X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpg.ma;h=377d39ea6cfc129ddf1f8628bfbb16c2b5feacdc;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=ff20a9b2a65f0dc16bce8d5e7b0ae0e902905ea0;hpb=c3904c007394068ed823575e3be3d73a9ad92cce;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 ff20a9b2a..377d39ea6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpg.ma
@@ -12,265 +12,288 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "ground_2/steps/rtc_shift.ma".
+include "ground_2/xoa/ex_3_3.ma".
+include "ground_2/xoa/ex_4_2.ma".
+include "ground_2/xoa/ex_4_4.ma".
+include "ground_2/xoa/ex_5_2.ma".
+include "ground_2/xoa/ex_6_9.ma".
+include "ground_2/xoa/ex_7_10.ma".
+include "ground_2/xoa/or_5.ma".
+include "ground_2/steps/rtc_max.ma".
 include "ground_2/steps/rtc_plus.ma".
-include "basic_2/notation/relations/pred_6.ma".
-include "basic_2/grammar/lenv.ma".
-include "basic_2/grammar/genv.ma".
-include "basic_2/relocation/lifts.ma".
-include "basic_2/static/sh.ma".
+include "basic_2/notation/relations/predty_7.ma".
+include "static_2/syntax/sh.ma".
+include "static_2/syntax/lenv.ma".
+include "static_2/syntax/genv.ma".
+include "static_2/relocation/lifts.ma".
 
-(* CONTEXT-SENSITIVE GENERIC PARALLEL RT-TRANSITION FOR TERMS ***************)
+(* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************)
 
 (* avtivate genv *)
-inductive cpg (h): rtc → relation4 genv lenv term term ≝
-| cpg_atom : ∀I,G,L. cpg h (𝟘𝟘) G L (⓪{I}) (⓪{I})
-| cpg_ess  : ∀G,L,s. cpg h (𝟘𝟙) G L (⋆s) (⋆(next h s))
-| cpg_delta: ∀r,G,L,V1,V2,W2. cpg h r G L V1 V2 →
-             ⬆*[1] V2 ≡ W2 → cpg h (↓r) G (L.ⓓV1) (#0) W2
-| cpg_ell  : ∀r,G,L,V1,V2,W2. cpg h r G L V1 V2 →
-             ⬆*[1] V2 ≡ W2 → cpg h ((↓r)+𝟘𝟙) G (L.ⓛV1) (#0) W2
-| cpt_lref : ∀r,I,G,L,V,T,U,i. cpg h r G L (#i) T → 
-             ⬆*[1] T ≡ U → cpg h r G (L.ⓑ{I}V) (#⫯i) U
-| cpg_bind : ∀rV,rT,p,I,G,L,V1,V2,T1,T2.
-             cpg h rV G L V1 V2 → cpg h rT G (L.ⓑ{I}V1) T1 T2 →
-             cpg h ((↓rV)+rT) G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
-| cpg_flat : ∀rV,rT,I,G,L,V1,V2,T1,T2.
-             cpg h rV G L V1 V2 → cpg h rT G L T1 T2 →
-             cpg h ((↓rV)+rT) G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-| cpg_zeta : ∀r,G,L,V,T1,T,T2. cpg h r G (L.ⓓV) T1 T →
-             ⬆*[1] T2 ≡ T → cpg h ((↓r)+𝟙𝟘) G L (+ⓓV.T1) T2
-| cpg_eps  : ∀r,G,L,V,T1,T2. cpg h r G L T1 T2 → cpg h ((↓r)+𝟙𝟘) G L (ⓝV.T1) T2
-| cpg_ee   : ∀r,G,L,V1,V2,T. cpg h r G L V1 V2 → cpg h ((↓r)+𝟘𝟙) G L (ⓝV1.T) V2
-| cpg_beta : ∀rV,rW,rT,p,G,L,V1,V2,W1,W2,T1,T2.
-             cpg h rV G L V1 V2 → cpg h rW G L W1 W2 → cpg h rT G (L.ⓛW1) T1 T2 →
-             cpg h ((↓rV)+(↓rW)+(↓rT)+𝟙𝟘) G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
-| cpg_theta: ∀rV,rW,rT,p,G,L,V1,V,V2,W1,W2,T1,T2.
-             cpg h rV G L V1 V → ⬆*[1] V ≡ V2 → cpg h rW G L W1 W2 →
-             cpg h rT G (L.ⓓW1) T1 T2 →
-             cpg h ((↓rV)+(↓rW)+(↓rT)+𝟙𝟘) G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+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_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
+| 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_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)
+| cpg_cast : ∀cU,cT,G,L,U1,U2,T1,T2. Rt cU cT →
+             cpg Rt h cU G L U1 U2 → cpg Rt h cT G L T1 T2 →
+             cpg Rt h (cU∨cT) G L (ⓝU1.T1) (ⓝU2.T2)
+| cpg_zeta : ∀c,G,L,V,T1,T,T2. ⇧*[1] T ≘ T1 → cpg Rt h c G L T T2 →
+             cpg Rt h (c+𝟙𝟘) G L (+ⓓV.T1) T2
+| cpg_eps  : ∀c,G,L,V,T1,T2. cpg Rt h c G L T1 T2 → cpg Rt h (c+𝟙𝟘) G L (ⓝV.T1) T2
+| 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_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)
 .
 
 interpretation
-   "context-sensitive generic parallel rt-transition (term)"
-   'PRed h r G L T1 T2 = (cpg h r G L T1 T2).
+   "bound context-sensitive parallel rt-transition (term)"
+   'PRedTy Rt c h G L T1 T2 = (cpg Rt h c G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-(* Note: this is "∀h,g,L. reflexive … (cpg h (𝟘𝟘) L)" *)
-lemma cpg_refl: ∀h,G,T,L. ⦃G, L⦄ ⊢ T ➡[h, 𝟘𝟘] T.
-#h #G #T elim T -T // * /2 width=1 by cpg_bind, cpg_flat/
-qed.
-
-lemma cpg_pair_sn: ∀h,r,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, r] V2 →
-                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h, ↓r] ②{I}V2.T.
-#h #r * /2 width=1 by cpg_bind, cpg_flat/
+(* Note: this is "∀Rt. reflexive … Rt → ∀h,g,L. reflexive … (cpg Rt h (𝟘𝟘) L)" *)
+lemma cpg_refl: ∀Rt. reflexive … Rt → ∀h,G,T,L. ❪G,L❫ ⊢ 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: ∀h,r,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, r] T2 → ∀J. T1 = ⓪{J} →
-                        ∨∨ T2 = ⓪{J}
-                         | ∃∃s. J = Sort s & T2 = ⋆(next h s)
-                         | ∃∃rV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h, rV] V2 & ⬆*[1] V2 ≡ T2 &
-                                         L = K.ⓓV1 & J = LRef 0 & r = ↓rV
-                         | ∃∃rV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h, rV] V2 & ⬆*[1] V2 ≡ T2 &
-                                         L = K.ⓛV1 & J = LRef 0 & r = (↓rV)+𝟘𝟙
-                         | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[h, r] T & ⬆*[1] T ≡ T2 &
-                                        L = K.ⓑ{I}V & J = LRef (⫯i).
-#h #r #G #L #T1 #T2 * -r -G -L -T1 -T2
-[ #I #G #L #J #H destruct /2 width=1 by or5_intro0/
-| #G #L #s #J #H destruct /3 width=3 by or5_intro1, ex2_intro/
-| #r #G #L #V1 #V2 #W2 #HV12 #VW2 #J #H destruct /3 width=8 by or5_intro2, ex5_4_intro/
-| #r #G #L #V1 #V2 #W2 #HV12 #VW2 #J #H destruct /3 width=8 by or5_intro3, ex5_4_intro/
-| #r #I #G #L #V #T #U #i #HT #HTU #J #H destruct /3 width=9 by or5_intro4, ex4_5_intro/
-| #rV #rT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #rV #rT #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
-| #r #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
-| #r #G #L #V #T1 #T2 #_ #J #H destruct
-| #r #G #L #V1 #V2 #T #_ #J #H destruct
-| #rV #rW #rT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
-| #rV #rW #rT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
+fact cpg_inv_atom1_aux: ∀Rt,c,h,G,L,T1,T2. ❪G,L❫ ⊢ T1 ⬈[Rt,c,h] T2 → ∀J. T1 = ⓪[J] →
+                        ∨∨ T2 = ⓪[J] ∧ c = 𝟘𝟘
+                         | ∃∃s. J = Sort s & T2 = ⋆(⫯[h]s) & c = 𝟘𝟙
+                         | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                                         L = K.ⓓV1 & J = LRef 0 & c = cV
+                         | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                                         L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
+                         | ∃∃I,K,T,i. ❪G,K❫ ⊢ #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/
+| #c #G #L #V1 #V2 #W2 #HV12 #VW2 #J #H destruct /3 width=8 by or5_intro2, ex5_4_intro/
+| #c #G #L #V1 #V2 #W2 #HV12 #VW2 #J #H destruct /3 width=8 by or5_intro3, ex5_4_intro/
+| #c #I #G #L #T #U #i #HT #HTU #J #H destruct /3 width=8 by or5_intro4, ex4_4_intro/
+| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
+| #cU #cT #G #L #U1 #U2 #T1 #T2 #_ #_ #_ #J #H destruct
+| #c #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
+| #c #G #L #V #T1 #T2 #_ #J #H destruct
+| #c #G #L #V1 #V2 #T #_ #J #H destruct
+| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
+| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
 ]
 qed-.
 
-lemma cpg_inv_atom1: ∀h,r,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, r] T2 →
-                     ∨∨ T2 = ⓪{J}
-                      | ∃∃s. J = Sort s & T2 = ⋆(next h s)
-                      | ∃∃rV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h, rV] V2 & ⬆*[1] V2 ≡ T2 &
-                                      L = K.ⓓV1 & J = LRef 0 & r = ↓rV
-                      | ∃∃rV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h, rV] V2 & ⬆*[1] V2 ≡ T2 &
-                                      L = K.ⓛV1 & J = LRef 0 & r = (↓rV)+𝟘𝟙
-                      | ∃∃I,K,V,T,i. ⦃G, K⦄ ⊢ #i ➡[h, r] T & ⬆*[1] T ≡ T2 &
-                                     L = K.ⓑ{I}V & J = LRef (⫯i).
+lemma cpg_inv_atom1: ∀Rt,c,h,J,G,L,T2. ❪G,L❫ ⊢ ⓪[J] ⬈[Rt,c,h] T2 →
+                     ∨∨ T2 = ⓪[J] ∧ c = 𝟘𝟘
+                      | ∃∃s. J = Sort s & T2 = ⋆(⫯[h]s) & c = 𝟘𝟙
+                      | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                                      L = K.ⓓV1 & J = LRef 0 & c = cV
+                      | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                                      L = K.ⓛV1 & J = LRef 0 & c = cV+𝟘𝟙
+                      | ∃∃I,K,T,i. ❪G,K❫ ⊢ #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: ∀h,r,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h, r] T2 →
-                     T2 = ⋆s ∨ T2 = ⋆(next h s).
-#h #r #G #L #T2 #s #H
-elim (cpg_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-[ #s0 #H destruct /2 width=1 by or_intror/
-|2,3: #rV #K #V1 #V2 #_ #_ #_ #H destruct
-| #I #K #V1 #V2 #i #_ #_ #_ #H destruct
+lemma cpg_inv_sort1: ∀Rt,c,h,G,L,T2,s. ❪G,L❫ ⊢ ⋆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/
+[ #s0 #H destruct /3 width=1 by or_intror, conj/
+|2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct
+| #I #K #T #i #_ #_ #_ #H destruct
 ]
 qed-.
 
-lemma cpg_inv_zero1: ∀h,r,G,L,T2. ⦃G, L⦄ ⊢ #0 ➡[h, r] T2 →
-                     ∨∨ T2 = #0 
-                      | ∃∃rV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h, rV] V2 & ⬆*[1] V2 ≡ T2 &
-                                      L = K.ⓓV1 & r = ↓rV
-                      | ∃∃rV,K,V1,V2. ⦃G, K⦄ ⊢ V1 ➡[h, rV] V2 & ⬆*[1] V2 ≡ T2 &
-                                      L = K.ⓛV1 & r = (↓rV)+𝟘𝟙.
-#h #r #G #L #T2 #H
-elim (cpg_inv_atom1 … H) -H /2 width=1 by or3_intro0/ *
+lemma cpg_inv_zero1: ∀Rt,c,h,G,L,T2. ❪G,L❫ ⊢ #0 ⬈[Rt,c,h] T2 →
+                     ∨∨ T2 = #0 ∧ c = 𝟘𝟘
+                      | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                                      L = K.ⓓV1 & c = cV
+                      | ∃∃cV,K,V1,V2. ❪G,K❫ ⊢ 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/
 [ #s #H destruct
-|2,3: #rV #K #V1 #V2 #HV12 #HVT2 #H1 #_ #H2 destruct /3 width=8 by or3_intro1, or3_intro2, ex4_4_intro/
-| #I #K #V1 #V2 #i #_ #_ #_ #H destruct
+|2,3: #cV #K #V1 #V2 #HV12 #HVT2 #H1 #_ #H2 destruct /3 width=8 by or3_intro1, or3_intro2, ex4_4_intro/
+| #I #K #T #i #_ #_ #_ #H destruct
 ]
 qed-.
 
-lemma cpg_inv_lref1: ∀h,r,G,L,T2,i. ⦃G, L⦄ ⊢ #⫯i ➡[h, r] T2 →
-                     (T2 = #⫯i) ∨
-                     ∃∃I,K,V,T. ⦃G, K⦄ ⊢ #i ➡[h, r] T & ⬆*[1] T ≡ T2 & L = K.ⓑ{I}V.
-#h #r #G #L #T2 #i #H
-elim (cpg_inv_atom1 … H) -H /2 width=1 by or_introl/ *
+lemma cpg_inv_lref1: ∀Rt,c,h,G,L,T2,i. ❪G,L❫ ⊢ #↑i ⬈[Rt,c,h] T2 →
+                     ∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘
+                      | ∃∃I,K,T. ❪G,K❫ ⊢ #i ⬈[Rt,c,h] T & ⇧*[1] T ≘ T2 & L = K.ⓘ[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
-|2,3: #rV #K #V1 #V2 #_ #_ #_ #H destruct
-| #I #K #V1 #V2 #j #HV2 #HVT2 #H1 #H2 destruct /3 width=7 by ex3_4_intro, or_intror/
+|2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct
+| #I #K #T #j #HT #HT2 #H1 #H2 destruct /3 width=6 by ex3_3_intro, or_intror/
 ]
 qed-.
 
-lemma cpg_inv_gref1: ∀h,r,G,L,T2,l.  ⦃G, L⦄ ⊢ §l ➡[h, r] T2 → T2 = §l.
-#h #r #G #L #T2 #l #H
-elim (cpg_inv_atom1 … H) -H // *
+lemma cpg_inv_gref1: ∀Rt,c,h,G,L,T2,l. ❪G,L❫ ⊢ §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
-|2,3: #rV #K #V1 #V2 #_ #_ #_ #H destruct
-| #I #K #V1 #V2 #i #_ #_ #_ #H destruct
+|2,3: #cV #K #V1 #V2 #_ #_ #_ #H destruct
+| #I #K #T #i #_ #_ #_ #H destruct
 ]
 qed-.
 
-fact cpg_inv_bind1_aux: ∀h,r,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, r] U2 →
-                        ∀p,J,V1,U1. U = ⓑ{p,J}V1.U1 → (
-                        ∃∃rV,rT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ U1 ➡[h, rT] T2 &
-                                       U2 = ⓑ{p,J}V2.T2 & r = (↓rV)+rT
-                        ) ∨
-                        ∃∃rT,T. ⦃G, L.ⓓV1⦄ ⊢ U1 ➡[h, rT] T & ⬆*[1] U2 ≡ T &
-                                p = true & J = Abbr & r = (↓rT)+𝟙𝟘.
-#h #r #G #L #U #U2 * -r -G -L -U -U2
+fact cpg_inv_bind1_aux: ∀Rt,c,h,G,L,U,U2. ❪G,L❫ ⊢ U ⬈[Rt,c,h] U2 →
+                        ∀p,J,V1,U1. U = ⓑ[p,J]V1.U1 →
+                        ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L.ⓑ[J]V1❫ ⊢ U1 ⬈[Rt,cT,h] T2 &
+                                          U2 = ⓑ[p,J]V2.T2 & c = ((↕*cV)∨cT)
+                         | ∃∃cT,T. ⇧*[1] T ≘ U1 & ❪G,L❫ ⊢ 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
 | #G #L #s #q #J #W #U1 #H destruct
-| #r #G #L #V1 #V2 #W2 #_ #_ #q #J #W #U1 #H destruct
-| #r #G #L #V1 #V2 #W2 #_ #_ #q #J #W #U1 #H destruct
-| #r #I #G #L #V #T #U #i #_ #_ #q #J #W #U1 #H destruct
-| #rv #rT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W #U1 #H destruct /3 width=8 by ex4_4_intro, or_introl/
-| #rv #rT #I #G #L #V1 #V2 #T1 #T2 #_ #_ #q #J #W #U1 #H destruct
-| #r #G #L #V #T1 #T #T2 #HT1 #HT2 #q #J #W #U1 #H destruct /3 width=5 by ex5_2_intro, or_intror/
-| #r #G #L #V #T1 #T2 #_ #q #J #W #U1 #H destruct
-| #r #G #L #V1 #V2 #T #_ #q #J #W #U1 #H destruct
-| #rv #rW #rT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #q #J #W #U1 #H destruct
-| #rv #rW #rT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #q #J #W #U1 #H destruct
+| #c #G #L #V1 #V2 #W2 #_ #_ #q #J #W #U1 #H destruct
+| #c #G #L #V1 #V2 #W2 #_ #_ #q #J #W #U1 #H destruct
+| #c #I #G #L #T #U #i #_ #_ #q #J #W #U1 #H destruct
+| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W #U1 #H destruct /3 width=8 by ex4_4_intro, or_introl/
+| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #q #J #W #U1 #H destruct
+| #cU #cT #G #L #U1 #U2 #T1 #T2 #_ #_ #_ #q #J #W #U1 #H destruct
+| #c #G #L #V #T1 #T #T2 #HT1 #HT2 #q #J #W #U1 #H destruct /3 width=5 by ex5_2_intro, or_intror/
+| #c #G #L #V #T1 #T2 #_ #q #J #W #U1 #H destruct
+| #c #G #L #V1 #V2 #T #_ #q #J #W #U1 #H destruct
+| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #q #J #W #U1 #H destruct
+| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #q #J #W #U1 #H destruct
 ]
 qed-.
 
-lemma cpg_inv_bind1: ∀h,r,p,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{p,I}V1.T1 ➡[h, r] U2 → (
-                     ∃∃rV,rT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, rT] T2 &
-                                    U2 = ⓑ{p,I}V2.T2 & r = (↓rV)+rT
-                     ) ∨
-                     ∃∃rT,T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, rT] T & ⬆*[1] U2 ≡ T &
-                             p = true & I = Abbr & r = (↓rT)+𝟙𝟘.
+lemma cpg_inv_bind1: ∀Rt,c,h,p,I,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[Rt,c,h] U2 →
+                     ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈[Rt,cT,h] T2 &
+                                       U2 = ⓑ[p,I]V2.T2 & c = ((↕*cV)∨cT)
+                      | ∃∃cT,T. ⇧*[1] T ≘ T1 & ❪G,L❫ ⊢ 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: ∀h,r,p,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{p}V1.T1 ➡[h, r] U2 → (
-                     ∃∃rV,rT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, rT] T2 &
-                                    U2 = ⓓ{p}V2.T2 & r = (↓rV)+rT
-                     ) ∨
-                     ∃∃rT,T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, rT] T & ⬆*[1] U2 ≡ T &
-                             p = true & r = (↓rT)+𝟙𝟘.
-#h #r #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H *
+lemma cpg_inv_abbr1: ∀Rt,c,h,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ⬈[Rt,c,h] U2 →
+                     ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ⬈[Rt,cT,h] T2 &
+                                       U2 = ⓓ[p]V2.T2 & c = ((↕*cV)∨cT)
+                      | ∃∃cT,T. ⇧*[1] T ≘ T1 & ❪G,L❫ ⊢ 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: ∀h,r,p,G,L,V1,T1,U2.  ⦃G, L⦄ ⊢ ⓛ{p}V1.T1 ➡[h, r] U2 →
-                     ∃∃rV,rT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[h, rT] T2 &
-                                    U2 = ⓛ{p} V2. T2 & r = (↓rV)+rT.
-#h #r #p #G #L #V1 #T1 #U2 #H elim (cpg_inv_bind1 … H) -H * 
+lemma cpg_inv_abst1: ∀Rt,c,h,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ⬈[Rt,c,h] U2 →
+                     ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L.ⓛV1❫ ⊢ 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/
-| #r #T #_ #_ #_ #H destruct
+| #c #T #_ #_ #_ #H destruct
 ]
 qed-.
 
-fact cpg_inv_flat1_aux: ∀h,r,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, r] U2 →
-                        ∀J,V1,U1. U = ⓕ{J}V1.U1 →
-                        ∨∨ ∃∃rV,rT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, rT] T2 &
-                                          U2 = ⓕ{J}V2.T2 & r = (↓rV)+rT
-                         | ∃∃rT. ⦃G, L⦄ ⊢ U1 ➡[h, rT] U2 & J = Cast & r = (↓rT)+𝟙𝟘
-                         | ∃∃rV. ⦃G, L⦄ ⊢ V1 ➡[h, rV] U2 & J = Cast & r = (↓rV)+𝟘𝟙
-                         | ∃∃rV,rW,rT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, rW] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, rT] T2 &
-                                                        J = Appl & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & r = (↓rV)+(↓rW)+(↓rT)+𝟙𝟘
-                         | ∃∃rV,rW,rT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ➡[h, rW] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, rT] T2 &
-                                                          J = Appl & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & r = (↓rV)+(↓rW)+(↓rT)+𝟙𝟘.
-#h #r #G #L #U #U2 * -r -G -L -U -U2
-[ #I #G #L #J #W #U1 #H destruct
-| #G #L #s #J #W #U1 #H destruct
-| #r #G #L #V1 #V2 #W2 #_ #_ #J #W #U1 #H destruct
-| #r #G #L #V1 #V2 #W2 #_ #_ #J #W #U1 #H destruct
-| #r #I #G #L #V #T #U #i #_ #_ #J #W #U1 #H destruct
-| #rv #rT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
-| #rv #rT #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=8 by or5_intro0, ex4_4_intro/
-| #r #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
-| #r #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=3 by or5_intro1, ex3_intro/
-| #r #G #L #V1 #V2 #T #HV12 #J #W #U1 #H destruct /3 width=3 by or5_intro2, ex3_intro/
-| #rv #rW #rT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=15 by or5_intro3, ex7_9_intro/
-| #rv #rW #rT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=17 by or5_intro4, ex8_10_intro/
+fact cpg_inv_appl1_aux: ∀Rt,c,h,G,L,U,U2. ❪G,L❫ ⊢ U ⬈[Rt,c,h] U2 →
+                        ∀V1,U1. U = ⓐV1.U1 →
+                        ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] T2 &
+                                          U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
+                         | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ W1 ⬈[Rt,cW,h] W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈[Rt,cT,h] 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 ⬈[Rt,cV,h] V & ⇧*[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈[Rt,cW,h] W2 & ❪G,L.ⓓW1❫ ⊢ 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
+| #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct
+| #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct
+| #c #I #G #L #T #U #i #_ #_ #W #U1 #H destruct
+| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #W #U1 #H destruct
+| #cV #cT #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #W #U1 #H destruct /3 width=8 by or3_intro0, ex4_4_intro/
+| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #_ #W #U1 #H destruct
+| #c #G #L #V #T1 #T #T2 #_ #_ #W #U1 #H destruct
+| #c #G #L #V #T1 #T2 #_ #W #U1 #H destruct
+| #c #G #L #V1 #V2 #T #_ #W #U1 #H destruct
+| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #W #U1 #H destruct /3 width=15 by or3_intro1, ex6_9_intro/
+| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #W #U1 #H destruct /3 width=17 by or3_intro2, ex7_10_intro/
 ]
 qed-.
 
-lemma cpg_inv_flat1: ∀h,r,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h, r] U2 →
-                     ∨∨ ∃∃rV,rT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, rT] T2 &
-                                       U2 = ⓕ{I}V2.T2 & r = (↓rV)+rT
-                      | ∃∃rT. ⦃G, L⦄ ⊢ U1 ➡[h, rT] U2 & I = Cast & r = (↓rT)+𝟙𝟘
-                      | ∃∃rV. ⦃G, L⦄ ⊢ V1 ➡[h, rV] U2 & I = Cast & r = (↓rV)+𝟘𝟙
-                      | ∃∃rV,rW,rT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, rW] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, rT] T2 &
-                                                     I = Appl & U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & r = (↓rV)+(↓rW)+(↓rT)+𝟙𝟘
-                      | ∃∃rV,rW,rT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ➡[h, rW] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, rT] T2 &
-                                                       I = Appl & U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & r = (↓rV)+(↓rW)+(↓rT)+𝟙𝟘.
-/2 width=3 by cpg_inv_flat1_aux/ qed-.
+lemma cpg_inv_appl1: ∀Rt,c,h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓐV1.U1 ⬈[Rt,c,h] U2 →
+                     ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] T2 &
+                                       U2 = ⓐV2.T2 & c = ((↕*cV)∨cT)
+                      | ∃∃cV,cW,cT,p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ W1 ⬈[Rt,cW,h] W2 & ❪G,L.ⓛW1❫ ⊢ T1 ⬈[Rt,cT,h] 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 ⬈[Rt,cV,h] V & ⇧*[1] V ≘ V2 & ❪G,L❫ ⊢ W1 ⬈[Rt,cW,h] W2 & ❪G,L.ⓓW1❫ ⊢ 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-.
 
-lemma cpg_inv_appl1: ∀h,r,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐV1.U1 ➡[h, r] U2 →
-                     ∨∨ ∃∃rV,rT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, rT] T2 &
-                                       U2 = ⓐV2.T2 & r = (↓rV)+rT
-                      | ∃∃rV,rW,rT,p,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, rW] W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, rT] T2 &
-                                                     U1 = ⓛ{p}W1.T1 & U2 = ⓓ{p}ⓝW2.V2.T2 & r = (↓rV)+(↓rW)+(↓rT)+𝟙𝟘
-                      | ∃∃rV,rW,rT,p,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V & ⬆*[1] V ≡ V2 & ⦃G, L⦄ ⊢ W1 ➡[h, rW] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, rT] T2 &
-                                                       U1 = ⓓ{p}W1.T1 & U2 = ⓓ{p}W2.ⓐV2.T2 & r = (↓rV)+(↓rW)+(↓rT)+𝟙𝟘.
-#h #r #G #L #V1 #U1 #U2 #H elim (cpg_inv_flat1 … H) -H *
-[ /3 width=8 by or3_intro0, ex4_4_intro/
-|2,3: #r #_ #H destruct
-| /3 width=15 by or3_intro1, ex6_9_intro/
-| /3 width=17 by or3_intro2, ex7_10_intro/
+fact cpg_inv_cast1_aux: ∀Rt,c,h,G,L,U,U2. ❪G,L❫ ⊢ U ⬈[Rt,c,h] U2 →
+                        ∀V1,U1. U = ⓝV1.U1 →
+                        ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] T2 &
+                                          Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
+                         | ∃∃cT. ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘
+                         | ∃∃cV. ❪G,L❫ ⊢ 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
+| #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct
+| #c #G #L #V1 #V2 #W2 #_ #_ #W #U1 #H destruct
+| #c #I #G #L #T #U #i #_ #_ #W #U1 #H destruct
+| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #_ #_ #W #U1 #H destruct
+| #cV #cT #G #L #V1 #V2 #T1 #T2 #_ #_ #W #U1 #H destruct
+| #cV #cT #G #L #V1 #V2 #T1 #T2 #HRt #HV12 #HT12 #W #U1 #H destruct /3 width=9 by or3_intro0, ex5_4_intro/
+| #c #G #L #V #T1 #T #T2 #_ #_ #W #U1 #H destruct
+| #c #G #L #V #T1 #T2 #HT12 #W #U1 #H destruct /3 width=3 by or3_intro1, ex2_intro/
+| #c #G #L #V1 #V2 #T #HV12 #W #U1 #H destruct /3 width=3 by or3_intro2, ex2_intro/
+| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #W #U1 #H destruct
+| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #W #U1 #H destruct
 ]
 qed-.
 
-lemma cpg_inv_cast1: ∀h,r,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[h, r] U2 →
-                     ∨∨ ∃∃rV,rT,V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, rV] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, rT] T2 &
-                                       U2 = ⓝV2.T2 & r = (↓rV)+rT
-                      | ∃∃rT. ⦃G, L⦄ ⊢ U1 ➡[h, rT] U2 & r = (↓rT)+𝟙𝟘
-                      | ∃∃rV. ⦃G, L⦄ ⊢ V1 ➡[h, rV] U2 & r = (↓rV)+𝟘𝟙.
-#h #r #G #L #V1 #U1 #U2 #H elim (cpg_inv_flat1 … H) -H *
-[ /3 width=8 by or3_intro0, ex4_4_intro/
-|2,3: /3 width=3 by or3_intro1, or3_intro2, ex2_intro/
-| #rv #rW #rT #p #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #H destruct
-| #rv #rW #rT #p #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #H destruct
-]
+lemma cpg_inv_cast1: ∀Rt,c,h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ⬈[Rt,c,h] U2 →
+                     ∨∨ ∃∃cV,cT,V2,T2. ❪G,L❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] T2 &
+                                       Rt cV cT & U2 = ⓝV2.T2 & c = (cV∨cT)
+                      | ∃∃cT. ❪G,L❫ ⊢ U1 ⬈[Rt,cT,h] U2 & c = cT+𝟙𝟘
+                      | ∃∃cV. ❪G,L❫ ⊢ 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: ∀Rt,c,h,I,G,K,V1,T2. ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈[Rt,c,h] T2 →
+                          ∨∨ T2 = #0 ∧ c = 𝟘𝟘
+                           | ∃∃cV,V2. ❪G,K❫ ⊢ V1 ⬈[Rt,cV,h] V2 & ⇧*[1] V2 ≘ T2 &
+                                      I = Abbr & c = cV
+                           | ∃∃cV,V2. ❪G,K❫ ⊢ 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: ∀Rt,c,h,I,G,K,T2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈[Rt,c,h] T2 →
+                          ∨∨ T2 = #(↑i) ∧ c = 𝟘𝟘
+                           | ∃∃T. ❪G,K❫ ⊢ #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: ∀h,r,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[h, r] T → ∀b.
-                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[h, r] ⓑ{b,I}V2.T2 &
-                                    T = -ⓑ{I}V2.T2.
-#h #r #I #G #L #V1 #T1 #T #H #b elim (cpg_inv_bind1 … H) -H *
-[ #rV #rT #V2 #T2 #HV12 #HT12 #H1 #H2 destruct /3 width=4 by cpg_bind, ex2_2_intro/
-| #r #T2 #_ #_ #H destruct
+lemma cpg_fwd_bind1_minus: ∀Rt,c,h,I,G,L,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ⬈[Rt,c,h] T → ∀p.
+                           ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[Rt,c,h] ⓑ[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
 ]
 qed-.