X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpx.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpx.ma;h=b69c9ebdefa75db660f1feeddb4dd9b41264a426;hb=8ec019202bff90959cf1a7158b309e7f83fa222e;hp=9e9dcf7b2e23a9e95e5616f1a221e5c72649d2f3;hpb=33d0a7a9029859be79b25b5a495e0f30dab11f37;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
index 9e9dcf7b2..b69c9ebde 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
@@ -25,7 +25,7 @@ include "basic_2/rt_transition/cpg.ma".
 (* EXTENDED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS **************)
 
 definition cpx (G) (L): relation2 term term ≝
-           λT1,T2. ∃c. ❪G,L❫ ⊢ T1 ⬈[sfull,rtc_eq_f,c] T2.
+           λT1,T2. ∃c. ❨G,L❩ ⊢ T1 ⬈[sfull,rtc_eq_f,c] T2.
 
 interpretation
   "extended context-sensitive parallel rt-transition (term)"
@@ -34,71 +34,71 @@ interpretation
 (* Basic properties *********************************************************)
 
 (* Basic_2A1: uses: cpx_st *)
-lemma cpx_qu (G) (L): ∀s1,s2. ❪G,L❫ ⊢ ⋆s1 ⬈ ⋆s2.
+lemma cpx_qu (G) (L): ∀s1,s2. ❨G,L❩ ⊢ ⋆s1 ⬈ ⋆s2.
 /3 width=2 by cpg_ess, ex_intro/ qed.
 
 lemma cpx_delta (G) (K):
-      ∀I,V1,V2,W2. ❪G,K❫ ⊢ V1 ⬈ V2 →
-      ⇧[1] V2 ≘ W2 → ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈ W2.
+      ∀I,V1,V2,W2. ❨G,K❩ ⊢ V1 ⬈ V2 →
+      ⇧[1] V2 ≘ W2 → ❨G,K.ⓑ[I]V1❩ ⊢ #0 ⬈ W2.
 #G #K * #V1 #V2 #W2 *
 /3 width=4 by cpg_delta, cpg_ell, ex_intro/
 qed.
 
 lemma cpx_lref (G) (K):
-      ∀I,T,U,i. ❪G,K❫ ⊢ #i ⬈ T →
-      ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈ U.
+      ∀I,T,U,i. ❨G,K❩ ⊢ #i ⬈ T →
+      ⇧[1] T ≘ U → ❨G,K.ⓘ[I]❩ ⊢ #↑i ⬈ U.
 #G #K #I #T #U #i *
 /3 width=4 by cpg_lref, ex_intro/
 qed.
 
 lemma cpx_bind (G) (L):
       ∀p,I,V1,V2,T1,T2.
-      ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈ T2 →
-      ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈ ⓑ[p,I]V2.T2.
+      ❨G,L❩ ⊢ V1 ⬈ V2 → ❨G,L.ⓑ[I]V1❩ ⊢ T1 ⬈ T2 →
+      ❨G,L❩ ⊢ ⓑ[p,I]V1.T1 ⬈ ⓑ[p,I]V2.T2.
 #G #L #p #I #V1 #V2 #T1 #T2 * #cV #HV12 *
 /3 width=2 by cpg_bind, ex_intro/
 qed.
 
 lemma cpx_flat (G) (L):
       ∀I,V1,V2,T1,T2.
-      ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L❫ ⊢ T1 ⬈ T2 →
-      ❪G,L❫ ⊢ ⓕ[I]V1.T1 ⬈ ⓕ[I]V2.T2.
+      ❨G,L❩ ⊢ V1 ⬈ V2 → ❨G,L❩ ⊢ T1 ⬈ T2 →
+      ❨G,L❩ ⊢ ⓕ[I]V1.T1 ⬈ ⓕ[I]V2.T2.
 #G #L * #V1 #V2 #T1 #T2 * #cV #HV12 *
 /3 width=5 by cpg_appl, cpg_cast, ex_intro/
 qed.
 
 lemma cpx_zeta (G) (L):
-      ∀T1,T. ⇧[1] T ≘ T1 → ∀T2. ❪G,L❫ ⊢ T ⬈ T2 →
-      ∀V. ❪G,L❫ ⊢ +ⓓV.T1 ⬈ T2.
+      ∀T1,T. ⇧[1] T ≘ T1 → ∀T2. ❨G,L❩ ⊢ T ⬈ T2 →
+      ∀V. ❨G,L❩ ⊢ +ⓓV.T1 ⬈ T2.
 #G #L #T1 #T #HT1 #T2 *
 /3 width=4 by cpg_zeta, ex_intro/
 qed.
 
 lemma cpx_eps (G) (L):
-      ∀V,T1,T2. ❪G,L❫ ⊢ T1 ⬈ T2 → ❪G,L❫ ⊢ ⓝV.T1 ⬈ T2.
+      ∀V,T1,T2. ❨G,L❩ ⊢ T1 ⬈ T2 → ❨G,L❩ ⊢ ⓝV.T1 ⬈ T2.
 #G #L #V #T1 #T2 *
 /3 width=2 by cpg_eps, ex_intro/
 qed.
 
 (* Basic_2A1: was: cpx_ct *)
 lemma cpx_ee (G) (L):
-      ∀V1,V2,T. ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L❫ ⊢ ⓝV1.T ⬈ V2.
+      ∀V1,V2,T. ❨G,L❩ ⊢ V1 ⬈ V2 → ❨G,L❩ ⊢ ⓝV1.T ⬈ V2.
 #G #L #V1 #V2 #T *
 /3 width=2 by cpg_ee, ex_intro/
 qed.
 
 lemma cpx_beta (G) (L):
       ∀p,V1,V2,W1,W2,T1,T2.
-      ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L.ⓛW1❫ ⊢ T1 ⬈ T2 →
-      ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ⬈ ⓓ[p]ⓝW2.V2.T2.
+      ❨G,L❩ ⊢ V1 ⬈ V2 → ❨G,L❩ ⊢ W1 ⬈ W2 → ❨G,L.ⓛW1❩ ⊢ T1 ⬈ T2 →
+      ❨G,L❩ ⊢ ⓐV1.ⓛ[p]W1.T1 ⬈ ⓓ[p]ⓝW2.V2.T2.
 #G #L #p #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 *
 /3 width=2 by cpg_beta, ex_intro/
 qed.
 
 lemma cpx_theta (G) (L):
       ∀p,V1,V,V2,W1,W2,T1,T2.
-      ❪G,L❫ ⊢ V1 ⬈ V → ⇧[1] V ≘ V2 → ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L.ⓓW1❫ ⊢ T1 ⬈ T2 →
-      ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ⬈ ⓓ[p]W2.ⓐV2.T2.
+      ❨G,L❩ ⊢ V1 ⬈ V → ⇧[1] V ≘ V2 → ❨G,L❩ ⊢ W1 ⬈ W2 → ❨G,L.ⓓW1❩ ⊢ T1 ⬈ T2 →
+      ❨G,L❩ ⊢ ⓐV1.ⓓ[p]W1.T1 ⬈ ⓓ[p]W2.ⓐV2.T2.
 #G #L #p #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
 /3 width=4 by cpg_theta, ex_intro/
 qed.
@@ -110,13 +110,13 @@ lemma cpx_refl (G) (L): reflexive … (cpx G L).
 (* Advanced properties ******************************************************)
 
 lemma cpx_pair_sn (G) (L):
-      ∀I,V1,V2. ❪G,L❫ ⊢ V1 ⬈ V2 →
-      ∀T. ❪G,L❫ ⊢ ②[I]V1.T ⬈ ②[I]V2.T.
+      ∀I,V1,V2. ❨G,L❩ ⊢ V1 ⬈ V2 →
+      ∀T. ❨G,L❩ ⊢ ②[I]V1.T ⬈ ②[I]V2.T.
 #G #L * /2 width=2 by cpx_flat, cpx_bind/
 qed.
 
 lemma cpg_cpx (Rs) (Rk) (c) (G) (L):
-      ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈[Rs,Rk,c] T2 → ❪G,L❫ ⊢ T1 ⬈ T2.
+      ∀T1,T2. ❨G,L❩ ⊢ T1 ⬈[Rs,Rk,c] T2 → ❨G,L❩ ⊢ T1 ⬈ T2.
 #Rs #Rk #c #G #L #T1 #T2 #H elim H -c -G -L -T1 -T2
 /2 width=3 by cpx_theta, cpx_beta, cpx_ee, cpx_eps, cpx_zeta, cpx_flat, cpx_bind, cpx_lref, cpx_delta/
 qed.
@@ -124,80 +124,80 @@ qed.
 (* Basic inversion lemmas ***************************************************)
 
 lemma cpx_inv_atom1 (G) (L):
-      ∀J,T2. ❪G,L❫ ⊢ ⓪[J] ⬈ T2 →
+      ∀J,T2. ❨G,L❩ ⊢ ⓪[J] ⬈ T2 →
       ∨∨ T2 = ⓪[J]
        | ∃∃s1,s2. T2 = ⋆s2 & J = Sort s1
-       | ∃∃I,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1 & J = LRef 0
-       | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ⬈ T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
+       | ∃∃I,K,V1,V2. ❨G,K❩ ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1 & J = LRef 0
+       | ∃∃I,K,T,i. ❨G,K❩ ⊢ #i ⬈ T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
 #G #L #J #T2 * #c #H elim (cpg_inv_atom1 … H) -H *
 /4 width=8 by or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex4_4_intro, ex2_2_intro, ex_intro/
 qed-.
 
 lemma cpx_inv_sort1 (G) (L):
-      ∀T2,s1. ❪G,L❫ ⊢ ⋆s1 ⬈ T2 →
+      ∀T2,s1. ❨G,L❩ ⊢ ⋆s1 ⬈ T2 →
       ∃s2. T2 = ⋆s2.
 #G #L #T2 #s1 * #c #H elim (cpg_inv_sort1 … H) -H *
 /2 width=2 by ex_intro/
 qed-.
 
 lemma cpx_inv_zero1 (G) (L):
-      ∀T2. ❪G,L❫ ⊢ #0 ⬈ T2 →
+      ∀T2. ❨G,L❩ ⊢ #0 ⬈ T2 →
       ∨∨ T2 = #0
-       | ∃∃I,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1.
+       | ∃∃I,K,V1,V2. ❨G,K❩ ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1.
 #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
 /4 width=7 by ex3_4_intro, ex_intro, or_introl, or_intror/
 qed-.
 
 lemma cpx_inv_lref1 (G) (L):
-      ∀T2,i. ❪G,L❫ ⊢ #↑i ⬈ T2 →
+      ∀T2,i. ❨G,L❩ ⊢ #↑i ⬈ T2 →
       ∨∨ T2 = #(↑i)
-       | ∃∃I,K,T. ❪G,K❫ ⊢ #i ⬈ T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
+       | ∃∃I,K,T. ❨G,K❩ ⊢ #i ⬈ T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
 #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
 /4 width=6 by ex3_3_intro, ex_intro, or_introl, or_intror/
 qed-.
 
 lemma cpx_inv_gref1 (G) (L):
-      ∀T2,l. ❪G,L❫ ⊢ §l ⬈ T2 → T2 = §l.
+      ∀T2,l. ❨G,L❩ ⊢ §l ⬈ T2 → T2 = §l.
 #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
 qed-.
 
 lemma cpx_inv_bind1 (G) (L):
-      ∀p,I,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈ U2 →
-      ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈ T2 & U2 = ⓑ[p,I]V2.T2
-       | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈ U2 & p = true & I = Abbr.
+      ∀p,I,V1,T1,U2. ❨G,L❩ ⊢ ⓑ[p,I]V1.T1 ⬈ U2 →
+      ∨∨ ∃∃V2,T2. ❨G,L❩ ⊢ V1 ⬈ V2 & ❨G,L.ⓑ[I]V1❩ ⊢ T1 ⬈ T2 & U2 = ⓑ[p,I]V2.T2
+       | ∃∃T. ⇧[1] T ≘ T1 & ❨G,L❩ ⊢ T ⬈ U2 & p = true & I = Abbr.
 #G #L #p #I #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
 /4 width=5 by ex4_intro, ex3_2_intro, ex_intro, or_introl, or_intror/
 qed-.
 
 lemma cpx_inv_abbr1 (G) (L):
-      ∀p,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ⬈ U2 →
-      ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L.ⓓV1❫ ⊢ T1 ⬈ T2 & U2 = ⓓ[p]V2.T2
-       | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈ U2 & p = true.
+      ∀p,V1,T1,U2. ❨G,L❩ ⊢ ⓓ[p]V1.T1 ⬈ U2 →
+      ∨∨ ∃∃V2,T2. ❨G,L❩ ⊢ V1 ⬈ V2 & ❨G,L.ⓓV1❩ ⊢ T1 ⬈ T2 & U2 = ⓓ[p]V2.T2
+       | ∃∃T. ⇧[1] T ≘ T1 & ❨G,L❩ ⊢ T ⬈ U2 & p = true.
 #G #L #p #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
 /4 width=5 by ex3_2_intro, ex3_intro, ex_intro, or_introl, or_intror/
 qed-.
 
 lemma cpx_inv_abst1 (G) (L):
-      ∀p,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ⬈ U2 →
-      ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L.ⓛV1❫ ⊢ T1 ⬈ T2 & U2 = ⓛ[p]V2.T2.
+      ∀p,V1,T1,U2. ❨G,L❩ ⊢ ⓛ[p]V1.T1 ⬈ U2 →
+      ∃∃V2,T2. ❨G,L❩ ⊢ V1 ⬈ V2 & ❨G,L.ⓛV1❩ ⊢ T1 ⬈ T2 & U2 = ⓛ[p]V2.T2.
 #G #L #p #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H
 /3 width=5 by ex3_2_intro, ex_intro/
 qed-.
 
 lemma cpx_inv_appl1 (G) (L):
-      ∀V1,U1,U2. ❪G,L❫ ⊢ ⓐ V1.U1 ⬈ U2 →
-      ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L❫ ⊢ U1 ⬈ T2 & U2 = ⓐV2.T2
-       | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L❫ ⊢ W1 ⬈ W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈ T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2
-       | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V & ⇧[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈ W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈ T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2.
+      ∀V1,U1,U2. ❨G,L❩ ⊢ ⓐ V1.U1 ⬈ U2 →
+      ∨∨ ∃∃V2,T2. ❨G,L❩ ⊢ V1 ⬈ V2 & ❨G,L❩ ⊢ U1 ⬈ T2 & U2 = ⓐV2.T2
+       | ∃∃p,V2,W1,W2,T1,T2. ❨G,L❩ ⊢ V1 ⬈ V2 & ❨G,L❩ ⊢ W1 ⬈ W2 & ❨G,L.ⓛW1❩ ⊢ T1 ⬈ T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2
+       | ∃∃p,V,V2,W1,W2,T1,T2. ❨G,L❩ ⊢ V1 ⬈ V & ⇧[1] V ≘ V2 & ❨G,L❩ ⊢ W1 ⬈ W2 & ❨G,L.ⓓW1❩ ⊢ T1 ⬈ T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2.
 #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
 /4 width=13 by or3_intro0, or3_intro1, or3_intro2, ex6_7_intro, ex5_6_intro, ex3_2_intro, ex_intro/
 qed-.
 
 lemma cpx_inv_cast1 (G) (L):
-      ∀V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ⬈ U2 →
-      ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L❫ ⊢ U1 ⬈ T2 & U2 = ⓝV2.T2
-       | ❪G,L❫ ⊢ U1 ⬈ U2
-       | ❪G,L❫ ⊢ V1 ⬈ U2.
+      ∀V1,U1,U2. ❨G,L❩ ⊢ ⓝV1.U1 ⬈ U2 →
+      ∨∨ ∃∃V2,T2. ❨G,L❩ ⊢ V1 ⬈ V2 & ❨G,L❩ ⊢ U1 ⬈ T2 & U2 = ⓝV2.T2
+       | ❨G,L❩ ⊢ U1 ⬈ U2
+       | ❨G,L❩ ⊢ V1 ⬈ U2.
 #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H *
 /4 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex_intro/
 qed-.
@@ -205,28 +205,28 @@ qed-.
 (* Advanced inversion lemmas ************************************************)
 
 lemma cpx_inv_zero1_pair (G) (K):
-      ∀I,V1,T2. ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈ T2 →
+      ∀I,V1,T2. ❨G,K.ⓑ[I]V1❩ ⊢ #0 ⬈ T2 →
       ∨∨ T2 = #0
-       | ∃∃V2. ❪G,K❫ ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2.
+       | ∃∃V2. ❨G,K❩ ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2.
 #G #K #I #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
 /4 width=3 by ex2_intro, ex_intro, or_intror, or_introl/
 qed-.
 
 lemma cpx_inv_lref1_bind (G) (K):
-      ∀I,T2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈ T2 →
+      ∀I,T2,i. ❨G,K.ⓘ[I]❩ ⊢ #↑i ⬈ T2 →
       ∨∨ T2 = #(↑i)
-       | ∃∃T. ❪G,K❫ ⊢ #i ⬈ T & ⇧[1] T ≘ T2.
+       | ∃∃T. ❨G,K❩ ⊢ #i ⬈ T & ⇧[1] T ≘ T2.
 #G #K #I #T2 #i * #c #H elim (cpg_inv_lref1_bind … H) -H *
 /4 width=3 by ex2_intro, ex_intro, or_introl, or_intror/
 qed-.
 
 lemma cpx_inv_flat1 (G) (L):
-      ∀I,V1,U1,U2. ❪G,L❫ ⊢ ⓕ[I]V1.U1 ⬈ U2 →
-      ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L❫ ⊢ U1 ⬈ T2 & U2 = ⓕ[I]V2.T2
-       | (❪G,L❫ ⊢ U1 ⬈ U2 ∧ I = Cast)
-       | (❪G,L❫ ⊢ V1 ⬈ U2 ∧ I = Cast)
-       | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V2 & ❪G,L❫ ⊢ W1 ⬈ W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈ T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & I = Appl
-       | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V & ⇧[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈ W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈ T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & I = Appl.
+      ∀I,V1,U1,U2. ❨G,L❩ ⊢ ⓕ[I]V1.U1 ⬈ U2 →
+      ∨∨ ∃∃V2,T2. ❨G,L❩ ⊢ V1 ⬈ V2 & ❨G,L❩ ⊢ U1 ⬈ T2 & U2 = ⓕ[I]V2.T2
+       | (❨G,L❩ ⊢ U1 ⬈ U2 ∧ I = Cast)
+       | (❨G,L❩ ⊢ V1 ⬈ U2 ∧ I = Cast)
+       | ∃∃p,V2,W1,W2,T1,T2. ❨G,L❩ ⊢ V1 ⬈ V2 & ❨G,L❩ ⊢ W1 ⬈ W2 & ❨G,L.ⓛW1❩ ⊢ T1 ⬈ T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & I = Appl
+       | ∃∃p,V,V2,W1,W2,T1,T2. ❨G,L❩ ⊢ V1 ⬈ V & ⇧[1] V ≘ V2 & ❨G,L❩ ⊢ W1 ⬈ W2 & ❨G,L.ⓓW1❩ ⊢ T1 ⬈ T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & I = Appl.
 #G #L * #V1 #U1 #U2 #H
 [ elim (cpx_inv_appl1 … H) -H *
   /3 width=14 by or5_intro0, or5_intro3, or5_intro4, ex7_7_intro, ex6_6_intro, ex3_2_intro/
@@ -238,8 +238,8 @@ qed-.
 (* Basic forward lemmas *****************************************************)
 
 lemma cpx_fwd_bind1_minus (G) (L):
-      ∀I,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ⬈ T → ∀p.
-      ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈ ⓑ[p,I]V2.T2 & T = -ⓑ[I]V2.T2.
+      ∀I,V1,T1,T. ❨G,L❩ ⊢ -ⓑ[I]V1.T1 ⬈ T → ∀p.
+      ∃∃V2,T2. ❨G,L❩ ⊢ ⓑ[p,I]V1.T1 ⬈ ⓑ[p,I]V2.T2 & T = -ⓑ[I]V2.T2.
 #G #L #I #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
 /3 width=4 by ex2_2_intro, ex_intro/
 qed-.
@@ -249,28 +249,28 @@ qed-.
 lemma cpx_ind (Q:relation4 …):
       (∀I,G,L. Q G L (⓪[I]) (⓪[I])) →
       (∀G,L,s1,s2. Q G L (⋆s1) (⋆s2)) →
-      (∀I,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ⬈ V2 → Q G K V1 V2 →
+      (∀I,G,K,V1,V2,W2. ❨G,K❩ ⊢ V1 ⬈ V2 → Q G K V1 V2 →
         ⇧[1] V2 ≘ W2 → Q G (K.ⓑ[I]V1) (#0) W2
-      ) → (∀I,G,K,T,U,i. ❪G,K❫ ⊢ #i ⬈ T → Q G K (#i) T →
+      ) → (∀I,G,K,T,U,i. ❨G,K❩ ⊢ #i ⬈ T → Q G K (#i) T →
         ⇧[1] T ≘ U → Q G (K.ⓘ[I]) (#↑i) (U)
-      ) → (∀p,I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈ T2 →
+      ) → (∀p,I,G,L,V1,V2,T1,T2. ❨G,L❩ ⊢ V1 ⬈ V2 → ❨G,L.ⓑ[I]V1❩ ⊢ T1 ⬈ T2 →
         Q G L V1 V2 → Q G (L.ⓑ[I]V1) T1 T2 → Q G L (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2)
-      ) → (∀I,G,L,V1,V2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L❫ ⊢ T1 ⬈ T2 →
+      ) → (∀I,G,L,V1,V2,T1,T2. ❨G,L❩ ⊢ V1 ⬈ V2 → ❨G,L❩ ⊢ T1 ⬈ T2 →
         Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ[I]V1.T1) (ⓕ[I]V2.T2)
-      ) → (∀G,L,V,T1,T,T2. ⇧[1] T ≘ T1 → ❪G,L❫ ⊢ T ⬈ T2 → Q G L T T2 →
+      ) → (∀G,L,V,T1,T,T2. ⇧[1] T ≘ T1 → ❨G,L❩ ⊢ T ⬈ T2 → Q G L T T2 →
         Q G L (+ⓓV.T1) T2
-      ) → (∀G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ⬈ T2 → Q G L T1 T2 →
+      ) → (∀G,L,V,T1,T2. ❨G,L❩ ⊢ T1 ⬈ T2 → Q G L T1 T2 →
         Q G L (ⓝV.T1) T2
-      ) → (∀G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ⬈ V2 → Q G L V1 V2 →
+      ) → (∀G,L,V1,V2,T. ❨G,L❩ ⊢ V1 ⬈ V2 → Q G L V1 V2 →
         Q G L (ⓝV1.T) V2
-      ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V2 → ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L.ⓛW1❫ ⊢ T1 ⬈ T2 →
+      ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ❨G,L❩ ⊢ V1 ⬈ V2 → ❨G,L❩ ⊢ W1 ⬈ W2 → ❨G,L.ⓛW1❩ ⊢ T1 ⬈ T2 →
         Q G L V1 V2 → Q G L W1 W2 → Q G (L.ⓛW1) T1 T2 →
         Q G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2)
-      ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈ V → ❪G,L❫ ⊢ W1 ⬈ W2 → ❪G,L.ⓓW1❫ ⊢ T1 ⬈ T2 →
+      ) → (∀p,G,L,V1,V,V2,W1,W2,T1,T2. ❨G,L❩ ⊢ V1 ⬈ V → ❨G,L❩ ⊢ W1 ⬈ W2 → ❨G,L.ⓓW1❩ ⊢ T1 ⬈ T2 →
         Q G L V1 V → Q G L W1 W2 → Q G (L.ⓓW1) T1 T2 →
         ⇧[1] V ≘ V2 → Q G L (ⓐV1.ⓓ[p]W1.T1) (ⓓ[p]W2.ⓐV2.T2)
       ) →
-      ∀G,L,T1,T2. ❪G,L❫ ⊢ T1 ⬈ T2 → Q G L T1 T2.
+      ∀G,L,T1,T2. ❨G,L❩ ⊢ T1 ⬈ T2 → Q G L T1 T2.
 #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2
 * #c #H elim H -c -G -L -T1 -T2 /3 width=4 by ex_intro/
 qed-.