src/HOL/Reflection/cooper_tac.ML

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Reflection/cooper_tac.ML	Tue Feb 03 16:50:41 2009 +0100
@@ -0,0 +1,139 @@
+(*  Title:      HOL/Reflection/cooper_tac.ML
+    Author:     Amine Chaieb, TU Muenchen
+*)
+
+structure Cooper_Tac =
+struct
+
+val trace = ref false;
+fun trace_msg s = if !trace then tracing s else ();
+
+val cooper_ss = @{simpset};
+
+val nT = HOLogic.natT;
+val binarith = @{thms normalize_bin_simps};
+val comp_arith = binarith @ simp_thms
+
+val zdvd_int = @{thm zdvd_int};
+val zdiff_int_split = @{thm zdiff_int_split};
+val all_nat = @{thm all_nat};
+val ex_nat = @{thm ex_nat};
+val number_of1 = @{thm number_of1};
+val number_of2 = @{thm number_of2};
+val split_zdiv = @{thm split_zdiv};
+val split_zmod = @{thm split_zmod};
+val mod_div_equality' = @{thm mod_div_equality'};
+val split_div' = @{thm split_div'};
+val Suc_plus1 = @{thm Suc_plus1};
+val imp_le_cong = @{thm imp_le_cong};
+val conj_le_cong = @{thm conj_le_cong};
+val nat_mod_add_eq = @{thm mod_add1_eq} RS sym;
+val nat_mod_add_left_eq = @{thm mod_add_left_eq} RS sym;
+val nat_mod_add_right_eq = @{thm mod_add_right_eq} RS sym;
+val int_mod_add_eq = @{thm zmod_zadd1_eq} RS sym;
+val int_mod_add_left_eq = @{thm zmod_zadd_left_eq} RS sym;
+val int_mod_add_right_eq = @{thm zmod_zadd_right_eq} RS sym;
+val nat_div_add_eq = @{thm div_add1_eq} RS sym;
+val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym;
+
+fun prepare_for_linz q fm = 
+  let
+    val ps = Logic.strip_params fm
+    val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm)
+    val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm)
+    fun mk_all ((s, T), (P,n)) =
+      if 0 mem loose_bnos P then
+        (HOLogic.all_const T $ Abs (s, T, P), n)
+      else (incr_boundvars ~1 P, n-1)
+    fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t;
+    val rhs = hs
+    val np = length ps
+    val (fm',np) =  foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n)))
+      (foldr HOLogic.mk_imp c rhs, np) ps
+    val (vs, _) = List.partition (fn t => q orelse (type_of t) = nT)
+      (OldTerm.term_frees fm' @ OldTerm.term_vars fm');
+    val fm2 = foldr mk_all2 fm' vs
+  in (fm2, np + length vs, length rhs) end;
+
+(*Object quantifier to meta --*)
+fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ;
+
+(* object implication to meta---*)
+fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp;
+
+
+fun linz_tac ctxt q i = ObjectLogic.atomize_prems_tac i THEN (fn st =>
+  let
+    val g = List.nth (prems_of st, i - 1)
+    val thy = ProofContext.theory_of ctxt
+    (* Transform the term*)
+    val (t,np,nh) = prepare_for_linz q g
+    (* Some simpsets for dealing with mod div abs and nat*)
+    val mod_div_simpset = HOL_basic_ss 
+			addsimps [refl,nat_mod_add_eq, nat_mod_add_left_eq, 
+				  nat_mod_add_right_eq, int_mod_add_eq, 
+				  int_mod_add_right_eq, int_mod_add_left_eq,
+				  nat_div_add_eq, int_div_add_eq,
+				  @{thm mod_self}, @{thm "zmod_self"},
+				  @{thm mod_by_0}, @{thm div_by_0},
+				  @{thm "zdiv_zero"}, @{thm "zmod_zero"}, @{thm "div_0"}, @{thm "mod_0"},
+				  @{thm "zdiv_1"}, @{thm "zmod_1"}, @{thm "div_1"}, @{thm "mod_1"},
+				  Suc_plus1]
+			addsimps @{thms add_ac}
+			addsimprocs [cancel_div_mod_proc]
+    val simpset0 = HOL_basic_ss
+      addsimps [mod_div_equality', Suc_plus1]
+      addsimps comp_arith
+      addsplits [split_zdiv, split_zmod, split_div', @{thm "split_min"}, @{thm "split_max"}]
+    (* Simp rules for changing (n::int) to int n *)
+    val simpset1 = HOL_basic_ss
+      addsimps [nat_number_of_def, zdvd_int] @ map (fn r => r RS sym)
+        [@{thm int_int_eq}, @{thm zle_int}, @{thm zless_int}, @{thm zadd_int}, @{thm zmult_int}]
+      addsplits [zdiff_int_split]
+    (*simp rules for elimination of int n*)
+
+    val simpset2 = HOL_basic_ss
+      addsimps [@{thm nat_0_le}, @{thm all_nat}, @{thm ex_nat}, @{thm number_of1}, @{thm number_of2}, @{thm int_0}, @{thm int_1}]
+      addcongs [@{thm conj_le_cong}, @{thm imp_le_cong}]
+    (* simp rules for elimination of abs *)
+    val simpset3 = HOL_basic_ss addsplits [@{thm abs_split}]
+    val ct = cterm_of thy (HOLogic.mk_Trueprop t)
+    (* Theorem for the nat --> int transformation *)
+    val pre_thm = Seq.hd (EVERY
+      [simp_tac mod_div_simpset 1, simp_tac simpset0 1,
+       TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1),
+       TRY (simp_tac simpset3 1), TRY (simp_tac cooper_ss 1)]
+      (trivial ct))
+    fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i)
+    (* The result of the quantifier elimination *)
+    val (th, tac) = case (prop_of pre_thm) of
+        Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ =>
+    let val pth = linzqe_oracle (cterm_of thy (Pattern.eta_long [] t1))
+    in 
+          ((pth RS iffD2) RS pre_thm,
+            assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))
+    end
+      | _ => (pre_thm, assm_tac i)
+  in (rtac (((mp_step nh) o (spec_step np)) th) i 
+      THEN tac) st
+  end handle Subscript => no_tac st);
+
+fun linz_args meth =
+ let val parse_flag = 
+         Args.$$$ "no_quantify" >> (K (K false));
+ in
+   Method.simple_args 
+  (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >>
+    curry (Library.foldl op |>) true)
+    (fn q => fn ctxt => meth ctxt q 1)
+  end;
+
+fun linz_method ctxt q i = Method.METHOD (fn facts =>
+  Method.insert_tac facts 1 THEN linz_tac ctxt q i);
+
+val setup =
+  Method.add_method ("cooper",
+     linz_args linz_method,
+     "decision procedure for linear integer arithmetic");
+
+end