(* *)
(**************************************************************************)
-include "basic_2/static/ssta.ma".
+include "basic_2/notation/relations/pred_6.ma".
+include "basic_2/static/sd.ma".
include "basic_2/reduction/cpr.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
-inductive cpx (h) (g): lenv → relation term ≝
-| cpx_cpr : ∀I,L,U2. L ⊢ ⓪{I} ➡ U2 → cpx h g L (⓪{I}) U2
-| cpx_ssta: ∀I,L,U2,l. ⦃h, L⦄ ⊢ ⓪{I} •[g] ⦃l+1, U2⦄ → cpx h g L (⓪{I}) U2
-| cpx_bind: ∀a,I,L,V1,V2,T1,T2,U2. cpx h g L V1 V2 → cpx h g (L.ⓑ{I}V1) T1 T2 →
- L ⊢ ⓑ{a,I}V2.T2 ➡ U2 → cpx h g L (ⓑ{a,I}V1.T1) U2
-| cpx_flat: ∀I,L,V1,V2,T1,T2,U2. cpx h g L V1 V2 → cpx h g L T1 T2 →
- L ⊢ ⓕ{I}V2.T2 ➡ U2 → cpx h g L (ⓕ{I}V1.T1) U2
+(* avtivate genv *)
+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 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 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 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 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 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 L T1 T2 = (cpx h 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,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
+ elim (lsubr_fwd_drop2_pair … HL12 … HLK1) -HL12 -HLK1 *
+ /4 width=7 by cpx_delta, cpx_ct/
+|4,9: /4 width=1 by cpx_bind, cpx_beta, lsubr_pair/
+|5,7,8: /3 width=1 by cpx_flat, cpx_eps, cpx_ct/
+|6,10: /4 width=3 by cpx_zeta, cpx_theta, lsubr_pair/
+]
+qed-.
+
(* Note: this is "∀h,g,L. reflexive … (cpx h g L)" *)
-lemma cpx_refl: ∀h,g,T,L. ⦃h, L⦄ ⊢ T ➡[g] T.
-#h #g #T elim T -T /2 width=1/ * /2 width=5/
+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,T1,L,T2. L ⊢ T1 ➡ T2 → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
-#h #g #T1 elim T1 -T1 /2 width=1/ * /2 width=5/
+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 ssta_cpx: ∀h,g,T1,L,T2,l. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, T2⦄ → ⦃h, L⦄ ⊢ T1 ➡[g] T2.
-#h #g #T1 elim T1 -T1 /2 width=2/ * [|*]
-[ #a #I #V1 #T1 #_ #IHT1 #L #X #l #H
- elim (ssta_inv_bind1 … H) -H #T2 #HT12 #H destruct /3 width=5/
-| #V1 #T1 #_ #IHT1 #L #X #l #H
- elim (ssta_inv_appl1 … H) -H #T2 #HT12 #H destruct /3 width=5/
-| #V1 #T1 #_ #IHT1 #L #X #l #H
- lapply (ssta_inv_cast1 … H) -H /3 width=5/
-]
+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,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
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split … HVW i i) /3 width=7 by cpx_delta, ex2_2_intro/
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L. ⓑ{I} W1) (l+1)) -IHU1 /3 width=9 by cpx_bind, drop_drop, lift_bind, ex2_2_intro/
+ | elim (IHU1 … HLK) -IHU1 -HLK /3 width=8 by cpx_flat, lift_flat, ex2_2_intro/
+ ]
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpx_inv_atom1_aux: ∀h,o,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, o] T2 → ∀J. T1 = ⓪{J} →
+ ∨∨ T2 = ⓪{J}
+ | ∃∃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 #o #L #T1 #T2 * -L -T1 -T2
+[ #I #G #L #J #H destruct /2 width=1 by or3_intro0/
+| #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
+| #G #L #V #T1 #T #T2 #_ #_ #J #H destruct
+| #G #L #V #T1 #T2 #_ #J #H destruct
+| #G #L #V1 #V2 #T #_ #J #H destruct
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #J #H destruct
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #J #H destruct
+]
+qed-.
+
+lemma cpx_inv_atom1: ∀h,o,J,G,L,T2. ⦃G, L⦄ ⊢ ⓪{J} ➡[h, o] T2 →
+ ∨∨ T2 = ⓪{J}
+ | ∃∃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,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/ *
+[ #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,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, o] V2 &
+ ⬆[O, i+1] V2 ≡ T2.
+#h #o #G #L #T2 #i #H
+elim (cpx_inv_atom1 … H) -H /2 width=1 by or_introl/ *
+[ #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,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,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 // *
+[ #s #d #_ #_ #H destruct
+| #I #K #V #V2 #i #_ #_ #_ #H destruct
+]
+qed-.
+
+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, o] V2 & ⦃G, L.ⓑ{J}V1⦄ ⊢ T1 ➡[h, o] T2 &
+ U2 = ⓑ{a,J}V2.T2
+ ) ∨
+ ∃∃T. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡[h, o] T & ⬆[0, 1] U2 ≡ T &
+ a = true & J = Abbr.
+#h #o #G #L #U1 #U2 * -L -U1 -U2
+[ #I #G #L #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
+| #G #L #V #T1 #T #T2 #HT1 #HT2 #b #J #W #U1 #H destruct /3 width=3 by ex4_intro, or_intror/
+| #G #L #V #T1 #T2 #_ #b #J #W #U1 #H destruct
+| #G #L #V1 #V2 #T #_ #b #J #W #U1 #H destruct
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #b #J #W #U1 #H destruct
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #b #J #W #U1 #H destruct
+]
+qed-.
+
+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, o] T & ⬆[0, 1] U2 ≡ T &
+ a = true & I = Abbr.
+/2 width=3 by cpx_inv_bind1_aux/ qed-.
+
+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, 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,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 #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,o,G,L,U,U2. ⦃G, L⦄ ⊢ U ➡[h, o] U2 →
+ ∀J,V1,U1. U = ⓕ{J}V1.U1 →
+ ∨∨ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡[h, o] V2 & ⦃G, L⦄ ⊢ U1 ➡[h, o] T2 &
+ U2 = ⓕ{J}V2.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, 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 #o #G #L #U #U2 * -L -U -U2
+[ #I #G #L #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/
+| #G #L #V #T1 #T #T2 #_ #_ #J #W #U1 #H destruct
+| #G #L #V #T1 #T2 #HT12 #J #W #U1 #H destruct /3 width=1 by or5_intro1, conj/
+| #G #L #V1 #V2 #T #HV12 #J #W #U1 #H destruct /3 width=1 by or5_intro2, conj/
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #J #W #U1 #H destruct /3 width=11 by or5_intro3, ex6_6_intro/
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #J #W #U1 #H destruct /3 width=13 by or5_intro4, ex7_7_intro/
+]
+qed-.
+
+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, 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, 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,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, 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, 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 #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/
+| /3 width=13 by or3_intro2, ex6_7_intro/
+]
+qed-.
+
+(* Note: the main property of simple terms *)
+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 #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
+ elim (simple_inv_bind … HT1)
+| #a #V #V2 #W1 #W2 #U1 #U2 #_ #_ #_ #_ #H #_ destruct
+ elim (simple_inv_bind … HT1)
+]
+qed-.
+
+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, 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
+| #a #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+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 #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
+]
+qed-.
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