(* Copyright (c) 2012-2015, Robbert Krebbers. *) (* This file is distributed under the terms of the BSD license. *) (** This files defines a lexicographic order on various common data structures and proves that it is a partial order having a strong variant of trichotomy. *) Require Import numbers. Notation cast_trichotomy T := match T with | inleft (left _) => inleft (left _) | inleft (right _) => inleft (right _) | inright _ => inright _ end. Instance prod_lexico `{Lexico A, Lexico B} : Lexico (A * B) := λ p1 p2, (**i 1.) *) lexico (p1.1) (p2.1) ∨ (**i 2.) *) p1.1 = p2.1 ∧ lexico (p1.2) (p2.2). Instance bool_lexico : Lexico bool := λ b1 b2, match b1, b2 with false, true => True | _, _ => False end. Instance nat_lexico : Lexico nat := (<). Instance N_lexico : Lexico N := (<)%N. Instance Z_lexico : Lexico Z := (<)%Z. Typeclasses Opaque bool_lexico nat_lexico N_lexico Z_lexico. Instance list_lexico `{Lexico A} : Lexico (list A) := fix go l1 l2 := let _ : Lexico (list A) := @go in match l1, l2 with | [], _ :: _ => True | x1 :: l1, x2 :: l2 => lexico (x1,l1) (x2,l2) | _, _ => False end. Instance sig_lexico `{Lexico A} (P : A → Prop) `{∀ x, ProofIrrel (P x)} : Lexico (sig P) := λ x1 x2, lexico (`x1) (`x2). Lemma prod_lexico_irreflexive `{Lexico A, Lexico B, !Irreflexive (@lexico A _)} (x : A) (y : B) : complement lexico y y → complement lexico (x,y) (x,y). Proof. intros ? [?|[??]]. by apply (irreflexivity lexico x). done. Qed. Lemma prod_lexico_transitive `{Lexico A, Lexico B, !Transitive (@lexico A _)} (x1 x2 x3 : A) (y1 y2 y3 : B) : lexico (x1,y1) (x2,y2) → lexico (x2,y2) (x3,y3) → (lexico y1 y2 → lexico y2 y3 → lexico y1 y3) → lexico (x1,y1) (x3,y3). Proof. intros Hx12 Hx23 ?; revert Hx12 Hx23. unfold lexico, prod_lexico. intros [|[??]] [?|[??]]; simplify_equality'; auto. by left; transitivity x2. Qed. Instance prod_lexico_po `{Lexico A, Lexico B, !StrictOrder (@lexico A _)} `{!StrictOrder (@lexico B _)} : StrictOrder (@lexico (A * B) _). Proof. split. * intros [x y]. apply prod_lexico_irreflexive. by apply (irreflexivity lexico y). * intros [??] [??] [??] ??. eapply prod_lexico_transitive; eauto. apply transitivity. Qed. Instance prod_lexico_trichotomyT `{Lexico A, tA : !TrichotomyT (@lexico A _)} `{Lexico B, tB : !TrichotomyT (@lexico B _)}: TrichotomyT (@lexico (A * B) _). Proof. red; refine (λ p1 p2, match trichotomyT lexico (p1.1) (p2.1) with | inleft (left _) => inleft (left _) | inleft (right _) => cast_trichotomy (trichotomyT lexico (p1.2) (p2.2)) | inright _ => inright _ end); clear tA tB; abstract (unfold lexico, prod_lexico; auto using injective_projections). Defined. Instance bool_lexico_po : StrictOrder (@lexico bool _). Proof. split. by intros [] ?. by intros [] [] [] ??. Qed. Instance bool_lexico_trichotomy: TrichotomyT (@lexico bool _). Proof. red; refine (λ b1 b2, match b1, b2 with | false, false => inleft (right _) | false, true => inleft (left _) | true, false => inright _ | true, true => inleft (right _) end); abstract (unfold strict, lexico, bool_lexico; naive_solver). Defined. Instance nat_lexico_po : StrictOrder (@lexico nat _). Proof. unfold lexico, nat_lexico. apply _. Qed. Instance nat_lexico_trichotomy: TrichotomyT (@lexico nat _). Proof. red; refine (λ n1 n2, match Nat.compare n1 n2 as c return Nat.compare n1 n2 = c → _ with | Lt => λ H, inleft (left (nat_compare_Lt_lt _ _ H)) | Eq => λ H, inleft (right (nat_compare_eq _ _ H)) | Gt => λ H, inright (nat_compare_Gt_gt _ _ H) end eq_refl). Defined. Instance N_lexico_po : StrictOrder (@lexico N _). Proof. unfold lexico, N_lexico. apply _. Qed. Instance N_lexico_trichotomy: TrichotomyT (@lexico N _). Proof. red; refine (λ n1 n2, match N.compare n1 n2 as c return N.compare n1 n2 = c → _ with | Lt => λ H, inleft (left (proj2 (N.compare_lt_iff _ _) H)) | Eq => λ H, inleft (right (N.compare_eq _ _ H)) | Gt => λ H, inright (proj1 (N.compare_gt_iff _ _) H) end eq_refl). Defined. Instance Z_lexico_po : StrictOrder (@lexico Z _). Proof. unfold lexico, Z_lexico. apply _. Qed. Instance Z_lexico_trichotomy: TrichotomyT (@lexico Z _). Proof. red; refine (λ n1 n2, match Z.compare n1 n2 as c return Z.compare n1 n2 = c → _ with | Lt => λ H, inleft (left (proj2 (Z.compare_lt_iff _ _) H)) | Eq => λ H, inleft (right (Z.compare_eq _ _ H)) | Gt => λ H, inright (proj1 (Z.compare_gt_iff _ _) H) end eq_refl). Defined. Instance list_lexico_po `{Lexico A, !StrictOrder (@lexico A _)} : StrictOrder (@lexico (list A) _). Proof. split. * intros l. induction l. by intros ?. by apply prod_lexico_irreflexive. * intros l1. induction l1 as [|x1 l1]; intros [|x2 l2] [|x3 l3] ??; try done. eapply prod_lexico_transitive; eauto. Qed. Instance list_lexico_trichotomy `{Lexico A, tA : !TrichotomyT (@lexico A _)} : TrichotomyT (@lexico (list A) _). Proof. refine ( fix go l1 l2 := let go' : TrichotomyT (@lexico (list A) _) := @go in match l1, l2 with | [], [] => inleft (right _) | [], _ :: _ => inleft (left _) | _ :: _, [] => inright _ | x1 :: l1, x2 :: l2 => cast_trichotomy (trichotomyT lexico (x1,l1) (x2,l2)) end); clear tA go go'; abstract (repeat (done || constructor || congruence || by inversion 1)). Defined. Instance sig_lexico_po `{Lexico A, !StrictOrder (@lexico A _)} (P : A → Prop) `{∀ x, ProofIrrel (P x)} : StrictOrder (@lexico (sig P) _). Proof. unfold lexico, sig_lexico. split. * intros [x ?] ?. by apply (irreflexivity lexico x). * intros [x1 ?] [x2 ?] [x3 ?] ??. by transitivity x2. Qed. Instance sig_lexico_trichotomy `{Lexico A, tA : !TrichotomyT (@lexico A _)} (P : A → Prop) `{∀ x, ProofIrrel (P x)} : TrichotomyT (@lexico (sig P) _). Proof. red; refine (λ x1 x2, cast_trichotomy (trichotomyT lexico (`x1) (`x2))); abstract (repeat (done || constructor || apply (sig_eq_pi P))). Defined.