(* RELOCATION ***************************************************************)
+(* Basic-1: includes:
+ lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
+*)
inductive lift: term → nat → nat → term → Prop ≝
| lift_sort : ∀k,d,e. lift (⋆k) d e (⋆k)
| lift_lref_lt: ∀i,d,e. i < d → lift (#i) d e (#i)
(* Basic properties *********************************************************)
+(* Basic-1: was: lift_lref_gt *)
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.
+(* Basic-1: was: lift_r *)
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/
+[ * #i // #d elim (lt_or_ge i d) /2/
+| * /2/
]
qed.
lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2.
#T1 elim T1 -T1
-[ /2/
-| #i #d #e elim (lt_or_ge i d) /3/
+[ * #i /2/ #d #e elim (lt_or_ge i d) /3/
| * #I #V1 #T1 #IHV1 #IHT1 #d #e
elim (IHV1 d e) -IHV1 #V2 #HV12
[ elim (IHT1 (d+1) e) -IHT1 /3/
]
qed.
+(* Basic-1: was: lift_free (right to left) *)
lemma lift_split: ∀d1,e2,T1,T2. ↑[d1, e2] T1 ≡ T2 → ∀d2,e1.
d1 ≤ d2 → d2 ≤ d1 + e1 → e1 ≤ e2 →
∃∃T. ↑[d1, e1] T1 ≡ T & ↑[d2, e2 - e1] T ≡ T2.
(* 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
]
qed.
+(* Basic-1: was: lift_gen_sort *)
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/
]
qed.
+(* Basic-1: was: lift_gen_lref *)
lemma lift_inv_lref2: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
(i < d ∧ T1 = #i) ∨ (d + e ≤ i ∧ T1 = #(i - e)).
/2/ qed.
+(* Basic-1: was: lift_gen_lref_lt *)
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.
+(* Basic-1: was: lift_gen_lref_false *)
+
+(* Basic-1: was: lift_gen_lref_ge *)
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 →
- ∀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
]
qed.
+(* Basic-1: was: lift_gen_bind *)
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.
/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
]
qed.
+(* Basic-1: was: lift_gen_flat *)
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.
/2/ qed.
+
+(* Basic-1: removed theorems 7:
+ lift_head lift_gen_head
+ lift_weight_map lift_weight lift_weight_add lift_weight_add_O
+ lift_tlt_dx
+*)