
diff -r fc19de122712 -r 8a48e18f081e src/HOL/Divides.thy
--- a/src/HOL/Divides.thy	Fri Sep 30 21:03:58 2022 +0200
+++ b/src/HOL/Divides.thy	Sat Oct 01 07:56:53 2022 +0000
@@ -1,25 +1,32 @@
 (*  Title:      HOL/Divides.thy
 *)
 
-section \<open>Lemmas of doubtful value\<close>
+section \<open>Misc lemmas on division, to be sorted out finally\<close>
 
 theory Divides
 imports Parity
 begin
 
 class unique_euclidean_semiring_numeral = unique_euclidean_semiring_with_nat + linordered_semidom +
-  assumes div_less: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
-    and mod_less: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
-    and div_positive: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
-    and mod_less_eq_dividend: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
-    and pos_mod_bound: "0 < b \<Longrightarrow> a mod b < b"
-    and pos_mod_sign: "0 < b \<Longrightarrow> 0 \<le> a mod b"
-    and mod_mult2_eq: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
-    and div_mult2_eq: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
-  assumes discrete: "a < b \<longleftrightarrow> a + 1 \<le> b"
+  assumes div_less [no_atp]: "0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a div b = 0"
+    and mod_less [no_atp]: " 0 \<le> a \<Longrightarrow> a < b \<Longrightarrow> a mod b = a"
+    and div_positive [no_atp]: "0 < b \<Longrightarrow> b \<le> a \<Longrightarrow> a div b > 0"
+    and mod_less_eq_dividend [no_atp]: "0 \<le> a \<Longrightarrow> a mod b \<le> a"
+    and pos_mod_bound [no_atp]: "0 < b \<Longrightarrow> a mod b < b"
+    and pos_mod_sign [no_atp]: "0 < b \<Longrightarrow> 0 \<le> a mod b"
+    and mod_mult2_eq [no_atp]: "0 \<le> c \<Longrightarrow> a mod (b * c) = b * (a div b mod c) + a mod b"
+    and div_mult2_eq [no_atp]: "0 \<le> c \<Longrightarrow> a div (b * c) = a div b div c"
+  assumes discrete [no_atp]: "a < b \<longleftrightarrow> a + 1 \<le> b"
+
+hide_fact (open) div_less mod_less mod_less_eq_dividend mod_mult2_eq div_mult2_eq
+
+context unique_euclidean_semiring_numeral
 begin
 
-lemma divmod_digit_1:
+context
+begin
+
+lemma divmod_digit_1 [no_atp]:
   assumes "0 \<le> a" "0 < b" and "b \<le> a mod (2 * b)"
   shows "2 * (a div (2 * b)) + 1 = a div b" (is "?P")
     and "a mod (2 * b) - b = a mod b" (is "?Q")
@@ -43,7 +50,7 @@
     by (simp_all add: div mod add_implies_diff [symmetric])
 qed
 
-lemma divmod_digit_0:
+lemma divmod_digit_0 [no_atp]:
   assumes "0 < b" and "a mod (2 * b) < b"
   shows "2 * (a div (2 * b)) = a div b" (is "?P")
     and "a mod (2 * b) = a mod b" (is "?Q")
@@ -68,7 +75,7 @@
     by (simp_all add: div mod)
 qed
 
-lemma mod_double_modulus:
+lemma mod_double_modulus: \<comment>\<open>This is actually useful and should be transferred to a suitable type class\<close>
   assumes "m > 0" "x \<ge> 0"
   shows   "x mod (2 * m) = x mod m \<or> x mod (2 * m) = x mod m + m"
 proof (cases "x mod (2 * m) < m")
@@ -85,7 +92,7 @@
 
 end
 
-hide_fact (open) div_less mod_less mod_less_eq_dividend mod_mult2_eq div_mult2_eq
+end
 
 instance nat :: unique_euclidean_semiring_numeral
   by standard
@@ -95,29 +102,29 @@
   by standard (auto intro: zmod_le_nonneg_dividend simp add:
     pos_imp_zdiv_pos_iff zmod_zmult2_eq zdiv_zmult2_eq)
 
-lemma div_geq: "m div n = Suc ((m - n) div n)" if "0 < n" and " \<not> m < n" for m n :: nat
-  by (rule le_div_geq) (use that in \<open>simp_all add: not_less\<close>)
-
-lemma mod_geq: "m mod n = (m - n) mod n" if "\<not> m < n" for m n :: nat
-  by (rule le_mod_geq) (use that in \<open>simp add: not_less\<close>)
-
-lemma mod_eq_0D: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: nat
-  using that by (auto simp add: mod_eq_0_iff_dvd)
-
-lemma pos_mod_conj: "0 < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b" for a b :: int
-  by simp
-
-lemma neg_mod_conj: "b < 0 \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b" for a b :: int
-  by simp
-
-lemma zmod_eq_0_iff: "m mod d = 0 \<longleftrightarrow> (\<exists>q. m = d * q)" for m d :: int
-  by (auto simp add: mod_eq_0_iff_dvd)
-
 (* REVISIT: should this be generalized to all semiring_div types? *)
 lemma zmod_eq_0D [dest!]: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: int
   using that by auto
 
-lemma div_positive_int:
+lemma div_geq [no_atp]: "m div n = Suc ((m - n) div n)" if "0 < n" and " \<not> m < n" for m n :: nat
+  by (rule le_div_geq) (use that in \<open>simp_all add: not_less\<close>)
+
+lemma mod_geq [no_atp]: "m mod n = (m - n) mod n" if "\<not> m < n" for m n :: nat
+  by (rule le_mod_geq) (use that in \<open>simp add: not_less\<close>)
+
+lemma mod_eq_0D [no_atp]: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: nat
+  using that by (auto simp add: mod_eq_0_iff_dvd)
+
+lemma pos_mod_conj [no_atp]: "0 < b \<Longrightarrow> 0 \<le> a mod b \<and> a mod b < b" for a b :: int
+  by simp
+
+lemma neg_mod_conj [no_atp]: "b < 0 \<Longrightarrow> a mod b \<le> 0 \<and> b < a mod b" for a b :: int
+  by simp
+
+lemma zmod_eq_0_iff [no_atp]: "m mod d = 0 \<longleftrightarrow> (\<exists>q. m = d * q)" for m d :: int
+  by (auto simp add: mod_eq_0_iff_dvd)
+
+lemma div_positive_int [no_atp]:
   "k div l > 0" if "k \<ge> l" and "l > 0" for k l :: int
   using that by (simp add: nonneg1_imp_zdiv_pos_iff)
 




