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
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```     4     Copyright   1997  University of Cambridge
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```     5
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```     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
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```    14  * the correctness and termination properties are not mingled as much as in
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```    15  * previous proofs of this algorithm.
```
```    16  *
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```    17  * Our approach for nested recursive functions is as follows:
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```    18  *
```
```    19  *    0. Prove the wellfoundedness of the termination relation.
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```    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
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```    24  *       theorem from (2) and by using the recursion equations from (2).
```
```    25  *       These are constrained by the nested termination conditions, but
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```    26  *       things work out magically (by wellfoundedness of the termination
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```    27  *       relation).
```
```    28  *    4. Eliminate the nested TCs from the results of (2).
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```    29  *    5. Prove further correctness properties using the results of (4).
```
```    30  *
```
```    31  * Deeper nestings require iteration of steps (3) and (4).
```
```    32  *---------------------------------------------------------------------------*)
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```    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
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```    52 (*---------------------------------------------------------------------------
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```    53  * Termination proof.
```
```    54  *---------------------------------------------------------------------------*)
```
```    55
```
```    56 Goalw [unifyRel_def, measure_def] "trans unifyRel";
```
```    57 by (REPEAT (resolve_tac [trans_inv_image, trans_lex_prod,
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```    58 			 trans_finite_psubset, trans_less_than,
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```    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
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```    65  * the nested call in Unify.  Loosely, it says that unifyRel doesn't care
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```    66  * about term structure.
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```    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 \
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```    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,
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```   101 			measure_def, inv_image_def]) 1);
```
```   102 by (simp_tac (simpset() addsimps [finite_psubset_def, finite_vars_of,
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```   103 				 psubset_def, set_eq_subset]) 1);
```
```   104 by (Blast_tac 1);
```
```   105 (** LEVEL 9 **)
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```   106 (*Final case, also finite_psubset*)
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```   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]
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```   118 	               addss (simpset() addsimps [srange_iff]))));
```
```   119 qed "var_elimR";
```
```   120
```
```   121
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```   122 (*---------------------------------------------------------------------------
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```   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 (*---------------------------------------------------------------------------
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```   139  * The extracted TC needs the scope of its quantifiers adjusted, so our
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```   140  * first step is to restrict the scopes of N1 and N2.
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```   141  *---------------------------------------------------------------------------*)
```
```   142 by (subgoal_tac "!M1 M2 theta.  \
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```   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,
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```   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) =      \
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```   174 \      (case unify(M1,M2)               \
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```   175 \        of None => None                \
```
```   176 \         | Some theta => (case unify(N1 <| theta, N2 <| theta)        \
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```   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",
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```   189 	  rule_by_tactic
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```   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
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```   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
```