X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpg.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpg.ma;h=3062e81202db3091e6332676e3e6cf3a88131490;hb=8ec019202bff90959cf1a7158b309e7f83fa222e;hp=3bdc93e975173a1cace87a5f801e8efa425b7c60;hpb=33d0a7a9029859be79b25b5a495e0f30dab11f37;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma
index 3bdc93e97..3062e8120 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma
@@ -69,7 +69,7 @@ interpretation
 
 (* Note: this is "∀Rs,Rk. reflexive … Rk → ∀G,L. reflexive … (cpg Rs Rk (𝟘𝟘) G L)" *)
 lemma cpg_refl (Rs) (Rk):
-      reflexive … Rk → ∀G,T,L. ❪G,L❫ ⊢ T ⬈[Rs,Rk,𝟘𝟘] T.
+      reflexive … Rk → ∀G,T,L. ❨G,L❩ ⊢ T ⬈[Rs,Rk,𝟘𝟘] T.
 #Rk #HRk #h #G #T elim T -T // * /2 width=1 by cpg_bind/
 * /2 width=1 by cpg_appl, cpg_cast/
 qed.
@@ -77,12 +77,12 @@ qed.
 (* Basic inversion lemmas ***************************************************)
 
 fact cpg_inv_atom1_aux (Rs) (Rk) (c) (G) (L):
-     ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈[Rs,Rk,c] T2 → ∀J. T1 = ⓪[J] →
+     ∀T1,T2. ❨G,L❩ ⊢ T1 ⬈[Rs,Rk,c] T2 → ∀J. T1 = ⓪[J] →
      ∨∨ ∧∧ T2 = ⓪[J] & c = 𝟘𝟘
       | ∃∃s1,s2. Rs s1 s2 & J = Sort s1 & T2 = ⋆s2 & c = 𝟘𝟙
-      | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV
-      | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
-      | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
+      | ∃∃cV,K,V1,V2. ❨G,K❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV
+      | ∃∃cV,K,V1,V2. ❨G,K❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
+      | ∃∃I,K,T,i. ❨G,K❩ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
 #Rs #Rk #c #G #L #T1 #T2 * -c -G -L -T1 -T2
 [ #I #G #L #J #H destruct /3 width=1 by or5_intro0, conj/
 | #G #L #s1 #s2 #HRs #J #H destruct /3 width=5 by or5_intro1, ex4_2_intro/
@@ -101,16 +101,16 @@ fact cpg_inv_atom1_aux (Rs) (Rk) (c) (G) (L):
 qed-.
 
 lemma cpg_inv_atom1 (Rs) (Rk) (c) (G) (L):
-      ∀J,T2. ❪G,L❫ ⊢ ⓪[J] ⬈[Rs,Rk,c] T2 →
+      ∀J,T2. ❨G,L❩ ⊢ ⓪[J] ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = ⓪[J] & c = 𝟘𝟘
        | ∃∃s1,s2. Rs s1 s2 & J = Sort s1 & T2 = ⋆s2 & c = 𝟘𝟙
-       | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV
-       | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
-       | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
+       | ∃∃cV,K,V1,V2. ❨G,K❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV
+       | ∃∃cV,K,V1,V2. ❨G,K❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
+       | ∃∃I,K,T,i. ❨G,K❩ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
 /2 width=3 by cpg_inv_atom1_aux/ qed-.
 
 lemma cpg_inv_sort1 (Rs) (Rk) (c) (G) (L):
-      ∀T2,s1. ❪G,L❫ ⊢ ⋆s1 ⬈[Rs,Rk,c] T2 →
+      ∀T2,s1. ❨G,L❩ ⊢ ⋆s1 ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = ⋆s1 & c = 𝟘𝟘
        | ∃∃s2. Rs s1 s2 & T2 = ⋆s2 & c = 𝟘𝟙.
 #Rs #Rk #c #G #L #T2 #s #H
@@ -122,10 +122,10 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
 qed-.
 
 lemma cpg_inv_zero1 (Rs) (Rk) (c) (G) (L):
-      ∀T2. ❪G,L❫ ⊢ #0 ⬈[Rs,Rk,c] T2 →
+      ∀T2. ❨G,L❩ ⊢ #0 ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = #0 & c = 𝟘𝟘
-       | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & c = cV
-       | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & c = cV+𝟘𝟙.
+       | ∃∃cV,K,V1,V2. ❨G,K❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & c = cV
+       | ∃∃cV,K,V1,V2. ❨G,K❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & c = cV+𝟘𝟙.
 #Rs #Rk #c #G #L #T2 #H
 elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
 [ #s1 #s2 #_ #H destruct
@@ -135,9 +135,9 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
 qed-.
 
 lemma cpg_inv_lref1 (Rs) (Rk) (c) (G) (L):
-      ∀T2,i. ❪G,L❫ ⊢ #↑i ⬈[Rs,Rk,c] T2 →
+      ∀T2,i. ❨G,L❩ ⊢ #↑i ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = #(↑i) & c = 𝟘𝟘
-       | ∃∃I,K,T. ❪G,K❫ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
+       | ∃∃I,K,T. ❨G,K❩ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
 #Rs #Rk #c #G #L #T2 #i #H
 elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
 [ #s1 #s2 #_ #H destruct
@@ -147,7 +147,7 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
 qed-.
 
 lemma cpg_inv_gref1 (Rs) (Rk) (c) (G) (L):
-      ∀T2,l. ❪G,L❫ ⊢ §l ⬈[Rs,Rk,c] T2 → ∧∧ T2 = §l & c = 𝟘𝟘.
+      ∀T2,l. ❨G,L❩ ⊢ §l ⬈[Rs,Rk,c] T2 → ∧∧ T2 = §l & c = 𝟘𝟘.
 #Rs #Rk #c #G #L #T2 #l #H
 elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/
 [ #s1 #s2 #_ #H destruct
@@ -157,10 +157,10 @@ elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/
 qed-.
 
 fact cpg_inv_bind1_aux (Rs) (Rk) (c) (G) (L):
-     ∀U,U2. ❪G,L❫ ⊢ U ⬈[Rs,Rk,c] U2 →
+     ∀U,U2. ❨G,L❩ ⊢ U ⬈[Rs,Rk,c] U2 →
      ∀p,J,V1,U1. U = ⓑ[p,J]V1.U1 →
-     ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L.ⓑ[J]V1❫ ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓑ[p,J]V2.T2 & c = ((↕*cV)∨cT)
-      | ∃∃cT,T. ⇧[1] T ≘ U1 & ❪G,L❫ ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & J = Abbr & c = cT+𝟙𝟘.
+     ∨∨ ∃∃cV,cT,V2,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❨G,L.ⓑ[J]V1❩ ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓑ[p,J]V2.T2 & c = ((↕*cV)∨cT)
+      | ∃∃cT,T. ⇧[1] T ≘ U1 & ❨G,L❩ ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & J = Abbr & c = cT+𝟙𝟘.
 #Rs #Rk #c #G #L #U #U2 * -c -G -L -U -U2
 [ #I #G #L #q #J #W #U1 #H destruct
 | #G #L #s1 #s2 #_ #q #J #W #U1 #H destruct
@@ -179,22 +179,22 @@ fact cpg_inv_bind1_aux (Rs) (Rk) (c) (G) (L):
 qed-.
 
 lemma cpg_inv_bind1 (Rs) (Rk) (c) (G) (L):
-      ∀p,I,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[Rs,Rk,c] U2 →
-      ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓑ[p,I]V2.T2 & c = ((↕*cV)∨cT)
-       | ∃∃cT,T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & I = Abbr & c = cT+𝟙𝟘.
+      ∀p,I,V1,T1,U2. ❨G,L❩ ⊢ ⓑ[p,I]V1.T1 ⬈[Rs,Rk,c] U2 →
+      ∨∨ ∃∃cV,cT,V2,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❨G,L.ⓑ[I]V1❩ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓑ[p,I]V2.T2 & c = ((↕*cV)∨cT)
+       | ∃∃cT,T. ⇧[1] T ≘ T1 & ❨G,L❩ ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & I = Abbr & c = cT+𝟙𝟘.
 /2 width=3 by cpg_inv_bind1_aux/ qed-.
 
 lemma cpg_inv_abbr1 (Rs) (Rk) (c) (G) (L):
-      ∀p,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ⬈[Rs,Rk,c] U2 →
-      ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓓ[p]V2.T2 & c = ((↕*cV)∨cT)
-       | ∃∃cT,T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & c = cT+𝟙𝟘.
+      ∀p,V1,T1,U2. ❨G,L❩ ⊢ ⓓ[p]V1.T1 ⬈[Rs,Rk,c] U2 →
+      ∨∨ ∃∃cV,cT,V2,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❨G,L.ⓓV1❩ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓓ[p]V2.T2 & c = ((↕*cV)∨cT)
+       | ∃∃cT,T. ⇧[1] T ≘ T1 & ❨G,L❩ ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & c = cT+𝟙𝟘.
 #Rs #Rk #c #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
 /3 width=8 by ex4_4_intro, ex4_2_intro, or_introl, or_intror/
 qed-.
 
 lemma cpg_inv_abst1 (Rs) (Rk) (c) (G) (L):
-      ∀p,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ⬈[Rs,Rk,c] U2 →
-      ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓛ[p]V2.T2 & c = ((↕*cV)∨cT).
+      ∀p,V1,T1,U2. ❨G,L❩ ⊢ ⓛ[p]V1.T1 ⬈[Rs,Rk,c] U2 →
+      ∃∃cV,cT,V2,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❨G,L.ⓛV1❩ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓛ[p]V2.T2 & c = ((↕*cV)∨cT).
 #Rs #Rk #c #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
 [ /3 width=8 by ex4_4_intro/
 | #c #T #_ #_ #_ #H destruct
@@ -202,11 +202,11 @@ lemma cpg_inv_abst1 (Rs) (Rk) (c) (G) (L):
 qed-.
 
 fact cpg_inv_appl1_aux (Rs) (Rk) (c) (G) (L):
-     ∀U,U2. ❪G,L❫ ⊢ U ⬈[Rs,Rk,c] U2 →
+     ∀U,U2. ❨G,L❩ ⊢ U ⬈[Rs,Rk,c] U2 →
      ∀V1,U1. U = ⓐV1.U1 →
-     ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
-      | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
-      | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V & ⇧[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
+     ∨∨ ∃∃cV,cT,V2,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❨G,L❩ ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
+      | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❨G,L❩ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❨G,L.ⓛW1❩ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
+      | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V & ⇧[1] V ≘ V2 & ❨G,L❩ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❨G,L.ⓓW1❩ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
 #Rs #Rk #c #G #L #U #U2 * -c -G -L -U -U2
 [ #I #G #L #W #U1 #H destruct
 | #G #L #s1 #s2 #_ #W #U1 #H destruct
@@ -225,18 +225,18 @@ fact cpg_inv_appl1_aux (Rs) (Rk) (c) (G) (L):
 qed-.
 
 lemma cpg_inv_appl1 (Rs) (Rk) (c) (G) (L):
-      ∀V1,U1,U2. ❪G,L❫ ⊢ ⓐV1.U1 ⬈[Rs,Rk,c] U2 →
-      ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
-       | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
-       | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V & ⇧[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
+      ∀V1,U1,U2. ❨G,L❩ ⊢ ⓐV1.U1 ⬈[Rs,Rk,c] U2 →
+      ∨∨ ∃∃cV,cT,V2,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❨G,L❩ ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
+       | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❨G,L❩ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❨G,L.ⓛW1❩ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
+       | ∃∃cV,cW,cT,p,V,V2,W1,W2,T1,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V & ⇧[1] V ≘ V2 & ❨G,L❩ ⊢ W1 ⬈[Rs,Rk,cW] W2 & ❨G,L.ⓓW1❩ ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
 /2 width=3 by cpg_inv_appl1_aux/ qed-.
 
 fact cpg_inv_cast1_aux (Rs) (Rk) (c) (G) (L):
-     ∀U,U2. ❪G,L❫ ⊢ U ⬈[Rs,Rk,c] U2 →
+     ∀U,U2. ❨G,L❩ ⊢ U ⬈[Rs,Rk,c] U2 →
      ∀V1,U1. U = ⓝV1.U1 →
-     ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] T2 & Rk cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
-      | ∃∃cT. ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] U2 & c = cT+𝟙𝟘
-      | ∃∃cV. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] U2 & c = cV+𝟘𝟙.
+     ∨∨ ∃∃cV,cT,V2,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❨G,L❩ ⊢ U1 ⬈[Rs,Rk,cT] T2 & Rk cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
+      | ∃∃cT. ❨G,L❩ ⊢ U1 ⬈[Rs,Rk,cT] U2 & c = cT+𝟙𝟘
+      | ∃∃cV. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] U2 & c = cV+𝟘𝟙.
 #Rs #Rk #c #G #L #U #U2 * -c -G -L -U -U2
 [ #I #G #L #W #U1 #H destruct
 | #G #L #s1 #s2 #_ #W #U1 #H destruct
@@ -255,27 +255,27 @@ fact cpg_inv_cast1_aux (Rs) (Rk) (c) (G) (L):
 qed-.
 
 lemma cpg_inv_cast1 (Rs) (Rk) (c) (G) (L):
-      ∀V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ⬈[Rs,Rk,c] U2 →
-      ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] T2 & Rk cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
-       | ∃∃cT. ❪G,L❫ ⊢ U1 ⬈[Rs,Rk,cT] U2 & c = cT+𝟙𝟘
-       | ∃∃cV. ❪G,L❫ ⊢ V1 ⬈[Rs,Rk,cV] U2 & c = cV+𝟘𝟙.
+      ∀V1,U1,U2. ❨G,L❩ ⊢ ⓝV1.U1 ⬈[Rs,Rk,c] U2 →
+      ∨∨ ∃∃cV,cT,V2,T2. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ❨G,L❩ ⊢ U1 ⬈[Rs,Rk,cT] T2 & Rk cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
+       | ∃∃cT. ❨G,L❩ ⊢ U1 ⬈[Rs,Rk,cT] U2 & c = cT+𝟙𝟘
+       | ∃∃cV. ❨G,L❩ ⊢ V1 ⬈[Rs,Rk,cV] U2 & c = cV+𝟘𝟙.
 /2 width=3 by cpg_inv_cast1_aux/ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
 lemma cpg_inv_zero1_pair (Rs) (Rk) (c) (G) (K):
-      ∀I,V1,T2. ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈[Rs,Rk,c] T2 →
+      ∀I,V1,T2. ❨G,K.ⓑ[I]V1❩ ⊢ #0 ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = #0 & c = 𝟘𝟘
-       | ∃∃cV,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & I = Abbr & c = cV
-       | ∃∃cV,V2. ❪G,K❫ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & I = Abst & c = cV+𝟘𝟙.
+       | ∃∃cV,V2. ❨G,K❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & I = Abbr & c = cV
+       | ∃∃cV,V2. ❨G,K❩ ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & I = Abst & c = cV+𝟘𝟙.
 #Rs #Rk #c #G #K #I #V1 #T2 #H elim (cpg_inv_zero1 … H) -H /2 width=1 by or3_intro0/
 * #z #Y #X1 #X2 #HX12 #HXT2 #H1 #H2 destruct /3 width=5 by or3_intro1, or3_intro2, ex4_2_intro/
 qed-.
 
 lemma cpg_inv_lref1_bind (Rs) (Rk) (c) (G) (K):
-      ∀I,T2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈[Rs,Rk,c] T2 →
+      ∀I,T2,i. ❨G,K.ⓘ[I]❩ ⊢ #↑i ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = #(↑i) & c = 𝟘𝟘
-       | ∃∃T. ❪G,K❫ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2.
+       | ∃∃T. ❨G,K❩ ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2.
 #Rs #Rk #c #G #K #I #T2 #i #H elim (cpg_inv_lref1 … H) -H /2 width=1 by or_introl/
 * #Z #Y #T #HT #HT2 #H destruct /3 width=3 by ex2_intro, or_intror/
 qed-.
@@ -283,8 +283,8 @@ qed-.
 (* Basic forward lemmas *****************************************************)
 
 lemma cpg_fwd_bind1_minus (Rs) (Rk) (c) (G) (L):
-      ∀I,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ⬈[Rs,Rk,c] T → ∀p.
-      ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[Rs,Rk,c] ⓑ[p,I]V2.T2 & T = -ⓑ[I]V2.T2.
+      ∀I,V1,T1,T. ❨G,L❩ ⊢ -ⓑ[I]V1.T1 ⬈[Rs,Rk,c] T → ∀p.
+      ∃∃V2,T2. ❨G,L❩ ⊢ ⓑ[p,I]V1.T1 ⬈[Rs,Rk,c] ⓑ[p,I]V2.T2 & T = -ⓑ[I]V2.T2.
 #Rs #Rk #c #G #L #I #V1 #T1 #T #H #p elim (cpg_inv_bind1 … H) -H *
 [ #cV #cT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct /3 width=4 by cpg_bind, ex2_2_intro/
 | #c #T2 #_ #_ #H destruct