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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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header {* Partial Fraction Decomposition *}
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theory Partial_Fractions imports RootRatEq begin
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ML{*
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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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*}
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subsection {* eval_ functions *}
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consts
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factors_from_solution :: "bool list => real"
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drop_questionmarks :: "'a => 'a"
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text {* these might be used for variants of fac_from_sol *}
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ML {*
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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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*}
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text {* from solutions (e.g. [z = 1, z = -2]) make linear factors (e.g. (z - 1)*(z - -2)) *}
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ML {*
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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 = e_term then error "mk_prod called with []" else prod
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| mk_prod prod (t :: []) =
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if prod = e_term 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 = e_term
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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 e_term (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 (mk_thmid thmid "" (term_to_string''' thy prod) "",
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Trueprop $ (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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*}
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text {* 'ansatz' introduces '?Vars' (questionable design); drop these again *}
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ML {*
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(*("drop_questionmarks", ("Partial_Fractions.drop_questionmarks", eval_drop_questionmarks ""))*)
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fun eval_drop_questionmarks (thmid:string) _
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(t as Const ("Partial_Fractions.drop_questionmarks", _) $ tm) thy =
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if contains_Var tm
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then
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let
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val tm' = var2free tm
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in SOME (mk_thmid thmid "" (term_to_string''' thy tm') "",
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Trueprop $ (mk_equality (t, tm')))
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end
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else NONE
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| eval_drop_questionmarks _ _ _ _ = NONE;
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*}
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text {* store eval_ functions for calls from Scripts *}
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setup {* KEStore_Elems.add_calcs
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[("drop_questionmarks", ("Partial_Fractions.drop'_questionmarks", eval_drop_questionmarks ""))] *}
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subsection {* 'ansatz' for partial fractions *}
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axiomatization where
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ansatz_2nd_order: "n / (a*b) = A/a + B/b" and
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ansatz_3rd_order: "n / (a*b*c) = A/a + B/b + C/c" and
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ansatz_4th_order: "n / (a*b*c*d) = A/a + B/b + C/c + D/d" and
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(*version 1*)
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equival_trans_2nd_order: "(n/(a*b) = A/a + B/b) = (n = A*b + B*a)" and
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equival_trans_3rd_order: "(n/(a*b*c) = A/a + B/b + C/c) = (n = A*b*c + B*a*c + C*a*b)" and
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equival_trans_4th_order: "(n/(a*b*c*d) = A/a + B/b + C/c + D/d) =
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(n = A*b*c*d + B*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 = A/a + B/b) = (a*b*n/x = A*b + B*a)" and
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multiply_3rd_order: "(n/x = A/a + B/b + C/c) = (a*b*c*n/x = A*b*c + B*a*c + C*a*b)" and
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multiply_4th_order:
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"(n/x = A/a + B/b + C/c + D/d) = (a*b*c*d*n/x = A*b*c*d + B*a*c*d + C*a*b*d + D*a*b*c)"
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text {* Probably the optimal formalization woudl be ...
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multiply_2nd_order: "x = a*b ==> (n/x = A/a + B/b) = (a*b*n/x = A*b + B*a)" and
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multiply_3rd_order: "x = a*b*c ==>
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(n/x = A/a + B/b + C/c) = (a*b*c*n/x = A*b*c + B*a*c + C*a*b)" and
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multiply_4th_order: "x = a*b*c*d ==>
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(n/x = A/a + B/b + C/c + D/d) = (a*b*c*d*n/x = A*b*c*d + B*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 = A * (z - -1 / 4) + B * (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 ar in partial_fractions.sml "--- progr.vers.2: "
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*}
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ML {*
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val ansatz_rls = prep_rls'(
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Rls {id = "ansatz_rls", preconds = [], rew_ord = ("dummy_ord",dummy_ord),
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erls = Erls, srls = Erls, calc = [], errpatts = [],
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rules =
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[Thm ("ansatz_2nd_order",num_str @{thm ansatz_2nd_order}),
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Thm ("ansatz_3rd_order",num_str @{thm ansatz_3rd_order})
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],
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scr = EmptyScr}:rls);
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val equival_trans = prep_rls'(
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Rls {id = "equival_trans", preconds = [], rew_ord = ("dummy_ord",dummy_ord),
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erls = Erls, srls = Erls, calc = [], errpatts = [],
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rules =
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[Thm ("equival_trans_2nd_order",num_str @{thm equival_trans_2nd_order}),
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Thm ("equival_trans_3rd_order",num_str @{thm equival_trans_3rd_order})
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],
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scr = EmptyScr}:rls);
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val multiply_ansatz = prep_rls'(
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Rls {id = "multiply_ansatz", preconds = [], rew_ord = ("dummy_ord",dummy_ord),
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erls = Erls,
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srls = Erls, calc = [], errpatts = [],
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rules =
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[Thm ("multiply_2nd_order",num_str @{thm multiply_2nd_order})
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],
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scr = EmptyScr}:rls);
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*}
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text {*store the rule set for math engine*}
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setup {* 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))] *}
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subsection {* Specification *}
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consts
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decomposedFunction :: "real => una"
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ML {*
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check_guhs_unique := false; (*WN120307 REMOVE after editing*)
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*}
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setup {* KEStore_Elems.add_pbts
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[(prep_pbt @{theory} "pbl_simp_rat_partfrac" [] e_pblID
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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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append_rls "e_rls" e_rls [(*for preds in where_ TODO*)],
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NONE,
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[["simplification","of_rationals","to_partial_fraction"]]))] *}
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subsection {* Method *}
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consts
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PartFracScript :: "[real,real, real] => real"
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("((Script PartFracScript (_ _ =))// (_))" 9)
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text {* rule set for functions called in the Script *}
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ML {*
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val srls_partial_fraction = Rls {id="srls_partial_fraction",
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preconds = [],
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rew_ord = ("termlessI",termlessI),
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erls = append_rls "erls_in_srls_partial_fraction" e_rls
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[(*for asm in NTH_CONS ...*)
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Calc ("Orderings.ord_class.less",eval_equ "#less_"),
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(*2nd NTH_CONS pushes n+-1 into asms*)
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Calc("Groups.plus_class.plus", eval_binop "#add_")],
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srls = Erls, calc = [], errpatts = [],
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rules = [
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Thm ("NTH_CONS",num_str @{thm NTH_CONS}),
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Calc("Groups.plus_class.plus", eval_binop "#add_"),
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Thm ("NTH_NIL",num_str @{thm NTH_NIL}),
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Calc("Tools.lhs", eval_lhs "eval_lhs_"),
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Calc("Tools.rhs", eval_rhs"eval_rhs_"),
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Calc("Atools.argument'_in", eval_argument_in "Atools.argument'_in"),
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Calc("Rational.get_denominator", eval_get_denominator "#get_denominator"),
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Calc("Rational.get_numerator", eval_get_numerator "#get_numerator"),
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Calc("Partial_Fractions.factors_from_solution",
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eval_factors_from_solution "#factors_from_solution"),
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Calc("Partial_Fractions.drop_questionmarks", eval_drop_questionmarks "#drop_?")],
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scr = EmptyScr};
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*}
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ML {*
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eval_drop_questionmarks;
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*}
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ML {*
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val ctxt = Proof_Context.init_global @{theory};
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val SOME t = parseNEW ctxt "eqr = drop_questionmarks eqr";
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*}
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ML {*
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parseNEW ctxt "decomposedFunction p_p'''";
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parseNEW ctxt "decomposedFunction";
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*}
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(* current version, error outcommented *)
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setup {* KEStore_Elems.add_mets
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[prep_met @{theory} "met_partial_fraction" [] e_metID
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(["simplification","of_rationals","to_partial_fraction"],
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[("#Given" ,["functionTerm t_t", "solveFor v_v"]),
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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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(*f_f = 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)), zzz: z*)
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{rew_ord'="tless_true", rls'= e_rls, calc = [], srls = srls_partial_fraction, prls = e_rls,
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crls = e_rls, errpats = [], nrls = e_rls},
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(*([], Frm), Problem (Partial_Fractions, [partial_fraction, rational, simplification])*)
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"Script PartFracScript (f_f::real) (zzz::real) = " ^
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(*([1], Frm), 3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z)))*)
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"(let f_f = Take f_f; " ^
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(* num_orig = 3*)
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" (num_orig::real) = get_numerator f_f; " ^
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(*([1], Res), 24 / (-1 + -2 * z + 8 * z ^^^ 2)*)
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" f_f = (Rewrite_Set norm_Rational False) f_f; " ^
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(* denom = -1 + -2 * z + 8 * z ^^^ 2*)
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" (denom::real) = get_denominator f_f; " ^
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(* equ = -1 + -2 * z + 8 * z ^^^ 2 = 0*)
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" (equ::bool) = (denom = (0::real)); " ^
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250 |
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(*([2], Pbl), solve (-1 + -2 * z + 8 * z ^^^ 2 = 0, z)*)
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" (L_L::bool list) = (SubProblem (PolyEq', " ^
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253 |
" [abcFormula, degree_2, polynomial, univariate, equation], " ^
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(*([2], Res), [z = 1 / 2, z = -1 / 4]*)
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" [no_met]) [BOOL equ, REAL zzz]); " ^
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(* facs: (z - 1 / 2) * (z - -1 / 4)*)
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257 |
" (facs::real) = factors_from_solution L_L; " ^
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s1210629013@55373
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(*([3], Frm), 33 / ((z - 1 / 2) * (z - -1 / 4)) *)
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s1210629013@55373
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259 |
" (eql::real) = Take (num_orig / facs); " ^
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s1210629013@55373
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260 |
(*([3], Res), ?A / (z - 1 / 2) + ?B / (z - -1 / 4)*)
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s1210629013@55373
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261 |
" (eqr::real) = (Try (Rewrite_Set ansatz_rls False)) eql; " ^
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s1210629013@55373
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262 |
(*([4], Frm), 3 / ((z - 1 / 2) * (z - -1 / 4)) = ?A / (z - 1 / 2) + ?B / (z - -1 / 4)*)
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s1210629013@55373
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263 |
" (eq::bool) = Take (eql = eqr); " ^
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s1210629013@55373
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264 |
(*([4], Res), 3 = ?A * (z - -1 / 4) + ?B * (z - 1 / 2)*)
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s1210629013@55373
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265 |
" eq = (Try (Rewrite_Set equival_trans False)) eq;" ^
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s1210629013@55373
|
266 |
(* eq = 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
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|
267 |
" eq = drop_questionmarks eq; " ^
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s1210629013@55373
|
268 |
(* z1 = 1 / 2*)
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|
269 |
" (z1::real) = (rhs (NTH 1 L_L)); " ^
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s1210629013@55373
|
270 |
(* z2 = -1 / 4*)
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|
271 |
" (z2::real) = (rhs (NTH 2 L_L)); " ^
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s1210629013@55373
|
272 |
(*([5], Frm), 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
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s1210629013@55373
|
273 |
" (eq_a::bool) = Take eq; " ^
|
s1210629013@55373
|
274 |
(*([5], Res), 3 = A * (1 / 2 - -1 / 4) + B * (1 / 2 - 1 / 2)*)
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s1210629013@55373
|
275 |
" eq_a = (Substitute [zzz = z1]) eq; " ^
|
s1210629013@55373
|
276 |
(*([6], Res), 3 = 3 * A / 4*)
|
s1210629013@55373
|
277 |
" eq_a = (Rewrite_Set norm_Rational False) eq_a; " ^
|
s1210629013@55373
|
278 |
|
s1210629013@55373
|
279 |
(*([7], Pbl), solve (3 = 3 * A / 4, A)*)
|
s1210629013@55373
|
280 |
" (sol_a::bool list) = " ^
|
s1210629013@55373
|
281 |
" (SubProblem (Isac', [univariate,equation], [no_met]) " ^
|
s1210629013@55373
|
282 |
(*([7], Res), [A = 4]*)
|
s1210629013@55373
|
283 |
" [BOOL eq_a, REAL (A::real)]); " ^
|
s1210629013@55373
|
284 |
(* a = 4*)
|
s1210629013@55373
|
285 |
" (a::real) = (rhs (NTH 1 sol_a)); " ^
|
s1210629013@55373
|
286 |
(*([8], Frm), 3 = A * (z - -1 / 4) + B * (z - 1 / 2)*)
|
s1210629013@55373
|
287 |
" (eq_b::bool) = Take eq; " ^
|
s1210629013@55373
|
288 |
(*([8], Res), 3 = A * (-1 / 4 - -1 / 4) + B * (-1 / 4 - 1 / 2)*)
|
s1210629013@55373
|
289 |
" eq_b = (Substitute [zzz = z2]) eq_b; " ^
|
s1210629013@55373
|
290 |
(*([9], Res), 3 = -3 * B / 4*)
|
s1210629013@55373
|
291 |
" eq_b = (Rewrite_Set norm_Rational False) eq_b; " ^
|
s1210629013@55373
|
292 |
(*([10], Pbl), solve (3 = -3 * B / 4, B)*)
|
s1210629013@55373
|
293 |
" (sol_b::bool list) = " ^
|
s1210629013@55373
|
294 |
" (SubProblem (Isac', [univariate,equation], [no_met]) " ^
|
s1210629013@55373
|
295 |
(*([10], Res), [B = -4]*)
|
s1210629013@55373
|
296 |
" [BOOL eq_b, REAL (B::real)]); " ^
|
s1210629013@55373
|
297 |
(* b = -4*)
|
s1210629013@55373
|
298 |
" (b::real) = (rhs (NTH 1 sol_b)); " ^
|
s1210629013@55373
|
299 |
(* eqr = A / (z - 1 / 2) + B / (z - -1 / 4)*)
|
s1210629013@55373
|
300 |
" eqr = drop_questionmarks eqr; " ^
|
s1210629013@55373
|
301 |
(*([11], Frm), A / (z - 1 / 2) + B / (z - -1 / 4)*)
|
s1210629013@55373
|
302 |
" (pbz::real) = Take eqr; " ^
|
s1210629013@55373
|
303 |
(*([11], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
|
s1210629013@55373
|
304 |
" pbz = ((Substitute [A = a, B = b]) pbz) " ^
|
s1210629013@55373
|
305 |
(*([], Res), 4 / (z - 1 / 2) + -4 / (z - -1 / 4)*)
|
s1210629013@55373
|
306 |
"in pbz)"
|
s1210629013@55373
|
307 |
)]
|
s1210629013@55373
|
308 |
*}
|
neuper@42376
|
309 |
ML {*
|
neuper@42376
|
310 |
(*
|
neuper@42376
|
311 |
val fmz =
|
neuper@42376
|
312 |
["functionTerm (3 / (z * (z - 1 / 4 + -1 / 8 * (1 / z))))",
|
neuper@42376
|
313 |
"solveFor z", "functionTerm p_p"];
|
neuper@42376
|
314 |
val (dI',pI',mI') =
|
neuper@42376
|
315 |
("Partial_Fractions",
|
neuper@42376
|
316 |
["partial_fraction", "rational", "simplification"],
|
neuper@42376
|
317 |
["simplification","of_rationals","to_partial_fraction"]);
|
neuper@42376
|
318 |
val (p,_,f,nxt,_,pt) = CalcTreeTEST [(fmz, (dI',pI',mI'))];
|
neuper@42376
|
319 |
*)
|
neuper@42376
|
320 |
*}
|
neuper@42289
|
321 |
|
jan@42295
|
322 |
|
neuper@42376
|
323 |
|
neuper@42376
|
324 |
subsection {**}
|
neuper@42376
|
325 |
|
neuper@42289
|
326 |
end |