--- /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/substitution/drop.ma".
+include "basic_2/substitution/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* alternative definition of lpx_sn *)
+definition lpx_sn_alt: relation3 lenv term term → relation lenv ≝
+ λR,L1,L2. |L1| = |L2| ∧
+ (∀I1,I2,K1,K2,V1,V2,i.
+ ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R K1 V1 V2
+ ).
+
+(* Basic forward lemmas ******************************************************)
+
+lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
+#R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
+normalize /2 width=1 by length_inv_zero_sn/
+qed-.
+
+lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
+ ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+#R #I1 #L2 #K1 #V1 #H elim H -H
+#H #IH elim (length_inv_pos_sn … H) -H
+#I2 #K2 #V2 #HK12 #H destruct
+elim (IH I1 I2 K1 K2 V1 V2 0) //
+#H #HV12 destruct @(ex3_2_intro … K2 V2) // -HV12
+@conj // -HK12
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH
+/2 width=1 by drop_drop, conj/
+qed-.
+
+lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
+#R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H
+normalize /2 width=1 by length_inv_zero_dx/
+qed-.
+
+lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
+ ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+#R #I2 #L1 #K2 #V2 #H elim H -H
+#H #IH elim (length_inv_pos_dx … H) -H
+#I1 #K1 #V1 #HK12 #H destruct
+elim (IH I1 I2 K1 K2 V1 V2 0) //
+#H #HV12 destruct @(ex3_2_intro … K1 V1) // -HV12
+@conj // -HK12
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH
+/2 width=1 by drop_drop, conj/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
+#R @conj //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (drop_inv_atom1 … HLK1) -HLK1
+#H destruct
+qed.
+
+lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
+ lpx_sn_alt R L1 L2 → R L1 V1 V2 →
+ lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
+#R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #W1 #W2 #i @(ynat_ind … i) -i
+[ #HLK1 #HLK2
+ lapply (drop_inv_O2 … HLK1) -HLK1 #H destruct
+ lapply (drop_inv_O2 … HLK2) -HLK2 #H destruct
+ /2 width=1 by conj/
+| -HL12 -HV12 /3 width=6 by drop_inv_drop1/
+| #H lapply (drop_fwd_Y2 … H) -H
+ #H elim (ylt_yle_false … H) -H //
+]
+qed.
+
+(* Main properties **********************************************************)
+
+theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2.
+#R #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/
+qed.
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2.
+#R #L1 elim L1 -L1
+[ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H //
+| #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H
+ #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/
+]
+qed-.
+
+(* alternative definition of lpx_sn *****************************************)
+
+lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
+ (∀I1,I2,K1,K2,V1,V2,i.
+ ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R K1 V1 V2
+ ) → lpx_sn R L1 L2.
+/4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed.
+
+lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
+ |L1| = |L2| ∧
+ ∀I1,I2,K1,K2,V1,V2,i.
+ ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
+ I1 = I2 ∧ R K1 V1 V2.
+#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
+#H elim H -H /3 width=4 by conj/
+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 "basic_2/substitution/drop.ma".
-include "basic_2/substitution/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-(* alternative definition of lpx_sn *)
-definition lpx_sn_alt: relation3 lenv term term → relation lenv ≝
- λR,L1,L2. |L1| = |L2| ∧
- (∀I1,I2,K1,K2,V1,V2,i.
- ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R K1 V1 V2
- ).
-
-(* Basic forward lemmas ******************************************************)
-
-lemma lpx_sn_alt_fwd_length: ∀R,L1,L2. lpx_sn_alt R L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H //
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lpx_sn_alt_inv_atom1: ∀R,L2. lpx_sn_alt R (⋆) L2 → L2 = ⋆.
-#R #L2 #H lapply (lpx_sn_alt_fwd_length … H) -H
-normalize /2 width=1 by length_inv_zero_sn/
-qed-.
-
-lemma lpx_sn_alt_inv_pair1: ∀R,I,L2,K1,V1. lpx_sn_alt R (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
-#R #I1 #L2 #K1 #V1 #H elim H -H
-#H #IH elim (length_inv_pos_sn … H) -H
-#I2 #K2 #V2 #HK12 #H destruct
-elim (IH I1 I2 K1 K2 V1 V2 0) //
-#H #HV12 destruct @(ex3_2_intro … K2 V2) // -HV12
-@conj // -HK12
-#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH
-/2 width=1 by drop_drop, conj/
-qed-.
-
-lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
-#R #L1 #H lapply (lpx_sn_alt_fwd_length … H) -H
-normalize /2 width=1 by length_inv_zero_dx/
-qed-.
-
-lemma lpx_sn_alt_inv_pair2: ∀R,I,L1,K2,V2. lpx_sn_alt R L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. lpx_sn_alt R K1 K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
-#R #I2 #L1 #K2 #V2 #H elim H -H
-#H #IH elim (length_inv_pos_dx … H) -H
-#I1 #K1 #V1 #HK12 #H destruct
-elim (IH I1 I2 K1 K2 V1 V2 0) //
-#H #HV12 destruct @(ex3_2_intro … K1 V1) // -HV12
-@conj // -HK12
-#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH
-/2 width=1 by drop_drop, conj/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
-#R @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (drop_inv_atom1 … HLK1) -HLK1
-#H destruct
-qed.
-
-lemma lpx_sn_alt_pair: ∀R,I,L1,L2,V1,V2.
- lpx_sn_alt R L1 L2 → R L1 V1 V2 →
- lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
-#R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H
-#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #W1 #W2 #i @(ynat_ind … i) -i
-[ #HLK1 #HLK2
- lapply (drop_inv_O2 … HLK1) -HLK1 #H destruct
- lapply (drop_inv_O2 … HLK2) -HLK2 #H destruct
- /2 width=1 by conj/
-| -HL12 -HV12 /3 width=6 by drop_inv_drop1/
-| #H lapply (drop_fwd_Y2 … H) -H
- #H elim (ylt_yle_false … H) -H //
-]
-qed.
-
-(* Main properties **********************************************************)
-
-theorem lpx_sn_lpx_sn_alt: ∀R,L1,L2. lpx_sn R L1 L2 → lpx_sn_alt R L1 L2.
-#R #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by lpx_sn_alt_atom, lpx_sn_alt_pair/
-qed.
-
-(* Main inversion lemmas ****************************************************)
-
-theorem lpx_sn_alt_inv_lpx_sn: ∀R,L1,L2. lpx_sn_alt R L1 L2 → lpx_sn R L1 L2.
-#R #L1 elim L1 -L1
-[ #L2 #H lapply (lpx_sn_alt_inv_atom1 … H) -H //
-| #L1 #I #V1 #IH #X #H elim (lpx_sn_alt_inv_pair1 … H) -H
- #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by lpx_sn_pair/
-]
-qed-.
-
-(* alternative definition of lpx_sn *****************************************)
-
-lemma lpx_sn_intro_alt: ∀R,L1,L2. |L1| = |L2| →
- (∀I1,I2,K1,K2,V1,V2,i.
- ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R K1 V1 V2
- ) → lpx_sn R L1 L2.
-/4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed.
-
-lemma lpx_sn_inv_alt: ∀R,L1,L2. lpx_sn R L1 L2 →
- |L1| = |L2| ∧
- ∀I1,I2,K1,K2,V1,V2,i.
- ⬇[i] L1 ≡ K1.ⓑ{I1}V1 → ⬇[i] L2 ≡ K2.ⓑ{I2}V2 →
- I1 = I2 ∧ R K1 V1 V2.
-#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
-#H elim H -H /3 width=4 by conj/
-qed-.
--- /dev/null
+definition dropable_dx: predicate (relation lenv) ≝
+ λR. ∀L1,L2. R L1 L2 → ∀K2,c,k. ⬇[c, 0, k] L2 ≡ K2 →
+ ∃∃K1. ⬇[c, 0, k] L1 ≡ K1 & R K1 K2.
+
+lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
+#R #HR #L1 #L2 #H elim H -L2
+[ #L2 #HL12 #K2 #c #k #HLK2 elim (HR … HL12 … HLK2) -HR -L2
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 #IHL1 #K2 #c #k #HLK2 elim (HR … HL2 … HLK2) -HR -L2
+ #K #HLK #HK2 elim (IHL1 … HLK) -L
+ /3 width=5 by step, ex2_intro/
+]
+qed-.
+
+
+fact lpx_sn_dropable_aux: ∀R,L2,K2,c,l,k. ⬇[c, l, k] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
+ l = 0 → ∃∃K1. ⬇[c, 0, k] L1 ≡ K1 & lpx_sn R K1 K2.
+#R #L2 #K2 #c #l #k #H elim H -L2 -K2 -l -k
+[ #l #k #Hm #X #H >(lpx_sn_inv_atom2 … H) -H
+ /4 width=3 by drop_atom, lpx_sn_atom, ex2_intro/
+| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
+ #K1 #V1 #HK12 #HV12 #H destruct
+ /3 width=5 by drop_pair, lpx_sn_pair, ex2_intro/
+| #I #L2 #K2 #V2 #k #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
+ #L1 #V1 #HL12 #HV12 #H destruct
+ elim (IHLK2 … HL12) -L2 /3 width=3 by drop_drop, ex2_intro/
+| #I #L2 #K2 #V2 #W2 #l #k #_ #_ #_ #L1 #_ #H elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lpx_sn_dropable: ∀R. dropable_dx (lpx_sn R).
+/2 width=5 by lpx_sn_dropable_aux/ qed-.
+
+lemma lpx_sn_drop_trans: ∀R,L1,L2. lpx_sn R L1 L2 →
+ ∀I,K2,V2,i. ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & lpx_sn R K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #H elim H -L1 -L2
+[ #I0 #K0 #V0 #i #H elim (drop_inv_atom1 … H) -H #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (drop_inv_O1_pair1 … H) * -H
+ [ -IHK12 #H1 #H2 destruct /3 width=5 by drop_pair, ex3_2_intro/
+ | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
+ /3 width=5 by drop_drop_lt, ex3_2_intro/
+ ]
+]
+qed-.
+
+lemma lpx_sn_drop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
+ ∀I,K1,V1,i. ⬇[i] L1 ≡ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 & lpx_sn R K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #H elim H -L1 -L2
+[ #I0 #K0 #V0 #i #H elim (drop_inv_atom1 … H) -H #H destruct
+| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (drop_inv_O1_pair1 … H) * -H
+ [ -IHK12 #H1 #H2 destruct /3 width=5 by drop_pair, ex3_2_intro/
+ | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
+ /3 width=5 by drop_drop_lt, ex3_2_intro/
+ ]
+]
+qed-.
+++ /dev/null
-definition dropable_dx: predicate (relation lenv) ≝
- λR. ∀L1,L2. R L1 L2 → ∀K2,c,k. ⬇[c, 0, k] L2 ≡ K2 →
- ∃∃K1. ⬇[c, 0, k] L1 ≡ K1 & R K1 K2.
-
-lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
-#R #HR #L1 #L2 #H elim H -L2
-[ #L2 #HL12 #K2 #c #k #HLK2 elim (HR … HL12 … HLK2) -HR -L2
- /3 width=3 by inj, ex2_intro/
-| #L #L2 #_ #HL2 #IHL1 #K2 #c #k #HLK2 elim (HR … HL2 … HLK2) -HR -L2
- #K #HLK #HK2 elim (IHL1 … HLK) -L
- /3 width=5 by step, ex2_intro/
-]
-qed-.
-
-
-fact lpx_sn_dropable_aux: ∀R,L2,K2,c,l,k. ⬇[c, l, k] L2 ≡ K2 → ∀L1. lpx_sn R L1 L2 →
- l = 0 → ∃∃K1. ⬇[c, 0, k] L1 ≡ K1 & lpx_sn R K1 K2.
-#R #L2 #K2 #c #l #k #H elim H -L2 -K2 -l -k
-[ #l #k #Hm #X #H >(lpx_sn_inv_atom2 … H) -H
- /4 width=3 by drop_atom, lpx_sn_atom, ex2_intro/
-| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 … H) -H
- #K1 #V1 #HK12 #HV12 #H destruct
- /3 width=5 by drop_pair, lpx_sn_pair, ex2_intro/
-| #I #L2 #K2 #V2 #k #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 … H) -H
- #L1 #V1 #HL12 #HV12 #H destruct
- elim (IHLK2 … HL12) -L2 /3 width=3 by drop_drop, ex2_intro/
-| #I #L2 #K2 #V2 #W2 #l #k #_ #_ #_ #L1 #_ #H elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lpx_sn_dropable: ∀R. dropable_dx (lpx_sn R).
-/2 width=5 by lpx_sn_dropable_aux/ qed-.
-
-lemma lpx_sn_drop_trans: ∀R,L1,L2. lpx_sn R L1 L2 →
- ∀I,K2,V2,i. ⬇[i] L2 ≡ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇[i] L1 ≡ K1.ⓑ{I}V1 & lpx_sn R K1 K2 & R K1 V1 V2.
-#R #L1 #L2 #H elim H -L1 -L2
-[ #I0 #K0 #V0 #i #H elim (drop_inv_atom1 … H) -H #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (drop_inv_O1_pair1 … H) * -H
- [ -IHK12 #H1 #H2 destruct /3 width=5 by drop_pair, ex3_2_intro/
- | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
- /3 width=5 by drop_drop_lt, ex3_2_intro/
- ]
-]
-qed-.
-
-lemma lpx_sn_drop_conf: ∀R,L1,L2. lpx_sn R L1 L2 →
- ∀I,K1,V1,i. ⬇[i] L1 ≡ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⬇[i] L2 ≡ K2.ⓑ{I}V2 & lpx_sn R K1 K2 & R K1 V1 V2.
-#R #L1 #L2 #H elim H -L1 -L2
-[ #I0 #K0 #V0 #i #H elim (drop_inv_atom1 … H) -H #H destruct
-| #I #K1 #K2 #V1 #V2 #HK12 #HV12 #IHK12 #I0 #K0 #V0 #i #H elim (drop_inv_O1_pair1 … H) * -H
- [ -IHK12 #H1 #H2 destruct /3 width=5 by drop_pair, ex3_2_intro/
- | -HK12 -HV12 #Hi #HK10 elim (IHK12 … HK10) -IHK12 -HK10
- /3 width=5 by drop_drop_lt, ex3_2_intro/
- ]
-]
-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 "basic_2/substitution/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+definition lpx_sn_confluent: relation (relation3 lenv term term) ≝ λR1,R2.
+ ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. lpx_sn R1 L0 L1 → ∀L2. lpx_sn R2 L0 L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+definition lpx_sn_transitive: relation (relation3 lenv term term) ≝ λR1,R2.
+ ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn R1 L1 L2 →
+ ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
+
+(* Main properties **********************************************************)
+
+theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn R).
+#R #HR #L1 #L #H elim H -L1 -L //
+#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
+elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5 by lpx_sn_pair/
+qed-.
+
+theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
+ confluent2 … (lpx_sn R1) (lpx_sn R2).
+#R1 #R2 #HR12 #L0 @(ynat_f_ind … length … L0) -L0 #x #IH *
+[ #_ #X1 #H1 #X2 #H2 -x
+ >(lpx_sn_inv_atom1 … H1) -X1
+ >(lpx_sn_inv_atom1 … H2) -X2 /2 width=3 by lpx_sn_atom, ex2_intro/
+| #L0 #I #V0 #Hx #X1 #H1 #X2 #H2 destruct
+ elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
+ elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
+ elim (IH … HL01 … HL02) -IH /2 width=2 by ylt_succ2_refl/ #L #HL1 #HL2
+ elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lpx_sn_pair, ex2_intro/
+]
+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 "basic_2/substitution/lpx_sn.ma".
+
+(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
+
+(* Properties on transitive_closure *****************************************)
+
+lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V).
+#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
+[ /2 width=1 by lpx_sn_refl/
+| /3 width=1 by TC_reflexive, lpx_sn_refl/
+| /3 width=5 by lpx_sn_pair, step/
+]
+qed-.
+
+lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
+ ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀V1,V2. LTC … R L1 V1 V2 →
+ TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
+#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
+[ /2 width=1 by TC_lpx_sn_pair_refl/
+| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
+]
+qed-.
+
+lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
+ ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
+ TC … (lpx_sn R) L1 L2.
+#R #HR #L1 #L2 #H elim H -L1 -L2
+/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
+qed-.
+
+(* Inversion lemmas on transitive closure ***********************************)
+
+lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
+#R #L1 #H @(TC_ind_dx … L1 H) -L1
+[ /2 width=2 by lpx_sn_inv_atom2/
+| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair2: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
+ ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
+ ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
+#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
+[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
+| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
+ elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
+ lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_ind: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
+ ∀S:relation lenv.
+ S (⋆) (⋆) → (
+ ∀I,K1,K2,V1,V2.
+ TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
+ S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+ ) →
+ ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
+#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
+[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
+| #L2 #I #V2 #IHL2 #X #H
+ elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
+ #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
+#R #L2 #H elim H -L2
+[ /2 width=2 by lpx_sn_inv_atom1/
+| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
+]
+qed-.
+
+fact TC_lpx_sn_inv_pair1_aux: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
+[ #J #K #W #H destruct
+| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma TC_lpx_sn_inv_pair1: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
+ ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
+ ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
+/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
+
+lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
+ ∀L1,L2. TC … (lpx_sn R) L1 L2 →
+ lpx_sn (LTC … R) L1 L2.
+/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
+
+(* Forward lemmas on transitive closure *************************************)
+
+lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #H elim H -L2
+[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
+| #L #L2 #_ #HL2 #IHL1
+ >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
+]
+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 "ground_2/ynat/ynat_lt.ma".
-include "basic_2/notation/relations/midiso_4.ma".
-include "basic_2/grammar/lenv_length.ma".
-
-(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
-
-inductive lreq: relation4 ynat ynat lenv lenv ≝
-| lreq_atom: ∀l,m. lreq l m (⋆) (⋆)
-| lreq_zero: ∀I1,I2,L1,L2,V1,V2.
- lreq 0 0 L1 L2 → lreq 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
-| lreq_pair: ∀I,L1,L2,V,m. lreq 0 m L1 L2 →
- lreq 0 (⫯m) (L1.ⓑ{I}V) (L2.ⓑ{I}V)
-| lreq_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
- lreq l m L1 L2 → lreq (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
-.
-
-interpretation
- "equivalence (local environment)"
- 'MidIso l m L1 L2 = (lreq l m L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lreq_pair_lt: ∀I,L1,L2,V,m. L1 ⩬[0, ⫰m] L2 → 0 < m →
- L1.ⓑ{I}V ⩬[0, m] L2.ⓑ{I}V.
-#I #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lreq_pair/
-qed.
-
-lemma lreq_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⩬[⫰l, m] L2 → 0 < l →
- L1.ⓑ{I1}V1 ⩬[l, m] L2. ⓑ{I2}V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lreq_succ/
-qed.
-
-lemma lreq_pair_O_Y: ∀L1,L2. L1 ⩬[0, ∞] L2 →
- ∀I,V. L1.ⓑ{I}V ⩬[0, ∞] L2.ⓑ{I}V.
-#L1 #L2 #HL12 #I #V lapply (lreq_pair I … V … HL12) -HL12 //
-qed.
-
-lemma lreq_refl: ∀L,l,m. L ⩬[l, m] L.
-#L elim L -L //
-#L #I #V #IHL #l elim (ynat_cases … l) [| * #x ]
-#Hl destruct /2 width=1 by lreq_succ/
-#m elim (ynat_cases … m) [| * #x ]
-#Hm destruct /2 width=1 by lreq_zero, lreq_pair/
-qed.
-
-lemma lreq_O2: ∀L1,L2,l. |L1| = |L2| → L1 ⩬[l, 0] L2.
-#L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ]
-* // [1,3: #L2 #I2 #V2 ] #l
-[ #H elim (ysucc_inv_O_sn … H)
-| >length_pair >length_pair #H
- lapply (ysucc_inv_inj … H) -H #HL12
- elim (ynat_cases l) /3 width=1 by lreq_zero/
- * /3 width=1 by lreq_succ/
-| #H elim (ysucc_inv_O_dx … H)
-]
-qed.
-
-lemma lreq_sym: ∀l,m. symmetric … (lreq l m).
-#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
-/2 width=1 by lreq_zero, lreq_pair, lreq_succ/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lreq_inv_atom1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 #l #m * -L1 -L2 -l -m //
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
-| #I #L1 #L2 #V #m #_ #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
-]
-qed-.
-
-lemma lreq_inv_atom1: ∀L2,l,m. ⋆ ⩬[l, m] L2 → L2 = ⋆.
-/2 width=5 by lreq_inv_atom1_aux/ qed-.
-
-fact lreq_inv_zero1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
- ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 →
- ∃∃J2,K2,W2. K1 ⩬[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
-#L1 #L2 #l #m * -L1 -L2 -l -m
-[ #l #m #J1 #K1 #W1 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
- /2 width=5 by ex2_3_intro/
-| #I #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
- elim (ysucc_inv_O_dx … H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lreq_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⩬[0, 0] L2 →
- ∃∃I2,K2,V2. K1 ⩬[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=9 by lreq_inv_zero1_aux/ qed-.
-
-fact lreq_inv_pair1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
- ∀J,K1,W. L1 = K1.ⓑ{J}W → l = 0 → 0 < m →
- ∃∃K2. K1 ⩬[0, ⫰m] K2 & L2 = K2.ⓑ{J}W.
-#L1 #L2 #l #m * -L1 -L2 -l -m
-[ #l #m #J #K1 #W #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J #K1 #W #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I #L1 #L2 #V #m #HL12 #J #K1 #W #H #_ #_ destruct
- /2 width=3 by ex2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J #K1 #W #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lreq_inv_pair1: ∀I,K1,L2,V,m. K1.ⓑ{I}V ⩬[0, m] L2 → 0 < m →
- ∃∃K2. K1 ⩬[0, ⫰m] K2 & L2 = K2.ⓑ{I}V.
-/2 width=6 by lreq_inv_pair1_aux/ qed-.
-
-lemma lreq_inv_pair: ∀I1,I2,L1,L2,V1,V2,m. L1.ⓑ{I1}V1 ⩬[0, m] L2.ⓑ{I2}V2 → 0 < m →
- ∧∧ L1 ⩬[0, ⫰m] L2 & I1 = I2 & V1 = V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #m #H #Hm elim (lreq_inv_pair1 … H) -H //
-#Y #HL12 #H destruct /2 width=1 by and3_intro/
-qed-.
-
-fact lreq_inv_succ1_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
- ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
- ∃∃J2,K2,W2. K1 ⩬[⫰l, m] K2 & L2 = K2.ⓑ{J2}W2.
-#L1 #L2 #l #m * -L1 -L2 -l -m
-[ #l #m #J1 #K1 #W1 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false … H) //
-| #I #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
- /2 width=5 by ex2_3_intro/
-]
-qed-.
-
-lemma lreq_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⩬[l, m] L2 → 0 < l →
- ∃∃I2,K2,V2. K1 ⩬[⫰l, m] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=5 by lreq_inv_succ1_aux/ qed-.
-
-lemma lreq_inv_atom2: ∀L1,l,m. L1 ⩬[l, m] ⋆ → L1 = ⋆.
-/3 width=3 by lreq_inv_atom1, lreq_sym/
-qed-.
-
-lemma lreq_inv_succ: ∀I1,I2,L1,L2,V1,V2,l,m. L1.ⓑ{I1}V1 ⩬[l, m] L2.ⓑ{I2}V2 → 0 < l →
- L1 ⩬[⫰l, m] L2.
-#I1 #I2 #L1 #L2 #V1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 … H) -H //
-#Z #Y #X #HL12 #H destruct //
-qed-.
-
-lemma lreq_inv_zero2: ∀I2,K2,L1,V2. L1 ⩬[0, 0] K2.ⓑ{I2}V2 →
- ∃∃I1,K1,V1. K1 ⩬[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
-#I2 #K2 #L1 #V2 #H elim (lreq_inv_zero1 … (lreq_sym … H)) -H
-/3 width=5 by lreq_sym, ex2_3_intro/
-qed-.
-
-lemma lreq_inv_pair2: ∀I,K2,L1,V,m. L1 ⩬[0, m] K2.ⓑ{I}V → 0 < m →
- ∃∃K1. K1 ⩬[0, ⫰m] K2 & L1 = K1.ⓑ{I}V.
-#I #K2 #L1 #V #m #H #Hm elim (lreq_inv_pair1 … (lreq_sym … H)) -H
-/3 width=3 by lreq_sym, ex2_intro/
-qed-.
-
-lemma lreq_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⩬[l, m] K2.ⓑ{I2}V2 → 0 < l →
- ∃∃I1,K1,V1. K1 ⩬[⫰l, m] K2 & L1 = K1.ⓑ{I1}V1.
-#I2 #K2 #L1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 … (lreq_sym … H)) -H
-/3 width=5 by lreq_sym, ex2_3_intro/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lreq_fwd_length: ∀L1,L2,l,m. L1 ⩬[l, m] L2 → |L1| = |L2|.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m //
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-fact lreq_inv_O_Y_aux: ∀L1,L2,l,m. L1 ⩬[l, m] L2 → l = 0 → m = ∞ → L1 = L2.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m //
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #_ #H destruct
-| /4 width=1 by eq_f3, ysucc_inv_Y_dx/
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #_ #H elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lreq_inv_O_Y: ∀L1,L2. L1 ⩬[0, ∞] L2 → L1 = L2.
-/2 width=5 by lreq_inv_O_Y_aux/ 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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ≡ break [ term 46 f ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyEq $f $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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ≡ break [ term 46 T , break term 46 f ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyEq $T $f $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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ⩬ break [ term 46 f , break term 46 m ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'MidIso $f $k $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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L1 ⦻ * break [ term 46 R , break term 46 f ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'RelationStar $R $f $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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L1 ⦻ * break [ term 46 R1 , break term 46 R2 , break term 46 f ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'RelationStar $R1 $R2 $f $L1 $L2 }.
(**************************************************************************)
include "ground_2/relocation/nstream_sle.ma".
-include "basic_2/notation/relations/relationstar_4.ma".
+include "basic_2/notation/relations/relationstar_5.ma".
include "basic_2/grammar/lenv.ma".
(* GENERAL ENTRYWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
(* Basic_2A1: includes: lpx_sn_atom lpx_sn_pair *)
-inductive lexs (R:relation4 bool lenv term term): rtmap → relation lenv ≝
-| lexs_atom: ∀f. lexs R f (⋆) (⋆)
+inductive lexs (RN,RP:relation3 lenv term term): rtmap → relation lenv ≝
+| lexs_atom: ∀f. lexs RN RP f (⋆) (⋆)
| lexs_next: ∀I,L1,L2,V1,V2,f.
- lexs R f L1 L2 → R (Ⓣ) L1 V1 V2 →
- lexs R (⫯f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
+ lexs RN RP f L1 L2 → RN L1 V1 V2 →
+ lexs RN RP (⫯f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
| lexs_push: ∀I,L1,L2,V1,V2,f.
- lexs R f L1 L2 → R (Ⓕ) L1 V1 V2 →
- lexs R (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
+ lexs RN RP f L1 L2 → RP L1 V1 V2 →
+ lexs RN RP (↑f) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
.
interpretation "general entrywise extension (local environment)"
- 'RelationStar R f L1 L2 = (lexs R f L1 L2).
+ 'RelationStar RN RP f L1 L2 = (lexs RN RP f L1 L2).
(* Basic inversion lemmas ***************************************************)
-fact lexs_inv_atom1_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → X = ⋆ → Y = ⋆.
-#R #X #Y #f * -X -Y -f //
+fact lexs_inv_atom1_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → X = ⋆ → Y = ⋆.
+#RN #RP #X #Y #f * -X -Y -f //
#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_atom1 *)
-lemma lexs_inv_atom1: ∀R,Y,f. ⋆ ⦻*[R, f] Y → Y = ⋆.
+lemma lexs_inv_atom1: ∀RN,RP,Y,f. ⋆ ⦻*[RN, RP, f] Y → Y = ⋆.
/2 width=6 by lexs_inv_atom1_aux/ qed-.
-fact lexs_inv_next1_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ⫯g →
- ∃∃K2,W2. K1 ⦻*[R, g] K2 & R (Ⓣ) K1 W1 W2 & Y = K2.ⓑ{J}W2.
-#R #X #Y #f * -X -Y -f
+fact lexs_inv_next1_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ⫯g →
+ ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
+#RN #RP #X #Y #f * -X -Y -f
[ #f #J #K1 #W1 #g #H destruct
-| #I #L1 #L2 #V1 #V2 #f #HL #HS #J #K1 #W1 #g #H1 #H2 <(injective_next … H2) -g destruct
+| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_next … H2) -g destruct
/2 width=5 by ex3_2_intro/
| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K1 #W1 #g #_ #H elim (discr_push_next … H)
]
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma lexs_inv_next1: ∀R,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[R, ⫯g] Y →
- ∃∃K2,W2. K1 ⦻*[R, g] K2 & R (Ⓣ) K1 W1 W2 & Y = K2.ⓑ{J}W2.
+lemma lexs_inv_next1: ∀RN,RP,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[RN, RP, ⫯g] Y →
+ ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & Y = K2.ⓑ{J}W2.
/2 width=7 by lexs_inv_next1_aux/ qed-.
-fact lexs_inv_push1_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ↑g →
- ∃∃K2,W2. K1 ⦻*[R, g] K2 & R (Ⓕ) K1 W1 W2 & Y = K2.ⓑ{J}W2.
-#R #X #Y #f * -X -Y -f
+fact lexs_inv_push1_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K1,W1,g. X = K1.ⓑ{J}W1 → f = ↑g →
+ ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
+#RN #RP #X #Y #f * -X -Y -f
[ #f #J #K1 #W1 #g #H destruct
| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K1 #W1 #g #_ #H elim (discr_next_push … H)
-| #I #L1 #L2 #V1 #V2 #f #HL #HO #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct
+| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K1 #W1 #g #H1 #H2 <(injective_push … H2) -g destruct
/2 width=5 by ex3_2_intro/
]
qed-.
-lemma lexs_inv_push1: ∀R,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[R, ↑g] Y →
- ∃∃K2,W2. K1 ⦻*[R, g] K2 & R (Ⓕ) K1 W1 W2 & Y = K2.ⓑ{J}W2.
+lemma lexs_inv_push1: ∀RN,RP,J,K1,Y,W1,g. K1.ⓑ{J}W1 ⦻*[RN, RP, ↑g] Y →
+ ∃∃K2,W2. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & Y = K2.ⓑ{J}W2.
/2 width=7 by lexs_inv_push1_aux/ qed-.
-fact lexs_inv_atom2_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → Y = ⋆ → X = ⋆.
-#R #X #Y #f * -X -Y -f //
+fact lexs_inv_atom2_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → Y = ⋆ → X = ⋆.
+#RN #RP #X #Y #f * -X -Y -f //
#I #L1 #L2 #V1 #V2 #f #_ #_ #H destruct
qed-.
(* Basic_2A1: includes lpx_sn_inv_atom2 *)
-lemma lexs_inv_atom2: ∀R,X,f. X ⦻*[R, f] ⋆ → X = ⋆.
+lemma lexs_inv_atom2: ∀RN,RP,X,f. X ⦻*[RN, RP, f] ⋆ → X = ⋆.
/2 width=6 by lexs_inv_atom2_aux/ qed-.
-fact lexs_inv_next2_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ⫯g →
- ∃∃K1,W1. K1 ⦻*[R, g] K2 & R (Ⓣ) K1 W1 W2 & X = K1.ⓑ{J}W1.
-#R #X #Y #f * -X -Y -f
+fact lexs_inv_next2_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ⫯g →
+ ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
+#RN #RP #X #Y #f * -X -Y -f
[ #f #J #K2 #W2 #g #H destruct
-| #I #L1 #L2 #V1 #V2 #f #HL #HS #J #K2 #W2 #g #H1 #H2 <(injective_next … H2) -g destruct
+| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_next … H2) -g destruct
/2 width=5 by ex3_2_intro/
| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K2 #W2 #g #_ #H elim (discr_push_next … H)
]
qed-.
(* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma lexs_inv_next2: ∀R,J,X,K2,W2,g. X ⦻*[R, ⫯g] K2.ⓑ{J}W2 →
- ∃∃K1,W1. K1 ⦻*[R, g] K2 & R (Ⓣ) K1 W1 W2 & X = K1.ⓑ{J}W1.
+lemma lexs_inv_next2: ∀RN,RP,J,X,K2,W2,g. X ⦻*[RN, RP, ⫯g] K2.ⓑ{J}W2 →
+ ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RN K1 W1 W2 & X = K1.ⓑ{J}W1.
/2 width=7 by lexs_inv_next2_aux/ qed-.
-fact lexs_inv_push2_aux: ∀R,X,Y,f. X ⦻*[R, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ↑g →
- ∃∃K1,W1. K1 ⦻*[R, g] K2 & R (Ⓕ) K1 W1 W2 & X = K1.ⓑ{J}W1.
-#R #X #Y #f * -X -Y -f
+fact lexs_inv_push2_aux: ∀RN,RP,X,Y,f. X ⦻*[RN, RP, f] Y → ∀J,K2,W2,g. Y = K2.ⓑ{J}W2 → f = ↑g →
+ ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
+#RN #RP #X #Y #f * -X -Y -f
[ #f #J #K2 #W2 #g #H destruct
| #I #L1 #L2 #V1 #V2 #f #_ #_ #J #K2 #W2 #g #_ #H elim (discr_next_push … H)
-| #I #L1 #L2 #V1 #V2 #f #HL #HO #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct
+| #I #L1 #L2 #V1 #V2 #f #HL #HV #J #K2 #W2 #g #H1 #H2 <(injective_push … H2) -g destruct
/2 width=5 by ex3_2_intro/
]
qed-.
-lemma lexs_inv_push2: ∀R,J,X,K2,W2,g. X ⦻*[R, ↑g] K2.ⓑ{J}W2 →
- ∃∃K1,W1. K1 ⦻*[R, g] K2 & R (Ⓕ) K1 W1 W2 & X = K1.ⓑ{J}W1.
+lemma lexs_inv_push2: ∀RN,RP,J,X,K2,W2,g. X ⦻*[RN, RP, ↑g] K2.ⓑ{J}W2 →
+ ∃∃K1,W1. K1 ⦻*[RN, RP, g] K2 & RP K1 W1 W2 & X = K1.ⓑ{J}W1.
/2 width=7 by lexs_inv_push2_aux/ qed-.
(* Basic_2A1: includes lpx_sn_inv_pair *)
-lemma lexs_inv_next: ∀R,I1,I2,L1,L2,V1,V2,f.
- L1.ⓑ{I1}V1 ⦻*[R, ⫯f] (L2.ⓑ{I2}V2) →
- ∧∧ L1 ⦻*[R, f] L2 & R (Ⓣ) L1 V1 V2 & I1 = I2.
-#R #I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_next1 … H) -H
+lemma lexs_inv_next: ∀RN,RP,I1,I2,L1,L2,V1,V2,f.
+ L1.ⓑ{I1}V1 ⦻*[RN, RP, ⫯f] (L2.ⓑ{I2}V2) →
+ ∧∧ L1 ⦻*[RN, RP, f] L2 & RN L1 V1 V2 & I1 = I2.
+#RN #RP #I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_next1 … H) -H
#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
qed-.
-lemma lexs_inv_push: ∀R,I1,I2,L1,L2,V1,V2,f.
- L1.ⓑ{I1}V1 ⦻*[R, ↑f] (L2.ⓑ{I2}V2) →
- ∧∧ L1 ⦻*[R, f] L2 & R (Ⓕ) L1 V1 V2 & I1 = I2.
-#R #I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_push1 … H) -H
+lemma lexs_inv_push: ∀RN,RP,I1,I2,L1,L2,V1,V2,f.
+ L1.ⓑ{I1}V1 ⦻*[RN, RP, ↑f] (L2.ⓑ{I2}V2) →
+ ∧∧ L1 ⦻*[RN, RP, f] L2 & RP L1 V1 V2 & I1 = I2.
+#RN #RP #I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_push1 … H) -H
#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
qed-.
(* Basic properties *********************************************************)
-lemma lexs_eq_repl_back: ∀R,L1,L2. eq_stream_repl_back … (λf. L1 ⦻*[R, f] L2).
-#R #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
-[ #I #L1 #L2 #V1 #v2 #f1 #_ #HS #IH #f2 #H elim (next_inv_sn … H) -H /3 width=1 by lexs_next/
-| #I #L1 #L2 #V1 #v2 #f1 #_ #HO #IH #f2 #H elim (push_inv_sn … H) -H /3 width=1 by lexs_push/
+lemma lexs_eq_repl_back: ∀RN,RP,L1,L2. eq_stream_repl_back … (λf. L1 ⦻*[RN, RP, f] L2).
+#RN #RP #L1 #L2 #f1 #H elim H -L1 -L2 -f1 //
+#I #L1 #L2 #V1 #v2 #f1 #_ #HV #IH #f2 #H
+[ elim (next_inv_sn … H) -H /3 width=1 by lexs_next/
+| elim (push_inv_sn … H) -H /3 width=1 by lexs_push/
]
qed-.
-lemma lexs_eq_repl_fwd: ∀R,L1,L2. eq_stream_repl_fwd … (λf. L1 ⦻*[R, f] L2).
-#R #L1 #L2 @eq_stream_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *)
+lemma lexs_eq_repl_fwd: ∀RN,RP,L1,L2. eq_stream_repl_fwd … (λf. L1 ⦻*[RN, RP, f] L2).
+#RN #RP #L1 #L2 @eq_stream_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *)
qed-.
+(* Note: fexs_sym and fexs_trans hold, but lexs_sym and lexs_trans do not *)
(* Basic_2A1: includes: lpx_sn_refl *)
-lemma lexs_refl: ∀R,f.
- (∀b,L. reflexive … (R b L)) →
- reflexive … (lexs R f).
-#R #f #HR #L generalize in match f; -f elim L -L //
+lemma lexs_refl: ∀RN,RP,f.
+ (∀L. reflexive … (RN L)) →
+ (∀L. reflexive … (RP L)) →
+ reflexive … (lexs RN RP f).
+#RN #RP #f #HRN #HRP #L generalize in match f; -f elim L -L //
#L #I #V #IH * * /2 width=1 by lexs_next, lexs_push/
qed.
-lemma sle_lexs_trans: ∀R. (∀L,T1,T2. R (Ⓣ) L T1 T2 → R (Ⓕ) L T1 T2) →
- ∀L1,L2,f2. L1 ⦻*[R, f2] L2 →
- ∀f1. f1 ⊆ f2 → L1 ⦻*[R, f1] L2.
-#R #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 //
+lemma sle_lexs_trans: ∀RN,RP. (∀L,T1,T2. RN L T1 T2 → RP L T1 T2) →
+ ∀L1,L2,f2. L1 ⦻*[RN, RP, f2] L2 →
+ ∀f1. f1 ⊆ f2 → L1 ⦻*[RN, RP, f1] L2.
+#RN #RP #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 //
#I #L1 #L2 #V1 #V2 #f2 #_ #HV12 #IH
[ * * [2: #n1 ] ] #f1 #H
[ /4 width=5 by lexs_next, sle_inv_SS_aux/
]
qed-.
-lemma sle_lexs_conf: ∀R. (∀L,T1,T2. R (Ⓕ) L T1 T2 → R (Ⓣ) L T1 T2) →
- ∀L1,L2,f1. L1 ⦻*[R, f1] L2 →
- ∀f2. f1 ⊆ f2 → L1 ⦻*[R, f2] L2.
-#R #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 //
+lemma sle_lexs_conf: ∀RN,RP. (∀L,T1,T2. RP L T1 T2 → RN L T1 T2) →
+ ∀L1,L2,f1. L1 ⦻*[RN, RP, f1] L2 →
+ ∀f2. f1 ⊆ f2 → L1 ⦻*[RN, RP, f2] L2.
+#RN #RP #HR #L1 #L2 #f2 #H elim H -L1 -L2 -f2 //
#I #L1 #L2 #V1 #V2 #f1 #_ #HV12 #IH
[2: * * [2: #n2 ] ] #f2 #H
[ /4 width=5 by lexs_next, sle_inv_OS_aux/
]
qed-.
-lemma lexs_co: ∀R1,R2.
- (∀b,L1,T1,T2. R1 b L1 T1 T2 → R2 b L1 T1 T2) →
- ∀L1,L2,f. L1 ⦻*[R1, f] L2 → L1 ⦻*[R2, f] L2.
-#R1 #R2 #HR #L1 #L2 #f #H elim H -L1 -L2 -f
+lemma lexs_co: ∀RN1,RP1,RN2,RP2.
+ (∀L1,T1,T2. RN1 L1 T1 T2 → RN2 L1 T1 T2) →
+ (∀L1,T1,T2. RP1 L1 T1 T2 → RP2 L1 T1 T2) →
+ ∀L1,L2,f. L1 ⦻*[RN1, RP1, f] L2 → L1 ⦻*[RN2, RP2, f] L2.
+#RN1 #RP1 #RN2 #RP2 #HRN #HRP #L1 #L2 #f #H elim H -L1 -L2 -f
/3 width=1 by lexs_atom, lexs_next, lexs_push/
qed-.
(* Forward lemmas on length for local environments **************************)
(* Basic_2A1: includes: lpx_sn_fwd_length *)
-lemma lexs_fwd_length: ∀R,L1,L2,f. L1 ⦻*[R, f] L2 → |L1| = |L2|.
-#R #L1 #L2 #f #H elim H -L1 -L2 -f //
+lemma lexs_fwd_length: ∀RN,RP,L1,L2,f. L1 ⦻*[RN, RP, f] L2 → |L1| = |L2|.
+#RM #RP #L1 #L2 #f #H elim H -L1 -L2 -f //
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 "basic_2/notation/relations/lazyeq_3.ma".
+include "basic_2/grammar/ceq.ma".
+include "basic_2/relocation/lexs.ma".
+
+(* RANGED EQUIVALENCE FOR LOCAL ENVIRONMENTS ********************************)
+
+(* Basic_2A1: includes: lreq_atom lreq_zero lreq_pair lreq_succ *)
+definition lreq: relation3 rtmap lenv lenv ≝ lexs ceq cfull.
+
+interpretation
+ "ranged equivalence (local environment)"
+ 'LazyEq f L1 L2 = (lreq f L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lreq_eq_repl_back: ∀L1,L2. eq_stream_repl_back … (λf. L1 ≡[f] L2).
+/2 width=3 by lexs_eq_repl_back/ qed-.
+
+lemma lreq_eq_repl_fwd: ∀L1,L2. eq_stream_repl_fwd … (λf. L1 ≡[f] L2).
+/2 width=3 by lexs_eq_repl_fwd/ qed-.
+
+lemma sle_lreq_trans: ∀L1,L2,f2. L1 ≡[f2] L2 →
+ ∀f1. f1 ⊆ f2 → L1 ≡[f1] L2.
+/2 width=3 by sle_lexs_trans/ qed-.
+
+(* Basic_2A1: includes: lreq_refl *)
+lemma lreq_refl: ∀f. reflexive … (lreq f).
+/2 width=1 by lexs_refl/ qed.
+
+(* Basic_2A1: includes: lreq_sym *)
+lemma lreq_sym: ∀f. symmetric … (lreq f).
+#f #L1 #L2 #H elim H -L1 -L2 -f
+/2 width=1 by lexs_next, lexs_push/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: includes: lreq_inv_atom1 *)
+lemma lreq_inv_atom1: ∀Y,f. ⋆ ≡[f] Y → Y = ⋆.
+/2 width=4 by lexs_inv_atom1/ qed-.
+
+(* Basic_2A1: includes: lreq_inv_pair1 *)
+lemma lreq_inv_next1: ∀J,K1,Y,W1,g. K1.ⓑ{J}W1 ≡[⫯g] Y →
+ ∃∃K2. K1 ≡[g] K2 & Y = K2.ⓑ{J}W1.
+#J #K1 #Y #W1 #g #H elim (lexs_inv_next1 … H) -H /2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_zero1 lreq_inv_succ1 *)
+lemma lreq_inv_push1: ∀J,K1,Y,W1,g. K1.ⓑ{J}W1 ≡[↑g] Y →
+ ∃∃K2,W2. K1 ≡[g] K2 & Y = K2.ⓑ{J}W2.
+#J #K1 #Y #W1 #g #H elim (lexs_inv_push1 … H) -H /2 width=4 by ex2_2_intro/ qed-.
+
+(* Basic_2A1: includes: lreq_inv_atom2 *)
+lemma lreq_inv_atom2: ∀X,f. X ≡[f] ⋆ → X = ⋆.
+/2 width=4 by lexs_inv_atom2/ qed-.
+
+(* Basic_2A1: includes: lreq_inv_pair2 *)
+lemma lreq_inv_next2: ∀J,X,K2,W2,g. X ≡[⫯g] K2.ⓑ{J}W2 →
+ ∃∃K1. K1 ≡[g] K2 & X = K1.ⓑ{J}W2.
+#J #X #K2 #W2 #g #H elim (lexs_inv_next2 … H) -H /2 width=3 by ex2_intro/ qed-.
+
+(* Basic_2A1: includes: lreq_inv_zero2 lreq_inv_succ2 *)
+lemma lreq_inv_push2: ∀J,X,K2,W2,g. X ≡[↑g] K2.ⓑ{J}W2 →
+ ∃∃K1,W1. K1 ≡[g] K2 & X = K1.ⓑ{J}W1.
+#J #X #K2 #W2 #g #H elim (lexs_inv_push2 … H) -H /2 width=4 by ex2_2_intro/ qed-.
+
+(* Basic_2A1: includes: lreq_inv_pair *)
+lemma lreq_inv_next: ∀I1,I2,L1,L2,V1,V2,f.
+ L1.ⓑ{I1}V1 ≡[⫯f] (L2.ⓑ{I2}V2) →
+ ∧∧ L1 ≡[f] L2 & V1 = V2 & I1 = I2.
+/2 width=1 by lexs_inv_next/ qed-.
+
+(* Basic_2A1: includes: lreq_inv_succ *)
+lemma lreq_inv_push: ∀I1,I2,L1,L2,V1,V2,f.
+ L1.ⓑ{I1}V1 ≡[↑f] (L2.ⓑ{I2}V2) →
+ L1 ≡[f] L2 ∧ I1 = I2.
+#I1 #I2 #L1 #L2 #V1 #V2 #f #H elim (lexs_inv_push … H) -H /2 width=1 by conj/
+qed-.
+
+(* Basic_2A1: removed theorems 5:
+ lreq_pair_lt lreq_succ_lt lreq_pair_O_Y lreq_O2 lreq_inv_O_Y
+*)
--- /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/relocation/lexs_length.ma".
+include "basic_2/relocation/lreq.ma".
+
+(* RANGED EQUIVALENCE FOR LOCAL ENVIRONMENTS ********************************)
+
+(* Forward lemmas on length for local environments **************************)
+
+(* Basic_2A1: includes: lreq_fwd_length *)
+lemma lreq_fwd_length: ∀L1,L2,f. L1 ≡[f] L2 → |L1| = |L2|.
+/2 width=4 by lexs_fwd_length/ 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 "basic_2/substitution/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-definition lpx_sn_confluent: relation (relation3 lenv term term) ≝ λR1,R2.
- ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. lpx_sn R1 L0 L1 → ∀L2. lpx_sn R2 L0 L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-definition lpx_sn_transitive: relation (relation3 lenv term term) ≝ λR1,R2.
- ∀L1,T1,T. R1 L1 T1 T → ∀L2. lpx_sn R1 L1 L2 →
- ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
-
-(* Main properties **********************************************************)
-
-theorem lpx_sn_trans: ∀R. lpx_sn_transitive R R → Transitive … (lpx_sn R).
-#R #HR #L1 #L #H elim H -L1 -L //
-#I #L1 #L #V1 #V #HL1 #HV1 #IHL1 #X #H
-elim (lpx_sn_inv_pair1 … H) -H #L2 #V2 #HL2 #HV2 #H destruct /3 width=5 by lpx_sn_pair/
-qed-.
-
-theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
- confluent2 … (lpx_sn R1) (lpx_sn R2).
-#R1 #R2 #HR12 #L0 @(ynat_f_ind … length … L0) -L0 #x #IH *
-[ #_ #X1 #H1 #X2 #H2 -x
- >(lpx_sn_inv_atom1 … H1) -X1
- >(lpx_sn_inv_atom1 … H2) -X2 /2 width=3 by lpx_sn_atom, ex2_intro/
-| #L0 #I #V0 #Hx #X1 #H1 #X2 #H2 destruct
- elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
- elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
- elim (IH … HL01 … HL02) -IH /2 width=2 by ylt_succ2_refl/ #L #HL1 #HL2
- elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lpx_sn_pair, ex2_intro/
-]
-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 "basic_2/substitution/lpx_sn.ma".
-
-(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
-
-(* Properties on transitive_closure *****************************************)
-
-lemma TC_lpx_sn_pair_refl: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,V. TC … (lpx_sn R) (L1. ⓑ{I} V) (L2. ⓑ{I} V).
-#R #HR #L1 #L2 #H @(TC_star_ind … L2 H) -L2
-[ /2 width=1 by lpx_sn_refl/
-| /3 width=1 by TC_reflexive, lpx_sn_refl/
-| /3 width=5 by lpx_sn_pair, step/
-]
-qed-.
-
-lemma TC_lpx_sn_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀V1,V2. LTC … R L1 V1 V2 →
- TC … (lpx_sn R) (L1. ⓑ{I} V1) (L2. ⓑ{I} V2).
-#R #HR #I #L1 #L2 #HL12 #V1 #V2 #H @(TC_star_ind_dx … V1 H) -V1 //
-[ /2 width=1 by TC_lpx_sn_pair_refl/
-| /4 width=3 by TC_strap, lpx_sn_pair, lpx_sn_refl/
-]
-qed-.
-
-lemma lpx_sn_LTC_TC_lpx_sn: ∀R. (∀L. reflexive … (R L)) →
- ∀L1,L2. lpx_sn (LTC … R) L1 L2 →
- TC … (lpx_sn R) L1 L2.
-#R #HR #L1 #L2 #H elim H -L1 -L2
-/2 width=1 by TC_lpx_sn_pair, lpx_sn_atom, inj/
-qed-.
-
-(* Inversion lemmas on transitive closure ***********************************)
-
-lemma TC_lpx_sn_inv_atom2: ∀R,L1. TC … (lpx_sn R) L1 (⋆) → L1 = ⋆.
-#R #L1 #H @(TC_ind_dx … L1 H) -L1
-[ /2 width=2 by lpx_sn_inv_atom2/
-| #L1 #L #HL1 #_ #IHL2 destruct /2 width=2 by lpx_sn_inv_atom2/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair2: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
- ∀I,L1,K2,V2. TC … (lpx_sn R) L1 (K2.ⓑ{I}V2) →
- ∃∃K1,V1. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L1 = K1. ⓑ{I} V1.
-#R #HR #I #L1 #K2 #V2 #H @(TC_ind_dx … L1 H) -L1
-[ #L1 #H elim (lpx_sn_inv_pair2 … H) -H /3 width=5 by inj, ex3_2_intro/
-| #L1 #L #HL1 #_ * #K #V #HK2 #HV2 #H destruct
- elim (lpx_sn_inv_pair2 … HL1) -HL1 #K1 #V1 #HK1 #HV1 #H destruct
- lapply (HR … HV2 … HK1) -HR -HV2 /3 width=5 by TC_strap, ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_ind: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
- ∀S:relation lenv.
- S (⋆) (⋆) → (
- ∀I,K1,K2,V1,V2.
- TC … (lpx_sn R) K1 K2 → LTC … R K1 V1 V2 →
- S K1 K2 → S (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
- ) →
- ∀L2,L1. TC … (lpx_sn R) L1 L2 → S L1 L2.
-#R #HR #S #IH1 #IH2 #L2 elim L2 -L2
-[ #X #H >(TC_lpx_sn_inv_atom2 … H) -X //
-| #L2 #I #V2 #IHL2 #X #H
- elim (TC_lpx_sn_inv_pair2 … H) // -H -HR
- #L1 #V1 #HL12 #HV12 #H destruct /3 width=1 by/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_atom1: ∀R,L2. TC … (lpx_sn R) (⋆) L2 → L2 = ⋆.
-#R #L2 #H elim H -L2
-[ /2 width=2 by lpx_sn_inv_atom1/
-| #L #L2 #_ #HL2 #IHL1 destruct /2 width=2 by lpx_sn_inv_atom1/
-]
-qed-.
-
-fact TC_lpx_sn_inv_pair1_aux: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-#R #HR #L1 #L2 #H @(TC_lpx_sn_ind … H) // -HR -L1 -L2
-[ #J #K #W #H destruct
-| #I #L1 #L2 #V1 #V2 #HL12 #HV12 #_ #J #K #W #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma TC_lpx_sn_inv_pair1: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
- ∀I,K1,L2,V1. TC … (lpx_sn R) (K1.ⓑ{I}V1) L2 →
- ∃∃K2,V2. TC … (lpx_sn R) K1 K2 & LTC … R K1 V1 V2 & L2 = K2. ⓑ{I} V2.
-/2 width=3 by TC_lpx_sn_inv_pair1_aux/ qed-.
-
-lemma TC_lpx_sn_inv_lpx_sn_LTC: ∀R. c_rs_transitive … R (λ_. lpx_sn R) →
- ∀L1,L2. TC … (lpx_sn R) L1 L2 →
- lpx_sn (LTC … R) L1 L2.
-/3 width=4 by TC_lpx_sn_ind, lpx_sn_pair/ qed-.
-
-(* Forward lemmas on transitive closure *************************************)
-
-lemma TC_lpx_sn_fwd_length: ∀R,L1,L2. TC … (lpx_sn R) L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L2
-[ #L2 #HL12 >(lpx_sn_fwd_length … HL12) -HL12 //
-| #L #L2 #_ #HL2 #IHL1
- >IHL1 -L1 >(lpx_sn_fwd_length … HL2) -HL2 //
-]
-qed-.
projects / helm.git / commitdiff