X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fsubstitution%2Fcpys_lift.ma;h=b6d9e45a1080b515395b3ddbe3bfacdb96ecc65e;hb=944b1f7b762774a6f8d99a2c2846f865b6788712;hp=3cf6f8b31ff67ba044fdfd47221b9453b946e988;hpb=32bdf7f107be22a121fab8225c5fae4eb6b33633;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
index 3cf6f8b31..b6d9e45a1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
@@ -21,28 +21,36 @@ include "basic_2/substitution/cpys.ma".
 
 lemma cpys_subst: ∀I,G,L,K,V,U1,i,d,e.
                   d ≤ yinj i → i < d + e →
-                  ⇩[0, i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*×[0, ⫰(d+e-i)] U1 →
-                  ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*×[d, e] U2.
+                  ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ⫰(d+e-i)] U1 →
+                  ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[d, e] U2.
 #I #G #L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(cpys_ind … H) -U1
 [ /3 width=5 by cpy_cpys, cpy_subst/
 | #U #U1 #_ #HU1 #IHU #U2 #HU12
   elim (lift_total U 0 (i+1)) #U0 #HU0
   lapply (IHU … HU0) -IHU #H
-  lapply (ldrop_fwd_ldrop2 … HLK) -HLK #HLK
+  lapply (ldrop_fwd_drop2 … HLK) -HLK #HLK
   lapply (cpy_lift_ge … HU1 … HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
   lapply (cpy_weak … HU02 d e ? ?) -HU02
   [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
-  >yplus_O_sn <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
+  >yplus_O1 <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
 ]
 qed.
 
+lemma cpys_subst_Y2: ∀I,G,L,K,V,U1,i,d.
+                     d ≤ yinj i →
+                     ⇩[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ▶*[0, ∞] U1 →
+                     ∀U2. ⇧[0, i+1] U1 ≡ U2 → ⦃G, L⦄ ⊢ #i ▶*[d, ∞] U2.
+#I #G #L #K #V #U1 #i #d #Hdi #HLK #HVU1 #U2 #HU12
+@(cpys_subst … HLK … HU12) >yminus_Y_inj //
+qed.
+
 (* Advanced inverion lemmas *************************************************)
 
-lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 →
+lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*[d, e] T2 →
                       T2 = ⓪{I} ∨
                       ∃∃J,K,V1,V2,i. d ≤ yinj i & i < d + e &
-                                    ⇩[O, i] L ≡ K.ⓑ{J}V1 &
-                                     ⦃G, K⦄ ⊢ V1 ▶*×[0, d+e-i-1] V2 &
+                                    ⇩[i] L ≡ K.ⓑ{J}V1 &
+                                     ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(d+e-i)] V2 &
                                      ⇧[O, i+1] V2 ≡ T2 &
                                      I = LRef i.
 #I #G #L #T2 #d #e #H @(cpys_ind … H) -T2
@@ -51,7 +59,7 @@ lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 →
   [ #H destruct
     elim (cpy_inv_atom1 … HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
   | * #J #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
-    lapply (ldrop_fwd_ldrop2 … HLK) #H
+    lapply (ldrop_fwd_drop2 … HLK) #H
     elim (cpy_inv_lift1_ge_up … HT2 … H … HVT) -HT2 -H -HVT
     [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
     /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
@@ -59,23 +67,38 @@ lemma cpys_inv_atom1: ∀I,G,L,T2,d,e. ⦃G, L⦄ ⊢ ⓪{I} ▶*×[d, e] T2 →
 ]
 qed-.
 
-lemma cpys_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*×[d, e] T2 →
+lemma cpys_inv_lref1: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*[d, e] T2 →
                       T2 = #i ∨
                       ∃∃I,K,V1,V2. d ≤ i & i < d + e &
-                                   ⇩[O, i] L ≡ K.ⓑ{I}V1 &
-                                   ⦃G, K⦄ ⊢ V1 ▶*×[0, d+e-i-1] V2 &
+                                   ⇩[i] L ≡ K.ⓑ{I}V1 &
+                                   ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(d+e-i)] V2 &
                                    ⇧[O, i+1] V2 ≡ T2.
 #G #L #T2 #i #d #e #H elim (cpys_inv_atom1 … H) -H /2 width=1 by or_introl/
 * #I #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
 qed-.
 
-(* Relocation properties ****************************************************)
+lemma cpys_inv_lref1_ldrop: ∀G,L,T2,i,d,e. ⦃G, L⦄ ⊢ #i ▶*[d, e] T2 →
+                            ∀I,K,V1. ⇩[i] L ≡ K.ⓑ{I}V1 →
+                            ∀V2. ⇧[O, i+1] V2 ≡ T2 →
+                            ∧∧ ⦃G, K⦄ ⊢ V1 ▶*[0, ⫰(d+e-i)] V2
+                             & d ≤ i
+                             & i < d + e.
+#G #L #T2 #i #d #e #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 … H) -H
+[ #H destruct elim (lift_inv_lref2_be … HVT2) -HVT2 -HLK //
+| * #Z #Y #X1 #X2 #Hdi #Hide #HLY #HX12 #HXT2
+  lapply (lift_inj … HXT2 … HVT2) -T2 #H destruct
+  lapply (ldrop_mono … HLY … HLK) -L #H destruct
+  /2 width=1 by and3_intro/
+]
+qed-.
+
+(* Properties on relocation *************************************************)
 
-lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
-                    ∀L,U1,d,e. dt + et ≤ yinj d → ⇩[d, e] L ≡ K →
+lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 →
+                    ∀L,U1,s,d,e. dt + et ≤ yinj d → ⇩[s, d, e] L ≡ K →
                     ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
-                    ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2.
-#G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdetd #HLK #HTU1 @(cpys_ind … H) -T2
+                    ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2.
+#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdetd #HLK #HTU1 @(cpys_ind … H) -T2
 [ #U2 #H >(lift_mono … HTU1 … H) -H //
 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
   elim (lift_total T d e) #U #HTU
@@ -84,11 +107,11 @@ lemma cpys_lift_le: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
 ]
 qed-.
 
-lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
-                    ∀L,U1,d,e. dt ≤ yinj d → d ≤ dt + et →
-                    ⇩[d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
-                    ∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*×[dt, et + e] U2.
-#G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind … H) -T2
+lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 →
+                    ∀L,U1,s,d,e. dt ≤ yinj d → d ≤ dt + et →
+                    ⇩[s, d, e] L ≡ K → ⇧[d, e] T1 ≡ U1 →
+                    ∀U2. ⇧[d, e] T2 ≡ U2 → ⦃G, L⦄ ⊢ U1 ▶*[dt, et + e] U2.
+#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind … H) -T2
 [ #U2 #H >(lift_mono … HTU1 … H) -H //
 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
   elim (lift_total T d e) #U #HTU
@@ -97,11 +120,11 @@ lemma cpys_lift_be: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
 ]
 qed-.
 
-lemma cpys_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
-                    ∀L,U1,d,e. yinj d ≤ dt → ⇩[d, e] L ≡ K →
+lemma cpys_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 →
+                    ∀L,U1,s,d,e. yinj d ≤ dt → ⇩[s, d, e] L ≡ K →
                     ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 →
-                    ⦃G, L⦄ ⊢ U1 ▶*×[dt+e, et] U2.
-#G #K #T1 #T2 #dt #et #H #L #U1 #d #e #Hddt #HLK #HTU1 @(cpys_ind … H) -T2
+                    ⦃G, L⦄ ⊢ U1 ▶*[dt+e, et] U2.
+#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hddt #HLK #HTU1 @(cpys_ind … H) -T2
 [ #U2 #H >(lift_mono … HTU1 … H) -H //
 | -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
   elim (lift_total T d e) #U #HTU
@@ -110,76 +133,86 @@ lemma cpys_lift_ge: ∀G,K,T1,T2,dt,et. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 →
 ]
 qed-.
 
-lemma cpys_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
-                         ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+(* Inversion lemmas for relocation ******************************************)
+
+lemma cpys_inv_lift1_le: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+                         ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                          dt + et ≤ d →
-                         ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdetd @(cpys_ind … H) -U2
+                         ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdetd @(cpys_ind … H) -U2
 [ /2 width=3 by ex2_intro/
 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
   elim (cpy_inv_lift1_le … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
 ]
 qed-.
 
-lemma cpys_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
-                         ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
-                         dt ≤ d → d + e ≤ dt + et →
-                         ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(cpys_ind … H) -U2
+lemma cpys_inv_lift1_be: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+                         ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+                         dt ≤ d → yinj d + e ≤ dt + et →
+                         ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, et - e] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(cpys_ind … H) -U2
 [ /2 width=3 by ex2_intro/
 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
   elim (cpy_inv_lift1_be … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
 ]
 qed-.
 
-lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
-                         ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
-                         d + e ≤ dt →
-                         ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind … H) -U2
+lemma cpys_inv_lift1_ge: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+                         ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+                         yinj d + e ≤ dt →
+                         ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt - e, et] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind … H) -U2
 [ /2 width=3 by ex2_intro/
 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
   elim (cpy_inv_lift1_ge … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
 ]
 qed-.
 
-lemma cpys_inv_lift1_eq: ∀G,L,U1,U2. ∀d,e:nat.
-                         ⦃G, L⦄ ⊢ U1 ▶*×[d, e] U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
-#G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind … H) -U2 //
-#U #U2 #_ #HU2 #IHU destruct
-<(cpy_inv_lift1_eq … HTU1 … HU2) -HU2 -HTU1 //
-qed-.
+(* Advanced inversion lemmas on relocation **********************************)
 
-lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
-                            ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
-                            d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
-                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[d, dt + et - (d + e)] T2 &
+lemma cpys_inv_lift1_ge_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+                            ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+                            d ≤ dt → dt ≤ yinj d + e → yinj d + e ≤ dt + et →
+                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[d, dt + et - (yinj d + e)] T2 &
                                  ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(cpys_ind … H) -U2
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(cpys_ind … H) -U2
 [ /2 width=3 by ex2_intro/
 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
   elim (cpy_inv_lift1_ge_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
 ]
 qed-.
 
-lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
-                            ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
-                            dt ≤ d → dt + et ≤ d + e →
-                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(cpys_ind … H) -U2
+lemma cpys_inv_lift1_be_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+                            ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+                            dt ≤ d → dt + et ≤ yinj d + e →
+                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(cpys_ind … H) -U2
 [ /2 width=3 by ex2_intro/
 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
   elim (cpy_inv_lift1_be_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
 ]
 qed-.
 
-lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*×[dt, et] U2 →
-                            ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
-                            dt ≤ d → d ≤ dt + et → dt + et ≤ d + e →
-                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*×[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
-#G #L #U1 #U2 #dt #et #H #K #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(cpys_ind … H) -U2
+lemma cpys_inv_lift1_le_up: ∀G,L,U1,U2,dt,et. ⦃G, L⦄ ⊢ U1 ▶*[dt, et] U2 →
+                            ∀K,s,d,e. ⇩[s, d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
+                            dt ≤ d → d ≤ dt + et → dt + et ≤ yinj d + e →
+                            ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[dt, d - dt] T2 & ⇧[d, e] T2 ≡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(cpys_ind … H) -U2
 [ /2 width=3 by ex2_intro/
 | -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
   elim (cpy_inv_lift1_le_up … HU2 … HLK … HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
 ]
 qed-.
+
+lemma cpys_inv_lift1_subst: ∀G,L,W1,W2,d,e. ⦃G, L⦄ ⊢ W1 ▶*[d, e] W2 →
+                            ∀K,V1,i. ⇩[i+1] L ≡ K → ⇧[O, i+1] V1 ≡ W1 → 
+                            d ≤ yinj i → i < d + e →
+                            ∃∃V2.  ⦃G, K⦄ ⊢ V1 ▶*[O, ⫰(d+e-i)] V2 & ⇧[O, i+1] V2 ≡ W2.
+#G #L #W1 #W2 #d #e #HW12 #K #V1 #i #HLK #HVW1 #Hdi #Hide
+elim (cpys_inv_lift1_ge_up … HW12 … HLK … HVW1 ? ? ?) //
+>yplus_O1 <yplus_inj >yplus_SO2
+[ >yminus_succ2 /2 width=3 by ex2_intro/
+| /2 width=1 by ylt_fwd_le_succ1/
+| /2 width=3 by yle_trans/
+]
+qed-.