(* Forward lemmas with r-equivalence ****************************************)
lemma cnv_cpms_conf_eq (h) (a) (n) (G) (L):
- â\88\80T. â¦\83G,Lâ¦\84 ⊢ T ![h,a] →
- â\88\80T1. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[n,h] T1 â\86\92 â\88\80T2. â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[n,h] T2 â\86\92 â¦\83G,Lâ¦\84 ⊢ T1 ⬌*[h] T2.
+ â\88\80T. â\9dªG,Lâ\9d« ⊢ T ![h,a] →
+ â\88\80T1. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[n,h] T1 â\86\92 â\88\80T2. â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[n,h] T2 â\86\92 â\9dªG,Lâ\9d« ⊢ T1 ⬌*[h] T2.
#h #a #n #G #L #T #HT #T1 #HT1 #T2 #HT2
elim (cnv_cpms_conf … HT … HT1 … HT2) -T <minus_n_n #T #HT1 #HT2
/2 width=3 by cprs_div/
qed-.
lemma cnv_cpms_fwd_appl_sn_decompose (h) (a) (G) (L):
- â\88\80V,T. â¦\83G,Lâ¦\84 â\8a¢ â\93\90V.T ![h,a] â\86\92 â\88\80n,X. â¦\83G,Lâ¦\84 ⊢ ⓐV.T ➡*[n,h] X →
- â\88\83â\88\83U. â¦\83G,Lâ¦\84 â\8a¢ T ![h,a] & â¦\83G,Lâ¦\84 â\8a¢ T â\9e¡*[n,h] U & â¦\83G,Lâ¦\84 ⊢ ⓐV.U ⬌*[h] X.
+ â\88\80V,T. â\9dªG,Lâ\9d« â\8a¢ â\93\90V.T ![h,a] â\86\92 â\88\80n,X. â\9dªG,Lâ\9d« ⊢ ⓐV.T ➡*[n,h] X →
+ â\88\83â\88\83U. â\9dªG,Lâ\9d« â\8a¢ T ![h,a] & â\9dªG,Lâ\9d« â\8a¢ T â\9e¡*[n,h] U & â\9dªG,Lâ\9d« ⊢ ⓐV.U ⬌*[h] X.
#h #a #G #L #V #T #H0 #n #X #HX
elim (cnv_inv_appl … H0) #m #p #W #U #_ #_ #HT #_ #_ -m -p -W -U
elim (cnv_fwd_cpms_total h … n … HT) #U #HTU