more lemmas to prove and a correction in subst

[helm.git] / matita / matita / lib / lambda-delta / reduction / pr.ma

diff --git a/matita/matita/lib/lambda-delta/reduction/pr.ma b/matita/matita/lib/lambda-delta/reduction/pr.ma
index 3abebc8484b1a03950d38d46f955b659f06f97a0..3c5f4ca96ec8cf31da927810216412f4b5b2acd9 100644 (file)
--- a/matita/matita/lib/lambda-delta/reduction/pr.ma
+++ b/matita/matita/lib/lambda-delta/reduction/pr.ma
@@ -11,7 +11,7 @@
 
 include "lambda-delta/substitution/thin.ma".
 
-(* SINGLE STEP PARALLEL REDUCTION *******************************************)
+(* SINGLE STEP PARALLEL REDUCTION ON TERMS **********************************)
 
 inductive pr: lenv → term → term → Prop ≝
 | pr_sort : ∀L,k. pr L (⋆k) (⋆k)
@@ -34,69 +34,26 @@ inductive pr: lenv → term → term → Prop ≝
 | pr_tau  : ∀L,V,T1,T2. pr L T1 T2 → pr L (𝕚{Cast} V. T1) T2
 .
 
-interpretation "single step parallel reduction" 'PR L T1 T2 = (pr L T1 T2).
+interpretation
+   "single step parallel reduction (term)"
+   'PR L T1 T2 = (pr L T1 T2).
 
-(* The three main lemmas on reduction ***************************************)
+(* The basic properties *****************************************************)
 
-lemma pr_inv_lift: ∀L,T1,T2. L ⊢ T1 ⇒ T2 →
-                   ∀d,e,K. ↓[d,e] L ≡ K → ∀U1. ↑[d,e] U1 ≡ T1 →
-                   ∃∃U2. ↑[d,e] U2 ≡ T2 & K ⊢ U1 ⇒ U2.
-#L #T1 #T2 #H elim H -H L T1 T2
-[ #L #k #d #e #K #_ #U1 #HU1
-  lapply (lift_inv_sort2 … HU1) -HU1 #H destruct -U1 /2/
-| #L #i #d #e #K #_ #U1 #HU1
-  lapply (lift_inv_lref2 … HU1) -HU1 * * #Hid #H destruct -U1 /3/
-| #L #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #K #HLK #X #HX
-  lapply (lift_inv_bind2 … HX) -HX * #V0 #T0 #HV01 #HT01 #HX destruct -X;
-  elim (IHV12 … HLK … HV01) -IHV12 #V3 #HV32 #HV03
-  elim (IHT12 … HT01) -IHT12 HT01 [2,3: -HV32 HV03 /3/] -HLK HV01 /3 width=5/
-| #L #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #K #HLK #X #HX
-  elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct -X;
-  elim (IHV12 … HLK … HV01) -IHV12 HV01 #V3 #HV32 #HV03
-  elim (IHT12 … HLK … HT01) -IHT12 HT01 HLK /3 width=5/
-| #L #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #K #HLK #X #HX
-  elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X;
-  elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct -Y;
-  elim (IHV12 … HLK … HV01) -IHV12 HV01 #V3 #HV32 #HV03
-  elim (IHT12 … HT01) -IHT12 HT01
-    [3: -HV32 HV03 @(thin_skip … HLK) /2/ |2: skip ] (**) (* /3 width=5/ is too slow *)
-    -HLK HW01
-  /3 width=5/
-| #L #K0 #V1 #V2 #V0 #i #HLK0 #HV12 #HV20 #IHV12 #d #e #K #HLK #X #HX
-  lapply (lift_inv_lref2 … HX) -HX * * #Hid #HX destruct -X;
-  [ -HV12;
-    elim (thin_conf_lt … HLK … HLK0 Hid) -HLK HLK0 L #L #V3 #HKL #HK0L #HV31
-    elim (IHV12 … HK0L … HV31) -IHV12 HK0L HV31 #V4 #HV42 #HV34
-    elim (lift_trans_le … HV42 … HV20 ?) -HV42 HV20 V2 // #V2 #HV42
-    >arith5 // -Hid #HV20  
-    @(ex2_1_intro … V2) /2 width=6/ (**) (* /3 width=8/ is slow *)
-  | -IHV12;
-    lapply (thin_conf_ge … HLK … HLK0 Hid) -HLK HLK0 L #HK
-    elim (lift_free … HV20 d (i - e + 1) ? ? ?) -HV20 /2/
-    >arith3 /2/ -Hid /3 width=8/ (**) (* just /3 width=8/ is a bit slow *)
-  ]
-| #L #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #K #HLK #X #HX
-  elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct -X;
-  elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct -Y;
-  elim (IHV12 ? ? ? HLK ? HV01) -IHV12 HV01 #V3 #HV32 #HV03
-  elim (IHW12 ? ? ? HLK ? HW01) -IHW12 #W3 #HW32 #HW03
-  elim (IHT12 … HT01) -IHT12 HT01
-    [3: -HV2 HV32 HV03 HW32 HW03 @(thin_skip … HLK) /2/ |2: skip ] (**) (* /3/ is too slow *)
-    -HLK HW01 #T3 #HT32 #HT03
-  elim (lift_trans_le … HV32 … HV2 ?) -HV32 HV2 V2 // #V2 #HV32 #HV2
-  @(ex2_1_intro … (𝕓{Abbr}W3.𝕗{Appl}V2.T3)) /3/ (**) (* /4/ loops *)
-| #L #V #T #T1 #T2 #HT1 #_ #IHT12 #d #e #K #HLK #X #HX
-  elim (lift_inv_bind2 … HX) -HX #V0 #T0 #_ #HT0 #H destruct -X;
-  elim (lift_conf_rev … HT1 … HT0 ?) -HT1 HT0 T // #T #HT0 #HT1
-  elim (IHT12 … HLK … HT1) -IHT12 HLK HT1 /3 width=5/
-| #L #V #T1 #T2 #_ #IHT12 #d #e #K #HLK #X #HX
-  elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct -X;
-  elim (IHT12 … HLK … HT01) -IHT12 HLK HT01 /3/
-]
-qed.
-
-(* this may be moved *)
 lemma pr_refl: ∀T,L. L ⊢ T ⇒ T.
 #T elim T -T //
 #I elim I -I /2/
 qed.
+(*
+lemma subst_pr: ∀d,e,L,T1,U2. L ⊢ ↓[d,e] T1 ≡ U2 → ∀T2. ↑[d,e] U2 ≡ T2 →
+                L ⊢ T1 ⇒ T2.
+#d #e #L #T1 #U2 #H elim H -H d e L T1 U2
+[ #L #k #d #e #X #HX lapply (lift_inv_sort1 … HX) -HX #HX destruct -X // 
+| #L #i #d #e #Hid #X #HX lapply (lift_inv_sort1 … HX) -HX #HX destruct -X //
+| #L #V1 #V2 #e #HV12 * #V #HV2 #HV1
+  elim (lift_total 0 1 V1) #W1 #HVW1
+  @(ex2_1_intro … W1)
+  [
+  | /2 width=6/  
+
+*)
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