X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Fsubstitution%2Flift.ma;h=0d5f1dfe4e1825b67020695128d70a7fff6c83ad;hb=658c000ee2ea2da04cf29efc0acdaf16364fbf5e;hp=1d02e6b8bcc46691df71bb6f3f3a4e4fa3c9a55b;hpb=1994fe8e6355243652770f53a02db5fdf26915f0;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 1d02e6b8b..0d5f1dfe4 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 "ground_2/ynat/ynat_plus.ma".
 include "basic_2/notation/relations/rlift_4.ma".
 include "basic_2/grammar/term_weight.ma".
 include "basic_2/grammar/term_simple.ma".
@@ -21,13 +22,13 @@ include "basic_2/grammar/term_simple.ma".
 (* Basic_1: includes:
             lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
 *)
-inductive lift: relation4 nat nat term term ≝
+inductive lift: relation4 ynat nat term term ≝
 | 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_lref_lt: ∀i,l,m. yinj i < l → lift l m (#i) (#i)
+| lift_lref_ge: ∀i,l,m. l ≤ yinj 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 V1 V2 → lift (⫯l) 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 →
@@ -45,7 +46,7 @@ qed-.
 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: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
+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
@@ -53,38 +54,36 @@ fact lift_inv_sort1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T1 = ⋆k 
 ]
 qed-.
 
-lemma lift_inv_sort1: ∀l,m,T2,k. ⬆[l,m] ⋆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: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀i. T1 = #i →
+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/
+| #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: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 →
+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: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 → i < l → T2 = #i.
+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)
+#Hli #_ #Hil elim (ylt_yle_false … Hli) -Hli //
 qed-.
 
-lemma lift_inv_lref1_ge: ∀l,m,T2,i. ⬆[l,m] #i ≡ T2 → l ≤ i → T2 = #(i + m).
+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)
+#Hil #_ #Hli elim (ylt_yle_false … Hli) -Hli //
 qed-.
 
-fact lift_inv_gref1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
+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
@@ -92,13 +91,13 @@ fact lift_inv_gref1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T1 = §p →
 ]
 qed-.
 
-lemma lift_inv_gref1: ∀l,m,T2,p. ⬆[l,m] §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: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
-                         ∀a,I,V1,U1. T1 = ⓑ{a,I} V1.U1 →
-                         ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
-                                  T2 = ⓑ{a,I} V2. U2.
+fact lift_inv_bind1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
+                         ∀a,I,V1,U1. T1 = ⓑ{a,I}V1.U1 →
+                         ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & ⬆[⫯l, m] U1 ≡ U2 &
+                                  T2 = ⓑ{a,I}V2.U2.
 #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
@@ -109,15 +108,15 @@ fact lift_inv_bind1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
 ]
 qed-.
 
-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.
+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, m] U1 ≡ U2 &
+                               T2 = ⓑ{a,I}V2.U2.
 /2 width=3 by lift_inv_bind1_aux/ qed-.
 
-fact lift_inv_flat1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
-                         ∀I,V1,U1. T1 = ⓕ{I} V1.U1 →
-                         ∃∃V2,U2. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
-                                  T2 = ⓕ{I} V2. U2.
+fact lift_inv_flat1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
+                         ∀I,V1,U1. T1 = ⓕ{I}V1.U1 →
+                         ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & ⬆[l, m] U1 ≡ U2 &
+                                  T2 = ⓕ{I}V2.U2.
 #l #m #T1 #T2 * -l -m -T1 -T2
 [ #k #l #m #I #V1 #U1 #H destruct
 | #i #l #m #_ #I #V1 #U1 #H destruct
@@ -128,12 +127,12 @@ fact lift_inv_flat1_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
 ]
 qed-.
 
-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.
+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: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
+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
@@ -142,15 +141,15 @@ fact lift_inv_sort2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀k. T2 = ⋆k 
 qed-.
 
 (* Basic_1: was: lift_gen_sort *)
-lemma lift_inv_sort2: ∀l,m,T1,k. ⬆[l,m] 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: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀i. T2 = #i →
+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/
+| #j #l #m #Hj #i #Hi destruct <minus_plus_m_m /4 width=1 by monotonic_yle_plus_dx, 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
@@ -158,36 +157,33 @@ fact lift_inv_lref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀i. T2 = #i →
 qed-.
 
 (* Basic_1: was: lift_gen_lref *)
-lemma lift_inv_lref2: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i →
+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: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i → i < l → T1 = #i.
+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)
+#H #_ #Hil lapply (yle_fwd_plus_sn1 … H) -H
+#Hli elim (ylt_yle_false … Hli) -Hli //
 qed-.
 
 (* Basic_1: was: lift_gen_lref_false *)
-lemma lift_inv_lref2_be: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i →
+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)
+elim (ylt_yle_false … H2) -H2 //
 qed-.
 
 (* Basic_1: was: lift_gen_lref_ge *)
-lemma lift_inv_lref2_ge: ∀l,m,T1,i. ⬆[l,m] T1 ≡ #i → l + m ≤ i → T1 = #(i - m).
+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)
+#Hil #_ #H lapply (yle_fwd_plus_sn1 … H) -H
+#Hli elim (ylt_yle_false … Hli) -Hli //
 qed-.
 
-fact lift_inv_gref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
+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
@@ -195,13 +191,13 @@ fact lift_inv_gref2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 → ∀p. T2 = §p →
 ]
 qed-.
 
-lemma lift_inv_gref2: ∀l,m,T1,p. ⬆[l,m] 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: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
-                         ∀a,I,V2,U2. T2 = ⓑ{a,I} V2.U2 →
-                         ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l+1,m] U1 ≡ U2 &
-                                  T1 = ⓑ{a,I} V1. U1.
+fact lift_inv_bind2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
+                         ∀a,I,V2,U2. T2 = ⓑ{a,I}V2.U2 →
+                         ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & ⬆[⫯l, m] U1 ≡ U2 &
+                                  T1 = ⓑ{a,I}V1.U1.
 #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
@@ -213,15 +209,15 @@ fact lift_inv_bind2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
 qed-.
 
 (* Basic_1: was: lift_gen_bind *)
-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.
+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, m] U1 ≡ U2 &
+                               T1 = ⓑ{a,I}V1.U1.
 /2 width=3 by lift_inv_bind2_aux/ qed-.
 
-fact lift_inv_flat2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
-                         ∀I,V2,U2. T2 = ⓕ{I} V2.U2 →
-                         ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & ⬆[l,m] U1 ≡ U2 &
-                                  T1 = ⓕ{I} V1. U1.
+fact lift_inv_flat2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
+                         ∀I,V2,U2. T2 = ⓕ{I}V2.U2 →
+                         ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & ⬆[l, m] U1 ≡ U2 &
+                                  T1 = ⓕ{I}V1.U1.
 #l #m #T1 #T2 * -l -m -T1 -T2
 [ #k #l #m #I #V2 #U2 #H destruct
 | #i #l #m #_ #I #V2 #U2 #H destruct
@@ -233,12 +229,24 @@ fact lift_inv_flat2_aux: ∀l,m,T1,T2. ⬆[l,m] T1 ≡ T2 →
 qed-.
 
 (* Basic_1: was: lift_gen_flat *)
-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.
+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: ∀l,m,I,V,T. ⬆[l, m] ②{I} V. T ≡ V → ⊥.
+lemma lift_inv_Y1: ∀T1,T2,m. ⬆[∞, m] T1 ≡ T2 → T1 = T2.
+#T1 elim T1 -T1 *
+[ #k #X #m #H lapply (lift_inv_sort1 … H) -H //
+| #i #X #m #H lapply (lift_inv_lref1_lt … H ?) -H //
+| #p #X #m #H lapply (lift_inv_gref1 … H) -H //
+| #a #I #V1 #T1 #IHV1 #IHT1 #X #m #H elim (lift_inv_bind1 … H) -H
+  #V2 #T2 #HV12 #HT12 #H destruct /3 width=2 by eq_f2/
+| #I #V1 #T1 #IHV1 #IHT1 #X #m #H elim (lift_inv_flat1 … H) -H
+  #V2 #T2 #HV12 #HT12 #H destruct /3 width=2 by eq_f2/
+]
+qed-.
+
+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
@@ -253,7 +261,7 @@ lemma lift_inv_pair_xy_x: ∀l,m,I,V,T. ⬆[l, m] ②{I} V. T ≡ V → ⊥.
 qed-.
 
 (* Basic_1: was: thead_x_lift_y_y *)
-lemma lift_inv_pair_xy_y: ∀I,T,V,l,m. ⬆[l, m] 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 #l #m #H
   [ lapply (lift_inv_sort2 … H) -H #H destruct
@@ -269,16 +277,16 @@ qed-.
 
 (* Basic forward lemmas *****************************************************)
 
-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.
+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,l,m. ⬆[l,m] T1 ≡ ②{I}V2.U2 →
-                      ∃∃V1,U1. ⬆[l,m] V1 ≡ V2 & T1 = ②{I}V1.U1.
+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/
@@ -304,28 +312,35 @@ qed-.
 (* Basic properties *********************************************************)
 
 (* Basic_1: was: lift_lref_gt *)
-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/
+lemma lift_lref_ge_minus: ∀l,m,i. l + yinj m ≤ yinj i → ⬆[l, m] #(i - m) ≡ #i.
+#l #m #i #H >(plus_minus_m_m i m) in ⊢ (? ? ? ? %);
+elim (yle_inv_plus_inj2 … H) -H #Hlim #H
+lapply (yle_inv_inj … H) -H /2 width=1 by lift_lref_ge/
 qed.
 
-lemma lift_lref_ge_minus_eq: ∀l,m,i,j. l + m ≤ i → j = i - m → ⬆[l, m] #j ≡ #i.
+lemma lift_lref_ge_minus_eq: ∀l,m,i,j. l + yinj m ≤ yinj i → j = i - m → ⬆[l, m] #j ≡ #i.
 /2 width=1 by lift_lref_ge_minus/ qed-.
 
 (* Basic_1: was: lift_r *)
 lemma lift_refl: ∀T,l. ⬆[l, 0] T ≡ T.
 #T elim T -T
-[ * #i // #l elim (lt_or_ge i l) /2 width=1 by lift_lref_lt, lift_lref_ge/
+[ * #i // #l elim (ylt_split 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,l,m. ∃T2. ⬆[l,m] T1 ≡ T2.
+(* Basic_2b: first lemma *)
+lemma lift_Y1: ∀T,m. ⬆[∞, m] T ≡ T.
+#T elim T -T * /2 width=1 by lift_lref_lt, lift_bind, lift_flat/
+qed.
+
+lemma lift_total: ∀T1,l,m. ∃T2. ⬆[l, m] T1 ≡ T2.
 #T1 elim T1 -T1
 [ * #i /2 width=2 by lift_sort, lift_gref, ex_intro/
-  #l #m elim (lt_or_ge i l) /3 width=2 by lift_lref_lt, lift_lref_ge, ex_intro/
+  #l #m elim (ylt_split 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_bind, ex_intro/
   | elim (IHT1 l m) -IHT1 /3 width=2 by lift_flat, ex_intro/
   ]
 ]
@@ -333,19 +348,19 @@ qed.
 
 (* Basic_1: was: lift_free (right to left) *)
 lemma lift_split: ∀l1,m2,T1,T2. ⬆[l1, m2] T1 ≡ T2 →
-                  ∀l2,m1. l1 ≤ l2 → l2 ≤ l1 + m1 → m1 ≤ m2 →
+                  ∀l2,m1. l1 ≤ l2 → l2 ≤ l1 + yinj 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 by lift_sort, 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/
+  lapply (ylt_yle_trans … Hl12 Hil1) -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
+  lapply (yle_trans … Hl21 (i+m1) ?) /2 width=1 by monotonic_yle_plus_dx/ -Hl21 #Hl21
   >(plus_minus_m_m m2 m1 ?) /3 width=3 by lift_lref_ge, ex2_intro/
 | /3 width=3 by lift_gref, 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/
+  elim (IHT (⫯l2) … ? ? Hm12) /3 width=5 by lift_bind, yle_succ, 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/
@@ -353,19 +368,19 @@ lemma lift_split: ∀l1,m2,T1,T2. ⬆[l1, m2] T1 ≡ T2 →
 qed.
 
 (* Basic_1: was only: dnf_dec2 dnf_dec *)
-lemma is_lift_dec: ∀T2,l,m. Decidable (∃T1. ⬆[l,m] 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 #l #m
-  elim (lt_or_ge i l) #Hli
+  elim (ylt_split i l) #Hli
   [ /4 width=3 by lift_lref_lt, ex_intro, or_introl/
-  | elim (lt_or_ge i (l + m)) #Hilm
+  | elim (ylt_split 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 #l #m
   [ elim (IHV2 l m) -IHV2
-    [ * #V1 #HV12 elim (IHT2 (l+1) m) -IHT2
+    [ * #V1 #HV12 elim (IHT2 (⫯l) 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/