more lemmas to prove and a correction in subst

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/  
+
+*)
\ No newline at end of file

diff --git a/matita/matita/lib/lambda-delta/substitution/lift.ma b/matita/matita/lib/lambda-delta/substitution/lift.ma
index acd84df08b05038921a2a7e9f7c27e5743157285..60b1b62e74481cbe2876661f8ac448bd59423995 100644 (file)
--- a/matita/matita/lib/lambda-delta/substitution/lift.ma
+++ b/matita/matita/lib/lambda-delta/substitution/lift.ma
@@ -169,6 +169,8 @@ qed.
 
 (* the main properies *******************************************************)
 
+axiom lift_total: ∀d,e,T1. ∃T2. ↑[d,e] T1 ≡ T2.
+
 axiom lift_mono:  ∀d,e,T,U1. ↑[d,e] T ≡ U1 → ∀U2. ↑[d,e] T ≡ U2 → U1 = U2.
 
 theorem lift_conf_rev: ∀d1,e1,T1,T. ↑[d1,e1] T1 ≡ T →

diff --git a/matita/matita/lib/lambda-delta/substitution/subst.ma b/matita/matita/lib/lambda-delta/substitution/subst.ma
index a9732529ddcc0bb4f1f34ceef5fffc508303f5b6..18eb6c5269e0f52be05dd2eee640b4e3998b5c08 100644 (file)
--- a/matita/matita/lib/lambda-delta/substitution/subst.ma
+++ b/matita/matita/lib/lambda-delta/substitution/subst.ma
@@ -17,7 +17,8 @@ include "lambda-delta/substitution/lift.ma".
 inductive subst: lenv → term → nat → nat → term → Prop ≝
 | subst_sort   : ∀L,k,d,e. subst L (⋆k) d e (⋆k)
 | subst_lref_lt: ∀L,i,d,e. i < d → subst L (#i) d e (#i)
-| subst_lref_O : ∀L,V,e. 0 < e → subst (L. 𝕓{Abbr} V) #0 0 e V
+| subst_lref_O : ∀L,V1,V2,e. subst L V1 0 e V2 →
+                 subst (L. 𝕓{Abbr} V1) #0 0 (e + 1) V2
 | subst_lref_S : ∀L,I,V,i,T1,T2,d,e.
                  d ≤ i → i < d + e → subst L #i d e T1 → ↑[d,1] T1 ≡ T2 →
                  subst (L. 𝕓{I} V) #(i + 1) (d + 1) e T2

diff --git a/matita/matita/lib/lambda-delta/substitution/thin.ma b/matita/matita/lib/lambda-delta/substitution/thin.ma
index 96432922dc5d1dd29ededb2e0d2fdd912ed754cd..be795742294b920ba6b3164c81312a5445a47f95 100644 (file)
--- a/matita/matita/lib/lambda-delta/substitution/thin.ma
+++ b/matita/matita/lib/lambda-delta/substitution/thin.ma
@@ -23,6 +23,26 @@ inductive thin: lenv → nat → nat → lenv → Prop ≝
 
 interpretation "thinning" 'RSubst L1 d e L2 = (thin L1 d e L2).
 
+(* the basic inversion lemmas ***********************************************)
+
+lemma thin_inv_skip2_aux: ∀d,e,L1,L2. ↓[d, e] L1 ≡ L2 → 0 < d →
+                          ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
+                          ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 &
+                                   K1 ⊢ ↓[d - 1, e] V1 ≡ V2 & 
+                                   L1 = K1. 𝕓{I} V1.
+#d #e #L1 #L2 #H elim H -H d e L1 L2
+[ #L #H elim (lt_false … H)
+| #L1 #L2 #I #V #e #_ #_ #H elim (lt_false … H)
+| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #_ #I #L2 #V2 #H destruct -X Y Z;
+  /2 width=5/
+]
+qed.
+
+lemma thin_inv_skip2: ∀d,e,I,L1,K2,V2. ↓[d, e] L1 ≡ K2. 𝕓{I} V2 → 0 < d →
+                      ∃∃K1,V1. ↓[d - 1, e] K1 ≡ K2 & K1 ⊢ ↓[d - 1, e] V1 ≡ V2 &
+                               L1 = K1. 𝕓{I} V1.
+/2/ qed.
+
 (* the main properties ******************************************************)
 
 axiom thin_conf_ge: ∀d1,e1,L,L1. ↓[d1,e1] L ≡ L1 →
@@ -32,3 +52,10 @@ axiom thin_conf_lt: ∀d1,e1,L,L1. ↓[d1,e1] L ≡ L1 →
                     ∀e2,K2,I,V2. ↓[0,e2] L ≡ K2. 𝕓{I} V2 →
                     e2 < d1 → let d ≝ d1 - e2 - 1 in
                     ∃∃K1,V1. ↓[0,e2] L1 ≡ K1. 𝕓{I} V1 & ↓[d,e1] K2 ≡ K1 & ↑[d,e1] V1 ≡ V2.
+
+axiom thin_trans_le: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
+                     ∀e2,L2. ↓[0, e2] L ≡ L2 → e2 ≤ d1 →
+                     ∃∃L0. ↓[0, e2] L1 ≡ L0 & ↓[d1 - e2, e1] L0 ≡ L2.
+
+axiom thin_trans_ge: ∀d1,e1,L1,L. ↓[d1, e1] L1 ≡ L →
+                     ∀e2,L2. ↓[0, e2] L ≡ L2 → d1 ≤ e2 → ↓[0, e1 + e2] L1 ≡ L2.