section ‹Preorders with explicit equivalence relation›
theory Preorder
imports Orderings
begin
class preorder_equiv = preorder
begin
definition equiv :: "'a ⇒ 'a ⇒ bool" where
"equiv x y ⟷ x ≤ y ∧ y ≤ x"
notation
equiv ("op ≈") and
equiv ("(_/ ≈ _)" [51, 51] 50)
lemma refl [iff]:
"x ≈ x"
unfolding equiv_def by simp
lemma trans:
"x ≈ y ⟹ y ≈ z ⟹ x ≈ z"
unfolding equiv_def by (auto intro: order_trans)
lemma antisym:
"x ≤ y ⟹ y ≤ x ⟹ x ≈ y"
unfolding 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_eq: "x ≤ y ⟹ x < y ∨ x ≈ y"
by (simp add: less_le)
lemma less_imp_not_eq: "x < y ⟹ x ≈ y ⟷ False"
by (simp add: less_le)
lemma less_imp_not_eq2: "x < y ⟹ y ≈ x ⟷ False"
by (simp add: equiv_def less_le)
lemma neq_le_trans: "¬ a ≈ b ⟹ a ≤ b ⟹ a < b"
by (simp add: less_le)
lemma le_neq_trans: "a ≤ b ⟹ ¬ a ≈ b ⟹ a < b"
by (simp add: less_le)
lemma antisym_conv: "y ≤ x ⟹ x ≤ y ⟷ x ≈ y"
by (simp add: equiv_def)
end
end