interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
-(* 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/
-| * /2/
-]
-qed.
-
-lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2.
-#T1 elim T1 -T1
-[ * #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/
- | elim (IHT1 d 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.
-#d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2
-[ /3/
-| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
- lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
-| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
- lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
- <(arith_d1 i e2 e1) // /3/
-| /3/
-| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
- elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
- elim (IHT (d2+1) … ? ? He12) /3 width = 5/
-| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
- elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
- elim (IHT d2 … ? ? He12) /3 width = 5/
-]
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-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 ***************************************************)
fact lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
qed.
lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
-/2/ qed.
+/2/ qed-.
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 //
qed.
lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
-/2 width=5/ qed.
+/2 width=5/ qed-.
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)).
lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
(i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-/2/ qed.
+/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_refl_false … Hdd)
-qed.
+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_refl_false … Hdd)
-qed.
+qed-.
fact lift_inv_gref1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
#d #e #T1 #T2 * -d e T1 T2 //
qed.
lemma lift_inv_gref1: ∀d,e,T2,p. ↑[d,e] §p ≡ T2 → T2 = §p.
-/2 width=5/ qed.
+/2 width=5/ qed-.
fact lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
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.
-/2/ qed.
+/2/ qed-.
fact 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.
-/2/ qed.
+/2/ qed-.
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 //
(* Basic_1: was: lift_gen_sort *)
lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
-/2 width=5/ qed.
+/2 width=5/ qed-.
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)).
(* 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.
+/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.
+qed-.
(* Basic_1: was: lift_gen_lref_false *)
+lemma lift_inv_lref2_be: ∀d,e,T1,i. ↑[d,e] T1 ≡ #i →
+ d ≤ i → i < d + e → False.
+#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H *
+[ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
+lapply (le_to_lt_to_lt … H2 H1) -H2 H1 #H
+elim (lt_refl_false … H)
+qed-.
(* 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.
+qed-.
fact lift_inv_gref2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
#d #e #T1 #T2 * -d e T1 T2 //
qed.
lemma lift_inv_gref2: ∀d,e,T1,p. ↑[d,e] T1 ≡ §p → T1 = §p.
-/2 width=5/ qed.
+/2 width=5/ qed-.
fact lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
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.
+/2/ qed-.
fact 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.
-/2/ qed.
+/2/ qed-.
+
+lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ↑[d, e] 𝕔{I} V. T ≡ V → False.
+#d #e #J #V elim V -V
+[ * #i #T #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct
+ | elim (lift_inv_lref2 … H) -H * #_ #H destruct
+ | lapply (lift_inv_gref2 … H) -H #H destruct
+ ]
+| * #I #W2 #U2 #IHW2 #_ #T #H
+ [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct -J T W1 /2/
+ | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct -J T W1 /2/
+ ]
+]
+qed-.
+
+lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ↑[d, e] 𝕔{I} V. T ≡ T → False.
+#J #T elim T -T
+[ * #i #V #d #e #H
+ [ lapply (lift_inv_sort2 … H) -H #H destruct
+ | elim (lift_inv_lref2 … H) -H * #_ #H destruct
+ | lapply (lift_inv_gref2 … H) -H #H destruct
+ ]
+| * #I #W2 #U2 #_ #IHU2 #V #d #e #H
+ [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct -J U1 W1 /2/
+ | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct -J U1 W1 /2/
+ ]
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+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 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 width=2/
+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/
+| * /2/
+]
+qed.
+
+lemma lift_total: ∀T1,d,e. ∃T2. ↑[d,e] T1 ≡ T2.
+#T1 elim T1 -T1
+[ * #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/
+ | elim (IHT1 d 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.
+#d1 #e2 #T1 #T2 #H elim H -H d1 e2 T1 T2
+[ /3/
+| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
+ lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4/
+| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
+ lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
+ <(arith_d1 i e2 e1) // /3/
+| /3/
+| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
+ elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
+ elim (IHT (d2+1) … ? ? He12) /3 width=5/
+| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
+ elim (IHV … Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
+ elim (IHT d2 … ? ? He12) /3 width=5/
+]
+qed.
+
+(* Basic_1: was only: dnf_dec2 dnf_dec *)
+lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ↑[d,e] T1 ≡ T2).
+#T1 elim T1 -T1
+[ * [1,3: /3 width=2/ ] #i #d #e
+ elim (lt_dec i d) #Hid
+ [ /4 width=2/
+ | lapply (false_lt_to_le … Hid) -Hid #Hid
+ elim (lt_dec i (d + e)) #Hide
+ [ @or_intror * #T1 #H
+ elim (lift_inv_lref2_be … H Hid Hide)
+ | lapply (false_lt_to_le … Hide) -Hide /4 width=2/
+ ]
+ ]
+| * #I #V2 #T2 #IHV2 #IHT2 #d #e
+ [ elim (IHV2 d e) -IHV2
+ [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2
+ [ * #T1 #HT12 @or_introl /3/
+ | -V1 #HT2 @or_intror * #X #H
+ elim (lift_inv_bind2 … H) -H /3 width=2/
+ ]
+ | -IHT2 #HV2 @or_intror * #X #H
+ elim (lift_inv_bind2 … H) -H /3 width=2/
+ ]
+ | elim (IHV2 d e) -IHV2
+ [ * #V1 #HV12 elim (IHT2 d e) -IHT2
+ [ * #T1 #HT12 /4 width=2/
+ | -V1 #HT2 @or_intror * #X #H
+ elim (lift_inv_flat2 … H) -H /3 width=2/
+ ]
+ | -IHT2 #HV2 @or_intror * #X #H
+ elim (lift_inv_flat2 … H) -H /3 width=2/
+ ]
+ ]
+]
+qed.
(* Basic_1: removed theorems 7:
lift_head lift_gen_head
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