src/HOL/simpdata.ML
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(*  Title:      HOL/simpdata.ML
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    ID:         $Id$
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    Author:     Tobias Nipkow
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    Copyright   1991  University of Cambridge
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Instantiation of the generic simplifier for HOL.
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*)
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(** tools setup **)
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structure Quantifier1 = Quantifier1Fun
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(struct
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  (*abstract syntax*)
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  fun dest_eq ((c as Const("op =",_)) $ s $ t) = SOME (c, s, t)
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    | dest_eq _ = NONE;
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  fun dest_conj ((c as Const("op &",_)) $ s $ t) = SOME (c, s, t)
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    | dest_conj _ = NONE;
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  fun dest_imp ((c as Const("op -->",_)) $ s $ t) = SOME (c, s, t)
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    | dest_imp _ = NONE;
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  val conj = HOLogic.conj
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  val imp  = HOLogic.imp
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  (*rules*)
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  val iff_reflection = HOL.eq_reflection
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  val iffI = HOL.iffI
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  val iff_trans = HOL.trans
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  val conjI= HOL.conjI
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  val conjE= HOL.conjE
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  val impI = HOL.impI
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  val mp   = HOL.mp
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  val uncurry = thm "uncurry"
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  val exI  = HOL.exI
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  val exE  = HOL.exE
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  val iff_allI = thm "iff_allI"
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  val iff_exI = thm "iff_exI"
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  val all_comm = thm "all_comm"
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  val ex_comm = thm "ex_comm"
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end);
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structure HOL =
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struct
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open HOL;
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val Eq_FalseI = thm "Eq_FalseI";
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val Eq_TrueI = thm "Eq_TrueI";
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val simp_implies_def = thm "simp_implies_def";
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val simp_impliesI = thm "simp_impliesI";
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fun mk_meta_eq r = r RS eq_reflection;
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fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r;
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fun mk_eq thm = case concl_of thm
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  (*expects Trueprop if not == *)
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  of Const ("==",_) $ _ $ _ => thm
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   | _ $ (Const ("op =", _) $ _ $ _) => mk_meta_eq thm
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   | _ $ (Const ("Not", _) $ _) => thm RS Eq_FalseI
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   | _ => thm RS Eq_TrueI;
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fun mk_eq_True r =
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  SOME (r RS meta_eq_to_obj_eq RS Eq_TrueI) handle Thm.THM _ => NONE;
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(* Produce theorems of the form
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  (P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y)
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*)
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fun lift_meta_eq_to_obj_eq i st =
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  let
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    fun count_imp (Const ("HOL.simp_implies", _) $ P $ Q) = 1 + count_imp Q
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      | count_imp _ = 0;
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    val j = count_imp (Logic.strip_assums_concl (List.nth (prems_of st, i - 1)))
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  in if j = 0 then meta_eq_to_obj_eq
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    else
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      let
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        val Ps = map (fn k => Free ("P" ^ string_of_int k, propT)) (1 upto j);
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        fun mk_simp_implies Q = foldr (fn (R, S) =>
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          Const ("HOL.simp_implies", propT --> propT --> propT) $ R $ S) Q Ps
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        val aT = TFree ("'a", HOLogic.typeS);
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        val x = Free ("x", aT);
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        val y = Free ("y", aT)
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      in Goal.prove_global (Thm.theory_of_thm st) []
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        [mk_simp_implies (Logic.mk_equals (x, y))]
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        (mk_simp_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y))))
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        (fn prems => EVERY
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         [rewrite_goals_tac [simp_implies_def],
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          REPEAT (ares_tac (meta_eq_to_obj_eq :: map (rewrite_rule [simp_implies_def]) prems) 1)])
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      end
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  end;
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(*Congruence rules for = (instead of ==)*)
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fun mk_meta_cong rl = zero_var_indexes
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  (let val rl' = Seq.hd (TRYALL (fn i => fn st =>
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     rtac (lift_meta_eq_to_obj_eq i st) i st) rl)
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   in mk_meta_eq rl' handle THM _ =>
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     if can Logic.dest_equals (concl_of rl') then rl'
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     else error "Conclusion of congruence rules must be =-equality"
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   end);
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(*
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val mk_atomize:      (string * thm list) list -> thm -> thm list
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looks too specific to move it somewhere else
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*)
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fun mk_atomize pairs =
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  let
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    fun atoms thm = case concl_of thm
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     of Const("Trueprop", _) $ p => (case head_of p
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        of Const(a, _) => (case AList.lookup (op =) pairs a
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           of SOME rls => maps atoms ([thm] RL rls)
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            | NONE => [thm])
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         | _ => [thm])
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      | _ => [thm]
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  in atoms end;
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fun mksimps pairs =
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  (map_filter (try mk_eq) o mk_atomize pairs o gen_all);
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fun unsafe_solver_tac prems =
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  (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
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  FIRST'[resolve_tac(reflexive_thm :: TrueI :: refl :: prems), atac, etac FalseE];
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val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac;
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(*No premature instantiation of variables during simplification*)
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fun safe_solver_tac prems =
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  (fn i => REPEAT_DETERM (match_tac [simp_impliesI] i)) THEN'
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  FIRST'[match_tac(reflexive_thm :: TrueI :: refl :: prems),
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         eq_assume_tac, ematch_tac [FalseE]];
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val safe_solver = mk_solver "HOL safe" safe_solver_tac;
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end;
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structure SplitterData =
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struct
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  structure Simplifier = Simplifier
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  val mk_eq           = HOL.mk_eq
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  val meta_eq_to_iff  = HOL.meta_eq_to_obj_eq
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  val iffD            = HOL.iffD2
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  val disjE           = HOL.disjE
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  val conjE           = HOL.conjE
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  val exE             = HOL.exE
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  val contrapos       = HOL.contrapos_nn
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  val contrapos2      = HOL.contrapos_pp
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  val notnotD         = HOL.notnotD
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end;
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structure Splitter = SplitterFun(SplitterData);
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(* integration of simplifier with classical reasoner *)
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structure Clasimp = ClasimpFun
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 (structure Simplifier = Simplifier and Splitter = Splitter
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  and Classical  = Classical and Blast = Blast
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  val iffD1 = HOL.iffD1 val iffD2 = HOL.iffD2 val notE = HOL.notE);
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structure HOL =
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struct
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open HOL;
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val mksimps_pairs =
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  [("op -->", [mp]), ("op &", [thm "conjunct1", thm "conjunct2"]),
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   ("All", [spec]), ("True", []), ("False", []),
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   ("HOL.If", [thm "if_bool_eq_conj" RS iffD1])];
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val simpset_basic =
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  Simplifier.theory_context (the_context ()) empty_ss
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    setsubgoaler asm_simp_tac
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    setSSolver safe_solver
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    setSolver unsafe_solver
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    setmksimps (mksimps mksimps_pairs)
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    setmkeqTrue mk_eq_True
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    setmkcong mk_meta_cong;
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fun simplify rews = Simplifier.full_simplify (simpset_basic addsimps rews);
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fun unfold_tac ths =
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  let val ss0 = Simplifier.clear_ss simpset_basic addsimps ths
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  in fn ss => ALLGOALS (full_simp_tac (Simplifier.inherit_context ss ss0)) end;
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(** simprocs **)
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(* simproc for proving "(y = x) == False" from premise "~(x = y)" *)
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val use_neq_simproc = ref true;
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local
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  val thy = the_context ();
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  val neq_to_EQ_False = thm "not_sym" RS HOL.Eq_FalseI;
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  fun neq_prover sg ss (eq $ lhs $ rhs) =
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    let
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      fun test thm = (case #prop (rep_thm thm) of
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                    _ $ (Not $ (eq' $ l' $ r')) =>
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                      Not = HOLogic.Not andalso eq' = eq andalso
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                      r' aconv lhs andalso l' aconv rhs
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                  | _ => false)
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    in if !use_neq_simproc then case find_first test (prems_of_ss ss)
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     of NONE => NONE
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      | SOME thm => SOME (thm RS neq_to_EQ_False)
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     else NONE
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    end
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in
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val neq_simproc = Simplifier.simproc thy "neq_simproc" ["x = y"] neq_prover;
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end; (*local*)
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(* simproc for Let *)
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val use_let_simproc = ref true;
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local
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  val thy = the_context ();
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  val Let_folded = thm "Let_folded";
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  val Let_unfold = thm "Let_unfold";
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  val (f_Let_unfold, x_Let_unfold) =
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      let val [(_$(f$x)$_)] = prems_of Let_unfold
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      in (cterm_of thy f, cterm_of thy x) end
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  val (f_Let_folded, x_Let_folded) =
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      let val [(_$(f$x)$_)] = prems_of Let_folded
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      in (cterm_of thy f, cterm_of thy x) end;
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  val g_Let_folded =
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      let val [(_$_$(g$_))] = prems_of Let_folded in cterm_of thy g end;
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in
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val let_simproc =
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  Simplifier.simproc thy "let_simp" ["Let x f"]
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   (fn sg => fn ss => fn t =>
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     let val ctxt = Simplifier.the_context ss;
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         val ([t'], ctxt') = Variable.import_terms false [t] ctxt;
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     in Option.map (hd o Variable.export ctxt' ctxt o single)
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      (case t' of (Const ("Let",_)$x$f) => (* x and f are already in normal form *)
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         if not (!use_let_simproc) then NONE
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         else if is_Free x orelse is_Bound x orelse is_Const x
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         then SOME (thm "Let_def")
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         else
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          let
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             val n = case f of (Abs (x,_,_)) => x | _ => "x";
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             val cx = cterm_of sg x;
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             val {T=xT,...} = rep_cterm cx;
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             val cf = cterm_of sg f;
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             val fx_g = Simplifier.rewrite ss (Thm.capply cf cx);
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             val (_$_$g) = prop_of fx_g;
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             val g' = abstract_over (x,g);
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           in (if (g aconv g')
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               then
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                  let
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                    val rl = cterm_instantiate [(f_Let_unfold,cf),(x_Let_unfold,cx)] Let_unfold;
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                  in SOME (rl OF [fx_g]) end
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               else if Term.betapply (f,x) aconv g then NONE (*avoid identity conversion*)
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               else let
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                     val abs_g'= Abs (n,xT,g');
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                     val g'x = abs_g'$x;
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                     val g_g'x = symmetric (beta_conversion false (cterm_of sg g'x));
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                     val rl = cterm_instantiate
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                               [(f_Let_folded,cterm_of sg f),(x_Let_folded,cx),
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                                (g_Let_folded,cterm_of sg abs_g')]
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                               Let_folded;
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                   in SOME (rl OF [transitive fx_g g_g'x])
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                   end)
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           end
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        | _ => NONE)
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     end)
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end; (*local*)
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(* generic refutation procedure *)
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(* parameters:
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   test: term -> bool
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   tests if a term is at all relevant to the refutation proof;
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   if not, then it can be discarded. Can improve performance,
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   esp. if disjunctions can be discarded (no case distinction needed!).
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   prep_tac: int -> tactic
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   A preparation tactic to be applied to the goal once all relevant premises
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   have been moved to the conclusion.
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   ref_tac: int -> tactic
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   the actual refutation tactic. Should be able to deal with goals
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   [| A1; ...; An |] ==> False
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   where the Ai are atomic, i.e. no top-level &, | or EX
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*)
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local
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  val nnf_simpset =
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    empty_ss setmkeqTrue mk_eq_True
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    setmksimps (mksimps mksimps_pairs)
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    addsimps [thm "imp_conv_disj", thm "iff_conv_conj_imp", thm "de_Morgan_disj", thm "de_Morgan_conj",
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      thm "not_all", thm "not_ex", thm "not_not"];
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  fun prem_nnf_tac i st =
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    full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st;
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in
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fun refute_tac test prep_tac ref_tac =
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  let val refute_prems_tac =
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        REPEAT_DETERM
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              (eresolve_tac [conjE, exE] 1 ORELSE
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               filter_prems_tac test 1 ORELSE
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               etac disjE 1) THEN
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        ((etac notE 1 THEN eq_assume_tac 1) ORELSE
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         ref_tac 1);
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  in EVERY'[TRY o filter_prems_tac test,
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            REPEAT_DETERM o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
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            SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
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  end;
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end; (*local*)
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val defALL_regroup =
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  Simplifier.simproc (the_context ())
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    "defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all;
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val defEX_regroup =
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  Simplifier.simproc (the_context ())
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    "defined EX" ["EX x. P x"] Quantifier1.rearrange_ex;
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val simpset_simprocs = simpset_basic
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  addsimprocs [defALL_regroup, defEX_regroup, neq_simproc, let_simproc]
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end; (*struct*)