(* GENERIC RELOCATION WITH PAIRS ********************************************)
inductive at: list2 nat nat → relation nat ≝
-| at_nil: ∀i. at ⟠ i i
+| at_nil: ∀i. at (⟠) i i
| at_lt : ∀des,d,e,i1,i2. i1 < d →
at des i1 i2 → at ({d, e} @ des) i1 i2
| at_ge : ∀des,d,e,i1,i2. d ≤ i1 →
| #des #d #e #i1 #i2 #_ #_ #H destruct
| #des #d #e #i1 #i2 #_ #_ #H destruct
]
-qed.
+qed-.
lemma at_inv_nil: ∀i1,i2. @⦃i1, ⟠⦄ ≡ i2 → i1 = i2.
-/2 width=3/ qed-.
+/2 width=3 by at_inv_nil_aux/ qed-.
fact at_inv_cons_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 →
∀d,e,des0. des = {d, e} @ des0 →
d ≤ i1 ∧ @⦃i1 + e, des0⦄ ≡ i2.
#des #i1 #i2 * -des -i1 -i2
[ #i #d #e #des #H destruct
-| #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/
-| #des1 #d1 #e1 #i1 #i2 #Hdi1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1/
+| #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1 by or_introl, conj/
+| #des1 #d1 #e1 #i1 #i2 #Hdi1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1 by or_intror, conj/
]
-qed.
+qed-.
lemma at_inv_cons: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
i1 < d ∧ @⦃i1, des⦄ ≡ i2 ∨
d ≤ i1 ∧ @⦃i1 + e, des⦄ ≡ i2.
-/2 width=3/ qed-.
+/2 width=3 by at_inv_cons_aux/ qed-.
lemma at_inv_cons_lt: ∀des,d,e,i1,i2. @⦃i1, {d, e} @ des⦄ ≡ i2 →
i1 < d → @⦃i1, des⦄ ≡ i2.