- update in Basic_2

[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 ff42a7ac490ca55b0ce33ea5cafd683d714c32ab..e453c4099cd4a8b06f7f1a63f507672fba04e989 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
@@ -19,8 +19,8 @@ include "Basic_2/substitution/tps.ma".
 
 (* Advanced inversion lemmas ************************************************)
 
-fact tps_inv_refl_SO2_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\89« T2 → e = 1 →
-                           â\88\80K,V. â\86\93[0, d] L â\89¡ K. ð\9d\95\93{Abst} V → T1 = T2.
+fact tps_inv_refl_SO2_aux: â\88\80L,T1,T2,d,e. L â\8a¢ T1 [d, e] â\96¶ T2 → e = 1 →
+                           â\88\80K,V. â\87©[0, d] L â\89¡ K. â\93\9bV → T1 = T2.
 #L #T1 #T2 #d #e #H elim H -L -T1 -T2 -d -e
 [ //
 | #L #K0 #V0 #W #i #d #e #Hdi #Hide #HLK0 #_ #H destruct
@@ -33,18 +33,18 @@ fact tps_inv_refl_SO2_aux: ∀L,T1,T2,d,e. L ⊢ T1 [d, e] ≫ T2 → e = 1 →
 ]
 qed.
 
-lemma tps_inv_refl_SO2: â\88\80L,T1,T2,d. L â\8a¢ T1 [d, 1] â\89« T2 →
-                        â\88\80K,V. â\86\93[0, d] L â\89¡ K. ð\9d\95\93{Abst} V → T1 = T2.
+lemma tps_inv_refl_SO2: â\88\80L,T1,T2,d. L â\8a¢ T1 [d, 1] â\96¶ T2 →
+                        â\88\80K,V. â\87©[0, d] L â\89¡ K. â\93\9bV → T1 = T2.
 /2 width=8/ qed-.
 
 (* Relocation properties ****************************************************)
 
 (* Basic_1: was: subst1_lift_lt *)
-lemma tps_lift_le: â\88\80K,T1,T2,dt,et. K â\8a¢ T1 [dt, et] â\89« T2 →
-                   â\88\80L,U1,U2,d,e. â\86\93[d, e] L ≡ K →
-                   â\86\91[d, e] T1 â\89¡ U1 â\86\92 â\86\91[d, e] T2 ≡ U2 →
+lemma tps_lift_le: â\88\80K,T1,T2,dt,et. K â\8a¢ T1 [dt, et] â\96¶ T2 →
+                   â\88\80L,U1,U2,d,e. â\87©[d, e] L ≡ K →
+                   â\87§[d, e] T1 â\89¡ U1 â\86\92 â\87§[d, e] T2 ≡ U2 →
                    dt + et ≤ d →
-                   L â\8a¢ U1 [dt, et] â\89« U2.
+                   L â\8a¢ U1 [dt, et] â\96¶ U2.
 #K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
 [ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 -H2 //
@@ -52,7 +52,7 @@ lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
   lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
   lapply (lift_inv_lref1_lt … H … Hid) -H #H destruct
   elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
-  elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=1/ #X #HLK #H
+  elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
   elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
   >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=4/
 | #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
@@ -65,11 +65,11 @@ lemma tps_lift_le: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
 ]
 qed.
 
-lemma tps_lift_be: â\88\80K,T1,T2,dt,et. K â\8a¢ T1 [dt, et] â\89« T2 →
-                   â\88\80L,U1,U2,d,e. â\86\93[d, e] L ≡ K →
-                   â\86\91[d, e] T1 â\89¡ U1 â\86\92 â\86\91[d, e] T2 ≡ U2 →
+lemma tps_lift_be: â\88\80K,T1,T2,dt,et. K â\8a¢ T1 [dt, et] â\96¶ T2 →
+                   â\88\80L,U1,U2,d,e. â\87©[d, e] L ≡ K →
+                   â\87§[d, e] T1 â\89¡ U1 â\86\92 â\87§[d, e] T2 ≡ U2 →
                    dt ≤ d → d ≤ dt + et →
-                   L â\8a¢ U1 [dt, et + e] â\89« U2.
+                   L â\8a¢ U1 [dt, et + e] â\96¶ U2.
 #K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
 [ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_ #_
   >(lift_mono … H1 … H2) -H1 -H2 //
@@ -78,7 +78,7 @@ lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
   [ -Hdtd
     lapply (lt_to_le_to_lt … (dt+et+e) Hidet ?) // -Hidet #Hidete
     elim (lift_trans_ge … HVW … HWU2 ?) -W // <minus_plus #W #HVW #HWU2
-    elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=1/ #X #HLK #H
+    elim (ldrop_trans_le … HLK … HKV ?) -K /2 width=2/ #X #HLK #H
     elim (ldrop_inv_skip2 … H ?) -H /2 width=1/ -Hid #K #Y #_ #HVY
     >(lift_mono … HVY … HVW) -V #H destruct /2 width=4/
   | -Hdti
@@ -90,7 +90,7 @@ lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
   elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
   @tps_bind [ /2 width=6/ | @IHT12 [3,4: // | skip |5,6: /2 width=1/ | /2 width=1/ ] 
-            ] (**) (* /3 width=6/ is too slow, arith3 needed to avoid crash, simplification like tps_lift_le is too slow *)
+            ] (**) (* /3 width=6/ is too slow, simplification like tps_lift_le is too slow *)
 | #K #I #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #d #e #HLK #H1 #H2 #Hdetd
   elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
   elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6/
@@ -98,11 +98,11 @@ lemma tps_lift_be: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
 qed.
 
 (* Basic_1: was: subst1_lift_ge *)
-lemma tps_lift_ge: â\88\80K,T1,T2,dt,et. K â\8a¢ T1 [dt, et] â\89« T2 →
-                   â\88\80L,U1,U2,d,e. â\86\93[d, e] L ≡ K →
-                   â\86\91[d, e] T1 â\89¡ U1 â\86\92 â\86\91[d, e] T2 ≡ U2 →
+lemma tps_lift_ge: â\88\80K,T1,T2,dt,et. K â\8a¢ T1 [dt, et] â\96¶ T2 →
+                   â\88\80L,U1,U2,d,e. â\87©[d, e] L ≡ K →
+                   â\87§[d, e] T1 â\89¡ U1 â\86\92 â\87§[d, e] T2 ≡ U2 →
                    d ≤ dt →
-                   L â\8a¢ U1 [dt + e, et] â\89« U2.
+                   L â\8a¢ U1 [dt + e, et] â\96¶ U2.
 #K #T1 #T2 #dt #et #H elim H -K -T1 -T2 -dt -et
 [ #K #I #dt #et #L #U1 #U2 #d #e #_ #H1 #H2 #_
   >(lift_mono … H1 … H2) -H1 -H2 //
@@ -122,10 +122,10 @@ lemma tps_lift_ge: ∀K,T1,T2,dt,et. K ⊢ T1 [dt, et] ≫ T2 →
 qed.
 
 (* Basic_1: was: subst1_gen_lift_lt *)
-lemma tps_inv_lift1_le: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\89« U2 →
-                        â\88\80K,d,e. â\86\93[d, e] L â\89¡ K â\86\92 â\88\80T1. â\86\91[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_le: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\96¶ U2 →
+                        â\88\80K,d,e. â\87©[d, e] L â\89¡ K â\86\92 â\88\80T1. â\87§[d, e] T1 ≡ U1 →
                         dt + et ≤ d →
-                        â\88\83â\88\83T2. K â\8a¢ T1 [dt, et] â\89« T2 & â\86\91[d, e] T2 ≡ U2.
+                        â\88\83â\88\83T2. K â\8a¢ T1 [dt, et] â\96¶ T2 & â\87§[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
 [ #L * #i #dt #et #K #d #e #_ #T1 #H #_
   [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
@@ -136,7 +136,7 @@ lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
   lapply (lt_to_le_to_lt … Hidet … Hdetd) -Hdetd #Hid
   lapply (lift_inv_lref2_lt … H … Hid) -H #H destruct
   elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
-  elim (lift_trans_le … HUV … HVW ?) -V // >arith_a2 // -Hid /3 width=4/
+  elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
 | #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
   elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
@@ -149,10 +149,10 @@ lemma tps_inv_lift1_le: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
 ]
 qed.
 
-lemma tps_inv_lift1_be: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\89« U2 →
-                        â\88\80K,d,e. â\86\93[d, e] L â\89¡ K â\86\92 â\88\80T1. â\86\91[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_be: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\96¶ U2 →
+                        â\88\80K,d,e. â\87©[d, e] L â\89¡ K â\86\92 â\88\80T1. â\87§[d, e] T1 ≡ U1 →
                         dt ≤ d → d + e ≤ dt + et →
-                        â\88\83â\88\83T2. K â\8a¢ T1 [dt, et - e] â\89« T2 & â\86\91[d, e] T2 ≡ U2.
+                        â\88\83â\88\83T2. K â\8a¢ T1 [dt, et - e] â\96¶ T2 & â\87§[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
 [ #L * #i #dt #et #K #d #e #_ #T1 #H #_
   [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
@@ -165,17 +165,19 @@ lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
   [ -Hdtd -Hidet
     lapply (lt_to_le_to_lt … (dt + (et - e)) Hid ?) [ <le_plus_minus /2 width=1/ ] -Hdedet #Hidete
     elim (ldrop_conf_lt … HLK … HLKV ?) -L // #L #U #HKL #_ #HUV
-    elim (lift_trans_le … HUV … HVW ?) -V // >arith_a2 // -Hid /3 width=4/
+    elim (lift_trans_le … HUV … HVW ?) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=4/
   | -Hdti -Hdedet
     lapply (transitive_le … (i - e) Hdtd ?) /2 width=1/ -Hdtd #Hdtie
-    lapply (plus_le_weak … Hid) #Hei
+    elim (le_inv_plus_l … Hid) #Hdie #Hei
     lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
-    elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |2,3: /2 width=1/ ] -Hid >arith_e2 // /4 width=4/
+    elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hid -Hdie
+    #V1 #HV1 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+    @ex2_1_intro [3: @H | skip | @tps_subst [3,5,6: // |1,2: skip | >commutative_plus >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
   ]
 | #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
   elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
   elim (IHV12 … HLK … HWV1 ? ?) -V1 // #W2 #HW12 #HWV2
-  elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ]
+  elim (IHU12 … HTU1 ? ?) -U1 [5: @ldrop_skip // |2: skip |3: >plus_plus_comm_23 >(plus_plus_comm_23 dt) /2 width=1/ |4: /2 width=1/ ] (**) (* 29s without the rewrites *)
   /3 width=5/
 | #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdtd #Hdedet
   elim (lift_inv_flat2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
@@ -185,10 +187,10 @@ lemma tps_inv_lift1_be: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
 qed.
 
 (* Basic_1: was: subst1_gen_lift_ge *)
-lemma tps_inv_lift1_ge: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\89« U2 →
-                        â\88\80K,d,e. â\86\93[d, e] L â\89¡ K â\86\92 â\88\80T1. â\86\91[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_ge: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\96¶ U2 →
+                        â\88\80K,d,e. â\87©[d, e] L â\89¡ K â\86\92 â\88\80T1. â\87§[d, e] T1 ≡ U1 →
                         d + e ≤ dt →
-                        â\88\83â\88\83T2. K â\8a¢ T1 [dt - e, et] â\89« T2 & â\86\91[d, e] T2 ≡ U2.
+                        â\88\83â\88\83T2. K â\8a¢ T1 [dt - e, et] â\96¶ T2 & â\87§[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #H elim H -L -U1 -U2 -dt -et
 [ #L * #i #dt #et #K #d #e #_ #T1 #H #_
   [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3/
@@ -197,15 +199,16 @@ lemma tps_inv_lift1_ge: ∀L,U1,U2,dt,et. L ⊢ U1 [dt, et] ≫ U2 →
   ]
 | #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
-  lapply (plus_le_weak … Hdei) #Hei  
+  elim (le_inv_plus_l … Hdedt) -Hdedt #_ #Hedt
+  elim (le_inv_plus_l … Hdei) #Hdie #Hei
   lapply (lift_inv_lref2_ge … H … Hdei) -H #H destruct
   lapply (ldrop_conf_ge … HLK … HLKV ?) -L // #HKV
-  elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |2,3: normalize /2 width=1/ ] -Hdei >arith_e2 // #V0 #HV10 #HV02
-  @ex2_1_intro /3 width=4/ (**) (* explicitc constructors *)
+  elim (lift_split … HVW d (i - e + 1) ? ? ?) -HVW [4: // |3: /2 width=1/ |2: /3 width=1/ ] -Hdei -Hdie 
+  #V0 #HV10 >plus_minus // <minus_minus // /2 width=1/ <minus_n_n <plus_n_O #H
+  @ex2_1_intro [3: @H | skip | @tps_subst [5,6: // |1,2: skip | /2 width=1/ | >plus_minus // /2 width=1/ ] ] (**) (* explicit constructor, uses monotonic_lt_minus_l *)
 | #L #I #V1 #V2 #U1 #U2 #dt #et #_ #_ #IHV12 #IHU12 #K #d #e #HLK #X #H #Hdetd
   elim (lift_inv_bind2 … H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
-  lapply (plus_le_weak … Hdetd) #Hedt
+  elim (le_inv_plus_l … Hdetd) #_ #Hedt
   elim (IHV12 … HLK … HWV1 ?) -V1 // #W2 #HW12 #HWV2
   elim (IHU12 … HTU1 ?) -U1 [4: @ldrop_skip // |2: skip |3: /2 width=1/ ]
   <plus_minus // /3 width=5/
@@ -218,7 +221,7 @@ qed.
 
 (* Basic_1: was: subst1_gen_lift_eq *)
 lemma tps_inv_lift1_eq: ∀L,U1,U2,d,e.
-                        L â\8a¢ U1 [d, e] â\89« U2 â\86\92 â\88\80T1. â\86\91[d, e] T1 ≡ U1 → U1 = U2.
+                        L â\8a¢ U1 [d, e] â\96¶ U2 â\86\92 â\88\80T1. â\87§[d, e] T1 ≡ U1 → U1 = U2.
 #L #U1 #U2 #d #e #H elim H -L -U1 -U2 -d -e
 [ //
 | #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #T1 #H
@@ -250,21 +253,21 @@ 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: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\89« U2 →
-                        â\88\80K,d,e. â\86\93[d, e] L â\89¡ K â\86\92 â\88\80T1. â\86\91[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_up: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\96¶ U2 →
+                        â\88\80K,d,e. â\87©[d, e] L â\89¡ K â\86\92 â\88\80T1. â\87§[d, e] T1 ≡ U1 →
                         d ≤ dt → dt ≤ d + e → d + e ≤ dt + et →
-                        â\88\83â\88\83T2. K â\8a¢ T1 [d, dt + et - (d + e)] â\89« T2 & â\86\91[d, e] T2 ≡ U2.
+                        â\88\83â\88\83T2. K â\8a¢ T1 [d, dt + et - (d + e)] â\96¶ T2 & â\87§[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_weak … HU1 d e ? ?) -HU1 // [ >commutative_plus /2 width=1/ ] -Hddt -Hdtde #HU1
 lapply (tps_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
 elim (tps_inv_lift1_ge … HU2 … HLK … HTU1 ?) -U -L // <minus_plus_m_m /2 width=3/
 qed.
 
-lemma tps_inv_lift1_be_up: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\89« U2 →
-                           â\88\80K,d,e. â\86\93[d, e] L â\89¡ K â\86\92 â\88\80T1. â\86\91[d, e] T1 ≡ U1 →
+lemma tps_inv_lift1_be_up: â\88\80L,U1,U2,dt,et. L â\8a¢ U1 [dt, et] â\96¶ U2 →
+                           â\88\80K,d,e. â\87©[d, e] L â\89¡ K â\86\92 â\88\80T1. â\87§[d, e] T1 ≡ U1 →
                            dt ≤ d → dt + et ≤ d + e →
-                           â\88\83â\88\83T2. K â\8a¢ T1 [dt, d - dt] â\89« T2 & â\86\91[d, e] T2 ≡ U2.
+                           â\88\83â\88\83T2. K â\8a¢ T1 [dt, d - dt] â\96¶ T2 & â\87§[d, e] T2 ≡ U2.
 #L #U1 #U2 #dt #et #HU12 #K #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
 lapply (tps_weak … HU12 dt (d + e - dt) ? ?) -HU12 // /2 width=3/ -Hdetde #HU12
 elim (tps_inv_lift1_be … HU12 … HLK … HTU1 ? ?) -U1 -L // /2 width=3/