- dynamic type assignment dismissed for now

[helm.git] / matita / matita / contribs / lambda_delta / basic_2 / reducibility / cpr_lift.ma

diff --git a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_lift.ma b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_lift.ma
index eb5fb1471a24bc17b5e62fdc35f2d19f3991a903..b3dbcf40c3d99a932058f61a042a22e3b87883d1 100644 (file)
--- a/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_lift.ma
+++ b/matita/matita/contribs/lambda_delta/basic_2/reducibility/cpr_lift.ma
@@ -21,7 +21,7 @@ include "basic_2/reducibility/cpr.ma".
 (* Advanced properties ******************************************************)
 
 lemma cpr_cdelta: ∀L,K,V1,W1,W2,i.
-                  ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 [0, |L| - i - 1] ▶* W1 →
+                  ⇩[0, i] L ≡ K. ⓓV1 → K ⊢ V1 ▶* [0, |L| - i - 1] W1 →
                   ⇧[0, i + 1] W1 ≡ W2 → L ⊢ #i ➡ W2.
 #L #K #V1 #W1 #W2 #i #HLK #HVW1 #HW12
 lapply (ldrop_fwd_ldrop2_length … HLK) #Hi
@@ -32,18 +32,29 @@ lemma cpr_abst: ∀L,V1,V2. L ⊢ V1 ➡ V2 → ∀V,T1,T2.
                 L.ⓛV ⊢ T1 ➡ T2 → L ⊢ ⓛV1. T1 ➡ ⓛV2. T2.
 #L #V1 #V2 * #V0 #HV10 #HV02 #V #T1 #T2 * #T0 #HT10 #HT02
 lapply (tpss_inv_S2 … HT02 L V ?) -HT02 // #HT02
-@(ex2_1_intro … (ⓛV0.T0)) /2 width=1/ -V1 -T1 (**) (* explicit constructors *)
-@tpss_bind // -V0
-@(tpss_lsubs_conf (L.ⓛV)) // -T0 -T2 /2 width=1/
+lapply (tpss_lsubs_trans … HT02 (L.ⓛV2) ?) -HT02 /2 width=1/ #HT02
+@(ex2_1_intro … (ⓛV0.T0)) /2 width=1/ (* explicit constructors *)
 qed.
 
+lemma cpr_beta: ∀L,V1,V2,W,T1,T2.
+                L ⊢ V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2.
+#L #V1 #V2 #W #T1 #T2 * #V #HV1 #HV2 * #T #HT1 #HT2
+lapply (tpss_inv_S2 … HT2 L W ?) -HT2 // #HT2
+lapply (tpss_lsubs_trans … HT2 (L.ⓓV2) ?) -HT2 /2 width=1/ #HT2
+@(ex2_1_intro … (ⓓV.T)) /2 width=1/ (**) (* explicit constructor, /3/ is too slow *)
+qed.
+
+lemma cpr_beta_dx: ∀L,V1,V2,W,T1,T2.
+                   V1 ➡ V2 → L.ⓛW ⊢ T1 ➡ T2 → L ⊢ ⓐV1.ⓛW.T1 ➡ ⓓV2.T2.
+/3 width=1/ qed.
+
 (* Advanced inversion lemmas ************************************************)
 
 (* Basic_1: was: pr2_gen_lref *)
 lemma cpr_inv_lref1: ∀L,T2,i. L ⊢ #i ➡ T2 →
                      T2 = #i ∨
                      ∃∃K,V1,T1. ⇩[0, i] L ≡ K. ⓓV1 &
-                                K ⊢ V1 [0, |L| - i - 1] ▶* T1 &
+                                K ⊢ V1 ▶* [0, |L| - i - 1] T1 &
                                 ⇧[0, i + 1] T1 ≡ T2 &
                                 i < |L|.
 #L #T2 #i * #X #H
@@ -52,13 +63,36 @@ elim (tpss_inv_lref1 … H) -H /2 width=1/
 * /3 width=6/
 qed-.
 
+(* Basic_1: was pr2_gen_abbr *)
+lemma cpr_inv_abbr1: ∀L,V1,T1,U2. L ⊢ ⓓV1. T1 ➡ U2 →
+                     (∃∃V,V2,T2. V1 ➡ V & L ⊢ V ▶* [O, |L|] V2 &
+                                 L. ⓓV ⊢ T1 ➡ T2 &
+                                 U2 = ⓓV2. T2
+                      ) ∨
+                      ∃∃T2. L.ⓓV1 ⊢ T1 ➡ T2 & ⇧[0,1] U2 ≡ T2.
+#L #V1 #T1 #Y * #X #H1 #H2
+elim (tpr_inv_abbr1 … H1) -H1 *
+[ #V #T #T0 #HV1 #HT1 #HT0 #H destruct
+  elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT02 #H destruct
+  lapply (tps_lsubs_trans … HT0 (L. ⓓV) ?) -HT0 /2 width=1/ #HT0
+  lapply (tps_weak_all … HT0) -HT0 #HT0
+  lapply (tpss_lsubs_trans … HT02 (L. ⓓV) ?) -HT02 /2 width=1/ #HT02
+  lapply (tpss_weak_all … HT02) -HT02 #HT02
+  lapply (tpss_strap2 … HT0 HT02) -T0 /4 width=7/
+| #T2 #HT12 #HXT2
+  elim (lift_total Y 0 1) #Z #HYZ
+  lapply (tpss_lift_ge … H2 (L.ⓓV1) … HXT2 … HYZ) -X // /2 width=1/ #H
+  lapply (cpr_intro … HT12 … H) -T2 /3 width=3/
+]
+qed-.
+
 (* Basic_1: was: pr2_gen_abst *)
 lemma cpr_inv_abst1: ∀L,V1,T1,U2. L ⊢ ⓛV1. T1 ➡ U2 → ∀I,W.
                      ∃∃V2,T2. L ⊢ V1 ➡ V2 & L. ⓑ{I} W ⊢ T1 ➡ T2 & U2 = ⓛV2. T2.
 #L #V1 #T1 #Y * #X #H1 #H2 #I #W
 elim (tpr_inv_abst1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
 elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
-lapply (tpss_lsubs_conf … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
+lapply (tpss_lsubs_trans … HT2 (L. ⓑ{I} W) ?) -HT2 /2 width=1/ /4 width=5/
 qed-.
 
 (* Basic_1: was pr2_gen_appl *)
@@ -88,7 +122,7 @@ elim (tpr_inv_appl1 … H1) -H1 *
 qed-.
 
 (* Note: the main property of simple terms *)
-lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒[T1] →
+lemma cpr_inv_appl1_simple: ∀L,V1,T1,U. L ⊢ ⓐV1. T1 ➡ U → 𝐒⦃T1⦄ →
                             ∃∃V2,T2. L ⊢ V1 ➡ V2 & L ⊢ T1 ➡ T2 &
                                      U = ⓐV2. T2.
 #L #V1 #T1 #U #H #HT1
@@ -121,7 +155,7 @@ lemma cpr_inv_lift: ∀L,K,d,e. ⇩[d, e] L ≡ K →
                     ∀T1,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. L ⊢ U1 ➡ U2 →
                     ∃∃T2. ⇧[d, e] T2 ≡ U2 & K ⊢ T1 ➡ T2.
 #L #K #d #e #HLK #T1 #U1 #HTU1 #U2 * #U #HU1 #HU2
-elim (tpr_inv_lift … HU1 … HTU1) -U1 #T #HTU #T1
+elim (tpr_inv_lift1 … HU1 … HTU1) -U1 #T #HTU #T1
 elim (lt_or_ge (|L|) d) #HLd
 [ elim (tpss_inv_lift1_le … HU2 … HLK … HTU ?) -U -HLK [ /5 width=4/ | /2 width=2/ ]
 | elim (lt_or_ge (|L|) (d + e)) #HLde