diff --git a/_CoqProject b/_CoqProject index dc2c81e2524d146fba8bab6be6186c9254b25837..123fc871923a271f2fbc98061111c01969952ef6 100644 --- a/_CoqProject +++ b/_CoqProject @@ -44,3 +44,4 @@ theories/infinite.v theories/nat_cancel.v theories/namespaces.v theories/telescopes.v +theories/binders.v diff --git a/theories/binders.v b/theories/binders.v new file mode 100644 index 0000000000000000000000000000000000000000..72c0f7e8929fd984cf8951e152416af74adc9757 --- /dev/null +++ b/theories/binders.v @@ -0,0 +1,49 @@ +(* Copyright (c) 2012-2019, Coq-std++ developers. *) +(* This file is distributed under the terms of the BSD license. *) +(** This file implements a type [binder] with elements [BAnon] for the +anonymous binder, and [BNamed] for named binders. This type is isomorphic to +[option string], but we use a special type so that we can define [BNamed] as +a coercion. + +This library is used in various Iris developments, like heap-lang, LambdaRust, +Iron, Fairis. *) +From stdpp Require Export strings. +From stdpp Require Import sets countable finite. + +Inductive binder := BAnon | BNamed :> string → binder. +Bind Scope binder_scope with binder. +Delimit Scope binder_scope with binder. +Notation "<>" := BAnon : binder_scope. + +Instance binder_dec_eq : EqDecision binder. +Proof. solve_decision. Defined. +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 []. +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 +added. *) +Definition cons_binder (mx : binder) (X : list string) : list string := + match mx with BAnon => X | BNamed x => x :: X 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. +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). +Proof. + constructor. rewrite <-(set_unfold (x ∈ X) P). + destruct mx; 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). +Proof. + constructor. rewrite <-(set_unfold (x ∈ X) P). induction mxl; set_solver. +Qed.