--- /dev/null
+(*#* #stop file *)
+
+Require Export contexts_defs.
+Require Export subst0_defs.
+Require Export drop_defs.
+
+ Inductive csubst0 : nat -> T -> C -> C -> Prop :=
+ | csubst0_fst : (k:?; i:?; u,v,w:?) (subst0 (r k i) v u w) -> (c:?)
+ (csubst0 (S i) v (CTail c k u) (CTail c k w))
+ | csubst0_snd : (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:?; u,v,w:?) (subst0 (r k i) v u w) ->
+ (c1,c2:?) (csubst0 (r k i) v c1 c2) ->
+ (csubst0 (S i) v (CTail c1 k u) (CTail c2 k w)).
+
+ Hint csubst0 : ltlc := Constructors csubst0.
+
+ Inductive fsubst0 [i:nat; v:T; c1:C; t1:T] : C -> T -> Prop :=
+ | fsubst0_t : (t2:?) (subst0 i v t1 t2) -> (fsubst0 i v c1 t1 c1 t2)
+ | fsubst0_c : (c2:?) (csubst0 i v c1 c2) -> (fsubst0 i v c1 t1 c2 t1)
+ | fsubst0_b : (t2:?) (subst0 i v t1 t2) ->
+ (c2:?) (csubst0 i v c1 c2) -> (fsubst0 i v c1 t1 c2 t2).
+
+ Hint fsubst0 : ltlc := Constructors fsubst0.
+
+ 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_fst *)
+ Inversion H0; Rewrite H3 in H; Rewrite H5 in H; Rewrite H6 in H; XEAuto.
+(* case 2: csubst0_snd *)
+ 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_fst *)
+ XAuto.
+(* case 2.2.2: csubst0_snd *)
+ 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_fst *)
+ XEAuto.
+(* case 2.2.2: csubst0_snd *)
+ 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_fst *)
+ XAuto.
+(* case 2.2.2 : csubst0_snd *)
+ 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 ].
+
projects / helm.git / blobdiff