X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frelocation%2Fllpx_sn_alt.ma;h=bc8528408f5466dd7e016f36dbddf78e8e73dbc1;hb=c69a33bba2ae2f37953737940fb45149136cf054;hp=4356553ff82f494888893f2180d56358a38b8b85;hpb=916c53e11fbf6dda073c8796cd97881c84ec5834;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt.ma
index 4356553ff..bc8528408 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_alt.ma
@@ -18,50 +18,83 @@ include "basic_2/relocation/llpx_sn.ma".
 
 (* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
 
-(* alternative definition of llpx_sn_alt *)
+(* alternative definition of llpx_sn *)
 inductive llpx_sn_alt (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
 | llpx_sn_alt_intro: ∀L1,L2,T,d.
                      (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
+                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
                      ) →
                      (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                       ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt R 0 V1 K1 K2
+                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt R 0 V1 K1 K2
                      ) → |L1| = |L2| → llpx_sn_alt R d T L1 L2
 .
 
-(* Basic forward lemmas ******************************************************)
+(* Compact definition of llpx_sn_alt ****************************************)
 
-lemma llpx_sn_alt_fwd_gen: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 →
+lemma llpx_sn_alt_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
+                             (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                                ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                                ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2
+                             ) → llpx_sn_alt R d T L1 L2.
+#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_intro // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
+qed.
+
+lemma llpx_sn_alt_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
+                           (∀L1,L2,T,d. |L1| = |L2| → (
+                              ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                              ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                              ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2 & S 0 V1 K1 K2
+                           ) → S d T L1 L2) →
+                           ∀L1,L2,T,d. llpx_sn_alt R d T L1 L2 → S d T L1 L2.
+#R #S #IH #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+#L1 #L2 #T #d #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
+qed-.
+
+lemma llpx_sn_alt_inv_alt: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 →
                            |L1| = |L2| ∧
                            ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
                            ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
                            ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2.
-#R #L1 #L2 #T #d * -L1 -L2 -T -d
-#L1 #L2 #T #d #IH1 #IH2 #HL12 @conj //
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH1 … HnT HLK1 HLK2) -IH1 /4 width=8 by and3_intro/
+#R #L1 #L2 #T #d #H @(llpx_sn_alt_ind_alt … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
 qed-.
 
-lemma llpx_sn_alt_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #T #d * -L1 -L2 -T -d //
+(* Basic inversion lemmas ***************************************************)
+
+lemma llpx_sn_alt_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓕ{I}V.T) L1 L2 →
+                            llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
+#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
+/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
 qed-.
 
-fact llpx_sn_alt_fwd_lref_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 → ∀i. X = #i →
-                               ∨∨ |L1| ≤ i ∧ |L2| ≤ i
-                                | yinj i < d
-                                | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
-                                                   ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
-                                                   llpx_sn_alt R (yinj 0) V1 K1 K2 &
-                                                   R K1 V1 V2 & d ≤ yinj i.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-#L1 #L2 #X #d #H1X #H2X #HL12 #i #H destruct
-elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
-elim (ylt_split i d) /3 width=1 by or3_intro1/
-#Hdi #HL1 elim (ldrop_O1_lt … HL1) #I1 #K1 #V1 #HLK1
-elim (ldrop_O1_lt L2 i) // #I2 #K2 #V2 #HLK2
-elim (H1X … HLK1 HLK2) -H1X /2 width=3 by nlift_lref_be_SO/ #H #HV12 destruct
-lapply (H2X … HLK1 HLK2) -H2X /2 width=3 by nlift_lref_be_SO/
-/3 width=9 by or3_intro2, ex5_5_intro/
+lemma llpx_sn_alt_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2 →
+                            llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_intro_alt [1,3: normalize // ] -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
+  /3 width=9 by nlift_bind_sn, and3_intro/
+| lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
+  lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
+  lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
+  elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
+  @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+]
+qed-.
+
+(* Basic forward lemmas ******************************************************)
+
+lemma llpx_sn_alt_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #T #d #H elim (llpx_sn_alt_inv_alt … H) -H //
 qed-.
 
 lemma llpx_sn_alt_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt R d (#i) L1 L2 →
@@ -71,75 +104,36 @@ lemma llpx_sn_alt_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt R d (#i) L1 L2 →
                                                 ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
                                                 llpx_sn_alt R (yinj 0) V1 K1 K2 &
                                                 R K1 V1 V2 & d ≤ yinj i.
-/2 width=3 by llpx_sn_alt_fwd_lref_aux/ qed-.
-
-(* Basic inversion lemmas ****************************************************)
-
-fact llpx_sn_alt_inv_flat_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 →
-                               ∀I,V,T. X = ⓕ{I}V.T →
-                               llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-#L1 #L2 #X #d #H1X #H2X #HL12
-#I #V #T #H destruct
-@conj @llpx_sn_alt_intro // -HL12
-/4 width=8 by nlift_flat_sn, nlift_flat_dx/
-qed-.
-
-lemma llpx_sn_alt_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓕ{I}V.T) L1 L2 →
-                            llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
-/2 width=4 by llpx_sn_alt_inv_flat_aux/ qed-.
-
-fact llpx_sn_alt_inv_bind_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 →
-                               ∀a,I,V,T. X = ⓑ{a,I}V.T →
-                               llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-#L1 #L2 #X #d #H1X #H2X #HL12
-#a #I #V #T #H destruct
-@conj @llpx_sn_alt_intro [3,6: normalize /2 width=1 by eq_f2/ ] -HL12
-#I1 #I2 #K1 #K2 #W1 #W2 #i #Hdi #H #HLK1 #HLK2
-[1,2: /4 width=9 by nlift_bind_sn/ ]
-lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
-lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
-lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
-[ @(H1X … HLK1 HLK2) | @(H2X … HLK1 HLK2) ] // -I1 -I2 -L1 -L2 -K1 -K2 -W1 -W2
-@nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+#R #L1 #L2 #d #i #H elim (llpx_sn_alt_inv_alt … H) -H
+#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
+elim (ylt_split i d) /3 width=1 by or3_intro1/
+#Hdi #HL1 elim (ldrop_O1_lt … HL1)
+#I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt L2 i) //
+#I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
+/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
 qed-.
 
-lemma llpx_sn_alt_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2 →
-                            llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-/2 width=4 by llpx_sn_alt_inv_bind_aux/ qed-.
-
 (* Basic properties **********************************************************)
 
-lemma llpx_sn_alt_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
-                             (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
-                                ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
-                                ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2
-                             ) → llpx_sn_alt R d T L1 L2.
-#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_intro // -HL12
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
-qed.
-
 lemma llpx_sn_alt_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt R d (⋆k) L1 L2.
-#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_intro_alt // -HL12
 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
 qed.
 
 lemma llpx_sn_alt_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt R d (§p) L1 L2.
-#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_intro_alt // -HL12
 #I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
 qed.
 
 lemma llpx_sn_alt_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_intro_alt // -HL12
 #I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
 /4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
 qed.
 
 lemma llpx_sn_alt_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
                         llpx_sn_alt R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_intro_alt // -HL12
 #I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
 lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
 /3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
@@ -149,53 +143,47 @@ lemma llpx_sn_alt_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
                         ⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
                         llpx_sn_alt R 0 V1 K1 K2 → R K1 V1 V2 →
                         llpx_sn_alt R d (#i) L1 L2.
-#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_intro
-[1,2: #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
+#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_intro_alt
+[ lapply (llpx_sn_alt_fwd_length … HK12) -HK12 #HK12
+  @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
+| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
   elim (lt_or_eq_or_gt i j) #Hij destruct
-  [1,4: elim (H (#i)) -H /2 width=1 by lift_lref_lt/
-  |2,5: lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
-    lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by conj/
-  |3,6: elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
+  [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
+  | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
+    lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
+  | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
   ]
-| lapply (llpx_sn_alt_fwd_length … HK12) -HK12 #HK12
-  @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize /2 width=1 by eq_f2/
 ]
 qed.
 
-fact llpx_sn_alt_flat_aux: ∀R,I,L1,L2,V,d. llpx_sn_alt R d V L1 L2 →
-                           ∀Y1,Y2,T,m. llpx_sn_alt R m T Y1 Y2 →
-                           Y1 = L1 → Y2 = L2 → m = d →
-                           llpx_sn_alt R d (ⓕ{I}V.T) L1 L2.
-#R #I #L1 #L2 #V #d * -L1 -L2 -V -d #L1 #L2 #V #d #H1V #H2V #HL12
-#Y1 #Y2 #T #m * -Y1 -Y2 -T -m #Y1 #Y2 #T #m #H1T #H2T #_
-#HT1 #HY2 #Hm destruct
-@llpx_sn_alt_intro // -HL12
-#J1 #J2 #K1 #K2 #W1 #W2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_flat … HnVT) -HnVT /3 width=8 by/
-qed-.
-
 lemma llpx_sn_alt_flat: ∀R,I,L1,L2,V,T,d.
                         llpx_sn_alt R d V L1 L2 → llpx_sn_alt R d T L1 L2 →
                         llpx_sn_alt R d (ⓕ{I}V.T) L1 L2.
-/2 width=7 by llpx_sn_alt_flat_aux/ qed.
-
-fact llpx_sn_alt_bind_aux: ∀R,a,I,L1,L2,V,d. llpx_sn_alt R d V L1 L2 →
-                           ∀Y1,Y2,T,m. llpx_sn_alt R m T Y1 Y2 →
-                           Y1 = L1.ⓑ{I}V → Y2 = L2.ⓑ{I}V → m = ⫯d →
-                           llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2.
-#R #a #I #L1 #L2 #V #d * -L1 -L2 -V -d #L1 #L2 #V #d #H1V #H2V #HL12
-#Y1 #Y2 #T #m * -Y1 -Y2 -T -m #Y1 #Y2 #T #m #H1T #H2T #_
-#HT1 #HY2 #Hm destruct
-@llpx_sn_alt_intro // -HL12
-#J1 #J2 #K1 #K2 #W1 #W2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_bind … HnVT) -HnVT /3 width=8 by ldrop_drop, yle_succ/
-qed-.
+#R #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_flat … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
+]
+qed.
 
 lemma llpx_sn_alt_bind: ∀R,a,I,L1,L2,V,T,d.
                         llpx_sn_alt R d V L1 L2 →
                         llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
                         llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2.
-/2 width=7 by llpx_sn_alt_bind_aux/ qed.
+#R #a #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_bind … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
+]
+qed.
 
 (* Main properties **********************************************************)
 
@@ -222,7 +210,7 @@ theorem llpx_sn_alt_inv_lpx_sn: ∀R,T,L1,L2,d. llpx_sn_alt R d T L1 L2 → llpx
 ]
 qed-.
 
-(* Advanced properties ******************************************************)
+(* Alternative definition of llpx_sn ****************************************)
 
 lemma llpx_sn_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
                          (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
@@ -231,20 +219,32 @@ lemma llpx_sn_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
                          ) → llpx_sn R d T L1 L2.
 #R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_inv_lpx_sn
 @llpx_sn_alt_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt, and3_intro/
 qed.
 
-(* Advanced forward lemmas lemmas *******************************************)
+lemma llpx_sn_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
+                       (∀L1,L2,T,d. |L1| = |L2| → (
+                          ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+                          ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+                          ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
+                       ) → S d T L1 L2) →
+                       ∀L1,L2,T,d. llpx_sn R d T L1 L2 → S d T L1 L2.
+#R #S #IH1 #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt … H) -H
+#H @(llpx_sn_alt_ind_alt … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_inv_lpx_sn, and4_intro/
+qed-.
 
-lemma llpx_sn_fwd_alt: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
+lemma llpx_sn_inv_alt: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
                        |L1| = |L2| ∧
                        ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
                        ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
                        ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
 #R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt … H) -H
-#H elim (llpx_sn_alt_fwd_gen … H) -H
+#H elim (llpx_sn_alt_inv_alt … H) -H
 #HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
 elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_inv_lpx_sn, and3_intro/
 qed-.