X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fstatic_2%2Fstatic%2Ffrees_drops.ma;h=215ba6e1aab18d91fed4eba2a6c3dc8f5d3ab263;hb=98e786e1a6bd7b621e37ba7cd4098d4a0a6f8278;hp=3c64660863240b8654d66b9f8aa03e58b7815299;hpb=ff612dc35167ec0c145864c9aa8ae5e1ebe20a48;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma b/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma
index 3c6466086..215ba6e1a 100644
--- a/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma
+++ b/matita/matita/contribs/lambdadelta/static_2/static/frees_drops.ma
@@ -12,7 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground_2/relocation/nstream_coafter.ma".
+include "ground/relocation/nstream_coafter.ma".
 include "static_2/relocation/drops_drops.ma".
 include "static_2/static/frees_fqup.ma".
 
@@ -20,8 +20,9 @@ include "static_2/static/frees_fqup.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≘ ⋆ →
-                        ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i]↑f.
+lemma frees_atom_drops:
+      ∀b,L,i. ⇩*[b,𝐔❨i❩] L ≘ ⋆ →
+      ∀f. 𝐈❪f❫ → L ⊢ 𝐅+❪#i❫ ≘ ⫯*[i]↑f.
 #b #L elim L -L /2 width=1 by frees_atom/
 #L #I #IH *
 [ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
@@ -29,55 +30,42 @@ lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≘ ⋆ →
 ]
 qed.
 
-lemma frees_pair_drops: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≘ f → 
-                        ∀i,I,L. ⬇*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
+lemma frees_pair_drops:
+      ∀f,K,V. K ⊢ 𝐅+❪V❫ ≘ f →
+      ∀i,I,L. ⇩[i] L ≘ K.ⓑ[I]V → L ⊢ 𝐅+❪#i❫ ≘ ⫯*[i] ↑f.
 #f #K #V #Hf #i elim i -i
 [ #I #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_pair/
 | #i #IH #I #L #H elim (drops_inv_succ … H) -H /3 width=2 by frees_lref/
 ]
 qed.
 
-lemma frees_unit_drops: ∀f.  𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≘ K.ⓤ{I} →
-                       L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
+lemma frees_unit_drops:
+      ∀f.  𝐈❪f❫ → ∀I,K,i,L. ⇩[i] L ≘ K.ⓤ[I] →
+      L ⊢ 𝐅+❪#i❫ ≘ ⫯*[i] ↑f.
 #f #Hf #I #K #i elim i -i
 [ #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_unit/
 | #i #IH #Y #H elim (drops_inv_succ … H) -H
   #J #L #HLK #H destruct /3 width=1 by frees_lref/
 ]
 qed.
-(*
-lemma frees_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≘ f →
-                        ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃⋆s⦄ ≘ ⫯*[i] f.
-#f #K #s #Hf #i elim i -i
-[ #L #H lapply (drops_fwd_isid … H ?) -H //
-| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_sort/
-]
-qed.
-*)
-lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≘ f →
-                        ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃#(i+j)⦄ ≘ ⫯*[i] f.
+
+lemma frees_lref_pushs:
+      ∀f,K,j. K ⊢ 𝐅+❪#j❫ ≘ f →
+      ∀i,L. ⇩[i] L ≘ K → L ⊢ 𝐅+❪#(i+j)❫ ≘ ⫯*[i] f.
 #f #K #j #Hf #i elim i -i
 [ #L #H lapply (drops_fwd_isid … H ?) -H //
 | #i #IH #L #H elim (drops_inv_succ … H) -H
   #I #Y #HYK #H destruct /3 width=1 by frees_lref/
 ]
 qed.
-(*
-lemma frees_gref_pushs: ∀f,K,l. K ⊢ 𝐅*⦃§l⦄ ≘ f →
-                        ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃§l⦄ ≘ ⫯*[i] f.
-#f #K #l #Hf #i elim i -i
-[ #L #H lapply (drops_fwd_isid … H ?) -H //
-| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_gref/
-]
-qed.
-*)
+
 (* Advanced inversion lemmas ************************************************)
 
-lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≘ f →
-                            ∨∨ ∃∃g. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
-                             | ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≘ g &
-                                          ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
-                             | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
+lemma frees_inv_lref_drops:
+      ∀L,i,f. L ⊢ 𝐅+❪#i❫ ≘ f →
+      ∨∨ ∃∃g. ⇩*[Ⓕ,𝐔❨i❩] L ≘ ⋆ & 𝐈❪g❫ & f = ⫯*[i] ↑g
+       | ∃∃g,I,K,V. K ⊢ 𝐅+❪V❫ ≘ g & ⇩[i] L ≘ K.ⓑ[I]V & f = ⫯*[i] ↑g
+       | ∃∃g,I,K. ⇩[i] L ≘ K.ⓤ[I] & 𝐈❪g❫ & f = ⫯*[i] ↑g.
 #L elim L -L
 [ #i #g | #L #I #IH * [ #g cases I -I [ #I | #I #V ] -IH | #i #g ] ] #H
 [ elim (frees_inv_atom … H) -H #f #Hf #H destruct
@@ -97,23 +85,24 @@ qed-.
 
 (* Properties with generic slicing for local environments *******************)
 
-lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 →
-                   ∀f,L. ⬇*[b, f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
-                   ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2.
+lemma frees_lifts:
+      ∀b,f1,K,T. K ⊢ 𝐅+❪T❫ ≘ f1 →
+      ∀f,L. ⇩*[b,f] L ≘ K → ∀U. ⇧*[f] T ≘ U →
+      ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅+❪U❫ ≘ f2.
 #b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
 [ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
-  lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
+  lapply (pr_coafter_isi_inv_dx … H3 … Hf1) -f1 #Hf2
   >(lifts_inv_sort1 … H2) -U /2 width=1 by frees_sort/
 | #f1 #i #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
   elim (lifts_inv_lref1 … H2) -H2 #j #Hij #H destruct
   elim (coafter_fwd_xnx_pushs … Hij H3) -H3 #g2 #Hg2 #H2 destruct
-  lapply (coafter_isid_inv_dx … Hg2 … Hf1) -f1 #Hf2
+  lapply (pr_coafter_isi_inv_dx … Hg2 … Hf1) -f1 #Hf2
   elim (drops_inv_atom2 … H1) -H1 #n #g #H1 #Hf
-  elim (after_at_fwd … Hij … Hf) -f #x #_ #Hj -g -i
-  lapply (at_inv_uni … Hj) -Hj #H destruct
+  elim (pr_after_pat_des … Hij … Hf) -f #x #_ #Hj -g -i
+  lapply (pr_pat_inv_uni … Hj) -Hj #H destruct
   /3 width=8 by frees_atom_drops, drops_trans/
 | #f1 #I #K #V #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
-  lapply (isfin_inv_next … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
+  lapply (pr_isf_inv_next … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
   lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
   elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #H
   elim (liftsb_inv_pair_sn … H) -H #W #HVW #H destruct
@@ -123,123 +112,129 @@ lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 →
 | #f1 #I #K #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
   lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
   elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
-  lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hg2
+  lapply (pr_coafter_isi_inv_dx … H3 … Hf1) -f1 #Hg2
   elim (drops_split_trans_bind2 … H1 … Hf) -H1 -Hf #Z #Y #HLY #_ #H
   lapply (liftsb_inv_unit_sn … H) -H #H destruct
   /2 width=3 by frees_unit_drops/
 | #f1 #I #K #i #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
-  lapply (isfin_inv_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
+  lapply (pr_isf_inv_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
   lapply (lifts_inv_lref1 … H2) -H2 * #x #Hf #H destruct
-  elim (at_inv_nxx … Hf) -Hf [ |*: // ] #j #Hf #H destruct
+  elim (pr_pat_inv_succ_sn … Hf) -Hf [ |*: // ] #j #Hf #H destruct
   elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #_
   elim (coafter_fwd_xpx_pushs … 0 … H3) [ |*: // ] #g2 #H3 #H2 destruct
   lapply (drops_isuni_fwd_drop2 … HLY) -HLY // #HLY
   lapply (IH … HYK … H3) -IH -H3 -HYK [4: |*: /2 width=2 by lifts_lref/ ]
-  >plus_S1 /2 width=3 by frees_lref_pushs/ (**) (* full auto fails *)
+  >nplus_succ_sn /2 width=3 by frees_lref_pushs/ (**) (* full auto fails *)
 | #f1 #K #l #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
-  lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
+  lapply (pr_coafter_isi_inv_dx … H3 … Hf1) -f1 #Hf2
   >(lifts_inv_gref1 … H2) -U /2 width=1 by frees_gref/
 | #f1V #f1T #f1 #p #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
-  elim (sor_inv_isfin3 … H1f1) // #Hf1V #H
-  lapply (isfin_inv_tl … H) -H
+  elim (pr_sor_inv_isf … H1f1) // #Hf1V #H
+  lapply (pr_isf_inv_tl … H) -H
   elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct
-  elim (coafter_sor … H3 … H1f1) /2 width=5 by coafter_isfin2_fwd/ -H3 -H1f1 #f2V #f2T #Hf2V #H
-  elim (coafter_inv_tl1 … H) -H
+  elim (pr_sor_coafter_dx_tans … H3 … H1f1) /2 width=5 by pr_coafter_des_ist_isf/ -H3 -H1f1 #f2V #f2T #Hf2V #H
+  elim (pr_coafter_inv_tl_dx … H) -H
   /5 width=5 by frees_bind, drops_skip, ext2_pair/
 | #f1V #f1T #f1 #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
-  elim (sor_inv_isfin3 … H1f1) //
+  elim (pr_sor_inv_isf … H1f1) //
   elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
-  elim (coafter_sor … H3 … H1f1)
-  /3 width=5 by coafter_isfin2_fwd, frees_flat/
+  elim (pr_sor_coafter_dx_tans … H3 … H1f1)
+  /3 width=5 by pr_coafter_des_ist_isf, frees_flat/
 ]
 qed-.
 
-lemma frees_lifts_SO: ∀b,L,K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
-                      ∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f.
+lemma frees_lifts_SO:
+      ∀b,L,K. ⇩*[b,𝐔❨1❩] L ≘ K → ∀T,U. ⇧[1] T ≘ U →
+      ∀f. K ⊢ 𝐅+❪T❫ ≘ f → L ⊢ 𝐅+❪U❫ ≘ ⫯f.
 #b #L #K #HLK #T #U #HTU #f #Hf
 @(frees_lifts b … Hf … HTU) //  (**) (* auto fails *)
 qed.
 
 (* Forward lemmas with generic slicing for local environments ***************)
 
-lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
-                         ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
-                         ∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
-/4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
+lemma frees_fwd_coafter:
+      ∀b,f2,L,U. L ⊢ 𝐅+❪U❫ ≘ f2 →
+      ∀f,K. ⇩*[b,f] L ≘ K → ∀T. ⇧*[f] T ≘ U →
+      ∀f1. K ⊢ 𝐅+❪T❫ ≘ f1 → f ~⊚ f1 ≘ f2.
+/4 width=11 by frees_lifts, frees_mono, pr_coafter_eq_repl_back/ qed-.
 
 (* Inversion lemmas with generic slicing for local environments *************)
 
-lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
-                          ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
-                          ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅*⦃T⦄ ≘ f1.
+lemma frees_inv_lifts_ex:
+      ∀b,f2,L,U. L ⊢ 𝐅+❪U❫ ≘ f2 →
+      ∀f,K. ⇩*[b,f] L ≘ K → ∀T. ⇧*[f] T ≘ U →
+      ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅+❪T❫ ≘ f1.
 #b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T)
 /3 width=9 by frees_fwd_coafter, ex2_intro/
 qed-.
 
-lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f →
-                          ∀K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
-                          K ⊢ 𝐅*⦃T⦄ ≘ ⫱f.
+lemma frees_inv_lifts_SO:
+      ∀b,f,L,U. L ⊢ 𝐅+❪U❫ ≘ f →
+      ∀K. ⇩*[b,𝐔❨1❩] L ≘ K → ∀T. ⇧[1] T ≘ U →
+      K ⊢ 𝐅+❪T❫ ≘ ⫰f.
 #b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
-#f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf
-/3 width=5 by frees_eq_repl_back, coafter_isid_inv_sn/
+#f1 #Hf #Hf1 elim (pr_coafter_inv_next_sn … Hf) -Hf
+/3 width=5 by frees_eq_repl_back, pr_coafter_isi_inv_sn/
 qed-.
 
-lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
-                       ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
-                       ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅*⦃T⦄ ≘ f1.
+lemma frees_inv_lifts:
+      ∀b,f2,L,U. L ⊢ 𝐅+❪U❫ ≘ f2 →
+      ∀f,K. ⇩*[b,f] L ≘ K → ∀T. ⇧*[f] T ≘ U →
+      ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅+❪T❫ ≘ f1.
 #b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
-/3 width=7 by frees_eq_repl_back, coafter_inj/
+/3 width=7 by frees_eq_repl_back, pr_coafter_inj/
 qed-.
 
 (* Note: this is used by rex_conf and might be modified *)
-lemma frees_inv_drops_next: ∀f1,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 →
-                            ∀I2,L2,V2,n. ⬇*[n] L1 ≘ L2.ⓑ{I2}V2 →
-                            ∀g1. ↑g1 = ⫱*[n] f1 →
-                            ∃∃g2. L2 ⊢ 𝐅*⦃V2⦄ ≘ g2 & g2 ⊆ g1.
+lemma frees_inv_drops_next:
+      ∀f1,L1,T1. L1 ⊢ 𝐅+❪T1❫ ≘ f1 →
+      ∀I2,L2,V2,i. ⇩[i] L1 ≘ L2.ⓑ[I2]V2 →
+      ∀g1. ↑g1 = ⫰*[i] f1 →
+      ∃∃g2. L2 ⊢ 𝐅+❪V2❫ ≘ g2 & g2 ⊆ g1.
 #f1 #L1 #T1 #H elim H -f1 -L1 -T1
-[ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -s
-  lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
-  elim (isid_inv_next … Hf1) -Hf1 //
-| #f1 #i #_ #I2 #L2 #V2 #n #H
+[ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #j #_ #g1 #H1 -I2 -L1 -s
+  lapply (pr_isi_tls j … Hf1) -Hf1 <H1 -f1 #Hf1
+  elim (pr_isi_inv_next … Hf1) -Hf1 //
+| #f1 #i #_ #I2 #L2 #V2 #j #H
   elim (drops_inv_atom1 … H) -H #H destruct
 | #f1 #I1 #L1 #V1 #Hf1 #IH #I2 #L2 #V2 *
   [ -IH #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 //
-    #H destruct #g1 #Hgf1 >(injective_next … Hgf1) -g1
+    #H destruct #g1 #Hgf1 >(eq_inv_pr_next_bi … Hgf1) -g1
     /2 width=3 by ex2_intro/
-  | -Hf1 #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
-    #HL12 #g1 <tls_xn <tl_next_rew #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
+  | -Hf1 #j #HL12 lapply (drops_inv_drop1 … HL12) -HL12
+    #HL12 #g1 <pr_tls_swap <pr_tl_next #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
     /2 width=3 by ex2_intro/
   ]
 | #f1 #I1 #L1 #Hf1 #I2 #L2 #V2 *
   [ #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct
-  | #n #_ #g1 #Hgf1 elim (isid_inv_next … Hgf1) -Hgf1 <tls_xn /2 width=1 by isid_tls/
+  | #j #_ #g1 #Hgf1 elim (pr_isi_inv_next … Hgf1) -Hgf1 <pr_tls_swap /2 width=1 by pr_isi_tls/
   ]
 | #f1 #I1 #L1 #i #_ #IH #I2 #L2 #V2 *
-  [ -IH #_ #g1 #Hgf1 elim (discr_next_push … Hgf1)
-  | #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
-    #HL12 #g1 <tls_xn #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
+  [ -IH #_ #g1 #Hgf1 elim (eq_inv_pr_next_push … Hgf1)
+  | #j #HL12 lapply (drops_inv_drop1 … HL12) -HL12
+    #HL12 #g1 <pr_tls_swap #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
     /2 width=3 by ex2_intro/
   ]
-| #f1 #L1 #l #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -l
-  lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
-  elim (isid_inv_next … Hf1) -Hf1 //
-| #fV1 #fT1 #f1 #p #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
-  lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
-  elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
+| #f1 #L1 #l #Hf1 #I2 #L2 #V2 #j #_ #g1 #H1 -I2 -L1 -l
+  lapply (pr_isi_tls j … Hf1) -Hf1 <H1 -f1 #Hf1
+  elim (pr_isi_inv_next … Hf1) -Hf1 //
+| #fV1 #fT1 #f1 #p #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #j #HL12 #g1 #Hgf1
+  lapply (pr_sor_tls … Hf1 j) -Hf1 <Hgf1 -Hgf1 #Hf1
+  elim (pr_sor_next_tl … Hf1) [1,2: * |*: // ] -Hf1
   #gV1 #gT1 #Hg1
   [ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
-    /3 width=6 by sor_inv_sle_sn_trans, ex2_intro/
-  | -IHV1 #_ >tls_xn #H2 elim (IHT1 … H2) -IHT1 -H2
-    /3 width=6 by drops_drop, sor_inv_sle_dx_trans, ex2_intro/
+    /3 width=6 by pr_sor_inv_sle_sn_trans, ex2_intro/
+  | -IHV1 #_ >pr_tls_swap #H2 elim (IHT1 … H2) -IHT1 -H2
+    /3 width=6 by drops_drop, pr_sor_inv_sle_dx_trans, ex2_intro/
   ]
-| #fV1 #fT1 #f1 #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
-  lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
-  elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
+| #fV1 #fT1 #f1 #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #j #HL12 #g1 #Hgf1
+  lapply (pr_sor_tls … Hf1 j) -Hf1 <Hgf1 -Hgf1 #Hf1
+  elim (pr_sor_next_tl … Hf1) [1,2: * |*: // ] -Hf1
   #gV1 #gT1 #Hg1
   [ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
-    /3 width=6 by sor_inv_sle_sn_trans, ex2_intro/
+    /3 width=6 by pr_sor_inv_sle_sn_trans, ex2_intro/
   | -IHV1 #_ #H2 elim (IHT1 … HL12 … H2) -IHT1 -HL12 -H2
-    /3 width=6 by sor_inv_sle_dx_trans, ex2_intro/
+    /3 width=6 by pr_sor_inv_sle_dx_trans, ex2_intro/
   ]
 ]
 qed-.