(* *)
(**************************************************************************)
+include "basic_2/notation/relations/rlift_4.ma".
include "basic_2/grammar/term_weight.ma".
include "basic_2/grammar/term_simple.ma".
(* Basic_1: includes:
lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
*)
-inductive lift: nat → nat → relation term ≝
+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))
(* Basic inversion lemmas ***************************************************)
-fact lift_inv_refl_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
+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/
-qed.
+qed-.
-lemma lift_inv_refl_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2.
-/2 width=4/ qed-.
+lemma lift_inv_O2: ∀d,T1,T2. ⇧[d, 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 //
| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
]
-qed.
+qed-.
lemma lift_inv_sort1: ∀d,e,T2,k. ⇧[d,e] ⋆k ≡ T2 → T2 = ⋆k.
-/2 width=5/ qed-.
+/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)).
| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
]
-qed.
+qed-.
lemma lift_inv_lref1: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 →
(i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-/2 width=3/ qed-.
+/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 * //
| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
]
-qed.
+qed-.
lemma lift_inv_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p.
-/2 width=5/ qed-.
+/2 width=5 by lift_inv_gref1_aux/ qed-.
fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V1 #U1 #H destruct /2 width=5/
| #J #W1 #W2 #T1 #T2 #d #e #_ #HT #a #I #V1 #U1 #H destruct
]
-qed.
+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 &
T2 = ⓑ{a,I} V2. U2.
-/2 width=3/ qed-.
+/2 width=3 by lift_inv_bind1_aux/ qed-.
fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
| #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/
]
-qed.
+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 &
T2 = ⓕ{I} V2. U2.
-/2 width=3/ qed-.
+/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 //
| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
]
-qed.
+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-.
+/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)).
| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
]
-qed.
+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 width=3/ qed-.
+/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.
| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
]
-qed.
+qed-.
lemma lift_inv_gref2: ∀d,e,T1,p. ⇧[d,e] T1 ≡ §p → T1 = §p.
-/2 width=5/ qed-.
+/2 width=5 by lift_inv_gref2_aux/ qed-.
fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V2 #U2 #H destruct /2 width=5/
| #J #W1 #W2 #T1 #T2 #d #e #_ #_ #a #I #V2 #U2 #H destruct
]
-qed.
+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 &
T1 = ⓑ{a,I} V1. U1.
-/2 width=3/ qed-.
+/2 width=3 by lift_inv_bind2_aux/ qed-.
fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
| #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/
]
-qed.
+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 &
T1 = ⓕ{I} V1. U1.
-/2 width=3/ qed-.
+/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
(* 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
+[ elim (lift_inv_bind1 … H) -H /2 width=4/
+| elim (lift_inv_flat1 … H) -H /2 width=4/
+]
+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
+[ elim (lift_inv_bind2 … H) -H /2 width=4/
+| elim (lift_inv_flat2 … H) -H /2 width=4/
+]
+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 //
qed-.