milestone update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / etc / lpx_sn / lpx_sn_alt.etc

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_alt.etc
index 8a0aa39317b06207086a899c6bcb15b316a229f1..5099ffa1ab609fc85ac57736e4fdf3cb0e3b8c9d 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_alt.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_alt.etc
@@ -12,17 +12,17 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/relocation/ldrop.ma".
-include "basic_2/relocation/lpx_sn.ma".
+include "basic_2/substitution/drop.ma".
+include "basic_2/substitution/lpx_sn.ma".
 
 (* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
 
 (* alternative definition of lpx_sn *)
-definition lpx_sn_alt: relation4 bind2 lenv term term → relation lenv ≝
+definition lpx_sn_alt: relation3 lenv term term → relation lenv ≝
                        λR,L1,L2. |L1| = |L2| ∧
                        (∀I1,I2,K1,K2,V1,V2,i.
-                          â\87©[i] L1 â\89¡ K1.â\93\91{I1}V1 â\86\92 â\87©[i] L2 ≡ K2.ⓑ{I2}V2 →
-                          I1 = I2 ∧ R I1 K1 V1 V2
+                          â¬\87[i] L1 â\89¡ K1.â\93\91{I1}V1 â\86\92 â¬\87[i] L2 ≡ K2.ⓑ{I2}V2 →
+                          I1 = I2 ∧ R K1 V1 V2
                        ).
 
 (* Basic forward lemmas ******************************************************)
@@ -39,15 +39,15 @@ normalize /2 width=1 by length_inv_zero_sn/
 qed-.
 
 lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
-                            ∃∃K2,V2. lpx_sn_alt R K1 K2 & R I K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+                            ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
 #R #I1 #L2 #K1 #V1 #H elim H -H
 #H #IH elim (length_inv_pos_sn … H) -H
 #I2 #K2 #V2 #HK12 #H destruct
 elim (IH I1 I2 K1 K2 V1 V2 0) //
 #H #HV12 destruct @(ex3_2_intro … K2 V2) // -HV12
 @conj // -HK12
-#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
-/2 width=1 by ldrop_drop, conj/
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH
+/2 width=1 by drop_drop, conj/
 qed-.
 
 lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
@@ -56,36 +56,38 @@ normalize /2 width=1 by length_inv_zero_dx/
 qed-.
 
 lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
-                            ∃∃K1,V1. lpx_sn_alt R K1 K2 & R I K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+                            ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
 #R #I2 #L1 #K2 #V2 #H elim H -H
 #H #IH elim (length_inv_pos_dx … H) -H
 #I1 #K1 #V1 #HK12 #H destruct
 elim (IH I1 I2 K1 K2 V1 V2 0) //
 #H #HV12 destruct @(ex3_2_intro … K1 V1) // -HV12
 @conj // -HK12
-#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (i+1)) -IH
-/2 width=1 by ldrop_drop, conj/
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH
+/2 width=1 by drop_drop, conj/
 qed-.
 
 (* Basic properties *********************************************************)
 
 lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
 #R @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (ldrop_inv_atom1 … HLK1) -HLK1
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (drop_inv_atom1 … HLK1) -HLK1
 #H destruct
 qed.
 
 lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
-                       lpx_sn_alt R L1 L2 → R I L1 V1 V2 →
+                       lpx_sn_alt R L1 L2 → R L1 V1 V2 →
                        lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
 #R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H
-#HL12 #IH @conj normalize //
-#I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #W1 #W2 #i @(ynat_ind … i) -i
 [ #HLK1 #HLK2
-  lapply (ldrop_inv_O2 … HLK1) -HLK1 #H destruct
-  lapply (ldrop_inv_O2 … HLK2) -HLK2 #H destruct
+  lapply (drop_inv_O2 … HLK1) -HLK1 #H destruct
+  lapply (drop_inv_O2 … HLK2) -HLK2 #H destruct
   /2 width=1 by conj/
-| -HL12 -HV12 /3 width=6 by ldrop_inv_drop1/
+| -HL12 -HV12 /3 width=6 by drop_inv_drop1/
+| #H lapply (drop_fwd_Y2 … H) -H
+  #H elim (ylt_yle_false … H) -H //
 ]
 qed.
 
@@ -110,16 +112,16 @@ qed-.
 
 lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
                         (∀I1,I2,K1,K2,V1,V2,i.
-                           â\87©[i] L1 â\89¡ K1.â\93\91{I1}V1 â\86\92 â\87©[i] L2 ≡ K2.ⓑ{I2}V2 →
-                           I1 = I2 ∧ R I1 K1 V1 V2
+                           â¬\87[i] L1 â\89¡ K1.â\93\91{I1}V1 â\86\92 â¬\87[i] L2 ≡ K2.ⓑ{I2}V2 →
+                           I1 = I2 ∧ R K1 V1 V2
                         ) → lpx_sn R L1 L2.
 /4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed.
 
 lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
                       |L1| = |L2| ∧
                       ∀I1,I2,K1,K2,V1,V2,i.
-                      â\87©[i] L1 â\89¡ K1.â\93\91{I1}V1 â\86\92 â\87©[i] L2 ≡ K2.ⓑ{I2}V2 →
-                      I1 = I2 ∧ R I1 K1 V1 V2.
+                      â¬\87[i] L1 â\89¡ K1.â\93\91{I1}V1 â\86\92 â¬\87[i] L2 ≡ K2.ⓑ{I2}V2 →
+                      I1 = I2 ∧ R K1 V1 V2.
 #R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
 #H elim H -H /3 width=4 by conj/
 qed-.