### Proofs about binders

parent aa051883
 ... ... @@ -21,29 +21,50 @@ Instance binder_inhabited : Inhabited binder := populate BAnon. Instance binder_countable : Countable binder. Proof. refine (inj_countable' (λ mx, match mx with BAnon => None | BNamed x => Some x end) (λ mx, match mx with None => BAnon | Some x => BNamed x end) _); by intros []. (λ b, match b with BAnon => None | BNamed s => Some s end) (λ b, match b with None => BAnon | Some s => BNamed s end) _); by intros []. Qed. (** The functions [cons_binder mx X] and [app_binder mxs X] are typically used to collect the free variables of an expression. Here [X] is the current list of free variables, and [mx], respectively [mxs], are the binders that are being (** The functions [cons_binder b ss] and [app_binder bs ss] are typically used to collect the free variables of an expression. Here [ss] is the current list of free variables, and [b], respectively [bs], are the binders that are being added. *) Definition cons_binder (mx : binder) (X : list string) : list string := match mx with BAnon => X | BNamed x => x :: X end. Definition cons_binder (b : binder) (ss : list string) : list string := match b with BAnon => ss | BNamed s => s :: ss end. Infix ":b:" := cons_binder (at level 60, right associativity). Fixpoint app_binder (mxs : list binder) (X : list string) : list string := match mxs with [] => X | b :: mxs => b :b: app_binder mxs X end. Fixpoint app_binder (bs : list binder) (ss : list string) : list string := match bs with [] => ss | b :: bs => b :b: app_binder bs ss end. Infix "+b+" := app_binder (at level 60, right associativity). Instance set_unfold_cons_binder x mx X P : SetUnfoldElemOf x X P → SetUnfoldElemOf x (mx :b: X) (BNamed x = mx ∨ P). Instance set_unfold_cons_binder s b ss P : SetUnfoldElemOf s ss P → SetUnfoldElemOf s (b :b: ss) (BNamed s = b ∨ P). Proof. constructor. rewrite <-(set_unfold (x ∈ X) P). destruct mx; simpl; rewrite ?elem_of_cons; naive_solver. constructor. rewrite <-(set_unfold (s ∈ ss) P). destruct b; simpl; rewrite ?elem_of_cons; naive_solver. Qed. Instance set_unfold_app_binder x mxl X P : SetUnfoldElemOf x X P → SetUnfoldElemOf x (mxl +b+ X) (BNamed x ∈ mxl ∨ P). Instance set_unfold_app_binder s bs ss P Q : SetUnfoldElemOf (BNamed s) bs P → SetUnfoldElemOf s ss Q → SetUnfoldElemOf s (bs +b+ ss) (P ∨ Q). Proof. constructor. rewrite <-(set_unfold (x ∈ X) P). induction mxl; set_solver. intros HinP HinQ. constructor. rewrite <-(set_unfold (s ∈ ss) Q), <-(set_unfold (BNamed s ∈ bs) P). clear HinP HinQ. induction bs; set_solver. Qed. Lemma app_binder_named ss1 ss2 : (BNamed <\$> ss1) +b+ ss2 = ss1 ++ ss2. Proof. induction ss1; by f_equal/=. Qed. Lemma app_binder_snoc bs s ss : bs +b+ (s :: ss) = (bs ++ [BNamed s]) +b+ ss. Proof. induction bs; by f_equal/=. Qed. Instance cons_binder_Permutation b : Proper ((≡ₚ) ==> (≡ₚ)) (cons_binder b). Proof. intros ss1 ss2 Hss. destruct b; csimpl; by rewrite Hss. Qed. Instance app_binder_Permutation : Proper ((≡ₚ) ==> (≡ₚ) ==> (≡ₚ)) app_binder. Proof. assert (∀ bs, Proper ((≡ₚ) ==> (≡ₚ)) (app_binder bs)). { induction bs as [|[]]; intros ss1 ss2; simpl; by intros ->. } induction 1 as [|[]|[] []|]; intros ss1 ss2 Hss; simpl; first [by eauto using perm_trans|by rewrite 1?perm_swap, Hss]. Qed.
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