closure property S4 added to abstract candidates of reducibility ...

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

diff --git a/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps_tps.ma b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps_tps.ma
index e07a701bfe10bf2fad70a598a55fbce22a1a4aa3..473f813b6aafb5fa283be7ecce7883ff557d56b2 100644 (file)
--- a/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps_tps.ma
+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps_tps.ma
@@ -17,96 +17,96 @@ include "Basic_2/substitution/ltps_ldrop.ma".
 
 (* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
 
-lemma ltps_tps_conf_ge: â\88\80L0,T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\89« U2 →
-                        â\88\80L1,d1,e1. L0 [d1, e1] â\89« L1 → d1 + e1 ≤ d2 →
-                        L1 â\8a¢ T2 [d2, e2] â\89« U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -H L0 T2 U2 d2 e2
+lemma ltps_tps_conf_ge: â\88\80L0,T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\96¶ U2 →
+                        â\88\80L1,d1,e1. L0 [d1, e1] â\96¶ L1 → d1 + e1 ≤ d2 →
+                        L1 â\8a¢ T2 [d2, e2] â\96¶ U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
 [ //
 | #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01 #Hde1d2
   lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
-  lapply (ltps_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /2/
+  lapply (ltps_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /2 width=4/
 | #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01 #Hde1d2
-  @tps_bind [ /2/ | @IHTU2 [3: /2/ |1,2: skip | /2/ ] ] (**) (* /3/ is too slow *)
-| /3/
+  @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltps_tps1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
 ]
 qed.
 
 (* Basic_1: was: subst1_subst1_back *)
-lemma ltps_tps_conf: â\88\80L0,T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\89« U2 →
-                     â\88\80L1,d1,e1. L0 [d1, e1] â\89« L1 →
-                     â\88\83â\88\83T. L1 â\8a¢ T2 [d2, e2] â\89« T &
-                          L1 â\8a¢ U2 [d1, e1] â\89« T.
-#L0 #T2 #U2 #d2 #e2 #H elim H -H L0 T2 U2 d2 e2
-[ /2/
+lemma ltps_tps_conf: â\88\80L0,T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\96¶ U2 →
+                     â\88\80L1,d1,e1. L0 [d1, e1] â\96¶ L1 →
+                     â\88\83â\88\83T. L1 â\8a¢ T2 [d2, e2] â\96¶ T &
+                          L1 â\8a¢ U2 [d1, e1] â\96¶ T.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
 | #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
   elim (lt_or_ge i2 d1) #Hi2d1
-  [ elim (ltps_ldrop_conf_le … HL01 … HLK0 ?) -HL01 HLK0 /2/ #X #H #HLK1
-    elim (ltps_inv_tps11 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
+  [ elim (ltps_ldrop_conf_le … HL01 … HLK0 ?) -L0 /2 width=2/ #X #H #HLK1
+    elim (ltps_inv_tps11 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
     lapply (ldrop_fwd_ldrop2 … HLK1) #H
     elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
-    lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -H HV01 HVW0 // >arith_a2 /3/
+    lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -V0 -H // >minus_plus <plus_minus_m_m // /3 width=4/
   | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
-    [ elim (ltps_ldrop_conf_be … HL01 … HLK0 ? ?) -HL01 HLK0 [2,3: /2/ ] #X #H #HLK1
-      elim (ltps_inv_tps21 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
+    [ elim (ltps_ldrop_conf_be … HL01 … HLK0 ? ?) -L0 // /2 width=2/ #X #H #HLK1
+      elim (ltps_inv_tps21 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
       lapply (ldrop_fwd_ldrop2 … HLK1) #H
       elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
-      lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -H HV01 HVW0 // normalize #HW01
-      lapply (tps_weak … HW01 d1 e1 ? ?) [2,3: /3/ ] >arith_i2 //
-    | lapply (ltps_ldrop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /3/
+      lapply (tps_lift_ge … HV01 … H HVW0 HVW1 ?) -V0 -H // normalize #HW01
+      lapply (tps_weak … HW01 d1 e1 ? ?) [2: /2 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+    | lapply (ltps_ldrop_conf_ge … HL01 … HLK0 ?) -L0 // /3 width=4/
     ]
   ]
 | #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
   elim (IHVW2 … HL01) -IHVW2 #V #HV2 #HVW2
-  elim (IHTU2 (L1. 𝕓{I} V) (d1 + 1) e1 ?) -IHTU2 [2: /2/ ] -HL01 /3 width=5/
+  elim (IHTU2 (L1. 𝕓{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL01 /3 width=5/
 | #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL01
-  elim (IHVW2 … HL01) -IHVW2;
-  elim (IHTU2 … HL01) -IHTU2 HL01 /3 width=5/
+  elim (IHVW2 … HL01) -IHVW2
+  elim (IHTU2 … HL01) -IHTU2 -HL01 /3 width=5/
 ]
 qed.
 
-lemma ltps_tps_trans_ge: â\88\80L0,T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\89« U2 →
-                         â\88\80L1,d1,e1. L1 [d1, e1] â\89« L0 → d1 + e1 ≤ d2 →
-                         L1 â\8a¢ T2 [d2, e2] â\89« U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -H L0 T2 U2 d2 e2
+lemma ltps_tps_trans_ge: â\88\80L0,T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\96¶ U2 →
+                         â\88\80L1,d1,e1. L1 [d1, e1] â\96¶ L0 → d1 + e1 ≤ d2 →
+                         L1 â\8a¢ T2 [d2, e2] â\96¶ U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
 [ //
 | #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10 #Hde1d2
   lapply (transitive_le … Hde1d2 Hdi2) -Hde1d2 #Hde1i2
-  lapply (ltps_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 HLK0 /2/
+  lapply (ltps_ldrop_trans_ge … HL10 … HLK0 ?) -L0 // /2 width=4/
 | #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10 #Hde1d2
-  @tps_bind [ /2/ | @IHTU2 [3: /2/ |1,2: skip | /2/ ] ] (**) (* /3/ is too slow *)
-| /3/
+  @tps_bind [ /2 width=4/ | @IHTU2 -IHTU2 -IHVW2 [3: /2 by ltps_tps1/ |1,2: skip | /2 width=1/ ] ] (**) (* explicit constructor *)
+| /3 width=4/
 ]
 qed.
 
 (* Basic_1: was: subst1_subst1 *)
-lemma ltps_tps_trans: â\88\80L0,T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\89« U2 →
-                      â\88\80L1,d1,e1. L1 [d1, e1] â\89« L0 →
-                      â\88\83â\88\83T. L1 â\8a¢ T2 [d2, e2] â\89« T &
-                           L0 â\8a¢ T [d1, e1] â\89« U2.
-#L0 #T2 #U2 #d2 #e2 #H elim H -H L0 T2 U2 d2 e2
-[ /2/
+lemma ltps_tps_trans: â\88\80L0,T2,U2,d2,e2. L0 â\8a¢ T2 [d2, e2] â\96¶ U2 →
+                      â\88\80L1,d1,e1. L1 [d1, e1] â\96¶ L0 →
+                      â\88\83â\88\83T. L1 â\8a¢ T2 [d2, e2] â\96¶ T &
+                           L0 â\8a¢ T [d1, e1] â\96¶ U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -L0 -T2 -U2 -d2 -e2
+[ /2 width=3/
 | #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
   elim (lt_or_ge i2 d1) #Hi2d1
-  [ elim (ltps_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2/ #X #H #HLK1
-    elim (ltps_inv_tps12 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
+  [ elim (ltps_ldrop_trans_le … HL10 … HLK0 ?) -HL10 /2 width=2/ #X #H #HLK1
+    elim (ltps_inv_tps12 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
     lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
     elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
-    lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -H HV01 HVW0 // >arith_a2 /3/
+    lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -V0 -H // >minus_plus <plus_minus_m_m /2 width=1/ /3 width=4/
   | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
-    [ elim (ltps_ldrop_trans_be … HL10 … HLK0 ? ?) -HL10 [2,3: /2/ ] #X #H #HLK1
-      elim (ltps_inv_tps22 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
+    [ elim (ltps_ldrop_trans_be … HL10 … HLK0 ? ?) -HL10 // /2 width=2/ #X #H #HLK1
+      elim (ltps_inv_tps22 … H ?) -H /2 width=1/ #K1 #V1 #_ #HV01 #H destruct
       lapply (ldrop_fwd_ldrop2 … HLK0) -HLK0 #H
       elim (lift_total V1 0 (i2 + 1)) #W1 #HVW1
-      lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -H HV01 HVW0 // normalize #HW01
-      lapply (tps_weak … HW01 d1 e1 ? ?) [2,3: /3/ ] >arith_i2 //
-    | lapply (ltps_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 HLK0 /3/
+      lapply (tps_lift_ge … HV01 … H HVW1 HVW0 ?) -V0 -H // normalize #HW01
+      lapply (tps_weak … HW01 d1 e1 ? ?) [2: /3 width=1/ |3: /3 width=4/ ] >minus_plus >commutative_plus /2 width=1/
+    | lapply (ltps_ldrop_trans_ge … HL10 … HLK0 ?) -HL10 -HLK0 // /3 width=4/
     ]
   ]
 | #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
   elim (IHVW2 … HL10) -IHVW2 #V #HV2 #HVW2
-  elim (IHTU2 (L1. 𝕓{I} V) (d1 + 1) e1 ?) -IHTU2 [2: /2/ ] -HL10 /3 width=5/
+  elim (IHTU2 (L1. 𝕓{I} V) (d1 + 1) e1 ?) -IHTU2 /2 width=1/ -HL10 /3 width=5/
 | #L0 #I #V2 #W2 #T2 #U2 #d2 #e2 #_ #_ #IHVW2 #IHTU2 #L1 #d1 #e1 #HL10
-  elim (IHVW2 … HL10) -IHVW2;
-  elim (IHTU2 … HL10) -IHTU2 HL10 /3 width=5/
+  elim (IHVW2 … HL10) -IHVW2
+  elim (IHTU2 … HL10) -IHTU2 -HL10 /3 width=5/
 ]
 qed.