Seems to work but takes a million years to type check

Makefile

@@ -16,7 +16,7 @@ uninstall: $(COQMAKEFILE)
 	$(MAKE) -f $(COQMAKEFILE) uninstall
 
 theories/Autosubst2/syntax.v theories/Autosubst2/core.v theories/Autosubst2/fintype.v : syntax.sig
-	autosubst -f -v ge813 -s coq -o theories/Autosubst2/syntax.v syntax.sig
+	autosubst -f -v ge813 -s ucoq -o theories/Autosubst2/syntax.v syntax.sig
 
 .PHONY: clean FORCE
 

theories/Autosubst2/unscoped.v

@@ -0,0 +1,213 @@
+(** * Autosubst Header for Unnamed Syntax
+
+Version: December 11, 2019.
+ *)
+
+(* Adrian:
+ I changed this library a bit to work better with my generated code.
+ 1. I use nat directly instead of defining fin to be nat and using Some/None as S/O
+ 2. I removed the "s, sigma" notation for scons because it interacts with dependent function types "forall x, A"*)
+Require Import core.
+Require Import Setoid Morphisms Relation_Definitions.
+
+Definition ap {X Y} (f : X -> Y) {x y : X} (p : x = y) : f x = f y :=
+  match p with eq_refl => eq_refl end.
+
+Definition apc {X Y} {f g : X -> Y} {x y : X} (p : f = g) (q : x = y) : f x = g y :=
+  match q with eq_refl => match p with eq_refl => eq_refl end end.
+
+(** ** Primitives of the Sigma Calculus. *)
+
+Definition shift  := S.
+
+Definition var_zero := 0.
+
+Definition id {X} := @Datatypes.id X.
+
+Definition scons {X: Type} (x : X) (xi : nat -> X) :=
+  fun n => match n with
+        | 0 => x
+        | S n => xi n
+        end.
+
+#[ export ]
+Hint Opaque scons : rewrite.
+
+(** ** Type Class Instances for Notation
+Required to make notation work. *)
+
+(** *** Type classes for renamings. *)
+
+Class Ren1 (X1  : Type) (Y Z : Type) :=
+  ren1 : X1 -> Y -> Z.
+
+Class Ren2 (X1 X2 : Type) (Y Z : Type) :=
+  ren2 : X1 -> X2 -> Y -> Z.
+
+Class Ren3 (X1 X2 X3 : Type) (Y Z : Type) :=
+  ren3 : X1 -> X2 -> X3 -> Y -> Z.
+
+Class Ren4 (X1 X2 X3 X4 : Type) (Y Z : Type) :=
+  ren4 : X1 -> X2 -> X3 -> X4 -> Y -> Z.
+
+Class Ren5 (X1 X2 X3 X4 X5 : Type) (Y Z : Type) :=
+  ren5 : X1 -> X2 -> X3 -> X4 -> X5 -> Y -> Z.
+
+Module RenNotations.
+  Notation "s ⟨ xi1 ⟩" := (ren1 xi1 s) (at level 7, left associativity, format "s  ⟨ xi1 ⟩") : subst_scope.
+
+  Notation "s ⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2 s) (at level 7, left associativity, format "s  ⟨ xi1 ; xi2 ⟩") : subst_scope.
+
+  Notation "s ⟨ xi1 ; xi2 ; xi3 ⟩" := (ren3 xi1 xi2 xi3 s) (at level 7, left associativity, format "s  ⟨ xi1 ; xi2 ; xi3 ⟩") : subst_scope.
+
+  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.
+
+  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.
+
+  Notation "⟨ xi ⟩" := (ren1 xi) (at level 1, left associativity, format "⟨ xi ⟩") : fscope.
+
+  Notation "⟨ xi1 ; xi2 ⟩" := (ren2 xi1 xi2) (at level 1, left associativity, format "⟨ xi1 ; xi2 ⟩") : fscope.
+End RenNotations.
+
+(** *** Type Classes for Substiution *)
+
+Class Subst1 (X1 : Type) (Y Z: Type) :=
+  subst1 : X1 -> Y -> Z.
+
+Class Subst2 (X1 X2 : Type) (Y Z: Type) :=
+  subst2 : X1 -> X2 -> Y  -> Z.
+
+Class Subst3 (X1 X2 X3 : Type) (Y Z: Type) :=
+  subst3 : X1 -> X2 -> X3 ->  Y  -> Z.
+
+Class Subst4 (X1 X2 X3 X4: Type) (Y Z: Type) :=
+  subst4 : X1 -> X2 -> X3 -> X4 -> Y  -> Z.
+
+Class Subst5 (X1 X2 X3 X4 X5 : Type) (Y Z: Type) :=
+  subst5 : X1 -> X2 -> X3 -> X4 -> X5  -> Y  -> Z.
+
+Module SubstNotations.
+  Notation "s [ sigma ]" := (subst1 sigma s) (at level 7, left associativity, format "s '/' [ sigma ]") : subst_scope.
+
+  Notation "s [ sigma ; tau ]" := (subst2 sigma tau s) (at level 7, left associativity, format "s '/' [ sigma ; '/'  tau ]") : subst_scope.
+End SubstNotations.
+
+(** *** Type Class for Variables *)
+
+Class Var X Y :=
+  ids : X -> Y.
+
+#[export] Instance idsRen : Var nat nat := id.
+
+(** ** Proofs for the substitution primitives. *)
+
+Arguments funcomp {X Y Z} (g)%fscope (f)%fscope.
+
+Module CombineNotations.
+  Notation "f >> g" := (funcomp g f) (at level 50) : fscope.
+
+  Notation "s .: sigma" := (scons s sigma) (at level 55, sigma at next level, right associativity) : subst_scope.
+
+  #[ global ]
+  Open Scope fscope.
+  #[ global ]
+  Open Scope subst_scope.
+End CombineNotations.
+
+Import CombineNotations.
+
+
+(** A generic lifting of a renaming. *)
+Definition up_ren (xi : nat -> nat) :=
+  0 .: (xi >> S).
+
+(** A generic proof that lifting of renamings composes. *)
+Lemma up_ren_ren (xi: nat -> nat) (zeta : nat -> nat) (rho: nat -> nat) (E: forall x, (xi >> zeta) x = rho x) :
+  forall x, (up_ren xi >> up_ren zeta) x = up_ren rho x.
+Proof.
+  intros [|x].
+  - reflexivity.
+  - unfold up_ren. cbn. unfold funcomp. f_equal. apply E.
+Qed.
+
+(** Eta laws. *)
+Lemma scons_eta' {T} (f : nat -> T) :
+  pointwise_relation _ eq (f var_zero .: (funcomp f shift)) f.
+Proof. intros x. destruct x; reflexivity. Qed.
+
+Lemma scons_eta_id' :
+  pointwise_relation _ eq (var_zero .: shift) id.
+Proof. intros x. destruct x; reflexivity. Qed.
+
+Lemma scons_comp' (T: Type) {U} (s: T) (sigma: nat -> T) (tau: T -> U) :
+  pointwise_relation _ eq (funcomp tau (s .: sigma)) ((tau s) .: (funcomp tau sigma)).
+Proof. intros x. destruct x; reflexivity. Qed.
+
+(* Morphism for Setoid Rewriting. The only morphism that can be defined statically. *)
+#[export] Instance scons_morphism {X: Type} :
+  Proper (eq ==> pointwise_relation _ eq ==> pointwise_relation _ eq) (@scons X).
+Proof.
+  intros ? t -> sigma tau H.
+  intros [|x].
+  cbn. reflexivity.
+  apply H.
+Qed.
+
+#[export] Instance scons_morphism2 {X: Type} :
+  Proper (eq ==> pointwise_relation _ eq ==> eq ==> eq) (@scons X).
+Proof.
+  intros ? t -> sigma tau H ? x ->.
+  destruct x as [|x].
+  cbn. reflexivity.
+  apply H.
+Qed.
+
+(** ** Generic lifting of an allfv predicate *)
+Definition up_allfv (p: nat -> Prop) : nat -> Prop := scons True p.
+
+(** ** Notations for unscoped syntax *)
+Module UnscopedNotations.
+  Include RenNotations.
+  Include SubstNotations.
+  Include CombineNotations.
+  
+  (* Notation "s , sigma" := (scons s sigma) (at level 60, format "s ,  sigma", right associativity) : subst_scope. *)
+
+  Notation "s '..'" := (scons s ids) (at level 1, format "s ..") : subst_scope.
+
+  Notation "↑" := (shift) : subst_scope.
+
+  #[global]
+  Open Scope fscope.
+  #[global]
+  Open Scope subst_scope.
+End UnscopedNotations.
+
+(** ** Tactics for unscoped syntax *)
+
+(** Automatically does a case analysis on a natural number, useful for proofs with context renamings/context morphisms. *)
+Tactic Notation "auto_case" tactic(t) :=  (match goal with
+                                           | [|- forall (i : nat), _] => intros []; t
+                                           end).
+
+
+(** Generic fsimpl tactic: simplifies the above primitives in a goal. *)
+Ltac fsimpl :=
+  repeat match goal with
+         | [|- context[id >> ?f]] => change (id >> f) with f (* AsimplCompIdL *)
+         | [|- context[?f >> id]] => change (f >> id) with f (* AsimplCompIdR *)
+         | [|- context [id ?s]] => change (id s) with s
+         | [|- context[(?f >> ?g) >> ?h]] => change ((f >> g) >> h) with (f >> (g >> h))
+         | [|- context[(?v .: ?g) var_zero]] => change ((v .: g) var_zero) with v
+         | [|- context[(?v .: ?g) 0]] => change ((v .: g) 0) with v
+         | [|- context[(?v .: ?g) (S ?n)]] => change ((v .: g) (S n)) with (g n)
+         | [|- context[?f >> (?x .: ?g)]] => change (f >> (x .: g)) with g (* f should evaluate to shift *)
+         | [|- context[var_zero]] =>  change var_zero with 0
+         | [|- context[?x2 .: (funcomp ?f shift)]] => change (scons x2 (funcomp f shift)) with (scons (f var_zero) (funcomp f shift)); setoid_rewrite (@scons_eta' _ _ f)
+         | [|- context[?f var_zero .: ?g]] => change (scons (f var_zero) g) with (scons (f var_zero) (funcomp f shift)); rewrite scons_eta'
+         | [|- _ =  ?h (?f ?s)] => change (h (f s)) with ((f >> h) s)
+         | [|-  ?h (?f ?s) = _] => change (h (f s)) with ((f >> h) s)
+         (* DONE had to put an underscore as the last argument to scons. This might be an argument against unfolding funcomp *)
+         | [|- context[funcomp _ (scons _ _)]] => setoid_rewrite scons_comp'; eta_reduce
+         | [|- context[scons var_zero shift]] => setoid_rewrite scons_eta_id'; eta_reduce
+                        end.

theories/common.v

@@ -4,6 +4,7 @@ Import Ltac2.Notations.
 Import Ltac2.Control.
 From Hammer Require Import Tactics.
 
+
 Definition renaming_ok {n m} (Γ : fin n -> PTm n) (Δ : fin m -> PTm m) (ξ : fin m -> fin n) :=
   forall (i : fin m), ren_PTm ξ (Δ i) = Γ (ξ i).
 

theories/executable.v

@@ -1,13 +1,64 @@
 From Equations Require Import Equations.
-Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax
+Require Import Autosubst2.core Autosubst2.unscoped Autosubst2.syntax.
-  common typing preservation admissible fp_red structural soundness.
+Derive NoConfusion for nat PTag BTag PTm.
-Require Import algorithmic.
+
-From stdpp Require Import relations (rtc(..), nsteps(..)).
 Require Import ssreflect ssrbool.
 From Hammer Require Import Tactics.
 
 
-Inductive algo_dom {n} : PTm n -> PTm n -> Prop :=
+Definition ishf (a : PTm) :=
+  match a with
+  | PPair _ _ => true
+  | PAbs _ => true
+  | PUniv _ => true
+  | PBind _ _ _ => true
+  | PNat => true
+  | PSuc _ => true
+  | PZero => true
+  | _ => false
+  end.
+
+Fixpoint ishne (a : PTm) :=
+  match a with
+  | VarPTm _ => true
+  | PApp a _ => ishne a
+  | PProj _ a => ishne a
+  | PBot => true
+  | PInd _ n _ _ => ishne n
+  | _ => false
+  end.
+
+Module HRed.
+  Inductive R : PTm -> PTm ->  Prop :=
+  (****************** Beta ***********************)
+  | AppAbs a b :
+    R (PApp (PAbs a) b)  (subst_PTm (scons b VarPTm) a)
+
+  | ProjPair p a b :
+    R (PProj p (PPair a b)) (if p is PL then a else b)
+
+  | IndZero P b c :
+    R (PInd P PZero b c) b
+
+  | IndSuc P a b c :
+    R (PInd P (PSuc a) b c) (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c)
+
+  (*************** Congruence ********************)
+  | AppCong a0 a1 b :
+    R a0 a1 ->
+    R (PApp a0 b) (PApp a1 b)
+  | ProjCong p a0 a1 :
+    R a0 a1 ->
+    R (PProj p a0) (PProj p a1)
+  | IndCong P a0 a1 b c :
+    R a0 a1 ->
+    R (PInd P a0 b c) (PInd P a1 b c).
+
+  Definition nf a := forall b, ~ R a b.
+End HRed.
+
+
+Inductive algo_dom : PTm -> PTm -> Prop :=
 | A_AbsAbs a b :
   algo_dom_r a b ->
   (* --------------------- *)
@@ -74,7 +125,7 @@ Inductive algo_dom {n} : PTm n -> PTm n -> Prop :=
   (* ------------------------- *)
   algo_dom (PApp u0 a0) (PApp u1 a1)
 
-with algo_dom_r {n} : PTm n -> PTm n -> Prop :=
+with algo_dom_r : PTm  -> PTm -> Prop :=
 | A_NfNf a b :
   algo_dom a b ->
   algo_dom_r a b
@@ -92,67 +143,26 @@ with algo_dom_r {n} : PTm n -> PTm n -> Prop :=
   (* ----------------------- *)
   algo_dom_r a b.
 
-Derive Signature for algo_dom algo_dom_r.
+Lemma algo_dom_hf_hne (a b : PTm) :
-Derive NoConfusion for PTm.
-Next Obligation.
-Admitted.
-Next Obligation.
-Admitted.
-
-Derive Dependent Inversion adom_inv with (forall n (a b : PTm n), algo_dom a b) Sort Prop.
-
-Lemma algo_dom_hf_hne n (a b : PTm n) :
   algo_dom a b ->
   (ishf a \/ ishne a) /\ (ishf b \/ ishne b).
 Proof.
   induction 1 =>//=; hauto lq:on.
 Qed.
 
-Lemma hf_no_hred n (a b : PTm n) :
+Lemma hf_no_hred (a b : PTm) :
   ishf a ->
   HRed.R a b ->
   False.
 Proof. hauto l:on inv:HRed.R. Qed.
 
-Lemma hne_no_hred n (a b : PTm n) :
+Lemma hne_no_hred (a b : PTm) :
   ishne a ->
   HRed.R a b ->
   False.
 Proof. elim : a b => //=; hauto l:on inv:HRed.R. Qed.
 
-Definition fin_beq {n} (i j : fin n) : bool.
+Derive Signature for algo_dom.
-Proof.
-  induction n.
-  - by exfalso.
-  - refine (match i , j with
-            | None, None => true
-            | Some i, Some j => IHn i j
-            | _, _ => false
-            end).
-Defined.
-
-Lemma fin_eq_dec {n} (i j : fin n) :
-  Bool.reflect (i = j) (fin_beq i j).
-Proof.
-  revert i j. induction n.
-  - destruct i.
-  - destruct i; destruct j.
-    + specialize (IHn f f0).
-      inversion IHn; subst.
-      simpl. rewrite -H.
-      apply ReflectT.
-      reflexivity.
-      simpl. rewrite -H.
-      apply ReflectF.
-      injection. tauto.
-    + by apply ReflectF.
-    + by apply ReflectF.
-    + by apply ReflectT.
-Defined.
-
-Scheme Equality for PTag.
-Scheme Equality for BTag.
-
 (* Fixpoint PTm_eqb {n} (a b : PTm n) := *)
 (*   match a, b with *)
 (*   | VarPTm i, VarPTm j => fin_eq i j *)
@@ -171,7 +181,7 @@ Scheme Equality for BTag.
 (*     destruct IHa1. *)
 (*     destruct a1. *)
 
-Fixpoint hred {n} (a : PTm n) : option (PTm n) :=
+Fixpoint hred (a : PTm) : option (PTm) :=
     match a with
     | VarPTm i => None
     | PAbs a => None
@@ -204,31 +214,31 @@ Fixpoint hred {n} (a : PTm n) : option (PTm n) :=
         end
     end.
 
-Lemma hred_complete n (a b : PTm n) :
+Lemma hred_complete (a b : PTm) :
   HRed.R a b -> hred a = Some b.
 Proof.
   induction 1; hauto lq:on rew:off inv:HRed.R b:on.
 Qed.
 
-Lemma hred_sound n (a b : PTm n):
+Lemma hred_sound (a b : PTm):
   hred a = Some b -> HRed.R a b.
 Proof.
   elim : a b; hauto q:on dep:on ctrs:HRed.R.
 Qed.
 
-Lemma hred_deter n (a b0 b1 : PTm n) :
+Lemma hred_deter (a b0 b1 : PTm) :
   HRed.R a b0 -> HRed.R a b1 -> b0 = b1.
 Proof.
   move /hred_complete => + /hred_complete. congruence.
 Qed.
 
-Definition hred_fancy n (a : PTm n) :
+Definition hred_fancy  (a : PTm) :
-  relations.nf HRed.R a + {x | HRed.R a x}.
+  HRed.nf a + {x | HRed.R a x}.
 Proof.
   destruct (hred a) as [a'|] eqn:eq .
   - right. exists a'. hauto q:on use:hred_sound.
   - left.
-    move => [a' h].
+    move => a' h.
     move /hred_complete in h.
     congruence.
 Defined.
@@ -241,6 +251,8 @@ Ltac check_equal_triv :=
   | _ => idtac
   end.
 
+Scheme Equality for nat. Scheme Equality for PTag.
+Scheme Equality for BTag. Scheme Equality for PTm.
 (* Program Fixpoint check_equal {n} (a b : PTm n) (h : algo_dom a b) {struct h} : bool := *)
 (*   match a, b with *)
 (*   | VarPTm i, VarPTm j => fin_beq i j *)
@@ -255,9 +267,9 @@ Ltac check_equal_triv :=
 (* 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);
   check_equal (PAbs a) b h := check_equal_r a (PApp (ren_PTm shift b) (VarPTm var_zero)) ltac:(check_equal_triv);
   check_equal a (PAbs b) h := check_equal_r (PApp (ren_PTm shift a) (VarPTm var_zero)) b ltac:(check_equal_triv);
@@ -274,26 +286,24 @@ Equations check_equal {n} (a b : PTm n) (h : algo_dom a b) :
   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 h; subst=>//=.
-  inversion 1; subst=>//=.
   exfalso. sfirstorder.
   exfalso. sfirstorder.
 Defined.
 
 Next Obligation.
   simpl.
-  move => /= ih ihr n a nfa b [b' hb'].
+  inversion h; subst =>//=.
-  inversion 1; 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 h; subst => //=.
-  inversion 1; 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.