-lemma cprs_eps: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h] T2 → ∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h] T2.
-#G #L #T1 #T2 #H @(cprs_ind … H) -T2
-/3 width=3 by cprs_strap1, cpr_cprs, cpr_eps/
-qed.
-
-lemma cprs_beta_dx: ∀a,G,L,V1,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h] T2 →
- ⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h] ⓓ{a}ⓝW2.V2.T2.
-#a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 * -T2
-/4 width=7 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_beta/
-qed.
-
-lemma cprs_theta_dx: ∀a,G,L,V1,V,V2,W1,W2,T1,T2.
- ⦃G, L⦄ ⊢ V1 ➡[h] V → ⬆[0, 1] V ≘ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h] T2 →
- ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h] ⓓ{a}W2.ⓐV2.T2.
-#a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2
-/4 width=9 by cprs_strap1, cpr_cprs, cprs_bind_dx, cprs_flat_dx, cpr_theta/
+lemma cprs_flat_sn (h) (I) (G) (L):
+ ∀T1,T2. ❪G,L❫ ⊢ T1 ➡[h,0] T2 → ∀V1,V2. ❪G,L❫ ⊢ V1 ➡*[h,0] V2 →
+ ❪G,L❫ ⊢ ⓕ[I] V1. T1 ➡*[h,0] ⓕ[I] V2. T2.
+#h #I #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cprs_ind_sn … H) -V1
+/3 width=3 by cprs_step_sn, cpm_cpms, cpr_flat/