Sep_Cancel.thy

(*
 * Copyright 2020, Data61, CSIRO (ABN 41 687 119 230)
 *
 * SPDX-License-Identifier: BSD-2-Clause
 *)

theory Sep_Cancel
imports Sep_Provers Sep_Tactic_Helpers Sep_Cancel_Set
begin

(* Sep_Cancel performs cancellative elimination of conjuncts *)


lemma sep_curry': "\<lbrakk>(P \<and>* F) s; \<And>s. (Q \<and>* P \<and>* F) s \<Longrightarrow> R s\<rbrakk> \<Longrightarrow> (Q \<longrightarrow>* R) s"
  by (metis (full_types) sep.mult_commute sep_curry)

lemma sep_conj_sep_impl_safe:
  "(P \<longrightarrow>* P') s \<Longrightarrow> (\<And>s. ((P \<longrightarrow>* P') \<and>* Q) s \<Longrightarrow> (Q') s) \<Longrightarrow> (Q \<longrightarrow>* Q') s"
  by (rule sep_curry)

lemma  sep_conj_sep_impl_safe': "P s \<Longrightarrow> (\<And>s. (P \<and>* Q) s \<Longrightarrow> (P \<and>* R) s) \<Longrightarrow> (Q \<longrightarrow>* P \<and>* R) s"
  by (rule sep_curry)

lemma sep_wand_lens_simple: "(\<And>s. T s = (Q \<and>* R) s) \<Longrightarrow> (P \<longrightarrow>* T) s \<Longrightarrow> (P \<longrightarrow>* Q \<and>* R) s"
  by (clarsimp simp: sep_impl_def)

schematic_goal schem_impAny: " (?C \<and>* B) s \<Longrightarrow> A s" by (erule sep_mp)

ML \<open>
  fun sep_cancel_tactic ctxt concl  =
    let val thms = rev (SepCancel_Rules.get ctxt)
        val tac  = assume_tac ctxt ORELSE'
                   eresolve_tac ctxt [@{thm sep_mp}, @{thm sep_conj_empty}, @{thm sep_empty_conj}] ORELSE'
                   sep_erule_tactic ctxt thms
        val direct_tac = eresolve_tac ctxt thms
        val safe_sep_wand_tac = rotator' ctxt (resolve0_tac [@{thm sep_wand_lens_simple}]) (eresolve0_tac [@{thm sep_conj_sep_impl_safe'}])
        fun sep_cancel_tactic_inner true   = sep_erule_full_tac' tac ctxt
          | sep_cancel_tactic_inner false  = sep_erule_full_tac tac ctxt
  in sep_cancel_tactic_inner concl ORELSE'
      eresolve_tac ctxt [@{thm sep_curry'}, @{thm sep_conj_sep_impl_safe}, @{thm sep_imp_empty}, @{thm sep_empty_imp'}] ORELSE'
      safe_sep_wand_tac ORELSE'
      direct_tac
  end

  fun sep_cancel_tactic' ctxt concl =
    let
      val sep_cancel = sep_cancel_tactic ctxt
    in
      (sep_flatten ctxt THEN_ALL_NEW sep_cancel concl) ORELSE' sep_cancel concl
    end

  fun sep_cancel_method (concl,_) ctxt = SIMPLE_METHOD' (sep_cancel_tactic' ctxt concl)

  val sep_cancel_syntax =
    Method.sections [Args.add -- Args.colon >> K (Method.modifier SepCancel_Rules.add @{here})];

  val sep_cancel_syntax' =
    Scan.lift (Args.mode "concl") -- sep_cancel_syntax
\<close>

method_setup sep_cancel =
  \<open>sep_cancel_syntax' >> sep_cancel_method\<close>  \<open>Simple elimination of conjuncts\<close>

end