Add new definition of algo_dom
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4 changed files with 599 additions and 648 deletions
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@ -11,26 +11,6 @@ Set Default Proof Mode "Classic".
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Require Import ssreflect ssrbool.
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From Hammer Require Import Tactics.
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Definition tm_nonconf (a b : PTm) : bool :=
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match a, b with
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| PAbs _, _ => (~~ ishf b) || isabs b
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| _, PAbs _ => ~~ ishf a
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| VarPTm _, VarPTm _ => true
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| PPair _ _, _ => (~~ ishf b) || ispair b
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| _, PPair _ _ => ~~ ishf a
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| PZero, PZero => true
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| PSuc _, PSuc _ => true
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| PApp _ _, PApp _ _ => (~~ ishf a) && (~~ ishf b)
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| PProj _ _, PProj _ _ => (~~ ishf a) && (~~ ishf b)
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| PInd _ _ _ _, PInd _ _ _ _ => (~~ ishf a) && (~~ ishf b)
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| PNat, PNat => true
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| PUniv _, PUniv _ => true
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| PBind _ _ _, PBind _ _ _ => true
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| _,_=> false
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end.
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Definition tm_conf (a b : PTm) := ~~ tm_nonconf a b.
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Inductive eq_view : PTm -> PTm -> Type :=
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| V_AbsAbs a b :
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eq_view (PAbs a) (PAbs b)
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@ -102,121 +82,6 @@ Equations tm_to_eq_view (a b : PTm) : eq_view a b :=
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tm_to_eq_view (PBind p0 A0 B0) (PBind p1 A1 B1) := V_BindBind p0 A0 B0 p1 A1 B1;
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tm_to_eq_view a b := V_Others a b _.
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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_ZeroZero :
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algo_dom PZero PZero
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| A_SucSuc a0 a1 :
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algo_dom_r a0 a1 ->
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algo_dom (PSuc a0) (PSuc 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_NatCong :
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algo_dom PNat PNat
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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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| A_IndCong P0 P1 u0 u1 b0 b1 c0 c1 :
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ishne u0 ->
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ishne u1 ->
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algo_dom_r P0 P1 ->
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algo_dom u0 u1 ->
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algo_dom_r b0 b1 ->
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algo_dom_r c0 c1 ->
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algo_dom (PInd P0 u0 b0 c0) (PInd P1 u1 b1 c1)
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| A_Conf a b :
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HRed.nf a ->
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HRed.nf b ->
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tm_conf a b ->
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algo_dom a b
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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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HRed.nf 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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Scheme algo_ind := Induction for algo_dom Sort Prop
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with algor_ind := Induction for algo_dom_r Sort Prop.
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Combined Scheme algo_dom_mutual from algo_ind, algor_ind.
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(* Inductive salgo_dom : PTm -> PTm -> Prop := *)
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(* | S_UnivCong i j : *)
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@ -273,83 +138,6 @@ Combined Scheme algo_dom_mutual from algo_ind, algor_ind.
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(* with algor_ind := Induction for salgo_dom_r Sort Prop. *)
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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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Lemma algo_dom_no_hred (a b : PTm) :
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algo_dom a b ->
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HRed.nf a /\ HRed.nf b.
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Proof.
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induction 1 =>//=; try hauto inv:HRed.R use:hne_no_hred, hf_no_hred lq:on unfold:HRed.nf.
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Qed.
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Derive Signature for algo_dom algo_dom_r.
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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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| 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 fancy_hred (a : PTm) : HRed.nf a + {b | HRed.R a b}.
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destruct (hred a) eqn:eq.
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right. exists p. by apply hred_sound in eq.
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left. move => b /hred_complete. congruence.
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Defined.
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Ltac2 destruct_algo () :=
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lazy_match! goal with
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| [h : algo_dom ?a ?b |- _ ] =>
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