- some pending conjectures closed in basic_2 and ground_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / substitution / lift.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma
index f90b349b3de0b7fe55dc57e698833d5f666ce3dc..f8848629202d42750d885d59d179636b93995ace 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma
@@ -39,7 +39,7 @@ interpretation "relocation" 'RLift d e T1 T2 = (lift d e 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/
+#d #e #T1 #T2 #H elim H -d -e -T1 -T2 /3 width=1 by eq_f2/
 qed-.
 
 lemma lift_inv_O2: ∀d,T1,T2. ⇧[d, 0] T1 ≡ T2 → T1 = T2.
@@ -60,8 +60,8 @@ 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
@@ -104,7 +104,7 @@ fact lift_inv_bind1_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
 | #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-.
@@ -124,7 +124,7 @@ 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-.
 
@@ -149,8 +149,8 @@ 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
@@ -207,7 +207,7 @@ fact lift_inv_bind2_aux: ∀d,e,T1,T2. ⇧[d,e] T1 ≡ T2 →
 | #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-.
@@ -228,7 +228,7 @@ 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-.
 
@@ -246,8 +246,8 @@ 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-.
@@ -261,8 +261,8 @@ 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-.
@@ -272,16 +272,16 @@ qed-.
 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/
+[ 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/
-|  elim (lift_inv_flat2 … H) -H /2 width=4/
+[ 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-.
 
@@ -305,7 +305,7 @@ qed-.
 
 (* 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/
+#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.
@@ -314,18 +314,18 @@ lemma lift_lref_ge_minus_eq: ∀d,e,i,j. d + e ≤ i → j = i - e → ⇧[d, e]
 (* 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 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.
 #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.
@@ -337,51 +337,49 @@ lemma lift_split: ∀d1,e2,T1,T2. ⇧[d1, e2] T1 ≡ 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).
 #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/
     ]
   ]
 ]