--- a/NEWS Thu Jun 27 16:52:17 2024 +0000
+++ b/NEWS Sun Jun 30 06:30:08 2024 +0000
@@ -37,6 +37,11 @@
wfP_multp ~> wfp_multp
wfP_subset_mset ~> wfp_subset_mset
+* Transitioinal theory "Divides" moved to "HOL-Library.Divides" and supposed to
+be removed in a furure release. Minor INCOMPATIBILITY. Import
+"HOL-Library.Divides" and keep an eye on theorems prefixed with "Divides." to
+ease transition.
+
New in Isabelle2024 (May 2024)
------------------------------
--- a/src/Doc/Main/Main_Doc.thy Thu Jun 27 16:52:17 2024 +0000
+++ b/src/Doc/Main/Main_Doc.thy Sun Jun 30 06:30:08 2024 +0000
@@ -306,7 +306,9 @@
\section*{Algebra}
-Theories \<^theory>\<open>HOL.Groups\<close>, \<^theory>\<open>HOL.Rings\<close>, \<^theory>\<open>HOL.Fields\<close> and \<^theory>\<open>HOL.Divides\<close> define a large collection of classes describing common algebraic
+Theories \<^theory>\<open>HOL.Groups\<close>, \<^theory>\<open>HOL.Rings\<close>,
+\<^theory>\<open>HOL.Euclidean_Rings\<close> and \<^theory>\<open>HOL.Fields\<close>
+define a large collection of classes describing common algebraic
structures from semigroups up to fields. Everything is done in terms of
overloaded operators:
--- a/src/HOL/Divides.thy Thu Jun 27 16:52:17 2024 +0000
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,141 +0,0 @@
-(* Title: HOL/Divides.thy
-*)
-
-section \<open>Misc lemmas on division, to be sorted out finally\<close>
-
-theory Divides
-imports Parity
-begin
-
-class unique_euclidean_semiring_numeral = linordered_euclidean_semiring + discrete_linordered_semidom +
- 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"
-
-hide_fact (open) div_less mod_less div_positive mod_less_eq_dividend pos_mod_bound pos_mod_sign mod_mult2_eq div_mult2_eq
-
-context unique_euclidean_semiring_numeral
-begin
-
-context
-begin
-
-qualified lemma discrete [no_atp]:
- "a < b \<longleftrightarrow> a + 1 \<le> b"
- by (fact less_iff_succ_less_eq)
-
-qualified 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")
-proof -
- from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
- by (auto intro: trans)
- with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
- then have [simp]: "1 \<le> a div b" by (simp add: discrete)
- with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
- define w where "w = a div b mod 2"
- then have w_exhaust: "w = 0 \<or> w = 1" by auto
- have mod_w: "a mod (2 * b) = a mod b + b * w"
- by (simp add: w_def mod_mult2_eq ac_simps)
- from assms w_exhaust have "w = 1"
- using mod_less by (auto simp add: mod_w)
- with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
- have "2 * (a div (2 * b)) = a div b - w"
- by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
- with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
- then show ?P and ?Q
- by (simp_all add: div mod add_implies_diff [symmetric])
-qed
-
-qualified 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")
-proof -
- define w where "w = a div b mod 2"
- then have w_exhaust: "w = 0 \<or> w = 1" by auto
- have mod_w: "a mod (2 * b) = a mod b + b * w"
- by (simp add: w_def mod_mult2_eq ac_simps)
- moreover have "b \<le> a mod b + b"
- proof -
- from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
- then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
- then show ?thesis by simp
- qed
- moreover note assms w_exhaust
- ultimately have "w = 0" by auto
- with mod_w have mod: "a mod (2 * b) = a mod b" by simp
- have "2 * (a div (2 * b)) = a div b - w"
- by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
- with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
- then show ?P and ?Q
- by (simp_all add: div mod)
-qed
-
-qualified lemma mod_double_modulus [no_atp]:
- 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")
- case True
- thus ?thesis using assms using divmod_digit_0(2)[of m x] by auto
-next
- case False
- hence *: "x mod (2 * m) - m = x mod m"
- using assms by (intro divmod_digit_1) auto
- hence "x mod (2 * m) = x mod m + m"
- by (subst * [symmetric], subst le_add_diff_inverse2) (use False in auto)
- thus ?thesis by simp
-qed
-
-end
-
-end
-
-instance nat :: unique_euclidean_semiring_numeral
- by standard
- (auto simp add: div_greater_zero_iff div_mult2_eq mod_mult2_eq)
-
-instance int :: unique_euclidean_semiring_numeral
- by standard (auto intro: zmod_le_nonneg_dividend simp add:
- pos_imp_zdiv_pos_iff zmod_zmult2_eq zdiv_zmult2_eq)
-
-context
-begin
-
-qualified lemma zmod_eq_0D: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: int
- using that by auto
-
-qualified 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>)
-
-qualified 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>)
-
-qualified 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)
-
-qualified 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
-
-qualified 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
-
-qualified 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)
-
-qualified 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)
-
-end
-
-code_identifier
- code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
-
-end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Divides.thy Sun Jun 30 06:30:08 2024 +0000
@@ -0,0 +1,141 @@
+(* Title: HOL/Library/Divides.thy
+*)
+
+section \<open>Misc lemmas on division, to be sorted out finally\<close>
+
+theory Divides
+imports Main
+begin
+
+class unique_euclidean_semiring_numeral = linordered_euclidean_semiring + discrete_linordered_semidom +
+ 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"
+
+hide_fact (open) div_less mod_less div_positive mod_less_eq_dividend pos_mod_bound pos_mod_sign mod_mult2_eq div_mult2_eq
+
+context unique_euclidean_semiring_numeral
+begin
+
+context
+begin
+
+qualified lemma discrete [no_atp]:
+ "a < b \<longleftrightarrow> a + 1 \<le> b"
+ by (fact less_iff_succ_less_eq)
+
+qualified 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")
+proof -
+ from assms mod_less_eq_dividend [of a "2 * b"] have "b \<le> a"
+ by (auto intro: trans)
+ with \<open>0 < b\<close> have "0 < a div b" by (auto intro: div_positive)
+ then have [simp]: "1 \<le> a div b" by (simp add: discrete)
+ with \<open>0 < b\<close> have mod_less: "a mod b < b" by (simp add: pos_mod_bound)
+ define w where "w = a div b mod 2"
+ then have w_exhaust: "w = 0 \<or> w = 1" by auto
+ have mod_w: "a mod (2 * b) = a mod b + b * w"
+ by (simp add: w_def mod_mult2_eq ac_simps)
+ from assms w_exhaust have "w = 1"
+ using mod_less by (auto simp add: mod_w)
+ with mod_w have mod: "a mod (2 * b) = a mod b + b" by simp
+ have "2 * (a div (2 * b)) = a div b - w"
+ by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
+ with \<open>w = 1\<close> have div: "2 * (a div (2 * b)) = a div b - 1" by simp
+ then show ?P and ?Q
+ by (simp_all add: div mod add_implies_diff [symmetric])
+qed
+
+qualified 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")
+proof -
+ define w where "w = a div b mod 2"
+ then have w_exhaust: "w = 0 \<or> w = 1" by auto
+ have mod_w: "a mod (2 * b) = a mod b + b * w"
+ by (simp add: w_def mod_mult2_eq ac_simps)
+ moreover have "b \<le> a mod b + b"
+ proof -
+ from \<open>0 < b\<close> pos_mod_sign have "0 \<le> a mod b" by blast
+ then have "0 + b \<le> a mod b + b" by (rule add_right_mono)
+ then show ?thesis by simp
+ qed
+ moreover note assms w_exhaust
+ ultimately have "w = 0" by auto
+ with mod_w have mod: "a mod (2 * b) = a mod b" by simp
+ have "2 * (a div (2 * b)) = a div b - w"
+ by (simp add: w_def div_mult2_eq minus_mod_eq_mult_div ac_simps)
+ with \<open>w = 0\<close> have div: "2 * (a div (2 * b)) = a div b" by simp
+ then show ?P and ?Q
+ by (simp_all add: div mod)
+qed
+
+qualified lemma mod_double_modulus [no_atp]:
+ 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")
+ case True
+ thus ?thesis using assms using divmod_digit_0(2)[of m x] by auto
+next
+ case False
+ hence *: "x mod (2 * m) - m = x mod m"
+ using assms by (intro divmod_digit_1) auto
+ hence "x mod (2 * m) = x mod m + m"
+ by (subst * [symmetric], subst le_add_diff_inverse2) (use False in auto)
+ thus ?thesis by simp
+qed
+
+end
+
+end
+
+instance nat :: unique_euclidean_semiring_numeral
+ by standard
+ (auto simp add: div_greater_zero_iff div_mult2_eq mod_mult2_eq)
+
+instance int :: unique_euclidean_semiring_numeral
+ by standard (auto intro: zmod_le_nonneg_dividend simp add:
+ pos_imp_zdiv_pos_iff zmod_zmult2_eq zdiv_zmult2_eq)
+
+context
+begin
+
+qualified lemma zmod_eq_0D: "\<exists>q. m = d * q" if "m mod d = 0" for m d :: int
+ using that by auto
+
+qualified 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>)
+
+qualified 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>)
+
+qualified 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)
+
+qualified 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
+
+qualified 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
+
+qualified 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)
+
+qualified 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)
+
+end
+
+code_identifier
+ code_module Divides \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
+
+end
--- a/src/HOL/Main.thy Thu Jun 27 16:52:17 2024 +0000
+++ b/src/HOL/Main.thy Sun Jun 30 06:30:08 2024 +0000
@@ -17,7 +17,6 @@
Conditionally_Complete_Lattices
Binomial
GCD
- Divides
begin
subsection \<open>Namespace cleanup\<close>
--- a/src/HOL/ROOT Thu Jun 27 16:52:17 2024 +0000
+++ b/src/HOL/ROOT Sun Jun 30 06:30:08 2024 +0000
@@ -95,6 +95,8 @@
Old_Recdef
Realizers
Refute
+ (*transitional theories*)
+ Divides
document_files "root.bib" "root.tex"
session "HOL-Analysis" (main timing) in Analysis = HOL +