\ /
V_______________________________________________________________ *)
-include "lambda-delta/substitution/thin_defs.ma".
+include "lambda-delta/substitution/drop_defs.ma".
(* SINGLE STEP PARALLEL REDUCTION ON TERMS **********************************)
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.
- â\86\93[0,i] L â\89¡ K. ð\9d\95\93{Abbr} V1 → pr K V1 V2 → ↑[0,i+1] V2 ≡ V →
+ â\86\91[0,i] K. ð\9d\95\93{Abbr} V1 â\89¡ L → 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*)
#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
--- /dev/null
+(*
+ ||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/syntax/lenv.ma".
+include "lambda-delta/substitution/lift_defs.ma".
+
+(* DROPPING *****************************************************************)
+
+inductive drop: lenv → nat → nat → lenv → Prop ≝
+| drop_refl: ∀L. drop L 0 0 L
+| drop_drop: ∀L1,L2,I,V,e. drop L1 0 e L2 → drop (L1. 𝕓{I} V) 0 (e + 1) L2
+| drop_skip: ∀L1,L2,I,V1,V2,d,e.
+ drop L1 d e L2 → ↑[d,e] V2 ≡ V1 →
+ drop (L1. 𝕓{I} V1) (d + 1) e (L2. 𝕓{I} V2)
+.
+
+interpretation "dropping" 'RLift L2 d e L1 = (drop L1 d e L2).
+
+(* the basic inversion lemmas ***********************************************)
+
+lemma drop_inv_skip2_aux: ∀d,e,L1,L2. ↑[d, e] L2 ≡ L1 → 0 < d →
+ ∀I,K2,V2. L2 = K2. 𝕓{I} V2 →
+ ∃∃K1,V1. ↑[d - 1, e] K2 ≡ K1 &
+ ↑[d - 1, e] V2 ≡ V1 &
+ L1 = K1. 𝕓{I} V1.
+#d #e #L1 #L2 #H elim H -H d e L1 L2
+[ #L #H elim (lt_false … H)
+| #L1 #L2 #I #V #e #_ #_ #H elim (lt_false … H)
+| #L1 #X #Y #V1 #Z #d #e #HL12 #HV12 #_ #_ #I #L2 #V2 #H destruct -X Y Z;
+ /2 width=5/
+]
+qed.
+
+lemma drop_inv_skip2: ∀d,e,I,L1,K2,V2. ↑[d, e] K2. 𝕓{I} V2 ≡ L1 → 0 < d →
+ ∃∃K1,V1. ↑[d - 1, e] K2 ≡ K1 & ↑[d - 1, e] V2 ≡ V1 &
+ L1 = K1. 𝕓{I} V1.
+/2/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||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 "lambda-delta/substitution/drop_defs.ma".
+
+(* DROPPING *****************************************************************)
+
+(* the main properties ******************************************************)
+
+axiom drop_conf_ge: ∀d1,e1,L,L1. ↑[d1, e1] L1 ≡ L →
+ ∀e2,L2. ↑[0, e2] L2 ≡ L → d1 + e1 ≤ e2 →
+ ↑[0, e2 - e1] L2 ≡ L1.
+
+axiom drop_conf_lt: ∀d1,e1,L,L1. ↑[d1, e1] L1 ≡ L →
+ ∀e2,K2,I,V2. ↑[0, e2] K2. 𝕓{I} V2 ≡ L →
+ e2 < d1 → let d ≝ d1 - e2 - 1 in
+ ∃∃K1,V1. ↑[0, e2] K1. 𝕓{I} V1 ≡ L1 &
+ ↑[d, e1] K2 ≡ K1 & ↑[d,e1] V1 ≡ V2.
+
+axiom drop_trans_le: ∀d1,e1,L1. ∀L:lenv. ↑[d1, e1] L ≡ L1 →
+ ∀e2,L2. ↑[0, e2] L2 ≡ L → e2 ≤ d1 →
+ ∃∃L0. ↑[0, e2] L0 ≡ L1 & ↑[d1 - e2, e1] L2 ≡ L0.
+
+axiom drop_trans_ge: ∀d1,e1,L1,L. ↑[d1, e1] L ≡ L1 →
+ ∀e2,L2. ↑[0, e2] L2 ≡ L → d1 ≤ e2 → ↑[0, e1 + e2] L2 ≡ L1.
\ /
V_______________________________________________________________ *)
-include "lambda-delta/syntax/lenv.ma".
-include "lambda-delta/substitution/lift_defs.ma".
+include "lambda-delta/substitution/drop_defs.ma".
(* TELESCOPIC SUBSTITUTION **************************************************)
inductive subst: lenv → term → nat → nat → term → Prop ≝
| subst_sort : ∀L,k,d,e. subst L (⋆k) d e (⋆k)
| subst_lref_lt: ∀L,i,d,e. i < d → subst L (#i) d e (#i)
-| subst_lref_O : ∀L,V1,V2,e. subst L V1 0 e V2 →
- subst (L. 𝕓{Abbr} V1) #0 0 (e + 1) V2
-| subst_lref_S : ∀L,I,V,i,T1,T2,d,e.
- d ≤ i → i < d + e → subst L #i d e T1 → ↑[d,1] T1 ≡ T2 →
- subst (L. 𝕓{I} V) #(i + 1) (d + 1) e T2
+| subst_lref_be: ∀L,K,V,U,i,d,e.
+ d ≤ i → i < d + e →
+ ↑[0, i] K. 𝕓{Abbr} V ≡ L → subst K V d (d + e - i - 1) U →
+ subst L (#i) d e U
| subst_lref_ge: ∀L,i,d,e. d + e ≤ i → subst L (#i) d e (#(i - e))
| subst_bind : ∀L,I,V1,V2,T1,T2,d,e.
subst L V1 d e V2 → subst (L. 𝕓{I} V1) T1 (d + 1) e T2 →
interpretation "telescopic substritution" 'RSubst L T1 d e T2 = (subst L T1 d e T2).
+(* The basic inversion lemmas ***********************************************)
+
+lemma subst_inv_lref1_be_aux: ∀d,e,L,T,U. L ⊢ ↓[d, e] T ≡ U →
+ ∀i. d ≤ i → i < d + e → T = #i →
+ ∃∃K,V. ↑[0, i] K. 𝕓{Abbr} V ≡ L &
+ K ⊢ ↓[d, d + e - i - 1] V ≡ U.
+#d #e #L #T #U #H elim H -H d e L T U
+[ #L #k #d #e #i #_ #_ #H destruct
+| #L #j #d #e #Hid #i #Hdi #_ #H destruct -j;
+ lapply (le_to_lt_to_lt … Hdi … Hid) -Hdi Hid #Hdd
+ elim (lt_false … Hdd)
+| #L #K #V #U #j #d #e #_ #_ #HLK #HVU #_ #i #Hdi #Hide #H destruct -j /2/
+| #L #j #d #e #Hdei #i #_ #Hide #H destruct -j;
+ lapply (le_to_lt_to_lt … Hdei … Hide) -Hdei Hide #Hdede
+ elim (lt_false … Hdede)
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #_ #_ #H destruct
+| #L #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #i #_ #_ #H destruct
+]
+qed.
+
+lemma subst_inv_lref1_be: ∀d,e,i,L,U. L ⊢ ↓[d, e] #i ≡ U →
+ d ≤ i → i < d + e →
+ ∃∃K,V. ↑[0, i] K. 𝕓{Abbr} V ≡ L &
+ K ⊢ ↓[d, d + e - i - 1] V ≡ U.
+/2/ qed.
+
(* The basic properties *****************************************************)
lemma subst_lift_inv: ∀d,e,T1,T2. ↑[d,e] T1 ≡ T2 → ∀L. L ⊢ ↓[d,e] T2 ≡ T1.
#d #e #T1 #T2 #H elim H -H d e T1 T2 /2/
-#i #d #e #Hdi #L >(minus_plus_m_m i e) in ⊢ (? ? ? ? ? %) /3/ (**) (* use \ldots *)
+#i #d #e #Hdi #L >(minus_plus_m_m i e) in ⊢ (? ? ? ? ? %) /3/
qed.
+(*
+| subst_lref_O : ∀L,V1,V2,e. subst L V1 0 e V2 →
+ subst (L. 𝕓{Abbr} V1) #0 0 (e + 1) V2
+| subst_lref_S : ∀L,I,V,i,T1,T2,d,e.
+ d ≤ i → i < d + e → subst L #i d e T1 → ↑[d,1] T2 ≡ T1 →
+ subst (L. 𝕓{I} V) #(i + 1) (d + 1) e T2
+*)
projects / helm.git / commitdiff