src/HOL/Inductive.thy
author haftmann
Wed Jan 30 10:57:44 2008 +0100 (2008-01-30)
changeset 26013 8764a1f1253b
parent 25557 ea6b11021e79
child 26793 e36a92ff543e
permissions -rw-r--r--
Theorem Inductive.lfp_ordinal_induct generalized to complete lattices
     1 (*  Title:      HOL/Inductive.thy
     2     ID:         $Id$
     3     Author:     Markus Wenzel, TU Muenchen
     4 *)
     5 
     6 header {* Knaster-Tarski Fixpoint Theorem and inductive definitions *}
     7 
     8 theory Inductive 
     9 imports Lattices Sum_Type
    10 uses
    11   ("Tools/inductive_package.ML")
    12   "Tools/dseq.ML"
    13   ("Tools/inductive_codegen.ML")
    14   ("Tools/datatype_aux.ML")
    15   ("Tools/datatype_prop.ML")
    16   ("Tools/datatype_rep_proofs.ML")
    17   ("Tools/datatype_abs_proofs.ML")
    18   ("Tools/datatype_case.ML")
    19   ("Tools/datatype_package.ML")
    20   ("Tools/old_primrec_package.ML")
    21   ("Tools/primrec_package.ML")
    22   ("Tools/datatype_codegen.ML")
    23 begin
    24 
    25 subsection {* Least and greatest fixed points *}
    26 
    27 context complete_lattice
    28 begin
    29 
    30 definition
    31   lfp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" where
    32   "lfp f = Inf {u. f u \<le> u}"    --{*least fixed point*}
    33 
    34 definition
    35   gfp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" where
    36   "gfp f = Sup {u. u \<le> f u}"    --{*greatest fixed point*}
    37 
    38 
    39 subsection{* Proof of Knaster-Tarski Theorem using @{term lfp} *}
    40 
    41 text{*@{term "lfp f"} is the least upper bound of 
    42       the set @{term "{u. f(u) \<le> u}"} *}
    43 
    44 lemma lfp_lowerbound: "f A \<le> A ==> lfp f \<le> A"
    45   by (auto simp add: lfp_def intro: Inf_lower)
    46 
    47 lemma lfp_greatest: "(!!u. f u \<le> u ==> A \<le> u) ==> A \<le> lfp f"
    48   by (auto simp add: lfp_def intro: Inf_greatest)
    49 
    50 end
    51 
    52 lemma lfp_lemma2: "mono f ==> f (lfp f) \<le> lfp f"
    53   by (iprover intro: lfp_greatest order_trans monoD lfp_lowerbound)
    54 
    55 lemma lfp_lemma3: "mono f ==> lfp f \<le> f (lfp f)"
    56   by (iprover intro: lfp_lemma2 monoD lfp_lowerbound)
    57 
    58 lemma lfp_unfold: "mono f ==> lfp f = f (lfp f)"
    59   by (iprover intro: order_antisym lfp_lemma2 lfp_lemma3)
    60 
    61 lemma lfp_const: "lfp (\<lambda>x. t) = t"
    62   by (rule lfp_unfold) (simp add:mono_def)
    63 
    64 
    65 subsection {* General induction rules for least fixed points *}
    66 
    67 theorem lfp_induct:
    68   assumes mono: "mono f" and ind: "f (inf (lfp f) P) <= P"
    69   shows "lfp f <= P"
    70 proof -
    71   have "inf (lfp f) P <= lfp f" by (rule inf_le1)
    72   with mono have "f (inf (lfp f) P) <= f (lfp f)" ..
    73   also from mono have "f (lfp f) = lfp f" by (rule lfp_unfold [symmetric])
    74   finally have "f (inf (lfp f) P) <= lfp f" .
    75   from this and ind have "f (inf (lfp f) P) <= inf (lfp f) P" by (rule le_infI)
    76   hence "lfp f <= inf (lfp f) P" by (rule lfp_lowerbound)
    77   also have "inf (lfp f) P <= P" by (rule inf_le2)
    78   finally show ?thesis .
    79 qed
    80 
    81 lemma lfp_induct_set:
    82   assumes lfp: "a: lfp(f)"
    83       and mono: "mono(f)"
    84       and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)"
    85   shows "P(a)"
    86   by (rule lfp_induct [THEN subsetD, THEN CollectD, OF mono _ lfp])
    87     (auto simp: inf_set_eq intro: indhyp)
    88 
    89 lemma lfp_ordinal_induct:
    90   fixes f :: "'a\<Colon>complete_lattice \<Rightarrow> 'a"
    91   assumes mono: "mono f"
    92   and P_f: "\<And>S. P S \<Longrightarrow> P (f S)"
    93   and P_Union: "\<And>M. \<forall>S\<in>M. P S \<Longrightarrow> P (Sup M)"
    94   shows "P (lfp f)"
    95 proof -
    96   let ?M = "{S. S \<le> lfp f \<and> P S}"
    97   have "P (Sup ?M)" using P_Union by simp
    98   also have "Sup ?M = lfp f"
    99   proof (rule antisym)
   100     show "Sup ?M \<le> lfp f" by (blast intro: Sup_least)
   101     hence "f (Sup ?M) \<le> f (lfp f)" by (rule mono [THEN monoD])
   102     hence "f (Sup ?M) \<le> lfp f" using mono [THEN lfp_unfold] by simp
   103     hence "f (Sup ?M) \<in> ?M" using P_f P_Union by simp
   104     hence "f (Sup ?M) \<le> Sup ?M" by (rule Sup_upper)
   105     thus "lfp f \<le> Sup ?M" by (rule lfp_lowerbound)
   106   qed
   107   finally show ?thesis .
   108 qed 
   109 
   110 lemma lfp_ordinal_induct_set: 
   111   assumes mono: "mono f"
   112   and P_f: "!!S. P S ==> P(f S)"
   113   and P_Union: "!!M. !S:M. P S ==> P(Union M)"
   114   shows "P(lfp f)"
   115   using assms unfolding Sup_set_def [symmetric]
   116   by (rule lfp_ordinal_induct) 
   117 
   118 
   119 text{*Definition forms of @{text lfp_unfold} and @{text lfp_induct}, 
   120     to control unfolding*}
   121 
   122 lemma def_lfp_unfold: "[| h==lfp(f);  mono(f) |] ==> h = f(h)"
   123 by (auto intro!: lfp_unfold)
   124 
   125 lemma def_lfp_induct: 
   126     "[| A == lfp(f); mono(f);
   127         f (inf A P) \<le> P
   128      |] ==> A \<le> P"
   129   by (blast intro: lfp_induct)
   130 
   131 lemma def_lfp_induct_set: 
   132     "[| A == lfp(f);  mono(f);   a:A;                    
   133         !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)         
   134      |] ==> P(a)"
   135   by (blast intro: lfp_induct_set)
   136 
   137 (*Monotonicity of lfp!*)
   138 lemma lfp_mono: "(!!Z. f Z \<le> g Z) ==> lfp f \<le> lfp g"
   139   by (rule lfp_lowerbound [THEN lfp_greatest], blast intro: order_trans)
   140 
   141 
   142 subsection {* Proof of Knaster-Tarski Theorem using @{term gfp} *}
   143 
   144 text{*@{term "gfp f"} is the greatest lower bound of 
   145       the set @{term "{u. u \<le> f(u)}"} *}
   146 
   147 lemma gfp_upperbound: "X \<le> f X ==> X \<le> gfp f"
   148   by (auto simp add: gfp_def intro: Sup_upper)
   149 
   150 lemma gfp_least: "(!!u. u \<le> f u ==> u \<le> X) ==> gfp f \<le> X"
   151   by (auto simp add: gfp_def intro: Sup_least)
   152 
   153 lemma gfp_lemma2: "mono f ==> gfp f \<le> f (gfp f)"
   154   by (iprover intro: gfp_least order_trans monoD gfp_upperbound)
   155 
   156 lemma gfp_lemma3: "mono f ==> f (gfp f) \<le> gfp f"
   157   by (iprover intro: gfp_lemma2 monoD gfp_upperbound)
   158 
   159 lemma gfp_unfold: "mono f ==> gfp f = f (gfp f)"
   160   by (iprover intro: order_antisym gfp_lemma2 gfp_lemma3)
   161 
   162 
   163 subsection {* Coinduction rules for greatest fixed points *}
   164 
   165 text{*weak version*}
   166 lemma weak_coinduct: "[| a: X;  X \<subseteq> f(X) |] ==> a : gfp(f)"
   167 by (rule gfp_upperbound [THEN subsetD], auto)
   168 
   169 lemma weak_coinduct_image: "!!X. [| a : X; g`X \<subseteq> f (g`X) |] ==> g a : gfp f"
   170 apply (erule gfp_upperbound [THEN subsetD])
   171 apply (erule imageI)
   172 done
   173 
   174 lemma coinduct_lemma:
   175      "[| X \<le> f (sup X (gfp f));  mono f |] ==> sup X (gfp f) \<le> f (sup X (gfp f))"
   176   apply (frule gfp_lemma2)
   177   apply (drule mono_sup)
   178   apply (rule le_supI)
   179   apply assumption
   180   apply (rule order_trans)
   181   apply (rule order_trans)
   182   apply assumption
   183   apply (rule sup_ge2)
   184   apply assumption
   185   done
   186 
   187 text{*strong version, thanks to Coen and Frost*}
   188 lemma coinduct_set: "[| mono(f);  a: X;  X \<subseteq> f(X Un gfp(f)) |] ==> a : gfp(f)"
   189 by (blast intro: weak_coinduct [OF _ coinduct_lemma, simplified sup_set_eq])
   190 
   191 lemma coinduct: "[| mono(f); X \<le> f (sup X (gfp f)) |] ==> X \<le> gfp(f)"
   192   apply (rule order_trans)
   193   apply (rule sup_ge1)
   194   apply (erule gfp_upperbound [OF coinduct_lemma])
   195   apply assumption
   196   done
   197 
   198 lemma gfp_fun_UnI2: "[| mono(f);  a: gfp(f) |] ==> a: f(X Un gfp(f))"
   199 by (blast dest: gfp_lemma2 mono_Un)
   200 
   201 
   202 subsection {* Even Stronger Coinduction Rule, by Martin Coen *}
   203 
   204 text{* Weakens the condition @{term "X \<subseteq> f(X)"} to one expressed using both
   205   @{term lfp} and @{term gfp}*}
   206 
   207 lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un X Un B)"
   208 by (iprover intro: subset_refl monoI Un_mono monoD)
   209 
   210 lemma coinduct3_lemma:
   211      "[| X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)));  mono(f) |]
   212       ==> lfp(%x. f(x) Un X Un gfp(f)) \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)))"
   213 apply (rule subset_trans)
   214 apply (erule coinduct3_mono_lemma [THEN lfp_lemma3])
   215 apply (rule Un_least [THEN Un_least])
   216 apply (rule subset_refl, assumption)
   217 apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption)
   218 apply (rule monoD, assumption)
   219 apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto)
   220 done
   221 
   222 lemma coinduct3: 
   223   "[| mono(f);  a:X;  X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)"
   224 apply (rule coinduct3_lemma [THEN [2] weak_coinduct])
   225 apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst], auto)
   226 done
   227 
   228 
   229 text{*Definition forms of @{text gfp_unfold} and @{text coinduct}, 
   230     to control unfolding*}
   231 
   232 lemma def_gfp_unfold: "[| A==gfp(f);  mono(f) |] ==> A = f(A)"
   233 by (auto intro!: gfp_unfold)
   234 
   235 lemma def_coinduct:
   236      "[| A==gfp(f);  mono(f);  X \<le> f(sup X A) |] ==> X \<le> A"
   237 by (iprover intro!: coinduct)
   238 
   239 lemma def_coinduct_set:
   240      "[| A==gfp(f);  mono(f);  a:X;  X \<subseteq> f(X Un A) |] ==> a: A"
   241 by (auto intro!: coinduct_set)
   242 
   243 (*The version used in the induction/coinduction package*)
   244 lemma def_Collect_coinduct:
   245     "[| A == gfp(%w. Collect(P(w)));  mono(%w. Collect(P(w)));   
   246         a: X;  !!z. z: X ==> P (X Un A) z |] ==>  
   247      a : A"
   248 apply (erule def_coinduct_set, auto) 
   249 done
   250 
   251 lemma def_coinduct3:
   252     "[| A==gfp(f); mono(f);  a:X;  X \<subseteq> f(lfp(%x. f(x) Un X Un A)) |] ==> a: A"
   253 by (auto intro!: coinduct3)
   254 
   255 text{*Monotonicity of @{term gfp}!*}
   256 lemma gfp_mono: "(!!Z. f Z \<le> g Z) ==> gfp f \<le> gfp g"
   257   by (rule gfp_upperbound [THEN gfp_least], blast intro: order_trans)
   258 
   259 
   260 subsection {* Inductive predicates and sets *}
   261 
   262 text {* Inversion of injective functions. *}
   263 
   264 constdefs
   265   myinv :: "('a => 'b) => ('b => 'a)"
   266   "myinv (f :: 'a => 'b) == \<lambda>y. THE x. f x = y"
   267 
   268 lemma myinv_f_f: "inj f ==> myinv f (f x) = x"
   269 proof -
   270   assume "inj f"
   271   hence "(THE x'. f x' = f x) = (THE x'. x' = x)"
   272     by (simp only: inj_eq)
   273   also have "... = x" by (rule the_eq_trivial)
   274   finally show ?thesis by (unfold myinv_def)
   275 qed
   276 
   277 lemma f_myinv_f: "inj f ==> y \<in> range f ==> f (myinv f y) = y"
   278 proof (unfold myinv_def)
   279   assume inj: "inj f"
   280   assume "y \<in> range f"
   281   then obtain x where "y = f x" ..
   282   hence x: "f x = y" ..
   283   thus "f (THE x. f x = y) = y"
   284   proof (rule theI)
   285     fix x' assume "f x' = y"
   286     with x have "f x' = f x" by simp
   287     with inj show "x' = x" by (rule injD)
   288   qed
   289 qed
   290 
   291 hide const myinv
   292 
   293 
   294 text {* Package setup. *}
   295 
   296 theorems basic_monos =
   297   subset_refl imp_refl disj_mono conj_mono ex_mono all_mono if_bool_eq_conj
   298   Collect_mono in_mono vimage_mono
   299   imp_conv_disj not_not de_Morgan_disj de_Morgan_conj
   300   not_all not_ex
   301   Ball_def Bex_def
   302   induct_rulify_fallback
   303 
   304 ML {*
   305 val def_lfp_unfold = @{thm def_lfp_unfold}
   306 val def_gfp_unfold = @{thm def_gfp_unfold}
   307 val def_lfp_induct = @{thm def_lfp_induct}
   308 val def_coinduct = @{thm def_coinduct}
   309 val inf_bool_eq = @{thm inf_bool_eq} RS @{thm eq_reflection}
   310 val inf_fun_eq = @{thm inf_fun_eq} RS @{thm eq_reflection}
   311 val sup_bool_eq = @{thm sup_bool_eq} RS @{thm eq_reflection}
   312 val sup_fun_eq = @{thm sup_fun_eq} RS @{thm eq_reflection}
   313 val le_boolI = @{thm le_boolI}
   314 val le_boolI' = @{thm le_boolI'}
   315 val le_funI = @{thm le_funI}
   316 val le_boolE = @{thm le_boolE}
   317 val le_funE = @{thm le_funE}
   318 val le_boolD = @{thm le_boolD}
   319 val le_funD = @{thm le_funD}
   320 val le_bool_def = @{thm le_bool_def} RS @{thm eq_reflection}
   321 val le_fun_def = @{thm le_fun_def} RS @{thm eq_reflection}
   322 *}
   323 
   324 use "Tools/inductive_package.ML"
   325 setup InductivePackage.setup
   326 
   327 theorems [mono] =
   328   imp_refl disj_mono conj_mono ex_mono all_mono if_bool_eq_conj
   329   imp_conv_disj not_not de_Morgan_disj de_Morgan_conj
   330   not_all not_ex
   331   Ball_def Bex_def
   332   induct_rulify_fallback
   333 
   334 
   335 subsection {* Inductive datatypes and primitive recursion *}
   336 
   337 text {* Package setup. *}
   338 
   339 use "Tools/datatype_aux.ML"
   340 use "Tools/datatype_prop.ML"
   341 use "Tools/datatype_rep_proofs.ML"
   342 use "Tools/datatype_abs_proofs.ML"
   343 use "Tools/datatype_case.ML"
   344 use "Tools/datatype_package.ML"
   345 setup DatatypePackage.setup
   346 use "Tools/old_primrec_package.ML"
   347 use "Tools/primrec_package.ML"
   348 
   349 use "Tools/datatype_codegen.ML"
   350 setup DatatypeCodegen.setup
   351 
   352 use "Tools/inductive_codegen.ML"
   353 setup InductiveCodegen.setup
   354 
   355 text{* Lambda-abstractions with pattern matching: *}
   356 
   357 syntax
   358   "_lam_pats_syntax" :: "cases_syn => 'a => 'b"               ("(%_)" 10)
   359 syntax (xsymbols)
   360   "_lam_pats_syntax" :: "cases_syn => 'a => 'b"               ("(\<lambda>_)" 10)
   361 
   362 parse_translation (advanced) {*
   363 let
   364   fun fun_tr ctxt [cs] =
   365     let
   366       val x = Free (Name.variant (add_term_free_names (cs, [])) "x", dummyT);
   367       val ft = DatatypeCase.case_tr true DatatypePackage.datatype_of_constr
   368                  ctxt [x, cs]
   369     in lambda x ft end
   370 in [("_lam_pats_syntax", fun_tr)] end
   371 *}
   372 
   373 end