X-Git-Url: http://matita.cs.unibo.it/gitweb/?p=helm.git;a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_transition%2Fcpx.ma;h=13e3a4230fb5e03800ae041d72cc55b19a018d14;hp=2da498aa0f136cbf73dd5b55332738da4445f98e;hb=3c7b4071a9ac096b02334c1d47468776b948e2de;hpb=2f6f2b7c01d47d23f61dd48d767bcb37aecdcfea

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 2da498aa0..13e3a4230 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
@@ -19,214 +19,217 @@ include "ground/xoa/ex_6_6.ma".
 include "ground/xoa/ex_6_7.ma".
 include "ground/xoa/ex_7_7.ma".
 include "ground/xoa/or_4.ma".
-include "basic_2/notation/relations/predty_5.ma".
+include "basic_2/notation/relations/predty_4.ma".
 include "basic_2/rt_transition/cpg.ma".
 
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+(* EXTENDED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS **************)
 
-definition cpx (h): relation4 genv lenv term term ≝
-                    λG,L,T1,T2. ∃c. ❪G,L❫ ⊢ T1 ⬈[eq_f,c,h] T2.
+definition sort_eq_f: relation nat ≝ λs1,s2. ⊤.
+
+definition cpx (G) (L): relation2 term term ≝
+           λT1,T2. ∃c. ❪G,L❫ ⊢ T1 ⬈[sort_eq_f,rtc_eq_f,c] T2.
 
 interpretation
-   "unbound context-sensitive parallel rt-transition (term)"
-   'PRedTy h G L T1 T2 = (cpx h G L T1 T2).
+  "extended context-sensitive parallel rt-transition (term)"
+  'PRedTy G L T1 T2 = (cpx G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-(* Basic_2A1: was: cpx_st *)
-lemma cpx_ess: ∀h,G,L,s. ❪G,L❫ ⊢ ⋆s ⬈[h] ⋆(⫯[h]s).
-/2 width=2 by cpg_ess, ex_intro/ qed.
+(* Basic_2A1: uses: cpx_st *)
+lemma cpx_qu (G) (L): ∀s1,s2. ❪G,L❫ ⊢ ⋆s1 ⬈ ⋆s2.
+/3 width=2 by cpg_ess, ex_intro/ qed.
 
-lemma cpx_delta: ∀h,I,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ⬈[h] V2 →
-                 ⇧[1] V2 ≘ W2 → ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈[h] W2.
-#h * #G #K #V1 #V2 #W2 *
+lemma cpx_delta (G) (K):
+      ∀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: ∀h,I,G,K,T,U,i. ❪G,K❫ ⊢ #i ⬈[h] T →
-                ⇧[1] T ≘ U → ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈[h] U.
-#h #I #G #K #T #U #i *
+lemma cpx_lref (G) (K):
+      ∀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: ∀h,p,I,G,L,V1,V2,T1,T2.
-                ❪G,L❫ ⊢ V1 ⬈[h] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈[h] T2 →
-                ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[h] ⓑ[p,I]V2.T2.
-#h #p #I #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
+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 #p #I #V1 #V2 #T1 #T2 * #cV #HV12 *
 /3 width=2 by cpg_bind, ex_intro/
 qed.
 
-lemma cpx_flat: ∀h,I,G,L,V1,V2,T1,T2.
-                ❪G,L❫ ⊢ V1 ⬈[h] V2 → ❪G,L❫ ⊢ T1 ⬈[h] T2 →
-                ❪G,L❫ ⊢ ⓕ[I]V1.T1 ⬈[h] ⓕ[I]V2.T2.
-#h * #G #L #V1 #V2 #T1 #T2 * #cV #HV12 *
+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 #T1 #T2 * #cV #HV12 *
 /3 width=5 by cpg_appl, cpg_cast, ex_intro/
 qed.
 
-lemma cpx_zeta (h) (G) (L):
-               ∀T1,T. ⇧[1] T ≘ T1 → ∀T2. ❪G,L❫ ⊢ T ⬈[h] T2 →
-               ∀V. ❪G,L❫ ⊢ +ⓓV.T1 ⬈[h] T2.
-#h #G #L #T1 #T #HT1 #T2 *
+lemma cpx_zeta (G) (L):
+      ∀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: ∀h,G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ⬈[h] T2 → ❪G,L❫ ⊢ ⓝV.T1 ⬈[h] T2.
-#h #G #L #V #T1 #T2 *
+lemma cpx_eps (G) (L):
+      ∀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: ∀h,G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ⬈[h] V2 → ❪G,L❫ ⊢ ⓝV1.T ⬈[h] V2.
-#h #G #L #V1 #V2 #T *
+lemma cpx_ee (G) (L):
+      ∀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: ∀h,p,G,L,V1,V2,W1,W2,T1,T2.
-                ❪G,L❫ ⊢ V1 ⬈[h] V2 → ❪G,L❫ ⊢ W1 ⬈[h] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ⬈[h] T2 →
-                ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ⬈[h] ⓓ[p]ⓝW2.V2.T2.
-#h #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 *
+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 #p #V1 #V2 #W1 #W2 #T1 #T2 * #cV #HV12 * #cW #HW12 *
 /3 width=2 by cpg_beta, ex_intro/
 qed.
 
-lemma cpx_theta: ∀h,p,G,L,V1,V,V2,W1,W2,T1,T2.
-                 ❪G,L❫ ⊢ V1 ⬈[h] V → ⇧[1] V ≘ V2 → ❪G,L❫ ⊢ W1 ⬈[h] W2 →
-                 ❪G,L.ⓓW1❫ ⊢ T1 ⬈[h] T2 →
-                 ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ⬈[h] ⓓ[p]W2.ⓐV2.T2.
-#h #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
+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 #p #V1 #V #V2 #W1 #W2 #T1 #T2 * #cV #HV1 #HV2 * #cW #HW12 *
 /3 width=4 by cpg_theta, ex_intro/
 qed.
 
 (* Basic_2A1: includes: cpx_atom *)
-lemma cpx_refl: ∀h,G,L. reflexive … (cpx h G L).
+lemma cpx_refl (G) (L): reflexive … (cpx G L).
 /3 width=2 by cpg_refl, ex_intro/ qed.
 
 (* Advanced properties ******************************************************)
 
-lemma cpx_pair_sn: ∀h,I,G,L,V1,V2. ❪G,L❫ ⊢ V1 ⬈[h] V2 →
-                   ∀T. ❪G,L❫ ⊢ ②[I]V1.T ⬈[h] ②[I]V2.T.
-#h * /2 width=2 by cpx_flat, cpx_bind/
+lemma cpx_pair_sn (G) (L):
+      ∀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 (h) (Rt) (c) (G) (L):
-              ∀T1,T2. ❪G,L❫ ⊢ T1 ⬈[Rt,c,h] T2 → ❪G,L❫ ⊢ T1 ⬈[h] T2.
-#h #Rt #c #G #L #T1 #T2 #H elim H -c -G -L -T1 -T2
+lemma cpg_cpx (Rs) (Rk) (c) (G) (L):
+      ∀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.
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma cpx_inv_atom1: ∀h,J,G,L,T2. ❪G,L❫ ⊢ ⓪[J] ⬈[h] T2 →
-                     ∨∨ T2 = ⓪[J]
-                      | ∃∃s. T2 = ⋆(⫯[h]s) & J = Sort s
-                      | ∃∃I,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[h] V2 & ⇧[1] V2 ≘ T2 &
-                                     L = K.ⓑ[I]V1 & J = LRef 0
-                      | ∃∃I,K,T,i. ❪G,K❫ ⊢ #i ⬈[h] T & ⇧[1] T ≘ T2 &
-                                   L = K.ⓘ[I] & J = LRef (↑i).
-#h #J #G #L #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_intro, ex_intro/
+lemma cpx_inv_atom1 (G) (L):
+      ∀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).
+#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: ∀h,G,L,T2,s. ❪G,L❫ ⊢ ⋆s ⬈[h] T2 →
-                     ∨∨ T2 = ⋆s | T2 = ⋆(⫯[h]s).
-#h #G #L #T2 #s * #c #H elim (cpg_inv_sort1 … H) -H *
-/2 width=1 by or_introl, or_intror/
+lemma cpx_inv_sort1 (G) (L):
+      ∀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: ∀h,G,L,T2. ❪G,L❫ ⊢ #0 ⬈[h] T2 →
-                     ∨∨ T2 = #0
-                      | ∃∃I,K,V1,V2. ❪G,K❫ ⊢ V1 ⬈[h] V2 & ⇧[1] V2 ≘ T2 &
-                                     L = K.ⓑ[I]V1.
-#h #G #L #T2 * #c #H elim (cpg_inv_zero1 … H) -H *
+lemma cpx_inv_zero1 (G) (L):
+      ∀T2. ❪G,L❫ ⊢ #0 ⬈ T2 →
+      ∨∨ T2 = #0
+       | ∃∃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: ∀h,G,L,T2,i. ❪G,L❫ ⊢ #↑i ⬈[h] T2 →
-                     ∨∨ T2 = #(↑i)
-                      | ∃∃I,K,T. ❪G,K❫ ⊢ #i ⬈[h] T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
-#h #G #L #T2 #i * #c #H elim (cpg_inv_lref1 … H) -H *
+lemma cpx_inv_lref1 (G) (L):
+      ∀T2,i. ❪G,L❫ ⊢ #↑i ⬈ T2 →
+      ∨∨ T2 = #(↑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: ∀h,G,L,T2,l. ❪G,L❫ ⊢ §l ⬈[h] T2 → T2 = §l.
-#h #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
+lemma cpx_inv_gref1 (G) (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: ∀h,p,I,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[h] U2 →
-                     ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈[h] T2 &
-                                 U2 = ⓑ[p,I]V2.T2
-                      | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈[h] U2 &
-                             p = true & I = Abbr.
-#h #p #I #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_bind1 … H) -H *
+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.
+#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: ∀h,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓓ[p]V1.T1 ⬈[h] U2 →
-                     ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L.ⓓV1❫ ⊢ T1 ⬈[h] T2 &
-                                 U2 = ⓓ[p]V2.T2
-                      | ∃∃T. ⇧[1] T ≘ T1 & ❪G,L❫ ⊢ T ⬈[h] U2 & p = true.
-#h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abbr1 … H) -H *
+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.
+#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: ∀h,p,G,L,V1,T1,U2. ❪G,L❫ ⊢ ⓛ[p]V1.T1 ⬈[h] U2 →
-                     ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L.ⓛV1❫ ⊢ T1 ⬈[h] T2 &
-                              U2 = ⓛ[p]V2.T2.
-#h #p #G #L #V1 #T1 #U2 * #c #H elim (cpg_inv_abst1 … H) -H
+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.
+#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: ∀h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓐ V1.U1 ⬈[h] U2 →
-                     ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L❫ ⊢ U1 ⬈[h] T2 &
-                                 U2 = ⓐV2.T2
-                      | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L❫ ⊢ W1 ⬈[h] W2 &
-                                            ❪G,L.ⓛW1❫ ⊢ T1 ⬈[h] T2 &
-                                            U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2
-                      | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[h] V & ⇧[1] V ≘ V2 &
-                                              ❪G,L❫ ⊢ W1 ⬈[h] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈[h] T2 &
-                                              U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2.
-#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_appl1 … H) -H *
+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.
+#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: ∀h,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓝV1.U1 ⬈[h] U2 →
-                     ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L❫ ⊢ U1 ⬈[h] T2 &
-                                 U2 = ⓝV2.T2
-                      | ❪G,L❫ ⊢ U1 ⬈[h] U2
-                      | ❪G,L❫ ⊢ V1 ⬈[h] U2.
-#h #G #L #V1 #U1 #U2 * #c #H elim (cpg_inv_cast1 … H) -H *
+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.
+#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-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma cpx_inv_zero1_pair: ∀h,I,G,K,V1,T2. ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈[h] T2 →
-                          ∨∨ T2 = #0
-                           | ∃∃V2. ❪G,K❫ ⊢ V1 ⬈[h] V2 & ⇧[1] V2 ≘ T2.
-#h #I #G #L #V1 #T2 * #c #H elim (cpg_inv_zero1_pair … H) -H *
+lemma cpx_inv_zero1_pair (G) (K):
+      ∀I,V1,T2. ❪G,K.ⓑ[I]V1❫ ⊢ #0 ⬈ T2 →
+      ∨∨ T2 = #0
+       | ∃∃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: ∀h,I,G,K,T2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈[h] T2 →
-                          ∨∨ T2 = #(↑i)
-                           | ∃∃T. ❪G,K❫ ⊢ #i ⬈[h] T & ⇧[1] T ≘ T2.
-#h #I #G #L #T2 #i * #c #H elim (cpg_inv_lref1_bind … H) -H *
+lemma cpx_inv_lref1_bind (G) (K):
+      ∀I,T2,i. ❪G,K.ⓘ[I]❫ ⊢ #↑i ⬈ T2 →
+      ∨∨ T2 = #(↑i)
+       | ∃∃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: ∀h,I,G,L,V1,U1,U2. ❪G,L❫ ⊢ ⓕ[I]V1.U1 ⬈[h] U2 →
-                     ∨∨ ∃∃V2,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L❫ ⊢ U1 ⬈[h] T2 &
-                                 U2 = ⓕ[I]V2.T2
-                      | (❪G,L❫ ⊢ U1 ⬈[h] U2 ∧ I = Cast)
-                      | (❪G,L❫ ⊢ V1 ⬈[h] U2 ∧ I = Cast)
-                      | ∃∃p,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 & ❪G,L❫ ⊢ W1 ⬈[h] W2 &
-                                            ❪G,L.ⓛW1❫ ⊢ T1 ⬈[h] T2 &
-                                            U1 = ⓛ[p]W1.T1 &
-                                            U2 = ⓓ[p]ⓝW2.V2.T2 & I = Appl
-                      | ∃∃p,V,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[h] V & ⇧[1] V ≘ V2 &
-                                              ❪G,L❫ ⊢ W1 ⬈[h] W2 & ❪G,L.ⓓW1❫ ⊢ T1 ⬈[h] T2 &
-                                              U1 = ⓓ[p]W1.T1 &
-                                              U2 = ⓓ[p]W2.ⓐV2.T2 & I = Appl.
-#h * #G #L #V1 #U1 #U2 #H
+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.
+#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/
 | elim (cpx_inv_cast1 … H) -H [ * ]
@@ -236,40 +239,40 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpx_fwd_bind1_minus: ∀h,I,G,L,V1,T1,T. ❪G,L❫ ⊢ -ⓑ[I]V1.T1 ⬈[h] T → ∀p.
-                           ∃∃V2,T2. ❪G,L❫ ⊢ ⓑ[p,I]V1.T1 ⬈[h] ⓑ[p,I]V2.T2 &
-                                    T = -ⓑ[I]V2.T2.
-#h #I #G #L #V1 #T1 #T * #c #H #p elim (cpg_fwd_bind1_minus … H p) -H
+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.
+#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-.
 
 (* Basic eliminators ********************************************************)
 
-lemma cpx_ind: ∀h. ∀Q:relation4 genv lenv term term.
-               (∀I,G,L. Q G L (⓪[I]) (⓪[I])) →
-               (∀G,L,s. Q G L (⋆s) (⋆(⫯[h]s))) →
-               (∀I,G,K,V1,V2,W2. ❪G,K❫ ⊢ V1 ⬈[h] 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 ⬈[h] 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 ⬈[h] V2 → ❪G,L.ⓑ[I]V1❫ ⊢ T1 ⬈[h] 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 ⬈[h] V2 → ❪G,L❫ ⊢ T1 ⬈[h] 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 ⬈[h] T2 → Q G L T T2 →
-                  Q G L (+ⓓV.T1) T2
-               ) → (∀G,L,V,T1,T2. ❪G,L❫ ⊢ T1 ⬈[h] T2 → Q G L T1 T2 →
-                  Q G L (ⓝV.T1) T2
-               ) → (∀G,L,V1,V2,T. ❪G,L❫ ⊢ V1 ⬈[h] V2 → Q G L V1 V2 →
-                  Q G L (ⓝV1.T) V2
-               ) → (∀p,G,L,V1,V2,W1,W2,T1,T2. ❪G,L❫ ⊢ V1 ⬈[h] V2 → ❪G,L❫ ⊢ W1 ⬈[h] W2 → ❪G,L.ⓛW1❫ ⊢ T1 ⬈[h] 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 ⬈[h] V → ❪G,L❫ ⊢ W1 ⬈[h] W2 → ❪G,L.ⓓW1❫ ⊢ T1 ⬈[h] 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 ⬈[h] T2 → Q G L T1 T2.
-#h #Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #G #L #T1 #T2
+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 →
+        ⇧[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 →
+        ⇧[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 →
+        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 →
+        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 →
+        Q G L (+ⓓV.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 →
+        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 →
+        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 →
+        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.
+#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-.