/3 width=5 by cpms_fwd_cpxs, cpxs_aaa_conf/ qed-.
lemma cpms_total_aaa (h) (G) (L) (n) (A):
- ∀T. ❪G,L❫ ⊢ T ⁝ A → ∃U. ❪G,L❫ ⊢ T ➡*[n,h] U.
+ ∀T. ❪G,L❫ ⊢ T ⁝ A → ∃U. ❪G,L❫ ⊢ T ➡*[h,n] U.
#h #G #L #n elim n -n
[ /2 width=3 by ex_intro/
| #n #IH #A #T1 #HT1 <plus_SO_dx
lemma cpms_abst_dx_le_aaa (h) (G) (L) (T) (W) (p):
∀A. ❪G,L❫ ⊢ T ⁝ A →
- ∀n1,U1. ❪G,L❫ ⊢ T ➡*[n1,h] ⓛ[p]W.U1 → ∀n2. n1 ≤ n2 →
- ∃∃U2. ❪G,L❫ ⊢ T ➡*[n2,h] ⓛ[p]W.U2 & ❪G,L.ⓛW❫ ⊢ U1 ➡*[n2-n1,h] U2.
+ ∀n1,U1. ❪G,L❫ ⊢ T ➡*[h,n1] ⓛ[p]W.U1 → ∀n2. n1 ≤ n2 →
+ ∃∃U2. ❪G,L❫ ⊢ T ➡*[h,n2] ⓛ[p]W.U2 & ❪G,L.ⓛW❫ ⊢ U1 ➡*[h,n2-n1] U2.
#h #G #L #T #W #p #A #HA #n1 #U1 #HTU1 #n2 #Hn12
lapply (cpms_aaa_conf … HA … HTU1) -HA #HA
elim (cpms_total_aaa h … (n2-n1) … HA) -HA #X #H