update in ground static_2 basic_2 apps_2

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

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 9e9dcf7b2e23a9e95e5616f1a221e5c72649d2f3..b69c9ebdefa75db660f1feeddb4dd9b41264a426 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx.ma
@@ -25,7 +25,7 @@ include "basic_2/rt_transition/cpg.ma".
 (* EXTENDED CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS **************)
 
 definition cpx (G) (L): relation2 term term ≝
-           λT1,T2. â\88\83c. â\9dªG,Lâ\9d« ⊢ T1 ⬈[sfull,rtc_eq_f,c] T2.
+           λT1,T2. â\88\83c. â\9d¨G,Lâ\9d© ⊢ T1 ⬈[sfull,rtc_eq_f,c] T2.
 
 interpretation
   "extended context-sensitive parallel rt-transition (term)"
@@ -34,71 +34,71 @@ interpretation
 (* Basic properties *********************************************************)
 
 (* Basic_2A1: uses: cpx_st *)
-lemma cpx_qu (G) (L): â\88\80s1,s2. â\9dªG,Lâ\9d« ⊢ ⋆s1 ⬈ ⋆s2.
+lemma cpx_qu (G) (L): â\88\80s1,s2. â\9d¨G,Lâ\9d© ⊢ ⋆s1 ⬈ ⋆s2.
 /3 width=2 by cpg_ess, ex_intro/ qed.
 
 lemma cpx_delta (G) (K):
-      â\88\80I,V1,V2,W2. â\9dªG,Kâ\9d« ⊢ V1 ⬈ V2 →
-      â\87§[1] V2 â\89\98 W2 â\86\92 â\9dªG,K.â\93\91[I]V1â\9d« ⊢ #0 ⬈ W2.
+      â\88\80I,V1,V2,W2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈ V2 →
+      â\87§[1] V2 â\89\98 W2 â\86\92 â\9d¨G,K.â\93\91[I]V1â\9d© ⊢ #0 ⬈ W2.
 #G #K * #V1 #V2 #W2 *
 /3 width=4 by cpg_delta, cpg_ell, ex_intro/
 qed.
 
 lemma cpx_lref (G) (K):
-      â\88\80I,T,U,i. â\9dªG,Kâ\9d« ⊢ #i ⬈ T →
-      â\87§[1] T â\89\98 U â\86\92 â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬈ U.
+      â\88\80I,T,U,i. â\9d¨G,Kâ\9d© ⊢ #i ⬈ T →
+      â\87§[1] T â\89\98 U â\86\92 â\9d¨G,K.â\93\98[I]â\9d© ⊢ #↑i ⬈ U.
 #G #K #I #T #U #i *
 /3 width=4 by cpg_lref, ex_intro/
 qed.
 
 lemma cpx_bind (G) (L):
       ∀p,I,V1,V2,T1,T2.
-      â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 â\86\92 â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ⬈ T2 →
-      â\9dªG,Lâ\9d« ⊢ ⓑ[p,I]V1.T1 ⬈ ⓑ[p,I]V2.T2.
+      â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 â\86\92 â\9d¨G,L.â\93\91[I]V1â\9d© ⊢ T1 ⬈ T2 →
+      â\9d¨G,Lâ\9d© ⊢ ⓑ[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 (G) (L):
       ∀I,V1,V2,T1,T2.
-      â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ⬈ T2 →
-      â\9dªG,Lâ\9d« ⊢ ⓕ[I]V1.T1 ⬈ ⓕ[I]V2.T2.
+      â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 â\86\92 â\9d¨G,Lâ\9d© ⊢ T1 ⬈ T2 →
+      â\9d¨G,Lâ\9d© ⊢ ⓕ[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 (G) (L):
-      â\88\80T1,T. â\87§[1] T â\89\98 T1 â\86\92 â\88\80T2. â\9dªG,Lâ\9d« ⊢ T ⬈ T2 →
-      â\88\80V. â\9dªG,Lâ\9d« ⊢ +ⓓV.T1 ⬈ T2.
+      â\88\80T1,T. â\87§[1] T â\89\98 T1 â\86\92 â\88\80T2. â\9d¨G,Lâ\9d© ⊢ T ⬈ T2 →
+      â\88\80V. â\9d¨G,Lâ\9d© ⊢ +ⓓV.T1 ⬈ T2.
 #G #L #T1 #T #HT1 #T2 *
 /3 width=4 by cpg_zeta, ex_intro/
 qed.
 
 lemma cpx_eps (G) (L):
-      â\88\80V,T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88 T2 â\86\92 â\9dªG,Lâ\9d« ⊢ ⓝV.T1 ⬈ T2.
+      â\88\80V,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â¬\88 T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ ⓝ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 (G) (L):
-      â\88\80V1,V2,T. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 â\86\92 â\9dªG,Lâ\9d« ⊢ ⓝV1.T ⬈ V2.
+      â\88\80V1,V2,T. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 â\86\92 â\9d¨G,Lâ\9d© ⊢ ⓝV1.T ⬈ V2.
 #G #L #V1 #V2 #T *
 /3 width=2 by cpg_ee, ex_intro/
 qed.
 
 lemma cpx_beta (G) (L):
       ∀p,V1,V2,W1,W2,T1,T2.
-      â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88 W2 â\86\92 â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈ T2 →
-      â\9dªG,Lâ\9d« ⊢ ⓐV1.ⓛ[p]W1.T1 ⬈ ⓓ[p]ⓝW2.V2.T2.
+      â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 â\86\92 â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88 W2 â\86\92 â\9d¨G,L.â\93\9bW1â\9d© ⊢ T1 ⬈ T2 →
+      â\9d¨G,Lâ\9d© ⊢ ⓐ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 (G) (L):
       ∀p,V1,V,V2,W1,W2,T1,T2.
-      â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V â\86\92 â\87§[1] V â\89\98 V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88 W2 â\86\92 â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ⬈ T2 →
-      â\9dªG,Lâ\9d« ⊢ ⓐV1.ⓓ[p]W1.T1 ⬈ ⓓ[p]W2.ⓐV2.T2.
+      â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V â\86\92 â\87§[1] V â\89\98 V2 â\86\92 â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88 W2 â\86\92 â\9d¨G,L.â\93\93W1â\9d© ⊢ T1 ⬈ T2 →
+      â\9d¨G,Lâ\9d© ⊢ ⓐ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.
@@ -110,13 +110,13 @@ lemma cpx_refl (G) (L): reflexive … (cpx G L).
 (* Advanced properties ******************************************************)
 
 lemma cpx_pair_sn (G) (L):
-      â\88\80I,V1,V2. â\9dªG,Lâ\9d« ⊢ V1 ⬈ V2 →
-      â\88\80T. â\9dªG,Lâ\9d« ⊢ ②[I]V1.T ⬈ ②[I]V2.T.
+      â\88\80I,V1,V2. â\9d¨G,Lâ\9d© ⊢ V1 ⬈ V2 →
+      â\88\80T. â\9d¨G,Lâ\9d© ⊢ ②[I]V1.T ⬈ ②[I]V2.T.
 #G #L * /2 width=2 by cpx_flat, cpx_bind/
 qed.
 
 lemma cpg_cpx (Rs) (Rk) (c) (G) (L):
-      â\88\80T1,T2. â\9dªG,Lâ\9d« â\8a¢ T1 â¬\88[Rs,Rk,c] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ⬈ T2.
+      â\88\80T1,T2. â\9d¨G,Lâ\9d© â\8a¢ T1 â¬\88[Rs,Rk,c] T2 â\86\92 â\9d¨G,Lâ\9d© ⊢ 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.
@@ -124,80 +124,80 @@ qed.
 (* Basic inversion lemmas ***************************************************)
 
 lemma cpx_inv_atom1 (G) (L):
-      â\88\80J,T2. â\9dªG,Lâ\9d« ⊢ ⓪[J] ⬈ T2 →
+      â\88\80J,T2. â\9d¨G,Lâ\9d© ⊢ ⓪[J] ⬈ T2 →
       ∨∨ T2 = ⓪[J]
        | ∃∃s1,s2. T2 = ⋆s2 & J = Sort s1
-       | â\88\83â\88\83I,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1 & J = LRef 0
-       | â\88\83â\88\83I,K,T,i. â\9dªG,Kâ\9d« ⊢ #i ⬈ T & ⇧[1] T ≘ T2 & L = K.ⓘ[I] & J = LRef (↑i).
+       | â\88\83â\88\83I,K,V1,V2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1 & J = LRef 0
+       | â\88\83â\88\83I,K,T,i. â\9d¨G,Kâ\9d© ⊢ #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 (G) (L):
-      â\88\80T2,s1. â\9dªG,Lâ\9d« ⊢ ⋆s1 ⬈ T2 →
+      â\88\80T2,s1. â\9d¨G,Lâ\9d© ⊢ ⋆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 (G) (L):
-      â\88\80T2. â\9dªG,Lâ\9d« ⊢ #0 ⬈ T2 →
+      â\88\80T2. â\9d¨G,Lâ\9d© ⊢ #0 ⬈ T2 →
       ∨∨ T2 = #0
-       | â\88\83â\88\83I,K,V1,V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2 & L = K.ⓑ[I]V1.
+       | â\88\83â\88\83I,K,V1,V2. â\9d¨G,Kâ\9d© ⊢ 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 (G) (L):
-      â\88\80T2,i. â\9dªG,Lâ\9d« ⊢ #↑i ⬈ T2 →
+      â\88\80T2,i. â\9d¨G,Lâ\9d© ⊢ #↑i ⬈ T2 →
       ∨∨ T2 = #(↑i)
-       | â\88\83â\88\83I,K,T. â\9dªG,Kâ\9d« ⊢ #i ⬈ T & ⇧[1] T ≘ T2 & L = K.ⓘ[I].
+       | â\88\83â\88\83I,K,T. â\9d¨G,Kâ\9d© ⊢ #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 (G) (L):
-      â\88\80T2,l. â\9dªG,Lâ\9d« ⊢ §l ⬈ T2 → T2 = §l.
+      â\88\80T2,l. â\9d¨G,Lâ\9d© ⊢ §l ⬈ T2 → T2 = §l.
 #G #L #T2 #l * #c #H elim (cpg_inv_gref1 … H) -H //
 qed-.
 
 lemma cpx_inv_bind1 (G) (L):
-      â\88\80p,I,V1,T1,U2. â\9dªG,Lâ\9d« ⊢ ⓑ[p,I]V1.T1 ⬈ U2 →
-      â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 & â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ⬈ T2 & U2 = ⓑ[p,I]V2.T2
-       | â\88\83â\88\83T. â\87§[1] T â\89\98 T1 & â\9dªG,Lâ\9d« ⊢ T ⬈ U2 & p = true & I = Abbr.
+      â\88\80p,I,V1,T1,U2. â\9d¨G,Lâ\9d© ⊢ ⓑ[p,I]V1.T1 ⬈ U2 →
+      â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 & â\9d¨G,L.â\93\91[I]V1â\9d© ⊢ T1 ⬈ T2 & U2 = ⓑ[p,I]V2.T2
+       | â\88\83â\88\83T. â\87§[1] T â\89\98 T1 & â\9d¨G,Lâ\9d© ⊢ 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 (G) (L):
-      â\88\80p,V1,T1,U2. â\9dªG,Lâ\9d« ⊢ ⓓ[p]V1.T1 ⬈ U2 →
-      â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 & â\9dªG,L.â\93\93V1â\9d« ⊢ T1 ⬈ T2 & U2 = ⓓ[p]V2.T2
-       | â\88\83â\88\83T. â\87§[1] T â\89\98 T1 & â\9dªG,Lâ\9d« ⊢ T ⬈ U2 & p = true.
+      â\88\80p,V1,T1,U2. â\9d¨G,Lâ\9d© ⊢ ⓓ[p]V1.T1 ⬈ U2 →
+      â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 & â\9d¨G,L.â\93\93V1â\9d© ⊢ T1 ⬈ T2 & U2 = ⓓ[p]V2.T2
+       | â\88\83â\88\83T. â\87§[1] T â\89\98 T1 & â\9d¨G,Lâ\9d© ⊢ 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 (G) (L):
-      â\88\80p,V1,T1,U2. â\9dªG,Lâ\9d« ⊢ ⓛ[p]V1.T1 ⬈ U2 →
-      â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 & â\9dªG,L.â\93\9bV1â\9d« ⊢ T1 ⬈ T2 & U2 = ⓛ[p]V2.T2.
+      â\88\80p,V1,T1,U2. â\9d¨G,Lâ\9d© ⊢ ⓛ[p]V1.T1 ⬈ U2 →
+      â\88\83â\88\83V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 & â\9d¨G,L.â\93\9bV1â\9d© ⊢ 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 (G) (L):
-      â\88\80V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓐ V1.U1 ⬈ U2 →
-      â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈ T2 & U2 = ⓐV2.T2
-       | â\88\83â\88\83p,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88 W2 & â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈ T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2
-       | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V & â\87§[1] V â\89\98 V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88 W2 & â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ⬈ T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2.
+      â\88\80V1,U1,U2. â\9d¨G,Lâ\9d© ⊢ ⓐ V1.U1 ⬈ U2 →
+      â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 & â\9d¨G,Lâ\9d© ⊢ U1 ⬈ T2 & U2 = ⓐV2.T2
+       | â\88\83â\88\83p,V2,W1,W2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 & â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88 W2 & â\9d¨G,L.â\93\9bW1â\9d© ⊢ T1 ⬈ T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2
+       | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V & â\87§[1] V â\89\98 V2 & â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88 W2 & â\9d¨G,L.â\93\93W1â\9d© ⊢ 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 (G) (L):
-      â\88\80V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓝV1.U1 ⬈ U2 →
-      â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈ T2 & U2 = ⓝV2.T2
-       | â\9dªG,Lâ\9d« ⊢ U1 ⬈ U2
-       | â\9dªG,Lâ\9d« ⊢ V1 ⬈ U2.
+      â\88\80V1,U1,U2. â\9d¨G,Lâ\9d© ⊢ ⓝV1.U1 ⬈ U2 →
+      â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 & â\9d¨G,Lâ\9d© ⊢ U1 ⬈ T2 & U2 = ⓝV2.T2
+       | â\9d¨G,Lâ\9d© ⊢ U1 ⬈ U2
+       | â\9d¨G,Lâ\9d© ⊢ 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-.
@@ -205,28 +205,28 @@ qed-.
 (* Advanced inversion lemmas ************************************************)
 
 lemma cpx_inv_zero1_pair (G) (K):
-      â\88\80I,V1,T2. â\9dªG,K.â\93\91[I]V1â\9d« ⊢ #0 ⬈ T2 →
+      â\88\80I,V1,T2. â\9d¨G,K.â\93\91[I]V1â\9d© ⊢ #0 ⬈ T2 →
       ∨∨ T2 = #0
-       | â\88\83â\88\83V2. â\9dªG,Kâ\9d« ⊢ V1 ⬈ V2 & ⇧[1] V2 ≘ T2.
+       | â\88\83â\88\83V2. â\9d¨G,Kâ\9d© ⊢ 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 (G) (K):
-      â\88\80I,T2,i. â\9dªG,K.â\93\98[I]â\9d« ⊢ #↑i ⬈ T2 →
+      â\88\80I,T2,i. â\9d¨G,K.â\93\98[I]â\9d© ⊢ #↑i ⬈ T2 →
       ∨∨ T2 = #(↑i)
-       | â\88\83â\88\83T. â\9dªG,Kâ\9d« ⊢ #i ⬈ T & ⇧[1] T ≘ T2.
+       | â\88\83â\88\83T. â\9d¨G,Kâ\9d© ⊢ #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 (G) (L):
-      â\88\80I,V1,U1,U2. â\9dªG,Lâ\9d« ⊢ ⓕ[I]V1.U1 ⬈ U2 →
-      â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 & â\9dªG,Lâ\9d« ⊢ U1 ⬈ T2 & U2 = ⓕ[I]V2.T2
-       | (â\9dªG,Lâ\9d« ⊢ U1 ⬈ U2 ∧ I = Cast)
-       | (â\9dªG,Lâ\9d« ⊢ V1 ⬈ U2 ∧ I = Cast)
-       | â\88\83â\88\83p,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88 W2 & â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈ T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & I = Appl
-       | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V & â\87§[1] V â\89\98 V2 & â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88 W2 & â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ⬈ T2 & U1 = ⓓ[p]W1.T1 & U2 = ⓓ[p]W2.ⓐV2.T2 & I = Appl.
+      â\88\80I,V1,U1,U2. â\9d¨G,Lâ\9d© ⊢ ⓕ[I]V1.U1 ⬈ U2 →
+      â\88¨â\88¨ â\88\83â\88\83V2,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 & â\9d¨G,Lâ\9d© ⊢ U1 ⬈ T2 & U2 = ⓕ[I]V2.T2
+       | (â\9d¨G,Lâ\9d© ⊢ U1 ⬈ U2 ∧ I = Cast)
+       | (â\9d¨G,Lâ\9d© ⊢ V1 ⬈ U2 ∧ I = Cast)
+       | â\88\83â\88\83p,V2,W1,W2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 & â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88 W2 & â\9d¨G,L.â\93\9bW1â\9d© ⊢ T1 ⬈ T2 & U1 = ⓛ[p]W1.T1 & U2 = ⓓ[p]ⓝW2.V2.T2 & I = Appl
+       | â\88\83â\88\83p,V,V2,W1,W2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V & â\87§[1] V â\89\98 V2 & â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88 W2 & â\9d¨G,L.â\93\93W1â\9d© ⊢ 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/
@@ -238,8 +238,8 @@ qed-.
 (* Basic forward lemmas *****************************************************)
 
 lemma cpx_fwd_bind1_minus (G) (L):
-      â\88\80I,V1,T1,T. â\9dªG,Lâ\9d« ⊢ -ⓑ[I]V1.T1 ⬈ T → ∀p.
-      â\88\83â\88\83V2,T2. â\9dªG,Lâ\9d« ⊢ ⓑ[p,I]V1.T1 ⬈ ⓑ[p,I]V2.T2 & T = -ⓑ[I]V2.T2.
+      â\88\80I,V1,T1,T. â\9d¨G,Lâ\9d© ⊢ -ⓑ[I]V1.T1 ⬈ T → ∀p.
+      â\88\83â\88\83V2,T2. â\9d¨G,Lâ\9d© ⊢ ⓑ[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-.
@@ -249,28 +249,28 @@ qed-.
 lemma cpx_ind (Q:relation4 …):
       (∀I,G,L. Q G L (⓪[I]) (⓪[I])) →
       (∀G,L,s1,s2. Q G L (⋆s1) (⋆s2)) →
-      (â\88\80I,G,K,V1,V2,W2. â\9dªG,Kâ\9d« ⊢ V1 ⬈ V2 → Q G K V1 V2 →
+      (â\88\80I,G,K,V1,V2,W2. â\9d¨G,Kâ\9d© ⊢ V1 ⬈ V2 → Q G K V1 V2 →
         ⇧[1] V2 ≘ W2 → Q G (K.ⓑ[I]V1) (#0) W2
-      ) â\86\92 (â\88\80I,G,K,T,U,i. â\9dªG,Kâ\9d« ⊢ #i ⬈ T → Q G K (#i) T →
+      ) â\86\92 (â\88\80I,G,K,T,U,i. â\9d¨G,Kâ\9d© ⊢ #i ⬈ T → Q G K (#i) T →
         ⇧[1] T ≘ U → Q G (K.ⓘ[I]) (#↑i) (U)
-      ) â\86\92 (â\88\80p,I,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 â\86\92 â\9dªG,L.â\93\91[I]V1â\9d« ⊢ T1 ⬈ T2 →
+      ) â\86\92 (â\88\80p,I,G,L,V1,V2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 â\86\92 â\9d¨G,L.â\93\91[I]V1â\9d© ⊢ 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)
-      ) â\86\92 (â\88\80I,G,L,V1,V2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ⬈ T2 →
+      ) â\86\92 (â\88\80I,G,L,V1,V2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 â\86\92 â\9d¨G,Lâ\9d© ⊢ T1 ⬈ T2 →
         Q G L V1 V2 → Q G L T1 T2 → Q G L (ⓕ[I]V1.T1) (ⓕ[I]V2.T2)
-      ) â\86\92 (â\88\80G,L,V,T1,T,T2. â\87§[1] T â\89\98 T1 â\86\92 â\9dªG,Lâ\9d« ⊢ T ⬈ T2 → Q G L T T2 →
+      ) â\86\92 (â\88\80G,L,V,T1,T,T2. â\87§[1] T â\89\98 T1 â\86\92 â\9d¨G,Lâ\9d© ⊢ T ⬈ T2 → Q G L T T2 →
         Q G L (+ⓓV.T1) T2
-      ) â\86\92 (â\88\80G,L,V,T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ⬈ T2 → Q G L T1 T2 →
+      ) â\86\92 (â\88\80G,L,V,T1,T2. â\9d¨G,Lâ\9d© ⊢ T1 ⬈ T2 → Q G L T1 T2 →
         Q G L (ⓝV.T1) T2
-      ) â\86\92 (â\88\80G,L,V1,V2,T. â\9dªG,Lâ\9d« ⊢ V1 ⬈ V2 → Q G L V1 V2 →
+      ) â\86\92 (â\88\80G,L,V1,V2,T. â\9d¨G,Lâ\9d© ⊢ V1 ⬈ V2 → Q G L V1 V2 →
         Q G L (ⓝV1.T) V2
-      ) â\86\92 (â\88\80p,G,L,V1,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V2 â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88 W2 â\86\92 â\9dªG,L.â\93\9bW1â\9d« ⊢ T1 ⬈ T2 →
+      ) â\86\92 (â\88\80p,G,L,V1,V2,W1,W2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V2 â\86\92 â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88 W2 â\86\92 â\9d¨G,L.â\93\9bW1â\9d© ⊢ 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)
-      ) â\86\92 (â\88\80p,G,L,V1,V,V2,W1,W2,T1,T2. â\9dªG,Lâ\9d« â\8a¢ V1 â¬\88 V â\86\92 â\9dªG,Lâ\9d« â\8a¢ W1 â¬\88 W2 â\86\92 â\9dªG,L.â\93\93W1â\9d« ⊢ T1 ⬈ T2 →
+      ) â\86\92 (â\88\80p,G,L,V1,V,V2,W1,W2,T1,T2. â\9d¨G,Lâ\9d© â\8a¢ V1 â¬\88 V â\86\92 â\9d¨G,Lâ\9d© â\8a¢ W1 â¬\88 W2 â\86\92 â\9d¨G,L.â\93\93W1â\9d© ⊢ 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)
       ) →
-      â\88\80G,L,T1,T2. â\9dªG,Lâ\9d« ⊢ T1 ⬈ T2 → Q G L T1 T2.
+      â\88\80G,L,T1,T2. â\9d¨G,Lâ\9d© ⊢ 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-.