X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambda_delta%2FBasic_2%2Fsubstitution%2Ftps_lift.ma;h=e453c4099cd4a8b06f7f1a63f507672fba04e989;hb=48b202cd4ccd3ffc10f9a134314f747fdee30d36;hp=35cff33723f7565bc03d2c17ef31917a1f1e94fa;hpb=035e3f52f8da3cb3cdb493aa20568ad673cc2cf5;p=helm.git

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps_lift.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps_lift.ma
index 35cff3372..e453c4099 100644
--- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps_lift.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/tps_lift.ma
@@ -19,231 +19,228 @@ include "Basic_2/substitution/tps.ma".
 
 (* Advanced inversion lemmas ************************************************)
 
-fact tps_inv_refl_SO2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 1 →
-                           ∀K,V. ↓[0, d] L ≡ K. 𝕓{Abst} V → T1 = T2.
-#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
+fact tps_inv_refl_SO2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ▶ T2 → e = 1 →
+                           ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
 [ //
-| #L #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct -e;
-  >(le_to_le_to_eq … Hdi ?) /2/ -d #K #V #HLK
+| #L #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct
+  >(le_to_le_to_eq … Hdi ?) /2 width=1/ -d #K #V #HLK
   lapply (ldrop_mono … HLK0 … HLK) #H destruct
 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
-  >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 K V) -IHT12 // /2/
+  >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 K V) -IHT12 // /2 width=1/
 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
   >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 … HLK) -IHT12 //
 ]
 qed.
 
-lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ≫ T2 →
-                        ∀K,V. ↓[0, d] L ≡ K. 𝕓{Abst} V → T1 = T2.
-/2 width=8/ qed.
+lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ▶ T2 →
+                        ∀K,V. ⇩[0, d] L ≡ K. ⓛV → T1 = T2.
+/2 width=8/ qed-.
 
 (* Relocation properties ****************************************************)
 
 (* Basic_1: was: subst1_lift_lt *)
-lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
-                   ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
-                   ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
+lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
+                   ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+                   ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
                    dt + et ≤ d →
-                   L ⊢ U1 [dt, et] ≫ U2.
-#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
+                   L ⊢ U1 [dt, et] ▶ U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
 [ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
-  >(lift_mono … H1 … H2) -H1 H2 //
+  >(lift_mono … H1 … H2) -H1 -H2 //
 | #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdetd
   lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
-  lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct -U1;
-  elim (lift_trans_ge … HVW … HWU2 ?) -HVW HWU2 W // <minus_plus #W #HVW #HWU2
-  elim (ldrop_trans_le … HLK … HKV ?) -HLK HKV K [2: /2/] #X #HLK #H
-  elim (ldrop_inv_skip2 … H ?) -H [2: /2/] -Hid #K #Y #_ #HVY
-  >(lift_mono … HVY … HVW) -HVY HVW Y #H destruct -X /2/
+  lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
+  elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
+  elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
+  elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+  >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
 | #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
-  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
-  @tps_bind [ /2 width=6/ | @IHT12 [4,5: // |1,2: skip | /2/ | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  @tps_bind [ /2 width=6/ | @IHT12 /2 width=6/ ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
 | #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
   elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
-  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
-  /3 width=6/
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
 ]
 qed.
 
-lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
-                   ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
-                   ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
+lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
+                   ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+                   ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
                    dt ≤ d → d ≤ dt + et →
-                   L ⊢ U1 [dt, et + e] ≫ U2.
-#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
+                   L ⊢ U1 [dt, et + e] ▶ U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
 [ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
-  >(lift_mono … H1 … H2) -H1 H2 //
+  >(lift_mono … H1 … H2) -H1 -H2 //
 | #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hdtd #_
-  elim (lift_inv_lref1 … H) -H * #Hid #H destruct -U1;
-  [ -Hdtd;
+  elim (lift_inv_lref1 … H) -H * #Hid #H destruct
+  [ -Hdtd
     lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
     elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
-    elim (ldrop_trans_le … HLK … HKV ?) -K [2: /2/] #X #HLK #H
-    elim (ldrop_inv_skip2 … H ?) -H [2: /2/] -Hid #K #Y #_ #HVY
-    >(lift_mono … HVY … HVW) -V #H destruct -X /2/
-  | -Hdti;
+    elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
+    elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
+    >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
+  | -Hdti
     lapply (transitive_le … Hdtd Hid) -Hdtd #Hdti
-    lapply (lift_trans_be … HVW … HWU2 ? ?) -W // [ /2/ ] >plus_plus_comm_23 #HVU2
-    lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3/
+    lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+    lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
   ]
 | #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdtd #Hddet
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
-  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
-  @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2/ | /2/ ] ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash *)
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+  @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ] 
+            ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
 | #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
   elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
-  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
-  /3 width=6/
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
 ]
 qed.
 
 (* Basic_1: was: subst1_lift_ge *)
-lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
-                   ∀L,U1,U2,d,e. ↓[d, e] L ≡ K →
-                   ↑[d, e] T1 ≡ U1 → ↑[d, e] T2 ≡ U2 →
+lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ▶ T2 →
+                   ∀L,U1,U2,d,e. ⇩[d, e] L ≡ K →
+                   ⇧[d, e] T1 ≡ U1 → ⇧[d, e] T2 ≡ U2 →
                    d ≤ dt →
-                   L ⊢ U1 [dt + e, et] ≫ U2.
-#K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
+                   L ⊢ U1 [dt + e, et] ▶ U2.
+#K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
 [ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
-  >(lift_mono … H1 … H2) -H1 H2 //
+  >(lift_mono … H1 … H2) -H1 -H2 //
 | #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
   lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
-  lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct -U1;
-  lapply (lift_trans_be … HVW … HWU2 ? ?) -HVW HWU2 W // [ /2/ ] >plus_plus_comm_23 #HVU2
-  lapply (ldrop_trans_ge_comm … HLK … HKV ?) -HLK HKV K // -Hid /3/
+  lapply (lift_inv_lref1_ge … H … Hid) -H #H destruct
+  lapply (lift_trans_be … HVW … HWU2 ? ?) -W // /2 width=1/ >plus_plus_comm_23 #HVU2
+  lapply (ldrop_trans_ge_comm … HLK … HKV ?) -K // -Hid /3 width=4/
 | #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
-  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
+  elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
   @tps_bind [ /2 width=5/ | /3 width=5/ ] (**) (* explicit constructor *)
 | #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hddt
   elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
-  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct -U1 U2;
-  /3 width=5/
+  elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=5/
 ]
 qed.
 
 (* Basic_1: was: subst1_gen_lift_lt *)
-lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
-                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+                        ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                         dt + et ≤ d →
-                        ∃∃T2. K ⊢ T1 [dt, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
+                        ∃∃T2. K ⊢ T1 [dt, et] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
 [ #L * #i #dt #et #K #d #e #_ #T1 #H #_
-  [ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
-  | lapply (lift_inv_gref2 … H) -H #H destruct -T1 /2/
+  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
   ]
 | #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
   lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
-  lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct -T1;
-  elim (ldrop_conf_lt … HLK … HLKV ?) -HLK HLKV L // #L #U #HKL #_ #HUV
-  elim (lift_trans_le … HUV … HVW ?) -HUV HVW V // >arith_a2 // -Hid /3/
+  lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
+  elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
+  elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
 | #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
-  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
-  elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 // #W2 #HW12 #HWV2
-  elim (IHU12 … HTU1 ?) -IHU12 HTU1 [3: /2/ |4: @ldrop_skip // |2: skip ] -HLK Hdetd (**) (* /3 width=5/ is too slow *)
+  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+  elim (IHU12 … HTU1 ?) -IHU12 -HTU1 [3: /2 width=1/ |4: @ldrop_skip // |2: skip ] -HLK -Hdetd (**) (* /3 width=5/ is too slow *)
   /3 width=5/
 | #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
-  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
-  elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
-  elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHV12 … HLK … HWV1 ?) -V1 //
+  elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
 ]
 qed.
 
-lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
-                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+                        ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                         dt ≤ d → d + e ≤ dt + et →
-                        ∃∃T2. K ⊢ T1 [dt, et - e] ≫ T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
+                        ∃∃T2. K ⊢ T1 [dt, et - e] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
 [ #L * #i #dt #et #K #d #e #_ #T1 #H #_
-  [ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
-  | lapply (lift_inv_gref2 … H) -H #H destruct -T1 /2/
+  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
   ]
 | #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdtd #Hdedet
   lapply (le_fwd_plus_plus_ge … Hdtd … Hdedet) #Heet
-  elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1
-  [ -Hdtd Hidet;
-    lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2/ ] -Hdedet #Hidete
+  elim (lift_inv_lref2 … H) -H * #Hid #H destruct
+  [ -Hdtd -Hidet
+    lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -Hdedet #Hidete
     elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
-    elim (lift_trans_le … HUV … HVW ?) -V // >arith_a2 // -Hid /3/
-  | -Hdti Hdedet;
-    lapply (transitive_le … (i - e) Hdtd ?) [ /2/ ] -Hdtd #Hdtie
-    lapply (plus_le_weak … Hid) #Hei
+    elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
+  | -Hdti -Hdedet
+    lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
+    elim (le_inv_plus_l … Hid) #Hdie #Hei
     lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
-    elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW; [4: // |2,3: /2/ ] -Hid >arith_e2 // /4/
+    elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+    #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+    @ex2_1_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
   ]
 | #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
-  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
-  elim (IHV12 … HLK … HWV1 ? ?) -IHV12 HWV1 // #W2 #HW12 #HWV2
-  elim (IHU12 … HTU1 ? ?) -IHU12 HTU1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2/ ]
+  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
+  elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
   /3 width=5/
 | #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
-  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
-  elim (IHV12 … HLK … HWV1 ? ?) -IHV12 HWV1 //
-  elim (IHU12 … HLK … HTU1 ? ?) -IHU12 HLK HTU1 // /3 width=5/
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHV12 … HLK … HWV1 ? ?) -V1 //
+  elim (IHU12 … HLK … HTU1 ? ?) -U1 -HLK // /3 width=5/
 ]
 qed.
 
 (* Basic_1: was: subst1_gen_lift_ge *)
-lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
-                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+                        ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                         d + e ≤ dt →
-                        ∃∃T2. K ⊢ T1 [dt - e, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
-#L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
+                        ∃∃T2. K ⊢ T1 [dt - e, et] ▶ T2 & ⇧[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
 [ #L * #i #dt #et #K #d #e #_ #T1 #H #_
-  [ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
-  | lapply (lift_inv_gref2 … H) -H #H destruct -T1 /2/
+  [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
+  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct /3 width=3/
+  | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3/
   ]
-| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt  
+| #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt
   lapply (transitive_le … Hdedt … Hdti) #Hdei
-  lapply (plus_le_weak … Hdedt) -Hdedt #Hedt
-  lapply (plus_le_weak … Hdei) #Hei  
-  lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct -T1;
-  lapply (ldrop_conf_ge … HLK … HLKV ?) -HLK HLKV L // #HKV
-  elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW; [4: // | 2,3: normalize /2/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02
-  @ex2_1_intro
-  [2: @tps_subst [3: /2/ |5,6: // |1,2: skip |4: @arith5 // ]
-  |1: skip
-  | //
-  ] (**) (* explicitc constructors *)
+  elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
+  elim (le_inv_plus_l … Hdei) #Hdie #Hei
+  lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
+  lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
+  elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie 
+  #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+  @ex2_1_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
 | #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
-  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
-  lapply (plus_le_weak … Hdetd) #Hedt
-  elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 // #W2 #HW12 #HWV2
-  elim (IHU12 … HTU1 ?) -IHU12 HTU1 [4: @ldrop_skip // |2: skip |3: /2/ ]
+  elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (le_inv_plus_l … Hdetd) #_ #Hedt
+  elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
+  elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
   <plus_minus // /3 width=5/
 | #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
-  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct -X;
-  elim (IHV12 … HLK … HWV1 ?) -IHV12 HWV1 //
-  elim (IHU12 … HLK … HTU1 ?) -IHU12 HLK HTU1 // /3 width=5/
+  elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
+  elim (IHV12 … HLK … HWV1 ?) -V1 //
+  elim (IHU12 … HLK … HTU1 ?) -U1 -HLK // /3 width=5/
 ]
 qed.
 
 (* Basic_1: was: subst1_gen_lift_eq *)
 lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
-                        L ⊢ U1 [d, e] ≫ U2 → ∀T1. ↑[d, e] T1 ≡ U1 → U1 = U2.
-#L #U1 #U2 #d #e #H elim H -H L U1 U2 d e
+                        L ⊢ U1 [d, e] ▶ U2 → ∀T1. ⇧[d, e] T1 ≡ U1 → U1 = U2.
+#L #U1 #U2 #d #e #H elim H -L -U1 -U2 -d -e
 [ //
 | #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
   elim (lift_inv_lref2 … H) -H * #H
-  [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi H #H
+  [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi -H #H
     elim (lt_refl_false … H)
-  | lapply (lt_to_le_to_lt … Hide … H) -Hide H #H
+  | lapply (lt_to_le_to_lt … Hide … H) -Hide -H #H
     elim (lt_refl_false … H)
   ]
 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
-  elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
+  elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #H destruct
   >IHV12 // >IHT12 //
 | #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
-  elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct -X
+  elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #H destruct
   >IHV12 // >IHT12 //
 ]
 qed.
 (*
-      Theorem subst0_gen_lift_rev_ge: (t1,v,u2:?; i,h,d:?) 
+      Theorem subst0_gen_lift_rev_ge: (t1,v,u2,i,h,d:?) 
                                       (subst0 i v t1 (lift h d u2)) ->
                                       (le (plus d h) i) ->
                                       (EX u1 | (subst0 (minus i h) v u1 u2) &
@@ -251,28 +248,27 @@ qed.
 		                      ).
 
 
-      Theorem subst0_gen_lift_rev_lelt: (t1,v,u2:?; i,h,d:?)
+      Theorem subst0_gen_lift_rev_lelt: (t1,v,u2,i,h,d:?)
                                         (subst0 i v t1 (lift h d u2)) ->
                                         (le d i) -> (lt i (plus d h)) ->
 				        (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
 *)
-
-lemma tps_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
-                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_up: ∀L,U1,U2,dt,et. 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. K ⊢ T1 [d, dt + et - (d + e)] ≫ T2 & ↑[d, e] T2 ≡ U2.
+                        ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ▶ T2 & ⇧[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
 elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (tps_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
-lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
-elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
+lapply (tps_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
+lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
 qed.
 
-lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
-                           ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_be_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ▶ U2 →
+                           ∀K,d,e. ⇩[d, e] L ≡ K → ∀T1. ⇧[d, e] T1 ≡ U1 →
                            dt ≤ d → dt + et ≤ d + e →
-                           ∃∃T2. K ⊢ T1 [dt, d - dt] ≫ T2 & ↑[d, e] T2 ≡ U2.
+                           ∃∃T2. K ⊢ T1 [dt, d - dt] ▶ T2 & ⇧[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
-lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // [ /2/ ] -Hdetde #HU12
-elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 L /2/
+lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
+elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/
 qed.