Ziele Untersuchung
mit Columbo Integrität von
Datenbanken Interaktion und
Portierbarkeit Ergonomie der
Schnittstellen

Angebot Produkte Projekt Beratung

Mittel Analytik Modellierung Sprachen Algebra Logik Hardware Denken Kreativität

Zusammenhänge Gesellschaft Wirtschaft Branche Firma


products/sources/formale Sprachen/Isabelle/HOL/Analysis/ (Beweissystem Isabelle Version 2025-1^©) Datei vom 16.11.2025 mit Größe 11 kB

Quelle metric_arith.ML Sprache: SML

(*  Title:      HOL/Analysis/metric_arith.ML
    Author:     Maximilian Schäffeler (port from HOL Light)

A decision procedure for metric spaces.
*)

signature METRIC_ARITH =
sig
  val trace: bool Config.T
  val argo_timeout: real Config.T
  val metric_arith_tac : Proof.context -> int -> tactic
end

structure Metric_Arith : METRIC_ARITH =
struct

(* apply f to both cterms in ct_pair, merge results *)
fun app_union_ct_pair f ct_pair = uncurry (union (op aconvc)) (apply2 f ct_pair)

val trace = Attrib.setup_config_bool \<^binding>\<open>metric_trace\<close> (K false)

fun trace_tac ctxt msg =
  if Config.get ctxt trace then print_tac ctxt msg else all_tac

val argo_timeout = Attrib.setup_config_real \<^binding>\<open>metric_argo_timeout\<close> (K 20.0)

fun argo_ctxt ctxt =
  let
    val ctxt1 =
      if Config.get ctxt trace
      then Config.map (Argo_Tactic.trace) (K "basic") ctxt
      else ctxt
  in Config.put Argo_Tactic.timeout (Config.get ctxt1 argo_timeout) ctxt1 end

fun free_in v t =
  Term.exists_subterm (fn u => u aconv Thm.term_of v) (Thm.term_of t);

(* build a cterm set with elements cts of type ty *)
fun mk_ct_set ctxt ty =
  map Thm.term_of #>
  HOLogic.mk_set ty #>
  Thm.cterm_of ctxt

fun prenex_tac ctxt =
  let
    val prenex_simps = Proof_Context.get_thms ctxt @{named_theorems metric_prenex}
    val prenex_ctxt = put_simpset HOL_basic_ss ctxt |> Simplifier.add_simps prenex_simps
  in
    simp_tac prenex_ctxt THEN'
    K (trace_tac ctxt "Prenex form")
  end

fun nnf_tac ctxt =
  let
    val nnf_simps = Proof_Context.get_thms ctxt @{named_theorems metric_nnf}
    val nnf_ctxt = put_simpset HOL_basic_ss ctxt |> Simplifier.add_simps nnf_simps
  in
    simp_tac nnf_ctxt THEN'
    K (trace_tac ctxt "NNF form")
  end

fun unfold_tac ctxt =
  asm_full_simp_tac (put_simpset HOL_basic_ss ctxt
    |> Simplifier.add_simps (Proof_Context.get_thms ctxt @{named_theorems metric_unfold}))

fun pre_arith_tac ctxt =
  simp_tac (put_simpset HOL_basic_ss ctxt
    |> Simplifier.add_simps (Proof_Context.get_thms ctxt @{named_theorems metric_pre_arith})) THEN'
    K (trace_tac ctxt "Prepared for decision procedure")

fun dist_refl_sym_tac ctxt =
  let
    val refl_sym_simps = @{thms dist_self dist_commute add_0_right add_0_left simp_thms}
    val refl_sym_ctxt = put_simpset HOL_basic_ss ctxt |> Simplifier.add_simps refl_sym_simps
  in
    simp_tac refl_sym_ctxt THEN'
    K (trace_tac ctxt ("Simplified using " ^ @{make_string} refl_sym_simps))
  end

fun is_exists \<^Const_>\<open>Ex _ for _\<close> = true
  | is_exists \<^Const_>\<open>Trueprop for t\<close> = is_exists t
  | is_exists _ = false

fun is_forall \<^Const_>\<open>All _ for _\<close> = true
  | is_forall \<^Const_>\<open>Trueprop for t\<close> = is_forall t
  | is_forall _ = false

(* find all free points in ct of type metric_ty *)
fun find_points ctxt metric_ty ct =
  let
    fun find ct =
      (if Thm.typ_of_cterm ct = metric_ty then [ct] else []) @
      (case Thm.term_of ct of
        _ $ _ => app_union_ct_pair find (Thm.dest_comb ct)
      | Abs _ =>
          (*ensure the point doesn't contain the bound variable*)
          let val (x, body) = Thm.dest_abs_global ct
          in filter_out (free_in x) (find body) end
      | _ => [])
  in
    (case find ct of
      [] =>
        (*if no point can be found, invent one*)
        let
          val names = Variable.names_of ctxt |> Term.declare_free_names (Thm.term_of ct)
          val x = Free (#1 (Name.variant "x" names), metric_ty)
        in [Thm.cterm_of ctxt x] end
    | points => points)
  end

(* find all cterms "dist x y" in ct, where x and y have type metric_ty *)
fun find_dist metric_ty =
  let
    fun find ct =
      (case Thm.term_of ct of
        \<^Const_>\<open>dist T for _ _\<close> => if T = metric_ty then [ct] else []
      | _ $ _ => app_union_ct_pair find (Thm.dest_comb ct)
      | Abs _ =>
          let val (x, body) = Thm.dest_abs_global ct
          in filter_out (free_in x) (find body) end
      | _ => [])
  in find end

(* find all "x=y", where x has type metric_ty *)
fun find_eq metric_ty =
  let
    fun find ct =
      (case Thm.term_of ct of
        \<^Const_>\<open>HOL.eq T for _ _\<close> =>
          if T = metric_ty then [ct]
          else app_union_ct_pair find (Thm.dest_binop ct)
      | _ $ _ => app_union_ct_pair find (Thm.dest_comb ct)
      | Abs _ =>
          let val (x, body) = Thm.dest_abs_global ct
          in filter_out (free_in x) (find body) end
      | _ => [])
  in find end

(* rewrite ct of the form "dist x y" using maxdist_thm *)
fun maxdist_conv ctxt fset_ct ct =
  let
    val (x, y) = Thm.dest_binop ct
    val solve_prems =
      rule_by_tactic ctxt (ALLGOALS (simp_tac (put_simpset HOL_ss ctxt
        |> Simplifier.add_simps @{thms finite.emptyI finite_insert empty_iff insert_iff})))
    val image_simp =
      Simplifier.rewrite (put_simpset HOL_ss ctxt
        |> Simplifier.add_simps @{thms image_empty image_insert})
    val dist_refl_sym_simp =
      Simplifier.rewrite (put_simpset HOL_ss ctxt
        |> Simplifier.add_simps @{thms dist_commute dist_self})
    val algebra_simp =
      Simplifier.rewrite (put_simpset HOL_ss ctxt |> Simplifier.add_simps
        @{thms diff_self diff_0_right diff_0 abs_zero abs_minus_cancel abs_minus_commute})
    val insert_simp =
      Simplifier.rewrite (put_simpset HOL_ss ctxt
        |> Simplifier.add_simps @{thms insert_absorb2 insert_commute})
    val sup_simp =
      Simplifier.rewrite (put_simpset HOL_ss ctxt
        |> Simplifier.add_simps @{thms cSup_singleton Sup_insert_insert})
    val real_abs_dist_simp =
      Simplifier.rewrite (put_simpset HOL_ss ctxt
        |> Simplifier.add_simps @{thms real_abs_dist})
    val maxdist_thm =
      \<^instantiate>\<open>'a = \Thm.ctyp_of_cterm x\ and x and y and s = fset_ct in
        lemma \<open>finite s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> dist x y \<equiv> SUP a\<in>s. \<bar>dist x a - dist a y\<bar>\<close>
          for x y :: \<open>'a::metric_space\
          by (fact maxdist_thm)\<close>
      |> solve_prems
  in
    ((Conv.rewr_conv maxdist_thm) then_conv
    (* SUP to Sup *)
    image_simp then_conv
    dist_refl_sym_simp then_conv
    algebra_simp then_conv
    (* eliminate duplicate terms in set *)
    insert_simp then_conv
    (* Sup to max *)
    sup_simp then_conv
    real_abs_dist_simp) ct
  end

(* rewrite ct of the form "x=y" using metric_eq_thm *)
fun metric_eq_conv ctxt fset_ct ct =
  let
    val (x, y) = Thm.dest_binop ct
    val solve_prems =
      rule_by_tactic ctxt (ALLGOALS (simp_tac (put_simpset HOL_ss ctxt
        |> Simplifier.add_simps @{thms empty_iff insert_iff})))
    val ball_simp =
      Simplifier.rewrite (put_simpset HOL_ss ctxt
        |> Simplifier.add_simps @{thms Set.ball_empty ball_insert})
    val dist_refl_sym_simp =
      Simplifier.rewrite (put_simpset HOL_ss ctxt
        |> Simplifier.add_simps @{thms dist_commute dist_self})
    val metric_eq_thm =
      \<^instantiate>\<open>'a = \Thm.ctyp_of_cterm x\ and x and y and s = fset_ct in
        lemma \<open>x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x = y \<equiv> \<forall>a\<in>s. dist x a = dist y a\<close>
          for x y :: \<open>'a::metric_space\
          by (fact metric_eq_thm)\<close>
      |> solve_prems
  in
    ((Conv.rewr_conv metric_eq_thm) then_conv
    (*convert \<forall>x\<in>{x\<^sub>1,...,x\<^sub>n}. P x to P x\<^sub>1 \<and> ... \<and> P x\<^sub>n*)
    ball_simp then_conv
    dist_refl_sym_simp) ct
  end

(* build list of theorems "0 \<le> dist x y" for all dist terms in ct *)
fun augment_dist_pos metric_ty ct =
  let fun inst dist_ct =
    let val (x, y) = Thm.dest_binop dist_ct in
      \<^instantiate>\<open>'a = \Thm.ctyp_of_cterm x\ and x and y
        in lemma \<open>dist x y \<ge> 0\<close> for x y :: \<open>'a::metric_space\ by simp\
    end
  in map inst (find_dist metric_ty ct) end

(* apply maxdist_conv and metric_eq_conv to the goal, thereby embedding the goal in (\<real>\<^sup>n,dist\<^sub>\<infinity>) *)
fun embedding_tac ctxt metric_ty = CSUBGOAL (fn (goal, i) =>
  let
    val points = find_points ctxt metric_ty goal
    val fset_ct = mk_ct_set ctxt metric_ty points
    (*embed all subterms of the form "dist x y" in (\<real>\<^sup>n,dist\<^sub>\<infinity>)*)
    val eq1 = map (maxdist_conv ctxt fset_ct) (find_dist metric_ty goal)
    (*replace point equality by equality of components in \<real>\<^sup>n*)
    val eq2 = map (metric_eq_conv ctxt fset_ct) (find_eq metric_ty goal)
  in
    (K (trace_tac ctxt "Embedding into \\<^sup>n") THEN'
      CONVERSION (Conv.top_sweep_rewrs_conv (eq1 @ eq2) ctxt)) i
  end)

(* decision procedure for linear real arithmetic *)
fun lin_real_arith_tac ctxt metric_ty = CSUBGOAL (fn (goal, i) =>
  let val dist_thms = augment_dist_pos metric_ty goal
  in Argo_Tactic.argo_tac (argo_ctxt ctxt) dist_thms i end)

fun basic_metric_arith_tac ctxt metric_ty =
  SELECT_GOAL (
    dist_refl_sym_tac ctxt 1 THEN
    IF_UNSOLVED (embedding_tac ctxt metric_ty 1) THEN
    IF_UNSOLVED (pre_arith_tac ctxt 1) THEN
    IF_UNSOLVED (lin_real_arith_tac ctxt metric_ty 1))

(* tries to infer the metric space from ct from dist terms,
   if no dist terms are present, equality terms will be used *)
fun guess_metric ctxt tm =
  let
    val thy = Proof_Context.theory_of ctxt
    fun find_dist t =
      (case t of
        \<^Const_>\<open>dist T for _ _\<close>  => SOME T
      | t1 $ t2 => (case find_dist t1 of NONE => find_dist t2 | some => some)
      | Abs _ => find_dist (#2 (Term.dest_abs_global t))
      | _ => NONE)
    fun find_eq t =
      (case t of
        \<^Const_>\<open>HOL.eq T for l r\<close> =>
          if Sign.of_sort thy (T, \<^sort>\<open>metric_space\<close>) then SOME T
          else (case find_eq l of NONE => find_eq r | some => some)
      | t1 $ t2 => (case find_eq t1 of NONE => find_eq t2 | some => some)
      | Abs _ => find_eq (#2 (Term.dest_abs_global t))
      | _ => NONE)
    in
      (case find_dist tm of
        SOME ty => ty
      | NONE =>
          case find_eq tm of
            SOME ty => ty
          | NONE => error "No Metric Space was found")
    end

(* solve \<exists> by proving the goal for a single witness from the metric space *)
fun exists_tac ctxt = CSUBGOAL (fn (goal, i) =>
  let
    val _ = \<^assert> (i = 1)
    val metric_ty = guess_metric ctxt (Thm.term_of goal)
    val points = find_points ctxt metric_ty goal

    fun try_point_tac ctxt pt =
      let
        val ex_rule =
          \<^instantiate>\<open>'a = \Thm.ctyp_of_cterm pt\ and x = pt in
            lemma (schematic) \<open>P x \<Longrightarrow> \<exists>x::'a. P x\ by (rule exI)\
      in
        HEADGOAL (resolve_tac ctxt [ex_rule] ORELSE'
        (*variable doesn't occur in body*)
        resolve_tac ctxt @{thms exI}) THEN
        trace_tac ctxt ("Removed existential quantifier, try " ^ @{make_string} pt) THEN
        HEADGOAL (try_points_tac ctxt)
      end
    and try_points_tac ctxt = SUBGOAL (fn (g, _) =>
      if is_exists g then
        FIRST (map (try_point_tac ctxt) points)
      else if is_forall g then
        resolve_tac ctxt @{thms HOL.allI} 1 THEN
        Subgoal.FOCUS (fn {context = ctxt', ...} =>
          trace_tac ctxt "Removed universal quantifier" THEN
          try_points_tac ctxt' 1) ctxt 1
      else basic_metric_arith_tac ctxt metric_ty 1)
  in try_points_tac ctxt 1 end)

fun metric_arith_tac ctxt =
  SELECT_GOAL (
    (*unfold common definitions to get rid of sets*)
    unfold_tac ctxt 1 THEN
    (*remove all meta-level connectives*)
    IF_UNSOLVED (Object_Logic.full_atomize_tac ctxt 1) THEN
    (*convert goal to prenex form*)
    IF_UNSOLVED (prenex_tac ctxt 1) THEN
    (*and NNF to ?*)
    IF_UNSOLVED (nnf_tac ctxt 1) THEN
    (*turn all universally quantified variables into free variables, by focusing the subgoal*)
    REPEAT (HEADGOAL (resolve_tac ctxt @{thms HOL.allI})) THEN
    IF_UNSOLVED (SUBPROOF (fn {context = ctxt', ...} =>
      trace_tac ctxt' "Focused on subgoal" THEN
      exists_tac ctxt' 1) ctxt 1))

end

quality100%

¤ Dauer der Verarbeitung: 0.25 Sekunden (vorverarbeitet) ¤

Wurzel

Suchen

Beweissystem der NASA

Beweissystem Isabelle

NIST Cobol Testsuite

Cephes Mathematical Library

Wiener Entwicklungsmethode

Haftungshinweis

Die Informationen auf dieser Webseite wurden nach bestem Wissen sorgfältig zusammengestellt. Es wird jedoch weder Vollständigkeit, noch Richtigkeit, noch Qualität der bereit gestellten Informationen zugesichert.