*c*n/x = AA*b*c + BB*a*c + C*a*b)" and
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multiply_4th_order:
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"(n/x = AA/a + BB/b + C/c + D/d) = (a*b*c*d*n/x = AA*b*c*d + BB*a*c*d + C*a*b*d + D*a*b*c)"
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text \<open>Probably the optimal formalization woudl be ...
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multiply_2nd_order: "x = a*b ==> (n/x = AA/a + BB/b) = (a*b*n/x = AA*b + BB*a)" and
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multiply_3rd_order: "x = a*b*c ==>
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(n/x = AA/a + BB/b + C/c) = (a*b*c*n/x = AA*b*c + BB*a*c + C*a*b)" and
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multiply_4th_order: "x = a*b*c*d ==>
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(n/x = AA/a + BB/b + C/c + D/d) = (a*b*c*d*n/x = AA*b*c*d + BB*a*c*d + C*a*b*d + D*a*b*c)"
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... because it would allow to start the ansatz as follows
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(1) 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))) = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))
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(2) 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))) = AA / (z - 1 / 2) + BB / (z - -1 / 4)
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(3) (z - 1 / 2) * (z - -1 / 4) * 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))) =
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(z - 1 / 2) * (z - -1 / 4) * AA / (z - 1 / 2) + BB / (z - -1 / 4)
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(4) 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)
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... (1==>2) ansatz
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(2==>3) multiply_*
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(3==>4) norm_Rational
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TODOs for this version are in partial_fractions.sml "--- progr.vers.2: "
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\<close>
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ML \<open>
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val ansatz_rls = prep_rls'(
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Rule_Def.Repeat {id = "ansatz_rls", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
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erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
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rules =
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[Rule.Thm ("ansatz_2nd_order",ThmC.numerals_to_Free @{thm ansatz_2nd_order}),
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Rule.Thm ("ansatz_3rd_order",ThmC.numerals_to_Free @{thm ansatz_3rd_order})
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],
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scr = Rule.Empty_Prog});
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val equival_trans = prep_rls'(
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Rule_Def.Repeat {id = "equival_trans", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
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erls = Rule_Set.Empty, srls = Rule_Set.Empty, calc = [], errpatts = [],
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rules =
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[Rule.Thm ("equival_trans_2nd_order",ThmC.numerals_to_Free @{thm equival_trans_2nd_order}),
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Rule.Thm ("equival_trans_3rd_order",ThmC.numerals_to_Free @{thm equival_trans_3rd_order})
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],
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scr = Rule.Empty_Prog});
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val multiply_ansatz = prep_rls'(
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Rule_Def.Repeat {id = "multiply_ansatz", preconds = [], rew_ord = ("dummy_ord",Rewrite_Ord.dummy_ord),
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erls = Rule_Set.Empty,
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srls = Rule_Set.Empty, calc = [], errpatts = [],
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rules =
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[Rule.Thm ("multiply_2nd_order",ThmC.numerals_to_Free @{thm multiply_2nd_order})
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],
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scr = Rule.Empty_Prog});
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\<close>
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text \<open>store the rule set for math engine\<close>
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setup \<open>KEStore_Elems.add_rlss
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[("ansatz_rls", (Context.theory_name @{theory}, ansatz_rls)),
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("multiply_ansatz", (Context.theory_name @{theory}, multiply_ansatz)),
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("equival_trans", (Context.theory_name @{theory}, equival_trans))]\<close>
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subsection \<open>Specification\<close>
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consts
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decomposedFunction :: "real => una"
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ML \<open>
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Check_Unique.on := false; (*WN120307 REMOVE after editing*)
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\<close>
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setup \<open>KEStore_Elems.add_pbts
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[(Specify.prep_pbt @{theory} "pbl_simp_rat_partfrac" [] Problem.id_empty
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(["partial_fraction", "rational", "simplification"],
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[("#Given" ,["functionTerm t_t", "solveFor v_v"]),
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(* TODO: call this sub-problem with appropriate functionTerm:
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leading coefficient of the denominator is 1: to be checked here! and..
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("#Where" ,["((get_numerator t_t) has_degree_in v_v) <
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((get_denominator t_t) has_degree_in v_v)"]), TODO*)
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("#Find" ,["decomposedFunction p_p'''"])],
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Rule_Set.append_rules "empty" Rule_Set.empty [(*for preds in where_ TODO*)],
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NONE,
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[["simplification","of_rationals","to_partial_fraction"]]))]\<close>
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subsection \<open>Method\<close>
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text \<open>rule set for functions called in the Program\<close>
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ML \<open>
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val srls_partial_fraction = Rule_Def.Repeat {id="srls_partial_fraction",
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preconds = [],
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rew_ord = ("termlessI",termlessI),
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erls = Rule_Set.append_rules "erls_in_srls_partial_fraction" Rule_Set.empty
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neuper@42376
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[(*for asm in NTH_CONS ...*)
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Rule.Eval ("Orderings.ord_class.less", Prog_Expr.eval_equ "#less_"),
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(*2nd NTH_CONS pushes n+-1 into asms*)
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Rule.Eval("Groups.plus_class.plus", (**)eval_binop "#add_")],
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srls = Rule_Set.Empty, calc = [], errpatts = [],
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rules = [
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Rule.Thm ("NTH_CONS",ThmC.numerals_to_Free @{thm NTH_CONS}),
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Rule.Eval("Groups.plus_class.plus", (**)eval_binop "#add_"),
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Rule.Thm ("NTH_NIL",ThmC.numerals_to_Free @{thm NTH_NIL}),
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Rule.Eval("Prog_Expr.lhs", Prog_Expr.eval_lhs "eval_lhs_"),
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Rule.Eval("Prog_Expr.rhs", Prog_Expr.eval_rhs"eval_rhs_"),
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Rule.Eval("Prog_Expr.argument'_in", Prog_Expr.eval_argument_in "Prog_Expr.argument'_in"),
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Rule.Eval("Rational.get_denominator", eval_get_denominator "#get_denominator"),
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Rule.Eval("Rational.get_numerator", eval_get_numerator "#get_numerator"),
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Rule.Eval("Partial_Fractions.factors_from_solution",
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eval_factors_from_solution "#factors_from_solution")
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],
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scr = Rule.Empty_Prog};
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\<close>
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(* current version, error outcommented *)
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partial_function (tailrec) partial_fraction :: "real \<Rightarrow> real \<Rightarrow> real"
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where
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"partial_fraction f_f zzz = \<comment> \<open>([1], Frm), 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))\<close>
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(let f_f = Take f_f; \<comment> \<open>num_orig = 3\<close>
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num_orig = get_numerator f_f; \<comment> \<open>([1], Res), 24 / (-1 + -2 * z + 8 * z ^^^ 2)\<close>
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walther@59635
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f_f = (Rewrite_Set ''norm_Rational'') f_f; \<comment> \<open>denom = -1 + -2 * z + 8 * z ^^^ 2\<close>
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denom = get_denominator f_f; \<comment> \<open>equ = -1 + -2 * z + 8 * z ^^^ 2 = 0\<close>
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equ = denom = (0::real);
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\<comment> \<open>----- ([2], Pbl), solve (-1 + -2 * z + 8 * z ^^^ 2 = 0, z)\<close>
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wneuper@59513
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L_L = SubProblem (''Partial_Fractions'', \<comment> \<open>([2], Res), [z = 1 / 2, z = -1 / 4\<close>
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[''abcFormula'', ''degree_2'', ''polynomial'', ''univariate'', ''equation''],
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[''no_met'']) [BOOL equ, REAL zzz]; \<comment> \<open>facs: (z - 1 / 2) * (z - -1 / 4)\<close>
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facs = factors_from_solution L_L; \<comment> \<open>([3], Frm), 33 / ((z - 1 / 2) * (z - -1 / 4))\<close>
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eql = Take (num_orig / facs); \<comment> \<open>([3], Res), ?A / (z - 1 / 2) + ?B / (z - -1 / 4)\<close>
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walther@59635
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eqr = (Try (Rewrite_Set ''ansatz_rls'')) eql;
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\<comment> \<open>([4], Frm), 3 / ((z - 1 / 2) * (z - -1 / 4)) = ?A / (z - 1 / 2) + ?B / (z - -1 / 4)\<close>
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wneuper@59536
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eq = Take (eql = eqr); \<comment> \<open>([4], Res), 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)\<close>
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walther@59635
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eq = (Try (Rewrite_Set ''equival_trans'')) eq;
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\<comment> \<open>eq = 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)\<close>
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z1 = rhs (NTH 1 L_L); \<comment> \<open>z2 = -1 / 4\<close>
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z2 = rhs (NTH 2 L_L); \<comment> \<open>([5], Frm), 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)\<close>
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wneuper@59536
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eq_a = Take eq; \<comment> \<open>([5], Res), 3 = AA * (1 / 2 - -1 / 4) + BB * (1 / 2 - 1 / 2)\<close>
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wneuper@59536
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eq_a = Substitute [zzz = z1] eq; \<comment> \<open>([6], Res), 3 = 3 * AA / 4\<close>
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walther@59635
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eq_a = (Rewrite_Set ''norm_Rational'') eq_a;
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wneuper@59536
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\<comment> \<open>----- ([7], Pbl), solve (3 = 3 * AA / 4, AA)\<close>
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\<comment> \<open>([7], Res), [AA = 4]\<close>
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wneuper@59592
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sol_a = SubProblem (''Isac_Knowledge'', [''univariate'',''equation''], [''no_met''])
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[BOOL eq_a, REAL (AA::real)] ; \<comment> \<open>a = 4\<close>
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a = rhs (NTH 1 sol_a); \<comment> \<open>([8], Frm), 3 = AA * (z - -1 / 4) + BB * (z - 1 / 2)\<close>
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eq_b = Take eq; \<comment> \<open>([8], Res), 3 = AA * (-1 / 4 - -1 / 4) + BB * (-1 / 4 - 1 / 2)\<close>
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wneuper@59536
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221 |
eq_b = Substitute [zzz = z2] eq_b; \<comment> \<open>([9], Res), 3 = -3 * BB / 4\<close>
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walther@59635
|
222 |
eq_b = (Rewrite_Set ''norm_Rational'') eq_b; \<comment> \<open>([10], Pbl), solve (3 = -3 * BB / 4, BB)\<close>
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walther@59635
|
223 |
sol_b = SubProblem (''Isac_Knowledge'', \<comment> \<open>([10], Res), [BB = -4]\<close>
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wneuper@59504
|
224 |
[''univariate'',''equation''], [''no_met''])
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wneuper@59536
|
225 |
[BOOL eq_b, REAL (BB::real)]; \<comment> \<open>b = -4\<close>
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wneuper@59536
|
226 |
b = rhs (NTH 1 sol_b); \<comment> \<open>eqr = AA / (z - 1 / 2) + BB / (z - -1 / 4)\<close>
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wneuper@59504
|
227 |
pbz = Take eqr; \<comment> \<open>([11], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)\<close>
|
wneuper@59512
|
228 |
pbz = Substitute [AA = a, BB = b] pbz \<comment> \<open>([], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)\<close>
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wneuper@59504
|
229 |
in pbz) "
|
wneuper@59472
|
230 |
setup \<open>KEStore_Elems.add_mets
|
walther@59903
|
231 |
[Specify.prep_met @{theory} "met_partial_fraction" [] Method.id_empty
|
s1210629013@55373
|
232 |
(["simplification","of_rationals","to_partial_fraction"],
|
s1210629013@55373
|
233 |
[("#Given" ,["functionTerm t_t", "solveFor v_v"]),
|
s1210629013@55373
|
234 |
(*("#Where" ,["((get_numerator t_t) has_degree_in v_v) <
|
s1210629013@55373
|
235 |
((get_denominator t_t) has_degree_in v_v)"]), TODO*)
|
s1210629013@55373
|
236 |
("#Find" ,["decomposedFunction p_p'''"])],
|
s1210629013@55373
|
237 |
(*f_f = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)), zzz: z*)
|
walther@59852
|
238 |
{rew_ord'="tless_true", rls'= Rule_Set.empty, calc = [], srls = srls_partial_fraction, prls = Rule_Set.empty,
|
walther@59852
|
239 |
crls = Rule_Set.empty, errpats = [], nrls = Rule_Set.empty},
|
s1210629013@55373
|
240 |
(*([], Frm), Problem (Partial_Fractions, [partial_fraction, rational, simplification])*)
|
wneuper@59551
|
241 |
@{thm partial_fraction.simps})]
|
wneuper@59472
|
242 |
\<close>
|
wneuper@59472
|
243 |
ML \<open>
|
neuper@42376
|
244 |
(*
|
neuper@42376
|
245 |
val fmz =
|
neuper@42376
|
246 |
["functionTerm (3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))))",
|
neuper@42376
|
247 |
"solveFor z", "functionTerm p_p"];
|
neuper@42376
|
248 |
val (dI',pI',mI') =
|
neuper@42376
|
249 |
("Partial_Fractions",
|
neuper@42376
|
250 |
["partial_fraction", "rational", "simplification"],
|
neuper@42376
|
251 |
["simplification","of_rationals","to_partial_fraction"]);
|
neuper@42376
|
252 |
val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
|
neuper@42376
|
253 |
*)
|
wneuper@59472
|
254 |
\<close>
|
neuper@42289
|
255 |
|
neuper@42289
|
256 |
end |