inductive minuss: nat → relation mr2 ≝
| minuss_nil: ∀i. minuss i (◊) (◊)
| minuss_lt : ∀cs1,cs2,l,m,i. i < l → minuss i cs1 cs2 →
- minuss i ({l, m} ⨮ cs1) ({l - i, m} ⨮ cs2)
+ minuss i ({l, m};cs1) ({l - i, m};cs2)
| minuss_ge : ∀cs1,cs2,l,m,i. l ≤ i → minuss (m + i) cs1 cs2 →
- minuss i ({l, m} ⨮ cs1) cs2
+ minuss i ({l, m};cs1) cs2
.
interpretation "minus (multiple relocation with pairs)"
/2 width=4 by minuss_inv_nil1_aux/ qed-.
fact minuss_inv_cons1_aux: ∀cs1,cs2,i. cs1 ▭ i ≘ cs2 →
- ∀l,m,cs. cs1 = {l, m} ⨮ cs →
+ ∀l,m,cs. cs1 = {l, m};cs →
l ≤ i ∧ cs ▭ m + i ≘ cs2 ∨
∃∃cs0. i < l & cs ▭ i ≘ cs0 &
- cs2 = {l - i, m} ⨮ cs0.
+ cs2 = {l - i, m};cs0.
#cs1 #cs2 #i * -cs1 -cs2 -i
[ #i #l #m #cs #H destruct
| #cs1 #cs #l1 #m1 #i1 #Hil1 #Hcs #l2 #m2 #cs2 #H destruct /3 width=3 by ex3_intro, or_intror/
]
qed-.
-lemma minuss_inv_cons1: ∀cs1,cs2,l,m,i. {l, m} ⨮ cs1 ▭ i ≘ cs2 →
+lemma minuss_inv_cons1: ∀cs1,cs2,l,m,i. {l, m};cs1 ▭ i ≘ cs2 →
l ≤ i ∧ cs1 ▭ m + i ≘ cs2 ∨
∃∃cs. i < l & cs1 ▭ i ≘ cs &
- cs2 = {l - i, m} ⨮ cs.
+ cs2 = {l - i, m};cs.
/2 width=3 by minuss_inv_cons1_aux/ qed-.
-lemma minuss_inv_cons1_ge: ∀cs1,cs2,l,m,i. {l, m} ⨮ cs1 ▭ i ≘ cs2 →
+lemma minuss_inv_cons1_ge: ∀cs1,cs2,l,m,i. {l, m};cs1 ▭ i ≘ cs2 →
l ≤ i → cs1 ▭ m + i ≘ cs2.
#cs1 #cs2 #l #m #i #H
elim (minuss_inv_cons1 … H) -H * // #cs #Hil #_ #_ #Hli
elim (lt_le_false … Hil Hli)
qed-.
-lemma minuss_inv_cons1_lt: ∀cs1,cs2,l,m,i. {l, m} ⨮ cs1 ▭ i ≘ cs2 →
+lemma minuss_inv_cons1_lt: ∀cs1,cs2,l,m,i. {l, m};cs1 ▭ i ≘ cs2 →
i < l →
- ∃∃cs. cs1 ▭ i ≘ cs & cs2 = {l - i, m} ⨮ cs.
+ ∃∃cs. cs1 ▭ i ≘ cs & cs2 = {l - i, m};cs.
#cs1 #cs2 #l #m #i #H elim (minuss_inv_cons1 … H) -H * /2 width=3 by ex2_intro/
#Hli #_ #Hil elim (lt_le_false … Hil Hli)
qed-.