include "basic_2/substitution/cofrees_lift.ma".
include "basic_2/substitution/llpx_sn_alt1.ma".
+lemma le_plus_xSy_O_false: ∀x,y. x + S y ≤ 0 → ⊥.
+#x #y #H lapply (le_n_O_to_eq … H) -H <plus_n_Sm #H destruct
+qed-.
+
+lemma ldrop_fwd_length_ge: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → |L1| ≤ d → |L2| = |L1|.
+#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e // normalize
+[ #I #L1 #L2 #V #e #_ #_ #H elim (le_plus_xSy_O_false … H)
+| /4 width=2 by le_plus_to_le_r, eq_f/
+]
+qed-.
+
+lemma ldrop_fwd_length_le_le: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → d ≤ |L1| → e ≤ |L1| - d → |L2| = |L1| - e.
+#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e // normalize
+[ /3 width=2 by le_plus_to_le_r/
+| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #IHL12 >minus_plus_plus_l
+ #Hd #He lapply (le_plus_to_le_r … Hd) -Hd
+ #Hd >IHL12 // -L2 >plus_minus /2 width=3 by transitive_le/
+]
+qed-.
+
+lemma ldrop_fwd_length_le_ge: ∀L1,L2,d,e,s. ⇩[s, d, e] L1 ≡ L2 → d ≤ |L1| → |L1| - d ≤ e → |L2| = d.
+#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e normalize
+[ /2 width=1 by le_n_O_to_eq/
+| #I #L #V #_ <minus_n_O #H elim (le_plus_xSy_O_false … H)
+| /3 width=2 by le_plus_to_le_r/
+| /4 width=2 by le_plus_to_le_r, eq_f/
+]
+qed-.
+
+lemma ldrop_O1_le: ∀s,e,L. e ≤ |L| → ∃K. ⇩[s, 0, e] L ≡ K.
+#s #e @(nat_ind_plus … e) -e /2 width=2 by ex_intro/
+#e #IHe *
+[ #H elim (le_plus_xSy_O_false … H)
+| #L #I #V normalize #H elim (IHe L) -IHe /3 width=2 by ldrop_drop, monotonic_pred, ex_intro/
+]
+qed-.
+
+lemma ldrop_O1_lt: ∀s,L,e. e < |L| → ∃∃I,K,V. ⇩[s, 0, e] L ≡ K.ⓑ{I}V.
+#s #L elim L -L
+[ #e #H elim (lt_zero_false … H)
+| #L #I #V #IHL #e @(nat_ind_plus … e) -e /2 width=4 by ldrop_pair, ex1_3_intro/
+ #e #_ normalize #H elim (IHL e) -IHL /3 width=4 by ldrop_drop, lt_plus_to_minus_r, lt_plus_to_lt_l, ex1_3_intro/
+]
+qed-.
+
+lemma ldrop_O1_pair: ∀L,K,e,s. ⇩[s, 0, e] L ≡ K → e ≤ |L| → ∀I,V.
+ ∃∃J,W. ⇩[s, 0, e] L.ⓑ{I}V ≡ K.ⓑ{J}W.
+#L elim L -L [| #L #Z #X #IHL ] #K #e #s #H normalize #He #I #V
+[ elim (ldrop_inv_atom1 … H) -H #H <(le_n_O_to_eq … He) -e
+ #Hs destruct /2 width=3 by ex1_2_intro/
+| elim (ldrop_inv_O1_pair1 … H) -H * #He #HLK destruct /2 width=3 by ex1_2_intro/
+ elim (IHL … HLK … Z X) -IHL -HLK
+ /3 width=3 by ldrop_drop_lt, le_plus_to_minus, ex1_2_intro/
+]
+qed-.
+
+lemma ldrop_inv_O1_gt: ∀L,K,e,s. ⇩[s, 0, e] L ≡ K → |L| < e →
+ s = Ⓣ ∧ K = ⋆.
+#L elim L -L [| #L #Z #X #IHL ] #K #e #s #H normalize in ⊢ (?%?→?); #H1e
+[ elim (ldrop_inv_atom1 … H) -H elim s -s /2 width=1 by conj/
+ #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (lt_zero_false … H1e)
+| elim (ldrop_inv_O1_pair1 … H) -H * #H2e #HLK destruct
+ [ elim (lt_zero_false … H1e)
+ | elim (IHL … HLK) -IHL -HLK /2 width=1 by lt_plus_to_minus_r, conj/
+ ]
+]
+qed-.
+
+lemma ldrop_O1_ge: ∀L,e. |L| ≤ e → ⇩[Ⓣ, 0, e] L ≡ ⋆.
+#L elim L -L [ #e #_ @ldrop_atom #H destruct ]
+#L #I #V #IHL #e @(nat_ind_plus … e) -e [ #H elim (le_plus_xSy_O_false … H) ]
+normalize /4 width=1 by ldrop_drop, monotonic_pred/
+qed.
+
+lemma ldrop_split: ∀L1,L2,d,e2,s. ⇩[s, d, e2] L1 ≡ L2 → ∀e1. e1 ≤ e2 →
+ ∃∃L. ⇩[s, d, e2 - e1] L1 ≡ L & ⇩[s, d, e1] L ≡ L2.
+#L1 #L2 #d #e2 #s #H elim H -L1 -L2 -d -e2
+[ #d #e2 #Hs #e1 #He12 @(ex2_intro … (⋆))
+ @ldrop_atom #H lapply (Hs H) -s #H destruct /2 width=1 by le_n_O_to_eq/
+| #I #L1 #V #e1 #He1 lapply (le_n_O_to_eq … He1) -He1
+ #H destruct /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #e2 #HL12 #IHL12 #e1 @(nat_ind_plus … e1) -e1
+ [ /3 width=3 by ldrop_drop, ex2_intro/
+ | -HL12 #e1 #_ #He12 lapply (le_plus_to_le_r … He12) -He12
+ #He12 elim (IHL12 … He12) -IHL12 >minus_plus_plus_l
+ #L #HL1 #HL2 elim (lt_or_ge (|L1|) (e2-e1)) #H0
+ [ elim (ldrop_inv_O1_gt … HL1 H0) -HL1 #H1 #H2 destruct
+ elim (ldrop_inv_atom1 … HL2) -HL2 #H #_ destruct
+ @(ex2_intro … (⋆)) [ @ldrop_O1_ge normalize // ]
+ @ldrop_atom #H destruct
+ | elim (ldrop_O1_pair … HL1 H0 I V) -HL1 -H0 /3 width=5 by ldrop_drop, ex2_intro/
+ ]
+ ]
+| #I #L1 #L2 #V1 #V2 #d #e2 #_ #HV21 #IHL12 #e1 #He12 elim (IHL12 … He12) -IHL12
+ #L #HL1 #HL2 elim (lift_split … HV21 d e1) -HV21 /3 width=5 by ldrop_skip, ex2_intro/
+]
+qed-.
+
(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
(* alternative definition of llpx_sn (not recursive) *)
elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) /2 width=2 by/
qed-.
+lemma cpy_inv_nlift2_ge: ∀G,L,U1,U2,d,e. ⦃G, L⦄ ⊢ U1 ▶[d, e] U2 →
+ ∀i. d ≤ yinj i → (* yinj i + yinj 1 ≤ d + e → *)
+ (∀T2. ⇧[i, 1] T2 ≡ U2 → ⊥) →
+ ∃∃j. d ≤ yinj j & j ≤ i & (∀T1. ⇧[j, 1] T1 ≡ U1 → ⊥).
+#G #L #U1 #U2 #d #e #H elim H -G -L -U1 -U2 -d -e
+[ #I #G #L #d #e #i #Hdi (* #Hide *) #H @(ex3_intro … i) /2 width=2 by/
+| #I #G #L #K #V #W #j #d #e #Hdj #Hjde #HLK #HVW #i #Hdi (* #Hide *) #HnW
+ elim (le_or_ge i j) #Hij [2: @(ex3_intro … j) /2 width=7 by lift_inv_lref2_be/ ]
+ elim (lift_split … HVW i j) -HVW //
+ #X #_ #H elim HnW -HnW //
+| #a #I #G #L #W1 #W2 #U1 #U2 #d #e #_ #_ #IHW12 #IHU12 #i #Hdi #H elim (nlift_inv_bind … H) -H
+ [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
+ #j #Hdj #Hji #HnW1 @(ex3_intro … j) /3 width=9 by nlift_bind_sn/
+ | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/
+ #j #Hdj #Hji
+ >(plus_minus_m_m j 1) in ⊢ (%→?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ]
+ #HnW1 @(ex3_intro … (j-1)) /3 width=9 by nlift_bind_dx, yle_pred, monotonic_pred/
+ ]
+| #I #G #L #W1 #W2 #U1 #U2 #d #e #_ #_ #IHW12 #IHU12 #i #Hdi #H elim (nlift_inv_flat … H) -H
+ [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
+ #j #Hdj #Hji #HnW1 @(ex3_intro … j) /3 width=8 by nlift_flat_sn/
+ | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 //
+ #j #Hdj #Hji #HnW1 @(ex3_intro … j) /3 width=8 by nlift_flat_dx/
+ ]
+]
+qed-.
+
+axiom frees_fwd_nlift_ge: ∀L,U,d,i. (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → d ≤ yinj i →
+ ∃∃j. d ≤ yinj j & j ≤ i & (∀T. ⇧[j, 1] T ≡ U → ⊥).
+
+(*
+lemma frees_ind_nlift: ∀L,d. ∀R:relation2 term nat.
+ (∀U1,i. d ≤ yinj i → (∀T1. ⇧[i, 1] T1 ≡ U1 → ⊥) → R U1 i) →
+ (∀U1,U2,i,j. d ≤ yinj j → j ≤ i → ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 → (∀T2. ⇧[i, 1] T2 ≡ U2 → ⊥) → R U2 i → R U1 j) →
+ ∀U,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R U i.
+#L #d #R #IH1 #IH2 #U1 #i #Hdi #H @(frees_ind … H) -U1 /3 width=4 by/
+#U1 #U2 #HU12 #HnU2 #HU2 @(IH2 … HU12 … HU2)
+
+qed-.
+*)(*
+lemma frees_fwd_nlift: ∀L,d. ∀R:relation2 term nat. (
+ ∀U1,j. (∀T1. ⇧[j, 1] T1 ≡ U1 → ⊥) ∨
+ (∃∃U2,i. d ≤ yinj j → j < i & (L ⊢ j ~ϵ 𝐅*[d]⦃U1⦄ → ⊥) & ⦃⋆, L⦄ ⊢ U1 ▶[d, ∞] U2 & R U1 i
+ ) →
+ d ≤ yinj j → R U1 j
+ ) →
+ ∀U,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) → R U i.
+#L #d #R #IHR #U1 #j #Hdj #H elim (frees_inv_gen … H) -H
+#U2 #H generalize in match Hdj; -Hdj generalize in match j; -j @(cpys_ind … H) -U2
+[ #j #Hdj #HnU1 @IHR -IHR /3 width=2 by or_introl/
+| #U0 #U2 #HU10 #HU02 #IHU10 #j #Hdj #HnU2 elim (cpy_inv_nlift2_ge … HU02 … Hdj HnU2) -HU02 -HnU2
+ #i #Hdi #Hij #HnU0 lapply (IHU10 … HnU0) // -IHU10
+ #Hi @IHR -IHR // -Hdj @or_intror
+
+lemma frees_fwd_nlift: ∀L,U,d,i. d ≤ yinj i → (L ⊢ i ~ϵ 𝐅*[d]⦃U⦄ → ⊥) →
+ ∃∃j. d ≤ yinj j & j ≤ i & (∀T. ⇧[j, 1] T ≡ U → ⊥).
+#L #U1 #d #i #Hdi #H
+#U2 #H #HnU2 @(cpys_ind_dx … H) -U1 [ @(ex3_intro … i) /2 width=2 by/ ] -Hdi -HnU2
+#U1 #U0 #HU10 #_ * #j #Hdj #Hji #HnU0 elim (cpy_inv_nlift2_ge … HU10 … Hdj HnU0) -U0 -Hdj
+/3 width=5 by transitive_le, ex3_intro/
+qed-.
+*)
+
+theorem llpx_sn_llpx_sn_alt2: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt2 R d T L1 L2.
+#R #L1 #L2 #U1 #d #H elim (llpx_sn_inv_alt1 … H) -H
+#HL12 #IH @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+elim (frees_fwd_nlift_ge … H Hdi) -H -Hdi #j #Hdj #Hji #HnU1
+lapply (ldrop_fwd_length_lt2 … HLK1) #HL1
+lapply (ldrop_fwd_length_lt2 … HLK2) #HL2
+lapply (le_to_lt_to_lt … Hji HL1) -HL1 #HL1
+lapply (le_to_lt_to_lt … Hji HL2) -HL2 #HL2
+elim (ldrop_O1_lt L1 j) // #Z1 #Y1 #X1 #HLY1
+elim (ldrop_O1_lt L2 j) // #Z2 #Y2 #X2 #HLY2
+
+
+
+
+generalize in match V2; -V2 generalize in match V1; -V1
+generalize in match K2; -K2 generalize in match K1; -K1
+generalize in match I2; -I2 generalize in match I1; -I1
+generalize in match IH; -IH
+@(frees_ind_nlift … Hdi H) -U1 -i
+[ #U1 #i #Hdi #HnU1 #IH #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2 elim (IH … HnU1 HLK1 HLK2) -IH -HnU1 -HLK1 -HLK2 /2 width=1 by conj/
+| #U1 #U2 #i #j #Hdj #Hji #HU12 #HnU2 #IHU12 #IH #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
+(*
+*)
+ @(IHU12) … HLK1 HLK2)
+
+ @(IHU12 … HLK1 HLK2) -IHU02 -I1 -I2 -K1 -K2 -V1 -V2
+ #I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #HnU0 #HLK1 #HLK2 @(IH … HLK1 HLK2) -IH -HLK1 -HLK2 //
+
+
+
+ elim (frees_fwd_nlift … HnU1) // -HnU1 -Hdi
+#j #Hdj #Hji #HnU1
+lapply (ldrop_fwd_length_lt2 … HLK1) #HL1
+lapply (ldrop_fwd_length_lt2 … HLK2) #HL2
+lapply (le_to_lt_to_lt … Hji HL1) -HL1 #HL1
+lapply (le_to_lt_to_lt … Hji HL2) -HL2 #HL2
+elim (ldrop_O1_lt L1 j) // #Z1 #Y1 #X1 #HLY1
+elim (ldrop_O1_lt L2 j) // #Z2 #Y2 #X2 #HLY2
+elim (IH … HnU1 HLY1 HLY2) // #H #HX12 #HY12 destruct
+
+
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+
+theorem llpx_sn_llpx_sn_alt2: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt2 R d T L1 L2.
+#R #L1 #L2 #U1 #d #H @(llpx_sn_ind_alt1 … H) -L1 -L2 -U1 -d
+#L1 #L2 #U1 #d #HL12 #IH @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU1 #HLK1 #HLK2 elim (frees_inv_gen … HnU1) -HnU1
+#U2 #H generalize in match IH; -IH @(cpys_ind_dx … H) -U1
+[ #IH #HnU2 elim (IH … HnU2 … HLK1 HLK2) -L1 -L2 -U2 /2 width=1 by conj/
+| #U1 #U0 #HU10 #_ #IHU02 #IH #HnU2 @IHU02 /2 width=2 by/ -I1 -I2 -K1 -K2 -V1 -V2 -U2 -i
+ #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU0 #HLK1 #HLK2 @(IH … HLK1 HLK2) -IH // -R -I2 -L2 -K2 -V2
+ @(cpy_inv_nlift2_be … HU10) /2 width=3 by/
+
+theorem llpx_sn_llpx_sn_alt2: ∀R,L1,L2,T2,d. llpx_sn R d T2 L1 L2 →
+ ∀T1. ⦃⋆, L1⦄ ⊢ T1 ▶*[d, ∞] T2 → llpx_sn_alt2 R d T1 L1 L2.
+#R #L1 #L2 #U2 #d #H elim (llpx_sn_inv_alt1 … H) -H
+#HL12 #IH #U1 #HU12 @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU1 #HLK1 #HLK2 elim (frees_inv_gen … HnU1) -HnU1
+#U2 #H generalize in match IH; -IH @(cpys_ind_dx … H) -U1
+[ #IH #HnU2 elim (IH … HnU2 … HLK1 HLK2) -L1 -L2 -U2 /2 width=1 by conj/
+| #U1 #U0 #HU10 #_ #IHU02 #IH #HnU2 @IHU02 /2 width=2 by/ -I1 -I2 -K1 -K2 -V1 -V2 -U2 -i
+ #I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU0 #HLK1 #HLK2 @(IH … HLK1 HLK2) -IH // -R -I2 -L2 -K2 -V2
+ @(cpy_inv_nlift2_be … HU10) /2 width=3 by/
+
+
theorem llpx_sn_llpx_sn_alt2: ∀R,L1,L2,T,d. llpx_sn R d T L1 L2 → llpx_sn_alt2 R d T L1 L2.
#R #L1 #L2 #U1 #d #H elim (llpx_sn_inv_alt1 … H) -H
#HL12 #IH @conj // -HL12
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