--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/psubst_6.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/substitution/lsuby.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* activate genv *)
+inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
+| cpy_atom : ∀I,G,L,l,m. cpy l m G L (⓪{I}) (⓪{I})
+| cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ i → i < l+m →
+ ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, ⫯i] V ≡ W → cpy l m G L (#i) W
+| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
+ cpy l m G L V1 V2 → cpy (⫯l) m G (L.ⓑ{I}V1) T1 T2 →
+ cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
+| cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
+ cpy l m G L V1 V2 → cpy l m G L T1 T2 →
+ cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation "context-sensitive extended ordinary substritution (term)"
+ 'PSubst G L T1 l m T2 = (cpy l m G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy l m G) (lsuby l m).
+#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
+[ //
+| #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12
+ elim (ylt_inv_plus_dx … Hilm) #m0 #H0 #_
+ elim (lsuby_drop_trans_be … HL12 … HLK1 … H0) -HL12 -HLK1 -H0 /2 width=5 by cpy_subst/
+| /4 width=1 by lsuby_succ, cpy_bind/
+| /3 width=1 by cpy_flat/
+]
+qed-.
+
+lemma cpy_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] T.
+#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
+qed.
+
+(* Basic_1: was: subst1_ex *)
+lemma cpy_full: ∀I,G,K,V,T1,L,l. ⬇[l] L ≡ K.ⓑ{I}V →
+ ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2.
+#I #G #K #V #T1 elim T1 -T1
+[ * #i #L #l #HLK
+ /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
+ elim (ylt_split_eq i l) /3 width=4 by lift_lref_pred, lift_lref_lt, ex2_2_intro/
+ #H destruct lapply (drop_fwd_Y2 … HLK) #Hi
+ elim (lift_total (⫯i) … V 0) /2 width=1 by ylt_succ/
+ #W #HVW elim (lift_split … HVW i i … 1)
+ /4 width=6 by cpy_subst, monotonic_ylt_plus_sn, ex2_2_intro/
+| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK
+ elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L.ⓑ{J}W1) (⫯l)) -IHU1
+ /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/
+ | elim (IHU1 … HLK) -IHU1 -HLK
+ /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
+ ]
+]
+qed-.
+
+lemma cpy_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T2 →
+ ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
+ ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T2.
+#G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 //
+[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
+| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
+| /3 width=1 by cpy_flat/
+]
+qed-.
+(*
+lemma cpy_weak_top: ∀G,L,T1,T2,l,m.
+ ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, ∞] T2.
+/2 width=5 by cpy_weak/ qed-.
+
+lemma cpy_weak_full: ∀G,L,T1,T2,l,m.
+ ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, ∞] T2.
+/2 width=5 by cpy_weak/ qed-.
+*)
+lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 →
+ ∀i,m2. i + m2 = l + m →
+ ∀m1. i ≤ l + m1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l, m1] T & ⦃G, L⦄ ⊢ T ▶[i, m2] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #m2 #H2 #m1 #H1
+ elim (ylt_split i j) [ -Hilm -H2 | -Hli ]
+ /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #m2 #H2 #m1 #H1
+ elim (IHV12 … H2 … H1) -IHV12 #V
+ elim (IHT12 (⫯i) … m2 … m1) -IHT12 /2 width=1 by yle_succ/ -H2 -H1
+ #T #HT1 #HT2 lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
+ /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #m2 #H2 #m1 #H1
+ elim (IHV12 … H2 … H1) -IHV12 elim (IHT12 … H2 … H1) -IHT12 -H2 -H1
+ /3 width=5 by ex2_intro, cpy_flat/
+]
+qed-.
+
+lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 →
+ ∀m1,m2. m = m1 + m2 →
+ ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[l+m2, m1] T & ⦃G, L⦄ ⊢ T ▶[l, m2] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #m1 #m2 #H destruct
+ elim (ylt_split i (l+m2)) [ -Hilm | -Hli ]
+ /3 width=9 by cpy_subst, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #m1 #m2 #H destruct
+ elim (IHV12 m1 m2) -IHV12 // #V
+ elim (IHT12 m1 m2) -IHT12 //
+ >yplus_succ1 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
+ /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #m1 #m2 #H destruct
+ elim (IHV12 m1 m2) -IHV12 // elim (IHT12 m1 m2) -IHT12 //
+ /3 width=5 by ex2_intro, cpy_flat/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpy_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
+ ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
+ l ≤ lt → l + m ≤ lt + mt →
+ ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #T1 #l #m #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt
+ elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ]
+ [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/
+ | elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi
+ elim (lift_split … HVW l (⫯(i-m)) ? ? ?) [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hlim
+ #T2 #_ >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by le_S, yle_inv_inj/ <minus_n_n <plus_n_O #H -Hmi
+ @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HVW1) -V1 -IHW12 //
+ elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
+ <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
+ /3 width=2 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
+ /3 width=2 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+lemma cpy_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize
+/3 width=1 by monotonic_le_plus_l, le_plus/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} →
+ T2 = ⓪{J} ∨
+ ∃∃I,K,V,i. l ≤ yinj i & i < l + m &
+ ⬇[i] L ≡ K.ⓑ{I}V &
+ ⬆[O, ⫯i] V ≡ T2 &
+ J = LRef i.
+#G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m
+[ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/
+| #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
+]
+qed-.
+
+lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 →
+ T2 = ⓪{I} ∨
+ ∃∃J,K,V,i. l ≤ yinj i & i < l + m &
+ ⬇[i] L ≡ K.ⓑ{J}V &
+ ⬆[O, ⫯i] V ≡ T2 &
+ I = LRef i.
+/2 width=4 by cpy_inv_atom1_aux/ qed-.
+
+(* Basic_1: was: subst1_gen_sort *)
+lemma cpy_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k.
+#G #L #T2 #k #l #m #H
+elim (cpy_inv_atom1 … H) -H //
+* #I #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: subst1_gen_lref *)
+lemma cpy_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 →
+ T2 = #i ∨
+ ∃∃I,K,V. l ≤ i & i < l + m &
+ ⬇[i] L ≡ K.ⓑ{I}V &
+ ⬆[O, ⫯i] V ≡ T2.
+#G #L #T2 #i #l #m #H
+elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/
+* #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
+qed-.
+
+lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p.
+#G #L #T2 #p #l #m #H
+elim (cpy_inv_atom1 … H) -H //
+* #I #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
+ ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+ ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
+ U2 = ⓑ{a,I}V2.T2.
+#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
+[ #I #G #L #l #m #b #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct
+]
+qed-.
+
+lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[l, m] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+ ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
+ U2 = ⓑ{a,I}V2.T2.
+/2 width=3 by cpy_inv_bind1_aux/ qed-.
+
+fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
+ ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+ ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
+ U2 = ⓕ{I}V2.T2.
+#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
+[ #I #G #L #l #m #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[l, m] U2 →
+ ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
+ ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
+ U2 = ⓕ{I}V2.T2.
+/2 width=3 by cpy_inv_flat1_aux/ qed-.
+
+
+fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
+[ //
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct
+ elim (ylt_yle_false … Hli) -Hli //
+| /3 width=1 by eq_f2/
+| /3 width=1 by eq_f2/
+]
+qed-.
+
+lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 0] T2 → T1 = T2.
+/2 width=6 by cpy_inv_refl_O2_aux/ qed-.
+
+(* Basic_1: was: subst1_gen_lift_eq *)
+lemma cpy_inv_lift1_eq: ∀G,T1,U1,l,m. ⬆[l, m] T1 ≡ U1 →
+ ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2.
+#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
+/2 width=4 by cpy_inv_refl_O2/
+qed-.
+
+(* Basic_1: removed theorems 25:
+ subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
+ subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
+ subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
+ subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
+ subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
+ subst0_confluence_lift subst0_tlt
+ subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpy_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: subst1_confluence_eq *)
+theorem cpy_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.
+#G #L #T0 #T1 #l1 #m1 #H elim H -G -L -T0 -T1 -l1 -m1
+[ /2 width=3 by ex2_intro/
+| #I1 #G #L #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #T2 #l2 #m2 #H
+ elim (cpy_inv_lref1 … H) -H
+ [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
+ | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
+ lapply (drop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
+ >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
+ ]
+| #a #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
+ elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2
+ /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+| #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
+ elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02) -V0
+ elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: subst1_confluence_neq *)
+theorem cpy_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.
+#G #L1 #T0 #T1 #l1 #m1 #H elim H -G -L1 -T0 -T1 -l1 -m1
+[ /2 width=3 by ex2_intro/
+| #I1 #G #L1 #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #L2 #T2 #l2 #m2 #H1 #H2
+ elim (cpy_inv_lref1 … H1) -H1
+ [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
+ | -HLK1 -HVT1 * #I2 #K2 #V2 #Hl2 #Hlm2 #_ #_ elim H2 -H2 #Hlml [ -Hl1 -Hlm2 | -Hl2 -Hlm1 ]
+ [ elim (ylt_yle_false … Hlm1) -Hlm1 /2 width=3 by yle_trans/
+ | elim (ylt_yle_false … Hlm2) -Hlm2 /2 width=3 by yle_trans/
+ ]
+ ]
+| #a #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
+ elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2
+ elim (IHT01 … HT02) -T0
+ [ -H #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+ lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+ | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
+ ]
+| #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
+ elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 … HV02 H) -V0
+ elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
+]
+qed-.
+
+(* Note: the constant 1 comes from cpy_subst *)
+(* Basic_1: was: subst1_trans *)
+theorem cpy_trans_ge: ∀G,L,T1,T0,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T0 →
+ ∀T2. ⦃G, L⦄ ⊢ T0 ▶[l, 1] T2 → 1 ≤ m → ⦃G, L⦄ ⊢ T1 ▶[l, m] T2.
+#G #L #T1 #T0 #l #m #H elim H -G -L -T1 -T0 -l -m
+[ #I #G #L #l #m #T2 #H #Hm
+ elim (cpy_inv_atom1 … H) -H
+ [ #H destruct //
+ | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct
+ lapply (ylt_yle_trans … (l+m) … Hilm2)
+ /2 width=5 by cpy_subst, monotonic_yle_plus_sn/
+ ]
+| #I #G #L #K #V #V2 #i #l #m #Hli #Hilm #HLK #HVW #T2 #HVT2 #Hm
+ lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
+ >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
+| #a #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
+ elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+ lapply (IHT10 … HT02 Hm) -T0 /3 width=1 by cpy_bind/
+| #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
+ elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
+]
+qed-.
+
+theorem cpy_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.
+#G #L #T1 #T0 #l1 #m1 #H elim H -G -L -T1 -T0 -l1 -m1
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i1 #l1 #m1 #Hli1 #Hilm1 #HLK #HVW #T2 #l2 #m2 #HWT2 #Hlm2l1
+ lapply (yle_trans … Hlm2l1 … Hli1) -Hlm2l1 #Hlm2i1
+ lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1
+ >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
+| #a #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
+ elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+ elim (IHV10 … HV02) -IHV10 -HV02 // #V
+ elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
+ lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
+| #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
+ elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV10 … HV02) -V0 //
+ elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/drop_drop.ma".
+include "basic_2/substitution/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Properties on relocation *************************************************)
+
+(* Basic_1: was: subst1_lift_lt *)
+lemma cpy_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+ ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+ ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+ lt + mt ≤ l → ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hlmtl
+ lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
+ lapply (lift_inv_lref1_lt … H … Hil) -H #H destruct
+ elim (lift_trans_ge … HVW … HWU2) -W /2 width=1 by ylt_fwd_le_succ1/
+ <yplus_inj >yplus_SO2 >yminus_succ2 #W #HVW #HWU2
+ elim (drop_trans_le … HLK … HKV) -K /2 width=2 by ylt_fwd_le/ #X #HLK #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=7 by cpy_bind, drop_skip, yle_succ/
+| #G #I #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=7 by cpy_flat/
+]
+qed-.
+
+lemma cpy_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+ ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+ ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+ lt ≤ l → l ≤ lt + mt → ⦃G, L⦄ ⊢ U1 ▶[lt, mt + m] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hltl #_
+ elim (lift_inv_lref1 … H) -H * #Hil #H destruct
+ [ -Hltl
+ lapply (ylt_yle_trans … (lt+mt+m) … Hilmt) // -Hilmt #Hilmtm
+ elim (lift_trans_ge … HVW … HWU2) -W <yplus_inj >yplus_SO2
+ [2: >yplus_O1 /2 width=1 by ylt_fwd_le_succ1/ ] >yminus_succ2 #W #HVW #HWU2
+ elim (drop_trans_le … HLK … HKV) -K /2 width=1 by ylt_fwd_le/ #X #HLK #H
+ elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K #Y #_ #HVY
+ >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
+ | -Hlti
+ lapply (yle_trans … Hltl … Hil) -Hltl #Hlti
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ >plus_plus_comm_23 #HVU2
+ lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil
+ /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans/
+ ]
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hltl #Hllmt
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=7 by cpy_bind, drop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=7 by cpy_flat/
+]
+qed-.
+
+(* Basic_1: was: subst1_lift_ge *)
+lemma cpy_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
+ ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
+ ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
+ l ≤ lt → ⦃G, L⦄ ⊢ U1 ▶[lt+m, mt] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
+ >(lift_mono … H1 … H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hllt
+ lapply (yle_trans … Hllt … Hlti) -Hllt #Hil
+ lapply (lift_inv_lref1_ge … H … Hil) -H #H destruct
+ lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ >plus_plus_comm_23 #HVU2
+ lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil
+ /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
+ elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=6 by cpy_bind, drop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
+ elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=6 by cpy_flat/
+]
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+(* Basic_1: was: subst1_gen_lift_lt *)
+lemma cpy_inv_lift1_le: ∀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 →
+ lt + mt ≤ l →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmtl
+ lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
+ lapply (lift_inv_lref2_lt … H … Hil) -H #H destruct
+ elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m //
+ <yminus_succ2 <yplus_inj >yplus_SO2 >ymax_pre_sn /2 width=1 by ylt_fwd_le_succ1/ -Hil
+ /3 width=5 by cpy_subst, ex2_intro/
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 … HTU1) -IHU12 -HTU1
+ /3 width=6 by cpy_bind, yle_succ, drop_skip, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -W1 //
+ elim (IHU12 … HLK … HTU1) -U1 -HLK
+ /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+lemma cpy_inv_lift1_be: ∀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 →
+ lt ≤ l → l + m ≤ lt + mt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt-m] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #KV #V #W #i #x #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hltl #Hlmlmt
+ elim (yle_inv_inj2 … Hlti) -Hlti #lt #Hlti #H destruct
+ lapply (yle_fwd_plus_yge … Hltl Hlmlmt) #Hmmt
+ elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -Hltl -Hilmt | -Hlti -Hlmlmt ]
+ [ lapply (ylt_yle_trans i l (lt+(mt-m)) ? ?) //
+ [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm
+ elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le … HUV … HVW) -V //
+ <yminus_succ2 <yplus_inj >yplus_SO2 >ymax_pre_sn /2 width=1 by ylt_fwd_le_succ1/ -Hil
+ /4 width=5 by cpy_subst, ex2_intro, yle_inj/
+ | elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi
+ lapply (yle_inv_inj … Hmi) -Hmi #Hmi
+ lapply (yle_trans … Hltl (i-m) ?) // -Hltl #Hltim
+ lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hil -Hlim
+ #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
+ @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
+ >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 … HTU1) -U1
+ /3 width=6 by cpy_bind, drop_skip, lift_bind, yle_succ, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -W1 //
+ elim (IHU12 … HLK … HTU1) -U1 -HLK //
+ /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: subst1_gen_lift_ge *)
+lemma cpy_inv_lift1_ge: ∀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 + m ≤ lt →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt-m, mt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmlt
+ lapply (yle_trans … Hlmlt … Hlti) #Hlmi
+ elim (yle_inv_plus_inj2 … Hlmlt) -Hlmlt #_ #Hmlt
+ elim (yle_inv_plus_inj2 … Hlmi) #Hlim #Hmi
+ lapply (yle_inv_inj … Hmi) -Hmi #Hmi
+ lapply (lift_inv_lref2_ge … H ?) -H // #H destruct
+ lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV
+ elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /3 width=1 by yle_succ, yle_pred_sn, le_S_S/ ] -Hlmi -Hlim
+ #V0 #HV10 >plus_minus // <minus_minus /3 width=1 by le_S/ <minus_n_n <plus_n_O #H
+ @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
+ [ /2 width=1 by monotonic_yle_minus_dx/
+ | <yminus_inj <yplus_minus_comm_inj // /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+ elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (yle_inv_plus_inj2 … Hlmtl) #_ #Hmlt
+ elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 … HTU1) -U1 [4: @drop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
+ >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
+ elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 … HLK … HVW1) -W1 //
+ elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas on relocation ***********************************)
+
+lemma cpy_inv_lift1_ge_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 (cpy_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
+lapply (cpy_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
+lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
+elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
+qed-.
+
+lemma cpy_inv_lift1_be_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 →
+ lt ≤ l → lt + mt ≤ l + m →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l-lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm
+lapply (cpy_weak … HU12 lt (l+m-lt) ? ?) -HU12 //
+[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hlmtlm #HU12
+elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
+qed-.
+
+lemma cpy_inv_lift1_le_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 →
+ lt ≤ l → l ≤ lt + mt → lt + mt ≤ l + m →
+ ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm
+elim (cpy_split_up … HU12 l) -HU12 // #U #HU1 #HU2
+elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
+[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU
+lapply (cpy_weak … HU2 l m ? ?) -HU2 //
+[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2
+lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lift_neg.ma".
+include "basic_2/substitution/lift_lift.ma".
+include "basic_2/substitution/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Inversion lemmas on negated relocation ***********************************)
+
+lemma cpy_fwd_nlift2_ge: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
+ ∀i. l ≤ i → (∀T2. ⬆[i, 1] T2 ≡ U2 → ⊥) →
+ (∀T1. ⬆[i, 1] T1 ≡ U1 → ⊥) ∨
+ ∃∃I,K,W,j. l ≤ yinj j & j < i & ⬇[j]L ≡ K.ⓑ{I}W &
+ (∀V. ⬆[⫰(i-j), 1] V ≡ W → ⊥) & (∀T1. ⬆[j, 1] T1 ≡ U1 → ⊥).
+#G #L #U1 #U2 #l #m #H elim H -G -L -U1 -U2 -l -m
+[ /3 width=2 by or_introl/
+| #I #G #L #K #V #W #j #l #m #Hlj #Hjlm #HLK #HVW #i #Hli #HnW
+ elim (ylt_split j i) #Hij
+ [ @or_intror @(ex5_4_intro … HLK) // -HLK
+ [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V // #Y #HXY
+ <yminus_succ2 <yplus_inj >yplus_SO2 >ymax_pre_sn /2 width=2 by ylt_fwd_le_succ1/
+ | -HnW /3 width=7 by lift_inv_lref2_be, ylt_inj/
+ ]
+ | elim (lift_split … HVW i j) -HVW //
+ #X #_ #H elim HnW -HnW //
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_bind … H) -H
+ [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
+ [ /4 width=9 by nlift_bind_sn, or_introl/
+ | * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/
+ ]
+ | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/
+ [ /4 width=9 by nlift_bind_dx, or_introl/
+ | * #J #K #W #j #Hlj elim (yle_inv_succ1 … Hlj) -Hlj #Hlj #Hj
+ <Hj >yminus_succ #Hji #HLK #HnW
+ lapply (drop_inv_drop1_lt … HLK ?) /2 width=1 by ylt_O/ -HLK #HLK
+ <yminus_SO2 in Hlj; #Hlj #H4
+ @or_intror @(ex5_4_intro … HLK) (**) (* explicit constructors *)
+ /3 width=9 by nlift_bind_dx, ylt_pred, ylt_inj/
+ ]
+ ]
+| #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_flat … H) -H
+ [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
+ [ /4 width=9 by nlift_flat_sn, or_introl/
+ | * /5 width=9 by nlift_flat_sn, ex5_4_intro, or_intror/
+ ]
+ | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 //
+ [ /4 width=9 by nlift_flat_dx, or_introl/
+ | * /5 width=9 by nlift_flat_dx, ex5_4_intro, or_intror/
+ ]
+]
+qed-.
--- /dev/null
+lemma drop_refl_atom_O2: ∀s,l. ⬇[s, l, O] ⋆ ≡ ⋆.
+/2 width=1 by drop_atom/ qed.
+
+(* Basic_1: was by definition: drop_refl *)
+lemma drop_refl: ∀L,l,s. ⬇[s, l, 0] L ≡ L.
+#L elim L -L //
+#L #I #V #IHL #l #s @(nat_ind_plus … l) -l /2 width=1 by drop_pair, drop_skip/
+qed.
+
+lemma drop_split: ∀L1,L2,l,m2,s. ⬇[s, l, m2] L1 ≡ L2 → ∀m1. m1 ≤ m2 →
+ ∃∃L. ⬇[s, l, m2 - m1] L1 ≡ L & ⬇[s, l, m1] L ≡ L2.
+#L1 #L2 #l #m2 #s #H elim H -L1 -L2 -l -m2
+[ #l #m2 #Hs #m1 #Hm12 @(ex2_intro … (⋆))
+ @drop_atom #H lapply (Hs H) -s #H destruct /2 width=1 by le_n_O_to_eq/
+| #I #L1 #V #m1 #Hm1 lapply (le_n_O_to_eq … Hm1) -Hm1
+ #H destruct /2 width=3 by ex2_intro/
+| #I #L1 #L2 #V #m2 #HL12 #IHL12 #m1 @(nat_ind_plus … m1) -m1
+ [ /3 width=3 by drop_drop, ex2_intro/
+ | -HL12 #m1 #_ #Hm12 lapply (le_plus_to_le_r … Hm12) -Hm12
+ #Hm12 elim (IHL12 … Hm12) -IHL12 >minus_plus_plus_l
+ #L #HL1 #HL2 elim (lt_or_ge (|L1|) (m2-m1)) #H0
+ [ elim (drop_inv_O1_gt … HL1 H0) -HL1 #H1 #H2 destruct
+ elim (drop_inv_atom1 … HL2) -HL2 #H #_ destruct
+ @(ex2_intro … (⋆)) [ @drop_O1_ge normalize // ]
+ @drop_atom #H destruct
+ | elim (drop_O1_pair … HL1 H0 I V) -HL1 -H0 /3 width=5 by drop_drop, ex2_intro/
+ ]
+ ]
+| #I #L1 #L2 #V1 #V2 #l #m2 #_ #HV21 #IHL12 #m1 #Hm12 elim (IHL12 … Hm12) -IHL12
+ #L #HL1 #HL2 elim (lift_split … HV21 l m1) -HV21 /3 width=5 by drop_skip, ex2_intro/
+]
+qed-.
+
+(* Basic_2A1: includes: drop_split *)
+lemma drops_split_trans: ∀L1,L2,t. ⬇*[t] L1 ≡ L2 → ∀t1,t2. t1 ⊚ t2 ≡ t →
+ ∃∃L. ⬇*[t1] L1 ≡ L & ⬇*[t2] L ≡ L2.
+#L1 #L2 #t #H elim H -L1 -L2 -t
+[ #t1 #t2 #H elim (after_inv_empty3 … H) -H
+ /2 width=3 by ex2_intro, drops_atom/
+| #I #L1 #L2 #V #t #HL12 #IHL12 #t1 #t2 #H elim (after_inv_false3 … H) -H *
+ [ #tl1 #tl2 #H1 #H2 #Ht destruct elim (IHL12 … Ht) -t
+ #tl #H1 #H2
+ | #tl1 #H #Ht destruct elim (IHL12 … Ht) -t
+ /3 width=5 by ex2_intro, drops_drop/
+ ]
+| #I #L1 #L2 #V1 #V2 #t #_ #HV21 #IHL12 #t1 #t2 #H elim (after_inv_true3 … H) -H
+ #tl1 #tl2 #H1 #H2 #Ht destruct elim (lifts_split_trans … HV21 … Ht) -HV21
+ elim (IHL12 … Ht) -t /3 width=5 by ex2_intro, drops_skip/
+]
+qed-.
+
+lemma drop_inv_refl: ∀L,l,m. ⬇[Ⓕ, l, m] L ≡ L → m = 0.
+/2 width=5 by drop_inv_length_eq/ qed-.
+
+fact drop_inv_FT_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 →
+ ∀I,K,V. L2 = K.ⓑ{I}V → s = Ⓣ → l = 0 →
+ ⬇[Ⓕ, l, m] L1 ≡ K.ⓑ{I}V.
+#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m
+[ #l #m #_ #J #K #W #H destruct
+| #I #L #V #J #K #W #H destruct //
+| #I #L1 #L2 #V #m #_ #IHL12 #J #K #W #H1 #H2 destruct
+ /3 width=1 by drop_drop/
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #_ #J #K #W #_ #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma drop_inv_FT: ∀I,L,K,V,m. ⬇[Ⓣ, 0, m] L ≡ K.ⓑ{I}V → ⬇[m] L ≡
+K.ⓑ{I}V.
+/2 width=5 by drop_inv_FT_aux/ qed.
+
+lemma drop_inv_gen: ∀I,L,K,V,s,m. ⬇[s, 0, m] L ≡ K.ⓑ{I}V → ⬇[m] L ≡
+K.ⓑ{I}V.
+#I #L #K #V * /2 width=1 by drop_inv_FT/
+qed-.
+
+lemma drop_inv_T: ∀I,L,K,V,s,m. ⬇[Ⓣ, 0, m] L ≡ K.ⓑ{I}V → ⬇[s, 0, m] L
+≡ K.ⓑ{I}V.
+#I #L #K #V * /2 width=1 by drop_inv_FT/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lenv_append.ma".
+include "basic_2/substitution/drop.ma".
+
+(* DROPPING *****************************************************************)
+
+(* Properties on append for local environments ******************************)
+
+fact drop_O1_append_sn_le_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 →
+ l = 0 → m ≤ |L1| →
+ ∀L. ⬇[s, 0, m] L @@ L1 ≡ L @@ L2.
+#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m //
+[ #l #m #_ #_ #H >(yle_inv_O2 … H) -m //
+| /4 width=1 by drop_drop, yle_inv_succ/
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #_ #H elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma drop_O1_append_sn_le: ∀L1,L2,s,m. ⬇[s, yinj 0, m] L1 ≡ L2 → m ≤ |L1| →
+ ∀L. ⬇[s, 0, m] L @@ L1 ≡ L @@ L2.
+/2 width=3 by drop_O1_append_sn_le_aux/ qed.
+
+(* Inversion lemmas on append for local environments ************************)
+
+lemma drop_O1_inv_append1_ge: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 @@ L2 ≡ K →
+ ∀m0. |L2| + m0 = m → ⬇[s, 0, m0] L1 ≡ K.
+#K #L1 #L2 elim L2 -L2
+[ #s #m #H #m0 >yplus_O1 #H0 destruct //
+| #L2 #I #V #IHL2 #s #m #H #m0 >yplus_succ1
+ #H0 elim (drop_inv_O1_pair1 … H) -H * #Hm #HL12 destruct
+ [ elim (ysucc_inv_O_dx … Hm)
+ | /2 width=3 by/
+ ]
+]
+qed-.
+
+lemma drop_O1_inv_append1_le: ∀K,L1,L2,s,m. ⬇[s, 0, m] L1 @@ L2 ≡ K → m ≤ |L2| →
+ ∀K2. ⬇[s, 0, m] L2 ≡ K2 → K = L1 @@ K2.
+#K #L1 #L2 elim L2 -L2
+[ #s #m #H1 #H2 #K2 #H3 lapply (yle_inv_O2 … H2) -H2
+ #H2 elim (drop_inv_atom1 … H3) -H3 #H3 #_ destruct
+ >(drop_inv_O2 … H1) -H1 //
+| #L2 #I #V #IHL2 #s #m @(ynat_ind … m) -m [ -IHL2 || -IHL2 ]
+ [ #H1 #_ #K2 #H2
+ lapply (drop_inv_O2 … H1) -H1 #H1
+ lapply (drop_inv_O2 … H2) -H2 #H2 destruct //
+ | /3 width=7 by drop_inv_drop1, yle_inv_succ/
+ | #_ #H lapply (yle_inv_Y1 … H) -H
+ #H elim (ylt_yle_false (|L2.ⓑ{I}V|) (∞)) //
+ ]
+]
+qed-.
--- /dev/null
+include "basic_2/grammar/lenv_length.ma".
+
+lemma drop_inv_O1_gt: ∀L,K,m,s. ⬇[s, 0, m] L ≡ K → |L| < m →
+ s = Ⓣ ∧ K = ⋆.
+#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H normalize in ⊢ (?%?→?); #H1m
+[ elim (drop_inv_atom1 … H) -H elim s -s /2 width=1 by conj/
+ #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (ylt_yle_false … H1m) -H1m //
+| elim (drop_inv_O1_pair1 … H) -H * #H2m #HLK destruct
+ [ elim (ylt_yle_false … H1m) -H1m //
+ | elim (IHL … HLK) -IHL -HLK /2 width=1 by ylt_pred, conj/
+ ]
+]
+qed-.
+
+lemma drop_O1_le: ∀s,m,L. m ≤ |L| → ∃K. ⬇[s, 0, m] L ≡ K.
+#s #m @(ynat_ind … m) -m /2 width=2 by ex_intro/
+[ #m #IHm *
+ [ #H elim (ylt_yle_false … H) -H //
+ | #L #I #V #H elim (IHm L) -IHm /3 width=2 by drop_drop, yle_inv_succ, ex_intro/
+ ]
+| #L #H elim (ylt_yle_false … H) -H //
+]
+qed-.
+
+lemma drop_O1_lt: ∀s,L,m. m < |L| → ∃∃I,K,V. ⬇[s, 0, m] L ≡ K.ⓑ{I}V.
+#s #L elim L -L
+[ #m #H elim (ylt_yle_false … H) -H //
+| #L #I #V #IHL #m @(ynat_ind … m) -m /2 width=4 by drop_pair, ex1_3_intro/
+ [ #m #_#H elim (IHL m) -IHL /3 width=4 by drop_drop, ylt_inv_succ, ex1_3_intro/
+ | #H elim (ylt_yle_false … H) -H //
+ ]
+]
+qed-.
+
+lemma drop_O1_pair: ∀L,K,m,s. ⬇[s, 0, m] L ≡ K → m ≤ |L| → ∀I,V.
+ ∃∃J,W. ⬇[s, 0, m] L.ⓑ{I}V ≡ K.ⓑ{J}W.
+#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H #Hm #I #V
+[ elim (drop_inv_atom1 … H) -H #H >(yle_inv_O2 … Hm) -m
+ #Hs destruct /2 width=3 by ex1_2_intro/
+| elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK destruct /2 width=3 by ex1_2_intro/
+ elim (IHL … HLK … Z X) -IHL -HLK
+ /3 width=3 by yle_pred, drop_drop_lt, ex1_2_intro/
+]
+qed-.
+
+lemma drop_O1_ge: ∀L,m. |L| ≤ m → ⬇[Ⓣ, 0, m] L ≡ ⋆.
+#L elim L -L [ #m #_ @drop_atom #H destruct ]
+#L #I #V #IHL #m @(ynat_ind … m) -m //
+[ #H elim (ylt_yle_false … H) -H /2 width=1 by ylt_inj/
+| /4 width=1 by drop_drop, yle_inv_succ/
+]
+qed.
+
+lemma drop_O1_eq: ∀L,s. ⬇[s, 0, |L|] L ≡ ⋆.
+#L elim L -L /2 width=1 by drop_drop/
+qed.
+
+lemma drop_fwd_length_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → |L1| ≤ l → |L2| = |L1|.
+#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m //
+[ #I #L1 #L2 #V #m #_ #_ #H elim (ylt_yle_false … H) -H //
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IH #H
+ lapply (yle_inv_succ … H) -H #H
+ >length_pair >length_pair /3 width=1 by eq_f/
+]
+qed-.
+
+lemma drop_fwd_length_le_le: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 →
+ ∀l0. l + m + l0 = |L1| → |L2| = l + l0.
+#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m //
+[ #l #m #Hm #l0 #H elim (yplus_inv_O … H) -H
+ #H #H0 elim (yplus_inv_O … H) -H
+ #H1 #_ destruct //
+| #I #L1 #L2 #V #m #_ >yplus_O1 >yplus_O1 #IH #l0
+ /3 width=1 by ysucc_inv_inj/
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IHL12 #l0 >yplus_succ1 >yplus_succ1 #H
+ lapply (ysucc_inv_inj … H) -H #Hl1
+ >yplus_succ1 /3 width=1 by eq_f/
+]
+qed-.
+
+lemma drop_fwd_length_le_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → l ≤ |L1| → |L1| ≤ l + m → |L2| = l.
+#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m
+[ #l #m #_ #H #_ /2 width=1 by yle_inv_O2/
+| #I #L #V #_ #H elim (ylt_yle_false … H) -H //
+| #I #L1 #L2 #V #m #_ >yplus_O1 >yplus_O1
+ /3 width=1 by yle_inv_succ/
+| #I #L1 #L2 #V1 #v2 #l #m #_ #_ #IH
+ >yplus_SO2 >yplus_SO2
+ /4 width=1 by yle_inv_succ, eq_f/
+]
+qed-.
+
+lemma drop_fwd_length: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L1| = |L2| + m.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m //
+#l #m #H >H -m //
+qed-.
+
+lemma drop_fwd_length_le2: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → m ≤ |L1|.
+#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H //
+qed-.
+
+lemma drop_fwd_length_le4: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L2| ≤ |L1|.
+#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H //
+qed-.
+
+lemma drop_fwd_length_lt2: ∀L1,I2,K2,V2,l,m.
+ ⬇[Ⓕ, l, m] L1 ≡ K2. ⓑ{I2} V2 → m < |L1|.
+#L1 #I2 #K2 #V2 #l #m #H
+lapply (drop_fwd_Y2 … H) #Hm
+lapply (drop_fwd_length … H) -l #H <(yplus_O2 m) >H -L1
+/2 width=1 by monotonic_ylt_plus_sn/
+qed-.
+
+lemma drop_fwd_length_lt4: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → 0 < m → |L2| < |L1|.
+#L1 #L2 #l #m #H
+lapply (drop_fwd_Y2 … H) #Hm
+lapply (drop_fwd_length … H) -l
+/2 width=1 by monotonic_ylt_plus_sn/
+qed-.
+
+lemma drop_fwd_length_eq1: ∀L1,L2,K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+ |L1| = |L2| → |K1| = |K2|.
+#L1 #L2 #K1 #K2 #l #m #HLK1 #HLK2 #HL12
+lapply (drop_fwd_Y2 … HLK1) #Hm
+lapply (drop_fwd_length … HLK1) -HLK1
+lapply (drop_fwd_length … HLK2) -HLK2
+#H #H0 >H in HL12; -H >H0 -H0 #H
+@(yplus_inv_monotonic_dx … H) -H // (**) (* auto fails *)
+qed-.
+
+lemma drop_fwd_length_eq2: ∀L1,L2,K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
+ |K1| = |K2| → |L1| = |L2|.
+#L1 #L2 #K1 #K2 #l #m #HLK1 #HLK2 #HL12
+lapply (drop_fwd_length … HLK1) -HLK1
+lapply (drop_fwd_length … HLK2) -HLK2 //
+qed-.
+
+lemma drop_inv_length_eq: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 →
+ |L1| = |L2| → m = 0.
+#L1 #L2 #l #m #H #HL12 lapply (drop_fwd_length … H) -H
+>HL12 -L1 #H elim (discr_yplus_x_xy … H) -H //
+#H elim (ylt_yle_false (|L2|) (∞)) //
+qed-.
+
+lemma drop_fwd_be: ∀L,K,s,l,m,i. ⬇[s, l, m] L ≡ K → |K| ≤ i → i < l → |L| ≤ i.
+#L #K #s #l #m #i #HLK #HK #Hl elim (ylt_split i (|L|)) //
+#HL elim (drop_O1_lt (Ⓕ) … HL) #I #K0 #V #HLK0 -HL
+elim (ylt_inv_plus_sn … Hl) -Hl #l0 #H0
+elim (drop_conf_lt … HLK … HLK0 … H0) -HLK -HLK0 -H0
+#K1 #V1 #HK1 #_ #_ lapply (drop_fwd_length_lt2 … HK1) -I -K1 -V1
+#H elim (ylt_yle_false … H) -H //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬇ * [ term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'RDropStar $m $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/rdrop_3.ma".
+include "basic_2/grammar/genv.ma".
+
+(* GLOBAL ENVIRONMENT READING ***********************************************)
+
+inductive gget (m:nat): relation genv ≝
+| gget_gt: ∀G. |G| ≤ m → gget m G (⋆)
+| gget_eq: ∀G. |G| = m + 1 → gget m G G
+| gget_lt: ∀I,G1,G2,V. m < |G1| → gget m G1 G2 → gget m (G1. ⓑ{I} V) G2
+.
+
+interpretation "global reading"
+ 'RDrop m G1 G2 = (gget m G1 G2).
+
+(* basic inversion lemmas ***************************************************)
+
+lemma gget_inv_gt: ∀G1,G2,m. ⬇[m] G1 ≡ G2 → |G1| ≤ m → G2 = ⋆.
+#G1 #G2 #m * -G1 -G2 //
+[ #G #H >H -H >commutative_plus #H (**) (* lemma needed here *)
+ lapply (le_plus_to_le_r … 0 H) -H #H
+ lapply (le_n_O_to_eq … H) -H #H destruct
+| #I #G1 #G2 #V #H1 #_ #H2
+ lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H
+ lapply (lt_plus_to_lt_l … 0 H) -H #H
+ elim (lt_zero_false … H)
+]
+qed-.
+
+lemma gget_inv_eq: ∀G1,G2,m. ⬇[m] G1 ≡ G2 → |G1| = m + 1 → G1 = G2.
+#G1 #G2 #m * -G1 -G2 //
+[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H (**) (* lemma needed here *)
+ lapply (le_plus_to_le_r … 0 H) -H #H
+ lapply (le_n_O_to_eq … H) -H #H destruct
+| #I #G1 #G2 #V #H1 #_ normalize #H2
+ <(injective_plus_l … H2) in H1; -H2 #H
+ elim (lt_refl_false … H)
+]
+qed-.
+
+fact gget_inv_lt_aux: ∀I,G,G1,G2,V,m. ⬇[m] G ≡ G2 → G = G1. ⓑ{I} V →
+ m < |G1| → ⬇[m] G1 ≡ G2.
+#I #G #G1 #G2 #V #m * -G -G2
+[ #G #H1 #H destruct #H2
+ lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H
+ lapply (lt_plus_to_lt_l … 0 H) -H #H
+ elim (lt_zero_false … H)
+| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H
+ elim (lt_refl_false … H)
+| #J #G #G2 #W #_ #HG2 #H destruct //
+]
+qed-.
+
+lemma gget_inv_lt: ∀I,G1,G2,V,m.
+ ⬇[m] G1. ⓑ{I} V ≡ G2 → m < |G1| → ⬇[m] G1 ≡ G2.
+/2 width=5 by gget_inv_lt_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma gget_total: ∀m,G1. ∃G2. ⬇[m] G1 ≡ G2.
+#m #G1 elim G1 -G1 /3 width=2 by gget_gt, ex_intro/
+#I #V #G1 * #G2 #HG12
+elim (lt_or_eq_or_gt m (|G1|)) #Hm
+[ /3 width=2 by gget_lt, ex_intro/
+| destruct /3 width=2 by gget_eq, ex_intro/
+| @ex_intro [2: @gget_gt normalize /2 width=1 by/ | skip ] (**) (* explicit constructor *)
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/gget.ma".
+
+(* GLOBAL ENVIRONMENT READING ***********************************************)
+
+(* Main properties **********************************************************)
+
+theorem gget_mono: ∀G,G1,m. ⬇[m] G ≡ G1 → ∀G2. ⬇[m] G ≡ G2 → G1 = G2.
+#G #G1 #m #H elim H -G -G1
+[ #G #Hm #G2 #H
+ >(gget_inv_gt … H Hm) -H -Hm //
+| #G #Hm #G2 #H
+ >(gget_inv_eq … H Hm) -H -Hm //
+| #I #G #G1 #V #Hm #_ #IHG1 #G2 #H
+ lapply (gget_inv_lt … H Hm) -H -Hm /2 width=1 by/
+]
+qed-.
+
+lemma gget_dec: ∀G1,G2,m. Decidable (⬇[m] G1 ≡ G2).
+#G1 #G2 #m
+elim (gget_total m G1) #G #HG1
+elim (eq_genv_dec G G2) #HG2
+[ destruct /2 width=1 by or_introl/
+| @or_intror #HG12
+ lapply (gget_mono … HG1 … HG12) -HG1 -HG12 /2 width=1 by/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬇ [ term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'RDrop $m $L1 $L2 }.
--- /dev/null
+(* Basic_1: was: lift_r *)
+lemma lift_refl: ∀T,l. ⬆[l, 0] T ≡ T.
+#T elim T -T
+[ * #i // #l elim (lt_or_ge i l) /2 width=1 by lift_lref_lt,
+lift_lref_ge/
+| * /2 width=1 by lift_bind, lift_flat/
+]
+qed.
+
+lemma lift_total: ∀T1,l,m. ∃T2. ⬆[l,m] T1 ≡ T2.
+#T1 elim T1 -T1
+[ * #i /2 width=2 by lift_sort, lift_gref, ex_intro/
+ #l #m elim (lt_or_ge i l) /3 width=2 by lift_lref_lt, lift_lref_ge,
+ex_intro/
+| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #l #m
+ elim (IHV1 l m) -IHV1 #V2 #HV12
+ [ elim (IHT1 (l+1) m) -IHT1 /3 width=2 by lift_bind, ex_intro/
+ | elim (IHT1 l m) -IHT1 /3 width=2 by lift_flat, ex_intro/
+ ]
+]
+qed.
+
+lemma liftv_total: ∀l,m. ∀T1s:list term. ∃T2s. ⬆[l, m] T1s ≡ T2s.
+#l #m #T1s elim T1s -T1s
+[ /2 width=2 by liftv_nil, ex_intro/
+| #T1 #T1s * #T2s #HT12s
+ elim (lift_total T1 l m) /3 width=2 by liftv_cons, ex_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lift.ma".
+
+(* BASIC TERM RELOCATION ****************************************************)
+
+(* Properties on negated basic relocation ***********************************)
+
+lemma nlift_lref_be_SO: ∀X,j. j < ∞ → ⬆[j, 1] X ≡ #j → ⊥.
+#X #j #Hj #H elim (lift_inv_lref2 … H) -H *
+[ #H elim (ylt_yle_false … H) -H //
+| #i #Hij #_ #H1 #H2 destruct
+ elim (ylt_inv_plus_Y … Hj) -Hj #Hi #_
+ elim (ylt_yle_false … Hij) -Hij /2 width=1 by monotonic_ylt_plus_sn/
+]
+qed-.
+
+lemma nlift_bind_sn: ∀W,l,m. (∀V. ⬆[l, m] V ≡ W → ⊥) →
+ ∀a,I,U. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥).
+#W #l #m #HW #a #I #U #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/
+qed-.
+
+lemma nlift_bind_dx: ∀U,l,m. (∀T. ⬆[⫯l, m] T ≡ U → ⊥) →
+ ∀a,I,W. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥).
+#U #l #m #HU #a #I #W #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/
+qed-.
+
+lemma nlift_flat_sn: ∀W,l,m. (∀V. ⬆[l, m] V ≡ W → ⊥) →
+ ∀I,U. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥).
+#W #l #m #HW #I #U #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/
+qed-.
+
+lemma nlift_flat_dx: ∀U,l,m. (∀T. ⬆[l, m] T ≡ U → ⊥) →
+ ∀I,W. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥).
+#U #l #m #HU #I #W #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/
+qed-.
+
+(* Inversion lemmas on negated basic relocation *****************************)
+
+lemma nlift_inv_lref_be_SO: ∀i,j. (∀X. ⬆[i, 1] X ≡ #j → ⊥) → j = i ∧ j < ∞.
+#i #j elim (ylt_split_eq i j) #Hij #H destruct
+[ elim (H (#⫰j)) -H /2 width=1 by lift_lref_pred/
+| elim (yle_split_eq i (∞)) /2 width=1 by conj/ #H0 destruct
+ elim (H (#∞)) -H /2 width=1 by lift_lref_plus, ylt_Y/
+| elim (H (#j)) -H /2 width=1 by lift_lref_lt/
+]
+qed-.
+
+lemma nlift_inv_bind: ∀a,I,W,U,l,m. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥) →
+ (∀V. ⬆[l, m] V ≡ W → ⊥) ∨ (∀T. ⬆[⫯l, m] T ≡ U → ⊥).
+#a #I #W #U #l #m #H elim (is_lift_dec W l m)
+[ * /4 width=2 by lift_bind, or_intror/
+| /4 width=2 by ex_intro, or_introl/
+]
+qed-.
+
+lemma nlift_inv_flat: ∀I,W,U,l,m. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥) →
+ (∀V. ⬆[l, m] V ≡ W → ⊥) ∨ (∀T. ⬆[l, m] T ≡ U → ⊥).
+#I #W #U #l #m #H elim (is_lift_dec W l m)
+[ * /4 width=2 by lift_flat, or_intror/
+| /4 width=2 by ex_intro, or_introl/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/lrsubeq_4.ma".
+include "basic_2/substitution/drop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+inductive lsuby: relation4 ynat ynat lenv lenv ≝
+| lsuby_atom: ∀L,l,m. lsuby l m L (⋆)
+| lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
+ lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+| lsuby_pair: ∀I1,I2,L1,L2,V,m. lsuby 0 m L1 L2 →
+ lsuby 0 (⫯m) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
+| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
+ lsuby l m L1 L2 → lsuby (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
+.
+
+interpretation
+ "local environment refinement (extended substitution)"
+ 'LRSubEq L1 l m L2 = (lsuby l m L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,m. L1 ⊆[0, ⫰m] L2 → 0 < m →
+ L1.ⓑ{I1}V ⊆[0, m] L2.ⓑ{I2}V.
+#I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lsuby_pair/
+qed.
+
+lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⊆[⫰l, m] L2 → 0 < l →
+ L1.ⓑ{I1}V1 ⊆[l, m] L2. ⓑ{I2}V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lsuby_succ/
+qed.
+
+lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 →
+ ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V.
+#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 //
+qed.
+
+lemma lsuby_refl: ∀L,l,m. L ⊆[l, m] L.
+#L elim L -L //
+#L #I #V #IHL #l elim (ynat_cases … l) [| * #x ]
+#Hl destruct /2 width=1 by lsuby_succ/
+#m elim (ynat_cases … m) [| * #x ]
+#Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/
+qed.
+
+lemma lsuby_O2: ∀L2,L1,l. |L2| ≤ |L1| → L1 ⊆[l, 0] L2.
+#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 *
+[ #l #H elim (ylt_yle_false … H) -H //
+| #L1 #I1 #V1 #l
+ #H lapply (yle_inv_succ … H) -H #HL12
+ elim (ynat_cases l) /3 width=1 by lsuby_zero/
+ * /3 width=1 by lsuby_succ/
+]
+qed.
+
+lemma lsuby_sym: ∀l,m,L1,L2. L1 ⊆[l, m] L2 → |L1| = |L2| → L2 ⊆[l, m] L1.
+#l #m #L1 #L2 #H elim H -l -m -L1 -L2
+[ #L1 #l #m #H >(length_inv_zero_dx … H) -L1 //
+| /2 width=1 by lsuby_O2/
+| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (ysucc_inv_inj … H) -H
+ /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (ysucc_inv_inj … H) -H
+ /3 width=1 by lsuby_succ/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuby_inv_atom1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 #l #m * -L1 -L2 -l -m //
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #m #_ #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
+]
+qed-.
+
+lemma lsuby_inv_atom1: ∀L2,l,m. ⋆ ⊆[l, m] L2 → L2 = ⋆.
+/2 width=5 by lsuby_inv_atom1_aux/ qed-.
+
+fact lsuby_inv_zero1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 →
+ L2 = ⋆ ∨
+ ∃∃J2,K2,W2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
+ /3 width=5 by ex2_3_intro, or_intror/
+| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
+ elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊆[0, 0] L2 →
+ L2 = ⋆ ∨
+ ∃∃I2,K2,V2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=9 by lsuby_inv_zero1_aux/ qed-.
+
+fact lsuby_inv_pair1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J1,K1,W. L1 = K1.ⓑ{J1}W → l = 0 → 0 < m →
+ L2 = ⋆ ∨
+ ∃∃J2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{J2}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #HL12 #J1 #K1 #W #H #_ #_ destruct
+ /3 width=4 by ex2_2_intro, or_intror/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_pair1: ∀I1,K1,L2,V,m. K1.ⓑ{I1}V ⊆[0, m] L2 → 0 < m →
+ L2 = ⋆ ∨
+ ∃∃I2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{I2}V.
+/2 width=6 by lsuby_inv_pair1_aux/ qed-.
+
+fact lsuby_inv_succ1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
+ L2 = ⋆ ∨
+ ∃∃J2,K2,W2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
+ /3 width=5 by ex2_3_intro, or_intror/
+]
+qed-.
+
+lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⊆[l, m] L2 → 0 < l →
+ L2 = ⋆ ∨
+ ∃∃I2,K2,V2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{I2}V2.
+/2 width=5 by lsuby_inv_succ1_aux/ qed-.
+
+fact lsuby_inv_zero2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → l = 0 → m = 0 →
+ ∃∃J1,K1,W1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W1 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
+ /2 width=5 by ex2_3_intro/
+| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K2 #W2 #_ #_ #H
+ elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊆[0, 0] K2.ⓑ{I2}V2 →
+ ∃∃I1,K1,V1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
+/2 width=9 by lsuby_inv_zero2_aux/ qed-.
+
+fact lsuby_inv_pair2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J2,K2,W. L2 = K2.ⓑ{J2}W → l = 0 → 0 < m →
+ ∃∃J1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{J1}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #HL12 #J2 #K2 #W #H #_ #_ destruct
+ /2 width=4 by ex2_2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W #_ #H
+ elim (ysucc_inv_O_dx … H)
+]
+qed-.
+
+lemma lsuby_inv_pair2: ∀I2,K2,L1,V,m. L1 ⊆[0, m] K2.ⓑ{I2}V → 0 < m →
+ ∃∃I1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{I1}V.
+/2 width=6 by lsuby_inv_pair2_aux/ qed-.
+
+fact lsuby_inv_succ2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < l →
+ ∃∃J1,K1,W1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{J1}W1.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W2 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H
+ elim (ylt_yle_false … H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J2 #K2 #W2 #H #_ destruct
+ /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⊆[l, m] K2.ⓑ{I2}V2 → 0 < l →
+ ∃∃I1,K1,V1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{I1}V1.
+/2 width=5 by lsuby_inv_succ2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsuby_fwd_length: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → |L2| ≤ |L1|.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m /2 width=1 by yle_succ/
+qed-.
+
+(* Properties on basic slicing **********************************************)
+
+lemma lsuby_drop_trans_be: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
+ ∀I2,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
+ l ≤ i → ∀m0. i + ⫯m0 = l + m →
+ ∃∃I1,K1. K1 ⊆[0, m0] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W #s #i #H
+ elim (drop_inv_atom1 … H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #m0
+ >yplus_O2 >yplus_succ2 #H elim (ysucc_inv_O_dx … H)
+| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ #m0
+ >yplus_succ2 >yplus_succ2 #H0 lapply (ysucc_inv_inj … H0) -H0
+ elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+ [ destruct -I2 /2 width=4 by drop_pair, ex2_2_intro/
+ | lapply (ylt_inv_O1 … Hi)
+ #H <H -H <yplus_succ_swap #Him elim (IHL12 … HLK1 … Him) -IHL12 -HLK1 -Him
+ /3 width=4 by drop_drop_lt, ex2_2_intro/
+ ]
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hli #m0
+ elim (yle_inv_succ1 … Hli) -Hli #Hli #Hi
+ lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O1/ #HLK2
+ >yplus_succ1 >yplus_succ2 #H lapply (ysucc_inv_inj … H) -H
+ <Hi <yplus_succ_swap #H elim (IHL12 … HLK2 … H) -IHL12 -HLK2 -H
+ /3 width=4 by drop_drop, ex2_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lsuby.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+(* Main properties **********************************************************)
+
+theorem lsuby_trans: ∀l,m. Transitive … (lsuby l m).
+#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
+[ #L1 #l #m #X #H lapply (lsuby_inv_atom1 … H) -H
+ #H destruct //
+| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 … H) -H //
+ * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_zero/
+| #I1 #I2 #L1 #L2 #V #m #_ #IHL1 #X #H elim (lsuby_inv_pair1 … H) -H //
+ * #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL1 #X #H elim (lsuby_inv_succ1 … H) -H //
+ * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/
+]
+qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⬇ [ term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'RDrop $m $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⬇ [ term 46 l , break term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'RDrop $l $m $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⬇ [ term 46 s , break term 46 l , break term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'RDrop $s $l $m $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⬇ * [ term 46 m ] break term 46 L1 ≡ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'RDropStar $m $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/supterm_6.ma".
+include "basic_2/grammar/cl_weight.ma".
+include "basic_2/substitution/drop.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* activate genv *)
+inductive fqu: tri_relation genv lenv term ≝
+| fqu_lref_O : ∀I,G,L,V. fqu G (L.ⓑ{I}V) (#0) G L V
+| fqu_pair_sn: ∀I,G,L,V,T. fqu G L (②{I}V.T) G L V
+| fqu_bind_dx: ∀a,I,G,L,V,T. fqu G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
+| fqu_flat_dx: ∀I,G,L,V,T. fqu G L (ⓕ{I}V.T) G L T
+| fqu_drop : ∀G,L,K,T,U,m.
+ ⬇[⫯m] L ≡ K → ⬆[0, ⫯m] T ≡ U → fqu G L U G K T
+.
+
+interpretation
+ "structural successor (closure)"
+ 'SupTerm G1 L1 T1 G2 L2 T2 = (fqu G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fqu_drop_lt: ∀G,L,K,T,U,m. 0 < m →
+ ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → ⦃G, L, U⦄ ⊐ ⦃G, K, T⦄.
+#G #L #K #T #U #m #Hm lapply (ylt_inv_O1 … Hm) -Hm
+#Hm <Hm -Hm /2 width=3 by fqu_drop/
+qed.
+
+lemma fqu_lref_S_lt: ∀I,G,L,V,i. yinj 0 < i → ⦃G, L.ⓑ{I}V, #i⦄ ⊐ ⦃G, L, #(⫰i)⦄.
+/4 width=3 by drop_drop, lift_lref_pred, fqu_drop/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fqu_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
+#G #L #K #T #U #m #HLK #HTU
+lapply (drop_fwd_lw_lt … HLK ?) -HLK // #HKL
+lapply (lift_fwd_tw … HTU) -m #H
+normalize in ⊢ (?%%); /2 width=1 by lt_minus_to_plus/
+qed-.
+
+fact fqu_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀i. T1 = #i → |L2| < |L1|.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[1: /2 width=1 by monotonic_ylt_plus_sn/
+|3: #a
+|5: /2 width=4 by drop_fwd_length_lt4/
+] #I #G #L #V #T #j #H destruct
+qed-.
+
+lemma fqu_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐ ⦃G2, L2, T2⦄ →
+ |L2| < |L1|.
+/2 width=7 by fqu_fwd_length_lref1_aux/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact fqu_inv_eq_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ G1 = G2 → |L1| = |L2| → T1 = T2 → ⊥.
+#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #V #_ #H lapply (ysucc_inv_refl … H) -H
+ #H elim (ylt_yle_false (|L|) (∞)) //
+|5: #G #L #K #T #U #m #HLK #_ #_ #H #_ -G -T -U >(drop_fwd_length … HLK) in H; -L
+ #H elim (discr_yplus_xy_x … H) -H /2 width=2 by ysucc_inv_O_sn/
+ #H elim (ylt_yle_false (|K|) (∞)) //
+]
+/2 width=4 by discr_tpair_xy_y, discr_tpair_xy_x/
+qed-.
+
+lemma fqu_inv_eq: ∀G,L1,L2,T. ⦃G, L1, T⦄ ⊐ ⦃G, L2, T⦄ → |L1| = |L2| → ⊥.
+#G #L1 #L2 #T #H #H0 @(fqu_inv_eq_aux … H … H0) // (**) (* full auto fails *)
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fqu_wf_ind: ∀R:relation3 …. (
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) → ∀G1,L1,T1. R G1 L1 T1.
+#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=1 by fqu_fwd_fw/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/suptermopt_6.ma".
+include "basic_2/substitution/fqu.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* activate genv *)
+inductive fquq: tri_relation genv lenv term ≝
+| fquq_lref_O : ∀I,G,L,V. fquq G (L.ⓑ{I}V) (#0) G L V
+| fquq_pair_sn: ∀I,G,L,V,T. fquq G L (②{I}V.T) G L V
+| fquq_bind_dx: ∀a,I,G,L,V,T. fquq G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
+| fquq_flat_dx: ∀I,G, L,V,T. fquq G L (ⓕ{I}V.T) G L T
+| fquq_drop : ∀G,L,K,T,U,m.
+ ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → fquq G L U G K T
+.
+
+interpretation
+ "optional structural successor (closure)"
+ 'SupTermOpt G1 L1 T1 G2 L2 T2 = (fquq G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fquq_refl: tri_reflexive … fquq.
+/2 width=3 by fquq_drop/ qed.
+
+lemma fqu_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 /2 width=3 by fquq_drop/
+qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fquq_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 /2 width=1 by lt_to_le/
+#G1 #L1 #K1 #T1 #U1 #m #HLK1 #HTU1
+lapply (drop_fwd_lw … HLK1) -HLK1
+lapply (lift_fwd_tw … HTU1) -HTU1
+/2 width=1 by le_plus, le_n/
+qed-.
+
+fact fquq_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀i. T1 = #i → |L2| ≤ |L1|.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
+[ #a #I #G #L #V #T #j #H destruct
+| #G1 #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #i #H destruct
+ /2 width=3 by drop_fwd_length_le4/
+]
+qed-.
+
+lemma fquq_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮ ⦃G2, L2, T2⦄ → |L2| ≤ |L1|.
+/2 width=7 by fquq_fwd_length_lref1_aux/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/notation/relations/suptermoptalt_6.ma".
+include "basic_2/substitution/fquq.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* alternative definition of fquq *)
+definition fquqa: tri_relation genv lenv term ≝ tri_RC … fqu.
+
+interpretation
+ "optional structural successor (closure) alternative"
+ 'SupTermOptAlt G1 L1 T1 G2 L2 T2 = (fquqa G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fquqa_refl: tri_reflexive … fquqa.
+// qed.
+
+lemma fquqa_drop: ∀G,L,K,T,U,m.
+ ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → ⦃G, L, U⦄ ⊐⊐⸮ ⦃G, K, T⦄.
+#G #L #K #T #U #m @(ynat_ind … m) -m /3 width=3 by fqu_drop, or_introl/
+#HLK #HTU >(drop_inv_O2 … HLK) -L >(lift_inv_O2 … HTU) -T //
+qed.
+
+(* Main properties **********************************************************)
+
+theorem fquq_fquqa: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+/2 width=3 by fquqa_drop, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, or_introl/
+qed.
+
+(* Main inversion properties ************************************************)
+
+theorem fquqa_inv_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=1 by fqu_fquq/
+* #H1 #H2 #H3 destruct //
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fquq_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_fquqa … H) -H [| * ]
+/2 width=1 by or_introl/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lreq_lreq.ma".
+include "basic_2/substitution/drop.ma".
+
+(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
+
+definition dedropable_sn: predicate (relation lenv) ≝
+ λR. ∀L1,K1,s,l,m. ⬇[s, l, m] L1 ≡ K1 → ∀K2. R K1 K2 →
+ ∃∃L2. R L1 L2 & ⬇[s, l, m] L2 ≡ K2 & L1 ⩬[l, m] L2.
+
+(* Properties on equivalence ************************************************)
+
+lemma lreq_drop_trans_be: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
+ ∀I,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W →
+ l ≤ i → ∀m0. i + ⫯m0 = l + m →
+ ∃∃K1. K1 ⩬[0, m0] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+[ #l #m #J #K2 #W #s #i #H
+ elim (drop_inv_atom1 … H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J #K2 #W #s #i #_ #_ #m0
+ >yplus_succ2 #H elim (ysucc_inv_O_dx … H)
+| #I #L1 #L2 #V #m #HL12 #IHL12 #J #K2 #W #s #i #H #_ >yplus_O1 #m0 #H0
+ elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+ [ destruct
+ /2 width=3 by drop_pair, ex2_intro/
+ | lapply (ylt_inv_O1 … Hi) #H <H in H0; -H
+ >yplus_succ1 #H lapply (ysucc_inv_inj … H) -H <(yplus_O1 m)
+ #H0 elim (IHL12 … HLK1 … H0) -IHL12 -HLK1 -H0 //
+ /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J #K2 #W #s #i #HLK2 #Hli #m0
+ elim (yle_inv_succ1 … Hli) -Hli
+ #Hli #Hi <Hi >yplus_succ1 >yplus_succ1 #H lapply (ysucc_inv_inj … H) -H
+ #H0 lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O1/
+ #HLK1 elim (IHL12 … HLK1 … H0) -IHL12 -HLK1 -H0
+ /4 width=3 by ylt_O1, drop_drop_lt, ex2_intro/
+]
+qed-.
+
+lemma lreq_drop_conf_be: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
+ ∀I,K1,W,s,i. ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W →
+ l ≤ i → ∀m0. i + ⫯m0 = l + m →
+ ∃∃K2. K1 ⩬[0, m0] K2 & ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W.
+#L1 #L2 #l #m #HL12 #I #K1 #W #s #i #HLK1 #Hli #m0 #H0
+elim (lreq_drop_trans_be … (lreq_sym … HL12) … HLK1 … H0) // -L1 -Hli -H0
+/3 width=3 by lreq_sym, ex2_intro/
+qed-.
+
+lemma drop_O1_ex: ∀K2,i,L1. |L1| = |K2| + i →
+ ∃∃L2. L1 ⩬[0, i] L2 & ⬇[i] L2 ≡ K2.
+#K2 #i @(ynat_ind … i) -i
+[ /3 width=3 by lreq_O2, ex2_intro/
+| #i #IHi #Y >yplus_succ2 #Hi
+ elim (drop_O1_lt (Ⓕ) Y 0) [2: >Hi // ]
+ #I #L1 #V #H lapply (drop_inv_O2 … H) -H #H destruct
+ >length_pair in Hi; #H lapply (ysucc_inv_inj … H) -H
+ #HL1K2 elim (IHi L1) -IHi // -HL1K2
+ /3 width=5 by lreq_pair, drop_drop, ex2_intro/
+| #L1 >yplus_Y2 #H elim (ylt_yle_false (|L1|) (∞)) //
+]
+qed-.
+
+lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R).
+#R #HR #L1 #K1 #s #l #m #HLK1 #K2 #H elim H -K2
+[ #K2 #HK12 elim (HR … HLK1 … HK12) -HR -K1
+ /3 width=4 by inj, ex3_intro/
+| #K #K2 #_ #HK2 * #L #H1L1 #HLK #H2L1 elim (HR … HLK … HK2) -HR -K
+ /3 width=6 by lreq_trans, step, ex3_intro/
+]
+qed-.
+
+(* Inversion lemmas on equivalence ******************************************)
+
+lemma drop_O1_inj: ∀i,L1,L2,K. ⬇[i] L1 ≡ K → ⬇[i] L2 ≡ K → L1 ⩬[i, ∞] L2.
+#i @(ynat_ind … i) -i
+[ #L1 #L2 #K #H <(drop_inv_O2 … H) -K #H <(drop_inv_O2 … H) -L1 //
+| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 //
+ lapply (drop_fwd_length … HLK1)
+ <(drop_fwd_length … HLK2) [ /4 width=5 by drop_inv_drop1, lreq_succ/ ]
+ #H [ elim (ysucc_inv_O_sn … H) | elim (ysucc_inv_O_dx … H) ]
+| #L1 #L2 #K #H1 lapply (drop_fwd_Y2 … H1) -H1
+ #H elim (ylt_yle_false … H) //
+]
+qed-.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/rdropstar_3.ma".
include "basic_2/notation/relations/rdropstar_4.ma".
-include "basic_2/substitution/drop.ma".
-include "basic_2/multiple/mr2_minus.ma".
-include "basic_2/multiple/lifts_vector.ma".
+include "basic_2/grammar/lenv.ma".
+include "basic_2/relocation/lifts.ma".
-(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+(* GENERAL SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-inductive drops (s:bool): list2 ynat nat → relation lenv ≝
-| drops_nil : ∀L. drops s (◊) L L
-| drops_cons: ∀L1,L,L2,cs,l,m.
- drops s cs L1 L → ⬇[s, l, m] L ≡ L2 → drops s ({l, m} @ cs) L1 L2
+(* Basic_1: includes: drop_skip_bind drop1_skip_bind *)
+(* Basic_2A1: includes: drop_atom drop_pair drop_drop drop_skip
+ drop_refl_atom_O2 drop_drop_lt drop_skip_lt
+*)
+inductive drops (s:bool): trace → relation lenv ≝
+| drops_atom: ∀t. (s = Ⓕ → 𝐈⦃t⦄) → drops s (t) (⋆) (⋆)
+| drops_drop: ∀I,L1,L2,V,t. drops s t L1 L2 → drops s (Ⓕ@t) (L1.ⓑ{I}V) L2
+| drops_skip: ∀I,L1,L2,V1,V2,t.
+ drops s t L1 L2 → ⬆*[t] V2 ≡ V1 →
+ drops s (Ⓣ@t) (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
.
-interpretation "iterated slicing (local environment) abstract"
- 'RDropStar s cs T1 T2 = (drops s cs T1 T2).
-(*
-interpretation "iterated slicing (local environment) general"
- 'RDropStar des T1 T2 = (drops true des T1 T2).
-*)
+interpretation "general slicing (local environment)"
+ 'RDropStar s t L1 L2 = (drops s t L1 L2).
definition d_liftable1: relation2 lenv term → predicate bool ≝
- λR,s. ∀K,T. R K T → ∀L,l,m. ⬇[s, l, m] L ≡ K →
- ∀U. ⬆[l, m] T ≡ U → R L U.
+ λR,s. ∀L,K,t. ⬇*[s, t] L ≡ K →
+ ∀T,U. ⬆*[t] T ≡ U → R K T → R L U.
+
+definition d_liftable2: predicate (lenv → relation term) ≝
+ λR. ∀K,T1,T2. R K T1 T2 → ∀L,s,t. ⬇*[s, t] L ≡ K →
+ ∀U1. ⬆*[t] T1 ≡ U1 →
+ ∃∃U2. ⬆*[t] T2 ≡ U2 & R L U1 U2.
+
+definition d_deliftable2_sn: predicate (lenv → relation term) ≝
+ λR. ∀L,U1,U2. R L U1 U2 → ∀K,s,t. ⬇*[s, t] L ≡ K →
+ ∀T1. ⬆*[t] T1 ≡ U1 →
+ ∃∃T2. ⬆*[t] T2 ≡ U2 & R K T1 T2.
+
+definition dropable_sn: predicate (relation lenv) ≝
+ λR. ∀L1,K1,s,t. ⬇*[s, t] L1 ≡ K1 → ∀L2. R L1 L2 →
+ ∃∃K2. R K1 K2 & ⬇*[s, t] L2 ≡ K2.
+(*
+definition dropable_dx: predicate (relation lenv) ≝
+ λR. ∀L1,L2. R L1 L2 → ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
+ ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & R K1 K2.
+*)
+(* Basic inversion lemmas ***************************************************)
-definition d_liftables1: relation2 lenv term → predicate bool ≝
- λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K →
- ∀T,U. ⬆*[cs] T ≡ U → R K T → R L U.
+fact drops_inv_atom1_aux: ∀X,Y,s,t. ⬇*[s, t] X ≡ Y → X = ⋆ →
+ Y = ⋆ ∧ (s = Ⓕ → 𝐈⦃t⦄).
+#X #Y #s #t * -X -Y -t
+[ /3 width=1 by conj/
+| #I #L1 #L2 #V #t #_ #H destruct
+| #I #L1 #L2 #V1 #V2 #t #_ #_ #H destruct
+]
+qed-.
-definition d_liftables1_all: relation2 lenv term → predicate bool ≝
- λR,s. ∀L,K,cs. ⬇*[s, cs] L ≡ K →
- ∀Ts,Us. ⬆*[cs] Ts ≡ Us →
- all … (R K) Ts → all … (R L) Us.
+(* Basic_1: includes: drop_gen_sort *)
+(* Basic_2A1: includes: drop_inv_atom1 *)
+lemma drops_inv_atom1: ∀Y,s,t. ⬇*[s, t] ⋆ ≡ Y → Y = ⋆ ∧ (s = Ⓕ → 𝐈⦃t⦄).
+/2 width=3 by drops_inv_atom1_aux/ qed-.
+
+fact drops_inv_drop1_aux: ∀X,Y,s,t. ⬇*[s, t] X ≡ Y → ∀I,K,V,tl. X = K.ⓑ{I}V → t = Ⓕ@tl →
+ ⬇*[s, tl] K ≡ Y.
+#X #Y #s #t * -X -Y -t
+[ #t #Ht #J #K #W #tl #H destruct
+| #I #L1 #L2 #V #t #HL #J #K #W #tl #H1 #H2 destruct //
+| #I #L1 #L2 #V1 #V2 #t #_ #_ #J #K #W #tl #_ #H2 destruct
+]
+qed-.
-(* Basic inversion lemmas ***************************************************)
+(* Basic_1: includes: drop_gen_drop *)
+(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
+lemma drops_inv_drop1: ∀I,K,Y,V,s,t. ⬇*[s, Ⓕ@t] K.ⓑ{I}V ≡ Y → ⬇*[s, t] K ≡ Y.
+/2 width=7 by drops_inv_drop1_aux/ qed-.
-fact drops_inv_nil_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → cs = ◊ → L1 = L2.
-#L1 #L2 #s #cs * -L1 -L2 -cs //
-#L1 #L #L2 #l #m #cs #_ #_ #H destruct
-qed-.
-(* Basic_1: was: drop1_gen_pnil *)
-lemma drops_inv_nil: ∀L1,L2,s. ⬇*[s, ◊] L1 ≡ L2 → L1 = L2.
-/2 width=4 by drops_inv_nil_aux/ qed-.
-
-fact drops_inv_cons_aux: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 →
- ∀l,m,tl. cs = {l, m} @ tl →
- ∃∃L. ⬇*[s, tl] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
-#L1 #L2 #s #cs * -L1 -L2 -cs
-[ #L #l #m #tl #H destruct
-| #L1 #L #L2 #cs #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct
- /2 width=3 by ex2_intro/
+fact drops_inv_skip1_aux: ∀X,Y,s,t. ⬇*[s, t] X ≡ Y → ∀I,K1,V1,tl. X = K1.ⓑ{I}V1 → t = Ⓣ@tl →
+ ∃∃K2,V2. ⬇*[s, tl] K1 ≡ K2 & ⬆*[tl] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+#X #Y #s #t * -X -Y -t
+[ #t #Ht #J #K1 #W1 #tl #H destruct
+| #I #L1 #L2 #V #t #_ #J #K1 #W1 #tl #_ #H2 destruct
+| #I #L1 #L2 #V1 #V2 #t #HL #HV #J #K1 #W1 #tl #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
]
qed-.
-(* Basic_1: was: drop1_gen_pcons *)
-lemma drops_inv_cons: ∀L1,L2,s,l,m,cs. ⬇*[s, {l, m} @ cs] L1 ≡ L2 →
- ∃∃L. ⬇*[s, cs] L1 ≡ L & ⬇[s, l, m] L ≡ L2.
-/2 width=3 by drops_inv_cons_aux/ qed-.
-
-lemma drops_inv_skip2: ∀I,s,cs,cs2,i. cs ▭ i ≡ cs2 →
- ∀L1,K2,V2. ⬇*[s, cs2] L1 ≡ K2. ⓑ{I} V2 →
- ∃∃K1,V1,cs1. cs + 1 ▭ i + 1 ≡ cs1 + 1 &
- ⬇*[s, cs1] K1 ≡ K2 &
- ⬆*[cs1] V2 ≡ V1 &
- L1 = K1. ⓑ{I} V1.
-#I #s #cs #cs2 #i #H elim H -cs -cs2 -i
-[ #i #L1 #K2 #V2 #H
- >(drops_inv_nil … H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, drops_nil/
-| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
- elim (drops_inv_cons … H) -H #L #HL1 #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ #K #V <yminus_succ2 #HK2 #HV2 #H destruct
- elim (IHcs2 … HL1) -IHcs2 -HL1 #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
- @(ex4_3_intro … K1 V1 … ) // [3,4: /2 width=7 by lifts_cons, drops_cons/ | skip ]
- >pluss_SO2 >pluss_SO2
- >yminus_succ2 >ylt_inv_O1 /2 width=1 by ylt_to_minus/ <yminus_succ >commutative_plus (**) (* <yminus_succ1_inj does not work *)
- /3 width=1 by minuss_lt, ylt_succ/
-| #cs #cs2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
- elim (IHcs2 … H) -IHcs2 -H #K1 #V1 #cs1 #Hcs1 #HK1 #HV1 #X destruct
- /4 width=7 by minuss_ge, yle_succ, ex4_3_intro/
+(* Basic_1: includes: drop_gen_skip_l *)
+(* Basic_2A1: includes: drop_inv_skip1 *)
+lemma drops_inv_skip1: ∀I,K1,V1,Y,s,t. ⬇*[s, Ⓣ@t] K1.ⓑ{I}V1 ≡ Y →
+ ∃∃K2,V2. ⬇*[s, t] K1 ≡ K2 & ⬆*[t] V2 ≡ V1 & Y = K2.ⓑ{I}V2.
+/2 width=5 by drops_inv_skip1_aux/ qed-.
+
+fact drops_inv_skip2_aux: ∀X,Y,s,t. ⬇*[s, t] X ≡ Y → ∀I,K2,V2,tl. Y = K2.ⓑ{I}V2 → t = Ⓣ@tl →
+ ∃∃K1,V1. ⬇*[s, tl] K1 ≡ K2 & ⬆*[tl] V2 ≡ V1 & X = K1.ⓑ{I}V1.
+#X #Y #s #t * -X -Y -t
+[ #t #Ht #J #K2 #W2 #tl #H destruct
+| #I #L1 #L2 #V #t #_ #J #K2 #W2 #tl #_ #H2 destruct
+| #I #L1 #L2 #V1 #V2 #t #HL #HV #J #K2 #W2 #tl #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
]
qed-.
+(* Basic_1: includes: drop_gen_skip_r *)
+(* Basic_2A1: includes: drop_inv_skip2 *)
+lemma drops_inv_skip2: ∀I,X,K2,V2,s,t. ⬇*[s, Ⓣ@t] X ≡ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[s, t] K1 ≡ K2 & ⬆*[t] V2 ≡ V1 & X = K1.ⓑ{I}V1.
+/2 width=5 by drops_inv_skip2_aux/ qed-.
+
(* Basic properties *********************************************************)
-(* Basic_1: was: drop1_skip_bind *)
-lemma drops_skip: ∀L1,L2,s,cs. ⬇*[s, cs] L1 ≡ L2 → ∀V1,V2. ⬆*[cs] V2 ≡ V1 →
- ∀I. ⬇*[s, cs + 1] L1.ⓑ{I}V1 ≡ L2.ⓑ{I}V2.
-#L1 #L2 #s #cs #H elim H -L1 -L2 -cs
-[ #L #V1 #V2 #HV12 #I
- >(lifts_inv_nil … HV12) -HV12 //
-| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #V1 #V2 #H #I
- elim (lifts_inv_cons … H) -H /3 width=5 by drop_skip, drops_cons/
-].
+(* Basic_2A1: includes: drop_FT *)
+lemma drops_FT: ∀L1,L2,t. ⬇*[Ⓕ, t] L1 ≡ L2 → ⬇*[Ⓣ, t] L1 ≡ L2.
+#L1 #L2 #t #H elim H -L1 -L2 -t
+/3 width=1 by drops_atom, drops_drop, drops_skip/
qed.
-lemma d1_liftable_liftables: ∀R,s. d_liftable1 R s → d_liftables1 R s.
-#R #s #HR #L #K #cs #H elim H -L -K -cs
-[ #L #T #U #H #HT <(lifts_inv_nil … H) -H //
-| #L1 #L #L2 #cs #l #m #_ #HL2 #IHL #T2 #T1 #H #HLT2
- elim (lifts_inv_cons … H) -H /3 width=10 by/
+(* Basic_2A1: includes: drop_gen *)
+lemma drops_gen: ∀L1,L2,s,t. ⬇*[Ⓕ, t] L1 ≡ L2 → ⬇*[s, t] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by drops_FT/
+qed-.
+
+(* Basic_2A1: includes: drop_T *)
+lemma drops_T: ∀L1,L2,s,t. ⬇*[s, t] L1 ≡ L2 → ⬇*[Ⓣ, t] L1 ≡ L2.
+#L1 #L2 * /2 width=1 by drops_FT/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+fact drops_fwd_drop2_aux: ∀X,Y,s,t2. ⬇*[s, t2] X ≡ Y → ∀I,K,V. Y = K.ⓑ{I}V →
+ ∃∃t1,t. 𝐈⦃t1⦄ & t2 ⊚ Ⓕ@t1 ≡ t & ⬇*[s, t] X ≡ K.
+#X #Y #s #t2 #H elim H -X -Y -t2
+[ #t2 #Ht2 #J #K #W #H destruct
+| #I #L1 #L2 #V #t2 #_ #IHL #J #K #W #H elim (IHL … H) -IHL
+ /3 width=5 by after_false, ex3_2_intro, drops_drop/
+| #I #L1 #L2 #V1 #V2 #t2 #HL #_ #_ #J #K #W #H destruct
+ elim (isid_after_dx t2) /3 width=5 by after_true, ex3_2_intro, drops_drop/
]
-qed.
+qed-.
-lemma d1_liftables_liftables_all: ∀R,s. d_liftables1 R s → d_liftables1_all R s.
-#R #s #HR #L #K #cs #HLK #Ts #Us #H elim H -Ts -Us normalize //
-#Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/
-qed.
+(* Basic_1: includes: drop_S *)
+(* Basic_2A1: includes: drop_fwd_drop2 *)
+lemma drops_fwd_drop2: ∀I,X,K,V,s,t2. ⬇*[s, t2] X ≡ K.ⓑ{I}V →
+ ∃∃t1,t. 𝐈⦃t1⦄ & t2 ⊚ Ⓕ@t1 ≡ t & ⬇*[s, t] X ≡ K.
+/2 width=5 by drops_fwd_drop2_aux/ qed-.
+
+fact drops_after_fwd_drop2_aux: ∀X,Y,s,t2. ⬇*[s, t2] X ≡ Y → ∀I,K,V. Y = K.ⓑ{I}V →
+ ∀t1,t. 𝐈⦃t1⦄ → t2 ⊚ Ⓕ@t1 ≡ t → ⬇*[s, t] X ≡ K.
+#X #Y #s #t2 #H elim H -X -Y -t2
+[ #t2 #Ht2 #J #K #W #H destruct
+| #I #L1 #L2 #V #t2 #_ #IHL #J #K #W #H #t1 #t #Ht1 #Ht elim (after_inv_false1 … Ht) -Ht
+ /3 width=3 by drops_drop/
+| #I #L1 #L2 #V1 #V2 #t2 #HL #_ #_ #J #K #W #H #t1 #t #Ht1 #Ht elim (after_inv_true1 … Ht) -Ht
+ #u1 #u #b #H1 #H2 #Hu destruct >(after_isid_inv_dx … Hu) -Hu /2 width=1 by drops_drop/
+]
+qed-.
+
+lemma drops_after_fwd_drop2: ∀I,X,K,V,s,t2. ⬇*[s, t2] X ≡ K.ⓑ{I}V →
+ ∀t1,t. 𝐈⦃t1⦄ → t2 ⊚ Ⓕ@t1 ≡ t → ⬇*[s, t] X ≡ K.
+/2 width=9 by drops_after_fwd_drop2_aux/ qed-.
-(* Basic_1: removed theorems 1: drop1_getl_trans *)
+(* Basic_1: includes: drop_gen_refl *)
+(* Basic_2A1: includes: drop_inv_O2 *)
+lemma drops_fwd_isid: ∀L1,L2,s,t. ⬇*[s, t] L1 ≡ L2 → 𝐈⦃t⦄ → L1 = L2.
+#L1 #L2 #s #t #H elim H -L1 -L2 -t //
+[ #I #L1 #L2 #V #t #_ #_ #H elim (isid_inv_false … H)
+| /5 width=3 by isid_inv_true, lifts_fwd_isid, eq_f3, sym_eq/
+]
+qed-.
+
+(* Basic_2A1: removed theorems 13:
+ drops_inv_nil drops_inv_cons d1_liftable_liftables
+ drop_inv_O1_pair1 drop_inv_pair1 drop_inv_O1_pair2
+ drop_inv_Y1 drop_Y1 drop_O_Y drop_fwd_Y2
+ drop_fwd_length_minus2 drop_fwd_length_minus4
+*)
+(* Basic_1: removed theorems 53:
+ drop1_gen_pnil drop1_gen_pcons drop1_getl_trans
+ drop_ctail drop_skip_flat
+ cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
+ drop_clear drop_clear_O drop_clear_S
+ clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
+ clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
+ getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
+ getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
+ getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
+ drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
+ getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+ getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
+ getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
+*)
(* *)
(**************************************************************************)
-include "basic_2/multiple/drops_drop.ma".
+include "basic_2/relocation/lifts_lifts.ma".
+include "basic_2/relocation/drops.ma".
-(* ITERATED LOCAL ENVIRONMENT SLICING ***************************************)
+(* GENERAL SLICING FOR LOCAL ENVIRONMENTS ***********************************)
(* Main properties **********************************************************)
+(* Basic_2A1: includes: drop_conf_ge drop_conf_be drop_conf_le *)
+theorem drops_conf: ∀L1,L,s1,t1. ⬇*[s1, t1] L1 ≡ L →
+ ∀L2,s2,t. ⬇*[s2, t] L1 ≡ L2 →
+ ∀t2. t1 ⊚ t2 ≡ t → ⬇*[s2, t2] L ≡ L2.
+#L1 #L #s1 #t1 #H elim H -L1 -L -t1
+[ #t1 #_ #L2 #s2 #t #H #t2 #Ht12 elim (drops_inv_atom1 … H) -s1 -H
+ #H #Ht destruct @drops_atom
+ #H elim (after_inv_isid3 … Ht12) -Ht12 /2 width=1 by/
+| #I #K1 #K #V1 #t1 #_ #IH #L2 #s2 #t #H12 #t2 #Ht elim (after_inv_false1 … Ht) -Ht
+ #u #H #Hu destruct /3 width=3 by drops_inv_drop1/
+| #I #K1 #K #V1 #V #t1 #_ #HV1 #IH #L2 #s2 #t #H #t2 #Ht elim (after_inv_true1 … Ht) -Ht
+ #u2 #u * #H1 #H2 #Hu destruct
+ [ elim (drops_inv_skip1 … H) -H /3 width=6 by drops_skip, lifts_div/
+ | /4 width=3 by drops_inv_drop1, drops_drop/
+ ]
+]
+qed-.
+
(* Basic_1: was: drop1_trans *)
-theorem drops_trans: ∀L,L2,s,cs2. ⬇*[s, cs2] L ≡ L2 → ∀L1,cs1. ⬇*[s, cs1] L1 ≡ L →
- ⬇*[s, cs2 @@ cs1] L1 ≡ L2.
-#L #L2 #s #cs2 #H elim H -L -L2 -cs2 /3 width=3 by drops_cons/
+(* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm
+ drops_drop_trans
+*)
+theorem drops_trans: ∀L1,L,s1,t1. ⬇*[s1, t1] L1 ≡ L →
+ ∀L2,s2,t2. ⬇*[s2, t2] L ≡ L2 →
+ ∀t. t1 ⊚ t2 ≡ t → ⬇*[s1∨s2, t] L1 ≡ L2.
+#L1 #L #s1 #t1 #H elim H -L1 -L -t1
+[ #t1 #Ht1 #L2 #s2 #t2 #H #t #Ht elim (drops_inv_atom1 … H) -H
+ #H #Ht2 destruct @drops_atom #H elim (orb_false_r … H) -H
+ #H1 #H2 >(after_isid_inv_sn … Ht) -Ht /2 width=1 by/
+| #I #K1 #K #V1 #t1 #_ #IH #L #s2 #t2 #HKL #t #Ht elim (after_inv_false1 … Ht) -Ht
+ /3 width=3 by drops_drop/
+| #I #K1 #K #V1 #V #t1 #_ #HV1 #IH #L #s2 #t2 #H #t #Ht elim (after_inv_true1 … Ht) -Ht
+ #u2 #u * #H1 #H2 #Hu destruct
+ [ elim (drops_inv_skip1 … H) -H /3 width=6 by drops_skip, lifts_trans/
+ | /4 width=3 by drops_inv_drop1, drops_drop/
+ ]
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: includes: drop_mono *)
+lemma drops_mono: ∀L,L1,s1,t. ⬇*[s1, t] L ≡ L1 →
+ ∀L2,s2. ⬇*[s2, t] L ≡ L2 → L1 = L2.
+#L #L1 #s1 #t elim (isid_after_dx t)
+/3 width=8 by drops_conf, drops_fwd_isid/
+qed-.
+
+(* Basic_2A1: includes: drop_conf_lt *)
+lemma drops_conf_skip1: ∀L,L2,s2,t. ⬇*[s2, t] L ≡ L2 →
+ ∀I,K1,V1,s1,t1. ⬇*[s1, t1] L ≡ K1.ⓑ{I}V1 →
+ ∀t2. t1 ⊚ Ⓣ@t2 ≡ t →
+ ∃∃K2,V2. L2 = K2.ⓑ{I}V2 &
+ ⬇*[s2, t2] K1 ≡ K2 & ⬆*[t2] V2 ≡ V1.
+#L #L2 #s2 #t #H2 #I #K1 #V1 #s1 #t1 #H1 #t2 #Ht lapply (drops_conf … H1 … H2 … Ht) -L -Ht
+#H elim (drops_inv_skip1 … H) -H /2 width=5 by ex3_2_intro/
+qed-.
+
+(* Basic_2A1: includes: drop_trans_lt *)
+lemma drops_trans_skip2: ∀L1,L,s1,t1. ⬇*[s1, t1] L1 ≡ L →
+ ∀I,K2,V2,s2,t2. ⬇*[s2, t2] L ≡ K2.ⓑ{I}V2 →
+ ∀t. t1 ⊚ t2 ≡ Ⓣ@t →
+ ∃∃K1,V1. L1 = K1.ⓑ{I}V1 &
+ ⬇*[s1∨s2, t] K1 ≡ K2 & ⬆*[t] V2 ≡ V1.
+#L1 #L #s1 #t1 #H1 #I #K2 #V2 #s2 #t2 #H2 #t #Ht
+lapply (drops_trans … H1 … H2 … Ht) -L -Ht
+#H elim (drops_inv_skip2 … H) -H /2 width=5 by ex3_2_intro/
qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/lib/lstar.ma".
+include "basic_2/relocation/drops.ma".
+
+(* GENERAL SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Properties on reflexive and transitive closure ***************************)
+
+(* Basic_2A1: was: d_liftable_LTC *)
+lemma d2_liftable_LTC: ∀R. d_liftable2 R → d_liftable2 (LTC … R).
+#R #HR #K #T1 #T2 #H elim H -T2
+[ #T2 #HT12 #L #s #t #HLK #U1 #HTU1
+ elim (HR … HT12 … HLK … HTU1) /3 width=3 by inj, ex2_intro/
+| #T #T2 #_ #HT2 #IHT1 #L #s #t #HLK #U1 #HTU1
+ elim (IHT1 … HLK … HTU1) -T1 #U #HTU #HU1
+ elim (HR … HT2 … HLK … HTU) -HR -K -T /3 width=5 by step, ex2_intro/
+]
+qed-.
+
+(* Basic_2A1: was: d_deliftable_sn_LTC *)
+lemma d2_deliftable_sn_LTC: ∀R. d_deliftable2_sn R → d_deliftable2_sn (LTC … R).
+#R #HR #L #U1 #U2 #H elim H -U2
+[ #U2 #HU12 #K #s #t #HLK #T1 #HTU1
+ elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3 by inj, ex2_intro/
+| #U #U2 #_ #HU2 #IHU1 #K #s #t #HLK #T1 #HTU1
+ elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+ elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by step, ex2_intro/
+]
+qed-.
+
+lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R).
+#R #HR #L1 #K1 #s #t #HLK1 #L2 #H elim H -L2
+[ #L2 #HL12 elim (HR … HLK1 … HL12) -HR -L1
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 * #K #HK1 #HLK elim (HR … HLK … HL2) -HR -L
+ /3 width=3 by step, ex2_intro/
+]
+qed-.
+(*
+lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
+#R #HR #L1 #L2 #H elim H -L2
+[ #L2 #HL12 #K2 #s #m #HLK2 elim (HR … HL12 … HLK2) -HR -L2
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 #IHL1 #K2 #s #m #HLK2 elim (HR … HL2 … HLK2) -HR -L2
+ #K #HLK #HK2 elim (IHL1 … HLK) -L
+ /3 width=5 by step, ex2_intro/
+]
+qed-.
+*)
+(* Basic_2A1: was: d_liftable_llstar *)
+lemma d2_liftable_llstar: ∀R. d_liftable2 R → ∀d. d_liftable2 (llstar … R d).
+#R #HR #d #K #T1 #T2 #H @(lstar_ind_r … d T2 H) -d -T2
+[ #L #s #t #_ #U1 #HTU1 -HR -K -s /2 width=3 by ex2_intro/
+| #d #T #T2 #_ #HT2 #IHT1 #L #s #t #HLK #U1 #HTU1
+ elim (IHT1 … HLK … HTU1) -T1 #U #HTU #HU1
+ elim (HR … HT2 … HLK … HTU) -T /3 width=5 by lstar_dx, ex2_intro/
+]
+qed-.
+
+(* Basic_2A1: was: d_deliftable_sn_llstar *)
+lemma d2_deliftable_sn_llstar: ∀R. d_deliftable2_sn R →
+ ∀d. d_deliftable2_sn (llstar … R d).
+#R #HR #d #L #U1 #U2 #H @(lstar_ind_r … d U2 H) -d -U2
+[ /2 width=3 by lstar_O, ex2_intro/
+| #d #U #U2 #_ #HU2 #IHU1 #K #s #t #HLK #T1 #HTU1
+ elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+ elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by lstar_dx, ex2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lifts_vector.ma".
+include "basic_2/relocation/drops.ma".
+
+(* GENERAL SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+definition d_liftable1_all: relation2 lenv term → predicate bool ≝
+ λR,s. ∀L,K,t. ⬇*[s, t] L ≡ K →
+ ∀Ts,Us. ⬆*[t] Ts ≡ Us →
+ all … (R K) Ts → all … (R L) Us.
+
+(* Properties on general relocation for term vectors ************************)
+
+(* Basic_2A1: was: d1_liftables_liftables_all *)
+lemma d1_liftable_liftable_all: ∀R,s. d_liftable1 R s → d_liftable1_all R s.
+#R #s #HR #L #K #t #HLK #Ts #Us #H elim H -Ts -Us normalize //
+#Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/cl_restricted_weight.ma".
+include "basic_2/relocation/lifts_weight.ma".
+include "basic_2/relocation/drops.ma".
+
+(* GENERAL SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Forward lemmas on weight for local environments **************************)
+
+(* Basic_2A1: includes: drop_fwd_lw *)
+lemma drops_fwd_lw: ∀L1,L2,s,t. ⬇*[s, t] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
+#L1 #L2 #s #t #H elim H -L1 -L2 -t //
+[ /2 width=3 by transitive_le/
+| #I #L1 #L2 #V1 #V2 #t #_ #HV21 #IHL12 normalize
+ >(lifts_fwd_tw … HV21) -HV21 /2 width=1 by monotonic_le_plus_l/
+]
+qed-.
+
+(* Basic_2A1: includes: drop_fwd_lw_lt *)
+(* Note: 𝐈⦃t⦄ → ⊥ is ∥l∥ < |L| *)
+lemma drops_fwd_lw_lt: ∀L1,L2,t. ⬇*[Ⓕ, t] L1 ≡ L2 →
+ (𝐈⦃t⦄ → ⊥) → ♯{L2} < ♯{L1}.
+#L1 #L2 #t #H elim H -L1 -L2 -t
+[ #t #Ht #Hnt elim Hnt -Hnt /2 width=1 by/
+| /3 width=3 by drops_fwd_lw, le_to_lt_to_lt/
+| #I #L1 #L2 #V1 #V2 #t #_ #HV21 #IHL12 #H normalize in ⊢ (?%%); -I
+ >(lifts_fwd_tw … HV21) -V2 /4 width=1 by monotonic_lt_plus_l/
+]
+qed-.
+
+(* Forward lemmas on restricted weight for closures *************************)
+
+(* Basic_2A1: includes: drop_fwd_rfw *)
+lemma drops_pair2_fwd_rfw: ∀I,L,K,V,s,t. ⬇*[s, t] L ≡ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
+#I #L #K #V #s #t #HLK lapply (drops_fwd_lw … HLK) -HLK
+normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt/
+qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_max.ma".
-include "basic_2/notation/relations/psubst_6.ma".
-include "basic_2/grammar/genv.ma".
-include "basic_2/substitution/lsuby.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* activate genv *)
-inductive cpy: ynat → ynat → relation4 genv lenv term term ≝
-| cpy_atom : ∀I,G,L,l,m. cpy l m G L (⓪{I}) (⓪{I})
-| cpy_subst: ∀I,G,L,K,V,W,i,l,m. l ≤ yinj i → i < l+m →
- ⬇[i] L ≡ K.ⓑ{I}V → ⬆[0, i+1] V ≡ W → cpy l m G L (#i) W
-| cpy_bind : ∀a,I,G,L,V1,V2,T1,T2,l,m.
- cpy l m G L V1 V2 → cpy (⫯l) m G (L.ⓑ{I}V1) T1 T2 →
- cpy l m G L (ⓑ{a,I}V1.T1) (ⓑ{a,I}V2.T2)
-| cpy_flat : ∀I,G,L,V1,V2,T1,T2,l,m.
- cpy l m G L V1 V2 → cpy l m G L T1 T2 →
- cpy l m G L (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-.
-
-interpretation "context-sensitive extended ordinary substritution (term)"
- 'PSubst G L T1 l m T2 = (cpy l m G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_cpy_trans: ∀G,l,m. lsub_trans … (cpy l m G) (lsuby l m).
-#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
-[ //
-| #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12
- elim (lsuby_drop_trans_be … HL12 … HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
-| /4 width=1 by lsuby_succ, cpy_bind/
-| /3 width=1 by cpy_flat/
-]
-qed-.
-
-lemma cpy_refl: ∀G,T,L,l,m. ⦃G, L⦄ ⊢ T ▶[l, m] T.
-#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
-qed.
-
-(* Basic_1: was: subst1_ex *)
-lemma cpy_full: ∀I,G,K,V,T1,L,l. ⬇[l] L ≡ K.ⓑ{I}V →
- ∃∃T2,T. ⦃G, L⦄ ⊢ T1 ▶[l, 1] T2 & ⬆[l, 1] T ≡ T2.
-#I #G #K #V #T1 elim T1 -T1
-[ * #i #L #l #HLK
- /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
- elim (lt_or_eq_or_gt i l) #Hil
- /4 width=4 by lift_lref_ge_minus, lift_lref_lt, ex2_2_intro, ylt_inj, yle_inj/ (**) (* was /3 width=4/ without inj *)
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split … HVW i i)
- /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
-| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK
- elim (IHW1 … HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L.ⓑ{J}W1) (l+1)) -IHU1
- /3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/
- | elim (IHU1 … HLK) -IHU1 -HLK
- /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
- ]
-]
-qed-.
-
-lemma cpy_weak: ∀G,L,T1,T2,l1,m1. ⦃G, L⦄ ⊢ T1 ▶[l1, m1] T2 →
- ∀l2,m2. l2 ≤ l1 → l1 + m1 ≤ l2 + m2 →
- ⦃G, L⦄ ⊢ T1 ▶[l2, m2] T2.
-#G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 //
-[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
-| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
-| /3 width=1 by cpy_flat/
-]
-qed-.
-
-lemma cpy_weak_top: ∀G,L,T1,T2,l,m.
- ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[l, |L| - l] T2.
-#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m //
-[ #I #G #L #K #V #W #i #l #m #Hli #_ #HLK #HVW
- lapply (drop_fwd_length_lt2 … HLK)
- /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
-| #a #I #G #L #V1 #V2 normalize in match (|L.ⓑ{I}V2|); (**) (* |?| does not work *)
- <yplus_inj >yplus_SO2 >yminus_succ
- /2 width=1 by cpy_bind/
-| /2 width=1 by cpy_flat/
-]
-qed-.
-
-lemma cpy_weak_full: ∀G,L,T1,T2,l,m.
- ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ⦃G, L⦄ ⊢ T1 ▶[0, |L|] T2.
-#G #L #T1 #T2 #l #m #HT12
-lapply (cpy_weak … HT12 0 (l + m) ? ?) -HT12
-/2 width=2 by cpy_weak_top/
-qed-.
-
-lemma cpy_split_up: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀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 elim H -G -L -T1 -T2 -l -m
-[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
- elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
- /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
-| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
- elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
- >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
- /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
- elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
- /3 width=5 by ex2_intro, cpy_flat/
-]
-qed-.
-
-lemma cpy_split_down: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀i. i ≤ l + m →
- ∃∃T. ⦃G, L⦄ ⊢ T1 ▶[i, l+m-i] T & ⦃G, L⦄ ⊢ T ▶[l, i-l] T2.
-#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
-[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
- elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
- /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
-| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
- elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
- >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2
- /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
- elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
- /3 width=5 by ex2_intro, cpy_flat/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpy_fwd_up: ∀G,L,U1,U2,lt,mt. ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2 →
- ∀T1,l,m. ⬆[l, m] T1 ≡ U1 →
- l ≤ lt → l + m ≤ lt + mt →
- ∃∃T2. ⦃G, L⦄ ⊢ U1 ▶[l+m, lt+mt-(l+m)] U2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
-[ * #i #G #L #lt #mt #T1 #l #m #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt
- elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ]
- [ elim (ylt_yle_false … Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/
- | elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi
- elim (lift_split … HVW l (i-m+1) ? ? ?) [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hlim
- #T2 #_ >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by le_S, yle_inv_inj/ <minus_n_n <plus_n_O #H -Hmi
- @(ex2_intro … H) -H @(cpy_subst … HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HVW1) -V1 -IHW12 //
- elim (IHU12 … HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
- <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
- /3 width=2 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HVW1) -V1 -IHW12 // elim (IHU12 … HTU1) -T1 -IHU12
- /3 width=2 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-lemma cpy_fwd_tw: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ♯{T1} ≤ ♯{T2}.
-#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize
-/3 width=1 by monotonic_le_plus_l, le_plus/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact cpy_inv_atom1_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → ∀J. T1 = ⓪{J} →
- T2 = ⓪{J} ∨
- ∃∃I,K,V,i. l ≤ yinj i & i < l + m &
- ⬇[i] L ≡ K.ⓑ{I}V &
- ⬆[O, i+1] V ≡ T2 &
- J = LRef i.
-#G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m
-[ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/
-| #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
-| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
-]
-qed-.
-
-lemma cpy_inv_atom1: ∀I,G,L,T2,l,m. ⦃G, L⦄ ⊢ ⓪{I} ▶[l, m] T2 →
- T2 = ⓪{I} ∨
- ∃∃J,K,V,i. l ≤ yinj i & i < l + m &
- ⬇[i] L ≡ K.ⓑ{J}V &
- ⬆[O, i+1] V ≡ T2 &
- I = LRef i.
-/2 width=4 by cpy_inv_atom1_aux/ qed-.
-
-(* Basic_1: was: subst1_gen_sort *)
-lemma cpy_inv_sort1: ∀G,L,T2,k,l,m. ⦃G, L⦄ ⊢ ⋆k ▶[l, m] T2 → T2 = ⋆k.
-#G #L #T2 #k #l #m #H
-elim (cpy_inv_atom1 … H) -H //
-* #I #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was: subst1_gen_lref *)
-lemma cpy_inv_lref1: ∀G,L,T2,i,l,m. ⦃G, L⦄ ⊢ #i ▶[l, m] T2 →
- T2 = #i ∨
- ∃∃I,K,V. l ≤ i & i < l + m &
- ⬇[i] L ≡ K.ⓑ{I}V &
- ⬆[O, i+1] V ≡ T2.
-#G #L #T2 #i #l #m #H
-elim (cpy_inv_atom1 … H) -H /2 width=1 by or_introl/
-* #I #K #V #j #Hlj #Hjlm #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
-qed-.
-
-lemma cpy_inv_gref1: ∀G,L,T2,p,l,m. ⦃G, L⦄ ⊢ §p ▶[l, m] T2 → T2 = §p.
-#G #L #T2 #p #l #m #H
-elim (cpy_inv_atom1 … H) -H //
-* #I #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-fact cpy_inv_bind1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
- ∀a,I,V1,T1. U1 = ⓑ{a,I}V1.T1 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
- ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
- U2 = ⓑ{a,I}V2.T2.
-#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
-[ #I #G #L #l #m #b #J #W1 #U1 #H destruct
-| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct
-]
-qed-.
-
-lemma cpy_inv_bind1: ∀a,I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓑ{a,I} V1. T1 ▶[l, m] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
- ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ▶[⫯l, m] T2 &
- U2 = ⓑ{a,I}V2.T2.
-/2 width=3 by cpy_inv_bind1_aux/ qed-.
-
-fact cpy_inv_flat1_aux: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
- ∀I,V1,T1. U1 = ⓕ{I}V1.T1 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
- ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
- U2 = ⓕ{I}V2.T2.
-#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
-[ #I #G #L #l #m #J #W1 #U1 #H destruct
-| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma cpy_inv_flat1: ∀I,G,L,V1,T1,U2,l,m. ⦃G, L⦄ ⊢ ⓕ{I} V1. T1 ▶[l, m] U2 →
- ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ▶[l, m] V2 &
- ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 &
- U2 = ⓕ{I}V2.T2.
-/2 width=3 by cpy_inv_flat1_aux/ qed-.
-
-
-fact cpy_inv_refl_O2_aux: ∀G,L,T1,T2,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T2 → m = 0 → T1 = T2.
-#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
-[ //
-| #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct
- elim (ylt_yle_false … Hli) -Hli //
-| /3 width=1 by eq_f2/
-| /3 width=1 by eq_f2/
-]
-qed-.
-
-lemma cpy_inv_refl_O2: ∀G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▶[l, 0] T2 → T1 = T2.
-/2 width=6 by cpy_inv_refl_O2_aux/ qed-.
-
-(* Basic_1: was: subst1_gen_lift_eq *)
-lemma cpy_inv_lift1_eq: ∀G,T1,U1,l,m. ⬆[l, m] T1 ≡ U1 →
- ∀L,U2. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 → U1 = U2.
-#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up … HU12 … HTU1) -HU12 -HTU1
-/2 width=4 by cpy_inv_refl_O2/
-qed-.
-
-(* Basic_1: removed theorems 25:
- subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
- subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
- subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
- subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
- subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
- subst0_confluence_lift subst0_tlt
- subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpy_lift.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: subst1_confluence_eq *)
-theorem cpy_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.
-#G #L #T0 #T1 #l1 #m1 #H elim H -G -L -T0 -T1 -l1 -m1
-[ /2 width=3 by ex2_intro/
-| #I1 #G #L #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #T2 #l2 #m2 #H
- elim (cpy_inv_lref1 … H) -H
- [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
- | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
- lapply (drop_mono … HLK1 … HLK2) -HLK1 -HLK2 #H destruct
- >(lift_mono … HVT1 … HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
- ]
-| #a #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
- elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02) -IHV01 -HV02 #V #HV1 #HV2
- elim (IHT01 … HT02) -T0 #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V2) ?) -HT2
- /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
-| #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
- elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02) -V0
- elim (IHT01 … HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
-]
-qed-.
-
-(* Basic_1: was: subst1_confluence_neq *)
-theorem cpy_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.
-#G #L1 #T0 #T1 #l1 #m1 #H elim H -G -L1 -T0 -T1 -l1 -m1
-[ /2 width=3 by ex2_intro/
-| #I1 #G #L1 #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #L2 #T2 #l2 #m2 #H1 #H2
- elim (cpy_inv_lref1 … H1) -H1
- [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
- | -HLK1 -HVT1 * #I2 #K2 #V2 #Hl2 #Hlm2 #_ #_ elim H2 -H2 #Hlml [ -Hl1 -Hlm2 | -Hl2 -Hlm1 ]
- [ elim (ylt_yle_false … Hlm1) -Hlm1 /2 width=3 by yle_trans/
- | elim (ylt_yle_false … Hlm2) -Hlm2 /2 width=3 by yle_trans/
- ]
- ]
-| #a #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
- elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -IHV01 -HV02 #V #HV1 #HV2
- elim (IHT01 … HT02) -T0
- [ -H #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT1 (L2.ⓑ{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans … HT2 (L1.ⓑ{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
- | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
- ]
-| #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
- elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 … HV02 H) -V0
- elim (IHT01 … HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
-]
-qed-.
-
-(* Note: the constant 1 comes from cpy_subst *)
-(* Basic_1: was: subst1_trans *)
-theorem cpy_trans_ge: ∀G,L,T1,T0,l,m. ⦃G, L⦄ ⊢ T1 ▶[l, m] T0 →
- ∀T2. ⦃G, L⦄ ⊢ T0 ▶[l, 1] T2 → 1 ≤ m → ⦃G, L⦄ ⊢ T1 ▶[l, m] T2.
-#G #L #T1 #T0 #l #m #H elim H -G -L -T1 -T0 -l -m
-[ #I #G #L #l #m #T2 #H #Hm
- elim (cpy_inv_atom1 … H) -H
- [ #H destruct //
- | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct
- lapply (ylt_yle_trans … (l+m) … Hilm2)
- /2 width=5 by cpy_subst, monotonic_yle_plus_sn/
- ]
-| #I #G #L #K #V #V2 #i #l #m #Hli #Hilm #HLK #HVW #T2 #HVT2 #Hm
- lapply (cpy_weak … HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
- >yplus_inj #HVT2 <(cpy_inv_lift1_eq … HVW … HVT2) -HVT2 /2 width=5 by cpy_subst/
-| #a #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
- elim (cpy_inv_bind1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct
- lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- lapply (IHT10 … HT02 Hm) -T0 /3 width=1 by cpy_bind/
-| #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
- elim (cpy_inv_flat1 … H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
-]
-qed-.
-
-theorem cpy_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.
-#G #L #T1 #T0 #l1 #m1 #H elim H -G -L -T1 -T0 -l1 -m1
-[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i1 #l1 #m1 #Hli1 #Hilm1 #HLK #HVW #T2 #l2 #m2 #HWT2 #Hlm2l1
- lapply (yle_trans … Hlm2l1 … Hli1) -Hlm2l1 #Hlm2i1
- lapply (cpy_weak … HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1
- >yplus_inj #HWT2 <(cpy_inv_lift1_eq … HVW … HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
-| #a #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
- elim (cpy_inv_bind1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- lapply (lsuby_cpy_trans … HT02 (L.ⓑ{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- elim (IHV10 … HV02) -IHV10 -HV02 // #V
- elim (IHT10 … HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
- lapply (lsuby_cpy_trans … HT2 (L.ⓑ{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
-| #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
- elim (cpy_inv_flat1 … HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV10 … HV02) -V0 //
- elim (IHT10 … HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/drop_drop.ma".
-include "basic_2/substitution/cpy.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Properties on relocation *************************************************)
-
-(* Basic_1: was: subst1_lift_lt *)
-lemma cpy_lift_le: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
- ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
- ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
- lt + mt ≤ l → ⦃G, L⦄ ⊢ U1 ▶[lt, mt] U2.
-#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
-[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hlmtl
- lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
- lapply (lift_inv_lref1_lt … H … Hil) -H #H destruct
- elim (lift_trans_ge … HVW … HWU2) -W /2 width=1 by ylt_fwd_le_succ1/
- <yplus_inj >yplus_SO2 >yminus_succ2 #W #HVW #HWU2
- elim (drop_trans_le … HLK … HKV) -K /2 width=2 by ylt_fwd_le/ #X #HLK #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
-| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=7 by cpy_bind, drop_skip, yle_succ/
-| #G #I #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=7 by cpy_flat/
-]
-qed-.
-
-lemma cpy_lift_be: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
- ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
- ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
- lt ≤ l → l ≤ lt + mt → ⦃G, L⦄ ⊢ U1 ▶[lt, mt + m] U2.
-#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
-[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hltl #_
- elim (lift_inv_lref1 … H) -H * #Hil #H destruct
- [ -Hltl
- lapply (ylt_yle_trans … (lt+mt+m) … Hilmt) // -Hilmt #Hilmtm
- elim (lift_trans_ge … HVW … HWU2) -W <yplus_inj >yplus_SO2
- [2: >yplus_O1 /2 width=1 by ylt_fwd_le_succ1/ ] >yminus_succ2 #W #HVW #HWU2
- elim (drop_trans_le … HLK … HKV) -K /2 width=1 by ylt_fwd_le/ #X #HLK #H
- elim (drop_inv_skip2 … H) -H /2 width=1 by ylt_to_minus/ -Hil #K #Y #_ #HVY
- >(lift_mono … HVY … HVW) -V #H destruct /2 width=5 by cpy_subst/
- | -Hlti
- lapply (yle_trans … Hltl … Hil) -Hltl #Hlti
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ >plus_plus_comm_23 #HVU2
- lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil
- /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans/
- ]
-| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hltl #Hllmt
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=7 by cpy_bind, drop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=7 by cpy_flat/
-]
-qed-.
-
-(* Basic_1: was: subst1_lift_ge *)
-lemma cpy_lift_ge: ∀G,K,T1,T2,lt,mt. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 →
- ∀L,U1,U2,s,l,m. ⬇[s, l, m] L ≡ K →
- ⬆[l, m] T1 ≡ U1 → ⬆[l, m] T2 ≡ U2 →
- l ≤ lt → ⦃G, L⦄ ⊢ U1 ▶[lt+m, mt] U2.
-#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
-[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
- >(lift_mono … H1 … H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hllt
- lapply (yle_trans … Hllt … Hlti) -Hllt #Hil
- lapply (lift_inv_lref1_ge … H … Hil) -H #H destruct
- lapply (lift_trans_be … HVW … HWU2 ? ?) -W /2 width=1 by yle_succ_dx/ >plus_plus_comm_23 #HVU2
- lapply (drop_trans_ge_comm … HLK … HKV ?) -K // -Hil
- /3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
-| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
- elim (lift_inv_bind1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=6 by cpy_bind, drop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
- elim (lift_inv_flat1 … H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 … H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=6 by cpy_flat/
-]
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-(* Basic_1: was: subst1_gen_lift_lt *)
-lemma cpy_inv_lift1_le: ∀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 →
- lt + mt ≤ l →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
-[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmtl
- lapply (ylt_yle_trans … Hlmtl … Hilmt) -Hlmtl #Hil
- lapply (lift_inv_lref2_lt … H … Hil) -H #H destruct
- elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW) -V // >minus_plus <plus_minus_m_m //
- <yminus_succ2 <yplus_inj >yplus_SO2 >ymax_pre_sn /2 width=1 by ylt_fwd_le_succ1/ -Hil
- /3 width=5 by cpy_subst, ex2_intro/
-| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 … HTU1) -IHU12 -HTU1
- /3 width=6 by cpy_bind, yle_succ, drop_skip, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -W1 //
- elim (IHU12 … HLK … HTU1) -U1 -HLK
- /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-lemma cpy_inv_lift1_be: ∀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 →
- lt ≤ l → l + m ≤ lt + mt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, mt-m] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
-[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #KV #V #W #i #x #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hltl #Hlmlmt
- elim (yle_inv_inj2 … Hlti) -Hlti #lt #Hlti #H destruct
- lapply (yle_fwd_plus_yge … Hltl Hlmlmt) #Hmmt
- elim (lift_inv_lref2 … H) -H * #Hil #H destruct [ -Hltl -Hilmt | -Hlti -Hlmlmt ]
- [ lapply (ylt_yle_trans i l (lt+(mt-m)) ? ?) //
- [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm
- elim (drop_conf_lt … HLK … HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le … HUV … HVW) -V //
- <yminus_succ2 <yplus_inj >yplus_SO2 >ymax_pre_sn /2 width=1 by ylt_fwd_le_succ1/ -Hil
- /4 width=5 by cpy_subst, ex2_intro, yle_inj/
- | elim (yle_inv_plus_inj2 … Hil) #Hlim #Hmi
- lapply (yle_inv_inj … Hmi) -Hmi #Hmi
- lapply (yle_trans … Hltl (i-m) ?) // -Hltl #Hltim
- lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /2 width=1 by yle_succ_dx, le_S_S/ ] -Hil -Hlim
- #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
- @(ex2_intro … H) @(cpy_subst … HKV HV1) // (**) (* explicit constructor *)
- >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 … HTU1) -U1
- /3 width=6 by cpy_bind, drop_skip, lift_bind, yle_succ, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -W1 //
- elim (IHU12 … HLK … HTU1) -U1 -HLK //
- /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-(* Basic_1: was: subst1_gen_lift_ge *)
-lemma cpy_inv_lift1_ge: ∀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 + m ≤ lt →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt-m, mt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
-[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
- [ lapply (lift_inv_sort2 … H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 … H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 … H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmlt
- lapply (yle_trans … Hlmlt … Hlti) #Hlmi
- elim (yle_inv_plus_inj2 … Hlmlt) -Hlmlt #_ #Hmlt
- elim (yle_inv_plus_inj2 … Hlmi) #Hlim #Hmi
- lapply (yle_inv_inj … Hmi) -Hmi #Hmi
- lapply (lift_inv_lref2_ge … H ?) -H // #H destruct
- lapply (drop_conf_ge … HLK … HLKV ?) -L // #HKV
- elim (lift_split … HVW l (i-m+1)) -HVW [2,3,4: /3 width=1 by yle_succ, yle_pred_sn, le_S_S/ ] -Hlmi -Hlim
- #V0 #HV10 >plus_minus // <minus_minus /3 width=1 by le_S/ <minus_n_n <plus_n_O #H
- @(ex2_intro … H) @(cpy_subst … HKV HV10) (**) (* explicit constructor *)
- [ /2 width=1 by monotonic_yle_minus_dx/
- | <yminus_inj <yplus_minus_comm_inj // /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
- elim (lift_inv_bind2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (yle_inv_plus_inj2 … Hlmtl) #_ #Hmlt
- elim (IHW12 … HLK … HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 … HTU1) -U1 [4: @drop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
- >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
- elim (lift_inv_flat2 … H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 … HLK … HVW1) -W1 //
- elim (IHU12 … HLK … HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-(* Advanced inversion lemmas on relocation ***********************************)
-
-lemma cpy_inv_lift1_ge_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 (cpy_split_up … HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
-lapply (cpy_weak … HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
-lapply (cpy_inv_lift1_eq … HTU1 … HU1) -HU1 #HU1 destruct
-elim (cpy_inv_lift1_ge … HU2 … HLK … HTU1) -U -L /2 width=3 by ex2_intro/
-qed-.
-
-lemma cpy_inv_lift1_be_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 →
- lt ≤ l → lt + mt ≤ l + m →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l-lt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm
-lapply (cpy_weak … HU12 lt (l+m-lt) ? ?) -HU12 //
-[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hlmtlm #HU12
-elim (cpy_inv_lift1_be … HU12 … HLK … HTU1) -U1 -L /2 width=3 by ex2_intro/
-qed-.
-
-lemma cpy_inv_lift1_le_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 →
- lt ≤ l → l ≤ lt + mt → lt + mt ≤ l + m →
- ∃∃T2. ⦃G, K⦄ ⊢ T1 ▶[lt, l - lt] T2 & ⬆[l, m] T2 ≡ U2.
-#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm
-elim (cpy_split_up … HU12 l) -HU12 // #U #HU1 #HU2
-elim (cpy_inv_lift1_le … HU1 … HLK … HTU1) -U1
-[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU
-lapply (cpy_weak … HU2 l m ? ?) -HU2 //
-[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2
-lapply (cpy_inv_lift1_eq … HTU … HU2) -L #H destruct /2 width=3 by ex2_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lift_neg.ma".
-include "basic_2/substitution/lift_lift.ma".
-include "basic_2/substitution/cpy.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Inversion lemmas on negated relocation ***********************************)
-
-lemma cpy_fwd_nlift2_ge: ∀G,L,U1,U2,l,m. ⦃G, L⦄ ⊢ U1 ▶[l, m] U2 →
- ∀i. l ≤ i → (∀T2. ⬆[i, 1] T2 ≡ U2 → ⊥) →
- (∀T1. ⬆[i, 1] T1 ≡ U1 → ⊥) ∨
- ∃∃I,K,W,j. l ≤ yinj j & j < i & ⬇[j]L ≡ K.ⓑ{I}W &
- (∀V. ⬆[⫰(i-j), 1] V ≡ W → ⊥) & (∀T1. ⬆[j, 1] T1 ≡ U1 → ⊥).
-#G #L #U1 #U2 #l #m #H elim H -G -L -U1 -U2 -l -m
-[ /3 width=2 by or_introl/
-| #I #G #L #K #V #W #j #l #m #Hlj #Hjlm #HLK #HVW #i #Hli #HnW
- elim (ylt_split j i) #Hij
- [ @or_intror @(ex5_4_intro … HLK) // -HLK
- [ #X #HXV elim (lift_trans_le … HXV … HVW ?) -V // #Y #HXY
- <yminus_succ2 <yplus_inj >yplus_SO2 >ymax_pre_sn /2 width=2 by ylt_fwd_le_succ1/
- | -HnW /3 width=7 by lift_inv_lref2_be, ylt_inj/
- ]
- | elim (lift_split … HVW i j) -HVW //
- #X #_ #H elim HnW -HnW //
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_bind … H) -H
- [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
- [ /4 width=9 by nlift_bind_sn, or_introl/
- | * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/
- ]
- | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/
- [ /4 width=9 by nlift_bind_dx, or_introl/
- | * #J #K #W #j #Hlj elim (yle_inv_succ1 … Hlj) -Hlj #Hlj #Hj
- <Hj >yminus_succ #Hji #HLK #HnW
- lapply (drop_inv_drop1_lt … HLK ?) /2 width=1 by ylt_O/ -HLK #HLK
- <yminus_SO2 in Hlj; #Hlj #H4
- @or_intror @(ex5_4_intro … HLK) (**) (* explicit constructors *)
- /3 width=9 by nlift_bind_dx, ylt_pred, ylt_inj/
- ]
- ]
-| #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_flat … H) -H
- [ #HnW2 elim (IHW12 … HnW2) -IHW12 -HnW2 -IHU12 //
- [ /4 width=9 by nlift_flat_sn, or_introl/
- | * /5 width=9 by nlift_flat_sn, ex5_4_intro, or_intror/
- ]
- | #HnU2 elim (IHU12 … HnU2) -IHU12 -HnU2 -IHW12 //
- [ /4 width=9 by nlift_flat_dx, or_introl/
- | * /5 width=9 by nlift_flat_dx, ex5_4_intro, or_intror/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/lib/lstar.ma".
-include "basic_2/notation/relations/rdrop_5.ma".
-include "basic_2/notation/relations/rdrop_4.ma".
-include "basic_2/notation/relations/rdrop_3.ma".
-include "basic_2/grammar/lenv_length.ma".
-include "basic_2/grammar/cl_restricted_weight.ma".
-include "basic_2/substitution/lift.ma".
-
-(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
-
-(* Basic_1: includes: drop_skip_bind *)
-inductive drop (s:bool): relation4 ynat nat lenv lenv ≝
-| drop_atom: ∀l,m. (s = Ⓕ → m = 0) → drop s l m (⋆) (⋆)
-| drop_pair: ∀I,L,V. drop s 0 0 (L.ⓑ{I}V) (L.ⓑ{I}V)
-| drop_drop: ∀I,L1,L2,V,m. drop s 0 m L1 L2 → drop s 0 (m+1) (L1.ⓑ{I}V) L2
-| drop_skip: ∀I,L1,L2,V1,V2,l,m.
- drop s l m L1 L2 → ⬆[l, m] V2 ≡ V1 →
- drop s (⫯l) m (L1.ⓑ{I}V1) (L2.ⓑ{I}V2)
-.
-
-interpretation
- "basic slicing (local environment) abstract"
- 'RDrop s l m L1 L2 = (drop s l m L1 L2).
-(*
-interpretation
- "basic slicing (local environment) general"
- 'RDrop d e L1 L2 = (drop true d e L1 L2).
-*)
-interpretation
- "basic slicing (local environment) lget"
- 'RDrop m L1 L2 = (drop false (yinj O) m L1 L2).
-
-definition d_liftable: predicate (lenv → relation term) ≝
- λR. ∀K,T1,T2. R K T1 T2 → ∀L,s,l,m. ⬇[s, l, m] L ≡ K →
- ∀U1. ⬆[l, m] T1 ≡ U1 → ∀U2. ⬆[l, m] T2 ≡ U2 → R L U1 U2.
-
-definition d_deliftable_sn: predicate (lenv → relation term) ≝
- λR. ∀L,U1,U2. R L U1 U2 → ∀K,s,l,m. ⬇[s, l, m] L ≡ K →
- ∀T1. ⬆[l, m] T1 ≡ U1 →
- ∃∃T2. ⬆[l, m] T2 ≡ U2 & R K T1 T2.
-
-definition dropable_sn: predicate (relation lenv) ≝
- λR. ∀L1,K1,s,l,m. ⬇[s, l, m] L1 ≡ K1 → ∀L2. R L1 L2 →
- ∃∃K2. R K1 K2 & ⬇[s, l, m] L2 ≡ K2.
-
-definition dropable_dx: predicate (relation lenv) ≝
- λR. ∀L1,L2. R L1 L2 → ∀K2,s,m. ⬇[s, 0, m] L2 ≡ K2 →
- ∃∃K1. ⬇[s, 0, m] L1 ≡ K1 & R K1 K2.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact drop_inv_atom1_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → L1 = ⋆ →
- L2 = ⋆ ∧ (s = Ⓕ → m = 0).
-#L1 #L2 #s #l #m * -L1 -L2 -l -m
-[ /3 width=1 by conj/
-| #I #L #V #H destruct
-| #I #L1 #L2 #V #m #_ #H destruct
-| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #H destruct
-]
-qed-.
-
-(* Basic_1: was: drop_gen_sort *)
-lemma drop_inv_atom1: ∀L2,s,l,m. ⬇[s, l, m] ⋆ ≡ L2 → L2 = ⋆ ∧ (s = Ⓕ → m = 0).
-/2 width=4 by drop_inv_atom1_aux/ qed-.
-
-fact drop_inv_O1_pair1_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → l = yinj 0 →
- ∀K,I,V. L1 = K.ⓑ{I}V →
- (m = 0 ∧ L2 = K.ⓑ{I}V) ∨
- (0 < m ∧ ⬇[s, l, m-1] K ≡ L2).
-#L1 #L2 #s #l #m * -L1 -L2 -l -m
-[ #l #m #_ #_ #K #J #W #H destruct
-| #I #L #V #_ #K #J #W #HX destruct /3 width=1 by or_introl, conj/
-| #I #L1 #L2 #V #m #HL12 #_ #K #J #W #H destruct /3 width=1 by or_intror, conj/
-| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #H elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma drop_inv_O1_pair1: ∀I,K,L2,V,s,m. ⬇[s, yinj 0, m] K. ⓑ{I} V ≡ L2 →
- (m = 0 ∧ L2 = K.ⓑ{I}V) ∨
- (0 < m ∧ ⬇[s, yinj 0, m-1] K ≡ L2).
-/2 width=3 by drop_inv_O1_pair1_aux/ qed-.
-
-lemma drop_inv_pair1: ∀I,K,L2,V,s. ⬇[s, 0, 0] K.ⓑ{I}V ≡ L2 → L2 = K.ⓑ{I}V.
-#I #K #L2 #V #s #H
-elim (drop_inv_O1_pair1 … H) -H * // #H destruct
-elim (lt_refl_false … H)
-qed-.
-
-(* Basic_1: was: drop_gen_drop *)
-lemma drop_inv_drop1_lt: ∀I,K,L2,V,s,m.
- ⬇[s, yinj 0, m] K.ⓑ{I}V ≡ L2 → 0 < m → ⬇[s, yinj 0, m-1] K ≡ L2.
-#I #K #L2 #V #s #m #H #Hm
-elim (drop_inv_O1_pair1 … H) -H * // #H destruct
-elim (lt_refl_false … Hm)
-qed-.
-
-lemma drop_inv_drop1: ∀I,K,L2,V,s,m.
- ⬇[s, 0, m+1] K.ⓑ{I}V ≡ L2 → ⬇[s, 0, m] K ≡ L2.
-#I #K #L2 #V #s #m #H lapply (drop_inv_drop1_lt … H ?) -H //
-qed-.
-
-fact drop_inv_skip1_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → 0 < l →
- ∀I,K1,V1. L1 = K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⬇[s, ⫰l, m] K1 ≡ K2 &
- ⬆[⫰l, m] V2 ≡ V1 &
- L2 = K2.ⓑ{I}V2.
-#L1 #L2 #s #l #m * -L1 -L2 -l -m
-[ #l #m #_ #_ #J #K1 #W1 #H destruct
-| #I #L #V #H elim (ylt_yle_false … H) -H //
-| #I #L1 #L2 #V #m #_ #H elim (ylt_yle_false … H) -H //
-| #I #L1 #L2 #V1 #V2 #l #m #HL12 #HV21 #_ #J #K1 #W1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_1: was: drop_gen_skip_l *)
-lemma drop_inv_skip1: ∀I,K1,V1,L2,s,l,m. ⬇[s, l, m] K1.ⓑ{I}V1 ≡ L2 → 0 < l →
- ∃∃K2,V2. ⬇[s, ⫰l, m] K1 ≡ K2 &
- ⬆[⫰l, m] V2 ≡ V1 &
- L2 = K2.ⓑ{I}V2.
-/2 width=3 by drop_inv_skip1_aux/ qed-.
-
-lemma drop_inv_O1_pair2: ∀I,K,V,s,m,L1. ⬇[s, yinj 0, m] L1 ≡ K.ⓑ{I}V →
- (m = 0 ∧ L1 = K.ⓑ{I}V) ∨
- ∃∃I1,K1,V1. ⬇[s, yinj 0, m-1] K1 ≡ K.ⓑ{I}V & L1 = K1.ⓑ{I1}V1 & 0 < m.
-#I #K #V #s #m *
-[ #H elim (drop_inv_atom1 … H) -H #H destruct
-| #L1 #I1 #V1 #H
- elim (drop_inv_O1_pair1 … H) -H *
- [ #H1 #H2 destruct /3 width=1 by or_introl, conj/
- | /3 width=5 by ex3_3_intro, or_intror/
- ]
-]
-qed-.
-
-fact drop_inv_skip2_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → 0 < l →
- ∀I,K2,V2. L2 = K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇[s, ⫰l, m] K1 ≡ K2 &
- ⬆[⫰l, m] V2 ≡ V1 &
- L1 = K1.ⓑ{I}V1.
-#L1 #L2 #s #l #m * -L1 -L2 -l -m
-[ #l #m #_ #_ #J #K2 #W2 #H destruct
-| #I #L #V #H elim (ylt_yle_false … H) -H //
-| #I #L1 #L2 #V #m #_ #H elim (ylt_yle_false … H) -H //
-| #I #L1 #L2 #V1 #V2 #l #m #HL12 #HV21 #_ #J #K2 #W2 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_1: was: drop_gen_skip_r *)
-lemma drop_inv_skip2: ∀I,L1,K2,V2,s,l,m. ⬇[s, l, m] L1 ≡ K2.ⓑ{I}V2 → 0 < l →
- ∃∃K1,V1. ⬇[s, ⫰l, m] K1 ≡ K2 & ⬆[⫰l, m] V2 ≡ V1 &
- L1 = K1.ⓑ{I}V1.
-/2 width=3 by drop_inv_skip2_aux/ qed-.
-
-lemma drop_inv_O1_gt: ∀L,K,m,s. ⬇[s, yinj 0, m] L ≡ K → |L| < m →
- s = Ⓣ ∧ K = ⋆.
-#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H normalize in ⊢ (?%?→?); #H1m
-[ elim (drop_inv_atom1 … H) -H elim s -s /2 width=1 by conj/
- #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (lt_zero_false … H1m)
-| elim (drop_inv_O1_pair1 … H) -H * #H2m #HLK destruct
- [ elim (lt_zero_false … H1m)
- | elim (IHL … HLK) -IHL -HLK /2 width=1 by lt_plus_to_minus_r, conj/
- ]
-]
-qed-.
-
-lemma drop_inv_Y1: ∀L,K,m,s. ⬇[s, ∞, m] L ≡ K →
- L = K ∧ (s = Ⓕ → m = 0).
-#L elim L -L
-[ #Y #m #s #H elim (drop_inv_atom1 … H) -H /3 width=1 by conj/
-| #L #I #V #IHL #Y #m #s #H elim (drop_inv_skip1 … H) -H //
- #K #W #HLK #HWV #H destruct
- lapply (lift_inv_Y1 … HWV) -HWV #H destruct
- elim (IHL … HLK) -IHL -HLK /3 width=1 by conj/
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma drop_refl_atom_O2: ∀s,l. ⬇[s, l, O] ⋆ ≡ ⋆.
-/2 width=1 by drop_atom/ qed.
-
-lemma drop_Y1: ∀m,s. (s = Ⓕ → m = 0) → ∀L. ⬇[s, ∞, m] L ≡ L.
-#m #s #H #L elim L -L /3 width=3 by drop_atom, drop_skip/
-qed.
-
-(* Basic_1: was by definition: drop_refl *)
-lemma drop_refl: ∀L,l,s. ⬇[s, l, 0] L ≡ L.
-#L elim L -L //
-#L #I #V #IHL #x #s elim (ynat_cases x)
-[ #H destruct //
-| * #l #H destruct /2 width=1 by drop_skip/
-]
-qed.
-
-lemma drop_drop_lt: ∀I,L1,L2,V,s,m.
- ⬇[s, yinj 0, m-1] L1 ≡ L2 → 0 < m → ⬇[s, yinj 0, m] L1.ⓑ{I}V ≡ L2.
-#I #L1 #L2 #V #s #m #HL12 #Hm >(plus_minus_m_m m 1) /2 width=1 by drop_drop/
-qed.
-
-lemma drop_skip_lt: ∀I,L1,L2,V1,V2,s,l,m.
- ⬇[s, ⫰l, m] L1 ≡ L2 → ⬆[⫰l, m] V2 ≡ V1 → 0 < l →
- ⬇[s, l, m] L1. ⓑ{I} V1 ≡ L2.ⓑ{I}V2.
-#I #L1 #L2 #V1 #V2 #s #l #m #HL12 #HV21 #Hl <(ylt_inv_O1 … Hl) -Hl
-/2 width=1 by drop_skip/
-qed.
-
-lemma drop_O1_le: ∀s,m,L. m ≤ |L| → ∃K. ⬇[s, 0, m] L ≡ K.
-#s #m @(nat_ind_plus … m) -m /2 width=2 by ex_intro/
-#m #IHm *
-[ #H elim (le_plus_xSy_O_false … H)
-| #L #I #V normalize #H elim (IHm L) -IHm /3 width=2 by drop_drop, monotonic_pred, ex_intro/
-]
-qed-.
-
-lemma drop_O1_lt: ∀s,L,m. m < |L| → ∃∃I,K,V. ⬇[s, 0, m] L ≡ K.ⓑ{I}V.
-#s #L elim L -L
-[ #m #H elim (lt_zero_false … H)
-| #L #I #V #IHL #m @(nat_ind_plus … m) -m /2 width=4 by drop_pair, ex1_3_intro/
- #m #_ normalize #H elim (IHL m) -IHL /3 width=4 by drop_drop, lt_plus_to_minus_r, lt_plus_to_lt_l, ex1_3_intro/
-]
-qed-.
-
-lemma drop_O1_pair: ∀L,K,m,s. ⬇[s, yinj 0, m] L ≡ K → m ≤ |L| → ∀I,V.
- ∃∃J,W. ⬇[s, yinj 0, m] L.ⓑ{I}V ≡ K.ⓑ{J}W.
-#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H normalize #Hm #I #V
-[ elim (drop_inv_atom1 … H) -H #H <(le_n_O_to_eq … Hm) -m
- #Hs destruct /2 width=3 by ex1_2_intro/
-| elim (drop_inv_O1_pair1 … H) -H * #Hm #HLK destruct /2 width=3 by ex1_2_intro/
- elim (IHL … HLK … Z X) -IHL -HLK
- /3 width=3 by drop_drop_lt, le_plus_to_minus, ex1_2_intro/
-]
-qed-.
-
-lemma drop_O1_ge: ∀L,m. |L| ≤ m → ⬇[Ⓣ, 0, m] L ≡ ⋆.
-#L elim L -L [ #m #_ @drop_atom #H destruct ]
-#L #I #V #IHL #m @(nat_ind_plus … m) -m [ #H elim (le_plus_xSy_O_false … H) ]
-normalize /4 width=1 by drop_drop, monotonic_pred/
-qed.
-
-lemma drop_O1_eq: ∀L,s. ⬇[s, 0, |L|] L ≡ ⋆.
-#L elim L -L /2 width=1 by drop_drop, drop_atom/
-qed.
-
-lemma drop_split: ∀L1,L2,l,m2,s. ⬇[s, l, m2] L1 ≡ L2 → ∀m1. m1 ≤ m2 →
- ∃∃L. ⬇[s, l, m2 - m1] L1 ≡ L & ⬇[s, l, m1] L ≡ L2.
-#L1 #L2 #l #m2 #s #H elim H -L1 -L2 -l -m2
-[ #l #m2 #Hs #m1 #Hm12 @(ex2_intro … (⋆))
- @drop_atom #H lapply (Hs H) -s #H destruct /2 width=1 by le_n_O_to_eq/
-| #I #L1 #V #m1 #Hm1 lapply (le_n_O_to_eq … Hm1) -Hm1
- #H destruct /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #m2 #HL12 #IHL12 #m1 @(nat_ind_plus … m1) -m1
- [ /3 width=3 by drop_drop, ex2_intro/
- | -HL12 #m1 #_ #Hm12 lapply (le_plus_to_le_r … Hm12) -Hm12
- #Hm12 elim (IHL12 … Hm12) -IHL12 >minus_plus_plus_l
- #L #HL1 #HL2 elim (lt_or_ge (|L1|) (m2-m1)) #H0
- [ elim (drop_inv_O1_gt … HL1 H0) -HL1 #H1 #H2 destruct
- elim (drop_inv_atom1 … HL2) -HL2 #H #_ destruct
- @(ex2_intro … (⋆)) [ @drop_O1_ge normalize // ]
- @drop_atom #H destruct
- | elim (drop_O1_pair … HL1 H0 I V) -HL1 -H0 /3 width=5 by drop_drop, ex2_intro/
- ]
- ]
-| #I #L1 #L2 #V1 #V2 #l #m2 #_ #HV21 #IHL12 #m1 #Hm12 elim (IHL12 … Hm12) -IHL12
- #L #HL1 #HL2 elim (lift_split … HV21 l m1) -HV21 /3 width=5 by drop_skip, ex2_intro/
-]
-qed-.
-
-lemma drop_FT: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → ⬇[Ⓣ, l, m] L1 ≡ L2.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
-/3 width=1 by drop_atom, drop_drop, drop_skip/
-qed.
-
-lemma drop_gen: ∀L1,L2,s,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → ⬇[s, l, m] L1 ≡ L2.
-#L1 #L2 * /2 width=1 by drop_FT/
-qed-.
-
-lemma drop_T: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → ⬇[Ⓣ, l, m] L1 ≡ L2.
-#L1 #L2 * /2 width=1 by drop_FT/
-qed-.
-
-lemma d_liftable_LTC: ∀R. d_liftable R → d_liftable (LTC … R).
-#R #HR #K #T1 #T2 #H elim H -T2
-[ /3 width=10 by inj/
-| #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #HTU1 #U2 #HTU2
- elim (lift_total T l m) /4 width=12 by step/
-]
-qed-.
-
-lemma d_deliftable_sn_LTC: ∀R. d_deliftable_sn R → d_deliftable_sn (LTC … R).
-#R #HR #L #U1 #U2 #H elim H -U2
-[ #U2 #HU12 #K #s #l #m #HLK #T1 #HTU1
- elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3 by inj, ex2_intro/
-| #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #T1 #HTU1
- elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
- elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by step, ex2_intro/
-]
-qed-.
-
-lemma dropable_sn_TC: ∀R. dropable_sn R → dropable_sn (TC … R).
-#R #HR #L1 #K1 #s #l #m #HLK1 #L2 #H elim H -L2
-[ #L2 #HL12 elim (HR … HLK1 … HL12) -HR -L1
- /3 width=3 by inj, ex2_intro/
-| #L #L2 #_ #HL2 * #K #HK1 #HLK elim (HR … HLK … HL2) -HR -L
- /3 width=3 by step, ex2_intro/
-]
-qed-.
-
-lemma dropable_dx_TC: ∀R. dropable_dx R → dropable_dx (TC … R).
-#R #HR #L1 #L2 #H elim H -L2
-[ #L2 #HL12 #K2 #s #m #HLK2 elim (HR … HL12 … HLK2) -HR -L2
- /3 width=3 by inj, ex2_intro/
-| #L #L2 #_ #HL2 #IHL1 #K2 #s #m #HLK2 elim (HR … HL2 … HLK2) -HR -L2
- #K #HLK #HK2 elim (IHL1 … HLK) -L
- /3 width=5 by step, ex2_intro/
-]
-qed-.
-
-lemma d_deliftable_sn_llstar: ∀R. d_deliftable_sn R →
- ∀d. d_deliftable_sn (llstar … R d).
-#R #HR #d #L #U1 #U2 #H @(lstar_ind_r … d U2 H) -d -U2
-[ /2 width=3 by lstar_O, ex2_intro/
-| #d #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #T1 #HTU1
- elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
- elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by lstar_dx, ex2_intro/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-(* Basic_1: was: drop_S *)
-lemma drop_fwd_drop2: ∀L1,I2,K2,V2,s,m. ⬇[s, O, m] L1 ≡ K2. ⓑ{I2} V2 →
- ⬇[s, O, m + 1] L1 ≡ K2.
-#L1 elim L1 -L1
-[ #I2 #K2 #V2 #s #m #H lapply (drop_inv_atom1 … H) -H * #H destruct
-| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #s #m #H
- elim (drop_inv_O1_pair1 … H) -H * #Hm #H
- [ -IHL1 destruct /2 width=1 by drop_drop/
- | @drop_drop >(plus_minus_m_m m 1) /2 width=3 by/
- ]
-]
-qed-.
-
-lemma drop_fwd_length_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → |L1| ≤ l → |L2| = |L1|.
-#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m //
-[ #I #L1 #L2 #V #m #_ #_ #H elim (ylt_yle_false … H) -H normalize //
-| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IH <yplus_inj >yplus_SO2 #H
- lapply (yle_inv_succ … H) -H #H normalize /3 width=1 by eq_f2/
-]
-qed-.
-
-lemma drop_fwd_length_le_le: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → l ≤ |L1| → m ≤ |L1| - l → |L2| = |L1| - m.
-#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m //
-[ #I #L1 #L2 #V #m #_ <yplus_inj normalize #IH #_
- >minus_plus_plus_l <yplus_inj >yplus_SO2 /3 width=1 by yle_inv_succ/
-| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IHL12 <yplus_inj >yplus_SO2 #H
- lapply (yle_inv_succ … H) -H #Hl1 >yminus_succ #Hml1 normalize
- <plus_minus /3 width=3 by yle_trans, yle_inv_inj, eq_f2/
-]
-qed-.
-
-lemma drop_fwd_length_le_ge: ∀L1,L2,l,m,s. ⬇[s, l, m] L1 ≡ L2 → l ≤ |L1| → |L1| - l ≤ m → yinj (|L2|) = l.
-#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m
-[ #l #m #_ #H #_ normalize /2 width=1 by yle_inv_O2/
-| #I #L #V #_ normalize <yplus_inj #H elim (ylt_yle_false … H) -H //
-| #I #L1 #L2 #V #m #_ >yminus_inj >yminus_inj <minus_n_O <minus_n_O #IH #_ normalize
- /3 width=2 by yle_inv_monotonic_plus_dx/
-| #I #L1 #L2 #V1 #v2 #l #m #_ #_ #IH
- <yplus_inj <yplus_inj >yplus_SO2 >yplus_SO2 >yminus_succ
- /4 width=1 by yle_inv_succ, eq_f/
-]
-qed-.
-
-lemma drop_fwd_length: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L1| = |L2| + m.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m // normalize /2 width=1 by/
-qed-.
-
-lemma drop_fwd_length_minus2: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L2| = |L1| - m.
-#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H /2 width=1 by plus_minus, le_n/
-qed-.
-
-lemma drop_fwd_length_minus4: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → m = |L1| - |L2|.
-#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H //
-qed-.
-
-lemma drop_fwd_length_le2: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → m ≤ |L1|.
-#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H //
-qed-.
-
-lemma drop_fwd_length_le4: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L2| ≤ |L1|.
-#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H //
-qed-.
-
-lemma drop_fwd_length_lt2: ∀L1,I2,K2,V2,l,m.
- ⬇[Ⓕ, l, m] L1 ≡ K2. ⓑ{I2} V2 → m < |L1|.
-#L1 #I2 #K2 #V2 #l #m #H
-lapply (drop_fwd_length … H) normalize in ⊢ (%→?); -I2 -V2 //
-qed-.
-
-lemma drop_fwd_length_lt4: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → 0 < m → |L2| < |L1|.
-#L1 #L2 #l #m #H lapply (drop_fwd_length … H) -H /2 width=1 by lt_minus_to_plus_r/
-qed-.
-
-lemma drop_fwd_length_eq1: ∀L1,L2,K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
- |L1| = |L2| → |K1| = |K2|.
-#L1 #L2 #K1 #K2 #l #m #HLK1 #HLK2 #HL12
-lapply (drop_fwd_length … HLK1) -HLK1
-lapply (drop_fwd_length … HLK2) -HLK2
-/2 width=2 by injective_plus_r/
-qed-.
-
-lemma drop_fwd_length_eq2: ∀L1,L2,K1,K2,l,m. ⬇[Ⓕ, l, m] L1 ≡ K1 → ⬇[Ⓕ, l, m] L2 ≡ K2 →
- |K1| = |K2| → |L1| = |L2|.
-#L1 #L2 #K1 #K2 #l #m #HLK1 #HLK2 #HL12
-lapply (drop_fwd_length … HLK1) -HLK1
-lapply (drop_fwd_length … HLK2) -HLK2 //
-qed-.
-
-lemma drop_fwd_lw: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → ♯{L2} ≤ ♯{L1}.
-#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m // normalize
-[ /2 width=3 by transitive_le/
-| #I #L1 #L2 #V1 #V2 #l #m #_ #HV21 #IHL12
- >(lift_fwd_tw … HV21) -HV21 /2 width=1 by monotonic_le_plus_l/
-]
-qed-.
-
-lemma drop_fwd_lw_lt: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → 0 < m → ♯{L2} < ♯{L1}.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
-[ #l #m #H >H -H //
-| #I #L #V #H elim (lt_refl_false … H)
-| #I #L1 #L2 #V #m #HL12 #_ #_
- lapply (drop_fwd_lw … HL12) -HL12 #HL12
- @(le_to_lt_to_lt … HL12) -HL12 //
-| #I #L1 #L2 #V1 #V2 #l #m #_ #HV21 #IHL12 #H normalize in ⊢ (?%%); -I
- >(lift_fwd_tw … HV21) -V2 /3 by lt_minus_to_plus/
-]
-qed-.
-
-lemma drop_fwd_rfw: ∀I,L,K,V,i. ⬇[i] L ≡ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
-#I #L #K #V #i #HLK lapply (drop_fwd_lw … HLK) -HLK
-normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-fact drop_inv_O2_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 → m = 0 → L1 = L2.
-#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m
-[ //
-| //
-| #I #L1 #L2 #V #m #_ #_ >commutative_plus normalize #H destruct
-| #I #L1 #L2 #V1 #V2 #l #m #_ #HV21 #IHL12 #H
- >(IHL12 H) -L1 >(lift_inv_O2_aux … HV21 … H) -V2 -l -m //
-]
-qed-.
-
-(* Basic_1: was: drop_gen_refl *)
-lemma drop_inv_O2: ∀L1,L2,s,l. ⬇[s, l, 0] L1 ≡ L2 → L1 = L2.
-/2 width=5 by drop_inv_O2_aux/ qed-.
-
-lemma drop_inv_length_eq: ∀L1,L2,l,m. ⬇[Ⓕ, l, m] L1 ≡ L2 → |L1| = |L2| → m = 0.
-#L1 #L2 #l #m #H #HL12 lapply (drop_fwd_length_minus4 … H) //
-qed-.
-
-lemma drop_inv_refl: ∀L,l,m. ⬇[Ⓕ, l, m] L ≡ L → m = 0.
-/2 width=5 by drop_inv_length_eq/ qed-.
-
-fact drop_inv_FT_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 →
- ∀I,K,V. L2 = K.ⓑ{I}V → s = Ⓣ → l = 0 →
- ⬇[Ⓕ, l, m] L1 ≡ K.ⓑ{I}V.
-#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m
-[ #l #m #_ #J #K #W #H destruct
-| #I #L #V #J #K #W #H destruct //
-| #I #L1 #L2 #V #m #_ #IHL12 #J #K #W #H1 #H2 destruct
- /3 width=1 by drop_drop/
-| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #_ #J #K #W #_ #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma drop_inv_FT: ∀I,L,K,V,m. ⬇[Ⓣ, 0, m] L ≡ K.ⓑ{I}V → ⬇[m] L ≡ K.ⓑ{I}V.
-/2 width=5 by drop_inv_FT_aux/ qed.
-
-lemma drop_inv_gen: ∀I,L,K,V,s,m. ⬇[s, 0, m] L ≡ K.ⓑ{I}V → ⬇[m] L ≡ K.ⓑ{I}V.
-#I #L #K #V * /2 width=1 by drop_inv_FT/
-qed-.
-
-lemma drop_inv_T: ∀I,L,K,V,s,m. ⬇[Ⓣ, 0, m] L ≡ K.ⓑ{I}V → ⬇[s, 0, m] L ≡ K.ⓑ{I}V.
-#I #L #K #V * /2 width=1 by drop_inv_FT/
-qed-.
-
-(* Basic_1: removed theorems 50:
- drop_ctail drop_skip_flat
- cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
- drop_clear drop_clear_O drop_clear_S
- clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
- clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
- getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
- getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
- getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
- drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
- getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
- getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
- getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lenv_append.ma".
-include "basic_2/substitution/drop.ma".
-
-(* DROPPING *****************************************************************)
-
-(* Properties on append for local environments ******************************)
-
-fact drop_O1_append_sn_le_aux: ∀L1,L2,s,l,m. ⬇[s, l, m] L1 ≡ L2 →
- l = yinj 0 → m ≤ |L1| →
- ∀L. ⬇[s, yinj 0, m] L @@ L1 ≡ L @@ L2.
-#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m //
-[ #l #m #_ #_ #H <(le_n_O_to_eq … H) -H //
-| normalize /4 width=1 by drop_drop, monotonic_pred/
-| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #_ #H elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma drop_O1_append_sn_le: ∀L1,L2,s,m. ⬇[s, yinj 0, m] L1 ≡ L2 → m ≤ |L1| →
- ∀L. ⬇[s, yinj 0, m] L @@ L1 ≡ L @@ L2.
-/2 width=3 by drop_O1_append_sn_le_aux/ qed.
-
-(* Inversion lemmas on append for local environments ************************)
-
-lemma drop_O1_inv_append1_ge: ∀K,L1,L2,s,m. ⬇[s, yinj 0, m] L1 @@ L2 ≡ K →
- |L2| ≤ m → ⬇[s, yinj 0, m - |L2|] L1 ≡ K.
-#K #L1 #L2 elim L2 -L2 normalize //
-#L2 #I #V #IHL2 #s #m #H #H1m
-elim (drop_inv_O1_pair1 … H) -H * #H2m #HL12 destruct
-[ lapply (le_n_O_to_eq … H1m) -H1m -IHL2 <plus_n_Sm #H destruct
-| <minus_plus >minus_minus_comm /3 width=1 by monotonic_pred/
-]
-qed-.
-
-lemma drop_O1_inv_append1_le: ∀K,L1,L2,s,m. ⬇[s, yinj 0, m] L1 @@ L2 ≡ K → m ≤ |L2| →
- ∀K2. ⬇[s, yinj 0, m] L2 ≡ K2 → K = L1 @@ K2.
-#K #L1 #L2 elim L2 -L2 normalize
-[ #s #m #H1 #H2 #K2 #H3 lapply (le_n_O_to_eq … H2) -H2
- #H2 elim (drop_inv_atom1 … H3) -H3 #H3 #_ destruct
- >(drop_inv_O2 … H1) -H1 //
-| #L2 #I #V #IHL2 #s #m @(nat_ind_plus … m) -m [ -IHL2 ]
- [ #H1 #_ #K2 #H2
- lapply (drop_inv_O2 … H1) -H1 #H1
- lapply (drop_inv_O2 … H2) -H2 #H2 destruct //
- | /4 width=7 by drop_inv_drop1, le_plus_to_le_r/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lift_lift.ma".
-include "basic_2/substitution/drop.ma".
-
-(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: drop_mono *)
-theorem drop_mono: ∀L,L1,s1,l,m. ⬇[s1, l, m] L ≡ L1 →
- ∀L2,s2. ⬇[s2, l, m] L ≡ L2 → L1 = L2.
-#L #L1 #s1 #l #m #H elim H -L -L1 -l -m
-[ #l #m #Hm #L2 #s2 #H elim (drop_inv_atom1 … H) -H //
-| #I #K #V #L2 #s2 #HL12 <(drop_inv_O2 … HL12) -L2 //
-| #I #L #K #V #m #_ #IHLK #L2 #s2 #H
- lapply (drop_inv_drop1 … H) -H /2 width=2 by/
-| #I #L #K1 #T #V1 #l #m #_ #HVT1 #IHLK1 #X #s2 #H
- elim (drop_inv_skip1 … H) -H // >ypred_succ #K2 #V2 #HLK2 #HVT2 #H destruct
- >(lift_inj … HVT1 … HVT2) -HVT1 -HVT2
- >(IHLK1 … HLK2) -IHLK1 -HLK2 //
-]
-qed-.
-
-(* Basic_1: was: drop_conf_ge *)
-theorem drop_conf_ge: ∀L,L1,s1,l1,m1. ⬇[s1, l1, m1] L ≡ L1 →
- ∀L2,s2,m2. ⬇[s2, 0, m2] L ≡ L2 → l1 + m1 ≤ m2 →
- ⬇[s2, 0, m2 - m1] L1 ≡ L2.
-#L #L1 #s1 #l1 #m1 #H elim H -L -L1 -l1 -m1 //
-[ #l #m #_ #L2 #s2 #m2 #H #_ elim (drop_inv_atom1 … H) -H
- #H #Hm destruct
- @drop_atom #H >Hm // (**) (* explicit constructor *)
-| #I #L #K #V #m #_ #IHLK #L2 #s2 #m2 #H >yplus_O1 <yplus_inj #Hm2
- lapply (drop_inv_drop1_lt … H ?) -H /3 width=2 by yle_inv_inj, ltn_to_ltO/ #HL2
- lapply (yle_plus1_to_minus_inj2 … Hm2) -Hm2 #Hm2
- <minus_plus >minus_minus_comm @IHLK //
-| #I #L #K #V1 #V2 #l #m #_ #_ #IHLK #L2 #s2 #m2 #H #Hlmm2
- lapply (yle_plus1_to_minus_inj2 … Hlmm2) #Hlm2m
- lapply (ylt_yle_trans 0 … Hlm2m ?) // -Hlm2m #Hm2m
- >yplus_succ1 in Hlmm2; #Hlmm2
- elim (yle_inv_succ1 … Hlmm2) -Hlmm2 #Hlmm2 #Hm2
- lapply (drop_inv_drop1_lt … H ?) -H /2 width=1 by ylt_inv_inj/ -Hm2 #HL2
- @drop_drop_lt /2 width=1 by ylt_inv_inj/ >minus_minus_comm
- <yminus_SO2 in Hlmm2; /2 width=1 by/
-]
-qed.
-(*
-(* Note: apparently this was missing in basic_1 *)
-theorem drop_conf_be: ∀L0,L1,s1,l1,m1. ⬇[s1, yinj l1, m1] L0 ≡ L1 →
- ∀L2,m2. ⬇[m2] L0 ≡ L2 → l1 ≤ m2 → m2 ≤ l1 + m1 →
- ∃∃L. ⬇[s1, yinj 0, l1 + m1 - m2] L2 ≡ L & ⬇[l1] L1 ≡ L.
-#L0 #L1 #s1 #l1 #m1 #H elim H -L0 -L1 -l1 -m1
-[ #l1 #m1 #Hm1 #L2 #m2 #H #Hl1 #_ elim (drop_inv_atom1 … H) -H #H #Hm2 destruct
- >(Hm2 ?) in Hl1; // -Hm2 #Hl1 <(le_n_O_to_eq … Hl1) -l1
- /4 width=3 by drop_atom, ex2_intro/
-| normalize #I #L #V #L2 #m2 #HL2 #_ #Hm2
- lapply (le_n_O_to_eq … Hm2) -Hm2 #H destruct
- lapply (drop_inv_O2 … HL2) -HL2 #H destruct /2 width=3 by drop_pair, ex2_intro/
-| normalize #I #L0 #K0 #V1 #m1 #HLK0 #IHLK0 #L2 #m2 #H #_ #Hm21
- lapply (drop_inv_O1_pair1 … H) -H * * #Hm2 #HL20
- [ -IHLK0 -Hm21 destruct <minus_n_O /3 width=3 by drop_drop, ex2_intro/
- | -HLK0 <minus_le_minus_minus_comm //
- elim (IHLK0 … HL20) -L0 /2 width=3 by monotonic_pred, ex2_intro/
- ]
-| #I #L0 #K0 #V0 #V1 #l1 #m1 >plus_plus_comm_23 #_ #_ #IHLK0 #L2 #m2 #H #Hl1m2 #Hm2lm1
- elim (le_inv_plus_l … Hl1m2) #_ #Hm2
- <minus_le_minus_minus_comm //
- lapply (drop_inv_drop1_lt … H ?) -H // #HL02
- elim (IHLK0 … HL02) -L0 /3 width=3 by drop_drop, monotonic_pred, ex2_intro/
-]
-qed-.
-*)
-(* Note: apparently this was missing in basic_1 *)
-theorem drop_conf_le: ∀L0,L1,s1,l1,m1. ⬇[s1, l1, m1] L0 ≡ L1 →
- ∀L2,s2,m2. ⬇[s2, 0, m2] L0 ≡ L2 → m2 ≤ l1 →
- ∃∃L. ⬇[s2, 0, m2] L1 ≡ L & ⬇[s1, l1 - m2, m1] L2 ≡ L.
-#L0 #L1 #s1 #l1 #m1 #H elim H -L0 -L1 -l1 -m1
-[ #l1 #m1 #Hm1 #L2 #s2 #m2 #H elim (drop_inv_atom1 … H) -H
- #H #Hm2 #_ destruct /4 width=3 by drop_atom, ex2_intro/
-| #I #K0 #V0 #L2 #s2 #m2 #H1 #H2
- lapply (yle_inv_O2 … H2) -H2 #H destruct
- lapply (drop_inv_pair1 … H1) -H1 #H destruct /2 width=3 by drop_pair, ex2_intro/
-| #I #K0 #K1 #V0 #m1 #HK01 #_ #L2 #s2 #m2 #H1 #H2
- lapply (yle_inv_O2 … H2) -H2 #H destruct
- lapply (drop_inv_pair1 … H1) -H1 #H destruct /3 width=3 by drop_drop, ex2_intro/
-| #I #K0 #K1 #V0 #V1 #l1 #m1 #HK01 #HV10 #IHK01 #L2 #s2 #m2 #H #Hm2l1
- elim (drop_inv_O1_pair1 … H) -H *
- [ -IHK01 -Hm2l1 #H1 #H2 destruct /3 width=5 by drop_pair, drop_skip, ex2_intro/
- | -HK01 -HV10 #Hm2 #HK0L2
- lapply (yle_inv_succ2 … Hm2l1) -Hm2l1 <yminus_SO2 #Hm2l1
- elim (IHK01 … HK0L2) -IHK01 -HK0L2 //
- <yminus_inj >yplus_minus_assoc_comm_inj /2 width=1 by yle_inj/
- >yplus_SO2 /3 width=3 by drop_drop_lt, ex2_intro/
- ]
-]
-qed-.
-
-(* Note: with "s2", the conclusion parameter is "s1 ∨ s2" *)
-(* Basic_1: was: drop_trans_ge *)
-theorem drop_trans_ge: ∀L1,L,s1,l1,m1. ⬇[s1, l1, m1] L1 ≡ L →
- ∀L2,m2. ⬇[m2] L ≡ L2 → l1 ≤ m2 → ⬇[s1, 0, m1 + m2] L1 ≡ L2.
-#L1 #L #s1 #l1 #m1 #H elim H -L1 -L -l1 -m1
-[ #l1 #m1 #Hm1 #L2 #m2 #H #_ elim (drop_inv_atom1 … H) -H
- #H #Hm2 destruct /4 width=1 by drop_atom, eq_f2/
-| /2 width=1 by drop_gen/
-| /3 width=1 by drop_drop/
-| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IHL12 #L #m2 #H #Hlm2
- elim (yle_inv_succ1 … Hlm2) -Hlm2 #Hlm2 #Hm2
- lapply (ylt_O … Hm2) -Hm2 #Hm2
- lapply (lt_to_le_to_lt … (m + m2) Hm2 ?) // #Hmm2
- lapply (drop_inv_drop1_lt … H ?) -H // #HL2
- @drop_drop_lt // >le_plus_minus <yminus_SO2 in Hlm2; /2 width=1 by/
-]
-qed.
-
-(* Basic_1: was: drop_trans_le *)
-theorem drop_trans_le: ∀L1,L,s1,l1,m1. ⬇[s1, l1, m1] L1 ≡ L →
- ∀L2,s2,m2. ⬇[s2, 0, m2] L ≡ L2 → m2 ≤ l1 →
- ∃∃L0. ⬇[s2, 0, m2] L1 ≡ L0 & ⬇[s1, l1 - m2, m1] L0 ≡ L2.
-#L1 #L #s1 #l1 #m1 #H elim H -L1 -L -l1 -m1
-[ #l1 #m1 #Hm1 #L2 #s2 #m2 #H #_ elim (drop_inv_atom1 … H) -H
- #H #Hm2 destruct /4 width=3 by drop_atom, ex2_intro/
-| #I #K #V #L2 #s2 #m2 #HL2 #H lapply (yle_inv_O2 … H) -H
- #H destruct /2 width=3 by drop_pair, ex2_intro/
-| #I #L1 #L2 #V #m #_ #IHL12 #L #s2 #m2 #HL2 #H lapply (yle_inv_O2 … H) -H
- #H destruct elim (IHL12 … HL2) -IHL12 -HL2 //
- #L0 #H #HL0 lapply (drop_inv_O2 … H) -H #H destruct
- /3 width=5 by drop_pair, drop_drop, ex2_intro/
-| #I #L1 #L2 #V1 #V2 #l #m #HL12 #HV12 #IHL12 #L #s2 #m2 #H #Hm2l
- elim (drop_inv_O1_pair1 … H) -H *
- [ -Hm2l -IHL12 #H1 #H2 destruct /3 width=5 by drop_pair, drop_skip, ex2_intro/
- | -HL12 -HV12 #Hm2 #HL2
- lapply (yle_inv_succ2 … Hm2l) -Hm2l <yminus_SO2 #Hm2l
- elim (IHL12 … HL2) -L2 //
- <yminus_inj >yplus_minus_assoc_comm_inj /3 width=3 by drop_drop_lt, yle_inj, ex2_intro/
- ]
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma d_liftable_llstar: ∀R. d_liftable R → ∀d. d_liftable (llstar … R d).
-#R #HR #d #K #T1 #T2 #H @(lstar_ind_r … d T2 H) -d -T2
-[ #L #s #l #m #_ #U1 #HTU1 #U2 #HTU2 -HR -K
- >(lift_mono … HTU2 … HTU1) -T1 -U2 -l -m //
-| #d #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #HTU1 #U2 #HTU2
- elim (lift_total T l m) /3 width=12 by lstar_dx/
-]
-qed-.
-
-(* Basic_1: was: drop_conf_lt *)
-lemma drop_conf_lt: ∀L,L1,s1,l1,m1. ⬇[s1, l1, m1] L ≡ L1 →
- ∀I,K2,V2,s2,m2. ⬇[s2, 0, m2] L ≡ K2.ⓑ{I}V2 →
- m2 < l1 → let l ≝ ⫰(l1 - m2) in
- ∃∃K1,V1. ⬇[s2, 0, m2] L1 ≡ K1.ⓑ{I}V1 &
- ⬇[s1, l, m1] K2 ≡ K1 & ⬆[l, m1] V1 ≡ V2.
-#L #L1 #s1 #l1 #m1 #H1 #I #K2 #V2 #s2 #m2 #H2 #Hm2l1
-elim (drop_conf_le … H1 … H2) -L /2 width=2 by ylt_fwd_le/ #K #HL1K #HK2
-elim (drop_inv_skip1 … HK2) -HK2 /2 width=1 by ylt_to_minus/
-#K1 #V1 #HK21 #HV12 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-(* Note: apparently this was missing in basic_1 *)
-lemma drop_trans_lt: ∀L1,L,s1,l1,m1. ⬇[s1, l1, m1] L1 ≡ L →
- ∀I,L2,V2,s2,m2. ⬇[s2, 0, m2] L ≡ L2.ⓑ{I}V2 →
- m2 < l1 → let l ≝ l1 - m2 - 1 in
- ∃∃L0,V0. ⬇[s2, 0, m2] L1 ≡ L0.ⓑ{I}V0 &
- ⬇[s1, l, m1] L0 ≡ L2 & ⬆[l, m1] V2 ≡ V0.
-#L1 #L #s1 #l1 #m1 #HL1 #I #L2 #V2 #s2 #m2 #HL2 #Hl21
-elim (drop_trans_le … HL1 … HL2) -L /2 width=1 by ylt_fwd_le/ #L0 #HL10 #HL02
-elim (drop_inv_skip2 … HL02) -HL02 /2 width=1 by ylt_to_minus/
-#L #V1 #HL2 #HV21 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma drop_trans_ge_comm: ∀L1,L,L2,s1,l1,m1,m2.
- ⬇[s1, l1, m1] L1 ≡ L → ⬇[m2] L ≡ L2 → l1 ≤ m2 →
- ⬇[s1, 0, m2 + m1] L1 ≡ L2.
-#L1 #L #L2 #s1 #l1 #m1 #m2
->commutative_plus /2 width=5 by drop_trans_ge/
-qed.
-
-lemma drop_conf_div: ∀I1,L,K,V1,m1. ⬇[m1] L ≡ K.ⓑ{I1}V1 →
- ∀I2,V2,m2. ⬇[m2] L ≡ K.ⓑ{I2}V2 →
- ∧∧ m1 = m2 & I1 = I2 & V1 = V2.
-#I1 #L #K #V1 #m1 #HLK1 #I2 #V2 #m2 #HLK2
-elim (le_or_ge m1 m2) #Hm
-[ lapply (drop_conf_ge … HLK1 … HLK2 ?)
-| lapply (drop_conf_ge … HLK2 … HLK1 ?)
-] -HLK1 -HLK2 /2 width=1 by yle_inj/ #HK
-lapply (drop_fwd_length_minus2 … HK) #H
-elim (discr_minus_x_xy … H) -H
-[1,3: normalize <plus_n_Sm #H destruct ]
-#H >H in HK; #HK
-lapply (drop_inv_O2 … HK) -HK #H destruct
-lapply (inv_eq_minus_O … H) -H /3 width=1 by le_to_le_to_eq, and3_intro/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma drop_fwd_be: ∀L,K,s,l,m,i. ⬇[s, l, m] L ≡ K → |K| ≤ i → yinj i < l → |L| ≤ i.
-#L #K #s #l #m #i #HLK #HK #Hl elim (lt_or_ge i (|L|)) //
-#HL elim (drop_O1_lt (Ⓕ) … HL) #I #K0 #V #HLK0 -HL
-elim (drop_conf_lt … HLK … HLK0) // -HLK -HLK0 -Hl
-#K1 #V1 #HK1 #_ #_ lapply (drop_fwd_length_lt2 … HK1) -I -K1 -V1
-#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/lreq_lreq.ma".
-include "basic_2/substitution/drop.ma".
-
-(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
-
-definition dedropable_sn: predicate (relation lenv) ≝
- λR. ∀L1,K1,s,l,m. ⬇[s, l, m] L1 ≡ K1 → ∀K2. R K1 K2 →
- ∃∃L2. R L1 L2 & ⬇[s, l, m] L2 ≡ K2 & L1 ⩬[l, m] L2.
-
-(* Properties on equivalence ************************************************)
-
-lemma lreq_drop_trans_be: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
- ∀I,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W →
- l ≤ i → i < l + m →
- ∃∃K1. K1 ⩬[0, ⫰(l+m-i)] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
-[ #l #m #J #K2 #W #s #i #H
- elim (drop_inv_atom1 … H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J #K2 #W #s #i #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I #L1 #L2 #V #m #HL12 #IHL12 #J #K2 #W #s #i #H #_ >yplus_O1
- elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
- [ #_ destruct >ypred_succ
- /2 width=3 by drop_pair, ex2_intro/
- | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
- #H <H -H #H lapply (ylt_inv_succ … H) -H <yminus_SO2
- #Him elim (IHL12 … HLK1) -IHL12 -HLK1 // -Him
- >yminus_succ <yminus_inj /3 width=3 by drop_drop_lt, ex2_intro/
- ]
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J #K2 #W #s #i #HLK2 #Hli
- elim (yle_inv_succ1 … Hli) -Hli
- #Hli #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
- #Hilm lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
- #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
- /4 width=3 by ylt_O, drop_drop_lt, ex2_intro/
-]
-qed-.
-
-lemma lreq_drop_conf_be: ∀L1,L2,l,m. L1 ⩬[l, m] L2 →
- ∀I,K1,W,s,i. ⬇[s, 0, i] L1 ≡ K1.ⓑ{I}W →
- l ≤ i → i < l + m →
- ∃∃K2. K1 ⩬[0, ⫰(l+m-i)] K2 & ⬇[s, 0, i] L2 ≡ K2.ⓑ{I}W.
-#L1 #L2 #l #m #HL12 #I #K1 #W #s #i #HLK1 #Hli #Hilm
-elim (lreq_drop_trans_be … (lreq_sym … HL12) … HLK1) // -L1 -Hli -Hilm
-/3 width=3 by lreq_sym, ex2_intro/
-qed-.
-
-lemma drop_O1_ex: ∀K2,i,L1. |L1| = |K2| + i →
- ∃∃L2. L1 ⩬[0, i] L2 & ⬇[i] L2 ≡ K2.
-#K2 #i @(nat_ind_plus … i) -i
-[ /3 width=3 by lreq_O2, ex2_intro/
-| #i #IHi #Y #Hi elim (drop_O1_lt (Ⓕ) Y 0) //
- #I #L1 #V #H lapply (drop_inv_O2 … H) -H #H destruct
- normalize in Hi; elim (IHi L1) -IHi
- /3 width=5 by drop_drop, lreq_pair, injective_plus_l, ex2_intro/
-]
-qed-.
-
-lemma dedropable_sn_TC: ∀R. dedropable_sn R → dedropable_sn (TC … R).
-#R #HR #L1 #K1 #s #l #m #HLK1 #K2 #H elim H -K2
-[ #K2 #HK12 elim (HR … HLK1 … HK12) -HR -K1
- /3 width=4 by inj, ex3_intro/
-| #K #K2 #_ #HK2 * #L #H1L1 #HLK #H2L1 elim (HR … HLK … HK2) -HR -K
- /3 width=6 by lreq_trans, step, ex3_intro/
-]
-qed-.
-
-(* Inversion lemmas on equivalence ******************************************)
-
-lemma drop_O1_inj: ∀i,L1,L2,K. ⬇[i] L1 ≡ K → ⬇[i] L2 ≡ K → L1 ⩬[i, ∞] L2.
-#i @(nat_ind_plus … i) -i
-[ #L1 #L2 #K #H <(drop_inv_O2 … H) -K #H <(drop_inv_O2 … H) -L1 //
-| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 //
- lapply (drop_fwd_length … HLK1)
- <(drop_fwd_length … HLK2) [ /4 width=5 by drop_inv_drop1, lreq_succ/ ]
- normalize <plus_n_Sm #H destruct
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/supterm_6.ma".
-include "basic_2/grammar/cl_weight.ma".
-include "basic_2/substitution/drop.ma".
-
-(* SUPCLOSURE ***************************************************************)
-
-(* activate genv *)
-inductive fqu: tri_relation genv lenv term ≝
-| fqu_lref_O : ∀I,G,L,V. fqu G (L.ⓑ{I}V) (#0) G L V
-| fqu_pair_sn: ∀I,G,L,V,T. fqu G L (②{I}V.T) G L V
-| fqu_bind_dx: ∀a,I,G,L,V,T. fqu G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
-| fqu_flat_dx: ∀I,G,L,V,T. fqu G L (ⓕ{I}V.T) G L T
-| fqu_drop : ∀G,L,K,T,U,m.
- ⬇[m+1] L ≡ K → ⬆[0, m+1] T ≡ U → fqu G L U G K T
-.
-
-interpretation
- "structural successor (closure)"
- 'SupTerm G1 L1 T1 G2 L2 T2 = (fqu G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fqu_drop_lt: ∀G,L,K,T,U,m. 0 < m →
- ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → ⦃G, L, U⦄ ⊐ ⦃G, K, T⦄.
-#G #L #K #T #U #m #Hm >(plus_minus_m_m m 1) /2 width=3 by fqu_drop/
-qed.
-
-lemma fqu_lref_S_lt: ∀I,G,L,V,i. 0 < i → ⦃G, L.ⓑ{I}V, #i⦄ ⊐ ⦃G, L, #(i-1)⦄.
-/4 width=3 by fqu_drop, drop_drop, lift_lref_ge_minus, yle_inj/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fqu_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} < ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
-#G #L #K #T #U #m #HLK #HTU
-lapply (drop_fwd_lw_lt … HLK ?) -HLK // #HKL
-lapply (lift_fwd_tw … HTU) -m #H
-normalize in ⊢ (?%%); /2 width=1 by lt_minus_to_plus/
-qed-.
-
-fact fqu_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀i. T1 = #i → |L2| < |L1|.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[1: normalize //
-|3: #a
-|5: /2 width=4 by drop_fwd_length_lt4/
-] #I #G #L #V #T #j #H destruct
-qed-.
-
-lemma fqu_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐ ⦃G2, L2, T2⦄ → |L2| < |L1|.
-/2 width=7 by fqu_fwd_length_lref1_aux/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact fqu_inv_eq_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- G1 = G2 → |L1| = |L2| → T1 = T2 → ⊥.
-#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 normalize
-/2 width=4 by discr_tpair_xy_y, discr_tpair_xy_x, plus_xSy_x_false/
-#G #L #K #T #U #m #HLK #_ #_ #H #_ -G -T -U >(drop_fwd_length … HLK) in H; -L
-/2 width=4 by plus_xySz_x_false/
-qed-.
-
-lemma fqu_inv_eq: ∀G,L1,L2,T. ⦃G, L1, T⦄ ⊐ ⦃G, L2, T⦄ → |L1| = |L2| → ⊥.
-#G #L1 #L2 #T #H #H0 @(fqu_inv_eq_aux … H … H0) // (**) (* full auto fails *)
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma fqu_wf_ind: ∀R:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) → ∀G1,L1,T1. R G1 L1 T1.
-#R #HR @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=1 by fqu_fwd_fw/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/suptermopt_6.ma".
-include "basic_2/substitution/fqu.ma".
-
-(* OPTIONAL SUPCLOSURE ******************************************************)
-
-(* activate genv *)
-inductive fquq: tri_relation genv lenv term ≝
-| fquq_lref_O : ∀I,G,L,V. fquq G (L.ⓑ{I}V) (#0) G L V
-| fquq_pair_sn: ∀I,G,L,V,T. fquq G L (②{I}V.T) G L V
-| fquq_bind_dx: ∀a,I,G,L,V,T. fquq G L (ⓑ{a,I}V.T) G (L.ⓑ{I}V) T
-| fquq_flat_dx: ∀I,G, L,V,T. fquq G L (ⓕ{I}V.T) G L T
-| fquq_drop : ∀G,L,K,T,U,m.
- ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → fquq G L U G K T
-.
-
-interpretation
- "optional structural successor (closure)"
- 'SupTermOpt G1 L1 T1 G2 L2 T2 = (fquq G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fquq_refl: tri_reflexive … fquq.
-/2 width=3 by fquq_drop/ qed.
-
-lemma fqu_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -L1 -L2 -T1 -T2 /2 width=3 by fquq_drop/
-qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fquq_fwd_fw: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 /2 width=1 by lt_to_le/
-#G1 #L1 #K1 #T1 #U1 #m #HLK1 #HTU1
-lapply (drop_fwd_lw … HLK1) -HLK1
-lapply (lift_fwd_tw … HTU1) -HTU1
-/2 width=1 by le_plus, le_n/
-qed-.
-
-fact fquq_fwd_length_lref1_aux: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀i. T1 = #i → |L2| ≤ |L1|.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
-[ #a #I #G #L #V #T #j #H destruct
-| #G1 #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #i #H destruct
- /2 width=3 by drop_fwd_length_le4/
-]
-qed-.
-
-lemma fquq_fwd_length_lref1: ∀G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮ ⦃G2, L2, T2⦄ → |L2| ≤ |L1|.
-/2 width=7 by fquq_fwd_length_lref1_aux/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/suptermoptalt_6.ma".
-include "basic_2/substitution/fquq.ma".
-
-(* OPTIONAL SUPCLOSURE ******************************************************)
-
-(* alternative definition of fquq *)
-definition fquqa: tri_relation genv lenv term ≝ tri_RC … fqu.
-
-interpretation
- "optional structural successor (closure) alternative"
- 'SupTermOptAlt G1 L1 T1 G2 L2 T2 = (fquqa G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fquqa_refl: tri_reflexive … fquqa.
-// qed.
-
-lemma fquqa_drop: ∀G,L,K,T,U,m.
- ⬇[m] L ≡ K → ⬆[0, m] T ≡ U → ⦃G, L, U⦄ ⊐⊐⸮ ⦃G, K, T⦄.
-#G #L #K #T #U #m #HLK #HTU elim (eq_or_gt m)
-/3 width=5 by fqu_drop_lt, or_introl/ #H destruct
->(drop_inv_O2 … HLK) -L >(lift_inv_O2 … HTU) -T //
-qed.
-
-(* Main properties **********************************************************)
-
-theorem fquq_fquqa: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-/2 width=3 by fquqa_drop, fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, or_introl/
-qed.
-
-(* Main inversion properties ************************************************)
-
-theorem fquqa_inv_fquq: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⊐⸮ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=1 by fqu_fquq/
-* #H1 #H2 #H3 destruct //
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fquq_inv_gen: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim (fquq_fquqa … H) -H [| * ]
-/2 width=1 by or_introl/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/rdrop_3.ma".
-include "basic_2/grammar/genv.ma".
-
-(* GLOBAL ENVIRONMENT READING ***********************************************)
-
-inductive gget (m:nat): relation genv ≝
-| gget_gt: ∀G. |G| ≤ m → gget m G (⋆)
-| gget_eq: ∀G. |G| = m + 1 → gget m G G
-| gget_lt: ∀I,G1,G2,V. m < |G1| → gget m G1 G2 → gget m (G1. ⓑ{I} V) G2
-.
-
-interpretation "global reading"
- 'RDrop m G1 G2 = (gget m G1 G2).
-
-(* basic inversion lemmas ***************************************************)
-
-lemma gget_inv_gt: ∀G1,G2,m. ⬇[m] G1 ≡ G2 → |G1| ≤ m → G2 = ⋆.
-#G1 #G2 #m * -G1 -G2 //
-[ #G #H >H -H >commutative_plus #H (**) (* lemma needed here *)
- lapply (le_plus_to_le_r … 0 H) -H #H
- lapply (le_n_O_to_eq … H) -H #H destruct
-| #I #G1 #G2 #V #H1 #_ #H2
- lapply (le_to_lt_to_lt … H2 H1) -H2 -H1 normalize in ⊢ (? % ? → ?); >commutative_plus #H
- lapply (lt_plus_to_lt_l … 0 H) -H #H
- elim (lt_zero_false … H)
-]
-qed-.
-
-lemma gget_inv_eq: ∀G1,G2,m. ⬇[m] G1 ≡ G2 → |G1| = m + 1 → G1 = G2.
-#G1 #G2 #m * -G1 -G2 //
-[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H (**) (* lemma needed here *)
- lapply (le_plus_to_le_r … 0 H) -H #H
- lapply (le_n_O_to_eq … H) -H #H destruct
-| #I #G1 #G2 #V #H1 #_ normalize #H2
- <(injective_plus_l … H2) in H1; -H2 #H
- elim (lt_refl_false … H)
-]
-qed-.
-
-fact gget_inv_lt_aux: ∀I,G,G1,G2,V,m. ⬇[m] G ≡ G2 → G = G1. ⓑ{I} V →
- m < |G1| → ⬇[m] G1 ≡ G2.
-#I #G #G1 #G2 #V #m * -G -G2
-[ #G #H1 #H destruct #H2
- lapply (le_to_lt_to_lt … H1 H2) -H1 -H2 normalize in ⊢ (? % ? → ?); >commutative_plus #H
- lapply (lt_plus_to_lt_l … 0 H) -H #H
- elim (lt_zero_false … H)
-| #G #H1 #H2 destruct >(injective_plus_l … H1) -H1 #H
- elim (lt_refl_false … H)
-| #J #G #G2 #W #_ #HG2 #H destruct //
-]
-qed-.
-
-lemma gget_inv_lt: ∀I,G1,G2,V,m.
- ⬇[m] G1. ⓑ{I} V ≡ G2 → m < |G1| → ⬇[m] G1 ≡ G2.
-/2 width=5 by gget_inv_lt_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma gget_total: ∀m,G1. ∃G2. ⬇[m] G1 ≡ G2.
-#m #G1 elim G1 -G1 /3 width=2 by gget_gt, ex_intro/
-#I #V #G1 * #G2 #HG12
-elim (lt_or_eq_or_gt m (|G1|)) #Hm
-[ /3 width=2 by gget_lt, ex_intro/
-| destruct /3 width=2 by gget_eq, ex_intro/
-| @ex_intro [2: @gget_gt normalize /2 width=1 by/ | skip ] (**) (* explicit constructor *)
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/gget.ma".
-
-(* GLOBAL ENVIRONMENT READING ***********************************************)
-
-(* Main properties **********************************************************)
-
-theorem gget_mono: ∀G,G1,m. ⬇[m] G ≡ G1 → ∀G2. ⬇[m] G ≡ G2 → G1 = G2.
-#G #G1 #m #H elim H -G -G1
-[ #G #Hm #G2 #H
- >(gget_inv_gt … H Hm) -H -Hm //
-| #G #Hm #G2 #H
- >(gget_inv_eq … H Hm) -H -Hm //
-| #I #G #G1 #V #Hm #_ #IHG1 #G2 #H
- lapply (gget_inv_lt … H Hm) -H -Hm /2 width=1 by/
-]
-qed-.
-
-lemma gget_dec: ∀G1,G2,m. Decidable (⬇[m] G1 ≡ G2).
-#G1 #G2 #m
-elim (gget_total m G1) #G #HG1
-elim (eq_genv_dec G G2) #HG2
-[ destruct /2 width=1 by or_introl/
-| @or_intror #HG12
- lapply (gget_mono … HG1 … HG12) -HG1 -HG12 /2 width=1 by/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_plus.ma".
-include "basic_2/notation/relations/rlift_4.ma".
-include "basic_2/grammar/term_weight.ma".
-include "basic_2/grammar/term_simple.ma".
-
-(* BASIC TERM RELOCATION ****************************************************)
-
-(* Basic_1: includes:
- lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
-*)
-inductive lift: relation4 ynat nat term term ≝
-| lift_sort : ∀k,l,m. lift l m (⋆k) (⋆k)
-| lift_lref_lt: ∀i,l,m. yinj i < l → lift l m (#i) (#i)
-| lift_lref_ge: ∀i,l,m. l ≤ yinj i → lift l m (#i) (#(i + m))
-| lift_gref : ∀p,l,m. lift l m (§p) (§p)
-| lift_bind : ∀a,I,V1,V2,T1,T2,l,m.
- lift l m V1 V2 → lift (⫯l) m T1 T2 →
- lift l m (ⓑ{a,I} V1. T1) (ⓑ{a,I} V2. T2)
-| lift_flat : ∀I,V1,V2,T1,T2,l,m.
- lift l m V1 V2 → lift l m T1 T2 →
- lift l m (ⓕ{I} V1. T1) (ⓕ{I} V2. T2)
-.
-
-interpretation "relocation" 'RLift l m T1 T2 = (lift l m T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lift_inv_O2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → m = 0 → T1 = T2.
-#l #m #T1 #T2 #H elim H -l -m -T1 -T2 /3 width=1 by eq_f2/
-qed-.
-
-lemma lift_inv_O2: ∀l,T1,T2. ⬆[l, 0] T1 ≡ T2 → T1 = T2.
-/2 width=4 by lift_inv_O2_aux/ qed-.
-
-fact lift_inv_sort1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀k. T1 = ⋆k → T2 = ⋆k.
-#l #m #T1 #T2 * -l -m -T1 -T2 //
-[ #i #l #m #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
-]
-qed-.
-
-lemma lift_inv_sort1: ∀l,m,T2,k. ⬆[l, m] ⋆k ≡ T2 → T2 = ⋆k.
-/2 width=5 by lift_inv_sort1_aux/ qed-.
-
-fact lift_inv_lref1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀i. T1 = #i →
- (i < l ∧ T2 = #i) ∨ (l ≤ i ∧ T2 = #(i + m)).
-#l #m #T1 #T2 * -l -m -T1 -T2
-[ #k #l #m #i #H destruct
-| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl,conj/
-| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_intror,conj/
-| #p #l #m #i #H destruct
-| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
-]
-qed-.
-
-lemma lift_inv_lref1: ∀l,m,T2,i. ⬆[l, m] #i ≡ T2 →
- (i < l ∧ T2 = #i) ∨ (l ≤ i ∧ T2 = #(i + m)).
-/2 width=3 by lift_inv_lref1_aux/ qed-.
-
-lemma lift_inv_lref1_lt: ∀l,m,T2,i. ⬆[l, m] #i ≡ T2 → i < l → T2 = #i.
-#l #m #T2 #i #H elim (lift_inv_lref1 … H) -H * //
-#Hli #_ #Hil elim (ylt_yle_false … Hli) -Hli //
-qed-.
-
-lemma lift_inv_lref1_ge: ∀l,m,T2,i. ⬆[l, m] #i ≡ T2 → l ≤ i → T2 = #(i + m).
-#l #m #T2 #i #H elim (lift_inv_lref1 … H) -H * //
-#Hil #_ #Hli elim (ylt_yle_false … Hli) -Hli //
-qed-.
-
-fact lift_inv_gref1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀p. T1 = §p → T2 = §p.
-#l #m #T1 #T2 * -l -m -T1 -T2 //
-[ #i #l #m #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
-]
-qed-.
-
-lemma lift_inv_gref1: ∀l,m,T2,p. ⬆[l, m] §p ≡ T2 → T2 = §p.
-/2 width=5 by lift_inv_gref1_aux/ qed-.
-
-fact lift_inv_bind1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
- ∀a,I,V1,U1. T1 = ⓑ{a,I}V1.U1 →
- ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & ⬆[⫯l, m] U1 ≡ U2 &
- T2 = ⓑ{a,I}V2.U2.
-#l #m #T1 #T2 * -l -m -T1 -T2
-[ #k #l #m #a #I #V1 #U1 #H destruct
-| #i #l #m #_ #a #I #V1 #U1 #H destruct
-| #i #l #m #_ #a #I #V1 #U1 #H destruct
-| #p #l #m #a #I #V1 #U1 #H destruct
-| #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-| #J #W1 #W2 #T1 #T2 #l #m #_ #HT #a #I #V1 #U1 #H destruct
-]
-qed-.
-
-lemma lift_inv_bind1: ∀l,m,T2,a,I,V1,U1. ⬆[l, m] ⓑ{a,I}V1.U1 ≡ T2 →
- ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & ⬆[⫯l, m] U1 ≡ U2 &
- T2 = ⓑ{a,I}V2.U2.
-/2 width=3 by lift_inv_bind1_aux/ qed-.
-
-fact lift_inv_flat1_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
- ∀I,V1,U1. T1 = ⓕ{I}V1.U1 →
- ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & ⬆[l, m] U1 ≡ U2 &
- T2 = ⓕ{I}V2.U2.
-#l #m #T1 #T2 * -l -m -T1 -T2
-[ #k #l #m #I #V1 #U1 #H destruct
-| #i #l #m #_ #I #V1 #U1 #H destruct
-| #i #l #m #_ #I #V1 #U1 #H destruct
-| #p #l #m #I #V1 #U1 #H destruct
-| #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V1 #U1 #H destruct
-| #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lift_inv_flat1: ∀l,m,T2,I,V1,U1. ⬆[l, m] ⓕ{I}V1.U1 ≡ T2 →
- ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & ⬆[l, m] U1 ≡ U2 &
- T2 = ⓕ{I}V2.U2.
-/2 width=3 by lift_inv_flat1_aux/ qed-.
-
-fact lift_inv_sort2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀k. T2 = ⋆k → T1 = ⋆k.
-#l #m #T1 #T2 * -l -m -T1 -T2 //
-[ #i #l #m #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
-]
-qed-.
-
-(* Basic_1: was: lift_gen_sort *)
-lemma lift_inv_sort2: ∀l,m,T1,k. ⬆[l, m] T1 ≡ ⋆k → T1 = ⋆k.
-/2 width=5 by lift_inv_sort2_aux/ qed-.
-
-fact lift_inv_lref2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀i. T2 = #i →
- (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
-#l #m #T1 #T2 * -l -m -T1 -T2
-[ #k #l #m #i #H destruct
-| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
-| #j #l #m #Hj #i #Hi destruct <minus_plus_m_m /4 width=1 by monotonic_yle_plus_dx, or_intror, conj/
-| #p #l #m #i #H destruct
-| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
-]
-qed-.
-
-(* Basic_1: was: lift_gen_lref *)
-lemma lift_inv_lref2: ∀l,m,T1,i. ⬆[l, m] T1 ≡ #i →
- (i < l ∧ T1 = #i) ∨ (l + m ≤ i ∧ T1 = #(i - m)).
-/2 width=3 by lift_inv_lref2_aux/ qed-.
-
-(* Basic_1: was: lift_gen_lref_lt *)
-lemma lift_inv_lref2_lt: ∀l,m,T1,i. ⬆[l, m] T1 ≡ #i → i < l → T1 = #i.
-#l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
-#H #_ #Hil lapply (yle_fwd_plus_sn1 … H) -H
-#Hli elim (ylt_yle_false … Hli) -Hli //
-qed-.
-
-(* Basic_1: was: lift_gen_lref_false *)
-lemma lift_inv_lref2_be: ∀l,m,T1,i. ⬆[l, m] T1 ≡ #i →
- l ≤ i → i < l + m → ⊥.
-#l #m #T1 #i #H elim (lift_inv_lref2 … H) -H *
-[ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
-elim (ylt_yle_false … H2) -H2 //
-qed-.
-
-(* Basic_1: was: lift_gen_lref_ge *)
-lemma lift_inv_lref2_ge: ∀l,m,T1,i. ⬆[l, m] T1 ≡ #i → l + m ≤ i → T1 = #(i - m).
-#l #m #T1 #i #H elim (lift_inv_lref2 … H) -H * //
-#Hil #_ #H lapply (yle_fwd_plus_sn1 … H) -H
-#Hli elim (ylt_yle_false … Hli) -Hli //
-qed-.
-
-fact lift_inv_gref2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ∀p. T2 = §p → T1 = §p.
-#l #m #T1 #T2 * -l -m -T1 -T2 //
-[ #i #l #m #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
-]
-qed-.
-
-lemma lift_inv_gref2: ∀l,m,T1,p. ⬆[l, m] T1 ≡ §p → T1 = §p.
-/2 width=5 by lift_inv_gref2_aux/ qed-.
-
-fact lift_inv_bind2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
- ∀a,I,V2,U2. T2 = ⓑ{a,I}V2.U2 →
- ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & ⬆[⫯l, m] U1 ≡ U2 &
- T1 = ⓑ{a,I}V1.U1.
-#l #m #T1 #T2 * -l -m -T1 -T2
-[ #k #l #m #a #I #V2 #U2 #H destruct
-| #i #l #m #_ #a #I #V2 #U2 #H destruct
-| #i #l #m #_ #a #I #V2 #U2 #H destruct
-| #p #l #m #a #I #V2 #U2 #H destruct
-| #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
-| #J #W1 #W2 #T1 #T2 #l #m #_ #_ #a #I #V2 #U2 #H destruct
-]
-qed-.
-
-(* Basic_1: was: lift_gen_bind *)
-lemma lift_inv_bind2: ∀l,m,T1,a,I,V2,U2. ⬆[l, m] T1 ≡ ⓑ{a,I}V2.U2 →
- ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & ⬆[⫯l, m] U1 ≡ U2 &
- T1 = ⓑ{a,I}V1.U1.
-/2 width=3 by lift_inv_bind2_aux/ qed-.
-
-fact lift_inv_flat2_aux: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 →
- ∀I,V2,U2. T2 = ⓕ{I}V2.U2 →
- ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & ⬆[l, m] U1 ≡ U2 &
- T1 = ⓕ{I}V1.U1.
-#l #m #T1 #T2 * -l -m -T1 -T2
-[ #k #l #m #I #V2 #U2 #H destruct
-| #i #l #m #_ #I #V2 #U2 #H destruct
-| #i #l #m #_ #I #V2 #U2 #H destruct
-| #p #l #m #I #V2 #U2 #H destruct
-| #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_1: was: lift_gen_flat *)
-lemma lift_inv_flat2: ∀l,m,T1,I,V2,U2. ⬆[l, m] T1 ≡ ⓕ{I}V2.U2 →
- ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & ⬆[l, m] U1 ≡ U2 &
- T1 = ⓕ{I}V1.U1.
-/2 width=3 by lift_inv_flat2_aux/ qed-.
-
-lemma lift_inv_Y1: ∀T1,T2,m. ⬆[∞, m] T1 ≡ T2 → T1 = T2.
-#T1 elim T1 -T1 *
-[ #k #X #m #H lapply (lift_inv_sort1 … H) -H //
-| #i #X #m #H lapply (lift_inv_lref1_lt … H ?) -H //
-| #p #X #m #H lapply (lift_inv_gref1 … H) -H //
-| #a #I #V1 #T1 #IHV1 #IHT1 #X #m #H elim (lift_inv_bind1 … H) -H
- #V2 #T2 #HV12 #HT12 #H destruct /3 width=2 by eq_f2/
-| #I #V1 #T1 #IHV1 #IHT1 #X #m #H elim (lift_inv_flat1 … H) -H
- #V2 #T2 #HV12 #HT12 #H destruct /3 width=2 by eq_f2/
-]
-qed-.
-
-lemma lift_inv_pair_xy_x: ∀l,m,I,V,T. ⬆[l, m] ②{I}V.T ≡ V → ⊥.
-#l #m #J #V elim V -V
-[ * #i #T #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct
- | elim (lift_inv_lref2 … H) -H * #_ #H destruct
- | lapply (lift_inv_gref2 … H) -H #H destruct
- ]
-| * [ #a ] #I #W2 #U2 #IHW2 #_ #T #H
- [ elim (lift_inv_bind2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
- | elim (lift_inv_flat2 … H) -H #W1 #U1 #HW12 #_ #H destruct /2 width=2 by/
- ]
-]
-qed-.
-
-(* Basic_1: was: thead_x_lift_y_y *)
-lemma lift_inv_pair_xy_y: ∀I,T,V,l,m. ⬆[l, m] ②{I}V.T ≡ T → ⊥.
-#J #T elim T -T
-[ * #i #V #l #m #H
- [ lapply (lift_inv_sort2 … H) -H #H destruct
- | elim (lift_inv_lref2 … H) -H * #_ #H destruct
- | lapply (lift_inv_gref2 … H) -H #H destruct
- ]
-| * [ #a ] #I #W2 #U2 #_ #IHU2 #V #l #m #H
- [ elim (lift_inv_bind2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
- | elim (lift_inv_flat2 … H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
- ]
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lift_fwd_pair1: ∀I,T2,V1,U1,l,m. ⬆[l, m] ②{I}V1.U1 ≡ T2 →
- ∃∃V2,U2. ⬆[l, m] V1 ≡ V2 & T2 = ②{I}V2.U2.
-* [ #a ] #I #T2 #V1 #U1 #l #m #H
-[ elim (lift_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
-| elim (lift_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lift_fwd_pair2: ∀I,T1,V2,U2,l,m. ⬆[l, m] T1 ≡ ②{I}V2.U2 →
- ∃∃V1,U1. ⬆[l, m] V1 ≡ V2 & T1 = ②{I}V1.U1.
-* [ #a ] #I #T1 #V2 #U2 #l #m #H
-[ elim (lift_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
-| elim (lift_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lift_fwd_tw: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → ♯{T1} = ♯{T2}.
-#l #m #T1 #T2 #H elim H -l -m -T1 -T2 normalize //
-qed-.
-
-lemma lift_simple_dx: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
-#a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
-elim (simple_inv_bind … H)
-qed-.
-
-lemma lift_simple_sn: ∀l,m,T1,T2. ⬆[l, m] T1 ≡ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-#l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
-#a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
-elim (simple_inv_bind … H)
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: lift_lref_gt *)
-lemma lift_lref_ge_minus: ∀l,m,i. l + yinj m ≤ yinj i → ⬆[l, m] #(i - m) ≡ #i.
-#l #m #i #H >(plus_minus_m_m i m) in ⊢ (? ? ? ? %);
-elim (yle_inv_plus_inj2 … H) -H #Hlim #H
-lapply (yle_inv_inj … H) -H /2 width=1 by lift_lref_ge/
-qed.
-
-lemma lift_lref_ge_minus_eq: ∀l,m,i,j. l + yinj m ≤ yinj i → j = i - m → ⬆[l, m] #j ≡ #i.
-/2 width=1 by lift_lref_ge_minus/ qed-.
-
-(* Basic_1: was: lift_r *)
-lemma lift_refl: ∀T,l. ⬆[l, 0] T ≡ T.
-#T elim T -T
-[ * #i // #l elim (ylt_split i l) /2 width=1 by lift_lref_lt, lift_lref_ge/
-| * /2 width=1 by lift_bind, lift_flat/
-]
-qed.
-
-(* Basic_2b: first lemma *)
-lemma lift_Y1: ∀T,m. ⬆[∞, m] T ≡ T.
-#T elim T -T * /2 width=1 by lift_lref_lt, lift_bind, lift_flat/
-qed.
-
-lemma lift_total: ∀T1,l,m. ∃T2. ⬆[l, m] T1 ≡ T2.
-#T1 elim T1 -T1
-[ * #i /2 width=2 by lift_sort, lift_gref, ex_intro/
- #l #m elim (ylt_split i l) /3 width=2 by lift_lref_lt, lift_lref_ge, ex_intro/
-| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #l #m
- elim (IHV1 l m) -IHV1 #V2 #HV12
- [ elim (IHT1 (⫯l) m) -IHT1 /3 width=2 by lift_bind, ex_intro/
- | elim (IHT1 l m) -IHT1 /3 width=2 by lift_flat, ex_intro/
- ]
-]
-qed.
-
-(* Basic_1: was: lift_free (right to left) *)
-lemma lift_split: ∀l1,m2,T1,T2. ⬆[l1, m2] T1 ≡ T2 →
- ∀l2,m1. l1 ≤ l2 → l2 ≤ l1 + yinj m1 → m1 ≤ m2 →
- ∃∃T. ⬆[l1, m1] T1 ≡ T & ⬆[l2, m2 - m1] T ≡ T2.
-#l1 #m2 #T1 #T2 #H elim H -l1 -m2 -T1 -T2
-[ /3 width=3 by lift_sort, ex2_intro/
-| #i #l1 #m2 #Hil1 #l2 #m1 #Hl12 #_ #_
- lapply (ylt_yle_trans … Hl12 Hil1) -Hl12 #Hil2 /4 width=3 by lift_lref_lt, ex2_intro/
-| #i #l1 #m2 #Hil1 #l2 #m1 #_ #Hl21 #Hm12
- lapply (yle_trans … Hl21 (i+m1) ?) /2 width=1 by monotonic_yle_plus_dx/ -Hl21 #Hl21
- >(plus_minus_m_m m2 m1 ?) /3 width=3 by lift_lref_ge, ex2_intro/
-| /3 width=3 by lift_gref, ex2_intro/
-| #a #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
- elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
- elim (IHT (⫯l2) … ? ? Hm12) /3 width=5 by lift_bind, yle_succ, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
- elim (IHV … Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
- elim (IHT l2 … ? ? Hm12) /3 width=5 by lift_flat, ex2_intro/
-]
-qed.
-
-(* Basic_1: was only: dnf_dec2 dnf_dec *)
-lemma is_lift_dec: ∀T2,l,m. Decidable (∃T1. ⬆[l, m] T1 ≡ T2).
-#T1 elim T1 -T1
-[ * [1,3: /3 width=2 by lift_sort, lift_gref, ex_intro, or_introl/ ] #i #l #m
- elim (ylt_split i l) #Hli
- [ /4 width=3 by lift_lref_lt, ex_intro, or_introl/
- | elim (ylt_split i (l + m)) #Hilm
- [ @or_intror * #T1 #H elim (lift_inv_lref2_be … H Hli Hilm)
- | -Hli /4 width=2 by lift_lref_ge_minus, ex_intro, or_introl/
- ]
- ]
-| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #l #m
- [ elim (IHV2 l m) -IHV2
- [ * #V1 #HV12 elim (IHT2 (⫯l) m) -IHT2
- [ * #T1 #HT12 @or_introl /3 width=2 by lift_bind, ex_intro/
- | -V1 #HT2 @or_intror * #X #H
- elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
- ]
- | -IHT2 #HV2 @or_intror * #X #H
- elim (lift_inv_bind2 … H) -H /3 width=2 by ex_intro/
- ]
- | elim (IHV2 l m) -IHV2
- [ * #V1 #HV12 elim (IHT2 l m) -IHT2
- [ * #T1 #HT12 /4 width=2 by lift_flat, ex_intro, or_introl/
- | -V1 #HT2 @or_intror * #X #H
- elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
- ]
- | -IHT2 #HV2 @or_intror * #X #H
- elim (lift_inv_flat2 … H) -H /3 width=2 by ex_intro/
- ]
- ]
-]
-qed.
-
-(* Basic_1: removed theorems 7:
- lift_head lift_gen_head
- lift_weight_map lift_weight lift_weight_add lift_weight_add_O
- lift_tlt_dx
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lift.ma".
-
-(* BASIC TERM RELOCATION ****************************************************)
-
-(* Main properties ***********************************************************)
-
-(* Basic_1: was: lift_inj *)
-theorem lift_inj: ∀l,m,T1,U. ⬆[l, m] T1 ≡ U → ∀T2. ⬆[l, m] T2 ≡ U → T1 = T2.
-#l #m #T1 #U #H elim H -l -m -T1 -U
-[ #k #l #m #X #HX
- lapply (lift_inv_sort2 … HX) -HX //
-| #i #l #m #Hil #X #HX
- lapply (lift_inv_lref2_lt … HX ?) -HX //
-| #i #l #m #Hli #X #HX
- lapply (lift_inv_lref2_ge … HX ?) -HX /2 width=1 by monotonic_yle_plus_dx/
-| #p #l #m #X #HX
- lapply (lift_inv_gref2 … HX) -HX //
-| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat2 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-]
-qed-.
-
-(* Basic_1: was: lift_gen_lift *)
-theorem lift_div_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
- ∀l2,m2,T2. ⬆[l2 + m1, m2] T2 ≡ T →
- l1 ≤ l2 →
- ∃∃T0. ⬆[l1, m1] T0 ≡ T2 & ⬆[l2, m2] T0 ≡ T1.
-#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
-[ #k #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
- lapply (lift_inv_sort2 … Hk) -Hk #Hk destruct /3 width=3 by lift_sort, ex2_intro/
-| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
- lapply (ylt_yle_trans … Hl12 Hil1) -Hl12 #Hil2
- lapply (lift_inv_lref2_lt … Hi ?) -Hi /3 width=3 by lift_lref_lt, ylt_plus_dx1_trans, ex2_intro/
-| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
- elim (lift_inv_lref2 … Hi) -Hi * <yplus_inj #Hil2 #H destruct
- [ -Hl12 lapply (ylt_inv_monotonic_plus_dx … Hil2) -Hil2 #Hil2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
- | -Hil1 >yplus_comm_23 in Hil2; #H lapply ( yle_inv_monotonic_plus_dx … H) -H #H
- elim (yle_inv_plus_inj2 … H) -H >yminus_inj #Hl2im2 #H
- lapply (yle_inv_inj … H) -H #Hm2i
- lapply (yle_trans … Hl12 … Hl2im2) -Hl12 #Hl1im2
- >le_plus_minus_comm // >(plus_minus_m_m i m2) in ⊢ (? ? ? %);
- /3 width=3 by lift_lref_ge, ex2_intro/
- ]
-| #p #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
- lapply (lift_inv_gref2 … Hk) -Hk #Hk destruct /3 width=3 by lift_gref, ex2_intro/
-| #a #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
- lapply (lift_inv_bind2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
- elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
- <yplus_succ1 in HU2; #HU2 elim (IHU … HU2) /3 width=5 by yle_succ, lift_bind, ex2_intro/
-| #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
- lapply (lift_inv_flat2 … H) -H * #W2 #U2 #HW2 #HU2 #H destruct
- elim (IHW … HW2) // -IHW -HW2 #W0 #HW2 #HW1
- elim (IHU … HU2) /3 width=5 by lift_flat, ex2_intro/
-]
-qed.
-
-(* Note: apparently this was missing in basic_1 *)
-theorem lift_div_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
- ∀m,m2,T2. ⬆[l1 + yinj m, m2] T2 ≡ T →
- m ≤ m1 → m1 ≤ m + m2 →
- ∃∃T0. ⬆[l1, m] T0 ≡ T2 & ⬆[l1, m + m2 - m1] T0 ≡ T1.
-#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
-[ #k #l1 #m1 #m #m2 #T2 #H >(lift_inv_sort2 … H) -H /2 width=3 by lift_sort, ex2_intro/
-| #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
- >(lift_inv_lref2_lt … H) -H /3 width=3 by ylt_plus_dx1_trans, lift_lref_lt, ex2_intro/
-| #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
- elim (ylt_split (i+m1) (l1+m+m2)) #H0
- [ elim (lift_inv_lref2_be … H) -H /3 width=2 by monotonic_yle_plus, yle_inj/
- | >(lift_inv_lref2_ge … H ?) -H //
- lapply (yle_plus2_to_minus_inj2 … H0) #Hl1m21i
- elim (yle_inv_plus_inj2 … H0) -H0 #Hl1m12 #Hm2im1
- @ex2_intro [2: /2 width=1 by lift_lref_ge_minus/ | skip ]
- @lift_lref_ge_minus_eq
- [ <yminus_inj <yplus_inj >yplus_minus_assoc_inj /2 width=1 by yle_inj/
- | /2 width=1 by minus_le_minus_minus_comm/
- ]
- ]
-| #p #l1 #m1 #m #m2 #T2 #H >(lift_inv_gref2 … H) -H /2 width=3 by lift_gref, ex2_intro/
-| #a #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
- elim (lift_inv_bind2 … H) -H #V2 #T2 #HV2 <yplus_succ1 #HT2 #H destruct
- elim (IHV1 … HV2) -V //
- elim (IHT1 … HT2) -T /3 width=5 by lift_bind, ex2_intro/
-| #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
- elim (lift_inv_flat2 … H) -H #V2 #T2 #HV2 #HT2 #H destruct
- elim (IHV1 … HV2) -V //
- elim (IHT1 … HT2) -T /3 width=5 by lift_flat, ex2_intro/
-]
-qed.
-
-theorem lift_mono: ∀l,m,T,U1. ⬆[l,m] T ≡ U1 → ∀U2. ⬆[l,m] T ≡ U2 → U1 = U2.
-#l #m #T #U1 #H elim H -l -m -T -U1
-[ #k #l #m #X #HX
- lapply (lift_inv_sort1 … HX) -HX //
-| #i #l #m #Hil #X #HX
- lapply (lift_inv_lref1_lt … HX ?) -HX //
-| #i #l #m #Hli #X #HX
- lapply (lift_inv_lref1_ge … HX ?) -HX //
-| #p #l #m #X #HX
- lapply (lift_inv_gref1 … HX) -HX //
-| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_bind1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
- elim (lift_inv_flat1 … HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-]
-qed-.
-
-(* Basic_1: was: lift_free (left to right) *)
-theorem lift_trans_be: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
- ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 →
- l1 ≤ l2 → l2 ≤ l1 + m1 → ⬆[l1, m1 + m2] T1 ≡ T2.
-#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
-[ #k #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
- >(lift_inv_sort1 … HT2) -HT2 //
-| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #Hl12 #_
- lapply (ylt_yle_trans … Hl12 Hil1) -Hl12 #Hil2
- lapply (lift_inv_lref1_lt … HT2 Hil2) /2 width=1 by lift_lref_lt/
-| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #_ #Hl21
- lapply (lift_inv_lref1_ge … HT2 ?) -HT2
- [ @(yle_trans … Hl21) -Hl21 /2 width=1 by monotonic_yle_plus_dx/
- | -Hl21 /2 width=1 by lift_lref_ge/
- ]
-| #p #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
- >(lift_inv_gref1 … HT2) -HT2 //
-| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_bind, yle_succ/ (**) (* full auto a bit slow *)
-| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- lapply (IHV12 … HV20 ? ?) // -IHV12 -HV20 #HV10
- lapply (IHT12 … HT20 ? ?) /2 width=1 by lift_flat/ (**) (* full auto a bit slow *)
-]
-qed.
-
-(* Basic_1: was: lift_d (right to left) *)
-theorem lift_trans_le: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
- ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l2 ≤ l1 →
- ∃∃T0. ⬆[l2, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T0 ≡ T2.
-#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
-[ #k #l1 #m1 #l2 #m2 #X #HX #_
- >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
-| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
- lapply (ylt_yle_trans … (l1+m2) ? Hil1) // #Him2
- elim (lift_inv_lref1 … HX) -HX * #Hil2 #HX destruct
- /4 width=3 by monotonic_ylt_plus_dx, monotonic_yle_plus_dx, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
-| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hl21
- lapply (yle_trans … Hl21 … Hil1) -Hl21 #Hil2
- lapply (lift_inv_lref1_ge … HX ?) -HX /2 width=3 by yle_plus_dx1_trans/ #HX destruct
- >plus_plus_comm_23 /4 width=3 by monotonic_yle_plus_dx, lift_lref_ge_minus, lift_lref_ge, ex2_intro/
-| #p #l1 #m1 #l2 #m2 #X #HX #_
- >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
-| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- elim (IHV12 … HV20) -IHV12 -HV20 //
- elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_bind, yle_succ, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- elim (IHV12 … HV20) -IHV12 -HV20 //
- elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
-]
-qed.
-
-(* Basic_1: was: lift_d (left to right) *)
-theorem lift_trans_ge: ∀l1,m1,T1,T. ⬆[l1, m1] T1 ≡ T →
- ∀l2,m2,T2. ⬆[l2, m2] T ≡ T2 → l1 + m1 ≤ l2 →
- ∃∃T0. ⬆[l2 - m1, m2] T1 ≡ T0 & ⬆[l1, m1] T0 ≡ T2.
-#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
-[ #k #l1 #m1 #l2 #m2 #X #HX #_
- >(lift_inv_sort1 … HX) -HX /2 width=3 by lift_sort, ex2_intro/
-| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hlml
- lapply (ylt_yle_trans … (l1+m1) ? Hil1) // #Hil1m
- lapply (ylt_yle_trans … (l2-m1) ? Hil1) /2 width=1 by yle_plus1_to_minus_inj2/ #Hil2m
- lapply (ylt_yle_trans … Hlml Hil1m) -Hil1m -Hlml #Hil2
- lapply (lift_inv_lref1_lt … HX ?) -HX // #HX destruct /3 width=3 by lift_lref_lt, ex2_intro/
-| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
- elim (lift_inv_lref1 … HX) -HX * <yplus_inj #Himl #HX destruct
- [ /4 width=3 by lift_lref_lt, lift_lref_ge, ylt_plus1_to_minus_inj2, ex2_intro/
- | /4 width=3 by lift_lref_ge, yle_plus_dx1_trans, monotonic_yle_minus_dx, ex2_intro/
- ]
-| #p #l1 #m1 #l2 #m2 #X #HX #_
- >(lift_inv_gref1 … HX) -HX /2 width=3 by lift_gref, ex2_intro/
-| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
- elim (yle_inv_plus_inj2 … Hlml) #Hlm #Hml
- elim (lift_inv_bind1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- elim (IHV12 … HV20) -IHV12 -HV20 //
- elim (IHT12 … HT20) -IHT12 -HT20 /2 width=1 by yle_succ/ -Hlml
- #T >yminus_succ1_inj /3 width=5 by lift_bind, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
- elim (lift_inv_flat1 … HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
- elim (IHV12 … HV20) -IHV12 -HV20 //
- elim (IHT12 … HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
-]
-qed.
-
-(* Advanced properties ******************************************************)
-
-lemma lift_conf_O1: ∀T,T1,l1,m1. ⬆[l1, m1] T ≡ T1 → ∀T2,m2. ⬆[0, m2] T ≡ T2 →
- ∃∃T0. ⬆[0, m2] T1 ≡ T0 & ⬆[l1 + m2, m1] T2 ≡ T0.
-#T #T1 #l1 #m1 #HT1 #T2 #m2 #HT2
-elim (lift_total T1 0 m2) #T0 #HT10
-elim (lift_trans_le … HT1 … HT10) -HT1 // #X #HTX #HT20
-lapply (lift_mono … HTX … HT2) -T #H destruct /2 width=3 by ex2_intro/
-qed.
-
-lemma lift_conf_be: ∀T,T1,l,m1. ⬆[l, m1] T ≡ T1 → ∀T2,m2. ⬆[l, m2] T ≡ T2 →
- m1 ≤ m2 → ⬆[l + yinj m1, m2 - m1] T1 ≡ T2.
-#T #T1 #l #m1 #HT1 #T2 #m2 #HT2 #Hm12
-elim (lift_split … HT2 (l+m1) m1) -HT2 // #X #H
->(lift_mono … H … HT1) -T //
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lift_lift.ma".
-include "basic_2/substitution/lift_vector.ma".
-
-(* BASIC TERM VECTOR RELOCATION *********************************************)
-
-(* Main properties ***********************************************************)
-
-theorem liftv_mono: ∀Ts,U1s,l,m. ⬆[l,m] Ts ≡ U1s →
- ∀U2s:list term. ⬆[l,m] Ts ≡ U2s → U1s = U2s.
-#Ts #U1s #l #m #H elim H -Ts -U1s
-[ #U2s #H >(liftv_inv_nil1 … H) -H //
-| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
- elim (liftv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
- >(lift_mono … HTU1 … HTU2) -T /3 width=1 by eq_f/
-]
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lift.ma".
-
-(* BASIC TERM RELOCATION ****************************************************)
-
-(* Properties on negated basic relocation ***********************************)
-
-lemma nlift_lref_be_SO: ∀X,i. ⬆[yinj i, 1] X ≡ #i → ⊥.
-/3 width=7 by lift_inv_lref2_be, ylt_inj/ qed-.
-
-lemma nlift_bind_sn: ∀W,l,m. (∀V. ⬆[l, m] V ≡ W → ⊥) →
- ∀a,I,U. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥).
-#W #l #m #HW #a #I #U #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/
-qed-.
-
-lemma nlift_bind_dx: ∀U,l,m. (∀T. ⬆[⫯l, m] T ≡ U → ⊥) →
- ∀a,I,W. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥).
-#U #l #m #HU #a #I #W #X #H elim (lift_inv_bind2 … H) -H /2 width=2 by/
-qed-.
-
-lemma nlift_flat_sn: ∀W,l,m. (∀V. ⬆[l, m] V ≡ W → ⊥) →
- ∀I,U. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥).
-#W #l #m #HW #I #U #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/
-qed-.
-
-lemma nlift_flat_dx: ∀U,l,m. (∀T. ⬆[l, m] T ≡ U → ⊥) →
- ∀I,W. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥).
-#U #l #m #HU #I #W #X #H elim (lift_inv_flat2 … H) -H /2 width=2 by/
-qed-.
-
-(* Inversion lemmas on negated basic relocation *****************************)
-
-lemma nlift_inv_lref_be_SO: ∀i,j. (∀X. ⬆[i, 1] X ≡ #j → ⊥) → yinj j = i.
-* [2: #j #H elim H -H // ]
-#i #j elim (lt_or_eq_or_gt i j) // #Hij #H
-[ elim (H (#(j-1))) -H /3 width=1 by lift_lref_ge_minus, yle_inj/
-| elim (H (#j)) -H /3 width=1 by lift_lref_lt, ylt_inj/
-]
-qed-.
-
-lemma nlift_inv_bind: ∀a,I,W,U,l,m. (∀X. ⬆[l, m] X ≡ ⓑ{a,I}W.U → ⊥) →
- (∀V. ⬆[l, m] V ≡ W → ⊥) ∨ (∀T. ⬆[⫯l, m] T ≡ U → ⊥).
-#a #I #W #U #l #m #H elim (is_lift_dec W l m)
-[ * /4 width=2 by lift_bind, or_intror/
-| /4 width=2 by ex_intro, or_introl/
-]
-qed-.
-
-lemma nlift_inv_flat: ∀I,W,U,l,m. (∀X. ⬆[l, m] X ≡ ⓕ{I}W.U → ⊥) →
- (∀V. ⬆[l, m] V ≡ W → ⊥) ∨ (∀T. ⬆[l, m] T ≡ U → ⊥).
-#I #W #U #l #m #H elim (is_lift_dec W l m)
-[ * /4 width=2 by lift_flat, or_intror/
-| /4 width=2 by ex_intro, or_introl/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/grammar/term_vector.ma".
-include "basic_2/substitution/lift.ma".
-
-(* BASIC TERM VECTOR RELOCATION *********************************************)
-
-inductive liftv (l) (m): relation (list term) ≝
-| liftv_nil : liftv l m (◊) (◊)
-| liftv_cons: ∀T1s,T2s,T1,T2.
- ⬆[l, m] T1 ≡ T2 → liftv l m T1s T2s →
- liftv l m (T1 @ T1s) (T2 @ T2s)
-.
-
-interpretation "relocation (vector)" 'RLift l m T1s T2s = (liftv l m T1s T2s).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact liftv_inv_nil1_aux: ∀T1s,T2s,l,m. ⬆[l, m] T1s ≡ T2s → T1s = ◊ → T2s = ◊.
-#T1s #T2s #l #m * -T1s -T2s //
-#T1s #T2s #T1 #T2 #_ #_ #H destruct
-qed-.
-
-lemma liftv_inv_nil1: ∀T2s,l,m. ⬆[l, m] ◊ ≡ T2s → T2s = ◊.
-/2 width=5 by liftv_inv_nil1_aux/ qed-.
-
-fact liftv_inv_cons1_aux: ∀T1s,T2s,l,m. ⬆[l, m] T1s ≡ T2s →
- ∀U1,U1s. T1s = U1 @ U1s →
- ∃∃U2,U2s. ⬆[l, m] U1 ≡ U2 & ⬆[l, m] U1s ≡ U2s &
- T2s = U2 @ U2s.
-#T1s #T2s #l #m * -T1s -T2s
-[ #U1 #U1s #H destruct
-| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma liftv_inv_cons1: ∀U1,U1s,T2s,l,m. ⬆[l, m] U1 @ U1s ≡ T2s →
- ∃∃U2,U2s. ⬆[l, m] U1 ≡ U2 & ⬆[l, m] U1s ≡ U2s &
- T2s = U2 @ U2s.
-/2 width=3 by liftv_inv_cons1_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma liftv_total: ∀l,m. ∀T1s:list term. ∃T2s. ⬆[l, m] T1s ≡ T2s.
-#l #m #T1s elim T1s -T1s
-[ /2 width=2 by liftv_nil, ex_intro/
-| #T1 #T1s * #T2s #HT12s
- elim (lift_total T1 l m) /3 width=2 by liftv_cons, ex_intro/
-]
-qed-.
(* Basic forward lemmas *****************************************************)
lemma lpx_sn_fwd_length: ∀R,L1,L2. lpx_sn R L1 L2 → |L1| = |L2|.
-#R #L1 #L2 #H elim H -L1 -L2 normalize //
+#R #L1 #L2 #H elim H -L1 -L2 //
qed-.
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
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH
/2 width=1 by drop_drop, conj/
qed-.
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
+#J1 #J2 #L1 #L2 #W1 #W2 #i #HKL1 #HKL2 elim (IH J1 J2 L1 L2 W1 W2 (⫯i)) -IH
/2 width=1 by drop_drop, conj/
qed-.
lpx_sn_alt R L1 L2 → R L1 V1 V2 →
lpx_sn_alt R (L1.ⓑ{I}V1) (L2.ⓑ{I}V2).
#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
+#HL12 #IH @conj //
+#I1 #I2 #K1 #K2 #W1 #W2 #i @(ynat_ind … i) -i
[ #HLK1 #HLK2
lapply (drop_inv_O2 … HLK1) -HLK1 #H destruct
lapply (drop_inv_O2 … HLK2) -HLK2 #H destruct
/2 width=1 by conj/
| -HL12 -HV12 /3 width=6 by drop_inv_drop1/
+| #H lapply (drop_fwd_Y2 … H) -H
+ #H elim (ylt_yle_false … H) -H //
]
qed.
#K2 #V2 #HK12 #HV12 #H destruct
lapply (lpx_sn_fwd_length … HK12)
#H @(ex3_intro … (K2.ⓑ{I}V2)) (**) (* explicit constructor *)
- /3 width=1 by lpx_sn_pair, monotonic_le_plus_l/
- @lreq_O2 normalize //
+ /3 width=1 by lpx_sn_pair, lreq_O2/
| #I #L1 #K1 #V1 #m #_ #IHLK1 #K2 #HK12 elim (IHLK1 … HK12) -K1
/3 width=5 by drop_drop, lreq_pair, lpx_sn_pair, ex3_intro/
| #I #L1 #K1 #V1 #W1 #l #m #HLK1 #HWV1 #IHLK1 #X #H
elim (lpx_sn_inv_pair1 … H) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (lift_total W2 l m) #V2 #HWV2
- lapply (H2R … HW12 … HLK1 … HWV1 … HWV2) -W1
+ elim (H2R … HW12 … HLK1 … HWV1) -W1
elim (IHLK1 … HK12) -K1
/3 width=6 by drop_skip, lreq_succ, lpx_sn_pair, ex3_intro/
]
theorem lpx_sn_conf: ∀R1,R2. lpx_sn_confluent R1 R2 →
confluent2 … (lpx_sn R1) (lpx_sn R2).
-#R1 #R2 #HR12 #L0 @(f_ind … length … L0) -L0 #x #IH *
+#R1 #R2 #HR12 #L0 @(ynat_f_ind … length … L0) -L0 #x #IH *
[ #_ #X1 #H1 #X2 #H2 -x
>(lpx_sn_inv_atom1 … H1) -X1
>(lpx_sn_inv_atom1 … H2) -X2 /2 width=3 by lpx_sn_atom, ex2_intro/
| #L0 #I #V0 #Hx #X1 #H1 #X2 #H2 destruct
elim (lpx_sn_inv_pair1 … H1) -H1 #L1 #V1 #HL01 #HV01 #H destruct
elim (lpx_sn_inv_pair1 … H2) -H2 #L2 #V2 #HL02 #HV02 #H destruct
- elim (IH … HL01 … HL02) -IH normalize // #L #HL1 #HL2
+ elim (IH … HL01 … HL02) -IH /2 width=2 by ylt_succ2_refl/ #L #HL1 #HL2
elim (HR12 … HV01 … HV02 … HL01 … HL02) -L0 -V0 /3 width=5 by lpx_sn_pair, ex2_intro/
]
qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_plus.ma".
-include "basic_2/notation/relations/lrsubeq_4.ma".
-include "basic_2/substitution/drop.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-
-inductive lsuby: relation4 ynat ynat lenv lenv ≝
-| lsuby_atom: ∀L,l,m. lsuby l m L (⋆)
-| lsuby_zero: ∀I1,I2,L1,L2,V1,V2.
- lsuby 0 0 L1 L2 → lsuby 0 0 (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
-| lsuby_pair: ∀I1,I2,L1,L2,V,m. lsuby 0 m L1 L2 →
- lsuby 0 (⫯m) (L1.ⓑ{I1}V) (L2.ⓑ{I2}V)
-| lsuby_succ: ∀I1,I2,L1,L2,V1,V2,l,m.
- lsuby l m L1 L2 → lsuby (⫯l) m (L1.ⓑ{I1}V1) (L2.ⓑ{I2}V2)
-.
-
-interpretation
- "local environment refinement (extended substitution)"
- 'LRSubEq L1 l m L2 = (lsuby l m L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_pair_lt: ∀I1,I2,L1,L2,V,m. L1 ⊆[0, ⫰m] L2 → 0 < m →
- L1.ⓑ{I1}V ⊆[0, m] L2.ⓑ{I2}V.
-#I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 … Hm) /2 width=1 by lsuby_pair/
-qed.
-
-lemma lsuby_succ_lt: ∀I1,I2,L1,L2,V1,V2,l,m. L1 ⊆[⫰l, m] L2 → 0 < l →
- L1.ⓑ{I1}V1 ⊆[l, m] L2. ⓑ{I2}V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 … Hl) /2 width=1 by lsuby_succ/
-qed.
-
-lemma lsuby_pair_O_Y: ∀L1,L2. L1 ⊆[0, ∞] L2 →
- ∀I1,I2,V. L1.ⓑ{I1}V ⊆[0,∞] L2.ⓑ{I2}V.
-#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 … V … HL12) -HL12 //
-qed.
-
-lemma lsuby_refl: ∀L,l,m. L ⊆[l, m] L.
-#L elim L -L //
-#L #I #V #IHL #l elim (ynat_cases … l) [| * #x ]
-#Hl destruct /2 width=1 by lsuby_succ/
-#m elim (ynat_cases … m) [| * #x ]
-#Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/
-qed.
-
-lemma lsuby_O2: ∀L2,L1,l. |L2| ≤ |L1| → L1 ⊆[l, yinj 0] L2.
-#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
-[ #l #H elim (le_plus_xSy_O_false … H)
-| #L1 #I1 #V1 #l #H lapply (le_plus_to_le_r … H) -H #HL12
- elim (ynat_cases l) /3 width=1 by lsuby_zero/
- * /3 width=1 by lsuby_succ/
-]
-qed.
-
-lemma lsuby_sym: ∀l,m,L1,L2. L1 ⊆[l, m] L2 → |L1| = |L2| → L2 ⊆[l, m] L1.
-#l #m #L1 #L2 #H elim H -l -m -L1 -L2
-[ #L1 #l #m #H >(length_inv_zero_dx … H) -L1 //
-| /2 width=1 by lsuby_O2/
-| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (injective_plus_l … H)
- /3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (injective_plus_l … H)
- /3 width=1 by lsuby_succ/
-]
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsuby_inv_atom1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 #l #m * -L1 -L2 -l -m //
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #m #_ #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
-]
-qed-.
-
-lemma lsuby_inv_atom1: ∀L2,l,m. ⋆ ⊆[l, m] L2 → L2 = ⋆.
-/2 width=5 by lsuby_inv_atom1_aux/ qed-.
-
-fact lsuby_inv_zero1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → l = 0 → m = 0 →
- L2 = ⋆ ∨
- ∃∃J2,K2,W2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{J2}W2.
-#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
- /3 width=5 by ex2_3_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
- elim (ysucc_inv_O_dx … H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_zero1: ∀I1,K1,L2,V1. K1.ⓑ{I1}V1 ⊆[0, 0] L2 →
- L2 = ⋆ ∨
- ∃∃I2,K2,V2. K1 ⊆[0, 0] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=9 by lsuby_inv_zero1_aux/ qed-.
-
-fact lsuby_inv_pair1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J1,K1,W. L1 = K1.ⓑ{J1}W → l = 0 → 0 < m →
- L2 = ⋆ ∨
- ∃∃J2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{J2}W.
-#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #m #HL12 #J1 #K1 #W #H #_ #_ destruct
- /3 width=4 by ex2_2_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_pair1: ∀I1,K1,L2,V,m. K1.ⓑ{I1}V ⊆[0, m] L2 → 0 < m →
- L2 = ⋆ ∨
- ∃∃I2,K2. K1 ⊆[0, ⫰m] K2 & L2 = K2.ⓑ{I2}V.
-/2 width=6 by lsuby_inv_pair1_aux/ qed-.
-
-fact lsuby_inv_succ1_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J1,K1,W1. L1 = K1.ⓑ{J1}W1 → 0 < l →
- L2 = ⋆ ∨
- ∃∃J2,K2,W2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{J2}W2.
-#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
- /3 width=5 by ex2_3_intro, or_intror/
-]
-qed-.
-
-lemma lsuby_inv_succ1: ∀I1,K1,L2,V1,l,m. K1.ⓑ{I1}V1 ⊆[l, m] L2 → 0 < l →
- L2 = ⋆ ∨
- ∃∃I2,K2,V2. K1 ⊆[⫰l, m] K2 & L2 = K2.ⓑ{I2}V2.
-/2 width=5 by lsuby_inv_succ1_aux/ qed-.
-
-fact lsuby_inv_zero2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → l = 0 → m = 0 →
- ∃∃J1,K1,W1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{J1}W1.
-#L1 #L2 #l #m * -L1 -L2 -l -m
-[ #L1 #l #m #J2 #K2 #W1 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
- /2 width=5 by ex2_3_intro/
-| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K2 #W2 #_ #_ #H
- elim (ysucc_inv_O_dx … H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_zero2: ∀I2,K2,L1,V2. L1 ⊆[0, 0] K2.ⓑ{I2}V2 →
- ∃∃I1,K1,V1. K1 ⊆[0, 0] K2 & L1 = K1.ⓑ{I1}V1.
-/2 width=9 by lsuby_inv_zero2_aux/ qed-.
-
-fact lsuby_inv_pair2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J2,K2,W. L2 = K2.ⓑ{J2}W → l = 0 → 0 < m →
- ∃∃J1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{J1}W.
-#L1 #L2 #l #m * -L1 -L2 -l -m
-[ #L1 #l #m #J2 #K2 #W #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #m #HL12 #J2 #K2 #W #H #_ #_ destruct
- /2 width=4 by ex2_2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W #_ #H
- elim (ysucc_inv_O_dx … H)
-]
-qed-.
-
-lemma lsuby_inv_pair2: ∀I2,K2,L1,V,m. L1 ⊆[0, m] K2.ⓑ{I2}V → 0 < m →
- ∃∃I1,K1. K1 ⊆[0, ⫰m] K2 & L1 = K1.ⓑ{I1}V.
-/2 width=6 by lsuby_inv_pair2_aux/ qed-.
-
-fact lsuby_inv_succ2_aux: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀J2,K2,W2. L2 = K2.ⓑ{J2}W2 → 0 < l →
- ∃∃J1,K1,W1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{J1}W1.
-#L1 #L2 #l #m * -L1 -L2 -l -m
-[ #L1 #l #m #J2 #K2 #W2 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J2 #K2 #W2 #H #_ destruct
- /2 width=5 by ex2_3_intro/
-]
-qed-.
-
-lemma lsuby_inv_succ2: ∀I2,K2,L1,V2,l,m. L1 ⊆[l, m] K2.ⓑ{I2}V2 → 0 < l →
- ∃∃I1,K1,V1. K1 ⊆[⫰l, m] K2 & L1 = K1.ⓑ{I1}V1.
-/2 width=5 by lsuby_inv_succ2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsuby_fwd_length: ∀L1,L2,l,m. L1 ⊆[l, m] L2 → |L2| ≤ |L1|.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m normalize /2 width=1 by le_S_S/
-qed-.
-
-(* Properties on basic slicing **********************************************)
-
-lemma lsuby_drop_trans_be: ∀L1,L2,l,m. L1 ⊆[l, m] L2 →
- ∀I2,K2,W,s,i. ⬇[s, 0, i] L2 ≡ K2.ⓑ{I2}W →
- l ≤ i → i < l + m →
- ∃∃I1,K1. K1 ⊆[0, ⫰(l+m-i)] K2 & ⬇[s, 0, i] L1 ≡ K1.ⓑ{I1}W.
-#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
-[ #L1 #l #m #J2 #K2 #W #s #i #H
- elim (drop_inv_atom1 … H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
- elim (ylt_yle_false … H) //
-| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
- elim (drop_inv_O1_pair1 … H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
- [ #_ destruct -I2 >ypred_succ
- /2 width=4 by drop_pair, ex2_2_intro/
- | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
- #H <H -H #H lapply (ylt_inv_succ … H) -H
- #Him elim (IHL12 … HLK1) -IHL12 -HLK1 [2: <yminus_inj ] // -Him
- >yminus_succ <yminus_inj /3 width=4 by drop_drop_lt, ex2_2_intro/
- ]
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hli
- elim (yle_inv_succ1 … Hli) -Hli
- #Hli #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ … H) -H
- #Hilm lapply (drop_inv_drop1_lt … HLK2 ?) -HLK2 /2 width=1 by ylt_O/
- #HLK1 elim (IHL12 … HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
- /4 width=4 by ylt_O, drop_drop_lt, ex2_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lsuby.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-
-(* Main properties **********************************************************)
-
-theorem lsuby_trans: ∀l,m. Transitive … (lsuby l m).
-#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
-[ #L1 #l #m #X #H lapply (lsuby_inv_atom1 … H) -H
- #H destruct //
-| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 … H) -H //
- * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_zero/
-| #I1 #I2 #L1 #L2 #V #m #_ #IHL1 #X #H elim (lsuby_inv_pair1 … H) -H //
- * #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL1 #X #H elim (lsuby_inv_succ1 … H) -H //
- * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/
-]
-qed-.
lemma after_inv_inh3: ∀cs1,cs2,tl,b. cs1 ⊚ cs2 ≡ b @ tl →
(∃∃tl1,tl2. cs1 = Ⓣ @ tl1 & cs2 = b @ tl2 & tl1 ⊚ tl2 ≡ tl) ∨
- ∃∃tl1. cs1 = Ⓕ @ tl1 & b = Ⓕ & tl1 ⊚ cs2 ≡ tl.
+ ∃∃tl1. cs1 = Ⓕ @ tl1 & b = Ⓕ & tl1 ⊚ cs2 ≡ tl.
/2 width=3 by after_inv_inh3_aux/ qed-.
+lemma after_inv_true3: ∀cs1,cs2,tl. cs1 ⊚ cs2 ≡ Ⓣ @ tl →
+ ∃∃tl1,tl2. cs1 = Ⓣ @ tl1 & cs2 = Ⓣ @ tl2 & tl1 ⊚ tl2 ≡ tl.
+#cs1 #cs2 #tl #H elim (after_inv_inh3 … H) -H // *
+#tl1 #_ #H destruct
+qed-.
+
+lemma after_inv_false3: ∀cs1,cs2,tl. cs1 ⊚ cs2 ≡ Ⓕ @ tl →
+ (∃∃tl1,tl2. cs1 = Ⓣ @ tl1 & cs2 = Ⓕ @ tl2 & tl1 ⊚ tl2 ≡ tl) ∨
+ ∃∃tl1. cs1 = Ⓕ @ tl1 & tl1 ⊚ cs2 ≡ tl.
+#cs1 #cs2 #tl #H elim (after_inv_inh3 … H) -H /2 width=1 by or_introl/ * /3 width=3 by ex2_intro, or_intror/
+qed-.
+
lemma after_inv_length: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs →
∧∧ ∥cs1∥ = |cs2| & |cs| = |cs1| & ∥cs∥ = ∥cs2∥.
#cs1 #cs2 #cs #H elim H -cs1 -cs2 -cs /2 width=1 by and3_intro/
(* Properties on composition ************************************************)
-lemma isid_after_sn: ∀cs1,cs2. cs1 ⊚ cs2 ≡ cs2 → 𝐈⦃cs1⦄ .
+lemma isid_after_sn: ∀cs2. ∃∃cs1. 𝐈⦃cs1⦄ & cs1 ⊚ cs2 ≡ cs2.
+#cs2 elim cs2 -cs2 /2 width=3 by after_empty, ex2_intro/
+#b #cs2 * /3 width=3 by isid_true, after_true, ex2_intro/
+qed-.
+
+lemma isid_after_dx: ∀cs1. ∃∃cs2. 𝐈⦃cs2⦄ & cs1 ⊚ cs2 ≡ cs1.
+#cs1 elim cs1 -cs1 /2 width=3 by after_empty, ex2_intro/
+* #cs1 * /3 width=3 by isid_true, after_true, after_false, ex2_intro/
+qed-.
+
+lemma after_isid_sn: ∀cs1,cs2. cs1 ⊚ cs2 ≡ cs2 → 𝐈⦃cs1⦄ .
#cs1 #cs2 #H elim (after_inv_length … H) -H //
qed.
-lemma isid_after_dx: ∀cs1,cs2. cs1 ⊚ cs2 ≡ cs1 → 𝐈⦃cs2⦄ .
+lemma after_isid_dx: ∀cs1,cs2. cs1 ⊚ cs2 ≡ cs1 → 𝐈⦃cs2⦄ .
#cs1 #cs2 #H elim (after_inv_length … H) -H //
qed.
(* Inversion lemmas on composition ******************************************)
-lemma isid_inv_after_sn: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → 𝐈⦃cs1⦄ → cs = cs2.
+lemma after_isid_inv_sn: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → 𝐈⦃cs1⦄ → cs = cs2.
#cs1 #cs2 #cs #H elim H -cs1 -cs2 -cs //
#cs1 #cs2 #cs #_ [ #b ] #IH #H
[ >IH -IH // | elim (isid_inv_false … H) ]
qed-.
-lemma isid_inv_after_dx: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → 𝐈⦃cs2⦄ → cs = cs1.
+lemma after_isid_inv_dx: ∀cs1,cs2,cs. cs1 ⊚ cs2 ≡ cs → 𝐈⦃cs2⦄ → cs = cs1.
#cs1 #cs2 #cs #H elim H -cs1 -cs2 -cs //
#cs1 #cs2 #cs #_ [ #b ] #IH #H
[ elim (isid_inv_cons … H) -H #H >IH -IH // | >IH -IH // ]
qed-.
+
+lemma after_inv_isid3: ∀t1,t2,t. t1 ⊚ t2 ≡ t → 𝐈⦃t⦄ → 𝐈⦃t1⦄ ∧ 𝐈⦃t2⦄.
+#t1 #t2 #t #H elim H -t1 -t2 -t
+[ /2 width=1 by conj/
+| #t1 #t2 #t #_ #b #IHt #H elim (isid_inv_cons … H) -H
+ #Ht #H elim (IHt Ht) -t /2 width=1 by isid_true, conj/
+| #t1 #t2 #t #_ #_ #H elim (isid_inv_false … H)
+]
+qed-.
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