Require Export contexts_defs.
Require Export subst0_defs.
Require Export drop_defs.

(*#* #caption "current axioms for strict substitution in contexts",
   "substituted tail item: second operand", 
   "substituted tail item: first operand", 
   "substituted tail item: both operands"
*)
(*#* #cap #cap c, c1, c2 #alpha v in W, u in V, u1 in V1, u2 in V2, k in z, r in q *)

      Inductive csubst0 : nat -> T -> C -> C -> Prop :=
         | csubst0_snd  : (k:?; i:?; v,u1,u2:?) (subst0 (r k i) v u1 u2) -> (c:?)
                          (csubst0 (S i) v (CTail c k u1) (CTail c k u2))
         | csubst0_fst  : (k:?; i:?; c1,c2:?; v:?) (csubst0 (r k i) v c1 c2) ->
                          (u:?) (csubst0 (S i) v (CTail c1 k u) (CTail c2 k u))
         | csubst0_both : (k:?; i:?; v,u1,u2:?) (subst0 (r k i) v u1 u2) ->
                          (c1,c2:?) (csubst0 (r k i) v c1 c2) ->
                          (csubst0 (S i) v (CTail c1 k u1) (CTail c2 k u2)).

(*#* #stop file *)

      Hint csubst0 : ltlc := Constructors csubst0.

   Section csubst0_gen_base. (***********************************************)

      Theorem csubst0_gen_tail: (k:?; c1,x:?; u1,v:?; i:?)
                                (csubst0 (S i) v (CTail c1 k u1) x) -> (OR
                                (EX u2 | x = (CTail c1 k u2) &
                                         (subst0 (r k i) v u1 u2)
                                ) |
                                (EX c2 | x = (CTail c2 k u1) &
                                         (csubst0 (r k i) v c1 c2)
                                ) |
                                (EX u2 c2 | x = (CTail c2 k u2) &
                                            (subst0 (r k i) v u1 u2) &
                                            (csubst0 (r k i) v c1 c2)
                                )).
      Intros until 1; InsertEq H '(S i); InsertEq H '(CTail c1 k u1).
      XCase H; Clear x v y y0; Intros; Inversion H1.
(* case 1: csubst0_snd *)
      Inversion H0; Rewrite H3 in H; Rewrite H5 in H; Rewrite H6 in H; XEAuto.
(* case 2: csubst0_fst *)
      Inversion H0; Rewrite H3 in H; Rewrite H4 in H; Rewrite H5 in H; XEAuto.
(* case 2: csubst0_both *)
      Inversion H2; Rewrite H5 in H; Rewrite H6 in H; Rewrite H7 in H;
      Rewrite H4 in H0; Rewrite H5 in H0; Rewrite H7 in H0; XEAuto.
      Qed.

   End csubst0_gen_base.

      Tactic Definition CSubst0GenBase :=
         Match Context With
            | [ H: (csubst0 (S ?1) ?2 (CTail ?3 ?4 ?5) ?6) |- ? ] ->
               LApply (csubst0_gen_tail ?4 ?3 ?6 ?5 ?2 ?1); [ Clear H; Intros H | XAuto ];
               XElim H; Intros H; XElim H; Intros.

   Section csubst0_drop. (***************************************************)

      Theorem csubst0_drop_ge : (i,n:?) (le i n) ->
                                (c1,c2:?; v:?) (csubst0 i v c1 c2) ->
                                (e:?) (drop n (0) c1 e) ->
                                (drop n (0) c2 e).
      XElim i.
(* case 1: i = 0 *)
      Intros; Inversion H0.
(* case 2: i > 0 *)
      Intros i; XElim n.
(* case 2.1: n = 0 *)
      Intros; Inversion H0.
(* case 2.2: n > 0 *)
      Intros until 3; Clear H0; InsertEq H2 '(S i); XElim H0; Intros.
      DropGenBase.
(* case 2.2.1: csubst0_snd *)
      XAuto.
(* case 2.2.2: csubst0_fst *)
      XReplaceIn H0 i0 i; DropGenBase; NewInduction k; XEAuto.
(* case 2.2.3: csubst0_both *)
      XReplaceIn H0 i0 i; XReplaceIn H2 i0 i.
      DropGenBase; NewInduction k; XEAuto.
      Qed.

      Tactic Definition IH :=
         Match Context With
            | [ H0: (n:?) (lt n ?1) -> (c1,c2:?; v:?) (csubst0 ?1 v c1 c2) -> (e:C) (drop n (0) c1 e) -> ?;
                H1: (csubst0 ?1 ?2 ?3 ?4); H2: (drop ?5 (0) ?3 ?6) |- ? ] ->
               LApply (H0 ?5); [ Clear H0; Intros H0 | XAuto ];
               LApply (H0 ?3 ?4 ?2); [ Clear H0 H1; Intros H0 | XAuto ];
               LApply (H0 ?6); [ Clear H0 H2; Intros H0 | XAuto ];
               XElim H0; Intros H0; [ Idtac | XElim H0 | XElim H0 | XElim H0 ]; Intros
            | [ H0: (r ? ?1) = (S ?1) -> (e:?) (drop (S ?2) (0) ?3 e) -> ?;
                H1: (drop (S ?2) (0) ?3 ?4) |- ? ] ->
               LApply H0; [ Clear H0; Intros H0 | XAuto ];
               LApply (H0 ?4); [ Clear H0 H1; Intros H0 | XAuto ];
               XElim H0; Intros H0; [ Idtac | XElim H0 | XElim H0 | XElim H0 ]; Intros.

      Theorem csubst0_drop_lt : (i,n:?) (lt n i) ->
                                (c1,c2:?; v:?) (csubst0 i v c1 c2) ->
                                (e:?) (drop n (0) c1 e) -> (OR
                                (drop n (0) c2 e) |
                                (EX k e0 u w | e = (CTail e0 k u) &
                                               (drop n (0) c2 (CTail e0 k w)) &
                                               (subst0 (minus (r k i) (S n)) v u w)
                                ) |
                                (EX k e1 e2 u | e = (CTail e1 k u) &
                                                (drop n (0) c2 (CTail e2 k u)) &
                                                (csubst0 (minus (r k i) (S n)) v e1 e2)
                                ) |
                                (EX k e1 e2 u w | e = (CTail e1 k u) &
                                                 (drop n (0) c2 (CTail e2 k w)) &
                                                 (subst0 (minus (r k i) (S n)) v u w) &
                                                 (csubst0 (minus (r k i) (S n)) v e1 e2)
                                )).
      XElim i.
(* case 1: i = 0 *)
      Intros; Inversion H.
(* case 2: i > 0 *)
      Intros i; XElim n.
(* case 2.1: n = 0 *)
      Intros H0; Clear H0; Intros until 1; InsertEq H0 '(S i); XElim H0;
      Clear H c1 c2 v y; Intros; DropGenBase; XRewrite e;
      Rewrite <- r_arith0 in H; Try Rewrite <- r_arith0 in H0; Replace i with i0; XEAuto.
(* case 2.2: n > 0 *)
      Intros until 3; Clear H0; InsertEq H2 '(S i); XElim H0; Clear c1 c2 v y;
      Intros; DropGenBase.
(* case 2.2.1: csubst0_snd *)
      XEAuto.
(* case 2.2.2: csubst0_fst *)
      Replace i0 with i; XAuto; XReplaceIn H0 i0 i; XReplaceIn H2 i0 i; Clear H3 i0.
      Apply (r_dis k); Intros; Rewrite (H3 i) in H0; Rewrite (H3 n) in H4.
(* case 2.2.2.1: bind *)
      IH; XRewrite e; Try Rewrite <- (H3 n) in H; Try Rewrite <- (H3 n) in H0;
      Try Rewrite <- r_arith1 in H4; Try Rewrite <- r_arith1 in H5; XEAuto.
(* case 2.2.2.2: flat *)
      IH; XRewrite e; Try Rewrite <- (H3 n) in H2; Try Rewrite <- (H3 n) in H4; XEAuto.
(* case 2.2.3: csubst0_both *)
      Replace i0 with i; XAuto; XReplaceIn H0 i0 i; XReplaceIn H2 i0 i; XReplaceIn H3 i0 i; Clear H4 i0.
      Apply (r_dis k); Intros; Rewrite (H4 i) in H2; Rewrite (H4 n) in H5.
(* case 2.2.2.1: bind *)
      IH; XRewrite e; Try Rewrite <- (H4 n) in H; Try Rewrite <- (H4 n) in H2;
      Try Rewrite <- r_arith1 in H5; Try Rewrite <- r_arith1 in H6; XEAuto.
(* case 2.2.3.2: flat *)
      IH; XRewrite e; Try Rewrite <- (H4 n) in H3; Try Rewrite <- (H4 n) in H5; XEAuto.
      Qed.

      Theorem csubst0_drop_ge_back : (i,n:?) (le i n) ->
                                     (c1,c2:?; v:?) (csubst0 i v c1 c2) ->
                                     (e:?) (drop n (0) c2 e) ->
                                     (drop n (0) c1 e).
      XElim i.
(* case 1 : i = 0 *)
      Intros; Inversion H0.
(* case 2 : i > 0 *)
      Intros i; XElim n.
(* case 2.1 : n = 0 *)
      Intros; Inversion H0.
(* case 2.2 : n > 0 *)
      Intros until 3; Clear H0; InsertEq H2 '(S i); XElim H0; Intros;
      DropGenBase.
(* case 2.2.1 : csubst0_snd *)
      XAuto.
(* case 2.2.2 : csubst0_fst *)
      XReplaceIn H0 i0 i; NewInduction k; XEAuto.
(* case 2.2.3 : csubst0_both *)
      XReplaceIn H0 i0 i; XReplaceIn H2 i0 i; NewInduction k; XEAuto.
      Qed.

   End csubst0_drop.

      Tactic Definition CSubst0Drop :=
         Match Context With
            | [ H1: (lt ?2 ?1);
                H2: (csubst0 ?1 ?3 ?4 ?5); H3: (drop ?2 (0) ?4 ?6) |- ? ] ->
               LApply (csubst0_drop_lt ?1 ?2); [ Intros H_x | XAuto ];
               LApply (H_x ?4 ?5 ?3); [ Clear H_x; Intros H_x | XAuto ];
               LApply (H_x ?6); [ Clear H_x H3; Intros H3 | XAuto ];
               XElim H3;
               [ Intros | Intros H3; XElim H3; Intros
               | Intros H3; XElim H3; Intros | Intros H3; XElim H3; Intros ]
            | [ H1: (le ?1 ?2);
                H2: (csubst0 ?1 ?3 ?4 ?5); H3: (drop ?2 (0) ?4 ?6) |- ? ] ->
               LApply (csubst0_drop_ge ?1 ?2); [ Intros H_x | XAuto ];
               LApply (H_x ?4 ?5 ?3); [ Clear H_x; Intros H_x | XAuto ];
               LApply (H_x ?6); [ Clear H_x H3; Intros | XAuto ]
            | [H2: (csubst0 ?1 ?3 ?4 ?5); H3: (drop ?1 (0) ?4 ?6) |- ? ] ->
               LApply (csubst0_drop_ge ?1 ?1); [ Intros H_x | XAuto ];
               LApply (H_x ?4 ?5 ?3); [ Clear H_x H2; Intros H2 | XAuto ];
               LApply (H2 ?6); [ Clear H2 H3; Intros | XAuto ]
            | [H2: (csubst0 ?1 ?3 ?4 ?5); H3: (drop ?1 (0) ?5 ?6) |- ? ] ->
               LApply (csubst0_drop_ge_back ?1 ?1); [ Intros H_x | XAuto ];
               LApply (H_x ?4 ?5 ?3); [ Clear H_x; Intros H_x | XAuto ];
               LApply (H_x ?6); [ Clear H_x H3; Intros | XAuto ]
            | [H2: (csubst0 ?1 ?3 ?4 ?5); H3: (drop ?2 (0) ?5 ?6) |- ? ] ->
               LApply (csubst0_drop_ge_back ?1 ?2); [ Intros H_x | XAuto ];
               LApply (H_x ?4 ?5 ?3); [ Clear H_x; Intros H_x | XAuto ];
               LApply (H_x ?6); [ Clear H_x H3; Intros | XAuto ].