+lemma cpms_appl_dx (n) (h) (G) (L):
+ ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
+ ❪G,L❫ ⊢ ⓐV1.T1 ➡*[n,h] ⓐV2.T2.
+#n #h #G #L #V1 #V2 #HV12 #T1 #T2 #H @(cpms_ind_sn … H) -T1
+/3 width=3 by cpms_step_sn, cpm_cpms, cpm_appl/
+qed.
+
+lemma cpms_zeta (n) (h) (G) (L):
+ ∀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.
+
+(* Basic_2A1: uses: cprs_eps *)
+lemma cpms_eps (n) (h) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡*[n,h] T2 →
+ ∀V. ❪G,L❫ ⊢ ⓝV.T1 ➡*[n,h] T2.
+#n #h #G #L #T1 #T2 #H @(cpms_ind_sn … H) -T1
+/3 width=3 by cpms_step_sn, cpm_cpms, cpm_eps/
+qed.
+
+lemma cpms_ee (n) (h) (G) (L):
+ ∀U1,U2. ❪G,L❫ ⊢ U1 ➡*[n,h] U2 →
+ ∀T. ❪G,L❫ ⊢ ⓝU1.T ➡*[↑n,h] U2.
+#n #h #G #L #U1 #U2 #H @(cpms_ind_sn … H) -U1 -n
+[ /3 width=1 by cpm_cpms, cpm_ee/
+| #n1 #n2 #U1 #U #HU1 #HU2 #_ #T >plus_S1
+ /3 width=3 by cpms_step_sn, cpm_ee/
+]
+qed.
+
+(* Basic_2A1: uses: cprs_beta_dx *)
+lemma cpms_beta_dx (n) (h) (G) (L):
+ ∀V1,V2. ❪G,L❫ ⊢ V1 ➡[h] V2 →
+ ∀W1,W2. ❪G,L❫ ⊢ W1 ➡[h] W2 →
+ ∀T1,T2. ❪G,L.ⓛW1❫ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ❪G,L❫ ⊢ ⓐV1.ⓛ[p]W1.T1 ➡*[n,h] ⓓ[p]ⓝW2.V2.T2.
+#n #h #G #L #V1 #V2 #HV12 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
+/4 width=7 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_beta/
+qed.
+
+(* Basic_2A1: uses: cprs_theta_dx *)
+lemma cpms_theta_dx (n) (h) (G) (L):
+ ∀V1,V. ❪G,L❫ ⊢ V1 ➡[h] V →
+ ∀V2. ⇧[1] V ≘ V2 →
+ ∀W1,W2. ❪G,L❫ ⊢ W1 ➡[h] W2 →
+ ∀T1,T2. ❪G,L.ⓓW1❫ ⊢ T1 ➡*[n,h] T2 →
+ ∀p. ❪G,L❫ ⊢ ⓐV1.ⓓ[p]W1.T1 ➡*[n,h] ⓓ[p]W2.ⓐV2.T2.
+#n #h #G #L #V1 #V #HV1 #V2 #HV2 #W1 #W2 #HW12 #T1 #T2 #H @(cpms_ind_dx … H) -T2
+/4 width=9 by cpms_step_dx, cpm_cpms, cpms_bind_dx, cpms_appl_dx, cpm_theta/
+qed.
+