(* RELOCATION MAP ***********************************************************)
-definition istot: predicate rtmap â\89\9d λf. â\88\80i. â\88\83j. @â¦\83i, fâ¦\84 â\89¡ j.
+definition istot: predicate rtmap â\89\9d λf. â\88\80i. â\88\83j. @â¦\83i, fâ¦\84 â\89\98 j.
interpretation "test for totality (rtmap)"
'IsTotal f = (istot f).
(* Properties on tl *********************************************************)
-lemma istot_tl: â\88\80f. ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83â\86\93f⦄.
+lemma istot_tl: â\88\80f. ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83⫱f⦄.
#f cases (pn_split f) *
#g * -f /2 width=3 by istot_inv_next, istot_inv_push/
qed.
-(* Properties on minus ******************************************************)
+(* Properties on tls ********************************************************)
-lemma istot_minus: ∀n,f. 𝐓⦃f⦄ → 𝐓⦃f-n⦄.
+lemma istot_tls: ∀n,f. 𝐓⦃f⦄ → 𝐓⦃⫱*[n]f⦄.
#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: â\88\80f,i1,i2. ð\9d\90\93â¦\83fâ¦\84 â\86\92 Decidable (@â¦\83i1, fâ¦\84 â\89¡ i2).
+lemma at_dec: â\88\80f,i1,i2. ð\9d\90\93â¦\83fâ¦\84 â\86\92 Decidable (@â¦\83i1, fâ¦\84 â\89\98 i2).
#f #i1 #i2 #Hf lapply (Hf i1) -Hf *
#j2 #Hf elim (eq_nat_dec i2 j2)
[ #H destruct /2 width=1 by or_introl/
]
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/
]
qed-.
-lemma is_at_dec: â\88\80f,i2. ð\9d\90\93â¦\83fâ¦\84 â\86\92 Decidable (â\88\83i1. @â¦\83i1, fâ¦\84 â\89¡ i2).
+lemma is_at_dec: â\88\80f,i2. ð\9d\90\93â¦\83fâ¦\84 â\86\92 Decidable (â\88\83i1. @â¦\83i1, fâ¦\84 â\89\98 i2).
#f #i2 #Hf @(is_at_dec_le ?? (⫯i2)) /2 width=4 by lt_le_false/
qed-.
(* Advanced properties on isid **********************************************)
-lemma isid_at_total: â\88\80f. ð\9d\90\93â¦\83fâ¦\84 â\86\92 (â\88\80i1,i2. @â¦\83i1, fâ¦\84 â\89¡ i2 → i1 = i2) → 𝐈⦃f⦄.
+lemma isid_at_total: â\88\80f. ð\9d\90\93â¦\83fâ¦\84 â\86\92 (â\88\80i1,i2. @â¦\83i1, fâ¦\84 â\89\98 i2 → i1 = i2) → 𝐈⦃f⦄.
#f #H1f #H2f @isid_at
#i lapply (H1f i) -H1f *
#j #Hf >(H2f … Hf) in ⊢ (???%); -H2f //