(* *)
(**************************************************************************)
-include "basic_2/grammar/lpx_sn_lpx_sn.ma".
-include "basic_2/substitution/fqup.ma".
-include "basic_2/substitution/lpys_ldrop.ma".
+include "basic_2/substitution/cpy_cpy.ma".
+include "basic_2/multiple/cpys_alt.ma".
(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-(* Main properties **********************************************************)
+(* Advanced inversion lemmas ************************************************)
-theorem cpys_antisym: ∀G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ▶*× T2 → ⦃G, L⦄ ⊢ T2 ▶*× T1 → T1 = T2.
-#G #L #T1 #T2 #H elim H -G -L -T1 -T2 //
-[ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #_ #HW2 lapply (ldrop_fwd_ldrop2 … HLK) -I -V1
- #HLK elim (cpys_inv_lift1 … HW2 … HLK … HVW2) -L -HVW2
- #X #H #_ elim (lift_inv_lref2_be … H) -G -K -V2 -W2 -X //
-| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #H elim (cpys_inv_bind1 … H) -H
- #V #T #HV2 #HT2 #H destruct
- lapply (IHV12 HV2) #H destruct -IHV12 -HV2 /3 width=1 by eq_f2/
-| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #H elim (cpys_inv_flat1 … H) -H
- #V #T #HV2 #HT2 #H destruct /3 width=1 by eq_f2/
-]
+lemma cpys_inv_SO2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶*[l, 1] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2.
+#G #L #T1 #T2 #l #H @(cpys_ind … H) -T2 /2 width=3 by cpy_trans_ge/
qed-.
-theorem cpys_trans_lpys: ∀G. lpx_sn_transitive (cpys G) (cpys G).
-#G0 #L0 #T0 @(fqup_wf_ind_eq … G0 L0 T0) -G0 -L0 -T0 #G0 #L0 #T0 #IH #G1 #L1 * [|*]
-[ #I #HG #HL #HT #T #H1 #L2 #HL12 #T2 #HT2 destruct
- elim (cpys_inv_atom1 … H1) -H1
- [ #H destruct
- elim (cpys_inv_atom1 … HT2) -HT2
- [ #H destruct //
- | * #I2 #K2 #V #V2 #i #HLK2 #HV2 #HVT2 #H destruct
- elim (lpys_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (lpys_inv_pair2 … H) -H #K1 #V1 #HK12 #HV1 #H destruct
- lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
- ]
- | * #I1 #K1 #V1 #V #i #HLK1 #HV1 #HVT #H destruct
- elim (lpys_ldrop_conf … HLK1 … HL12) -HL12 #X #H #HLK2
- elim (lpys_inv_pair1 … H) -H #K2 #W2 #HK12 #_ #H destruct
- lapply (ldrop_fwd_ldrop2 … HLK2) -W2 #HLK2
- elim (cpys_inv_lift1 … HT2 … HLK2 … HVT) -L2 -T
- lapply (fqup_lref … G1 … HLK1) /3 width=10 by cpys_delta/
- ]
-| #a #I #V1 #T1 #HG #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpys_inv_bind1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
- elim (cpys_inv_bind1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- /4 width=5 by cpys_bind, lpys_pair/
-| #I #V1 #T1 #HG #HL #HT #X1 #H1 #L2 #HL12 #X2 #H2
- elim (cpys_inv_flat1 … H1) -H1 #V #T #HV1 #HT1 #H destruct
- elim (cpys_inv_flat1 … H2) -H2 #V2 #T2 #HV2 #HT2 #H destruct
- /3 width=5 by cpys_flat/
+(* Advanced properties ******************************************************)
+
+lemma cpys_strip_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-.
+
+lemma cpys_strip_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 →
+ ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶[l2, m2] T2 →
+ (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
+ ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-.
+
+lemma cpys_strap1_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶[l2, m2] T2 → l2 + m2 ≤ l1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed.
+
+lemma cpys_strap2_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T0 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-.
+
+lemma cpys_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2 →
+ ∀i. l ≤ i → i ≤ l + m →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l, i - l] T & ⦃G, L⦄ ⊢ T ▶*[i, l + m - i] T2.
+#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(cpys_ind … H) -T2
+[ /2 width=3 by ex2_intro/
+| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
+ elim (cpy_split_up … HT12 … Hilm) -HT12 -Hilm #T0 #HT0 #HT02
+ elim (cpys_strap1_down … HT3 … HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
+ >ymax_pre_sn_comm //
]
qed-.
-theorem cpys_trans: ∀G,L. Transitive … (cpys G L).
-/2 width=5 by cpys_trans_lpys/ qed-.
+lemma cpys_inv_lift1_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶*[lt, mt] U2 →
+ ∀K,s,l,m. ⬇[s, l, m] L ≡ K → ∀T1. ⬆[l, m] T1 ≡ U1 →
+ l ≤ lt → lt ≤ l + m → l + m ≤ lt + mt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶*[l, lt + mt - (l + m)] T2 &
+ ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt
+elim (cpys_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
+lapply (cpys_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
+lapply (cpys_inv_lift1_eq … HU1 … HTU1) -HU1 #HU1 destruct
+elim (cpys_inv_lift1_ge … HU2 … HLK … HTU1) -HU2 -HLK -HTU1 //
+>yplus_minus_inj /2 width=3 by ex2_intro/
+qed-.
-(* Advanced properties ******************************************************)
+(* Main properties **********************************************************)
+
+theorem cpys_conf_eq: ∀G,L,T0,T1,l1,m1. ⦃G, L⦄ ⊢ T0 ▶*[l1, m1] T1 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-.
+
+theorem cpys_conf_neq: ∀G,L1,T0,T1,l1,m1. ⦃G, L1⦄ ⊢ T0 ▶*[l1, m1] T1 →
+ ∀L2,T2,l2,m2. ⦃G, L2⦄ ⊢ T0 ▶*[l2, m2] T2 →
+ (l1 + m1 ≤ l2 ∨ l2 + m2 ≤ l1) →
+ ∃∃T. ⦃G, L2⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L1⦄ ⊢ T2 ▶*[l1, m1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-.
+
+theorem cpys_trans_eq: ∀G,L,T1,T,T2,l,m.
+ ⦃G, L⦄ ⊢ T1 ▶*[l, m] T → ⦃G, L⦄ ⊢ T ▶*[l, m] T2 →
+ ⦃G, L⦄ ⊢ T1 ▶*[l, m] T2.
+normalize /2 width=3 by trans_TC/ qed-.
-lemma lpys_cpys_trans: ∀G. lsub_trans … (cpys G) (lpys G).
-/2 width=5 by cpys_trans_lpys/ qed-.
+theorem cpys_trans_down: ∀G,L,T1,T0,l1,m1. ⦃G, L⦄ ⊢ T1 ▶*[l1, m1] T0 →
+ ∀T2,l2,m2. ⦃G, L⦄ ⊢ T0 ▶*[l2, m2] T2 → l2 + m2 ≤ l1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶*[l2, m2] T & ⦃G, L⦄ ⊢ T ▶*[l1, m1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-.
+
+theorem cpys_antisym_eq: ∀G,L1,T1,T2,l,m. ⦃G, L1⦄ ⊢ T1 ▶*[l, m] T2 →
+ ∀L2. ⦃G, L2⦄ ⊢ T2 ▶*[l, m] T1 → T1 = T2.
+#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt … H) -G -L1 -T1 -T2 //
+[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #_ #_ #HVW2 #_ #L2 #HW2
+ elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hli -Hilm | ]
+ [ lapply (cpys_weak_full … HW2) -HW2 #HW2
+ lapply (cpys_weak … HW2 0 (i+1) ? ?) -HW2 //
+ [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi
+ #HW2 >(cpys_inv_lift1_eq … HW2) -HW2 //
+ | elim (drop_O1_le (Ⓕ) … Hi) -Hi #K2 #HLK2
+ elim (cpys_inv_lift1_ge_up … HW2 … HLK2 … HVW2 ? ? ?) -HW2 -HLK2 -HVW2
+ /2 width=1 by ylt_fwd_le_succ1, yle_succ_dx/ -Hli -Hilm
+ #X #_ #H elim (lift_inv_lref2_be … H) -H /2 width=1 by ylt_inj/
+ ]
+| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 … H) -H
+ #V #T #HV2 #HT2 #H destruct
+ lapply (IHV12 … HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/
+| #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 … H) -H
+ #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/
+]
+qed-.
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