update in ground_2, static_2, basic_2, apps_2, alpha_1

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / rt_computation / lpxs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma
index 507e0c3effa0153a61415a01301eaf7438839d41..ade37fde5a7aa19e2b53c0be3824591792928263 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma
@@ -13,28 +13,69 @@
 (**************************************************************************)
 
 include "basic_2/notation/relations/predtysnstar_4.ma".
-include "basic_2/relocation/lex.ma".
+include "static_2/relocation/lex.ma".
 include "basic_2/rt_computation/cpxs_ext.ma".
 
-(* UNCOUNTED PARALLEL RT-COMPUTATION FOR LOCAL ENVIRONMENTS *****************)
+(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
 
-definition lpxs: ∀h. relation3 genv lenv lenv ≝
-                 λh,G. lex (cpxs h G).
+definition lpxs (h) (G): relation lenv ≝
+                         lex (cpxs h G).
 
 interpretation
-   "uncounted parallel rt-computation (local environment)"
+   "unbound parallel rt-computation on all entries (local environment)"
    'PRedTySnStar h G L1 L2 = (lpxs h G L1 L2).
 
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: uses: lpxs_pair_refl *)
+lemma lpxs_bind_refl_dx (h) (G): ∀L1,L2. ❪G,L1❫ ⊢ ⬈*[h] L2 →
+                                 ∀I. ❪G,L1.ⓘ[I]❫ ⊢ ⬈*[h] L2.ⓘ[I].
+/2 width=1 by lex_bind_refl_dx/ qed.
+
+lemma lpxs_pair (h) (G): ∀L1,L2. ❪G,L1❫ ⊢ ⬈*[h] L2 →
+                         ∀V1,V2. ❪G,L1❫ ⊢ V1 ⬈*[h] V2 →
+                         ∀I. ❪G,L1.ⓑ[I]V1❫ ⊢ ⬈*[h] L2.ⓑ[I]V2.
+/2 width=1 by lex_pair/ qed.
+
+lemma lpxs_refl (h) (G): reflexive … (lpxs h G).
+/2 width=1 by lex_refl/ qed.
+
 (* Basic inversion lemmas ***************************************************)
 
-lemma lpxs_inv_bind_sn: ∀h,G,I1,L2,K1. ⦃G, K1.ⓘ{I1}⦄ ⊢⬈*[h] L2 →
-                        ∃∃I2,K2. ⦃G, K1⦄ ⊢⬈*[h] K2 & ⦃G, K1⦄ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ{I2}.
+(* Basic_2A1: was: lpxs_inv_atom1 *)
+lemma lpxs_inv_atom_sn (h) (G): ∀L2. ❪G,⋆❫ ⊢ ⬈*[h] L2 → L2 = ⋆.
+/2 width=2 by lex_inv_atom_sn/ qed-.
+
+lemma lpxs_inv_bind_sn (h) (G): ∀I1,L2,K1. ❪G,K1.ⓘ[I1]❫ ⊢ ⬈*[h] L2 →
+                                ∃∃I2,K2. ❪G,K1❫ ⊢ ⬈*[h] K2 & ❪G,K1❫ ⊢ I1 ⬈*[h] I2 & L2 = K2.ⓘ[I2].
 /2 width=1 by lex_inv_bind_sn/ qed-.
 
-lemma lpxs_inv_pair_sn: ∀h,G,I,L2,K1,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢⬈*[h] L2 →
-                        ∃∃K2,V2. ⦃G, K1⦄ ⊢⬈*[h] K2 & ⦃G, K1⦄ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ{I}V2.
-#h #G #I #L2 #K1 #V1 #H
-elim (lpxs_inv_bind_sn … H) -H #Y #K2 #HK12 #H0 #H destruct
-elim (ext2_inv_pair_sn … H0) -H0 #V2 #HV12 #H destruct
-/2 width=5 by ex3_2_intro/
-qed-.
+(* Basic_2A1: was: lpxs_inv_pair1 *)
+lemma lpxs_inv_pair_sn (h) (G): ∀I,L2,K1,V1. ❪G,K1.ⓑ[I]V1❫ ⊢ ⬈*[h] L2 →
+                                ∃∃K2,V2. ❪G,K1❫ ⊢ ⬈*[h] K2 & ❪G,K1❫ ⊢ V1 ⬈*[h] V2 & L2 = K2.ⓑ[I]V2.
+/2 width=1 by lex_inv_pair_sn/ qed-.
+
+(* Basic_2A1: was: lpxs_inv_atom2 *)
+lemma lpxs_inv_atom_dx (h) (G): ∀L1. ❪G,L1❫ ⊢ ⬈*[h] ⋆ → L1 = ⋆.
+/2 width=2 by lex_inv_atom_dx/ qed-.
+
+(* Basic_2A1: was: lpxs_inv_pair2 *)
+lemma lpxs_inv_pair_dx (h) (G): ∀I,L1,K2,V2. ❪G,L1❫ ⊢ ⬈*[h] K2.ⓑ[I]V2 →
+                                ∃∃K1,V1. ❪G,K1❫ ⊢ ⬈*[h] K2 & ❪G,K1❫ ⊢ V1 ⬈*[h] V2 & L1 = K1.ⓑ[I]V1.
+/2 width=1 by lex_inv_pair_dx/ qed-.
+
+(* Basic eliminators ********************************************************)
+
+(* Basic_2A1: was: lpxs_ind_alt *)
+lemma lpxs_ind (h) (G): ∀Q:relation lenv.
+                        Q (⋆) (⋆) → (
+                          ∀I,K1,K2.
+                          ❪G,K1❫ ⊢ ⬈*[h] K2 →
+                          Q K1 K2 → Q (K1.ⓘ[I]) (K2.ⓘ[I])
+                        ) → (
+                          ∀I,K1,K2,V1,V2.
+                          ❪G,K1❫ ⊢ ⬈*[h] K2 → ❪G,K1❫ ⊢ V1 ⬈*[h] V2 →
+                          Q K1 K2 → Q (K1.ⓑ[I]V1) (K2.ⓑ[I]V2)
+                        ) →
+                        ∀L1,L2. ❪G,L1❫ ⊢ ⬈*[h] L2 → Q L1 L2.
+/3 width=4 by lex_ind/ qed-.