Seems to work but takes a million years to type check
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Yiyun Liu
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213
theories/Autosubst2/unscoped.v
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213
theories/Autosubst2/unscoped.v
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(** * Autosubst Header for Unnamed Syntax
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Version: December 11, 2019.
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*)
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(* Adrian:
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I changed this library a bit to work better with my generated code.
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1. I use nat directly instead of defining fin to be nat and using Some/None as S/O
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2. I removed the "s, sigma" notation for scons because it interacts with dependent function types "forall x, A"*)
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Require Import core.
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Require Import Setoid Morphisms Relation_Definitions.
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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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Definition shift := S.
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Definition var_zero := 0.
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Definition id {X} := @Datatypes.id X.
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Definition scons {X: Type} (x : X) (xi : nat -> X) :=
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fun n => match n with
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| 0 => x
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| S n => xi n
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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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Required to make notation work. *)
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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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#[export] Instance idsRen : Var nat nat := id.
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(** ** Proofs for the substitution primitives. *)
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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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(** A generic lifting of a renaming. *)
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Definition up_ren (xi : nat -> nat) :=
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0 .: (xi >> S).
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(** A generic proof that lifting of renamings composes. *)
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Lemma up_ren_ren (xi: nat -> nat) (zeta : nat -> nat) (rho: nat -> nat) (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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- reflexivity.
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- unfold up_ren. cbn. unfold funcomp. f_equal. apply E.
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Qed.
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(** Eta laws. *)
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Lemma scons_eta' {T} (f : nat -> 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' :
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pointwise_relation _ 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} (s: T) (sigma: nat -> 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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(* Morphism for Setoid Rewriting. The only morphism that can be defined statically. *)
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#[export] Instance scons_morphism {X: Type} :
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Proper (eq ==> pointwise_relation _ eq ==> pointwise_relation _ eq) (@scons X).
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Proof.
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intros ? t -> sigma tau H.
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intros [|x].
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cbn. reflexivity.
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apply H.
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Qed.
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#[export] Instance scons_morphism2 {X: Type} :
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Proper (eq ==> pointwise_relation _ eq ==> eq ==> eq) (@scons X).
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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. reflexivity.
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apply H.
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Qed.
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(** ** Generic lifting of an allfv predicate *)
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Definition up_allfv (p: nat -> Prop) : nat -> Prop := scons True p.
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(** ** Notations for unscoped syntax *)
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Module UnscopedNotations.
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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 UnscopedNotations.
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(** ** Tactics for unscoped syntax *)
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(** Automatically does a case analysis on a natural number, useful for proofs with context renamings/context morphisms. *)
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Tactic Notation "auto_case" tactic(t) := (match goal with
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| [|- forall (i : nat), _] => intros []; t
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end).
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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 *)
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| [|- context [id ?s]] => change (id s) with s
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| [|- context[(?f >> ?g) >> ?h]] => change ((f >> g) >> h) with (f >> (g >> h))
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| [|- context[(?v .: ?g) var_zero]] => change ((v .: g) var_zero) with v
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| [|- context[(?v .: ?g) 0]] => change ((v .: g) 0) with v
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| [|- context[(?v .: ?g) (S ?n)]] => change ((v .: g) (S n)) with (g n)
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| [|- context[?f >> (?x .: ?g)]] => change (f >> (x .: g)) with g (* f should evaluate to shift *)
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| [|- context[var_zero]] => change var_zero with 0
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| [|- context[?x2 .: (funcomp ?f shift)]] => change (scons x2 (funcomp f shift)) with (scons (f var_zero) (funcomp f shift)); setoid_rewrite (@scons_eta' _ _ f)
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| [|- context[?f var_zero .: ?g]] => change (scons (f var_zero) g) with (scons (f var_zero) (funcomp f shift)); rewrite scons_eta'
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| [|- _ = ?h (?f ?s)] => change (h (f s)) with ((f >> h) s)
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| [|- ?h (?f ?s) = _] => change (h (f s)) with ((f >> h) s)
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(* DONE had to put an underscore as the last argument to scons. This might be an argument against unfolding funcomp *)
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| [|- context[funcomp _ (scons _ _)]] => setoid_rewrite scons_comp'; eta_reduce
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| [|- context[scons var_zero shift]] => setoid_rewrite scons_eta_id'; eta_reduce
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end.
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@ -4,6 +4,7 @@ Import Ltac2.Notations.
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Import Ltac2.Control.
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From Hammer Require Import Tactics.
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Definition renaming_ok {n m} (Γ : fin n -> PTm n) (Δ : fin m -> PTm m) (ξ : fin m -> fin n) :=
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forall (i : fin m), ren_PTm ξ (Δ i) = Γ (ξ i).
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@ -1,13 +1,64 @@
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From Equations Require Import Equations.
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Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax
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common typing preservation admissible fp_red structural soundness.
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Require Import algorithmic.
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From stdpp Require Import relations (rtc(..), nsteps(..)).
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Require Import Autosubst2.core Autosubst2.unscoped Autosubst2.syntax.
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Derive NoConfusion for nat PTag BTag PTm.
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Require Import ssreflect ssrbool.
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From Hammer Require Import Tactics.
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Inductive algo_dom {n} : PTm n -> PTm n -> Prop :=
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Definition ishf (a : PTm) :=
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match a with
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| PPair _ _ => true
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| PAbs _ => true
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| PUniv _ => true
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| PBind _ _ _ => true
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| PNat => true
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| PSuc _ => true
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| PZero => true
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| _ => false
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end.
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Fixpoint ishne (a : PTm) :=
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match a with
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| VarPTm _ => true
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| PApp a _ => ishne a
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| PProj _ a => ishne a
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| PBot => true
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| PInd _ n _ _ => ishne n
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| _ => false
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end.
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Module HRed.
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Inductive R : PTm -> PTm -> Prop :=
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(****************** Beta ***********************)
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| AppAbs a b :
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R (PApp (PAbs a) b) (subst_PTm (scons b VarPTm) a)
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| ProjPair p a b :
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R (PProj p (PPair a b)) (if p is PL then a else b)
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| IndZero P b c :
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R (PInd P PZero b c) b
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| IndSuc P a b c :
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R (PInd P (PSuc a) b c) (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c)
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(*************** Congruence ********************)
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| AppCong a0 a1 b :
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R a0 a1 ->
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R (PApp a0 b) (PApp a1 b)
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| ProjCong p a0 a1 :
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R a0 a1 ->
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R (PProj p a0) (PProj p a1)
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| IndCong P a0 a1 b c :
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R a0 a1 ->
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R (PInd P a0 b c) (PInd P a1 b c).
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Definition nf a := forall b, ~ R a b.
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End HRed.
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Inductive algo_dom : PTm -> PTm -> Prop :=
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| A_AbsAbs a b :
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algo_dom_r a b ->
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(* --------------------- *)
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@ -74,7 +125,7 @@ Inductive algo_dom {n} : PTm n -> PTm n -> Prop :=
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(* ------------------------- *)
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algo_dom (PApp u0 a0) (PApp u1 a1)
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with algo_dom_r {n} : PTm n -> PTm n -> Prop :=
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with algo_dom_r : PTm -> PTm -> Prop :=
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| A_NfNf a b :
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algo_dom a b ->
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algo_dom_r a b
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@ -92,67 +143,26 @@ with algo_dom_r {n} : PTm n -> PTm n -> Prop :=
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(* ----------------------- *)
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algo_dom_r a b.
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Derive Signature for algo_dom algo_dom_r.
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Derive NoConfusion for PTm.
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Next Obligation.
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Admitted.
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Next Obligation.
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Admitted.
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Derive Dependent Inversion adom_inv with (forall n (a b : PTm n), algo_dom a b) Sort Prop.
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Lemma algo_dom_hf_hne n (a b : PTm n) :
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Lemma algo_dom_hf_hne (a b : PTm) :
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algo_dom a b ->
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(ishf a \/ ishne a) /\ (ishf b \/ ishne b).
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Proof.
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induction 1 =>//=; hauto lq:on.
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Qed.
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Lemma hf_no_hred n (a b : PTm n) :
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Lemma hf_no_hred (a b : PTm) :
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ishf a ->
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HRed.R a b ->
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False.
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Proof. hauto l:on inv:HRed.R. Qed.
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Lemma hne_no_hred n (a b : PTm n) :
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Lemma hne_no_hred (a b : PTm) :
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ishne a ->
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HRed.R a b ->
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False.
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Proof. elim : a b => //=; hauto l:on inv:HRed.R. Qed.
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Definition fin_beq {n} (i j : fin n) : bool.
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Proof.
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induction n.
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- by exfalso.
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- refine (match i , j with
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| None, None => true
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| Some i, Some j => IHn i j
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| _, _ => false
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end).
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Defined.
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Lemma fin_eq_dec {n} (i j : fin n) :
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Bool.reflect (i = j) (fin_beq i j).
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Proof.
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revert i j. induction n.
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- destruct i.
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- destruct i; destruct j.
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+ specialize (IHn f f0).
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inversion IHn; subst.
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simpl. rewrite -H.
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apply ReflectT.
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reflexivity.
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simpl. rewrite -H.
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apply ReflectF.
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injection. tauto.
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+ by apply ReflectF.
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+ by apply ReflectF.
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+ by apply ReflectT.
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Defined.
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Scheme Equality for PTag.
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Scheme Equality for BTag.
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Derive Signature for algo_dom.
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(* Fixpoint PTm_eqb {n} (a b : PTm n) := *)
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(* match a, b with *)
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(* | VarPTm i, VarPTm j => fin_eq i j *)
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@ -171,7 +181,7 @@ Scheme Equality for BTag.
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(* destruct IHa1. *)
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(* destruct a1. *)
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Fixpoint hred {n} (a : PTm n) : option (PTm n) :=
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Fixpoint hred (a : PTm) : option (PTm) :=
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match a with
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| VarPTm i => None
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| PAbs a => None
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@ -204,31 +214,31 @@ Fixpoint hred {n} (a : PTm n) : option (PTm n) :=
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end
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end.
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Lemma hred_complete n (a b : PTm n) :
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Lemma hred_complete (a b : PTm) :
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HRed.R a b -> hred a = Some b.
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Proof.
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induction 1; hauto lq:on rew:off inv:HRed.R b:on.
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Qed.
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Lemma hred_sound n (a b : PTm n):
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Lemma hred_sound (a b : PTm):
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hred a = Some b -> HRed.R a b.
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Proof.
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elim : a b; hauto q:on dep:on ctrs:HRed.R.
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Qed.
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Lemma hred_deter n (a b0 b1 : PTm n) :
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Lemma hred_deter (a b0 b1 : PTm) :
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HRed.R a b0 -> HRed.R a b1 -> b0 = b1.
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Proof.
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move /hred_complete => + /hred_complete. congruence.
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Qed.
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Definition hred_fancy n (a : PTm n) :
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relations.nf HRed.R a + {x | HRed.R a x}.
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Definition hred_fancy (a : PTm) :
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HRed.nf a + {x | HRed.R a x}.
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Proof.
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destruct (hred a) as [a'|] eqn:eq .
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- right. exists a'. hauto q:on use:hred_sound.
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- left.
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move => [a' h].
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move => a' h.
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move /hred_complete in h.
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congruence.
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Defined.
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|
@ -241,6 +251,8 @@ Ltac check_equal_triv :=
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| _ => idtac
|
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end.
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Scheme Equality for nat. Scheme Equality for PTag.
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Scheme Equality for BTag. Scheme Equality for PTm.
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(* Program Fixpoint check_equal {n} (a b : PTm n) (h : algo_dom a b) {struct h} : bool := *)
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(* match a, b with *)
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(* | VarPTm i, VarPTm j => fin_beq i j *)
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|
@ -255,9 +267,9 @@ Ltac check_equal_triv :=
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|||
(* Next Obligation. *)
|
||||
(* simpl. *)
|
||||
|
||||
Equations check_equal {n} (a b : PTm n) (h : algo_dom a b) :
|
||||
Equations check_equal (a b : PTm) (h : algo_dom a b) :
|
||||
bool by struct h :=
|
||||
check_equal (VarPTm i) (VarPTm j) h := fin_beq i j;
|
||||
check_equal (VarPTm i) (VarPTm j) h := nat_beq i j;
|
||||
check_equal (PAbs a) (PAbs b) h := check_equal_r a b ltac:(check_equal_triv);
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check_equal (PAbs a) b h := check_equal_r a (PApp (ren_PTm shift b) (VarPTm var_zero)) ltac:(check_equal_triv);
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check_equal a (PAbs b) h := check_equal_r (PApp (ren_PTm shift a) (VarPTm var_zero)) b ltac:(check_equal_triv);
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||||
|
|
@ -274,26 +286,24 @@ Equations check_equal {n} (a b : PTm n) (h : algo_dom a b) :
|
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check_equal (PSuc a) (PSuc b) h := check_equal_r a b ltac:(check_equal_triv);
|
||||
check_equal (PUniv i) (PUniv j) _ := Nat.eqb i j;
|
||||
check_equal a b h := false;
|
||||
with check_equal_r {n} (a b : PTm n) (h : algo_dom_r a b) :
|
||||
with check_equal_r (a b : PTm) (h : algo_dom_r a b) :
|
||||
bool by struct h :=
|
||||
check_equal_r a b h with hred_fancy _ a =>
|
||||
check_equal_r a b h with hred_fancy a =>
|
||||
{ check_equal_r a b h (inr a') := check_equal_r (proj1_sig a') b _;
|
||||
check_equal_r a b h (inl _) with hred_fancy _ b =>
|
||||
check_equal_r a b h (inl _) with hred_fancy b =>
|
||||
{ check_equal_r a b h (inl _) (inl _) := check_equal a b _;
|
||||
check_equal_r a b h (inl _) (inr b') := check_equal_r a (proj1_sig b') _}} .
|
||||
|
||||
|
||||
Next Obligation.
|
||||
move => /= ih ihr n a nfa b nfb.
|
||||
inversion 1; subst=>//=.
|
||||
inversion h; subst=>//=.
|
||||
exfalso. sfirstorder.
|
||||
exfalso. sfirstorder.
|
||||
Defined.
|
||||
|
||||
Next Obligation.
|
||||
simpl.
|
||||
move => /= ih ihr n a nfa b [b' hb'].
|
||||
inversion 1; subst =>//=.
|
||||
inversion h; subst =>//=.
|
||||
exfalso. hauto lq:on use:algo_dom_hf_hne, hf_no_hred, hne_no_hred.
|
||||
exfalso. sfirstorder.
|
||||
have ? : b' = b'0 by eauto using hred_deter.
|
||||
|
|
@ -302,8 +312,7 @@ Next Obligation.
|
|||
Defined.
|
||||
|
||||
Next Obligation.
|
||||
simpl => ih ihr n a [a' ha'] b.
|
||||
inversion 1; subst => //=.
|
||||
inversion h; subst => //=.
|
||||
exfalso. hauto lq:on use:algo_dom_hf_hne, hf_no_hred, hne_no_hred.
|
||||
suff ? : a'0 = a' by subst; assumption.
|
||||
by eauto using hred_deter.
|
||||
|
|
|
|||
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