component "substitution" updated to new syntax ...

[helm.git] / matita / matita / contribs / lambda_delta / Basic_2 / substitution / lift.ma

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma
index 7d0f43c8188b42d29f392aa4775808fe440f37a3..f58c1906e449b098da320dad10d28ec99bef31e6 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/lift.ma
@@ -37,14 +37,14 @@ interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
 (* Basic inversion lemmas ***************************************************)
 
 fact lift_inv_refl_aux: ∀d,e,T1,T2. ↑[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
-#d #e #T1 #T2 #H elim H -H d e T1 T2 /3/
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 // /3 width=1/
 qed.
 
 lemma lift_inv_refl: ∀d,T1,T2. ↑[d, 0] T1 ≡ T2 → T1 = T2.
-/2/ qed-.
+/2 width=4/ 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 //
+#d #e #T1 #T2 * -d -e -T1 -T2 //
 [ #i #d #e #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
@@ -56,10 +56,10 @@ lemma lift_inv_sort1: ∀d,e,T2,k. ↑[d,e] ⋆k ≡ T2 → T2 = ⋆k.
 
 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
+#d #e #T1 #T2 * -d -e -T1 -T2
 [ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3/
-| #j #d #e #Hj #i #Hi destruct /3/
+| #j #d #e #Hj #i #Hi destruct /3 width=1/
+| #j #d #e #Hj #i #Hi destruct /3 width=1/
 | #p #d #e #i #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
@@ -68,22 +68,22 @@ qed.
 
 lemma lift_inv_lref1: ∀d,e,T2,i. ↑[d,e] #i ≡ T2 →
                       (i < d ∧ T2 = #i) ∨ (d ≤ i ∧ T2 = #(i + e)).
-/2/ qed-.
+/2 width=3/ 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
+#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).
 #d #e #T2 #i #H elim (lift_inv_lref1 … H) -H * //
-#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi Hid #Hdd
+#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.
-#d #e #T1 #T2 * -d e T1 T2 //
+#d #e #T1 #T2 * -d -e -T1 -T2 //
 [ #i #d #e #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
@@ -97,7 +97,7 @@ fact lift_inv_bind1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
                          ∀I,V1,U1. T1 = 𝕓{I} V1.U1 →
                          ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
                                   T2 = 𝕓{I} V2. U2.
-#d #e #T1 #T2 * -d e T1 T2
+#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
@@ -110,13 +110,13 @@ qed.
 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 width=3/ qed-.
 
 fact lift_inv_flat1_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
                          ∀I,V1,U1. T1 = 𝕗{I} V1.U1 →
                          ∃∃V2,U2. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
                                   T2 = 𝕗{I} V2. U2.
-#d #e #T1 #T2 * -d e T1 T2
+#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
@@ -129,10 +129,10 @@ qed.
 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 width=3/ 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 //
+#d #e #T1 #T2 * -d -e -T1 -T2 //
 [ #i #d #e #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
@@ -145,10 +145,10 @@ lemma lift_inv_sort2: ∀d,e,T1,k. ↑[d,e] T1 ≡ ⋆k → T1 = ⋆k.
 
 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
+#d #e #T1 #T2 * -d -e -T1 -T2
 [ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3/
-| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4/
+| #j #d #e #Hj #i #Hi destruct /3 width=1/
+| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4 width=1/
 | #p #d #e #i #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
@@ -158,12 +158,12 @@ 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/ qed-.
+/2 width=3/ 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
+#Hdi #_ #Hid lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
 elim (plus_lt_false … Hdd)
 qed-.
 
@@ -172,19 +172,19 @@ 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
+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
+#Hid #_ #Hdi lapply (le_to_lt_to_lt … Hdi Hid) -Hdi -Hid #Hdd
 elim (plus_lt_false … Hdd)
 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 //
+#d #e #T1 #T2 * -d -e -T1 -T2 //
 [ #i #d #e #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
 | #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
@@ -198,7 +198,7 @@ fact lift_inv_bind2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
                          ∀I,V2,U2. T2 = 𝕓{I} V2.U2 →
                          ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d+1,e] U1 ≡ U2 &
                                   T1 = 𝕓{I} V1. U1.
-#d #e #T1 #T2 * -d e T1 T2
+#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
@@ -212,19 +212,19 @@ qed.
 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 width=3/ qed-.
 
 fact lift_inv_flat2_aux: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 →
                          ∀I,V2,U2. T2 = 𝕗{I} V2.U2 →
                          ∃∃V1,U1. ↑[d,e] V1 ≡ V2 & ↑[d,e] U1 ≡ U2 &
                                   T1 = 𝕗{I} V1. U1.
-#d #e #T1 #T2 * -d e T1 T2
+#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
 | #J #W1 #W2 #T1 #T2 #d #e #HW #HT #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/
 ]
 qed.
 
@@ -232,7 +232,7 @@ qed.
 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 width=3/ 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
@@ -242,8 +242,8 @@ lemma lift_inv_pair_xy_x: ∀d,e,I,V,T. ↑[d, e] 𝕔{I} V. T ≡ V → False.
   | 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/
+  [ 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/
   ]
 ]
 qed-.
@@ -256,8 +256,8 @@ lemma lift_inv_pair_xy_y: ∀I,T,V,d,e. ↑[d, e] 𝕔{I} V. T ≡ T → False.
   | 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/
+  [ 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/
   ]
 ]
 qed-.
@@ -265,31 +265,31 @@ 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 //
+#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/
+#d #e #i #H >(plus_minus_m_m i e) in ⊢ (? ? ? ? %); /2 width=2/ /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/
+[ * #i // #d elim (lt_or_ge i d) /2 width=1/
+| * /2 width=1/
 ]
 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 /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2/
 | * #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/
+  [ elim (IHT1 (d+1) e) -IHT1 /3 width=2/
+  | elim (IHT1 d e) -IHT1 /3 width=2/
   ]
 ]
 qed.
@@ -298,20 +298,20 @@ qed.
 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/
+#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/
+  lapply (lt_to_le_to_lt … Hid1 Hd12) -Hd12 #Hid2 /4 width=3/
 | #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
-  lapply (transitive_le …(i+e1) Hd21 ?) /2/ -Hd21 #Hd21
-  <(arith_d1 i e2 e1) // /3/
-| /3/
+  lapply (transitive_le …(i+e1) Hd21 ?) /2 width=1/ -Hd21 #Hd21
+  <(arith_d1 i e2 e1) // /3 width=3/
+| /3 width=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/
+  elim (IHT (d2+1) … ? ? He12) /2 width=1/ /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/
+  elim (IHT d2 … ? ? He12) // /3 width=5/
 ]
 qed.
 
@@ -324,14 +324,14 @@ lemma is_lift_dec: ∀T2,d,e. Decidable (∃T1. ↑[d,e] T1 ≡ T2).
   | 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) 
+      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/
+      [ * #T1 #HT12 @or_introl /3 width=2/
       | -V1 #HT2 @or_intror * #X #H
         elim (lift_inv_bind2 … H) -H /3 width=2/
       ]