(* GENERIC RELOCATION WITH PAIRS ********************************************)
inductive minuss: nat → relation (list2 nat nat) ≝
-| minuss_nil: ∀i. minuss i ⟠ ⟠
+| 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_ge : ∀des1,des2,d,e,i. d ≤ i → minuss (e + i) des1 des2 →
| #des1 #des2 #d #e #i #_ #_ #H destruct
| #des1 #des2 #d #e #i #_ #_ #H destruct
]
-qed.
+qed-.
lemma minuss_inv_nil1: ∀des2,i. ⟠ ▭ i ≡ des2 → des2 = ⟠.
-/2 width=4/ qed-.
+/2 width=4 by minuss_inv_nil1_aux/ qed-.
fact minuss_inv_cons1_aux: ∀des1,des2,i. des1 ▭ i ≡ des2 →
∀d,e,des. des1 = {d, e} @ des →
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/
-| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1/
+| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3 by ex3_intro, or_intror/
+| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1 by or_introl, conj/
]
-qed.
+qed-.
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.
-/2 width=3/ qed-.
+/2 width=3 by minuss_inv_cons1_aux/ qed-.
lemma minuss_inv_cons1_ge: ∀des1,des2,d,e,i. {d, e} @ des1 ▭ i ≡ des2 →
d ≤ i → des1 ▭ e + 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.
-#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
-elim (lt_refl_false … Hi)
+#des1 #des2 #d #e #i #H elim (minuss_inv_cons1 … H) -H * /2 width=3 by ex2_intro/
+#Hdi #_ #Hid lapply (lt_to_le_to_lt … Hid Hdi) -Hid -Hdi
+#Hi elim (lt_refl_false … Hi)
qed-.