src/HOL/Inductive.thy
 author haftmann Mon Oct 08 22:03:25 2007 +0200 (2007-10-08) changeset 24915 fc90277c0dd7 parent 24845 abcd15369ffa child 25510 38c15efe603b permissions -rw-r--r--
integrated FixedPoint into Inductive
1 (*  Title:      HOL/Inductive.thy
2     ID:         \$Id\$
3     Author:     Markus Wenzel, TU Muenchen
4 *)
6 header {* Knaster-Tarski Fixpoint Theorem and inductive definitions *}
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/primrec_package.ML")
21 begin
23 subsection {* Least and greatest fixed points *}
25 definition
26   lfp :: "('a\<Colon>complete_lattice \<Rightarrow> 'a) \<Rightarrow> 'a" where
27   "lfp f = Inf {u. f u \<le> u}"    --{*least fixed point*}
29 definition
30   gfp :: "('a\<Colon>complete_lattice \<Rightarrow> 'a) \<Rightarrow> 'a" where
31   "gfp f = Sup {u. u \<le> f u}"    --{*greatest fixed point*}
34 subsection{* Proof of Knaster-Tarski Theorem using @{term lfp} *}
36 text{*@{term "lfp f"} is the least upper bound of
37       the set @{term "{u. f(u) \<le> u}"} *}
39 lemma lfp_lowerbound: "f A \<le> A ==> lfp f \<le> A"
40   by (auto simp add: lfp_def intro: Inf_lower)
42 lemma lfp_greatest: "(!!u. f u \<le> u ==> A \<le> u) ==> A \<le> lfp f"
43   by (auto simp add: lfp_def intro: Inf_greatest)
45 lemma lfp_lemma2: "mono f ==> f (lfp f) \<le> lfp f"
46   by (iprover intro: lfp_greatest order_trans monoD lfp_lowerbound)
48 lemma lfp_lemma3: "mono f ==> lfp f \<le> f (lfp f)"
49   by (iprover intro: lfp_lemma2 monoD lfp_lowerbound)
51 lemma lfp_unfold: "mono f ==> lfp f = f (lfp f)"
52   by (iprover intro: order_antisym lfp_lemma2 lfp_lemma3)
54 lemma lfp_const: "lfp (\<lambda>x. t) = t"
55   by (rule lfp_unfold) (simp add:mono_def)
58 subsection {* General induction rules for least fixed points *}
60 theorem lfp_induct:
61   assumes mono: "mono f" and ind: "f (inf (lfp f) P) <= P"
62   shows "lfp f <= P"
63 proof -
64   have "inf (lfp f) P <= lfp f" by (rule inf_le1)
65   with mono have "f (inf (lfp f) P) <= f (lfp f)" ..
66   also from mono have "f (lfp f) = lfp f" by (rule lfp_unfold [symmetric])
67   finally have "f (inf (lfp f) P) <= lfp f" .
68   from this and ind have "f (inf (lfp f) P) <= inf (lfp f) P" by (rule le_infI)
69   hence "lfp f <= inf (lfp f) P" by (rule lfp_lowerbound)
70   also have "inf (lfp f) P <= P" by (rule inf_le2)
71   finally show ?thesis .
72 qed
74 lemma lfp_induct_set:
75   assumes lfp: "a: lfp(f)"
76       and mono: "mono(f)"
77       and indhyp: "!!x. [| x: f(lfp(f) Int {x. P(x)}) |] ==> P(x)"
78   shows "P(a)"
79   by (rule lfp_induct [THEN subsetD, THEN CollectD, OF mono _ lfp])
80     (auto simp: inf_set_eq intro: indhyp)
82 lemma lfp_ordinal_induct:
83   assumes mono: "mono f"
84   and P_f: "!!S. P S ==> P(f S)"
85   and P_Union: "!!M. !S:M. P S ==> P(Union M)"
86   shows "P(lfp f)"
87 proof -
88   let ?M = "{S. S \<subseteq> lfp f & P S}"
89   have "P (Union ?M)" using P_Union by simp
90   also have "Union ?M = lfp f"
91   proof
92     show "Union ?M \<subseteq> lfp f" by blast
93     hence "f (Union ?M) \<subseteq> f (lfp f)" by (rule mono [THEN monoD])
94     hence "f (Union ?M) \<subseteq> lfp f" using mono [THEN lfp_unfold] by simp
95     hence "f (Union ?M) \<in> ?M" using P_f P_Union by simp
96     hence "f (Union ?M) \<subseteq> Union ?M" by (rule Union_upper)
97     thus "lfp f \<subseteq> Union ?M" by (rule lfp_lowerbound)
98   qed
99   finally show ?thesis .
100 qed
103 text{*Definition forms of @{text lfp_unfold} and @{text lfp_induct},
104     to control unfolding*}
106 lemma def_lfp_unfold: "[| h==lfp(f);  mono(f) |] ==> h = f(h)"
107 by (auto intro!: lfp_unfold)
109 lemma def_lfp_induct:
110     "[| A == lfp(f); mono(f);
111         f (inf A P) \<le> P
112      |] ==> A \<le> P"
113   by (blast intro: lfp_induct)
115 lemma def_lfp_induct_set:
116     "[| A == lfp(f);  mono(f);   a:A;
117         !!x. [| x: f(A Int {x. P(x)}) |] ==> P(x)
118      |] ==> P(a)"
119   by (blast intro: lfp_induct_set)
121 (*Monotonicity of lfp!*)
122 lemma lfp_mono: "(!!Z. f Z \<le> g Z) ==> lfp f \<le> lfp g"
123   by (rule lfp_lowerbound [THEN lfp_greatest], blast intro: order_trans)
126 subsection {* Proof of Knaster-Tarski Theorem using @{term gfp} *}
128 text{*@{term "gfp f"} is the greatest lower bound of
129       the set @{term "{u. u \<le> f(u)}"} *}
131 lemma gfp_upperbound: "X \<le> f X ==> X \<le> gfp f"
132   by (auto simp add: gfp_def intro: Sup_upper)
134 lemma gfp_least: "(!!u. u \<le> f u ==> u \<le> X) ==> gfp f \<le> X"
135   by (auto simp add: gfp_def intro: Sup_least)
137 lemma gfp_lemma2: "mono f ==> gfp f \<le> f (gfp f)"
138   by (iprover intro: gfp_least order_trans monoD gfp_upperbound)
140 lemma gfp_lemma3: "mono f ==> f (gfp f) \<le> gfp f"
141   by (iprover intro: gfp_lemma2 monoD gfp_upperbound)
143 lemma gfp_unfold: "mono f ==> gfp f = f (gfp f)"
144   by (iprover intro: order_antisym gfp_lemma2 gfp_lemma3)
147 subsection {* Coinduction rules for greatest fixed points *}
149 text{*weak version*}
150 lemma weak_coinduct: "[| a: X;  X \<subseteq> f(X) |] ==> a : gfp(f)"
151 by (rule gfp_upperbound [THEN subsetD], auto)
153 lemma weak_coinduct_image: "!!X. [| a : X; g`X \<subseteq> f (g`X) |] ==> g a : gfp f"
154 apply (erule gfp_upperbound [THEN subsetD])
155 apply (erule imageI)
156 done
158 lemma coinduct_lemma:
159      "[| X \<le> f (sup X (gfp f));  mono f |] ==> sup X (gfp f) \<le> f (sup X (gfp f))"
160   apply (frule gfp_lemma2)
161   apply (drule mono_sup)
162   apply (rule le_supI)
163   apply assumption
164   apply (rule order_trans)
165   apply (rule order_trans)
166   apply assumption
167   apply (rule sup_ge2)
168   apply assumption
169   done
171 text{*strong version, thanks to Coen and Frost*}
172 lemma coinduct_set: "[| mono(f);  a: X;  X \<subseteq> f(X Un gfp(f)) |] ==> a : gfp(f)"
173 by (blast intro: weak_coinduct [OF _ coinduct_lemma, simplified sup_set_eq])
175 lemma coinduct: "[| mono(f); X \<le> f (sup X (gfp f)) |] ==> X \<le> gfp(f)"
176   apply (rule order_trans)
177   apply (rule sup_ge1)
178   apply (erule gfp_upperbound [OF coinduct_lemma])
179   apply assumption
180   done
182 lemma gfp_fun_UnI2: "[| mono(f);  a: gfp(f) |] ==> a: f(X Un gfp(f))"
183 by (blast dest: gfp_lemma2 mono_Un)
186 subsection {* Even Stronger Coinduction Rule, by Martin Coen *}
188 text{* Weakens the condition @{term "X \<subseteq> f(X)"} to one expressed using both
189   @{term lfp} and @{term gfp}*}
191 lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un X Un B)"
192 by (iprover intro: subset_refl monoI Un_mono monoD)
194 lemma coinduct3_lemma:
195      "[| X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)));  mono(f) |]
196       ==> lfp(%x. f(x) Un X Un gfp(f)) \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f)))"
197 apply (rule subset_trans)
198 apply (erule coinduct3_mono_lemma [THEN lfp_lemma3])
199 apply (rule Un_least [THEN Un_least])
200 apply (rule subset_refl, assumption)
201 apply (rule gfp_unfold [THEN equalityD1, THEN subset_trans], assumption)
202 apply (rule monoD, assumption)
203 apply (subst coinduct3_mono_lemma [THEN lfp_unfold], auto)
204 done
206 lemma coinduct3:
207   "[| mono(f);  a:X;  X \<subseteq> f(lfp(%x. f(x) Un X Un gfp(f))) |] ==> a : gfp(f)"
208 apply (rule coinduct3_lemma [THEN  weak_coinduct])
209 apply (rule coinduct3_mono_lemma [THEN lfp_unfold, THEN ssubst], auto)
210 done
213 text{*Definition forms of @{text gfp_unfold} and @{text coinduct},
214     to control unfolding*}
216 lemma def_gfp_unfold: "[| A==gfp(f);  mono(f) |] ==> A = f(A)"
217 by (auto intro!: gfp_unfold)
219 lemma def_coinduct:
220      "[| A==gfp(f);  mono(f);  X \<le> f(sup X A) |] ==> X \<le> A"
221 by (iprover intro!: coinduct)
223 lemma def_coinduct_set:
224      "[| A==gfp(f);  mono(f);  a:X;  X \<subseteq> f(X Un A) |] ==> a: A"
225 by (auto intro!: coinduct_set)
227 (*The version used in the induction/coinduction package*)
228 lemma def_Collect_coinduct:
229     "[| A == gfp(%w. Collect(P(w)));  mono(%w. Collect(P(w)));
230         a: X;  !!z. z: X ==> P (X Un A) z |] ==>
231      a : A"
232 apply (erule def_coinduct_set, auto)
233 done
235 lemma def_coinduct3:
236     "[| A==gfp(f); mono(f);  a:X;  X \<subseteq> f(lfp(%x. f(x) Un X Un A)) |] ==> a: A"
237 by (auto intro!: coinduct3)
239 text{*Monotonicity of @{term gfp}!*}
240 lemma gfp_mono: "(!!Z. f Z \<le> g Z) ==> gfp f \<le> gfp g"
241   by (rule gfp_upperbound [THEN gfp_least], blast intro: order_trans)
244 subsection {* Inductive predicates and sets *}
246 text {* Inversion of injective functions. *}
248 constdefs
249   myinv :: "('a => 'b) => ('b => 'a)"
250   "myinv (f :: 'a => 'b) == \<lambda>y. THE x. f x = y"
252 lemma myinv_f_f: "inj f ==> myinv f (f x) = x"
253 proof -
254   assume "inj f"
255   hence "(THE x'. f x' = f x) = (THE x'. x' = x)"
256     by (simp only: inj_eq)
257   also have "... = x" by (rule the_eq_trivial)
258   finally show ?thesis by (unfold myinv_def)
259 qed
261 lemma f_myinv_f: "inj f ==> y \<in> range f ==> f (myinv f y) = y"
262 proof (unfold myinv_def)
263   assume inj: "inj f"
264   assume "y \<in> range f"
265   then obtain x where "y = f x" ..
266   hence x: "f x = y" ..
267   thus "f (THE x. f x = y) = y"
268   proof (rule theI)
269     fix x' assume "f x' = y"
270     with x have "f x' = f x" by simp
271     with inj show "x' = x" by (rule injD)
272   qed
273 qed
275 hide const myinv
278 text {* Package setup. *}
280 theorems basic_monos =
281   subset_refl imp_refl disj_mono conj_mono ex_mono all_mono if_bool_eq_conj
282   Collect_mono in_mono vimage_mono
283   imp_conv_disj not_not de_Morgan_disj de_Morgan_conj
284   not_all not_ex
285   Ball_def Bex_def
286   induct_rulify_fallback
288 ML {*
289 val def_lfp_unfold = @{thm def_lfp_unfold}
290 val def_gfp_unfold = @{thm def_gfp_unfold}
291 val def_lfp_induct = @{thm def_lfp_induct}
292 val def_coinduct = @{thm def_coinduct}
293 val inf_bool_eq = @{thm inf_bool_eq}
294 val inf_fun_eq = @{thm inf_fun_eq}
295 val le_boolI = @{thm le_boolI}
296 val le_boolI' = @{thm le_boolI'}
297 val le_funI = @{thm le_funI}
298 val le_boolE = @{thm le_boolE}
299 val le_funE = @{thm le_funE}
300 val le_boolD = @{thm le_boolD}
301 val le_funD = @{thm le_funD}
302 val le_bool_def = @{thm le_bool_def}
303 val le_fun_def = @{thm le_fun_def}
304 *}
306 use "Tools/inductive_package.ML"
307 setup InductivePackage.setup
309 theorems [mono] =
310   imp_refl disj_mono conj_mono ex_mono all_mono if_bool_eq_conj
311   imp_conv_disj not_not de_Morgan_disj de_Morgan_conj
312   not_all not_ex
313   Ball_def Bex_def
314   induct_rulify_fallback
317 subsection {* Inductive datatypes and primitive recursion *}
319 text {* Package setup. *}
321 use "Tools/datatype_aux.ML"
322 use "Tools/datatype_prop.ML"
323 use "Tools/datatype_rep_proofs.ML"
324 use "Tools/datatype_abs_proofs.ML"
325 use "Tools/datatype_case.ML"
326 use "Tools/datatype_package.ML"
327 setup DatatypePackage.setup
328 use "Tools/primrec_package.ML"
330 use "Tools/inductive_codegen.ML"
331 setup InductiveCodegen.setup
333 text{* Lambda-abstractions with pattern matching: *}
335 syntax
336   "_lam_pats_syntax" :: "cases_syn => 'a => 'b"               ("(%_)" 10)
337 syntax (xsymbols)
338   "_lam_pats_syntax" :: "cases_syn => 'a => 'b"               ("(\<lambda>_)" 10)
340 parse_translation (advanced) {*
341 let
342   fun fun_tr ctxt [cs] =
343     let
344       val x = Free (Name.variant (add_term_free_names (cs, [])) "x", dummyT);
345       val ft = DatatypeCase.case_tr true DatatypePackage.datatype_of_constr
346                  ctxt [x, cs]
347     in lambda x ft end
348 in [("_lam_pats_syntax", fun_tr)] end
349 *}
351 end