inductive at: list2 nat nat → relation nat ≝
| at_nil: ∀i. at (◊) i i
-| at_lt : ∀des,l,m,i1,i2. i1 < l →
- at des i1 i2 → at ({l, m} @ des) i1 i2
-| at_ge : ∀des,l,m,i1,i2. l ≤ i1 →
- at des (i1 + m) i2 → at ({l, m} @ des) i1 i2
+| at_lt : ∀cs,l,m,i1,i2. i1 < l →
+ at cs i1 i2 → at ({l, m} @ cs) i1 i2
+| at_ge : ∀cs,l,m,i1,i2. l ≤ i1 →
+ at cs (i1 + m) i2 → at ({l, m} @ cs) i1 i2
.
interpretation "application (multiple relocation with pairs)"
- 'RAt i1 des i2 = (at des i1 i2).
+ 'RAt i1 cs i2 = (at cs i1 i2).
(* Basic inversion lemmas ***************************************************)
-fact at_inv_nil_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 → des = ◊ → i1 = i2.
-#des #i1 #i2 * -des -i1 -i2
+fact at_inv_nil_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 → cs = ◊ → i1 = i2.
+#cs #i1 #i2 * -cs -i1 -i2
[ //
-| #des #l #m #i1 #i2 #_ #_ #H destruct
-| #des #l #m #i1 #i2 #_ #_ #H destruct
+| #cs #l #m #i1 #i2 #_ #_ #H destruct
+| #cs #l #m #i1 #i2 #_ #_ #H destruct
]
qed-.
lemma at_inv_nil: ∀i1,i2. @⦃i1, ◊⦄ ≡ i2 → i1 = i2.
/2 width=3 by at_inv_nil_aux/ qed-.
-fact at_inv_cons_aux: ∀des,i1,i2. @⦃i1, des⦄ ≡ i2 →
- ∀l,m,des0. des = {l, m} @ des0 →
- i1 < l ∧ @⦃i1, des0⦄ ≡ i2 ∨
- l ≤ i1 ∧ @⦃i1 + m, des0⦄ ≡ i2.
-#des #i1 #i2 * -des -i1 -i2
-[ #i #l #m #des #H destruct
-| #des1 #l1 #m1 #i1 #i2 #Hil1 #Hi12 #l2 #m2 #des2 #H destruct /3 width=1 by or_introl, conj/
-| #des1 #l1 #m1 #i1 #i2 #Hli1 #Hi12 #l2 #m2 #des2 #H destruct /3 width=1 by or_intror, conj/
+fact at_inv_cons_aux: ∀cs,i1,i2. @⦃i1, cs⦄ ≡ i2 →
+ ∀l,m,cs0. cs = {l, m} @ cs0 →
+ i1 < l ∧ @⦃i1, cs0⦄ ≡ i2 ∨
+ l ≤ i1 ∧ @⦃i1 + m, cs0⦄ ≡ i2.
+#cs #i1 #i2 * -cs -i1 -i2
+[ #i #l #m #cs #H destruct
+| #cs1 #l1 #m1 #i1 #i2 #Hil1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_introl, conj/
+| #cs1 #l1 #m1 #i1 #i2 #Hli1 #Hi12 #l2 #m2 #cs2 #H destruct /3 width=1 by or_intror, conj/
]
qed-.
-lemma at_inv_cons: ∀des,l,m,i1,i2. @⦃i1, {l, m} @ des⦄ ≡ i2 →
- i1 < l ∧ @⦃i1, des⦄ ≡ i2 ∨
- l ≤ i1 ∧ @⦃i1 + m, des⦄ ≡ i2.
+lemma at_inv_cons: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
+ i1 < l ∧ @⦃i1, cs⦄ ≡ i2 ∨
+ l ≤ i1 ∧ @⦃i1 + m, cs⦄ ≡ i2.
/2 width=3 by at_inv_cons_aux/ qed-.
-lemma at_inv_cons_lt: ∀des,l,m,i1,i2. @⦃i1, {l, m} @ des⦄ ≡ i2 →
- i1 < l → @⦃i1, des⦄ ≡ i2.
-#des #l #m #i1 #m2 #H
+lemma at_inv_cons_lt: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
+ i1 < l → @⦃i1, cs⦄ ≡ i2.
+#cs #l #m #i1 #m2 #H
elim (at_inv_cons … H) -H * // #Hli1 #_ #Hi1l
lapply (le_to_lt_to_lt … Hli1 Hi1l) -Hli1 -Hi1l #Hl
elim (lt_refl_false … Hl)
qed-.
-lemma at_inv_cons_ge: ∀des,l,m,i1,i2. @⦃i1, {l, m} @ des⦄ ≡ i2 →
- l ≤ i1 → @⦃i1 + m, des⦄ ≡ i2.
-#des #l #m #i1 #m2 #H
+lemma at_inv_cons_ge: ∀cs,l,m,i1,i2. @⦃i1, {l, m} @ cs⦄ ≡ i2 →
+ l ≤ i1 → @⦃i1 + m, cs⦄ ≡ i2.
+#cs #l #m #i1 #m2 #H
elim (at_inv_cons … H) -H * // #Hi1l #_ #Hli1
lapply (le_to_lt_to_lt … Hli1 Hi1l) -Hli1 -Hi1l #Hl
elim (lt_refl_false … Hl)