#h #G #L #T1 #Q @ltc_ind_dx_refl //
qed-.
-(* Basic inversion lemmas ***************************************************)
-
-lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[n, h] X2 → X2 = ⋆(((next h)^n) s).
-#n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
-#n1 #n2 #X #X2 #_ #IH #HX2 destruct
-elim (cpm_inv_sort1 … HX2) -HX2 * // #H1 #H2 destruct
-/2 width=3 by refl, trans_eq/
-qed-.
-
(* Basic properties *********************************************************)
(* Basic_1: includes: pr1_pr0 *)
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
qed.
-(* Basic_2A1: uses: cprs_zeta *)
lemma cpms_zeta (n) (h) (G) (L):
- ∀T2,T. ⬆*[1] T2 ≘ T →
- ∀V,T1. ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[n, h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
+ ∀T1,T. ⬆*[1] T ≘ T1 →
+ ∀V,T2. ⦃G, L⦄ ⊢ T ➡*[n, h] T2 → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
+#n #h #G #L #T1 #T #HT1 #V #T2 #H @(cpms_ind_dx … H) -T2
+/3 width=3 by cpms_step_dx, cpm_cpms, cpm_zeta/
+qed.
+
+(* Basic_2A1: uses: cprs_zeta *)
+lemma cpms_zeta_dx (n) (h) (G) (L):
+ ∀T2,T. ⬆*[1] T2 ≘ T →
+ ∀V,T1. ⦃G, L.ⓓV⦄ ⊢ T1 ➡*[n, h] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[n, h] T2.
#n #h #G #L #T2 #T #HT2 #V #T1 #H @(cpms_ind_sn … H) -T1
/3 width=3 by cpms_step_sn, cpm_cpms, cpm_bind, cpm_zeta/
qed.
lemma cprs_refl: ∀h,G,L. reflexive … (cpms h G L 0).
/2 width=1 by cpm_cpms/ qed.
+(* Advanced properties ******************************************************)
+
+lemma cpms_sort (h) (G) (L) (n):
+ ∀s. ⦃G,L⦄ ⊢ ⋆s ➡*[n,h] ⋆((next h)^n s).
+#h #G #L #n elim n -n [ // ]
+#n #IH #s <plus_SO
+/3 width=3 by cpms_step_dx, cpm_sort/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpms_inv_sort1 (n) (h) (G) (L): ∀X2,s. ⦃G, L⦄ ⊢ ⋆s ➡*[n, h] X2 → X2 = ⋆(((next h)^n) s).
+#n #h #G #L #X2 #s #H @(cpms_ind_dx … H) -X2 //
+#n1 #n2 #X #X2 #_ #IH #HX2 destruct
+elim (cpm_inv_sort1 … HX2) -HX2 #H #_ destruct //
+qed-.
+
+lemma cpms_inv_cast1 (h) (n) (G) (L):
+ ∀W1,T1,X2. ⦃G, L⦄ ⊢ ⓝW1.T1 ➡*[n,h] X2 →
+ ∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[n,h] W2 & ⦃G, L⦄ ⊢ T1 ➡*[n,h] T2 & X2 = ⓝW2.T2
+ | ⦃G, L⦄ ⊢ T1 ➡*[n,h] X2
+ | ∃∃m. ⦃G, L⦄ ⊢ W1 ➡*[m,h] X2 & n = ↑m.
+#h #n #G #L #W1 #T1 #X2 #H @(cpms_ind_dx … H) -n -X2
+[ /3 width=5 by or3_intro0, ex3_2_intro/
+| #n1 #n2 #X #X2 #_ * [ * || * ]
+ [ #W #T #HW1 #HT1 #H #HX2 destruct
+ elim (cpm_inv_cast1 … HX2) -HX2 [ * || * ]
+ [ #W2 #T2 #HW2 #HT2 #H destruct
+ /4 width=5 by cpms_step_dx, ex3_2_intro, or3_intro0/
+ | #HX2 /3 width=3 by cpms_step_dx, or3_intro1/
+ | #m #HX2 #H destruct <plus_n_Sm
+ /4 width=3 by cpms_step_dx, ex2_intro, or3_intro2/
+ ]
+ | #HX #HX2 /3 width=3 by cpms_step_dx, or3_intro1/
+ | #m #HX #H #HX2 destruct
+ /4 width=3 by cpms_step_dx, ex2_intro, or3_intro2/
+ ]
+]
+qed-.
+
(* Basic_2A1: removed theorems 5:
sta_cprs_scpds lstas_scpds scpds_strap1 scpds_fwd_cprs
scpds_inv_lstas_eq