refactoring ...

[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
new file mode 100644 (file)
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+++ b/matita/matita/contribs/lambda_delta/Basic_2/substitution/ltps_tps.ma
@@ -0,0 +1,112 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "Basic-2/substitution/tps_lift.ma".
+include "Basic-2/substitution/ltps_drop.ma".
+
+(* PARALLEL SUBSTITUTION ON LOCAL ENVIRONMENTS ******************************)
+
+lemma ltps_tps_conf_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ≫ U2 →
+                        ∀L1,d1,e1. L0 [d1, e1] ≫ L1 → d1 + e1 ≤ d2 →
+                        L1 ⊢ T2 [d2, e2] ≫ U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -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_drop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /2/
+| #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/
+]
+qed.
+
+(* Basic-1: was: subst1_subst1_back *)
+lemma ltps_tps_conf: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ≫ U2 →
+                     ∀L1,d1,e1. L0 [d1, e1] ≫ L1 →
+                     ∃∃T. L1 ⊢ T2 [d2, e2] ≫ T &
+                          L1 ⊢ U2 [d1, e1] ≫ T.
+#L0 #T2 #U2 #d2 #e2 #H elim H -H L0 T2 U2 d2 e2
+[ /2/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL01
+  elim (lt_or_ge i2 d1) #Hi2d1
+  [ elim (ltps_drop_conf_le … HL01 … HLK0 ?) -HL01 HLK0 /2/ #X #H #HLK1
+    elim (ltps_inv_tps11 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
+    lapply (drop_fwd_drop2 … 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/
+  | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+    [ elim (ltps_drop_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;
+      lapply (drop_fwd_drop2 … 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_drop_conf_ge … HL01 … HLK0 ?) -HL01 HLK0 /3/
+    ]
+  ]
+| #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/
+| #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/
+]
+qed.
+
+lemma ltps_tps_trans_ge: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ≫ U2 →
+                         ∀L1,d1,e1. L1 [d1, e1] ≫ L0 → d1 + e1 ≤ d2 →
+                         L1 ⊢ T2 [d2, e2] ≫ U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -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_drop_trans_ge … HL10 … HLK0 ?) -HL10 HLK0 /2/
+| #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/
+]
+qed.
+
+(* Basic-1: was: subst1_subst1 *)
+lemma ltps_tps_trans: ∀L0,T2,U2,d2,e2. L0 ⊢ T2 [d2, e2] ≫ U2 →
+                      ∀L1,d1,e1. L1 [d1, e1] ≫ L0 →
+                      ∃∃T. L1 ⊢ T2 [d2, e2] ≫ T &
+                           L0 ⊢ T [d1, e1] ≫ U2.
+#L0 #T2 #U2 #d2 #e2 #H elim H -H L0 T2 U2 d2 e2
+[ /2/
+| #L0 #K0 #V0 #W0 #i2 #d2 #e2 #Hdi2 #Hide2 #HLK0 #HVW0 #L1 #d1 #e1 #HL10
+  elim (lt_or_ge i2 d1) #Hi2d1
+  [ elim (ltps_drop_trans_le … HL10 … HLK0 ?) -HL10 /2/ #X #H #HLK1
+    elim (ltps_inv_tps12 … H ?) -H [2: /2/ ] #K1 #V1 #_ #HV01 #H destruct -X;
+    lapply (drop_fwd_drop2 … 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/
+  | elim (lt_or_ge i2 (d1 + e1)) #Hde1i2
+    [ elim (ltps_drop_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;
+      lapply (drop_fwd_drop2 … 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_drop_trans_ge … HL10 … HLK0 ?) -HL10 HLK0 /3/
+    ]
+  ]
+| #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/
+| #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/
+]
+qed.