Commit 2f9f3d3f by Robbert Krebbers

### Introduce `RelDecision` for decidable relations and define `EqDecision` using it.

`This allows for more control over `Hint Mode`.`
parent 929d64cc
 ... ... @@ -150,13 +150,30 @@ Hint Extern 0 (_ ≡ _) => symmetry; assumption. (** * Type classes *) (** ** Decidable propositions *) (** This type class by (Spitters/van der Weegen, 2011) collects decidable propositions. For example to declare a parameter expressing decidable equality on a type [A] we write [`{∀ x y : A, Decision (x = y)}] and use it by writing [decide (x = y)]. *) propositions. *) Class Decision (P : Prop) := decide : {P} + {¬P}. Hint Mode Decision ! : typeclass_instances. Arguments decide _ {_} : assert. Notation EqDecision A := (∀ x y : A, Decision (x = y)). Arguments decide _ {_} : simpl never, assert. (** Although [RelDecision R] is just [∀ x y, Decision (R x y)], we make this an explicit class instead of a notation for two reasons: - It allows us to control [Hint Mode] more precisely. In particular, if it were defined as a notation, the above [Hint Mode] for [Decision] would not prevent diverging instance search when looking for [RelDecision (@eq ?A)], which would result in it looking for [Decision (@eq ?A x y)], i.e. an instance where the head position of [Decision] is not en evar. - We use it to avoid inefficient computation due to eager evaluation of propositions by [vm_compute]. This inefficiency arises for example if [(x = y) := (f x = f y)]. Since [decide (x = y)] evaluates to [decide (f x = f y)], this would then lead to evaluation of [f x] and [f y]. Using the [RelDecision], the [f] is hidden under a lambda, which prevents unnecessary evaluation. *) Class RelDecision {A B} (R : A → B → Prop) := decide_rel x y :> Decision (R x y). Hint Mode RelDecision ! ! ! : typeclass_instances. Arguments decide_rel {_ _} _ {_} _ _ : simpl never, assert. Notation EqDecision A := (RelDecision (@eq A)). (** ** Inhabited types *) (** This type class collects types that are inhabited. *) ... ...
 ... ... @@ -29,7 +29,8 @@ Proof. - intros X Y x; unfold elem_of, bset_elem_of; simpl. destruct (bset_car X x), (bset_car Y x); simpl; tauto. Qed. Instance bset_elem_of_dec {A} x (X : bset A) : Decision (x ∈ X) := _. Instance bset_elem_of_dec {A} : RelDecision (@elem_of _ (bset A) _). Proof. refine (λ x X, cast_if (decide (bset_car X x))); done. Defined. Typeclasses Opaque bset_elem_of. Global Opaque bset_empty bset_singleton bset_union ... ...
 ... ... @@ -190,17 +190,18 @@ Proof. intros p; apply eq_bool_prop_intro, (HXY p). Qed. Instance coPset_elem_of_dec (p : positive) (X : coPset) : Decision (p ∈ X) := _. Instance coPset_equiv_dec (X Y : coPset) : Decision (X ≡ Y). Proof. refine (cast_if (decide (X = Y))); abstract (by fold_leibniz). Defined. Instance mapset_disjoint_dec (X Y : coPset) : Decision (X ⊥ Y). Instance coPset_elem_of_dec : RelDecision (@elem_of _ coPset _). Proof. solve_decision. Defined. Instance coPset_equiv_dec : RelDecision (@equiv coPset _). Proof. refine (λ X Y, cast_if (decide (X = Y))); abstract (by fold_leibniz). Defined. Instance mapset_disjoint_dec : RelDecision (@disjoint coPset _). Proof. refine (cast_if (decide (X ∩ Y = ∅))); refine (λ X Y, cast_if (decide (X ∩ Y = ∅))); abstract (by rewrite disjoint_intersection_L). Defined. Instance mapset_subseteq_dec (X Y : coPset) : Decision (X ⊆ Y). Instance mapset_subseteq_dec : RelDecision (@subseteq coPset _). Proof. refine (cast_if (decide (X ∪ Y = Y))); abstract (by rewrite subseteq_union_L). refine (λ X Y, cast_if (decide (X ∪ Y = Y))); abstract (by rewrite subseteq_union_L). Defined. (** * Top *) ... ...
 ... ... @@ -527,7 +527,7 @@ Section simple_collection. End leibniz. Section dec. Context `{∀ (X Y : C), Decision (X ≡ Y)}. Context `{!RelDecision (@equiv C _)}. Lemma collection_subseteq_inv X Y : X ⊆ Y → X ⊂ Y ∨ X ≡ Y. Proof. destruct (decide (X ≡ Y)); [by right|left;set_solver]. Qed. Lemma collection_not_subset_inv X Y : X ⊄ Y → X ⊈ Y ∨ X ≡ Y. ... ... @@ -692,7 +692,7 @@ Section collection. End leibniz. Section dec. Context `{∀ (x : A) (X : C), Decision (x ∈ X)}. Context `{!RelDecision (@elem_of A C _)}. Lemma not_elem_of_intersection x X Y : x ∉ X ∩ Y ↔ x ∉ X ∨ x ∉ Y. Proof. rewrite elem_of_intersection. destruct (decide (x ∈ X)); tauto. Qed. Lemma not_elem_of_difference x X Y : x ∉ X ∖ Y ↔ x ∉ X ∨ x ∈ Y. ... ...
 ... ... @@ -10,6 +10,7 @@ Class Countable A `{EqDecision A} := { decode : positive → option A; decode_encode x : decode (encode x) = Some x }. Hint Mode Countable ! - : typeclass_instances. Arguments encode : simpl never. Arguments decode : simpl never. ... ... @@ -36,8 +37,8 @@ Section choice. Context `{Countable A} (P : A → Prop). Inductive choose_step: relation positive := | choose_step_None {p} : decode p = None → choose_step (Psucc p) p | choose_step_Some {p x} : | choose_step_None {p} : decode (A:=A) p = None → choose_step (Psucc p) p | choose_step_Some {p} {x : A} : decode p = Some x → ¬P x → choose_step (Psucc p) p. Lemma choose_step_acc : (∃ x, P x) → Acc choose_step 1%positive. Proof. ... ... @@ -320,7 +321,7 @@ Arguments GenNode {_} _ _ : assert. Instance gen_tree_dec `{EqDecision T} : EqDecision (gen_tree T). Proof. refine ( fix go t1 t2 := fix go t1 t2 := let _ : EqDecision _ := @go in match t1, t2 with | GenLeaf x1, GenLeaf x2 => cast_if (decide (x1 = x2)) | GenNode n1 ts1, GenNode n2 ts2 => ... ...
 ... ... @@ -6,8 +6,6 @@ type class. *) From stdpp Require Export proof_irrel. Set Default Proof Using "Type*". Hint Extern 200 (Decision _) => progress (lazy beta) : typeclass_instances. Lemma dec_stable `{Decision P} : ¬¬P → P. Proof. firstorder. Qed. ... ... @@ -16,17 +14,6 @@ Proof. destruct b. left; constructor. right. intros []. Qed. Instance: Inj (=) (↔) Is_true. Proof. intros [] []; simpl; intuition. Qed. (** We introduce [decide_rel] to avoid inefficienct computation due to eager evaluation of propositions by [vm_compute]. This inefficiency occurs if [(x = y) := (f x = f y)] as [decide (x = y)] evaluates to [decide (f x = f y)] which then might lead to evaluation of [f x] and [f y]. Using [decide_rel] we hide [f] under a lambda abstraction to avoid this unnecessary evaluation. *) Definition decide_rel {A B} (R : A → B → Prop) {dec : ∀ x y, Decision (R x y)} (x : A) (y : B) : Decision (R x y) := dec x y. Lemma decide_rel_correct {A B} (R : A → B → Prop) `{∀ x y, Decision (R x y)} (x : A) (y : B) : decide_rel R x y = decide (R x y). Proof. reflexivity. Qed. Lemma decide_True {A} `{Decision P} (x y : A) : P → (if decide P then x else y) = x. Proof. destruct (decide P); tauto. Qed. ... ... @@ -75,9 +62,10 @@ Ltac solve_trivial_decision := | |- Decision (?P) => apply _ | |- sumbool ?P (¬?P) => change (Decision P); apply _ end. Ltac solve_decision := intros; first [ solve_trivial_decision | unfold Decision; decide equality; solve_trivial_decision ]. Ltac solve_decision := unfold EqDecision; intros; first [ solve_trivial_decision | unfold Decision; decide equality; solve_trivial_decision ]. (** The following combinators are useful to create Decision proofs in combination with the [refine] tactic. *) ... ...
 ... ... @@ -23,9 +23,9 @@ Implicit Types X Y : C. Lemma fin_collection_finite X : set_finite X. Proof. by exists (elements X); intros; rewrite elem_of_elements. Qed. Instance elem_of_dec_slow (x : A) (X : C) : Decision (x ∈ X) | 100. Instance elem_of_dec_slow : RelDecision (@elem_of A C _) | 100. Proof. refine (cast_if (decide_rel (∈) x (elements X))); refine (λ x X, cast_if (decide_rel (∈) x (elements X))); by rewrite <-(elem_of_elements _). Defined. ... ...
 ... ... @@ -1238,10 +1238,9 @@ Proof. destruct (m1 !! i), (m2 !! i); simplify_eq/=; auto; by eapply bool_decide_unpack, Hm. Qed. Global Instance map_relation_dec `{∀ x y, Decision (R x y), ∀ x, Decision (P x), ∀ y, Decision (Q y)} (m1 : M A) (m2 : M B) : Decision (map_relation R P Q m1 m2). Global Instance map_relation_dec : RelDecision (map_relation (M:=M) R P Q). Proof. refine (cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2)))); refine (λ m1 m2, cast_if (decide (map_Forall (λ _, Is_true) (merge f m1 m2)))); abstract by rewrite map_relation_alt. Defined. (** Due to the finiteness of finite maps, we can extract a witness if the ... ...
 ... ... @@ -8,6 +8,7 @@ Class Finite A `{EqDecision A} := { NoDup_enum : NoDup enum; elem_of_enum x : x ∈ enum }. Hint Mode Finite ! - : typeclass_instances. Arguments enum : clear implicits. Arguments enum _ {_ _} : assert. Arguments NoDup_enum : clear implicits. ... ...
 ... ... @@ -7,18 +7,14 @@ Record gmultiset A `{Countable A} := GMultiSet { gmultiset_car : gmap A nat }. Arguments GMultiSet {_ _ _} _ : assert. Arguments gmultiset_car {_ _ _} _ : assert. Lemma gmultiset_eq_dec `{Countable A} : EqDecision (gmultiset A). Instance gmultiset_eq_dec `{Countable A} : EqDecision (gmultiset A). Proof. solve_decision. Defined. Hint Extern 1 (Decision (@eq (gmultiset _) _ _)) => eapply @gmultiset_eq_dec : typeclass_instances. Program Definition gmultiset_countable `{Countable A} : Program Instance gmultiset_countable `{Countable A} : Countable (gmultiset A) := {| encode X := encode (gmultiset_car X); decode p := GMultiSet <\$> decode p |}. Next Obligation. intros A ?? [X]; simpl. by rewrite decode_encode. Qed. Hint Extern 1 (Countable (gmultiset _)) => eapply @gmultiset_countable : typeclass_instances. Section definitions. Context `{Countable A}. ... ... @@ -102,8 +98,8 @@ Proof. by split; auto with lia. - intros X Y x. rewrite !elem_of_multiplicity, multiplicity_union. omega. Qed. Global Instance gmultiset_elem_of_dec x X : Decision (x ∈ X). Proof. unfold elem_of, gmultiset_elem_of. apply _. Defined. Global Instance gmultiset_elem_of_dec : RelDecision (@elem_of _ (gmultiset A) _). Proof. refine (λ x X, cast_if (decide (0 < multiplicity x X))); done. Defined. (* Algebraic laws *) Global Instance gmultiset_comm : Comm (@eq (gmultiset A)) (∪). ... ... @@ -144,8 +140,9 @@ Qed. Lemma gmultiset_elements_empty_inv X : elements X = [] → X = ∅. Proof. destruct X as [X]; unfold elements, gmultiset_elements; simpl. intros; apply (f_equal GMultiSet). destruct (map_to_list X) as [|[]] eqn:?; naive_solver eauto using map_to_list_empty_inv. intros; apply (f_equal GMultiSet). destruct (map_to_list X) as [|[]] eqn:?. - by apply map_to_list_empty_inv. - naive_solver. Qed. Lemma gmultiset_elements_empty' X : elements X = [] ↔ X = ∅. Proof. ... ... @@ -249,9 +246,9 @@ Proof. apply forall_proper; intros x. unfold multiplicity. destruct (gmultiset_car X !! x), (gmultiset_car Y !! x); naive_solver omega. Qed. Global Instance gmultiset_subseteq_dec X Y : Decision (X ⊆ Y). Global Instance gmultiset_subseteq_dec : RelDecision (@subseteq (gmultiset A) _). Proof. refine (cast_if (decide (map_relation (≤) refine (λ X Y, cast_if (decide (map_relation (≤) (λ _, False) (λ _, True) (gmultiset_car X) (gmultiset_car Y)))); by rewrite gmultiset_subseteq_alt. Defined. ... ...
 ... ... @@ -311,13 +311,13 @@ Instance list_subseteq {A} : SubsetEq (list A) := λ l1 l2, ∀ x, x ∈ l1 → Section list_set. Context `{dec : EqDecision A}. Global Instance elem_of_list_dec (x : A) : ∀ l : list A, Decision (x ∈ l). Global Instance elem_of_list_dec : RelDecision (@elem_of A (list A) _). Proof. refine ( fix go l := fix go x l := match l return Decision (x ∈ l) with | [] => right _ | y :: l => cast_if_or (decide (x = y)) (go l) | y :: l => cast_if_or (decide (x = y)) (go x l) end); clear go dec; subst; try (by constructor); abstract by inversion 1. Defined. Fixpoint remove_dups (l : list A) : list A := ... ... @@ -1505,9 +1505,9 @@ Proof. - intros ?. by eexists []. - intros ???[k1->] [k2->]. exists (k2 ++ k1). by rewrite (assoc_L (++)). Qed. Global Instance prefix_dec `{!EqDecision A} : ∀ l1 l2, Decision (l1 `prefix_of` l2) := fix go l1 l2 := match l1, l2 return { l1 `prefix_of` l2 } + { ¬l1 `prefix_of` l2 } with Global Instance prefix_dec `{!EqDecision A} : RelDecision prefix := fix go l1 l2 := match l1, l2 with | [], _ => left (prefix_nil _) | _, [] => right (prefix_nil_not _ _) | x :: l1, y :: l2 => ... ... @@ -1639,10 +1639,9 @@ Lemma suffix_length l1 l2 : l1 `suffix_of` l2 → length l1 ≤ length l2. Proof. intros [? ->]. rewrite app_length. lia. Qed. Lemma suffix_cons_not x l : ¬x :: l `suffix_of` l. Proof. intros [??]. discriminate_list. Qed. Global Instance suffix_dec `{!EqDecision A} l1 l2 : Decision (l1 `suffix_of` l2). Global Instance suffix_dec `{!EqDecision A} : RelDecision (@suffix A). Proof. refine (cast_if (decide_rel prefix (reverse l1) (reverse l2))); refine (λ l1 l2, cast_if (decide_rel prefix (reverse l1) (reverse l2))); abstract (by rewrite suffix_prefix_reverse). Defined. ... ... @@ -2087,14 +2086,14 @@ Section submseteq_dec. destruct (list_remove x l2) as [k'|] eqn:?; intros; simplify_eq. rewrite submseteq_cons_l. eauto using list_remove_Some. Qed. Global Instance submseteq_dec l1 l2 : Decision (l1 ⊆+ l2). Global Instance submseteq_dec : RelDecision (submseteq : relation (list A)). Proof. refine (cast_if (decide (is_Some (list_remove_list l1 l2)))); refine (λ l1 l2, cast_if (decide (is_Some (list_remove_list l1 l2)))); abstract (rewrite list_remove_list_submseteq; tauto). Defined. Global Instance Permutation_dec l1 l2 : Decision (l1 ≡ₚ l2). Global Instance Permutation_dec : RelDecision (Permutation : relation (list A)). Proof. refine (cast_if_and refine (λ l1 l2, cast_if_and (decide (length l1 = length l2)) (decide (l1 ⊆+ l2))); abstract (rewrite Permutation_alt; tauto). Defined. ... ... @@ -2621,7 +2620,7 @@ Section Forall2. Qed. Global Instance Forall2_dec `{dec : ∀ x y, Decision (P x y)} : ∀ l k, Decision (Forall2 P l k). RelDecision (Forall2 P). Proof. refine ( fix go l k : Decision (Forall2 P l k) := ... ...
 ... ... @@ -76,18 +76,18 @@ Section deciders. match X1, X2 with Mapset m1, Mapset m2 => cast_if (decide (m1 = m2)) end); abstract congruence. Defined. Global Instance mapset_equiv_dec (X1 X2 : mapset M) : Decision (X1 ≡ X2) | 1. Proof. refine (cast_if (decide (X1 = X2))); abstract (by fold_leibniz). Defined. Global Instance mapset_elem_of_dec x (X : mapset M) : Decision (x ∈ X) | 1. Proof. solve_decision. Defined. Global Instance mapset_disjoint_dec (X1 X2 : mapset M) : Decision (X1 ⊥ X2). Global Instance mapset_equiv_dec : RelDecision (@equiv (mapset M)_) | 1. Proof. refine (λ X1 X2, cast_if (decide (X1 = X2))); abstract (by fold_leibniz). Defined. Global Instance mapset_elem_of_dec : RelDecision (@elem_of _ (mapset M) _) | 1. Proof. refine (λ x X, cast_if (decide (mapset_car X !! x = Some ()))); done. Defined. Global Instance mapset_disjoint_dec : RelDecision (@disjoint (mapset M) _). Proof. refine (cast_if (decide (X1 ∩ X2 = ∅))); refine (λ X1 X2, cast_if (decide (X1 ∩ X2 = ∅))); abstract (by rewrite disjoint_intersection_L). Defined. Global Instance mapset_subseteq_dec (X1 X2 : mapset M) : Decision (X1 ⊆ X2). Global Instance mapset_subseteq_dec : RelDecision (@subseteq (mapset M) _). Proof. refine (cast_if (decide (X1 ∪ X2 = X2))); refine (λ X1 X2, cast_if (decide (X1 ∪ X2 = X2))); abstract (by rewrite subseteq_union_L). Defined. End deciders. ... ...
 ... ... @@ -36,8 +36,8 @@ Infix "`max`" := Nat.max (at level 35) : nat_scope. Infix "`min`" := Nat.min (at level 35) : nat_scope. Instance nat_eq_dec: EqDecision nat := eq_nat_dec. Instance nat_le_dec: ∀ x y : nat, Decision (x ≤ y) := le_dec. Instance nat_lt_dec: ∀ x y : nat, Decision (x < y) := lt_dec. Instance nat_le_dec: RelDecision le := le_dec. Instance nat_lt_dec: RelDecision lt := lt_dec. Instance nat_inhabited: Inhabited nat := populate 0%nat. Instance S_inj: Inj (=) (=) S. Proof. by injection 1. Qed. ... ... @@ -77,9 +77,9 @@ Proof. intros. destruct (Nat_mul_split_l n x2 x1 y2 y1); auto with lia. Qed. Notation lcm := Nat.lcm. Notation divide := Nat.divide. Notation "( x | y )" := (divide x y) : nat_scope. Instance Nat_divide_dec x y : Decision (x | y). Instance Nat_divide_dec : RelDecision Nat.divide. Proof. refine (cast_if (decide (lcm x y = y))); by rewrite Nat.divide_lcm_iff. refine (λ x y, cast_if (decide (lcm x y = y))); by rewrite Nat.divide_lcm_iff. Defined. Instance: PartialOrder divide. Proof. ... ... @@ -130,6 +130,10 @@ Arguments Pos.of_nat : simpl never. Arguments Pmult : simpl never. Instance positive_eq_dec: EqDecision positive := Pos.eq_dec. Instance positive_le_dec: RelDecision Pos.le. Proof. refine (λ x y, decide ((x ?= y) ≠ Gt)). Defined. Instance positive_lt_dec: RelDecision Pos.lt. Proof. refine (λ x y, decide ((x ?= y) = Lt)). Defined. Instance positive_inhabited: Inhabited positive := populate 1. Instance maybe_xO : Maybe xO := λ p, match p with p~0 => Some p | _ => None end. ... ... @@ -218,10 +222,10 @@ Instance Npos_inj : Inj (=) (=) Npos. Proof. by injection 1. Qed. Instance N_eq_dec: EqDecision N := N.eq_dec. Program Instance N_le_dec (x y : N) : Decision (x ≤ y)%N := Program Instance N_le_dec : RelDecision N.le := λ x y, match Ncompare x y with Gt => right _ | _ => left _ end. Solve Obligations with naive_solver. Program Instance N_lt_dec (x y : N) : Decision (x < y)%N := Program Instance N_lt_dec : RelDecision N.lt := λ x y, match Ncompare x y with Lt => left _ | _ => right _ end. Solve Obligations with naive_solver. Instance N_inhabited: Inhabited N := populate 1%N. ... ... @@ -259,8 +263,8 @@ Instance Z_of_nat_inj : Inj (=) (=) Z.of_nat. Proof. intros n1 n2. apply Nat2Z.inj. Qed. Instance Z_eq_dec: EqDecision Z := Z.eq_dec. Instance Z_le_dec: ∀ x y : Z, Decision (x ≤ y) := Z_le_dec. Instance Z_lt_dec: ∀ x y : Z, Decision (x < y) := Z_lt_dec. Instance Z_le_dec: RelDecision Z.le := Z_le_dec. Instance Z_lt_dec: RelDecision Z.lt := Z_lt_dec. Instance Z_inhabited: Inhabited Z := populate 1. Instance Z_le_po : PartialOrder (≤). Proof. ... ... @@ -365,11 +369,11 @@ Hint Extern 1 (_ ≤ _) => reflexivity || discriminate. Arguments Qred : simpl never. Instance Qc_eq_dec: EqDecision Qc := Qc_eq_dec. Program Instance Qc_le_dec (x y : Qc) : Decision (x ≤ y) := Program Instance Qc_le_dec: RelDecision Qcle := λ x y, if Qclt_le_dec y x then right _ else left _. Next Obligation. intros x y; apply Qclt_not_le. Qed. Next Obligation. done. Qed. Program Instance Qc_lt_dec (x y : Qc) : Decision (x < y) := Program Instance Qc_lt_dec: RelDecision Qclt := λ x y, if Qclt_le_dec x y then left _ else right _. Solve Obligations with done. Next Obligation. intros x y; apply Qcle_not_lt. Qed. ... ...
 ... ... @@ -41,15 +41,16 @@ Section orders. 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 := _. Global Instance preorder_subset_dec_slow `{!RelDecision R} : RelDecision (strict R) | 100. Proof. solve_decision. Defined. Lemma strict_spec_alt `{!AntiSymm (=) R} X Y : X ⊂ Y ↔ X ⊆ Y ∧ X ≠ Y. Proof. split. - intros [? HYX]. split. done. by intros <-. - intros [? HXY]. split. done. by contradict HXY; apply (anti_symm R). Qed. Lemma po_eq_dec `{!PartialOrder R, ∀ X Y, Decision (X ⊆ Y)} : EqDecision A. Lemma po_eq_dec `{!PartialOrder R, !RelDecision R} : EqDecision A. Proof. refine (λ X Y, cast_if_and (decide (X ⊆ Y)) (decide (Y ⊆ X))); abstract (rewrite anti_symm_iff; tauto). ... ... @@ -76,8 +77,8 @@ Section strict_orders. Lemma strict_anti_symm `{!StrictOrder R} X Y : X ⊂ Y → Y ⊂ X → False. Proof. intros. apply (irreflexivity R X). by trans Y. Qed. Global Instance trichotomyT_dec `{!TrichotomyT R, !StrictOrder R} X Y : Decision (X ⊂ Y) := Global Instance trichotomyT_dec `{!TrichotomyT R, !StrictOrder R} : RelDecision R := λ X Y, match trichotomyT R X Y with | inleft (left H) => left H | inleft (right H) => right (irreflexive_eq _ _ H) ... ...
 ... ... @@ -15,7 +15,7 @@ Section merge_sort. | [], _ => l2 | _, [] => l1 | x1 :: l1, x2 :: l2 => if decide_rel R x1 x2 then x1 :: list_merge l1 (x2 :: l2) if decide (R x1 x2) then x1 :: list_merge l1 (x2 :: l2) else x2 :: list_merge_aux l2 end. Global Arguments list_merge !_ !_ / : assert. ... ...
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