(* Main properies ***********************************************************)
-lemma lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2.
+theorem lift_inj: ∀d,e,T1,U. ↑[d,e] T1 ≡ U → ∀T2. ↑[d,e] T2 ≡ U → T1 = T2.
#d #e #T1 #U #H elim H -H d e T1 U
[ #k #d #e #X #HX
lapply (lift_inv_sort2 … HX) -HX //
]
qed.
-lemma lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
- d1 ≤ d2 →
- ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1.
+theorem lift_div_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ↑[d2 + e1, e2] T2 ≡ T →
+ d1 ≤ d2 →
+ ∃∃T0. ↑[d1, e1] T0 ≡ T2 & ↑[d2, e2] T0 ≡ T1.
#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct -T2 /3/
]
qed.
-lemma lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
+theorem lift_mono: ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
#d #e #T #U1 #H elim H -H d e T U1
[ #k #d #e #X #HX
lapply (lift_inv_sort1 … HX) -HX //
]
qed.
-lemma lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 →
- d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2.
+theorem lift_trans_be: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 →
+ d1 ≤ d2 → d2 ≤ d1 + e1 → ↑[d1, e1 + e2] T1 ≡ T2.
#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
>(lift_inv_sort1 … HT2) -HT2 //
]
qed.
-lemma lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 →
- ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2.
+theorem lift_trans_le: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d2 ≤ d1 →
+ ∃∃T0. ↑[d2, e2] T1 ≡ T0 & ↑[d1 + e2, e1] T0 ≡ T2.
#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
[ #k #d1 #e1 #d2 #e2 #X #HX #_
>(lift_inv_sort1 … HX) -HX /2/
]
qed.
-lemma lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
- ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
- ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2.
+theorem lift_trans_ge: ∀d1,e1,T1,T. ↑[d1, e1] T1 ≡ T →
+ ∀d2,e2,T2. ↑[d2, e2] T ≡ T2 → d1 + e1 ≤ d2 →
+ ∃∃T0. ↑[d2 - e1, e2] T1 ≡ T0 & ↑[d1, e1] T0 ≡ T2.
#d1 #e1 #T1 #T #H elim H -H d1 e1 T1 T
[ #k #d1 #e1 #d2 #e2 #X #HX #_
>(lift_inv_sort1 … HX) -HX /2/