update in ground_2, static_2, basic_2, apps_2, alpha_1

[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 c33b21973f94da02b7745e130145ff79ee91282a..377d39ea6cfc129ddf1f8628bfbb16c2b5feacdc 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma
@@ -31,17 +31,17 @@ include "static_2/relocation/lifts.ma".
 
 (* avtivate genv *)
 inductive cpg (Rt:relation rtc) (h): rtc → relation4 genv lenv term term ≝
-| cpg_atom : ∀I,G,L. cpg Rt h (𝟘𝟘) G L (⓪{I}) (⓪{I})
+| cpg_atom : ∀I,G,L. cpg Rt h (𝟘𝟘) G L (⓪[I]) (⓪[I])
 | cpg_ess  : ∀G,L,s. cpg Rt h (𝟘𝟙) G L (⋆s) (⋆(⫯[h]s))
 | cpg_delta: ∀c,G,L,V1,V2,W2. cpg Rt h c G L V1 V2 →
              ⇧*[1] V2 ≘ W2 → cpg Rt h c G (L.ⓓV1) (#0) W2
 | cpg_ell  : ∀c,G,L,V1,V2,W2. cpg Rt h c G L V1 V2 →
              ⇧*[1] V2 ≘ W2 → cpg Rt h (c+𝟘𝟙) G (L.ⓛV1) (#0) W2
 | cpg_lref : ∀c,I,G,L,T,U,i. cpg Rt h c G L (#i) T →
-             ⇧*[1] T ≘ U → cpg Rt h c G (L.ⓘ{I}) (#↑i) U
+             ⇧*[1] T ≘ U → cpg Rt h c G (L.ⓘ[I]) (#↑i) U
 | cpg_bind : ∀cV,cT,p,I,G,L,V1,V2,T1,T2.
-             cpg Rt h cV G L V1 V2 → cpg Rt h cT G (L.ⓑ{I}V1) T1 T2 →
-             cpg Rt h ((↕*cV)∨cT) G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
+             cpg Rt h cV G L V1 V2 → cpg Rt h cT G (L.ⓑ[I]V1) T1 T2 →
+             cpg Rt h ((↕*cV)∨cT) G L (ⓑ[p,I]V1.T1) (ⓑ[p,I]V2.T2)
 | cpg_appl : ∀cV,cT,G,L,V1,V2,T1,T2.
              cpg Rt h cV G L V1 V2 → cpg Rt h cT G L T1 T2 →
              cpg Rt h ((↕*cV)∨cT) G L (ⓐV1.T1) (ⓐV2.T2)
@@ -54,11 +54,11 @@ inductive cpg (Rt:relation rtc) (h): rtc → relation4 genv lenv term term ≝
 | cpg_ee   : ∀c,G,L,V1,V2,T. cpg Rt h c G L V1 V2 → cpg Rt h (c+𝟘𝟙) G L (ⓝV1.T) V2
 | cpg_beta : ∀cV,cW,cT,p,G,L,V1,V2,W1,W2,T1,T2.
              cpg Rt h cV G L V1 V2 → cpg Rt h cW G L W1 W2 → cpg Rt h cT G (L.ⓛW1) T1 T2 →
-             cpg Rt h (((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
+             cpg Rt h (((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓛ[p]W1.T1) (ⓓ[p]ⓝW2.V2.T2)
 | cpg_theta: ∀cV,cW,cT,p,G,L,V1,V,V2,W1,W2,T1,T2.
              cpg Rt h cV G L V1 V → ⇧*[1] V ≘ V2 → cpg Rt h cW G L W1 W2 →
              cpg Rt h cT G (L.ⓓW1) T1 T2 →
-             cpg Rt h (((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+             cpg Rt h (((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘) G L (ⓐV1.ⓓ[p]W1.T1) (ⓓ[p]W2.ⓐV2.T2)
 .
 
 interpretation
@@ -68,22 +68,22 @@ interpretation
 (* Basic properties *********************************************************)
 
 (* Note: this is "∀Rt. reflexive … Rt → ∀h,g,L. reflexive … (cpg Rt h (𝟘𝟘) L)" *)
-lemma cpg_refl: â\88\80Rt. reflexive â\80¦ Rt â\86\92 â\88\80h,G,T,L. â¦\83G,Lâ¦\84 ⊢ T ⬈[Rt,𝟘𝟘,h] T.
+lemma cpg_refl: â\88\80Rt. reflexive â\80¦ Rt â\86\92 â\88\80h,G,T,L. â\9dªG,Lâ\9d« ⊢ T ⬈[Rt,𝟘𝟘,h] T.
 #Rt #HRt #h #G #T elim T -T // * /2 width=1 by cpg_bind/
 * /2 width=1 by cpg_appl, cpg_cast/
 qed.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact cpg_inv_atom1_aux: â\88\80Rt,c,h,G,L,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ T1 â¬\88[Rt,c,h] T2 â\86\92 â\88\80J. T1 = â\93ª{J} →
-                        ∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘
+fact cpg_inv_atom1_aux: â\88\80Rt,c,h,G,L,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88[Rt,c,h] T2 â\86\92 â\88\80J. T1 = â\93ª[J] →
+                        ∨∨ T2 = ⓪[J] ∧ c = 𝟘𝟘
                          | ∃∃s. J = Sort s & T2 = ⋆(⫯[h]s) & c = 𝟘𝟙
-                         | â\88\83â\88\83cV,K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                         | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
                                          L = K.ⓓV1 & J = LRef 0 & c = cV
-                         | â\88\83â\88\83cV,K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                         | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
                                          L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
-                         | â\88\83â\88\83I,K,T,i. â¦\83G,Kâ¦\84 ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 &
-                                      L = K.ⓘ{I} & J = LRef (↑i).
+                         | â\88\83â\88\83I,K,T,i. â\9dªG,Kâ\9d« ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 &
+                                      L = K.ⓘ[I] & J = LRef (↑i).
 #Rt #c #h #G #L #T1 #T2 * -c -G -L -T1 -T2
 [ #I #G #L #J #H destruct /3 width=1 by or5_intro0, conj/
 | #G #L #s #J #H destruct /3 width=3 by or5_intro1, ex3_intro/
@@ -101,18 +101,18 @@ fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ⦃G,L⦄ ⊢ T1 ⬈[Rt,c,h] T2 →
 ]
 qed-.
 
-lemma cpg_inv_atom1: â\88\80Rt,c,h,J,G,L,T2. â¦\83G,Lâ¦\84 â\8a¢ â\93ª{J} ⬈[Rt,c,h] T2 →
-                     ∨∨ T2 = ⓪{J} ∧ c = 𝟘𝟘
+lemma cpg_inv_atom1: â\88\80Rt,c,h,J,G,L,T2. â\9dªG,Lâ\9d« â\8a¢ â\93ª[J] ⬈[Rt,c,h] T2 →
+                     ∨∨ T2 = ⓪[J] ∧ c = 𝟘𝟘
                       | ∃∃s. J = Sort s & T2 = ⋆(⫯[h]s) & c = 𝟘𝟙
-                      | â\88\83â\88\83cV,K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                      | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
                                       L = K.ⓓV1 & J = LRef 0 & c = cV
-                      | â\88\83â\88\83cV,K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                      | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
                                       L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
-                      | â\88\83â\88\83I,K,T,i. â¦\83G,Kâ¦\84 ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 &
-                                   L = K.ⓘ{I} & J = LRef (↑i).
+                      | â\88\83â\88\83I,K,T,i. â\9dªG,Kâ\9d« ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 &
+                                   L = K.ⓘ[I] & J = LRef (↑i).
 /2 width=3 by cpg_inv_atom1_aux/ qed-.
 
-lemma cpg_inv_sort1: â\88\80Rt,c,h,G,L,T2,s. â¦\83G,Lâ¦\84 ⊢ ⋆s ⬈[Rt,c,h] T2 →
+lemma cpg_inv_sort1: â\88\80Rt,c,h,G,L,T2,s. â\9dªG,Lâ\9d« ⊢ ⋆s ⬈[Rt,c,h] T2 →
                      ∨∨ T2 = ⋆s ∧ c = 𝟘𝟘 | T2 = ⋆(⫯[h]s) ∧ c = 𝟘𝟙.
 #Rt #c #h #G #L #T2 #s #H
 elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
@@ -122,11 +122,11 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
 ]
 qed-.
 
-lemma cpg_inv_zero1: â\88\80Rt,c,h,G,L,T2. â¦\83G,Lâ¦\84 ⊢ #0 ⬈[Rt,c,h] T2 →
+lemma cpg_inv_zero1: â\88\80Rt,c,h,G,L,T2. â\9dªG,Lâ\9d« ⊢ #0 ⬈[Rt,c,h] T2 →
                      ∨∨ T2 = #0 ∧ c = 𝟘𝟘
-                      | â\88\83â\88\83cV,K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                      | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
                                       L = K.ⓓV1 & c = cV
-                      | â\88\83â\88\83cV,K,V1,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                      | â\88\83â\88\83cV,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
                                       L = K.ⓛV1 & c = cV+𝟘𝟙.
 #Rt #c #h #G #L #T2 #H
 elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
@@ -136,9 +136,9 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or3_intro0, conj/
 ]
 qed-.
 
-lemma cpg_inv_lref1: â\88\80Rt,c,h,G,L,T2,i. â¦\83G,Lâ¦\84 ⊢ #↑i ⬈[Rt,c,h] T2 →
+lemma cpg_inv_lref1: â\88\80Rt,c,h,G,L,T2,i. â\9dªG,Lâ\9d« ⊢ #↑i ⬈[Rt,c,h] T2 →
                      ∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘
-                      | â\88\83â\88\83I,K,T. â¦\83G,Kâ¦\84 â\8a¢ #i â¬\88[Rt,c,h] T & â\87§*[1] T â\89\98 T2 & L = K.â\93\98{I}.
+                      | â\88\83â\88\83I,K,T. â\9dªG,Kâ\9d« â\8a¢ #i â¬\88[Rt,c,h] T & â\87§*[1] T â\89\98 T2 & L = K.â\93\98[I].
 #Rt #c #h #G #L #T2 #i #H
 elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
 [ #s #H destruct
@@ -147,7 +147,7 @@ elim (cpg_inv_atom1 … H) -H * /3 width=1 by or_introl, conj/
 ]
 qed-.
 
-lemma cpg_inv_gref1: â\88\80Rt,c,h,G,L,T2,l. â¦\83G,Lâ¦\84 ⊢ §l ⬈[Rt,c,h] T2 → T2 = §l ∧ c = 𝟘𝟘.
+lemma cpg_inv_gref1: â\88\80Rt,c,h,G,L,T2,l. â\9dªG,Lâ\9d« ⊢ §l ⬈[Rt,c,h] T2 → T2 = §l ∧ c = 𝟘𝟘.
 #Rt #c #h #G #L #T2 #l #H
 elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/
 [ #s #H destruct
@@ -156,11 +156,11 @@ elim (cpg_inv_atom1 … H) -H * /2 width=1 by conj/
 ]
 qed-.
 
-fact cpg_inv_bind1_aux: â\88\80Rt,c,h,G,L,U,U2. â¦\83G,Lâ¦\84 ⊢ U ⬈[Rt,c,h] U2 →
-                        ∀p,J,V1,U1. U = ⓑ{p,J}V1.U1 →
-                        â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â¦\83G,L.â\93\91{J}V1â¦\84 ⊢ U1 ⬈[Rt,cT,h] T2 &
-                                          U2 = ⓑ{p,J}V2.T2 & c = ((↕*cV)∨cT)
-                         | â\88\83â\88\83cT,T. â\87§*[1] T â\89\98 U1 & â¦\83G,Lâ¦\84 ⊢ T ⬈[Rt,cT,h] U2 &
+fact cpg_inv_bind1_aux: â\88\80Rt,c,h,G,L,U,U2. â\9dªG,Lâ\9d« ⊢ U ⬈[Rt,c,h] 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[Rt,cV,h] V2 & â\9dªG,L.â\93\91[J]V1â\9d« ⊢ U1 ⬈[Rt,cT,h] 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 ⬈[Rt,cT,h] U2 &
                                    p = true & J = Abbr & c = cT+𝟙𝟘.
 #Rt #c #h #G #L #U #U2 * -c -G -L -U -U2
 [ #I #G #L #q #J #W #U1 #H destruct
@@ -179,39 +179,39 @@ fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 →
 ]
 qed-.
 
-lemma cpg_inv_bind1: â\88\80Rt,c,h,p,I,G,L,V1,T1,U2. â¦\83G,Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 ⬈[Rt,c,h] U2 →
-                     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â¦\83G,L.â\93\91{I}V1â¦\84 ⊢ T1 ⬈[Rt,cT,h] T2 &
-                                       U2 = ⓑ{p,I}V2.T2 & c = ((↕*cV)∨cT)
-                      | â\88\83â\88\83cT,T. â\87§*[1] T â\89\98 T1 & â¦\83G,Lâ¦\84 ⊢ T ⬈[Rt,cT,h] U2 &
+lemma cpg_inv_bind1: â\88\80Rt,c,h,p,I,G,L,V1,T1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 ⬈[Rt,c,h] U2 →
+                     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ⬈[Rt,cT,h] 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 ⬈[Rt,cT,h] U2 &
                                 p = true & I = Abbr & c = cT+𝟙𝟘.
 /2 width=3 by cpg_inv_bind1_aux/ qed-.
 
-lemma cpg_inv_abbr1: â\88\80Rt,c,h,p,G,L,V1,T1,U2. â¦\83G,Lâ¦\84 â\8a¢ â\93\93{p}V1.T1 ⬈[Rt,c,h] U2 →
-                     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â¦\83G,L.â\93\93V1â¦\84 ⊢ T1 ⬈[Rt,cT,h] T2 &
-                                       U2 = ⓓ{p}V2.T2 & c = ((↕*cV)∨cT)
-                      | â\88\83â\88\83cT,T. â\87§*[1] T â\89\98 T1 & â¦\83G,Lâ¦\84 ⊢ T ⬈[Rt,cT,h] U2 &
+lemma cpg_inv_abbr1: â\88\80Rt,c,h,p,G,L,V1,T1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\93[p]V1.T1 ⬈[Rt,c,h] U2 →
+                     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â\9dªG,L.â\93\93V1â\9d« ⊢ T1 ⬈[Rt,cT,h] 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 ⬈[Rt,cT,h] U2 &
                                 p = true & c = cT+𝟙𝟘.
 #Rt #c #h #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: â\88\80Rt,c,h,p,G,L,V1,T1,U2. â¦\83G,Lâ¦\84 â\8a¢ â\93\9b{p}V1.T1 ⬈[Rt,c,h] U2 →
-                     â\88\83â\88\83cV,cT,V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â¦\83G,L.â\93\9bV1â¦\84 ⊢ T1 ⬈[Rt,cT,h] T2 &
-                                    U2 = ⓛ{p}V2.T2 & c = ((↕*cV)∨cT).
+lemma cpg_inv_abst1: â\88\80Rt,c,h,p,G,L,V1,T1,U2. â\9dªG,Lâ\9d« â\8a¢ â\93\9b[p]V1.T1 ⬈[Rt,c,h] U2 →
+                     â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â\9dªG,L.â\93\9bV1â\9d« ⊢ T1 ⬈[Rt,cT,h] T2 &
+                                    U2 = ⓛ[p]V2.T2 & c = ((↕*cV)∨cT).
 #Rt #c #h #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
 [ /3 width=8 by ex4_4_intro/
 | #c #T #_ #_ #_ #H destruct
 ]
 qed-.
 
-fact cpg_inv_appl1_aux: â\88\80Rt,c,h,G,L,U,U2. â¦\83G,Lâ¦\84 ⊢ U ⬈[Rt,c,h] U2 →
+fact cpg_inv_appl1_aux: â\88\80Rt,c,h,G,L,U,U2. â\9dªG,Lâ\9d« ⊢ U ⬈[Rt,c,h] U2 →
                         ∀V1,U1. U = ⓐV1.U1 →
-                        â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â¦\83G,Lâ¦\84 ⊢ U1 ⬈[Rt,cT,h] T2 &
+                        â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rt,cT,h] T2 &
                                           U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
-                         | â\88\83â\88\83cV,cW,cT,p,V2,W1,W2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â¦\83G,Lâ¦\84 â\8a¢ W1 â¬\88[Rt,cW,h] W2 & â¦\83G,L.â\93\9bW1â¦\84 ⊢ T1 ⬈[Rt,cT,h] 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¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V & â\87§*[1] V â\89\98 V2 & â¦\83G,Lâ¦\84 â\8a¢ W1 â¬\88[Rt,cW,h] W2 & â¦\83G,L.â\93\93W1â¦\84 ⊢ T1 ⬈[Rt,cT,h] T2 &
-                                                          U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
+                         | â\88\83â\88\83cV,cW,cT,p,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[Rt,cW,h] W2 & â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈[Rt,cT,h] 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[Rt,cV,h] V & â\87§*[1] V â\89\98 V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[Rt,cW,h] W2 & â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ⬈[Rt,cT,h] T2 &
+                                                          U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
 #Rt #c #h #G #L #U #U2 * -c -G -L -U -U2
 [ #I #G #L #W #U1 #H destruct
 | #G #L #s #W #U1 #H destruct
@@ -229,21 +229,21 @@ fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 →
 ]
 qed-.
 
-lemma cpg_inv_appl1: â\88\80Rt,c,h,G,L,V1,U1,U2. â¦\83G,Lâ¦\84 ⊢ ⓐV1.U1 ⬈[Rt,c,h] U2 →
-                     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â¦\83G,Lâ¦\84 ⊢ U1 ⬈[Rt,cT,h] T2 &
+lemma cpg_inv_appl1: â\88\80Rt,c,h,G,L,V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓐV1.U1 ⬈[Rt,c,h] U2 →
+                     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rt,cT,h] T2 &
                                        U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
-                      | â\88\83â\88\83cV,cW,cT,p,V2,W1,W2,T1,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â¦\83G,Lâ¦\84 â\8a¢ W1 â¬\88[Rt,cW,h] W2 & â¦\83G,L.â\93\9bW1â¦\84 ⊢ T1 ⬈[Rt,cT,h] 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¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V & â\87§*[1] V â\89\98 V2 & â¦\83G,Lâ¦\84 â\8a¢ W1 â¬\88[Rt,cW,h] W2 & â¦\83G,L.â\93\93W1â¦\84 ⊢ T1 ⬈[Rt,cT,h] T2 &
-                                                       U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & c = ((↕*cV)∨(↕*cW)∨cT)+𝟙𝟘.
+                      | â\88\83â\88\83cV,cW,cT,p,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[Rt,cW,h] W2 & â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈[Rt,cT,h] 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[Rt,cV,h] V & â\87§*[1] V â\89\98 V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88[Rt,cW,h] W2 & â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ⬈[Rt,cT,h] 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: â\88\80Rt,c,h,G,L,U,U2. â¦\83G,Lâ¦\84 ⊢ U ⬈[Rt,c,h] U2 →
+fact cpg_inv_cast1_aux: â\88\80Rt,c,h,G,L,U,U2. â\9dªG,Lâ\9d« ⊢ U ⬈[Rt,c,h] U2 →
                         ∀V1,U1. U = ⓝV1.U1 →
-                        â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â¦\83G,Lâ¦\84 ⊢ U1 ⬈[Rt,cT,h] T2 &
+                        â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rt,cT,h] T2 &
                                           Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
-                         | â\88\83â\88\83cT. â¦\83G,Lâ¦\84 ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘
-                         | â\88\83â\88\83cV. â¦\83G,Lâ¦\84 ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙.
+                         | â\88\83â\88\83cT. â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘
+                         | â\88\83â\88\83cV. â\9dªG,Lâ\9d« ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙.
 #Rt #c #h #G #L #U #U2 * -c -G -L -U -U2
 [ #I #G #L #W #U1 #H destruct
 | #G #L #s #W #U1 #H destruct
@@ -261,37 +261,37 @@ fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ⦃G,L⦄ ⊢ U ⬈[Rt,c,h] U2 →
 ]
 qed-.
 
-lemma cpg_inv_cast1: â\88\80Rt,c,h,G,L,V1,U1,U2. â¦\83G,Lâ¦\84 ⊢ ⓝV1.U1 ⬈[Rt,c,h] U2 →
-                     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â¦\83G,Lâ¦\84 â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â¦\83G,Lâ¦\84 ⊢ U1 ⬈[Rt,cT,h] T2 &
+lemma cpg_inv_cast1: â\88\80Rt,c,h,G,L,V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓝV1.U1 ⬈[Rt,c,h] U2 →
+                     â\88¨â\88¨ â\88\83â\88\83cV,cT,V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88[Rt,cV,h] V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rt,cT,h] T2 &
                                        Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
-                      | â\88\83â\88\83cT. â¦\83G,Lâ¦\84 ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘
-                      | â\88\83â\88\83cV. â¦\83G,Lâ¦\84 ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙.
+                      | â\88\83â\88\83cT. â\9dªG,Lâ\9d« ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘
+                      | â\88\83â\88\83cV. â\9dªG,Lâ\9d« ⊢ V1 ⬈[Rt,cV,h] U2 & c = cV+𝟘𝟙.
 /2 width=3 by cpg_inv_cast1_aux/ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpg_inv_zero1_pair: â\88\80Rt,c,h,I,G,K,V1,T2. â¦\83G,K.â\93\91{I}V1â¦\84 ⊢ #0 ⬈[Rt,c,h] T2 →
+lemma cpg_inv_zero1_pair: â\88\80Rt,c,h,I,G,K,V1,T2. â\9dªG,K.â\93\91[I]V1â\9d« ⊢ #0 ⬈[Rt,c,h] T2 →
                           ∨∨ T2 = #0 ∧ c = 𝟘𝟘
-                           | â\88\83â\88\83cV,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                           | â\88\83â\88\83cV,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
                                       I = Abbr & c = cV
-                           | â\88\83â\88\83cV,V2. â¦\83G,Kâ¦\84 ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                           | â\88\83â\88\83cV,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
                                       I = Abst & c = cV+𝟘𝟙.
 #Rt #c #h #I #G #K #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: â\88\80Rt,c,h,I,G,K,T2,i. â¦\83G,K.â\93\98{I}â¦\84 ⊢ #↑i ⬈[Rt,c,h] T2 →
+lemma cpg_inv_lref1_bind: â\88\80Rt,c,h,I,G,K,T2,i. â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬈[Rt,c,h] T2 →
                           ∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘
-                           | â\88\83â\88\83T. â¦\83G,Kâ¦\84 ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2.
+                           | â\88\83â\88\83T. â\9dªG,Kâ\9d« ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2.
 #Rt #c #h #I #G #L #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-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpg_fwd_bind1_minus: â\88\80Rt,c,h,I,G,L,V1,T1,T. â¦\83G,Lâ¦\84 â\8a¢ -â\93\91{I}V1.T1 ⬈[Rt,c,h] T → ∀p.
-                           â\88\83â\88\83V2,T2. â¦\83G,Lâ¦\84 â\8a¢ â\93\91{p,I}V1.T1 â¬\88[Rt,c,h] â\93\91{p,I}V2.T2 &
-                                    T = -ⓑ{I}V2.T2.
+lemma cpg_fwd_bind1_minus: â\88\80Rt,c,h,I,G,L,V1,T1,T. â\9dªG,Lâ\9d« â\8a¢ -â\93\91[I]V1.T1 ⬈[Rt,c,h] T → ∀p.
+                           â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ â\93\91[p,I]V1.T1 â¬\88[Rt,c,h] â\93\91[p,I]V2.T2 &
+                                    T = -ⓑ[I]V2.T2.
 #Rt #c #h #I #G #L #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