lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
*)
inductive lift: relation4 nat nat term term ≝
-| lift_sort : ∀k,d,e. lift d e (⋆k) (⋆k)
-| lift_lref_lt: ∀i,d,e. i < d → lift d e (#i) (#i)
-| lift_lref_ge: ∀i,d,e. d ≤ i → lift d e (#i) (#(i + e))
-| lift_gref : ∀p,d,e. lift d e (§p) (§p)
-| lift_bind : ∀a,I,V1,V2,T1,T2,d,e.
- lift d e V1 V2 → lift (d + 1) e T1 T2 →
- lift d e (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| lift_flat : ∀I,V1,V2,T1,T2,d,e.
- lift d e V1 V2 → lift d e T1 T2 →
- lift d e (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
+| lift_sort : ∀k,l,m. lift l m (⋆k) (⋆k)
+| lift_lref_lt: ∀i,l,m. i < l → lift l m (#i) (#i)
+| lift_lref_ge: ∀i,l,m. l ≤ i → lift l m (#i) (#(i + m))
+| lift_gref : ∀p,l,m. lift l m (§p) (§p)
+| lift_bind : ∀a,I,V1,V2,T1,T2,l,m.
+ lift l m V1 V2 → lift (l + 1) m T1 T2 →
+ lift l m (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
+| lift_flat : ∀I,V1,V2,T1,T2,l,m.
+ lift l m V1 V2 → lift l m T1 T2 →
+ lift l m (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
.
-interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
+interpretation "relocation" 'RLift l m T1 T2 = (lift l m T1 T2).
(* Basic inversion lemmas ***************************************************)
-fact lift_inv_O2_aux: ∀d,e,T1,T2. ⬆[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 /3 width=1 by eq_f2/
+fact lift_inv_O2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → m = 0 → T1 = T2.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 /3 width=1 by eq_f2/
qed-.
-lemma lift_inv_O2: ∀d,T1,T2. ⬆[d, 0] T1 ≡ T2 → T1 = T2.
+lemma lift_inv_O2: ∀l,T1,T2. ⬆[l, 0] T1 ≡ T2 → T1 = T2.
/2 width=4 by lift_inv_O2_aux/ 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 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+fact lift_inv_sort1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
]
qed-.
-lemma lift_inv_sort1: ∀d,e,T2,k. ⬆[d,e] ⋆k ≡ T2 → T2 = ⋆k.
+lemma lift_inv_sort1: ∀l,m,T2,k. ⬆[l,m] ⋆k ≡ T2 → T2 = ⋆k.
/2 width=5 by lift_inv_sort1_aux/ 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)).
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
-| #j #d #e #Hj #i #Hi destruct /3 width=1 by or_intror, conj/
-| #p #d #e #i #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+fact lift_inv_lref1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀i. T1 = #i →
+ (i < l ∧ T2 = #i) ∨ (l ≤ i ∧ T2 = #(i + m)).
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #i #H destruct
+| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
+| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_intror, conj/
+| #p #l #m #i #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
]
qed-.
-lemma lift_inv_lref1: ∀d,e,T2,i. ⬆[d,e] #i ≡ T2 →
- (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
+lemma lift_inv_lref1: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 →
+ (i < l ∧ T2 = #i) ∨ (l ≤ i ∧ T2 = #(i + m)).
/2 width=3 by lift_inv_lref1_aux/ 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)
+lemma lift_inv_lref1_lt: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 → i < l → T2 = #i.
+#l #m #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hli #_ #Hil lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
+elim (lt_refl_false … Hll)
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)
+lemma lift_inv_lref1_ge: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 → l ≤ i → T2 = #(i + m).
+#l #m #T2 #i #H elim (lift_inv_lref1 … H) -H * //
+#Hil #_ #Hli lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
+elim (lt_refl_false … Hll)
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 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+fact lift_inv_gref1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
]
qed-.
-lemma lift_inv_gref1: ∀d,e,T2,p. ⬆[d,e] §p ≡ T2 → T2 = §p.
+lemma lift_inv_gref1: ∀l,m,T2,p. ⬆[l,m] §p ≡ T2 → T2 = §p.
/2 width=5 by lift_inv_gref1_aux/ qed-.
-fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⬆[d,e] T1 ≡ T2 →
+fact lift_inv_bind1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
- ∃∃V2,U2. ⬆[d,e] V1 ≡ V2 & ⬆[d+1,e] U1 ≡ U2 &
+ ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
T2 = ⓑ{a,I} V2. U2.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #a #I #V1 #U1 #H destruct
-| #i #d #e #_ #a #I #V1 #U1 #H destruct
-| #i #d #e #_ #a #I #V1 #U1 #H destruct
-| #p #d #e #a #I #V1 #U1 #H destruct
-| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-| #J #W1 #W2 #T1 #T2 #d #e #_ #HT #a #I #V1 #U1 #H destruct
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #a #I #V1 #U1 #H destruct
+| #i #l #m #_ #a #I #V1 #U1 #H destruct
+| #i #l #m #_ #a #I #V1 #U1 #H destruct
+| #p #l #m #a #I #V1 #U1 #H destruct
+| #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #J #W1 #W2 #T1 #T2 #l #m #_ #HT #a #I #V1 #U1 #H destruct
]
qed-.
-lemma lift_inv_bind1: ∀d,e,T2,a,I,V1,U1. ⬆[d,e] ⓑ{a,I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ⬆[d,e] V1 ≡ V2 & ⬆[d+1,e] U1 ≡ U2 &
+lemma lift_inv_bind1: ∀l,m,T2,a,I,V1,U1. ⬆[l,m] ⓑ{a,I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
T2 = ⓑ{a,I} V2. U2.
/2 width=3 by lift_inv_bind1_aux/ qed-.
-fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⬆[d,e] T1 ≡ T2 →
+fact lift_inv_flat1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
- ∃∃V2,U2. ⬆[d,e] V1 ≡ V2 & ⬆[d,e] U1 ≡ U2 &
+ ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
T2 = ⓕ{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
-| #i #d #e #_ #I #V1 #U1 #H destruct
-| #p #d #e #I #V1 #U1 #H destruct
-| #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V1 #U1 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #I #V1 #U1 #H destruct
+| #i #l #m #_ #I #V1 #U1 #H destruct
+| #i #l #m #_ #I #V1 #U1 #H destruct
+| #p #l #m #I #V1 #U1 #H destruct
+| #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V1 #U1 #H destruct
+| #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma lift_inv_flat1: ∀d,e,T2,I,V1,U1. ⬆[d,e] ⓕ{I} V1. U1 ≡ T2 →
- ∃∃V2,U2. ⬆[d,e] V1 ≡ V2 & ⬆[d,e] U1 ≡ U2 &
+lemma lift_inv_flat1: ∀l,m,T2,I,V1,U1. ⬆[l,m] ⓕ{I} V1. U1 ≡ T2 →
+ ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
T2 = ⓕ{I} V2. U2.
/2 width=3 by lift_inv_flat1_aux/ 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 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+fact lift_inv_sort2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
]
qed-.
(* Basic_1: was: lift_gen_sort *)
-lemma lift_inv_sort2: ∀d,e,T1,k. ⬆[d,e] T1 ≡ ⋆k → T1 = ⋆k.
+lemma lift_inv_sort2: ∀l,m,T1,k. ⬆[l,m] T1 ≡ ⋆k → T1 = ⋆k.
/2 width=5 by lift_inv_sort2_aux/ 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)).
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
-| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4 width=1 by monotonic_le_plus_l, or_intror, conj/
-| #p #d #e #i #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+fact lift_inv_lref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀i. T2 = #i →
+ (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #i #H destruct
+| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
+| #j #l #m #Hj #i #Hi destruct <minus_plus_m_m /4 width=1 by monotonic_le_plus_l, or_intror, conj/
+| #p #l #m #i #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
]
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)).
+lemma lift_inv_lref2: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i →
+ (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
/2 width=3 by lift_inv_lref2_aux/ 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 (lt_inv_plus_l … Hdd) -Hdd #Hdd
-elim (lt_refl_false … Hdd)
+lemma lift_inv_lref2_lt: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i → i < l → T1 = #i.
+#l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hli #_ #Hil lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
+elim (lt_inv_plus_l … Hll) -Hll #Hll
+elim (lt_refl_false … Hll)
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 → ⊥.
-#d #e #T1 #i #H elim (lift_inv_lref2 … H) -H *
+lemma lift_inv_lref2_be: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i →
+ l ≤ i → i < l + m → ⊥.
+#l #m #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 (lt_inv_plus_l … Hdd) -Hdd #Hdd
-elim (lt_refl_false … Hdd)
+lemma lift_inv_lref2_ge: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i → l + m ≤ i → T1 = #(i - m).
+#l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
+#Hil #_ #Hli lapply (le_to_lt_to_lt … Hli Hil) -Hli -Hil #Hll
+elim (lt_inv_plus_l … Hll) -Hll #Hll
+elim (lt_refl_false … Hll)
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 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+fact lift_inv_gref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
]
qed-.
-lemma lift_inv_gref2: ∀d,e,T1,p. ⬆[d,e] T1 ≡ §p → T1 = §p.
+lemma lift_inv_gref2: ∀l,m,T1,p. ⬆[l,m] T1 ≡ §p → T1 = §p.
/2 width=5 by lift_inv_gref2_aux/ qed-.
-fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⬆[d,e] T1 ≡ T2 →
+fact lift_inv_bind2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
- ∃∃V1,U1. ⬆[d,e] V1 ≡ V2 & ⬆[d+1,e] U1 ≡ U2 &
+ ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
T1 = ⓑ{a,I} V1. U1.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #a #I #V2 #U2 #H destruct
-| #i #d #e #_ #a #I #V2 #U2 #H destruct
-| #i #d #e #_ #a #I #V2 #U2 #H destruct
-| #p #d #e #a #I #V2 #U2 #H destruct
-| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
-| #J #W1 #W2 #T1 #T2 #d #e #_ #_ #a #I #V2 #U2 #H destruct
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #a #I #V2 #U2 #H destruct
+| #i #l #m #_ #a #I #V2 #U2 #H destruct
+| #i #l #m #_ #a #I #V2 #U2 #H destruct
+| #p #l #m #a #I #V2 #U2 #H destruct
+| #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
+| #J #W1 #W2 #T1 #T2 #l #m #_ #_ #a #I #V2 #U2 #H destruct
]
qed-.
(* Basic_1: was: lift_gen_bind *)
-lemma lift_inv_bind2: ∀d,e,T1,a,I,V2,U2. ⬆[d,e] T1 ≡ ⓑ{a,I} V2. U2 →
- ∃∃V1,U1. ⬆[d,e] V1 ≡ V2 & ⬆[d+1,e] U1 ≡ U2 &
+lemma lift_inv_bind2: ∀l,m,T1,a,I,V2,U2. ⬆[l,m] T1 ≡ ⓑ{a,I} V2. U2 →
+ ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
T1 = ⓑ{a,I} V1. U1.
/2 width=3 by lift_inv_bind2_aux/ qed-.
-fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⬆[d,e] T1 ≡ T2 →
+fact lift_inv_flat2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
- ∃∃V1,U1. ⬆[d,e] V1 ≡ V2 & ⬆[d,e] U1 ≡ U2 &
+ ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
T1 = ⓕ{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
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #p #d #e #I #V2 #U2 #H destruct
-| #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #I #V2 #U2 #H destruct
+| #i #l #m #_ #I #V2 #U2 #H destruct
+| #i #l #m #_ #I #V2 #U2 #H destruct
+| #p #l #m #I #V2 #U2 #H destruct
+| #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V2 #U2 #H destruct
+| #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_1: was: lift_gen_flat *)
-lemma lift_inv_flat2: ∀d,e,T1,I,V2,U2. ⬆[d,e] T1 ≡ ⓕ{I} V2. U2 →
- ∃∃V1,U1. ⬆[d,e] V1 ≡ V2 & ⬆[d,e] U1 ≡ U2 &
+lemma lift_inv_flat2: ∀l,m,T1,I,V2,U2. ⬆[l,m] T1 ≡ ⓕ{I} V2. U2 →
+ ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
T1 = ⓕ{I} V1. U1.
/2 width=3 by lift_inv_flat2_aux/ qed-.
-lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ⬆[d, e] ②{I} V. T ≡ V → ⊥.
-#d #e #J #V elim V -V
+lemma lift_inv_pair_xy_x: ∀l,m,I,V,T. ⬆[l, m] ②{I} V. T ≡ V → ⊥.
+#l #m #J #V elim V -V
[ * #i #T #H
[ lapply (lift_inv_sort2 … H) -H #H destruct
| elim (lift_inv_lref2 … H) -H * #_ #H destruct
qed-.
(* Basic_1: was: thead_x_lift_y_y *)
-lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ⬆[d, e] ②{I} V. T ≡ T → ⊥.
+lemma lift_inv_pair_xy_y: ∀I,T,V,l,m. ⬆[l, m] ②{I} V. T ≡ T → ⊥.
#J #T elim T -T
-[ * #i #V #d #e #H
+[ * #i #V #l #m #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
]
-| * [ #a ] #I #W2 #U2 #_ #IHU2 #V #d #e #H
+| * [ #a ] #I #W2 #U2 #_ #IHU2 #V #l #m #H
[ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
| elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
]
(* Basic forward lemmas *****************************************************)
-lemma lift_fwd_pair1: ∀I,T2,V1,U1,d,e. ⬆[d,e] ②{I}V1.U1 ≡ T2 →
- ∃∃V2,U2. ⬆[d,e] V1 ≡ V2 & T2 = ②{I}V2.U2.
-* [ #a ] #I #T2 #V1 #U1 #d #e #H
+lemma lift_fwd_pair1: ∀I,T2,V1,U1,l,m. ⬆[l,m] ②{I}V1.U1 ≡ T2 →
+ ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & T2 = ②{I}V2.U2.
+* [ #a ] #I #T2 #V1 #U1 #l #m #H
[ elim (lift_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
| elim (lift_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lift_fwd_pair2: ∀I,T1,V2,U2,d,e. ⬆[d,e] T1 ≡ ②{I}V2.U2 →
- ∃∃V1,U1. ⬆[d,e] V1 ≡ V2 & T1 = ②{I}V1.U1.
-* [ #a ] #I #T1 #V2 #U2 #d #e #H
+lemma lift_fwd_pair2: ∀I,T1,V2,U2,l,m. ⬆[l,m] T1 ≡ ②{I}V2.U2 →
+ ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & T1 = ②{I}V1.U1.
+* [ #a ] #I #T1 #V2 #U2 #l #m #H
[ elim (lift_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
| elim (lift_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lift_fwd_tw: ∀d,e,T1,T2. ⬆[d, e] T1 ≡ T2 → ♯{T1} = ♯{T2}.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize //
+lemma lift_fwd_tw: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ♯{T1} = ♯{T2}.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 normalize //
qed-.
-lemma lift_simple_dx: ∀d,e,T1,T2. ⬆[d, e] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
-#a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
+lemma lift_simple_dx: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
+#a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
elim (simple_inv_bind … H)
qed-.
-lemma lift_simple_sn: ∀d,e,T1,T2. ⬆[d, e] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
-#a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
+lemma lift_simple_sn: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
+#a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
elim (simple_inv_bind … H)
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 by lift_lref_ge, le_plus_to_minus_r, le_plus_b/
+lemma lift_lref_ge_minus: ∀l,m,i. l + m ≤ i → ⬆[l, m] #(i - m) ≡ #i.
+#l #m #i #H >(plus_minus_m_m i m) in ⊢ (? ? ? ? %); /3 width=2 by lift_lref_ge, le_plus_to_minus_r, le_plus_b/
qed.
-lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⬆[d, e] #j ≡ #i.
+lemma lift_lref_ge_minus_eq: ∀l,m,i,j. l + m ≤ i → j = i - m → ⬆[l, m] #j ≡ #i.
/2 width=1/ qed-.
(* Basic_1: was: lift_r *)
-lemma lift_refl: ∀T,d. ⬆[d, 0] T ≡ T.
+lemma lift_refl: ∀T,l. ⬆[l, 0] T ≡ T.
#T elim T -T
-[ * #i // #d elim (lt_or_ge i d) /2 width=1 by lift_lref_lt, lift_lref_ge/
+[ * #i // #l elim (lt_or_ge i l) /2 width=1 by lift_lref_lt, lift_lref_ge/
| * /2 width=1 by lift_bind, lift_flat/
]
qed.
-lemma lift_total: ∀T1,d,e. ∃T2. ⬆[d,e] T1 ≡ T2.
+lemma lift_total: ∀T1,l,m. ∃T2. ⬆[l,m] T1 ≡ T2.
#T1 elim T1 -T1
-[ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2 by lift_lref_lt, lift_lref_ge, ex_intro/
-| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #d #e
- elim (IHV1 d e) -IHV1 #V2 #HV12
- [ elim (IHT1 (d+1) e) -IHT1 /3 width=2 by lift_bind, ex_intro/
- | elim (IHT1 d e) -IHT1 /3 width=2 by lift_flat, ex_intro/
+[ * #i /2 width=2/ #l #m elim (lt_or_ge i l) /3 width=2 by lift_lref_lt, lift_lref_ge, ex_intro/
+| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #l #m
+ elim (IHV1 l m) -IHV1 #V2 #HV12
+ [ elim (IHT1 (l+1) m) -IHT1 /3 width=2 by lift_bind, ex_intro/
+ | elim (IHT1 l m) -IHT1 /3 width=2 by lift_flat, ex_intro/
]
]
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 -d1 -e2 -T1 -T2
+lemma lift_split: ∀l1,m2,T1,T2. ⬆[l1, m2] T1 ≡ T2 →
+ ∀l2,m1. l1 ≤ l2 → l2 ≤ l1 + m1 → m1 ≤ m2 →
+ ∃∃T. ⬆[l1, m1] T1 ≡ T & ⬆[l2, m2 - m1] T ≡ T2.
+#l1 #m2 #T1 #T2 #H elim H -l1 -m2 -T1 -T2
[ /3 width=3/
-| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
- lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4 width=3 by lift_lref_lt, ex2_intro/
-| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
- lapply (transitive_le … (i+e1) Hd21 ?) /2 width=1 by monotonic_le_plus_l/ -Hd21 #Hd21
- >(plus_minus_m_m e2 e1 ?) /3 width=3 by lift_lref_ge, ex2_intro/
+| #i #l1 #m2 #Hil1 #l2 #m1 #Hl12 #_ #_
+ lapply (lt_to_le_to_lt … Hil1 Hl12) -Hl12 #Hil2 /4 width=3 by lift_lref_lt, ex2_intro/
+| #i #l1 #m2 #Hil1 #l2 #m1 #_ #Hl21 #Hm12
+ lapply (transitive_le … (i+m1) Hl21 ?) /2 width=1 by monotonic_le_plus_l/ -Hl21 #Hl21
+ >(plus_minus_m_m m2 m1 ?) /3 width=3 by lift_lref_ge, ex2_intro/
| /3 width=3/
-| #a #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 by lift_bind, le_S_S, ex2_intro/
-| #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 by lift_flat, ex2_intro/
+| #a #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
+ elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
+ elim (IHT (l2+1) … ? ? Hm12) /3 width=5 by lift_bind, le_S_S, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
+ elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
+ elim (IHT l2 … ? ? Hm12) /3 width=5 by lift_flat, ex2_intro/
]
qed.
(* Basic_1: was only: dnf_dec2 dnf_dec *)
-lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ⬆[d,e] T1 ≡ T2).
+lemma is_lift_dec: ∀T2,l,m. Decidable (∃T1. ⬆[l,m] T1 ≡ T2).
#T1 elim T1 -T1
-[ * [1,3: /3 width=2 by lift_sort, lift_gref, ex_intro, or_introl/ ] #i #d #e
- elim (lt_or_ge i d) #Hdi
+[ * [1,3: /3 width=2 by lift_sort, lift_gref, ex_intro, or_introl/ ] #i #l #m
+ elim (lt_or_ge i l) #Hli
[ /4 width=3 by lift_lref_lt, ex_intro, or_introl/
- | elim (lt_or_ge i (d + e)) #Hide
- [ @or_intror * #T1 #H elim (lift_inv_lref2_be … H Hdi Hide)
- | -Hdi /4 width=2 by lift_lref_ge_minus, ex_intro, or_introl/
+ | elim (lt_or_ge i (l + m)) #Hilm
+ [ @or_intror * #T1 #H elim (lift_inv_lref2_be … H Hli Hilm)
+ | -Hli /4 width=2 by lift_lref_ge_minus, ex_intro, or_introl/
]
]
-| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #d #e
- [ elim (IHV2 d e) -IHV2
- [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2
+| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #l #m
+ [ elim (IHV2 l m) -IHV2
+ [ * #V1 #HV12 elim (IHT2 (l+1) m) -IHT2
[ * #T1 #HT12 @or_introl /3 width=2 by lift_bind, ex_intro/
| -V1 #HT2 @or_intror * #X #H
elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
| -IHT2 #HV2 @or_intror * #X #H
elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
]
- | elim (IHV2 d e) -IHV2
- [ * #V1 #HV12 elim (IHT2 d e) -IHT2
+ | elim (IHV2 l m) -IHV2
+ [ * #V1 #HV12 elim (IHT2 l m) -IHT2
[ * #T1 #HT12 /4 width=2/
| -V1 #HT2 @or_intror * #X #H
elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
projects / helm.git / blobdiff