src/HOL/Groups.thy
 author wenzelm Sat, 18 Jul 2015 22:58:50 +0200 changeset 60758 d8d85a8172b5 parent 59815 cce82e360c2f child 60762 bf0c76ccee8d permissions -rw-r--r--
isabelle update_cartouches;

(*  Title:   HOL/Groups.thy
Author:  Gertrud Bauer, Steven Obua, Lawrence C Paulson, Markus Wenzel, Jeremy Avigad
*)

section \<open>Groups, also combined with orderings\<close>

theory Groups
imports Orderings
begin

subsection \<open>Dynamic facts\<close>

named_theorems ac_simps "associativity and commutativity simplification rules"

text\<open>The rewrites accumulated in @{text algebra_simps} deal with the
classical algebraic structures of groups, rings and family. They simplify
terms by multiplying everything out (in case of a ring) and bringing sums and
products into a canonical form (by ordered rewriting). As a result it decides
group and ring equalities but also helps with inequalities.

Of course it also works for fields, but it knows nothing about multiplicative
inverses or division. This is catered for by @{text field_simps}.\<close>

named_theorems algebra_simps "algebra simplification rules"

text\<open>Lemmas @{text field_simps} multiply with denominators in (in)equations
if they can be proved to be non-zero (for equations) or positive/negative
(for inequations). Can be too aggressive and is therefore separate from the
more benign @{text algebra_simps}.\<close>

named_theorems field_simps "algebra simplification rules for fields"

subsection \<open>Abstract structures\<close>

text \<open>
These locales provide basic structures for interpretation into
bigger structures;  extensions require careful thinking, otherwise
undesired effects may occur due to interpretation.
\<close>

locale semigroup =
fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "*" 70)
assumes assoc [ac_simps]: "a * b * c = a * (b * c)"

locale abel_semigroup = semigroup +
assumes commute [ac_simps]: "a * b = b * a"
begin

lemma left_commute [ac_simps]:
"b * (a * c) = a * (b * c)"
proof -
have "(b * a) * c = (a * b) * c"
by (simp only: commute)
then show ?thesis
by (simp only: assoc)
qed

end

locale monoid = semigroup +
fixes z :: 'a ("1")
assumes left_neutral [simp]: "1 * a = a"
assumes right_neutral [simp]: "a * 1 = a"

locale comm_monoid = abel_semigroup +
fixes z :: 'a ("1")
assumes comm_neutral: "a * 1 = a"
begin

sublocale monoid
by default (simp_all add: commute comm_neutral)

end

subsection \<open>Generic operations\<close>

class zero =
fixes zero :: 'a  ("0")

class one =
fixes one  :: 'a  ("1")

hide_const (open) zero one

lemma Let_0 [simp]: "Let 0 f = f 0"
unfolding Let_def ..

lemma Let_1 [simp]: "Let 1 f = f 1"
unfolding Let_def ..

setup \<open>
(fn Const(@{const_name Groups.zero}, _) => true
| Const(@{const_name Groups.one}, _) => true
| _ => false)
\<close>

simproc_setup reorient_zero ("0 = x") = Reorient_Proc.proc
simproc_setup reorient_one ("1 = x") = Reorient_Proc.proc

typed_print_translation \<open>
let
fun tr' c = (c, fn ctxt => fn T => fn ts =>
if null ts andalso Printer.type_emphasis ctxt T then
Syntax.const @{syntax_const "_constrain"} $Syntax.const c$
Syntax_Phases.term_of_typ ctxt T
else raise Match);
in map tr' [@{const_syntax Groups.one}, @{const_syntax Groups.zero}] end;
\<close> -- \<open>show types that are presumably too general\<close>

class plus =
fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "+" 65)

class minus =
fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "-" 65)

class uminus =
fixes uminus :: "'a \<Rightarrow> 'a"  ("- _" [81] 80)

class times =
fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"  (infixl "*" 70)

subsection \<open>Semigroups and Monoids\<close>

assumes add_assoc [algebra_simps, field_simps]: "(a + b) + c = a + (b + c)"
begin

end

assumes add_commute [algebra_simps, field_simps]: "a + b = b + a"
begin

end

class semigroup_mult = times +
assumes mult_assoc [algebra_simps, field_simps]: "(a * b) * c = a * (b * c)"
begin

sublocale mult!: semigroup times
by default (fact mult_assoc)

end

hide_fact mult_assoc

class ab_semigroup_mult = semigroup_mult +
assumes mult_commute [algebra_simps, field_simps]: "a * b = b * a"
begin

sublocale mult!: abel_semigroup times
by default (fact mult_commute)

declare mult.left_commute [algebra_simps, field_simps]

theorems mult_ac = mult.assoc mult.commute mult.left_commute

end

hide_fact mult_commute

theorems mult_ac = mult.assoc mult.commute mult.left_commute

assumes add_0_left: "0 + a = a"
and add_0_right: "a + 0 = a"
begin

end

lemma zero_reorient: "0 = x \<longleftrightarrow> x = 0"
by (fact eq_commute)

assumes add_0: "0 + a = a"
begin

end

class monoid_mult = one + semigroup_mult +
assumes mult_1_left: "1 * a  = a"
and mult_1_right: "a * 1 = a"
begin

sublocale mult!: monoid times 1
by default (fact mult_1_left mult_1_right)+

end

lemma one_reorient: "1 = x \<longleftrightarrow> x = 1"
by (fact eq_commute)

class comm_monoid_mult = one + ab_semigroup_mult +
assumes mult_1: "1 * a = a"
begin

subclass monoid_mult
by default (simp_all add: mult_1 mult.commute [of _ 1])

sublocale mult!: comm_monoid times 1

end

assumes add_left_imp_eq: "a + b = a + c \<Longrightarrow> b = c"
assumes add_right_imp_eq: "b + a = c + a \<Longrightarrow> b = c"
begin

"a + b = a + c \<longleftrightarrow> b = c"

"b + a = c + a \<longleftrightarrow> b = c"

end

assumes add_diff_cancel_left' [simp]: "(a + b) - a = b"
assumes diff_diff_add [algebra_simps, field_simps]: "a - b - c = a - (b + c)"
begin

"(a + b) - b = a"

proof
fix a b c :: 'a
assume "a + b = a + c"
then have "a + b - a = a + c - a"
by simp
then show "b = c"
by simp
next
fix a b c :: 'a
assume "b + a = c + a"
then have "b + a - a = c + a - a"
by simp
then show "b = c"
by simp
qed

"(c + a) - (c + b) = a - b"

"(a + c) - (b + c) = a - b"

lemma diff_right_commute:
"a - c - b = a - b - c"

end

begin

lemma diff_zero [simp]:
"a - 0 = a"
using add_diff_cancel_right' [of a 0] by simp

lemma diff_cancel [simp]:
"a - a = 0"
proof -
have "(a + 0) - (a + 0) = 0" by (simp only: add_diff_cancel_left diff_zero)
then show ?thesis by simp
qed

assumes "c + b = a"
shows "c = a - b"
proof -
from assms have "(b + c) - (b + 0) = a - b" by (simp add: add.commute)
then show "c = a - b" by simp
qed

end

assumes zero_diff [simp]: "0 - a = 0"
begin

"a - (a + b) = 0"
proof -
have "a - (a + b) = (a + 0) - (a + b)" by simp
also have "\<dots> = 0" by (simp only: add_diff_cancel_left zero_diff)
finally show ?thesis .
qed

end

subsection \<open>Groups\<close>

assumes left_minus [simp]: "- a + a = 0"
assumes add_uminus_conv_diff [simp]: "a + (- b) = a - b"
begin

"a - b = a + (- b)"
by simp

lemma minus_unique:
assumes "a + b = 0" shows "- a = b"
proof -
have "- a = - a + (a + b)" using assms by simp
finally show ?thesis .
qed

lemma minus_zero [simp]: "- 0 = 0"
proof -
have "0 + 0 = 0" by (rule add_0_right)
thus "- 0 = 0" by (rule minus_unique)
qed

lemma minus_minus [simp]: "- (- a) = a"
proof -
have "- a + a = 0" by (rule left_minus)
thus "- (- a) = a" by (rule minus_unique)
qed

lemma right_minus: "a + - a = 0"
proof -
have "a + - a = - (- a) + - a" by simp
also have "\<dots> = 0" by (rule left_minus)
finally show ?thesis .
qed

lemma diff_self [simp]:
"a - a = 0"
using right_minus [of a] by simp

proof
fix a b c :: 'a
assume "a + b = a + c"
then have "- a + a + b = - a + a + c"
then show "b = c" by simp
next
fix a b c :: 'a
assume "b + a = c + a"
then have "b + a + - a = c + a  + - a" by simp
then show "b = c" unfolding add.assoc by simp
qed

"- a + (a + b) = b"

"a + (- a + b) = b"

"a - b + b = a"

"a + b - b = a"

"- (a + b) = - b + - a"
proof -
have "(a + b) + (- b + - a) = 0"
then show "- (a + b) = - b + - a"
by (rule minus_unique)
qed

lemma right_minus_eq [simp]:
"a - b = 0 \<longleftrightarrow> a = b"
proof
assume "a - b = 0"
have "a = (a - b) + b" by (simp add: add.assoc)
also have "\<dots> = b" using \<open>a - b = 0\<close> by simp
finally show "a = b" .
next
assume "a = b" thus "a - b = 0" by simp
qed

lemma eq_iff_diff_eq_0:
"a = b \<longleftrightarrow> a - b = 0"
by (fact right_minus_eq [symmetric])

lemma diff_0 [simp]:
"0 - a = - a"

lemma diff_0_right [simp]:
"a - 0 = a"

"a - - b = a + b"

lemma neg_equal_iff_equal [simp]:
"- a = - b \<longleftrightarrow> a = b"
proof
assume "- a = - b"
hence "- (- a) = - (- b)" by simp
thus "a = b" by simp
next
assume "a = b"
thus "- a = - b" by simp
qed

lemma neg_equal_0_iff_equal [simp]:
"- a = 0 \<longleftrightarrow> a = 0"
by (subst neg_equal_iff_equal [symmetric]) simp

lemma neg_0_equal_iff_equal [simp]:
"0 = - a \<longleftrightarrow> 0 = a"
by (subst neg_equal_iff_equal [symmetric]) simp

text\<open>The next two equations can make the simplifier loop!\<close>

lemma equation_minus_iff:
"a = - b \<longleftrightarrow> b = - a"
proof -
have "- (- a) = - b \<longleftrightarrow> - a = b" by (rule neg_equal_iff_equal)
thus ?thesis by (simp add: eq_commute)
qed

lemma minus_equation_iff:
"- a = b \<longleftrightarrow> - b = a"
proof -
have "- a = - (- b) \<longleftrightarrow> a = -b" by (rule neg_equal_iff_equal)
thus ?thesis by (simp add: eq_commute)
qed

"a = - b \<longleftrightarrow> a + b = 0"
proof
assume "a = - b" then show "a + b = 0" by simp
next
assume "a + b = 0"
moreover have "a + (b + - b) = (a + b) + - b"
ultimately show "a = - b" by simp
qed

"a + b = 0 \<longleftrightarrow> a = - b"

"- a = b \<longleftrightarrow> a + b = 0"

"a + b = 0 \<longleftrightarrow> b = - a"

lemma minus_diff_eq [simp]:
"- (a - b) = b - a"

"a + (b - c) = (a + b) - c"

"a - (b + c) = a - c - b"

lemma diff_eq_eq [algebra_simps, field_simps]:
"a - b = c \<longleftrightarrow> a = c + b"
by auto

lemma eq_diff_eq [algebra_simps, field_simps]:
"a = c - b \<longleftrightarrow> a + b = c"
by auto

lemma diff_diff_eq2 [algebra_simps, field_simps]:
"a - (b - c) = (a + c) - b"

lemma diff_eq_diff_eq:
"a - b = c - d \<Longrightarrow> a = b \<longleftrightarrow> c = d"
by (simp only: eq_iff_diff_eq_0 [of a b] eq_iff_diff_eq_0 [of c d])

end

assumes ab_left_minus: "- a + a = 0"
assumes ab_diff_conv_add_uminus: "a - b = a + (- b)"
begin

proof
fix a b c :: 'a
have "b + a - a = b"
by simp
then show "a + b - a = b"
show "a - b - c = a - (b + c)"
qed

"- a + b = b - a"

"- (a + b) = - a + - b"

"(a - b) + c = (a + c) - b"

end

subsection \<open>(Partially) Ordered Groups\<close>

text \<open>
The theory of partially ordered groups 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}
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}
\<close>

assumes add_left_mono: "a \<le> b \<Longrightarrow> c + a \<le> c + b"
begin

"a \<le> b \<Longrightarrow> a + c \<le> b + c"

text \<open>non-strict, in both arguments\<close>
"a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c \<le> b + d"
done

end

begin

"a < b \<Longrightarrow> c + a < c + b"

"a < b \<Longrightarrow> a + c < b + c"

text\<open>Strict monotonicity in both arguments\<close>
"a < b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
done

"a < b \<Longrightarrow> c \<le> d \<Longrightarrow> a + c < b + d"
done

"a \<le> b \<Longrightarrow> c < d \<Longrightarrow> a + c < b + d"
done

end

assumes add_le_imp_le_left: "c + a \<le> c + b \<Longrightarrow> a \<le> b"
begin

assumes less: "c + a < c + b" shows "a < b"
proof -
from less have le: "c + a <= c + b" by (simp add: order_le_less)
have "a <= b"
apply (insert le)
by (insert le, drule add_le_imp_le_left, assumption)
moreover have "a \<noteq> b"
proof (rule ccontr)
assume "~(a \<noteq> b)"
then have "a = b" by simp
then have "c + a = c + b" by simp
with less show "False"by simp
qed
ultimately show "a < b" by (simp add: order_le_less)
qed

"a + c < b + c \<Longrightarrow> a < b"
done

"c + a < c + b \<longleftrightarrow> a < b"

"a + c < b + c \<longleftrightarrow> a < b"

"c + a \<le> c + b \<longleftrightarrow> a \<le> b"

"a + c \<le> b + c \<longleftrightarrow> a \<le> b"

"a + c \<le> b + c \<Longrightarrow> a \<le> b"
by simp

"max x y + z = max (x + z) (y + z)"
unfolding max_def by auto

"min x y + z = min (x + z) (y + z)"
unfolding min_def by auto

"x + max y z = max (x + y) (x + z)"
unfolding max_def by auto

"x + min y z = min (x + y) (x + z)"
unfolding min_def by auto

end

class ordered_cancel_comm_monoid_diff = comm_monoid_diff + ordered_ab_semigroup_add_imp_le +
assumes le_iff_add: "a \<le> b \<longleftrightarrow> (\<exists>c. b = a + c)"
begin

context
fixes a b
assumes "a \<le> b"
begin

"a + (b - a) = b"

"c + (b - a) = c + b - a"

"b - a + c = b + c - a"

"c + b - a = c + (b - a)"

"b + c - a = b - a + c"

lemma diff_diff_right:
"c - (b - a) = c + a - b"

"b - a + a = b"

"c \<le> b + c - a"

"a \<le> b \<Longrightarrow> b - a = c \<longleftrightarrow> b = c + a"

lemma le_diff_conv2:
"c \<le> b - a \<longleftrightarrow> c + a \<le> b" (is "?P \<longleftrightarrow> ?Q")
proof
assume ?P
then have "c + a \<le> b - a + a" by (rule add_right_mono)
next
assume ?Q
then show ?P by simp
qed

end

end

subsection \<open>Support for reasoning about signs\<close>

begin

assumes "0 < a" and "0 \<le> b" shows "0 < a + b"
proof -
have "0 + 0 < a + b"
then show ?thesis by simp
qed

assumes "0 < a" and "0 < b" shows "0 < a + b"
by (rule add_pos_nonneg) (insert assms, auto)

assumes "0 \<le> a" and "0 < b" shows "0 < a + b"
proof -
have "0 + 0 < a + b"
then show ?thesis by simp
qed

assumes "0 \<le> a" and "0 \<le> b" shows "0 \<le> a + b"
proof -
have "0 + 0 \<le> a + b"
then show ?thesis by simp
qed

assumes "a < 0" and "b \<le> 0" shows "a + b < 0"
proof -
have "a + b < 0 + 0"
then show ?thesis by simp
qed

assumes "a < 0" and "b < 0" shows "a + b < 0"
by (rule add_neg_nonpos) (insert assms, auto)

assumes "a \<le> 0" and "b < 0" shows "a + b < 0"
proof -
have "a + b < 0 + 0"
then show ?thesis by simp
qed

assumes "a \<le> 0" and "b \<le> 0" shows "a + b \<le> 0"
proof -
have "a + b \<le> 0 + 0"
then show ?thesis by simp
qed

assumes x: "0 \<le> x" and y: "0 \<le> y"
shows "x + y = 0 \<longleftrightarrow> x = 0 \<and> y = 0"
proof (intro iffI conjI)
have "x = x + 0" by simp
also have "x + 0 \<le> x + y" using y by (rule add_left_mono)
also assume "x + y = 0"
also have "0 \<le> x" using x .
finally show "x = 0" .
next
have "y = 0 + y" by simp
also have "0 + y \<le> x + y" using x by (rule add_right_mono)
also assume "x + y = 0"
also have "0 \<le> y" using y .
finally show "y = 0" .
next
assume "x = 0 \<and> y = 0"
then show "x + y = 0" by simp
qed

"0 \<le> a \<Longrightarrow> b \<le> c \<Longrightarrow> b \<le> a + c"
by (insert add_mono [of 0 a b c], simp)

"0 \<le> c \<Longrightarrow> b \<le> a \<Longrightarrow> b \<le> a + c"

"0 < a \<Longrightarrow> b \<le> c \<Longrightarrow> b < a + c"
by (insert add_less_le_mono [of 0 a b c], simp)

"0 \<le> a \<Longrightarrow> b < c \<Longrightarrow> b < a + c"
by (insert add_le_less_mono [of 0 a b c], simp)

end

begin

proof
fix a b c :: 'a
assume "c + a \<le> c + b"
hence "(-c) + (c + a) \<le> (-c) + (c + b)" by (rule add_left_mono)
hence "((-c) + c) + a \<le> ((-c) + c) + b" by (simp only: add.assoc)
thus "a \<le> b" by simp
qed

"b + a < b \<longleftrightarrow> a < 0"
using add_less_cancel_left [of _ _ 0] by simp

"a + b < b \<longleftrightarrow> a < 0"
using add_less_cancel_right [of _ _ 0] by simp

"a < a + b \<longleftrightarrow> 0 < b"
using add_less_cancel_left [of _ 0] by simp

"a < b + a \<longleftrightarrow> 0 < b"
using add_less_cancel_right [of 0] by simp

"b + a \<le> b \<longleftrightarrow> a \<le> 0"
using add_le_cancel_left [of _ _ 0] by simp

"a + b \<le> b \<longleftrightarrow> a \<le> 0"
using add_le_cancel_right [of _ _ 0] by simp

"a \<le> a + b \<longleftrightarrow> 0 \<le> b"
using add_le_cancel_left [of _ 0] by simp

"a \<le> b + a \<longleftrightarrow> 0 \<le> b"
using add_le_cancel_right [of 0] by simp

lemma max_diff_distrib_left:
shows "max x y - z = max (x - z) (y - z)"
using max_add_distrib_left [of x y "- z"] by simp

lemma min_diff_distrib_left:
shows "min x y - z = min (x - z) (y - z)"
using min_add_distrib_left [of x y "- z"] by simp

lemma le_imp_neg_le:
assumes "a \<le> b" shows "-b \<le> -a"
proof -
have "-a+a \<le> -a+b" using \<open>a \<le> b\<close> by (rule add_left_mono)
then have "0 \<le> -a+b" by simp
then have "0 + (-b) \<le> (-a + b) + (-b)" by (rule add_right_mono)
then show ?thesis by (simp add: algebra_simps)
qed

lemma neg_le_iff_le [simp]: "- b \<le> - a \<longleftrightarrow> a \<le> b"
proof
assume "- b \<le> - a"
hence "- (- a) \<le> - (- b)" by (rule le_imp_neg_le)
thus "a\<le>b" by simp
next
assume "a\<le>b"
thus "-b \<le> -a" by (rule le_imp_neg_le)
qed

lemma neg_le_0_iff_le [simp]: "- a \<le> 0 \<longleftrightarrow> 0 \<le> a"
by (subst neg_le_iff_le [symmetric], simp)

lemma neg_0_le_iff_le [simp]: "0 \<le> - a \<longleftrightarrow> a \<le> 0"
by (subst neg_le_iff_le [symmetric], simp)

lemma neg_less_iff_less [simp]: "- b < - a \<longleftrightarrow> a < b"

lemma neg_less_0_iff_less [simp]: "- a < 0 \<longleftrightarrow> 0 < a"
by (subst neg_less_iff_less [symmetric], simp)

lemma neg_0_less_iff_less [simp]: "0 < - a \<longleftrightarrow> a < 0"
by (subst neg_less_iff_less [symmetric], simp)

text\<open>The next several equations can make the simplifier loop!\<close>

lemma less_minus_iff: "a < - b \<longleftrightarrow> b < - a"
proof -
have "(- (-a) < - b) = (b < - a)" by (rule neg_less_iff_less)
thus ?thesis by simp
qed

lemma minus_less_iff: "- a < b \<longleftrightarrow> - b < a"
proof -
have "(- a < - (-b)) = (- b < a)" by (rule neg_less_iff_less)
thus ?thesis by simp
qed

lemma le_minus_iff: "a \<le> - b \<longleftrightarrow> b \<le> - a"
proof -
have mm: "!! a (b::'a). (-(-a)) < -b \<Longrightarrow> -(-b) < -a" by (simp only: minus_less_iff)
have "(- (- a) <= -b) = (b <= - a)"
apply (auto simp only: le_less)
apply (drule mm)
apply (simp_all)
apply (drule mm[simplified], assumption)
done
then show ?thesis by simp
qed

lemma minus_le_iff: "- a \<le> b \<longleftrightarrow> - b \<le> a"
by (auto simp add: le_less minus_less_iff)

lemma diff_less_0_iff_less [simp]:
"a - b < 0 \<longleftrightarrow> a < b"
proof -
have "a - b < 0 \<longleftrightarrow> a + (- b) < b + (- b)" by simp
also have "... \<longleftrightarrow> a < b" by (simp only: add_less_cancel_right)
finally show ?thesis .
qed

lemmas less_iff_diff_less_0 = diff_less_0_iff_less [symmetric]

lemma diff_less_eq [algebra_simps, field_simps]:
"a - b < c \<longleftrightarrow> a < c + b"
apply (subst less_iff_diff_less_0 [of a])
apply (rule less_iff_diff_less_0 [of _ c, THEN ssubst])
done

lemma less_diff_eq[algebra_simps, field_simps]:
"a < c - b \<longleftrightarrow> a + b < c"
apply (subst less_iff_diff_less_0 [of "a + b"])
apply (subst less_iff_diff_less_0 [of a])
done

lemma diff_le_eq[algebra_simps, field_simps]: "a - b \<le> c \<longleftrightarrow> a \<le> c + b"
by (auto simp add: le_less diff_less_eq )

lemma le_diff_eq[algebra_simps, field_simps]: "a \<le> c - b \<longleftrightarrow> a + b \<le> c"
by (auto simp add: le_less less_diff_eq)

lemma diff_le_0_iff_le [simp]:
"a - b \<le> 0 \<longleftrightarrow> a \<le> b"

lemmas le_iff_diff_le_0 = diff_le_0_iff_le [symmetric]

lemma diff_eq_diff_less:
"a - b = c - d \<Longrightarrow> a < b \<longleftrightarrow> c < d"
by (auto simp only: less_iff_diff_less_0 [of a b] less_iff_diff_less_0 [of c d])

lemma diff_eq_diff_less_eq:
"a - b = c - d \<Longrightarrow> a \<le> b \<longleftrightarrow> c \<le> d"
by (auto simp only: le_iff_diff_le_0 [of a b] le_iff_diff_le_0 [of c d])

lemma diff_mono: "a \<le> b \<Longrightarrow> d \<le> c \<Longrightarrow> a - c \<le> b - d"

lemma diff_left_mono: "b \<le> a \<Longrightarrow> c - a \<le> c - b"

lemma diff_right_mono: "a \<le> b \<Longrightarrow> a - c \<le> b - c"

lemma diff_strict_mono: "a < b \<Longrightarrow> d < c \<Longrightarrow> a - c < b - d"

lemma diff_strict_left_mono: "b < a \<Longrightarrow> c - a < c - b"

lemma diff_strict_right_mono: "a < b \<Longrightarrow> a - c < b - c"

end

ML_file "Tools/group_cancel.ML"

\<open>fn phi => fn ss => try Group_Cancel.cancel_add_conv\<close>

simproc_setup group_cancel_diff ("a - b::'a::ab_group_add") =
\<open>fn phi => fn ss => try Group_Cancel.cancel_diff_conv\<close>

simproc_setup group_cancel_eq ("a = (b::'a::ab_group_add)") =
\<open>fn phi => fn ss => try Group_Cancel.cancel_eq_conv\<close>

simproc_setup group_cancel_le ("a \<le> (b::'a::ordered_ab_group_add)") =
\<open>fn phi => fn ss => try Group_Cancel.cancel_le_conv\<close>

simproc_setup group_cancel_less ("a < (b::'a::ordered_ab_group_add)") =
\<open>fn phi => fn ss => try Group_Cancel.cancel_less_conv\<close>

begin

proof
fix a b c :: 'a
assume le: "c + a <= c + b"
show "a <= b"
proof (rule ccontr)
assume w: "~ a \<le> b"
hence "b <= a" by (simp add: linorder_not_le)
hence le2: "c + b <= c + a" by (rule add_left_mono)
have "a = b"
apply (insert le)
apply (insert le2)
apply (drule antisym, simp_all)
done
with w show False
qed
qed

end

begin

lemma equal_neg_zero [simp]:
"a = - a \<longleftrightarrow> a = 0"
proof
assume "a = 0" then show "a = - a" by simp
next
assume A: "a = - a" show "a = 0"
proof (cases "0 \<le> a")
case True with A have "0 \<le> - a" by auto
with le_minus_iff have "a \<le> 0" by simp
with True show ?thesis by (auto intro: order_trans)
next
case False then have B: "a \<le> 0" by auto
with A have "- a \<le> 0" by auto
with B show ?thesis by (auto intro: order_trans)
qed
qed

lemma neg_equal_zero [simp]:
"- a = a \<longleftrightarrow> a = 0"
by (auto dest: sym)

lemma neg_less_eq_nonneg [simp]:
"- a \<le> a \<longleftrightarrow> 0 \<le> a"
proof
assume A: "- a \<le> a" show "0 \<le> a"
proof (rule classical)
assume "\<not> 0 \<le> a"
then have "a < 0" by auto
with A have "- a < 0" by (rule le_less_trans)
then show ?thesis by auto
qed
next
assume A: "0 \<le> a" show "- a \<le> a"
proof (rule order_trans)
show "- a \<le> 0" using A by (simp add: minus_le_iff)
next
show "0 \<le> a" using A .
qed
qed

lemma neg_less_pos [simp]:
"- a < a \<longleftrightarrow> 0 < a"

lemma less_eq_neg_nonpos [simp]:
"a \<le> - a \<longleftrightarrow> a \<le> 0"
using neg_less_eq_nonneg [of "- a"] by simp

lemma less_neg_neg [simp]:
"a < - a \<longleftrightarrow> a < 0"
using neg_less_pos [of "- a"] by simp

lemma double_zero [simp]:
"a + a = 0 \<longleftrightarrow> a = 0"
proof
assume assm: "a + a = 0"
then have a: "- a = a" by (rule minus_unique)
then show "a = 0" by (simp only: neg_equal_zero)
qed simp

lemma double_zero_sym [simp]:
"0 = a + a \<longleftrightarrow> a = 0"
by (rule, drule sym) simp_all

"0 < a + a \<longleftrightarrow> 0 < a"
proof
assume "0 < a + a"
then have "0 - a < a" by (simp only: diff_less_eq)
then have "- a < a" by simp
then show "0 < a" by simp
next
assume "0 < a"
with this have "0 + 0 < a + a"
then show "0 < a + a" by simp
qed

"0 \<le> a + a \<longleftrightarrow> 0 \<le> a"

"a + a < 0 \<longleftrightarrow> a < 0"
proof -
have "\<not> a + a < 0 \<longleftrightarrow> \<not> a < 0"
then show ?thesis by simp
qed

"a + a \<le> 0 \<longleftrightarrow> a \<le> 0"
proof -
have "\<not> a + a \<le> 0 \<longleftrightarrow> \<not> a \<le> 0"
then show ?thesis by simp
qed

lemma minus_max_eq_min:
"- max x y = min (-x) (-y)"
by (auto simp add: max_def min_def)

lemma minus_min_eq_max:
"- min x y = max (-x) (-y)"
by (auto simp add: max_def min_def)

end

class abs =
fixes abs :: "'a \<Rightarrow> 'a"
begin

notation (xsymbols)
abs  ("\<bar>_\<bar>")

notation (HTML output)
abs  ("\<bar>_\<bar>")

end

class sgn =
fixes sgn :: "'a \<Rightarrow> 'a"

class abs_if = minus + uminus + ord + zero + abs +
assumes abs_if: "\<bar>a\<bar> = (if a < 0 then - a else a)"

class sgn_if = minus + uminus + zero + one + ord + sgn +
assumes sgn_if: "sgn x = (if x = 0 then 0 else if 0 < x then 1 else - 1)"
begin

lemma sgn0 [simp]: "sgn 0 = 0"

end

assumes abs_ge_zero [simp]: "\<bar>a\<bar> \<ge> 0"
and abs_ge_self: "a \<le> \<bar>a\<bar>"
and abs_leI: "a \<le> b \<Longrightarrow> - a \<le> b \<Longrightarrow> \<bar>a\<bar> \<le> b"
and abs_minus_cancel [simp]: "\<bar>-a\<bar> = \<bar>a\<bar>"
and abs_triangle_ineq: "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
begin

lemma abs_minus_le_zero: "- \<bar>a\<bar> \<le> 0"
unfolding neg_le_0_iff_le by simp

lemma abs_of_nonneg [simp]:
assumes nonneg: "0 \<le> a" shows "\<bar>a\<bar> = a"
proof (rule antisym)
from nonneg le_imp_neg_le have "- a \<le> 0" by simp
from this nonneg have "- a \<le> a" by (rule order_trans)
then show "\<bar>a\<bar> \<le> a" by (auto intro: abs_leI)
qed (rule abs_ge_self)

lemma abs_idempotent [simp]: "\<bar>\<bar>a\<bar>\<bar> = \<bar>a\<bar>"
by (rule antisym)
(auto intro!: abs_ge_self abs_leI order_trans [of "- \<bar>a\<bar>" 0 "\<bar>a\<bar>"])

lemma abs_eq_0 [simp]: "\<bar>a\<bar> = 0 \<longleftrightarrow> a = 0"
proof -
have "\<bar>a\<bar> = 0 \<Longrightarrow> a = 0"
proof (rule antisym)
assume zero: "\<bar>a\<bar> = 0"
with abs_ge_self show "a \<le> 0" by auto
from zero have "\<bar>-a\<bar> = 0" by simp
with abs_ge_self [of "- a"] have "- a \<le> 0" by auto
with neg_le_0_iff_le show "0 \<le> a" by auto
qed
then show ?thesis by auto
qed

lemma abs_zero [simp]: "\<bar>0\<bar> = 0"
by simp

lemma abs_0_eq [simp]: "0 = \<bar>a\<bar> \<longleftrightarrow> a = 0"
proof -
have "0 = \<bar>a\<bar> \<longleftrightarrow> \<bar>a\<bar> = 0" by (simp only: eq_ac)
thus ?thesis by simp
qed

lemma abs_le_zero_iff [simp]: "\<bar>a\<bar> \<le> 0 \<longleftrightarrow> a = 0"
proof
assume "\<bar>a\<bar> \<le> 0"
then have "\<bar>a\<bar> = 0" by (rule antisym) simp
thus "a = 0" by simp
next
assume "a = 0"
thus "\<bar>a\<bar> \<le> 0" by simp
qed

lemma zero_less_abs_iff [simp]: "0 < \<bar>a\<bar> \<longleftrightarrow> a \<noteq> 0"

lemma abs_not_less_zero [simp]: "\<not> \<bar>a\<bar> < 0"
proof -
have a: "\<And>x y. x \<le> y \<Longrightarrow> \<not> y < x" by auto
show ?thesis by (simp add: a)
qed

lemma abs_ge_minus_self: "- a \<le> \<bar>a\<bar>"
proof -
have "- a \<le> \<bar>-a\<bar>" by (rule abs_ge_self)
then show ?thesis by simp
qed

lemma abs_minus_commute:
"\<bar>a - b\<bar> = \<bar>b - a\<bar>"
proof -
have "\<bar>a - b\<bar> = \<bar>- (a - b)\<bar>" by (simp only: abs_minus_cancel)
also have "... = \<bar>b - a\<bar>" by simp
finally show ?thesis .
qed

lemma abs_of_pos: "0 < a \<Longrightarrow> \<bar>a\<bar> = a"
by (rule abs_of_nonneg, rule less_imp_le)

lemma abs_of_nonpos [simp]:
assumes "a \<le> 0" shows "\<bar>a\<bar> = - a"
proof -
let ?b = "- a"
have "- ?b \<le> 0 \<Longrightarrow> \<bar>- ?b\<bar> = - (- ?b)"
unfolding abs_minus_cancel [of "?b"]
unfolding neg_le_0_iff_le [of "?b"]
unfolding minus_minus by (erule abs_of_nonneg)
then show ?thesis using assms by auto
qed

lemma abs_of_neg: "a < 0 \<Longrightarrow> \<bar>a\<bar> = - a"
by (rule abs_of_nonpos, rule less_imp_le)

lemma abs_le_D1: "\<bar>a\<bar> \<le> b \<Longrightarrow> a \<le> b"
by (insert abs_ge_self, blast intro: order_trans)

lemma abs_le_D2: "\<bar>a\<bar> \<le> b \<Longrightarrow> - a \<le> b"
by (insert abs_le_D1 [of "- a"], simp)

lemma abs_le_iff: "\<bar>a\<bar> \<le> b \<longleftrightarrow> a \<le> b \<and> - a \<le> b"
by (blast intro: abs_leI dest: abs_le_D1 abs_le_D2)

lemma abs_triangle_ineq2: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>a - b\<bar>"
proof -
have "\<bar>a\<bar> = \<bar>b + (a - b)\<bar>"
then have "\<bar>a\<bar> \<le> \<bar>b\<bar> + \<bar>a - b\<bar>"
then show ?thesis
qed

lemma abs_triangle_ineq2_sym: "\<bar>a\<bar> - \<bar>b\<bar> \<le> \<bar>b - a\<bar>"
by (simp only: abs_minus_commute [of b] abs_triangle_ineq2)

lemma abs_triangle_ineq3: "\<bar>\<bar>a\<bar> - \<bar>b\<bar>\<bar> \<le> \<bar>a - b\<bar>"
by (simp add: abs_le_iff abs_triangle_ineq2 abs_triangle_ineq2_sym)

lemma abs_triangle_ineq4: "\<bar>a - b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
proof -
have "\<bar>a - b\<bar> = \<bar>a + - b\<bar>" by (simp add: algebra_simps)
also have "... \<le> \<bar>a\<bar> + \<bar>- b\<bar>" by (rule abs_triangle_ineq)
finally show ?thesis by simp
qed

lemma abs_diff_triangle_ineq: "\<bar>a + b - (c + d)\<bar> \<le> \<bar>a - c\<bar> + \<bar>b - d\<bar>"
proof -
have "\<bar>a + b - (c+d)\<bar> = \<bar>(a-c) + (b-d)\<bar>" by (simp add: algebra_simps)
also have "... \<le> \<bar>a-c\<bar> + \<bar>b-d\<bar>" by (rule abs_triangle_ineq)
finally show ?thesis .
qed

"\<bar>\<bar>a\<bar> + \<bar>b\<bar>\<bar> = \<bar>a\<bar> + \<bar>b\<bar>" (is "?L = ?R")
proof (rule antisym)
show "?L \<ge> ?R" by(rule abs_ge_self)
next
have "?L \<le> \<bar>\<bar>a\<bar>\<bar> + \<bar>\<bar>b\<bar>\<bar>" by(rule abs_triangle_ineq)
also have "\<dots> = ?R" by simp
finally show "?L \<le> ?R" .
qed

end

lemmas mult_1 = mult_1_left -- \<open>FIXME duplicate\<close>
lemmas ab_left_minus = left_minus -- \<open>FIXME duplicate\<close>
lemmas diff_diff_eq = diff_diff_add -- \<open>FIXME duplicate\<close>

subsection \<open>Tools setup\<close>

fixes i j k :: "'a\<Colon>ordered_ab_semigroup_add"
shows "i \<le> j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
and "i = j \<and> k \<le> l \<Longrightarrow> i + k \<le> j + l"
and "i \<le> j \<and> k = l \<Longrightarrow> i + k \<le> j + l"
and "i = j \<and> k = l \<Longrightarrow> i + k = j + l"

fixes i j k :: "'a\<Colon>ordered_cancel_ab_semigroup_add"
shows "i < j \<and> k = l \<Longrightarrow> i + k < j + l"
and "i = j \<and> k < l \<Longrightarrow> i + k < j + l"
and "i < j \<and> k \<le> l \<Longrightarrow> i + k < j + l"
and "i \<le> j \<and> k < l \<Longrightarrow> i + k < j + l"
and "i < j \<and> k < l \<Longrightarrow> i + k < j + l"