contribution about \lambda-\delta

[helm.git] / helm / coq-contribs / LAMBDA-TYPES / lift_gen.v

diff --git a/helm/coq-contribs/LAMBDA-TYPES/lift_gen.v b/helm/coq-contribs/LAMBDA-TYPES/lift_gen.v
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+(*#* #stop file *)
+
+Require lift_defs.
+
+   Section lift_ini. (*******************************************************)
+
+      Tactic Definition IH :=
+         Match Context With
+            | [ H1: (lift ?1 ?2 t) = (lift ?1 ?2 ?3) |- ? ] ->
+               LApply (H ?3 ?1 ?2); [ Clear H H1; Intros | XAuto ]
+            | [ H1: (lift ?1 ?2 t0) = (lift ?1 ?2 ?3) |- ? ] ->
+               LApply (H0 ?3 ?1 ?2); [ Clear H0 H1; Intros | XAuto ].
+
+(*#* #start file *)
+
+(*#* #caption "main properties of lift" #clauses lift_props *)
+
+(*#* #caption "injectivity" *)
+(*#* #cap #alpha x in T1, t in T2, d in i *)
+
+      Theorem lift_inj : (x,t:?; h,d:?) (lift h d x) = (lift h d t) -> x = t.
+
+(*#* #stop file *)
+
+      XElim x.
+(* case 1 : TSort *)
+      Intros; Rewrite lift_sort in H; LiftGenBase; XAuto.
+(* case 2 : TBRef n *)
+      Intros; Apply (lt_le_e n d); Intros.
+(* case 2.1 : n < d *)
+      Rewrite lift_bref_lt in H; [ LiftGenBase; XAuto | XAuto ].
+(* case 2.2 : n >= d *)
+      Rewrite lift_bref_ge in H; [ LiftGenBase; XAuto | XAuto ].
+(* case 3 : TTail k *)
+      XElim k; Intros; [ Rewrite lift_bind in H1 | Rewrite lift_flat in H1 ];
+      LiftGenBase; Rewrite H1; IH; IH; XAuto.
+      Qed.
+
+   End lift_ini.
+
+   Section lift_gen_lift. (**************************************************)
+
+      Tactic Definition IH :=
+         Match Context With
+            | [ H_x: (lift ?0 ?1 t) = (lift ?2 (plus ?3 ?0) ?4) |- ? ] ->
+               LApply (H ?4 ?0 ?2 ?1 ?3); [ Clear H; Intros H | XAuto ];
+               LApply H; [ Clear H H_x; Intros H | XAuto ];
+               XElim H; Intros
+            | [ H_x: (lift ?0 ?1 t0) = (lift ?2 (plus ?3 ?0) ?4) |- ? ] ->
+               LApply (H0 ?4 ?0 ?2 ?1 ?3); [ Clear H0; Intros H0 | XAuto ];
+               LApply H0; [ Clear H0 H_x; Intros H0 | XAuto ];
+               XElim H0; Intros.
+
+(*#* #start file *)
+
+(*#* #caption "generation lemma for lift" *)
+(*#* #cap #cap t1 #alpha t2 in T, x in T2, d1 in i1, d2 in i2 *)
+
+      Theorem lift_gen_lift : (t1,x:?; h1,h2,d1,d2:?) (le d1 d2) ->
+                              (lift h1 d1 t1) = (lift h2 (plus d2 h1) x) ->
+                              (EX t2 | x = (lift h1 d1 t2) &
+                                       t1 = (lift h2 d2 t2)
+                              ).
+
+(*#* #stop file *)
+
+      XElim t1; Intros.
+(* case 1 : TSort *)
+      Rewrite lift_sort in H0.
+      LiftGenBase; Rewrite H0; Clear H0 x.
+      EApply ex2_intro; Rewrite lift_sort; XAuto.
+(* case 2 : TBRef n *)
+      Apply (lt_le_e n d1); Intros.
+(* case 2.1 : n < d1 *)
+      Rewrite lift_bref_lt in H0; [ Idtac | XAuto ].
+      LiftGenBase; Rewrite H0; Clear H0 x.
+      EApply ex2_intro; Rewrite lift_bref_lt; XEAuto.
+(* case 2.2 : n >= d1 *)
+      Rewrite lift_bref_ge in H0; [ Idtac | XAuto ].
+      Apply (lt_le_e n d2); Intros.
+(* case 2.2.1 : n < d2 *)
+      LiftGenBase; Rewrite H0; Clear H0 x.
+      EApply ex2_intro; [ Rewrite lift_bref_ge | Rewrite lift_bref_lt ]; XEAuto.
+(* case 2.2.2 : n >= d2 *)
+      Apply (lt_le_e n (plus d2 h2)); Intros.
+(* case 2.2.2.1 : n < d2 + h2 *)
+      EApply lift_gen_bref_false; [ Idtac | Idtac | Apply H0 ];
+      [ XAuto | Rewrite plus_permute_2_in_3; XAuto ].
+(* case 2.2.2.2 : n >= d2 + h2 *)
+      Rewrite (le_plus_minus_sym h2 (plus n h1)) in H0; [ Idtac | XEAuto ].
+      LiftGenBase; Rewrite H0; Clear H0 x.
+      EApply ex2_intro;
+      [ Rewrite le_minus_plus; [ Idtac | XEAuto ]
+      | Rewrite (le_plus_minus_sym h2 n); [ Idtac | XEAuto ] ];
+      Rewrite lift_bref_ge; XEAuto.
+(* case 3 : TTail k *)
+      NewInduction k.
+(* case 3.1 : Bind *)
+      Rewrite lift_bind in H2.
+      LiftGenBase; Rewrite H2; Clear H2 x.
+      IH; Rewrite H; Rewrite H2; Clear H H2 x0.
+      Arith4In H4 d2 h1; IH; Rewrite H; Rewrite H0; Clear H H0 x1 t t0.
+      EApply ex2_intro; Rewrite lift_bind; XAuto.
+(* case 3.2 : Flat *)
+      Rewrite lift_flat in H2.
+      LiftGenBase; Rewrite H2; Clear H2 x.
+      IH; Rewrite H; Rewrite H2; Clear H H2 x0.
+      IH; Rewrite H; Rewrite H0; Clear H H0 x1 t t0.
+      EApply ex2_intro; Rewrite lift_flat; XAuto.
+      Qed.
+
+   End lift_gen_lift.
+
+      Tactic Definition LiftGen :=
+         Match Context With
+            | [ H: (lift ?1 ?2 ?3) = (lift ?1 ?2 ?4) |- ? ] ->
+               LApply (lift_inj ?3 ?4 ?1 ?2); [ Clear H; Intros | XAuto ]
+            | [ H: (lift ?0 ?1 ?2) = (lift ?3 (plus ?4 ?0) ?5) |- ? ] ->
+               LApply (lift_gen_lift ?2 ?5 ?0 ?3 ?1 ?4); [ Intros H_x | XAuto ];
+               LApply H_x; [ Clear H H_x; Intros H | XAuto ];
+               XElim H; Intros
+            | [ H: (lift (1) (0) ?1) = (lift (1) (S ?2) ?3) |- ? ] ->
+               LApply (lift_gen_lift ?1 ?3 (1) (1) (0) ?2); [ Intros H_x | XAuto ];
+               LApply H_x; [ Clear H_x H; Intros H | Arith7' ?2; XAuto ];
+               XElim H; Intros
+            | _ -> LiftGenBase.