(* Basic forward lemmas *****************************************************)
-lemma tw_lift: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → #T1 = #T2.
+lemma tw_lift: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → #[T1] = #[T2].
#d #e #T1 #T2 #H elim H -d e T1 T2; normalize //
qed.
(* Basic inversion lemmas ***************************************************)
-lemma lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
+fact 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.
+fact lift_inv_sort1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
#d #e #T1 #T2 * -d e T1 T2 //
[ #i #d #e #_ #k #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
/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)).
+fact lift_inv_lref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀i. T1 = #i →
+ (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
#d #e #T1 #T2 * -d e T1 T2
[ #k #d #e #i #H destruct
| #j #d #e #Hj #i #Hi destruct /3/
elim (lt_refl_false … Hdd)
qed.
-lemma lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
- ∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
- ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
- T2 = 𝕓{I} V2. U2.
+fact lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
+ ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+ T2 = 𝕓{I} V2. U2.
#d #e #T1 #T2 * -d e T1 T2
[ #k #d #e #I #V1 #U1 #H destruct
| #i #d #e #_ #I #V1 #U1 #H destruct
T2 = 𝕓{I} V2. U2.
/2/ qed.
-lemma lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
- ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
- ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
- T2 = 𝕗{I} V2. U2.
+fact lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
+ ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+ T2 = 𝕗{I} V2. U2.
#d #e #T1 #T2 * -d e T1 T2
[ #k #d #e #I #V1 #U1 #H destruct
| #i #d #e #_ #I #V1 #U1 #H destruct
T2 = 𝕗{I} V2. U2.
/2/ qed.
-lemma lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
+fact lift_inv_sort2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
#d #e #T1 #T2 * -d e T1 T2 //
[ #i #d #e #_ #k #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
/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)).
+fact 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)).
#d #e #T1 #T2 * -d e T1 T2
[ #k #d #e #i #H destruct
| #j #d #e #Hj #i #Hi destruct /3/
elim (plus_lt_false … Hdd)
qed.
-lemma lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
- ∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
- ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
- T1 = 𝕓{I} V1. U1.
+fact lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
+ ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
+ T1 = 𝕓{I} V1. U1.
#d #e #T1 #T2 * -d e T1 T2
[ #k #d #e #I #V2 #U2 #H destruct
| #i #d #e #_ #I #V2 #U2 #H destruct
T1 = 𝕓{I} V1. U1.
/2/ qed.
-lemma lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
- ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
- ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
- T1 = 𝕗{I} V1. U1.
+fact lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
+ ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
+ ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
+ T1 = 𝕗{I} V1. U1.
#d #e #T1 #T2 * -d e T1 T2
[ #k #d #e #I #V2 #U2 #H destruct
| #i #d #e #_ #I #V2 #U2 #H destruct