- some work on extensions:

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / relocation / lpx_sn_alt.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_alt.ma
index c9c381a32b300fc7456b6736ea4d4a662216450b..1fb1858b5ab28637ff7cda12f80262831e6a4493 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lpx_sn_alt.ma
@@ -17,7 +17,7 @@ include "basic_2/relocation/lpx_sn.ma".
 
 (* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
 
-(* alternative definition of lpx_sn_alt *)
+(* alternative definition of lpx_sn *)
 inductive lpx_sn_alt (R:relation3 lenv term term): relation lenv ≝
 | lpx_sn_alt_intro: ∀L1,L2.
                     (∀I1,I2,K1,K2,V1,V2,i.
@@ -28,25 +28,50 @@ inductive lpx_sn_alt (R:relation3 lenv term term): relation lenv ≝
                     ) → |L1| = |L2| → lpx_sn_alt R L1 L2
 .
 
-(* Basic forward lemmas ******************************************************)
-
-lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
-#R #L1 #L2 * -L1 -L2 //
+(* compact definition of lpx_sn_alt *****************************************)
+
+lemma lpx_sn_alt_ind_alt: ∀R. ∀S:relation lenv.
+                          (∀L1,L2. |L1| = |L2| → (
+                             ∀I1,I2,K1,K2,V1,V2,i.
+                             ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                             ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2 & S K1 K2
+                          ) → S L1 L2) →
+                          ∀L1,L2. lpx_sn_alt R L1 L2 → S L1 L2.
+#R #S #IH #L1 #L2 #H elim H -L1 -L2
+#L1 #L2 #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (H1 … HLK1 HLK2) -H1
+/3 width=7 by and4_intro/
 qed-.
 
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpx_sn_alt_inv_gen: ∀R,L1,L2. lpx_sn_alt R L1 L2 →
+lemma lpx_sn_alt_inv_alt: ∀R,L1,L2. lpx_sn_alt R L1 L2 →
                           |L1| = |L2| ∧
                           ∀I1,I2,K1,K2,V1,V2,i.
                           ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
                           ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2.
-#R #L1 #L2 * -L1 -L2
-#L1 #L2 #IH1 #IH2 #HL12 @conj //
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #HLK1 #HLK2
-elim (IH1 … HLK1 HLK2) -IH1 /3 width=7 by and3_intro/
+#R #L1 #L2 #H @(lpx_sn_alt_ind_alt … H) -L1 -L2
+#L1 #L2 #HL12 #IH @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
+/2 width=1 by and3_intro/
 qed-.
 
+lemma lpx_sn_alt_intro_alt: ∀R,L1,L2. |L1| = |L2| →
+                            (∀I1,I2,K1,K2,V1,V2,i.
+                               ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                               ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2
+                            ) → lpx_sn_alt R L1 L2.
+#R #L1 #L2 #HL12 #IH @lpx_sn_alt_intro // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by conj/
+qed.
+
+(* Basic forward lemmas ******************************************************)
+
+lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim (lpx_sn_alt_inv_alt … H) //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
 lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
 #R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
 normalize /2 width=1 by length_inv_zero_sn/
@@ -54,7 +79,7 @@ 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 K1 V1 V2 & L2 = K2.ⓑ{I}V2.
-#R #I1 #L2 #K1 #V1 #H elim (lpx_sn_alt_inv_gen … H) -H
+#R #I1 #L2 #K1 #V1 #H elim (lpx_sn_alt_inv_alt … 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) -IH /2 width=5 by ex3_2_intro/
@@ -67,7 +92,7 @@ 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 K1 V1 V2 & L1 = K1.ⓑ{I}V1.
-#R #I2 #L1 #K2 #V2 #H elim (lpx_sn_alt_inv_gen … H) -H
+#R #I2 #L1 #K2 #V2 #H elim (lpx_sn_alt_inv_alt … 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) -IH /2 width=5 by ex3_2_intro/
@@ -75,16 +100,6 @@ qed-.
 
 (* Basic properties *********************************************************)
 
-lemma lpx_sn_alt_intro_alt: ∀R,L1,L2. |L1| = |L2| →
-                            (∀I1,I2,K1,K2,V1,V2,i.
-                               ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                               ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn_alt R K1 K2
-                            ) → lpx_sn_alt R L1 L2.
-#R #L1 #L2 #HL12 #IH @lpx_sn_alt_intro // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
-elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by conj/
-qed.
-
 lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
 #R @lpx_sn_alt_intro_alt //
 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (ldrop_inv_atom1 … HLK1) -HLK1
@@ -94,7 +109,7 @@ qed.
 lemma lpx_sn_alt_pair: ∀R,I,L1,L2,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 (lpx_sn_alt_inv_gen … H) -H
+#R #I #L1 #L2 #V1 #V2 #H #HV12 elim (lpx_sn_alt_inv_alt … H) -H
 #HL12 #IH @lpx_sn_alt_intro_alt normalize //
 #I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
 [ #HLK1 #HLK2
@@ -122,7 +137,7 @@ theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2
 ]
 qed-.
 
-(* Advanced properties ******************************************************)
+(* alternative definition of lpx_sn *****************************************)
 
 lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
                         (∀I1,I2,K1,K2,V1,V2,i.
@@ -135,15 +150,27 @@ lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=1 by lpx_sn_lpx_sn_alt, and3_intro/
 qed.
 
-(* Advanced inversion lemmas ************************************************)
+lemma lpx_sn_ind_alt: ∀R. ∀S:relation lenv.
+                      (∀L1,L2. |L1| = |L2| → (
+                         ∀I1,I2,K1,K2,V1,V2,i.
+                         ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                         ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn R K1 K2 & S K1 K2
+                      ) → S L1 L2) →
+                      ∀L1,L2. lpx_sn R L1 L2 → S L1 L2.
+#R #S #IH1 #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
+#H @(lpx_sn_alt_ind_alt … H) -L1 -L2
+#L1 #L2 #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (IH2 … HLK1 HLK2) -IH2 -HLK1 -HLK2
+/3 width=1 by lpx_sn_alt_inv_lpx_sn, and4_intro/
+qed-.
 
-lemma lpx_sn_inv_gen: ∀R,L1,L2. lpx_sn R L1 L2 →
+lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
                       |L1| = |L2| ∧
                       ∀I1,I2,K1,K2,V1,V2,i.
                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
                       ∧∧ I1 = I2 & R K1 V1 V2 & lpx_sn R K1 K2.
 #R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
-#H elim (lpx_sn_alt_inv_gen … H) -H
+#H elim (lpx_sn_alt_inv_alt … H) -H
 #HL12 #IH @conj //
 #I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=1 by lpx_sn_alt_inv_lpx_sn, and3_intro/