orders.v 24.3 KB
Newer Older
Robbert Krebbers's avatar
Robbert Krebbers committed
1
(* Copyright (c) 2012-2014, Robbert Krebbers. *)
2 3 4
(* 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. *)
5
Require Export Sorted.
6
Require Export base decidable tactics list.
Robbert Krebbers's avatar
Robbert Krebbers committed
7

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
(** * Arbitrary pre-, parial and total orders *)
(** Properties about arbitrary pre-, partial, and total orders. We do not use
the relation [⊆] because we often have multiple orders on the same structure *)
Section orders.
  Context {A} {R : relation A}.
  Implicit Types X Y : A.
  Infix "⊆" := R.
  Notation "X ⊈ Y" := (¬X  Y).
  Infix "⊂" := (strict R).

  Lemma reflexive_eq `{!Reflexive R} X Y : X = Y  X  Y.
  Proof. by intros <-. Qed.
  Lemma anti_symmetric_iff `{!PartialOrder R} X Y : X = Y  R X Y  R Y X.
  Proof. intuition (subst; auto). Qed.
  Lemma strict_spec X Y : X  Y  X  Y  Y  X.
  Proof. done. Qed.
  Lemma strict_include X Y : X  Y  X  Y.
  Proof. by intros [? _]. Qed.
  Lemma strict_ne X Y : X  Y  X  Y.
  Proof. by intros [??] <-. Qed.
  Lemma strict_ne_sym X Y : X  Y  Y  X.
  Proof. by intros [??] <-. Qed.
  Lemma strict_transitive_l `{!Transitive R} X Y Z : X  Y  Y  Z  X  Z.
  Proof.
    intros [? HXY] ?. split.
    * by transitivity Y.
    * contradict HXY. by transitivity Z.
  Qed.
  Lemma strict_transitive_r `{!Transitive R} X Y Z : X  Y  Y  Z  X  Z.
  Proof.
    intros ? [? HYZ]. split.
    * by transitivity Y.
    * contradict HYZ. by transitivity X.
  Qed.
  Global Instance: Irreflexive (strict R).
  Proof. firstorder. Qed.
  Global Instance: Transitive R  StrictOrder (strict R).
  Proof.
    split; try apply _.
    eauto using strict_transitive_r, strict_include.
  Qed.
  Global Instance preorder_subset_dec_slow `{ X Y, Decision (X  Y)}
    (X Y : A) : Decision (X  Y) | 100 := _.
51
  Lemma strict_spec_alt `{!AntiSymmetric (=) R} X Y : X  Y  X  Y  X  Y.
52 53 54 55 56 57 58 59 60 61 62 63 64 65 66
  Proof.
    split.
    * intros [? HYX]. split. done. by intros <-.
    * intros [? HXY]. split. done. by contradict HXY; apply (anti_symmetric R).
  Qed.
  Lemma po_eq_dec `{!PartialOrder R} `{ X Y, Decision (X  Y)} (X Y : A) :
    Decision (X = Y).
  Proof.
    refine (cast_if_and (decide (X  Y)) (decide (Y  X)));
     abstract (rewrite anti_symmetric_iff; tauto).
  Defined.
  Lemma total_not `{!Total R} X Y : X  Y  Y  X.
  Proof. intros. destruct (total R X Y); tauto. Qed.
  Lemma total_not_strict `{!Total R} X Y : X  Y  Y  X.
  Proof. red; auto using total_not. Qed.
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87
  Global Instance trichotomy_total
    `{!Trichotomy (strict R)} `{!Reflexive R} : Total R.
  Proof.
    intros X Y.
    destruct (trichotomy (strict R) X Y) as [[??]|[<-|[??]]]; intuition.
  Qed.
End orders.

Section strict_orders.
  Context {A} {R : relation A}.
  Implicit Types X Y : A.
  Infix "⊂" := R.

  Lemma irreflexive_eq `{!Irreflexive R} X Y : X = Y  ¬X  Y.
  Proof. intros ->. apply (irreflexivity R). Qed.
  Lemma strict_anti_symmetric `{!StrictOrder R} X Y :
    X  Y  Y  X  False.
  Proof. intros. apply (irreflexivity R X). by transitivity Y. Qed.

  Global Instance trichotomyT_dec `{!TrichotomyT R}
      `{!StrictOrder R} X Y : Decision (X  Y) :=
88
    match trichotomyT R X Y with
89 90 91
    | inleft (left H) => left H
    | inleft (right H) => right (irreflexive_eq _ _ H)
    | inright H => right (strict_anti_symmetric _ _ H)
92
    end.
93

94 95
  Global Instance trichotomyT_trichotomy `{!TrichotomyT R} : Trichotomy R.
  Proof. intros X Y. destruct (trichotomyT R X Y) as [[|]|]; tauto. Qed.
96
End strict_orders.
97 98 99 100

Ltac simplify_order := repeat
  match goal with
  | _ => progress simplify_equality
101
  | H : ?R ?x ?x |- _ => by destruct (irreflexivity _ _ H)
102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216
  | H1 : ?R ?x ?y |- _ =>
    match goal with
    | H2 : R y x |- _ =>
      assert (x = y) by (by apply (anti_symmetric R)); clear H1 H2
    | H2 : R y ?z |- _ =>
      unless (R x z) by done;
      assert (R x z) by (by transitivity y)
    end
  end.

(** * Sorting *)
(** Merge sort. Adapted from the implementation of Hugo Herbelin in the Coq
standard library, but without using the module system. *)
Section merge_sort.
  Context  {A} (R : relation A) `{ x y, Decision (R x y)}.

  Fixpoint list_merge (l1 : list A) : list A  list A :=
    fix list_merge_aux l2 :=
    match l1, l2 with
    | [], _ => l2
    | _, [] => l1
    | x1 :: l1, x2 :: l2 =>
       if decide_rel R x1 x2 then x1 :: list_merge l1 (x2 :: l2)
       else x2 :: list_merge_aux l2
    end.
  Global Arguments list_merge !_ !_ /.

  Local Notation stack := (list (option (list A))).
  Fixpoint merge_list_to_stack (st : stack) (l : list A) : stack :=
    match st with
    | [] => [Some l]
    | None :: st => Some l :: st
    | Some l' :: st => None :: merge_list_to_stack st (list_merge l' l)
    end.
  Fixpoint merge_stack (st : stack) : list A :=
    match st with
    | [] => []
    | None :: st => merge_stack st
    | Some l :: st => list_merge l (merge_stack st)
    end.
  Fixpoint merge_sort_aux (st : stack) (l : list A) : list A :=
    match l with
    | [] => merge_stack st
    | x :: l => merge_sort_aux (merge_list_to_stack st [x]) l
    end.
  Definition merge_sort : list A  list A := merge_sort_aux [].
End merge_sort.

(** ** Properties of the [Sorted] and [StronglySorted] predicate *)
Section sorted.
  Context {A} (R : relation A).

  Lemma Sorted_StronglySorted `{!Transitive R} l :
    Sorted R l  StronglySorted R l.
  Proof. by apply Sorted.Sorted_StronglySorted. Qed.
  Lemma StronglySorted_unique `{!AntiSymmetric (=) R} l1 l2 :
    StronglySorted R l1  StronglySorted R l2  l1  l2  l1 = l2.
  Proof.
    intros Hl1; revert l2. induction Hl1 as [|x1 l1 ? IH Hx1]; intros l2 Hl2 E.
    { symmetry. by apply Permutation_nil. }
    destruct Hl2 as [|x2 l2 ? Hx2].
    { by apply Permutation_nil in E. }
    assert (x1 = x2); subst.
    { rewrite Forall_forall in Hx1, Hx2.
      assert (x2  x1 :: l1) as Hx2' by (by rewrite E; left).
      assert (x1  x2 :: l2) as Hx1' by (by rewrite <-E; left).
      inversion Hx1'; inversion Hx2'; simplify_equality; auto. }
    f_equal. by apply IH, (injective (x2 ::)).
  Qed.
  Lemma Sorted_unique `{!Transitive R} `{!AntiSymmetric (=) R} l1 l2 :
    Sorted R l1  Sorted R l2  l1  l2  l1 = l2.
  Proof. auto using StronglySorted_unique, Sorted_StronglySorted. Qed.

  Global Instance HdRel_dec x `{ y, Decision (R x y)} l :
    Decision (HdRel R x l).
  Proof.
   refine
    match l with
    | [] => left _
    | y :: l => cast_if (decide (R x y))
    end; abstract first [by constructor | by inversion 1].
  Defined.

  Global Instance Sorted_dec `{ x y, Decision (R x y)} :  l,
    Decision (Sorted R l).
  Proof.
   refine
    (fix go l :=
    match l return Decision (Sorted R l) with
    | [] => left _
    | x :: l => cast_if_and (decide (HdRel R x l)) (go l)
    end); clear go; abstract first [by constructor | by inversion 1].
  Defined.
  Global Instance StronglySorted_dec `{ x y, Decision (R x y)} :  l,
    Decision (StronglySorted R l).
  Proof.
   refine
    (fix go l :=
    match l return Decision (StronglySorted R l) with
    | [] => left _
    | x :: l => cast_if_and (decide (Forall (R x) l)) (go l)
    end); clear go; abstract first [by constructor | by inversion 1].
  Defined.

  Context {B} (f : A  B).
  Lemma HdRel_fmap (R1 : relation A) (R2 : relation B) x l :
    ( y, R1 x y  R2 (f x) (f y))  HdRel R1 x l  HdRel R2 (f x) (f <$> l).
  Proof. destruct 2; constructor; auto. Qed.
  Lemma Sorted_fmap (R1 : relation A) (R2 : relation B) l :
    ( x y, R1 x y  R2 (f x) (f y))  Sorted R1 l  Sorted R2 (f <$> l).
  Proof. induction 2; simpl; constructor; eauto using HdRel_fmap. Qed.
  Lemma StronglySorted_fmap (R1 : relation A) (R2 : relation B) l :
    ( x y, R1 x y  R2 (f x) (f y)) 
    StronglySorted R1 l  StronglySorted R2 (f <$> l).
  Proof.
217
    induction 2; csimpl; constructor;
218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317
      rewrite ?Forall_fmap; eauto using Forall_impl.
  Qed.
End sorted.

(** ** Correctness of merge sort *)
Section merge_sort_correct.
  Context  {A} (R : relation A) `{ x y, Decision (R x y)} `{!Total R}.

  Lemma list_merge_cons x1 x2 l1 l2 :
    list_merge R (x1 :: l1) (x2 :: l2) =
      if decide (R x1 x2) then x1 :: list_merge R l1 (x2 :: l2)
      else x2 :: list_merge R (x1 :: l1) l2.
  Proof. done. Qed.

  Lemma HdRel_list_merge x l1 l2 :
    HdRel R x l1  HdRel R x l2  HdRel R x (list_merge R l1 l2).
  Proof.
    destruct 1 as [|x1 l1 IH1], 1 as [|x2 l2 IH2];
      rewrite ?list_merge_cons; simpl; repeat case_decide; auto.
  Qed.
  Lemma Sorted_list_merge l1 l2 :
    Sorted R l1  Sorted R l2  Sorted R (list_merge R l1 l2).
  Proof.
    intros Hl1. revert l2. induction Hl1 as [|x1 l1 IH1];
      induction 1 as [|x2 l2 IH2]; rewrite ?list_merge_cons; simpl;
      repeat case_decide;
      constructor; eauto using HdRel_list_merge, HdRel_cons, total_not.
  Qed.
  Lemma merge_Permutation l1 l2 : list_merge R l1 l2  l1 ++ l2.
  Proof.
    revert l2. induction l1 as [|x1 l1 IH1]; intros l2;
      induction l2 as [|x2 l2 IH2]; rewrite ?list_merge_cons; simpl;
      repeat case_decide; auto.
    * by rewrite (right_id_L [] (++)).
    * by rewrite IH2, Permutation_middle.
  Qed.

  Local Notation stack := (list (option (list A))).
  Inductive merge_stack_Sorted : stack  Prop :=
    | merge_stack_Sorted_nil : merge_stack_Sorted []
    | merge_stack_Sorted_cons_None st :
       merge_stack_Sorted st  merge_stack_Sorted (None :: st)
    | merge_stack_Sorted_cons_Some l st :
       Sorted R l  merge_stack_Sorted st  merge_stack_Sorted (Some l :: st).
  Fixpoint merge_stack_flatten (st : stack) : list A :=
    match st with
    | [] => []
    | None :: st => merge_stack_flatten st
    | Some l :: st => l ++ merge_stack_flatten st
    end.

  Lemma Sorted_merge_list_to_stack st l :
    merge_stack_Sorted st  Sorted R l 
    merge_stack_Sorted (merge_list_to_stack R st l).
  Proof.
    intros Hst. revert l.
    induction Hst; repeat constructor; naive_solver auto using Sorted_list_merge.
  Qed.
  Lemma merge_list_to_stack_Permutation st l :
    merge_stack_flatten (merge_list_to_stack R st l) 
      l ++ merge_stack_flatten st.
  Proof.
    revert l. induction st as [|[l'|] st IH]; intros l; simpl; auto.
    by rewrite IH, merge_Permutation, (associative_L _), (commutative (++) l).
  Qed.
  Lemma Sorted_merge_stack st :
    merge_stack_Sorted st  Sorted R (merge_stack R st).
  Proof. induction 1; simpl; auto using Sorted_list_merge. Qed.
  Lemma merge_stack_Permutation st : merge_stack R st  merge_stack_flatten st.
  Proof.
    induction st as [|[] ? IH]; intros; simpl; auto.
    by rewrite merge_Permutation, IH.
  Qed.
  Lemma Sorted_merge_sort_aux st l :
    merge_stack_Sorted st  Sorted R (merge_sort_aux R st l).
  Proof.
    revert st. induction l; simpl;
      auto using Sorted_merge_stack, Sorted_merge_list_to_stack.
  Qed.
  Lemma merge_sort_aux_Permutation st l :
    merge_sort_aux R st l  merge_stack_flatten st ++ l.
  Proof.
    revert st. induction l as [|?? IH]; simpl; intros.
    * by rewrite (right_id_L [] (++)), merge_stack_Permutation.
    * rewrite IH, merge_list_to_stack_Permutation; simpl.
      by rewrite Permutation_middle.
  Qed.
  Lemma Sorted_merge_sort l : Sorted R (merge_sort R l).
  Proof. apply Sorted_merge_sort_aux. by constructor. Qed.
  Lemma merge_sort_Permutation l : merge_sort R l  l.
  Proof. unfold merge_sort. by rewrite merge_sort_aux_Permutation. Qed.
  Lemma StronglySorted_merge_sort `{!Transitive R} l :
    StronglySorted R (merge_sort R l).
  Proof. auto using Sorted_StronglySorted, Sorted_merge_sort. Qed.
End merge_sort_correct.

(** * Canonical pre and partial orders *)
(** We extend the canonical pre-order [⊆] to a partial order by defining setoid
equality as [λ X Y, X ⊆ Y ∧ Y ⊆ X]. We prove that this indeed gives rise to a
setoid. *)
318 319
Instance preorder_equiv `{SubsetEq A} : Equiv A := λ X Y, X  Y  Y  X.

Robbert Krebbers's avatar
Robbert Krebbers committed
320
Section preorder.
321
  Context `{SubsetEq A} `{!PreOrder (@subseteq A _)}.
Robbert Krebbers's avatar
Robbert Krebbers committed
322 323

  Instance preorder_equivalence: @Equivalence A ().
324 325
  Proof.
    split.
326 327
    * done.
    * by intros ?? [??].
328
    * by intros X Y Z [??] [??]; split; transitivity Y.
329
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
330 331
  Global Instance: Proper (() ==> () ==> iff) ().
  Proof.
332 333 334
    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
335
  Qed.
336
  Lemma subset_spec X Y : X  Y  X  Y  X  Y.
337 338
  Proof.
    split.
339
    * intros [? HYX]. split. done. contradict HYX. by rewrite <-HYX.
340 341
    * intros [? HXY]. split. done. by contradict HXY.
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
342

343 344 345 346 347
  Section dec.
    Context `{ X Y : A, Decision (X  Y)}.
    Global Instance preorder_equiv_dec_slow (X Y : A) :
      Decision (X  Y) | 100 := _.
    Lemma subseteq_inv X Y : X  Y  X  Y  X  Y.
348
    Proof. rewrite subset_spec. destruct (decide (X  Y)); tauto. Qed.
349
    Lemma not_subset_inv X Y : X  Y  X  Y  X  Y.
350
    Proof. rewrite subset_spec. destruct (decide (X  Y)); tauto. Qed.
351
  End dec.
352
End preorder.
Robbert Krebbers's avatar
Robbert Krebbers committed
353

354 355 356 357 358 359 360 361 362 363 364 365
Section preorder_leibniz.
  Context `{SubsetEq A} `{!PreOrder (@subseteq A _)} `{!LeibnizEquiv A}.
  Lemma subset_spec_L X Y : X  Y  X  Y  X  Y.
  Proof. unfold_leibniz. apply subset_spec. Qed.

  Context `{ X Y : A, Decision (X  Y)}.
  Lemma subseteq_inv_L X Y : X  Y  X  Y  X = Y.
  Proof. unfold_leibniz. apply subseteq_inv. Qed.
  Lemma not_subset_inv_L X Y : X  Y  X  Y  X = Y.
  Proof. unfold_leibniz. apply not_subset_inv. Qed.
End preorder_leibniz.

366
Typeclasses Opaque preorder_equiv.
367 368
Hint Extern 0 (@Equivalence _ ()) =>
  class_apply preorder_equivalence : typeclass_instances.
Robbert Krebbers's avatar
Robbert Krebbers committed
369

370 371
(** * Partial orders *)
Section partialorder.
372
  Context `{SubsetEq A} `{!PartialOrder (@subseteq A _)}.
373
  Global Instance: LeibnizEquiv A.
374
  Proof. split. intros [??]. by apply (anti_symmetric ()). by intros ->. Qed.
375 376
End partialorder.

377 378
(** * Join semi lattices *)
(** General purpose theorems on join semi lattices. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
379 380
Section bounded_join_sl.
  Context `{BoundedJoinSemiLattice A}.
381 382
  Implicit Types X Y : A.
  Implicit Types Xs Ys : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
383

384
  Hint Resolve subseteq_empty union_subseteq_l union_subseteq_r union_least.
385 386 387 388 389 390
  Lemma union_subseteq_l_transitive X1 X2 Y : X1  X2  X1  X2  Y.
  Proof. intros. transitivity X2; auto. Qed.
  Lemma union_subseteq_r_transitive X1 X2 Y : X1  X2  X1  Y  X2.
  Proof. intros. transitivity X2; auto. Qed.
  Hint Resolve union_subseteq_l_transitive union_subseteq_r_transitive.
  Lemma union_preserving_l X Y1 Y2 : Y1  Y2  X  Y1  X  Y2.
391
  Proof. auto. Qed.
392
  Lemma union_preserving_r X1 X2 Y : X1  X2  X1  Y  X2  Y.
393
  Proof. auto. Qed.
394
  Lemma union_preserving X1 X2 Y1 Y2 : X1  X2  Y1  Y2  X1  Y1  X2  Y2.
Robbert Krebbers's avatar
Robbert Krebbers committed
395
  Proof. auto. Qed.
396
  Lemma union_empty X : X    X.
397
  Proof. by apply union_least. Qed.
398
  Global Instance union_proper : Proper (() ==> () ==> ()) ().
399
  Proof.
400 401
    unfold equiv, preorder_equiv.
    split; apply union_preserving; simpl in *; tauto.
402
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
403 404 405 406 407 408 409
  Global Instance: Idempotent () ().
  Proof. split; eauto. Qed.
  Global Instance: LeftId ()  ().
  Proof. split; eauto. Qed.
  Global Instance: RightId ()  ().
  Proof. split; eauto. Qed.
  Global Instance: Commutative () ().
410
  Proof. split; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
411
  Global Instance: Associative () ().
412
  Proof. split; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
413
  Lemma subseteq_union X Y : X  Y  X  Y  Y.
414
  Proof. repeat split; eauto. intros HXY. rewrite <-HXY. auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
415 416 417 418 419 420
  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.
421 422 423
  Global Instance union_list_proper:
    Proper (Forall2 () ==> ()) union_list.
  Proof. induction 1; simpl. done. by apply union_proper. Qed.
424 425
  Lemma union_list_nil :  @nil A = .
  Proof. done. Qed.
426
  Lemma union_list_cons X Xs :  (X :: Xs) = X   Xs.
427
  Proof. done. Qed.
428
  Lemma union_list_singleton X :  [X]  X.
429
  Proof. simpl. by rewrite (right_id  _). Qed.
430
  Lemma union_list_app Xs1 Xs2 :  (Xs1 ++ Xs2)   Xs1   Xs2.
431
  Proof.
432 433
    induction Xs1 as [|X Xs1 IH]; simpl; [by rewrite (left_id  _)|].
    by rewrite IH, (associative _).
434
  Qed.
435
  Lemma union_list_reverse Xs :  (reverse Xs)   Xs.
436 437 438 439 440
  Proof.
    induction Xs as [|X Xs IH]; simpl; [done |].
    by rewrite reverse_cons, union_list_app,
      union_list_singleton, (commutative _), IH.
  Qed.
441
  Lemma union_list_preserving Xs Ys : Xs * Ys   Xs   Ys.
442
  Proof. induction 1; simpl; auto using union_preserving. Qed.
443 444 445
  Lemma empty_union X Y : X  Y    X    Y  .
  Proof.
    split.
446 447 448
    * intros HXY. split; apply equiv_empty;
        by transitivity (X  Y); [auto | rewrite HXY].
    * intros [HX HY]. by rewrite HX, HY, (left_id _ _).
449
  Qed.
450
  Lemma empty_union_list Xs :  Xs    Forall ( ) Xs.
451 452 453 454 455 456
  Proof.
    split.
    * induction Xs; simpl; rewrite ?empty_union; intuition.
    * induction 1 as [|?? E1 ? E2]; simpl. done. by apply empty_union.
  Qed.

457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484
  Section leibniz.
    Context `{!LeibnizEquiv A}.
    Global Instance: Idempotent (=) ().
    Proof. intros ?. unfold_leibniz. apply (idempotent _). Qed.
    Global Instance: LeftId (=)  ().
    Proof. intros ?. unfold_leibniz. apply (left_id _ _). Qed.
    Global Instance: RightId (=)  ().
    Proof. intros ?. unfold_leibniz. apply (right_id _ _). Qed.
    Global Instance: Commutative (=) ().
    Proof. intros ??. unfold_leibniz. apply (commutative _). Qed.
    Global Instance: Associative (=) ().
    Proof. intros ???. unfold_leibniz. apply (associative _). Qed.
    Lemma subseteq_union_L X Y : X  Y  X  Y = Y.
    Proof. unfold_leibniz. apply subseteq_union. Qed.
    Lemma subseteq_union_1_L X Y : X  Y  X  Y = Y.
    Proof. unfold_leibniz. apply subseteq_union_1. Qed.
    Lemma subseteq_union_2_L X Y : X  Y = Y  X  Y.
    Proof. unfold_leibniz. apply subseteq_union_2. Qed.
    Lemma equiv_empty_L X : X    X = .
    Proof. unfold_leibniz. apply equiv_empty. Qed.
    Lemma union_list_singleton_L (X : A) :  [X] = X.
    Proof. unfold_leibniz. apply union_list_singleton. Qed.
    Lemma union_list_app_L (Xs1 Xs2 : list A) :  (Xs1 ++ Xs2) =  Xs1   Xs2.
    Proof. unfold_leibniz. apply union_list_app. Qed.
    Lemma union_list_reverse_L (Xs : list A) :  (reverse Xs) =  Xs.
    Proof. unfold_leibniz. apply union_list_reverse. Qed.
    Lemma empty_union_L X Y : X  Y =   X =   Y = .
    Proof. unfold_leibniz. apply empty_union. Qed.
485
    Lemma empty_union_list_L Xs :  Xs =   Forall (= ) Xs.
486
    Proof. unfold_leibniz. by rewrite empty_union_list. Qed. 
487 488 489 490
  End leibniz.

  Section dec.
    Context `{ X Y : A, Decision (X  Y)}.
491
    Lemma non_empty_union X Y : X  Y    X    Y  .
492
    Proof. rewrite empty_union. destruct (decide (X  )); intuition. Qed.
493 494
    Lemma non_empty_union_list Xs :  Xs    Exists ( ) Xs.
    Proof. rewrite empty_union_list. apply (not_Forall_Exists _). Qed.
495
    Context `{!LeibnizEquiv A}.
496
    Lemma non_empty_union_L X Y : X  Y    X    Y  .
497
    Proof. unfold_leibniz. apply non_empty_union. Qed.
498 499
    Lemma non_empty_union_list_L Xs :  Xs    Exists ( ) Xs.
    Proof. unfold_leibniz. apply non_empty_union_list. Qed.
500
  End dec.
Robbert Krebbers's avatar
Robbert Krebbers committed
501 502
End bounded_join_sl.

503 504
(** * Meet semi lattices *)
(** The dual of the above section, but now for meet semi lattices. *)
Robbert Krebbers's avatar
Robbert Krebbers committed
505 506
Section meet_sl.
  Context `{MeetSemiLattice A}.
507 508
  Implicit Types X Y : A.
  Implicit Types Xs Ys : list A.
Robbert Krebbers's avatar
Robbert Krebbers committed
509

510
  Hint Resolve intersection_subseteq_l intersection_subseteq_r
511
    intersection_greatest.
512 513 514 515 516 517 518
  Lemma intersection_subseteq_l_transitive X1 X2 Y : X1  X2  X1  Y  X2.
  Proof. intros. transitivity X1; auto. Qed.
  Lemma intersection_subseteq_r_transitive X1 X2 Y : X1  X2  Y  X1  X2.
  Proof. intros. transitivity X1; auto. Qed.
  Hint Resolve intersection_subseteq_l_transitive
    intersection_subseteq_r_transitive.
  Lemma intersection_preserving_l X Y1 Y2 : Y1  Y2  X  Y1  X  Y2.
Robbert Krebbers's avatar
Robbert Krebbers committed
519
  Proof. auto. Qed.
520
  Lemma intersection_preserving_r X1 X2 Y : X1  X2  X1  Y  X2  Y.
Robbert Krebbers's avatar
Robbert Krebbers committed
521
  Proof. auto. Qed.
522 523
  Lemma intersection_preserving X1 X2 Y1 Y2 :
    X1  X2  Y1  Y2  X1  Y1  X2  Y2.
Robbert Krebbers's avatar
Robbert Krebbers committed
524 525
  Proof. auto. Qed.
  Global Instance: Proper (() ==> () ==> ()) ().
526 527
  Proof.
    unfold equiv, preorder_equiv. split;
528
      apply intersection_preserving; simpl in *; tauto.
529
  Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
530 531 532
  Global Instance: Idempotent () ().
  Proof. split; eauto. Qed.
  Global Instance: Commutative () ().
533
  Proof. split; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
534
  Global Instance: Associative () ().
535
  Proof. split; auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
536
  Lemma subseteq_intersection X Y : X  Y  X  Y  X.
537
  Proof. repeat split; eauto. intros HXY. rewrite <-HXY. auto. Qed.
Robbert Krebbers's avatar
Robbert Krebbers committed
538 539 540 541
  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.
542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557

  Section leibniz.
    Context `{!LeibnizEquiv A}.
    Global Instance: Idempotent (=) ().
    Proof. intros ?. unfold_leibniz. apply (idempotent _). Qed.
    Global Instance: Commutative (=) ().
    Proof. intros ??. unfold_leibniz. apply (commutative _). Qed.
    Global Instance: Associative (=) ().
    Proof. intros ???. unfold_leibniz. apply (associative _). Qed.
    Lemma subseteq_intersection_L X Y : X  Y  X  Y = X.
    Proof. unfold_leibniz. apply subseteq_intersection. Qed.
    Lemma subseteq_intersection_1_L X Y : X  Y  X  Y = X.
    Proof. unfold_leibniz. apply subseteq_intersection_1. Qed.
    Lemma subseteq_intersection_2_L X Y : X  Y = X  X  Y.
    Proof. unfold_leibniz. apply subseteq_intersection_2. Qed.
  End leibniz.
Robbert Krebbers's avatar
Robbert Krebbers committed
558
End meet_sl.
559 560 561 562 563 564 565

(** * Lower bounded lattices *)
Section lower_bounded_lattice.
  Context `{LowerBoundedLattice A}.

  Global Instance: LeftAbsorb ()  ().
  Proof.
566
    split. by apply intersection_subseteq_l. by apply subseteq_empty.
567 568 569
  Qed.
  Global Instance: RightAbsorb ()  ().
  Proof. intros ?. by rewrite (commutative _), (left_absorb _ _). Qed.
570 571
  Global Instance: LeftDistr () () ().
  Proof.
572
    intros X Y Z. split.
573 574 575
    * apply union_least.
      { apply intersection_greatest; auto using union_subseteq_l. }
      apply intersection_greatest.
576 577
      + apply union_subseteq_r_transitive, intersection_subseteq_l.
      + apply union_subseteq_r_transitive, intersection_subseteq_r.
578 579 580
    * apply lbl_distr.
  Qed.
  Global Instance: RightDistr () () ().
581
  Proof. intros X Y Z. by rewrite !(commutative _ _ Z), (left_distr _ _). Qed.
582 583
  Global Instance: LeftDistr () () ().
  Proof.
584
    intros X Y Z. split.
585 586
    * rewrite (left_distr () ()).
      apply intersection_greatest.
587
      { apply union_subseteq_r_transitive, intersection_subseteq_l. }
588 589 590 591 592 593
      rewrite (right_distr () ()). apply intersection_preserving.
      + apply union_subseteq_l.
      + done.
    * apply intersection_greatest.
      { apply union_least; auto using intersection_subseteq_l. }
      apply union_least.
594 595
      + apply intersection_subseteq_r_transitive, union_subseteq_l.
      + apply intersection_subseteq_r_transitive, union_subseteq_r.
596 597
  Qed.
  Global Instance: RightDistr () () ().
598
  Proof. intros X Y Z. by rewrite !(commutative _ _ Z), (left_distr _ _). Qed.
599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614

  Section leibniz.
    Context `{!LeibnizEquiv A}.
    Global Instance: LeftAbsorb (=)  ().
    Proof. intros ?. unfold_leibniz. apply (left_absorb _ _). Qed.
    Global Instance: RightAbsorb (=)  ().
    Proof. intros ?. unfold_leibniz. apply (right_absorb _ _). Qed.
    Global Instance: LeftDistr (=) () ().
    Proof. intros ???. unfold_leibniz. apply (left_distr _ _). Qed.
    Global Instance: RightDistr (=) () ().
    Proof. intros ???. unfold_leibniz. apply (right_distr _ _). Qed.
    Global Instance: LeftDistr (=) () ().
    Proof. intros ???. unfold_leibniz. apply (left_distr _ _). Qed.
    Global Instance: RightDistr (=) () ().
    Proof. intros ???. unfold_leibniz. apply (right_distr _ _). Qed.
  End leibniz.
615
End lower_bounded_lattice.