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src/HOL/Enum.thy

changeset 45117 | 3911cf09899a |

parent 41115 | 2c362ff5daf4 |

child 45119 | 055c6ff9c5c3 |

--- a/src/HOL/Enum.thy Mon Oct 03 14:43:13 2011 +0200 +++ b/src/HOL/Enum.thy Mon Oct 03 14:43:14 2011 +0200 @@ -758,12 +758,78 @@ unfolding enum_the_def by (auto split: list.split) qed +subsection {* An executable card operator on finite types *} + +lemma + [code]: "card R = length (filter R enum)" +by (simp add: distinct_length_filter[OF enum_distinct] enum_UNIV Collect_def) + +subsection {* An executable (reflexive) transitive closure on finite relations *} + +text {* Definitions could be moved to Transitive_Closure if they are of more general use. *} + +definition ntrancl :: "('a * 'a => bool) => nat => ('a * 'a => bool)" +where + [code del]: "ntrancl R n = (UN i : {i. 0 < i & i <= (Suc n)}. R ^^ i)" + +lemma [code]: + "ntrancl (R :: 'a * 'a => bool) 0 = R" +proof + show "R <= ntrancl R 0" + unfolding ntrancl_def by fastforce +next + { + fix i have "(0 < i & i <= Suc 0) = (i = 1)" by auto + } + from this show "ntrancl R 0 <= R" + unfolding ntrancl_def by auto +qed + +lemma [code]: + "ntrancl (R :: 'a * 'a => bool) (Suc n) = (ntrancl R n) O (Id Un R)" +proof + { + fix a b + assume "(a, b) : ntrancl R (Suc n)" + from this obtain i where "0 < i" "i <= Suc (Suc n)" "(a, b) : R ^^ i" + unfolding ntrancl_def by auto + have "(a, b) : ntrancl R n O (Id Un R)" + proof (cases "i = 1") + case True + from this `(a, b) : R ^^ i` show ?thesis + unfolding ntrancl_def by auto + next + case False + from this `0 < i` obtain j where j: "i = Suc j" "0 < j" + by (cases i) auto + from this `(a, b) : R ^^ i` obtain c where c1: "(a, c) : R ^^ j" and c2:"(c, b) : R" + by auto + from c1 j `i <= Suc (Suc n)` have "(a, c): ntrancl R n" + unfolding ntrancl_def by fastforce + from this c2 show ?thesis by fastforce + qed + } + from this show "ntrancl R (Suc n) <= ntrancl R n O (Id Un R)" by auto +next + show "ntrancl R n O (Id Un R) <= ntrancl R (Suc n)" + unfolding ntrancl_def by fastforce +qed + +lemma [code]: "trancl (R :: ('a :: finite) * 'a => bool) = ntrancl R (card R - 1)" +by (cases "card R") (auto simp add: trancl_finite_eq_rel_pow rel_pow_empty ntrancl_def) + +(* a copy of Nitpick.rtrancl_unfold, should be moved to Transitive_Closure *) +lemma [code]: "r^* = (r^+)^=" +by simp + +subsection {* Closing up *} + code_abort enum_the hide_const (open) a\<^isub>1 a\<^isub>2 a\<^isub>3 a\<^isub>4 a\<^isub>5 hide_type (open) finite_1 finite_2 finite_3 finite_4 finite_5 -hide_const (open) enum enum_all enum_ex n_lists all_n_lists ex_n_lists product +hide_const (open) enum enum_all enum_ex n_lists all_n_lists ex_n_lists product ntrancl end