Require Import Autosubst2.core Autosubst2.fintype Autosubst2.syntax.
Require Import fp_red.
From Hammer Require Import Tactics.
From Equations Require Import Equations.
Require Import ssreflect ssrbool.
Require Import Logic.PropExtensionality (propositional_extensionality).
From stdpp Require Import relations (rtc(..), rtc_subrel).
Import Psatz.

Definition ProdSpace {n} (PA : Tm n -> Prop)
  (PF : Tm n -> (Tm n -> Prop) -> Prop) b : Prop :=
  forall a PB, PA a -> PF a PB -> PB (App b a).

Definition SumSpace {n} (PA : Tm n -> Prop)
  (PF : Tm n -> (Tm n -> Prop) -> Prop) t : Prop :=
  wne t \/ exists a b, rtc RPar'.R t (Pair a b) /\ PA a /\ (forall PB, PF a PB -> PB b).

Definition BindSpace {n} p := if p is TPi then @ProdSpace n else SumSpace.

Reserved Notation "⟦ A ⟧ i ;; I ↘ S" (at level 70).
Inductive InterpExt {n} i (I : nat -> Tm n -> Prop) : Tm n -> (Tm n -> Prop) -> Prop :=
| InterpExt_Ne A :
  ne A ->
  ⟦ A ⟧ i ;; I ↘ wne
| InterpExt_Bind p A B PA PF :
  ⟦ A ⟧ i ;; I ↘ PA ->
  (forall a, PA a -> exists PB, PF a PB) ->
  (forall a PB, PF a PB -> ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘ PB) ->
  ⟦ TBind p A B ⟧ i ;; I ↘ BindSpace p PA PF

| InterpExt_Univ j :
  j < i ->
  ⟦ Univ j ⟧ i ;; I ↘ (I j)

| InterpExt_Step A A0 PA :
  RPar'.R A A0 ->
  ⟦ A0 ⟧ i ;; I ↘ PA ->
  ⟦ A ⟧ i ;; I ↘ PA
where "⟦ A ⟧ i ;; I ↘ S" := (InterpExt i I A S).

Lemma InterpExt_Univ' n i  I j (PF : Tm n -> Prop) :
  PF = I j ->
  j < i ->
  ⟦ Univ j ⟧ i ;; I ↘ PF.
Proof. hauto lq:on ctrs:InterpExt. Qed.

Infix "<?" := Compare_dec.lt_dec (at level 60).

Equations InterpUnivN n (i : nat) : Tm n -> (Tm n -> Prop) -> Prop by wf i lt :=
  InterpUnivN n i := @InterpExt n i
                     (fun j A =>
                        match j <? i  with
                        | left _ => exists PA, InterpUnivN n j A PA
                        | right _ => False
                        end).
Arguments InterpUnivN {n}.

Lemma InterpExt_lt_impl n i I I' A (PA : Tm n -> Prop) :
  (forall j, j < i -> I j = I' j) ->
  ⟦ A ⟧ i ;; I ↘ PA ->
  ⟦ A ⟧ i ;; I' ↘ PA.
Proof.
  move => hI h.
  elim : A PA /h.
  - hauto q:on ctrs:InterpExt.
  - hauto lq:on rew:off ctrs:InterpExt.
  - hauto q:on ctrs:InterpExt.
  - hauto lq:on ctrs:InterpExt.
Qed.

Lemma InterpExt_lt_eq n i I I' A (PA : Tm n -> Prop) :
  (forall j, j < i -> I j = I' j) ->
  ⟦ A ⟧ i ;; I ↘ PA =
  ⟦ A ⟧ i ;; I' ↘ PA.
Proof.
  move => hI. apply propositional_extensionality.
  have : forall j, j < i -> I' j = I j by sfirstorder.
  firstorder using InterpExt_lt_impl.
Qed.

Notation "⟦ A ⟧ i ↘ S" := (InterpUnivN i A S) (at level 70).

Lemma InterpUnivN_nolt n i :
  @InterpUnivN n i = @InterpExt n i (fun j (A : Tm n) => exists PA, ⟦ A ⟧ j ↘ PA).
Proof.
  simp InterpUnivN.
  extensionality A. extensionality PA.
  set I0 := (fun _ => _).
  set I1 := (fun _ => _).
  apply InterpExt_lt_eq.
  hauto q:on.
Qed.

#[export]Hint Rewrite @InterpUnivN_nolt : InterpUniv.

Lemma RPar_substone n (a b : Tm (S n)) (c : Tm n):
    RPar'.R a b -> RPar'.R (subst_Tm (scons c VarTm) a) (subst_Tm (scons c VarTm) b).
  Proof. hauto l:on inv:option use:RPar'.substing, RPar'.refl. Qed.

Lemma InterpExt_Bind_inv n p i I (A : Tm n) B P
  (h :  ⟦ TBind p A B ⟧ i ;; I ↘ P) :
  exists (PA : Tm n -> Prop) (PF : Tm n -> (Tm n -> Prop) -> Prop),
     ⟦ A ⟧ i ;; I ↘ PA /\
    (forall a, PA a -> exists PB, PF a PB) /\
    (forall a PB, PF a PB -> ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘ PB) /\
    P = BindSpace p PA PF.
Proof.
  move E : (TBind p A B) h => T h.
  move : A B E.
  elim : T P / h => //.
  - move => //= *. scongruence.
  - hauto l:on.
  - move => A A0 PA hA hA0 hPi A1 B ?. subst.
    elim /RPar'.inv : hA => //= _ p0 A2 A3 B0 B1 hA1 hB0 [*]. subst.
    hauto lq:on ctrs:InterpExt use:RPar_substone.
Qed.

Lemma InterpExt_Ne_inv n i A I P
  (h :  ⟦ A : Tm n  ⟧ i ;; I ↘ P) :
  ne A ->
  P = wne.
Proof.
  elim : A P / h => //=.
  qauto l:on ctrs:prov inv:prov use:nf_refl.
Qed.

Lemma InterpExt_Univ_inv n i I j P
  (h :  ⟦ Univ j : Tm n  ⟧ i ;; I ↘ P) :
  P = I j /\ j < i.
Proof.
  move : h.
  move E : (Univ j) => T h. move : j E.
  elim : T P /h => //.
  - move => //= *. scongruence.
  - hauto l:on.
  - hauto lq:on rew:off inv:RPar'.R.
Qed.

Lemma InterpExt_Bind_nopf n p i I (A : Tm n) B PA  :
  ⟦ A ⟧ i ;; I ↘ PA ->
  (forall a, PA a -> exists PB, ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘  PB) ->
  ⟦ TBind p A B ⟧ i ;; I ↘ (BindSpace p PA (fun a PB => ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘ PB)).
Proof.
  move => h0 h1. apply InterpExt_Bind =>//.
Qed.

Lemma InterpUnivN_Fun_nopf n p i (A : Tm n) B PA :
  ⟦ A ⟧ i ↘ PA ->
  (forall a, PA a -> exists PB, ⟦ subst_Tm (scons a VarTm) B ⟧ i ↘ PB) ->
  ⟦ TBind p A B ⟧ i ↘ (BindSpace p PA (fun a PB => ⟦ subst_Tm (scons a VarTm) B ⟧ i ↘ PB)).
Proof.
  hauto l:on use:InterpExt_Bind_nopf rew:db:InterpUniv.
Qed.

Lemma InterpExt_cumulative n i j I (A : Tm n) PA :
  i <= j ->
   ⟦ A ⟧ i ;; I ↘ PA ->
   ⟦ A ⟧ j ;; I ↘ PA.
Proof.
  move => h h0.
  elim : A PA /h0;
    hauto l:on ctrs:InterpExt solve+:(by lia).
Qed.

Lemma InterpUnivN_cumulative n i (A : Tm n) PA :
   ⟦ A ⟧ i ↘ PA -> forall j, i <= j ->
   ⟦ A ⟧ j ↘ PA.
Proof.
  hauto l:on rew:db:InterpUniv use:InterpExt_cumulative.
Qed.

Lemma InterpExt_preservation n i I (A : Tm n) B P (h : InterpExt i I A P) :
  RPar'.R A B ->
  ⟦ B ⟧ i ;; I ↘ P.
Proof.
  move : B.
  elim : A P / h; auto.
  - hauto lq:on use:nf_refl ctrs:InterpExt.
  - move => p A B PA PF hPA ihPA hPB hPB' ihPB T hT.
    elim /RPar'.inv :  hT => //.
    move => hPar p0 A0 A1 B0 B1 h0 h1 [? ?] ? ?; subst.
    apply InterpExt_Bind; auto => a PB hPB0.
    apply : ihPB; eauto.
    sfirstorder use:RPar'.cong, RPar'.refl.
  - hauto lq:on inv:RPar'.R ctrs:InterpExt.
  - move => A B P h0 h1 ih1 C hC.
    have [D [h2 h3]] := RPar'_diamond  _ _ _ _ h0 hC.
    hauto lq:on ctrs:InterpExt.
Qed.

Lemma InterpUnivN_preservation n i (A : Tm n) B P (h : ⟦ A ⟧ i ↘ P) :
  RPar'.R A B ->
  ⟦ B ⟧ i ↘ P.
Proof. hauto l:on rew:db:InterpUnivN use: InterpExt_preservation. Qed.

Lemma InterpExt_back_preservation_star n i I (A : Tm n) B P (h : ⟦ B ⟧ i ;; I ↘ P) :
  rtc RPar'.R A B ->
  ⟦ A ⟧ i ;; I ↘ P.
Proof. induction 1; hauto l:on ctrs:InterpExt. Qed.

Lemma InterpExt_preservation_star n i I (A : Tm n) B P (h : ⟦ A ⟧ i ;; I ↘ P) :
  rtc RPar'.R A B ->
  ⟦ B ⟧ i ;; I ↘ P.
Proof. induction 1; hauto l:on use:InterpExt_preservation. Qed.

Lemma InterpUnivN_preservation_star n i (A : Tm n) B P (h : ⟦ A ⟧ i ↘ P) :
  rtc RPar'.R A B ->
  ⟦ B ⟧ i ↘ P.
Proof. hauto l:on rew:db:InterpUnivN use:InterpExt_preservation_star. Qed.

Lemma InterpUnivN_back_preservation_star n i (A : Tm n) B P (h : ⟦ B ⟧ i ↘ P) :
  rtc RPar'.R A B ->
  ⟦ A ⟧ i ↘ P.
Proof. hauto l:on rew:db:InterpUnivN use:InterpExt_back_preservation_star. Qed.

Function hfb {n} (A : Tm n) :=
  match A with
  | TBind _ _ _ => true
  | Univ _ => true
  | _ => ne A
  end.

Inductive hfb_case {n} : Tm n -> Prop :=
| hfb_bind p A B :
  hfb_case (TBind p A B)
| hfb_univ i :
  hfb_case (Univ i)
| hfb_ne A :
  ne A ->
  hfb_case A.

Derive Dependent Inversion hfb_inv with (forall n (a : Tm n), hfb_case a) Sort Prop.

Lemma ne_hfb {n} (A : Tm n) : ne A -> hfb A.
Proof. case : A => //=. Qed.

Lemma hfb_caseP {n} (A : Tm n) : hfb A -> hfb_case A.
Proof. hauto lq:on ctrs:hfb_case inv:Tm use:ne_hfb. Qed.

Lemma InterpExtInv n i I (A : Tm n) PA :
  ⟦ A ⟧ i ;; I ↘ PA ->
  exists B, hfb B /\ rtc RPar'.R A B /\  ⟦ B ⟧ i ;; I ↘ PA.
Proof.
  move => h. elim : A PA /h.
  - hauto q:on ctrs:InterpExt, rtc use:ne_hfb.
  - move => p A B PA PF hPA _ hPF hPF0 _.
    exists (TBind p A B). repeat split => //=.
    apply rtc_refl.
    hauto l:on ctrs:InterpExt.
  - move => j ?. exists (Univ j).
    hauto l:on ctrs:InterpExt.
  - hauto lq:on ctrs:rtc.
Qed.

Lemma RPar'_Par n (A B : Tm n) :
  RPar'.R A B ->
  Par.R A B.
Proof. induction 1; hauto lq:on ctrs:Par.R. Qed.

Lemma RPar's_Pars n (A B : Tm n) :
  rtc RPar'.R A B ->
  rtc Par.R A B.
Proof. hauto lq:on use:RPar'_Par, rtc_subrel. Qed.

Lemma RPar's_join n (A B : Tm n) :
  rtc RPar'.R A B -> join A B.
Proof. hauto lq:on ctrs:rtc use:RPar's_Pars. Qed.

Lemma bindspace_iff n p (PA : Tm n -> Prop) PF PF0 b  :
  (forall (a : Tm n) (PB PB0 : Tm n -> Prop), PF a PB -> PF0 a PB0 -> PB = PB0) ->
  (forall a, PA a -> exists PB, PF a PB) ->
  (forall a, PA a -> exists PB0, PF0 a PB0) ->
  (BindSpace p PA PF b <-> BindSpace p PA PF0 b).
Proof.
  rewrite /BindSpace => h hPF hPF0.
  case : p => /=.
  - rewrite /ProdSpace.
    split.
    move => h1 a PB ha hPF'.
    specialize hPF with (1 := ha).
    specialize hPF0 with (1 := ha).
    sblast.
    move => ? a PB ha.
    specialize hPF with (1 := ha).
    specialize hPF0 with (1 := ha).
    sblast.
  - rewrite /SumSpace.
    hauto lq:on rew:off.
Qed.

Lemma ne_prov_inv n (a : Tm n) :
  ne a -> (exists i, prov (VarTm i) a) \/ prov Bot a.
Proof.
  elim : n /a => //=.
  - hauto lq:on ctrs:prov.
  - hauto lq:on rew:off ctrs:prov b:on.
  - hauto lq:on ctrs:prov.
  - move => n.
    have : @prov n Bot Bot by auto using P_Bot.
    tauto.
Qed.

Lemma ne_pars_inv n (a b : Tm n) :
  ne a -> rtc Par.R a b -> (exists i, prov (VarTm i) b) \/ prov Bot b.
Proof.
  move /ne_prov_inv.
  sfirstorder use:prov_pars.
Qed.

Lemma ne_pars_extract  n (a b : Tm n) :
  ne a -> rtc Par.R a b -> (exists i, extract b = (VarTm i)) \/ extract b = Bot.
Proof. hauto lq:on rew:off use:ne_pars_inv, prov_extract. Qed.

Lemma join_bind_ne_contra n p (A : Tm n) B C :
  ne C ->
  join (TBind p A B) C -> False.
Proof.
  move => hC [D [h0 h1]].
  move /pars_pi_inv : h0 => [A0 [B0 [h2 [h3 h4]]]].
  have : (exists i, extract D = (VarTm i)) \/ extract D = Bot by eauto using ne_pars_extract.
  sfirstorder.
Qed.

Lemma join_univ_ne_contra n i C :
  ne C ->
  join (Univ i : Tm n) C -> False.
Proof.
  move => hC [D [h0 h1]].
  move /pars_univ_inv : h0 => ?.
  have : (exists i, extract D = (VarTm i)) \/ extract D = Bot by eauto using ne_pars_extract.
  sfirstorder.
Qed.

#[export]Hint Resolve join_univ_ne_contra join_bind_ne_contra join_univ_pi_contra join_symmetric join_transitive : join.

Lemma InterpExt_Join n i I (A B : Tm n) PA PB :
  ⟦ A ⟧ i ;; I ↘ PA ->
  ⟦ B ⟧ i ;; I ↘ PB ->
  join A B ->
  PA = PB.
Proof.
  move => h. move : B PB. elim : A PA /h.
  - move => A hA B PB /InterpExtInv.
    move => [B0 []].
    move /hfb_caseP. elim/hfb_inv => _.
    + move => p A0 B1 ? [/RPar's_join h0 h1] h2. subst. exfalso.
      eauto with join.
    + move => ? ? [/RPar's_join *]. subst. exfalso.
      eauto with join.
    + hauto lq:on use:InterpExt_Ne_inv.
  - move => p A B PA PF hPA ihPA hTot hRes ihPF U PU /InterpExtInv.
    move => [B0 []].
    move /hfb_caseP.
    elim /hfb_inv => _.
    rename B0 into B00.
    + move => p0 A0 B0 ?  [hr hPi]. subst.
      move /InterpExt_Bind_inv : hPi.
      move => [PA0][PF0][hPA0][hTot0][hRes0]?. subst.
      move => hjoin.
      have{}hr : join U (TBind p0 A0 B0) by auto using RPar's_join.
      have hj : join (TBind p A B) (TBind p0 A0 B0) by eauto using join_transitive.
      have {hj} : p0 = p /\ join A A0 /\ join B B0 by hauto l:on use:join_pi_inj.
      move => [? [h0 h1]]. subst.
      have ? : PA0 = PA by hauto l:on. subst.
      rewrite /ProdSpace.
      extensionality b.
      apply propositional_extensionality.
      apply bindspace_iff; eauto.
      move => a PB PB0 hPB hPB0.
      apply : ihPF; eauto.
      by apply join_substing.
    + move => j ?. subst.
      move => [h0 h1] h.
      have ? : join U (Univ j) by eauto using RPar's_join.
      have : join (TBind p A B) (Univ j) by eauto using join_transitive.
      move => ?. exfalso.
      eauto using join_univ_pi_contra.
    + move => A0 ? ? [/RPar's_join ?]. subst.
      move => _ ?. exfalso. eauto with join.
  - move => j ? B PB /InterpExtInv.
    move => [? []]. move/hfb_caseP.
    elim /hfb_inv => //= _.
    + move => p A0 B0 _ [].
      move /RPar's_join => *.
      exfalso. eauto with join.
    + move => m _ [/RPar's_join h0 + h1].
      have /join_univ_inj {h0 h1} ?  : join (Univ j : Tm n) (Univ m) by eauto using join_transitive.
      subst.
      move /InterpExt_Univ_inv. firstorder.
    + move => A ? ? [/RPar's_join] *. subst. exfalso. eauto with join.
  - move => A A0 PA h.
    have /join_symmetric {}h : join A A0 by hauto lq:on ctrs:rtc use:RPar'_Par, relations.rtc_once.
    eauto using join_transitive.
Qed.

Lemma InterpUniv_Join n i (A B : Tm n) PA PB :
  ⟦ A ⟧ i ↘ PA ->
  ⟦ B ⟧ i ↘ PB ->
  join A B ->
  PA = PB.
Proof. hauto l:on use:InterpExt_Join rew:db:InterpUniv. Qed.

Lemma InterpUniv_Bind_inv n p i  (A : Tm n) B P
  (h :  ⟦ TBind p A B ⟧ i ↘ P) :
  exists (PA : Tm n -> Prop) (PF : Tm n -> (Tm n -> Prop) -> Prop),
     ⟦ A ⟧ i ↘ PA /\
    (forall a, PA a -> exists PB, PF a PB) /\
    (forall a PB, PF a PB -> ⟦ subst_Tm (scons a VarTm) B ⟧ i ↘ PB) /\
    P = BindSpace p PA PF.
Proof. hauto l:on use:InterpExt_Bind_inv rew:db:InterpUniv. Qed.

Lemma InterpUniv_Univ_inv n i j P
  (h :  ⟦ Univ j  ⟧ i ↘ P) :
  P = (fun (A : Tm n) => exists PA, ⟦ A ⟧ j ↘ PA) /\ j < i.
Proof. hauto l:on use:InterpExt_Univ_inv rew:db:InterpUniv. Qed.

Lemma InterpExt_Functional n i I (A B : Tm n) PA PB :
  ⟦ A ⟧ i ;; I ↘ PA ->
  ⟦ A ⟧ i ;; I ↘ PB ->
  PA = PB.
Proof. hauto use:InterpExt_Join, join_refl. Qed.

Lemma InterpUniv_Functional n i (A : Tm n) PA PB :
  ⟦ A ⟧ i ↘ PA ->
  ⟦ A ⟧ i ↘ PB ->
  PA = PB.
Proof. hauto use:InterpExt_Functional rew:db:InterpUniv. Qed.

Lemma InterpUniv_Join' n i j (A B : Tm n) PA PB :
  ⟦ A ⟧ i ↘ PA ->
  ⟦ B ⟧ j ↘ PB ->
  join A B ->
  PA = PB.
Proof.
  have [? ?] : i <= max i j /\ j <= max i j by lia.
  move => hPA hPB.
  have : ⟦ A ⟧ (max i j) ↘ PA by eauto using InterpUnivN_cumulative.
  have : ⟦ B ⟧ (max i j) ↘ PB by eauto using InterpUnivN_cumulative.
  eauto using InterpUniv_Join.
Qed.

Lemma InterpUniv_Functional' n i j A PA PB :
  ⟦ A : Tm n ⟧ i ↘ PA ->
  ⟦ A ⟧ j ↘ PB ->
  PA = PB.
Proof.
  hauto l:on use:InterpUniv_Join', join_refl.
Qed.

Lemma InterpExt_Bind_inv_nopf i n I p A B P (h :  ⟦TBind p A B ⟧ i ;; I ↘ P) :
  exists (PA : Tm n -> Prop),
     ⟦ A ⟧ i ;; I ↘ PA /\
    (forall a, PA a -> exists PB, ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘ PB) /\
      P = BindSpace p PA (fun a PB => ⟦ subst_Tm (scons a VarTm) B ⟧ i ;; I ↘ PB).
Proof.
  move /InterpExt_Bind_inv : h. intros (PA & PF & hPA & hPF & hPF' & ?); subst.
  exists PA. repeat split => //.
  - sfirstorder.
  - extensionality b.
    case : p => /=.
    + extensionality a.
      extensionality PB.
      extensionality ha.
      apply propositional_extensionality.
      split.
      * hecrush use:InterpExt_Functional.
      * sfirstorder.
    + rewrite /SumSpace. apply propositional_extensionality.
      split; hauto q:on use:InterpExt_Functional.
Qed.

Lemma InterpUniv_Bind_inv_nopf n i p A B P (h :  ⟦TBind p A B ⟧ i ↘ P) :
  exists (PA : Tm n -> Prop),
     ⟦ A ⟧ i ↘ PA /\
    (forall a, PA a -> exists PB, ⟦ subst_Tm (scons a VarTm) B ⟧ i ↘ PB) /\
      P = BindSpace p PA (fun a PB => ⟦ subst_Tm (scons a VarTm) B ⟧ i ↘ PB).
Proof. hauto l:on use:InterpExt_Bind_inv_nopf rew:db:InterpUniv. Qed.

Lemma InterpExt_back_clos n i I (A : Tm n) PA :
  (forall j,  j < i -> forall a b, (RPar'.R a b) ->  I j b -> I j a) ->
    ⟦ A ⟧ i ;; I ↘ PA ->
    forall a b, (RPar'.R a b) ->
           PA b -> PA a.
Proof.
  move => hI h.
  elim : A PA /h.
  - hauto q:on ctrs:rtc unfold:wne.
  - move => p A B PA PF hPA ihPA hTot hRes ihPF a b hr.
    case : p => //=.
    + have  : forall b0 b1 a, RPar'.R b0 b1 -> RPar'.R (App b0 a) (App b1 a)
          by hauto lq:on ctrs:RPar'.R use:RPar'.refl.
      hauto lq:on rew:off unfold:ProdSpace.
    + hauto lq:on ctrs:rtc unfold:SumSpace.
  - eauto.
  - eauto.
Qed.

Lemma InterpExt_back_clos_star n i I (A : Tm n) PA :
  (forall j,  j < i -> forall a b, (RPar'.R a b) ->  I j b -> I j a) ->
    ⟦ A ⟧ i ;; I ↘ PA ->
    forall a b, (rtc RPar'.R a b) ->
           PA b -> PA a.
Proof. induction 3; hauto l:on use:InterpExt_back_clos. Qed.

Lemma InterpUniv_back_clos n i (A : Tm n) PA :
    ⟦ A ⟧ i ↘ PA ->
    forall a b, (RPar'.R a b) ->
           PA b -> PA a.
Proof.
  simp InterpUniv.
  apply InterpExt_back_clos.
  hauto lq:on ctrs:rtc use:InterpUnivN_back_preservation_star.
Qed.

Lemma InterpUniv_back_clos_star n i (A : Tm n) PA :
    ⟦ A ⟧ i ↘ PA ->
    forall a b, rtc RPar'.R a b ->
           PA b -> PA a.
Proof.
  move => h  a b.
  induction 1=> //.
  hauto lq:on use:InterpUniv_back_clos.
Qed.

Lemma pars'_wn {n} a b :
  rtc RPar'.R a b ->
  @wn n b ->
  wn a.
Proof. sfirstorder unfold:wn use:@relations.rtc_transitive. Qed.

(* P identifies a set of "reducibility candidates" *)
Definition CR {n} (P : Tm n -> Prop) :=
  (forall a, P a -> wn a) /\
    (forall a, ne a -> P a).

Lemma adequacy_ext i n I A PA
  (hI0 : forall j,  j < i -> forall a b, (RPar'.R a b) ->  I j b -> I j a)
  (hI : forall j, j < i -> CR (I j))
  (h :  ⟦ A : Tm n ⟧ i ;; I ↘ PA) :
  CR PA /\ wn A.
Proof.
  elim : A PA / h.
  - hauto unfold:wne use:wne_wn.
  - move => p A B PA PF hA hPA hTot hRes ihPF.
    rewrite /CR.
    have hb : PA Bot by firstorder.
    repeat split.
    + case : p => /=.
      * qauto l:on use:ext_wn unfold:ProdSpace, CR.
      * rewrite /SumSpace => a []; first by eauto with nfne.
        move => [q0][q1]*.
        have : wn q0 /\ wn q1 by hauto q:on.
        qauto l:on use:wn_pair, pars'_wn.
    + case : p => /=.
      * rewrite /ProdSpace.
        move => a ha c PB hc hPB.
        have hc' : wn c by sfirstorder.
        have : wne (App a c) by hauto lq:on use:wne_app ctrs:rtc.
        have h : (forall a, ne a -> PB a) by sfirstorder.
        suff : (forall a, wne a -> PB a) by hauto l:on.
        move => a0 [a1 [h0 h1]].
        eapply InterpExt_back_clos_star with (b := a1); eauto.
      * rewrite /SumSpace.
        move => a ha. left.
        sfirstorder ctrs:rtc.
    + have wnA : wn A by firstorder.
      apply wn_bind => //.
      apply wn_antirenaming with (ρ := scons Bot VarTm);first by  hauto q:on inv:option.
      hauto lq:on.
  - hauto l:on.
  - hauto lq:on rew:off ctrs:rtc.
Qed.

Lemma adequacy i n A PA
  (h :  ⟦ A : Tm n ⟧ i ↘ PA) :
  CR PA /\ wn A.
Proof.
  move : i A PA h.
  elim /Wf_nat.lt_wf_ind => i ih A PA.
  simp InterpUniv.
  apply adequacy_ext.
  hauto lq:on ctrs:rtc use:InterpUnivN_back_preservation_star.
  hauto l:on use:InterpExt_Ne rew:db:InterpUniv.
Qed.

Lemma adequacy_wne i n A PA a : ⟦ A : Tm n ⟧ i ↘ PA -> wne a -> PA a.
Proof. qauto l:on use:InterpUniv_back_clos_star, adequacy unfold:CR. Qed.

Lemma adequacy_wn i n A PA (h :  ⟦ A : Tm n ⟧ i ↘ PA) a : PA a -> wn a.
Proof. hauto q:on use:adequacy. Qed.

Definition ρ_ok {n} (Γ : fin n -> Tm n) (ρ : fin n -> Tm 0) := forall i k PA,
  ⟦ subst_Tm ρ (Γ i) ⟧ k ↘ PA -> PA (ρ i).

Definition SemWt {n} Γ (a A : Tm n) := forall ρ, ρ_ok Γ ρ -> exists k PA, ⟦ subst_Tm ρ A ⟧ k ↘ PA /\ PA (subst_Tm ρ a).
Notation "Γ ⊨ a ∈ A" := (SemWt Γ a A) (at level 70).

(* Semantic context wellformedness *)
Definition SemWff {n} Γ := forall (i : fin n), exists j, Γ ⊨ Γ i ∈ Univ j.
Notation "⊨ Γ" := (SemWff Γ) (at level 70).

Lemma ρ_ok_bot n (Γ : fin n -> Tm n)  :
  ρ_ok Γ (fun _ => Bot).
Proof.
  rewrite /ρ_ok.
  hauto q:on use:adequacy.
Qed.

Lemma ρ_ok_cons n i (Γ : fin n -> Tm n) ρ a PA A :
  ⟦ subst_Tm ρ A ⟧ i ↘ PA -> PA a ->
  ρ_ok Γ ρ ->
  ρ_ok (funcomp (ren_Tm shift) (scons A Γ)) (scons a ρ).
Proof.
  move => h0 h1 h2.
  rewrite /ρ_ok.
  move => j.
  destruct j as [j|].
  - move => m PA0. asimpl => ?.
    asimpl.
    firstorder.
  - move => m PA0. asimpl => h3.
    have ? : PA0 = PA by eauto using InterpUniv_Functional'.
    by subst.
Qed.

Definition renaming_ok {n m} (Γ : fin n -> Tm n) (Δ : fin m -> Tm m) (ξ : fin m -> fin n) :=
  forall (i : fin m), ren_Tm ξ (Δ i) = Γ (ξ i).

Lemma ρ_ok_renaming n m (Γ : fin n -> Tm n) ρ :
  forall (Δ : fin m -> Tm m) ξ,
    renaming_ok Γ Δ ξ ->
    ρ_ok Γ ρ ->
    ρ_ok Δ (funcomp ρ ξ).
Proof.
  move => Δ ξ hξ hρ.
  rewrite /ρ_ok => i m' PA.
  rewrite /renaming_ok in hξ.
  rewrite /ρ_ok in hρ.
  move => h.
  rewrite /funcomp.
  apply hρ with (k := m').
  move : h. rewrite -hξ.
  by asimpl.
Qed.

Lemma renaming_SemWt {n} Γ a A :
  Γ ⊨ a ∈ A ->
  forall {m} Δ (ξ : fin n -> fin m),
    renaming_ok Δ Γ ξ ->
    Δ ⊨ ren_Tm ξ a ∈ ren_Tm ξ A.
Proof.
  rewrite /SemWt => h m Δ ξ hξ ρ hρ.
  have /h hρ' : (ρ_ok Γ (funcomp ρ ξ)) by eauto using ρ_ok_renaming.
  hauto q:on solve+:(by asimpl).
Qed.

Lemma weakening_Sem n Γ (a : Tm n) A B i
  (h0 : Γ ⊨ B ∈ Univ i)
  (h1 : Γ ⊨ a ∈ A) :
   funcomp (ren_Tm shift) (scons B Γ) ⊨ ren_Tm shift a ∈ ren_Tm shift A.
Proof.
  apply : renaming_SemWt; eauto.
  hauto lq:on inv:option unfold:renaming_ok.
Qed.

Lemma SemWt_Wn n Γ (a : Tm n) A :
  Γ ⊨ a ∈ A ->
  wn a /\ wn A.
Proof.
  move => h.
  have {}/h := ρ_ok_bot _ Γ => h.
  have h0 : wn (subst_Tm (fun _ : fin n => (Bot : Tm 0)) A) by hauto l:on use:adequacy.
  have h1 : wn (subst_Tm (fun _ : fin n => (Bot : Tm 0)) a)by hauto l:on use:adequacy_wn.
  move {h}. hauto lq:on use:wn_antirenaming.
Qed.

Lemma SemWt_Univ n Γ (A : Tm n) i  :
  Γ ⊨ A ∈ Univ i <->
  forall ρ, ρ_ok Γ ρ -> exists S, ⟦ subst_Tm ρ A ⟧ i ↘ S.
Proof.
  rewrite /SemWt.
  split.
  - hauto lq:on rew:off use:InterpUniv_Univ_inv.
  - move => /[swap] ρ /[apply].
    move => [PA hPA].
    exists (S i). eexists.
    split.
    + simp InterpUniv. apply InterpExt_Univ. lia.
    + simpl. eauto.
Qed.

(* Structural laws for Semantic context wellformedness *)
Lemma SemWff_nil : SemWff null.
Proof. case. Qed.

Lemma SemWff_cons n Γ (A : Tm n) i :
    ⊨ Γ ->
    Γ ⊨ A ∈ Univ i ->
    (* -------------- *)
    ⊨ funcomp (ren_Tm shift) (scons A Γ).
Proof.
  move => h h0.
  move => j. destruct j as [j|].
  - move /(_ j) : h => [k hk].
    exists k. change (Univ k) with (ren_Tm shift (Univ k : Tm n)).
    eauto using weakening_Sem.
  - hauto q:on use:weakening_Sem.
Qed.

(* Semantic typing rules *)
Lemma ST_Var n Γ (i : fin n) :
  ⊨ Γ ->
  Γ ⊨ VarTm i ∈ Γ i.
Proof.
  move /(_ i) => [j /SemWt_Univ h].
  rewrite /SemWt => ρ /[dup] hρ {}/h [S hS].
  exists j, S.
  asimpl. firstorder.
Qed.

Lemma ST_Bind n Γ i j p (A : Tm n) (B : Tm (S n)) :
  Γ ⊨ A ∈ Univ i ->
  funcomp (ren_Tm shift) (scons A Γ) ⊨ B ∈ Univ j ->
  Γ ⊨ TBind p A B ∈ Univ (max i j).
Proof.
  move => /SemWt_Univ h0 /SemWt_Univ h1.
  apply SemWt_Univ => ρ hρ.
  move /h0 : (hρ){h0} => [S hS].
  eexists => /=.
  have ? : i <= Nat.max i j by lia.
  apply InterpUnivN_Fun_nopf.
  - eauto using InterpUnivN_cumulative.
  - move => *. asimpl. hauto l:on use:InterpUnivN_cumulative, ρ_ok_cons.
Qed.

Lemma ST_Conv n Γ (a : Tm n) A B i :
  Γ ⊨ a ∈ A ->
  Γ ⊨ B ∈ Univ i ->
  join A B ->
  Γ ⊨ a ∈ B.
Proof.
  move => ha /SemWt_Univ h h0.
  move => ρ hρ.
  have {}h0 : join (subst_Tm ρ A) (subst_Tm ρ B) by eauto using join_substing.
  move /ha : (hρ){ha} => [m [PA [h1 h2]]].
  move /h : (hρ){h} => [S hS].
  have ? : PA = S by eauto using InterpUniv_Join'. subst.
  eauto.
Qed.

Lemma ST_Abs n Γ (a : Tm (S n)) A B i :
  Γ ⊨ TBind TPi A B ∈ (Univ i) ->
  funcomp (ren_Tm shift) (scons A Γ) ⊨ a ∈ B ->
  Γ ⊨ Abs a ∈ TBind TPi A B.
Proof.
  rename a into b.
  move /SemWt_Univ => + hb ρ hρ.
  move /(_ _ hρ) => [PPi hPPi].
  exists i, PPi. split => //.
  simpl in hPPi.
  move /InterpUniv_Bind_inv_nopf : hPPi.
  move => [PA [hPA [hTot ?]]]. subst=>/=.
  move => a PB ha. asimpl => hPB.
  move : ρ_ok_cons (hPA) (hρ) (ha). repeat move/[apply].
  move /hb.
  intros (m & PB0 & hPB0 & hPB0').
  replace PB0 with PB in * by hauto l:on use:InterpUniv_Functional'.
  apply : InterpUniv_back_clos; eauto.
  apply : RPar'.AppAbs'; eauto using RPar'.refl.
  by asimpl.
Qed.

Lemma ST_App n Γ (b a : Tm n) A B :
  Γ ⊨ b ∈ TBind TPi A B ->
  Γ ⊨ a ∈ A ->
  Γ ⊨ App b a ∈ subst_Tm (scons a VarTm) B.
Proof.
  move => hf hb ρ hρ.
  move /(_ ρ hρ) : hf; intros (i & PPi & hPi & hf).
  move /(_ ρ hρ) : hb; intros (j & PA & hPA & hb).
  simpl in hPi.
  move /InterpUniv_Bind_inv_nopf : hPi. intros (PA0 & hPA0 & hTot & ?). subst.
  have ? : PA0 = PA by eauto using InterpUniv_Functional'. subst.
  move  : hf (hb). move/[apply].
  move : hTot hb. move/[apply].
  asimpl. hauto lq:on.
Qed.

Lemma ST_Pair n Γ (a b : Tm n) A B i :
  Γ ⊨ TBind TSig A B ∈ (Univ i) ->
  Γ ⊨ a ∈ A ->
  Γ ⊨ b ∈ subst_Tm (scons a VarTm) B ->
  Γ ⊨ Pair a b ∈ TBind TSig A B.
Proof.
  move /SemWt_Univ => + ha hb ρ hρ.
  move /(_ _ hρ) => [PPi hPPi].
  exists i, PPi. split => //.
  simpl in hPPi.
  move /InterpUniv_Bind_inv_nopf : hPPi.
  move => [PA [hPA [hTot ?]]]. subst=>/=.
  rewrite /SumSpace. right.
  exists (subst_Tm ρ a), (subst_Tm ρ b).
  split.
  - hauto l:on use:Pars.substing.
  - move /ha : (hρ){ha}.
    move => [m][PA0][h0]h1.
    move /hb : (hρ){hb}.
    move => [k][PB][h2]h3.
    have ? : PA0 = PA by eauto using InterpUniv_Functional'. subst.
    split => // PB0.
    move : h2. asimpl => *.
    have ? : PB0 = PB by eauto using InterpUniv_Functional'. by subst.
Qed.

Lemma ST_Proj1 n Γ (a : Tm n) A B :
  Γ ⊨ a ∈ TBind TSig A B ->
  Γ ⊨ Proj PL a ∈ A.
Proof.
  move => h ρ /[dup]hρ {}/h [m][PA][/= /InterpUniv_Bind_inv_nopf h0]h1.
  move : h0 => [S][h2][h3]?. subst.
  move : h1 => /=.
  rewrite /SumSpace.
  case; first by hauto lq:on use:adequacy_wne, wne_proj.
  move => [a0 [b0 [h4 [h5 h6]]]].
  exists m, S. split => //=.
  have {}h4 : rtc RPar'.R (Proj PL (subst_Tm ρ a)) (Proj PL (Pair a0 b0)) by eauto using RPars'.ProjCong.
  have ? : RPar'.R (Proj PL (Pair a0 b0)) a0 by hauto l:on use:RPar'.refl, RPar'.ProjPair'.
  have : rtc RPar'.R (Proj PL (subst_Tm ρ a)) a0 by eauto using @relations.rtc_r.
  move => h.
  apply : InterpUniv_back_clos_star; eauto.
Qed.

Lemma substing_RPar' n m (A : Tm (S n)) ρ (B : Tm m) C :
  RPar'.R B C ->
  RPar'.R (subst_Tm (scons B ρ) A) (subst_Tm (scons C ρ) A).
Proof. hauto lq:on inv:option use:RPar'.morphing, RPar'.refl. Qed.

Lemma substing_RPar's n m (A : Tm (S n)) ρ (B : Tm m) C :
  rtc RPar'.R B C ->
  rtc RPar'.R (subst_Tm (scons B ρ) A) (subst_Tm (scons C ρ) A).
Proof. induction 1; hauto lq:on ctrs:rtc use:substing_RPar'. Qed.

Lemma ST_Proj2 n Γ (a : Tm n) A B :
  Γ ⊨ a ∈ TBind TSig A B ->
  Γ ⊨ Proj PR a ∈ subst_Tm (scons (Proj PL a) VarTm) B.
Proof.
  move => h ρ hρ.
  move : (hρ) => {}/h [m][PA][/= /InterpUniv_Bind_inv_nopf h0]h1.
  move : h0 => [S][h2][h3]?. subst.
  move : h1 => /=.
  rewrite /SumSpace.
  case.
  - move => h.
    have hp : forall p, wne (Proj p (subst_Tm ρ a)) by auto using wne_proj.
    have hp0 := hp PL. have hp1 := hp PR => {hp}.
    have : S (Proj PL (subst_Tm ρ a)) by hauto q:on use:adequacy_wne.
    move /h3 => [PB]. asimpl. hauto lq:on use:adequacy_wne.
  - move => [a0 [b0 [h4 [h5 h6]]]].
    specialize h3 with (1 := h5).
    move : h3 => [PB hPB].
    have hr : forall p, rtc RPar'.R (Proj p (subst_Tm ρ a)) (Proj p (Pair a0 b0)) by eauto using RPars'.ProjCong.
    have hrl : RPar'.R (Proj PL (Pair a0 b0)) a0 by hauto l:on use:RPar'.ProjPair', RPar'.refl.
    have hrr : RPar'.R (Proj PR (Pair a0 b0)) b0 by hauto l:on use:RPar'.ProjPair', RPar'.refl.
    exists m, PB.
    asimpl. split.
    + have h : rtc RPar'.R (Proj PL (subst_Tm ρ a)) a0 by eauto using @relations.rtc_r.
      have {}h : rtc RPar'.R (subst_Tm (scons (Proj PL (subst_Tm ρ a)) ρ) B) (subst_Tm (scons a0 ρ) B) by eauto using substing_RPar's.
      move : hPB. asimpl.
      eauto using InterpUnivN_back_preservation_star.
    + hauto lq:on use:@relations.rtc_r, InterpUniv_back_clos_star.
Qed.

Lemma ne_nf_preservation n (a b : Tm n) : ERed.R b a -> (ne a -> ne b) /\ (nf a -> nf b).
Proof.
  move => h. elim : n b a /h => //=.
  - move => n a.
    split => //=.
    hauto lqb:on use:ne_nf_ren db:nfne.
  - hauto lqb:on db:nfne.
  - hauto lqb:on db:nfne.
  - hauto lqb:on db:nfne.
  - hauto lqb:on db:nfne.
  - hauto lqb:on db:nfne.
  - hauto lqb:on db:nfne.
  - hauto lqb:on db:nfne.
  - hauto lqb:on db:nfne.
  - hauto lqb:on db:nfne.
Qed.

Fixpoint size_tm {n} (a : Tm n) :=
  match a with
  | VarTm _ => 1
  | TBind _ A B => 1 + Nat.add (size_tm A) (size_tm B)
  | Abs a => 1 + size_tm a
  | App a b => 1 + Nat.add (size_tm a) (size_tm b)
  | Proj p a => 1 + size_tm a
  | Pair a b => 1 + Nat.add (size_tm a) (size_tm b)
  | Bot => 1
  | Univ _ => 1
  end.

Lemma size_tm_ren n m (ξ : fin n -> fin m) a : size_tm (ren_Tm ξ a) = size_tm a.
Proof.
  move : m ξ. elim : n / a => //=; scongruence.
Qed.

#[export]Hint Rewrite size_tm_ren : size_tm.

Lemma size_η_lt n (a b : Tm n) :
  ERed.R b a ->
  size_tm b < size_tm a.
Proof.
  move => h. elim : b a / h => //=; hauto l:on rew:db:size_tm.
Qed.

Lemma ered_local_confluence n (a b c : Tm n) :
  ERed.R b a ->
  ERed.R c a ->
  exists d, rtc ERed.R d b  /\ rtc ERed.R d c.
Proof.
  move => h. move : c.
  elim : n b a / h => n.
  - move => a c.
    elim /ERed.inv => //= _.
    + move => ? ? [*]. subst.
      have : subst_Tm (scons Bot VarTm) (ren_Tm shift c) = (subst_Tm (scons Bot VarTm) (ren_Tm shift a))
        by congruence.
      asimpl => ?. subst.
      eauto using rtc_refl.
    + move => a0 a1 ha ? [*]. subst.
      elim /ERed.inv : ha => //= _.
      * move => a1 a2 b0 ha ? [*]. subst.
        have [a2 [h0 h1]] : exists a2, ERed.R a2 a /\ a1 = ren_Tm shift a2 by admit. subst.
        eexists. split; cycle 1.
        apply : relations.rtc_r; cycle 1.
        apply ERed.AppEta.
        apply rtc_refl.
        eauto using relations.rtc_once.
      * hauto q:on ctrs:rtc, ERed.R inv:ERed.R.
  - move => a c ha.
    elim /ERed.inv : ha => //= _.
    + hauto l:on.
    + move => a0 a1 b0 ha ? [*]. subst.
      elim /ERed.inv : ha => //= _.
      move => p a1 a2 ha ? [*]. subst.
      exists a1. split. by apply relations.rtc_once.
      apply : rtc_l. apply ERed.PairEta.
      apply : rtc_l. apply ERed.PairCong1. eauto using ERed.ProjCong.
      apply rtc_refl.
    + move => a0 b0 b1 ha ? [*]. subst.
      elim /ERed.inv : ha => //= _ p a0 a1 h ? [*]. subst.
      exists a0. split; first by apply relations.rtc_once.
      apply : rtc_l; first by apply ERed.PairEta.
      apply relations.rtc_once.
      hauto lq:on ctrs:ERed.R.
  - move => a0 a1 ha iha c.
    elim /ERed.inv => //= _.
    + move => a2 ? [*]. subst.
      elim /ERed.inv : ha => //=_.
      * move => a1 a2 b0 ha ? [*] {iha}. subst.
        have [a0 [h0 h1]] : exists a0, ERed.R a0 c /\ a1 = ren_Tm shift a0 by admit. subst.
        exists a0. split; last by apply relations.rtc_once.
        apply relations.rtc_once. apply ERed.AppEta.
      * hauto q:on inv:ERed.R.
    + hauto l:on use:EReds.AbsCong.
  - move => a0 a1 b ha iha c.
    elim /ERed.inv => //= _.
    + hauto lq:on ctrs:rtc use:EReds.AppCong.
    + hauto lq:on use:@relations.rtc_once ctrs:ERed.R.
  - move => a b0 b1 hb ihb c.
    elim /ERed.inv => //=_.
    + move => a0 a1 a2 ha ? [*]. subst.
      move {ihb}. exists (App a0 b0).
      hauto lq:on use:@relations.rtc_once ctrs:ERed.R.
    + hauto lq:on ctrs:rtc use:EReds.AppCong.
  - move => a0 a1 b ha iha c.
    elim /ERed.inv => //= _.
    + move => ? ?[*]. subst.
      elim /ERed.inv : ha => //= _ p a1 a2 h ? [*]. subst.
      exists a1. split; last by apply relations.rtc_once.
      apply : rtc_l. apply ERed.PairEta.
      apply relations.rtc_once. hauto lq:on ctrs:ERed.R.
    + hauto lq:on ctrs:rtc use:EReds.PairCong.
    + hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
  - move => a b0 b1 hb hc c. elim /ERed.inv => //= _.
    + move => ? ? [*]. subst.
      elim /ERed.inv : hb => //= _ p a0 a1 ha ? [*]. subst.
      move {hc}.
      exists a0. split; last by apply relations.rtc_once.
      apply : rtc_l; first by apply ERed.PairEta.
      hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
    + hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
    + hauto lq:on ctrs:rtc use:EReds.PairCong.
  - qauto l:on inv:ERed.R use:EReds.ProjCong.
  - move => p A0 A1 B hA ihA.
    move => c. elim/ERed.inv => //=.
    + hauto lq:on ctrs:rtc use:EReds.BindCong.
    + hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
  - move => p A B0 B1 hB ihB c.
    elim /ERed.inv => //=.
    + hauto lq:on ctrs:ERed.R use:@relations.rtc_once.
    + hauto lq:on ctrs:rtc use:EReds.BindCong.
Admitted.