src/HOL/Library/Order_Continuity.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 56020 f92479477c52 child 58881 b9556a055632 permissions -rw-r--r--
simpler proof
```     1 (*  Title:      HOL/Library/Order_Continuity.thy
```
```     2     Author:     David von Oheimb, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Continuity and iterations (of set transformers) *}
```
```     6
```
```     7 theory Order_Continuity
```
```     8 imports Main
```
```     9 begin
```
```    10
```
```    11 (* TODO: Generalize theory to chain-complete partial orders *)
```
```    12
```
```    13 lemma SUP_nat_binary:
```
```    14   "(SUP n::nat. if n = 0 then A else B) = (sup A B::'a::complete_lattice)"
```
```    15   apply (auto intro!: antisym SUP_least)
```
```    16   apply (rule SUP_upper2[where i=0])
```
```    17   apply simp_all
```
```    18   apply (rule SUP_upper2[where i=1])
```
```    19   apply simp_all
```
```    20   done
```
```    21
```
```    22 lemma INF_nat_binary:
```
```    23   "(INF n::nat. if n = 0 then A else B) = (inf A B::'a::complete_lattice)"
```
```    24   apply (auto intro!: antisym INF_greatest)
```
```    25   apply (rule INF_lower2[where i=0])
```
```    26   apply simp_all
```
```    27   apply (rule INF_lower2[where i=1])
```
```    28   apply simp_all
```
```    29   done
```
```    30
```
```    31 subsection {* Continuity for complete lattices *}
```
```    32
```
```    33 definition
```
```    34   continuous :: "('a::complete_lattice \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
```
```    35   "continuous F \<longleftrightarrow> (\<forall>M::nat \<Rightarrow> 'a. mono M \<longrightarrow> F (SUP i. M i) = (SUP i. F (M i)))"
```
```    36
```
```    37 lemma continuousD: "continuous F \<Longrightarrow> mono M \<Longrightarrow> F (SUP i::nat. M i) = (SUP i. F (M i))"
```
```    38   by (auto simp: continuous_def)
```
```    39
```
```    40 lemma continuous_mono:
```
```    41   fixes F :: "'a::complete_lattice \<Rightarrow> 'a::complete_lattice"
```
```    42   assumes [simp]: "continuous F" shows "mono F"
```
```    43 proof
```
```    44   fix A B :: "'a" assume [simp]: "A \<le> B"
```
```    45   have "F B = F (SUP n::nat. if n = 0 then A else B)"
```
```    46     by (simp add: sup_absorb2 SUP_nat_binary)
```
```    47   also have "\<dots> = (SUP n::nat. if n = 0 then F A else F B)"
```
```    48     by (auto simp: continuousD mono_def intro!: SUP_cong)
```
```    49   finally show "F A \<le> F B"
```
```    50     by (simp add: SUP_nat_binary le_iff_sup)
```
```    51 qed
```
```    52
```
```    53 lemma continuous_lfp:
```
```    54   assumes "continuous F" shows "lfp F = (SUP i. (F ^^ i) bot)" (is "lfp F = ?U")
```
```    55 proof (rule antisym)
```
```    56   note mono = continuous_mono[OF `continuous F`]
```
```    57   show "?U \<le> lfp F"
```
```    58   proof (rule SUP_least)
```
```    59     fix i show "(F ^^ i) bot \<le> lfp F"
```
```    60     proof (induct i)
```
```    61       case (Suc i)
```
```    62       have "(F ^^ Suc i) bot = F ((F ^^ i) bot)" by simp
```
```    63       also have "\<dots> \<le> F (lfp F)" by (rule monoD[OF mono Suc])
```
```    64       also have "\<dots> = lfp F" by (simp add: lfp_unfold[OF mono, symmetric])
```
```    65       finally show ?case .
```
```    66     qed simp
```
```    67   qed
```
```    68   show "lfp F \<le> ?U"
```
```    69   proof (rule lfp_lowerbound)
```
```    70     have "mono (\<lambda>i::nat. (F ^^ i) bot)"
```
```    71     proof -
```
```    72       { fix i::nat have "(F ^^ i) bot \<le> (F ^^ (Suc i)) bot"
```
```    73         proof (induct i)
```
```    74           case 0 show ?case by simp
```
```    75         next
```
```    76           case Suc thus ?case using monoD[OF mono Suc] by auto
```
```    77         qed }
```
```    78       thus ?thesis by (auto simp add: mono_iff_le_Suc)
```
```    79     qed
```
```    80     hence "F ?U = (SUP i. (F ^^ Suc i) bot)" using `continuous F` by (simp add: continuous_def)
```
```    81     also have "\<dots> \<le> ?U" by (fast intro: SUP_least SUP_upper)
```
```    82     finally show "F ?U \<le> ?U" .
```
```    83   qed
```
```    84 qed
```
```    85
```
```    86 definition
```
```    87   down_continuous :: "('a::complete_lattice \<Rightarrow> 'a::complete_lattice) \<Rightarrow> bool" where
```
```    88   "down_continuous F \<longleftrightarrow> (\<forall>M::nat \<Rightarrow> 'a. antimono M \<longrightarrow> F (INF i. M i) = (INF i. F (M i)))"
```
```    89
```
```    90 lemma down_continuousD: "down_continuous F \<Longrightarrow> antimono M \<Longrightarrow> F (INF i::nat. M i) = (INF i. F (M i))"
```
```    91   by (auto simp: down_continuous_def)
```
```    92
```
```    93 lemma down_continuous_mono:
```
```    94   fixes F :: "'a::complete_lattice \<Rightarrow> 'a::complete_lattice"
```
```    95   assumes [simp]: "down_continuous F" shows "mono F"
```
```    96 proof
```
```    97   fix A B :: "'a" assume [simp]: "A \<le> B"
```
```    98   have "F A = F (INF n::nat. if n = 0 then B else A)"
```
```    99     by (simp add: inf_absorb2 INF_nat_binary)
```
```   100   also have "\<dots> = (INF n::nat. if n = 0 then F B else F A)"
```
```   101     by (auto simp: down_continuousD antimono_def intro!: INF_cong)
```
```   102   finally show "F A \<le> F B"
```
```   103     by (simp add: INF_nat_binary le_iff_inf inf_commute)
```
```   104 qed
```
```   105
```
```   106 lemma down_continuous_gfp:
```
```   107   assumes "down_continuous F" shows "gfp F = (INF i. (F ^^ i) top)" (is "gfp F = ?U")
```
```   108 proof (rule antisym)
```
```   109   note mono = down_continuous_mono[OF `down_continuous F`]
```
```   110   show "gfp F \<le> ?U"
```
```   111   proof (rule INF_greatest)
```
```   112     fix i show "gfp F \<le> (F ^^ i) top"
```
```   113     proof (induct i)
```
```   114       case (Suc i)
```
```   115       have "gfp F = F (gfp F)" by (simp add: gfp_unfold[OF mono, symmetric])
```
```   116       also have "\<dots> \<le> F ((F ^^ i) top)" by (rule monoD[OF mono Suc])
```
```   117       also have "\<dots> = (F ^^ Suc i) top" by simp
```
```   118       finally show ?case .
```
```   119     qed simp
```
```   120   qed
```
```   121   show "?U \<le> gfp F"
```
```   122   proof (rule gfp_upperbound)
```
```   123     have *: "antimono (\<lambda>i::nat. (F ^^ i) top)"
```
```   124     proof -
```
```   125       { fix i::nat have "(F ^^ Suc i) top \<le> (F ^^ i) top"
```
```   126         proof (induct i)
```
```   127           case 0 show ?case by simp
```
```   128         next
```
```   129           case Suc thus ?case using monoD[OF mono Suc] by auto
```
```   130         qed }
```
```   131       thus ?thesis by (auto simp add: antimono_iff_le_Suc)
```
```   132     qed
```
```   133     have "?U \<le> (INF i. (F ^^ Suc i) top)"
```
```   134       by (fast intro: INF_greatest INF_lower)
```
```   135     also have "\<dots> \<le> F ?U"
```
```   136       by (simp add: down_continuousD `down_continuous F` *)
```
```   137     finally show "?U \<le> F ?U" .
```
```   138   qed
```
```   139 qed
```
```   140
```
```   141 end
```