- we shared the atomic term constructions

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

diff --git a/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps_tps.ma b/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps_tps.ma
index 1e3c3629a189e6ba42f96c5f3e130de5e235f5e1..65b1b86d448ad2ea48207969fbf3140d3556eb72 100644 (file)
--- a/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps_tps.ma
+++ b/matita/matita/contribs/lambda-delta/Basic-2/substitution/tps_tps.ma
@@ -12,73 +12,60 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "Basic-2/substitution/tps_split.ma".
+include "Basic-2/substitution/tps_lift.ma".
 
-(* PARTIAL SUBSTITUTION ON TERMS ********************************************)
+(* PARALLEL SUBSTITUTION ON TERMS *******************************************)
 
 (* Main properties **********************************************************)
-(*
-theorem tps_trans: ∀L,T1,T,d,e. L ⊢ T1 [d, e] ≫ T → ∀T2. L ⊢ T [d, e] ≫ T2 →
-                   L ⊢ T1 [d, e] ≫ T2.
-#L #T1 #T #d #e #H elim H -L T1 T d e
-[ //
-| //
-| #L #K #V #V1 #V2 #i #d #e #Hdi #Hide #HLK #_ #HV12 #IHV12 #T2 #HVT2
-  lapply (drop_fwd_drop2 … HLK) #H
-  elim (tps_inv_lift1_up … HVT2 … H … HV12 ? ? ?) -HVT2 H HV12 // normalize [2: /2/ ] #V #HV1 #HVT2
-  @tps_subst [4,5,6,8: // |1,2,3: skip | /2/ ]
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
-  elim (tps_inv_bind1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X;
-  @tps_bind /2/ @IHT12 @(tps_leq_repl … HT2) /2/
-| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
-  elim (tps_inv_flat1 … HX) -HX #V #T #HV2 #HT2 #HX destruct -X /3/
-]
-qed.
-*)
-
-axiom tps_conf_subst_subst_lt: ∀L,K1,V1,T1,i1,d,e,T2,K2,V2,i2.
-   ↓[O, i1] L ≡ K1. 𝕓{Abbr} V1 → ↓[O, i2] L≡ K2. 𝕓{Abbr} V2 →
-   ↑[O, i1 + 1] V1 ≡ T1 → ↑[O, i2 + 1] V2 ≡ T2 → 
-   d ≤ i1 → i1 < d + e → d ≤ i2 → i2 < d + e → i1 < i2 →
-   ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.  
-(*
-#L #K1 #V1 #T1 #i1 #d #e #T2 #K2 #V2 #i2
-#HLK1 #HLK2 #HVW1 #HVW2 #HWT1 #HWT2 #Hdi1 #Hi1de #Hdi2 #Hi2de #Hi12
-*)
 
 theorem tps_conf: ∀L,T0,T1,d,e. L ⊢ T0 [d, e] ≫ T1 → ∀T2. L ⊢ T0 [d, e] ≫ T2 →
                   ∃∃T. L ⊢ T1 [d, e] ≫ T & L ⊢ T2 [d, e] ≫ T.
 #L #T0 #T1 #d #e #H elim H -H L T0 T1 d e
 [ /2/
-| /2/
 | #L #K1 #V1 #T1 #i1 #d #e #Hdi1 #Hi1de #HLK1 #HVT1 #T2 #H
   elim (tps_inv_lref1 … H) -H
   [ #HX destruct -T2 /4/
-  | * #K2 #V2 #i2 #Hdi2 #Hi2de #HLK2 #HVT2
-    elim (lt_or_eq_or_gt i1 i2) #Hi12
-    [ @tps_conf_subst_subst_lt /width=15/
-    | -Hdi2 Hi2de; destruct -i2;
-      lapply (drop_mono … HLK1 … HLK2) -HLK1 #H destruct -V1 K1
-      >(lift_mono … HVT1 … HVT2) -HVT1 /2/
-    | @ex2_1_comm @tps_conf_subst_subst_lt /width=15/
-    ]
+  | * #K2 #V2 #_ #_ #HLK2 #HVT2
+    lapply (drop_mono … HLK1 … HLK2) -HLK1 #H destruct -V1 K1
+    >(lift_mono … HVT1 … HVT2) -HVT1 /2/
   ]
 | #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
-  elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; 
+  elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
   elim (IHV01 … HV02) -IHV01 HV02 #V #HV1 #HV2
-  elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2 
+  elim (IHT01 … HT02) -IHT01 HT02 #T #HT1 #HT2
   @ex2_1_intro
   [2: @tps_bind [4: @(tps_leq_repl … HT1) /2/ |2: skip ]
   |1: skip
   |3: @tps_bind [2: @(tps_leq_repl … HT2) /2/ ]
   ] // (**) (* /5/ is too slow *)
 | #L #I #V0 #V1 #T0 #T1 #d #e #_ #_ #IHV01 #IHT01 #X #HX
-  elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X; 
+  elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
   elim (IHV01 … HV02) -IHV01 HV02;
   elim (IHT01 … HT02) -IHT01 HT02 /3 width=5/
 ]
 qed.
 
+theorem tps_trans_down: ∀L,T1,T0,d1,e1. L ⊢ T1 [d1, e1] ≫ T0 →
+                        ∀T2,d2,e2. L ⊢ T0 [d2, e2] ≫ T2 → d2 + e2 ≤ d1 →
+                        ∃∃T. L ⊢ T1 [d2, e2] ≫ T & L ⊢ T [d1, e1] ≫ T2.
+#L #T1 #T0 #d1 #e1 #H elim H -L T1 T0 d1 e1
+[ /2/
+| #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
+  lapply (transitive_le … Hde2d1 Hdi1) -Hde2d1 #Hde2i1
+  lapply (tps_weak … HWT2 0 (i1 + 1) ? ?) -HWT2; normalize /2/ -Hde2i1 #HWT2
+  <(tps_inv_lift1_eq … HWT2 … HVW) -HWT2 /4/
+| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+  elim (tps_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
+  lapply (tps_leq_repl … HT02 (L. 𝕓{I} V1) ?) -HT02 /2/ #HT02
+  elim (IHV10 … HV02 ?) -IHV10 HV02 // #V
+  elim (IHT10 … HT02 ?) -IHT10 HT02 [2: /2/ ] #T #HT1 #HT2
+  lapply (tps_leq_repl … HT2 (L. 𝕓{I} V) ?) -HT2 /3 width=6/
+| #L #I #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+  elim (tps_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct -X;
+  elim (IHV10 … HV02 ?) -IHV10 HV02 //
+  elim (IHT10 … HT02 ?) -IHT10 HT02 // /3 width=6/
+]
+qed.
 (*
       Theorem subst0_subst0: (t1,t2,u2:?; j:?) (subst0 j u2 t1 t2) ->
                              (u1,u:?; i:?) (subst0 i u u1 u2) ->