inductive minuss: nat → relation (list2 nat nat) ≝
| minuss_nil: ∀i. minuss i ⟠ ⟠
| minuss_lt : ∀des1,des2,d,e,i. i < d → minuss i des1 des2 →
- minuss i ({d, e} :: des1) ({d - i, e} :: des2)
+ minuss i ({d, e} @ des1) ({d - i, e} @ des2)
| minuss_ge : ∀des1,des2,d,e,i. d ≤ i → minuss (e + i) des1 des2 →
- minuss i ({d, e} :: des1) des2
+ minuss i ({d, e} @ des1) des2
.
interpretation "minus (generic relocation with pairs)"
/2 width=4/ qed-.
fact minuss_inv_cons1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 →
- ∀d,e,des. des1 = {d, e} :: des →
+ ∀d,e,des. des1 = {d, e} @ des →
d ≤ i ∧ des ▭ e + i ≡ des2 ∨
∃∃des0. i < d & des ▭ i ≡ des0 &
- des2 = {d - i, e} :: des0.
+ des2 = {d - i, e} @ des0.
#des1 #des2 #i * -des1 -des2 -i
[ #i #d #e #des #H destruct
| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3/
]
qed.
-lemma minuss_inv_cons1: ∀des1,des2,d,e,i. {d, e} :: des1 ▭ i ≡ des2 →
+lemma minuss_inv_cons1: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
d ≤ i ∧ des1 ▭ e + i ≡ des2 ∨
∃∃des. i < d & des1 ▭ i ≡ des &
- des2 = {d - i, e} :: des.
+ des2 = {d - i, e} @ des.
/2 width=3/ qed-.
-lemma minuss_inv_cons1_ge: ∀des1,des2,d,e,i. {d, e} :: des1 ▭ i ≡ des2 →
+lemma minuss_inv_cons1_ge: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
d ≤ i → des1 ▭ e + i ≡ des2.
#des1 #des2 #d #e #i #H
elim (minuss_inv_cons1 … H) -H * // #des #Hid #_ #_ #Hdi
elim (lt_refl_false … Hi)
qed-.
-lemma minuss_inv_cons1_lt: ∀des1,des2,d,e,i. {d, e} :: des1 ▭ i ≡ des2 →
+lemma minuss_inv_cons1_lt: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
i < d →
- ∃∃des. des1 ▭ i ≡ des & des2 = {d - i, e} :: des.
+ ∃∃des. des1 ▭ i ≡ des & des2 = {d - i, e} @ des.
#des1 #des2 #d #e #i #H
elim (minuss_inv_cons1 … H) -H * /2 width=3/ #Hdi #_ #Hid
lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi #Hi