X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fequivalence%2Fcpcs_cpcs.ma;h=7b9645afa39f8f845e6c8c52cb368ba9363e4a77;hb=e62715437a9c39244c9809c00585a5ef44a39797;hp=125cc70feaa77829b63db3bafc5a5a2695093da0;hpb=29973426e0227ee48368d1c24dc0c17bf2baef77;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
index 125cc70fe..7b9645afa 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
@@ -20,194 +20,193 @@ include "basic_2/equivalence/cpcs_cprs.ma".
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpcs_inv_cprs: ∀L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+lemma cpcs_inv_cprs: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
                      ∃∃T. ⦃G, L⦄ ⊢ T1 ➡* T & ⦃G, L⦄ ⊢ T2 ➡* T.
-#L #T1 #T2 #H @(cpcs_ind … H) -T2
-[ /3 width=3/
+#G #L #T1 #T2 #H @(cpcs_ind … H) -T2
+[ /3 width=3 by ex2_intro/
 | #T #T2 #_ #HT2 * #T0 #HT10 elim HT2 -HT2 #HT2 #HT0
-  [ elim (cprs_strip … HT0 … HT2) -T #T #HT0 #HT2
-    lapply (cprs_strap1 … HT10 … HT0) -T0 /2 width=3/
-  | lapply (cprs_strap2 … HT2 … HT0) -T /2 width=3/
+  [ elim (cprs_strip … HT0 … HT2) -T /3 width=3 by cprs_strap1, ex2_intro/
+  | /3 width=5 by cprs_strap2, ex2_intro/
   ]
 ]
 qed-.
 
 (* Basic_1: was: pc3_gen_sort *)
-lemma cpcs_inv_sort: ∀L,k1,k2. ⦃G, L⦄ ⊢ ⋆k1 ⬌* ⋆k2 → k1 = k2.
-#L #k1 #k2 #H
-elim (cpcs_inv_cprs … H) -H #T #H1
->(cprs_inv_sort1 … H1) -T #H2
+lemma cpcs_inv_sort: ∀G,L,s1,s2. ⦃G, L⦄ ⊢ ⋆s1 ⬌* ⋆s2 → s1 = s2.
+#G #L #s1 #s2 #H elim (cpcs_inv_cprs … H) -H
+#T #H1 >(cprs_inv_sort1 … H1) -T #H2
 lapply (cprs_inv_sort1 … H2) -L #H destruct //
 qed-.
 
-lemma cpcs_inv_abst1: ∀a,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T →
+lemma cpcs_inv_abst1: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ⬌* T →
                       ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
-#a #L #W1 #T1 #T #H
+#a #G #L #W1 #T1 #T #H
 elim (cpcs_inv_cprs … H) -H #X #H1 #H2
 elim (cprs_inv_abst1 … H1) -H1 #W2 #T2 #HW12 #HT12 #H destruct
-@(ex2_2_intro … H2) -H2 /2 width=2/ (**) (* explicit constructor, /3 width=6/ is slow *)
+/3 width=6 by cprs_bind, ex2_2_intro/
 qed-.
 
-lemma cpcs_inv_abst2: ∀a,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 →
+lemma cpcs_inv_abst2: ∀a,G,L,W1,T1,T. ⦃G, L⦄ ⊢ T ⬌* ⓛ{a}W1.T1 →
                       ∃∃W2,T2. ⦃G, L⦄ ⊢ T ➡* ⓛ{a}W2.T2 & ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2.
 /3 width=1 by cpcs_inv_abst1, cpcs_sym/ qed-.
 
 (* Basic_1: was: pc3_gen_sort_abst *)
-lemma cpcs_inv_sort_abst: ∀a,L,W,T,k. ⦃G, L⦄ ⊢ ⋆k ⬌* ⓛ{a}W.T → ⊥.
-#a #L #W #T #k #H
+lemma cpcs_inv_sort_abst: ∀a,G,L,W,T,s. ⦃G, L⦄ ⊢ ⋆s ⬌* ⓛ{a}W.T → ⊥.
+#a #G #L #W #T #s #H
 elim (cpcs_inv_cprs … H) -H #X #H1
 >(cprs_inv_sort1 … H1) -X #H2
 elim (cprs_inv_abst1 … H2) -H2 #W0 #T0 #_ #_ #H destruct
 qed-.
 
 (* Basic_1: was: pc3_gen_lift *)
-lemma cpcs_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
-                     ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
-                     ⦃G, L⦄ ⊢ U1 ⬌* U2 → K ⊢ T1 ⬌* T2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
+lemma cpcs_inv_lift: ∀G,L,K,b,l,k. ⬇[b, l, k] L ≘ K →
+                     ∀T1,U1. ⬆[l, k] T1 ≘ U1 → ∀T2,U2. ⬆[l, k] T2 ≘ U2 →
+                     ⦃G, L⦄ ⊢ U1 ⬌* U2 → ⦃G, K⦄ ⊢ T1 ⬌* T2.
+#G #L #K #b #l #k #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
 elim (cpcs_inv_cprs … HU12) -HU12 #U #HU1 #HU2
 elim (cprs_inv_lift1 … HU1 … HLK … HTU1) -U1 #T #HTU #HT1
 elim (cprs_inv_lift1 … HU2 … HLK … HTU2) -L -U2 #X #HXU
->(lift_inj … HXU … HTU) -X -U -d -e /2 width=3/
+>(lift_inj … HXU … HTU) -X -U -l -k /2 width=3 by cprs_div/
 qed-.
 
 (* Advanced properties ******************************************************)
 
-lemma lpr_cpcs_trans: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
-#L1 #L2 #HL12 #T1 #T2 #H
-elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
-lapply (lpr_cprs_trans … HT1 … HL12) -HT1
-lapply (lpr_cprs_trans … HT2 … HL12) -L2 /2 width=3/
+lemma lpr_cpcs_trans: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                      ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
+/4 width=5 by cprs_div, lpr_cprs_trans/
 qed-.
 
-lemma lprs_cpcs_trans: ∀L1,L2. L1 ⊢ ➡* L2 → ∀T1,T2. L2 ⊢ T1 ⬌* T2 → L1 ⊢ T1 ⬌* T2.
-#L1 #L2 #HL12 #T1 #T2 #H
-elim (cpcs_inv_cprs … H) -H #T #HT1 #HT2
-lapply (lprs_cprs_trans … HT1 … HL12) -HT1
-lapply (lprs_cprs_trans … HT2 … HL12) -L2 /2 width=3/
+lemma lprs_cpcs_trans: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
+                       ∀T1,T2. ⦃G, L2⦄ ⊢ T1 ⬌* T2 → ⦃G, L1⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
+/4 width=5 by cprs_div, lprs_cprs_trans/
 qed-.
 
-lemma cpr_cprs_conf_cpcs: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-#L #T #T1 #T2 #HT1 #HT2
-elim (cprs_strip … HT1 … HT2) /2 width=3 by cpr_cprs_div/
+lemma cpr_cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cpr_cprs_div/
 qed-.
 
-lemma cprs_cpr_conf_cpcs: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1.
-#L #T #T1 #T2 #HT1 #HT2
-elim (cprs_strip … HT1 … HT2) /2 width=3 by cprs_cpr_div/
+lemma cprs_cpr_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡ T2 → ⦃G, L⦄ ⊢ T2 ⬌* T1.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_strip … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cprs_cpr_div/
 qed-.
 
-lemma cprs_conf_cpcs: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-#L #T #T1 #T2 #HT1 #HT2
-elim (cprs_conf … HT1 … HT2) /2 width=3/
+lemma cprs_conf_cpcs: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ➡* T1 → ⦃G, L⦄ ⊢ T ➡* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T #T1 #T2 #HT1 #HT2 elim (cprs_conf … HT1 … HT2) -HT1 -HT2
+/2 width=3 by cprs_div/
 qed-.
 
-lemma lprs_cprs_conf: ∀L1,L2. L1 ⊢ ➡* L2 → ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #HL12 #T1 #T2 #HT12
-elim (lprs_cprs_conf_dx … HT12 … HL12) -L1 /2 width=3/
+lemma lprs_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
+                      ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (lprs_cprs_conf_dx … HT12 … HL12) -L1
+/2 width=3 by cprs_div/
 qed-.
 
 (* Basic_1: was: pc3_wcpr0_t *)
 (* Basic_1: note: pc3_wcpr0_t should be renamed *)
-lemma lpr_cprs_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡* T2 → L2 ⊢ T1 ⬌* T2.
-/3 width=5 by lprs_cprs_conf, lpr_lprs/ qed-.
+(* Note: alternative proof /3 width=5 by lprs_cprs_conf, lpr_lprs/ *)
+lemma lpr_cprs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                     ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #HL12 #T1 #T2 #HT12 elim (cprs_lpr_conf_dx … HT12 … HL12) -L1
+/2 width=3 by cprs_div/
+qed-.
 
 (* Basic_1: was only: pc3_pr0_pr2_t *)
 (* Basic_1: note: pc3_pr0_pr2_t should be renamed *)
-lemma lpr_cpr_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ➡ T2 → L2 ⊢ T1 ⬌* T2.
+lemma lpr_cpr_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                    ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ➡ T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
 /3 width=5 by lpr_cprs_conf, cpr_cprs/ qed-.
 
 (* Basic_1: was only: pc3_thin_dx *)
-lemma cpcs_flat: ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
-                 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
-#L #V1 #V2 #HV12 #T1 #T2 #HT12 #I
-elim (cpcs_inv_cprs … HV12) -HV12 #V #HV1 #HV2
-elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_flat, cprs_div/ (**) (* /3 width=5/ is too slow *)
+lemma cpcs_flat: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                 ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
+#G #L #V1 #V2 #HV12 #T1 #T2 #HT12
+elim (cpcs_inv_cprs … HV12) -HV12
+elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_flat, cprs_div/
 qed.
 
-lemma cpcs_flat_dx_cpr_rev: ∀L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
-                            ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1. T1 ⬌* ⓕ{I}V2. T2.
-/3 width=1/ qed.
+lemma cpcs_flat_dx_cpr_rev: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V2 ➡ V1 → ∀T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T2 →
+                            ∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ⬌* ⓕ{I}V2.T2.
+/3 width=1 by cpr_cpcs_sn, cpcs_flat/ qed.
 
-lemma cpcs_bind_dx: ∀a,I,L,V,T1,T2. L.ⓑ{I}V ⊢ T1 ⬌* T2 →
-                    ⦃G, L⦄ ⊢ ⓑ{a,I}V. T1 ⬌* ⓑ{a,I}V. T2.
-#a #I #L #V #T1 #T2 #HT12
-elim (cpcs_inv_cprs … HT12) -HT12 /3 width=5 by cprs_div, cprs_bind/ (**) (* /3 width=5/ is a bit slow *)
+lemma cpcs_bind_dx: ∀a,I,G,L,V,T1,T2. ⦃G, L.ⓑ{I}V⦄ ⊢ T1 ⬌* T2 →
+                    ⦃G, L⦄ ⊢ ⓑ{a,I}V.T1 ⬌* ⓑ{a,I}V.T2.
+#a #I #G #L #V #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_div, cprs_bind/
 qed.
 
-lemma cpcs_bind_sn: ∀a,I,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
-#a #I #L #V1 #V2 #T #HV12
-elim (cpcs_inv_cprs … HV12) -HV12 /3 width=5 by cprs_div, cprs_bind/ (**) (* /3 width=5/ is a bit slow *)
+lemma cpcs_bind_sn: ∀a,I,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T ⬌* ⓑ{a,I}V2. T.
+#a #I #G #L #V1 #V2 #T #HV12 elim (cpcs_inv_cprs … HV12) -HV12
+/3 width=5 by cprs_div, cprs_bind/
 qed.
 
-lemma lsubr_cpcs_trans: ∀L1,T1,T2. L1 ⊢ T1 ⬌* T2 →
-                        ∀L2. L2 ⊑ L1 → L2 ⊢ T1 ⬌* T2.
-#L1 #T1 #T2 #HT12
-elim (cpcs_inv_cprs … HT12) -HT12
-/3 width=5 by cprs_div, lsubr_cprs_trans/ (**) (* /3 width=5/ is a bit slow *)
+lemma lsubr_cpcs_trans: ∀G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 →
+                        ∀L2. L2 ⫃ L1 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #T1 #T2 #HT12 elim (cpcs_inv_cprs … HT12) -HT12
+/3 width=5 by cprs_div, lsubr_cprs_trans/
 qed-.
 
 (* Basic_1: was: pc3_lift *)
-lemma cpcs_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
-                 ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀T2,U2. ⇧[d, e] T2 ≡ U2 →
-                 K ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
-#L #K #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
+lemma cpcs_lift: ∀G,L,K,b,l,k. ⬇[b, l, k] L ≘ K →
+                 ∀T1,U1. ⬆[l, k] T1 ≘ U1 → ∀T2,U2. ⬆[l, k] T2 ≘ U2 →
+                 ⦃G, K⦄ ⊢ T1 ⬌* T2 → ⦃G, L⦄ ⊢ U1 ⬌* U2.
+#G #L #K #b #l #k #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
 elim (cpcs_inv_cprs … HT12) -HT12 #T #HT1 #HT2
-elim (lift_total T d e) #U #HTU
-lapply (cprs_lift … HT1 … HLK … HTU1 … HTU) -T1 #HU1
-lapply (cprs_lift … HT2 … HLK … HTU2 … HTU) -K -T2 -T -d -e /2 width=3/
+elim (lift_total T l k) /3 width=12 by cprs_div, cprs_lift/
 qed.
 
-lemma cpcs_strip: ∀L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
+lemma cpcs_strip: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T ⬌* T1 → ∀T2. ⦃G, L⦄ ⊢ T ⬌ T2 →
                   ∃∃T0. ⦃G, L⦄ ⊢ T1 ⬌ T0 & ⦃G, L⦄ ⊢ T2 ⬌* T0.
-#L #T1 #T @TC_strip1 /2 width=3/ qed-.
+#G #L #T1 #T @TC_strip1 /2 width=3 by cpc_conf/ qed-.
 
 (* More inversion lemmas ****************************************************)
 
-lemma cpcs_inv_abst_sn: ∀a1,a2,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
-                        ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & L.ⓛW1 ⊢ T1 ⬌* T2 & a1 = a2.
-#a1 #a2 #L #W1 #W2 #T1 #T2 #H
+(* Note: there must be a proof suitable for llpr *)
+lemma cpcs_inv_abst_sn: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+                        ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ⬌* T2 & a1 = a2.
+#a1 #a2 #G #L #W1 #W2 #T1 #T2 #H
 elim (cpcs_inv_cprs … H) -H #T #H1 #H2
 elim (cprs_inv_abst1 … H1) -H1 #W0 #T0 #HW10 #HT10 #H destruct
 elim (cprs_inv_abst1 … H2) -H2 #W #T #HW2 #HT2 #H destruct
-lapply (lprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1/ -HT2 #HT2
-lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1/ -HT2 #HT2
+lapply (lprs_cprs_conf … (L.ⓛW) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2
+lapply (lprs_cpcs_trans … (L.ⓛW1) … HT2) /2 width=1 by lprs_pair/ -HT2 #HT2
 /4 width=3 by and3_intro, cprs_div, cpcs_cprs_div, cpcs_sym/
 qed-.
 
-lemma cpcs_inv_abst_dx: ∀a1,a2,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
-                        ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & L. ⓛW2 ⊢ T1 ⬌* T2 & a1 = a2.
-#a1 #a2 #L #W1 #W2 #T1 #T2 #HT12
-lapply (cpcs_sym … HT12) -HT12 #HT12
-elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1/
+lemma cpcs_inv_abst_dx: ∀a1,a2,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a1}W1.T1 ⬌* ⓛ{a2}W2.T2 →
+                        ∧∧ ⦃G, L⦄ ⊢ W1 ⬌* W2 & ⦃G, L.ⓛW2⦄ ⊢ T1 ⬌* T2 & a1 = a2.
+#a1 #a2 #G #L #W1 #W2 #T1 #T2 #HT12 lapply (cpcs_sym … HT12) -HT12
+#HT12 elim (cpcs_inv_abst_sn … HT12) -HT12 /3 width=1 by cpcs_sym, and3_intro/
 qed-.
 
 (* Main properties **********************************************************)
 
 (* Basic_1: was pc3_t *)
-theorem cpcs_trans: ∀L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-#L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
+theorem cpcs_trans: ∀G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬌* T → ∀T2. ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+#G #L #T1 #T #HT1 #T2 @(trans_TC … HT1) qed-.
 
-theorem cpcs_canc_sn: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-/3 width=3 by cpcs_trans, cpcs_sym/ qed-. (**) (* /3 width=3/ is too slow *)
+theorem cpcs_canc_sn: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T ⬌* T1 → ⦃G, L⦄ ⊢ T ⬌* T2 → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
 
-theorem cpcs_canc_dx: ∀L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
-/3 width=3 by cpcs_trans, cpcs_sym/ qed-. (**) (* /3 width=3/ is too slow *)
+theorem cpcs_canc_dx: ∀G,L,T,T1,T2. ⦃G, L⦄ ⊢ T1 ⬌* T → ⦃G, L⦄ ⊢ T2 ⬌* T → ⦃G, L⦄ ⊢ T1 ⬌* T2.
+/3 width=3 by cpcs_trans, cpcs_sym/ qed-.
 
-lemma cpcs_bind1: ∀a,I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V1 ⊢ T1 ⬌* T2 →
+lemma cpcs_bind1: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
+                  ∀T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ⬌* T2 →
                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
-#a #I #L #V1 #V2 #HV12 #T1 #T2 #HT12
-@(cpcs_trans … (ⓑ{a,I}V1.T2)) /2 width=1/
-qed.
+/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
 
-lemma cpcs_bind2: ∀a,I,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 → ∀T1,T2. L.ⓑ{I}V2 ⊢ T1 ⬌* T2 →
+lemma cpcs_bind2: ∀a,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬌* V2 →
+                  ∀T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ⬌* T2 →
                   ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ⬌* ⓑ{a,I}V2. T2.
-#a #I #L #V1 #V2 #HV12 #T1 #T2 #HT12
-@(cpcs_trans … (ⓑ{a,I}V2.T1)) /2 width=1/
-qed.
+/3 width=3 by cpcs_trans, cpcs_bind_sn, cpcs_bind_dx/ qed.
 
 (* Basic_1: was: pc3_wcpr0 *)
-lemma lpr_cpcs_conf: ∀L1,L2. L1 ⊢ ➡ L2 → ∀T1,T2. L1 ⊢ T1 ⬌* T2 → L2 ⊢ T1 ⬌* T2.
-#L1 #L2 #HL12 #T1 #T2 #H
-elim (cpcs_inv_cprs … H) -H /3 width=5 by cpcs_canc_dx, lpr_cprs_conf/
+lemma lpr_cpcs_conf: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 →
+                     ∀T1,T2. ⦃G, L1⦄ ⊢ T1 ⬌* T2 → ⦃G, L2⦄ ⊢ T1 ⬌* T2.
+#G #L1 #L2 #HL12 #T1 #T2 #H elim (cpcs_inv_cprs … H) -H
+/3 width=5 by cpcs_canc_dx, lpr_cprs_conf/
 qed-.