1 (* Title: HOL/BNF/BNF_LFP.thy
2 Author: Dmitriy Traytel, TU Muenchen
3 Author: Lorenz Panny, TU Muenchen
4 Author: Jasmin Blanchette, TU Muenchen
7 Least fixed point operation on bounded natural functors.
10 header {* Least Fixed Point Operation on Bounded Natural Functors *}
15 "datatype_new" :: thy_decl and
16 "datatype_new_compat" :: thy_decl and
17 "primrec_new" :: thy_decl
20 lemma subset_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> {}"
23 lemma image_Collect_subsetI:
24 "(\<And>x. P x \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` {x. P x} \<subseteq> B"
27 lemma Collect_restrict: "{x. x \<in> X \<and> P x} \<subseteq> X"
30 lemma prop_restrict: "\<lbrakk>x \<in> Z; Z \<subseteq> {x. x \<in> X \<and> P x}\<rbrakk> \<Longrightarrow> P x"
33 lemma underS_I: "\<lbrakk>i \<noteq> j; (i, j) \<in> R\<rbrakk> \<Longrightarrow> i \<in> rel.underS R j"
34 unfolding rel.underS_def by simp
36 lemma underS_E: "i \<in> rel.underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R"
37 unfolding rel.underS_def by simp
39 lemma underS_Field: "i \<in> rel.underS R j \<Longrightarrow> i \<in> Field R"
40 unfolding rel.underS_def Field_def by auto
42 lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
43 unfolding Field_def by auto
45 lemma fst_convol': "fst (<f, g> x) = f x"
46 using fst_convol unfolding convol_def by simp
48 lemma snd_convol': "snd (<f, g> x) = g x"
49 using snd_convol unfolding convol_def by simp
51 lemma convol_expand_snd: "fst o f = g \<Longrightarrow> <g, snd o f> = f"
52 unfolding convol_def by auto
54 lemma convol_expand_snd': "(fst o f = g) \<Longrightarrow> (h = snd o f) \<longleftrightarrow> (<g, h> = f)"
55 by (metis convol_expand_snd snd_convol)
57 definition inver where
58 "inver g f A = (ALL a : A. g (f a) = a)"
60 lemma bij_betw_iff_ex:
61 "bij_betw f A B = (EX g. g ` B = A \<and> inver g f A \<and> inver f g B)" (is "?L = ?R")
64 hence f: "f ` A = B" and inj_f: "inj_on f A" unfolding bij_betw_def by auto
65 let ?phi = "% b a. a : A \<and> f a = b"
66 have "ALL b : B. EX a. ?phi b a" using f by blast
67 then obtain g where g: "ALL b : B. g b : A \<and> f (g b) = b"
68 using bchoice[of B ?phi] by blast
69 hence gg: "ALL b : f ` A. g b : A \<and> f (g b) = b" using f by blast
70 have gf: "inver g f A" unfolding inver_def
71 by (metis (no_types) gg imageI[of _ A f] the_inv_into_f_f[OF inj_f])
72 moreover have "g ` B \<le> A \<and> inver f g B" using g unfolding inver_def by blast
73 moreover have "A \<le> g ` B"
75 fix a assume a: "a : A"
76 hence "f a : B" using f by auto
77 moreover have "a = g (f a)" using a gf unfolding inver_def by auto
78 ultimately show "a : g ` B" by blast
80 ultimately show ?R by blast
83 then obtain g where g: "g ` B = A \<and> inver g f A \<and> inver f g B" by blast
84 show ?L unfolding bij_betw_def
86 show "inj_on f A" unfolding inj_on_def
88 fix a1 a2 assume a: "a1 : A" "a2 : A" and "f a1 = f a2"
89 hence "g (f a1) = g (f a2)" by simp
90 thus "a1 = a2" using a g unfolding inver_def by simp
94 then obtain b where b: "b : B" and a: "a = g b" using g by blast
95 hence "b = f (g b)" using g unfolding inver_def by auto
96 thus "f a : B" unfolding a using b by simp
99 hence "g b : A \<and> b = f (g b)" using g unfolding inver_def by auto
100 thus "b : f ` A" by auto
104 lemma bij_betw_ex_weakE:
105 "\<lbrakk>bij_betw f A B\<rbrakk> \<Longrightarrow> \<exists>g. g ` B \<subseteq> A \<and> inver g f A \<and> inver f g B"
106 by (auto simp only: bij_betw_iff_ex)
108 lemma inver_surj: "\<lbrakk>g ` B \<subseteq> A; f ` A \<subseteq> B; inver g f A\<rbrakk> \<Longrightarrow> g ` B = A"
109 unfolding inver_def by auto (rule rev_image_eqI, auto)
111 lemma inver_mono: "\<lbrakk>A \<subseteq> B; inver f g B\<rbrakk> \<Longrightarrow> inver f g A"
112 unfolding inver_def by auto
114 lemma inver_pointfree: "inver f g A = (\<forall>a \<in> A. (f o g) a = a)"
115 unfolding inver_def by simp
117 lemma bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B"
118 unfolding bij_betw_def by auto
120 lemma bij_betw_imageE: "bij_betw f A B \<Longrightarrow> f ` A = B"
121 unfolding bij_betw_def by auto
123 lemma inverE: "\<lbrakk>inver f f' A; x \<in> A\<rbrakk> \<Longrightarrow> f (f' x) = x"
124 unfolding inver_def by auto
126 lemma bij_betw_inver1: "bij_betw f A B \<Longrightarrow> inver (inv_into A f) f A"
127 unfolding bij_betw_def inver_def by auto
129 lemma bij_betw_inver2: "bij_betw f A B \<Longrightarrow> inver f (inv_into A f) B"
130 unfolding bij_betw_def inver_def by auto
132 lemma bij_betwI: "\<lbrakk>bij_betw g B A; inver g f A; inver f g B\<rbrakk> \<Longrightarrow> bij_betw f A B"
133 by (drule bij_betw_imageE, unfold bij_betw_iff_ex) blast
136 "\<lbrakk>\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y);
137 \<And>x. x \<in> X \<Longrightarrow> f x \<in> Y;
138 \<And>y. y \<in> Y \<Longrightarrow> \<exists>x \<in> X. y = f x\<rbrakk> \<Longrightarrow> bij_betw f X Y"
139 unfolding bij_betw_def inj_on_def
142 by (erule thin_rl) blast
145 assumes surj_on: "f ` X = UNIV" and eq_on: "\<forall>x \<in> X. (g1 o f) x = (g2 o f) x"
149 from surj_on obtain x where "x \<in> X" and "y = f x" by blast
150 thus "g1 y = g2 y" using eq_on by simp
153 lemma Card_order_wo_rel: "Card_order r \<Longrightarrow> wo_rel r"
154 unfolding wo_rel_def card_order_on_def by blast
156 lemma Cinfinite_limit: "\<lbrakk>x \<in> Field r; Cinfinite r\<rbrakk> \<Longrightarrow>
157 \<exists>y \<in> Field r. x \<noteq> y \<and> (x, y) \<in> r"
158 unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)
160 lemma Card_order_trans:
161 "\<lbrakk>Card_order r; x \<noteq> y; (x, y) \<in> r; y \<noteq> z; (y, z) \<in> r\<rbrakk> \<Longrightarrow> x \<noteq> z \<and> (x, z) \<in> r"
162 unfolding card_order_on_def well_order_on_def linear_order_on_def
163 partial_order_on_def preorder_on_def trans_def antisym_def by blast
165 lemma Cinfinite_limit2:
166 assumes x1: "x1 \<in> Field r" and x2: "x2 \<in> Field r" and r: "Cinfinite r"
167 shows "\<exists>y \<in> Field r. (x1 \<noteq> y \<and> (x1, y) \<in> r) \<and> (x2 \<noteq> y \<and> (x2, y) \<in> r)"
169 from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
170 unfolding card_order_on_def well_order_on_def linear_order_on_def
171 partial_order_on_def preorder_on_def by auto
172 obtain y1 where y1: "y1 \<in> Field r" "x1 \<noteq> y1" "(x1, y1) \<in> r"
173 using Cinfinite_limit[OF x1 r] by blast
174 obtain y2 where y2: "y2 \<in> Field r" "x2 \<noteq> y2" "(x2, y2) \<in> r"
175 using Cinfinite_limit[OF x2 r] by blast
177 proof (cases "y1 = y2")
178 case True with y1 y2 show ?thesis by blast
181 with y1(1) y2(1) total have "(y1, y2) \<in> r \<or> (y2, y1) \<in> r"
182 unfolding total_on_def by auto
185 assume *: "(y1, y2) \<in> r"
186 with trans y1(3) have "(x1, y2) \<in> r" unfolding trans_def by blast
187 with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
189 assume *: "(y2, y1) \<in> r"
190 with trans y2(3) have "(x2, y1) \<in> r" unfolding trans_def by blast
191 with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
196 lemma Cinfinite_limit_finite: "\<lbrakk>finite X; X \<subseteq> Field r; Cinfinite r\<rbrakk>
197 \<Longrightarrow> \<exists>y \<in> Field r. \<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)"
198 proof (induct X rule: finite_induct)
199 case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
202 then obtain y where y: "y \<in> Field r" "\<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)" by blast
203 then obtain z where z: "z \<in> Field r" "x \<noteq> z \<and> (x, z) \<in> r" "y \<noteq> z \<and> (y, z) \<in> r"
204 using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
206 apply (intro bexI ballI)
207 apply (erule insertE)
210 using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3)
216 lemma insert_subsetI: "\<lbrakk>x \<in> A; X \<subseteq> A\<rbrakk> \<Longrightarrow> insert x X \<subseteq> A"
220 lemma well_order_induct_imp:
221 "wo_rel r \<Longrightarrow> (\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> y \<in> Field r \<longrightarrow> P y \<Longrightarrow> x \<in> Field r \<longrightarrow> P x) \<Longrightarrow>
222 x \<in> Field r \<longrightarrow> P x"
223 by (erule wo_rel.well_order_induct)
226 assumes "(\<And>x y. PROP P x y)"
228 by (rule `(\<And>x y. PROP P x y)`)
230 lemma vimage2p_fun_rel: "(fun_rel (vimage2p f g R) R) f g"
231 unfolding fun_rel_def vimage2p_def by auto
233 lemma predicate2D_vimage2p: "\<lbrakk>R \<le> vimage2p f g S; R x y\<rbrakk> \<Longrightarrow> S (f x) (g y)"
234 unfolding vimage2p_def by auto
236 ML_file "Tools/bnf_lfp_util.ML"
237 ML_file "Tools/bnf_lfp_tactics.ML"
238 ML_file "Tools/bnf_lfp.ML"
239 ML_file "Tools/bnf_lfp_compat.ML"