partial update update in basic_2

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / etc / lpx_sn / lpx_sn_tc.etc

diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_tc.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_tc.etc
index cb772458bdb089360ab6ba3476f0036cc92d412e..51b09380f28920c2f1ac6bca6e559d6e52bc3834 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_tc.etc
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lpx_sn/lpx_sn_tc.etc
@@ -18,32 +18,74 @@ include "basic_2/substitution/lpx_sn.ma".
 
 (* Properties on transitive_closure *****************************************)
 
-lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
-                            ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
-                            TC … (lpx_sn R) L1 L2.
-#R #HR #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
+lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) →
+                           ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+                           ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V).
+#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
+[ /2 width=1 by lpx_sn_refl/
+| /3 width=1 by TC_reflexive, lpx_sn_refl/
+| /3 width=5 by lpx_sn_pair, step/
+]
+qed-.
+
+lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
+                      ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
+                      ∀V1,V2. LTC … R L1 V1 V2 →
+                      TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
+#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
+[ /2 width=1 by TC_lpx_sn_pair_refl/
+| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
+]
 qed-.
 
 (* Inversion lemmas on transitive closure ***********************************)
 
-lemma TC_lpx_sn_ind: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
-                     ∀S:relation lenv.
-                     S (⋆) (⋆) → (
-                        ∀I,K1,K2,V1,V2.
-                        TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
-                        S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
-                     ) →
-                     ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
-#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
-[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
-| #L2 #I #V2 #IHL2 #X #H
-  elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
-  #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
+lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
+#R #L1 #H @(TC_ind_dx … L1 H) -L1
+[ /2 width=2 by lpx_sn_inv_atom2/
+| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
 ]
 qed-.
 
-lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
-                                ∀L1,L2. TC … (lpx_sn R) L1 L2 →
-                                lpx_sn (LTC … R) L1 L2.
-/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
+lemma TC_lpx_sn_inv_pair2: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+                           ∀I,L1,K2,V2. TC  … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
+                           ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
+[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
+| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
+  elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
+  lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
+#R #L2 #H elim H -L2
+[ /2 width=2 by lpx_sn_inv_atom1/
+| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
+]
+qed-.
+
+fact TC_lpx_sn_inv_pair1_aux: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+                              ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+                              ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+                              ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
+[ #J #K #W #H destruct
+| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair1: ∀R. s_rs_transitive … R (λ_. lpx_sn R) →
+                           ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
+                           ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
+
+(* Forward lemmas on transitive closure *************************************)
+
+lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L2
+[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
+| #L #L2 #_ #HL2 #IHL1
+  >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
+]
+qed-.