--- a/src/HOL/Rings.thy Mon Jun 20 17:51:47 2016 +0200
+++ b/src/HOL/Rings.thy Mon Jun 20 21:40:48 2016 +0200
@@ -18,10 +18,9 @@
assumes distrib_left[algebra_simps]: "a * (b + c) = a * b + a * c"
begin
-text\<open>For the \<open>combine_numerals\<close> simproc\<close>
-lemma combine_common_factor:
- "a * e + (b * e + c) = (a + b) * e + c"
-by (simp add: distrib_right ac_simps)
+text \<open>For the \<open>combine_numerals\<close> simproc\<close>
+lemma combine_common_factor: "a * e + (b * e + c) = (a + b) * e + c"
+ by (simp add: distrib_right ac_simps)
end
@@ -30,8 +29,7 @@
assumes mult_zero_right [simp]: "a * 0 = 0"
begin
-lemma mult_not_zero:
- "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
+lemma mult_not_zero: "a * b \<noteq> 0 \<Longrightarrow> a \<noteq> 0 \<and> b \<noteq> 0"
by auto
end
@@ -45,11 +43,9 @@
proof
fix a :: 'a
have "0 * a + 0 * a = 0 * a + 0" by (simp add: distrib_right [symmetric])
- thus "0 * a = 0" by (simp only: add_left_cancel)
-next
- fix a :: 'a
+ then show "0 * a = 0" by (simp only: add_left_cancel)
have "a * 0 + a * 0 = a * 0 + 0" by (simp add: distrib_left [symmetric])
- thus "a * 0 = 0" by (simp only: add_left_cancel)
+ then show "a * 0 = 0" by (simp only: add_left_cancel)
qed
end
@@ -63,8 +59,8 @@
fix a b c :: 'a
show "(a + b) * c = a * c + b * c" by (simp add: distrib)
have "a * (b + c) = (b + c) * a" by (simp add: ac_simps)
- also have "... = b * a + c * a" by (simp only: distrib)
- also have "... = a * b + a * c" by (simp add: ac_simps)
+ also have "\<dots> = b * a + c * a" by (simp only: distrib)
+ also have "\<dots> = a * b + a * c" by (simp add: ac_simps)
finally show "a * (b + c) = a * b + a * c" by blast
qed
@@ -91,27 +87,23 @@
begin
lemma one_neq_zero [simp]: "1 \<noteq> 0"
-by (rule not_sym) (rule zero_neq_one)
+ by (rule not_sym) (rule zero_neq_one)
definition of_bool :: "bool \<Rightarrow> 'a"
-where
- "of_bool p = (if p then 1 else 0)"
+ where "of_bool p = (if p then 1 else 0)"
lemma of_bool_eq [simp, code]:
"of_bool False = 0"
"of_bool True = 1"
by (simp_all add: of_bool_def)
-lemma of_bool_eq_iff:
- "of_bool p = of_bool q \<longleftrightarrow> p = q"
+lemma of_bool_eq_iff: "of_bool p = of_bool q \<longleftrightarrow> p = q"
by (simp add: of_bool_def)
-lemma split_of_bool [split]:
- "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
+lemma split_of_bool [split]: "P (of_bool p) \<longleftrightarrow> (p \<longrightarrow> P 1) \<and> (\<not> p \<longrightarrow> P 0)"
by (cases p) simp_all
-lemma split_of_bool_asm:
- "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
+lemma split_of_bool_asm: "P (of_bool p) \<longleftrightarrow> \<not> (p \<and> \<not> P 1 \<or> \<not> p \<and> \<not> P 0)"
by (cases p) simp_all
end
@@ -123,8 +115,8 @@
class dvd = times
begin
-definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50) where
- "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
+definition dvd :: "'a \<Rightarrow> 'a \<Rightarrow> bool" (infix "dvd" 50)
+ where "b dvd a \<longleftrightarrow> (\<exists>k. a = b * k)"
lemma dvdI [intro?]: "a = b * k \<Longrightarrow> b dvd a"
unfolding dvd_def ..
@@ -139,8 +131,7 @@
subclass dvd .
-lemma dvd_refl [simp]:
- "a dvd a"
+lemma dvd_refl [simp]: "a dvd a"
proof
show "a = a * 1" by simp
qed
@@ -155,32 +146,25 @@
then show ?thesis ..
qed
-lemma subset_divisors_dvd:
- "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
+lemma subset_divisors_dvd: "{c. c dvd a} \<subseteq> {c. c dvd b} \<longleftrightarrow> a dvd b"
by (auto simp add: subset_iff intro: dvd_trans)
-lemma strict_subset_divisors_dvd:
- "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
+lemma strict_subset_divisors_dvd: "{c. c dvd a} \<subset> {c. c dvd b} \<longleftrightarrow> a dvd b \<and> \<not> b dvd a"
by (auto simp add: subset_iff intro: dvd_trans)
-lemma one_dvd [simp]:
- "1 dvd a"
+lemma one_dvd [simp]: "1 dvd a"
by (auto intro!: dvdI)
-lemma dvd_mult [simp]:
- "a dvd c \<Longrightarrow> a dvd (b * c)"
+lemma dvd_mult [simp]: "a dvd c \<Longrightarrow> a dvd (b * c)"
by (auto intro!: mult.left_commute dvdI elim!: dvdE)
-lemma dvd_mult2 [simp]:
- "a dvd b \<Longrightarrow> a dvd (b * c)"
+lemma dvd_mult2 [simp]: "a dvd b \<Longrightarrow> a dvd (b * c)"
using dvd_mult [of a b c] by (simp add: ac_simps)
-lemma dvd_triv_right [simp]:
- "a dvd b * a"
+lemma dvd_triv_right [simp]: "a dvd b * a"
by (rule dvd_mult) (rule dvd_refl)
-lemma dvd_triv_left [simp]:
- "a dvd a * b"
+lemma dvd_triv_left [simp]: "a dvd a * b"
by (rule dvd_mult2) (rule dvd_refl)
lemma mult_dvd_mono:
@@ -194,12 +178,10 @@
then show ?thesis ..
qed
-lemma dvd_mult_left:
- "a * b dvd c \<Longrightarrow> a dvd c"
+lemma dvd_mult_left: "a * b dvd c \<Longrightarrow> a dvd c"
by (simp add: dvd_def mult.assoc) blast
-lemma dvd_mult_right:
- "a * b dvd c \<Longrightarrow> b dvd c"
+lemma dvd_mult_right: "a * b dvd c \<Longrightarrow> b dvd c"
using dvd_mult_left [of b a c] by (simp add: ac_simps)
end
@@ -209,18 +191,15 @@
subclass semiring_1 ..
-lemma dvd_0_left_iff [simp]:
- "0 dvd a \<longleftrightarrow> a = 0"
+lemma dvd_0_left_iff [simp]: "0 dvd a \<longleftrightarrow> a = 0"
by (auto intro: dvd_refl elim!: dvdE)
-lemma dvd_0_right [iff]:
- "a dvd 0"
+lemma dvd_0_right [iff]: "a dvd 0"
proof
show "0 = a * 0" by simp
qed
-lemma dvd_0_left:
- "0 dvd a \<Longrightarrow> a = 0"
+lemma dvd_0_left: "0 dvd a \<Longrightarrow> a = 0"
by simp
lemma dvd_add [simp]:
@@ -245,8 +224,8 @@
end
-class comm_semiring_1_cancel = comm_semiring + cancel_comm_monoid_add +
- zero_neq_one + comm_monoid_mult +
+class comm_semiring_1_cancel =
+ comm_semiring + cancel_comm_monoid_add + zero_neq_one + comm_monoid_mult +
assumes right_diff_distrib' [algebra_simps]: "a * (b - c) = a * b - a * c"
begin
@@ -254,16 +233,15 @@
subclass comm_semiring_0_cancel ..
subclass comm_semiring_1 ..
-lemma left_diff_distrib' [algebra_simps]:
- "(b - c) * a = b * a - c * a"
+lemma left_diff_distrib' [algebra_simps]: "(b - c) * a = b * a - c * a"
by (simp add: algebra_simps)
-lemma dvd_add_times_triv_left_iff [simp]:
- "a dvd c * a + b \<longleftrightarrow> a dvd b"
+lemma dvd_add_times_triv_left_iff [simp]: "a dvd c * a + b \<longleftrightarrow> a dvd b"
proof -
have "a dvd a * c + b \<longleftrightarrow> a dvd b" (is "?P \<longleftrightarrow> ?Q")
proof
- assume ?Q then show ?P by simp
+ assume ?Q
+ then show ?P by simp
next
assume ?P
then obtain d where "a * c + b = a * d" ..
@@ -275,23 +253,21 @@
then show "a dvd c * a + b \<longleftrightarrow> a dvd b" by (simp add: ac_simps)
qed
-lemma dvd_add_times_triv_right_iff [simp]:
- "a dvd b + c * a \<longleftrightarrow> a dvd b"
+lemma dvd_add_times_triv_right_iff [simp]: "a dvd b + c * a \<longleftrightarrow> a dvd b"
using dvd_add_times_triv_left_iff [of a c b] by (simp add: ac_simps)
-lemma dvd_add_triv_left_iff [simp]:
- "a dvd a + b \<longleftrightarrow> a dvd b"
+lemma dvd_add_triv_left_iff [simp]: "a dvd a + b \<longleftrightarrow> a dvd b"
using dvd_add_times_triv_left_iff [of a 1 b] by simp
-lemma dvd_add_triv_right_iff [simp]:
- "a dvd b + a \<longleftrightarrow> a dvd b"
+lemma dvd_add_triv_right_iff [simp]: "a dvd b + a \<longleftrightarrow> a dvd b"
using dvd_add_times_triv_right_iff [of a b 1] by simp
lemma dvd_add_right_iff:
assumes "a dvd b"
shows "a dvd b + c \<longleftrightarrow> a dvd c" (is "?P \<longleftrightarrow> ?Q")
proof
- assume ?P then obtain d where "b + c = a * d" ..
+ assume ?P
+ then obtain d where "b + c = a * d" ..
moreover from \<open>a dvd b\<close> obtain e where "b = a * e" ..
ultimately have "a * e + c = a * d" by simp
then have "a * e + c - a * e = a * d - a * e" by simp
@@ -299,13 +275,12 @@
then have "c = a * (d - e)" by (simp add: algebra_simps)
then show ?Q ..
next
- assume ?Q with assms show ?P by simp
+ assume ?Q
+ with assms show ?P by simp
qed
-lemma dvd_add_left_iff:
- assumes "a dvd c"
- shows "a dvd b + c \<longleftrightarrow> a dvd b"
- using assms dvd_add_right_iff [of a c b] by (simp add: ac_simps)
+lemma dvd_add_left_iff: "a dvd c \<Longrightarrow> a dvd b + c \<longleftrightarrow> a dvd b"
+ using dvd_add_right_iff [of a c b] by (simp add: ac_simps)
end
@@ -317,44 +292,38 @@
text \<open>Distribution rules\<close>
lemma minus_mult_left: "- (a * b) = - a * b"
-by (rule minus_unique) (simp add: distrib_right [symmetric])
+ by (rule minus_unique) (simp add: distrib_right [symmetric])
lemma minus_mult_right: "- (a * b) = a * - b"
-by (rule minus_unique) (simp add: distrib_left [symmetric])
+ by (rule minus_unique) (simp add: distrib_left [symmetric])
-text\<open>Extract signs from products\<close>
+text \<open>Extract signs from products\<close>
lemmas mult_minus_left [simp] = minus_mult_left [symmetric]
lemmas mult_minus_right [simp] = minus_mult_right [symmetric]
lemma minus_mult_minus [simp]: "- a * - b = a * b"
-by simp
+ by simp
lemma minus_mult_commute: "- a * b = a * - b"
-by simp
+ by simp
-lemma right_diff_distrib [algebra_simps]:
- "a * (b - c) = a * b - a * c"
+lemma right_diff_distrib [algebra_simps]: "a * (b - c) = a * b - a * c"
using distrib_left [of a b "-c "] by simp
-lemma left_diff_distrib [algebra_simps]:
- "(a - b) * c = a * c - b * c"
+lemma left_diff_distrib [algebra_simps]: "(a - b) * c = a * c - b * c"
using distrib_right [of a "- b" c] by simp
-lemmas ring_distribs =
- distrib_left distrib_right left_diff_distrib right_diff_distrib
+lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
-lemma eq_add_iff1:
- "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
-by (simp add: algebra_simps)
+lemma eq_add_iff1: "a * e + c = b * e + d \<longleftrightarrow> (a - b) * e + c = d"
+ by (simp add: algebra_simps)
-lemma eq_add_iff2:
- "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
-by (simp add: algebra_simps)
+lemma eq_add_iff2: "a * e + c = b * e + d \<longleftrightarrow> c = (b - a) * e + d"
+ by (simp add: algebra_simps)
end
-lemmas ring_distribs =
- distrib_left distrib_right left_diff_distrib right_diff_distrib
+lemmas ring_distribs = distrib_left distrib_right left_diff_distrib right_diff_distrib
class comm_ring = comm_semiring + ab_group_add
begin
@@ -362,8 +331,7 @@
subclass ring ..
subclass comm_semiring_0_cancel ..
-lemma square_diff_square_factored:
- "x * x - y * y = (x + y) * (x - y)"
+lemma square_diff_square_factored: "x * x - y * y = (x + y) * (x - y)"
by (simp add: algebra_simps)
end
@@ -373,8 +341,7 @@
subclass semiring_1_cancel ..
-lemma square_diff_one_factored:
- "x * x - 1 = (x + 1) * (x - 1)"
+lemma square_diff_one_factored: "x * x - 1 = (x + 1) * (x - 1)"
by (simp add: algebra_simps)
end
@@ -410,8 +377,7 @@
then show "- x dvd y" ..
qed
-lemma dvd_diff [simp]:
- "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
+lemma dvd_diff [simp]: "x dvd y \<Longrightarrow> x dvd z \<Longrightarrow> x dvd (y - z)"
using dvd_add [of x y "- z"] by simp
end
@@ -424,19 +390,20 @@
assumes "a * b = 0"
shows "a = 0 \<or> b = 0"
proof (rule classical)
- assume "\<not> (a = 0 \<or> b = 0)"
+ assume "\<not> ?thesis"
then have "a \<noteq> 0" and "b \<noteq> 0" by auto
with no_zero_divisors have "a * b \<noteq> 0" by blast
with assms show ?thesis by simp
qed
-lemma mult_eq_0_iff [simp]:
- shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
+lemma mult_eq_0_iff [simp]: "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
proof (cases "a = 0 \<or> b = 0")
- case False then have "a \<noteq> 0" and "b \<noteq> 0" by auto
+ case False
+ then have "a \<noteq> 0" and "b \<noteq> 0" by auto
then show ?thesis using no_zero_divisors by simp
next
- case True then show ?thesis by auto
+ case True
+ then show ?thesis by auto
qed
end
@@ -448,12 +415,10 @@
and mult_cancel_left [simp]: "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
begin
-lemma mult_left_cancel:
- "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
+lemma mult_left_cancel: "c \<noteq> 0 \<Longrightarrow> c * a = c * b \<longleftrightarrow> a = b"
by simp
-lemma mult_right_cancel:
- "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
+lemma mult_right_cancel: "c \<noteq> 0 \<Longrightarrow> a * c = b * c \<longleftrightarrow> a = b"
by simp
end
@@ -483,32 +448,27 @@
subclass semiring_1_no_zero_divisors ..
-lemma square_eq_1_iff:
- "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
+lemma square_eq_1_iff: "x * x = 1 \<longleftrightarrow> x = 1 \<or> x = - 1"
proof -
have "(x - 1) * (x + 1) = x * x - 1"
by (simp add: algebra_simps)
- hence "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
+ then have "x * x = 1 \<longleftrightarrow> (x - 1) * (x + 1) = 0"
by simp
- thus ?thesis
+ then show ?thesis
by (simp add: eq_neg_iff_add_eq_0)
qed
-lemma mult_cancel_right1 [simp]:
- "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
-by (insert mult_cancel_right [of 1 c b], force)
+lemma mult_cancel_right1 [simp]: "c = b * c \<longleftrightarrow> c = 0 \<or> b = 1"
+ using mult_cancel_right [of 1 c b] by auto
-lemma mult_cancel_right2 [simp]:
- "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
-by (insert mult_cancel_right [of a c 1], simp)
+lemma mult_cancel_right2 [simp]: "a * c = c \<longleftrightarrow> c = 0 \<or> a = 1"
+ using mult_cancel_right [of a c 1] by simp
-lemma mult_cancel_left1 [simp]:
- "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
-by (insert mult_cancel_left [of c 1 b], force)
+lemma mult_cancel_left1 [simp]: "c = c * b \<longleftrightarrow> c = 0 \<or> b = 1"
+ using mult_cancel_left [of c 1 b] by force
-lemma mult_cancel_left2 [simp]:
- "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
-by (insert mult_cancel_left [of c a 1], simp)
+lemma mult_cancel_left2 [simp]: "c * a = c \<longleftrightarrow> c = 0 \<or> a = 1"
+ using mult_cancel_left [of c a 1] by simp
end
@@ -526,8 +486,7 @@
subclass ring_1_no_zero_divisors ..
-lemma dvd_mult_cancel_right [simp]:
- "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
+lemma dvd_mult_cancel_right [simp]: "a * c dvd b * c \<longleftrightarrow> c = 0 \<or> a dvd b"
proof -
have "a * c dvd b * c \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
unfolding dvd_def by (simp add: ac_simps)
@@ -536,8 +495,7 @@
finally show ?thesis .
qed
-lemma dvd_mult_cancel_left [simp]:
- "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
+lemma dvd_mult_cancel_left [simp]: "c * a dvd c * b \<longleftrightarrow> c = 0 \<or> a dvd b"
proof -
have "c * a dvd c * b \<longleftrightarrow> (\<exists>k. b * c = (a * k) * c)"
unfolding dvd_def by (simp add: ac_simps)
@@ -562,15 +520,12 @@
text \<open>
The theory of partially ordered rings is taken from the books:
- \begin{itemize}
- \item \emph{Lattice Theory} by Garret Birkhoff, American Mathematical Society 1979
- \item \emph{Partially Ordered Algebraic Systems}, Pergamon Press 1963
- \end{itemize}
+ \<^item> \<^emph>\<open>Lattice Theory\<close> by Garret Birkhoff, American Mathematical Society, 1979
+ \<^item> \<^emph>\<open>Partially Ordered Algebraic Systems\<close>, Pergamon Press, 1963
+
Most of the used notions can also be looked up in
- \begin{itemize}
- \item @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
- \item \emph{Algebra I} by van der Waerden, Springer.
- \end{itemize}
+ \<^item> @{url "http://www.mathworld.com"} by Eric Weisstein et. al.
+ \<^item> \<^emph>\<open>Algebra I\<close> by van der Waerden, Springer
\<close>
class divide =
@@ -605,49 +560,45 @@
assumes divide_zero [simp]: "a div 0 = 0"
begin
-lemma nonzero_mult_divide_cancel_left [simp]:
- "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
+lemma nonzero_mult_divide_cancel_left [simp]: "a \<noteq> 0 \<Longrightarrow> (a * b) div a = b"
using nonzero_mult_divide_cancel_right [of a b] by (simp add: ac_simps)
subclass semiring_no_zero_divisors_cancel
proof
- fix a b c
- { fix a b c
- show "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b"
- proof (cases "c = 0")
- case True then show ?thesis by simp
- next
- case False
- { assume "a * c = b * c"
- then have "a * c div c = b * c div c"
- by simp
- with False have "a = b"
- by simp
- } then show ?thesis by auto
- qed
- }
- from this [of a c b]
- show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b"
- by (simp add: ac_simps)
+ show *: "a * c = b * c \<longleftrightarrow> c = 0 \<or> a = b" for a b c
+ proof (cases "c = 0")
+ case True
+ then show ?thesis by simp
+ next
+ case False
+ {
+ assume "a * c = b * c"
+ then have "a * c div c = b * c div c"
+ by simp
+ with False have "a = b"
+ by simp
+ }
+ then show ?thesis by auto
+ qed
+ show "c * a = c * b \<longleftrightarrow> c = 0 \<or> a = b" for a b c
+ using * [of a c b] by (simp add: ac_simps)
qed
-lemma div_self [simp]:
- assumes "a \<noteq> 0"
- shows "a div a = 1"
- using assms nonzero_mult_divide_cancel_left [of a 1] by simp
+lemma div_self [simp]: "a \<noteq> 0 \<Longrightarrow> a div a = 1"
+ using nonzero_mult_divide_cancel_left [of a 1] by simp
-lemma divide_zero_left [simp]:
- "0 div a = 0"
+lemma divide_zero_left [simp]: "0 div a = 0"
proof (cases "a = 0")
- case True then show ?thesis by simp
+ case True
+ then show ?thesis by simp
next
- case False then have "a * 0 div a = 0"
+ case False
+ then have "a * 0 div a = 0"
by (rule nonzero_mult_divide_cancel_left)
then show ?thesis by simp
qed
-lemma divide_1 [simp]:
- "a div 1 = a"
+lemma divide_1 [simp]: "a div 1 = a"
using nonzero_mult_divide_cancel_left [of 1 a] by simp
end
@@ -668,11 +619,13 @@
assumes "a \<noteq> 0"
shows "a * b dvd a * c \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
proof
- assume ?P then obtain d where "a * c = a * b * d" ..
+ assume ?P
+ then obtain d where "a * c = a * b * d" ..
with assms have "c = b * d" by (simp add: ac_simps)
then show ?Q ..
next
- assume ?Q then obtain d where "c = b * d" ..
+ assume ?Q
+ then obtain d where "c = b * d" ..
then have "a * c = a * b * d" by (simp add: ac_simps)
then show ?P ..
qed
@@ -680,7 +633,7 @@
lemma dvd_times_right_cancel_iff [simp]:
assumes "a \<noteq> 0"
shows "b * a dvd c * a \<longleftrightarrow> b dvd c" (is "?P \<longleftrightarrow> ?Q")
-using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
+ using dvd_times_left_cancel_iff [of a b c] assms by (simp add: ac_simps)
lemma div_dvd_iff_mult:
assumes "b \<noteq> 0" and "b dvd a"
@@ -702,7 +655,8 @@
assumes "a dvd b" and "a dvd c"
shows "b div a dvd c div a \<longleftrightarrow> b dvd c"
proof (cases "a = 0")
- case True with assms show ?thesis by simp
+ case True
+ with assms show ?thesis by simp
next
case False
moreover from assms obtain k l where "b = a * k" and "c = a * l"
@@ -714,7 +668,8 @@
assumes "c dvd a" and "c dvd b"
shows "(a + b) div c = a div c + b div c"
proof (cases "c = 0")
- case True then show ?thesis by simp
+ case True
+ then show ?thesis by simp
next
case False
moreover from assms obtain k l where "a = c * k" and "b = c * l"
@@ -729,7 +684,8 @@
assumes "b dvd a" and "d dvd c"
shows "(a div b) * (c div d) = (a * c) div (b * d)"
proof (cases "b = 0 \<or> c = 0")
- case True with assms show ?thesis by auto
+ case True
+ with assms show ?thesis by auto
next
case False
moreover from assms obtain k l where "a = b * k" and "c = d * l"
@@ -748,42 +704,39 @@
next
assume "b div a = c"
then have "b div a * a = c * a" by simp
- moreover from \<open>a \<noteq> 0\<close> \<open>a dvd b\<close> have "b div a * a = b"
+ moreover from assms have "b div a * a = b"
by (auto elim!: dvdE simp add: ac_simps)
ultimately show "b = c * a" by simp
qed
-lemma dvd_div_mult_self [simp]:
- "a dvd b \<Longrightarrow> b div a * a = b"
+lemma dvd_div_mult_self [simp]: "a dvd b \<Longrightarrow> b div a * a = b"
by (cases "a = 0") (auto elim: dvdE simp add: ac_simps)
-lemma dvd_mult_div_cancel [simp]:
- "a dvd b \<Longrightarrow> a * (b div a) = b"
+lemma dvd_mult_div_cancel [simp]: "a dvd b \<Longrightarrow> a * (b div a) = b"
using dvd_div_mult_self [of a b] by (simp add: ac_simps)
lemma div_mult_swap:
assumes "c dvd b"
shows "a * (b div c) = (a * b) div c"
proof (cases "c = 0")
- case True then show ?thesis by simp
+ case True
+ then show ?thesis by simp
next
- case False from assms obtain d where "b = c * d" ..
+ case False
+ from assms obtain d where "b = c * d" ..
moreover from False have "a * divide (d * c) c = ((a * d) * c) div c"
by simp
ultimately show ?thesis by (simp add: ac_simps)
qed
-lemma dvd_div_mult:
- assumes "c dvd b"
- shows "b div c * a = (b * a) div c"
- using assms div_mult_swap [of c b a] by (simp add: ac_simps)
+lemma dvd_div_mult: "c dvd b \<Longrightarrow> b div c * a = (b * a) div c"
+ using div_mult_swap [of c b a] by (simp add: ac_simps)
lemma dvd_div_mult2_eq:
assumes "b * c dvd a"
shows "a div (b * c) = a div b div c"
-using assms proof
- fix k
- assume "a = b * c * k"
+proof -
+ from assms obtain k where "a = b * c * k" ..
then show ?thesis
by (cases "b = 0 \<or> c = 0") (auto, simp add: ac_simps)
qed
@@ -808,15 +761,12 @@
text \<open>Units: invertible elements in a ring\<close>
abbreviation is_unit :: "'a \<Rightarrow> bool"
-where
- "is_unit a \<equiv> a dvd 1"
+ where "is_unit a \<equiv> a dvd 1"
-lemma not_is_unit_0 [simp]:
- "\<not> is_unit 0"
+lemma not_is_unit_0 [simp]: "\<not> is_unit 0"
by simp
-lemma unit_imp_dvd [dest]:
- "is_unit b \<Longrightarrow> b dvd a"
+lemma unit_imp_dvd [dest]: "is_unit b \<Longrightarrow> b dvd a"
by (rule dvd_trans [of _ 1]) simp_all
lemma unit_dvdE:
@@ -829,8 +779,7 @@
ultimately show thesis using that by blast
qed
-lemma dvd_unit_imp_unit:
- "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
+lemma dvd_unit_imp_unit: "a dvd b \<Longrightarrow> is_unit b \<Longrightarrow> is_unit a"
by (rule dvd_trans)
lemma unit_div_1_unit [simp, intro]:
@@ -849,27 +798,24 @@
proof (rule that)
define b where "b = 1 div a"
then show "1 div a = b" by simp
- from b_def \<open>is_unit a\<close> show "is_unit b" by simp
- from \<open>is_unit a\<close> and \<open>is_unit b\<close> show "a \<noteq> 0" and "b \<noteq> 0" by auto
- from b_def \<open>is_unit a\<close> show "a * b = 1" by simp
+ from assms b_def show "is_unit b" by simp
+ with assms show "a \<noteq> 0" and "b \<noteq> 0" by auto
+ from assms b_def show "a * b = 1" by simp
then have "1 = a * b" ..
with b_def \<open>b \<noteq> 0\<close> show "1 div b = a" by simp
- from \<open>is_unit a\<close> have "a dvd c" ..
+ from assms have "a dvd c" ..
then obtain d where "c = a * d" ..
with \<open>a \<noteq> 0\<close> \<open>a * b = 1\<close> show "c div a = c * b"
by (simp add: mult.assoc mult.left_commute [of a])
qed
-lemma unit_prod [intro]:
- "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
+lemma unit_prod [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a * b)"
by (subst mult_1_left [of 1, symmetric]) (rule mult_dvd_mono)
-lemma is_unit_mult_iff:
- "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b" (is "?P \<longleftrightarrow> ?Q")
+lemma is_unit_mult_iff: "is_unit (a * b) \<longleftrightarrow> is_unit a \<and> is_unit b"
by (auto dest: dvd_mult_left dvd_mult_right)
-lemma unit_div [intro]:
- "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
+lemma unit_div [intro]: "is_unit a \<Longrightarrow> is_unit b \<Longrightarrow> is_unit (a div b)"
by (erule is_unitE [of b a]) (simp add: ac_simps unit_prod)
lemma mult_unit_dvd_iff:
@@ -894,7 +840,8 @@
assume "a dvd c * b"
with assms have "c * b dvd c * (b * (1 div b))"
by (subst mult_assoc [symmetric]) simp
- also from \<open>is_unit b\<close> have "b * (1 div b) = 1" by (rule is_unitE) simp
+ also from assms have "b * (1 div b) = 1"
+ by (rule is_unitE) simp
finally have "c * b dvd c" by simp
with \<open>a dvd c * b\<close> show "a dvd c" by (rule dvd_trans)
next
@@ -902,52 +849,40 @@
then show "a dvd c * b" by simp
qed
-lemma div_unit_dvd_iff:
- "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
+lemma div_unit_dvd_iff: "is_unit b \<Longrightarrow> a div b dvd c \<longleftrightarrow> a dvd c"
by (erule is_unitE [of _ a]) (auto simp add: mult_unit_dvd_iff)
-lemma dvd_div_unit_iff:
- "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
+lemma dvd_div_unit_iff: "is_unit b \<Longrightarrow> a dvd c div b \<longleftrightarrow> a dvd c"
by (erule is_unitE [of _ c]) (simp add: dvd_mult_unit_iff)
lemmas unit_dvd_iff = mult_unit_dvd_iff div_unit_dvd_iff
- dvd_mult_unit_iff dvd_div_unit_iff \<comment> \<open>FIXME consider fact collection\<close>
+ dvd_mult_unit_iff dvd_div_unit_iff (* FIXME consider named_theorems *)
-lemma unit_mult_div_div [simp]:
- "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
+lemma unit_mult_div_div [simp]: "is_unit a \<Longrightarrow> b * (1 div a) = b div a"
by (erule is_unitE [of _ b]) simp
-lemma unit_div_mult_self [simp]:
- "is_unit a \<Longrightarrow> b div a * a = b"
+lemma unit_div_mult_self [simp]: "is_unit a \<Longrightarrow> b div a * a = b"
by (rule dvd_div_mult_self) auto
-lemma unit_div_1_div_1 [simp]:
- "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
+lemma unit_div_1_div_1 [simp]: "is_unit a \<Longrightarrow> 1 div (1 div a) = a"
by (erule is_unitE) simp
-lemma unit_div_mult_swap:
- "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
+lemma unit_div_mult_swap: "is_unit c \<Longrightarrow> a * (b div c) = (a * b) div c"
by (erule unit_dvdE [of _ b]) (simp add: mult.left_commute [of _ c])
-lemma unit_div_commute:
- "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
+lemma unit_div_commute: "is_unit b \<Longrightarrow> (a div b) * c = (a * c) div b"
using unit_div_mult_swap [of b c a] by (simp add: ac_simps)
-lemma unit_eq_div1:
- "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
+lemma unit_eq_div1: "is_unit b \<Longrightarrow> a div b = c \<longleftrightarrow> a = c * b"
by (auto elim: is_unitE)
-lemma unit_eq_div2:
- "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
+lemma unit_eq_div2: "is_unit b \<Longrightarrow> a = c div b \<longleftrightarrow> a * b = c"
using unit_eq_div1 [of b c a] by auto
-lemma unit_mult_left_cancel:
- assumes "is_unit a"
- shows "a * b = a * c \<longleftrightarrow> b = c" (is "?P \<longleftrightarrow> ?Q")
- using assms mult_cancel_left [of a b c] by auto
+lemma unit_mult_left_cancel: "is_unit a \<Longrightarrow> a * b = a * c \<longleftrightarrow> b = c"
+ using mult_cancel_left [of a b c] by auto
-lemma unit_mult_right_cancel:
- "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
+lemma unit_mult_right_cancel: "is_unit a \<Longrightarrow> b * a = c * a \<longleftrightarrow> b = c"
using unit_mult_left_cancel [of a b c] by (auto simp add: ac_simps)
lemma unit_div_cancel:
@@ -964,7 +899,8 @@
assumes "is_unit b" and "is_unit c"
shows "a div (b * c) = a div b div c"
proof -
- from assms have "is_unit (b * c)" by (simp add: unit_prod)
+ from assms have "is_unit (b * c)"
+ by (simp add: unit_prod)
then have "b * c dvd a"
by (rule unit_imp_dvd)
then show ?thesis
@@ -1015,58 +951,57 @@
values rather than associated elements.
\<close>
-lemma unit_factor_dvd [simp]:
- "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
+lemma unit_factor_dvd [simp]: "a \<noteq> 0 \<Longrightarrow> unit_factor a dvd b"
by (rule unit_imp_dvd) simp
-lemma unit_factor_self [simp]:
- "unit_factor a dvd a"
+lemma unit_factor_self [simp]: "unit_factor a dvd a"
by (cases "a = 0") simp_all
-lemma normalize_mult_unit_factor [simp]:
- "normalize a * unit_factor a = a"
+lemma normalize_mult_unit_factor [simp]: "normalize a * unit_factor a = a"
using unit_factor_mult_normalize [of a] by (simp add: ac_simps)
-lemma normalize_eq_0_iff [simp]:
- "normalize a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
+lemma normalize_eq_0_iff [simp]: "normalize a = 0 \<longleftrightarrow> a = 0"
+ (is "?P \<longleftrightarrow> ?Q")
proof
assume ?P
moreover have "unit_factor a * normalize a = a" by simp
ultimately show ?Q by simp
next
- assume ?Q then show ?P by simp
+ assume ?Q
+ then show ?P by simp
qed
-lemma unit_factor_eq_0_iff [simp]:
- "unit_factor a = 0 \<longleftrightarrow> a = 0" (is "?P \<longleftrightarrow> ?Q")
+lemma unit_factor_eq_0_iff [simp]: "unit_factor a = 0 \<longleftrightarrow> a = 0"
+ (is "?P \<longleftrightarrow> ?Q")
proof
assume ?P
moreover have "unit_factor a * normalize a = a" by simp
ultimately show ?Q by simp
next
- assume ?Q then show ?P by simp
+ assume ?Q
+ then show ?P by simp
qed
lemma is_unit_unit_factor:
- assumes "is_unit a" shows "unit_factor a = a"
+ assumes "is_unit a"
+ shows "unit_factor a = a"
proof -
from assms have "normalize a = 1" by (rule is_unit_normalize)
moreover from unit_factor_mult_normalize have "unit_factor a * normalize a = a" .
ultimately show ?thesis by simp
qed
-lemma unit_factor_1 [simp]:
- "unit_factor 1 = 1"
+lemma unit_factor_1 [simp]: "unit_factor 1 = 1"
by (rule is_unit_unit_factor) simp
-lemma normalize_1 [simp]:
- "normalize 1 = 1"
+lemma normalize_1 [simp]: "normalize 1 = 1"
by (rule is_unit_normalize) simp
-lemma normalize_1_iff:
- "normalize a = 1 \<longleftrightarrow> is_unit a" (is "?P \<longleftrightarrow> ?Q")
+lemma normalize_1_iff: "normalize a = 1 \<longleftrightarrow> is_unit a"
+ (is "?P \<longleftrightarrow> ?Q")
proof
- assume ?Q then show ?P by (rule is_unit_normalize)
+ assume ?Q
+ then show ?P by (rule is_unit_normalize)
next
assume ?P
then have "a \<noteq> 0" by auto
@@ -1079,32 +1014,34 @@
ultimately show ?Q by simp
qed
-lemma div_normalize [simp]:
- "a div normalize a = unit_factor a"
+lemma div_normalize [simp]: "a div normalize a = unit_factor a"
proof (cases "a = 0")
- case True then show ?thesis by simp
+ case True
+ then show ?thesis by simp
next
- case False then have "normalize a \<noteq> 0" by simp
+ case False
+ then have "normalize a \<noteq> 0" by simp
with nonzero_mult_divide_cancel_right
have "unit_factor a * normalize a div normalize a = unit_factor a" by blast
then show ?thesis by simp
qed
-lemma div_unit_factor [simp]:
- "a div unit_factor a = normalize a"
+lemma div_unit_factor [simp]: "a div unit_factor a = normalize a"
proof (cases "a = 0")
- case True then show ?thesis by simp
+ case True
+ then show ?thesis by simp
next
- case False then have "unit_factor a \<noteq> 0" by simp
+ case False
+ then have "unit_factor a \<noteq> 0" by simp
with nonzero_mult_divide_cancel_left
have "unit_factor a * normalize a div unit_factor a = normalize a" by blast
then show ?thesis by simp
qed
-lemma normalize_div [simp]:
- "normalize a div a = 1 div unit_factor a"
+lemma normalize_div [simp]: "normalize a div a = 1 div unit_factor a"
proof (cases "a = 0")
- case True then show ?thesis by simp
+ case True
+ then show ?thesis by simp
next
case False
have "normalize a div a = normalize a div (unit_factor a * normalize a)"
@@ -1114,62 +1051,64 @@
finally show ?thesis .
qed
-lemma mult_one_div_unit_factor [simp]:
- "a * (1 div unit_factor b) = a div unit_factor b"
+lemma mult_one_div_unit_factor [simp]: "a * (1 div unit_factor b) = a div unit_factor b"
by (cases "b = 0") simp_all
-lemma normalize_mult:
- "normalize (a * b) = normalize a * normalize b"
+lemma normalize_mult: "normalize (a * b) = normalize a * normalize b"
proof (cases "a = 0 \<or> b = 0")
- case True then show ?thesis by auto
+ case True
+ then show ?thesis by auto
next
case False
from unit_factor_mult_normalize have "unit_factor (a * b) * normalize (a * b) = a * b" .
- then have "normalize (a * b) = a * b div unit_factor (a * b)" by simp
- also have "\<dots> = a * b div unit_factor (b * a)" by (simp add: ac_simps)
+ then have "normalize (a * b) = a * b div unit_factor (a * b)"
+ by simp
+ also have "\<dots> = a * b div unit_factor (b * a)"
+ by (simp add: ac_simps)
also have "\<dots> = a * b div unit_factor b div unit_factor a"
using False by (simp add: unit_factor_mult is_unit_div_mult2_eq [symmetric])
also have "\<dots> = a * (b div unit_factor b) div unit_factor a"
using False by (subst unit_div_mult_swap) simp_all
also have "\<dots> = normalize a * normalize b"
- using False by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
+ using False
+ by (simp add: mult.commute [of a] mult.commute [of "normalize a"] unit_div_mult_swap [symmetric])
finally show ?thesis .
qed
-lemma unit_factor_idem [simp]:
- "unit_factor (unit_factor a) = unit_factor a"
+lemma unit_factor_idem [simp]: "unit_factor (unit_factor a) = unit_factor a"
by (cases "a = 0") (auto intro: is_unit_unit_factor)
-lemma normalize_unit_factor [simp]:
- "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
+lemma normalize_unit_factor [simp]: "a \<noteq> 0 \<Longrightarrow> normalize (unit_factor a) = 1"
by (rule is_unit_normalize) simp
-lemma normalize_idem [simp]:
- "normalize (normalize a) = normalize a"
+lemma normalize_idem [simp]: "normalize (normalize a) = normalize a"
proof (cases "a = 0")
- case True then show ?thesis by simp
+ case True
+ then show ?thesis by simp
next
case False
- have "normalize a = normalize (unit_factor a * normalize a)" by simp
+ have "normalize a = normalize (unit_factor a * normalize a)"
+ by simp
also have "\<dots> = normalize (unit_factor a) * normalize (normalize a)"
by (simp only: normalize_mult)
- finally show ?thesis using False by simp_all
+ finally show ?thesis
+ using False by simp_all
qed
lemma unit_factor_normalize [simp]:
assumes "a \<noteq> 0"
shows "unit_factor (normalize a) = 1"
proof -
- from assms have "normalize a \<noteq> 0" by simp
+ from assms have *: "normalize a \<noteq> 0"
+ by simp
have "unit_factor (normalize a) * normalize (normalize a) = normalize a"
by (simp only: unit_factor_mult_normalize)
then have "unit_factor (normalize a) * normalize a = normalize a"
by simp
- with \<open>normalize a \<noteq> 0\<close>
- have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
+ with * have "unit_factor (normalize a) * normalize a div normalize a = normalize a div normalize a"
by simp
- with \<open>normalize a \<noteq> 0\<close>
- show ?thesis by simp
+ with * show ?thesis
+ by simp
qed
lemma dvd_unit_factor_div:
@@ -1196,8 +1135,7 @@
by (cases "b = 0") (simp_all add: normalize_mult)
qed
-lemma normalize_dvd_iff [simp]:
- "normalize a dvd b \<longleftrightarrow> a dvd b"
+lemma normalize_dvd_iff [simp]: "normalize a dvd b \<longleftrightarrow> a dvd b"
proof -
have "normalize a dvd b \<longleftrightarrow> unit_factor a * normalize a dvd b"
using mult_unit_dvd_iff [of "unit_factor a" "normalize a" b]
@@ -1205,8 +1143,7 @@
then show ?thesis by simp
qed
-lemma dvd_normalize_iff [simp]:
- "a dvd normalize b \<longleftrightarrow> a dvd b"
+lemma dvd_normalize_iff [simp]: "a dvd normalize b \<longleftrightarrow> a dvd b"
proof -
have "a dvd normalize b \<longleftrightarrow> a dvd normalize b * unit_factor b"
using dvd_mult_unit_iff [of "unit_factor b" a "normalize b"]
@@ -1226,36 +1163,38 @@
assumes "a dvd b" and "b dvd a"
shows "normalize a = normalize b"
proof (cases "a = 0 \<or> b = 0")
- case True with assms show ?thesis by auto
+ case True
+ with assms show ?thesis by auto
next
case False
from \<open>a dvd b\<close> obtain c where b: "b = a * c" ..
moreover from \<open>b dvd a\<close> obtain d where a: "a = b * d" ..
- ultimately have "b * 1 = b * (c * d)" by (simp add: ac_simps)
+ ultimately have "b * 1 = b * (c * d)"
+ by (simp add: ac_simps)
with False have "1 = c * d"
unfolding mult_cancel_left by simp
- then have "is_unit c" and "is_unit d" by auto
- with a b show ?thesis by (simp add: normalize_mult is_unit_normalize)
+ then have "is_unit c" and "is_unit d"
+ by auto
+ with a b show ?thesis
+ by (simp add: normalize_mult is_unit_normalize)
qed
-lemma associatedD1:
- "normalize a = normalize b \<Longrightarrow> a dvd b"
+lemma associatedD1: "normalize a = normalize b \<Longrightarrow> a dvd b"
using dvd_normalize_iff [of _ b, symmetric] normalize_dvd_iff [of a _, symmetric]
by simp
-lemma associatedD2:
- "normalize a = normalize b \<Longrightarrow> b dvd a"
+lemma associatedD2: "normalize a = normalize b \<Longrightarrow> b dvd a"
using dvd_normalize_iff [of _ a, symmetric] normalize_dvd_iff [of b _, symmetric]
by simp
-lemma associated_unit:
- "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
+lemma associated_unit: "normalize a = normalize b \<Longrightarrow> is_unit a \<Longrightarrow> is_unit b"
using dvd_unit_imp_unit by (auto dest!: associatedD1 associatedD2)
-lemma associated_iff_dvd:
- "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a" (is "?P \<longleftrightarrow> ?Q")
+lemma associated_iff_dvd: "normalize a = normalize b \<longleftrightarrow> a dvd b \<and> b dvd a"
+ (is "?P \<longleftrightarrow> ?Q")
proof
- assume ?Q then show ?P by (auto intro!: associatedI)
+ assume ?Q
+ then show ?P by (auto intro!: associatedI)
next
assume ?P
then have "unit_factor a * normalize a = unit_factor a * normalize b"
@@ -1264,7 +1203,8 @@
by (simp add: ac_simps)
show ?Q
proof (cases "a = 0 \<or> b = 0")
- case True with \<open>?P\<close> show ?thesis by auto
+ case True
+ with \<open>?P\<close> show ?thesis by auto
next
case False
then have "b dvd normalize b * unit_factor a" and "normalize b * unit_factor a dvd b"
@@ -1291,38 +1231,38 @@
assumes mult_right_mono: "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
begin
-lemma mult_mono:
- "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
-apply (erule mult_right_mono [THEN order_trans], assumption)
-apply (erule mult_left_mono, assumption)
-done
+lemma mult_mono: "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
+ apply (erule (1) mult_right_mono [THEN order_trans])
+ apply (erule (1) mult_left_mono)
+ done
-lemma mult_mono':
- "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
-apply (rule mult_mono)
-apply (fast intro: order_trans)+
-done
+lemma mult_mono': "a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> 0 \<le> a \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * d"
+ apply (rule mult_mono)
+ apply (fast intro: order_trans)+
+ done
end
class ordered_semiring_0 = semiring_0 + ordered_semiring
begin
-lemma mult_nonneg_nonneg[simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
-using mult_left_mono [of 0 b a] by simp
+lemma mult_nonneg_nonneg [simp]: "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a * b"
+ using mult_left_mono [of 0 b a] by simp
lemma mult_nonneg_nonpos: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> a * b \<le> 0"
-using mult_left_mono [of b 0 a] by simp
+ using mult_left_mono [of b 0 a] by simp
lemma mult_nonpos_nonneg: "a \<le> 0 \<Longrightarrow> 0 \<le> b \<Longrightarrow> a * b \<le> 0"
-using mult_right_mono [of a 0 b] by simp
+ using mult_right_mono [of a 0 b] by simp
text \<open>Legacy - use \<open>mult_nonpos_nonneg\<close>\<close>
lemma mult_nonneg_nonpos2: "0 \<le> a \<Longrightarrow> b \<le> 0 \<Longrightarrow> b * a \<le> 0"
-by (drule mult_right_mono [of b 0], auto)
+ apply (drule mult_right_mono [of b 0])
+ apply auto
+ done
lemma split_mult_neg_le: "(0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b) \<Longrightarrow> a * b \<le> 0"
-by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
+ by (auto simp add: mult_nonneg_nonpos mult_nonneg_nonpos2)
end
@@ -1341,44 +1281,34 @@
subclass ordered_cancel_comm_monoid_add ..
-lemma mult_left_less_imp_less:
- "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
-by (force simp add: mult_left_mono not_le [symmetric])
+lemma mult_left_less_imp_less: "c * a < c * b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
+ by (force simp add: mult_left_mono not_le [symmetric])
-lemma mult_right_less_imp_less:
- "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
-by (force simp add: mult_right_mono not_le [symmetric])
+lemma mult_right_less_imp_less: "a * c < b * c \<Longrightarrow> 0 \<le> c \<Longrightarrow> a < b"
+ by (force simp add: mult_right_mono not_le [symmetric])
-lemma less_eq_add_cancel_left_greater_eq_zero [simp]:
- "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
+lemma less_eq_add_cancel_left_greater_eq_zero [simp]: "a \<le> a + b \<longleftrightarrow> 0 \<le> b"
using add_le_cancel_left [of a 0 b] by simp
-lemma less_eq_add_cancel_left_less_eq_zero [simp]:
- "a + b \<le> a \<longleftrightarrow> b \<le> 0"
+lemma less_eq_add_cancel_left_less_eq_zero [simp]: "a + b \<le> a \<longleftrightarrow> b \<le> 0"
using add_le_cancel_left [of a b 0] by simp
-lemma less_eq_add_cancel_right_greater_eq_zero [simp]:
- "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
+lemma less_eq_add_cancel_right_greater_eq_zero [simp]: "a \<le> b + a \<longleftrightarrow> 0 \<le> b"
using add_le_cancel_right [of 0 a b] by simp
-lemma less_eq_add_cancel_right_less_eq_zero [simp]:
- "b + a \<le> a \<longleftrightarrow> b \<le> 0"
+lemma less_eq_add_cancel_right_less_eq_zero [simp]: "b + a \<le> a \<longleftrightarrow> b \<le> 0"
using add_le_cancel_right [of b a 0] by simp
-lemma less_add_cancel_left_greater_zero [simp]:
- "a < a + b \<longleftrightarrow> 0 < b"
+lemma less_add_cancel_left_greater_zero [simp]: "a < a + b \<longleftrightarrow> 0 < b"
using add_less_cancel_left [of a 0 b] by simp
-lemma less_add_cancel_left_less_zero [simp]:
- "a + b < a \<longleftrightarrow> b < 0"
+lemma less_add_cancel_left_less_zero [simp]: "a + b < a \<longleftrightarrow> b < 0"
using add_less_cancel_left [of a b 0] by simp
-lemma less_add_cancel_right_greater_zero [simp]:
- "a < b + a \<longleftrightarrow> 0 < b"
+lemma less_add_cancel_right_greater_zero [simp]: "a < b + a \<longleftrightarrow> 0 < b"
using add_less_cancel_right [of 0 a b] by simp
-lemma less_add_cancel_right_less_zero [simp]:
- "b + a < a \<longleftrightarrow> b < 0"
+lemma less_add_cancel_right_less_zero [simp]: "b + a < a \<longleftrightarrow> b < 0"
using add_less_cancel_right [of b a 0] by simp
end
@@ -1392,7 +1322,8 @@
proof-
from assms have "u * x + v * y \<le> u * a + v * a"
by (simp add: add_mono mult_left_mono)
- thus ?thesis using assms unfolding distrib_right[symmetric] by simp
+ with assms show ?thesis
+ unfolding distrib_right[symmetric] by simp
qed
end
@@ -1416,80 +1347,79 @@
using mult_strict_right_mono by (cases "c = 0") auto
qed
-lemma mult_left_le_imp_le:
- "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
-by (force simp add: mult_strict_left_mono _not_less [symmetric])
+lemma mult_left_le_imp_le: "c * a \<le> c * b \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
+ by (auto simp add: mult_strict_left_mono _not_less [symmetric])
-lemma mult_right_le_imp_le:
- "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
-by (force simp add: mult_strict_right_mono not_less [symmetric])
+lemma mult_right_le_imp_le: "a * c \<le> b * c \<Longrightarrow> 0 < c \<Longrightarrow> a \<le> b"
+ by (auto simp add: mult_strict_right_mono not_less [symmetric])
lemma mult_pos_pos[simp]: "0 < a \<Longrightarrow> 0 < b \<Longrightarrow> 0 < a * b"
-using mult_strict_left_mono [of 0 b a] by simp
+ using mult_strict_left_mono [of 0 b a] by simp
lemma mult_pos_neg: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> a * b < 0"
-using mult_strict_left_mono [of b 0 a] by simp
+ using mult_strict_left_mono [of b 0 a] by simp
lemma mult_neg_pos: "a < 0 \<Longrightarrow> 0 < b \<Longrightarrow> a * b < 0"
-using mult_strict_right_mono [of a 0 b] by simp
+ using mult_strict_right_mono [of a 0 b] by simp
text \<open>Legacy - use \<open>mult_neg_pos\<close>\<close>
lemma mult_pos_neg2: "0 < a \<Longrightarrow> b < 0 \<Longrightarrow> b * a < 0"
-by (drule mult_strict_right_mono [of b 0], auto)
-
-lemma zero_less_mult_pos:
- "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
-apply (cases "b\<le>0")
- apply (auto simp add: le_less not_less)
-apply (drule_tac mult_pos_neg [of a b])
- apply (auto dest: less_not_sym)
-done
+ apply (drule mult_strict_right_mono [of b 0])
+ apply auto
+ done
-lemma zero_less_mult_pos2:
- "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
-apply (cases "b\<le>0")
- apply (auto simp add: le_less not_less)
-apply (drule_tac mult_pos_neg2 [of a b])
- apply (auto dest: less_not_sym)
-done
+lemma zero_less_mult_pos: "0 < a * b \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
+ apply (cases "b \<le> 0")
+ apply (auto simp add: le_less not_less)
+ apply (drule_tac mult_pos_neg [of a b])
+ apply (auto dest: less_not_sym)
+ done
-text\<open>Strict monotonicity in both arguments\<close>
+lemma zero_less_mult_pos2: "0 < b * a \<Longrightarrow> 0 < a \<Longrightarrow> 0 < b"
+ apply (cases "b \<le> 0")
+ apply (auto simp add: le_less not_less)
+ apply (drule_tac mult_pos_neg2 [of a b])
+ apply (auto dest: less_not_sym)
+ done
+
+text \<open>Strict monotonicity in both arguments\<close>
lemma mult_strict_mono:
assumes "a < b" and "c < d" and "0 < b" and "0 \<le> c"
shows "a * c < b * d"
- using assms apply (cases "c=0")
- apply (simp)
+ using assms
+ apply (cases "c = 0")
+ apply simp
apply (erule mult_strict_right_mono [THEN less_trans])
- apply (force simp add: le_less)
- apply (erule mult_strict_left_mono, assumption)
+ apply (auto simp add: le_less)
+ apply (erule (1) mult_strict_left_mono)
done
-text\<open>This weaker variant has more natural premises\<close>
+text \<open>This weaker variant has more natural premises\<close>
lemma mult_strict_mono':
assumes "a < b" and "c < d" and "0 \<le> a" and "0 \<le> c"
shows "a * c < b * d"
-by (rule mult_strict_mono) (insert assms, auto)
+ by (rule mult_strict_mono) (insert assms, auto)
lemma mult_less_le_imp_less:
assumes "a < b" and "c \<le> d" and "0 \<le> a" and "0 < c"
shows "a * c < b * d"
- using assms apply (subgoal_tac "a * c < b * c")
+ using assms
+ apply (subgoal_tac "a * c < b * c")
apply (erule less_le_trans)
apply (erule mult_left_mono)
apply simp
- apply (erule mult_strict_right_mono)
- apply assumption
+ apply (erule (1) mult_strict_right_mono)
done
lemma mult_le_less_imp_less:
assumes "a \<le> b" and "c < d" and "0 < a" and "0 \<le> c"
shows "a * c < b * d"
- using assms apply (subgoal_tac "a * c \<le> b * c")
+ using assms
+ apply (subgoal_tac "a * c \<le> b * c")
apply (erule le_less_trans)
apply (erule mult_strict_left_mono)
apply simp
- apply (erule mult_right_mono)
- apply simp
+ apply (erule (1) mult_right_mono)
done
end
@@ -1504,9 +1434,9 @@
shows "u * x + v * y < a"
proof -
from assms have "u * x + v * y < u * a + v * a"
- by (cases "u = 0")
- (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
- thus ?thesis using assms unfolding distrib_right[symmetric] by simp
+ by (cases "u = 0") (auto intro!: add_less_le_mono mult_strict_left_mono mult_left_mono)
+ with assms show ?thesis
+ unfolding distrib_right[symmetric] by simp
qed
end
@@ -1519,8 +1449,8 @@
proof
fix a b c :: 'a
assume "a \<le> b" "0 \<le> c"
- thus "c * a \<le> c * b" by (rule comm_mult_left_mono)
- thus "a * c \<le> b * c" by (simp only: mult.commute)
+ then show "c * a \<le> c * b" by (rule comm_mult_left_mono)
+ then show "a * c \<le> b * c" by (simp only: mult.commute)
qed
end
@@ -1542,15 +1472,15 @@
proof
fix a b c :: 'a
assume "a < b" "0 < c"
- thus "c * a < c * b" by (rule comm_mult_strict_left_mono)
- thus "a * c < b * c" by (simp only: mult.commute)
+ then show "c * a < c * b" by (rule comm_mult_strict_left_mono)
+ then show "a * c < b * c" by (simp only: mult.commute)
qed
subclass ordered_cancel_comm_semiring
proof
fix a b c :: 'a
assume "a \<le> b" "0 \<le> c"
- thus "c * a \<le> c * b"
+ then show "c * a \<le> c * b"
unfolding le_less
using mult_strict_left_mono by (cases "c = 0") auto
qed
@@ -1562,40 +1492,33 @@
subclass ordered_ab_group_add ..
-lemma less_add_iff1:
- "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
-by (simp add: algebra_simps)
+lemma less_add_iff1: "a * e + c < b * e + d \<longleftrightarrow> (a - b) * e + c < d"
+ by (simp add: algebra_simps)
-lemma less_add_iff2:
- "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
-by (simp add: algebra_simps)
+lemma less_add_iff2: "a * e + c < b * e + d \<longleftrightarrow> c < (b - a) * e + d"
+ by (simp add: algebra_simps)
-lemma le_add_iff1:
- "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
-by (simp add: algebra_simps)
+lemma le_add_iff1: "a * e + c \<le> b * e + d \<longleftrightarrow> (a - b) * e + c \<le> d"
+ by (simp add: algebra_simps)
-lemma le_add_iff2:
- "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
-by (simp add: algebra_simps)
+lemma le_add_iff2: "a * e + c \<le> b * e + d \<longleftrightarrow> c \<le> (b - a) * e + d"
+ by (simp add: algebra_simps)
-lemma mult_left_mono_neg:
- "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
+lemma mult_left_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> c * a \<le> c * b"
apply (drule mult_left_mono [of _ _ "- c"])
apply simp_all
done
-lemma mult_right_mono_neg:
- "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
+lemma mult_right_mono_neg: "b \<le> a \<Longrightarrow> c \<le> 0 \<Longrightarrow> a * c \<le> b * c"
apply (drule mult_right_mono [of _ _ "- c"])
apply simp_all
done
lemma mult_nonpos_nonpos: "a \<le> 0 \<Longrightarrow> b \<le> 0 \<Longrightarrow> 0 \<le> a * b"
-using mult_right_mono_neg [of a 0 b] by simp
+ using mult_right_mono_neg [of a 0 b] by simp
-lemma split_mult_pos_le:
- "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
-by (auto simp add: mult_nonpos_nonpos)
+lemma split_mult_pos_le: "(0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0) \<Longrightarrow> 0 \<le> a * b"
+ by (auto simp add: mult_nonpos_nonpos)
end
@@ -1608,12 +1531,12 @@
proof
fix a b
show "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
- by (auto simp add: abs_if not_le not_less algebra_simps simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
+ by (auto simp add: abs_if not_le not_less algebra_simps
+ simp del: add.commute dest: add_neg_neg add_nonneg_nonneg)
qed (auto simp add: abs_if)
lemma zero_le_square [simp]: "0 \<le> a * a"
- using linear [of 0 a]
- by (auto simp add: mult_nonpos_nonpos)
+ using linear [of 0 a] by (auto simp add: mult_nonpos_nonpos)
lemma not_square_less_zero [simp]: "\<not> (a * a < 0)"
by (simp add: not_less)
@@ -1621,12 +1544,10 @@
proposition abs_eq_iff: "\<bar>x\<bar> = \<bar>y\<bar> \<longleftrightarrow> x = y \<or> x = -y"
by (auto simp add: abs_if split: if_split_asm)
-lemma sum_squares_ge_zero:
- "0 \<le> x * x + y * y"
+lemma sum_squares_ge_zero: "0 \<le> x * x + y * y"
by (intro add_nonneg_nonneg zero_le_square)
-lemma not_sum_squares_lt_zero:
- "\<not> x * x + y * y < 0"
+lemma not_sum_squares_lt_zero: "\<not> x * x + y * y < 0"
by (simp add: not_less sum_squares_ge_zero)
end
@@ -1638,40 +1559,49 @@
subclass linordered_ring ..
lemma mult_strict_left_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> c * a < c * b"
-using mult_strict_left_mono [of b a "- c"] by simp
+ using mult_strict_left_mono [of b a "- c"] by simp
lemma mult_strict_right_mono_neg: "b < a \<Longrightarrow> c < 0 \<Longrightarrow> a * c < b * c"
-using mult_strict_right_mono [of b a "- c"] by simp
+ using mult_strict_right_mono [of b a "- c"] by simp
lemma mult_neg_neg: "a < 0 \<Longrightarrow> b < 0 \<Longrightarrow> 0 < a * b"
-using mult_strict_right_mono_neg [of a 0 b] by simp
+ using mult_strict_right_mono_neg [of a 0 b] by simp
subclass ring_no_zero_divisors
proof
fix a b
- assume "a \<noteq> 0" then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
- assume "b \<noteq> 0" then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
+ assume "a \<noteq> 0"
+ then have A: "a < 0 \<or> 0 < a" by (simp add: neq_iff)
+ assume "b \<noteq> 0"
+ then have B: "b < 0 \<or> 0 < b" by (simp add: neq_iff)
have "a * b < 0 \<or> 0 < a * b"
proof (cases "a < 0")
- case True note A' = this
- show ?thesis proof (cases "b < 0")
- case True with A'
- show ?thesis by (auto dest: mult_neg_neg)
+ case A': True
+ show ?thesis
+ proof (cases "b < 0")
+ case True
+ with A' show ?thesis by (auto dest: mult_neg_neg)
next
- case False with B have "0 < b" by auto
+ case False
+ with B have "0 < b" by auto
with A' show ?thesis by (auto dest: mult_strict_right_mono)
qed
next
- case False with A have A': "0 < a" by auto
- show ?thesis proof (cases "b < 0")
- case True with A'
- show ?thesis by (auto dest: mult_strict_right_mono_neg)
+ case False
+ with A have A': "0 < a" by auto
+ show ?thesis
+ proof (cases "b < 0")
+ case True
+ with A' show ?thesis
+ by (auto dest: mult_strict_right_mono_neg)
next
- case False with B have "0 < b" by auto
+ case False
+ with B have "0 < b" by auto
with A' show ?thesis by auto
qed
qed
- then show "a * b \<noteq> 0" by (simp add: neq_iff)
+ then show "a * b \<noteq> 0"
+ by (simp add: neq_iff)
qed
lemma zero_less_mult_iff: "0 < a * b \<longleftrightarrow> 0 < a \<and> 0 < b \<or> a < 0 \<and> b < 0"
@@ -1681,84 +1611,66 @@
lemma zero_le_mult_iff: "0 \<le> a * b \<longleftrightarrow> 0 \<le> a \<and> 0 \<le> b \<or> a \<le> 0 \<and> b \<le> 0"
by (auto simp add: eq_commute [of 0] le_less not_less zero_less_mult_iff)
-lemma mult_less_0_iff:
- "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
- apply (insert zero_less_mult_iff [of "-a" b])
- apply force
- done
+lemma mult_less_0_iff: "a * b < 0 \<longleftrightarrow> 0 < a \<and> b < 0 \<or> a < 0 \<and> 0 < b"
+ using zero_less_mult_iff [of "- a" b] by auto
-lemma mult_le_0_iff:
- "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
- apply (insert zero_le_mult_iff [of "-a" b])
- apply force
- done
+lemma mult_le_0_iff: "a * b \<le> 0 \<longleftrightarrow> 0 \<le> a \<and> b \<le> 0 \<or> a \<le> 0 \<and> 0 \<le> b"
+ using zero_le_mult_iff [of "- a" b] by auto
-text\<open>Cancellation laws for @{term "c*a < c*b"} and @{term "a*c < b*c"},
- also with the relations \<open>\<le>\<close> and equality.\<close>
+text \<open>
+ Cancellation laws for @{term "c * a < c * b"} and @{term "a * c < b * c"},
+ also with the relations \<open>\<le>\<close> and equality.
+\<close>
-text\<open>These ``disjunction'' versions produce two cases when the comparison is
- an assumption, but effectively four when the comparison is a goal.\<close>
+text \<open>
+ These ``disjunction'' versions produce two cases when the comparison is
+ an assumption, but effectively four when the comparison is a goal.
+\<close>
-lemma mult_less_cancel_right_disj:
- "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a"
+lemma mult_less_cancel_right_disj: "a * c < b * c \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a"
apply (cases "c = 0")
- apply (auto simp add: neq_iff mult_strict_right_mono
- mult_strict_right_mono_neg)
- apply (auto simp add: not_less
- not_le [symmetric, of "a*c"]
- not_le [symmetric, of a])
+ apply (auto simp add: neq_iff mult_strict_right_mono mult_strict_right_mono_neg)
+ apply (auto simp add: not_less not_le [symmetric, of "a*c"] not_le [symmetric, of a])
apply (erule_tac [!] notE)
- apply (auto simp add: less_imp_le mult_right_mono
- mult_right_mono_neg)
+ apply (auto simp add: less_imp_le mult_right_mono mult_right_mono_neg)
done
-lemma mult_less_cancel_left_disj:
- "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a"
+lemma mult_less_cancel_left_disj: "c * a < c * b \<longleftrightarrow> 0 < c \<and> a < b \<or> c < 0 \<and> b < a"
apply (cases "c = 0")
- apply (auto simp add: neq_iff mult_strict_left_mono
- mult_strict_left_mono_neg)
- apply (auto simp add: not_less
- not_le [symmetric, of "c*a"]
- not_le [symmetric, of a])
+ apply (auto simp add: neq_iff mult_strict_left_mono mult_strict_left_mono_neg)
+ apply (auto simp add: not_less not_le [symmetric, of "c * a"] not_le [symmetric, of a])
apply (erule_tac [!] notE)
- apply (auto simp add: less_imp_le mult_left_mono
- mult_left_mono_neg)
+ apply (auto simp add: less_imp_le mult_left_mono mult_left_mono_neg)
done
-text\<open>The ``conjunction of implication'' lemmas produce two cases when the
-comparison is a goal, but give four when the comparison is an assumption.\<close>
+text \<open>
+ The ``conjunction of implication'' lemmas produce two cases when the
+ comparison is a goal, but give four when the comparison is an assumption.
+\<close>
-lemma mult_less_cancel_right:
- "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
+lemma mult_less_cancel_right: "a * c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
using mult_less_cancel_right_disj [of a c b] by auto
-lemma mult_less_cancel_left:
- "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
+lemma mult_less_cancel_left: "c * a < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < b) \<and> (c \<le> 0 \<longrightarrow> b < a)"
using mult_less_cancel_left_disj [of c a b] by auto
-lemma mult_le_cancel_right:
- "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
-by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
+lemma mult_le_cancel_right: "a * c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
+ by (simp add: not_less [symmetric] mult_less_cancel_right_disj)
-lemma mult_le_cancel_left:
- "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
-by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
+lemma mult_le_cancel_left: "c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> a)"
+ by (simp add: not_less [symmetric] mult_less_cancel_left_disj)
-lemma mult_le_cancel_left_pos:
- "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
-by (auto simp: mult_le_cancel_left)
+lemma mult_le_cancel_left_pos: "0 < c \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> a \<le> b"
+ by (auto simp: mult_le_cancel_left)
-lemma mult_le_cancel_left_neg:
- "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
-by (auto simp: mult_le_cancel_left)
+lemma mult_le_cancel_left_neg: "c < 0 \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> b \<le> a"
+ by (auto simp: mult_le_cancel_left)
-lemma mult_less_cancel_left_pos:
- "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
-by (auto simp: mult_less_cancel_left)
+lemma mult_less_cancel_left_pos: "0 < c \<Longrightarrow> c * a < c * b \<longleftrightarrow> a < b"
+ by (auto simp: mult_less_cancel_left)
-lemma mult_less_cancel_left_neg:
- "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
-by (auto simp: mult_less_cancel_left)
+lemma mult_less_cancel_left_neg: "c < 0 \<Longrightarrow> c * a < c * b \<longleftrightarrow> b < a"
+ by (auto simp: mult_less_cancel_left)
end
@@ -1783,19 +1695,19 @@
begin
subclass zero_neq_one
- proof qed (insert zero_less_one, blast)
+ by standard (insert zero_less_one, blast)
subclass comm_semiring_1
- proof qed (rule mult_1_left)
+ by standard (rule mult_1_left)
lemma zero_le_one [simp]: "0 \<le> 1"
-by (rule zero_less_one [THEN less_imp_le])
+ by (rule zero_less_one [THEN less_imp_le])
lemma not_one_le_zero [simp]: "\<not> 1 \<le> 0"
-by (simp add: not_le)
+ by (simp add: not_le)
lemma not_one_less_zero [simp]: "\<not> 1 < 0"
-by (simp add: not_less)
+ by (simp add: not_less)
lemma mult_left_le: "c \<le> 1 \<Longrightarrow> 0 \<le> a \<Longrightarrow> a * c \<le> a"
using mult_left_mono[of c 1 a] by simp
@@ -1812,8 +1724,7 @@
assumes le_add_diff_inverse2 [simp]: "b \<le> a \<Longrightarrow> a - b + b = a"
begin
-subclass linordered_nonzero_semiring
- proof qed
+subclass linordered_nonzero_semiring ..
text \<open>Addition is the inverse of subtraction.\<close>
@@ -1823,31 +1734,31 @@
lemma add_diff_inverse: "\<not> a < b \<Longrightarrow> b + (a - b) = a"
by simp
-lemma add_le_imp_le_diff:
- shows "i + k \<le> n \<Longrightarrow> i \<le> n - k"
+lemma add_le_imp_le_diff: "i + k \<le> n \<Longrightarrow> i \<le> n - k"
apply (subst add_le_cancel_right [where c=k, symmetric])
apply (frule le_add_diff_inverse2)
apply (simp only: add.assoc [symmetric])
- using add_implies_diff by fastforce
+ using add_implies_diff apply fastforce
+ done
lemma add_le_add_imp_diff_le:
- assumes a1: "i + k \<le> n"
- and a2: "n \<le> j + k"
- shows "\<lbrakk>i + k \<le> n; n \<le> j + k\<rbrakk> \<Longrightarrow> n - k \<le> j"
+ assumes 1: "i + k \<le> n"
+ and 2: "n \<le> j + k"
+ shows "i + k \<le> n \<Longrightarrow> n \<le> j + k \<Longrightarrow> n - k \<le> j"
proof -
have "n - (i + k) + (i + k) = n"
- using a1 by simp
+ using 1 by simp
moreover have "n - k = n - k - i + i"
- using a1 by (simp add: add_le_imp_le_diff)
+ using 1 by (simp add: add_le_imp_le_diff)
ultimately show ?thesis
- using a2
+ using 2
apply (simp add: add.assoc [symmetric])
- apply (rule add_le_imp_le_diff [of _ k "j+k", simplified add_diff_cancel_right'])
- by (simp add: add.commute diff_diff_add)
+ apply (rule add_le_imp_le_diff [of _ k "j + k", simplified add_diff_cancel_right'])
+ apply (simp add: add.commute diff_diff_add)
+ done
qed
-lemma less_1_mult:
- "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n"
+lemma less_1_mult: "1 < m \<Longrightarrow> 1 < n \<Longrightarrow> 1 < m * n"
using mult_strict_mono [of 1 m 1 n] by (simp add: less_trans [OF zero_less_one])
end
@@ -1864,90 +1775,73 @@
subclass linordered_semidom
proof
have "0 \<le> 1 * 1" by (rule zero_le_square)
- thus "0 < 1" by (simp add: le_less)
- show "\<And>b a. b \<le> a \<Longrightarrow> a - b + b = a"
+ then show "0 < 1" by (simp add: le_less)
+ show "b \<le> a \<Longrightarrow> a - b + b = a" for a b
by simp
qed
lemma linorder_neqE_linordered_idom:
- assumes "x \<noteq> y" obtains "x < y" | "y < x"
+ assumes "x \<noteq> y"
+ obtains "x < y" | "y < x"
using assms by (rule neqE)
text \<open>These cancellation simprules also produce two cases when the comparison is a goal.\<close>
-lemma mult_le_cancel_right1:
- "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
-by (insert mult_le_cancel_right [of 1 c b], simp)
+lemma mult_le_cancel_right1: "c \<le> b * c \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
+ using mult_le_cancel_right [of 1 c b] by simp
-lemma mult_le_cancel_right2:
- "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
-by (insert mult_le_cancel_right [of a c 1], simp)
+lemma mult_le_cancel_right2: "a * c \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
+ using mult_le_cancel_right [of a c 1] by simp
-lemma mult_le_cancel_left1:
- "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
-by (insert mult_le_cancel_left [of c 1 b], simp)
+lemma mult_le_cancel_left1: "c \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> 1 \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 1)"
+ using mult_le_cancel_left [of c 1 b] by simp
-lemma mult_le_cancel_left2:
- "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
-by (insert mult_le_cancel_left [of c a 1], simp)
+lemma mult_le_cancel_left2: "c * a \<le> c \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> 1) \<and> (c < 0 \<longrightarrow> 1 \<le> a)"
+ using mult_le_cancel_left [of c a 1] by simp
-lemma mult_less_cancel_right1:
- "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
-by (insert mult_less_cancel_right [of 1 c b], simp)
+lemma mult_less_cancel_right1: "c < b * c \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
+ using mult_less_cancel_right [of 1 c b] by simp
-lemma mult_less_cancel_right2:
- "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
-by (insert mult_less_cancel_right [of a c 1], simp)
+lemma mult_less_cancel_right2: "a * c < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
+ using mult_less_cancel_right [of a c 1] by simp
-lemma mult_less_cancel_left1:
- "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
-by (insert mult_less_cancel_left [of c 1 b], simp)
+lemma mult_less_cancel_left1: "c < c * b \<longleftrightarrow> (0 \<le> c \<longrightarrow> 1 < b) \<and> (c \<le> 0 \<longrightarrow> b < 1)"
+ using mult_less_cancel_left [of c 1 b] by simp
-lemma mult_less_cancel_left2:
- "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
-by (insert mult_less_cancel_left [of c a 1], simp)
+lemma mult_less_cancel_left2: "c * a < c \<longleftrightarrow> (0 \<le> c \<longrightarrow> a < 1) \<and> (c \<le> 0 \<longrightarrow> 1 < a)"
+ using mult_less_cancel_left [of c a 1] by simp
-lemma sgn_sgn [simp]:
- "sgn (sgn a) = sgn a"
-unfolding sgn_if by simp
+lemma sgn_sgn [simp]: "sgn (sgn a) = sgn a"
+ unfolding sgn_if by simp
-lemma sgn_0_0:
- "sgn a = 0 \<longleftrightarrow> a = 0"
-unfolding sgn_if by simp
+lemma sgn_0_0: "sgn a = 0 \<longleftrightarrow> a = 0"
+ unfolding sgn_if by simp
-lemma sgn_1_pos:
- "sgn a = 1 \<longleftrightarrow> a > 0"
-unfolding sgn_if by simp
+lemma sgn_1_pos: "sgn a = 1 \<longleftrightarrow> a > 0"
+ unfolding sgn_if by simp
-lemma sgn_1_neg:
- "sgn a = - 1 \<longleftrightarrow> a < 0"
-unfolding sgn_if by auto
+lemma sgn_1_neg: "sgn a = - 1 \<longleftrightarrow> a < 0"
+ unfolding sgn_if by auto
-lemma sgn_pos [simp]:
- "0 < a \<Longrightarrow> sgn a = 1"
-unfolding sgn_1_pos .
+lemma sgn_pos [simp]: "0 < a \<Longrightarrow> sgn a = 1"
+ by (simp only: sgn_1_pos)
-lemma sgn_neg [simp]:
- "a < 0 \<Longrightarrow> sgn a = - 1"
-unfolding sgn_1_neg .
+lemma sgn_neg [simp]: "a < 0 \<Longrightarrow> sgn a = - 1"
+ by (simp only: sgn_1_neg)
-lemma sgn_times:
- "sgn (a * b) = sgn a * sgn b"
-by (auto simp add: sgn_if zero_less_mult_iff)
+lemma sgn_times: "sgn (a * b) = sgn a * sgn b"
+ by (auto simp add: sgn_if zero_less_mult_iff)
lemma abs_sgn: "\<bar>k\<bar> = k * sgn k"
-unfolding sgn_if abs_if by auto
+ unfolding sgn_if abs_if by auto
-lemma sgn_greater [simp]:
- "0 < sgn a \<longleftrightarrow> 0 < a"
+lemma sgn_greater [simp]: "0 < sgn a \<longleftrightarrow> 0 < a"
unfolding sgn_if by auto
-lemma sgn_less [simp]:
- "sgn a < 0 \<longleftrightarrow> a < 0"
+lemma sgn_less [simp]: "sgn a < 0 \<longleftrightarrow> a < 0"
unfolding sgn_if by auto
-lemma abs_sgn_eq:
- "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
+lemma abs_sgn_eq: "\<bar>sgn a\<bar> = (if a = 0 then 0 else 1)"
by (simp add: sgn_if)
lemma abs_dvd_iff [simp]: "\<bar>m\<bar> dvd k \<longleftrightarrow> m dvd k"
@@ -1956,36 +1850,31 @@
lemma dvd_abs_iff [simp]: "m dvd \<bar>k\<bar> \<longleftrightarrow> m dvd k"
by (simp add: abs_if)
-lemma dvd_if_abs_eq:
- "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
-by(subst abs_dvd_iff[symmetric]) simp
+lemma dvd_if_abs_eq: "\<bar>l\<bar> = \<bar>k\<bar> \<Longrightarrow> l dvd k"
+ by (subst abs_dvd_iff [symmetric]) simp
-text \<open>The following lemmas can be proven in more general structures, but
-are dangerous as simp rules in absence of @{thm neg_equal_zero},
-@{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.\<close>
+text \<open>
+ The following lemmas can be proven in more general structures, but
+ are dangerous as simp rules in absence of @{thm neg_equal_zero},
+ @{thm neg_less_pos}, @{thm neg_less_eq_nonneg}.
+\<close>
-lemma equation_minus_iff_1 [simp, no_atp]:
- "1 = - a \<longleftrightarrow> a = - 1"
+lemma equation_minus_iff_1 [simp, no_atp]: "1 = - a \<longleftrightarrow> a = - 1"
by (fact equation_minus_iff)
-lemma minus_equation_iff_1 [simp, no_atp]:
- "- a = 1 \<longleftrightarrow> a = - 1"
+lemma minus_equation_iff_1 [simp, no_atp]: "- a = 1 \<longleftrightarrow> a = - 1"
by (subst minus_equation_iff, auto)
-lemma le_minus_iff_1 [simp, no_atp]:
- "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
+lemma le_minus_iff_1 [simp, no_atp]: "1 \<le> - b \<longleftrightarrow> b \<le> - 1"
by (fact le_minus_iff)
-lemma minus_le_iff_1 [simp, no_atp]:
- "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
+lemma minus_le_iff_1 [simp, no_atp]: "- a \<le> 1 \<longleftrightarrow> - 1 \<le> a"
by (fact minus_le_iff)
-lemma less_minus_iff_1 [simp, no_atp]:
- "1 < - b \<longleftrightarrow> b < - 1"
+lemma less_minus_iff_1 [simp, no_atp]: "1 < - b \<longleftrightarrow> b < - 1"
by (fact less_minus_iff)
-lemma minus_less_iff_1 [simp, no_atp]:
- "- a < 1 \<longleftrightarrow> - 1 < a"
+lemma minus_less_iff_1 [simp, no_atp]: "- a < 1 \<longleftrightarrow> - 1 < a"
by (fact minus_less_iff)
end
@@ -1993,15 +1882,16 @@
text \<open>Simprules for comparisons where common factors can be cancelled.\<close>
lemmas mult_compare_simps =
- mult_le_cancel_right mult_le_cancel_left
- mult_le_cancel_right1 mult_le_cancel_right2
- mult_le_cancel_left1 mult_le_cancel_left2
- mult_less_cancel_right mult_less_cancel_left
- mult_less_cancel_right1 mult_less_cancel_right2
- mult_less_cancel_left1 mult_less_cancel_left2
- mult_cancel_right mult_cancel_left
- mult_cancel_right1 mult_cancel_right2
- mult_cancel_left1 mult_cancel_left2
+ mult_le_cancel_right mult_le_cancel_left
+ mult_le_cancel_right1 mult_le_cancel_right2
+ mult_le_cancel_left1 mult_le_cancel_left2
+ mult_less_cancel_right mult_less_cancel_left
+ mult_less_cancel_right1 mult_less_cancel_right2
+ mult_less_cancel_left1 mult_less_cancel_left2
+ mult_cancel_right mult_cancel_left
+ mult_cancel_right1 mult_cancel_right2
+ mult_cancel_left1 mult_cancel_left2
+
text \<open>Reasoning about inequalities with division\<close>
@@ -2012,7 +1902,7 @@
proof -
have "a + 0 < a + 1"
by (blast intro: zero_less_one add_strict_left_mono)
- thus ?thesis by simp
+ then show ?thesis by simp
qed
end
@@ -2020,12 +1910,10 @@
context linordered_idom
begin
-lemma mult_right_le_one_le:
- "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
+lemma mult_right_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> x * y \<le> x"
by (rule mult_left_le)
-lemma mult_left_le_one_le:
- "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
+lemma mult_left_le_one_le: "0 \<le> x \<Longrightarrow> 0 \<le> y \<Longrightarrow> y \<le> 1 \<Longrightarrow> y * x \<le> x"
by (auto simp add: mult_le_cancel_right2)
end
@@ -2035,12 +1923,10 @@
context linordered_idom
begin
-lemma mult_sgn_abs:
- "sgn x * \<bar>x\<bar> = x"
+lemma mult_sgn_abs: "sgn x * \<bar>x\<bar> = x"
unfolding abs_if sgn_if by auto
-lemma abs_one [simp]:
- "\<bar>1\<bar> = 1"
+lemma abs_one [simp]: "\<bar>1\<bar> = 1"
by (simp add: abs_if)
end
@@ -2052,57 +1938,54 @@
context linordered_idom
begin
-subclass ordered_ring_abs proof
-qed (auto simp add: abs_if not_less mult_less_0_iff)
+subclass ordered_ring_abs
+ by standard (auto simp add: abs_if not_less mult_less_0_iff)
-lemma abs_mult:
- "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
+lemma abs_mult: "\<bar>a * b\<bar> = \<bar>a\<bar> * \<bar>b\<bar>"
by (rule abs_eq_mult) auto
-lemma abs_mult_self [simp]:
- "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
+lemma abs_mult_self [simp]: "\<bar>a\<bar> * \<bar>a\<bar> = a * a"
by (simp add: abs_if)
lemma abs_mult_less:
- "\<bar>a\<bar> < c \<Longrightarrow> \<bar>b\<bar> < d \<Longrightarrow> \<bar>a\<bar> * \<bar>b\<bar> < c * d"
+ assumes ac: "\<bar>a\<bar> < c"
+ and bd: "\<bar>b\<bar> < d"
+ shows "\<bar>a\<bar> * \<bar>b\<bar> < c * d"
proof -
- assume ac: "\<bar>a\<bar> < c"
- hence cpos: "0<c" by (blast intro: le_less_trans abs_ge_zero)
- assume "\<bar>b\<bar> < d"
- thus ?thesis by (simp add: ac cpos mult_strict_mono)
+ from ac have "0 < c"
+ by (blast intro: le_less_trans abs_ge_zero)
+ with bd show ?thesis by (simp add: ac mult_strict_mono)
qed
-lemma abs_less_iff:
- "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
+lemma abs_less_iff: "\<bar>a\<bar> < b \<longleftrightarrow> a < b \<and> - a < b"
by (simp add: less_le abs_le_iff) (auto simp add: abs_if)
-lemma abs_mult_pos:
- "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
+lemma abs_mult_pos: "0 \<le> x \<Longrightarrow> \<bar>y\<bar> * x = \<bar>y * x\<bar>"
by (simp add: abs_mult)
-lemma abs_diff_less_iff:
- "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
+lemma abs_diff_less_iff: "\<bar>x - a\<bar> < r \<longleftrightarrow> a - r < x \<and> x < a + r"
by (auto simp add: diff_less_eq ac_simps abs_less_iff)
-lemma abs_diff_le_iff:
- "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
+lemma abs_diff_le_iff: "\<bar>x - a\<bar> \<le> r \<longleftrightarrow> a - r \<le> x \<and> x \<le> a + r"
by (auto simp add: diff_le_eq ac_simps abs_le_iff)
lemma abs_add_one_gt_zero: "0 < 1 + \<bar>x\<bar>"
- by (force simp: abs_if not_less intro: zero_less_one add_strict_increasing less_trans)
+ by (auto simp: abs_if not_less intro: zero_less_one add_strict_increasing less_trans)
end
subsection \<open>Dioids\<close>
-text \<open>Dioids are the alternative extensions of semirings, a semiring can either be a ring or a dioid
-but never both.\<close>
+text \<open>
+ Dioids are the alternative extensions of semirings, a semiring can
+ either be a ring or a dioid but never both.
+\<close>
class dioid = semiring_1 + canonically_ordered_monoid_add
begin
subclass ordered_semiring
- proof qed (auto simp: le_iff_add distrib_left distrib_right)
+ by standard (auto simp: le_iff_add distrib_left distrib_right)
end