Require Export terms_defs.
- Fixpoint bref_map [g:nat->nat; d:nat; t:T] : T := Cases t of
+ Fixpoint lref_map [g:nat->nat; d:nat; t:T] : T := Cases t of
| (TSort n) => (TSort n)
- | (TBRef n) =>
- if (blt n d) then (TBRef n) else (TBRef (g n))
+ | (TLRef n) =>
+ if (blt n d) then (TLRef n) else (TLRef (g n))
| (TTail k u t) =>
- (TTail k (bref_map g d u) (bref_map g (s k d) t))
+ (TTail k (lref_map g d u) (lref_map g (s k d) t))
end.
Definition lift : nat -> nat -> T -> T :=
- [h](bref_map [x](plus x h)).
+ [h](lref_map [x](plus x h)).
Section lift_rw. (********************************************************)
XAuto.
Qed.
- Theorem lift_bref_lt: (n:?; h,d:?) (lt n d) ->
- (lift h d (TBRef n)) = (TBRef n).
+ Theorem lift_lref_lt: (n:?; h,d:?) (lt n d) ->
+ (lift h d (TLRef n)) = (TLRef n).
Intros; Unfold lift; Simpl.
Replace (blt n d) with true; XAuto.
Qed.
- Theorem lift_bref_ge: (n:?; h,d:?) (le d n) ->
- (lift h d (TBRef n)) = (TBRef (plus n h)).
+ Theorem lift_lref_ge: (n:?; h,d:?) (le d n) ->
+ (lift h d (TLRef n)) = (TLRef (plus n h)).
Intros; Unfold lift; Simpl.
Replace (blt n d) with false; XAuto.
End lift_rw.
- Hints Resolve lift_bref_lt lift_bind lift_flat : ltlc.
+ Hints Resolve lift_lref_lt lift_bind lift_flat : ltlc.
Tactic Definition LiftTailRw :=
Repeat (Rewrite lift_tail Orelse Rewrite lift_bind Orelse Rewrite lift_flat).
XElim t; Intros.
(* case 1 : TSort *)
XAuto.
-(* case 2 : TBRef n0 *)
+(* case 2 : TLRef n0 *)
Apply (lt_le_e n0 d); Intros.
(* case 2.1 : n0 < d *)
- Rewrite lift_bref_lt in H; [ Inversion H | XAuto ].
+ Rewrite lift_lref_lt in H; [ Inversion H | XAuto ].
(* case 2.2 : n0 >= d *)
- Rewrite lift_bref_ge in H; [ Inversion H | XAuto ].
+ Rewrite lift_lref_ge in H; [ Inversion H | XAuto ].
(* case 3 : TTail k *)
Rewrite lift_tail in H1; Inversion H1.
Qed.
- Theorem lift_gen_bref_lt: (h,d,n:?) (lt n d) ->
- (t:?) (TBRef n) = (lift h d t) ->
- t = (TBRef n).
+ Theorem lift_gen_lref_lt: (h,d,n:?) (lt n d) ->
+ (t:?) (TLRef n) = (lift h d t) ->
+ t = (TLRef n).
XElim t; Intros.
(* case 1 : TSort *)
XAuto.
-(* case 2 : TBRef n0 *)
+(* case 2 : TLRef n0 *)
Apply (lt_le_e n0 d); Intros.
(* case 2.1 : n0 < d *)
- Rewrite lift_bref_lt in H0; XAuto.
+ Rewrite lift_lref_lt in H0; XAuto.
(* case 2.2 : n0 >= d *)
- Rewrite lift_bref_ge in H0; [ Inversion H0; Clear H0 | XAuto ].
+ Rewrite lift_lref_ge in H0; [ Inversion H0; Clear H0 | XAuto ].
Rewrite H3 in H; Clear H3 n.
EApply le_false; [ Apply H1 | XEAuto ].
(* case 3 : TTail k *)
Rewrite lift_tail in H2; Inversion H2.
Qed.
- Theorem lift_gen_bref_false: (h,d,n:?) (le d n) -> (lt n (plus d h)) ->
- (t:?) (TBRef n) = (lift h d t) ->
+ Theorem lift_gen_lref_false: (h,d,n:?) (le d n) -> (lt n (plus d h)) ->
+ (t:?) (TLRef n) = (lift h d t) ->
(P:Prop) P.
XElim t; Intros.
(* case 1 : TSort *)
Inversion H1.
-(* case 2 : TBRef n0 *)
+(* case 2 : TLRef n0 *)
Apply (lt_le_e n0 d); Intros.
(* case 2.1 : n0 < d *)
- Rewrite lift_bref_lt in H1; [ Inversion H1; Clear H1 | XAuto ].
+ Rewrite lift_lref_lt in H1; [ Inversion H1; Clear H1 | XAuto ].
Rewrite <- H4 in H2; Clear H4 n0.
EApply le_false; [ Apply H | XEAuto ].
(* case 2.2 : n0 >= d *)
- Rewrite lift_bref_ge in H1; [ Inversion H1; Clear H1 | XAuto ].
+ Rewrite lift_lref_ge in H1; [ Inversion H1; Clear H1 | XAuto ].
Rewrite H4 in H0; Clear H4.
EApply le_false; [ Apply H2 | XEAuto ].
(* case 3 : TTail k *)
Rewrite lift_tail in H3; Inversion H3.
Qed.
- Theorem lift_gen_bref_ge: (h,d,n:?) (le d n) ->
- (t:?) (TBRef (plus n h)) = (lift h d t) ->
- t = (TBRef n).
+ Theorem lift_gen_lref_ge: (h,d,n:?) (le d n) ->
+ (t:?) (TLRef (plus n h)) = (lift h d t) ->
+ t = (TLRef n).
XElim t; Intros.
(* case 1 : TSort *)
Inversion H0.
-(* case 2 : TBRef n0 *)
+(* case 2 : TLRef n0 *)
Apply (lt_le_e n0 d); Intros.
(* case 2.1 : n0 < d *)
- Rewrite lift_bref_lt in H0; [ Inversion H0; Clear H0 | XAuto ].
+ Rewrite lift_lref_lt in H0; [ Inversion H0; Clear H0 | XAuto ].
Rewrite <- H3 in H1; Clear H3 n0.
EApply le_false; [ Apply H | XEAuto ].
(* case 2.2 : n0 >= d *)
- Rewrite lift_bref_ge in H0; [ Inversion H0; XEAuto | XAuto ].
+ Rewrite lift_lref_ge in H0; [ Inversion H0; XEAuto | XAuto ].
(* case 3 : TTail k *)
Rewrite lift_tail in H2; Inversion H2.
Qed.
XElim x; Intros.
(* case 1 : TSort *)
Inversion H.
-(* case 2 : TBRef n *)
+(* case 2 : TLRef n *)
Apply (lt_le_e n d); Intros.
(* case 2.1 : n < d *)
- Rewrite lift_bref_lt in H; [ Inversion H | XAuto ].
+ Rewrite lift_lref_lt in H; [ Inversion H | XAuto ].
(* case 2.2 : n >= d *)
- Rewrite lift_bref_ge in H; [ Inversion H | XAuto ].
+ Rewrite lift_lref_ge in H; [ Inversion H | XAuto ].
(* case 3 : TTail k *)
Rewrite lift_tail in H1; Inversion H1.
XEAuto.
XElim x; Intros.
(* case 1 : TSort *)
Inversion H.
-(* case 2 : TBRef n *)
+(* case 2 : TLRef n *)
Apply (lt_le_e n d); Intros.
(* case 2.1 : n < d *)
- Rewrite lift_bref_lt in H; [ Inversion H | XAuto ].
+ Rewrite lift_lref_lt in H; [ Inversion H | XAuto ].
(* case 2.2 : n >= d *)
- Rewrite lift_bref_ge in H; [ Inversion H | XAuto ].
+ Rewrite lift_lref_ge in H; [ Inversion H | XAuto ].
(* case 3 : TTail k *)
Rewrite lift_tail in H1; Inversion H1.
XEAuto.
| [ H: (TSort ?0) = (lift ?1 ?2 ?3) |- ? ] ->
LApply (lift_gen_sort ?1 ?2 ?0 ?3); [ Clear H; Intros | XAuto ]
| [ H1: (le ?1 ?2); H2: (lt ?2 (plus ?1 ?3));
- H3: (TBRef ?2) = (lift ?3 ?1 ?4) |- ? ] ->
- Apply (lift_gen_bref_false ?3 ?1 ?2 H1 H2 ?4 H3); XAuto
- | [ H: (TBRef ?1) = (lift (1) ?1 ?2) |- ? ] ->
- LApply (lift_gen_bref_false (1) ?1 ?1); [ Intros H_x | XAuto ];
+ H3: (TLRef ?2) = (lift ?3 ?1 ?4) |- ? ] ->
+ Apply (lift_gen_lref_false ?3 ?1 ?2 H1 H2 ?4 H3); XAuto
+ | [ _: (TLRef ?1) = (lift (S ?1) (0) ?2) |- ? ] ->
+ EApply lift_gen_lref_false; [ Idtac | Idtac | XEAuto ]; XEAuto
+ | [ H: (TLRef ?1) = (lift (1) ?1 ?2) |- ? ] ->
+ LApply (lift_gen_lref_false (1) ?1 ?1); [ Intros H_x | XAuto ];
LApply H_x; [ Clear H_x; Intros H_x | Arith7' ?1; XAuto ];
LApply (H_x ?2); [ Clear H_x; Intros H_x | XAuto ];
Apply H_x
- | [ H: (TBRef (plus ?0 ?1)) = (lift ?1 ?2 ?3) |- ? ] ->
- LApply (lift_gen_bref_ge ?1 ?2 ?0); [ Intros H_x | XAuto ];
+ | [ H: (TLRef (plus ?0 ?1)) = (lift ?1 ?2 ?3) |- ? ] ->
+ LApply (lift_gen_lref_ge ?1 ?2 ?0); [ Intros H_x | XAuto ];
LApply (H_x ?3); [ Clear H_x H; Intros | XAuto ]
- | [ H1: (TBRef ?0) = (lift ?1 ?2 ?3); H2: (lt ?0 ?4) |- ? ] ->
- LApply (lift_gen_bref_lt ?1 ?2 ?0);
+ | [ H1: (TLRef ?0) = (lift ?1 ?2 ?3); H2: (lt ?0 ?4) |- ? ] ->
+ LApply (lift_gen_lref_lt ?1 ?2 ?0);
[ Intros H_x | Apply lt_le_trans with m:=?4; XEAuto ];
LApply (H_x ?3); [ Clear H_x H1; Intros | XAuto ]
- | [ H: (TBRef ?0) = (lift ?1 ?2 ?3) |- ? ] ->
- LApply (lift_gen_bref_lt ?1 ?2 ?0); [ Intros H_x | XEAuto ];
+ | [ H: (TLRef ?0) = (lift ?1 ?2 ?3) |- ? ] ->
+ LApply (lift_gen_lref_lt ?1 ?2 ?0); [ Intros H_x | XEAuto ];
LApply (H_x ?3); [ Clear H_x H; Intros | XAuto ]
| [ H: (TTail (Bind ?0) ?1 ?2) = (lift ?3 ?4 ?5) |- ? ] ->
LApply (lift_gen_bind ?0 ?1 ?2 ?5 ?3 ?4); [ Clear H; Intros H | XAuto ];
XElim t; Intros.
(* case 1: TSort *)
XAuto.
-(* case 2: TBRef n *)
+(* case 2: TLRef n *)
Apply (lt_le_e n d); Intros.
(* case 2.1: n < d *)
- Rewrite lift_bref_lt; XAuto.
+ Rewrite lift_lref_lt; XAuto.
(* case 2.2: n >= d *)
- Rewrite lift_bref_ge; XAuto.
+ Rewrite lift_lref_ge; XAuto.
(* case 3: TTail *)
LiftTailRw; XAuto.
Qed.
- Theorem lift_bref_gt : (d,n:?) (lt d n) ->
- (lift (1) d (TBRef (pred n))) = (TBRef n).
+ Theorem lift_lref_gt : (d,n:?) (lt d n) ->
+ (lift (1) d (TLRef (pred n))) = (TLRef n).
Intros.
- Rewrite lift_bref_ge.
+ Rewrite lift_lref_ge.
(* case 1: first branch *)
Rewrite <- plus_sym; Simpl; Rewrite <- (S_pred n d); XAuto.
(* case 2: second branch *)
End lift_props.
- Hints Resolve lift_r lift_bref_gt : ltlc.
+ Hints Resolve lift_r lift_lref_gt : ltlc.