330 lines
8.9 KiB
Coq
330 lines
8.9 KiB
Coq
From Equations Require Import Equations.
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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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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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algo_dom (PAbs a) (PAbs b)
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| A_AbsNeu a u :
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ishne u ->
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algo_dom_r a (PApp (ren_PTm shift u) (VarPTm var_zero)) ->
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(* --------------------- *)
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algo_dom (PAbs a) u
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| A_NeuAbs a u :
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ishne u ->
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algo_dom_r (PApp (ren_PTm shift u) (VarPTm var_zero)) a ->
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(* --------------------- *)
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algo_dom u (PAbs a)
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| A_PairPair a0 a1 b0 b1 :
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algo_dom_r a0 a1 ->
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algo_dom_r b0 b1 ->
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(* ---------------------------- *)
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algo_dom (PPair a0 b0) (PPair a1 b1)
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| A_PairNeu a0 a1 u :
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ishne u ->
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algo_dom_r a0 (PProj PL u) ->
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algo_dom_r a1 (PProj PR u) ->
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(* ----------------------- *)
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algo_dom (PPair a0 a1) u
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| A_NeuPair a0 a1 u :
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ishne u ->
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algo_dom_r (PProj PL u) a0 ->
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algo_dom_r (PProj PR u) a1 ->
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(* ----------------------- *)
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algo_dom u (PPair a0 a1)
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| A_UnivCong i j :
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(* -------------------------- *)
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algo_dom (PUniv i) (PUniv j)
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| A_BindCong p0 p1 A0 A1 B0 B1 :
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algo_dom_r A0 A1 ->
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algo_dom_r B0 B1 ->
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(* ---------------------------- *)
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algo_dom (PBind p0 A0 B0) (PBind p1 A1 B1)
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| A_VarCong i j :
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(* -------------------------- *)
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algo_dom (VarPTm i) (VarPTm j)
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| A_ProjCong p0 p1 u0 u1 :
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ishne u0 ->
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ishne u1 ->
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algo_dom u0 u1 ->
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(* --------------------- *)
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algo_dom (PProj p0 u0) (PProj p1 u1)
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| A_AppCong u0 u1 a0 a1 :
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ishne u0 ->
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ishne u1 ->
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algo_dom u0 u1 ->
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algo_dom_r a0 a1 ->
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(* ------------------------- *)
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algo_dom (PApp u0 a0) (PApp u1 a1)
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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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| A_HRedL a a' b :
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HRed.R a a' ->
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algo_dom_r a' b ->
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(* ----------------------- *)
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algo_dom_r a b
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| A_HRedR a b b' :
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ishne a \/ ishf a ->
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HRed.R b b' ->
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algo_dom_r a b' ->
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(* ----------------------- *)
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algo_dom_r a b.
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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 (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 (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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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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(* | PAbs a, PAbs b => PTm_eqb a b *)
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(* | PApp a0 b0, PApp a1 b1 => PTm_eqb a0 a1 && PTm_eqb b0 b1 *)
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(* | PBind p0 A0 B0, PBind p1 A1 B1 => BTag_beq p0 p1 && PTm_eqb A0 A1 && PTm_eqb B0 B1 *)
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(* | PPair a0 b0, PPair a1 b1 => PTm_eqb a0 a1 && PTm_eqb b0 b1 *)
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(* | *)
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(* Lemma hred {n} (a : PTm n) : (relations.nf HRed.R a) + {x | HRed.R a x}. *)
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(* Proof. *)
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(* induction a. *)
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(* - hauto lq:on inv:HRed.R unfold:relations.nf. *)
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(* - hauto lq:on inv:HRed.R unfold:relations.nf. *)
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(* - clear IHa2. *)
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(* destruct IHa1. *)
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(* destruct a1. *)
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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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| PApp (PAbs a) b => Some (subst_PTm (scons b VarPTm) a)
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| PApp a b =>
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match hred a with
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| Some a => Some (PApp a b)
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| None => None
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end
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| PPair a b => None
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| PProj p (PPair a b) => if p is PL then Some a else Some b
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| PProj p a =>
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match hred a with
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| Some a => Some (PProj p a)
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| None => None
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end
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| PUniv i => None
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| PBind p A B => None
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| PBot => None
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| PNat => None
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| PZero => None
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| PSuc a => None
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| PInd P PZero b c => Some b
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| PInd P (PSuc a) b c =>
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Some (subst_PTm (scons (PInd P a b c) (scons a VarPTm)) c)
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| PInd P a b c =>
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match hred a with
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| Some a => Some (PInd P a b c)
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| None => None
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end
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end.
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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 (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 (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 (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 /hred_complete in h.
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congruence.
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Defined.
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Ltac check_equal_triv :=
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intros;subst;
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lazymatch goal with
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(* | [h : algo_dom (VarPTm _) (PAbs _) |- _] => idtac *)
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| [h : algo_dom _ _ |- _] => try inversion h; by subst
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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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(* | PAbs a, PAbs b => check_equal_r a b ltac:(check_equal_triv) *)
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(* | _, _ => false *)
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(* end *)
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(* with check_equal_r {n} (a b : PTm n) (h : algo_dom_r a b) {struct h} : bool := *)
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(* match hred_fancy _ a with *)
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(* | inr a' => check_equal_r (proj1_sig a') b _ *)
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(* | _ => false *)
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(* end. *)
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(* Next Obligation. *)
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(* simpl. *)
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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 := 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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check_equal (PPair a0 b0) (PPair a1 b1) h :=
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check_equal_r a0 a1 ltac:(check_equal_triv) && check_equal_r b0 b1 ltac:(check_equal_triv);
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check_equal (PPair a0 b0) u h :=
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check_equal_r a0 (PProj PL u) ltac:(check_equal_triv) && check_equal_r b0 (PProj PR u) ltac:(check_equal_triv);
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check_equal u (PPair a1 b1) h :=
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check_equal_r (PProj PL u) a1 ltac:(check_equal_triv) && check_equal_r (PProj PR u) b1 ltac:(check_equal_triv);
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check_equal (PBind p0 A0 B0) (PBind p1 A1 B1) h :=
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BTag_beq p0 p1 && check_equal_r A0 A1 ltac:(check_equal_triv) && check_equal_r B0 B1 ltac:(check_equal_triv);
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check_equal PNat PNat _ := true;
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check_equal PZero PZero _ := true;
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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 (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 (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 _) (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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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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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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subst.
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assumption.
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Defined.
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Next Obligation.
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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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exfalso. hauto lq:on use:hf_no_hred, hne_no_hred.
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Defined.
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Search (Bool.reflect (is_true _) _).
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Check idP.
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Definition metric {n} k (a b : PTm n) :=
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exists i j va vb, nsteps LoRed.R i a va /\ nsteps LoRed.R j b vb /\
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nf va /\ nf vb /\ size_PTm va + size_PTm vb + i + j <= k.
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Search (nat -> nat -> bool).
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