(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
(* alternative definition of lpx_sn *)
-inductive lpx_sn_alt (R:relation3 lenv term term): relation lenv ≝
-| lpx_sn_alt_intro: ∀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,K1,K2,V1,V2,i.
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → lpx_sn_alt R K1 K2
- ) → |L1| = |L2| → lpx_sn_alt R L1 L2
-.
-
-(* compact definition of lpx_sn_alt *****************************************)
-
-lemma lpx_sn_alt_ind_alt: ∀R. ∀S:relation lenv.
- (∀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_alt R K1 K2 & S K1 K2
- ) → S L1 L2) →
- ∀L1,L2. lpx_sn_alt R L1 L2 → S L1 L2.
-#R #S #IH #L1 #L2 #H elim H -L1 -L2
-#L1 #L2 #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (H1 … HLK1 HLK2) -H1
-/3 width=7 by and4_intro/
-qed-.
-
-lemma lpx_sn_alt_inv_alt: ∀R,L1,L2. lpx_sn_alt R L1 L2 →
- |L1| = |L2| ∧
- ∀I1,I2,K1,K2,V1,V2,i.
+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 & lpx_sn_alt R K1 K2.
-#R #L1 #L2 #H @(lpx_sn_alt_ind_alt … H) -L1 -L2
-#L1 #L2 #HL12 #IH @conj // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
-/2 width=1 by and3_intro/
-qed-.
-
-lemma lpx_sn_alt_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_alt R K1 K2
- ) → lpx_sn_alt R L1 L2.
-#R #L1 #L2 #HL12 #IH @lpx_sn_alt_intro // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
-elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by conj/
-qed.
+ I1 = I2 ∧ R I1 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 (lpx_sn_alt_inv_alt … H) //
+#R #L1 #L2 #H elim H //
qed-.
(* Basic inversion 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.
-#R #I1 #L2 #K1 #V1 #H elim (lpx_sn_alt_inv_alt … H) -H
+ ∃∃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
-elim (IH I1 I2 K1 K2 V1 V2 0) -IH /2 width=5 by ex3_2_intro/
+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+1)) -IH
+/2 width=1 by ldrop_drop, conj/
qed-.
lemma lpx_sn_alt_inv_atom2: ∀R,L1. lpx_sn_alt R L1 (⋆) → L1 = ⋆.
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 (lpx_sn_alt_inv_alt … H) -H
+ ∃∃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
-elim (IH I1 I2 K1 K2 V1 V2 0) -IH /2 width=5 by ex3_2_intro/
+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+1)) -IH
+/2 width=1 by ldrop_drop, conj/
qed-.
(* Basic properties *********************************************************)
lemma lpx_sn_alt_atom: ∀R. lpx_sn_alt R (⋆) (⋆).
-#R @lpx_sn_alt_intro_alt //
+#R @conj //
#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 elim (ldrop_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 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 (lpx_sn_alt_inv_alt … H) -H
-#HL12 #IH @lpx_sn_alt_intro_alt normalize //
+#R #I #L1 #L2 #V1 #V2 #H #HV12 elim H -H
+#HL12 #IH @conj normalize //
#I1 #I2 #K1 #K2 #W1 #W2 #i @(nat_ind_plus … i) -i
[ #HLK1 #HLK2
lapply (ldrop_inv_O2 … HLK1) -HLK1 #H destruct
lapply (ldrop_inv_O2 … HLK2) -HLK2 #H destruct
- /4 width=3 by lpx_sn_alt_intro_alt, and3_intro/
-| -HL12 -HV12 /3 width=5 by ldrop_inv_drop1/
+ /2 width=1 by conj/
+| -HL12 -HV12 /3 width=6 by ldrop_inv_drop1/
]
qed.
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 K1 K2
+ I1 = I2 ∧ R I1 K1 V1 V2
) → lpx_sn R L1 L2.
-#R #L1 #L2 #HL12 #IH @lpx_sn_alt_inv_lpx_sn
-@lpx_sn_alt_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
-elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=1 by lpx_sn_lpx_sn_alt, and3_intro/
-qed.
-
-lemma lpx_sn_ind_alt: ∀R. ∀S:relation lenv.
- (∀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 K1 K2 & S K1 K2
- ) → S L1 L2) →
- ∀L1,L2. lpx_sn R L1 L2 → S L1 L2.
-#R #S #IH1 #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
-#H @(lpx_sn_alt_ind_alt … H) -L1 -L2
-#L1 #L2 #HL12 #IH2 @IH1 -IH1 // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2 elim (IH2 … HLK1 HLK2) -IH2 -HLK1 -HLK2
-/3 width=1 by lpx_sn_alt_inv_lpx_sn, and4_intro/
-qed-.
+/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 & lpx_sn R K1 K2.
+ I1 = I2 ∧ R I1 K1 V1 V2.
#R #L1 #L2 #H lapply (lpx_sn_lpx_sn_alt … H) -H
-#H elim (lpx_sn_alt_inv_alt … H) -H
-#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #HLK1 #HLK2
-elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /3 width=1 by lpx_sn_alt_inv_lpx_sn, and3_intro/
+#H elim H -H /3 width=4 by conj/
qed-.