#n elim n -n /3 width=1 by istot_tl/
qed.
-(* Advanced forward lemmas on at ********************************************)
+(* Main forward lemmas on at ************************************************)
-let corec at_ext: ∀f1,f2. 𝐓⦃f1⦄ → 𝐓⦃f2⦄ →
- (∀i,i1,i2. @⦃i, f1⦄ ≡ i1 → @⦃i, f2⦄ ≡ i2 → i1 = i2) → f1 ≗ f2 ≝ ?.
+corec theorem at_ext: ∀f1,f2. 𝐓⦃f1⦄ → 𝐓⦃f2⦄ →
+ (∀i,i1,i2. @⦃i, f1⦄ ≡ i1 → @⦃i, f2⦄ ≡ i2 → i1 = i2) →
+ f1 ≗ f2.
#f1 cases (pn_split f1) * #g1 #H1
#f2 cases (pn_split f2) * #g2 #H2
#Hf1 #Hf2 #Hi
]
qed-.
-(* Main properties on at ****************************************************)
+(* Advanced properties on at ************************************************)
lemma at_dec: ∀f,i1,i2. 𝐓⦃f⦄ → Decidable (@⦃i1, f⦄ ≡ i2).
#f #i1 #i2 #Hf lapply (Hf i1) -Hf *
]
qed-.
-lemma is_at_dec_le: ∀f,i2,i. 𝐓⦃f⦄ → (∀i1. i1 + i ≤ i2 → @⦃i1, f⦄ ≡ i2 → ⊥) → Decidable (∃i1. @⦃i1, f⦄ ≡ i2).
+lemma is_at_dec_le: ∀f,i2,i. 𝐓⦃f⦄ → (∀i1. i1 + i ≤ i2 → @⦃i1, f⦄ ≡ i2 → ⊥) →
+ Decidable (∃i1. @⦃i1, f⦄ ≡ i2).
#f #i2 #i #Hf elim i -i
[ #Ht @or_intror * /3 width=3 by at_increasing/
| #i #IH #Ht elim (at_dec f (i2-i) i2) /3 width=2 by ex_intro, or_introl/