author wenzelm Wed, 10 Aug 2016 22:03:58 +0200 changeset 63652 804b80a80016 parent 63651 f8e79d14d61f child 63653 4453cfb745e5
misc tuning and modernization;
 src/HOL/Int.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/Int.thy	Wed Aug 10 16:55:49 2016 +0200
+++ b/src/HOL/Int.thy	Wed Aug 10 22:03:58 2016 +0200
@@ -6,13 +6,13 @@
section \<open>The Integers as Equivalence Classes over Pairs of Natural Numbers\<close>

theory Int
-imports Equiv_Relations Power Quotient Fun_Def
+  imports Equiv_Relations Power Quotient Fun_Def
begin

subsection \<open>Definition of integers as a quotient type\<close>

-definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool" where
-  "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"
+definition intrel :: "(nat \<times> nat) \<Rightarrow> (nat \<times> nat) \<Rightarrow> bool"
+  where "intrel = (\<lambda>(x, y) (u, v). x + v = u + y)"

lemma intrel_iff [simp]: "intrel (x, y) (u, v) \<longleftrightarrow> x + v = u + y"
@@ -20,17 +20,15 @@
quotient_type int = "nat \<times> nat" / "intrel"
morphisms Rep_Integ Abs_Integ
proof (rule equivpI)
-  show "reflp intrel"
-    unfolding reflp_def by auto
-  show "symp intrel"
-    unfolding symp_def by auto
-  show "transp intrel"
-    unfolding transp_def by auto
+  show "reflp intrel" by (auto simp: reflp_def)
+  show "symp intrel" by (auto simp: symp_def)
+  show "transp intrel" by (auto simp: transp_def)
qed

lemma eq_Abs_Integ [case_names Abs_Integ, cases type: int]:
-     "(!!x y. z = Abs_Integ (x, y) ==> P) ==> P"
-by (induct z) auto
+  "(\<And>x y. z = Abs_Integ (x, y) \<Longrightarrow> P) \<Longrightarrow> P"
+  by (induct z) auto
+

subsection \<open>Integers form a commutative ring\<close>

@@ -58,33 +56,31 @@
proof (clarsimp)
fix s t u v w x y z :: nat
assume "s + v = u + t" and "w + z = y + x"
-  hence "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x)
-       = (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
+  then have "(s + v) * w + (u + t) * x + u * (w + z) + v * (y + x) =
+    (u + t) * w + (s + v) * x + u * (y + x) + v * (w + z)"
by simp
-  thus "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
+  then show "(s * w + t * x) + (u * z + v * y) = (u * y + v * z) + (s * x + t * w)"
qed

instance
-  by standard (transfer, clarsimp simp: algebra_simps)+
+  by standard (transfer; clarsimp simp: algebra_simps)+

end

-abbreviation int :: "nat \<Rightarrow> int" where
-  "int \<equiv> of_nat"
+abbreviation int :: "nat \<Rightarrow> int"
+  where "int \<equiv> of_nat"

lemma int_def: "int n = Abs_Integ (n, 0)"
-  by (induct n, simp add: zero_int.abs_eq,

-lemma int_transfer [transfer_rule]:
-  "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int"
-  unfolding rel_fun_def int.pcr_cr_eq cr_int_def int_def by simp
+lemma int_transfer [transfer_rule]: "(rel_fun (op =) pcr_int) (\<lambda>n. (n, 0)) int"
+  by (simp add: rel_fun_def int.pcr_cr_eq cr_int_def int_def)

-lemma int_diff_cases:
-  obtains (diff) m n where "z = int m - int n"
+lemma int_diff_cases: obtains (diff) m n where "z = int m - int n"
by transfer clarsimp

+
subsection \<open>Integers are totally ordered\<close>

instantiation int :: linorder
@@ -106,18 +102,16 @@
instantiation int :: distrib_lattice
begin

-definition
-  "(inf :: int \<Rightarrow> int \<Rightarrow> int) = min"
+definition "(inf :: int \<Rightarrow> int \<Rightarrow> int) = min"

-definition
-  "(sup :: int \<Rightarrow> int \<Rightarrow> int) = max"
+definition "(sup :: int \<Rightarrow> int \<Rightarrow> int) = max"

instance
-  by intro_classes
-    (auto simp add: inf_int_def sup_int_def max_min_distrib2)
+  by standard (auto simp add: inf_int_def sup_int_def max_min_distrib2)

end

+
subsection \<open>Ordering properties of arithmetic operations\<close>

@@ -127,46 +121,52 @@
by transfer clarsimp
qed

-text\<open>Strict Monotonicity of Multiplication\<close>
+text \<open>Strict Monotonicity of Multiplication.\<close>

-text\<open>strict, in 1st argument; proof is by induction on k>0\<close>
-lemma zmult_zless_mono2_lemma:
-     "(i::int)<j ==> 0<k ==> int k * i < int k * j"
-apply (induct k)
-apply simp
-apply (case_tac "k=0")
-done
+text \<open>Strict, in 1st argument; proof is by induction on \<open>k > 0\<close>.\<close>
+lemma zmult_zless_mono2_lemma: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> int k * i < int k * j"
+  for i j :: int
+proof (induct k)
+  case 0
+  then show ?case by simp
+next
+  case (Suc k)
+  then show ?case
+qed

-lemma zero_le_imp_eq_int: "(0::int) \<le> k ==> \<exists>n. k = int n"
-apply transfer
-apply clarsimp
-apply (rule_tac x="a - b" in exI, simp)
-done
+lemma zero_le_imp_eq_int: "0 \<le> k \<Longrightarrow> \<exists>n. k = int n"
+  for k :: int
+  apply transfer
+  apply clarsimp
+  apply (rule_tac x="a - b" in exI)
+  apply simp
+  done

-lemma zero_less_imp_eq_int: "(0::int) < k ==> \<exists>n>0. k = int n"
-apply transfer
-apply clarsimp
-apply (rule_tac x="a - b" in exI, simp)
-done
+lemma zero_less_imp_eq_int: "0 < k \<Longrightarrow> \<exists>n>0. k = int n"
+  for k :: int
+  apply transfer
+  apply clarsimp
+  apply (rule_tac x="a - b" in exI)
+  apply simp
+  done

-lemma zmult_zless_mono2: "[| i<j;  (0::int) < k |] ==> k*i < k*j"
-apply (drule zero_less_imp_eq_int)
-done
+lemma zmult_zless_mono2: "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
+  for i j k :: int
+  by (drule zero_less_imp_eq_int) (auto simp add: zmult_zless_mono2_lemma)

-text\<open>The integers form an ordered integral domain\<close>
+
+text \<open>The integers form an ordered integral domain.\<close>
+
instantiation int :: linordered_idom
begin

-definition
-  zabs_def: "\<bar>i::int\<bar> = (if i < 0 then - i else i)"
+definition zabs_def: "\<bar>i::int\<bar> = (if i < 0 then - i else i)"

-definition
-  zsgn_def: "sgn (i::int) = (if i=0 then 0 else if 0<i then 1 else - 1)"
+definition zsgn_def: "sgn (i::int) = (if i = 0 then 0 else if 0 < i then 1 else - 1)"

-instance proof
+instance
+proof
fix i j k :: int
show "i < j \<Longrightarrow> 0 < k \<Longrightarrow> k * i < k * j"
by (rule zmult_zless_mono2)
@@ -178,31 +178,29 @@

end

-lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + (1::int) \<le> z"
+lemma zless_imp_add1_zle: "w < z \<Longrightarrow> w + 1 \<le> z"
+  for w z :: int
by transfer clarsimp

-  "(w :: int) < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
-apply transfer
-apply auto
-apply (rename_tac a b c d)
-apply (rule_tac x="c+b - Suc(a+d)" in exI)
-apply arith
-done
+lemma zless_iff_Suc_zadd: "w < z \<longleftrightarrow> (\<exists>n. z = w + int (Suc n))"
+  for w z :: int
+  apply transfer
+  apply auto
+  apply (rename_tac a b c d)
+  apply (rule_tac x="c+b - Suc(a+d)" in exI)
+  apply arith
+  done

-lemma zabs_less_one_iff [simp]:
-  fixes z :: int
-  shows "\<bar>z\<bar> < 1 \<longleftrightarrow> z = 0" (is "?P \<longleftrightarrow> ?Q")
+lemma zabs_less_one_iff [simp]: "\<bar>z\<bar> < 1 \<longleftrightarrow> z = 0" (is "?lhs \<longleftrightarrow> ?rhs")
+  for z :: int
proof
-  assume ?Q then show ?P by simp
+  assume ?rhs
+  then show ?lhs by simp
next
-  assume ?P
-  with zless_imp_add1_zle [of "\<bar>z\<bar>" 1] have "\<bar>z\<bar> + 1 \<le> 1"
-    by simp
-  then have "\<bar>z\<bar> \<le> 0"
-    by simp
-  then show ?Q
-    by simp
+  assume ?lhs
+  with zless_imp_add1_zle [of "\<bar>z\<bar>" 1] have "\<bar>z\<bar> + 1 \<le> 1" by simp
+  then have "\<bar>z\<bar> \<le> 0" by simp
+  then show ?rhs by simp
qed

lemmas int_distrib =
@@ -218,9 +216,10 @@
context ring_1
begin

-lift_definition of_int :: "int \<Rightarrow> 'a" is "\<lambda>(i, j). of_nat i - of_nat j"
+lift_definition of_int :: "int \<Rightarrow> 'a"
+  is "\<lambda>(i, j). of_nat i - of_nat j"

lemma of_int_0 [simp]: "of_int 0 = 0"
by transfer simp
@@ -228,24 +227,24 @@
lemma of_int_1 [simp]: "of_int 1 = 1"
by transfer simp

-lemma of_int_add [simp]: "of_int (w+z) = of_int w + of_int z"
+lemma of_int_add [simp]: "of_int (w + z) = of_int w + of_int z"
by transfer (clarsimp simp add: algebra_simps)

-lemma of_int_minus [simp]: "of_int (-z) = - (of_int z)"
+lemma of_int_minus [simp]: "of_int (- z) = - (of_int z)"
by (transfer fixing: uminus) clarsimp

lemma of_int_diff [simp]: "of_int (w - z) = of_int w - of_int z"
using of_int_add [of w "- z"] by simp

lemma of_int_mult [simp]: "of_int (w*z) = of_int w * of_int z"
-  by (transfer fixing: times) (clarsimp simp add: algebra_simps of_nat_mult)
+  by (transfer fixing: times) (clarsimp simp add: algebra_simps)

lemma mult_of_int_commute: "of_int x * y = y * of_int x"
by (transfer fixing: times) (auto simp: algebra_simps mult_of_nat_commute)

-text\<open>Collapse nested embeddings\<close>
+text \<open>Collapse nested embeddings.\<close>
lemma of_int_of_nat_eq [simp]: "of_int (int n) = of_nat n"
-by (induct n) auto
+  by (induct n) auto

lemma of_int_numeral [simp, code_post]: "of_int (numeral k) = numeral k"
by (simp add: of_nat_numeral [symmetric] of_int_of_nat_eq [symmetric])
@@ -253,8 +252,7 @@
lemma of_int_neg_numeral [code_post]: "of_int (- numeral k) = - numeral k"
by simp

-lemma of_int_power [simp]:
-  "of_int (z ^ n) = of_int z ^ n"
+lemma of_int_power [simp]: "of_int (z ^ n) = of_int z ^ n"
by (induct n) simp_all

end
@@ -262,22 +260,17 @@
context ring_char_0
begin

-lemma of_int_eq_iff [simp]:
-   "of_int w = of_int z \<longleftrightarrow> w = z"
-  by transfer (clarsimp simp add: algebra_simps
+lemma of_int_eq_iff [simp]: "of_int w = of_int z \<longleftrightarrow> w = z"

-text\<open>Special cases where either operand is zero\<close>
-lemma of_int_eq_0_iff [simp]:
-  "of_int z = 0 \<longleftrightarrow> z = 0"
+text \<open>Special cases where either operand is zero.\<close>
+lemma of_int_eq_0_iff [simp]: "of_int z = 0 \<longleftrightarrow> z = 0"
using of_int_eq_iff [of z 0] by simp

-lemma of_int_0_eq_iff [simp]:
-  "0 = of_int z \<longleftrightarrow> z = 0"
+lemma of_int_0_eq_iff [simp]: "0 = of_int z \<longleftrightarrow> z = 0"
using of_int_eq_iff [of 0 z] by simp

-lemma of_int_eq_1_iff [iff]:
-   "of_int z = 1 \<longleftrightarrow> z = 1"
+lemma of_int_eq_1_iff [iff]: "of_int z = 1 \<longleftrightarrow> z = 1"
using of_int_eq_iff [of z 1] by simp

end
@@ -285,48 +278,38 @@
context linordered_idom
begin

-text\<open>Every \<open>linordered_idom\<close> has characteristic zero.\<close>
+text \<open>Every \<open>linordered_idom\<close> has characteristic zero.\<close>
subclass ring_char_0 ..

-lemma of_int_le_iff [simp]:
-  "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
-  by (transfer fixing: less_eq) (clarsimp simp add: algebra_simps
+lemma of_int_le_iff [simp]: "of_int w \<le> of_int z \<longleftrightarrow> w \<le> z"
+  by (transfer fixing: less_eq)

-lemma of_int_less_iff [simp]:
-  "of_int w < of_int z \<longleftrightarrow> w < z"
+lemma of_int_less_iff [simp]: "of_int w < of_int z \<longleftrightarrow> w < z"

-lemma of_int_0_le_iff [simp]:
-  "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
+lemma of_int_0_le_iff [simp]: "0 \<le> of_int z \<longleftrightarrow> 0 \<le> z"
using of_int_le_iff [of 0 z] by simp

-lemma of_int_le_0_iff [simp]:
-  "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
+lemma of_int_le_0_iff [simp]: "of_int z \<le> 0 \<longleftrightarrow> z \<le> 0"
using of_int_le_iff [of z 0] by simp

-lemma of_int_0_less_iff [simp]:
-  "0 < of_int z \<longleftrightarrow> 0 < z"
+lemma of_int_0_less_iff [simp]: "0 < of_int z \<longleftrightarrow> 0 < z"
using of_int_less_iff [of 0 z] by simp

-lemma of_int_less_0_iff [simp]:
-  "of_int z < 0 \<longleftrightarrow> z < 0"
+lemma of_int_less_0_iff [simp]: "of_int z < 0 \<longleftrightarrow> z < 0"
using of_int_less_iff [of z 0] by simp

-lemma of_int_1_le_iff [simp]:
-  "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z"
+lemma of_int_1_le_iff [simp]: "1 \<le> of_int z \<longleftrightarrow> 1 \<le> z"
using of_int_le_iff [of 1 z] by simp

-lemma of_int_le_1_iff [simp]:
-  "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1"
+lemma of_int_le_1_iff [simp]: "of_int z \<le> 1 \<longleftrightarrow> z \<le> 1"
using of_int_le_iff [of z 1] by simp

-lemma of_int_1_less_iff [simp]:
-  "1 < of_int z \<longleftrightarrow> 1 < z"
+lemma of_int_1_less_iff [simp]: "1 < of_int z \<longleftrightarrow> 1 < z"
using of_int_less_iff [of 1 z] by simp

-lemma of_int_less_1_iff [simp]:
-  "of_int z < 1 \<longleftrightarrow> z < 1"
+lemma of_int_less_1_iff [simp]: "of_int z < 1 \<longleftrightarrow> z < 1"
using of_int_less_iff [of z 1] by simp

lemma of_int_pos: "z > 0 \<Longrightarrow> of_int z > 0"
@@ -335,15 +318,15 @@
lemma of_int_nonneg: "z \<ge> 0 \<Longrightarrow> of_int z \<ge> 0"
by simp

-lemma of_int_abs [simp]:
-  "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>"
+lemma of_int_abs [simp]: "of_int \<bar>x\<bar> = \<bar>of_int x\<bar>"

lemma of_int_lessD:
assumes "\<bar>of_int n\<bar> < x"
shows "n = 0 \<or> x > 1"
proof (cases "n = 0")
-  case True then show ?thesis by simp
+  case True
+  then show ?thesis by simp
next
case False
then have "\<bar>n\<bar> \<noteq> 0" by simp
@@ -360,7 +343,8 @@
assumes "\<bar>of_int n\<bar> \<le> x"
shows "n = 0 \<or> 1 \<le> x"
proof (cases "n = 0")
-  case True then show ?thesis by simp
+  case True
+  then show ?thesis by simp
next
case False
then have "\<bar>n\<bar> \<noteq> 0" by simp
@@ -378,40 +362,34 @@

text \<open>Comparisons involving @{term of_int}.\<close>

-lemma of_int_eq_numeral_iff [iff]:
-   "of_int z = (numeral n :: 'a::ring_char_0)
-   \<longleftrightarrow> z = numeral n"
+lemma of_int_eq_numeral_iff [iff]: "of_int z = (numeral n :: 'a::ring_char_0) \<longleftrightarrow> z = numeral n"
using of_int_eq_iff by fastforce

lemma of_int_le_numeral_iff [simp]:
-   "of_int z \<le> (numeral n :: 'a::linordered_idom)
-   \<longleftrightarrow> z \<le> numeral n"
+  "of_int z \<le> (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z \<le> numeral n"
using of_int_le_iff [of z "numeral n"] by simp

lemma of_int_numeral_le_iff [simp]:
-   "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
+  "(numeral n :: 'a::linordered_idom) \<le> of_int z \<longleftrightarrow> numeral n \<le> z"
using of_int_le_iff [of "numeral n"] by simp

lemma of_int_less_numeral_iff [simp]:
-   "of_int z < (numeral n :: 'a::linordered_idom)
-   \<longleftrightarrow> z < numeral n"
+  "of_int z < (numeral n :: 'a::linordered_idom) \<longleftrightarrow> z < numeral n"
using of_int_less_iff [of z "numeral n"] by simp

lemma of_int_numeral_less_iff [simp]:
-   "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
+  "(numeral n :: 'a::linordered_idom) < of_int z \<longleftrightarrow> numeral n < z"
using of_int_less_iff [of "numeral n" z] by simp

-lemma of_nat_less_of_int_iff:
-  "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
+lemma of_nat_less_of_int_iff: "(of_nat n::'a::linordered_idom) < of_int x \<longleftrightarrow> int n < x"
by (metis of_int_of_nat_eq of_int_less_iff)

lemma of_int_eq_id [simp]: "of_int = id"
proof
-  fix z show "of_int z = id z"
-    by (cases z rule: int_diff_cases, simp)
+  show "of_int z = id z" for z
+    by (cases z rule: int_diff_cases) simp
qed

-
instance int :: no_top
apply standard
apply (rule_tac x="x + 1" in exI)
@@ -424,6 +402,7 @@
apply simp
done

+
subsection \<open>Magnitude of an Integer, as a Natural Number: \<open>nat\<close>\<close>

lift_definition nat :: "int \<Rightarrow> nat" is "\<lambda>(x, y). x - y"
@@ -435,23 +414,23 @@
lemma int_nat_eq [simp]: "int (nat z) = (if 0 \<le> z then z else 0)"
by transfer clarsimp

-corollary nat_0_le: "0 \<le> z ==> int (nat z) = z"
-by simp
+lemma nat_0_le: "0 \<le> z \<Longrightarrow> int (nat z) = z"
+  by simp

-lemma nat_le_0 [simp]: "z \<le> 0 ==> nat z = 0"
+lemma nat_le_0 [simp]: "z \<le> 0 \<Longrightarrow> nat z = 0"
by transfer clarsimp

-lemma nat_le_eq_zle: "0 < w | 0 \<le> z ==> (nat w \<le> nat z) = (w\<le>z)"
+lemma nat_le_eq_zle: "0 < w \<or> 0 \<le> z \<Longrightarrow> nat w \<le> nat z \<longleftrightarrow> w \<le> z"
by transfer (clarsimp, arith)

-text\<open>An alternative condition is @{term "0 \<le> w"}\<close>
-corollary nat_mono_iff: "0 < z ==> (nat w < nat z) = (w < z)"
-by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
+text \<open>An alternative condition is @{term "0 \<le> w"}.\<close>
+lemma nat_mono_iff: "0 < z \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
+  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])

-corollary nat_less_eq_zless: "0 \<le> w ==> (nat w < nat z) = (w<z)"
-by (simp add: nat_le_eq_zle linorder_not_le [symmetric])
+lemma nat_less_eq_zless: "0 \<le> w \<Longrightarrow> nat w < nat z \<longleftrightarrow> w < z"
+  by (simp add: nat_le_eq_zle linorder_not_le [symmetric])

-lemma zless_nat_conj [simp]: "(nat w < nat z) = (0 < z & w < z)"
+lemma zless_nat_conj [simp]: "nat w < nat z \<longleftrightarrow> 0 < z \<and> w < z"
by transfer (clarsimp, arith)

lemma nonneg_eq_int:
@@ -460,34 +439,28 @@
shows P
using assms by (blast dest: nat_0_le sym)

-lemma nat_eq_iff:
-  "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
+lemma nat_eq_iff: "nat w = m \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"

-corollary nat_eq_iff2:
-  "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
+lemma nat_eq_iff2: "m = nat w \<longleftrightarrow> (if 0 \<le> w then w = int m else m = 0)"
using nat_eq_iff [of w m] by auto

-lemma nat_0 [simp]:
-  "nat 0 = 0"
+lemma nat_0 [simp]: "nat 0 = 0"

-lemma nat_1 [simp]:
-  "nat 1 = Suc 0"
+lemma nat_1 [simp]: "nat 1 = Suc 0"

-lemma nat_numeral [simp]:
-  "nat (numeral k) = numeral k"
+lemma nat_numeral [simp]: "nat (numeral k) = numeral k"

-lemma nat_neg_numeral [simp]:
-  "nat (- numeral k) = 0"
+lemma nat_neg_numeral [simp]: "nat (- numeral k) = 0"
by simp

lemma nat_2: "nat 2 = Suc (Suc 0)"
by simp

-lemma nat_less_iff: "0 \<le> w ==> (nat w < m) = (w < of_nat m)"
+lemma nat_less_iff: "0 \<le> w \<Longrightarrow> nat w < m \<longleftrightarrow> w < of_nat m"
by transfer (clarsimp, arith)

lemma nat_le_iff: "nat x \<le> n \<longleftrightarrow> x \<le> int n"
@@ -496,133 +469,127 @@
lemma nat_mono: "x \<le> y \<Longrightarrow> nat x \<le> nat y"
by transfer auto

-lemma nat_0_iff[simp]: "nat(i::int) = 0 \<longleftrightarrow> i\<le>0"
+lemma nat_0_iff[simp]: "nat i = 0 \<longleftrightarrow> i \<le> 0"
+  for i :: int
by transfer clarsimp

-lemma int_eq_iff: "(of_nat m = z) = (m = nat z & 0 \<le> z)"
+lemma int_eq_iff: "of_nat m = z \<longleftrightarrow> m = nat z \<and> 0 \<le> z"
+  by (auto simp add: nat_eq_iff2)

-lemma zero_less_nat_eq [simp]: "(0 < nat z) = (0 < z)"
-by (insert zless_nat_conj [of 0], auto)
+lemma zero_less_nat_eq [simp]: "0 < nat z \<longleftrightarrow> 0 < z"
+  using zless_nat_conj [of 0] by auto

-  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
+lemma nat_add_distrib: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat (z + z') = nat z + nat z'"
by transfer clarsimp

-lemma nat_diff_distrib':
-  "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
+lemma nat_diff_distrib': "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> nat (x - y) = nat x - nat y"
by transfer clarsimp

-lemma nat_diff_distrib:
-  "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
+lemma nat_diff_distrib: "0 \<le> z' \<Longrightarrow> z' \<le> z \<Longrightarrow> nat (z - z') = nat z - nat z'"
by (rule nat_diff_distrib') auto

lemma nat_zminus_int [simp]: "nat (- int n) = 0"
by transfer simp

-lemma le_nat_iff:
-  "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
+lemma le_nat_iff: "k \<ge> 0 \<Longrightarrow> n \<le> nat k \<longleftrightarrow> int n \<le> k"
by transfer auto

-lemma zless_nat_eq_int_zless: "(m < nat z) = (int m < z)"
+lemma zless_nat_eq_int_zless: "m < nat z \<longleftrightarrow> int m < z"
by transfer (clarsimp simp add: less_diff_conv)

-context ring_1
-begin
-
-lemma of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
+lemma (in ring_1) of_nat_nat [simp]: "0 \<le> z \<Longrightarrow> of_nat (nat z) = of_int z"
by transfer (clarsimp simp add: of_nat_diff)

-end
-
-lemma diff_nat_numeral [simp]:
-  "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
+lemma diff_nat_numeral [simp]: "(numeral v :: nat) - numeral v' = nat (numeral v - numeral v')"
by (simp only: nat_diff_distrib' zero_le_numeral nat_numeral)

text \<open>For termination proofs:\<close>
-lemma measure_function_int[measure_function]: "is_measure (nat o abs)" ..
+lemma measure_function_int[measure_function]: "is_measure (nat \<circ> abs)" ..

-subsection\<open>Lemmas about the Function @{term of_nat} and Orderings\<close>
+subsection \<open>Lemmas about the Function @{term of_nat} and Orderings\<close>

lemma negative_zless_0: "- (int (Suc n)) < (0 :: int)"
-by (simp add: order_less_le del: of_nat_Suc)
+  by (simp add: order_less_le del: of_nat_Suc)

lemma negative_zless [iff]: "- (int (Suc n)) < int m"
-by (rule negative_zless_0 [THEN order_less_le_trans], simp)
+  by (rule negative_zless_0 [THEN order_less_le_trans], simp)

lemma negative_zle_0: "- int n \<le> 0"

lemma negative_zle [iff]: "- int n \<le> int m"
-by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])
+  by (rule order_trans [OF negative_zle_0 of_nat_0_le_iff])

-lemma not_zle_0_negative [simp]: "~ (0 \<le> - (int (Suc n)))"
-by (subst le_minus_iff, simp del: of_nat_Suc)
+lemma not_zle_0_negative [simp]: "\<not> 0 \<le> - int (Suc n)"
+  by (subst le_minus_iff) (simp del: of_nat_Suc)

-lemma int_zle_neg: "(int n \<le> - int m) = (n = 0 & m = 0)"
+lemma int_zle_neg: "int n \<le> - int m \<longleftrightarrow> n = 0 \<and> m = 0"
by transfer simp

-lemma not_int_zless_negative [simp]: "~ (int n < - int m)"
+lemma not_int_zless_negative [simp]: "\<not> int n < - int m"

-lemma negative_eq_positive [simp]: "(- int n = of_nat m) = (n = 0 & m = 0)"
-by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)
+lemma negative_eq_positive [simp]: "- int n = of_nat m \<longleftrightarrow> n = 0 \<and> m = 0"
+  by (force simp add: order_eq_iff [of "- of_nat n"] int_zle_neg)

-  "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)" (is "?P \<longleftrightarrow> ?Q")
+lemma zle_iff_zadd: "w \<le> z \<longleftrightarrow> (\<exists>n. z = w + int n)"
+  (is "?lhs \<longleftrightarrow> ?rhs")
proof
-  assume ?Q
-  then show ?P by auto
+  assume ?rhs
+  then show ?lhs by auto
next
-  assume ?P
+  assume ?lhs
then have "0 \<le> z - w" by simp
then obtain n where "z - w = int n"
using zero_le_imp_eq_int [of "z - w"] by blast
-  then have "z = w + int n"
-    by simp
-  then show ?Q ..
+  then have "z = w + int n" by simp
+  then show ?rhs ..
qed

lemma zadd_int_left: "int m + (int n + z) = int (m + n) + z"
-by simp
+  by simp

-text\<open>This version is proved for all ordered rings, not just integers!
-      It is proved here because attribute \<open>arith_split\<close> is not available
-      in theory \<open>Rings\<close>.
-      But is it really better than just rewriting with \<open>abs_if\<close>?\<close>
-lemma abs_split [arith_split, no_atp]:
-     "P \<bar>a::'a::linordered_idom\<bar> = ((0 \<le> a --> P a) & (a < 0 --> P(-a)))"
-by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)
+text \<open>
+  This version is proved for all ordered rings, not just integers!
+  It is proved here because attribute \<open>arith_split\<close> is not available
+  in theory \<open>Rings\<close>.
+  But is it really better than just rewriting with \<open>abs_if\<close>?
+\<close>
+lemma abs_split [arith_split, no_atp]: "P \<bar>a\<bar> \<longleftrightarrow> (0 \<le> a \<longrightarrow> P a) \<and> (a < 0 \<longrightarrow> P (- a))"
+  for a :: "'a::linordered_idom"
+  by (force dest: order_less_le_trans simp add: abs_if linorder_not_less)

lemma negD: "x < 0 \<Longrightarrow> \<exists>n. x = - (int (Suc n))"
-apply transfer
-apply clarsimp
-apply (rule_tac x="b - Suc a" in exI, arith)
-done
+  apply transfer
+  apply clarsimp
+  apply (rule_tac x="b - Suc a" in exI)
+  apply arith
+  done
+

subsection \<open>Cases and induction\<close>

-text\<open>Now we replace the case analysis rule by a more conventional one:
-whether an integer is negative or not.\<close>
+text \<open>
+  Now we replace the case analysis rule by a more conventional one:
+  whether an integer is negative or not.
+\<close>

-text\<open>This version is symmetric in the two subgoals.\<close>
-theorem int_cases2 [case_names nonneg nonpos, cases type: int]:
-  "\<lbrakk>!! n. z = int n \<Longrightarrow> P;  !! n. z = - (int n) \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
-apply (cases "z < 0")
-apply (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])
-done
+text \<open>This version is symmetric in the two subgoals.\<close>
+lemma int_cases2 [case_names nonneg nonpos, cases type: int]:
+  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int n) \<Longrightarrow> P) \<Longrightarrow> P"
+  by (cases "z < 0") (auto simp add: linorder_not_less dest!: negD nat_0_le [THEN sym])

-text\<open>This is the default, with a negative case.\<close>
-theorem int_cases [case_names nonneg neg, cases type: int]:
-  "[|!! n. z = int n ==> P;  !! n. z = - (int (Suc n)) ==> P |] ==> P"
-apply (cases "z < 0")
-apply (blast dest!: negD)
-apply (simp add: linorder_not_less del: of_nat_Suc)
-apply auto
-apply (blast dest: nat_0_le [THEN sym])
-done
+text \<open>This is the default, with a negative case.\<close>
+lemma int_cases [case_names nonneg neg, cases type: int]:
+  "(\<And>n. z = int n \<Longrightarrow> P) \<Longrightarrow> (\<And>n. z = - (int (Suc n)) \<Longrightarrow> P) \<Longrightarrow> P"
+  apply (cases "z < 0")
+   apply (blast dest!: negD)
+  apply (simp add: linorder_not_less del: of_nat_Suc)
+  apply auto
+  apply (blast dest: nat_0_le [THEN sym])
+  done

lemma int_cases3 [case_names zero pos neg]:
fixes k :: int
@@ -630,7 +597,8 @@
and "\<And>n. k = - int n \<Longrightarrow> n > 0 \<Longrightarrow> P"
shows "P"
proof (cases k "0::int" rule: linorder_cases)
-  case equal with assms(1) show P by simp
+  case equal
+  with assms(1) show P by simp
next
case greater
then have *: "nat k > 0" by simp
@@ -643,12 +611,13 @@
ultimately show P using assms(3) by blast
qed

-theorem int_of_nat_induct [case_names nonneg neg, induct type: int]:
-     "[|!! n. P (int n);  !!n. P (- (int (Suc n))) |] ==> P z"
+lemma int_of_nat_induct [case_names nonneg neg, induct type: int]:
+  "(\<And>n. P (int n)) \<Longrightarrow> (\<And>n. P (- (int (Suc n)))) \<Longrightarrow> P z"
by (cases z) auto

lemma nonneg_int_cases:
-  assumes "0 \<le> k" obtains n where "k = int n"
+  assumes "0 \<le> k"
+  obtains n where "k = int n"
using assms by (rule nonneg_eq_int)

lemma Let_numeral [simp]: "Let (numeral v) f = f (numeral v)"
@@ -679,50 +648,54 @@
text \<open>Preliminaries\<close>

lemma le_imp_0_less:
+  fixes z :: int
assumes le: "0 \<le> z"
-  shows "(0::int) < 1 + z"
+  shows "0 < 1 + z"
proof -
have "0 \<le> z" by fact
-  also have "... < z + 1" by (rule less_add_one)
-  also have "... = 1 + z" by (simp add: ac_simps)
+  also have "\<dots> < z + 1" by (rule less_add_one)
+  also have "\<dots> = 1 + z" by (simp add: ac_simps)
finally show "0 < 1 + z" .
qed

-lemma odd_less_0_iff:
-  "(1 + z + z < 0) = (z < (0::int))"
+lemma odd_less_0_iff: "1 + z + z < 0 \<longleftrightarrow> z < 0"
+  for z :: int
proof (cases z)
case (nonneg n)
-                             le_imp_0_less [THEN order_less_imp_le])
+  then show ?thesis
next
case (neg n)
-  thus ?thesis by (simp del: of_nat_Suc of_nat_add of_nat_1
+  then show ?thesis
+    by (simp del: of_nat_Suc of_nat_add of_nat_1
qed

+
subsubsection \<open>Comparisons, for Ordered Rings\<close>

lemmas double_eq_0_iff = double_zero

-lemma odd_nonzero:
-  "1 + z + z \<noteq> (0::int)"
+lemma odd_nonzero: "1 + z + z \<noteq> 0"
+  for z :: int
proof (cases z)
case (nonneg n)
-  thus ?thesis using  le_imp_0_less [OF le]
+  have le: "0 \<le> z + z"
+  then show ?thesis
+    using  le_imp_0_less [OF le] by (auto simp: add.assoc)
next
case (neg n)
show ?thesis
proof
assume eq: "1 + z + z = 0"
-    have "(0::int) < 1 + (int n + int n)"
+    have "0 < 1 + (int n + int n)"
-    also have "... = - (1 + z + z)"
+    also have "\<dots> = - (1 + z + z)"
-    also have "... = 0" by (simp add: eq)
+    also have "\<dots> = 0" by (simp add: eq)
finally have "0<0" ..
-    thus False by blast
+    then show False by blast
qed
qed

@@ -751,31 +724,31 @@
by (subst of_nat_numeral [symmetric], rule Ints_of_nat)

lemma Ints_add [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a + b \<in> \<int>"
-apply (rule range_eqI)
-done
+  apply (auto simp add: Ints_def)
+  apply (rule range_eqI)
+  done

lemma Ints_minus [simp]: "a \<in> \<int> \<Longrightarrow> -a \<in> \<int>"
-apply (rule range_eqI)
-apply (rule of_int_minus [symmetric])
-done
+  apply (auto simp add: Ints_def)
+  apply (rule range_eqI)
+  apply (rule of_int_minus [symmetric])
+  done

lemma Ints_diff [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a - b \<in> \<int>"
-apply (rule range_eqI)
-apply (rule of_int_diff [symmetric])
-done
+  apply (auto simp add: Ints_def)
+  apply (rule range_eqI)
+  apply (rule of_int_diff [symmetric])
+  done

lemma Ints_mult [simp]: "a \<in> \<int> \<Longrightarrow> b \<in> \<int> \<Longrightarrow> a * b \<in> \<int>"
-apply (rule range_eqI)
-apply (rule of_int_mult [symmetric])
-done
+  apply (auto simp add: Ints_def)
+  apply (rule range_eqI)
+  apply (rule of_int_mult [symmetric])
+  done

lemma Ints_power [simp]: "a \<in> \<int> \<Longrightarrow> a ^ n \<in> \<int>"
-by (induct n) simp_all
+  by (induct n) simp_all

lemma Ints_cases [cases set: Ints]:
assumes "q \<in> \<int>"
@@ -797,16 +770,22 @@

lemma Nats_altdef1: "\<nat> = {of_int n |n. n \<ge> 0}"
proof (intro subsetI equalityI)
-  fix x :: 'a assume "x \<in> {of_int n |n. n \<ge> 0}"
-  then obtain n where "x = of_int n" "n \<ge> 0" by (auto elim!: Ints_cases)
-  hence "x = of_nat (nat n)" by (subst of_nat_nat) simp_all
-  thus "x \<in> \<nat>" by simp
+  fix x :: 'a
+  assume "x \<in> {of_int n |n. n \<ge> 0}"
+  then obtain n where "x = of_int n" "n \<ge> 0"
+    by (auto elim!: Ints_cases)
+  then have "x = of_nat (nat n)"
+    by (subst of_nat_nat) simp_all
+  then show "x \<in> \<nat>"
+    by simp
next
-  fix x :: 'a assume "x \<in> \<nat>"
-  then obtain n where "x = of_nat n" by (auto elim!: Nats_cases)
-  hence "x = of_int (int n)" by simp
+  fix x :: 'a
+  assume "x \<in> \<nat>"
+  then obtain n where "x = of_nat n"
+    by (auto elim!: Nats_cases)
+  then have "x = of_int (int n)" by simp
also have "int n \<ge> 0" by simp
-  hence "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
+  then have "of_int (int n) \<in> {of_int n |n. n \<ge> 0}" by blast
finally show "x \<in> {of_int n |n. n \<ge> 0}" .
qed

@@ -814,45 +793,54 @@

lemma (in linordered_idom) Nats_altdef2: "\<nat> = {n \<in> \<int>. n \<ge> 0}"
proof (intro subsetI equalityI)
-  fix x :: 'a assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
-  then obtain n where "x = of_int n" "n \<ge> 0" by (auto elim!: Ints_cases)
-  hence "x = of_nat (nat n)" by (subst of_nat_nat) simp_all
-  thus "x \<in> \<nat>" by simp
+  fix x :: 'a
+  assume "x \<in> {n \<in> \<int>. n \<ge> 0}"
+  then obtain n where "x = of_int n" "n \<ge> 0"
+    by (auto elim!: Ints_cases)
+  then have "x = of_nat (nat n)"
+    by (subst of_nat_nat) simp_all
+  then show "x \<in> \<nat>"
+    by simp
qed (auto elim!: Nats_cases)

text \<open>The premise involving @{term Ints} prevents @{term "a = 1/2"}.\<close>

lemma Ints_double_eq_0_iff:
+  fixes a :: "'a::ring_char_0"
assumes in_Ints: "a \<in> \<int>"
-  shows "(a + a = 0) = (a = (0::'a::ring_char_0))"
+  shows "a + a = 0 \<longleftrightarrow> a = 0"
+    (is "?lhs \<longleftrightarrow> ?rhs")
proof -
-  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
+  from in_Ints have "a \<in> range of_int"
+    unfolding Ints_def [symmetric] .
then obtain z where a: "a = of_int z" ..
show ?thesis
proof
-    assume "a = 0"
-    thus "a + a = 0" by simp
+    assume ?rhs
+    then show ?lhs by simp
next
-    assume eq: "a + a = 0"
-    hence "of_int (z + z) = (of_int 0 :: 'a)" by (simp add: a)
-    hence "z + z = 0" by (simp only: of_int_eq_iff)
-    hence "z = 0" by (simp only: double_eq_0_iff)
-    thus "a = 0" by (simp add: a)
+    assume ?lhs
+    with a have "of_int (z + z) = (of_int 0 :: 'a)" by simp
+    then have "z + z = 0" by (simp only: of_int_eq_iff)
+    then have "z = 0" by (simp only: double_eq_0_iff)
+    with a show ?rhs by simp
qed
qed

lemma Ints_odd_nonzero:
+  fixes a :: "'a::ring_char_0"
assumes in_Ints: "a \<in> \<int>"
-  shows "1 + a + a \<noteq> (0::'a::ring_char_0)"
+  shows "1 + a + a \<noteq> 0"
proof -
-  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
+  from in_Ints have "a \<in> range of_int"
+    unfolding Ints_def [symmetric] .
then obtain z where a: "a = of_int z" ..
show ?thesis
proof
-    assume eq: "1 + a + a = 0"
-    hence "of_int (1 + z + z) = (of_int 0 :: 'a)" by (simp add: a)
-    hence "1 + z + z = 0" by (simp only: of_int_eq_iff)
+    assume "1 + a + a = 0"
+    with a have "of_int (1 + z + z) = (of_int 0 :: 'a)" by simp
+    then have "1 + z + z = 0" by (simp only: of_int_eq_iff)
with odd_nonzero show False by blast
qed
qed
@@ -861,15 +849,19 @@
using of_nat_in_Nats [of "numeral w"] by simp

lemma Ints_odd_less_0:
+  fixes a :: "'a::linordered_idom"
assumes in_Ints: "a \<in> \<int>"
-  shows "(1 + a + a < 0) = (a < (0::'a::linordered_idom))"
+  shows "1 + a + a < 0 \<longleftrightarrow> a < 0"
proof -
-  from in_Ints have "a \<in> range of_int" unfolding Ints_def [symmetric] .
+  from in_Ints have "a \<in> range of_int"
+    unfolding Ints_def [symmetric] .
then obtain z where a: "a = of_int z" ..
-  hence "((1::'a) + a + a < 0) = (of_int (1 + z + z) < (of_int 0 :: 'a))"
+  with a have "1 + a + a < 0 \<longleftrightarrow> of_int (1 + z + z) < (of_int 0 :: 'a)"
+    by simp
+  also have "\<dots> \<longleftrightarrow> z < 0"
+    by (simp only: of_int_less_iff odd_less_0_iff)
+  also have "\<dots> \<longleftrightarrow> a < 0"
-  also have "... = (z < 0)" by (simp only: of_int_less_iff odd_less_0_iff)
-  also have "... = (a < 0)" by (simp add: a)
finally show ?thesis .
qed

@@ -877,24 +869,16 @@
subsection \<open>@{term setsum} and @{term setprod}\<close>

lemma of_nat_setsum [simp]: "of_nat (setsum f A) = (\<Sum>x\<in>A. of_nat(f x))"
-  apply (cases "finite A")
-  apply (erule finite_induct, auto)
-  done
+  by (induct A rule: infinite_finite_induct) auto

lemma of_int_setsum [simp]: "of_int (setsum f A) = (\<Sum>x\<in>A. of_int(f x))"
-  apply (cases "finite A")
-  apply (erule finite_induct, auto)
-  done
+  by (induct A rule: infinite_finite_induct) auto

lemma of_nat_setprod [simp]: "of_nat (setprod f A) = (\<Prod>x\<in>A. of_nat(f x))"
-  apply (cases "finite A")
-  apply (erule finite_induct, auto simp add: of_nat_mult)
-  done
+  by (induct A rule: infinite_finite_induct) auto

lemma of_int_setprod [simp]: "of_int (setprod f A) = (\<Prod>x\<in>A. of_int(f x))"
-  apply (cases "finite A")
-  apply (erule finite_induct, auto)
-  done
+  by (induct A rule: infinite_finite_induct) auto

lemmas int_setsum = of_nat_setsum [where 'a=int]
lemmas int_setprod = of_nat_setprod [where 'a=int]
@@ -905,6 +889,7 @@
lemmas zle_int = of_nat_le_iff [where 'a=int]
lemmas int_int_eq = of_nat_eq_iff [where 'a=int]

+
subsection \<open>Setting up simplification procedures\<close>

lemmas of_int_simps =
@@ -913,53 +898,64 @@
ML_file "Tools/int_arith.ML"
declaration \<open>K Int_Arith.setup\<close>

-simproc_setup fast_arith ("(m::'a::linordered_idom) < n" |
-  "(m::'a::linordered_idom) \<le> n" |
-  "(m::'a::linordered_idom) = n") =
+simproc_setup fast_arith
+  ("(m::'a::linordered_idom) < n" |
+    "(m::'a::linordered_idom) \<le> n" |
+    "(m::'a::linordered_idom) = n") =
\<open>K Lin_Arith.simproc\<close>

subsection\<open>More Inequality Reasoning\<close>

-lemma zless_add1_eq: "(w < z + (1::int)) = (w<z | w=z)"
-by arith
+lemma zless_add1_eq: "w < z + 1 \<longleftrightarrow> w < z \<or> w = z"
+  for w z :: int
+  by arith

-lemma add1_zle_eq: "(w + (1::int) \<le> z) = (w<z)"
-by arith
+lemma add1_zle_eq: "w + 1 \<le> z \<longleftrightarrow> w < z"
+  for w z :: int
+  by arith

-lemma zle_diff1_eq [simp]: "(w \<le> z - (1::int)) = (w<z)"
-by arith
+lemma zle_diff1_eq [simp]: "w \<le> z - 1 \<longleftrightarrow> w < z"
+  for w z :: int
+  by arith

-lemma zle_add1_eq_le [simp]: "(w < z + (1::int)) = (w\<le>z)"
-by arith
+lemma zle_add1_eq_le [simp]: "w < z + 1 \<longleftrightarrow> w \<le> z"
+  for w z :: int
+  by arith

-lemma int_one_le_iff_zero_less: "((1::int) \<le> z) = (0 < z)"
-by arith
+lemma int_one_le_iff_zero_less: "1 \<le> z \<longleftrightarrow> 0 < z"
+  for z :: int
+  by arith

-subsection\<open>The functions @{term nat} and @{term int}\<close>
+subsection \<open>The functions @{term nat} and @{term int}\<close>

-text\<open>Simplify the term @{term "w + - z"}\<close>
+text \<open>Simplify the term @{term "w + - z"}.\<close>

-lemma one_less_nat_eq [simp]: "(Suc 0 < nat z) = (1 < z)"
+lemma one_less_nat_eq [simp]: "Suc 0 < nat z \<longleftrightarrow> 1 < z"
using zless_nat_conj [of 1 z] by auto

-text\<open>This simplifies expressions of the form @{term "int n = z"} where
-      z is an integer literal.\<close>
+text \<open>
+  This simplifies expressions of the form @{term "int n = z"} where
+  \<open>z\<close> is an integer literal.
+\<close>
lemmas int_eq_iff_numeral [simp] = int_eq_iff [of _ "numeral v"] for v

-lemma split_nat [arith_split]:
-  "P(nat(i::int)) = ((\<forall>n. i = int n \<longrightarrow> P n) & (i < 0 \<longrightarrow> P 0))"
-  (is "?P = (?L & ?R)")
+lemma split_nat [arith_split]: "P (nat i) = ((\<forall>n. i = int n \<longrightarrow> P n) \<and> (i < 0 \<longrightarrow> P 0))"
+  (is "?P = (?L \<and> ?R)")
+  for i :: int
proof (cases "i < 0")
-  case True thus ?thesis by auto
+  case True
+  then show ?thesis by auto
next
case False
have "?P = ?L"
proof
-    assume ?P thus ?L using False by clarsimp
+    assume ?P
+    then show ?L using False by auto
next
-    assume ?L thus ?P using False by simp
+    assume ?L
+    then show ?P using False by simp
qed
with False show ?thesis by simp
qed
@@ -976,11 +972,13 @@
lemma of_int_of_nat [nitpick_simp]:
"of_int k = (if k < 0 then - of_nat (nat (- k)) else of_nat (nat k))"
proof (cases "k < 0")
-  case True then have "0 \<le> - k" by simp
+  case True
+  then have "0 \<le> - k" by simp
then have "of_nat (nat (- k)) = of_int (- k)" by (rule of_nat_nat)
with True show ?thesis by simp
next
-  case False then show ?thesis by (simp add: not_less of_nat_nat)
+  case False
+  then show ?thesis by (simp add: not_less)
qed

end
@@ -990,39 +988,39 @@
assumes "0 \<le> z"
shows "nat (z * z') = nat z * nat z'"
proof (cases "0 \<le> z'")
-  case False with assms have "z * z' \<le> 0"
+  case False
+  with assms have "z * z' \<le> 0"
then have "nat (z * z') = 0" by simp
moreover from False have "nat z' = 0" by simp
ultimately show ?thesis by simp
next
-  case True with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
+  case True
+  with assms have ge_0: "z * z' \<ge> 0" by (simp add: zero_le_mult_iff)
show ?thesis
by (rule injD [of "of_nat :: nat \<Rightarrow> int", OF inj_of_nat])
(simp only: of_nat_mult of_nat_nat [OF True]
of_nat_nat [OF assms] of_nat_nat [OF ge_0], simp)
qed

-lemma nat_mult_distrib_neg: "z \<le> (0::int) ==> nat(z*z') = nat(-z) * nat(-z')"
-apply (rule trans)
-apply (rule_tac  nat_mult_distrib, auto)
-done
+lemma nat_mult_distrib_neg: "z \<le> 0 \<Longrightarrow> nat (z * z') = nat (- z) * nat (- z')"
+  for z z' :: int
+  apply (rule trans)
+   apply (rule_tac  nat_mult_distrib)
+   apply auto
+  done

lemma nat_abs_mult_distrib: "nat \<bar>w * z\<bar> = nat \<bar>w\<bar> * nat \<bar>z\<bar>"
-apply (cases "z=0 | w=0")
-apply (auto simp add: abs_if nat_mult_distrib [symmetric]
-                      nat_mult_distrib_neg [symmetric] mult_less_0_iff)
-done
+  by (cases "z = 0 \<or> w = 0")
+    (auto simp add: abs_if nat_mult_distrib [symmetric]
+      nat_mult_distrib_neg [symmetric] mult_less_0_iff)

-lemma int_in_range_abs [simp]:
-  "int n \<in> range abs"
+lemma int_in_range_abs [simp]: "int n \<in> range abs"
proof (rule range_eqI)
-  show "int n = \<bar>int n\<bar>"
-    by simp
+  show "int n = \<bar>int n\<bar>" by simp
qed

-lemma range_abs_Nats [simp]:
-  "range abs = (\<nat> :: int set)"
+lemma range_abs_Nats [simp]: "range abs = (\<nat> :: int set)"
proof -
have "\<bar>k\<bar> \<in> \<nat>" for k :: int
by (cases k) simp_all
@@ -1031,130 +1029,123 @@
ultimately show ?thesis by blast
qed

-lemma Suc_nat_eq_nat_zadd1: "(0::int) <= z ==> Suc (nat z) = nat (1 + z)"
-apply (rule sym)
-done
+lemma Suc_nat_eq_nat_zadd1: "0 \<le> z \<Longrightarrow> Suc (nat z) = nat (1 + z)"
+  for z :: int
+  by (rule sym) (simp add: nat_eq_iff)

lemma diff_nat_eq_if:
-     "nat z - nat z' =
-        (if z' < 0 then nat z
-         else let d = z-z' in
-              if d < 0 then 0 else nat d)"
-by (simp add: Let_def nat_diff_distrib [symmetric])
+  "nat z - nat z' =
+    (if z' < 0 then nat z
+     else
+      let d = z - z'
+      in if d < 0 then 0 else nat d)"
+  by (simp add: Let_def nat_diff_distrib [symmetric])

-lemma nat_numeral_diff_1 [simp]:
-  "numeral v - (1::nat) = nat (numeral v - 1)"
+lemma nat_numeral_diff_1 [simp]: "numeral v - (1::nat) = nat (numeral v - 1)"
using diff_nat_numeral [of v Num.One] by simp

-subsection "Induction principles for int"
+subsection \<open>Induction principles for int\<close>

-text\<open>Well-founded segments of the integers\<close>
+text \<open>Well-founded segments of the integers.\<close>

-definition
-  int_ge_less_than  ::  "int => (int * int) set"
-where
-  "int_ge_less_than d = {(z',z). d \<le> z' & z' < z}"
+definition int_ge_less_than :: "int \<Rightarrow> (int \<times> int) set"
+  where "int_ge_less_than d = {(z', z). d \<le> z' \<and> z' < z}"

-theorem wf_int_ge_less_than: "wf (int_ge_less_than d)"
+lemma wf_int_ge_less_than: "wf (int_ge_less_than d)"
proof -
-  have "int_ge_less_than d \<subseteq> measure (%z. nat (z-d))"
+  have "int_ge_less_than d \<subseteq> measure (\<lambda>z. nat (z - d))"
-  thus ?thesis
+  then show ?thesis
by (rule wf_subset [OF wf_measure])
qed

-text\<open>This variant looks odd, but is typical of the relations suggested
-by RankFinder.\<close>
+text \<open>
+  This variant looks odd, but is typical of the relations suggested
+  by RankFinder.\<close>

-definition
-  int_ge_less_than2 ::  "int => (int * int) set"
-where
-  "int_ge_less_than2 d = {(z',z). d \<le> z & z' < z}"
+definition int_ge_less_than2 :: "int \<Rightarrow> (int \<times> int) set"
+  where "int_ge_less_than2 d = {(z',z). d \<le> z \<and> z' < z}"

-theorem wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
+lemma wf_int_ge_less_than2: "wf (int_ge_less_than2 d)"
proof -
-  have "int_ge_less_than2 d \<subseteq> measure (%z. nat (1+z-d))"
+  have "int_ge_less_than2 d \<subseteq> measure (\<lambda>z. nat (1 + z - d))"
-  thus ?thesis
+  then show ?thesis
by (rule wf_subset [OF wf_measure])
qed

(* `set:int': dummy construction *)
theorem int_ge_induct [case_names base step, induct set: int]:
fixes i :: int
-  assumes ge: "k \<le> i" and
-    base: "P k" and
-    step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
+  assumes ge: "k \<le> i"
+    and base: "P k"
+    and step: "\<And>i. k \<le> i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
shows "P i"
proof -
-  { fix n
-    have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i"
-    proof (induct n)
-      case 0
-      hence "i = k" by arith
-      thus "P i" using base by simp
-    next
-      case (Suc n)
-      then have "n = nat((i - 1) - k)" by arith
-      moreover
-      have ki1: "k \<le> i - 1" using Suc.prems by arith
-      ultimately
-      have "P (i - 1)" by (rule Suc.hyps)
-      from step [OF ki1 this] show ?case by simp
-    qed
-  }
+  have "\<And>i::int. n = nat (i - k) \<Longrightarrow> k \<le> i \<Longrightarrow> P i" for n
+  proof (induct n)
+    case 0
+    then have "i = k" by arith
+    with base show "P i" by simp
+  next
+    case (Suc n)
+    then have "n = nat ((i - 1) - k)" by arith
+    moreover have k: "k \<le> i - 1" using Suc.prems by arith
+    ultimately have "P (i - 1)" by (rule Suc.hyps)
+    from step [OF k this] show ?case by simp
+  qed
with ge show ?thesis by fast
qed

(* `set:int': dummy construction *)
theorem int_gr_induct [case_names base step, induct set: int]:
-  assumes gr: "k < (i::int)" and
-        base: "P(k+1)" and
-        step: "\<And>i. \<lbrakk>k < i; P i\<rbrakk> \<Longrightarrow> P(i+1)"
+  fixes i k :: int
+  assumes gr: "k < i"
+    and base: "P (k + 1)"
+    and step: "\<And>i. k < i \<Longrightarrow> P i \<Longrightarrow> P (i + 1)"
shows "P i"
-apply(rule int_ge_induct[of "k + 1"])
+  apply (rule int_ge_induct[of "k + 1"])
using gr apply arith
- apply(rule base)
-apply (rule step, simp+)
-done
+   apply (rule base)
+  apply (rule step)
+   apply simp_all
+  done

theorem int_le_induct [consumes 1, case_names base step]:
-  assumes le: "i \<le> (k::int)" and
-        base: "P(k)" and
-        step: "\<And>i. \<lbrakk>i \<le> k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
+  fixes i k :: int
+  assumes le: "i \<le> k"
+    and base: "P k"
+    and step: "\<And>i. i \<le> k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
shows "P i"
proof -
-  { fix n
-    have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i"
-    proof (induct n)
-      case 0
-      hence "i = k" by arith
-      thus "P i" using base by simp
-    next
-      case (Suc n)
-      hence "n = nat (k - (i + 1))" by arith
-      moreover
-      have ki1: "i + 1 \<le> k" using Suc.prems by arith
-      ultimately
-      have "P (i + 1)" by(rule Suc.hyps)
-      from step[OF ki1 this] show ?case by simp
-    qed
-  }
+  have "\<And>i::int. n = nat(k-i) \<Longrightarrow> i \<le> k \<Longrightarrow> P i" for n
+  proof (induct n)
+    case 0
+    then have "i = k" by arith
+    with base show "P i" by simp
+  next
+    case (Suc n)
+    then have "n = nat (k - (i + 1))" by arith
+    moreover have k: "i + 1 \<le> k" using Suc.prems by arith
+    ultimately have "P (i + 1)" by (rule Suc.hyps)
+    from step[OF k this] show ?case by simp
+  qed
with le show ?thesis by fast
qed

theorem int_less_induct [consumes 1, case_names base step]:
-  assumes less: "(i::int) < k" and
-        base: "P(k - 1)" and
-        step: "\<And>i. \<lbrakk>i < k; P i\<rbrakk> \<Longrightarrow> P(i - 1)"
+  fixes i k :: int
+  assumes less: "i < k"
+    and base: "P (k - 1)"
+    and step: "\<And>i. i < k \<Longrightarrow> P i \<Longrightarrow> P (i - 1)"
shows "P i"
-apply(rule int_le_induct[of _ "k - 1"])
+  apply (rule int_le_induct[of _ "k - 1"])
using less apply arith
- apply(rule base)
-apply (rule step, simp+)
-done
+   apply (rule base)
+  apply (rule step)
+   apply simp_all
+  done

theorem int_induct [case_names base step1 step2]:
fixes k :: int
@@ -1167,88 +1158,93 @@
then show ?thesis
proof
assume "i \<ge> k"
-    then show ?thesis using base
-      by (rule int_ge_induct) (fact step1)
+    then show ?thesis
+      using base by (rule int_ge_induct) (fact step1)
next
assume "i \<le> k"
-    then show ?thesis using base
-      by (rule int_le_induct) (fact step2)
+    then show ?thesis
+      using base by (rule int_le_induct) (fact step2)
qed
qed

-subsection\<open>Intermediate value theorems\<close>
+
+subsection \<open>Intermediate value theorems\<close>

-lemma int_val_lemma:
-     "(\<forall>i<n::nat. \<bar>f(i+1) - f i\<bar> \<le> 1) -->
-      f 0 \<le> k --> k \<le> f n --> (\<exists>i \<le> n. f i = (k::int))"
-unfolding One_nat_def
-apply (induct n)
-apply simp
-apply (intro strip)
-apply (erule impE, simp)
-apply (erule_tac x = n in allE, simp)
-apply (case_tac "k = f (Suc n)")
-apply force
-apply (erule impE)
- apply (simp add: abs_if split: if_split_asm)
-apply (blast intro: le_SucI)
-done
+lemma int_val_lemma: "(\<forall>i<n. \<bar>f (i + 1) - f i\<bar> \<le> 1) \<longrightarrow> f 0 \<le> k \<longrightarrow> k \<le> f n \<longrightarrow> (\<exists>i \<le> n. f i = k)"
+  for n :: nat and k :: int
+  unfolding One_nat_def
+  apply (induct n)
+   apply simp
+  apply (intro strip)
+  apply (erule impE)
+   apply simp
+  apply (erule_tac x = n in allE)
+  apply simp
+  apply (case_tac "k = f (Suc n)")
+   apply force
+  apply (erule impE)
+   apply (simp add: abs_if split: if_split_asm)
+  apply (blast intro: le_SucI)
+  done

lemmas nat0_intermed_int_val = int_val_lemma [rule_format (no_asm)]

lemma nat_intermed_int_val:
-     "[| \<forall>i. m \<le> i & i < n --> \<bar>f(i + 1::nat) - f i\<bar> \<le> 1; m < n;
-         f m \<le> k; k \<le> f n |] ==> ? i. m \<le> i & i \<le> n & f i = (k::int)"
-apply (cut_tac n = "n-m" and f = "%i. f (i+m) " and k = k
-       in int_val_lemma)
-unfolding One_nat_def
-apply simp
-apply (erule exE)
-apply (rule_tac x = "i+m" in exI, arith)
-done
+  "\<forall>i. m \<le> i \<and> i < n \<longrightarrow> \<bar>f (i + 1) - f i\<bar> \<le> 1 \<Longrightarrow> m < n \<Longrightarrow>
+    f m \<le> k \<Longrightarrow> k \<le> f n \<Longrightarrow> \<exists>i. m \<le> i \<and> i \<le> n \<and> f i = k"
+    for f :: "nat \<Rightarrow> int" and k :: int
+  apply (cut_tac n = "n-m" and f = "\<lambda>i. f (i + m)" and k = k in int_val_lemma)
+  unfolding One_nat_def
+  apply simp
+  apply (erule exE)
+  apply (rule_tac x = "i+m" in exI)
+  apply arith
+  done

-subsection\<open>Products and 1, by T. M. Rasmussen\<close>
+subsection \<open>Products and 1, by T. M. Rasmussen\<close>

lemma abs_zmult_eq_1:
+  fixes m n :: int
assumes mn: "\<bar>m * n\<bar> = 1"
-  shows "\<bar>m\<bar> = (1::int)"
+  shows "\<bar>m\<bar> = 1"
proof -
-  have 0: "m \<noteq> 0 & n \<noteq> 0" using mn
-    by auto
-  have "~ (2 \<le> \<bar>m\<bar>)"
+  from mn have 0: "m \<noteq> 0" "n \<noteq> 0" by auto
+  have "\<not> 2 \<le> \<bar>m\<bar>"
proof
assume "2 \<le> \<bar>m\<bar>"
-    hence "2*\<bar>n\<bar> \<le> \<bar>m\<bar>*\<bar>n\<bar>"
-      by (simp add: mult_mono 0)
-    also have "... = \<bar>m*n\<bar>"
-    also have "... = 1"
-    finally have "2*\<bar>n\<bar> \<le> 1" .
-    thus "False" using 0
-      by arith
+    then have "2 * \<bar>n\<bar> \<le> \<bar>m\<bar> * \<bar>n\<bar>" by (simp add: mult_mono 0)
+    also have "\<dots> = \<bar>m * n\<bar>" by (simp add: abs_mult)
+    also from mn have "\<dots> = 1" by simp
+    finally have "2 * \<bar>n\<bar> \<le> 1" .
+    with 0 show "False" by arith
qed
-  thus ?thesis using 0
-    by auto
+  with 0 show ?thesis by auto
qed

-lemma pos_zmult_eq_1_iff_lemma: "(m * n = 1) ==> m = (1::int) | m = -1"
-by (insert abs_zmult_eq_1 [of m n], arith)
+lemma pos_zmult_eq_1_iff_lemma: "m * n = 1 \<Longrightarrow> m = 1 \<or> m = - 1"
+  for m n :: int
+  using abs_zmult_eq_1 [of m n] by arith

lemma pos_zmult_eq_1_iff:
-  assumes "0 < (m::int)" shows "(m * n = 1) = (m = 1 & n = 1)"
+  fixes m n :: int
+  assumes "0 < m"
+  shows "m * n = 1 \<longleftrightarrow> m = 1 \<and> n = 1"
proof -
-  from assms have "m * n = 1 ==> m = 1" by (auto dest: pos_zmult_eq_1_iff_lemma)
-  thus ?thesis by (auto dest: pos_zmult_eq_1_iff_lemma)
+  from assms have "m * n = 1 \<Longrightarrow> m = 1"
+    by (auto dest: pos_zmult_eq_1_iff_lemma)
+  then show ?thesis
+    by (auto dest: pos_zmult_eq_1_iff_lemma)
qed

-lemma zmult_eq_1_iff: "(m*n = (1::int)) = ((m = 1 & n = 1) | (m = -1 & n = -1))"
-apply (rule iffI)
- apply (frule pos_zmult_eq_1_iff_lemma)
- apply (simp add: mult.commute [of m])
- apply (frule pos_zmult_eq_1_iff_lemma, auto)
-done
+lemma zmult_eq_1_iff: "m * n = 1 \<longleftrightarrow> (m = 1 \<and> n = 1) \<or> (m = - 1 \<and> n = - 1)"
+  for m n :: int
+  apply (rule iffI)
+   apply (frule pos_zmult_eq_1_iff_lemma)
+   apply (simp add: mult.commute [of m])
+   apply (frule pos_zmult_eq_1_iff_lemma)
+   apply auto
+  done

lemma infinite_UNIV_int: "\<not> finite (UNIV::int set)"
proof
@@ -1264,16 +1260,16 @@

subsection \<open>Further theorems on numerals\<close>

-subsubsection\<open>Special Simplification for Constants\<close>
+subsubsection \<open>Special Simplification for Constants\<close>

-text\<open>These distributive laws move literals inside sums and differences.\<close>
+text \<open>These distributive laws move literals inside sums and differences.\<close>

lemmas distrib_right_numeral [simp] = distrib_right [of _ _ "numeral v"] for v
lemmas distrib_left_numeral [simp] = distrib_left [of "numeral v"] for v
lemmas left_diff_distrib_numeral [simp] = left_diff_distrib [of _ _ "numeral v"] for v
lemmas right_diff_distrib_numeral [simp] = right_diff_distrib [of "numeral v"] for v

-text\<open>These are actually for fields, like real: but where else to put them?\<close>
+text \<open>These are actually for fields, like real: but where else to put them?\<close>

lemmas zero_less_divide_iff_numeral [simp, no_atp] = zero_less_divide_iff [of "numeral w"] for w
lemmas divide_less_0_iff_numeral [simp, no_atp] = divide_less_0_iff [of "numeral w"] for w
@@ -1291,7 +1287,7 @@
inverse_eq_divide [of "- numeral w"] for w

text \<open>These laws simplify inequalities, moving unary minus from a term
-into the literal.\<close>
+  into the literal.\<close>

lemmas equation_minus_iff_numeral [no_atp] =
equation_minus_iff [of "numeral v"] for v
@@ -1311,7 +1307,7 @@
lemmas minus_less_iff_numeral [no_atp] =
minus_less_iff [of _ "numeral v"] for v

-\<comment> \<open>FIXME maybe simproc\<close>
+(* FIXME maybe simproc *)

text \<open>Cancellation of constant factors in comparisons (\<open><\<close> and \<open>\<le>\<close>)\<close>
@@ -1351,9 +1347,9 @@
divide_eq_eq [of _ "- numeral w"] for w

-subsubsection\<open>Optional Simplification Rules Involving Constants\<close>
+subsubsection \<open>Optional Simplification Rules Involving Constants\<close>

-text\<open>Simplify quotients that are compared with a literal constant.\<close>
+text \<open>Simplify quotients that are compared with a literal constant.\<close>

lemmas le_divide_eq_numeral [divide_const_simps] =
le_divide_eq [of "numeral w"]
@@ -1380,37 +1376,44 @@
divide_eq_eq [of _ _ "- numeral w"] for w

-text\<open>Not good as automatic simprules because they cause case splits.\<close>
-lemmas [divide_const_simps] = le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1
+text \<open>Not good as automatic simprules because they cause case splits.\<close>
+lemmas [divide_const_simps] =
+  le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1

subsection \<open>The divides relation\<close>

-lemma zdvd_antisym_nonneg:
-    "0 <= m ==> 0 <= n ==> m dvd n ==> n dvd m ==> m = (n::int)"
-  apply (simp add: dvd_def, auto)
-  apply (auto simp add: mult.assoc zero_le_mult_iff zmult_eq_1_iff)
-  done
+lemma zdvd_antisym_nonneg: "0 \<le> m \<Longrightarrow> 0 \<le> n \<Longrightarrow> m dvd n \<Longrightarrow> n dvd m \<Longrightarrow> m = n"
+  for m n :: int
+  by (auto simp add: dvd_def mult.assoc zero_le_mult_iff zmult_eq_1_iff)

-lemma zdvd_antisym_abs: assumes "(a::int) dvd b" and "b dvd a"
+lemma zdvd_antisym_abs:
+  fixes a b :: int
+  assumes "a dvd b" and "b dvd a"
shows "\<bar>a\<bar> = \<bar>b\<bar>"
-proof cases
-  assume "a = 0" with assms show ?thesis by simp
+proof (cases "a = 0")
+  case True
+  with assms show ?thesis by simp
next
-  assume "a \<noteq> 0"
-  from \<open>a dvd b\<close> obtain k where k:"b = a*k" unfolding dvd_def by blast
-  from \<open>b dvd a\<close> obtain k' where k':"a = b*k'" unfolding dvd_def by blast
-  from k k' have "a = a*k*k'" by simp
-  with mult_cancel_left1[where c="a" and b="k*k'"]
-  have kk':"k*k' = 1" using \<open>a\<noteq>0\<close> by (simp add: mult.assoc)
-  hence "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1" by (simp add: zmult_eq_1_iff)
-  thus ?thesis using k k' by auto
+  case False
+  from \<open>a dvd b\<close> obtain k where k: "b = a * k"
+    unfolding dvd_def by blast
+  from \<open>b dvd a\<close> obtain k' where k': "a = b * k'"
+    unfolding dvd_def by blast
+  from k k' have "a = a * k * k'" by simp
+  with mult_cancel_left1[where c="a" and b="k*k'"] have kk': "k * k' = 1"
+    using \<open>a \<noteq> 0\<close> by (simp add: mult.assoc)
+  then have "k = 1 \<and> k' = 1 \<or> k = -1 \<and> k' = -1"
+  with k k' show ?thesis by auto
qed

-lemma zdvd_zdiffD: "k dvd m - n ==> k dvd n ==> k dvd (m::int)"
+lemma zdvd_zdiffD: "k dvd m - n \<Longrightarrow> k dvd n \<Longrightarrow> k dvd m"
+  for k m n :: int
using dvd_add_right_iff [of k "- n" m] by simp

-lemma zdvd_reduce: "(k dvd n + k * m) = (k dvd (n::int))"
+lemma zdvd_reduce: "k dvd n + k * m \<longleftrightarrow> k dvd n"
+  for k m n :: int

lemma dvd_imp_le_int:
@@ -1438,167 +1441,184 @@
with \<open>0 < n\<close> \<open>0 < k\<close> show False unfolding mult_less_cancel_left by auto
qed

-lemma zdvd_mult_cancel: assumes d:"k * m dvd k * n" and kz:"k \<noteq> (0::int)"
+lemma zdvd_mult_cancel:
+  fixes k m n :: int
+  assumes d: "k * m dvd k * n"
+    and "k \<noteq> 0"
shows "m dvd n"
-proof-
-  from d obtain h where h: "k*n = k*m * h" unfolding dvd_def by blast
-  {assume "n \<noteq> m*h" hence "k* n \<noteq> k* (m*h)" using kz by simp
-    with h have False by (simp add: mult.assoc)}
-  hence "n = m * h" by blast
-  thus ?thesis by simp
+proof -
+  from d obtain h where h: "k * n = k * m * h"
+    unfolding dvd_def by blast
+  have "n = m * h"
+  proof (rule ccontr)
+    assume "\<not> ?thesis"
+    with \<open>k \<noteq> 0\<close> have "k * n \<noteq> k * (m * h)" by simp
+    with h show False
+  qed
+  then show ?thesis by simp
qed

-theorem zdvd_int: "(x dvd y) = (int x dvd int y)"
+theorem zdvd_int: "x dvd y \<longleftrightarrow> int x dvd int y"
proof -
-  have "\<And>k. int y = int x * k \<Longrightarrow> x dvd y"
-  proof -
-    fix k
-    assume A: "int y = int x * k"
-    then show "x dvd y"
-    proof (cases k)
-      case (nonneg n)
-      with A have "y = x * n" by (simp del: of_nat_mult add: of_nat_mult [symmetric])
-      then show ?thesis ..
-    next
-      case (neg n)
-      with A have "int y = int x * (- int (Suc n))" by simp
-      also have "\<dots> = - (int x * int (Suc n))" by (simp only: mult_minus_right)
-      also have "\<dots> = - int (x * Suc n)" by (simp only: of_nat_mult [symmetric])
-      finally have "- int (x * Suc n) = int y" ..
-      then show ?thesis by (simp only: negative_eq_positive) auto
-    qed
+  have "x dvd y" if "int y = int x * k" for k
+  proof (cases k)
+    case (nonneg n)
+    with that have "y = x * n"
+      by (simp del: of_nat_mult add: of_nat_mult [symmetric])
+    then show ?thesis ..
+  next
+    case (neg n)
+    with that have "int y = int x * (- int (Suc n))"
+      by simp
+    also have "\<dots> = - (int x * int (Suc n))"
+      by (simp only: mult_minus_right)
+    also have "\<dots> = - int (x * Suc n)"
+      by (simp only: of_nat_mult [symmetric])
+    finally have "- int (x * Suc n) = int y" ..
+    then show ?thesis
+      by (simp only: negative_eq_positive) auto
qed
-  then show ?thesis by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
+  then show ?thesis
+    by (auto elim!: dvdE simp only: dvd_triv_left of_nat_mult)
qed

-lemma zdvd1_eq[simp]: "(x::int) dvd 1 = (\<bar>x\<bar> = 1)"
+lemma zdvd1_eq[simp]: "x dvd 1 \<longleftrightarrow> \<bar>x\<bar> = 1"
+  (is "?lhs \<longleftrightarrow> ?rhs")
+  for x :: int
proof
-  assume d: "x dvd 1" hence "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
-  hence "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
-  hence "nat \<bar>x\<bar> = 1"  by simp
-  thus "\<bar>x\<bar> = 1" by (cases "x < 0") auto
+  assume ?lhs
+  then have "int (nat \<bar>x\<bar>) dvd int (nat 1)" by simp
+  then have "nat \<bar>x\<bar> dvd 1" by (simp add: zdvd_int)
+  then have "nat \<bar>x\<bar> = 1" by simp
+  then show ?rhs by (cases "x < 0") auto
next
-  assume "\<bar>x\<bar>=1"
-  then have "x = 1 \<or> x = -1" by auto
-  then show "x dvd 1" by (auto intro: dvdI)
+  assume ?rhs
+  then have "x = 1 \<or> x = - 1" by auto
+  then show ?lhs by (auto intro: dvdI)
qed

lemma zdvd_mult_cancel1:
-  assumes mp:"m \<noteq>(0::int)" shows "(m * n dvd m) = (\<bar>n\<bar> = 1)"
+  fixes m :: int
+  assumes mp: "m \<noteq> 0"
+  shows "m * n dvd m \<longleftrightarrow> \<bar>n\<bar> = 1"
+    (is "?lhs \<longleftrightarrow> ?rhs")
proof
-  assume n1: "\<bar>n\<bar> = 1" thus "m * n dvd m"
-    by (cases "n >0") (auto simp add: minus_equation_iff)
+  assume ?rhs
+  then show ?lhs
+    by (cases "n > 0") (auto simp add: minus_equation_iff)
next
-  assume H: "m * n dvd m" hence H2: "m * n dvd m * 1" by simp
-  from zdvd_mult_cancel[OF H2 mp] show "\<bar>n\<bar> = 1" by (simp only: zdvd1_eq)
+  assume ?lhs
+  then have "m * n dvd m * 1" by simp
+  from zdvd_mult_cancel[OF this mp] show ?rhs
+    by (simp only: zdvd1_eq)
qed

-lemma int_dvd_iff: "(int m dvd z) = (m dvd nat \<bar>z\<bar>)"
-  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
-
-lemma dvd_int_iff: "(z dvd int m) = (nat \<bar>z\<bar> dvd m)"
-  unfolding zdvd_int by (cases "z \<ge> 0") simp_all
+lemma int_dvd_iff: "int m dvd z \<longleftrightarrow> m dvd nat \<bar>z\<bar>"
+  by (cases "z \<ge> 0") (simp_all add: zdvd_int)

-lemma dvd_int_unfold_dvd_nat:
-  "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
-  unfolding dvd_int_iff [symmetric] by simp
+lemma dvd_int_iff: "z dvd int m \<longleftrightarrow> nat \<bar>z\<bar> dvd m"
+  by (cases "z \<ge> 0") (simp_all add: zdvd_int)

-lemma nat_dvd_iff: "(nat z dvd m) = (if 0 \<le> z then (z dvd int m) else m = 0)"
+lemma dvd_int_unfold_dvd_nat: "k dvd l \<longleftrightarrow> nat \<bar>k\<bar> dvd nat \<bar>l\<bar>"
+  by (simp add: dvd_int_iff [symmetric])
+
+lemma nat_dvd_iff: "nat z dvd m \<longleftrightarrow> (if 0 \<le> z then z dvd int m else m = 0)"

-lemma eq_nat_nat_iff:
-  "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
+lemma eq_nat_nat_iff: "0 \<le> z \<Longrightarrow> 0 \<le> z' \<Longrightarrow> nat z = nat z' \<longleftrightarrow> z = z'"
by (auto elim!: nonneg_eq_int)

-lemma nat_power_eq:
-  "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
+lemma nat_power_eq: "0 \<le> z \<Longrightarrow> nat (z ^ n) = nat z ^ n"
by (induct n) (simp_all add: nat_mult_distrib)

-lemma zdvd_imp_le: "[| z dvd n; 0 < n |] ==> z \<le> (n::int)"
+lemma zdvd_imp_le: "z dvd n \<Longrightarrow> 0 < n \<Longrightarrow> z \<le> n"
+  for n z :: int
apply (cases n)
-  apply (auto simp add: dvd_int_iff)
+   apply (auto simp add: dvd_int_iff)
apply (cases z)
-  apply (auto simp add: dvd_imp_le)
+   apply (auto simp add: dvd_imp_le)
done

lemma zdvd_period:
fixes a d :: int
assumes "a dvd d"
shows "a dvd (x + t) \<longleftrightarrow> a dvd ((x + c * d) + t)"
+    (is "?lhs \<longleftrightarrow> ?rhs")
proof -
from assms obtain k where "d = a * k" by (rule dvdE)
show ?thesis
proof
-    assume "a dvd (x + t)"
+    assume ?lhs
then obtain l where "x + t = a * l" by (rule dvdE)
then have "x = a * l - t" by simp
-    with \<open>d = a * k\<close> show "a dvd x + c * d + t" by simp
+    with \<open>d = a * k\<close> show ?rhs by simp
next
-    assume "a dvd x + c * d + t"
+    assume ?rhs
then obtain l where "x + c * d + t = a * l" by (rule dvdE)
then have "x = a * l - c * d - t" by simp
-    with \<open>d = a * k\<close> show "a dvd (x + t)" by simp
+    with \<open>d = a * k\<close> show ?lhs by simp
qed
qed

subsection \<open>Finiteness of intervals\<close>

-lemma finite_interval_int1 [iff]: "finite {i :: int. a <= i & i <= b}"
-proof (cases "a <= b")
+lemma finite_interval_int1 [iff]: "finite {i :: int. a \<le> i \<and> i \<le> b}"
+proof (cases "a \<le> b")
case True
-  from this show ?thesis
+  then show ?thesis
proof (induct b rule: int_ge_induct)
case base
-    have "{i. a <= i & i <= a} = {a}" by auto
-    from this show ?case by simp
+    have "{i. a \<le> i \<and> i \<le> a} = {a}" by auto
+    then show ?case by simp
next
case (step b)
-    from this have "{i. a <= i & i <= b + 1} = {i. a <= i & i <= b} \<union> {b + 1}" by auto
-    from this step show ?case by simp
+    then have "{i. a \<le> i \<and> i \<le> b + 1} = {i. a \<le> i \<and> i \<le> b} \<union> {b + 1}" by auto
+    with step show ?case by simp
qed
next
-  case False from this show ?thesis
+  case False
+  then show ?thesis
by (metis (lifting, no_types) Collect_empty_eq finite.emptyI order_trans)
qed

-lemma finite_interval_int2 [iff]: "finite {i :: int. a <= i & i < b}"
-by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
+lemma finite_interval_int2 [iff]: "finite {i :: int. a \<le> i \<and> i < b}"
+  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto

-lemma finite_interval_int3 [iff]: "finite {i :: int. a < i & i <= b}"
-by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
+lemma finite_interval_int3 [iff]: "finite {i :: int. a < i \<and> i \<le> b}"
+  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto

-lemma finite_interval_int4 [iff]: "finite {i :: int. a < i & i < b}"
-by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto
+lemma finite_interval_int4 [iff]: "finite {i :: int. a < i \<and> i < b}"
+  by (rule rev_finite_subset[OF finite_interval_int1[of "a" "b"]]) auto

subsection \<open>Configuration of the code generator\<close>

text \<open>Constructors\<close>

-definition Pos :: "num \<Rightarrow> int" where
-  [simp, code_abbrev]: "Pos = numeral"
+definition Pos :: "num \<Rightarrow> int"
+  where [simp, code_abbrev]: "Pos = numeral"

-definition Neg :: "num \<Rightarrow> int" where
-  [simp, code_abbrev]: "Neg n = - (Pos n)"
+definition Neg :: "num \<Rightarrow> int"
+  where [simp, code_abbrev]: "Neg n = - (Pos n)"

code_datatype "0::int" Pos Neg

-text \<open>Auxiliary operations\<close>
+text \<open>Auxiliary operations.\<close>

-definition dup :: "int \<Rightarrow> int" where
-  [simp]: "dup k = k + k"
+definition dup :: "int \<Rightarrow> int"
+  where [simp]: "dup k = k + k"

lemma dup_code [code]:
"dup 0 = 0"
"dup (Pos n) = Pos (Num.Bit0 n)"
"dup (Neg n) = Neg (Num.Bit0 n)"
-  unfolding Pos_def Neg_def

-definition sub :: "num \<Rightarrow> num \<Rightarrow> int" where
-  [simp]: "sub m n = numeral m - numeral n"
+definition sub :: "num \<Rightarrow> num \<Rightarrow> int"
+  where [simp]: "sub m n = numeral m - numeral n"

lemma sub_code [code]:
"sub Num.One Num.One = 0"
@@ -1610,25 +1630,25 @@
"sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
"sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
"sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
-  apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
-  apply (simp_all only: algebra_simps minus_diff_eq)
+          apply (simp_all only: sub_def dup_def numeral.simps Pos_def Neg_def numeral_BitM)
+        apply (simp_all only: algebra_simps minus_diff_eq)
apply (simp_all only: add.commute [of _ "- (numeral n + numeral n)"])
done

-text \<open>Implementations\<close>
+text \<open>Implementations.\<close>

-lemma one_int_code [code, code_unfold]:
-  "1 = Pos Num.One"
+lemma one_int_code [code, code_unfold]: "1 = Pos Num.One"
by simp

lemma plus_int_code [code]:
-  "k + 0 = (k::int)"
-  "0 + l = (l::int)"
+  "k + 0 = k"
+  "0 + l = l"
"Pos m + Pos n = Pos (m + n)"
"Pos m + Neg n = sub m n"
"Neg m + Pos n = sub n m"
"Neg m + Neg n = Neg (m + n)"
+  for k l :: int
by simp_all

lemma uminus_int_code [code]:
@@ -1638,28 +1658,29 @@
by simp_all

lemma minus_int_code [code]:
-  "k - 0 = (k::int)"
-  "0 - l = uminus (l::int)"
+  "k - 0 = k"
+  "0 - l = uminus l"
"Pos m - Pos n = sub m n"
"Pos m - Neg n = Pos (m + n)"
"Neg m - Pos n = Neg (m + n)"
"Neg m - Neg n = sub n m"
+  for k l :: int
by simp_all

lemma times_int_code [code]:
-  "k * 0 = (0::int)"
-  "0 * l = (0::int)"
+  "k * 0 = 0"
+  "0 * l = 0"
"Pos m * Pos n = Pos (m * n)"
"Pos m * Neg n = Neg (m * n)"
"Neg m * Pos n = Neg (m * n)"
"Neg m * Neg n = Pos (m * n)"
+  for k l :: int
by simp_all

instantiation int :: equal
begin

-definition
-  "HOL.equal k l \<longleftrightarrow> k = (l::int)"
+definition "HOL.equal k l \<longleftrightarrow> k = (l::int)"

instance
by standard (rule equal_int_def)
@@ -1678,8 +1699,8 @@
"HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"

-lemma equal_int_refl [code nbe]:
-  "HOL.equal (k::int) k \<longleftrightarrow> True"
+lemma equal_int_refl [code nbe]: "HOL.equal k k \<longleftrightarrow> True"
+  for k :: int
by (fact equal_refl)

lemma less_eq_int_code [code]:
@@ -1719,7 +1740,7 @@
by simp_all

-text \<open>Serializer setup\<close>
+text \<open>Serializer setup.\<close>

code_identifier
code_module Int \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith```