some restructuring

[helm.git] / matita / matita / lib / lambda-delta / reduction / pr_defs.ma

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+(*
+    ||M||  This file is part of HELM, an Hypertextual, Electronic
+    ||A||  Library of Mathematics, developed at the Computer Science
+    ||T||  Department of the University of Bologna, Italy.
+    ||I||
+    ||T||
+    ||A||  This file is distributed under the terms of the
+    \   /  GNU General Public License Version 2
+     \ /
+      V_______________________________________________________________ *)
+
+include "lambda-delta/substitution/thin_defs.ma".
+
+(* SINGLE STEP PARALLEL REDUCTION ON TERMS **********************************)
+
+inductive pr: lenv → term → term → Prop ≝
+| pr_sort : ∀L,k. pr L (⋆k) (⋆k)
+| pr_lref : ∀L,i. pr L (#i) (#i)
+| pr_bind : ∀L,I,V1,V2,T1,T2. pr L V1 V2 → pr (L. 𝕓{I} V1) T1 T2 →
+            pr L (𝕓{I} V1. T1) (𝕓{I} V2. T2)
+| pr_flat : ∀L,I,V1,V2,T1,T2. pr L V1 V2 → pr L T1 T2 →
+            pr L (𝕗{I} V1. T1) (𝕗{I} V2. T2)
+| pr_beta : ∀L,V1,V2,W,T1,T2.
+            pr L V1 V2 → pr (L. 𝕓{Abst} W) T1 T2 → (*𝕓*)
+            pr L (𝕚{Appl} V1. 𝕚{Abst} W. T1) (𝕚{Abbr} V2. T2)
+| pr_delta: ∀L,K,V1,V2,V,i.
+            ↓[0,i] L ≡ K. 𝕓{Abbr} V1 → pr K V1 V2 → ↑[0,i+1] V2 ≡ V →
+            pr L (#i) V
+| pr_theta: ∀L,V,V1,V2,W1,W2,T1,T2.
+            pr L V1 V2 → ↑[0,1] V2 ≡ V → pr L W1 W2 → pr (L. 𝕓{Abbr} W1) T1 T2 → (*𝕓*)
+            pr L (𝕚{Appl} V1. 𝕚{Abbr} W1. T1) (𝕚{Abbr} W2. 𝕚{Appl} V. T2)
+| pr_zeta : ∀L,V,T,T1,T2. ↑[0,1] T1 ≡ T → pr L T1 T2 →
+            pr L (𝕚{Abbr} V. T) T2
+| pr_tau  : ∀L,V,T1,T2. pr L T1 T2 → pr L (𝕚{Cast} V. T1) T2
+.
+
+interpretation
+   "single step parallel reduction (term)"
+   'PR L T1 T2 = (pr L T1 T2).
+
+(* The basic properties *****************************************************)
+
+lemma pr_refl: ∀T,L. L ⊢ T ⇒ T.
+#T elim T -T //
+#I elim I -I /2/
+qed.
+(*
+lemma subst_pr: ∀d,e,L,T1,U2. L ⊢ ↓[d,e] T1 ≡ U2 → ∀T2. ↑[d,e] U2 ≡ T2 →
+                L ⊢ T1 ⇒ T2.
+#d #e #L #T1 #U2 #H elim H -H d e L T1 U2
+[ #L #k #d #e #X #HX lapply (lift_inv_sort1 … HX) -HX #HX destruct -X // 
+| #L #i #d #e #Hid #X #HX lapply (lift_inv_sort1 … HX) -HX #HX destruct -X //
+| #L #V1 #V2 #e #HV12 * #V #HV2 #HV1
+  elim (lift_total 0 1 V1) #W1 #HVW1
+  @(ex2_1_intro … W1)
+  [
+  | /2 width=6/  
+
+*)
\ No newline at end of file