X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Freduction%2Fcpx.ma;h=f7fd702e50cbaf7852b0d2356bb886d1b933c22b;hb=93bba1c94779e83184d111cd077d4167e42a74aa;hp=840bf0866c06eb991a1f43207a72fb7cb299ca7b;hpb=9a023f554e56d6edbbb2eeaf17ce61e31857ef4a;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
index 840bf0866..f7fd702e5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
@@ -19,39 +19,39 @@ include "basic_2/reduction/cpr.ma".
 (* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
 
 (* avtivate genv *)
-inductive cpx (h) (g): relation4 genv lenv term term ≝
-| cpx_atom : ∀I,G,L. cpx h g G L (⓪{I}) (⓪{I})
-| cpx_st   : ∀G,L,k,d. deg h g k (d+1) → cpx h g G L (⋆k) (⋆(next h k))
+inductive cpx (h) (o): relation4 genv lenv term term ≝
+| cpx_atom : ∀I,G,L. cpx h o G L (⓪{I}) (⓪{I})
+| cpx_st   : ∀G,L,s,d. deg h o s (d+1) → cpx h o G L (⋆s) (⋆(next h s))
 | cpx_delta: ∀I,G,L,K,V,V2,W2,i.
-             ⬇[i] L ≡ K.ⓑ{I}V → cpx h g G K V V2 →
-             ⬆[0, i+1] V2 ≡ W2 → cpx h g G L (#i) W2
+             ⬇[i] L ≡ K.ⓑ{I}V → cpx h o G K V V2 →
+             ⬆[0, i+1] V2 ≡ W2 → cpx h o G L (#i) W2
 | cpx_bind : ∀a,I,G,L,V1,V2,T1,T2.
-             cpx h g G L V1 V2 → cpx h g G (L.ⓑ{I}V1) T1 T2 →
-             cpx h g G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+             cpx h o G L V1 V2 → cpx h o G (L.ⓑ{I}V1) T1 T2 →
+             cpx h o G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
 | cpx_flat : ∀I,G,L,V1,V2,T1,T2.
-             cpx h g G L V1 V2 → cpx h g G L T1 T2 →
-             cpx h g G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-| cpx_zeta : ∀G,L,V,T1,T,T2. cpx h g G (L.ⓓV) T1 T →
-             ⬆[0, 1] T2 ≡ T → cpx h g G L (+ⓓV.T1) T2
-| cpx_eps  : ∀G,L,V,T1,T2. cpx h g G L T1 T2 → cpx h g G L (ⓝV.T1) T2
-| cpx_ct   : ∀G,L,V1,V2,T. cpx h g G L V1 V2 → cpx h g G L (ⓝV1.T) V2
+             cpx h o G L V1 V2 → cpx h o G L T1 T2 →
+             cpx h o G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+| cpx_zeta : ∀G,L,V,T1,T,T2. cpx h o G (L.ⓓV) T1 T →
+             ⬆[0, 1] T2 ≡ T → cpx h o G L (+ⓓV.T1) T2
+| cpx_eps  : ∀G,L,V,T1,T2. cpx h o G L T1 T2 → cpx h o G L (ⓝV.T1) T2
+| cpx_ct   : ∀G,L,V1,V2,T. cpx h o G L V1 V2 → cpx h o G L (ⓝV1.T) V2
 | cpx_beta : ∀a,G,L,V1,V2,W1,W2,T1,T2.
-             cpx h g G L V1 V2 → cpx h g G L W1 W2 → cpx h g G (L.ⓛW1) T1 T2 →
-             cpx h g G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
+             cpx h o G L V1 V2 → cpx h o G L W1 W2 → cpx h o G (L.ⓛW1) T1 T2 →
+             cpx h o G L (ⓐV1.ⓛ{a}W1.T1) (ⓓ{a}ⓝW2.V2.T2)
 | cpx_theta: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
-             cpx h g G L V1 V → ⬆[0, 1] V ≡ V2 → cpx h g G L W1 W2 →
-             cpx h g G (L.ⓓW1) T1 T2 →
-             cpx h g G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
+             cpx h o G L V1 V → ⬆[0, 1] V ≡ V2 → cpx h o G L W1 W2 →
+             cpx h o G (L.ⓓW1) T1 T2 →
+             cpx h o G L (ⓐV1.ⓓ{a}W1.T1) (ⓓ{a}W2.ⓐV2.T2)
 .
 
 interpretation
    "context-sensitive extended parallel reduction (term)"
-   'PRed h g G L T1 T2 = (cpx h g G L T1 T2).
+   'PRed h o G L T1 T2 = (cpx h o G L T1 T2).
 
 (* Basic properties *********************************************************)
 
-lemma lsubr_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) lsubr.
-#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+lemma lsubr_cpx_trans: ∀h,o,G. lsub_trans … (cpx h o G) lsubr.
+#h #o #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
 [ //
 | /2 width=2 by cpx_st/
 | #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
@@ -64,23 +64,23 @@ lemma lsubr_cpx_trans: ∀h,g,G. lsub_trans … (cpx h g G) lsubr.
 qed-.
 
 (* Note: this is "∀h,g,L. reflexive … (cpx h g L)" *)
-lemma cpx_refl: ∀h,g,G,T,L. ⦃G, L⦄ ⊢ T ➡[h, g] T.
-#h #g #G #T elim T -T // * /2 width=1 by cpx_bind, cpx_flat/
+lemma cpx_refl: ∀h,o,G,T,L. ⦃G, L⦄ ⊢ T ➡[h, o] T.
+#h #o #G #T elim T -T // * /2 width=1 by cpx_bind, cpx_flat/
 qed.
 
-lemma cpr_cpx: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h, g] T2.
-#h #g #G #L #T1 #T2 #H elim H -L -T1 -T2
+lemma cpr_cpx: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → ⦃G, L⦄ ⊢ T1 ➡[h, o] T2.
+#h #o #G #L #T1 #T2 #H elim H -L -T1 -T2
 /2 width=7 by cpx_delta, cpx_bind, cpx_flat, cpx_zeta, cpx_eps, cpx_beta, cpx_theta/
 qed.
 
-lemma cpx_pair_sn: ∀h,g,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
-                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h, g] ②{I}V2.T.
-#h #g * /2 width=1 by cpx_bind, cpx_flat/
+lemma cpx_pair_sn: ∀h,o,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 →
+                   ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡[h, o] ②{I}V2.T.
+#h #o * /2 width=1 by cpx_bind, cpx_flat/
 qed.
 
-lemma cpx_delift: ∀h,g,I,G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓑ{I}V) →
-                  ∃∃T2,T.  ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 & ⬆[l, 1] T ≡ T2.
-#h #g #I #G #K #V #T1 elim T1 -T1
+lemma cpx_delift: ∀h,o,I,G,K,V,T1,L,l. ⬇[l] L ≡ (K.ⓑ{I}V) →
+                  ∃∃T2,T.  ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 & ⬆[l, 1] T ≡ T2.
+#h #o #I #G #K #V #T1 elim T1 -T1
 [ * #i #L #l /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
   elim (lt_or_eq_or_gt i l) #Hil [1,3: /4 width=4 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ylt_inj, yle_inj, ex2_2_intro/ ]
   destruct
@@ -96,14 +96,14 @@ qed-.
 
 (* Basic inversion lemmas ***************************************************)
 
-fact cpx_inv_atom1_aux: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀J. T1 = ⓪{J} →
+fact cpx_inv_atom1_aux: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ∀J. T1 = ⓪{J} →
                         ∨∨ T2 = ⓪{J}
-                         | ∃∃k,d. deg h g k (d+1) & T2 = ⋆(next h k) & J = Sort k
-                         | ∃∃I,K,V,V2,i. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                         | ∃∃s,d. deg h o s (d+1) & T2 = ⋆(next h s) & J = Sort s
+                         | ∃∃I,K,V,V2,i. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, o] V2 &
                                          ⬆[O, i+1] V2 ≡ T2 & J = LRef i.
-#G #h #g #L #T1 #T2 * -L -T1 -T2
+#G #h #o #L #T1 #T2 * -L -T1 -T2
 [ #I #G #L #J #H destruct /2 width=1 by or3_intro0/
-| #G #L #k #d #Hkd #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
+| #G #L #s #d #Hkd #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
 | #I #G #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=9 by or3_intro2, ex4_5_intro/
 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
@@ -115,57 +115,57 @@ fact cpx_inv_atom1_aux: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ∀
 ]
 qed-.
 
-lemma cpx_inv_atom1: ∀h,g,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, g] T2 →
+lemma cpx_inv_atom1: ∀h,o,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, o] T2 →
                      ∨∨ T2 = ⓪{J}
-                      | ∃∃k,d. deg h g k (d+1) & T2 = ⋆(next h k) & J = Sort k
-                      | ∃∃I,K,V,V2,i. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                      | ∃∃s,d. deg h o s (d+1) & T2 = ⋆(next h s) & J = Sort s
+                      | ∃∃I,K,V,V2,i. ⬇[i] L ≡ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, o] V2 &
                                       ⬆[O, i+1] V2 ≡ T2 & J = LRef i.
 /2 width=3 by cpx_inv_atom1_aux/ qed-.
 
-lemma cpx_inv_sort1: ∀h,g,G,L,T2,k. ⦃G, L⦄ ⊢ ⋆k ➡[h, g] T2 → T2 = ⋆k ∨
-                     ∃∃d. deg h g k (d+1) & T2 = ⋆(next h k).
-#h #g #G #L #T2 #k #H
+lemma cpx_inv_sort1: ∀h,o,G,L,T2,s. ⦃G, L⦄ ⊢ ⋆s ➡[h, o] T2 → T2 = ⋆s ∨
+                     ∃∃d. deg h o s (d+1) & T2 = ⋆(next h s).
+#h #o #G #L #T2 #s #H
 elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-[ #k0 #d0 #Hkd0 #H1 #H2 destruct /3 width=4 by ex2_intro, or_intror/
+[ #s0 #d0 #Hkd0 #H1 #H2 destruct /3 width=4 by ex2_intro, or_intror/
 | #I #K #V #V2 #i #_ #_ #_ #H destruct
 ]
 qed-.
 
-lemma cpx_inv_lref1: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 →
+lemma cpx_inv_lref1: ∀h,o,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, o] T2 →
                      T2 = #i ∨
-                     ∃∃I,K,V,V2. ⬇[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, g] V2 &
+                     ∃∃I,K,V,V2. ⬇[i] L ≡ K. ⓑ{I}V & ⦃G, K⦄ ⊢ V ➡[h, o] V2 &
                                  ⬆[O, i+1] V2 ≡ T2.
-#h #g #G #L #T2 #i #H
+#h #o #G #L #T2 #i #H
 elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
-[ #k #d #_ #_ #H destruct
+[ #s #d #_ #_ #H destruct
 | #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
 ]
 qed-.
 
-lemma cpx_inv_lref1_ge: ∀h,g,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, g] T2 → |L| ≤ i → T2 = #i.
-#h #g #G #L #T2 #i #H elim (cpx_inv_lref1 … H) -H // *
+lemma cpx_inv_lref1_ge: ∀h,o,G,L,T2,i. ⦃G, L⦄ ⊢ #i ➡[h, o] T2 → |L| ≤ i → T2 = #i.
+#h #o #G #L #T2 #i #H elim (cpx_inv_lref1 … H) -H // *
 #I #K #V1 #V2 #HLK #_ #_ #HL -h -G -V2 lapply (drop_fwd_length_lt2 … HLK) -K -I -V1
 #H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
 qed-.
 
-lemma cpx_inv_gref1: ∀h,g,G,L,T2,p.  ⦃G, L⦄ ⊢ §p ➡[h, g] T2 → T2 = §p.
-#h #g #G #L #T2 #p #H
+lemma cpx_inv_gref1: ∀h,o,G,L,T2,p.  ⦃G, L⦄ ⊢ §p ➡[h, o] T2 → T2 = §p.
+#h #o #G #L #T2 #p #H
 elim (cpx_inv_atom1 … H) -H // *
-[ #k #d #_ #_ #H destruct
+[ #s #d #_ #_ #H destruct
 | #I #K #V #V2 #i #_ #_ #_ #H destruct
 ]
 qed-.
 
-fact cpx_inv_bind1_aux: ∀h,g,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 →
+fact cpx_inv_bind1_aux: ∀h,o,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, o] U2 →
                         ∀a,J,V1,T1. U1 = ⓑ{a,J}V1.T1 → (
-                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, g] T2 &
+                        ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, o] T2 &
                                  U2 = ⓑ{a,J}V2.T2
                         ) ∨
-                        ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⬆[0, 1] U2 ≡ T &
+                        ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, o] T & ⬆[0, 1] U2 ≡ T &
                              a = true & J = Abbr.
-#h #g #G #L #U1 #U2 * -L -U1 -U2
+#h #o #G #L #U1 #U2 * -L -U1 -U2
 [ #I #G #L #b #J #W #U1 #H destruct
-| #G #L #k #d #_ #b #J #W #U1 #H destruct
+| #G #L #s #d #_ #b #J #W #U1 #H destruct
 | #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
 | #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
 | #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
@@ -177,50 +177,50 @@ fact cpx_inv_bind1_aux: ∀h,g,G,L,U1,U2. ⦃G, L⦄ ⊢ U1 ➡[h, g] U2 →
 ]
 qed-.
 
-lemma cpx_inv_bind1: ∀h,g,a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡[h, g] U2 → (
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, g] T2 &
+lemma cpx_inv_bind1: ∀h,o,a,I,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡[h, o] U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[h, o] T2 &
                               U2 = ⓑ{a,I} V2. T2
                      ) ∨
-                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⬆[0, 1] U2 ≡ T &
+                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, o] T & ⬆[0, 1] U2 ≡ T &
                           a = true & I = Abbr.
 /2 width=3 by cpx_inv_bind1_aux/ qed-.
 
-lemma cpx_inv_abbr1: ∀h,g,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡[h, g] U2 → (
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T2 &
+lemma cpx_inv_abbr1: ∀h,o,a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡[h, o] U2 → (
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, o] T2 &
                               U2 = ⓓ{a} V2. T2
                      ) ∨
-                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, g] T & ⬆[0, 1] U2 ≡ T & a = true.
-#h #g #a #G #L #V1 #T1 #U2 #H
+                     ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, o] T & ⬆[0, 1] U2 ≡ T & a = true.
+#h #o #a #G #L #V1 #T1 #U2 #H
 elim (cpx_inv_bind1 … H) -H * /3 width=5 by ex3_2_intro, ex3_intro, or_introl, or_intror/
 qed-.
 
-lemma cpx_inv_abst1: ∀h,g,a,G,L,V1,T1,U2.  ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡[h, g] U2 →
-                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 &  ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[h, g] T2 &
+lemma cpx_inv_abst1: ∀h,o,a,G,L,V1,T1,U2.  ⦃G, L⦄ ⊢ ⓛ{a}V1.T1 ➡[h, o] U2 →
+                     ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 &  ⦃G, L.ⓛV1⦄ ⊢ T1 ➡[h, o] T2 &
                               U2 = ⓛ{a} V2. T2.
-#h #g #a #G #L #V1 #T1 #U2 #H
+#h #o #a #G #L #V1 #T1 #U2 #H
 elim (cpx_inv_bind1 … H) -H *
 [ /3 width=5 by ex3_2_intro/
 | #T #_ #_ #_ #H destruct
 ]
 qed-.
 
-fact cpx_inv_flat1_aux: ∀h,g,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, g] U2 →
+fact cpx_inv_flat1_aux: ∀h,o,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, o] U2 →
                         ∀J,V1,U1. U = ⓕ{J}V1.U1 →
-                        ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+                        ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, o] T2 &
                                     U2 = ⓕ{J}V2.T2
-                         | (⦃G, L⦄ ⊢ U1 ➡[h, g] U2 ∧ J = Cast)
-                         | (⦃G, L⦄ ⊢ V1 ➡[h, g] U2 ∧ J = Cast)
-                         | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
-                                               ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
+                         | (⦃G, L⦄ ⊢ U1 ➡[h, o] U2 ∧ J = Cast)
+                         | (⦃G, L⦄ ⊢ V1 ➡[h, o] U2 ∧ J = Cast)
+                         | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 &
+                                               ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, o] T2 &
                                                U1 = ⓛ{a}W1.T1 &
                                                U2 = ⓓ{a}ⓝW2.V2.T2 & J = Appl
-                         | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⬆[0,1] V ≡ V2 &
-                                                 ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
+                         | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V & ⬆[0,1] V ≡ V2 &
+                                                 ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, o] T2 &
                                                  U1 = ⓓ{a}W1.T1 &
                                                  U2 = ⓓ{a}W2.ⓐV2.T2 & J = Appl.
-#h #g #G #L #U #U2 * -L -U -U2
+#h #o #G #L #U #U2 * -L -U -U2
 [ #I #G #L #J #W #U1 #H destruct
-| #G #L #k #d #_ #J #W #U1 #H destruct
+| #G #L #s #d #_ #J #W #U1 #H destruct
 | #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
 | #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
 | #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or5_intro0, ex3_2_intro/
@@ -232,31 +232,31 @@ fact cpx_inv_flat1_aux: ∀h,g,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, g] U2 →
 ]
 qed-.
 
-lemma cpx_inv_flat1: ∀h,g,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h, g] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+lemma cpx_inv_flat1: ∀h,o,I,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓕ{I}V1.U1 ➡[h, o] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, o] T2 &
                                  U2 = ⓕ{I} V2. T2
-                      | (⦃G, L⦄ ⊢ U1 ➡[h, g] U2 ∧ I = Cast)
-                      | (⦃G, L⦄ ⊢ V1 ➡[h, g] U2 ∧ I = Cast)
-                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
-                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
+                      | (⦃G, L⦄ ⊢ U1 ➡[h, o] U2 ∧ I = Cast)
+                      | (⦃G, L⦄ ⊢ V1 ➡[h, o] U2 ∧ I = Cast)
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, o] T2 &
                                             U1 = ⓛ{a}W1.T1 &
                                             U2 = ⓓ{a}ⓝW2.V2.T2 & I = Appl
-                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⬆[0,1] V ≡ V2 &
-                                              ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V & ⬆[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, o] T2 &
                                               U1 = ⓓ{a}W1.T1 &
                                               U2 = ⓓ{a}W2.ⓐV2.T2 & I = Appl.
 /2 width=3 by cpx_inv_flat1_aux/ qed-.
 
-lemma cpx_inv_appl1: ∀h,g,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[h, g] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+lemma cpx_inv_appl1: ∀h,o,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[h, o] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, o] T2 &
                                  U2 = ⓐ V2. T2
-                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 &
-                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, g] T2 &
+                      | ∃∃a,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 &
+                                            ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[h, o] T2 &
                                             U1 = ⓛ{a}W1.T1 & U2 = ⓓ{a}ⓝW2.V2.T2
-                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V & ⬆[0,1] V ≡ V2 &
-                                              ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, g] T2 &
+                      | ∃∃a,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V & ⬆[0,1] V ≡ V2 &
+                                              ⦃G, L⦄ ⊢ W1 ➡[h, o] W2 & ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[h, o] T2 &
                                               U1 = ⓓ{a}W1.T1 & U2 = ⓓ{a}W2. ⓐV2. T2.
-#h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
+#h #o #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
 [ /3 width=5 by or3_intro0, ex3_2_intro/
 |2,3: #_ #H destruct
 | /3 width=11 by or3_intro1, ex5_6_intro/
@@ -265,10 +265,10 @@ lemma cpx_inv_appl1: ∀h,g,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓐ V1.U1 ➡[h, g] U2
 qed-.
 
 (* Note: the main property of simple terms *)
-lemma cpx_inv_appl1_simple: ∀h,g,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[h, g] U → 𝐒⦃T1⦄ →
-                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 &
+lemma cpx_inv_appl1_simple: ∀h,o,G,L,V1,T1,U. ⦃G, L⦄ ⊢ ⓐV1.T1 ➡[h, o] U → 𝐒⦃T1⦄ →
+                            ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 &
                                      U = ⓐV2.T2.
-#h #g #G #L #V1 #T1 #U #H #HT1
+#h #o #G #L #V1 #T1 #U #H #HT1
 elim (cpx_inv_appl1 … H) -H *
 [ /2 width=5 by ex3_2_intro/
 | #a #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #H #_ destruct
@@ -278,12 +278,12 @@ elim (cpx_inv_appl1 … H) -H *
 ]
 qed-.
 
-lemma cpx_inv_cast1: ∀h,g,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[h, g] U2 →
-                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, g] T2 &
+lemma cpx_inv_cast1: ∀h,o,G,L,V1,U1,U2. ⦃G, L⦄ ⊢ ⓝV1.U1 ➡[h, o] U2 →
+                     ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, o] T2 &
                                  U2 = ⓝ V2. T2
-                      | ⦃G, L⦄ ⊢ U1 ➡[h, g] U2
-                      | ⦃G, L⦄ ⊢ V1 ➡[h, g] U2.
-#h #g #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
+                      | ⦃G, L⦄ ⊢ U1 ➡[h, o] U2
+                      | ⦃G, L⦄ ⊢ V1 ➡[h, o] U2.
+#h #o #G #L #V1 #U1 #U2 #H elim (cpx_inv_flat1 … H) -H *
 [ /3 width=5 by or3_intro0, ex3_2_intro/
 |2,3: /2 width=1 by or3_intro1, or3_intro2/
 | #a #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #H destruct
@@ -293,10 +293,10 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma cpx_fwd_bind1_minus: ∀h,g,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[h, g] T → ∀b.
-                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[h, g] ⓑ{b,I}V2.T2 &
+lemma cpx_fwd_bind1_minus: ∀h,o,I,G,L,V1,T1,T. ⦃G, L⦄ ⊢ -ⓑ{I}V1.T1 ➡[h, o] T → ∀b.
+                           ∃∃V2,T2. ⦃G, L⦄ ⊢ ⓑ{b,I}V1.T1 ➡[h, o] ⓑ{b,I}V2.T2 &
                                     T = -ⓑ{I}V2.T2.
-#h #g #I #G #L #V1 #T1 #T #H #b
+#h #o #I #G #L #V1 #T1 #T #H #b
 elim (cpx_inv_bind1 … H) -H *
 [ #V2 #T2 #HV12 #HT12 #H destruct /3 width=4 by cpx_bind, ex2_2_intro/
 | #T2 #_ #_ #H destruct