(* THE FORMAL SYSTEM λδ: MATITA SOURCE FILES
* Suggested invocation to start formal specifications with:
* - Patience on me to gain peace and perfection! -
+ * 2012 July 26:
+ * term binders polarized to control ζ reduction.
* 2012 April 16 (anniversary milestone):
* context-sensitive subject equivalence for atomic arity assignment.
* 2012 March 15:
--- /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 "basic_2/grammar/lenv_length.ma".
+
+(* POINTWISE EXTENSION OF A CONTEXT-FREE REALTION FOR TERMS *****************)
+
+inductive lpx (R:relation term): relation lenv ≝
+| lpx_stom: lpx R (⋆) (⋆)
+| lpx_pair: ∀K1,K2,I,V1,V2.
+ lpx R K1 K2 → R V1 V2 → lpx R (K1. ⓑ{I} V1) (K2. ⓑ{I} V2)
+.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_refl: ∀R. reflexive ? R → reflexive … (lpx R).
+#R #HR #L elim L -L // /2 width=1/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lpx_inv_atom1_aux: ∀R,L1,L2. lpx R L1 L2 → L1 = ⋆ → L2 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #K1 #K2 #I #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_inv_atom1: ∀R,L2. lpx R (⋆) L2 → L2 = ⋆.
+/2 width=4 by lpx_inv_atom1_aux/ qed-.
+
+fact lpx_inv_pair1_aux: ∀R,L1,L2. lpx R L1 L2 → ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
+ ∃∃K2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #L1 #L2 * -L1 -L2
+[ #K1 #I #V1 #H destruct
+| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #L #J #W #H destruct /2 width=5/
+]
+qed-.
+
+lemma lpx_inv_pair1: ∀R,K1,I,V1,L2. lpx R (K1. ⓑ{I} V1) L2 →
+ ∃∃K2,V2. lpx R K1 K2 & R V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by lpx_inv_pair1_aux/ qed-.
+
+fact lpx_inv_atom2_aux: ∀R,L1,L2. lpx R L1 L2 → L2 = ⋆ → L1 = ⋆.
+#R #L1 #L2 * -L1 -L2
+[ //
+| #K1 #K2 #I #V1 #V2 #_ #_ #H destruct
+]
+qed-.
+
+lemma lpx_inv_atom2: ∀R,L1. lpx R L1 (⋆) → L1 = ⋆.
+/2 width=4 by lpx_inv_atom2_aux/ qed-.
+
+fact lpx_inv_pair2_aux: ∀R,L1,L2. lpx R L1 L2 → ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
+ ∃∃K1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #L1 #L2 * -L1 -L2
+[ #K2 #I #V2 #H destruct
+| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #K #J #W #H destruct /2 width=5/
+]
+qed-.
+
+lemma lpx_inv_pair2: ∀R,L1,K2,I,V2. lpx R L1 (K2. ⓑ{I} V2) →
+ ∃∃K1,V1. lpx R K1 K2 & R V1 V2 & L1 = K1. ⓑ{I} V1.
+/2 width=3 by lpx_inv_pair2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lpx_fwd_length: ∀R,L1,L2. lpx R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L1 -L2 normalize //
+qed-.
(* *)
(**************************************************************************)
+include "basic_2/grammar/lenv_px.ma".
include "basic_2/reducibility/tpr.ma".
(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
-inductive ltpr: relation lenv ≝
-| ltpr_stom: ltpr (⋆) (⋆)
-| ltpr_pair: ∀K1,K2,I,V1,V2.
- ltpr K1 K2 → V1 ➡ V2 → ltpr (K1. ⓑ{I} V1) (K2. ⓑ{I} V2) (*ⓑ*)
-.
+definition ltpr: relation lenv ≝ lpx tpr.
interpretation
"context-free parallel reduction (environment)"
(* Basic properties *********************************************************)
-lemma ltpr_refl: ∀L:lenv. L ➡ L.
-#L elim L -L // /2 width=1/
-qed.
+lemma ltpr_refl: reflexive … ltpr.
+/2 width=1/ qed.
(* Basic inversion lemmas ***************************************************)
-fact ltpr_inv_atom1_aux: ∀L1,L2. L1 ➡ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 * -L1 -L2
-[ //
-| #K1 #K2 #I #V1 #V2 #_ #_ #H destruct
-]
-qed.
-
(* Basic_1: was: wcpr0_gen_sort *)
lemma ltpr_inv_atom1: ∀L2. ⋆ ➡ L2 → L2 = ⋆.
-/2 width=3/ qed-.
-
-fact ltpr_inv_pair1_aux: ∀L1,L2. L1 ➡ L2 → ∀K1,I,V1. L1 = K1. ⓑ{I} V1 →
- ∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
-#L1 #L2 * -L1 -L2
-[ #K1 #I #V1 #H destruct
-| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #L #J #W #H destruct /2 width=5/
-]
-qed.
+/2 width=2 by lpx_inv_atom1/ qed-.
(* Basic_1: was: wcpr0_gen_head *)
lemma ltpr_inv_pair1: ∀K1,I,V1,L2. K1. ⓑ{I} V1 ➡ L2 →
∃∃K2,V2. K1 ➡ K2 & V1 ➡ V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3/ qed-.
-
-fact ltpr_inv_atom2_aux: ∀L1,L2. L1 ➡ L2 → L2 = ⋆ → L1 = ⋆.
-#L1 #L2 * -L1 -L2
-[ //
-| #K1 #K2 #I #V1 #V2 #_ #_ #H destruct
-]
-qed.
+/2 width=1 by lpx_inv_pair1/ qed-.
lemma ltpr_inv_atom2: ∀L1. L1 ➡ ⋆ → L1 = ⋆.
-/2 width=3/ qed-.
-
-fact ltpr_inv_pair2_aux: ∀L1,L2. L1 ➡ L2 → ∀K2,I,V2. L2 = K2. ⓑ{I} V2 →
- ∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
-#L1 #L2 * -L1 -L2
-[ #K2 #I #V2 #H destruct
-| #K1 #K2 #I #V1 #V2 #HK12 #HV12 #K #J #W #H destruct /2 width=5/
-]
-qed.
+/2 width=2 by lpx_inv_atom2/ qed-.
lemma ltpr_inv_pair2: ∀L1,K2,I,V2. L1 ➡ K2. ⓑ{I} V2 →
∃∃K1,V1. K1 ➡ K2 & V1 ➡ V2 & L1 = K1. ⓑ{I} V1.
-/2 width=3/ qed-.
+/2 width=1 by lpx_inv_pair2/ qed-.
(* Basic forward lemmas *****************************************************)
lemma ltpr_fwd_length: ∀L1,L2. L1 ➡ L2 → |L1| = |L2|.
-#L1 #L2 #H elim H -L1 -L2 normalize //
-qed-.
+/2 width=2 by lpx_fwd_length/ qed-.
(* Basic_1: removed theorems 2: wcpr0_getl wcpr0_getl_back *)
| #I #L1 #K1 #V1 #B #i #HLK1 #HK1 #H1 #H2 #L2 #HL12 #T2 #H destruct
>(tpr_inv_atom1 … H) -T2
lapply (ldrop_pair2_fwd_cw … HLK1 (#i)) #HKV1
- elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #HLK2 #H
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -HLK1 -HL12 #X #H #HLK2
elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct
lapply (IH … HKV1 … HK1 … HK12 … HV12) // -L1 -K1 -V1 /2 width=5/
| #a #L1 #V1 #T1 #B #A #HB #HA #H1 #H2 #L2 #HL12 #X #H destruct
(* *)
(**************************************************************************)
+include "basic_2/substitution/ldrop_lpx.ma".
include "basic_2/reducibility/tpr_lift.ma".
include "basic_2/reducibility/ltpr.ma".
(* CONTEXT-FREE PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ********************)
(* Basic_1: was: wcpr0_drop *)
-lemma ltpr_ldrop_conf: ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀L2. L1 ➡ L2 →
- ∃∃K2. ⇩[d, e] L2 ≡ K2 & K1 ➡ K2.
-#L1 #K1 #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (IHLK1 … HL12) -L1 /3 width=3/
-| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
- elim (tpr_inv_lift1 … HV12 … HWV1) -V1
- elim (IHLK1 … HL12) -L1 /3 width=5/
-]
-qed.
+lemma ltpr_ldrop_conf: dropable_sn ltpr.
+/3 width=3 by lpx_deliftable_dropable, tpr_inv_lift1/ qed.
(* Basic_1: was: wcpr0_drop_back *)
-lemma ldrop_ltpr_trans: ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. K1 ➡ K2 →
- ∃∃L2. ⇩[d, e] L2 ≡ K2 & L1 ➡ L2.
-#L1 #K1 #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #X #H >(ltpr_inv_atom1 … H) -H /2 width=3/
-| #K1 #I #V1 #X #H
- elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/
-| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
- elim (IHLK1 … HK12) -K1 /3 width=5/
-| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
- elim (ltpr_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (lift_total W2 d e) #V2 #HWV2
- lapply (tpr_lift … HW12 … HWV1 … HWV2) -W1
- elim (IHLK1 … HK12) -K1 /3 width=5/
-]
-qed.
+lemma ldrop_ltpr_trans: dedropable_sn ltpr.
+/2 width=3/ qed.
-fact ltpr_ldrop_trans_O1_aux: ∀L2,K2,d,e. ⇩[d, e] L2 ≡ K2 → ∀L1. L1 ➡ L2 →
- d = 0 → ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 ➡ K2.
-#L2 #K2 #d #e #H elim H -L2 -K2 -d -e
-[ #d #e #X #H >(ltpr_inv_atom2 … H) -H /2 width=3/
-| #K2 #I #V2 #X #H
- elim (ltpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/
-| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_
- elim (ltpr_inv_pair2 … H) -H #L1 #V1 #HL12 #HV12 #H destruct
- elim (IHLK2 … HL12 ?) -L2 // /3 width=3/
-| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_
- >commutative_plus normalize #H destruct
-]
-qed.
-
-lemma ltpr_ldrop_trans_O1: ∀L1,L2. L1 ➡ L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
- ∃∃K1. ⇩[0, e] L1 ≡ K1 & K1 ➡ K2.
-/2 width=5/ qed.
+lemma ltpr_ldrop_trans_O1: dropable_dx ltpr.
+/2 width=3/ qed.
#L1 #T1 #T2 #d #e #H elim H -L1 -T1 -T2 -d -e
[ /2 width=3/
| #L1 #K1 #V1 #W1 #i #d #e #Hdi #Hide #HLK1 #HVW1 #L2 #HL12
- elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #HLK2 #H
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
elim (ltpr_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct -K1
elim (lift_total V2 0 (i+1)) #W2 #HVW2
lapply (tpr_lift … HV12 … HVW1 … HVW2) -V1 /3 width=4/
/2 width=3/ qed.
(* Basic_1: was by definition: pr0_refl *)
-lemma tpr_refl: ∀T. T ➡ T.
+lemma tpr_refl: reflexive … tpr.
#T elim T -T //
#I elim I -I /2 width=1/
qed.
(* Relocation properties ****************************************************)
(* Basic_1: was: pr0_lift *)
-lemma tpr_lift: ∀T1,T2. T1 ➡ T2 →
- ∀d,e,U1. ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → U1 ➡ U2.
+lemma tpr_lift: t_liftable tpr.
#T1 #T2 #H elim H -T1 -T2
-[ * #i #d #e #U1 #HU1 #U2 #HU2
+[ * #i #U1 #d #e #HU1 #U2 #HU2
lapply (lift_mono … HU1 … HU2) -HU1 #H destruct
[ lapply (lift_inv_sort1 … HU2) -HU2 #H destruct //
| lapply (lift_inv_lref1 … HU2) * * #Hid #H destruct //
| lapply (lift_inv_gref1 … HU2) -HU2 #H destruct //
]
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
elim (lift_inv_flat1 … HX2) -HX2 #W2 #U2 #HVW2 #HTU2 #HX2 destruct /3 width=4/
-| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
+| #a #V1 #V2 #W #T1 #T2 #_ #_ #IHV12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_inv_bind1 … HX2) -HX2 #V3 #T3 #HV23 #HT23 #HX2 destruct /3 width=4/
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #d #e #X1 #HX1 #X2 #HX2
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X1 #d #e #HX1 #X2 #HX2
elim (lift_inv_bind1 … HX1) -HX1 #W1 #U1 #HVW1 #HTU1 #HX1 destruct
elim (lift_inv_bind1 … HX2) -HX2 #W2 #U0 #HVW2 #HTU0 #HX2 destruct
elim (lift_total T (d + 1) e) #U #HTU
@tpr_delta
[4: @(tps_lift_le … HT2 … HTU HTU0 ?) /2 width=1/ |1: skip |2: /2 width=4/ |3: /2 width=4/ ] (**) (*/3. is too slow *)
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X1 #HX1 #X2 #HX2
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X1 #d #e #HX1 #X2 #HX2
elim (lift_inv_flat1 … HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_inv_bind1 … HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
elim (lift_inv_flat1 … HX) -HX #V3 #T3 #HV3 #HT23 #HX destruct
elim (lift_trans_ge … HV2 … HV3 ?) -V // /3 width=4/
-| #V #T1 #T #T2 #_ #HT2 #IHT1 #d #e #X #H #U2 #HTU2
+| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #H #U2 #HTU2
elim (lift_inv_bind1 … H) -H #V3 #T3 #_ #HT13 #H destruct -V
elim (lift_conf_O1 … HTU2 … HT2) -T2 /3 width=4/
-| #V #T1 #T2 #_ #IHT12 #d #e #X #HX #T #HT2
+| #V #T1 #T2 #_ #IHT12 #X #d #e #HX #T #HT2
elim (lift_inv_flat1 … HX) -HX #V0 #T0 #_ #HT0 #HX destruct /3 width=4/
]
qed.
(* Basic_1: was: pr0_gen_lift *)
-lemma tpr_inv_lift1: ∀T1,T2. T1 ➡ T2 →
- ∀d,e,U1. ⇧[d, e] U1 ≡ T1 →
- ∃∃U2. ⇧[d, e] U2 ≡ T2 & U1 ➡ U2.
+lemma tpr_inv_lift1: t_deliftable_sn tpr.
#T1 #T2 #H elim H -T1 -T2
-[ * #i #d #e #U1 #HU1
- [ lapply (lift_inv_sort2 … HU1) -HU1 #H destruct /2 width=3/
- | lapply (lift_inv_lref2 … HU1) -HU1 * * #Hid #H destruct /3 width=3/
- | lapply (lift_inv_gref2 … HU1) -HU1 #H destruct /2 width=3/
+[ * #i #X #d #e #HX
+ [ lapply (lift_inv_sort2 … HX) -HX #H destruct /2 width=3/
+ | lapply (lift_inv_lref2 … HX) -HX * * #Hid #H destruct /3 width=3/
+ | lapply (lift_inv_gref2 … HX) -HX #H destruct /2 width=3/
]
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
elim (lift_inv_flat2 … HX) -HX #V0 #T0 #HV01 #HT01 #HX destruct
elim (IHV12 … HV01) -V1
elim (IHT12 … HT01) -T1 /3 width=5/
-| #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #e #X #HX
+| #a #V1 #V2 #W1 #T1 #T2 #_ #_ #IHV12 #IHT12 #X #d #e #HX
elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
elim (IHV12 … HV01) -V1
elim (IHT12 … HT01) -T1 /3 width=5/
-| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #d #e #X #HX
+| #a #I #V1 #V2 #T1 #T #T2 #_ #_ #HT2 #IHV12 #IHT1 #X #d #e #HX
elim (lift_inv_bind2 … HX) -HX #W1 #U1 #HWV1 #HUT1 #HX destruct
elim (IHV12 … HWV1) -V1 #W2 #HWV2 #HW12
elim (IHT1 … HUT1) -T1 #U #HUT #HU1
elim (tps_inv_lift1_le … HT2 … HUT ?) -T // [3: /2 width=5/ |2: skip ] #U2 #HU2 #HUT2
@ex2_1_intro [2: /2 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /3 width=5/ is slow *)
-| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #d #e #X #HX
+| #a #V #V1 #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV12 #IHW12 #IHT12 #X #d #e #HX
elim (lift_inv_flat2 … HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 … HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
elim (IHV12 … HV01) -V1 #V3 #HV32 #HV03
elim (IHT12 … HT01) -T1 #T3 #HT32 #HT03
elim (lift_trans_le … HV32 … HV2 ?) -V2 // #V2 #HV32 #HV2
@ex2_1_intro [2: /3 width=2/ |1: skip |3: /2 width=3/ ] (**) (* /4 width=5/ is slow *)
-| #V #T1 #T #T2 #_ #HT2 #IHT1 #d #e #X #HX
+| #V #T1 #T #T2 #_ #HT2 #IHT1 #X #d #e #HX
elim (lift_inv_bind2 … HX) -HX #V3 #T3 #_ #HT31 #H destruct
elim (IHT1 … HT31) -T1 #T1 #HT1 #HT31
elim (lift_div_le … HT2 … HT1 ?) -T // /3 width=5/
-| #V #T1 #T2 #_ #IHT12 #d #e #X #HX
+| #V #T1 #T2 #_ #IHT12 #X #d #e #HX
elim (lift_inv_flat2 … HX) -HX #V0 #T0 #_ #HT01 #H destruct
elim (IHT12 … HT01) -T1 /3 width=3/
]
elim (tps_inv_atom1 … H) -H
[ #H destruct /2 width=3/
| * #K1 #V1 #i #Hdi #Hide #HLK1 #HVU1 #H #L2 #HL12 destruct
- elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #HLK2 #H
+ elim (ltpr_ldrop_conf … HLK1 … HL12) -L1 #X #H #HLK2
elim (ltpr_inv_pair1 … H) -H #K2 #V2 #_ #HV12 #H destruct
elim (lift_total V2 0 (i+1)) #U2 #HVU2
lapply (tpr_lift … HV12 … HVU1 … HVU2) -V1 #HU12
interpretation "local slicing" 'RDrop d e L1 L2 = (ldrop d e L1 L2).
+definition l_liftable: (lenv → relation term) → Prop ≝
+ λR. ∀K,T1,T2. R K T1 T2 → ∀L,d,e. ⇩[d, e] L ≡ K →
+ ∀U1. ⇧[d, e] T1 ≡ U1 → ∀U2. ⇧[d, e] T2 ≡ U2 → R L U1 U2.
+
+definition l_deliftable_sn: (lenv → relation term) → Prop ≝
+ λR. ∀L,U1,U2. R L U1 U2 → ∀K,d,e. ⇩[d, e] L ≡ K →
+ ∀T1. ⇧[d, e] T1 ≡ U1 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & R K T1 T2.
+
+definition dropable_sn: relation lenv → Prop ≝
+ λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀L2. R L1 L2 →
+ ∃∃K2. R K1 K2 & ⇩[d, e] L2 ≡ K2.
+
+definition dedropable_sn: relation lenv → Prop ≝
+ λR. ∀L1,K1,d,e. ⇩[d, e] L1 ≡ K1 → ∀K2. R K1 K2 →
+ ∃∃L2. R L1 L2 & ⇩[d, e] L2 ≡ K2.
+
+definition dropable_dx: relation lenv → Prop ≝
+ λR. ∀L1,L2. R L1 L2 → ∀K2,e. ⇩[0, e] L2 ≡ K2 →
+ ∃∃K1. ⇩[0, e] L1 ≡ K1 & R K1 K2.
+
(* Basic inversion lemmas ***************************************************)
fact ldrop_inv_refl_aux: ∀d,e,L1,L2. ⇩[d, e] L1 ≡ L2 → d = 0 → e = 0 → L1 = L2.
--- /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 "basic_2/grammar/lenv_px.ma".
+include "basic_2/substitution/ldrop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Properties on pointwise extension ****************************************)
+
+lemma lpx_deliftable_dropable: ∀R. t_deliftable_sn R → dropable_sn (lpx R).
+#R #HR #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #X #H
+ elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #e #_ #IHLK1 #X #H
+ elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (IHLK1 … HL12) -L1 /3 width=3/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
+ elim (lpx_inv_pair1 … H) -H #L2 #V2 #HL12 #HV12 #H destruct
+ elim (HR … HV12 … HWV1) -V1
+ elim (IHLK1 … HL12) -L1 /3 width=5/
+]
+qed.
+
+lemma lpx_liftable_dedropable: ∀R. reflexive ? R →
+ t_liftable R → dedropable_sn (lpx R).
+#R #H1R #H2R #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
+[ #d #e #X #H >(lpx_inv_atom1 … H) -H /2 width=3/
+| #K1 #I #V1 #X #H
+ elim (lpx_inv_pair1 … H) -H #K2 #V2 #HK12 #HV12 #H destruct /3 width=5/
+| #L1 #K1 #I #V1 #e #_ #IHLK1 #K2 #HK12
+ elim (IHLK1 … HK12) -K1 /3 width=5/
+| #L1 #K1 #I #V1 #W1 #d #e #_ #HWV1 #IHLK1 #X #H
+ elim (lpx_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
+ elim (lift_total W2 d e) #V2 #HWV2
+ lapply (H2R … HW12 … HWV1 … HWV2) -W1
+ elim (IHLK1 … HK12) -K1 /3 width=5/
+]
+qed.
+
+fact lpx_dropable_aux: ∀R,L2,K2,d,e. ⇩[d, e] L2 ≡ K2 → ∀L1. lpx R L1 L2 →
+ d = 0 → ∃∃K1. ⇩[0, e] L1 ≡ K1 & lpx R K1 K2.
+#R #L2 #K2 #d #e #H elim H -L2 -K2 -d -e
+[ #d #e #X #H >(lpx_inv_atom2 … H) -H /2 width=3/
+| #K2 #I #V2 #X #H
+ elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV12 #H destruct /3 width=5/
+| #L2 #K2 #I #V2 #e #_ #IHLK2 #X #H #_
+ elim (lpx_inv_pair2 … H) -H #L1 #V1 #HL12 #HV12 #H destruct
+ elim (IHLK2 … HL12 ?) -L2 // /3 width=3/
+| #L2 #K2 #I #V2 #W2 #d #e #_ #_ #_ #L1 #_
+ >commutative_plus normalize #H destruct
+]
+qed.
+
+lemma ltpr_dropable: ∀R. dropable_dx (lpx R).
+/2 width=5/ qed.
interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
+definition t_liftable: relation term → Prop ≝
+ λR. ∀T1,T2. R T1 T2 → ∀U1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∀U2. ⇧[d, e] T2 ≡ U2 → R U1 U2.
+
+definition t_deliftable_sn: relation term → Prop ≝
+ λR. ∀U1,U2. R U1 U2 → ∀T1,d,e. ⇧[d, e] T1 ≡ U1 →
+ ∃∃T2. ⇧[d, e] T2 ≡ U2 & R T1 T2.
+
(* Basic inversion lemmas ***************************************************)
fact lift_inv_refl_O2_aux: ∀d,e,T1,T2. ⇧[d, e] T1 ≡ T2 → e = 0 → T1 = T2.
]
qed.
+lemma t_liftable_TC: ∀R. t_liftable R → t_liftable (TC … R).
+#R #HR #T1 #T2 #H elim H -T2
+[ /3 width=7/
+| #T #T2 #_ #HT2 #IHT1 #U1 #d #e #HTU1 #U2 #HTU2
+ elim (lift_total T d e) /3 width=9/
+]
+qed.
+
+lemma t_deliftable_sn_TC: ∀R. t_deliftable_sn R → t_deliftable_sn (TC … R).
+#R #HR #U1 #U2 #H elim H -U2
+[ #U2 #HU12 #T1 #d #e #HTU1
+ elim (HR … HU12 … HTU1) -U1 /3 width=3/
+| #U #U2 #_ #HU2 #IHU1 #T1 #d #e #HTU1
+ elim (IHU1 … HTU1) -U1 #T #HTU #HT1
+ elim (HR … HU2 … HTU) -U /3 width=5/
+]
+qed-.
+
(* Basic_1: removed theorems 7:
lift_head lift_gen_head
lift_weight_map lift_weight lift_weight_add lift_weight_add_O
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