--- /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/star.ma".
+include "basic_2/notation/relations/relationstar_4.ma".
+include "basic_2/static/rex.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+definition rexs (R): term → relation lenv ≝ CTC … (rex R).
+
+interpretation "iterated extension on referred entries (local environment)"
+ 'RelationStar R T L1 L2 = (rexs R T L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma rexs_step_dx: ∀R,L1,L,T. L1 ⪤*[R, T] L →
+ ∀L2. L ⪤[R, T] L2 → L1 ⪤*[R, T] L2.
+#R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *)
+qed-.
+
+lemma rexs_step_sn: ∀R,L1,L,T. L1 ⪤[R, T] L →
+ ∀L2. L ⪤*[R, T] L2 → L1 ⪤*[R, T] L2.
+#R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *)
+qed-.
+
+lemma rexs_atom: ∀R,I. ⋆ ⪤*[R, ⓪{I}] ⋆.
+/2 width=1 by inj/ qed.
+
+lemma rexs_sort: ∀R,I,L1,L2,V1,V2,s.
+ L1 ⪤*[R, ⋆s] L2 → L1.ⓑ{I}V1 ⪤*[R, ⋆s] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
+/3 width=4 by rex_sort, rexs_step_dx, inj/
+qed.
+
+lemma rexs_pair: ∀R. (∀L. reflexive … (R L)) →
+ ∀I,L1,L2,V. L1 ⪤*[R, V] L2 →
+ L1.ⓑ{I}V ⪤*[R, #0] L2.ⓑ{I}V.
+#R #HR #I #L1 #L2 #V #H elim H -L2
+/3 width=5 by rex_pair, rexs_step_dx, inj/
+qed.
+
+lemma rexs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 →
+ L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
+/3 width=3 by rex_unit, inj/ qed.
+
+lemma rexs_lref: ∀R,I,L1,L2,V1,V2,i.
+ L1 ⪤*[R, #i] L2 → L1.ⓑ{I}V1 ⪤*[R, #↑i] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
+/3 width=4 by rex_lref, rexs_step_dx, inj/
+qed.
+
+lemma rexs_gref: ∀R,I,L1,L2,V1,V2,l.
+ L1 ⪤*[R, §l] L2 → L1.ⓑ{I}V1 ⪤*[R, §l] L2.ⓑ{I}V2.
+#R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
+/3 width=4 by rex_gref, rexs_step_dx, inj/
+qed.
+
+lemma rexs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
+ ∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2.
+#R1 #R2 #HR #L1 #L2 #T #H elim H -L2
+/4 width=5 by rex_co, rexs_step_dx, inj/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *)
+lemma rexs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤*[R, ⓪{I}] Y2 → Y2 = ⋆.
+#R #I #Y2 #H elim H -Y2 /3 width=3 by inj, rex_inv_atom_sn/
+qed-.
+
+(* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *)
+lemma rexs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤*[R, ⓪{I}] ⋆ → Y1 = ⋆.
+#R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
+/3 width=3 by inj, rex_inv_atom_dx/
+qed-.
+
+lemma rexs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤*[R, ⋆s] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+#R #Y1 #Y2 #s #H elim H -Y2
+[ #Y2 #H elim (rex_inv_sort … H) -H *
+ /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/
+| #Y #Y2 #_ #H elim (rex_inv_sort … H) -H *
+ [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
+ | #I #I2 #L #L2 #HL2 #H #H2 * *
+ [ #H1 #H0 destruct
+ | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct
+ /4 width=7 by ex3_4_intro, rexs_step_dx, or_intror/
+ ]
+ ]
+]
+qed-.
+
+lemma rexs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤*[R, §l] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+#R #Y1 #Y2 #l #H elim H -Y2
+[ #Y2 #H elim (rex_inv_gref … H) -H *
+ /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/
+| #Y #Y2 #_ #H elim (rex_inv_gref … H) -H *
+ [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
+ | #I #I2 #L #L2 #HL2 #H #H2 * *
+ [ #H1 #H0 destruct
+ | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct
+ /4 width=7 by ex3_4_intro, rexs_step_dx, or_intror/
+ ]
+ ]
+]
+qed-.
+
+lemma rexs_inv_bind: ∀R. (∀L. reflexive … (R L)) →
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ⪤*[R, V] L2 & L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V.
+#R #HR #p #I #L1 #L2 #V #T #H elim H -L2
+[ #L2 #H elim (rex_inv_bind … V ? H) -H /3 width=1 by inj, conj/
+| #L #L2 #_ #H * elim (rex_inv_bind … V ? H) -H /3 width=3 by rexs_step_dx, conj/
+]
+qed-.
+
+lemma rexs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 →
+ ∧∧ L1 ⪤*[R, V] L2 & L1 ⪤*[R, T] L2.
+#R #I #L1 #L2 #V #T #H elim H -L2
+[ #L2 #H elim (rex_inv_flat … H) -H /3 width=1 by inj, conj/
+| #L #L2 #_ #H * elim (rex_inv_flat … H) -H /3 width=3 by rexs_step_dx, conj/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rexs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤*[R, ⋆s] Y2 →
+ ∃∃I2,L2. L1 ⪤*[R, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
+#R #I1 #Y2 #L1 #s #H elim (rexs_inv_sort … H) -H *
+[ #H destruct
+| #Z #I2 #Y1 #L2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma rexs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤*[R, ⋆s] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ⪤*[R, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
+#R #I2 #Y1 #L2 #s #H elim (rexs_inv_sort … H) -H *
+[ #_ #H destruct
+| #I1 #Z #L1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma rexs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤*[R, §l] Y2 →
+ ∃∃I2,L2. L1 ⪤*[R, §l] L2 & Y2 = L2.ⓘ{I2}.
+#R #I1 #Y2 #L1 #l #H elim (rexs_inv_gref … H) -H *
+[ #H destruct
+| #Z #I2 #Y1 #L2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma rexs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤*[R, §l] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ⪤*[R, §l] L2 & Y1 = L1.ⓘ{I1}.
+#R #I2 #Y1 #L2 #l #H elim (rexs_inv_gref … H) -H *
+[ #_ #H destruct
+| #I1 #Z #L1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma rexs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ②{I}V.T] L2 → L1 ⪤*[R, V] L2.
+#R #I #L1 #L2 #V #T #H elim H -L2
+/3 width=5 by rex_fwd_pair_sn, rexs_step_dx, inj/
+qed-.
+
+lemma rexs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) →
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
+ L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V.
+#R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind … H) -H //
+qed-.
+
+lemma rexs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → L1 ⪤*[R, T] L2.
+#R #I #L1 #L2 #V #T #H elim (rexs_inv_flat … H) -H //
+qed-.
+
+(* Basic_2A1: removed theorems 2:
+ TC_lpx_sn_inv_pair1 TC_lpx_sn_inv_pair2
+*)
--- /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/seq_seq.ma".
+include "basic_2/static/rex_drops.ma".
+include "basic_2/i_static/rexs.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+definition tc_f_dedropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
+ ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
+ ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+
+definition tc_f_dropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
+
+definition tc_f_dropable_dx: predicate (relation3 lenv term term) ≝
+ λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
+ ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2.
+
+(* Properties with generic slicing for local environments *******************)
+
+lemma dedropable_sn_CTC: ∀R. f_dedropable_sn R → tc_f_dedropable_sn R.
+#R #HR #b #f #L1 #K1 #HLK1 #K2 #T #H elim H -K2
+[ #K2 #HK12 #U #HTU elim (HR … HLK1 … HK12 … HTU) -K1 -T -HR
+ /3 width=4 by ex3_intro, inj/
+| #K #K2 #_ #HK2 #IH #U #HTU -HLK1
+ elim (IH … HTU) -IH #L #HL1 #HLK
+ elim (HR … HLK … HK2 … HTU) -K -T -HR
+ /3 width=6 by seq_trans, rexs_step_dx, ex3_intro/
+]
+qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+lemma dropable_sn_CTC: ∀R. f_dropable_sn R → tc_f_dropable_sn R.
+#R #HR #b #f #L1 #K1 #HLK1 #Hf #L2 #U #H elim H -L2
+[ #L2 #HL12 #T #HTU elim (HR … HLK1 … HL12 … HTU) -L1 -U -HR
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 #IH #T #HTU -HLK1
+ elim (IH … HTU) -IH #K #HK1 #HLK
+ elim (HR … HLK … HL2 … HTU) -L -U -HR
+ /3 width=3 by rexs_step_dx, ex2_intro/
+]
+qed-.
+
+lemma dropable_dx_CTC: ∀R. f_dropable_dx R → tc_f_dropable_dx R.
+#R #HR #L1 #L2 #U #H elim H -L2
+[ #L2 #HL12 #b #f #K2 #HLK2 #Hf #T #HTU
+ elim (HR … HL12 … HLK2 … HTU) -L2 -U -HR
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 #IH #b #f #K2 #HLK2 #Hf #T #HTU
+ elim (HR … HL2 … HLK2 … HTU) -L2 -HR // #K #HLK #HK2
+ elim (IH … HLK … HTU) -IH -L -U
+ /3 width=5 by rexs_step_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/static/rex_fqup.ma".
+include "basic_2/i_static/rexs.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+(* Advanced properties ******************************************************)
+
+lemma rexs_refl: ∀R. c_reflexive … R →
+ ∀T. reflexive … (rexs R T).
+/3 width=1 by rex_refl, inj/ qed.
+
+(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
+lemma rexs_pair_refl: ∀R. c_reflexive … R →
+ ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R, T] L.ⓑ{I}V2.
+#R #HR #L #V1 #V2 #H elim H -V2
+/3 width=3 by rexs_step_dx, rex_pair_refl, inj/
+qed.
+
+lemma rexs_tc: ∀R,L1,L2,T,f. 𝐈⦃f⦄ → TC … (sex cfull (cext2 R) f) L1 L2 →
+ L1 ⪤*[R, T] L2.
+#R #L1 #L2 #T #f #Hf #H elim H -L2
+[ elim (frees_total L1 T) | #L elim (frees_total L T) ]
+/5 width=7 by sex_sdj, rexs_step_dx, sdj_isid_sn, inj, ex2_intro/
+qed.
+
+(* Advanced eliminators *****************************************************)
+
+lemma rexs_ind_sn: ∀R. c_reflexive … R →
+ ∀L1,T. ∀Q:predicate …. Q L1 →
+ (∀L,L2. L1 ⪤*[R, T] L → L ⪤[R, T] L2 → Q L → Q L2) →
+ ∀L2. L1 ⪤*[R, T] L2 → Q L2.
+#R #HR #L1 #T #Q #HL1 #IHL1 #L2 #HL12
+@(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by rex_refl/
+qed-.
+
+lemma rexs_ind_dx: ∀R. c_reflexive … R →
+ ∀L2,T. ∀Q:predicate …. Q L2 →
+ (∀L1,L. L1 ⪤[R, T] L → L ⪤*[R, T] L2 → Q L → Q L1) →
+ ∀L1. L1 ⪤*[R, T] L2 → Q L1.
+#R #HR #L2 #Q #HL2 #IHL2 #L1 #HL12
+@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by rex_refl/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rexs_inv_bind_void: ∀R. c_reflexive … R →
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ⪤*[R, V] L2 & L1.ⓧ ⪤*[R, T] L2.ⓧ.
+#R #HR #p #I #L1 #L2 #V #T #H @(rexs_ind_sn … HR … H) -L2
+[ /3 width=1 by rexs_refl, conj/
+| #L #L2 #_ #H * elim (rex_inv_bind_void … H) -H /3 width=3 by rexs_step_dx, conj/
+]
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma rexs_fwd_bind_dx_void: ∀R. c_reflexive … R →
+ ∀p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
+ L1.ⓧ ⪤*[R, T] L2.ⓧ.
+#R #HR #p #I #L1 #L2 #V #T #H elim (rexs_inv_bind_void … 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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/static/rex_length.ma".
+include "basic_2/i_static/rexs.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Basic_2A1: uses: TC_lpx_sn_fwd_length *)
+lemma rexs_fwd_length: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 → |L1| = |L2|.
+#R #L1 #L2 #T #H elim H -L2
+[ #L2 #HL12 >(rex_fwd_length … HL12) -HL12 //
+| #L #L2 #_ #HL2 #IHL1
+ >IHL1 -L1 >(rex_fwd_length … HL2) -HL2 //
+]
+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/lex_tc.ma".
+include "basic_2/static/req_fqup.ma".
+include "basic_2/static/req_fsle.ma".
+include "basic_2/i_static/rexs_fqup.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+(* Properties with generic extension of a context sensitive relation ********)
+
+lemma rexs_lex: ∀R. c_reflexive … R →
+ ∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤*[R, T] L2.
+#R #HR #L1 #L2 #T *
+/5 width=7 by rexs_tc, sex_inv_tc_dx, sex_co, ext2_inv_tc, ext2_refl/
+qed.
+
+lemma rexs_lex_req: ∀R. c_reflexive … R →
+ ∀L1,L. L1 ⪤[CTC … R] L → ∀L2,T. L ≡[T] L2 →
+ L1 ⪤*[R, T] L2.
+/3 width=3 by rexs_lex, rexs_step_dx, req_fwd_rex/ qed.
+
+(* Inversion lemmas with generic extension of a context sensitive relation **)
+
+(* Note: s_rs_transitive_lex_inv_isid could be invoked in the last auto but makes it too slow *)
+lemma rexs_inv_lex_req: ∀R. c_reflexive … R →
+ rex_fsge_compatible R →
+ s_rs_transitive … R (λ_.lex R) →
+ req_transitive R →
+ ∀L1,L2,T. L1 ⪤*[R, T] L2 →
+ ∃∃L. L1 ⪤[CTC … R] L & L ≡[T] L2.
+#R #H1R #H2R #H3R #H4R #L1 #L2 #T #H
+lapply (s_rs_transitive_lex_inv_isid … H3R) -H3R #H3R
+@(rexs_ind_sn … H1R … H) -H -L2
+[ /4 width=3 by req_refl, lex_refl, inj, ex2_intro/
+| #L0 #L2 #_ #HL02 * #L * #f0 #Hf0 #HL1 #HL0
+ lapply (req_rex_trans … HL0 … HL02) -L0 // * #f1 #Hf1 #HL2
+ elim (sex_sdj_split … ceq_ext … HL2 f0 ?) -HL2
+ [ #L0 #HL0 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ]
+ lapply (sex_sdj … HL0 f1 ?) /2 width=1 by sdj_isid_sn/ #H
+ elim (frees_sex_conf … Hf1 … H) // -H2R -H #f2 #Hf2 #Hf21
+ lapply (sle_sex_trans … HL02 … Hf21) -f1 // #HL02
+ lapply (sex_co ?? cfull (CTC … (cext2 R)) … HL1) -HL1 /2 width=1 by ext2_inv_tc/ #HL1
+ /8 width=11 by sex_inv_tc_dx, sex_tc_dx, sex_co, ext2_tc, ext2_refl, step, ex2_intro/ (**) (* full auto too slow *)
+]
+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/static/rex_fsle.ma".
+include "basic_2/i_static/rexs.ma".
+
+(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
+
+(* Advanced properties ******************************************************)
+
+lemma rexs_sym: ∀R. rex_fsge_compatible R →
+ (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+ ∀T. symmetric … (rexs R T).
+#R #H1R #H2R #T #L1 #L2 #H elim H -L2
+/4 width=3 by rex_sym, rexs_step_sn, inj/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem rexs_trans: ∀R,T. Transitive … (rexs R T).
+#R #T #L1 #L #HL1 #L2 #HL2 @(trans_TC … HL1 HL2) (**) (* auto fails *)
+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/star.ma".
-include "basic_2/notation/relations/relationstarstar_4.ma".
-include "basic_2/static/lfxs.ma".
-
-(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
-
-definition tc_lfxs (R): term → relation lenv ≝ CTC … (lfxs R).
-
-interpretation "iterated extension on referred entries (local environment)"
- 'RelationStarStar R T L1 L2 = (tc_lfxs R T L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma tc_lfxs_step_dx: ∀R,L1,L,T. L1 ⪤**[R, T] L →
- ∀L2. L ⪤*[R, T] L2 → L1 ⪤**[R, T] L2.
-#R #L1 #L2 #T #HL1 #L2 @step @HL1 (**) (* auto fails *)
-qed-.
-
-lemma tc_lfxs_step_sn: ∀R,L1,L,T. L1 ⪤*[R, T] L →
- ∀L2. L ⪤**[R, T] L2 → L1 ⪤**[R, T] L2.
-#R #L1 #L2 #T #HL1 #L2 @TC_strap @HL1 (**) (* auto fails *)
-qed-.
-
-lemma tc_lfxs_atom: ∀R,I. ⋆ ⪤**[R, ⓪{I}] ⋆.
-/2 width=1 by inj/ qed.
-
-lemma tc_lfxs_sort: ∀R,I,L1,L2,V1,V2,s.
- L1 ⪤**[R, ⋆s] L2 → L1.ⓑ{I}V1 ⪤**[R, ⋆s] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #s #H elim H -L2
-/3 width=4 by lfxs_sort, tc_lfxs_step_dx, inj/
-qed.
-
-lemma tc_lfxs_pair: ∀R. (∀L. reflexive … (R L)) →
- ∀I,L1,L2,V. L1 ⪤**[R, V] L2 →
- L1.ⓑ{I}V ⪤**[R, #0] L2.ⓑ{I}V.
-#R #HR #I #L1 #L2 #V #H elim H -L2
-/3 width=5 by lfxs_pair, tc_lfxs_step_dx, inj/
-qed.
-
-lemma tc_lfxs_unit: ∀R,f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cext2 R, cfull, f] L2 →
- L1.ⓤ{I} ⪤**[R, #0] L2.ⓤ{I}.
-/3 width=3 by lfxs_unit, inj/ qed.
-
-lemma tc_lfxs_lref: ∀R,I,L1,L2,V1,V2,i.
- L1 ⪤**[R, #i] L2 → L1.ⓑ{I}V1 ⪤**[R, #↑i] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #i #H elim H -L2
-/3 width=4 by lfxs_lref, tc_lfxs_step_dx, inj/
-qed.
-
-lemma tc_lfxs_gref: ∀R,I,L1,L2,V1,V2,l.
- L1 ⪤**[R, §l] L2 → L1.ⓑ{I}V1 ⪤**[R, §l] L2.ⓑ{I}V2.
-#R #I #L1 #L2 #V1 #V2 #l #H elim H -L2
-/3 width=4 by lfxs_gref, tc_lfxs_step_dx, inj/
-qed.
-
-lemma tc_lfxs_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- ∀L1,L2,T. L1 ⪤**[R1, T] L2 → L1 ⪤**[R2, T] L2.
-#R1 #R2 #HR #L1 #L2 #T #H elim H -L2
-/4 width=5 by lfxs_co, tc_lfxs_step_dx, inj/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_2A1: uses: TC_lpx_sn_inv_atom1 *)
-lemma tc_lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⪤**[R, ⓪{I}] Y2 → Y2 = ⋆.
-#R #I #Y2 #H elim H -Y2 /3 width=3 by inj, lfxs_inv_atom_sn/
-qed-.
-
-(* Basic_2A1: uses: TC_lpx_sn_inv_atom2 *)
-lemma tc_lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⪤**[R, ⓪{I}] ⋆ → Y1 = ⋆.
-#R #I #Y1 #H @(TC_ind_dx ??????? H) -Y1
-/3 width=3 by inj, lfxs_inv_atom_dx/
-qed-.
-
-lemma tc_lfxs_inv_sort: ∀R,Y1,Y2,s. Y1 ⪤**[R, ⋆s] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤**[R, ⋆s] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-#R #Y1 #Y2 #s #H elim H -Y2
-[ #Y2 #H elim (lfxs_inv_sort … H) -H *
- /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/
-| #Y #Y2 #_ #H elim (lfxs_inv_sort … H) -H *
- [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
- | #I #I2 #L #L2 #HL2 #H #H2 * *
- [ #H1 #H0 destruct
- | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct
- /4 width=7 by ex3_4_intro, tc_lfxs_step_dx, or_intror/
- ]
- ]
-]
-qed-.
-
-lemma tc_lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⪤**[R, §l] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤**[R, §l] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-#R #Y1 #Y2 #l #H elim H -Y2
-[ #Y2 #H elim (lfxs_inv_gref … H) -H *
- /4 width=8 by ex3_4_intro, inj, or_introl, or_intror, conj/
-| #Y #Y2 #_ #H elim (lfxs_inv_gref … H) -H *
- [ #H #H2 * * /3 width=7 by ex3_4_intro, or_introl, or_intror, conj/
- | #I #I2 #L #L2 #HL2 #H #H2 * *
- [ #H1 #H0 destruct
- | #I1 #I0 #L1 #L0 #HL10 #H1 #H0 destruct
- /4 width=7 by ex3_4_intro, tc_lfxs_step_dx, or_intror/
- ]
- ]
-]
-qed-.
-
-lemma tc_lfxs_inv_bind: ∀R. (∀L. reflexive … (R L)) →
- ∀p,I,L1,L2,V,T. L1 ⪤**[R, ⓑ{p,I}V.T] L2 →
- L1 ⪤**[R, V] L2 ∧ L1.ⓑ{I}V ⪤**[R, T] L2.ⓑ{I}V.
-#R #HR #p #I #L1 #L2 #V #T #H elim H -L2
-[ #L2 #H elim (lfxs_inv_bind … V ? H) -H /3 width=1 by inj, conj/
-| #L #L2 #_ #H * elim (lfxs_inv_bind … V ? H) -H /3 width=3 by tc_lfxs_step_dx, conj/
-]
-qed-.
-
-lemma tc_lfxs_inv_flat: ∀R,I,L1,L2,V,T. L1 ⪤**[R, ⓕ{I}V.T] L2 →
- L1 ⪤**[R, V] L2 ∧ L1 ⪤**[R, T] L2.
-#R #I #L1 #L2 #V #T #H elim H -L2
-[ #L2 #H elim (lfxs_inv_flat … H) -H /3 width=1 by inj, conj/
-| #L #L2 #_ #H * elim (lfxs_inv_flat … H) -H /3 width=3 by tc_lfxs_step_dx, conj/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma tc_lfxs_inv_sort_bind_sn: ∀R,I1,Y2,L1,s. L1.ⓘ{I1} ⪤**[R, ⋆s] Y2 →
- ∃∃I2,L2. L1 ⪤**[R, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
-#R #I1 #Y2 #L1 #s #H elim (tc_lfxs_inv_sort … H) -H *
-[ #H destruct
-| #Z #I2 #Y1 #L2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma tc_lfxs_inv_sort_bind_dx: ∀R,I2,Y1,L2,s. Y1 ⪤**[R, ⋆s] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ⪤**[R, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
-#R #I2 #Y1 #L2 #s #H elim (tc_lfxs_inv_sort … H) -H *
-[ #_ #H destruct
-| #I1 #Z #L1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma tc_lfxs_inv_gref_bind_sn: ∀R,I1,Y2,L1,l. L1.ⓘ{I1} ⪤**[R, §l] Y2 →
- ∃∃I2,L2. L1 ⪤**[R, §l] L2 & Y2 = L2.ⓘ{I2}.
-#R #I1 #Y2 #L1 #l #H elim (tc_lfxs_inv_gref … H) -H *
-[ #H destruct
-| #Z #I2 #Y1 #L2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma tc_lfxs_inv_gref_bind_dx: ∀R,I2,Y1,L2,l. Y1 ⪤**[R, §l] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ⪤**[R, §l] L2 & Y1 = L1.ⓘ{I1}.
-#R #I2 #Y1 #L2 #l #H elim (tc_lfxs_inv_gref … H) -H *
-[ #_ #H destruct
-| #I1 #Z #L1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tc_lfxs_fwd_pair_sn: ∀R,I,L1,L2,V,T. L1 ⪤**[R, ②{I}V.T] L2 → L1 ⪤**[R, V] L2.
-#R #I #L1 #L2 #V #T #H elim H -L2
-/3 width=5 by lfxs_fwd_pair_sn, tc_lfxs_step_dx, inj/
-qed-.
-
-lemma tc_lfxs_fwd_bind_dx: ∀R. (∀L. reflexive … (R L)) →
- ∀p,I,L1,L2,V,T. L1 ⪤**[R, ⓑ{p,I}V.T] L2 →
- L1.ⓑ{I}V ⪤**[R, T] L2.ⓑ{I}V.
-#R #HR #p #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_bind … H) -H //
-qed-.
-
-lemma tc_lfxs_fwd_flat_dx: ∀R,I,L1,L2,V,T. L1 ⪤**[R, ⓕ{I}V.T] L2 → L1 ⪤**[R, T] L2.
-#R #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_flat … H) -H //
-qed-.
-
-(* Basic_2A1: removed theorems 2:
- TC_lpx_sn_inv_pair1 TC_lpx_sn_inv_pair2
-*)
+++ /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/lreq_lreq.ma".
-include "basic_2/static/lfxs_drops.ma".
-include "basic_2/i_static/tc_lfxs.ma".
-
-(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
-
-definition tc_f_dedropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
- ∀K2,T. K1 ⪤**[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
- ∃∃L2. L1 ⪤**[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
-
-definition tc_f_dropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
- ∀L2,U. L1 ⪤**[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
- ∃∃K2. K1 ⪤**[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
-
-definition tc_f_dropable_dx: predicate (relation3 lenv term term) ≝
- λR. ∀L1,L2,U. L1 ⪤**[R, U] L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤**[R, T] K2.
-
-(* Properties with generic slicing for local environments *******************)
-
-lemma dedropable_sn_CTC: ∀R. f_dedropable_sn R → tc_f_dedropable_sn R.
-#R #HR #b #f #L1 #K1 #HLK1 #K2 #T #H elim H -K2
-[ #K2 #HK12 #U #HTU elim (HR … HLK1 … HK12 … HTU) -K1 -T -HR
- /3 width=4 by ex3_intro, inj/
-| #K #K2 #_ #HK2 #IH #U #HTU -HLK1
- elim (IH … HTU) -IH #L #HL1 #HLK
- elim (HR … HLK … HK2 … HTU) -K -T -HR
- /3 width=6 by lreq_trans, tc_lfxs_step_dx, ex3_intro/
-]
-qed-.
-
-(* Inversion lemmas with generic slicing for local environments *************)
-
-lemma dropable_sn_CTC: ∀R. f_dropable_sn R → tc_f_dropable_sn R.
-#R #HR #b #f #L1 #K1 #HLK1 #Hf #L2 #U #H elim H -L2
-[ #L2 #HL12 #T #HTU elim (HR … HLK1 … HL12 … HTU) -L1 -U -HR
- /3 width=3 by inj, ex2_intro/
-| #L #L2 #_ #HL2 #IH #T #HTU -HLK1
- elim (IH … HTU) -IH #K #HK1 #HLK
- elim (HR … HLK … HL2 … HTU) -L -U -HR
- /3 width=3 by tc_lfxs_step_dx, ex2_intro/
-]
-qed-.
-
-lemma dropable_dx_CTC: ∀R. f_dropable_dx R → tc_f_dropable_dx R.
-#R #HR #L1 #L2 #U #H elim H -L2
-[ #L2 #HL12 #b #f #K2 #HLK2 #Hf #T #HTU
- elim (HR … HL12 … HLK2 … HTU) -L2 -U -HR
- /3 width=3 by inj, ex2_intro/
-| #L #L2 #_ #HL2 #IH #b #f #K2 #HLK2 #Hf #T #HTU
- elim (HR … HL2 … HLK2 … HTU) -L2 -HR // #K #HLK #HK2
- elim (IH … HLK … HTU) -IH -L -U
- /3 width=5 by tc_lfxs_step_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/static/lfxs_fqup.ma".
-include "basic_2/i_static/tc_lfxs.ma".
-
-(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
-
-(* Advanced properties ******************************************************)
-
-lemma tc_lfxs_refl: ∀R. c_reflexive … R →
- ∀T. reflexive … (tc_lfxs R T).
-/3 width=1 by lfxs_refl, inj/ qed.
-
-(* Basic_2A1: uses: TC_lpx_sn_pair TC_lpx_sn_pair_refl *)
-lemma tc_lfxs_pair_refl: ∀R. c_reflexive … R →
- ∀L,V1,V2. CTC … R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤**[R, T] L.ⓑ{I}V2.
-#R #HR #L #V1 #V2 #H elim H -V2
-/3 width=3 by tc_lfxs_step_dx, lfxs_pair_refl, inj/
-qed.
-
-lemma tc_lfxs_tc: ∀R,L1,L2,T,f. 𝐈⦃f⦄ → TC … (lexs cfull (cext2 R) f) L1 L2 →
- L1 ⪤**[R, T] L2.
-#R #L1 #L2 #T #f #Hf #H elim H -L2
-[ elim (frees_total L1 T) | #L elim (frees_total L T) ]
-/5 width=7 by lexs_sdj, tc_lfxs_step_dx, sdj_isid_sn, inj, ex2_intro/
-qed.
-
-(* Advanced eliminators *****************************************************)
-
-lemma tc_lfxs_ind_sn: ∀R. c_reflexive … R →
- ∀L1,T. ∀Q:predicate …. Q L1 →
- (∀L,L2. L1 ⪤**[R, T] L → L ⪤*[R, T] L2 → Q L → Q L2) →
- ∀L2. L1 ⪤**[R, T] L2 → Q L2.
-#R #HR #L1 #T #Q #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) /2 width=1 by lfxs_refl/
-qed-.
-
-lemma tc_lfxs_ind_dx: ∀R. c_reflexive … R →
- ∀L2,T. ∀Q:predicate …. Q L2 →
- (∀L1,L. L1 ⪤*[R, T] L → L ⪤**[R, T] L2 → Q L → Q L1) →
- ∀L1. L1 ⪤**[R, T] L2 → Q L1.
-#R #HR #L2 #Q #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) /2 width=4 by lfxs_refl/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma tc_lfxs_inv_bind_void: ∀R. c_reflexive … R →
- ∀p,I,L1,L2,V,T. L1 ⪤**[R, ⓑ{p,I}V.T] L2 →
- L1 ⪤**[R, V] L2 ∧ L1.ⓧ ⪤**[R, T] L2.ⓧ.
-#R #HR #p #I #L1 #L2 #V #T #H @(tc_lfxs_ind_sn … HR … H) -L2
-[ /3 width=1 by tc_lfxs_refl, conj/
-| #L #L2 #_ #H * elim (lfxs_inv_bind_void … H) -H /3 width=3 by tc_lfxs_step_dx, conj/
-]
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma tc_lfxs_fwd_bind_dx_void: ∀R. c_reflexive … R →
- ∀p,I,L1,L2,V,T. L1 ⪤**[R, ⓑ{p,I}V.T] L2 →
- L1.ⓧ ⪤**[R, T] L2.ⓧ.
-#R #HR #p #I #L1 #L2 #V #T #H elim (tc_lfxs_inv_bind_void … 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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/lfxs_length.ma".
-include "basic_2/i_static/tc_lfxs.ma".
-
-(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
-
-(* Forward lemmas with length for local environments ************************)
-
-(* Basic_2A1: uses: TC_lpx_sn_fwd_length *)
-lemma tc_lfxs_fwd_length: ∀R,L1,L2,T. L1 ⪤**[R, T] L2 → |L1| = |L2|.
-#R #L1 #L2 #T #H elim H -L2
-[ #L2 #HL12 >(lfxs_fwd_length … HL12) -HL12 //
-| #L #L2 #_ #HL2 #IHL1
- >IHL1 -L1 >(lfxs_fwd_length … HL2) -HL2 //
-]
-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/lex_tc.ma".
-include "basic_2/static/lfeq_fqup.ma".
-include "basic_2/static/lfeq_fsle.ma".
-include "basic_2/i_static/tc_lfxs_fqup.ma".
-
-(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
-
-(* Properties with generic extension of a context sensitive relation ********)
-
-lemma tc_lfxs_lex: ∀R. c_reflexive … R →
- ∀L1,L2,T. L1 ⪤[CTC … R] L2 → L1 ⪤**[R, T] L2.
-#R #HR #L1 #L2 #T *
-/5 width=7 by tc_lfxs_tc, lexs_inv_tc_dx, lexs_co, ext2_inv_tc, ext2_refl/
-qed.
-
-lemma tc_lfxs_lex_lfeq: ∀R. c_reflexive … R →
- ∀L1,L. L1 ⪤[CTC … R] L → ∀L2,T. L ≡[T] L2 →
- L1 ⪤**[R, T] L2.
-/3 width=3 by tc_lfxs_lex, tc_lfxs_step_dx, lfeq_fwd_lfxs/ qed.
-
-(* Inversion lemmas with generic extension of a context sensitive relation **)
-
-(* Note: s_rs_transitive_lex_inv_isid could be invoked in the last auto but makes it too slow *)
-lemma tc_lfxs_inv_lex_lfeq: ∀R. c_reflexive … R →
- lfxs_fsge_compatible R →
- s_rs_transitive … R (λ_.lex R) →
- lfeq_transitive R →
- ∀L1,L2,T. L1 ⪤**[R, T] L2 →
- ∃∃L. L1 ⪤[CTC … R] L & L ≡[T] L2.
-#R #H1R #H2R #H3R #H4R #L1 #L2 #T #H
-lapply (s_rs_transitive_lex_inv_isid … H3R) -H3R #H3R
-@(tc_lfxs_ind_sn … H1R … H) -H -L2
-[ /4 width=3 by lfeq_refl, lex_refl, inj, ex2_intro/
-| #L0 #L2 #_ #HL02 * #L * #f0 #Hf0 #HL1 #HL0
- lapply (lfeq_lfxs_trans … HL0 … HL02) -L0 // * #f1 #Hf1 #HL2
- elim (lexs_sdj_split … ceq_ext … HL2 f0 ?) -HL2
- [ #L0 #HL0 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ]
- lapply (lexs_sdj … HL0 f1 ?) /2 width=1 by sdj_isid_sn/ #H
- elim (frees_lexs_conf … Hf1 … H) // -H2R -H #f2 #Hf2 #Hf21
- lapply (sle_lexs_trans … HL02 … Hf21) -f1 // #HL02
- lapply (lexs_co ?? cfull (CTC … (cext2 R)) … HL1) -HL1 /2 width=1 by ext2_inv_tc/ #HL1
- /8 width=11 by lexs_inv_tc_dx, lexs_tc_dx, lexs_co, ext2_tc, ext2_refl, step, ex2_intro/ (**) (* full auto too slow *)
-]
-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/static/lfxs_fsle.ma".
-include "basic_2/i_static/tc_lfxs.ma".
-
-(* ITERATED EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ***)
-
-(* Advanced properties ******************************************************)
-
-lemma tc_lfxs_sym: ∀R. lfxs_fsge_compatible R →
- (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀T. symmetric … (tc_lfxs R T).
-#R #H1R #H2R #T #L1 #L2 #H elim H -L2
-/4 width=3 by lfxs_sym, tc_lfxs_step_sn, inj/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem tc_lfxs_trans: ∀R,T. Transitive … (tc_lfxs R T).
-#R #T #L1 #L #HL1 #L2 #HL2 @(trans_TC … HL1 HL2) (**) (* auto fails *)
-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( L1 ⪤[ break term 46 R, break term 46 T ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'Relation $R $T $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( L1 ⪤[ break term 46 R1, break term 46 R2, break term 46 f ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'Relation $R1 $R2 $f $L1 $L2 }.
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L1 ⪤ * [ break term 46 R, break term 46 T ] break term 46 L2 )"
+notation "hvbox( L1 ⪤*[ break term 46 R, break term 46 T ] break term 46 L2 )"
non associative with precedence 45
for @{ 'RelationStar $R $T $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( L1 ⪤ * [ break term 46 R1, break term 46 R2, break term 46 f ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'RelationStar $R1 $R2 $f $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( L1 ⪤ * * [ break term 46 R, break term 46 T ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'RelationStarStar $R $T $L1 $L2 }.
include "ground_2/relocation/rtmap_coafter.ma".
include "basic_2/notation/relations/rdropstar_3.ma".
include "basic_2/notation/relations/rdropstar_4.ma".
-include "basic_2/relocation/lreq.ma".
+include "basic_2/relocation/seq.ma".
include "basic_2/relocation/lifts_bind.ma".
(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
(**************************************************************************)
include "ground_2/lib/star.ma".
-include "basic_2/relocation/lreq_lreq.ma".
+include "basic_2/relocation/seq_seq.ma".
(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
/3 width=4 by inj, ex3_intro/
| #K #K2 #_ #HK2 #IH #f2 #Hf elim (IH … Hf) -IH -K1
#L #H1L1 #HLK #H2L1 elim (HR … HLK … HK2 … Hf) -HR -f1 -K
- /3 width=6 by lreq_trans, step, ex3_intro/
+ /3 width=6 by seq_trans, step, ex3_intro/
]
qed-.
include "basic_2/relocation/lex.ma".
include "basic_2/relocation/drops_cext2.ma".
-include "basic_2/relocation/drops_lexs.ma".
+include "basic_2/relocation/drops_sex.ma".
(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
lemma lex_liftable_dedropable_sn (R): c_reflexive … R →
d_liftable2_sn … lifts R → dedropable_sn R.
#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 * #f1 #Hf1 #HK12
-elim (lexs_liftable_co_dedropable_sn … HLK1 … HK12) -K1
+elim (sex_liftable_co_dedropable_sn … HLK1 … HK12) -K1
/3 width=6 by cext2_d_liftable2_sn, cfull_lift_sn, ext2_refl, coafter_isid_dx, ex3_intro, ex2_intro/
qed-.
(* Basic_2A1: was: lpx_sn_deliftable_dropable *)
lemma lex_dropable_sn (R): dropable_sn R.
#R #b #f #L1 #K1 #HLK1 #H1f #L2 * #f2 #Hf2 #HL12
-elim (lexs_co_dropable_sn … HLK1 … HL12) -L1
+elim (sex_co_dropable_sn … HLK1 … HL12) -L1
/3 width=3 by coafter_isid_dx, ex2_intro/
qed-.
(* Basic_2A1: was: lpx_sn_dropable *)
lemma lex_dropable_dx (R): dropable_dx R.
#R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #Hf
-elim (lexs_co_dropable_dx … HL12 … HLK2) -L2
+elim (sex_co_dropable_dx … HL12 … HLK2) -L2
/3 width=5 by coafter_isid_dx, ex2_intro/
qed-.
∀b,f,I,K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 → 𝐔⦃f⦄ →
∃∃K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R] K2 & R K1 V1 V2.
#R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K1 #V1 #HLK1 #Hf
-elim (lexs_drops_conf_push … HL12 … HLK1 Hf f2) -L1 -Hf
+elim (sex_drops_conf_push … HL12 … HLK1 Hf f2) -L1 -Hf
[ #Z2 #K2 #HLK2 #HK12 #H
elim (ext2_inv_pair_sn … H) -H #V2 #HV12 #H destruct
/3 width=5 by ex3_2_intro, ex2_intro/
∀b,f,I,K2,V2. ⬇*[b, f] L2 ≘ K2.ⓑ{I}V2 → 𝐔⦃f⦄ →
∃∃K1,V1. ⬇*[b, f] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R] K2 & R K1 V1 V2.
#R #L1 #L2 * #f2 #Hf2 #HL12 #b #f #I #K2 #V2 #HLK2 #Hf
-elim (lexs_drops_trans_push … HL12 … HLK2 Hf f2) -L2 -Hf
+elim (sex_drops_trans_push … HL12 … HLK2 Hf f2) -L2 -Hf
[ #Z1 #K1 #HLK1 #HK12 #H
elim (ext2_inv_pair_dx … H) -H #V1 #HV12 #H destruct
/3 width=5 by ex3_2_intro, ex2_intro/
+++ /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_lifts_bind.ma".
-include "basic_2/relocation/drops.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Properties with entrywise extension of context-sensitive relations *******)
-
-(**) (* changed after commit 13218 *)
-lemma lexs_co_dropable_sn: ∀RN,RP. co_dropable_sn (lexs RN RP).
-#RN #RP #b #f #L1 #K1 #H elim H -f -L1 -K1
-[ #f #Hf #_ #f2 #X #H #f1 #Hf2 >(lexs_inv_atom1 … H) -X
- /4 width=3 by lexs_atom, drops_atom, ex2_intro/
-| #f #I1 #L1 #K1 #_ #IH #Hf #f2 #X #H #f1 #Hf2
- elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H2 destruct
- elim (lexs_inv_push1 … H) -H #I2 #L2 #HL12 #HI12 #H destruct
- elim (IH … HL12 … Hg2) -g2
- /3 width=3 by isuni_inv_next, drops_drop, ex2_intro/
-| #f #I1 #J1 #L1 #K1 #HLK #HJI1 #IH #Hf #f2 #X #H #f1 #Hf2
- lapply (isuni_inv_push … Hf ??) -Hf [3: |*: // ] #Hf
- lapply (drops_fwd_isid … HLK … Hf) -HLK #H0 destruct
- lapply (liftsb_fwd_isid … HJI1 … Hf) -HJI1 #H0 destruct
- elim (coafter_inv_pxx … Hf2) -Hf2 [1,3:* |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
- [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H #I2 #L2 #HL12 #HI12 #H destruct
- elim (IH … HL12 … Hg2) -g2 -IH /2 width=1 by isuni_isid/ #K2 #HK12 #HLK2
- lapply (drops_fwd_isid … HLK2 … Hf) -HLK2 #H0 destruct
- /4 width=3 by drops_refl, lexs_next, lexs_push, isid_push, ex2_intro/
-]
-qed-.
-
-lemma lexs_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f2,L1,L2. L1 ⪤*[cfull, RP, f2] L2 → ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
- ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- f ~⊚ f1 ≘ f2 → L1 ⪤*[RN, RP, f2] L2.
-#RN #RP #HRN #HRP #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
-#g2 #I1 #I2 #L1 #L2 #HL12 #HI12 #IH #f1 #Y1 #Y2 #HK12 #b #f #HY1 #HY2 #H
-[ elim (coafter_inv_xxn … H) [ |*: // ] -H #g #g1 #Hg2 #H1 #H2 destruct
- elim (drops_inv_skip1 … HY1) -HY1 #J1 #K1 #HLK1 #HJI1 #H destruct
- elim (drops_inv_skip1 … HY2) -HY2 #J2 #K2 #HLK2 #HJI2 #H destruct
- elim (lexs_inv_next … HK12) -HK12 #HK12 #HJ12
- elim (HRN … HJ12 … HLK1 … HJI1) -HJ12 -HJI1 #Z #Hz
- >(liftsb_mono … Hz … HJI2) -Z /3 width=9 by lexs_next/
-| elim (coafter_inv_xxp … H) [1,2: |*: // ] -H *
- [ #g #g1 #Hg2 #H1 #H2 destruct
- elim (drops_inv_skip1 … HY1) -HY1 #J1 #K1 #HLK1 #HJI1 #H destruct
- elim (drops_inv_skip1 … HY2) -HY2 #J2 #K2 #HLK2 #HJI2 #H destruct
- elim (lexs_inv_push … HK12) -HK12 #HK12 #HJ12
- elim (HRP … HJ12 … HLK1 … HJI1) -HJ12 -HJI1 #Z #Hz
- >(liftsb_mono … Hz … HJI2) -Z /3 width=9 by lexs_push/
- | #g #Hg2 #H destruct
- lapply (drops_inv_drop1 … HY1) -HY1 #HLK1
- lapply (drops_inv_drop1 … HY2) -HY2 #HLK2
- /3 width=9 by lexs_push/
- ]
-]
-qed-.
-
-lemma lexs_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
- d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- co_dedropable_sn (lexs RN RP).
-#RN #RP #H1RN #H1RP #H2RN #H2RP #b #f #L1 #K1 #H elim H -f -L1 -K1
-[ #f #Hf #X #f1 #H #f2 #Hf2 >(lexs_inv_atom1 … H) -X
- /4 width=4 by drops_atom, lexs_atom, ex3_intro/
-| #f #I1 #L1 #K1 #_ #IHLK1 #K2 #f1 #HK12 #f2 #Hf2
- elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
- elim (IHLK1 … HK12 … Hg2) -K1
- /3 width=6 by drops_drop, lexs_next, lexs_push, ex3_intro/
-| #f #I1 #J1 #L1 #K1 #HLK1 #HJI1 #IHLK1 #X #f1 #H #f2 #Hf2
- elim (coafter_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
- [ elim (lexs_inv_push1 … H) | elim (lexs_inv_next1 … H) ] -H #J2 #K2 #HK12 #HJ12 #H destruct
- [ elim (H2RP … HJ12 … HLK1 … HJI1) | elim (H2RN … HJ12 … HLK1 … HJI1) ] -J1
- elim (IHLK1 … HK12 … Hg2) -K1
- /3 width=6 by drops_skip, lexs_next, lexs_push, ex3_intro/
-]
-qed-.
-
-fact lexs_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
- ∀f2,L1. L1 ⪤*[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[RN, RP, f1] K2.
-#RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
-[ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (lexs_inv_atom2 … H) -H
- #H destruct /4 width=3 by lexs_atom, drops_atom, ex2_intro/
-| #f #I2 #L2 #K2 #_ #IH #Hf #f2 #X #HX #f1 #Hf2
- elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
- elim (lexs_inv_push2 … HX) -HX #I1 #L1 #HL12 #HI12 #H destruct
- elim (IH … HL12 … Hg2) -L2 -I2 -g2
- /3 width=3 by drops_drop, isuni_inv_next, ex2_intro/
-| #f #I2 #J2 #L2 #K2 #_ #HJI2 #IH #Hf #f2 #X #HX #f1 #Hf2
- elim (coafter_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
- [ elim (lexs_inv_push2 … HX) | elim (lexs_inv_next2 … HX) ] -HX #I1 #L1 #HL12 #HI12 #H destruct
- elim (IH … HL12 … Hg2) -L2 -g2 /2 width=3 by isuni_fwd_push/ #K1 #HLK1 #HK12
- lapply (isuni_inv_push … Hf ??) -Hf [3,6: |*: // ] #Hf
- lapply (liftsb_fwd_isid … HJI2 … Hf) #H destruct -HJI2
- lapply (drops_fwd_isid … HLK1 … Hf) #H destruct -HLK1
- /4 width=5 by lexs_next, lexs_push, drops_refl, isid_push, ex2_intro/
-]
-qed-.
-
-lemma lexs_co_dropable_dx: ∀RN,RP. co_dropable_dx (lexs RN RP).
-/2 width=5 by lexs_dropable_dx_aux/ qed-.
-
-lemma lexs_drops_conf_next: ∀RN,RP.
- ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
- ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
- ∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤*[RN, RP, f1] K2 & RN K1 I1 I2.
-#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
-elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
-#X #HX #HLK2 elim (lexs_inv_next1 … HX) -HX
-#I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma lexs_drops_conf_push: ∀RN,RP.
- ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
- ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
- ∀f1. f ~⊚ ⫯f1 ≘ f2 →
- ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤*[RN, RP, f1] K2 & RP K1 I1 I2.
-#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
-elim (lexs_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
-#X #HX #HLK2 elim (lexs_inv_push1 … HX) -HX
-#I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma lexs_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
- ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
- ∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤*[RN, RP, f1] K2 & RN K1 I1 I2.
-#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
-elim (lexs_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
-#X #HLK1 #HX elim (lexs_inv_next2 … HX) -HX
-#I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma lexs_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
- ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
- ∀f1. f ~⊚ ⫯f1 ≘ f2 →
- ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤*[RN, RP, f1] K2 & RP K1 I1 I2.
-#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
-elim (lexs_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
-#X #HLK1 #HX elim (lexs_inv_push2 … HX) -HX
-#I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma drops_lexs_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
- d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
- ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
- ∀f2. f ~⊚ f1 ≘ ↑f2 →
- ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤*[RN, RP, f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
-#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
-elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
-#X #HX #HLK2 #H1L12 elim (lexs_inv_next1 … HX) -HX
-#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
-qed-.
-
-lemma drops_lexs_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
- d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
- ∀f1,K1,K2. K1 ⪤*[RN, RP, f1] K2 →
- ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
- ∀f2. f ~⊚ f1 ≘ ⫯f2 →
- ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤*[RN, RP, f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
-#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
-elim (lexs_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
-#X #HX #HLK2 #H1L12 elim (lexs_inv_push1 … HX) -HX
-#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
-qed-.
-
-lemma drops_atom2_lexs_conf: ∀RN,RP,b,f1,L1. ⬇*[b, f1] L1 ≘ ⋆ → 𝐔⦃f1⦄ →
- ∀f,L2. L1 ⪤*[RN, RP, f] L2 →
- ∀f2. f1 ~⊚ f2 ≘f → ⬇*[b, f1] L2 ≘ ⋆.
-#RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3
-elim (lexs_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1
-#L #H #HL2 lapply (lexs_inv_atom1 … 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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/drops_lexs.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Properties with ranged equivalence for local environments ****************)
-
-lemma lreq_co_dedropable_sn: co_dedropable_sn lreq.
-@lexs_liftable_co_dedropable_sn
-/2 width=6 by cfull_lift_sn, ceq_lift_sn/ qed-.
-
-lemma lreq_co_dropable_sn: co_dropable_sn lreq.
-@lexs_co_dropable_sn qed-.
-
-lemma lreq_co_dropable_dx: co_dropable_dx lreq.
-@lexs_co_dropable_dx qed-.
-
-(* Basic_2A1: includes: lreq_drop_trans_be *)
-lemma lreq_drops_trans_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
- ∀b,f,I,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} → 𝐔⦃f⦄ →
- ∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} & K1 ≡[f1] K2.
-#f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
-elim (lexs_drops_trans_next … HL12 … HLK2 Hf … Hf2) -f2 -L2 -Hf
-#I1 #K1 #HLK1 #HK12 #H <(ceq_ext_inv_eq … H) -I2
-/2 width=3 by ex2_intro/
-qed-.
-
-(* Basic_2A1: includes: lreq_drop_conf_be *)
-lemma lreq_drops_conf_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
- ∀b,f,I,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} → 𝐔⦃f⦄ →
- ∀f1. f ~⊚ ↑f1 ≘ f2 →
- ∃∃K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} & K1 ≡[f1] K2.
-#f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
-elim (lreq_drops_trans_next … (lreq_sym … HL12) … HLK1 … Hf2) // -f2 -L1 -Hf
-/3 width=3 by lreq_sym, ex2_intro/
-qed-.
-
-lemma drops_lreq_trans_next: ∀f1,K1,K2. K1 ≡[f1] K2 →
- ∀b,f,I,L1. ⬇*[b, f] L1.ⓘ{I} ≘ K1 →
- ∀f2. f ~⊚ f1 ≘ ↑f2 →
- ∃∃L2. ⬇*[b, f] L2.ⓘ{I} ≘ K2 & L1 ≡[f2] L2 & L1.ⓘ{I} ≡[f] L2.ⓘ{I}.
-#f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
-elim (drops_lexs_trans_next … HK12 … HLK1 … Hf2) -f1 -K1
-/2 width=6 by cfull_lift_sn, ceq_lift_sn/
-#I2 #L2 #HLK2 #HL12 #H >(ceq_ext_inv_eq … H) -I1
-/2 width=4 by ex3_intro/
-qed-.
(**************************************************************************)
include "ground_2/lib/ltc.ma".
-include "basic_2/relocation/lreq_lreq.ma".
+include "basic_2/relocation/seq_seq.ma".
(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
| #n1 #n2 #K #K2 #_ #IH #HK2 #f2 #Hf
elim (IH … Hf) -K1 -IH #L #H1L1 #HLK #H2L1
elim (HR … HLK … HK2 … Hf) -f1 -K -HR
- /3 width=6 by lreq_trans, ltc_dx, ex3_intro/
+ /3 width=6 by seq_trans, ltc_dx, ex3_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/drops_sex.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Properties with syntactic equivalence for selected local environments ****)
+
+lemma seq_co_dedropable_sn: co_dedropable_sn seq.
+@sex_liftable_co_dedropable_sn
+/2 width=6 by cfull_lift_sn, ceq_lift_sn/ qed-.
+
+lemma seq_co_dropable_sn: co_dropable_sn seq.
+@sex_co_dropable_sn qed-.
+
+lemma seq_co_dropable_dx: co_dropable_dx seq.
+@sex_co_dropable_dx qed-.
+
+(* Basic_2A1: includes: lreq_drop_trans_be *)
+lemma seq_drops_trans_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
+ ∀b,f,I,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ↑f1 ≘ f2 →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} & K1 ≡[f1] K2.
+#f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
+elim (sex_drops_trans_next … HL12 … HLK2 Hf … Hf2) -f2 -L2 -Hf
+#I1 #K1 #HLK1 #HK12 #H <(ceq_ext_inv_eq … H) -I2
+/2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: lreq_drop_conf_be *)
+lemma seq_drops_conf_next: ∀f2,L1,L2. L1 ≡[f2] L2 →
+ ∀b,f,I,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ↑f1 ≘ f2 →
+ ∃∃K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I} & K1 ≡[f1] K2.
+#f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
+elim (seq_drops_trans_next … (seq_sym … HL12) … HLK1 … Hf2) // -f2 -L1 -Hf
+/3 width=3 by seq_sym, ex2_intro/
+qed-.
+
+lemma drops_seq_trans_next: ∀f1,K1,K2. K1 ≡[f1] K2 →
+ ∀b,f,I,L1. ⬇*[b, f] L1.ⓘ{I} ≘ K1 →
+ ∀f2. f ~⊚ f1 ≘ ↑f2 →
+ ∃∃L2. ⬇*[b, f] L2.ⓘ{I} ≘ K2 & L1 ≡[f2] L2 & L1.ⓘ{I} ≡[f] L2.ⓘ{I}.
+#f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
+elim (drops_sex_trans_next … HK12 … HLK1 … Hf2) -f1 -K1
+/2 width=6 by cfull_lift_sn, ceq_lift_sn/
+#I2 #L2 #HLK2 #HL12 #H >(ceq_ext_inv_eq … H) -I1
+/2 width=4 by ex3_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_lifts_bind.ma".
+include "basic_2/relocation/drops.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Properties with entrywise extension of context-sensitive relations *******)
+
+(**) (* changed after commit 13218 *)
+lemma sex_co_dropable_sn: ∀RN,RP. co_dropable_sn (sex RN RP).
+#RN #RP #b #f #L1 #K1 #H elim H -f -L1 -K1
+[ #f #Hf #_ #f2 #X #H #f1 #Hf2 >(sex_inv_atom1 … H) -X
+ /4 width=3 by sex_atom, drops_atom, ex2_intro/
+| #f #I1 #L1 #K1 #_ #IH #Hf #f2 #X #H #f1 #Hf2
+ elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H2 destruct
+ elim (sex_inv_push1 … H) -H #I2 #L2 #HL12 #HI12 #H destruct
+ elim (IH … HL12 … Hg2) -g2
+ /3 width=3 by isuni_inv_next, drops_drop, ex2_intro/
+| #f #I1 #J1 #L1 #K1 #HLK #HJI1 #IH #Hf #f2 #X #H #f1 #Hf2
+ lapply (isuni_inv_push … Hf ??) -Hf [3: |*: // ] #Hf
+ lapply (drops_fwd_isid … HLK … Hf) -HLK #H0 destruct
+ lapply (liftsb_fwd_isid … HJI1 … Hf) -HJI1 #H0 destruct
+ elim (coafter_inv_pxx … Hf2) -Hf2 [1,3:* |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
+ [ elim (sex_inv_push1 … H) | elim (sex_inv_next1 … H) ] -H #I2 #L2 #HL12 #HI12 #H destruct
+ elim (IH … HL12 … Hg2) -g2 -IH /2 width=1 by isuni_isid/ #K2 #HK12 #HLK2
+ lapply (drops_fwd_isid … HLK2 … Hf) -HLK2 #H0 destruct
+ /4 width=3 by drops_refl, sex_next, sex_push, isid_push, ex2_intro/
+]
+qed-.
+
+lemma sex_liftable_co_dedropable_bi: ∀RN,RP. d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+ ∀f2,L1,L2. L1 ⪤[cfull, RP, f2] L2 → ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
+ ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ f ~⊚ f1 ≘ f2 → L1 ⪤[RN, RP, f2] L2.
+#RN #RP #HRN #HRP #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
+#g2 #I1 #I2 #L1 #L2 #HL12 #HI12 #IH #f1 #Y1 #Y2 #HK12 #b #f #HY1 #HY2 #H
+[ elim (coafter_inv_xxn … H) [ |*: // ] -H #g #g1 #Hg2 #H1 #H2 destruct
+ elim (drops_inv_skip1 … HY1) -HY1 #J1 #K1 #HLK1 #HJI1 #H destruct
+ elim (drops_inv_skip1 … HY2) -HY2 #J2 #K2 #HLK2 #HJI2 #H destruct
+ elim (sex_inv_next … HK12) -HK12 #HK12 #HJ12
+ elim (HRN … HJ12 … HLK1 … HJI1) -HJ12 -HJI1 #Z #Hz
+ >(liftsb_mono … Hz … HJI2) -Z /3 width=9 by sex_next/
+| elim (coafter_inv_xxp … H) [1,2: |*: // ] -H *
+ [ #g #g1 #Hg2 #H1 #H2 destruct
+ elim (drops_inv_skip1 … HY1) -HY1 #J1 #K1 #HLK1 #HJI1 #H destruct
+ elim (drops_inv_skip1 … HY2) -HY2 #J2 #K2 #HLK2 #HJI2 #H destruct
+ elim (sex_inv_push … HK12) -HK12 #HK12 #HJ12
+ elim (HRP … HJ12 … HLK1 … HJI1) -HJ12 -HJI1 #Z #Hz
+ >(liftsb_mono … Hz … HJI2) -Z /3 width=9 by sex_push/
+ | #g #Hg2 #H destruct
+ lapply (drops_inv_drop1 … HY1) -HY1 #HLK1
+ lapply (drops_inv_drop1 … HY2) -HY2 #HLK2
+ /3 width=9 by sex_push/
+ ]
+]
+qed-.
+
+lemma sex_liftable_co_dedropable_sn: ∀RN,RP. (∀L. reflexive … (RN L)) → (∀L. reflexive … (RP L)) →
+ d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+ co_dedropable_sn (sex RN RP).
+#RN #RP #H1RN #H1RP #H2RN #H2RP #b #f #L1 #K1 #H elim H -f -L1 -K1
+[ #f #Hf #X #f1 #H #f2 #Hf2 >(sex_inv_atom1 … H) -X
+ /4 width=4 by drops_atom, sex_atom, ex3_intro/
+| #f #I1 #L1 #K1 #_ #IHLK1 #K2 #f1 #HK12 #f2 #Hf2
+ elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
+ elim (IHLK1 … HK12 … Hg2) -K1
+ /3 width=6 by drops_drop, sex_next, sex_push, ex3_intro/
+| #f #I1 #J1 #L1 #K1 #HLK1 #HJI1 #IHLK1 #X #f1 #H #f2 #Hf2
+ elim (coafter_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
+ [ elim (sex_inv_push1 … H) | elim (sex_inv_next1 … H) ] -H #J2 #K2 #HK12 #HJ12 #H destruct
+ [ elim (H2RP … HJ12 … HLK1 … HJI1) | elim (H2RN … HJ12 … HLK1 … HJI1) ] -J1
+ elim (IHLK1 … HK12 … Hg2) -K1
+ /3 width=6 by drops_skip, sex_next, sex_push, ex3_intro/
+]
+qed-.
+
+fact sex_dropable_dx_aux: ∀RN,RP,b,f,L2,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∀f2,L1. L1 ⪤[RN, RP, f2] L2 → ∀f1. f ~⊚ f1 ≘ f2 →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[RN, RP, f1] K2.
+#RN #RP #b #f #L2 #K2 #H elim H -f -L2 -K2
+[ #f #Hf #_ #f2 #X #H #f1 #Hf2 lapply (sex_inv_atom2 … H) -H
+ #H destruct /4 width=3 by sex_atom, drops_atom, ex2_intro/
+| #f #I2 #L2 #K2 #_ #IH #Hf #f2 #X #HX #f1 #Hf2
+ elim (coafter_inv_nxx … Hf2) -Hf2 [2,3: // ] #g2 #Hg2 #H destruct
+ elim (sex_inv_push2 … HX) -HX #I1 #L1 #HL12 #HI12 #H destruct
+ elim (IH … HL12 … Hg2) -L2 -I2 -g2
+ /3 width=3 by drops_drop, isuni_inv_next, ex2_intro/
+| #f #I2 #J2 #L2 #K2 #_ #HJI2 #IH #Hf #f2 #X #HX #f1 #Hf2
+ elim (coafter_inv_pxx … Hf2) -Hf2 [1,3: * |*: // ] #g1 #g2 #Hg2 #H1 #H2 destruct
+ [ elim (sex_inv_push2 … HX) | elim (sex_inv_next2 … HX) ] -HX #I1 #L1 #HL12 #HI12 #H destruct
+ elim (IH … HL12 … Hg2) -L2 -g2 /2 width=3 by isuni_fwd_push/ #K1 #HLK1 #HK12
+ lapply (isuni_inv_push … Hf ??) -Hf [3,6: |*: // ] #Hf
+ lapply (liftsb_fwd_isid … HJI2 … Hf) #H destruct -HJI2
+ lapply (drops_fwd_isid … HLK1 … Hf) #H destruct -HLK1
+ /4 width=5 by sex_next, sex_push, drops_refl, isid_push, ex2_intro/
+]
+qed-.
+
+lemma sex_co_dropable_dx: ∀RN,RP. co_dropable_dx (sex RN RP).
+/2 width=5 by sex_dropable_dx_aux/ qed-.
+
+lemma sex_drops_conf_next: ∀RN,RP.
+ ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
+ ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ↑f1 ≘ f2 →
+ ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN, RP, f1] K2 & RN K1 I1 I2.
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
+elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
+#X #HX #HLK2 elim (sex_inv_next1 … HX) -HX
+#I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+lemma sex_drops_conf_push: ∀RN,RP.
+ ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
+ ∀b,f,I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ⫯f1 ≘ f2 →
+ ∃∃I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} & K1 ⪤[RN, RP, f1] K2 & RP K1 I1 I2.
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I1 #K1 #HLK1 #Hf #f1 #Hf2
+elim (sex_co_dropable_sn … HLK1 … Hf … HL12 … Hf2) -L1 -f2 -Hf
+#X #HX #HLK2 elim (sex_inv_push1 … HX) -HX
+#I2 #K2 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+lemma sex_drops_trans_next: ∀RN,RP,f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
+ ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ↑f1 ≘ f2 →
+ ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN, RP, f1] K2 & RN K1 I1 I2.
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
+elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
+#X #HLK1 #HX elim (sex_inv_next2 … HX) -HX
+#I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+lemma sex_drops_trans_push: ∀RN,RP,f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
+ ∀b,f,I2,K2. ⬇*[b, f] L2 ≘ K2.ⓘ{I2} → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ ⫯f1 ≘ f2 →
+ ∃∃I1,K1. ⬇*[b, f] L1 ≘ K1.ⓘ{I1} & K1 ⪤[RN, RP, f1] K2 & RP K1 I1 I2.
+#RN #RP #f2 #L1 #L2 #HL12 #b #f #I2 #K2 #HLK2 #Hf #f1 #Hf2
+elim (sex_co_dropable_dx … HL12 … HLK2 … Hf … Hf2) -L2 -f2 -Hf
+#X #HLK1 #HX elim (sex_inv_push2 … HX) -HX
+#I1 #K1 #HK12 #HI12 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+lemma drops_sex_trans_next: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
+ d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+ ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
+ ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
+ ∀f2. f ~⊚ f1 ≘ ↑f2 →
+ ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN, RP, f2] L2 & RN L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
+elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
+#X #HX #HLK2 #H1L12 elim (sex_inv_next1 … HX) -HX
+#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
+qed-.
+
+lemma drops_sex_trans_push: ∀RN,RP. (∀L. reflexive ? (RN L)) → (∀L. reflexive ? (RP L)) →
+ d_liftable2_sn … liftsb RN → d_liftable2_sn … liftsb RP →
+ ∀f1,K1,K2. K1 ⪤[RN, RP, f1] K2 →
+ ∀b,f,I1,L1. ⬇*[b, f] L1.ⓘ{I1} ≘ K1 →
+ ∀f2. f ~⊚ f1 ≘ ⫯f2 →
+ ∃∃I2,L2. ⬇*[b, f] L2.ⓘ{I2} ≘ K2 & L1 ⪤[RN, RP, f2] L2 & RP L1 I1 I2 & L1.ⓘ{I1} ≡[f] L2.ⓘ{I2}.
+#RN #RP #H1RN #H1RP #H2RN #H2RP #f1 #K1 #K2 #HK12 #b #f #I1 #L1 #HLK1 #f2 #Hf2
+elim (sex_liftable_co_dedropable_sn … H1RN H1RP H2RN H2RP … HLK1 … HK12 … Hf2) -K1 -f1 -H1RN -H1RP -H2RN -H2RP
+#X #HX #HLK2 #H1L12 elim (sex_inv_push1 … HX) -HX
+#I2 #L2 #H2L12 #HI12 #H destruct /2 width=6 by ex4_2_intro/
+qed-.
+
+lemma drops_atom2_sex_conf: ∀RN,RP,b,f1,L1. ⬇*[b, f1] L1 ≘ ⋆ → 𝐔⦃f1⦄ →
+ ∀f,L2. L1 ⪤[RN, RP, f] L2 →
+ ∀f2. f1 ~⊚ f2 ≘f → ⬇*[b, f1] L2 ≘ ⋆.
+#RN #RP #b #f1 #L1 #H1 #Hf1 #f #L2 #H2 #f2 #H3
+elim (sex_co_dropable_sn … H1 … H2 … H3) // -H1 -H2 -H3 -Hf1
+#L #H #HL2 lapply (sex_inv_atom1 … H) -H //
+qed-.
include "ground_2/relocation/rtmap_uni.ma".
include "basic_2/notation/relations/relation_3.ma".
include "basic_2/syntax/cext2.ma".
-include "basic_2/relocation/lexs.ma".
+include "basic_2/relocation/sex.ma".
(* GENERIC EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **************)
definition lex (R): relation lenv ≝
- λL1,L2. ∃∃f. 𝐈⦃f⦄ & L1 ⪤*[cfull, cext2 R, f] L2.
+ λL1,L2. ∃∃f. 𝐈⦃f⦄ & L1 ⪤[cfull, cext2 R, f] L2.
interpretation "generic extension (local environment)"
'Relation R L1 L2 = (lex R L1 L2).
(* Basic_2A1: was: lpx_sn_atom *)
lemma lex_atom (R): ⋆ ⪤[R] ⋆.
-/2 width=3 by lexs_atom, ex2_intro/ qed.
+/2 width=3 by sex_atom, ex2_intro/ qed.
lemma lex_bind (R): ∀I1,I2,K1,K2. K1 ⪤[R] K2 → cext2 R K1 I1 I2 →
K1.ⓘ{I1} ⪤[R] K2.ⓘ{I2}.
#R #I1 #I2 #K1 #K2 * #f #Hf #HK12 #HI12
-/3 width=3 by lexs_push, isid_push, ex2_intro/
+/3 width=3 by sex_push, isid_push, ex2_intro/
qed.
(* Basic_2A1: was: lpx_sn_refl *)
lemma lex_refl (R): c_reflexive … R → reflexive … (lex R).
-/4 width=3 by lexs_refl, ext2_refl, ex2_intro/ qed.
+/4 width=3 by sex_refl, ext2_refl, ex2_intro/ qed.
lemma lex_co (R1) (R2): (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
∀L1,L2. L1 ⪤[R1] L2 → L1 ⪤[R2] L2.
-#R1 #R2 #HR #L1 #L2 * /5 width=7 by lexs_co, cext2_co, ex2_intro/
+#R1 #R2 #HR #L1 #L2 * /5 width=7 by sex_co, cext2_co, ex2_intro/
qed-.
(* Advanced properties ******************************************************)
(* Basic_2A1: was: lpx_sn_inv_atom1: *)
lemma lex_inv_atom_sn (R): ∀L2. ⋆ ⪤[R] L2 → L2 = ⋆.
-#R #L2 * #f #Hf #H >(lexs_inv_atom1 … H) -L2 //
+#R #L2 * #f #Hf #H >(sex_inv_atom1 … H) -L2 //
qed-.
lemma lex_inv_bind_sn (R): ∀I1,L2,K1. K1.ⓘ{I1} ⪤[R] L2 →
∃∃I2,K2. K1 ⪤[R] K2 & cext2 R K1 I1 I2 & L2 = K2.ⓘ{I2}.
#R #I1 #L2 #K1 * #f #Hf #H
-lapply (lexs_eq_repl_fwd … H (⫯f) ?) -H /2 width=1 by eq_push_inv_isid/ #H
-elim (lexs_inv_push1 … H) -H #I2 #K2 #HK12 #HI12 #H destruct
+lapply (sex_eq_repl_fwd … H (⫯f) ?) -H /2 width=1 by eq_push_inv_isid/ #H
+elim (sex_inv_push1 … H) -H #I2 #K2 #HK12 #HI12 #H destruct
/3 width=5 by ex2_intro, ex3_2_intro/
qed-.
(* Basic_2A1: was: lpx_sn_inv_atom2 *)
lemma lex_inv_atom_dx (R): ∀L1. L1 ⪤[R] ⋆ → L1 = ⋆.
-#R #L1 * #f #Hf #H >(lexs_inv_atom2 … H) -L1 //
+#R #L1 * #f #Hf #H >(sex_inv_atom2 … H) -L1 //
qed-.
lemma lex_inv_bind_dx (R): ∀I2,L1,K2. L1 ⪤[R] K2.ⓘ{I2} →
∃∃I1,K1. K1 ⪤[R] K2 & cext2 R K1 I1 I2 & L1 = K1.ⓘ{I1}.
#R #I2 #L1 #K2 * #f #Hf #H
-lapply (lexs_eq_repl_fwd … H (⫯f) ?) -H /2 width=1 by eq_push_inv_isid/ #H
-elim (lexs_inv_push2 … H) -H #I1 #K1 #HK12 #HI12 #H destruct
+lapply (sex_eq_repl_fwd … H (⫯f) ?) -H /2 width=1 by eq_push_inv_isid/ #H
+elim (sex_inv_push2 … H) -H #I1 #K1 #HK12 #HI12 #H destruct
/3 width=5 by ex3_2_intro, ex2_intro/
qed-.
(* *)
(**************************************************************************)
-include "basic_2/relocation/lexs_length.ma".
+include "basic_2/relocation/sex_length.ma".
include "basic_2/relocation/lex.ma".
(* GENERIC EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **************)
(* Basic_2A1: was: lpx_sn_fwd_length *)
lemma lex_fwd_length: ∀R,L1,L2. L1 ⪤[R] L2 → |L1| = |L2|.
-#R #L1 #L2 * /2 width=4 by lexs_fwd_length/
+#R #L1 #L2 * /2 width=4 by sex_fwd_length/
qed-.
(**************************************************************************)
include "basic_2/syntax/ext2_tc.ma".
-include "basic_2/relocation/lexs_tc.ma".
+include "basic_2/relocation/sex_tc.ma".
include "basic_2/relocation/lex.ma".
alias symbol "subseteq" = "relation inclusion".
lemma lex_inv_CTC (R): c_reflexive … R →
lex (CTC … R) ⊆ TC … (lex R).
#R #HR #L1 #L2 *
-/5 width=11 by lexs_inv_tc_dx, lexs_co, ext2_inv_tc, ext2_refl, monotonic_TC, ex2_intro/
+/5 width=11 by sex_inv_tc_dx, sex_co, ext2_inv_tc, ext2_refl, monotonic_TC, ex2_intro/
qed-.
lemma s_rs_transitive_lex_inv_isid (R): s_rs_transitive … R (λ_.lex R) →
lemma lex_CTC (R): s_rs_transitive … R (λ_. lex R) →
TC … (lex R) ⊆ lex (CTC … R).
#R #HR #L1 #L2 #HL12
-lapply (monotonic_TC … (lexs cfull (cext2 R) 𝐈𝐝) … HL12) -HL12
-[ #L1 #L2 * /3 width=3 by lexs_eq_repl_fwd, eq_id_inv_isid/
-| /5 width=9 by s_rs_transitive_lex_inv_isid, lexs_tc_dx, lexs_co, ext2_tc, ex2_intro/
+lapply (monotonic_TC … (sex cfull (cext2 R) 𝐈𝐝) … HL12) -HL12
+[ #L1 #L2 * /3 width=3 by sex_eq_repl_fwd, eq_id_inv_isid/
+| /5 width=9 by s_rs_transitive_lex_inv_isid, sex_tc_dx, sex_co, ext2_tc, 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/relocation/rtmap_sle.ma".
-include "ground_2/relocation/rtmap_sdj.ma".
-include "basic_2/notation/relations/relationstar_5.ma".
-include "basic_2/syntax/lenv.ma".
-
-(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
-
-inductive lexs (RN,RP:relation3 lenv bind bind): rtmap → relation lenv ≝
-| lexs_atom: ∀f. lexs RN RP f (⋆) (⋆)
-| lexs_next: ∀f,I1,I2,L1,L2.
- lexs RN RP f L1 L2 → RN L1 I1 I2 →
- lexs RN RP (↑f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
-| lexs_push: ∀f,I1,I2,L1,L2.
- lexs RN RP f L1 L2 → RP L1 I1 I2 →
- lexs RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
-.
-
-interpretation "generic entrywise extension (local environment)"
- 'RelationStar RN RP f L1 L2 = (lexs RN RP f L1 L2).
-
-definition R_pw_confluent2_lexs: relation3 lenv bind bind → relation3 lenv bind bind →
- relation3 lenv bind bind → relation3 lenv bind bind →
- relation3 lenv bind bind → relation3 lenv bind bind →
- relation3 rtmap lenv bind ≝
- λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
- ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
- ∀L1. L0 ⪤*[RN1, RP1, f] L1 → ∀L2. L0 ⪤*[RN2, RP2, f] L2 →
- ∃∃I. R2 L1 I1 I & R1 L2 I2 I.
-
-definition lexs_transitive: relation3 lenv bind bind → relation3 lenv bind bind →
- relation3 lenv bind bind →
- relation3 lenv bind bind → relation3 lenv bind bind →
- relation3 rtmap lenv bind ≝
- λR1,R2,R3,RN,RP,f,L1,I1.
- ∀I. R1 L1 I1 I → ∀L2. L1 ⪤*[RN, RP, f] L2 →
- ∀I2. R2 L2 I I2 → R3 L1 I1 I2.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lexs_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → X = ⋆ → Y = ⋆.
-#RN #RP #f #X #Y * -f -X -Y //
-#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
-qed-.
-
-(* Basic_2A1: includes lpx_sn_inv_atom1 *)
-lemma lexs_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤*[RN, RP, f] Y → Y = ⋆.
-/2 width=6 by lexs_inv_atom1_aux/ qed-.
-
-fact lexs_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g →
- ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
-#RN #RP #f #X #Y * -f -X -Y
-[ #f #g #J1 #K1 #H destruct
-| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_push_next … H)
-]
-qed-.
-
-(* Basic_2A1: includes lpx_sn_inv_pair1 *)
-lemma lexs_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤*[RN, RP, ↑g] Y →
- ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
-/2 width=7 by lexs_inv_next1_aux/ qed-.
-
-fact lexs_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g →
- ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
-#RN #RP #f #X #Y * -f -X -Y
-[ #f #g #J1 #K1 #H destruct
-| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_next_push … H)
-| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_push … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lexs_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤*[RN, RP, ⫯g] Y →
- ∃∃J2,K2. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
-/2 width=7 by lexs_inv_push1_aux/ qed-.
-
-fact lexs_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → Y = ⋆ → X = ⋆.
-#RN #RP #f #X #Y * -f -X -Y //
-#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
-qed-.
-
-(* Basic_2A1: includes lpx_sn_inv_atom2 *)
-lemma lexs_inv_atom2: ∀RN,RP,f,X. X ⪤*[RN, RP, f] ⋆ → X = ⋆.
-/2 width=6 by lexs_inv_atom2_aux/ qed-.
-
-fact lexs_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g →
- ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
-#RN #RP #f #X #Y * -f -X -Y
-[ #f #g #J2 #K2 #H destruct
-| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_push_next … H)
-]
-qed-.
-
-(* Basic_2A1: includes lpx_sn_inv_pair2 *)
-lemma lexs_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ↑g] K2.ⓘ{J2} →
- ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
-/2 width=7 by lexs_inv_next2_aux/ qed-.
-
-fact lexs_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤*[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g →
- ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
-#RN #RP #f #X #Y * -f -X -Y
-[ #f #J2 #K2 #g #H destruct
-| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_next_push … H)
-| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_push … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lexs_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤*[RN, RP, ⫯g] K2.ⓘ{J2} →
- ∃∃J1,K1. K1 ⪤*[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
-/2 width=7 by lexs_inv_push2_aux/ qed-.
-
-(* Basic_2A1: includes lpx_sn_inv_pair *)
-lemma lexs_inv_next: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤*[RN, RP, ↑f] L2.ⓘ{I2} →
- L1 ⪤*[RN, RP, f] L2 ∧ RN L1 I1 I2.
-#RN #RP #f #I1 #I2 #L1 #L2 #H elim (lexs_inv_next1 … H) -H
-#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
-qed-.
-
-lemma lexs_inv_push: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤*[RN, RP, ⫯f] L2.ⓘ{I2} →
- L1 ⪤*[RN, RP, f] L2 ∧ RP L1 I1 I2.
-#RN #RP #f #I1 #I2 #L1 #L2 #H elim (lexs_inv_push1 … H) -H
-#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
-qed-.
-
-lemma lexs_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤*[RN, RP, ⫱f] L2 →
- RN L1 I1 I2 → RP L1 I1 I2 →
- L1.ⓘ{I1} ⪤*[RN, RP, f] L2.ⓘ{I2}.
-#RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) *
-/2 width=1 by lexs_next, lexs_push/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lexs_fwd_bind: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤*[RN, RP, f] L2.ⓘ{I2} →
- L1 ⪤*[RN, RP, ⫱f] L2.
-#RN #RP #f #I1 #I2 #L1 #L2 #Hf
-elim (pn_split f) * #g #H destruct
-[ elim (lexs_inv_push … Hf) | elim (lexs_inv_next … Hf) ] -Hf //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lexs_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤*[RN, RP, f] L2).
-#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
-#f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H
-[ elim (eq_inv_nx … H) -H /3 width=3 by lexs_next/
-| elim (eq_inv_px … H) -H /3 width=3 by lexs_push/
-]
-qed-.
-
-lemma lexs_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤*[RN, RP, f] L2).
-#RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by lexs_eq_repl_back/ (**) (* full auto fails *)
-qed-.
-
-lemma lexs_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f.reflexive … (lexs RN RP f).
-#RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L //
-#L #I #IH #f elim (pn_split f) *
-#g #H destruct /2 width=1 by lexs_next, lexs_push/
-qed.
-
-lemma lexs_sym: ∀RN,RP.
- (∀L1,L2,I1,I2. RN L1 I1 I2 → RN L2 I2 I1) →
- (∀L1,L2,I1,I2. RP L1 I1 I2 → RP L2 I2 I1) →
- ∀f. symmetric … (lexs RN RP f).
-#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -L1 -L2 -f
-/3 width=2 by lexs_next, lexs_push/
-qed-.
-
-lemma lexs_pair_repl: ∀RN,RP,f,I1,I2,L1,L2.
- L1.ⓘ{I1} ⪤*[RN, RP, f] L2.ⓘ{I2} →
- ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 →
- L1.ⓘ{J1} ⪤*[RN, RP, f] L2.ⓘ{J2}.
-/3 width=3 by lexs_inv_tl, lexs_fwd_bind/ qed-.
-
-lemma lexs_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 →
- ∀f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 → L1 ⪤*[RN2, RP2, f] L2.
-#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
-/3 width=1 by lexs_atom, lexs_next, lexs_push/
-qed-.
-
-lemma lexs_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 →
- ∀f,L1,L2. L1 ⪤*[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
- L1 ⪤*[RN2, RP2, f] L2.
-#RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 //
-#f #I1 #I2 #K1 #K2 #_ #HI12 #IH #H
-[ elim (isid_inv_next … H) -H //
-| /4 width=3 by lexs_push, isid_inv_push/
-]
-qed-.
-
-lemma lexs_sdj: ∀RN,RP. RP ⊆ RN →
- ∀f1,L1,L2. L1 ⪤*[RN, RP, f1] L2 →
- ∀f2. f1 ∥ f2 → L1 ⪤*[RP, RN, f2] L2.
-#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
-#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
-[ elim (sdj_inv_nx … H12) -H12 [2,3: // ]
- #g2 #H #H2 destruct /3 width=1 by lexs_push/
-| elim (sdj_inv_px … H12) -H12 [2,4: // ] *
- #g2 #H #H2 destruct /3 width=1 by lexs_next, lexs_push/
-]
-qed-.
-
-lemma sle_lexs_trans: ∀RN,RP. RN ⊆ RP →
- ∀f2,L1,L2. L1 ⪤*[RN, RP, f2] L2 →
- ∀f1. f1 ⊆ f2 → L1 ⪤*[RN, RP, f1] L2.
-#RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
-#f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12
-[ elim (pn_split f1) * ]
-[ /4 width=5 by lexs_push, sle_inv_pn/
-| /4 width=5 by lexs_next, sle_inv_nn/
-| elim (sle_inv_xp … H12) -H12 [2,3: // ]
- #g1 #H #H1 destruct /3 width=5 by lexs_push/
-]
-qed-.
-
-lemma sle_lexs_conf: ∀RN,RP. RP ⊆ RN →
- ∀f1,L1,L2. L1 ⪤*[RN, RP, f1] L2 →
- ∀f2. f1 ⊆ f2 → L1 ⪤*[RN, RP, f2] L2.
-#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
-#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
-[2: elim (pn_split f2) * ]
-[ /4 width=5 by lexs_push, sle_inv_pp/
-| /4 width=5 by lexs_next, sle_inv_pn/
-| elim (sle_inv_nx … H12) -H12 [2,3: // ]
- #g2 #H #H2 destruct /3 width=5 by lexs_next/
-]
-qed-.
-
-lemma lexs_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
- ∀f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ⊆ g →
- ∃∃L. L1 ⪤*[R1, RP, g] L & L ⪤*[R2, cfull, f] L2.
-#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
-[ /2 width=3 by lexs_atom, ex2_intro/ ]
-#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
-[ elim (sle_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
- elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, ex2_intro/
-| elim (sle_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
- elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
-]
-qed-.
-
-lemma lexs_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
- ∀f,L1,L2. L1 ⪤*[R1, RP, f] L2 → ∀g. f ∥ g →
- ∃∃L. L1 ⪤*[RP, R1, g] L & L ⪤*[R2, cfull, f] L2.
-#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
-[ /2 width=3 by lexs_atom, ex2_intro/ ]
-#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
-[ elim (sdj_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
- elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
-| elim (sdj_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
- elim (IH … Hfg) -IH -Hfg /3 width=5 by lexs_next, lexs_push, ex2_intro/
-]
-qed-.
-
-lemma lexs_dec: ∀RN,RP.
- (∀L,I1,I2. Decidable (RN L I1 I2)) →
- (∀L,I1,I2. Decidable (RP L I1 I2)) →
- ∀L1,L2,f. Decidable (L1 ⪤*[RN, RP, f] L2).
-#RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ]
-[ /2 width=1 by lexs_atom, or_introl/
-| #L2 #I2 #f @or_intror #H
- lapply (lexs_inv_atom1 … H) -H #H destruct
-| #f @or_intror #H
- lapply (lexs_inv_atom2 … H) -H #H destruct
-| #L2 #I2 #f elim (IH L2 (⫱f)) -IH #HL12
- [2: /4 width=3 by lexs_fwd_bind, or_intror/ ]
- elim (pn_split f) * #g #H destruct
- [ elim (HRP L1 I1 I2) | elim (HRN L1 I1 I2) ] -HRP -HRN #HV12
- [1,3: /3 width=1 by lexs_push, lexs_next, or_introl/ ]
- @or_intror #H
- [ elim (lexs_inv_push … H) | elim (lexs_inv_next … H) ] -H
- /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 "basic_2/syntax/lenv_length.ma".
-include "basic_2/relocation/lexs.ma".
-
-(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
-
-(* Forward lemmas with length for local environments ************************)
-
-lemma lexs_fwd_length: ∀RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → |L1| = |L2|.
-#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2 //
-#f #I1 #I2 #L1 #L2 >length_bind >length_bind //
-qed-.
-
-(* Properties with length for local environments ****************************)
-
-lemma lexs_length_cfull: ∀L1,L2. |L1| = |L2| → ∀f. L1 ⪤*[cfull, cfull, f] L2.
-#L1 elim L1 -L1
-[ #Y2 #H >(length_inv_zero_sn … H) -Y2 //
-| #L1 #I1 #IH #Y2 #H #f
- elim (length_inv_succ_sn … H) -H #I2 #L2 #HL12 #H destruct
- elim (pn_split f) * #g #H destruct /3 width=1 by lexs_next, lexs_push/
-]
-qed.
-
-lemma lexs_length_isid: ∀R,L1,L2. |L1| = |L2| →
- ∀f. 𝐈⦃f⦄ → L1 ⪤*[R, cfull, f] L2.
-#R #L1 elim L1 -L1
-[ #Y2 #H >(length_inv_zero_sn … H) -Y2 //
-| #L1 #I1 #IH #Y2 #H #f #Hf
- elim (length_inv_succ_sn … H) -H #I2 #L2 #HL12 #H destruct
- elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct /3 width=1 by lexs_push/
-]
-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/relocation/rtmap_sand.ma".
-include "basic_2/relocation/drops.ma".
-
-(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
-
-(* Main properties **********************************************************)
-
-theorem lexs_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP):
- ∀L1,f.
- (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → lexs_transitive RN1 RN2 RN RN1 RP1 g K I) →
- (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → lexs_transitive RP1 RP2 RP RN1 RP1 g K I) →
- ∀L0. L1 ⪤*[RN1, RP1, f] L0 →
- ∀L2. L0 ⪤*[RN2, RP2, f] L2 →
- L1 ⪤*[RN, RP, f] L2.
-#RN1 #RP1 #RN2 #RP2 #RN #RP #L1 elim L1 -L1
-[ #f #_ #_ #L0 #H1 #L2 #H2
- lapply (lexs_inv_atom1 … H1) -H1 #H destruct
- lapply (lexs_inv_atom1 … H2) -H2 #H destruct
- /2 width=1 by lexs_atom/
-| #K1 #I1 #IH #f elim (pn_split f) * #g #H destruct
- #HN #HP #L0 #H1 #L2 #H2
- [ elim (lexs_inv_push1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
- elim (lexs_inv_push1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct
- lapply (HP … 0 … HI10 … HK10 … HI02) -HI10 -HI02 /2 width=2 by drops_refl/ #HI12
- lapply (IH … HK10 … HK02) -IH -K0 /3 width=3 by lexs_push, drops_drop/
- | elim (lexs_inv_next1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
- elim (lexs_inv_next1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct
- lapply (HN … 0 … HI10 … HK10 … HI02) -HI10 -HI02 /2 width=2 by drops_refl/ #HI12
- lapply (IH … HK10 … HK02) -IH -K0 /3 width=3 by lexs_next, drops_drop/
- ]
-]
-qed-.
-
-theorem lexs_trans (RN) (RP) (f): (∀g,I,K. lexs_transitive RN RN RN RN RP g K I) →
- (∀g,I,K. lexs_transitive RP RP RP RN RP g K I) →
- Transitive … (lexs RN RP f).
-/2 width=9 by lexs_trans_gen/ qed-.
-
-theorem lexs_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤*[R1, cfull, f] L → 𝐈⦃f⦄ →
- ∀L2. L ⪤*[R2, cfull, f] L2 → L1 ⪤*[R3, cfull, f] L2.
-#R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f
-[ #f #Hf #L2 #H >(lexs_inv_atom1 … H) -L2 // ]
-#f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H
-[ elim (isid_inv_next … Hf) | lapply (isid_inv_push … Hf ??) ] -Hf [5: |*: // ] #Hf
-elim (lexs_inv_push1 … H) -H #I2 #K2 #HK2 #_ #H destruct
-/3 width=1 by lexs_push/
-qed-.
-
-theorem lexs_conf (RN1) (RP1) (RN2) (RP2):
- ∀L,f.
- (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → R_pw_confluent2_lexs RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
- (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → R_pw_confluent2_lexs RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
- pw_confluent2 … (lexs RN1 RP1 f) (lexs RN2 RP2 f) L.
-#RN1 #RP1 #RN2 #RP2 #L elim L -L
-[ #f #_ #_ #L1 #H1 #L2 #H2 >(lexs_inv_atom1 … H1) >(lexs_inv_atom1 … H2) -H2 -H1
- /2 width=3 by lexs_atom, ex2_intro/
-| #L #I0 #IH #f elim (pn_split f) * #g #H destruct
- #HN #HP #Y1 #H1 #Y2 #H2
- [ elim (lexs_inv_push1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
- elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
- elim (HP … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
- elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_push, ex2_intro/
- | elim (lexs_inv_next1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
- elim (lexs_inv_next1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
- elim (HN … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
- elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, lexs_next, ex2_intro/
- ]
-]
-qed-.
-
-theorem lexs_canc_sn: ∀RN,RP,f. Transitive … (lexs RN RP f) →
- symmetric … (lexs RN RP f) →
- left_cancellable … (lexs RN RP f).
-/3 width=3 by/ qed-.
-
-theorem lexs_canc_dx: ∀RN,RP,f. Transitive … (lexs RN RP f) →
- symmetric … (lexs RN RP f) →
- right_cancellable … (lexs RN RP f).
-/3 width=3 by/ qed-.
-
-lemma lexs_meet: ∀RN,RP,L1,L2.
- ∀f1. L1 ⪤*[RN, RP, f1] L2 →
- ∀f2. L1 ⪤*[RN, RP, f2] L2 →
- ∀f. f1 ⋒ f2 ≘ f → L1 ⪤*[RN, RP, f] L2.
-#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
-#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
-elim (pn_split f2) * #g2 #H2 destruct
-try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
-[ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
-| elim (sand_inv_ppx … Hf) | elim (sand_inv_pnx … Hf)
-] -Hf /3 width=5 by lexs_next, lexs_push/
-qed-.
-
-lemma lexs_join: ∀RN,RP,L1,L2.
- ∀f1. L1 ⪤*[RN, RP, f1] L2 →
- ∀f2. L1 ⪤*[RN, RP, f2] L2 →
- ∀f. f1 ⋓ f2 ≘ f → L1 ⪤*[RN, RP, f] L2.
-#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
-#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
-elim (pn_split f2) * #g2 #H2 destruct
-try elim (lexs_inv_push … H) try elim (lexs_inv_next … H) -H
-[ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)
-| elim (sor_inv_ppx … Hf) | elim (sor_inv_pnx … Hf)
-] -Hf /3 width=5 by lexs_next, lexs_push/
-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/star.ma".
-include "basic_2/relocation/lexs.ma".
-
-(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
-
-definition s_rs_transitive_isid: relation (relation3 lenv bind bind) ≝ λRN,RP.
- ∀f. 𝐈⦃f⦄ → s_rs_transitive … RP (λ_.lexs RN RP f).
-
-(* Properties with transitive closure ***************************************)
-
-lemma lexs_tc_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f. reflexive … (TC … (lexs RN RP f)).
-/3 width=1 by lexs_refl, TC_reflexive/ qed.
-
-lemma lexs_tc_next_sn: ∀RN,RP. c_reflexive … RN →
- ∀f,I2,L1,L2. TC … (lexs RN RP f) L1 L2 → ∀I1. RN L1 I1 I2 →
- TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRN #f #I2 #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
-/3 width=3 by lexs_next, TC_strap, inj/
-qed.
-
-lemma lexs_tc_next_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
- TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
-/4 width=5 by lexs_refl, lexs_next, step, inj/
-qed.
-
-lemma lexs_tc_push_sn: ∀RN,RP. c_reflexive … RP →
- ∀f,I2,L1,L2. TC … (lexs RN RP f) L1 L2 → ∀I1. RP L1 I1 I2 →
- TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRP #f #I2 #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
-/3 width=3 by lexs_push, TC_strap, inj/
-qed.
-
-lemma lexs_tc_push_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤*[RN, RP, f] L2 →
- TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
-/4 width=5 by lexs_refl, lexs_push, step, inj/
-qed.
-
-lemma lexs_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → L1 ⪤*[CTC … RN, RP, f] L2.
-#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
-/3 width=1 by lexs_push, lexs_next, inj/
-qed.
-
-lemma lexs_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤*[RN, RP, f] L2 → L1 ⪤*[RN, CTC … RP, f] L2.
-#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
-/3 width=1 by lexs_push, lexs_next, inj/
-qed.
-
-(* Main properties with transitive closure **********************************)
-
-theorem lexs_tc_next: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
- TC … (lexs RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
-/4 width=5 by lexs_tc_next_sn, lexs_tc_refl, trans_TC/
-qed.
-
-theorem lexs_tc_push: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. TC … (lexs RN RP f) L1 L2 →
- TC … (lexs RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
-#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
-/4 width=5 by lexs_tc_push_sn, lexs_tc_refl, trans_TC/
-qed.
-
-(* Basic_2A1: uses: TC_lpx_sn_ind *)
-theorem lexs_tc_step_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
- ∀f,L1,L. L1 ⪤*[RN, RP, f] L → 𝐈⦃f⦄ →
- ∀L2. L ⪤*[RN, CTC … RP, f] L2 → L1⪤* [RN, CTC … RP, f] L2.
-#RN #RP #HRP #f #L1 #L #H elim H -f -L1 -L
-[ #f #_ #Y #H -HRP >(lexs_inv_atom1 … H) -Y // ]
-#f #I1 #I #L1 #L #HL1 #HI1 #IH #Hf #Y #H
-[ elim (isid_inv_next … Hf) -Hf //
-| lapply (isid_inv_push … Hf ??) -Hf [3: |*: // ] #Hf
- elim (lexs_inv_push1 … H) -H #I2 #L2 #HL2 #HI2 #H destruct
- @lexs_push [ /2 width=1 by/ ] -L2 -IH
- @(TC_strap … HI1) -HI1
- @(HRP … HL1) // (**) (* auto fails *)
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lexs_tc_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
- ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (lexs RN RP f) L1 L2 → L1 ⪤*[RN, CTC … RP, f] L2.
-#RN #RP #HRP #f #Hf #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
-/3 width=3 by lexs_tc_step_dx, lexs_tc_inj_dx/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lexs_inv_tc_sn: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,L1,L2. L1 ⪤*[CTC … RN, RP, f] L2 → TC … (lexs RN RP f) L1 L2.
-#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
-/2 width=1 by lexs_tc_next, lexs_tc_push_sn, lexs_atom, inj/
-qed-.
-
-lemma lexs_inv_tc_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
- ∀f,L1,L2. L1 ⪤*[RN, CTC … RP, f] L2 → TC … (lexs RN RP f) L1 L2.
-#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
-/2 width=1 by lexs_tc_push, lexs_tc_next_sn, lexs_atom, inj/
-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/ideqsn_3.ma".
-include "basic_2/syntax/ceq_ext.ma".
-include "basic_2/relocation/lexs.ma".
-
-(* RANGED EQUIVALENCE FOR LOCAL ENVIRONMENTS ********************************)
-
-(* Basic_2A1: includes: lreq_atom lreq_zero lreq_pair lreq_succ *)
-definition lreq: relation3 rtmap lenv lenv ≝ lexs ceq_ext cfull.
-
-interpretation
- "ranged equivalence (local environment)"
- 'IdEqSn f L1 L2 = (lreq f L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lreq_eq_repl_back: ∀L1,L2. eq_repl_back … (λf. L1 ≡[f] L2).
-/2 width=3 by lexs_eq_repl_back/ qed-.
-
-lemma lreq_eq_repl_fwd: ∀L1,L2. eq_repl_fwd … (λf. L1 ≡[f] L2).
-/2 width=3 by lexs_eq_repl_fwd/ qed-.
-
-lemma sle_lreq_trans: ∀f2,L1,L2. L1 ≡[f2] L2 →
- ∀f1. f1 ⊆ f2 → L1 ≡[f1] L2.
-/2 width=3 by sle_lexs_trans/ qed-.
-
-(* Basic_2A1: includes: lreq_refl *)
-lemma lreq_refl: ∀f. reflexive … (lreq f).
-/2 width=1 by lexs_refl/ qed.
-
-(* Basic_2A1: includes: lreq_sym *)
-lemma lreq_sym: ∀f. symmetric … (lreq f).
-/3 width=2 by lexs_sym, cext2_sym/ qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_2A1: includes: lreq_inv_atom1 *)
-lemma lreq_inv_atom1: ∀f,Y. ⋆ ≡[f] Y → Y = ⋆.
-/2 width=4 by lexs_inv_atom1/ qed-.
-
-(* Basic_2A1: includes: lreq_inv_pair1 *)
-lemma lreq_inv_next1: ∀g,J,K1,Y. K1.ⓘ{J} ≡[↑g] Y →
- ∃∃K2. K1 ≡[g] K2 & Y = K2.ⓘ{J}.
-#g #J #K1 #Y #H
-elim (lexs_inv_next1 … H) -H #Z #K2 #HK12 #H1 #H2 destruct
-<(ceq_ext_inv_eq … H1) -Z /2 width=3 by ex2_intro/
-qed-.
-
-(* Basic_2A1: includes: lreq_inv_zero1 lreq_inv_succ1 *)
-lemma lreq_inv_push1: ∀g,J1,K1,Y. K1.ⓘ{J1} ≡[⫯g] Y →
- ∃∃J2,K2. K1 ≡[g] K2 & Y = K2.ⓘ{J2}.
-#g #J1 #K1 #Y #H elim (lexs_inv_push1 … H) -H /2 width=4 by ex2_2_intro/
-qed-.
-
-(* Basic_2A1: includes: lreq_inv_atom2 *)
-lemma lreq_inv_atom2: ∀f,X. X ≡[f] ⋆ → X = ⋆.
-/2 width=4 by lexs_inv_atom2/ qed-.
-
-(* Basic_2A1: includes: lreq_inv_pair2 *)
-lemma lreq_inv_next2: ∀g,J,X,K2. X ≡[↑g] K2.ⓘ{J} →
- ∃∃K1. K1 ≡[g] K2 & X = K1.ⓘ{J}.
-#g #J #X #K2 #H
-elim (lexs_inv_next2 … H) -H #Z #K1 #HK12 #H1 #H2 destruct
-<(ceq_ext_inv_eq … H1) -J /2 width=3 by ex2_intro/
-qed-.
-
-(* Basic_2A1: includes: lreq_inv_zero2 lreq_inv_succ2 *)
-lemma lreq_inv_push2: ∀g,J2,X,K2. X ≡[⫯g] K2.ⓘ{J2} →
- ∃∃J1,K1. K1 ≡[g] K2 & X = K1.ⓘ{J1}.
-#g #J2 #X #K2 #H elim (lexs_inv_push2 … H) -H /2 width=4 by ex2_2_intro/
-qed-.
-
-(* Basic_2A1: includes: lreq_inv_pair *)
-lemma lreq_inv_next: ∀f,I1,I2,L1,L2. L1.ⓘ{I1} ≡[↑f] L2.ⓘ{I2} →
- L1 ≡[f] L2 ∧ I1 = I2.
-#f #I1 #I2 #L1 #L2 #H elim (lexs_inv_next … H) -H
-/3 width=3 by ceq_ext_inv_eq, conj/
-qed-.
-
-(* Basic_2A1: includes: lreq_inv_succ *)
-lemma lreq_inv_push: ∀f,I1,I2,L1,L2. L1.ⓘ{I1} ≡[⫯f] L2.ⓘ{I2} → L1 ≡[f] L2.
-#f #I1 #I2 #L1 #L2 #H elim (lexs_inv_push … H) -H /2 width=1 by conj/
-qed-.
-
-lemma lreq_inv_tl: ∀f,I,L1,L2. L1 ≡[⫱f] L2 → L1.ⓘ{I} ≡[f] L2.ⓘ{I}.
-/2 width=1 by lexs_inv_tl/ qed-.
-
-(* Basic_2A1: removed theorems 5:
- lreq_pair_lt lreq_succ_lt lreq_pair_O_Y lreq_O2 lreq_inv_O_Y
-*)
+++ /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/lexs_length.ma".
-include "basic_2/relocation/lreq.ma".
-
-(* RANGED EQUIVALENCE FOR LOCAL ENVIRONMENTS ********************************)
-
-(* Forward lemmas with length for local environments ************************)
-
-(* Basic_2A1: includes: lreq_fwd_length *)
-lemma lreq_fwd_length: ∀f,L1,L2. L1 ≡[f] L2 → |L1| = |L2|.
-/2 width=4 by lexs_fwd_length/ 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/syntax/ceq_ext_ceq_ext.ma".
-include "basic_2/relocation/lexs_lexs.ma".
-
-(* RANGED EQUIVALENCE FOR LOCAL ENVIRONMENTS ********************************)
-
-(* Main properties **********************************************************)
-
-theorem lreq_trans: ∀f. Transitive … (lreq f).
-/3 width=5 by lexs_trans, ceq_ext_trans/ qed-.
-
-theorem lreq_canc_sn: ∀f. left_cancellable … (lreq f).
-/3 width=3 by lexs_canc_sn, lreq_trans, lreq_sym/ qed-.
-
-theorem lreq_canc_dx: ∀f. right_cancellable … (lreq f).
-/3 width=3 by lexs_canc_dx, lreq_trans, lreq_sym/ qed-.
-
-theorem lreq_join: ∀f1,L1,L2. L1 ≡[f1] L2 → ∀f2. L1 ≡[f2] L2 →
- ∀f. f1 ⋓ f2 ≘ f → L1 ≡[f] L2.
-/2 width=5 by lexs_join/ qed-.
-
-theorem lreq_meet: ∀f1,L1,L2. L1 ≡[f1] L2 → ∀f2. L1 ≡[f2] L2 →
- ∀f. f1 ⋒ f2 ≘ f → L1 ≡[f] L2.
-/2 width=5 by lexs_meet/ 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/ideqsn_3.ma".
+include "basic_2/syntax/ceq_ext.ma".
+include "basic_2/relocation/sex.ma".
+
+(* SYNTACTIC EQUIVALENCE FOR SELECTED LOCAL ENVIRONMENTS ********************)
+
+(* Basic_2A1: includes: lreq_atom lreq_zero lreq_pair lreq_succ *)
+definition seq: relation3 rtmap lenv lenv ≝ sex ceq_ext cfull.
+
+interpretation
+ "syntactic equivalence on selected entries (local environment)"
+ 'IdEqSn f L1 L2 = (seq f L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma seq_eq_repl_back: ∀L1,L2. eq_repl_back … (λf. L1 ≡[f] L2).
+/2 width=3 by sex_eq_repl_back/ qed-.
+
+lemma seq_eq_repl_fwd: ∀L1,L2. eq_repl_fwd … (λf. L1 ≡[f] L2).
+/2 width=3 by sex_eq_repl_fwd/ qed-.
+
+lemma sle_seq_trans: ∀f2,L1,L2. L1 ≡[f2] L2 →
+ ∀f1. f1 ⊆ f2 → L1 ≡[f1] L2.
+/2 width=3 by sle_sex_trans/ qed-.
+
+(* Basic_2A1: includes: lreq_refl *)
+lemma seq_refl: ∀f. reflexive … (seq f).
+/2 width=1 by sex_refl/ qed.
+
+(* Basic_2A1: includes: lreq_sym *)
+lemma seq_sym: ∀f. symmetric … (seq f).
+/3 width=2 by sex_sym, cext2_sym/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: includes: lreq_inv_atom1 *)
+lemma seq_inv_atom1: ∀f,Y. ⋆ ≡[f] Y → Y = ⋆.
+/2 width=4 by sex_inv_atom1/ qed-.
+
+(* Basic_2A1: includes: lreq_inv_pair1 *)
+lemma seq_inv_next1: ∀g,J,K1,Y. K1.ⓘ{J} ≡[↑g] Y →
+ ∃∃K2. K1 ≡[g] K2 & Y = K2.ⓘ{J}.
+#g #J #K1 #Y #H
+elim (sex_inv_next1 … H) -H #Z #K2 #HK12 #H1 #H2 destruct
+<(ceq_ext_inv_eq … H1) -Z /2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_zero1 lreq_inv_succ1 *)
+lemma seq_inv_push1: ∀g,J1,K1,Y. K1.ⓘ{J1} ≡[⫯g] Y →
+ ∃∃J2,K2. K1 ≡[g] K2 & Y = K2.ⓘ{J2}.
+#g #J1 #K1 #Y #H elim (sex_inv_push1 … H) -H /2 width=4 by ex2_2_intro/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_atom2 *)
+lemma seq_inv_atom2: ∀f,X. X ≡[f] ⋆ → X = ⋆.
+/2 width=4 by sex_inv_atom2/ qed-.
+
+(* Basic_2A1: includes: lreq_inv_pair2 *)
+lemma seq_inv_next2: ∀g,J,X,K2. X ≡[↑g] K2.ⓘ{J} →
+ ∃∃K1. K1 ≡[g] K2 & X = K1.ⓘ{J}.
+#g #J #X #K2 #H
+elim (sex_inv_next2 … H) -H #Z #K1 #HK12 #H1 #H2 destruct
+<(ceq_ext_inv_eq … H1) -J /2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_zero2 lreq_inv_succ2 *)
+lemma seq_inv_push2: ∀g,J2,X,K2. X ≡[⫯g] K2.ⓘ{J2} →
+ ∃∃J1,K1. K1 ≡[g] K2 & X = K1.ⓘ{J1}.
+#g #J2 #X #K2 #H elim (sex_inv_push2 … H) -H /2 width=4 by ex2_2_intro/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_pair *)
+lemma seq_inv_next: ∀f,I1,I2,L1,L2. L1.ⓘ{I1} ≡[↑f] L2.ⓘ{I2} →
+ ∧∧ L1 ≡[f] L2 & I1 = I2.
+#f #I1 #I2 #L1 #L2 #H elim (sex_inv_next … H) -H
+/3 width=3 by ceq_ext_inv_eq, conj/
+qed-.
+
+(* Basic_2A1: includes: lreq_inv_succ *)
+lemma seq_inv_push: ∀f,I1,I2,L1,L2. L1.ⓘ{I1} ≡[⫯f] L2.ⓘ{I2} → L1 ≡[f] L2.
+#f #I1 #I2 #L1 #L2 #H elim (sex_inv_push … H) -H /2 width=1 by conj/
+qed-.
+
+lemma seq_inv_tl: ∀f,I,L1,L2. L1 ≡[⫱f] L2 → L1.ⓘ{I} ≡[f] L2.ⓘ{I}.
+/2 width=1 by sex_inv_tl/ qed-.
+
+(* Basic_2A1: removed theorems 5:
+ lreq_pair_lt lreq_succ_lt lreq_pair_O_Y lreq_O2 lreq_inv_O_Y
+*)
--- /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/sex_length.ma".
+include "basic_2/relocation/seq.ma".
+
+(* SYNTACTIC EQUIVALENCE FOR SELECTED LOCAL ENVIRONMENTS ********************)
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Basic_2A1: includes: lreq_fwd_length *)
+lemma seq_fwd_length: ∀f,L1,L2. L1 ≡[f] L2 → |L1| = |L2|.
+/2 width=4 by sex_fwd_length/ 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/syntax/ceq_ext_ceq_ext.ma".
+include "basic_2/relocation/sex_sex.ma".
+
+(* SYNTACTIC EQUIVALENCE FOR SELECTED LOCAL ENVIRONMENTS ********************)
+
+(* Main properties **********************************************************)
+
+theorem seq_trans: ∀f. Transitive … (seq f).
+/3 width=5 by sex_trans, ceq_ext_trans/ qed-.
+
+theorem seq_canc_sn: ∀f. left_cancellable … (seq f).
+/3 width=3 by sex_canc_sn, seq_trans, seq_sym/ qed-.
+
+theorem seq_canc_dx: ∀f. right_cancellable … (seq f).
+/3 width=3 by sex_canc_dx, seq_trans, seq_sym/ qed-.
+
+theorem seq_join: ∀f1,L1,L2. L1 ≡[f1] L2 → ∀f2. L1 ≡[f2] L2 →
+ ∀f. f1 ⋓ f2 ≘ f → L1 ≡[f] L2.
+/2 width=5 by sex_join/ qed-.
+
+theorem seq_meet: ∀f1,L1,L2. L1 ≡[f1] L2 → ∀f2. L1 ≡[f2] L2 →
+ ∀f. f1 ⋒ f2 ≘ f → L1 ≡[f] L2.
+/2 width=5 by sex_meet/ 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/relocation/rtmap_sle.ma".
+include "ground_2/relocation/rtmap_sdj.ma".
+include "basic_2/notation/relations/relation_5.ma".
+include "basic_2/syntax/lenv.ma".
+
+(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
+
+inductive sex (RN,RP:relation3 lenv bind bind): rtmap → relation lenv ≝
+| sex_atom: ∀f. sex RN RP f (⋆) (⋆)
+| sex_next: ∀f,I1,I2,L1,L2.
+ sex RN RP f L1 L2 → RN L1 I1 I2 →
+ sex RN RP (↑f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+| sex_push: ∀f,I1,I2,L1,L2.
+ sex RN RP f L1 L2 → RP L1 I1 I2 →
+ sex RN RP (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+.
+
+interpretation "generic entrywise extension (local environment)"
+ 'Relation RN RP f L1 L2 = (sex RN RP f L1 L2).
+
+definition R_pw_confluent2_sex: relation3 lenv bind bind → relation3 lenv bind bind →
+ relation3 lenv bind bind → relation3 lenv bind bind →
+ relation3 lenv bind bind → relation3 lenv bind bind →
+ relation3 rtmap lenv bind ≝
+ λR1,R2,RN1,RP1,RN2,RP2,f,L0,I0.
+ ∀I1. R1 L0 I0 I1 → ∀I2. R2 L0 I0 I2 →
+ ∀L1. L0 ⪤[RN1, RP1, f] L1 → ∀L2. L0 ⪤[RN2, RP2, f] L2 →
+ ∃∃I. R2 L1 I1 I & R1 L2 I2 I.
+
+definition sex_transitive: relation3 lenv bind bind → relation3 lenv bind bind →
+ relation3 lenv bind bind →
+ relation3 lenv bind bind → relation3 lenv bind bind →
+ relation3 rtmap lenv bind ≝
+ λR1,R2,R3,RN,RP,f,L1,I1.
+ ∀I. R1 L1 I1 I → ∀L2. L1 ⪤[RN, RP, f] L2 →
+ ∀I2. R2 L2 I I2 → R3 L1 I1 I2.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact sex_inv_atom1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → X = ⋆ → Y = ⋆.
+#RN #RP #f #X #Y * -f -X -Y //
+#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
+qed-.
+
+(* Basic_2A1: includes lpx_sn_inv_atom1 *)
+lemma sex_inv_atom1: ∀RN,RP,f,Y. ⋆ ⪤[RN, RP, f] Y → Y = ⋆.
+/2 width=6 by sex_inv_atom1_aux/ qed-.
+
+fact sex_inv_next1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ↑g →
+ ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
+#RN #RP #f #X #Y * -f -X -Y
+[ #f #g #J1 #K1 #H destruct
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_next … H2) -g destruct
+ /2 width=5 by ex3_2_intro/
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_push_next … H)
+]
+qed-.
+
+(* Basic_2A1: includes lpx_sn_inv_pair1 *)
+lemma sex_inv_next1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN, RP, ↑g] Y →
+ ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & Y = K2.ⓘ{J2}.
+/2 width=7 by sex_inv_next1_aux/ qed-.
+
+fact sex_inv_push1_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J1,K1. X = K1.ⓘ{J1} → f = ⫯g →
+ ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
+#RN #RP #f #X #Y * -f -X -Y
+[ #f #g #J1 #K1 #H destruct
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J1 #K1 #_ #H elim (discr_next_push … H)
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J1 #K1 #H1 #H2 <(injective_push … H2) -g destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma sex_inv_push1: ∀RN,RP,g,J1,K1,Y. K1.ⓘ{J1} ⪤[RN, RP, ⫯g] Y →
+ ∃∃J2,K2. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & Y = K2.ⓘ{J2}.
+/2 width=7 by sex_inv_push1_aux/ qed-.
+
+fact sex_inv_atom2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → Y = ⋆ → X = ⋆.
+#RN #RP #f #X #Y * -f -X -Y //
+#f #I1 #I2 #L1 #L2 #_ #_ #H destruct
+qed-.
+
+(* Basic_2A1: includes lpx_sn_inv_atom2 *)
+lemma sex_inv_atom2: ∀RN,RP,f,X. X ⪤[RN, RP, f] ⋆ → X = ⋆.
+/2 width=6 by sex_inv_atom2_aux/ qed-.
+
+fact sex_inv_next2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ↑g →
+ ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
+#RN #RP #f #X #Y * -f -X -Y
+[ #f #g #J2 #K2 #H destruct
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_next … H2) -g destruct
+ /2 width=5 by ex3_2_intro/
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_push_next … H)
+]
+qed-.
+
+(* Basic_2A1: includes lpx_sn_inv_pair2 *)
+lemma sex_inv_next2: ∀RN,RP,g,J2,X,K2. X ⪤[RN, RP, ↑g] K2.ⓘ{J2} →
+ ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RN K1 J1 J2 & X = K1.ⓘ{J1}.
+/2 width=7 by sex_inv_next2_aux/ qed-.
+
+fact sex_inv_push2_aux: ∀RN,RP,f,X,Y. X ⪤[RN, RP, f] Y → ∀g,J2,K2. Y = K2.ⓘ{J2} → f = ⫯g →
+ ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
+#RN #RP #f #X #Y * -f -X -Y
+[ #f #J2 #K2 #g #H destruct
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J2 #K2 #_ #H elim (discr_next_push … H)
+| #f #I1 #I2 #L1 #L2 #HL #HI #g #J2 #K2 #H1 #H2 <(injective_push … H2) -g destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma sex_inv_push2: ∀RN,RP,g,J2,X,K2. X ⪤[RN, RP, ⫯g] K2.ⓘ{J2} →
+ ∃∃J1,K1. K1 ⪤[RN, RP, g] K2 & RP K1 J1 J2 & X = K1.ⓘ{J1}.
+/2 width=7 by sex_inv_push2_aux/ qed-.
+
+(* Basic_2A1: includes lpx_sn_inv_pair *)
+lemma sex_inv_next: ∀RN,RP,f,I1,I2,L1,L2.
+ L1.ⓘ{I1} ⪤[RN, RP, ↑f] L2.ⓘ{I2} →
+ L1 ⪤[RN, RP, f] L2 ∧ RN L1 I1 I2.
+#RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_next1 … H) -H
+#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
+qed-.
+
+lemma sex_inv_push: ∀RN,RP,f,I1,I2,L1,L2.
+ L1.ⓘ{I1} ⪤[RN, RP, ⫯f] L2.ⓘ{I2} →
+ L1 ⪤[RN, RP, f] L2 ∧ RP L1 I1 I2.
+#RN #RP #f #I1 #I2 #L1 #L2 #H elim (sex_inv_push1 … H) -H
+#I0 #L0 #HL10 #HI10 #H destruct /2 width=1 by conj/
+qed-.
+
+lemma sex_inv_tl: ∀RN,RP,f,I1,I2,L1,L2. L1 ⪤[RN, RP, ⫱f] L2 →
+ RN L1 I1 I2 → RP L1 I1 I2 →
+ L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2}.
+#RN #RP #f #I1 #I2 #L2 #L2 elim (pn_split f) *
+/2 width=1 by sex_next, sex_push/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma sex_fwd_bind: ∀RN,RP,f,I1,I2,L1,L2.
+ L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} →
+ L1 ⪤[RN, RP, ⫱f] L2.
+#RN #RP #f #I1 #I2 #L1 #L2 #Hf
+elim (pn_split f) * #g #H destruct
+[ elim (sex_inv_push … Hf) | elim (sex_inv_next … Hf) ] -Hf //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma sex_eq_repl_back: ∀RN,RP,L1,L2. eq_repl_back … (λf. L1 ⪤[RN, RP, f] L2).
+#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
+#f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H
+[ elim (eq_inv_nx … H) -H /3 width=3 by sex_next/
+| elim (eq_inv_px … H) -H /3 width=3 by sex_push/
+]
+qed-.
+
+lemma sex_eq_repl_fwd: ∀RN,RP,L1,L2. eq_repl_fwd … (λf. L1 ⪤[RN, RP, f] L2).
+#RN #RP #L1 #L2 @eq_repl_sym /2 width=3 by sex_eq_repl_back/ (**) (* full auto fails *)
+qed-.
+
+lemma sex_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
+ ∀f.reflexive … (sex RN RP f).
+#RN #RP #HRN #HRP #f #L generalize in match f; -f elim L -L //
+#L #I #IH #f elim (pn_split f) *
+#g #H destruct /2 width=1 by sex_next, sex_push/
+qed.
+
+lemma sex_sym: ∀RN,RP.
+ (∀L1,L2,I1,I2. RN L1 I1 I2 → RN L2 I2 I1) →
+ (∀L1,L2,I1,I2. RP L1 I1 I2 → RP L2 I2 I1) →
+ ∀f. symmetric … (sex RN RP f).
+#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -L1 -L2 -f
+/3 width=2 by sex_next, sex_push/
+qed-.
+
+lemma sex_pair_repl: ∀RN,RP,f,I1,I2,L1,L2.
+ L1.ⓘ{I1} ⪤[RN, RP, f] L2.ⓘ{I2} →
+ ∀J1,J2. RN L1 J1 J2 → RP L1 J1 J2 →
+ L1.ⓘ{J1} ⪤[RN, RP, f] L2.ⓘ{J2}.
+/3 width=3 by sex_inv_tl, sex_fwd_bind/ qed-.
+
+lemma sex_co: ∀RN1,RP1,RN2,RP2. RN1 ⊆ RN2 → RP1 ⊆ RP2 →
+ ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → L1 ⪤[RN2, RP2, f] L2.
+#RN1 #RP1 #RN2 #RP2 #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
+/3 width=1 by sex_atom, sex_next, sex_push/
+qed-.
+
+lemma sex_co_isid: ∀RN1,RP1,RN2,RP2. RP1 ⊆ RP2 →
+ ∀f,L1,L2. L1 ⪤[RN1, RP1, f] L2 → 𝐈⦃f⦄ →
+ L1 ⪤[RN2, RP2, f] L2.
+#RN1 #RP1 #RN2 #RP2 #HR #f #L1 #L2 #H elim H -f -L1 -L2 //
+#f #I1 #I2 #K1 #K2 #_ #HI12 #IH #H
+[ elim (isid_inv_next … H) -H //
+| /4 width=3 by sex_push, isid_inv_push/
+]
+qed-.
+
+lemma sex_sdj: ∀RN,RP. RP ⊆ RN →
+ ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 →
+ ∀f2. f1 ∥ f2 → L1 ⪤[RP, RN, f2] L2.
+#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
+#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
+[ elim (sdj_inv_nx … H12) -H12 [2,3: // ]
+ #g2 #H #H2 destruct /3 width=1 by sex_push/
+| elim (sdj_inv_px … H12) -H12 [2,4: // ] *
+ #g2 #H #H2 destruct /3 width=1 by sex_next, sex_push/
+]
+qed-.
+
+lemma sle_sex_trans: ∀RN,RP. RN ⊆ RP →
+ ∀f2,L1,L2. L1 ⪤[RN, RP, f2] L2 →
+ ∀f1. f1 ⊆ f2 → L1 ⪤[RN, RP, f1] L2.
+#RN #RP #HR #f2 #L1 #L2 #H elim H -f2 -L1 -L2 //
+#f2 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f1 #H12
+[ elim (pn_split f1) * ]
+[ /4 width=5 by sex_push, sle_inv_pn/
+| /4 width=5 by sex_next, sle_inv_nn/
+| elim (sle_inv_xp … H12) -H12 [2,3: // ]
+ #g1 #H #H1 destruct /3 width=5 by sex_push/
+]
+qed-.
+
+lemma sle_sex_conf: ∀RN,RP. RP ⊆ RN →
+ ∀f1,L1,L2. L1 ⪤[RN, RP, f1] L2 →
+ ∀f2. f1 ⊆ f2 → L1 ⪤[RN, RP, f2] L2.
+#RN #RP #HR #f1 #L1 #L2 #H elim H -f1 -L1 -L2 //
+#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H12
+[2: elim (pn_split f2) * ]
+[ /4 width=5 by sex_push, sle_inv_pp/
+| /4 width=5 by sex_next, sle_inv_pn/
+| elim (sle_inv_nx … H12) -H12 [2,3: // ]
+ #g2 #H #H2 destruct /3 width=5 by sex_next/
+]
+qed-.
+
+lemma sex_sle_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
+ ∀f,L1,L2. L1 ⪤[R1, RP, f] L2 → ∀g. f ⊆ g →
+ ∃∃L. L1 ⪤[R1, RP, g] L & L ⪤[R2, cfull, f] L2.
+#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
+[ /2 width=3 by sex_atom, ex2_intro/ ]
+#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
+[ elim (sle_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
+ elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, ex2_intro/
+| elim (sle_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
+ elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, sex_push, ex2_intro/
+]
+qed-.
+
+lemma sex_sdj_split: ∀R1,R2,RP. c_reflexive … R1 → c_reflexive … R2 →
+ ∀f,L1,L2. L1 ⪤[R1, RP, f] L2 → ∀g. f ∥ g →
+ ∃∃L. L1 ⪤[RP, R1, g] L & L ⪤[R2, cfull, f] L2.
+#R1 #R2 #RP #HR1 #HR2 #f #L1 #L2 #H elim H -f -L1 -L2
+[ /2 width=3 by sex_atom, ex2_intro/ ]
+#f #I1 #I2 #L1 #L2 #_ #HI12 #IH #y #H
+[ elim (sdj_inv_nx … H ??) -H [ |*: // ] #g #Hfg #H destruct
+ elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, sex_push, ex2_intro/
+| elim (sdj_inv_px … H ??) -H [1,3: * |*: // ] #g #Hfg #H destruct
+ elim (IH … Hfg) -IH -Hfg /3 width=5 by sex_next, sex_push, ex2_intro/
+]
+qed-.
+
+lemma sex_dec: ∀RN,RP.
+ (∀L,I1,I2. Decidable (RN L I1 I2)) →
+ (∀L,I1,I2. Decidable (RP L I1 I2)) →
+ ∀L1,L2,f. Decidable (L1 ⪤[RN, RP, f] L2).
+#RN #RP #HRN #HRP #L1 elim L1 -L1 [ * | #L1 #I1 #IH * ]
+[ /2 width=1 by sex_atom, or_introl/
+| #L2 #I2 #f @or_intror #H
+ lapply (sex_inv_atom1 … H) -H #H destruct
+| #f @or_intror #H
+ lapply (sex_inv_atom2 … H) -H #H destruct
+| #L2 #I2 #f elim (IH L2 (⫱f)) -IH #HL12
+ [2: /4 width=3 by sex_fwd_bind, or_intror/ ]
+ elim (pn_split f) * #g #H destruct
+ [ elim (HRP L1 I1 I2) | elim (HRN L1 I1 I2) ] -HRP -HRN #HV12
+ [1,3: /3 width=1 by sex_push, sex_next, or_introl/ ]
+ @or_intror #H
+ [ elim (sex_inv_push … H) | elim (sex_inv_next … H) ] -H
+ /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 "basic_2/syntax/lenv_length.ma".
+include "basic_2/relocation/sex.ma".
+
+(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma sex_fwd_length: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → |L1| = |L2|.
+#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2 //
+#f #I1 #I2 #L1 #L2 >length_bind >length_bind //
+qed-.
+
+(* Properties with length for local environments ****************************)
+
+lemma sex_length_cfull: ∀L1,L2. |L1| = |L2| → ∀f. L1 ⪤[cfull, cfull, f] L2.
+#L1 elim L1 -L1
+[ #Y2 #H >(length_inv_zero_sn … H) -Y2 //
+| #L1 #I1 #IH #Y2 #H #f
+ elim (length_inv_succ_sn … H) -H #I2 #L2 #HL12 #H destruct
+ elim (pn_split f) * #g #H destruct /3 width=1 by sex_next, sex_push/
+]
+qed.
+
+lemma sex_length_isid: ∀R,L1,L2. |L1| = |L2| →
+ ∀f. 𝐈⦃f⦄ → L1 ⪤[R, cfull, f] L2.
+#R #L1 elim L1 -L1
+[ #Y2 #H >(length_inv_zero_sn … H) -Y2 //
+| #L1 #I1 #IH #Y2 #H #f #Hf
+ elim (length_inv_succ_sn … H) -H #I2 #L2 #HL12 #H destruct
+ elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct /3 width=1 by sex_push/
+]
+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/relocation/rtmap_sand.ma".
+include "basic_2/relocation/drops.ma".
+
+(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
+
+(* Main properties **********************************************************)
+
+theorem sex_trans_gen (RN1) (RP1) (RN2) (RP2) (RN) (RP):
+ ∀L1,f.
+ (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → sex_transitive RN1 RN2 RN RN1 RP1 g K I) →
+ (∀g,I,K,n. ⬇*[n] L1 ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → sex_transitive RP1 RP2 RP RN1 RP1 g K I) →
+ ∀L0. L1 ⪤[RN1, RP1, f] L0 →
+ ∀L2. L0 ⪤[RN2, RP2, f] L2 →
+ L1 ⪤[RN, RP, f] L2.
+#RN1 #RP1 #RN2 #RP2 #RN #RP #L1 elim L1 -L1
+[ #f #_ #_ #L0 #H1 #L2 #H2
+ lapply (sex_inv_atom1 … H1) -H1 #H destruct
+ lapply (sex_inv_atom1 … H2) -H2 #H destruct
+ /2 width=1 by sex_atom/
+| #K1 #I1 #IH #f elim (pn_split f) * #g #H destruct
+ #HN #HP #L0 #H1 #L2 #H2
+ [ elim (sex_inv_push1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
+ elim (sex_inv_push1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct
+ lapply (HP … 0 … HI10 … HK10 … HI02) -HI10 -HI02 /2 width=2 by drops_refl/ #HI12
+ lapply (IH … HK10 … HK02) -IH -K0 /3 width=3 by sex_push, drops_drop/
+ | elim (sex_inv_next1 … H1) -H1 #I0 #K0 #HK10 #HI10 #H destruct
+ elim (sex_inv_next1 … H2) -H2 #I2 #K2 #HK02 #HI02 #H destruct
+ lapply (HN … 0 … HI10 … HK10 … HI02) -HI10 -HI02 /2 width=2 by drops_refl/ #HI12
+ lapply (IH … HK10 … HK02) -IH -K0 /3 width=3 by sex_next, drops_drop/
+ ]
+]
+qed-.
+
+theorem sex_trans (RN) (RP) (f): (∀g,I,K. sex_transitive RN RN RN RN RP g K I) →
+ (∀g,I,K. sex_transitive RP RP RP RN RP g K I) →
+ Transitive … (sex RN RP f).
+/2 width=9 by sex_trans_gen/ qed-.
+
+theorem sex_trans_id_cfull: ∀R1,R2,R3,L1,L,f. L1 ⪤[R1, cfull, f] L → 𝐈⦃f⦄ →
+ ∀L2. L ⪤[R2, cfull, f] L2 → L1 ⪤[R3, cfull, f] L2.
+#R1 #R2 #R3 #L1 #L #f #H elim H -L1 -L -f
+[ #f #Hf #L2 #H >(sex_inv_atom1 … H) -L2 // ]
+#f #I1 #I #K1 #K #HK1 #_ #IH #Hf #L2 #H
+[ elim (isid_inv_next … Hf) | lapply (isid_inv_push … Hf ??) ] -Hf [5: |*: // ] #Hf
+elim (sex_inv_push1 … H) -H #I2 #K2 #HK2 #_ #H destruct
+/3 width=1 by sex_push/
+qed-.
+
+theorem sex_conf (RN1) (RP1) (RN2) (RP2):
+ ∀L,f.
+ (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f → R_pw_confluent2_sex RN1 RN2 RN1 RP1 RN2 RP2 g K I) →
+ (∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ⫯g = ⫱*[n] f → R_pw_confluent2_sex RP1 RP2 RN1 RP1 RN2 RP2 g K I) →
+ pw_confluent2 … (sex RN1 RP1 f) (sex RN2 RP2 f) L.
+#RN1 #RP1 #RN2 #RP2 #L elim L -L
+[ #f #_ #_ #L1 #H1 #L2 #H2 >(sex_inv_atom1 … H1) >(sex_inv_atom1 … H2) -H2 -H1
+ /2 width=3 by sex_atom, ex2_intro/
+| #L #I0 #IH #f elim (pn_split f) * #g #H destruct
+ #HN #HP #Y1 #H1 #Y2 #H2
+ [ elim (sex_inv_push1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
+ elim (sex_inv_push1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
+ elim (HP … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
+ elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, sex_push, ex2_intro/
+ | elim (sex_inv_next1 … H1) -H1 #I1 #L1 #HL1 #HI01 #H destruct
+ elim (sex_inv_next1 … H2) -H2 #I2 #L2 #HL2 #HI02 #H destruct
+ elim (HN … 0 … HI01 … HI02 … HL1 … HL2) -HI01 -HI02 /2 width=2 by drops_refl/ #I #HI1 #HI2
+ elim (IH … HL1 … HL2) -IH -HL1 -HL2 /3 width=5 by drops_drop, sex_next, ex2_intro/
+ ]
+]
+qed-.
+
+theorem sex_canc_sn: ∀RN,RP,f. Transitive … (sex RN RP f) →
+ symmetric … (sex RN RP f) →
+ left_cancellable … (sex RN RP f).
+/3 width=3 by/ qed-.
+
+theorem sex_canc_dx: ∀RN,RP,f. Transitive … (sex RN RP f) →
+ symmetric … (sex RN RP f) →
+ right_cancellable … (sex RN RP f).
+/3 width=3 by/ qed-.
+
+lemma sex_meet: ∀RN,RP,L1,L2.
+ ∀f1. L1 ⪤[RN, RP, f1] L2 →
+ ∀f2. L1 ⪤[RN, RP, f2] L2 →
+ ∀f. f1 ⋒ f2 ≘ f → L1 ⪤[RN, RP, f] L2.
+#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
+#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
+elim (pn_split f2) * #g2 #H2 destruct
+try elim (sex_inv_push … H) try elim (sex_inv_next … H) -H
+[ elim (sand_inv_npx … Hf) | elim (sand_inv_nnx … Hf)
+| elim (sand_inv_ppx … Hf) | elim (sand_inv_pnx … Hf)
+] -Hf /3 width=5 by sex_next, sex_push/
+qed-.
+
+lemma sex_join: ∀RN,RP,L1,L2.
+ ∀f1. L1 ⪤[RN, RP, f1] L2 →
+ ∀f2. L1 ⪤[RN, RP, f2] L2 →
+ ∀f. f1 ⋓ f2 ≘ f → L1 ⪤[RN, RP, f] L2.
+#RN #RP #L1 #L2 #f1 #H elim H -f1 -L1 -L2 //
+#f1 #I1 #I2 #L1 #L2 #_ #HI12 #IH #f2 #H #f #Hf
+elim (pn_split f2) * #g2 #H2 destruct
+try elim (sex_inv_push … H) try elim (sex_inv_next … H) -H
+[ elim (sor_inv_npx … Hf) | elim (sor_inv_nnx … Hf)
+| elim (sor_inv_ppx … Hf) | elim (sor_inv_pnx … Hf)
+] -Hf /3 width=5 by sex_next, sex_push/
+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/star.ma".
+include "basic_2/relocation/sex.ma".
+
+(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
+
+definition s_rs_transitive_isid: relation (relation3 lenv bind bind) ≝ λRN,RP.
+ ∀f. 𝐈⦃f⦄ → s_rs_transitive … RP (λ_.sex RN RP f).
+
+(* Properties with transitive closure ***************************************)
+
+lemma sex_tc_refl: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
+ ∀f. reflexive … (TC … (sex RN RP f)).
+/3 width=1 by sex_refl, TC_reflexive/ qed.
+
+lemma sex_tc_next_sn: ∀RN,RP. c_reflexive … RN →
+ ∀f,I2,L1,L2. TC … (sex RN RP f) L1 L2 → ∀I1. RN L1 I1 I2 →
+ TC … (sex RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRN #f #I2 #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
+/3 width=3 by sex_next, TC_strap, inj/
+qed.
+
+lemma sex_tc_next_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
+ ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. L1 ⪤[RN, RP, f] L2 →
+ TC … (sex RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
+/4 width=5 by sex_refl, sex_next, step, inj/
+qed.
+
+lemma sex_tc_push_sn: ∀RN,RP. c_reflexive … RP →
+ ∀f,I2,L1,L2. TC … (sex RN RP f) L1 L2 → ∀I1. RP L1 I1 I2 →
+ TC … (sex RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRP #f #I2 #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
+/3 width=3 by sex_push, TC_strap, inj/
+qed.
+
+lemma sex_tc_push_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
+ ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. L1 ⪤[RN, RP, f] L2 →
+ TC … (sex RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
+/4 width=5 by sex_refl, sex_push, step, inj/
+qed.
+
+lemma sex_tc_inj_sn: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → L1 ⪤[CTC … RN, RP, f] L2.
+#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
+/3 width=1 by sex_push, sex_next, inj/
+qed.
+
+lemma sex_tc_inj_dx: ∀RN,RP,f,L1,L2. L1 ⪤[RN, RP, f] L2 → L1 ⪤[RN, CTC … RP, f] L2.
+#RN #RP #f #L1 #L2 #H elim H -f -L1 -L2
+/3 width=1 by sex_push, sex_next, inj/
+qed.
+
+(* Main properties with transitive closure **********************************)
+
+theorem sex_tc_next: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
+ ∀f,I1,I2,L1. (CTC … RN) L1 I1 I2 → ∀L2. TC … (sex RN RP f) L1 L2 →
+ TC … (sex RN RP (↑f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
+/4 width=5 by sex_tc_next_sn, sex_tc_refl, trans_TC/
+qed.
+
+theorem sex_tc_push: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
+ ∀f,I1,I2,L1. (CTC … RP) L1 I1 I2 → ∀L2. TC … (sex RN RP f) L1 L2 →
+ TC … (sex RN RP (⫯f)) (L1.ⓘ{I1}) (L2.ⓘ{I2}).
+#RN #RP #HRN #HRP #f #I1 #I2 #L1 #H elim H -I2
+/4 width=5 by sex_tc_push_sn, sex_tc_refl, trans_TC/
+qed.
+
+(* Basic_2A1: uses: TC_lpx_sn_ind *)
+theorem sex_tc_step_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
+ ∀f,L1,L. L1 ⪤[RN, RP, f] L → 𝐈⦃f⦄ →
+ ∀L2. L ⪤[RN, CTC … RP, f] L2 → L1⪤ [RN, CTC … RP, f] L2.
+#RN #RP #HRP #f #L1 #L #H elim H -f -L1 -L
+[ #f #_ #Y #H -HRP >(sex_inv_atom1 … H) -Y // ]
+#f #I1 #I #L1 #L #HL1 #HI1 #IH #Hf #Y #H
+[ elim (isid_inv_next … Hf) -Hf //
+| lapply (isid_inv_push … Hf ??) -Hf [3: |*: // ] #Hf
+ elim (sex_inv_push1 … H) -H #I2 #L2 #HL2 #HI2 #H destruct
+ @sex_push [ /2 width=1 by/ ] -L2 -IH
+ @(TC_strap … HI1) -HI1
+ @(HRP … HL1) // (**) (* auto fails *)
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma sex_tc_dx: ∀RN,RP. s_rs_transitive_isid RN RP →
+ ∀f. 𝐈⦃f⦄ → ∀L1,L2. TC … (sex RN RP f) L1 L2 → L1 ⪤[RN, CTC … RP, f] L2.
+#RN #RP #HRP #f #Hf #L1 #L2 #H @(TC_ind_dx ??????? H) -L1
+/3 width=3 by sex_tc_step_dx, sex_tc_inj_dx/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma sex_inv_tc_sn: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
+ ∀f,L1,L2. L1 ⪤[CTC … RN, RP, f] L2 → TC … (sex RN RP f) L1 L2.
+#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
+/2 width=1 by sex_tc_next, sex_tc_push_sn, sex_atom, inj/
+qed-.
+
+lemma sex_inv_tc_dx: ∀RN,RP. c_reflexive … RN → c_reflexive … RP →
+ ∀f,L1,L2. L1 ⪤[RN, CTC … RP, f] L2 → TC … (sex RN RP f) L1 L2.
+#RN #RP #HRN #HRP #f #L1 #L2 #H elim H -f -L1 -L2
+/2 width=1 by sex_tc_push, sex_tc_next_sn, sex_atom, inj/
+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/static/fdeq.ma".
+include "basic_2/rt_computation/cpxs_rdeq.ma".
+
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Properties with degree-based equivalence for closures ********************)
+
+lemma fdeq_cpxs_trans: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T⦄ →
+ ∀T2. ⦃G2, L2⦄ ⊢ T ⬈*[h] T2 →
+ ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T0 & ⦃G1, L1, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
+elim (fdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
+elim (rdeq_cpxs_trans … HT2 … HL12) #T0 #HT0 #HT02
+lapply (cpxs_rdeq_conf_dx … HT2 … HL12) -HL12 #HL12
+elim (tdeq_cpxs_trans … HT1 … HT0) -T #T #HT1 #HT0
+/4 width=5 by fdeq_intro_dx, tdeq_trans, 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/static/ffdeq.ma".
-include "basic_2/rt_computation/cpxs_lfdeq.ma".
-
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-
-(* Properties with degree-based equivalence for closures ********************)
-
-lemma ffdeq_cpxs_trans: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T⦄ →
- ∀T2. ⦃G2, L2⦄ ⊢ T ⬈*[h] T2 →
- ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈*[h] T0 & ⦃G1, L1, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
-elim (ffdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
-elim (lfdeq_cpxs_trans … HT2 … HL12) #T0 #HT0 #HT02
-lapply (cpxs_lfdeq_conf_dx … HT2 … HL12) -HL12 #HL12
-elim (tdeq_cpxs_trans … HT1 … HT0) -T #T #HT1 #HT0
-/4 width=5 by ffdeq_intro_dx, tdeq_trans, 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/rt_transition/cpx_lfdeq.ma".
-include "basic_2/rt_computation/cpxs_tdeq.ma".
-
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
-
-(* Properties with degree-based equivalence for local environments **********)
-
-(* Basic_2A1: was just: lleq_cpxs_trans *)
-lemma lfdeq_cpxs_trans: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
- ∀L2. L2 ≛[h, o, T0] L0 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛[h, o] T1.
-#h #o #G #L0 #T0 #T1 #H @(cpxs_ind_dx … H) -T0 /2 width=3 by ex2_intro/
-#T0 #T #HT0 #_ #IH #L2 #HL2
-elim (lfdeq_cpx_trans … HL2 … HT0) #U1 #H1 #H2
-elim (IH L2) -IH /2 width=4 by cpx_lfdeq_conf_dx/ -L0 #U2 #H3 #H4
-elim (tdeq_cpxs_trans … H2 … H3) -T #U0 #H2 #H3
-/3 width=5 by cpxs_strap2, tdeq_trans, ex2_intro/
-qed-.
-
-(* Basic_2A1: was just: cpxs_lleq_conf *)
-lemma cpxs_lfdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
- ∀L2. L0 ≛[h, o, T0] L2 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛[h, o] T1.
-/3 width=3 by lfdeq_cpxs_trans, lfdeq_sym/ qed-.
-
-(* Basic_2A1: was just: cpxs_lleq_conf_dx *)
-lemma cpxs_lfdeq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈*[h] T2 →
- ∀L1. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
-#h #o #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_lfdeq_conf_dx/
-qed-.
-
-(* Basic_2A1: was just: lleq_conf_sn *)
-lemma cpxs_lfdeq_conf_sn: ∀h,o,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬈*[h] T2 →
- ∀L2. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
-/4 width=6 by cpxs_lfdeq_conf_dx, lfdeq_sym/ 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/rt_transition/cpx_rdeq.ma".
+include "basic_2/rt_computation/cpxs_tdeq.ma".
+
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Properties with degree-based equivalence for local environments **********)
+
+(* Basic_2A1: was just: lleq_cpxs_trans *)
+lemma rdeq_cpxs_trans: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
+ ∀L2. L2 ≛[h, o, T0] L0 →
+ ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛[h, o] T1.
+#h #o #G #L0 #T0 #T1 #H @(cpxs_ind_dx … H) -T0 /2 width=3 by ex2_intro/
+#T0 #T #HT0 #_ #IH #L2 #HL2
+elim (rdeq_cpx_trans … HL2 … HT0) #U1 #H1 #H2
+elim (IH L2) -IH /2 width=4 by cpx_rdeq_conf_dx/ -L0 #U2 #H3 #H4
+elim (tdeq_cpxs_trans … H2 … H3) -T #U0 #H2 #H3
+/3 width=5 by cpxs_strap2, tdeq_trans, ex2_intro/
+qed-.
+
+(* Basic_2A1: was just: cpxs_lleq_conf *)
+lemma cpxs_rdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈*[h] T1 →
+ ∀L2. L0 ≛[h, o, T0] L2 →
+ ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈*[h] T & T ≛[h, o] T1.
+/3 width=3 by rdeq_cpxs_trans, rdeq_sym/ qed-.
+
+(* Basic_2A1: was just: cpxs_lleq_conf_dx *)
+lemma cpxs_rdeq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀L1. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
+#h #o #G #L2 #T1 #T2 #H @(cpxs_ind … H) -T2 /3 width=6 by cpx_rdeq_conf_dx/
+qed-.
+
+(* Basic_2A1: was just: lleq_conf_sn *)
+lemma cpxs_rdeq_conf_sn: ∀h,o,G,L1,T1,T2. ⦃G, L1⦄ ⊢ T1 ⬈*[h] T2 →
+ ∀L2. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
+/4 width=6 by cpxs_rdeq_conf_dx, rdeq_sym/ qed-.
(* *)
(**************************************************************************)
-include "basic_2/rt_transition/lfpx_lfdeq.ma".
+include "basic_2/rt_transition/rpx_rdeq.ma".
include "basic_2/rt_computation/cpxs.ma".
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
(* *)
(**************************************************************************)
-include "basic_2/rt_transition/lpx_lfdeq.ma".
+include "basic_2/rt_transition/lpx_rdeq.ma".
include "basic_2/rt_computation/csx_drops.ma".
(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
--- /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/static/fdeq.ma".
+include "basic_2/rt_computation/csx_rdeq.ma".
+
+(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
+
+(* Properties with degree-based equivalence for closures ********************)
+
+lemma csx_fdeq_conf: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
+#h #o #G1 #L1 #T1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
+/3 width=3 by csx_rdeq_conf, csx_tdeq_trans/
+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/static/ffdeq.ma".
-include "basic_2/rt_computation/csx_lfdeq.ma".
-
-(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
-
-(* Properties with degree-based equivalence for closures ********************)
-
-lemma csx_ffdeq_conf: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
-#h #o #G1 #L1 #T1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
-/3 width=3 by csx_lfdeq_conf, csx_tdeq_trans/
-qed-.
include "basic_2/rt_transition/fpbq.ma".
include "basic_2/rt_computation/csx_fqus.ma".
-include "basic_2/rt_computation/csx_ffdeq.ma".
+include "basic_2/rt_computation/csx_fdeq.ma".
include "basic_2/rt_computation/csx_lpx.ma".
(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
lemma csx_fpbq_conf: ∀h,o,G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T1⦄ →
∀G2,L2,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ → ⦃G2, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T2⦄.
#h #o #G1 #L1 #T1 #HT1 #G2 #L2 #T2 *
-/2 width=6 by csx_cpx_trans, csx_fquq_conf, csx_lpx_conf, csx_ffdeq_conf/
+/2 width=6 by csx_cpx_trans, csx_fquq_conf, csx_lpx_conf, csx_fdeq_conf/
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/rt_transition/cpx_lfdeq.ma".
-include "basic_2/rt_computation/csx_csx.ma".
-
-(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
-
-(* Properties with degree-based equivalence for local environments **********)
-
-(* Basic_2A1: uses: csx_lleq_conf *)
-lemma csx_lfdeq_conf: ∀h,o,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ →
- ∀L2. L1 ≛[h, o, T] L2 → ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-#h #o #G #L1 #T #H
-@(csx_ind … H) -T #T1 #_ #IH #L2 #HL12
-@csx_intro #T2 #HT12 #HnT12
-elim (lfdeq_cpx_trans … HL12 … HT12) -HT12
-/5 width=4 by cpx_lfdeq_conf_sn, csx_tdeq_trans, tdeq_trans/
-qed-.
-
-(* Basic_2A1: uses: csx_lleq_conf *)
-lemma csx_lfdeq_trans: ∀h,o,L1,L2,T. L1 ≛[h, o, T] L2 →
- ∀G. ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
-/3 width=3 by csx_lfdeq_conf, lfdeq_sym/ 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/rt_transition/cpx_rdeq.ma".
+include "basic_2/rt_computation/csx_csx.ma".
+
+(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
+
+(* Properties with degree-based equivalence for local environments **********)
+
+(* Basic_2A1: uses: csx_lleq_conf *)
+lemma csx_rdeq_conf: ∀h,o,G,L1,T. ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ →
+ ∀L2. L1 ≛[h, o, T] L2 → ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+#h #o #G #L1 #T #H
+@(csx_ind … H) -T #T1 #_ #IH #L2 #HL12
+@csx_intro #T2 #HT12 #HnT12
+elim (rdeq_cpx_trans … HL12 … HT12) -HT12
+/5 width=4 by cpx_rdeq_conf_sn, csx_tdeq_trans, tdeq_trans/
+qed-.
+
+(* Basic_2A1: uses: csx_lleq_conf *)
+lemma csx_rdeq_trans: ∀h,o,L1,L2,T. L1 ≛[h, o, T] L2 →
+ ∀G. ⦃G, L2⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄ → ⦃G, L1⦄ ⊢ ⬈*[h, o] 𝐒⦃T⦄.
+/3 width=3 by csx_rdeq_conf, rdeq_sym/ qed-.
qed-.
(* Basic_2A1: uses: fpbg_fleq_trans *)
-lemma fpbg_ffdeq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbg_fpbq_trans, fpbq_ffdeq/ qed-.
+lemma fpbg_fdeq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ >[h, o] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbg_fpbq_trans, fpbq_fdeq/ qed-.
(* *)
(**************************************************************************)
-include "basic_2/static/ffdeq_fqup.ma".
-include "basic_2/static/ffdeq_ffdeq.ma".
+include "basic_2/static/fdeq_fqup.ma".
+include "basic_2/static/fdeq_fdeq.ma".
include "basic_2/rt_transition/fpbq_fpb.ma".
include "basic_2/rt_computation/fpbg.ma".
(* Advanced properties with degree-based equivalence on closures ************)
(* Basic_2A1: uses: fleq_fpbg_trans *)
-lemma ffdeq_fpbg_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >[h, o] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
+lemma fdeq_fpbg_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ >[h, o] ⦃G2, L2, T2⦄ →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
#h #o #G #G2 #L #L2 #T #T2 * #G0 #L0 #T0 #H0 #H02 #G1 #L1 #T1 #H1
-elim (ffdeq_fpb_trans … H1 … H0) -G -L -T
-/4 width=9 by fpbs_strap2, fpbq_ffdeq, ex2_3_intro/
+elim (fdeq_fpb_trans … H1 … H0) -G -L -T
+/4 width=9 by fpbs_strap2, fpbq_fdeq, ex2_3_intro/
qed-.
(* Properties with parallel proper rst-reduction on closures ****************)
⦃G1, L1, T1⦄ >[h, o] ⦃G2, L2, T2⦄.
#h #o #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2
elim (fpbq_inv_fpb … H1) -H1
-/2 width=5 by ffdeq_fpbg_trans, fpb_fpbg_trans/
+/2 width=5 by fdeq_fpbg_trans, fpb_fpbg_trans/
qed-.
(* Properties with parallel rst-compuutation on closures ********************)
[ /2 width=1 by or_introl/
| #G #G2 #L #L2 #T #T2 #_ #H2 * #H1
elim (fpbq_inv_fpb … H2) -H2 #H2
- [ /3 width=5 by ffdeq_trans, or_introl/
- | elim (ffdeq_fpb_trans … H1 … H2) -G -L -T
- /4 width=5 by ex2_3_intro, or_intror, ffdeq_fpbs/
- | /3 width=5 by fpbg_ffdeq_trans, or_intror/
+ [ /3 width=5 by fdeq_trans, or_introl/
+ | elim (fdeq_fpb_trans … H1 … H2) -G -L -T
+ /4 width=5 by ex2_3_intro, or_intror, fdeq_fpbs/
+ | /3 width=5 by fpbg_fdeq_trans, or_intror/
| /4 width=5 by fpbg_fpbq_trans, fpb_fpbq, or_intror/
]
]
∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, o] ⦃G2, L2, U2⦄ →
∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, o] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≥[h, o] ⦃G2, L2, U2⦄.
#h #o #F1 #F2 #K1 #K2 #T1 #T2 #H elim (fpbs_inv_fpbg … H) -H
-[ #H12 #G2 #L2 #U2 #H2 elim (ffdeq_fpb_trans … H12 … H2) -F2 -K2 -T2
- /3 width=5 by ffdeq_fpbs, ex2_3_intro/
+[ #H12 #G2 #L2 #U2 #H2 elim (fdeq_fpb_trans … H12 … H2) -F2 -K2 -T2
+ /3 width=5 by fdeq_fpbs, ex2_3_intro/
| * #H1 #H2 #H3 #H4 #H5 #H6 #H7 #H8 #H9
@(ex2_3_intro … H4) -H4 /3 width=5 by fpbs_strap1, fpb_fpbq/
]
(* Properties with unbound rt-computation on full local environments ********)
(* Basic_2A1: uses: lpxs_fpbg *)
-lemma lpxs_lfdneq_fpbg: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
- (L1 ≛[h, o, T] L2 → ⊥) → ⦃G, L1, T⦄ >[h, o] ⦃G, L2, T⦄.
+lemma lpxs_rdneq_fpbg: ∀h,o,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈*[h] L2 →
+ (L1 ≛[h, o, T] L2 → ⊥) → ⦃G, L1, T⦄ >[h, o] ⦃G, L2, T⦄.
#h #o #G #L1 #L2 #T #H #H0
-elim (lpxs_lfdneq_inv_step_sn … H … H0) -H -H0
-/4 width=7 by fpb_lpx, lpxs_ffdeq_fpbs, ffdeq_intro_sn, ex2_3_intro/
+elim (lpxs_rdneq_inv_step_sn … H … H0) -H -H0
+/4 width=7 by fpb_lpx, lpxs_fdeq_fpbs, fdeq_intro_sn, ex2_3_intro/
qed.
/2 width=5 by tri_TC_strap/ qed-.
(* Basic_2A1: uses: lleq_fpbs fleq_fpbs *)
-lemma ffdeq_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=1 by fpbq_fpbs, fpbq_ffdeq/ qed.
+lemma fdeq_fpbs: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=1 by fpbq_fpbs, fpbq_fdeq/ qed.
(* Basic_2A1: uses: fpbs_lleq_trans *)
-lemma fpbs_ffdeq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=9 by fpbs_strap1, fpbq_ffdeq/ qed-.
+lemma fpbs_fdeq_trans: ∀h,o,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=9 by fpbs_strap1, fpbq_fdeq/ qed-.
(* Basic_2A1: uses: lleq_fpbs_trans *)
-lemma ffdeq_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_strap2, fpbq_ffdeq/ qed-.
+lemma fdeq_fpbs_trans: ∀h,o,G,G2,L,L2,T,T2. ⦃G, L, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_strap2, fpbq_fdeq/ qed-.
-lemma tdeq_lfdeq_lpx_fpbs: ∀h,o,T1,T2. T1 ≛[h, o] T2 → ∀L1,L0. L1 ≛[h, o, T2] L0 →
- ∀G,L2. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃G, L1, T1⦄ ≥[h, o] ⦃G, L2, T2⦄.
-/4 width=5 by ffdeq_fpbs, fpbs_strap1, fpbq_lpx, ffdeq_intro_dx/ qed.
+lemma tdeq_rdeq_lpx_fpbs: ∀h,o,T1,T2. T1 ≛[h, o] T2 → ∀L1,L0. L1 ≛[h, o, T2] L0 →
+ ∀G,L2. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃G, L1, T1⦄ ≥[h, o] ⦃G, L2, T2⦄.
+/4 width=5 by fdeq_fpbs, fpbs_strap1, fpbq_lpx, fdeq_intro_dx/ qed.
(* Basic_2A1: removed theorems 3:
fpb_fpbsa_trans fpbs_fpbsa fpbsa_inv_fpbs
(* *)
(**************************************************************************)
-include "basic_2/rt_transition/cpx_ffdeq.ma".
+include "basic_2/rt_transition/cpx_fdeq.ma".
include "basic_2/rt_computation/lpxs_cpxs.ma".
include "basic_2/rt_computation/fpbs_lpxs.ma".
∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈[h] U1 & T1 ≛[h, o] U1 → ⊥ & ⦃G1, L1, U1⦄ ≥[h, o] ⦃G2, L2, U2⦄.
#h #o #G1 #G2 #L1 #L2 #T1 #T2 #H #U2 #HTU2 #HnTU2
elim (fpbs_inv_star … H) -H #G0 #L0 #L3 #T0 #T3 #HT10 #H10 #HL03 #H32
-elim (ffdeq_cpx_trans … H32 … HTU2) -HTU2 #T4 #HT34 #H42
-lapply (ffdeq_tdneq_repl_dx … H32 … H42 … HnTU2) -T2 #HnT34
+elim (fdeq_cpx_trans … H32 … HTU2) -HTU2 #T4 #HT34 #H42
+lapply (fdeq_tdneq_repl_dx … H32 … H42 … HnTU2) -T2 #HnT34
lapply (lpxs_cpx_trans … HT34 … HL03) -HT34 #HT34
elim (fqus_cpxs_trans_tdneq … H10 … HT34 HnT34) -T3 #T2 #HT02 #HnT02 #H24
elim (tdeq_dec h o T1 T0) [ #H10 | -HnT02 #HnT10 ]
lemma cpxs_tdeq_fpbs: ∀h,o,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ⬈*[h] T →
∀T2. T ≛[h, o] T2 → ⦃G, L, T1⦄ ≥[h, o] ⦃G, L, T2⦄.
-/4 width=3 by cpxs_fpbs_trans, ffdeq_fpbs, tdeq_ffdeq/ qed.
+/4 width=3 by cpxs_fpbs_trans, fdeq_fpbs, tdeq_fdeq/ qed.
(* Properties with star-iterated structural successor for closures **********)
(**************************************************************************)
include "basic_2/s_computation/fqus_fqup.ma".
-include "basic_2/static/ffdeq_fqup.ma".
+include "basic_2/static/fdeq_fqup.ma".
include "basic_2/rt_computation/fpbs_fqus.ma".
(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
lemma tdeq_fpbs_trans: ∀h,o,T1,T. T1 ≛[h, o] T →
∀G1,G2,L1,L2,T2. ⦃G1, L1, T⦄ ≥[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by ffdeq_fpbs_trans, tdeq_ffdeq/ qed-.
+/3 width=5 by fdeq_fpbs_trans, tdeq_fdeq/ qed-.
lemma fpbs_tdeq_trans: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T⦄ →
∀T2. T ≛[h, o] T2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by fpbs_ffdeq_trans, tdeq_ffdeq/ qed-.
+/3 width=5 by fpbs_fdeq_trans, tdeq_fdeq/ qed-.
(* Properties with plus-iterated structural successor for closures **********)
(* *)
(**************************************************************************)
-include "basic_2/static/ffdeq_fqus.ma".
-include "basic_2/static/ffdeq_ffdeq.ma".
+include "basic_2/static/fdeq_fqus.ma".
+include "basic_2/static/fdeq_fdeq.ma".
include "basic_2/rt_computation/cpxs_fqus.ma".
-include "basic_2/rt_computation/cpxs_ffdeq.ma".
-include "basic_2/rt_computation/lpxs_ffdeq.ma".
+include "basic_2/rt_computation/cpxs_fdeq.ma".
+include "basic_2/rt_computation/lpxs_fdeq.ma".
include "basic_2/rt_computation/fpbs_cpxs.ma".
(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
qed-.
(* Basic_2A1: uses: lpxs_lleq_fpbs *)
-lemma lpxs_ffdeq_fpbs: ∀h,o,G1,L1,L,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] L →
- ∀G2,L2,T2. ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
-/3 width=3 by lpxs_fpbs_trans, ffdeq_fpbs/ qed.
+lemma lpxs_fdeq_fpbs: ∀h,o,G1,L1,L,T1. ⦃G1, L1⦄ ⊢ ⬈*[h] L →
+ ∀G2,L2,T2. ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
+/3 width=3 by lpxs_fpbs_trans, fdeq_fpbs/ qed.
lemma fpbs_lpx_trans: ∀h,o,G1,G2,L1,L,T1,T2. ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L, T2⦄ →
∀L2. ⦃G2, L⦄ ⊢ ⬈[h] L2 → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄.
∀G,L,T0. ⦃G1, L1, T⦄ ⊐* ⦃G, L, T0⦄ →
∀L0. ⦃G, L⦄ ⊢ ⬈*[h] L0 →
∀G2,L2,T2. ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, o] ⦃G2, L2, T2⦄ .
-/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_ffdeq/ qed.
+/3 width=5 by cpxs_fqus_lpxs_fpbs, fpbs_strap1, fpbq_fdeq/ qed.
(* Advanced inversion lemmas *************************************************)
elim (lpx_fqus_trans … H34 … HL10) -L0
/3 width=9 by lpxs_step_sn, cpxs_trans, ex4_5_intro/
| #G0 #L0 #T0 #H10 #_ * #G3 #L3 #L4 #T3 #T4 #HT03 #H34 #HL34 #H42
- elim (ffdeq_cpxs_trans … H10 … HT03) -T0 #T0 #HT10 #H03
- elim (ffdeq_fqus_trans … H03 … H34) -G0 -L0 -T3 #G0 #L0 #T3 #H03 #H34
- elim (ffdeq_lpxs_trans … H34 … HL34) -L3 #L3 #HL03 #H34
- /3 width=13 by ffdeq_trans, ex4_5_intro/
+ elim (fdeq_cpxs_trans … H10 … HT03) -T0 #T0 #HT10 #H03
+ elim (fdeq_fqus_trans … H03 … H34) -G0 -L0 -T3 #G0 #L0 #T3 #H03 #H34
+ elim (fdeq_lpxs_trans … H34 … HL34) -L3 #L3 #HL03 #H34
+ /3 width=13 by fdeq_trans, ex4_5_intro/
]
]
qed-.
elim (lpx_fqup_trans … H03 … HL02) -L2 #L4 #T4 #HT04 #H43 #HL43
elim (tdeq_dec h o T0 T4) [ -IHc -HT04 #HT04 | -IHu #HnT04 ]
[ elim (tdeq_fqup_trans … H43 … HT04) -T4 #L2 #T4 #H04 #HT43 #HL24
- /4 width=7 by fsb_fpbs_trans, tdeq_lfdeq_lpx_fpbs, fpbs_fqup_trans/
+ /4 width=7 by fsb_fpbs_trans, tdeq_rdeq_lpx_fpbs, fpbs_fqup_trans/
| elim (cpxs_tdneq_fwd_step_sn … HT04 HnT04) -HT04 -HnT04 #T2 #T5 #HT02 #HnT02 #HT25 #HT54
elim (fpbs_cpx_tdneq_trans … H10 … HT02 HnT02) -T0 #T0 #HT10 #HnT10 #H02
/3 width=14 by fpbs_cpxs_tdeq_fqup_lpx_trans/
--- /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/rt_transition/fpb_fdeq.ma".
+include "basic_2/rt_computation/fsb.ma".
+
+(* STRONGLY NORMALIZING CLOSURES FOR PARALLEL RST-TRANSITION ****************)
+
+(* Properties with degree-based equivalence for closures ********************)
+
+lemma fsb_fdeq_trans: ∀h,o,G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ≥[h, o] 𝐒⦃G2, L2, T2⦄.
+#h #o #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
+#G1 #L1 #T1 #_ #IH #G2 #L2 #T2 #H12
+@fsb_intro #G #L #T #H2
+elim (fdeq_fpb_trans … H12 … H2) -G2 -L2 -T2
+/2 width=5 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 "basic_2/rt_transition/fpb_ffdeq.ma".
-include "basic_2/rt_computation/fsb.ma".
-
-(* STRONGLY NORMALIZING CLOSURES FOR PARALLEL RST-TRANSITION ****************)
-
-(* Properties with degree-based equivalence for closures ********************)
-
-lemma fsb_ffdeq_trans: ∀h,o,G1,L1,T1. ≥[h, o] 𝐒⦃G1, L1, T1⦄ →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ≥[h, o] 𝐒⦃G2, L2, T2⦄.
-#h #o #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
-#G1 #L1 #T1 #_ #IH #G2 #L2 #T2 #H12
-@fsb_intro #G #L #T #H2
-elim (ffdeq_fpb_trans … H12 … H2) -G2 -L2 -T2
-/2 width=5 by/
-qed-.
(**************************************************************************)
include "basic_2/rt_computation/fpbg_fpbs.ma".
-include "basic_2/rt_computation/fsb_ffdeq.ma".
+include "basic_2/rt_computation/fsb_fdeq.ma".
(* STRONGLY NORMALIZING CLOSURES FOR PARALLEL RST-TRANSITION ****************)
#h #o #G1 #L1 #T1 #H @(fsb_ind_alt … H) -G1 -L1 -T1
#G1 #L1 #T1 #H1 #IH #G2 #L2 #T2 #H12
elim (fpbs_inv_fpbg … H12) -H12
-[ -IH /2 width=5 by fsb_ffdeq_trans/
+[ -IH /2 width=5 by fsb_fdeq_trans/
| -H1 * /2 width=5 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 "basic_2/static/fdeq.ma".
+include "basic_2/rt_computation/lpxs_rdeq.ma".
+
+(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
+
+(* Properties with degree-based equivalence on closures *********************)
+
+lemma fdeq_lpxs_trans (h) (o): ∀G1,G2,L1,L0,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L0, T2⦄ →
+ ∀L2. ⦃G2, L0⦄ ⊢⬈*[h] L2 →
+ ∃∃L. ⦃G1, L1⦄ ⊢⬈*[h] L & ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L0 #T1 #T2 #H1 #L2 #HL02
+elim (fdeq_inv_gen_dx … H1) -H1 #HG #HL10 #HT12 destruct
+elim (rdeq_lpxs_trans … HL02 … HL10) -L0 #L0 #HL10 #HL02
+/3 width=3 by fdeq_intro_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/static/ffdeq.ma".
-include "basic_2/rt_computation/lpxs_lfdeq.ma".
-
-(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
-
-(* Properties with degree-based equivalence on closures *********************)
-
-lemma ffdeq_lpxs_trans (h) (o): ∀G1,G2,L1,L0,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L0, T2⦄ →
- ∀L2. ⦃G2, L0⦄ ⊢⬈*[h] L2 →
- ∃∃L. ⦃G1, L1⦄ ⊢⬈*[h] L & ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L0 #T1 #T2 #H1 #L2 #HL02
-elim (ffdeq_inv_gen_dx … H1) -H1 #HG #HL10 #HT12 destruct
-elim (lfdeq_lpxs_trans … HL02 … HL10) -L0 #L0 #HL10 #HL02
-/3 width=3 by ffdeq_intro_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/rt_transition/lpx_lfdeq.ma".
-include "basic_2/rt_computation/lpxs_lpx.ma".
-
-(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
-
-(* Properties with degree-based equivalence on referred entries *************)
-
-(* Basic_2A1: uses: lleq_lpxs_trans *)
-lemma lfdeq_lpxs_trans (h) (o) (G) (T:term): ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈*[h] K2 →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ⬈*[h] K1 & K1 ≛[h, o, T] K2.
-#h #o #G #T #L2 #K2 #H @(lpxs_ind_sn … H) -L2 /2 width=3 by ex2_intro/
-#L #L2 #HL2 #_ #IH #L1 #HT
-elim (lfdeq_lpx_trans … HL2 … HT) -L #L #HL1 #HT
-elim (IH … HT) -L2 #K #HLK #HT
-/3 width=3 by lpxs_step_sn, ex2_intro/
-qed-.
-
-(* Basic_2A1: uses: lpxs_nlleq_inv_step_sn *)
-lemma lpxs_lfdneq_inv_step_sn (h) (o) (G) (T:term):
- ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) →
- ∃∃L,L0. ⦃G, L1⦄ ⊢ ⬈[h] L & L1 ≛[h, o, T] L → ⊥ &
- ⦃G, L⦄ ⊢ ⬈*[h] L0 & L0 ≛[h, o, T] L2.
-#h #o #G #T #L1 #L2 #H @(lpxs_ind_sn … H) -L1
-[ #H elim H -H //
-| #L1 #L #H1 #H2 #IH2 #H12 elim (lfdeq_dec h o L1 L T) #H
- [ -H1 -H2 elim IH2 -IH2 /3 width=3 by lfdeq_trans/ -H12
- #L0 #L3 #H1 #H2 #H3 #H4 lapply (lfdeq_lfdneq_trans … H … H2) -H2
- #H2 elim (lfdeq_lpx_trans … H1 … H) -L
- #L #H1 #H lapply (lfdneq_lfdeq_div … H … H2) -H2
- #H2 elim (lfdeq_lpxs_trans … H3 … H) -L0
- /3 width=8 by lfdeq_trans, ex4_2_intro/
- | -H12 -IH2 /3 width=6 by ex4_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/rt_transition/lpx_rdeq.ma".
+include "basic_2/rt_computation/lpxs_lpx.ma".
+
+(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
+
+(* Properties with degree-based equivalence on referred entries *************)
+
+(* Basic_2A1: uses: lleq_lpxs_trans *)
+lemma rdeq_lpxs_trans (h) (o) (G) (T:term): ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈*[h] K2 →
+ ∀L1. L1 ≛[h, o, T] L2 →
+ ∃∃K1. ⦃G, L1⦄ ⊢ ⬈*[h] K1 & K1 ≛[h, o, T] K2.
+#h #o #G #T #L2 #K2 #H @(lpxs_ind_sn … H) -L2 /2 width=3 by ex2_intro/
+#L #L2 #HL2 #_ #IH #L1 #HT
+elim (rdeq_lpx_trans … HL2 … HT) -L #L #HL1 #HT
+elim (IH … HT) -L2 #K #HLK #HT
+/3 width=3 by lpxs_step_sn, ex2_intro/
+qed-.
+
+(* Basic_2A1: uses: lpxs_nlleq_inv_step_sn *)
+lemma lpxs_rdneq_inv_step_sn (h) (o) (G) (T:term):
+ ∀L1,L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) →
+ ∃∃L,L0. ⦃G, L1⦄ ⊢ ⬈[h] L & L1 ≛[h, o, T] L → ⊥ &
+ ⦃G, L⦄ ⊢ ⬈*[h] L0 & L0 ≛[h, o, T] L2.
+#h #o #G #T #L1 #L2 #H @(lpxs_ind_sn … H) -L1
+[ #H elim H -H //
+| #L1 #L #H1 #H2 #IH2 #H12 elim (rdeq_dec h o L1 L T) #H
+ [ -H1 -H2 elim IH2 -IH2 /3 width=3 by rdeq_trans/ -H12
+ #L0 #L3 #H1 #H2 #H3 #H4 lapply (rdeq_rdneq_trans … H … H2) -H2
+ #H2 elim (rdeq_lpx_trans … H1 … H) -L
+ #L #H1 #H lapply (rdneq_rdeq_div … H … H2) -H2
+ #H2 elim (rdeq_lpxs_trans … H3 … H) -L0
+ /3 width=8 by rdeq_trans, ex4_2_intro/
+ | -H12 -IH2 /3 width=6 by ex4_2_intro/
+ ]
+]
+qed-.
(* CLEAR OF STRONGLY NORMALIZING ENTRIES FOR UNBOUND RT-TRANSITION **********)
-(* Note: this should be an instance of a more general lexs *)
+(* Note: this should be an instance of a more general sex *)
(* Basic_2A1: uses: lcosx *)
inductive lsubsx (h) (o) (G): rtmap → relation lenv ≝
| lsubsx_atom: ∀f. lsubsx h o G f (⋆) (⋆)
(**************************************************************************)
include "basic_2/notation/relations/predtysnstrong_5.ma".
-include "basic_2/static/lfdeq.ma".
+include "basic_2/static/rdeq.ma".
include "basic_2/rt_transition/lpx.ma".
(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
definition rdsx (h) (o) (G) (T): predicate lenv ≝
- SN … (lpx h G) (lfdeq h o T).
+ SN … (lpx h G) (rdeq h o T).
interpretation
"strong normalization for unbound context-sensitive parallel rt-transition on referred entries (local environment)"
#h #o #G #I #L #V #T #H
@(rdsx_ind … H) -L #L1 #_ #IHL1
@rdsx_intro #L2 #HL12 #HnL12
-/4 width=3 by lfdeq_fwd_pair_sn/
+/4 width=3 by rdeq_fwd_pair_sn/
qed-.
(* Basic_2A1: uses: lsx_fwd_flat_dx *)
#h #o #G #I #L #V #T #H
@(rdsx_ind … H) -L #L1 #_ #IHL1
@rdsx_intro #L2 #HL12 #HnL12
-/4 width=3 by lfdeq_fwd_flat_dx/
+/4 width=3 by rdeq_fwd_flat_dx/
qed-.
fact rdsx_fwd_pair_aux (h) (o) (G): ∀L. G ⊢ ⬈*[h, o, #0] 𝐒⦃L⦄ →
∀I,K,V. L = K.ⓑ{I}V → G ⊢ ⬈*[h, o, V] 𝐒⦃K⦄.
#h #o #G #L #H
@(rdsx_ind … H) -L #L1 #_ #IH #I #K1 #V #H destruct
-/5 width=5 by lpx_pair, rdsx_intro, lfdeq_fwd_zero_pair/
+/5 width=5 by lpx_pair, rdsx_intro, rdeq_fwd_zero_pair/
qed-.
lemma rdsx_fwd_pair (h) (o) (G):
@rdsx_intro #Y #HY #HnY
elim (lpx_inv_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
elim (tdeq_dec h o V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnY ]
-[ /5 width=5 by rdsx_lfdeq_trans, lpxs_step_dx, lfdeq_pair/
+[ /5 width=5 by rdsx_rdeq_trans, lpxs_step_dx, rdeq_pair/
| @(IHV0 … HnV02) -IHV0 -HnV02
[ /2 width=3 by lpxs_cpx_trans/
| /3 width=3 by rdsx_lpx_trans, rdsx_cpx_trans/
(* *)
(**************************************************************************)
-include "basic_2/static/lfdeq_drops.ma".
+include "basic_2/static/rdeq_drops.ma".
include "basic_2/rt_transition/lpx_drops.ma".
include "basic_2/rt_computation/rdsx_length.ma".
include "basic_2/rt_computation/rdsx_fqup.ma".
#h #o #G #K #T #H @(rdsx_ind … H) -K
#K1 #_ #IH #b #f #L1 #HLK1 #Hf #U #HTU @rdsx_intro
#L2 #HL12 #HnL12 elim (lpx_drops_conf … HLK1 … HL12)
-/5 width=9 by lfdeq_lifts_bi, lpx_fwd_length/
+/5 width=9 by rdeq_lifts_bi, lpx_fwd_length/
qed-.
(* Inversion lemmas on relocation *******************************************)
#h #o #G #L #U #H @(rdsx_ind … H) -L
#L1 #_ #IH #b #f #K1 #HLK1 #Hf #T #HTU @rdsx_intro
#K2 #HK12 #HnK12 elim (drops_lpx_trans … HLK1 … HK12) -HK12
-/4 width=10 by lfdeq_inv_lifts_bi/
+/4 width=10 by rdeq_inv_lifts_bi/
qed-.
(* Advanced properties ******************************************************)
(* *)
(**************************************************************************)
-include "basic_2/static/lfdeq_fqup.ma".
+include "basic_2/static/rdeq_fqup.ma".
include "basic_2/rt_computation/rdsx.ma".
(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
@(rdsx_ind … H) -L #L1 #_ #IH
@rdsx_intro #Y #H #HT
elim (lpx_inv_unit_sn … H) -H #L2 #HL12 #H destruct
-/4 width=4 by lfdeq_fwd_bind_dx_void/
+/4 width=4 by rdeq_fwd_bind_dx_void/
qed-.
(* Advanced inversion lemmas ************************************************)
(* *)
(**************************************************************************)
-include "basic_2/static/lfdeq_length.ma".
+include "basic_2/static/rdeq_length.ma".
include "basic_2/rt_transition/lpx_length.ma".
include "basic_2/rt_computation/rdsx.ma".
(* Basic_2A1: uses: lsx_sort *)
lemma rdsx_sort (h) (o) (G): ∀L,s. G ⊢ ⬈*[h, o, ⋆s] 𝐒⦃L⦄.
#h #o #G #L1 #s @rdsx_intro #L2 #H #Hs
-elim Hs -Hs /3 width=3 by lpx_fwd_length, lfdeq_sort_length/
+elim Hs -Hs /3 width=3 by lpx_fwd_length, rdeq_sort_length/
qed.
(* Basic_2A1: uses: lsx_gref *)
lemma rdsx_gref (h) (o) (G): ∀L,l. G ⊢ ⬈*[h, o, §l] 𝐒⦃L⦄.
#h #o #G #L1 #s @rdsx_intro #L2 #H #Hs
-elim Hs -Hs /3 width=3 by lpx_fwd_length, lfdeq_gref_length/
+elim Hs -Hs /3 width=3 by lpx_fwd_length, rdeq_gref_length/
qed.
lemma rdsx_unit (h) (o) (G): ∀I,L. G ⊢ ⬈*[h, o, #0] 𝐒⦃L.ⓤ{I}⦄.
#h #o #G #I #L1 @rdsx_intro
#Y #HY #HnY elim HnY -HnY
elim (lpx_inv_unit_sn … HY) -HY #L2 #HL12 #H destruct
-/3 width=3 by lpx_fwd_length, lfdeq_unit_length/
+/3 width=3 by lpx_fwd_length, rdeq_unit_length/
qed.
(* *)
(**************************************************************************)
-include "basic_2/rt_computation/lpxs_lfdeq.ma".
+include "basic_2/rt_computation/lpxs_rdeq.ma".
include "basic_2/rt_computation/lpxs_lpxs.ma".
include "basic_2/rt_computation/rdsx_rdsx.ma".
(* Eliminators with unbound rt-computation for full local environments ******)
-lemma rdsx_ind_lpxs_lfdeq (h) (o) (G):
- ∀T. ∀Q:predicate lenv.
- (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → Q L2) →
- Q L1
- ) →
- ∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- ∀L0. ⦃G, L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[h, o, T] L2 → Q L2.
+lemma rdsx_ind_lpxs_rdeq (h) (o) (G):
+ ∀T. ∀Q:predicate lenv.
+ (∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
+ (∀L2. ⦃G, L1⦄ ⊢ ⬈*[h] L2 → (L1 ≛[h, o, T] L2 → ⊥) → Q L2) →
+ Q L1
+ ) →
+ ∀L1. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
+ ∀L0. ⦃G, L1⦄ ⊢ ⬈*[h] L0 → ∀L2. L0 ≛[h, o, T] L2 → Q L2.
#h #o #G #T #Q #IH #L1 #H @(rdsx_ind … H) -L1
#L1 #HL1 #IH1 #L0 #HL10 #L2 #HL02
-@IH -IH /3 width=3 by rdsx_lpxs_trans, rdsx_lfdeq_trans/ -HL1 #K2 #HLK2 #HnLK2
-lapply (lfdeq_lfdneq_trans … HL02 … HnLK2) -HnLK2 #H
-elim (lfdeq_lpxs_trans … HLK2 … HL02) -L2 #K0 #HLK0 #HK02
-lapply (lfdneq_lfdeq_canc_dx … H … HK02) -H #HnLK0
-elim (lfdeq_dec h o L1 L0 T) #H
-[ lapply (lfdeq_lfdneq_trans … H … HnLK0) -H -HnLK0 #Hn10
+@IH -IH /3 width=3 by rdsx_lpxs_trans, rdsx_rdeq_trans/ -HL1 #K2 #HLK2 #HnLK2
+lapply (rdeq_rdneq_trans … HL02 … HnLK2) -HnLK2 #H
+elim (rdeq_lpxs_trans … HLK2 … HL02) -L2 #K0 #HLK0 #HK02
+lapply (rdneq_rdeq_canc_dx … H … HK02) -H #HnLK0
+elim (rdeq_dec h o L1 L0 T) #H
+[ lapply (rdeq_rdneq_trans … H … HnLK0) -H -HnLK0 #Hn10
lapply (lpxs_trans … HL10 … HLK0) -L0 #H10
- elim (lpxs_lfdneq_inv_step_sn … H10 … Hn10) -H10 -Hn10
- /3 width=8 by lfdeq_trans/
-| elim (lpxs_lfdneq_inv_step_sn … HL10 … H) -HL10 -H #L #K #HL1 #HnL1 #HLK #HKL0
- elim (lfdeq_lpxs_trans … HLK0 … HKL0) -L0
- /3 width=8 by lpxs_trans, lfdeq_trans/
+ elim (lpxs_rdneq_inv_step_sn … H10 … Hn10) -H10 -Hn10
+ /3 width=8 by rdeq_trans/
+| elim (lpxs_rdneq_inv_step_sn … HL10 … H) -HL10 -H #L #K #HL1 #HnL1 #HLK #HKL0
+ elim (rdeq_lpxs_trans … HLK0 … HKL0) -L0
+ /3 width=8 by lpxs_trans, rdeq_trans/
]
qed-.
) →
∀L. G ⊢ ⬈*[h, o, T] 𝐒⦃L⦄ → Q L.
#h #o #G #T #Q #IH #L #HL
-@(rdsx_ind_lpxs_lfdeq … IH … HL) -IH -HL // (**) (* full auto fails *)
+@(rdsx_ind_lpxs_rdeq … IH … HL) -IH -HL // (**) (* full auto fails *)
qed-.
(* Advanced properties ******************************************************)
#Y #HY #IHY #L2 #H #HL12 destruct
@rdsx_intro_lpxs #L0 #HL20
lapply (lpxs_trans … HL12 … HL20) #HL10 #H
-elim (lfdneq_inv_bind … H) -H [ -IHY | -HY -IHL1 -HL12 ]
-[ #HnV elim (lfdeq_dec h o L1 L2 V)
+elim (rdneq_inv_bind … H) -H [ -IHY | -HY -IHL1 -HL12 ]
+[ #HnV elim (rdeq_dec h o L1 L2 V)
[ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
- /3 width=4 by rdsx_lpxs_trans, lpxs_bind_refl_dx, lfdeq_canc_sn/ (**) (* full auto too slow *)
+ /3 width=4 by rdsx_lpxs_trans, lpxs_bind_refl_dx, rdeq_canc_sn/ (**) (* full auto too slow *)
| -HnV -HL10 /4 width=4 by rdsx_lpxs_trans, lpxs_bind_refl_dx/
]
| /3 width=4 by lpxs_bind_refl_dx/
#L1 #HL1 #IHL1 #L2 #T #H @(rdsx_ind_lpxs … H) -L2
#L2 #HL2 #IHL2 #HL12 @rdsx_intro_lpxs
#L0 #HL20 lapply (lpxs_trans … HL12 … HL20)
-#HL10 #H elim (lfdneq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
-[ #HnV elim (lfdeq_dec h o L1 L2 V)
+#HL10 #H elim (rdneq_inv_flat … H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
+[ #HnV elim (rdeq_dec h o L1 L2 V)
[ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
- /3 width=5 by rdsx_lpxs_trans, lfdeq_canc_sn/ (**) (* full auto too slow: 47s *)
+ /3 width=5 by rdsx_lpxs_trans, rdeq_canc_sn/ (**) (* full auto too slow: 47s *)
| -HnV -HL10 /3 width=4 by rdsx_lpxs_trans/
]
| /3 width=3 by/
#Y #HY #IHY #L2 #H #HL12 destruct
@rdsx_intro_lpxs #L0 #HL20
lapply (lpxs_trans … HL12 … HL20) #HL10 #H
-elim (lfdneq_inv_bind_void … H) -H [ -IHY | -HY -IHL1 -HL12 ]
-[ #HnV elim (lfdeq_dec h o L1 L2 V)
+elim (rdneq_inv_bind_void … H) -H [ -IHY | -HY -IHL1 -HL12 ]
+[ #HnV elim (rdeq_dec h o L1 L2 V)
[ #HV @(IHL1 … HL10) -IHL1 -HL12 -HL10
- /3 width=6 by rdsx_lpxs_trans, lpxs_bind_refl_dx, lfdeq_canc_sn/ (**) (* full auto too slow *)
+ /3 width=6 by rdsx_lpxs_trans, lpxs_bind_refl_dx, rdeq_canc_sn/ (**) (* full auto too slow *)
| -HnV -HL10 /4 width=4 by rdsx_lpxs_trans, lpxs_bind_refl_dx/
]
| /3 width=4 by lpxs_bind_refl_dx/
(* *)
(**************************************************************************)
-include "basic_2/rt_transition/lpx_lfdeq.ma".
+include "basic_2/rt_transition/lpx_rdeq.ma".
include "basic_2/rt_computation/rdsx.ma".
(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
(* Advanced properties ******************************************************)
(* Basic_2A1: uses: lsx_lleq_trans *)
-lemma rdsx_lfdeq_trans (h) (o) (G):
- ∀L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
- ∀L2. L1 ≛[h, o, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
+lemma rdsx_rdeq_trans (h) (o) (G):
+ ∀L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
+ ∀L2. L1 ≛[h, o, T] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
#h #o #G #L1 #T #H @(rdsx_ind … H) -L1
#L1 #_ #IHL1 #L2 #HL12 @rdsx_intro
-#L #HL2 #HnL2 elim (lfdeq_lpx_trans … HL2 … HL12) -HL2
-/4 width=5 by lfdeq_repl/
+#L #HL2 #HnL2 elim (rdeq_lpx_trans … HL2 … HL12) -HL2
+/4 width=5 by rdeq_repl/
qed-.
(* Basic_2A1: uses: lsx_lpx_trans *)
∀L1,T. G ⊢ ⬈*[h, o, T] 𝐒⦃L1⦄ →
∀L2. ⦃G, L1⦄ ⊢ ⬈[h] L2 → G ⊢ ⬈*[h, o, T] 𝐒⦃L2⦄.
#h #o #G #L1 #T #H @(rdsx_ind … H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
-elim (lfdeq_dec h o L1 L2 T) /3 width=4 by rdsx_lfdeq_trans/
+elim (rdeq_dec h o L1 L2 T) /3 width=4 by rdsx_rdeq_trans/
qed-.
(* *)
(**************************************************************************)
-include "basic_2/rt_transition/lfpx_lfdeq.ma".
+include "basic_2/rt_transition/rpx_rdeq.ma".
include "basic_2/rt_transition/cnx.ma".
(* NORMAL TERMS FOR UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ********)
(* *)
(**************************************************************************)
-include "basic_2/rt_transition/lfpx_fsle.ma".
+include "basic_2/rt_transition/rpx_fsle.ma".
include "basic_2/rt_transition/cpm_cpx.ma".
(* CONTEXT-SENSITIVE PARALLEL REDUCTION FOR TERMS ***************************)
lemma cpm_fsge_comp: ∀n,h,G. R_fsge_compatible (λL. cpm h G L n).
/3 width=6 by cpm_fwd_cpx, cpx_fsge_comp/ qed-.
-lemma lfpr_fsge_comp: ∀h,G. lfxs_fsge_compatible (λL. cpm h G L 0).
-/4 width=5 by cpm_fwd_cpx, lfpx_fsge_comp, lfxs_co/ qed-.
+lemma rpr_fsge_comp: ∀h,G. rex_fsge_compatible (λL. cpm h G L 0).
+/4 width=5 by cpm_fwd_cpx, rpx_fsge_comp, rex_co/ qed-.
(* Properties with generic extension on referred entries ********************)
(* Basic_2A1: was just: cpr_llpx_sn_conf *)
-lemma cpm_lfxs_conf: ∀R,n,h,G. s_r_confluent1 … (λL. cpm h G L n) (lfxs R).
-/3 width=5 by cpm_fwd_cpx, cpx_lfxs_conf/ qed-.
+lemma cpm_rex_conf: ∀R,n,h,G. s_r_confluent1 … (λL. cpm h G L n) (rex R).
+/3 width=5 by cpm_fwd_cpx, cpx_rex_conf/ 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/static/fdeq.ma".
+include "basic_2/rt_transition/cpx_rdeq.ma".
+include "basic_2/rt_transition/rpx_rdeq.ma".
+
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+
+(* Properties with degree-based equivalence for closures ********************)
+
+lemma fdeq_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T⦄ →
+ ∀T2. ⦃G2, L2⦄ ⊢ T ⬈[h] T2 →
+ ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T0 & ⦃G1, L1, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+#h #o #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
+elim (fdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
+elim (rdeq_cpx_trans … HL12 … HT2) #T0 #HT0 #HT02
+lapply (cpx_rdeq_conf_dx … HT2 … HL12) -HL12 #HL12
+elim (tdeq_cpx_trans … HT1 … HT0) -T #T #HT1 #HT0
+/4 width=5 by fdeq_intro_dx, tdeq_trans, 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/static/ffdeq.ma".
-include "basic_2/rt_transition/cpx_lfdeq.ma".
-include "basic_2/rt_transition/lfpx_lfdeq.ma".
-
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
-
-(* Properties with degree-based equivalence for closures ********************)
-
-lemma ffdeq_cpx_trans: ∀h,o,G1,G2,L1,L2,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T⦄ →
- ∀T2. ⦃G2, L2⦄ ⊢ T ⬈[h] T2 →
- ∃∃T0. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T0 & ⦃G1, L1, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #G1 #G2 #L1 #L2 #T1 #T #H #T2 #HT2
-elim (ffdeq_inv_gen_dx … H) -H #H #HL12 #HT1 destruct
-elim (lfdeq_cpx_trans … HL12 … HT2) #T0 #HT0 #HT02
-lapply (cpx_lfdeq_conf_dx … HT2 … HL12) -HL12 #HL12
-elim (tdeq_cpx_trans … HT1 … HT0) -T #T #HT1 #HT0
-/4 width=5 by ffdeq_intro_dx, tdeq_trans, 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/static/lfdeq_lfdeq.ma".
-include "basic_2/rt_transition/lfpx_fsle.ma".
-
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
-
-(* Properties with degree-based equivalence for local environments **********)
-
-(* Basic_2A1: was just: cpx_lleq_conf_sn *)
-lemma cpx_lfdeq_conf_sn: ∀h,o,G. s_r_confluent1 … (cpx h G) (lfdeq h o).
-/3 width=6 by cpx_lfxs_conf/ qed-.
-
-(* Basic_2A1: was just: cpx_lleq_conf_dx *)
-lemma cpx_lfdeq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
- ∀L1. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
-/4 width=4 by cpx_lfdeq_conf_sn, lfdeq_sym/ 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/static/lfeq.ma".
-include "basic_2/rt_transition/lfpx_fsle.ma".
-
-(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
-
-(* Properties with syntactic equivalence for lenvs on referred entries ******)
-
-(* Basic_2A1: was: lleq_cpx_trans *)
-lemma lfeq_cpx_trans: ∀h,G. lfeq_transitive (cpx h G).
-#h #G #L2 #T1 #T2 #H @(cpx_ind … H) -G -L2 -T1 -T2 /2 width=2 by cpx_ess/
-[ #I #G #K2 #V1 #V2 #W2 #_ #IH #HVW2 #L1 #H
- elim (lfeq_inv_zero_pair_dx … H) -H #K1 #HK12 #H destruct
- /3 width=3 by cpx_delta/
-| #I2 #G #K2 #T #U #i #_ #IH #HTU #L1 #H
- elim (lfeq_inv_lref_bind_dx … H) -H #I1 #K1 #HK12 #H destruct
- /3 width=3 by cpx_lref/
-| #p #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H
- elim (lfeq_inv_bind … H) -H /3 width=1 by cpx_bind/
-| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H
- elim (lfeq_inv_flat … H) -H /3 width=1 by cpx_flat/
-| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IH #L1 #H
- elim (lfeq_inv_bind … H) -H /3 width=3 by cpx_zeta/
-| #G #L2 #W1 #T1 #T2 #_ #IH #L1 #H
- elim (lfeq_inv_flat … H) -H /3 width=1 by cpx_eps/
-| #G #L2 #W1 #W2 #T1 #_ #IH #L1 #H
- elim (lfeq_inv_flat … H) -H /3 width=1 by cpx_ee/
-| #p #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #H
- elim (lfeq_inv_flat … H) -H #HV1 #H
- elim (lfeq_inv_bind … H) -H /3 width=1 by cpx_beta/
-| #p #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV1 #IHW12 #IHT12 #HV2 #L1 #H
- elim (lfeq_inv_flat … H) -H #HV1 #H
- elim (lfeq_inv_bind … H) -H /3 width=3 by cpx_theta/
-]
-qed-.
-(*
-(* Basic_2A1: was: cpx_lleq_conf *)
-lemma cpx_lfeq_conf: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
- ∀L1. L2 ≘[T1] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h] T2.
-/3 width=3 by lfeq_cpx_trans, lfeq_sym/ qed-.
-*)
-(* Basic_2A1: was: cpx_lleq_conf_sn *)
-lemma cpx_lfeq_conf_sn: ∀h,G. s_r_confluent1 … (cpx h G) lfeq.
-/2 width=5 by cpx_lfxs_conf/ qed-.
-(*
-(* Basic_2A1: was: cpx_lleq_conf_dx *)
-lemma cpx_lfeq_conf_dx: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
- ∀L1. L1 ≘[T1] L2 → L1 ≘[T2] L2.
-/4 width=6 by cpx_lfeq_conf_sn, lfeq_sym/ 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/static/rdeq_rdeq.ma".
+include "basic_2/rt_transition/rpx_fsle.ma".
+
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+
+(* Properties with degree-based equivalence for local environments **********)
+
+(* Basic_2A1: was just: cpx_lleq_conf_sn *)
+lemma cpx_rdeq_conf_sn: ∀h,o,G. s_r_confluent1 … (cpx h G) (rdeq h o).
+/3 width=6 by cpx_rex_conf/ qed-.
+
+(* Basic_2A1: was just: cpx_lleq_conf_dx *)
+lemma cpx_rdeq_conf_dx: ∀h,o,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L1. L1 ≛[h, o, T1] L2 → L1 ≛[h, o, T2] L2.
+/4 width=4 by cpx_rdeq_conf_sn, rdeq_sym/ 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/static/req.ma".
+include "basic_2/rt_transition/rpx_fsle.ma".
+
+(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
+
+(* Properties with syntactic equivalence for lenvs on referred entries ******)
+
+(* Basic_2A1: was: lleq_cpx_trans *)
+lemma req_cpx_trans: ∀h,G. req_transitive (cpx h G).
+#h #G #L2 #T1 #T2 #H @(cpx_ind … H) -G -L2 -T1 -T2 /2 width=2 by cpx_ess/
+[ #I #G #K2 #V1 #V2 #W2 #_ #IH #HVW2 #L1 #H
+ elim (req_inv_zero_pair_dx … H) -H #K1 #HK12 #H destruct
+ /3 width=3 by cpx_delta/
+| #I2 #G #K2 #T #U #i #_ #IH #HTU #L1 #H
+ elim (req_inv_lref_bind_dx … H) -H #I1 #K1 #HK12 #H destruct
+ /3 width=3 by cpx_lref/
+| #p #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H
+ elim (req_inv_bind … H) -H /3 width=1 by cpx_bind/
+| #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #H
+ elim (req_inv_flat … H) -H /3 width=1 by cpx_flat/
+| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IH #L1 #H
+ elim (req_inv_bind … H) -H /3 width=3 by cpx_zeta/
+| #G #L2 #W1 #T1 #T2 #_ #IH #L1 #H
+ elim (req_inv_flat … H) -H /3 width=1 by cpx_eps/
+| #G #L2 #W1 #W2 #T1 #_ #IH #L1 #H
+ elim (req_inv_flat … H) -H /3 width=1 by cpx_ee/
+| #p #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #H
+ elim (req_inv_flat … H) -H #HV1 #H
+ elim (req_inv_bind … H) -H /3 width=1 by cpx_beta/
+| #p #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV1 #IHW12 #IHT12 #HV2 #L1 #H
+ elim (req_inv_flat … H) -H #HV1 #H
+ elim (req_inv_bind … H) -H /3 width=3 by cpx_theta/
+]
+qed-.
+(*
+(* Basic_2A1: was: cpx_lleq_conf *)
+lemma cpx_req_conf: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L1. L2 ≘[T1] L1 → ⦃G, L1⦄ ⊢ T1 ⬈[h] T2.
+/3 width=3 by req_cpx_trans, req_sym/ qed-.
+*)
+(* Basic_2A1: was: cpx_lleq_conf_sn *)
+lemma cpx_req_conf_sn: ∀h,G. s_r_confluent1 … (cpx h G) req.
+/2 width=5 by cpx_rex_conf/ qed-.
+(*
+(* Basic_2A1: was: cpx_lleq_conf_dx *)
+lemma cpx_req_conf_dx: ∀h,G,L2,T1,T2. ⦃G, L2⦄ ⊢ T1 ⬈[h] T2 →
+ ∀L1. L1 ≘[T1] L2 → L1 ≘[T2] L2.
+/4 width=6 by cpx_req_conf_sn, req_sym/ qed-.
+*)
include "basic_2/notation/relations/predsubtyproper_8.ma".
include "basic_2/s_transition/fqu.ma".
-include "basic_2/static/lfdeq.ma".
+include "basic_2/static/rdeq.ma".
include "basic_2/rt_transition/lpr_lpx.ma".
(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
--- /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/s_transition/fqu_tdeq.ma".
+include "basic_2/static/fdeq.ma".
+include "basic_2/rt_transition/fpb_rdeq.ma".
+
+(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
+
+(* Properties with degree-based equivalence for closures ********************)
+
+(* Basic_2A1: uses: fleq_fpb_trans *)
+lemma fdeq_fpb_trans: ∀h,o,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≛[h, o] ⦃F2, K2, T2⦄ →
+ ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, o] ⦃G2, L2, U2⦄ →
+ ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, o] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≛[h, o] ⦃G2, L2, U2⦄.
+#h #o #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
+#K2 #T2 #HK12 #HT12 #G2 #L2 #U2 #H12
+elim (tdeq_fpb_trans … HT12 … H12) -T2 #K0 #T0 #H #HT0 #HK0
+elim (rdeq_fpb_trans … HK12 … H) -K2 #L0 #U0 #H #HUT0 #HLK0
+@(ex2_3_intro … H) -H (**) (* full auto too slow *)
+/4 width=3 by fdeq_intro_dx, rdeq_trans, tdeq_rdeq_conf, tdeq_trans/
+qed-.
+
+(* Inversion lemmas with degree-based equivalence for closures **************)
+
+(* Basic_2A1: uses: fpb_inv_fleq *)
+lemma fpb_inv_fdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+[ #G2 #L2 #T2 #H12 #H elim (fdeq_inv_gen_sn … H) -H
+ /3 width=11 by rdeq_fwd_length, fqu_inv_tdeq/
+| #T2 #_ #HnT #H elim (fdeq_inv_gen_sn … H) -H /2 width=1 by/
+| #L2 #_ #HnL #H elim (fdeq_inv_gen_sn … H) -H /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 "basic_2/s_transition/fqu_tdeq.ma".
-include "basic_2/static/ffdeq.ma".
-include "basic_2/rt_transition/fpb_lfdeq.ma".
-
-(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
-
-(* Properties with degree-based equivalence for closures ********************)
-
-(* Basic_2A1: uses: fleq_fpb_trans *)
-lemma ffdeq_fpb_trans: ∀h,o,F1,F2,K1,K2,T1,T2. ⦃F1, K1, T1⦄ ≛[h, o] ⦃F2, K2, T2⦄ →
- ∀G2,L2,U2. ⦃F2, K2, T2⦄ ≻[h, o] ⦃G2, L2, U2⦄ →
- ∃∃G1,L1,U1. ⦃F1, K1, T1⦄ ≻[h, o] ⦃G1, L1, U1⦄ & ⦃G1, L1, U1⦄ ≛[h, o] ⦃G2, L2, U2⦄.
-#h #o #F1 #F2 #K1 #K2 #T1 #T2 * -F2 -K2 -T2
-#K2 #T2 #HK12 #HT12 #G2 #L2 #U2 #H12
-elim (tdeq_fpb_trans … HT12 … H12) -T2 #K0 #T0 #H #HT0 #HK0
-elim (lfdeq_fpb_trans … HK12 … H) -K2 #L0 #U0 #H #HUT0 #HLK0
-@(ex2_3_intro … H) -H (**) (* full auto too slow *)
-/4 width=3 by ffdeq_intro_dx, lfdeq_trans, tdeq_lfdeq_conf, tdeq_trans/
-qed-.
-
-(* Inversion lemmas with degree-based equivalence for closures **************)
-
-(* Basic_2A1: uses: fpb_inv_fleq *)
-lemma fpb_inv_ffdeq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-[ #G2 #L2 #T2 #H12 #H elim (ffdeq_inv_gen_sn … H) -H
- /3 width=11 by lfdeq_fwd_length, fqu_inv_tdeq/
-| #T2 #_ #HnT #H elim (ffdeq_inv_gen_sn … H) -H /2 width=1 by/
-| #L2 #_ #HnL #H elim (ffdeq_inv_gen_sn … H) -H /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 "basic_2/static/lfdeq_fqus.ma".
-include "basic_2/rt_transition/cpx_lfdeq.ma".
-include "basic_2/rt_transition/lpx_lfdeq.ma".
-include "basic_2/rt_transition/fpb.ma".
-
-(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
-
-(* Properties with degree-based equivalence for local environments **********)
-
-lemma tdeq_fpb_trans: ∀h,o,U2,U1. U2 ≛[h, o] U1 →
- ∀G1,G2,L1,L2,T1. ⦃G1, L1, U1⦄ ≻[h, o] ⦃G2, L2, T1⦄ →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ≻[h, o] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
-#h #o #U2 #U1 #HU21 #G1 #G2 #L1 #L2 #T1 * -G2 -L2 -T1
-[ #G2 #L2 #T1 #H
- elim (tdeq_fqu_trans … H … HU21) -H
- /3 width=5 by fpb_fqu, ex3_2_intro/
-| #T1 #HUT1 #HnUT1
- elim (tdeq_cpx_trans … HU21 … HUT1) -HUT1
- /6 width=5 by fpb_cpx, tdeq_canc_sn, tdeq_trans, ex3_2_intro/
-| /6 width=5 by fpb_lpx, lfpx_tdeq_div, tdeq_lfdeq_conf, ex3_2_intro/
-]
-qed-.
-
-(* Basic_2A1: was just: lleq_fpb_trans *)
-lemma lfdeq_fpb_trans: ∀h,o,F,K1,K2,T. K1 ≛[h, o, T] K2 →
- ∀G,L2,U. ⦃F, K2, T⦄ ≻[h, o] ⦃G, L2, U⦄ →
- ∃∃L1,U0. ⦃F, K1, T⦄ ≻[h, o] ⦃G, L1, U0⦄ & U0 ≛[h, o] U & L1 ≛[h, o, U] L2.
-#h #o #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
-[ #G #L2 #U #H2 elim (lfdeq_fqu_trans … H2 … HT) -K2
- /3 width=5 by fpb_fqu, ex3_2_intro/
-| #U #HTU #HnTU elim (lfdeq_cpx_trans … HT … HTU) -HTU
- /5 width=10 by fpb_cpx, cpx_lfdeq_conf_sn, tdeq_trans, tdeq_lfdeq_conf, ex3_2_intro/
-| #L2 #HKL2 #HnKL2 elim (lfdeq_lpx_trans … HKL2 … HT) -HKL2
- /6 width=5 by fpb_lpx, (* 2x *) lfdeq_canc_sn, 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 "basic_2/static/rdeq_fqus.ma".
+include "basic_2/rt_transition/cpx_rdeq.ma".
+include "basic_2/rt_transition/lpx_rdeq.ma".
+include "basic_2/rt_transition/fpb.ma".
+
+(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
+
+(* Properties with degree-based equivalence for local environments **********)
+
+lemma tdeq_fpb_trans: ∀h,o,U2,U1. U2 ≛[h, o] U1 →
+ ∀G1,G2,L1,L2,T1. ⦃G1, L1, U1⦄ ≻[h, o] ⦃G2, L2, T1⦄ →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ≻[h, o] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #U2 #U1 #HU21 #G1 #G2 #L1 #L2 #T1 * -G2 -L2 -T1
+[ #G2 #L2 #T1 #H
+ elim (tdeq_fqu_trans … H … HU21) -H
+ /3 width=5 by fpb_fqu, ex3_2_intro/
+| #T1 #HUT1 #HnUT1
+ elim (tdeq_cpx_trans … HU21 … HUT1) -HUT1
+ /6 width=5 by fpb_cpx, tdeq_canc_sn, tdeq_trans, ex3_2_intro/
+| /6 width=5 by fpb_lpx, rpx_tdeq_div, tdeq_rdeq_conf, ex3_2_intro/
+]
+qed-.
+
+(* Basic_2A1: was just: lleq_fpb_trans *)
+lemma rdeq_fpb_trans: ∀h,o,F,K1,K2,T. K1 ≛[h, o, T] K2 →
+ ∀G,L2,U. ⦃F, K2, T⦄ ≻[h, o] ⦃G, L2, U⦄ →
+ ∃∃L1,U0. ⦃F, K1, T⦄ ≻[h, o] ⦃G, L1, U0⦄ & U0 ≛[h, o] U & L1 ≛[h, o, U] L2.
+#h #o #F #K1 #K2 #T #HT #G #L2 #U * -G -L2 -U
+[ #G #L2 #U #H2 elim (rdeq_fqu_trans … H2 … HT) -K2
+ /3 width=5 by fpb_fqu, ex3_2_intro/
+| #U #HTU #HnTU elim (rdeq_cpx_trans … HT … HTU) -HTU
+ /5 width=10 by fpb_cpx, cpx_rdeq_conf_sn, tdeq_trans, tdeq_rdeq_conf, ex3_2_intro/
+| #L2 #HKL2 #HnKL2 elim (rdeq_lpx_trans … HKL2 … HT) -HKL2
+ /6 width=5 by fpb_lpx, (* 2x *) rdeq_canc_sn, ex3_2_intro/
+]
+qed-.
(**************************************************************************)
include "basic_2/notation/relations/predsubty_8.ma".
-include "basic_2/static/ffdeq.ma".
+include "basic_2/static/fdeq.ma".
include "basic_2/s_transition/fquq.ma".
include "basic_2/rt_transition/lpr_lpx.ma".
(* Basic_2A1: includes: fleq_fpbq fpbq_lleq *)
inductive fpbq (h) (o) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpbq_fquq : ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
-| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h o G1 L1 T1 G1 L1 T2
-| fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → fpbq h o G1 L1 T1 G1 L2 T1
-| fpbq_ffdeq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
+| fpbq_fquq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
+| fpbq_cpx : ∀T2. ⦃G1, L1⦄ ⊢ T1 ⬈[h] T2 → fpbq h o G1 L1 T1 G1 L1 T2
+| fpbq_lpx : ∀L2. ⦃G1, L1⦄ ⊢ ⬈[h] L2 → fpbq h o G1 L1 T1 G1 L2 T1
+| fpbq_fdeq: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → fpbq h o G1 L1 T1 G2 L2 T2
.
interpretation
(**************************************************************************)
include "basic_2/static/aaa_fqus.ma".
-include "basic_2/static/aaa_ffdeq.ma".
+include "basic_2/static/aaa_fdeq.ma".
include "basic_2/rt_transition/lpx_aaa.ma".
include "basic_2/rt_transition/fpbq.ma".
lemma fpbq_aaa_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ →
∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=8 by lpx_aaa_conf, cpx_aaa_conf, aaa_ffdeq_conf, aaa_fquq_conf, ex_intro/
+/3 width=8 by lpx_aaa_conf, cpx_aaa_conf, aaa_fdeq_conf, aaa_fquq_conf, ex_intro/
qed-.
(* *)
(**************************************************************************)
-include "basic_2/rt_transition/fpb_ffdeq.ma".
+include "basic_2/rt_transition/fpb_fdeq.ma".
include "basic_2/rt_transition/fpbq.ma".
(* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
(* Basic_2A1: fpb_fpbq_alt *)
lemma fpb_fpbq_ffdneq: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄ →
∧∧ ⦃G1, L1, T1⦄ ≽[h, o] ⦃G2, L2, T2⦄ & (⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⊥).
-/3 width=10 by fpb_fpbq, fpb_inv_ffdeq, conj/ qed-.
+/3 width=10 by fpb_fpbq, fpb_inv_fdeq, conj/ qed-.
(* Inversrion lemmas with proper parallel rst-transition for closures *******)
| ⦃G1, L1, T1⦄ ≻[h, o] ⦃G2, L2, T2⦄.
#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
[ #G2 #L2 #T2 * [2: * #H1 #H2 #H3 destruct ]
- /3 width=1 by fpb_fqu, ffdeq_intro_sn, or_intror, or_introl/
+ /3 width=1 by fpb_fqu, fdeq_intro_sn, or_intror, or_introl/
| #T2 #H elim (tdeq_dec h o T1 T2)
- /4 width=1 by fpb_cpx, ffdeq_intro_sn, or_intror, or_introl/
-| #L2 elim (lfdeq_dec h o L1 L2 T1)
- /4 width=1 by fpb_lpx, ffdeq_intro_sn, or_intror, or_introl/
+ /4 width=1 by fpb_cpx, fdeq_intro_sn, or_intror, or_introl/
+| #L2 elim (rdeq_dec h o L1 L2 T1)
+ /4 width=1 by fpb_lpx, fdeq_intro_sn, or_intror, or_introl/
| /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/predtysn_5.ma".
-include "basic_2/static/lfxs.ma".
-include "basic_2/rt_transition/cpx_ext.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-
-definition lfpx (h) (G): relation3 term lenv lenv ≝
- lfxs (cpx h G).
-
-interpretation
- "unbound parallel rt-transition on referred entries (local environment)"
- 'PRedTySn h T G L1 L2 = (lfpx h G T L1 L2).
-
-(* Basic properties ***********************************************************)
-
-lemma lfpx_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] ⋆.
-/2 width=1 by lfxs_atom/ qed.
-
-lemma lfpx_sort: ∀h,I1,I2,G,L1,L2,s.
- ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2}.
-/2 width=1 by lfxs_sort/ qed.
-
-lemma lfpx_pair: ∀h,I,G,L1,L2,V1,V2.
- ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 → ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2.
-/2 width=1 by lfxs_pair/ qed.
-
-lemma lfpx_lref: ∀h,I1,I2,G,L1,L2,i.
- ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2}.
-/2 width=1 by lfxs_lref/ qed.
-
-lemma lfpx_gref: ∀h,I1,I2,G,L1,L2,l.
- ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2}.
-/2 width=1 by lfxs_gref/ qed.
-
-lemma lfpx_bind_repl_dx: ∀h,I,I1,G,L1,L2,T.
- ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I1} →
- ∀I2. ⦃G, L1⦄ ⊢ I ⬈[h] I2 →
- ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I2}.
-/2 width=2 by lfxs_bind_repl_dx/ qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lfpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G, ⋆⦄ ⊢ ⬈[h, T] Y2 → Y2 = ⋆.
-/2 width=3 by lfxs_inv_atom_sn/ qed-.
-
-lemma lfpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G, Y1⦄ ⊢ ⬈[h, T] ⋆ → Y1 = ⋆.
-/2 width=3 by lfxs_inv_atom_dx/ qed-.
-
-lemma lfpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-/2 width=1 by lfxs_inv_sort/ qed-.
-
-lemma lfpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-/2 width=1 by lfxs_inv_lref/ qed-.
-
-lemma lfpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-/2 width=1 by lfxs_inv_gref/ qed-.
-
-lemma lfpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
-/2 width=2 by lfxs_inv_bind/ qed-.
-
-lemma lfpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
-/2 width=2 by lfxs_inv_flat/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] Y2 →
- ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
-/2 width=2 by lfxs_inv_sort_bind_sn/ qed-.
-
-lemma lfpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2} →
- ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
-/2 width=2 by lfxs_inv_sort_bind_dx/ qed-.
-
-lemma lfpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] Y2 →
- ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
- Y2 = L2.ⓑ{I}V2.
-/2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
-
-lemma lfpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2 →
- ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
- Y1 = L1.ⓑ{I}V1.
-/2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
-
-lemma lfpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] Y2 →
- ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y2 = L2.ⓘ{I2}.
-/2 width=2 by lfxs_inv_lref_bind_sn/ qed-.
-
-lemma lfpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2} →
- ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓘ{I1}.
-/2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
-
-lemma lfpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] Y2 →
- ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y2 = L2.ⓘ{I2}.
-/2 width=2 by lfxs_inv_gref_bind_sn/ qed-.
-
-lemma lfpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2} →
- ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y1 = L1.ⓘ{I1}.
-/2 width=2 by lfxs_inv_gref_bind_dx/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lfpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
-/2 width=3 by lfxs_fwd_pair_sn/ qed-.
-
-lemma lfpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
-/2 width=2 by lfxs_fwd_bind_dx/ qed-.
-
-lemma lfpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
-/2 width=3 by lfxs_fwd_flat_dx/ 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/static/lfxs_fqup.ma".
-include "basic_2/rt_transition/lfpx.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-
-(* Advanced properties ******************************************************)
-
-lemma lfpx_refl: ∀h,G,T. reflexive … (lfpx h G T).
-/2 width=1 by lfxs_refl/ qed.
-
-lemma lfpx_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
- ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L.ⓑ{I}V2.
-/2 width=1 by lfxs_pair_refl/ qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfpx_inv_bind_void: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
- ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ.
-/2 width=3 by lfxs_inv_bind_void/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lfpx_fwd_bind_dx_void: ∀h,p,I,G,L1,L2,V,T.
- ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ.
-/2 width=4 by lfxs_fwd_bind_dx_void/ 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/static/fsle_drops.ma".
-include "basic_2/static/lfxs_fsle.ma".
-include "basic_2/rt_transition/lfpx_length.ma".
-include "basic_2/rt_transition/lfpx_fqup.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-
-(* Forward lemmas with free variables inclusion for restricted closures *****)
-
-(* Note: "⦃L2, T1⦄ ⊆ ⦃L2, T0⦄" does not hold *)
-(* Note: Take L0 = K0.ⓓ(ⓝW.V), L2 = K0.ⓓW, T0 = #0, T1 = ⬆*[1]V *)
-(* Note: This invalidates lfpxs_cpx_conf: "∀h,G. s_r_confluent1 … (cpx h G) (lfpxs h G)" *)
-lemma lfpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
-#h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓕ) … G0 L0 T0) -G0 -L0 -T0
-#G #L #T #IH #G0 #L0 * *
-[ #s #HG #HL #HT #X #HX #Y #HY destruct -IH
- elim (cpx_inv_sort1 … HX) -HX #H destruct
- lapply (lfpx_fwd_length … HY) -HY #H0
- /2 width=1 by fsle_sort_bi/
-| * [| #i ] #HG #HL #HT #X #HX #Y #HY destruct
- [ elim (cpx_inv_zero1 … HX) -HX
- [ #H destruct
- elim (lfpx_inv_zero_length … HY) -HY *
- [ #H1 #H2 destruct -IH //
- | #I #K0 #K2 #V0 #V2 #HK02 #HV02 #H1 #H2 destruct
- lapply (lfpx_fwd_length … HK02) #H0
- /4 width=4 by fsle_pair_bi, fqu_fqup, fqu_lref_O/
- | #I #K0 #K2 #HK02 #H1 #H2 destruct -IH
- /2 width=1 by fsle_unit_bi/
- ]
- | * #I0 #K0 #V0 #V1 #HV01 #HV1X #H destruct
- elim (lfpx_inv_zero_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
- lapply (lfpx_fwd_length … HK02) #H0
- /4 width=4 by fsle_lifts_SO_sn, fqu_fqup, fqu_lref_O/
- ]
- | elim (cpx_inv_lref1 … HX) -HX
- [ #H destruct
- elim (lfpx_inv_lref … HY) -HY *
- [ #H0 #H1 destruct //
- | #I0 #I2 #K0 #K2 #HK02 #H1 #H2 destruct
- lapply (lfpx_fwd_length … HK02) #H0
- /4 width=5 by fsle_lifts_SO, fqu_fqup/
- ]
- | * #I0 #K0 #V1 #HV1 #HV1X #H0 destruct
- elim (lfpx_inv_lref_bind_sn … HY) -HY #I2 #K2 #HK02 #H destruct
- lapply (lfpx_fwd_length … HK02) #H0
- /4 width=5 by fsle_lifts_SO, fqu_fqup/
- ]
- ]
-| #l #HG #HL #HT #X #HX #Y #HY destruct -IH
- >(cpx_inv_gref1 … HX) -X
- lapply (lfpx_fwd_length … HY) -HY #H0
- /2 width=1 by fsle_gref_bi/
-| #p #I #V0 #T0 #HG #HL #HT #X #HX #Y #HY destruct
- elim (lfpx_inv_bind … V0 ? HY) -HY #HV0 #HT0
- elim (cpx_inv_bind1 … HX) -HX *
- [ #V1 #T1 #HV01 #HT01 #H destruct
- lapply (lfpx_fwd_length … HV0) #H0
- /4 width=6 by fsle_bind_eq, fsle_fwd_pair_sn/
- | #T #HT #HXT #H1 #H2 destruct
- lapply (lfpx_fwd_length … HV0) #H0
- /3 width=8 by fsle_inv_lifts_sn/
- ]
-| #I #V0 #X0 #HG #HL #HT #X #HX #Y #HY destruct
- elim (lfxs_inv_flat … HY) -HY #HV0 #HX0
- elim (cpx_inv_flat1 … HX) -HX *
- [ #V1 #T1 #HV01 #HT01 #H destruct
- /3 width=4 by fsle_flat/
- | #HX #H destruct
- /4 width=4 by fsle_flat_dx_dx/
- | #HX #H destruct
- /4 width=4 by fsle_flat_dx_sn/
- | #p #V1 #W0 #W1 #T0 #T1 #HV01 #HW01 #HT01 #H1 #H2 #H3 destruct
- elim (lfpx_inv_bind … W0 ? HX0) -HX0 #HW0 #HT0
- lapply (lfpx_fwd_length … HV0) #H0
- lapply (IH … HV01 … HV0) -HV01 -HV0 // #HV
- lapply (IH … HW01 … HW0) -HW01 -HW0 // #HW
- lapply (IH … HT01 … HT0) -HT01 -HT0 -IH // #HT
- lapply (fsle_fwd_pair_sn … HT) -HT #HT
- @fsle_bind_sn_ge //
- [ /4 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn, fsle_bind_dx_sn/
- | /3 width=1 by fsle_flat_dx_dx, fsle_shift/
- ]
- | #p #V1 #X1 #W0 #W1 #T0 #T1 #HV01 #HVX1 #HW01 #HT01 #H1 #H2 #H3 destruct
- elim (lfpx_inv_bind … W0 ? HX0) -HX0 #HW0 #HT0
- lapply (lfpx_fwd_length … HV0) #H0
- lapply (IH … HV01 … HV0) -HV01 -HV0 // #HV
- lapply (IH … HW01 … HW0) -HW01 -HW0 // #HW
- lapply (IH … HT01 … HT0) -HT01 -HT0 -IH // #HT
- lapply (fsle_fwd_pair_sn … HT) -HT #HT
- @fsle_bind_sn_ge //
- [ /3 width=1 by fsle_flat_dx_dx, fsle_bind_dx_sn/
- | /4 width=3 by fsle_flat_sn, fsle_flat_dx_sn, fsle_flat_dx_dx, fsle_shift, fsle_lifts_sn/
- ]
- ]
-]
-qed-.
-
-lemma lfpx_fsge_comp (h) (G): lfxs_fsge_compatible (cpx h G).
-/2 width=4 by lfpx_cpx_conf_fsge/ qed-.
-
-(**) (* this section concerns cpx *)
-(* Properties with generic extension on referred entries ********************)
-
-(* Basic_2A1: uses: cpx_frees_trans *)
-lemma cpx_fsge_comp (h) (G): R_fsge_compatible (cpx h G).
-/2 width=4 by lfpx_cpx_conf_fsge/ qed-.
-
-(* Note: lemma 1000 *)
-(* Basic_2A1: uses: cpx_llpx_sn_conf *)
-lemma cpx_lfxs_conf (R) (h) (G): s_r_confluent1 … (cpx h G) (lfxs R).
-/3 width=3 by fsge_lfxs_trans, cpx_fsge_comp/ qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lfpx_cpx_conf (h) (G): s_r_confluent1 … (cpx h G) (lfpx h G).
-/2 width=5 by cpx_lfxs_conf/ qed-.
-
-lemma lfpx_cpx_conf_fsge_dx (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄.
-/3 width=5 by lfpx_cpx_conf, lfpx_fsge_comp/ 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/static/lfxs_length.ma".
-include "basic_2/rt_transition/lfpx.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-
-(* Forward lemmas with length for local environments ************************)
-
-lemma lfpx_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → |L1| = |L2|.
-/2 width=3 by lfxs_fwd_length/ qed-.
-
-(* Inversion lemmas with length for local environments **********************)
-
-lemma lfpx_inv_zero_length: ∀h,G,Y1,Y2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] Y2 →
- ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 &
- ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
- |∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
-/2 width=1 by lfxs_inv_zero_length/ 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/static/lfdeq_fqup.ma".
-include "basic_2/static/lfdeq_lfdeq.ma".
-include "basic_2/rt_transition/lfpx_fsle.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-
-(* Properties with degree-based equivalence for local environments **********)
-
-lemma lfpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀o,I,T.
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[h, o, V] L2.
-/3 width=5 by lfpx_fsge_comp, lfxs_pair_sn_split/ qed-.
-
-lemma lfpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀o,I,V.
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[h, o, T] L2.
-/3 width=5 by lfpx_fsge_comp, lfxs_flat_dx_split/ qed-.
-
-lemma lfpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀o,p.
- ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[h, o, T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
-/3 width=5 by lfpx_fsge_comp, lfxs_bind_dx_split/ qed-.
-
-lemma lfpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀o,p,I,V.
- ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[h, o, T] L2.
-/3 width=5 by lfpx_fsge_comp, lfxs_bind_dx_split_void/ qed-.
-
-lemma lfpx_tdeq_conf: ∀h,o,G. s_r_confluent1 … (cdeq h o) (lfpx h G).
-/2 width=5 by tdeq_lfxs_conf/ qed-.
-
-lemma lfpx_tdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
- ∀G,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T2] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T1] L2.
-/2 width=5 by tdeq_lfxs_div/ qed-.
-
-lemma cpx_tdeq_conf_lexs: ∀h,o,G. R_confluent2_lfxs … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
-#h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
-[ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
- elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct
- /4 width=3 by tdeq_sort, deg_next, ex2_intro/
-| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
- >(tdeq_inv_lref1 … H0) -H0
- elim (lfpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
- elim (lfdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
- elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
- elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
-| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
- >(tdeq_inv_lref1 … H0) -H0
- elim (lfpx_inv_lref_bind_sn … H1) -H1 #I1 #K1 #HK01 #H destruct
- elim (lfdeq_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK02 #H destruct
- elim (IH … HK01 … HK02) [|*: //] -K0 #V #HV1 #HV2
- elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
-| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
- elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
- lapply (lfdeq_bind_repl_dx … H2 (BPair I V2) ?) -H2 /2 width=1 by ext2_pair/ #H2
- elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
- elim (IHT … HT02 … H1 … H2) -L0 -T0
- /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
-| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
- elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
- elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
- elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
- /3 width=5 by cpx_flat, tdeq_pair, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #U1 #_ #IH #HUT1 #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
- elim (lfpx_inv_bind … H1) -H1 #HL01 #H1
- elim (lfdeq_inv_bind … H2) -H2 #HL02 #H2
- lapply (lfdeq_bind_repl_dx … H2 (BPair Abbr V2) ?) -H2 /2 width=1 by ext2_pair/ -HV02 #H2
- elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
- elim (tdeq_inv_lifts_sn … HT1 … HUT1) -T1
- /3 width=5 by cpx_zeta, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
- elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
- elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
- elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
- /3 width=3 by cpx_eps, ex2_intro/
-| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
- elim (lfpx_inv_flat … H1) -H1 #HL01 #H1
- elim (lfdeq_inv_flat … H2) -H2 #HL02 #H2
- elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
- /3 width=3 by cpx_ee, ex2_intro/
-| #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
- elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
- elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
- elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
- elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
- elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
- lapply (lfdeq_bind_repl_dx … H2LT0 (BPair Abst W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
- elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
- elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
- elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
- /4 width=7 by cpx_beta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
-| #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
- elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
- elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
- elim (lfpx_inv_flat … H1) -H1 #H1LV0 #H1
- elim (lfpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
- elim (lfdeq_inv_flat … H2) -H2 #H2LV0 #H2
- elim (lfdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
- lapply (lfdeq_bind_repl_dx … H2LT0 (BPair Abbr W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
- elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
- elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
- elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
- elim (tdeq_lifts_sn … HV1 … HVU1) -V1
- /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
-]
-qed-.
-
-lemma cpx_tdeq_conf: ∀h,o,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
- ∀T2. T0 ≛[h, o] T2 →
- ∃∃T. T1 ≛[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
-#h #o #G #L #T0 #T1 #HT01 #T2 #HT02
-elim (cpx_tdeq_conf_lexs … HT01 … HT02 L … L) -HT01 -HT02
-/2 width=3 by lfxs_refl, ex2_intro/
-qed-.
-
-lemma tdeq_cpx_trans: ∀h,o,G,L,T2. ∀T0:term. T2 ≛[h, o] T0 →
- ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
- ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≛[h, o] T1.
-#h #o #G #L #T2 #T0 #HT20 #T1 #HT01
-elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
-qed-.
-
-(* Basic_2A1: uses: cpx_lleq_conf *)
-lemma cpx_lfdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∀L2. L0 ≛[h, o, T0] L2 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≛[h, o] T.
-#h #o #G #L0 #T0 #T1 #HT01 #L2 #HL02
-elim (cpx_tdeq_conf_lexs … HT01 T0 … L0 … HL02) -HT01 -HL02
-/2 width=3 by lfxs_refl, ex2_intro/
-qed-.
-
-(* Basic_2A1: uses: lleq_cpx_trans *)
-lemma lfdeq_cpx_trans: ∀h,o,G,L2,L0,T0. L2 ≛[h, o, T0] L0 →
- ∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
- ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≛[h, o] T1.
-#h #o #G #L2 #L0 #T0 #HL20 #T1 #HT01
-elim (cpx_lfdeq_conf … o … HT01 L2) -HT01
-/3 width=3 by lfdeq_sym, tdeq_sym, ex2_intro/
-qed-.
-
-lemma lfpx_lfdeq_conf: ∀h,o,G,T. confluent2 … (lfpx h G T) (lfdeq h o T).
-/3 width=6 by lfpx_fsge_comp, lfdeq_fsge_comp, cpx_tdeq_conf_lexs, lfxs_conf/ qed-.
-
-lemma lfdeq_lfpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≛[h, o, T] K2.
-#h #o #G #T #L2 #K2 #HLK2 #L1 #HL12
-elim (lfpx_lfdeq_conf … o … HLK2 L1)
-/3 width=3 by lfdeq_sym, 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/static/lfxs_lfxs.ma".
-include "basic_2/rt_transition/lfpx.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-
-(* Main properties **********************************************************)
-
-theorem lfpx_bind: ∀h,G,L1,L2,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 →
- ∀I,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V2 →
- ∀p. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L2.
-/2 width=2 by lfxs_bind/ qed.
-
-theorem lfpx_flat: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 →
- ∀I,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2.
-/2 width=1 by lfxs_flat/ qed.
-
-theorem lfpx_bind_void: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 →
- ∀T. ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ →
- ∀p,I. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2.
-/2 width=1 by lfxs_bind_void/ 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/static/lfxs_lex.ma".
-include "basic_2/rt_transition/lfpx_fsle.ma".
-include "basic_2/rt_transition/lpx.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
-
-(* Properties with syntactic equivalence for referred local environments ****)
-
-lemma fleq_lfpx (h) (G): ∀L1,L2,T. L1 ≡[T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
-/2 width=1 by lfeq_fwd_lfxs/ qed.
-
-(* Properties with unbound parallel rt-transition for full local envs *******)
-
-lemma lpx_lfpx: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
-/2 width=1 by lfxs_lex/ qed.
-
-(* Inversion lemmas with unbound parallel rt-transition for full local envs *)
-
-lemma lfpx_inv_lpx_lfeq: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 →
- ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h] L & L ≡[T] L2.
-/3 width=3 by lfpx_fsge_comp, lfxs_inv_lex_lfeq/ qed-.
(* *)
(**************************************************************************)
-include "basic_2/rt_transition/lfpx_lpx.ma".
+include "basic_2/rt_transition/rpx_lpx.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
(* Basic_2A1: uses: lpx_cpx_frees_trans *)
lemma lpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
∀L2. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
-/3 width=4 by lfpx_cpx_conf_fsge, lpx_lfpx/ qed-.
+/3 width=4 by rpx_cpx_conf_fsge, lpx_rpx/ qed-.
(* Basic_2A1: uses: lpx_frees_trans *)
lemma lpx_fsge_comp (h) (G): ∀L0,L2,T0. ⦃G, L0⦄ ⊢ ⬈[h] L2 → ⦃L2, T0⦄ ⊆ ⦃L0, T0⦄.
+++ /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/static/lfdeq_lfeq.ma".
-include "basic_2/rt_transition/lfpx_lfdeq.ma".
-include "basic_2/rt_transition/lfpx_lpx.ma".
-
-(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
-
-(* Properties with degree-based equivalence for local environments **********)
-
-(* Basic_2A1: uses: lleq_lpx_trans *)
-lemma lfdeq_lpx_trans (h) (o) (G): ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈[h] K2 →
- ∀L1. ∀T:term. L1 ≛[h, o, T] L2 →
- ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h] K1 & K1 ≛[h, o, T] K2.
-#h #o #G #L2 #K2 #HLK2 #L1 #T #HL12
-lapply (lpx_lfpx … T HLK2) -HLK2 #HLK2
-elim (lfdeq_lfpx_trans … HLK2 … HL12) -L2 #K #H #HK2
-elim (lfpx_inv_lpx_lfeq … H) -H #K1 #HLK1 #HK1
-/3 width=5 by lfeq_lfdeq_trans, 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/static/rdeq_req.ma".
+include "basic_2/rt_transition/rpx_rdeq.ma".
+include "basic_2/rt_transition/rpx_lpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
+
+(* Properties with degree-based equivalence for local environments **********)
+
+(* Basic_2A1: uses: lleq_lpx_trans *)
+lemma rdeq_lpx_trans (h) (o) (G): ∀L2,K2. ⦃G, L2⦄ ⊢ ⬈[h] K2 →
+ ∀L1. ∀T:term. L1 ≛[h, o, T] L2 →
+ ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h] K1 & K1 ≛[h, o, T] K2.
+#h #o #G #L2 #K2 #HLK2 #L1 #T #HL12
+lapply (lpx_rpx … T HLK2) -HLK2 #HLK2
+elim (rdeq_rpx_trans … HLK2 … HL12) -L2 #K #H #HK2
+elim (rpx_inv_lpx_req … H) -H #K1 #HLK1 #HK1
+/3 width=5 by req_rdeq_trans, 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/notation/relations/predtysn_5.ma".
+include "basic_2/static/rex.ma".
+include "basic_2/rt_transition/cpx_ext.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
+
+definition rpx (h) (G): relation3 term lenv lenv ≝
+ rex (cpx h G).
+
+interpretation
+ "unbound parallel rt-transition on referred entries (local environment)"
+ 'PRedTySn h T G L1 L2 = (rpx h G T L1 L2).
+
+(* Basic properties ***********************************************************)
+
+lemma rpx_atom: ∀h,I,G. ⦃G, ⋆⦄ ⊢ ⬈[h, ⓪{I}] ⋆.
+/2 width=1 by rex_atom/ qed.
+
+lemma rpx_sort: ∀h,I1,I2,G,L1,L2,s.
+ ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2}.
+/2 width=1 by rex_sort/ qed.
+
+lemma rpx_pair: ∀h,I,G,L1,L2,V1,V2.
+ ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 → ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2.
+/2 width=1 by rex_pair/ qed.
+
+lemma rpx_lref: ∀h,I1,I2,G,L1,L2,i.
+ ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2}.
+/2 width=1 by rex_lref/ qed.
+
+lemma rpx_gref: ∀h,I1,I2,G,L1,L2,l.
+ ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 → ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2}.
+/2 width=1 by rex_gref/ qed.
+
+lemma rpx_bind_repl_dx: ∀h,I,I1,G,L1,L2,T.
+ ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I1} →
+ ∀I2. ⦃G, L1⦄ ⊢ I ⬈[h] I2 →
+ ⦃G, L1.ⓘ{I}⦄ ⊢ ⬈[h, T] L2.ⓘ{I2}.
+/2 width=2 by rex_bind_repl_dx/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma rpx_inv_atom_sn: ∀h,G,Y2,T. ⦃G, ⋆⦄ ⊢ ⬈[h, T] Y2 → Y2 = ⋆.
+/2 width=3 by rex_inv_atom_sn/ qed-.
+
+lemma rpx_inv_atom_dx: ∀h,G,Y1,T. ⦃G, Y1⦄ ⊢ ⬈[h, T] ⋆ → Y1 = ⋆.
+/2 width=3 by rex_inv_atom_dx/ qed-.
+
+lemma rpx_inv_sort: ∀h,G,Y1,Y2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+/2 width=1 by rex_inv_sort/ qed-.
+
+lemma rpx_inv_lref: ∀h,G,Y1,Y2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+/2 width=1 by rex_inv_lref/ qed-.
+
+lemma rpx_inv_gref: ∀h,G,Y1,Y2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+/2 width=1 by rex_inv_gref/ qed-.
+
+lemma rpx_inv_bind: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
+ ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
+/2 width=2 by rex_inv_bind/ qed-.
+
+lemma rpx_inv_flat: ∀h,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 →
+ ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+/2 width=2 by rex_inv_flat/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rpx_inv_sort_bind_sn: ∀h,I1,G,Y2,L1,s. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, ⋆s] Y2 →
+ ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y2 = L2.ⓘ{I2}.
+/2 width=2 by rex_inv_sort_bind_sn/ qed-.
+
+lemma rpx_inv_sort_bind_dx: ∀h,I2,G,Y1,L2,s. ⦃G, Y1⦄ ⊢ ⬈[h, ⋆s] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, ⋆s] L2 & Y1 = L1.ⓘ{I1}.
+/2 width=2 by rex_inv_sort_bind_dx/ qed-.
+
+lemma rpx_inv_zero_pair_sn: ∀h,I,G,Y2,L1,V1. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, #0] Y2 →
+ ∃∃L2,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+ Y2 = L2.ⓑ{I}V2.
+/2 width=1 by rex_inv_zero_pair_sn/ qed-.
+
+lemma rpx_inv_zero_pair_dx: ∀h,I,G,Y1,L2,V2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+ Y1 = L1.ⓑ{I}V1.
+/2 width=1 by rex_inv_zero_pair_dx/ qed-.
+
+lemma rpx_inv_lref_bind_sn: ∀h,I1,G,Y2,L1,i. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, #↑i] Y2 →
+ ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y2 = L2.ⓘ{I2}.
+/2 width=2 by rex_inv_lref_bind_sn/ qed-.
+
+lemma rpx_inv_lref_bind_dx: ∀h,I2,G,Y1,L2,i. ⦃G, Y1⦄ ⊢ ⬈[h, #↑i] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, #i] L2 & Y1 = L1.ⓘ{I1}.
+/2 width=2 by rex_inv_lref_bind_dx/ qed-.
+
+lemma rpx_inv_gref_bind_sn: ∀h,I1,G,Y2,L1,l. ⦃G, L1.ⓘ{I1}⦄ ⊢ ⬈[h, §l] Y2 →
+ ∃∃I2,L2. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y2 = L2.ⓘ{I2}.
+/2 width=2 by rex_inv_gref_bind_sn/ qed-.
+
+lemma rpx_inv_gref_bind_dx: ∀h,I2,G,Y1,L2,l. ⦃G, Y1⦄ ⊢ ⬈[h, §l] L2.ⓘ{I2} →
+ ∃∃I1,L1. ⦃G, L1⦄ ⊢ ⬈[h, §l] L2 & Y1 = L1.ⓘ{I1}.
+/2 width=2 by rex_inv_gref_bind_dx/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma rpx_fwd_pair_sn: ∀h,I,G,L1,L2,V,T.
+ ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, V] L2.
+/2 width=3 by rex_fwd_pair_sn/ qed-.
+
+lemma rpx_fwd_bind_dx: ∀h,p,I,G,L1,L2,V,T.
+ ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓑ{I}V⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V.
+/2 width=2 by rex_fwd_bind_dx/ qed-.
+
+lemma rpx_fwd_flat_dx: ∀h,I,G,L1,L2,V,T.
+ ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+/2 width=3 by rex_fwd_flat_dx/ 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/static/rex_fqup.ma".
+include "basic_2/rt_transition/rpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
+
+(* Advanced properties ******************************************************)
+
+lemma rpx_refl: ∀h,G,T. reflexive … (rpx h G T).
+/2 width=1 by rex_refl/ qed.
+
+lemma rpx_pair_refl: ∀h,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ⬈[h] V2 →
+ ∀I,T. ⦃G, L.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L.ⓑ{I}V2.
+/2 width=1 by rex_pair_refl/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rpx_inv_bind_void: ∀h,p,I,G,L1,L2,V,T. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 →
+ ∧∧ ⦃G, L1⦄ ⊢ ⬈[h, V] L2 & ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ.
+/2 width=3 by rex_inv_bind_void/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma rpx_fwd_bind_dx_void: ∀h,p,I,G,L1,L2,V,T.
+ ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2 → ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ.
+/2 width=4 by rex_fwd_bind_dx_void/ 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/static/fsle_drops.ma".
+include "basic_2/static/rex_fsle.ma".
+include "basic_2/rt_transition/rpx_length.ma".
+include "basic_2/rt_transition/rpx_fqup.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
+
+(* Forward lemmas with free variables inclusion for restricted closures *****)
+
+(* Note: "⦃L2, T1⦄ ⊆ ⦃L2, T0⦄" does not hold *)
+(* Note: Take L0 = K0.ⓓ(ⓝW.V), L2 = K0.ⓓW, T0 = #0, T1 = ⬆*[1]V *)
+(* Note: This invalidates rpxs_cpx_conf: "∀h,G. s_r_confluent1 … (cpx h G) (rpxs h G)" *)
+lemma rpx_cpx_conf_fsge (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T0⦄.
+#h #G0 #L0 #T0 @(fqup_wf_ind_eq (Ⓕ) … G0 L0 T0) -G0 -L0 -T0
+#G #L #T #IH #G0 #L0 * *
+[ #s #HG #HL #HT #X #HX #Y #HY destruct -IH
+ elim (cpx_inv_sort1 … HX) -HX #H destruct
+ lapply (rpx_fwd_length … HY) -HY #H0
+ /2 width=1 by fsle_sort_bi/
+| * [| #i ] #HG #HL #HT #X #HX #Y #HY destruct
+ [ elim (cpx_inv_zero1 … HX) -HX
+ [ #H destruct
+ elim (rpx_inv_zero_length … HY) -HY *
+ [ #H1 #H2 destruct -IH //
+ | #I #K0 #K2 #V0 #V2 #HK02 #HV02 #H1 #H2 destruct
+ lapply (rpx_fwd_length … HK02) #H0
+ /4 width=4 by fsle_pair_bi, fqu_fqup, fqu_lref_O/
+ | #I #K0 #K2 #HK02 #H1 #H2 destruct -IH
+ /2 width=1 by fsle_unit_bi/
+ ]
+ | * #I0 #K0 #V0 #V1 #HV01 #HV1X #H destruct
+ elim (rpx_inv_zero_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
+ lapply (rpx_fwd_length … HK02) #H0
+ /4 width=4 by fsle_lifts_SO_sn, fqu_fqup, fqu_lref_O/
+ ]
+ | elim (cpx_inv_lref1 … HX) -HX
+ [ #H destruct
+ elim (rpx_inv_lref … HY) -HY *
+ [ #H0 #H1 destruct //
+ | #I0 #I2 #K0 #K2 #HK02 #H1 #H2 destruct
+ lapply (rpx_fwd_length … HK02) #H0
+ /4 width=5 by fsle_lifts_SO, fqu_fqup/
+ ]
+ | * #I0 #K0 #V1 #HV1 #HV1X #H0 destruct
+ elim (rpx_inv_lref_bind_sn … HY) -HY #I2 #K2 #HK02 #H destruct
+ lapply (rpx_fwd_length … HK02) #H0
+ /4 width=5 by fsle_lifts_SO, fqu_fqup/
+ ]
+ ]
+| #l #HG #HL #HT #X #HX #Y #HY destruct -IH
+ >(cpx_inv_gref1 … HX) -X
+ lapply (rpx_fwd_length … HY) -HY #H0
+ /2 width=1 by fsle_gref_bi/
+| #p #I #V0 #T0 #HG #HL #HT #X #HX #Y #HY destruct
+ elim (rpx_inv_bind … V0 ? HY) -HY #HV0 #HT0
+ elim (cpx_inv_bind1 … HX) -HX *
+ [ #V1 #T1 #HV01 #HT01 #H destruct
+ lapply (rpx_fwd_length … HV0) #H0
+ /4 width=6 by fsle_bind_eq, fsle_fwd_pair_sn/
+ | #T #HT #HXT #H1 #H2 destruct
+ lapply (rpx_fwd_length … HV0) #H0
+ /3 width=8 by fsle_inv_lifts_sn/
+ ]
+| #I #V0 #X0 #HG #HL #HT #X #HX #Y #HY destruct
+ elim (rex_inv_flat … HY) -HY #HV0 #HX0
+ elim (cpx_inv_flat1 … HX) -HX *
+ [ #V1 #T1 #HV01 #HT01 #H destruct
+ /3 width=4 by fsle_flat/
+ | #HX #H destruct
+ /4 width=4 by fsle_flat_dx_dx/
+ | #HX #H destruct
+ /4 width=4 by fsle_flat_dx_sn/
+ | #p #V1 #W0 #W1 #T0 #T1 #HV01 #HW01 #HT01 #H1 #H2 #H3 destruct
+ elim (rpx_inv_bind … W0 ? HX0) -HX0 #HW0 #HT0
+ lapply (rpx_fwd_length … HV0) #H0
+ lapply (IH … HV01 … HV0) -HV01 -HV0 // #HV
+ lapply (IH … HW01 … HW0) -HW01 -HW0 // #HW
+ lapply (IH … HT01 … HT0) -HT01 -HT0 -IH // #HT
+ lapply (fsle_fwd_pair_sn … HT) -HT #HT
+ @fsle_bind_sn_ge //
+ [ /4 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_sn, fsle_bind_dx_sn/
+ | /3 width=1 by fsle_flat_dx_dx, fsle_shift/
+ ]
+ | #p #V1 #X1 #W0 #W1 #T0 #T1 #HV01 #HVX1 #HW01 #HT01 #H1 #H2 #H3 destruct
+ elim (rpx_inv_bind … W0 ? HX0) -HX0 #HW0 #HT0
+ lapply (rpx_fwd_length … HV0) #H0
+ lapply (IH … HV01 … HV0) -HV01 -HV0 // #HV
+ lapply (IH … HW01 … HW0) -HW01 -HW0 // #HW
+ lapply (IH … HT01 … HT0) -HT01 -HT0 -IH // #HT
+ lapply (fsle_fwd_pair_sn … HT) -HT #HT
+ @fsle_bind_sn_ge //
+ [ /3 width=1 by fsle_flat_dx_dx, fsle_bind_dx_sn/
+ | /4 width=3 by fsle_flat_sn, fsle_flat_dx_sn, fsle_flat_dx_dx, fsle_shift, fsle_lifts_sn/
+ ]
+ ]
+]
+qed-.
+
+lemma rpx_fsge_comp (h) (G): rex_fsge_compatible (cpx h G).
+/2 width=4 by rpx_cpx_conf_fsge/ qed-.
+
+(**) (* this section concerns cpx *)
+(* Properties with generic extension on referred entries ********************)
+
+(* Basic_2A1: uses: cpx_frees_trans *)
+lemma cpx_fsge_comp (h) (G): R_fsge_compatible (cpx h G).
+/2 width=4 by rpx_cpx_conf_fsge/ qed-.
+
+(* Note: lemma 1000 *)
+(* Basic_2A1: uses: cpx_llpx_sn_conf *)
+lemma cpx_rex_conf (R) (h) (G): s_r_confluent1 … (cpx h G) (rex R).
+/3 width=3 by fsge_rex_trans, cpx_fsge_comp/ qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma rpx_cpx_conf (h) (G): s_r_confluent1 … (cpx h G) (rpx h G).
+/2 width=5 by cpx_rex_conf/ qed-.
+
+lemma rpx_cpx_conf_fsge_dx (h) (G): ∀L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. ⦃G, L0⦄ ⊢⬈[h, T0] L2 → ⦃L2, T1⦄ ⊆ ⦃L0, T1⦄.
+/3 width=5 by rpx_cpx_conf, rpx_fsge_comp/ 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/static/rex_length.ma".
+include "basic_2/rt_transition/rpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma rpx_fwd_length: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → |L1| = |L2|.
+/2 width=3 by rex_fwd_length/ qed-.
+
+(* Inversion lemmas with length for local environments **********************)
+
+lemma rpx_inv_zero_length: ∀h,G,Y1,Y2. ⦃G, Y1⦄ ⊢ ⬈[h, #0] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I,L1,L2,V1,V2. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 &
+ ⦃G, L1⦄ ⊢ V1 ⬈[h] V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
+ | ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
+/2 width=1 by rex_inv_zero_length/ 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/static/rex_lex.ma".
+include "basic_2/rt_transition/rpx_fsle.ma".
+include "basic_2/rt_transition/lpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
+
+(* Properties with syntactic equivalence for referred local environments ****)
+
+lemma fleq_rpx (h) (G): ∀L1,L2,T. L1 ≡[T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+/2 width=1 by req_fwd_rex/ qed.
+
+(* Properties with unbound parallel rt-transition for full local envs *******)
+
+lemma lpx_rpx: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T] L2.
+/2 width=1 by rex_lex/ qed.
+
+(* Inversion lemmas with unbound parallel rt-transition for full local envs *)
+
+lemma rpx_inv_lpx_req: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 →
+ ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h] L & L ≡[T] L2.
+/3 width=3 by rpx_fsge_comp, rex_inv_lex_req/ 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/static/rdeq_fqup.ma".
+include "basic_2/static/rdeq_rdeq.ma".
+include "basic_2/rt_transition/rpx_fsle.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
+
+(* Properties with degree-based equivalence for local environments **********)
+
+lemma rpx_pair_sn_split: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 → ∀o,I,T.
+ ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ②{I}V.T] L & L ≛[h, o, V] L2.
+/3 width=5 by rpx_fsge_comp, rex_pair_sn_split/ qed-.
+
+lemma rpx_flat_dx_split: ∀h,G,L1,L2,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ∀o,I,V.
+ ∃∃L. ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L & L ≛[h, o, T] L2.
+/3 width=5 by rpx_fsge_comp, rex_flat_dx_split/ qed-.
+
+lemma rpx_bind_dx_split: ∀h,I,G,L1,L2,V1,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2 → ∀o,p.
+ ∃∃L,V. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ≛[h, o, T] L2 & ⦃G, L1⦄ ⊢ V1 ⬈[h] V.
+/3 width=5 by rpx_fsge_comp, rex_bind_dx_split/ qed-.
+
+lemma rpx_bind_dx_split_void: ∀h,G,K1,L2,T. ⦃G, K1.ⓧ⦄ ⊢ ⬈[h, T] L2 → ∀o,p,I,V.
+ ∃∃K2. ⦃G, K1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] K2 & K2.ⓧ ≛[h, o, T] L2.
+/3 width=5 by rpx_fsge_comp, rex_bind_dx_split_void/ qed-.
+
+lemma rpx_tdeq_conf: ∀h,o,G. s_r_confluent1 … (cdeq h o) (rpx h G).
+/2 width=5 by tdeq_rex_conf/ qed-.
+
+lemma rpx_tdeq_div: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
+ ∀G,L1,L2. ⦃G, L1⦄ ⊢ ⬈[h, T2] L2 → ⦃G, L1⦄ ⊢ ⬈[h, T1] L2.
+/2 width=5 by tdeq_rex_div/ qed-.
+
+lemma cpx_tdeq_conf_sex: ∀h,o,G. R_confluent2_rex … (cpx h G) (cdeq h o) (cpx h G) (cdeq h o).
+#h #o #G #L0 #T0 #T1 #H @(cpx_ind … H) -G -L0 -T0 -T1 /2 width=3 by ex2_intro/
+[ #G #L0 #s0 #X0 #H0 #L1 #HL01 #L2 #HL02
+ elim (tdeq_inv_sort1 … H0) -H0 #s1 #d1 #Hs0 #Hs1 #H destruct
+ /4 width=3 by tdeq_sort, deg_next, ex2_intro/
+| #I #G #K0 #V0 #V1 #W1 #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
+ >(tdeq_inv_lref1 … H0) -H0
+ elim (rpx_inv_zero_pair_sn … H1) -H1 #K1 #X1 #HK01 #HX1 #H destruct
+ elim (rdeq_inv_zero_pair_sn … H2) -H2 #K2 #X2 #HK02 #HX2 #H destruct
+ elim (IH X2 … HK01 … HK02) // -K0 -V0 #V #HV1 #HV2
+ elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_delta, ex2_intro/
+| #I0 #G #K0 #V1 #W1 #i #_ #IH #HVW1 #T2 #H0 #L1 #H1 #L2 #H2
+ >(tdeq_inv_lref1 … H0) -H0
+ elim (rpx_inv_lref_bind_sn … H1) -H1 #I1 #K1 #HK01 #H destruct
+ elim (rdeq_inv_lref_bind_sn … H2) -H2 #I2 #K2 #HK02 #H destruct
+ elim (IH … HK01 … HK02) [|*: //] -K0 #V #HV1 #HV2
+ elim (tdeq_lifts_sn … HV1 … HVW1) -V1 /3 width=5 by cpx_lref, ex2_intro/
+| #p #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
+ elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (rpx_inv_bind … H1) -H1 #HL01 #H1
+ elim (rdeq_inv_bind … H2) -H2 #HL02 #H2
+ lapply (rdeq_bind_repl_dx … H2 (BPair I V2) ?) -H2 /2 width=1 by ext2_pair/ #H2
+ elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
+ elim (IHT … HT02 … H1 … H2) -L0 -T0
+ /3 width=5 by cpx_bind, tdeq_pair, ex2_intro/
+| #I #G #L0 #V0 #V1 #T0 #T1 #_ #_ #IHV #IHT #X0 #H0 #L1 #H1 #L2 #H2
+ elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (rpx_inv_flat … H1) -H1 #HL01 #H1
+ elim (rdeq_inv_flat … H2) -H2 #HL02 #H2
+ elim (IHV … HV02 … HL01 … HL02) -IHV -HV02 -HL01 -HL02
+ elim (IHT … HT02 … H1 … H2) -L0 -V0 -T0
+ /3 width=5 by cpx_flat, tdeq_pair, ex2_intro/
+| #G #L0 #V0 #T0 #T1 #U1 #_ #IH #HUT1 #X0 #H0 #L1 #H1 #L2 #H2
+ elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #HT02 #H destruct
+ elim (rpx_inv_bind … H1) -H1 #HL01 #H1
+ elim (rdeq_inv_bind … H2) -H2 #HL02 #H2
+ lapply (rdeq_bind_repl_dx … H2 (BPair Abbr V2) ?) -H2 /2 width=1 by ext2_pair/ -HV02 #H2
+ elim (IH … HT02 … H1 … H2) -L0 -T0 #T #HT1
+ elim (tdeq_inv_lifts_sn … HT1 … HUT1) -T1
+ /3 width=5 by cpx_zeta, ex2_intro/
+| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
+ elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #_ #HT02 #H destruct
+ elim (rpx_inv_flat … H1) -H1 #HL01 #H1
+ elim (rdeq_inv_flat … H2) -H2 #HL02 #H2
+ elim (IH … HT02 … H1 … H2) -L0 -V0 -T0
+ /3 width=3 by cpx_eps, ex2_intro/
+| #G #L0 #V0 #T0 #T1 #_ #IH #X0 #H0 #L1 #H1 #L2 #H2
+ elim (tdeq_inv_pair1 … H0) -H0 #V2 #T2 #HV02 #_ #H destruct
+ elim (rpx_inv_flat … H1) -H1 #HL01 #H1
+ elim (rdeq_inv_flat … H2) -H2 #HL02 #H2
+ elim (IH … HV02 … HL01 … HL02) -L0 -V0 -T1
+ /3 width=3 by cpx_ee, ex2_intro/
+| #p #G #L0 #V0 #V1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #X0 #H0 #L1 #H1 #L2 #H2
+ elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
+ elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
+ elim (rpx_inv_flat … H1) -H1 #H1LV0 #H1
+ elim (rpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
+ elim (rdeq_inv_flat … H2) -H2 #H2LV0 #H2
+ elim (rdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
+ lapply (rdeq_bind_repl_dx … H2LT0 (BPair Abst W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
+ elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0
+ elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
+ elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
+ /4 width=7 by cpx_beta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
+| #p #G #L0 #V0 #V1 #U1 #W0 #W1 #T0 #T1 #_ #_ #_ #IHV #IHW #IHT #HVU1 #X0 #H0 #L1 #H1 #L2 #H2
+ elim (tdeq_inv_pair1 … H0) -H0 #V2 #X #HV02 #H0 #H destruct
+ elim (tdeq_inv_pair1 … H0) -H0 #W2 #T2 #HW02 #HT02 #H destruct
+ elim (rpx_inv_flat … H1) -H1 #H1LV0 #H1
+ elim (rpx_inv_bind … H1) -H1 #H1LW0 #H1LT0
+ elim (rdeq_inv_flat … H2) -H2 #H2LV0 #H2
+ elim (rdeq_inv_bind … H2) -H2 #H2LW0 #H2LT0
+ lapply (rdeq_bind_repl_dx … H2LT0 (BPair Abbr W2) ?) -H2LT0 /2 width=1 by ext2_pair/ #H2LT0
+ elim (IHV … HV02 … H1LV0 … H2LV0) -IHV -HV02 -H1LV0 -H2LV0 #V #HV1
+ elim (IHW … HW02 … H1LW0 … H2LW0) -IHW -HW02 -H1LW0 -H2LW0
+ elim (IHT … HT02 … H1LT0 … H2LT0) -L0 -V0 -T0
+ elim (tdeq_lifts_sn … HV1 … HVU1) -V1
+ /4 width=9 by cpx_theta, tdeq_pair, ex2_intro/ (* note: 2 tdeq_pair *)
+]
+qed-.
+
+lemma cpx_tdeq_conf: ∀h,o,G,L. ∀T0:term. ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
+ ∀T2. T0 ≛[h, o] T2 →
+ ∃∃T. T1 ≛[h, o] T & ⦃G, L⦄ ⊢ T2 ⬈[h] T.
+#h #o #G #L #T0 #T1 #HT01 #T2 #HT02
+elim (cpx_tdeq_conf_sex … HT01 … HT02 L … L) -HT01 -HT02
+/2 width=3 by rex_refl, ex2_intro/
+qed-.
+
+lemma tdeq_cpx_trans: ∀h,o,G,L,T2. ∀T0:term. T2 ≛[h, o] T0 →
+ ∀T1. ⦃G, L⦄ ⊢ T0 ⬈[h] T1 →
+ ∃∃T. ⦃G, L⦄ ⊢ T2 ⬈[h] T & T ≛[h, o] T1.
+#h #o #G #L #T2 #T0 #HT20 #T1 #HT01
+elim (cpx_tdeq_conf … HT01 T2) -HT01 /3 width=3 by tdeq_sym, ex2_intro/
+qed-.
+
+(* Basic_2A1: uses: cpx_lleq_conf *)
+lemma cpx_rdeq_conf: ∀h,o,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∀L2. L0 ≛[h, o, T0] L2 →
+ ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T1 ≛[h, o] T.
+#h #o #G #L0 #T0 #T1 #HT01 #L2 #HL02
+elim (cpx_tdeq_conf_sex … HT01 T0 … L0 … HL02) -HT01 -HL02
+/2 width=3 by rex_refl, ex2_intro/
+qed-.
+
+(* Basic_2A1: uses: lleq_cpx_trans *)
+lemma rdeq_cpx_trans: ∀h,o,G,L2,L0,T0. L2 ≛[h, o, T0] L0 →
+ ∀T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+ ∃∃T. ⦃G, L2⦄ ⊢ T0 ⬈[h] T & T ≛[h, o] T1.
+#h #o #G #L2 #L0 #T0 #HL20 #T1 #HT01
+elim (cpx_rdeq_conf … o … HT01 L2) -HT01
+/3 width=3 by rdeq_sym, tdeq_sym, ex2_intro/
+qed-.
+
+lemma rpx_rdeq_conf: ∀h,o,G,T. confluent2 … (rpx h G T) (rdeq h o T).
+/3 width=6 by rpx_fsge_comp, rdeq_fsge_comp, cpx_tdeq_conf_sex, rex_conf/ qed-.
+
+lemma rdeq_rpx_trans: ∀h,o,G,T,L2,K2. ⦃G, L2⦄ ⊢ ⬈[h, T] K2 →
+ ∀L1. L1 ≛[h, o, T] L2 →
+ ∃∃K1. ⦃G, L1⦄ ⊢ ⬈[h, T] K1 & K1 ≛[h, o, T] K2.
+#h #o #G #T #L2 #K2 #HLK2 #L1 #HL12
+elim (rpx_rdeq_conf … o … HLK2 L1)
+/3 width=3 by rdeq_sym, 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/static/rex_rex.ma".
+include "basic_2/rt_transition/rpx.ma".
+
+(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
+
+(* Main properties **********************************************************)
+
+theorem rpx_bind: ∀h,G,L1,L2,V1. ⦃G, L1⦄ ⊢ ⬈[h, V1] L2 →
+ ∀I,V2,T. ⦃G, L1.ⓑ{I}V1⦄ ⊢ ⬈[h, T] L2.ⓑ{I}V2 →
+ ∀p. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V1.T] L2.
+/2 width=2 by rex_bind/ qed.
+
+theorem rpx_flat: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 →
+ ∀I,T. ⦃G, L1⦄ ⊢ ⬈[h, T] L2 → ⦃G, L1⦄ ⊢ ⬈[h, ⓕ{I}V.T] L2.
+/2 width=1 by rex_flat/ qed.
+
+theorem rpx_bind_void: ∀h,G,L1,L2,V. ⦃G, L1⦄ ⊢ ⬈[h, V] L2 →
+ ∀T. ⦃G, L1.ⓧ⦄ ⊢ ⬈[h, T] L2.ⓧ →
+ ∀p,I. ⦃G, L1⦄ ⊢ ⬈[h, ⓑ{p,I}V.T] L2.
+/2 width=1 by rex_bind_void/ 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/static/fdeq.ma".
+include "basic_2/static/aaa_rdeq.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties with degree-based equivalence on referred entries *************)
+
+lemma aaa_fdeq_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
+ ∀A. ⦃G1, L1⦄ ⊢ T1 ⁝ A → ⦃G2, L2⦄ ⊢ T2 ⁝ A.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/2 width=7 by aaa_tdeq_conf_rdeq/ 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/static/ffdeq.ma".
-include "basic_2/static/aaa_lfdeq.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties with degree-based equivalence on referred entries *************)
-
-lemma aaa_ffdeq_conf: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
- ∀A. ⦃G1, L1⦄ ⊢ T1 ⁝ A → ⦃G2, L2⦄ ⊢ T2 ⁝ A.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/2 width=7 by aaa_tdeq_conf_lfdeq/ 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/static/lfdeq.ma".
-include "basic_2/static/aaa.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties with degree-based equivalence on referred entries *************)
-
-lemma aaa_tdeq_conf_lfdeq: ∀h,o,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀T2. T1 ≛[h, o] T2 →
- ∀L2. L1 ≛[h, o, T1] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-#h #o #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
-[ #G #L1 #s1 #X #H1 elim (tdeq_inv_sort1 … H1) -H1 //
-| #I #G #L1 #V1 #B #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1
- #Y #H2 elim (lfdeq_inv_zero_pair_sn … H2) -H2
- #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by aaa_zero/
-| #I #G #L1 #A #i #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1
- #Y #H2 elim (lfdeq_inv_lref_bind_sn … H2) -H2
- #J #L2 #HL12 #H destruct /3 width=1 by aaa_lref/
-| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1
- #V2 #T2 #HV12 #HT12 #H #L2 #H2 elim (lfdeq_inv_bind … H2) -H2 destruct
- /5 width=2 by aaa_abbr, lfdeq_bind_repl_dx, ext2_pair/
-| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1
- #V2 #T2 #HV12 #HT12 #H #L2 #H2 elim (lfdeq_inv_bind … H2) -H2 destruct
- /5 width=2 by aaa_abst, lfdeq_bind_repl_dx, ext2_pair/
-| #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1
- #V2 #T2 #HV12 #HT12 #H #L2 #H2 elim (lfdeq_inv_flat … H2) -H2 destruct
- /3 width=3 by aaa_appl/
-| #G #L1 #V1 #T1 #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1
- #V2 #T2 #HV12 #HT12 #H #L2 #H2 elim (lfdeq_inv_flat … H2) -H2 destruct
- /3 width=1 by aaa_cast/
-]
-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/static/rdeq.ma".
+include "basic_2/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties with degree-based equivalence on referred entries *************)
+
+lemma aaa_tdeq_conf_rdeq: ∀h,o,G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀T2. T1 ≛[h, o] T2 →
+ ∀L2. L1 ≛[h, o, T1] L2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+#h #o #G #L1 #T1 #A #H elim H -G -L1 -T1 -A
+[ #G #L1 #s1 #X #H1 elim (tdeq_inv_sort1 … H1) -H1 //
+| #I #G #L1 #V1 #B #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1
+ #Y #H2 elim (rdeq_inv_zero_pair_sn … H2) -H2
+ #L2 #V2 #HL12 #HV12 #H destruct /3 width=1 by aaa_zero/
+| #I #G #L1 #A #i #_ #IH #X #H1 >(tdeq_inv_lref1 … H1) -H1
+ #Y #H2 elim (rdeq_inv_lref_bind_sn … H2) -H2
+ #J #L2 #HL12 #H destruct /3 width=1 by aaa_lref/
+| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1
+ #V2 #T2 #HV12 #HT12 #H #L2 #H2 elim (rdeq_inv_bind … H2) -H2 destruct
+ /5 width=2 by aaa_abbr, rdeq_bind_repl_dx, ext2_pair/
+| #p #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1
+ #V2 #T2 #HV12 #HT12 #H #L2 #H2 elim (rdeq_inv_bind … H2) -H2 destruct
+ /5 width=2 by aaa_abst, rdeq_bind_repl_dx, ext2_pair/
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1
+ #V2 #T2 #HV12 #HT12 #H #L2 #H2 elim (rdeq_inv_flat … H2) -H2 destruct
+ /3 width=3 by aaa_appl/
+| #G #L1 #V1 #T1 #A #_ #_ #IHV #IHT #X #H1 elim (tdeq_inv_pair1 … H1) -H1
+ #V2 #T2 #HV12 #HT12 #H #L2 #H2 elim (rdeq_inv_flat … H2) -H2 destruct
+ /3 width=1 by aaa_cast/
+]
+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/stareqsn_8.ma".
+include "basic_2/syntax/genv.ma".
+include "basic_2/static/rdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+
+inductive fdeq (h) (o) (G) (L1) (T1): relation3 genv lenv term ≝
+| fdeq_intro_sn: ∀L2,T2. L1 ≛[h, o, T1] L2 → T1 ≛[h, o] T2 →
+ fdeq h o G L1 T1 G L2 T2
+.
+
+interpretation
+ "degree-based equivalence on referred entries (closure)"
+ 'StarEqSn h o G1 L1 T1 G2 L2 T2 = (fdeq h o G1 L1 T1 G2 L2 T2).
+
+(* Basic_properties *********************************************************)
+
+lemma fdeq_intro_dx (h) (o) (G): ∀L1,L2,T2. L1 ≛[h, o, T2] L2 →
+ ∀T1. T1 ≛[h, o] T2 → ⦃G, L1, T1⦄ ≛[h, o] ⦃G, L2, T2⦄.
+/3 width=3 by fdeq_intro_sn, tdeq_rdeq_div/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fdeq_inv_gen_sn: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L1 ≛[h, o, T1] L2 & T1 ≛[h, o] T2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/
+qed-.
+
+lemma fdeq_inv_gen_dx: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L1 ≛[h, o, T2] L2 & T1 ≛[h, o] T2.
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/3 width=3 by tdeq_rdeq_conf, and3_intro/
+qed-.
+
+(* Basic_2A1: removed theorems 6:
+ fleq_refl fleq_sym fleq_inv_gen
+ fleq_trans fleq_canc_sn fleq_canc_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/static/rdeq_rdeq.ma".
+include "basic_2/static/fdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+
+(* Advanced properties ******************************************************)
+
+lemma fdeq_sym: ∀h,o. tri_symmetric … (fdeq h o).
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1
+/3 width=1 by fdeq_intro_dx, rdeq_sym, tdeq_sym/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem fdeq_trans: ∀h,o. tri_transitive … (fdeq h o).
+#h #o #G1 #G #L1 #L #T1 #T * -G -L -T
+#L #T #HL1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
+/4 width=5 by fdeq_intro_sn, rdeq_trans, tdeq_rdeq_div, tdeq_trans/
+qed-.
+
+theorem fdeq_canc_sn: ∀h,o,G,G1,G2,L,L1,L2,T,T1,T2.
+ ⦃G, L, T⦄ ≛[h, o] ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+/3 width=5 by fdeq_trans, fdeq_sym/ qed-.
+
+theorem fdeq_canc_dx: ∀h,o,G1,G2,G,L1,L2,L,T1,T2,T.
+ ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+/3 width=5 by fdeq_trans, fdeq_sym/ qed-.
+
+(* Main inversion lemmas with degree-based equivalence on terms *************)
+
+theorem fdeq_tdneq_repl_dx: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
+ ∀U1,U2. ⦃G1, L1, U1⦄ ≛[h, o] ⦃G2, L2, U2⦄ →
+ (T2 ≛[h, o] U2 → ⊥) → (T1 ≛[h, o] U1 → ⊥).
+#h #o #G1 #G2 #L1 #L2 #T1 #T2 #HT #U1 #U2 #HU #HnTU2 #HTU1
+elim (fdeq_inv_gen_sn … HT) -HT #_ #_ #HT
+elim (fdeq_inv_gen_sn … HU) -HU #_ #_ #HU
+/3 width=5 by tdeq_repl/
+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/static/rdeq_fqup.ma".
+include "basic_2/static/fdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+
+(* Properties with degree-based equivalence for terms ***********************)
+
+lemma tdeq_fdeq: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
+ ∀G,L. ⦃G, L, T1⦄ ≛[h, o] ⦃G, L, T2⦄.
+/2 width=1 by fdeq_intro_sn/ qed.
+
+(* Advanced properties ******************************************************)
+
+lemma fdeq_refl: ∀h,o. tri_reflexive … (fdeq h o).
+/2 width=1 by fdeq_intro_sn/ 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/static/rdeq_fqus.ma".
+include "basic_2/static/fdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+
+(* Properties with star-iterated structural successor for closures **********)
+
+lemma fdeq_fqus_trans: ∀h,o,b,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ →
+ ∀G2,L2,T2. ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ ∃∃G,L0,T0. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L0, T0⦄ & ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+#h #o #b #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H2
+elim(fdeq_inv_gen_dx … H1) -H1 #HG #HL1 #HT1 destruct
+elim (rdeq_fqus_trans … H2 … HL1) -L #L #T0 #H2 #HT02 #HL2
+elim (tdeq_fqus_trans … H2 … HT1) -T #L0 #T #H2 #HT0 #HL0
+lapply (tdeq_rdeq_conf … HT02 … HL0) -HL0 #HL0
+/4 width=7 by fdeq_intro_dx, rdeq_trans, tdeq_trans, ex2_3_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/static/rdeq_req.ma".
+include "basic_2/static/fdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
+
+(* Properties with syntactic equivalence on referred entries ****************)
+
+lemma req_rdeq_trans: ∀h,o,L1,L,T1. L1 ≡[T1] L →
+ ∀G1,G2,L2,T2. ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
+#h #o #L1 #L #T1 #HL1 #G1 #G2 #L2 #T2 #H
+elim (fdeq_inv_gen_sn … H) -H #H #HL2 #T12 destruct
+/3 width=3 by fdeq_intro_sn, req_rdeq_trans/
+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/stareqsn_8.ma".
-include "basic_2/syntax/genv.ma".
-include "basic_2/static/lfdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
-
-inductive ffdeq (h) (o) (G) (L1) (T1): relation3 genv lenv term ≝
-| ffdeq_intro_sn: ∀L2,T2. L1 ≛[h, o, T1] L2 → T1 ≛[h, o] T2 →
- ffdeq h o G L1 T1 G L2 T2
-.
-
-interpretation
- "degree-based equivalence on referred entries (closure)"
- 'StarEqSn h o G1 L1 T1 G2 L2 T2 = (ffdeq h o G1 L1 T1 G2 L2 T2).
-
-(* Basic_properties *********************************************************)
-
-lemma ffdeq_intro_dx (h) (o) (G): ∀L1,L2,T2. L1 ≛[h, o, T2] L2 →
- ∀T1. T1 ≛[h, o] T2 → ⦃G, L1, T1⦄ ≛[h, o] ⦃G, L2, T2⦄.
-/3 width=3 by ffdeq_intro_sn, tdeq_lfdeq_div/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma ffdeq_inv_gen_sn: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L1 ≛[h, o, T1] L2 & T1 ≛[h, o] T2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2 /2 width=1 by and3_intro/
-qed-.
-
-lemma ffdeq_inv_gen_dx: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L1 ≛[h, o, T2] L2 & T1 ≛[h, o] T2.
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=3 by tdeq_lfdeq_conf, and3_intro/
-qed-.
-
-(* Basic_2A1: removed theorems 6:
- fleq_refl fleq_sym fleq_inv_gen
- fleq_trans fleq_canc_sn fleq_canc_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/static/lfdeq_lfdeq.ma".
-include "basic_2/static/ffdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
-
-(* Advanced properties ******************************************************)
-
-lemma ffdeq_sym: ∀h,o. tri_symmetric … (ffdeq h o).
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -L1 -T1
-/3 width=1 by ffdeq_intro_dx, lfdeq_sym, tdeq_sym/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem ffdeq_trans: ∀h,o. tri_transitive … (ffdeq h o).
-#h #o #G1 #G #L1 #L #T1 #T * -G -L -T
-#L #T #HL1 #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
-/4 width=5 by ffdeq_intro_sn, lfdeq_trans, tdeq_lfdeq_div, tdeq_trans/
-qed-.
-
-theorem ffdeq_canc_sn: ∀h,o,G,G1,G2,L,L1,L2,T,T1,T2.
- ⦃G, L, T⦄ ≛[h, o] ⦃G1, L1, T1⦄→ ⦃G, L, T⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by ffdeq_trans, ffdeq_sym/ qed-.
-
-theorem ffdeq_canc_dx: ∀h,o,G1,G2,G,L1,L2,L,T1,T2,T.
- ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G2, L2, T2⦄ ≛[h, o] ⦃G, L, T⦄ → ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-/3 width=5 by ffdeq_trans, ffdeq_sym/ qed-.
-
-(* Main inversion lemmas with degree-based equivalence on terms *************)
-
-theorem ffdeq_tdneq_repl_dx: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ →
- ∀U1,U2. ⦃G1, L1, U1⦄ ≛[h, o] ⦃G2, L2, U2⦄ →
- (T2 ≛[h, o] U2 → ⊥) → (T1 ≛[h, o] U1 → ⊥).
-#h #o #G1 #G2 #L1 #L2 #T1 #T2 #HT #U1 #U2 #HU #HnTU2 #HTU1
-elim (ffdeq_inv_gen_sn … HT) -HT #_ #_ #HT
-elim (ffdeq_inv_gen_sn … HU) -HU #_ #_ #HU
-/3 width=5 by tdeq_repl/
-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/static/lfdeq_fqup.ma".
-include "basic_2/static/ffdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
-
-(* Properties with degree-based equivalence for terms ***********************)
-
-lemma tdeq_ffdeq: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
- ∀G,L. ⦃G, L, T1⦄ ≛[h, o] ⦃G, L, T2⦄.
-/2 width=1 by ffdeq_intro_sn/ qed.
-
-(* Advanced properties ******************************************************)
-
-lemma ffdeq_refl: ∀h,o. tri_reflexive … (ffdeq h o).
-/2 width=1 by ffdeq_intro_sn/ 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/static/lfdeq_fqus.ma".
-include "basic_2/static/ffdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
-
-(* Properties with star-iterated structural successor for closures **********)
-
-lemma ffdeq_fqus_trans: ∀h,o,b,G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ≛[h, o] ⦃G, L, T⦄ →
- ∀G2,L2,T2. ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∃∃G,L0,T0. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L0, T0⦄ & ⦃G, L0, T0⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #b #G1 #G #L1 #L #T1 #T #H1 #G2 #L2 #T2 #H2
-elim(ffdeq_inv_gen_dx … H1) -H1 #HG #HL1 #HT1 destruct
-elim (lfdeq_fqus_trans … H2 … HL1) -L #L #T0 #H2 #HT02 #HL2
-elim (tdeq_fqus_trans … H2 … HT1) -T #L0 #T #H2 #HT0 #HL0
-lapply (tdeq_lfdeq_conf … HT02 … HL0) -HL0 #HL0
-/4 width=7 by ffdeq_intro_dx, lfdeq_trans, tdeq_trans, ex2_3_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/static/lfdeq_lfeq.ma".
-include "basic_2/static/ffdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR CLOSURES ON REFERRED ENTRIES ****************)
-
-(* Properties with syntactic equivalence on referred entries ****************)
-
-lemma lfeq_lfdeq_trans: ∀h,o,L1,L,T1. L1 ≡[T1] L →
- ∀G1,G2,L2,T2. ⦃G1, L, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≛[h, o] ⦃G2, L2, T2⦄.
-#h #o #L1 #L #T1 #HL1 #G1 #G2 #L2 #T2 #H
-elim (ffdeq_inv_gen_sn … H) -H #H #HL2 #T12 destruct
-/3 width=3 by ffdeq_intro_sn, lfeq_lfdeq_trans/
-qed-.
/3 width=7 by frees_eq_repl_back, coafter_inj/
qed-.
-(* Note: this is used by lfxs_conf and might be modified *)
+(* Note: this is used by rex_conf and might be modified *)
lemma frees_inv_drops_next: ∀f1,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 →
∀I2,L2,V2,n. ⬇*[n] L1 ≘ L2.ⓑ{I2}V2 →
∀g1. ↑g1 = ⫱*[n] f1 →
+++ /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/stareqsn_5.ma".
-include "basic_2/syntax/tdeq_ext.ma".
-include "basic_2/static/lfxs.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-
-definition lfdeq (h) (o): relation3 term lenv lenv ≝
- lfxs (cdeq h o).
-
-interpretation
- "degree-based equivalence on referred entries (local environment)"
- 'StarEqSn h o T L1 L2 = (lfdeq h o T L1 L2).
-
-interpretation
- "degree-based ranged equivalence (local environment)"
- 'StarEqSn h o f L1 L2 = (lexs (cdeq_ext h o) cfull f L1 L2).
-
-(* Basic properties ***********************************************************)
-
-lemma frees_tdeq_conf_lfdeq (h) (o): ∀f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → ∀T2. T1 ≛[h, o] T2 →
- ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≘ f.
-#h #o #f #L1 #T1 #H elim H -f -L1 -T1
-[ #f #L1 #s1 #Hf #X #H1 #L2 #_
- elim (tdeq_inv_sort1 … H1) -H1 #s2 #d #_ #_ #H destruct
- /2 width=3 by frees_sort/
-| #f #i #Hf #X #H1
- >(tdeq_inv_lref1 … H1) -X #Y #H2
- >(lexs_inv_atom1 … H2) -Y
- /2 width=1 by frees_atom/
-| #f #I #L1 #V1 #_ #IH #X #H1
- >(tdeq_inv_lref1 … H1) -X #Y #H2
- elim (lexs_inv_next1 … H2) -H2 #Z #L2 #HL12 #HZ #H destruct
- elim (ext2_inv_pair_sn … HZ) -HZ #V2 #HV12 #H destruct
- /3 width=1 by frees_pair/
-| #f #I #L1 #Hf #X #H1
- >(tdeq_inv_lref1 … H1) -X #Y #H2
- elim (lexs_inv_next1 … H2) -H2 #Z #L2 #_ #HZ #H destruct
- >(ext2_inv_unit_sn … HZ) -Z /2 width=1 by frees_unit/
-| #f #I #L1 #i #_ #IH #X #H1
- >(tdeq_inv_lref1 … H1) -X #Y #H2
- elim (lexs_inv_push1 … H2) -H2 #J #L2 #HL12 #_ #H destruct
- /3 width=1 by frees_lref/
-| #f #L1 #l #Hf #X #H1 #L2 #_
- >(tdeq_inv_gref1 … H1) -X /2 width=1 by frees_gref/
-| #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1
- elim (tdeq_inv_pair1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct
- /6 width=5 by frees_bind, lexs_inv_tl, ext2_pair, sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn/
-| #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1
- elim (tdeq_inv_pair1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct
- /5 width=5 by frees_flat, sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn/
-]
-qed-.
-
-lemma frees_tdeq_conf (h) (o): ∀f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≘ f →
- ∀T2. T1 ≛[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≘ f.
-/4 width=7 by frees_tdeq_conf_lfdeq, lexs_refl, ext2_refl/ qed-.
-
-lemma frees_lfdeq_conf (h) (o): ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f →
- ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f.
-/2 width=7 by frees_tdeq_conf_lfdeq, tdeq_refl/ qed-.
-
-lemma tdeq_lfxs_conf (R) (h) (o): s_r_confluent1 … (cdeq h o) (lfxs R).
-#R #h #o #L1 #T1 #T2 #HT12 #L2 *
-/3 width=5 by frees_tdeq_conf, ex2_intro/
-qed-.
-
-lemma tdeq_lfxs_div (R) (h) (o): ∀T1,T2. T1 ≛[h, o] T2 →
- ∀L1,L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
-/3 width=5 by tdeq_lfxs_conf, tdeq_sym/ qed-.
-
-lemma tdeq_lfdeq_conf (h) (o): s_r_confluent1 … (cdeq h o) (lfdeq h o).
-/2 width=5 by tdeq_lfxs_conf/ qed-.
-
-lemma tdeq_lfdeq_div (h) (o): ∀T1,T2. T1 ≛[h, o] T2 →
- ∀L1,L2. L1 ≛[h, o, T2] L2 → L1 ≛[h, o, T1] L2.
-/2 width=5 by tdeq_lfxs_div/ qed-.
-
-lemma lfdeq_atom (h) (o): ∀I. ⋆ ≛[h, o, ⓪{I}] ⋆.
-/2 width=1 by lfxs_atom/ qed.
-
-lemma lfdeq_sort (h) (o): ∀I1,I2,L1,L2,s.
- L1 ≛[h, o, ⋆s] L2 → L1.ⓘ{I1} ≛[h, o, ⋆s] L2.ⓘ{I2}.
-/2 width=1 by lfxs_sort/ qed.
-
-lemma lfdeq_pair (h) (o): ∀I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 → V1 ≛[h, o] V2 →
- L1.ⓑ{I}V1 ≛[h, o, #0] L2.ⓑ{I}V2.
-/2 width=1 by lfxs_pair/ qed.
-(*
-lemma lfdeq_unit (h) (o): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cdeq_ext h o, cfull, f] L2 →
- L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
-/2 width=3 by lfxs_unit/ qed.
-*)
-lemma lfdeq_lref (h) (o): ∀I1,I2,L1,L2,i.
- L1 ≛[h, o, #i] L2 → L1.ⓘ{I1} ≛[h, o, #↑i] L2.ⓘ{I2}.
-/2 width=1 by lfxs_lref/ qed.
-
-lemma lfdeq_gref (h) (o): ∀I1,I2,L1,L2,l.
- L1 ≛[h, o, §l] L2 → L1.ⓘ{I1} ≛[h, o, §l] L2.ⓘ{I2}.
-/2 width=1 by lfxs_gref/ qed.
-
-lemma lfdeq_bind_repl_dx (h) (o): ∀I,I1,L1,L2.∀T:term.
- L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I1} →
- ∀I2. I ≛[h, o] I2 →
- L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I2}.
-/2 width=2 by lfxs_bind_repl_dx/ qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lfdeq_inv_atom_sn (h) (o): ∀Y2. ∀T:term. ⋆ ≛[h, o, T] Y2 → Y2 = ⋆.
-/2 width=3 by lfxs_inv_atom_sn/ qed-.
-
-lemma lfdeq_inv_atom_dx (h) (o): ∀Y1. ∀T:term. Y1 ≛[h, o, T] ⋆ → Y1 = ⋆.
-/2 width=3 by lfxs_inv_atom_dx/ qed-.
-(*
-lemma lfdeq_inv_zero (h) (o): ∀Y1,Y2. Y1 ≛[h, o, #0] Y2 →
- ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
- | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤*[cdeq_ext h o, cfull, f] L2 &
- Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
-#h #o #Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
-/3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/
-qed-.
-*)
-lemma lfdeq_inv_lref (h) (o): ∀Y1,Y2,i. Y1 ≛[h, o, #↑i] Y2 →
- ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ≛[h, o, #i] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-/2 width=1 by lfxs_inv_lref/ qed-.
-
-(* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *)
-lemma lfdeq_inv_bind (h) (o): ∀p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
- ∧∧ L1 ≛[h, o, V] L2 & L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
-/2 width=2 by lfxs_inv_bind/ qed-.
-
-(* Basic_2A1: uses: lleq_inv_flat *)
-lemma lfdeq_inv_flat (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 →
- ∧∧ L1 ≛[h, o, V] L2 & L1 ≛[h, o, T] L2.
-/2 width=2 by lfxs_inv_flat/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfdeq_inv_zero_pair_sn (h) (o): ∀I,Y2,L1,V1. L1.ⓑ{I}V1 ≛[h, o, #0] Y2 →
- ∃∃L2,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y2 = L2.ⓑ{I}V2.
-/2 width=1 by lfxs_inv_zero_pair_sn/ qed-.
-
-lemma lfdeq_inv_zero_pair_dx (h) (o): ∀I,Y1,L2,V2. Y1 ≛[h, o, #0] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y1 = L1.ⓑ{I}V1.
-/2 width=1 by lfxs_inv_zero_pair_dx/ qed-.
-
-lemma lfdeq_inv_lref_bind_sn (h) (o): ∀I1,Y2,L1,i. L1.ⓘ{I1} ≛[h, o, #↑i] Y2 →
- ∃∃I2,L2. L1 ≛[h, o, #i] L2 & Y2 = L2.ⓘ{I2}.
-/2 width=2 by lfxs_inv_lref_bind_sn/ qed-.
-
-lemma lfdeq_inv_lref_bind_dx (h) (o): ∀I2,Y1,L2,i. Y1 ≛[h, o, #↑i] L2.ⓘ{I2} →
- ∃∃I1,L1. L1 ≛[h, o, #i] L2 & Y1 = L1.ⓘ{I1}.
-/2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lfdeq_fwd_zero_pair (h) (o): ∀I,K1,K2,V1,V2.
- K1.ⓑ{I}V1 ≛[h, o, #0] K2.ⓑ{I}V2 → K1 ≛[h, o, V1] K2.
-/2 width=3 by lfxs_fwd_zero_pair/ qed-.
-
-(* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *)
-lemma lfdeq_fwd_pair_sn (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ②{I}V.T] L2 → L1 ≛[h, o, V] L2.
-/2 width=3 by lfxs_fwd_pair_sn/ qed-.
-
-(* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *)
-lemma lfdeq_fwd_bind_dx (h) (o): ∀p,I,L1,L2,V,T.
- L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
-/2 width=2 by lfxs_fwd_bind_dx/ qed-.
-
-(* Basic_2A1: uses: lleq_fwd_flat_dx *)
-lemma lfdeq_fwd_flat_dx (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 → L1 ≛[h, o, T] L2.
-/2 width=3 by lfxs_fwd_flat_dx/ qed-.
-
-lemma lfdeq_fwd_dx (h) (o): ∀I2,L1,K2. ∀T:term. L1 ≛[h, o, T] K2.ⓘ{I2} →
- ∃∃I1,K1. L1 = K1.ⓘ{I1}.
-/2 width=5 by lfxs_fwd_dx/ 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_tdeq.ma".
-include "basic_2/static/lfxs_drops.ma".
-include "basic_2/static/lfdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-
-(* Properties with generic slicing for local environments *******************)
-
-lemma lfdeq_lifts_sn: ∀h,o. f_dedropable_sn (cdeq h o).
-/3 width=5 by lfxs_liftable_dedropable_sn, tdeq_lifts_sn/ qed-.
-
-(* Inversion lemmas with generic slicing for local environments *************)
-
-lemma lfdeq_inv_lifts_sn: ∀h,o. f_dropable_sn (cdeq h o).
-/2 width=5 by lfxs_dropable_sn/ qed-.
-
-(* Note: missing in basic_2A1 *)
-lemma lfdeq_inv_lifts_dx: ∀h,o. f_dropable_dx (cdeq h o).
-/2 width=5 by lfxs_dropable_dx/ qed-.
-
-(* Basic_2A1: uses: lleq_inv_lift_le lleq_inv_lift_be lleq_inv_lift_ge *)
-lemma lfdeq_inv_lifts_bi: ∀h,o,L1,L2,U. L1 ≛[h, o, U] L2 → ∀b,f. 𝐔⦃f⦄ →
- ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- ∀T. ⬆*[f] T ≘ U → K1 ≛[h, o, T] K2.
-/2 width=10 by lfxs_inv_lifts_bi/ qed-.
-
-lemma lfdeq_inv_lref_pair_sn: ∀h,o,L1,L2,i. L1 ≛[h, o, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ≛[h, o, V1] K2 & V1 ≛[h, o] V2.
-/2 width=3 by lfxs_inv_lref_pair_sn/ qed-.
-
-lemma lfdeq_inv_lref_pair_dx: ∀h,o,L1,L2,i. L1 ≛[h, o, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ≛[h, o, V1] K2 & V1 ≛[h, o] V2.
-/2 width=3 by lfxs_inv_lref_pair_dx/ 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/static/lfxs_fqup.ma".
-include "basic_2/static/lfdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-
-(* Advanced properties ******************************************************)
-
-lemma lfdeq_refl: ∀h,o,T. reflexive … (lfdeq h o T).
-/2 width=1 by lfxs_refl/ qed.
-
-lemma lfdeq_pair_refl: ∀h,o,V1,V2. V1 ≛[h, o] V2 →
- ∀I,L. ∀T:term. L.ⓑ{I}V1 ≛[h, o, T] L.ⓑ{I}V2.
-/2 width=1 by lfxs_pair_refl/ qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfdeq_inv_bind_void: ∀h,o,p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
- L1 ≛[h, o, V] L2 ∧ L1.ⓧ ≛[h, o, T] L2.ⓧ.
-/2 width=3 by lfxs_inv_bind_void/ qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lfdeq_fwd_bind_dx_void: ∀h,o,p,I,L1,L2,V,T.
- L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓧ ≛[h, o, T] L2.ⓧ.
-/2 width=4 by lfxs_fwd_bind_dx_void/ 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/s_computation/fqus_fqup.ma".
-include "basic_2/static/lfdeq_drops.ma".
-include "basic_2/static/lfdeq_fqup.ma".
-include "basic_2/static/lfdeq_lfdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-
-(* Properties with extended structural successor for closures ***************)
-
-lemma fqu_tdeq_conf: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
- ∀U2. U1 ≛[h, o] U2 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & L2 ≛[h, o, T1] L & T1 ≛[h, o] T2.
-#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -G1 -G2 -L1 -L2 -U1 -T1
-[ #I #G #L #W #X #H >(tdeq_inv_lref1 … H) -X
- /2 width=5 by fqu_lref_O, ex3_2_intro/
-| #I #G #L #W1 #U1 #X #H
- elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #_ #H destruct
- /2 width=5 by fqu_pair_sn, ex3_2_intro/
-| #p #I #G #L #W1 #U1 #X #H
- elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #HU12 #H destruct
- /3 width=5 by lfdeq_pair_refl, fqu_bind_dx, ex3_2_intro/
-| #p #I #G #L #W1 #U1 #Hb #X #H
- elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #HU12 #H destruct
- /3 width=5 by fqu_clear, ex3_2_intro/
-| #I #G #L #W1 #U1 #X #H
- elim (tdeq_inv_pair1 … H) -H #W2 #U2 #_ #HU12 #H destruct
- /2 width=5 by fqu_flat_dx, ex3_2_intro/
-| #I #G #L #T1 #U1 #HTU1 #U2 #HU12
- elim (tdeq_inv_lifts_sn … HU12 … HTU1) -U1
- /3 width=5 by fqu_drop, ex3_2_intro/
-]
-qed-.
-
-lemma tdeq_fqu_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
- ∀U2. U2 ≛[h, o] U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
-#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H12 #U2 #HU21
-elim (fqu_tdeq_conf … o … H12 U2) -H12
-/3 width=5 by lfdeq_sym, tdeq_sym, ex3_2_intro/
-qed-.
-
-(* Basic_2A1: uses: lleq_fqu_trans *)
-lemma lfdeq_fqu_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐[b] ⦃G2, K2, U⦄ →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
-#h #o #b #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
-[ #I #G #L2 #V2 #L1 #H elim (lfdeq_inv_zero_pair_dx … H) -H
- #K1 #V1 #HV1 #HV12 #H destruct
- /3 width=7 by tdeq_lfdeq_conf, fqu_lref_O, ex3_2_intro/
-| * [ #p ] #I #G #L2 #V #T #L1 #H
- [ elim (lfdeq_inv_bind … H)
- | elim (lfdeq_inv_flat … H)
- ] -H
- /2 width=5 by fqu_pair_sn, ex3_2_intro/
-| #p #I #G #L2 #V #T #L1 #H elim (lfdeq_inv_bind … H) -H
- /2 width=5 by fqu_bind_dx, ex3_2_intro/
-| #p #I #G #L2 #V #T #Hb #L1 #H elim (lfdeq_inv_bind_void … H) -H
- /3 width=5 by fqu_clear, ex3_2_intro/
-| #I #G #L2 #V #T #L1 #H elim (lfdeq_inv_flat … H) -H
- /2 width=5 by fqu_flat_dx, ex3_2_intro/
-| #I #G #L2 #T #U #HTU #Y #HU
- elim (lfdeq_fwd_dx … HU) #L1 #V1 #H destruct
- /5 width=14 by lfdeq_inv_lifts_bi, fqu_drop, drops_refl, drops_drop, ex3_2_intro/
-]
-qed-.
-
-(* Properties with optional structural successor for closures ***************)
-
-lemma tdeq_fquq_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, T1⦄ →
- ∀U2. U2 ≛[h, o] U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐⸮[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
-#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -H
-[ #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
- /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_2A1: was just: lleq_fquq_trans *)
-lemma lfdeq_fquq_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮[b] ⦃G2, K2, U⦄ →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐⸮[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
-#h #o #b #G1 #G2 #L2 #K2 #T #U #H elim H -H
-[ #H #L1 #HL12 elim (lfdeq_fqu_trans … H … HL12) -L2 /3 width=5 by fqu_fquq, ex3_2_intro/
-| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Properties with plus-iterated structural successor for closures **********)
-
-(* Basic_2A1: was just: lleq_fqup_trans *)
-lemma lfdeq_fqup_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+[b] ⦃G2, K2, U⦄ →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐+[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
-#h #o #b #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
-[ #G2 #K2 #U #HTU #L1 #HL12 elim (lfdeq_fqu_trans … HTU … HL12) -L2
- /3 width=5 by fqu_fqup, ex3_2_intro/
-| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12
- elim (IHTU … HL12) -L2 #K0 #U0 #HTU #HU0 #HK0
- elim (lfdeq_fqu_trans … HU2 … HK0) -K #K1 #U1 #HU1 #HU12 #HK12
- elim (tdeq_fqu_trans … HU1 … HU0) -U #K3 #U3 #HU03 #HU31 #HK31
- @(ex3_2_intro … K3 U3) (**) (* full auto too slow *)
- /3 width=5 by lfdeq_trans, tdeq_lfdeq_conf, fqup_strap1, tdeq_trans/
-]
-qed-.
-
-lemma tdeq_fqup_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, T1⦄ →
- ∀U2. U2 ≛[h, o] U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐+[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
-#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -U1
-[ #G1 #L1 #U1 #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
- /3 width=5 by fqu_fqup, ex3_2_intro/
-| #G1 #G #L1 #L #U1 #U #H #_ #IH #U2 #HU21
- elim (tdeq_fqu_trans … H … HU21) -U1 #L0 #T #H1 #HTU #HL0
- lapply (tdeq_lfdeq_div … HTU … HL0) -HL0 #HL0
- elim (IH … HTU) -U #K2 #U1 #H2 #HUT1 #HKL2
- elim (lfdeq_fqup_trans … H2 … HL0) -L #K #U #H2 #HU1 #HK2
- lapply (tdeq_lfdeq_conf … HUT1 … HK2) -HK2 #HK2
- /3 width=7 by lfdeq_trans, fqup_strap2, tdeq_trans, ex3_2_intro/
-]
-qed-.
-
-(* Properties with star-iterated structural successor for closures **********)
-
-lemma tdeq_fqus_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, T1⦄ →
- ∀U2. U2 ≛[h, o] U1 →
- ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐*[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
-#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H #U2 #HU21 elim(fqus_inv_fqup … H) -H
-[ #H elim (tdeq_fqup_trans … H … HU21) -U1 /3 width=5 by fqup_fqus, ex3_2_intro/
-| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_2A1: was just: lleq_fqus_trans *)
-lemma lfdeq_fqus_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐*[b] ⦃G2, K2, U⦄ →
- ∀L1. L1 ≛[h, o, T] L2 →
- ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐*[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
-#h #o #b #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_fqup … H) -H
-[ #H elim (lfdeq_fqup_trans … H … HL12) -L2 /3 width=5 by fqup_fqus, ex3_2_intro/
-| * #HG #HL #HT destruct /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 "basic_2/relocation/lifts_tdeq.ma".
-include "basic_2/static/lfxs_length.ma".
-include "basic_2/static/lfxs_fsle.ma".
-include "basic_2/static/lfdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-
-(* Advanved properties with free variables inclusion ************************)
-
-lemma lfdeq_fsge_comp (h) (o): lfxs_fsge_compatible (cdeq h o).
-#h #o #L1 #L2 #T * #f1 #Hf1 #HL12
-lapply (frees_lfdeq_conf h o … Hf1 … HL12)
-lapply (lexs_fwd_length … HL12)
-/3 width=8 by lveq_length_eq, ex4_4_intro/ (**) (* full auto fails *)
-qed-.
-
-(* Properties with length for local environments ****************************)
-
-(* Basic_2A1: uses: lleq_sort *)
-lemma lfdeq_sort_length (h) (o): ∀L1,L2. |L1| = |L2| → ∀s. L1 ≛[h, o, ⋆s] L2.
-/2 width=1 by lfxs_sort_length/ qed.
-
-(* Basic_2A1: uses: lleq_gref *)
-lemma lfdeq_gref_length (h) (o): ∀L1,L2. |L1| = |L2| → ∀l. L1 ≛[h, o, §l] L2.
-/2 width=1 by lfxs_gref_length/ qed.
-
-lemma lfdeq_unit_length (h) (o): ∀L1,L2. |L1| = |L2| →
- ∀I. L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
-/2 width=1 by lfxs_unit_length/ qed.
-
-(* Basic_2A1: uses: lleq_lift_le lleq_lift_ge *)
-lemma lfdeq_lifts_bi (h) (o): ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ≛[h, o, T] K2 →
- ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- ∀U. ⬆*[f] T ≘ U → L1 ≛[h, o, U] L2.
-/3 width=9 by lfxs_lifts_bi, tdeq_lifts_sn/ qed-.
-
-(* Forward lemmas with length for local environments ************************)
-
-(* Basic_2A1: lleq_fwd_length *)
-lemma lfdeq_fwd_length (h) (o): ∀L1,L2. ∀T:term. L1 ≛[h, o, T] L2 → |L1| = |L2|.
-/2 width=3 by lfxs_fwd_length/ 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/syntax/ext2_ext2.ma".
-include "basic_2/syntax/tdeq_tdeq.ma".
-include "basic_2/static/lfdeq_length.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: uses: lleq_sym *)
-lemma lfdeq_sym: ∀h,o,T. symmetric … (lfdeq h o T).
-/3 width=3 by lfdeq_fsge_comp, lfxs_sym, tdeq_sym/ qed-.
-
-(* Basic_2A1: uses: lleq_dec *)
-lemma lfdeq_dec: ∀h,o,L1,L2. ∀T:term. Decidable (L1 ≛[h, o, T] L2).
-/3 width=1 by lfxs_dec, tdeq_dec/ qed-.
-
-(* Main properties **********************************************************)
-
-(* Basic_2A1: uses: lleq_bind lleq_bind_O *)
-theorem lfdeq_bind: ∀h,o,p,I,L1,L2,V1,V2,T.
- L1 ≛[h, o, V1] L2 → L1.ⓑ{I}V1 ≛[h, o, T] L2.ⓑ{I}V2 →
- L1 ≛[h, o, ⓑ{p,I}V1.T] L2.
-/2 width=2 by lfxs_bind/ qed.
-
-(* Basic_2A1: uses: lleq_flat *)
-theorem lfdeq_flat: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, V] L2 → L1 ≛[h, o, T] L2 →
- L1 ≛[h, o, ⓕ{I}V.T] L2.
-/2 width=1 by lfxs_flat/ qed.
-
-theorem lfdeq_bind_void: ∀h,o,p,I,L1,L2,V,T.
- L1 ≛[h, o, V] L2 → L1.ⓧ ≛[h, o, T] L2.ⓧ →
- L1 ≛[h, o, ⓑ{p,I}V.T] L2.
-/2 width=1 by lfxs_bind_void/ qed.
-
-(* Basic_2A1: uses: lleq_trans *)
-theorem lfdeq_trans: ∀h,o,T. Transitive … (lfdeq h o T).
-#h #o #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
-lapply (frees_tdeq_conf_lfdeq … Hf1 T … HL1) // #H0
-lapply (frees_mono … Hf2 … H0) -Hf2 -H0
-/5 width=7 by lexs_trans, lexs_eq_repl_back, tdeq_trans, ext2_trans, ex2_intro/
-qed-.
-
-(* Basic_2A1: uses: lleq_canc_sn *)
-theorem lfdeq_canc_sn: ∀h,o,T. left_cancellable … (lfdeq h o T).
-/3 width=3 by lfdeq_trans, lfdeq_sym/ qed-.
-
-(* Basic_2A1: uses: lleq_canc_dx *)
-theorem lfdeq_canc_dx: ∀h,o,T. right_cancellable … (lfdeq h o T).
-/3 width=3 by lfdeq_trans, lfdeq_sym/ qed-.
-
-theorem lfdeq_repl: ∀h,o,L1,L2. ∀T:term. L1 ≛[h, o, T] L2 →
- ∀K1. L1 ≛[h, o, T] K1 → ∀K2. L2 ≛[h, o, T] K2 → K1 ≛[h, o, T] K2.
-/3 width=3 by lfdeq_canc_sn, lfdeq_trans/ qed-.
-
-(* Negated properties *******************************************************)
-
-(* Note: auto works with /4 width=8/ so lfdeq_canc_sn is preferred **********)
-(* Basic_2A1: uses: lleq_nlleq_trans *)
-lemma lfdeq_lfdneq_trans: ∀h,o.∀T:term.∀L1,L. L1 ≛[h, o, T] L →
- ∀L2. (L ≛[h, o, T] L2 → ⊥) → (L1 ≛[h, o, T] L2 → ⊥).
-/3 width=3 by lfdeq_canc_sn/ qed-.
-
-(* Basic_2A1: uses: nlleq_lleq_div *)
-lemma lfdneq_lfdeq_div: ∀h,o.∀T:term.∀L2,L. L2 ≛[h, o, T] L →
- ∀L1. (L1 ≛[h, o, T] L → ⊥) → (L1 ≛[h, o, T] L2 → ⊥).
-/3 width=3 by lfdeq_trans/ qed-.
-
-theorem lfdneq_lfdeq_canc_dx: ∀h,o,L1,L. ∀T:term. (L1 ≛[h, o, T] L → ⊥) →
- ∀L2. L2 ≛[h, o, T] L → L1 ≛[h, o, T] L2 → ⊥.
-/3 width=3 by lfdeq_trans/ qed-.
-
-(* Negated inversion lemmas *************************************************)
-
-(* Basic_2A1: uses: nlleq_inv_bind nlleq_inv_bind_O *)
-lemma lfdneq_inv_bind: ∀h,o,p,I,L1,L2,V,T. (L1 ≛[h, o, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ≛[h, o, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V → ⊥).
-/3 width=2 by lfnxs_inv_bind, tdeq_dec/ qed-.
-
-(* Basic_2A1: uses: nlleq_inv_flat *)
-lemma lfdneq_inv_flat: ∀h,o,I,L1,L2,V,T. (L1 ≛[h, o, ⓕ{I}V.T] L2 → ⊥) →
- (L1 ≛[h, o, V] L2 → ⊥) ∨ (L1 ≛[h, o, T] L2 → ⊥).
-/3 width=2 by lfnxs_inv_flat, tdeq_dec/ qed-.
-
-lemma lfdneq_inv_bind_void: ∀h,o,p,I,L1,L2,V,T. (L1 ≛[h, o, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ≛[h, o, V] L2 → ⊥) ∨ (L1.ⓧ ≛[h, o, T] L2.ⓧ → ⊥).
-/3 width=3 by lfnxs_inv_bind_void, tdeq_dec/ 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/static/lfeq_fsle.ma".
-include "basic_2/static/lfdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
-
-(* Properties with syntactic equivalence on referred entries ****************)
-
-lemma lfeq_lfdeq: ∀h,o,L1,L2. ∀T:term. L1 ≡[T] L2 → L1 ≛[h, o, T] L2.
-/2 width=3 by lfxs_co/ qed.
-
-lemma lfeq_lfdeq_trans: ∀h,o,L1,L. ∀T:term. L1 ≡[T] L →
- ∀L2. L ≛[h, o, T] L2 → L1 ≛[h, o, T] L2.
-/2 width=3 by lfeq_lfxs_trans/ 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/ideqsn_3.ma".
-include "basic_2/static/lfxs.ma".
-
-(* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********)
-
-(* Basic_2A1: was: lleq *)
-definition lfeq: relation3 term lenv lenv ≝
- lfxs ceq.
-
-interpretation
- "syntactic equivalence on referred entries (local environment)"
- 'IdEqSn T L1 L2 = (lfeq T L1 L2).
-
-(* Note: "lfeq_transitive R" is equivalent to "lfxs_transitive ceq R R" *)
-(* Basic_2A1: uses: lleq_transitive *)
-definition lfeq_transitive: predicate (relation3 lenv term term) ≝
- λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1] L2 → R L1 T1 T2.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lfeq_inv_bind: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 →
- ∧∧ L1 ≡[V] L2 & L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
-/2 width=2 by lfxs_inv_bind/ qed-.
-
-lemma lfeq_inv_flat: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 →
- ∧∧ L1 ≡[V] L2 & L1 ≡[T] L2.
-/2 width=2 by lfxs_inv_flat/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfeq_inv_zero_pair_sn: ∀I,L2,K1,V. K1.ⓑ{I}V ≡[#0] L2 →
- ∃∃K2. K1 ≡[V] K2 & L2 = K2.ⓑ{I}V.
-#I #L2 #K1 #V #H
-elim (lfxs_inv_zero_pair_sn … H) -H #K2 #X #HK12 #HX #H destruct
-/2 width=3 by ex2_intro/
-qed-.
-
-lemma lfeq_inv_zero_pair_dx: ∀I,L1,K2,V. L1 ≡[#0] K2.ⓑ{I}V →
- ∃∃K1. K1 ≡[V] K2 & L1 = K1.ⓑ{I}V.
-#I #L1 #K2 #V #H
-elim (lfxs_inv_zero_pair_dx … H) -H #K1 #X #HK12 #HX #H destruct
-/2 width=3 by ex2_intro/
-qed-.
-
-lemma lfeq_inv_lref_bind_sn: ∀I1,K1,L2,i. K1.ⓘ{I1} ≡[#↑i] L2 →
- ∃∃I2,K2. K1 ≡[#i] K2 & L2 = K2.ⓘ{I2}.
-/2 width=2 by lfxs_inv_lref_bind_sn/ qed-.
-
-lemma lfeq_inv_lref_bind_dx: ∀I2,K2,L1,i. L1 ≡[#↑i] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ≡[#i] K2 & L1 = K1.ⓘ{I1}.
-/2 width=2 by lfxs_inv_lref_bind_dx/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-(* Basic_2A1: was: llpx_sn_lrefl *)
-(* Note: this should have been lleq_fwd_llpx_sn *)
-lemma lfeq_fwd_lfxs: ∀R. c_reflexive … R →
- ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤*[R, T] L2.
-#R #HR #L1 #L2 #T * #f #Hf #HL12
-/4 width=7 by lexs_co, cext2_co, ex2_intro/
-qed-.
-
-(* Basic_properties *********************************************************)
-
-lemma frees_lfeq_conf: ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f →
- ∀L2. L1 ≡[T] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f.
-#f #L1 #T #H elim H -f -L1 -T
-[ /2 width=3 by frees_sort/
-| #f #i #Hf #L2 #H2
- >(lfxs_inv_atom_sn … H2) -L2
- /2 width=1 by frees_atom/
-| #f #I #L1 #V1 #_ #IH #Y #H2
- elim (lfeq_inv_zero_pair_sn … H2) -H2 #L2 #HL12 #H destruct
- /3 width=1 by frees_pair/
-| #f #I #L1 #Hf #Y #H2
- elim (lfxs_inv_zero_unit_sn … H2) -H2 #g #L2 #_ #_ #H destruct
- /2 width=1 by frees_unit/
-| #f #I #L1 #i #_ #IH #Y #H2
- elim (lfeq_inv_lref_bind_sn … H2) -H2 #J #L2 #HL12 #H destruct
- /3 width=1 by frees_lref/
-| /2 width=1 by frees_gref/
-| #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #L2 #H2
- elim (lfeq_inv_bind … H2) -H2 /3 width=5 by frees_bind/
-| #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #L2 #H2
- elim (lfeq_inv_flat … H2) -H2 /3 width=5 by frees_flat/
-]
-qed-.
-
-(* Basic_2A1: removed theorems 10:
- lleq_ind lleq_fwd_lref
- lleq_fwd_drop_sn lleq_fwd_drop_dx
- lleq_skip lleq_lref lleq_free
- lleq_Y lleq_ge_up lleq_ge
-
-*)
+++ /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/static/lfxs_fqup.ma".
-include "basic_2/static/lfeq.ma".
-
-(* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: was: lleq_refl *)
-lemma lfeq_refl: ∀T. reflexive … (lfeq T).
-/2 width=1 by lfxs_refl/ 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/static/lfxs_length.ma".
-include "basic_2/static/lfxs_fsle.ma".
-include "basic_2/static/lfeq.ma".
-
-(* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********)
-
-(* Properties with free variables inclusion for restricted closures *********)
-
-lemma lfeq_fsle_comp: lfxs_fsle_compatible ceq.
-#L1 #L2 #T #HL12
-elim (frees_total L1 T)
-/4 width=8 by frees_lfeq_conf, lfxs_fwd_length, lveq_length_eq, sle_refl, ex4_4_intro/
-qed.
-
-(* Forward lemmas with free variables inclusion for restricted closures *****)
-
-lemma lfeq_lfxs_trans: ∀R. lfeq_transitive R →
- ∀L1,L,T. L1 ≡[T] L → ∀L2. L ⪤*[R, T] L2 → L1 ⪤*[R, T] L2.
-/4 width=16 by lfeq_fsle_comp, lfxs_trans_fsle, lfxs_trans_next/ 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/relocation/rtmap_id.ma".
-include "basic_2/notation/relations/relationstar_4.ma".
-include "basic_2/syntax/cext2.ma".
-include "basic_2/relocation/lexs.ma".
-include "basic_2/static/frees.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-definition lfxs (R) (T): relation lenv ≝
- λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≘ f & L1 ⪤*[cext2 R, cfull, f] L2.
-
-interpretation "generic extension on referred entries (local environment)"
- 'RelationStar R T L1 L2 = (lfxs R T L1 L2).
-
-definition R_confluent2_lfxs: relation4 (relation3 lenv term term)
- (relation3 lenv term term) … ≝
- λR1,R2,RP1,RP2.
- ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. L0 ⪤*[RP1, T0] L1 → ∀L2. L0 ⪤*[RP2, T0] L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-definition lfxs_confluent: relation … ≝
- λR1,R2.
- ∀K1,K,V1. K1 ⪤*[R1, V1] K → ∀V. R1 K1 V1 V →
- ∀K2. K ⪤*[R2, V] K2 → K ⪤*[R2, V1] K2.
-
-definition lfxs_transitive: relation3 ? (relation3 ?? term) … ≝
- λR1,R2,R3.
- ∀K1,K,V1. K1 ⪤*[R1, V1] K →
- ∀V. R1 K1 V1 V → ∀V2. R2 K V V2 → R3 K1 V1 V2.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lfxs_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤*[R, T] Y2 → Y2 = ⋆.
-#R #Y2 #T * /2 width=4 by lexs_inv_atom1/
-qed-.
-
-lemma lfxs_inv_atom_dx (R): ∀Y1,T. Y1 ⪤*[R, T] ⋆ → Y1 = ⋆.
-#R #I #Y1 * /2 width=4 by lexs_inv_atom2/
-qed-.
-
-lemma lfxs_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤*[R, ⋆s] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤*[R, ⋆s] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-#R * [ | #Y1 #I1 ] #Y2 #s * #f #H1 #H2
-[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
-| lapply (frees_inv_sort … H1) -H1 #Hf
- elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
- elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct
- /5 width=7 by frees_sort, ex3_4_intro, ex2_intro, or_intror/
-]
-qed-.
-
-lemma lfxs_inv_zero (R): ∀Y1,Y2. Y1 ⪤*[R, #0] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
- | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤*[cext2 R, cfull, f] L2 &
- Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
-#R * [ | #Y1 * #I1 [ | #X ] ] #Y2 * #f #H1 #H2
-[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/
-| elim (frees_inv_unit … H1) -H1 #g #HX #H destruct
- elim (lexs_inv_next1 … H2) -H2 #I2 #L2 #HL12 #H #H2 destruct
- >(ext2_inv_unit_sn … H) -H /3 width=8 by or3_intro2, ex4_4_intro/
-| elim (frees_inv_pair … H1) -H1 #g #Hg #H destruct
- elim (lexs_inv_next1 … H2) -H2 #Z2 #L2 #HL12 #H
- elim (ext2_inv_pair_sn … H) -H
- /4 width=9 by or3_intro1, ex4_5_intro, ex2_intro/
-]
-qed-.
-
-lemma lfxs_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤*[R, #↑i] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤*[R, #i] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-#R * [ | #Y1 #I1 ] #Y2 #i * #f #H1 #H2
-[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
-| elim (frees_inv_lref … H1) -H1 #g #Hg #H destruct
- elim (lexs_inv_push1 … H2) -H2
- /4 width=7 by ex3_4_intro, ex2_intro, or_intror/
-]
-qed-.
-
-lemma lfxs_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤*[R, §l] Y2 →
- ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
- | ∃∃I1,I2,L1,L2. L1 ⪤*[R, §l] L2 &
- Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
-#R * [ | #Y1 #I1 ] #Y2 #l * #f #H1 #H2
-[ lapply (lexs_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
-| lapply (frees_inv_gref … H1) -H1 #Hf
- elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
- elim (lexs_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct
- /5 width=7 by frees_gref, ex3_4_intro, ex2_intro, or_intror/
-]
-qed-.
-
-(* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *)
-lemma lfxs_inv_bind (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤*[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
- ∧∧ L1 ⪤*[R, V1] L2 & L1.ⓑ{I}V1 ⪤*[R, T] L2.ⓑ{I}V2.
-#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
-/6 width=6 by sle_lexs_trans, lexs_inv_tl, ext2_pair, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
-qed-.
-
-(* Basic_2A1: uses: llpx_sn_inv_flat *)
-lemma lfxs_inv_flat (R): ∀I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 →
- ∧∧ L1 ⪤*[R, V] L2 & L1 ⪤*[R, T] L2.
-#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
-/5 width=6 by sle_lexs_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfxs_inv_sort_bind_sn (R): ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤*[R, ⋆s] L2 →
- ∃∃I2,K2. K1 ⪤*[R, ⋆s] K2 & L2 = K2.ⓘ{I2}.
-#R #I1 #K1 #L2 #s #H elim (lfxs_inv_sort … H) -H *
-[ #H destruct
-| #Z1 #I2 #Y1 #K2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_sort_bind_dx (R): ∀I2,K2,L1,s. L1 ⪤*[R, ⋆s] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⪤*[R, ⋆s] K2 & L1 = K1.ⓘ{I1}.
-#R #I2 #K2 #L1 #s #H elim (lfxs_inv_sort … H) -H *
-[ #_ #H destruct
-| #I1 #Z2 #K1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤*[R, #0] L2 →
- ∃∃K2,V2. K1 ⪤*[R, V1] K2 & R K1 V1 V2 &
- L2 = K2.ⓑ{I}V2.
-#R #I #L2 #K1 #V1 #H elim (lfxs_inv_zero … H) -H *
-[ #H destruct
-| #Z #Y1 #K2 #X1 #V2 #HK12 #HV12 #H1 #H2 destruct
- /2 width=5 by ex3_2_intro/
-| #f #Z #Y1 #Y2 #_ #_ #H destruct
-]
-qed-.
-
-lemma lfxs_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤*[R, #0] K2.ⓑ{I}V2 →
- ∃∃K1,V1. K1 ⪤*[R, V1] K2 & R K1 V1 V2 &
- L1 = K1.ⓑ{I}V1.
-#R #I #L1 #K2 #V2 #H elim (lfxs_inv_zero … H) -H *
-[ #_ #H destruct
-| #Z #K1 #Y2 #V1 #X2 #HK12 #HV12 #H1 #H2 destruct
- /2 width=5 by ex3_2_intro/
-| #f #Z #Y1 #Y2 #_ #_ #_ #H destruct
-]
-qed-.
-
-lemma lfxs_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤*[R, #0] L2 →
- ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤*[cext2 R, cfull, f] K2 &
- L2 = K2.ⓤ{I}.
-#R #I #K1 #L2 #H elim (lfxs_inv_zero … H) -H *
-[ #H destruct
-| #Z #Y1 #Y2 #X1 #X2 #_ #_ #H destruct
-| #f #Z #Y1 #K2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤*[R, #0] K2.ⓤ{I} →
- ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤*[cext2 R, cfull, f] K2 &
- L1 = K1.ⓤ{I}.
-#R #I #L1 #K2 #H elim (lfxs_inv_zero … H) -H *
-[ #_ #H destruct
-| #Z #Y1 #Y2 #X1 #X2 #_ #_ #_ #H destruct
-| #f #Z #K1 #Y2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_lref_bind_sn (R): ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤*[R, #↑i] L2 →
- ∃∃I2,K2. K1 ⪤*[R, #i] K2 & L2 = K2.ⓘ{I2}.
-#R #I1 #K1 #L2 #i #H elim (lfxs_inv_lref … H) -H *
-[ #H destruct
-| #Z1 #I2 #Y1 #K2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_lref_bind_dx (R): ∀I2,K2,L1,i. L1 ⪤*[R, #↑i] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⪤*[R, #i] K2 & L1 = K1.ⓘ{I1}.
-#R #I2 #K2 #L1 #i #H elim (lfxs_inv_lref … H) -H *
-[ #_ #H destruct
-| #I1 #Z2 #K1 #Y2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_gref_bind_sn (R): ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤*[R, §l] L2 →
- ∃∃I2,K2. K1 ⪤*[R, §l] K2 & L2 = K2.ⓘ{I2}.
-#R #I1 #K1 #L2 #l #H elim (lfxs_inv_gref … H) -H *
-[ #H destruct
-| #Z1 #I2 #Y1 #K2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_gref_bind_dx (R): ∀I2,K2,L1,l. L1 ⪤*[R, §l] K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⪤*[R, §l] K2 & L1 = K1.ⓘ{I1}.
-#R #I2 #K2 #L1 #l #H elim (lfxs_inv_gref … H) -H *
-[ #_ #H destruct
-| #I1 #Z2 #K1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lfxs_fwd_zero_pair (R): ∀I,K1,K2,V1,V2.
- K1.ⓑ{I}V1 ⪤*[R, #0] K2.ⓑ{I}V2 → K1 ⪤*[R, V1] K2.
-#R #I #K1 #K2 #V1 #V2 #H
-elim (lfxs_inv_zero_pair_sn … H) -H #Y #X #HK12 #_ #H destruct //
-qed-.
-
-(* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *)
-lemma lfxs_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤*[R, ②{I}V.T] L2 → L1 ⪤*[R, V] L2.
-#R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL
-[ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf
-/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
-qed-.
-
-(* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *)
-lemma lfxs_fwd_bind_dx (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤*[R, ⓑ{p,I}V1.T] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⪤*[R, T] L2.ⓑ{I}V2.
-#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (lfxs_inv_bind … H HV) -H -HV //
-qed-.
-
-(* Basic_2A1: uses: llpx_sn_fwd_flat_dx *)
-lemma lfxs_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤*[R, ⓕ{I}V.T] L2 → L1 ⪤*[R, T] L2.
-#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
-qed-.
-
-lemma lfxs_fwd_dx (R): ∀I2,L1,K2,T. L1 ⪤*[R, T] K2.ⓘ{I2} →
- ∃∃I1,K1. L1 = K1.ⓘ{I1}.
-#R #I2 #L1 #K2 #T * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
-[ elim (lexs_inv_push2 … Hf) | elim (lexs_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct
-/2 width=3 by ex1_2_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lfxs_atom (R): ∀I. ⋆ ⪤*[R, ⓪{I}] ⋆.
-#R * /3 width=3 by frees_sort, frees_atom, frees_gref, lexs_atom, ex2_intro/
-qed.
-
-lemma lfxs_sort (R): ∀I1,I2,L1,L2,s.
- L1 ⪤*[R, ⋆s] L2 → L1.ⓘ{I1} ⪤*[R, ⋆s] L2.ⓘ{I2}.
-#R #I1 #I2 #L1 #L2 #s * #f #Hf #H12
-lapply (frees_inv_sort … Hf) -Hf
-/4 width=3 by frees_sort, lexs_push, isid_push, ex2_intro/
-qed.
-
-lemma lfxs_pair (R): ∀I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 →
- R L1 V1 V2 → L1.ⓑ{I}V1 ⪤*[R, #0] L2.ⓑ{I}V2.
-#R #I1 #I2 #L1 #L2 #V1 *
-/4 width=3 by ext2_pair, frees_pair, lexs_next, ex2_intro/
-qed.
-
-lemma lfxs_unit (R): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤*[cext2 R, cfull, f] L2 →
- L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
-/4 width=3 by frees_unit, lexs_next, ext2_unit, ex2_intro/ qed.
-
-lemma lfxs_lref (R): ∀I1,I2,L1,L2,i.
- L1 ⪤*[R, #i] L2 → L1.ⓘ{I1} ⪤*[R, #↑i] L2.ⓘ{I2}.
-#R #I1 #I2 #L1 #L2 #i * /3 width=3 by lexs_push, frees_lref, ex2_intro/
-qed.
-
-lemma lfxs_gref (R): ∀I1,I2,L1,L2,l.
- L1 ⪤*[R, §l] L2 → L1.ⓘ{I1} ⪤*[R, §l] L2.ⓘ{I2}.
-#R #I1 #I2 #L1 #L2 #l * #f #Hf #H12
-lapply (frees_inv_gref … Hf) -Hf
-/4 width=3 by frees_gref, lexs_push, isid_push, ex2_intro/
-qed.
-
-lemma lfxs_bind_repl_dx (R): ∀I,I1,L1,L2,T.
- L1.ⓘ{I} ⪤*[R, T] L2.ⓘ{I1} →
- ∀I2. cext2 R L1 I I2 →
- L1.ⓘ{I} ⪤*[R, T] L2.ⓘ{I2}.
-#R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR
-/3 width=5 by lexs_pair_repl, ex2_intro/
-qed-.
-
-(* Basic_2A1: uses: llpx_sn_co *)
-lemma lfxs_co (R1) (R2): (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- ∀L1,L2,T. L1 ⪤*[R1, T] L2 → L1 ⪤*[R2, T] L2.
-#R1 #R2 #HR #L1 #L2 #T * /5 width=7 by lexs_co, cext2_co, ex2_intro/
-qed-.
-
-lemma lfxs_isid (R1) (R2): ∀L1,L2,T1,T2.
- (∀f. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → 𝐈⦃f⦄) →
- (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≘ f) →
- L1 ⪤*[R1, T1] L2 → L1 ⪤*[R2, T2] L2.
-#R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 *
-/4 width=7 by lexs_co_isid, ex2_intro/
-qed-.
-
-lemma lfxs_unit_sn (R1) (R2):
- ∀I,K1,L2. K1.ⓤ{I} ⪤*[R1, #0] L2 → K1.ⓤ{I} ⪤*[R2, #0] L2.
-#R1 #R2 #I #K1 #L2 #H
-elim (lfxs_inv_zero_unit_sn … H) -H #f #K2 #Hf #HK12 #H destruct
-/3 width=7 by lfxs_unit, lexs_co_isid/
-qed-.
-
-(* Basic_2A1: removed theorems 9:
- llpx_sn_skip llpx_sn_lref llpx_sn_free
- llpx_sn_fwd_lref
- llpx_sn_Y llpx_sn_ge_up llpx_sn_ge
- llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_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/relocation/drops_cext2.ma".
-include "basic_2/relocation/drops_lexs.ma".
-include "basic_2/static/frees_drops.ma".
-include "basic_2/static/lfxs.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-definition f_dedropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
- ∀K2,T. K1 ⪤*[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
- ∃∃L2. L1 ⪤*[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
-
-definition f_dropable_sn: predicate (relation3 lenv term term) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
- ∀L2,U. L1 ⪤*[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
- ∃∃K2. K1 ⪤*[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
-
-definition f_dropable_dx: predicate (relation3 lenv term term) ≝
- λR. ∀L1,L2,U. L1 ⪤*[R, U] L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤*[R, T] K2.
-
-definition f_transitive_next: relation3 … ≝ λR1,R2,R3.
- ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
- ∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f →
- lexs_transitive (cext2 R1) (cext2 R2) (cext2 R3) (cext2 R1) cfull g K I.
-
-(* Properties with generic slicing for local environments *******************)
-
-lemma lfxs_liftable_dedropable_sn: ∀R. (∀L. reflexive ? (R L)) →
- d_liftable2_sn … lifts R → f_dedropable_sn R.
-#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU
-elim (frees_total L1 U) #f2 #Hf2
-lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
-elim (lexs_liftable_co_dedropable_sn … HLK1 … HK12 … Hf) -f1 -K1
-/3 width=6 by cext2_d_liftable2_sn, cfull_lift_sn, ext2_refl, ex3_intro, ex2_intro/
-qed-.
-
-lemma lfxs_trans_next: ∀R1,R2,R3. lfxs_transitive R1 R2 R3 → f_transitive_next R1 R2 R3.
-#R1 #R2 #R3 #HR #f #L1 #T #Hf #g #I1 #K1 #n #HLK #Hgf #I #H
-generalize in match HLK; -HLK elim H -I1 -I
-[ #I #_ #L2 #_ #I2 #H
- lapply (ext2_inv_unit_sn … H) -H #H destruct
- /2 width=1 by ext2_unit/
-| #I #V1 #V #HV1 #HLK1 #L2 #HL12 #I2 #H
- elim (ext2_inv_pair_sn … H) -H #V2 #HV2 #H destruct
- elim (frees_inv_drops_next … Hf … HLK1 … Hgf) -f -HLK1 #f #Hf #Hfg
- /5 width=5 by ext2_pair, sle_lexs_trans, ex2_intro/
-]
-qed.
-
-(* Inversion lemmas with generic slicing for local environments *************)
-
-(* Basic_2A1: uses: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *)
-(* Basic_2A1: was: llpx_sn_drop_conf_O *)
-lemma lfxs_dropable_sn: ∀R. f_dropable_sn R.
-#R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU
-elim (frees_total K1 T) #f1 #Hf1
-lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f
-elim (lexs_co_dropable_sn … HLK1 … HL12 … H2f) -f2 -L1
-/3 width=3 by ex2_intro/
-qed-.
-
-(* Basic_2A1: was: llpx_sn_drop_trans_O *)
-(* Note: the proof might be simplified *)
-lemma lfxs_dropable_dx: ∀R. f_dropable_dx R.
-#R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU
-elim (drops_isuni_ex … H1f L1) #K1 #HLK1
-elim (frees_total K1 T) #f1 #Hf1
-lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -K1 #H2f
-elim (lexs_co_dropable_dx … HL12 … HLK2 … H2f) -L2
-/4 width=9 by frees_inv_lifts, ex2_intro/
-qed-.
-
-(* Basic_2A1: uses: llpx_sn_inv_lift_O *)
-lemma lfxs_inv_lifts_bi: ∀R,L1,L2,U. L1 ⪤*[R, U] L2 → ∀b,f. 𝐔⦃f⦄ →
- ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- ∀T. ⬆*[f] T ≘ U → K1 ⪤*[R, T] K2.
-#R #L1 #L2 #U #HL12 #b #f #Hf #K1 #K2 #HLK1 #HLK2 #T #HTU
-elim (lfxs_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY
-lapply (drops_mono … HY … HLK2) -b -f -L2 #H destruct //
-qed-.
-
-lemma lfxs_inv_lref_pair_sn: ∀R,L1,L2,i. L1 ⪤*[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
- ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤*[R, V1] K2 & R K1 V1 V2.
-#R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
-#Y #HY #HLK2 elim (lfxs_inv_zero_pair_sn … HY) -HY
-#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma lfxs_inv_lref_pair_dx: ∀R,L1,L2,i. L1 ⪤*[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
- ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤*[R, V1] K2 & R K1 V1 V2.
-#R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
-#Y #HLK1 #HY elim (lfxs_inv_zero_pair_dx … HY) -HY
-#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma lfxs_inv_lref_unit_sn: ∀R,L1,L2,i. L1 ⪤*[R, #i] L2 → ∀I,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} →
- ∃∃f,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} & K1 ⪤*[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
-#R #L1 #L2 #i #HL12 #I #K1 #HLK1 elim (lfxs_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
-#Y #HY #HLK2 elim (lfxs_inv_zero_unit_sn … HY) -HY
-#f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/
-qed-.
-
-lemma lfxs_inv_lref_unit_dx: ∀R,L1,L2,i. L1 ⪤*[R, #i] L2 → ∀I,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} →
- ∃∃f,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} & K1 ⪤*[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
-#R #L1 #L2 #i #HL12 #I #K2 #HLK2 elim (lfxs_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
-#Y #HLK1 #HY elim (lfxs_inv_zero_unit_dx … HY) -HY
-#f #K2 #Hf #HK12 #H destruct /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 "basic_2/static/frees_fqup.ma".
-include "basic_2/static/lfxs.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: uses: llpx_sn_refl *)
-lemma lfxs_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⪤*[R, T] L.
-#R #HR #L #T elim (frees_total L T)
-/4 width=3 by lexs_refl, ext2_refl, ex2_intro/
-qed.
-
-lemma lfxs_pair_refl: ∀R. (∀L. reflexive … (R L)) →
- ∀L,V1,V2. R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤*[R, T] L.ⓑ{I}V2.
-#R #HR #L #V1 #V2 #HV12 #I #T
-elim (frees_total (L.ⓑ{I}V1) T) #f #Hf
-elim (pn_split f) * #g #H destruct
-/5 width=3 by lexs_refl, lexs_next, lexs_push, ext2_refl, ext2_pair, ex2_intro/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfxs_inv_bind_void: ∀R,p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- L1 ⪤*[R, V] L2 ∧ L1.ⓧ ⪤*[R, T] L2.ⓧ.
-#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind_void … Hf) -Hf
-/6 width=6 by sle_lexs_trans, lexs_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lfxs_fwd_bind_dx_void: ∀R,p,I,L1,L2,V,T. L1 ⪤*[R, ⓑ{p,I}V.T] L2 →
- L1.ⓧ ⪤*[R, T] L2.ⓧ.
-#R #p #I #L1 #L2 #V #T #H elim (lfxs_inv_bind_void … 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 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lexs_length.ma".
-include "basic_2/static/fsle_fsle.ma".
-include "basic_2/static/lfxs_drops.ma".
-include "basic_2/static/lfxs_lfxs.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-definition R_fsge_compatible: predicate (relation3 …) ≝ λRN.
- ∀L,T1,T2. RN L T1 T2 → ⦃L, T2⦄ ⊆ ⦃L, T1⦄.
-
-definition lfxs_fsge_compatible: predicate (relation3 …) ≝ λRN.
- ∀L1,L2,T. L1 ⪤*[RN, T] L2 → ⦃L2, T⦄ ⊆ ⦃L1, T⦄.
-
-definition lfxs_fsle_compatible: predicate (relation3 …) ≝ λRN.
- ∀L1,L2,T. L1 ⪤*[RN, T] L2 → ⦃L1, T⦄ ⊆ ⦃L2, T⦄.
-
-(* Basic inversions with free variables inclusion for restricted closures ***)
-
-lemma frees_lexs_conf: ∀R. lfxs_fsge_compatible R →
- ∀L1,T,f1. L1 ⊢ 𝐅*⦃T⦄ ≘ f1 →
- ∀L2. L1 ⪤*[cext2 R, cfull, f1] L2 →
- ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 & f2 ⊆ f1.
-#R #HR #L1 #T #f1 #Hf1 #L2 #H1L
-lapply (HR L1 L2 T ?) /2 width=3 by ex2_intro/ #H2L
-@(fsle_frees_trans_eq … H2L … Hf1) /3 width=4 by lexs_fwd_length, sym_eq/
-qed-.
-
-(* Properties with free variables inclusion for restricted closures *********)
-
-(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
-lemma fsge_lfxs_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
- ∀L2. L1 ⪤*[R, T2] L2 → L1 ⪤*[R, T1] L2.
-#R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12
-elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct
-/4 width=5 by lfxs_inv_frees, sle_lexs_trans, ex2_intro/
-qed-.
-
-lemma lfxs_sym: ∀R. lfxs_fsge_compatible R →
- (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀T. symmetric … (lfxs R T).
-#R #H1R #H2R #T #L1 #L2
-* #f1 #Hf1 #HL12
-elim (frees_lexs_conf … Hf1 … HL12) -Hf1 //
-/5 width=5 by sle_lexs_trans, lexs_sym, cext2_sym, ex2_intro/
-qed-.
-
-lemma lfxs_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lfxs_fsge_compatible R1 →
- ∀L1,L2,V. L1 ⪤*[R1, V] L2 → ∀I,T.
- ∃∃L. L1 ⪤*[R1, ②{I}V.T] L & L ⪤*[R2, V] L2.
-#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T
-[ elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
- elim (frees_inv_bind … Hg) #y1 #y2 #H #_ #Hy
-| elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg
- elim (frees_inv_flat … Hg) #y1 #y2 #H #_ #Hy
-]
-lapply(frees_mono … H … Hf) -H #H1
-lapply (sor_eq_repl_back1 … Hy … H1) -y1 #Hy
-lapply (sor_inv_sle_sn … Hy) -y2 #Hfg
-elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #L #HL1 #HL2
-lapply (sle_lexs_trans … HL1 … Hfg) // #H
-elim (frees_lexs_conf … Hf … H) -Hf -H
-/4 width=7 by sle_lexs_trans, ex2_intro/
-qed-.
-
-lemma lfxs_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lfxs_fsge_compatible R1 →
- ∀L1,L2,T. L1 ⪤*[R1, T] L2 → ∀I,V.
- ∃∃L. L1 ⪤*[R1, ⓕ{I}V.T] L & L ⪤*[R2, T] L2.
-#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V
-elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg
-elim (frees_inv_flat … Hg) #y1 #y2 #_ #H #Hy
-lapply(frees_mono … H … Hf) -H #H2
-lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy
-lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
-elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #L #HL1 #HL2
-lapply (sle_lexs_trans … HL1 … Hfg) // #H
-elim (frees_lexs_conf … Hf … H) -Hf -H
-/4 width=7 by sle_lexs_trans, ex2_intro/
-qed-.
-
-lemma lfxs_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lfxs_fsge_compatible R1 →
- ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤*[R1, T] L2 → ∀p.
- ∃∃L,V. L1 ⪤*[R1, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤*[R2, T] L2 & R1 L1 V1 V.
-#R1 #R2 #HR1 #HR2 #HR #I #L1 #L2 #V1 #T * #f #Hf #HL12 #p
-elim (frees_total L1 (ⓑ{p,I}V1.T)) #g #Hg
-elim (frees_inv_bind … Hg) #y1 #y2 #_ #H #Hy
-lapply(frees_mono … H … Hf) -H #H2
-lapply (tl_eq_repl … H2) -H2 #H2
-lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy
-lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
-lapply (sle_inv_tl_sn … Hfg) -Hfg #Hfg
-elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #Y #H #HL2
-lapply (sle_lexs_trans … H … Hfg) // #H0
-elim (lexs_inv_next1 … H) -H #Z #L #HL1 #H
-elim (ext2_inv_pair_sn … H) -H #V #HV #H1 #H2 destruct
-elim (frees_lexs_conf … Hf … H0) -Hf -H0
-/4 width=7 by sle_lexs_trans, ex3_2_intro, ex2_intro/
-qed-.
-
-lemma lfxs_bind_dx_split_void: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
- lfxs_fsge_compatible R1 →
- ∀L1,L2,T. L1.ⓧ ⪤*[R1, T] L2 → ∀p,I,V.
- ∃∃L. L1 ⪤*[R1, ⓑ{p,I}V.T] L & L.ⓧ ⪤*[R2, T] L2.
-#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #p #I #V
-elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
-elim (frees_inv_bind_void … Hg) #y1 #y2 #_ #H #Hy
-lapply(frees_mono … H … Hf) -H #H2
-lapply (tl_eq_repl … H2) -H2 #H2
-lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy
-lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
-lapply (sle_inv_tl_sn … Hfg) -Hfg #Hfg
-elim (lexs_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #Y #H #HL2
-lapply (sle_lexs_trans … H … Hfg) // #H0
-elim (lexs_inv_next1 … H) -H #Z #L #HL1 #H
-elim (ext2_inv_unit_sn … H) -H #H destruct
-elim (frees_lexs_conf … Hf … H0) -Hf -H0
-/4 width=7 by sle_lexs_trans, ex2_intro/ (* note: 2 ex2_intro *)
-qed-.
-
-(* Main properties with free variables inclusion for restricted closures ****)
-
-theorem lfxs_conf: ∀R1,R2.
- lfxs_fsge_compatible R1 →
- lfxs_fsge_compatible R2 →
- R_confluent2_lfxs R1 R2 R1 R2 →
- ∀T. confluent2 … (lfxs R1 T) (lfxs R2 T).
-#R1 #R2 #HR1 #HR2 #HR12 #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02
-lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12
-lapply (lexs_eq_repl_back … HL01 … Hf12) -f1 #HL01
-elim (lexs_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ]
-[ #L #HL1 #HL2
- elim (frees_lexs_conf … Hf … HL01) // -HR1 -HL01 #f1 #Hf1 #H1
- elim (frees_lexs_conf … Hf … HL02) // -HR2 -HL02 #f2 #Hf2 #H2
- lapply (sle_lexs_trans … HL1 … H1) // -HL1 -H1 #HL1
- lapply (sle_lexs_trans … HL2 … H2) // -HL2 -H2 #HL2
- /3 width=5 by ex2_intro/
-| #g * #I0 [2: #V0 ] #K0 #n #HLK0 #Hgf #Z1 #H1 #Z2 #H2 #K1 #HK01 #K2 #HK02
- [ elim (ext2_inv_pair_sn … H1) -H1 #V1 #HV01 #H destruct
- elim (ext2_inv_pair_sn … H2) -H2 #V2 #HV02 #H destruct
- elim (frees_inv_drops_next … Hf … HLK0 … Hgf) -Hf -HLK0 -Hgf #g0 #Hg0 #H0
- lapply (sle_lexs_trans … HK01 … H0) // -HK01 #HK01
- lapply (sle_lexs_trans … HK02 … H0) // -HK02 #HK02
- elim (HR12 … HV01 … HV02 K1 … K2) /3 width=3 by ext2_pair, ex2_intro/
- | lapply (ext2_inv_unit_sn … H1) -H1 #H destruct
- lapply (ext2_inv_unit_sn … H2) -H2 #H destruct
- /3 width=3 by ext2_unit, ex2_intro/
- ]
-]
-qed-.
-
-theorem lfxs_trans_fsle: ∀R1,R2,R3.
- lfxs_fsle_compatible R1 → f_transitive_next R1 R2 R3 →
- ∀L1,L,T. L1 ⪤*[R1, T] L →
- ∀L2. L ⪤*[R2, T] L2 → L1 ⪤*[R3, T] L2.
-#R1 #R2 #R3 #H1R #H2R #L1 #L #T #H
-lapply (H1R … H) -H1R #H0
-cases H -H #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
-lapply (fsle_inv_frees_eq … H0 … Hf1 … Hf2) -H0 -Hf2
-/4 width=14 by lexs_trans_gen, lexs_fwd_length, sle_lexs_trans, 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/lexs_length.ma".
-include "basic_2/static/lfxs_drops.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-(* Forward lemmas with length for local environments ************************)
-
-(* Basic_2A1: uses: llpx_sn_fwd_length *)
-lemma lfxs_fwd_length (R): ∀L1,L2,T. L1 ⪤*[R, T] L2 → |L1| = |L2|.
-#R #L1 #L2 #T * /2 width=4 by lexs_fwd_length/
-qed-.
-
-(* Properties with length for local environments ****************************)
-
-(* Basic_2A1: uses: llpx_sn_sort *)
-lemma lfxs_sort_length (R): ∀L1,L2. |L1| = |L2| → ∀s. L1 ⪤*[R, ⋆s] L2.
-#R #L1 elim L1 -L1
-[ #Y #H #s >(length_inv_zero_sn … H) -H //
-| #K1 #I1 #IH #Y #H #s
- elim (length_inv_succ_sn … H) -H #I2 #K2 #HK12 #H destruct
- /3 width=1 by lfxs_sort/
-]
-qed.
-
-(* Basic_2A1: uses: llpx_sn_gref *)
-lemma lfxs_gref_length (R): ∀L1,L2. |L1| = |L2| → ∀l. L1 ⪤*[R, §l] L2.
-#R #L1 elim L1 -L1
-[ #Y #H #s >(length_inv_zero_sn … H) -H //
-| #K1 #I1 #IH #Y #H #s
- elim (length_inv_succ_sn … H) -H #I2 #K2 #HK12 #H destruct
- /3 width=1 by lfxs_gref/
-]
-qed.
-
-lemma lfxs_unit_length (R): ∀L1,L2. |L1| = |L2| → ∀I. L1.ⓤ{I} ⪤*[R, #0] L2.ⓤ{I}.
-/3 width=3 by lfxs_unit, lexs_length_isid/ qed.
-
-(* Basic_2A1: uses: llpx_sn_lift_le llpx_sn_lift_ge *)
-lemma lfxs_lifts_bi (R): d_liftable2_sn … lifts R →
- ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ⪤*[R, T] K2 →
- ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
- ∀U. ⬆*[f] T ≘ U → L1 ⪤*[R, U] L2.
-#R #HR #L1 #L2 #HL12 #K1 #K2 #T * #f1 #Hf1 #HK12 #b #f #HLK1 #HLK2 #U #HTU
-elim (frees_total L1 U) #f2 #Hf2
-lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
-/4 width=12 by lexs_length_cfull, lexs_liftable_co_dedropable_bi, cext2_d_liftable2_sn, cfull_lift_sn, ex2_intro/
-qed-.
-
-(* Inversion lemmas with length for local environment ***********************)
-
-lemma lfxs_inv_zero_length (R): ∀Y1,Y2. Y1 ⪤*[R, #0] Y2 →
- ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
- | ∃∃I,L1,L2,V1,V2. L1 ⪤*[R, V1] L2 & R L1 V1 V2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
- | ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
-#R #Y1 #Y2 #H elim (lfxs_inv_zero … H) -H *
-/4 width=9 by lexs_fwd_length, ex4_5_intro, ex3_3_intro, or3_intro2, or3_intro1, or3_intro0, 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/relocation/lex.ma".
-include "basic_2/static/lfxs_fsle.ma".
-include "basic_2/static/lfeq.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-(* Properties with generic extension of a context-sensitive relation ********)
-
-lemma lfxs_lex: ∀R,L1,L2. L1 ⪤[R] L2 → ∀T. L1 ⪤*[R, T] L2.
-#R #L1 #L2 * #f #Hf #HL12 #T
-elim (frees_total L1 T) #g #Hg
-/4 width=5 by lexs_sdj, sdj_isid_sn, ex2_intro/
-qed.
-
-(* Inversion lemmas with generic extension of a context sensitive relation **)
-
-lemma lfxs_inv_lex_lfeq: ∀R. c_reflexive … R →
- lfxs_fsge_compatible R →
- ∀L1,L2,T. L1 ⪤*[R, T] L2 →
- ∃∃L. L1 ⪤[R] L & L ≡[T] L2.
-#R #H1R #H2R #L1 #L2 #T * #f1 #Hf1 #HL
-elim (lexs_sdj_split … ceq_ext … HL 𝐈𝐝 ?) -HL
-[ #L0 #HL10 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ] -H1R
-lapply (lexs_sdj … HL10 f1 ?) /2 width=1 by sdj_isid_sn/ #H
-elim (frees_lexs_conf … Hf1 … H) // -H2R -H #f0 #Hf0 #Hf01
-/4 width=7 by sle_lexs_trans, (* 2x *) 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/lexs_lexs.ma".
-include "basic_2/static/frees_fqup.ma".
-include "basic_2/static/lfxs.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lfxs_inv_frees: ∀R,L1,L2,T. L1 ⪤*[R, T] L2 →
- ∀f. L1 ⊢ 𝐅*⦃T⦄ ≘ f → L1 ⪤*[cext2 R, cfull, f] L2.
-#R #L1 #L2 #T * /3 width=6 by frees_mono, lexs_eq_repl_back/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: uses: llpx_sn_dec *)
-lemma lfxs_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀L1,L2,T. Decidable (L1 ⪤*[R, T] L2).
-#R #HR #L1 #L2 #T
-elim (frees_total L1 T) #f #Hf
-elim (lexs_dec (cext2 R) cfull … L1 L2 f)
-/4 width=3 by lfxs_inv_frees, cfull_dec, ext2_dec, ex2_intro, or_intror, or_introl/
-qed-.
-
-(* Main properties **********************************************************)
-
-(* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *)
-theorem lfxs_bind: ∀R,p,I,L1,L2,V1,V2,T.
- L1 ⪤*[R, V1] L2 → L1.ⓑ{I}V1 ⪤*[R, T] L2.ⓑ{I}V2 →
- L1 ⪤*[R, ⓑ{p,I}V1.T] L2.
-#R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
-lapply (lexs_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
-/3 width=7 by frees_fwd_isfin, frees_bind, lexs_join, isfin_tl, ex2_intro/
-qed.
-
-(* Basic_2A1: llpx_sn_flat *)
-theorem lfxs_flat: ∀R,I,L1,L2,V,T.
- L1 ⪤*[R, V] L2 → L1 ⪤*[R, T] L2 →
- L1 ⪤*[R, ⓕ{I}V.T] L2.
-#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
-/3 width=7 by frees_fwd_isfin, frees_flat, lexs_join, ex2_intro/
-qed.
-
-theorem lfxs_bind_void: ∀R,p,I,L1,L2,V,T.
- L1 ⪤*[R, V] L2 → L1.ⓧ ⪤*[R, T] L2.ⓧ →
- L1 ⪤*[R, ⓑ{p,I}V.T] L2.
-#R #p #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
-lapply (lexs_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
-/3 width=7 by frees_fwd_isfin, frees_bind_void, lexs_join, isfin_tl, ex2_intro/
-qed.
-
-(* Negated inversion lemmas *************************************************)
-
-(* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *)
-lemma lfnxs_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀p,I,L1,L2,V,T. (L1 ⪤*[R, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ⪤*[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤*[R, T] L2.ⓑ{I}V → ⊥).
-#R #HR #p #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V)
-/4 width=2 by lfxs_bind, or_intror, or_introl/
-qed-.
-
-(* Basic_2A1: uses: nllpx_sn_inv_flat *)
-lemma lfnxs_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀I,L1,L2,V,T. (L1 ⪤*[R, ⓕ{I}V.T] L2 → ⊥) →
- (L1 ⪤*[R, V] L2 → ⊥) ∨ (L1 ⪤*[R, T] L2 → ⊥).
-#R #HR #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V)
-/4 width=1 by lfxs_flat, or_intror, or_introl/
-qed-.
-
-lemma lfnxs_inv_bind_void: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
- ∀p,I,L1,L2,V,T. (L1 ⪤*[R, ⓑ{p,I}V.T] L2 → ⊥) →
- (L1 ⪤*[R, V] L2 → ⊥) ∨ (L1.ⓧ ⪤*[R, T] L2.ⓧ → ⊥).
-#R #HR #p #I #L1 #L2 #V #T #H elim (lfxs_dec … HR L1 L2 V)
-/4 width=2 by lfxs_bind_void, or_intror, 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/stareqsn_5.ma".
+include "basic_2/syntax/tdeq_ext.ma".
+include "basic_2/static/rex.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+definition rdeq (h) (o): relation3 term lenv lenv ≝
+ rex (cdeq h o).
+
+interpretation
+ "degree-based equivalence on referred entries (local environment)"
+ 'StarEqSn h o T L1 L2 = (rdeq h o T L1 L2).
+
+interpretation
+ "degree-based ranged equivalence (local environment)"
+ 'StarEqSn h o f L1 L2 = (sex (cdeq_ext h o) cfull f L1 L2).
+
+(* Basic properties ***********************************************************)
+
+lemma frees_tdeq_conf_rdeq (h) (o): ∀f,L1,T1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → ∀T2. T1 ≛[h, o] T2 →
+ ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T2⦄ ≘ f.
+#h #o #f #L1 #T1 #H elim H -f -L1 -T1
+[ #f #L1 #s1 #Hf #X #H1 #L2 #_
+ elim (tdeq_inv_sort1 … H1) -H1 #s2 #d #_ #_ #H destruct
+ /2 width=3 by frees_sort/
+| #f #i #Hf #X #H1
+ >(tdeq_inv_lref1 … H1) -X #Y #H2
+ >(sex_inv_atom1 … H2) -Y
+ /2 width=1 by frees_atom/
+| #f #I #L1 #V1 #_ #IH #X #H1
+ >(tdeq_inv_lref1 … H1) -X #Y #H2
+ elim (sex_inv_next1 … H2) -H2 #Z #L2 #HL12 #HZ #H destruct
+ elim (ext2_inv_pair_sn … HZ) -HZ #V2 #HV12 #H destruct
+ /3 width=1 by frees_pair/
+| #f #I #L1 #Hf #X #H1
+ >(tdeq_inv_lref1 … H1) -X #Y #H2
+ elim (sex_inv_next1 … H2) -H2 #Z #L2 #_ #HZ #H destruct
+ >(ext2_inv_unit_sn … HZ) -Z /2 width=1 by frees_unit/
+| #f #I #L1 #i #_ #IH #X #H1
+ >(tdeq_inv_lref1 … H1) -X #Y #H2
+ elim (sex_inv_push1 … H2) -H2 #J #L2 #HL12 #_ #H destruct
+ /3 width=1 by frees_lref/
+| #f #L1 #l #Hf #X #H1 #L2 #_
+ >(tdeq_inv_gref1 … H1) -X /2 width=1 by frees_gref/
+| #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1
+ elim (tdeq_inv_pair1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct
+ /6 width=5 by frees_bind, sex_inv_tl, ext2_pair, sle_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn/
+| #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #X #H1
+ elim (tdeq_inv_pair1 … H1) -H1 #V2 #T2 #HV12 #HT12 #H1 #L2 #HL12 destruct
+ /5 width=5 by frees_flat, sle_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn/
+]
+qed-.
+
+lemma frees_tdeq_conf (h) (o): ∀f,L,T1. L ⊢ 𝐅*⦃T1⦄ ≘ f →
+ ∀T2. T1 ≛[h, o] T2 → L ⊢ 𝐅*⦃T2⦄ ≘ f.
+/4 width=7 by frees_tdeq_conf_rdeq, sex_refl, ext2_refl/ qed-.
+
+lemma frees_rdeq_conf (h) (o): ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f →
+ ∀L2. L1 ≛[h, o, f] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f.
+/2 width=7 by frees_tdeq_conf_rdeq, tdeq_refl/ qed-.
+
+lemma tdeq_rex_conf (R) (h) (o): s_r_confluent1 … (cdeq h o) (rex R).
+#R #h #o #L1 #T1 #T2 #HT12 #L2 *
+/3 width=5 by frees_tdeq_conf, ex2_intro/
+qed-.
+
+lemma tdeq_rex_div (R) (h) (o): ∀T1,T2. T1 ≛[h, o] T2 →
+ ∀L1,L2. L1 ⪤[R, T2] L2 → L1 ⪤[R, T1] L2.
+/3 width=5 by tdeq_rex_conf, tdeq_sym/ qed-.
+
+lemma tdeq_rdeq_conf (h) (o): s_r_confluent1 … (cdeq h o) (rdeq h o).
+/2 width=5 by tdeq_rex_conf/ qed-.
+
+lemma tdeq_rdeq_div (h) (o): ∀T1,T2. T1 ≛[h, o] T2 →
+ ∀L1,L2. L1 ≛[h, o, T2] L2 → L1 ≛[h, o, T1] L2.
+/2 width=5 by tdeq_rex_div/ qed-.
+
+lemma rdeq_atom (h) (o): ∀I. ⋆ ≛[h, o, ⓪{I}] ⋆.
+/2 width=1 by rex_atom/ qed.
+
+lemma rdeq_sort (h) (o): ∀I1,I2,L1,L2,s.
+ L1 ≛[h, o, ⋆s] L2 → L1.ⓘ{I1} ≛[h, o, ⋆s] L2.ⓘ{I2}.
+/2 width=1 by rex_sort/ qed.
+
+lemma rdeq_pair (h) (o): ∀I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 → V1 ≛[h, o] V2 →
+ L1.ⓑ{I}V1 ≛[h, o, #0] L2.ⓑ{I}V2.
+/2 width=1 by rex_pair/ qed.
+(*
+lemma rdeq_unit (h) (o): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cdeq_ext h o, cfull, f] L2 →
+ L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
+/2 width=3 by rex_unit/ qed.
+*)
+lemma rdeq_lref (h) (o): ∀I1,I2,L1,L2,i.
+ L1 ≛[h, o, #i] L2 → L1.ⓘ{I1} ≛[h, o, #↑i] L2.ⓘ{I2}.
+/2 width=1 by rex_lref/ qed.
+
+lemma rdeq_gref (h) (o): ∀I1,I2,L1,L2,l.
+ L1 ≛[h, o, §l] L2 → L1.ⓘ{I1} ≛[h, o, §l] L2.ⓘ{I2}.
+/2 width=1 by rex_gref/ qed.
+
+lemma rdeq_bind_repl_dx (h) (o): ∀I,I1,L1,L2.∀T:term.
+ L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I1} →
+ ∀I2. I ≛[h, o] I2 →
+ L1.ⓘ{I} ≛[h, o, T] L2.ⓘ{I2}.
+/2 width=2 by rex_bind_repl_dx/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma rdeq_inv_atom_sn (h) (o): ∀Y2. ∀T:term. ⋆ ≛[h, o, T] Y2 → Y2 = ⋆.
+/2 width=3 by rex_inv_atom_sn/ qed-.
+
+lemma rdeq_inv_atom_dx (h) (o): ∀Y1. ∀T:term. Y1 ≛[h, o, T] ⋆ → Y1 = ⋆.
+/2 width=3 by rex_inv_atom_dx/ qed-.
+(*
+lemma rdeq_inv_zero (h) (o): ∀Y1,Y2. Y1 ≛[h, o, #0] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I,L1,L2,V1,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
+ | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cdeq_ext h o, cfull, f] L2 &
+ Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
+#h #o #Y1 #Y2 #H elim (rex_inv_zero … H) -H *
+/3 width=9 by or3_intro0, or3_intro1, or3_intro2, ex4_5_intro, ex4_4_intro, conj/
+qed-.
+*)
+lemma rdeq_inv_lref (h) (o): ∀Y1,Y2,i. Y1 ≛[h, o, #↑i] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ≛[h, o, #i] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+/2 width=1 by rex_inv_lref/ qed-.
+
+(* Basic_2A1: uses: lleq_inv_bind lleq_inv_bind_O *)
+lemma rdeq_inv_bind (h) (o): ∀p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ≛[h, o, V] L2 & L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
+/2 width=2 by rex_inv_bind/ qed-.
+
+(* Basic_2A1: uses: lleq_inv_flat *)
+lemma rdeq_inv_flat (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 →
+ ∧∧ L1 ≛[h, o, V] L2 & L1 ≛[h, o, T] L2.
+/2 width=2 by rex_inv_flat/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rdeq_inv_zero_pair_sn (h) (o): ∀I,Y2,L1,V1. L1.ⓑ{I}V1 ≛[h, o, #0] Y2 →
+ ∃∃L2,V2. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y2 = L2.ⓑ{I}V2.
+/2 width=1 by rex_inv_zero_pair_sn/ qed-.
+
+lemma rdeq_inv_zero_pair_dx (h) (o): ∀I,Y1,L2,V2. Y1 ≛[h, o, #0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ≛[h, o, V1] L2 & V1 ≛[h, o] V2 & Y1 = L1.ⓑ{I}V1.
+/2 width=1 by rex_inv_zero_pair_dx/ qed-.
+
+lemma rdeq_inv_lref_bind_sn (h) (o): ∀I1,Y2,L1,i. L1.ⓘ{I1} ≛[h, o, #↑i] Y2 →
+ ∃∃I2,L2. L1 ≛[h, o, #i] L2 & Y2 = L2.ⓘ{I2}.
+/2 width=2 by rex_inv_lref_bind_sn/ qed-.
+
+lemma rdeq_inv_lref_bind_dx (h) (o): ∀I2,Y1,L2,i. Y1 ≛[h, o, #↑i] L2.ⓘ{I2} →
+ ∃∃I1,L1. L1 ≛[h, o, #i] L2 & Y1 = L1.ⓘ{I1}.
+/2 width=2 by rex_inv_lref_bind_dx/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma rdeq_fwd_zero_pair (h) (o): ∀I,K1,K2,V1,V2.
+ K1.ⓑ{I}V1 ≛[h, o, #0] K2.ⓑ{I}V2 → K1 ≛[h, o, V1] K2.
+/2 width=3 by rex_fwd_zero_pair/ qed-.
+
+(* Basic_2A1: uses: lleq_fwd_bind_sn lleq_fwd_flat_sn *)
+lemma rdeq_fwd_pair_sn (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ②{I}V.T] L2 → L1 ≛[h, o, V] L2.
+/2 width=3 by rex_fwd_pair_sn/ qed-.
+
+(* Basic_2A1: uses: lleq_fwd_bind_dx lleq_fwd_bind_O_dx *)
+lemma rdeq_fwd_bind_dx (h) (o): ∀p,I,L1,L2,V,T.
+ L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V.
+/2 width=2 by rex_fwd_bind_dx/ qed-.
+
+(* Basic_2A1: uses: lleq_fwd_flat_dx *)
+lemma rdeq_fwd_flat_dx (h) (o): ∀I,L1,L2,V,T. L1 ≛[h, o, ⓕ{I}V.T] L2 → L1 ≛[h, o, T] L2.
+/2 width=3 by rex_fwd_flat_dx/ qed-.
+
+lemma rdeq_fwd_dx (h) (o): ∀I2,L1,K2. ∀T:term. L1 ≛[h, o, T] K2.ⓘ{I2} →
+ ∃∃I1,K1. L1 = K1.ⓘ{I1}.
+/2 width=5 by rex_fwd_dx/ 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_tdeq.ma".
+include "basic_2/static/rex_drops.ma".
+include "basic_2/static/rdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+(* Properties with generic slicing for local environments *******************)
+
+lemma rdeq_lifts_sn: ∀h,o. f_dedropable_sn (cdeq h o).
+/3 width=5 by rex_liftable_dedropable_sn, tdeq_lifts_sn/ qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+lemma rdeq_inv_lifts_sn: ∀h,o. f_dropable_sn (cdeq h o).
+/2 width=5 by rex_dropable_sn/ qed-.
+
+(* Note: missing in basic_2A1 *)
+lemma rdeq_inv_lifts_dx: ∀h,o. f_dropable_dx (cdeq h o).
+/2 width=5 by rex_dropable_dx/ qed-.
+
+(* Basic_2A1: uses: lleq_inv_lift_le lleq_inv_lift_be lleq_inv_lift_ge *)
+lemma rdeq_inv_lifts_bi: ∀h,o,L1,L2,U. L1 ≛[h, o, U] L2 → ∀b,f. 𝐔⦃f⦄ →
+ ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ ∀T. ⬆*[f] T ≘ U → K1 ≛[h, o, T] K2.
+/2 width=10 by rex_inv_lifts_bi/ qed-.
+
+lemma rdeq_inv_lref_pair_sn: ∀h,o,L1,L2,i. L1 ≛[h, o, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ≛[h, o, V1] K2 & V1 ≛[h, o] V2.
+/2 width=3 by rex_inv_lref_pair_sn/ qed-.
+
+lemma rdeq_inv_lref_pair_dx: ∀h,o,L1,L2,i. L1 ≛[h, o, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ≛[h, o, V1] K2 & V1 ≛[h, o] V2.
+/2 width=3 by rex_inv_lref_pair_dx/ 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/static/rex_fqup.ma".
+include "basic_2/static/rdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+(* Advanced properties ******************************************************)
+
+lemma rdeq_refl: ∀h,o,T. reflexive … (rdeq h o T).
+/2 width=1 by rex_refl/ qed.
+
+lemma rdeq_pair_refl: ∀h,o,V1,V2. V1 ≛[h, o] V2 →
+ ∀I,L. ∀T:term. L.ⓑ{I}V1 ≛[h, o, T] L.ⓑ{I}V2.
+/2 width=1 by rex_pair_refl/ qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rdeq_inv_bind_void: ∀h,o,p,I,L1,L2,V,T. L1 ≛[h, o, ⓑ{p,I}V.T] L2 →
+ L1 ≛[h, o, V] L2 ∧ L1.ⓧ ≛[h, o, T] L2.ⓧ.
+/2 width=3 by rex_inv_bind_void/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma rdeq_fwd_bind_dx_void: ∀h,o,p,I,L1,L2,V,T.
+ L1 ≛[h, o, ⓑ{p,I}V.T] L2 → L1.ⓧ ≛[h, o, T] L2.ⓧ.
+/2 width=4 by rex_fwd_bind_dx_void/ 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/s_computation/fqus_fqup.ma".
+include "basic_2/static/rdeq_drops.ma".
+include "basic_2/static/rdeq_fqup.ma".
+include "basic_2/static/rdeq_rdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+(* Properties with extended structural successor for closures ***************)
+
+lemma fqu_tdeq_conf: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U1 ≛[h, o] U2 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & L2 ≛[h, o, T1] L & T1 ≛[h, o] T2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -G1 -G2 -L1 -L2 -U1 -T1
+[ #I #G #L #W #X #H >(tdeq_inv_lref1 … H) -X
+ /2 width=5 by fqu_lref_O, ex3_2_intro/
+| #I #G #L #W1 #U1 #X #H
+ elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #_ #H destruct
+ /2 width=5 by fqu_pair_sn, ex3_2_intro/
+| #p #I #G #L #W1 #U1 #X #H
+ elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #HU12 #H destruct
+ /3 width=5 by rdeq_pair_refl, fqu_bind_dx, ex3_2_intro/
+| #p #I #G #L #W1 #U1 #Hb #X #H
+ elim (tdeq_inv_pair1 … H) -H #W2 #U2 #HW12 #HU12 #H destruct
+ /3 width=5 by fqu_clear, ex3_2_intro/
+| #I #G #L #W1 #U1 #X #H
+ elim (tdeq_inv_pair1 … H) -H #W2 #U2 #_ #HU12 #H destruct
+ /2 width=5 by fqu_flat_dx, ex3_2_intro/
+| #I #G #L #T1 #U1 #HTU1 #U2 #HU12
+ elim (tdeq_inv_lifts_sn … HU12 … HTU1) -U1
+ /3 width=5 by fqu_drop, ex3_2_intro/
+]
+qed-.
+
+lemma tdeq_fqu_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H12 #U2 #HU21
+elim (fqu_tdeq_conf … o … H12 U2) -H12
+/3 width=5 by rdeq_sym, tdeq_sym, ex3_2_intro/
+qed-.
+
+(* Basic_2A1: uses: lleq_fqu_trans *)
+lemma rdeq_fqu_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐[b] ⦃G2, K2, U⦄ →
+ ∀L1. L1 ≛[h, o, T] L2 →
+ ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
+#h #o #b #G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
+[ #I #G #L2 #V2 #L1 #H elim (rdeq_inv_zero_pair_dx … H) -H
+ #K1 #V1 #HV1 #HV12 #H destruct
+ /3 width=7 by tdeq_rdeq_conf, fqu_lref_O, ex3_2_intro/
+| * [ #p ] #I #G #L2 #V #T #L1 #H
+ [ elim (rdeq_inv_bind … H)
+ | elim (rdeq_inv_flat … H)
+ ] -H
+ /2 width=5 by fqu_pair_sn, ex3_2_intro/
+| #p #I #G #L2 #V #T #L1 #H elim (rdeq_inv_bind … H) -H
+ /2 width=5 by fqu_bind_dx, ex3_2_intro/
+| #p #I #G #L2 #V #T #Hb #L1 #H elim (rdeq_inv_bind_void … H) -H
+ /3 width=5 by fqu_clear, ex3_2_intro/
+| #I #G #L2 #V #T #L1 #H elim (rdeq_inv_flat … H) -H
+ /2 width=5 by fqu_flat_dx, ex3_2_intro/
+| #I #G #L2 #T #U #HTU #Y #HU
+ elim (rdeq_fwd_dx … HU) #L1 #V1 #H destruct
+ /5 width=14 by rdeq_inv_lifts_bi, fqu_drop, drops_refl, drops_drop, ex3_2_intro/
+]
+qed-.
+
+(* Properties with optional structural successor for closures ***************)
+
+lemma tdeq_fquq_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐⸮[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐⸮[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H elim H -H
+[ #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
+ /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_2A1: was just: lleq_fquq_trans *)
+lemma rdeq_fquq_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐⸮[b] ⦃G2, K2, U⦄ →
+ ∀L1. L1 ≛[h, o, T] L2 →
+ ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐⸮[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
+#h #o #b #G1 #G2 #L2 #K2 #T #U #H elim H -H
+[ #H #L1 #HL12 elim (rdeq_fqu_trans … H … HL12) -L2 /3 width=5 by fqu_fquq, ex3_2_intro/
+| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Properties with plus-iterated structural successor for closures **********)
+
+(* Basic_2A1: was just: lleq_fqup_trans *)
+lemma rdeq_fqup_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐+[b] ⦃G2, K2, U⦄ →
+ ∀L1. L1 ≛[h, o, T] L2 →
+ ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐+[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
+#h #o #b #G1 #G2 #L2 #K2 #T #U #H @(fqup_ind … H) -G2 -K2 -U
+[ #G2 #K2 #U #HTU #L1 #HL12 elim (rdeq_fqu_trans … HTU … HL12) -L2
+ /3 width=5 by fqu_fqup, ex3_2_intro/
+| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12
+ elim (IHTU … HL12) -L2 #K0 #U0 #HTU #HU0 #HK0
+ elim (rdeq_fqu_trans … HU2 … HK0) -K #K1 #U1 #HU1 #HU12 #HK12
+ elim (tdeq_fqu_trans … HU1 … HU0) -U #K3 #U3 #HU03 #HU31 #HK31
+ @(ex3_2_intro … K3 U3) (**) (* full auto too slow *)
+ /3 width=5 by rdeq_trans, tdeq_rdeq_conf, fqup_strap1, tdeq_trans/
+]
+qed-.
+
+lemma tdeq_fqup_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐+[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H @(fqup_ind_dx … H) -G1 -L1 -U1
+[ #G1 #L1 #U1 #H #U2 #HU21 elim (tdeq_fqu_trans … H … HU21) -U1
+ /3 width=5 by fqu_fqup, ex3_2_intro/
+| #G1 #G #L1 #L #U1 #U #H #_ #IH #U2 #HU21
+ elim (tdeq_fqu_trans … H … HU21) -U1 #L0 #T #H1 #HTU #HL0
+ lapply (tdeq_rdeq_div … HTU … HL0) -HL0 #HL0
+ elim (IH … HTU) -U #K2 #U1 #H2 #HUT1 #HKL2
+ elim (rdeq_fqup_trans … H2 … HL0) -L #K #U #H2 #HU1 #HK2
+ lapply (tdeq_rdeq_conf … HUT1 … HK2) -HK2 #HK2
+ /3 width=7 by rdeq_trans, fqup_strap2, tdeq_trans, ex3_2_intro/
+]
+qed-.
+
+(* Properties with star-iterated structural successor for closures **********)
+
+lemma tdeq_fqus_trans: ∀h,o,b,G1,G2,L1,L2,U1,T1. ⦃G1, L1, U1⦄ ⊐*[b] ⦃G2, L2, T1⦄ →
+ ∀U2. U2 ≛[h, o] U1 →
+ ∃∃L,T2. ⦃G1, L1, U2⦄ ⊐*[b] ⦃G2, L, T2⦄ & T2 ≛[h, o] T1 & L ≛[h, o, T1] L2.
+#h #o #b #G1 #G2 #L1 #L2 #U1 #T1 #H #U2 #HU21 elim(fqus_inv_fqup … H) -H
+[ #H elim (tdeq_fqup_trans … H … HU21) -U1 /3 width=5 by fqup_fqus, ex3_2_intro/
+| * #HG #HL #HT destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_2A1: was just: lleq_fqus_trans *)
+lemma rdeq_fqus_trans: ∀h,o,b,G1,G2,L2,K2,T,U. ⦃G1, L2, T⦄ ⊐*[b] ⦃G2, K2, U⦄ →
+ ∀L1. L1 ≛[h, o, T] L2 →
+ ∃∃K1,U0. ⦃G1, L1, T⦄ ⊐*[b] ⦃G2, K1, U0⦄ & U0 ≛[h, o] U & K1 ≛[h, o, U] K2.
+#h #o #b #G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_fqup … H) -H
+[ #H elim (rdeq_fqup_trans … H … HL12) -L2 /3 width=5 by fqup_fqus, ex3_2_intro/
+| * #HG #HL #HT destruct /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 "basic_2/relocation/lifts_tdeq.ma".
+include "basic_2/static/rex_length.ma".
+include "basic_2/static/rex_fsle.ma".
+include "basic_2/static/rdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+(* Advanved properties with free variables inclusion ************************)
+
+lemma rdeq_fsge_comp (h) (o): rex_fsge_compatible (cdeq h o).
+#h #o #L1 #L2 #T * #f1 #Hf1 #HL12
+lapply (frees_rdeq_conf h o … Hf1 … HL12)
+lapply (sex_fwd_length … HL12)
+/3 width=8 by lveq_length_eq, ex4_4_intro/ (**) (* full auto fails *)
+qed-.
+
+(* Properties with length for local environments ****************************)
+
+(* Basic_2A1: uses: lleq_sort *)
+lemma rdeq_sort_length (h) (o): ∀L1,L2. |L1| = |L2| → ∀s. L1 ≛[h, o, ⋆s] L2.
+/2 width=1 by rex_sort_length/ qed.
+
+(* Basic_2A1: uses: lleq_gref *)
+lemma rdeq_gref_length (h) (o): ∀L1,L2. |L1| = |L2| → ∀l. L1 ≛[h, o, §l] L2.
+/2 width=1 by rex_gref_length/ qed.
+
+lemma rdeq_unit_length (h) (o): ∀L1,L2. |L1| = |L2| →
+ ∀I. L1.ⓤ{I} ≛[h, o, #0] L2.ⓤ{I}.
+/2 width=1 by rex_unit_length/ qed.
+
+(* Basic_2A1: uses: lleq_lift_le lleq_lift_ge *)
+lemma rdeq_lifts_bi (h) (o): ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ≛[h, o, T] K2 →
+ ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ ∀U. ⬆*[f] T ≘ U → L1 ≛[h, o, U] L2.
+/3 width=9 by rex_lifts_bi, tdeq_lifts_sn/ qed-.
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Basic_2A1: lleq_fwd_length *)
+lemma rdeq_fwd_length (h) (o): ∀L1,L2. ∀T:term. L1 ≛[h, o, T] L2 → |L1| = |L2|.
+/2 width=3 by rex_fwd_length/ 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/syntax/ext2_ext2.ma".
+include "basic_2/syntax/tdeq_tdeq.ma".
+include "basic_2/static/rdeq_length.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: lleq_sym *)
+lemma rdeq_sym: ∀h,o,T. symmetric … (rdeq h o T).
+/3 width=3 by rdeq_fsge_comp, rex_sym, tdeq_sym/ qed-.
+
+(* Basic_2A1: uses: lleq_dec *)
+lemma rdeq_dec: ∀h,o,L1,L2. ∀T:term. Decidable (L1 ≛[h, o, T] L2).
+/3 width=1 by rex_dec, tdeq_dec/ qed-.
+
+(* Main properties **********************************************************)
+
+(* Basic_2A1: uses: lleq_bind lleq_bind_O *)
+theorem rdeq_bind: ∀h,o,p,I,L1,L2,V1,V2,T.
+ L1 ≛[h, o, V1] L2 → L1.ⓑ{I}V1 ≛[h, o, T] L2.ⓑ{I}V2 →
+ L1 ≛[h, o, ⓑ{p,I}V1.T] L2.
+/2 width=2 by rex_bind/ qed.
+
+(* Basic_2A1: uses: lleq_flat *)
+theorem rdeq_flat: ∀h,o,I,L1,L2,V,T. L1 ≛[h, o, V] L2 → L1 ≛[h, o, T] L2 →
+ L1 ≛[h, o, ⓕ{I}V.T] L2.
+/2 width=1 by rex_flat/ qed.
+
+theorem rdeq_bind_void: ∀h,o,p,I,L1,L2,V,T.
+ L1 ≛[h, o, V] L2 → L1.ⓧ ≛[h, o, T] L2.ⓧ →
+ L1 ≛[h, o, ⓑ{p,I}V.T] L2.
+/2 width=1 by rex_bind_void/ qed.
+
+(* Basic_2A1: uses: lleq_trans *)
+theorem rdeq_trans: ∀h,o,T. Transitive … (rdeq h o T).
+#h #o #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
+lapply (frees_tdeq_conf_rdeq … Hf1 T … HL1) // #H0
+lapply (frees_mono … Hf2 … H0) -Hf2 -H0
+/5 width=7 by sex_trans, sex_eq_repl_back, tdeq_trans, ext2_trans, ex2_intro/
+qed-.
+
+(* Basic_2A1: uses: lleq_canc_sn *)
+theorem rdeq_canc_sn: ∀h,o,T. left_cancellable … (rdeq h o T).
+/3 width=3 by rdeq_trans, rdeq_sym/ qed-.
+
+(* Basic_2A1: uses: lleq_canc_dx *)
+theorem rdeq_canc_dx: ∀h,o,T. right_cancellable … (rdeq h o T).
+/3 width=3 by rdeq_trans, rdeq_sym/ qed-.
+
+theorem rdeq_repl: ∀h,o,L1,L2. ∀T:term. L1 ≛[h, o, T] L2 →
+ ∀K1. L1 ≛[h, o, T] K1 → ∀K2. L2 ≛[h, o, T] K2 → K1 ≛[h, o, T] K2.
+/3 width=3 by rdeq_canc_sn, rdeq_trans/ qed-.
+
+(* Negated properties *******************************************************)
+
+(* Note: auto works with /4 width=8/ so rdeq_canc_sn is preferred **********)
+(* Basic_2A1: uses: lleq_nlleq_trans *)
+lemma rdeq_rdneq_trans: ∀h,o.∀T:term.∀L1,L. L1 ≛[h, o, T] L →
+ ∀L2. (L ≛[h, o, T] L2 → ⊥) → (L1 ≛[h, o, T] L2 → ⊥).
+/3 width=3 by rdeq_canc_sn/ qed-.
+
+(* Basic_2A1: uses: nlleq_lleq_div *)
+lemma rdneq_rdeq_div: ∀h,o.∀T:term.∀L2,L. L2 ≛[h, o, T] L →
+ ∀L1. (L1 ≛[h, o, T] L → ⊥) → (L1 ≛[h, o, T] L2 → ⊥).
+/3 width=3 by rdeq_trans/ qed-.
+
+theorem rdneq_rdeq_canc_dx: ∀h,o,L1,L. ∀T:term. (L1 ≛[h, o, T] L → ⊥) →
+ ∀L2. L2 ≛[h, o, T] L → L1 ≛[h, o, T] L2 → ⊥.
+/3 width=3 by rdeq_trans/ qed-.
+
+(* Negated inversion lemmas *************************************************)
+
+(* Basic_2A1: uses: nlleq_inv_bind nlleq_inv_bind_O *)
+lemma rdneq_inv_bind: ∀h,o,p,I,L1,L2,V,T. (L1 ≛[h, o, ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ≛[h, o, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ≛[h, o, T] L2.ⓑ{I}V → ⊥).
+/3 width=2 by rnex_inv_bind, tdeq_dec/ qed-.
+
+(* Basic_2A1: uses: nlleq_inv_flat *)
+lemma rdneq_inv_flat: ∀h,o,I,L1,L2,V,T. (L1 ≛[h, o, ⓕ{I}V.T] L2 → ⊥) →
+ (L1 ≛[h, o, V] L2 → ⊥) ∨ (L1 ≛[h, o, T] L2 → ⊥).
+/3 width=2 by rnex_inv_flat, tdeq_dec/ qed-.
+
+lemma rdneq_inv_bind_void: ∀h,o,p,I,L1,L2,V,T. (L1 ≛[h, o, ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ≛[h, o, V] L2 → ⊥) ∨ (L1.ⓧ ≛[h, o, T] L2.ⓧ → ⊥).
+/3 width=3 by rnex_inv_bind_void, tdeq_dec/ 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/static/req_fsle.ma".
+include "basic_2/static/rdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES ******)
+
+(* Properties with syntactic equivalence on referred entries ****************)
+
+lemma req_rdeq: ∀h,o,L1,L2. ∀T:term. L1 ≡[T] L2 → L1 ≛[h, o, T] L2.
+/2 width=3 by rex_co/ qed.
+
+lemma req_rdeq_trans: ∀h,o,L1,L. ∀T:term. L1 ≡[T] L →
+ ∀L2. L ≛[h, o, T] L2 → L1 ≛[h, o, T] L2.
+/2 width=3 by req_rex_trans/ 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/ideqsn_3.ma".
+include "basic_2/static/rex.ma".
+
+(* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********)
+
+(* Basic_2A1: was: lleq *)
+definition req: relation3 term lenv lenv ≝
+ rex ceq.
+
+interpretation
+ "syntactic equivalence on referred entries (local environment)"
+ 'IdEqSn T L1 L2 = (req T L1 L2).
+
+(* Note: "req_transitive R" is equivalent to "rex_transitive ceq R R" *)
+(* Basic_2A1: uses: lleq_transitive *)
+definition req_transitive: predicate (relation3 lenv term term) ≝
+ λR. ∀L2,T1,T2. R L2 T1 T2 → ∀L1. L1 ≡[T1] L2 → R L1 T1 T2.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma req_inv_bind: ∀p,I,L1,L2,V,T. L1 ≡[ⓑ{p,I}V.T] L2 →
+ ∧∧ L1 ≡[V] L2 & L1.ⓑ{I}V ≡[T] L2.ⓑ{I}V.
+/2 width=2 by rex_inv_bind/ qed-.
+
+lemma req_inv_flat: ∀I,L1,L2,V,T. L1 ≡[ⓕ{I}V.T] L2 →
+ ∧∧ L1 ≡[V] L2 & L1 ≡[T] L2.
+/2 width=2 by rex_inv_flat/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma req_inv_zero_pair_sn: ∀I,L2,K1,V. K1.ⓑ{I}V ≡[#0] L2 →
+ ∃∃K2. K1 ≡[V] K2 & L2 = K2.ⓑ{I}V.
+#I #L2 #K1 #V #H
+elim (rex_inv_zero_pair_sn … H) -H #K2 #X #HK12 #HX #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma req_inv_zero_pair_dx: ∀I,L1,K2,V. L1 ≡[#0] K2.ⓑ{I}V →
+ ∃∃K1. K1 ≡[V] K2 & L1 = K1.ⓑ{I}V.
+#I #L1 #K2 #V #H
+elim (rex_inv_zero_pair_dx … H) -H #K1 #X #HK12 #HX #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma req_inv_lref_bind_sn: ∀I1,K1,L2,i. K1.ⓘ{I1} ≡[#↑i] L2 →
+ ∃∃I2,K2. K1 ≡[#i] K2 & L2 = K2.ⓘ{I2}.
+/2 width=2 by rex_inv_lref_bind_sn/ qed-.
+
+lemma req_inv_lref_bind_dx: ∀I2,K2,L1,i. L1 ≡[#↑i] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ≡[#i] K2 & L1 = K1.ⓘ{I1}.
+/2 width=2 by rex_inv_lref_bind_dx/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+(* Basic_2A1: was: llpx_sn_lrefl *)
+(* Basic_2A1: this should have been lleq_fwd_llpx_sn *)
+lemma req_fwd_rex: ∀R. c_reflexive … R →
+ ∀L1,L2,T. L1 ≡[T] L2 → L1 ⪤[R, T] L2.
+#R #HR #L1 #L2 #T * #f #Hf #HL12
+/4 width=7 by sex_co, cext2_co, ex2_intro/
+qed-.
+
+(* Basic_properties *********************************************************)
+
+lemma frees_req_conf: ∀f,L1,T. L1 ⊢ 𝐅*⦃T⦄ ≘ f →
+ ∀L2. L1 ≡[T] L2 → L2 ⊢ 𝐅*⦃T⦄ ≘ f.
+#f #L1 #T #H elim H -f -L1 -T
+[ /2 width=3 by frees_sort/
+| #f #i #Hf #L2 #H2
+ >(rex_inv_atom_sn … H2) -L2
+ /2 width=1 by frees_atom/
+| #f #I #L1 #V1 #_ #IH #Y #H2
+ elim (req_inv_zero_pair_sn … H2) -H2 #L2 #HL12 #H destruct
+ /3 width=1 by frees_pair/
+| #f #I #L1 #Hf #Y #H2
+ elim (rex_inv_zero_unit_sn … H2) -H2 #g #L2 #_ #_ #H destruct
+ /2 width=1 by frees_unit/
+| #f #I #L1 #i #_ #IH #Y #H2
+ elim (req_inv_lref_bind_sn … H2) -H2 #J #L2 #HL12 #H destruct
+ /3 width=1 by frees_lref/
+| /2 width=1 by frees_gref/
+| #f1V #f1T #f1 #p #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #L2 #H2
+ elim (req_inv_bind … H2) -H2 /3 width=5 by frees_bind/
+| #f1V #f1T #f1 #I #L1 #V1 #T1 #_ #_ #Hf1 #IHV #IHT #L2 #H2
+ elim (req_inv_flat … H2) -H2 /3 width=5 by frees_flat/
+]
+qed-.
+
+(* Basic_2A1: removed theorems 10:
+ lleq_ind lleq_fwd_lref
+ lleq_fwd_drop_sn lleq_fwd_drop_dx
+ lleq_skip lleq_lref lleq_free
+ lleq_Y lleq_ge_up lleq_ge
+
+*)
--- /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/static/rex_fqup.ma".
+include "basic_2/static/req.ma".
+
+(* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: was: lleq_refl *)
+lemma req_refl: ∀T. reflexive … (req T).
+/2 width=1 by rex_refl/ 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/static/rex_length.ma".
+include "basic_2/static/rex_fsle.ma".
+include "basic_2/static/req.ma".
+
+(* SYNTACTIC EQUIVALENCE FOR LOCAL ENVIRONMENTS ON REFERRED ENTRIES *********)
+
+(* Properties with free variables inclusion for restricted closures *********)
+
+lemma req_fsle_comp: rex_fsle_compatible ceq.
+#L1 #L2 #T #HL12
+elim (frees_total L1 T)
+/4 width=8 by frees_req_conf, rex_fwd_length, lveq_length_eq, sle_refl, ex4_4_intro/
+qed.
+
+(* Forward lemmas with free variables inclusion for restricted closures *****)
+
+lemma req_rex_trans: ∀R. req_transitive R →
+ ∀L1,L,T. L1 ≡[T] L → ∀L2. L ⪤[R, T] L2 → L1 ⪤[R, T] L2.
+/4 width=16 by req_fsle_comp, rex_trans_fsle, rex_trans_next/ 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/relocation/rtmap_id.ma".
+include "basic_2/notation/relations/relation_4.ma".
+include "basic_2/syntax/cext2.ma".
+include "basic_2/relocation/sex.ma".
+include "basic_2/static/frees.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+definition rex (R) (T): relation lenv ≝
+ λL1,L2. ∃∃f. L1 ⊢ 𝐅*⦃T⦄ ≘ f & L1 ⪤[cext2 R, cfull, f] L2.
+
+interpretation "generic extension on referred entries (local environment)"
+ 'Relation R T L1 L2 = (rex R T L1 L2).
+
+definition R_confluent2_rex: relation4 (relation3 lenv term term)
+ (relation3 lenv term term) … ≝
+ λR1,R2,RP1,RP2.
+ ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. L0 ⪤[RP1, T0] L1 → ∀L2. L0 ⪤[RP2, T0] L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+definition rex_confluent: relation … ≝
+ λR1,R2.
+ ∀K1,K,V1. K1 ⪤[R1, V1] K → ∀V. R1 K1 V1 V →
+ ∀K2. K ⪤[R2, V] K2 → K ⪤[R2, V1] K2.
+
+definition rex_transitive: relation3 ? (relation3 ?? term) … ≝
+ λR1,R2,R3.
+ ∀K1,K,V1. K1 ⪤[R1, V1] K →
+ ∀V. R1 K1 V1 V → ∀V2. R2 K V V2 → R3 K1 V1 V2.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma rex_inv_atom_sn (R): ∀Y2,T. ⋆ ⪤[R, T] Y2 → Y2 = ⋆.
+#R #Y2 #T * /2 width=4 by sex_inv_atom1/
+qed-.
+
+lemma rex_inv_atom_dx (R): ∀Y1,T. Y1 ⪤[R, T] ⋆ → Y1 = ⋆.
+#R #I #Y1 * /2 width=4 by sex_inv_atom2/
+qed-.
+
+lemma rex_inv_sort (R): ∀Y1,Y2,s. Y1 ⪤[R, ⋆s] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤[R, ⋆s] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+#R * [ | #Y1 #I1 ] #Y2 #s * #f #H1 #H2
+[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
+| lapply (frees_inv_sort … H1) -H1 #Hf
+ elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
+ elim (sex_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct
+ /5 width=7 by frees_sort, ex3_4_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma rex_inv_zero (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
+ | ∃∃f,I,L1,L2. 𝐈⦃f⦄ & L1 ⪤[cext2 R, cfull, f] L2 &
+ Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
+#R * [ | #Y1 * #I1 [ | #X ] ] #Y2 * #f #H1 #H2
+[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or3_intro0, conj/
+| elim (frees_inv_unit … H1) -H1 #g #HX #H destruct
+ elim (sex_inv_next1 … H2) -H2 #I2 #L2 #HL12 #H #H2 destruct
+ >(ext2_inv_unit_sn … H) -H /3 width=8 by or3_intro2, ex4_4_intro/
+| elim (frees_inv_pair … H1) -H1 #g #Hg #H destruct
+ elim (sex_inv_next1 … H2) -H2 #Z2 #L2 #HL12 #H
+ elim (ext2_inv_pair_sn … H) -H
+ /4 width=9 by or3_intro1, ex4_5_intro, ex2_intro/
+]
+qed-.
+
+lemma rex_inv_lref (R): ∀Y1,Y2,i. Y1 ⪤[R, #↑i] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤[R, #i] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+#R * [ | #Y1 #I1 ] #Y2 #i * #f #H1 #H2
+[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
+| elim (frees_inv_lref … H1) -H1 #g #Hg #H destruct
+ elim (sex_inv_push1 … H2) -H2
+ /4 width=7 by ex3_4_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma rex_inv_gref (R): ∀Y1,Y2,l. Y1 ⪤[R, §l] Y2 →
+ ∨∨ Y1 = ⋆ ∧ Y2 = ⋆
+ | ∃∃I1,I2,L1,L2. L1 ⪤[R, §l] L2 &
+ Y1 = L1.ⓘ{I1} & Y2 = L2.ⓘ{I2}.
+#R * [ | #Y1 #I1 ] #Y2 #l * #f #H1 #H2
+[ lapply (sex_inv_atom1 … H2) -H2 /3 width=1 by or_introl, conj/
+| lapply (frees_inv_gref … H1) -H1 #Hf
+ elim (isid_inv_gen … Hf) -Hf #g #Hg #H destruct
+ elim (sex_inv_push1 … H2) -H2 #I2 #L2 #H12 #_ #H destruct
+ /5 width=7 by frees_gref, ex3_4_intro, ex2_intro, or_intror/
+]
+qed-.
+
+(* Basic_2A1: uses: llpx_sn_inv_bind llpx_sn_inv_bind_O *)
+lemma rex_inv_bind (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R, ⓑ{p,I}V1.T] L2 → R L1 V1 V2 →
+ ∧∧ L1 ⪤[R, V1] L2 & L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2.
+#R #p #I #L1 #L2 #V1 #V2 #T * #f #Hf #HL #HV elim (frees_inv_bind … Hf) -Hf
+/6 width=6 by sle_sex_trans, sex_inv_tl, ext2_pair, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+qed-.
+
+(* Basic_2A1: uses: llpx_sn_inv_flat *)
+lemma rex_inv_flat (R): ∀I,L1,L2,V,T. L1 ⪤[R, ⓕ{I}V.T] L2 →
+ ∧∧ L1 ⪤[R, V] L2 & L1 ⪤[R, T] L2.
+#R #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_flat … Hf) -Hf
+/5 width=6 by sle_sex_trans, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rex_inv_sort_bind_sn (R): ∀I1,K1,L2,s. K1.ⓘ{I1} ⪤[R, ⋆s] L2 →
+ ∃∃I2,K2. K1 ⪤[R, ⋆s] K2 & L2 = K2.ⓘ{I2}.
+#R #I1 #K1 #L2 #s #H elim (rex_inv_sort … H) -H *
+[ #H destruct
+| #Z1 #I2 #Y1 #K2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma rex_inv_sort_bind_dx (R): ∀I2,K2,L1,s. L1 ⪤[R, ⋆s] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤[R, ⋆s] K2 & L1 = K1.ⓘ{I1}.
+#R #I2 #K2 #L1 #s #H elim (rex_inv_sort … H) -H *
+[ #_ #H destruct
+| #I1 #Z2 #K1 #Y2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma rex_inv_zero_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R, #0] L2 →
+ ∃∃K2,V2. K1 ⪤[R, V1] K2 & R K1 V1 V2 &
+ L2 = K2.ⓑ{I}V2.
+#R #I #L2 #K1 #V1 #H elim (rex_inv_zero … H) -H *
+[ #H destruct
+| #Z #Y1 #K2 #X1 #V2 #HK12 #HV12 #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| #f #Z #Y1 #Y2 #_ #_ #H destruct
+]
+qed-.
+
+lemma rex_inv_zero_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R, #0] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. K1 ⪤[R, V1] K2 & R K1 V1 V2 &
+ L1 = K1.ⓑ{I}V1.
+#R #I #L1 #K2 #V2 #H elim (rex_inv_zero … H) -H *
+[ #_ #H destruct
+| #Z #K1 #Y2 #V1 #X2 #HK12 #HV12 #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+| #f #Z #Y1 #Y2 #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma rex_inv_zero_unit_sn (R): ∀I,K1,L2. K1.ⓤ{I} ⪤[R, #0] L2 →
+ ∃∃f,K2. 𝐈⦃f⦄ & K1 ⪤[cext2 R, cfull, f] K2 &
+ L2 = K2.ⓤ{I}.
+#R #I #K1 #L2 #H elim (rex_inv_zero … H) -H *
+[ #H destruct
+| #Z #Y1 #Y2 #X1 #X2 #_ #_ #H destruct
+| #f #Z #Y1 #K2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma rex_inv_zero_unit_dx (R): ∀I,L1,K2. L1 ⪤[R, #0] K2.ⓤ{I} →
+ ∃∃f,K1. 𝐈⦃f⦄ & K1 ⪤[cext2 R, cfull, f] K2 &
+ L1 = K1.ⓤ{I}.
+#R #I #L1 #K2 #H elim (rex_inv_zero … H) -H *
+[ #_ #H destruct
+| #Z #Y1 #Y2 #X1 #X2 #_ #_ #_ #H destruct
+| #f #Z #K1 #Y2 #Hf #HK12 #H1 #H2 destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma rex_inv_lref_bind_sn (R): ∀I1,K1,L2,i. K1.ⓘ{I1} ⪤[R, #↑i] L2 →
+ ∃∃I2,K2. K1 ⪤[R, #i] K2 & L2 = K2.ⓘ{I2}.
+#R #I1 #K1 #L2 #i #H elim (rex_inv_lref … H) -H *
+[ #H destruct
+| #Z1 #I2 #Y1 #K2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma rex_inv_lref_bind_dx (R): ∀I2,K2,L1,i. L1 ⪤[R, #↑i] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤[R, #i] K2 & L1 = K1.ⓘ{I1}.
+#R #I2 #K2 #L1 #i #H elim (rex_inv_lref … H) -H *
+[ #_ #H destruct
+| #I1 #Z2 #K1 #Y2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma rex_inv_gref_bind_sn (R): ∀I1,K1,L2,l. K1.ⓘ{I1} ⪤[R, §l] L2 →
+ ∃∃I2,K2. K1 ⪤[R, §l] K2 & L2 = K2.ⓘ{I2}.
+#R #I1 #K1 #L2 #l #H elim (rex_inv_gref … H) -H *
+[ #H destruct
+| #Z1 #I2 #Y1 #K2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma rex_inv_gref_bind_dx (R): ∀I2,K2,L1,l. L1 ⪤[R, §l] K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⪤[R, §l] K2 & L1 = K1.ⓘ{I1}.
+#R #I2 #K2 #L1 #l #H elim (rex_inv_gref … H) -H *
+[ #_ #H destruct
+| #I1 #Z2 #K1 #Y2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma rex_fwd_zero_pair (R): ∀I,K1,K2,V1,V2.
+ K1.ⓑ{I}V1 ⪤[R, #0] K2.ⓑ{I}V2 → K1 ⪤[R, V1] K2.
+#R #I #K1 #K2 #V1 #V2 #H
+elim (rex_inv_zero_pair_sn … H) -H #Y #X #HK12 #_ #H destruct //
+qed-.
+
+(* Basic_2A1: uses: llpx_sn_fwd_pair_sn llpx_sn_fwd_bind_sn llpx_sn_fwd_flat_sn *)
+lemma rex_fwd_pair_sn (R): ∀I,L1,L2,V,T. L1 ⪤[R, ②{I}V.T] L2 → L1 ⪤[R, V] L2.
+#R * [ #p ] #I #L1 #L2 #V #T * #f #Hf #HL
+[ elim (frees_inv_bind … Hf) | elim (frees_inv_flat … Hf) ] -Hf
+/4 width=6 by sle_sex_trans, sor_inv_sle_sn, ex2_intro/
+qed-.
+
+(* Basic_2A1: uses: llpx_sn_fwd_bind_dx llpx_sn_fwd_bind_O_dx *)
+lemma rex_fwd_bind_dx (R): ∀p,I,L1,L2,V1,V2,T. L1 ⪤[R, ⓑ{p,I}V1.T] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2.
+#R #p #I #L1 #L2 #V1 #V2 #T #H #HV elim (rex_inv_bind … H HV) -H -HV //
+qed-.
+
+(* Basic_2A1: uses: llpx_sn_fwd_flat_dx *)
+lemma rex_fwd_flat_dx (R): ∀I,L1,L2,V,T. L1 ⪤[R, ⓕ{I}V.T] L2 → L1 ⪤[R, T] L2.
+#R #I #L1 #L2 #V #T #H elim (rex_inv_flat … H) -H //
+qed-.
+
+lemma rex_fwd_dx (R): ∀I2,L1,K2,T. L1 ⪤[R, T] K2.ⓘ{I2} →
+ ∃∃I1,K1. L1 = K1.ⓘ{I1}.
+#R #I2 #L1 #K2 #T * #f elim (pn_split f) * #g #Hg #_ #Hf destruct
+[ elim (sex_inv_push2 … Hf) | elim (sex_inv_next2 … Hf) ] -Hf #I1 #K1 #_ #_ #H destruct
+/2 width=3 by ex1_2_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma rex_atom (R): ∀I. ⋆ ⪤[R, ⓪{I}] ⋆.
+#R * /3 width=3 by frees_sort, frees_atom, frees_gref, sex_atom, ex2_intro/
+qed.
+
+lemma rex_sort (R): ∀I1,I2,L1,L2,s.
+ L1 ⪤[R, ⋆s] L2 → L1.ⓘ{I1} ⪤[R, ⋆s] L2.ⓘ{I2}.
+#R #I1 #I2 #L1 #L2 #s * #f #Hf #H12
+lapply (frees_inv_sort … Hf) -Hf
+/4 width=3 by frees_sort, sex_push, isid_push, ex2_intro/
+qed.
+
+lemma rex_pair (R): ∀I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 →
+ R L1 V1 V2 → L1.ⓑ{I}V1 ⪤[R, #0] L2.ⓑ{I}V2.
+#R #I1 #I2 #L1 #L2 #V1 *
+/4 width=3 by ext2_pair, frees_pair, sex_next, ex2_intro/
+qed.
+
+lemma rex_unit (R): ∀f,I,L1,L2. 𝐈⦃f⦄ → L1 ⪤[cext2 R, cfull, f] L2 →
+ L1.ⓤ{I} ⪤[R, #0] L2.ⓤ{I}.
+/4 width=3 by frees_unit, sex_next, ext2_unit, ex2_intro/ qed.
+
+lemma rex_lref (R): ∀I1,I2,L1,L2,i.
+ L1 ⪤[R, #i] L2 → L1.ⓘ{I1} ⪤[R, #↑i] L2.ⓘ{I2}.
+#R #I1 #I2 #L1 #L2 #i * /3 width=3 by sex_push, frees_lref, ex2_intro/
+qed.
+
+lemma rex_gref (R): ∀I1,I2,L1,L2,l.
+ L1 ⪤[R, §l] L2 → L1.ⓘ{I1} ⪤[R, §l] L2.ⓘ{I2}.
+#R #I1 #I2 #L1 #L2 #l * #f #Hf #H12
+lapply (frees_inv_gref … Hf) -Hf
+/4 width=3 by frees_gref, sex_push, isid_push, ex2_intro/
+qed.
+
+lemma rex_bind_repl_dx (R): ∀I,I1,L1,L2,T.
+ L1.ⓘ{I} ⪤[R, T] L2.ⓘ{I1} →
+ ∀I2. cext2 R L1 I I2 →
+ L1.ⓘ{I} ⪤[R, T] L2.ⓘ{I2}.
+#R #I #I1 #L1 #L2 #T * #f #Hf #HL12 #I2 #HR
+/3 width=5 by sex_pair_repl, ex2_intro/
+qed-.
+
+(* Basic_2A1: uses: llpx_sn_co *)
+lemma rex_co (R1) (R2): (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
+ ∀L1,L2,T. L1 ⪤[R1, T] L2 → L1 ⪤[R2, T] L2.
+#R1 #R2 #HR #L1 #L2 #T * /5 width=7 by sex_co, cext2_co, ex2_intro/
+qed-.
+
+lemma rex_isid (R1) (R2): ∀L1,L2,T1,T2.
+ (∀f. L1 ⊢ 𝐅*⦃T1⦄ ≘ f → 𝐈⦃f⦄) →
+ (∀f. 𝐈⦃f⦄ → L1 ⊢ 𝐅*⦃T2⦄ ≘ f) →
+ L1 ⪤[R1, T1] L2 → L1 ⪤[R2, T2] L2.
+#R1 #R2 #L1 #L2 #T1 #T2 #H1 #H2 *
+/4 width=7 by sex_co_isid, ex2_intro/
+qed-.
+
+lemma rex_unit_sn (R1) (R2):
+ ∀I,K1,L2. K1.ⓤ{I} ⪤[R1, #0] L2 → K1.ⓤ{I} ⪤[R2, #0] L2.
+#R1 #R2 #I #K1 #L2 #H
+elim (rex_inv_zero_unit_sn … H) -H #f #K2 #Hf #HK12 #H destruct
+/3 width=7 by rex_unit, sex_co_isid/
+qed-.
+
+(* Basic_2A1: removed theorems 9:
+ llpx_sn_skip llpx_sn_lref llpx_sn_free
+ llpx_sn_fwd_lref
+ llpx_sn_Y llpx_sn_ge_up llpx_sn_ge
+ llpx_sn_fwd_drop_sn llpx_sn_fwd_drop_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/relocation/drops_cext2.ma".
+include "basic_2/relocation/drops_sex.ma".
+include "basic_2/static/frees_drops.ma".
+include "basic_2/static/rex.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+definition f_dedropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 →
+ ∀K2,T. K1 ⪤[R, T] K2 → ∀U. ⬆*[f] T ≘ U →
+ ∃∃L2. L1 ⪤[R, U] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+
+definition f_dropable_sn: predicate (relation3 lenv term term) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ ∀L2,U. L1 ⪤[R, U] L2 → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K2. K1 ⪤[R, T] K2 & ⬇*[b, f] L2 ≘ K2.
+
+definition f_dropable_dx: predicate (relation3 lenv term term) ≝
+ λR. ∀L1,L2,U. L1 ⪤[R, U] L2 →
+ ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ → ∀T. ⬆*[f] T ≘ U →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R, T] K2.
+
+definition f_transitive_next: relation3 … ≝ λR1,R2,R3.
+ ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
+ ∀g,I,K,n. ⬇*[n] L ≘ K.ⓘ{I} → ↑g = ⫱*[n] f →
+ sex_transitive (cext2 R1) (cext2 R2) (cext2 R3) (cext2 R1) cfull g K I.
+
+(* Properties with generic slicing for local environments *******************)
+
+lemma rex_liftable_dedropable_sn: ∀R. (∀L. reflexive ? (R L)) →
+ d_liftable2_sn … lifts R → f_dedropable_sn R.
+#R #H1R #H2R #b #f #L1 #K1 #HLK1 #K2 #T * #f1 #Hf1 #HK12 #U #HTU
+elim (frees_total L1 U) #f2 #Hf2
+lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
+elim (sex_liftable_co_dedropable_sn … HLK1 … HK12 … Hf) -f1 -K1
+/3 width=6 by cext2_d_liftable2_sn, cfull_lift_sn, ext2_refl, ex3_intro, ex2_intro/
+qed-.
+
+lemma rex_trans_next: ∀R1,R2,R3. rex_transitive R1 R2 R3 → f_transitive_next R1 R2 R3.
+#R1 #R2 #R3 #HR #f #L1 #T #Hf #g #I1 #K1 #n #HLK #Hgf #I #H
+generalize in match HLK; -HLK elim H -I1 -I
+[ #I #_ #L2 #_ #I2 #H
+ lapply (ext2_inv_unit_sn … H) -H #H destruct
+ /2 width=1 by ext2_unit/
+| #I #V1 #V #HV1 #HLK1 #L2 #HL12 #I2 #H
+ elim (ext2_inv_pair_sn … H) -H #V2 #HV2 #H destruct
+ elim (frees_inv_drops_next … Hf … HLK1 … Hgf) -f -HLK1 #f #Hf #Hfg
+ /5 width=5 by ext2_pair, sle_sex_trans, ex2_intro/
+]
+qed.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_2A1: uses: llpx_sn_inv_lift_le llpx_sn_inv_lift_be llpx_sn_inv_lift_ge *)
+(* Basic_2A1: was: llpx_sn_drop_conf_O *)
+lemma rex_dropable_sn: ∀R. f_dropable_sn R.
+#R #b #f #L1 #K1 #HLK1 #H1f #L2 #U * #f2 #Hf2 #HL12 #T #HTU
+elim (frees_total K1 T) #f1 #Hf1
+lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #H2f
+elim (sex_co_dropable_sn … HLK1 … HL12 … H2f) -f2 -L1
+/3 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: was: llpx_sn_drop_trans_O *)
+(* Note: the proof might be simplified *)
+lemma rex_dropable_dx: ∀R. f_dropable_dx R.
+#R #L1 #L2 #U * #f2 #Hf2 #HL12 #b #f #K2 #HLK2 #H1f #T #HTU
+elim (drops_isuni_ex … H1f L1) #K1 #HLK1
+elim (frees_total K1 T) #f1 #Hf1
+lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -K1 #H2f
+elim (sex_co_dropable_dx … HL12 … HLK2 … H2f) -L2
+/4 width=9 by frees_inv_lifts, ex2_intro/
+qed-.
+
+(* Basic_2A1: uses: llpx_sn_inv_lift_O *)
+lemma rex_inv_lifts_bi: ∀R,L1,L2,U. L1 ⪤[R, U] L2 → ∀b,f. 𝐔⦃f⦄ →
+ ∀K1,K2. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ ∀T. ⬆*[f] T ≘ U → K1 ⪤[R, T] K2.
+#R #L1 #L2 #U #HL12 #b #f #Hf #K1 #K2 #HLK1 #HLK2 #T #HTU
+elim (rex_dropable_sn … HLK1 … HL12 … HTU) -L1 -U // #Y #HK12 #HY
+lapply (drops_mono … HY … HLK2) -b -f -L2 #H destruct //
+qed-.
+
+lemma rex_inv_lref_pair_sn: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 →
+ ∃∃K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 & K1 ⪤[R, V1] K2 & R K1 V1 V2.
+#R #L1 #L2 #i #HL12 #I #K1 #V1 #HLK1 elim (rex_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
+#Y #HY #HLK2 elim (rex_inv_zero_pair_sn … HY) -HY
+#K2 #V2 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+lemma rex_inv_lref_pair_dx: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K2,V2. ⬇*[i] L2 ≘ K2.ⓑ{I}V2 →
+ ∃∃K1,V1. ⬇*[i] L1 ≘ K1.ⓑ{I}V1 & K1 ⪤[R, V1] K2 & R K1 V1 V2.
+#R #L1 #L2 #i #HL12 #I #K2 #V2 #HLK2 elim (rex_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
+#Y #HLK1 #HY elim (rex_inv_zero_pair_dx … HY) -HY
+#K1 #V1 #HK12 #HV12 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+lemma rex_inv_lref_unit_sn: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} →
+ ∃∃f,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} & K1 ⪤[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
+#R #L1 #L2 #i #HL12 #I #K1 #HLK1 elim (rex_dropable_sn … HLK1 … HL12 (#0)) -HLK1 -HL12 //
+#Y #HY #HLK2 elim (rex_inv_zero_unit_sn … HY) -HY
+#f #K2 #Hf #HK12 #H destruct /2 width=5 by ex3_2_intro/
+qed-.
+
+lemma rex_inv_lref_unit_dx: ∀R,L1,L2,i. L1 ⪤[R, #i] L2 → ∀I,K2. ⬇*[i] L2 ≘ K2.ⓤ{I} →
+ ∃∃f,K1. ⬇*[i] L1 ≘ K1.ⓤ{I} & K1 ⪤[cext2 R, cfull, f] K2 & 𝐈⦃f⦄.
+#R #L1 #L2 #i #HL12 #I #K2 #HLK2 elim (rex_dropable_dx … HL12 … HLK2 … (#0)) -HLK2 -HL12 //
+#Y #HLK1 #HY elim (rex_inv_zero_unit_dx … HY) -HY
+#f #K2 #Hf #HK12 #H destruct /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 "basic_2/static/frees_fqup.ma".
+include "basic_2/static/rex.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: llpx_sn_refl *)
+lemma rex_refl: ∀R. (∀L. reflexive … (R L)) → ∀L,T. L ⪤[R, T] L.
+#R #HR #L #T elim (frees_total L T)
+/4 width=3 by sex_refl, ext2_refl, ex2_intro/
+qed.
+
+lemma rex_pair_refl: ∀R. (∀L. reflexive … (R L)) →
+ ∀L,V1,V2. R L V1 V2 → ∀I,T. L.ⓑ{I}V1 ⪤[R, T] L.ⓑ{I}V2.
+#R #HR #L #V1 #V2 #HV12 #I #T
+elim (frees_total (L.ⓑ{I}V1) T) #f #Hf
+elim (pn_split f) * #g #H destruct
+/5 width=3 by sex_refl, sex_next, sex_push, ext2_refl, ext2_pair, ex2_intro/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rex_inv_bind_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R, ⓑ{p,I}V.T] L2 →
+ L1 ⪤[R, V] L2 ∧ L1.ⓧ ⪤[R, T] L2.ⓧ.
+#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind_void … Hf) -Hf
+/6 width=6 by sle_sex_trans, sex_inv_tl, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma rex_fwd_bind_dx_void: ∀R,p,I,L1,L2,V,T. L1 ⪤[R, ⓑ{p,I}V.T] L2 →
+ L1.ⓧ ⪤[R, T] L2.ⓧ.
+#R #p #I #L1 #L2 #V #T #H elim (rex_inv_bind_void … 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 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/sex_length.ma".
+include "basic_2/static/fsle_fsle.ma".
+include "basic_2/static/rex_drops.ma".
+include "basic_2/static/rex_rex.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+definition R_fsge_compatible: predicate (relation3 …) ≝ λRN.
+ ∀L,T1,T2. RN L T1 T2 → ⦃L, T2⦄ ⊆ ⦃L, T1⦄.
+
+definition rex_fsge_compatible: predicate (relation3 …) ≝ λRN.
+ ∀L1,L2,T. L1 ⪤[RN, T] L2 → ⦃L2, T⦄ ⊆ ⦃L1, T⦄.
+
+definition rex_fsle_compatible: predicate (relation3 …) ≝ λRN.
+ ∀L1,L2,T. L1 ⪤[RN, T] L2 → ⦃L1, T⦄ ⊆ ⦃L2, T⦄.
+
+(* Basic inversions with free variables inclusion for restricted closures ***)
+
+lemma frees_sex_conf: ∀R. rex_fsge_compatible R →
+ ∀L1,T,f1. L1 ⊢ 𝐅*⦃T⦄ ≘ f1 →
+ ∀L2. L1 ⪤[cext2 R, cfull, f1] L2 →
+ ∃∃f2. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 & f2 ⊆ f1.
+#R #HR #L1 #T #f1 #Hf1 #L2 #H1L
+lapply (HR L1 L2 T ?) /2 width=3 by ex2_intro/ #H2L
+@(fsle_frees_trans_eq … H2L … Hf1) /3 width=4 by sex_fwd_length, sym_eq/
+qed-.
+
+(* Properties with free variables inclusion for restricted closures *********)
+
+(* Note: we just need lveq_inv_refl: ∀L,n1,n2. L ≋ⓧ*[n1, n2] L → ∧∧ 0 = n1 & 0 = n2 *)
+lemma fsge_rex_trans: ∀R,L1,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L1, T2⦄ →
+ ∀L2. L1 ⪤[R, T2] L2 → L1 ⪤[R, T1] L2.
+#R #L1 #T1 #T2 * #n1 #n2 #f1 #f2 #Hf1 #Hf2 #Hn #Hf #L2 #HL12
+elim (lveq_inj_length … Hn ?) // #H1 #H2 destruct
+/4 width=5 by rex_inv_frees, sle_sex_trans, ex2_intro/
+qed-.
+
+lemma rex_sym: ∀R. rex_fsge_compatible R →
+ (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+ ∀T. symmetric … (rex R T).
+#R #H1R #H2R #T #L1 #L2
+* #f1 #Hf1 #HL12
+elim (frees_sex_conf … Hf1 … HL12) -Hf1 //
+/5 width=5 by sle_sex_trans, sex_sym, cext2_sym, ex2_intro/
+qed-.
+
+lemma rex_pair_sn_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
+ rex_fsge_compatible R1 →
+ ∀L1,L2,V. L1 ⪤[R1, V] L2 → ∀I,T.
+ ∃∃L. L1 ⪤[R1, ②{I}V.T] L & L ⪤[R2, V] L2.
+#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #V * #f #Hf #HL12 * [ #p ] #I #T
+[ elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
+ elim (frees_inv_bind … Hg) #y1 #y2 #H #_ #Hy
+| elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg
+ elim (frees_inv_flat … Hg) #y1 #y2 #H #_ #Hy
+]
+lapply(frees_mono … H … Hf) -H #H1
+lapply (sor_eq_repl_back1 … Hy … H1) -y1 #Hy
+lapply (sor_inv_sle_sn … Hy) -y2 #Hfg
+elim (sex_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #L #HL1 #HL2
+lapply (sle_sex_trans … HL1 … Hfg) // #H
+elim (frees_sex_conf … Hf … H) -Hf -H
+/4 width=7 by sle_sex_trans, ex2_intro/
+qed-.
+
+lemma rex_flat_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
+ rex_fsge_compatible R1 →
+ ∀L1,L2,T. L1 ⪤[R1, T] L2 → ∀I,V.
+ ∃∃L. L1 ⪤[R1, ⓕ{I}V.T] L & L ⪤[R2, T] L2.
+#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #I #V
+elim (frees_total L1 (ⓕ{I}V.T)) #g #Hg
+elim (frees_inv_flat … Hg) #y1 #y2 #_ #H #Hy
+lapply(frees_mono … H … Hf) -H #H2
+lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy
+lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
+elim (sex_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #L #HL1 #HL2
+lapply (sle_sex_trans … HL1 … Hfg) // #H
+elim (frees_sex_conf … Hf … H) -Hf -H
+/4 width=7 by sle_sex_trans, ex2_intro/
+qed-.
+
+lemma rex_bind_dx_split: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
+ rex_fsge_compatible R1 →
+ ∀I,L1,L2,V1,T. L1.ⓑ{I}V1 ⪤[R1, T] L2 → ∀p.
+ ∃∃L,V. L1 ⪤[R1, ⓑ{p,I}V1.T] L & L.ⓑ{I}V ⪤[R2, T] L2 & R1 L1 V1 V.
+#R1 #R2 #HR1 #HR2 #HR #I #L1 #L2 #V1 #T * #f #Hf #HL12 #p
+elim (frees_total L1 (ⓑ{p,I}V1.T)) #g #Hg
+elim (frees_inv_bind … Hg) #y1 #y2 #_ #H #Hy
+lapply(frees_mono … H … Hf) -H #H2
+lapply (tl_eq_repl … H2) -H2 #H2
+lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy
+lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
+lapply (sle_inv_tl_sn … Hfg) -Hfg #Hfg
+elim (sex_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #Y #H #HL2
+lapply (sle_sex_trans … H … Hfg) // #H0
+elim (sex_inv_next1 … H) -H #Z #L #HL1 #H
+elim (ext2_inv_pair_sn … H) -H #V #HV #H1 #H2 destruct
+elim (frees_sex_conf … Hf … H0) -Hf -H0
+/4 width=7 by sle_sex_trans, ex3_2_intro, ex2_intro/
+qed-.
+
+lemma rex_bind_dx_split_void: ∀R1,R2. (∀L. reflexive … (R1 L)) → (∀L. reflexive … (R2 L)) →
+ rex_fsge_compatible R1 →
+ ∀L1,L2,T. L1.ⓧ ⪤[R1, T] L2 → ∀p,I,V.
+ ∃∃L. L1 ⪤[R1, ⓑ{p,I}V.T] L & L.ⓧ ⪤[R2, T] L2.
+#R1 #R2 #HR1 #HR2 #HR #L1 #L2 #T * #f #Hf #HL12 #p #I #V
+elim (frees_total L1 (ⓑ{p,I}V.T)) #g #Hg
+elim (frees_inv_bind_void … Hg) #y1 #y2 #_ #H #Hy
+lapply(frees_mono … H … Hf) -H #H2
+lapply (tl_eq_repl … H2) -H2 #H2
+lapply (sor_eq_repl_back2 … Hy … H2) -y2 #Hy
+lapply (sor_inv_sle_dx … Hy) -y1 #Hfg
+lapply (sle_inv_tl_sn … Hfg) -Hfg #Hfg
+elim (sex_sle_split (cext2 R1) (cext2 R2) … HL12 … Hfg) -HL12 /2 width=1 by ext2_refl/ #Y #H #HL2
+lapply (sle_sex_trans … H … Hfg) // #H0
+elim (sex_inv_next1 … H) -H #Z #L #HL1 #H
+elim (ext2_inv_unit_sn … H) -H #H destruct
+elim (frees_sex_conf … Hf … H0) -Hf -H0
+/4 width=7 by sle_sex_trans, ex2_intro/ (* note: 2 ex2_intro *)
+qed-.
+
+(* Main properties with free variables inclusion for restricted closures ****)
+
+theorem rex_conf: ∀R1,R2.
+ rex_fsge_compatible R1 →
+ rex_fsge_compatible R2 →
+ R_confluent2_rex R1 R2 R1 R2 →
+ ∀T. confluent2 … (rex R1 T) (rex R2 T).
+#R1 #R2 #HR1 #HR2 #HR12 #T #L0 #L1 * #f1 #Hf1 #HL01 #L2 * #f #Hf #HL02
+lapply (frees_mono … Hf1 … Hf) -Hf1 #Hf12
+lapply (sex_eq_repl_back … HL01 … Hf12) -f1 #HL01
+elim (sex_conf … HL01 … HL02) /2 width=3 by ex2_intro/ [ | -HL01 -HL02 ]
+[ #L #HL1 #HL2
+ elim (frees_sex_conf … Hf … HL01) // -HR1 -HL01 #f1 #Hf1 #H1
+ elim (frees_sex_conf … Hf … HL02) // -HR2 -HL02 #f2 #Hf2 #H2
+ lapply (sle_sex_trans … HL1 … H1) // -HL1 -H1 #HL1
+ lapply (sle_sex_trans … HL2 … H2) // -HL2 -H2 #HL2
+ /3 width=5 by ex2_intro/
+| #g * #I0 [2: #V0 ] #K0 #n #HLK0 #Hgf #Z1 #H1 #Z2 #H2 #K1 #HK01 #K2 #HK02
+ [ elim (ext2_inv_pair_sn … H1) -H1 #V1 #HV01 #H destruct
+ elim (ext2_inv_pair_sn … H2) -H2 #V2 #HV02 #H destruct
+ elim (frees_inv_drops_next … Hf … HLK0 … Hgf) -Hf -HLK0 -Hgf #g0 #Hg0 #H0
+ lapply (sle_sex_trans … HK01 … H0) // -HK01 #HK01
+ lapply (sle_sex_trans … HK02 … H0) // -HK02 #HK02
+ elim (HR12 … HV01 … HV02 K1 … K2) /3 width=3 by ext2_pair, ex2_intro/
+ | lapply (ext2_inv_unit_sn … H1) -H1 #H destruct
+ lapply (ext2_inv_unit_sn … H2) -H2 #H destruct
+ /3 width=3 by ext2_unit, ex2_intro/
+ ]
+]
+qed-.
+
+theorem rex_trans_fsle: ∀R1,R2,R3.
+ rex_fsle_compatible R1 → f_transitive_next R1 R2 R3 →
+ ∀L1,L,T. L1 ⪤[R1, T] L →
+ ∀L2. L ⪤[R2, T] L2 → L1 ⪤[R3, T] L2.
+#R1 #R2 #R3 #H1R #H2R #L1 #L #T #H
+lapply (H1R … H) -H1R #H0
+cases H -H #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
+lapply (fsle_inv_frees_eq … H0 … Hf1 … Hf2) -H0 -Hf2
+/4 width=14 by sex_trans_gen, sex_fwd_length, sle_sex_trans, 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/sex_length.ma".
+include "basic_2/static/rex_drops.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Basic_2A1: uses: llpx_sn_fwd_length *)
+lemma rex_fwd_length (R): ∀L1,L2,T. L1 ⪤[R, T] L2 → |L1| = |L2|.
+#R #L1 #L2 #T * /2 width=4 by sex_fwd_length/
+qed-.
+
+(* Properties with length for local environments ****************************)
+
+(* Basic_2A1: uses: llpx_sn_sort *)
+lemma rex_sort_length (R): ∀L1,L2. |L1| = |L2| → ∀s. L1 ⪤[R, ⋆s] L2.
+#R #L1 elim L1 -L1
+[ #Y #H #s >(length_inv_zero_sn … H) -H //
+| #K1 #I1 #IH #Y #H #s
+ elim (length_inv_succ_sn … H) -H #I2 #K2 #HK12 #H destruct
+ /3 width=1 by rex_sort/
+]
+qed.
+
+(* Basic_2A1: uses: llpx_sn_gref *)
+lemma rex_gref_length (R): ∀L1,L2. |L1| = |L2| → ∀l. L1 ⪤[R, §l] L2.
+#R #L1 elim L1 -L1
+[ #Y #H #s >(length_inv_zero_sn … H) -H //
+| #K1 #I1 #IH #Y #H #s
+ elim (length_inv_succ_sn … H) -H #I2 #K2 #HK12 #H destruct
+ /3 width=1 by rex_gref/
+]
+qed.
+
+lemma rex_unit_length (R): ∀L1,L2. |L1| = |L2| → ∀I. L1.ⓤ{I} ⪤[R, #0] L2.ⓤ{I}.
+/3 width=3 by rex_unit, sex_length_isid/ qed.
+
+(* Basic_2A1: uses: llpx_sn_lift_le llpx_sn_lift_ge *)
+lemma rex_lifts_bi (R): d_liftable2_sn … lifts R →
+ ∀L1,L2. |L1| = |L2| → ∀K1,K2,T. K1 ⪤[R, T] K2 →
+ ∀b,f. ⬇*[b, f] L1 ≘ K1 → ⬇*[b, f] L2 ≘ K2 →
+ ∀U. ⬆*[f] T ≘ U → L1 ⪤[R, U] L2.
+#R #HR #L1 #L2 #HL12 #K1 #K2 #T * #f1 #Hf1 #HK12 #b #f #HLK1 #HLK2 #U #HTU
+elim (frees_total L1 U) #f2 #Hf2
+lapply (frees_fwd_coafter … Hf2 … HLK1 … HTU … Hf1) -HTU #Hf
+/4 width=12 by sex_length_cfull, sex_liftable_co_dedropable_bi, cext2_d_liftable2_sn, cfull_lift_sn, ex2_intro/
+qed-.
+
+(* Inversion lemmas with length for local environment ***********************)
+
+lemma rex_inv_zero_length (R): ∀Y1,Y2. Y1 ⪤[R, #0] Y2 →
+ ∨∨ ∧∧ Y1 = ⋆ & Y2 = ⋆
+ | ∃∃I,L1,L2,V1,V2. L1 ⪤[R, V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2
+ | ∃∃I,L1,L2. |L1| = |L2| & Y1 = L1.ⓤ{I} & Y2 = L2.ⓤ{I}.
+#R #Y1 #Y2 #H elim (rex_inv_zero … H) -H *
+/4 width=9 by sex_fwd_length, ex4_5_intro, ex3_3_intro, or3_intro2, or3_intro1, or3_intro0, 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/relocation/lex.ma".
+include "basic_2/static/rex_fsle.ma".
+include "basic_2/static/req.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+(* Properties with generic extension of a context-sensitive relation ********)
+
+lemma rex_lex: ∀R,L1,L2. L1 ⪤[R] L2 → ∀T. L1 ⪤[R, T] L2.
+#R #L1 #L2 * #f #Hf #HL12 #T
+elim (frees_total L1 T) #g #Hg
+/4 width=5 by sex_sdj, sdj_isid_sn, ex2_intro/
+qed.
+
+(* Inversion lemmas with generic extension of a context sensitive relation **)
+
+lemma rex_inv_lex_req: ∀R. c_reflexive … R →
+ rex_fsge_compatible R →
+ ∀L1,L2,T. L1 ⪤[R, T] L2 →
+ ∃∃L. L1 ⪤[R] L & L ≡[T] L2.
+#R #H1R #H2R #L1 #L2 #T * #f1 #Hf1 #HL
+elim (sex_sdj_split … ceq_ext … HL 𝐈𝐝 ?) -HL
+[ #L0 #HL10 #HL02 |*: /2 width=1 by ext2_refl, sdj_isid_dx/ ] -H1R
+lapply (sex_sdj … HL10 f1 ?) /2 width=1 by sdj_isid_sn/ #H
+elim (frees_sex_conf … Hf1 … H) // -H2R -H #f0 #Hf0 #Hf01
+/4 width=7 by sle_sex_trans, (* 2x *) 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/sex_sex.ma".
+include "basic_2/static/frees_fqup.ma".
+include "basic_2/static/rex.ma".
+
+(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma rex_inv_frees: ∀R,L1,L2,T. L1 ⪤[R, T] L2 →
+ ∀f. L1 ⊢ 𝐅*⦃T⦄ ≘ f → L1 ⪤[cext2 R, cfull, f] L2.
+#R #L1 #L2 #T * /3 width=6 by frees_mono, sex_eq_repl_back/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: uses: llpx_sn_dec *)
+lemma rex_dec: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀L1,L2,T. Decidable (L1 ⪤[R, T] L2).
+#R #HR #L1 #L2 #T
+elim (frees_total L1 T) #f #Hf
+elim (sex_dec (cext2 R) cfull … L1 L2 f)
+/4 width=3 by rex_inv_frees, cfull_dec, ext2_dec, ex2_intro, or_intror, or_introl/
+qed-.
+
+(* Main properties **********************************************************)
+
+(* Basic_2A1: uses: llpx_sn_bind llpx_sn_bind_O *)
+theorem rex_bind: ∀R,p,I,L1,L2,V1,V2,T.
+ L1 ⪤[R, V1] L2 → L1.ⓑ{I}V1 ⪤[R, T] L2.ⓑ{I}V2 →
+ L1 ⪤[R, ⓑ{p,I}V1.T] L2.
+#R #p #I #L1 #L2 #V1 #V2 #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
+lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
+/3 width=7 by frees_fwd_isfin, frees_bind, sex_join, isfin_tl, ex2_intro/
+qed.
+
+(* Basic_2A1: llpx_sn_flat *)
+theorem rex_flat: ∀R,I,L1,L2,V,T.
+ L1 ⪤[R, V] L2 → L1 ⪤[R, T] L2 →
+ L1 ⪤[R, ⓕ{I}V.T] L2.
+#R #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2 elim (sor_isfin_ex f1 f2)
+/3 width=7 by frees_fwd_isfin, frees_flat, sex_join, ex2_intro/
+qed.
+
+theorem rex_bind_void: ∀R,p,I,L1,L2,V,T.
+ L1 ⪤[R, V] L2 → L1.ⓧ ⪤[R, T] L2.ⓧ →
+ L1 ⪤[R, ⓑ{p,I}V.T] L2.
+#R #p #I #L1 #L2 #V #T * #f1 #HV #Hf1 * #f2 #HT #Hf2
+lapply (sex_fwd_bind … Hf2) -Hf2 #Hf2 elim (sor_isfin_ex f1 (⫱f2))
+/3 width=7 by frees_fwd_isfin, frees_bind_void, sex_join, isfin_tl, ex2_intro/
+qed.
+
+(* Negated inversion lemmas *************************************************)
+
+(* Basic_2A1: uses: nllpx_sn_inv_bind nllpx_sn_inv_bind_O *)
+lemma rnex_inv_bind: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ⪤[R, T] L2.ⓑ{I}V → ⊥).
+#R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
+/4 width=2 by rex_bind, or_intror, or_introl/
+qed-.
+
+(* Basic_2A1: uses: nllpx_sn_inv_flat *)
+lemma rnex_inv_flat: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀I,L1,L2,V,T. (L1 ⪤[R, ⓕ{I}V.T] L2 → ⊥) →
+ (L1 ⪤[R, V] L2 → ⊥) ∨ (L1 ⪤[R, T] L2 → ⊥).
+#R #HR #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
+/4 width=1 by rex_flat, or_intror, or_introl/
+qed-.
+
+lemma rnex_inv_bind_void: ∀R. (∀L,T1,T2. Decidable (R L T1 T2)) →
+ ∀p,I,L1,L2,V,T. (L1 ⪤[R, ⓑ{p,I}V.T] L2 → ⊥) →
+ (L1 ⪤[R, V] L2 → ⊥) ∨ (L1.ⓧ ⪤[R, T] L2.ⓧ → ⊥).
+#R #HR #p #I #L1 #L2 #V #T #H elim (rex_dec … HR L1 L2 V)
+/4 width=2 by rex_bind_void, or_intror, or_introl/
+qed-.
}
]
[ { "unbound context-sensitive parallel rst-computation" * } {
- [ [ "strongly normalizing for closures" ] "fsb" + "( ≥[?,?] 𝐒⦃?,?,?⦄ )" "fsb_ffdeq" + "fsb_aaa" + "fsb_csx" + "fsb_fpbg" * ]
+ [ [ "strongly normalizing for closures" ] "fsb" + "( ≥[?,?] 𝐒⦃?,?,?⦄ )" "fsb_fdeq" + "fsb_aaa" + "fsb_csx" + "fsb_fpbg" * ]
[ [ "proper for closures" ] "fpbg" + "( ⦃?,?,?⦄ >[?,?] ⦃?,?,?⦄ )" "fpbg_fqup" + "fpbg_cpxs" + "fpbg_lpxs" + "fpbg_fpbs" + "fpbg_fpbg" * ]
[ [ "for closures" ] "fpbs" + "( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_fqup" + "fpbs_fqus" + "fpbs_aaa" + "fpbs_cpx" + "fpbs_fpb" + "fpbs_cpxs" + "fpbs_lpxs" + "fpbs_csx" + "fpbs_fpbs" * ]
}
[ [ "refinement for lenvs on selected entries" ] "lsubsx" + "( ? ⊢ ? ⊆ⓧ[?,?,?] ? )" "lsubsx_lfsx" + "lsubsx_lsubsx" * ]
[ [ "strongly normalizing for lenvs on referred entries" ] "rdsx" + "( ? ⊢ ⬈*[?,?,?] 𝐒⦃?⦄ )" "rdsx_length" + "rdsx_drops" + "rdsx_fqup" + "rdsx_cpxs" + "rdsx_csx" + "rdsx_rdsx" * ]
[ [ "strongly normalizing for term vectors" ] "csx_vector" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_cnx_vector" + "csx_csx_vector" * ]
- [ [ "strongly normalizing for terms" ] "csx" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_simple" + "csx_simple_theq" + "csx_drops" + "csx_fqus" + "csx_lsubr" + "csx_lfdeq" + "csx_ffdeq" + "csx_aaa" + "csx_gcp" + "csx_gcr" + "csx_lpx" + "csx_cnx" + "csx_fpbq" + "csx_cpxs" + "csx_lpxs" + "csx_csx" * ]
- [ [ "for lenvs on all entries" ] "lpxs" + "( ⦃?,?⦄ ⊢ ⬈*[?] ? )" "lpxs_length" + "lpxs_drops" + "lpxs_lfdeq" + "lpxs_ffdeq" + "lpxs_aaa" + "lpxs_lpx" + "lpxs_cpxs" + "lpxs_lpxs" * ]
+ [ [ "strongly normalizing for terms" ] "csx" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_simple" + "csx_simple_theq" + "csx_drops" + "csx_fqus" + "csx_lsubr" + "csx_rdeq" + "csx_fdeq" + "csx_aaa" + "csx_gcp" + "csx_gcr" + "csx_lpx" + "csx_cnx" + "csx_fpbq" + "csx_cpxs" + "csx_lpxs" + "csx_csx" * ]
+ [ [ "for lenvs on all entries" ] "lpxs" + "( ⦃?,?⦄ ⊢ ⬈*[?] ? )" "lpxs_length" + "lpxs_drops" + "lpxs_rdeq" + "lpxs_fdeq" + "lpxs_aaa" + "lpxs_lpx" + "lpxs_cpxs" + "lpxs_lpxs" * ]
[ [ "for binders" ] "cpxs_ext" + "( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" * ]
- [ [ "for terms" ] "cpxs" + "( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_theq" + "cpxs_theq_vector" + "cpxs_drops" + "cpxs_fqus" + "cpxs_lsubr" + "cpxs_lfdeq" + "cpxs_ffdeq" + "cpxs_aaa" + "cpxs_lpx" + "cpxs_cnx" + "cpxs_cpxs" * ]
+ [ [ "for terms" ] "cpxs" + "( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_theq" + "cpxs_theq_vector" + "cpxs_drops" + "cpxs_fqus" + "cpxs_lsubr" + "cpxs_rdeq" + "cpxs_fdeq" + "cpxs_aaa" + "cpxs_lpx" + "cpxs_cnx" + "cpxs_cpxs" * ]
}
]
}
[ { "rt-transition" * } {
[ { "unbound parallel rst-transition" * } {
[ [ "for closures" ] "fpbq" + "( ⦃?,?,?⦄ ≽[?,?] ⦃?,?,?⦄ )" "fpbq_aaa" + "fpbq_fpb" * ]
- [ [ "proper for closures" ] "fpb" + "( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpb_lfdeq" + "fpb_ffdeq" * ]
+ [ [ "proper for closures" ] "fpb" + "( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpb_rdeq" + "fpb_fdeq" * ]
}
]
[ { "context-sensitive parallel r-transition" * } {
]
[ { "unbound context-sensitive parallel rt-transition" * } {
[ [ "normal form for terms" ] "cnx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] 𝐍⦃?⦄ )" "cnx_simple" + "cnx_drops" + "cnx_cnx" * ]
- [ [ "for lenvs on referred entries" ] "lfpx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "lfpx_length" + "lfpx_fqup" + "lfpx_fsle" + "lfpx_lfdeq" + "lfpx_lpx" + "lfpx_lfpx" * ]
- [ [ "for lenvs on all entries" ] "lpx" + "( ⦃?,?⦄ ⊢ ⬈[?] ? )" "lpx_length" + "lpx_drops" + "lpx_fquq" + "lpx_fsle" + "lpx_lfdeq" + "lpx_aaa" * ]
+ [ [ "for lenvs on referred entries" ] "rpx" + "( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "rpx_length" + "rpx_fqup" + "rpx_fsle" + "rpx_rdeq" + "rpx_lpx" + "rpx_rpx" * ]
+ [ [ "for lenvs on all entries" ] "lpx" + "( ⦃?,?⦄ ⊢ ⬈[?] ? )" "lpx_length" + "lpx_drops" + "lpx_fquq" + "lpx_fsle" + "lpx_rdeq" + "lpx_aaa" * ]
[ [ "for binders" ] "cpx_ext" + "( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" * ]
- [ [ "for terms" ] "cpx" + "( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_fqus" + "cpx_lsubr" + "cpx_lfeq" + "cpx_lfdeq" + "cpx_ffdeq" * ]
+ [ [ "for terms" ] "cpx" + "( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_fqus" + "cpx_lsubr" + "cpx_req" + "cpx_rdeq" + "cpx_fdeq" * ]
}
]
[ { "bound context-sensitive parallel rt-transition" * } {
class "water"
[ { "iterated static typing" * } {
[ { "iterated generic extension of a context-sensitive relation" * } {
- [ [ "for lenvs on referred entries" ] "tc_lfxs" + "( ? ⦻**[?,?] ? )" "tc_lfxs_length" + "tc_lfxs_lex" + "tc_lfxs_drops" + "tc_lfxs_fqup" + "tc_lfxs_tc_lfxs" * ]
+ [ [ "for lenvs on referred entries" ] "rexs" + "( ? ⪤*[?,?] ? )" "rexs_length" + "rexs_lex" + "rexs_drops" + "rexs_fqup" + "rexs_rexs" * ]
}
]
}
]
[ { "atomic arity assignment" * } {
[ [ "restricted refinement for lenvs" ] "lsuba" + "( ? ⊢ ? ⫃⁝ ? )" "lsuba_drops" + "lsuba_lsubr" + "lsuba_aaa" + "lsuba_lsuba" * ]
- [ [ "for terms" ] "aaa" + "( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + "aaa_lfdeq" + "aaa_ffdeq" + "aaa_aaa" * ]
+ [ [ "for terms" ] "aaa" + "( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + "aaa_rdeq" + "aaa_fdeq" + "aaa_aaa" * ]
}
]
[ { "degree-based equivalence" * } {
- [ [ "for closures on referred entries" ] "ffdeq" + "( ⦃?,?,?⦄ ≛[?,?] ⦃?,?,?⦄ )" "ffdeq_fqup" + "ffdeq_fqus" + "ffdeq_lfeq" + "ffdeq_ffdeq" * ]
- [ [ "for lenvs on referred entries" ] "lfdeq" + "( ? ≛[?,?,?] ? )" "lfdeq_length" + "lfdeq_drops" + "lfdeq_fqup" + "lfdeq_fqus" + "lfdeq_lfeq" + "lfdeq_lfdeq" * ]
+ [ [ "for closures on referred entries" ] "fdeq" + "( ⦃?,?,?⦄ ≛[?,?] ⦃?,?,?⦄ )" "fdeq_fqup" + "fdeq_fqus" + "fdeq_req" + "fdeq_fdeq" * ]
+ [ [ "for lenvs on referred entries" ] "rdeq" + "( ? ≛[?,?,?] ? )" "rdeq_length" + "rdeq_drops" + "rdeq_fqup" + "rdeq_fqus" + "rdeq_req" + "rdeq_rdeq" * ]
}
]
[ { "syntactic equivalence" * } {
- [ [ "for lenvs on referred entries" ] "lfeq" + "( ? ≡[?] ? )" "lfeq_fqup" + "lfeq_fsle" * ]
+ [ [ "for lenvs on referred entries" ] "req" + "( ? ≡[?] ? )" "req_fqup" + "req_fsle" * ]
}
]
[ { "generic extension of a context-sensitive relation" * } {
- [ [ "for lenvs on referred entries" ] "lfxs" + "( ? ⦻*[?,?] ? )" "lfxs_length" + "lfxs_lex" + "lfxs_drops" + "lfxs_fqup" + "lfxs_fsle" + "lfxs_lfxs" * ]
+ [ [ "for lenvs on referred entries" ] "rex" + "( ? ⪤[?,?] ? )" "rex_length" + "rex_lex" + "rex_drops" + "rex_fqup" + "rex_fsle" + "rex_rex" * ]
}
]
[ { "context-sensitive free variables" * } {
class "orange"
[ { "relocation" * } {
[ { "generic slicing" * } {
- [ [ "for lenvs" ] "drops" + "( ⬇*[?,?] ? ≘ ? )" + "( ⬇*[?] ? ≘ ? )" "drops_ctc" + "drops_ltc" + "drops_weight" + "drops_length" + "drops_cext2" + "drops_lexs" + "drops_lex" + "drops_lreq" + "drops_drops" + "drops_vector" * ]
+ [ [ "for lenvs" ] "drops" + "( ⬇*[?,?] ? ≘ ? )" + "( ⬇*[?] ? ≘ ? )" "drops_ctc" + "drops_ltc" + "drops_weight" + "drops_length" + "drops_cext2" + "drops_sex" + "drops_lex" + "drops_seq" + "drops_drops" + "drops_vector" * ]
}
]
[ { "generic relocation" * } {
}
]
[ { "syntactic equivalence" * } {
- [ [ "for lenvs on selected entries" ] "lreq" + "( ? ≡[?] ? )" "lreq_length" + "lreq_lreq" * ]
+ [ [ "for lenvs on selected entries" ] "seq" + "( ? ≡[?] ? )" "seq_length" + "seq_seq" * ]
}
]
[ { "generic entrywise extension" * } {
- [ [ "for lenvs of one contex-sensitive relation" ] "lex" + "( ? ⦻[?] ? )" "lex_tc" + "lex_length" + "lex_lex" * ]
- [ [ "for lenvs of two contex-sensitive relations" ] "lexs" + "( ? ⦻*[?,?,?] ? )" "lexs_tc" + "lexs_length" + "lexs_lexs" * ]
+ [ [ "for lenvs of one contex-sensitive relation" ] "lex" + "( ? ⪤[?] ? )" "lex_tc" + "lex_length" + "lex_lex" * ]
+ [ [ "for lenvs of two contex-sensitive relations" ] "sex" + "( ? ⪤[?,?,?] ? )" "sex_tc" + "sex_length" + "sex_sex" * ]
}
]
}
[ [ "" ] "scpes ( ⦃?,?⦄ ⊢ ? •*⬌*[?,?,?,?] ? )" "scpes_aaa" + "scpes_cpcs" + "scpes_scpes" * ]
}
]
- [ [ "for lenvs on referred entries" ] "lfpxs" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] ? )" "lfpxs_length" + "lfpxs_drops" + "lfpxs_fqup" + "lfpxs_lfdeq" + "lfpxs_ffdeq" + "lfpxs_aaa" + "lfpxs_cpxs" + "lfpxs_lpxs" + "lfpxs_lfpxs" * ]
+ [ [ "for lenvs on referred entries" ] "rpxs" + "( ⦃?,?⦄ ⊢ ⬈*[?,?] ? )" "rpxs_length" + "rpxs_drops" + "rpxs_fqup" + "rpxs_rdeq" + "rpxs_fdeq" + "rpxs_aaa" + "rpxs_cpxs" + "rpxs_lpxs" + "rpxs_rpxs" * ]
[ [ "for lenvs on referred entries" ]
- "lfpr" + "( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lfpr_length" + "lfpr_drops" + "lfpr_fquq" + "lfpr_fqup" + "lfpr_aaa" + "lfpr_lfpx" + "lfpr_lfpr" * ]
+ "lfpr" + "( ⦃?,?⦄ ⊢ ➡[?,?] ? )" "lfpr_length" + "lfpr_drops" + "lfpr_fquq" + "lfpr_fqup" + "lfpr_aaa" + "lfpr_rpx" + "lfpr_lfpr" * ]
[ { "evaluation for context-sensitive rt-reduction" * } {
[ [ "" ] "cpxe ( ⦃?,?⦄ ⊢ ➡*[?,?] 𝐍⦃?⦄ )" * ]
}
#!/bin/sh
-for MA in `find -name "*.ma"`; do
- if [ -s ${MA}.old ];
- then echo ${MA}; mv -f ${MA}.old ${MA};
+for SRC in `find -name "*.ma" -or -name "*.tbl"`; do
+ if [ -s ${SRC}.old ];
+ then echo ${SRC}; mv -f ${SRC}.old ${SRC};
fi
done
-unset MA
+unset SRC
projects / helm.git / commitdiff