- support for transitive closures started

[helm.git] / matita / matita / contribs / lambda-delta / Basic-2 / substitution / drop.ma

diff --git a/matita/matita/contribs/lambda-delta/Basic-2/substitution/drop.ma b/matita/matita/contribs/lambda-delta/Basic-2/substitution/drop.ma
index a6752fac6bb48c4f7524dbf1b1809a4010f77b1e..4d31115b276a340c00b21686bb0f014175824e2c 100644 (file)
--- a/matita/matita/contribs/lambda-delta/Basic-2/substitution/drop.ma
+++ b/matita/matita/contribs/lambda-delta/Basic-2/substitution/drop.ma
@@ -12,60 +12,61 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Basic-2/substitution/leq.ma".
+include "Basic-2/grammar/leq.ma".
 include "Basic-2/substitution/lift.ma".
 
 (* DROPPING *****************************************************************)
 
-inductive drop: lenv → nat → nat → lenv → Prop ≝
-| drop_sort: ∀d,e. drop (⋆) d e (⋆)
-| drop_comp: ∀L1,L2,I,V. drop L1 0 0 L2 → drop (L1. 𝕓{I} V) 0 0 (L2. 𝕓{I} V)
-| drop_drop: ∀L1,L2,I,V,e. drop L1 0 e L2 → drop (L1. 𝕓{I} V) 0 (e + 1) L2
+(* Basic-1: includes: drop_skip_bind *)
+inductive drop: nat → nat → relation lenv ≝
+| drop_atom: ∀d,e. drop d e (⋆) (⋆)
+| drop_pair: ∀L,I,V. drop 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V)
+| drop_drop: ∀L1,L2,I,V,e. drop 0 e L1 L2 → drop 0 (e + 1) (L1. 𝕓{I} V) L2
 | drop_skip: ∀L1,L2,I,V1,V2,d,e.
-             drop L1 d e L2 → ↑[d,e] V2 ≡ V1 →
-             drop (L1. 𝕓{I} V1) (d + 1) e (L2. 𝕓{I} V2)
+             drop d e L1 L2 → ↑[d,e] V2 ≡ V1 →
+             drop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
 .
 
-interpretation "dropping" 'RDrop L1 d e L2 = (drop L1 d e L2).
+interpretation "dropping" 'RDrop d e L1 L2 = (drop d e L1 L2).
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
-#d #e #L1 #L2 #H elim H -H d e L1 L2
+fact drop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
+#d #e #L1 #L2 * -d e L1 L2
 [ //
-| #L1 #L2 #I #V #_ #IHL12 #H1 #H2
-  >(IHL12 H1 H2) -IHL12 H1 H2 L1 //
-| #L1 #L2 #I #V #e #_ #_ #_ #H
+| //
+| #L1 #L2 #I #V #e #_ #_ #H
   elim (plus_S_eq_O_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #_ #H
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H
   elim (plus_S_eq_O_false … H)
 ]
 qed.
 
+(* Basic-1: was: drop_gen_refl *)
 lemma drop_inv_refl: ∀L1,L2. ↓[0, 0] L1 ≡ L2 → L1 = L2.
 /2 width=5/ qed.
 
-lemma drop_inv_sort1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
-                          L2 = ⋆.
+fact drop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
+                         L2 = ⋆.
 #d #e #L1 #L2 * -d e L1 L2
 [ //
-| #L1 #L2 #I #V #_ #H destruct
+| #L #I #V #H destruct
 | #L1 #L2 #I #V #e #_ #H destruct
 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H destruct
 ]
 qed.
 
-lemma drop_inv_sort1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
+(* Basic-1: was: drop_gen_sort *)
+lemma drop_inv_atom1: ∀d,e,L2. ↓[d, e] ⋆ ≡ L2 → L2 = ⋆.
 /2 width=5/ qed.
 
-lemma drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
-                       ∀K,I,V. L1 = K. 𝕓{I} V → 
-                       (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
-                       (0 < e ∧ ↓[d, e - 1] K ≡ L2).
+fact drop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
+                      ∀K,I,V. L1 = K. 𝕓{I} V → 
+                      (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
+                      (0 < e ∧ ↓[d, e - 1] K ≡ L2).
 #d #e #L1 #L2 * -d e L1 L2
 [ #d #e #_ #K #I #V #H destruct
-| #L1 #L2 #I #V #HL12 #H #K #J #W #HX destruct -L1 I V
-  >(drop_inv_refl … HL12) -HL12 K /3/
+| #L #I #V #_ #K #J #W #HX destruct -L I V /3/
 | #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct -L1 I V /3/
 | #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
 ]
@@ -76,6 +77,7 @@ lemma drop_inv_O1: ∀e,K,I,V,L2. ↓[0, e] K. 𝕓{I} V ≡ L2 →
                    (0 < e ∧ ↓[0, e - 1] K ≡ L2).
 /2/ qed.
 
+(* Basic-1: was: drop_gen_drop *)
 lemma drop_inv_drop1: ∀e,K,I,V,L2.
                       ↓[0, e] K. 𝕓{I} V ≡ L2 → 0 < e → ↓[0, e - 1] K ≡ L2.
 #e #K #I #V #L2 #H #He
@@ -83,40 +85,42 @@ elim (drop_inv_O1 … H) -H * // #H destruct -e;
 elim (lt_refl_false … He)
 qed.
 
-lemma drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
-                          ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
-                          ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
-                                   ↑[d - 1, e] V2 ≡ V1 & 
-                                   L2 = K2. 𝕓{I} V2.
+fact drop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+                         ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
+                         ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
+                                  ↑[d - 1, e] V2 ≡ V1 & 
+                                  L2 = K2. 𝕓{I} V2.
 #d #e #L1 #L2 * -d e L1 L2
 [ #d #e #_ #I #K #V #H destruct
-| #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
+| #L #I #V #H elim (lt_refl_false … H)
 | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
 | #X #L2 #Y #Z #V2 #d #e #HL12 #HV12 #_ #I #L1 #V1 #H destruct -X Y Z
   /2 width=5/
 ]
 qed.
 
+(* Basic-1: was: drop_gen_skip_l *)
 lemma drop_inv_skip1: ∀d,e,I,K1,V1,L2. ↓[d, e] K1. 𝕓{I} V1 ≡ L2 → 0 < d →
                       ∃∃K2,V2. ↓[d - 1, e] K1 ≡ K2 &
                                ↑[d - 1, e] V2 ≡ V1 & 
                                L2 = K2. 𝕓{I} V2.
 /2/ qed.
 
-lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
-                          ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
-                          ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
-                                   ↑[d - 1, e] V2 ≡ V1 & 
-                                   L1 = K1. 𝕓{I} V1.
+fact drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+                         ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
+                         ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
+                                  ↑[d - 1, e] V2 ≡ V1 & 
+                                  L1 = K1. 𝕓{I} V1.
 #d #e #L1 #L2 * -d e L1 L2
 [ #d #e #_ #I #K #V #H destruct
-| #L1 #L2 #I #V #_ #H elim (lt_refl_false … H)
+| #L #I #V #H elim (lt_refl_false … H)
 | #L1 #L2 #I #V #e #_ #H elim (lt_refl_false … H)
 | #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #I #L2 #V2 #H destruct -X Y Z
   /2 width=5/
 ]
 qed.
 
+(* Basic-1: was: drop_gen_skip_r *)
 lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
                       ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
                                L1 = K1. 𝕓{I} V1.
@@ -124,8 +128,9 @@ lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 <
 
 (* Basic properties *********************************************************)
 
+(* Basic-1: was by definition: drop_refl *)
 lemma drop_refl: ∀L. ↓[0, 0] L ≡ L.
-#L elim L -L /2/
+#L elim L -L //
 qed.
 
 lemma drop_drop_lt: ∀L1,L2,I,V,e.
@@ -140,7 +145,7 @@ lemma drop_leq_drop1: ∀L1,L2,d,e. L1 [d, e] ≈ L2 →
                             ↓[0, i] L2 ≡ K2. 𝕓{I} V.
 #L1 #L2 #d #e #H elim H -H L1 L2 d e
 [ #d #e #I #K1 #V #i #H
-  lapply (drop_inv_sort1 … H) -H #H destruct
+  lapply (drop_inv_atom1 … H) -H #H destruct
 | #L1 #L2 #I #K1 #V #i #_ #_ #H
   elim (lt_zero_false … H)
 | #L1 #L2 #I #V #e #HL12 #IHL12 #J #K1 #W #i #H #_ #Hie
@@ -159,10 +164,11 @@ qed.
 
 (* Basic forvard lemmas *****************************************************)
 
+(* Basic-1: was: drop_S *)
 lemma drop_fwd_drop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
                       ↓[O, e + 1] L1 ≡ K2.
 #L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
+[ #I2 #K2 #V2 #e #H lapply (drop_inv_atom1 … H) -H #H destruct
 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
   elim (drop_inv_O1 … H) -H * #He #H
   [ -IHL1; destruct -e K2 I2 V2 /2/
@@ -174,7 +180,7 @@ qed.
 lemma drop_fwd_drop2_length: ∀L1,I2,K2,V2,e. 
                              ↓[0, e] L1 ≡ K2. 𝕓{I2} V2 → e < |L1|.
 #L1 elim L1 -L1
-[ #I2 #K2 #V2 #e #H lapply (drop_inv_sort1 … H) -H #H destruct
+[ #I2 #K2 #V2 #e #H lapply (drop_inv_atom1 … H) -H #H destruct
 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
   elim (drop_inv_O1 … H) -H * #He #H
   [ -IHL1; destruct -e K2 I2 V2 //
@@ -185,7 +191,7 @@ qed.
 
 lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
 #L1 elim L1 -L1
-[ #L2 #e #H >(drop_inv_sort1 … H) -H //
+[ #L2 #e #H >(drop_inv_atom1 … H) -H //
 | #K1 #I1 #V1 #IHL1 #L2 #e #H
   elim (drop_inv_O1 … H) -H * #He #H
   [ -IHL1; destruct -e L2 //
@@ -194,3 +200,18 @@ lemma drop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
   ]
 ]
 qed.
+
+(* Basic-1: removed theorems 49:
+            drop_skip_flat
+            cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
+            drop_clear drop_clear_O drop_clear_S
+            clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
+            clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
+            getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
+            getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
+            getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
+            drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
+            getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+            getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
+            getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
+*)