src/HOL/Subst/Unify.ML
author nipkow
Wed Aug 18 11:09:40 2004 +0200 (2004-08-18)
changeset 15140 322485b816ac
parent 13630 a013a9dd370f
child 15197 19e735596e51
permissions -rw-r--r--
import -> imports
     1 (*  Title:      Subst/Unify
     2     ID:         $Id$
     3     Author:     Konrad Slind, Cambridge University Computer Laboratory
     4     Copyright   1997  University of Cambridge
     5 
     6 Unification algorithm
     7 *)
     8 
     9 (*---------------------------------------------------------------------------
    10  * This file defines a nested unification algorithm, then proves that it 
    11  * terminates, then proves 2 correctness theorems: that when the algorithm
    12  * succeeds, it 1) returns an MGU; and 2) returns an idempotent substitution.
    13  * Although the proofs may seem long, they are actually quite direct, in that
    14  * the correctness and termination properties are not mingled as much as in 
    15  * previous proofs of this algorithm. 
    16  *
    17  * Our approach for nested recursive functions is as follows: 
    18  *
    19  *    0. Prove the wellfoundedness of the termination relation.
    20  *    1. Prove the non-nested termination conditions.
    21  *    2. Eliminate (0) and (1) from the recursion equations and the 
    22  *       induction theorem.
    23  *    3. Prove the nested termination conditions by using the induction 
    24  *       theorem from (2) and by using the recursion equations from (2). 
    25  *       These are constrained by the nested termination conditions, but 
    26  *       things work out magically (by wellfoundedness of the termination 
    27  *       relation).
    28  *    4. Eliminate the nested TCs from the results of (2).
    29  *    5. Prove further correctness properties using the results of (4).
    30  *
    31  * Deeper nestings require iteration of steps (3) and (4).
    32  *---------------------------------------------------------------------------*)
    33 
    34 (*---------------------------------------------------------------------------
    35  * The non-nested TC plus the wellfoundedness of unifyRel.
    36  *---------------------------------------------------------------------------*)
    37 Tfl.tgoalw Unify.thy [] unify.simps;
    38 (* Wellfoundedness of unifyRel *)
    39 by (simp_tac (simpset() addsimps [unifyRel_def,
    40 				 wf_inv_image, wf_lex_prod, wf_finite_psubset,
    41 				 wf_measure]) 1);
    42 (* TC *)
    43 by Safe_tac;
    44 by (simp_tac (simpset() addsimps [finite_psubset_def, finite_vars_of,
    45 				 lex_prod_def, measure_def, inv_image_def]) 1);
    46 by (rtac (monotone_vars_of RS (subset_iff_psubset_eq RS iffD1) RS disjE) 1);
    47 by (Blast_tac 1);
    48 by (asm_simp_tac (simpset() addsimps [less_eq, less_add_Suc1]) 1);
    49 qed "tc0";
    50 
    51 
    52 (*---------------------------------------------------------------------------
    53  * Termination proof.
    54  *---------------------------------------------------------------------------*)
    55 
    56 Goalw [unifyRel_def, measure_def] "trans unifyRel";
    57 by (REPEAT (resolve_tac [trans_inv_image, trans_lex_prod, 
    58 			 trans_finite_psubset, trans_less_than,
    59 			 trans_inv_image] 1));
    60 qed "trans_unifyRel";
    61 
    62 
    63 (*---------------------------------------------------------------------------
    64  * The following lemma is used in the last step of the termination proof for 
    65  * the nested call in Unify.  Loosely, it says that unifyRel doesn't care
    66  * about term structure.
    67  *---------------------------------------------------------------------------*)
    68 Goalw [unifyRel_def,lex_prod_def, inv_image_def]
    69   "((X,Y), (Comb A (Comb B C), Comb D (Comb E F))) : unifyRel  ==> \
    70 \  ((X,Y), (Comb (Comb A B) C, Comb (Comb D E) F)) : unifyRel";
    71 by (asm_full_simp_tac (simpset() addsimps [measure_def, 
    72                           less_eq, inv_image_def,add_assoc]) 1);
    73 by (subgoal_tac "(vars_of A Un vars_of B Un vars_of C Un \
    74                 \  (vars_of D Un vars_of E Un vars_of F)) = \
    75                 \ (vars_of A Un (vars_of B Un vars_of C) Un \
    76                 \  (vars_of D Un (vars_of E Un vars_of F)))" 1);
    77 by (Blast_tac 2);
    78 by (Asm_simp_tac 1);
    79 qed "Rassoc";
    80 
    81 
    82 (*---------------------------------------------------------------------------
    83  * This lemma proves the nested termination condition for the base cases 
    84  * 3, 4, and 6. 
    85  *---------------------------------------------------------------------------*)
    86 Goal "~(Var x <: M) ==> \
    87 \   ((N1 <| [(x,M)], N2 <| [(x,M)]), (Comb M N1, Comb(Var x) N2)) : unifyRel \
    88 \ & ((N1 <| [(x,M)], N2 <| [(x,M)]), (Comb(Var x) N1, Comb M N2)) : unifyRel";
    89 by (case_tac "Var x = M" 1);
    90 by (hyp_subst_tac 1);
    91 by (Simp_tac 1);
    92 by (case_tac "x: (vars_of N1 Un vars_of N2)" 1);
    93 (*uterm_less case*)
    94 by (asm_simp_tac
    95     (simpset() addsimps [less_eq, unifyRel_def, lex_prod_def,
    96 			measure_def, inv_image_def]) 1);
    97 by (Blast_tac 1);
    98 (*finite_psubset case*)
    99 by (simp_tac
   100     (simpset() addsimps [unifyRel_def, lex_prod_def,
   101 			measure_def, inv_image_def]) 1);
   102 by (simp_tac (simpset() addsimps [finite_psubset_def, finite_vars_of,
   103 				 psubset_def, set_eq_subset]) 1);
   104 by (Blast_tac 1);
   105 (** LEVEL 9 **)
   106 (*Final case, also finite_psubset*)
   107 by (simp_tac
   108     (simpset() addsimps [finite_vars_of, unifyRel_def, finite_psubset_def,
   109 			lex_prod_def, measure_def, inv_image_def]) 1);
   110 by (cut_inst_tac [("s","[(x,M)]"), ("v", "x"), ("t","N2")] Var_elim 1);
   111 by (cut_inst_tac [("s","[(x,M)]"), ("v", "x"), ("t","N1")] Var_elim 3);
   112 by (ALLGOALS (asm_simp_tac(simpset() addsimps [srange_iff, vars_iff_occseq])));
   113 by (REPEAT_FIRST (resolve_tac [conjI, disjI1, psubsetI]));
   114 by (ALLGOALS (asm_full_simp_tac 
   115 	      (simpset() addsimps [srange_iff, set_eq_subset]))); 
   116 by (ALLGOALS
   117     (fast_tac (claset() addEs [Var_intro RS disjE]
   118 	               addss (simpset() addsimps [srange_iff]))));
   119 qed "var_elimR";
   120 
   121 
   122 (*---------------------------------------------------------------------------
   123  * Eliminate tc0 from the recursion equations and the induction theorem.
   124  *---------------------------------------------------------------------------*)
   125 val wfr = tc0 RS conjunct1
   126 and tc  = tc0 RS conjunct2;
   127 val unifyRules0 = map (rule_by_tactic (rtac wfr 1 THEN TRY(rtac tc 1)))
   128                      unify.simps;
   129 
   130 val unifyInduct0 = [wfr,tc] MRS unify.induct;
   131 
   132 
   133 (*---------------------------------------------------------------------------
   134  * The nested TC. Proved by recursion induction.
   135  *---------------------------------------------------------------------------*)
   136 val [_,_,tc3] = unify.tcs;
   137 goalw_cterm [] (cterm_of (sign_of Unify.thy) (HOLogic.mk_Trueprop tc3));
   138 (*---------------------------------------------------------------------------
   139  * The extracted TC needs the scope of its quantifiers adjusted, so our 
   140  * first step is to restrict the scopes of N1 and N2.
   141  *---------------------------------------------------------------------------*)
   142 by (subgoal_tac "!M1 M2 theta.  \
   143  \   unify(M1, M2) = Some theta --> \
   144  \   (!N1 N2. ((N1<|theta, N2<|theta), (Comb M1 N1, Comb M2 N2)) : unifyRel)" 1);
   145 by (Blast_tac 1);
   146 by (rtac allI 1); 
   147 by (rtac allI 1);
   148 (* Apply induction *)
   149 by (res_inst_tac [("u","M1"),("v","M2")] unifyInduct0 1);
   150 by (ALLGOALS 
   151     (asm_simp_tac (simpset() addsimps (var_elimR::unifyRules0))));
   152 (*Const-Const case*)
   153 by (simp_tac
   154     (simpset() addsimps [unifyRel_def, lex_prod_def, measure_def,
   155 			inv_image_def, less_eq]) 1);
   156 (** LEVEL 7 **)
   157 (*Comb-Comb case*)
   158 by (asm_simp_tac (simpset() addsplits [option.split]) 1);
   159 by (strip_tac 1);
   160 by (rtac (trans_unifyRel RS transD) 1);
   161 by (Blast_tac 1);
   162 by (simp_tac (HOL_ss addsimps [subst_Comb RS sym]) 1);
   163 by (rtac Rassoc 1);
   164 by (Blast_tac 1);
   165 qed_spec_mp "unify_TC";
   166 
   167 
   168 (*---------------------------------------------------------------------------
   169  * Now for elimination of nested TC from unify.simps and induction. 
   170  *---------------------------------------------------------------------------*)
   171 
   172 (*Desired rule, copied from the theory file.  Could it be made available?*)
   173 Goal "unify(Comb M1 N1, Comb M2 N2) =      \
   174 \      (case unify(M1,M2)               \
   175 \        of None => None                \
   176 \         | Some theta => (case unify(N1 <| theta, N2 <| theta)        \
   177 \                            of None => None    \
   178 \                             | Some sigma => Some (theta <> sigma)))";
   179 by (asm_simp_tac (simpset() addsimps (unify_TC::unifyRules0)
   180 			   addsplits [option.split]) 1);
   181 qed "unifyCombComb";
   182 
   183 
   184 val unifyRules = rev (unifyCombComb :: tl (rev unifyRules0));
   185 
   186 Addsimps unifyRules;
   187 
   188 bind_thm ("unifyInduct",
   189 	  rule_by_tactic
   190 	     (ALLGOALS (full_simp_tac (simpset() addsimps [unify_TC])))
   191 	     unifyInduct0);
   192 
   193 
   194 (*---------------------------------------------------------------------------
   195  * Correctness. Notice that idempotence is not needed to prove that the 
   196  * algorithm terminates and is not needed to prove the algorithm correct, 
   197  * if you are only interested in an MGU.  This is in contrast to the
   198  * approach of M&W, who used idempotence and MGU-ness in the termination proof.
   199  *---------------------------------------------------------------------------*)
   200 
   201 Goal "!theta. unify(M,N) = Some theta --> MGUnifier theta M N";
   202 by (res_inst_tac [("u","M"),("v","N")] unifyInduct 1);
   203 by (ALLGOALS Asm_simp_tac);
   204 (*Const-Const case*)
   205 by (simp_tac (simpset() addsimps [MGUnifier_def,Unifier_def]) 1);
   206 (*Const-Var case*)
   207 by (stac mgu_sym 1);
   208 by (simp_tac (simpset() addsimps [MGUnifier_Var]) 1);
   209 (*Var-M case*)
   210 by (simp_tac (simpset() addsimps [MGUnifier_Var]) 1);
   211 (*Comb-Var case*)
   212 by (stac mgu_sym 1);
   213 by (simp_tac (simpset() addsimps [MGUnifier_Var]) 1);
   214 (** LEVEL 8 **)
   215 (*Comb-Comb case*)
   216 by (asm_simp_tac (simpset() addsplits [option.split]) 1);
   217 by (strip_tac 1);
   218 by (asm_full_simp_tac 
   219     (simpset() addsimps [MGUnifier_def, Unifier_def, MoreGeneral_def]) 1);
   220 by (Safe_tac THEN rename_tac "theta sigma gamma" 1);
   221 by (eres_inst_tac [("x","gamma")] allE 1 THEN mp_tac 1);
   222 by (etac exE 1 THEN rename_tac "delta" 1);
   223 by (eres_inst_tac [("x","delta")] allE 1);
   224 by (subgoal_tac "N1 <| theta <| delta = N2 <| theta <| delta" 1);
   225 (*Proving the subgoal*)
   226 by (full_simp_tac (simpset() addsimps [subst_eq_iff]) 2
   227     THEN blast_tac (claset() addIs [trans,sym] delrules [impCE]) 2);
   228 by (blast_tac (claset() addIs [subst_trans, subst_cong, 
   229 			      comp_assoc RS subst_sym]) 1);
   230 qed_spec_mp "unify_gives_MGU";
   231 
   232 
   233 (*---------------------------------------------------------------------------
   234  * Unify returns idempotent substitutions, when it succeeds.
   235  *---------------------------------------------------------------------------*)
   236 Goal "!theta. unify(M,N) = Some theta --> Idem theta";
   237 by (res_inst_tac [("u","M"),("v","N")] unifyInduct 1);
   238 by (ALLGOALS 
   239     (asm_simp_tac 
   240        (simpset() addsimps [Var_Idem] addsplits [option.split])));
   241 (*Comb-Comb case*)
   242 by Safe_tac;
   243 by (REPEAT (dtac spec 1 THEN mp_tac 1));
   244 by (safe_tac (claset() addSDs [rewrite_rule [MGUnifier_def] unify_gives_MGU]));
   245 by (rtac Idem_comp 1);
   246 by (atac 1);
   247 by (atac 1);
   248 by (best_tac (claset() addss (simpset() addsimps 
   249 			     [MoreGeneral_def, subst_eq_iff, Idem_def])) 1);
   250 qed_spec_mp "unify_gives_Idem";
   251