68630
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signature MULTISERIES_EXPANSION = sig
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type expansion_thm = thm
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type trimmed_thm = thm
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type expr = Exp_Log_Expression.expr
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type basis = Asymptotic_Basis.basis
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datatype trim_mode = Simple_Trim | Pos_Trim | Neg_Trim | Sgn_Trim
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datatype zeroness = IsZero | IsNonZero | IsPos | IsNeg
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datatype intyness = Nat of thm | Neg_Nat of thm | No_Nat
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datatype parity = Even of thm | Odd of thm | Unknown_Parity
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datatype limit =
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Zero_Limit of bool option
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| Finite_Limit of term
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| Infinite_Limit of bool option
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datatype trim_result =
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Trimmed of zeroness * trimmed_thm option
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| Aborted of order
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val get_intyness : Proof.context -> cterm -> intyness
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val get_parity : cterm -> parity
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val get_expansion : thm -> term
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val get_coeff : term -> term
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val get_exponent : term -> term
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val get_expanded_fun : thm -> term
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val get_eval : term -> term
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val expands_to_hd : thm -> thm
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val mk_eval_ctxt : Proof.context -> Lazy_Eval.eval_ctxt
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val expand : Lazy_Eval.eval_ctxt -> expr -> basis -> expansion_thm * basis
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val expand_term : Lazy_Eval.eval_ctxt -> term -> basis -> expansion_thm * basis
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val expand_terms : Lazy_Eval.eval_ctxt -> term list -> basis -> expansion_thm list * basis
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val limit_of_expansion : bool * bool -> Lazy_Eval.eval_ctxt -> thm * basis -> limit * thm
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val compute_limit : Lazy_Eval.eval_ctxt -> term -> limit * thm
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val compare_expansions :
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Lazy_Eval.eval_ctxt -> expansion_thm * expansion_thm * basis ->
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order * thm * expansion_thm * expansion_thm
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(* TODO DEBUG *)
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datatype comparison_result =
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Cmp_Dominated of order * thm list * zeroness * trimmed_thm * expansion_thm * expansion_thm
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| Cmp_Asymp_Equiv of thm * thm
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val compare_expansions' :
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Lazy_Eval.eval_ctxt ->
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thm * thm * Asymptotic_Basis.basis ->
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comparison_result
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val prove_at_infinity : Lazy_Eval.eval_ctxt -> thm * basis -> thm
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val prove_at_top : Lazy_Eval.eval_ctxt -> thm * basis -> thm
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val prove_at_bot : Lazy_Eval.eval_ctxt -> thm * basis -> thm
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val prove_nhds : Lazy_Eval.eval_ctxt -> thm * basis -> thm
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val prove_at_0 : Lazy_Eval.eval_ctxt -> thm * basis -> thm
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val prove_at_left_0 : Lazy_Eval.eval_ctxt -> thm * basis -> thm
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val prove_at_right_0 : Lazy_Eval.eval_ctxt -> thm * basis -> thm
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val prove_smallo : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
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val prove_bigo : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
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val prove_bigtheta : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
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val prove_asymp_equiv : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
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val prove_asymptotic_relation : Lazy_Eval.eval_ctxt -> thm * thm * basis -> order * thm
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val prove_eventually_less : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
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val prove_eventually_greater : Lazy_Eval.eval_ctxt -> thm * thm * basis -> thm
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val prove_eventually_nonzero : Lazy_Eval.eval_ctxt -> thm * basis -> thm
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val extract_terms : int * bool -> Lazy_Eval.eval_ctxt -> basis -> term -> term * term option
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(* Internal functions *)
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val check_expansion : Exp_Log_Expression.expr -> expansion_thm -> expansion_thm
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val zero_expansion : basis -> expansion_thm
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val const_expansion : Lazy_Eval.eval_ctxt -> basis -> term -> expansion_thm
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val ln_expansion :
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Lazy_Eval.eval_ctxt -> trimmed_thm -> expansion_thm -> basis -> expansion_thm * basis
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val exp_expansion : Lazy_Eval.eval_ctxt -> expansion_thm -> basis -> expansion_thm * basis
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val powr_expansion :
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Lazy_Eval.eval_ctxt -> expansion_thm * expansion_thm * basis -> expansion_thm * basis
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val powr_const_expansion :
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Lazy_Eval.eval_ctxt -> expansion_thm * term * basis -> expansion_thm
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val powr_nat_expansion :
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Lazy_Eval.eval_ctxt -> expansion_thm * expansion_thm * basis -> expansion_thm * basis
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val power_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * term * basis -> expansion_thm
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val root_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * term * basis -> expansion_thm
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val sgn_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * basis -> expansion_thm
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val min_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * expansion_thm * basis -> expansion_thm
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val max_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * expansion_thm * basis -> expansion_thm
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val arctan_expansion : Lazy_Eval.eval_ctxt -> basis -> expansion_thm -> expansion_thm
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val ev_zeroness_oracle : Lazy_Eval.eval_ctxt -> term -> thm option
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val zeroness_oracle : bool -> trim_mode option -> Lazy_Eval.eval_ctxt -> term -> zeroness * thm option
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val whnf_expansion : Lazy_Eval.eval_ctxt -> expansion_thm -> term option * expansion_thm * thm
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val simplify_expansion : Lazy_Eval.eval_ctxt -> expansion_thm -> expansion_thm
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val simplify_term : Lazy_Eval.eval_ctxt -> term -> term
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val trim_expansion_while_greater :
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bool -> term list option -> bool -> trim_mode option -> Lazy_Eval.eval_ctxt ->
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thm * Asymptotic_Basis.basis -> thm * trim_result * (zeroness * thm) list
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val trim_expansion : bool -> trim_mode option -> Lazy_Eval.eval_ctxt -> expansion_thm * basis ->
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expansion_thm * zeroness * trimmed_thm option
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val try_drop_leading_term_ex : bool -> Lazy_Eval.eval_ctxt -> expansion_thm -> expansion_thm option
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val try_prove_real_eq : bool -> Lazy_Eval.eval_ctxt -> term * term -> thm option
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val try_prove_ev_eq : Lazy_Eval.eval_ctxt -> term * term -> thm option
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val prove_compare_expansions : order -> thm list -> thm
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val simplify_trimmed_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * trimmed_thm ->
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expansion_thm * trimmed_thm
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val retrim_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * basis -> expansion_thm * thm
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val retrim_pos_expansion : Lazy_Eval.eval_ctxt -> expansion_thm * basis * trimmed_thm ->
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expansion_thm * thm * trimmed_thm
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val register_sign_oracle :
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binding * (Proof.context -> int -> tactic) -> Context.generic -> Context.generic
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val get_sign_oracles :
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Context.generic -> (string * (Proof.context -> int -> tactic)) list
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val solve_eval_eq : thm -> thm
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end
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structure Multiseries_Expansion : MULTISERIES_EXPANSION = struct
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open Asymptotic_Basis
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open Exp_Log_Expression
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open Lazy_Eval
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structure Data = Generic_Data
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(
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type T = (Proof.context -> int -> tactic) Name_Space.table;
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val empty : T = Name_Space.empty_table "sign oracle tactics";
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val extend = I;
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fun merge (tactics1, tactics2) : T = Name_Space.merge_tables (tactics1, tactics2);
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);
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fun register_sign_oracle (s, tac) ctxt =
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Data.map (Name_Space.define ctxt false (s, tac) #> snd) ctxt
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fun get_sign_oracles ctxt = Name_Space.fold_table cons (Data.get ctxt) []
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fun apply_sign_oracles ctxt tac =
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let
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val oracles = get_sign_oracles (Context.Proof ctxt)
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fun tac' {context = ctxt, concl, ...} =
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if Thm.term_of concl = @{term "HOL.Trueprop HOL.False"} then
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no_tac
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else
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FIRST (map (fn tac => HEADGOAL (snd tac ctxt)) oracles)
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in
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tac THEN_ALL_NEW (Subgoal.FOCUS_PREMS tac' ctxt)
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end
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type expansion_thm = thm
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type trimmed_thm = thm
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val dest_fun = dest_comb #> fst
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val dest_arg = dest_comb #> snd
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val concl_of' = Thm.concl_of #> HOLogic.dest_Trueprop
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fun get_expansion thm =
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thm |> Thm.concl_of |> HOLogic.dest_Trueprop |> Term.dest_comb |> fst |> Term.dest_comb |> snd
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fun get_expanded_fun thm = thm |> concl_of' |> dest_fun |> dest_fun |> dest_arg
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(*
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The following function is useful in order to detect whether a given real constant is
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an integer, which allows us to use the "f(x) ^ n" operation instead of "f(x) powr n".
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This usually leads to nicer results.
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*)
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datatype intyness = Nat of thm | Neg_Nat of thm | No_Nat
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fun get_intyness ctxt ct =
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if Thm.typ_of_cterm ct = @{typ Real.real} then
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let
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val ctxt' = put_simpset HOL_basic_ss ctxt addsimps @{thms intyness_simps}
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val conv =
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Simplifier.rewrite ctxt then_conv Simplifier.rewrite ctxt'
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fun flip (Nat thm) = Neg_Nat (thm RS @{thm intyness_uminus})
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| flip _ = No_Nat
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fun get_intyness' ct =
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case Thm.term_of ct of
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@{term "0::real"} => Nat @{thm intyness_0}
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| @{term "1::real"} => Nat @{thm intyness_1}
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| Const (@{const_name numeral}, _) $ _ =>
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Nat (Thm.reflexive (Thm.dest_arg ct) RS @{thm intyness_numeral})
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| Const (@{const_name uminus}, _) $ _ => flip (get_intyness' (Thm.dest_arg ct))
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| Const (@{const_name of_nat}, _) $ _ =>
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Nat (Thm.reflexive (Thm.dest_arg ct) RS @{thm intyness_of_nat})
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| _ => No_Nat
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val thm = conv ct
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val ct' = thm |> Thm.cprop_of |> Thm.dest_equals_rhs
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in
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case get_intyness' ct' of
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Nat thm' => Nat (Thm.transitive thm thm' RS @{thm HOL.meta_eq_to_obj_eq})
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| Neg_Nat thm' => Neg_Nat (Thm.transitive thm thm' RS @{thm HOL.meta_eq_to_obj_eq})
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| No_Nat => No_Nat
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end
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handle CTERM _ => No_Nat
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else
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No_Nat
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datatype parity = Even of thm | Odd of thm | Unknown_Parity
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(* TODO: powers *)
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fun get_parity ct =
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let
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fun inst thm cts =
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let
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val tvs = Term.add_tvars (Thm.concl_of thm) []
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in
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case tvs of
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[v] =>
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let
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val thm' = Thm.instantiate ([(v, Thm.ctyp_of_cterm ct)], []) thm
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val vs = take (length cts) (rev (Term.add_vars (Thm.concl_of thm') []))
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in
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Thm.instantiate ([], vs ~~ cts) thm'
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end
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| _ => raise THM ("get_parity", 0, [thm])
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end
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val get_num = Thm.dest_arg o Thm.dest_arg
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in
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case Thm.term_of ct of
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Const (@{const_name Groups.zero}, _) => Even (inst @{thm even_zero} [])
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| Const (@{const_name Groups.one}, _) => Odd (inst @{thm odd_one} [])
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| Const (@{const_name Num.numeral_class.numeral}, _) $ @{term "Num.One"} =>
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Odd (inst @{thm odd_Numeral1} [])
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| Const (@{const_name Num.numeral_class.numeral}, _) $ (@{term "Num.Bit0"} $ _) =>
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Even (inst @{thm even_numeral} [get_num ct])
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| Const (@{const_name Num.numeral_class.numeral}, _) $ (@{term "Num.Bit1"} $ _) =>
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Odd (inst @{thm odd_numeral} [get_num ct])
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| Const (@{const_name Groups.uminus}, _) $ _ => (
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case get_parity (Thm.dest_arg ct) of
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Even thm => Even (@{thm even_uminusI} OF [thm])
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| Odd thm => Odd (@{thm odd_uminusI} OF [thm])
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| _ => Unknown_Parity)
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| Const (@{const_name Groups.plus}, _) $ _ $ _ => (
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case apply2 get_parity (Thm.dest_binop ct) of
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(Even thm1, Even thm2) => Even (@{thm even_addI(1)} OF [thm1, thm2])
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| (Odd thm1, Odd thm2) => Even (@{thm even_addI(2)} OF [thm1, thm2])
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| (Even thm1, Odd thm2) => Odd (@{thm odd_addI(1)} OF [thm1, thm2])
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| (Odd thm1, Even thm2) => Odd (@{thm odd_addI(2)} OF [thm1, thm2])
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| _ => Unknown_Parity)
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| Const (@{const_name Groups.minus}, _) $ _ $ _ => (
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case apply2 get_parity (Thm.dest_binop ct) of
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(Even thm1, Even thm2) => Even (@{thm even_diffI(1)} OF [thm1, thm2])
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| (Odd thm1, Odd thm2) => Even (@{thm even_diffI(2)} OF [thm1, thm2])
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| (Even thm1, Odd thm2) => Odd (@{thm odd_diffI(1)} OF [thm1, thm2])
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| (Odd thm1, Even thm2) => Odd (@{thm odd_diffI(2)} OF [thm1, thm2])
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| _ => Unknown_Parity)
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| Const (@{const_name Groups.times}, _) $ _ $ _ => (
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case apply2 get_parity (Thm.dest_binop ct) of
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(Even thm1, _) => Even (@{thm even_multI(1)} OF [thm1])
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| (_, Even thm2) => Even (@{thm even_multI(2)} OF [thm2])
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| (Odd thm1, Odd thm2) => Odd (@{thm odd_multI} OF [thm1, thm2])
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| _ => Unknown_Parity)
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| Const (@{const_name Power.power}, _) $ _ $ _ =>
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let
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val (a, n) = Thm.dest_binop ct
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in
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case get_parity a of
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Odd thm => Odd (inst @{thm odd_powerI} [a, n] OF [thm])
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| _ => Unknown_Parity
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end
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| _ => Unknown_Parity
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end
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fun simplify_term' facts ctxt =
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let
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val ctxt = Simplifier.add_prems facts ctxt
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in
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Thm.cterm_of ctxt #> Simplifier.rewrite ctxt #>
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Thm.concl_of #> Logic.dest_equals #> snd
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end
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fun simplify_term ectxt = simplify_term' (get_facts ectxt) (get_ctxt ectxt)
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fun simplify_eval ctxt =
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simplify_term' [] (put_simpset HOL_basic_ss ctxt addsimps @{thms eval_simps})
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datatype zeroness = IsZero | IsNonZero | IsPos | IsNeg
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(* Caution: The following functions assume that the given expansion is in normal form already
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as far as needed. *)
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(* Returns the leading coefficient of the given expansion. This coefficient is a multiseries. *)
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fun try_get_coeff expr =
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case expr of
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Const (@{const_name MS}, _) $ (
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Const (@{const_name MSLCons}, _) $ (
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Const (@{const_name Pair}, _) $ c $ _) $ _) $ _ =>
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SOME c
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| _ => NONE
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fun get_coeff expr =
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expr |> dest_comb |> fst |> dest_comb |> snd |> dest_comb |> fst |> dest_comb |> snd
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|> dest_comb |> fst |> dest_comb |> snd
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(* Returns the coefficient of the leading term in the expansion (i.e. a real number) *)
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fun get_lead_coeff expr =
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case try_get_coeff expr of
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NONE => expr
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| SOME c => get_lead_coeff c
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(* Returns the exponent (w.r.t. the fastest-growing basis element) of the leading term *)
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fun get_exponent expr =
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expr |> dest_comb |> fst |> dest_comb |> snd |> dest_comb |> fst |> dest_comb |> snd
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|> dest_comb |> snd
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(* Returns the list of exponents of the leading term *)
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fun get_exponents exp =
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if fastype_of exp = @{typ real} then
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[]
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else
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get_exponent exp :: get_exponents (get_coeff exp)
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(* Returns the function that the expansion corresponds to *)
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fun get_eval expr =
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if fastype_of expr = @{typ real} then
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Abs ("x", @{typ real}, expr)
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|
330 |
else
|
|
331 |
expr |> dest_comb |> snd
|
|
332 |
|
|
333 |
val eval_simps = @{thms eval_simps [THEN eq_reflection]}
|
|
334 |
|
|
335 |
(* Tries to prove that the given function is eventually zero *)
|
|
336 |
fun ev_zeroness_oracle ectxt t =
|
|
337 |
let
|
|
338 |
val ctxt = Lazy_Eval.get_ctxt ectxt
|
|
339 |
val goal =
|
|
340 |
betapply (@{term "\<lambda>f::real \<Rightarrow> real. eventually (\<lambda>x. f x = 0) at_top"}, t)
|
|
341 |
|> HOLogic.mk_Trueprop
|
|
342 |
fun tac {context = ctxt, ...} =
|
|
343 |
HEADGOAL (Method.insert_tac ctxt (get_facts ectxt))
|
|
344 |
THEN Local_Defs.unfold_tac ctxt eval_simps
|
|
345 |
THEN HEADGOAL (Simplifier.asm_full_simp_tac ctxt)
|
|
346 |
in
|
|
347 |
try (Goal.prove ctxt [] [] goal) tac
|
|
348 |
end
|
|
349 |
|
|
350 |
(*
|
|
351 |
Encodes the kind of trimming/zeroness checking operation to be performed.
|
|
352 |
Simple_Trim only checks for zeroness/non-zeroness. Pos_Trim/Neg_Trim try to prove either
|
|
353 |
zeroness or positivity (resp. negativity). Sgn_Trim tries all three possibilities (positive,
|
|
354 |
negative, zero). *)
|
|
355 |
datatype trim_mode = Simple_Trim | Pos_Trim | Neg_Trim | Sgn_Trim
|
|
356 |
|
|
357 |
(*
|
|
358 |
Checks (and proves) whether the given term (assumed to be a real number) is zero, positive,
|
|
359 |
or negative, depending on given flags. The "fail" flag determines whether an exception is
|
|
360 |
thrown if this fails.
|
|
361 |
*)
|
|
362 |
fun zeroness_oracle fail mode ectxt exp =
|
|
363 |
let
|
|
364 |
val ctxt = Lazy_Eval.get_ctxt ectxt
|
|
365 |
val eq = (exp, @{term "0::real"}) |> HOLogic.mk_eq
|
|
366 |
val goal1 = (IsZero, eq |> HOLogic.mk_Trueprop)
|
|
367 |
val goal2 =
|
|
368 |
case mode of
|
|
369 |
SOME Pos_Trim =>
|
|
370 |
(IsPos, @{term "(<) (0::real)"} $ exp |> HOLogic.mk_Trueprop)
|
|
371 |
| SOME Sgn_Trim =>
|
|
372 |
(IsPos, @{term "(<) (0::real)"} $ exp |> HOLogic.mk_Trueprop)
|
|
373 |
| SOME Neg_Trim =>
|
|
374 |
(IsNeg, betapply (@{term "\<lambda>x. x < (0::real)"}, exp) |> HOLogic.mk_Trueprop)
|
|
375 |
| _ =>
|
|
376 |
(IsNonZero, eq |> HOLogic.mk_not |> HOLogic.mk_Trueprop)
|
|
377 |
val goals =
|
|
378 |
(if mode = SOME Sgn_Trim then
|
|
379 |
[(IsNeg, betapply (@{term "\<lambda>x. x < (0::real)"}, exp) |> HOLogic.mk_Trueprop)]
|
|
380 |
else
|
|
381 |
[])
|
|
382 |
val goals = goal2 :: goals
|
|
383 |
fun tac {context = ctxt, ...} =
|
|
384 |
HEADGOAL (Method.insert_tac ctxt (get_facts ectxt))
|
|
385 |
THEN Local_Defs.unfold_tac ctxt eval_simps
|
|
386 |
THEN HEADGOAL (apply_sign_oracles ctxt (Simplifier.asm_full_simp_tac ctxt))
|
|
387 |
fun prove (res, goal) = try (fn goal => (res, SOME (Goal.prove ctxt [] [] goal tac))) goal
|
|
388 |
fun err () =
|
|
389 |
let
|
|
390 |
val mode_msg =
|
|
391 |
case mode of
|
|
392 |
SOME Simple_Trim => "whether the following constant is zero"
|
|
393 |
| SOME Pos_Trim => "whether the following constant is zero or positive"
|
|
394 |
| SOME Neg_Trim => "whether the following constant is zero or negative"
|
|
395 |
| SOME Sgn_Trim => "the sign of the following constant"
|
|
396 |
| _ => raise Match
|
|
397 |
val t = simplify_term' (get_facts ectxt) ctxt exp
|
|
398 |
val _ =
|
|
399 |
if #verbose (#ctxt ectxt) then
|
|
400 |
let
|
|
401 |
val p = Pretty.str ("real_asymp failed to determine " ^ mode_msg ^ ":")
|
|
402 |
val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t)]
|
|
403 |
in
|
|
404 |
Pretty.writeln p
|
|
405 |
end else ()
|
|
406 |
in
|
|
407 |
raise TERM ("zeroness_oracle", [t])
|
|
408 |
end
|
|
409 |
in
|
|
410 |
case prove goal1 of
|
|
411 |
SOME res => res
|
|
412 |
| NONE =>
|
|
413 |
if mode = NONE then
|
|
414 |
(IsNonZero, NONE)
|
|
415 |
else
|
|
416 |
case get_first prove (goal2 :: goals) of
|
|
417 |
NONE => if fail then err () else (IsNonZero, NONE)
|
|
418 |
| SOME res => res
|
|
419 |
end
|
|
420 |
|
|
421 |
(* Tries to prove a given equality of real numbers. *)
|
|
422 |
fun try_prove_real_eq fail ectxt (lhs, rhs) =
|
|
423 |
case zeroness_oracle false NONE ectxt (@{term "(-) :: real => _"} $ lhs $ rhs) of
|
|
424 |
(IsZero, SOME thm) => SOME (thm RS @{thm real_eqI})
|
|
425 |
| _ =>
|
|
426 |
if not fail then NONE else
|
|
427 |
let
|
|
428 |
val ctxt = get_ctxt ectxt
|
|
429 |
val ts = map (simplify_term' (get_facts ectxt) ctxt) [lhs, rhs]
|
|
430 |
val _ =
|
|
431 |
if #verbose (#ctxt ectxt) then
|
|
432 |
let
|
|
433 |
val p =
|
|
434 |
Pretty.str ("real_asymp failed to prove that the following two numbers are equal:")
|
|
435 |
val p = Pretty.chunks (p :: map (Pretty.indent 2 o Syntax.pretty_term ctxt) ts)
|
|
436 |
in
|
|
437 |
Pretty.writeln p
|
|
438 |
end else ()
|
|
439 |
in
|
|
440 |
raise TERM ("try_prove_real_eq", [lhs, rhs])
|
|
441 |
end
|
|
442 |
|
|
443 |
(* Tries to prove a given eventual equality of real functions. *)
|
|
444 |
fun try_prove_ev_eq ectxt (f, g) =
|
|
445 |
let
|
|
446 |
val t = Envir.beta_eta_contract (@{term "\<lambda>(f::real=>real) g x. f x - g x"} $ f $ g)
|
|
447 |
in
|
|
448 |
Option.map (fn thm => thm RS @{thm eventually_diff_zero_imp_eq}) (ev_zeroness_oracle ectxt t)
|
|
449 |
end
|
|
450 |
|
|
451 |
fun real_less a b = @{term "(<) :: real \<Rightarrow> real \<Rightarrow> bool"} $ a $ b
|
|
452 |
fun real_eq a b = @{term "(=) :: real \<Rightarrow> real \<Rightarrow> bool"} $ a $ b
|
|
453 |
fun real_neq a b = @{term "(\<noteq>) :: real \<Rightarrow> real \<Rightarrow> bool"} $ a $ b
|
|
454 |
|
|
455 |
(* The hook that is called by the Lazy_Eval module whenever two real numbers have to be compared *)
|
|
456 |
fun real_sgn_hook ({pctxt = ctxt, facts, verbose, ...}) t =
|
|
457 |
let
|
|
458 |
val get_rhs = Thm.concl_of #> Logic.dest_equals #> snd
|
|
459 |
fun tac {context = ctxt, ...} =
|
|
460 |
HEADGOAL (Method.insert_tac ctxt (Net.content facts)
|
|
461 |
THEN' (apply_sign_oracles ctxt (Simplifier.asm_full_simp_tac ctxt)))
|
|
462 |
fun prove_first err [] [] =
|
|
463 |
if not verbose then raise TERM ("real_sgn_hook", [t])
|
|
464 |
else let val _ = err () in raise TERM ("real_sgn_hook", [t]) end
|
|
465 |
| prove_first err (goal :: goals) (thm :: thms) =
|
|
466 |
(case try (Goal.prove ctxt [] [] goal) tac of
|
|
467 |
SOME thm' =>
|
|
468 |
let val thm'' = thm' RS thm in SOME (get_rhs thm'', Conv.rewr_conv thm'') end
|
|
469 |
| NONE => prove_first err goals thms)
|
|
470 |
| prove_first _ _ _ = raise Match
|
|
471 |
in
|
|
472 |
case t of
|
|
473 |
@{term "(=) :: real => _"} $ a $ @{term "0 :: real"} =>
|
|
474 |
let
|
|
475 |
val goals =
|
|
476 |
map (fn c => HOLogic.mk_Trueprop (c a @{term "0 :: real"})) [real_neq, real_eq]
|
|
477 |
fun err () =
|
|
478 |
let
|
|
479 |
val facts' = Net.content facts
|
|
480 |
val a' = simplify_term' facts' ctxt a
|
|
481 |
val p = Pretty.str ("real_asymp failed to determine whether the following " ^
|
|
482 |
"constant is zero: ")
|
|
483 |
val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt a')]
|
|
484 |
in
|
|
485 |
Pretty.writeln p
|
|
486 |
end
|
|
487 |
in
|
|
488 |
prove_first err goals @{thms Eq_FalseI Eq_TrueI}
|
|
489 |
end
|
|
490 |
| Const (@{const_name COMPARE}, _) $ a $ b =>
|
|
491 |
let
|
|
492 |
val goals = map HOLogic.mk_Trueprop [real_less a b, real_less b a, real_eq a b]
|
|
493 |
fun err () =
|
|
494 |
let
|
|
495 |
val facts' = Net.content facts
|
|
496 |
val (a', b') = apply2 (simplify_term' facts' ctxt) (a, b)
|
|
497 |
val p = Pretty.str ("real_asymp failed to compare" ^
|
|
498 |
"the following two constants: ")
|
|
499 |
val p = Pretty.chunks (p :: map (Pretty.indent 2 o Syntax.pretty_term ctxt) [a', b'])
|
|
500 |
in
|
|
501 |
Pretty.writeln p
|
|
502 |
end
|
|
503 |
in
|
|
504 |
prove_first err goals @{thms COMPARE_intros}
|
|
505 |
end
|
|
506 |
| _ => NONE
|
|
507 |
end
|
|
508 |
|
|
509 |
(*
|
|
510 |
Returns the datatype constructors registered for use with the Lazy_Eval package.
|
|
511 |
All constructors on which pattern matching is performed need to be registered for evaluation
|
|
512 |
to work. It should be rare for users to add additional ones.
|
|
513 |
*)
|
|
514 |
fun get_constructors ctxt =
|
|
515 |
let
|
|
516 |
val thms = Named_Theorems.get ctxt @{named_theorems exp_log_eval_constructor}
|
|
517 |
fun go _ [] acc = rev acc
|
|
518 |
| go f (x :: xs) acc =
|
|
519 |
case f x of
|
|
520 |
NONE => go f xs acc
|
|
521 |
| SOME y => go f xs (y :: acc)
|
|
522 |
fun map_option f xs = go f xs []
|
|
523 |
fun dest_constructor thm =
|
|
524 |
case Thm.concl_of thm of
|
|
525 |
Const (@{const_name HOL.Trueprop}, _) $
|
|
526 |
(Const (@{const_name REAL_ASYMP_EVAL_CONSTRUCTOR}, _) $ Const (c, T)) =>
|
|
527 |
SOME (c, length (fst (strip_type T)))
|
|
528 |
| _ => NONE
|
|
529 |
in
|
|
530 |
thms |> map_option dest_constructor
|
|
531 |
end
|
|
532 |
|
|
533 |
(*
|
|
534 |
Creates an evaluation context with the correct setup of constructors, equations, and hooks.
|
|
535 |
*)
|
|
536 |
fun mk_eval_ctxt ctxt =
|
|
537 |
let
|
|
538 |
val eval_eqs = (Named_Theorems.get ctxt @{named_theorems real_asymp_eval_eqs})
|
|
539 |
val constructors = get_constructors ctxt
|
|
540 |
in
|
|
541 |
Lazy_Eval.mk_eval_ctxt ctxt constructors eval_eqs
|
|
542 |
|> add_hook real_sgn_hook
|
|
543 |
end
|
|
544 |
|
|
545 |
(* A pattern for determining the leading coefficient of a multiseries *)
|
|
546 |
val exp_pat =
|
|
547 |
let
|
|
548 |
val anypat = AnyPat ("_", 0)
|
|
549 |
in
|
|
550 |
ConsPat (@{const_name MS},
|
|
551 |
[ConsPat (@{const_name MSLCons},
|
|
552 |
[ConsPat (@{const_name Pair}, [anypat, anypat]), anypat]), anypat])
|
|
553 |
end
|
|
554 |
|
|
555 |
(*
|
|
556 |
Evaluates an expansion to (weak) head normal form, so that the leading coefficient and
|
|
557 |
exponent can be read off.
|
|
558 |
*)
|
|
559 |
fun whnf_expansion ectxt thm =
|
|
560 |
let
|
|
561 |
val ctxt = get_ctxt ectxt
|
|
562 |
val exp = get_expansion thm
|
|
563 |
val (_, _, conv) = match ectxt exp_pat exp (SOME [])
|
|
564 |
val eq_thm = conv (Thm.cterm_of ctxt exp)
|
|
565 |
val exp' = eq_thm |> Thm.concl_of |> Logic.dest_equals |> snd
|
|
566 |
in
|
|
567 |
case exp' of
|
|
568 |
Const (@{const_name MS}, _) $ (Const (@{const_name MSLCons}, _) $
|
|
569 |
(Const (@{const_name Pair}, _) $ c $ _) $ _) $ _ =>
|
|
570 |
(SOME c, @{thm expands_to_meta_eq_cong} OF [thm, eq_thm], eq_thm)
|
|
571 |
| Const (@{const_name MS}, _) $ Const (@{const_name MSLNil}, _) $ _ =>
|
|
572 |
(NONE, @{thm expands_to_meta_eq_cong} OF [thm, eq_thm], eq_thm)
|
|
573 |
| _ => raise TERM ("whnf_expansion", [exp'])
|
|
574 |
end
|
|
575 |
|
|
576 |
fun try_lift_function ectxt (thm, SEmpty) _ = (NONE, thm)
|
|
577 |
| try_lift_function ectxt (thm, basis) cont =
|
|
578 |
case whnf_expansion ectxt thm of
|
|
579 |
(SOME c, thm, _) =>
|
|
580 |
let
|
|
581 |
val f = get_expanded_fun thm
|
|
582 |
val T = fastype_of c
|
|
583 |
val t = Const (@{const_name eval}, T --> @{typ "real \<Rightarrow> real"}) $ c
|
|
584 |
val t = Term.betapply (Term.betapply (@{term "\<lambda>(f::real\<Rightarrow>real) g x. f x - g x"}, f), t)
|
|
585 |
in
|
|
586 |
case ev_zeroness_oracle ectxt t of
|
|
587 |
NONE => (NONE, thm)
|
|
588 |
| SOME zero_thm =>
|
|
589 |
let
|
|
590 |
val thm' = cont ectxt (thm RS @{thm expands_to_hd''}, tl_basis basis)
|
|
591 |
val thm'' = @{thm expands_to_lift_function} OF [zero_thm, thm']
|
|
592 |
in
|
|
593 |
(SOME (lift basis thm''), thm)
|
|
594 |
end
|
|
595 |
end
|
|
596 |
| _ => (NONE, thm)
|
|
597 |
|
|
598 |
(* Turns an expansion theorem into an expansion theorem for the leading coefficient. *)
|
|
599 |
fun expands_to_hd thm = thm RS
|
|
600 |
(if fastype_of (get_expansion thm) = @{typ "real ms"} then
|
|
601 |
@{thm expands_to_hd'}
|
|
602 |
else
|
|
603 |
@{thm expands_to_hd})
|
|
604 |
|
|
605 |
fun simplify_expansion ectxt thm =
|
|
606 |
let
|
|
607 |
val exp = get_expansion thm
|
|
608 |
val ctxt = get_ctxt ectxt
|
|
609 |
val eq_thm = Simplifier.rewrite ctxt (Thm.cterm_of ctxt exp)
|
|
610 |
in
|
|
611 |
@{thm expands_to_meta_eq_cong} OF [thm, eq_thm]
|
|
612 |
end
|
|
613 |
|
|
614 |
(*
|
|
615 |
Simplifies a trimmed expansion and returns the simplified expansion theorem and
|
|
616 |
the trimming theorem for that simplified expansion.
|
|
617 |
*)
|
|
618 |
fun simplify_trimmed_expansion ectxt (thm, trimmed_thm) =
|
|
619 |
let
|
|
620 |
val exp = get_expansion thm
|
|
621 |
val ctxt = get_ctxt ectxt
|
|
622 |
val eq_thm = Simplifier.rewrite ctxt (Thm.cterm_of ctxt exp)
|
|
623 |
val trimmed_cong_thm =
|
|
624 |
case trimmed_thm |> concl_of' |> dest_fun of
|
|
625 |
Const (@{const_name trimmed}, _) => @{thm trimmed_eq_cong}
|
|
626 |
| Const (@{const_name trimmed_pos}, _) => @{thm trimmed_pos_eq_cong}
|
|
627 |
| Const (@{const_name trimmed_neg}, _) => @{thm trimmed_neg_eq_cong}
|
|
628 |
| _ => raise THM ("simplify_trimmed_expansion", 2, [thm, trimmed_thm])
|
|
629 |
in
|
|
630 |
(@{thm expands_to_meta_eq_cong} OF [thm, eq_thm],
|
|
631 |
trimmed_cong_thm OF [trimmed_thm, eq_thm])
|
|
632 |
end
|
|
633 |
|
|
634 |
(*
|
|
635 |
Re-normalises a trimmed expansion (so that the leading term with its (real) coefficient and
|
|
636 |
all exponents can be read off. This may be necessary after lifting a trimmed expansion to
|
|
637 |
a larger basis.
|
|
638 |
*)
|
|
639 |
fun retrim_expansion ectxt (thm, basis) =
|
|
640 |
let
|
|
641 |
val (c, thm, eq_thm) = whnf_expansion ectxt thm
|
|
642 |
in
|
|
643 |
case c of
|
|
644 |
NONE => (thm, eq_thm)
|
|
645 |
| SOME c =>
|
|
646 |
if fastype_of c = @{typ real} then
|
|
647 |
(thm, eq_thm)
|
|
648 |
else
|
|
649 |
let
|
|
650 |
val c_thm = thm RS @{thm expands_to_hd''}
|
|
651 |
val (c_thm', eq_thm') = retrim_expansion ectxt (c_thm, tl_basis basis)
|
|
652 |
val thm = @{thm expands_to_trim_cong} OF [thm, c_thm']
|
|
653 |
in
|
|
654 |
(thm, @{thm trim_lift_eq} OF [eq_thm, eq_thm'])
|
|
655 |
end
|
|
656 |
end
|
|
657 |
|
|
658 |
fun retrim_pos_expansion ectxt (thm, basis, trimmed_thm) =
|
|
659 |
let
|
|
660 |
val (thm', eq_thm) = retrim_expansion ectxt (thm, basis)
|
|
661 |
in
|
|
662 |
(thm', eq_thm, @{thm trimmed_pos_eq_cong} OF [trimmed_thm, eq_thm])
|
|
663 |
end
|
|
664 |
|
|
665 |
(*
|
|
666 |
Tries to determine whether the leading term is (identically) zero and drops it if it is.
|
|
667 |
If "fail" is set, an exception is thrown when that term is a real number and zeroness cannot
|
|
668 |
be determined. (Which typically indicates missing facts or case distinctions)
|
|
669 |
*)
|
|
670 |
fun try_drop_leading_term_ex fail ectxt thm =
|
|
671 |
let
|
|
672 |
val exp = get_expansion thm
|
|
673 |
in
|
|
674 |
if fastype_of exp = @{typ real} then
|
|
675 |
NONE
|
|
676 |
else if fastype_of (get_coeff exp) = @{typ real} then
|
|
677 |
case zeroness_oracle fail (SOME Simple_Trim) ectxt (get_coeff exp) of
|
|
678 |
(IsZero, SOME zero_thm) => SOME (@{thm drop_zero_ms'} OF [zero_thm, thm])
|
|
679 |
| _ => NONE
|
|
680 |
else
|
|
681 |
let
|
|
682 |
val c = get_coeff exp
|
|
683 |
val T = fastype_of c
|
|
684 |
val t = Const (@{const_name eval}, T --> @{typ "real \<Rightarrow> real"}) $ c
|
|
685 |
in
|
|
686 |
case ev_zeroness_oracle ectxt t of
|
|
687 |
SOME zero_thm => SOME (@{thm expands_to_drop_zero} OF [zero_thm, thm])
|
|
688 |
| _ => NONE
|
|
689 |
end
|
|
690 |
end
|
|
691 |
|
|
692 |
(*
|
|
693 |
Tries to drop the leading term of an expansion. If this is not possible, an exception
|
|
694 |
is thrown and an informative error message is printed.
|
|
695 |
*)
|
|
696 |
fun try_drop_leading_term ectxt thm =
|
|
697 |
let
|
|
698 |
fun err () =
|
|
699 |
let
|
|
700 |
val ctxt = get_ctxt ectxt
|
|
701 |
val exp = get_expansion thm
|
|
702 |
val c = get_coeff exp
|
|
703 |
val t =
|
|
704 |
if fastype_of c = @{typ real} then c else c |> dest_arg
|
|
705 |
val t = simplify_term' (get_facts ectxt) ctxt t
|
|
706 |
val _ =
|
|
707 |
if #verbose (#ctxt ectxt) then
|
|
708 |
let
|
|
709 |
val p = Pretty.str ("real_asymp failed to prove that the following term is zero: ")
|
|
710 |
val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t)]
|
|
711 |
in
|
|
712 |
Pretty.writeln p
|
|
713 |
end else ()
|
|
714 |
in
|
|
715 |
raise TERM ("try_drop_leading_term", [t])
|
|
716 |
end
|
|
717 |
in
|
|
718 |
case try_drop_leading_term_ex true ectxt thm of
|
|
719 |
NONE => err ()
|
|
720 |
| SOME thm => thm
|
|
721 |
end
|
|
722 |
|
|
723 |
|
|
724 |
datatype trim_result =
|
|
725 |
Trimmed of zeroness * trimmed_thm option
|
|
726 |
| Aborted of order
|
|
727 |
|
|
728 |
fun cstrip_assms ct =
|
|
729 |
case Thm.term_of ct of
|
|
730 |
@{term "(==>)"} $ _ $ _ => cstrip_assms (snd (Thm.dest_implies ct))
|
|
731 |
| _ => ct
|
|
732 |
|
|
733 |
(*
|
|
734 |
Trims an expansion (i.e. drops leading zero terms) and provides a trimmedness theorem.
|
|
735 |
Optionally, a list of exponents can be given to instruct the function to only trim until
|
|
736 |
the exponents of the leading term are lexicographically less than (or less than or equal) than
|
|
737 |
the given ones. This is useful to avoid unnecessary trimming.
|
|
738 |
|
|
739 |
The "strict" flag indicates whether the trimming should already be aborted when the
|
|
740 |
exponents are lexicographically equal or not.
|
|
741 |
|
|
742 |
The "fail" flag is passed on to the zeroness oracle and determines whether a failure to determine
|
|
743 |
the sign of a real number leads to an exception.
|
|
744 |
|
|
745 |
"mode" indicates what kind of trimmedness theorem will be returned: Simple_Trim only gives the
|
|
746 |
default trimmedness theorem, whereas Pos_Trim/Neg_Trim/Sgn_Trim will give trimmed_pos or
|
|
747 |
trimmed_neg. Giving "None" as mode will produce no trimmedness theorem; it will only drop
|
|
748 |
leading zero terms until zeroness cannot be proven anymore, upon which it will stop.
|
|
749 |
|
|
750 |
The main result of the function is the trimmed expansion theorem.
|
|
751 |
|
|
752 |
The function returns whether the trimming has been aborted or not. If was aborted, either
|
|
753 |
LESS or EQUAL will be returned, indicating whether the exponents of the leading term are
|
|
754 |
now lexicographically smaller or equal to the given ones. In the other case, the zeroness
|
|
755 |
of the leading coefficient is returned (zero, non-zero, positive, negative) together with a
|
|
756 |
trimmedness theorem.
|
|
757 |
|
|
758 |
Lastly, a list of the exponent comparison results and associated theorems is also returned, so
|
|
759 |
that the caller can reconstruct the result of the lexicographic ordering without doing the
|
|
760 |
exponent comparisons again.
|
|
761 |
*)
|
|
762 |
fun trim_expansion_while_greater strict es fail mode ectxt (thm, basis) =
|
|
763 |
let
|
|
764 |
val (_, thm, _) = whnf_expansion ectxt thm
|
|
765 |
val thm = simplify_expansion ectxt thm
|
|
766 |
val cexp = thm |> Thm.cprop_of |> cstrip_assms |> Thm.dest_arg |> Thm.dest_fun |> Thm.dest_arg
|
|
767 |
val c = try_get_coeff (get_expansion thm)
|
|
768 |
fun lift_trimmed_thm nz thm =
|
|
769 |
let
|
|
770 |
val cexp = thm |> Thm.cprop_of |> cstrip_assms |> Thm.dest_arg |> Thm.dest_fun |> Thm.dest_arg
|
|
771 |
val lift_thm =
|
|
772 |
case nz of
|
|
773 |
IsNonZero => @{thm trimmed_eq_cong[rotated, OF _ lift_trimmed]}
|
|
774 |
| IsPos => @{thm trimmed_pos_eq_cong[rotated, OF _ lift_trimmed_pos]}
|
|
775 |
| IsNeg => @{thm trimmed_neg_eq_cong[rotated, OF _ lift_trimmed_neg]}
|
|
776 |
| _ => raise TERM ("Unexpected zeroness result in trim_expansion", [])
|
|
777 |
in
|
|
778 |
Thm.reflexive cexp RS lift_thm
|
|
779 |
end
|
|
780 |
fun trimmed_real_thm nz = Thm.reflexive cexp RS (
|
|
781 |
case nz of
|
|
782 |
IsNonZero => @{thm trimmed_eq_cong[rotated, OF _ lift_trimmed[OF trimmed_realI]]}
|
|
783 |
| IsPos => @{thm trimmed_pos_eq_cong[rotated, OF _ lift_trimmed_pos[OF trimmed_pos_realI]]}
|
|
784 |
| IsNeg => @{thm trimmed_neg_eq_cong[rotated, OF _ lift_trimmed_neg[OF trimmed_neg_realI]]}
|
|
785 |
| _ => raise TERM ("Unexpected zeroness result in trim_expansion", []))
|
|
786 |
fun do_trim es =
|
|
787 |
let
|
|
788 |
val c = the c
|
|
789 |
val T = fastype_of c
|
|
790 |
val t = Const (@{const_name eval}, T --> @{typ "real \<Rightarrow> real"}) $ c
|
|
791 |
in
|
|
792 |
if T = @{typ real} then (
|
|
793 |
case zeroness_oracle fail mode ectxt c of
|
|
794 |
(IsZero, SOME zero_thm) =>
|
|
795 |
trim_expansion_while_greater strict es fail mode ectxt
|
|
796 |
(@{thm drop_zero_ms'} OF [zero_thm, thm], basis)
|
|
797 |
| (nz, SOME nz_thm) => (thm, Trimmed (nz, SOME (nz_thm RS trimmed_real_thm nz)), [])
|
|
798 |
| (nz, NONE) => (thm, Trimmed (nz, NONE), []))
|
|
799 |
else
|
|
800 |
case trim_expansion_while_greater strict (Option.map tl es) fail mode ectxt
|
|
801 |
(thm RS @{thm expands_to_hd''}, tl_basis basis) of
|
|
802 |
(c_thm', Aborted ord, thms) =>
|
|
803 |
(@{thm expands_to_trim_cong} OF [thm, c_thm'], Aborted ord, thms)
|
|
804 |
| (c_thm', Trimmed (nz, trimmed_thm), thms) =>
|
|
805 |
let
|
|
806 |
val thm = (@{thm expands_to_trim_cong} OF [thm, c_thm'])
|
|
807 |
fun err () =
|
|
808 |
raise TERM ("trim_expansion: zero coefficient should have been trimmed", [c])
|
|
809 |
in
|
|
810 |
case (nz, trimmed_thm) of
|
|
811 |
(IsZero, _) =>
|
|
812 |
if #verbose (#ctxt ectxt) then
|
|
813 |
let
|
|
814 |
val ctxt = get_ctxt ectxt
|
|
815 |
val t' = t |> simplify_eval ctxt |> simplify_term' (get_facts ectxt) ctxt
|
|
816 |
val p = Pretty.str ("trim_expansion failed to recognise zeroness of " ^
|
|
817 |
"the following term:")
|
|
818 |
val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t')]
|
|
819 |
val _ = Pretty.writeln p
|
|
820 |
in
|
|
821 |
err ()
|
|
822 |
end
|
|
823 |
else err ()
|
|
824 |
| (_, SOME trimmed_thm) =>
|
|
825 |
(thm, Trimmed (nz, SOME (trimmed_thm RS lift_trimmed_thm nz thm)), thms)
|
|
826 |
| (_, NONE) => (thm, Trimmed (nz, NONE), thms)
|
|
827 |
end
|
|
828 |
end
|
|
829 |
val minus = @{term "(-) :: real => real => real"}
|
|
830 |
in
|
|
831 |
case (c, es) of
|
|
832 |
(NONE, _) => (thm, Trimmed (IsZero, NONE), [])
|
|
833 |
| (SOME c, SOME (e' :: _)) =>
|
|
834 |
let
|
|
835 |
val e = get_exponent (get_expansion thm)
|
|
836 |
in
|
|
837 |
case zeroness_oracle true (SOME Sgn_Trim) ectxt (minus $ e $ e') of
|
|
838 |
(IsPos, SOME pos_thm) => (
|
|
839 |
case try_drop_leading_term_ex false ectxt thm of
|
|
840 |
SOME thm =>
|
|
841 |
trim_expansion_while_greater strict es fail mode ectxt (thm, basis)
|
|
842 |
| NONE => do_trim NONE |> @{apply 3(3)} (fn thms => (IsPos, pos_thm) :: thms))
|
|
843 |
| (IsNeg, SOME neg_thm) => (thm, Aborted LESS, [(IsNeg, neg_thm)])
|
|
844 |
| (IsZero, SOME zero_thm) =>
|
|
845 |
if not strict andalso fastype_of c = @{typ real} then
|
|
846 |
(thm, Aborted EQUAL, [(IsZero, zero_thm)])
|
|
847 |
else (
|
|
848 |
case try_drop_leading_term_ex false ectxt thm of
|
|
849 |
SOME thm => trim_expansion_while_greater strict es fail mode ectxt (thm, basis)
|
|
850 |
| NONE => (do_trim es |> @{apply 3(3)} (fn thms => (IsZero, zero_thm) :: thms)))
|
|
851 |
| _ => do_trim NONE
|
|
852 |
end
|
|
853 |
| _ => (
|
|
854 |
case try_drop_leading_term_ex false ectxt thm of
|
|
855 |
SOME thm => trim_expansion_while_greater strict es fail mode ectxt (thm, basis)
|
|
856 |
| NONE => do_trim NONE)
|
|
857 |
end
|
|
858 |
|
|
859 |
(*
|
|
860 |
Trims an expansion without any stopping criterion.
|
|
861 |
*)
|
|
862 |
fun trim_expansion fail mode ectxt (thm, basis) =
|
|
863 |
case trim_expansion_while_greater false NONE fail mode ectxt (thm, basis) of
|
|
864 |
(thm, Trimmed (zeroness, trimmed_thm), _) => (thm, zeroness, trimmed_thm)
|
|
865 |
| _ => raise Match
|
|
866 |
|
|
867 |
(*
|
|
868 |
Determines the sign of an expansion that has already been trimmed.
|
|
869 |
*)
|
|
870 |
fun determine_trimmed_sgn ectxt exp =
|
|
871 |
if fastype_of exp = @{typ real} then
|
|
872 |
(case zeroness_oracle true (SOME Sgn_Trim) ectxt exp of
|
|
873 |
(IsPos, SOME thm) => (IsPos, thm RS @{thm trimmed_pos_realI})
|
|
874 |
| (IsNeg, SOME thm) => (IsNeg, thm RS @{thm trimmed_neg_realI})
|
|
875 |
| _ => raise TERM ("determine_trimmed_sgn", []))
|
|
876 |
else
|
|
877 |
let
|
|
878 |
val ct = Thm.cterm_of (get_ctxt ectxt) exp
|
|
879 |
in
|
|
880 |
(case determine_trimmed_sgn ectxt (get_coeff exp) of
|
|
881 |
(IsPos, thm) => (IsPos, @{thm lift_trimmed_pos'} OF [thm, Thm.reflexive ct])
|
|
882 |
| (IsNeg, thm) => (IsNeg, @{thm lift_trimmed_neg'} OF [thm, Thm.reflexive ct])
|
|
883 |
| _ => raise TERM ("determine_trimmed_sgn", []))
|
|
884 |
end
|
|
885 |
|
|
886 |
fun mk_compare_expansions_const T =
|
|
887 |
Const (@{const_name compare_expansions},
|
|
888 |
T --> T --> @{typ "cmp_result \<times> real \<times> real"})
|
|
889 |
|
|
890 |
datatype comparison_result =
|
|
891 |
Cmp_Dominated of order * thm list * zeroness * trimmed_thm * expansion_thm * expansion_thm
|
|
892 |
| Cmp_Asymp_Equiv of thm * thm
|
|
893 |
|
|
894 |
fun compare_expansions' _ (thm1, thm2, SEmpty) = Cmp_Asymp_Equiv (thm1, thm2)
|
|
895 |
| compare_expansions' ectxt (thm1, thm2, basis) =
|
|
896 |
let
|
|
897 |
fun lift_trimmed_thm nz =
|
|
898 |
case nz of
|
|
899 |
IsPos => @{thm lift_trimmed_pos}
|
|
900 |
| IsNeg => @{thm lift_trimmed_neg}
|
|
901 |
| _ => raise TERM ("Unexpected zeroness result in compare_expansions'", [])
|
|
902 |
val (e1, e2) = apply2 (get_expansion #> get_exponent) (thm1, thm2)
|
|
903 |
val e = @{term "(-) :: real => _"} $ e1 $ e2
|
|
904 |
fun trim thm = trim_expansion true (SOME Sgn_Trim) ectxt (thm, basis)
|
|
905 |
val try_drop = Option.map (whnf_expansion ectxt #> #2) o try_drop_leading_term_ex false ectxt
|
|
906 |
fun handle_result ord zeroness trimmed_thm thm1 thm2 =
|
|
907 |
let
|
|
908 |
val (e1, e2) = apply2 (get_expansion #> get_exponent) (thm1, thm2)
|
|
909 |
val e = @{term "(-) :: real => _"} $ e1 $ e2
|
|
910 |
val mode = if ord = LESS then Neg_Trim else Pos_Trim
|
|
911 |
in
|
|
912 |
case zeroness_oracle true (SOME mode) ectxt e of
|
|
913 |
(_, SOME e_thm) => Cmp_Dominated (ord, [e_thm], zeroness, trimmed_thm, thm1, thm2)
|
|
914 |
| _ => raise Match
|
|
915 |
end
|
|
916 |
fun recurse e_zero_thm =
|
|
917 |
case basis of
|
|
918 |
SNE (SSng _) => Cmp_Asymp_Equiv (thm1, thm2)
|
|
919 |
| _ =>
|
|
920 |
let
|
|
921 |
val (thm1', thm2') = apply2 (fn thm => thm RS @{thm expands_to_hd''}) (thm1, thm2)
|
|
922 |
val (thm1', thm2') = apply2 (whnf_expansion ectxt #> #2) (thm1', thm2')
|
|
923 |
in
|
|
924 |
case compare_expansions' ectxt (thm1', thm2', tl_basis basis) of
|
|
925 |
Cmp_Dominated (order, e_thms, zeroness, trimmed_thm, thm1', thm2') =>
|
|
926 |
Cmp_Dominated (order, e_zero_thm :: e_thms, zeroness,
|
|
927 |
trimmed_thm RS lift_trimmed_thm zeroness,
|
|
928 |
@{thm expands_to_trim_cong} OF [thm1, thm1'],
|
|
929 |
@{thm expands_to_trim_cong} OF [thm2, thm2'])
|
|
930 |
| Cmp_Asymp_Equiv (thm1', thm2') => Cmp_Asymp_Equiv
|
|
931 |
(@{thm expands_to_trim_cong} OF [thm1, thm1'],
|
|
932 |
@{thm expands_to_trim_cong} OF [thm2, thm2'])
|
|
933 |
end
|
|
934 |
in
|
|
935 |
case zeroness_oracle false (SOME Sgn_Trim) ectxt e of
|
|
936 |
(IsPos, SOME _) => (
|
|
937 |
case try_drop thm1 of
|
|
938 |
SOME thm1 => compare_expansions' ectxt (thm1, thm2, basis)
|
|
939 |
| NONE => (
|
|
940 |
case trim thm1 of
|
|
941 |
(thm1, zeroness, SOME trimmed_thm) =>
|
|
942 |
handle_result GREATER zeroness trimmed_thm thm1 thm2
|
|
943 |
| _ => raise TERM ("compare_expansions", map get_expansion [thm1, thm2])))
|
|
944 |
| (IsNeg, SOME _) => (
|
|
945 |
case try_drop thm2 of
|
|
946 |
SOME thm2 => compare_expansions' ectxt (thm1, thm2, basis)
|
|
947 |
| NONE => (
|
|
948 |
case trim thm2 of
|
|
949 |
(thm2, zeroness, SOME trimmed_thm) =>
|
|
950 |
handle_result LESS zeroness trimmed_thm thm1 thm2
|
|
951 |
| _ => raise TERM ("compare_expansions", map get_expansion [thm1, thm2])))
|
|
952 |
| (IsZero, SOME e_zero_thm) => (
|
|
953 |
case try_drop thm1 of
|
|
954 |
SOME thm1 => compare_expansions' ectxt (thm1, thm2, basis)
|
|
955 |
| NONE => (
|
|
956 |
case try_drop thm2 of
|
|
957 |
SOME thm2 => compare_expansions' ectxt (thm1, thm2, basis)
|
|
958 |
| NONE => recurse e_zero_thm))
|
|
959 |
| _ =>
|
|
960 |
case try_drop thm1 of
|
|
961 |
SOME thm1 => compare_expansions' ectxt (thm1, thm2, basis)
|
|
962 |
| NONE => (
|
|
963 |
case try_drop thm2 of
|
|
964 |
SOME thm2 => compare_expansions' ectxt (thm1, thm2, basis)
|
|
965 |
| NONE => raise TERM ("compare_expansions", [e1, e2]))
|
|
966 |
end
|
|
967 |
|
|
968 |
(* Uses a list of exponent comparison results to show that compare_expansions has a given result.*)
|
|
969 |
fun prove_compare_expansions ord [thm] = (
|
|
970 |
case ord of
|
|
971 |
LESS => @{thm compare_expansions_LT_I} OF [thm]
|
|
972 |
| GREATER => @{thm compare_expansions_GT_I} OF [thm]
|
|
973 |
| EQUAL => @{thm compare_expansions_same_exp[OF _ compare_expansions_real]} OF [thm])
|
|
974 |
| prove_compare_expansions ord (thm :: thms) =
|
|
975 |
@{thm compare_expansions_same_exp} OF [thm, prove_compare_expansions ord thms]
|
|
976 |
| prove_compare_expansions _ [] = raise Match
|
|
977 |
|
|
978 |
val ev_zero_pos_thm = Eventuallize.eventuallize @{context}
|
|
979 |
@{lemma "\<forall>x::real. f x = 0 \<longrightarrow> g x > 0 \<longrightarrow> f x < g x" by auto} NONE
|
|
980 |
OF @{thms _ expands_to_imp_eventually_pos}
|
|
981 |
|
|
982 |
val ev_zero_neg_thm = Eventuallize.eventuallize @{context}
|
|
983 |
@{lemma "\<forall>x::real. f x = 0 \<longrightarrow> g x < 0 \<longrightarrow> f x > g x" by auto} NONE
|
|
984 |
OF @{thms _ expands_to_imp_eventually_neg}
|
|
985 |
|
|
986 |
val ev_zero_zero_thm = Eventuallize.eventuallize @{context}
|
|
987 |
@{lemma "\<forall>x::real. f x = 0 \<longrightarrow> g x = 0 \<longrightarrow> f x = g x" by auto} NONE
|
|
988 |
|
|
989 |
fun compare_expansions_trivial ectxt (thm1, thm2, basis) =
|
|
990 |
case try_prove_ev_eq ectxt (apply2 get_expanded_fun (thm1, thm2)) of
|
|
991 |
SOME thm => SOME (EQUAL, thm, thm1, thm2)
|
|
992 |
| NONE =>
|
|
993 |
case apply2 (ev_zeroness_oracle ectxt o get_expanded_fun) (thm1, thm2) of
|
|
994 |
(NONE, NONE) => NONE
|
|
995 |
| (SOME zero1_thm, NONE) => (
|
|
996 |
case trim_expansion true (SOME Sgn_Trim) ectxt (thm2, basis) of
|
|
997 |
(thm2, IsPos, SOME trimmed2_thm) =>
|
|
998 |
SOME (LESS, ev_zero_pos_thm OF
|
|
999 |
[zero1_thm, get_basis_wf_thm basis, thm2, trimmed2_thm], thm1, thm2)
|
|
1000 |
| (thm2, IsNeg, SOME trimmed2_thm) =>
|
|
1001 |
SOME (GREATER, ev_zero_neg_thm OF
|
|
1002 |
[zero1_thm, get_basis_wf_thm basis, thm2, trimmed2_thm], thm1, thm2)
|
|
1003 |
| _ => raise TERM ("Unexpected zeroness result in compare_expansions", []))
|
|
1004 |
| (NONE, SOME zero2_thm) => (
|
|
1005 |
case trim_expansion true (SOME Sgn_Trim) ectxt (thm1, basis) of
|
|
1006 |
(thm1, IsPos, SOME trimmed1_thm) =>
|
|
1007 |
SOME (GREATER, ev_zero_pos_thm OF
|
|
1008 |
[zero2_thm, get_basis_wf_thm basis, thm1, trimmed1_thm], thm1, thm2)
|
|
1009 |
| (thm1, IsNeg, SOME trimmed1_thm) =>
|
|
1010 |
SOME (LESS, ev_zero_neg_thm OF
|
|
1011 |
[zero2_thm, get_basis_wf_thm basis, thm1, trimmed1_thm], thm1, thm2)
|
|
1012 |
| _ => raise TERM ("Unexpected zeroness result in compare_expansions", []))
|
|
1013 |
| (SOME zero1_thm, SOME zero2_thm) =>
|
|
1014 |
SOME (EQUAL, ev_zero_zero_thm OF [zero1_thm, zero2_thm] , thm1, thm2)
|
|
1015 |
|
|
1016 |
fun compare_expansions ectxt (thm1, thm2, basis) =
|
|
1017 |
case compare_expansions_trivial ectxt (thm1, thm2, basis) of
|
|
1018 |
SOME res => res
|
|
1019 |
| NONE =>
|
|
1020 |
let
|
|
1021 |
val (_, thm1, _) = whnf_expansion ectxt thm1
|
|
1022 |
val (_, thm2, _) = whnf_expansion ectxt thm2
|
|
1023 |
in
|
|
1024 |
case compare_expansions' ectxt (thm1, thm2, basis) of
|
|
1025 |
Cmp_Dominated (order, e_thms, zeroness, trimmed_thm, thm1, thm2) =>
|
|
1026 |
let
|
|
1027 |
val wf_thm = get_basis_wf_thm basis
|
|
1028 |
val cmp_thm = prove_compare_expansions order e_thms
|
|
1029 |
val trimmed_thm' = trimmed_thm RS
|
|
1030 |
(if zeroness = IsPos then @{thm trimmed_pos_imp_trimmed}
|
|
1031 |
else @{thm trimmed_neg_imp_trimmed})
|
|
1032 |
val smallo_thm =
|
|
1033 |
(if order = LESS then @{thm compare_expansions_LT} else @{thm compare_expansions_GT}) OF
|
|
1034 |
[cmp_thm, trimmed_thm', thm1, thm2, wf_thm]
|
|
1035 |
val thm' =
|
|
1036 |
if zeroness = IsPos then @{thm smallo_trimmed_imp_eventually_less}
|
|
1037 |
else @{thm smallo_trimmed_imp_eventually_greater}
|
|
1038 |
val result_thm =
|
|
1039 |
thm' OF [smallo_thm, if order = LESS then thm2 else thm1, wf_thm, trimmed_thm]
|
|
1040 |
in
|
|
1041 |
(order, result_thm, thm1, thm2)
|
|
1042 |
end
|
|
1043 |
| Cmp_Asymp_Equiv (thm1, thm2) =>
|
|
1044 |
let
|
|
1045 |
val thm = @{thm expands_to_minus} OF [get_basis_wf_thm basis, thm1, thm2]
|
|
1046 |
val (order, result_thm) =
|
|
1047 |
case trim_expansion true (SOME Sgn_Trim) ectxt (thm, basis) of
|
|
1048 |
(thm, IsPos, SOME pos_thm) => (GREATER,
|
|
1049 |
@{thm expands_to_imp_eventually_gt} OF [get_basis_wf_thm basis, thm, pos_thm])
|
|
1050 |
| (thm, IsNeg, SOME neg_thm) => (LESS,
|
|
1051 |
@{thm expands_to_imp_eventually_lt} OF [get_basis_wf_thm basis, thm, neg_thm])
|
|
1052 |
| _ => raise TERM ("Unexpected zeroness result in prove_eventually_less", [])
|
|
1053 |
in
|
|
1054 |
(order, result_thm, thm1, thm2)
|
|
1055 |
end
|
|
1056 |
end
|
|
1057 |
|
|
1058 |
|
|
1059 |
|
|
1060 |
(*
|
|
1061 |
Throws an exception and prints an error message indicating that the leading term could
|
|
1062 |
not be determined to be either zero or non-zero.
|
|
1063 |
*)
|
|
1064 |
fun raise_trimming_error ectxt thm =
|
|
1065 |
let
|
|
1066 |
val ctxt = get_ctxt ectxt
|
|
1067 |
fun lead_coeff exp =
|
|
1068 |
if fastype_of exp = @{typ real} then exp else lead_coeff (get_coeff exp)
|
|
1069 |
val c = lead_coeff (get_expansion thm)
|
|
1070 |
fun err () =
|
|
1071 |
let
|
|
1072 |
val t = simplify_term' (get_facts ectxt) ctxt c
|
|
1073 |
val _ =
|
|
1074 |
if #verbose (#ctxt ectxt) then
|
|
1075 |
let
|
|
1076 |
val p = Pretty.str
|
|
1077 |
("real_asymp failed to determine whether the following constant is zero:")
|
|
1078 |
val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t)]
|
|
1079 |
in
|
|
1080 |
Pretty.writeln p
|
|
1081 |
end else ()
|
|
1082 |
in
|
|
1083 |
raise TERM ("zeroness_oracle", [t])
|
|
1084 |
end
|
|
1085 |
in
|
|
1086 |
err ()
|
|
1087 |
end
|
|
1088 |
|
|
1089 |
|
|
1090 |
(* TODO Here be dragons *)
|
|
1091 |
fun solve_eval_eq thm =
|
|
1092 |
case try (fn _ => @{thm refl} RS thm) () of
|
|
1093 |
SOME thm' => thm'
|
|
1094 |
| NONE =>
|
|
1095 |
case try (fn _ => @{thm eval_real_def} RS thm) () of
|
|
1096 |
SOME thm' => thm'
|
|
1097 |
| NONE => @{thm eval_ms.simps} RS thm
|
|
1098 |
|
|
1099 |
(*
|
|
1100 |
Returns an expansion theorem for the logarithm of the given expansion.
|
|
1101 |
May add one additional element to the basis at the end.
|
|
1102 |
*)
|
|
1103 |
fun ln_expansion _ _ _ SEmpty = raise TERM ("ln_expansion: empty basis", [])
|
|
1104 |
| ln_expansion ectxt trimmed_thm thm (SNE basis) =
|
|
1105 |
let
|
|
1106 |
fun trailing_exponent expr (SSng _) = get_exponent expr
|
|
1107 |
| trailing_exponent expr (SCons (_, _, tl)) = trailing_exponent (get_coeff expr) tl
|
|
1108 |
val e = trailing_exponent (get_expansion thm) basis
|
|
1109 |
fun ln_expansion_aux trimmed_thm zero_thm thm basis =
|
|
1110 |
let
|
|
1111 |
val t = betapply (@{term "\<lambda>(f::real \<Rightarrow> real) x. f x - 1 :: real"}, get_expanded_fun thm)
|
|
1112 |
in
|
|
1113 |
case ev_zeroness_oracle ectxt t of
|
|
1114 |
NONE => ln_expansion_aux' trimmed_thm zero_thm thm basis
|
|
1115 |
| SOME zero_thm =>
|
|
1116 |
@{thm expands_to_ln_eventually_1} OF
|
|
1117 |
[get_basis_wf_thm' basis, mk_expansion_level_eq_thm' basis, zero_thm]
|
|
1118 |
end
|
|
1119 |
and ln_expansion_aux' trimmed_thm zero_thm thm (SSng {wf_thm, ...}) =
|
|
1120 |
( @{thm expands_to_ln} OF
|
|
1121 |
[trimmed_thm, wf_thm, thm,
|
|
1122 |
@{thm expands_to_ln_aux_0} OF [zero_thm, @{thm expands_to_ln_const}]])
|
|
1123 |
|> solve_eval_eq
|
|
1124 |
| ln_expansion_aux' trimmed_thm zero_thm thm (SCons ({wf_thm, ...}, {ln_thm, ...}, basis')) =
|
|
1125 |
let
|
|
1126 |
val c_thm =
|
|
1127 |
ln_expansion_aux (trimmed_thm RS @{thm trimmed_pos_hd_coeff}) zero_thm
|
|
1128 |
(expands_to_hd thm) basis'
|
|
1129 |
val e = get_exponent (get_expansion thm)
|
|
1130 |
val c_thm' =
|
|
1131 |
case zeroness_oracle true NONE ectxt e of
|
|
1132 |
(IsZero, SOME thm) =>
|
|
1133 |
@{thm expands_to_ln_to_expands_to_ln_eval [OF expands_to_ln_aux_0]} OF [thm,c_thm]
|
|
1134 |
| _ =>
|
|
1135 |
case try_prove_real_eq false ectxt (e, @{term "1::real"}) of
|
|
1136 |
SOME thm =>
|
|
1137 |
@{thm expands_to_ln_to_expands_to_ln_eval [OF expands_to_ln_aux_1]}
|
|
1138 |
OF [thm, wf_thm, c_thm, ln_thm]
|
|
1139 |
| NONE =>
|
|
1140 |
@{thm expands_to_ln_to_expands_to_ln_eval [OF expands_to_ln_aux]}
|
|
1141 |
OF [wf_thm, c_thm, ln_thm]
|
|
1142 |
in
|
|
1143 |
(@{thm expands_to_ln} OF [trimmed_thm, wf_thm, thm, c_thm'])
|
|
1144 |
|> solve_eval_eq
|
|
1145 |
end
|
|
1146 |
in
|
|
1147 |
case zeroness_oracle true NONE ectxt e of
|
|
1148 |
(IsZero, SOME zero_thm) => (ln_expansion_aux trimmed_thm zero_thm thm basis, SNE basis)
|
|
1149 |
| _ =>
|
|
1150 |
let
|
|
1151 |
val basis' = insert_ln (SNE basis)
|
|
1152 |
val lifting = mk_lifting (get_basis_list' basis) basis'
|
|
1153 |
val thm' = lift_expands_to_thm lifting thm
|
|
1154 |
val trimmed_thm' = lift_trimmed_pos_thm lifting trimmed_thm
|
|
1155 |
val (thm'', eq_thm) = retrim_expansion ectxt (thm', basis')
|
|
1156 |
val trimmed_thm'' = @{thm trimmed_pos_eq_cong} OF [trimmed_thm', eq_thm]
|
|
1157 |
in
|
|
1158 |
ln_expansion ectxt trimmed_thm'' thm'' basis'
|
|
1159 |
end
|
|
1160 |
end
|
|
1161 |
|
|
1162 |
(*
|
|
1163 |
Handles a possible basis change after expanding exp(c(x)) for an expansion of the form
|
|
1164 |
f(x) = c(x) + g(x). Expanding exp(c(x)) may have inserted an additional basis element. If the
|
|
1165 |
old basis was b :: bs (i.e. c is an expansion w.r.t. bs) and the updated one is bs' (which
|
|
1166 |
agrees with bs except for one additional element b'), we need to argue that b :: bs' is still
|
|
1167 |
well-formed. This may require us to show that ln(b') is o(ln(b)), which the function takes
|
|
1168 |
as an argument.
|
|
1169 |
*)
|
|
1170 |
fun adjust_exp_basis basis basis' ln_smallo_thm =
|
|
1171 |
if length (get_basis_list basis) = length (get_basis_list basis') + 1 then
|
|
1172 |
basis
|
|
1173 |
else
|
|
1174 |
let
|
|
1175 |
val SNE (SCons (info, ln_info, tail)) = basis
|
|
1176 |
val SNE tail' = basis'
|
|
1177 |
val wf_thms = map get_basis_wf_thm [basis, basis']
|
|
1178 |
val wf_thm' =
|
|
1179 |
case
|
|
1180 |
get_first (fn f => try f ())
|
|
1181 |
[fn _ => @{thm basis_wf_lift_modification} OF wf_thms,
|
|
1182 |
fn _ => @{thm basis_wf_insert_exp_near} OF (wf_thms @ [ln_smallo_thm]),
|
|
1183 |
fn _ => @{thm basis_wf_insert_exp_near} OF (wf_thms @
|
|
1184 |
[ln_smallo_thm RS @{thm basis_wf_insert_exp_uminus'}])] of
|
|
1185 |
SOME wf_thm => wf_thm
|
|
1186 |
| _ => raise TERM ("Lifting basis modification in exp_expansion failed.", map Thm.concl_of (wf_thms @ [ln_smallo_thm]))
|
|
1187 |
val info' = {wf_thm = wf_thm', head = #head info}
|
|
1188 |
val lifting = mk_lifting (get_basis_list' tail) basis'
|
|
1189 |
val ln_info' =
|
|
1190 |
{trimmed_thm = lift_trimmed_pos_thm lifting (#trimmed_thm ln_info),
|
|
1191 |
ln_thm = lift_expands_to_thm lifting (#ln_thm ln_info)}
|
|
1192 |
in
|
|
1193 |
SNE (SCons (info', ln_info', tail'))
|
|
1194 |
end
|
|
1195 |
|
|
1196 |
(* inserts the exponential of a given function at the beginning of the given basis *)
|
|
1197 |
fun insert_exp _ _ _ _ _ SEmpty = raise TERM ("insert_exp", [])
|
|
1198 |
| insert_exp t ln_thm ln_smallo_thm ln_trimmed_thm lim_thm (SNE basis) =
|
|
1199 |
let
|
|
1200 |
val head = Envir.beta_eta_contract (@{term "\<lambda>(f::real\<Rightarrow>real) x. exp (f x)"} $ t)
|
|
1201 |
val ln_smallo_thm = ln_smallo_thm RS @{thm ln_smallo_ln_exp}
|
|
1202 |
val wf_thm = @{thm basis_wf_manyI} OF [lim_thm, ln_smallo_thm, get_basis_wf_thm' basis]
|
|
1203 |
val basis' = SNE (SCons ({wf_thm = wf_thm, head = head},
|
|
1204 |
{ln_thm = ln_thm, trimmed_thm = ln_trimmed_thm} , basis))
|
|
1205 |
in
|
|
1206 |
check_basis basis'
|
|
1207 |
end
|
|
1208 |
|
|
1209 |
(*
|
|
1210 |
Returns an expansion of the exponential of the given expansion. This may add several
|
|
1211 |
new basis elements at any position of the basis (except at the very end
|
|
1212 |
*)
|
|
1213 |
fun exp_expansion _ thm SEmpty = (thm RS @{thm expands_to_exp_real}, SEmpty)
|
|
1214 |
| exp_expansion ectxt thm basis =
|
|
1215 |
let
|
|
1216 |
val (_, thm, _) = whnf_expansion ectxt thm
|
|
1217 |
in
|
|
1218 |
case ev_zeroness_oracle ectxt (get_eval (get_expansion thm)) of
|
|
1219 |
SOME zero_thm =>
|
|
1220 |
(@{thm expands_to_exp_zero} OF
|
|
1221 |
[thm, zero_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis], basis)
|
|
1222 |
| NONE =>
|
|
1223 |
let
|
|
1224 |
val ln =
|
|
1225 |
Option.map (fn x => (#ln_thm x, #trimmed_thm x)) (get_ln_info basis)
|
|
1226 |
val ln = Option.map (fn (x, y) => retrim_pos_expansion ectxt (x, basis, y)) ln
|
|
1227 |
val es' = @{term "0::real"} :: (
|
|
1228 |
case ln of
|
|
1229 |
NONE => []
|
|
1230 |
| SOME (ln_thm, _, _) => get_exponents (get_expansion ln_thm))
|
|
1231 |
val trim_result =
|
|
1232 |
trim_expansion_while_greater true (SOME es') false (SOME Simple_Trim) ectxt (thm, basis)
|
|
1233 |
in
|
|
1234 |
exp_expansion' ectxt trim_result ln basis
|
|
1235 |
end
|
|
1236 |
end
|
|
1237 |
and exp_expansion' _ (thm, _, _) _ SEmpty = (thm RS @{thm expands_to_exp_real}, SEmpty)
|
|
1238 |
| exp_expansion' ectxt (thm, trim_result, e_thms) ln basis =
|
|
1239 |
let
|
|
1240 |
val exp = get_expansion thm
|
|
1241 |
val wf_thm = get_basis_wf_thm basis
|
|
1242 |
val f = get_expanded_fun thm
|
|
1243 |
fun exp_expansion_insert ln_smallo_thm = (
|
|
1244 |
case determine_trimmed_sgn ectxt exp of
|
|
1245 |
(IsPos, trimmed_thm) =>
|
|
1246 |
let
|
|
1247 |
val [lim_thm, ln_thm', thm'] =
|
|
1248 |
@{thms expands_to_exp_insert_pos}
|
|
1249 |
|> map (fn thm' => thm' OF [thm, wf_thm, trimmed_thm, ln_smallo_thm])
|
|
1250 |
val basis' = insert_exp f ln_thm' ln_smallo_thm trimmed_thm lim_thm basis
|
|
1251 |
in
|
|
1252 |
(thm', basis')
|
|
1253 |
end
|
|
1254 |
| (IsNeg, trimmed_thm) =>
|
|
1255 |
let
|
|
1256 |
val [lim_thm, ln_thm', ln_trimmed_thm, thm'] =
|
|
1257 |
@{thms expands_to_exp_insert_neg}
|
|
1258 |
|> map (fn thm' => thm' OF [thm, wf_thm, trimmed_thm, ln_smallo_thm])
|
|
1259 |
val ln_smallo_thm = ln_smallo_thm RS @{thm basis_wf_insert_exp_uminus}
|
|
1260 |
val f' = Envir.beta_eta_contract (@{term "\<lambda>(f::real\<Rightarrow>real) x. -f x"} $ f)
|
|
1261 |
val basis' = insert_exp f' ln_thm' ln_smallo_thm ln_trimmed_thm lim_thm basis
|
|
1262 |
in
|
|
1263 |
(thm', basis')
|
|
1264 |
end
|
|
1265 |
| _ => raise TERM ("Unexpected zeroness result in exp_expansion", []))
|
|
1266 |
fun lexord (IsNeg :: _) = LESS
|
|
1267 |
| lexord (IsPos :: _) = GREATER
|
|
1268 |
| lexord (IsZero :: xs) = lexord xs
|
|
1269 |
| lexord [] = EQUAL
|
|
1270 |
| lexord _ = raise Match
|
|
1271 |
val compare_result = lexord (map fst e_thms)
|
|
1272 |
in
|
|
1273 |
case (trim_result, e_thms, compare_result) of
|
|
1274 |
(Aborted _, (IsNeg, e_neg_thm) :: _, _) =>
|
|
1275 |
(* leading exponent is negative; we can simply Taylor-expand exp(x) around 0 *)
|
|
1276 |
(@{thm expands_to_exp_neg} OF [thm, get_basis_wf_thm basis, e_neg_thm], basis)
|
|
1277 |
| (Trimmed (_, SOME trimmed_thm), (IsPos, e_pos_thm) :: _, GREATER) =>
|
|
1278 |
(* leading exponent is positive; exp(f(x)) or exp(-f(x)) is new basis element *)
|
|
1279 |
let
|
|
1280 |
val ln_smallo_thm =
|
|
1281 |
@{thm basis_wf_insert_exp_pos} OF [thm, get_basis_wf_thm basis, trimmed_thm, e_pos_thm]
|
|
1282 |
in
|
|
1283 |
exp_expansion_insert ln_smallo_thm
|
|
1284 |
end
|
|
1285 |
| (Trimmed (_, SOME trimmed_thm), _, GREATER) =>
|
|
1286 |
(* leading exponent is zero, but f(x) grows faster than ln(b(x)), so
|
|
1287 |
exp(f(x)) or exp(-f(x)) must still be new basis elements *)
|
|
1288 |
let
|
|
1289 |
val ln_thm =
|
|
1290 |
case ln of
|
|
1291 |
SOME (ln_thm, _, _) => ln_thm
|
|
1292 |
| NONE => raise TERM ("TODO blubb", [])
|
|
1293 |
val ln_thm = @{thm expands_to_lift''} OF [get_basis_wf_thm basis, ln_thm]
|
|
1294 |
val ln_smallo_thm =
|
|
1295 |
@{thm compare_expansions_GT} OF [prove_compare_expansions GREATER (map snd e_thms),
|
|
1296 |
trimmed_thm, thm, ln_thm, get_basis_wf_thm basis]
|
|
1297 |
in
|
|
1298 |
exp_expansion_insert ln_smallo_thm
|
|
1299 |
end
|
|
1300 |
| (Aborted LESS, (IsZero, e_zero_thm) :: e_thms', _) =>
|
|
1301 |
(* leading exponent is zero and f(x) grows more slowly than ln(b(x)), so
|
|
1302 |
we can write f(x) = c(x) + g(x) and therefore exp(f(x)) = exp(c(x)) * exp(g(x)).
|
|
1303 |
The former is treated by a recursive call; the latter by Taylor expansion. *)
|
|
1304 |
let
|
|
1305 |
val (ln_thm, trimmed_thm) =
|
|
1306 |
case ln of
|
|
1307 |
SOME (ln_thm, _, trimmed_thm) =>
|
|
1308 |
(ln_thm, trimmed_thm RS @{thm trimmed_pos_imp_trimmed})
|
|
1309 |
| NONE => raise TERM ("TODO foo", [])
|
|
1310 |
val c_thm = expands_to_hd thm
|
|
1311 |
val ln_smallo_thm =
|
|
1312 |
@{thm compare_expansions_LT} OF [prove_compare_expansions LESS (map snd e_thms'),
|
|
1313 |
trimmed_thm, c_thm, ln_thm, get_basis_wf_thm (tl_basis basis)]
|
|
1314 |
val (c_thm, c_basis) = exp_expansion ectxt c_thm (tl_basis basis)
|
|
1315 |
val basis' = adjust_exp_basis basis c_basis ln_smallo_thm
|
|
1316 |
val wf_thm = get_basis_wf_thm basis'
|
|
1317 |
val thm' = lift basis' thm
|
|
1318 |
val (thm'', _) = retrim_expansion ectxt (thm', basis')
|
|
1319 |
in
|
|
1320 |
(@{thm expands_to_exp_0} OF [thm'', wf_thm, e_zero_thm, c_thm], basis')
|
|
1321 |
end
|
|
1322 |
| (Trimmed _, [(IsZero, e_zero_thm)], EQUAL) =>
|
|
1323 |
(* f(x) can be written as c + g(x) where c is just a real constant.
|
|
1324 |
We can therefore write exp(f(x)) = exp(c) * exp(g(x)), where the latter is
|
|
1325 |
a simple Taylor expansion. *)
|
|
1326 |
(@{thm expands_to_exp_0_real} OF [thm, wf_thm, e_zero_thm], basis)
|
|
1327 |
| (Trimmed _, (_, e_zero_thm) :: _, EQUAL) =>
|
|
1328 |
(* f(x) is asymptotically equivalent to c * ln(b(x)), so we can write f(x) as
|
|
1329 |
c * ln(b(x)) + g(x) and therefore exp(f(x)) = b(x)^c * exp(g(x)). The second
|
|
1330 |
factor is handled by a recursive call *)
|
|
1331 |
let
|
|
1332 |
val ln_thm =
|
|
1333 |
case ln of
|
|
1334 |
SOME (ln_thm, _, _) => ln_thm
|
|
1335 |
| NONE => raise TERM ("TODO blargh", [])
|
|
1336 |
val c =
|
|
1337 |
case (thm, ln_thm) |> apply2 (get_expansion #> get_lead_coeff) of
|
|
1338 |
(c1, c2) => @{term "(/) :: real => _"} $ c1 $ c2
|
|
1339 |
val c = Thm.cterm_of (get_ctxt ectxt) c
|
|
1340 |
|
|
1341 |
val thm' =
|
|
1342 |
@{thm expands_to_exp_0_pull_out1}
|
|
1343 |
OF [thm, ln_thm, wf_thm, e_zero_thm, Thm.reflexive c]
|
|
1344 |
val (thm'', basis') = exp_expansion ectxt thm' basis
|
|
1345 |
val pat = ConsPat ("MS", [AnyPat ("_", 0), AnyPat ("_", 0)])
|
|
1346 |
val (_, _, conv) = match ectxt pat (get_expansion thm'') (SOME [])
|
|
1347 |
val eq_thm = conv (Thm.cterm_of (get_ctxt ectxt) (get_expansion thm''))
|
|
1348 |
val thm''' = @{thm expands_to_meta_eq_cong} OF [thm'', eq_thm]
|
|
1349 |
val thm'''' =
|
|
1350 |
case get_intyness (get_ctxt ectxt) c of
|
|
1351 |
No_Nat =>
|
|
1352 |
@{thm expands_to_exp_0_pull_out2} OF [thm''', get_basis_wf_thm basis']
|
|
1353 |
| Nat nat_thm =>
|
|
1354 |
@{thm expands_to_exp_0_pull_out2_nat} OF
|
|
1355 |
[thm''', get_basis_wf_thm basis', nat_thm]
|
|
1356 |
| Neg_Nat nat_thm =>
|
|
1357 |
@{thm expands_to_exp_0_pull_out2_neg_nat} OF
|
|
1358 |
[thm''', get_basis_wf_thm basis', nat_thm]
|
|
1359 |
in
|
|
1360 |
(thm'''', basis')
|
|
1361 |
end
|
|
1362 |
| (Trimmed (IsZero, _), [], _) =>
|
|
1363 |
(* Expansion is empty, i.e. f(x) is identically zero *)
|
|
1364 |
(@{thm expands_to_exp_MSLNil} OF [thm, get_basis_wf_thm basis], basis)
|
|
1365 |
| (Trimmed (_, NONE), _, GREATER) =>
|
|
1366 |
(* We could not determine whether f(x) grows faster than ln(b(x)) or not. *)
|
|
1367 |
raise_trimming_error ectxt thm
|
|
1368 |
| _ => raise Match
|
|
1369 |
end
|
|
1370 |
|
|
1371 |
fun powr_expansion ectxt (thm1, thm2, basis) =
|
|
1372 |
case ev_zeroness_oracle ectxt (get_expanded_fun thm1) of
|
|
1373 |
SOME zero_thm =>
|
|
1374 |
(@{thm expands_to_powr_0} OF
|
|
1375 |
[zero_thm, Thm.reflexive (Thm.cterm_of (get_ctxt ectxt) (get_expanded_fun thm2)),
|
|
1376 |
get_basis_wf_thm basis, mk_expansion_level_eq_thm basis],
|
|
1377 |
basis)
|
|
1378 |
| NONE =>
|
|
1379 |
let
|
|
1380 |
val (thm1, _, SOME trimmed_thm) =
|
|
1381 |
trim_expansion true (SOME Pos_Trim) ectxt (thm1, basis)
|
|
1382 |
val (ln_thm, basis') = ln_expansion ectxt trimmed_thm thm1 basis
|
|
1383 |
val thm2' = lift basis' thm2 |> simplify_expansion ectxt
|
|
1384 |
val mult_thm = @{thm expands_to_mult} OF [get_basis_wf_thm basis', ln_thm, thm2']
|
|
1385 |
val (exp_thm, basis'') = exp_expansion ectxt mult_thm basis'
|
|
1386 |
val thm = @{thm expands_to_powr} OF
|
|
1387 |
[trimmed_thm, get_basis_wf_thm basis, thm1, exp_thm]
|
|
1388 |
in
|
|
1389 |
(thm, basis'')
|
|
1390 |
end
|
|
1391 |
|
|
1392 |
fun powr_nat_expansion ectxt (thm1, thm2, basis) =
|
|
1393 |
case ev_zeroness_oracle ectxt (get_expanded_fun thm1) of
|
|
1394 |
SOME zero_thm => (
|
|
1395 |
case ev_zeroness_oracle ectxt (get_expanded_fun thm2) of
|
|
1396 |
SOME zero'_thm => (@{thm expands_to_powr_nat_0_0} OF
|
|
1397 |
[zero_thm, zero'_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis], basis)
|
|
1398 |
| NONE => (
|
|
1399 |
case trim_expansion true (SOME Simple_Trim) ectxt (thm2, basis) of
|
|
1400 |
(thm2, _, SOME trimmed_thm) =>
|
|
1401 |
(@{thm expands_to_powr_nat_0} OF [zero_thm, thm2, trimmed_thm,
|
|
1402 |
get_basis_wf_thm basis, mk_expansion_level_eq_thm basis], basis)))
|
|
1403 |
| NONE =>
|
|
1404 |
let
|
|
1405 |
val (thm1, _, SOME trimmed_thm) =
|
|
1406 |
trim_expansion true (SOME Pos_Trim) ectxt (thm1, basis)
|
|
1407 |
val (ln_thm, basis') = ln_expansion ectxt trimmed_thm thm1 basis
|
|
1408 |
val thm2' = lift basis' thm2 |> simplify_expansion ectxt
|
|
1409 |
val mult_thm = @{thm expands_to_mult} OF [get_basis_wf_thm basis', ln_thm, thm2']
|
|
1410 |
val (exp_thm, basis'') = exp_expansion ectxt mult_thm basis'
|
|
1411 |
val thm = @{thm expands_to_powr_nat} OF
|
|
1412 |
[trimmed_thm, get_basis_wf_thm basis, thm1, exp_thm]
|
|
1413 |
in
|
|
1414 |
(thm, basis'')
|
|
1415 |
end
|
|
1416 |
|
|
1417 |
fun is_numeral t =
|
|
1418 |
let
|
|
1419 |
val _ = HOLogic.dest_number t
|
|
1420 |
in
|
|
1421 |
true
|
|
1422 |
end
|
|
1423 |
handle TERM _ => false
|
|
1424 |
|
|
1425 |
fun power_expansion ectxt (thm, n, basis) =
|
|
1426 |
case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
|
|
1427 |
SOME zero_thm => @{thm expands_to_power_0} OF
|
|
1428 |
[zero_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis,
|
|
1429 |
Thm.reflexive (Thm.cterm_of (get_ctxt ectxt) n)]
|
|
1430 |
| NONE => (
|
|
1431 |
case trim_expansion true (SOME Simple_Trim) ectxt (thm, basis) of
|
|
1432 |
(thm', _, SOME trimmed_thm) =>
|
|
1433 |
let
|
|
1434 |
val ctxt = get_ctxt ectxt
|
|
1435 |
val thm =
|
|
1436 |
if is_numeral n then @{thm expands_to_power[where abort = True]}
|
|
1437 |
else @{thm expands_to_power[where abort = False]}
|
|
1438 |
val thm =
|
|
1439 |
Drule.infer_instantiate' ctxt [NONE, NONE, NONE, SOME (Thm.cterm_of ctxt n)] thm
|
|
1440 |
in
|
|
1441 |
thm OF [trimmed_thm, get_basis_wf_thm basis, thm']
|
|
1442 |
end
|
|
1443 |
| _ => raise TERM ("Unexpected zeroness result in power_expansion", []))
|
|
1444 |
|
|
1445 |
fun powr_const_expansion ectxt (thm, p, basis) =
|
|
1446 |
let
|
|
1447 |
val pthm = Thm.reflexive (Thm.cterm_of (get_ctxt ectxt) p)
|
|
1448 |
in
|
|
1449 |
case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
|
|
1450 |
SOME zero_thm => @{thm expands_to_powr_const_0} OF
|
|
1451 |
[zero_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis, pthm]
|
|
1452 |
| NONE =>
|
|
1453 |
case trim_expansion true (SOME Pos_Trim) ectxt (thm, basis) of
|
|
1454 |
(_, _, NONE) => raise TERM ("Unexpected zeroness result for powr", [])
|
|
1455 |
| (thm, _, SOME trimmed_thm) =>
|
|
1456 |
(if is_numeral p then @{thm expands_to_powr_const[where abort = True]}
|
|
1457 |
else @{thm expands_to_powr_const[where abort = False]})
|
|
1458 |
OF [trimmed_thm, get_basis_wf_thm basis, thm, pthm]
|
|
1459 |
end
|
|
1460 |
|
|
1461 |
fun sgn_expansion ectxt (thm, basis) =
|
|
1462 |
let
|
|
1463 |
val thms = [get_basis_wf_thm basis, mk_expansion_level_eq_thm basis]
|
|
1464 |
in
|
|
1465 |
case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
|
|
1466 |
SOME zero_thm => @{thm expands_to_sgn_zero} OF (zero_thm :: thms)
|
|
1467 |
| NONE =>
|
|
1468 |
case trim_expansion true (SOME Sgn_Trim) ectxt (thm, basis) of
|
|
1469 |
(thm, IsPos, SOME trimmed_thm) =>
|
|
1470 |
@{thm expands_to_sgn_pos} OF ([trimmed_thm, thm] @ thms)
|
|
1471 |
| (thm, IsNeg, SOME trimmed_thm) =>
|
|
1472 |
@{thm expands_to_sgn_neg} OF ([trimmed_thm, thm] @ thms)
|
|
1473 |
| _ => raise TERM ("Unexpected zeroness result in sgn_expansion", [])
|
|
1474 |
end
|
|
1475 |
|
|
1476 |
(*
|
|
1477 |
Returns an expansion of the sine and cosine of the given expansion. Fails if that function
|
|
1478 |
goes to infinity.
|
|
1479 |
*)
|
|
1480 |
fun sin_cos_expansion _ thm SEmpty =
|
|
1481 |
(thm RS @{thm expands_to_sin_real}, thm RS @{thm expands_to_cos_real})
|
|
1482 |
| sin_cos_expansion ectxt thm basis =
|
|
1483 |
let
|
|
1484 |
val exp = get_expansion thm
|
|
1485 |
val e = get_exponent exp
|
|
1486 |
in
|
|
1487 |
case zeroness_oracle true (SOME Sgn_Trim) ectxt e of
|
|
1488 |
(IsPos, _) => raise THM ("sin_cos_expansion", 0, [thm])
|
|
1489 |
| (IsNeg, SOME e_thm) =>
|
|
1490 |
let
|
|
1491 |
val [thm1, thm2] =
|
|
1492 |
map (fn thm' => thm' OF [e_thm, get_basis_wf_thm basis, thm])
|
|
1493 |
@{thms expands_to_sin_ms_neg_exp expands_to_cos_ms_neg_exp}
|
|
1494 |
in
|
|
1495 |
(thm1, thm2)
|
|
1496 |
end
|
|
1497 |
| (IsZero, SOME e_thm) =>
|
|
1498 |
let
|
|
1499 |
val (sin_thm, cos_thm) = (sin_cos_expansion ectxt (expands_to_hd thm) (tl_basis basis))
|
|
1500 |
fun mk_thm thm' =
|
|
1501 |
(thm' OF [e_thm, get_basis_wf_thm basis, thm, sin_thm, cos_thm]) |> solve_eval_eq
|
|
1502 |
val [thm1, thm2] =
|
|
1503 |
map mk_thm @{thms expands_to_sin_ms_zero_exp expands_to_cos_ms_zero_exp}
|
|
1504 |
in
|
|
1505 |
(thm1, thm2)
|
|
1506 |
end
|
|
1507 |
| _ => raise TERM ("Unexpected zeroness result in sin_exp_expansion", [])
|
|
1508 |
end
|
|
1509 |
|
|
1510 |
fun abconv (t, t') = Envir.beta_eta_contract t aconv Envir.beta_eta_contract t'
|
|
1511 |
|
|
1512 |
(*
|
|
1513 |
Makes sure that an expansion theorem really talks about the right function.
|
|
1514 |
This is basically a sanity check to make things fail early and in the right place.
|
|
1515 |
*)
|
|
1516 |
fun check_expansion e thm =
|
|
1517 |
if abconv (expr_to_term e, get_expanded_fun thm) then
|
|
1518 |
thm
|
|
1519 |
else
|
|
1520 |
(* TODO Remove Debugging stuff *)
|
|
1521 |
let val _ = @{print} e
|
|
1522 |
val _ = @{print} (get_expanded_fun thm)
|
|
1523 |
in
|
|
1524 |
raise TERM ("check_expansion", [Thm.concl_of thm, expr_to_term e])
|
|
1525 |
end
|
|
1526 |
|
|
1527 |
fun minmax_expansion max [less_thm, eq_thm, gt_thm] ectxt (thm1, thm2, basis) = (
|
|
1528 |
case compare_expansions ectxt (thm1, thm2, basis) of
|
|
1529 |
(LESS, less_thm', thm1, thm2) => less_thm OF [if max then thm2 else thm1, less_thm']
|
|
1530 |
| (GREATER, gt_thm', thm1, thm2) => gt_thm OF [if max then thm1 else thm2, gt_thm']
|
|
1531 |
| (EQUAL, eq_thm', thm1, _) => eq_thm OF [thm1, eq_thm'])
|
|
1532 |
| minmax_expansion _ _ _ _ = raise Match
|
|
1533 |
|
|
1534 |
val min_expansion =
|
|
1535 |
minmax_expansion false @{thms expands_to_min_lt expands_to_min_eq expands_to_min_gt}
|
|
1536 |
val max_expansion =
|
|
1537 |
minmax_expansion true @{thms expands_to_max_lt expands_to_max_eq expands_to_max_gt}
|
|
1538 |
|
|
1539 |
fun zero_expansion basis =
|
|
1540 |
@{thm expands_to_zero} OF [get_basis_wf_thm basis, mk_expansion_level_eq_thm basis]
|
|
1541 |
|
|
1542 |
fun const_expansion _ basis @{term "0 :: real"} = zero_expansion basis
|
|
1543 |
| const_expansion ectxt basis t =
|
|
1544 |
let
|
|
1545 |
val ctxt = get_ctxt ectxt
|
|
1546 |
val thm = Drule.infer_instantiate' ctxt [NONE, SOME (Thm.cterm_of ctxt t)]
|
|
1547 |
@{thm expands_to_const}
|
|
1548 |
in
|
|
1549 |
thm OF [get_basis_wf_thm basis, mk_expansion_level_eq_thm basis]
|
|
1550 |
end
|
|
1551 |
|
|
1552 |
fun root_expansion ectxt (thm, n, basis) =
|
|
1553 |
let
|
|
1554 |
val ctxt = get_ctxt ectxt
|
|
1555 |
fun tac {context = ctxt, ...} =
|
|
1556 |
HEADGOAL (Method.insert_tac ctxt (get_facts ectxt))
|
|
1557 |
THEN Local_Defs.unfold_tac ctxt eval_simps
|
|
1558 |
THEN HEADGOAL (Simplifier.asm_full_simp_tac ctxt)
|
|
1559 |
fun prove goal =
|
|
1560 |
try (Goal.prove ctxt [] [] (HOLogic.mk_Trueprop (Term.betapply (goal, n)))) tac
|
|
1561 |
fun err () =
|
|
1562 |
let
|
|
1563 |
val t = simplify_term' (get_facts ectxt) ctxt n
|
|
1564 |
val _ =
|
|
1565 |
if #verbose (#ctxt ectxt) then
|
|
1566 |
let
|
|
1567 |
val p = Pretty.str ("real_asymp failed to determine whether the following constant " ^
|
|
1568 |
"is zero or not:")
|
|
1569 |
val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t)]
|
|
1570 |
in
|
|
1571 |
Pretty.writeln p
|
|
1572 |
end else ()
|
|
1573 |
in
|
|
1574 |
raise TERM ("zeroness_oracle", [n])
|
|
1575 |
end
|
|
1576 |
fun aux nz_thm =
|
|
1577 |
case trim_expansion true (SOME Sgn_Trim) ectxt (thm, basis) of
|
|
1578 |
(thm, IsPos, SOME trimmed_thm) =>
|
|
1579 |
@{thm expands_to_root} OF [nz_thm, trimmed_thm, get_basis_wf_thm basis, thm]
|
|
1580 |
| (thm, IsNeg, SOME trimmed_thm) =>
|
|
1581 |
@{thm expands_to_root_neg} OF [nz_thm, trimmed_thm, get_basis_wf_thm basis, thm]
|
|
1582 |
| _ => raise TERM ("Unexpected zeroness result in root_expansion", [])
|
|
1583 |
in
|
|
1584 |
case prove @{term "\<lambda>n::nat. n = 0"} of
|
|
1585 |
SOME zero_thm =>
|
|
1586 |
@{thm expands_to_0th_root} OF
|
|
1587 |
[zero_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis,
|
|
1588 |
Thm.reflexive (Thm.cterm_of ctxt (get_expanded_fun thm))]
|
|
1589 |
| NONE =>
|
|
1590 |
case prove @{term "\<lambda>n::nat. n > 0"} of
|
|
1591 |
NONE => err ()
|
|
1592 |
| SOME nz_thm =>
|
|
1593 |
case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
|
|
1594 |
SOME zero_thm => @{thm expands_to_root_0} OF
|
|
1595 |
[nz_thm, zero_thm, get_basis_wf_thm basis, mk_expansion_level_eq_thm basis]
|
|
1596 |
| NONE => aux nz_thm
|
|
1597 |
end
|
|
1598 |
|
|
1599 |
|
|
1600 |
fun arctan_expansion _ SEmpty thm =
|
|
1601 |
@{thm expands_to_real_compose[where g = arctan]} OF [thm]
|
|
1602 |
| arctan_expansion ectxt basis thm =
|
|
1603 |
case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
|
|
1604 |
SOME zero_thm => @{thm expands_to_arctan_zero} OF [zero_expansion basis, zero_thm]
|
|
1605 |
| NONE =>
|
|
1606 |
let
|
|
1607 |
val (thm, _, _) = trim_expansion true (SOME Simple_Trim) ectxt (thm, basis)
|
|
1608 |
val e = get_exponent (get_expansion thm)
|
|
1609 |
fun cont ectxt (thm, basis) = arctan_expansion ectxt basis thm
|
|
1610 |
in
|
|
1611 |
case zeroness_oracle true (SOME Sgn_Trim) ectxt e of
|
|
1612 |
(IsNeg, SOME neg_thm) =>
|
|
1613 |
@{thm expands_to_arctan_ms_neg_exp} OF [neg_thm, get_basis_wf_thm basis, thm]
|
|
1614 |
| (IsPos, SOME e_pos_thm) => (
|
|
1615 |
case determine_trimmed_sgn ectxt (get_expansion thm) of
|
|
1616 |
(IsPos, trimmed_thm) =>
|
|
1617 |
@{thm expands_to_arctan_ms_pos_exp_pos} OF
|
|
1618 |
[e_pos_thm, trimmed_thm, get_basis_wf_thm basis, thm]
|
|
1619 |
| (IsNeg, trimmed_thm) =>
|
|
1620 |
@{thm expands_to_arctan_ms_pos_exp_neg} OF
|
|
1621 |
[e_pos_thm, trimmed_thm, get_basis_wf_thm basis, thm]
|
|
1622 |
| _ => raise TERM ("Unexpected trim result during expansion of arctan", []))
|
|
1623 |
| (IsZero, _) => (
|
|
1624 |
case try_lift_function ectxt (thm, basis) cont of
|
|
1625 |
(SOME thm', _) => thm'
|
|
1626 |
| _ =>
|
|
1627 |
let
|
|
1628 |
val _ = if get_verbose ectxt then
|
|
1629 |
writeln "Unsupported occurrence of arctan" else ()
|
|
1630 |
in
|
|
1631 |
raise TERM ("Unsupported occurence of arctan", [])
|
|
1632 |
end)
|
|
1633 |
| _ => raise TERM ("Unexpected trim result during expansion of arctan", [])
|
|
1634 |
end
|
|
1635 |
|
|
1636 |
(* Returns an expansion theorem for a function that is already a basis element *)
|
|
1637 |
fun expand_basic _ t SEmpty = raise TERM ("expand_basic", [t])
|
|
1638 |
| expand_basic thm t basis =
|
|
1639 |
if abconv (get_basis_head basis, t) then
|
|
1640 |
thm (get_basis_wf_thm basis) (mk_expansion_level_eq_thm (tl_basis basis))
|
|
1641 |
else
|
|
1642 |
@{thm expands_to_lift'} OF [get_basis_wf_thm basis, expand_basic thm t (tl_basis basis)]
|
|
1643 |
|
|
1644 |
fun expand_unary ectxt thm e basis =
|
|
1645 |
let
|
|
1646 |
val (thm', basis') = expand' ectxt e basis |> apfst (simplify_expansion ectxt)
|
|
1647 |
in
|
|
1648 |
(thm OF [get_basis_wf_thm basis', thm'], basis')
|
|
1649 |
end
|
|
1650 |
and expand_binary ectxt thm (e1, e2) basis =
|
|
1651 |
let
|
|
1652 |
val (thm1, basis') = expand' ectxt e1 basis |> apfst (simplify_expansion ectxt)
|
|
1653 |
val (thm2, basis'') = expand' ectxt e2 basis' |> apfst (simplify_expansion ectxt)
|
|
1654 |
val thm1 = lift basis'' thm1 |> simplify_expansion ectxt
|
|
1655 |
in
|
|
1656 |
(thm OF [get_basis_wf_thm basis'', thm1, thm2], basis'')
|
|
1657 |
end
|
|
1658 |
and trim_nz mode ectxt e basis =
|
|
1659 |
let
|
|
1660 |
val (thm, basis') = expand' ectxt e basis |> apfst (simplify_expansion ectxt)
|
|
1661 |
val (thm', nz, trimmed_thm) = trim_expansion true (SOME mode) ectxt (thm, basis')
|
|
1662 |
in
|
|
1663 |
case trimmed_thm of
|
|
1664 |
NONE => raise TERM ("expand: zero denominator", [get_expansion thm])
|
|
1665 |
| SOME trimmed_thm => (thm', basis', nz, trimmed_thm)
|
|
1666 |
end
|
|
1667 |
and expand'' ectxt (ConstExpr c) basis = (const_expansion ectxt basis c, basis)
|
|
1668 |
| expand'' _ X basis = (lift basis @{thm expands_to_X}, basis)
|
|
1669 |
| expand'' ectxt (Uminus e) basis = expand_unary ectxt @{thm expands_to_uminus} e basis
|
|
1670 |
| expand'' ectxt (Add e12) basis = expand_binary ectxt @{thm expands_to_add} e12 basis
|
|
1671 |
| expand'' ectxt (Minus e12) basis = expand_binary ectxt @{thm expands_to_minus} e12 basis
|
|
1672 |
| expand'' ectxt (Mult e12) basis = expand_binary ectxt @{thm expands_to_mult} e12 basis
|
|
1673 |
| expand'' ectxt (Powr' (e, p)) basis = (* TODO zero basis *)
|
|
1674 |
let
|
|
1675 |
val (thm, basis') = expand' ectxt e basis |> apfst (simplify_expansion ectxt)
|
|
1676 |
in
|
|
1677 |
(powr_const_expansion ectxt (thm, p, basis'), basis')
|
|
1678 |
end
|
|
1679 |
| expand'' ectxt (Powr (e1, e2)) basis =
|
|
1680 |
let
|
|
1681 |
val (thm2, basis1) = expand' ectxt e2 basis |> apfst (simplify_expansion ectxt)
|
|
1682 |
val (thm1, basis2) = expand' ectxt e1 basis1 |> apfst (simplify_expansion ectxt)
|
|
1683 |
in
|
|
1684 |
powr_expansion ectxt (thm1, thm2, basis2)
|
|
1685 |
end
|
|
1686 |
| expand'' ectxt (Powr_Nat (e1, e2)) basis =
|
|
1687 |
let
|
|
1688 |
val (thm2, basis1) = expand' ectxt e2 basis |> apfst (simplify_expansion ectxt)
|
|
1689 |
val (thm1, basis2) = expand' ectxt e1 basis1 |> apfst (simplify_expansion ectxt)
|
|
1690 |
in
|
|
1691 |
powr_nat_expansion ectxt (thm1, thm2, basis2)
|
|
1692 |
end
|
|
1693 |
| expand'' ectxt (LnPowr (e1, e2)) basis =
|
|
1694 |
let (* TODO zero base *)
|
|
1695 |
val (thm2, basis1) = expand' ectxt e2 basis |> apfst (simplify_expansion ectxt)
|
|
1696 |
val (thm1, basis2, _, trimmed_thm) = trim_nz Pos_Trim ectxt e1 basis1
|
|
1697 |
val (ln_thm, basis3) = ln_expansion ectxt trimmed_thm thm1 basis2
|
|
1698 |
val thm2' = lift basis3 thm2 |> simplify_expansion ectxt
|
|
1699 |
val mult_thm = @{thm expands_to_mult} OF [get_basis_wf_thm basis3, ln_thm, thm2']
|
|
1700 |
val thm = @{thm expands_to_ln_powr} OF
|
|
1701 |
[trimmed_thm, get_basis_wf_thm basis2, thm1, mult_thm]
|
|
1702 |
in
|
|
1703 |
(thm, basis3)
|
|
1704 |
end
|
|
1705 |
| expand'' ectxt (ExpLn e) basis =
|
|
1706 |
let
|
|
1707 |
val (thm, basis', _, trimmed_thm) = trim_nz Pos_Trim ectxt e basis
|
|
1708 |
val thm = @{thm expands_to_exp_ln} OF [trimmed_thm, get_basis_wf_thm basis', thm]
|
|
1709 |
in
|
|
1710 |
(thm, basis')
|
|
1711 |
end
|
|
1712 |
| expand'' ectxt (Power (e, n)) basis =
|
|
1713 |
let
|
|
1714 |
val (thm, basis') = expand' ectxt e basis |> apfst (simplify_expansion ectxt)
|
|
1715 |
in
|
|
1716 |
(power_expansion ectxt (thm, n, basis'), basis')
|
|
1717 |
end
|
|
1718 |
| expand'' ectxt (Root (e, n)) basis =
|
|
1719 |
let
|
|
1720 |
val (thm, basis') = expand' ectxt e basis |> apfst (simplify_expansion ectxt)
|
|
1721 |
in
|
|
1722 |
(root_expansion ectxt (thm, n, basis'), basis')
|
|
1723 |
end
|
|
1724 |
| expand'' ectxt (Inverse e) basis =
|
|
1725 |
(case trim_nz Simple_Trim ectxt e basis of
|
|
1726 |
(thm, basis', _, trimmed_thm) =>
|
|
1727 |
(@{thm expands_to_inverse} OF [trimmed_thm, get_basis_wf_thm basis', thm], basis'))
|
|
1728 |
| expand'' ectxt (Div (e1, e2)) basis =
|
|
1729 |
let
|
|
1730 |
val (thm1, basis') = expand' ectxt e1 basis
|
|
1731 |
val (thm2, basis'', _, trimmed_thm) = trim_nz Simple_Trim ectxt e2 basis'
|
|
1732 |
val thm1 = lift basis'' thm1
|
|
1733 |
in
|
|
1734 |
(@{thm expands_to_divide} OF [trimmed_thm, get_basis_wf_thm basis'', thm1, thm2], basis'')
|
|
1735 |
end
|
|
1736 |
| expand'' ectxt (Ln e) basis =
|
|
1737 |
let
|
|
1738 |
val (thm, basis', _, trimmed_thm) = trim_nz Pos_Trim ectxt e basis
|
|
1739 |
in
|
|
1740 |
ln_expansion ectxt trimmed_thm thm basis'
|
|
1741 |
end
|
|
1742 |
| expand'' ectxt (Exp e) basis =
|
|
1743 |
let
|
|
1744 |
val (thm, basis') = expand' ectxt e basis
|
|
1745 |
in
|
|
1746 |
exp_expansion ectxt thm basis'
|
|
1747 |
end
|
|
1748 |
| expand'' ectxt (Absolute e) basis =
|
|
1749 |
let
|
|
1750 |
val (thm, basis', nz, trimmed_thm) = trim_nz Sgn_Trim ectxt e basis
|
|
1751 |
val thm' =
|
|
1752 |
case nz of
|
|
1753 |
IsPos => @{thm expands_to_abs_pos}
|
|
1754 |
| IsNeg => @{thm expands_to_abs_neg}
|
|
1755 |
| _ => raise TERM ("Unexpected trim result during expansion of abs", [])
|
|
1756 |
in
|
|
1757 |
(thm' OF [trimmed_thm, get_basis_wf_thm basis', thm], basis')
|
|
1758 |
end
|
|
1759 |
| expand'' ectxt (Sgn e) basis =
|
|
1760 |
let
|
|
1761 |
val (thm, basis') = expand' ectxt e basis
|
|
1762 |
in
|
|
1763 |
(sgn_expansion ectxt (thm, basis'), basis')
|
|
1764 |
end
|
|
1765 |
| expand'' ectxt (Min (e1, e2)) basis = (
|
|
1766 |
case try_prove_ev_eq ectxt (apply2 expr_to_term (e1, e2)) of
|
|
1767 |
SOME eq_thm =>
|
|
1768 |
expand' ectxt e1 basis
|
|
1769 |
|> apfst (fn thm => @{thm expands_to_min_eq} OF [thm, eq_thm])
|
|
1770 |
| NONE =>
|
|
1771 |
let
|
|
1772 |
val (thm1, basis') = expand' ectxt e1 basis
|
|
1773 |
val (thm2, basis'') = expand' ectxt e2 basis'
|
|
1774 |
val thm1' = lift basis'' thm1
|
|
1775 |
in
|
|
1776 |
(min_expansion ectxt (thm1', thm2, basis''), basis'')
|
|
1777 |
end)
|
|
1778 |
| expand'' ectxt (Max (e1, e2)) basis = (
|
|
1779 |
case try_prove_ev_eq ectxt (apply2 expr_to_term (e1, e2)) of
|
|
1780 |
SOME eq_thm =>
|
|
1781 |
expand' ectxt e1 basis
|
|
1782 |
|> apfst (fn thm => @{thm expands_to_max_eq} OF [thm, eq_thm])
|
|
1783 |
| NONE =>
|
|
1784 |
let
|
|
1785 |
val (thm1, basis') = expand' ectxt e1 basis
|
|
1786 |
val (thm2, basis'') = expand' ectxt e2 basis'
|
|
1787 |
val thm1' = lift basis'' thm1
|
|
1788 |
in
|
|
1789 |
(max_expansion ectxt (thm1', thm2, basis''), basis'')
|
|
1790 |
end)
|
|
1791 |
| expand'' ectxt (Sin e) basis =
|
|
1792 |
let
|
|
1793 |
val (thm, basis', _, _) = trim_nz Simple_Trim ectxt e basis (* TODO could be relaxed *)
|
|
1794 |
in
|
|
1795 |
(sin_cos_expansion ectxt thm basis' |> fst, basis')
|
|
1796 |
end
|
|
1797 |
| expand'' ectxt (Cos e) basis =
|
|
1798 |
let
|
|
1799 |
val (thm, basis', _, _) = trim_nz Simple_Trim ectxt e basis (* TODO could be relaxed *)
|
|
1800 |
in
|
|
1801 |
(sin_cos_expansion ectxt thm basis' |> snd, basis')
|
|
1802 |
end
|
|
1803 |
| expand'' _ (Floor _) _ =
|
|
1804 |
raise TERM ("floor not supported.", [])
|
|
1805 |
| expand'' _ (Ceiling _) _ =
|
|
1806 |
raise TERM ("ceiling not supported.", [])
|
|
1807 |
| expand'' _ (Frac _) _ =
|
|
1808 |
raise TERM ("frac not supported.", [])
|
|
1809 |
| expand'' _ (NatMod _) _ =
|
|
1810 |
raise TERM ("mod not supported.", [])
|
|
1811 |
| expand'' ectxt (ArcTan e) basis =
|
|
1812 |
let
|
|
1813 |
(* TODO: what if it's zero *)
|
|
1814 |
val (thm, basis') = expand' ectxt e basis
|
|
1815 |
in
|
|
1816 |
(arctan_expansion ectxt basis' thm, basis')
|
|
1817 |
end
|
|
1818 |
| expand'' ectxt (Custom (name, t, args)) basis =
|
|
1819 |
let
|
|
1820 |
fun expand_args acc basis [] = (rev acc, basis)
|
|
1821 |
| expand_args acc basis (arg :: args) =
|
|
1822 |
case expand' ectxt arg basis of
|
|
1823 |
(thm, basis') => expand_args (thm :: acc) basis' args
|
|
1824 |
in
|
|
1825 |
case expand_custom (get_ctxt ectxt) name of
|
|
1826 |
NONE => raise TERM ("Unsupported custom function: " ^ name, [t])
|
|
1827 |
| SOME e => e ectxt t (expand_args [] basis args)
|
|
1828 |
end
|
|
1829 |
|
|
1830 |
and expand' ectxt (e' as (Inverse e)) basis =
|
|
1831 |
let
|
|
1832 |
val t = expr_to_term e
|
|
1833 |
fun thm wf_thm len_thm =
|
|
1834 |
@{thm expands_to_basic_inverse} OF [wf_thm, len_thm]
|
|
1835 |
in
|
|
1836 |
if member abconv (get_basis_list basis) t then
|
|
1837 |
(expand_basic thm t basis, basis) |> apfst (check_expansion e')
|
|
1838 |
else
|
|
1839 |
expand'' ectxt e' basis |> apfst (check_expansion e')
|
|
1840 |
end
|
|
1841 |
| expand' ectxt (Div (e1, e2)) basis =
|
|
1842 |
let
|
|
1843 |
val (thm1, basis') = expand' ectxt e1 basis
|
|
1844 |
val t = expr_to_term e2
|
|
1845 |
fun thm wf_thm len_thm =
|
|
1846 |
@{thm expands_to_basic_inverse} OF [wf_thm, len_thm]
|
|
1847 |
in
|
|
1848 |
if member abconv (get_basis_list basis') t then
|
|
1849 |
(@{thm expands_to_div'} OF [get_basis_wf_thm basis', thm1, expand_basic thm t basis'],
|
|
1850 |
basis')
|
|
1851 |
else
|
|
1852 |
let
|
|
1853 |
val (thm2, basis'', _, trimmed_thm) = trim_nz Simple_Trim ectxt e2 basis'
|
|
1854 |
val thm1 = lift basis'' thm1
|
|
1855 |
in
|
|
1856 |
(@{thm expands_to_divide} OF [trimmed_thm, get_basis_wf_thm basis'', thm1, thm2],
|
|
1857 |
basis'')
|
|
1858 |
end
|
|
1859 |
end
|
|
1860 |
| expand' ectxt (e' as (Powr' (e, p))) basis =
|
|
1861 |
let
|
|
1862 |
val t = expr_to_term e
|
|
1863 |
val ctxt = get_ctxt ectxt
|
|
1864 |
fun thm wf_thm len_thm =
|
|
1865 |
(Drule.infer_instantiate' ctxt [NONE, NONE, SOME (Thm.cterm_of ctxt p)]
|
|
1866 |
@{thm expands_to_basic_powr}) OF [wf_thm, len_thm]
|
|
1867 |
in
|
|
1868 |
if member abconv (get_basis_list basis) t then
|
|
1869 |
(expand_basic thm t basis, basis) |> apfst (check_expansion e')
|
|
1870 |
else
|
|
1871 |
expand'' ectxt e' basis |> apfst (check_expansion e')
|
|
1872 |
end
|
|
1873 |
| expand' ectxt e basis =
|
|
1874 |
let
|
|
1875 |
val t = expr_to_term e
|
|
1876 |
fun thm wf_thm len_thm = @{thm expands_to_basic} OF [wf_thm, len_thm]
|
|
1877 |
in
|
|
1878 |
if member abconv (get_basis_list basis) t then
|
|
1879 |
(expand_basic thm t basis, basis) |> apfst (check_expansion e)
|
|
1880 |
else
|
|
1881 |
expand'' ectxt e basis |> apfst (check_expansion e)
|
|
1882 |
end
|
|
1883 |
|
|
1884 |
fun expand ectxt e basis =
|
|
1885 |
expand' ectxt e basis
|
|
1886 |
|> apfst (simplify_expansion ectxt)
|
|
1887 |
|> apfst (check_expansion e)
|
|
1888 |
|
|
1889 |
fun expand_term ectxt t basis =
|
|
1890 |
let
|
|
1891 |
val ctxt = get_ctxt ectxt
|
|
1892 |
val (e, eq_thm) = reify ctxt t
|
|
1893 |
val (thm, basis) = expand ectxt e basis
|
|
1894 |
in
|
|
1895 |
(@{thm expands_to_meta_eq_cong'} OF [thm, eq_thm], basis)
|
|
1896 |
end
|
|
1897 |
|
|
1898 |
fun expand_terms ectxt ts basis =
|
|
1899 |
let
|
|
1900 |
val ctxt = get_ctxt ectxt
|
|
1901 |
val e_eq_thms = map (reify ctxt) ts
|
|
1902 |
fun step (e, eq_thm) (thms, basis) =
|
|
1903 |
let
|
|
1904 |
val (thm, basis) = expand' ectxt e basis
|
|
1905 |
val thm = @{thm expands_to_meta_eq_cong'} OF [simplify_expansion ectxt thm, eq_thm]
|
|
1906 |
in
|
|
1907 |
(thm :: thms, basis)
|
|
1908 |
end
|
|
1909 |
val (thms, basis) = fold step e_eq_thms ([], basis)
|
|
1910 |
fun lift thm = lift_expands_to_thm (mk_lifting (extract_basis_list thm) basis) thm
|
|
1911 |
in
|
|
1912 |
(map lift (rev thms), basis)
|
|
1913 |
end
|
|
1914 |
|
|
1915 |
datatype limit =
|
|
1916 |
Zero_Limit of bool option
|
|
1917 |
| Finite_Limit of term
|
|
1918 |
| Infinite_Limit of bool option
|
|
1919 |
|
|
1920 |
fun is_empty_expansion (Const (@{const_name MS}, _) $ Const (@{const_name MSLNil}, _) $ _) = true
|
|
1921 |
| is_empty_expansion _ = false
|
|
1922 |
|
|
1923 |
fun limit_of_expansion_aux ectxt basis thm =
|
|
1924 |
let
|
|
1925 |
val n = length (get_basis_list basis)
|
|
1926 |
val (thm, res, e_thms) =
|
|
1927 |
trim_expansion_while_greater false (SOME (replicate n @{term "0::real"})) true
|
|
1928 |
(SOME Simple_Trim) ectxt (thm, basis)
|
|
1929 |
val trimmed_thm = case res of Trimmed (_, trimmed_thm) => trimmed_thm | _ => NONE
|
|
1930 |
val res = case res of Trimmed _ => GREATER | Aborted res => res
|
|
1931 |
val exp = get_expansion thm
|
|
1932 |
val _ = if res = GREATER andalso is_none trimmed_thm andalso not (is_empty_expansion exp) then
|
|
1933 |
raise TERM ("limit_of_expansion", [get_expansion thm]) else ()
|
|
1934 |
fun go thm _ _ [] = (
|
|
1935 |
case zeroness_oracle false (SOME Simple_Trim) ectxt (get_expansion thm) of
|
|
1936 |
(IsZero, _) => (Zero_Limit NONE, @{thm expands_to_real_imp_filterlim} OF [thm])
|
|
1937 |
| _ => (Finite_Limit @{term "0::real"}, @{thm expands_to_real_imp_filterlim} OF [thm]))
|
|
1938 |
| go thm _ basis ((IsNeg, neg_thm) :: _) = (Zero_Limit NONE,
|
|
1939 |
@{thm expands_to_neg_exponent_imp_filterlim} OF
|
|
1940 |
[thm, get_basis_wf_thm basis, neg_thm RS @{thm compare_reals_diff_sgnD(1)}])
|
|
1941 |
| go thm trimmed_thm basis ((IsPos, pos_thm) :: _) = (Infinite_Limit NONE,
|
|
1942 |
@{thm expands_to_pos_exponent_imp_filterlim} OF
|
|
1943 |
[thm, the trimmed_thm, get_basis_wf_thm basis,
|
|
1944 |
pos_thm RS @{thm compare_reals_diff_sgnD(3)}])
|
|
1945 |
| go thm trimmed_thm basis ((IsZero, zero_thm) :: e_thms) =
|
|
1946 |
let
|
|
1947 |
val thm' = thm RS @{thm expands_to_hd''}
|
|
1948 |
val trimmed_thm' = Option.map (fn thm => thm RS @{thm trimmed_hd}) trimmed_thm
|
|
1949 |
val (lim, lim_thm) = go thm' trimmed_thm' (tl_basis basis) e_thms
|
|
1950 |
val lim_lift_thm =
|
|
1951 |
case lim of
|
|
1952 |
Infinite_Limit _ => @{thm expands_to_zero_exponent_imp_filterlim(1)}
|
|
1953 |
| _ => @{thm expands_to_zero_exponent_imp_filterlim(2)}
|
|
1954 |
val lim_thm' =
|
|
1955 |
lim_lift_thm OF [thm, get_basis_wf_thm basis,
|
|
1956 |
zero_thm RS @{thm compare_reals_diff_sgnD(2)}, lim_thm]
|
|
1957 |
in
|
|
1958 |
(lim, lim_thm')
|
|
1959 |
end
|
|
1960 |
| go _ _ _ _ = raise Match
|
|
1961 |
in
|
|
1962 |
if is_empty_expansion exp then
|
|
1963 |
(Zero_Limit NONE, thm RS @{thm expands_to_MSLNil_imp_filterlim}, thm)
|
|
1964 |
else
|
|
1965 |
case go thm trimmed_thm basis e_thms of
|
|
1966 |
(lim, lim_thm) => (lim, lim_thm, thm)
|
|
1967 |
end
|
|
1968 |
|
|
1969 |
(*
|
|
1970 |
Determines the limit of a function from its expansion. The two flags control whether the
|
|
1971 |
the sign of the approach should be determined for the infinite case (i.e. at_top/at_bot instead
|
|
1972 |
of just at_infinity) and the zero case (i.e. at_right 0/at_left 0 instead of just nhds 0)
|
|
1973 |
*)
|
|
1974 |
fun limit_of_expansion (sgn_zero, sgn_inf) ectxt (thm, basis) =
|
|
1975 |
let
|
|
1976 |
val (lim, lim_thm, thm) = limit_of_expansion_aux ectxt basis thm
|
|
1977 |
in
|
|
1978 |
case lim of
|
|
1979 |
Zero_Limit _ => (
|
|
1980 |
if sgn_zero then
|
|
1981 |
case trim_expansion false (SOME Sgn_Trim) ectxt (thm, basis) of
|
|
1982 |
(thm, IsPos, SOME pos_thm) => (Zero_Limit (SOME true),
|
|
1983 |
@{thm tendsto_imp_filterlim_at_right[OF _ expands_to_imp_eventually_pos]} OF
|
|
1984 |
[lim_thm, get_basis_wf_thm basis, thm, pos_thm])
|
|
1985 |
| (thm, IsNeg, SOME neg_thm) => (Zero_Limit (SOME false),
|
|
1986 |
@{thm tendsto_imp_filterlim_at_left[OF _ expands_to_imp_eventually_neg]} OF
|
|
1987 |
[lim_thm, get_basis_wf_thm basis, thm, neg_thm])
|
|
1988 |
| _ => (Zero_Limit NONE, lim_thm)
|
|
1989 |
else (Zero_Limit NONE, lim_thm))
|
|
1990 |
| Infinite_Limit _ => (
|
|
1991 |
if sgn_inf then
|
|
1992 |
case trim_expansion false (SOME Sgn_Trim) ectxt (thm, basis) of
|
|
1993 |
(thm, IsPos, SOME pos_thm) => (Infinite_Limit (SOME true),
|
|
1994 |
(@{thm filterlim_at_infinity_imp_filterlim_at_top[OF _ expands_to_imp_eventually_pos]} OF
|
|
1995 |
[lim_thm, get_basis_wf_thm basis, thm, pos_thm]))
|
|
1996 |
| (thm, IsNeg, SOME neg_thm) => (Infinite_Limit (SOME false),
|
|
1997 |
@{thm filterlim_at_infinity_imp_filterlim_at_bot[OF _ expands_to_imp_eventually_neg]} OF
|
|
1998 |
[lim_thm, get_basis_wf_thm basis, thm, neg_thm])
|
|
1999 |
| _ => (Infinite_Limit NONE, lim_thm)
|
|
2000 |
else (Infinite_Limit NONE, lim_thm))
|
|
2001 |
| Finite_Limit c => (Finite_Limit c, lim_thm)
|
|
2002 |
end
|
|
2003 |
|
|
2004 |
fun compute_limit ectxt t =
|
|
2005 |
case expand_term ectxt t default_basis of
|
|
2006 |
(thm, basis) => limit_of_expansion (true, true) ectxt (thm, basis)
|
|
2007 |
|
|
2008 |
fun prove_at_infinity ectxt (thm, basis) =
|
|
2009 |
let
|
|
2010 |
fun err () = raise TERM ("prove_at_infinity", [get_expanded_fun thm])
|
|
2011 |
val (thm, _, SOME trimmed_thm) = trim_expansion true (SOME Simple_Trim) ectxt (thm, basis)
|
|
2012 |
fun go basis thm trimmed_thm =
|
|
2013 |
if fastype_of (get_expansion thm) = @{typ "real"} then
|
|
2014 |
err ()
|
|
2015 |
else
|
|
2016 |
case zeroness_oracle true (SOME Pos_Trim) ectxt (get_exponent (get_expansion thm)) of
|
|
2017 |
(IsPos, SOME pos_thm) =>
|
|
2018 |
@{thm expands_to_pos_exponent_imp_filterlim} OF
|
|
2019 |
[thm, trimmed_thm, get_basis_wf_thm basis, pos_thm]
|
|
2020 |
| (IsZero, SOME zero_thm) =>
|
|
2021 |
@{thm expands_to_zero_exponent_imp_filterlim(1)} OF
|
|
2022 |
[thm, get_basis_wf_thm basis, zero_thm,
|
|
2023 |
go (tl_basis basis) (thm RS @{thm expands_to_hd''})
|
|
2024 |
(trimmed_thm RS @{thm trimmed_hd})]
|
|
2025 |
| _ => err ()
|
|
2026 |
in
|
|
2027 |
go basis thm trimmed_thm
|
|
2028 |
end
|
|
2029 |
|
|
2030 |
fun prove_at_top_at_bot mode ectxt (thm, basis) =
|
|
2031 |
let
|
|
2032 |
val s = if mode = Pos_Trim then "prove_at_top" else "prove_at_bot"
|
|
2033 |
fun err () = raise TERM (s, [get_expanded_fun thm])
|
|
2034 |
val (thm, _, SOME trimmed_thm) = trim_expansion true (SOME mode) ectxt (thm, basis)
|
|
2035 |
val trimmed_thm' = trimmed_thm RS
|
|
2036 |
(if mode = Pos_Trim then @{thm trimmed_pos_imp_trimmed} else @{thm trimmed_neg_imp_trimmed})
|
|
2037 |
fun go basis thm trimmed_thm =
|
|
2038 |
if fastype_of (get_expansion thm) = @{typ "real"} then
|
|
2039 |
err ()
|
|
2040 |
else
|
|
2041 |
case zeroness_oracle true (SOME Pos_Trim) ectxt (get_exponent (get_expansion thm)) of
|
|
2042 |
(IsPos, SOME pos_thm) =>
|
|
2043 |
@{thm expands_to_pos_exponent_imp_filterlim} OF
|
|
2044 |
[thm, trimmed_thm, get_basis_wf_thm basis, pos_thm]
|
|
2045 |
| (IsZero, SOME zero_thm) =>
|
|
2046 |
@{thm expands_to_zero_exponent_imp_filterlim(1)} OF
|
|
2047 |
[thm, get_basis_wf_thm basis, zero_thm,
|
|
2048 |
go (tl_basis basis) (thm RS @{thm expands_to_hd''})
|
|
2049 |
(trimmed_thm RS @{thm trimmed_hd})]
|
|
2050 |
| _ => err ()
|
|
2051 |
val lim_thm = go basis thm trimmed_thm'
|
|
2052 |
val add_sign_thm =
|
|
2053 |
if mode = Pos_Trim then
|
|
2054 |
@{thm filterlim_at_infinity_imp_filterlim_at_top[OF _ expands_to_imp_eventually_pos]}
|
|
2055 |
else
|
|
2056 |
@{thm filterlim_at_infinity_imp_filterlim_at_bot[OF _ expands_to_imp_eventually_neg]}
|
|
2057 |
in
|
|
2058 |
add_sign_thm OF [lim_thm, get_basis_wf_thm basis, thm, trimmed_thm]
|
|
2059 |
end
|
|
2060 |
|
|
2061 |
val prove_at_top = prove_at_top_at_bot Pos_Trim
|
|
2062 |
val prove_at_bot = prove_at_top_at_bot Neg_Trim
|
|
2063 |
|
|
2064 |
|
|
2065 |
fun prove_at_aux mode ectxt (thm, basis) =
|
|
2066 |
let
|
|
2067 |
val (s, add_sign_thm) =
|
|
2068 |
case mode of
|
|
2069 |
Simple_Trim =>
|
|
2070 |
("prove_at_0", @{thm Topological_Spaces.filterlim_atI[OF _ expands_to_imp_eventually_nz]})
|
|
2071 |
| Pos_Trim =>
|
|
2072 |
("prove_at_right_0",
|
|
2073 |
@{thm tendsto_imp_filterlim_at_right[OF _ expands_to_imp_eventually_pos]})
|
|
2074 |
| Neg_Trim =>
|
|
2075 |
("prove_at_left_0",
|
|
2076 |
@{thm tendsto_imp_filterlim_at_left[OF _ expands_to_imp_eventually_neg]})
|
|
2077 |
fun err () = raise TERM (s, [get_expanded_fun thm])
|
|
2078 |
val (thm, _, SOME trimmed_thm) = trim_expansion true (SOME mode) ectxt (thm, basis)
|
|
2079 |
fun go basis thm =
|
|
2080 |
if fastype_of (get_expansion thm) = @{typ "real"} then
|
|
2081 |
err ()
|
|
2082 |
else
|
|
2083 |
case zeroness_oracle true (SOME Neg_Trim) ectxt (get_exponent (get_expansion thm)) of
|
|
2084 |
(IsNeg, SOME neg_thm) =>
|
|
2085 |
@{thm expands_to_neg_exponent_imp_filterlim} OF
|
|
2086 |
[thm, get_basis_wf_thm basis, neg_thm]
|
|
2087 |
| (IsZero, SOME zero_thm) =>
|
|
2088 |
@{thm expands_to_zero_exponent_imp_filterlim(2)} OF
|
|
2089 |
[thm, get_basis_wf_thm basis, zero_thm,
|
|
2090 |
go (tl_basis basis) (thm RS @{thm expands_to_hd''})]
|
|
2091 |
| _ => err ()
|
|
2092 |
val lim_thm = go basis thm
|
|
2093 |
in
|
|
2094 |
add_sign_thm OF [lim_thm, get_basis_wf_thm basis, thm, trimmed_thm]
|
|
2095 |
end
|
|
2096 |
|
|
2097 |
val prove_at_0 = prove_at_aux Simple_Trim
|
|
2098 |
val prove_at_left_0 = prove_at_aux Neg_Trim
|
|
2099 |
val prove_at_right_0 = prove_at_aux Pos_Trim
|
|
2100 |
|
|
2101 |
|
|
2102 |
fun prove_nhds ectxt (thm, basis) =
|
|
2103 |
let
|
|
2104 |
fun simplify (a, b, c) = (a, simplify_expansion ectxt b, c)
|
|
2105 |
fun go thm basis =
|
|
2106 |
if fastype_of (get_expansion thm) = @{typ "real"} then
|
|
2107 |
@{thm expands_to_real_imp_filterlim} OF [thm]
|
|
2108 |
else
|
|
2109 |
case whnf_expansion ectxt thm |> simplify of
|
|
2110 |
(NONE, thm, _) => @{thm expands_to_MSLNil_imp_filterlim} OF [thm]
|
|
2111 |
| (SOME _, thm, _) => (
|
|
2112 |
case zeroness_oracle true (SOME Sgn_Trim) ectxt (get_exponent (get_expansion thm)) of
|
|
2113 |
(IsZero, SOME zero_thm) =>
|
|
2114 |
@{thm expands_to_zero_exponent_imp_filterlim(2)} OF
|
|
2115 |
[thm, get_basis_wf_thm basis, zero_thm,
|
|
2116 |
go (thm RS @{thm expands_to_hd''}) (tl_basis basis)]
|
|
2117 |
| (IsNeg, SOME neg_thm) =>
|
|
2118 |
@{thm expands_to_neg_exponent_imp_filterlim} OF
|
|
2119 |
[thm, get_basis_wf_thm basis, neg_thm]
|
|
2120 |
| (IsPos, _) =>
|
|
2121 |
go (try_drop_leading_term ectxt thm) basis
|
|
2122 |
| _ => raise TERM ("Unexpected zeroness result in prove_nhds",
|
|
2123 |
[get_exponent (get_expansion thm)]))
|
|
2124 |
in
|
|
2125 |
go thm basis
|
|
2126 |
end
|
|
2127 |
|
|
2128 |
fun prove_equivalent theta ectxt (thm1, thm2, basis) =
|
|
2129 |
let
|
|
2130 |
val ((thm1, _, SOME trimmed_thm1), (thm2, _, SOME trimmed_thm2)) =
|
|
2131 |
apply2 (trim_expansion true (SOME Simple_Trim) ectxt) ((thm1, basis), (thm2, basis))
|
|
2132 |
val pat = ConsPat (@{const_name Pair}, [ConsPat (@{const_name Lazy_Eval.cmp_result.EQ}, []),
|
|
2133 |
ConsPat (@{const_name Pair}, [AnyPat ("_", 0), AnyPat ("_", 0)])])
|
|
2134 |
val (exp1, exp2) = apply2 get_expansion (thm1, thm2)
|
|
2135 |
val T = fastype_of exp1
|
|
2136 |
val t = mk_compare_expansions_const T $ exp1 $ exp2
|
|
2137 |
fun eq_thm conv = HOLogic.mk_obj_eq (conv (Thm.cterm_of (get_ctxt ectxt) t))
|
|
2138 |
val imp_thm =
|
|
2139 |
if theta then @{thm compare_expansions_EQ_imp_bigtheta}
|
|
2140 |
else @{thm compare_expansions_EQ_same}
|
|
2141 |
in
|
|
2142 |
case match ectxt pat t (SOME []) of
|
|
2143 |
(SOME _, t, conv) =>
|
|
2144 |
let
|
|
2145 |
val [_, c1, c2] = HOLogic.strip_tuple t
|
|
2146 |
val c12_thm = if theta then [] else [the (try_prove_real_eq true ectxt (c1, c2))]
|
|
2147 |
in
|
|
2148 |
imp_thm OF ([eq_thm conv, trimmed_thm1, trimmed_thm2, thm1, thm2, get_basis_wf_thm basis]
|
|
2149 |
@ c12_thm)
|
|
2150 |
end
|
|
2151 |
| _ => raise TERM ("prove_equivalent", map get_expanded_fun [thm1, thm2])
|
|
2152 |
end
|
|
2153 |
|
|
2154 |
val prove_bigtheta = prove_equivalent true
|
|
2155 |
val prove_asymp_equiv = prove_equivalent false
|
|
2156 |
|
|
2157 |
fun print_trimming_error s ectxt exp =
|
|
2158 |
let
|
|
2159 |
val c = get_coeff exp
|
|
2160 |
val t = if fastype_of c = @{typ real} then c else get_eval c
|
|
2161 |
in
|
|
2162 |
if #verbose (#ctxt ectxt) then
|
|
2163 |
let
|
|
2164 |
val ctxt = get_ctxt ectxt
|
|
2165 |
val p = Pretty.str "real_asymp failed to show zeroness of the following expression:"
|
|
2166 |
val p = Pretty.chunks [p, Pretty.indent 2 (Syntax.pretty_term ctxt t)]
|
|
2167 |
val _ = Pretty.writeln p
|
|
2168 |
in
|
|
2169 |
raise TERM (s, [t])
|
|
2170 |
end
|
|
2171 |
else
|
|
2172 |
raise TERM (s, [t])
|
|
2173 |
end
|
|
2174 |
|
|
2175 |
fun prove_smallo ectxt (thm1, thm2, basis) =
|
|
2176 |
let
|
|
2177 |
val (thm2, _, SOME trimmed_thm) = trim_expansion true (SOME Simple_Trim) ectxt (thm2, basis)
|
|
2178 |
val es = get_exponents (get_expansion thm2)
|
|
2179 |
in
|
|
2180 |
case trim_expansion_while_greater true (SOME es) false NONE ectxt (thm1, basis) of
|
|
2181 |
(thm1, Aborted LESS, thms) =>
|
|
2182 |
@{thm compare_expansions_LT} OF [prove_compare_expansions LESS (map snd thms),
|
|
2183 |
trimmed_thm, thm1, thm2, get_basis_wf_thm basis]
|
|
2184 |
| (thm1, Aborted _, _) =>
|
|
2185 |
print_trimming_error "prove_smallo" ectxt (get_expansion thm1)
|
|
2186 |
| (thm1, Trimmed _, _) =>
|
|
2187 |
print_trimming_error "prove_smallo" ectxt (get_expansion thm1)
|
|
2188 |
end
|
|
2189 |
|
|
2190 |
fun prove_bigo ectxt (thm1, thm2, basis) =
|
|
2191 |
let
|
|
2192 |
val (thm2, _, SOME trimmed_thm) = trim_expansion true (SOME Simple_Trim) ectxt (thm2, basis)
|
|
2193 |
val es = get_exponents (get_expansion thm2)
|
|
2194 |
in
|
|
2195 |
case trim_expansion_while_greater false (SOME es) false NONE ectxt (thm1, basis) of
|
|
2196 |
(thm1, Aborted LESS, thms) =>
|
|
2197 |
@{thm landau_o.small_imp_big[OF compare_expansions_LT]} OF
|
|
2198 |
[prove_compare_expansions LESS (map snd thms), trimmed_thm, thm1, thm2,
|
|
2199 |
get_basis_wf_thm basis]
|
|
2200 |
| (thm1, Aborted EQ, thms) =>
|
|
2201 |
@{thm compare_expansions_EQ_imp_bigo} OF [prove_compare_expansions EQ (map snd thms),
|
|
2202 |
trimmed_thm, thm1, thm2, get_basis_wf_thm basis]
|
|
2203 |
| (thm1, Trimmed _, _) =>
|
|
2204 |
print_trimming_error "prove_bigo" ectxt (get_expansion thm1)
|
|
2205 |
end
|
|
2206 |
|
|
2207 |
|
|
2208 |
fun prove_asymptotic_relation_aux mode f ectxt (thm1, thm2, basis) = f (
|
|
2209 |
let
|
|
2210 |
val thm = @{thm expands_to_minus} OF [get_basis_wf_thm basis, thm1, thm2]
|
|
2211 |
in
|
|
2212 |
case ev_zeroness_oracle ectxt (get_expanded_fun thm) of
|
|
2213 |
SOME zero_thm => (EQUAL, zero_thm RS @{thm eventually_diff_zero_imp_eq})
|
|
2214 |
| _ => (
|
|
2215 |
case trim_expansion true (SOME mode) ectxt (thm, basis) of
|
|
2216 |
(thm, IsPos, SOME pos_thm) =>
|
|
2217 |
(GREATER, @{thm expands_to_imp_eventually_gt} OF [get_basis_wf_thm basis, thm, pos_thm])
|
|
2218 |
| (thm, IsNeg, SOME neg_thm) =>
|
|
2219 |
(LESS, @{thm expands_to_imp_eventually_lt} OF [get_basis_wf_thm basis, thm, neg_thm])
|
|
2220 |
| _ => raise TERM ("Unexpected zeroness result in prove_asymptotic_relation", []))
|
|
2221 |
end)
|
|
2222 |
|
|
2223 |
val prove_eventually_greater = prove_asymptotic_relation_aux Pos_Trim snd
|
|
2224 |
val prove_eventually_less = prove_asymptotic_relation_aux Neg_Trim snd
|
|
2225 |
val prove_asymptotic_relation = prove_asymptotic_relation_aux Sgn_Trim I
|
|
2226 |
|
|
2227 |
fun prove_eventually_nonzero ectxt (thm, basis) =
|
|
2228 |
case trim_expansion true (SOME Simple_Trim) ectxt (thm, basis) of
|
|
2229 |
(thm, _, SOME trimmed_thm) =>
|
|
2230 |
@{thm expands_to_imp_eventually_nz} OF [get_basis_wf_thm basis, thm, trimmed_thm]
|
|
2231 |
| _ => raise TERM ("prove_eventually_nonzero", [get_expanded_fun thm])
|
|
2232 |
|
|
2233 |
fun extract_terms (n, strict) ectxt basis t =
|
|
2234 |
let
|
|
2235 |
val bs = get_basis_list basis
|
|
2236 |
fun mk_constfun c = (Abs ("x", @{typ real}, c))
|
|
2237 |
val const_0 = mk_constfun @{term "0 :: real"}
|
|
2238 |
val const_1 = mk_constfun @{term "1 :: real"}
|
|
2239 |
fun uminus t = Term.betapply (@{term "\<lambda>(f::real\<Rightarrow>real) x. -f x"}, t)
|
|
2240 |
fun betapply2 a b c = Term.betapply (Term.betapply (a, b), c)
|
|
2241 |
|
|
2242 |
fun mk_sum' [] acc = acc
|
|
2243 |
| mk_sum' ((t, sgn) :: ts) acc = mk_sum' ts (
|
|
2244 |
if sgn then
|
|
2245 |
betapply2 @{term "%(f::real=>real) g x. f x - g x"} acc t
|
|
2246 |
else
|
|
2247 |
betapply2 @{term "%(f::real=>real) g x. f x + g x"} acc t)
|
|
2248 |
fun mk_sum [] = const_0
|
|
2249 |
| mk_sum ((t, sgn) :: ts) = mk_sum' ts (if sgn then uminus t else t)
|
|
2250 |
|
|
2251 |
fun mk_mult a b =
|
|
2252 |
if a aconv const_0 then
|
|
2253 |
const_0
|
|
2254 |
else if b aconv const_0 then
|
|
2255 |
const_0
|
|
2256 |
else if a aconv @{term "\<lambda>_::real. 1 :: real"} then
|
|
2257 |
b
|
|
2258 |
else if b aconv @{term "\<lambda>_::real. 1 :: real"} then
|
|
2259 |
a
|
|
2260 |
else if a aconv @{term "\<lambda>_::real. -1 :: real"} then
|
|
2261 |
Term.betapply (@{term "\<lambda>(f::real\<Rightarrow>real) x. -f x"}, b)
|
|
2262 |
else if b aconv @{term "\<lambda>_::real. -1 :: real"} then
|
|
2263 |
Term.betapply (@{term "\<lambda>(f::real\<Rightarrow>real) x. -f x"}, a)
|
|
2264 |
else
|
|
2265 |
Abs ("x", @{typ real}, @{term "( *) :: real => _"} $
|
|
2266 |
(Term.betapply (a, Bound 0)) $ (Term.betapply (b, Bound 0)))
|
|
2267 |
|
|
2268 |
fun mk_powr b e =
|
|
2269 |
if e = @{term "0 :: real"} then
|
|
2270 |
const_1
|
|
2271 |
else
|
|
2272 |
let
|
|
2273 |
val n = HOLogic.dest_number e |> snd
|
|
2274 |
in
|
|
2275 |
if n >= 0 then
|
|
2276 |
Term.betapply (Term.betapply (@{term "%(b::real=>real) e x. b x ^ e"}, b),
|
|
2277 |
HOLogic.mk_number @{typ nat} n)
|
|
2278 |
else
|
|
2279 |
Term.betapply (Term.betapply (@{term "%(b::real=>real) e x. b x powr e"}, b), e)
|
|
2280 |
end
|
|
2281 |
handle TERM _ =>
|
|
2282 |
Term.betapply (Term.betapply (@{term "%(b::real=>real) e x. b x powr e"}, b), e)
|
|
2283 |
|
|
2284 |
fun mk_scale_elem b e acc =
|
|
2285 |
let
|
|
2286 |
val e = simplify_term ectxt e
|
|
2287 |
in
|
|
2288 |
if e = @{term "0 :: real"} then
|
|
2289 |
acc
|
|
2290 |
else if e = @{term "1 :: real"} then
|
|
2291 |
mk_mult acc b
|
|
2292 |
else
|
|
2293 |
mk_mult acc (mk_powr b e)
|
|
2294 |
end
|
|
2295 |
|
|
2296 |
fun mk_scale_elems [] _ acc = acc
|
|
2297 |
| mk_scale_elems (b :: bs) (e :: es) acc =
|
|
2298 |
mk_scale_elems bs es (mk_scale_elem b e acc)
|
|
2299 |
| mk_scale_elems _ _ _ = raise Match
|
|
2300 |
|
|
2301 |
fun mk_summand c es =
|
|
2302 |
let
|
|
2303 |
val es = mk_scale_elems bs es @{term "\<lambda>_::real. 1 :: real"}
|
|
2304 |
in
|
|
2305 |
case c of
|
|
2306 |
Const (@{const_name uminus}, _) $ c => ((c, true), es)
|
|
2307 |
| _ => ((c, false), es)
|
|
2308 |
end
|
|
2309 |
|
|
2310 |
fun go _ _ _ acc 0 = (acc, 0)
|
|
2311 |
| go 0 es t acc n =
|
|
2312 |
let
|
|
2313 |
val c = simplify_term ectxt t
|
|
2314 |
in
|
|
2315 |
if strict andalso c = @{term "0 :: real"} then
|
|
2316 |
(acc, n)
|
|
2317 |
else
|
|
2318 |
(mk_summand c (rev es) :: acc, n - 1)
|
|
2319 |
end
|
|
2320 |
| go m es t acc n =
|
|
2321 |
case Lazy_Eval.whnf ectxt t |> fst of
|
|
2322 |
Const (@{const_name MS}, _) $ cs $ _ =>
|
|
2323 |
go' m es (simplify_term ectxt cs) acc n
|
|
2324 |
| _ => raise TERM("extract_terms", [t])
|
|
2325 |
and go' _ _ _ acc 0 = (acc, 0)
|
|
2326 |
| go' m es cs acc n =
|
|
2327 |
case Lazy_Eval.whnf ectxt cs |> fst of
|
|
2328 |
Const (@{const_name MSLNil}, _) => (acc, n)
|
|
2329 |
| Const (@{const_name MSLCons}, _) $ c $ cs => (
|
|
2330 |
case Lazy_Eval.whnf ectxt c |> fst |> HOLogic.dest_prod of
|
|
2331 |
(c, e) =>
|
|
2332 |
case go (m - 1) (e :: es) c acc n of
|
|
2333 |
(acc, n) => go' m es (simplify_term ectxt cs) acc n)
|
|
2334 |
| _ => raise TERM("extract_terms", [t])
|
|
2335 |
val (summands, remaining) = go (basis_size basis) [] t [] (n + 1)
|
|
2336 |
val (summands, error) =
|
|
2337 |
if remaining = 0 then (rev (tl summands), SOME (snd (hd summands))) else (rev summands, NONE)
|
|
2338 |
val summands = map (fn ((c, sgn), es) => (mk_mult (mk_constfun c) es, sgn)) summands
|
|
2339 |
val error = Option.map (fn err => Term.betapply (@{term "\<lambda>f::real\<Rightarrow>real. O(f)"}, err)) error
|
|
2340 |
val expansion = mk_sum summands
|
|
2341 |
in
|
|
2342 |
(expansion, error)
|
|
2343 |
end
|
|
2344 |
|
|
2345 |
end
|
|
2346 |
|
|
2347 |
|
|
2348 |
structure Multiseries_Expansion_Basic : EXPANSION_INTERFACE =
|
|
2349 |
struct
|
|
2350 |
open Multiseries_Expansion;
|
|
2351 |
|
|
2352 |
type T = expansion_thm
|
|
2353 |
|
|
2354 |
val expand_term = expand_term
|
|
2355 |
val expand_terms = expand_terms
|
|
2356 |
|
|
2357 |
val prove_nhds = prove_nhds
|
|
2358 |
val prove_at_infinity = prove_at_infinity
|
|
2359 |
val prove_at_top = prove_at_top
|
|
2360 |
val prove_at_bot = prove_at_bot
|
|
2361 |
val prove_at_0 = prove_at_0
|
|
2362 |
val prove_at_right_0 = prove_at_right_0
|
|
2363 |
val prove_at_left_0 = prove_at_left_0
|
|
2364 |
val prove_eventually_nonzero = prove_eventually_nonzero
|
|
2365 |
|
|
2366 |
val prove_eventually_less = prove_eventually_less
|
|
2367 |
val prove_eventually_greater = prove_eventually_greater
|
|
2368 |
|
|
2369 |
val prove_smallo = prove_smallo
|
|
2370 |
val prove_bigo = prove_bigo
|
|
2371 |
val prove_bigtheta = prove_bigtheta
|
|
2372 |
val prove_asymp_equiv = prove_asymp_equiv
|
|
2373 |
|
|
2374 |
end
|