(* Note: this requires ↑ on first n *)
(*** at_pxx_tls *)
lemma gr_pat_unit_succ_tls (n) (f):
- @â\9dªð\9d\9f\8f,fâ\9d« â\89\98 â\86\91n â\86\92 @â\9dªð\9d\9f\8f,⫱*[n]f❫ ≘ 𝟏.
+ @â\9dªð\9d\9f\8f,fâ\9d« â\89\98 â\86\91n â\86\92 @â\9dªð\9d\9f\8f,â«°*[n]f❫ ≘ 𝟏.
#n @(nat_ind_succ … n) -n //
#n #IH #f #Hf
elim (gr_pat_inv_unit_succ … Hf) -Hf [|*: // ] #g #Hg #H0 destruct
(* Note: this requires ↑ on third n2 *)
(*** at_tls *)
-lemma gr_pat_tls (n2) (f): ⫯⫱*[â\86\91n2]f â\89¡ ⫱*[n2]f → ∃i1. @❪i1,f❫ ≘ ↑n2.
+lemma gr_pat_tls (n2) (f): ⫯⫰*[â\86\91n2]f â\89¡ â«°*[n2]f → ∃i1. @❪i1,f❫ ≘ ↑n2.
#n2 @(nat_ind_succ … n2) -n2
[ /4 width=4 by gr_pat_eq_repl_back, gr_pat_refl, ex_intro/
| #n2 #IH #f <gr_tls_swap <gr_tls_swap in ⊢ (??%→?); #H
(*** at_inv_nxx *)
lemma gr_pat_inv_succ_sn (p) (g) (i1) (j2):
@❪↑i1,g❫ ≘ j2 → @❪𝟏,g❫ ≘ p →
- â\88\83â\88\83i2. @â\9dªi1,⫱*[p]g❫ ≘ i2 & p+i2 = j2.
+ â\88\83â\88\83i2. @â\9dªi1,â«°*[p]g❫ ≘ i2 & p+i2 = j2.
#p elim p -p
[ #g #i1 #j2 #Hg #H
elim (gr_pat_inv_unit_bi … H) -H [|*: // ] #f #H0
(* Note: this requires ↑ on first n2 *)
(*** at_inv_tls *)
lemma gr_pat_inv_succ_dx_tls (n2) (i1) (f):
- @â\9dªi1,fâ\9d« â\89\98 â\86\91n2 â\86\92 ⫯⫱*[â\86\91n2]f â\89¡ ⫱*[n2]f.
+ @â\9dªi1,fâ\9d« â\89\98 â\86\91n2 â\86\92 ⫯⫰*[â\86\91n2]f â\89¡ â«°*[n2]f.
#n2 @(nat_ind_succ … n2) -n2
[ #i1 #f #Hf elim (gr_pat_inv_unit_dx … Hf) -Hf // #g #H1 #H destruct
/2 width=1 by gr_eq_refl/