- property S6 of stronfly normalizing terms proved

[helm.git] / matita / matita / contribs / lambda_delta / Basic_2 / substitution / ldrop.ma

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/ldrop.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ldrop.ma
index 2b8f3f9535ba442e2fda7f3e7cdfb4a9aa7e606d..c00819a7454644157340366ca2dd9e59c0f2bef8 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/ldrop.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ldrop.ma
@@ -18,36 +18,34 @@ include "Basic_2/substitution/lift.ma".
 
 (* LOCAL ENVIRONMENT SLICING ************************************************)
 
-(* Basic_1: includes: ldrop_skip_bind *)
+(* Basic_1: includes: drop_skip_bind *)
 inductive ldrop: nat → nat → relation lenv ≝
-| ldrop_atom: ∀d,e. ldrop d e (⋆) (⋆)
-| ldrop_pair: ∀L,I,V. ldrop 0 0 (L. 𝕓{I} V) (L. 𝕓{I} V)
-| ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. 𝕓{I} V) L2
-| ldrop_skip: ∀L1,L2,I,V1,V2,d,e.
-              ldrop d e L1 L2 → ↑[d,e] V2 ≡ V1 →
-              ldrop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
+| ldrop_atom : ∀d,e. ldrop d e (⋆) (⋆)
+| ldrop_pair : ∀L,I,V. ldrop 0 0 (L. ⓑ{I} V) (L. ⓑ{I} V)
+| ldrop_ldrop: ∀L1,L2,I,V,e. ldrop 0 e L1 L2 → ldrop 0 (e + 1) (L1. ⓑ{I} V) L2
+| ldrop_skip : ∀L1,L2,I,V1,V2,d,e.
+               ldrop d e L1 L2 → ⇧[d,e] V2 ≡ V1 →
+               ldrop (d + 1) e (L1. ⓑ{I} V1) (L2. ⓑ{I} V2)
 .
 
-interpretation "ldropping" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
+interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
 
 (* Basic inversion lemmas ***************************************************)
 
-fact ldrop_inv_refl_aux: â\88\80d,e,L1,L2. â\86\93[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
+fact ldrop_inv_refl_aux: â\88\80d,e,L1,L2. â\87©[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
 #d #e #L1 #L2 * -d -e -L1 -L2
 [ //
 | //
-| #L1 #L2 #I #V #e #_ #_ #H
-  elim (plus_S_eq_O_false … H)
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H
-  elim (plus_S_eq_O_false … H)
+| #L1 #L2 #I #V #e #_ #_ >commutative_plus normalize #H destruct
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
 ]
 qed.
 
-(* Basic_1: was: ldrop_gen_refl *)
-lemma ldrop_inv_refl: â\88\80L1,L2. â\86\93[0, 0] L1 ≡ L2 → L1 = L2.
+(* Basic_1: was: drop_gen_refl *)
+lemma ldrop_inv_refl: â\88\80L1,L2. â\87©[0, 0] L1 ≡ L2 → L1 = L2.
 /2 width=5/ qed-.
 
-fact ldrop_inv_atom1_aux: â\88\80d,e,L1,L2. â\86\93[d, e] L1 ≡ L2 → L1 = ⋆ →
+fact ldrop_inv_atom1_aux: â\88\80d,e,L1,L2. â\87©[d, e] L1 ≡ L2 → L1 = ⋆ →
                           L2 = ⋆.
 #d #e #L1 #L2 * -d -e -L1 -L2
 [ //
@@ -57,40 +55,40 @@ fact ldrop_inv_atom1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → L1 = ⋆ →
 ]
 qed.
 
-(* Basic_1: was: ldrop_gen_sort *)
-lemma ldrop_inv_atom1: â\88\80d,e,L2. â\86\93[d, e] ⋆ ≡ L2 → L2 = ⋆.
+(* Basic_1: was: drop_gen_sort *)
+lemma ldrop_inv_atom1: â\88\80d,e,L2. â\87©[d, e] ⋆ ≡ L2 → L2 = ⋆.
 /2 width=5/ qed-.
 
-fact ldrop_inv_O1_aux: â\88\80d,e,L1,L2. â\86\93[d, e] L1 ≡ L2 → d = 0 →
-                       ∀K,I,V. L1 = K. 𝕓{I} V → 
-                       (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
-                       (0 < e â\88§ â\86\93[d, e - 1] K ≡ L2).
+fact ldrop_inv_O1_aux: â\88\80d,e,L1,L2. â\87©[d, e] L1 ≡ L2 → d = 0 →
+                       ∀K,I,V. L1 = K. ⓑ{I} V → 
+                       (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
+                       (0 < e â\88§ â\87©[d, e - 1] K ≡ L2).
 #d #e #L1 #L2 * -d -e -L1 -L2
 [ #d #e #_ #K #I #V #H destruct
 | #L #I #V #_ #K #J #W #HX destruct /3 width=1/
 | #L1 #L2 #I #V #e #HL12 #_ #K #J #W #H destruct /3 width=1/
-| #L1 #L2 #I #V1 #V2 #d #e #_ #_ #H elim (plus_S_eq_O_false … H)
+| #L1 #L2 #I #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
 ]
 qed.
 
-lemma ldrop_inv_O1: â\88\80e,K,I,V,L2. â\86\93[0, e] K. ð\9d\95\93{I} V ≡ L2 →
-                    (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
-                    (0 < e â\88§ â\86\93[0, e - 1] K ≡ L2).
+lemma ldrop_inv_O1: â\88\80e,K,I,V,L2. â\87©[0, e] K. â\93\91{I} V ≡ L2 →
+                    (e = 0 ∧ L2 = K. ⓑ{I} V) ∨
+                    (0 < e â\88§ â\87©[0, e - 1] K ≡ L2).
 /2 width=3/ qed-.
 
-(* Basic_1: was: ldrop_gen_ldrop *)
+(* Basic_1: was: drop_gen_drop *)
 lemma ldrop_inv_ldrop1: ∀e,K,I,V,L2.
-                        â\86\93[0, e] K. ð\9d\95\93{I} V â\89¡ L2 â\86\92 0 < e â\86\92 â\86\93[0, e - 1] K ≡ L2.
+                        â\87©[0, e] K. â\93\91{I} V â\89¡ L2 â\86\92 0 < e â\86\92 â\87©[0, e - 1] K ≡ L2.
 #e #K #I #V #L2 #H #He
 elim (ldrop_inv_O1 … H) -H * // #H destruct
 elim (lt_refl_false … He)
 qed-.
 
-fact ldrop_inv_skip1_aux: â\88\80d,e,L1,L2. â\86\93[d, e] L1 ≡ L2 → 0 < d →
-                          ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
-                          â\88\83â\88\83K2,V2. â\86\93[d - 1, e] K1 ≡ K2 &
-                                   â\86\91[d - 1, e] V2 ≡ V1 & 
-                                   L2 = K2. 𝕓{I} V2.
+fact ldrop_inv_skip1_aux: â\88\80d,e,L1,L2. â\87©[d, e] L1 ≡ L2 → 0 < d →
+                          ∀I,K1,V1. L1 = K1. ⓑ{I} V1 →
+                          â\88\83â\88\83K2,V2. â\87©[d - 1, e] K1 ≡ K2 &
+                                   â\87§[d - 1, e] V2 ≡ V1 & 
+                                   L2 = K2. ⓑ{I} V2.
 #d #e #L1 #L2 * -d -e -L1 -L2
 [ #d #e #_ #I #K #V #H destruct
 | #L #I #V #H elim (lt_refl_false … H)
@@ -99,18 +97,18 @@ fact ldrop_inv_skip1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
 ]
 qed.
 
-(* Basic_1: was: ldrop_gen_skip_l *)
-lemma ldrop_inv_skip1: â\88\80d,e,I,K1,V1,L2. â\86\93[d, e] K1. ð\9d\95\93{I} V1 ≡ L2 → 0 < d →
-                       â\88\83â\88\83K2,V2. â\86\93[d - 1, e] K1 ≡ K2 &
-                                â\86\91[d - 1, e] V2 ≡ V1 & 
-                                L2 = K2. 𝕓{I} V2.
+(* Basic_1: was: drop_gen_skip_l *)
+lemma ldrop_inv_skip1: â\88\80d,e,I,K1,V1,L2. â\87©[d, e] K1. â\93\91{I} V1 ≡ L2 → 0 < d →
+                       â\88\83â\88\83K2,V2. â\87©[d - 1, e] K1 ≡ K2 &
+                                â\87§[d - 1, e] V2 ≡ V1 & 
+                                L2 = K2. ⓑ{I} V2.
 /2 width=3/ qed-.
 
-fact ldrop_inv_skip2_aux: â\88\80d,e,L1,L2. â\86\93[d, e] L1 ≡ L2 → 0 < d →
-                          ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
-                          â\88\83â\88\83K1,V1. â\86\93[d - 1, e] K1 ≡ K2 &
-                                   â\86\91[d - 1, e] V2 ≡ V1 & 
-                                   L1 = K1. 𝕓{I} V1.
+fact ldrop_inv_skip2_aux: â\88\80d,e,L1,L2. â\87©[d, e] L1 ≡ L2 → 0 < d →
+                          ∀I,K2,V2. L2 = K2. ⓑ{I} V2 →
+                          â\88\83â\88\83K1,V1. â\87©[d - 1, e] K1 ≡ K2 &
+                                   â\87§[d - 1, e] V2 ≡ V1 & 
+                                   L1 = K1. ⓑ{I} V1.
 #d #e #L1 #L2 * -d -e -L1 -L2
 [ #d #e #_ #I #K #V #H destruct
 | #L #I #V #H elim (lt_refl_false … H)
@@ -119,29 +117,29 @@ fact ldrop_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
 ]
 qed.
 
-(* Basic_1: was: ldrop_gen_skip_r *)
-lemma ldrop_inv_skip2: â\88\80d,e,I,L1,K2,V2. â\86\93[d, e] L1 â\89¡ K2. ð\9d\95\93{I} V2 → 0 < d →
-                       â\88\83â\88\83K1,V1. â\86\93[d - 1, e] K1 â\89¡ K2 & â\86\91[d - 1, e] V2 ≡ V1 &
-                                L1 = K1. 𝕓{I} V1.
+(* Basic_1: was: drop_gen_skip_r *)
+lemma ldrop_inv_skip2: â\88\80d,e,I,L1,K2,V2. â\87©[d, e] L1 â\89¡ K2. â\93\91{I} V2 → 0 < d →
+                       â\88\83â\88\83K1,V1. â\87©[d - 1, e] K1 â\89¡ K2 & â\87§[d - 1, e] V2 ≡ V1 &
+                                L1 = K1. ⓑ{I} V1.
 /2 width=3/ qed-.
 
 (* Basic properties *********************************************************)
 
-(* Basic_1: was by definition: ldrop_refl *)
-lemma ldrop_refl: â\88\80L. â\86\93[0, 0] L ≡ L.
+(* Basic_1: was by definition: drop_refl *)
+lemma ldrop_refl: â\88\80L. â\87©[0, 0] L ≡ L.
 #L elim L -L //
 qed.
 
 lemma ldrop_ldrop_lt: ∀L1,L2,I,V,e.
-                      â\86\93[0, e - 1] L1 â\89¡ L2 â\86\92 0 < e â\86\92 â\86\93[0, e] L1. ð\9d\95\93{I} V ≡ L2.
+                      â\87©[0, e - 1] L1 â\89¡ L2 â\86\92 0 < e â\86\92 â\87©[0, e] L1. â\93\91{I} V ≡ L2.
 #L1 #L2 #I #V #e #HL12 #He >(plus_minus_m_m e 1) // /2 width=1/
 qed.
 
 lemma ldrop_lsubs_ldrop1_abbr: ∀L1,L2,d,e. L1 [d, e] ≼ L2 →
-                               â\88\80K1,V,i. â\86\93[0, i] L1 â\89¡ K1. ð\9d\95\93{Abbr} V →
+                               â\88\80K1,V,i. â\87©[0, i] L1 â\89¡ K1. â\93\93V →
                                d ≤ i → i < d + e →
                                ∃∃K2. K1 [0, d + e - i - 1] ≼ K2 &
-                                     â\86\93[0, i] L2 â\89¡ K2. ð\9d\95\93{Abbr} V.
+                                     â\87©[0, i] L2 â\89¡ K2. â\93\93V.
 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e
 [ #d #e #K1 #V #i #H
   lapply (ldrop_inv_atom1 … H) -H #H destruct
@@ -168,9 +166,9 @@ qed.
 
 (* Basic forvard lemmas *****************************************************)
 
-(* Basic_1: was: ldrop_S *)
-lemma ldrop_fwd_ldrop2: â\88\80L1,I2,K2,V2,e. â\86\93[O, e] L1 â\89¡ K2. ð\9d\95\93{I2} V2 →
-                        â\86\93[O, e + 1] L1 ≡ K2.
+(* Basic_1: was: drop_S *)
+lemma ldrop_fwd_ldrop2: â\88\80L1,I2,K2,V2,e. â\87©[O, e] L1 â\89¡ K2. â\93\91{I2} V2 →
+                        â\87©[O, e + 1] L1 ≡ K2.
 #L1 elim L1 -L1
 [ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
@@ -181,16 +179,16 @@ lemma ldrop_fwd_ldrop2: ∀L1,I2,K2,V2,e. ↓[O, e] L1 ≡ K2. 𝕓{I2} V2 →
 ]
 qed-.
 
-lemma ldrop_fwd_lw: â\88\80L1,L2,d,e. â\86\93[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
+lemma ldrop_fwd_lw: â\88\80L1,L2,d,e. â\87©[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize
-[ /2 width=1/
+[ /2 width=3/
 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
   >(tw_lift … HV21) -HV21 /2 width=1/
 ]
 qed-. 
 
 lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
-                               â\86\93[0, e] L1 â\89¡ K2. ð\9d\95\93{I2} V2 → e < |L1|.
+                               â\87©[0, e] L1 â\89¡ K2. â\93\91{I2} V2 → e < |L1|.
 #L1 elim L1 -L1
 [ #I2 #K2 #V2 #e #H lapply (ldrop_inv_atom1 … H) -H #H destruct
 | #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #e #H
@@ -201,7 +199,7 @@ lemma ldrop_fwd_ldrop2_length: ∀L1,I2,K2,V2,e.
 ]
 qed-.
 
-lemma ldrop_fwd_O1_length: â\88\80L1,L2,e. â\86\93[0, e] L1 ≡ L2 → |L2| = |L1| - e.
+lemma ldrop_fwd_O1_length: â\88\80L1,L2,e. â\87©[0, e] L1 ≡ L2 → |L2| = |L1| - e.
 #L1 elim L1 -L1
 [ #L2 #e #H >(ldrop_inv_atom1 … H) -H //
 | #K1 #I1 #V1 #IHL1 #L2 #e #H
@@ -214,16 +212,16 @@ lemma ldrop_fwd_O1_length: ∀L1,L2,e. ↓[0, e] L1 ≡ L2 → |L2| = |L1| - e.
 qed-.
 
 (* Basic_1: removed theorems 49:
-            ldrop_skip_flat
+            drop_skip_flat
             cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
-            ldrop_clear ldrop_clear_O ldrop_clear_S
+            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_ldrop getl_ldrop_conf_lt
-            getl_ldrop_conf_ge getl_conf_ge_ldrop getl_ldrop_conf_rev
-            ldrop_getl_trans_lt ldrop_getl_trans_le ldrop_getl_trans_ge
-            getl_ldrop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+            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
 *)