(* Author: Florian Haftmann, TU Muenchen *)

section ‹Preorders with explicit equivalence relation›

theory Preorder
imports Main
begin

class preorder_equiv = preorder
begin

definition equiv :: "'a → 'a → bool"
  where "equiv x y ⟷ x ≤ y ∧ y ≤ x"

notation
  equiv (‹'(≈')›) and
  equiv (‹(‹notation=‹infix ≈››_/ ≈ _)› [51, 51] 50)

lemma equivD1: "x ≤ y" if "x ≈ y"
  using that by (simp add: equiv_def)

lemma equivD2: "y ≤ x" if "x ≈ y"
  using that by (simp add: equiv_def)

lemma equiv_refl [iff]: "x ≈ x"
  by (simp add: equiv_def)

lemma equiv_sym: "x ≈ y ⟷ y ≈ x"
  by (auto simp add: equiv_def)

lemma equiv_trans: "x ≈ y ==> y ≈ z ==> x ≈ z"
  by (auto simp: equiv_def intro: order_trans)

lemma equiv_antisym: "x ≤ y ==> y ≤ x ==> x ≈ y"
  by (simp only: equiv_def)

lemma less_le: "x < y ⟷ x ≤ y ∧ ¬ x ≈ y"
  by (auto simp add: equiv_def less_le_not_le)

lemma le_less: "x ≤ y ⟷ x < y ∨ x ≈ y"
  by (auto simp add: equiv_def less_le)

lemma le_imp_less_or_equiv: "x ≤ y ==> x < y ∨ x ≈ y"
  by (simp add: less_le)

lemma less_imp_not_equiv: "x < y ==> ¬ x ≈ y"
  by (simp add: less_le)

lemma not_equiv_le_trans: "¬ a ≈ b ==> a ≤ b ==> a < b"
  by (simp add: less_le)

lemma le_not_equiv_trans: "a ≤ b ==> ¬ a ≈ b ==> a < b"
  by (rule not_equiv_le_trans)

lemma antisym_conv: "y ≤ x ==> x ≤ y ⟷ x ≈ y"
  by (simp add: equiv_def)

end

ML_file ‹~~/src/Provers/preorder.ML›

ML ‹
  Quasi = Quasi_Tac(
 

  le_trans = @{thm order_trans};
  le_refl = @{thm order_refl};
  eqD1 = @{thm equivD1};
  eqD2 = @{thm equivD2};
  less_reflE = @{thm less_irrefl};
  less_imp_le = @{thm less_imp_le};
  le_neq_trans = @{thm le_not_equiv_trans};
  neq_le_trans = @{thm not_equiv_le_trans};
  less_imp_neq = @{thm less_imp_not_equiv};

  decomp_quasi thy (Const (@{const_name less_eq}, _) $ t1 $ t2) = SOME (t1, "<=", t2)
 | decomp_quasi thy (Const (@{const_name less}, _) $ t1 $ t2) = SOME (t1, "<", t2)
 | decomp_quasi thy (Const (@{const_name equiv}, _) $ t1 $ t2) = SOME (t1, "=", t2)
 | decomp_quasi thy (Const (@{const_name Not}, _) $ (Const (@{const_name equiv}, _) $ t1 $ t2)) = SOME (t1, "~=", t2)
 | decomp_quasi thy _ = NONE;

  decomp_trans thy t = case decomp_quasi thy t of
 x as SOME (t1, "<=", t2) => x
 | _ => NONE;

 
 ;
 ›

end