(* The basic properties *****************************************************)
-lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d,e] #(i - e) ≡ #i.
+lemma lift_lref_ge_minus: ∀d,e,i. d + e ≤ i → ↑[d, e] #(i - e) ≡ #i.
#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %) /3/
qed.
+lemma lift_refl: ∀T,d. ↑[d, 0] T ≡ T.
+#T elim T -T
+[ //
+| #i #d elim (lt_or_ge i d) /2/
+| #I elim I -I /2/
+]
+qed.
+
(* The basic inversion lemmas ***********************************************)
+lemma lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
+#d #e #T1 #T2 #H elim H -H d e T1 T2 /3/
+qed.
+
+lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
+/2/ qed.
+
lemma lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
#d #e #T1 #T2 #H elim H -H d e T1 T2 //
[ #i #d #e #_ #k #H destruct
qed.
lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
-#d #e #T2 #k #H lapply (lift_inv_sort1_aux … H) /2/
-qed.
+/2 width=5/ qed.
lemma lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
(i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
(i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-#d #e #T2 #i #H lapply (lift_inv_lref1_aux … H) /2/
+/2/ qed.
+
+lemma lift_inv_lref1_lt: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → i < d → T2 = #i.
+#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (lt_false … Hdd)
+qed.
+
+lemma lift_inv_lref1_ge: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
+#d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (lt_false … Hdd)
qed.
lemma lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
lemma lift_inv_bind1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕓{I} V1. U1 ≡ T2 →
∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
T2 = 𝕓{I} V2. U2.
-#d #e #T2 #I #V1 #U1 #H lapply (lift_inv_bind1_aux … H) /2/
-qed.
+/2/ qed.
lemma lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ↑[d,e] 𝕗{I} V1. U1 ≡ T2 →
∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
T2 = 𝕗{I} V2. U2.
-#d #e #T2 #I #V1 #U1 #H lapply (lift_inv_flat1_aux … H) /2/
-qed.
+/2/ qed.
lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
#d #e #T1 #T2 #H elim H -H d e T1 T2 //
qed.
lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
-#d #e #T1 #k #H lapply (lift_inv_sort2_aux … H) /2/
-qed.
+/2 width=5/ qed.
lemma lift_inv_lref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T2 = #i →
(i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
(i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
-#d #e #T1 #i #H lapply (lift_inv_lref2_aux … H) /2/
+/2/ qed.
+
+lemma lift_inv_lref2_lt: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → i < d → T1 = #i.
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (plus_lt_false … Hdd)
+qed.
+
+lemma lift_inv_lref2_ge: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i → d + e ≤ i → T1 = #(i - e).
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+elim (plus_lt_false … Hdd)
qed.
lemma lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
lemma lift_inv_bind2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕓{I} V2. U2 →
∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
T1 = 𝕓{I} V1. U1.
-#d #e #T1 #I #V2 #U2 #H lapply (lift_inv_bind2_aux … H) /2/
-qed.
+/2/ qed.
lemma lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ↑[d,e] T1 ≡ 𝕗{I} V2. U2 →
∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
T1 = 𝕗{I} V1. U1.
-#d #e #T1 #I #V2 #U2 #H lapply (lift_inv_flat2_aux … H) /2/
-qed.
+/2/ qed.