- support for candidates of reducibility continues ...

[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 b4d3355d6bd4c38d249a9f9246b08343f37d18fe..c3219f4f46903d059f139f7179fc771beaf96f4c 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/ldrop.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ldrop.ma
@@ -20,19 +20,19 @@ include "Basic_2/substitution/lift.ma".
 
 (* Basic_1: includes: ldrop_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_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 â\86\92 â\86\91[d,e] V2 â\89¡ V1 â\86\92
-              ldrop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
+| ldrop_skip : ∀L1,L2,I,V1,V2,d,e.
+               ldrop d e L1 L2 â\86\92 â\87\91[d,e] V2 â\89¡ V1 â\86\92
+               ldrop (d + 1) e (L1. 𝕓{I} V1) (L2. 𝕓{I} V2)
 .
 
-interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
+interpretation "local slicing" 'RLDrop 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 â\89¡ L2 â\86\92 d = 0 â\86\92 e = 0 â\86\92 L1 = L2.
+fact ldrop_inv_refl_aux: â\88\80d,e,L1,L2. â\87\93[d, e] L1 â\89¡ L2 â\86\92 d = 0 â\86\92 e = 0 â\86\92 L1 = L2.
 #d #e #L1 #L2 * -d -e -L1 -L2
 [ //
 | //
@@ -42,10 +42,10 @@ fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 → e = 0 
 qed.
 
 (* Basic_1: was: ldrop_gen_refl *)
-lemma ldrop_inv_refl: â\88\80L1,L2. â\86\93[0, 0] L1 â\89¡ L2 â\86\92 L1 = L2.
+lemma ldrop_inv_refl: â\88\80L1,L2. â\87\93[0, 0] L1 â\89¡ L2 â\86\92 L1 = L2.
 /2 width=5/ qed-.
 
-fact ldrop_inv_atom1_aux: â\88\80d,e,L1,L2. â\86\93[d, e] L1 â\89¡ L2 â\86\92 L1 = â\8b\86 â\86\92
+fact ldrop_inv_atom1_aux: â\88\80d,e,L1,L2. â\87\93[d, e] L1 â\89¡ L2 â\86\92 L1 = â\8b\86 â\86\92
                           L2 = ⋆.
 #d #e #L1 #L2 * -d -e -L1 -L2
 [ //
@@ -56,13 +56,13 @@ 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] â\8b\86 â\89¡ L2 â\86\92 L2 = â\8b\86.
+lemma ldrop_inv_atom1: â\88\80d,e,L2. â\87\93[d, e] â\8b\86 â\89¡ L2 â\86\92 L2 = â\8b\86.
 /2 width=5/ qed-.
 
-fact ldrop_inv_O1_aux: â\88\80d,e,L1,L2. â\86\93[d, e] L1 â\89¡ L2 â\86\92 d = 0 â\86\92
+fact ldrop_inv_O1_aux: â\88\80d,e,L1,L2. â\87\93[d, e] L1 â\89¡ L2 â\86\92 d = 0 â\86\92
                        ∀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 â\89¡ L2).
+                       (0 < e â\88§ â\87\93[d, e - 1] K â\89¡ 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/
@@ -71,23 +71,23 @@ fact ldrop_inv_O1_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → d = 0 →
 ]
 qed.
 
-lemma ldrop_inv_O1: â\88\80e,K,I,V,L2. â\86\93[0, e] K. ð\9d\95\93{I} V â\89¡ L2 â\86\92
+lemma ldrop_inv_O1: â\88\80e,K,I,V,L2. â\87\93[0, e] K. ð\9d\95\93{I} V â\89¡ L2 â\86\92
                     (e = 0 ∧ L2 = K. 𝕓{I} V) ∨
-                    (0 < e â\88§ â\86\93[0, e - 1] K â\89¡ L2).
+                    (0 < e â\88§ â\87\93[0, e - 1] K â\89¡ L2).
 /2 width=3/ qed-.
 
 (* Basic_1: was: ldrop_gen_ldrop *)
 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 â\89¡ L2.
+                        â\87\93[0, e] K. ð\9d\95\93{I} V â\89¡ L2 â\86\92 0 < e â\86\92 â\87\93[0, e - 1] K â\89¡ 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 â\89¡ L2 â\86\92 0 < d â\86\92
+fact ldrop_inv_skip1_aux: â\88\80d,e,L1,L2. â\87\93[d, e] L1 â\89¡ L2 â\86\92 0 < d â\86\92
                           ∀I,K1,V1. L1 = K1. 𝕓{I} V1 →
-                          â\88\83â\88\83K2,V2. â\86\93[d - 1, e] K1 â\89¡ K2 &
-                                   â\86\91[d - 1, e] V2 â\89¡ V1 & 
+                          â\88\83â\88\83K2,V2. â\87\93[d - 1, e] K1 â\89¡ K2 &
+                                   â\87\91[d - 1, e] V2 â\89¡ V1 & 
                                    L2 = K2. 𝕓{I} V2.
 #d #e #L1 #L2 * -d -e -L1 -L2
 [ #d #e #_ #I #K #V #H destruct
@@ -98,16 +98,16 @@ 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 â\89¡ L2 â\86\92 0 < d â\86\92
-                       â\88\83â\88\83K2,V2. â\86\93[d - 1, e] K1 â\89¡ K2 &
-                                â\86\91[d - 1, e] V2 â\89¡ V1 & 
+lemma ldrop_inv_skip1: â\88\80d,e,I,K1,V1,L2. â\87\93[d, e] K1. ð\9d\95\93{I} V1 â\89¡ L2 â\86\92 0 < d â\86\92
+                       â\88\83â\88\83K2,V2. â\87\93[d - 1, e] K1 â\89¡ K2 &
+                                â\87\91[d - 1, e] V2 â\89¡ V1 & 
                                 L2 = K2. 𝕓{I} V2.
 /2 width=3/ qed-.
 
-fact ldrop_inv_skip2_aux: â\88\80d,e,L1,L2. â\86\93[d, e] L1 â\89¡ L2 â\86\92 0 < d â\86\92
+fact ldrop_inv_skip2_aux: â\88\80d,e,L1,L2. â\87\93[d, e] L1 â\89¡ L2 â\86\92 0 < d â\86\92
                           ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
-                          â\88\83â\88\83K1,V1. â\86\93[d - 1, e] K1 â\89¡ K2 &
-                                   â\86\91[d - 1, e] V2 â\89¡ V1 & 
+                          â\88\83â\88\83K1,V1. â\87\93[d - 1, e] K1 â\89¡ K2 &
+                                   â\87\91[d - 1, e] V2 â\89¡ V1 & 
                                    L1 = K1. 𝕓{I} V1.
 #d #e #L1 #L2 * -d -e -L1 -L2
 [ #d #e #_ #I #K #V #H destruct
@@ -118,28 +118,28 @@ 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 â\86\92 0 < d â\86\92
-                       â\88\83â\88\83K1,V1. â\86\93[d - 1, e] K1 â\89¡ K2 & â\86\91[d - 1, e] V2 â\89¡ V1 &
+lemma ldrop_inv_skip2: â\88\80d,e,I,L1,K2,V2. â\87\93[d, e] L1 â\89¡ K2. ð\9d\95\93{I} V2 â\86\92 0 < d â\86\92
+                       â\88\83â\88\83K1,V1. â\87\93[d - 1, e] K1 â\89¡ K2 & â\87\91[d - 1, e] V2 â\89¡ V1 &
                                 L1 = K1. 𝕓{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 â\89¡ L.
+lemma ldrop_refl: â\88\80L. â\87\93[0, 0] L â\89¡ 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 â\89¡ L2.
+                      â\87\93[0, e - 1] L1 â\89¡ L2 â\86\92 0 < e â\86\92 â\87\93[0, e] L1. ð\9d\95\93{I} V â\89¡ 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\86\92
+                               â\88\80K1,V,i. â\87\93[0, i] L1 â\89¡ K1. ð\9d\95\93{Abbr} V â\86\92
                                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\93[0, i] L2 â\89¡ K2. ð\9d\95\93{Abbr} V.
 #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
@@ -167,8 +167,8 @@ 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\92
-                        â\86\93[O, e + 1] L1 â\89¡ K2.
+lemma ldrop_fwd_ldrop2: â\88\80L1,I2,K2,V2,e. â\87\93[O, e] L1 â\89¡ K2. ð\9d\95\93{I2} V2 â\86\92
+                        â\87\93[O, e + 1] L1 â\89¡ 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
@@ -179,7 +179,7 @@ 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 â\89¡ L2 â\86\92 #[L2] â\89¤ #[L1].
+lemma ldrop_fwd_lw: â\88\80L1,L2,d,e. â\87\93[d, e] L1 â\89¡ L2 â\86\92 #[L2] â\89¤ #[L1].
 #L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize
 [ /2 width=3/
 | #L1 #L2 #I #V1 #V2 #d #e #_ #HV21 #IHL12
@@ -188,7 +188,7 @@ lemma ldrop_fwd_lw: ∀L1,L2,d,e. ↓[d, e] L1 ≡ L2 → #[L2] ≤ #[L1].
 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 â\86\92 e < |L1|.
+                               â\87\93[0, e] L1 â\89¡ K2. ð\9d\95\93{I2} V2 â\86\92 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
@@ -199,7 +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 â\89¡ L2 â\86\92 |L2| = |L1| - e.
+lemma ldrop_fwd_O1_length: â\88\80L1,L2,e. â\87\93[0, e] L1 â\89¡ L2 â\86\92 |L2| = |L1| - e.
 #L1 elim L1 -L1
 [ #L2 #e #H >(ldrop_inv_atom1 … H) -H //
 | #K1 #I1 #V1 #IHL1 #L2 #e #H