orders.v 4.65 KB
Newer Older
1 2 3 4
(* Copyright (c) 2012, Robbert Krebbers. *)
(* This file is distributed under the terms of the BSD license. *)
(** This file collects common properties of pre-orders and semi lattices. This
theory will mainly be used for the theory on collections and finite maps. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
5 6
Require Export base.

7 8 9
(** * Pre-orders *)
(** We extend a pre-order to a partial order by defining equality as
[λ X Y, X ⊆ Y ∧ Y ⊆ X]. We prove that this indeed gives rise to a setoid. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
10 11 12 13 14
Section preorder.
  Context `{SubsetEq A} `{!PreOrder ()}.

  Global Instance preorder_equiv: Equiv A := λ X Y, X  Y  Y  X.
  Instance preorder_equivalence: @Equivalence A ().
15 16 17 18 19 20
  Proof.
    split.
    * firstorder.
    * firstorder.
    * intros x y z; split; transitivity y; firstorder.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
21 22 23

  Global Instance: Proper (() ==> () ==> iff) ().
  Proof.
24 25 26 27
    unfold equiv, preorder_equiv.
    intros x1 y1 ? x2 y2 ?. split; intro.
    * transitivity x1. tauto. transitivity x2; tauto.
    * transitivity y1. tauto. transitivity y2; tauto.
Robbert Krebbers's avatar
Robbert Krebbers committed
28 29 30
  Qed.
End preorder.

31 32
Hint Extern 0 (@Equivalence _ ()) =>
  class_apply preorder_equivalence : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
33

34 35
(** * Join semi lattices *)
(** General purpose theorems on join semi lattices. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58
Section bounded_join_sl.
  Context `{BoundedJoinSemiLattice A}.

  Hint Resolve subseteq_empty subseteq_union_l subseteq_union_r union_least.

  Lemma union_compat_l x1 x2 y : x1  x2  x1  x2  y.
  Proof. intros. transitivity x2; auto. Qed.
  Lemma union_compat_r x1 x2 y : x1  x2  x1  y  x2.
  Proof. intros. transitivity x2; auto. Qed.
  Hint Resolve union_compat_l union_compat_r.

  Lemma union_compat x1 x2 y1 y2 : x1  x2  y1  y2  x1  y1  x2  y2.
  Proof. auto. Qed.
  Lemma union_empty x : x    x.
  Proof. apply union_least. easy. auto. Qed.
  Lemma union_comm x y : x  y  y  x.
  Proof. auto. Qed.
  Lemma union_assoc_1 x y z : (x  y)  z  x  (y  z).
  Proof. auto. Qed.
  Lemma union_assoc_2 x y z : x  (y  z)  (x  y)  z.
  Proof. auto. Qed.

  Global Instance: Proper (() ==> () ==> ()) ().
59 60 61
  Proof.
    unfold equiv, preorder_equiv. split; apply union_compat; simpl in *; tauto.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83
  Global Instance: Idempotent () ().
  Proof. split; eauto. Qed.
  Global Instance: LeftId ()  ().
  Proof. split; eauto. Qed.
  Global Instance: RightId ()  ().
  Proof. split; eauto. Qed.
  Global Instance: Commutative () ().
  Proof. split; apply union_comm. Qed.
  Global Instance: Associative () ().
  Proof. split. apply union_assoc_2. apply union_assoc_1. Qed.

  Lemma subseteq_union X Y : X  Y  X  Y  Y.
  Proof. repeat split; eauto. intros E. rewrite <-E. auto. Qed.
  Lemma subseteq_union_1 X Y : X  Y  X  Y  Y.
  Proof. apply subseteq_union. Qed.
  Lemma subseteq_union_2 X Y : X  Y  Y  X  Y.
  Proof. apply subseteq_union. Qed.

  Lemma equiv_empty X : X    X  .
  Proof. split; eauto. Qed.
End bounded_join_sl.

84 85
(** * Meet semi lattices *)
(** The dual of the above section, but now for meet semi lattices. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
86 87 88
Section meet_sl.
  Context `{MeetSemiLattice A}.

89 90
  Hint Resolve subseteq_intersection_l subseteq_intersection_r
    intersection_greatest.
Robbert Krebbers's avatar
Robbert Krebbers committed
91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

  Lemma intersection_compat_l x1 x2 y : x1  x2  x1  y  x2.
  Proof. intros. transitivity x1; auto. Qed.
  Lemma intersection_compat_r x1 x2 y : x1  x2  y  x1  x2.
  Proof. intros. transitivity x1; auto. Qed.
  Hint Resolve intersection_compat_l intersection_compat_r.

  Lemma intersection_compat x1 x2 y1 y2 : x1  x2  y1  y2  x1  y1  x2  y2.
  Proof. auto. Qed.
  Lemma intersection_comm x y : x  y  y  x.
  Proof. auto. Qed.
  Lemma intersection_assoc_1 x y z : (x  y)  z  x  (y  z).
  Proof. auto. Qed.
  Lemma intersection_assoc_2 x y z : x  (y  z)  (x  y)  z.
  Proof. auto. Qed.

  Global Instance: Proper (() ==> () ==> ()) ().
108 109 110 111
  Proof.
    unfold equiv, preorder_equiv. split;
      apply intersection_compat; simpl in *; tauto.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
112 113 114 115 116 117 118 119 120 121 122 123 124 125
  Global Instance: Idempotent () ().
  Proof. split; eauto. Qed.
  Global Instance: Commutative () ().
  Proof. split; apply intersection_comm. Qed.
  Global Instance: Associative () ().
  Proof. split. apply intersection_assoc_2. apply intersection_assoc_1. Qed.

  Lemma subseteq_intersection X Y : X  Y  X  Y  X.
  Proof. repeat split; eauto. intros E. rewrite <-E. auto. Qed.
  Lemma subseteq_intersection_1 X Y : X  Y  X  Y  X.
  Proof. apply subseteq_intersection. Qed.
  Lemma subseteq_intersection_2 X Y : X  Y  X  X  Y.
  Proof. apply subseteq_intersection. Qed.
End meet_sl.