- lambda_delta: programmed renaming to lambdadelta

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / equivalence / cpcs.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+include "basic_2/conversion/cpc.ma".
+
+(* CONTEXT-SENSITIVE PARALLEL EQUIVALENCE ON TERMS **************************)
+
+definition cpcs: lenv → relation term ≝
+                 λL. TC … (cpc L).
+
+interpretation "context-sensitive parallel equivalence (term)"
+   'PConvStar L T1 T2 = (cpcs L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma cpcs_ind: ∀L,T1. ∀R:predicate term. R T1 →
+                (∀T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → R T → R T2) →
+                ∀T2. L ⊢ T1 ⬌* T2 → R T2.
+#L #T1 #R #HT1 #IHT1 #T2 #HT12 @(TC_star_ind … HT1 IHT1 … HT12) //
+qed-.
+
+lemma cpcs_ind_dx: ∀L,T2. ∀R:predicate term. R T2 →
+                   (∀T1,T. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → R T → R T1) →
+                   ∀T1. L ⊢ T1 ⬌* T2 → R T1.
+#L #T2 #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx … HT2 IHT2 … HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: pc3_refl *)
+lemma cpcs_refl: ∀L. reflexive … (cpcs L).
+/2 width=1/ qed.
+
+(* Basic_1: was: pc3_s *)
+lemma cpcs_sym: ∀L. symmetric … (cpcs L).
+/3 width=1/ qed.
+
+lemma cpcs_strap1: ∀L,T1,T,T2. L ⊢ T1 ⬌* T → L ⊢ T ⬌ T2 → L ⊢ T1 ⬌* T2.
+/2 width=3/ qed.
+
+lemma cpcs_strap2: ∀L,T1,T,T2. L ⊢ T1 ⬌ T → L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+/2 width=3/ qed.
+
+(* Basic_1: was: pc3_pr2_r *)
+lemma cpcs_cpr_dx: ∀L,T1,T2. L ⊢ T1 ➡ T2 → L ⊢ T1 ⬌* T2.
+/3 width=1/ qed.
+
+lemma cpcs_tpr_dx: ∀L,T1,T2. T1 ➡ T2 → L ⊢ T1 ⬌* T2.
+/3 width=1/ qed.
+
+(* Basic_1: was: pc3_pr2_x *)
+lemma cpcs_cpr_sn: ∀L,T1,T2. L ⊢ T2 ➡ T1 → L ⊢ T1 ⬌* T2.
+/3 width=1/ qed.
+
+lemma cpcs_tpr_sn: ∀L,T1,T2. T2 ➡ T1 → L ⊢ T1 ⬌* T2.
+/3 width=1/ qed.
+
+lemma cpcs_cpr_strap1: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T ➡ T2 → L ⊢ T1 ⬌* T2.
+/3 width=3/ qed.
+
+(* Basic_1: was: pc3_pr2_u *)
+lemma cpcs_cpr_strap2: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+/3 width=3/ qed.
+
+lemma cpcs_cpr_div: ∀L,T1,T. L ⊢ T1 ⬌* T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
+/3 width=3/ qed.
+
+lemma cpr_div: ∀L,T1,T. L ⊢ T1 ➡ T → ∀T2. L ⊢ T2 ➡ T → L ⊢ T1 ⬌* T2.
+/3 width=3/ qed-.
+
+(* Basic_1: was: pc3_pr2_u2 *)
+lemma cpcs_cpr_conf: ∀L,T1,T. L ⊢ T ➡ T1 → ∀T2. L ⊢ T ⬌* T2 → L ⊢ T1 ⬌* T2.
+/3 width=3/ qed.
+
+(* Basic_1: removed theorems 9:
+            clear_pc3_trans pc3_ind_left
+            pc3_head_1 pc3_head_2 pc3_head_12 pc3_head_21
+            pc3_pr2_fsubst0 pc3_pr2_fsubst0_back pc3_fsubst0
+   Basic_1: removed local theorems 6:
+            pc3_left_pr3 pc3_left_trans pc3_left_sym pc3_left_pc3 pc3_pc3_left
+            pc3_wcpr0_t_aux
+*)