src/HOL/Orderings.thy
changeset 30298 abefe1dfadbb
parent 30107 f3b3b0e3d184
child 30510 4120fc59dd85
     1.1 --- a/src/HOL/Orderings.thy	Fri Mar 06 09:35:43 2009 +0100
     1.2 +++ b/src/HOL/Orderings.thy	Fri Mar 06 14:33:19 2009 +0100
     1.3 @@ -968,9 +968,7 @@
     1.4  context order
     1.5  begin
     1.6  
     1.7 -definition
     1.8 -  mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool"
     1.9 -where
    1.10 +definition mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
    1.11    "mono f \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> f x \<le> f y)"
    1.12  
    1.13  lemma monoI [intro?]:
    1.14 @@ -983,11 +981,76 @@
    1.15    shows "mono f \<Longrightarrow> x \<le> y \<Longrightarrow> f x \<le> f y"
    1.16    unfolding mono_def by iprover
    1.17  
    1.18 +definition strict_mono :: "('a \<Rightarrow> 'b\<Colon>order) \<Rightarrow> bool" where
    1.19 +  "strict_mono f \<longleftrightarrow> (\<forall>x y. x < y \<longrightarrow> f x < f y)"
    1.20 +
    1.21 +lemma strict_monoI [intro?]:
    1.22 +  assumes "\<And>x y. x < y \<Longrightarrow> f x < f y"
    1.23 +  shows "strict_mono f"
    1.24 +  using assms unfolding strict_mono_def by auto
    1.25 +
    1.26 +lemma strict_monoD [dest?]:
    1.27 +  "strict_mono f \<Longrightarrow> x < y \<Longrightarrow> f x < f y"
    1.28 +  unfolding strict_mono_def by auto
    1.29 +
    1.30 +lemma strict_mono_mono [dest?]:
    1.31 +  assumes "strict_mono f"
    1.32 +  shows "mono f"
    1.33 +proof (rule monoI)
    1.34 +  fix x y
    1.35 +  assume "x \<le> y"
    1.36 +  show "f x \<le> f y"
    1.37 +  proof (cases "x = y")
    1.38 +    case True then show ?thesis by simp
    1.39 +  next
    1.40 +    case False with `x \<le> y` have "x < y" by simp
    1.41 +    with assms strict_monoD have "f x < f y" by auto
    1.42 +    then show ?thesis by simp
    1.43 +  qed
    1.44 +qed
    1.45 +
    1.46  end
    1.47  
    1.48  context linorder
    1.49  begin
    1.50  
    1.51 +lemma strict_mono_eq:
    1.52 +  assumes "strict_mono f"
    1.53 +  shows "f x = f y \<longleftrightarrow> x = y"
    1.54 +proof
    1.55 +  assume "f x = f y"
    1.56 +  show "x = y" proof (cases x y rule: linorder_cases)
    1.57 +    case less with assms strict_monoD have "f x < f y" by auto
    1.58 +    with `f x = f y` show ?thesis by simp
    1.59 +  next
    1.60 +    case equal then show ?thesis .
    1.61 +  next
    1.62 +    case greater with assms strict_monoD have "f y < f x" by auto
    1.63 +    with `f x = f y` show ?thesis by simp
    1.64 +  qed
    1.65 +qed simp
    1.66 +
    1.67 +lemma strict_mono_less_eq:
    1.68 +  assumes "strict_mono f"
    1.69 +  shows "f x \<le> f y \<longleftrightarrow> x \<le> y"
    1.70 +proof
    1.71 +  assume "x \<le> y"
    1.72 +  with assms strict_mono_mono monoD show "f x \<le> f y" by auto
    1.73 +next
    1.74 +  assume "f x \<le> f y"
    1.75 +  show "x \<le> y" proof (rule ccontr)
    1.76 +    assume "\<not> x \<le> y" then have "y < x" by simp
    1.77 +    with assms strict_monoD have "f y < f x" by auto
    1.78 +    with `f x \<le> f y` show False by simp
    1.79 +  qed
    1.80 +qed
    1.81 +  
    1.82 +lemma strict_mono_less:
    1.83 +  assumes "strict_mono f"
    1.84 +  shows "f x < f y \<longleftrightarrow> x < y"
    1.85 +  using assms
    1.86 +    by (auto simp add: less_le Orderings.less_le strict_mono_eq strict_mono_less_eq)
    1.87 +
    1.88  lemma min_of_mono:
    1.89    fixes f :: "'a \<Rightarrow> 'b\<Colon>linorder"
    1.90    shows "mono f \<Longrightarrow> min (f m) (f n) = f (min m n)"