- degree assignment, static type assignment, iterated static type

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / relocation / ldrop_ldrop.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_ldrop.ma
index 88f37fcfba78463e5ec35a4324e91bcb8a2c6e02..8b1e5d29206caa710c569a3b348f394f1b64405b 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_ldrop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_ldrop.ma
@@ -23,14 +23,12 @@ include "basic_2/relocation/ldrop.ma".
 theorem ldrop_mono: ∀d,e,L,L1. ⇩[d, e] L ≡ L1 →
                     ∀L2. ⇩[d, e] L ≡ L2 → L1 = L2.
 #d #e #L #L1 #H elim H -d -e -L -L1
-[ #d #e #L2 #H
-  >(ldrop_inv_atom1 … H) -L2 //
-| #K #I #V #L2 #HL12
-   <(ldrop_inv_refl … HL12) -L2 //
+[ #d #L2 #H elim (ldrop_inv_atom1 … H) -H //
+| #K #I #V #L2 #HL12 <(ldrop_inv_O2 … HL12) -L2 //
 | #L #K #I #V #e #_ #IHLK #L2 #H
   lapply (ldrop_inv_ldrop1 … H ?) -H // /2 width=1/
 | #L #K1 #I #T #V1 #d #e #_ #HVT1 #IHLK1 #X #H
-  elim (ldrop_inv_skip1 … H ?) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
+  elim (ldrop_inv_skip1 … H) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
   >(lift_inj … HVT1 … HVT2) -HVT1 -HVT2
   >(IHLK1 … HLK2) -IHLK1 -HLK2 //
 ]
@@ -40,11 +38,8 @@ qed-.
 theorem ldrop_conf_ge: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
                        ∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 + e1 ≤ e2 →
                        ⇩[0, e2 - e1] L1 ≡ L2.
-#d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1
-[ #d #e #e2 #L2 #H
-  >(ldrop_inv_atom1 … H) -L2 //
-| //
-| #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
+#d1 #e1 #L #L1 #H elim H -d1 -e1 -L -L1 //
+[ #L #K #I #V #e #_ #IHLK #e2 #L2 #H #He2
   lapply (ldrop_inv_ldrop1 … H ?) -H /2 width=2/ #HL2
   <minus_plus >minus_minus_comm /3 width=1/
 | #L #K #I #V1 #V2 #d #e #_ #_ #IHLK #e2 #L2 #H #Hdee2
@@ -60,21 +55,22 @@ theorem ldrop_conf_be: ∀L0,L1,d1,e1. ⇩[d1, e1] L0 ≡ L1 →
                        ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → d1 ≤ e2 → e2 ≤ d1 + e1 →
                        ∃∃L. ⇩[0, d1 + e1 - e2] L2 ≡ L & ⇩[0, d1] L1 ≡ L.
 #L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H >(ldrop_inv_atom1 … H) -H /2 width=3/
+[ #d1 #L2 #e2 #H #Hd1 #_ elim (ldrop_inv_atom1 … H) -H #H1 #H2 destruct
+  <(le_n_O_to_eq … Hd1) -d1 /2 width=3/
 | normalize #L #I #V #L2 #e2 #HL2 #_ #He2
   lapply (le_n_O_to_eq … He2) -He2 #H destruct
-  lapply (ldrop_inv_refl … HL2) -HL2 #H destruct /2 width=3/
+  lapply (ldrop_inv_O2 … HL2) -HL2 #H destruct /2 width=3/
 | normalize #L0 #K0 #I #V1 #e1 #HLK0 #IHLK0 #L2 #e2 #H #_ #He21
-  lapply (ldrop_inv_O1 … H) -H * * #He2 #HL20
+  lapply (ldrop_inv_O1_pair1 … H) -H * * #He2 #HL20
   [ -IHLK0 -He21 destruct <minus_n_O /3 width=3/
   | -HLK0 <minus_le_minus_minus_comm //
-    elim (IHLK0 … HL20 ? ?) -L0 // /2 width=1/ /2 width=3/
+    elim (IHLK0 … HL20) -L0 // /2 width=1/ /2 width=3/
   ]
 | #L0 #K0 #I #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHLK0 #L2 #e2 #H #Hd1e2 #He2de1
   elim (le_inv_plus_l … Hd1e2) #_ #He2
   <minus_le_minus_minus_comm //
   lapply (ldrop_inv_ldrop1 … H ?) -H // #HL02
-  elim (IHLK0 … HL02 ? ?) -L0 /2 width=1/ /3 width=3/
+  elim (IHLK0 … HL02) -L0 /2 width=1/ /3 width=3/
 ]
 qed.
 
@@ -83,8 +79,8 @@ theorem ldrop_conf_le: ∀L0,L1,d1,e1. ⇩[d1, e1] L0 ≡ L1 →
                        ∀L2,e2. ⇩[0, e2] L0 ≡ L2 → e2 ≤ d1 →
                        ∃∃L. ⇩[0, e2] L1 ≡ L & ⇩[d1 - e2, e1] L2 ≡ L.
 #L0 #L1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #L2 #e2 #H
-  lapply (ldrop_inv_atom1 … H) -H #H destruct /2 width=3/
+[ #d1 #L2 #e2 #H
+  elim (ldrop_inv_atom1 … H) -H #H destruct /2 width=3/
 | #K0 #I #V0 #L2 #e2 #H1 #H2
   lapply (le_n_O_to_eq … H2) -H2 #H destruct
   lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /2 width=3/
@@ -92,10 +88,10 @@ theorem ldrop_conf_le: ∀L0,L1,d1,e1. ⇩[d1, e1] L0 ≡ L1 →
   lapply (le_n_O_to_eq … H2) -H2 #H destruct
   lapply (ldrop_inv_pair1 … H1) -H1 #H destruct /3 width=3/
 | #K0 #K1 #I #V0 #V1 #d1 #e1 #HK01 #HV10 #IHK01 #L2 #e2 #H #He2d1
-  elim (ldrop_inv_O1 … H) -H *
+  elim (ldrop_inv_O1_pair1 … H) -H *
   [ -IHK01 -He2d1 #H1 #H2 destruct /3 width=5/
   | -HK01 -HV10 #He2 #HK0L2
-    elim (IHK01 … HK0L2 ?) -IHK01 -HK0L2 /2 width=1/ >minus_le_minus_minus_comm // /3 width=3/
+    elim (IHK01 … HK0L2) -IHK01 -HK0L2 /2 width=1/ >minus_le_minus_minus_comm // /3 width=3/
   ]
 ]
 qed.
@@ -103,11 +99,8 @@ qed.
 (* Basic_1: was: drop_trans_ge *)
 theorem ldrop_trans_ge: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L →
                         ∀e2,L2. ⇩[0, e2] L ≡ L2 → d1 ≤ e2 → ⇩[0, e1 + e2] L1 ≡ L2.
-#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L
-[ #d #e #e2 #L2 #H
-  >(ldrop_inv_atom1 … H) -H -L2 //
-| //
-| /3 width=1/
+#d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L //
+[ /3 width=1/
 | #L1 #L2 #I #V1 #V2 #d #e #H_ #_ #IHL12 #e2 #L #H #Hde2
   lapply (lt_to_le_to_lt 0 … Hde2) // #He2
   lapply (lt_to_le_to_lt … (e + e2) He2 ?) // #Hee2
@@ -121,19 +114,19 @@ theorem ldrop_trans_le: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L →
                         ∀e2,L2. ⇩[0, e2] L ≡ L2 → e2 ≤ d1 →
                         ∃∃L0. ⇩[0, e2] L1 ≡ L0 & ⇩[d1 - e2, e1] L0 ≡ L2.
 #d1 #e1 #L1 #L #H elim H -d1 -e1 -L1 -L
-[ #d #e #e2 #L2 #H
-  >(ldrop_inv_atom1 … H) -L2 /2 width=3/
+[ #d #e2 #L2 #H
+  elim (ldrop_inv_atom1 … H) -H /2 width=3/
 | #K #I #V #e2 #L2 #HL2 #H
   lapply (le_n_O_to_eq … H) -H #H destruct /2 width=3/
 | #L1 #L2 #I #V #e #_ #IHL12 #e2 #L #HL2 #H
   lapply (le_n_O_to_eq … H) -H #H destruct
-  elim (IHL12 … HL2 ?) -IHL12 -HL2 // #L0 #H #HL0
-  lapply (ldrop_inv_refl … H) -H #H destruct /3 width=5/
+  elim (IHL12 … HL2) -IHL12 -HL2 // #L0 #H #HL0
+  lapply (ldrop_inv_O2 … H) -H #H destruct /3 width=5/
 | #L1 #L2 #I #V1 #V2 #d #e #HL12 #HV12 #IHL12 #e2 #L #H #He2d
-  elim (ldrop_inv_O1 … H) -H *
+  elim (ldrop_inv_O1_pair1 … H) -H *
   [ -He2d -IHL12 #H1 #H2 destruct /3 width=5/
   | -HL12 -HV12 #He2 #HL2
-    elim (IHL12 … HL2 ?) -L2 [ >minus_le_minus_minus_comm // /3 width=3/ | /2 width=1/ ]
+    elim (IHL12 … HL2) -L2 [ >minus_le_minus_minus_comm // /3 width=3/ | /2 width=1/ ]
   ]
 ]
 qed.
@@ -142,6 +135,17 @@ qed.
 axiom ldrop_div: ∀e1,L1,L. ⇩[0, e1] L1 ≡ L → ∀e2,L2. ⇩[0, e2] L2 ≡ L →
                  ∃∃L0. ⇩[0, e1] L0 ≡ L2 & ⇩[e1, e2] L0 ≡ L1.
 
+(* Advanced properties ******************************************************)
+
+lemma l_liftable_llstar: ∀R. l_liftable R → ∀l. l_liftable (llstar … R l).
+#R #HR #l #K #T1 #T2 #H @(lstar_ind_r … l T2 H) -l -T2
+[ #L #d #e #_ #U1 #HTU1 #U2 #HTU2 -HR -K
+  >(lift_mono … HTU2 … HTU1) -T1 -U2 -d -e //
+| #l #T #T2 #_ #HT2 #IHT1 #L #d #e #HLK #U1 #HTU1 #U2 #HTU2
+  elim (lift_total T d e) /3 width=11 by lstar_dx/ (**) (* auto too slow without trace *)
+]
+qed.
+
 (* Basic_1: was: drop_conf_lt *)
 lemma ldrop_conf_lt: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
                      ∀e2,K2,I,V2. ⇩[0, e2] L ≡ K2. ⓑ{I} V2 →
@@ -149,10 +153,21 @@ lemma ldrop_conf_lt: ∀d1,e1,L,L1. ⇩[d1, e1] L ≡ L1 →
                      ∃∃K1,V1. ⇩[0, e2] L1 ≡ K1. ⓑ{I} V1 &
                               ⇩[d, e1] K2 ≡ K1 & ⇧[d, e1] V1 ≡ V2.
 #d1 #e1 #L #L1 #H1 #e2 #K2 #I #V2 #H2 #He2d1
-elim (ldrop_conf_le … H1 … H2 ?) -L [2: /2 width=2/] #K #HL1K #HK2
-elim (ldrop_inv_skip1 … HK2 ?) -HK2 [2: /2 width=1/] #K1 #V1 #HK21 #HV12 #H destruct /2 width=5/
+elim (ldrop_conf_le … H1 … H2) -L [2: /2 width=2/] #K #HL1K #HK2
+elim (ldrop_inv_skip1 … HK2) -HK2 [2: /2 width=1/] #K1 #V1 #HK21 #HV12 #H destruct /2 width=5/
 qed.
 
+(* Note: apparently this was missing in basic_1 *)
+lemma ldrop_trans_lt: ∀d1,e1,L1,L. ⇩[d1, e1] L1 ≡ L →
+                      ∀e2,L2,I,V2. ⇩[0, e2] L ≡ L2.ⓑ{I}V2 →
+                      e2 < d1 → let d ≝ d1 - e2 - 1 in
+                      ∃∃L0,V0. ⇩[0, e2] L1 ≡ L0.ⓑ{I}V0 &
+                               ⇩[d, e1] L0 ≡ L2 & ⇧[d, e1] V2 ≡ V0.
+#d1 #e1 #L1 #L #HL1 #e2 #L2 #I #V2 #HL2 #Hd21
+elim (ldrop_trans_le … HL1 … HL2) -L [2: /2 width=1/ ] #L0 #HL10 #HL02
+elim (ldrop_inv_skip2 … HL02) -HL02 [2: /2 width=1/ ] #L #V1 #HL2 #HV21 #H destruct /2 width=5/
+qed-.
+
 lemma ldrop_trans_ge_comm: ∀d1,e1,e2,L1,L2,L.
                            ⇩[d1, e1] L1 ≡ L → ⇩[0, e2] L ≡ L2 → d1 ≤ e2 →
                            ⇩[0, e2 + e1] L1 ≡ L2.
@@ -167,10 +182,10 @@ elim (le_or_ge e1 e2) #He
 [ lapply (ldrop_conf_ge … HLK1 … HLK2 ?)
 | lapply (ldrop_conf_ge … HLK2 … HLK1 ?)
 ] -HLK1 -HLK2 // #HK
-lapply (ldrop_fwd_O1_length … HK) #H
+lapply (ldrop_fwd_length_minus2 … HK) #H
 elim (discr_minus_x_xy … H) -H
 [1,3: normalize <plus_n_Sm #H destruct ]
 #H >H in HK; #HK
-lapply (ldrop_inv_refl … HK) -HK #H destruct
+lapply (ldrop_inv_O2 … HK) -HK #H destruct
 lapply (inv_eq_minus_O … H) -H /3 width=1/
 qed-.