(* 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 ≝
+definition lpx_sn_alt: relation4 bind2 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
+ I1 = I2 ∧ R I1 K1 V1 V2
).
(* Basic forward lemmas ******************************************************)
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.
+ ∃∃K2,V2. lpx_sn_alt R K1 K2 & R I 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
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.
+ ∃∃K1,V1. lpx_sn_alt R K1 K2 & R I 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
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 L2 → R I 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 normalize //
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
+ I1 = I2 ∧ R I1 K1 V1 V2
) → lpx_sn R L1 L2.
/4 width=4 by lpx_sn_alt_inv_lpx_sn, conj/ qed.
|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.
+ I1 = I2 ∧ R I1 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-.