(* Properties concerning basic term relocation ******************************)
(* Basic_1: was: lift1_xhg (right to left) *)
-lemma lifts_lift_trans_le: ∀T1,T,des. ⬆*[des] T1 ≡ T → ∀T2. ⬆[0, 1] T ≡ T2 →
- ∃∃T0. ⬆[0, 1] T1 ≡ T0 & ⬆*[des + 1] T0 ≡ T2.
-#T1 #T #des #H elim H -T1 -T -des
-[ /2 width=3/
-| #T1 #T3 #T #des #l #m #HT13 #_ #IHT13 #T2 #HT2
+lemma lifts_lift_trans_le: ∀T1,T,cs. ⬆*[cs] T1 ≡ T → ∀T2. ⬆[0, 1] T ≡ T2 →
+ ∃∃T0. ⬆[0, 1] T1 ≡ T0 & ⬆*[cs + 1] T0 ≡ T2.
+#T1 #T #cs #H elim H -T1 -T -cs
+[ /2 width=3 by lifts_nil, ex2_intro/
+| #T1 #T3 #T #cs #l #m #HT13 #_ #IHT13 #T2 #HT2
elim (IHT13 … HT2) -T #T #HT3 #HT2
- elim (lift_trans_le … HT13 … HT3) -T3 // /3 width=5/
+ elim (lift_trans_le … HT13 … HT3) -T3 /3 width=5 by lifts_cons, ex2_intro/
]
qed-.
(* Basic_1: was: lift1_free (right to left) *)
-lemma lifts_lift_trans: ∀des,i,i0. @⦃i, des⦄ ≡ i0 →
- ∀des0. des + 1 ▭ i + 1 ≡ des0 + 1 →
- ∀T1,T0. ⬆*[des0] T1 ≡ T0 →
+lemma lifts_lift_trans: ∀cs,i,i0. @⦃i, cs⦄ ≡ i0 →
+ ∀cs0. cs + 1 ▭ i + 1 ≡ cs0 + 1 →
+ ∀T1,T0. ⬆*[cs0] T1 ≡ T0 →
∀T2. ⬆[O, i0 + 1] T0 ≡ T2 →
- ∃∃T. ⬆[0, i + 1] T1 ≡ T & ⬆*[des] T ≡ T2.
-#des elim des -des normalize
-[ #i #x #H1 #des0 #H2 #T1 #T0 #HT10 #T2
+ ∃∃T. ⬆[0, i + 1] T1 ≡ T & ⬆*[cs] T ≡ T2.
+#cs elim cs -cs normalize
+[ #i #x #H1 #cs0 #H2 #T1 #T0 #HT10 #T2
<(at_inv_nil … H1) -x #HT02
lapply (minuss_inv_nil1 … H2) -H2 #H
- >(pluss_inv_nil2 … H) in HT10; -des0 #H
- >(lifts_inv_nil … H) -T1 /2 width=3/
-| #l #m #des #IHcs #i #i0 #H1 #des0 #H2 #T1 #T0 #HT10 #T2 #HT02
+ >(pluss_inv_nil2 … H) in HT10; -cs0 #H
+ >(lifts_inv_nil … H) -T1 /2 width=3 by lifts_nil, ex2_intro/
+| #l #m #cs #IHcs #i #i0 #H1 #cs0 #H2 #T1 #T0 #HT10 #T2 #HT02
elim (at_inv_cons … H1) -H1 * #Hil #Hi0
- [ elim (minuss_inv_cons1_lt … H2) -H2 [2: /2 width=1/ ] #des1 #Hcs1 <minus_le_minus_minus_comm // <minus_plus_m_m #H
- elim (pluss_inv_cons2 … H) -H #des2 #H1 #H2 destruct
+ [ elim (minuss_inv_cons1_lt … H2) -H2 [2: /2 width=1 by lt_minus_to_plus/ ] #cs1 #Hcs1 <minus_le_minus_minus_comm // <minus_plus_m_m #H
+ elim (pluss_inv_cons2 … H) -H #cs2 #H1 #H2 destruct
elim (lifts_inv_cons … HT10) -HT10 #T >minus_plus #HT1 #HT0
elim (IHcs … Hi0 … Hcs1 … HT0 … HT02) -IHcs -Hi0 -Hcs1 -T0 #T0 #HT0 #HT02
- elim (lift_trans_le … HT1 … HT0) -T /2 width=1/ #T #HT1 <plus_minus_m_m /2 width=1/ /3 width=5/
+ elim (lift_trans_le … HT1 … HT0) -T /2 width=1 by/ #T #HT1 <plus_minus_m_m /3 width=5 by lifts_cons, ex2_intro/
| >commutative_plus in Hi0; #Hi0
- lapply (minuss_inv_cons1_ge … H2 ?) -H2 [ /2 width=1/ ] <associative_plus #Hcs0
+ lapply (minuss_inv_cons1_ge … H2 ?) -H2 [ /2 width=1 by le_S_S/ ] <associative_plus #Hcs0
elim (IHcs … Hi0 … Hcs0 … HT10 … HT02) -IHcs -Hi0 -Hcs0 -T0 #T0 #HT0 #HT02
- elim (lift_split … HT0 l (i+1)) -HT0 /2 width=1/ /3 width=5/
+ elim (lift_split … HT0 l (i+1)) -HT0 /3 width=5 by lifts_cons, le_S, ex2_intro/
]
]
qed-.