src/HOL/Library/Preorder.thy
author paulson <lp15@cam.ac.uk>
Mon Feb 22 14:37:56 2016 +0000 (2016-02-22)
changeset 62379 340738057c8c
parent 61384 9f5145281888
child 63465 d7610beb98bc
permissions -rw-r--r--
An assortment of useful lemmas about sums, norm, etc. Also: norm_conv_dist [symmetric] is now a simprule!
     1 (* Author: Florian Haftmann, TU Muenchen *)
     2 
     3 section \<open>Preorders with explicit equivalence relation\<close>
     4 
     5 theory Preorder
     6 imports Orderings
     7 begin
     8 
     9 class preorder_equiv = preorder
    10 begin
    11 
    12 definition equiv :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where
    13   "equiv x y \<longleftrightarrow> x \<le> y \<and> y \<le> x"
    14 
    15 notation
    16   equiv ("op \<approx>") and
    17   equiv ("(_/ \<approx> _)"  [51, 51] 50)
    18 
    19 lemma refl [iff]:
    20   "x \<approx> x"
    21   unfolding equiv_def by simp
    22 
    23 lemma trans:
    24   "x \<approx> y \<Longrightarrow> y \<approx> z \<Longrightarrow> x \<approx> z"
    25   unfolding equiv_def by (auto intro: order_trans)
    26 
    27 lemma antisym:
    28   "x \<le> y \<Longrightarrow> y \<le> x \<Longrightarrow> x \<approx> y"
    29   unfolding equiv_def ..
    30 
    31 lemma less_le: "x < y \<longleftrightarrow> x \<le> y \<and> \<not> x \<approx> y"
    32   by (auto simp add: equiv_def less_le_not_le)
    33 
    34 lemma le_less: "x \<le> y \<longleftrightarrow> x < y \<or> x \<approx> y"
    35   by (auto simp add: equiv_def less_le)
    36 
    37 lemma le_imp_less_or_eq: "x \<le> y \<Longrightarrow> x < y \<or> x \<approx> y"
    38   by (simp add: less_le)
    39 
    40 lemma less_imp_not_eq: "x < y \<Longrightarrow> x \<approx> y \<longleftrightarrow> False"
    41   by (simp add: less_le)
    42 
    43 lemma less_imp_not_eq2: "x < y \<Longrightarrow> y \<approx> x \<longleftrightarrow> False"
    44   by (simp add: equiv_def less_le)
    45 
    46 lemma neq_le_trans: "\<not> a \<approx> b \<Longrightarrow> a \<le> b \<Longrightarrow> a < b"
    47   by (simp add: less_le)
    48 
    49 lemma le_neq_trans: "a \<le> b \<Longrightarrow> \<not> a \<approx> b \<Longrightarrow> a < b"
    50   by (simp add: less_le)
    51 
    52 lemma antisym_conv: "y \<le> x \<Longrightarrow> x \<le> y \<longleftrightarrow> x \<approx> y"
    53   by (simp add: equiv_def)
    54 
    55 end
    56 
    57 end