syntactic components detached from basic_2 become static_2

[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 b6c91c8b124135cc7625cb1789b3e4bf9e7a8748..322f437124e4b37efefbde394cc60b118da5110f 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/lpxs.ma
@@ -12,63 +12,70 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "basic_2/notation/relations/predsnstar_5.ma".
-include "basic_2/reduction/lpx.ma".
-include "basic_2/computation/lprs.ma".
+include "basic_2/notation/relations/predtysnstar_4.ma".
+include "static_2/relocation/lex.ma".
+include "basic_2/rt_computation/cpxs_ext.ma".
 
-(* SN EXTENDED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *******************)
+(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
 
-definition lpxs: ∀h. sd h → relation3 genv lenv lenv ≝
-                 λh,o,G. TC … (lpx h o G).
+definition lpxs (h) (G): relation lenv ≝
+                         lex (cpxs h G).
 
-interpretation "extended parallel computation (local environment, sn variant)"
-   'PRedSnStar h o G L1 L2 = (lpxs h o G L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma lpxs_ind: ∀h,o,G,L1. ∀R:predicate lenv. R L1 →
-                (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L → ⦃G, L⦄ ⊢ ➡[h, o] L2 → R L → R L2) →
-                ∀L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → R L2.
-#h #o #G #L1 #R #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma lpxs_ind_dx: ∀h,o,G,L2. ∀R:predicate lenv. R L2 →
-                   (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h, o] L → ⦃G, L⦄ ⊢ ➡*[h, o] L2 → R L → R L1) →
-                   ∀L1. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → R L1.
-#h #o #G #L2 #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
+interpretation
+   "unbound parallel rt-computation on all entries (local environment)"
+   'PRedTySnStar h G L1 L2 = (lpxs h G L1 L2).
 
 (* Basic properties *********************************************************)
 
-lemma lprs_lpxs: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/3 width=3 by lpr_lpx, monotonic_TC/ qed.
+(* 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 lpx_lpxs: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/2 width=1 by inj/ 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,o,G,L. ⦃G, L⦄ ⊢ ➡*[h, o] L.
-/2 width=1 by lprs_lpxs/ qed.
+lemma lpxs_refl (h) (G): reflexive … (lpxs h G).
+/2 width=1 by lex_refl/ qed.
 
-lemma lpxs_strap1: ∀h,o,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L → ⦃G, L⦄ ⊢ ➡[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/2 width=3 by step/ qed.
+(* Basic inversion lemmas ***************************************************)
 
-lemma lpxs_strap2: ∀h,o,G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡[h, o] L → ⦃G, L⦄ ⊢ ➡*[h, o] L2 → ⦃G, L1⦄ ⊢ ➡*[h, o] L2.
-/2 width=3 by TC_strap/ qed.
+(* 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_pair_refl: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡*[h, o] L2.ⓑ{I}V.
-/2 width=1 by TC_lpx_sn_pair_refl/ 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-.
 
-(* Basic inversion lemmas ***************************************************)
+(* 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-.
 
-lemma lpxs_inv_atom1: ∀h,o,G,L2. ⦃G, ⋆⦄ ⊢ ➡*[h, o] L2 → L2 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom1/ 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-.
 
-lemma lpxs_inv_atom2: ∀h,o,G,L1. ⦃G, L1⦄ ⊢ ➡*[h, o] ⋆ → L1 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom2/ 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 forward lemmas *****************************************************)
+(* Basic eliminators ********************************************************)
 
-lemma lpxs_fwd_length: ∀h,o,G,L1,L2. ⦃G, L1⦄ ⊢ ➡*[h, o] L2 → |L1| = |L2|.
-/2 width=2 by TC_lpx_sn_fwd_length/ qed-.
+(* 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-.