# HG changeset patch
# User haftmann
# Date 1588919188 0
# Node ID 67cc2319104f8ec65348e4eb3576a93621f50272
# Parent  541e68d1a964c31f9adc05168e936bd4a06655cd
prefer _ mod 2 over of_bool (odd _)

diff -r 541e68d1a964 -r 67cc2319104f src/HOL/Parity.thy
--- a/src/HOL/Parity.thy	Fri May 08 06:26:27 2020 +0000
+++ b/src/HOL/Parity.thy	Fri May 08 06:26:28 2020 +0000
@@ -934,6 +934,10 @@
   qed
 qed
 
+lemma bit_mod_2_iff [simp]:
+  \<open>bit (a mod 2) n \<longleftrightarrow> n = 0 \<and> odd a\<close>
+  by (cases a rule: parity_cases) (simp_all add: bit_def)
+
 lemma bit_mask_iff:
   \<open>bit (2 ^ m - 1) n \<longleftrightarrow> 2 ^ n \<noteq> 0 \<and> n < m\<close>
   by (simp add: bit_def even_mask_div_iff not_le)
@@ -1204,7 +1208,7 @@
   by (simp add: take_bit_eq_mod)
 
 lemma take_bit_Suc:
-  \<open>take_bit (Suc n) a = take_bit n (a div 2) * 2 + of_bool (odd a)\<close>
+  \<open>take_bit (Suc n) a = take_bit n (a div 2) * 2 + a mod 2\<close>
 proof -
   have \<open>take_bit (Suc n) (a div 2 * 2 + of_bool (odd a)) = take_bit n (a div 2) * 2 + of_bool (odd a)\<close>
     using even_succ_mod_exp [of \<open>2 * (a div 2)\<close> \<open>Suc n\<close>]
@@ -1215,7 +1219,7 @@
 qed
 
 lemma take_bit_rec:
-  \<open>take_bit n a = (if n = 0 then 0 else take_bit (n - 1) (a div 2) * 2 + of_bool (odd a))\<close>
+  \<open>take_bit n a = (if n = 0 then 0 else take_bit (n - 1) (a div 2) * 2 + a mod 2)\<close>
   by (cases n) (simp_all add: take_bit_Suc)
 
 lemma take_bit_Suc_0 [simp]:
@@ -1442,7 +1446,7 @@
 
 lemma take_bit_Suc_bit1 [simp]:
   \<open>take_bit (Suc n) (numeral (Num.Bit1 k)) = take_bit n (numeral k) * 2 + 1\<close>
-  by (simp add: take_bit_Suc numeral_Bit1_div_2)
+  by (simp add: take_bit_Suc numeral_Bit1_div_2 mod_2_eq_odd)
 
 lemma take_bit_numeral_bit0 [simp]:
   \<open>take_bit (numeral l) (numeral (Num.Bit0 k)) = take_bit (pred_numeral l) (numeral k) * 2\<close>
@@ -1450,7 +1454,7 @@
 
 lemma take_bit_numeral_bit1 [simp]:
   \<open>take_bit (numeral l) (numeral (Num.Bit1 k)) = take_bit (pred_numeral l) (numeral k) * 2 + 1\<close>
-  by (simp add: take_bit_rec numeral_Bit1_div_2)
+  by (simp add: take_bit_rec numeral_Bit1_div_2 mod_2_eq_odd)
 
 lemma take_bit_of_nat:
   "take_bit n (of_nat m) = of_nat (take_bit n m)"
diff -r 541e68d1a964 -r 67cc2319104f src/HOL/Set_Interval.thy
--- a/src/HOL/Set_Interval.thy	Fri May 08 06:26:27 2020 +0000
+++ b/src/HOL/Set_Interval.thy	Fri May 08 06:26:28 2020 +0000
@@ -2087,7 +2087,7 @@
     = (\<Sum>k = 0..<n. push_bit k (of_bool (bit (a div 2) k))) * 2"
     by (simp only: sum.atLeast_Suc_lessThan_Suc_shift) (simp add: sum_distrib_right push_bit_double drop_bit_Suc bit_Suc)
   finally show ?case
-    using Suc [of "a div 2"] by (simp add: ac_simps take_bit_Suc)
+    using Suc [of "a div 2"] by (simp add: ac_simps take_bit_Suc mod_2_eq_odd)
 qed
 
 end
diff -r 541e68d1a964 -r 67cc2319104f src/HOL/String.thy
--- a/src/HOL/String.thy	Fri May 08 06:26:27 2020 +0000
+++ b/src/HOL/String.thy	Fri May 08 06:26:28 2020 +0000
@@ -90,7 +90,7 @@
     also have \<open>\<dots> = b # map (bit a) [0..<m]\<close>
       by (simp only: flip: map_Suc_upt) (simp add: bit_Suc rec.hyps)
     finally show ?thesis
-      using Suc rec.IH [of m] by (simp add: take_bit_Suc rec.hyps, simp add: ac_simps)
+      using Suc rec.IH [of m] by (simp add: take_bit_Suc rec.hyps, simp add: ac_simps mod_2_eq_odd)
   qed
 qed
 
diff -r 541e68d1a964 -r 67cc2319104f src/HOL/ex/Bit_Operations.thy
--- a/src/HOL/ex/Bit_Operations.thy	Fri May 08 06:26:27 2020 +0000
+++ b/src/HOL/ex/Bit_Operations.thy	Fri May 08 06:26:28 2020 +0000
@@ -46,26 +46,26 @@
   by (simp add: bit_eq_iff bit_and_iff)
 
 lemma one_and_eq [simp]:
-  "1 AND a = of_bool (odd a)"
+  "1 AND a = a mod 2"
   by (simp add: bit_eq_iff bit_and_iff) (auto simp add: bit_1_iff)
 
 lemma and_one_eq [simp]:
-  "a AND 1 = of_bool (odd a)"
+  "a AND 1 = a mod 2"
   using one_and_eq [of a] by (simp add: ac_simps)
 
-lemma one_or_eq [simp]:
+lemma one_or_eq:
   "1 OR a = a + of_bool (even a)"
   by (simp add: bit_eq_iff bit_or_iff add.commute [of _ 1] even_bit_succ_iff) (auto simp add: bit_1_iff)
 
-lemma or_one_eq [simp]:
+lemma or_one_eq:
   "a OR 1 = a + of_bool (even a)"
   using one_or_eq [of a] by (simp add: ac_simps)
 
-lemma one_xor_eq [simp]:
+lemma one_xor_eq:
   "1 XOR a = a + of_bool (even a) - of_bool (odd a)"
   by (simp add: bit_eq_iff bit_xor_iff add.commute [of _ 1] even_bit_succ_iff) (auto simp add: bit_1_iff odd_bit_iff_bit_pred elim: oddE)
 
-lemma xor_one_eq [simp]:
+lemma xor_one_eq:
   "a XOR 1 = a + of_bool (even a) - of_bool (odd a)"
   using one_xor_eq [of a] by (simp add: ac_simps)
 
@@ -533,26 +533,26 @@
   by (simp add: xor_nat_def xor_int_rec [of \<open>int m\<close> \<open>int n\<close>] zdiv_int nat_add_distrib nat_mult_distrib)
 
 lemma Suc_0_and_eq [simp]:
-  \<open>Suc 0 AND n = of_bool (odd n)\<close>
+  \<open>Suc 0 AND n = n mod 2\<close>
   using one_and_eq [of n] by simp
 
 lemma and_Suc_0_eq [simp]:
-  \<open>n AND Suc 0 = of_bool (odd n)\<close>
+  \<open>n AND Suc 0 = n mod 2\<close>
   using and_one_eq [of n] by simp
 
-lemma Suc_0_or_eq [simp]:
+lemma Suc_0_or_eq:
   \<open>Suc 0 OR n = n + of_bool (even n)\<close>
   using one_or_eq [of n] by simp
 
-lemma or_Suc_0_eq [simp]:
+lemma or_Suc_0_eq:
   \<open>n OR Suc 0 = n + of_bool (even n)\<close>
   using or_one_eq [of n] by simp
 
-lemma Suc_0_xor_eq [simp]:
+lemma Suc_0_xor_eq:
   \<open>Suc 0 XOR n = n + of_bool (even n) - of_bool (odd n)\<close>
   using one_xor_eq [of n] by simp
 
-lemma xor_Suc_0_eq [simp]:
+lemma xor_Suc_0_eq:
   \<open>n XOR Suc 0 = n + of_bool (even n) - of_bool (odd n)\<close>
   using xor_one_eq [of n] by simp
 
diff -r 541e68d1a964 -r 67cc2319104f src/HOL/ex/Word.thy
--- a/src/HOL/ex/Word.thy	Fri May 08 06:26:27 2020 +0000
+++ b/src/HOL/ex/Word.thy	Fri May 08 06:26:28 2020 +0000
@@ -592,7 +592,8 @@
   show \<open>(1 + a) div 2 = a div 2\<close>
     if \<open>even a\<close>
     for a :: \<open>'a word\<close>
-    using that by transfer (auto dest: le_Suc_ex simp add: take_bit_Suc)
+    using that by transfer
+      (auto dest: le_Suc_ex simp add: mod_2_eq_odd take_bit_Suc elim!: evenE)
   show \<open>(2 :: 'a word) ^ m div 2 ^ n = of_bool ((2 :: 'a word) ^ m \<noteq> 0 \<and> n \<le> m) * 2 ^ (m - n)\<close>
     for m n :: nat
     by transfer (simp, simp add: exp_div_exp_eq)