(* Title: HOL/ex/Word_Msb.thy Author: Florian Haftmann, TU Muenchen
*)
section \<open>An attempt for msb on word\<close>
theory Word_Msb imports"HOL-Library.Word" begin
class word = ring_bit_operations + fixes word_length :: \<open>'a itself \<Rightarrow> nat\<close> assumes word_length_positive [simp]: \<open>0 < word_length TYPE('a)\<close> and possible_bit_msb: \<open>possible_bit TYPE('a) (word_length TYPE('a) - Suc 0)\<close> and not_possible_bit_length: \<open>\<not> possible_bit TYPE('a) (word_length TYPE('a))\<close> begin
lemma word_length_not_0 [simp]: \<open>word_length TYPE('a) \<noteq> 0\<close> using word_length_positive by simp
lemma possible_bit_iff_less_word_length: \<open>possible_bit TYPE('a) n \<longleftrightarrow> n < word_length TYPE('a)\<close> (is \<open>?P \<longleftrightarrow> ?Q\<close>) proof assume\<open>?P\<close> show ?Q proof (rule ccontr) assume\<open>\<not> n < word_length TYPE('a)\<close> thenhave\<open>word_length TYPE('a) \<le> n\<close> by simp with\<open>?P\<close> have \<open>possible_bit TYPE('a) (word_length TYPE('a))\<close> by (rule possible_bit_less_imp) with not_possible_bit_length show False .. qed next assume\<open>?Q\<close> thenhave\<open>n \<le> word_length TYPE('a) - Suc 0\<close> by simp with possible_bit_msb show ?P by (rule possible_bit_less_imp) qed
end
instantiation word :: (len) word begin
definition word_length_word :: \<open>'a word itself \<Rightarrow> nat\<close> where [simp, code_unfold]: \<open>word_length_word _ = LENGTH('a)\<close>
instance by standard simp_all
end
context word begin
context includes bit_operations_syntax begin
definition msb :: \<open>'a \<Rightarrow> bool\<close> where\<open>msb w = bit w (word_length TYPE('a) - Suc 0)\<close>
lemma not_msb_0 [simp]: \<open>\<not> msb 0\<close> by (simp add: msb_def)
lemma msb_push_bit_iff: \<open>msb (push_bit n w) \<longleftrightarrow> n < word_length TYPE('a) \<and> bit w (word_length TYPE('a) - Suc n)\<close> by (simp add: msb_def bit_simps le_diff_conv2 Suc_le_eq possible_bit_iff_less_word_length)
lemma msb_drop_bit_iff [simp]: \<open>msb (drop_bit n w) \<longleftrightarrow> n = 0 \<and> msb w\<close> by (cases n)
(auto simp add: msb_def bit_simps possible_bit_iff_less_word_length intro!: impossible_bit)
lemma msb_take_bit_iff [simp]: \<open>msb (take_bit n w) \<longleftrightarrow> word_length TYPE('a) \<le> n \<and> msb w\<close> by (simp add: take_bit_eq_mask ac_simps)
lemma msb_signed_take_bit_iff: \<open>msb (signed_take_bit n w) \<longleftrightarrow> bit w (min n (word_length TYPE('a) - Suc 0))\<close> by (simp add: msb_def bit_simps possible_bit_iff_less_word_length)
definition signed_drop_bit :: \<open>nat \<Rightarrow> 'a \<Rightarrow> 'a\<close> where\<open>signed_drop_bit n w = drop_bit n w
OR (of_bool (bit w (word_length TYPE('a) - Suc 0)) * NOT (mask (word_length TYPE('a) - Suc n)))\<close>
lemma msb_signed_drop_bit_iff [simp]: \<open>msb (signed_drop_bit n w) \<longleftrightarrow> msb w\<close> by (simp add: signed_drop_bit_def bit_simps not_le not_less)
(simp add: msb_def)
end
end
end
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