(* CONTEXT-SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
-definition IH_cnv_cpm_tdeq_conf_lpr (a) (h): relation3 genv lenv term ≝
- λG,L0,T0. ⦃G, L0⦄ ⊢ T0 ![a,h] →
- ∀n1,T1. ⦃G, L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
- ∀n2,T2. ⦃G, L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
- ∀L1. ⦃G, L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G, L0⦄ ⊢ ➡[h] L2 →
- ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛ T & ⦃G, L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛ T.
+definition IH_cnv_cpm_tdeq_conf_lpr (h) (a): relation3 genv lenv term ≝
+ λG,L0,T0. ⦃G,L0⦄ ⊢ T0 ![h,a] →
+ ∀n1,T1. ⦃G,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
+ ∀n2,T2. ⦃G,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
+ ∀L1. ⦃G,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G,L0⦄ ⊢ ➡[h] L2 →
+ ∃∃T. ⦃G,L1⦄ ⊢ T1 ➡[n2-n1,h] T & T1 ≛ T & ⦃G,L2⦄ ⊢ T2 ➡[n1-n2,h] T & T2 ≛ T.
(* Diamond propery with restricted rt-transition for terms ******************)
fact cnv_cpm_tdeq_conf_lpr_atom_atom_aux (h) (G0) (L1) (L2) (I):
- ∃∃T. ⦃G0,L1⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T & ⦃G0, L2⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T.
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T & ⦃G0,L2⦄ ⊢ ⓪{I} ➡[h] T & ⓪{I} ≛ T.
#h #G0 #L1 #L2 #I
/2 width=5 by ex4_intro/
qed-.
fact cnv_cpm_tdeq_conf_lpr_atom_ess_aux (h) (G0) (L1) (L2) (s):
- ∃∃T. ⦃G0,L1⦄ ⊢ ⋆s ➡[1,h] T & ⋆s ≛ T & ⦃G0,L2⦄ ⊢ ⋆(next h s) ➡[h] T & ⋆(next h s) ≛ T.
+ ∃∃T. ⦃G0,L1⦄ ⊢ ⋆s ➡[1,h] T & ⋆s ≛ T & ⦃G0,L2⦄ ⊢ ⋆(⫯[h]s) ➡[h] T & ⋆(⫯[h]s) ≛ T.
#h #G0 #L1 #L2 #s
/3 width=5 by tdeq_sort, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_bind_bind_aux (a) (h) (p) (I) (G0) (L0) (V0) (T0):
- (â\88\80G,L,T. â¦\83G0,L0,â\93\91{p,I}V0.T0â¦\84 â\8a\90+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ ⓑ{p,I}V0.T0 ![a,h] →
+fact cnv_cpm_tdeq_conf_lpr_bind_bind_aux (h) (a) (p) (I) (G0) (L0) (V0) (T0):
+ (â\88\80G,L,T. â¦\83G0,L0,â\93\91{p,I}V0.T0â¦\84 â¬\82+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ ⓑ{p,I}V0.T0 ![h,a] →
∀n1,T1. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
∀n2,T2. ⦃G0,L0.ⓑ{I}V0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ ⓑ{p,I}V0.T1 ➡[n2-n1,h] T & ⓑ{p,I}V0.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓑ{p,I}V0.T2 ➡[n1-n2,h] T & ⓑ{p,I}V0.T2 ≛ T.
-#a #h #p #I #G0 #L0 #V0 #T0 #IH #H0
+#h #a #p #I #G0 #L0 #V0 #T0 #IH #H0
#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_bind … H0) -H0 #_ #HT0
/3 width=7 by cpm_bind, tdeq_pair, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_appl_appl_aux (a) (h) (G0) (L0) (V0) (T0):
- (â\88\80G,L,T. â¦\83G0,L0,â\93\90V0.T0â¦\84 â\8a\90+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ ⓐV0.T0 ![a,h] →
+fact cnv_cpm_tdeq_conf_lpr_appl_appl_aux (h) (a) (G0) (L0) (V0) (T0):
+ (â\88\80G,L,T. â¦\83G0,L0,â\93\90V0.T0â¦\84 â¬\82+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ ⓐV0.T0 ![h,a] →
∀n1,T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
∀n2,T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ ⓐV0.T1 ➡[n2-n1,h] T & ⓐV0.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓐV0.T2 ➡[n1-n2,h] T & ⓐV0.T2 ≛ T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #T1 #H1T01 #H2T01 #n2 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_appl … H0) -H0 #n0 #p0 #X01 #X02 #_ #_ #HT0 #_ #_ -n0 -p0 -X01 -X02
/3 width=7 by cpm_appl, tdeq_pair, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_cast_cast_aux (a) (h) (G0) (L0) (V0) (T0):
- (â\88\80G,L,T. â¦\83G0,L0,â\93\9dV0.T0â¦\84 â\8a\90+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
- ⦃G0,L0⦄ ⊢ ⓝV0.T0 ![a,h] →
+fact cnv_cpm_tdeq_conf_lpr_cast_cast_aux (h) (a) (G0) (L0) (V0) (T0):
+ (â\88\80G,L,T. â¦\83G0,L0,â\93\9dV0.T0â¦\84 â¬\82+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ⦃G0,L0⦄ ⊢ ⓝV0.T0 ![h,a] →
∀n1,V1. ⦃G0,L0⦄ ⊢ V0 ➡[n1,h] V1 → V0 ≛ V1 →
∀n2,V2. ⦃G0,L0⦄ ⊢ V0 ➡[n2,h] V2 → V0 ≛ V2 →
∀T1. ⦃G0,L0⦄ ⊢ T0 ➡[n1,h] T1 → T0 ≛ T1 →
∀T2. ⦃G0,L0⦄ ⊢ T0 ➡[n2,h] T2 → T0 ≛ T2 →
∀L1. ⦃G0,L0⦄ ⊢ ➡[h] L1 → ∀L2. ⦃G0,L0⦄ ⊢ ➡[h] L2 →
∃∃T. ⦃G0,L1⦄ ⊢ ⓝV1.T1 ➡[n2-n1,h] T & ⓝV1.T1 ≛ T & ⦃G0,L2⦄ ⊢ ⓝV2.T2 ➡[n1-n2,h] T & ⓝV2.T2 ≛ T.
-#a #h #G0 #L0 #V0 #T0 #IH #H0
+#h #a #G0 #L0 #V0 #T0 #IH #H0
#n1 #V1 #H1V01 #H2V01 #n2 #V2 #H1V02 #H2V02 #T1 #H1T01 #H2T01 #T2 #H1T02 #H2T02
#L1 #HL01 #L2 #HL02
elim (cnv_inv_cast … H0) -H0 #X0 #HV0 #HT0 #_ #_ -X0
/3 width=7 by cpm_cast, tdeq_pair, ex4_intro/
qed-.
-fact cnv_cpm_tdeq_conf_lpr_aux (a) (h) (G0) (L0) (T0):
- (â\88\80G,L,T. â¦\83G0,L0,T0â¦\84 â\8a\90+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr a h G L T) →
- ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpm_tdeq_conf_lpr a h G L T.
-#a #h #G0 #L0 #T0 #IH1 #G #L * [| * [| * ]]
+fact cnv_cpm_tdeq_conf_lpr_aux (h) (a) (G0) (L0) (T0):
+ (â\88\80G,L,T. â¦\83G0,L0,T0â¦\84 â¬\82+ â¦\83G,L,Tâ¦\84 â\86\92 IH_cnv_cpm_tdeq_conf_lpr h a G L T) →
+ ∀G,L,T. G0 = G → L0 = L → T0 = T → IH_cnv_cpm_tdeq_conf_lpr h a G L T.
+#h #a #G0 #L0 #T0 #IH1 #G #L * [| * [| * ]]
[ #I #HG0 #HL0 #HT0 #HT #n1 #X1 #H1X1 #H2X1 #n2 #X2 #H1X2 #H2X2 #L1 #HL1 #L2 #HL2 destruct
elim (cpm_tdeq_inv_atom_sn … H1X1 H2X1) -H1X1 -H2X1 *
elim (cpm_tdeq_inv_atom_sn … H1X2 H2X2) -H1X2 -H2X2 *
]
qed-.
-lemma cnv_cpm_tdeq_conf_lpr (a) (h) (G0) (L0) (T0):
- IH_cnv_cpm_tdeq_conf_lpr a h G0 L0 T0.
-#a #h #G0 #L0 #T0
+lemma cnv_cpm_tdeq_conf_lpr (h) (a) (G0) (L0) (T0):
+ IH_cnv_cpm_tdeq_conf_lpr h a G0 L0 T0.
+#h #a #G0 #L0 #T0
@(fqup_wf_ind (Ⓣ) … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH
/3 width=17 by cnv_cpm_tdeq_conf_lpr_aux/
qed-.