+++ /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/relationstarstar_4.ma".
-include "basic_2/static/lfxs.ma".
-
-(* GENERIC EXTENSION ON REFERRED ENTRIES OF A CONTEXT-SENSITIVE REALTION ****)
-
-definition lfxss (R) (T): relation lenv ≝ TC … (lfxs R T).
-
-interpretation "tc of generic extension on referred entries (local environment)"
- 'RelationStarStar R T L1 L2 = (lfxss R T L1 L2).
-
-(* Basic properties ***********************************************************)
-
-lemma lfxss_atom: ∀R,I. ⋆ ⦻**[R, ⓪{I}] ⋆.
-/2 width=1 by inj/ qed.
-
-lemma lfxss_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, step, inj/
-qed.
-
-lemma lfxss_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, step, inj/
-qed.
-
-lemma lfxss_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, step, inj/
-qed.
-
-lemma lfxss_sym: ∀R. lexs_frees_confluent R cfull →
- (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀T. symmetric … (lfxss R T).
-#R #H1R #H2R #T #L1 #L2 #H elim H -L2
-/4 width=3 by lfxs_sym, TC_strap, inj/
-qed-.
-
-lemma lfxss_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, step, inj/
-qed-.
-(*
-(* Basic inversion lemmas ***************************************************)
-
-lemma lfxs_inv_atom_sn: ∀R,I,Y2. ⋆ ⦻*[R, ⓪{I}] Y2 → Y2 = ⋆.
-#R #I #Y2 * /2 width=4 by lexs_inv_atom1/
-qed-.
-
-lemma lfxs_inv_atom_dx: ∀R,I,Y1. Y1 ⦻*[R, ⓪{I}] ⋆ → 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 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, ⋆s] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R * [ | #Y1 #I #V1 ] #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 #L2 #V2 #H12 #_ #H destruct
- /5 width=8 by frees_sort_gen, ex3_5_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.
-#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
-[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
-| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
- /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
-]
-qed-.
-
-lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
-[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
-| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
- /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
-]
-qed-.
-
-lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
- (Y1 = ⋆ ∧ Y2 = ⋆) ∨
- ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
- Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
-#R * [ | #Y1 #I #V1 ] #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 #L2 #V2 #H12 #_ #H destruct
- /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
-]
-qed-.
-
-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, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
-qed-.
-
-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_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 →
- ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
-[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
-[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
- ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
- Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
-[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
- Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
-[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 →
- ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
-[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
-[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 →
- ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
-#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
-[ #H destruct
-| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
- ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
-#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
-[ #_ #H destruct
-| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
-#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
-/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
-qed-.
-
-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-.
-
-lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
-#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
-qed-.
-
-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_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
-#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
-qed-.
-
-(* Basic_2A1: removed theorems 24:
- llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
- llpx_sn_bind llpx_sn_flat
- llpx_sn_inv_bind llpx_sn_inv_flat
- llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
- llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
- llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
- 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/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) (T): relation lenv ≝ TC … (lfxs R T).
+
+interpretation "iterated extension on referred entries (local environment)"
+ 'RelationStarStar R T L1 L2 = (tc_lfxs R T L1 L2).
+
+(* Basic properties *********************************************************)
+
+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, step, 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, step, 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, step, inj/
+qed.
+
+lemma tc_lfxs_sym: ∀R. lexs_frees_confluent R cfull →
+ (∀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_strap, 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, step, inj/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+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-.
+
+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 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻**[R, ⋆s] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R #Y1 #Y2 #s #H elim H -Y2
+[ #Y2 #H elim (lfxs_inv_sort … H) -H *
+ /4 width=8 by ex3_5_intro, inj, or_introl, or_intror, conj/
+| #Y #Y2 #_ #H elim (lfxs_inv_sort … H) -H *
+ [ #H #H2 * * /3 width=8 by ex3_5_intro, or_introl, or_intror, conj/
+ | #I #L #L2 #V #V2 #HL2 #H #H2 * *
+ [ #H1 #H0 destruct
+ | #I0 #L1 #L0 #V1 #V0 #HL10 #H1 #H0 destruct
+ /4 width=8 by ex3_5_intro, step, or_intror/
+ ]
+ ]
+]
+qed-.
+(*
+lemma tc_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.
+#R #Y1 #Y2 #H elim H -Y2
+[ #Y2 #H elim (lfxs_inv_zero … H) -H *
+ /4 width=9 by ex4_5_intro, inj, or_introl, or_intror, conj/
+| #Y #Y2 #_ #H elim (lfxs_inv_zero … H) -H *
+ [ #H #H2 * * /3 width=9 by ex4_5_intro, or_introl, or_intror, conj/
+ | #I #L #L2 #V #V2 #HL2 #HV2 #H #H2 * *
+ [ #H1 #H0 destruct
+ | #I0 #L1 #L0 #V1 #V0 #HL10 #HV10 #H1 #H0 destruct
+ @or_intror @ex4_5_intro [6,7: |*: /width=7/ ]
+
+ /4 width=8 by ex3_5_intro, step, or_intror/
+ ]
+ ]
+]
+qed-.
+
+
+
+
+
+#R #Y1 #Y2 * #f #H1 #H2 elim (frees_inv_zero … H1) -H1 *
+[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
+| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_next1_aux … H2 … HY1 Hg) -H2 -Hg
+ /4 width=9 by ex4_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_lref: ∀R,Y1,Y2,i. Y1 ⦻*[R, #⫯i] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, #i] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R #Y1 #Y2 #i * #f #H1 #H2 elim (frees_inv_lref … H1) -H1 *
+[ #H #_ lapply (lexs_inv_atom1_aux … H2 H) -H2 /3 width=1 by or_introl, conj/
+| #I1 #L1 #V1 #g #HV1 #HY1 #Hg elim (lexs_inv_push1_aux … H2 … HY1 Hg) -H2 -Hg
+ /4 width=8 by ex3_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+lemma lfxs_inv_gref: ∀R,Y1,Y2,l. Y1 ⦻*[R, §l] Y2 →
+ (Y1 = ⋆ ∧ Y2 = ⋆) ∨
+ ∃∃I,L1,L2,V1,V2. L1 ⦻*[R, §l] L2 &
+ Y1 = L1.ⓑ{I}V1 & Y2 = L2.ⓑ{I}V2.
+#R * [ | #Y1 #I #V1 ] #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 #L2 #V2 #H12 #_ #H destruct
+ /5 width=8 by frees_gref_gen, ex3_5_intro, ex2_intro, or_intror/
+]
+qed-.
+
+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, sor_inv_sle_dx, sor_inv_sle_sn, ex2_intro, conj/
+qed-.
+
+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_pair_sn: ∀R,I,Y2,L1,V1,s. L1.ⓑ{I}V1 ⦻*[R, ⋆s] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, ⋆s] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #s #H elim (lfxs_inv_sort … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_sort_pair_dx: ∀R,I,Y1,L2,V2,s. Y1 ⦻*[R, ⋆s] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, ⋆s] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #s #H elim (lfxs_inv_sort … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #Hs #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_zero_pair_sn: ∀R,I,Y2,L1,V1. L1.ⓑ{I}V1 ⦻*[R, #0] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #H elim (lfxs_inv_zero … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #HV1 #HV12 #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_zero_pair_dx: ∀R,I,Y1,L2,V2. Y1 ⦻*[R, #0] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, V1] L2 & R L1 V1 V2 &
+ Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #H elim (lfxs_inv_zero … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #HV1 #HV12 #H1 #H2 destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_lref_pair_sn: ∀R,I,Y2,L1,V1,i. L1.ⓑ{I}V1 ⦻*[R, #⫯i] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, #i] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #i #H elim (lfxs_inv_lref … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_lref_pair_dx: ∀R,I,Y1,L2,V2,i. Y1 ⦻*[R, #⫯i] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, #i] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #i #H elim (lfxs_inv_lref … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #Hi #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_gref_pair_sn: ∀R,I,Y2,L1,V1,l. L1.ⓑ{I}V1 ⦻*[R, §l] Y2 →
+ ∃∃L2,V2. L1 ⦻*[R, §l] L2 & Y2 = L2.ⓑ{I}V2.
+#R #I #Y2 #L1 #V1 #l #H elim (lfxs_inv_gref … H) -H *
+[ #H destruct
+| #J #Y1 #L2 #X1 #V2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lfxs_inv_gref_pair_dx: ∀R,I,Y1,L2,V2,l. Y1 ⦻*[R, §l] L2.ⓑ{I}V2 →
+ ∃∃L1,V1. L1 ⦻*[R, §l] L2 & Y1 = L1.ⓑ{I}V1.
+#R #I #Y1 #L2 #V2 #l #H elim (lfxs_inv_gref … H) -H *
+[ #_ #H destruct
+| #J #L1 #Y2 #V1 #X2 #Hl #H1 #H2 destruct /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lfxs_fwd_bind_sn: ∀R,p,I,L1,L2,V,T. L1 ⦻*[R, ⓑ{p,I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R #p #I #L1 #L2 #V #T * #f #Hf #HL elim (frees_inv_bind … Hf) -Hf
+/4 width=6 by sle_lexs_trans, sor_inv_sle_sn, ex2_intro/
+qed-.
+
+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-.
+
+lemma lfxs_fwd_flat_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ⓕ{I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R #I #L1 #L2 #V #T #H elim (lfxs_inv_flat … H) -H //
+qed-.
+
+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_pair_sn: ∀R,I,L1,L2,V,T. L1 ⦻*[R, ②{I}V.T] L2 → L1 ⦻*[R, V] L2.
+#R * /2 width=4 by lfxs_fwd_flat_sn, lfxs_fwd_bind_sn/
+qed-.
+
+(* Basic_2A1: removed theorems 24:
+ llpx_sn_sort llpx_sn_skip llpx_sn_lref llpx_sn_free llpx_sn_gref
+ llpx_sn_bind llpx_sn_flat
+ llpx_sn_inv_bind llpx_sn_inv_flat
+ llpx_sn_fwd_lref llpx_sn_fwd_pair_sn llpx_sn_fwd_length
+ llpx_sn_fwd_bind_sn llpx_sn_fwd_bind_dx llpx_sn_fwd_flat_sn llpx_sn_fwd_flat_dx
+ llpx_sn_refl llpx_sn_Y llpx_sn_bind_O llpx_sn_ge_up llpx_sn_ge llpx_sn_co
+ 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/rt_transition/cpx_drops.ma".
+include "basic_2/rt_computation/cpxs.ma".
+
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpxs_delta: ∀h,I,G,K,V1,V2. ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀W2. ⬆*[1] V2 ≡ W2 → ⦃G, K.ⓑ{I}V1⦄ ⊢ #0 ⬈*[h] W2.
+#h #I #G #K #V1 #V2 #H @(cpxs_ind … H) -V2
+[ /3 width=3 by cpx_cpxs, cpx_delta/
+| #V #V2 #_ #HV2 #IH #W2 #HVW2
+ elim (lifts_total V (𝐔❴1❵)) #W #HVW
+ elim (cpx_lifts … HV2 (Ⓣ) … (K.ⓑ{I}V1) … HVW) -HV2
+ [ #V0 #HV20 <(lifts_mono … HVW2 … HV20) -V2 -V0 /3 width=3 by cpxs_strap1/
+ | /3 width=1 by drops_refl, drops_drop/
+ ]
+]
+qed.
+
+lemma cpxs_lref: ∀h,I,G,K,V,T,i. ⦃G, K⦄ ⊢ #i ⬈*[h] T →
+ ∀U. ⬆*[1] T ≡ U → ⦃G, K.ⓑ{I}V⦄ ⊢ #⫯i ⬈*[h] U.
+#h #I #G #K #V #T #i #H @(cpxs_ind … H) -T
+[ /3 width=3 by cpx_cpxs, cpx_lref/
+| #T0 #T #_ #HT2 #IH #U #HTU
+ elim (lifts_total T0 (𝐔❴1❵)) #U0 #HTU0
+ elim (cpx_lifts … HT2 (Ⓣ) … (K.ⓑ{I}V) … HTU0) -HT2
+ [ #X #HTX <(lifts_mono … HTU … HTX) -T -X /3 width=3 by cpxs_strap1/
+ | /3 width=1 by drops_refl, drops_drop/
+ ]
+]
+qed.
+
+(* Basic_2A1: was: cpxs_delta *)
+lemma cpxs_delta_drops: ∀h,I,G,L,K,V1,V2,i.
+ ⬇*[i] L ≡ K.ⓑ{I}V1 → ⦃G, K⦄ ⊢ V1 ⬈*[h] V2 →
+ ∀W2. ⬆*[⫯i] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ⬈*[h] W2.
+#h #I #G #L #K #V1 #V2 #i #HLK #H @(cpxs_ind … H) -V2
+[ /3 width=7 by cpx_cpxs, cpx_delta_drops/
+| #V #V2 #_ #HV2 #IH #W2 #HVW2
+ elim (lifts_total V (𝐔❴⫯i❵)) #W #HVW
+ elim (cpx_lifts … HV2 (Ⓣ) … L … HVW) -HV2
+ [ #V0 #HV20 <(lifts_mono … HVW2 … HV20) -V2 -V0 /3 width=3 by cpxs_strap1/
+ | /2 width=3 by drops_isuni_fwd_drop2/
+ ]
+]
+qed.
+(*
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpxs_inv_lref1: ∀h,o,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈*[h, o] T2 →
+ T2 = #i ∨
+ ∃∃I,K,V1,T1. ⬇[i] L ≡ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ⬈*[h, o] T1 &
+ ⬆[0, i+1] T1 ≡ T2.
+#h #o #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
+#T #T2 #_ #HT2 *
+[ #H destruct
+ elim (cpx_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
+ * /4 width=7 by cpx_cpxs, ex3_4_intro, or_intror/
+| * #I #K #V1 #T1 #HLK #HVT1 #HT1
+ lapply (drop_fwd_drop2 … HLK) #H0LK
+ elim (cpx_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
+ /4 width=7 by cpxs_strap1, ex3_4_intro, or_intror/
+]
+qed-.
+
+(* Relocation properties ****************************************************)
+
+lemma cpxs_lift: ∀h,o,G. d_liftable (cpxs h o G).
+/3 width=10 by cpx_lift, cpxs_strap1, d_liftable_LTC/ qed.
+
+lemma cpxs_inv_lift1: ∀h,o,G. d_deliftable_sn (cpxs h o G).
+/3 width=6 by d_deliftable_sn_LTC, cpx_inv_lift1/
+qed-.
+*)
--- /dev/null
+include "basic_2/computation/cpxs_lift.ma".
+include "basic_2/multiple/fqus_fqus.ma".
+
+(* Advanced properties ******************************************************)
+
+lemma lstas_cpxs: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
+ ∀d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2.
+#h #o #G #L #T1 #T2 #d2 #H elim H -G -L -T1 -T2 -d2 //
+[ /3 width=3 by cpxs_sort, da_inv_sort/
+| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
+ elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
+ lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpxs_delta/
+| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
+ elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
+ lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
+ #HV1 #H destruct lapply (le_plus_to_le_c … Hd21) -Hd21
+ /3 width=7 by cpxs_delta/
+| /4 width=3 by cpxs_bind_dx, da_inv_bind/
+| /4 width=3 by cpxs_flat_dx, da_inv_flat/
+| /4 width=3 by cpxs_eps, da_inv_flat/
+]
+qed.
+
+(* Properties on supclosure *************************************************)
+
+lemma fqu_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
+ ∀T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
+#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
+#T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
+qed-.
+
+lemma fquq_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
+ ∀T1. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fquq_inv_gen … H) -H
+[ #HT12 elim (fqu_cpxs_trans … HTU2 … HT12) /3 width=3 by fqu_fquq, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma fquq_lstas_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
+ ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d2 → d1 ≤ d2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
+/3 width=5 by fquq_cpxs_trans, lstas_cpxs/ qed-.
+
+lemma fqup_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
+ ∀T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
+#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
+#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
+qed-.
+
+lemma fqus_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
+ ∀T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+#h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fqus_inv_gen … H) -H
+[ #HT12 elim (fqup_cpxs_trans … HTU2 … HT12) /3 width=3 by fqup_fqus, ex2_intro/
+| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma fqus_lstas_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
+ ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d2 → d1 ≤ d2 →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
+/3 width=6 by fqus_cpxs_trans, lstas_cpxs/ 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/lfpx_lfpx.ma".
+include "basic_2/rt_computation/cpxs_drops.ma".
+include "basic_2/rt_computation/cpxs_cpxs.ma".
+
+(* UNCOUNTED CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS ************)
+
+(* Properties with uncounted parallel rt-transition on referred entries *****)
+
+lemma lfpx_cpxs_conf: ∀h,G. s_r_confluent1 … (cpxs h G) (lfpx h G).
+/3 width=5 by lfpx_cpx_conf, s_r_conf1_LTC1/ qed-.
+
+lemma lfpx_cpx_trans: ∀h,G. s_r_transitive … (cpx h G) (lfpx h G).
+#h #G #L2 #T1 #T2 #H @(cpx_ind … H) -G -L2 -T1 -T2 //
+[ #G #L2 #s1 #L1 #H elim (lfpx_inv_sort … H) -H * /2 width=1 by cpx_cpxs/
+| #I #G #L2 #V #V2 #W2 #_ #IH #HVW2 #Y1 #H
+ elim (lfpx_inv_zero_pair_dx … H) -H #L1 #V1 #HL1 #HV1 #H destruct
+ /5 width=3 by lfpx_cpx_conf, cpxs_delta, cpxs_strap2/
+| #I #G #L2 #V #V2 #W2 #i #_ #IH #HVW2 #Y1 #H
+ elim (lfpx_inv_lref_pair_dx … H) -H #L1 #V1 #HL1 #HV1
+ /4 width=3 by cpxs_lref, cpxs_strap2/
+| #p #I #G #L2 #V #V2 #T #T2 #_ #_ #IHV #IHT #L1 #H
+ elim (lfpx_inv_bind … H) -H /3 width=1 by cpxs_bind/
+| #I #G #L2 #V #V2 #T #T2 #_ #_ #IHV #IHT #L1 #H
+ elim (lfpx_inv_flat … H) -H /3 width=1 by cpxs_flat/
+| #G #L2 #V #T #T2 #T0 #_ #IH #HT02 #L1 #H
+ elim (lfpx_inv_bind … H) -H /3 width=3 by cpxs_zeta/
+| #G #L2 #V #T #T2 #_ #IH #L1 #H
+ elim (lfpx_inv_flat … H) -H /3 width=1 by cpxs_eps/
+| #G #L2 #V #V2 #T #_ #IH #L1 #H
+ elim (lfpx_inv_flat … H) -H /3 width=1 by cpxs_ee/
+| #p #G #L2 #V #V2 #W #W2 #T #T2 #_ #_ #_ #IHV #IHW #IHT #L1 #H
+ elim (lfpx_inv_flat … H) -H #HV #H
+ elim (lfpx_inv_bind … H) -H /3 width=1 by cpxs_beta/
+| #p #G #L2 #V #V2 #V0 #W #W2 #T #T2 #_ #_ #_ #IHV #IHW #IHT #HV20 #L1 #H
+ elim (lfpx_inv_flat … H) -H #HV #H
+ elim (lfpx_inv_bind … H) -H /3 width=3 by cpxs_theta/
+]
+qed.
+
+lemma lfpx_cpxs_trans: ∀h,G. s_rs_transitive … (cpx h G) (lfpx h G).
+#h #G #L2 #T1 #T2 #H #L1 #HL12 @(cpxs_ind … H) -T2
+/4 width=7 by lfpx_cpx_trans, cpxs_trans, lfpx_cpxs_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/multiple/fqus_fqus.ma".
-include "basic_2/reduction/cpx_lift.ma".
-include "basic_2/computation/cpxs.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
-
-(* Advanced properties ******************************************************)
-
-lemma cpxs_delta: ∀h,o,I,G,L,K,V,V2,i.
- ⬇[i] L ≡ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⬈*[h, o] V2 →
- ∀W2. ⬆[0, i+1] V2 ≡ W2 → ⦃G, L⦄ ⊢ #i ⬈*[h, o] W2.
-#h #o #I #G #L #K #V #V2 #i #HLK #H elim H -V2
-[ /3 width=9 by cpx_cpxs, cpx_delta/
-| #V1 lapply (drop_fwd_drop2 … HLK) -HLK
- elim (lift_total V1 0 (i+1)) /4 width=12 by cpx_lift, cpxs_strap1/
-]
-qed.
-
-lemma lstas_cpxs: ∀h,o,G,L,T1,T2,d2. ⦃G, L⦄ ⊢ T1 •*[h, d2] T2 →
- ∀d1. ⦃G, L⦄ ⊢ T1 ▪[h, o] d1 → d2 ≤ d1 → ⦃G, L⦄ ⊢ T1 ⬈*[h, o] T2.
-#h #o #G #L #T1 #T2 #d2 #H elim H -G -L -T1 -T2 -d2 //
-[ /3 width=3 by cpxs_sort, da_inv_sort/
-| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
- elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
- lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct /3 width=7 by cpxs_delta/
-| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
- elim (da_inv_lref … H) -H * #K0 #V0 [| #d0 ] #HLK0
- lapply (drop_mono … HLK0 … HLK) -HLK0 #H destruct
- #HV1 #H destruct lapply (le_plus_to_le_c … Hd21) -Hd21
- /3 width=7 by cpxs_delta/
-| /4 width=3 by cpxs_bind_dx, da_inv_bind/
-| /4 width=3 by cpxs_flat_dx, da_inv_flat/
-| /4 width=3 by cpxs_eps, da_inv_flat/
-]
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cpxs_inv_lref1: ∀h,o,G,L,T2,i. ⦃G, L⦄ ⊢ #i ⬈*[h, o] T2 →
- T2 = #i ∨
- ∃∃I,K,V1,T1. ⬇[i] L ≡ K.ⓑ{I}V1 & ⦃G, K⦄ ⊢ V1 ⬈*[h, o] T1 &
- ⬆[0, i+1] T1 ≡ T2.
-#h #o #G #L #T2 #i #H @(cpxs_ind … H) -T2 /2 width=1 by or_introl/
-#T #T2 #_ #HT2 *
-[ #H destruct
- elim (cpx_inv_lref1 … HT2) -HT2 /2 width=1 by or_introl/
- * /4 width=7 by cpx_cpxs, ex3_4_intro, or_intror/
-| * #I #K #V1 #T1 #HLK #HVT1 #HT1
- lapply (drop_fwd_drop2 … HLK) #H0LK
- elim (cpx_inv_lift1 … HT2 … H0LK … HT1) -H0LK -T
- /4 width=7 by cpxs_strap1, ex3_4_intro, or_intror/
-]
-qed-.
-
-(* Relocation properties ****************************************************)
-
-lemma cpxs_lift: ∀h,o,G. d_liftable (cpxs h o G).
-/3 width=10 by cpx_lift, cpxs_strap1, d_liftable_LTC/ qed.
-
-lemma cpxs_inv_lift1: ∀h,o,G. d_deliftable_sn (cpxs h o G).
-/3 width=6 by d_deliftable_sn_LTC, cpx_inv_lift1/
-qed-.
-
-(* Properties on supclosure *************************************************)
-
-lemma fqu_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
-#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqu_cpx_trans … HT1 … HT2) -T
-#T #HT1 #HT2 elim (IHTU2 … HT2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
-qed-.
-
-lemma fquq_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fquq_inv_gen … H) -H
-[ #HT12 elim (fqu_cpxs_trans … HTU2 … HT12) /3 width=3 by fqu_fquq, ex2_intro/
-| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma fquq_lstas_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
- ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d2 → d1 ≤ d2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐⸮ ⦃G2, L2, U2⦄.
-/3 width=5 by fquq_cpxs_trans, lstas_cpxs/ qed-.
-
-lemma fqup_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐+ ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T2 #U2 #H @(cpxs_ind_dx … H) -T2 /2 width=3 by ex2_intro/
-#T #T2 #HT2 #_ #IHTU2 #T1 #HT1 elim (fqup_cpx_trans … HT1 … HT2) -T
-#U1 #HTU1 #H2 elim (IHTU2 … H2) -T2 /3 width=3 by cpxs_strap2, ex2_intro/
-qed-.
-
-lemma fqus_cpxs_trans: ∀h,o,G1,G2,L1,L2,T2,U2. ⦃G2, L2⦄ ⊢ T2 ⬈*[h, o] U2 →
- ∀T1. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-#h #o #G1 #G2 #L1 #L2 #T2 #U2 #HTU2 #T1 #H elim (fqus_inv_gen … H) -H
-[ #HT12 elim (fqup_cpxs_trans … HTU2 … HT12) /3 width=3 by fqup_fqus, ex2_intro/
-| * #H1 #H2 #H3 destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma fqus_lstas_trans: ∀h,o,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀U2,d1. ⦃G2, L2⦄ ⊢ T2 •*[h, d1] U2 →
- ∀d2. ⦃G2, L2⦄ ⊢ T2 ▪[h, o] d2 → d1 ≤ d2 →
- ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ⬈*[h, o] U1 & ⦃G1, L1, U1⦄ ⊐* ⦃G2, L2, U2⦄.
-/3 width=6 by fqus_cpxs_trans, lstas_cpxs/ qed-.
-cpxs.ma cpxs_tdeq.ma cpxs_cpxs.ma
+cpxs.ma cpxs_tdeq.ma cpxs_drops.ma cpxs_lfpx.ma cpxs_cpxs.ma
lfpxs.ma lfpxs_fqup.ma
csx.ma csx_cnx.ma csx_cpxs.ma csx_csx.ma
csx_vector.ma
--- /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_lfxs.ma".
+include "basic_2/rt_transition/lfpx.ma".
+
+(* UNCOUNTED PARALLEL RT-TRANSITION FOR LOCAL ENV.S ON REFERRED ENTRIES *****)
+
+(* 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-.
cpg.ma cpg_simple.ma cpg_drops.ma cpg_lsubr.ma
cpx.ma cpx_simple.ma cpx_drops.ma cpx_fqus.ma cpx_lsubr.ma cpx_lfxs.ma
-lfpx.ma lfpx_length.ma lfpx_drops.ma lfpx_fqup.ma lfpx_frees.ma lfpx_lfdeq.ma lfpx_aaa.ma
+lfpx.ma lfpx_length.ma lfpx_drops.ma lfpx_fqup.ma lfpx_frees.ma lfpx_lfdeq.ma lfpx_aaa.ma lfpx_lfpx.ma
cnx.ma cnx_simple.ma cnx_drops.ma
cpm.ma cpm_simple.ma cpm_drops.ma cpm_lsubr.ma cpm_lfxs.ma cpm_cpx.ma
cpr.ma cpr_drops.ma
∀L1. L0 ⦻*[RP1, T0] L1 → ∀L2. L0 ⦻*[RP2, T0] L2 →
∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-(* Basic properties ***********************************************************)
+(* Basic properties *********************************************************)
lemma lfxs_atom: ∀R,I. ⋆ ⦻*[R, ⓪{I}] ⋆.
/3 width=3 by lexs_atom, frees_atom, ex2_intro/
[ "csx_vector ( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" * ]
[ "csx ( ⦃?,?⦄ ⊢ ⬈*[?,?] 𝐒⦃?⦄ )" "csx_cnx" + "csx_cpxs" + "csx_csx" * ]
[ "lfpxs ( ⦃?,?⦄ ⊢ ⬈*[?,?] ? )" "lfpxs_length" + "lfpxs_fqup" * ]
- [ "cpxs ( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_cpxs" * ]
+ [ "cpxs ( ⦃?,?⦄ ⊢ ? ⬈*[?] ? )" "cpxs_tdeq" + "cpxs_drops" + "cpxs_lfpx" + "cpxs_cpxs" * ]
}
]
}
]
[ { "uncounted context-sensitive rt-transition" * } {
[ "cnx ( ⦃?,?⦄ ⊢ ⬈[?,?] 𝐍⦃?⦄ )" "cnx_simple" + "cnx_drops" * ]
- [ "lfpx ( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "lfpx_length" + "lfpx_drops" + "lfpx_fqup" + "lfpx_frees" + "lfpx_lfdeq" + "lfpx_aaa" * ]
+ [ "lfpx ( ⦃?,?⦄ ⊢ ⬈[?,?] ? )" "lfpx_length" + "lfpx_drops" + "lfpx_fqup" + "lfpx_frees" + "lfpx_lfdeq" + "lfpx_aaa" + "lfpx_lfpx" * ]
[ "cpx ( ⦃?,?⦄ ⊢ ? ⬈[?] ? )" "cpx_simple" + "cpx_drops" + "cpx_fqus" + "cpx_lsubr" + "cpx_lfxs" * ]
}
]
basic_2/s_computation
basic_2/static
basic_2/i_static
+apps_2/examples/ex_cpr_omega.ma
projects / helm.git / commitdiff