Finish defining the algorithm
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Yiyun Liu
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4 changed files with 84 additions and 472 deletions
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(** * Autosubst Header for Scoped Syntax
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Our development utilises well-scoped de Bruijn syntax. This means that the de Bruijn indices are taken from finite types. As a consequence, any kind of substitution or environment used in conjunction with well-scoped syntax takes the form of a mapping from some finite type _I^n_. In particular, _renamings_ are mappings _I^n -> I^m_. Here we develop the theory of how these parts interact.
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Version: December 11, 2019.
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*)
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Require Import core.
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Require Import Setoid Morphisms Relation_Definitions.
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Set Implicit Arguments.
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Unset Strict Implicit.
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Definition ap {X Y} (f : X -> Y) {x y : X} (p : x = y) : f x = f y :=
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match p with eq_refl => eq_refl end.
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Definition apc {X Y} {f g : X -> Y} {x y : X} (p : f = g) (q : x = y) : f x = g y :=
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match q with eq_refl => match p with eq_refl => eq_refl end end.
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(** ** Primitives of the Sigma Calculus
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We implement the finite type with _n_ elements, _I^n_, as the _n_-fold iteration of the Option Type. _I^0_ is implemented as the empty type.
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*)
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Fixpoint fin (n : nat) : Type :=
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match n with
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| 0 => False
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| S m => option (fin m)
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end.
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(** Renamings and Injective Renamings
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_Renamings_ are mappings between finite types.
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*)
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Definition ren (m n : nat) : Type := fin m -> fin n.
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Definition id {X} := @Datatypes.id X.
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Definition idren {k: nat} : ren k k := @Datatypes.id (fin k).
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(** We give a special name, to the newest element in a non-empty finite type, as it usually corresponds to a freshly bound variable. *)
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Definition var_zero {n : nat} : fin (S n) := None.
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Definition null {T} (i : fin 0) : T := match i with end.
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Definition shift {n : nat} : ren n (S n) :=
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Some.
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(** Extension of Finite Mappings
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Assume we are given a mapping _f_ from _I^n_ to some type _X_, then we can _extend_ this mapping with a new value from _x : X_ to a mapping from _I^n+1_ to _X_. We denote this operation by _x . f_ and define it as follows:
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*)
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Definition scons {X : Type} {n : nat} (x : X) (f : fin n -> X) (m : fin (S n)) : X :=
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match m with
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| None => x
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| Some i => f i
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end.
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#[ export ]
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Hint Opaque scons : rewrite.
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(** ** Type Class Instances for Notation *)
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(** *** Type classes for renamings. *)
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Class Ren1 (X1 : Type) (Y Z : Type) :=
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ren1 : X1 -> Y -> Z.
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Class Ren2 (X1 X2 : Type) (Y Z : Type) :=
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ren2 : X1 -> X2 -> Y -> Z.
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Class Ren3 (X1 X2 X3 : Type) (Y Z : Type) :=
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ren3 : X1 -> X2 -> X3 -> Y -> Z.
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Class Ren4 (X1 X2 X3 X4 : Type) (Y Z : Type) :=
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ren4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
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Class Ren5 (X1 X2 X3 X4 X5 : Type) (Y Z : Type) :=
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ren5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
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Module RenNotations.
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Notation "s ⟨ xi1 ⟩" := (ren1 xi1 s) (at level 7, left associativity, format "s ⟨ xi1 ⟩") : subst_scope.
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Notation "s ⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ⟩") : subst_scope.
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Notation "s ⟨ xi1 ; xi2 ; xi3 ⟩" := (ren3 xi1 xi2 xi3 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ⟩") : subst_scope.
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Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩" := (ren4 xi1 xi2 xi3 xi4 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ⟩") : subst_scope.
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Notation "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩" := (ren5 xi1 xi2 xi3 xi4 xi5 s) (at level 7, left associativity, format "s ⟨ xi1 ; xi2 ; xi3 ; xi4 ; xi5 ⟩") : subst_scope.
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Notation "⟨ xi ⟩" := (ren1 xi) (at level 1, left associativity, format "⟨ xi ⟩") : fscope.
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Notation "⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2) (at level 1, left associativity, format "⟨ xi1 ; xi2 ⟩") : fscope.
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End RenNotations.
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(** *** Type Classes for Substiution *)
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Class Subst1 (X1 : Type) (Y Z: Type) :=
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subst1 : X1 -> Y -> Z.
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Class Subst2 (X1 X2 : Type) (Y Z: Type) :=
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subst2 : X1 -> X2 -> Y -> Z.
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Class Subst3 (X1 X2 X3 : Type) (Y Z: Type) :=
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subst3 : X1 -> X2 -> X3 -> Y -> Z.
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Class Subst4 (X1 X2 X3 X4: Type) (Y Z: Type) :=
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subst4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
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Class Subst5 (X1 X2 X3 X4 X5 : Type) (Y Z: Type) :=
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subst5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
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Module SubstNotations.
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Notation "s [ sigma ]" := (subst1 sigma s) (at level 7, left associativity, format "s '/' [ sigma ]") : subst_scope.
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Notation "s [ sigma ; tau ]" := (subst2 sigma tau s) (at level 7, left associativity, format "s '/' [ sigma ; '/' tau ]") : subst_scope.
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End SubstNotations.
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(** ** Type Class for Variables *)
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Class Var X Y :=
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ids : X -> Y.
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(** ** Proofs for substitution primitives *)
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(** Forward Function Composition
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Substitutions represented as functions are ubiquitious in this development and we often have to compose them, without talking about their pointwise behaviour.
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That is, we are interested in the forward compostion of functions, _f o g_, for which we introduce a convenient notation, "f >> g". The direction of the arrow serves as a reminder of the _forward_ nature of this composition, that is first apply _f_, then _g_. *)
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Arguments funcomp {X Y Z} (g)%fscope (f)%fscope.
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Module CombineNotations.
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Notation "f >> g" := (funcomp g f) (at level 50) : fscope.
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Notation "s .: sigma" := (scons s sigma) (at level 55, sigma at next level, right associativity) : subst_scope.
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#[ global ]
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Open Scope fscope.
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#[ global ]
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Open Scope subst_scope.
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End CombineNotations.
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Import CombineNotations.
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(** Generic lifting operation for renamings *)
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Definition up_ren {m n} (xi : ren m n) : ren (S m) (S n) :=
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var_zero .: xi >> shift.
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(** Generic proof that lifting of renamings composes. *)
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Lemma up_ren_ren k l m (xi: ren k l) (zeta : ren l m) (rho: ren k m) (E: forall x, (xi >> zeta) x = rho x) :
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forall x, (up_ren xi >> up_ren zeta) x = up_ren rho x.
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Proof.
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intros [x|].
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- cbn. unfold funcomp. now rewrite <- E.
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- reflexivity.
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Qed.
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Arguments up_ren_ren {k l m} xi zeta rho E.
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Lemma fin_eta {X} (f g : fin 0 -> X) :
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pointwise_relation _ eq f g.
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Proof. intros []. Qed.
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(** Eta laws *)
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Lemma scons_eta' {T} {n : nat} (f : fin (S n) -> T) :
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pointwise_relation _ eq (f var_zero .: (funcomp f shift)) f.
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Proof. intros x. destruct x; reflexivity. Qed.
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Lemma scons_eta_id' {n : nat} :
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pointwise_relation (fin (S n)) eq (var_zero .: shift) id.
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Proof. intros x. destruct x; reflexivity. Qed.
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Lemma scons_comp' {T:Type} {U} {m} (s: T) (sigma: fin m -> T) (tau: T -> U) :
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pointwise_relation _ eq (funcomp tau (s .: sigma)) ((tau s) .: (funcomp tau sigma)).
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Proof. intros x. destruct x; reflexivity. Qed.
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(* Lemma scons_tail'_pointwise {X} {n} (s : X) (f : fin n -> X) : *)
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(* pointwise_relation _ eq (funcomp (scons s f) shift) f. *)
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(* Proof. *)
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(* reflexivity. *)
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(* Qed. *)
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(* Lemma scons_tail' {X} {n} (s : X) (f : fin n -> X) x : *)
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(* (scons s f (shift x)) = f x. *)
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(* Proof. *)
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(* reflexivity. *)
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(* Qed. *)
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(* Morphism for Setoid Rewriting. The only morphism that can be defined statically. *)
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#[export] Instance scons_morphism {X: Type} {n:nat} :
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Proper (eq ==> pointwise_relation _ eq ==> pointwise_relation _ eq) (@scons X n).
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Proof.
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intros t t' -> sigma tau H.
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intros [x|].
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cbn. apply H.
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reflexivity.
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Qed.
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#[export] Instance scons_morphism2 {X: Type} {n: nat} :
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Proper (eq ==> pointwise_relation _ eq ==> eq ==> eq) (@scons X n).
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Proof.
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intros ? t -> sigma tau H ? x ->.
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destruct x as [x|].
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cbn. apply H.
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reflexivity.
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Qed.
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(** ** Variadic Substitution Primitives *)
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Fixpoint shift_p (p : nat) {n} : ren n (p + n) :=
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fun n => match p with
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| 0 => n
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| S p => Some (shift_p p n)
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end.
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Fixpoint scons_p {X: Type} {m : nat} : forall {n} (f : fin m -> X) (g : fin n -> X), fin (m + n) -> X.
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Proof.
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destruct m.
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- intros n f g. exact g.
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- intros n f g. cbn. apply scons.
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+ exact (f var_zero).
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+ apply scons_p.
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* intros z. exact (f (Some z)).
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* exact g.
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Defined.
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#[export] Hint Opaque scons_p : rewrite.
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#[export] Instance scons_p_morphism {X: Type} {m n:nat} :
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Proper (pointwise_relation _ eq ==> pointwise_relation _ eq ==> pointwise_relation _ eq) (@scons_p X m n).
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Proof.
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intros sigma sigma' Hsigma tau tau' Htau.
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intros x.
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induction m.
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- cbn. apply Htau.
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- cbn. change (fin (S m + n)) with (fin (S (m + n))) in x.
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destruct x as [x|].
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+ cbn. apply IHm.
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intros ?. apply Hsigma.
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+ cbn. apply Hsigma.
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Qed.
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#[export] Instance scons_p_morphism2 {X: Type} {m n:nat} :
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Proper (pointwise_relation _ eq ==> pointwise_relation _ eq ==> eq ==> eq) (@scons_p X m n).
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Proof.
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intros sigma sigma' Hsigma tau tau' Htau ? x ->.
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induction m.
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- cbn. apply Htau.
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- cbn. change (fin (S m + n)) with (fin (S (m + n))) in x.
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destruct x as [x|].
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+ cbn. apply IHm.
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intros ?. apply Hsigma.
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+ cbn. apply Hsigma.
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Qed.
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Definition zero_p {m : nat} {n} : fin m -> fin (m + n).
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Proof.
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induction m.
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- intros [].
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- intros [x|].
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+ exact (shift_p 1 (IHm x)).
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+ exact var_zero.
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Defined.
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Lemma scons_p_head' {X} {m n} (f : fin m -> X) (g : fin n -> X) z:
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(scons_p f g) (zero_p z) = f z.
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Proof.
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induction m.
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- inversion z.
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- destruct z.
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+ simpl. simpl. now rewrite IHm.
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+ reflexivity.
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Qed.
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Lemma scons_p_head_pointwise {X} {m n} (f : fin m -> X) (g : fin n -> X) :
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pointwise_relation _ eq (funcomp (scons_p f g) zero_p) f.
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Proof.
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intros z.
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unfold funcomp.
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induction m.
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- inversion z.
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- destruct z.
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+ simpl. now rewrite IHm.
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+ reflexivity.
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Qed.
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Lemma scons_p_tail' X m n (f : fin m -> X) (g : fin n -> X) z :
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scons_p f g (shift_p m z) = g z.
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Proof. induction m; cbn; eauto. Qed.
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Lemma scons_p_tail_pointwise X m n (f : fin m -> X) (g : fin n -> X) :
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pointwise_relation _ eq (funcomp (scons_p f g) (shift_p m)) g.
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Proof. intros z. induction m; cbn; eauto. Qed.
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Lemma destruct_fin {m n} (x : fin (m + n)):
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(exists x', x = zero_p x') \/ exists x', x = shift_p m x'.
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Proof.
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induction m; simpl in *.
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- right. eauto.
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- destruct x as [x|].
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+ destruct (IHm x) as [[x' ->] |[x' ->]].
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* left. now exists (Some x').
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* right. eauto.
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+ left. exists None. eauto.
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Qed.
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Lemma scons_p_comp' X Y m n (f : fin m -> X) (g : fin n -> X) (h : X -> Y) :
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pointwise_relation _ eq (funcomp h (scons_p f g)) (scons_p (f >> h) (g >> h)).
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Proof.
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intros x.
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destruct (destruct_fin x) as [[x' ->]|[x' ->]].
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(* TODO better way to solve this? *)
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- revert x'.
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apply pointwise_forall.
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change (fun x => (scons_p f g >> h) (zero_p x)) with (zero_p >> scons_p f g >> h).
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now setoid_rewrite scons_p_head_pointwise.
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- revert x'.
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apply pointwise_forall.
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change (fun x => (scons_p f g >> h) (shift_p m x)) with (shift_p m >> scons_p f g >> h).
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change (fun x => scons_p (f >> h) (g >> h) (shift_p m x)) with (shift_p m >> scons_p (f >> h) (g >> h)).
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now rewrite !scons_p_tail_pointwise.
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Qed.
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Lemma scons_p_congr {X} {m n} (f f' : fin m -> X) (g g': fin n -> X) z:
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(forall x, f x = f' x) -> (forall x, g x = g' x) -> scons_p f g z = scons_p f' g' z.
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Proof. intros H1 H2. induction m; eauto. cbn. destruct z; eauto. Qed.
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(** Generic n-ary lifting operation. *)
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Definition upRen_p p { m : nat } { n : nat } (xi : (fin) (m) -> (fin) (n)) : fin (p + m) -> fin (p + n) :=
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scons_p (zero_p ) (xi >> shift_p _).
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Arguments upRen_p p {m n} xi.
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(** Generic proof for composition of n-ary lifting. *)
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Lemma up_ren_ren_p p k l m (xi: ren k l) (zeta : ren l m) (rho: ren k m) (E: forall x, (xi >> zeta) x = rho x) :
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forall x, (upRen_p p xi >> upRen_p p zeta) x = upRen_p p rho x.
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Proof.
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intros x. destruct (destruct_fin x) as [[? ->]|[? ->]].
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- unfold upRen_p. unfold funcomp. now repeat rewrite scons_p_head'.
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- unfold upRen_p. unfold funcomp. repeat rewrite scons_p_tail'.
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now rewrite <- E.
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Qed.
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Arguments zero_p m {n}.
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Arguments scons_p {X} m {n} f g.
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Lemma scons_p_eta {X} {m n} {f : fin m -> X}
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{g : fin n -> X} (h: fin (m + n) -> X) {z: fin (m + n)}:
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(forall x, g x = h (shift_p m x)) -> (forall x, f x = h (zero_p m x)) -> scons_p m f g z = h z.
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Proof.
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intros H1 H2. destruct (destruct_fin z) as [[? ->] |[? ->]].
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- rewrite scons_p_head'. eauto.
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- rewrite scons_p_tail'. eauto.
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Qed.
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Arguments scons_p_eta {X} {m n} {f g} h {z}.
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Arguments scons_p_congr {X} {m n} {f f'} {g g'} {z}.
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(** ** Notations for Scoped Syntax *)
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Module ScopedNotations.
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Include RenNotations.
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Include SubstNotations.
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Include CombineNotations.
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(* Notation "s , sigma" := (scons s sigma) (at level 60, format "s , sigma", right associativity) : subst_scope. *)
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Notation "s '..'" := (scons s ids) (at level 1, format "s ..") : subst_scope.
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Notation "↑" := (shift) : subst_scope.
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#[global]
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Open Scope fscope.
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#[global]
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Open Scope subst_scope.
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End ScopedNotations.
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(** ** Tactics for Scoped Syntax *)
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Tactic Notation "auto_case" tactic(t) := (match goal with
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| [|- forall (i : fin 0), _] => intros []; t
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| [|- forall (i : fin (S (S (S (S _))))), _] => intros [[[[|]|]|]|]; t
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| [|- forall (i : fin (S (S (S _)))), _] => intros [[[|]|]|]; t
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| [|- forall (i : fin (S (S _))), _] => intros [[?|]|]; t
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| [|- forall (i : fin (S _)), _] => intros [?|]; t
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end).
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#[export] Hint Rewrite @scons_p_head' @scons_p_tail' : FunctorInstances.
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(** Generic fsimpl tactic: simplifies the above primitives in a goal. *)
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Ltac fsimpl :=
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repeat match goal with
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| [|- context[id >> ?f]] => change (id >> f) with f (* AsimplCompIdL *)
|
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| [|- context[?f >> id]] => change (f >> id) with f (* AsimplCompIdR *)
|
||||
| [|- context [id ?s]] => change (id s) with s
|
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| [|- context[(?f >> ?g) >> ?h]] => change ((f >> g) >> h) with (f >> (g >> h)) (* AsimplComp *)
|
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(* | [|- zero_p >> scons_p ?f ?g] => rewrite scons_p_head *)
|
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| [|- context[(?s .: ?sigma) var_zero]] => change ((s.:sigma) var_zero) with s
|
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| [|- context[(?s .: ?sigma) (shift ?m)]] => change ((s.:sigma) (shift m)) with (sigma m)
|
||||
(* first [progress setoid_rewrite scons_tail' | progress setoid_rewrite scons_tail'_pointwise ] *)
|
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| [|- context[idren >> ?f]] => change (idren >> f) with f
|
||||
| [|- context[?f >> idren]] => change (f >> idren) with f
|
||||
| [|- context[?f >> (?x .: ?g)]] => change (f >> (x .: g)) with g (* f should evaluate to shift *)
|
||||
| [|- context[?x2 .: (funcomp ?f shift)]] => change (scons x2 (funcomp f shift)) with (scons (f var_zero) (funcomp f shift)); setoid_rewrite (@scons_eta' _ _ f); eta_reduce
|
||||
| [|- context[?f var_zero .: ?g]] => change (scons (f var_zero) g) with (scons (f var_zero) (funcomp f shift)); setoid_rewrite scons_eta'; eta_reduce
|
||||
| [|- _ = ?h (?f ?s)] => change (h (f s)) with ((f >> h) s)
|
||||
| [|- ?h (?f ?s) = _] => change (h (f s)) with ((f >> h) s)
|
||||
| [|- context[funcomp _ (scons _ _)]] => setoid_rewrite scons_comp'; eta_reduce
|
||||
| [|- context[funcomp _ (scons_p _ _ _)]] => setoid_rewrite scons_p_comp'; eta_reduce
|
||||
| [|- context[scons (@var_zero _) shift]] => setoid_rewrite scons_eta_id'; eta_reduce
|
||||
(* | _ => progress autorewrite with FunctorInstances *)
|
||||
| [|- context[funcomp (scons_p _ _ _) (zero_p _)]] =>
|
||||
first [progress setoid_rewrite scons_p_head_pointwise | progress setoid_rewrite scons_p_head' ]
|
||||
| [|- context[scons_p _ _ _ (zero_p _ _)]] => setoid_rewrite scons_p_head'
|
||||
| [|- context[funcomp (scons_p _ _ _) (shift_p _)]] =>
|
||||
first [progress setoid_rewrite scons_p_tail_pointwise | progress setoid_rewrite scons_p_tail' ]
|
||||
| [|- context[scons_p _ _ _ (shift_p _ _)]] => setoid_rewrite scons_p_tail'
|
||||
| _ => first [progress minimize | progress cbn [shift scons scons_p] ]
|
||||
end.
|
||||
|
|
@ -711,11 +711,11 @@ Qed.
|
|||
Class Up_PTm X Y :=
|
||||
up_PTm : X -> Y.
|
||||
|
||||
#[global]Instance Subst_PTm : (Subst1 _ _ _) := @subst_PTm.
|
||||
#[global] Instance Subst_PTm : (Subst1 _ _ _) := @subst_PTm.
|
||||
|
||||
#[global]Instance Up_PTm_PTm : (Up_PTm _ _) := @up_PTm_PTm.
|
||||
#[global] Instance Up_PTm_PTm : (Up_PTm _ _) := @up_PTm_PTm.
|
||||
|
||||
#[global]Instance Ren_PTm : (Ren1 _ _ _) := @ren_PTm.
|
||||
#[global] Instance Ren_PTm : (Ren1 _ _ _) := @ren_PTm.
|
||||
|
||||
#[global]
|
||||
Instance VarInstance_PTm : (Var _ _) := @VarPTm.
|
||||
|
|
@ -833,9 +833,9 @@ Module Extra.
|
|||
|
||||
Import Core.
|
||||
|
||||
#[global]Hint Opaque subst_PTm: rewrite.
|
||||
#[global] Hint Opaque subst_PTm: rewrite.
|
||||
|
||||
#[global]Hint Opaque ren_PTm: rewrite.
|
||||
#[global] Hint Opaque ren_PTm: rewrite.
|
||||
|
||||
End Extra.
|
||||
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue