* #n #f cases i -i
[ @n
| #i lapply (apply i f) -apply -i -f
- #i @(⫯(n+i))
+ #i @(â\86\91(n+i))
]
defined.
#i2 elim i2 -i2 /2 width=5 by at_refl, at_next/
qed.
-lemma at_S1: â\88\80n,f,i1,i2. @â¦\83i1, fâ¦\84 â\89\98 i2 â\86\92 @â¦\83⫯i1, n@fâ¦\84 â\89\98 ⫯(n+i2).
+lemma at_S1: â\88\80n,f,i1,i2. @â¦\83i1, fâ¦\84 â\89\98 i2 â\86\92 @â¦\83â\86\91i1, n@fâ¦\84 â\89\98 â\86\91(n+i2).
#n elim n -n /3 width=7 by at_push, at_next/
qed.
#j2 #Hj * -i2 /3 width=1 by eq_f/
qed-.
-lemma at_inv_S1: â\88\80f,n,j1,i2. @â¦\83⫯j1, n@f⦄ ≘ i2 →
- â\88\83â\88\83j2. @â¦\83j1, fâ¦\84 â\89\98 j2 & ⫯(n+j2) = i2.
+lemma at_inv_S1: â\88\80f,n,j1,i2. @â¦\83â\86\91j1, n@f⦄ ≘ i2 →
+ â\88\83â\88\83j2. @â¦\83j1, fâ¦\84 â\89\98 j2 & â\86\91(n+j2) = i2.
#f #n elim n -n /2 width=5 by at_inv_npx/
#n #IH #j1 #i2 #H elim (at_inv_xnx … H) -H [2,3: // ]
#j2 #Hj * -i2 elim (IH … Hj) -IH -Hj
#Hn #Hf /4 width=1 by eq_f2, eq_f/
qed.
-lemma apply_S1: â\88\80f,i. (⫯f)@â\9d´iâ\9dµ = ⫯(f@❴i❵).
+lemma apply_S1: â\88\80f,i. (â\86\91f)@â\9d´iâ\9dµ = â\86\91(f@❴i❵).
* #n #f * //
qed.