+++ /dev/null
-(*#* #stop file *)
-
-Require Export terms_defs.
-
- Fixpoint lref_map [g:nat->nat; d:nat; t:T] : T := Cases t of
- | (TSort n) => (TSort n)
- | (TLRef n) =>
- if (blt n d) then (TLRef n) else (TLRef (g n))
- | (TTail k u t) =>
- (TTail k (lref_map g d u) (lref_map g (s k d) t))
- end.
-
- Definition lift : nat -> nat -> T -> T :=
- [h](lref_map [x](plus x h)).
-
- Section lift_rw. (********************************************************)
-
- Theorem lift_sort: (n:?; h,d:?) (lift h d (TSort n)) = (TSort n).
- XAuto.
- Qed.
-
- 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_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.
- Qed.
-
- Theorem lift_tail: (k:?; u,t:?; h,d:?)
- (lift h d (TTail k u t)) =
- (TTail k (lift h d u) (lift h (s k d) t)).
- XAuto.
- Qed.
-
- Theorem lift_bind: (b:?; u,t:?; h,d:?)
- (lift h d (TTail (Bind b) u t)) =
- (TTail (Bind b) (lift h d u) (lift h (S d) t)).
- XAuto.
- Qed.
-
- Theorem lift_flat: (f:?; u,t:?; h,d:?)
- (lift h d (TTail (Flat f) u t)) =
- (TTail (Flat f) (lift h d u) (lift h d t)).
- XAuto.
- Qed.
-
- End lift_rw.
-
- 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).
-
- Tactic Definition LiftTailRwBack :=
- Repeat (Rewrite <- lift_tail Orelse Rewrite <- lift_bind Orelse Rewrite <- lift_flat).
-
- Section lift_gen. (*******************************************************)
-
- Theorem lift_gen_sort: (h,d,n:?; t:?) (TSort n) = (lift h d t) ->
- t = (TSort n).
- XElim t; Intros.
-(* case 1 : TSort *)
- XAuto.
-(* case 2 : TLRef n0 *)
- Apply (lt_le_e n0 d); Intros.
-(* case 2.1 : n0 < d *)
- Rewrite lift_lref_lt in H; [ Inversion H | XAuto ].
-(* case 2.2 : n0 >= d *)
- Rewrite lift_lref_ge in H; [ Inversion H | XAuto ].
-(* case 3 : TTail k *)
- Rewrite lift_tail in H1; Inversion H1.
- Qed.
-
- 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 : TLRef n0 *)
- Apply (lt_le_e n0 d); Intros.
-(* case 2.1 : n0 < d *)
- Rewrite lift_lref_lt in H0; XAuto.
-(* case 2.2 : n0 >= d *)
- 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_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 : TLRef n0 *)
- Apply (lt_le_e n0 d); Intros.
-(* case 2.1 : n0 < d *)
- 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_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_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 : TLRef n0 *)
- Apply (lt_le_e n0 d); Intros.
-(* case 2.1 : n0 < d *)
- 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_lref_ge in H0; [ Inversion H0; XEAuto | XAuto ].
-(* case 3 : TTail k *)
- Rewrite lift_tail in H2; Inversion H2.
- Qed.
-
-(* NOTE: lift_gen_tail should be used instead of these two *) (**)
- Theorem lift_gen_bind: (b:?; u,t,x:?; h,d:?)
- (TTail (Bind b) u t) = (lift h d x) ->
- (EX y z | x = (TTail (Bind b) y z) &
- u = (lift h d y) &
- t = (lift h (S d) z)
- ).
- XElim x; Intros.
-(* case 1 : TSort *)
- Inversion H.
-(* case 2 : TLRef n *)
- Apply (lt_le_e n d); Intros.
-(* case 2.1 : n < d *)
- Rewrite lift_lref_lt in H; [ Inversion H | XAuto ].
-(* case 2.2 : n >= d *)
- Rewrite lift_lref_ge in H; [ Inversion H | XAuto ].
-(* case 3 : TTail k *)
- Rewrite lift_tail in H1; Inversion H1.
- XEAuto.
- Qed.
-
- Theorem lift_gen_flat: (f:?; u,t,x:?; h,d:?)
- (TTail (Flat f) u t) = (lift h d x) ->
- (EX y z | x = (TTail (Flat f) y z) &
- u = (lift h d y) &
- t = (lift h d z)
- ).
- XElim x; Intros.
-(* case 1 : TSort *)
- Inversion H.
-(* case 2 : TLRef n *)
- Apply (lt_le_e n d); Intros.
-(* case 2.1 : n < d *)
- Rewrite lift_lref_lt in H; [ Inversion H | XAuto ].
-(* case 2.2 : n >= d *)
- Rewrite lift_lref_ge in H; [ Inversion H | XAuto ].
-(* case 3 : TTail k *)
- Rewrite lift_tail in H1; Inversion H1.
- XEAuto.
- Qed.
-
- End lift_gen.
-
- Tactic Definition LiftGenBase :=
- Match Context With
- | [ 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: (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: (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: (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: (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 H; Intros
- | [ H: (TTail (Flat ?0) ?1 ?2) = (lift ?3 ?4 ?5) |- ? ] ->
- LApply (lift_gen_flat ?0 ?1 ?2 ?5 ?3 ?4); [ Clear H; Intros H | XAuto ];
- XElim H; Intros.
-
- Section lift_props. (*****************************************************)
-
- Theorem lift_r: (t:?; d:?) (lift (0) d t) = t.
- XElim t; Intros.
-(* case 1: TSort *)
- XAuto.
-(* case 2: TLRef n *)
- Apply (lt_le_e n d); Intros.
-(* case 2.1: n < d *)
- Rewrite lift_lref_lt; XAuto.
-(* case 2.2: n >= d *)
- Rewrite lift_lref_ge; XAuto.
-(* case 3: TTail *)
- LiftTailRw; XAuto.
- Qed.
-
- Theorem lift_lref_gt : (d,n:?) (lt d n) ->
- (lift (1) d (TLRef (pred n))) = (TLRef n).
- Intros.
- Rewrite lift_lref_ge.
-(* case 1: first branch *)
- Rewrite <- plus_sym; Simpl; Rewrite <- (S_pred n d); XAuto.
-(* case 2: second branch *)
- Apply le_S_n; Rewrite <- (S_pred n d); XAuto.
- Qed.
-
- End lift_props.
-
- Hints Resolve lift_r lift_lref_gt : ltlc.
projects / helm.git / blobdiff