- transitivity of parallel telescopic substitution closed!

[helm.git] / matita / matita / lib / lambda-delta / substitution / drop_defs.ma

diff --git a/matita/matita/lib/lambda-delta/substitution/drop_defs.ma b/matita/matita/lib/lambda-delta/substitution/drop_defs.ma
index 1bf8e278d33ef4962ae8bfff4da40f094e74b7a8..51c657615b07456f5af74037852704952e751c31 100644 (file)
--- a/matita/matita/lib/lambda-delta/substitution/drop_defs.ma
+++ b/matita/matita/lib/lambda-delta/substitution/drop_defs.ma
@@ -9,7 +9,7 @@
      \ /
       V_______________________________________________________________ *)
 
-include "lambda-delta/syntax/lenv.ma".
+include "lambda-delta/substitution/leq_defs.ma".
 include "lambda-delta/substitution/lift_defs.ma".
 
 (* DROPPING *****************************************************************)
@@ -24,21 +24,14 @@ inductive drop: lenv → nat → nat → lenv → Prop ≝
 
 interpretation "dropping" 'RDrop L1 d e L2 = (drop L1 d e L2).
 
-(* Basic properties *********************************************************) 
-
-lemma drop_drop_lt: ∀L1,L2,I,V,e. 
-                    ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
-#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
-qed.
-
 (* 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
+#d #e #L1 #L2 * -d e L1 L2
 [ //
-| #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.
@@ -46,14 +39,27 @@ qed.
 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 = ⋆ & d = 0 & e = 0.
+#d #e #L1 #L2 * -d e L1 L2
+[ /2/
+| #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 = ⋆ & d = 0 & e = 0.
+/2/ 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).
-#d #e #L1 #L2 #H elim H -H d e L1 L2
+#d #e #L1 #L2 * -d e L1 L2
 [ /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)
+| #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)
 ]
 qed.
 
@@ -74,10 +80,10 @@ lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
                           ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
                                    ↑[d - 1, e] V2 ≡ V1 & 
                                    L1 = K1. 𝕓{I} V1.
-#d #e #L1 #L2 #H elim H -H d e L1 L2
+#d #e #L1 #L2 * -d e L1 L2
 [ #L #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;
+| #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.
@@ -86,3 +92,46 @@ lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 <
                       ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & ↑[d - 1, e] V2 ≡ V1 &
                                L1 = K1. 𝕓{I} V1.
 /2/ qed.
+
+(* Basic properties *********************************************************)
+
+lemma drop_drop_lt: ∀L1,L2,I,V,e.
+                    ↓[0, e - 1] L1 ≡ L2 → 0 < e → ↓[0, e] L1. 𝕓{I} V ≡ L2.
+#L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) /2/
+qed.
+
+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 elim (drop_inv_sort1 … 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/
+  | @drop_drop >(plus_minus_m_m e 1) /2/
+  ]
+]
+qed.
+
+lemma drop_leq_drop1: ∀L1,L2,d,e. L1 [d, e] ≈ L2 →
+                      ∀I,K1,V,i. ↓[0, i] L1 ≡ K1. 𝕓{I} V →
+                      d ≤ i → i < d + e →
+                      ∃∃K2. K1 [0, d + e - i - 1] ≈ K2 &
+                            ↓[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
+  elim (drop_inv_sort1 … H) -H #H destruct
+| #L1 #L2 #I1 #I2 #V1 #V2 #_ #_ #I #K1 #V #i #_ #_ #H
+  elim (lt_zero_false … H)
+| #L1 #L2 #I #V #e #HL12 #IHL12 #J #K1 #W #i #H #_ #Hie
+  elim (drop_inv_O1 … H) -H * #Hi #HLK1
+  [ -IHL12 Hie; destruct -i K1 J W;
+    <minus_n_O <minus_plus_m_m /2/
+  | -HL12;
+    elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 // [2: /2/ ] -Hie >arith_g1 // /3/
+  ]
+| #L1 #L2 #I1 #I2 #V1 #V2 #d #e #_ #IHL12 #I #K1 #V #i #H #Hdi >plus_plus_comm_23 #Hide
+  lapply (plus_S_le_to_pos … Hdi) #Hi
+  lapply (drop_inv_drop1 … H ?) -H // #HLK1
+  elim (IHL12 … HLK1 ? ?) -IHL12 HLK1 [2: /2/ |3: /2/ ] -Hdi Hide >arith_g1 // /3/
+]
+qed.