- the theory of parallel substitution of local environments (ltps) is ready

[helm.git] / matita / matita / contribs / lambda-delta / Basic-2 / substitution / tps_lift.ma

diff --git a/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps_lift.ma b/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps_lift.ma
index e5ad25ff13ebf98ea6fccfb4df3a3560389e39fc..e3b6ee39eaae82f12e23c97c5cdb27c2996a9ae3 100644 (file)
--- a/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps_lift.ma
+++ b/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps_lift.ma
@@ -25,9 +25,7 @@ lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
                    dt + et ≤ d →
                    L ⊢ U1 [dt, et] ≫ U2.
 #K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
-[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
-  >(lift_mono … H1 … H2) -H1 H2 //
-| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 H2 //
 | #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HVU2 #Hdetd
   lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
@@ -53,9 +51,7 @@ lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
                    d ≤ dt →
                    L ⊢ U1 [dt + e, et] ≫ U2.
 #K #T1 #T2 #dt #et #H elim H -H K T1 T2 dt et
-[ #K #k #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
-  >(lift_mono … H1 … H2) -H1 H2 //
-| #K #i #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
+[ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 H2 //
 | #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #d #e #HLK #H #HWU2 #Hddt
   lapply (transitive_le … Hddt … Hdti) -Hddt #Hid
@@ -78,10 +74,10 @@ lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
                         dt + et ≤ d →
                         ∃∃T2. K ⊢ T1 [dt, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
-[ #L #k #dt #et #K #d #e #_ #T1 #H #_
-  lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-| #L #i #dt #et #K #d #e #_ #T1 #H #_
-  elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+  [ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
+  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+  ]
 | #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdetd
   lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
   lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct -T1;
@@ -103,10 +99,10 @@ lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
                         d + e ≤ dt →
                         ∃∃T2. K ⊢ T1 [dt - e, et] ≫ T2 & ↑[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #H elim H -H L U1 U2 dt et
-[ #L #k #dt #et #K #d #e #_ #T1 #H #_
-  lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
-| #L #i #dt #et #K #d #e #_ #T1 #H #_
-  elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+[ #L * #i #dt #et #K #d #e #_ #T1 #H #_
+  [ lapply (lift_inv_sort2 … H) -H #H destruct -T1 /2/
+  | elim (lift_inv_lref2 … H) -H * #Hid #H destruct -T1 /3/
+  ]
 | #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #d #e #HLK #T1 #H #Hdedt  
   lapply (transitive_le … Hdedt … Hdti) #Hdei
   lapply (plus_le_weak … Hdedt) -Hdedt #Hedt
@@ -136,7 +132,6 @@ lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
                         L ⊢ U1 [d, e] ≫ U2 → ∀T1. ↑[d, e] T1 ≡ U1 → U1 = U2.
 #L #U1 #U2 #d #e #H elim H -H L U1 U2 d e
 [ //
-| //
 | #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
   elim (lift_inv_lref2 … H) -H * #H
   [ lapply (le_to_lt_to_lt … Hdi … H) -Hdi H #H
@@ -170,3 +165,34 @@ qed.
                                         (le d i) -> (lt i (plus d h)) ->
                                        (EX u1 | t1 = (lift (minus (plus d h) (S i)) (S i) u1)).
 *)
+
+lemma tps_inv_lift1_up: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
+                        ∀K,d,e. ↓[d, e] L ≡ K → ∀T1. ↑[d, e] T1 ≡ U1 →
+                        d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
+                        ∃∃T2. K ⊢ T1 [d, dt + et - (d + e)] ≫ T2 & ↑[d, e] T2 ≡ U2.
+#L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (tps_split_up … HU12 (d + e) ? ?) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (tps_weak … HU1 d e ? ?) -HU1 // <plus_minus_m_m_comm // -Hddt Hdtde #HU1
+lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct -U1;
+elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -HU2 HLK HTU1 // <minus_plus_m_m /2/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact tps_inv_refl_SO2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 1 →
+                           ∀K,V. ↓[0, d] L ≡ K. 𝕓{Abst} V → T1 = T2.
+#L #T1 #T2 #d #e #H elim H -H L T1 T2 d e
+[ //
+| #L #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct -e;
+  >(le_to_le_to_eq … Hdi ?) /2/ -d #K #V #HLK
+  lapply (drop_mono … HLK0 … HLK) #H destruct
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
+  >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 K V) -IHT12 /2/
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #H1 #K #V #HLK
+  >(IHV12 H1 … HLK) -IHV12 >(IHT12 H1 … HLK) -IHT12 //
+]
+qed.
+
+lemma tps_inv_refl_SO2: ∀L,T1,T2,d. L ⊢ T1 [d, 1] ≫ T2 →
+                        ∀K,V. ↓[0, d] L ≡ K. 𝕓{Abst} V → T1 = T2.
+/2 width=8/ qed.