(*** coafter_tls_succ *)
lemma pr_coafter_tls_tl_tls:
∀g2,g1,g. g2 ~⊚ g1 ≘ g →
- â\88\80j. @â\9dªð\9d\9f\8f, g2â\9d« ≘ j → ⫰*[j]g2 ~⊚ ⫰g1 ≘ ⫰*[j]g.
+ â\88\80j. @â\9d¨ð\9d\9f\8f, g2â\9d© ≘ j → ⫰*[j]g2 ~⊚ ⫰g1 ≘ ⫰*[j]g.
#g2 #g1 #g #Hg #j #Hg2
lapply (pr_nat_pred_bi … Hg2) -Hg2 #Hg2
lapply (pr_coafter_tls_bi_tls … Hg2 … Hg) -Hg #Hg
(* Note: parked for now
lemma coafter_fwd_xpx_pushs:
- â\88\80g2,f1,g,i,j. @â\9dªi, g2â\9d« ≘ j → g2 ~⊚ ⫯*[i]⫯f1 ≘ g →
+ â\88\80g2,f1,g,i,j. @â\9d¨i, g2â\9d© ≘ j → g2 ~⊚ ⫯*[i]⫯f1 ≘ g →
∃∃f. ⫰*[↑j]g2 ~⊚ f1 ≘ f & ⫯*[j]⫯f = g.
#g2 #g1 #g #i #j #Hg2 <pushs_xn #Hg(coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs <tls_pushs #Hf
qed-.
lemma coafter_fwd_xnx_pushs:
- â\88\80g2,f1,g,i,j. @â\9dªi, g2â\9d« ≘ j → g2 ~⊚ ⫯*[i]↑f1 ≘ g →
+ â\88\80g2,f1,g,i,j. @â\9d¨i, g2â\9d© ≘ j → g2 ~⊚ ⫯*[i]↑f1 ≘ g →
∃∃f. ⫰*[↑j]g2 ~⊚ f1 ≘ f & ⫯*[j] ↑f = g.
#g2 #g1 #g #i #j #Hg2 #Hg
elim (coafter_fwd_pushs … Hg Hg2) #f #H0 destruct