Seems to work but takes a million years to type check
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Yiyun Liu
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5 changed files with 580 additions and 572 deletions
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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. *)
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(* simpl. *)
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Equations check_equal {n} (a b : PTm n) (h : algo_dom a b) :
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Equations check_equal (a b : PTm) (h : algo_dom a b) :
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bool by struct h :=
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check_equal (VarPTm i) (VarPTm j) h := fin_beq i j;
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check_equal (VarPTm i) (VarPTm j) h := nat_beq i j;
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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);
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check_equal (PUniv i) (PUniv j) _ := Nat.eqb i j;
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check_equal a b h := false;
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with check_equal_r {n} (a b : PTm n) (h : algo_dom_r a b) :
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with check_equal_r (a b : PTm) (h : algo_dom_r a b) :
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bool by struct h :=
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check_equal_r a b h with hred_fancy _ a =>
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check_equal_r a b h with hred_fancy a =>
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{ check_equal_r a b h (inr a') := check_equal_r (proj1_sig a') b _;
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check_equal_r a b h (inl _) with hred_fancy _ b =>
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check_equal_r a b h (inl _) with hred_fancy b =>
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{ check_equal_r a b h (inl _) (inl _) := check_equal a b _;
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check_equal_r a b h (inl _) (inr b') := check_equal_r a (proj1_sig b') _}} .
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Next Obligation.
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move => /= ih ihr n a nfa b nfb.
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inversion 1; subst=>//=.
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inversion h; subst=>//=.
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exfalso. sfirstorder.
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exfalso. sfirstorder.
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Defined.
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Next Obligation.
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simpl.
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move => /= ih ihr n a nfa b [b' hb'].
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inversion 1; subst =>//=.
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inversion h; subst =>//=.
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exfalso. hauto lq:on use:algo_dom_hf_hne, hf_no_hred, hne_no_hred.
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exfalso. sfirstorder.
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have ? : b' = b'0 by eauto using hred_deter.
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@ -302,8 +312,7 @@ Next Obligation.
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Defined.
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Next Obligation.
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simpl => ih ihr n a [a' ha'] b.
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inversion 1; subst => //=.
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inversion h; subst => //=.
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exfalso. hauto lq:on use:algo_dom_hf_hne, hf_no_hred, hne_no_hred.
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suff ? : a'0 = a' by subst; assumption.
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by eauto using hred_deter.
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