X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fsubstitution%2Flift.ma;h=d6537046caf4051af2f50853896f467d95f232ed;hb=6f1f9e20aa2775d41bba64289fc903e6612baaf3;hp=84babbdc8fa8c3fb48b4643952221f93e55b9323;hpb=08cb57944c0df08611d4f35d286e46c0d13e4813;p=helm.git

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma
index 84babbdc8..d6537046c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma
@@ -12,6 +12,7 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "basic_2/notation/relations/rlift_4.ma".
 include "basic_2/grammar/term_weight.ma".
 include "basic_2/grammar/term_simple.ma".
 
@@ -20,7 +21,7 @@ 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))
@@ -35,95 +36,87 @@ inductive lift: nat → nat → relation term ≝
 
 interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
 
-definition t_liftable: relation term → Prop ≝
-                       λR. ∀T1,T2. R T1 T2 → ∀U1,d,e. ⇧[d, e] T1 ≡ U1 →
-                       ∀U2. ⇧[d, e] T2 ≡ U2 → R U1 U2.
-
-definition t_deliftable_sn: relation term → Prop ≝
-                            λR. ∀U1,U2. R U1 U2 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
-                            ∃∃T2. ⇧[d, e] T2 ≡ U2 & R T1 T2.
-
 (* Basic inversion lemmas ***************************************************)
 
-fact lift_inv_refl_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.
+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/
+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.
+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
 ]
-qed.
+qed-.
 
-lemma lift_inv_sort1: ∀d,e,T2,k. ⇧[d,e] ⋆k ≡ T2 → T2 = ⋆k.
-/2 width=5/ qed-.
+lemma lift_inv_sort1: ∀d,e,T2,k. ⬆[d,e] ⋆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 →
+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/
-| #j #d #e #Hj #i #Hi destruct /3 width=1/
+| #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
 ]
-qed.
+qed-.
 
-lemma lift_inv_lref1: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 →
+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.
+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-.
 
-lemma lift_inv_lref1_ge: ∀d,e,T2,i. ⇧[d,e] #i ≡ T2 → d ≤ i → T2 = #(i + e).
+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-.
 
-fact lift_inv_gref1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
+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
 ]
-qed.
+qed-.
 
-lemma lift_inv_gref1: ∀d,e,T2,p. ⇧[d,e] §p ≡ T2 → T2 = §p.
-/2 width=5/ qed-.
+lemma lift_inv_gref1: ∀d,e,T2,p. ⬆[d,e] §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: ∀d,e,T1,T2. ⬆[d,e] T1 ≡ T2 →
                          ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
-                         ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
+                         ∃∃V2,U2. ⬆[d,e] V1 ≡ V2 & ⬆[d+1,e] 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/
+| #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
 ]
-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 &
+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 →
+fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⬆[d,e] T1 ≡ T2 →
                          ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
-                         ∃∃V2,U2. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
+                         ∃∃V2,U2. ⬆[d,e] V1 ≡ V2 & ⬆[d,e] U1 ≡ U2 &
                                   T2 = ⓕ{I} V2. U2.
 #d #e #T1 #T2 * -d -e -T1 -T2
 [ #k #d #e #I #V1 #U1 #H destruct
@@ -131,46 +124,46 @@ fact lift_inv_flat1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
 | #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/
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
 ]
-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 &
+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.
+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
 ]
-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-.
+lemma lift_inv_sort2: ∀d,e,T1,k. ⬆[d,e] 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 →
+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/
-| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4 width=1/
+| #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
 ]
-qed.
+qed-.
 
 (* Basic_1: was: lift_gen_lref *)
-lemma lift_inv_lref2: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
+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.
+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
@@ -178,7 +171,7 @@ elim (lt_refl_false … Hdd)
 qed-.
 
 (* Basic_1: was: lift_gen_lref_false *)
-lemma lift_inv_lref2_be: ∀d,e,T1,i. ⇧[d,e] T1 ≡ #i →
+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 *
 [ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
@@ -187,47 +180,47 @@ 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).
+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)
 qed-.
 
-fact lift_inv_gref2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
+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
 ]
-qed.
+qed-.
 
-lemma lift_inv_gref2: ∀d,e,T1,p. ⇧[d,e] T1 ≡ §p → T1 = §p.
-/2 width=5/ qed-.
+lemma lift_inv_gref2: ∀d,e,T1,p. ⬆[d,e] 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: ∀d,e,T1,T2. ⬆[d,e] T1 ≡ T2 →
                          ∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
-                         ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d+1,e] U1 ≡ U2 &
+                         ∃∃V1,U1. ⬆[d,e] V1 ≡ V2 & ⬆[d+1,e] 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/
+| #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
 ]
-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 &
+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 →
+fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⬆[d,e] T1 ≡ T2 →
                          ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
-                         ∃∃V1,U1. ⇧[d,e] V1 ≡ V2 & ⇧[d,e] U1 ≡ U2 &
+                         ∃∃V1,U1. ⬆[d,e] V1 ≡ V2 & ⬆[d,e] U1 ≡ U2 &
                                   T1 = ⓕ{I} V1. U1.
 #d #e #T1 #T2 * -d -e -T1 -T2
 [ #k #d #e #I #V2 #U2 #H destruct
@@ -235,17 +228,17 @@ fact lift_inv_flat2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
 | #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/
+| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
 ]
-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 &
+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 → ⊥.
+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
 [ * #i #T #H
   [ lapply (lift_inv_sort2 … H) -H #H destruct
@@ -253,14 +246,14 @@ lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ⇧[d, e] ②{I} V. T ≡ V → ⊥.
   | lapply (lift_inv_gref2 … H) -H #H destruct
   ]
 | * [ #a ] #I #W2 #U2 #IHW2 #_ #T #H
-  [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
-  | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2/
+  [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
+  | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
   ]
 ]
 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,d,e. ⬆[d, e] ②{I} V. T ≡ T → ⊥.
 #J #T elim T -T
 [ * #i #V #d #e #H
   [ lapply (lift_inv_sort2 … H) -H #H destruct
@@ -268,25 +261,41 @@ lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ⇧[d, e] ②{I} V. T ≡ T → ⊥.
   | lapply (lift_inv_gref2 … H) -H #H destruct
   ]
 | * [ #a ] #I #W2 #U2 #_ #IHU2 #V #d #e #H
-  [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
-  | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4/
+  [ 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/
   ]
 ]
 qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-lemma tw_lift: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → ♯{T1} = ♯{T2}.
+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 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
+[ 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 //
 qed-.
 
-lemma lift_simple_dx: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+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
 elim (simple_inv_bind … H)
 qed-.
 
-lemma lift_simple_sn: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+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
 elim (simple_inv_bind … H)
@@ -295,107 +304,87 @@ 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 ⊢ (? ? ? ? %); /2 width=2/ /3 width=2/
+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/
 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: ∀d,e,i,j. d + e ≤ i → j = i - e → ⬆[d, e] #j ≡ #i.
 /2 width=1/ qed-.
 
 (* Basic_1: was: lift_r *)
-lemma lift_refl: ∀T,d. ⇧[d, 0] T ≡ T.
+lemma lift_refl: ∀T,d. ⬆[d, 0] T ≡ T.
 #T elim T -T
-[ * #i // #d elim (lt_or_ge i d) /2 width=1/
-| * /2 width=1/
+[ * #i // #d elim (lt_or_ge i d) /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,d,e. ∃T2. ⬆[d,e] T1 ≡ T2.
 #T1 elim T1 -T1
-[ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/
+[ * #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/
-  | elim (IHT1 d e) -IHT1 /3 width=2/
+  [ 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/
   ]
 ]
 qed.
 
 (* Basic_1: was: lift_free (right to left) *)
-lemma lift_split: ∀d1,e2,T1,T2. ⇧[d1, e2] T1 ≡ T2 →
+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.
+                  ∃∃T. ⬆[d1, e1] T1 ≡ T & ⬆[d2, e2 - e1] T ≡ T2.
 #d1 #e2 #T1 #T2 #H elim H -d1 -e2 -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/
+  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/ -Hd21 #Hd21
-  >(plus_minus_m_m e2 e1 ?) // /3 width=3/
+  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/
 | /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) /2 width=1/ /3 width=5/
+  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/
+  elim (IHT d2 … ? ? He12) /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,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/
+[ * [1,3: /3 width=2 by lift_sort, lift_gref, ex_intro, or_introl/ ] #i #d #e
+  elim (lt_or_ge i d) #Hdi
+  [ /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/
     ]
   ]
 | * [ #a ] #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 width=2/
+      [ * #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/
+        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/
+      elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
     ]
   | 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/
+        elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
       ]
     | -IHT2 #HV2 @or_intror * #X #H
-      elim (lift_inv_flat2 … H) -H /3 width=2/
+      elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
     ]
   ]
 ]
 qed.
 
-lemma t_liftable_TC: ∀R. t_liftable R → t_liftable (TC … R).
-#R #HR #T1 #T2 #H elim H -T2
-[ /3 width=7/
-| #T #T2 #_ #HT2 #IHT1 #U1 #d #e #HTU1 #U2 #HTU2
-  elim (lift_total T d e) /3 width=9/
-]
-qed.
-
-lemma t_deliftable_sn_TC: ∀R. t_deliftable_sn R → t_deliftable_sn (TC … R).
-#R #HR #U1 #U2 #H elim H -U2
-[ #U2 #HU12 #T1 #d #e #HTU1
-  elim (HR … HU12 … HTU1) -U1 /3 width=3/
-| #U #U2 #_ #HU2 #IHU1 #T1 #d #e #HTU1
-  elim (IHU1 … HTU1) -U1 #T #HTU #HT1
-  elim (HR … HU2 … HTU) -U /3 width=5/
-]
-qed-.
-
 (* Basic_1: removed theorems 7:
             lift_head lift_gen_head
             lift_weight_map lift_weight lift_weight_add lift_weight_add_O