(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-(* alternative definition of llpx_sn_alt *)
-inductive llpx_sn_alt (R:relation3 lenv term term): relation4 ynat term lenv lenv ≝
+(* alternative definition of llpx_sn *)
+inductive llpx_sn_alt (R:relation4 bind2 lenv term term): relation4 ynat term lenv lenv ≝
| llpx_sn_alt_intro: ∀L1,L2,T,d.
(∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R K1 V1 V2
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → I1 = I2 ∧ R I1 K1 V1 V2
) →
(∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt R 0 V1 K1 K2
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 → llpx_sn_alt R 0 V1 K1 K2
) → |L1| = |L2| → llpx_sn_alt R d T L1 L2
.
-(* Basic forward lemmas ******************************************************)
+(* Compact definition of llpx_sn_alt ****************************************)
-lemma llpx_sn_alt_fwd_gen: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 →
+lemma llpx_sn_alt_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
+ (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2
+ ) → llpx_sn_alt R d T L1 L2.
+#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_intro // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
+qed.
+
+lemma llpx_sn_alt_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
+ (∀L1,L2,T,d. |L1| = |L2| → (
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2 & S 0 V1 K1 K2
+ ) → S d T L1 L2) →
+ ∀L1,L2,T,d. llpx_sn_alt R d T L1 L2 → S d T L1 L2.
+#R #S #IH #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+#L1 #L2 #T #d #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (H1 … HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
+qed-.
+
+lemma llpx_sn_alt_inv_alt: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 →
|L1| = |L2| ∧
∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2.
-#R #L1 #L2 #T #d * -L1 -L2 -T -d
-#L1 #L2 #T #d #IH1 #IH2 #HL12 @conj //
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH1 … HnT HLK1 HLK2) -IH1 /4 width=8 by and3_intro/
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2.
+#R #L1 #L2 #T #d #H @(llpx_sn_alt_ind_alt … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
qed-.
-lemma llpx_sn_alt_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #T #d * -L1 -L2 -T -d //
+(* Basic inversion lemmas ***************************************************)
+
+lemma llpx_sn_alt_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓕ{I}V.T) L1 L2 →
+ llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
+#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
+/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
qed-.
-fact llpx_sn_alt_fwd_lref_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 → ∀i. X = #i →
- ∨∨ |L1| ≤ i ∧ |L2| ≤ i
- | yinj i < d
- | ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
- ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
- llpx_sn_alt R (yinj 0) V1 K1 K2 &
- R K1 V1 V2 & d ≤ yinj i.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-#L1 #L2 #X #d #H1X #H2X #HL12 #i #H destruct
-elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
-elim (ylt_split i d) /3 width=1 by or3_intro1/
-#Hdi #HL1 elim (ldrop_O1_lt … HL1) #I1 #K1 #V1 #HLK1
-elim (ldrop_O1_lt L2 i) // #I2 #K2 #V2 #HLK2
-elim (H1X … HLK1 HLK2) -H1X /2 width=3 by nlift_lref_be_SO/ #H #HV12 destruct
-lapply (H2X … HLK1 HLK2) -H2X /2 width=3 by nlift_lref_be_SO/
-/3 width=9 by or3_intro2, ex5_5_intro/
+lemma llpx_sn_alt_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2 →
+ llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
+#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_inv_alt … H) -H
+#HL12 #IH @conj @llpx_sn_alt_intro_alt [1,3: normalize // ] -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+[ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2
+ /3 width=9 by nlift_bind_sn, and3_intro/
+| lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
+ lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
+ lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
+ elim (IH … HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
+ @nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+]
+qed-.
+
+(* Basic forward lemmas ******************************************************)
+
+lemma llpx_sn_alt_fwd_length: ∀R,L1,L2,T,d. llpx_sn_alt R d T L1 L2 → |L1| = |L2|.
+#R #L1 #L2 #T #d #H elim (llpx_sn_alt_inv_alt … H) -H //
qed-.
lemma llpx_sn_alt_fwd_lref: ∀R,L1,L2,d,i. llpx_sn_alt R d (#i) L1 L2 →
| ∃∃I,K1,K2,V1,V2. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
⇩[i] L2 ≡ K2.ⓑ{I}V2 &
llpx_sn_alt R (yinj 0) V1 K1 K2 &
- R K1 V1 V2 & d ≤ yinj i.
-/2 width=3 by llpx_sn_alt_fwd_lref_aux/ qed-.
-
-(* Basic inversion lemmas ****************************************************)
-
-fact llpx_sn_alt_inv_flat_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 →
- ∀I,V,T. X = ⓕ{I}V.T →
- llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-#L1 #L2 #X #d #H1X #H2X #HL12
-#I #V #T #H destruct
-@conj @llpx_sn_alt_intro // -HL12
-/4 width=8 by nlift_flat_sn, nlift_flat_dx/
-qed-.
-
-lemma llpx_sn_alt_inv_flat: ∀R,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓕ{I}V.T) L1 L2 →
- llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R d T L1 L2.
-/2 width=4 by llpx_sn_alt_inv_flat_aux/ qed-.
-
-fact llpx_sn_alt_inv_bind_aux: ∀R,L1,L2,X,d. llpx_sn_alt R d X L1 L2 →
- ∀a,I,V,T. X = ⓑ{a,I}V.T →
- llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-#L1 #L2 #X #d #H1X #H2X #HL12
-#a #I #V #T #H destruct
-@conj @llpx_sn_alt_intro [3,6: normalize /2 width=1 by eq_f2/ ] -HL12
-#I1 #I2 #K1 #K2 #W1 #W2 #i #Hdi #H #HLK1 #HLK2
-[1,2: /4 width=9 by nlift_bind_sn/ ]
-lapply (yle_inv_succ1 … Hdi) -Hdi * #Hdi #Hi
-lapply (ldrop_inv_drop1_lt … HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
-lapply (ldrop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
-[ @(H1X … HLK1 HLK2) | @(H2X … HLK1 HLK2) ] // -I1 -I2 -L1 -L2 -K1 -K2 -W1 -W2
-@nlift_bind_dx <plus_minus_m_m /2 width=2 by ylt_O/
+ R I K1 V1 V2 & d ≤ yinj i.
+#R #L1 #L2 #d #i #H elim (llpx_sn_alt_inv_alt … H) -H
+#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
+elim (ylt_split i d) /3 width=1 by or3_intro1/
+#Hdi #HL1 elim (ldrop_O1_lt … HL1)
+#I1 #K1 #V1 #HLK1 elim (ldrop_O1_lt L2 i) //
+#I2 #K2 #V2 #HLK2 elim (IH … HLK1 HLK2) -IH
+/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
qed-.
-lemma llpx_sn_alt_inv_bind: ∀R,a,I,L1,L2,V,T,d. llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2 →
- llpx_sn_alt R d V L1 L2 ∧ llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
-/2 width=4 by llpx_sn_alt_inv_bind_aux/ qed-.
-
(* Basic properties **********************************************************)
-lemma llpx_sn_alt_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
- (∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
- ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn_alt R 0 V1 K1 K2
- ) → llpx_sn_alt R d T L1 L2.
-#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_intro // -HL12
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
-elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
-qed.
-
lemma llpx_sn_alt_sort: ∀R,L1,L2,d,k. |L1| = |L2| → llpx_sn_alt R d (⋆k) L1 L2.
-#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_intro_alt // -HL12
#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (⋆k)) //
qed.
lemma llpx_sn_alt_gref: ∀R,L1,L2,d,p. |L1| = |L2| → llpx_sn_alt R d (§p) L1 L2.
-#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_intro_alt // -HL12
#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
qed.
lemma llpx_sn_alt_skip: ∀R,L1,L2,d,i. |L1| = |L2| → yinj i < d → llpx_sn_alt R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_intro_alt // -HL12
#I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
/4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
qed.
lemma llpx_sn_alt_free: ∀R,L1,L2,d,i. |L1| ≤ i → |L2| ≤ i → |L1| = |L2| →
llpx_sn_alt R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_intro // -HL12
+#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_intro_alt // -HL12
#I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
/3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
lemma llpx_sn_alt_lref: ∀R,I,L1,L2,K1,K2,V1,V2,d,i. d ≤ yinj i →
⇩[i] L1 ≡ K1.ⓑ{I}V1 → ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
- llpx_sn_alt R 0 V1 K1 K2 → R K1 V1 V2 →
+ llpx_sn_alt R 0 V1 K1 K2 → R I K1 V1 V2 →
llpx_sn_alt R d (#i) L1 L2.
-#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_intro
-[1,2: #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
+#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_intro_alt
+[ lapply (llpx_sn_alt_fwd_length … HK12) -HK12 #HK12
+ @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize //
+| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
elim (lt_or_eq_or_gt i j) #Hij destruct
- [1,4: elim (H (#i)) -H /2 width=1 by lift_lref_lt/
- |2,5: lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
- lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by conj/
- |3,6: elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
+ [ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
+ | lapply (ldrop_mono … HLY1 … HLK1) -HLY1 -HLK1 #H destruct
+ lapply (ldrop_mono … HLY2 … HLK2) -HLY2 -HLK2 #H destruct /2 width=1 by and3_intro/
+ | elim (H (#(i-1))) -H /2 width=1 by lift_lref_ge_minus/
]
-| lapply (llpx_sn_alt_fwd_length … HK12) -HK12 #HK12
- @(ldrop_fwd_length_eq2 … HLK1 HLK2) normalize /2 width=1 by eq_f2/
]
qed.
-fact llpx_sn_alt_flat_aux: ∀R,I,L1,L2,V,d. llpx_sn_alt R d V L1 L2 →
- ∀Y1,Y2,T,m. llpx_sn_alt R m T Y1 Y2 →
- Y1 = L1 → Y2 = L2 → m = d →
- llpx_sn_alt R d (ⓕ{I}V.T) L1 L2.
-#R #I #L1 #L2 #V #d * -L1 -L2 -V -d #L1 #L2 #V #d #H1V #H2V #HL12
-#Y1 #Y2 #T #m * -Y1 -Y2 -T -m #Y1 #Y2 #T #m #H1T #H2T #_
-#HT1 #HY2 #Hm destruct
-@llpx_sn_alt_intro // -HL12
-#J1 #J2 #K1 #K2 #W1 #W2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_flat … HnVT) -HnVT /3 width=8 by/
-qed-.
-
lemma llpx_sn_alt_flat: ∀R,I,L1,L2,V,T,d.
llpx_sn_alt R d V L1 L2 → llpx_sn_alt R d T L1 L2 →
llpx_sn_alt R d (ⓕ{I}V.T) L1 L2.
-/2 width=7 by llpx_sn_alt_flat_aux/ qed.
-
-fact llpx_sn_alt_bind_aux: ∀R,a,I,L1,L2,V,d. llpx_sn_alt R d V L1 L2 →
- ∀Y1,Y2,T,m. llpx_sn_alt R m T Y1 Y2 →
- Y1 = L1.ⓑ{I}V → Y2 = L2.ⓑ{I}V → m = ⫯d →
- llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2.
-#R #a #I #L1 #L2 #V #d * -L1 -L2 -V -d #L1 #L2 #V #d #H1V #H2V #HL12
-#Y1 #Y2 #T #m * -Y1 -Y2 -T -m #Y1 #Y2 #T #m #H1T #H2T #_
-#HT1 #HY2 #Hm destruct
-@llpx_sn_alt_intro // -HL12
-#J1 #J2 #K1 #K2 #W1 #W2 #i #Hdi #HnVT #HLK1 #HLK2
-elim (nlift_inv_bind … HnVT) -HnVT /3 width=8 by ldrop_drop, yle_succ/
-qed-.
+#R #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_flat … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim (IHT … HLK1 … HLK2) -IHT /3 width=2 by and3_intro/
+]
+qed.
lemma llpx_sn_alt_bind: ∀R,a,I,L1,L2,V,T,d.
llpx_sn_alt R d V L1 L2 →
llpx_sn_alt R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) →
llpx_sn_alt R d (ⓑ{a,I}V.T) L1 L2.
-/2 width=7 by llpx_sn_alt_bind_aux/ qed.
+#R #a #I #L1 #L2 #V #T #d #HV #HT
+elim (llpx_sn_alt_inv_alt … HV) -HV #HL12 #IHV
+elim (llpx_sn_alt_inv_alt … HT) -HT #_ #IHT
+@llpx_sn_alt_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+elim (nlift_inv_bind … HnVT) -HnVT #H
+[ elim (IHV … HLK1 … HLK2) -IHV /2 width=2 by and3_intro/
+| elim IHT -IHT /2 width=12 by ldrop_drop, yle_succ, and3_intro/
+]
+qed.
(* Main properties **********************************************************)
]
qed-.
-(* Advanced properties ******************************************************)
+(* Alternative definition of llpx_sn ****************************************)
lemma llpx_sn_intro_alt: ∀R,L1,L2,T,d. |L1| = |L2| →
(∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2
) → llpx_sn R d T L1 L2.
#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_inv_lpx_sn
@llpx_sn_alt_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt, and3_intro/
qed.
-(* Advanced forward lemmas lemmas *******************************************)
+lemma llpx_sn_ind_alt: ∀R. ∀S:relation4 ynat term lenv lenv.
+ (∀L1,L2,T,d. |L1| = |L2| → (
+ ∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
+ ⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
+ ) → S d T L1 L2) →
+ ∀L1,L2,T,d. llpx_sn R d T L1 L2 → S d T L1 L2.
+#R #S #IH1 #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt … H) -H
+#H @(llpx_sn_alt_ind_alt … H) -L1 -L2 -T -d
+#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+elim (IH2 … HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_inv_lpx_sn, and4_intro/
+qed-.
-lemma llpx_sn_fwd_alt: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
+lemma llpx_sn_inv_alt: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 →
|L1| = |L2| ∧
∀I1,I2,K1,K2,V1,V2,i. d ≤ yinj i → (∀U. ⇧[i, 1] U ≡ T → ⊥) →
⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
- ∧∧ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
+ ∧∧ I1 = I2 & R I1 K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
#R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt … H) -H
-#H elim (llpx_sn_alt_fwd_gen … H) -H
+#H elim (llpx_sn_alt_inv_alt … H) -H
#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #HLK1 #HLK2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
elim (IH … HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_inv_lpx_sn, and3_intro/
qed-.
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