update in ground static_2 basic_2 apps_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_transition / cpg.ma

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 3bdc93e975173a1cace87a5f801e8efa425b7c60..3062e81202db3091e6332676e3e6cf3a88131490 100644 (file)
--- 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 â\80¦ Rk â\86\92 â\88\80G,T,L. â\9dªG,Lâ\9d« ⊢ T ⬈[Rs,Rk,𝟘𝟘] T.
+      reflexive â\80¦ Rk â\86\92 â\88\80G,T,L. â\9d¨G,Lâ\9d© ⊢ 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):
-     â\88\80T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ⬈[Rs,Rk,c] T2 → ∀J. T1 = ⓪[J] →
+     â\88\80T1,T2. â\9d¨G,Lâ\9d© ⊢ 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
-      | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV
-      | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
-      | â\88\83â\88\83I,K,T,i. â\9dªG,Kâ\9d« ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
+      | â\88\83â\88\83cV,K,V1,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV
+      | â\88\83â\88\83cV,K,V1,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
+      | â\88\83â\88\83I,K,T,i. â\9d¨G,Kâ\9d© ⊢ #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):
-      â\88\80J,T2. â\9dªG,Lâ\9d« ⊢ ⓪[J] ⬈[Rs,Rk,c] T2 →
+      â\88\80J,T2. â\9d¨G,Lâ\9d© ⊢ ⓪[J] ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = ⓪[J] & c = 𝟘𝟘
        | ∃∃s1,s2. Rs s1 s2 & J = Sort s1 & T2 = ⋆s2 & c = 𝟘𝟙
-       | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV
-       | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
-       | â\88\83â\88\83I,K,T,i. â\9dªG,Kâ\9d« ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
+       | â\88\83â\88\83cV,K,V1,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & J = LRef 0 & c = cV
+       | â\88\83â\88\83cV,K,V1,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
+       | â\88\83â\88\83I,K,T,i. â\9d¨G,Kâ\9d© ⊢ #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):
-      â\88\80T2,s1. â\9dªG,Lâ\9d« ⊢ ⋆s1 ⬈[Rs,Rk,c] T2 →
+      â\88\80T2,s1. â\9d¨G,Lâ\9d© ⊢ ⋆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):
-      â\88\80T2. â\9dªG,Lâ\9d« ⊢ #0 ⬈[Rs,Rk,c] T2 →
+      â\88\80T2. â\9d¨G,Lâ\9d© ⊢ #0 ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = #0 & c = 𝟘𝟘
-       | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & c = cV
-       | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓛV1 & c = cV+𝟘𝟙.
+       | â\88\83â\88\83cV,K,V1,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & L = K.ⓓV1 & c = cV
+       | â\88\83â\88\83cV,K,V1,V2. â\9d¨G,Kâ\9d© ⊢ 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):
-      â\88\80T2,i. â\9dªG,Lâ\9d« ⊢ #↑i ⬈[Rs,Rk,c] T2 →
+      â\88\80T2,i. â\9d¨G,Lâ\9d© ⊢ #↑i ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = #(↑i) & c = 𝟘𝟘
-       | â\88\83â\88\83I,K,T. â\9dªG,Kâ\9d« ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
+       | â\88\83â\88\83I,K,T. â\9d¨G,Kâ\9d© ⊢ #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):
-      â\88\80T2,l. â\9dªG,Lâ\9d« ⊢ §l ⬈[Rs,Rk,c] T2 → ∧∧ T2 = §l & c = 𝟘𝟘.
+      â\88\80T2,l. â\9d¨G,Lâ\9d© ⊢ §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):
-     â\88\80U,U2. â\9dªG,Lâ\9d« ⊢ U ⬈[Rs,Rk,c] U2 →
+     â\88\80U,U2. â\9d¨G,Lâ\9d© ⊢ U ⬈[Rs,Rk,c] U2 →
      ∀p,J,V1,U1. U = ⓑ[p,J]V1.U1 →
-     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9dªG,L.â\93\91[J]V1â\9d« ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓑ[p,J]V2.T2 & c = ((↕*cV)∨cT)
-      | â\88\83â\88\83cT,T. â\87§[1] T â\89\98 U1 & â\9dªG,Lâ\9d« ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & J = Abbr & c = cT+𝟙𝟘.
+     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9d¨G,L.â\93\91[J]V1â\9d© ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓑ[p,J]V2.T2 & c = ((↕*cV)∨cT)
+      | â\88\83â\88\83cT,T. â\87§[1] T â\89\98 U1 & â\9d¨G,Lâ\9d© ⊢ 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):
-      â\88\80p,I,V1,T1,U2. â\9dªG,Lâ\9d« ⊢ ⓑ[p,I]V1.T1 ⬈[Rs,Rk,c] U2 →
-      â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓑ[p,I]V2.T2 & c = ((↕*cV)∨cT)
-       | â\88\83â\88\83cT,T. â\87§[1] T â\89\98 T1 & â\9dªG,Lâ\9d« ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & I = Abbr & c = cT+𝟙𝟘.
+      â\88\80p,I,V1,T1,U2. â\9d¨G,Lâ\9d© ⊢ ⓑ[p,I]V1.T1 ⬈[Rs,Rk,c] U2 →
+      â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9d¨G,L.â\93\91[I]V1â\9d© ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓑ[p,I]V2.T2 & c = ((↕*cV)∨cT)
+       | â\88\83â\88\83cT,T. â\87§[1] T â\89\98 T1 & â\9d¨G,Lâ\9d© ⊢ 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):
-      â\88\80p,V1,T1,U2. â\9dªG,Lâ\9d« ⊢ ⓓ[p]V1.T1 ⬈[Rs,Rk,c] U2 →
-      â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9dªG,L.â\93\93V1â\9d« ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓓ[p]V2.T2 & c = ((↕*cV)∨cT)
-       | â\88\83â\88\83cT,T. â\87§[1] T â\89\98 T1 & â\9dªG,Lâ\9d« ⊢ T ⬈[Rs,Rk,cT] U2 & p = true & c = cT+𝟙𝟘.
+      â\88\80p,V1,T1,U2. â\9d¨G,Lâ\9d© ⊢ ⓓ[p]V1.T1 ⬈[Rs,Rk,c] U2 →
+      â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9d¨G,L.â\93\93V1â\9d© ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓓ[p]V2.T2 & c = ((↕*cV)∨cT)
+       | â\88\83â\88\83cT,T. â\87§[1] T â\89\98 T1 & â\9d¨G,Lâ\9d© ⊢ 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):
-      â\88\80p,V1,T1,U2. â\9dªG,Lâ\9d« ⊢ ⓛ[p]V1.T1 ⬈[Rs,Rk,c] U2 →
-      â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9dªG,L.â\93\9bV1â\9d« ⊢ T1 ⬈[Rs,Rk,cT] T2 & U2 = ⓛ[p]V2.T2 & c = ((↕*cV)∨cT).
+      â\88\80p,V1,T1,U2. â\9d¨G,Lâ\9d© ⊢ ⓛ[p]V1.T1 ⬈[Rs,Rk,c] U2 →
+      â\88\83â\88\83cV,cT,V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9d¨G,L.â\93\9bV1â\9d© ⊢ 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):
-     â\88\80U,U2. â\9dªG,Lâ\9d« ⊢ U ⬈[Rs,Rk,c] U2 →
+     â\88\80U,U2. â\9d¨G,Lâ\9d© ⊢ U ⬈[Rs,Rk,c] U2 →
      ∀V1,U1. U = ⓐV1.U1 →
-     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
-      | â\88\83â\88\83cV,cW,cT,p,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[Rs,Rk,cW] W2 & â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
-      | â\88\83â\88\83cV,cW,cT,p,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V & â\87§[1] V â\89\98 V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[Rs,Rk,cW] W2 & â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
+     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9d¨G,Lâ\9d© ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
+      | â\88\83â\88\83cV,cW,cT,p,V2,W1,W2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88[Rs,Rk,cW] W2 & â\9d¨G,L.â\93\9bW1â\9d© ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
+      | â\88\83â\88\83cV,cW,cT,p,V,V2,W1,W2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V & â\87§[1] V â\89\98 V2 & â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88[Rs,Rk,cW] W2 & â\9d¨G,L.â\93\93W1â\9d© ⊢ 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):
-      â\88\80V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓐV1.U1 ⬈[Rs,Rk,c] U2 →
-      â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
-       | â\88\83â\88\83cV,cW,cT,p,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[Rs,Rk,cW] W2 & â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
-       | â\88\83â\88\83cV,cW,cT,p,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V & â\87§[1] V â\89\98 V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[Rs,Rk,cW] W2 & â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
+      â\88\80V1,U1,U2. â\9d¨G,Lâ\9d© ⊢ ⓐV1.U1 ⬈[Rs,Rk,c] U2 →
+      â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9d¨G,Lâ\9d© ⊢ U1 ⬈[Rs,Rk,cT] T2 & U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
+       | â\88\83â\88\83cV,cW,cT,p,V2,W1,W2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88[Rs,Rk,cW] W2 & â\9d¨G,L.â\93\9bW1â\9d© ⊢ T1 ⬈[Rs,Rk,cT] T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘
+       | â\88\83â\88\83cV,cW,cT,p,V,V2,W1,W2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V & â\87§[1] V â\89\98 V2 & â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88[Rs,Rk,cW] W2 & â\9d¨G,L.â\93\93W1â\9d© ⊢ 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):
-     â\88\80U,U2. â\9dªG,Lâ\9d« ⊢ U ⬈[Rs,Rk,c] U2 →
+     â\88\80U,U2. â\9d¨G,Lâ\9d© ⊢ U ⬈[Rs,Rk,c] U2 →
      ∀V1,U1. U = ⓝV1.U1 →
-     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rs,Rk,cT] T2 & Rk cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
-      | â\88\83â\88\83cT. â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rs,Rk,cT] U2 & c = cT+𝟙𝟘
-      | â\88\83â\88\83cV. â\9dªG,Lâ\9d« ⊢ V1 ⬈[Rs,Rk,cV] U2 & c = cV+𝟘𝟙.
+     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9d¨G,Lâ\9d© ⊢ U1 ⬈[Rs,Rk,cT] T2 & Rk cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
+      | â\88\83â\88\83cT. â\9d¨G,Lâ\9d© ⊢ U1 ⬈[Rs,Rk,cT] U2 & c = cT+𝟙𝟘
+      | â\88\83â\88\83cV. â\9d¨G,Lâ\9d© ⊢ 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):
-      â\88\80V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓝV1.U1 ⬈[Rs,Rk,c] U2 →
-      â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rs,Rk,cT] T2 & Rk cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
-       | â\88\83â\88\83cT. â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rs,Rk,cT] U2 & c = cT+𝟙𝟘
-       | â\88\83â\88\83cV. â\9dªG,Lâ\9d« ⊢ V1 ⬈[Rs,Rk,cV] U2 & c = cV+𝟘𝟙.
+      â\88\80V1,U1,U2. â\9d¨G,Lâ\9d© ⊢ ⓝV1.U1 ⬈[Rs,Rk,c] U2 →
+      â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88[Rs,Rk,cV] V2 & â\9d¨G,Lâ\9d© ⊢ U1 ⬈[Rs,Rk,cT] T2 & Rk cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
+       | â\88\83â\88\83cT. â\9d¨G,Lâ\9d© ⊢ U1 ⬈[Rs,Rk,cT] U2 & c = cT+𝟙𝟘
+       | â\88\83â\88\83cV. â\9d¨G,Lâ\9d© ⊢ 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):
-      â\88\80I,V1,T2. â\9dªG,K.â\93\91[I]V1â\9d« ⊢ #0 ⬈[Rs,Rk,c] T2 →
+      â\88\80I,V1,T2. â\9d¨G,K.â\93\91[I]V1â\9d© ⊢ #0 ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = #0 & c = 𝟘𝟘
-       | â\88\83â\88\83cV,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & I = Abbr & c = cV
-       | â\88\83â\88\83cV,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & I = Abst & c = cV+𝟘𝟙.
+       | â\88\83â\88\83cV,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈[Rs,Rk,cV] V2 & ⇧[1] V2 ≘ T2 & I = Abbr & c = cV
+       | â\88\83â\88\83cV,V2. â\9d¨G,Kâ\9d© ⊢ 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):
-      â\88\80I,T2,i. â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬈[Rs,Rk,c] T2 →
+      â\88\80I,T2,i. â\9d¨G,K.â\93\98[I]â\9d© ⊢ #↑i ⬈[Rs,Rk,c] T2 →
       ∨∨ ∧∧ T2 = #(↑i) & c = 𝟘𝟘
-       | â\88\83â\88\83T. â\9dªG,Kâ\9d« ⊢ #i ⬈[Rs,Rk,c] T & ⇧[1] T ≘ T2.
+       | â\88\83â\88\83T. â\9d¨G,Kâ\9d© ⊢ #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):
-      â\88\80I,V1,T1,T. â\9dªG,Lâ\9d« ⊢ -ⓑ[I]V1.T1 ⬈[Rs,Rk,c] T → ∀p.
-      â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« ⊢ ⓑ[p,I]V1.T1 ⬈[Rs,Rk,c] ⓑ[p,I]V2.T2 & T = -ⓑ[I]V2.T2.
+      â\88\80I,V1,T1,T. â\9d¨G,Lâ\9d© ⊢ -ⓑ[I]V1.T1 ⬈[Rs,Rk,c] T → ∀p.
+      â\88\83â\88\83V2,T2. â\9d¨G,Lâ\9d© ⊢ ⓑ[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