src/HOL/Inductive.thy
 author wenzelm Wed Dec 28 20:03:13 2011 +0100 (2011-12-28) changeset 46008 c296c75f4cf4 parent 45907 4b41967bd77e child 46947 b8c7eb0c2f89 permissions -rw-r--r--
reverted some changes for set->predicate transition, according to "hg log -u berghofe -r Isabelle2007:Isabelle2008";
tuned proofs;
```     1 (*  Title:      HOL/Inductive.thy
```
```     2     Author:     Markus Wenzel, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Knaster-Tarski Fixpoint Theorem and inductive definitions *}
```
```     6
```
```     7 theory Inductive
```
```     8 imports Complete_Lattices
```
```     9 uses
```
```    10   "Tools/dseq.ML"
```
```    11   ("Tools/inductive.ML")
```
```    12   ("Tools/Datatype/datatype_aux.ML")
```
```    13   ("Tools/Datatype/datatype_prop.ML")
```
```    14   ("Tools/Datatype/datatype_data.ML")
```
```    15   ("Tools/Datatype/datatype_case.ML")
```
```    16   ("Tools/Datatype/rep_datatype.ML")
```
```    17   ("Tools/Datatype/datatype_codegen.ML")
```
```    18   ("Tools/Datatype/primrec.ML")
```
```    19 begin
```
```    20
```
```    21 subsection {* Least and greatest fixed points *}
```
```    22
```
```    23 context complete_lattice
```
```    24 begin
```
```    25
```
```    26 definition
```
```    27   lfp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```    28   "lfp f = Inf {u. f u \<le> u}"    --{*least fixed point*}
```
```    29
```
```    30 definition
```
```    31   gfp :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a" where
```
```    32   "gfp f = Sup {u. u \<le> f u}"    --{*greatest fixed point*}
```
```    33
```
```    34
```
```    35 subsection{* Proof of Knaster-Tarski Theorem using @{term lfp} *}
```
```    36
```
```    37 text{*@{term "lfp f"} is the least upper bound of
```
```    38       the set @{term "{u. f(u) \<le> u}"} *}
```
```    39
```
```    40 lemma lfp_lowerbound: "f A \<le> A ==> lfp f \<le> A"
```
```    41   by (auto simp add: lfp_def intro: Inf_lower)
```
```    42
```
```    43 lemma lfp_greatest: "(!!u. f u \<le> u ==> A \<le> u) ==> A \<le> lfp f"
```
```    44   by (auto simp add: lfp_def intro: Inf_greatest)
```
```    45
```
```    46 end
```
```    47
```
```    48 lemma lfp_lemma2: "mono f ==> f (lfp f) \<le> lfp f"
```
```    49   by (iprover intro: lfp_greatest order_trans monoD lfp_lowerbound)
```
```    50
```
```    51 lemma lfp_lemma3: "mono f ==> lfp f \<le> f (lfp f)"
```
```    52   by (iprover intro: lfp_lemma2 monoD lfp_lowerbound)
```
```    53
```
```    54 lemma lfp_unfold: "mono f ==> lfp f = f (lfp f)"
```
```    55   by (iprover intro: order_antisym lfp_lemma2 lfp_lemma3)
```
```    56
```
```    57 lemma lfp_const: "lfp (\<lambda>x. t) = t"
```
```    58   by (rule lfp_unfold) (simp add:mono_def)
```
```    59
```
```    60
```
```    61 subsection {* General induction rules for least fixed points *}
```
```    62
```
```    63 theorem lfp_induct:
```
```    64   assumes mono: "mono f" and ind: "f (inf (lfp f) P) <= P"
```
```    65   shows "lfp f <= P"
```
```    66 proof -
```
```    67   have "inf (lfp f) P <= lfp f" by (rule inf_le1)
```
```    68   with mono have "f (inf (lfp f) P) <= f (lfp f)" ..
```
```    69   also from mono have "f (lfp f) = lfp f" by (rule lfp_unfold [symmetric])
```
```    70   finally have "f (inf (lfp f) P) <= lfp f" .
```
```    71   from this and ind have "f (inf (lfp f) P) <= inf (lfp f) P" by (rule le_infI)
```
```    72   hence "lfp f <= inf (lfp f) P" by (rule lfp_lowerbound)
```
```    73   also have "inf (lfp f) P <= P" by (rule inf_le2)
```
```    74   finally show ?thesis .
```
```    75 qed
```
```    76
```
```    77 lemma lfp_induct_set:
```
```    78   assumes lfp: "a: lfp(f)"
```
```    79       and mono: "mono(f)"
```
```    80       and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)"
```
```    81   shows "P(a)"
```
```    82   by (rule lfp_induct [THEN subsetD, THEN CollectD, OF mono _ lfp])
```
```    83     (auto simp: intro: indhyp)
```
```    84
```
```    85 lemma lfp_ordinal_induct:
```
```    86   fixes f :: "'a\<Colon>complete_lattice \<Rightarrow> 'a"
```
```    87   assumes mono: "mono f"
```
```    88   and P_f: "\<And>S. P S \<Longrightarrow> P (f S)"
```
```    89   and P_Union: "\<And>M. \<forall>S\<in>M. P S \<Longrightarrow> P (Sup M)"
```
```    90   shows "P (lfp f)"
```
```    91 proof -
```
```    92   let ?M = "{S. S \<le> lfp f \<and> P S}"
```
```    93   have "P (Sup ?M)" using P_Union by simp
```
```    94   also have "Sup ?M = lfp f"
```
```    95   proof (rule antisym)
```
```    96     show "Sup ?M \<le> lfp f" by (blast intro: Sup_least)
```
```    97     hence "f (Sup ?M) \<le> f (lfp f)" by (rule mono [THEN monoD])
```
```    98     hence "f (Sup ?M) \<le> lfp f" using mono [THEN lfp_unfold] by simp
```
```    99     hence "f (Sup ?M) \<in> ?M" using P_f P_Union by simp
```
```   100     hence "f (Sup ?M) \<le> Sup ?M" by (rule Sup_upper)
```
```   101     thus "lfp f \<le> Sup ?M" by (rule lfp_lowerbound)
```
```   102   qed
```
```   103   finally show ?thesis .
```
```   104 qed
```
```   105
```
```   106 lemma lfp_ordinal_induct_set:
```
```   107   assumes mono: "mono f"
```
```   108   and P_f: "!!S. P S ==> P(f S)"
```
```   109   and P_Union: "!!M. !S:M. P S ==> P(Union M)"
```
```   110   shows "P(lfp f)"
```
```   111   using assms by (rule lfp_ordinal_induct)
```
```   112
```
```   113
```
```   114 text{*Definition forms of @{text lfp_unfold} and @{text lfp_induct},
```
```   115     to control unfolding*}
```
```   116
```
```   117 lemma def_lfp_unfold: "[| h==lfp(f);  mono(f) |] ==> h = f(h)"
```
```   118   by (auto intro!: lfp_unfold)
```
```   119
```
```   120 lemma def_lfp_induct:
```
```   121     "[| A == lfp(f); mono(f);
```
```   122         f (inf A P) \<le> P
```
```   123      |] ==> A \<le> P"
```
```   124   by (blast intro: lfp_induct)
```
```   125
```
```   126 lemma def_lfp_induct_set:
```
```   127     "[| A == lfp(f);  mono(f);   a:A;
```
```   128         !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)
```
```   129      |] ==> P(a)"
```
```   130   by (blast intro: lfp_induct_set)
```
```   131
```
```   132 (*Monotonicity of lfp!*)
```
```   133 lemma lfp_mono: "(!!Z. f Z \<le> g Z) ==> lfp f \<le> lfp g"
```
```   134   by (rule lfp_lowerbound [THEN lfp_greatest], blast intro: order_trans)
```
```   135
```
```   136
```
```   137 subsection {* Proof of Knaster-Tarski Theorem using @{term gfp} *}
```
```   138
```
```   139 text{*@{term "gfp f"} is the greatest lower bound of
```
```   140       the set @{term "{u. u \<le> f(u)}"} *}
```
```   141
```
```   142 lemma gfp_upperbound: "X \<le> f X ==> X \<le> gfp f"
```
```   143   by (auto simp add: gfp_def intro: Sup_upper)
```
```   144
```
```   145 lemma gfp_least: "(!!u. u \<le> f u ==> u \<le> X) ==> gfp f \<le> X"
```
```   146   by (auto simp add: gfp_def intro: Sup_least)
```
```   147
```
```   148 lemma gfp_lemma2: "mono f ==> gfp f \<le> f (gfp f)"
```
```   149   by (iprover intro: gfp_least order_trans monoD gfp_upperbound)
```
```   150
```
```   151 lemma gfp_lemma3: "mono f ==> f (gfp f) \<le> gfp f"
```
```   152   by (iprover intro: gfp_lemma2 monoD gfp_upperbound)
```
```   153
```
```   154 lemma gfp_unfold: "mono f ==> gfp f = f (gfp f)"
```
```   155   by (iprover intro: order_antisym gfp_lemma2 gfp_lemma3)
```
```   156
```
```   157
```
```   158 subsection {* Coinduction rules for greatest fixed points *}
```
```   159
```
```   160 text{*weak version*}
```
```   161 lemma weak_coinduct: "[| a: X;  X \<subseteq> f(X) |] ==> a : gfp(f)"
```
```   162   by (rule gfp_upperbound [THEN subsetD]) auto
```
```   163
```
```   164 lemma weak_coinduct_image: "!!X. [| a : X; g`X \<subseteq> f (g`X) |] ==> g a : gfp f"
```
```   165   apply (erule gfp_upperbound [THEN subsetD])
```
```   166   apply (erule imageI)
```
```   167   done
```
```   168
```
```   169 lemma coinduct_lemma:
```
```   170      "[| X \<le> f (sup X (gfp f));  mono f |] ==> sup X (gfp f) \<le> f (sup X (gfp f))"
```
```   171   apply (frule gfp_lemma2)
```
```   172   apply (drule mono_sup)
```
```   173   apply (rule le_supI)
```
```   174   apply assumption
```
```   175   apply (rule order_trans)
```
```   176   apply (rule order_trans)
```
```   177   apply assumption
```
```   178   apply (rule sup_ge2)
```
```   179   apply assumption
```
```   180   done
```
```   181
```
```   182 text{*strong version, thanks to Coen and Frost*}
```
```   183 lemma coinduct_set: "[| mono(f);  a: X;  X \<subseteq> f(X Un gfp(f)) |] ==> a : gfp(f)"
```
```   184   by (blast intro: weak_coinduct [OF _ coinduct_lemma])
```
```   185
```
```   186 lemma coinduct: "[| mono(f); X \<le> f (sup X (gfp f)) |] ==> X \<le> gfp(f)"
```
```   187   apply (rule order_trans)
```
```   188   apply (rule sup_ge1)
```
```   189   apply (erule gfp_upperbound [OF coinduct_lemma])
```
```   190   apply assumption
```
```   191   done
```
```   192
```
```   193 lemma gfp_fun_UnI2: "[| mono(f);  a: gfp(f) |] ==> a: f(X Un gfp(f))"
```
```   194   by (blast dest: gfp_lemma2 mono_Un)
```
```   195
```
```   196
```
```   197 subsection {* Even Stronger Coinduction Rule, by Martin Coen *}
```
```   198
```
```   199 text{* Weakens the condition @{term "X \<subseteq> f(X)"} to one expressed using both
```
```   200   @{term lfp} and @{term gfp}*}
```
```   201
```
```   202 lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un X Un B)"
```
```   203 by (iprover intro: subset_refl monoI Un_mono monoD)
```
```   204
```
```   205 lemma coinduct3_lemma:
```
```   206      "[| X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)));  mono(f) |]
```
```   207       ==> lfp(%x. f(x) Un X Un gfp(f)) \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)))"
```
```   208 apply (rule subset_trans)
```
```   209 apply (erule coinduct3_mono_lemma [THEN lfp_lemma3])
```
```   210 apply (rule Un_least [THEN Un_least])
```
```   211 apply (rule subset_refl, assumption)
```
```   212 apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption)
```
```   213 apply (rule monoD, assumption)
```
```   214 apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto)
```
```   215 done
```
```   216
```
```   217 lemma coinduct3:
```
```   218   "[| mono(f);  a:X;  X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)"
```
```   219 apply (rule coinduct3_lemma [THEN [2] weak_coinduct])
```
```   220 apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst])
```
```   221 apply (simp_all)
```
```   222 done
```
```   223
```
```   224
```
```   225 text{*Definition forms of @{text gfp_unfold} and @{text coinduct},
```
```   226     to control unfolding*}
```
```   227
```
```   228 lemma def_gfp_unfold: "[| A==gfp(f);  mono(f) |] ==> A = f(A)"
```
```   229   by (auto intro!: gfp_unfold)
```
```   230
```
```   231 lemma def_coinduct:
```
```   232      "[| A==gfp(f);  mono(f);  X \<le> f(sup X A) |] ==> X \<le> A"
```
```   233   by (iprover intro!: coinduct)
```
```   234
```
```   235 lemma def_coinduct_set:
```
```   236      "[| A==gfp(f);  mono(f);  a:X;  X \<subseteq> f(X Un A) |] ==> a: A"
```
```   237   by (auto intro!: coinduct_set)
```
```   238
```
```   239 (*The version used in the induction/coinduction package*)
```
```   240 lemma def_Collect_coinduct:
```
```   241     "[| A == gfp(%w. Collect(P(w)));  mono(%w. Collect(P(w)));
```
```   242         a: X;  !!z. z: X ==> P (X Un A) z |] ==>
```
```   243      a : A"
```
```   244   by (erule def_coinduct_set) auto
```
```   245
```
```   246 lemma def_coinduct3:
```
```   247     "[| A==gfp(f); mono(f);  a:X;  X \<subseteq> f(lfp(%x. f(x) Un X Un A)) |] ==> a: A"
```
```   248   by (auto intro!: coinduct3)
```
```   249
```
```   250 text{*Monotonicity of @{term gfp}!*}
```
```   251 lemma gfp_mono: "(!!Z. f Z \<le> g Z) ==> gfp f \<le> gfp g"
```
```   252   by (rule gfp_upperbound [THEN gfp_least], blast intro: order_trans)
```
```   253
```
```   254
```
```   255 subsection {* Inductive predicates and sets *}
```
```   256
```
```   257 text {* Package setup. *}
```
```   258
```
```   259 theorems basic_monos =
```
```   260   subset_refl imp_refl disj_mono conj_mono ex_mono all_mono if_bool_eq_conj
```
```   261   Collect_mono in_mono vimage_mono
```
```   262
```
```   263 use "Tools/inductive.ML"
```
```   264 setup Inductive.setup
```
```   265
```
```   266 theorems [mono] =
```
```   267   imp_refl disj_mono conj_mono ex_mono all_mono if_bool_eq_conj
```
```   268   imp_mono not_mono
```
```   269   Ball_def Bex_def
```
```   270   induct_rulify_fallback
```
```   271
```
```   272
```
```   273 subsection {* Inductive datatypes and primitive recursion *}
```
```   274
```
```   275 text {* Package setup. *}
```
```   276
```
```   277 use "Tools/Datatype/datatype_aux.ML"
```
```   278 use "Tools/Datatype/datatype_prop.ML"
```
```   279 use "Tools/Datatype/datatype_data.ML" setup Datatype_Data.setup
```
```   280 use "Tools/Datatype/datatype_case.ML" setup Datatype_Case.setup
```
```   281 use "Tools/Datatype/rep_datatype.ML"
```
```   282 use "Tools/Datatype/datatype_codegen.ML" setup Datatype_Codegen.setup
```
```   283 use "Tools/Datatype/primrec.ML"
```
```   284
```
```   285 text{* Lambda-abstractions with pattern matching: *}
```
```   286
```
```   287 syntax
```
```   288   "_lam_pats_syntax" :: "cases_syn => 'a => 'b"               ("(%_)" 10)
```
```   289 syntax (xsymbols)
```
```   290   "_lam_pats_syntax" :: "cases_syn => 'a => 'b"               ("(\<lambda>_)" 10)
```
```   291
```
```   292 parse_translation (advanced) {*
```
```   293 let
```
```   294   fun fun_tr ctxt [cs] =
```
```   295     let
```
```   296       val x = Syntax.free (fst (Name.variant "x" (Term.declare_term_frees cs Name.context)));
```
```   297       val ft = Datatype_Case.case_tr true ctxt [x, cs];
```
```   298     in lambda x ft end
```
```   299 in [(@{syntax_const "_lam_pats_syntax"}, fun_tr)] end
```
```   300 *}
```
```   301
```
```   302 end
```