∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
i < d ∨
∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & llpx_sn R 0 V1 K1 K2 &
∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
i < d ∨
∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 & llpx_sn R 0 V1 K1 K2 &
#R #L1 #L2 #d #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
[ #_ #H elim (lt_refl_false i)
lapply (ldrop_fwd_length_lt2 … HLK2) -HLK2
#R #L1 #L2 #d #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
[ #_ #H elim (lt_refl_false i)
lapply (ldrop_fwd_length_lt2 … HLK2) -HLK2
∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
i < d ∨
∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & llpx_sn R 0 V1 K1 K2 &
∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
i < d ∨
∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 & llpx_sn R 0 V1 K1 K2 &
#R #L1 #L2 #d #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
[ #H #_ elim (lt_refl_false i)
lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
#R #L1 #L2 #d #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref … H) -H [ * || * ]
[ #H #_ elim (lt_refl_false i)
lapply (ldrop_fwd_length_lt2 … HLK1) -HLK1
lemma llpx_sn_inv_lref_ge_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
lemma llpx_sn_inv_lref_ge_dx: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
∀I,K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 →
∃∃K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 &
#R #L1 #L2 #d #i #H #Hdi #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
[ #H elim (ylt_yle_false … H Hdi)
| * /2 width=5 by ex3_2_intro/
#R #L1 #L2 #d #i #H #Hdi #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx … H … HLK2) -L2
[ #H elim (ylt_yle_false … H Hdi)
| * /2 width=5 by ex3_2_intro/
lemma llpx_sn_inv_lref_ge_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
lemma llpx_sn_inv_lref_ge_sn: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
∀I,K1,V1. ⇩[i] L1 ≡ K1.ⓑ{I}V1 →
∃∃K2,V2. ⇩[i] L2 ≡ K2.ⓑ{I}V2 &
#R #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
[ #H elim (ylt_yle_false … H Hdi)
| * /2 width=5 by ex3_2_intro/
#R #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn … H … HLK1) -L1
[ #H elim (ylt_yle_false … H Hdi)
| * /2 width=5 by ex3_2_intro/
lemma llpx_sn_inv_lref_ge_bi: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
∀I1,I2,K1,K2,V1,V2.
⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
lemma llpx_sn_inv_lref_ge_bi: ∀R,L1,L2,d,i. llpx_sn R d (#i) L1 L2 → d ≤ i →
∀I1,I2,K1,K2,V1,V2.
⇩[i] L1 ≡ K1.ⓑ{I1}V1 → ⇩[i] L2 ≡ K2.ⓑ{I2}V2 →
#R #L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
elim (llpx_sn_inv_lref_ge_sn … HL12 … HLK1) // -L1 -d
#J #Y #HY lapply (ldrop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by and3_intro/
#R #L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
elim (llpx_sn_inv_lref_ge_sn … HL12 … HLK1) // -L1 -d
#J #Y #HY lapply (ldrop_mono … HY … HLK2) -L2 -i #H destruct /2 width=1 by and3_intro/
fact llpx_sn_inv_S_aux: ∀R,L1,L2,T,d0. llpx_sn R d0 T L1 L2 → ∀d. d0 = d + 1 →
∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
fact llpx_sn_inv_S_aux: ∀R,L1,L2,T,d0. llpx_sn R d0 T L1 L2 → ∀d. d0 = d + 1 →
∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
#R #L1 #L2 #T #d0 #H elim H -L1 -L2 -T -d0
/2 width=1 by llpx_sn_gref, llpx_sn_free, llpx_sn_sort/
[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct
#R #L1 #L2 #T #d0 #H elim H -L1 -L2 -T -d0
/2 width=1 by llpx_sn_gref, llpx_sn_free, llpx_sn_sort/
[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct
lemma llpx_sn_inv_S: ∀R,L1,L2,T,d. llpx_sn R (d + 1) T L1 L2 →
∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
lemma llpx_sn_inv_S: ∀R,L1,L2,T,d. llpx_sn R (d + 1) T L1 L2 →
∀K1,K2,I,V1,V2. ⇩[d] L1 ≡ K1.ⓑ{I}V1 → ⇩[d] L2 ≡ K2.ⓑ{I}V2 →
∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
llpx_sn R 0 V L1 L2 ∧ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind … H) -H
∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
llpx_sn R 0 V L1 L2 ∧ llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind … H) -H
∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind_O … H) -H //
∀a,I,L1,L2,V,T. llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 →
llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V).
#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind_O … H) -H //
(* Advanced properties ******************************************************)
lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) →
(* Advanced properties ******************************************************)
lemma llpx_sn_bind_repl_O: ∀R,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.ⓑ{I}V1) (L2.ⓑ{I}V2) →
-lemma llpx_sn_dec: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
+lemma llpx_sn_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
∀T,L1,L2,d. Decidable (llpx_sn R d T L1 L2).
#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
#n #IH #L1 * *
∀T,L1,L2,d. Decidable (llpx_sn R d T L1 L2).
#R #HR #T #L1 @(f2_ind … rfw … L1 T) -L1 -T
#n #IH #L1 * *
elim (ldrop_O1_lt (Ⓕ) … HiL2) #I2 #K2 #V2 #HLK2
elim (ldrop_O1_lt (Ⓕ) … HiL1) #I1 #K1 #V1 #HLK1
elim (eq_bind2_dec I2 I1)
elim (ldrop_O1_lt (Ⓕ) … HiL2) #I2 #K2 #V2 #HLK2
elim (ldrop_O1_lt (Ⓕ) … HiL1) #I1 #K1 #V1 #HLK1
elim (eq_bind2_dec I2 I1)
[ #H3 elim (IH K1 V1 … K2 0) destruct
/3 width=9 by llpx_sn_lref, ldrop_fwd_rfw, or_introl/
]
[ #H3 elim (IH K1 V1 … K2 0) destruct
/3 width=9 by llpx_sn_lref, ldrop_fwd_rfw, or_introl/
]
∀K1,K2,T,d0. llpx_sn R d0 T K1 K2 →
∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
∀U. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 U L1 L2.
∀K1,K2,T,d0. llpx_sn R d0 T K1 K2 →
∀L1,L2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
∀U. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 U L1 L2.
∀L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
∀T. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 T K1 K2.
∀L1,L2,U,d0. llpx_sn R d0 U L1 L2 →
∀K1,K2,d,e. ⇩[Ⓕ, d, e] L1 ≡ K1 → ⇩[Ⓕ, d, e] L2 ≡ K2 →
∀T. ⇧[d, e] T ≡ U → d0 ≤ d → llpx_sn R d0 T K1 K2.
-lemma nllpx_sn_inv_bind: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
+lemma nllpx_sn_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
∀a,I,L1,L2,V,T,d. (llpx_sn R d (ⓑ{a,I}V.T) L1 L2 → ⊥) →
(llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
#R #HR #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
/4 width=1 by llpx_sn_bind, or_intror, or_introl/
qed-.
∀a,I,L1,L2,V,T,d. (llpx_sn R d (ⓑ{a,I}V.T) L1 L2 → ⊥) →
(llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R (⫯d) T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
#R #HR #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
/4 width=1 by llpx_sn_bind, or_intror, or_introl/
qed-.
-lemma nllpx_sn_inv_flat: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
+lemma nllpx_sn_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
∀I,L1,L2,V,T,d. (llpx_sn R d (ⓕ{I}V.T) L1 L2 → ⊥) →
(llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R d T L1 L2 → ⊥).
#R #HR #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
/4 width=1 by llpx_sn_flat, or_intror, or_introl/
qed-.
∀I,L1,L2,V,T,d. (llpx_sn R d (ⓕ{I}V.T) L1 L2 → ⊥) →
(llpx_sn R d V L1 L2 → ⊥) ∨ (llpx_sn R d T L1 L2 → ⊥).
#R #HR #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec … HR V L1 L2 d)
/4 width=1 by llpx_sn_flat, or_intror, or_introl/
qed-.
-lemma nllpx_sn_inv_bind_O: ∀R. (∀I,L,T1,T2. Decidable (R I L T1 T2)) →
+lemma nllpx_sn_inv_bind_O: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
∀a,I,L1,L2,V,T. (llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 → ⊥) →
(llpx_sn R 0 V L1 L2 → ⊥) ∨ (llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_dec … HR V L1 L2 0)
∀a,I,L1,L2,V,T. (llpx_sn R 0 (ⓑ{a,I}V.T) L1 L2 → ⊥) →
(llpx_sn R 0 V L1 L2 → ⊥) ∨ (llpx_sn R 0 T (L1.ⓑ{I}V) (L2.ⓑ{I}V) → ⊥).
#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_dec … HR V L1 L2 0)