+fact cpm_fwd_plus_aux (n) (h): ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+ ∀n1,n2. n1+n2 = n →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ➡[n1, h] T & ⦃G, L⦄ ⊢ T ➡[n2, h] T2.
+#n #h #G #L #T1 #T2 #H @(cpm_ind … H) -G -L -T1 -T2 -n
+[ #I #G #L #n1 #n2 #H
+ elim (plus_inv_O3 … H) -H #H1 #H2 destruct
+ /2 width=3 by ex2_intro/
+| #G #L #s #x1 #n2 #H
+ elim (plus_inv_S3_sn … H) -H *
+ [ #H1 #H2 destruct /2 width=3 by ex2_intro/
+ | #n1 #H1 #H elim (plus_inv_O3 … H) -H #H2 #H3 destruct
+ /2 width=3 by ex2_intro/
+ ]
+| #n #G #K #V1 #V2 #W2 #_ #IH #HVW2 #n1 #n2 #H destruct
+ elim IH [|*: // ] -IH #V #HV1 #HV2
+ elim (lifts_total V 𝐔❴↑O❵) #W #HVW
+ /5 width=11 by cpm_lifts_bi, cpm_delta, drops_refl, drops_drop, ex2_intro/
+| #n #G #K #V1 #V2 #W2 #HV12 #IH #HVW2 #x1 #n2 #H
+ elim (plus_inv_S3_sn … H) -H *
+ [ #H1 #H2 destruct -IH /3 width=3 by cpm_ell, ex2_intro/
+ | #n1 #H1 #H2 destruct -HV12
+ elim (IH n1) [|*: // ] -IH #V #HV1 #HV2
+ elim (lifts_total V 𝐔❴↑O❵) #W #HVW
+ /5 width=11 by cpm_lifts_bi, cpm_ell, drops_refl, drops_drop, ex2_intro/
+ ]
+| #n #I #G #K #T2 #U2 #i #_ #IH #HTU2 #n1 #n2 #H destruct
+ elim IH [|*: // ] -IH #T #HT1 #HT2
+ elim (lifts_total T 𝐔❴↑O❵) #U #HTU
+ /5 width=11 by cpm_lifts_bi, cpm_lref, drops_refl, drops_drop, ex2_intro/
+| #n #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #_ #_ #IHT #n1 #n2 #H destruct
+ elim IHT [|*: // ] -IHT #T #HT1 #HT2
+ /3 width=5 by cpm_bind, ex2_intro/
+| #n #G #L #V1 #V2 #T1 #T2 #HV12 #_ #_ #IHT #n1 #n2 #H destruct
+ elim IHT [|*: // ] -IHT #T #HT1 #HT2
+ /3 width=5 by cpm_appl, ex2_intro/
+| #n #G #L #U1 #U2 #T1 #T2 #_ #_ #IHU #IHT #n1 #n2 #H destruct
+ elim IHU [|*: // ] -IHU #U #HU1 #HU2
+ elim IHT [|*: // ] -IHT #T #HT1 #HT2
+ /3 width=5 by cpm_cast, ex2_intro/
+| #n #G #K #V #U1 #T1 #T2 #HTU1 #_ #IH #n1 #n2 #H destruct
+ elim IH [|*: // ] -IH #T #HT1 #HT2
+ /3 width=3 by cpm_zeta, ex2_intro/
+| #n #G #L #U #T1 #T2 #_ #IH #n1 #n2 #H destruct
+ elim IH [|*: // ] -IH #T #HT1 #HT2
+ /3 width=3 by cpm_eps, ex2_intro/
+| #n #G #L #U1 #U2 #T #HU12 #IH #x1 #n2 #H
+ elim (plus_inv_S3_sn … H) -H *
+ [ #H1 #H2 destruct -IH /3 width=4 by cpm_ee, cpm_cast, ex2_intro/
+ | #n1 #H1 #H2 destruct -HU12
+ elim (IH n1) [|*: // ] -IH #U #HU1 #HU2
+ /3 width=3 by cpm_ee, ex2_intro/
+ ]
+| #n #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #_ #_ #_ #IH #n1 #n2 #H destruct
+ elim IH [|*: // ] -IH #T #HT1 #HT2
+ /4 width=7 by cpm_beta, cpm_appl, cpm_bind, ex2_intro/
+| #n #p #G #L #V1 #V2 #U2 #W1 #W2 #T1 #T2 #HV12 #HW12 #_ #_ #_ #IH #HVU2 #n1 #n2 #H destruct
+ elim IH [|*: // ] -IH #T #HT1 #HT2
+ /4 width=7 by cpm_theta, cpm_appl, cpm_bind, ex2_intro/
+]