helm/www/lambdadelta/xslt/basic_2_blk.xsl
helm/www/lambdadelta/xslt/basic_2_src.xsl
helm/www/lambdadelta/xslt/basic_2_sum.xsl
+helm/www/lambdadelta/xslt/static_2_src.xsl
+helm/www/lambdadelta/xslt/static_2_sum.xsl
helm/www/lambdadelta/xslt/ground_2_src.xsl
helm/www/lambdadelta/xslt/ground_2_sum.xsl
helm/www/lambdadelta/xslt/basic_1_blk.xsl
class "white" {
[ @@("html/specification" "specification") * ]
[ @@("html/specification#v2" "version 2")
- "(" ^ @@("html/ground_2" "background") + "-" +
+ "(" ^ @@("html/ground_2" "background") + "-" +
+ @@("html/static_2" "syntax") + "-" +
@@("html/basic_2" "core") + "-" +
@@("html/apps_2" "applications") ^ ")"
* ]
<body>
Informational pages on the parts of the specification:
<rlink to="html/ground_2.html">Background</rlink>,
+ <rlink to="html/static_2.html">Syntax</rlink>,
<rlink to="html/basic_2.html">Core</rlink>,
<rlink to="html/apps_2.html">Applications</rlink>.
</body>
TAGS := all xoa xoa2 orig elim deps top leaf stats tbls trim contrib clean \
www up-html
-PACKAGES := ground_2 basic_2 apps_2 alpha_1
+PACKAGES := ground_2 basic_2 static_2 apps_2 alpha_1
XPACKAGES := ground_2 basic_2
LDWS := $(shell find -name "*.ldw.xml")
include "ground_2/notation/relations/ringeq_3.ma".
include "apps_2/notation/models/at_3.ma".
include "apps_2/notation/models/wbrackets_4.ma".
-include "basic_2/syntax/term.ma".
+include "static_2/syntax/term.ma".
(* MODEL ********************************************************************)
(* *)
(**************************************************************************)
-include "basic_2/syntax/genv.ma".
+include "static_2/syntax/genv.ma".
(* GLOBAL ENVIRONMENT INTERPRETATION ***************************************)
(* *)
(**************************************************************************)
-include "basic_2/syntax/lenv.ma".
+include "static_2/syntax/lenv.ma".
include "apps_2/models/model_vlift.ma".
include "apps_2/notation/models/inwbrackets_4.ma".
(**************************************************************************)
include "ground_2/relocation/rtmap_basic.ma".
-include "basic_2/relocation/lifts.ma".
+include "static_2/relocation/lifts.ma".
include "apps_2/models/veq.ma".
(* EVALUATION EQUIVALENCE **************************************************)
(* *)
(**************************************************************************)
-include "basic_2/s_computation/fqus_fqup.ma".
+include "static_2/s_computation/fqus_fqup.ma".
include "basic_2/dynamic/cnv_drops.ma".
(* CONTEXT_SENSITIVE NATIVE VALIDITY FOR TERMS ******************************)
(* *)
(**************************************************************************)
-include "basic_2/static/lsubd.ma".
+include "static_2/static/lsubd.ma".
include "basic_2/dynamic/lsubsv.ma".
(* LOCAL ENVIRONMENT REFINEMENT FOR STRATIFIED NATIVE VALIDITY **************)
(* *)
(**************************************************************************)
-include "basic_2/static/lsubd_da.ma".
+include "static_2/static/lsubd_da.ma".
include "basic_2/dynamic/lsubsv_lsubd.ma".
include "basic_2/dynamic/lsubsv_lstas.ma".
(* *)
(**************************************************************************)
-include "basic_2/static/lsubd_da.ma".
+include "static_2/static/lsubd_da.ma".
include "basic_2/dynamic/snv_aaa.ma".
include "basic_2/dynamic/snv_scpes.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 "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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( L. break ⓓ T1 )"
- left associative with precedence 50
- for @{ 'DxAbbr $L $T1 }.
+++ /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( L. break ⓛ T1 )"
- left associative with precedence 51
- for @{ 'DxAbst $L $T1 }.
+++ /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( L. break ⓤ { term 46 I } )"
- non associative with precedence 47
- for @{ 'DxBind1 $L $I }.
+++ /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( L. break ⓑ { term 46 I } break term 49 T1 )"
- non associative with precedence 48
- for @{ 'DxBind2 $L $I $T1 }.
+++ /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( L. ⓘ { break term 46 I } )"
- non associative with precedence 46
- for @{'DxItem $L $I }.
+++ /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( L. ⓧ )"
- non associative with precedence 49
- for @{ 'DxVoid $L }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( § term 90 l )"
- non associative with precedence 55
- for @{ 'GRef $l }.
+++ /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 "⓪"
- non associative with precedence 55
- for @{ 'Item0 }.
+++ /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( ⓪ { term 46 I } )"
- non associative with precedence 55
- for @{ 'Item0 $I }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( # term 90 i )"
- non associative with precedence 55
- for @{ 'LRef $i }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓓ { term 46 p } break term 55 T1. break term 55 T2 )"
- non associative with precedence 55
- for @{ 'SnAbbr $p $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( - ⓓ term 55 T1. break term 55 T2 )"
- non associative with precedence 55
- for @{ 'SnAbbrNeg $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( + ⓓ term 55 T1. break term 55 T2 )"
- non associative with precedence 55
- for @{ 'SnAbbrPos $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓛ { term 46 p } break term 55 T1. break term 55 T2 )"
- non associative with precedence 55
- for @{ 'SnAbst $p $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( - ⓛ term 55 T1. break term 55 T2 )"
- non associative with precedence 55
- for @{ 'SnAbstNeg $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( + ⓛ term 55 T1. break term 55 T2 )"
- non associative with precedence 55
- for @{ 'SnAbstPos $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓐ term 55 T1. break term 55 T2 )"
- non associative with precedence 55
- for @{ 'SnAppl $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓑ { term 46 p, break term 46 I } break term 55 T1. break term 55 T )"
- non associative with precedence 55
- for @{ 'SnBind2 $p $I $T1 $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( - ⓑ { term 46 I } break term 55 T1. break term 55 T )"
- non associative with precedence 55
- for @{ 'SnBind2Neg $I $T1 $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( + ⓑ { term 46 I } break term 55 T1. break term 55 T )"
- non associative with precedence 55
- for @{ 'SnBind2Pos $I $T1 $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓝ term 55 T1. break term 55 T2 )"
- non associative with precedence 55
- for @{ 'SnCast $T1 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓕ { term 46 I } break term 55 T1. break term 55 T )"
- non associative with precedence 55
- for @{ 'SnFlat2 $I $T1 $T }.
+++ /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( ② term 55 T1. break term 55 T )"
- non associative with precedence 55
- for @{ 'SnItem2 $T1 $T }.
+++ /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( ② { term 46 I } break term 55 T1. break term 55 T )"
- non associative with precedence 55
- for @{ 'SnItem2 $I $T1 $T }.
+++ /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 "⋆"
- non associative with precedence 46
- for @{ 'Star }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⋆ term 90 s )"
- non associative with precedence 55
- for @{ 'Star $s }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓓ term 55 T. break term 55 L )"
- non associative with precedence 55
- for @{ 'SnAbbr $T $L }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓛ term 55 T. break term 55 L )"
- non associative with precedence 55
- for @{ 'SnAbst $T $L }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( Ⓐ term 55 T1. break term 55 T )"
- non associative with precedence 55
- for @{ 'SnApplVector $T1 $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓤ { term 46 I }. break term 55 L )"
- non associative with precedence 55
- for @{ 'SnBind1 $I $L }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓑ { term 46 I } break term 55 T. break term 55 L )"
- non associative with precedence 55
- for @{ 'SnBind2 $I $T $L }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓘ { term 46 I }. break term 55 L )"
- non associative with precedence 55
- for @{ 'SnItem $I $L }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⓧ. term 55 L )"
- non associative with precedence 55
- for @{ 'SnVoid $L }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ♯ { term 46 X } )"
- non associative with precedence 90
- for @{ 'Weight $X }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ♯ { term 46 L, break term 46 T } )"
- non associative with precedence 90
- for @{ 'Weight $L $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ♯ { term 46 G, break term 46 L, break term 46 T } )"
- non associative with precedence 90
- for @{ 'Weight $G $L $T }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G, break term 46 L⦄ ⊢ break term 46 T ⁝ break term 46 A )"
- non associative with precedence 45
- for @{ 'AtomicArity $G $L $T $A }.
+++ /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( L ⊢ 𝐅 * ⦃ break term 46 T ⦄ ≘ break term 46 f )"
- non associative with precedence 45
- for @{ 'FreeStar $L $T $f }.
+++ /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 f ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'IdEqSn $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( ⦃ term 46 G, break term 46 L, break term 46 T ⦄ ϵ [ break term 46 R ] 〚 break term 46 A 〛 )"
- non associative with precedence 45
- for @{ 'InEInt $R $G $L $T $A }.
+++ /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( G ⊢ break term 46 L1 ⫃ ⁝ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LRSubEqA $G $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 L2 )"
- non associative with precedence 45
- for @{ 'LRSubEqC $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( G ⊢ break term 46 L1 ⫃ [ break term 46 R ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LRSubEqC $R $G $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 L1, break term 46 f1 ⦄ ⫃ 𝐅* ⦃ break term 46 L2, break term 46 f2 ⦄ )"
- non associative with precedence 45
- for @{ 'LRSubEqF $L1 $f1 $L2 $f2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⬇ * [ term 46 i ] break term 46 L1 ≘ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'RDropStar $i $L1 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⬇ * [ term 46 b, break term 46 f ] break term 46 L1 ≘ break term 46 L2 )"
- non associative with precedence 45
- for @{ 'RDropStar $b $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 L2 )"
- non associative with precedence 45
- for @{ 'Relation $R $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 @{ '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 }.
+++ /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 @{ '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( ⬆ * [ term 46 f ] break term 46 T1 ≘ break term 46 T2 )"
- non associative with precedence 45
- for @{ 'RLiftStar $f $T1 $T2 }.
+++ /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( 𝐒 ⦃ term 46 T ⦄ )"
- non associative with precedence 45
- for @{ 'Simple $T }.
+++ /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( T1 ≛ [ break term 46 h, break term 46 o ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'StarEq $h $o $T1 $T2 }.
+++ /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( L ⊢ break term 46 T1 ≛ [ break term 46 h, break term 46 o ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'StarEq $h $o $L $T1 $T2 }.
+++ /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 h, break term 46 o, break term 46 T ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'StarEqSn $h $o $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( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≛ [ break term 46 h, break term 46 o ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'StarEqSn $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊆ ⦃ break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SubSetEq $L1 $T1 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SupTerm $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SupTerm $b $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐⸮ ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SupTermOpt $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐⸮ [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SupTermOpt $b $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + ⦃ break term 46 G2, term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SupTermPlus $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + [ break term 46 b ] ⦃ break term 46 G2, term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SupTermPlus $b $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SupTermStar $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'SupTermStar $b $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /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( T1 ⩳ [ break term 46 h, break term 46 o ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'TopIso $h $o $T1 $T2 }.
+++ /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 n1, break term 46 n2 ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'VoidStarEq $L1 $n1 $n2 $L2 }.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "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/seq.ma".
-include "basic_2/relocation/lifts_bind.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Basic_1: includes: drop_skip_bind drop1_skip_bind *)
-(* Basic_2A1: includes: drop_atom drop_pair drop_drop drop_skip
- drop_refl_atom_O2 drop_drop_lt drop_skip_lt
-*)
-inductive drops (b:bool): rtmap → relation lenv ≝
-| drops_atom: ∀f. (b = Ⓣ → 𝐈⦃f⦄) → drops b (f) (⋆) (⋆)
-| drops_drop: ∀f,I,L1,L2. drops b f L1 L2 → drops b (↑f) (L1.ⓘ{I}) L2
-| drops_skip: ∀f,I1,I2,L1,L2.
- drops b f L1 L2 → ⬆*[f] I2 ≘ I1 →
- drops b (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
-.
-
-interpretation "uniform slicing (local environment)"
- 'RDropStar i L1 L2 = (drops true (uni i) L1 L2).
-
-interpretation "generic slicing (local environment)"
- 'RDropStar b f L1 L2 = (drops b f L1 L2).
-
-definition d_liftable1: predicate (relation2 lenv term) ≝
- λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K →
- ∀U. ⬆*[f] T ≘ U → R L U.
-
-definition d_liftable1_isuni: predicate (relation2 lenv term) ≝
- λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ →
- ∀U. ⬆*[f] T ≘ U → R L U.
-
-definition d_deliftable1: predicate (relation2 lenv term) ≝
- λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K →
- ∀T. ⬆*[f] T ≘ U → R K T.
-
-definition d_deliftable1_isuni: predicate (relation2 lenv term) ≝
- λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ →
- ∀T. ⬆*[f] T ≘ U → R K T.
-
-definition d_liftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
- predicate (lenv → relation C) ≝
- λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≘ K →
- ∀U1. S f T1 U1 →
- ∃∃U2. S f T2 U2 & R L U1 U2.
-
-definition d_deliftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
- predicate (lenv → relation C) ≝
- λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≘ K →
- ∀T1. S f T1 U1 →
- ∃∃T2. S f T2 U2 & R K T1 T2.
-
-definition d_liftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
- predicate (lenv → relation C) ≝
- λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≘ K →
- ∀U1. S f T1 U1 →
- ∀U2. S f T2 U2 → R L U1 U2.
-
-definition d_deliftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
- predicate (lenv → relation C) ≝
- λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≘ K →
- ∀T1. S f T1 U1 →
- ∀T2. S f T2 U2 → R K T1 T2.
-
-definition co_dropable_sn: predicate (rtmap → relation lenv) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
- ∀f2,L2. R f2 L1 L2 → ∀f1. f ~⊚ f1 ≘ f2 →
- ∃∃K2. R f1 K1 K2 & ⬇*[b, f] L2 ≘ K2.
-
-definition co_dropable_dx: predicate (rtmap → relation lenv) ≝
- λR. ∀f2,L1,L2. R f2 L1 L2 →
- ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
- ∀f1. f ~⊚ f1 ≘ f2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & R f1 K1 K2.
-
-definition co_dedropable_sn: predicate (rtmap → relation lenv) ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀f1,K2. R f1 K1 K2 →
- ∀f2. f ~⊚ f1 ≘ f2 →
- ∃∃L2. R f2 L1 L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
-
-(* Basic properties *********************************************************)
-
-lemma drops_atom_F: ∀f. ⬇*[Ⓕ, f] ⋆ ≘ ⋆.
-#f @drops_atom #H destruct
-qed.
-
-lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⬇*[b, f] L1 ≘ L2).
-#b #L1 #L2 #f1 #H elim H -f1 -L1 -L2
-[ /4 width=3 by drops_atom, isid_eq_repl_back/
-| #f1 #I #L1 #L2 #_ #IH #f2 #H elim (eq_inv_nx … H) -H
- /3 width=3 by drops_drop/
-| #f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H elim (eq_inv_px … H) -H
- /3 width=3 by drops_skip, liftsb_eq_repl_back/
-]
-qed-.
-
-lemma drops_eq_repl_fwd: ∀b,L1,L2. eq_repl_fwd … (λf. ⬇*[b, f] L1 ≘ L2).
-#b #L1 #L2 @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *)
-qed-.
-
-(* Basic_2A1: includes: drop_FT *)
-lemma drops_TF: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2.
-#f #L1 #L2 #H elim H -f -L1 -L2
-/3 width=1 by drops_atom, drops_drop, drops_skip/
-qed.
-
-(* Basic_2A1: includes: drop_gen *)
-lemma drops_gen: ∀b,f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[b, f] L1 ≘ L2.
-* /2 width=1 by drops_TF/
-qed-.
-
-(* Basic_2A1: includes: drop_T *)
-lemma drops_F: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2.
-* /2 width=1 by drops_TF/
-qed-.
-
-lemma d_liftable2_sn_bi: ∀C,S. (∀f,c. is_mono … (S f c)) →
- ∀R. d_liftable2_sn C S R → d_liftable2_bi C S R.
-#C #S #HS #R #HR #K #T1 #T2 #HT12 #b #f #L #HLK #U1 #HTU1 #U2 #HTU2
-elim (HR … HT12 … HLK … HTU1) -HR -K -T1 #X #HTX #HUX
-<(HS … HTX … HTU2) -T2 -U2 -b -f //
-qed-.
-
-lemma d_deliftable2_sn_bi: ∀C,S. (∀f. is_inj2 … (S f)) →
- ∀R. d_deliftable2_sn C S R → d_deliftable2_bi C S R.
-#C #S #HS #R #HR #L #U1 #U2 #HU12 #b #f #K #HLK #T1 #HTU1 #T2 #HTU2
-elim (HR … HU12 … HLK … HTU1) -HR -L -U1 #X #HUX #HTX
-<(HS … HUX … HTU2) -U2 -T2 -b -f //
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact drops_inv_atom1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → X = ⋆ →
- Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
-#b #f #X #Y * -f -X -Y
-[ /3 width=1 by conj/
-| #f #I #L1 #L2 #_ #H destruct
-| #f #I1 #I2 #L1 #L2 #_ #_ #H destruct
-]
-qed-.
-
-(* Basic_1: includes: drop_gen_sort *)
-(* Basic_2A1: includes: drop_inv_atom1 *)
-lemma drops_inv_atom1: ∀b,f,Y. ⬇*[b, f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
-/2 width=3 by drops_inv_atom1_aux/ qed-.
-
-fact drops_inv_drop1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I,K. X = K.ⓘ{I} → f = ↑g →
- ⬇*[b, g] K ≘ Y.
-#b #f #X #Y * -f -X -Y
-[ #f #Hf #g #J #K #H destruct
-| #f #I #L1 #L2 #HL #g #J #K #H1 #H2 <(injective_next … H2) -g destruct //
-| #f #I1 #I2 #L1 #L2 #_ #_ #g #J #K #_ #H2 elim (discr_push_next … H2)
-]
-qed-.
-
-(* Basic_1: includes: drop_gen_drop *)
-(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
-lemma drops_inv_drop1: ∀b,f,I,K,Y. ⬇*[b, ↑f] K.ⓘ{I} ≘ Y → ⬇*[b, f] K ≘ Y.
-/2 width=6 by drops_inv_drop1_aux/ qed-.
-
-fact drops_inv_skip1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ{I1} → f = ⫯g →
- ∃∃I2,K2. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}.
-#b #f #X #Y * -f -X -Y
-[ #f #Hf #g #J1 #K1 #H destruct
-| #f #I #L1 #L2 #_ #g #J1 #K1 #_ #H2 elim (discr_next_push … H2)
-| #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-.
-
-(* Basic_1: includes: drop_gen_skip_l *)
-(* Basic_2A1: includes: drop_inv_skip1 *)
-lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⬇*[b, ⫯f] K1.ⓘ{I1} ≘ Y →
- ∃∃I2,K2. ⬇*[b, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & Y = K2.ⓘ{I2}.
-/2 width=5 by drops_inv_skip1_aux/ qed-.
-
-fact drops_inv_skip2_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ{I2} → f = ⫯g →
- ∃∃I1,K1. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & X = K1.ⓘ{I1}.
-#b #f #X #Y * -f -X -Y
-[ #f #Hf #g #J2 #K2 #H destruct
-| #f #I #L1 #L2 #_ #g #J2 #K2 #_ #H2 elim (discr_next_push … H2)
-| #f #I1 #I2 #L1 #L2 #HL #HV #g #J2 #K2 #H1 #H2 <(injective_push … H2) -g destruct
- /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_1: includes: drop_gen_skip_r *)
-(* Basic_2A1: includes: drop_inv_skip2 *)
-lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⬇*[b, ⫯f] X ≘ K2.ⓘ{I2} →
- ∃∃I1,K1. ⬇*[b, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & X = K1.ⓘ{I1}.
-/2 width=5 by drops_inv_skip2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⬇*[b, f2] X ≘ Y → ∀I,K. Y = K.ⓘ{I} →
- ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b, f] X ≘ K.
-#b #f2 #X #Y #H elim H -f2 -X -Y
-[ #f2 #Hf2 #J #K #H destruct
-| #f2 #I #L1 #L2 #_ #IHL #J #K #H elim (IHL … H) -IHL
- /3 width=7 by after_next, ex3_2_intro, drops_drop/
-| #f2 #I1 #I2 #L1 #L2 #HL #_ #_ #J #K #H destruct
- lapply (after_isid_dx 𝐈𝐝 … f2) /3 width=9 by after_push, ex3_2_intro, drops_drop/
-]
-qed-.
-
-lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} →
- ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b, f] X ≘ K.
-/2 width=4 by drops_fwd_drop2_aux/ qed-.
-
-(* Properties with test for identity ****************************************)
-
-(* Basic_2A1: includes: drop_refl *)
-lemma drops_refl: ∀b,L,f. 𝐈⦃f⦄ → ⬇*[b, f] L ≘ L.
-#b #L elim L -L /2 width=1 by drops_atom/
-#L #I #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf
-/3 width=1 by drops_skip, liftsb_refl/
-qed.
-
-(* Forward lemmas test for identity *****************************************)
-
-(* Basic_1: includes: drop_gen_refl *)
-(* Basic_2A1: includes: drop_inv_O2 *)
-lemma drops_fwd_isid: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → 𝐈⦃f⦄ → L1 = L2.
-#b #f #L1 #L2 #H elim H -f -L1 -L2 //
-[ #f #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) //
-| /5 width=5 by isid_inv_push, liftsb_fwd_isid, eq_f2, sym_eq/
-]
-qed-.
-
-lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} →
- ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ↑f1 ≘ f → ⬇*[b, f] X ≘ K.
-#b #f2 #I #X #K #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H
-#g1 #g #Hg1 #Hg #HK lapply (after_mono_eq … Hg … Hf ??) -Hg -Hf
-/3 width=5 by drops_eq_repl_back, isid_inv_eq_repl, eq_next/
-qed-.
-
-(* Forward lemmas with test for finite colength *****************************)
-
-lemma drops_fwd_isfin: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐅⦃f⦄.
-#f #L1 #L2 #H elim H -f -L1 -L2
-/3 width=1 by isfin_next, isfin_push, isfin_isid/
-qed-.
-
-(* Properties with test for uniformity **************************************)
-
-lemma drops_isuni_ex: ∀f. 𝐔⦃f⦄ → ∀L. ∃K. ⬇*[Ⓕ, f] L ≘ K.
-#f #H elim H -f /4 width=2 by drops_refl, drops_TF, ex_intro/
-#f #_ #g #H #IH destruct * /2 width=2 by ex_intro/
-#L #I elim (IH L) -IH /3 width=2 by drops_drop, ex_intro/
-qed-.
-
-(* Inversion lemmas with test for uniformity ********************************)
-
-lemma drops_inv_isuni: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐔⦃f⦄ →
- (𝐈⦃f⦄ ∧ L1 = L2) ∨
- ∃∃g,I,K. ⬇*[Ⓣ, g] K ≘ L2 & 𝐔⦃g⦄ & L1 = K.ⓘ{I} & f = ↑g.
-#f #L1 #L2 * -f -L1 -L2
-[ /4 width=1 by or_introl, conj/
-| /4 width=7 by isuni_inv_next, ex4_3_intro, or_intror/
-| /7 width=6 by drops_fwd_isid, liftsb_fwd_isid, isuni_inv_push, isid_push, or_introl, conj, eq_f2, sym_eq/
-]
-qed-.
-
-(* Basic_2A1: was: drop_inv_O1_pair1 *)
-lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b, f] K.ⓘ{I} ≘ L2 →
- (𝐈⦃f⦄ ∧ L2 = K.ⓘ{I}) ∨
- ∃∃g. 𝐔⦃g⦄ & ⬇*[b, g] K ≘ L2 & f = ↑g.
-#b #f #I #K #L2 #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct
-[ lapply (drops_inv_skip1 … H) -H * #Z #Y #HY #HZ #H destruct
- <(drops_fwd_isid … HY Hg) -Y >(liftsb_fwd_isid … HZ Hg) -Z
- /4 width=3 by isid_push, or_introl, conj/
-| lapply (drops_inv_drop1 … H) -H /3 width=4 by ex3_intro, or_intror/
-]
-qed-.
-
-(* Basic_2A1: was: drop_inv_O1_pair2 *)
-lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K.ⓘ{I} →
- (𝐈⦃f⦄ ∧ L1 = K.ⓘ{I}) ∨
- ∃∃g,I1,K1. 𝐔⦃g⦄ & ⬇*[b, g] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1} & f = ↑g.
-#b #f #I #K *
-[ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct
-| #L1 #I1 #Hf #H elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
- [ #Hf #H destruct /3 width=1 by or_introl, conj/
- | /3 width=7 by ex4_3_intro, or_intror/
- ]
-]
-qed-.
-
-lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, ↑f] L1 ≘ K.ⓘ{I} →
- ∃∃I1,K1. ⬇*[b, f] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1}.
-#b #f #I #K #L1 #Hf #H elim (drops_inv_bind2_isuni … H) -H /2 width=3 by isuni_next/ -Hf *
-[ #H elim (isid_inv_next … H) -H //
-| /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ, f] L1 ≘ L2 → 𝐔⦃f⦄ →
- ∀I,K. L2 = K.ⓘ{I} → ⬇*[Ⓣ, f] L1 ≘ K.ⓘ{I}.
-#f #L1 #L2 #H elim H -f -L1 -L2
-[ #f #_ #_ #J #K #H destruct
-| #f #I #L1 #L2 #_ #IH #Hf #J #K #H destruct
- /4 width=3 by drops_drop, isuni_inv_next/
-| #f #I1 #I2 #L1 #L2 #HL12 #HI21 #_ #Hf #J #K #H destruct
- lapply (isuni_inv_push … Hf ??) -Hf [1,2: // ] #Hf
- <(drops_fwd_isid … HL12) -K // <(liftsb_fwd_isid … HI21) -I1
- /3 width=3 by drops_refl, isid_push/
-]
-qed-.
-
-(* Basic_2A1: includes: drop_inv_FT *)
-lemma drops_inv_TF: ∀f,I,L,K. ⬇*[Ⓕ, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≘ K.ⓘ{I}.
-/2 width=3 by drops_inv_TF_aux/ qed-.
-
-(* Basic_2A1: includes: drop_inv_gen *)
-lemma drops_inv_gen: ∀b,f,I,L,K. ⬇*[b, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≘ K.ⓘ{I}.
-* /2 width=1 by drops_inv_TF/
-qed-.
-
-(* Basic_2A1: includes: drop_inv_T *)
-lemma drops_inv_F: ∀b,f,I,L,K. ⬇*[Ⓕ, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[b, f] L ≘ K.ⓘ{I}.
-* /2 width=1 by drops_inv_TF/
-qed-.
-
-(* Forward lemmas with test for uniformity **********************************)
-
-(* Basic_1: was: drop_S *)
-(* Basic_2A1: was: drop_fwd_drop2 *)
-lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔⦃f⦄ → ⬇*[b, f] X ≘ K.ⓘ{I} → ⬇*[b, ↑f] X ≘ K.
-/3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
-
-(* Inversion lemmas with uniform relocations ********************************)
-
-lemma drops_inv_atom2: ∀b,L,f. ⬇*[b, f] L ≘ ⋆ →
- ∃∃n,f1. ⬇*[b, 𝐔❴n❵] L ≘ ⋆ & 𝐔❴n❵ ⊚ f1 ≘ f.
-#b #L elim L -L
-[ /3 width=4 by drops_atom, after_isid_sn, ex2_2_intro/
-| #L #I #IH #f #H elim (pn_split f) * #g #H0 destruct
- [ elim (drops_inv_skip1 … H) -H #J #K #_ #_ #H destruct
- | lapply (drops_inv_drop1 … H) -H #HL
- elim (IH … HL) -IH -HL /3 width=8 by drops_drop, after_next, ex2_2_intro/
- ]
-]
-qed-.
-
-lemma drops_inv_succ: ∀L1,L2,i. ⬇*[↑i] L1 ≘ L2 →
- ∃∃I,K. ⬇*[i] K ≘ L2 & L1 = K.ⓘ{I}.
-#L1 #L2 #i #H elim (drops_inv_isuni … H) -H // *
-[ #H elim (isid_inv_next … H) -H //
-| /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-(* Properties with uniform relocations **************************************)
-
-lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⬇*[i] L ≘ K.ⓘ{I}.
-#L elim L -L /2 width=1 by or_introl/
-#L #I #IH * /4 width=3 by drops_refl, ex1_2_intro, or_intror/
-#i elim (IH i) -IH /3 width=1 by drops_drop, or_introl/
-* /4 width=3 by drops_drop, ex1_2_intro, or_intror/
-qed-.
-
-(* Basic_2A1: includes: drop_split *)
-lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ f2 ≘ f → 𝐔⦃f1⦄ →
- ∃∃L. ⬇*[b, f1] L1 ≘ L & ⬇*[b, f2] L ≘ L2.
-#b #f #L1 #L2 #H elim H -f -L1 -L2
-[ #f #H0f #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom
- #H lapply (H0f H) -b
- #H elim (after_inv_isid3 … Hf H) -f //
-| #f #I #L1 #L2 #HL12 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxn … Hf) -Hf [1,3: * |*: // ]
- [ #g1 #g2 #Hf #H1 #H2 destruct
- lapply (isuni_inv_push … Hf1 ??) -Hf1 [1,2: // ] #Hg1
- elim (IHL12 … Hf) -f
- /4 width=5 by drops_drop, drops_skip, liftsb_refl, isuni_isid, ex2_intro/
- | #g1 #Hf #H destruct elim (IHL12 … Hf) -f
- /3 width=5 by ex2_intro, drops_drop, isuni_inv_next/
- ]
-| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxp … Hf) -Hf [2,3: // ]
- #g1 #g2 #Hf #H1 #H2 destruct elim (liftsb_split_trans … HI21 … Hf) -HI21
- elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip, isuni_fwd_push/
-]
-qed-.
-
-lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b, f1] L1 ≘ L → ∀f2,f. f1 ⊚ f2 ≘ f → 𝐔⦃f2⦄ →
- ∃∃L2. ⬇*[Ⓕ, f2] L ≘ L2 & ⬇*[Ⓕ, f] L1 ≘ L2.
-#b #f1 #L1 #L #H elim H -f1 -L1 -L
-[ #f1 #Hf1 #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct
-| #f1 #I #L1 #L #HL1 #IH #f2 #f #Hf #Hf2 elim (after_inv_nxx … Hf) -Hf [2,3: // ]
- #g #Hg #H destruct elim (IH … Hg) -IH -Hg /3 width=5 by drops_drop, ex2_intro/
-| #f1 #I1 #I #L1 #L #HL1 #HI1 #IH #f2 #f #Hf #Hf2
- elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ]
- #g2 #g #Hg #H2 #H0 destruct
- [ lapply (isuni_inv_push … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -IH
- lapply (after_isid_inv_dx … Hg … Hg2) -Hg #Hg
- /5 width=7 by drops_eq_repl_back, drops_F, drops_refl, drops_skip, liftsb_eq_repl_back, isid_push, ex2_intro/
- | lapply (isuni_inv_next … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -HL1 -HI1
- elim (IH … Hg) -f1 /3 width=3 by drops_drop, ex2_intro/
- ]
-]
-qed-.
-
-(* Properties with application **********************************************)
-
-lemma drops_tls_at: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
- ∀b,L1,L2. ⬇*[b,⫱*[i2]f] L1 ≘ L2 →
- ⬇*[b,⫯⫱*[↑i2]f] L1 ≘ L2.
-/3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-.
-
-lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⬇*[b, f] L ≘ K0.ⓘ{I} → ∀i. @⦃O, f⦄ ≘ i →
- ∃∃J,K. ⬇*[i]L ≘ K.ⓘ{J} & ⬇*[b, ⫱*[↑i]f] K ≘ K0 & ⬆*[⫱*[↑i]f] I ≘ J.
-#b #f #I #L #K0 #H #i #Hf
-elim (drops_split_trans … H) -H [ |5: @(after_uni_dx … Hf) |2,3: skip ] /2 width=1 by after_isid_dx/ #Y #HLY #H
-lapply (drops_tls_at … Hf … H) -H #H
-elim (drops_inv_skip2 … H) -H #J #K #HK0 #HIJ #H destruct
-/3 width=5 by drops_inv_gen, ex3_2_intro/
-qed-.
-
-(* Properties with context-sensitive equivalence for terms ******************)
-
-lemma ceq_lift_sn: d_liftable2_sn … liftsb ceq_ext.
-#K #I1 #I2 #H <(ceq_ext_inv_eq … H) -I2
-/2 width=3 by ex2_intro/ qed-.
-
-lemma ceq_inv_lift_sn: d_deliftable2_sn … liftsb ceq_ext.
-#L #J1 #J2 #H <(ceq_ext_inv_eq … H) -J2
-/2 width=3 by ex2_intro/ qed-.
-
-(* Note: d_deliftable2_sn cfull does not hold *)
-lemma cfull_lift_sn: d_liftable2_sn … liftsb cfull.
-#K #I1 #I2 #_ #b #f #L #_ #J1 #_ -K -I1 -b
-elim (liftsb_total I2 f) /2 width=3 by ex2_intro/
-qed-.
-
-(* Basic_2A1: removed theorems 12:
- drops_inv_nil drops_inv_cons d1_liftable_liftables
- drop_refl_atom_O2 drop_inv_pair1
- drop_inv_Y1 drop_Y1 drop_O_Y drop_fwd_Y2
- drop_fwd_length_minus2 drop_fwd_length_minus4
-*)
-(* Basic_1: removed theorems 53:
- drop1_gen_pnil drop1_gen_pcons drop1_getl_trans
- drop_ctail drop_skip_flat
- cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
- drop_clear drop_clear_O drop_clear_S
- clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
- clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
- getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
- getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
- getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
- drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
- getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
- getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
- getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
-*)
+++ /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.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Properties with the extension to binders of a context-sensitive relation *)
-
-lemma cext2_d_liftable2_sn: ∀R. d_liftable2_sn … lifts R →
- d_liftable2_sn … liftsb (cext2 R).
-#R #HR #K #I1 #I2 * -I1 -I2 #I [| #T1 #T2 #HT12 ]
-#b #f #L #HLK #Z1 #H
-[ lapply (liftsb_inv_unit_sn … H)
-| lapply (liftsb_inv_pair_sn … H) * #U1 #HTU1
-] -H #H destruct /3 width=3 by ext2_unit, ex2_intro/
-elim (HR … HT12 … HLK … HTU1) -HR -b -K -T1 /3 width=3 by ext2_pair, ex2_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/lib/star.ma".
-include "basic_2/relocation/seq_seq.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Properties with contextual transitive closure ****************************)
-
-(* Basic_2A1: was: d_liftable_LTC *)
-lemma d2_liftable_sn_CTC: ∀C,S,R. d_liftable2_sn C S R → d_liftable2_sn C S (CTC … R).
-#C #S #R #HR #K #T1 #T2 #H elim H -T2
-[ #T2 #HT12 #b #f #L #HLK #U1 #HTU1
- elim (HR … HT12 … HLK … HTU1) /3 width=3 by inj, ex2_intro/
-| #T #T2 #_ #HT2 #IHT1 #b #f #L #HLK #U1 #HTU1
- elim (IHT1 … HLK … HTU1) -T1 #U #HTU #HU1
- elim (HR … HT2 … HLK … HTU) -HR -K -T /3 width=5 by step, ex2_intro/
-]
-qed-.
-
-(* Basic_2A1: was: d_deliftable_sn_LTC *)
-lemma d2_deliftable_sn_CTC: ∀C,S,R. d_deliftable2_sn C S R → d_deliftable2_sn C S (CTC … R).
-#C #S #R #HR #L #U1 #U2 #H elim H -U2
-[ #U2 #HU12 #b #f #K #HLK #T1 #HTU1
- elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3 by inj, ex2_intro/
-| #U #U2 #_ #HU2 #IHU1 #b #f #K #HLK #T1 #HTU1
- elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
- elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by step, ex2_intro/
-]
-qed-.
-
-lemma co_dropable_sn_CTC: ∀R. co_dropable_sn R → co_dropable_sn (CTC … R).
-#R #HR #b #f #L1 #K1 #HLK1 #Hf #f2 #L2 #H elim H -L2
-[ #L2 #HL12 #f1 #H elim (HR … HLK1 … Hf … HL12 … H) -HR -Hf -f2 -L1
- /3 width=3 by inj, ex2_intro/
-| #L #L2 #_ #HL2 #IH #f1 #H elim (IH … H) -IH
- #K #HK1 #HLK elim (HR … HLK … HL2 … H) -HR -f2 -L
- /3 width=3 by step, ex2_intro/
-]
-qed-.
-
-lemma co_dropable_dx_CTC: ∀R. co_dropable_dx R → co_dropable_dx (CTC … R).
-#R #HR #f2 #L1 #L2 #H elim H -L2
-[ #L2 #HL12 #b #f #K2 #HLK2 #Hf #f1 #Hf2 elim (HR … HL12 … HLK2 … Hf … Hf2) -HR -Hf -f2 -L2
- /3 width=3 by inj, ex2_intro/
-| #L #L2 #_ #HL2 #IHL1 #b #f #K2 #HLK2 #Hf #f1 #Hf2 elim (HR … HL2 … HLK2 … Hf … Hf2) -HR -L2
- #K #HLK #HK2 elim (IHL1 … HLK … Hf … Hf2) -Hf -f2 -L
- /3 width=5 by step, ex2_intro/
-]
-qed-.
-
-lemma co_dedropable_sn_CTC: ∀R. co_dedropable_sn R → co_dedropable_sn (CTC … R).
-#R #HR #b #f #L1 #K1 #HLK1 #f1 #K2 #H elim H -K2
-[ #K2 #HK12 #f2 #Hf elim (HR … HLK1 … HK12 … Hf) -HR -f1 -K1
- /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 seq_trans, step, 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_weight.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_2A1: includes: drop_conf_ge drop_conf_be drop_conf_le *)
-theorem drops_conf: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f,L2. ⬇*[b2, f] L1 ≘ L2 →
- ∀f2. f1 ⊚ f2 ≘ f → ⬇*[b2, f2] L ≘ L2.
-#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
-[ #f1 #_ #b2 #f #L2 #HL2 #f2 #Hf12 elim (drops_inv_atom1 … HL2) -b1 -HL2
- #H #Hf destruct @drops_atom
- #H elim (after_inv_isid3 … Hf12) -Hf12 /2 width=1 by/
-| #f1 #I1 #K1 #K #_ #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_nxx … Hf) -Hf [2,3: // ]
- #g #Hg #H destruct /3 width=3 by drops_inv_drop1/
-| #f1 #I1 #I #K1 #K #_ #HI1 #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*:// ]
- #g2 #g #Hf #H1 #H2 destruct
- [ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, liftsb_div3/
- | /4 width=3 by drops_inv_drop1, drops_drop/
- ]
-]
-qed-.
-
-(* Basic_1: was: drop1_trans *)
-(* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm
- drops_drop_trans
-*)
-theorem drops_trans: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f2,L2. ⬇*[b2, f2] L ≘ L2 →
- ∀f. f1 ⊚ f2 ≘ f → ⬇*[b1∧b2, f] L1 ≘ L2.
-#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
-[ #f1 #Hf1 #b2 #f2 #L2 #HL2 #f #Hf elim (drops_inv_atom1 … HL2) -HL2
- #H #Hf2 destruct @drops_atom #H elim (andb_inv_true_dx … H) -H
- #H1 #H2 lapply (after_isid_inv_sn … Hf ?) -Hf
- /3 width=3 by isid_eq_repl_back/
-| #f1 #I1 #K1 #K #_ #IH #b2 #f2 #L2 #HL2 #f #Hf elim (after_inv_nxx … Hf) -Hf
- /3 width=3 by drops_drop/
-| #f1 #I1 #I #K1 #K #_ #HI1 #IH #b2 #f2 #L2 #HL2 #f #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ]
- #g2 #g #Hg #H1 #H2 destruct
- [ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, liftsb_trans/
- | /4 width=3 by drops_inv_drop1, drops_drop/
- ]
-]
-qed-.
-
-theorem drops_conf_div: ∀f1,L,K. ⬇*[Ⓣ,f1] L ≘ K → ∀f2. ⬇*[Ⓣ,f2] L ≘ K →
- 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2.
-#f1 #L #K #H elim H -f1 -L -K
-[ #f1 #Hf1 #f2 #Hf2 elim (drops_inv_atom1 … Hf2) -Hf2
- /3 width=1 by isid_inv_eq_repl/
-| #f1 #I #L #K #Hf1 #IH #f2 elim (pn_split f2) *
- #g2 #H2 #Hf2 #HU1 #HU2 destruct
- [ elim (drops_inv_skip1 … Hf2) -IH -HU1 -Hf2 #Y2 #X2 #HY2 #_ #H destruct
- lapply (drops_fwd_isid … HY2 ?) -HY2 /2 width=3 by isuni_inv_push/ -HU2
- #H destruct elim (drops_inv_x_bind_xy … Hf1)
- | /4 width=5 by drops_inv_drop1, isuni_inv_next, eq_next/
- ]
-| #f1 #I1 #I2 #L #K #Hf1 #_ #IH #f2 elim (pn_split f2) *
- #g2 #H2 #Hf2 #HU1 #HU2 destruct
- [ elim (drops_inv_skip1 … Hf2) -Hf2 #Y2 #X2 #HY2 #_ #H destruct -Hf1
- /4 width=5 by isuni_fwd_push, eq_push/
- | lapply (drops_inv_drop1 … Hf2) -Hf2 -IH -HU2 #Hg2
- lapply (drops_fwd_isid … Hf1 ?) -Hf1 /2 width=3 by isuni_inv_push/ -HU1
- #H destruct elim (drops_inv_x_bind_xy … Hg2)
- ]
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: includes: drop_mono *)
-lemma drops_mono: ∀b1,f,L,L1. ⬇*[b1, f] L ≘ L1 →
- ∀b2,L2. ⬇*[b2, f] L ≘ L2 → L1 = L2.
-#b1 #f #L #L1 lapply (after_isid_dx 𝐈𝐝 … f)
-/3 width=8 by drops_conf, drops_fwd_isid/
-qed-.
-
-(* Basic_2A1: includes: drop_conf_lt *)
-lemma drops_conf_skip1: ∀b2,f,L,L2. ⬇*[b2, f] L ≘ L2 →
- ∀b1,f1,I1,K1. ⬇*[b1, f1] L ≘ K1.ⓘ{I1} →
- ∀f2. f1 ⊚ ⫯f2 ≘ f →
- ∃∃I2,K2. L2 = K2.ⓘ{I2} &
- ⬇*[b2, f2] K1 ≘ K2 & ⬆*[f2] I2 ≘ I1.
-#b2 #f #L #L2 #H2 #b1 #f1 #I1 #K1 #H1 #f2 #Hf lapply (drops_conf … H1 … H2 … Hf) -L -Hf
-#H elim (drops_inv_skip1 … H) -H /2 width=5 by ex3_2_intro/
-qed-.
-
-(* Basic_2A1: includes: drop_trans_lt *)
-lemma drops_trans_skip2: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f2,I2,K2. ⬇*[b2, f2] L ≘ K2.ⓘ{I2} →
- ∀f. f1 ⊚ f2 ≘ ⫯f →
- ∃∃I1,K1. L1 = K1.ⓘ{I1} &
- ⬇*[b1∧b2, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1.
-#b1 #f1 #L1 #L #H1 #b2 #f2 #I2 #K2 #H2 #f #Hf
-lapply (drops_trans … H1 … H2 … Hf) -L -Hf
-#H elim (drops_inv_skip2 … H) -H /2 width=5 by ex3_2_intro/
-qed-.
-
-(* Basic_2A1: includes: drops_conf_div *)
-lemma drops_conf_div_bind: ∀f1,f2,I1,I2,L,K.
- ⬇*[Ⓣ, f1] L ≘ K.ⓘ{I1} → ⬇*[Ⓣ, f2] L ≘ K.ⓘ{I2} →
- 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2 ∧ I1 = I2.
-#f1 #f2 #I1 #I2 #L #K #Hf1 #Hf2 #HU1 #HU2
-lapply (drops_isuni_fwd_drop2 … Hf1) // #H1
-lapply (drops_isuni_fwd_drop2 … Hf2) // #H2
-lapply (drops_conf_div … H1 … H2 ??) /2 width=3 by isuni_next/ -H1 -H2 -HU1 -HU2 #H
-lapply (eq_inv_nn … H ????) -H [5: |*: // ] #H12
-lapply (drops_eq_repl_back … Hf1 … H12) -Hf1 #H0
-lapply (drops_mono … H0 … Hf2) -L #H
-destruct /2 width=1 by conj/
-qed-.
-
-lemma drops_inv_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ → ∀I,K. ⬇*[i] L ≘ K.ⓘ{I} → ⊥.
-#L #i #H1 #I #K #H2
-lapply (drops_F … H2) -H2 #H2
-lapply (drops_mono … H2 … H1) -L -i #H destruct
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/syntax/lenv_length.ma".
-include "basic_2/relocation/drops.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Forward lemmas with length for local environments ************************)
-
-(* Basic_2A1: includes: drop_fwd_length_le4 *)
-lemma drops_fwd_length_le4: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → |L2| ≤ |L1|.
-#b #f #L1 #L2 #H elim H -f -L1 -L2 /2 width=1 by le_S, le_S_S/
-qed-.
-
-(* Basic_2A1: includes: drop_fwd_length_eq1 *)
-theorem drops_fwd_length_eq1: ∀b1,b2,f,L1,K1. ⬇*[b1, f] L1 ≘ K1 →
- ∀L2,K2. ⬇*[b2, f] L2 ≘ K2 →
- |L1| = |L2| → |K1| = |K2|.
-#b1 #b2 #f #L1 #K1 #HLK1 elim HLK1 -f -L1 -K1
-[ #f #_ #L2 #K2 #HLK2 #H lapply (length_inv_zero_sn … H) -H
- #H destruct elim (drops_inv_atom1 … HLK2) -HLK2 //
-| #f #I1 #L1 #K1 #_ #IH #X2 #K2 #HX #H elim (length_inv_succ_sn … H) -H
- #I2 #L2 #H12 #H destruct lapply (drops_inv_drop1 … HX) -HX
- #HLK2 @(IH … HLK2 H12) (**) (* auto fails *)
-| #f #I1 #I2 #L1 #K1 #_ #_ #IH #X2 #Y2 #HX #H elim (length_inv_succ_sn … H) -H
- #I2 #L2 #H12 #H destruct elim (drops_inv_skip1 … HX) -HX
- #I2 #K2 #HLK2 #_ #H destruct
- lapply (IH … HLK2 H12) -f >length_bind >length_bind /2 width=1 by/ (**) (* full auto fails *)
-]
-qed-.
-
-(* forward lemmas with finite colength assignment ***************************)
-
-lemma drops_fwd_fcla: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
- ∃∃n. 𝐂⦃f⦄ ≘ n & |L1| = |L2| + n.
-#f #L1 #L2 #H elim H -f -L1 -L2
-[ /4 width=3 by fcla_isid, ex2_intro/
-| #f #I #L1 #L2 #_ * >length_bind /3 width=3 by fcla_next, ex2_intro, eq_f/
-| #f #I1 #I2 #L1 #L2 #_ #_ * >length_bind >length_bind /3 width=3 by fcla_push, ex2_intro/
-]
-qed-.
-
-(* Basic_2A1: includes: drop_fwd_length *)
-lemma drops_fcla_fwd: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
- |L1| = |L2| + n.
-#f #l1 #l2 #n #Hf #Hn elim (drops_fwd_fcla … Hf) -Hf
-#k #Hm #H <(fcla_mono … Hm … Hn) -f //
-qed-.
-
-lemma drops_fwd_fcla_le2: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
- ∃∃n. 𝐂⦃f⦄ ≘ n & n ≤ |L1|.
-#f #L1 #L2 #H elim (drops_fwd_fcla … H) -H /2 width=3 by ex2_intro/
-qed-.
-
-(* Basic_2A1: includes: drop_fwd_length_le2 *)
-lemma drops_fcla_fwd_le2: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
- n ≤ |L1|.
-#f #L1 #L2 #n #H #Hn elim (drops_fwd_fcla_le2 … H) -H
-#k #Hm #H <(fcla_mono … Hm … Hn) -f //
-qed-.
-
-lemma drops_fwd_fcla_lt2: ∀f,L1,I2,K2. ⬇*[Ⓣ, f] L1 ≘ K2.ⓘ{I2} →
- ∃∃n. 𝐂⦃f⦄ ≘ n & n < |L1|.
-#f #L1 #I2 #K2 #H elim (drops_fwd_fcla … H) -H
-#n #Hf #H >H -L1 /3 width=3 by le_S_S, ex2_intro/
-qed-.
-
-(* Basic_2A1: includes: drop_fwd_length_lt2 *)
-lemma drops_fcla_fwd_lt2: ∀f,L1,I2,K2,n.
- ⬇*[Ⓣ, f] L1 ≘ K2.ⓘ{I2} → 𝐂⦃f⦄ ≘ n →
- n < |L1|.
-#f #L1 #I2 #K2 #n #H #Hn elim (drops_fwd_fcla_lt2 … H) -H
-#k #Hm #H <(fcla_mono … Hm … Hn) -f //
-qed-.
-
-(* Basic_2A1: includes: drop_fwd_length_lt4 *)
-lemma drops_fcla_fwd_lt4: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → 0 < n →
- |L2| < |L1|.
-#f #L1 #L2 #n #H #Hf #Hn lapply (drops_fcla_fwd … H Hf) -f
-/2 width=1 by lt_minus_to_plus_r/ qed-.
-
-(* Basic_2A1: includes: drop_inv_length_eq *)
-lemma drops_inv_length_eq: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → |L1| = |L2| → 𝐈⦃f⦄.
-#f #L1 #L2 #H #HL12 elim (drops_fwd_fcla … H) -H
-#n #Hn <HL12 -L2 #H lapply (discr_plus_x_xy … H) -H
-/2 width=3 by fcla_inv_xp/
-qed-.
-
-(* Basic_2A1: includes: drop_fwd_length_eq2 *)
-theorem drops_fwd_length_eq2: ∀f,L1,L2,K1,K2. ⬇*[Ⓣ, f] L1 ≘ K1 → ⬇*[Ⓣ, f] L2 ≘ K2 →
- |K1| = |K2| → |L1| = |L2|.
-#f #L1 #L2 #K1 #K2 #HLK1 #HLK2 #HL12
-elim (drops_fwd_fcla … HLK1) -HLK1 #n1 #Hn1 #H1 >H1 -L1
-elim (drops_fwd_fcla … HLK2) -HLK2 #n2 #Hn2 #H2 >H2 -L2
-<(fcla_mono … Hn2 … Hn1) -f //
-qed-.
-
-theorem drops_conf_div: ∀f1,f2,L1,L2. ⬇*[Ⓣ, f1] L1 ≘ L2 → ⬇*[Ⓣ, f2] L1 ≘ L2 →
- ∃∃n. 𝐂⦃f1⦄ ≘ n & 𝐂⦃f2⦄ ≘ n.
-#f1 #f2 #L1 #L2 #H1 #H2
-elim (drops_fwd_fcla … H1) -H1 #n1 #Hf1 #H1
-elim (drops_fwd_fcla … H2) -H2 #n2 #Hf2 >H1 -L1 #H
-lapply (injective_plus_r … H) -L2 #H destruct /2 width=3 by ex2_intro/
-qed-.
-
-theorem drops_conf_div_fcla: ∀f1,f2,L1,L2,n1,n2.
- ⬇*[Ⓣ, f1] L1 ≘ L2 → ⬇*[Ⓣ, f2] L1 ≘ L2 → 𝐂⦃f1⦄ ≘ n1 → 𝐂⦃f2⦄ ≘ n2 →
- n1 = n2.
-#f1 #f2 #L1 #L2 #n1 #n2 #Hf1 #Hf2 #Hn1 #Hn2
-lapply (drops_fcla_fwd … Hf1 Hn1) -f1 #H1
-lapply (drops_fcla_fwd … Hf2 Hn2) -f2 >H1 -L1
-/2 width=1 by injective_plus_r/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lex.ma".
-include "basic_2/relocation/drops_cext2.ma".
-include "basic_2/relocation/drops_sex.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-definition dedropable_sn: predicate … ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. K1 ⪤[R] K2 →
- ∃∃L2. L1 ⪤[R] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
-
-definition dropable_sn: predicate … ≝
- λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀L2. L1 ⪤[R] L2 →
- ∃∃K2. K1 ⪤[R] K2 & ⬇*[b, f] L2 ≘ K2.
-
-definition dropable_dx: predicate … ≝
- λR. ∀L1,L2. L1 ⪤[R] L2 → ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R] K2.
-
-(* Properties with generic extension ****************************************)
-
-(* Basic_2A1: was: lpx_sn_liftable_dedropable *)
-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 (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-.
-
-(* Inversion lemmas with generic extension **********************************)
-
-(* 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 (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 (sex_co_dropable_dx … HL12 … HLK2) -L2
-/3 width=5 by coafter_isid_dx, ex2_intro/
-qed-.
-
-(* Basic_2A1: includes: lpx_sn_drop_conf *)
-lemma lex_drops_conf_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
- ∀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 (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/
-| /3 width=3 by coafter_isid_dx, isid_push/
-]
-qed-.
-
-(* Basic_2A1: includes: lpx_sn_drop_trans *)
-lemma lex_drops_trans_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
- ∀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 (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/
-| /3 width=3 by coafter_isid_dx, isid_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/ltc.ma".
-include "basic_2/relocation/seq_seq.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Properties with labeled transitive closure *******************************)
-
-lemma d2_liftable_sn_ltc: ∀A,f. associative … f →
- ∀C,S,R. (∀n. d_liftable2_sn C S (λL. R L n)) →
- ∀n. d_liftable2_sn C S (λL. ltc A f … (R L) n).
-#A #g #Hg #C #S #R #HR #n #K #T1 #T2 #H
-@(ltc_ind_dx … Hg ???? H) -n -T2
-[ #n #T2 #HT12 #b #g #L #HLK #U1 #HTU1
- elim (HR … HT12 … HLK … HTU1) -b -K -T1 -HR
- /3 width=3 by ltc_rc, ex2_intro/
-| #n1 #n2 #T #T2 #_ #IHT1 #HT2 #b #f #L #HLK #U1 #HTU1
- elim (IHT1 … HLK … HTU1) -T1 #U #HTU #HU1
- elim (HR … HT2 … HLK … HTU) -HR -K -T
- /3 width=5 by ltc_dx, ex2_intro/
-]
-qed-.
-
-lemma d2_deliftable_sn_ltc: ∀A,f. associative … f →
- ∀C,S,R. (∀n. d_deliftable2_sn C S (λL. R L n)) →
- ∀n. d_deliftable2_sn C S (λL. ltc A f … (R L) n).
-#A #g #Hg #C #S #R #HR #n #L #U1 #U2 #H
-@(ltc_ind_dx … Hg ???? H) -n -U2
-[ #n #U2 #HU12 #b #g #K #HLK #T1 #HTU1
- elim (HR … HU12 … HLK … HTU1) -b -L -U1 -HR
- /3 width=3 by ltc_rc, ex2_intro/
-| #n1 #n2 #U #U2 #_ #IHU1 #HU2 #b #f #K #HLK #T1 #HTU1
- elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
- elim (HR … HU2 … HLK … HTU) -L -U -HR
- /3 width=5 by ltc_dx, ex2_intro/
-]
-qed-.
-
-lemma co_dropable_sn_ltc: ∀A,f. associative … f →
- ∀R. (∀n. co_dropable_sn (λL. R L n)) →
- ∀n. co_dropable_sn (λL. ltc A f … (R L) n).
-#A #g #Hg #R #HR #n #b #f #L1 #K1 #HLK1 #Hf #f2 #L2 #H
-@(ltc_ind_dx … Hg ???? H) -n -L2
-[ #n #L2 #HL12 #g1 #H
- elim (HR … HLK1 … Hf … HL12 … H) -f2 -L1 -HR -Hf
- /3 width=3 by ltc_rc, ex2_intro/
-| #n1 #n2 #L #L2 #_ #IH #HL2 #f1 #H
- elim (IH … H) -IH #K #HK1 #HLK
- elim (HR … HLK … HL2 … H) -f2 -L -HR
- /3 width=3 by ltc_dx, ex2_intro/
-]
-qed-.
-
-lemma co_dropable_dx_ltc: ∀A,f. associative … f →
- ∀R. (∀n. co_dropable_dx (λL. R L n)) →
- ∀n. co_dropable_dx (λL. ltc A f … (R L) n).
-#A #g #Hg #R #HR #n #f2 #L1 #L2 #H
-@(ltc_ind_dx … Hg ???? H) -n -L2
-[ #n #L2 #HL12 #b #f #K2 #HLK2 #Hf #f1 #Hf2
- elim (HR … HL12 … HLK2 … Hf … Hf2) -f2 -L2 -HR -Hf
- /3 width=3 by ltc_rc, ex2_intro/
-| #n1 #n2 #L #L2 #_ #IHL1 #HL2 #b #f #K2 #HLK2 #Hf #f1 #Hf2
- elim (HR … HL2 … HLK2 … Hf … Hf2) -L2 -HR #K #HLK #HK2
- elim (IHL1 … HLK … Hf … Hf2) -Hf -f2 -L
- /3 width=5 by ltc_dx, ex2_intro/
-]
-qed-.
-
-lemma co_dedropable_sn_ltc: ∀A,f. associative … f →
- ∀R. (∀n. co_dedropable_sn (λL. R L n)) →
- ∀n. co_dedropable_sn (λL. ltc A f … (R L) n).
-#A #g #Hg #R #HR #n #b #f #L1 #K1 #HLK1 #f1 #K2 #H
-@(ltc_ind_dx … Hg ???? H) -n -K2
-[ #n #K2 #HK12 #f2 #Hf
- elim (HR … HLK1 … HK12 … Hf) -f1 -K1 -HR
- /3 width=4 by ltc_rc, ex3_intro/
-| #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 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-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lifts_vector.ma".
-include "basic_2/relocation/drops.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-definition d_liftable1_all: predicate (relation2 lenv term) ≝
- λR. ∀K,Ts. all … (R K) Ts →
- ∀b,f,L. ⬇*[b, f] L ≘ K →
- ∀Us. ⬆*[f] Ts ≘ Us → all … (R L) Us.
-
-(* Properties with generic relocation for term vectors **********************)
-
-(* Basic_2A1: was: d1_liftables_liftables_all *)
-lemma d1_liftable_liftable_all: ∀R. d_liftable1 R → d_liftable1_all R.
-#R #HR #K #Ts #HTs #b #f #L #HLK #Us #H
-generalize in match HTs; -HTs elim H -Ts -Us normalize //
-#Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/syntax/cl_restricted_weight.ma".
-include "basic_2/relocation/lifts_weight_bind.ma".
-include "basic_2/relocation/drops.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Forward lemmas with weight for local environments ************************)
-
-(* Basic_2A1: includes: drop_fwd_lw *)
-lemma drops_fwd_lw: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ♯{L2} ≤ ♯{L1}.
-#b #f #L1 #L2 #H elim H -f -L1 -L2 //
-[ /2 width=3 by transitive_le/
-| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 normalize
- >(liftsb_fwd_bw … HI21) -HI21 /2 width=1 by monotonic_le_plus_l/
-]
-qed-.
-
-(* Basic_2A1: includes: drop_fwd_lw_lt *)
-lemma drops_fwd_lw_lt: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
- (𝐈⦃f⦄ → ⊥) → ♯{L2} < ♯{L1}.
-#f #L1 #L2 #H elim H -f -L1 -L2
-[ #f #Hf #Hnf elim Hnf -Hnf /2 width=1 by/
-| /3 width=3 by drops_fwd_lw, le_to_lt_to_lt/
-| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 #H normalize in ⊢ (?%%);
- >(liftsb_fwd_bw … HI21) -I2 /5 width=3 by isid_push, monotonic_lt_plus_l/
-]
-qed-.
-
-(* Forward lemmas with restricted weight for closures ***********************)
-
-(* Basic_2A1: includes: drop_fwd_rfw *)
-lemma drops_bind2_fwd_rfw: ∀b,f,I,L,K,V. ⬇*[b, f] L ≘ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
-#b #f #I #L #K #V #HLK lapply (drops_fwd_lw … HLK) -HLK
-normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt, monotonic_lt_plus_r/
-qed-.
-
-(* Advanced inversion lemma *************************************************)
-
-lemma drops_inv_x_bind_xy: ∀b,f,I,L. ⬇*[b, f] L ≘ L.ⓘ{I} → ⊥.
-#b #f #I #L #H lapply (drops_fwd_lw … H) -b -f
-/2 width=4 by lt_le_false/ (**) (* full auto is a bit slow: 19s *)
-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/pull/pull_2.ma".
-include "ground_2/pull/pull_4.ma".
-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/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.
-
-interpretation "generic extension (local environment)"
- 'Relation R L1 L2 = (lex R L1 L2).
-
-definition lex_confluent: relation (relation3 …) ≝ λR1,R2.
- ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
- ∀L1. L0 ⪤[R1] L1 → ∀L2. L0 ⪤[R2] L2 →
- ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
-
-definition lex_transitive: relation (relation3 …) ≝ λR1,R2.
- ∀L1,T1,T. R1 L1 T1 T → ∀L2. L1 ⪤[R1] L2 →
- ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
-
-(* Basic properties *********************************************************)
-
-(* Basic_2A1: was: lpx_sn_atom *)
-lemma lex_atom (R): ⋆ ⪤[R] ⋆.
-/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 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 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 sex_co, cext2_co, ex2_intro/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lex_bind_refl_dx (R): c_reflexive … R →
- ∀I,K1,K2. K1 ⪤[R] K2 → K1.ⓘ{I} ⪤[R] K2.ⓘ{I}.
-/3 width=3 by ext2_refl, lex_bind/ qed.
-
-lemma lex_unit (R): ∀I,K1,K2. K1 ⪤[R] K2 → K1.ⓤ{I} ⪤[R] K2.ⓤ{I}.
-/3 width=1 by lex_bind, ext2_unit/ qed.
-
-(* Basic_2A1: was: lpx_sn_pair *)
-lemma lex_pair (R): ∀I,K1,K2,V1,V2. K1 ⪤[R] K2 → R K1 V1 V2 →
- K1.ⓑ{I}V1 ⪤[R] K2.ⓑ{I}V2.
-/3 width=1 by lex_bind, ext2_pair/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Basic_2A1: was: lpx_sn_inv_atom1: *)
-lemma lex_inv_atom_sn (R): ∀L2. ⋆ ⪤[R] L2 → 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 (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 >(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 (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-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lex_inv_unit_sn (R): ∀I,L2,K1. K1.ⓤ{I} ⪤[R] L2 →
- ∃∃K2. K1 ⪤[R] K2 & L2 = K2.ⓤ{I}.
-#R #I #L2 #K1 #H
-elim (lex_inv_bind_sn … H) -H #Z2 #K2 #HK12 #HZ2 #H destruct
-elim (ext2_inv_unit_sn … HZ2) -HZ2
-/2 width=3 by ex2_intro/
-qed-.
-
-(* Basic_2A1: was: lpx_sn_inv_pair1 *)
-lemma lex_inv_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R] L2 →
- ∃∃K2,V2. K1 ⪤[R] K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
-#R #I #L2 #K1 #V1 #H
-elim (lex_inv_bind_sn … H) -H #Z2 #K2 #HK12 #HZ2 #H destruct
-elim (ext2_inv_pair_sn … HZ2) -HZ2 #V2 #HV12 #H destruct
-/2 width=5 by ex3_2_intro/
-qed-.
-
-lemma lex_inv_unit_dx (R): ∀I,L1,K2. L1 ⪤[R] K2.ⓤ{I} →
- ∃∃K1. K1 ⪤[R] K2 & L1 = K1.ⓤ{I}.
-#R #I #L1 #K2 #H
-elim (lex_inv_bind_dx … H) -H #Z1 #K1 #HK12 #HZ1 #H destruct
-elim (ext2_inv_unit_dx … HZ1) -HZ1
-/2 width=3 by ex2_intro/
-qed-.
-
-(* Basic_2A1: was: lpx_sn_inv_pair2 *)
-lemma lex_inv_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R] K2.ⓑ{I}V2 →
- ∃∃K1,V1. K1 ⪤[R] K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
-#R #I #L1 #K2 #V2 #H
-elim (lex_inv_bind_dx … H) -H #Z1 #K1 #HK12 #HZ1 #H destruct
-elim (ext2_inv_pair_dx … HZ1) -HZ1 #V1 #HV12 #H destruct
-/2 width=5 by ex3_2_intro/
-qed-.
-
-(* Basic_2A1: was: lpx_sn_inv_pair *)
-lemma lex_inv_pair (R): ∀I1,I2,L1,L2,V1,V2.
- L1.ⓑ{I1}V1 ⪤[R] L2.ⓑ{I2}V2 →
- ∧∧ L1 ⪤[R] L2 & R L1 V1 V2 & I1 = I2.
-#R #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lex_inv_pair_sn … H) -H
-#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
-qed-.
-
-(* Basic eliminators ********************************************************)
-
-lemma lex_ind (R) (Q:relation2 …):
- Q (⋆) (⋆) →
- (
- ∀I,K1,K2. K1 ⪤[R] K2 → Q K1 K2 → Q (K1.ⓤ{I}) (K2.ⓤ{I})
- ) → (
- ∀I,K1,K2,V1,V2. K1 ⪤[R] K2 → Q K1 K2 → R K1 V1 V2 →Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
- ) →
- ∀L1,L2. L1 ⪤[R] L2 → Q L1 L2.
-#R #Q #IH1 #IH2 #IH3 #L1 #L2 * #f @pull_2 #H
-elim H -f -L1 -L2 // #f #I1 #I2 #K1 #K2 @pull_4 #H
-[ elim (isid_inv_next … H)
-| lapply (isid_inv_push … H ??)
-] -H [5:|*: // ] #Hf @pull_2 #H
-elim H -H /3 width=3 by ex2_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/sex_length.ma".
-include "basic_2/relocation/lex.ma".
-
-(* GENERIC EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **************)
-
-(* Forward lemmas with length for local environments ************************)
-
-(* 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 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/relocation/lex.ma".
-
-(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
-
-(* Main properties **********************************************************)
-
-(* Basic_2A1: was: lpx_sn_trans *)
-theorem lex_trans (R): lex_transitive R R → Transitive … (lex R).
-#R #HR #L1 #L #H @(lex_ind … H) -L1 -L //
-[ #I #K1 #K #HK1 #IH #Y #H
- elim (lex_inv_unit_sn … H) -H #K2 #K2 #H destruct
- /3 width=1 by lex_unit/
-| #I #K1 #L #V1 #V #HK1 #IH #HV1 #X #H
- elim (lex_inv_pair_sn … H) -H #K2 #V2 #HK2 #HV2 #H destruct
- /3 width=5 by lex_pair/
-]
-qed-.
-
-(* Basic_2A1: was: lpx_sn_conf *)
-theorem lex_conf (R1) (R2): lex_confluent R1 R2 → confluent2 … (lex R1) (lex R2).
-#R1 #R2 #HR12 #L0 elim L0 -L0 [| #K0 * ]
-[ #Y1 #H1 #Y2 #H2
- >(lex_inv_atom_sn … H1) -Y1
- >(lex_inv_atom_sn … H2) -Y2
- /2 width=3 by lex_atom, ex2_intro/
-| #I #IH #Y1 #H1 #Y2 #H2
- elim (lex_inv_unit_sn … H1) -H1 #K1 #HK01 #H destruct
- elim (lex_inv_unit_sn … H2) -H2 #K2 #HK02 #H destruct
- elim (IH … HK01 … HK02) -K0 #K #HK1 #HK2
- /3 width=3 by lex_unit, ex2_intro/
-| #I #V0 #IH #Y1 #H1 #Y2 #H2
- elim (lex_inv_pair_sn … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
- elim (lex_inv_pair_sn … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct
- elim (HR12 … HV01 … HV02 … HK01 … HK02) -V0 #V #HV1 #HV2
- elim (IH … HK01 … HK02) -K0 #K #HK1 #HK2
- /3 width=5 by lex_pair, ex2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/syntax/ext2_tc.ma".
-include "basic_2/relocation/sex_tc.ma".
-include "basic_2/relocation/lex.ma".
-
-alias symbol "subseteq" = "relation inclusion".
-
-(* GENERIC EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **************)
-
-(* Inversion lemmas with transitive closure *********************************)
-
-(* Basic_2A1: was: lpx_sn_LTC_TC_lpx_sn *)
-lemma lex_inv_CTC (R): c_reflexive … R →
- lex (CTC … R) ⊆ TC … (lex R).
-#R #HR #L1 #L2 *
-/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) →
- s_rs_transitive_isid cfull (cext2 R).
-#R #HR #f #Hf #L2 #T1 #T2 #H #L1 #HL12
-elim (ext2_tc … H) -H
-[ /3 width=1 by ext2_inv_tc, ext2_unit/
-| #I #V1 #V2 #HV12
- @ext2_inv_tc @ext2_pair
- @(HR … HV12) -HV12 /2 width=3 by ex2_intro/ (**) (* auto fails *)
-]
-qed-.
-
-(* Properties with transitive closure ***************************************)
-
-(* Basic_2A1: was: TC_lpx_sn_inv_lpx_sn_LTC *)
-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 … (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-.
-
-lemma lex_CTC_inj (R): s_rs_transitive … R (λ_. lex R) →
- (lex R) ⊆ lex (CTC … R).
-/3 width=1 by lex_CTC, inj/ qed-.
-
-lemma lex_CTC_step_dx (R): c_reflexive … R → s_rs_transitive … R (λ_. lex R) →
- ∀L1,L. lex (CTC … R) L1 L →
- ∀L2. lex R L L2 → lex (CTC … R) L1 L2.
-/4 width=3 by lex_CTC, lex_inv_CTC, step/ qed-.
-
-lemma lex_CTC_step_sn (R): c_reflexive … R → s_rs_transitive … R (λ_. lex R) →
- ∀L1,L. lex R L1 L →
- ∀L2. lex (CTC … R) L L2 → lex (CTC … R) L1 L2.
-/4 width=3 by lex_CTC, lex_inv_CTC, TC_strap/ qed-.
-
-(* Eliminators with transitive closure **************************************)
-
-lemma lex_CTC_ind_sn (R) (L2): c_reflexive … R → s_rs_transitive … R (λ_. lex R) →
- ∀Q:predicate lenv. Q L2 →
- (∀L1,L. L1 ⪤[R] L → L ⪤[CTC … R] L2 → Q L → Q L1) →
- ∀L1. L1 ⪤[CTC … R] L2 → Q L1.
-#R #L2 #H1R #H2R #Q #IH1 #IH2 #L1 #H
-lapply (lex_inv_CTC … H1R … H) -H #H
-@(TC_star_ind_dx ???????? H) -H
-/3 width=4 by lex_CTC, lex_refl/
-qed-.
-
-lemma lex_CTC_ind_dx (R) (L1): c_reflexive … R → s_rs_transitive … R (λ_. lex R) →
- ∀Q:predicate lenv. Q L1 →
- (∀L,L2. L1 ⪤[CTC … R] L → L ⪤[R] L2 → Q L → Q L2) →
- ∀L2. L1 ⪤[CTC … R] L2 → Q L2.
-#R #L1 #H1R #H2R #Q #IH1 #IH2 #L2 #H
-lapply (lex_inv_CTC … H1R … H) -H #H
-@(TC_star_ind ???????? H) -H
-/3 width=4 by lex_CTC, lex_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 "ground_2/relocation/nstream_after.ma".
-include "basic_2/notation/relations/rliftstar_3.ma".
-include "basic_2/syntax/term.ma".
-
-(* GENERIC RELOCATION FOR TERMS *********************************************)
-
-(* Basic_1: includes:
- lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
- lifts_nil lifts_cons
-*)
-inductive lifts: rtmap → relation term ≝
-| lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
-| lifts_lref: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → lifts f (#i1) (#i2)
-| lifts_gref: ∀f,l. lifts f (§l) (§l)
-| lifts_bind: ∀f,p,I,V1,V2,T1,T2.
- lifts f V1 V2 → lifts (⫯f) T1 T2 →
- lifts f (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
-| lifts_flat: ∀f,I,V1,V2,T1,T2.
- lifts f V1 V2 → lifts f T1 T2 →
- lifts f (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
-.
-
-interpretation "uniform relocation (term)"
- 'RLiftStar i T1 T2 = (lifts (uni i) T1 T2).
-
-interpretation "generic relocation (term)"
- 'RLiftStar f T1 T2 = (lifts f T1 T2).
-
-definition liftable2_sn: predicate (relation term) ≝
- λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≘ U1 →
- ∃∃U2. ⬆*[f] T2 ≘ U2 & R U1 U2.
-
-definition deliftable2_sn: predicate (relation term) ≝
- λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≘ U1 →
- ∃∃T2. ⬆*[f] T2 ≘ U2 & R T1 T2.
-
-definition liftable2_bi: predicate (relation term) ≝
- λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≘ U1 →
- ∀U2. ⬆*[f] T2 ≘ U2 → R U1 U2.
-
-definition deliftable2_bi: predicate (relation term) ≝
- λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≘ U1 →
- ∀T2. ⬆*[f] T2 ≘ U2 → R T1 T2.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lifts_inv_sort1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀s. X = ⋆s → Y = ⋆s.
-#f #X #Y * -f -X -Y //
-[ #f #i1 #i2 #_ #x #H destruct
-| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-]
-qed-.
-
-(* Basic_1: was: lift1_sort *)
-(* Basic_2A1: includes: lift_inv_sort1 *)
-lemma lifts_inv_sort1: ∀f,Y,s. ⬆*[f] ⋆s ≘ Y → Y = ⋆s.
-/2 width=4 by lifts_inv_sort1_aux/ qed-.
-
-fact lifts_inv_lref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i1. X = #i1 →
- ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
-#f #X #Y * -f -X -Y
-[ #f #s #x #H destruct
-| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
-| #f #l #x #H destruct
-| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-]
-qed-.
-
-(* Basic_1: was: lift1_lref *)
-(* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
-lemma lifts_inv_lref1: ∀f,Y,i1. ⬆*[f] #i1 ≘ Y →
- ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
-/2 width=3 by lifts_inv_lref1_aux/ qed-.
-
-fact lifts_inv_gref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. X = §l → Y = §l.
-#f #X #Y * -f -X -Y //
-[ #f #i1 #i2 #_ #x #H destruct
-| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-]
-qed-.
-
-(* Basic_2A1: includes: lift_inv_gref1 *)
-lemma lifts_inv_gref1: ∀f,Y,l. ⬆*[f] §l ≘ Y → Y = §l.
-/2 width=4 by lifts_inv_gref1_aux/ qed-.
-
-fact lifts_inv_bind1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y →
- ∀p,I,V1,T1. X = ⓑ{p,I}V1.T1 →
- ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
- Y = ⓑ{p,I}V2.T2.
-#f #X #Y * -f -X -Y
-[ #f #s #q #J #W1 #U1 #H destruct
-| #f #i1 #i2 #_ #q #J #W1 #U1 #H destruct
-| #f #l #b #J #W1 #U1 #H destruct
-| #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-| #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W1 #U1 #H destruct
-]
-qed-.
-
-(* Basic_1: was: lift1_bind *)
-(* Basic_2A1: includes: lift_inv_bind1 *)
-lemma lifts_inv_bind1: ∀f,p,I,V1,T1,Y. ⬆*[f] ⓑ{p,I}V1.T1 ≘ Y →
- ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
- Y = ⓑ{p,I}V2.T2.
-/2 width=3 by lifts_inv_bind1_aux/ qed-.
-
-fact lifts_inv_flat1_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
- ∀I,V1,T1. X = ⓕ{I}V1.T1 →
- ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
- Y = ⓕ{I}V2.T2.
-#f #X #Y * -f -X -Y
-[ #f #s #J #W1 #U1 #H destruct
-| #f #i1 #i2 #_ #J #W1 #U1 #H destruct
-| #f #l #J #W1 #U1 #H destruct
-| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
-| #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_1: was: lift1_flat *)
-(* Basic_2A1: includes: lift_inv_flat1 *)
-lemma lifts_inv_flat1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ⓕ{I}V1.T1 ≘ Y →
- ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
- Y = ⓕ{I}V2.T2.
-/2 width=3 by lifts_inv_flat1_aux/ qed-.
-
-fact lifts_inv_sort2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀s. Y = ⋆s → X = ⋆s.
-#f #X #Y * -f -X -Y //
-[ #f #i1 #i2 #_ #x #H destruct
-| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-]
-qed-.
-
-(* Basic_1: includes: lift_gen_sort *)
-(* Basic_2A1: includes: lift_inv_sort2 *)
-lemma lifts_inv_sort2: ∀f,X,s. ⬆*[f] X ≘ ⋆s → X = ⋆s.
-/2 width=4 by lifts_inv_sort2_aux/ qed-.
-
-fact lifts_inv_lref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i2. Y = #i2 →
- ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
-#f #X #Y * -f -X -Y
-[ #f #s #x #H destruct
-| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
-| #f #l #x #H destruct
-| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-]
-qed-.
-
-(* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *)
-(* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *)
-lemma lifts_inv_lref2: ∀f,X,i2. ⬆*[f] X ≘ #i2 →
- ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
-/2 width=3 by lifts_inv_lref2_aux/ qed-.
-
-fact lifts_inv_gref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. Y = §l → X = §l.
-#f #X #Y * -f -X -Y //
-[ #f #i1 #i2 #_ #x #H destruct
-| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
-]
-qed-.
-
-(* Basic_2A1: includes: lift_inv_gref1 *)
-lemma lifts_inv_gref2: ∀f,X,l. ⬆*[f] X ≘ §l → X = §l.
-/2 width=4 by lifts_inv_gref2_aux/ qed-.
-
-fact lifts_inv_bind2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y →
- ∀p,I,V2,T2. Y = ⓑ{p,I}V2.T2 →
- ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
- X = ⓑ{p,I}V1.T1.
-#f #X #Y * -f -X -Y
-[ #f #s #q #J #W2 #U2 #H destruct
-| #f #i1 #i2 #_ #q #J #W2 #U2 #H destruct
-| #f #l #q #J #W2 #U2 #H destruct
-| #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
-| #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W2 #U2 #H destruct
-]
-qed-.
-
-(* Basic_1: includes: lift_gen_bind *)
-(* Basic_2A1: includes: lift_inv_bind2 *)
-lemma lifts_inv_bind2: ∀f,p,I,V2,T2,X. ⬆*[f] X ≘ ⓑ{p,I}V2.T2 →
- ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
- X = ⓑ{p,I}V1.T1.
-/2 width=3 by lifts_inv_bind2_aux/ qed-.
-
-fact lifts_inv_flat2_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
- ∀I,V2,T2. Y = ⓕ{I}V2.T2 →
- ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
- X = ⓕ{I}V1.T1.
-#f #X #Y * -f -X -Y
-[ #f #s #J #W2 #U2 #H destruct
-| #f #i1 #i2 #_ #J #W2 #U2 #H destruct
-| #f #l #J #W2 #U2 #H destruct
-| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W2 #U2 #H destruct
-| #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_1: includes: lift_gen_flat *)
-(* Basic_2A1: includes: lift_inv_flat2 *)
-lemma lifts_inv_flat2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≘ ⓕ{I}V2.T2 →
- ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
- X = ⓕ{I}V1.T1.
-/2 width=3 by lifts_inv_flat2_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≘ Y →
- ∨∨ ∃∃s. I = Sort s & Y = ⋆s
- | ∃∃i,j. @⦃i, f⦄ ≘ j & I = LRef i & Y = #j
- | ∃∃l. I = GRef l & Y = §l.
-#f * #n #Y #H
-[ lapply (lifts_inv_sort1 … H)
-| elim (lifts_inv_lref1 … H)
-| lapply (lifts_inv_gref1 … H)
-] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
-qed-.
-
-lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≘ ⓪{I} →
- ∨∨ ∃∃s. X = ⋆s & I = Sort s
- | ∃∃i,j. @⦃i, f⦄ ≘ j & X = #i & I = LRef j
- | ∃∃l. X = §l & I = GRef l.
-#f * #n #X #H
-[ lapply (lifts_inv_sort2 … H)
-| elim (lifts_inv_lref2 … H)
-| lapply (lifts_inv_gref2 … H)
-] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
-qed-.
-
-(* Basic_2A1: includes: lift_inv_pair_xy_x *)
-lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⬆*[f] ②{I}V.T ≘ V → ⊥.
-#f #J #V elim V -V
-[ * #i #U #H
- [ lapply (lifts_inv_sort2 … H) -H #H destruct
- | elim (lifts_inv_lref2 … H) -H
- #x #_ #H destruct
- | lapply (lifts_inv_gref2 … H) -H #H destruct
- ]
-| * [ #p ] #I #V2 #T2 #IHV2 #_ #U #H
- [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
- | elim (lifts_inv_flat2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
- ]
-]
-qed-.
-
-(* Basic_1: includes: thead_x_lift_y_y *)
-(* Basic_2A1: includes: lift_inv_pair_xy_y *)
-lemma lifts_inv_pair_xy_y: ∀I,T,V,f. ⬆*[f] ②{I}V.T ≘ T → ⊥.
-#J #T elim T -T
-[ * #i #W #f #H
- [ lapply (lifts_inv_sort2 … H) -H #H destruct
- | elim (lifts_inv_lref2 … H) -H
- #x #_ #H destruct
- | lapply (lifts_inv_gref2 … H) -H #H destruct
- ]
-| * [ #p ] #I #V2 #T2 #_ #IHT2 #W #f #H
- [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
- | elim (lifts_inv_flat2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
- ]
-]
-qed-.
-
-(* Inversion lemmas with uniform relocations ********************************)
-
-lemma lifts_inv_lref1_uni: ∀l,Y,i. ⬆*[l] #i ≘ Y → Y = #(l+i).
-#l #Y #i1 #H elim (lifts_inv_lref1 … H) -H /4 width=4 by at_mono, eq_f/
-qed-.
-
-lemma lifts_inv_lref2_uni: ∀l,X,i2. ⬆*[l] X ≘ #i2 →
- ∃∃i1. X = #i1 & i2 = l + i1.
-#l #X #i2 #H elim (lifts_inv_lref2 … H) -H
-/3 width=3 by at_inv_uni, ex2_intro/
-qed-.
-
-lemma lifts_inv_lref2_uni_ge: ∀l,X,i. ⬆*[l] X ≘ #(l + i) → X = #i.
-#l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
-#i1 #H1 #H2 destruct /4 width=2 by injective_plus_r, eq_f, sym_eq/
-qed-.
-
-lemma lifts_inv_lref2_uni_lt: ∀l,X,i. ⬆*[l] X ≘ #i → i < l → ⊥.
-#l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
-#i1 #_ #H1 #H2 destruct /2 width=4 by lt_le_false/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-(* Basic_2A1: includes: lift_inv_O2 *)
-lemma lifts_fwd_isid: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐈⦃f⦄ → T1 = T2.
-#f #T1 #T2 #H elim H -f -T1 -T2
-/4 width=3 by isid_inv_at_mono, isid_push, eq_f2, eq_f/
-qed-.
-
-(* Basic_2A1: includes: lift_fwd_pair1 *)
-lemma lifts_fwd_pair1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ②{I}V1.T1 ≘ Y →
- ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & Y = ②{I}V2.T2.
-#f * [ #p ] #I #V1 #T1 #Y #H
-[ elim (lifts_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
-| elim (lifts_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-(* Basic_2A1: includes: lift_fwd_pair2 *)
-lemma lifts_fwd_pair2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≘ ②{I}V2.T2 →
- ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & X = ②{I}V1.T1.
-#f * [ #p ] #I #V2 #T2 #X #H
-[ elim (lifts_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
-| elim (lifts_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lifts_eq_repl_back: ∀T1,T2. eq_repl_back … (λf. ⬆*[f] T1 ≘ T2).
-#T1 #T2 #f1 #H elim H -T1 -T2 -f1
-/4 width=5 by lifts_flat, lifts_bind, lifts_lref, at_eq_repl_back, eq_push/
-qed-.
-
-lemma lifts_eq_repl_fwd: ∀T1,T2. eq_repl_fwd … (λf. ⬆*[f] T1 ≘ T2).
-#T1 #T2 @eq_repl_sym /2 width=3 by lifts_eq_repl_back/ (**) (* full auto fails *)
-qed-.
-
-(* Basic_1: includes: lift_r *)
-(* Basic_2A1: includes: lift_refl *)
-lemma lifts_refl: ∀T,f. 𝐈⦃f⦄ → ⬆*[f] T ≘ T.
-#T elim T -T *
-/4 width=3 by lifts_flat, lifts_bind, lifts_lref, isid_inv_at, isid_push/
-qed.
-
-(* Basic_2A1: includes: lift_total *)
-lemma lifts_total: ∀T1,f. ∃T2. ⬆*[f] T1 ≘ T2.
-#T1 elim T1 -T1 *
-/3 width=2 by lifts_lref, lifts_sort, lifts_gref, ex_intro/
-[ #p ] #I #V1 #T1 #IHV1 #IHT1 #f
-elim (IHV1 f) -IHV1 #V2 #HV12
-[ elim (IHT1 (⫯f)) -IHT1 /3 width=2 by lifts_bind, ex_intro/
-| elim (IHT1 f) -IHT1 /3 width=2 by lifts_flat, ex_intro/
-]
-qed-.
-
-lemma lift_lref_uni: ∀l,i. ⬆*[l] #i ≘ #(l+i).
-#l elim l -l /2 width=1 by lifts_lref/
-qed.
-
-(* Basic_1: includes: lift_free (right to left) *)
-(* Basic_2A1: includes: lift_split *)
-lemma lifts_split_trans: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 →
- ∀f1,f2. f2 ⊚ f1 ≘ f →
- ∃∃T. ⬆*[f1] T1 ≘ T & ⬆*[f2] T ≘ T2.
-#f #T1 #T2 #H elim H -f -T1 -T2
-[ /3 width=3 by lifts_sort, ex2_intro/
-| #f #i1 #i2 #Hi #f1 #f2 #Ht elim (after_at_fwd … Hi … Ht) -Hi -Ht
- /3 width=3 by lifts_lref, ex2_intro/
-| /3 width=3 by lifts_gref, ex2_intro/
-| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
- elim (IHV … Ht) elim (IHT (⫯f1) (⫯f2)) -IHV -IHT
- /3 width=5 by lifts_bind, after_O2, ex2_intro/
-| #f #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
- elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
- /3 width=5 by lifts_flat, ex2_intro/
-]
-qed-.
-
-(* Note: apparently, this was missing in Basic_2A1 *)
-lemma lifts_split_div: ∀f1,T1,T2. ⬆*[f1] T1 ≘ T2 →
- ∀f2,f. f2 ⊚ f1 ≘ f →
- ∃∃T. ⬆*[f2] T2 ≘ T & ⬆*[f] T1 ≘ T.
-#f1 #T1 #T2 #H elim H -f1 -T1 -T2
-[ /3 width=3 by lifts_sort, ex2_intro/
-| #f1 #i1 #i2 #Hi #f2 #f #Ht elim (after_at1_fwd … Hi … Ht) -Hi -Ht
- /3 width=3 by lifts_lref, ex2_intro/
-| /3 width=3 by lifts_gref, ex2_intro/
-| #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
- elim (IHV … Ht) elim (IHT (⫯f2) (⫯f)) -IHV -IHT
- /3 width=5 by lifts_bind, after_O2, ex2_intro/
-| #f1 #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
- elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
- /3 width=5 by lifts_flat, ex2_intro/
-]
-qed-.
-
-(* Basic_1: includes: dnf_dec2 dnf_dec *)
-(* Basic_2A1: includes: is_lift_dec *)
-lemma is_lifts_dec: ∀T2,f. Decidable (∃T1. ⬆*[f] T1 ≘ T2).
-#T1 elim T1 -T1
-[ * [1,3: /3 width=2 by lifts_sort, lifts_gref, ex_intro, or_introl/ ]
- #i2 #f elim (is_at_dec f i2) //
- [ * /4 width=3 by lifts_lref, ex_intro, or_introl/
- | #H @or_intror *
- #X #HX elim (lifts_inv_lref2 … HX) -HX
- /3 width=2 by ex_intro/
- ]
-| * [ #p ] #I #V2 #T2 #IHV2 #IHT2 #f
- [ elim (IHV2 f) -IHV2
- [ * #V1 #HV12 elim (IHT2 (⫯f)) -IHT2
- [ * #T1 #HT12 @or_introl /3 width=2 by lifts_bind, ex_intro/
- | -V1 #HT2 @or_intror * #X #H
- elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
- ]
- | -IHT2 #HV2 @or_intror * #X #H
- elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
- ]
- | elim (IHV2 f) -IHV2
- [ * #V1 #HV12 elim (IHT2 f) -IHT2
- [ * #T1 #HT12 /4 width=2 by lifts_flat, ex_intro, or_introl/
- | -V1 #HT2 @or_intror * #X #H
- elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
- ]
- | -IHT2 #HV2 @or_intror * #X #H
- elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
- ]
- ]
-]
-qed-.
-
-(* Properties with uniform relocation ***************************************)
-
-lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≘ U → ⬆*[n1+n2] T ≘ U.
-/3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
-
-(* Basic_2A1: removed theorems 14:
- lifts_inv_nil lifts_inv_cons
- lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1
- lift_lref_lt_eq lift_lref_ge_eq lift_lref_plus lift_lref_pred
- lift_lref_ge_minus lift_lref_ge_minus_eq
-*)
-(* Basic_1: removed theorems 8:
- lift_lref_gt
- lift_head lift_gen_head
- lift_weight_map lift_weight lift_weight_add lift_weight_add_O
- lift_tlt_dx
-*)
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/syntax/ext2.ma".
-include "basic_2/relocation/lifts.ma".
-
-(* GENERIC RELOCATION FOR BINDERS *******************************************)
-
-definition liftsb: rtmap → relation bind ≝
- λf. ext2 (lifts f).
-
-interpretation "uniform relocation (binder for local environments)"
- 'RLiftStar i I1 I2 = (liftsb (uni i) I1 I2).
-
-interpretation "generic relocation (binder for local environments)"
- 'RLiftStar f I1 I2 = (liftsb f I1 I2).
-
-(* Basic_inversion lemmas **************************************************)
-
-lemma liftsb_inv_unit_sn: ∀f,I,Z2. ⬆*[f] BUnit I ≘ Z2 → Z2 = BUnit I.
-/2 width=2 by ext2_inv_unit_sn/ qed-.
-
-lemma liftsb_inv_pair_sn: ∀f:rtmap. ∀Z2,I,V1. ⬆*[f] BPair I V1 ≘ Z2 →
- ∃∃V2. ⬆*[f] V1 ≘ V2 & Z2 = BPair I V2.
-/2 width=1 by ext2_inv_pair_sn/ qed-.
-
-lemma liftsb_inv_unit_dx: ∀f,I,Z1. ⬆*[f] Z1 ≘ BUnit I → Z1 = BUnit I.
-/2 width=2 by ext2_inv_unit_dx/ qed-.
-
-lemma liftsb_inv_pair_dx: ∀f:rtmap. ∀Z1,I,V2. ⬆*[f] Z1 ≘ BPair I V2 →
- ∃∃V1. ⬆*[f] V1 ≘ V2 & Z1 = BPair I V1.
-/2 width=1 by ext2_inv_pair_dx/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma liftsb_eq_repl_back: ∀I1,I2. eq_repl_back … (λf. ⬆*[f] I1 ≘ I2).
-#I1 #I2 #f1 * -I1 -I2 /3 width=3 by lifts_eq_repl_back, ext2_pair/
-qed-.
-
-lemma liftsb_refl: ∀f. 𝐈⦃f⦄ → reflexive … (liftsb f).
-/3 width=1 by lifts_refl, ext2_refl/ qed.
-
-lemma liftsb_total: ∀I1,f. ∃I2. ⬆*[f] I1 ≘ I2.
-* [2: #I #T1 #f elim (lifts_total T1 f) ]
-/3 width=2 by ext2_unit, ext2_pair, ex_intro/
-qed-.
-
-lemma liftsb_split_trans: ∀f,I1,I2. ⬆*[f] I1 ≘ I2 →
- ∀f1,f2. f2 ⊚ f1 ≘ f →
- ∃∃I. ⬆*[f1] I1 ≘ I & ⬆*[f2] I ≘ I2.
-#f #I1 #I2 * -I1 -I2 /2 width=3 by ext2_unit, ex2_intro/
-#I #V1 #V2 #HV12 #f1 #f2 #Hf elim (lifts_split_trans … HV12 … Hf) -f
-/3 width=3 by ext2_pair, ex2_intro/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma liftsb_fwd_isid: ∀f,I1,I2. ⬆*[f] I1 ≘ I2 → 𝐈⦃f⦄ → I1 = I2.
-#f #I1 #I2 * -I1 -I2 /3 width=3 by lifts_fwd_isid, eq_f2/
-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.ma".
-
-(* GENERIC RELOCATION FOR TERMS *********************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: includes: lift_gen_lift *)
-(* Basic_2A1: includes: lift_div_le lift_div_be *)
-theorem lifts_div4: ∀f2,Tf,T. ⬆*[f2] Tf ≘ T → ∀g2,Tg. ⬆*[g2] Tg ≘ T →
- ∀f1,g1. H_at_div f2 g2 f1 g1 →
- ∃∃T0. ⬆*[f1] T0 ≘ Tf & ⬆*[g1] T0 ≘ Tg.
-#f2 #Tf #T #H elim H -f2 -Tf -T
-[ #f2 #s #g2 #Tg #H #f1 #g1 #_
- lapply (lifts_inv_sort2 … H) -H #H destruct
- /2 width=3 by ex2_intro/
-| #f2 #jf #j #Hf2 #g2 #Tg #H #f1 #g1 #H0
- elim (lifts_inv_lref2 … H) -H #jg #Hg2 #H destruct
- elim (H0 … Hf2 Hg2) -H0 -j /3 width=3 by lifts_lref, ex2_intro/
-| #f2 #l #g2 #Tg #H #f1 #g1 #_
- lapply (lifts_inv_gref2 … H) -H #H destruct
- /2 width=3 by ex2_intro/
-| #f2 #p #I #Vf #V #Tf #T #_ #_ #IHV #IHT #g2 #X #H #f1 #g1 #H0
- elim (lifts_inv_bind2 … H) -H #Vg #Tg #HVg #HTg #H destruct
- elim (IHV … HVg … H0) -IHV -HVg
- elim (IHT … HTg) -IHT -HTg [ |*: /2 width=8 by at_div_pp/ ]
- /3 width=5 by lifts_bind, ex2_intro/
-| #f2 #I #Vf #V #Tf #T #_ #_ #IHV #IHT #g2 #X #H #f1 #g1 #H0
- elim (lifts_inv_flat2 … H) -H #Vg #Tg #HVg #HTg #H destruct
- elim (IHV … HVg … H0) -IHV -HVg
- elim (IHT … HTg … H0) -IHT -HTg -H0
- /3 width=5 by lifts_flat, ex2_intro/
-]
-qed-.
-
-lemma lifts_div4_one: ∀f,Tf,T. ⬆*[⫯f] Tf ≘ T →
- ∀T1. ⬆*[1] T1 ≘ T →
- ∃∃T0. ⬆*[1] T0 ≘ Tf & ⬆*[f] T0 ≘ T1.
-/4 width=6 by lifts_div4, at_div_id_dx, at_div_pn/ qed-.
-
-theorem lifts_div3: ∀f2,T,T2. ⬆*[f2] T2 ≘ T → ∀f,T1. ⬆*[f] T1 ≘ T →
- ∀f1. f2 ⊚ f1 ≘ f → ⬆*[f1] T1 ≘ T2.
-#f2 #T #T2 #H elim H -f2 -T -T2
-[ #f2 #s #f #T1 #H >(lifts_inv_sort2 … H) -T1 //
-| #f2 #i2 #i #Hi2 #f #T1 #H #f1 #Ht21 elim (lifts_inv_lref2 … H) -H
- #i1 #Hi1 #H destruct /3 width=6 by lifts_lref, after_fwd_at1/
-| #f2 #l #f #T1 #H >(lifts_inv_gref2 … H) -T1 //
-| #f2 #p #I #W2 #W #U2 #U #_ #_ #IHW #IHU #f #T1 #H
- elim (lifts_inv_bind2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
- /4 width=3 by lifts_bind, after_O2/
-| #f2 #I #W2 #W #U2 #U #_ #_ #IHW #IHU #f #T1 #H
- elim (lifts_inv_flat2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
- /3 width=3 by lifts_flat/
-]
-qed-.
-
-(* Basic_1: was: lift1_lift1 (left to right) *)
-(* Basic_1: includes: lift_free (left to right) lift_d lift1_xhg (right to left) lift1_free (right to left) *)
-(* Basic_2A1: includes: lift_trans_be lift_trans_le lift_trans_ge lifts_lift_trans_le lifts_lift_trans *)
-theorem lifts_trans: ∀f1,T1,T. ⬆*[f1] T1 ≘ T → ∀f2,T2. ⬆*[f2] T ≘ T2 →
- ∀f. f2 ⊚ f1 ≘ f → ⬆*[f] T1 ≘ T2.
-#f1 #T1 #T #H elim H -f1 -T1 -T
-[ #f1 #s #f2 #T2 #H >(lifts_inv_sort1 … H) -T2 //
-| #f1 #i1 #i #Hi1 #f2 #T2 #H #f #Ht21 elim (lifts_inv_lref1 … H) -H
- #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at/
-| #f1 #l #f2 #T2 #H >(lifts_inv_gref1 … H) -T2 //
-| #f1 #p #I #W1 #W #U1 #U #_ #_ #IHW #IHU #f2 #T2 #H
- elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
- /4 width=3 by lifts_bind, after_O2/
-| #f1 #I #W1 #W #U1 #U #_ #_ #IHW #IHU #f2 #T2 #H
- elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
- /3 width=3 by lifts_flat/
-]
-qed-.
-
-(* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
-theorem lifts_conf: ∀f1,T,T1. ⬆*[f1] T ≘ T1 → ∀f,T2. ⬆*[f] T ≘ T2 →
- ∀f2. f2 ⊚ f1 ≘ f → ⬆*[f2] T1 ≘ T2.
-#f1 #T #T1 #H elim H -f1 -T -T1
-[ #f1 #s #f #T2 #H >(lifts_inv_sort1 … H) -T2 //
-| #f1 #i #i1 #Hi1 #f #T2 #H #f2 #Ht21 elim (lifts_inv_lref1 … H) -H
- #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at2/
-| #f1 #l #f #T2 #H >(lifts_inv_gref1 … H) -T2 //
-| #f1 #p #I #W #W1 #U #U1 #_ #_ #IHW #IHU #f #T2 #H
- elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
- /4 width=3 by lifts_bind, after_O2/
-| #f1 #I #W #W1 #U #U1 #_ #_ #IHW #IHU #f #T2 #H
- elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
- /3 width=3 by lifts_flat/
-]
-qed-.
-
-(* Advanced proprerties *****************************************************)
-
-(* Basic_2A1: includes: lift_inj *)
-lemma lifts_inj: ∀f. is_inj2 … (lifts f).
-#f #T1 #U #H1 #T2 #H2 lapply (after_isid_dx 𝐈𝐝 … f)
-/3 width=6 by lifts_div3, lifts_fwd_isid/
-qed-.
-
-(* Basic_2A1: includes: lift_mono *)
-lemma lifts_mono: ∀f,T. is_mono … (lifts f T).
-#f #T #U1 #H1 #U2 #H2 lapply (after_isid_sn 𝐈𝐝 … f)
-/3 width=6 by lifts_conf, lifts_fwd_isid/
-qed-.
-
-lemma liftable2_sn_bi: ∀R. liftable2_sn R → liftable2_bi R.
-#R #HR #T1 #T2 #HT12 #f #U1 #HTU1 #U2 #HTU2
-elim (HR … HT12 … HTU1) -HR -T1 #X #HTX #HUX
-<(lifts_mono … HTX … HTU2) -T2 -U2 -f //
-qed-.
-
-lemma deliftable2_sn_bi: ∀R. deliftable2_sn R → deliftable2_bi R.
-#R #HR #U1 #U2 #HU12 #f #T1 #HTU1 #T2 #HTU2
-elim (HR … HU12 … HTU1) -HR -U1 #X #HUX #HTX
-<(lifts_inj … HUX … HTU2) -U2 -T2 -f //
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/lifts_lifts.ma".
-include "basic_2/relocation/lifts_bind.ma".
-
-(* GENERIC RELOCATION FOR BINDERS *******************************************)
-
-(* Main properties **********************************************************)
-
-theorem liftsb_div3: ∀f2,I,I2. ⬆*[f2] I2 ≘ I → ∀f,I1. ⬆*[f] I1 ≘ I →
- ∀f1. f2 ⊚ f1 ≘ f → ⬆*[f1] I1 ≘ I2.
-#f2 #I #I2 * -I -I2 #I [2: #V #V2 #HV2 ] #f #I1 #H
-[ elim (liftsb_inv_pair_dx … H) | lapply (liftsb_inv_unit_dx … H) ] -H
-/3 width=6 by lifts_div3, ext2_pair, ext2_unit/
-qed-.
-
-theorem liftsb_trans: ∀f1,I1,I. ⬆*[f1] I1 ≘ I → ∀f2,I2. ⬆*[f2] I ≘ I2 →
- ∀f. f2 ⊚ f1 ≘ f → ⬆*[f] I1 ≘ I2.
-#f1 #I1 #I * -I1 -I #I1 [2: #V1 #V #HV1 ] #f2 #I2 #H
-[ elim (liftsb_inv_pair_sn … H) | lapply (liftsb_inv_unit_sn … H) ] -H
-/3 width=6 by lifts_trans, ext2_pair, ext2_unit/
-qed-.
-
-theorem liftsb_conf: ∀f1,I,I1. ⬆*[f1] I ≘ I1 → ∀f,I2. ⬆*[f] I ≘ I2 →
- ∀f2. f2 ⊚ f1 ≘ f → ⬆*[f2] I1 ≘ I2.
-#f1 #I #I1 * -I -I1 #I [2: #V #V1 #HV1 ] #f2 #I2 #H
-[ elim (liftsb_inv_pair_sn … H) | lapply (liftsb_inv_unit_sn … H) ] -H
-/3 width=6 by lifts_conf, ext2_pair, ext2_unit/
-qed-.
-
-(* Advanced proprerties *****************************************************)
-
-lemma liftsb_inj: ∀f. is_inj2 … (liftsb f).
-#f #T1 #U #H1 #T2 #H2 lapply (after_isid_dx 𝐈𝐝 … f)
-/3 width=6 by liftsb_div3, liftsb_fwd_isid/
-qed-.
-
-lemma liftsb_mono: ∀f,T. is_mono … (liftsb f T).
-#f #T #U1 #H1 #U2 #H2 lapply (after_isid_sn 𝐈𝐝 … f)
-/3 width=6 by liftsb_conf, liftsb_fwd_isid/
-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.ma".
-include "basic_2/relocation/lifts_vector.ma".
-
-(* GENERIC RELOCATION FOR TERM VECTORS *************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: includes: lifts_inj *)
-theorem liftsv_inj: ∀f,T1s,Us. ⬆*[f] T1s ≘ Us →
- ∀T2s. ⬆*[f] T2s ≘ Us → T1s = T2s.
-#f #T1s #Us #H elim H -T1s -Us
-[ #T2s #H >(liftsv_inv_nil2 … H) -H //
-| #T1s #Us #T1 #U #HT1U #_ #IHT1Us #X #H destruct
- elim (liftsv_inv_cons2 … H) -H #T2 #T2s #HT2U #HT2Us #H destruct
- >(lifts_inj … HT1U … HT2U) -U /3 width=1 by eq_f/
-]
-qed-.
-
-(* Basic_2A1: includes: liftv_mono *)
-theorem liftsv_mono: ∀f,Ts,U1s. ⬆*[f] Ts ≘ U1s →
- ∀U2s. ⬆*[f] Ts ≘ U2s → U1s = U2s.
-#f #Ts #U1s #H elim H -Ts -U1s
-[ #U2s #H >(liftsv_inv_nil1 … H) -H //
-| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
- elim (liftsv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
- >(lifts_mono … HTU1 … HTU2) -T /3 width=1 by eq_f/
-]
-qed-.
-
-(* Basic_1: includes: lifts1_xhg (right to left) *)
-(* Basic_2A1: includes: liftsv_liftv_trans_le *)
-theorem liftsv_trans: ∀f1,T1s,Ts. ⬆*[f1] T1s ≘ Ts → ∀T2s,f2. ⬆*[f2] Ts ≘ T2s →
- ∀f. f2 ⊚ f1 ≘ f → ⬆*[f] T1s ≘ T2s.
-#f1 #T1s #Ts #H elim H -T1s -Ts
-[ #T2s #f2 #H >(liftsv_inv_nil1 … H) -T2s /2 width=3 by liftsv_nil/
-| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #f2 #H elim (liftsv_inv_cons1 … H) -H
- #T2 #T2s #HT2 #HT2s #H destruct /3 width=6 by lifts_trans, liftsv_cons/
-]
-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/term_simple.ma".
-include "basic_2/relocation/lifts.ma".
-
-(* GENERIC RELOCATION FOR TERMS *********************************************)
-
-(* Forward lemmas with simple terms *****************************************)
-
-(* Basic_2A1: includes: lift_simple_dx *)
-lemma lifts_simple_dx: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#f #T1 #T2 #H elim H -f -T1 -T2 //
-#f #p #I #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H elim (simple_inv_bind … H)
-qed-.
-
-(* Basic_2A1: includes: lift_simple_sn *)
-lemma lifts_simple_sn: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-#f #T1 #T2 #H elim H -f -T1 -T2 //
-#f #p #I #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H elim (simple_inv_bind … 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/syntax/tdeq.ma".
-include "basic_2/relocation/lifts_lifts.ma".
-
-(* GENERIC RELOCATION FOR TERMS *********************************************)
-
-(* Properties with degree-based equivalence for terms ***********************)
-
-lemma tdeq_lifts_sn: ∀h,o. liftable2_sn (tdeq h o).
-#h #o #T1 #T2 #H elim H -T1 -T2 [||| * ]
-[ #s1 #s2 #d #Hs1 #Hs2 #f #X #H >(lifts_inv_sort1 … H) -H
- /3 width=5 by lifts_sort, tdeq_sort, ex2_intro/
-| #i #f #X #H elim (lifts_inv_lref1 … H) -H
- /3 width=3 by lifts_lref, tdeq_lref, ex2_intro/
-| #l #f #X #H >(lifts_inv_gref1 … H) -H
- /2 width=3 by lifts_gref, tdeq_gref, ex2_intro/
-| #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f #X #H elim (lifts_inv_bind1 … H) -H
- #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (IHV … HVW1) -V1 elim (IHT … HTU1) -T1
- /3 width=5 by lifts_bind, tdeq_pair, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f #X #H elim (lifts_inv_flat1 … H) -H
- #W1 #U1 #HVW1 #HTU1 #H destruct
- elim (IHV … HVW1) -V1 elim (IHT … HTU1) -T1
- /3 width=5 by lifts_flat, tdeq_pair, ex2_intro/
-]
-qed-.
-
-lemma tdeq_lifts_bi: ∀h,o. liftable2_bi (tdeq h o).
-/3 width=6 by tdeq_lifts_sn, liftable2_sn_bi/ qed-.
-
-(* Inversion lemmas with degree-based equivalence for terms *****************)
-
-lemma tdeq_inv_lifts_sn: ∀h,o. deliftable2_sn (tdeq h o).
-#h #o #U1 #U2 #H elim H -U1 -U2 [||| * ]
-[ #s1 #s2 #d #Hs1 #Hs2 #f #X #H >(lifts_inv_sort2 … H) -H
- /3 width=5 by lifts_sort, tdeq_sort, ex2_intro/
-| #i #f #X #H elim (lifts_inv_lref2 … H) -H
- /3 width=3 by lifts_lref, tdeq_lref, ex2_intro/
-| #l #f #X #H >(lifts_inv_gref2 … H) -H
- /2 width=3 by lifts_gref, tdeq_gref, ex2_intro/
-| #p #I #W1 #W2 #U1 #U2 #_ #_ #IHW #IHU #f #X #H elim (lifts_inv_bind2 … H) -H
- #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW … HVW1) -W1 elim (IHU … HTU1) -U1
- /3 width=5 by lifts_bind, tdeq_pair, ex2_intro/
-| #I #W1 #W2 #U1 #U2 #_ #_ #IHW #IHU #f #X #H elim (lifts_inv_flat2 … H) -H
- #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW … HVW1) -W1 elim (IHU … HTU1) -U1
- /3 width=5 by lifts_flat, tdeq_pair, ex2_intro/
-]
-qed-.
-
-lemma tdeq_inv_lifts_bi: ∀h,o. deliftable2_bi (tdeq h o).
-/3 width=6 by tdeq_inv_lifts_sn, deliftable2_sn_bi/ 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/term_vector.ma".
-include "basic_2/relocation/lifts.ma".
-
-(* GENERIC RELOCATION FOR TERM VECTORS *************************************)
-
-(* Basic_2A1: includes: liftv_nil liftv_cons *)
-inductive liftsv (f:rtmap): relation (list term) ≝
-| liftsv_nil : liftsv f (Ⓔ) (Ⓔ)
-| liftsv_cons: ∀T1s,T2s,T1,T2.
- ⬆*[f] T1 ≘ T2 → liftsv f T1s T2s →
- liftsv f (T1 ⨮ T1s) (T2 ⨮ T2s)
-.
-
-interpretation "uniform relocation (term vector)"
- 'RLiftStar i T1s T2s = (liftsv (uni i) T1s T2s).
-
-interpretation "generic relocation (term vector)"
- 'RLiftStar f T1s T2s = (liftsv f T1s T2s).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact liftsv_inv_nil1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → X = Ⓔ → Y = Ⓔ.
-#f #X #Y * -X -Y //
-#T1s #T2s #T1 #T2 #_ #_ #H destruct
-qed-.
-
-(* Basic_2A1: includes: liftv_inv_nil1 *)
-lemma liftsv_inv_nil1: ∀f,Y. ⬆*[f] Ⓔ ≘ Y → Y = Ⓔ.
-/2 width=5 by liftsv_inv_nil1_aux/ qed-.
-
-fact liftsv_inv_cons1_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
- ∀T1,T1s. X = T1 ⨮ T1s →
- ∃∃T2,T2s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ T2s &
- Y = T2 ⨮ T2s.
-#f #X #Y * -X -Y
-[ #U1 #U1s #H destruct
-| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-(* Basic_2A1: includes: liftv_inv_cons1 *)
-lemma liftsv_inv_cons1: ∀f:rtmap. ∀T1,T1s,Y. ⬆*[f] T1 ⨮ T1s ≘ Y →
- ∃∃T2,T2s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ T2s &
- Y = T2 ⨮ T2s.
-/2 width=3 by liftsv_inv_cons1_aux/ qed-.
-
-fact liftsv_inv_nil2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → Y = Ⓔ → X = Ⓔ.
-#f #X #Y * -X -Y //
-#T1s #T2s #T1 #T2 #_ #_ #H destruct
-qed-.
-
-lemma liftsv_inv_nil2: ∀f,X. ⬆*[f] X ≘ Ⓔ → X = Ⓔ.
-/2 width=5 by liftsv_inv_nil2_aux/ qed-.
-
-fact liftsv_inv_cons2_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
- ∀T2,T2s. Y = T2 ⨮ T2s →
- ∃∃T1,T1s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ T2s &
- X = T1 ⨮ T1s.
-#f #X #Y * -X -Y
-[ #U2 #U2s #H destruct
-| #T1s #T2s #T1 #T2 #HT12 #HT12s #U2 #U2s #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma liftsv_inv_cons2: ∀f:rtmap. ∀X,T2,T2s. ⬆*[f] X ≘ T2 ⨮ T2s →
- ∃∃T1,T1s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ T2s &
- X = T1 ⨮ T1s.
-/2 width=3 by liftsv_inv_cons2_aux/ qed-.
-
-(* Basic_1: was: lifts1_flat (left to right) *)
-lemma lifts_inv_applv1: ∀f:rtmap. ∀V1s,U1,T2. ⬆*[f] Ⓐ V1s.U1 ≘ T2 →
- ∃∃V2s,U2. ⬆*[f] V1s ≘ V2s & ⬆*[f] U1 ≘ U2 &
- T2 = Ⓐ V2s.U2.
-#f #V1s elim V1s -V1s
-[ /3 width=5 by ex3_2_intro, liftsv_nil/
-| #V1 #V1s #IHV1s #T1 #X #H elim (lifts_inv_flat1 … H) -H
- #V2 #Y #HV12 #HY #H destruct elim (IHV1s … HY) -IHV1s -HY
- #V2s #T2 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
-]
-qed-.
-
-lemma lifts_inv_applv2: ∀f:rtmap. ∀V2s,U2,T1. ⬆*[f] T1 ≘ Ⓐ V2s.U2 →
- ∃∃V1s,U1. ⬆*[f] V1s ≘ V2s & ⬆*[f] U1 ≘ U2 &
- T1 = Ⓐ V1s.U1.
-#f #V2s elim V2s -V2s
-[ /3 width=5 by ex3_2_intro, liftsv_nil/
-| #V2 #V2s #IHV2s #T2 #X #H elim (lifts_inv_flat2 … H) -H
- #V1 #Y #HV12 #HY #H destruct elim (IHV2s … HY) -IHV2s -HY
- #V1s #T1 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_2A1: includes: liftv_total *)
-lemma liftsv_total: ∀f. ∀T1s:list term. ∃T2s. ⬆*[f] T1s ≘ T2s.
-#f #T1s elim T1s -T1s
-[ /2 width=2 by liftsv_nil, ex_intro/
-| #T1 #T1s * #T2s #HT12s
- elim (lifts_total T1 f) /3 width=2 by liftsv_cons, ex_intro/
-]
-qed-.
-
-(* Basic_1: was: lifts1_flat (right to left) *)
-lemma lifts_applv: ∀f:rtmap. ∀V1s,V2s. ⬆*[f] V1s ≘ V2s →
- ∀T1,T2. ⬆*[f] T1 ≘ T2 →
- ⬆*[f] Ⓐ V1s.T1 ≘ Ⓐ V2s.T2.
-#f #V1s #V2s #H elim H -V1s -V2s /3 width=1 by lifts_flat/
-qed.
-
-lemma liftsv_split_trans: ∀f,T1s,T2s. ⬆*[f] T1s ≘ T2s →
- ∀f1,f2. f2 ⊚ f1 ≘ f →
- ∃∃Ts. ⬆*[f1] T1s ≘ Ts & ⬆*[f2] Ts ≘ T2s.
-#f #T1s #T2s #H elim H -T1s -T2s
-[ /2 width=3 by liftsv_nil, ex2_intro/
-| #T1s #T2s #T1 #T2 #HT12 #_ #IH #f1 #f2 #Hf
- elim (IH … Hf) -IH
- elim (lifts_split_trans … HT12 … Hf) -HT12 -Hf
- /3 width=5 by liftsv_cons, ex2_intro/
-]
-qed-.
-
-(* Basic_1: removed theorems 2: lifts1_nil lifts1_cons *)
+++ /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/term_weight.ma".
-include "basic_2/relocation/lifts.ma".
-
-(* GENERIC RELOCATION FOR TERMS *********************************************)
-
-(* Forward lemmas with weight for terms *************************************)
-
-(* Basic_2A1: includes: lift_fwd_tw *)
-lemma lifts_fwd_tw: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → ♯{T1} = ♯{T2}.
-#f #T1 #T2 #H elim H -f -T1 -T2 normalize //
-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/bind_weight.ma".
-include "basic_2/relocation/lifts_weight.ma".
-include "basic_2/relocation/lifts_bind.ma".
-
-(* GENERIC RELOCATION FOR BINDERS *******************************************)
-
-(* Forward lemmas with weight for binders ***********************************)
-
-lemma liftsb_fwd_bw: ∀f,I1,I2. ⬆*[f] I1 ≘ I2 → ♯{I1} = ♯{I2}.
-#f #I1 #I2 * -I1 -I2 /2 width=2 by lifts_fwd_tw/
-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-.
(* *)
(**************************************************************************)
-include "basic_2/relocation/drops_ltc.ma".
+include "static_2/relocation/drops_ltc.ma".
include "basic_2/rt_transition/cpm_drops.ma".
include "basic_2/rt_computation/cpms.ma".
(* *)
(**************************************************************************)
-include "basic_2/syntax/cext2.ma".
+include "static_2/syntax/cext2.ma".
include "basic_2/rt_computation/cpms.ma".
(* CONTEXT-SENSITIVE PARALLEL R-COMPUTATION FOR BINDERS *********************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/drops_ctc.ma".
+include "static_2/relocation/drops_ctc.ma".
include "basic_2/rt_transition/cpx_drops.ma".
include "basic_2/rt_computation/cpxs.ma".
(* *)
(**************************************************************************)
-include "basic_2/syntax/cext2.ma".
+include "static_2/syntax/cext2.ma".
include "basic_2/rt_computation/cpxs.ma".
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR BINDERS ************)
(* *)
(**************************************************************************)
-include "basic_2/static/fdeq.ma".
+include "static_2/static/fdeq.ma".
include "basic_2/rt_computation/cpxs_rdeq.ma".
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
(* *)
(**************************************************************************)
-include "basic_2/syntax/theq_tdeq.ma".
+include "static_2/syntax/theq_tdeq.ma".
include "basic_2/rt_computation/cpxs_lsubr.ma".
include "basic_2/rt_computation/cpxs_cnx.ma".
include "basic_2/rt_computation/lpxs_cpxs.ma".
(* *)
(**************************************************************************)
-include "basic_2/syntax/theq_simple_vector.ma".
-include "basic_2/relocation/lifts_vector.ma".
+include "static_2/syntax/theq_simple_vector.ma".
+include "static_2/relocation/lifts_vector.ma".
include "basic_2/rt_computation/cpxs_theq.ma".
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
(**************************************************************************)
include "basic_2/notation/relations/predtystrong_5.ma".
-include "basic_2/syntax/tdeq.ma".
+include "static_2/syntax/tdeq.ma".
include "basic_2/rt_transition/cpx.ma".
(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
(* *)
(**************************************************************************)
-include "basic_2/static/gcp_aaa.ma".
+include "static_2/static/gcp_aaa.ma".
include "basic_2/rt_computation/cpxs_aaa.ma".
include "basic_2/rt_computation/csx_gcp.ma".
include "basic_2/rt_computation/csx_gcr.ma".
(* *)
(**************************************************************************)
-include "basic_2/relocation/lifts_tdeq.ma".
+include "static_2/relocation/lifts_tdeq.ma".
include "basic_2/rt_transition/cpx_drops.ma".
include "basic_2/rt_computation/csx.ma".
(* *)
(**************************************************************************)
-include "basic_2/static/fdeq.ma".
+include "static_2/static/fdeq.ma".
include "basic_2/rt_computation/csx_rdeq.ma".
(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
(* *)
(**************************************************************************)
-include "basic_2/s_computation/fqus.ma".
+include "static_2/s_computation/fqus.ma".
include "basic_2/rt_computation/csx_lsubr.ma".
(* STRONGLY NORMALIZING TERMS FOR UNBOUND PARALLEL RT-TRANSITION ************)
(* *)
(**************************************************************************)
-include "basic_2/static/gcp.ma".
+include "static_2/static/gcp.ma".
include "basic_2/rt_transition/cnx_drops.ma".
include "basic_2/rt_computation/csx_drops.ma".
(* *)
(**************************************************************************)
-include "basic_2/static/gcp_cr.ma".
+include "static_2/static/gcp_cr.ma".
include "basic_2/rt_computation/csx_cnx_vector.ma".
include "basic_2/rt_computation/csx_csx_vector.ma".
(* *)
(**************************************************************************)
-include "basic_2/syntax/theq_simple.ma".
-include "basic_2/syntax/theq_theq.ma".
+include "static_2/syntax/theq_simple.ma".
+include "static_2/syntax/theq_theq.ma".
include "basic_2/rt_transition/cpx_simple.ma".
include "basic_2/rt_computation/cpxs.ma".
include "basic_2/rt_computation/csx_csx.ma".
(* *)
(**************************************************************************)
-include "basic_2/syntax/term_vector.ma".
+include "static_2/syntax/term_vector.ma".
include "basic_2/rt_computation/csx.ma".
(* STRONGLY NORMALIZING TERMS VECTORS FOR UNBOUND PARALLEL RT-TRANSITION ****)
(* *)
(**************************************************************************)
-include "basic_2/static/fdeq_fqup.ma".
-include "basic_2/static/fdeq_fdeq.ma".
+include "static_2/static/fdeq_fqup.ma".
+include "static_2/static/fdeq_fdeq.ma".
include "basic_2/rt_transition/fpbq_fpb.ma".
include "basic_2/rt_computation/fpbg.ma".
(* *)
(**************************************************************************)
-include "basic_2/s_computation/fqus_fqup.ma".
-include "basic_2/static/fdeq_fqup.ma".
+include "static_2/s_computation/fqus_fqup.ma".
+include "static_2/static/fdeq_fqup.ma".
include "basic_2/rt_computation/fpbs_fqus.ma".
(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
(* *)
(**************************************************************************)
-include "basic_2/s_computation/fqus.ma".
+include "static_2/s_computation/fqus.ma".
include "basic_2/rt_computation/fpbs.ma".
(* PARALLEL RST-COMPUTATION FOR CLOSURES ************************************)
(* *)
(**************************************************************************)
-include "basic_2/static/fdeq_fqus.ma".
-include "basic_2/static/fdeq_fdeq.ma".
+include "static_2/static/fdeq_fqus.ma".
+include "static_2/static/fdeq_fdeq.ma".
include "basic_2/rt_computation/cpxs_fqus.ma".
include "basic_2/rt_computation/cpxs_fdeq.ma".
include "basic_2/rt_computation/lpxs_fdeq.ma".
(**************************************************************************)
include "basic_2/notation/relations/predsnstar_4.ma".
-include "basic_2/relocation/lex.ma".
+include "static_2/relocation/lex.ma".
include "basic_2/rt_computation/cprs_ext.ma".
(* PARALLEL R-COMPUTATION FOR FULL LOCAL ENVIRONMENTS ***********************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/drops_lex.ma".
+include "static_2/relocation/drops_lex.ma".
include "basic_2/rt_computation/cpms_drops.ma".
(* PARALLEL R-COMPUTATION FOR FULL LOCAL ENVIRONMENTS ***********************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/lex_length.ma".
+include "static_2/relocation/lex_length.ma".
include "basic_2/rt_computation/lprs.ma".
(* PARALLEL R-COMPUTATION FOR FULL LOCAL ENVIRONMENTS ***********************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/lex_tc.ma".
+include "static_2/relocation/lex_tc.ma".
include "basic_2/rt_computation/lprs_ctc.ma".
include "basic_2/rt_computation/cprs_lpr.ma".
(**************************************************************************)
include "basic_2/notation/relations/predtysnstar_4.ma".
-include "basic_2/relocation/lex.ma".
+include "static_2/relocation/lex.ma".
include "basic_2/rt_computation/cpxs_ext.ma".
(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/drops_lex.ma".
+include "static_2/relocation/drops_lex.ma".
include "basic_2/rt_computation/cpxs_drops.ma".
(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
(* *)
(**************************************************************************)
-include "basic_2/static/fdeq.ma".
+include "static_2/static/fdeq.ma".
include "basic_2/rt_computation/lpxs_rdeq.ma".
(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/lex_length.ma".
+include "static_2/relocation/lex_length.ma".
include "basic_2/rt_computation/lpxs.ma".
(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/lex_tc.ma".
+include "static_2/relocation/lex_tc.ma".
include "basic_2/rt_computation/cpxs_lpx.ma".
include "basic_2/rt_computation/lpxs.ma".
(* *)
(**************************************************************************)
-include "basic_2/relocation/lex_lex.ma".
+include "static_2/relocation/lex_lex.ma".
include "basic_2/rt_computation/lpxs_lpx.ma".
(* UNBOUND PARALLEL RT-COMPUTATION FOR FULL LOCAL ENVIRONMENTS **************)
(**************************************************************************)
include "basic_2/notation/relations/predtysnstrong_5.ma".
-include "basic_2/static/rdeq.ma".
+include "static_2/static/rdeq.ma".
include "basic_2/rt_transition/lpx.ma".
(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
(* *)
(**************************************************************************)
-include "basic_2/static/rdeq_drops.ma".
+include "static_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".
(* *)
(**************************************************************************)
-include "basic_2/static/rdeq_fqup.ma".
+include "static_2/static/rdeq_fqup.ma".
include "basic_2/rt_computation/rdsx.ma".
(* STRONGLY NORMALIZING REFERRED LOCAL ENV.S FOR UNBOUND RT-TRANSITION ******)
(* *)
(**************************************************************************)
-include "basic_2/static/rdeq_length.ma".
+include "static_2/static/rdeq_length.ma".
include "basic_2/rt_transition/lpx_length.ma".
include "basic_2/rt_computation/rdsx.ma".
(**************************************************************************)
include "basic_2/notation/relations/predtynormal_5.ma".
-include "basic_2/syntax/tdeq.ma".
+include "static_2/syntax/tdeq.ma".
include "basic_2/rt_transition/cpx.ma".
(* NORMAL TERMS FOR UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION ********)
(* *)
(**************************************************************************)
-include "basic_2/relocation/lifts_tdeq.ma".
+include "static_2/relocation/lifts_tdeq.ma".
include "basic_2/rt_transition/cpx_drops.ma".
include "basic_2/rt_transition/cnx.ma".
include "ground_2/steps/rtc_max.ma".
include "ground_2/steps/rtc_plus.ma".
include "basic_2/notation/relations/predty_7.ma".
-include "basic_2/syntax/item_sh.ma".
-include "basic_2/syntax/lenv.ma".
-include "basic_2/syntax/genv.ma".
-include "basic_2/relocation/lifts.ma".
+include "static_2/syntax/item_sh.ma".
+include "static_2/syntax/lenv.ma".
+include "static_2/syntax/genv.ma".
+include "static_2/relocation/lifts.ma".
(* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/drops_drops.ma".
-include "basic_2/s_computation/fqup_weight.ma".
-include "basic_2/s_computation/fqup_drops.ma".
+include "static_2/relocation/drops_drops.ma".
+include "static_2/s_computation/fqup_weight.ma".
+include "static_2/s_computation/fqup_drops.ma".
include "basic_2/rt_transition/cpg.ma".
(* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************)
(* *)
(**************************************************************************)
-include "basic_2/static/lsubr.ma".
+include "static_2/static/lsubr.ma".
include "basic_2/rt_transition/cpg.ma".
(* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************)
(* *)
(**************************************************************************)
-include "basic_2/syntax/term_simple.ma".
+include "static_2/syntax/term_simple.ma".
include "basic_2/rt_transition/cpg.ma".
(* BOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS *****************)
(* *)
(**************************************************************************)
-include "basic_2/syntax/cext2.ma".
+include "static_2/syntax/cext2.ma".
include "basic_2/rt_transition/cpm.ma".
(* CONTEXT-SENSITIVE PARALLEL R-TRANSITION FOR BINDERS **********************)
(* *)
(**************************************************************************)
-include "basic_2/syntax/cext2.ma".
+include "static_2/syntax/cext2.ma".
include "basic_2/rt_transition/cpx.ma".
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR BINDERS *************)
(* *)
(**************************************************************************)
-include "basic_2/static/fdeq.ma".
+include "static_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 ***************)
-include "basic_2/relocation/lifts_tdeq.ma".
-include "basic_2/s_computation/fqus_fqup.ma".
+include "static_2/relocation/lifts_tdeq.ma".
+include "static_2/s_computation/fqus_fqup.ma".
include "basic_2/rt_transition/cpx_drops.ma".
include "basic_2/rt_transition/cpx_lsubr.ma".
(* *)
(**************************************************************************)
-include "basic_2/static/rdeq_rdeq.ma".
+include "static_2/static/rdeq_rdeq.ma".
include "basic_2/rt_transition/rpx_fsle.ma".
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
(* *)
(**************************************************************************)
-include "basic_2/static/req.ma".
+include "static_2/static/req.ma".
include "basic_2/rt_transition/rpx_fsle.ma".
(* UNBOUND CONTEXT-SENSITIVE PARALLEL RT-TRANSITION FOR TERMS ***************)
(**************************************************************************)
include "basic_2/notation/relations/predsubtyproper_8.ma".
-include "basic_2/s_transition/fqu.ma".
-include "basic_2/static/rdeq.ma".
+include "static_2/s_transition/fqu.ma".
+include "static_2/static/rdeq.ma".
include "basic_2/rt_transition/lpr_lpx.ma".
(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
(* *)
(**************************************************************************)
-include "basic_2/s_transition/fqu_tdeq.ma".
-include "basic_2/static/fdeq.ma".
+include "static_2/s_transition/fqu_tdeq.ma".
+include "static_2/static/fdeq.ma".
include "basic_2/rt_transition/fpb_rdeq.ma".
(* PROPER PARALLEL RST-TRANSITION FOR CLOSURES ******************************)
(* *)
(**************************************************************************)
-include "basic_2/static/rdeq_fqus.ma".
+include "static_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".
(**************************************************************************)
include "basic_2/notation/relations/predsubty_8.ma".
-include "basic_2/static/fdeq.ma".
-include "basic_2/s_transition/fquq.ma".
+include "static_2/static/fdeq.ma".
+include "static_2/s_transition/fquq.ma".
include "basic_2/rt_transition/lpr_lpx.ma".
(* PARALLEL RST-TRANSITION FOR CLOSURES *************************************)
(* *)
(**************************************************************************)
-include "basic_2/static/aaa_fqus.ma".
-include "basic_2/static/aaa_fdeq.ma".
+include "static_2/static/aaa_fqus.ma".
+include "static_2/static/aaa_fdeq.ma".
include "basic_2/rt_transition/lpx_aaa.ma".
include "basic_2/rt_transition/fpbq.ma".
(**************************************************************************)
include "basic_2/notation/relations/predsn_4.ma".
-include "basic_2/relocation/lex.ma".
+include "static_2/relocation/lex.ma".
include "basic_2/rt_transition/cpr_ext.ma".
(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/drops_lex.ma".
+include "static_2/relocation/drops_lex.ma".
include "basic_2/rt_transition/cpm_drops.ma".
include "basic_2/rt_transition/lpr.ma".
(* *)
(**************************************************************************)
-include "basic_2/s_transition/fquq.ma".
+include "static_2/s_transition/fquq.ma".
include "basic_2/rt_transition/cpm_drops.ma".
include "basic_2/rt_transition/cpm_lsubr.ma".
include "basic_2/rt_transition/cpr.ma".
(* *)
(**************************************************************************)
-include "basic_2/relocation/lex_length.ma".
+include "static_2/relocation/lex_length.ma".
include "basic_2/rt_transition/lpr.ma".
(* PARALLEL R-TRANSITION FOR FULL LOCAL ENVIRONMENTS ************************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/lex_lex.ma".
+include "static_2/relocation/lex_lex.ma".
include "basic_2/rt_transition/cpm_lsubr.ma".
include "basic_2/rt_transition/cpr.ma".
include "basic_2/rt_transition/cpr_drops.ma".
(**************************************************************************)
include "basic_2/notation/relations/predtysn_4.ma".
-include "basic_2/relocation/lex.ma".
+include "static_2/relocation/lex.ma".
include "basic_2/rt_transition/cpx_ext.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
(* *)
(**************************************************************************)
-include "basic_2/static/aaa_drops.ma".
-include "basic_2/static/lsuba_aaa.ma".
+include "static_2/static/aaa_drops.ma".
+include "static_2/static/lsuba_aaa.ma".
include "basic_2/rt_transition/lpx_drops.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/drops_lex.ma".
+include "static_2/relocation/drops_lex.ma".
include "basic_2/rt_transition/cpx_drops.ma".
include "basic_2/rt_transition/lpx.ma".
(* *)
(**************************************************************************)
-include "basic_2/s_transition/fquq.ma".
+include "static_2/s_transition/fquq.ma".
include "basic_2/rt_transition/lpx.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
(* *)
(**************************************************************************)
-include "basic_2/relocation/lex_length.ma".
+include "static_2/relocation/lex_length.ma".
include "basic_2/rt_transition/lpx.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR FULL LOCAL ENVIRONMENTS ***************)
(* *)
(**************************************************************************)
-include "basic_2/static/rdeq_req.ma".
+include "static_2/static/rdeq_req.ma".
include "basic_2/rt_transition/rpx_rdeq.ma".
include "basic_2/rt_transition/rpx_lpx.ma".
(**************************************************************************)
include "basic_2/notation/relations/predtysn_5.ma".
-include "basic_2/static/rex.ma".
+include "static_2/static/rex.ma".
include "basic_2/rt_transition/cpx_ext.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
(* *)
(**************************************************************************)
-include "basic_2/static/rex_fqup.ma".
+include "static_2/static/rex_fqup.ma".
include "basic_2/rt_transition/rpx.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
(* *)
(**************************************************************************)
-include "basic_2/static/fsle_drops.ma".
-include "basic_2/static/rex_fsle.ma".
+include "static_2/static/fsle_drops.ma".
+include "static_2/static/rex_fsle.ma".
include "basic_2/rt_transition/rpx_length.ma".
include "basic_2/rt_transition/rpx_fqup.ma".
(* *)
(**************************************************************************)
-include "basic_2/static/rex_length.ma".
+include "static_2/static/rex_length.ma".
include "basic_2/rt_transition/rpx.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
(* *)
(**************************************************************************)
-include "basic_2/static/rex_lex.ma".
+include "static_2/static/rex_lex.ma".
include "basic_2/rt_transition/rpx_fsle.ma".
include "basic_2/rt_transition/lpx.ma".
(* *)
(**************************************************************************)
-include "basic_2/static/rdeq_fqup.ma".
-include "basic_2/static/rdeq_rdeq.ma".
+include "static_2/static/rdeq_fqup.ma".
+include "static_2/static/rdeq_rdeq.ma".
include "basic_2/rt_transition/rpx_fsle.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
(* *)
(**************************************************************************)
-include "basic_2/static/rex_rex.ma".
+include "static_2/static/rex_rex.ma".
include "basic_2/rt_transition/rpx.ma".
(* UNBOUND PARALLEL RT-TRANSITION FOR REFERRED LOCAL ENVIRONMENTS ***********)
+++ /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/suptermplus_6.ma".
-include "basic_2/notation/relations/suptermplus_7.ma".
-include "basic_2/s_transition/fqu.ma".
-
-(* PLUS-ITERATED SUPCLOSURE *************************************************)
-
-definition fqup: bool → tri_relation genv lenv term ≝
- λb. tri_TC … (fqu b).
-
-interpretation "extended plus-iterated structural successor (closure)"
- 'SupTermPlus b G1 L1 T1 G2 L2 T2 = (fqup b G1 L1 T1 G2 L2 T2).
-
-interpretation "plus-iterated structural successor (closure)"
- 'SupTermPlus G1 L1 T1 G2 L2 T2 = (fqup true G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
-/2 width=1 by tri_inj/ qed.
-
-lemma fqup_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
-/2 width=5 by tri_step/ qed.
-
-lemma fqup_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
- ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
-/2 width=5 by tri_TC_strap/ qed.
-
-lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+[b] ⦃G, L, V⦄.
-/2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
-
-lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G, L.ⓑ{I}V, T⦄.
-/2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
-
-lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[Ⓕ] ⦃G, L.ⓧ, T⦄.
-/3 width=1 by fqu_clear, fqu_fqup/ qed.
-
-lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+[b] ⦃G, L, T⦄.
-/2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
-
-lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+[b] ⦃G, L, V2⦄.
-/2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
-
-lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I1}V1, T⦄.
-/2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
-
-lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I2}V2, T⦄.
-/2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
-
-(* Basic eliminators ********************************************************)
-
-lemma fqup_ind: ∀b,G1,L1,T1. ∀Q:relation3 ….
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
-#b #G1 #L1 #T1 #Q #IH1 #IH2 #G2 #L2 #T2 #H
-@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
-qed-.
-
-lemma fqup_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 ….
- (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1) →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
-#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H
-@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
-qed-.
-
-(* Basic_2A1: removed theorems 1: fqup_drop *)
+++ /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.ma".
-include "basic_2/s_computation/fqup.ma".
-
-(* PLUS-ITERATED SUPCLOSURE *************************************************)
-
-(* Properties with generic slicing for local environments *******************)
-
-lemma fqup_drops_succ: ∀b,G,K,T,i,L,U. ⬇*[↑i] L ≘ K → ⬆*[↑i] T ≘ U →
- ⦃G, L, U⦄ ⊐+[b] ⦃G, K, T⦄.
-#b #G #K #T #i elim i -i
-[ #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK
- #I #Y #HY #H destruct <(drops_fwd_isid … HY) -K //
- /3 width=2 by fqu_fqup, fqu_drop/
-| #l #IH #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK
- #I #Y #HY #H destruct
- elim (lifts_split_trans … HTU … (𝐔❴↑l❵) (𝐔❴1❵)) -HTU
- /4 width=5 by fqup_strap2, fqu_drop/
-]
-qed.
-
-lemma fqup_drops_strap1: ∀b,G1,G2,L1,K1,K2,T1,T2,U1,i. ⬇*[i] L1 ≘ K1 → ⬆*[i] T1 ≘ U1 →
- ⦃G1, K1, T1⦄ ⊐[b] ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, K2, T2⦄.
-#b #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 *
-[ #HLK1 #HTU1 #HT12
- >(drops_fwd_isid … HLK1) -L1 //
- <(lifts_fwd_isid … HTU1) -U1 /2 width=1 by fqu_fqup/
-| /3 width=5 by fqup_strap1, fqup_drops_succ/
-]
-qed-.
-
-lemma fqup_lref: ∀b,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+[b] ⦃G, K, V⦄.
-/2 width=6 by fqup_drops_strap1/ 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/fqup.ma".
-
-(* PLUS-ITERATED SUPCLOSURE *************************************************)
-
-(* Main properties **********************************************************)
-
-theorem fqup_trans: ∀b. tri_transitive … (fqup b).
-/2 width=5 by tri_TC_transitive/ 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_weight.ma".
-include "basic_2/s_computation/fqup.ma".
-
-(* PLUS-ITERATED SUPCLOSURE *************************************************)
-
-(* Forward lemmas with weight for closures **********************************)
-
-lemma fqup_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} < ♯{G1, L1, T1}.
-#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/3 width=3 by fqu_fwd_fw, transitive_lt/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma fqup_wf_ind: ∀b. ∀Q:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
- Q G1 L1 T1
- ) → ∀G1,L1,T1. Q G1 L1 T1.
-#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct
-/4 width=2 by fqup_fwd_fw/
-qed-.
-
-lemma fqup_wf_ind_eq: ∀b. ∀Q:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
- ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → Q G2 L2 T2
- ) → ∀G1,L1,T1. Q G1 L1 T1.
-#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct
-/4 width=7 by fqup_fwd_fw/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/lib/star.ma".
-include "basic_2/notation/relations/suptermstar_6.ma".
-include "basic_2/notation/relations/suptermstar_7.ma".
-include "basic_2/s_transition/fquq.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-definition fqus: bool → tri_relation genv lenv term ≝
- λb. tri_TC … (fquq b).
-
-interpretation "extended star-iterated structural successor (closure)"
- 'SupTermStar b G1 L1 T1 G2 L2 T2 = (fqus b G1 L1 T1 G2 L2 T2).
-
-interpretation "star-iterated structural successor (closure)"
- 'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus true G1 L1 T1 G2 L2 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma fqus_ind: ∀b,G1,L1,T1. ∀Q:relation3 …. Q G1 L1 T1 →
- (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
-#b #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
-@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
-qed-.
-
-lemma fqus_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 …. Q G2 L2 T2 →
- (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
- ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
-#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H
-@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma fqus_refl: ∀b. tri_reflexive … (fqus b).
-/2 width=1 by tri_inj/ qed.
-
-lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
-/2 width=1 by tri_inj/ qed.
-
-lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
-/2 width=5 by tri_step/ qed-.
-
-lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
-/2 width=5 by tri_TC_strap/ qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1 = L2 & T1 = T2) ∨
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄.
-#b #G1 #G2 #L1 #L2 #T1 #T2 #H12 @(fqus_ind_dx … H12) -G1 -L1 -T1 /3 width=1 by and3_intro, or_introl/
-#G1 #G #L1 #L #T1 #T * /3 width=5 by ex2_3_intro, or_intror/
-* #HG #HL #HT #_ destruct //
-qed-.
-
-lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1 = L2 & ⋆s = T2) ∨
- ∃∃J,L. ⦃G1, L, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}.
-#b #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
-#G #L #T #H elim (fqu_inv_sort1 … H) -H /3 width=4 by ex2_2_intro, or_intror/
-qed-.
-
-lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∨∨ ∧∧ G1 = G2 & L1 = L2 & #i = T2
- | ∃∃J,L,V. ⦃G1, L, V⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = 0
- | ∃∃J,L,j. ⦃G1, L, #j⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J} & i = ↑j.
-#b #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/
-#G #L #T #H elim (fqu_inv_lref1 … H) -H * /3 width=7 by or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro/
-qed-.
-
-lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1 = L2 & §l = T2) ∨
- ∃∃J,L. ⦃G1, L, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}.
-#b #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
-#G #L #T #H elim (fqu_inv_gref1 … H) -H /3 width=4 by ex2_2_intro, or_intror/
-qed-.
-
-lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
- | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | ⦃G1, L1.ⓧ, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ ∧ b = Ⓕ
- | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
-#b #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or5_intro0/
-#G #L #T #H elim (fqu_inv_bind1 … H) -H *
-[4: #J ] #H1 #H2 #H3 [4: #Hb ] #H destruct
-/3 width=6 by or5_intro1, or5_intro2, or5_intro3, or5_intro4, ex3_3_intro, conj/
-qed-.
-
-
-lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
- | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
- | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄
- | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
-#p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_bind1 … H) -H [1,4: * ]
-/3 width=1 by and3_intro, or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex3_3_intro/
-#_ #H destruct
-qed-.
-
-lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2
- | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄
- | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}.
-#b #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/
-#G #L #T #H elim (fqu_inv_flat1 … H) -H *
-[3: #J ] #H1 #H2 #H3 #H destruct
-/3 width=6 by or4_intro1, or4_intro2, or4_intro3, ex3_3_intro/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = T2.
-#b #I #G1 #G2 #L2 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /2 width=1 by and3_intro/
-#G #L #T #H elim (fqu_inv_atom1 … H)
-qed-.
-
-lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1, L1.ⓘ{I}, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄.
-#b #I #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
-#G #L #T #H elim (fqu_inv_sort1_bind … H) -H
-#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
-qed-.
-
-lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
-#b #I #G1 #G2 #L1 #L2 #V1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
-#G #L #T #H elim (fqu_inv_zero1_pair … H) -H
-#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
-qed-.
-
-lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1, L1.ⓘ{I}, #↑i⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄.
-#b #I #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
-#G #L #T #H elim (fqu_inv_lref1_bind … H) -H
-#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
-qed-.
-
-lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1, L1.ⓘ{I}, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄.
-#b #I #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
-#G #L #T #H elim (fqu_inv_gref1_bind … H) -H
-#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
-qed-.
-
-(* Basic_2A1: removed theorems 1: fqus_drop *)
+++ /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/fqup_drops.ma".
-include "basic_2/s_computation/fqus_fqup.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-(* Properties with generic slicing for local environments *******************)
-
-lemma fqus_drops: ∀b,G,L,K,T,U,i. ⬇*[i] L ≘ K → ⬆*[i] T ≘ U →
- ⦃G, L, U⦄ ⊐*[b] ⦃G, K, T⦄.
-#b #G #L #K #T #U * /3 width=3 by fqup_drops_succ, fqup_fqus/
-#HLK #HTU <(lifts_fwd_isid … HTU) -U // <(drops_fwd_isid … HLK) -K //
-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/fqup.ma".
-include "basic_2/s_computation/fqus.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-(* Alternative definition with plus-iterated supclosure *********************)
-
-lemma fqup_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
-#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
-qed.
-
-(* Basic_2A1: was: fqus_inv_gen *)
-lemma fqus_inv_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
-#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 //
-#G #G2 #L #L2 #T #T2 #_ *
-[ #H2 * /3 width=5 by fqup_strap1, or_introl/
- * /3 width=1 by fqu_fqup, or_introl/
-| * #HG #HL #HT destruct //
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma fqus_strap1_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
-#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H1) -H1
-[ /2 width=5 by fqup_strap1/
-| * /2 width=1 by fqu_fqup/
-]
-qed-.
-
-lemma fqus_strap2_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
-#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H2) -H2
-[ /2 width=5 by fqup_strap2/
-| * /2 width=1 by fqu_fqup/
-]
-qed-.
-
-lemma fqus_fqup_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
-#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2
-/2 width=5 by fqus_strap1_fqu, fqup_strap1/
-qed-.
-
-lemma fqup_fqus_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ →
- ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
-#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1
-/3 width=5 by fqus_strap2_fqu, fqup_strap2/
-qed-.
-
-(* Advanced inversion lemmas for plus-iterated supclosure *******************)
-
-lemma fqup_inv_step_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄.
-#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
-#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, 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/s_computation/fqus.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-(* Main properties **********************************************************)
-
-theorem fqus_trans: ∀b. tri_transitive … (fqus b).
-/2 width=5 by tri_TC_transitive/ 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/fquq_weight.ma".
-include "basic_2/s_computation/fqus.ma".
-
-(* STAR-ITERATED SUPCLOSURE *************************************************)
-
-(* Forward lemmas with weight for closures **********************************)
-
-lemma fqus_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
-#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
-/3 width=3 by fquq_fwd_fw, transitive_le/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fqus_inv_refl_atom3: ∀b,I,G,L,X. ⦃G, L, ⓪{I}⦄ ⊐*[b] ⦃G, L, X⦄ → ⓪{I} = X.
-#b #I #G #L #X #H elim (fqus_inv_fqu_sn … H) -H * //
-#G0 #L0 #T0 #H1 #H2 lapply (fqu_fwd_fw … H1) lapply (fqus_fwd_fw … H2) -H2 -H1
-#H2 #H1 lapply (le_to_lt_to_lt … H2 H1) -G0 -L0 -T0
-#H elim (lt_le_false … H) -H /2 width=1 by monotonic_le_plus_r/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/supterm_6.ma".
-include "basic_2/notation/relations/supterm_7.ma".
-include "basic_2/syntax/lenv.ma".
-include "basic_2/syntax/genv.ma".
-include "basic_2/relocation/lifts.ma".
-
-(* SUPCLOSURE ***************************************************************)
-
-(* activate genv *)
-(* Note: frees_total requires fqu_drop for all atoms
- fqu_cpx_trans requires fqu_drop for all terms
- frees_fqus_drops requires fqu_drop restricted on atoms
-*)
-inductive fqu (b:bool): tri_relation genv lenv term ≝
-| fqu_lref_O : ∀I,G,L,V. fqu b G (L.ⓑ{I}V) (#0) G L V
-| fqu_pair_sn: ∀I,G,L,V,T. fqu b G L (②{I}V.T) G L V
-| fqu_bind_dx: ∀p,I,G,L,V,T. fqu b G L (ⓑ{p,I}V.T) G (L.ⓑ{I}V) T
-| fqu_clear : ∀p,I,G,L,V,T. b = Ⓕ → fqu b G L (ⓑ{p,I}V.T) G (L.ⓧ) T
-| fqu_flat_dx: ∀I,G,L,V,T. fqu b G L (ⓕ{I}V.T) G L T
-| fqu_drop : ∀I,G,L,T,U. ⬆*[1] T ≘ U → fqu b G (L.ⓘ{I}) U G L T
-.
-
-interpretation
- "extended structural successor (closure)"
- 'SupTerm b G1 L1 T1 G2 L2 T2 = (fqu b G1 L1 T1 G2 L2 T2).
-
-interpretation
- "structural successor (closure)"
- 'SupTerm G1 L1 T1 G2 L2 T2 = (fqu true G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma fqu_sort: ∀b,I,G,L,s. ⦃G, L.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G, L, ⋆s⦄.
-/2 width=1 by fqu_drop/ qed.
-
-lemma fqu_lref_S: ∀b,I,G,L,i. ⦃G, L.ⓘ{I}, #↑i⦄ ⊐[b] ⦃G, L, #i⦄.
-/2 width=1 by fqu_drop/ qed.
-
-lemma fqu_gref: ∀b,I,G,L,l. ⦃G, L.ⓘ{I}, §l⦄ ⊐[b] ⦃G, L, §l⦄.
-/2 width=1 by fqu_drop/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact fqu_inv_sort1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀s. T1 = ⋆s →
- ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s.
-#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #s #H destruct
-| #I #G #L #V #T #s #H destruct
-| #p #I #G #L #V #T #s #H destruct
-| #p #I #G #L #V #T #_ #s #H destruct
-| #I #G #L #V #T #s #H destruct
-| #I #G #L #T #U #HI12 #s #H destruct
- lapply (lifts_inv_sort2 … HI12) -HI12 /2 width=2 by ex3_intro/
-]
-qed-.
-
-lemma fqu_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s.
-/2 width=4 by fqu_inv_sort1_aux/ qed-.
-
-fact fqu_inv_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀i. T1 = #i →
- (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
- ∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j.
-#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #i #H destruct /3 width=4 by ex4_2_intro, or_introl/
-| #I #G #L #V #T #i #H destruct
-| #p #I #G #L #V #T #i #H destruct
-| #p #I #G #L #V #T #_ #i #H destruct
-| #I #G #L #V #T #i #H destruct
-| #I #G #L #T #U #HI12 #i #H destruct
- elim (lifts_inv_lref2_uni … HI12) -HI12 /3 width=3 by ex4_2_intro, or_intror/
-]
-qed-.
-
-lemma fqu_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
- ∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j.
-/2 width=4 by fqu_inv_lref1_aux/ qed-.
-
-fact fqu_inv_gref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀l. T1 = §l →
- ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l.
-#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #l #H destruct
-| #I #G #L #V #T #l #H destruct
-| #p #I #G #L #V #T #l #H destruct
-| #p #I #G #L #V #T #_ #l #H destruct
-| #I #G #L #V #T #s #H destruct
-| #I #G #L #T #U #HI12 #l #H destruct
- lapply (lifts_inv_gref2 … HI12) -HI12 /2 width=3 by ex3_intro/
-]
-qed-.
-
-lemma fqu_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l.
-/2 width=4 by fqu_inv_gref1_aux/ qed-.
-
-fact fqu_inv_bind1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀p,I,V1,U1. T1 = ⓑ{p,I}V1.U1 →
- ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
- | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
- | ∧∧ G1 = G2 & L1.ⓧ = L2 & U1 = T2 & b = Ⓕ
- | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
-#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #q #J #V0 #U0 #H destruct
-| #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or4_intro0/
-| #p #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or4_intro1/
-| #p #I #G #L #V #T #Hb #q #J #V0 #U0 #H destruct /3 width=1 by and4_intro, or4_intro2/
-| #I #G #L #V #T #q #J #V0 #U0 #H destruct
-| #I #G #L #T #U #HTU #q #J #V0 #U0 #H destruct /3 width=2 by or4_intro3, ex3_intro/
-]
-qed-.
-
-lemma fqu_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
- | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
- | ∧∧ G1 = G2 & L1.ⓧ = L2 & U1 = T2 & b = Ⓕ
- | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
-/2 width=4 by fqu_inv_bind1_aux/ qed-.
-
-lemma fqu_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
- | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
- | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
-#p #I #G1 #G2 #L1 #L2 #V1 #U1 #T2 #H elim (fqu_inv_bind1 … H) -H
-/3 width=1 by or3_intro0, or3_intro1, or3_intro2/
-* #_ #_ #_ #H destruct
-qed-.
-
-fact fqu_inv_flat1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀I,V1,U1. T1 = ⓕ{I}V1.U1 →
- ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
- | ∧∧ G1 = G2 & L1 = L2 & U1 = T2
- | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓕ{I}V1.U1.
-#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #J #V0 #U0 #H destruct
-| #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro0/
-| #p #I #G #L #V #T #J #V0 #U0 #H destruct
-| #p #I #G #L #V #T #_ #J #V0 #U0 #H destruct
-| #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro1/
-| #I #G #L #T #U #HTU #J #V0 #U0 #H destruct /3 width=2 by or3_intro2, ex3_intro/
-]
-qed-.
-
-lemma fqu_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓕ{I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
- | ∧∧ G1 = G2 & L1 = L2 & U1 = T2
- | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓕ{I}V1.U1.
-/2 width=4 by fqu_inv_flat1_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fqu_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⊥.
-#b * #x #G1 #G2 #L2 #T2 #H
-[ elim (fqu_inv_sort1 … H) | elim (fqu_inv_lref1 … H) * | elim (fqu_inv_gref1 … H) ] -H
-#I [2: #V |3: #i ] #_ #H destruct
-qed-.
-
-lemma fqu_inv_sort1_bind: ∀b,I,G1,G2,K,L2,T2,s. ⦃G1, K.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L2 = K & T2 = ⋆s.
-#b #I #G1 #G2 #K #L2 #T2 #s #H elim (fqu_inv_sort1 … H) -H
-#Z #X #H1 #H2 destruct /2 width=1 by and3_intro/
-qed-.
-
-lemma fqu_inv_zero1_pair: ∀b,I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L2 = K & T2 = V.
-#b #I #G1 #G2 #K #L2 #V #T2 #H elim (fqu_inv_lref1 … H) -H *
-#Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
-qed-.
-
-lemma fqu_inv_lref1_bind: ∀b,I,G1,G2,K,L2,T2,i. ⦃G1, K.ⓘ{I}, #(↑i)⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L2 = K & T2 = #i.
-#b #I #G1 #G2 #K #L2 #T2 #i #H elim (fqu_inv_lref1 … H) -H *
-#Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
-qed-.
-
-lemma fqu_inv_gref1_bind: ∀b,I,G1,G2,K,L2,T2,l. ⦃G1, K.ⓘ{I}, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∧∧ G1 = G2 & L2 = K & T2 = §l.
-#b #I #G1 #G2 #K #L2 #T2 #l #H elim (fqu_inv_gref1 … H) -H
-#Z #H1 #H2 #H3 destruct /2 width=1 by and3_intro/
-qed-.
-
-(* Basic_2A1: removed theorems 3:
- fqu_drop fqu_drop_lt fqu_lref_S_lt
-*)
+++ /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/s_transition/fqu.ma".
-
-(* SUPCLOSURE ***************************************************************)
-
-(* Forward lemmas with length for local environments ************************)
-
-fact fqu_fwd_length_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ∀i. T1 = #i → |L2| < |L1|.
-#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 // [2,3: #p]
-#I #G #L #V #T [2: #_ ] #j #H destruct
-qed-.
-
-lemma fqu_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- |L2| < |L1|.
-/2 width=8 by fqu_fwd_length_lref1_aux/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/syntax/tdeq.ma".
-include "basic_2/s_transition/fqu_length.ma".
-
-(* SUPCLOSURE ***************************************************************)
-
-(* Inversion lemmas with context-free degree-based equivalence for terms ****)
-
-fact fqu_inv_tdeq_aux: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- G1 = G2 → |L1| = |L2| → T1 ≛[h, o] T2 → ⊥.
-#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
-[1: #I #G #L #V #_ #H elim (succ_inv_refl_sn … H)
-|6: #I #G #L #T #U #_ #_ #H elim (succ_inv_refl_sn … H)
-]
-/2 width=6 by tdeq_inv_pair_xy_y, tdeq_inv_pair_xy_x/
-qed-.
-
-(* Basic_2A1: uses: fqu_inv_eq *)
-lemma fqu_inv_tdeq: ∀h,o,b,G,L1,L2,T1,T2. ⦃G, L1, T1⦄ ⊐[b] ⦃G, L2, T2⦄ →
- |L1| = |L2| → T1 ≛[h, o] T2 → ⊥.
-#h #o #b #G #L1 #L2 #T1 #T2 #H
-@(fqu_inv_tdeq_aux … H) // (**) (* full 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 "basic_2/syntax/cl_weight.ma".
-include "basic_2/relocation/lifts_weight.ma".
-include "basic_2/s_transition/fqu.ma".
-
-(* SUPCLOSURE ***************************************************************)
-
-(* Forward lemmas with weight for closures **********************************)
-
-lemma fqu_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} < ♯{G1, L1, T1}.
-#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
-#I #I1 #I2 #G #L #HI12 normalize in ⊢ (?%%); -I1
-<(lifts_fwd_tw … HI12) /3 width=1 by monotonic_lt_plus_r, monotonic_lt_plus_l/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma fqu_wf_ind: ∀b. ∀Q:relation3 …. (
- ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
- Q G1 L1 T1
- ) → ∀G1,L1,T1. Q G1 L1 T1.
-#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=2 by fqu_fwd_fw/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/relations/suptermopt_6.ma".
-include "basic_2/notation/relations/suptermopt_7.ma".
-include "basic_2/s_transition/fqu.ma".
-
-(* OPTIONAL SUPCLOSURE ******************************************************)
-
-(* Basic_2A1: was: fquqa *)
-(* Basic_2A1: includes: fquq_inv_gen *)
-definition fquq: bool → tri_relation genv lenv term ≝
- λb. tri_RC … (fqu b).
-
-interpretation
- "extended optional structural successor (closure)"
- 'SupTermOpt b G1 L1 T1 G2 L2 T2 = (fquq b G1 L1 T1 G2 L2 T2).
-
-interpretation
- "optional structural successor (closure)"
- 'SupTermOpt G1 L1 T1 G2 L2 T2 = (fquq true G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-(* Basic_2A1: includes: fquqa_refl *)
-lemma fquq_refl: ∀b. tri_reflexive … (fquq b).
-// qed.
-
-lemma fqu_fquq: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄.
-/2 width=1 by or_introl/ qed.
-
-(* Basic_2A1: removed theorems 8:
- fquq_lref_O fquq_pair_sn fquq_bind_dx fquq_flat_dx fquq_drop
- fquqa_drop fquq_fquqa fquqa_inv_fquq
-*)
+++ /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_length.ma".
-include "basic_2/s_transition/fquq.ma".
-
-(* OPTIONAL SUPCLOSURE ******************************************************)
-
-(* Forward lemmas with length for local environments ************************)
-
-lemma fquq_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- |L2| ≤ |L1|.
-#b #G1 #G2 #L1 #L2 #T2 #i #H elim H -H [2: * ]
-/3 width=6 by fqu_fwd_length_lref1, lt_to_le/
-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_weight.ma".
-include "basic_2/s_transition/fquq.ma".
-
-(* OPTIONAL SUPCLOSURE ******************************************************)
-
-(* Forward lemmas with weight for closures **********************************)
-
-lemma fquq_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
- ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
-#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [2: * ]
-/3 width=2 by fqu_fwd_fw, lt_to_le/
-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/atomicarity_4.ma".
-include "basic_2/syntax/aarity.ma".
-include "basic_2/syntax/lenv.ma".
-include "basic_2/syntax/genv.ma".
-
-(* ATONIC ARITY ASSIGNMENT FOR TERMS ****************************************)
-
-(* activate genv *)
-inductive aaa: relation4 genv lenv term aarity ≝
-| aaa_sort: ∀G,L,s. aaa G L (⋆s) (⓪)
-| aaa_zero: ∀I,G,L,V,B. aaa G L V B → aaa G (L.ⓑ{I}V) (#0) B
-| aaa_lref: ∀I,G,L,A,i. aaa G L (#i) A → aaa G (L.ⓘ{I}) (#↑i) A
-| aaa_abbr: ∀p,G,L,V,T,B,A.
- aaa G L V B → aaa G (L.ⓓV) T A → aaa G L (ⓓ{p}V.T) A
-| aaa_abst: ∀p,G,L,V,T,B,A.
- aaa G L V B → aaa G (L.ⓛV) T A → aaa G L (ⓛ{p}V.T) (②B.A)
-| aaa_appl: ∀G,L,V,T,B,A. aaa G L V B → aaa G L T (②B.A) → aaa G L (ⓐV.T) A
-| aaa_cast: ∀G,L,V,T,A. aaa G L V A → aaa G L T A → aaa G L (ⓝV.T) A
-.
-
-interpretation "atomic arity assignment (term)"
- 'AtomicArity G L T A = (aaa G L T A).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → A = ⓪.
-#G #L #T #A * -G -L -T -A //
-[ #I #G #L #V #B #_ #s #H destruct
-| #I #G #L #A #i #_ #s #H destruct
-| #p #G #L #V #T #B #A #_ #_ #s #H destruct
-| #p #G #L #V #T #B #A #_ #_ #s #H destruct
-| #G #L #V #T #B #A #_ #_ #s #H destruct
-| #G #L #V #T #A #_ #_ #s #H destruct
-]
-qed-.
-
-lemma aaa_inv_sort: ∀G,L,A,s. ⦃G, L⦄ ⊢ ⋆s ⁝ A → A = ⓪.
-/2 width=6 by aaa_inv_sort_aux/ qed-.
-
-fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → T = #0 →
- ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
-#G #L #T #A * -G -L -T -A /2 width=5 by ex2_3_intro/
-[ #G #L #s #H destruct
-| #I #G #L #A #i #_ #H destruct
-| #p #G #L #V #T #B #A #_ #_ #H destruct
-| #p #G #L #V #T #B #A #_ #_ #H destruct
-| #G #L #V #T #B #A #_ #_ #H destruct
-| #G #L #V #T #A #_ #_ #H destruct
-]
-qed-.
-
-lemma aaa_inv_zero: ∀G,L,A. ⦃G, L⦄ ⊢ #0 ⁝ A →
- ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
-/2 width=3 by aaa_inv_zero_aux/ qed-.
-
-fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #(↑i) →
- ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A.
-#G #L #T #A * -G -L -T -A
-[ #G #L #s #j #H destruct
-| #I #G #L #V #B #_ #j #H destruct
-| #I #G #L #A #i #HA #j #H destruct /2 width=4 by ex2_2_intro/
-| #p #G #L #V #T #B #A #_ #_ #j #H destruct
-| #p #G #L #V #T #B #A #_ #_ #j #H destruct
-| #G #L #V #T #B #A #_ #_ #j #H destruct
-| #G #L #V #T #A #_ #_ #j #H destruct
-]
-qed-.
-
-lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #↑i ⁝ A →
- ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A.
-/2 width=3 by aaa_inv_lref_aux/ qed-.
-
-fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀l. T = §l → ⊥.
-#G #L #T #A * -G -L -T -A
-[ #G #L #s #k #H destruct
-| #I #G #L #V #B #_ #k #H destruct
-| #I #G #L #A #i #_ #k #H destruct
-| #p #G #L #V #T #B #A #_ #_ #k #H destruct
-| #p #G #L #V #T #B #A #_ #_ #k #H destruct
-| #G #L #V #T #B #A #_ #_ #k #H destruct
-| #G #L #V #T #A #_ #_ #k #H destruct
-]
-qed-.
-
-lemma aaa_inv_gref: ∀G,L,A,l. ⦃G, L⦄ ⊢ §l ⁝ A → ⊥.
-/2 width=7 by aaa_inv_gref_aux/ qed-.
-
-fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{p}W.U →
- ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L.ⓓW⦄ ⊢ U ⁝ A.
-#G #L #T #A * -G -L -T -A
-[ #G #L #s #q #W #U #H destruct
-| #I #G #L #V #B #_ #q #W #U #H destruct
-| #I #G #L #A #i #_ #q #W #U #H destruct
-| #p #G #L #V #T #B #A #HV #HT #q #W #U #H destruct /2 width=2 by ex2_intro/
-| #p #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
-| #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
-| #G #L #V #T #A #_ #_ #q #W #U #H destruct
-]
-qed-.
-
-lemma aaa_inv_abbr: ∀p,G,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{p}V.T ⁝ A →
- ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L.ⓓV⦄ ⊢ T ⁝ A.
-/2 width=4 by aaa_inv_abbr_aux/ qed-.
-
-fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{p}W.U →
- ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2.
-#G #L #T #A * -G -L -T -A
-[ #G #L #s #q #W #U #H destruct
-| #I #G #L #V #B #_ #q #W #U #H destruct
-| #I #G #L #A #i #_ #q #W #U #H destruct
-| #p #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
-| #p #G #L #V #T #B #A #HV #HT #q #W #U #H destruct /2 width=5 by ex3_2_intro/
-| #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
-| #G #L #V #T #A #_ #_ #q #W #U #H destruct
-]
-qed-.
-
-lemma aaa_inv_abst: ∀p,G,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{p}W.T ⁝ A →
- ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2.
-/2 width=4 by aaa_inv_abst_aux/ qed-.
-
-fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U →
- ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L⦄ ⊢ U ⁝ ②B.A.
-#G #L #T #A * -G -L -T -A
-[ #G #L #s #W #U #H destruct
-| #I #G #L #V #B #_ #W #U #H destruct
-| #I #G #L #A #i #_ #W #U #H destruct
-| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #G #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3 by ex2_intro/
-| #G #L #V #T #A #_ #_ #W #U #H destruct
-]
-qed-.
-
-lemma aaa_inv_appl: ∀G,L,V,T,A. ⦃G, L⦄ ⊢ ⓐV.T ⁝ A →
- ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L⦄ ⊢ T ⁝ ②B.A.
-/2 width=3 by aaa_inv_appl_aux/ qed-.
-
-fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.U →
- ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ U ⁝ A.
-#G #L #T #A * -G -L -T -A
-[ #G #L #s #W #U #H destruct
-| #I #G #L #V #B #_ #W #U #H destruct
-| #I #G #L #A #i #_ #W #U #H destruct
-| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #G #L #V #T #B #A #_ #_ #W #U #H destruct
-| #G #L #V #T #A #HV #HT #W #U #H destruct /2 width=1 by conj/
-]
-qed-.
-
-lemma aaa_inv_cast: ∀G,L,W,T,A. ⦃G, L⦄ ⊢ ⓝW.T ⁝ A →
- ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ T ⁝ A.
-/2 width=3 by aaa_inv_cast_aux/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/aaa.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Main inversion lemmas ****************************************************)
-
-theorem aaa_mono: ∀G,L,T,A1. ⦃G, L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T ⁝ A2 → A1 = A2.
-#G #L #T #A1 #H elim H -G -L -T -A1
-[ #G #L #s #A2 #H >(aaa_inv_sort … H) -H //
-| #I1 #G #L #V1 #B #_ #IH #A2 #H
- elim (aaa_inv_zero … H) -H #I2 #K2 #V2 #H #HA2 destruct /2 width=1 by/
-| #I1 #G #L #B #i #_ #IH #A2 #H
- elim (aaa_inv_lref … H) -H #I2 #K2 #H #HA2 destruct /2 width=1 by/
-| #p #G #L #V #T #B1 #A1 #_ #_ #_ #IH #A2 #H
- elim (aaa_inv_abbr … H) -H /2 width=1 by/
-| #p #G #L #V1 #T1 #B1 #A1 #_ #_ #IHB1 #IHA1 #X #H
- elim (aaa_inv_abst … H) -H #B2 #A2 #HB2 #HA2 #H destruct /3 width=1 by eq_f2/
-| #G #L #V1 #T1 #B1 #A1 #_ #_ #_ #IHA1 #A2 #H
- elim (aaa_inv_appl … H) -H #B2 #_ #HA2
- lapply (IHA1 … HA2) -L #H destruct //
-| #G #L #V #T #A1 #_ #_ #_ #IHA1 #A2 #H
- elim (aaa_inv_cast … 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/relocation/drops_drops.ma".
-include "basic_2/s_computation/fqup_weight.ma".
-include "basic_2/s_computation/fqup_drops.ma".
-include "basic_2/static/aaa.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: was: aaa_lref *)
-lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
-#I #G #K #V #B #i elim i -i
-[ #L #H lapply (drops_fwd_isid … H ?) -H //
- #H destruct /2 width=1 by aaa_zero/
-| #i #IH #L <uni_succ #H #HB lapply (drops_inv_bind2_isuni_next … H) -H // *
- #Z #Y #HY #H destruct /3 width=1 by aaa_lref/
-]
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-(* Basic_2A1: was: aaa_inv_lref *)
-lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
- ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
-#G #A #i elim i -i
-[ #L #H elim (aaa_inv_zero … H) -H /3 width=5 by drops_refl, ex2_3_intro/
-| #i #IH #L #H elim (aaa_inv_lref … H) -H
- #I #K #H #HA destruct elim (IH … HA) -IH -HA /3 width=5 by drops_drop, ex2_3_intro/
-]
-qed-.
-
-(* Properties with generic slicing for local environments *******************)
-
-(* Basic_2A1: includes: aaa_lift *)
-(* Note: it should use drops_split_trans_pair2 *)
-lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b, f] L2 ≘ L1 →
- ∀T2. ⬆*[f] T1 ≘ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
-@(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L1 * *
-[ #s #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -IH
- lapply (aaa_inv_sort … H) -H #H destruct
- >(lifts_inv_sort1 … HX) -HX //
-| #i1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
- elim (aaa_inv_lref_drops … H) -H #J #K1 #V1 #HLK1 #HA
- elim (lifts_inv_lref1 … HX) -HX #i2 #Hf #H destruct
- lapply (drops_trans … HL21 … HLK1 ??) -HL21 [1,2: // ] #H
- elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK2 #HY
- lapply (drops_tls_at … Hf … HY) -HY #HY -Hf
- elim (drops_inv_skip2 … HY) -HY #Z #K2 #HK21 #HZ #H destruct
- elim (liftsb_inv_pair_sn … HZ) -HZ #V2 #HV12 #H destruct
- /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_gen/
-| #l #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -f -IH
- elim (aaa_inv_gref … H)
-| #p * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
- [ elim (aaa_inv_abbr … H) -H #B #HB #HA
- elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
- /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
- | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
- elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
- /5 width=8 by aaa_abst, drops_skip, ext2_pair/
- ]
-| * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
- [ elim (aaa_inv_appl … H) -H #B #HB #HA
- elim (lifts_inv_flat1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
- /3 width=10 by aaa_appl/
- | elim (aaa_inv_cast … H) -H #H1A #H2A
- elim (lifts_inv_flat1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
- /3 width=8 by aaa_cast/
- ]
-]
-qed-.
-
-(* Inversion lemmas with generic slicing for local environments *************)
-
-(* Basic_2A1: includes: aaa_inv_lift *)
-lemma aaa_inv_lifts: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀b,f,L1. ⬇*[b, f] L2 ≘ L1 →
- ∀T1. ⬆*[f] T1 ≘ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
-@(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L2 * *
-[ #s #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -b -IH
- lapply (aaa_inv_sort … H) -H #H destruct
- >(lifts_inv_sort2 … HX) -HX //
-| #i2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
- elim (aaa_inv_lref_drops … H) -H #J #K2 #V2 #HLK2 #HA
- elim (lifts_inv_lref2 … HX) -HX #i1 #Hf #H destruct
- lapply (drops_split_div … HL21 (𝐔❴i1❵) ???) -HL21 [4: * |*: // ] #Y #HLK1 #HY
- lapply (drops_conf … HLK2 … HY ??) -HY [1,2: /2 width=6 by after_uni_dx/ ] #HY
- lapply (drops_tls_at … Hf … HY) -HY #HY -Hf
- elim (drops_inv_skip1 … HY) -HY #Z #K1 #HK21 #HZ #H destruct
- elim (liftsb_inv_pair_dx … HZ) -HZ #V1 #HV12 #H destruct
- /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_F/
-| #l #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -IH -b -f
- elim (aaa_inv_gref … H)
-| #p * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
- [ elim (aaa_inv_abbr … H) -H #B #HB #HA
- elim (lifts_inv_bind2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
- /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
- | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
- elim (lifts_inv_bind2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
- /5 width=8 by aaa_abst, drops_skip, ext2_pair/
- ]
-| * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
- [ elim (aaa_inv_appl … H) -H #B #HB #HA
- elim (lifts_inv_flat2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
- /3 width=10 by aaa_appl/
- | elim (aaa_inv_cast … H) -H #H1A #H2A
- elim (lifts_inv_flat2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
- /3 width=8 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/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/s_computation/fqus_fqup.ma".
-include "basic_2/static/aaa_drops.ma".
-
-(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
-
-(* Properties on supclosure *************************************************)
-
-lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
-[ #I #G #L #T #A #H elim (aaa_inv_zero … H) -H
- #J #K #V #H #HA destruct /2 width=2 by ex_intro/
-| * [ #p ] * #G #L #V #T #X #H
- [ elim (aaa_inv_abbr … H)
- | elim (aaa_inv_abst … H)
- | elim (aaa_inv_appl … H)
- | elim (aaa_inv_cast … H)
- ] -H /2 width=2 by ex_intro/
-| #p * #G #L #V #T #X #H
- [ elim (aaa_inv_abbr … H)
- | elim (aaa_inv_abst … H)
- ] -H /2 width=2 by ex_intro/
-| #p #I #G #L #V #T #H destruct
-| * #G #L #V #T #X #H
- [ elim (aaa_inv_appl … H)
- | elim (aaa_inv_cast … H)
- ] -H /2 width=2 by ex_intro/
-| /5 width=8 by aaa_inv_lifts, drops_refl, drops_drop, ex_intro/
-]
-qed-.
-
-lemma aaa_fquq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=6 by aaa_fqu_conf/
-* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
-qed-.
-
-lemma aaa_fqup_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
-#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
-[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A ]
-/2 width=6 by aaa_fqu_conf/
-qed-.
-
-lemma aaa_fqus_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
- ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
-#G1 #G2 #L1 #L2 #T1 #T2 #H elim(fqus_inv_fqup … H) -H /2 width=6 by aaa_fqup_conf/
-* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/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 "ground_2/relocation/rtmap_sor.ma".
-include "basic_2/notation/relations/freestar_3.ma".
-include "basic_2/syntax/lenv.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-inductive frees: relation3 lenv term rtmap ≝
-| frees_sort: ∀f,L,s. 𝐈⦃f⦄ → frees L (⋆s) f
-| frees_atom: ∀f,i. 𝐈⦃f⦄ → frees (⋆) (#i) (⫯*[i]↑f)
-| frees_pair: ∀f,I,L,V. frees L V f →
- frees (L.ⓑ{I}V) (#0) (↑f)
-| frees_unit: ∀f,I,L. 𝐈⦃f⦄ → frees (L.ⓤ{I}) (#0) (↑f)
-| frees_lref: ∀f,I,L,i. frees L (#i) f →
- frees (L.ⓘ{I}) (#↑i) (⫯f)
-| frees_gref: ∀f,L,l. 𝐈⦃f⦄ → frees L (§l) f
-| frees_bind: ∀f1,f2,f,p,I,L,V,T. frees L V f1 → frees (L.ⓑ{I}V) T f2 →
- f1 ⋓ ⫱f2 ≘ f → frees L (ⓑ{p,I}V.T) f
-| frees_flat: ∀f1,f2,f,I,L,V,T. frees L V f1 → frees L T f2 →
- f1 ⋓ f2 ≘ f → frees L (ⓕ{I}V.T) f
-.
-
-interpretation
- "context-sensitive free variables (term)"
- 'FreeStar L T f = (frees L T f).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
-#L #X #f #H elim H -f -L -X //
-[ #f #i #_ #x #H destruct
-| #f #_ #L #V #_ #_ #x #H destruct
-| #f #_ #L #_ #x #H destruct
-| #f #_ #L #i #_ #_ #x #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
-]
-qed-.
-
-lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅*⦃⋆s⦄ ≘ f → 𝐈⦃f⦄.
-/2 width=5 by frees_inv_sort_aux/ qed-.
-
-fact frees_inv_atom_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀i. L = ⋆ → X = #i →
- ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
-#f #L #X #H elim H -f -L -X
-[ #f #L #s #_ #j #_ #H destruct
-| #f #i #Hf #j #_ #H destruct /2 width=3 by ex2_intro/
-| #f #I #L #V #_ #_ #j #H destruct
-| #f #I #L #_ #j #H destruct
-| #f #I #L #i #_ #_ #j #H destruct
-| #f #L #l #_ #j #_ #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #j #_ #H destruct
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #j #_ #H destruct
-]
-qed-.
-
-lemma frees_inv_atom: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
-/2 width=5 by frees_inv_atom_aux/ qed-.
-
-fact frees_inv_pair_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,V. L = K.ⓑ{I}V → X = #0 →
- ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
-#f #L #X * -f -L -X
-[ #f #L #s #_ #Z #Y #X #_ #H destruct
-| #f #i #_ #Z #Y #X #H destruct
-| #f #I #L #V #Hf #Z #Y #X #H #_ destruct /2 width=3 by ex2_intro/
-| #f #I #L #_ #Z #Y #X #H destruct
-| #f #I #L #i #_ #Z #Y #X #_ #H destruct
-| #f #L #l #_ #Z #Y #X #_ #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #X #_ #H destruct
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #X #_ #H destruct
-]
-qed-.
-
-lemma frees_inv_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
-/2 width=6 by frees_inv_pair_aux/ qed-.
-
-fact frees_inv_unit_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K. L = K.ⓤ{I} → X = #0 →
- ∃∃g. 𝐈⦃g⦄ & f = ↑g.
-#f #L #X * -f -L -X
-[ #f #L #s #_ #Z #Y #_ #H destruct
-| #f #i #_ #Z #Y #H destruct
-| #f #I #L #V #_ #Z #Y #H destruct
-| #f #I #L #Hf #Z #Y #H destruct /2 width=3 by ex2_intro/
-| #f #I #L #i #_ #Z #Y #_ #H destruct
-| #f #L #l #_ #Z #Y #_ #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #_ #H destruct
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #_ #H destruct
-]
-qed-.
-
-lemma frees_inv_unit: ∀f,I,K. K.ⓤ{I} ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ↑g.
-/2 width=7 by frees_inv_unit_aux/ qed-.
-
-fact frees_inv_lref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,j. L = K.ⓘ{I} → X = #(↑j) →
- ∃∃g. K ⊢ 𝐅*⦃#j⦄ ≘ g & f = ⫯g.
-#f #L #X * -f -L -X
-[ #f #L #s #_ #Z #Y #j #_ #H destruct
-| #f #i #_ #Z #Y #j #H destruct
-| #f #I #L #V #_ #Z #Y #j #_ #H destruct
-| #f #I #L #_ #Z #Y #j #_ #H destruct
-| #f #I #L #i #Hf #Z #Y #j #H1 #H2 destruct /2 width=3 by ex2_intro/
-| #f #L #l #_ #Z #Y #j #_ #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #j #_ #H destruct
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #j #_ #H destruct
-]
-qed-.
-
-lemma frees_inv_lref: ∀f,I,K,i. K.ⓘ{I} ⊢ 𝐅*⦃#(↑i)⦄ ≘ f →
- ∃∃g. K ⊢ 𝐅*⦃#i⦄ ≘ g & f = ⫯g.
-/2 width=6 by frees_inv_lref_aux/ qed-.
-
-fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = §x → 𝐈⦃f⦄.
-#f #L #X #H elim H -f -L -X //
-[ #f #i #_ #x #H destruct
-| #f #_ #L #V #_ #_ #x #H destruct
-| #f #_ #L #_ #x #H destruct
-| #f #_ #L #i #_ #_ #x #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
-]
-qed-.
-
-lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅*⦃§l⦄ ≘ f → 𝐈⦃f⦄.
-/2 width=5 by frees_inv_gref_aux/ qed-.
-
-fact frees_inv_bind_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀p,I,V,T. X = ⓑ{p,I}V.T →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
-#f #L #X * -f -L -X
-[ #f #L #s #_ #q #J #W #U #H destruct
-| #f #i #_ #q #J #W #U #H destruct
-| #f #I #L #V #_ #q #J #W #U #H destruct
-| #f #I #L #_ #q #J #W #U #H destruct
-| #f #I #L #i #_ #q #J #W #U #H destruct
-| #f #L #l #_ #q #J #W #U #H destruct
-| #f1 #f2 #f #p #I #L #V #T #HV #HT #Hf #q #J #W #U #H destruct /2 width=5 by ex3_2_intro/
-| #f1 #f2 #f #I #L #V #T #_ #_ #_ #q #J #W #U #H destruct
-]
-qed-.
-
-lemma frees_inv_bind: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
-/2 width=4 by frees_inv_bind_aux/ qed-.
-
-fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,V,T. X = ⓕ{I}V.T →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
-#f #L #X * -f -L -X
-[ #f #L #s #_ #J #W #U #H destruct
-| #f #i #_ #J #W #U #H destruct
-| #f #I #L #V #_ #J #W #U #H destruct
-| #f #I #L #_ #J #W #U #H destruct
-| #f #I #L #i #_ #J #W #U #H destruct
-| #f #L #l #_ #J #W #U #H destruct
-| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #J #W #U #H destruct
-| #f1 #f2 #f #I #L #V #T #HV #HT #Hf #J #W #U #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma frees_inv_flat: ∀f,I,L,V,T. L ⊢ 𝐅*⦃ⓕ{I}V.T⦄ ≘ f →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
-/2 width=4 by frees_inv_flat_aux/ qed-.
-
-(* Basic properties ********************************************************)
-
-lemma frees_eq_repl_back: ∀L,T. eq_repl_back … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
-#L #T #f1 #H elim H -f1 -L -T
-[ /3 width=3 by frees_sort, isid_eq_repl_back/
-| #f1 #i #Hf1 #g2 #H
- elim (eq_inv_pushs_sn … H) -H #g #Hg #H destruct
- elim (eq_inv_nx … Hg) -Hg
- /3 width=3 by frees_atom, isid_eq_repl_back/
-| #f1 #I #L #V #_ #IH #g2 #H
- elim (eq_inv_nx … H) -H
- /3 width=3 by frees_pair/
-| #f1 #I #L #Hf1 #g2 #H
- elim (eq_inv_nx … H) -H
- /3 width=3 by frees_unit, isid_eq_repl_back/
-| #f1 #I #L #i #_ #IH #g2 #H
- elim (eq_inv_px … H) -H /3 width=3 by frees_lref/
-| /3 width=3 by frees_gref, isid_eq_repl_back/
-| /3 width=7 by frees_bind, sor_eq_repl_back3/
-| /3 width=7 by frees_flat, sor_eq_repl_back3/
-]
-qed-.
-
-lemma frees_eq_repl_fwd: ∀L,T. eq_repl_fwd … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
-#L #T @eq_repl_sym /2 width=3 by frees_eq_repl_back/
-qed-.
-
-lemma frees_lref_push: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ⋆ ⊢ 𝐅*⦃#↑i⦄ ≘ ⫯f.
-#f #i #H
-elim (frees_inv_atom … H) -H #g #Hg #H destruct
-/2 width=1 by frees_atom/
-qed.
-
-(* Forward lemmas with test for finite colength *****************************)
-
-lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f → 𝐅⦃f⦄.
-#f #L #T #H elim H -f -L -T
-/4 width=5 by sor_isfin, isfin_isid, isfin_tl, isfin_pushs, isfin_push, isfin_next/
-qed-.
-
-(* Basic_2A1: removed theorems 30:
- frees_eq frees_be frees_inv
- frees_inv_sort frees_inv_gref frees_inv_lref frees_inv_lref_free
- frees_inv_lref_skip frees_inv_lref_ge frees_inv_lref_lt
- frees_inv_bind frees_inv_flat frees_inv_bind_O
- frees_lref_eq frees_lref_be frees_weak
- frees_bind_sn frees_bind_dx frees_flat_sn frees_flat_dx
- frees_lift_ge frees_inv_lift_be frees_inv_lift_ge
- lreq_frees_trans frees_lreq_conf
- llor_atom llor_skip llor_total
- llor_tail_frees llor_tail_cofrees
-*)
+++ /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/append.ma".
-include "basic_2/static/frees.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-(* Properties with append for local environments ****************************)
-
-lemma frees_append_void: ∀f,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f → ⓧ.K ⊢ 𝐅*⦃T⦄ ≘ f.
-#f #K #T #H elim H -f -K -T
-[ /2 width=1 by frees_sort/
-| #f * /3 width=1 by frees_atom, frees_unit, frees_lref/
-| /2 width=1 by frees_pair/
-| /2 width=1 by frees_unit/
-| /2 width=1 by frees_lref/
-| /2 width=1 by frees_gref/
-| /3 width=5 by frees_bind/
-| /3 width=5 by frees_flat/
-]
-qed.
-
-(* Inversion lemmas with append for local environments **********************)
-
-fact frees_inv_append_void_aux: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
- ∀K. L = ⓧ.K → K ⊢ 𝐅*⦃T⦄ ≘ f.
-#f #L #T #H elim H -f -L -T
-[ /2 width=1 by frees_sort/
-| #f #i #_ #K #H
- elim (append_inv_atom3_sn … H) -H #H1 #H2 destruct
-| #f #I #L #V #_ #IH #K #H
- elim (append_inv_bind3_sn … H) -H * [ | #Y ] #H1 #H2 destruct
- /3 width=1 by frees_pair/
-| #f #I #L #Hf #K #H
- elim (append_inv_bind3_sn … H) -H * [ | #Y ] #H1 #H2 destruct
- /2 width=1 by frees_atom, frees_unit/
-| #f #I #L #i #Hf #IH #K #H
- elim (append_inv_bind3_sn … H) -H * [ | #Y ] #H1 #H2 destruct
- /3 width=1 by frees_lref, frees_lref_push/
-| /2 width=1 by frees_gref/
-| /3 width=5 by frees_bind/
-| /3 width=5 by frees_flat/
-]
-qed-.
-
-lemma frees_inv_append_void: ∀f,K,T. ⓧ.K ⊢ 𝐅*⦃T⦄ ≘ f → K ⊢ 𝐅*⦃T⦄ ≘ f.
-/2 width=3 by frees_inv_append_void_aux/ qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/relocation/nstream_coafter.ma".
-include "basic_2/relocation/drops_drops.ma".
-include "basic_2/static/frees_fqup.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-(* Advanced properties ******************************************************)
-
-lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≘ ⋆ →
- ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i]↑f.
-#b #L elim L -L /2 width=1 by frees_atom/
-#L #I #IH *
-[ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
-| /4 width=3 by frees_lref, drops_inv_drop1/
-]
-qed.
-
-lemma frees_pair_drops: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≘ f →
- ∀i,I,L. ⬇*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
-#f #K #V #Hf #i elim i -i
-[ #I #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_pair/
-| #i #IH #I #L #H elim (drops_inv_succ … H) -H /3 width=2 by frees_lref/
-]
-qed.
-
-lemma frees_unit_drops: ∀f. 𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≘ K.ⓤ{I} →
- L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
-#f #Hf #I #K #i elim i -i
-[ #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_unit/
-| #i #IH #Y #H elim (drops_inv_succ … H) -H
- #J #L #HLK #H destruct /3 width=1 by frees_lref/
-]
-qed.
-(*
-lemma frees_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≘ f →
- ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃⋆s⦄ ≘ ⫯*[i] f.
-#f #K #s #Hf #i elim i -i
-[ #L #H lapply (drops_fwd_isid … H ?) -H //
-| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_sort/
-]
-qed.
-*)
-lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≘ f →
- ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃#(i+j)⦄ ≘ ⫯*[i] f.
-#f #K #j #Hf #i elim i -i
-[ #L #H lapply (drops_fwd_isid … H ?) -H //
-| #i #IH #L #H elim (drops_inv_succ … H) -H
- #I #Y #HYK #H destruct /3 width=1 by frees_lref/
-]
-qed.
-(*
-lemma frees_gref_pushs: ∀f,K,l. K ⊢ 𝐅*⦃§l⦄ ≘ f →
- ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃§l⦄ ≘ ⫯*[i] f.
-#f #K #l #Hf #i elim i -i
-[ #L #H lapply (drops_fwd_isid … H ?) -H //
-| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_gref/
-]
-qed.
-*)
-(* Advanced inversion lemmas ************************************************)
-
-lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≘ f →
- ∨∨ ∃∃g. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
- | ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≘ g &
- ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
- | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
-#L elim L -L
-[ #i #g | #L #I #IH * [ #g cases I -I [ #I | #I #V ] -IH | #i #g ] ] #H
-[ elim (frees_inv_atom … H) -H #f #Hf #H destruct
- /3 width=3 by or3_intro0, ex3_intro/
-| elim (frees_inv_unit … H) -H #f #Hf #H destruct
- /4 width=3 by drops_refl, or3_intro2, ex3_3_intro/
-| elim (frees_inv_pair … H) -H #f #Hf #H destruct
- /4 width=7 by drops_refl, or3_intro1, ex3_4_intro/
-| elim (frees_inv_lref … H) -H #f #Hf #H destruct
- elim (IH … Hf) -IH -Hf *
- [ /4 width=3 by drops_drop, or3_intro0, ex3_intro/
- | /4 width=7 by drops_drop, or3_intro1, ex3_4_intro/
- | /4 width=3 by drops_drop, or3_intro2, ex3_3_intro/
- ]
-]
-qed-.
-
-(* Properties with generic slicing for local environments *******************)
-
-lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 →
- ∀f,L. ⬇*[b, f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
- ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2.
-#b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
-[ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
- lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
- >(lifts_inv_sort1 … H2) -U /2 width=1 by frees_sort/
-| #f1 #i #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
- elim (lifts_inv_lref1 … H2) -H2 #j #Hij #H destruct
- elim (coafter_fwd_xnx_pushs … Hij H3) -H3 #g2 #Hg2 #H2 destruct
- lapply (coafter_isid_inv_dx … Hg2 … Hf1) -f1 #Hf2
- elim (drops_inv_atom2 … H1) -H1 #n #g #H1 #Hf
- elim (after_at_fwd … Hij … Hf) -f #x #_ #Hj -g -i
- lapply (at_inv_uni … Hj) -Hj #H destruct
- /3 width=8 by frees_atom_drops, drops_trans/
-| #f1 #I #K #V #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
- lapply (isfin_inv_next … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
- lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
- elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #H
- elim (liftsb_inv_pair_sn … H) -H #W #HVW #H destruct
- elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
- lapply (IH … HYK … HVW … H3) -IH -H3 -HYK -HVW //
- /2 width=5 by frees_pair_drops/
-| #f1 #I #K #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
- lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
- elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
- lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hg2
- elim (drops_split_trans_bind2 … H1 … Hf) -H1 -Hf #Z #Y #HLY #_ #H
- lapply (liftsb_inv_unit_sn … H) -H #H destruct
- /2 width=3 by frees_unit_drops/
-| #f1 #I #K #i #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
- lapply (isfin_inv_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
- lapply (lifts_inv_lref1 … H2) -H2 * #x #Hf #H destruct
- elim (at_inv_nxx … Hf) -Hf [ |*: // ] #j #Hf #H destruct
- elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #_
- elim (coafter_fwd_xpx_pushs … 0 … H3) [ |*: // ] #g2 #H3 #H2 destruct
- lapply (drops_isuni_fwd_drop2 … HLY) -HLY // #HLY
- lapply (IH … HYK … H3) -IH -H3 -HYK [4: |*: /2 width=2 by lifts_lref/ ]
- >plus_S1 /2 width=3 by frees_lref_pushs/ (**) (* full auto fails *)
-| #f1 #K #l #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
- lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
- >(lifts_inv_gref1 … H2) -U /2 width=1 by frees_gref/
-| #f1V #f1T #f1 #p #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
- elim (sor_inv_isfin3 … H1f1) // #Hf1V #H
- lapply (isfin_inv_tl … H) -H
- elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct
- elim (coafter_sor … H3 … H1f1) /2 width=5 by coafter_isfin2_fwd/ -H3 -H1f1 #f2V #f2T #Hf2V #H
- elim (coafter_inv_tl1 … H) -H
- /5 width=5 by frees_bind, drops_skip, ext2_pair/
-| #f1V #f1T #f1 #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
- elim (sor_inv_isfin3 … H1f1) //
- elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
- elim (coafter_sor … H3 … H1f1)
- /3 width=5 by coafter_isfin2_fwd, frees_flat/
-]
-qed-.
-
-lemma frees_lifts_SO: ∀b,L,K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
- ∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f.
-#b #L #K #HLK #T #U #HTU #f #Hf
-@(frees_lifts b … Hf … HTU) // (**) (* auto fails *)
-qed.
-
-(* Forward lemmas with generic slicing for local environments ***************)
-
-lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
- ∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
-/4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
-
-(* Inversion lemmas with generic slicing for local environments *************)
-
-lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
- ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅*⦃T⦄ ≘ f1.
-#b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T)
-/3 width=9 by frees_fwd_coafter, ex2_intro/
-qed-.
-
-lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f →
- ∀K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
- K ⊢ 𝐅*⦃T⦄ ≘ ⫱f.
-#b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
-#f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf
-/3 width=5 by frees_eq_repl_back, coafter_isid_inv_sn/
-qed-.
-
-lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
- ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
- ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅*⦃T⦄ ≘ f1.
-#b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
-/3 width=7 by frees_eq_repl_back, coafter_inj/
-qed-.
-
-(* 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 →
- ∃∃g2. L2 ⊢ 𝐅*⦃V2⦄ ≘ g2 & g2 ⊆ g1.
-#f1 #L1 #T1 #H elim H -f1 -L1 -T1
-[ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -s
- lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
- elim (isid_inv_next … Hf1) -Hf1 //
-| #f1 #i #_ #I2 #L2 #V2 #n #H
- elim (drops_inv_atom1 … H) -H #H destruct
-| #f1 #I1 #L1 #V1 #Hf1 #IH #I2 #L2 #V2 *
- [ -IH #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 //
- #H destruct #g1 #Hgf1 >(injective_next … Hgf1) -g1
- /2 width=3 by ex2_intro/
- | -Hf1 #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
- #HL12 #g1 <tls_xn <tl_next_rew #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
- /2 width=3 by ex2_intro/
- ]
-| #f1 #I1 #L1 #Hf1 #I2 #L2 #V2 *
- [ #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct
- | #n #_ #g1 #Hgf1 elim (isid_inv_next … Hgf1) -Hgf1 <tls_xn /2 width=1 by isid_tls/
- ]
-| #f1 #I1 #L1 #i #_ #IH #I2 #L2 #V2 *
- [ -IH #_ #g1 #Hgf1 elim (discr_next_push … Hgf1)
- | #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
- #HL12 #g1 <tls_xn #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
- /2 width=3 by ex2_intro/
- ]
-| #f1 #L1 #l #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -l
- lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
- elim (isid_inv_next … Hf1) -Hf1 //
-| #fV1 #fT1 #f1 #p #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
- lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
- elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
- #gV1 #gT1 #Hg1
- [ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
- /3 width=6 by sor_inv_sle_sn_trans, ex2_intro/
- | -IHV1 #_ >tls_xn #H2 elim (IHT1 … H2) -IHT1 -H2
- /3 width=6 by drops_drop, sor_inv_sle_dx_trans, ex2_intro/
- ]
-| #fV1 #fT1 #f1 #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
- lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
- elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
- #gV1 #gT1 #Hg1
- [ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
- /3 width=6 by sor_inv_sle_sn_trans, ex2_intro/
- | -IHV1 #_ #H2 elim (IHT1 … HL12 … H2) -IHT1 -HL12 -H2
- /3 width=6 by sor_inv_sle_dx_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/s_computation/fqup_weight.ma".
-include "basic_2/static/lsubf_lsubr.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-(* Advanced properties ******************************************************)
-
-(* Note: this replaces lemma 1400 concluding the "big tree" theorem *)
-lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅*⦃T⦄ ≘ f.
-#L #T @(fqup_wf_ind_eq (Ⓣ) … (⋆) L T) -L -T
-#G0 #L0 #T0 #IH #G #L * *
-[ /3 width=2 by frees_sort, ex_intro/
-| cases L -L /3 width=2 by frees_atom, ex_intro/
- #L #I *
- [ cases I -I #I [2: #V ] #HG #HL #HT destruct
- [ elim (IH G L V) -IH
- /3 width=2 by frees_pair, fqu_fqup, fqu_lref_O, ex_intro/
- | -IH /3 width=2 by frees_unit, ex_intro/
- ]
- | #i #HG #HL #HT destruct
- elim (IH G L (#i)) -IH
- /3 width=2 by frees_lref, fqu_fqup, fqu_drop, ex_intro/
- ]
-| /3 width=2 by frees_gref, ex_intro/
-| #p #I #V #T #HG #HL #HT destruct
- elim (IH G L V) // #f1 #HV
- elim (IH G (L.ⓑ{I}V) T) -IH // #f2 #HT
- elim (sor_isfin_ex f1 (⫱f2))
- /3 width=6 by frees_fwd_isfin, frees_bind, isfin_tl, ex_intro/
-| #I #V #T #HG #HL #HT destruct
- elim (IH G L V) // #f1 #HV
- elim (IH G L T) -IH // #f2 #HT
- elim (sor_isfin_ex f1 f2)
- /3 width=6 by frees_fwd_isfin, frees_flat, ex_intro/
-]
-qed-.
-
-(* Advanced main properties *************************************************)
-
-theorem frees_bind_void: ∀f1,L,V. L ⊢ 𝐅*⦃V⦄ ≘ f1 → ∀f2,T. L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 →
- ∀f. f1 ⋓ ⫱f2 ≘ f → ∀p,I. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f.
-#f1 #L #V #Hf1 #f2 #T #Hf2 #f #Hf #p #I
-elim (frees_total (L.ⓑ{I}V) T) #f0 #Hf0
-lapply (lsubr_lsubf … Hf2 … Hf0) -Hf2 /2 width=5 by lsubr_unit/ #H02
-elim (pn_split f2) * #g2 #H destruct
-[ elim (lsubf_inv_push2 … H02) -H02 #g0 #Z #Y #H02 #H0 #H destruct
- lapply (lsubf_inv_refl … H02) -H02 #H02
- lapply (sor_eq_repl_fwd2 … Hf … H02) -g2 #Hf
- /2 width=5 by frees_bind/
-| elim (lsubf_inv_unit2 … H02) -H02 * [ #g0 #Y #_ #_ #H destruct ]
- #z1 #g0 #z #Z #Y #X #H02 #Hz1 #Hz #H0 #H destruct
- lapply (lsubf_inv_refl … H02) -H02 #H02
- lapply (frees_mono … Hz1 … Hf1) -Hz1 #H1
- lapply (sor_eq_repl_back1 … Hz … H02) -g0 #Hz
- lapply (sor_eq_repl_back2 … Hz … H1) -z1 #Hz
- lapply (sor_comm … Hz) -Hz #Hz
- lapply (sor_mono … f Hz ?) // -Hz #H
- lapply (sor_inv_sle_sn … Hf) -Hf #Hf
- lapply (frees_eq_repl_back … Hf0 (↑f) ?) /2 width=5 by eq_next/ -z #Hf0
- @(frees_bind … Hf1 Hf0) -Hf1 -Hf0 (**) (* constructor needed *)
- /2 width=1 by sor_sle_dx/
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma frees_inv_bind_void: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
- ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
-#f #p #I #L #V #T #H
-elim (frees_inv_bind … H) -H #f1 #f2 #Hf1 #Hf2 #Hf
-elim (frees_total (L.ⓧ) T) #f0 #Hf0
-lapply (lsubr_lsubf … Hf0 … Hf2) -Hf2 /2 width=5 by lsubr_unit/ #H20
-elim (pn_split f0) * #g0 #H destruct
-[ elim (lsubf_inv_push2 … H20) -H20 #g2 #I #Y #H20 #H2 #H destruct
- lapply (lsubf_inv_refl … H20) -H20 #H20
- lapply (sor_eq_repl_back2 … Hf … H20) -g2 #Hf
- /2 width=5 by ex3_2_intro/
-| elim (lsubf_inv_unit2 … H20) -H20 * [ #g2 #Y #_ #_ #H destruct ]
- #z1 #z0 #g2 #Z #Y #X #H20 #Hz1 #Hg2 #H2 #H destruct
- lapply (lsubf_inv_refl … H20) -H20 #H0
- lapply (frees_mono … Hz1 … Hf1) -Hz1 #H1
- lapply (sor_eq_repl_back1 … Hg2 … H0) -z0 #Hg2
- lapply (sor_eq_repl_back2 … Hg2 … H1) -z1 #Hg2
- @(ex3_2_intro … Hf1 Hf0) -Hf1 -Hf0 (**) (* constructor needed *)
- /2 width=3 by sor_comm_23_idem/
-]
-qed-.
-
-lemma frees_ind_void: ∀Q:relation3 ….
- (
- ∀f,L,s. 𝐈⦃f⦄ → Q L (⋆s) f
- ) → (
- ∀f,i. 𝐈⦃f⦄ → Q (⋆) (#i) (⫯*[i]↑f)
- ) → (
- ∀f,I,L,V.
- L ⊢ 𝐅*⦃V⦄ ≘ f → Q L V f→ Q (L.ⓑ{I}V) (#O) (↑f)
- ) → (
- ∀f,I,L. 𝐈⦃f⦄ → Q (L.ⓤ{I}) (#O) (↑f)
- ) → (
- ∀f,I,L,i.
- L ⊢ 𝐅*⦃#i⦄ ≘ f → Q L (#i) f → Q (L.ⓘ{I}) (#(↑i)) (⫯f)
- ) → (
- ∀f,L,l. 𝐈⦃f⦄ → Q L (§l) f
- ) → (
- ∀f1,f2,f,p,I,L,V,T.
- L ⊢ 𝐅*⦃V⦄ ≘ f1 → L.ⓧ ⊢𝐅*⦃T⦄≘ f2 → f1 ⋓ ⫱f2 ≘ f →
- Q L V f1 → Q (L.ⓧ) T f2 → Q L (ⓑ{p,I}V.T) f
- ) → (
- ∀f1,f2,f,I,L,V,T.
- L ⊢ 𝐅*⦃V⦄ ≘ f1 → L ⊢𝐅*⦃T⦄ ≘ f2 → f1 ⋓ f2 ≘ f →
- Q L V f1 → Q L T f2 → Q L (ⓕ{I}V.T) f
- ) →
- ∀L,T,f. L ⊢ 𝐅*⦃T⦄ ≘ f → Q L T f.
-#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #L #T
-@(fqup_wf_ind_eq (Ⓕ) … (⋆) L T) -L -T #G0 #L0 #T0 #IH #G #L * *
-[ #s #HG #HL #HT #f #H destruct -IH
- lapply (frees_inv_sort … H) -H /2 width=1 by/
-| cases L -L
- [ #i #HG #HL #HT #f #H destruct -IH
- elim (frees_inv_atom … H) -H #g #Hg #H destruct /2 width=1 by/
- | #L #I * [ cases I -I #I [ | #V ] | #i ] #HG #HL #HT #f #H destruct
- [ elim (frees_inv_unit … H) -H #g #Hg #H destruct /2 width=1 by/
- | elim (frees_inv_pair … H) -H #g #Hg #H destruct
- /4 width=2 by fqu_fqup, fqu_lref_O/
- | elim (frees_inv_lref … H) -H #g #Hg #H destruct
- /4 width=2 by fqu_fqup/
- ]
- ]
-| #l #HG #HL #HT #f #H destruct -IH
- lapply (frees_inv_gref … H) -H /2 width=1 by/
-| #p #I #V #T #HG #HL #HT #f #H destruct
- elim (frees_inv_bind_void … H) -H /3 width=7 by/
-| #I #V #T #HG #HL #HT #f #H destruct
- elim (frees_inv_flat … H) -H /3 width=7 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/frees.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-(* Main inversion lemmas ****************************************************)
-
-theorem frees_mono: ∀f1,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f1 → ∀f2. L ⊢ 𝐅*⦃T⦄ ≘ f2 → f1 ≡ f2.
-#f1 #L #T #H elim H -f1 -L -T
-[ /3 width=3 by frees_inv_sort, isid_inv_eq_repl/
-| #f1 #i #Hf1 #g2 #H
- elim (frees_inv_atom … H) -H #f2 #Hf2 #H destruct
- /4 width=5 by isid_inv_eq_repl, pushs_eq_repl, eq_next/
-| #f1 #I #L #V #_ #IH #g2 #H elim (frees_inv_pair … H) -H
- #f2 #Hf2 #H destruct /3 width=5 by eq_next/
-| #f1 #I #L #Hf1 #g2 #H elim (frees_inv_unit … H) -H
- #f2 #Hf2 #H destruct /3 width=5 by isid_inv_eq_repl, eq_next/
-| #f1 #I #L #i #_ #IH #g2 #H elim (frees_inv_lref … H) -H
- #f2 #Hf2 #H destruct /3 width=5 by eq_push/
-| /3 width=3 by frees_inv_gref, isid_inv_eq_repl/
-| #f1V #f1T #f1 #p #I #L #V #T #_ #_ #Hf1 #IHV #IHT #f2 #H elim (frees_inv_bind … H) -H
- #f2V #f2T #HV #HT #Hf2 @(sor_mono … Hf1) -Hf1
- /5 width=3 by sor_eq_repl_fwd2, sor_eq_repl_fwd1, tl_eq_repl/ (**) (* full auto too slow *)
-| #f1V #f1T #f1 #I #L #V #T #_ #_ #Hf1 #IHV #IHT #f2 #H elim (frees_inv_flat … H) -H
- #f2V #f2T #HV #HT #Hf2 @(sor_mono … Hf1) -Hf1
- /4 width=3 by sor_eq_repl_fwd2, sor_eq_repl_fwd1/ (**) (* 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 "ground_2/relocation/rtmap_id.ma".
-include "basic_2/notation/relations/subseteq_4.ma".
-include "basic_2/syntax/lveq.ma".
-include "basic_2/static/frees.ma".
-
-(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
-
-definition fsle: bi_relation lenv term ≝ λL1,T1,L2,T2.
- ∃∃n1,n2,f1,f2. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 &
- L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
-
-interpretation "free variables inclusion (restricted closure)"
- 'SubSetEq L1 T1 L2 T2 = (fsle L1 T1 L2 T2).
-
-interpretation "free variables inclusion (term)"
- 'subseteq T1 T2 = (fsle LAtom T1 LAtom T2).
-
-(* Basic properties *********************************************************)
-
-lemma fsle_sort: ∀L,s1,s2. ⦃L, ⋆s1⦄ ⊆ ⦃L, ⋆s2⦄.
-/3 width=8 by frees_sort, sle_refl, ex4_4_intro/ qed.
-
-lemma fsle_gref: ∀L,l1,l2. ⦃L, §l1⦄ ⊆ ⦃L, §l2⦄.
-/3 width=8 by frees_gref, sle_refl, ex4_4_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_drops.ma".
-include "basic_2/static/fsle_length.ma".
-
-(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
-
-(* Advanced properties ******************************************************)
-
-lemma fsle_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≘ U1 → ∀L1,L2. |L2| ≤ |L1| →
- ∀T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ⦃L1.ⓧ, U1⦄ ⊆ ⦃L2, T2⦄.
-#T1 #U1 #HTU1 #L1 #L2 #H1L #T2
-* #n #m #f #g #Hf #Hg #H2L #Hfg
-lapply (lveq_length_fwd_dx … H2L ?) // -H1L #H destruct
-lapply (frees_lifts_SO (Ⓣ) (L1.ⓧ) … HTU1 … Hf)
-[ /3 width=4 by drops_refl, drops_drop/ ] -T1 #Hf
-@(ex4_4_intro … Hf Hg) /2 width=4 by lveq_void_sn/ (**) (* explict constructor *)
-qed-.
-
-lemma fsle_lifts_SO_sn: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ →
- ∀W1. ⬆*[1] V1 ≘ W1 → ∀I1,I2. ⦃K1.ⓘ{I1}, W1⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄.
-#K1 #K2 #HK #V1 #V2
-* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
-#W1 #HVW1 #I1 #I2
-elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
-/5 width=12 by frees_lifts_SO, frees_pair, drops_refl, drops_drop, lveq_bind, sle_weak, ex4_4_intro/
-qed.
-
-lemma fsle_lifts_SO: ∀K1,K2. |K1| = |K2| → ∀T1,T2. ⦃K1, T1⦄ ⊆ ⦃K2, T2⦄ →
- ∀U1,U2. ⬆*[1] T1 ≘ U1 → ⬆*[1] T2 ≘ U2 →
- ∀I1,I2. ⦃K1.ⓘ{I1}, U1⦄ ⊆ ⦃K2.ⓘ{I2}, U2⦄.
-#K1 #K2 #HK #T1 #T2
-* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
-#U1 #U2 #HTU1 #HTU2 #I1 #I2
-elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
-/5 width=12 by frees_lifts_SO, drops_refl, drops_drop, lveq_bind, sle_push, ex4_4_intro/
-qed.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fsle_inv_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≘ U1 →
- ∀I1,I2,L1,L2,V1,V2,U2. ⦃L1.ⓑ{I1}V1,U1⦄ ⊆ ⦃L2.ⓑ{I2}V2, U2⦄ →
- ∀p. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.U2⦄.
-#T1 #U1 #HTU1 #I1 #I2 #L1 #L2 #V1 #V2 #U2
-* #n #m #f2 #g2 #Hf2 #Hg2 #HL #Hfg2 #p
-elim (lveq_inv_pair_pair … HL) -HL #HL #H1 #H2 destruct
-elim (frees_total L2 V2) #g1 #Hg1
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
-lapply (frees_inv_lifts_SO (Ⓣ) … Hf2 … HTU1)
-[1,2: /3 width=4 by drops_refl, drops_drop/ ] -U1 #Hf2
-lapply (sor_inv_sle_dx … Hg) #H0g
-/5 width=10 by frees_bind, sle_tl, sle_trans, ex4_4_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/fsle_length.ma".
-
-(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
-
-(* Advanced properties ******************************************************)
-
-lemma fsle_refl: bi_reflexive … fsle.
-#L #T
-elim (frees_total L T) #f #Hf
-/2 width=8 by sle_refl, ex4_4_intro/
-qed.
-
-lemma fsle_shift: ∀L1,L2. |L1| = |L2| →
- ∀I,T1,T2,V. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I}V, T2⦄ →
- ∀p. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V.T2⦄.
-#L1 #L2 #H1L #I #T1 #T2 #V
-* #n #m #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
-elim (lveq_inj_length … H2L) // -H1L #H1 #H2 destruct
-lapply (lveq_inv_bind … H2L) -H2L #HL
-elim (frees_total L2 V) #g1 #Hg1
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
-lapply (sor_inv_sle_dx … Hg) #H0g
-/4 width=10 by frees_bind, lveq_void_sn, sle_tl, sle_trans, ex4_4_intro/
-qed.
-
-lemma fsle_bind_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀p,I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
-#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #p #I #T2
-elim (frees_total (L2.ⓧ) T2) #g2 #Hg2
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
-@(ex4_4_intro … g Hf1 … HL12) (**) (* full auto too slow *)
-/4 width=5 by frees_bind_void, sor_inv_sle_sn, sor_tls, sle_trans/
-qed.
-
-lemma fsle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2.ⓧ, T2⦄ → |L1| ≤ |L2| →
- ∀p,I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
-#L1 #L2 #T1 #T2 * #n1 #x1 #f2 #g2 #Hf2 #Hg2 #H #Hfg2 #HL12 #p #I #V2
-elim (lveq_inv_void_dx_length … H HL12) -H -HL12 #m1 #HL12 #H1 #H2 destruct
-<tls_xn in Hfg2; #Hfg2
-elim (frees_total L2 V2) #g1 #Hg1
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
-@(ex4_4_intro … g Hf2 … HL12) (**) (* full auto too slow *)
-/4 width=5 by frees_bind_void, sor_inv_sle_dx, sor_tls, sle_trans/
-qed.
-
-lemma fsle_flat_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
-#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #I #T2
-elim (frees_total L2 T2) #g2 #Hg2
-elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
-@(ex4_4_intro … g Hf1 … HL12) (**) (* full auto too slow *)
-/4 width=5 by frees_flat, sor_inv_sle_sn, sor_tls, sle_trans/
-qed.
-
-lemma fsle_flat_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
-#L1 #L2 #T1 #T2 * #n1 #m1 #f2 #g2 #Hf2 #Hg2 #HL12 #Hfg2 #I #V2
-elim (frees_total L2 V2) #g1 #Hg1
-elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
-@(ex4_4_intro … g Hf2 … HL12) (**) (* full auto too slow *)
-/4 width=5 by frees_flat, sor_inv_sle_dx, sor_tls, sle_trans/
-qed.
-
-(* Advanced forward lemmas ***************************************************)
-
-lemma fsle_fwd_pair_sn: ∀I1,I2,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄.
-#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 *
-#n1 #n2 #f1 #f2 #Hf1 #Hf2 #HL12 #Hf12
-elim (lveq_inv_pair_pair … HL12) -HL12 #HL12 #H1 #H2 destruct
-elim (frees_total (L1.ⓧ) T1) #g1 #Hg1
-lapply (lsubr_lsubf … Hg1 … Hf1) -Hf1 /2 width=1 by lsubr_unit/ #Hfg1
-/5 width=10 by lsubf_fwd_sle, lveq_bind, sle_trans, ex4_4_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/syntax/lveq_lveq.ma".
-include "basic_2/static/fsle_fqup.ma".
-
-(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma fsle_frees_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
- ∃∃n1,n2,f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 &
- L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
-#L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
-lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
-lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
-lapply (sle_eq_repl_back2 … Hn … Hgf2) -g2
-/2 width=6 by ex3_3_intro/
-qed-.
-
-lemma fsle_frees_trans_eq: ∀L1,L2. |L1| = |L2| →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
- ∃∃f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & f1 ⊆ f2.
-#L1 #L2 #H1L #T1 #T2 #H2L #f2 #Hf2
-elim (fsle_frees_trans … H2L … Hf2) -T2 #n1 #n2 #f1 #Hf1 #H2L #Hf12
-elim (lveq_inj_length … H2L) // -L2 #H1 #H2 destruct
-/2 width=3 by ex2_intro/
-qed-.
-
-lemma fsle_inv_frees_eq: ∀L1,L2. |L1| = |L2| →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
- f1 ⊆ f2.
-#L1 #L2 #H1L #T1 #T2 #H2L #f1 #Hf1 #f2 #Hf2
-elim (fsle_frees_trans_eq … H2L … Hf2) // -L2 -T2
-/3 width=6 by frees_mono, sle_eq_repl_back1/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem fsle_trans_sn: ∀L1,L2,T1,T. ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀T2. ⦃L2, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
-#L1 #L2 #T1 #T
-* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
-#T2
-* #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
-lapply (frees_mono … Hf0 … Hg0) -Hf0 -Hg0 #Hfg0
-elim (lveq_inj_length … Hn) // -Hn #H1 #H2 destruct
-lapply (sle_eq_repl_back1 … Hf … Hfg0) -f0
-/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
-qed-.
-
-theorem fsle_trans_dx: ∀L1,T1,T. ⦃L1, T1⦄ ⊆ ⦃L1, T⦄ →
- ∀L2,T2. ⦃L1, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
-#L1 #T1 #T
-* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
-#L2 #T2
-* #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
-lapply (frees_mono … Hg0 … Hf0) -Hg0 -Hf0 #Hgf0
-elim (lveq_inj_length … Hm) // -Hm #H1 #H2 destruct
-lapply (sle_eq_repl_back2 … Hg … Hgf0) -g0
-/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
-qed-.
-
-theorem fsle_trans_rc: ∀L1,L,T1,T. |L1| = |L| → ⦃L1, T1⦄ ⊆ ⦃L, T⦄ →
- ∀L2,T2. |L| = |L2| → ⦃L, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
-#L1 #L #T1 #T #HL1
-* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
-#L2 #T2 #HL2
-* #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
-lapply (frees_mono … Hg0 … Hf0) -Hg0 -Hf0 #Hgf0
-elim (lveq_inj_length … Hm) // -Hm #H1 #H2 destruct
-elim (lveq_inj_length … Hn) // -Hn #H1 #H2 destruct
-lapply (sle_eq_repl_back2 … Hg … Hgf0) -g0
-/3 width=10 by lveq_length_eq, sle_trans, ex4_4_intro/
-qed-.
-
-theorem fsle_bind_sn_ge: ∀L1,L2. |L2| ≤ |L1| →
- ∀V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀p,I. ⦃L1, ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2, T⦄.
-#L1 #L2 #HL #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #p #I
-elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
-elim (lveq_inj_void_sn_ge … H1n1 … H1n2) -H1n2 // #H1 #H2 #H3 destruct
-elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
-<tls_xn in H2n2; #H2n2
-/4 width=12 by frees_bind_void, sor_inv_sle, sor_tls, ex4_4_intro/
-qed.
-
-theorem fsle_flat_sn: ∀L1,L2,V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
- ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⊆ ⦃L2, T⦄.
-#L1 #L2 #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #I
-elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
-elim (lveq_inj … H1n1 … H1n2) -H1n2 #H1 #H2 destruct
-elim (sor_isfin_ex f1 f2) /2 width=3 by frees_fwd_isfin/ #f #Hf #_
-/4 width=12 by frees_flat, sor_inv_sle, sor_tls, ex4_4_intro/
-qed.
-
-theorem fsle_bind_eq: ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I2,T1,T2. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ∀p,I1. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
-#L1 #L2 #HL #V1 #V2
-* #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I2 #T1 #T2
-* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p #I1
-elim (lveq_inj_length … H1L) // #H1 #H2 destruct
-elim (lveq_inj_length … H2L) // -HL -H2L #H1 #H2 destruct
-elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
-/4 width=15 by frees_bind_void, frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
-qed.
-
-theorem fsle_bind: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
- ∀p. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
-#L1 #L2 #V1 #V2
-* #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I1 #I2 #T1 #T2
-* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
-elim (lveq_inv_pair_pair … H2L) -H2L #H2L #H1 #H2 destruct
-elim (lveq_inj … H2L … H1L) -H1L #H1 #H2 destruct
-elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
-elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
-/4 width=15 by frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
-qed.
-
-theorem fsle_flat: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
- ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
- ∀I1,I2. ⦃L1, ⓕ{I1}V1.T1⦄ ⊆ ⦃L2, ⓕ{I2}V2.T2⦄.
-/3 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_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/syntax/lveq_length.ma".
-include "basic_2/static/fsle.ma".
-
-(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
-
-(* Properties with length for local environments ****************************)
-
-lemma fsle_sort_bi: ∀L1,L2,s1,s2. |L1| = |L2| → ⦃L1, ⋆s1⦄ ⊆ ⦃L2, ⋆s2⦄.
-/3 width=8 by lveq_length_eq, frees_sort, sle_refl, ex4_4_intro/ qed.
-
-lemma fsle_gref_bi: ∀L1,L2,l1,l2. |L1| = |L2| → ⦃L1, §l1⦄ ⊆ ⦃L2, §l2⦄.
-/3 width=8 by lveq_length_eq, frees_gref, sle_refl, ex4_4_intro/ qed.
-
-lemma fsle_pair_bi: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ →
- ∀I1,I2. ⦃K1.ⓑ{I1}V1, #O⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄.
-#K1 #K2 #HK #V1 #V2
-* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
-#I1 #I2
-elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
-/3 width=12 by frees_pair, lveq_bind, sle_next, ex4_4_intro/
-qed.
-
-lemma fsle_unit_bi: ∀K1,K2. |K1| = |K2| →
- ∀I1,I2. ⦃K1.ⓤ{I1}, #O⦄ ⊆ ⦃K2.ⓤ{I2}, #O⦄.
-/3 width=8 by frees_unit, lveq_length_eq, sle_refl, ex4_4_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/syntax/genv.ma".
-include "basic_2/relocation/drops_vector.ma".
-
-(* GENERIC COMPUTATION PROPERTIES *******************************************)
-
-definition nf ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- λG,L,T. NF … (RR G L) RS T.
-
-definition candidate: Type[0] ≝ relation3 genv lenv term.
-
-definition CP0 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- ∀G. d_liftable1 (nf RR RS G).
-
-definition CP1 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
- ∀G,L. ∃s. NF … (RR G L) RS (⋆s).
-
-definition CP2 ≝ λRP:candidate. ∀G. d_liftable1 (RP G).
-
-definition CP3 ≝ λRP:candidate.
- ∀G,L,T,s. RP G L (ⓐ⋆s.T) → RP G L T.
-
-(* requirements for generic computation properties *)
-(* Basic_1: includes: nf2_lift1 *)
-(* Basic_2A1: includes: gcp0_lifts *)
-(* Basic_2A1: includes: gcp2_lifts *)
-record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate) : Prop ≝
-{ cp0: CP0 RR RS;
- cp1: CP1 RR RS;
- cp2: CP2 RP;
- cp3: CP3 RP
-}.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was only: sns3_lifts1 *)
-(* Basic_2A1: was: gcp2_lifts_all *)
-lemma gcp2_all: ∀RR,RS,RP. gcp RR RS RP → ∀G. d_liftable1_all (RP G).
-/3 width=7 by cp2, d1_liftable_liftable_all/ 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/aaa_aaa.ma".
-include "basic_2/static/lsubc_drops.ma".
-
-(* GENERIC COMPUTATION PROPERTIES *******************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: sc3_arity_csubc *)
-theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
- gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀b,f,L0. ⬇*[b, f] L0 ≘ L1 →
- ∀T0. ⬆*[f] T ≘ T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
- ⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
-#RR #RS #RP #H1RP #H2RP #G #L1 #T @(fqup_wf_ind_eq (Ⓣ) … G L1 T) -G -L1 -T
-#Z #Y #X #IH #G #L1 * [ * | * [ #p ] * ]
-[ #s #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct -IH
- lapply (aaa_inv_sort … HA) -HA #H destruct
- >(lifts_inv_sort1 … H0) -H0
- lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
- lapply (s4 … HAtom G L2 (Ⓔ)) /2 width=1 by/
-| #i #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
- elim (aaa_inv_lref_drops … HA) -HA #I #K1 #V1 #HLK1 #HKV1
- elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
- lapply (acr_gcr … H1RP H2RP A) #HA
- lapply (drops_trans … HL01 … HLK1 ??) -HL01 [3: |*: // ] #H
- elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK0 #HY
- lapply (drops_tls_at … Hf … HY) -Hf -HY #HY
- elim (drops_inv_skip2 … HY) -HY #Z #K0 #HK01 #HZ #H destruct
- elim (liftsb_inv_pair_sn … HZ) -HZ #V0 #HV10 #H destruct
- elim (lifts_total V0 (𝐔❴↑j❵)) #V #HV0
- elim (lsubc_drops_trans_isuni … HL20 … HLK0) -HL20 -HLK0 // #Y #HLK2 #H
- elim (lsubc_inv_bind2 … H) -H *
- [ #K2 #HK20 #H destruct
- lapply (drops_isuni_fwd_drop2 … HLK2) // #HLK2b
- lapply (s5 … HA ? G ? ? (Ⓔ) … HV0 ?) -HA
- /4 width=11 by acr_lifts, fqup_lref, drops_inv_gen/
- | #K2 #V2 #W2 #B #HKV2 #HK2V0 #HKV0B #_ #H1 #H2 destruct -IH -HLK1
- lapply (drops_isuni_fwd_drop2 … HLK2) // #HLK2b
- lapply (aaa_lifts … HKV1 … HK01 … HV10) -HKV1 -HK01 -HV10 #HKV0A
- lapply (aaa_mono … HKV0B … HKV0A) #H destruct -HKV0B -HKV0A
- elim (lifts_total V2 (𝐔❴↑j❵)) #V3 #HV23
- lapply (s5 … HA … G … (Ⓔ) … (ⓝW2.V2) (ⓝV.V3) ????)
- [3: |*: /2 width=9 by drops_inv_gen, lifts_flat/ ] -HLK2
- lapply (s7 … HA G L2 (Ⓔ)) -HA /3 width=7 by acr_lifts/
- ]
-| #l #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct -IH
- elim (aaa_inv_gref … HA)
-| #V #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
- elim (aaa_inv_abbr … HA) -HA #B #HV #HT
- elim (lifts_inv_bind1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
- lapply (acr_gcr … H1RP H2RP A) #HA
- lapply (acr_gcr … H1RP H2RP B) #HB
- lapply (s1 … HB) -HB #HB
- lapply (s6 … HA G L2 (Ⓔ) (Ⓔ)) /5 width=10 by lsubc_bind, liftsv_nil, drops_skip, ext2_pair/
-| #W #T #HG #HL #HT #Z0 #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
- elim (aaa_inv_abst … HA) -HA #B #A #HW #HT #H destruct
- elim (lifts_inv_bind1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
- @(acr_abst … H1RP H2RP) /2 width=10 by/
- #b3 #f3 #L3 #V3 #W3 #T3 #HL32 #HW03 #HT03 #H1B #H2B
- elim (drops_lsubc_trans … H1RP … HL32 … HL20) -L2 #L2 #HL32 #HL20
- lapply (aaa_lifts … HW … (f3∘f) L2 … W3 ?) -HW
- [4: |*: /2 width=8 by drops_trans, lifts_trans/ ] #HW3
- @(IH … ((⫯f3)∘⫯f) … (L2. ⓛW3)) -IH
- /4 width=12 by lsubc_beta, drops_trans, drops_skip, lifts_trans, ext2_pair/
-| #V #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
- elim (aaa_inv_appl … HA) -HA #B #HV #HT
- elim (lifts_inv_flat1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
- lapply (IH … HT … HL01 … HT0 … HL20) -HT -HT0
- /3 width=10 by drops_refl, lifts_refl/
-| #W #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
- elim (aaa_inv_cast … HA) -HA #HW #HT
- elim (lifts_inv_flat1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
- lapply (acr_gcr … H1RP H2RP A) #HA
- lapply (s7 … HA G L2 (Ⓔ)) /3 width=10 by/
-]
-qed.
-
-(* Basic_1: was: sc3_arity *)
-lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
-/3 width=9 by drops_refl, lifts_refl, acr_aaa_csubc_lifts/ qed.
-
-lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
-#RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
-lapply (acr_gcr … H1RP H2RP A) #HA
-@(s1 … HA) /2 width=4 by acr_aaa/
-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/ineint_5.ma".
-include "basic_2/syntax/aarity.ma".
-include "basic_2/relocation/lifts_simple.ma".
-include "basic_2/relocation/lifts_lifts_vector.ma".
-include "basic_2/relocation/drops_drops.ma".
-include "basic_2/static/gcp.ma".
-
-(* GENERIC COMPUTATION PROPERTIES *******************************************)
-
-(* Note: this is Girard's CR1 *)
-definition S1 ≝ λRP,C:candidate.
- ∀G,L,T. C G L T → RP G L T.
-
-(* Note: this is Tait's iii, or Girard's CR4 *)
-definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
- ∀G,L,Vs. all … (RP G L) Vs →
- ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
-
-(* Note: this generalizes Tait's ii *)
-definition S3 ≝ λC:candidate.
- ∀a,G,L,Vs,V,T,W.
- C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
-
-definition S4 ≝ λRP,C:candidate.
- ∀G,L,Vs. all … (RP G L) Vs → ∀s. C G L (ⒶVs.⋆s).
-
-definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
- C G L (ⒶVs.V2) → ⬆*[↑i] V1 ≘ V2 →
- ⬇*[i] L ≘ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
-
-definition S6 ≝ λRP,C:candidate.
- ∀G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
- ∀a,V,T. C G (L.ⓓV) (ⒶV2b.T) → RP G L V → C G L (ⒶV1b.ⓓ{a}V.T).
-
-definition S7 ≝ λC:candidate.
- ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
-
-(* requirements for the generic reducibility candidate *)
-record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
-{ s1: S1 RP C;
- s2: S2 RR RS RP C;
- s3: S3 C;
- s4: S4 RP C;
- s5: S5 C;
- s6: S6 RP C;
- s7: S7 C
-}.
-
-(* the functional construction for candidates *)
-definition cfun: candidate → candidate → candidate ≝
- λC1,C2,G,K,T. ∀f,L,W,U.
- ⬇*[Ⓕ, f] L ≘ K → ⬆*[f] T ≘ U → C1 G L W → C2 G L (ⓐW.U).
-
-(* the reducibility candidate associated to an atomic arity *)
-rec definition acr (RP:candidate) (A:aarity) on A: candidate ≝
-match A with
-[ AAtom ⇒ RP
-| APair B A ⇒ cfun (acr RP B) (acr RP A)
-].
-
-interpretation
- "reducibility candidate of an atomic arity (abstract)"
- 'InEInt RP G L T A = (acr RP A G L T).
-
-(* Basic properties *********************************************************)
-
-(* Note: this requires Ⓕ-slicing in cfun since b is unknown in d_liftable_1 *)
-(* Note: this requires multiple relocation *)
-(* Basic 1: includes: sc3_lift *)
-(* Basic 2A1: includes: gcr_lift *)
-(* Basic 2A1: note: gcr_lift should be acr_lift *)
-(* Basic_1: was: sc3_lift1 *)
-(* Basic 2A1: was: gcr_lifts *)
-(* Basic 2A1: note: gcr_lifts should be acr_lifts *)
-lemma acr_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftable1 (acr RP A G).
-#RR #RS #RP #H #A #G elim A -A
-[ /2 width=7 by cp2/
-| #B #A #HB #HA #K #T #HKT #b #f #L #HLK #U #HTU #f0 #L0 #W #U0 #HL0 #HU0 #HW
- lapply (drops_trans … HL0 … HLK ??) [3:|*: // ] -L #HL0K
- lapply (lifts_trans … HTU … HU0 ??) [3:|*: // ] -U #HTU0
- /2 width=3 by/ (**) (* full auto fails *)
-]
-qed-.
-
-(* Basic_1: was:
- sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast
-*)
-lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀A. gcr RR RS RP (acr RP A).
-#RR #RS #RP #H1RP #H2RP #A elim A -A //
-#B #A #IHB #IHA @mk_gcr
-[ #G #L #T #H
- elim (cp1 … H1RP G L) #s #HK
- lapply (s2 … IHB G L (Ⓔ) … HK) // #HB
- lapply (H (𝐈𝐝) L (⋆s) T ? ? ?) -H
- /3 width=6 by s1, cp3, drops_refl, lifts_refl/
-| #G #L #Vs #HVs #T #H1T #H2T #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
- lapply (s1 … IHB … HB) #HV0
- @(s2 … IHA … (V0⨮V0s)) /3 width=13 by cp0, gcp2_all, lifts_simple_dx, conj/
-| #p #G #L #Vs #U #T #W #HA #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
- elim (lifts_inv_flat1 … H0) -H0 #U0 #X #HU0 #HX #H destruct
- elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
- @(s3 … IHA … (V0⨮V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
-| #G #L #Vs #HVs #s #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
- >(lifts_inv_sort1 … H0) -X0
- lapply (s1 … IHB … HB) #HV0
- @(s4 … IHA … (V0⨮V0s)) /3 width=7 by gcp2_all, conj/
-| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
- elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
- lapply (drops_trans … HL0 … HLK ??) [3: |*: // ] -HLK #H
- elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK0 #HY
- lapply (drops_tls_at … Hf … HY) -HY #HY
- elim (drops_inv_skip2 … HY) -HY #Z #K0 #HK0 #HZ #H destruct
- elim (liftsb_inv_pair_sn … HZ) -HZ #W1 #HVW1 #H destruct
- elim (lifts_total W1 (𝐔❴↑j❵)) #W2 #HW12
- lapply (lifts_trans … HVW1 … HW12 ??) -HVW1 [3: |*: // ] #H
- lapply (lifts_conf … HV12 … H f ?) -V1 [ /2 width=3 by after_uni_succ_sn/ ] #HVW2
- @(s5 … IHA … (V0⨮V0s) … HW12) /3 width=4 by drops_inv_gen, lifts_applv/
-| #G #L #V1s #V2s #HV12s #p #V #T #HA #HV #f #L0 #V10 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V10s #X0 #HV10s #H0 #H destruct
- elim (lifts_inv_bind1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
- elim (lifts_total V10 (𝐔❴1❵)) #V20 #HV120
- elim (liftsv_total (𝐔❴1❵) V10s) #V20s #HV120s
- @(s6 … IHA … (V10⨮V10s) (V20⨮V20s)) /3 width=7 by cp2, liftsv_cons/
- @(HA … (⫯f)) /3 width=2 by drops_skip, ext2_pair/
- [ @lifts_applv //
- lapply (liftsv_trans … HV10s … HV120s ??) -V10s [3: |*: // ] #H
- elim (liftsv_split_trans … H (𝐔❴1❵) (⫯f)) /2 width=1 by after_uni_one_sn/ #V10s #HV10s #HV120s
- >(liftsv_mono … HV12s … HV10s) -V1s //
- | @(acr_lifts … H1RP … HB … HV120) /3 width=2 by drops_refl, drops_drop/
- ]
-| #G #L #Vs #T #W #HA #HW #f #L0 #V0 #X #HL0 #H #HB
- elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
- elim (lifts_inv_flat1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
- @(s7 … IHA … (V0⨮V0s)) /3 width=5 by lifts_applv/
-]
-qed.
-
-lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀p,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
- ∀b,f,L0,V0,W0,T0. ⬇*[b, f] L0 ≘ L → ⬆*[f] W ≘ W0 → ⬆*[⫯f] T ≘ T0 →
- ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
- ) →
- ⦃G, L, ⓛ{p}W.T⦄ ϵ[RP] 〚②B.A〛.
-#RR #RS #RP #H1RP #H2RP #p #G #L #W #T #A #B #HW #HA #f #L0 #V0 #X #HL0 #H #HB
-lapply (acr_gcr … H1RP H2RP A) #HCA
-lapply (acr_gcr … H1RP H2RP B) #HCB
-elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
-lapply (acr_lifts … H1RP … HW … HL0 … HW0) -HW #HW0
-lapply (s3 … HCA … p G L0 (Ⓔ)) #H @H -H
-lapply (s6 … HCA G L0 (Ⓔ) (Ⓔ) ?) // #H @H -H
-[ @(HA … HL0) //
-| lapply (s1 … HCB) -HCB #HCB
- lapply (s7 … H2RP G L0 (Ⓔ)) /3 width=1 by/
-]
-qed.
-
-(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
-(* Basic_1: removed local theorems 1: sc3_sn3_abst *)
+++ /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/lrsubeqa_3.ma".
-include "basic_2/static/aaa.ma".
-
-(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
-
-inductive lsuba (G:genv): relation lenv ≝
-| lsuba_atom: lsuba G (⋆) (⋆)
-| lsuba_bind: ∀I,L1,L2. lsuba G L1 L2 → lsuba G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsuba_beta: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
- lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
-.
-
-interpretation
- "local environment refinement (atomic arity assignment)"
- 'LRSubEqA G L1 L2 = (lsuba G L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsuba_inv_atom1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 = ⋆ → L2 = ⋆.
-#G #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #_ #H destruct
-| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
-]
-qed-.
-
-lemma lsuba_inv_atom1: ∀G,L2. G ⊢ ⋆ ⫃⁝ L2 → L2 = ⋆.
-/2 width=4 by lsuba_inv_atom1_aux/ qed-.
-
-fact lsuba_inv_bind1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
- (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⫃⁝ K2 & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
-#G #L1 #L2 * -L1 -L2
-[ #J #K1 #H destruct
-| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #H destruct /3 width=9 by ex5_4_intro, or_intror/
-]
-qed-.
-
-lemma lsuba_inv_bind1: ∀I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃⁝ L2 →
- (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
- I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
-/2 width=3 by lsuba_inv_bind1_aux/ qed-.
-
-fact lsuba_inv_atom2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L2 = ⋆ → L1 = ⋆.
-#G #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #_ #H destruct
-| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubc_inv_atom2: ∀G,L1. G ⊢ L1 ⫃⁝ ⋆ → L1 = ⋆.
-/2 width=4 by lsuba_inv_atom2_aux/ qed-.
-
-fact lsuba_inv_bind2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
- (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
- ∃∃K1,V,W, A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⫃⁝ K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
-#G #L1 #L2 * -L1 -L2
-[ #J #K2 #H destruct
-| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #H destruct /3 width=9 by ex5_4_intro, or_intror/
-]
-qed-.
-
-lemma lsuba_inv_bind2: ∀I,G,L1,K2. G ⊢ L1 ⫃⁝ K2.ⓘ{I} →
- (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
- I = BPair Abst W & L1 = K1.ⓓⓝW.V.
-/2 width=3 by lsuba_inv_bind2_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-lemma lsuba_refl: ∀G,L. G ⊢ L ⫃⁝ L.
-#G #L elim L -L /2 width=1 by lsuba_atom, lsuba_bind/
-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/aaa_aaa.ma".
-include "basic_2/static/lsuba.ma".
-
-(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
-
-(* Properties with atomic arity assignment **********************************)
-
-lemma lsuba_aaa_conf: ∀G,L1,V,A. ⦃G, L1⦄ ⊢ V ⁝ A →
- ∀L2. G ⊢ L1 ⫃⁝ L2 → ⦃G, L2⦄ ⊢ V ⁝ A.
-#G #L1 #V #A #H elim H -G -L1 -V -A
-[ //
-| #I #G #L1 #V #A #HA #IH #L2 #H
- elim (lsuba_inv_bind1 … H) -H * /3 width=1 by aaa_zero/
- #L0 #W0 #V0 #A0 #HV0 #HW0 #HL10 #H1 #H2 destruct
- lapply (aaa_mono … HV0 … HA) #H destruct -V0 -L1 /2 width=1 by aaa_zero/
-| #I #G #K1 #A #i #_ #IH #L2 #H
- elim (lsuba_inv_bind1 … H) -H * /3 width=1 by aaa_lref/
-| /4 width=2 by lsuba_bind, aaa_abbr/
-| /4 width=1 by lsuba_bind, aaa_abst/
-| /3 width=3 by aaa_appl/
-| /3 width=1 by aaa_cast/
-]
-qed-.
-
-lemma lsuba_aaa_trans: ∀G,L2,V,A. ⦃G, L2⦄ ⊢ V ⁝ A →
- ∀L1. G ⊢ L1 ⫃⁝ L2 → ⦃G, L1⦄ ⊢ V ⁝ A.
-#G #L2 #V #A #H elim H -G -L2 -V -A
-[ //
-| #I #G #L2 #V #A #HA #IH #L1 #H
- elim (lsuba_inv_bind2 … H) -H * /3 width=1 by aaa_zero/
- #L0 #V0 #W0 #A0 #HV0 #HW0 #HL02 #H1 #H2 destruct
- lapply (aaa_mono … HW0 … HA) #H destruct -L2 /2 width=1 by aaa_zero/
-| #I #G #K2 #A #i #_ #IH #L1 #H
- elim (lsuba_inv_bind2 … H) -H * /3 width=1 by aaa_lref/
-| /4 width=2 by lsuba_bind, aaa_abbr/
-| /4 width=1 by lsuba_bind, aaa_abst/
-| /3 width=3 by aaa_appl/
-| /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/relocation/drops.ma".
-include "basic_2/static/lsuba.ma".
-
-(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
-
-(* Properties with generic slicing for local environments *******************)
-
-(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
-(* Basic_2A1: includes: lsuba_drop_O1_conf *)
-lemma lsuba_drops_conf_isuni: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 →
- ∀b,f,K1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K1 →
- ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇*[b, f] L2 ≘ K2.
-#G #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #HL12 #IH #b #f #K1 #Hf #H
- elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
- [ #Hf #H destruct -IH
- /3 width=3 by lsuba_bind, drops_refl, ex2_intro/
- | #g #Hg #HLK1 #H destruct -HL12
- elim (IH … Hg HLK1) -L1 -Hg /3 width=3 by drops_drop, ex2_intro/
- ]
-| #L1 #L2 #W #V #A #HV #HW #HL12 #IH #b #f #K1 #Hf #H
- elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
- [ #Hf #H destruct -IH
- /3 width=3 by drops_refl, lsuba_beta, ex2_intro/
- | #g #Hg #HLK1 #H destruct -HL12
- elim (IH … Hg HLK1) -L1 -Hg /3 width=3 by drops_drop, ex2_intro/
- ]
-]
-qed-.
-
-(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
-(* Basic_2A1: includes: lsuba_drop_O1_trans *)
-lemma lsuba_drops_trans_isuni: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 →
- ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2 →
- ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇*[b, f] L1 ≘ K1.
-#G #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #HL12 #IH #b #f #K2 #Hf #H
- elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
- [ #Hf #H destruct -IH
- /3 width=3 by lsuba_bind, drops_refl, ex2_intro/
- | #g #Hg #HLK2 #H destruct -HL12
- elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
- ]
-| #L1 #L2 #W #V #A #HV #HW #HL12 #IH #b #f #K2 #Hf #H
- elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
- [ #Hf #H destruct -IH
- /3 width=3 by drops_refl, lsuba_beta, ex2_intro/
- | #g #Hg #HLK2 #H destruct -HL12
- elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, 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/lsuba_aaa.ma".
-
-(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
-
-(* Main properties **********************************************************)
-
-theorem lsuba_trans: ∀G. Transitive … (lsuba G).
-#G #L1 #L #H elim H -L1 -L
-[ #X #H >(lsuba_inv_atom1 … H) -H //
-| #I #L1 #L #HL1 #IHL1 #Y #H
- elim (lsuba_inv_bind1 … H) -H * #L2
- [ #HL2 #H destruct /3 width=1 by lsuba_bind/
- | #W #V #A #HV #HW #HL2 #H1 #H2 destruct
- /3 width=3 by lsuba_beta, lsuba_aaa_trans/
- ]
-| #L1 #L #W #V #A #HV #HW #HL1 #IHL1 #Y #H
- elim (lsuba_inv_bind1 … H) -H * #L2
- [ #HL2 #H destruct /3 width=5 by lsuba_beta, lsuba_aaa_conf/
- | #W0 #V0 #A0 #_ #_ #_ #H destruct
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/lsubr.ma".
-include "basic_2/static/lsuba.ma".
-
-(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
-
-(* Forward lemmas with restricted refinement for local environments *********)
-
-lemma lsuba_fwd_lsubr: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 ⫃ L2.
-#G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_beta/
-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/lrsubeqc_4.ma".
-include "basic_2/static/aaa.ma".
-include "basic_2/static/gcp_cr.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
-
-inductive lsubc (RP) (G): relation lenv ≝
-| lsubc_atom: lsubc RP G (⋆) (⋆)
-| lsubc_bind: ∀I,L1,L2. lsubc RP G L1 L2 → lsubc RP G (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubc_beta: ∀L1,L2,V,W,A. ⦃G, L1, V⦄ ϵ[RP] 〚A〛 → ⦃G, L1, W⦄ ϵ[RP] 〚A〛 → ⦃G, L2⦄ ⊢ W ⁝ A →
- lsubc RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
-.
-
-interpretation
- "local environment refinement (generic reducibility)"
- 'LRSubEqC RP G L1 L2 = (lsubc RP G L1 L2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubc_inv_atom1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 = ⋆ → L2 = ⋆.
-#RP #G #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-(* Basic_1: was just: csubc_gen_sort_r *)
-lemma lsubc_inv_atom1: ∀RP,G,L2. G ⊢ ⋆ ⫃[RP] L2 → L2 = ⋆.
-/2 width=5 by lsubc_inv_atom1_aux/ qed-.
-
-fact lsubc_inv_bind1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1. L1 = K1.ⓘ{I} →
- (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⫃[RP] K2 &
- L2 = K2. ⓛW & I = BPair Abbr (ⓝW.V).
-#RP #G #L1 #L2 * -L1 -L2
-[ #I #K1 #H destruct
-| #J #L1 #L2 #HL12 #I #K1 #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K1 #H destruct
- /3 width=10 by ex6_4_intro, or_intror/
-]
-qed-.
-
-(* Basic_1: was: csubc_gen_head_r *)
-lemma lsubc_inv_bind1: ∀RP,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃[RP] L2 →
- (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓘ{I}) ∨
- ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⫃[RP] K2 &
- L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V).
-/2 width=3 by lsubc_inv_bind1_aux/ qed-.
-
-fact lsubc_inv_atom2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L2 = ⋆ → L1 = ⋆.
-#RP #G #L1 #L2 * -L1 -L2
-[ //
-| #I #L1 #L2 #_ #H destruct
-| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-(* Basic_1: was just: csubc_gen_sort_l *)
-lemma lsubc_inv_atom2: ∀RP,G,L1. G ⊢ L1 ⫃[RP] ⋆ → L1 = ⋆.
-/2 width=5 by lsubc_inv_atom2_aux/ qed-.
-
-fact lsubc_inv_bind2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2. L2 = K2.ⓘ{I} →
- (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1. ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⫃[RP] K2 &
- L1 = K1.ⓓⓝW.V & I = BPair Abst W.
-#RP #G #L1 #L2 * -L1 -L2
-[ #I #K2 #H destruct
-| #J #L1 #L2 #HL12 #I #K2 #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K2 #H destruct
- /3 width=10 by ex6_4_intro, or_intror/
-]
-qed-.
-
-(* Basic_1: was just: csubc_gen_head_l *)
-lemma lsubc_inv_bind2: ∀RP,I,G,L1,K2. G ⊢ L1 ⫃[RP] K2.ⓘ{I} →
- (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1.ⓘ{I}) ∨
- ∃∃K1,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
- G ⊢ K1 ⫃[RP] K2 &
- L1 = K1.ⓓⓝW.V & I = BPair Abst W.
-/2 width=3 by lsubc_inv_bind2_aux/ qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was just: csubc_refl *)
-lemma lsubc_refl: ∀RP,G,L. G ⊢ L ⫃[RP] L.
-#RP #G #L elim L -L /2 width=1 by lsubc_bind/
-qed.
-
-(* Basic_1: removed theorems 3:
- csubc_clear_conf csubc_getl_conf csubc_csuba
-*)
+++ /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/aaa_drops.ma".
-include "basic_2/static/lsubc.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
-
-(* Properties with generic slicing ******************************************)
-
-(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
-(* Basic_1: includes: csubc_drop_conf_O *)
-(* Basic_2A1: includes: lsubc_drop_O1_trans *)
-lemma lsubc_drops_trans_isuni: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 →
- ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2 →
- ∃∃K1. ⬇*[b, f] L1 ≘ K1 & G ⊢ K1 ⫃[RP] K2.
-#RP #G #L1 #L2 #H elim H -L1 -L2
-[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #HL12 #IH #b #f #K2 #Hf #H
- elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
- [ #Hf #H destruct -IH
- /3 width=3 by lsubc_bind, drops_refl, ex2_intro/
- | #g #Hg #HLK2 #H destruct -HL12
- elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
- ]
-| #L1 #L2 #V #W #A #HV #H1W #H2W #HL12 #IH #b #f #K2 #Hf #H
- elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
- [ #Hf #H destruct -IH
- /3 width=8 by drops_refl, lsubc_beta, ex2_intro/
- | #g #Hg #HLK2 #H destruct -HL12
- elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
- ]
-]
-qed-.
-
-(* Basic_1: was: csubc_drop1_conf_rev *)
-(* Basic_1: includes: csubc_drop_conf_rev *)
-(* Basic_2A1: includes: drop_lsubc_trans *)
-lemma drops_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
- ∀b,f,G,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
- ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇*[b, f] L2 ≘ K2.
-#RR #RS #RP #HR #b #f #G #L1 #K1 #H elim H -f -L1 -K1
-[ #f #Hf #Y #H lapply (lsubc_inv_atom1 … H) -H
- #H destruct /4 width=3 by lsubc_atom, drops_atom, ex2_intro/
-| #f #I #L1 #K1 #_ #IH #K2 #HK12 elim (IH … HK12) -K1
- /3 width=5 by lsubc_bind, drops_drop, ex2_intro/
-| #f #Z #I #L1 #K1 #HLK1 #HZ #IH #Y #H elim (lsubc_inv_bind1 … H) -H *
- [ #K2 #HK12 #H destruct -HLK1
- elim (IH … HK12) -K1 /3 width=5 by lsubc_bind, drops_skip, ex2_intro/
- | #K2 #V2 #W2 #A #HV2 #H1W2 #H2W2 #HK12 #H1 #H2 destruct
- elim (liftsb_inv_pair_sn … HZ) -HZ #V1 #HV21 #H destruct
- elim (lifts_inv_flat1 … HV21) -HV21 #W3 #V3 #HW23 #HV3 #H destruct
- elim (IH … HK12) -IH -HK12 #K #HL1K #HK2
- lapply (acr_lifts … HR … HV2 … HLK1 … HV3) -HV2
- lapply (acr_lifts … HR … H1W2 … HLK1 … HW23) -H1W2
- /4 width=10 by lsubc_beta, aaa_lifts, drops_skip, ext2_pair, ex2_intro/
- ]
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/static/lsuba.ma".
-include "basic_2/static/gcp_aaa.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
-
-(* Properties with lenv refinement for atomic arity assignment **************)
-
-lemma lsuba_lsubc: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
- ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → G ⊢ L1 ⫃[RP] L2.
-#RR #RS #RP #H1RP #H2RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubc_bind/
-#L1 #L2 #W #V #A #H elim (aaa_inv_cast … H) -H /3 width=4 by acr_aaa, lsubc_beta/
-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/lsubr.ma".
-include "basic_2/static/lsubc.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
-
-(* Forward lemmas with restricted refinement for local environments *********)
-
-lemma lsubc_fwd_lsubr: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 ⫃ L2.
-#RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_beta/
-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/lrsubeqf_4.ma".
-include "ground_2/relocation/nstream_sor.ma".
-include "basic_2/static/frees.ma".
-
-(* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
-
-inductive lsubf: relation4 lenv rtmap lenv rtmap ≝
-| lsubf_atom: ∀f1,f2. f1 ≡ f2 → lsubf (⋆) f1 (⋆) f2
-| lsubf_push: ∀f1,f2,I1,I2,L1,L2. lsubf L1 (f1) L2 (f2) →
- lsubf (L1.ⓘ{I1}) (⫯f1) (L2.ⓘ{I2}) (⫯f2)
-| lsubf_bind: ∀f1,f2,I,L1,L2. lsubf L1 f1 L2 f2 →
- lsubf (L1.ⓘ{I}) (↑f1) (L2.ⓘ{I}) (↑f2)
-| lsubf_beta: ∀f,f0,f1,f2,L1,L2,W,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
- lsubf L1 f0 L2 f2 → lsubf (L1.ⓓⓝW.V) (↑f1) (L2.ⓛW) (↑f2)
-| lsubf_unit: ∀f,f0,f1,f2,I1,I2,L1,L2,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
- lsubf L1 f0 L2 f2 → lsubf (L1.ⓑ{I1}V) (↑f1) (L2.ⓤ{I2}) (↑f2)
-.
-
-interpretation
- "local environment refinement (context-sensitive free variables)"
- 'LRSubEqF L1 f1 L2 f2 = (lsubf L1 f1 L2 f2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 = ⋆ →
- f1 ≡ f2 ∧ L2 = ⋆.
-#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ /2 width=1 by conj/
-| #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
-| #f1 #f2 #I #L1 #L2 #_ #H destruct
-| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
-| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f1 ≡ f2 ∧ L2 = ⋆.
-/2 width=3 by lsubf_inv_atom1_aux/ qed-.
-
-fact lsubf_inv_push1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
- ∃∃g2,I2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
-#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ #f1 #f2 #_ #g1 #J1 #K1 #_ #H destruct
-| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J1 #K1 #H1 #H2 destruct
- <(injective_push … H1) -g1 /2 width=6 by ex3_3_intro/
-| #f1 #f2 #I #L1 #L2 #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
-| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
-| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
-]
-qed-.
-
-lemma lsubf_inv_push1: ∀g1,f2,I1,K1,L2. ⦃K1.ⓘ{I1}, ⫯g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∃∃g2,I2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
-/2 width=6 by lsubf_inv_push1_aux/ qed-.
-
-fact lsubf_inv_pair1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ{I}X →
- ∨∨ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
- | ∃∃g,g0,g2,K2,W,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
- I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
- | ∃∃g,g0,g2,J,K2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
- L2 = K2.ⓤ{J}.
-#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ #f1 #f2 #_ #g1 #J #K1 #X #_ #H destruct
-| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #X #H elim (discr_push_next … H)
-| #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #X #H1 #H2 destruct
- <(injective_next … H1) -g1 /3 width=5 by or3_intro0, ex3_2_intro/
-| #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
- <(injective_next … H1) -g1 /3 width=12 by or3_intro1, ex7_6_intro/
-| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
- <(injective_next … H1) -g1 /3 width=10 by or3_intro2, ex5_5_intro/
-]
-qed-.
-
-lemma lsubf_inv_pair1: ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∨∨ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
- | ∃∃g,g0,g2,K2,W,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
- I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
- | ∃∃g,g0,g2,J,K2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
- L2 = K2.ⓤ{J}.
-/2 width=5 by lsubf_inv_pair1_aux/ qed-.
-
-fact lsubf_inv_unit1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
- ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
-#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ #f1 #f2 #_ #g1 #J #K1 #_ #H destruct
-| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #H elim (discr_push_next … H)
-| #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #H1 #H2 destruct
- <(injective_next … H1) -g1 /2 width=5 by ex3_2_intro/
-| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
-| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
-]
-qed-.
-
-lemma lsubf_inv_unit1: ∀g1,f2,I,K1,L2. ⦃K1.ⓤ{I}, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
-/2 width=5 by lsubf_inv_unit1_aux/ qed-.
-
-fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L2 = ⋆ →
- f1 ≡ f2 ∧ L1 = ⋆.
-#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ /2 width=1 by conj/
-| #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
-| #f1 #f2 #I #L1 #L2 #_ #H destruct
-| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
-| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → f1 ≡ f2 ∧ L1 = ⋆.
-/2 width=3 by lsubf_inv_atom2_aux/ qed-.
-
-fact lsubf_inv_push2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
- ∃∃g1,I1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
-#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ #f1 #f2 #_ #g2 #J2 #K2 #_ #H destruct
-| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J2 #K2 #H1 #H2 destruct
- <(injective_push … H1) -g2 /2 width=6 by ex3_3_intro/
-| #f1 #f2 #I #L1 #L2 #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
-| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
-| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
-]
-qed-.
-
-lemma lsubf_inv_push2: ∀f1,g2,I2,L1,K2. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓘ{I2}, ⫯g2⦄ →
- ∃∃g1,I1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
-/2 width=6 by lsubf_inv_push2_aux/ qed-.
-
-fact lsubf_inv_pair2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ{I}W →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
- | ∃∃g,g0,g1,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
- I = Abst & L1 = K1.ⓓⓝW.V.
-#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ #f1 #f2 #_ #g2 #J #K2 #X #_ #H destruct
-| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #X #H elim (discr_push_next … H)
-| #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #X #H1 #H2 destruct
- <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
-| #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g2 #J #K2 #X #H1 #H2 destruct
- <(injective_next … H1) -g2 /3 width=10 by ex6_5_intro, or_intror/
-| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J #K2 #X #_ #H destruct
-]
-qed-.
-
-lemma lsubf_inv_pair2: ∀f1,g2,I,L1,K2,W. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓑ{I}W, ↑g2⦄ →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
- | ∃∃g,g0,g1,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
- I = Abst & L1 = K1.ⓓⓝW.V.
-/2 width=5 by lsubf_inv_pair2_aux/ qed-.
-
-fact lsubf_inv_unit2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ{I} →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
- | ∃∃g,g0,g1,J,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
- L1 = K1.ⓑ{J}V.
-#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
-[ #f1 #f2 #_ #g2 #J #K2 #_ #H destruct
-| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #H elim (discr_push_next … H)
-| #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #H1 #H2 destruct
- <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
-| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J #K2 #_ #H destruct
-| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g2 #J #K2 #H1 #H2 destruct
- <(injective_next … H1) -g2 /3 width=11 by ex5_6_intro, or_intror/
-]
-qed-.
-
-lemma lsubf_inv_unit2: ∀f1,g2,I,L1,K2. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓤ{I}, ↑g2⦄ →
- ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
- | ∃∃g,g0,g1,J,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
- K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
- L1 = K1.ⓑ{J}V.
-/2 width=5 by lsubf_inv_unit2_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lsubf_inv_atom: ∀f1,f2. ⦃⋆, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → f1 ≡ f2.
-#f1 #f2 #H elim (lsubf_inv_atom1 … H) -H //
-qed-.
-
-lemma lsubf_inv_push_sn: ∀g1,f2,I1,I2,K1,K2. ⦃K1.ⓘ{I1}, ⫯g1⦄ ⫃𝐅* ⦃K2.ⓘ{I2}, f2⦄ →
- ∃∃g2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2.
-#g1 #f2 #I #K1 #K2 #X #H elim (lsubf_inv_push1 … H) -H
-#g2 #I #Y #H0 #H2 #H destruct /2 width=3 by ex2_intro/
-qed-.
-
-lemma lsubf_inv_bind_sn: ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I}, ↑g1⦄ ⫃𝐅* ⦃K2.ⓘ{I}, f2⦄ →
- ∃∃g2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2.
-#g1 #f2 * #I [2: #X ] #K1 #K2 #H
-[ elim (lsubf_inv_pair1 … H) -H *
- [ #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
- | #z #z0 #z2 #Y2 #W #V #_ #_ #_ #_ #H0 #_ #H destruct
- | #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
- ]
-| elim (lsubf_inv_unit1 … H) -H
- #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma lsubf_inv_beta_sn: ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V, ↑g1⦄ ⫃𝐅* ⦃K2.ⓛW, f2⦄ →
- ∃∃g,g0,g2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
-#g1 #f2 #K1 #K2 #V #W #H elim (lsubf_inv_pair1 … H) -H *
-[ #z2 #Y2 #_ #_ #H destruct
-| #z #z0 #z2 #Y2 #X0 #X #H02 #Hz #Hg1 #H #_ #H0 #H1 destruct
- /2 width=7 by ex4_3_intro/
-| #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubf_inv_unit_sn: ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V, ↑g1⦄ ⫃𝐅* ⦃K2.ⓤ{J}, f2⦄ →
- ∃∃g,g0,g2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
-#g1 #f2 #I #J #K1 #K2 #V #H elim (lsubf_inv_pair1 … H) -H *
-[ #z2 #Y2 #_ #_ #H destruct
-| #z #z0 #z2 #Y2 #X0 #X #_ #_ #_ #_ #_ #_ #H destruct
-| #z #z0 #z2 #Z2 #Y2 #H02 #Hz #Hg1 #H0 #H1 destruct
- /2 width=7 by ex4_3_intro/
-]
-qed-.
-
-lemma lsubf_inv_refl: ∀L,f1,f2. ⦃L,f1⦄ ⫃𝐅* ⦃L,f2⦄ → f1 ≡ f2.
-#L elim L -L /2 width=1 by lsubf_inv_atom/
-#L #I #IH #f1 #f2 #H12
-elim (pn_split f1) * #g1 #H destruct
-[ elim (lsubf_inv_push_sn … H12) | elim (lsubf_inv_bind_sn … H12) ] -H12
-#g2 #H12 #H destruct /3 width=5 by eq_next, eq_push/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsubf_fwd_bind_tl: ∀f1,f2,I,L1,L2.
- ⦃L1.ⓘ{I}, f1⦄ ⫃𝐅* ⦃L2.ⓘ{I}, f2⦄ → ⦃L1, ⫱f1⦄ ⫃𝐅* ⦃L2, ⫱f2⦄.
-#f1 #f2 #I #L1 #L2 #H
-elim (pn_split f1) * #g1 #H0 destruct
-[ elim (lsubf_inv_push_sn … H) | elim (lsubf_inv_bind_sn … H) ] -H
-#g2 #H12 #H destruct //
-qed-.
-
-lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
-#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
-[ /2 width=3 by isid_eq_repl_fwd/
-| /4 width=3 by isid_inv_push, isid_push/
-| #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
-| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
-| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
-]
-qed-.
-
-lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → 𝐈⦃f1⦄ → 𝐈⦃f2⦄.
-#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
-[ /2 width=3 by isid_eq_repl_back/
-| /4 width=3 by isid_inv_push, isid_push/
-| #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
-| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
-| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
-]
-qed-.
-
-lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f2 ⊆ f1.
-#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
-/3 width=5 by sor_inv_sle_sn_trans, sle_next, sle_push, sle_refl_eq, eq_sym/
-qed-.
-
-(* Basic properties *********************************************************)
-
-axiom lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
-
-lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. eq_repl_fwd … (λf1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
-#f2 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back1/
-qed-.
-
-axiom lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
-
-lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. eq_repl_fwd … (λf2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
-#f1 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back2/
-qed-.
-
-lemma lsubf_refl: bi_reflexive … lsubf.
-#L elim L -L /2 width=1 by lsubf_atom, eq_refl/
-#L #I #IH #f elim (pn_split f) * #g #H destruct
-/2 width=1 by lsubf_push, lsubf_bind/
-qed.
-
-lemma lsubf_refl_eq: ∀f1,f2,L. f1 ≡ f2 → ⦃L, f1⦄ ⫃𝐅* ⦃L, f2⦄.
-/2 width=3 by lsubf_eq_repl_back2/ qed.
-
-lemma lsubf_bind_tl_dx: ∀g1,f2,I,L1,L2. ⦃L1, g1⦄ ⫃𝐅* ⦃L2, ⫱f2⦄ →
- ∃∃f1. ⦃L1.ⓘ{I}, f1⦄ ⫃𝐅* ⦃L2.ⓘ{I}, f2⦄ & g1 = ⫱f1.
-#g1 #f2 #I #L1 #L2 #H
-elim (pn_split f2) * #g2 #H2 destruct
-@ex2_intro [1,2,4,5: /2 width=2 by lsubf_push, lsubf_bind/ ] // (**) (* constructor needed *)
-qed-.
-
-lemma lsubf_beta_tl_dx: ∀f,f0,g1,L1,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ g1 →
- ∀f2,L2,W. ⦃L1, f0⦄ ⫃𝐅* ⦃L2, ⫱f2⦄ →
- ∃∃f1. ⦃L1.ⓓⓝW.V, f1⦄ ⫃𝐅* ⦃L2.ⓛW, f2⦄ & ⫱f1 ⊆ g1.
-#f #f0 #g1 #L1 #V #Hf #Hg1 #f2
-elim (pn_split f2) * #x2 #H2 #L2 #W #HL12 destruct
-[ /3 width=4 by lsubf_push, sor_inv_sle_sn, ex2_intro/
-| @(ex2_intro … (↑g1)) /2 width=5 by lsubf_beta/ (**) (* full auto fails *)
-]
-qed-.
-
-(* Note: this might be moved *)
-lemma lsubf_inv_sor_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- ∀f2l,f2r. f2l⋓f2r ≘ f2 →
- ∃∃f1l,f1r. ⦃L1, f1l⦄ ⫃𝐅* ⦃L2, f2l⦄ & ⦃L1, f1r⦄ ⫃𝐅* ⦃L2, f2r⦄ & f1l⋓f1r ≘ f1.
-#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
-[ /3 width=7 by sor_eq_repl_fwd3, ex3_2_intro/
-| #g1 #g2 #I1 #I2 #L1 #L2 #_ #IH #f2l #f2r #H
- elim (sor_inv_xxp … H) -H [|*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
- elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, sor_pp, ex3_2_intro/
-| #g1 #g2 #I #L1 #L2 #_ #IH #f2l #f2r #H
- elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
- elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, lsubf_bind, sor_np, sor_pn, sor_nn, ex3_2_intro/
-| #g #g0 #g1 #g2 #L1 #L2 #W #V #Hg #Hg1 #_ #IH #f2l #f2r #H
- elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
- elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
- [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
- /3 width=11 by lsubf_push, lsubf_beta, sor_np, ex3_2_intro/
- | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
- /3 width=11 by lsubf_push, lsubf_beta, sor_pn, ex3_2_intro/
- | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
- /3 width=11 by lsubf_beta, sor_nn, ex3_2_intro/
- ]
-| #g #g0 #g1 #g2 #I1 #I2 #L1 #L2 #V #Hg #Hg1 #_ #IH #f2l #f2r #H
- elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
- elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
- [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
- /3 width=11 by lsubf_push, lsubf_unit, sor_np, ex3_2_intro/
- | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
- /3 width=11 by lsubf_push, lsubf_unit, sor_pn, ex3_2_intro/
- | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
- /3 width=11 by lsubf_unit, sor_nn, 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/lsubf.ma".
-
-(* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
-
-(* Properties with context-sensitive free variables *************************)
-
-lemma lsubf_frees_trans: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 →
- ∀f1,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 ⊢ 𝐅*⦃T⦄ ≘ f1.
-#f2 #L2 #T #H elim H -f2 -L2 -T
-[ /3 width=5 by lsubf_fwd_isid_dx, frees_sort/
-| #f2 #i #Hf2 #g1 #Y1 #H
- elim (lsubf_inv_atom2 … H) -H #Hg1 #H destruct
- elim (eq_inv_pushs_dx … Hg1) -Hg1 #g #Hg #H destruct
- elim (eq_inv_xn … Hg) -Hg
- /3 width=3 by frees_atom, isid_eq_repl_fwd/
-| #f2 #I #K2 #W #_ #IH #g1 #Y1 #H elim (lsubf_inv_pair2 … H) -H *
- [ #f1 #K1 #H12 #H1 #H2 destruct /3 width=1 by frees_pair/
- | #f #f0 #f1 #K1 #V #H12 #Hf #Hf1 #H1 #H2 #H3 destruct
- /4 width=5 by frees_pair, frees_flat/
- ]
-| #f2 #I #L2 #Hf2 #g1 #Y1 #H elim (lsubf_inv_unit2 … H) -H *
- [ #f1 #L1 #H12 #H1 #H2 destruct
- /3 width=5 by lsubf_fwd_isid_dx, frees_unit/
- | #f #f0 #f1 #J #L1 #V #H12 #Hf #Hf1 #H1 #H2 destruct
- /5 width=9 by lsubf_fwd_isid_dx, frees_eq_repl_back, frees_pair, sor_isid_inv_sn/
- ]
-| #f2 #I #L2 #i #_ #IH #g1 #L1 #H elim (lsubf_inv_push2 … H) -H
- /3 width=1 by frees_lref/
-| /3 width=5 by lsubf_fwd_isid_dx, frees_gref/
-| #f2V #f2T #f2 #p #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f1 #L1 #H12
- elim (lsubf_inv_sor_dx … H12 … Hf2) -f2 #f1V #g1T #HV #HT #Hf1
- elim (lsubf_bind_tl_dx … (BPair I V) … HT) -HT #f1T #HT #H destruct
- /3 width=5 by frees_bind/
-| #f2V #f2T #f2 #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f1 #L1 #H12
- elim (lsubf_inv_sor_dx … H12 … Hf2) -f2 /3 width=5 by frees_flat/
-]
-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_frees.ma".
-include "basic_2/static/lsubf.ma".
-
-(* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
-
-(* Main properties **********************************************************)
-
-theorem lsubf_sor: ∀K,L,g1,f1. ⦃K, g1⦄ ⫃𝐅* ⦃L, f1⦄ →
- ∀g2,f2. ⦃K, g2⦄ ⫃𝐅* ⦃L, f2⦄ →
- ∀g. g1 ⋓ g2 ≘ g → ∀f. f1 ⋓ f2 ≘ f → ⦃K, g⦄ ⫃𝐅* ⦃L, f⦄.
-#K elim K -K
-[ #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
- elim (lsubf_inv_atom1 … H1) -H1 #H1 #H destruct
- lapply (lsubf_inv_atom … H2) -H2 #H2
- /5 width=4 by lsubf_atom, sor_mono, sor_eq_repl_back2, sor_eq_repl_back1/
-| #K #J #IH #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
- elim (pn_split g1) * #y1 #H destruct
- elim (pn_split g2) * #y2 #H destruct
- [ elim (sor_inv_ppx … Hg) -Hg [|*: // ] #y #Hy #H destruct
- elim (lsubf_inv_push1 … H1) -H1 #x1 #Z1 #Y1 #H1 #H #H0 destruct
- elim (lsubf_inv_push_sn … H2) -H2 #x2 #H2 #H destruct
- elim (sor_inv_ppx … Hf) -Hf [|*: // ] #x #Hx #H destruct
- /3 width=8 by lsubf_push/
- | elim (sor_inv_pnx … Hg) -Hg [|*: // ] #y #Hy #H destruct
- elim (lsubf_inv_push1 … H1) -H1 #x1 #Z1 #Y1 #H1 #H #H0 destruct
- generalize in match H2; -H2 cases J -J #J [| #V ] #H2
- [ elim (lsubf_inv_unit1 … H2) -H2 #x2 #Y2 #H2 #H #H0 destruct
- | elim (lsubf_inv_pair1 … H2) -H2 *
- [ #x2 #Z2 #H2 #H #H0 destruct
- | #y3 #y4 #x2 #Y2 #W #U #H2 #Hy3 #Hy2 #H #H0 #H3 #H4 destruct
- | #y3 #y4 #x2 #Z2 #Y2 #H2 #Hy3 #Hy2 #H #H0 destruct
- ]
- ]
- elim (sor_inv_pnx … Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
- /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_assoc_sn/
- | elim (sor_inv_npx … Hg) -Hg [|*: // ] #y #Hy #H destruct
- elim (lsubf_inv_push1 … H2) -H2 #x2 #Z2 #Y2 #H2 #H #H0 destruct
- generalize in match H1; -H1 cases J -J #J [| #V ] #H1
- [ elim (lsubf_inv_unit1 … H1) -H1 #x1 #Y1 #H1 #H #H0 destruct
- | elim (lsubf_inv_pair1 … H1) -H1 *
- [ #x1 #Z1 #H1 #H #H0 destruct
- | #y3 #y4 #x1 #Y1 #W #U #H1 #Hy3 #Hy1 #H #H0 #H3 #H4 destruct
- | #y3 #y4 #x1 #Z1 #Y1 #H1 #Hy3 #Hy1 #H #H0 destruct
- ]
- ]
- elim (sor_inv_npx … Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
- /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_comm_23/
- | elim (sor_inv_nnx … Hg) -Hg [|*: // ] #y #Hy #H destruct
- generalize in match H2; generalize in match H1; -H1 -H2 cases J -J #J [| #V ] #H1 #H2
- [ elim (lsubf_inv_unit1 … H1) -H1 #x1 #Y1 #H1 #H #H0 destruct
- elim (lsubf_inv_bind_sn … H2) -H2 #x2 #H2 #H destruct
- | elim (lsubf_inv_pair1 … H1) -H1 *
- [ #x1 #Z1 #H1 #H #H0 destruct
- elim (lsubf_inv_bind_sn … H2) -H2 #x2 #H2 #H destruct
- | #y3 #y4 #x1 #Y1 #W #U #H1 #Hy3 #Hy1 #H #H0 #H3 #H4 destruct
- elim (lsubf_inv_beta_sn … H2) -H2 #z3 #z4 #x2 #H2 #Hz3 #Hy2 #H destruct
- lapply (frees_mono … Hz3 … Hy3) -Hz3 #H3
- lapply (sor_eq_repl_back2 … Hy2 … H3) -z3 #Hy2
- | #y3 #y4 #x1 #Z1 #Y1 #H1 #Hy3 #Hy1 #H #H0 destruct
- elim (lsubf_inv_unit_sn … H2) -H2 #z3 #z4 #x2 #H2 #Hz3 #Hy2 #H destruct
- lapply (frees_mono … Hz3 … Hy3) -Hz3 #H3
- lapply (sor_eq_repl_back2 … Hy2 … H3) -z3 #Hy2
- ]
- ]
- elim (sor_inv_nnx … Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
- /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_coll_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/lsubr.ma".
-include "basic_2/static/lsubf_lsubf.ma".
-
-(* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
-
-(* Forward lemmas with restricted refinement for local environments *********)
-
-lemma lsubf_fwd_lsubr_isdiv: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
- 𝛀⦃f1⦄ → 𝛀⦃f2⦄ → L1 ⫃ L2.
-#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
-/4 width=3 by lsubr_bind, isdiv_inv_next/
-[ #f1 #f2 #I1 #I2 #L1 #L2 #_ #_ #H
- elim (isdiv_inv_push … H) //
-| /5 width=5 by lsubf_fwd_sle, lsubr_beta, sle_inv_isdiv_sn, isdiv_inv_next/
-| /5 width=5 by lsubf_fwd_sle, lsubr_unit, sle_inv_isdiv_sn, isdiv_inv_next/
-]
-qed-.
-
-(* Properties with restricted refinement for local environments *************)
-
-lemma lsubr_lsubf_isid: ∀L1,L2. L1 ⫃ L2 →
- ∀f1,f2. 𝐈⦃f1⦄ → 𝐈⦃f2⦄ → ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄.
-#L1 #L2 #H elim H -L1 -L2
-[ /3 width=1 by lsubf_atom, isid_inv_eq_repl/
-| #I #L1 #L2 | #L1 #L2 #V #W | #I1 #I2 #L1 #L2 #V
-]
-#_ #IH #f1 #f2 #Hf1 #Hf2
-elim (isid_inv_gen … Hf1) -Hf1 #g1 #Hg1 #H destruct
-elim (isid_inv_gen … Hf2) -Hf2 #g2 #Hg2 #H destruct
-/3 width=1 by lsubf_push/
-qed.
-
-lemma lsubr_lsubf: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 → ∀L1. L1 ⫃ L2 →
- ∀f1. L1 ⊢ 𝐅*⦃T⦄ ≘ f1 → ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄.
-#f2 #L2 #T #H elim H -f2 -L2 -T
-[ #f2 #L2 #s #Hf2 #L1 #HL12 #f1 #Hf1
- lapply (frees_inv_sort … Hf1) -Hf1 /2 width=1 by lsubr_lsubf_isid/
-| #f2 #i #Hf2 #Y1 #HY1
- >(lsubr_inv_atom2 … HY1) -Y1 #g1 #Hg1
- elim (frees_inv_atom … Hg1) -Hg1 #f1 #Hf1 #H destruct
- /5 width=5 by lsubf_atom, isid_inv_eq_repl, pushs_eq_repl, eq_next/
-| #f2 #Z #L2 #W #_ #IH #Y1 #HY1 #g1 #Hg1 elim (lsubr_inv_pair2 … HY1) -HY1 *
- [ #L1 #HL12 #H destruct
- elim (frees_inv_pair … Hg1) -Hg1 #f1 #Hf1 #H destruct
- /3 width=1 by lsubf_bind/
- | #L1 #V #HL12 #H1 #H2 destruct
- elim (frees_inv_pair … Hg1) -Hg1 #f1 #Hf1 #H destruct
- elim (frees_inv_flat … Hf1) -Hf1 /3 width=5 by lsubf_beta/
- ]
-| #f2 #I2 #L2 #Hf2 #Y1 #HY1 #g1 #Hg1 elim (lsubr_inv_unit2 … HY1) -HY1 *
- [ #L1 #HL12 #H destruct
- elim (frees_inv_unit … Hg1) -Hg1 #f1 #Hf1 #H destruct
- /3 width=1 by lsubf_bind, lsubr_lsubf_isid/
- | #I #L1 #V #HL12 #H destruct
- elim (frees_inv_pair … Hg1) -Hg1 #f1 #Hf1 #H destruct
- /3 width=5 by lsubf_unit, sor_isid_sn, lsubr_lsubf_isid/
- ]
-| #f2 #I2 #L2 #i #_ #IH #Y1 #HY1 #g1 #Hg1
- elim (lsubr_fwd_bind2 … HY1) -HY1 #I1 #L1 #HL12 #H destruct
- elim (frees_inv_lref … Hg1) -Hg1 #f1 #Hf1 #H destruct
- /3 width=1 by lsubf_push/
-| #f2 #L2 #l #Hf2 #L1 #HL12 #f1 #Hf1
- lapply (frees_inv_gref … Hf1) -Hf1 /2 width=1 by lsubr_lsubf_isid/
-| #f2V #f2T #f2 #p #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #L1 #HL12 #f1 #Hf1
- elim (frees_inv_bind … Hf1) -Hf1 #f1V #f1T #Hf1V #Hf1T #Hf1
- /5 width=8 by lsubf_sor, lsubf_fwd_bind_tl, lsubr_bind/
-| #f2V #f2T #f2 #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #L1 #HL12 #f1 #Hf1
- elim (frees_inv_flat … Hf1) -Hf1 #f1V #f1T #Hf1V #Hf1T #Hf1
- /3 width=8 by lsubf_sor/
-]
-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/lrsubeqc_2.ma".
-include "basic_2/syntax/lenv.ma".
-
-(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
-
-(* Basic_2A1: just tpr_cpr and tprs_cprs require the extended lsubr_atom *)
-(* Basic_2A1: includes: lsubr_pair *)
-inductive lsubr: relation lenv ≝
-| lsubr_atom: lsubr (⋆) (⋆)
-| lsubr_bind: ∀I,L1,L2. lsubr L1 L2 → lsubr (L1.ⓘ{I}) (L2.ⓘ{I})
-| lsubr_beta: ∀L1,L2,V,W. lsubr L1 L2 → lsubr (L1.ⓓⓝW.V) (L2.ⓛW)
-| lsubr_unit: ∀I1,I2,L1,L2,V. lsubr L1 L2 → lsubr (L1.ⓑ{I1}V) (L2.ⓤ{I2})
-.
-
-interpretation
- "restricted refinement (local environment)"
- 'LRSubEqC L1 L2 = (lsubr L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lsubr_refl: ∀L. L ⫃ L.
-#L elim L -L /2 width=1 by lsubr_atom, lsubr_bind/
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⫃ L2 → L1 = ⋆ → L2 = ⋆.
-#L1 #L2 * -L1 -L2 //
-[ #I #L1 #L2 #_ #H destruct
-| #L1 #L2 #V #W #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_atom1: ∀L2. ⋆ ⫃ L2 → L2 = ⋆.
-/2 width=3 by lsubr_inv_atom1_aux/ qed-.
-
-fact lsubr_inv_bind1_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
- I = BPair Abbr (ⓝW.V)
- | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
- I = BPair J1 V.
-#L1 #L2 * -L1 -L2
-[ #J #K1 #H destruct
-| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by or3_intro0, ex2_intro/
-| #L1 #L2 #V #W #HL12 #J #K1 #H destruct /3 width=6 by or3_intro1, ex3_3_intro/
-| #I1 #I2 #L1 #L2 #V #HL12 #J #K1 #H destruct /3 width=4 by or3_intro2, ex3_4_intro/
-]
-qed-.
-
-(* Basic_2A1: uses: lsubr_inv_pair1 *)
-lemma lsubr_inv_bind1: ∀I,K1,L2. K1.ⓘ{I} ⫃ L2 →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
- | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
- I = BPair Abbr (ⓝW.V)
- | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
- I = BPair J1 V.
-/2 width=3 by lsubr_inv_bind1_aux/ qed-.
-
-fact lsubr_inv_atom2_aux: ∀L1,L2. L1 ⫃ L2 → L2 = ⋆ → L1 = ⋆.
-#L1 #L2 * -L1 -L2 //
-[ #I #L1 #L2 #_ #H destruct
-| #L1 #L2 #V #W #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_atom2: ∀L1. L1 ⫃ ⋆ → L1 = ⋆.
-/2 width=3 by lsubr_inv_atom2_aux/ qed-.
-
-fact lsubr_inv_bind2_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
-#L1 #L2 * -L1 -L2
-[ #J #K2 #H destruct
-| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or3_intro0/
-| #L1 #L2 #V1 #V2 #HL12 #J #K2 #H destruct /3 width=6 by ex3_3_intro, or3_intro1/
-| #I1 #I2 #L1 #L2 #V #HL12 #J #K2 #H destruct /3 width=5 by ex3_4_intro, or3_intro2/
-]
-qed-.
-
-lemma lsubr_inv_bind2: ∀I,L1,K2. L1 ⫃ K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
-/2 width=3 by lsubr_inv_bind2_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⫃ L2 →
- ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW
- | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ{I2}.
-#K1 #L2 #W #H elim (lsubr_inv_bind1 … H) -H *
-/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
-#K2 #V2 #W2 #_ #_ #H destruct
-qed-.
-
-lemma lsubr_inv_unit1: ∀I,K1,L2. K1.ⓤ{I} ⫃ L2 →
- ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ{I}.
-#I #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
-[ #K2 #HK12 #H destruct /2 width=3 by ex2_intro/
-| #K2 #V #W #_ #_ #H destruct
-| #I1 #I2 #K2 #V #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_pair2: ∀I,L1,K2,W. L1 ⫃ K2.ⓑ{I}W →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ{I}W
- | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = Abst.
-#I #L1 #K2 #W #H elim (lsubr_inv_bind2 … H) -H *
-[ /3 width=3 by ex2_intro, or_introl/
-| #K2 #X #V #HK12 #H1 #H2 destruct /3 width=4 by ex3_2_intro, or_intror/
-| #I1 #I1 #K2 #V #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_abbr2: ∀L1,K2,V. L1 ⫃ K2.ⓓV →
- ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓV.
-#L1 #K2 #V #H elim (lsubr_inv_pair2 … H) -H *
-[ /2 width=3 by ex2_intro/
-| #K1 #X #_ #_ #H destruct
-]
-qed-.
-
-lemma lsubr_inv_abst2: ∀L1,K2,W. L1 ⫃ K2.ⓛW →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓛW
- | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V.
-#L1 #K2 #W #H elim (lsubr_inv_pair2 … H) -H *
-/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
-qed-.
-
-lemma lsubr_inv_unit2: ∀I,L1,K2. L1 ⫃ K2.ⓤ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓤ{I}
- | ∃∃J,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J}V.
-#I #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
-[ /3 width=3 by ex2_intro, or_introl/
-| #K1 #W #V #_ #_ #H destruct
-| #I1 #I2 #K1 #V #HK12 #H1 #H2 destruct /3 width=5 by ex2_3_intro, or_intror/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsubr_fwd_bind1: ∀I1,K1,L2. K1.ⓘ{I1} ⫃ L2 →
- ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓘ{I2}.
-#I1 #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
-[ #K2 #HK12 #H destruct /3 width=4 by ex2_2_intro/
-| #K2 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-| #I1 #I2 #K2 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lsubr_fwd_bind2: ∀I2,L1,K2. L1 ⫃ K2.ⓘ{I2} →
- ∃∃I1,K1. K1 ⫃ K2 & L1 = K1.ⓘ{I1}.
-#I2 #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
-[ #K1 #HK12 #H destruct /3 width=4 by ex2_2_intro/
-| #K1 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-| #I1 #I2 #K1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/relocation/drops.ma".
-include "basic_2/static/lsubr.ma".
-
-(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
-
-(* Forward lemmas with generic slicing for local environments ***************)
-
-(* Basic_2A1: includes: lsubr_fwd_drop2_pair *)
-lemma lsubr_fwd_drops2_bind: ∀L1,L2. L1 ⫃ L2 →
- ∀b,f,I,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2.ⓘ{I} →
- ∨∨ ∃∃K1. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓘ{I}
- | ∃∃K1,W,V. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓓⓝW.V & I = BPair Abst W
- | ∃∃J1,J2,K1,V. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓑ{J1}V & I = BUnit J2.
-#L1 #L2 #H elim H -L1 -L2
-[ #b #f #I #K2 #_ #H
- elim (drops_inv_atom1 … H) -H #H destruct
-| #J #L1 #L2 | #L1 #L2 #V #W | #I1 #I2 #L1 #L2 #V1
-]
-#HL12 #IH #b #f #I #K2 #Hf #H
-elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
-[1,3,5: #Hf #H destruct -IH
- /4 width=6 by drops_refl, or3_intro0, or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro, ex2_intro/
-|2,4,6: #g #Hg #HLK2 #H destruct -HL12
- elim (IH … Hg HLK2) -IH -Hg -HLK2 *
- /4 width=6 by drops_drop, or3_intro0, or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro, ex2_intro/
-]
-qed-.
-
-(* Basic_2A1: includes: lsubr_fwd_drop2_abbr *)
-lemma lsubr_fwd_drops2_abbr: ∀L1,L2. L1 ⫃ L2 →
- ∀b,f,K2,V. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2.ⓓV →
- ∃∃K1. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓓV.
-#L1 #L2 #HL12 #b #f #K2 #V #Hf #HLK2
-elim (lsubr_fwd_drops2_bind … HL12 … Hf HLK2) -L2 -Hf // *
-[ #K1 #W #V #_ #_ #H destruct
-| #I1 #I2 #K1 #V #_ #_ #H destruct
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/syntax/lenv_length.ma".
-include "basic_2/static/lsubr.ma".
-
-(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
-
-(* Forward lemmas with length for local environments ************************)
-
-lemma lsubr_fwd_length: ∀L1,L2. L1 ⫃ L2 → |L2| = |L1|.
-#L1 #L2 #H elim H -L1 -L2 normalize //
-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/lsubr.ma".
-
-(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
-
-(* Main properties **********************************************************)
-
-theorem lsubr_trans: Transitive … lsubr.
-#L1 #L #H elim H -L1 -L //
-[ #I #L1 #L #_ #IH #X #H elim (lsubr_inv_bind1 … H) -H *
- [ #L2 #HL2 #H | #L2 #V #W #HL2 #H1 #H2 | #I1 #I2 #L2 #V #Hl2 #H1 #H2 ]
- destruct /3 width=1 by lsubr_bind, lsubr_beta, lsubr_unit/
-| #L1 #L #V #W #_ #IH #X #H elim (lsubr_inv_abst1 … H) -H *
- [ #L2 #HL2 #H | #I #L2 #HL2 #H ]
- destruct /3 width=1 by lsubr_beta, lsubr_unit/
-| #I1 #I2 #L1 #L #V #_ #IH #X #H elim (lsubr_inv_unit1 … H) -H
- /4 width=1 by lsubr_unit/
-]
-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-.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* THE FORMAL SYSTEM λδ: MATITA SOURCE FILES
- * Initial invocation: - Patience on me to gain peace and perfection! -
- *)
-
-include "ground_2/lib/relations.ma".
-include "basic_2/notation/constructors/item0_0.ma".
-include "basic_2/notation/constructors/snitem2_2.ma".
-
-(* ATOMIC ARITY *************************************************************)
-
-inductive aarity: Type[0] ≝
- | AAtom: aarity (* atomic aarity construction *)
- | APair: aarity → aarity → aarity (* binary aarity construction *)
-.
-
-interpretation "atomic arity construction (atomic)"
- 'Item0 = AAtom.
-
-interpretation "atomic arity construction (binary)"
- 'SnItem2 A1 A2 = (APair A1 A2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact destruct_apair_apair_aux: ∀A1,A2,B1,B2. ②B1.A1 = ②B2.A2 → B1 = B2 ∧ A1 = A2.
-#A1 #A2 #B1 #B2 #H destruct /2 width=1 by conj/
-qed-.
-
-lemma discr_apair_xy_x: ∀A,B. ②B.A = B → ⊥.
-#A #B elim B -B
-[ #H destruct
-| #Y #X #IHY #_ #H elim (destruct_apair_apair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
-]
-qed-.
-
-(* Basic_2A1: was: discr_tpair_xy_y *)
-lemma discr_apair_xy_y: ∀B,A. ②B. A = A → ⊥.
-#B #A elim A -A
-[ #H destruct
-| #Y #X #_ #IHX #H elim (destruct_apair_apair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
-]
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma eq_aarity_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
-#A1 elim A1 -A1
-[ #A2 elim A2 -A2 /2 width=1 by or_introl/
- #B2 #A2 #_ #_ @or_intror #H destruct
-| #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
- [ -IHB1 -IHA1 @or_intror #H destruct
- | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
- [ #H destruct elim (IHA1 A2) -IHA1
- [ #H destruct /2 width=1 by or_introl/
- | #HA12 @or_intror #H destruct /2 width=1 by/
- ]
- | -IHA1 #HB12 @or_intror #H destruct /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/notation/functions/snitem_2.ma".
-include "basic_2/notation/functions/snbind1_2.ma".
-include "basic_2/notation/functions/snbind2_3.ma".
-include "basic_2/notation/functions/snvoid_1.ma".
-include "basic_2/notation/functions/snabbr_2.ma".
-include "basic_2/notation/functions/snabst_2.ma".
-include "basic_2/syntax/lenv.ma".
-
-(* APPEND FOR LOCAL ENVIRONMENTS ********************************************)
-
-rec definition append L K on K ≝ match K with
-[ LAtom ⇒ L
-| LBind K I ⇒ (append L K).ⓘ{I}
-].
-
-interpretation "append (local environment)" 'plus L1 L2 = (append L1 L2).
-
-interpretation "local environment tail binding construction (generic)"
- 'SnItem I L = (append (LBind LAtom I) L).
-
-interpretation "local environment tail binding construction (unary)"
- 'SnBind1 I L = (append (LBind LAtom (BUnit I)) L).
-
-interpretation "local environment tail binding construction (binary)"
- 'SnBind2 I T L = (append (LBind LAtom (BPair I T)) L).
-
-interpretation "tail exclusion (local environment)"
- 'SnVoid L = (append (LBind LAtom (BUnit Void)) L).
-
-interpretation "tail abbreviation (local environment)"
- 'SnAbbr T L = (append (LBind LAtom (BPair Abbr T)) L).
-
-interpretation "tail abstraction (local environment)"
- 'SnAbst L T = (append (LBind LAtom (BPair Abst T)) L).
-
-definition d_appendable_sn: predicate (lenv→relation term) ≝ λR.
- ∀K,T1,T2. R K T1 T2 → ∀L. R (L+K) T1 T2.
-
-(* Basic properties *********************************************************)
-
-lemma append_atom: ∀L. (L + ⋆) = L. (**) (* () should be redundant *)
-// qed.
-
-(* Basic_2A1: uses: append_pair *)
-lemma append_bind: ∀I,L,K. L+(K.ⓘ{I}) = (L+K).ⓘ{I}.
-// qed.
-
-lemma append_atom_sn: ∀L. ⋆ + L = L.
-#L elim L -L //
-#L #I >append_bind //
-qed.
-
-lemma append_assoc: associative … append.
-#L1 #L2 #L3 elim L3 -L3 //
-qed.
-
-lemma append_shift: ∀L,K,I. L+(ⓘ{I}.K) = (L.ⓘ{I})+K.
-#L #K #I <append_assoc //
-qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma append_inv_atom3_sn: ∀L,K. ⋆ = L + K → ∧∧ ⋆ = L & ⋆ = K.
-#L * /2 width=1 by conj/
-#K #I >append_bind #H destruct
-qed-.
-
-lemma append_inv_bind3_sn: ∀I0,L,L0,K. L0.ⓘ{I0} = L + K →
- ∨∨ ∧∧ L0.ⓘ{I0} = L & ⋆ = K
- | ∃∃K0. K = K0.ⓘ{I0} & L0 = L + K0.
-#I0 #L #L0 * /3 width=1 by or_introl, conj/
-#K #I >append_bind #H destruct /3 width=3 by ex2_intro, or_intror/
-qed-.
-
-lemma append_inj_sn: ∀K,L1,L2. L1+K = L2+K → L1 = L2.
-#K elim K -K //
-#K #I #IH #L1 #L2 >append_bind #H
-elim (destruct_lbind_lbind_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
-qed-.
-
-(* Basic_1: uses: chead_ctail *)
-(* Basic_2A1: uses: lpair_ltail *)
-lemma lenv_case_tail: ∀L. L = ⋆ ∨ ∃∃K,I. L = ⓘ{I}.K.
-#L elim L -L /2 width=1 by or_introl/
-#L #I * [2: * ] /3 width=3 by ex1_2_intro, or_intror/
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/syntax/lenv_length.ma".
-include "basic_2/syntax/append.ma".
-
-(* APPEND FOR LOCAL ENVIRONMENTS ********************************************)
-
-(* Properties with length for local environments ****************************)
-
-lemma append_length: ∀L1,L2. |L1 + L2| = |L1| + |L2|.
-#L1 #L2 elim L2 -L2 //
-#L2 #I >append_bind >length_bind >length_bind //
-qed.
-
-lemma ltail_length: ∀I,L. |ⓘ{I}.L| = ↑|L|.
-#I #L >append_length //
-qed.
-
-(* Advanced inversion lemmas on length for local environments ***************)
-
-(* Basic_2A1: was: length_inv_pos_dx_ltail *)
-lemma length_inv_succ_dx_ltail: ∀L,n. |L| = ↑n →
- ∃∃I,K. |K| = n & L = ⓘ{I}.K.
-#Y #n #H elim (length_inv_succ_dx … H) -H #I #L #Hn #HLK destruct
-elim (lenv_case_tail … L) [2: * #K #J ]
-#H destruct /2 width=4 by ex2_2_intro/
-qed-.
-
-(* Basic_2A1: was: length_inv_pos_sn_ltail *)
-lemma length_inv_succ_sn_ltail: ∀L,n. ↑n = |L| →
- ∃∃I,K. n = |K| & L = ⓘ{I}.K.
-#Y #n #H elim (length_inv_succ_sn … H) -H #I #L #Hn #HLK destruct
-elim (lenv_case_tail … L) [2: * #K #J ]
-#H destruct /2 width=4 by ex2_2_intro/
-qed-.
-
-(* Inversion lemmas with length for local environments **********************)
-
-(* Basic_2A1: was: append_inj_sn *)
-lemma append_inj_length_sn: ∀K1,K2,L1,L2. L1 + K1 = L2 + K2 → |K1| = |K2| →
- L1 = L2 ∧ K1 = K2.
-#K1 elim K1 -K1
-[ * /2 width=1 by conj/
- #K2 #I2 #L1 #L2 #_ >length_atom >length_bind
- #H destruct
-| #K1 #I1 #IH *
- [ #L1 #L2 #_ >length_atom >length_bind
- #H destruct
- | #K2 #I2 #L1 #L2 #H1 >length_bind >length_bind #H2
- elim (destruct_lbind_lbind_aux … H1) -H1 #H1 #H3 destruct (**) (* destruct lemma needed *)
- elim (IH … H1) -IH -H1 /3 width=4 by conj/
- ]
-]
-qed-.
-
-(* Note: lemma 750 *)
-(* Basic_2A1: was: append_inj_dx *)
-lemma append_inj_length_dx: ∀K1,K2,L1,L2. L1 + K1 = L2 + K2 → |L1| = |L2| →
- L1 = L2 ∧ K1 = K2.
-#K1 elim K1 -K1
-[ * /2 width=1 by conj/
- #K2 #I2 #L1 #L2 >append_atom >append_bind #H destruct
- >length_bind >append_length >plus_n_Sm
- #H elim (plus_xSy_x_false … H)
-| #K1 #I1 #IH *
- [ #L1 #L2 >append_bind >append_atom #H destruct
- >length_bind >append_length >plus_n_Sm #H
- lapply (discr_plus_x_xy … H) -H #H destruct
- | #K2 #I2 #L1 #L2 >append_bind >append_bind #H1 #H2
- elim (destruct_lbind_lbind_aux … H1) -H1 #H1 #H3 destruct (**) (* destruct lemma needed *)
- elim (IH … H1) -IH -H1 /2 width=1 by conj/
- ]
-]
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma append_inj_dx: ∀L,K1,K2. L+K1 = L+K2 → K1 = K2.
-#L #K1 #K2 #H elim (append_inj_length_dx … H) -H //
-qed-.
-
-lemma append_inv_refl_dx: ∀L,K. L+K = L → K = ⋆.
-#L #K #H elim (append_inj_dx … (⋆) … H) //
-qed-.
-
-lemma append_inv_pair_dx: ∀I,L,K,V. L+K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
-#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
-qed-.
-
-(* Basic eliminators ********************************************************)
-
-(* Basic_1: was: c_tail_ind *)
-(* Basic_2A1: was: lenv_ind_alt *)
-lemma lenv_ind_tail: ∀Q:predicate lenv.
- Q (⋆) → (∀I,L. Q L → Q (ⓘ{I}.L)) → ∀L. Q L.
-#Q #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * //
-#L #I -IH1 #H destruct
-elim (lenv_case_tail … L) [2: * #K #J ]
-#H destruct /3 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/term.ma".
-
-(* BINDERS FOR LOCAL ENVIRONMENTS ******************************************)
-
-inductive bind: Type[0] ≝
-| BUnit: bind1 → bind
-| BPair: bind2 → term → bind
-.
-
-(* Basic properties ********************************************************)
-
-lemma eq_bind_dec: ∀I1,I2:bind. Decidable (I1 = I2).
-* #I1 [2: #V1 ] * #I2 [2,4: #V2 ]
-[1: elim (eq_bind2_dec I1 I2) #HI
- [ elim (eq_term_dec V1 V2) #HV ]
-|4: elim (eq_bind1_dec I1 I2) #HI
-]
-/2 width=1 by or_introl/
-@or_intror #H destruct /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/term_weight.ma".
-include "basic_2/syntax/bind.ma".
-
-(* WEIGHT OF A BINDER FOR LOCAL ENVIRONMENTS *******************************)
-
-rec definition bw I ≝ match I with
-[ BUnit _ ⇒ 1
-| BPair _ V ⇒ ♯{V}
-].
-
-interpretation "weight (binder for local environments)" 'Weight I = (bw I).
-
-(* Basic properties *********************************************************)
-
-lemma bw_pos: ∀I. 1 ≤ ♯{I}.
-* //
-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/cext2.ma".
-
-(* CONTEXT-AWARE SYNTACTIC EQUIVALENCE FOR BINDERS **************************)
-
-definition ceq_ext: lenv → relation bind ≝
- cext2 ceq.
-
-(* Basic properties *********************************************************)
-
-lemma ceq_ext_refl (L): reflexive … (ceq_ext L).
-/2 width=1 by ext2_refl/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma ceq_ext_inv_eq: ∀L,I1,I2. ceq_ext L I1 I2 → I1 = I2.
-#L #I1 #I2 * -I1 -I2 //
-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.ma".
-
-(* CONTEXT-AWARE SYNTACTIC EQUIVALENCE FOR BINDERS **************************)
-
-(* Main properties **********************************************************)
-
-theorem ceq_ext_trans: ∀L1,I1,I. ceq_ext L1 I1 I →
- ∀L2,I2. ceq_ext L2 I I2 → ∀L3. ceq_ext L3 I1 I2.
-#L1 #I1 #I * -I1 -I //
-#I1 #V1 #V #HV1 #L2 #Z #H elim (ext2_inv_pair_sn … 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/syntax/ext2.ma".
-include "basic_2/syntax/lenv.ma".
-
-(* EXTENSION TO BINDERS OF A CONTEXT-SENSITIVE RELATION FOR TERMS ***********)
-
-definition cext2: (lenv → relation term) → lenv → relation bind ≝
- λR,L. ext2 (R L).
-
-(* Basic properties *********************************************************)
-
-lemma cext2_sym: ∀R. (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
- ∀L1,L2,I1,I2. cext2 R L1 I1 I2 → cext2 R L2 I2 I1.
-#R #HR #L1 #L2 #I1 #I2 * /3 width=2 by ext2_pair/
-qed-.
-
-lemma cext2_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
- ∀L,I1,I2. cext2 R1 L I1 I2 → cext2 R2 L I1 I2.
-#R1 #R2 #HR #L #I1 #I2 * /3 width=2 by ext2_unit, ext2_pair/
-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/functions/weight_2.ma".
-include "basic_2/syntax/lenv_weight.ma".
-
-(* WEIGHT OF A RESTRICTED CLOSURE *******************************************)
-
-definition rfw: lenv → term → ? ≝ λL,T. ♯{L} + ♯{T}.
-
-interpretation "weight (restricted closure)" 'Weight L T = (rfw L T).
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: flt_shift *)
-lemma rfw_shift: ∀p,I,K,V,T. ♯{K.ⓑ{I}V, T} < ♯{K, ⓑ{p,I}V.T}.
-normalize /2 width=1 by monotonic_le_plus_r/
-qed.
-
-lemma rfw_clear: ∀p,I1,I2,K,V,T. ♯{K.ⓤ{I1}, T} < ♯{K, ⓑ{p,I2}V.T}.
-normalize /4 width=1 by monotonic_le_plus_r, le_S_S/
-qed.
-
-lemma rfw_tpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L, ②{I}V.T}.
-normalize in ⊢ (?→?→?→?→?%%); //
-qed.
-
-lemma rfw_tpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L, ②{I}V.T}.
-normalize in ⊢ (?→?→?→?→?%%); //
-qed.
-
-lemma rfw_lpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L.ⓑ{I}V, T}.
-normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
-qed.
-
-lemma rfw_lpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L.ⓑ{I}V, T}.
-normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
-qed.
-
-(* Basic_1: removed theorems 7:
- flt_thead_sx flt_thead_dx flt_trans
- flt_arith0 flt_arith1 flt_arith2 flt_wf_ind
-*)
-(* Basic_1: removed local theorems 1: q_ind *)
+++ /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/functions/weight_3.ma".
-include "basic_2/syntax/lenv_weight.ma".
-include "basic_2/syntax/genv.ma".
-
-(* WEIGHT OF A CLOSURE ******************************************************)
-
-(* activate genv *)
-definition fw: genv → lenv → term → ? ≝ λG,L,T. ♯{L} + ♯{T}.
-
-interpretation "weight (closure)" 'Weight G L T = (fw G L T).
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: flt_shift *)
-lemma fw_shift: ∀p,I,G,K,V,T. ♯{G, K.ⓑ{I}V, T} < ♯{G, K, ⓑ{p,I}V.T}.
-normalize /2 width=1 by monotonic_le_plus_r/
-qed.
-
-lemma fw_clear: ∀p,I1,I2,G,K,V,T. ♯{G, K.ⓤ{I1}, T} < ♯{G, K, ⓑ{p,I2}V.T}.
-normalize /4 width=1 by monotonic_le_plus_r, le_S_S/
-qed.
-
-lemma fw_tpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L, ②{I}V.T}.
-normalize in ⊢ (?→?→?→?→?→?%%); //
-qed.
-
-lemma fw_tpair_dx: ∀I,G,L,V,T. ♯{G, L, T} < ♯{G, L, ②{I}V.T}.
-normalize in ⊢ (?→?→?→?→?→?%%); //
-qed.
-
-lemma fw_lpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L.ⓑ{I}V, T}.
-normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
-qed.
-
-(* Basic_1: removed theorems 7:
- flt_thead_sx flt_thead_dx flt_trans
- flt_arith0 flt_arith1 flt_arith2 flt_wf_ind
-*)
-(* Basic_1: removed local theorems 1: q_ind *)
+++ /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/bind.ma".
-
-(* EXTENSION TO BINDERS OF A RELATION FOR TERMS *****************************)
-
-inductive ext2 (R:relation term): relation bind ≝
-| ext2_unit: ∀I. ext2 R (BUnit I) (BUnit I)
-| ext2_pair: ∀I,V1,V2. R V1 V2 → ext2 R (BPair I V1) (BPair I V2)
-.
-
-(* Basic_inversion lemmas **************************************************)
-
-fact ext2_inv_unit_sn_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
- ∀I. Z1 = BUnit I → Z2 = BUnit I.
-#R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #_ ]
-#J #H destruct //
-qed-.
-
-lemma ext2_inv_unit_sn: ∀R,I,Z2. ext2 R (BUnit I) Z2 → Z2 = BUnit I.
-/2 width=4 by ext2_inv_unit_sn_aux/ qed-.
-
-fact ext2_inv_pair_sn_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
- ∀I,V1. Z1 = BPair I V1 →
- ∃∃V2. R V1 V2 & Z2 = BPair I V2.
-#R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #HV12 ]
-#J #W1 #H destruct /2 width=3 by ex2_intro/
-qed-.
-
-lemma ext2_inv_pair_sn: ∀R,Z2,I,V1. ext2 R (BPair I V1) Z2 →
- ∃∃V2. R V1 V2 & Z2 = BPair I V2.
-/2 width=3 by ext2_inv_pair_sn_aux/ qed-.
-
-fact ext2_inv_unit_dx_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
- ∀I. Z2 = BUnit I → Z1 = BUnit I.
-#R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #_ ]
-#J #H destruct //
-qed-.
-
-lemma ext2_inv_unit_dx: ∀R,I,Z1. ext2 R Z1 (BUnit I) → Z1 = BUnit I.
-/2 width=4 by ext2_inv_unit_dx_aux/ qed-.
-
-fact ext2_inv_pair_dx_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
- ∀I,V2. Z2 = BPair I V2 →
- ∃∃V1. R V1 V2 & Z1 = BPair I V1.
-#R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #HV12 ]
-#J #W2 #H destruct /2 width=3 by ex2_intro/
-qed-.
-
-lemma ext2_inv_pair_dx: ∀R,Z1,I,V2. ext2 R Z1 (BPair I V2) →
- ∃∃V1. R V1 V2 & Z1 = BPair I V1.
-/2 width=3 by ext2_inv_pair_dx_aux/ qed-.
-
-(* Advanced inversion lemmas ***********************************************)
-
-lemma ext2_inv_unit: ∀R,I1,I2. ext2 R (BUnit I1) (BUnit I2) → I1 = I2.
-#R #I1 #I2 #H lapply (ext2_inv_unit_sn … H) -H
-#H destruct //
-qed-.
-
-lemma ext2_inv_pair: ∀R,I1,I2,V1,V2. ext2 R (BPair I1 V1) (BPair I2 V2) →
- I1 = I2 ∧ R V1 V2.
-#R #I1 #I2 #V1 #V2 #H elim (ext2_inv_pair_sn … H) -H
-#V #HV #H destruct /2 width=1 by conj/
-qed-.
-
-(* Basic properties ********************************************************)
-
-lemma ext2_refl: ∀R. reflexive … R → reflexive … (ext2 R).
-#R #HR * /2 width=1 by ext2_pair/
-qed.
-
-lemma ext2_sym: ∀R. symmetric … R → symmetric … (ext2 R).
-#R #HR #T1 #T2 * /3 width=1 by ext2_unit, ext2_pair/
-qed-.
-
-lemma ext2_dec: ∀R. (∀T1,T2. Decidable (R T1 T2)) →
- ∀I1,I2. Decidable (ext2 R I1 I2).
-#R #HR * #I1 [2: #T1 ] * #I2 [2,4: #T2 ]
-[ elim (eq_bind2_dec I1 I2) #HI12 destruct
- [ elim (HR T1 T2) -HR #HT12 /3 width=1 by ext2_pair, or_introl/ ]
- @or_intror #H elim (ext2_inv_pair … H) -H /2 width=1 by/
-| @or_intror #H lapply (ext2_inv_unit_sn … H) -H
- #H destruct
-| @or_intror #H lapply (ext2_inv_unit_dx … H) -H
- #H destruct
-| elim (eq_bind1_dec I1 I2) #HI12 destruct
- /4 width=2 by ext2_inv_unit, ext2_unit, 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/syntax/ext2.ma".
-
-(* EXTENSION TO BINDERS OF A RELATION FOR TERMS *****************************)
-
-(* Main properties **********************************************************)
-
-theorem ext2_trans: ∀R. Transitive … R → Transitive … (ext2 R).
-#R #HR #I1 #I #H elim H -I1 -I
-[ #I1 #J #H >(ext2_inv_unit_sn … H) -J /2 width=1 by ext2_unit/
-| #I1 #V1 #V #HV1 #J #H elim (ext2_inv_pair_sn … H) -H
- #V2 #HV2 #H destruct /3 width=3 by ext2_pair/
-]
-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/syntax/ext2.ma".
-
-(* EXTENSION TO BINDERS OF A RELATION FOR TERMS *****************************)
-
-(* Properties with transitive closure ***************************************)
-
-lemma ext2_tc_pair: ∀R,I,V1,V2. TC … R V1 V2 →
- TC … (ext2 R) (BPair I V1) (BPair I V2).
-#R #I #V1 #V2 #H elim H -H -V2
-/3 width=3 by ext2_pair, step, inj/
-qed.
-
-lemma ext2_tc_inj: ∀R,I1,I2. ext2 R I1 I2 → ext2 (TC … R) I1 I2.
-#R #I1 #I2 * -I1 -I2
-/3 width=1 by ext2_unit, ext2_pair, inj/
-qed.
-
-(* Main properties with transitive closure **********************************)
-
-theorem ext2_tc_step: ∀R,I1,I. ext2 (TC … R) I1 I →
- ∀I2. ext2 R I I2 → ext2 (TC … R) I1 I2.
-#R #I1 #I * -I1 -I
-[ #I #Z #H >(ext2_inv_unit_sn … H) -Z /2 width=1 by ext2_unit/
-| #I #V1 #V #HV1 #Z #H
- elim (ext2_inv_pair_sn … H) -H #V2 #HV2 #H destruct
- /3 width=3 by ext2_pair, step/
-]
-qed-.
-
-(* Advanced properties with transitive closure ******************************)
-
-lemma ext2_tc: ∀R,I1,I2. TC … (ext2 R) I1 I2 → ext2 (TC … R) I1 I2.
-#R #I1 #I2 #H elim H -I2
-/2 width=3 by ext2_tc_step, ext2_tc_inj/
-qed.
-
-(* Advanced inversion lemmas with transitive closure ************************)
-
-lemma ext2_inv_tc: ∀R,I1,I2. ext2 (TC … R) I1 I2 → TC … (ext2 R) I1 I2.
-#R #I1 #I2 * -I1 -I2
-/3 width=1 by ext2_tc_pair, ext2_unit, 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/constructors/star_0.ma".
-include "basic_2/notation/constructors/dxbind2_3.ma".
-include "basic_2/notation/constructors/dxabbr_2.ma".
-include "basic_2/notation/constructors/dxabst_2.ma".
-include "basic_2/syntax/term.ma".
-
-(* GLOBAL ENVIRONMENTS ******************************************************)
-
-(* global environments *)
-inductive genv: Type[0] ≝
-| GAtom: genv (* empty *)
-| GPair: genv → bind2 → term → genv (* binary binding construction *)
-.
-
-interpretation "sort (global environment)"
- 'Star = (GAtom).
-
-interpretation "global environment binding construction (binary)"
- 'DxBind2 G I T = (GPair G I T).
-
-interpretation "abbreviation (global environment)"
- 'DxAbbr G T = (GPair G Abbr T).
-
-interpretation "abstraction (global environment)"
- 'DxAbst G T = (GPair G Abst T).
-
-(* Basic properties *********************************************************)
-
-lemma eq_genv_dec: ∀G1,G2:genv. Decidable (G1 = G2).
-#G1 elim G1 -G1 [| #G1 #I1 #T1 #IHG1 ] * [2,4: #G2 #I2 #T2 ]
-[3: /2 width=1 by or_introl/
-|2: elim (eq_bind2_dec I1 I2) #HI
- [ elim (IHG1 G2) -IHG1 #HG
- [ elim (eq_term_dec T1 T2) #HT /2 width=1 by or_introl/ ]
- ]
-]
-@or_intror #H destruct /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/genv.ma".
-
-(* LENGTH OF A GLOBAL ENVIRONMENT *******************************************)
-
-rec definition glength G on G ≝ match G with
-[ GAtom ⇒ 0
-| GPair G _ _ ⇒ ↑(glength G)
-].
-
-interpretation "length (global environment)"
- 'card G = (glength G).
+++ /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/term_weight.ma".
-include "basic_2/syntax/genv.ma".
-
-(* WEIGHT OF A GLOBAL ENVIRONMENT *******************************************)
-
-rec definition gw G ≝ match G with
-[ GAtom ⇒ 0
-| GPair G I T ⇒ gw G + ♯{T}
-].
-
-interpretation "weight (global environment)" 'Weight G = (gw G).
-
-(* Basic properties *********************************************************)
-
-lemma gw_pair: ∀I,G,T. ♯{G} < ♯{G.ⓑ{I}T}.
-normalize /2 width=1 by monotonic_le_plus_r/ qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/lib/bool.ma".
-include "ground_2/lib/arith.ma".
-
-(* ITEMS ********************************************************************)
-
-(* atomic items *)
-inductive item0: Type[0] ≝
- | Sort: nat → item0 (* sort: starting at 0 *)
- | LRef: nat → item0 (* reference by index: starting at 0 *)
- | GRef: nat → item0 (* reference by position: starting at 0 *)
-.
-
-(* unary binding items *)
-inductive bind1: Type[0] ≝
- | Void: bind1 (* exclusion *)
-.
-
-(* binary binding items *)
-inductive bind2: Type[0] ≝
- | Abbr: bind2 (* abbreviation *)
- | Abst: bind2 (* abstraction *)
-.
-
-(* binary non-binding items *)
-inductive flat2: Type[0] ≝
- | Appl: flat2 (* application *)
- | Cast: flat2 (* explicit type annotation *)
-.
-
-(* binary items *)
-inductive item2: Type[0] ≝
- | Bind2: bool → bind2 → item2 (* polarized binding item *)
- | Flat2: flat2 → item2 (* non-binding item *)
-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact destruct_sort_sort_aux: ∀s1,s2. Sort s1 = Sort s2 → s1 = s2.
-#s1 #s2 #H destruct //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma eq_item0_dec: ∀I1,I2:item0. Decidable (I1 = I2).
-* #i1 * #i2 [2,3,4,6,7,8: @or_intror #H destruct ]
-[2: elim (eq_nat_dec i1 i2) |1,3: elim (eq_nat_dec i1 i2) ] /2 width=1 by or_introl/
-#Hni12 @or_intror #H destruct /2 width=1 by/
-qed-.
-
-lemma eq_bind1_dec: ∀I1,I2:bind1. Decidable (I1 = I2).
-* * /2 width=1 by or_introl/
-qed-.
-
-(* Basic_1: was: bind_dec *)
-lemma eq_bind2_dec: ∀I1,I2:bind2. Decidable (I1 = I2).
-* * /2 width=1 by or_introl/
-@or_intror #H destruct
-qed-.
-
-(* Basic_1: was: flat_dec *)
-lemma eq_flat2_dec: ∀I1,I2:flat2. Decidable (I1 = I2).
-* * /2 width=1 by or_introl/
-@or_intror #H destruct
-qed-.
-
-(* Basic_1: was: kind_dec *)
-lemma eq_item2_dec: ∀I1,I2:item2. Decidable (I1 = I2).
-* [ #p1 ] #I1 * [1,3: #p2 ] #I2
-[2,3: @or_intror #H destruct
-| elim (eq_bool_dec p1 p2) #Hp
- [ elim (eq_bind2_dec I1 I2) /2 width=1 by or_introl/ #HI ]
- @or_intror #H destruct /2 width=1 by/
-| elim (eq_flat2_dec I1 I2) /2 width=1 by or_introl/ #HI
- @or_intror #H destruct /2 width=1 by/
-]
-qed-.
-
-(* Basic_1: removed theorems 21:
- s_S s_plus s_plus_sym s_minus minus_s_s s_le s_lt s_inj s_inc
- s_arith0 s_arith1
- r_S r_plus r_plus_sym r_minus r_dis s_r r_arith0 r_arith1
- not_abbr_abst bind_dec_not
-*)
+++ /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/item_sh.ma".
-
-(* SORT DEGREE **************************************************************)
-
-(* sort degree specification *)
-record sd (h:sh): Type[0] ≝ {
- deg : relation nat; (* degree of the sort *)
- deg_total: ∀s. ∃d. deg s d; (* functional relation axioms *)
- deg_mono : ∀s,d1,d2. deg s d1 → deg s d2 → d1 = d2;
- deg_next : ∀s,d. deg s d → deg (next h s) (↓d) (* compatibility condition *)
-}.
-
-(* Notable specifications ***************************************************)
-
-definition deg_O: relation nat ≝ λs,d. d = 0.
-
-definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O ….
-/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ defined.
-
-(* Basic_2A1: includes: deg_SO_pos *)
-inductive deg_SO (h:sh) (s:nat) (s0:nat): predicate nat ≝
-| deg_SO_succ : ∀n. (next h)^n s0 = s → deg_SO h s s0 (↑n)
-| deg_SO_zero: ((∃n. (next h)^n s0 = s) → ⊥) → deg_SO h s s0 0
-.
-
-fact deg_SO_inv_succ_aux: ∀h,s,s0,n0. deg_SO h s s0 n0 → ∀n. n0 = ↑n →
- (next h)^n s0 = s.
-#h #s #s0 #n0 * -n0
-[ #n #Hn #x #H destruct //
-| #_ #x #H destruct
-]
-qed-.
-
-(* Basic_2A1: was: deg_SO_inv_pos *)
-lemma deg_SO_inv_succ: ∀h,s,s0,n. deg_SO h s s0 (↑n) → (next h)^n s0 = s.
-/2 width=3 by deg_SO_inv_succ_aux/ qed-.
-
-lemma deg_SO_refl: ∀h,s. deg_SO h s s 1.
-#h #s @(deg_SO_succ … 0 ?) //
-qed.
-
-lemma deg_SO_gt: ∀h,s1,s2. s1 < s2 → deg_SO h s1 s2 0.
-#h #s1 #s2 #HK12 @deg_SO_zero * #n elim n -n normalize
-[ #H destruct
- elim (lt_refl_false … HK12)
-| #n #_ #H
- lapply (next_lt h ((next h)^n s2)) >H -H #H
- lapply (transitive_lt … H HK12) -s1 #H1
- lapply (nexts_le h s2 n) #H2
- lapply (le_to_lt_to_lt … H2 H1) -h -n #H
- elim (lt_refl_false … H)
-]
-qed.
-
-definition sd_SO: ∀h. nat → sd h ≝ λh,s. mk_sd h (deg_SO h s) ….
-[ #s0
- lapply (nexts_dec h s0 s) *
- [ * /3 width=2 by deg_SO_succ, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
-| #K0 #d1 #d2 * [ #n1 ] #H1 * [1,3: #n2 ] #H2 //
- [ < H2 in H1; -H2 #H
- lapply (nexts_inj … H) -H #H destruct //
- | elim H1 /2 width=2 by ex_intro/
- | elim H2 /2 width=2 by ex_intro/
- ]
-| #s0 #n *
- [ #d #H destruct elim d -d normalize
- /2 width=1 by deg_SO_gt, deg_SO_succ, next_lt/
- | #H1 @deg_SO_zero * #d #H2 destruct
- @H1 -H1 @(ex_intro … (↑d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
- ]
-]
-defined.
-
-rec definition sd_d (h:sh) (s:nat) (d:nat) on d : sd h ≝
- match d with
- [ O ⇒ sd_O h
- | S d ⇒ match d with
- [ O ⇒ sd_SO h s
- | _ ⇒ sd_d h (next h s) d
- ]
- ].
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma deg_inv_pred: ∀h,o,s,d. deg h o (next h s) (↑d) → deg h o s (↑↑d).
-#h #o #s #d #H1
-elim (deg_total h o s) #n #H0
-lapply (deg_next … H0) #H2
-lapply (deg_mono … H1 H2) -H1 -H2 #H >H >S_pred /2 width=2 by ltn_to_ltO/
-qed-.
-
-lemma deg_inv_prec: ∀h,o,s,n,d. deg h o ((next h)^n s) (↑d) → deg h o s (↑(d+n)).
-#h #o #s #n elim n -n normalize /3 width=1 by deg_inv_pred/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma deg_iter: ∀h,o,s,d,n. deg h o s d → deg h o ((next h)^n s) (d-n).
-#h #o #s #d #n elim n -n normalize /3 width=1 by deg_next/
-qed.
-
-lemma deg_next_SO: ∀h,o,s,d. deg h o s (↑d) → deg h o (next h s) d.
-/2 width=1 by deg_next/ qed-.
-
-lemma sd_d_SS: ∀h,s,d. sd_d h s (↑↑d) = sd_d h (next h s) (↑d).
-// qed.
-
-lemma sd_d_correct: ∀h,d,s. deg h (sd_d h s d) s d.
-#h #d elim d -d // #d elim d -d /3 width=1 by deg_inv_pred/
-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/arith.ma".
-
-(* SORT HIERARCHY ***********************************************************)
-
-(* sort hierarchy specification *)
-record sh: Type[0] ≝ {
- next : nat → nat; (* next sort in the hierarchy *)
- next_lt: ∀s. s < next s (* strict monotonicity condition *)
-}.
-
-definition sh_N: sh ≝ mk_sh S ….
-// defined.
-
-(* Basic properties *********************************************************)
-
-lemma nexts_le: ∀h,s,n. s ≤ (next h)^n s.
-#h #s #n elim n -n // normalize #n #IH
-lapply (next_lt h ((next h)^n s)) #H
-lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
-qed.
-
-lemma nexts_lt: ∀h,s,n. s < (next h)^(↑n) s.
-#h #s #n normalize
-lapply (nexts_le h s n) #H
-@(le_to_lt_to_lt … H) //
-qed.
-
-axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2).
-
-axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2.
+++ /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/constructors/star_0.ma".
-include "basic_2/notation/constructors/dxitem_2.ma".
-include "basic_2/notation/constructors/dxbind1_2.ma".
-include "basic_2/notation/constructors/dxbind2_3.ma".
-include "basic_2/notation/constructors/dxvoid_1.ma".
-include "basic_2/notation/constructors/dxabbr_2.ma".
-include "basic_2/notation/constructors/dxabst_2.ma".
-include "basic_2/syntax/bind.ma".
-
-(* LOCAL ENVIRONMENTS *******************************************************)
-
-(* local environments *)
-inductive lenv: Type[0] ≝
-| LAtom: lenv (* empty *)
-| LBind: lenv → bind → lenv (* binding construction *)
-.
-
-interpretation "sort (local environment)"
- 'Star = LAtom.
-
-interpretation "local environment binding construction (generic)"
- 'DxItem L I = (LBind L I).
-
-interpretation "local environment binding construction (unary)"
- 'DxBind1 L I = (LBind L (BUnit I)).
-
-interpretation "local environment binding construction (binary)"
- 'DxBind2 L I T = (LBind L (BPair I T)).
-
-interpretation "void (local environment)"
- 'DxVoid L = (LBind L (BUnit Void)).
-
-interpretation "abbreviation (local environment)"
- 'DxAbbr L T = (LBind L (BPair Abbr T)).
-
-interpretation "abstraction (local environment)"
- 'DxAbst L T = (LBind L (BPair Abst T)).
-
-definition cfull: relation3 lenv bind bind ≝ λL,I1,I2. ⊤.
-
-definition ceq: relation3 lenv term term ≝ λL. eq ….
-
-(* Basic properties *********************************************************)
-
-lemma eq_lenv_dec: ∀L1,L2:lenv. Decidable (L1 = L2).
-#L1 elim L1 -L1 [| #L1 #I1 #IHL1 ] * [2,4: #L2 #I2 ]
-[3: /2 width=1 by or_introl/
-|2: elim (eq_bind_dec I1 I2) #HI
- [ elim (IHL1 L2) -IHL1 #HL /2 width=1 by or_introl/ ]
-]
-@or_intror #H destruct /2 width=1 by/
-qed-.
-
-lemma cfull_dec: ∀L,T1,T2. Decidable (cfull L T1 T2).
-/2 width=1 by or_introl/ qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact destruct_lbind_lbind_aux: ∀I1,I2,L1,L2. L1.ⓘ{I1} = L2.ⓘ{I2} →
- L1 = L2 ∧ I1 = I2.
-#I1 #I2 #L1 #L2 #H destruct /2 width=1 by conj/
-qed-.
-
-(* Basic_2A1: uses: discr_lpair_x_xy *)
-lemma discr_lbind_x_xy: ∀I,L. L = L.ⓘ{I} → ⊥.
-#I #L elim L -L
-[ #H destruct
-| #L #J #IHL #H elim (destruct_lbind_lbind_aux … H) -H (**) (* destruct lemma needed *)
- #H1 #H2 destruct /2 width=1 by/
-]
-qed-.
-
-(* Basic_2A1: uses: discr_lpair_xy_x *)
-lemma discr_lbind_xy_x: ∀I,L. L.ⓘ{I} = L → ⊥.
-/2 width=4 by discr_lbind_x_xy/ 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.ma".
-
-(* LENGTH OF A LOCAL ENVIRONMENT ********************************************)
-
-rec definition length L ≝ match L with
-[ LAtom ⇒ 0
-| LBind L _ ⇒ ↑(length L)
-].
-
-interpretation "length (local environment)" 'card L = (length L).
-
-(* Basic properties *********************************************************)
-
-lemma length_atom: |⋆| = 0.
-// qed.
-
-(* Basic_2A1: uses: length_pair *)
-lemma length_bind: ∀I,L. |L.ⓘ{I}| = ↑|L|.
-// qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma length_inv_zero_dx: ∀L. |L| = 0 → L = ⋆.
-* // #L #I >length_bind
-#H destruct
-qed-.
-
-lemma length_inv_zero_sn: ∀L. 0 = |L| → L = ⋆.
-/2 width=1 by length_inv_zero_dx/ qed-.
-
-(* Basic_2A1: was: length_inv_pos_dx *)
-lemma length_inv_succ_dx: ∀n,L. |L| = ↑n →
- ∃∃I,K. |K| = n & L = K. ⓘ{I}.
-#n *
-[ >length_atom #H destruct
-| #L #I >length_bind /3 width=4 by ex2_2_intro, injective_S/
-]
-qed-.
-
-(* Basic_2A1: was: length_inv_pos_sn *)
-lemma length_inv_succ_sn: ∀n,L. ↑n = |L| →
- ∃∃I,K. n = |K| & L = K. ⓘ{I}.
-#n #L #H lapply (sym_eq ??? H) -H
-#H elim (length_inv_succ_dx … H) -H /2 width=4 by ex2_2_intro/
-qed-.
-
-(* Basic_2A1: removed theorems 1: length_inj *)
+++ /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/bind_weight.ma".
-include "basic_2/syntax/lenv.ma".
-
-(* WEIGHT OF A LOCAL ENVIRONMENT ********************************************)
-
-rec definition lw L ≝ match L with
-[ LAtom ⇒ 0
-| LBind L I ⇒ lw L + ♯{I}
-].
-
-interpretation "weight (local environment)" 'Weight L = (lw L).
-
-(* Basic properties *********************************************************)
-
-(* Basic_2A1: uses: lw_pair *)
-lemma lw_bind: ∀I,L. ♯{L} < ♯{L.ⓘ{I}}.
-normalize /2 width=1 by monotonic_le_plus_r/ qed.
-
-(* Basic_1: removed theorems 4: clt_cong clt_head clt_thead clt_wf_ind *)
-(* Basic_1: removed local theorems 1: clt_wf__q_ind *)
-(* Basic_1: note: clt_thead should be renamed clt_ctail *)
+++ /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/voidstareq_4.ma".
-include "basic_2/syntax/lenv.ma".
-
-(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
-
-inductive lveq: bi_relation nat lenv ≝
-| lveq_atom : lveq 0 (⋆) 0 (⋆)
-| lveq_bind : ∀I1,I2,K1,K2. lveq 0 K1 0 K2 →
- lveq 0 (K1.ⓘ{I1}) 0 (K2.ⓘ{I2})
-| lveq_void_sn: ∀K1,K2,n1. lveq n1 K1 0 K2 →
- lveq (↑n1) (K1.ⓧ) 0 K2
-| lveq_void_dx: ∀K1,K2,n2. lveq 0 K1 n2 K2 →
- lveq 0 K1 (↑n2) (K2.ⓧ)
-.
-
-interpretation "equivalence up to exclusion binders (local environment)"
- 'VoidStarEq L1 n1 n2 L2 = (lveq n1 L1 n2 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lveq_refl: ∀L. L ≋ⓧ*[0, 0] L.
-#L elim L -L /2 width=1 by lveq_atom, lveq_bind/
-qed.
-
-lemma lveq_sym: bi_symmetric … lveq.
-#n1 #n2 #L1 #L2 #H elim H -L1 -L2 -n1 -n2
-/2 width=1 by lveq_atom, lveq_bind, lveq_void_sn, lveq_void_dx/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- 0 = n1 → 0 = n2 →
- ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
- | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
-#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
-[1: /3 width=1 by or_introl, conj/
-|2: /3 width=7 by ex3_4_intro, or_intror/
-|*: #K1 #K2 #n #_ #H1 #H2 destruct
-]
-qed-.
-
-lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 →
- ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
- | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
-/2 width=5 by lveq_inv_zero_aux/ qed-.
-
-fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ∀m1. ↑m1 = n1 →
- ∃∃K1. K1 ≋ⓧ*[m1, 0] L2 & K1.ⓧ = L1 & 0 = n2.
-#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
-[1: #m #H destruct
-|2: #I1 #I2 #K1 #K2 #_ #m #H destruct
-|*: #K1 #K2 #n #HK #m #H destruct /2 width=3 by ex3_intro/
-]
-qed-.
-
-lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1, n2] K2 →
- ∃∃K1. K1 ≋ⓧ*[n1, 0] K2 & K1.ⓧ = L1 & 0 = n2.
-/2 width=3 by lveq_inv_succ_sn_aux/ qed-.
-
-lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1, ↑n2] L2 →
- ∃∃K2. K1 ≋ⓧ*[0, n2] K2 & K2.ⓧ = L2 & 0 = n1.
-#K1 #L2 #n1 #n2 #H
-lapply (lveq_sym … H) -H #H
-elim (lveq_inv_succ_sn … H) -H /3 width=3 by lveq_sym, ex3_intro/
-qed-.
-
-fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ∀m1,m2. ↑m1 = n1 → ↑m2 = n2 → ⊥.
-#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
-[1: #m1 #m2 #H1 #H2 destruct
-|2: #I1 #I2 #K1 #K2 #_ #m1 #m2 #H1 #H2 destruct
-|*: #K1 #K2 #n #_ #m1 #m2 #H1 #H2 destruct
-]
-qed-.
-
-lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1, ↑n2] L2 → ⊥.
-/2 width=9 by lveq_inv_succ_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lveq_inv_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0, 0] K2.ⓘ{I2} → K1 ≋ⓧ*[0, 0] K2.
-#I1 #I2 #K1 #K2 #H
-elim (lveq_inv_zero … H) -H * [| #Z1 #Z2 #Y1 #Y2 #HY ] #H1 #H2 destruct //
-qed-.
-
-lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1, n2] ⋆ → ∧∧ 0 = n1 & 0 = n2.
-* [2: #n1 ] * [2,4: #n2 ] #H
-[ elim (lveq_inv_succ … H)
-| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
-| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
-| /2 width=1 by conj/
-]
-qed-.
-
-lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1, n2] ⋆ →
- ∃∃m1. K1 ≋ⓧ*[m1, 0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2.
-#I1 #K1 * [2: #n1 ] * [2,4: #n2 ] #H
-[ elim (lveq_inv_succ … H)
-| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
-| elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=3 by ex4_intro/
-| elim (lveq_inv_zero … H) -H *
- [ #H1 #H2 destruct
- | #Z1 #Z2 #Y1 #Y2 #_ #H1 #H2 destruct
- ]
-]
-qed-.
-
-lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1, n2] K2.ⓘ{I2} →
- ∃∃m2. ⋆ ≋ⓧ*[0, m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2.
-#I2 #K2 #n1 #n2 #H
-lapply (lveq_sym … H) -H #H
-elim (lveq_inv_bind_atom … H) -H
-/3 width=3 by lveq_sym, ex4_intro/
-qed-.
-
-lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 →
- ∧∧ K1 ≋ⓧ*[0, 0] K2 & 0 = n1 & 0 = n2.
-#I1 #I2 #K1 #K2 #V1 #V2 * [2: #n1 ] * [2,4: #n2 ] #H
-[ elim (lveq_inv_succ … H)
-| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
-| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
-| elim (lveq_inv_zero … H) -H *
- [ #H1 #H2 destruct
- | #Z1 #Z2 #Y1 #Y2 #HY #H1 #H2 destruct /3 width=1 by and3_intro/
- ]
-]
-qed-.
-
-lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1, n2] L2 →
- ∧∧ L1 ≋ ⓧ*[n1, 0] L2 & 0 = n2.
-#L1 #L2 #n1 #n2 #H
-elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=1 by conj/
-qed-.
-
-lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, ↑n2] L2.ⓧ →
- ∧∧ L1 ≋ ⓧ*[0, n2] L2 & 0 = n1.
-#L1 #L2 #n1 #n2 #H
-lapply (lveq_sym … H) -H #H
-elim (lveq_inv_void_succ_sn … H) -H
-/3 width=1 by lveq_sym, conj/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ∨∨ 0 = n1 | 0 = n2.
-#L1 #L2 * [2: #n1 ] * [2,4: #n2 ] #H
-[ elim (lveq_inv_succ … H) ]
-/2 width=1 by or_introl, or_intror/
-qed-.
-
-lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2 → 0 = n1.
-#I1 #K1 #L2 #V1 * [2: #n1 ] // * [2: #n2 ] #H
-[ elim (lveq_inv_succ … H)
-| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
-]
-qed-.
-
-lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 → 0 = n2.
-/3 width=6 by lveq_fwd_pair_sn, lveq_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/syntax/lenv_length.ma".
-include "basic_2/syntax/lveq.ma".
-
-(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
-
-(* Properties with length for local environments ****************************)
-
-lemma lveq_length_eq: ∀L1,L2. |L1| = |L2| → L1 ≋ⓧ*[0, 0] L2.
-#L1 elim L1 -L1
-[ #Y2 #H >(length_inv_zero_sn … H) -Y2 /2 width=3 by lveq_atom, ex_intro/
-| #K1 #I1 #IH #Y2 #H
- elim (length_inv_succ_sn … H) -H #I2 #K2 #HK #H destruct
- /3 width=1 by lveq_bind/
-]
-qed.
-
-(* Forward lemmas with length for local environments ************************)
-
-lemma lveq_fwd_length_le_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → n1 ≤ |L1|.
-#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
-/2 width=1 by le_S_S/
-qed-.
-
-lemma lveq_fwd_length_le_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → n2 ≤ |L2|.
-#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
-/2 width=1 by le_S_S/
-qed-.
-
-lemma lveq_fwd_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ∧∧ |L1|-|L2| = n1 & |L2|-|L1| = n2.
-#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 /2 width=1 by conj/
-#K1 #K2 #n #_ * #H1 #H2 >length_bind /3 width=1 by minus_Sn_m, conj/
-qed-.
-
-lemma lveq_length_fwd_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → |L1| ≤ |L2| → 0 = n1.
-#L1 #L2 #n1 #n2 #H #HL
-elim (lveq_fwd_length … H) -H
->(eq_minus_O … HL) //
-qed-.
-
-lemma lveq_length_fwd_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → |L2| ≤ |L1| → 0 = n2.
-#L1 #L2 #n1 #n2 #H #HL
-elim (lveq_fwd_length … H) -H
->(eq_minus_O … HL) //
-qed-.
-
-lemma lveq_inj_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- |L1| = |L2| → ∧∧ 0 = n1 & 0 = n2.
-#L1 #L2 #n1 #n2 #H #HL
-elim (lveq_fwd_length … H) -H
->HL -HL /2 width=1 by conj/
-qed-.
-
-lemma lveq_fwd_length_plus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- |L1| + n2 = |L2| + n1.
-#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
-/2 width=2 by injective_plus_r/
-qed-.
-
-lemma lveq_fwd_length_eq: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 → |L1| = |L2|.
-/3 width=2 by lveq_fwd_length_plus, injective_plus_l/ qed-.
-
-lemma lveq_fwd_length_minus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- |L1| - n1 = |L2| - n2.
-/3 width=3 by lveq_fwd_length_plus, lveq_fwd_length_le_dx, lveq_fwd_length_le_sn, plus_to_minus_2/ qed-.
-
-lemma lveq_fwd_abst_bind_length_le: ∀I1,I2,L1,L2,V1,n1,n2.
- L1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2.ⓘ{I2} → |L1| ≤ |L2|.
-#I1 #I2 #L1 #L2 #V1 #n1 #n2 #HL
-lapply (lveq_fwd_pair_sn … HL) #H destruct
-elim (lveq_fwd_length … HL) -HL >length_bind >length_bind //
-qed-.
-
-lemma lveq_fwd_bind_abst_length_le: ∀I1,I2,L1,L2,V2,n1,n2.
- L1.ⓘ{I1} ≋ⓧ*[n1, n2] L2.ⓑ{I2}V2 → |L2| ≤ |L1|.
-/3 width=6 by lveq_fwd_abst_bind_length_le, lveq_sym/ qed-.
-
-(* Inversion lemmas with length for local environments **********************)
-
-lemma lveq_inv_void_dx_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2.ⓧ → |L1| ≤ |L2| →
- ∃∃m2. L1 ≋ ⓧ*[n1, m2] L2 & 0 = n1 & ↑m2 = n2.
-#L1 #L2 #n1 #n2 #H #HL12
-lapply (lveq_fwd_length_plus … H) normalize >plus_n_Sm #H0
-lapply (plus2_inv_le_sn … H0 HL12) -H0 -HL12 #H0
-elim (le_inv_S1 … H0) -H0 #m2 #_ #H0 destruct
-elim (lveq_inv_void_succ_dx … H) -H /2 width=3 by ex3_intro/
-qed-.
-
-lemma lveq_inv_void_sn_length: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[n1, n2] L2 → |L2| ≤ |L1| →
- ∃∃m1. L1 ≋ ⓧ*[m1, n2] L2 & ↑m1 = n1 & 0 = n2.
-#L1 #L2 #n1 #n2 #H #HL
-lapply (lveq_sym … H) -H #H
-elim (lveq_inv_void_dx_length … H HL) -H -HL
-/3 width=4 by lveq_sym, 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/syntax/lveq_length.ma".
-
-(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
-
-(* Main inversion lemmas ****************************************************)
-
-theorem lveq_inv_bind: ∀K1,K2. K1 ≋ⓧ*[0, 0] K2 →
- ∀I1,I2,m1,m2. K1.ⓘ{I1} ≋ⓧ*[m1, m2] K2.ⓘ{I2} →
- ∧∧ 0 = m1 & 0 = m2.
-#K1 #K2 #HK #I1 #I2 #m1 #m2 #H
-lapply (lveq_fwd_length_eq … HK) -HK #HK
-elim (lveq_inj_length … H) -H normalize /3 width=1 by conj, eq_f/
-qed-.
-
-theorem lveq_inj: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
- ∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 →
- ∧∧ n1 = m1 & n2 = m2.
-#L1 #L2 #n1 #n2 #Hn #m1 #m2 #Hm
-elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct
-elim (lveq_fwd_length … Hm) -Hm #H1 #H2 destruct
-/2 width=1 by conj/
-qed-.
-
-theorem lveq_inj_void_sn_ge: ∀K1,K2. |K2| ≤ |K1| →
- ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
- ∀m1,m2. K1.ⓧ ≋ⓧ*[m1, m2] K2 →
- ∧∧ ↑n1 = m1 & 0 = m2 & 0 = n2.
-#L1 #L2 #HL #n1 #n2 #Hn #m1 #m2 #Hm
-elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct
-elim (lveq_fwd_length … Hm) -Hm #H1 #H2 destruct
->length_bind >eq_minus_S_pred >(eq_minus_O … HL)
-/3 width=4 by plus_minus, and3_intro/
-qed-.
-
-theorem lveq_inj_void_dx_le: ∀K1,K2. |K1| ≤ |K2| →
- ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
- ∀m1,m2. K1 ≋ⓧ*[m1, m2] K2.ⓧ →
- ∧∧ ↑n2 = m2 & 0 = m1 & 0 = n1.
-/3 width=5 by lveq_inj_void_sn_ge, lveq_sym/ qed-. (* auto: 2x lveq_sym *)
+++ /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/stareq_4.ma".
-include "basic_2/syntax/item_sd.ma".
-include "basic_2/syntax/term.ma".
-
-(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************)
-
-inductive tdeq (h) (o): relation term ≝
-| tdeq_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → tdeq h o (⋆s1) (⋆s2)
-| tdeq_lref: ∀i. tdeq h o (#i) (#i)
-| tdeq_gref: ∀l. tdeq h o (§l) (§l)
-| tdeq_pair: ∀I,V1,V2,T1,T2. tdeq h o V1 V2 → tdeq h o T1 T2 → tdeq h o (②{I}V1.T1) (②{I}V2.T2)
-.
-
-interpretation
- "context-free degree-based equivalence (term)"
- 'StarEq h o T1 T2 = (tdeq h o T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact tdeq_inv_sort1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀s1. X = ⋆s1 →
- ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
-#h #o #X #Y * -X -Y
-[ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/
-| #i #s #H destruct
-| #l #s #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #s #H destruct
-]
-qed-.
-
-lemma tdeq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ≛[h, o] Y →
- ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
-/2 width=3 by tdeq_inv_sort1_aux/ qed-.
-
-fact tdeq_inv_lref1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀i. X = #i → Y = #i.
-#h #o #X #Y * -X -Y //
-[ #s1 #s2 #d #_ #_ #j #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
-]
-qed-.
-
-lemma tdeq_inv_lref1: ∀h,o,Y,i. #i ≛[h, o] Y → Y = #i.
-/2 width=5 by tdeq_inv_lref1_aux/ qed-.
-
-fact tdeq_inv_gref1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀l. X = §l → Y = §l.
-#h #o #X #Y * -X -Y //
-[ #s1 #s2 #d #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
-]
-qed-.
-
-lemma tdeq_inv_gref1: ∀h,o,Y,l. §l ≛[h, o] Y → Y = §l.
-/2 width=5 by tdeq_inv_gref1_aux/ qed-.
-
-fact tdeq_inv_pair1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀I,V1,T1. X = ②{I}V1.T1 →
- ∃∃V2,T2. V1 ≛[h, o] V2 & T1 ≛[h, o] T2 & Y = ②{I}V2.T2.
-#h #o #X #Y * -X -Y
-[ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct
-| #i #J #W1 #U1 #H destruct
-| #l #J #W1 #U1 #H destruct
-| #I #V1 #V2 #T1 #T2 #HV #HT #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma tdeq_inv_pair1: ∀h,o,I,V1,T1,Y. ②{I}V1.T1 ≛[h, o] Y →
- ∃∃V2,T2. V1 ≛[h, o] V2 & T1 ≛[h, o] T2 & Y = ②{I}V2.T2.
-/2 width=3 by tdeq_inv_pair1_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma tdeq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ≛[h, o] Y → ∀d. deg h o s1 d →
- ∃∃s2. deg h o s2 d & Y = ⋆s2.
-#h #o #Y #s1 #H #d #Hs1 elim (tdeq_inv_sort1 … H) -H
-#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/
-qed-.
-
-lemma tdeq_inv_sort_deg: ∀h,o,s1,s2. ⋆s1 ≛[h, o] ⋆s2 →
- ∀d1,d2. deg h o s1 d1 → deg h o s2 d2 →
- d1 = d2.
-#h #o #s1 #y #H #d1 #d2 #Hs1 #Hy
-elim (tdeq_inv_sort1_deg … H … Hs1) -s1 #s2 #Hs2 #H destruct
-<(deg_mono h o … Hy … Hs2) -s2 -d1 //
-qed-.
-
-lemma tdeq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ≛[h, o] ②{I2}V2.T2 →
- ∧∧ I1 = I2 & V1 ≛[h, o] V2 & T1 ≛[h, o] T2.
-#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H elim (tdeq_inv_pair1 … H) -H
-#V0 #T0 #HV #HT #H destruct /2 width=1 by and3_intro/
-qed-.
-
-lemma tdeq_inv_pair_xy_x: ∀h,o,I,V,T. ②{I}V.T ≛[h, o] V → ⊥.
-#h #o #I #V elim V -V
-[ #J #T #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
-| #J #X #Y #IHX #_ #T #H elim (tdeq_inv_pair … H) -H #H #HY #_ destruct /2 width=2 by/
-]
-qed-.
-
-lemma tdeq_inv_pair_xy_y: ∀h,o,I,T,V. ②{I}V.T ≛[h, o] T → ⊥.
-#h #o #I #T elim T -T
-[ #J #V #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
-| #J #X #Y #_ #IHY #V #H elim (tdeq_inv_pair … H) -H #H #_ #HY destruct /2 width=2 by/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma tdeq_fwd_atom1: ∀h,o,I,Y. ⓪{I} ≛[h, o] Y → ∃J. Y = ⓪{J}.
-#h #o * #x #Y #H [ elim (tdeq_inv_sort1 … H) -H ]
-/3 width=4 by tdeq_inv_gref1, tdeq_inv_lref1, ex_intro/
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma tdeq_refl: ∀h,o. reflexive … (tdeq h o).
-#h #o #T elim T -T /2 width=1 by tdeq_pair/
-* /2 width=1 by tdeq_lref, tdeq_gref/
-#s elim (deg_total h o s) /2 width=3 by tdeq_sort/
-qed.
-
-lemma tdeq_sym: ∀h,o. symmetric … (tdeq h o).
-#h #o #T1 #T2 #H elim H -T1 -T2
-/2 width=3 by tdeq_sort, tdeq_lref, tdeq_gref, tdeq_pair/
-qed-.
-
-lemma tdeq_dec: ∀h,o,T1,T2. Decidable (T1 ≛[h, o] T2).
-#h #o #T1 elim T1 -T1 [ * #s1 | #I1 #V1 #T1 #IHV #IHT ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
-[ elim (deg_total h o s1) #d1 #H1
- elim (deg_total h o s2) #d2 #H2
- elim (eq_nat_dec d1 d2) #Hd12 destruct /3 width=3 by tdeq_sort, or_introl/
- @or_intror #H
- lapply (tdeq_inv_sort_deg … H … H1 H2) -H -H1 -H2 /2 width=1 by/
-|2,3,13:
- @or_intror #H
- elim (tdeq_inv_sort1 … H) -H #x1 #x2 #_ #_ #H destruct
-|4,6,14:
- @or_intror #H
- lapply (tdeq_inv_lref1 … H) -H #H destruct
-|5:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (tdeq_inv_lref1 … H) -H #H destruct /2 width=1 by/
-|7,8,15:
- @or_intror #H
- lapply (tdeq_inv_gref1 … H) -H #H destruct
-|9:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (tdeq_inv_gref1 … H) -H #H destruct /2 width=1 by/
-|10,11,12:
- @or_intror #H
- elim (tdeq_inv_pair1 … H) -H #X1 #X2 #_ #_ #H destruct
-|16:
- elim (eq_item2_dec I1 I2) #HI12 destruct
- [ elim (IHV V2) -IHV #HV12
- elim (IHT T2) -IHT #HT12
- [ /3 width=1 by tdeq_pair, or_introl/ ]
- ]
- @or_intror #H
- elim (tdeq_inv_pair … H) -H /2 width=1 by/
-]
-qed-.
-
-(* Negated inversion lemmas *************************************************)
-
-lemma tdneq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2.
- (②{I1}V1.T1 ≛[h, o] ②{I2}V2.T2 → ⊥) →
- ∨∨ I1 = I2 → ⊥
- | (V1 ≛[h, o] V2 → ⊥)
- | (T1 ≛[h, o] T2 → ⊥).
-#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H12
-elim (eq_item2_dec I1 I2) /3 width=1 by or3_intro0/ #H destruct
-elim (tdeq_dec h o V1 V2) /3 width=1 by or3_intro1/
-elim (tdeq_dec h o T1 T2) /4 width=1 by tdeq_pair, or3_intro2/
-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/stareq_5.ma".
-include "basic_2/syntax/cext2.ma".
-include "basic_2/syntax/tdeq.ma".
-
-(* EXTENDED DEGREE-BASED EQUIVALENCE ****************************************)
-
-definition tdeq_ext: ∀h. sd h → relation bind ≝
- λh,o. ext2 (tdeq h o).
-
-definition cdeq: ∀h. sd h → relation3 lenv term term ≝
- λh,o,L. tdeq h o.
-
-definition cdeq_ext: ∀h. sd h → relation3 lenv bind bind ≝
- λh,o. cext2 (cdeq h o).
-
-interpretation
- "context-free degree-based equivalence (binder)"
- 'StarEq h o I1 I2 = (tdeq_ext h o I1 I2).
-
-interpretation
- "context-dependent degree-based equivalence (term)"
- 'StarEq h o L T1 T2 = (cdeq h o L T1 T2).
-
-interpretation
- "context-dependent degree-based equivalence (binder)"
- 'StarEq h o L I1 I2 = (cdeq_ext h o L I1 I2).
+++ /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/tdeq.ma".
-
-(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************)
-
-(* Main properties **********************************************************)
-
-theorem tdeq_trans: ∀h,o. Transitive … (tdeq h o).
-#h #o #T1 #T #H elim H -T1 -T
-[ #s1 #s #d #Hs1 #Hs #X #H
- elim (tdeq_inv_sort1_deg … H … Hs) -s /2 width=3 by tdeq_sort/
-| #i1 #i #H <(tdeq_inv_lref1 … H) -H //
-| #l1 #l #H <(tdeq_inv_gref1 … H) -H //
-| #I #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H
- elim (tdeq_inv_pair1 … H) -H /3 width=1 by tdeq_pair/
-]
-qed-.
-
-theorem tdeq_canc_sn: ∀h,o. left_cancellable … (tdeq h o).
-/3 width=3 by tdeq_trans, tdeq_sym/ qed-.
-
-theorem tdeq_canc_dx: ∀h,o. right_cancellable … (tdeq h o).
-/3 width=3 by tdeq_trans, tdeq_sym/ qed-.
-
-theorem tdeq_repl: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
- ∀U1. T1 ≛[h, o] U1 → ∀U2. T2 ≛[h, o] U2 → U1 ≛[h, o] U2.
-/3 width=3 by tdeq_canc_sn, tdeq_trans/ qed-.
-
-(* Negated main properies ***************************************************)
-
-theorem tdeq_tdneq_trans: ∀h,o,T1,T. T1 ≛[h, o] T → ∀T2. (T ≛[h, o] T2 → ⊥) →
- T1 ≛[h, o] T2 → ⊥.
-/3 width=3 by tdeq_canc_sn/ qed-.
-
-theorem tdneq_tdeq_canc_dx: ∀h,o,T1,T. (T1 ≛[h, o] T → ⊥) → ∀T2. T2 ≛[h, o] T →
- T1 ≛[h, o] T2 → ⊥.
-/3 width=3 by 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/notation/constructors/item0_1.ma".
-include "basic_2/notation/constructors/snitem2_3.ma".
-include "basic_2/notation/constructors/snbind2_4.ma".
-include "basic_2/notation/constructors/snbind2pos_3.ma".
-include "basic_2/notation/constructors/snbind2neg_3.ma".
-include "basic_2/notation/constructors/snflat2_3.ma".
-include "basic_2/notation/constructors/star_1.ma".
-include "basic_2/notation/constructors/lref_1.ma".
-include "basic_2/notation/constructors/gref_1.ma".
-include "basic_2/notation/constructors/snabbr_3.ma".
-include "basic_2/notation/constructors/snabbrpos_2.ma".
-include "basic_2/notation/constructors/snabbrneg_2.ma".
-include "basic_2/notation/constructors/snabst_3.ma".
-include "basic_2/notation/constructors/snabstpos_2.ma".
-include "basic_2/notation/constructors/snabstneg_2.ma".
-include "basic_2/notation/constructors/snappl_2.ma".
-include "basic_2/notation/constructors/sncast_2.ma".
-include "basic_2/syntax/item.ma".
-
-(* TERMS ********************************************************************)
-
-(* terms *)
-inductive term: Type[0] ≝
- | TAtom: item0 → term (* atomic item construction *)
- | TPair: item2 → term → term → term (* binary item construction *)
-.
-
-interpretation "term construction (atomic)"
- 'Item0 I = (TAtom I).
-
-interpretation "term construction (binary)"
- 'SnItem2 I T1 T2 = (TPair I T1 T2).
-
-interpretation "term binding construction (binary)"
- 'SnBind2 p I T1 T2 = (TPair (Bind2 p I) T1 T2).
-
-interpretation "term positive binding construction (binary)"
- 'SnBind2Pos I T1 T2 = (TPair (Bind2 true I) T1 T2).
-
-interpretation "term negative binding construction (binary)"
- 'SnBind2Neg I T1 T2 = (TPair (Bind2 false I) T1 T2).
-
-interpretation "term flat construction (binary)"
- 'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
-
-interpretation "sort (term)"
- 'Star s = (TAtom (Sort s)).
-
-interpretation "local reference (term)"
- 'LRef i = (TAtom (LRef i)).
-
-interpretation "global reference (term)"
- 'GRef l = (TAtom (GRef l)).
-
-interpretation "abbreviation (term)"
- 'SnAbbr p T1 T2 = (TPair (Bind2 p Abbr) T1 T2).
-
-interpretation "positive abbreviation (term)"
- 'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2).
-
-interpretation "negative abbreviation (term)"
- 'SnAbbrNeg T1 T2 = (TPair (Bind2 false Abbr) T1 T2).
-
-interpretation "abstraction (term)"
- 'SnAbst p T1 T2 = (TPair (Bind2 p Abst) T1 T2).
-
-interpretation "positive abstraction (term)"
- 'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2).
-
-interpretation "negative abstraction (term)"
- 'SnAbstNeg T1 T2 = (TPair (Bind2 false Abst) T1 T2).
-
-interpretation "application (term)"
- 'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
-
-interpretation "native type annotation (term)"
- 'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2).
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: term_dec *)
-lemma eq_term_dec: ∀T1,T2:term. Decidable (T1 = T2).
-#T1 elim T1 -T1 #I1 [| #V1 #T1 #IHV1 #IHT1 ] * #I2 [2,4: #V2 #T2 ]
-[1,4: @or_intror #H destruct
-| elim (eq_item2_dec I1 I2) #HI
- [ elim (IHV1 V2) -IHV1 #HV
- [ elim (IHT1 T2) -IHT1 /2 width=1 by or_introl/ #HT ]
- ]
- @or_intror #H destruct /2 width=1 by/
-| elim (eq_item0_dec I1 I2) /2 width=1 by or_introl/ #HI
- @or_intror #H destruct /2 width=1 by/
-]
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact destruct_tatom_tatom_aux: ∀I1,I2. ⓪{I1} = ⓪{I2} → I1 = I2.
-#I1 #I2 #H destruct //
-qed-.
-
-fact destruct_tpair_tpair_aux: ∀I1,I2,T1,T2,V1,V2. ②{I1}T1.V1 = ②{I2}T2.V2 →
- ∧∧T1 = T2 & I1 = I2 & V1 = V2.
-#I1 #I2 #T1 #T2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
-qed-.
-
-lemma discr_tpair_xy_x: ∀I,T,V. ②{I}V.T = V → ⊥.
-#I #T #V elim V -V
-[ #J #H destruct
-| #J #W #U #IHW #_ #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
-]
-qed-.
-
-(* Basic_1: was: thead_x_y_y *)
-lemma discr_tpair_xy_y: ∀I,V,T. ②{I}V.T = T → ⊥.
-#I #V #T elim T -T
-[ #J #H destruct
-| #J #W #U #_ #IHU #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
-]
-qed-.
-
-lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2.
- (②{I}V1.T1 = ②{I}V2.T2 → ⊥) →
- (V1 = V2 → ⊥) ∨ (V1 = V2 ∧ (T1 = T2 → ⊥)).
-#I #V1 #T1 #V2 #T2 #H
-elim (eq_term_dec V1 V2) /3 width=1 by or_introl/ #HV12 destruct
-@or_intror @conj // #HT12 destruct /2 width=1 by/
-qed-.
-
-lemma eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2.
- (②{I} V1. T1 = ②{I}V2.T2 → ⊥) →
- (T1 = T2 → ⊥) ∨ (T1 = T2 ∧ (V1 = V2 → ⊥)).
-#I #V1 #T1 #V2 #T2 #H
-elim (eq_term_dec T1 T2) /3 width=1 by or_introl/ #HT12 destruct
-@or_intror @conj // #HT12 destruct /2 width=1 by/
-qed-.
-
-(* Basic_1: removed theorems 3:
- not_void_abst not_abbr_void not_abst_void
-*)
+++ /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/simple_1.ma".
-include "basic_2/syntax/term.ma".
-
-(* SIMPLE (NEUTRAL) TERMS ***************************************************)
-
-inductive simple: predicate term ≝
- | simple_atom: ∀I. simple (⓪{I})
- | simple_flat: ∀I,V,T. simple (ⓕ{I}V.T)
-.
-
-interpretation "simple (term)" 'Simple T = (simple T).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact simple_inv_bind_aux: ∀T. 𝐒⦃T⦄ → ∀p,J,W,U. T = ⓑ{p,J}W.U → ⊥.
-#T * -T
-[ #I #p #J #W #U #H destruct
-| #I #V #T #a #J #W #U #H destruct
-]
-qed-.
-
-lemma simple_inv_bind: ∀p,I,V,T. 𝐒⦃ⓑ{p,I} V. T⦄ → ⊥.
-/2 width=7 by simple_inv_bind_aux/ qed-.
-
-lemma simple_inv_pair: ∀I,V,T. 𝐒⦃②{I}V.T⦄ → ∃J. I = Flat2 J.
-* /2 width=2 by ex_intro/
-#p #I #V #T #H elim (simple_inv_bind … 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 "ground_2/lib/list.ma".
-include "basic_2/notation/functions/snapplvector_2.ma".
-include "basic_2/syntax/term_simple.ma".
-
-(* TERMS ********************************************************************)
-
-rec definition applv Vs T on Vs ≝
- match Vs with
- [ nil ⇒ T
- | cons hd tl ⇒ ⓐhd. (applv tl T)
- ].
-
-interpretation "application to vector (term)"
- 'SnApplVector Vs T = (applv Vs T).
-
-(* Basic properties *********************************************************)
-
-lemma applv_nil: ∀T. ⒶⒺ.T = T.
-// qed.
-
-lemma applv_cons: ∀V,Vs,T. ⒶV⨮Vs.T = ⓐV.ⒶVs.T.
-// qed.
-
-(* Properties with simple terms *********************************************)
-
-lemma applv_simple: ∀T,Vs. 𝐒⦃T⦄ → 𝐒⦃ⒶVs.T⦄.
-#T * //
-qed.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/notation/functions/weight_1.ma".
-include "basic_2/syntax/term.ma".
-
-(* WEIGHT OF A TERM *********************************************************)
-
-rec definition tw T ≝ match T with
-[ TAtom _ ⇒ 1
-| TPair _ V T ⇒ ↑(tw V + tw T)
-].
-
-interpretation "weight (term)" 'Weight T = (tw T).
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: tweight_lt *)
-lemma tw_pos: ∀T. 1 ≤ ♯{T}.
-#T elim T -T //
-qed.
-
-(* Basic_1: removed theorems 11:
- wadd_le wadd_lt wadd_O weight_le weight_eq weight_add_O
- weight_add_S tlt_trans tlt_head_sx tlt_head_dx tlt_wf_ind
-*)
-(* Basic_1: removed local theorems 1: q_ind *)
+++ /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/topiso_4.ma".
-include "basic_2/syntax/item_sd.ma".
-include "basic_2/syntax/term.ma".
-
-(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
-
-(* Basic_2A1: includes: tsts_atom tsts_pair *)
-inductive theq (h) (o): relation term ≝
-| theq_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → theq h o (⋆s1) (⋆s2)
-| theq_lref: ∀i. theq h o (#i) (#i)
-| theq_gref: ∀l. theq h o (§l) (§l)
-| theq_pair: ∀I,V1,V2,T1,T2. theq h o (②{I}V1.T1) (②{I}V2.T2)
-.
-
-interpretation "head equivalence (term)" 'TopIso h o T1 T2 = (theq h o T1 T2).
-
-(* Basic inversion lemmas ***************************************************)
-
-fact theq_inv_sort1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀s1. X = ⋆s1 →
- ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
-#h #o #X #Y * -X -Y
-[ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/
-| #i #s #H destruct
-| #l #s #H destruct
-| #I #V1 #V2 #T1 #T2 #s #H destruct
-]
-qed-.
-
-(* Basic_1: was just: iso_gen_sort *)
-lemma theq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ⩳[h, o] Y →
- ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
-/2 width=3 by theq_inv_sort1_aux/ qed-.
-
-fact theq_inv_lref1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀i. X = #i → Y = #i.
-#h #o #X #Y * -X -Y //
-[ #s1 #s2 #d #_ #_ #j #H destruct
-| #I #V1 #V2 #T1 #T2 #j #H destruct
-]
-qed-.
-
-(* Basic_1: was: iso_gen_lref *)
-lemma theq_inv_lref1: ∀h,o,Y,i. #i ⩳[h, o] Y → Y = #i.
-/2 width=5 by theq_inv_lref1_aux/ qed-.
-
-fact theq_inv_gref1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀l. X = §l → Y = §l.
-#h #o #X #Y * -X -Y //
-[ #s1 #s2 #d #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #k #H destruct
-]
-qed-.
-
-lemma theq_inv_gref1: ∀h,o,Y,l. §l ⩳[h, o] Y → Y = §l.
-/2 width=5 by theq_inv_gref1_aux/ qed-.
-
-fact theq_inv_pair1_aux: ∀h,o,T1,T2. T1 ⩳[h, o] T2 →
- ∀J,W1,U1. T1 = ②{J}W1.U1 →
- ∃∃W2,U2. T2 = ②{J}W2.U2.
-#h #o #T1 #T2 * -T1 -T2
-[ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct
-| #i #J #W1 #U1 #H destruct
-| #l #J #W1 #U1 #H destruct
-| #I #V1 #V2 #T1 #T2 #J #W1 #U1 #H destruct /2 width=3 by ex1_2_intro/
-]
-qed-.
-
-(* Basic_1: was: iso_gen_head *)
-(* Basic_2A1: was: tsts_inv_pair1 *)
-lemma theq_inv_pair1: ∀h,o,J,W1,U1,T2. ②{J}W1.U1 ⩳[h, o] T2 →
- ∃∃W2,U2. T2 = ②{J}W2. U2.
-/2 width=7 by theq_inv_pair1_aux/ qed-.
-
-fact theq_inv_pair2_aux: ∀h,o,T1,T2. T1 ⩳[h, o] T2 →
- ∀J,W2,U2. T2 = ②{J}W2.U2 →
- ∃∃W1,U1. T1 = ②{J}W1.U1.
-#h #o #T1 #T2 * -T1 -T2
-[ #s1 #s2 #d #_ #_ #J #W2 #U2 #H destruct
-| #i #J #W2 #U2 #H destruct
-| #l #J #W2 #U2 #H destruct
-| #I #V1 #V2 #T1 #T2 #J #W2 #U2 #H destruct /2 width=3 by ex1_2_intro/
-]
-qed-.
-
-(* Basic_2A1: was: tsts_inv_pair2 *)
-lemma theq_inv_pair2: ∀h,o,J,T1,W2,U2. T1 ⩳[h, o] ②{J}W2.U2 →
- ∃∃W1,U1. T1 = ②{J}W1.U1.
-/2 width=7 by theq_inv_pair2_aux/ qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma theq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ⩳[h, o] Y → ∀d. deg h o s1 d →
- ∃∃s2. deg h o s2 d & Y = ⋆s2.
-#h #o #Y #s1 #H #d #Hs1 elim (theq_inv_sort1 … H) -H
-#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/
-qed-.
-
-lemma theq_inv_sort_deg: ∀h,o,s1,s2. ⋆s1 ⩳[h, o] ⋆s2 →
- ∀d1,d2. deg h o s1 d1 → deg h o s2 d2 →
- d1 = d2.
-#h #o #s1 #y #H #d1 #d2 #Hs1 #Hy
-elim (theq_inv_sort1_deg … H … Hs1) -s1 #s2 #Hs2 #H destruct
-<(deg_mono h o … Hy … Hs2) -s2 -d1 //
-qed-.
-
-lemma theq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ⩳[h, o] ②{I2}V2.T2 →
- I1 = I2.
-#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H elim (theq_inv_pair1 … H) -H
-#V0 #T0 #H destruct //
-qed-.
-
-(* Basic properties *********************************************************)
-
-(* Basic_1: was: iso_refl *)
-(* Basic_2A1: was: tsts_refl *)
-lemma theq_refl: ∀h,o. reflexive … (theq h o).
-#h #o * //
-* /2 width=1 by theq_lref, theq_gref/
-#s elim (deg_total h o s) /2 width=3 by theq_sort/
-qed.
-
-(* Basic_2A1: was: tsts_sym *)
-lemma theq_sym: ∀h,o. symmetric … (theq h o).
-#h #o #T1 #T2 * -T1 -T2 /2 width=3 by theq_sort/
-qed-.
-
-(* Basic_2A1: was: tsts_dec *)
-lemma theq_dec: ∀h,o,T1,T2. Decidable (T1 ⩳[h, o] T2).
-#h #o * [ * #s1 | #I1 #V1 #T1 ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
-[ elim (deg_total h o s1) #d1 #H1
- elim (deg_total h o s2) #d2 #H2
- elim (eq_nat_dec d1 d2) #Hd12 destruct /3 width=3 by theq_sort, or_introl/
- @or_intror #H
- lapply (theq_inv_sort_deg … H … H1 H2) -H -H1 -H2 /2 width=1 by/
-|2,3,13:
- @or_intror #H
- elim (theq_inv_sort1 … H) -H #x1 #x2 #_ #_ #H destruct
-|4,6,14:
- @or_intror #H
- lapply (theq_inv_lref1 … H) -H #H destruct
-|5:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (theq_inv_lref1 … H) -H #H destruct /2 width=1 by/
-|7,8,15:
- @or_intror #H
- lapply (theq_inv_gref1 … H) -H #H destruct
-|9:
- elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
- @or_intror #H
- lapply (theq_inv_gref1 … H) -H #H destruct /2 width=1 by/
-|10,11,12:
- @or_intror #H
- elim (theq_inv_pair1 … H) -H #X1 #X2 #H destruct
-|16:
- elim (eq_item2_dec I1 I2) #HI12 destruct
- [ /3 width=1 by theq_pair, or_introl/ ]
- @or_intror #H
- lapply (theq_inv_pair … H) -H /2 width=1 by/
-]
-qed-.
-
-(* Basic_2A1: removed theorems 2:
- tsts_inv_atom1 tsts_inv_atom2
-*)
+++ /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/term_simple.ma".
-include "basic_2/syntax/theq.ma".
-
-(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
-
-(* Properies with simple (neutral) terms ************************************)
-
-(* Basic_2A1: was: simple_tsts_repl_dx *)
-lemma simple_theq_repl_dx: ∀h,o,T1,T2. T1 ⩳[h, o] T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
-#h #o #T1 #T2 * -T1 -T2 //
-#I #V1 #V2 #T1 #T2 #H
-elim (simple_inv_pair … H) -H #J #H destruct //
-qed-.
-
-(* Basic_2A1: was: simple_tsts_repl_sn *)
-lemma simple_theq_repl_sn: ∀h,o,T1,T2. T1 ⩳[h, o] T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
-/3 width=5 by simple_theq_repl_dx, theq_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/syntax/term_vector.ma".
-include "basic_2/syntax/theq_simple.ma".
-
-(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
-
-(* Advanced inversion lemmas with simple (neutral) terms ********************)
-
-(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *)
-(* Basic_2A1: was: tsts_inv_bind_applv_simple *)
-lemma theq_inv_applv_bind_simple: ∀h,o,p,I,Vs,V2,T1,T2. ⒶVs.T1 ⩳[h, o] ⓑ{p,I}V2.T2 →
- 𝐒⦃T1⦄ → ⊥.
-#h #o #p #I #Vs #V2 #T1 #T2 #H elim (theq_inv_pair2 … H) -H
-#V0 #T0 elim Vs -Vs normalize
-[ #H destruct #H /2 width=5 by simple_inv_bind/
-| #V #Vs #_ #H destruct
-]
-qed-.
+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/syntax/tdeq.ma".
-include "basic_2/syntax/theq.ma".
-
-(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
-
-(* Properties with degree-based equivalence for terms ***********************)
-
-lemma tdeq_theq: ∀h,o,T1,T2. T1 ≛[h, o] T2 → T1 ⩳[h, o] T2.
-#h #o #T1 #T2 * -T1 -T2 /2 width=3 by theq_sort, theq_pair/
-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/theq.ma".
-
-(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: iso_trans *)
-(* Basic_2A1: was: tsts_trans *)
-theorem theq_trans: ∀h,o. Transitive … (theq h o).
-#h #o #T1 #T * -T1 -T
-[ #s1 #s #d #Hs1 #Hs #X #H
- elim (theq_inv_sort1_deg … H … Hs) -s /2 width=3 by theq_sort/
-| #i1 #i #H <(theq_inv_lref1 … H) -H //
-| #l1 #l #H <(theq_inv_gref1 … H) -H //
-| #I #V1 #V #T1 #T #X #H
- elim (theq_inv_pair1 … H) -H #V2 #T2 #H destruct //
-]
-qed-.
-
-(* Basic_2A1: was: tsts_canc_sn *)
-theorem theq_canc_sn: ∀h,o. left_cancellable … (theq h o).
-/3 width=3 by theq_trans, theq_sym/ qed-.
-
-(* Basic_2A1: was: tsts_canc_dx *)
-theorem theq_canc_dx: ∀h,o. right_cancellable … (theq h o).
-/3 width=3 by theq_trans, theq_sym/ qed-.
]
}
]
- class "water"
- [ { "iterated static typing" * } {
- [ { "iterated generic extension of a context-sensitive relation" * } {
- [ [ "for lenvs on referred entries" ] "rexs" + "( ? ⪤*[?,?] ? )" "rexs_length" + "rexs_lex" + "rexs_drops" + "rexs_fqup" + "rexs_rexs" * ]
- }
- ]
- }
- ]
- class "green"
- [ { "static typing" * } {
- [ { "generic reducibility" * } {
- [ [ "restricted refinement for lenvs" ] "lsubc" + "( ? ⊢ ? ⫃[?] ? )" "lsubc_drops" + "lsubc_lsubr" + "lsubc_lsuba" * ]
- [ [ "candidates" ] "gcp_cr" + "( ⦃?,?,?⦄ ϵ[?] 〚?〛 )" "gcp_aaa" * ]
- [ [ "computation properties" ] "gcp" *]
- }
- ]
- [ { "atomic arity assignment" * } {
- [ [ "restricted refinement for lenvs" ] "lsuba" + "( ? ⊢ ? ⫃⁝ ? )" "lsuba_drops" + "lsuba_lsubr" + "lsuba_aaa" + "lsuba_lsuba" * ]
- [ [ "for terms" ] "aaa" + "( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + "aaa_rdeq" + "aaa_fdeq" + "aaa_aaa" * ]
- }
- ]
- [ { "degree-based equivalence" * } {
- [ [ "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" ] "req" + "( ? ≡[?] ? )" "req_fqup" + "req_fsle" * ]
- }
- ]
- [ { "generic extension of a context-sensitive relation" * } {
- [ [ "for lenvs on referred entries" ] "rex" + "( ? ⪤[?,?] ? )" "rex_length" + "rex_lex" + "rex_drops" + "rex_fqup" + "rex_fsle" + "rex_rex" * ]
- }
- ]
- [ { "context-sensitive free variables" * } {
- [ [ "inclusion for restricted closures" ] "fsle" + "( ⦃?,?⦄ ⊆ ⦃?,?⦄ )" "fsle_length" + "fsle_drops" + "fsle_fqup" + "fsle_fsle" * ]
- [ [ "restricted refinement for lenvs" ] "lsubf" + "( ⦃?,?⦄ ⫃𝐅* ⦃?,?⦄ )" "lsubf_lsubr" + "lsubf_frees" + "lsubf_lsubf" * ]
- [ [ "for terms" ] "frees" + "( ? ⊢ 𝐅*⦃?⦄ ≘ ? )" "frees_append" + "frees_drops" + "frees_fqup" + "frees_frees" * ]
- }
- ]
- [ { "local environments" * } {
- [ [ "restricted refinement" ] "lsubr" + "( ? ⫃ ? )" "lsubr_length" + "lsubr_drops" + "lsubr_lsubr" * ]
- }
- ]
- }
- ]
- class "grass"
- [ { "s-computation" * } {
- [ { "iterated structural successor" * } {
- [ [ "for closures" ] "fqus" + "( ⦃?,?,?⦄ ⊐*[?] ⦃?,?,?⦄ )" + "( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )" "fqus_weight" + "fqus_drops" + "fqus_fqup" + "fqus_fqus" * ]
- [ [ "proper for closures" ] "fqup" + "( ⦃?,?,?⦄ ⊐+[?] ⦃?,?,?⦄ )" + "( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )" "fqup_weight" + "fqup_drops" + "fqup_fqup" * ]
- }
- ]
- }
- ]
- class "yellow"
- [ { "s-transition" * } {
- [ { "structural successor" * } {
- [ [ "for closures" ] "fquq" + "( ⦃?,?,?⦄ ⊐⸮[?] ⦃?,?,?⦄ )" + "( ⦃?,?,?⦄ ⊐⸮ ⦃?,?,?⦄ )" "fquq_length" + "fquq_weight" * ]
- [ [ "proper for closures" ] "fqu" + "( ⦃?,?,?⦄ ⊐[?] ⦃?,?,?⦄ )" + "( ⦃?,?,?⦄ ⊐ ⦃?,?,?⦄ )" "fqu_length" + "fqu_weight" + "fqu_tdeq" * ]
- }
- ]
- }
- ]
- class "orange"
- [ { "relocation" * } {
- [ { "generic slicing" * } {
- [ [ "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" * } {
- [ [ "for binders" ] "lifts_bind" + "( ⬆*[?] ? ≘ ? )" "lifts_weight_bind" + "lifts_lifts_bind" * ]
- [ [ "for term vectors" ] "lifts_vector" + "( ⬆*[?] ? ≘ ? )" "lifts_lifts_vector" * ]
- [ [ "for terms" ] "lifts" + "( ⬆*[?] ? ≘ ? )" "lifts_simple" + "lifts_weight" + "lifts_tdeq" + "lifts_lifts" * ]
- }
- ]
- [ { "syntactic equivalence" * } {
- [ [ "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" ] "sex" + "( ? ⪤[?,?,?] ? )" "sex_tc" + "sex_length" + "sex_sex" * ]
- }
- ]
- }
- ]
- class "red"
- [ { "syntax" * } {
- [ { "equivalence up to exclusion binders" * } {
- [ [ "for lenvs" ] "lveq" + "( ? ≋ⓧ*[?,?] ? )" "lveq_length" + "lveq_lveq" * ]
- }
- ]
- [ { "append" * } {
- [ [ "for lenvs" ] "append" + "( ? + ? )" "append_length" * ]
- }
- ]
- [ { "head equivalence" * } {
- [ [ "for terms" ] "theq" + "( ? ⩳[?,?] ? )" "theq_simple" + "theq_tdeq" + "theq_theq" + "theq_simple_vector" * ]
- }
- ]
- [ { "degree-based equivalence" * } {
- [ [ "" ] "tdeq_ext" + "( ? ≛[?,?] ? )" + "( ? ⊢ ? ≛[?,?] ? )" * ]
- [ [ "" ] "tdeq" + "( ? ≛[?,?] ? )" "tdeq_tdeq" * ]
- }
- ]
- [ { "closures" * } {
- [ [ "" ] "cl_weight" + "( ♯{?,?,?} )" * ]
- [ [ "" ] "cl_restricted_weight" + "( ♯{?,?} )" * ]
- }
- ]
- [ { "global environments" * } {
- [ [ "" ] "genv_length" + "( |?| )" * ]
- [ [ "" ] "genv_weight" + "( ♯{?} )" * ]
- [ [ "" ] "genv" * ]
- }
- ]
- [ { "local environments" * } {
- [ [ "" ] "ceq_ext" "ceq_ext_ceq_ext" * ]
- [ [ "" ] "cext2" * ]
- [ [ "" ] "lenv_length" + "( |?| )" * ]
- [ [ "" ] "lenv_weight" + "( ♯{?} )" * ]
- [ [ "" ] "lenv" * ]
- }
- ]
- [ { "binders for local environments" * } {
- [ [ "" ] "ext2" "ext2_tc" + "ext2_ext2" * ]
- [ [ "" ] "bind" "bind_weight" * ]
- }
- ]
- [ { "terms" * } {
- [ [ "" ] "term_vector" + "( Ⓐ?.? )" * ]
- [ [ "" ] "term_simple" + "( 𝐒⦃?⦄ )" * ]
- [ [ "" ] "term_weight" + "( ♯{?} )" * ]
- [ [ "" ] "term" * ]
- }
- ]
- [ { "items" * } {
- [ [ "" ] "item_sd" * ]
- [ [ "" ] "item_sh" * ]
- [ [ "" ] "item" * ]
- }
- ]
- [ { "atomic arities" * } {
- [ [ "" ] "aarity" * ]
- }
- ]
- }
- ]
}
class "top" { * }
+++ /dev/null
-../../matitac.opt `cat static.txt`
(* *)
(**************************************************************************)
-include "ground_2/notation/constructors/uparrow_1.ma".
+include "ground_2/notation/functions/uparrow_1.ma".
include "ground_2/notation/functions/downarrow_1.ma".
include "arithmetics/nat.ma".
include "ground_2/lib/relations.ma".
include "basics/bool.ma".
include "ground_2/lib/relations.ma".
-include "ground_2/notation/constructors/no_0.ma".
-include "ground_2/notation/constructors/yes_0.ma".
+include "ground_2/notation/functions/no_0.ma".
+include "ground_2/notation/functions/yes_0.ma".
(* BOOLEAN PROPERTIES *******************************************************)
(* *)
(**************************************************************************)
-include "ground_2/notation/constructors/circledE_1.ma".
-include "ground_2/notation/constructors/oplusright_3.ma".
+include "ground_2/notation/functions/circledE_1.ma".
+include "ground_2/notation/functions/oplusright_3.ma".
include "ground_2/lib/relations.ma".
(* LISTS ********************************************************************)
(* *)
(**************************************************************************)
-include "ground_2/notation/constructors/oplusright_3.ma".
+include "ground_2/notation/functions/oplusright_3.ma".
include "ground_2/lib/relations.ma".
(* STREAMS ******************************************************************)
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation < "hvbox( Ⓔ )"
- non associative with precedence 55
- for @{ 'CircledE $S }.
-
-notation > "hvbox( Ⓔ )"
- non associative with precedence 55
- for @{ 'CircledE ? }.
-
-notation > "hvbox( Ⓔ{ term 46 C } )"
- non associative with precedence 55
- for @{ 'CircledE $S }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "◊"
- non associative with precedence 55
- for @{ 'Diamond }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "∞"
- non associative with precedence 55
- for @{ 'Infinity }.
-
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "Ⓕ"
- non associative with precedence 55
- for @{'no}.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "𝟙𝟘"
- non associative with precedence 55
- for @{ 'OneZero }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation < "hvbox( hd ⨮ break tl )"
- right associative with precedence 47
- for @{ 'OPlusRight $S $hd $tl }.
-
-notation > "hvbox( hd ⨮ break tl )"
- right associative with precedence 47
- for @{ 'OPlusRight ? $hd $tl }.
-
-notation > "hvbox( hd ⨮{ break term 46 S } break term 46 tl )"
- non associative with precedence 47
- for @{ 'OPlusRight $S $hd $tl }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "hvbox( { term 46 hd1, break term 46 hd2 }; break term 46 tl )"
- non associative with precedence 47
- for @{ 'Semicolon $hd1 $hd2 $tl }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "hvbox ( 〈 term 46 x1, break term 46 x2 , break term 46 x3, break term 46 x4 〉 )"
- non associative with precedence 55
- for @{ 'Tuple $x1 $x2 $x3 $x4 }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "hvbox( ↑ term 70 T )"
- non associative with precedence 70
- for @{ 'UpArrow $T }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "Ⓣ"
- non associative with precedence 55
- for @{'yes}.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "𝟘𝟙"
- non associative with precedence 55
- for @{ 'ZeroOne }.
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
-
-notation "𝟘𝟘"
- non associative with precedence 55
- for @{ 'ZeroZero }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation < "hvbox( Ⓔ )"
+ non associative with precedence 55
+ for @{ 'CircledE $S }.
+
+notation > "hvbox( Ⓔ )"
+ non associative with precedence 55
+ for @{ 'CircledE ? }.
+
+notation > "hvbox( Ⓔ{ term 46 C } )"
+ non associative with precedence 55
+ for @{ 'CircledE $S }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "◊"
+ non associative with precedence 55
+ for @{ 'Diamond }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "∞"
+ non associative with precedence 55
+ for @{ 'Infinity }.
+
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "Ⓕ"
+ non associative with precedence 55
+ for @{'no}.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "𝟙𝟘"
+ non associative with precedence 55
+ for @{ 'OneZero }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation < "hvbox( hd ⨮ break tl )"
+ right associative with precedence 47
+ for @{ 'OPlusRight $S $hd $tl }.
+
+notation > "hvbox( hd ⨮ break tl )"
+ right associative with precedence 47
+ for @{ 'OPlusRight ? $hd $tl }.
+
+notation > "hvbox( hd ⨮{ break term 46 S } break term 46 tl )"
+ non associative with precedence 47
+ for @{ 'OPlusRight $S $hd $tl }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox( { term 46 hd1, break term 46 hd2 }; break term 46 tl )"
+ non associative with precedence 47
+ for @{ 'Semicolon $hd1 $hd2 $tl }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox ( 〈 term 46 x1, break term 46 x2 , break term 46 x3, break term 46 x4 〉 )"
+ non associative with precedence 55
+ for @{ 'Tuple $x1 $x2 $x3 $x4 }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "hvbox( ↑ term 70 T )"
+ non associative with precedence 70
+ for @{ 'UpArrow $T }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "Ⓣ"
+ non associative with precedence 55
+ for @{'yes}.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "𝟘𝟙"
+ non associative with precedence 55
+ for @{ 'ZeroOne }.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* GENERAL NOTATION USED BY THE FORMAL SYSTEM λδ ****************************)
+
+notation "𝟘𝟘"
+ non associative with precedence 55
+ for @{ 'ZeroZero }.
(* *)
(**************************************************************************)
-include "ground_2/notation/constructors/diamond_0.ma".
-include "ground_2/notation/constructors/semicolon_3.ma".
+include "ground_2/notation/functions/diamond_0.ma".
+include "ground_2/notation/functions/semicolon_3.ma".
include "ground_2/lib/arith.ma".
(* MULTIPLE RELOCATION WITH PAIRS *******************************************)
(* *)
(**************************************************************************)
-include "ground_2/notation/constructors/tuple_4.ma".
-include "ground_2/notation/constructors/zerozero_0.ma".
-include "ground_2/notation/constructors/zeroone_0.ma".
-include "ground_2/notation/constructors/onezero_0.ma".
+include "ground_2/notation/functions/tuple_4.ma".
+include "ground_2/notation/functions/zerozero_0.ma".
+include "ground_2/notation/functions/zeroone_0.ma".
+include "ground_2/notation/functions/onezero_0.ma".
include "ground_2/lib/arith.ma".
(* RT-TRANSITION COUNTER ****************************************************)
(**************************************************************************)
include "ground_2/lib/arith.ma".
-include "ground_2/notation/constructors/infinity_0.ma".
+include "ground_2/notation/functions/infinity_0.ma".
(* NATURAL NUMBERS WITH INFINITY ********************************************)
ground_2
-basic_2/syntax
-basic_2/relocation
-basic_2/s_transition
-basic_2/s_computation
-basic_2/static
-basic_2/i_static
+static_2
basic_2/rt_transition
basic_2/rt_computation
basic_2/rt_conversion
#!/bin/sh
-for SRC in `find ground_2 basic_2 apps_2 -name "*.ma" -or -name "*.tbl"`; do
+for SRC in `find grond_2 static_2 basic_2 apps_2 -name "*.ma" -or -name "*.tbl"`; do
sed "s!$1!$2!g" ${SRC} > ${SRC}.new
if [ ! -s ${SRC}.new ] || diff ${SRC} ${SRC}.new > /dev/null;
then rm -f ${SRC}.new;
+++ /dev/null
-ground_2
-basic_2/syntax
-basic_2/relocation
-basic_2/s_transition
-basic_2/s_computation
-basic_2/static
-basic_2/i_static
--- /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 "static_2/notation/relations/relationstar_4.ma".
+include "static_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 "static_2/relocation/seq_seq.ma".
+include "static_2/static/rex_drops.ma".
+include "static_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 "static_2/static/rex_fqup.ma".
+include "static_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 "static_2/static/rex_length.ma".
+include "static_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 "static_2/relocation/lex_tc.ma".
+include "static_2/static/req_fqup.ma".
+include "static_2/static/req_fsle.ma".
+include "static_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 "static_2/static/rex_fsle.ma".
+include "static_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 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( L. break ⓓ T1 )"
+ left associative with precedence 50
+ for @{ 'DxAbbr $L $T1 }.
--- /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( L. break ⓛ T1 )"
+ left associative with precedence 51
+ for @{ 'DxAbst $L $T1 }.
--- /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( L. break ⓤ { term 46 I } )"
+ non associative with precedence 47
+ for @{ 'DxBind1 $L $I }.
--- /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( L. break ⓑ { term 46 I } break term 49 T1 )"
+ non associative with precedence 48
+ for @{ 'DxBind2 $L $I $T1 }.
--- /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( L. ⓘ { break term 46 I } )"
+ non associative with precedence 46
+ for @{'DxItem $L $I }.
--- /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( L. ⓧ )"
+ non associative with precedence 49
+ for @{ 'DxVoid $L }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( § term 90 l )"
+ non associative with precedence 55
+ for @{ 'GRef $l }.
--- /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 "⓪"
+ non associative with precedence 55
+ for @{ 'Item0 }.
--- /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( ⓪ { term 46 I } )"
+ non associative with precedence 55
+ for @{ 'Item0 $I }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( # term 90 i )"
+ non associative with precedence 55
+ for @{ 'LRef $i }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓓ term 55 T. break term 55 L )"
+ non associative with precedence 55
+ for @{ 'SnAbbr $T $L }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓓ { term 46 p } break term 55 T1. break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbbr $p $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( - ⓓ term 55 T1. break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbbrNeg $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( + ⓓ term 55 T1. break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbbrPos $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓛ term 55 T. break term 55 L )"
+ non associative with precedence 55
+ for @{ 'SnAbst $T $L }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓛ { term 46 p } break term 55 T1. break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbst $p $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( - ⓛ term 55 T1. break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbstNeg $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( + ⓛ term 55 T1. break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAbstPos $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓐ term 55 T1. break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnAppl $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( Ⓐ term 55 T1. break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnApplVector $T1 $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓤ { term 46 I }. break term 55 L )"
+ non associative with precedence 55
+ for @{ 'SnBind1 $I $L }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓑ { term 46 I } break term 55 T. break term 55 L )"
+ non associative with precedence 55
+ for @{ 'SnBind2 $I $T $L }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓑ { term 46 p, break term 46 I } break term 55 T1. break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnBind2 $p $I $T1 $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( - ⓑ { term 46 I } break term 55 T1. break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnBind2Neg $I $T1 $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( + ⓑ { term 46 I } break term 55 T1. break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnBind2Pos $I $T1 $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓝ term 55 T1. break term 55 T2 )"
+ non associative with precedence 55
+ for @{ 'SnCast $T1 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓕ { term 46 I } break term 55 T1. break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnFlat2 $I $T1 $T }.
--- /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( ② term 55 T1. break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnItem2 $T1 $T }.
--- /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( ② { term 46 I } break term 55 T1. break term 55 T )"
+ non associative with precedence 55
+ for @{ 'SnItem2 $I $T1 $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓘ { term 46 I }. break term 55 L )"
+ non associative with precedence 55
+ for @{ 'SnItem $I $L }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⓧ. term 55 L )"
+ non associative with precedence 55
+ for @{ 'SnVoid $L }.
--- /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 "⋆"
+ non associative with precedence 46
+ for @{ 'Star }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⋆ term 90 s )"
+ non associative with precedence 55
+ for @{ 'Star $s }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ♯ { term 46 X } )"
+ non associative with precedence 90
+ for @{ 'Weight $X }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ♯ { term 46 L, break term 46 T } )"
+ non associative with precedence 90
+ for @{ 'Weight $L $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ♯ { term 46 G, break term 46 L, break term 46 T } )"
+ non associative with precedence 90
+ for @{ 'Weight $G $L $T }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G, break term 46 L⦄ ⊢ break term 46 T ⁝ break term 46 A )"
+ non associative with precedence 45
+ for @{ 'AtomicArity $G $L $T $A }.
--- /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( L ⊢ 𝐅 * ⦃ break term 46 T ⦄ ≘ break term 46 f )"
+ non associative with precedence 45
+ for @{ 'FreeStar $L $T $f }.
--- /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 f ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'IdEqSn $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( ⦃ term 46 G, break term 46 L, break term 46 T ⦄ ϵ [ break term 46 R ] 〚 break term 46 A 〛 )"
+ non associative with precedence 45
+ for @{ 'InEInt $R $G $L $T $A }.
--- /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( G ⊢ break term 46 L1 ⫃ ⁝ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LRSubEqA $G $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 L2 )"
+ non associative with precedence 45
+ for @{ 'LRSubEqC $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( G ⊢ break term 46 L1 ⫃ [ break term 46 R ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LRSubEqC $R $G $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 L1, break term 46 f1 ⦄ ⫃ 𝐅* ⦃ break term 46 L2, break term 46 f2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'LRSubEqF $L1 $f1 $L2 $f2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬇ * [ term 46 i ] break term 46 L1 ≘ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'RDropStar $i $L1 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⬇ * [ term 46 b, break term 46 f ] break term 46 L1 ≘ break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'RDropStar $b $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 L2 )"
+ non associative with precedence 45
+ for @{ 'Relation $R $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 @{ '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 }.
--- /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 @{ '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( ⬆ * [ term 46 f ] break term 46 T1 ≘ break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'RLiftStar $f $T1 $T2 }.
--- /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( 𝐒 ⦃ term 46 T ⦄ )"
+ non associative with precedence 45
+ for @{ 'Simple $T }.
--- /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( T1 ≛ [ break term 46 h, break term 46 o ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'StarEq $h $o $T1 $T2 }.
--- /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( L ⊢ break term 46 T1 ≛ [ break term 46 h, break term 46 o ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'StarEq $h $o $L $T1 $T2 }.
--- /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 h, break term 46 o, break term 46 T ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'StarEqSn $h $o $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( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ≛ [ break term 46 h, break term 46 o ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'StarEqSn $h $o $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 L1, break term 46 T1 ⦄ ⊆ ⦃ break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'SubSetEq $L1 $T1 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'SupTerm $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐ [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'SupTerm $b $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐⸮ ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'SupTermOpt $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⊐⸮ [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'SupTermOpt $b $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + ⦃ break term 46 G2, term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'SupTermPlus $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ + [ break term 46 b ] ⦃ break term 46 G2, term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'SupTermPlus $b $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'SupTermStar $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
+
+notation "hvbox( ⦃ term 46 G1, term 46 L1, break term 46 T1 ⦄ ⊐ * [ break term 46 b ] ⦃ break term 46 G2, break term 46 L2, break term 46 T2 ⦄ )"
+ non associative with precedence 45
+ for @{ 'SupTermStar $b $G1 $L1 $T1 $G2 $L2 $T2 }.
--- /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( T1 ⩳ [ break term 46 h, break term 46 o ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'TopIso $h $o $T1 $T2 }.
--- /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 n1, break term 46 n2 ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'VoidStarEq $L1 $n1 $n2 $L2 }.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/relocation/rtmap_coafter.ma".
+include "static_2/notation/relations/rdropstar_3.ma".
+include "static_2/notation/relations/rdropstar_4.ma".
+include "static_2/relocation/seq.ma".
+include "static_2/relocation/lifts_bind.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Basic_1: includes: drop_skip_bind drop1_skip_bind *)
+(* Basic_2A1: includes: drop_atom drop_pair drop_drop drop_skip
+ drop_refl_atom_O2 drop_drop_lt drop_skip_lt
+*)
+inductive drops (b:bool): rtmap → relation lenv ≝
+| drops_atom: ∀f. (b = Ⓣ → 𝐈⦃f⦄) → drops b (f) (⋆) (⋆)
+| drops_drop: ∀f,I,L1,L2. drops b f L1 L2 → drops b (↑f) (L1.ⓘ{I}) L2
+| drops_skip: ∀f,I1,I2,L1,L2.
+ drops b f L1 L2 → ⬆*[f] I2 ≘ I1 →
+ drops b (⫯f) (L1.ⓘ{I1}) (L2.ⓘ{I2})
+.
+
+interpretation "uniform slicing (local environment)"
+ 'RDropStar i L1 L2 = (drops true (uni i) L1 L2).
+
+interpretation "generic slicing (local environment)"
+ 'RDropStar b f L1 L2 = (drops b f L1 L2).
+
+definition d_liftable1: predicate (relation2 lenv term) ≝
+ λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K →
+ ∀U. ⬆*[f] T ≘ U → R L U.
+
+definition d_liftable1_isuni: predicate (relation2 lenv term) ≝
+ λR. ∀K,T. R K T → ∀b,f,L. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ →
+ ∀U. ⬆*[f] T ≘ U → R L U.
+
+definition d_deliftable1: predicate (relation2 lenv term) ≝
+ λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K →
+ ∀T. ⬆*[f] T ≘ U → R K T.
+
+definition d_deliftable1_isuni: predicate (relation2 lenv term) ≝
+ λR. ∀L,U. R L U → ∀b,f,K. ⬇*[b, f] L ≘ K → 𝐔⦃f⦄ →
+ ∀T. ⬆*[f] T ≘ U → R K T.
+
+definition d_liftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
+ predicate (lenv → relation C) ≝
+ λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≘ K →
+ ∀U1. S f T1 U1 →
+ ∃∃U2. S f T2 U2 & R L U1 U2.
+
+definition d_deliftable2_sn: ∀C:Type[0]. ∀S:rtmap → relation C.
+ predicate (lenv → relation C) ≝
+ λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≘ K →
+ ∀T1. S f T1 U1 →
+ ∃∃T2. S f T2 U2 & R K T1 T2.
+
+definition d_liftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
+ predicate (lenv → relation C) ≝
+ λC,S,R. ∀K,T1,T2. R K T1 T2 → ∀b,f,L. ⬇*[b, f] L ≘ K →
+ ∀U1. S f T1 U1 →
+ ∀U2. S f T2 U2 → R L U1 U2.
+
+definition d_deliftable2_bi: ∀C:Type[0]. ∀S:rtmap → relation C.
+ predicate (lenv → relation C) ≝
+ λC,S,R. ∀L,U1,U2. R L U1 U2 → ∀b,f,K. ⬇*[b, f] L ≘ K →
+ ∀T1. S f T1 U1 →
+ ∀T2. S f T2 U2 → R K T1 T2.
+
+definition co_dropable_sn: predicate (rtmap → relation lenv) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ →
+ ∀f2,L2. R f2 L1 L2 → ∀f1. f ~⊚ f1 ≘ f2 →
+ ∃∃K2. R f1 K1 K2 & ⬇*[b, f] L2 ≘ K2.
+
+definition co_dropable_dx: predicate (rtmap → relation lenv) ≝
+ λR. ∀f2,L1,L2. R f2 L1 L2 →
+ ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∀f1. f ~⊚ f1 ≘ f2 →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & R f1 K1 K2.
+
+definition co_dedropable_sn: predicate (rtmap → relation lenv) ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀f1,K2. R f1 K1 K2 →
+ ∀f2. f ~⊚ f1 ≘ f2 →
+ ∃∃L2. R f2 L1 L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+
+(* Basic properties *********************************************************)
+
+lemma drops_atom_F: ∀f. ⬇*[Ⓕ, f] ⋆ ≘ ⋆.
+#f @drops_atom #H destruct
+qed.
+
+lemma drops_eq_repl_back: ∀b,L1,L2. eq_repl_back … (λf. ⬇*[b, f] L1 ≘ L2).
+#b #L1 #L2 #f1 #H elim H -f1 -L1 -L2
+[ /4 width=3 by drops_atom, isid_eq_repl_back/
+| #f1 #I #L1 #L2 #_ #IH #f2 #H elim (eq_inv_nx … H) -H
+ /3 width=3 by drops_drop/
+| #f1 #I1 #I2 #L1 #L2 #_ #HI #IH #f2 #H elim (eq_inv_px … H) -H
+ /3 width=3 by drops_skip, liftsb_eq_repl_back/
+]
+qed-.
+
+lemma drops_eq_repl_fwd: ∀b,L1,L2. eq_repl_fwd … (λf. ⬇*[b, f] L1 ≘ L2).
+#b #L1 #L2 @eq_repl_sym /2 width=3 by drops_eq_repl_back/ (**) (* full auto fails *)
+qed-.
+
+(* Basic_2A1: includes: drop_FT *)
+lemma drops_TF: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2.
+#f #L1 #L2 #H elim H -f -L1 -L2
+/3 width=1 by drops_atom, drops_drop, drops_skip/
+qed.
+
+(* Basic_2A1: includes: drop_gen *)
+lemma drops_gen: ∀b,f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → ⬇*[b, f] L1 ≘ L2.
+* /2 width=1 by drops_TF/
+qed-.
+
+(* Basic_2A1: includes: drop_T *)
+lemma drops_F: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ⬇*[Ⓕ, f] L1 ≘ L2.
+* /2 width=1 by drops_TF/
+qed-.
+
+lemma d_liftable2_sn_bi: ∀C,S. (∀f,c. is_mono … (S f c)) →
+ ∀R. d_liftable2_sn C S R → d_liftable2_bi C S R.
+#C #S #HS #R #HR #K #T1 #T2 #HT12 #b #f #L #HLK #U1 #HTU1 #U2 #HTU2
+elim (HR … HT12 … HLK … HTU1) -HR -K -T1 #X #HTX #HUX
+<(HS … HTX … HTU2) -T2 -U2 -b -f //
+qed-.
+
+lemma d_deliftable2_sn_bi: ∀C,S. (∀f. is_inj2 … (S f)) →
+ ∀R. d_deliftable2_sn C S R → d_deliftable2_bi C S R.
+#C #S #HS #R #HR #L #U1 #U2 #HU12 #b #f #K #HLK #T1 #HTU1 #T2 #HTU2
+elim (HR … HU12 … HLK … HTU1) -HR -L -U1 #X #HUX #HTX
+<(HS … HUX … HTU2) -U2 -T2 -b -f //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact drops_inv_atom1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → X = ⋆ →
+ Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
+#b #f #X #Y * -f -X -Y
+[ /3 width=1 by conj/
+| #f #I #L1 #L2 #_ #H destruct
+| #f #I1 #I2 #L1 #L2 #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: includes: drop_gen_sort *)
+(* Basic_2A1: includes: drop_inv_atom1 *)
+lemma drops_inv_atom1: ∀b,f,Y. ⬇*[b, f] ⋆ ≘ Y → Y = ⋆ ∧ (b = Ⓣ → 𝐈⦃f⦄).
+/2 width=3 by drops_inv_atom1_aux/ qed-.
+
+fact drops_inv_drop1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I,K. X = K.ⓘ{I} → f = ↑g →
+ ⬇*[b, g] K ≘ Y.
+#b #f #X #Y * -f -X -Y
+[ #f #Hf #g #J #K #H destruct
+| #f #I #L1 #L2 #HL #g #J #K #H1 #H2 <(injective_next … H2) -g destruct //
+| #f #I1 #I2 #L1 #L2 #_ #_ #g #J #K #_ #H2 elim (discr_push_next … H2)
+]
+qed-.
+
+(* Basic_1: includes: drop_gen_drop *)
+(* Basic_2A1: includes: drop_inv_drop1_lt drop_inv_drop1 *)
+lemma drops_inv_drop1: ∀b,f,I,K,Y. ⬇*[b, ↑f] K.ⓘ{I} ≘ Y → ⬇*[b, f] K ≘ Y.
+/2 width=6 by drops_inv_drop1_aux/ qed-.
+
+fact drops_inv_skip1_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I1,K1. X = K1.ⓘ{I1} → f = ⫯g →
+ ∃∃I2,K2. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & Y = K2.ⓘ{I2}.
+#b #f #X #Y * -f -X -Y
+[ #f #Hf #g #J1 #K1 #H destruct
+| #f #I #L1 #L2 #_ #g #J1 #K1 #_ #H2 elim (discr_next_push … H2)
+| #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-.
+
+(* Basic_1: includes: drop_gen_skip_l *)
+(* Basic_2A1: includes: drop_inv_skip1 *)
+lemma drops_inv_skip1: ∀b,f,I1,K1,Y. ⬇*[b, ⫯f] K1.ⓘ{I1} ≘ Y →
+ ∃∃I2,K2. ⬇*[b, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & Y = K2.ⓘ{I2}.
+/2 width=5 by drops_inv_skip1_aux/ qed-.
+
+fact drops_inv_skip2_aux: ∀b,f,X,Y. ⬇*[b, f] X ≘ Y → ∀g,I2,K2. Y = K2.ⓘ{I2} → f = ⫯g →
+ ∃∃I1,K1. ⬇*[b, g] K1 ≘ K2 & ⬆*[g] I2 ≘ I1 & X = K1.ⓘ{I1}.
+#b #f #X #Y * -f -X -Y
+[ #f #Hf #g #J2 #K2 #H destruct
+| #f #I #L1 #L2 #_ #g #J2 #K2 #_ #H2 elim (discr_next_push … H2)
+| #f #I1 #I2 #L1 #L2 #HL #HV #g #J2 #K2 #H1 #H2 <(injective_push … H2) -g destruct
+ /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_1: includes: drop_gen_skip_r *)
+(* Basic_2A1: includes: drop_inv_skip2 *)
+lemma drops_inv_skip2: ∀b,f,I2,X,K2. ⬇*[b, ⫯f] X ≘ K2.ⓘ{I2} →
+ ∃∃I1,K1. ⬇*[b, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1 & X = K1.ⓘ{I1}.
+/2 width=5 by drops_inv_skip2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+fact drops_fwd_drop2_aux: ∀b,f2,X,Y. ⬇*[b, f2] X ≘ Y → ∀I,K. Y = K.ⓘ{I} →
+ ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b, f] X ≘ K.
+#b #f2 #X #Y #H elim H -f2 -X -Y
+[ #f2 #Hf2 #J #K #H destruct
+| #f2 #I #L1 #L2 #_ #IHL #J #K #H elim (IHL … H) -IHL
+ /3 width=7 by after_next, ex3_2_intro, drops_drop/
+| #f2 #I1 #I2 #L1 #L2 #HL #_ #_ #J #K #H destruct
+ lapply (after_isid_dx 𝐈𝐝 … f2) /3 width=9 by after_push, ex3_2_intro, drops_drop/
+]
+qed-.
+
+lemma drops_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} →
+ ∃∃f1,f. 𝐈⦃f1⦄ & f2 ⊚ ↑f1 ≘ f & ⬇*[b, f] X ≘ K.
+/2 width=4 by drops_fwd_drop2_aux/ qed-.
+
+(* Properties with test for identity ****************************************)
+
+(* Basic_2A1: includes: drop_refl *)
+lemma drops_refl: ∀b,L,f. 𝐈⦃f⦄ → ⬇*[b, f] L ≘ L.
+#b #L elim L -L /2 width=1 by drops_atom/
+#L #I #IHL #f #Hf elim (isid_inv_gen … Hf) -Hf
+/3 width=1 by drops_skip, liftsb_refl/
+qed.
+
+(* Forward lemmas test for identity *****************************************)
+
+(* Basic_1: includes: drop_gen_refl *)
+(* Basic_2A1: includes: drop_inv_O2 *)
+lemma drops_fwd_isid: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → 𝐈⦃f⦄ → L1 = L2.
+#b #f #L1 #L2 #H elim H -f -L1 -L2 //
+[ #f #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) //
+| /5 width=5 by isid_inv_push, liftsb_fwd_isid, eq_f2, sym_eq/
+]
+qed-.
+
+lemma drops_after_fwd_drop2: ∀b,f2,I,X,K. ⬇*[b, f2] X ≘ K.ⓘ{I} →
+ ∀f1,f. 𝐈⦃f1⦄ → f2 ⊚ ↑f1 ≘ f → ⬇*[b, f] X ≘ K.
+#b #f2 #I #X #K #H #f1 #f #Hf1 #Hf elim (drops_fwd_drop2 … H) -H
+#g1 #g #Hg1 #Hg #HK lapply (after_mono_eq … Hg … Hf ??) -Hg -Hf
+/3 width=5 by drops_eq_repl_back, isid_inv_eq_repl, eq_next/
+qed-.
+
+(* Forward lemmas with test for finite colength *****************************)
+
+lemma drops_fwd_isfin: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐅⦃f⦄.
+#f #L1 #L2 #H elim H -f -L1 -L2
+/3 width=1 by isfin_next, isfin_push, isfin_isid/
+qed-.
+
+(* Properties with test for uniformity **************************************)
+
+lemma drops_isuni_ex: ∀f. 𝐔⦃f⦄ → ∀L. ∃K. ⬇*[Ⓕ, f] L ≘ K.
+#f #H elim H -f /4 width=2 by drops_refl, drops_TF, ex_intro/
+#f #_ #g #H #IH destruct * /2 width=2 by ex_intro/
+#L #I elim (IH L) -IH /3 width=2 by drops_drop, ex_intro/
+qed-.
+
+(* Inversion lemmas with test for uniformity ********************************)
+
+lemma drops_inv_isuni: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐔⦃f⦄ →
+ (𝐈⦃f⦄ ∧ L1 = L2) ∨
+ ∃∃g,I,K. ⬇*[Ⓣ, g] K ≘ L2 & 𝐔⦃g⦄ & L1 = K.ⓘ{I} & f = ↑g.
+#f #L1 #L2 * -f -L1 -L2
+[ /4 width=1 by or_introl, conj/
+| /4 width=7 by isuni_inv_next, ex4_3_intro, or_intror/
+| /7 width=6 by drops_fwd_isid, liftsb_fwd_isid, isuni_inv_push, isid_push, or_introl, conj, eq_f2, sym_eq/
+]
+qed-.
+
+(* Basic_2A1: was: drop_inv_O1_pair1 *)
+lemma drops_inv_bind1_isuni: ∀b,f,I,K,L2. 𝐔⦃f⦄ → ⬇*[b, f] K.ⓘ{I} ≘ L2 →
+ (𝐈⦃f⦄ ∧ L2 = K.ⓘ{I}) ∨
+ ∃∃g. 𝐔⦃g⦄ & ⬇*[b, g] K ≘ L2 & f = ↑g.
+#b #f #I #K #L2 #Hf #H elim (isuni_split … Hf) -Hf * #g #Hg #H0 destruct
+[ lapply (drops_inv_skip1 … H) -H * #Z #Y #HY #HZ #H destruct
+ <(drops_fwd_isid … HY Hg) -Y >(liftsb_fwd_isid … HZ Hg) -Z
+ /4 width=3 by isid_push, or_introl, conj/
+| lapply (drops_inv_drop1 … H) -H /3 width=4 by ex3_intro, or_intror/
+]
+qed-.
+
+(* Basic_2A1: was: drop_inv_O1_pair2 *)
+lemma drops_inv_bind2_isuni: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K.ⓘ{I} →
+ (𝐈⦃f⦄ ∧ L1 = K.ⓘ{I}) ∨
+ ∃∃g,I1,K1. 𝐔⦃g⦄ & ⬇*[b, g] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1} & f = ↑g.
+#b #f #I #K *
+[ #Hf #H elim (drops_inv_atom1 … H) -H #H destruct
+| #L1 #I1 #Hf #H elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct /3 width=1 by or_introl, conj/
+ | /3 width=7 by ex4_3_intro, or_intror/
+ ]
+]
+qed-.
+
+lemma drops_inv_bind2_isuni_next: ∀b,f,I,K,L1. 𝐔⦃f⦄ → ⬇*[b, ↑f] L1 ≘ K.ⓘ{I} →
+ ∃∃I1,K1. ⬇*[b, f] K1 ≘ K.ⓘ{I} & L1 = K1.ⓘ{I1}.
+#b #f #I #K #L1 #Hf #H elim (drops_inv_bind2_isuni … H) -H /2 width=3 by isuni_next/ -Hf *
+[ #H elim (isid_inv_next … H) -H //
+| /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+fact drops_inv_TF_aux: ∀f,L1,L2. ⬇*[Ⓕ, f] L1 ≘ L2 → 𝐔⦃f⦄ →
+ ∀I,K. L2 = K.ⓘ{I} → ⬇*[Ⓣ, f] L1 ≘ K.ⓘ{I}.
+#f #L1 #L2 #H elim H -f -L1 -L2
+[ #f #_ #_ #J #K #H destruct
+| #f #I #L1 #L2 #_ #IH #Hf #J #K #H destruct
+ /4 width=3 by drops_drop, isuni_inv_next/
+| #f #I1 #I2 #L1 #L2 #HL12 #HI21 #_ #Hf #J #K #H destruct
+ lapply (isuni_inv_push … Hf ??) -Hf [1,2: // ] #Hf
+ <(drops_fwd_isid … HL12) -K // <(liftsb_fwd_isid … HI21) -I1
+ /3 width=3 by drops_refl, isid_push/
+]
+qed-.
+
+(* Basic_2A1: includes: drop_inv_FT *)
+lemma drops_inv_TF: ∀f,I,L,K. ⬇*[Ⓕ, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≘ K.ⓘ{I}.
+/2 width=3 by drops_inv_TF_aux/ qed-.
+
+(* Basic_2A1: includes: drop_inv_gen *)
+lemma drops_inv_gen: ∀b,f,I,L,K. ⬇*[b, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[Ⓣ, f] L ≘ K.ⓘ{I}.
+* /2 width=1 by drops_inv_TF/
+qed-.
+
+(* Basic_2A1: includes: drop_inv_T *)
+lemma drops_inv_F: ∀b,f,I,L,K. ⬇*[Ⓕ, f] L ≘ K.ⓘ{I} → 𝐔⦃f⦄ → ⬇*[b, f] L ≘ K.ⓘ{I}.
+* /2 width=1 by drops_inv_TF/
+qed-.
+
+(* Forward lemmas with test for uniformity **********************************)
+
+(* Basic_1: was: drop_S *)
+(* Basic_2A1: was: drop_fwd_drop2 *)
+lemma drops_isuni_fwd_drop2: ∀b,f,I,X,K. 𝐔⦃f⦄ → ⬇*[b, f] X ≘ K.ⓘ{I} → ⬇*[b, ↑f] X ≘ K.
+/3 width=7 by drops_after_fwd_drop2, after_isid_isuni/ qed-.
+
+(* Inversion lemmas with uniform relocations ********************************)
+
+lemma drops_inv_atom2: ∀b,L,f. ⬇*[b, f] L ≘ ⋆ →
+ ∃∃n,f1. ⬇*[b, 𝐔❴n❵] L ≘ ⋆ & 𝐔❴n❵ ⊚ f1 ≘ f.
+#b #L elim L -L
+[ /3 width=4 by drops_atom, after_isid_sn, ex2_2_intro/
+| #L #I #IH #f #H elim (pn_split f) * #g #H0 destruct
+ [ elim (drops_inv_skip1 … H) -H #J #K #_ #_ #H destruct
+ | lapply (drops_inv_drop1 … H) -H #HL
+ elim (IH … HL) -IH -HL /3 width=8 by drops_drop, after_next, ex2_2_intro/
+ ]
+]
+qed-.
+
+lemma drops_inv_succ: ∀L1,L2,i. ⬇*[↑i] L1 ≘ L2 →
+ ∃∃I,K. ⬇*[i] K ≘ L2 & L1 = K.ⓘ{I}.
+#L1 #L2 #i #H elim (drops_inv_isuni … H) -H // *
+[ #H elim (isid_inv_next … H) -H //
+| /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+(* Properties with uniform relocations **************************************)
+
+lemma drops_F_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ ∨ ∃∃I,K. ⬇*[i] L ≘ K.ⓘ{I}.
+#L elim L -L /2 width=1 by or_introl/
+#L #I #IH * /4 width=3 by drops_refl, ex1_2_intro, or_intror/
+#i elim (IH i) -IH /3 width=1 by drops_drop, or_introl/
+* /4 width=3 by drops_drop, ex1_2_intro, or_intror/
+qed-.
+
+(* Basic_2A1: includes: drop_split *)
+lemma drops_split_trans: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ∀f1,f2. f1 ⊚ f2 ≘ f → 𝐔⦃f1⦄ →
+ ∃∃L. ⬇*[b, f1] L1 ≘ L & ⬇*[b, f2] L ≘ L2.
+#b #f #L1 #L2 #H elim H -f -L1 -L2
+[ #f #H0f #f1 #f2 #Hf #Hf1 @(ex2_intro … (⋆)) @drops_atom
+ #H lapply (H0f H) -b
+ #H elim (after_inv_isid3 … Hf H) -f //
+| #f #I #L1 #L2 #HL12 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxn … Hf) -Hf [1,3: * |*: // ]
+ [ #g1 #g2 #Hf #H1 #H2 destruct
+ lapply (isuni_inv_push … Hf1 ??) -Hf1 [1,2: // ] #Hg1
+ elim (IHL12 … Hf) -f
+ /4 width=5 by drops_drop, drops_skip, liftsb_refl, isuni_isid, ex2_intro/
+ | #g1 #Hf #H destruct elim (IHL12 … Hf) -f
+ /3 width=5 by ex2_intro, drops_drop, isuni_inv_next/
+ ]
+| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 #f1 #f2 #Hf #Hf1 elim (after_inv_xxp … Hf) -Hf [2,3: // ]
+ #g1 #g2 #Hf #H1 #H2 destruct elim (liftsb_split_trans … HI21 … Hf) -HI21
+ elim (IHL12 … Hf) -f /3 width=5 by ex2_intro, drops_skip, isuni_fwd_push/
+]
+qed-.
+
+lemma drops_split_div: ∀b,f1,L1,L. ⬇*[b, f1] L1 ≘ L → ∀f2,f. f1 ⊚ f2 ≘ f → 𝐔⦃f2⦄ →
+ ∃∃L2. ⬇*[Ⓕ, f2] L ≘ L2 & ⬇*[Ⓕ, f] L1 ≘ L2.
+#b #f1 #L1 #L #H elim H -f1 -L1 -L
+[ #f1 #Hf1 #f2 #f #Hf #Hf2 @(ex2_intro … (⋆)) @drops_atom #H destruct
+| #f1 #I #L1 #L #HL1 #IH #f2 #f #Hf #Hf2 elim (after_inv_nxx … Hf) -Hf [2,3: // ]
+ #g #Hg #H destruct elim (IH … Hg) -IH -Hg /3 width=5 by drops_drop, ex2_intro/
+| #f1 #I1 #I #L1 #L #HL1 #HI1 #IH #f2 #f #Hf #Hf2
+ elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ]
+ #g2 #g #Hg #H2 #H0 destruct
+ [ lapply (isuni_inv_push … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -IH
+ lapply (after_isid_inv_dx … Hg … Hg2) -Hg #Hg
+ /5 width=7 by drops_eq_repl_back, drops_F, drops_refl, drops_skip, liftsb_eq_repl_back, isid_push, ex2_intro/
+ | lapply (isuni_inv_next … Hf2 ??) -Hf2 [1,2: // ] #Hg2 -HL1 -HI1
+ elim (IH … Hg) -f1 /3 width=3 by drops_drop, ex2_intro/
+ ]
+]
+qed-.
+
+(* Properties with application **********************************************)
+
+lemma drops_tls_at: ∀f,i1,i2. @⦃i1,f⦄ ≘ i2 →
+ ∀b,L1,L2. ⬇*[b,⫱*[i2]f] L1 ≘ L2 →
+ ⬇*[b,⫯⫱*[↑i2]f] L1 ≘ L2.
+/3 width=3 by drops_eq_repl_fwd, at_inv_tls/ qed-.
+
+lemma drops_split_trans_bind2: ∀b,f,I,L,K0. ⬇*[b, f] L ≘ K0.ⓘ{I} → ∀i. @⦃O, f⦄ ≘ i →
+ ∃∃J,K. ⬇*[i]L ≘ K.ⓘ{J} & ⬇*[b, ⫱*[↑i]f] K ≘ K0 & ⬆*[⫱*[↑i]f] I ≘ J.
+#b #f #I #L #K0 #H #i #Hf
+elim (drops_split_trans … H) -H [ |5: @(after_uni_dx … Hf) |2,3: skip ] /2 width=1 by after_isid_dx/ #Y #HLY #H
+lapply (drops_tls_at … Hf … H) -H #H
+elim (drops_inv_skip2 … H) -H #J #K #HK0 #HIJ #H destruct
+/3 width=5 by drops_inv_gen, ex3_2_intro/
+qed-.
+
+(* Properties with context-sensitive equivalence for terms ******************)
+
+lemma ceq_lift_sn: d_liftable2_sn … liftsb ceq_ext.
+#K #I1 #I2 #H <(ceq_ext_inv_eq … H) -I2
+/2 width=3 by ex2_intro/ qed-.
+
+lemma ceq_inv_lift_sn: d_deliftable2_sn … liftsb ceq_ext.
+#L #J1 #J2 #H <(ceq_ext_inv_eq … H) -J2
+/2 width=3 by ex2_intro/ qed-.
+
+(* Note: d_deliftable2_sn cfull does not hold *)
+lemma cfull_lift_sn: d_liftable2_sn … liftsb cfull.
+#K #I1 #I2 #_ #b #f #L #_ #J1 #_ -K -I1 -b
+elim (liftsb_total I2 f) /2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: removed theorems 12:
+ drops_inv_nil drops_inv_cons d1_liftable_liftables
+ drop_refl_atom_O2 drop_inv_pair1
+ drop_inv_Y1 drop_Y1 drop_O_Y drop_fwd_Y2
+ drop_fwd_length_minus2 drop_fwd_length_minus4
+*)
+(* Basic_1: removed theorems 53:
+ drop1_gen_pnil drop1_gen_pcons drop1_getl_trans
+ drop_ctail drop_skip_flat
+ cimp_flat_sx cimp_flat_dx cimp_bind cimp_getl_conf
+ drop_clear drop_clear_O drop_clear_S
+ clear_gen_sort clear_gen_bind clear_gen_flat clear_gen_flat_r
+ clear_gen_all clear_clear clear_mono clear_trans clear_ctail clear_cle
+ getl_ctail_clen getl_gen_tail clear_getl_trans getl_clear_trans
+ getl_clear_bind getl_clear_conf getl_dec getl_drop getl_drop_conf_lt
+ getl_drop_conf_ge getl_conf_ge_drop getl_drop_conf_rev
+ drop_getl_trans_lt drop_getl_trans_le drop_getl_trans_ge
+ getl_drop_trans getl_flt getl_gen_all getl_gen_sort getl_gen_O
+ getl_gen_S getl_gen_2 getl_gen_flat getl_gen_bind getl_conf_le
+ getl_trans getl_refl getl_head getl_flat getl_ctail getl_mono
+*)
--- /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 "static_2/relocation/drops.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Properties with the extension to binders of a context-sensitive relation *)
+
+lemma cext2_d_liftable2_sn: ∀R. d_liftable2_sn … lifts R →
+ d_liftable2_sn … liftsb (cext2 R).
+#R #HR #K #I1 #I2 * -I1 -I2 #I [| #T1 #T2 #HT12 ]
+#b #f #L #HLK #Z1 #H
+[ lapply (liftsb_inv_unit_sn … H)
+| lapply (liftsb_inv_pair_sn … H) * #U1 #HTU1
+] -H #H destruct /3 width=3 by ext2_unit, ex2_intro/
+elim (HR … HT12 … HLK … HTU1) -HR -b -K -T1 /3 width=3 by ext2_pair, ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/lib/star.ma".
+include "static_2/relocation/seq_seq.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Properties with contextual transitive closure ****************************)
+
+(* Basic_2A1: was: d_liftable_LTC *)
+lemma d2_liftable_sn_CTC: ∀C,S,R. d_liftable2_sn C S R → d_liftable2_sn C S (CTC … R).
+#C #S #R #HR #K #T1 #T2 #H elim H -T2
+[ #T2 #HT12 #b #f #L #HLK #U1 #HTU1
+ elim (HR … HT12 … HLK … HTU1) /3 width=3 by inj, ex2_intro/
+| #T #T2 #_ #HT2 #IHT1 #b #f #L #HLK #U1 #HTU1
+ elim (IHT1 … HLK … HTU1) -T1 #U #HTU #HU1
+ elim (HR … HT2 … HLK … HTU) -HR -K -T /3 width=5 by step, ex2_intro/
+]
+qed-.
+
+(* Basic_2A1: was: d_deliftable_sn_LTC *)
+lemma d2_deliftable_sn_CTC: ∀C,S,R. d_deliftable2_sn C S R → d_deliftable2_sn C S (CTC … R).
+#C #S #R #HR #L #U1 #U2 #H elim H -U2
+[ #U2 #HU12 #b #f #K #HLK #T1 #HTU1
+ elim (HR … HU12 … HLK … HTU1) -HR -L -U1 /3 width=3 by inj, ex2_intro/
+| #U #U2 #_ #HU2 #IHU1 #b #f #K #HLK #T1 #HTU1
+ elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+ elim (HR … HU2 … HLK … HTU) -HR -L -U /3 width=5 by step, ex2_intro/
+]
+qed-.
+
+lemma co_dropable_sn_CTC: ∀R. co_dropable_sn R → co_dropable_sn (CTC … R).
+#R #HR #b #f #L1 #K1 #HLK1 #Hf #f2 #L2 #H elim H -L2
+[ #L2 #HL12 #f1 #H elim (HR … HLK1 … Hf … HL12 … H) -HR -Hf -f2 -L1
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 #IH #f1 #H elim (IH … H) -IH
+ #K #HK1 #HLK elim (HR … HLK … HL2 … H) -HR -f2 -L
+ /3 width=3 by step, ex2_intro/
+]
+qed-.
+
+lemma co_dropable_dx_CTC: ∀R. co_dropable_dx R → co_dropable_dx (CTC … R).
+#R #HR #f2 #L1 #L2 #H elim H -L2
+[ #L2 #HL12 #b #f #K2 #HLK2 #Hf #f1 #Hf2 elim (HR … HL12 … HLK2 … Hf … Hf2) -HR -Hf -f2 -L2
+ /3 width=3 by inj, ex2_intro/
+| #L #L2 #_ #HL2 #IHL1 #b #f #K2 #HLK2 #Hf #f1 #Hf2 elim (HR … HL2 … HLK2 … Hf … Hf2) -HR -L2
+ #K #HLK #HK2 elim (IHL1 … HLK … Hf … Hf2) -Hf -f2 -L
+ /3 width=5 by step, ex2_intro/
+]
+qed-.
+
+lemma co_dedropable_sn_CTC: ∀R. co_dedropable_sn R → co_dedropable_sn (CTC … R).
+#R #HR #b #f #L1 #K1 #HLK1 #f1 #K2 #H elim H -K2
+[ #K2 #HK12 #f2 #Hf elim (HR … HLK1 … HK12 … Hf) -HR -f1 -K1
+ /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 seq_trans, step, 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 "static_2/relocation/lifts_lifts_bind.ma".
+include "static_2/relocation/drops_weight.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_2A1: includes: drop_conf_ge drop_conf_be drop_conf_le *)
+theorem drops_conf: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
+ ∀b2,f,L2. ⬇*[b2, f] L1 ≘ L2 →
+ ∀f2. f1 ⊚ f2 ≘ f → ⬇*[b2, f2] L ≘ L2.
+#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
+[ #f1 #_ #b2 #f #L2 #HL2 #f2 #Hf12 elim (drops_inv_atom1 … HL2) -b1 -HL2
+ #H #Hf destruct @drops_atom
+ #H elim (after_inv_isid3 … Hf12) -Hf12 /2 width=1 by/
+| #f1 #I1 #K1 #K #_ #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_nxx … Hf) -Hf [2,3: // ]
+ #g #Hg #H destruct /3 width=3 by drops_inv_drop1/
+| #f1 #I1 #I #K1 #K #_ #HI1 #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*:// ]
+ #g2 #g #Hf #H1 #H2 destruct
+ [ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, liftsb_div3/
+ | /4 width=3 by drops_inv_drop1, drops_drop/
+ ]
+]
+qed-.
+
+(* Basic_1: was: drop1_trans *)
+(* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm
+ drops_drop_trans
+*)
+theorem drops_trans: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
+ ∀b2,f2,L2. ⬇*[b2, f2] L ≘ L2 →
+ ∀f. f1 ⊚ f2 ≘ f → ⬇*[b1∧b2, f] L1 ≘ L2.
+#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
+[ #f1 #Hf1 #b2 #f2 #L2 #HL2 #f #Hf elim (drops_inv_atom1 … HL2) -HL2
+ #H #Hf2 destruct @drops_atom #H elim (andb_inv_true_dx … H) -H
+ #H1 #H2 lapply (after_isid_inv_sn … Hf ?) -Hf
+ /3 width=3 by isid_eq_repl_back/
+| #f1 #I1 #K1 #K #_ #IH #b2 #f2 #L2 #HL2 #f #Hf elim (after_inv_nxx … Hf) -Hf
+ /3 width=3 by drops_drop/
+| #f1 #I1 #I #K1 #K #_ #HI1 #IH #b2 #f2 #L2 #HL2 #f #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ]
+ #g2 #g #Hg #H1 #H2 destruct
+ [ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, liftsb_trans/
+ | /4 width=3 by drops_inv_drop1, drops_drop/
+ ]
+]
+qed-.
+
+theorem drops_conf_div: ∀f1,L,K. ⬇*[Ⓣ,f1] L ≘ K → ∀f2. ⬇*[Ⓣ,f2] L ≘ K →
+ 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2.
+#f1 #L #K #H elim H -f1 -L -K
+[ #f1 #Hf1 #f2 #Hf2 elim (drops_inv_atom1 … Hf2) -Hf2
+ /3 width=1 by isid_inv_eq_repl/
+| #f1 #I #L #K #Hf1 #IH #f2 elim (pn_split f2) *
+ #g2 #H2 #Hf2 #HU1 #HU2 destruct
+ [ elim (drops_inv_skip1 … Hf2) -IH -HU1 -Hf2 #Y2 #X2 #HY2 #_ #H destruct
+ lapply (drops_fwd_isid … HY2 ?) -HY2 /2 width=3 by isuni_inv_push/ -HU2
+ #H destruct elim (drops_inv_x_bind_xy … Hf1)
+ | /4 width=5 by drops_inv_drop1, isuni_inv_next, eq_next/
+ ]
+| #f1 #I1 #I2 #L #K #Hf1 #_ #IH #f2 elim (pn_split f2) *
+ #g2 #H2 #Hf2 #HU1 #HU2 destruct
+ [ elim (drops_inv_skip1 … Hf2) -Hf2 #Y2 #X2 #HY2 #_ #H destruct -Hf1
+ /4 width=5 by isuni_fwd_push, eq_push/
+ | lapply (drops_inv_drop1 … Hf2) -Hf2 -IH -HU2 #Hg2
+ lapply (drops_fwd_isid … Hf1 ?) -Hf1 /2 width=3 by isuni_inv_push/ -HU1
+ #H destruct elim (drops_inv_x_bind_xy … Hg2)
+ ]
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: includes: drop_mono *)
+lemma drops_mono: ∀b1,f,L,L1. ⬇*[b1, f] L ≘ L1 →
+ ∀b2,L2. ⬇*[b2, f] L ≘ L2 → L1 = L2.
+#b1 #f #L #L1 lapply (after_isid_dx 𝐈𝐝 … f)
+/3 width=8 by drops_conf, drops_fwd_isid/
+qed-.
+
+(* Basic_2A1: includes: drop_conf_lt *)
+lemma drops_conf_skip1: ∀b2,f,L,L2. ⬇*[b2, f] L ≘ L2 →
+ ∀b1,f1,I1,K1. ⬇*[b1, f1] L ≘ K1.ⓘ{I1} →
+ ∀f2. f1 ⊚ ⫯f2 ≘ f →
+ ∃∃I2,K2. L2 = K2.ⓘ{I2} &
+ ⬇*[b2, f2] K1 ≘ K2 & ⬆*[f2] I2 ≘ I1.
+#b2 #f #L #L2 #H2 #b1 #f1 #I1 #K1 #H1 #f2 #Hf lapply (drops_conf … H1 … H2 … Hf) -L -Hf
+#H elim (drops_inv_skip1 … H) -H /2 width=5 by ex3_2_intro/
+qed-.
+
+(* Basic_2A1: includes: drop_trans_lt *)
+lemma drops_trans_skip2: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
+ ∀b2,f2,I2,K2. ⬇*[b2, f2] L ≘ K2.ⓘ{I2} →
+ ∀f. f1 ⊚ f2 ≘ ⫯f →
+ ∃∃I1,K1. L1 = K1.ⓘ{I1} &
+ ⬇*[b1∧b2, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1.
+#b1 #f1 #L1 #L #H1 #b2 #f2 #I2 #K2 #H2 #f #Hf
+lapply (drops_trans … H1 … H2 … Hf) -L -Hf
+#H elim (drops_inv_skip2 … H) -H /2 width=5 by ex3_2_intro/
+qed-.
+
+(* Basic_2A1: includes: drops_conf_div *)
+lemma drops_conf_div_bind: ∀f1,f2,I1,I2,L,K.
+ ⬇*[Ⓣ, f1] L ≘ K.ⓘ{I1} → ⬇*[Ⓣ, f2] L ≘ K.ⓘ{I2} →
+ 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2 ∧ I1 = I2.
+#f1 #f2 #I1 #I2 #L #K #Hf1 #Hf2 #HU1 #HU2
+lapply (drops_isuni_fwd_drop2 … Hf1) // #H1
+lapply (drops_isuni_fwd_drop2 … Hf2) // #H2
+lapply (drops_conf_div … H1 … H2 ??) /2 width=3 by isuni_next/ -H1 -H2 -HU1 -HU2 #H
+lapply (eq_inv_nn … H ????) -H [5: |*: // ] #H12
+lapply (drops_eq_repl_back … Hf1 … H12) -Hf1 #H0
+lapply (drops_mono … H0 … Hf2) -L #H
+destruct /2 width=1 by conj/
+qed-.
+
+lemma drops_inv_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ → ∀I,K. ⬇*[i] L ≘ K.ⓘ{I} → ⊥.
+#L #i #H1 #I #K #H2
+lapply (drops_F … H2) -H2 #H2
+lapply (drops_mono … H2 … H1) -L -i #H destruct
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/lenv_length.ma".
+include "static_2/relocation/drops.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Basic_2A1: includes: drop_fwd_length_le4 *)
+lemma drops_fwd_length_le4: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → |L2| ≤ |L1|.
+#b #f #L1 #L2 #H elim H -f -L1 -L2 /2 width=1 by le_S, le_S_S/
+qed-.
+
+(* Basic_2A1: includes: drop_fwd_length_eq1 *)
+theorem drops_fwd_length_eq1: ∀b1,b2,f,L1,K1. ⬇*[b1, f] L1 ≘ K1 →
+ ∀L2,K2. ⬇*[b2, f] L2 ≘ K2 →
+ |L1| = |L2| → |K1| = |K2|.
+#b1 #b2 #f #L1 #K1 #HLK1 elim HLK1 -f -L1 -K1
+[ #f #_ #L2 #K2 #HLK2 #H lapply (length_inv_zero_sn … H) -H
+ #H destruct elim (drops_inv_atom1 … HLK2) -HLK2 //
+| #f #I1 #L1 #K1 #_ #IH #X2 #K2 #HX #H elim (length_inv_succ_sn … H) -H
+ #I2 #L2 #H12 #H destruct lapply (drops_inv_drop1 … HX) -HX
+ #HLK2 @(IH … HLK2 H12) (**) (* auto fails *)
+| #f #I1 #I2 #L1 #K1 #_ #_ #IH #X2 #Y2 #HX #H elim (length_inv_succ_sn … H) -H
+ #I2 #L2 #H12 #H destruct elim (drops_inv_skip1 … HX) -HX
+ #I2 #K2 #HLK2 #_ #H destruct
+ lapply (IH … HLK2 H12) -f >length_bind >length_bind /2 width=1 by/ (**) (* full auto fails *)
+]
+qed-.
+
+(* forward lemmas with finite colength assignment ***************************)
+
+lemma drops_fwd_fcla: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
+ ∃∃n. 𝐂⦃f⦄ ≘ n & |L1| = |L2| + n.
+#f #L1 #L2 #H elim H -f -L1 -L2
+[ /4 width=3 by fcla_isid, ex2_intro/
+| #f #I #L1 #L2 #_ * >length_bind /3 width=3 by fcla_next, ex2_intro, eq_f/
+| #f #I1 #I2 #L1 #L2 #_ #_ * >length_bind >length_bind /3 width=3 by fcla_push, ex2_intro/
+]
+qed-.
+
+(* Basic_2A1: includes: drop_fwd_length *)
+lemma drops_fcla_fwd: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
+ |L1| = |L2| + n.
+#f #l1 #l2 #n #Hf #Hn elim (drops_fwd_fcla … Hf) -Hf
+#k #Hm #H <(fcla_mono … Hm … Hn) -f //
+qed-.
+
+lemma drops_fwd_fcla_le2: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
+ ∃∃n. 𝐂⦃f⦄ ≘ n & n ≤ |L1|.
+#f #L1 #L2 #H elim (drops_fwd_fcla … H) -H /2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: drop_fwd_length_le2 *)
+lemma drops_fcla_fwd_le2: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n →
+ n ≤ |L1|.
+#f #L1 #L2 #n #H #Hn elim (drops_fwd_fcla_le2 … H) -H
+#k #Hm #H <(fcla_mono … Hm … Hn) -f //
+qed-.
+
+lemma drops_fwd_fcla_lt2: ∀f,L1,I2,K2. ⬇*[Ⓣ, f] L1 ≘ K2.ⓘ{I2} →
+ ∃∃n. 𝐂⦃f⦄ ≘ n & n < |L1|.
+#f #L1 #I2 #K2 #H elim (drops_fwd_fcla … H) -H
+#n #Hf #H >H -L1 /3 width=3 by le_S_S, ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: drop_fwd_length_lt2 *)
+lemma drops_fcla_fwd_lt2: ∀f,L1,I2,K2,n.
+ ⬇*[Ⓣ, f] L1 ≘ K2.ⓘ{I2} → 𝐂⦃f⦄ ≘ n →
+ n < |L1|.
+#f #L1 #I2 #K2 #n #H #Hn elim (drops_fwd_fcla_lt2 … H) -H
+#k #Hm #H <(fcla_mono … Hm … Hn) -f //
+qed-.
+
+(* Basic_2A1: includes: drop_fwd_length_lt4 *)
+lemma drops_fcla_fwd_lt4: ∀f,L1,L2,n. ⬇*[Ⓣ, f] L1 ≘ L2 → 𝐂⦃f⦄ ≘ n → 0 < n →
+ |L2| < |L1|.
+#f #L1 #L2 #n #H #Hf #Hn lapply (drops_fcla_fwd … H Hf) -f
+/2 width=1 by lt_minus_to_plus_r/ qed-.
+
+(* Basic_2A1: includes: drop_inv_length_eq *)
+lemma drops_inv_length_eq: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 → |L1| = |L2| → 𝐈⦃f⦄.
+#f #L1 #L2 #H #HL12 elim (drops_fwd_fcla … H) -H
+#n #Hn <HL12 -L2 #H lapply (discr_plus_x_xy … H) -H
+/2 width=3 by fcla_inv_xp/
+qed-.
+
+(* Basic_2A1: includes: drop_fwd_length_eq2 *)
+theorem drops_fwd_length_eq2: ∀f,L1,L2,K1,K2. ⬇*[Ⓣ, f] L1 ≘ K1 → ⬇*[Ⓣ, f] L2 ≘ K2 →
+ |K1| = |K2| → |L1| = |L2|.
+#f #L1 #L2 #K1 #K2 #HLK1 #HLK2 #HL12
+elim (drops_fwd_fcla … HLK1) -HLK1 #n1 #Hn1 #H1 >H1 -L1
+elim (drops_fwd_fcla … HLK2) -HLK2 #n2 #Hn2 #H2 >H2 -L2
+<(fcla_mono … Hn2 … Hn1) -f //
+qed-.
+
+theorem drops_conf_div: ∀f1,f2,L1,L2. ⬇*[Ⓣ, f1] L1 ≘ L2 → ⬇*[Ⓣ, f2] L1 ≘ L2 →
+ ∃∃n. 𝐂⦃f1⦄ ≘ n & 𝐂⦃f2⦄ ≘ n.
+#f1 #f2 #L1 #L2 #H1 #H2
+elim (drops_fwd_fcla … H1) -H1 #n1 #Hf1 #H1
+elim (drops_fwd_fcla … H2) -H2 #n2 #Hf2 >H1 -L1 #H
+lapply (injective_plus_r … H) -L2 #H destruct /2 width=3 by ex2_intro/
+qed-.
+
+theorem drops_conf_div_fcla: ∀f1,f2,L1,L2,n1,n2.
+ ⬇*[Ⓣ, f1] L1 ≘ L2 → ⬇*[Ⓣ, f2] L1 ≘ L2 → 𝐂⦃f1⦄ ≘ n1 → 𝐂⦃f2⦄ ≘ n2 →
+ n1 = n2.
+#f1 #f2 #L1 #L2 #n1 #n2 #Hf1 #Hf2 #Hn1 #Hn2
+lapply (drops_fcla_fwd … Hf1 Hn1) -f1 #H1
+lapply (drops_fcla_fwd … Hf2 Hn2) -f2 >H1 -L1
+/2 width=1 by injective_plus_r/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lex.ma".
+include "static_2/relocation/drops_cext2.ma".
+include "static_2/relocation/drops_sex.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+definition dedropable_sn: predicate … ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. K1 ⪤[R] K2 →
+ ∃∃L2. L1 ⪤[R] L2 & ⬇*[b, f] L2 ≘ K2 & L1 ≡[f] L2.
+
+definition dropable_sn: predicate … ≝
+ λR. ∀b,f,L1,K1. ⬇*[b, f] L1 ≘ K1 → 𝐔⦃f⦄ → ∀L2. L1 ⪤[R] L2 →
+ ∃∃K2. K1 ⪤[R] K2 & ⬇*[b, f] L2 ≘ K2.
+
+definition dropable_dx: predicate … ≝
+ λR. ∀L1,L2. L1 ⪤[R] L2 → ∀b,f,K2. ⬇*[b, f] L2 ≘ K2 → 𝐔⦃f⦄ →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & K1 ⪤[R] K2.
+
+(* Properties with generic extension ****************************************)
+
+(* Basic_2A1: was: lpx_sn_liftable_dedropable *)
+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 (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-.
+
+(* Inversion lemmas with generic extension **********************************)
+
+(* 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 (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 (sex_co_dropable_dx … HL12 … HLK2) -L2
+/3 width=5 by coafter_isid_dx, ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: lpx_sn_drop_conf *)
+lemma lex_drops_conf_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
+ ∀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 (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/
+| /3 width=3 by coafter_isid_dx, isid_push/
+]
+qed-.
+
+(* Basic_2A1: includes: lpx_sn_drop_trans *)
+lemma lex_drops_trans_pair (R): ∀L1,L2. L1 ⪤[R] L2 →
+ ∀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 (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/
+| /3 width=3 by coafter_isid_dx, isid_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/ltc.ma".
+include "static_2/relocation/seq_seq.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Properties with labeled transitive closure *******************************)
+
+lemma d2_liftable_sn_ltc: ∀A,f. associative … f →
+ ∀C,S,R. (∀n. d_liftable2_sn C S (λL. R L n)) →
+ ∀n. d_liftable2_sn C S (λL. ltc A f … (R L) n).
+#A #g #Hg #C #S #R #HR #n #K #T1 #T2 #H
+@(ltc_ind_dx … Hg ???? H) -n -T2
+[ #n #T2 #HT12 #b #g #L #HLK #U1 #HTU1
+ elim (HR … HT12 … HLK … HTU1) -b -K -T1 -HR
+ /3 width=3 by ltc_rc, ex2_intro/
+| #n1 #n2 #T #T2 #_ #IHT1 #HT2 #b #f #L #HLK #U1 #HTU1
+ elim (IHT1 … HLK … HTU1) -T1 #U #HTU #HU1
+ elim (HR … HT2 … HLK … HTU) -HR -K -T
+ /3 width=5 by ltc_dx, ex2_intro/
+]
+qed-.
+
+lemma d2_deliftable_sn_ltc: ∀A,f. associative … f →
+ ∀C,S,R. (∀n. d_deliftable2_sn C S (λL. R L n)) →
+ ∀n. d_deliftable2_sn C S (λL. ltc A f … (R L) n).
+#A #g #Hg #C #S #R #HR #n #L #U1 #U2 #H
+@(ltc_ind_dx … Hg ???? H) -n -U2
+[ #n #U2 #HU12 #b #g #K #HLK #T1 #HTU1
+ elim (HR … HU12 … HLK … HTU1) -b -L -U1 -HR
+ /3 width=3 by ltc_rc, ex2_intro/
+| #n1 #n2 #U #U2 #_ #IHU1 #HU2 #b #f #K #HLK #T1 #HTU1
+ elim (IHU1 … HLK … HTU1) -IHU1 -U1 #T #HTU #HT1
+ elim (HR … HU2 … HLK … HTU) -L -U -HR
+ /3 width=5 by ltc_dx, ex2_intro/
+]
+qed-.
+
+lemma co_dropable_sn_ltc: ∀A,f. associative … f →
+ ∀R. (∀n. co_dropable_sn (λL. R L n)) →
+ ∀n. co_dropable_sn (λL. ltc A f … (R L) n).
+#A #g #Hg #R #HR #n #b #f #L1 #K1 #HLK1 #Hf #f2 #L2 #H
+@(ltc_ind_dx … Hg ???? H) -n -L2
+[ #n #L2 #HL12 #g1 #H
+ elim (HR … HLK1 … Hf … HL12 … H) -f2 -L1 -HR -Hf
+ /3 width=3 by ltc_rc, ex2_intro/
+| #n1 #n2 #L #L2 #_ #IH #HL2 #f1 #H
+ elim (IH … H) -IH #K #HK1 #HLK
+ elim (HR … HLK … HL2 … H) -f2 -L -HR
+ /3 width=3 by ltc_dx, ex2_intro/
+]
+qed-.
+
+lemma co_dropable_dx_ltc: ∀A,f. associative … f →
+ ∀R. (∀n. co_dropable_dx (λL. R L n)) →
+ ∀n. co_dropable_dx (λL. ltc A f … (R L) n).
+#A #g #Hg #R #HR #n #f2 #L1 #L2 #H
+@(ltc_ind_dx … Hg ???? H) -n -L2
+[ #n #L2 #HL12 #b #f #K2 #HLK2 #Hf #f1 #Hf2
+ elim (HR … HL12 … HLK2 … Hf … Hf2) -f2 -L2 -HR -Hf
+ /3 width=3 by ltc_rc, ex2_intro/
+| #n1 #n2 #L #L2 #_ #IHL1 #HL2 #b #f #K2 #HLK2 #Hf #f1 #Hf2
+ elim (HR … HL2 … HLK2 … Hf … Hf2) -L2 -HR #K #HLK #HK2
+ elim (IHL1 … HLK … Hf … Hf2) -Hf -f2 -L
+ /3 width=5 by ltc_dx, ex2_intro/
+]
+qed-.
+
+lemma co_dedropable_sn_ltc: ∀A,f. associative … f →
+ ∀R. (∀n. co_dedropable_sn (λL. R L n)) →
+ ∀n. co_dedropable_sn (λL. ltc A f … (R L) n).
+#A #g #Hg #R #HR #n #b #f #L1 #K1 #HLK1 #f1 #K2 #H
+@(ltc_ind_dx … Hg ???? H) -n -K2
+[ #n #K2 #HK12 #f2 #Hf
+ elim (HR … HLK1 … HK12 … Hf) -f1 -K1 -HR
+ /3 width=4 by ltc_rc, ex3_intro/
+| #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 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 "static_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 "static_2/relocation/lifts_lifts_bind.ma".
+include "static_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-.
--- /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 "static_2/relocation/lifts_vector.ma".
+include "static_2/relocation/drops.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+definition d_liftable1_all: predicate (relation2 lenv term) ≝
+ λR. ∀K,Ts. all … (R K) Ts →
+ ∀b,f,L. ⬇*[b, f] L ≘ K →
+ ∀Us. ⬆*[f] Ts ≘ Us → all … (R L) Us.
+
+(* Properties with generic relocation for term vectors **********************)
+
+(* Basic_2A1: was: d1_liftables_liftables_all *)
+lemma d1_liftable_liftable_all: ∀R. d_liftable1 R → d_liftable1_all R.
+#R #HR #K #Ts #HTs #b #f #L #HLK #Us #H
+generalize in match HTs; -HTs elim H -Ts -Us normalize //
+#Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/cl_restricted_weight.ma".
+include "static_2/relocation/lifts_weight_bind.ma".
+include "static_2/relocation/drops.ma".
+
+(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
+
+(* Forward lemmas with weight for local environments ************************)
+
+(* Basic_2A1: includes: drop_fwd_lw *)
+lemma drops_fwd_lw: ∀b,f,L1,L2. ⬇*[b, f] L1 ≘ L2 → ♯{L2} ≤ ♯{L1}.
+#b #f #L1 #L2 #H elim H -f -L1 -L2 //
+[ /2 width=3 by transitive_le/
+| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 normalize
+ >(liftsb_fwd_bw … HI21) -HI21 /2 width=1 by monotonic_le_plus_l/
+]
+qed-.
+
+(* Basic_2A1: includes: drop_fwd_lw_lt *)
+lemma drops_fwd_lw_lt: ∀f,L1,L2. ⬇*[Ⓣ, f] L1 ≘ L2 →
+ (𝐈⦃f⦄ → ⊥) → ♯{L2} < ♯{L1}.
+#f #L1 #L2 #H elim H -f -L1 -L2
+[ #f #Hf #Hnf elim Hnf -Hnf /2 width=1 by/
+| /3 width=3 by drops_fwd_lw, le_to_lt_to_lt/
+| #f #I1 #I2 #L1 #L2 #_ #HI21 #IHL12 #H normalize in ⊢ (?%%);
+ >(liftsb_fwd_bw … HI21) -I2 /5 width=3 by isid_push, monotonic_lt_plus_l/
+]
+qed-.
+
+(* Forward lemmas with restricted weight for closures ***********************)
+
+(* Basic_2A1: includes: drop_fwd_rfw *)
+lemma drops_bind2_fwd_rfw: ∀b,f,I,L,K,V. ⬇*[b, f] L ≘ K.ⓑ{I}V → ∀T. ♯{K, V} < ♯{L, T}.
+#b #f #I #L #K #V #HLK lapply (drops_fwd_lw … HLK) -HLK
+normalize in ⊢ (%→?→?%%); /3 width=3 by le_to_lt_to_lt, monotonic_lt_plus_r/
+qed-.
+
+(* Advanced inversion lemma *************************************************)
+
+lemma drops_inv_x_bind_xy: ∀b,f,I,L. ⬇*[b, f] L ≘ L.ⓘ{I} → ⊥.
+#b #f #I #L #H lapply (drops_fwd_lw … H) -b -f
+/2 width=4 by lt_le_false/ (**) (* full auto is a bit slow: 19s *)
+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/pull/pull_2.ma".
+include "ground_2/pull/pull_4.ma".
+include "ground_2/relocation/rtmap_uni.ma".
+include "static_2/notation/relations/relation_3.ma".
+include "static_2/syntax/cext2.ma".
+include "static_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.
+
+interpretation "generic extension (local environment)"
+ 'Relation R L1 L2 = (lex R L1 L2).
+
+definition lex_confluent: relation (relation3 …) ≝ λR1,R2.
+ ∀L0,T0,T1. R1 L0 T0 T1 → ∀T2. R2 L0 T0 T2 →
+ ∀L1. L0 ⪤[R1] L1 → ∀L2. L0 ⪤[R2] L2 →
+ ∃∃T. R2 L1 T1 T & R1 L2 T2 T.
+
+definition lex_transitive: relation (relation3 …) ≝ λR1,R2.
+ ∀L1,T1,T. R1 L1 T1 T → ∀L2. L1 ⪤[R1] L2 →
+ ∀T2. R2 L2 T T2 → R1 L1 T1 T2.
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: was: lpx_sn_atom *)
+lemma lex_atom (R): ⋆ ⪤[R] ⋆.
+/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 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 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 sex_co, cext2_co, ex2_intro/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lex_bind_refl_dx (R): c_reflexive … R →
+ ∀I,K1,K2. K1 ⪤[R] K2 → K1.ⓘ{I} ⪤[R] K2.ⓘ{I}.
+/3 width=3 by ext2_refl, lex_bind/ qed.
+
+lemma lex_unit (R): ∀I,K1,K2. K1 ⪤[R] K2 → K1.ⓤ{I} ⪤[R] K2.ⓤ{I}.
+/3 width=1 by lex_bind, ext2_unit/ qed.
+
+(* Basic_2A1: was: lpx_sn_pair *)
+lemma lex_pair (R): ∀I,K1,K2,V1,V2. K1 ⪤[R] K2 → R K1 V1 V2 →
+ K1.ⓑ{I}V1 ⪤[R] K2.ⓑ{I}V2.
+/3 width=1 by lex_bind, ext2_pair/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Basic_2A1: was: lpx_sn_inv_atom1: *)
+lemma lex_inv_atom_sn (R): ∀L2. ⋆ ⪤[R] L2 → 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 (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 >(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 (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-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lex_inv_unit_sn (R): ∀I,L2,K1. K1.ⓤ{I} ⪤[R] L2 →
+ ∃∃K2. K1 ⪤[R] K2 & L2 = K2.ⓤ{I}.
+#R #I #L2 #K1 #H
+elim (lex_inv_bind_sn … H) -H #Z2 #K2 #HK12 #HZ2 #H destruct
+elim (ext2_inv_unit_sn … HZ2) -HZ2
+/2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: was: lpx_sn_inv_pair1 *)
+lemma lex_inv_pair_sn (R): ∀I,L2,K1,V1. K1.ⓑ{I}V1 ⪤[R] L2 →
+ ∃∃K2,V2. K1 ⪤[R] K2 & R K1 V1 V2 & L2 = K2.ⓑ{I}V2.
+#R #I #L2 #K1 #V1 #H
+elim (lex_inv_bind_sn … H) -H #Z2 #K2 #HK12 #HZ2 #H destruct
+elim (ext2_inv_pair_sn … HZ2) -HZ2 #V2 #HV12 #H destruct
+/2 width=5 by ex3_2_intro/
+qed-.
+
+lemma lex_inv_unit_dx (R): ∀I,L1,K2. L1 ⪤[R] K2.ⓤ{I} →
+ ∃∃K1. K1 ⪤[R] K2 & L1 = K1.ⓤ{I}.
+#R #I #L1 #K2 #H
+elim (lex_inv_bind_dx … H) -H #Z1 #K1 #HK12 #HZ1 #H destruct
+elim (ext2_inv_unit_dx … HZ1) -HZ1
+/2 width=3 by ex2_intro/
+qed-.
+
+(* Basic_2A1: was: lpx_sn_inv_pair2 *)
+lemma lex_inv_pair_dx (R): ∀I,L1,K2,V2. L1 ⪤[R] K2.ⓑ{I}V2 →
+ ∃∃K1,V1. K1 ⪤[R] K2 & R K1 V1 V2 & L1 = K1.ⓑ{I}V1.
+#R #I #L1 #K2 #V2 #H
+elim (lex_inv_bind_dx … H) -H #Z1 #K1 #HK12 #HZ1 #H destruct
+elim (ext2_inv_pair_dx … HZ1) -HZ1 #V1 #HV12 #H destruct
+/2 width=5 by ex3_2_intro/
+qed-.
+
+(* Basic_2A1: was: lpx_sn_inv_pair *)
+lemma lex_inv_pair (R): ∀I1,I2,L1,L2,V1,V2.
+ L1.ⓑ{I1}V1 ⪤[R] L2.ⓑ{I2}V2 →
+ ∧∧ L1 ⪤[R] L2 & R L1 V1 V2 & I1 = I2.
+#R #I1 #I2 #L1 #L2 #V1 #V2 #H elim (lex_inv_pair_sn … H) -H
+#L0 #V0 #HL10 #HV10 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+(* Basic eliminators ********************************************************)
+
+lemma lex_ind (R) (Q:relation2 …):
+ Q (⋆) (⋆) →
+ (
+ ∀I,K1,K2. K1 ⪤[R] K2 → Q K1 K2 → Q (K1.ⓤ{I}) (K2.ⓤ{I})
+ ) → (
+ ∀I,K1,K2,V1,V2. K1 ⪤[R] K2 → Q K1 K2 → R K1 V1 V2 →Q (K1.ⓑ{I}V1) (K2.ⓑ{I}V2)
+ ) →
+ ∀L1,L2. L1 ⪤[R] L2 → Q L1 L2.
+#R #Q #IH1 #IH2 #IH3 #L1 #L2 * #f @pull_2 #H
+elim H -f -L1 -L2 // #f #I1 #I2 #K1 #K2 @pull_4 #H
+[ elim (isid_inv_next … H)
+| lapply (isid_inv_push … H ??)
+] -H [5:|*: // ] #Hf @pull_2 #H
+elim H -H /3 width=3 by ex2_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/sex_length.ma".
+include "static_2/relocation/lex.ma".
+
+(* GENERIC EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **************)
+
+(* Forward lemmas with length for local environments ************************)
+
+(* 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 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 "static_2/relocation/lex.ma".
+
+(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
+
+(* Main properties **********************************************************)
+
+(* Basic_2A1: was: lpx_sn_trans *)
+theorem lex_trans (R): lex_transitive R R → Transitive … (lex R).
+#R #HR #L1 #L #H @(lex_ind … H) -L1 -L //
+[ #I #K1 #K #HK1 #IH #Y #H
+ elim (lex_inv_unit_sn … H) -H #K2 #K2 #H destruct
+ /3 width=1 by lex_unit/
+| #I #K1 #L #V1 #V #HK1 #IH #HV1 #X #H
+ elim (lex_inv_pair_sn … H) -H #K2 #V2 #HK2 #HV2 #H destruct
+ /3 width=5 by lex_pair/
+]
+qed-.
+
+(* Basic_2A1: was: lpx_sn_conf *)
+theorem lex_conf (R1) (R2): lex_confluent R1 R2 → confluent2 … (lex R1) (lex R2).
+#R1 #R2 #HR12 #L0 elim L0 -L0 [| #K0 * ]
+[ #Y1 #H1 #Y2 #H2
+ >(lex_inv_atom_sn … H1) -Y1
+ >(lex_inv_atom_sn … H2) -Y2
+ /2 width=3 by lex_atom, ex2_intro/
+| #I #IH #Y1 #H1 #Y2 #H2
+ elim (lex_inv_unit_sn … H1) -H1 #K1 #HK01 #H destruct
+ elim (lex_inv_unit_sn … H2) -H2 #K2 #HK02 #H destruct
+ elim (IH … HK01 … HK02) -K0 #K #HK1 #HK2
+ /3 width=3 by lex_unit, ex2_intro/
+| #I #V0 #IH #Y1 #H1 #Y2 #H2
+ elim (lex_inv_pair_sn … H1) -H1 #K1 #V1 #HK01 #HV01 #H destruct
+ elim (lex_inv_pair_sn … H2) -H2 #K2 #V2 #HK02 #HV02 #H destruct
+ elim (HR12 … HV01 … HV02 … HK01 … HK02) -V0 #V #HV1 #HV2
+ elim (IH … HK01 … HK02) -K0 #K #HK1 #HK2
+ /3 width=5 by lex_pair, ex2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/ext2_tc.ma".
+include "static_2/relocation/sex_tc.ma".
+include "static_2/relocation/lex.ma".
+
+alias symbol "subseteq" = "relation inclusion".
+
+(* GENERIC EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS **************)
+
+(* Inversion lemmas with transitive closure *********************************)
+
+(* Basic_2A1: was: lpx_sn_LTC_TC_lpx_sn *)
+lemma lex_inv_CTC (R): c_reflexive … R →
+ lex (CTC … R) ⊆ TC … (lex R).
+#R #HR #L1 #L2 *
+/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) →
+ s_rs_transitive_isid cfull (cext2 R).
+#R #HR #f #Hf #L2 #T1 #T2 #H #L1 #HL12
+elim (ext2_tc … H) -H
+[ /3 width=1 by ext2_inv_tc, ext2_unit/
+| #I #V1 #V2 #HV12
+ @ext2_inv_tc @ext2_pair
+ @(HR … HV12) -HV12 /2 width=3 by ex2_intro/ (**) (* auto fails *)
+]
+qed-.
+
+(* Properties with transitive closure ***************************************)
+
+(* Basic_2A1: was: TC_lpx_sn_inv_lpx_sn_LTC *)
+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 … (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-.
+
+lemma lex_CTC_inj (R): s_rs_transitive … R (λ_. lex R) →
+ (lex R) ⊆ lex (CTC … R).
+/3 width=1 by lex_CTC, inj/ qed-.
+
+lemma lex_CTC_step_dx (R): c_reflexive … R → s_rs_transitive … R (λ_. lex R) →
+ ∀L1,L. lex (CTC … R) L1 L →
+ ∀L2. lex R L L2 → lex (CTC … R) L1 L2.
+/4 width=3 by lex_CTC, lex_inv_CTC, step/ qed-.
+
+lemma lex_CTC_step_sn (R): c_reflexive … R → s_rs_transitive … R (λ_. lex R) →
+ ∀L1,L. lex R L1 L →
+ ∀L2. lex (CTC … R) L L2 → lex (CTC … R) L1 L2.
+/4 width=3 by lex_CTC, lex_inv_CTC, TC_strap/ qed-.
+
+(* Eliminators with transitive closure **************************************)
+
+lemma lex_CTC_ind_sn (R) (L2): c_reflexive … R → s_rs_transitive … R (λ_. lex R) →
+ ∀Q:predicate lenv. Q L2 →
+ (∀L1,L. L1 ⪤[R] L → L ⪤[CTC … R] L2 → Q L → Q L1) →
+ ∀L1. L1 ⪤[CTC … R] L2 → Q L1.
+#R #L2 #H1R #H2R #Q #IH1 #IH2 #L1 #H
+lapply (lex_inv_CTC … H1R … H) -H #H
+@(TC_star_ind_dx ???????? H) -H
+/3 width=4 by lex_CTC, lex_refl/
+qed-.
+
+lemma lex_CTC_ind_dx (R) (L1): c_reflexive … R → s_rs_transitive … R (λ_. lex R) →
+ ∀Q:predicate lenv. Q L1 →
+ (∀L,L2. L1 ⪤[CTC … R] L → L ⪤[R] L2 → Q L → Q L2) →
+ ∀L2. L1 ⪤[CTC … R] L2 → Q L2.
+#R #L1 #H1R #H2R #Q #IH1 #IH2 #L2 #H
+lapply (lex_inv_CTC … H1R … H) -H #H
+@(TC_star_ind ???????? H) -H
+/3 width=4 by lex_CTC, lex_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 "ground_2/relocation/nstream_after.ma".
+include "static_2/notation/relations/rliftstar_3.ma".
+include "static_2/syntax/term.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Basic_1: includes:
+ lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
+ lifts_nil lifts_cons
+*)
+inductive lifts: rtmap → relation term ≝
+| lifts_sort: ∀f,s. lifts f (⋆s) (⋆s)
+| lifts_lref: ∀f,i1,i2. @⦃i1, f⦄ ≘ i2 → lifts f (#i1) (#i2)
+| lifts_gref: ∀f,l. lifts f (§l) (§l)
+| lifts_bind: ∀f,p,I,V1,V2,T1,T2.
+ lifts f V1 V2 → lifts (⫯f) T1 T2 →
+ lifts f (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
+| lifts_flat: ∀f,I,V1,V2,T1,T2.
+ lifts f V1 V2 → lifts f T1 T2 →
+ lifts f (ⓕ{I}V1.T1) (ⓕ{I}V2.T2)
+.
+
+interpretation "uniform relocation (term)"
+ 'RLiftStar i T1 T2 = (lifts (uni i) T1 T2).
+
+interpretation "generic relocation (term)"
+ 'RLiftStar f T1 T2 = (lifts f T1 T2).
+
+definition liftable2_sn: predicate (relation term) ≝
+ λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≘ U1 →
+ ∃∃U2. ⬆*[f] T2 ≘ U2 & R U1 U2.
+
+definition deliftable2_sn: predicate (relation term) ≝
+ λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≘ U1 →
+ ∃∃T2. ⬆*[f] T2 ≘ U2 & R T1 T2.
+
+definition liftable2_bi: predicate (relation term) ≝
+ λR. ∀T1,T2. R T1 T2 → ∀f,U1. ⬆*[f] T1 ≘ U1 →
+ ∀U2. ⬆*[f] T2 ≘ U2 → R U1 U2.
+
+definition deliftable2_bi: predicate (relation term) ≝
+ λR. ∀U1,U2. R U1 U2 → ∀f,T1. ⬆*[f] T1 ≘ U1 →
+ ∀T2. ⬆*[f] T2 ≘ U2 → R T1 T2.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lifts_inv_sort1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀s. X = ⋆s → Y = ⋆s.
+#f #X #Y * -f -X -Y //
+[ #f #i1 #i2 #_ #x #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+]
+qed-.
+
+(* Basic_1: was: lift1_sort *)
+(* Basic_2A1: includes: lift_inv_sort1 *)
+lemma lifts_inv_sort1: ∀f,Y,s. ⬆*[f] ⋆s ≘ Y → Y = ⋆s.
+/2 width=4 by lifts_inv_sort1_aux/ qed-.
+
+fact lifts_inv_lref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i1. X = #i1 →
+ ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
+#f #X #Y * -f -X -Y
+[ #f #s #x #H destruct
+| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
+| #f #l #x #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+]
+qed-.
+
+(* Basic_1: was: lift1_lref *)
+(* Basic_2A1: includes: lift_inv_lref1 lift_inv_lref1_lt lift_inv_lref1_ge *)
+lemma lifts_inv_lref1: ∀f,Y,i1. ⬆*[f] #i1 ≘ Y →
+ ∃∃i2. @⦃i1, f⦄ ≘ i2 & Y = #i2.
+/2 width=3 by lifts_inv_lref1_aux/ qed-.
+
+fact lifts_inv_gref1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. X = §l → Y = §l.
+#f #X #Y * -f -X -Y //
+[ #f #i1 #i2 #_ #x #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+]
+qed-.
+
+(* Basic_2A1: includes: lift_inv_gref1 *)
+lemma lifts_inv_gref1: ∀f,Y,l. ⬆*[f] §l ≘ Y → Y = §l.
+/2 width=4 by lifts_inv_gref1_aux/ qed-.
+
+fact lifts_inv_bind1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y →
+ ∀p,I,V1,T1. X = ⓑ{p,I}V1.T1 →
+ ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
+ Y = ⓑ{p,I}V2.T2.
+#f #X #Y * -f -X -Y
+[ #f #s #q #J #W1 #U1 #H destruct
+| #f #i1 #i2 #_ #q #J #W1 #U1 #H destruct
+| #f #l #b #J #W1 #U1 #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W1 #U1 #H destruct
+]
+qed-.
+
+(* Basic_1: was: lift1_bind *)
+(* Basic_2A1: includes: lift_inv_bind1 *)
+lemma lifts_inv_bind1: ∀f,p,I,V1,T1,Y. ⬆*[f] ⓑ{p,I}V1.T1 ≘ Y →
+ ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
+ Y = ⓑ{p,I}V2.T2.
+/2 width=3 by lifts_inv_bind1_aux/ qed-.
+
+fact lifts_inv_flat1_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
+ ∀I,V1,T1. X = ⓕ{I}V1.T1 →
+ ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
+ Y = ⓕ{I}V2.T2.
+#f #X #Y * -f -X -Y
+[ #f #s #J #W1 #U1 #H destruct
+| #f #i1 #i2 #_ #J #W1 #U1 #H destruct
+| #f #l #J #W1 #U1 #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W1 #U1 #H destruct
+| #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_1: was: lift1_flat *)
+(* Basic_2A1: includes: lift_inv_flat1 *)
+lemma lifts_inv_flat1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ⓕ{I}V1.T1 ≘ Y →
+ ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
+ Y = ⓕ{I}V2.T2.
+/2 width=3 by lifts_inv_flat1_aux/ qed-.
+
+fact lifts_inv_sort2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀s. Y = ⋆s → X = ⋆s.
+#f #X #Y * -f -X -Y //
+[ #f #i1 #i2 #_ #x #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+]
+qed-.
+
+(* Basic_1: includes: lift_gen_sort *)
+(* Basic_2A1: includes: lift_inv_sort2 *)
+lemma lifts_inv_sort2: ∀f,X,s. ⬆*[f] X ≘ ⋆s → X = ⋆s.
+/2 width=4 by lifts_inv_sort2_aux/ qed-.
+
+fact lifts_inv_lref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀i2. Y = #i2 →
+ ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
+#f #X #Y * -f -X -Y
+[ #f #s #x #H destruct
+| #f #i1 #i2 #Hi12 #x #H destruct /2 width=3 by ex2_intro/
+| #f #l #x #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+]
+qed-.
+
+(* Basic_1: includes: lift_gen_lref lift_gen_lref_lt lift_gen_lref_false lift_gen_lref_ge *)
+(* Basic_2A1: includes: lift_inv_lref2 lift_inv_lref2_lt lift_inv_lref2_be lift_inv_lref2_ge lift_inv_lref2_plus *)
+lemma lifts_inv_lref2: ∀f,X,i2. ⬆*[f] X ≘ #i2 →
+ ∃∃i1. @⦃i1, f⦄ ≘ i2 & X = #i1.
+/2 width=3 by lifts_inv_lref2_aux/ qed-.
+
+fact lifts_inv_gref2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → ∀l. Y = §l → X = §l.
+#f #X #Y * -f -X -Y //
+[ #f #i1 #i2 #_ #x #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #x #H destruct
+]
+qed-.
+
+(* Basic_2A1: includes: lift_inv_gref1 *)
+lemma lifts_inv_gref2: ∀f,X,l. ⬆*[f] X ≘ §l → X = §l.
+/2 width=4 by lifts_inv_gref2_aux/ qed-.
+
+fact lifts_inv_bind2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y →
+ ∀p,I,V2,T2. Y = ⓑ{p,I}V2.T2 →
+ ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
+ X = ⓑ{p,I}V1.T1.
+#f #X #Y * -f -X -Y
+[ #f #s #q #J #W2 #U2 #H destruct
+| #f #i1 #i2 #_ #q #J #W2 #U2 #H destruct
+| #f #l #q #J #W2 #U2 #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #HV12 #HT12 #q #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #q #J #W2 #U2 #H destruct
+]
+qed-.
+
+(* Basic_1: includes: lift_gen_bind *)
+(* Basic_2A1: includes: lift_inv_bind2 *)
+lemma lifts_inv_bind2: ∀f,p,I,V2,T2,X. ⬆*[f] X ≘ ⓑ{p,I}V2.T2 →
+ ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[⫯f] T1 ≘ T2 &
+ X = ⓑ{p,I}V1.T1.
+/2 width=3 by lifts_inv_bind2_aux/ qed-.
+
+fact lifts_inv_flat2_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
+ ∀I,V2,T2. Y = ⓕ{I}V2.T2 →
+ ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
+ X = ⓕ{I}V1.T1.
+#f #X #Y * -f -X -Y
+[ #f #s #J #W2 #U2 #H destruct
+| #f #i1 #i2 #_ #J #W2 #U2 #H destruct
+| #f #l #J #W2 #U2 #H destruct
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #J #W2 #U2 #H destruct
+| #f #I #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W2 #U2 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_1: includes: lift_gen_flat *)
+(* Basic_2A1: includes: lift_inv_flat2 *)
+lemma lifts_inv_flat2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≘ ⓕ{I}V2.T2 →
+ ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & ⬆*[f] T1 ≘ T2 &
+ X = ⓕ{I}V1.T1.
+/2 width=3 by lifts_inv_flat2_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lifts_inv_atom1: ∀f,I,Y. ⬆*[f] ⓪{I} ≘ Y →
+ ∨∨ ∃∃s. I = Sort s & Y = ⋆s
+ | ∃∃i,j. @⦃i, f⦄ ≘ j & I = LRef i & Y = #j
+ | ∃∃l. I = GRef l & Y = §l.
+#f * #n #Y #H
+[ lapply (lifts_inv_sort1 … H)
+| elim (lifts_inv_lref1 … H)
+| lapply (lifts_inv_gref1 … H)
+] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
+qed-.
+
+lemma lifts_inv_atom2: ∀f,I,X. ⬆*[f] X ≘ ⓪{I} →
+ ∨∨ ∃∃s. X = ⋆s & I = Sort s
+ | ∃∃i,j. @⦃i, f⦄ ≘ j & X = #i & I = LRef j
+ | ∃∃l. X = §l & I = GRef l.
+#f * #n #X #H
+[ lapply (lifts_inv_sort2 … H)
+| elim (lifts_inv_lref2 … H)
+| lapply (lifts_inv_gref2 … H)
+] -H /3 width=5 by or3_intro0, or3_intro1, or3_intro2, ex3_2_intro, ex2_intro/
+qed-.
+
+(* Basic_2A1: includes: lift_inv_pair_xy_x *)
+lemma lifts_inv_pair_xy_x: ∀f,I,V,T. ⬆*[f] ②{I}V.T ≘ V → ⊥.
+#f #J #V elim V -V
+[ * #i #U #H
+ [ lapply (lifts_inv_sort2 … H) -H #H destruct
+ | elim (lifts_inv_lref2 … H) -H
+ #x #_ #H destruct
+ | lapply (lifts_inv_gref2 … H) -H #H destruct
+ ]
+| * [ #p ] #I #V2 #T2 #IHV2 #_ #U #H
+ [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
+ | elim (lifts_inv_flat2 … H) -H #V1 #T1 #HV12 #_ #H destruct /2 width=3 by/
+ ]
+]
+qed-.
+
+(* Basic_1: includes: thead_x_lift_y_y *)
+(* Basic_2A1: includes: lift_inv_pair_xy_y *)
+lemma lifts_inv_pair_xy_y: ∀I,T,V,f. ⬆*[f] ②{I}V.T ≘ T → ⊥.
+#J #T elim T -T
+[ * #i #W #f #H
+ [ lapply (lifts_inv_sort2 … H) -H #H destruct
+ | elim (lifts_inv_lref2 … H) -H
+ #x #_ #H destruct
+ | lapply (lifts_inv_gref2 … H) -H #H destruct
+ ]
+| * [ #p ] #I #V2 #T2 #_ #IHT2 #W #f #H
+ [ elim (lifts_inv_bind2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
+ | elim (lifts_inv_flat2 … H) -H #V1 #T1 #_ #HT12 #H destruct /2 width=4 by/
+ ]
+]
+qed-.
+
+(* Inversion lemmas with uniform relocations ********************************)
+
+lemma lifts_inv_lref1_uni: ∀l,Y,i. ⬆*[l] #i ≘ Y → Y = #(l+i).
+#l #Y #i1 #H elim (lifts_inv_lref1 … H) -H /4 width=4 by at_mono, eq_f/
+qed-.
+
+lemma lifts_inv_lref2_uni: ∀l,X,i2. ⬆*[l] X ≘ #i2 →
+ ∃∃i1. X = #i1 & i2 = l + i1.
+#l #X #i2 #H elim (lifts_inv_lref2 … H) -H
+/3 width=3 by at_inv_uni, ex2_intro/
+qed-.
+
+lemma lifts_inv_lref2_uni_ge: ∀l,X,i. ⬆*[l] X ≘ #(l + i) → X = #i.
+#l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
+#i1 #H1 #H2 destruct /4 width=2 by injective_plus_r, eq_f, sym_eq/
+qed-.
+
+lemma lifts_inv_lref2_uni_lt: ∀l,X,i. ⬆*[l] X ≘ #i → i < l → ⊥.
+#l #X #i2 #H elim (lifts_inv_lref2_uni … H) -H
+#i1 #_ #H1 #H2 destruct /2 width=4 by lt_le_false/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+(* Basic_2A1: includes: lift_inv_O2 *)
+lemma lifts_fwd_isid: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐈⦃f⦄ → T1 = T2.
+#f #T1 #T2 #H elim H -f -T1 -T2
+/4 width=3 by isid_inv_at_mono, isid_push, eq_f2, eq_f/
+qed-.
+
+(* Basic_2A1: includes: lift_fwd_pair1 *)
+lemma lifts_fwd_pair1: ∀f:rtmap. ∀I,V1,T1,Y. ⬆*[f] ②{I}V1.T1 ≘ Y →
+ ∃∃V2,T2. ⬆*[f] V1 ≘ V2 & Y = ②{I}V2.T2.
+#f * [ #p ] #I #V1 #T1 #Y #H
+[ elim (lifts_inv_bind1 … H) -H /2 width=4 by ex2_2_intro/
+| elim (lifts_inv_flat1 … H) -H /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+(* Basic_2A1: includes: lift_fwd_pair2 *)
+lemma lifts_fwd_pair2: ∀f:rtmap. ∀I,V2,T2,X. ⬆*[f] X ≘ ②{I}V2.T2 →
+ ∃∃V1,T1. ⬆*[f] V1 ≘ V2 & X = ②{I}V1.T1.
+#f * [ #p ] #I #V2 #T2 #X #H
+[ elim (lifts_inv_bind2 … H) -H /2 width=4 by ex2_2_intro/
+| elim (lifts_inv_flat2 … H) -H /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lifts_eq_repl_back: ∀T1,T2. eq_repl_back … (λf. ⬆*[f] T1 ≘ T2).
+#T1 #T2 #f1 #H elim H -T1 -T2 -f1
+/4 width=5 by lifts_flat, lifts_bind, lifts_lref, at_eq_repl_back, eq_push/
+qed-.
+
+lemma lifts_eq_repl_fwd: ∀T1,T2. eq_repl_fwd … (λf. ⬆*[f] T1 ≘ T2).
+#T1 #T2 @eq_repl_sym /2 width=3 by lifts_eq_repl_back/ (**) (* full auto fails *)
+qed-.
+
+(* Basic_1: includes: lift_r *)
+(* Basic_2A1: includes: lift_refl *)
+lemma lifts_refl: ∀T,f. 𝐈⦃f⦄ → ⬆*[f] T ≘ T.
+#T elim T -T *
+/4 width=3 by lifts_flat, lifts_bind, lifts_lref, isid_inv_at, isid_push/
+qed.
+
+(* Basic_2A1: includes: lift_total *)
+lemma lifts_total: ∀T1,f. ∃T2. ⬆*[f] T1 ≘ T2.
+#T1 elim T1 -T1 *
+/3 width=2 by lifts_lref, lifts_sort, lifts_gref, ex_intro/
+[ #p ] #I #V1 #T1 #IHV1 #IHT1 #f
+elim (IHV1 f) -IHV1 #V2 #HV12
+[ elim (IHT1 (⫯f)) -IHT1 /3 width=2 by lifts_bind, ex_intro/
+| elim (IHT1 f) -IHT1 /3 width=2 by lifts_flat, ex_intro/
+]
+qed-.
+
+lemma lift_lref_uni: ∀l,i. ⬆*[l] #i ≘ #(l+i).
+#l elim l -l /2 width=1 by lifts_lref/
+qed.
+
+(* Basic_1: includes: lift_free (right to left) *)
+(* Basic_2A1: includes: lift_split *)
+lemma lifts_split_trans: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 →
+ ∀f1,f2. f2 ⊚ f1 ≘ f →
+ ∃∃T. ⬆*[f1] T1 ≘ T & ⬆*[f2] T ≘ T2.
+#f #T1 #T2 #H elim H -f -T1 -T2
+[ /3 width=3 by lifts_sort, ex2_intro/
+| #f #i1 #i2 #Hi #f1 #f2 #Ht elim (after_at_fwd … Hi … Ht) -Hi -Ht
+ /3 width=3 by lifts_lref, ex2_intro/
+| /3 width=3 by lifts_gref, ex2_intro/
+| #f #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
+ elim (IHV … Ht) elim (IHT (⫯f1) (⫯f2)) -IHV -IHT
+ /3 width=5 by lifts_bind, after_O2, ex2_intro/
+| #f #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f1 #f2 #Ht
+ elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
+ /3 width=5 by lifts_flat, ex2_intro/
+]
+qed-.
+
+(* Note: apparently, this was missing in Basic_2A1 *)
+lemma lifts_split_div: ∀f1,T1,T2. ⬆*[f1] T1 ≘ T2 →
+ ∀f2,f. f2 ⊚ f1 ≘ f →
+ ∃∃T. ⬆*[f2] T2 ≘ T & ⬆*[f] T1 ≘ T.
+#f1 #T1 #T2 #H elim H -f1 -T1 -T2
+[ /3 width=3 by lifts_sort, ex2_intro/
+| #f1 #i1 #i2 #Hi #f2 #f #Ht elim (after_at1_fwd … Hi … Ht) -Hi -Ht
+ /3 width=3 by lifts_lref, ex2_intro/
+| /3 width=3 by lifts_gref, ex2_intro/
+| #f1 #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
+ elim (IHV … Ht) elim (IHT (⫯f2) (⫯f)) -IHV -IHT
+ /3 width=5 by lifts_bind, after_O2, ex2_intro/
+| #f1 #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f2 #f #Ht
+ elim (IHV … Ht) elim (IHT … Ht) -IHV -IHT -Ht
+ /3 width=5 by lifts_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: includes: dnf_dec2 dnf_dec *)
+(* Basic_2A1: includes: is_lift_dec *)
+lemma is_lifts_dec: ∀T2,f. Decidable (∃T1. ⬆*[f] T1 ≘ T2).
+#T1 elim T1 -T1
+[ * [1,3: /3 width=2 by lifts_sort, lifts_gref, ex_intro, or_introl/ ]
+ #i2 #f elim (is_at_dec f i2) //
+ [ * /4 width=3 by lifts_lref, ex_intro, or_introl/
+ | #H @or_intror *
+ #X #HX elim (lifts_inv_lref2 … HX) -HX
+ /3 width=2 by ex_intro/
+ ]
+| * [ #p ] #I #V2 #T2 #IHV2 #IHT2 #f
+ [ elim (IHV2 f) -IHV2
+ [ * #V1 #HV12 elim (IHT2 (⫯f)) -IHT2
+ [ * #T1 #HT12 @or_introl /3 width=2 by lifts_bind, ex_intro/
+ | -V1 #HT2 @or_intror * #X #H
+ elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
+ ]
+ | -IHT2 #HV2 @or_intror * #X #H
+ elim (lifts_inv_bind2 … H) -H /3 width=2 by ex_intro/
+ ]
+ | elim (IHV2 f) -IHV2
+ [ * #V1 #HV12 elim (IHT2 f) -IHT2
+ [ * #T1 #HT12 /4 width=2 by lifts_flat, ex_intro, or_introl/
+ | -V1 #HT2 @or_intror * #X #H
+ elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
+ ]
+ | -IHT2 #HV2 @or_intror * #X #H
+ elim (lifts_inv_flat2 … H) -H /3 width=2 by ex_intro/
+ ]
+ ]
+]
+qed-.
+
+(* Properties with uniform relocation ***************************************)
+
+lemma lifts_uni: ∀n1,n2,T,U. ⬆*[𝐔❴n1❵∘𝐔❴n2❵] T ≘ U → ⬆*[n1+n2] T ≘ U.
+/3 width=4 by lifts_eq_repl_back, after_inv_total/ qed.
+
+(* Basic_2A1: removed theorems 14:
+ lifts_inv_nil lifts_inv_cons
+ lift_inv_Y1 lift_inv_Y2 lift_inv_lref_Y1 lift_inv_lref_Y2 lift_lref_Y lift_Y1
+ lift_lref_lt_eq lift_lref_ge_eq lift_lref_plus lift_lref_pred
+ lift_lref_ge_minus lift_lref_ge_minus_eq
+*)
+(* Basic_1: removed theorems 8:
+ lift_lref_gt
+ lift_head lift_gen_head
+ lift_weight_map lift_weight lift_weight_add lift_weight_add_O
+ lift_tlt_dx
+*)
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/ext2.ma".
+include "static_2/relocation/lifts.ma".
+
+(* GENERIC RELOCATION FOR BINDERS *******************************************)
+
+definition liftsb: rtmap → relation bind ≝
+ λf. ext2 (lifts f).
+
+interpretation "uniform relocation (binder for local environments)"
+ 'RLiftStar i I1 I2 = (liftsb (uni i) I1 I2).
+
+interpretation "generic relocation (binder for local environments)"
+ 'RLiftStar f I1 I2 = (liftsb f I1 I2).
+
+(* Basic_inversion lemmas **************************************************)
+
+lemma liftsb_inv_unit_sn: ∀f,I,Z2. ⬆*[f] BUnit I ≘ Z2 → Z2 = BUnit I.
+/2 width=2 by ext2_inv_unit_sn/ qed-.
+
+lemma liftsb_inv_pair_sn: ∀f:rtmap. ∀Z2,I,V1. ⬆*[f] BPair I V1 ≘ Z2 →
+ ∃∃V2. ⬆*[f] V1 ≘ V2 & Z2 = BPair I V2.
+/2 width=1 by ext2_inv_pair_sn/ qed-.
+
+lemma liftsb_inv_unit_dx: ∀f,I,Z1. ⬆*[f] Z1 ≘ BUnit I → Z1 = BUnit I.
+/2 width=2 by ext2_inv_unit_dx/ qed-.
+
+lemma liftsb_inv_pair_dx: ∀f:rtmap. ∀Z1,I,V2. ⬆*[f] Z1 ≘ BPair I V2 →
+ ∃∃V1. ⬆*[f] V1 ≘ V2 & Z1 = BPair I V1.
+/2 width=1 by ext2_inv_pair_dx/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma liftsb_eq_repl_back: ∀I1,I2. eq_repl_back … (λf. ⬆*[f] I1 ≘ I2).
+#I1 #I2 #f1 * -I1 -I2 /3 width=3 by lifts_eq_repl_back, ext2_pair/
+qed-.
+
+lemma liftsb_refl: ∀f. 𝐈⦃f⦄ → reflexive … (liftsb f).
+/3 width=1 by lifts_refl, ext2_refl/ qed.
+
+lemma liftsb_total: ∀I1,f. ∃I2. ⬆*[f] I1 ≘ I2.
+* [2: #I #T1 #f elim (lifts_total T1 f) ]
+/3 width=2 by ext2_unit, ext2_pair, ex_intro/
+qed-.
+
+lemma liftsb_split_trans: ∀f,I1,I2. ⬆*[f] I1 ≘ I2 →
+ ∀f1,f2. f2 ⊚ f1 ≘ f →
+ ∃∃I. ⬆*[f1] I1 ≘ I & ⬆*[f2] I ≘ I2.
+#f #I1 #I2 * -I1 -I2 /2 width=3 by ext2_unit, ex2_intro/
+#I #V1 #V2 #HV12 #f1 #f2 #Hf elim (lifts_split_trans … HV12 … Hf) -f
+/3 width=3 by ext2_pair, ex2_intro/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma liftsb_fwd_isid: ∀f,I1,I2. ⬆*[f] I1 ≘ I2 → 𝐈⦃f⦄ → I1 = I2.
+#f #I1 #I2 * -I1 -I2 /3 width=3 by lifts_fwd_isid, eq_f2/
+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 "static_2/relocation/lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: includes: lift_gen_lift *)
+(* Basic_2A1: includes: lift_div_le lift_div_be *)
+theorem lifts_div4: ∀f2,Tf,T. ⬆*[f2] Tf ≘ T → ∀g2,Tg. ⬆*[g2] Tg ≘ T →
+ ∀f1,g1. H_at_div f2 g2 f1 g1 →
+ ∃∃T0. ⬆*[f1] T0 ≘ Tf & ⬆*[g1] T0 ≘ Tg.
+#f2 #Tf #T #H elim H -f2 -Tf -T
+[ #f2 #s #g2 #Tg #H #f1 #g1 #_
+ lapply (lifts_inv_sort2 … H) -H #H destruct
+ /2 width=3 by ex2_intro/
+| #f2 #jf #j #Hf2 #g2 #Tg #H #f1 #g1 #H0
+ elim (lifts_inv_lref2 … H) -H #jg #Hg2 #H destruct
+ elim (H0 … Hf2 Hg2) -H0 -j /3 width=3 by lifts_lref, ex2_intro/
+| #f2 #l #g2 #Tg #H #f1 #g1 #_
+ lapply (lifts_inv_gref2 … H) -H #H destruct
+ /2 width=3 by ex2_intro/
+| #f2 #p #I #Vf #V #Tf #T #_ #_ #IHV #IHT #g2 #X #H #f1 #g1 #H0
+ elim (lifts_inv_bind2 … H) -H #Vg #Tg #HVg #HTg #H destruct
+ elim (IHV … HVg … H0) -IHV -HVg
+ elim (IHT … HTg) -IHT -HTg [ |*: /2 width=8 by at_div_pp/ ]
+ /3 width=5 by lifts_bind, ex2_intro/
+| #f2 #I #Vf #V #Tf #T #_ #_ #IHV #IHT #g2 #X #H #f1 #g1 #H0
+ elim (lifts_inv_flat2 … H) -H #Vg #Tg #HVg #HTg #H destruct
+ elim (IHV … HVg … H0) -IHV -HVg
+ elim (IHT … HTg … H0) -IHT -HTg -H0
+ /3 width=5 by lifts_flat, ex2_intro/
+]
+qed-.
+
+lemma lifts_div4_one: ∀f,Tf,T. ⬆*[⫯f] Tf ≘ T →
+ ∀T1. ⬆*[1] T1 ≘ T →
+ ∃∃T0. ⬆*[1] T0 ≘ Tf & ⬆*[f] T0 ≘ T1.
+/4 width=6 by lifts_div4, at_div_id_dx, at_div_pn/ qed-.
+
+theorem lifts_div3: ∀f2,T,T2. ⬆*[f2] T2 ≘ T → ∀f,T1. ⬆*[f] T1 ≘ T →
+ ∀f1. f2 ⊚ f1 ≘ f → ⬆*[f1] T1 ≘ T2.
+#f2 #T #T2 #H elim H -f2 -T -T2
+[ #f2 #s #f #T1 #H >(lifts_inv_sort2 … H) -T1 //
+| #f2 #i2 #i #Hi2 #f #T1 #H #f1 #Ht21 elim (lifts_inv_lref2 … H) -H
+ #i1 #Hi1 #H destruct /3 width=6 by lifts_lref, after_fwd_at1/
+| #f2 #l #f #T1 #H >(lifts_inv_gref2 … H) -T1 //
+| #f2 #p #I #W2 #W #U2 #U #_ #_ #IHW #IHU #f #T1 #H
+ elim (lifts_inv_bind2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
+ /4 width=3 by lifts_bind, after_O2/
+| #f2 #I #W2 #W #U2 #U #_ #_ #IHW #IHU #f #T1 #H
+ elim (lifts_inv_flat2 … H) -H #W1 #U1 #HW1 #HU1 #H destruct
+ /3 width=3 by lifts_flat/
+]
+qed-.
+
+(* Basic_1: was: lift1_lift1 (left to right) *)
+(* Basic_1: includes: lift_free (left to right) lift_d lift1_xhg (right to left) lift1_free (right to left) *)
+(* Basic_2A1: includes: lift_trans_be lift_trans_le lift_trans_ge lifts_lift_trans_le lifts_lift_trans *)
+theorem lifts_trans: ∀f1,T1,T. ⬆*[f1] T1 ≘ T → ∀f2,T2. ⬆*[f2] T ≘ T2 →
+ ∀f. f2 ⊚ f1 ≘ f → ⬆*[f] T1 ≘ T2.
+#f1 #T1 #T #H elim H -f1 -T1 -T
+[ #f1 #s #f2 #T2 #H >(lifts_inv_sort1 … H) -T2 //
+| #f1 #i1 #i #Hi1 #f2 #T2 #H #f #Ht21 elim (lifts_inv_lref1 … H) -H
+ #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at/
+| #f1 #l #f2 #T2 #H >(lifts_inv_gref1 … H) -T2 //
+| #f1 #p #I #W1 #W #U1 #U #_ #_ #IHW #IHU #f2 #T2 #H
+ elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
+ /4 width=3 by lifts_bind, after_O2/
+| #f1 #I #W1 #W #U1 #U #_ #_ #IHW #IHU #f2 #T2 #H
+ elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
+ /3 width=3 by lifts_flat/
+]
+qed-.
+
+(* Basic_2A1: includes: lift_conf_O1 lift_conf_be *)
+theorem lifts_conf: ∀f1,T,T1. ⬆*[f1] T ≘ T1 → ∀f,T2. ⬆*[f] T ≘ T2 →
+ ∀f2. f2 ⊚ f1 ≘ f → ⬆*[f2] T1 ≘ T2.
+#f1 #T #T1 #H elim H -f1 -T -T1
+[ #f1 #s #f #T2 #H >(lifts_inv_sort1 … H) -T2 //
+| #f1 #i #i1 #Hi1 #f #T2 #H #f2 #Ht21 elim (lifts_inv_lref1 … H) -H
+ #i2 #Hi2 #H destruct /3 width=6 by lifts_lref, after_fwd_at2/
+| #f1 #l #f #T2 #H >(lifts_inv_gref1 … H) -T2 //
+| #f1 #p #I #W #W1 #U #U1 #_ #_ #IHW #IHU #f #T2 #H
+ elim (lifts_inv_bind1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
+ /4 width=3 by lifts_bind, after_O2/
+| #f1 #I #W #W1 #U #U1 #_ #_ #IHW #IHU #f #T2 #H
+ elim (lifts_inv_flat1 … H) -H #W2 #U2 #HW2 #HU2 #H destruct
+ /3 width=3 by lifts_flat/
+]
+qed-.
+
+(* Advanced proprerties *****************************************************)
+
+(* Basic_2A1: includes: lift_inj *)
+lemma lifts_inj: ∀f. is_inj2 … (lifts f).
+#f #T1 #U #H1 #T2 #H2 lapply (after_isid_dx 𝐈𝐝 … f)
+/3 width=6 by lifts_div3, lifts_fwd_isid/
+qed-.
+
+(* Basic_2A1: includes: lift_mono *)
+lemma lifts_mono: ∀f,T. is_mono … (lifts f T).
+#f #T #U1 #H1 #U2 #H2 lapply (after_isid_sn 𝐈𝐝 … f)
+/3 width=6 by lifts_conf, lifts_fwd_isid/
+qed-.
+
+lemma liftable2_sn_bi: ∀R. liftable2_sn R → liftable2_bi R.
+#R #HR #T1 #T2 #HT12 #f #U1 #HTU1 #U2 #HTU2
+elim (HR … HT12 … HTU1) -HR -T1 #X #HTX #HUX
+<(lifts_mono … HTX … HTU2) -T2 -U2 -f //
+qed-.
+
+lemma deliftable2_sn_bi: ∀R. deliftable2_sn R → deliftable2_bi R.
+#R #HR #U1 #U2 #HU12 #f #T1 #HTU1 #T2 #HTU2
+elim (HR … HU12 … HTU1) -HR -U1 #X #HUX #HTX
+<(lifts_inj … HUX … HTU2) -U2 -T2 -f //
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/lifts_lifts.ma".
+include "static_2/relocation/lifts_bind.ma".
+
+(* GENERIC RELOCATION FOR BINDERS *******************************************)
+
+(* Main properties **********************************************************)
+
+theorem liftsb_div3: ∀f2,I,I2. ⬆*[f2] I2 ≘ I → ∀f,I1. ⬆*[f] I1 ≘ I →
+ ∀f1. f2 ⊚ f1 ≘ f → ⬆*[f1] I1 ≘ I2.
+#f2 #I #I2 * -I -I2 #I [2: #V #V2 #HV2 ] #f #I1 #H
+[ elim (liftsb_inv_pair_dx … H) | lapply (liftsb_inv_unit_dx … H) ] -H
+/3 width=6 by lifts_div3, ext2_pair, ext2_unit/
+qed-.
+
+theorem liftsb_trans: ∀f1,I1,I. ⬆*[f1] I1 ≘ I → ∀f2,I2. ⬆*[f2] I ≘ I2 →
+ ∀f. f2 ⊚ f1 ≘ f → ⬆*[f] I1 ≘ I2.
+#f1 #I1 #I * -I1 -I #I1 [2: #V1 #V #HV1 ] #f2 #I2 #H
+[ elim (liftsb_inv_pair_sn … H) | lapply (liftsb_inv_unit_sn … H) ] -H
+/3 width=6 by lifts_trans, ext2_pair, ext2_unit/
+qed-.
+
+theorem liftsb_conf: ∀f1,I,I1. ⬆*[f1] I ≘ I1 → ∀f,I2. ⬆*[f] I ≘ I2 →
+ ∀f2. f2 ⊚ f1 ≘ f → ⬆*[f2] I1 ≘ I2.
+#f1 #I #I1 * -I -I1 #I [2: #V #V1 #HV1 ] #f2 #I2 #H
+[ elim (liftsb_inv_pair_sn … H) | lapply (liftsb_inv_unit_sn … H) ] -H
+/3 width=6 by lifts_conf, ext2_pair, ext2_unit/
+qed-.
+
+(* Advanced proprerties *****************************************************)
+
+lemma liftsb_inj: ∀f. is_inj2 … (liftsb f).
+#f #T1 #U #H1 #T2 #H2 lapply (after_isid_dx 𝐈𝐝 … f)
+/3 width=6 by liftsb_div3, liftsb_fwd_isid/
+qed-.
+
+lemma liftsb_mono: ∀f,T. is_mono … (liftsb f T).
+#f #T #U1 #H1 #U2 #H2 lapply (after_isid_sn 𝐈𝐝 … f)
+/3 width=6 by liftsb_conf, liftsb_fwd_isid/
+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 "static_2/relocation/lifts_lifts.ma".
+include "static_2/relocation/lifts_vector.ma".
+
+(* GENERIC RELOCATION FOR TERM VECTORS *************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: includes: lifts_inj *)
+theorem liftsv_inj: ∀f,T1s,Us. ⬆*[f] T1s ≘ Us →
+ ∀T2s. ⬆*[f] T2s ≘ Us → T1s = T2s.
+#f #T1s #Us #H elim H -T1s -Us
+[ #T2s #H >(liftsv_inv_nil2 … H) -H //
+| #T1s #Us #T1 #U #HT1U #_ #IHT1Us #X #H destruct
+ elim (liftsv_inv_cons2 … H) -H #T2 #T2s #HT2U #HT2Us #H destruct
+ >(lifts_inj … HT1U … HT2U) -U /3 width=1 by eq_f/
+]
+qed-.
+
+(* Basic_2A1: includes: liftv_mono *)
+theorem liftsv_mono: ∀f,Ts,U1s. ⬆*[f] Ts ≘ U1s →
+ ∀U2s. ⬆*[f] Ts ≘ U2s → U1s = U2s.
+#f #Ts #U1s #H elim H -Ts -U1s
+[ #U2s #H >(liftsv_inv_nil1 … H) -H //
+| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
+ elim (liftsv_inv_cons1 … H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
+ >(lifts_mono … HTU1 … HTU2) -T /3 width=1 by eq_f/
+]
+qed-.
+
+(* Basic_1: includes: lifts1_xhg (right to left) *)
+(* Basic_2A1: includes: liftsv_liftv_trans_le *)
+theorem liftsv_trans: ∀f1,T1s,Ts. ⬆*[f1] T1s ≘ Ts → ∀T2s,f2. ⬆*[f2] Ts ≘ T2s →
+ ∀f. f2 ⊚ f1 ≘ f → ⬆*[f] T1s ≘ T2s.
+#f1 #T1s #Ts #H elim H -T1s -Ts
+[ #T2s #f2 #H >(liftsv_inv_nil1 … H) -T2s /2 width=3 by liftsv_nil/
+| #T1s #Ts #T1 #T #HT1 #_ #IHT1s #X #f2 #H elim (liftsv_inv_cons1 … H) -H
+ #T2 #T2s #HT2 #HT2s #H destruct /3 width=6 by lifts_trans, liftsv_cons/
+]
+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 "static_2/syntax/term_simple.ma".
+include "static_2/relocation/lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Forward lemmas with simple terms *****************************************)
+
+(* Basic_2A1: includes: lift_simple_dx *)
+lemma lifts_simple_dx: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#f #T1 #T2 #H elim H -f -T1 -T2 //
+#f #p #I #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H elim (simple_inv_bind … H)
+qed-.
+
+(* Basic_2A1: includes: lift_simple_sn *)
+lemma lifts_simple_sn: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+#f #T1 #T2 #H elim H -f -T1 -T2 //
+#f #p #I #V1 #V2 #T1 #T2 #_ #_ #_ #_ #H elim (simple_inv_bind … 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 "static_2/syntax/tdeq.ma".
+include "static_2/relocation/lifts_lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Properties with degree-based equivalence for terms ***********************)
+
+lemma tdeq_lifts_sn: ∀h,o. liftable2_sn (tdeq h o).
+#h #o #T1 #T2 #H elim H -T1 -T2 [||| * ]
+[ #s1 #s2 #d #Hs1 #Hs2 #f #X #H >(lifts_inv_sort1 … H) -H
+ /3 width=5 by lifts_sort, tdeq_sort, ex2_intro/
+| #i #f #X #H elim (lifts_inv_lref1 … H) -H
+ /3 width=3 by lifts_lref, tdeq_lref, ex2_intro/
+| #l #f #X #H >(lifts_inv_gref1 … H) -H
+ /2 width=3 by lifts_gref, tdeq_gref, ex2_intro/
+| #p #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f #X #H elim (lifts_inv_bind1 … H) -H
+ #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HVW1) -V1 elim (IHT … HTU1) -T1
+ /3 width=5 by lifts_bind, tdeq_pair, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #f #X #H elim (lifts_inv_flat1 … H) -H
+ #W1 #U1 #HVW1 #HTU1 #H destruct
+ elim (IHV … HVW1) -V1 elim (IHT … HTU1) -T1
+ /3 width=5 by lifts_flat, tdeq_pair, ex2_intro/
+]
+qed-.
+
+lemma tdeq_lifts_bi: ∀h,o. liftable2_bi (tdeq h o).
+/3 width=6 by tdeq_lifts_sn, liftable2_sn_bi/ qed-.
+
+(* Inversion lemmas with degree-based equivalence for terms *****************)
+
+lemma tdeq_inv_lifts_sn: ∀h,o. deliftable2_sn (tdeq h o).
+#h #o #U1 #U2 #H elim H -U1 -U2 [||| * ]
+[ #s1 #s2 #d #Hs1 #Hs2 #f #X #H >(lifts_inv_sort2 … H) -H
+ /3 width=5 by lifts_sort, tdeq_sort, ex2_intro/
+| #i #f #X #H elim (lifts_inv_lref2 … H) -H
+ /3 width=3 by lifts_lref, tdeq_lref, ex2_intro/
+| #l #f #X #H >(lifts_inv_gref2 … H) -H
+ /2 width=3 by lifts_gref, tdeq_gref, ex2_intro/
+| #p #I #W1 #W2 #U1 #U2 #_ #_ #IHW #IHU #f #X #H elim (lifts_inv_bind2 … H) -H
+ #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW … HVW1) -W1 elim (IHU … HTU1) -U1
+ /3 width=5 by lifts_bind, tdeq_pair, ex2_intro/
+| #I #W1 #W2 #U1 #U2 #_ #_ #IHW #IHU #f #X #H elim (lifts_inv_flat2 … H) -H
+ #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW … HVW1) -W1 elim (IHU … HTU1) -U1
+ /3 width=5 by lifts_flat, tdeq_pair, ex2_intro/
+]
+qed-.
+
+lemma tdeq_inv_lifts_bi: ∀h,o. deliftable2_bi (tdeq h o).
+/3 width=6 by tdeq_inv_lifts_sn, deliftable2_sn_bi/ 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 "static_2/syntax/term_vector.ma".
+include "static_2/relocation/lifts.ma".
+
+(* GENERIC RELOCATION FOR TERM VECTORS *************************************)
+
+(* Basic_2A1: includes: liftv_nil liftv_cons *)
+inductive liftsv (f:rtmap): relation (list term) ≝
+| liftsv_nil : liftsv f (Ⓔ) (Ⓔ)
+| liftsv_cons: ∀T1s,T2s,T1,T2.
+ ⬆*[f] T1 ≘ T2 → liftsv f T1s T2s →
+ liftsv f (T1 ⨮ T1s) (T2 ⨮ T2s)
+.
+
+interpretation "uniform relocation (term vector)"
+ 'RLiftStar i T1s T2s = (liftsv (uni i) T1s T2s).
+
+interpretation "generic relocation (term vector)"
+ 'RLiftStar f T1s T2s = (liftsv f T1s T2s).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact liftsv_inv_nil1_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → X = Ⓔ → Y = Ⓔ.
+#f #X #Y * -X -Y //
+#T1s #T2s #T1 #T2 #_ #_ #H destruct
+qed-.
+
+(* Basic_2A1: includes: liftv_inv_nil1 *)
+lemma liftsv_inv_nil1: ∀f,Y. ⬆*[f] Ⓔ ≘ Y → Y = Ⓔ.
+/2 width=5 by liftsv_inv_nil1_aux/ qed-.
+
+fact liftsv_inv_cons1_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
+ ∀T1,T1s. X = T1 ⨮ T1s →
+ ∃∃T2,T2s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ T2s &
+ Y = T2 ⨮ T2s.
+#f #X #Y * -X -Y
+[ #U1 #U1s #H destruct
+| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+(* Basic_2A1: includes: liftv_inv_cons1 *)
+lemma liftsv_inv_cons1: ∀f:rtmap. ∀T1,T1s,Y. ⬆*[f] T1 ⨮ T1s ≘ Y →
+ ∃∃T2,T2s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ T2s &
+ Y = T2 ⨮ T2s.
+/2 width=3 by liftsv_inv_cons1_aux/ qed-.
+
+fact liftsv_inv_nil2_aux: ∀f,X,Y. ⬆*[f] X ≘ Y → Y = Ⓔ → X = Ⓔ.
+#f #X #Y * -X -Y //
+#T1s #T2s #T1 #T2 #_ #_ #H destruct
+qed-.
+
+lemma liftsv_inv_nil2: ∀f,X. ⬆*[f] X ≘ Ⓔ → X = Ⓔ.
+/2 width=5 by liftsv_inv_nil2_aux/ qed-.
+
+fact liftsv_inv_cons2_aux: ∀f:rtmap. ∀X,Y. ⬆*[f] X ≘ Y →
+ ∀T2,T2s. Y = T2 ⨮ T2s →
+ ∃∃T1,T1s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ T2s &
+ X = T1 ⨮ T1s.
+#f #X #Y * -X -Y
+[ #U2 #U2s #H destruct
+| #T1s #T2s #T1 #T2 #HT12 #HT12s #U2 #U2s #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma liftsv_inv_cons2: ∀f:rtmap. ∀X,T2,T2s. ⬆*[f] X ≘ T2 ⨮ T2s →
+ ∃∃T1,T1s. ⬆*[f] T1 ≘ T2 & ⬆*[f] T1s ≘ T2s &
+ X = T1 ⨮ T1s.
+/2 width=3 by liftsv_inv_cons2_aux/ qed-.
+
+(* Basic_1: was: lifts1_flat (left to right) *)
+lemma lifts_inv_applv1: ∀f:rtmap. ∀V1s,U1,T2. ⬆*[f] Ⓐ V1s.U1 ≘ T2 →
+ ∃∃V2s,U2. ⬆*[f] V1s ≘ V2s & ⬆*[f] U1 ≘ U2 &
+ T2 = Ⓐ V2s.U2.
+#f #V1s elim V1s -V1s
+[ /3 width=5 by ex3_2_intro, liftsv_nil/
+| #V1 #V1s #IHV1s #T1 #X #H elim (lifts_inv_flat1 … H) -H
+ #V2 #Y #HV12 #HY #H destruct elim (IHV1s … HY) -IHV1s -HY
+ #V2s #T2 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
+]
+qed-.
+
+lemma lifts_inv_applv2: ∀f:rtmap. ∀V2s,U2,T1. ⬆*[f] T1 ≘ Ⓐ V2s.U2 →
+ ∃∃V1s,U1. ⬆*[f] V1s ≘ V2s & ⬆*[f] U1 ≘ U2 &
+ T1 = Ⓐ V1s.U1.
+#f #V2s elim V2s -V2s
+[ /3 width=5 by ex3_2_intro, liftsv_nil/
+| #V2 #V2s #IHV2s #T2 #X #H elim (lifts_inv_flat2 … H) -H
+ #V1 #Y #HV12 #HY #H destruct elim (IHV2s … HY) -IHV2s -HY
+ #V1s #T1 #HV12s #HT12 #H destruct /3 width=5 by ex3_2_intro, liftsv_cons/
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: includes: liftv_total *)
+lemma liftsv_total: ∀f. ∀T1s:list term. ∃T2s. ⬆*[f] T1s ≘ T2s.
+#f #T1s elim T1s -T1s
+[ /2 width=2 by liftsv_nil, ex_intro/
+| #T1 #T1s * #T2s #HT12s
+ elim (lifts_total T1 f) /3 width=2 by liftsv_cons, ex_intro/
+]
+qed-.
+
+(* Basic_1: was: lifts1_flat (right to left) *)
+lemma lifts_applv: ∀f:rtmap. ∀V1s,V2s. ⬆*[f] V1s ≘ V2s →
+ ∀T1,T2. ⬆*[f] T1 ≘ T2 →
+ ⬆*[f] Ⓐ V1s.T1 ≘ Ⓐ V2s.T2.
+#f #V1s #V2s #H elim H -V1s -V2s /3 width=1 by lifts_flat/
+qed.
+
+lemma liftsv_split_trans: ∀f,T1s,T2s. ⬆*[f] T1s ≘ T2s →
+ ∀f1,f2. f2 ⊚ f1 ≘ f →
+ ∃∃Ts. ⬆*[f1] T1s ≘ Ts & ⬆*[f2] Ts ≘ T2s.
+#f #T1s #T2s #H elim H -T1s -T2s
+[ /2 width=3 by liftsv_nil, ex2_intro/
+| #T1s #T2s #T1 #T2 #HT12 #_ #IH #f1 #f2 #Hf
+ elim (IH … Hf) -IH
+ elim (lifts_split_trans … HT12 … Hf) -HT12 -Hf
+ /3 width=5 by liftsv_cons, ex2_intro/
+]
+qed-.
+
+(* Basic_1: removed theorems 2: lifts1_nil lifts1_cons *)
--- /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 "static_2/syntax/term_weight.ma".
+include "static_2/relocation/lifts.ma".
+
+(* GENERIC RELOCATION FOR TERMS *********************************************)
+
+(* Forward lemmas with weight for terms *************************************)
+
+(* Basic_2A1: includes: lift_fwd_tw *)
+lemma lifts_fwd_tw: ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → ♯{T1} = ♯{T2}.
+#f #T1 #T2 #H elim H -f -T1 -T2 normalize //
+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 "static_2/syntax/bind_weight.ma".
+include "static_2/relocation/lifts_weight.ma".
+include "static_2/relocation/lifts_bind.ma".
+
+(* GENERIC RELOCATION FOR BINDERS *******************************************)
+
+(* Forward lemmas with weight for binders ***********************************)
+
+lemma liftsb_fwd_bw: ∀f,I1,I2. ⬆*[f] I1 ≘ I2 → ♯{I1} = ♯{I2}.
+#f #I1 #I2 * -I1 -I2 /2 width=2 by lifts_fwd_tw/
+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 "static_2/notation/relations/ideqsn_3.ma".
+include "static_2/syntax/ceq_ext.ma".
+include "static_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 "static_2/relocation/sex_length.ma".
+include "static_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 "static_2/syntax/ceq_ext_ceq_ext.ma".
+include "static_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 "static_2/notation/relations/relation_5.ma".
+include "static_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 "static_2/syntax/lenv_length.ma".
+include "static_2/relocation/sex.ma".
+
+(* GENERIC ENTRYWISE EXTENSION OF CONTEXT-SENSITIVE REALTIONS FOR TERMS *****)
+
+(* Forward lemmas with length for local environments ************************)
+
+(* Note: "#f #I1 #I2 #L1 #L2 >length_bind >length_bind //" was needed to conclude *)
+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 //
+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 "static_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 "static_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 "ground_2/lib/star.ma".
+include "static_2/notation/relations/suptermplus_6.ma".
+include "static_2/notation/relations/suptermplus_7.ma".
+include "static_2/s_transition/fqu.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+definition fqup: bool → tri_relation genv lenv term ≝
+ λb. tri_TC … (fqu b).
+
+interpretation "extended plus-iterated structural successor (closure)"
+ 'SupTermPlus b G1 L1 T1 G2 L2 T2 = (fqup b G1 L1 T1 G2 L2 T2).
+
+interpretation "plus-iterated structural successor (closure)"
+ 'SupTermPlus G1 L1 T1 G2 L2 T2 = (fqup true G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fqu_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fqup_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed.
+
+lemma fqup_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2.
+ ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed.
+
+lemma fqup_pair_sn: ∀b,I,G,L,V,T. ⦃G, L, ②{I}V.T⦄ ⊐+[b] ⦃G, L, V⦄.
+/2 width=1 by fqu_pair_sn, fqu_fqup/ qed.
+
+lemma fqup_bind_dx: ∀b,p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[b] ⦃G, L.ⓑ{I}V, T⦄.
+/2 width=1 by fqu_bind_dx, fqu_fqup/ qed.
+
+lemma fqup_clear: ∀p,I,G,L,V,T. ⦃G, L, ⓑ{p,I}V.T⦄ ⊐+[Ⓕ] ⦃G, L.ⓧ, T⦄.
+/3 width=1 by fqu_clear, fqu_fqup/ qed.
+
+lemma fqup_flat_dx: ∀b,I,G,L,V,T. ⦃G, L, ⓕ{I}V.T⦄ ⊐+[b] ⦃G, L, T⦄.
+/2 width=1 by fqu_flat_dx, fqu_fqup/ qed.
+
+lemma fqup_flat_dx_pair_sn: ∀b,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.②{I2}V2.T⦄ ⊐+[b] ⦃G, L, V2⦄.
+/2 width=5 by fqu_pair_sn, fqup_strap1/ qed.
+
+lemma fqup_bind_dx_flat_dx: ∀b,p,G,I1,I2,L,V1,V2,T. ⦃G, L, ⓑ{p,I1}V1.ⓕ{I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I1}V1, T⦄.
+/2 width=5 by fqu_flat_dx, fqup_strap1/ qed.
+
+lemma fqup_flat_dx_bind_dx: ∀b,p,I1,I2,G,L,V1,V2,T. ⦃G, L, ⓕ{I1}V1.ⓑ{p,I2}V2.T⦄ ⊐+[b] ⦃G, L.ⓑ{I2}V2, T⦄.
+/2 width=5 by fqu_bind_dx, fqup_strap1/ qed.
+
+(* Basic eliminators ********************************************************)
+
+lemma fqup_ind: ∀b,G1,L1,T1. ∀Q:relation3 ….
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+#b #G1 #L1 #T1 #Q #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_ind … IH1 IH2 G2 L2 T2 H)
+qed-.
+
+lemma fqup_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 ….
+ (∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1) →
+ (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
+#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_ind_dx … IH1 IH2 G1 L1 T1 H)
+qed-.
+
+(* Basic_2A1: removed theorems 1: fqup_drop *)
--- /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 "static_2/relocation/drops.ma".
+include "static_2/s_computation/fqup.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+(* Properties with generic slicing for local environments *******************)
+
+lemma fqup_drops_succ: ∀b,G,K,T,i,L,U. ⬇*[↑i] L ≘ K → ⬆*[↑i] T ≘ U →
+ ⦃G, L, U⦄ ⊐+[b] ⦃G, K, T⦄.
+#b #G #K #T #i elim i -i
+[ #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK
+ #I #Y #HY #H destruct <(drops_fwd_isid … HY) -K //
+ /3 width=2 by fqu_fqup, fqu_drop/
+| #l #IH #L #U #HLK #HTU elim (drops_inv_succ … HLK) -HLK
+ #I #Y #HY #H destruct
+ elim (lifts_split_trans … HTU … (𝐔❴↑l❵) (𝐔❴1❵)) -HTU
+ /4 width=5 by fqup_strap2, fqu_drop/
+]
+qed.
+
+lemma fqup_drops_strap1: ∀b,G1,G2,L1,K1,K2,T1,T2,U1,i. ⬇*[i] L1 ≘ K1 → ⬆*[i] T1 ≘ U1 →
+ ⦃G1, K1, T1⦄ ⊐[b] ⦃G2, K2, T2⦄ → ⦃G1, L1, U1⦄ ⊐+[b] ⦃G2, K2, T2⦄.
+#b #G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 *
+[ #HLK1 #HTU1 #HT12
+ >(drops_fwd_isid … HLK1) -L1 //
+ <(lifts_fwd_isid … HTU1) -U1 /2 width=1 by fqu_fqup/
+| /3 width=5 by fqup_strap1, fqup_drops_succ/
+]
+qed-.
+
+lemma fqup_lref: ∀b,I,G,L,K,V,i. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, L, #i⦄ ⊐+[b] ⦃G, K, V⦄.
+/2 width=6 by fqup_drops_strap1/ 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 "static_2/s_computation/fqup.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+(* Main properties **********************************************************)
+
+theorem fqup_trans: ∀b. tri_transitive … (fqup b).
+/2 width=5 by tri_TC_transitive/ 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 "static_2/s_transition/fqu_weight.ma".
+include "static_2/s_computation/fqup.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+(* Forward lemmas with weight for closures **********************************)
+
+lemma fqup_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
+ ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=3 by fqu_fwd_fw, transitive_lt/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fqup_wf_ind: ∀b. ∀Q:relation3 …. (
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) → ∀G1,L1,T1. Q G1 L1 T1.
+#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct
+/4 width=2 by fqup_fwd_fw/
+qed-.
+
+lemma fqup_wf_ind_eq: ∀b. ∀Q:relation3 …. (
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ ∀G2,L2,T2. G1 = G2 → L1 = L2 → T1 = T2 → Q G2 L2 T2
+ ) → ∀G1,L1,T1. Q G1 L1 T1.
+#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct
+/4 width=7 by fqup_fwd_fw/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/lib/star.ma".
+include "static_2/notation/relations/suptermstar_6.ma".
+include "static_2/notation/relations/suptermstar_7.ma".
+include "static_2/s_transition/fquq.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+definition fqus: bool → tri_relation genv lenv term ≝
+ λb. tri_TC … (fquq b).
+
+interpretation "extended star-iterated structural successor (closure)"
+ 'SupTermStar b G1 L1 T1 G2 L2 T2 = (fqus b G1 L1 T1 G2 L2 T2).
+
+interpretation "star-iterated structural successor (closure)"
+ 'SupTermStar G1 L1 T1 G2 L2 T2 = (fqus true G1 L1 T1 G2 L2 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma fqus_ind: ∀b,G1,L1,T1. ∀Q:relation3 …. Q G1 L1 T1 →
+ (∀G,G2,L,L2,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → Q G L T → Q G2 L2 T2) →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2.
+#b #G1 #L1 #T1 #R #IH1 #IH2 #G2 #L2 #T2 #H
+@(tri_TC_star_ind … IH1 IH2 G2 L2 T2 H) //
+qed-.
+
+lemma fqus_ind_dx: ∀b,G2,L2,T2. ∀Q:relation3 …. Q G2 L2 T2 →
+ (∀G1,G,L1,L,T1,T. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G L T → Q G1 L1 T1) →
+ ∀G1,L1,T1. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ → Q G1 L1 T1.
+#b #G2 #L2 #T2 #Q #IH1 #IH2 #G1 #L1 #T1 #H
+@(tri_TC_star_ind_dx … IH1 IH2 G1 L1 T1 H) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma fqus_refl: ∀b. tri_reflexive … (fqus b).
+/2 width=1 by tri_inj/ qed.
+
+lemma fquq_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+/2 width=1 by tri_inj/ qed.
+
+lemma fqus_strap1: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_step/ qed-.
+
+lemma fqus_strap2: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+/2 width=5 by tri_TC_strap/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma fqus_inv_fqu_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ (∧∧ G1 = G2 & L1 = L2 & T1 = T2) ∨
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H12 @(fqus_ind_dx … H12) -G1 -L1 -T1 /3 width=1 by and3_intro, or_introl/
+#G1 #G #L1 #L #T1 #T * /3 width=5 by ex2_3_intro, or_intror/
+* #HG #HL #HT #_ destruct //
+qed-.
+
+lemma fqus_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ (∧∧ G1 = G2 & L1 = L2 & ⋆s = T2) ∨
+ ∃∃J,L. ⦃G1, L, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}.
+#b #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
+#G #L #T #H elim (fqu_inv_sort1 … H) -H /3 width=4 by ex2_2_intro, or_intror/
+qed-.
+
+lemma fqus_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & #i = T2
+ | ∃∃J,L,V. ⦃G1, L, V⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓑ{J}V & i = 0
+ | ∃∃J,L,j. ⦃G1, L, #j⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J} & i = ↑j.
+#b #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or3_intro0/
+#G #L #T #H elim (fqu_inv_lref1 … H) -H * /3 width=7 by or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro/
+qed-.
+
+lemma fqus_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ (∧∧ G1 = G2 & L1 = L2 & §l = T2) ∨
+ ∃∃J,L. ⦃G1, L, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ & L1 = L.ⓘ{J}.
+#b #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
+#G #L #T #H elim (fqu_inv_gref1 … H) -H /3 width=4 by ex2_2_intro, or_intror/
+qed-.
+
+lemma fqus_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
+ | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄
+ | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄
+ | ⦃G1, L1.ⓧ, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ ∧ b = Ⓕ
+ | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
+#b #p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or5_intro0/
+#G #L #T #H elim (fqu_inv_bind1 … H) -H *
+[4: #J ] #H1 #H2 #H3 [4: #Hb ] #H destruct
+/3 width=6 by or5_intro1, or5_intro2, or5_intro3, or5_intro4, ex3_3_intro, conj/
+qed-.
+
+
+lemma fqus_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓑ{p,I}V1.T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓑ{p,I}V1.T1 = T2
+ | ⦃G1, L1, V1⦄ ⊐* ⦃G2, L2, T2⦄
+ | ⦃G1, L1.ⓑ{I}V1, T1⦄ ⊐* ⦃G2, L2, T2⦄
+ | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐* ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓑ{p,I}V1.T1 & L1 = L.ⓘ{J}.
+#p #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_bind1 … H) -H [1,4: * ]
+/3 width=1 by and3_intro, or4_intro0, or4_intro1, or4_intro2, or4_intro3, ex3_3_intro/
+#_ #H destruct
+qed-.
+
+lemma fqus_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,T1,T2. ⦃G1, L1, ⓕ{I}V1.T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & ⓕ{I}V1.T1 = T2
+ | ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄
+ | ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄
+ | ∃∃J,L,T. ⦃G1, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ & ⬆*[1] T ≘ ⓕ{I}V1.T1 & L1 = L.ⓘ{J}.
+#b #I #G1 #G2 #L1 #L2 #V1 #T1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or4_intro0/
+#G #L #T #H elim (fqu_inv_flat1 … H) -H *
+[3: #J ] #H1 #H2 #H3 #H destruct
+/3 width=6 by or4_intro1, or4_intro2, or4_intro3, ex3_3_intro/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fqus_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & ⋆ = L2 & ⓪{I} = T2.
+#b #I #G1 #G2 #L2 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /2 width=1 by and3_intro/
+#G #L #T #H elim (fqu_inv_atom1 … H)
+qed-.
+
+lemma fqus_inv_sort1_bind: ∀b,I,G1,G2,L1,L2,T2,s. ⦃G1, L1.ⓘ{I}, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & ⋆s = T2) ∨ ⦃G1, L1, ⋆s⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+#b #I #G1 #G2 #L1 #L2 #T2 #s #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
+#G #L #T #H elim (fqu_inv_sort1_bind … H) -H
+#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
+qed-.
+
+lemma fqus_inv_zero1_pair: ∀b,I,G1,G2,L1,L2,V1,T2. ⦃G1, L1.ⓑ{I}V1, #0⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & #0 = T2) ∨ ⦃G1, L1, V1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+#b #I #G1 #G2 #L1 #L2 #V1 #T2 #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
+#G #L #T #H elim (fqu_inv_zero1_pair … H) -H
+#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
+qed-.
+
+lemma fqus_inv_lref1_bind: ∀b,I,G1,G2,L1,L2,T2,i. ⦃G1, L1.ⓘ{I}, #↑i⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & #(↑i) = T2) ∨ ⦃G1, L1, #i⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+#b #I #G1 #G2 #L1 #L2 #T2 #i #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
+#G #L #T #H elim (fqu_inv_lref1_bind … H) -H
+#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
+qed-.
+
+lemma fqus_inv_gref1_bind: ∀b,I,G1,G2,L1,L2,T2,l. ⦃G1, L1.ⓘ{I}, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ (∧∧ G1 = G2 & L1.ⓘ{I} = L2 & §l = T2) ∨ ⦃G1, L1, §l⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+#b #I #G1 #G2 #L1 #L2 #T2 #l #H elim (fqus_inv_fqu_sn … H) -H * /3 width=1 by and3_intro, or_introl/
+#G #L #T #H elim (fqu_inv_gref1_bind … H) -H
+#H1 #H2 #H3 #H destruct /2 width=1 by or_intror/
+qed-.
+
+(* Basic_2A1: removed theorems 1: fqus_drop *)
--- /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 "static_2/s_computation/fqup_drops.ma".
+include "static_2/s_computation/fqus_fqup.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Properties with generic slicing for local environments *******************)
+
+lemma fqus_drops: ∀b,G,L,K,T,U,i. ⬇*[i] L ≘ K → ⬆*[i] T ≘ U →
+ ⦃G, L, U⦄ ⊐*[b] ⦃G, K, T⦄.
+#b #G #L #K #T #U * /3 width=3 by fqup_drops_succ, fqup_fqus/
+#HLK #HTU <(lifts_fwd_isid … HTU) -U // <(drops_fwd_isid … HLK) -K //
+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 "static_2/s_computation/fqup.ma".
+include "static_2/s_computation/fqus.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Alternative definition with plus-iterated supclosure *********************)
+
+lemma fqup_fqus: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+/3 width=5 by fqus_strap1, fquq_fqus, fqu_fquq/
+qed.
+
+(* Basic_2A1: was: fqus_inv_gen *)
+lemma fqus_inv_fqup: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ ∨ (∧∧ G1 = G2 & L1 = L2 & T1 = T2).
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -G2 -L2 -T2 //
+#G #G2 #L #L2 #T #T2 #_ *
+[ #H2 * /3 width=5 by fqup_strap1, or_introl/
+ * /3 width=1 by fqu_fqup, or_introl/
+| * #HG #HL #HT destruct //
+]
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma fqus_strap1_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H1) -H1
+[ /2 width=5 by fqup_strap1/
+| * /2 width=1 by fqu_fqup/
+]
+qed-.
+
+lemma fqus_strap2_fqu: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim (fqus_inv_fqup … H2) -H2
+[ /2 width=5 by fqup_strap2/
+| * /2 width=1 by fqu_fqup/
+]
+qed-.
+
+lemma fqus_fqup_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G, L, T⦄ → ⦃G, L, T⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 #H2 @(fqup_ind … H2) -H2 -G2 -L2 -T2
+/2 width=5 by fqus_strap1_fqu, fqup_strap1/
+qed-.
+
+lemma fqup_fqus_trans: ∀b,G1,G,G2,L1,L,L2,T1,T,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G, L, T⦄ →
+ ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄.
+#b #G1 #G #G2 #L1 #L #L2 #T1 #T #T2 #H1 @(fqup_ind_dx … H1) -H1 -G1 -L1 -T1
+/3 width=5 by fqus_strap2_fqu, fqup_strap2/
+qed-.
+
+(* Advanced inversion lemmas for plus-iterated supclosure *******************)
+
+lemma fqup_inv_step_sn: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+[b] ⦃G2, L2, T2⦄ →
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ⊐[b] ⦃G, L, T⦄ & ⦃G, L, T⦄ ⊐*[b] ⦃G2, L2, T2⦄.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1 /2 width=5 by ex2_3_intro/
+#G1 #G #L1 #L #T1 #T #H1 #_ * /4 width=9 by fqus_strap2, fqu_fquq, 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 "static_2/s_computation/fqus.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Main properties **********************************************************)
+
+theorem fqus_trans: ∀b. tri_transitive … (fqus b).
+/2 width=5 by tri_TC_transitive/ 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 "static_2/s_transition/fquq_weight.ma".
+include "static_2/s_computation/fqus.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Forward lemmas with weight for closures **********************************)
+
+lemma fqus_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐*[b] ⦃G2, L2, T2⦄ →
+ ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqus_ind … H) -L2 -T2
+/3 width=3 by fquq_fwd_fw, transitive_le/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fqus_inv_refl_atom3: ∀b,I,G,L,X. ⦃G, L, ⓪{I}⦄ ⊐*[b] ⦃G, L, X⦄ → ⓪{I} = X.
+#b #I #G #L #X #H elim (fqus_inv_fqu_sn … H) -H * //
+#G0 #L0 #T0 #H1 #H2 lapply (fqu_fwd_fw … H1) lapply (fqus_fwd_fw … H2) -H2 -H1
+#H2 #H1 lapply (le_to_lt_to_lt … H2 H1) -G0 -L0 -T0
+#H elim (lt_le_false … H) -H /2 width=1 by monotonic_le_plus_r/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/notation/relations/supterm_6.ma".
+include "static_2/notation/relations/supterm_7.ma".
+include "static_2/syntax/lenv.ma".
+include "static_2/syntax/genv.ma".
+include "static_2/relocation/lifts.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* activate genv *)
+(* Note: frees_total requires fqu_drop for all atoms
+ fqu_cpx_trans requires fqu_drop for all terms
+ frees_fqus_drops requires fqu_drop restricted on atoms
+*)
+inductive fqu (b:bool): tri_relation genv lenv term ≝
+| fqu_lref_O : ∀I,G,L,V. fqu b G (L.ⓑ{I}V) (#0) G L V
+| fqu_pair_sn: ∀I,G,L,V,T. fqu b G L (②{I}V.T) G L V
+| fqu_bind_dx: ∀p,I,G,L,V,T. fqu b G L (ⓑ{p,I}V.T) G (L.ⓑ{I}V) T
+| fqu_clear : ∀p,I,G,L,V,T. b = Ⓕ → fqu b G L (ⓑ{p,I}V.T) G (L.ⓧ) T
+| fqu_flat_dx: ∀I,G,L,V,T. fqu b G L (ⓕ{I}V.T) G L T
+| fqu_drop : ∀I,G,L,T,U. ⬆*[1] T ≘ U → fqu b G (L.ⓘ{I}) U G L T
+.
+
+interpretation
+ "extended structural successor (closure)"
+ 'SupTerm b G1 L1 T1 G2 L2 T2 = (fqu b G1 L1 T1 G2 L2 T2).
+
+interpretation
+ "structural successor (closure)"
+ 'SupTerm G1 L1 T1 G2 L2 T2 = (fqu true G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma fqu_sort: ∀b,I,G,L,s. ⦃G, L.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G, L, ⋆s⦄.
+/2 width=1 by fqu_drop/ qed.
+
+lemma fqu_lref_S: ∀b,I,G,L,i. ⦃G, L.ⓘ{I}, #↑i⦄ ⊐[b] ⦃G, L, #i⦄.
+/2 width=1 by fqu_drop/ qed.
+
+lemma fqu_gref: ∀b,I,G,L,l. ⦃G, L.ⓘ{I}, §l⦄ ⊐[b] ⦃G, L, §l⦄.
+/2 width=1 by fqu_drop/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact fqu_inv_sort1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀s. T1 = ⋆s →
+ ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s.
+#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #T #s #H destruct
+| #I #G #L #V #T #s #H destruct
+| #p #I #G #L #V #T #s #H destruct
+| #p #I #G #L #V #T #_ #s #H destruct
+| #I #G #L #V #T #s #H destruct
+| #I #G #L #T #U #HI12 #s #H destruct
+ lapply (lifts_inv_sort2 … HI12) -HI12 /2 width=2 by ex3_intro/
+]
+qed-.
+
+lemma fqu_inv_sort1: ∀b,G1,G2,L1,L2,T2,s. ⦃G1, L1, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = ⋆s.
+/2 width=4 by fqu_inv_sort1_aux/ qed-.
+
+fact fqu_inv_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀i. T1 = #i →
+ (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
+ ∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j.
+#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #T #i #H destruct /3 width=4 by ex4_2_intro, or_introl/
+| #I #G #L #V #T #i #H destruct
+| #p #I #G #L #V #T #i #H destruct
+| #p #I #G #L #V #T #_ #i #H destruct
+| #I #G #L #V #T #i #H destruct
+| #I #G #L #T #U #HI12 #i #H destruct
+ elim (lifts_inv_lref2_uni … HI12) -HI12 /3 width=3 by ex4_2_intro, or_intror/
+]
+qed-.
+
+lemma fqu_inv_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ (∃∃J,V. G1 = G2 & L1 = L2.ⓑ{J}V & T2 = V & i = 0) ∨
+ ∃∃J,j. G1 = G2 & L1 = L2.ⓘ{J} & T2 = #j & i = ↑j.
+/2 width=4 by fqu_inv_lref1_aux/ qed-.
+
+fact fqu_inv_gref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀l. T1 = §l →
+ ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l.
+#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #T #l #H destruct
+| #I #G #L #V #T #l #H destruct
+| #p #I #G #L #V #T #l #H destruct
+| #p #I #G #L #V #T #_ #l #H destruct
+| #I #G #L #V #T #s #H destruct
+| #I #G #L #T #U #HI12 #l #H destruct
+ lapply (lifts_inv_gref2 … HI12) -HI12 /2 width=3 by ex3_intro/
+]
+qed-.
+
+lemma fqu_inv_gref1: ∀b,G1,G2,L1,L2,T2,l. ⦃G1, L1, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & T2 = §l.
+/2 width=4 by fqu_inv_gref1_aux/ qed-.
+
+fact fqu_inv_bind1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀p,I,V1,U1. T1 = ⓑ{p,I}V1.U1 →
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+ | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
+ | ∧∧ G1 = G2 & L1.ⓧ = L2 & U1 = T2 & b = Ⓕ
+ | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
+#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #T #q #J #V0 #U0 #H destruct
+| #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or4_intro0/
+| #p #I #G #L #V #T #q #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or4_intro1/
+| #p #I #G #L #V #T #Hb #q #J #V0 #U0 #H destruct /3 width=1 by and4_intro, or4_intro2/
+| #I #G #L #V #T #q #J #V0 #U0 #H destruct
+| #I #G #L #T #U #HTU #q #J #V0 #U0 #H destruct /3 width=2 by or4_intro3, ex3_intro/
+]
+qed-.
+
+lemma fqu_inv_bind1: ∀b,p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+ | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
+ | ∧∧ G1 = G2 & L1.ⓧ = L2 & U1 = T2 & b = Ⓕ
+ | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
+/2 width=4 by fqu_inv_bind1_aux/ qed-.
+
+lemma fqu_inv_bind1_true: ∀p,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓑ{p,I}V1.U1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+ | ∧∧ G1 = G2 & L1.ⓑ{I}V1 = L2 & U1 = T2
+ | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓑ{p,I}V1.U1.
+#p #I #G1 #G2 #L1 #L2 #V1 #U1 #T2 #H elim (fqu_inv_bind1 … H) -H
+/3 width=1 by or3_intro0, or3_intro1, or3_intro2/
+* #_ #_ #_ #H destruct
+qed-.
+
+fact fqu_inv_flat1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀I,V1,U1. T1 = ⓕ{I}V1.U1 →
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+ | ∧∧ G1 = G2 & L1 = L2 & U1 = T2
+ | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓕ{I}V1.U1.
+#b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #T #J #V0 #U0 #H destruct
+| #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro0/
+| #p #I #G #L #V #T #J #V0 #U0 #H destruct
+| #p #I #G #L #V #T #_ #J #V0 #U0 #H destruct
+| #I #G #L #V #T #J #V0 #U0 #H destruct /3 width=1 by and3_intro, or3_intro1/
+| #I #G #L #T #U #HTU #J #V0 #U0 #H destruct /3 width=2 by or3_intro2, ex3_intro/
+]
+qed-.
+
+lemma fqu_inv_flat1: ∀b,I,G1,G2,L1,L2,V1,U1,T2. ⦃G1, L1, ⓕ{I}V1.U1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∨∨ ∧∧ G1 = G2 & L1 = L2 & V1 = T2
+ | ∧∧ G1 = G2 & L1 = L2 & U1 = T2
+ | ∃∃J. G1 = G2 & L1 = L2.ⓘ{J} & ⬆*[1] T2 ≘ ⓕ{I}V1.U1.
+/2 width=4 by fqu_inv_flat1_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fqu_inv_atom1: ∀b,I,G1,G2,L2,T2. ⦃G1, ⋆, ⓪{I}⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⊥.
+#b * #x #G1 #G2 #L2 #T2 #H
+[ elim (fqu_inv_sort1 … H) | elim (fqu_inv_lref1 … H) * | elim (fqu_inv_gref1 … H) ] -H
+#I [2: #V |3: #i ] #_ #H destruct
+qed-.
+
+lemma fqu_inv_sort1_bind: ∀b,I,G1,G2,K,L2,T2,s. ⦃G1, K.ⓘ{I}, ⋆s⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L2 = K & T2 = ⋆s.
+#b #I #G1 #G2 #K #L2 #T2 #s #H elim (fqu_inv_sort1 … H) -H
+#Z #X #H1 #H2 destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma fqu_inv_zero1_pair: ∀b,I,G1,G2,K,L2,V,T2. ⦃G1, K.ⓑ{I}V, #0⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L2 = K & T2 = V.
+#b #I #G1 #G2 #K #L2 #V #T2 #H elim (fqu_inv_lref1 … H) -H *
+#Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma fqu_inv_lref1_bind: ∀b,I,G1,G2,K,L2,T2,i. ⦃G1, K.ⓘ{I}, #(↑i)⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L2 = K & T2 = #i.
+#b #I #G1 #G2 #K #L2 #T2 #i #H elim (fqu_inv_lref1 … H) -H *
+#Z #X #H1 #H2 #H3 #H4 destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma fqu_inv_gref1_bind: ∀b,I,G1,G2,K,L2,T2,l. ⦃G1, K.ⓘ{I}, §l⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∧∧ G1 = G2 & L2 = K & T2 = §l.
+#b #I #G1 #G2 #K #L2 #T2 #l #H elim (fqu_inv_gref1 … H) -H
+#Z #H1 #H2 #H3 destruct /2 width=1 by and3_intro/
+qed-.
+
+(* Basic_2A1: removed theorems 3:
+ fqu_drop fqu_drop_lt fqu_lref_S_lt
+*)
--- /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 "static_2/syntax/lenv_length.ma".
+include "static_2/s_transition/fqu.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* Forward lemmas with length for local environments ************************)
+
+fact fqu_fwd_length_lref1_aux: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ∀i. T1 = #i → |L2| < |L1|.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 // [2,3: #p]
+#I #G #L #V #T [2: #_ ] #j #H destruct
+qed-.
+
+lemma fqu_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ |L2| < |L1|.
+/2 width=8 by fqu_fwd_length_lref1_aux/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tdeq.ma".
+include "static_2/s_transition/fqu_length.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* Inversion lemmas with context-free degree-based equivalence for terms ****)
+
+fact fqu_inv_tdeq_aux: ∀h,o,b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ G1 = G2 → |L1| = |L2| → T1 ≛[h, o] T2 → ⊥.
+#h #o #b #G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2
+[1: #I #G #L #V #_ #H elim (succ_inv_refl_sn … H)
+|6: #I #G #L #T #U #_ #_ #H elim (succ_inv_refl_sn … H)
+]
+/2 width=6 by tdeq_inv_pair_xy_y, tdeq_inv_pair_xy_x/
+qed-.
+
+(* Basic_2A1: uses: fqu_inv_eq *)
+lemma fqu_inv_tdeq: ∀h,o,b,G,L1,L2,T1,T2. ⦃G, L1, T1⦄ ⊐[b] ⦃G, L2, T2⦄ →
+ |L1| = |L2| → T1 ≛[h, o] T2 → ⊥.
+#h #o #b #G #L1 #L2 #T1 #T2 #H
+@(fqu_inv_tdeq_aux … H) // (**) (* full 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 "static_2/syntax/cl_weight.ma".
+include "static_2/relocation/lifts_weight.ma".
+include "static_2/s_transition/fqu.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* Forward lemmas with weight for closures **********************************)
+
+lemma fqu_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ →
+ ♯{G2, L2, T2} < ♯{G1, L1, T1}.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
+#I #I1 #I2 #G #L #HI12 normalize in ⊢ (?%%); -I1
+<(lifts_fwd_tw … HI12) /3 width=1 by monotonic_lt_plus_r, monotonic_lt_plus_l/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma fqu_wf_ind: ∀b. ∀Q:relation3 …. (
+ ∀G1,L1,T1. (∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → Q G2 L2 T2) →
+ Q G1 L1 T1
+ ) → ∀G1,L1,T1. Q G1 L1 T1.
+#b #Q #HQ @(f3_ind … fw) #x #IHx #G1 #L1 #T1 #H destruct /4 width=2 by fqu_fwd_fw/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/notation/relations/suptermopt_6.ma".
+include "static_2/notation/relations/suptermopt_7.ma".
+include "static_2/s_transition/fqu.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* Basic_2A1: was: fquqa *)
+(* Basic_2A1: includes: fquq_inv_gen *)
+definition fquq: bool → tri_relation genv lenv term ≝
+ λb. tri_RC … (fqu b).
+
+interpretation
+ "extended optional structural successor (closure)"
+ 'SupTermOpt b G1 L1 T1 G2 L2 T2 = (fquq b G1 L1 T1 G2 L2 T2).
+
+interpretation
+ "optional structural successor (closure)"
+ 'SupTermOpt G1 L1 T1 G2 L2 T2 = (fquq true G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: includes: fquqa_refl *)
+lemma fquq_refl: ∀b. tri_reflexive … (fquq b).
+// qed.
+
+lemma fqu_fquq: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐[b] ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄.
+/2 width=1 by or_introl/ qed.
+
+(* Basic_2A1: removed theorems 8:
+ fquq_lref_O fquq_pair_sn fquq_bind_dx fquq_flat_dx fquq_drop
+ fquqa_drop fquq_fquqa fquqa_inv_fquq
+*)
--- /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 "static_2/s_transition/fqu_length.ma".
+include "static_2/s_transition/fquq.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma fquq_fwd_length_lref1: ∀b,G1,G2,L1,L2,T2,i. ⦃G1, L1, #i⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ |L2| ≤ |L1|.
+#b #G1 #G2 #L1 #L2 #T2 #i #H elim H -H [2: * ]
+/3 width=6 by fqu_fwd_length_lref1, lt_to_le/
+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 "static_2/s_transition/fqu_weight.ma".
+include "static_2/s_transition/fquq.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* Forward lemmas with weight for closures **********************************)
+
+lemma fquq_fwd_fw: ∀b,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮[b] ⦃G2, L2, T2⦄ →
+ ♯{G2, L2, T2} ≤ ♯{G1, L1, T1}.
+#b #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H [2: * ]
+/3 width=2 by fqu_fwd_fw, lt_to_le/
+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 "static_2/notation/relations/atomicarity_4.ma".
+include "static_2/syntax/aarity.ma".
+include "static_2/syntax/lenv.ma".
+include "static_2/syntax/genv.ma".
+
+(* ATONIC ARITY ASSIGNMENT FOR TERMS ****************************************)
+
+(* activate genv *)
+inductive aaa: relation4 genv lenv term aarity ≝
+| aaa_sort: ∀G,L,s. aaa G L (⋆s) (⓪)
+| aaa_zero: ∀I,G,L,V,B. aaa G L V B → aaa G (L.ⓑ{I}V) (#0) B
+| aaa_lref: ∀I,G,L,A,i. aaa G L (#i) A → aaa G (L.ⓘ{I}) (#↑i) A
+| aaa_abbr: ∀p,G,L,V,T,B,A.
+ aaa G L V B → aaa G (L.ⓓV) T A → aaa G L (ⓓ{p}V.T) A
+| aaa_abst: ∀p,G,L,V,T,B,A.
+ aaa G L V B → aaa G (L.ⓛV) T A → aaa G L (ⓛ{p}V.T) (②B.A)
+| aaa_appl: ∀G,L,V,T,B,A. aaa G L V B → aaa G L T (②B.A) → aaa G L (ⓐV.T) A
+| aaa_cast: ∀G,L,V,T,A. aaa G L V A → aaa G L T A → aaa G L (ⓝV.T) A
+.
+
+interpretation "atomic arity assignment (term)"
+ 'AtomicArity G L T A = (aaa G L T A).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact aaa_inv_sort_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀s. T = ⋆s → A = ⓪.
+#G #L #T #A * -G -L -T -A //
+[ #I #G #L #V #B #_ #s #H destruct
+| #I #G #L #A #i #_ #s #H destruct
+| #p #G #L #V #T #B #A #_ #_ #s #H destruct
+| #p #G #L #V #T #B #A #_ #_ #s #H destruct
+| #G #L #V #T #B #A #_ #_ #s #H destruct
+| #G #L #V #T #A #_ #_ #s #H destruct
+]
+qed-.
+
+lemma aaa_inv_sort: ∀G,L,A,s. ⦃G, L⦄ ⊢ ⋆s ⁝ A → A = ⓪.
+/2 width=6 by aaa_inv_sort_aux/ qed-.
+
+fact aaa_inv_zero_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → T = #0 →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+#G #L #T #A * -G -L -T -A /2 width=5 by ex2_3_intro/
+[ #G #L #s #H destruct
+| #I #G #L #A #i #_ #H destruct
+| #p #G #L #V #T #B #A #_ #_ #H destruct
+| #p #G #L #V #T #B #A #_ #_ #H destruct
+| #G #L #V #T #B #A #_ #_ #H destruct
+| #G #L #V #T #A #_ #_ #H destruct
+]
+qed-.
+
+lemma aaa_inv_zero: ∀G,L,A. ⦃G, L⦄ ⊢ #0 ⁝ A →
+ ∃∃I,K,V. L = K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+/2 width=3 by aaa_inv_zero_aux/ qed-.
+
+fact aaa_inv_lref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀i. T = #(↑i) →
+ ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #j #H destruct
+| #I #G #L #V #B #_ #j #H destruct
+| #I #G #L #A #i #HA #j #H destruct /2 width=4 by ex2_2_intro/
+| #p #G #L #V #T #B #A #_ #_ #j #H destruct
+| #p #G #L #V #T #B #A #_ #_ #j #H destruct
+| #G #L #V #T #B #A #_ #_ #j #H destruct
+| #G #L #V #T #A #_ #_ #j #H destruct
+]
+qed-.
+
+lemma aaa_inv_lref: ∀G,L,A,i. ⦃G, L⦄ ⊢ #↑i ⁝ A →
+ ∃∃I,K. L = K.ⓘ{I} & ⦃G, K⦄ ⊢ #i ⁝ A.
+/2 width=3 by aaa_inv_lref_aux/ qed-.
+
+fact aaa_inv_gref_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀l. T = §l → ⊥.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #k #H destruct
+| #I #G #L #V #B #_ #k #H destruct
+| #I #G #L #A #i #_ #k #H destruct
+| #p #G #L #V #T #B #A #_ #_ #k #H destruct
+| #p #G #L #V #T #B #A #_ #_ #k #H destruct
+| #G #L #V #T #B #A #_ #_ #k #H destruct
+| #G #L #V #T #A #_ #_ #k #H destruct
+]
+qed-.
+
+lemma aaa_inv_gref: ∀G,L,A,l. ⦃G, L⦄ ⊢ §l ⁝ A → ⊥.
+/2 width=7 by aaa_inv_gref_aux/ qed-.
+
+fact aaa_inv_abbr_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓓ{p}W.U →
+ ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L.ⓓW⦄ ⊢ U ⁝ A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #q #W #U #H destruct
+| #I #G #L #V #B #_ #q #W #U #H destruct
+| #I #G #L #A #i #_ #q #W #U #H destruct
+| #p #G #L #V #T #B #A #HV #HT #q #W #U #H destruct /2 width=2 by ex2_intro/
+| #p #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #G #L #V #T #A #_ #_ #q #W #U #H destruct
+]
+qed-.
+
+lemma aaa_inv_abbr: ∀p,G,L,V,T,A. ⦃G, L⦄ ⊢ ⓓ{p}V.T ⁝ A →
+ ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L.ⓓV⦄ ⊢ T ⁝ A.
+/2 width=4 by aaa_inv_abbr_aux/ qed-.
+
+fact aaa_inv_abst_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀p,W,U. T = ⓛ{p}W.U →
+ ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ U ⁝ B2 & A = ②B1.B2.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #q #W #U #H destruct
+| #I #G #L #V #B #_ #q #W #U #H destruct
+| #I #G #L #A #i #_ #q #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #p #G #L #V #T #B #A #HV #HT #q #W #U #H destruct /2 width=5 by ex3_2_intro/
+| #G #L #V #T #B #A #_ #_ #q #W #U #H destruct
+| #G #L #V #T #A #_ #_ #q #W #U #H destruct
+]
+qed-.
+
+lemma aaa_inv_abst: ∀p,G,L,W,T,A. ⦃G, L⦄ ⊢ ⓛ{p}W.T ⁝ A →
+ ∃∃B1,B2. ⦃G, L⦄ ⊢ W ⁝ B1 & ⦃G, L.ⓛW⦄ ⊢ T ⁝ B2 & A = ②B1.B2.
+/2 width=4 by aaa_inv_abst_aux/ qed-.
+
+fact aaa_inv_appl_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓐW.U →
+ ∃∃B. ⦃G, L⦄ ⊢ W ⁝ B & ⦃G, L⦄ ⊢ U ⁝ ②B.A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #W #U #H destruct
+| #I #G #L #V #B #_ #W #U #H destruct
+| #I #G #L #A #i #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #G #L #V #T #B #A #HV #HT #W #U #H destruct /2 width=3 by ex2_intro/
+| #G #L #V #T #A #_ #_ #W #U #H destruct
+]
+qed-.
+
+lemma aaa_inv_appl: ∀G,L,V,T,A. ⦃G, L⦄ ⊢ ⓐV.T ⁝ A →
+ ∃∃B. ⦃G, L⦄ ⊢ V ⁝ B & ⦃G, L⦄ ⊢ T ⁝ ②B.A.
+/2 width=3 by aaa_inv_appl_aux/ qed-.
+
+fact aaa_inv_cast_aux: ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ∀W,U. T = ⓝW.U →
+ ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ U ⁝ A.
+#G #L #T #A * -G -L -T -A
+[ #G #L #s #W #U #H destruct
+| #I #G #L #V #B #_ #W #U #H destruct
+| #I #G #L #A #i #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #p #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #G #L #V #T #B #A #_ #_ #W #U #H destruct
+| #G #L #V #T #A #HV #HT #W #U #H destruct /2 width=1 by conj/
+]
+qed-.
+
+lemma aaa_inv_cast: ∀G,L,W,T,A. ⦃G, L⦄ ⊢ ⓝW.T ⁝ A →
+ ⦃G, L⦄ ⊢ W ⁝ A ∧ ⦃G, L⦄ ⊢ T ⁝ A.
+/2 width=3 by aaa_inv_cast_aux/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Main inversion lemmas ****************************************************)
+
+theorem aaa_mono: ∀G,L,T,A1. ⦃G, L⦄ ⊢ T ⁝ A1 → ∀A2. ⦃G, L⦄ ⊢ T ⁝ A2 → A1 = A2.
+#G #L #T #A1 #H elim H -G -L -T -A1
+[ #G #L #s #A2 #H >(aaa_inv_sort … H) -H //
+| #I1 #G #L #V1 #B #_ #IH #A2 #H
+ elim (aaa_inv_zero … H) -H #I2 #K2 #V2 #H #HA2 destruct /2 width=1 by/
+| #I1 #G #L #B #i #_ #IH #A2 #H
+ elim (aaa_inv_lref … H) -H #I2 #K2 #H #HA2 destruct /2 width=1 by/
+| #p #G #L #V #T #B1 #A1 #_ #_ #_ #IH #A2 #H
+ elim (aaa_inv_abbr … H) -H /2 width=1 by/
+| #p #G #L #V1 #T1 #B1 #A1 #_ #_ #IHB1 #IHA1 #X #H
+ elim (aaa_inv_abst … H) -H #B2 #A2 #HB2 #HA2 #H destruct /3 width=1 by eq_f2/
+| #G #L #V1 #T1 #B1 #A1 #_ #_ #_ #IHA1 #A2 #H
+ elim (aaa_inv_appl … H) -H #B2 #_ #HA2
+ lapply (IHA1 … HA2) -L #H destruct //
+| #G #L #V #T #A1 #_ #_ #_ #IHA1 #A2 #H
+ elim (aaa_inv_cast … 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 "static_2/relocation/drops_drops.ma".
+include "static_2/s_computation/fqup_weight.ma".
+include "static_2/s_computation/fqup_drops.ma".
+include "static_2/static/aaa.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Advanced properties ******************************************************)
+
+(* Basic_2A1: was: aaa_lref *)
+lemma aaa_lref_drops: ∀I,G,K,V,B,i,L. ⬇*[i] L ≘ K.ⓑ{I}V → ⦃G, K⦄ ⊢ V ⁝ B → ⦃G, L⦄ ⊢ #i ⁝ B.
+#I #G #K #V #B #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H //
+ #H destruct /2 width=1 by aaa_zero/
+| #i #IH #L <uni_succ #H #HB lapply (drops_inv_bind2_isuni_next … H) -H // *
+ #Z #Y #HY #H destruct /3 width=1 by aaa_lref/
+]
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+(* Basic_2A1: was: aaa_inv_lref *)
+lemma aaa_inv_lref_drops: ∀G,A,i,L. ⦃G, L⦄ ⊢ #i ⁝ A →
+ ∃∃I,K,V. ⬇*[i] L ≘ K.ⓑ{I}V & ⦃G, K⦄ ⊢ V ⁝ A.
+#G #A #i elim i -i
+[ #L #H elim (aaa_inv_zero … H) -H /3 width=5 by drops_refl, ex2_3_intro/
+| #i #IH #L #H elim (aaa_inv_lref … H) -H
+ #I #K #H #HA destruct elim (IH … HA) -IH -HA /3 width=5 by drops_drop, ex2_3_intro/
+]
+qed-.
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Basic_2A1: includes: aaa_lift *)
+(* Note: it should use drops_split_trans_pair2 *)
+lemma aaa_lifts: ∀G,L1,T1,A. ⦃G, L1⦄ ⊢ T1 ⁝ A → ∀b,f,L2. ⬇*[b, f] L2 ≘ L1 →
+ ∀T2. ⬆*[f] T1 ≘ T2 → ⦃G, L2⦄ ⊢ T2 ⁝ A.
+@(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L1 * *
+[ #s #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -IH
+ lapply (aaa_inv_sort … H) -H #H destruct
+ >(lifts_inv_sort1 … HX) -HX //
+| #i1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
+ elim (aaa_inv_lref_drops … H) -H #J #K1 #V1 #HLK1 #HA
+ elim (lifts_inv_lref1 … HX) -HX #i2 #Hf #H destruct
+ lapply (drops_trans … HL21 … HLK1 ??) -HL21 [1,2: // ] #H
+ elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK2 #HY
+ lapply (drops_tls_at … Hf … HY) -HY #HY -Hf
+ elim (drops_inv_skip2 … HY) -HY #Z #K2 #HK21 #HZ #H destruct
+ elim (liftsb_inv_pair_sn … HZ) -HZ #V2 #HV12 #H destruct
+ /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_gen/
+| #l #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX -b -f -IH
+ elim (aaa_inv_gref … H)
+| #p * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
+ [ elim (aaa_inv_abbr … H) -H #B #HB #HA
+ elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
+ | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
+ elim (lifts_inv_bind1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /5 width=8 by aaa_abst, drops_skip, ext2_pair/
+ ]
+| * #V1 #T1 #HG #HL #HT #A #H #b #f #L2 #HL21 #X #HX
+ [ elim (aaa_inv_appl … H) -H #B #HB #HA
+ elim (lifts_inv_flat1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /3 width=10 by aaa_appl/
+ | elim (aaa_inv_cast … H) -H #H1A #H2A
+ elim (lifts_inv_flat1 … HX) -HX #V2 #T2 #HV12 #HT12 #H destruct
+ /3 width=8 by aaa_cast/
+ ]
+]
+qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+(* Basic_2A1: includes: aaa_inv_lift *)
+lemma aaa_inv_lifts: ∀G,L2,T2,A. ⦃G, L2⦄ ⊢ T2 ⁝ A → ∀b,f,L1. ⬇*[b, f] L2 ≘ L1 →
+ ∀T1. ⬆*[f] T1 ≘ T2 → ⦃G, L1⦄ ⊢ T1 ⁝ A.
+@(fqup_wf_ind_eq (Ⓣ)) #G0 #L0 #T0 #IH #G #L2 * *
+[ #s #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -b -IH
+ lapply (aaa_inv_sort … H) -H #H destruct
+ >(lifts_inv_sort2 … HX) -HX //
+| #i2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
+ elim (aaa_inv_lref_drops … H) -H #J #K2 #V2 #HLK2 #HA
+ elim (lifts_inv_lref2 … HX) -HX #i1 #Hf #H destruct
+ lapply (drops_split_div … HL21 (𝐔❴i1❵) ???) -HL21 [4: * |*: // ] #Y #HLK1 #HY
+ lapply (drops_conf … HLK2 … HY ??) -HY [1,2: /2 width=6 by after_uni_dx/ ] #HY
+ lapply (drops_tls_at … Hf … HY) -HY #HY -Hf
+ elim (drops_inv_skip1 … HY) -HY #Z #K1 #HK21 #HZ #H destruct
+ elim (liftsb_inv_pair_dx … HZ) -HZ #V1 #HV12 #H destruct
+ /4 width=12 by aaa_lref_drops, fqup_lref, drops_inv_F/
+| #l #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX -IH -b -f
+ elim (aaa_inv_gref … H)
+| #p * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
+ [ elim (aaa_inv_abbr … H) -H #B #HB #HA
+ elim (lifts_inv_bind2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
+ /5 width=9 by aaa_abbr, drops_skip, ext2_pair/
+ | elim (aaa_inv_abst … H) -H #B #A0 #HB #HA #H0
+ elim (lifts_inv_bind2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
+ /5 width=8 by aaa_abst, drops_skip, ext2_pair/
+ ]
+| * #V2 #T2 #HG #HL #HT #A #H #b #f #L1 #HL21 #X #HX
+ [ elim (aaa_inv_appl … H) -H #B #HB #HA
+ elim (lifts_inv_flat2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
+ /3 width=10 by aaa_appl/
+ | elim (aaa_inv_cast … H) -H #H1A #H2A
+ elim (lifts_inv_flat2 … HX) -HX #V1 #T1 #HV12 #HT12 #H destruct
+ /3 width=8 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 "static_2/static/fdeq.ma".
+include "static_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 "static_2/s_computation/fqus_fqup.ma".
+include "static_2/static/aaa_drops.ma".
+
+(* ATONIC ARITY ASSIGNMENT ON TERMS *****************************************)
+
+(* Properties on supclosure *************************************************)
+
+lemma aaa_fqu_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐ ⦃G2, L2, T2⦄ →
+ ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #T #A #H elim (aaa_inv_zero … H) -H
+ #J #K #V #H #HA destruct /2 width=2 by ex_intro/
+| * [ #p ] * #G #L #V #T #X #H
+ [ elim (aaa_inv_abbr … H)
+ | elim (aaa_inv_abst … H)
+ | elim (aaa_inv_appl … H)
+ | elim (aaa_inv_cast … H)
+ ] -H /2 width=2 by ex_intro/
+| #p * #G #L #V #T #X #H
+ [ elim (aaa_inv_abbr … H)
+ | elim (aaa_inv_abst … H)
+ ] -H /2 width=2 by ex_intro/
+| #p #I #G #L #V #T #H destruct
+| * #G #L #V #T #X #H
+ [ elim (aaa_inv_appl … H)
+ | elim (aaa_inv_cast … H)
+ ] -H /2 width=2 by ex_intro/
+| /5 width=8 by aaa_inv_lifts, drops_refl, drops_drop, ex_intro/
+]
+qed-.
+
+lemma aaa_fquq_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐⸮ ⦃G2, L2, T2⦄ →
+ ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -H /2 width=6 by aaa_fqu_conf/
+* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
+qed-.
+
+lemma aaa_fqup_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐+ ⦃G2, L2, T2⦄ →
+ ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind … H) -G2 -L2 -T2
+[2: #G #G2 #L #L2 #T #T2 #_ #H2 #IH1 #A #HA elim (IH1 … HA) -IH1 -A ]
+/2 width=6 by aaa_fqu_conf/
+qed-.
+
+lemma aaa_fqus_conf: ∀G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊐* ⦃G2, L2, T2⦄ →
+ ∀A1. ⦃G1, L1⦄ ⊢ T1 ⁝ A1 → ∃A2. ⦃G2, L2⦄ ⊢ T2 ⁝ A2.
+#G1 #G2 #L1 #L2 #T1 #T2 #H elim(fqus_inv_fqup … H) -H /2 width=6 by aaa_fqup_conf/
+* #H1 #H2 #H3 destruct /2 width=2 by ex_intro/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/static/rdeq.ma".
+include "static_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 "static_2/notation/relations/stareqsn_8.ma".
+include "static_2/syntax/genv.ma".
+include "static_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 "static_2/static/rdeq_rdeq.ma".
+include "static_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 "static_2/static/rdeq_fqup.ma".
+include "static_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 "static_2/static/rdeq_fqus.ma".
+include "static_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 "static_2/static/rdeq_req.ma".
+include "static_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 "ground_2/relocation/rtmap_sor.ma".
+include "static_2/notation/relations/freestar_3.ma".
+include "static_2/syntax/lenv.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+inductive frees: relation3 lenv term rtmap ≝
+| frees_sort: ∀f,L,s. 𝐈⦃f⦄ → frees L (⋆s) f
+| frees_atom: ∀f,i. 𝐈⦃f⦄ → frees (⋆) (#i) (⫯*[i]↑f)
+| frees_pair: ∀f,I,L,V. frees L V f →
+ frees (L.ⓑ{I}V) (#0) (↑f)
+| frees_unit: ∀f,I,L. 𝐈⦃f⦄ → frees (L.ⓤ{I}) (#0) (↑f)
+| frees_lref: ∀f,I,L,i. frees L (#i) f →
+ frees (L.ⓘ{I}) (#↑i) (⫯f)
+| frees_gref: ∀f,L,l. 𝐈⦃f⦄ → frees L (§l) f
+| frees_bind: ∀f1,f2,f,p,I,L,V,T. frees L V f1 → frees (L.ⓑ{I}V) T f2 →
+ f1 ⋓ ⫱f2 ≘ f → frees L (ⓑ{p,I}V.T) f
+| frees_flat: ∀f1,f2,f,I,L,V,T. frees L V f1 → frees L T f2 →
+ f1 ⋓ f2 ≘ f → frees L (ⓕ{I}V.T) f
+.
+
+interpretation
+ "context-sensitive free variables (term)"
+ 'FreeStar L T f = (frees L T f).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact frees_inv_sort_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = ⋆x → 𝐈⦃f⦄.
+#L #X #f #H elim H -f -L -X //
+[ #f #i #_ #x #H destruct
+| #f #_ #L #V #_ #_ #x #H destruct
+| #f #_ #L #_ #x #H destruct
+| #f #_ #L #i #_ #_ #x #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+]
+qed-.
+
+lemma frees_inv_sort: ∀f,L,s. L ⊢ 𝐅*⦃⋆s⦄ ≘ f → 𝐈⦃f⦄.
+/2 width=5 by frees_inv_sort_aux/ qed-.
+
+fact frees_inv_atom_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀i. L = ⋆ → X = #i →
+ ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+#f #L #X #H elim H -f -L -X
+[ #f #L #s #_ #j #_ #H destruct
+| #f #i #Hf #j #_ #H destruct /2 width=3 by ex2_intro/
+| #f #I #L #V #_ #_ #j #H destruct
+| #f #I #L #_ #j #H destruct
+| #f #I #L #i #_ #_ #j #H destruct
+| #f #L #l #_ #j #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #j #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #j #_ #H destruct
+]
+qed-.
+
+lemma frees_inv_atom: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ⫯*[i]↑g.
+/2 width=5 by frees_inv_atom_aux/ qed-.
+
+fact frees_inv_pair_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,V. L = K.ⓑ{I}V → X = #0 →
+ ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #Z #Y #X #_ #H destruct
+| #f #i #_ #Z #Y #X #H destruct
+| #f #I #L #V #Hf #Z #Y #X #H #_ destruct /2 width=3 by ex2_intro/
+| #f #I #L #_ #Z #Y #X #H destruct
+| #f #I #L #i #_ #Z #Y #X #_ #H destruct
+| #f #L #l #_ #Z #Y #X #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #X #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #X #_ #H destruct
+]
+qed-.
+
+lemma frees_inv_pair: ∀f,I,K,V. K.ⓑ{I}V ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. K ⊢ 𝐅*⦃V⦄ ≘ g & f = ↑g.
+/2 width=6 by frees_inv_pair_aux/ qed-.
+
+fact frees_inv_unit_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K. L = K.ⓤ{I} → X = #0 →
+ ∃∃g. 𝐈⦃g⦄ & f = ↑g.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #Z #Y #_ #H destruct
+| #f #i #_ #Z #Y #H destruct
+| #f #I #L #V #_ #Z #Y #H destruct
+| #f #I #L #Hf #Z #Y #H destruct /2 width=3 by ex2_intro/
+| #f #I #L #i #_ #Z #Y #_ #H destruct
+| #f #L #l #_ #Z #Y #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #_ #H destruct
+]
+qed-.
+
+lemma frees_inv_unit: ∀f,I,K. K.ⓤ{I} ⊢ 𝐅*⦃#0⦄ ≘ f → ∃∃g. 𝐈⦃g⦄ & f = ↑g.
+/2 width=7 by frees_inv_unit_aux/ qed-.
+
+fact frees_inv_lref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,K,j. L = K.ⓘ{I} → X = #(↑j) →
+ ∃∃g. K ⊢ 𝐅*⦃#j⦄ ≘ g & f = ⫯g.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #Z #Y #j #_ #H destruct
+| #f #i #_ #Z #Y #j #H destruct
+| #f #I #L #V #_ #Z #Y #j #_ #H destruct
+| #f #I #L #_ #Z #Y #j #_ #H destruct
+| #f #I #L #i #Hf #Z #Y #j #H1 #H2 destruct /2 width=3 by ex2_intro/
+| #f #L #l #_ #Z #Y #j #_ #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #Z #Y #j #_ #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #Z #Y #j #_ #H destruct
+]
+qed-.
+
+lemma frees_inv_lref: ∀f,I,K,i. K.ⓘ{I} ⊢ 𝐅*⦃#(↑i)⦄ ≘ f →
+ ∃∃g. K ⊢ 𝐅*⦃#i⦄ ≘ g & f = ⫯g.
+/2 width=6 by frees_inv_lref_aux/ qed-.
+
+fact frees_inv_gref_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀x. X = §x → 𝐈⦃f⦄.
+#f #L #X #H elim H -f -L -X //
+[ #f #i #_ #x #H destruct
+| #f #_ #L #V #_ #_ #x #H destruct
+| #f #_ #L #_ #x #H destruct
+| #f #_ #L #i #_ #_ #x #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #_ #_ #x #H destruct
+]
+qed-.
+
+lemma frees_inv_gref: ∀f,L,l. L ⊢ 𝐅*⦃§l⦄ ≘ f → 𝐈⦃f⦄.
+/2 width=5 by frees_inv_gref_aux/ qed-.
+
+fact frees_inv_bind_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀p,I,V,T. X = ⓑ{p,I}V.T →
+ ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #q #J #W #U #H destruct
+| #f #i #_ #q #J #W #U #H destruct
+| #f #I #L #V #_ #q #J #W #U #H destruct
+| #f #I #L #_ #q #J #W #U #H destruct
+| #f #I #L #i #_ #q #J #W #U #H destruct
+| #f #L #l #_ #q #J #W #U #H destruct
+| #f1 #f2 #f #p #I #L #V #T #HV #HT #Hf #q #J #W #U #H destruct /2 width=5 by ex3_2_intro/
+| #f1 #f2 #f #I #L #V #T #_ #_ #_ #q #J #W #U #H destruct
+]
+qed-.
+
+lemma frees_inv_bind: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
+ ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓑ{I}V ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+/2 width=4 by frees_inv_bind_aux/ qed-.
+
+fact frees_inv_flat_aux: ∀f,L,X. L ⊢ 𝐅*⦃X⦄ ≘ f → ∀I,V,T. X = ⓕ{I}V.T →
+ ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
+#f #L #X * -f -L -X
+[ #f #L #s #_ #J #W #U #H destruct
+| #f #i #_ #J #W #U #H destruct
+| #f #I #L #V #_ #J #W #U #H destruct
+| #f #I #L #_ #J #W #U #H destruct
+| #f #I #L #i #_ #J #W #U #H destruct
+| #f #L #l #_ #J #W #U #H destruct
+| #f1 #f2 #f #p #I #L #V #T #_ #_ #_ #J #W #U #H destruct
+| #f1 #f2 #f #I #L #V #T #HV #HT #Hf #J #W #U #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma frees_inv_flat: ∀f,I,L,V,T. L ⊢ 𝐅*⦃ⓕ{I}V.T⦄ ≘ f →
+ ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ f2 ≘ f.
+/2 width=4 by frees_inv_flat_aux/ qed-.
+
+(* Basic properties ********************************************************)
+
+lemma frees_eq_repl_back: ∀L,T. eq_repl_back … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
+#L #T #f1 #H elim H -f1 -L -T
+[ /3 width=3 by frees_sort, isid_eq_repl_back/
+| #f1 #i #Hf1 #g2 #H
+ elim (eq_inv_pushs_sn … H) -H #g #Hg #H destruct
+ elim (eq_inv_nx … Hg) -Hg
+ /3 width=3 by frees_atom, isid_eq_repl_back/
+| #f1 #I #L #V #_ #IH #g2 #H
+ elim (eq_inv_nx … H) -H
+ /3 width=3 by frees_pair/
+| #f1 #I #L #Hf1 #g2 #H
+ elim (eq_inv_nx … H) -H
+ /3 width=3 by frees_unit, isid_eq_repl_back/
+| #f1 #I #L #i #_ #IH #g2 #H
+ elim (eq_inv_px … H) -H /3 width=3 by frees_lref/
+| /3 width=3 by frees_gref, isid_eq_repl_back/
+| /3 width=7 by frees_bind, sor_eq_repl_back3/
+| /3 width=7 by frees_flat, sor_eq_repl_back3/
+]
+qed-.
+
+lemma frees_eq_repl_fwd: ∀L,T. eq_repl_fwd … (λf. L ⊢ 𝐅*⦃T⦄ ≘ f).
+#L #T @eq_repl_sym /2 width=3 by frees_eq_repl_back/
+qed-.
+
+lemma frees_lref_push: ∀f,i. ⋆ ⊢ 𝐅*⦃#i⦄ ≘ f → ⋆ ⊢ 𝐅*⦃#↑i⦄ ≘ ⫯f.
+#f #i #H
+elim (frees_inv_atom … H) -H #g #Hg #H destruct
+/2 width=1 by frees_atom/
+qed.
+
+(* Forward lemmas with test for finite colength *****************************)
+
+lemma frees_fwd_isfin: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f → 𝐅⦃f⦄.
+#f #L #T #H elim H -f -L -T
+/4 width=5 by sor_isfin, isfin_isid, isfin_tl, isfin_pushs, isfin_push, isfin_next/
+qed-.
+
+(* Basic_2A1: removed theorems 30:
+ frees_eq frees_be frees_inv
+ frees_inv_sort frees_inv_gref frees_inv_lref frees_inv_lref_free
+ frees_inv_lref_skip frees_inv_lref_ge frees_inv_lref_lt
+ frees_inv_bind frees_inv_flat frees_inv_bind_O
+ frees_lref_eq frees_lref_be frees_weak
+ frees_bind_sn frees_bind_dx frees_flat_sn frees_flat_dx
+ frees_lift_ge frees_inv_lift_be frees_inv_lift_ge
+ lreq_frees_trans frees_lreq_conf
+ llor_atom llor_skip llor_total
+ llor_tail_frees llor_tail_cofrees
+*)
--- /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 "static_2/syntax/append.ma".
+include "static_2/static/frees.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Properties with append for local environments ****************************)
+
+lemma frees_append_void: ∀f,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f → ⓧ.K ⊢ 𝐅*⦃T⦄ ≘ f.
+#f #K #T #H elim H -f -K -T
+[ /2 width=1 by frees_sort/
+| #f * /3 width=1 by frees_atom, frees_unit, frees_lref/
+| /2 width=1 by frees_pair/
+| /2 width=1 by frees_unit/
+| /2 width=1 by frees_lref/
+| /2 width=1 by frees_gref/
+| /3 width=5 by frees_bind/
+| /3 width=5 by frees_flat/
+]
+qed.
+
+(* Inversion lemmas with append for local environments **********************)
+
+fact frees_inv_append_void_aux: ∀f,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f →
+ ∀K. L = ⓧ.K → K ⊢ 𝐅*⦃T⦄ ≘ f.
+#f #L #T #H elim H -f -L -T
+[ /2 width=1 by frees_sort/
+| #f #i #_ #K #H
+ elim (append_inv_atom3_sn … H) -H #H1 #H2 destruct
+| #f #I #L #V #_ #IH #K #H
+ elim (append_inv_bind3_sn … H) -H * [ | #Y ] #H1 #H2 destruct
+ /3 width=1 by frees_pair/
+| #f #I #L #Hf #K #H
+ elim (append_inv_bind3_sn … H) -H * [ | #Y ] #H1 #H2 destruct
+ /2 width=1 by frees_atom, frees_unit/
+| #f #I #L #i #Hf #IH #K #H
+ elim (append_inv_bind3_sn … H) -H * [ | #Y ] #H1 #H2 destruct
+ /3 width=1 by frees_lref, frees_lref_push/
+| /2 width=1 by frees_gref/
+| /3 width=5 by frees_bind/
+| /3 width=5 by frees_flat/
+]
+qed-.
+
+lemma frees_inv_append_void: ∀f,K,T. ⓧ.K ⊢ 𝐅*⦃T⦄ ≘ f → K ⊢ 𝐅*⦃T⦄ ≘ f.
+/2 width=3 by frees_inv_append_void_aux/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/relocation/nstream_coafter.ma".
+include "static_2/relocation/drops_drops.ma".
+include "static_2/static/frees_fqup.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Advanced properties ******************************************************)
+
+lemma frees_atom_drops: ∀b,L,i. ⬇*[b, 𝐔❴i❵] L ≘ ⋆ →
+ ∀f. 𝐈⦃f⦄ → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i]↑f.
+#b #L elim L -L /2 width=1 by frees_atom/
+#L #I #IH *
+[ #H lapply (drops_fwd_isid … H ?) -H // #H destruct
+| /4 width=3 by frees_lref, drops_inv_drop1/
+]
+qed.
+
+lemma frees_pair_drops: ∀f,K,V. K ⊢ 𝐅*⦃V⦄ ≘ f →
+ ∀i,I,L. ⬇*[i] L ≘ K.ⓑ{I}V → L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
+#f #K #V #Hf #i elim i -i
+[ #I #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_pair/
+| #i #IH #I #L #H elim (drops_inv_succ … H) -H /3 width=2 by frees_lref/
+]
+qed.
+
+lemma frees_unit_drops: ∀f. 𝐈⦃f⦄ → ∀I,K,i,L. ⬇*[i] L ≘ K.ⓤ{I} →
+ L ⊢ 𝐅*⦃#i⦄ ≘ ⫯*[i] ↑f.
+#f #Hf #I #K #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H /2 width=1 by frees_unit/
+| #i #IH #Y #H elim (drops_inv_succ … H) -H
+ #J #L #HLK #H destruct /3 width=1 by frees_lref/
+]
+qed.
+(*
+lemma frees_sort_pushs: ∀f,K,s. K ⊢ 𝐅*⦃⋆s⦄ ≘ f →
+ ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃⋆s⦄ ≘ ⫯*[i] f.
+#f #K #s #Hf #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H //
+| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_sort/
+]
+qed.
+*)
+lemma frees_lref_pushs: ∀f,K,j. K ⊢ 𝐅*⦃#j⦄ ≘ f →
+ ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃#(i+j)⦄ ≘ ⫯*[i] f.
+#f #K #j #Hf #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H //
+| #i #IH #L #H elim (drops_inv_succ … H) -H
+ #I #Y #HYK #H destruct /3 width=1 by frees_lref/
+]
+qed.
+(*
+lemma frees_gref_pushs: ∀f,K,l. K ⊢ 𝐅*⦃§l⦄ ≘ f →
+ ∀i,L. ⬇*[i] L ≘ K → L ⊢ 𝐅*⦃§l⦄ ≘ ⫯*[i] f.
+#f #K #l #Hf #i elim i -i
+[ #L #H lapply (drops_fwd_isid … H ?) -H //
+| #i #IH #L #H elim (drops_inv_succ … H) -H /3 width=1 by frees_gref/
+]
+qed.
+*)
+(* Advanced inversion lemmas ************************************************)
+
+lemma frees_inv_lref_drops: ∀L,i,f. L ⊢ 𝐅*⦃#i⦄ ≘ f →
+ ∨∨ ∃∃g. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ & 𝐈⦃g⦄ & f = ⫯*[i] ↑g
+ | ∃∃g,I,K,V. K ⊢ 𝐅*⦃V⦄ ≘ g &
+ ⬇*[i] L ≘ K.ⓑ{I}V & f = ⫯*[i] ↑g
+ | ∃∃g,I,K. ⬇*[i] L ≘ K.ⓤ{I} & 𝐈⦃g⦄ & f = ⫯*[i] ↑g.
+#L elim L -L
+[ #i #g | #L #I #IH * [ #g cases I -I [ #I | #I #V ] -IH | #i #g ] ] #H
+[ elim (frees_inv_atom … H) -H #f #Hf #H destruct
+ /3 width=3 by or3_intro0, ex3_intro/
+| elim (frees_inv_unit … H) -H #f #Hf #H destruct
+ /4 width=3 by drops_refl, or3_intro2, ex3_3_intro/
+| elim (frees_inv_pair … H) -H #f #Hf #H destruct
+ /4 width=7 by drops_refl, or3_intro1, ex3_4_intro/
+| elim (frees_inv_lref … H) -H #f #Hf #H destruct
+ elim (IH … Hf) -IH -Hf *
+ [ /4 width=3 by drops_drop, or3_intro0, ex3_intro/
+ | /4 width=7 by drops_drop, or3_intro1, ex3_4_intro/
+ | /4 width=3 by drops_drop, or3_intro2, ex3_3_intro/
+ ]
+]
+qed-.
+
+(* Properties with generic slicing for local environments *******************)
+
+lemma frees_lifts: ∀b,f1,K,T. K ⊢ 𝐅*⦃T⦄ ≘ f1 →
+ ∀f,L. ⬇*[b, f] L ≘ K → ∀U. ⬆*[f] T ≘ U →
+ ∀f2. f ~⊚ f1 ≘ f2 → L ⊢ 𝐅*⦃U⦄ ≘ f2.
+#b #f1 #K #T #H lapply (frees_fwd_isfin … H) elim H -f1 -K -T
+[ #f1 #K #s #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
+ lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
+ >(lifts_inv_sort1 … H2) -U /2 width=1 by frees_sort/
+| #f1 #i #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
+ elim (lifts_inv_lref1 … H2) -H2 #j #Hij #H destruct
+ elim (coafter_fwd_xnx_pushs … Hij H3) -H3 #g2 #Hg2 #H2 destruct
+ lapply (coafter_isid_inv_dx … Hg2 … Hf1) -f1 #Hf2
+ elim (drops_inv_atom2 … H1) -H1 #n #g #H1 #Hf
+ elim (after_at_fwd … Hij … Hf) -f #x #_ #Hj -g -i
+ lapply (at_inv_uni … Hj) -Hj #H destruct
+ /3 width=8 by frees_atom_drops, drops_trans/
+| #f1 #I #K #V #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
+ lapply (isfin_inv_next … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
+ lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
+ elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #H
+ elim (liftsb_inv_pair_sn … H) -H #W #HVW #H destruct
+ elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
+ lapply (IH … HYK … HVW … H3) -IH -H3 -HYK -HVW //
+ /2 width=5 by frees_pair_drops/
+| #f1 #I #K #Hf1 #_ #f #L #H1 #U #H2 #f2 #H3
+ lapply (lifts_inv_lref1 … H2) -H2 * #j #Hf #H destruct
+ elim (coafter_fwd_xnx_pushs … Hf H3) -H3 #g2 #H3 #H2 destruct
+ lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hg2
+ elim (drops_split_trans_bind2 … H1 … Hf) -H1 -Hf #Z #Y #HLY #_ #H
+ lapply (liftsb_inv_unit_sn … H) -H #H destruct
+ /2 width=3 by frees_unit_drops/
+| #f1 #I #K #i #_ #IH #Hf1 #f #L #H1 #U #H2 #f2 #H3
+ lapply (isfin_inv_push … Hf1 ??) -Hf1 [3: |*: // ] #Hf1
+ lapply (lifts_inv_lref1 … H2) -H2 * #x #Hf #H destruct
+ elim (at_inv_nxx … Hf) -Hf [ |*: // ] #j #Hf #H destruct
+ elim (drops_split_trans_bind2 … H1) -H1 [ |*: // ] #Z #Y #HLY #HYK #_
+ elim (coafter_fwd_xpx_pushs … 0 … H3) [ |*: // ] #g2 #H3 #H2 destruct
+ lapply (drops_isuni_fwd_drop2 … HLY) -HLY // #HLY
+ lapply (IH … HYK … H3) -IH -H3 -HYK [4: |*: /2 width=2 by lifts_lref/ ]
+ >plus_S1 /2 width=3 by frees_lref_pushs/ (**) (* full auto fails *)
+| #f1 #K #l #Hf1 #_ #f #L #HLK #U #H2 #f2 #H3
+ lapply (coafter_isid_inv_dx … H3 … Hf1) -f1 #Hf2
+ >(lifts_inv_gref1 … H2) -U /2 width=1 by frees_gref/
+| #f1V #f1T #f1 #p #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
+ elim (sor_inv_isfin3 … H1f1) // #Hf1V #H
+ lapply (isfin_inv_tl … H) -H
+ elim (lifts_inv_bind1 … H2) -H2 #W #U #HVW #HTU #H destruct
+ elim (coafter_sor … H3 … H1f1) /2 width=5 by coafter_isfin2_fwd/ -H3 -H1f1 #f2V #f2T #Hf2V #H
+ elim (coafter_inv_tl1 … H) -H
+ /5 width=5 by frees_bind, drops_skip, ext2_pair/
+| #f1V #f1T #f1 #I #K #V #T #_ #_ #H1f1 #IHV #IHT #H2f1 #f #L #H1 #Y #H2 #f2 #H3
+ elim (sor_inv_isfin3 … H1f1) //
+ elim (lifts_inv_flat1 … H2) -H2 #W #U #HVW #HTU #H destruct
+ elim (coafter_sor … H3 … H1f1)
+ /3 width=5 by coafter_isfin2_fwd, frees_flat/
+]
+qed-.
+
+lemma frees_lifts_SO: ∀b,L,K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T,U. ⬆*[1] T ≘ U →
+ ∀f. K ⊢ 𝐅*⦃T⦄ ≘ f → L ⊢ 𝐅*⦃U⦄ ≘ ⫯f.
+#b #L #K #HLK #T #U #HTU #f #Hf
+@(frees_lifts b … Hf … HTU) // (**) (* auto fails *)
+qed.
+
+(* Forward lemmas with generic slicing for local environments ***************)
+
+lemma frees_fwd_coafter: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
+ ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∀f1. K ⊢ 𝐅*⦃T⦄ ≘ f1 → f ~⊚ f1 ≘ f2.
+/4 width=11 by frees_lifts, frees_mono, coafter_eq_repl_back0/ qed-.
+
+(* Inversion lemmas with generic slicing for local environments *************)
+
+lemma frees_inv_lifts_ex: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
+ ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∃∃f1. f ~⊚ f1 ≘ f2 & K ⊢ 𝐅*⦃T⦄ ≘ f1.
+#b #f2 #L #U #Hf2 #f #K #HLK #T elim (frees_total K T)
+/3 width=9 by frees_fwd_coafter, ex2_intro/
+qed-.
+
+lemma frees_inv_lifts_SO: ∀b,f,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f →
+ ∀K. ⬇*[b, 𝐔❴1❵] L ≘ K → ∀T. ⬆*[1] T ≘ U →
+ K ⊢ 𝐅*⦃T⦄ ≘ ⫱f.
+#b #f #L #U #H #K #HLK #T #HTU elim(frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
+#f1 #Hf #Hf1 elim (coafter_inv_nxx … Hf) -Hf
+/3 width=5 by frees_eq_repl_back, coafter_isid_inv_sn/
+qed-.
+
+lemma frees_inv_lifts: ∀b,f2,L,U. L ⊢ 𝐅*⦃U⦄ ≘ f2 →
+ ∀f,K. ⬇*[b, f] L ≘ K → ∀T. ⬆*[f] T ≘ U →
+ ∀f1. f ~⊚ f1 ≘ f2 → K ⊢ 𝐅*⦃T⦄ ≘ f1.
+#b #f2 #L #U #H #f #K #HLK #T #HTU #f1 #Hf2 elim (frees_inv_lifts_ex … H … HLK … HTU) -b -L -U
+/3 width=7 by frees_eq_repl_back, coafter_inj/
+qed-.
+
+(* 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 →
+ ∃∃g2. L2 ⊢ 𝐅*⦃V2⦄ ≘ g2 & g2 ⊆ g1.
+#f1 #L1 #T1 #H elim H -f1 -L1 -T1
+[ #f1 #L1 #s #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -s
+ lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
+ elim (isid_inv_next … Hf1) -Hf1 //
+| #f1 #i #_ #I2 #L2 #V2 #n #H
+ elim (drops_inv_atom1 … H) -H #H destruct
+| #f1 #I1 #L1 #V1 #Hf1 #IH #I2 #L2 #V2 *
+ [ -IH #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 //
+ #H destruct #g1 #Hgf1 >(injective_next … Hgf1) -g1
+ /2 width=3 by ex2_intro/
+ | -Hf1 #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
+ #HL12 #g1 <tls_xn <tl_next_rew #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
+ /2 width=3 by ex2_intro/
+ ]
+| #f1 #I1 #L1 #Hf1 #I2 #L2 #V2 *
+ [ #HL12 lapply (drops_fwd_isid … HL12 ?) -HL12 // #H destruct
+ | #n #_ #g1 #Hgf1 elim (isid_inv_next … Hgf1) -Hgf1 <tls_xn /2 width=1 by isid_tls/
+ ]
+| #f1 #I1 #L1 #i #_ #IH #I2 #L2 #V2 *
+ [ -IH #_ #g1 #Hgf1 elim (discr_next_push … Hgf1)
+ | #n #HL12 lapply (drops_inv_drop1 … HL12) -HL12
+ #HL12 #g1 <tls_xn #Hgf1 elim (IH … HL12 … Hgf1) -IH -HL12 -Hgf1
+ /2 width=3 by ex2_intro/
+ ]
+| #f1 #L1 #l #Hf1 #I2 #L2 #V2 #n #_ #g1 #H1 -I2 -L1 -l
+ lapply (isid_tls n … Hf1) -Hf1 <H1 -f1 #Hf1
+ elim (isid_inv_next … Hf1) -Hf1 //
+| #fV1 #fT1 #f1 #p #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
+ lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
+ elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
+ #gV1 #gT1 #Hg1
+ [ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
+ /3 width=6 by sor_inv_sle_sn_trans, ex2_intro/
+ | -IHV1 #_ >tls_xn #H2 elim (IHT1 … H2) -IHT1 -H2
+ /3 width=6 by drops_drop, sor_inv_sle_dx_trans, ex2_intro/
+ ]
+| #fV1 #fT1 #f1 #I1 #L1 #V1 #T1 #_ #_ #Hf1 #IHV1 #IHT1 #I2 #L2 #V2 #n #HL12 #g1 #Hgf1
+ lapply (sor_tls … Hf1 n) -Hf1 <Hgf1 -Hgf1 #Hf1
+ elim (sor_xxn_tl … Hf1) [1,2: * |*: // ] -Hf1
+ #gV1 #gT1 #Hg1
+ [ -IHT1 #H1 #_ elim (IHV1 … HL12 … H1) -IHV1 -HL12 -H1
+ /3 width=6 by sor_inv_sle_sn_trans, ex2_intro/
+ | -IHV1 #_ #H2 elim (IHT1 … HL12 … H2) -IHT1 -HL12 -H2
+ /3 width=6 by sor_inv_sle_dx_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 "static_2/s_computation/fqup_weight.ma".
+include "static_2/static/lsubf_lsubr.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Advanced properties ******************************************************)
+
+(* Note: this replaces lemma 1400 concluding the "big tree" theorem *)
+lemma frees_total: ∀L,T. ∃f. L ⊢ 𝐅*⦃T⦄ ≘ f.
+#L #T @(fqup_wf_ind_eq (Ⓣ) … (⋆) L T) -L -T
+#G0 #L0 #T0 #IH #G #L * *
+[ /3 width=2 by frees_sort, ex_intro/
+| cases L -L /3 width=2 by frees_atom, ex_intro/
+ #L #I *
+ [ cases I -I #I [2: #V ] #HG #HL #HT destruct
+ [ elim (IH G L V) -IH
+ /3 width=2 by frees_pair, fqu_fqup, fqu_lref_O, ex_intro/
+ | -IH /3 width=2 by frees_unit, ex_intro/
+ ]
+ | #i #HG #HL #HT destruct
+ elim (IH G L (#i)) -IH
+ /3 width=2 by frees_lref, fqu_fqup, fqu_drop, ex_intro/
+ ]
+| /3 width=2 by frees_gref, ex_intro/
+| #p #I #V #T #HG #HL #HT destruct
+ elim (IH G L V) // #f1 #HV
+ elim (IH G (L.ⓑ{I}V) T) -IH // #f2 #HT
+ elim (sor_isfin_ex f1 (⫱f2))
+ /3 width=6 by frees_fwd_isfin, frees_bind, isfin_tl, ex_intro/
+| #I #V #T #HG #HL #HT destruct
+ elim (IH G L V) // #f1 #HV
+ elim (IH G L T) -IH // #f2 #HT
+ elim (sor_isfin_ex f1 f2)
+ /3 width=6 by frees_fwd_isfin, frees_flat, ex_intro/
+]
+qed-.
+
+(* Advanced main properties *************************************************)
+
+theorem frees_bind_void: ∀f1,L,V. L ⊢ 𝐅*⦃V⦄ ≘ f1 → ∀f2,T. L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 →
+ ∀f. f1 ⋓ ⫱f2 ≘ f → ∀p,I. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f.
+#f1 #L #V #Hf1 #f2 #T #Hf2 #f #Hf #p #I
+elim (frees_total (L.ⓑ{I}V) T) #f0 #Hf0
+lapply (lsubr_lsubf … Hf2 … Hf0) -Hf2 /2 width=5 by lsubr_unit/ #H02
+elim (pn_split f2) * #g2 #H destruct
+[ elim (lsubf_inv_push2 … H02) -H02 #g0 #Z #Y #H02 #H0 #H destruct
+ lapply (lsubf_inv_refl … H02) -H02 #H02
+ lapply (sor_eq_repl_fwd2 … Hf … H02) -g2 #Hf
+ /2 width=5 by frees_bind/
+| elim (lsubf_inv_unit2 … H02) -H02 * [ #g0 #Y #_ #_ #H destruct ]
+ #z1 #g0 #z #Z #Y #X #H02 #Hz1 #Hz #H0 #H destruct
+ lapply (lsubf_inv_refl … H02) -H02 #H02
+ lapply (frees_mono … Hz1 … Hf1) -Hz1 #H1
+ lapply (sor_eq_repl_back1 … Hz … H02) -g0 #Hz
+ lapply (sor_eq_repl_back2 … Hz … H1) -z1 #Hz
+ lapply (sor_comm … Hz) -Hz #Hz
+ lapply (sor_mono … f Hz ?) // -Hz #H
+ lapply (sor_inv_sle_sn … Hf) -Hf #Hf
+ lapply (frees_eq_repl_back … Hf0 (↑f) ?) /2 width=5 by eq_next/ -z #Hf0
+ @(frees_bind … Hf1 Hf0) -Hf1 -Hf0 (**) (* constructor needed *)
+ /2 width=1 by sor_sle_dx/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma frees_inv_bind_void: ∀f,p,I,L,V,T. L ⊢ 𝐅*⦃ⓑ{p,I}V.T⦄ ≘ f →
+ ∃∃f1,f2. L ⊢ 𝐅*⦃V⦄ ≘ f1 & L.ⓧ ⊢ 𝐅*⦃T⦄ ≘ f2 & f1 ⋓ ⫱f2 ≘ f.
+#f #p #I #L #V #T #H
+elim (frees_inv_bind … H) -H #f1 #f2 #Hf1 #Hf2 #Hf
+elim (frees_total (L.ⓧ) T) #f0 #Hf0
+lapply (lsubr_lsubf … Hf0 … Hf2) -Hf2 /2 width=5 by lsubr_unit/ #H20
+elim (pn_split f0) * #g0 #H destruct
+[ elim (lsubf_inv_push2 … H20) -H20 #g2 #I #Y #H20 #H2 #H destruct
+ lapply (lsubf_inv_refl … H20) -H20 #H20
+ lapply (sor_eq_repl_back2 … Hf … H20) -g2 #Hf
+ /2 width=5 by ex3_2_intro/
+| elim (lsubf_inv_unit2 … H20) -H20 * [ #g2 #Y #_ #_ #H destruct ]
+ #z1 #z0 #g2 #Z #Y #X #H20 #Hz1 #Hg2 #H2 #H destruct
+ lapply (lsubf_inv_refl … H20) -H20 #H0
+ lapply (frees_mono … Hz1 … Hf1) -Hz1 #H1
+ lapply (sor_eq_repl_back1 … Hg2 … H0) -z0 #Hg2
+ lapply (sor_eq_repl_back2 … Hg2 … H1) -z1 #Hg2
+ @(ex3_2_intro … Hf1 Hf0) -Hf1 -Hf0 (**) (* constructor needed *)
+ /2 width=3 by sor_comm_23_idem/
+]
+qed-.
+
+lemma frees_ind_void: ∀Q:relation3 ….
+ (
+ ∀f,L,s. 𝐈⦃f⦄ → Q L (⋆s) f
+ ) → (
+ ∀f,i. 𝐈⦃f⦄ → Q (⋆) (#i) (⫯*[i]↑f)
+ ) → (
+ ∀f,I,L,V.
+ L ⊢ 𝐅*⦃V⦄ ≘ f → Q L V f→ Q (L.ⓑ{I}V) (#O) (↑f)
+ ) → (
+ ∀f,I,L. 𝐈⦃f⦄ → Q (L.ⓤ{I}) (#O) (↑f)
+ ) → (
+ ∀f,I,L,i.
+ L ⊢ 𝐅*⦃#i⦄ ≘ f → Q L (#i) f → Q (L.ⓘ{I}) (#(↑i)) (⫯f)
+ ) → (
+ ∀f,L,l. 𝐈⦃f⦄ → Q L (§l) f
+ ) → (
+ ∀f1,f2,f,p,I,L,V,T.
+ L ⊢ 𝐅*⦃V⦄ ≘ f1 → L.ⓧ ⊢𝐅*⦃T⦄≘ f2 → f1 ⋓ ⫱f2 ≘ f →
+ Q L V f1 → Q (L.ⓧ) T f2 → Q L (ⓑ{p,I}V.T) f
+ ) → (
+ ∀f1,f2,f,I,L,V,T.
+ L ⊢ 𝐅*⦃V⦄ ≘ f1 → L ⊢𝐅*⦃T⦄ ≘ f2 → f1 ⋓ f2 ≘ f →
+ Q L V f1 → Q L T f2 → Q L (ⓕ{I}V.T) f
+ ) →
+ ∀L,T,f. L ⊢ 𝐅*⦃T⦄ ≘ f → Q L T f.
+#Q #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #L #T
+@(fqup_wf_ind_eq (Ⓕ) … (⋆) L T) -L -T #G0 #L0 #T0 #IH #G #L * *
+[ #s #HG #HL #HT #f #H destruct -IH
+ lapply (frees_inv_sort … H) -H /2 width=1 by/
+| cases L -L
+ [ #i #HG #HL #HT #f #H destruct -IH
+ elim (frees_inv_atom … H) -H #g #Hg #H destruct /2 width=1 by/
+ | #L #I * [ cases I -I #I [ | #V ] | #i ] #HG #HL #HT #f #H destruct
+ [ elim (frees_inv_unit … H) -H #g #Hg #H destruct /2 width=1 by/
+ | elim (frees_inv_pair … H) -H #g #Hg #H destruct
+ /4 width=2 by fqu_fqup, fqu_lref_O/
+ | elim (frees_inv_lref … H) -H #g #Hg #H destruct
+ /4 width=2 by fqu_fqup/
+ ]
+ ]
+| #l #HG #HL #HT #f #H destruct -IH
+ lapply (frees_inv_gref … H) -H /2 width=1 by/
+| #p #I #V #T #HG #HL #HT #f #H destruct
+ elim (frees_inv_bind_void … H) -H /3 width=7 by/
+| #I #V #T #HG #HL #HT #f #H destruct
+ elim (frees_inv_flat … H) -H /3 width=7 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 "static_2/static/frees.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Main inversion lemmas ****************************************************)
+
+theorem frees_mono: ∀f1,L,T. L ⊢ 𝐅*⦃T⦄ ≘ f1 → ∀f2. L ⊢ 𝐅*⦃T⦄ ≘ f2 → f1 ≡ f2.
+#f1 #L #T #H elim H -f1 -L -T
+[ /3 width=3 by frees_inv_sort, isid_inv_eq_repl/
+| #f1 #i #Hf1 #g2 #H
+ elim (frees_inv_atom … H) -H #f2 #Hf2 #H destruct
+ /4 width=5 by isid_inv_eq_repl, pushs_eq_repl, eq_next/
+| #f1 #I #L #V #_ #IH #g2 #H elim (frees_inv_pair … H) -H
+ #f2 #Hf2 #H destruct /3 width=5 by eq_next/
+| #f1 #I #L #Hf1 #g2 #H elim (frees_inv_unit … H) -H
+ #f2 #Hf2 #H destruct /3 width=5 by isid_inv_eq_repl, eq_next/
+| #f1 #I #L #i #_ #IH #g2 #H elim (frees_inv_lref … H) -H
+ #f2 #Hf2 #H destruct /3 width=5 by eq_push/
+| /3 width=3 by frees_inv_gref, isid_inv_eq_repl/
+| #f1V #f1T #f1 #p #I #L #V #T #_ #_ #Hf1 #IHV #IHT #f2 #H elim (frees_inv_bind … H) -H
+ #f2V #f2T #HV #HT #Hf2 @(sor_mono … Hf1) -Hf1
+ /5 width=3 by sor_eq_repl_fwd2, sor_eq_repl_fwd1, tl_eq_repl/ (**) (* full auto too slow *)
+| #f1V #f1T #f1 #I #L #V #T #_ #_ #Hf1 #IHV #IHT #f2 #H elim (frees_inv_flat … H) -H
+ #f2V #f2T #HV #HT #Hf2 @(sor_mono … Hf1) -Hf1
+ /4 width=3 by sor_eq_repl_fwd2, sor_eq_repl_fwd1/ (**) (* 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 "ground_2/relocation/rtmap_id.ma".
+include "static_2/notation/relations/subseteq_4.ma".
+include "static_2/syntax/lveq.ma".
+include "static_2/static/frees.ma".
+
+(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
+
+definition fsle: bi_relation lenv term ≝ λL1,T1,L2,T2.
+ ∃∃n1,n2,f1,f2. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 &
+ L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
+
+interpretation "free variables inclusion (restricted closure)"
+ 'SubSetEq L1 T1 L2 T2 = (fsle L1 T1 L2 T2).
+
+interpretation "free variables inclusion (term)"
+ 'subseteq T1 T2 = (fsle LAtom T1 LAtom T2).
+
+(* Basic properties *********************************************************)
+
+lemma fsle_sort: ∀L,s1,s2. ⦃L, ⋆s1⦄ ⊆ ⦃L, ⋆s2⦄.
+/3 width=8 by frees_sort, sle_refl, ex4_4_intro/ qed.
+
+lemma fsle_gref: ∀L,l1,l2. ⦃L, §l1⦄ ⊆ ⦃L, §l2⦄.
+/3 width=8 by frees_gref, sle_refl, ex4_4_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 "static_2/static/frees_drops.ma".
+include "static_2/static/fsle_length.ma".
+
+(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
+
+(* Advanced properties ******************************************************)
+
+lemma fsle_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≘ U1 → ∀L1,L2. |L2| ≤ |L1| →
+ ∀T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ⦃L1.ⓧ, U1⦄ ⊆ ⦃L2, T2⦄.
+#T1 #U1 #HTU1 #L1 #L2 #H1L #T2
+* #n #m #f #g #Hf #Hg #H2L #Hfg
+lapply (lveq_length_fwd_dx … H2L ?) // -H1L #H destruct
+lapply (frees_lifts_SO (Ⓣ) (L1.ⓧ) … HTU1 … Hf)
+[ /3 width=4 by drops_refl, drops_drop/ ] -T1 #Hf
+@(ex4_4_intro … Hf Hg) /2 width=4 by lveq_void_sn/ (**) (* explict constructor *)
+qed-.
+
+lemma fsle_lifts_SO_sn: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ →
+ ∀W1. ⬆*[1] V1 ≘ W1 → ∀I1,I2. ⦃K1.ⓘ{I1}, W1⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄.
+#K1 #K2 #HK #V1 #V2
+* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
+#W1 #HVW1 #I1 #I2
+elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
+/5 width=12 by frees_lifts_SO, frees_pair, drops_refl, drops_drop, lveq_bind, sle_weak, ex4_4_intro/
+qed.
+
+lemma fsle_lifts_SO: ∀K1,K2. |K1| = |K2| → ∀T1,T2. ⦃K1, T1⦄ ⊆ ⦃K2, T2⦄ →
+ ∀U1,U2. ⬆*[1] T1 ≘ U1 → ⬆*[1] T2 ≘ U2 →
+ ∀I1,I2. ⦃K1.ⓘ{I1}, U1⦄ ⊆ ⦃K2.ⓘ{I2}, U2⦄.
+#K1 #K2 #HK #T1 #T2
+* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
+#U1 #U2 #HTU1 #HTU2 #I1 #I2
+elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
+/5 width=12 by frees_lifts_SO, drops_refl, drops_drop, lveq_bind, sle_push, ex4_4_intro/
+qed.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fsle_inv_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≘ U1 →
+ ∀I1,I2,L1,L2,V1,V2,U2. ⦃L1.ⓑ{I1}V1,U1⦄ ⊆ ⦃L2.ⓑ{I2}V2, U2⦄ →
+ ∀p. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.U2⦄.
+#T1 #U1 #HTU1 #I1 #I2 #L1 #L2 #V1 #V2 #U2
+* #n #m #f2 #g2 #Hf2 #Hg2 #HL #Hfg2 #p
+elim (lveq_inv_pair_pair … HL) -HL #HL #H1 #H2 destruct
+elim (frees_total L2 V2) #g1 #Hg1
+elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+lapply (frees_inv_lifts_SO (Ⓣ) … Hf2 … HTU1)
+[1,2: /3 width=4 by drops_refl, drops_drop/ ] -U1 #Hf2
+lapply (sor_inv_sle_dx … Hg) #H0g
+/5 width=10 by frees_bind, sle_tl, sle_trans, ex4_4_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 "static_2/static/frees_fqup.ma".
+include "static_2/static/fsle_length.ma".
+
+(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
+
+(* Advanced properties ******************************************************)
+
+lemma fsle_refl: bi_reflexive … fsle.
+#L #T
+elim (frees_total L T) #f #Hf
+/2 width=8 by sle_refl, ex4_4_intro/
+qed.
+
+lemma fsle_shift: ∀L1,L2. |L1| = |L2| →
+ ∀I,T1,T2,V. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I}V, T2⦄ →
+ ∀p. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V.T2⦄.
+#L1 #L2 #H1L #I #T1 #T2 #V
+* #n #m #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
+elim (lveq_inj_length … H2L) // -H1L #H1 #H2 destruct
+lapply (lveq_inv_bind … H2L) -H2L #HL
+elim (frees_total L2 V) #g1 #Hg1
+elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+lapply (sor_inv_sle_dx … Hg) #H0g
+/4 width=10 by frees_bind, lveq_void_sn, sle_tl, sle_trans, ex4_4_intro/
+qed.
+
+lemma fsle_bind_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
+ ∀p,I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
+#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #p #I #T2
+elim (frees_total (L2.ⓧ) T2) #g2 #Hg2
+elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+@(ex4_4_intro … g Hf1 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_bind_void, sor_inv_sle_sn, sor_tls, sle_trans/
+qed.
+
+lemma fsle_bind_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2.ⓧ, T2⦄ → |L1| ≤ |L2| →
+ ∀p,I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓑ{p,I}V2.T2⦄.
+#L1 #L2 #T1 #T2 * #n1 #x1 #f2 #g2 #Hf2 #Hg2 #H #Hfg2 #HL12 #p #I #V2
+elim (lveq_inv_void_dx_length … H HL12) -H -HL12 #m1 #HL12 #H1 #H2 destruct
+<tls_xn in Hfg2; #Hfg2
+elim (frees_total L2 V2) #g1 #Hg1
+elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+@(ex4_4_intro … g Hf2 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_bind_void, sor_inv_sle_dx, sor_tls, sle_trans/
+qed.
+
+lemma fsle_flat_dx_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
+ ∀I,T2. ⦃L1, V1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
+#L1 #L2 #V1 #V2 * #n1 #m1 #f1 #g1 #Hf1 #Hg1 #HL12 #Hfg1 #I #T2
+elim (frees_total L2 T2) #g2 #Hg2
+elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
+@(ex4_4_intro … g Hf1 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_flat, sor_inv_sle_sn, sor_tls, sle_trans/
+qed.
+
+lemma fsle_flat_dx_dx: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+ ∀I,V2. ⦃L1, T1⦄ ⊆ ⦃L2, ⓕ{I}V2.T2⦄.
+#L1 #L2 #T1 #T2 * #n1 #m1 #f2 #g2 #Hf2 #Hg2 #HL12 #Hfg2 #I #V2
+elim (frees_total L2 V2) #g1 #Hg1
+elim (sor_isfin_ex g1 g2) /2 width=3 by frees_fwd_isfin/ #g #Hg #_
+@(ex4_4_intro … g Hf2 … HL12) (**) (* full auto too slow *)
+/4 width=5 by frees_flat, sor_inv_sle_dx, sor_tls, sle_trans/
+qed.
+
+(* Advanced forward lemmas ***************************************************)
+
+lemma fsle_fwd_pair_sn: ∀I1,I2,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
+ ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄.
+#I1 #I2 #L1 #L2 #V1 #V2 #T1 #T2 *
+#n1 #n2 #f1 #f2 #Hf1 #Hf2 #HL12 #Hf12
+elim (lveq_inv_pair_pair … HL12) -HL12 #HL12 #H1 #H2 destruct
+elim (frees_total (L1.ⓧ) T1) #g1 #Hg1
+lapply (lsubr_lsubf … Hg1 … Hf1) -Hf1 /2 width=1 by lsubr_unit/ #Hfg1
+/5 width=10 by lsubf_fwd_sle, lveq_bind, sle_trans, ex4_4_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 "static_2/syntax/lveq_lveq.ma".
+include "static_2/static/fsle_fqup.ma".
+
+(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma fsle_frees_trans: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+ ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
+ ∃∃n1,n2,f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 &
+ L1 ≋ⓧ*[n1, n2] L2 & ⫱*[n1]f1 ⊆ ⫱*[n2]f2.
+#L1 #L2 #T1 #T2 * #n1 #n2 #f1 #g2 #Hf1 #Hg2 #HL #Hn #f2 #Hf2
+lapply (frees_mono … Hg2 … Hf2) -Hg2 -Hf2 #Hgf2
+lapply (tls_eq_repl n2 … Hgf2) -Hgf2 #Hgf2
+lapply (sle_eq_repl_back2 … Hn … Hgf2) -g2
+/2 width=6 by ex3_3_intro/
+qed-.
+
+lemma fsle_frees_trans_eq: ∀L1,L2. |L1| = |L2| →
+ ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
+ ∃∃f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 & f1 ⊆ f2.
+#L1 #L2 #H1L #T1 #T2 #H2L #f2 #Hf2
+elim (fsle_frees_trans … H2L … Hf2) -T2 #n1 #n2 #f1 #Hf1 #H2L #Hf12
+elim (lveq_inj_length … H2L) // -L2 #H1 #H2 destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma fsle_inv_frees_eq: ∀L1,L2. |L1| = |L2| →
+ ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+ ∀f1. L1 ⊢ 𝐅*⦃T1⦄ ≘ f1 → ∀f2. L2 ⊢ 𝐅*⦃T2⦄ ≘ f2 →
+ f1 ⊆ f2.
+#L1 #L2 #H1L #T1 #T2 #H2L #f1 #Hf1 #f2 #Hf2
+elim (fsle_frees_trans_eq … H2L … Hf2) // -L2 -T2
+/3 width=6 by frees_mono, sle_eq_repl_back1/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem fsle_trans_sn: ∀L1,L2,T1,T. ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
+ ∀T2. ⦃L2, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+#L1 #L2 #T1 #T
+* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
+#T2
+* #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
+lapply (frees_mono … Hf0 … Hg0) -Hf0 -Hg0 #Hfg0
+elim (lveq_inj_length … Hn) // -Hn #H1 #H2 destruct
+lapply (sle_eq_repl_back1 … Hf … Hfg0) -f0
+/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
+qed-.
+
+theorem fsle_trans_dx: ∀L1,T1,T. ⦃L1, T1⦄ ⊆ ⦃L1, T⦄ →
+ ∀L2,T2. ⦃L1, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+#L1 #T1 #T
+* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
+#L2 #T2
+* #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
+lapply (frees_mono … Hg0 … Hf0) -Hg0 -Hf0 #Hgf0
+elim (lveq_inj_length … Hm) // -Hm #H1 #H2 destruct
+lapply (sle_eq_repl_back2 … Hg … Hgf0) -g0
+/4 width=10 by sle_tls, sle_trans, ex4_4_intro/
+qed-.
+
+theorem fsle_trans_rc: ∀L1,L,T1,T. |L1| = |L| → ⦃L1, T1⦄ ⊆ ⦃L, T⦄ →
+ ∀L2,T2. |L| = |L2| → ⦃L, T⦄ ⊆ ⦃L2, T2⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄.
+#L1 #L #T1 #T #HL1
+* #m1 #m0 #g1 #g0 #Hg1 #Hg0 #Hm #Hg
+#L2 #T2 #HL2
+* #n0 #n2 #f0 #f2 #Hf0 #Hf2 #Hn #Hf
+lapply (frees_mono … Hg0 … Hf0) -Hg0 -Hf0 #Hgf0
+elim (lveq_inj_length … Hm) // -Hm #H1 #H2 destruct
+elim (lveq_inj_length … Hn) // -Hn #H1 #H2 destruct
+lapply (sle_eq_repl_back2 … Hg … Hgf0) -g0
+/3 width=10 by lveq_length_eq, sle_trans, ex4_4_intro/
+qed-.
+
+theorem fsle_bind_sn_ge: ∀L1,L2. |L2| ≤ |L1| →
+ ∀V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2, T⦄ →
+ ∀p,I. ⦃L1, ⓑ{p,I}V1.T1⦄ ⊆ ⦃L2, T⦄.
+#L1 #L2 #HL #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #p #I
+elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
+elim (lveq_inj_void_sn_ge … H1n1 … H1n2) -H1n2 // #H1 #H2 #H3 destruct
+elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
+<tls_xn in H2n2; #H2n2
+/4 width=12 by frees_bind_void, sor_inv_sle, sor_tls, ex4_4_intro/
+qed.
+
+theorem fsle_flat_sn: ∀L1,L2,V1,T1,T. ⦃L1, V1⦄ ⊆ ⦃L2, T⦄ → ⦃L1, T1⦄ ⊆ ⦃L2, T⦄ →
+ ∀I. ⦃L1, ⓕ{I}V1.T1⦄ ⊆ ⦃L2, T⦄.
+#L1 #L2 #V1 #T1 #T * #n1 #x #f1 #g #Hf1 #Hg #H1n1 #H2n1 #H #I
+elim (fsle_frees_trans … H … Hg) -H #n2 #n #f2 #Hf2 #H1n2 #H2n2
+elim (lveq_inj … H1n1 … H1n2) -H1n2 #H1 #H2 destruct
+elim (sor_isfin_ex f1 f2) /2 width=3 by frees_fwd_isfin/ #f #Hf #_
+/4 width=12 by frees_flat, sor_inv_sle, sor_tls, ex4_4_intro/
+qed.
+
+theorem fsle_bind_eq: ∀L1,L2. |L1| = |L2| → ∀V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
+ ∀I2,T1,T2. ⦃L1.ⓧ, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
+ ∀p,I1. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
+#L1 #L2 #HL #V1 #V2
+* #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I2 #T1 #T2
+* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p #I1
+elim (lveq_inj_length … H1L) // #H1 #H2 destruct
+elim (lveq_inj_length … H2L) // -HL -H2L #H1 #H2 destruct
+elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
+elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+/4 width=15 by frees_bind_void, frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
+qed.
+
+theorem fsle_bind: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
+ ∀I1,I2,T1,T2. ⦃L1.ⓑ{I1}V1, T1⦄ ⊆ ⦃L2.ⓑ{I2}V2, T2⦄ →
+ ∀p. ⦃L1, ⓑ{p,I1}V1.T1⦄ ⊆ ⦃L2, ⓑ{p,I2}V2.T2⦄.
+#L1 #L2 #V1 #V2
+* #n1 #m1 #f1 #g1 #Hf1 #Hg1 #H1L #Hfg1 #I1 #I2 #T1 #T2
+* #n2 #m2 #f2 #g2 #Hf2 #Hg2 #H2L #Hfg2 #p
+elim (lveq_inv_pair_pair … H2L) -H2L #H2L #H1 #H2 destruct
+elim (lveq_inj … H2L … H1L) -H1L #H1 #H2 destruct
+elim (sor_isfin_ex f1 (⫱f2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #f #Hf #_
+elim (sor_isfin_ex g1 (⫱g2)) /3 width=3 by frees_fwd_isfin, isfin_tl/ #g #Hg #_
+/4 width=15 by frees_bind, monotonic_sle_sor, sle_tl, ex4_4_intro/
+qed.
+
+theorem fsle_flat: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ⊆ ⦃L2, V2⦄ →
+ ∀T1,T2. ⦃L1, T1⦄ ⊆ ⦃L2, T2⦄ →
+ ∀I1,I2. ⦃L1, ⓕ{I1}V1.T1⦄ ⊆ ⦃L2, ⓕ{I2}V2.T2⦄.
+/3 width=1 by fsle_flat_sn, fsle_flat_dx_dx, fsle_flat_dx_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 "static_2/syntax/lveq_length.ma".
+include "static_2/static/fsle.ma".
+
+(* FREE VARIABLES INCLUSION FOR RESTRICTED CLOSURES *************************)
+
+(* Properties with length for local environments ****************************)
+
+lemma fsle_sort_bi: ∀L1,L2,s1,s2. |L1| = |L2| → ⦃L1, ⋆s1⦄ ⊆ ⦃L2, ⋆s2⦄.
+/3 width=8 by lveq_length_eq, frees_sort, sle_refl, ex4_4_intro/ qed.
+
+lemma fsle_gref_bi: ∀L1,L2,l1,l2. |L1| = |L2| → ⦃L1, §l1⦄ ⊆ ⦃L2, §l2⦄.
+/3 width=8 by lveq_length_eq, frees_gref, sle_refl, ex4_4_intro/ qed.
+
+lemma fsle_pair_bi: ∀K1,K2. |K1| = |K2| → ∀V1,V2. ⦃K1, V1⦄ ⊆ ⦃K2, V2⦄ →
+ ∀I1,I2. ⦃K1.ⓑ{I1}V1, #O⦄ ⊆ ⦃K2.ⓑ{I2}V2, #O⦄.
+#K1 #K2 #HK #V1 #V2
+* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
+#I1 #I2
+elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
+/3 width=12 by frees_pair, lveq_bind, sle_next, ex4_4_intro/
+qed.
+
+lemma fsle_unit_bi: ∀K1,K2. |K1| = |K2| →
+ ∀I1,I2. ⦃K1.ⓤ{I1}, #O⦄ ⊆ ⦃K2.ⓤ{I2}, #O⦄.
+/3 width=8 by frees_unit, lveq_length_eq, sle_refl, ex4_4_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 "static_2/syntax/genv.ma".
+include "static_2/relocation/drops_vector.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+definition nf ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+ λG,L,T. NF … (RR G L) RS T.
+
+definition candidate: Type[0] ≝ relation3 genv lenv term.
+
+definition CP0 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+ ∀G. d_liftable1 (nf RR RS G).
+
+definition CP1 ≝ λRR:relation4 genv lenv term term. λRS:relation term.
+ ∀G,L. ∃s. NF … (RR G L) RS (⋆s).
+
+definition CP2 ≝ λRP:candidate. ∀G. d_liftable1 (RP G).
+
+definition CP3 ≝ λRP:candidate.
+ ∀G,L,T,s. RP G L (ⓐ⋆s.T) → RP G L T.
+
+(* requirements for generic computation properties *)
+(* Basic_1: includes: nf2_lift1 *)
+(* Basic_2A1: includes: gcp0_lifts *)
+(* Basic_2A1: includes: gcp2_lifts *)
+record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate) : Prop ≝
+{ cp0: CP0 RR RS;
+ cp1: CP1 RR RS;
+ cp2: CP2 RP;
+ cp3: CP3 RP
+}.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was only: sns3_lifts1 *)
+(* Basic_2A1: was: gcp2_lifts_all *)
+lemma gcp2_all: ∀RR,RS,RP. gcp RR RS RP → ∀G. d_liftable1_all (RP G).
+/3 width=7 by cp2, d1_liftable_liftable_all/ 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 "static_2/static/aaa_aaa.ma".
+include "static_2/static/lsubc_drops.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: sc3_arity_csubc *)
+theorem acr_aaa_csubc_lifts: ∀RR,RS,RP.
+ gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L1,T,A. ⦃G, L1⦄ ⊢ T ⁝ A → ∀b,f,L0. ⬇*[b, f] L0 ≘ L1 →
+ ∀T0. ⬆*[f] T ≘ T0 → ∀L2. G ⊢ L2 ⫃[RP] L0 →
+ ⦃G, L2, T0⦄ ϵ[RP] 〚A〛.
+#RR #RS #RP #H1RP #H2RP #G #L1 #T @(fqup_wf_ind_eq (Ⓣ) … G L1 T) -G -L1 -T
+#Z #Y #X #IH #G #L1 * [ * | * [ #p ] * ]
+[ #s #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct -IH
+ lapply (aaa_inv_sort … HA) -HA #H destruct
+ >(lifts_inv_sort1 … H0) -H0
+ lapply (acr_gcr … H1RP H2RP (⓪)) #HAtom
+ lapply (s4 … HAtom G L2 (Ⓔ)) /2 width=1 by/
+| #i #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
+ elim (aaa_inv_lref_drops … HA) -HA #I #K1 #V1 #HLK1 #HKV1
+ elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
+ lapply (acr_gcr … H1RP H2RP A) #HA
+ lapply (drops_trans … HL01 … HLK1 ??) -HL01 [3: |*: // ] #H
+ elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK0 #HY
+ lapply (drops_tls_at … Hf … HY) -Hf -HY #HY
+ elim (drops_inv_skip2 … HY) -HY #Z #K0 #HK01 #HZ #H destruct
+ elim (liftsb_inv_pair_sn … HZ) -HZ #V0 #HV10 #H destruct
+ elim (lifts_total V0 (𝐔❴↑j❵)) #V #HV0
+ elim (lsubc_drops_trans_isuni … HL20 … HLK0) -HL20 -HLK0 // #Y #HLK2 #H
+ elim (lsubc_inv_bind2 … H) -H *
+ [ #K2 #HK20 #H destruct
+ lapply (drops_isuni_fwd_drop2 … HLK2) // #HLK2b
+ lapply (s5 … HA ? G ? ? (Ⓔ) … HV0 ?) -HA
+ /4 width=11 by acr_lifts, fqup_lref, drops_inv_gen/
+ | #K2 #V2 #W2 #B #HKV2 #HK2V0 #HKV0B #_ #H1 #H2 destruct -IH -HLK1
+ lapply (drops_isuni_fwd_drop2 … HLK2) // #HLK2b
+ lapply (aaa_lifts … HKV1 … HK01 … HV10) -HKV1 -HK01 -HV10 #HKV0A
+ lapply (aaa_mono … HKV0B … HKV0A) #H destruct -HKV0B -HKV0A
+ elim (lifts_total V2 (𝐔❴↑j❵)) #V3 #HV23
+ lapply (s5 … HA … G … (Ⓔ) … (ⓝW2.V2) (ⓝV.V3) ????)
+ [3: |*: /2 width=9 by drops_inv_gen, lifts_flat/ ] -HLK2
+ lapply (s7 … HA G L2 (Ⓔ)) -HA /3 width=7 by acr_lifts/
+ ]
+| #l #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct -IH
+ elim (aaa_inv_gref … HA)
+| #V #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
+ elim (aaa_inv_abbr … HA) -HA #B #HV #HT
+ elim (lifts_inv_bind1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
+ lapply (acr_gcr … H1RP H2RP A) #HA
+ lapply (acr_gcr … H1RP H2RP B) #HB
+ lapply (s1 … HB) -HB #HB
+ lapply (s6 … HA G L2 (Ⓔ) (Ⓔ)) /5 width=10 by lsubc_bind, liftsv_nil, drops_skip, ext2_pair/
+| #W #T #HG #HL #HT #Z0 #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
+ elim (aaa_inv_abst … HA) -HA #B #A #HW #HT #H destruct
+ elim (lifts_inv_bind1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
+ @(acr_abst … H1RP H2RP) /2 width=10 by/
+ #b3 #f3 #L3 #V3 #W3 #T3 #HL32 #HW03 #HT03 #H1B #H2B
+ elim (drops_lsubc_trans … H1RP … HL32 … HL20) -L2 #L2 #HL32 #HL20
+ lapply (aaa_lifts … HW … (f3∘f) L2 … W3 ?) -HW
+ [4: |*: /2 width=8 by drops_trans, lifts_trans/ ] #HW3
+ @(IH … ((⫯f3)∘⫯f) … (L2. ⓛW3)) -IH
+ /4 width=12 by lsubc_beta, drops_trans, drops_skip, lifts_trans, ext2_pair/
+| #V #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
+ elim (aaa_inv_appl … HA) -HA #B #HV #HT
+ elim (lifts_inv_flat1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
+ lapply (IH … HT … HL01 … HT0 … HL20) -HT -HT0
+ /3 width=10 by drops_refl, lifts_refl/
+| #W #T #HG #HL #HT #A #HA #b #f #L0 #HL01 #X0 #H0 #L2 #HL20 destruct
+ elim (aaa_inv_cast … HA) -HA #HW #HT
+ elim (lifts_inv_flat1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
+ lapply (acr_gcr … H1RP H2RP A) #HA
+ lapply (s7 … HA G L2 (Ⓔ)) /3 width=10 by/
+]
+qed.
+
+(* Basic_1: was: sc3_arity *)
+lemma acr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → ⦃G, L, T⦄ ϵ[RP] 〚A〛.
+/3 width=9 by drops_refl, lifts_refl, acr_aaa_csubc_lifts/ qed.
+
+lemma gcr_aaa: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L,T,A. ⦃G, L⦄ ⊢ T ⁝ A → RP G L T.
+#RR #RS #RP #H1RP #H2RP #G #L #T #A #HT
+lapply (acr_gcr … H1RP H2RP A) #HA
+@(s1 … HA) /2 width=4 by acr_aaa/
+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 "static_2/notation/relations/ineint_5.ma".
+include "static_2/syntax/aarity.ma".
+include "static_2/relocation/lifts_simple.ma".
+include "static_2/relocation/lifts_lifts_vector.ma".
+include "static_2/relocation/drops_drops.ma".
+include "static_2/static/gcp.ma".
+
+(* GENERIC COMPUTATION PROPERTIES *******************************************)
+
+(* Note: this is Girard's CR1 *)
+definition S1 ≝ λRP,C:candidate.
+ ∀G,L,T. C G L T → RP G L T.
+
+(* Note: this is Tait's iii, or Girard's CR4 *)
+definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate.
+ ∀G,L,Vs. all … (RP G L) Vs →
+ ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T).
+
+(* Note: this generalizes Tait's ii *)
+definition S3 ≝ λC:candidate.
+ ∀a,G,L,Vs,V,T,W.
+ C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T).
+
+definition S4 ≝ λRP,C:candidate.
+ ∀G,L,Vs. all … (RP G L) Vs → ∀s. C G L (ⒶVs.⋆s).
+
+definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i.
+ C G L (ⒶVs.V2) → ⬆*[↑i] V1 ≘ V2 →
+ ⬇*[i] L ≘ K.ⓑ{I}V1 → C G L (ⒶVs.#i).
+
+definition S6 ≝ λRP,C:candidate.
+ ∀G,L,V1b,V2b. ⬆*[1] V1b ≘ V2b →
+ ∀a,V,T. C G (L.ⓓV) (ⒶV2b.T) → RP G L V → C G L (ⒶV1b.ⓓ{a}V.T).
+
+definition S7 ≝ λC:candidate.
+ ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T).
+
+(* requirements for the generic reducibility candidate *)
+record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝
+{ s1: S1 RP C;
+ s2: S2 RR RS RP C;
+ s3: S3 C;
+ s4: S4 RP C;
+ s5: S5 C;
+ s6: S6 RP C;
+ s7: S7 C
+}.
+
+(* the functional construction for candidates *)
+definition cfun: candidate → candidate → candidate ≝
+ λC1,C2,G,K,T. ∀f,L,W,U.
+ ⬇*[Ⓕ, f] L ≘ K → ⬆*[f] T ≘ U → C1 G L W → C2 G L (ⓐW.U).
+
+(* the reducibility candidate associated to an atomic arity *)
+rec definition acr (RP:candidate) (A:aarity) on A: candidate ≝
+match A with
+[ AAtom ⇒ RP
+| APair B A ⇒ cfun (acr RP B) (acr RP A)
+].
+
+interpretation
+ "reducibility candidate of an atomic arity (abstract)"
+ 'InEInt RP G L T A = (acr RP A G L T).
+
+(* Basic properties *********************************************************)
+
+(* Note: this requires Ⓕ-slicing in cfun since b is unknown in d_liftable_1 *)
+(* Note: this requires multiple relocation *)
+(* Basic 1: includes: sc3_lift *)
+(* Basic 2A1: includes: gcr_lift *)
+(* Basic 2A1: note: gcr_lift should be acr_lift *)
+(* Basic_1: was: sc3_lift1 *)
+(* Basic 2A1: was: gcr_lifts *)
+(* Basic 2A1: note: gcr_lifts should be acr_lifts *)
+lemma acr_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftable1 (acr RP A G).
+#RR #RS #RP #H #A #G elim A -A
+[ /2 width=7 by cp2/
+| #B #A #HB #HA #K #T #HKT #b #f #L #HLK #U #HTU #f0 #L0 #W #U0 #HL0 #HU0 #HW
+ lapply (drops_trans … HL0 … HLK ??) [3:|*: // ] -L #HL0K
+ lapply (lifts_trans … HTU … HU0 ??) [3:|*: // ] -U #HTU0
+ /2 width=3 by/ (**) (* full auto fails *)
+]
+qed-.
+
+(* Basic_1: was:
+ sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast
+*)
+lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀A. gcr RR RS RP (acr RP A).
+#RR #RS #RP #H1RP #H2RP #A elim A -A //
+#B #A #IHB #IHA @mk_gcr
+[ #G #L #T #H
+ elim (cp1 … H1RP G L) #s #HK
+ lapply (s2 … IHB G L (Ⓔ) … HK) // #HB
+ lapply (H (𝐈𝐝) L (⋆s) T ? ? ?) -H
+ /3 width=6 by s1, cp3, drops_refl, lifts_refl/
+| #G #L #Vs #HVs #T #H1T #H2T #f #L0 #V0 #X #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V0s #T0 #HV0s #HT0 #H destruct
+ lapply (s1 … IHB … HB) #HV0
+ @(s2 … IHA … (V0⨮V0s)) /3 width=13 by cp0, gcp2_all, lifts_simple_dx, conj/
+| #p #G #L #Vs #U #T #W #HA #f #L0 #V0 #X #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
+ elim (lifts_inv_flat1 … H0) -H0 #U0 #X #HU0 #HX #H destruct
+ elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct
+ @(s3 … IHA … (V0⨮V0s)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/
+| #G #L #Vs #HVs #s #f #L0 #V0 #X #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
+ >(lifts_inv_sort1 … H0) -X0
+ lapply (s1 … IHB … HB) #HV0
+ @(s4 … IHA … (V0⨮V0s)) /3 width=7 by gcp2_all, conj/
+| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #f #L0 #V0 #X #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
+ elim (lifts_inv_lref1 … H0) -H0 #j #Hf #H destruct
+ lapply (drops_trans … HL0 … HLK ??) [3: |*: // ] -HLK #H
+ elim (drops_split_trans … H) -H [ |*: /2 width=6 by after_uni_dx/ ] #Y #HLK0 #HY
+ lapply (drops_tls_at … Hf … HY) -HY #HY
+ elim (drops_inv_skip2 … HY) -HY #Z #K0 #HK0 #HZ #H destruct
+ elim (liftsb_inv_pair_sn … HZ) -HZ #W1 #HVW1 #H destruct
+ elim (lifts_total W1 (𝐔❴↑j❵)) #W2 #HW12
+ lapply (lifts_trans … HVW1 … HW12 ??) -HVW1 [3: |*: // ] #H
+ lapply (lifts_conf … HV12 … H f ?) -V1 [ /2 width=3 by after_uni_succ_sn/ ] #HVW2
+ @(s5 … IHA … (V0⨮V0s) … HW12) /3 width=4 by drops_inv_gen, lifts_applv/
+| #G #L #V1s #V2s #HV12s #p #V #T #HA #HV #f #L0 #V10 #X #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V10s #X0 #HV10s #H0 #H destruct
+ elim (lifts_inv_bind1 … H0) -H0 #V0 #T0 #HV0 #HT0 #H destruct
+ elim (lifts_total V10 (𝐔❴1❵)) #V20 #HV120
+ elim (liftsv_total (𝐔❴1❵) V10s) #V20s #HV120s
+ @(s6 … IHA … (V10⨮V10s) (V20⨮V20s)) /3 width=7 by cp2, liftsv_cons/
+ @(HA … (⫯f)) /3 width=2 by drops_skip, ext2_pair/
+ [ @lifts_applv //
+ lapply (liftsv_trans … HV10s … HV120s ??) -V10s [3: |*: // ] #H
+ elim (liftsv_split_trans … H (𝐔❴1❵) (⫯f)) /2 width=1 by after_uni_one_sn/ #V10s #HV10s #HV120s
+ >(liftsv_mono … HV12s … HV10s) -V1s //
+ | @(acr_lifts … H1RP … HB … HV120) /3 width=2 by drops_refl, drops_drop/
+ ]
+| #G #L #Vs #T #W #HA #HW #f #L0 #V0 #X #HL0 #H #HB
+ elim (lifts_inv_applv1 … H) -H #V0s #X0 #HV0s #H0 #H destruct
+ elim (lifts_inv_flat1 … H0) -H0 #W0 #T0 #HW0 #HT0 #H destruct
+ @(s7 … IHA … (V0⨮V0s)) /3 width=5 by lifts_applv/
+]
+qed.
+
+lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀p,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → (
+ ∀b,f,L0,V0,W0,T0. ⬇*[b, f] L0 ≘ L → ⬆*[f] W ≘ W0 → ⬆*[⫯f] T ≘ T0 →
+ ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛
+ ) →
+ ⦃G, L, ⓛ{p}W.T⦄ ϵ[RP] 〚②B.A〛.
+#RR #RS #RP #H1RP #H2RP #p #G #L #W #T #A #B #HW #HA #f #L0 #V0 #X #HL0 #H #HB
+lapply (acr_gcr … H1RP H2RP A) #HCA
+lapply (acr_gcr … H1RP H2RP B) #HCB
+elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct
+lapply (acr_lifts … H1RP … HW … HL0 … HW0) -HW #HW0
+lapply (s3 … HCA … p G L0 (Ⓔ)) #H @H -H
+lapply (s6 … HCA G L0 (Ⓔ) (Ⓔ) ?) // #H @H -H
+[ @(HA … HL0) //
+| lapply (s1 … HCB) -HCB #HCB
+ lapply (s7 … H2RP G L0 (Ⓔ)) /3 width=1 by/
+]
+qed.
+
+(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *)
+(* Basic_1: removed local theorems 1: sc3_sn3_abst *)
--- /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 "static_2/notation/relations/lrsubeqa_3.ma".
+include "static_2/static/aaa.ma".
+
+(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
+
+inductive lsuba (G:genv): relation lenv ≝
+| lsuba_atom: lsuba G (⋆) (⋆)
+| lsuba_bind: ∀I,L1,L2. lsuba G L1 L2 → lsuba G (L1.ⓘ{I}) (L2.ⓘ{I})
+| lsuba_beta: ∀L1,L2,W,V,A. ⦃G, L1⦄ ⊢ ⓝW.V ⁝ A → ⦃G, L2⦄ ⊢ W ⁝ A →
+ lsuba G L1 L2 → lsuba G (L1.ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+ "local environment refinement (atomic arity assignment)"
+ 'LRSubEqA G L1 L2 = (lsuba G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuba_inv_atom1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 = ⋆ → L2 = ⋆.
+#G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #_ #H destruct
+| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsuba_inv_atom1: ∀G,L2. G ⊢ ⋆ ⫃⁝ L2 → L2 = ⋆.
+/2 width=4 by lsuba_inv_atom1_aux/ qed-.
+
+fact lsuba_inv_bind1_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
+ (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓘ{I}) ∨
+ ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃⁝ K2 & I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
+#G #L1 #L2 * -L1 -L2
+[ #J #K1 #H destruct
+| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K1 #H destruct /3 width=9 by ex5_4_intro, or_intror/
+]
+qed-.
+
+lemma lsuba_inv_bind1: ∀I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃⁝ L2 →
+ (∃∃K2. G ⊢ K1 ⫃⁝ K2 & L2 = K2.ⓘ{I}) ∨
+ ∃∃K2,W,V,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+ I = BPair Abbr (ⓝW.V) & L2 = K2.ⓛW.
+/2 width=3 by lsuba_inv_bind1_aux/ qed-.
+
+fact lsuba_inv_atom2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L2 = ⋆ → L1 = ⋆.
+#G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #_ #H destruct
+| #L1 #L2 #W #V #A #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubc_inv_atom2: ∀G,L1. G ⊢ L1 ⫃⁝ ⋆ → L1 = ⋆.
+/2 width=4 by lsuba_inv_atom2_aux/ qed-.
+
+fact lsuba_inv_bind2_aux: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
+ (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
+ ∃∃K1,V,W, A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃⁝ K2 & I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+#G #L1 #L2 * -L1 -L2
+[ #J #K2 #H destruct
+| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #W #V #A #HV #HW #HL12 #J #K2 #H destruct /3 width=9 by ex5_4_intro, or_intror/
+]
+qed-.
+
+lemma lsuba_inv_bind2: ∀I,G,L1,K2. G ⊢ L1 ⫃⁝ K2.ⓘ{I} →
+ (∃∃K1. G ⊢ K1 ⫃⁝ K2 & L1 = K1.ⓘ{I}) ∨
+ ∃∃K1,V,W,A. ⦃G, K1⦄ ⊢ ⓝW.V ⁝ A & ⦃G, K2⦄ ⊢ W ⁝ A & G ⊢ K1 ⫃⁝ K2 &
+ I = BPair Abst W & L1 = K1.ⓓⓝW.V.
+/2 width=3 by lsuba_inv_bind2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+lemma lsuba_refl: ∀G,L. G ⊢ L ⫃⁝ L.
+#G #L elim L -L /2 width=1 by lsuba_atom, lsuba_bind/
+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 "static_2/static/aaa_aaa.ma".
+include "static_2/static/lsuba.ma".
+
+(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
+
+(* Properties with atomic arity assignment **********************************)
+
+lemma lsuba_aaa_conf: ∀G,L1,V,A. ⦃G, L1⦄ ⊢ V ⁝ A →
+ ∀L2. G ⊢ L1 ⫃⁝ L2 → ⦃G, L2⦄ ⊢ V ⁝ A.
+#G #L1 #V #A #H elim H -G -L1 -V -A
+[ //
+| #I #G #L1 #V #A #HA #IH #L2 #H
+ elim (lsuba_inv_bind1 … H) -H * /3 width=1 by aaa_zero/
+ #L0 #W0 #V0 #A0 #HV0 #HW0 #HL10 #H1 #H2 destruct
+ lapply (aaa_mono … HV0 … HA) #H destruct -V0 -L1 /2 width=1 by aaa_zero/
+| #I #G #K1 #A #i #_ #IH #L2 #H
+ elim (lsuba_inv_bind1 … H) -H * /3 width=1 by aaa_lref/
+| /4 width=2 by lsuba_bind, aaa_abbr/
+| /4 width=1 by lsuba_bind, aaa_abst/
+| /3 width=3 by aaa_appl/
+| /3 width=1 by aaa_cast/
+]
+qed-.
+
+lemma lsuba_aaa_trans: ∀G,L2,V,A. ⦃G, L2⦄ ⊢ V ⁝ A →
+ ∀L1. G ⊢ L1 ⫃⁝ L2 → ⦃G, L1⦄ ⊢ V ⁝ A.
+#G #L2 #V #A #H elim H -G -L2 -V -A
+[ //
+| #I #G #L2 #V #A #HA #IH #L1 #H
+ elim (lsuba_inv_bind2 … H) -H * /3 width=1 by aaa_zero/
+ #L0 #V0 #W0 #A0 #HV0 #HW0 #HL02 #H1 #H2 destruct
+ lapply (aaa_mono … HW0 … HA) #H destruct -L2 /2 width=1 by aaa_zero/
+| #I #G #K2 #A #i #_ #IH #L1 #H
+ elim (lsuba_inv_bind2 … H) -H * /3 width=1 by aaa_lref/
+| /4 width=2 by lsuba_bind, aaa_abbr/
+| /4 width=1 by lsuba_bind, aaa_abst/
+| /3 width=3 by aaa_appl/
+| /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 "static_2/relocation/drops.ma".
+include "static_2/static/lsuba.ma".
+
+(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
+
+(* Properties with generic slicing for local environments *******************)
+
+(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
+(* Basic_2A1: includes: lsuba_drop_O1_conf *)
+lemma lsuba_drops_conf_isuni: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 →
+ ∀b,f,K1. 𝐔⦃f⦄ → ⬇*[b, f] L1 ≘ K1 →
+ ∃∃K2. G ⊢ K1 ⫃⁝ K2 & ⬇*[b, f] L2 ≘ K2.
+#G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #HL12 #IH #b #f #K1 #Hf #H
+ elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct -IH
+ /3 width=3 by lsuba_bind, drops_refl, ex2_intro/
+ | #g #Hg #HLK1 #H destruct -HL12
+ elim (IH … Hg HLK1) -L1 -Hg /3 width=3 by drops_drop, ex2_intro/
+ ]
+| #L1 #L2 #W #V #A #HV #HW #HL12 #IH #b #f #K1 #Hf #H
+ elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct -IH
+ /3 width=3 by drops_refl, lsuba_beta, ex2_intro/
+ | #g #Hg #HLK1 #H destruct -HL12
+ elim (IH … Hg HLK1) -L1 -Hg /3 width=3 by drops_drop, ex2_intro/
+ ]
+]
+qed-.
+
+(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
+(* Basic_2A1: includes: lsuba_drop_O1_trans *)
+lemma lsuba_drops_trans_isuni: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 →
+ ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2 →
+ ∃∃K1. G ⊢ K1 ⫃⁝ K2 & ⬇*[b, f] L1 ≘ K1.
+#G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #HL12 #IH #b #f #K2 #Hf #H
+ elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct -IH
+ /3 width=3 by lsuba_bind, drops_refl, ex2_intro/
+ | #g #Hg #HLK2 #H destruct -HL12
+ elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
+ ]
+| #L1 #L2 #W #V #A #HV #HW #HL12 #IH #b #f #K2 #Hf #H
+ elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct -IH
+ /3 width=3 by drops_refl, lsuba_beta, ex2_intro/
+ | #g #Hg #HLK2 #H destruct -HL12
+ elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, 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 "static_2/static/lsuba_aaa.ma".
+
+(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
+
+(* Main properties **********************************************************)
+
+theorem lsuba_trans: ∀G. Transitive … (lsuba G).
+#G #L1 #L #H elim H -L1 -L
+[ #X #H >(lsuba_inv_atom1 … H) -H //
+| #I #L1 #L #HL1 #IHL1 #Y #H
+ elim (lsuba_inv_bind1 … H) -H * #L2
+ [ #HL2 #H destruct /3 width=1 by lsuba_bind/
+ | #W #V #A #HV #HW #HL2 #H1 #H2 destruct
+ /3 width=3 by lsuba_beta, lsuba_aaa_trans/
+ ]
+| #L1 #L #W #V #A #HV #HW #HL1 #IHL1 #Y #H
+ elim (lsuba_inv_bind1 … H) -H * #L2
+ [ #HL2 #H destruct /3 width=5 by lsuba_beta, lsuba_aaa_conf/
+ | #W0 #V0 #A0 #_ #_ #_ #H destruct
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/static/lsubr.ma".
+include "static_2/static/lsuba.ma".
+
+(* RESTRICTED REFINEMENT FOR ATOMIC ARITY ASSIGNMENT ************************)
+
+(* Forward lemmas with restricted refinement for local environments *********)
+
+lemma lsuba_fwd_lsubr: ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → L1 ⫃ L2.
+#G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_beta/
+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 "static_2/notation/relations/lrsubeqc_4.ma".
+include "static_2/static/aaa.ma".
+include "static_2/static/gcp_cr.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+inductive lsubc (RP) (G): relation lenv ≝
+| lsubc_atom: lsubc RP G (⋆) (⋆)
+| lsubc_bind: ∀I,L1,L2. lsubc RP G L1 L2 → lsubc RP G (L1.ⓘ{I}) (L2.ⓘ{I})
+| lsubc_beta: ∀L1,L2,V,W,A. ⦃G, L1, V⦄ ϵ[RP] 〚A〛 → ⦃G, L1, W⦄ ϵ[RP] 〚A〛 → ⦃G, L2⦄ ⊢ W ⁝ A →
+ lsubc RP G L1 L2 → lsubc RP G (L1. ⓓⓝW.V) (L2.ⓛW)
+.
+
+interpretation
+ "local environment refinement (generic reducibility)"
+ 'LRSubEqC RP G L1 L2 = (lsubc RP G L1 L2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubc_inv_atom1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 = ⋆ → L2 = ⋆.
+#RP #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: was just: csubc_gen_sort_r *)
+lemma lsubc_inv_atom1: ∀RP,G,L2. G ⊢ ⋆ ⫃[RP] L2 → L2 = ⋆.
+/2 width=5 by lsubc_inv_atom1_aux/ qed-.
+
+fact lsubc_inv_bind1_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K1. L1 = K1.ⓘ{I} →
+ (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓘ{I}) ∨
+ ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
+ L2 = K2. ⓛW & I = BPair Abbr (ⓝW.V).
+#RP #G #L1 #L2 * -L1 -L2
+[ #I #K1 #H destruct
+| #J #L1 #L2 #HL12 #I #K1 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K1 #H destruct
+ /3 width=10 by ex6_4_intro, or_intror/
+]
+qed-.
+
+(* Basic_1: was: csubc_gen_head_r *)
+lemma lsubc_inv_bind1: ∀RP,I,G,K1,L2. G ⊢ K1.ⓘ{I} ⫃[RP] L2 →
+ (∃∃K2. G ⊢ K1 ⫃[RP] K2 & L2 = K2.ⓘ{I}) ∨
+ ∃∃K2,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
+ L2 = K2.ⓛW & I = BPair Abbr (ⓝW.V).
+/2 width=3 by lsubc_inv_bind1_aux/ qed-.
+
+fact lsubc_inv_atom2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L2 = ⋆ → L1 = ⋆.
+#RP #G #L1 #L2 * -L1 -L2
+[ //
+| #I #L1 #L2 #_ #H destruct
+| #L1 #L2 #V #W #A #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+(* Basic_1: was just: csubc_gen_sort_l *)
+lemma lsubc_inv_atom2: ∀RP,G,L1. G ⊢ L1 ⫃[RP] ⋆ → L1 = ⋆.
+/2 width=5 by lsubc_inv_atom2_aux/ qed-.
+
+fact lsubc_inv_bind2_aux: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → ∀I,K2. L2 = K2.ⓘ{I} →
+ (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1. ⓘ{I}) ∨
+ ∃∃K1,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
+ L1 = K1.ⓓⓝW.V & I = BPair Abst W.
+#RP #G #L1 #L2 * -L1 -L2
+[ #I #K2 #H destruct
+| #J #L1 #L2 #HL12 #I #K2 #H destruct /3 width=3 by ex2_intro, or_introl/
+| #L1 #L2 #V1 #W2 #A #HV1 #H1W2 #H2W2 #HL12 #I #K2 #H destruct
+ /3 width=10 by ex6_4_intro, or_intror/
+]
+qed-.
+
+(* Basic_1: was just: csubc_gen_head_l *)
+lemma lsubc_inv_bind2: ∀RP,I,G,L1,K2. G ⊢ L1 ⫃[RP] K2.ⓘ{I} →
+ (∃∃K1. G ⊢ K1 ⫃[RP] K2 & L1 = K1.ⓘ{I}) ∨
+ ∃∃K1,V,W,A. ⦃G, K1, V⦄ ϵ[RP] 〚A〛 & ⦃G, K1, W⦄ ϵ[RP] 〚A〛 & ⦃G, K2⦄ ⊢ W ⁝ A &
+ G ⊢ K1 ⫃[RP] K2 &
+ L1 = K1.ⓓⓝW.V & I = BPair Abst W.
+/2 width=3 by lsubc_inv_bind2_aux/ qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was just: csubc_refl *)
+lemma lsubc_refl: ∀RP,G,L. G ⊢ L ⫃[RP] L.
+#RP #G #L elim L -L /2 width=1 by lsubc_bind/
+qed.
+
+(* Basic_1: removed theorems 3:
+ csubc_clear_conf csubc_getl_conf csubc_csuba
+*)
--- /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 "static_2/static/aaa_drops.ma".
+include "static_2/static/lsubc.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+(* Properties with generic slicing ******************************************)
+
+(* Note: the premise 𝐔⦃f⦄ cannot be removed *)
+(* Basic_1: includes: csubc_drop_conf_O *)
+(* Basic_2A1: includes: lsubc_drop_O1_trans *)
+lemma lsubc_drops_trans_isuni: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 →
+ ∀b,f,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2 →
+ ∃∃K1. ⬇*[b, f] L1 ≘ K1 & G ⊢ K1 ⫃[RP] K2.
+#RP #G #L1 #L2 #H elim H -L1 -L2
+[ /2 width=3 by ex2_intro/
+| #I #L1 #L2 #HL12 #IH #b #f #K2 #Hf #H
+ elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct -IH
+ /3 width=3 by lsubc_bind, drops_refl, ex2_intro/
+ | #g #Hg #HLK2 #H destruct -HL12
+ elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
+ ]
+| #L1 #L2 #V #W #A #HV #H1W #H2W #HL12 #IH #b #f #K2 #Hf #H
+ elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
+ [ #Hf #H destruct -IH
+ /3 width=8 by drops_refl, lsubc_beta, ex2_intro/
+ | #g #Hg #HLK2 #H destruct -HL12
+ elim (IH … Hg HLK2) -L2 -Hg /3 width=3 by drops_drop, ex2_intro/
+ ]
+]
+qed-.
+
+(* Basic_1: was: csubc_drop1_conf_rev *)
+(* Basic_1: includes: csubc_drop_conf_rev *)
+(* Basic_2A1: includes: drop_lsubc_trans *)
+lemma drops_lsubc_trans: ∀RR,RS,RP. gcp RR RS RP →
+ ∀b,f,G,L1,K1. ⬇*[b, f] L1 ≘ K1 → ∀K2. G ⊢ K1 ⫃[RP] K2 →
+ ∃∃L2. G ⊢ L1 ⫃[RP] L2 & ⬇*[b, f] L2 ≘ K2.
+#RR #RS #RP #HR #b #f #G #L1 #K1 #H elim H -f -L1 -K1
+[ #f #Hf #Y #H lapply (lsubc_inv_atom1 … H) -H
+ #H destruct /4 width=3 by lsubc_atom, drops_atom, ex2_intro/
+| #f #I #L1 #K1 #_ #IH #K2 #HK12 elim (IH … HK12) -K1
+ /3 width=5 by lsubc_bind, drops_drop, ex2_intro/
+| #f #Z #I #L1 #K1 #HLK1 #HZ #IH #Y #H elim (lsubc_inv_bind1 … H) -H *
+ [ #K2 #HK12 #H destruct -HLK1
+ elim (IH … HK12) -K1 /3 width=5 by lsubc_bind, drops_skip, ex2_intro/
+ | #K2 #V2 #W2 #A #HV2 #H1W2 #H2W2 #HK12 #H1 #H2 destruct
+ elim (liftsb_inv_pair_sn … HZ) -HZ #V1 #HV21 #H destruct
+ elim (lifts_inv_flat1 … HV21) -HV21 #W3 #V3 #HW23 #HV3 #H destruct
+ elim (IH … HK12) -IH -HK12 #K #HL1K #HK2
+ lapply (acr_lifts … HR … HV2 … HLK1 … HV3) -HV2
+ lapply (acr_lifts … HR … H1W2 … HLK1 … HW23) -H1W2
+ /4 width=10 by lsubc_beta, aaa_lifts, drops_skip, ext2_pair, ex2_intro/
+ ]
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/static/lsuba.ma".
+include "static_2/static/gcp_aaa.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+(* Properties with lenv refinement for atomic arity assignment **************)
+
+lemma lsuba_lsubc: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP →
+ ∀G,L1,L2. G ⊢ L1 ⫃⁝ L2 → G ⊢ L1 ⫃[RP] L2.
+#RR #RS #RP #H1RP #H2RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubc_bind/
+#L1 #L2 #W #V #A #H elim (aaa_inv_cast … H) -H /3 width=4 by acr_aaa, lsubc_beta/
+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 "static_2/static/lsubr.ma".
+include "static_2/static/lsubc.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR GENERIC REDUCIBILITY ********************)
+
+(* Forward lemmas with restricted refinement for local environments *********)
+
+lemma lsubc_fwd_lsubr: ∀RP,G,L1,L2. G ⊢ L1 ⫃[RP] L2 → L1 ⫃ L2.
+#RP #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsubr_bind, lsubr_beta/
+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 "static_2/notation/relations/lrsubeqf_4.ma".
+include "ground_2/relocation/nstream_sor.ma".
+include "static_2/static/frees.ma".
+
+(* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
+
+inductive lsubf: relation4 lenv rtmap lenv rtmap ≝
+| lsubf_atom: ∀f1,f2. f1 ≡ f2 → lsubf (⋆) f1 (⋆) f2
+| lsubf_push: ∀f1,f2,I1,I2,L1,L2. lsubf L1 (f1) L2 (f2) →
+ lsubf (L1.ⓘ{I1}) (⫯f1) (L2.ⓘ{I2}) (⫯f2)
+| lsubf_bind: ∀f1,f2,I,L1,L2. lsubf L1 f1 L2 f2 →
+ lsubf (L1.ⓘ{I}) (↑f1) (L2.ⓘ{I}) (↑f2)
+| lsubf_beta: ∀f,f0,f1,f2,L1,L2,W,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
+ lsubf L1 f0 L2 f2 → lsubf (L1.ⓓⓝW.V) (↑f1) (L2.ⓛW) (↑f2)
+| lsubf_unit: ∀f,f0,f1,f2,I1,I2,L1,L2,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ f1 →
+ lsubf L1 f0 L2 f2 → lsubf (L1.ⓑ{I1}V) (↑f1) (L2.ⓤ{I2}) (↑f2)
+.
+
+interpretation
+ "local environment refinement (context-sensitive free variables)"
+ 'LRSubEqF L1 f1 L2 f2 = (lsubf L1 f1 L2 f2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubf_inv_atom1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 = ⋆ →
+ f1 ≡ f2 ∧ L2 = ⋆.
+#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
+[ /2 width=1 by conj/
+| #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
+| #f1 #f2 #I #L1 #L2 #_ #H destruct
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubf_inv_atom1: ∀f1,f2,L2. ⦃⋆, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f1 ≡ f2 ∧ L2 = ⋆.
+/2 width=3 by lsubf_inv_atom1_aux/ qed-.
+
+fact lsubf_inv_push1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ ∀g1,I1,K1. f1 = ⫯g1 → L1 = K1.ⓘ{I1} →
+ ∃∃g2,I2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
+#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
+[ #f1 #f2 #_ #g1 #J1 #K1 #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J1 #K1 #H1 #H2 destruct
+ <(injective_push … H1) -g1 /2 width=6 by ex3_3_intro/
+| #f1 #f2 #I #L1 #L2 #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J1 #K1 #H elim (discr_next_push … H)
+]
+qed-.
+
+lemma lsubf_inv_push1: ∀g1,f2,I1,K1,L2. ⦃K1.ⓘ{I1}, ⫯g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ ∃∃g2,I2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2 & L2 = K2.ⓘ{I2}.
+/2 width=6 by lsubf_inv_push1_aux/ qed-.
+
+fact lsubf_inv_pair1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ ∀g1,I,K1,X. f1 = ↑g1 → L1 = K1.ⓑ{I}X →
+ ∨∨ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
+ | ∃∃g,g0,g2,K2,W,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+ I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
+ | ∃∃g,g0,g2,J,K2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+ L2 = K2.ⓤ{J}.
+#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
+[ #f1 #f2 #_ #g1 #J #K1 #X #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #X #H elim (discr_push_next … H)
+| #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #X #H1 #H2 destruct
+ <(injective_next … H1) -g1 /3 width=5 by or3_intro0, ex3_2_intro/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
+ <(injective_next … H1) -g1 /3 width=12 by or3_intro1, ex7_6_intro/
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g1 #J #K1 #X #H1 #H2 destruct
+ <(injective_next … H1) -g1 /3 width=10 by or3_intro2, ex5_5_intro/
+]
+qed-.
+
+lemma lsubf_inv_pair1: ∀g1,f2,I,K1,L2,X. ⦃K1.ⓑ{I}X, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ ∨∨ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓑ{I}X
+ | ∃∃g,g0,g2,K2,W,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+ I = Abbr & X = ⓝW.V & L2 = K2.ⓛW
+ | ∃∃g,g0,g2,J,K2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ K1 ⊢ 𝐅*⦃X⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2 &
+ L2 = K2.ⓤ{J}.
+/2 width=5 by lsubf_inv_pair1_aux/ qed-.
+
+fact lsubf_inv_unit1_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ ∀g1,I,K1. f1 = ↑g1 → L1 = K1.ⓤ{I} →
+ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
+#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
+[ #f1 #f2 #_ #g1 #J #K1 #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g1 #J #K1 #H elim (discr_push_next … H)
+| #f1 #f2 #I #L1 #L2 #H12 #g1 #J #K1 #H1 #H2 destruct
+ <(injective_next … H1) -g1 /2 width=5 by ex3_2_intro/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g1 #J #K1 #_ #H destruct
+]
+qed-.
+
+lemma lsubf_inv_unit1: ∀g1,f2,I,K1,L2. ⦃K1.ⓤ{I}, ↑g1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ ∃∃g2,K2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2 & L2 = K2.ⓤ{I}.
+/2 width=5 by lsubf_inv_unit1_aux/ qed-.
+
+fact lsubf_inv_atom2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L2 = ⋆ →
+ f1 ≡ f2 ∧ L1 = ⋆.
+#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
+[ /2 width=1 by conj/
+| #f1 #f2 #I1 #I2 #L1 #L2 #_ #H destruct
+| #f1 #f2 #I #L1 #L2 #_ #H destruct
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubf_inv_atom2: ∀f1,f2,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → f1 ≡ f2 ∧ L1 = ⋆.
+/2 width=3 by lsubf_inv_atom2_aux/ qed-.
+
+fact lsubf_inv_push2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ ∀g2,I2,K2. f2 = ⫯g2 → L2 = K2.ⓘ{I2} →
+ ∃∃g1,I1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
+#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
+[ #f1 #f2 #_ #g2 #J2 #K2 #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J2 #K2 #H1 #H2 destruct
+ <(injective_push … H1) -g2 /2 width=6 by ex3_3_intro/
+| #f1 #f2 #I #L1 #L2 #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J2 #K2 #H elim (discr_next_push … H)
+]
+qed-.
+
+lemma lsubf_inv_push2: ∀f1,g2,I2,L1,K2. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓘ{I2}, ⫯g2⦄ →
+ ∃∃g1,I1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ⫯g1 & L1 = K1.ⓘ{I1}.
+/2 width=6 by lsubf_inv_push2_aux/ qed-.
+
+fact lsubf_inv_pair2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ ∀g2,I,K2,W. f2 = ↑g2 → L2 = K2.ⓑ{I}W →
+ ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
+ | ∃∃g,g0,g1,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+ I = Abst & L1 = K1.ⓓⓝW.V.
+#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
+[ #f1 #f2 #_ #g2 #J #K2 #X #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #X #H elim (discr_push_next … H)
+| #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #X #H1 #H2 destruct
+ <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #Hf #Hf1 #H12 #g2 #J #K2 #X #H1 #H2 destruct
+ <(injective_next … H1) -g2 /3 width=10 by ex6_5_intro, or_intror/
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #g2 #J #K2 #X #_ #H destruct
+]
+qed-.
+
+lemma lsubf_inv_pair2: ∀f1,g2,I,L1,K2,W. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓑ{I}W, ↑g2⦄ →
+ ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓑ{I}W
+ | ∃∃g,g0,g1,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+ I = Abst & L1 = K1.ⓓⓝW.V.
+/2 width=5 by lsubf_inv_pair2_aux/ qed-.
+
+fact lsubf_inv_unit2_aux: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ ∀g2,I,K2. f2 = ↑g2 → L2 = K2.ⓤ{I} →
+ ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
+ | ∃∃g,g0,g1,J,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+ L1 = K1.ⓑ{J}V.
+#f1 #f2 #L1 #L2 * -f1 -f2 -L1 -L2
+[ #f1 #f2 #_ #g2 #J #K2 #_ #H destruct
+| #f1 #f2 #I1 #I2 #L1 #L2 #H12 #g2 #J #K2 #H elim (discr_push_next … H)
+| #f1 #f2 #I #L1 #L2 #H12 #g2 #J #K2 #H1 #H2 destruct
+ <(injective_next … H1) -g2 /3 width=5 by ex3_2_intro, or_introl/
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #g2 #J #K2 #_ #H destruct
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #Hf #Hf1 #H12 #g2 #J #K2 #H1 #H2 destruct
+ <(injective_next … H1) -g2 /3 width=11 by ex5_6_intro, or_intror/
+]
+qed-.
+
+lemma lsubf_inv_unit2: ∀f1,g2,I,L1,K2. ⦃L1, f1⦄ ⫃𝐅* ⦃K2.ⓤ{I}, ↑g2⦄ →
+ ∨∨ ∃∃g1,K1. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f1 = ↑g1 & L1 = K1.ⓤ{I}
+ | ∃∃g,g0,g1,J,K1,V. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ &
+ K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f1 = ↑g1 &
+ L1 = K1.ⓑ{J}V.
+/2 width=5 by lsubf_inv_unit2_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lsubf_inv_atom: ∀f1,f2. ⦃⋆, f1⦄ ⫃𝐅* ⦃⋆, f2⦄ → f1 ≡ f2.
+#f1 #f2 #H elim (lsubf_inv_atom1 … H) -H //
+qed-.
+
+lemma lsubf_inv_push_sn: ∀g1,f2,I1,I2,K1,K2. ⦃K1.ⓘ{I1}, ⫯g1⦄ ⫃𝐅* ⦃K2.ⓘ{I2}, f2⦄ →
+ ∃∃g2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ⫯g2.
+#g1 #f2 #I #K1 #K2 #X #H elim (lsubf_inv_push1 … H) -H
+#g2 #I #Y #H0 #H2 #H destruct /2 width=3 by ex2_intro/
+qed-.
+
+lemma lsubf_inv_bind_sn: ∀g1,f2,I,K1,K2. ⦃K1.ⓘ{I}, ↑g1⦄ ⫃𝐅* ⦃K2.ⓘ{I}, f2⦄ →
+ ∃∃g2. ⦃K1, g1⦄ ⫃𝐅* ⦃K2, g2⦄ & f2 = ↑g2.
+#g1 #f2 * #I [2: #X ] #K1 #K2 #H
+[ elim (lsubf_inv_pair1 … H) -H *
+ [ #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
+ | #z #z0 #z2 #Y2 #W #V #_ #_ #_ #_ #H0 #_ #H destruct
+ | #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
+ ]
+| elim (lsubf_inv_unit1 … H) -H
+ #z2 #Y2 #H2 #H #H0 destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma lsubf_inv_beta_sn: ∀g1,f2,K1,K2,V,W. ⦃K1.ⓓⓝW.V, ↑g1⦄ ⫃𝐅* ⦃K2.ⓛW, f2⦄ →
+ ∃∃g,g0,g2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
+#g1 #f2 #K1 #K2 #V #W #H elim (lsubf_inv_pair1 … H) -H *
+[ #z2 #Y2 #_ #_ #H destruct
+| #z #z0 #z2 #Y2 #X0 #X #H02 #Hz #Hg1 #H #_ #H0 #H1 destruct
+ /2 width=7 by ex4_3_intro/
+| #z #z0 #z2 #Z2 #Y2 #_ #_ #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubf_inv_unit_sn: ∀g1,f2,I,J,K1,K2,V. ⦃K1.ⓑ{I}V, ↑g1⦄ ⫃𝐅* ⦃K2.ⓤ{J}, f2⦄ →
+ ∃∃g,g0,g2. ⦃K1, g0⦄ ⫃𝐅* ⦃K2, g2⦄ & K1 ⊢ 𝐅*⦃V⦄ ≘ g & g0 ⋓ g ≘ g1 & f2 = ↑g2.
+#g1 #f2 #I #J #K1 #K2 #V #H elim (lsubf_inv_pair1 … H) -H *
+[ #z2 #Y2 #_ #_ #H destruct
+| #z #z0 #z2 #Y2 #X0 #X #_ #_ #_ #_ #_ #_ #H destruct
+| #z #z0 #z2 #Z2 #Y2 #H02 #Hz #Hg1 #H0 #H1 destruct
+ /2 width=7 by ex4_3_intro/
+]
+qed-.
+
+lemma lsubf_inv_refl: ∀L,f1,f2. ⦃L,f1⦄ ⫃𝐅* ⦃L,f2⦄ → f1 ≡ f2.
+#L elim L -L /2 width=1 by lsubf_inv_atom/
+#L #I #IH #f1 #f2 #H12
+elim (pn_split f1) * #g1 #H destruct
+[ elim (lsubf_inv_push_sn … H12) | elim (lsubf_inv_bind_sn … H12) ] -H12
+#g2 #H12 #H destruct /3 width=5 by eq_next, eq_push/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubf_fwd_bind_tl: ∀f1,f2,I,L1,L2.
+ ⦃L1.ⓘ{I}, f1⦄ ⫃𝐅* ⦃L2.ⓘ{I}, f2⦄ → ⦃L1, ⫱f1⦄ ⫃𝐅* ⦃L2, ⫱f2⦄.
+#f1 #f2 #I #L1 #L2 #H
+elim (pn_split f1) * #g1 #H0 destruct
+[ elim (lsubf_inv_push_sn … H) | elim (lsubf_inv_bind_sn … H) ] -H
+#g2 #H12 #H destruct //
+qed-.
+
+lemma lsubf_fwd_isid_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → 𝐈⦃f2⦄ → 𝐈⦃f1⦄.
+#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
+[ /2 width=3 by isid_eq_repl_fwd/
+| /4 width=3 by isid_inv_push, isid_push/
+| #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+]
+qed-.
+
+lemma lsubf_fwd_isid_sn: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → 𝐈⦃f1⦄ → 𝐈⦃f2⦄.
+#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
+[ /2 width=3 by isid_eq_repl_back/
+| /4 width=3 by isid_inv_push, isid_push/
+| #f1 #f2 #I #L1 #L2 #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #L1 #L2 #W #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+| #f #f0 #f1 #f2 #I1 #I2 #L1 #L2 #V #_ #_ #_ #_ #H elim (isid_inv_next … H) -H //
+]
+qed-.
+
+lemma lsubf_fwd_sle: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → f2 ⊆ f1.
+#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
+/3 width=5 by sor_inv_sle_sn_trans, sle_next, sle_push, sle_refl_eq, eq_sym/
+qed-.
+
+(* Basic properties *********************************************************)
+
+axiom lsubf_eq_repl_back1: ∀f2,L1,L2. eq_repl_back … (λf1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+
+lemma lsubf_eq_repl_fwd1: ∀f2,L1,L2. eq_repl_fwd … (λf1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+#f2 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back1/
+qed-.
+
+axiom lsubf_eq_repl_back2: ∀f1,L1,L2. eq_repl_back … (λf2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+
+lemma lsubf_eq_repl_fwd2: ∀f1,L1,L2. eq_repl_fwd … (λf2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄).
+#f1 #L1 #L2 @eq_repl_sym /2 width=3 by lsubf_eq_repl_back2/
+qed-.
+
+lemma lsubf_refl: bi_reflexive … lsubf.
+#L elim L -L /2 width=1 by lsubf_atom, eq_refl/
+#L #I #IH #f elim (pn_split f) * #g #H destruct
+/2 width=1 by lsubf_push, lsubf_bind/
+qed.
+
+lemma lsubf_refl_eq: ∀f1,f2,L. f1 ≡ f2 → ⦃L, f1⦄ ⫃𝐅* ⦃L, f2⦄.
+/2 width=3 by lsubf_eq_repl_back2/ qed.
+
+lemma lsubf_bind_tl_dx: ∀g1,f2,I,L1,L2. ⦃L1, g1⦄ ⫃𝐅* ⦃L2, ⫱f2⦄ →
+ ∃∃f1. ⦃L1.ⓘ{I}, f1⦄ ⫃𝐅* ⦃L2.ⓘ{I}, f2⦄ & g1 = ⫱f1.
+#g1 #f2 #I #L1 #L2 #H
+elim (pn_split f2) * #g2 #H2 destruct
+@ex2_intro [1,2,4,5: /2 width=2 by lsubf_push, lsubf_bind/ ] // (**) (* constructor needed *)
+qed-.
+
+lemma lsubf_beta_tl_dx: ∀f,f0,g1,L1,V. L1 ⊢ 𝐅*⦃V⦄ ≘ f → f0 ⋓ f ≘ g1 →
+ ∀f2,L2,W. ⦃L1, f0⦄ ⫃𝐅* ⦃L2, ⫱f2⦄ →
+ ∃∃f1. ⦃L1.ⓓⓝW.V, f1⦄ ⫃𝐅* ⦃L2.ⓛW, f2⦄ & ⫱f1 ⊆ g1.
+#f #f0 #g1 #L1 #V #Hf #Hg1 #f2
+elim (pn_split f2) * #x2 #H2 #L2 #W #HL12 destruct
+[ /3 width=4 by lsubf_push, sor_inv_sle_sn, ex2_intro/
+| @(ex2_intro … (↑g1)) /2 width=5 by lsubf_beta/ (**) (* full auto fails *)
+]
+qed-.
+
+(* Note: this might be moved *)
+lemma lsubf_inv_sor_dx: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ ∀f2l,f2r. f2l⋓f2r ≘ f2 →
+ ∃∃f1l,f1r. ⦃L1, f1l⦄ ⫃𝐅* ⦃L2, f2l⦄ & ⦃L1, f1r⦄ ⫃𝐅* ⦃L2, f2r⦄ & f1l⋓f1r ≘ f1.
+#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
+[ /3 width=7 by sor_eq_repl_fwd3, ex3_2_intro/
+| #g1 #g2 #I1 #I2 #L1 #L2 #_ #IH #f2l #f2r #H
+ elim (sor_inv_xxp … H) -H [|*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+ elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, sor_pp, ex3_2_intro/
+| #g1 #g2 #I #L1 #L2 #_ #IH #f2l #f2r #H
+ elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+ elim (IH … Hg2) -g2 /3 width=11 by lsubf_push, lsubf_bind, sor_np, sor_pn, sor_nn, ex3_2_intro/
+| #g #g0 #g1 #g2 #L1 #L2 #W #V #Hg #Hg1 #_ #IH #f2l #f2r #H
+ elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+ elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
+ [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
+ /3 width=11 by lsubf_push, lsubf_beta, sor_np, ex3_2_intro/
+ | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
+ /3 width=11 by lsubf_push, lsubf_beta, sor_pn, ex3_2_intro/
+ | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
+ /3 width=11 by lsubf_beta, sor_nn, ex3_2_intro/
+ ]
+| #g #g0 #g1 #g2 #I1 #I2 #L1 #L2 #V #Hg #Hg1 #_ #IH #f2l #f2r #H
+ elim (sor_inv_xxn … H) -H [1,3,4: * |*: // ] #g2l #g2r #Hg2 #Hl #Hr destruct
+ elim (IH … Hg2) -g2 #g1l #g1r #Hl #Hr #Hg0
+ [ lapply (sor_comm_23 … Hg0 Hg1 ?) -g0 [3: |*: // ] #Hg1
+ /3 width=11 by lsubf_push, lsubf_unit, sor_np, ex3_2_intro/
+ | lapply (sor_assoc_dx … Hg1 … Hg0 ??) -g0 [3: |*: // ] #Hg1
+ /3 width=11 by lsubf_push, lsubf_unit, sor_pn, ex3_2_intro/
+ | lapply (sor_distr_dx … Hg0 … Hg1) -g0 [5: |*: // ] #Hg1
+ /3 width=11 by lsubf_unit, sor_nn, 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 "static_2/static/lsubf.ma".
+
+(* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
+
+(* Properties with context-sensitive free variables *************************)
+
+lemma lsubf_frees_trans: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 →
+ ∀f1,L1. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ → L1 ⊢ 𝐅*⦃T⦄ ≘ f1.
+#f2 #L2 #T #H elim H -f2 -L2 -T
+[ /3 width=5 by lsubf_fwd_isid_dx, frees_sort/
+| #f2 #i #Hf2 #g1 #Y1 #H
+ elim (lsubf_inv_atom2 … H) -H #Hg1 #H destruct
+ elim (eq_inv_pushs_dx … Hg1) -Hg1 #g #Hg #H destruct
+ elim (eq_inv_xn … Hg) -Hg
+ /3 width=3 by frees_atom, isid_eq_repl_fwd/
+| #f2 #I #K2 #W #_ #IH #g1 #Y1 #H elim (lsubf_inv_pair2 … H) -H *
+ [ #f1 #K1 #H12 #H1 #H2 destruct /3 width=1 by frees_pair/
+ | #f #f0 #f1 #K1 #V #H12 #Hf #Hf1 #H1 #H2 #H3 destruct
+ /4 width=5 by frees_pair, frees_flat/
+ ]
+| #f2 #I #L2 #Hf2 #g1 #Y1 #H elim (lsubf_inv_unit2 … H) -H *
+ [ #f1 #L1 #H12 #H1 #H2 destruct
+ /3 width=5 by lsubf_fwd_isid_dx, frees_unit/
+ | #f #f0 #f1 #J #L1 #V #H12 #Hf #Hf1 #H1 #H2 destruct
+ /5 width=9 by lsubf_fwd_isid_dx, frees_eq_repl_back, frees_pair, sor_isid_inv_sn/
+ ]
+| #f2 #I #L2 #i #_ #IH #g1 #L1 #H elim (lsubf_inv_push2 … H) -H
+ /3 width=1 by frees_lref/
+| /3 width=5 by lsubf_fwd_isid_dx, frees_gref/
+| #f2V #f2T #f2 #p #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f1 #L1 #H12
+ elim (lsubf_inv_sor_dx … H12 … Hf2) -f2 #f1V #g1T #HV #HT #Hf1
+ elim (lsubf_bind_tl_dx … (BPair I V) … HT) -HT #f1T #HT #H destruct
+ /3 width=5 by frees_bind/
+| #f2V #f2T #f2 #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #f1 #L1 #H12
+ elim (lsubf_inv_sor_dx … H12 … Hf2) -f2 /3 width=5 by frees_flat/
+]
+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 "static_2/static/frees_frees.ma".
+include "static_2/static/lsubf.ma".
+
+(* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
+
+(* Main properties **********************************************************)
+
+theorem lsubf_sor: ∀K,L,g1,f1. ⦃K, g1⦄ ⫃𝐅* ⦃L, f1⦄ →
+ ∀g2,f2. ⦃K, g2⦄ ⫃𝐅* ⦃L, f2⦄ →
+ ∀g. g1 ⋓ g2 ≘ g → ∀f. f1 ⋓ f2 ≘ f → ⦃K, g⦄ ⫃𝐅* ⦃L, f⦄.
+#K elim K -K
+[ #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
+ elim (lsubf_inv_atom1 … H1) -H1 #H1 #H destruct
+ lapply (lsubf_inv_atom … H2) -H2 #H2
+ /5 width=4 by lsubf_atom, sor_mono, sor_eq_repl_back2, sor_eq_repl_back1/
+| #K #J #IH #L #g1 #f1 #H1 #g2 #f2 #H2 #g #Hg #f #Hf
+ elim (pn_split g1) * #y1 #H destruct
+ elim (pn_split g2) * #y2 #H destruct
+ [ elim (sor_inv_ppx … Hg) -Hg [|*: // ] #y #Hy #H destruct
+ elim (lsubf_inv_push1 … H1) -H1 #x1 #Z1 #Y1 #H1 #H #H0 destruct
+ elim (lsubf_inv_push_sn … H2) -H2 #x2 #H2 #H destruct
+ elim (sor_inv_ppx … Hf) -Hf [|*: // ] #x #Hx #H destruct
+ /3 width=8 by lsubf_push/
+ | elim (sor_inv_pnx … Hg) -Hg [|*: // ] #y #Hy #H destruct
+ elim (lsubf_inv_push1 … H1) -H1 #x1 #Z1 #Y1 #H1 #H #H0 destruct
+ generalize in match H2; -H2 cases J -J #J [| #V ] #H2
+ [ elim (lsubf_inv_unit1 … H2) -H2 #x2 #Y2 #H2 #H #H0 destruct
+ | elim (lsubf_inv_pair1 … H2) -H2 *
+ [ #x2 #Z2 #H2 #H #H0 destruct
+ | #y3 #y4 #x2 #Y2 #W #U #H2 #Hy3 #Hy2 #H #H0 #H3 #H4 destruct
+ | #y3 #y4 #x2 #Z2 #Y2 #H2 #Hy3 #Hy2 #H #H0 destruct
+ ]
+ ]
+ elim (sor_inv_pnx … Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
+ /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_assoc_sn/
+ | elim (sor_inv_npx … Hg) -Hg [|*: // ] #y #Hy #H destruct
+ elim (lsubf_inv_push1 … H2) -H2 #x2 #Z2 #Y2 #H2 #H #H0 destruct
+ generalize in match H1; -H1 cases J -J #J [| #V ] #H1
+ [ elim (lsubf_inv_unit1 … H1) -H1 #x1 #Y1 #H1 #H #H0 destruct
+ | elim (lsubf_inv_pair1 … H1) -H1 *
+ [ #x1 #Z1 #H1 #H #H0 destruct
+ | #y3 #y4 #x1 #Y1 #W #U #H1 #Hy3 #Hy1 #H #H0 #H3 #H4 destruct
+ | #y3 #y4 #x1 #Z1 #Y1 #H1 #Hy3 #Hy1 #H #H0 destruct
+ ]
+ ]
+ elim (sor_inv_npx … Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
+ /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_comm_23/
+ | elim (sor_inv_nnx … Hg) -Hg [|*: // ] #y #Hy #H destruct
+ generalize in match H2; generalize in match H1; -H1 -H2 cases J -J #J [| #V ] #H1 #H2
+ [ elim (lsubf_inv_unit1 … H1) -H1 #x1 #Y1 #H1 #H #H0 destruct
+ elim (lsubf_inv_bind_sn … H2) -H2 #x2 #H2 #H destruct
+ | elim (lsubf_inv_pair1 … H1) -H1 *
+ [ #x1 #Z1 #H1 #H #H0 destruct
+ elim (lsubf_inv_bind_sn … H2) -H2 #x2 #H2 #H destruct
+ | #y3 #y4 #x1 #Y1 #W #U #H1 #Hy3 #Hy1 #H #H0 #H3 #H4 destruct
+ elim (lsubf_inv_beta_sn … H2) -H2 #z3 #z4 #x2 #H2 #Hz3 #Hy2 #H destruct
+ lapply (frees_mono … Hz3 … Hy3) -Hz3 #H3
+ lapply (sor_eq_repl_back2 … Hy2 … H3) -z3 #Hy2
+ | #y3 #y4 #x1 #Z1 #Y1 #H1 #Hy3 #Hy1 #H #H0 destruct
+ elim (lsubf_inv_unit_sn … H2) -H2 #z3 #z4 #x2 #H2 #Hz3 #Hy2 #H destruct
+ lapply (frees_mono … Hz3 … Hy3) -Hz3 #H3
+ lapply (sor_eq_repl_back2 … Hy2 … H3) -z3 #Hy2
+ ]
+ ]
+ elim (sor_inv_nnx … Hf) -Hf [1,6,11,16:|*: // ] #x #Hx #H destruct
+ /3 width=12 by lsubf_unit, lsubf_beta, lsubf_bind, sor_coll_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 "static_2/static/lsubr.ma".
+include "static_2/static/lsubf_lsubf.ma".
+
+(* RESTRICTED REFINEMENT FOR CONTEXT-SENSITIVE FREE VARIABLES ***************)
+
+(* Forward lemmas with restricted refinement for local environments *********)
+
+lemma lsubf_fwd_lsubr_isdiv: ∀f1,f2,L1,L2. ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄ →
+ 𝛀⦃f1⦄ → 𝛀⦃f2⦄ → L1 ⫃ L2.
+#f1 #f2 #L1 #L2 #H elim H -f1 -f2 -L1 -L2
+/4 width=3 by lsubr_bind, isdiv_inv_next/
+[ #f1 #f2 #I1 #I2 #L1 #L2 #_ #_ #H
+ elim (isdiv_inv_push … H) //
+| /5 width=5 by lsubf_fwd_sle, lsubr_beta, sle_inv_isdiv_sn, isdiv_inv_next/
+| /5 width=5 by lsubf_fwd_sle, lsubr_unit, sle_inv_isdiv_sn, isdiv_inv_next/
+]
+qed-.
+
+(* Properties with restricted refinement for local environments *************)
+
+lemma lsubr_lsubf_isid: ∀L1,L2. L1 ⫃ L2 →
+ ∀f1,f2. 𝐈⦃f1⦄ → 𝐈⦃f2⦄ → ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄.
+#L1 #L2 #H elim H -L1 -L2
+[ /3 width=1 by lsubf_atom, isid_inv_eq_repl/
+| #I #L1 #L2 | #L1 #L2 #V #W | #I1 #I2 #L1 #L2 #V
+]
+#_ #IH #f1 #f2 #Hf1 #Hf2
+elim (isid_inv_gen … Hf1) -Hf1 #g1 #Hg1 #H destruct
+elim (isid_inv_gen … Hf2) -Hf2 #g2 #Hg2 #H destruct
+/3 width=1 by lsubf_push/
+qed.
+
+lemma lsubr_lsubf: ∀f2,L2,T. L2 ⊢ 𝐅*⦃T⦄ ≘ f2 → ∀L1. L1 ⫃ L2 →
+ ∀f1. L1 ⊢ 𝐅*⦃T⦄ ≘ f1 → ⦃L1, f1⦄ ⫃𝐅* ⦃L2, f2⦄.
+#f2 #L2 #T #H elim H -f2 -L2 -T
+[ #f2 #L2 #s #Hf2 #L1 #HL12 #f1 #Hf1
+ lapply (frees_inv_sort … Hf1) -Hf1 /2 width=1 by lsubr_lsubf_isid/
+| #f2 #i #Hf2 #Y1 #HY1
+ >(lsubr_inv_atom2 … HY1) -Y1 #g1 #Hg1
+ elim (frees_inv_atom … Hg1) -Hg1 #f1 #Hf1 #H destruct
+ /5 width=5 by lsubf_atom, isid_inv_eq_repl, pushs_eq_repl, eq_next/
+| #f2 #Z #L2 #W #_ #IH #Y1 #HY1 #g1 #Hg1 elim (lsubr_inv_pair2 … HY1) -HY1 *
+ [ #L1 #HL12 #H destruct
+ elim (frees_inv_pair … Hg1) -Hg1 #f1 #Hf1 #H destruct
+ /3 width=1 by lsubf_bind/
+ | #L1 #V #HL12 #H1 #H2 destruct
+ elim (frees_inv_pair … Hg1) -Hg1 #f1 #Hf1 #H destruct
+ elim (frees_inv_flat … Hf1) -Hf1 /3 width=5 by lsubf_beta/
+ ]
+| #f2 #I2 #L2 #Hf2 #Y1 #HY1 #g1 #Hg1 elim (lsubr_inv_unit2 … HY1) -HY1 *
+ [ #L1 #HL12 #H destruct
+ elim (frees_inv_unit … Hg1) -Hg1 #f1 #Hf1 #H destruct
+ /3 width=1 by lsubf_bind, lsubr_lsubf_isid/
+ | #I #L1 #V #HL12 #H destruct
+ elim (frees_inv_pair … Hg1) -Hg1 #f1 #Hf1 #H destruct
+ /3 width=5 by lsubf_unit, sor_isid_sn, lsubr_lsubf_isid/
+ ]
+| #f2 #I2 #L2 #i #_ #IH #Y1 #HY1 #g1 #Hg1
+ elim (lsubr_fwd_bind2 … HY1) -HY1 #I1 #L1 #HL12 #H destruct
+ elim (frees_inv_lref … Hg1) -Hg1 #f1 #Hf1 #H destruct
+ /3 width=1 by lsubf_push/
+| #f2 #L2 #l #Hf2 #L1 #HL12 #f1 #Hf1
+ lapply (frees_inv_gref … Hf1) -Hf1 /2 width=1 by lsubr_lsubf_isid/
+| #f2V #f2T #f2 #p #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #L1 #HL12 #f1 #Hf1
+ elim (frees_inv_bind … Hf1) -Hf1 #f1V #f1T #Hf1V #Hf1T #Hf1
+ /5 width=8 by lsubf_sor, lsubf_fwd_bind_tl, lsubr_bind/
+| #f2V #f2T #f2 #I #L2 #V #T #_ #_ #Hf2 #IHV #IHT #L1 #HL12 #f1 #Hf1
+ elim (frees_inv_flat … Hf1) -Hf1 #f1V #f1T #Hf1V #Hf1T #Hf1
+ /3 width=8 by lsubf_sor/
+]
+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 "static_2/notation/relations/lrsubeqc_2.ma".
+include "static_2/syntax/lenv.ma".
+
+(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Basic_2A1: just tpr_cpr and tprs_cprs require the extended lsubr_atom *)
+(* Basic_2A1: includes: lsubr_pair *)
+inductive lsubr: relation lenv ≝
+| lsubr_atom: lsubr (⋆) (⋆)
+| lsubr_bind: ∀I,L1,L2. lsubr L1 L2 → lsubr (L1.ⓘ{I}) (L2.ⓘ{I})
+| lsubr_beta: ∀L1,L2,V,W. lsubr L1 L2 → lsubr (L1.ⓓⓝW.V) (L2.ⓛW)
+| lsubr_unit: ∀I1,I2,L1,L2,V. lsubr L1 L2 → lsubr (L1.ⓑ{I1}V) (L2.ⓤ{I2})
+.
+
+interpretation
+ "restricted refinement (local environment)"
+ 'LRSubEqC L1 L2 = (lsubr L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lsubr_refl: ∀L. L ⫃ L.
+#L elim L -L /2 width=1 by lsubr_atom, lsubr_bind/
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsubr_inv_atom1_aux: ∀L1,L2. L1 ⫃ L2 → L1 = ⋆ → L2 = ⋆.
+#L1 #L2 * -L1 -L2 //
+[ #I #L1 #L2 #_ #H destruct
+| #L1 #L2 #V #W #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_atom1: ∀L2. ⋆ ⫃ L2 → L2 = ⋆.
+/2 width=3 by lsubr_inv_atom1_aux/ qed-.
+
+fact lsubr_inv_bind1_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K1. L1 = K1.ⓘ{I} →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
+ I = BPair Abbr (ⓝW.V)
+ | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
+ I = BPair J1 V.
+#L1 #L2 * -L1 -L2
+[ #J #K1 #H destruct
+| #I #L1 #L2 #HL12 #J #K1 #H destruct /3 width=3 by or3_intro0, ex2_intro/
+| #L1 #L2 #V #W #HL12 #J #K1 #H destruct /3 width=6 by or3_intro1, ex3_3_intro/
+| #I1 #I2 #L1 #L2 #V #HL12 #J #K1 #H destruct /3 width=4 by or3_intro2, ex3_4_intro/
+]
+qed-.
+
+(* Basic_2A1: uses: lsubr_inv_pair1 *)
+lemma lsubr_inv_bind1: ∀I,K1,L2. K1.ⓘ{I} ⫃ L2 →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓘ{I}
+ | ∃∃K2,V,W. K1 ⫃ K2 & L2 = K2.ⓛW &
+ I = BPair Abbr (ⓝW.V)
+ | ∃∃J1,J2,K2,V. K1 ⫃ K2 & L2 = K2.ⓤ{J2} &
+ I = BPair J1 V.
+/2 width=3 by lsubr_inv_bind1_aux/ qed-.
+
+fact lsubr_inv_atom2_aux: ∀L1,L2. L1 ⫃ L2 → L2 = ⋆ → L1 = ⋆.
+#L1 #L2 * -L1 -L2 //
+[ #I #L1 #L2 #_ #H destruct
+| #L1 #L2 #V #W #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_atom2: ∀L1. L1 ⫃ ⋆ → L1 = ⋆.
+/2 width=3 by lsubr_inv_atom2_aux/ qed-.
+
+fact lsubr_inv_bind2_aux: ∀L1,L2. L1 ⫃ L2 → ∀I,K2. L2 = K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
+#L1 #L2 * -L1 -L2
+[ #J #K2 #H destruct
+| #I #L1 #L2 #HL12 #J #K2 #H destruct /3 width=3 by ex2_intro, or3_intro0/
+| #L1 #L2 #V1 #V2 #HL12 #J #K2 #H destruct /3 width=6 by ex3_3_intro, or3_intro1/
+| #I1 #I2 #L1 #L2 #V #HL12 #J #K2 #H destruct /3 width=5 by ex3_4_intro, or3_intro2/
+]
+qed-.
+
+lemma lsubr_inv_bind2: ∀I,L1,K2. L1 ⫃ K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J1}V & I = BUnit J2.
+/2 width=3 by lsubr_inv_bind2_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lsubr_inv_abst1: ∀K1,L2,W. K1.ⓛW ⫃ L2 →
+ ∨∨ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓛW
+ | ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓤ{I2}.
+#K1 #L2 #W #H elim (lsubr_inv_bind1 … H) -H *
+/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
+#K2 #V2 #W2 #_ #_ #H destruct
+qed-.
+
+lemma lsubr_inv_unit1: ∀I,K1,L2. K1.ⓤ{I} ⫃ L2 →
+ ∃∃K2. K1 ⫃ K2 & L2 = K2.ⓤ{I}.
+#I #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
+[ #K2 #HK12 #H destruct /2 width=3 by ex2_intro/
+| #K2 #V #W #_ #_ #H destruct
+| #I1 #I2 #K2 #V #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_pair2: ∀I,L1,K2,W. L1 ⫃ K2.ⓑ{I}W →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓑ{I}W
+ | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V & I = Abst.
+#I #L1 #K2 #W #H elim (lsubr_inv_bind2 … H) -H *
+[ /3 width=3 by ex2_intro, or_introl/
+| #K2 #X #V #HK12 #H1 #H2 destruct /3 width=4 by ex3_2_intro, or_intror/
+| #I1 #I1 #K2 #V #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_abbr2: ∀L1,K2,V. L1 ⫃ K2.ⓓV →
+ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓓV.
+#L1 #K2 #V #H elim (lsubr_inv_pair2 … H) -H *
+[ /2 width=3 by ex2_intro/
+| #K1 #X #_ #_ #H destruct
+]
+qed-.
+
+lemma lsubr_inv_abst2: ∀L1,K2,W. L1 ⫃ K2.ⓛW →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓛW
+ | ∃∃K1,V. K1 ⫃ K2 & L1 = K1.ⓓⓝW.V.
+#L1 #K2 #W #H elim (lsubr_inv_pair2 … H) -H *
+/3 width=4 by ex2_2_intro, ex2_intro, or_introl, or_intror/
+qed-.
+
+lemma lsubr_inv_unit2: ∀I,L1,K2. L1 ⫃ K2.ⓤ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & L1 = K1.ⓤ{I}
+ | ∃∃J,K1,V. K1 ⫃ K2 & L1 = K1.ⓑ{J}V.
+#I #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
+[ /3 width=3 by ex2_intro, or_introl/
+| #K1 #W #V #_ #_ #H destruct
+| #I1 #I2 #K1 #V #HK12 #H1 #H2 destruct /3 width=5 by ex2_3_intro, or_intror/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsubr_fwd_bind1: ∀I1,K1,L2. K1.ⓘ{I1} ⫃ L2 →
+ ∃∃I2,K2. K1 ⫃ K2 & L2 = K2.ⓘ{I2}.
+#I1 #K1 #L2 #H elim (lsubr_inv_bind1 … H) -H *
+[ #K2 #HK12 #H destruct /3 width=4 by ex2_2_intro/
+| #K2 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+| #I1 #I2 #K2 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lsubr_fwd_bind2: ∀I2,L1,K2. L1 ⫃ K2.ⓘ{I2} →
+ ∃∃I1,K1. K1 ⫃ K2 & L1 = K1.ⓘ{I1}.
+#I2 #L1 #K2 #H elim (lsubr_inv_bind2 … H) -H *
+[ #K1 #HK12 #H destruct /3 width=4 by ex2_2_intro/
+| #K1 #W1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+| #I1 #I2 #K1 #V1 #HK12 #H1 #H2 destruct /3 width=4 by ex2_2_intro/
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/relocation/drops.ma".
+include "static_2/static/lsubr.ma".
+
+(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Forward lemmas with generic slicing for local environments ***************)
+
+(* Basic_2A1: includes: lsubr_fwd_drop2_pair *)
+lemma lsubr_fwd_drops2_bind: ∀L1,L2. L1 ⫃ L2 →
+ ∀b,f,I,K2. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2.ⓘ{I} →
+ ∨∨ ∃∃K1. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓘ{I}
+ | ∃∃K1,W,V. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓓⓝW.V & I = BPair Abst W
+ | ∃∃J1,J2,K1,V. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓑ{J1}V & I = BUnit J2.
+#L1 #L2 #H elim H -L1 -L2
+[ #b #f #I #K2 #_ #H
+ elim (drops_inv_atom1 … H) -H #H destruct
+| #J #L1 #L2 | #L1 #L2 #V #W | #I1 #I2 #L1 #L2 #V1
+]
+#HL12 #IH #b #f #I #K2 #Hf #H
+elim (drops_inv_bind1_isuni … Hf H) -Hf -H *
+[1,3,5: #Hf #H destruct -IH
+ /4 width=6 by drops_refl, or3_intro0, or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro, ex2_intro/
+|2,4,6: #g #Hg #HLK2 #H destruct -HL12
+ elim (IH … Hg HLK2) -IH -Hg -HLK2 *
+ /4 width=6 by drops_drop, or3_intro0, or3_intro1, or3_intro2, ex3_4_intro, ex3_3_intro, ex2_intro/
+]
+qed-.
+
+(* Basic_2A1: includes: lsubr_fwd_drop2_abbr *)
+lemma lsubr_fwd_drops2_abbr: ∀L1,L2. L1 ⫃ L2 →
+ ∀b,f,K2,V. 𝐔⦃f⦄ → ⬇*[b, f] L2 ≘ K2.ⓓV →
+ ∃∃K1. K1 ⫃ K2 & ⬇*[b, f] L1 ≘ K1.ⓓV.
+#L1 #L2 #HL12 #b #f #K2 #V #Hf #HLK2
+elim (lsubr_fwd_drops2_bind … HL12 … Hf HLK2) -L2 -Hf // *
+[ #K1 #W #V #_ #_ #H destruct
+| #I1 #I2 #K1 #V #_ #_ #H destruct
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/lenv_length.ma".
+include "static_2/static/lsubr.ma".
+
+(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma lsubr_fwd_length: ∀L1,L2. L1 ⫃ L2 → |L2| = |L1|.
+#L1 #L2 #H elim H -L1 -L2 normalize //
+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 "static_2/static/lsubr.ma".
+
+(* RESTRICTED REFINEMENT FOR LOCAL ENVIRONMENTS *****************************)
+
+(* Main properties **********************************************************)
+
+theorem lsubr_trans: Transitive … lsubr.
+#L1 #L #H elim H -L1 -L //
+[ #I #L1 #L #_ #IH #X #H elim (lsubr_inv_bind1 … H) -H *
+ [ #L2 #HL2 #H | #L2 #V #W #HL2 #H1 #H2 | #I1 #I2 #L2 #V #Hl2 #H1 #H2 ]
+ destruct /3 width=1 by lsubr_bind, lsubr_beta, lsubr_unit/
+| #L1 #L #V #W #_ #IH #X #H elim (lsubr_inv_abst1 … H) -H *
+ [ #L2 #HL2 #H | #I #L2 #HL2 #H ]
+ destruct /3 width=1 by lsubr_beta, lsubr_unit/
+| #I1 #I2 #L1 #L #V #_ #IH #X #H elim (lsubr_inv_unit1 … H) -H
+ /4 width=1 by lsubr_unit/
+]
+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 "static_2/notation/relations/stareqsn_5.ma".
+include "static_2/syntax/tdeq_ext.ma".
+include "static_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 "static_2/relocation/lifts_tdeq.ma".
+include "static_2/static/rex_drops.ma".
+include "static_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 "static_2/static/rex_fqup.ma".
+include "static_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 "static_2/s_computation/fqus_fqup.ma".
+include "static_2/static/rdeq_drops.ma".
+include "static_2/static/rdeq_fqup.ma".
+include "static_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 "static_2/relocation/lifts_tdeq.ma".
+include "static_2/static/rex_length.ma".
+include "static_2/static/rex_fsle.ma".
+include "static_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 "static_2/syntax/ext2_ext2.ma".
+include "static_2/syntax/tdeq_tdeq.ma".
+include "static_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 "static_2/static/req_fsle.ma".
+include "static_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 "static_2/notation/relations/ideqsn_3.ma".
+include "static_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 "static_2/static/rex_fqup.ma".
+include "static_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 "static_2/static/rex_length.ma".
+include "static_2/static/rex_fsle.ma".
+include "static_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 "static_2/notation/relations/relation_4.ma".
+include "static_2/syntax/cext2.ma".
+include "static_2/relocation/sex.ma".
+include "static_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 "static_2/relocation/drops_cext2.ma".
+include "static_2/relocation/drops_sex.ma".
+include "static_2/static/frees_drops.ma".
+include "static_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 "static_2/static/frees_fqup.ma".
+include "static_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 "static_2/relocation/sex_length.ma".
+include "static_2/static/fsle_fsle.ma".
+include "static_2/static/rex_drops.ma".
+include "static_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 "static_2/relocation/sex_length.ma".
+include "static_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 "static_2/relocation/lex.ma".
+include "static_2/static/rex_fsle.ma".
+include "static_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 "static_2/relocation/sex_sex.ma".
+include "static_2/static/frees_fqup.ma".
+include "static_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-.
--- /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 *)
+(* *)
+(**************************************************************************)
+
+(* THE FORMAL SYSTEM λδ: MATITA SOURCE FILES
+ * Initial invocation: - Patience on me to gain peace and perfection! -
+ *)
+
+include "ground_2/lib/relations.ma".
+include "static_2/notation/functions/item0_0.ma".
+include "static_2/notation/functions/snitem2_2.ma".
+
+(* ATOMIC ARITY *************************************************************)
+
+inductive aarity: Type[0] ≝
+ | AAtom: aarity (* atomic aarity construction *)
+ | APair: aarity → aarity → aarity (* binary aarity construction *)
+.
+
+interpretation "atomic arity construction (atomic)"
+ 'Item0 = AAtom.
+
+interpretation "atomic arity construction (binary)"
+ 'SnItem2 A1 A2 = (APair A1 A2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact destruct_apair_apair_aux: ∀A1,A2,B1,B2. ②B1.A1 = ②B2.A2 → B1 = B2 ∧ A1 = A2.
+#A1 #A2 #B1 #B2 #H destruct /2 width=1 by conj/
+qed-.
+
+lemma discr_apair_xy_x: ∀A,B. ②B.A = B → ⊥.
+#A #B elim B -B
+[ #H destruct
+| #Y #X #IHY #_ #H elim (destruct_apair_apair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+]
+qed-.
+
+(* Basic_2A1: was: discr_tpair_xy_y *)
+lemma discr_apair_xy_y: ∀B,A. ②B. A = A → ⊥.
+#B #A elim A -A
+[ #H destruct
+| #Y #X #_ #IHX #H elim (destruct_apair_apair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+]
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma eq_aarity_dec: ∀A1,A2:aarity. Decidable (A1 = A2).
+#A1 elim A1 -A1
+[ #A2 elim A2 -A2 /2 width=1 by or_introl/
+ #B2 #A2 #_ #_ @or_intror #H destruct
+| #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
+ [ -IHB1 -IHA1 @or_intror #H destruct
+ | #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
+ [ #H destruct elim (IHA1 A2) -IHA1
+ [ #H destruct /2 width=1 by or_introl/
+ | #HA12 @or_intror #H destruct /2 width=1 by/
+ ]
+ | -IHA1 #HB12 @or_intror #H destruct /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 "static_2/notation/functions/snitem_2.ma".
+include "static_2/notation/functions/snbind1_2.ma".
+include "static_2/notation/functions/snbind2_3.ma".
+include "static_2/notation/functions/snvoid_1.ma".
+include "static_2/notation/functions/snabbr_2.ma".
+include "static_2/notation/functions/snabst_2.ma".
+include "static_2/syntax/lenv.ma".
+
+(* APPEND FOR LOCAL ENVIRONMENTS ********************************************)
+
+rec definition append L K on K ≝ match K with
+[ LAtom ⇒ L
+| LBind K I ⇒ (append L K).ⓘ{I}
+].
+
+interpretation "append (local environment)" 'plus L1 L2 = (append L1 L2).
+
+interpretation "local environment tail binding construction (generic)"
+ 'SnItem I L = (append (LBind LAtom I) L).
+
+interpretation "local environment tail binding construction (unary)"
+ 'SnBind1 I L = (append (LBind LAtom (BUnit I)) L).
+
+interpretation "local environment tail binding construction (binary)"
+ 'SnBind2 I T L = (append (LBind LAtom (BPair I T)) L).
+
+interpretation "tail exclusion (local environment)"
+ 'SnVoid L = (append (LBind LAtom (BUnit Void)) L).
+
+interpretation "tail abbreviation (local environment)"
+ 'SnAbbr T L = (append (LBind LAtom (BPair Abbr T)) L).
+
+interpretation "tail abstraction (local environment)"
+ 'SnAbst L T = (append (LBind LAtom (BPair Abst T)) L).
+
+definition d_appendable_sn: predicate (lenv→relation term) ≝ λR.
+ ∀K,T1,T2. R K T1 T2 → ∀L. R (L+K) T1 T2.
+
+(* Basic properties *********************************************************)
+
+lemma append_atom: ∀L. (L + ⋆) = L. (**) (* () should be redundant *)
+// qed.
+
+(* Basic_2A1: uses: append_pair *)
+lemma append_bind: ∀I,L,K. L+(K.ⓘ{I}) = (L+K).ⓘ{I}.
+// qed.
+
+lemma append_atom_sn: ∀L. ⋆ + L = L.
+#L elim L -L //
+#L #I >append_bind //
+qed.
+
+lemma append_assoc: associative … append.
+#L1 #L2 #L3 elim L3 -L3 //
+qed.
+
+lemma append_shift: ∀L,K,I. L+(ⓘ{I}.K) = (L.ⓘ{I})+K.
+#L #K #I <append_assoc //
+qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma append_inv_atom3_sn: ∀L,K. ⋆ = L + K → ∧∧ ⋆ = L & ⋆ = K.
+#L * /2 width=1 by conj/
+#K #I >append_bind #H destruct
+qed-.
+
+lemma append_inv_bind3_sn: ∀I0,L,L0,K. L0.ⓘ{I0} = L + K →
+ ∨∨ ∧∧ L0.ⓘ{I0} = L & ⋆ = K
+ | ∃∃K0. K = K0.ⓘ{I0} & L0 = L + K0.
+#I0 #L #L0 * /3 width=1 by or_introl, conj/
+#K #I >append_bind #H destruct /3 width=3 by ex2_intro, or_intror/
+qed-.
+
+lemma append_inj_sn: ∀K,L1,L2. L1+K = L2+K → L1 = L2.
+#K elim K -K //
+#K #I #IH #L1 #L2 >append_bind #H
+elim (destruct_lbind_lbind_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+qed-.
+
+(* Basic_1: uses: chead_ctail *)
+(* Basic_2A1: uses: lpair_ltail *)
+lemma lenv_case_tail: ∀L. L = ⋆ ∨ ∃∃K,I. L = ⓘ{I}.K.
+#L elim L -L /2 width=1 by or_introl/
+#L #I * [2: * ] /3 width=3 by ex1_2_intro, or_intror/
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/lenv_length.ma".
+include "static_2/syntax/append.ma".
+
+(* APPEND FOR LOCAL ENVIRONMENTS ********************************************)
+
+(* Properties with length for local environments ****************************)
+
+lemma append_length: ∀L1,L2. |L1 + L2| = |L1| + |L2|.
+#L1 #L2 elim L2 -L2 //
+#L2 #I >append_bind >length_bind >length_bind //
+qed.
+
+lemma ltail_length: ∀I,L. |ⓘ{I}.L| = ↑|L|.
+#I #L >append_length //
+qed.
+
+(* Advanced inversion lemmas on length for local environments ***************)
+
+(* Basic_2A1: was: length_inv_pos_dx_ltail *)
+lemma length_inv_succ_dx_ltail: ∀L,n. |L| = ↑n →
+ ∃∃I,K. |K| = n & L = ⓘ{I}.K.
+#Y #n #H elim (length_inv_succ_dx … H) -H #I #L #Hn #HLK destruct
+elim (lenv_case_tail … L) [2: * #K #J ]
+#H destruct /2 width=4 by ex2_2_intro/
+qed-.
+
+(* Basic_2A1: was: length_inv_pos_sn_ltail *)
+lemma length_inv_succ_sn_ltail: ∀L,n. ↑n = |L| →
+ ∃∃I,K. n = |K| & L = ⓘ{I}.K.
+#Y #n #H elim (length_inv_succ_sn … H) -H #I #L #Hn #HLK destruct
+elim (lenv_case_tail … L) [2: * #K #J ]
+#H destruct /2 width=4 by ex2_2_intro/
+qed-.
+
+(* Inversion lemmas with length for local environments **********************)
+
+(* Basic_2A1: was: append_inj_sn *)
+lemma append_inj_length_sn: ∀K1,K2,L1,L2. L1 + K1 = L2 + K2 → |K1| = |K2| →
+ L1 = L2 ∧ K1 = K2.
+#K1 elim K1 -K1
+[ * /2 width=1 by conj/
+ #K2 #I2 #L1 #L2 #_ >length_atom >length_bind
+ #H destruct
+| #K1 #I1 #IH *
+ [ #L1 #L2 #_ >length_atom >length_bind
+ #H destruct
+ | #K2 #I2 #L1 #L2 #H1 >length_bind >length_bind #H2
+ elim (destruct_lbind_lbind_aux … H1) -H1 #H1 #H3 destruct (**) (* destruct lemma needed *)
+ elim (IH … H1) -IH -H1 /3 width=4 by conj/
+ ]
+]
+qed-.
+
+(* Note: lemma 750 *)
+(* Basic_2A1: was: append_inj_dx *)
+lemma append_inj_length_dx: ∀K1,K2,L1,L2. L1 + K1 = L2 + K2 → |L1| = |L2| →
+ L1 = L2 ∧ K1 = K2.
+#K1 elim K1 -K1
+[ * /2 width=1 by conj/
+ #K2 #I2 #L1 #L2 >append_atom >append_bind #H destruct
+ >length_bind >append_length >plus_n_Sm
+ #H elim (plus_xSy_x_false … H)
+| #K1 #I1 #IH *
+ [ #L1 #L2 >append_bind >append_atom #H destruct
+ >length_bind >append_length >plus_n_Sm #H
+ lapply (discr_plus_x_xy … H) -H #H destruct
+ | #K2 #I2 #L1 #L2 >append_bind >append_bind #H1 #H2
+ elim (destruct_lbind_lbind_aux … H1) -H1 #H1 #H3 destruct (**) (* destruct lemma needed *)
+ elim (IH … H1) -IH -H1 /2 width=1 by conj/
+ ]
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma append_inj_dx: ∀L,K1,K2. L+K1 = L+K2 → K1 = K2.
+#L #K1 #K2 #H elim (append_inj_length_dx … H) -H //
+qed-.
+
+lemma append_inv_refl_dx: ∀L,K. L+K = L → K = ⋆.
+#L #K #H elim (append_inj_dx … (⋆) … H) //
+qed-.
+
+lemma append_inv_pair_dx: ∀I,L,K,V. L+K = L.ⓑ{I}V → K = ⋆.ⓑ{I}V.
+#I #L #K #V #H elim (append_inj_dx … (⋆.ⓑ{I}V) … H) //
+qed-.
+
+(* Basic eliminators ********************************************************)
+
+(* Basic_1: was: c_tail_ind *)
+(* Basic_2A1: was: lenv_ind_alt *)
+lemma lenv_ind_tail: ∀Q:predicate lenv.
+ Q (⋆) → (∀I,L. Q L → Q (ⓘ{I}.L)) → ∀L. Q L.
+#Q #IH1 #IH2 #L @(f_ind … length … L) -L #x #IHx * //
+#L #I -IH1 #H destruct
+elim (lenv_case_tail … L) [2: * #K #J ]
+#H destruct /3 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 "static_2/syntax/term.ma".
+
+(* BINDERS FOR LOCAL ENVIRONMENTS ******************************************)
+
+inductive bind: Type[0] ≝
+| BUnit: bind1 → bind
+| BPair: bind2 → term → bind
+.
+
+(* Basic properties ********************************************************)
+
+lemma eq_bind_dec: ∀I1,I2:bind. Decidable (I1 = I2).
+* #I1 [2: #V1 ] * #I2 [2,4: #V2 ]
+[1: elim (eq_bind2_dec I1 I2) #HI
+ [ elim (eq_term_dec V1 V2) #HV ]
+|4: elim (eq_bind1_dec I1 I2) #HI
+]
+/2 width=1 by or_introl/
+@or_intror #H destruct /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 "static_2/syntax/term_weight.ma".
+include "static_2/syntax/bind.ma".
+
+(* WEIGHT OF A BINDER FOR LOCAL ENVIRONMENTS *******************************)
+
+rec definition bw I ≝ match I with
+[ BUnit _ ⇒ 1
+| BPair _ V ⇒ ♯{V}
+].
+
+interpretation "weight (binder for local environments)" 'Weight I = (bw I).
+
+(* Basic properties *********************************************************)
+
+lemma bw_pos: ∀I. 1 ≤ ♯{I}.
+* //
+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 "static_2/syntax/cext2.ma".
+
+(* CONTEXT-AWARE SYNTACTIC EQUIVALENCE FOR BINDERS **************************)
+
+definition ceq_ext: lenv → relation bind ≝
+ cext2 ceq.
+
+(* Basic properties *********************************************************)
+
+lemma ceq_ext_refl (L): reflexive … (ceq_ext L).
+/2 width=1 by ext2_refl/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma ceq_ext_inv_eq: ∀L,I1,I2. ceq_ext L I1 I2 → I1 = I2.
+#L #I1 #I2 * -I1 -I2 //
+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 "static_2/syntax/ceq_ext.ma".
+
+(* CONTEXT-AWARE SYNTACTIC EQUIVALENCE FOR BINDERS **************************)
+
+(* Main properties **********************************************************)
+
+theorem ceq_ext_trans: ∀L1,I1,I. ceq_ext L1 I1 I →
+ ∀L2,I2. ceq_ext L2 I I2 → ∀L3. ceq_ext L3 I1 I2.
+#L1 #I1 #I * -I1 -I //
+#I1 #V1 #V #HV1 #L2 #Z #H elim (ext2_inv_pair_sn … 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 "static_2/syntax/ext2.ma".
+include "static_2/syntax/lenv.ma".
+
+(* EXTENSION TO BINDERS OF A CONTEXT-SENSITIVE RELATION FOR TERMS ***********)
+
+definition cext2: (lenv → relation term) → lenv → relation bind ≝
+ λR,L. ext2 (R L).
+
+(* Basic properties *********************************************************)
+
+lemma cext2_sym: ∀R. (∀L1,L2,T1,T2. R L1 T1 T2 → R L2 T2 T1) →
+ ∀L1,L2,I1,I2. cext2 R L1 I1 I2 → cext2 R L2 I2 I1.
+#R #HR #L1 #L2 #I1 #I2 * /3 width=2 by ext2_pair/
+qed-.
+
+lemma cext2_co: ∀R1,R2. (∀L,T1,T2. R1 L T1 T2 → R2 L T1 T2) →
+ ∀L,I1,I2. cext2 R1 L I1 I2 → cext2 R2 L I1 I2.
+#R1 #R2 #HR #L #I1 #I2 * /3 width=2 by ext2_unit, ext2_pair/
+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 "static_2/notation/functions/weight_2.ma".
+include "static_2/syntax/lenv_weight.ma".
+
+(* WEIGHT OF A RESTRICTED CLOSURE *******************************************)
+
+definition rfw: lenv → term → ? ≝ λL,T. ♯{L} + ♯{T}.
+
+interpretation "weight (restricted closure)" 'Weight L T = (rfw L T).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: flt_shift *)
+lemma rfw_shift: ∀p,I,K,V,T. ♯{K.ⓑ{I}V, T} < ♯{K, ⓑ{p,I}V.T}.
+normalize /2 width=1 by monotonic_le_plus_r/
+qed.
+
+lemma rfw_clear: ∀p,I1,I2,K,V,T. ♯{K.ⓤ{I1}, T} < ♯{K, ⓑ{p,I2}V.T}.
+normalize /4 width=1 by monotonic_le_plus_r, le_S_S/
+qed.
+
+lemma rfw_tpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L, ②{I}V.T}.
+normalize in ⊢ (?→?→?→?→?%%); //
+qed.
+
+lemma rfw_tpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L, ②{I}V.T}.
+normalize in ⊢ (?→?→?→?→?%%); //
+qed.
+
+lemma rfw_lpair_sn: ∀I,L,V,T. ♯{L, V} < ♯{L.ⓑ{I}V, T}.
+normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
+qed.
+
+lemma rfw_lpair_dx: ∀I,L,V,T. ♯{L, T} < ♯{L.ⓑ{I}V, T}.
+normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
+qed.
+
+(* Basic_1: removed theorems 7:
+ flt_thead_sx flt_thead_dx flt_trans
+ flt_arith0 flt_arith1 flt_arith2 flt_wf_ind
+*)
+(* Basic_1: removed local theorems 1: q_ind *)
--- /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 "static_2/notation/functions/weight_3.ma".
+include "static_2/syntax/lenv_weight.ma".
+include "static_2/syntax/genv.ma".
+
+(* WEIGHT OF A CLOSURE ******************************************************)
+
+(* activate genv *)
+definition fw: genv → lenv → term → ? ≝ λG,L,T. ♯{L} + ♯{T}.
+
+interpretation "weight (closure)" 'Weight G L T = (fw G L T).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: flt_shift *)
+lemma fw_shift: ∀p,I,G,K,V,T. ♯{G, K.ⓑ{I}V, T} < ♯{G, K, ⓑ{p,I}V.T}.
+normalize /2 width=1 by monotonic_le_plus_r/
+qed.
+
+lemma fw_clear: ∀p,I1,I2,G,K,V,T. ♯{G, K.ⓤ{I1}, T} < ♯{G, K, ⓑ{p,I2}V.T}.
+normalize /4 width=1 by monotonic_le_plus_r, le_S_S/
+qed.
+
+lemma fw_tpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L, ②{I}V.T}.
+normalize in ⊢ (?→?→?→?→?→?%%); //
+qed.
+
+lemma fw_tpair_dx: ∀I,G,L,V,T. ♯{G, L, T} < ♯{G, L, ②{I}V.T}.
+normalize in ⊢ (?→?→?→?→?→?%%); //
+qed.
+
+lemma fw_lpair_sn: ∀I,G,L,V,T. ♯{G, L, V} < ♯{G, L.ⓑ{I}V, T}.
+normalize /3 width=1 by monotonic_lt_plus_l, monotonic_le_plus_r/
+qed.
+
+(* Basic_1: removed theorems 7:
+ flt_thead_sx flt_thead_dx flt_trans
+ flt_arith0 flt_arith1 flt_arith2 flt_wf_ind
+*)
+(* Basic_1: removed local theorems 1: q_ind *)
--- /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 "static_2/syntax/bind.ma".
+
+(* EXTENSION TO BINDERS OF A RELATION FOR TERMS *****************************)
+
+inductive ext2 (R:relation term): relation bind ≝
+| ext2_unit: ∀I. ext2 R (BUnit I) (BUnit I)
+| ext2_pair: ∀I,V1,V2. R V1 V2 → ext2 R (BPair I V1) (BPair I V2)
+.
+
+(* Basic_inversion lemmas **************************************************)
+
+fact ext2_inv_unit_sn_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
+ ∀I. Z1 = BUnit I → Z2 = BUnit I.
+#R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #_ ]
+#J #H destruct //
+qed-.
+
+lemma ext2_inv_unit_sn: ∀R,I,Z2. ext2 R (BUnit I) Z2 → Z2 = BUnit I.
+/2 width=4 by ext2_inv_unit_sn_aux/ qed-.
+
+fact ext2_inv_pair_sn_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
+ ∀I,V1. Z1 = BPair I V1 →
+ ∃∃V2. R V1 V2 & Z2 = BPair I V2.
+#R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #HV12 ]
+#J #W1 #H destruct /2 width=3 by ex2_intro/
+qed-.
+
+lemma ext2_inv_pair_sn: ∀R,Z2,I,V1. ext2 R (BPair I V1) Z2 →
+ ∃∃V2. R V1 V2 & Z2 = BPair I V2.
+/2 width=3 by ext2_inv_pair_sn_aux/ qed-.
+
+fact ext2_inv_unit_dx_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
+ ∀I. Z2 = BUnit I → Z1 = BUnit I.
+#R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #_ ]
+#J #H destruct //
+qed-.
+
+lemma ext2_inv_unit_dx: ∀R,I,Z1. ext2 R Z1 (BUnit I) → Z1 = BUnit I.
+/2 width=4 by ext2_inv_unit_dx_aux/ qed-.
+
+fact ext2_inv_pair_dx_aux: ∀R,Z1,Z2. ext2 R Z1 Z2 →
+ ∀I,V2. Z2 = BPair I V2 →
+ ∃∃V1. R V1 V2 & Z1 = BPair I V1.
+#R #Z1 #Z2 * -Z1 -Z2 #I [2: #V1 #V2 #HV12 ]
+#J #W2 #H destruct /2 width=3 by ex2_intro/
+qed-.
+
+lemma ext2_inv_pair_dx: ∀R,Z1,I,V2. ext2 R Z1 (BPair I V2) →
+ ∃∃V1. R V1 V2 & Z1 = BPair I V1.
+/2 width=3 by ext2_inv_pair_dx_aux/ qed-.
+
+(* Advanced inversion lemmas ***********************************************)
+
+lemma ext2_inv_unit: ∀R,I1,I2. ext2 R (BUnit I1) (BUnit I2) → I1 = I2.
+#R #I1 #I2 #H lapply (ext2_inv_unit_sn … H) -H
+#H destruct //
+qed-.
+
+lemma ext2_inv_pair: ∀R,I1,I2,V1,V2. ext2 R (BPair I1 V1) (BPair I2 V2) →
+ I1 = I2 ∧ R V1 V2.
+#R #I1 #I2 #V1 #V2 #H elim (ext2_inv_pair_sn … H) -H
+#V #HV #H destruct /2 width=1 by conj/
+qed-.
+
+(* Basic properties ********************************************************)
+
+lemma ext2_refl: ∀R. reflexive … R → reflexive … (ext2 R).
+#R #HR * /2 width=1 by ext2_pair/
+qed.
+
+lemma ext2_sym: ∀R. symmetric … R → symmetric … (ext2 R).
+#R #HR #T1 #T2 * /3 width=1 by ext2_unit, ext2_pair/
+qed-.
+
+lemma ext2_dec: ∀R. (∀T1,T2. Decidable (R T1 T2)) →
+ ∀I1,I2. Decidable (ext2 R I1 I2).
+#R #HR * #I1 [2: #T1 ] * #I2 [2,4: #T2 ]
+[ elim (eq_bind2_dec I1 I2) #HI12 destruct
+ [ elim (HR T1 T2) -HR #HT12 /3 width=1 by ext2_pair, or_introl/ ]
+ @or_intror #H elim (ext2_inv_pair … H) -H /2 width=1 by/
+| @or_intror #H lapply (ext2_inv_unit_sn … H) -H
+ #H destruct
+| @or_intror #H lapply (ext2_inv_unit_dx … H) -H
+ #H destruct
+| elim (eq_bind1_dec I1 I2) #HI12 destruct
+ /4 width=2 by ext2_inv_unit, ext2_unit, 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 "static_2/syntax/ext2.ma".
+
+(* EXTENSION TO BINDERS OF A RELATION FOR TERMS *****************************)
+
+(* Main properties **********************************************************)
+
+theorem ext2_trans: ∀R. Transitive … R → Transitive … (ext2 R).
+#R #HR #I1 #I #H elim H -I1 -I
+[ #I1 #J #H >(ext2_inv_unit_sn … H) -J /2 width=1 by ext2_unit/
+| #I1 #V1 #V #HV1 #J #H elim (ext2_inv_pair_sn … H) -H
+ #V2 #HV2 #H destruct /3 width=3 by ext2_pair/
+]
+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 "static_2/syntax/ext2.ma".
+
+(* EXTENSION TO BINDERS OF A RELATION FOR TERMS *****************************)
+
+(* Properties with transitive closure ***************************************)
+
+lemma ext2_tc_pair: ∀R,I,V1,V2. TC … R V1 V2 →
+ TC … (ext2 R) (BPair I V1) (BPair I V2).
+#R #I #V1 #V2 #H elim H -H -V2
+/3 width=3 by ext2_pair, step, inj/
+qed.
+
+lemma ext2_tc_inj: ∀R,I1,I2. ext2 R I1 I2 → ext2 (TC … R) I1 I2.
+#R #I1 #I2 * -I1 -I2
+/3 width=1 by ext2_unit, ext2_pair, inj/
+qed.
+
+(* Main properties with transitive closure **********************************)
+
+theorem ext2_tc_step: ∀R,I1,I. ext2 (TC … R) I1 I →
+ ∀I2. ext2 R I I2 → ext2 (TC … R) I1 I2.
+#R #I1 #I * -I1 -I
+[ #I #Z #H >(ext2_inv_unit_sn … H) -Z /2 width=1 by ext2_unit/
+| #I #V1 #V #HV1 #Z #H
+ elim (ext2_inv_pair_sn … H) -H #V2 #HV2 #H destruct
+ /3 width=3 by ext2_pair, step/
+]
+qed-.
+
+(* Advanced properties with transitive closure ******************************)
+
+lemma ext2_tc: ∀R,I1,I2. TC … (ext2 R) I1 I2 → ext2 (TC … R) I1 I2.
+#R #I1 #I2 #H elim H -I2
+/2 width=3 by ext2_tc_step, ext2_tc_inj/
+qed.
+
+(* Advanced inversion lemmas with transitive closure ************************)
+
+lemma ext2_inv_tc: ∀R,I1,I2. ext2 (TC … R) I1 I2 → TC … (ext2 R) I1 I2.
+#R #I1 #I2 * -I1 -I2
+/3 width=1 by ext2_tc_pair, ext2_unit, 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 "static_2/notation/functions/star_0.ma".
+include "static_2/notation/functions/dxbind2_3.ma".
+include "static_2/notation/functions/dxabbr_2.ma".
+include "static_2/notation/functions/dxabst_2.ma".
+include "static_2/syntax/term.ma".
+
+(* GLOBAL ENVIRONMENTS ******************************************************)
+
+(* global environments *)
+inductive genv: Type[0] ≝
+| GAtom: genv (* empty *)
+| GPair: genv → bind2 → term → genv (* binary binding construction *)
+.
+
+interpretation "sort (global environment)"
+ 'Star = (GAtom).
+
+interpretation "global environment binding construction (binary)"
+ 'DxBind2 G I T = (GPair G I T).
+
+interpretation "abbreviation (global environment)"
+ 'DxAbbr G T = (GPair G Abbr T).
+
+interpretation "abstraction (global environment)"
+ 'DxAbst G T = (GPair G Abst T).
+
+(* Basic properties *********************************************************)
+
+lemma eq_genv_dec: ∀G1,G2:genv. Decidable (G1 = G2).
+#G1 elim G1 -G1 [| #G1 #I1 #T1 #IHG1 ] * [2,4: #G2 #I2 #T2 ]
+[3: /2 width=1 by or_introl/
+|2: elim (eq_bind2_dec I1 I2) #HI
+ [ elim (IHG1 G2) -IHG1 #HG
+ [ elim (eq_term_dec T1 T2) #HT /2 width=1 by or_introl/ ]
+ ]
+]
+@or_intror #H destruct /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 "static_2/syntax/genv.ma".
+
+(* LENGTH OF A GLOBAL ENVIRONMENT *******************************************)
+
+rec definition glength G on G ≝ match G with
+[ GAtom ⇒ 0
+| GPair G _ _ ⇒ ↑(glength G)
+].
+
+interpretation "length (global environment)"
+ 'card G = (glength G).
--- /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 "static_2/syntax/term_weight.ma".
+include "static_2/syntax/genv.ma".
+
+(* WEIGHT OF A GLOBAL ENVIRONMENT *******************************************)
+
+rec definition gw G ≝ match G with
+[ GAtom ⇒ 0
+| GPair G I T ⇒ gw G + ♯{T}
+].
+
+interpretation "weight (global environment)" 'Weight G = (gw G).
+
+(* Basic properties *********************************************************)
+
+lemma gw_pair: ∀I,G,T. ♯{G} < ♯{G.ⓑ{I}T}.
+normalize /2 width=1 by monotonic_le_plus_r/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/lib/bool.ma".
+include "ground_2/lib/arith.ma".
+
+(* ITEMS ********************************************************************)
+
+(* atomic items *)
+inductive item0: Type[0] ≝
+ | Sort: nat → item0 (* sort: starting at 0 *)
+ | LRef: nat → item0 (* reference by index: starting at 0 *)
+ | GRef: nat → item0 (* reference by position: starting at 0 *)
+.
+
+(* unary binding items *)
+inductive bind1: Type[0] ≝
+ | Void: bind1 (* exclusion *)
+.
+
+(* binary binding items *)
+inductive bind2: Type[0] ≝
+ | Abbr: bind2 (* abbreviation *)
+ | Abst: bind2 (* abstraction *)
+.
+
+(* binary non-binding items *)
+inductive flat2: Type[0] ≝
+ | Appl: flat2 (* application *)
+ | Cast: flat2 (* explicit type annotation *)
+.
+
+(* binary items *)
+inductive item2: Type[0] ≝
+ | Bind2: bool → bind2 → item2 (* polarized binding item *)
+ | Flat2: flat2 → item2 (* non-binding item *)
+.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact destruct_sort_sort_aux: ∀s1,s2. Sort s1 = Sort s2 → s1 = s2.
+#s1 #s2 #H destruct //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma eq_item0_dec: ∀I1,I2:item0. Decidable (I1 = I2).
+* #i1 * #i2 [2,3,4,6,7,8: @or_intror #H destruct ]
+[2: elim (eq_nat_dec i1 i2) |1,3: elim (eq_nat_dec i1 i2) ] /2 width=1 by or_introl/
+#Hni12 @or_intror #H destruct /2 width=1 by/
+qed-.
+
+lemma eq_bind1_dec: ∀I1,I2:bind1. Decidable (I1 = I2).
+* * /2 width=1 by or_introl/
+qed-.
+
+(* Basic_1: was: bind_dec *)
+lemma eq_bind2_dec: ∀I1,I2:bind2. Decidable (I1 = I2).
+* * /2 width=1 by or_introl/
+@or_intror #H destruct
+qed-.
+
+(* Basic_1: was: flat_dec *)
+lemma eq_flat2_dec: ∀I1,I2:flat2. Decidable (I1 = I2).
+* * /2 width=1 by or_introl/
+@or_intror #H destruct
+qed-.
+
+(* Basic_1: was: kind_dec *)
+lemma eq_item2_dec: ∀I1,I2:item2. Decidable (I1 = I2).
+* [ #p1 ] #I1 * [1,3: #p2 ] #I2
+[2,3: @or_intror #H destruct
+| elim (eq_bool_dec p1 p2) #Hp
+ [ elim (eq_bind2_dec I1 I2) /2 width=1 by or_introl/ #HI ]
+ @or_intror #H destruct /2 width=1 by/
+| elim (eq_flat2_dec I1 I2) /2 width=1 by or_introl/ #HI
+ @or_intror #H destruct /2 width=1 by/
+]
+qed-.
+
+(* Basic_1: removed theorems 21:
+ s_S s_plus s_plus_sym s_minus minus_s_s s_le s_lt s_inj s_inc
+ s_arith0 s_arith1
+ r_S r_plus r_plus_sym r_minus r_dis s_r r_arith0 r_arith1
+ not_abbr_abst bind_dec_not
+*)
--- /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 "static_2/syntax/item_sh.ma".
+
+(* SORT DEGREE **************************************************************)
+
+(* sort degree specification *)
+record sd (h:sh): Type[0] ≝ {
+ deg : relation nat; (* degree of the sort *)
+ deg_total: ∀s. ∃d. deg s d; (* functional relation axioms *)
+ deg_mono : ∀s,d1,d2. deg s d1 → deg s d2 → d1 = d2;
+ deg_next : ∀s,d. deg s d → deg (next h s) (↓d) (* compatibility condition *)
+}.
+
+(* Notable specifications ***************************************************)
+
+definition deg_O: relation nat ≝ λs,d. d = 0.
+
+definition sd_O: ∀h. sd h ≝ λh. mk_sd h deg_O ….
+/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ defined.
+
+(* Basic_2A1: includes: deg_SO_pos *)
+inductive deg_SO (h:sh) (s:nat) (s0:nat): predicate nat ≝
+| deg_SO_succ : ∀n. (next h)^n s0 = s → deg_SO h s s0 (↑n)
+| deg_SO_zero: ((∃n. (next h)^n s0 = s) → ⊥) → deg_SO h s s0 0
+.
+
+fact deg_SO_inv_succ_aux: ∀h,s,s0,n0. deg_SO h s s0 n0 → ∀n. n0 = ↑n →
+ (next h)^n s0 = s.
+#h #s #s0 #n0 * -n0
+[ #n #Hn #x #H destruct //
+| #_ #x #H destruct
+]
+qed-.
+
+(* Basic_2A1: was: deg_SO_inv_pos *)
+lemma deg_SO_inv_succ: ∀h,s,s0,n. deg_SO h s s0 (↑n) → (next h)^n s0 = s.
+/2 width=3 by deg_SO_inv_succ_aux/ qed-.
+
+lemma deg_SO_refl: ∀h,s. deg_SO h s s 1.
+#h #s @(deg_SO_succ … 0 ?) //
+qed.
+
+lemma deg_SO_gt: ∀h,s1,s2. s1 < s2 → deg_SO h s1 s2 0.
+#h #s1 #s2 #HK12 @deg_SO_zero * #n elim n -n normalize
+[ #H destruct
+ elim (lt_refl_false … HK12)
+| #n #_ #H
+ lapply (next_lt h ((next h)^n s2)) >H -H #H
+ lapply (transitive_lt … H HK12) -s1 #H1
+ lapply (nexts_le h s2 n) #H2
+ lapply (le_to_lt_to_lt … H2 H1) -h -n #H
+ elim (lt_refl_false … H)
+]
+qed.
+
+definition sd_SO: ∀h. nat → sd h ≝ λh,s. mk_sd h (deg_SO h s) ….
+[ #s0
+ lapply (nexts_dec h s0 s) *
+ [ * /3 width=2 by deg_SO_succ, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
+| #K0 #d1 #d2 * [ #n1 ] #H1 * [1,3: #n2 ] #H2 //
+ [ < H2 in H1; -H2 #H
+ lapply (nexts_inj … H) -H #H destruct //
+ | elim H1 /2 width=2 by ex_intro/
+ | elim H2 /2 width=2 by ex_intro/
+ ]
+| #s0 #n *
+ [ #d #H destruct elim d -d normalize
+ /2 width=1 by deg_SO_gt, deg_SO_succ, next_lt/
+ | #H1 @deg_SO_zero * #d #H2 destruct
+ @H1 -H1 @(ex_intro … (↑d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
+ ]
+]
+defined.
+
+rec definition sd_d (h:sh) (s:nat) (d:nat) on d : sd h ≝
+ match d with
+ [ O ⇒ sd_O h
+ | S d ⇒ match d with
+ [ O ⇒ sd_SO h s
+ | _ ⇒ sd_d h (next h s) d
+ ]
+ ].
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma deg_inv_pred: ∀h,o,s,d. deg h o (next h s) (↑d) → deg h o s (↑↑d).
+#h #o #s #d #H1
+elim (deg_total h o s) #n #H0
+lapply (deg_next … H0) #H2
+lapply (deg_mono … H1 H2) -H1 -H2 #H >H >S_pred /2 width=2 by ltn_to_ltO/
+qed-.
+
+lemma deg_inv_prec: ∀h,o,s,n,d. deg h o ((next h)^n s) (↑d) → deg h o s (↑(d+n)).
+#h #o #s #n elim n -n normalize /3 width=1 by deg_inv_pred/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma deg_iter: ∀h,o,s,d,n. deg h o s d → deg h o ((next h)^n s) (d-n).
+#h #o #s #d #n elim n -n normalize /3 width=1 by deg_next/
+qed.
+
+lemma deg_next_SO: ∀h,o,s,d. deg h o s (↑d) → deg h o (next h s) d.
+/2 width=1 by deg_next/ qed-.
+
+lemma sd_d_SS: ∀h,s,d. sd_d h s (↑↑d) = sd_d h (next h s) (↑d).
+// qed.
+
+lemma sd_d_correct: ∀h,d,s. deg h (sd_d h s d) s d.
+#h #d elim d -d // #d elim d -d /3 width=1 by deg_inv_pred/
+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/arith.ma".
+
+(* SORT HIERARCHY ***********************************************************)
+
+(* sort hierarchy specification *)
+record sh: Type[0] ≝ {
+ next : nat → nat; (* next sort in the hierarchy *)
+ next_lt: ∀s. s < next s (* strict monotonicity condition *)
+}.
+
+definition sh_N: sh ≝ mk_sh S ….
+// defined.
+
+(* Basic properties *********************************************************)
+
+lemma nexts_le: ∀h,s,n. s ≤ (next h)^n s.
+#h #s #n elim n -n // normalize #n #IH
+lapply (next_lt h ((next h)^n s)) #H
+lapply (le_to_lt_to_lt … IH H) -IH -H /2 width=2 by lt_to_le/
+qed.
+
+lemma nexts_lt: ∀h,s,n. s < (next h)^(↑n) s.
+#h #s #n normalize
+lapply (nexts_le h s n) #H
+@(le_to_lt_to_lt … H) //
+qed.
+
+axiom nexts_dec: ∀h,s1,s2. Decidable (∃n. (next h)^n s1 = s2).
+
+axiom nexts_inj: ∀h,s,n1,n2. (next h)^n1 s = (next h)^n2 s → n1 = n2.
--- /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 "static_2/notation/functions/star_0.ma".
+include "static_2/notation/functions/dxitem_2.ma".
+include "static_2/notation/functions/dxbind1_2.ma".
+include "static_2/notation/functions/dxbind2_3.ma".
+include "static_2/notation/functions/dxvoid_1.ma".
+include "static_2/notation/functions/dxabbr_2.ma".
+include "static_2/notation/functions/dxabst_2.ma".
+include "static_2/syntax/bind.ma".
+
+(* LOCAL ENVIRONMENTS *******************************************************)
+
+(* local environments *)
+inductive lenv: Type[0] ≝
+| LAtom: lenv (* empty *)
+| LBind: lenv → bind → lenv (* binding construction *)
+.
+
+interpretation "sort (local environment)"
+ 'Star = LAtom.
+
+interpretation "local environment binding construction (generic)"
+ 'DxItem L I = (LBind L I).
+
+interpretation "local environment binding construction (unary)"
+ 'DxBind1 L I = (LBind L (BUnit I)).
+
+interpretation "local environment binding construction (binary)"
+ 'DxBind2 L I T = (LBind L (BPair I T)).
+
+interpretation "void (local environment)"
+ 'DxVoid L = (LBind L (BUnit Void)).
+
+interpretation "abbreviation (local environment)"
+ 'DxAbbr L T = (LBind L (BPair Abbr T)).
+
+interpretation "abstraction (local environment)"
+ 'DxAbst L T = (LBind L (BPair Abst T)).
+
+definition cfull: relation3 lenv bind bind ≝ λL,I1,I2. ⊤.
+
+definition ceq: relation3 lenv term term ≝ λL. eq ….
+
+(* Basic properties *********************************************************)
+
+lemma eq_lenv_dec: ∀L1,L2:lenv. Decidable (L1 = L2).
+#L1 elim L1 -L1 [| #L1 #I1 #IHL1 ] * [2,4: #L2 #I2 ]
+[3: /2 width=1 by or_introl/
+|2: elim (eq_bind_dec I1 I2) #HI
+ [ elim (IHL1 L2) -IHL1 #HL /2 width=1 by or_introl/ ]
+]
+@or_intror #H destruct /2 width=1 by/
+qed-.
+
+lemma cfull_dec: ∀L,T1,T2. Decidable (cfull L T1 T2).
+/2 width=1 by or_introl/ qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact destruct_lbind_lbind_aux: ∀I1,I2,L1,L2. L1.ⓘ{I1} = L2.ⓘ{I2} →
+ L1 = L2 ∧ I1 = I2.
+#I1 #I2 #L1 #L2 #H destruct /2 width=1 by conj/
+qed-.
+
+(* Basic_2A1: uses: discr_lpair_x_xy *)
+lemma discr_lbind_x_xy: ∀I,L. L = L.ⓘ{I} → ⊥.
+#I #L elim L -L
+[ #H destruct
+| #L #J #IHL #H elim (destruct_lbind_lbind_aux … H) -H (**) (* destruct lemma needed *)
+ #H1 #H2 destruct /2 width=1 by/
+]
+qed-.
+
+(* Basic_2A1: uses: discr_lpair_xy_x *)
+lemma discr_lbind_xy_x: ∀I,L. L.ⓘ{I} = L → ⊥.
+/2 width=4 by discr_lbind_x_xy/ 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 "static_2/syntax/lenv.ma".
+
+(* LENGTH OF A LOCAL ENVIRONMENT ********************************************)
+
+rec definition length L ≝ match L with
+[ LAtom ⇒ 0
+| LBind L _ ⇒ ↑(length L)
+].
+
+interpretation "length (local environment)" 'card L = (length L).
+
+(* Basic properties *********************************************************)
+
+lemma length_atom: |⋆| = 0.
+// qed.
+
+(* Basic_2A1: uses: length_pair *)
+lemma length_bind: ∀I,L. |L.ⓘ{I}| = ↑|L|.
+// qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma length_inv_zero_dx: ∀L. |L| = 0 → L = ⋆.
+* // #L #I >length_bind
+#H destruct
+qed-.
+
+lemma length_inv_zero_sn: ∀L. 0 = |L| → L = ⋆.
+/2 width=1 by length_inv_zero_dx/ qed-.
+
+(* Basic_2A1: was: length_inv_pos_dx *)
+lemma length_inv_succ_dx: ∀n,L. |L| = ↑n →
+ ∃∃I,K. |K| = n & L = K. ⓘ{I}.
+#n *
+[ >length_atom #H destruct
+| #L #I >length_bind /3 width=4 by ex2_2_intro, injective_S/
+]
+qed-.
+
+(* Basic_2A1: was: length_inv_pos_sn *)
+lemma length_inv_succ_sn: ∀n,L. ↑n = |L| →
+ ∃∃I,K. n = |K| & L = K. ⓘ{I}.
+#n #L #H lapply (sym_eq ??? H) -H
+#H elim (length_inv_succ_dx … H) -H /2 width=4 by ex2_2_intro/
+qed-.
+
+(* Basic_2A1: removed theorems 1: length_inj *)
--- /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 "static_2/syntax/bind_weight.ma".
+include "static_2/syntax/lenv.ma".
+
+(* WEIGHT OF A LOCAL ENVIRONMENT ********************************************)
+
+rec definition lw L ≝ match L with
+[ LAtom ⇒ 0
+| LBind L I ⇒ lw L + ♯{I}
+].
+
+interpretation "weight (local environment)" 'Weight L = (lw L).
+
+(* Basic properties *********************************************************)
+
+(* Basic_2A1: uses: lw_pair *)
+lemma lw_bind: ∀I,L. ♯{L} < ♯{L.ⓘ{I}}.
+normalize /2 width=1 by monotonic_le_plus_r/ qed.
+
+(* Basic_1: removed theorems 4: clt_cong clt_head clt_thead clt_wf_ind *)
+(* Basic_1: removed local theorems 1: clt_wf__q_ind *)
+(* Basic_1: note: clt_thead should be renamed clt_ctail *)
--- /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 "static_2/notation/relations/voidstareq_4.ma".
+include "static_2/syntax/lenv.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
+
+inductive lveq: bi_relation nat lenv ≝
+| lveq_atom : lveq 0 (⋆) 0 (⋆)
+| lveq_bind : ∀I1,I2,K1,K2. lveq 0 K1 0 K2 →
+ lveq 0 (K1.ⓘ{I1}) 0 (K2.ⓘ{I2})
+| lveq_void_sn: ∀K1,K2,n1. lveq n1 K1 0 K2 →
+ lveq (↑n1) (K1.ⓧ) 0 K2
+| lveq_void_dx: ∀K1,K2,n2. lveq 0 K1 n2 K2 →
+ lveq 0 K1 (↑n2) (K2.ⓧ)
+.
+
+interpretation "equivalence up to exclusion binders (local environment)"
+ 'VoidStarEq L1 n1 n2 L2 = (lveq n1 L1 n2 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lveq_refl: ∀L. L ≋ⓧ*[0, 0] L.
+#L elim L -L /2 width=1 by lveq_atom, lveq_bind/
+qed.
+
+lemma lveq_sym: bi_symmetric … lveq.
+#n1 #n2 #L1 #L2 #H elim H -L1 -L2 -n1 -n2
+/2 width=1 by lveq_atom, lveq_bind, lveq_void_sn, lveq_void_dx/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lveq_inv_zero_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ 0 = n1 → 0 = n2 →
+ ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
+ | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
+#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
+[1: /3 width=1 by or_introl, conj/
+|2: /3 width=7 by ex3_4_intro, or_intror/
+|*: #K1 #K2 #n #_ #H1 #H2 destruct
+]
+qed-.
+
+lemma lveq_inv_zero: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 →
+ ∨∨ ∧∧ ⋆ = L1 & ⋆ = L2
+ | ∃∃I1,I2,K1,K2. K1 ≋ⓧ*[0, 0] K2 & K1.ⓘ{I1} = L1 & K2.ⓘ{I2} = L2.
+/2 width=5 by lveq_inv_zero_aux/ qed-.
+
+fact lveq_inv_succ_sn_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ ∀m1. ↑m1 = n1 →
+ ∃∃K1. K1 ≋ⓧ*[m1, 0] L2 & K1.ⓧ = L1 & 0 = n2.
+#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
+[1: #m #H destruct
+|2: #I1 #I2 #K1 #K2 #_ #m #H destruct
+|*: #K1 #K2 #n #HK #m #H destruct /2 width=3 by ex3_intro/
+]
+qed-.
+
+lemma lveq_inv_succ_sn: ∀L1,K2,n1,n2. L1 ≋ⓧ*[↑n1, n2] K2 →
+ ∃∃K1. K1 ≋ⓧ*[n1, 0] K2 & K1.ⓧ = L1 & 0 = n2.
+/2 width=3 by lveq_inv_succ_sn_aux/ qed-.
+
+lemma lveq_inv_succ_dx: ∀K1,L2,n1,n2. K1 ≋ⓧ*[n1, ↑n2] L2 →
+ ∃∃K2. K1 ≋ⓧ*[0, n2] K2 & K2.ⓧ = L2 & 0 = n1.
+#K1 #L2 #n1 #n2 #H
+lapply (lveq_sym … H) -H #H
+elim (lveq_inv_succ_sn … H) -H /3 width=3 by lveq_sym, ex3_intro/
+qed-.
+
+fact lveq_inv_succ_aux: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ ∀m1,m2. ↑m1 = n1 → ↑m2 = n2 → ⊥.
+#L1 #L2 #n1 #n2 * -L1 -L2 -n1 -n2
+[1: #m1 #m2 #H1 #H2 destruct
+|2: #I1 #I2 #K1 #K2 #_ #m1 #m2 #H1 #H2 destruct
+|*: #K1 #K2 #n #_ #m1 #m2 #H1 #H2 destruct
+]
+qed-.
+
+lemma lveq_inv_succ: ∀L1,L2,n1,n2. L1 ≋ⓧ*[↑n1, ↑n2] L2 → ⊥.
+/2 width=9 by lveq_inv_succ_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lveq_inv_bind: ∀I1,I2,K1,K2. K1.ⓘ{I1} ≋ⓧ*[0, 0] K2.ⓘ{I2} → K1 ≋ⓧ*[0, 0] K2.
+#I1 #I2 #K1 #K2 #H
+elim (lveq_inv_zero … H) -H * [| #Z1 #Z2 #Y1 #Y2 #HY ] #H1 #H2 destruct //
+qed-.
+
+lemma lveq_inv_atom_atom: ∀n1,n2. ⋆ ≋ⓧ*[n1, n2] ⋆ → ∧∧ 0 = n1 & 0 = n2.
+* [2: #n1 ] * [2,4: #n2 ] #H
+[ elim (lveq_inv_succ … H)
+| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
+| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
+| /2 width=1 by conj/
+]
+qed-.
+
+lemma lveq_inv_bind_atom: ∀I1,K1,n1,n2. K1.ⓘ{I1} ≋ⓧ*[n1, n2] ⋆ →
+ ∃∃m1. K1 ≋ⓧ*[m1, 0] ⋆ & BUnit Void = I1 & ↑m1 = n1 & 0 = n2.
+#I1 #K1 * [2: #n1 ] * [2,4: #n2 ] #H
+[ elim (lveq_inv_succ … H)
+| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
+| elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=3 by ex4_intro/
+| elim (lveq_inv_zero … H) -H *
+ [ #H1 #H2 destruct
+ | #Z1 #Z2 #Y1 #Y2 #_ #H1 #H2 destruct
+ ]
+]
+qed-.
+
+lemma lveq_inv_atom_bind: ∀I2,K2,n1,n2. ⋆ ≋ⓧ*[n1, n2] K2.ⓘ{I2} →
+ ∃∃m2. ⋆ ≋ⓧ*[0, m2] K2 & BUnit Void = I2 & 0 = n1 & ↑m2 = n2.
+#I2 #K2 #n1 #n2 #H
+lapply (lveq_sym … H) -H #H
+elim (lveq_inv_bind_atom … H) -H
+/3 width=3 by lveq_sym, ex4_intro/
+qed-.
+
+lemma lveq_inv_pair_pair: ∀I1,I2,K1,K2,V1,V2,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 →
+ ∧∧ K1 ≋ⓧ*[0, 0] K2 & 0 = n1 & 0 = n2.
+#I1 #I2 #K1 #K2 #V1 #V2 * [2: #n1 ] * [2,4: #n2 ] #H
+[ elim (lveq_inv_succ … H)
+| elim (lveq_inv_succ_dx … H) -H #Y #_ #H1 #H2 destruct
+| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
+| elim (lveq_inv_zero … H) -H *
+ [ #H1 #H2 destruct
+ | #Z1 #Z2 #Y1 #Y2 #HY #H1 #H2 destruct /3 width=1 by and3_intro/
+ ]
+]
+qed-.
+
+lemma lveq_inv_void_succ_sn: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[↑n1, n2] L2 →
+ ∧∧ L1 ≋ ⓧ*[n1, 0] L2 & 0 = n2.
+#L1 #L2 #n1 #n2 #H
+elim (lveq_inv_succ_sn … H) -H #Y #HY #H1 #H2 destruct /2 width=1 by conj/
+qed-.
+
+lemma lveq_inv_void_succ_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, ↑n2] L2.ⓧ →
+ ∧∧ L1 ≋ ⓧ*[0, n2] L2 & 0 = n1.
+#L1 #L2 #n1 #n2 #H
+lapply (lveq_sym … H) -H #H
+elim (lveq_inv_void_succ_sn … H) -H
+/3 width=1 by lveq_sym, conj/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lveq_fwd_gen: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ ∨∨ 0 = n1 | 0 = n2.
+#L1 #L2 * [2: #n1 ] * [2,4: #n2 ] #H
+[ elim (lveq_inv_succ … H) ]
+/2 width=1 by or_introl, or_intror/
+qed-.
+
+lemma lveq_fwd_pair_sn: ∀I1,K1,L2,V1,n1,n2. K1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2 → 0 = n1.
+#I1 #K1 #L2 #V1 * [2: #n1 ] // * [2: #n2 ] #H
+[ elim (lveq_inv_succ … H)
+| elim (lveq_inv_succ_sn … H) -H #Y #_ #H1 #H2 destruct
+]
+qed-.
+
+lemma lveq_fwd_pair_dx: ∀I2,L1,K2,V2,n1,n2. L1 ≋ⓧ*[n1, n2] K2.ⓑ{I2}V2 → 0 = n2.
+/3 width=6 by lveq_fwd_pair_sn, lveq_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 "static_2/syntax/lenv_length.ma".
+include "static_2/syntax/lveq.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
+
+(* Properties with length for local environments ****************************)
+
+lemma lveq_length_eq: ∀L1,L2. |L1| = |L2| → L1 ≋ⓧ*[0, 0] L2.
+#L1 elim L1 -L1
+[ #Y2 #H >(length_inv_zero_sn … H) -Y2 /2 width=3 by lveq_atom, ex_intro/
+| #K1 #I1 #IH #Y2 #H
+ elim (length_inv_succ_sn … H) -H #I2 #K2 #HK #H destruct
+ /3 width=1 by lveq_bind/
+]
+qed.
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma lveq_fwd_length_le_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → n1 ≤ |L1|.
+#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
+/2 width=1 by le_S_S/
+qed-.
+
+lemma lveq_fwd_length_le_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → n2 ≤ |L2|.
+#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
+/2 width=1 by le_S_S/
+qed-.
+
+lemma lveq_fwd_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ ∧∧ |L1|-|L2| = n1 & |L2|-|L1| = n2.
+#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 /2 width=1 by conj/
+#K1 #K2 #n #_ * #H1 #H2 >length_bind /3 width=1 by minus_Sn_m, conj/
+qed-.
+
+lemma lveq_length_fwd_sn: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → |L1| ≤ |L2| → 0 = n1.
+#L1 #L2 #n1 #n2 #H #HL
+elim (lveq_fwd_length … H) -H
+>(eq_minus_O … HL) //
+qed-.
+
+lemma lveq_length_fwd_dx: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 → |L2| ≤ |L1| → 0 = n2.
+#L1 #L2 #n1 #n2 #H #HL
+elim (lveq_fwd_length … H) -H
+>(eq_minus_O … HL) //
+qed-.
+
+lemma lveq_inj_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ |L1| = |L2| → ∧∧ 0 = n1 & 0 = n2.
+#L1 #L2 #n1 #n2 #H #HL
+elim (lveq_fwd_length … H) -H
+>HL -HL /2 width=1 by conj/
+qed-.
+
+lemma lveq_fwd_length_plus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ |L1| + n2 = |L2| + n1.
+#L1 #L2 #n1 #n2 #H elim H -L1 -L2 -n1 -n2 normalize
+/2 width=2 by injective_plus_r/
+qed-.
+
+lemma lveq_fwd_length_eq: ∀L1,L2. L1 ≋ⓧ*[0, 0] L2 → |L1| = |L2|.
+/3 width=2 by lveq_fwd_length_plus, injective_plus_l/ qed-.
+
+lemma lveq_fwd_length_minus: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ |L1| - n1 = |L2| - n2.
+/3 width=3 by lveq_fwd_length_plus, lveq_fwd_length_le_dx, lveq_fwd_length_le_sn, plus_to_minus_2/ qed-.
+
+lemma lveq_fwd_abst_bind_length_le: ∀I1,I2,L1,L2,V1,n1,n2.
+ L1.ⓑ{I1}V1 ≋ⓧ*[n1, n2] L2.ⓘ{I2} → |L1| ≤ |L2|.
+#I1 #I2 #L1 #L2 #V1 #n1 #n2 #HL
+lapply (lveq_fwd_pair_sn … HL) #H destruct
+elim (lveq_fwd_length … HL) -HL >length_bind >length_bind //
+qed-.
+
+lemma lveq_fwd_bind_abst_length_le: ∀I1,I2,L1,L2,V2,n1,n2.
+ L1.ⓘ{I1} ≋ⓧ*[n1, n2] L2.ⓑ{I2}V2 → |L2| ≤ |L1|.
+/3 width=6 by lveq_fwd_abst_bind_length_le, lveq_sym/ qed-.
+
+(* Inversion lemmas with length for local environments **********************)
+
+lemma lveq_inv_void_dx_length: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2.ⓧ → |L1| ≤ |L2| →
+ ∃∃m2. L1 ≋ ⓧ*[n1, m2] L2 & 0 = n1 & ↑m2 = n2.
+#L1 #L2 #n1 #n2 #H #HL12
+lapply (lveq_fwd_length_plus … H) normalize >plus_n_Sm #H0
+lapply (plus2_inv_le_sn … H0 HL12) -H0 -HL12 #H0
+elim (le_inv_S1 … H0) -H0 #m2 #_ #H0 destruct
+elim (lveq_inv_void_succ_dx … H) -H /2 width=3 by ex3_intro/
+qed-.
+
+lemma lveq_inv_void_sn_length: ∀L1,L2,n1,n2. L1.ⓧ ≋ⓧ*[n1, n2] L2 → |L2| ≤ |L1| →
+ ∃∃m1. L1 ≋ ⓧ*[m1, n2] L2 & ↑m1 = n1 & 0 = n2.
+#L1 #L2 #n1 #n2 #H #HL
+lapply (lveq_sym … H) -H #H
+elim (lveq_inv_void_dx_length … H HL) -H -HL
+/3 width=4 by lveq_sym, 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 "static_2/syntax/lveq_length.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS UP TO EXCLUSION BINDERS ***************)
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lveq_inv_bind: ∀K1,K2. K1 ≋ⓧ*[0, 0] K2 →
+ ∀I1,I2,m1,m2. K1.ⓘ{I1} ≋ⓧ*[m1, m2] K2.ⓘ{I2} →
+ ∧∧ 0 = m1 & 0 = m2.
+#K1 #K2 #HK #I1 #I2 #m1 #m2 #H
+lapply (lveq_fwd_length_eq … HK) -HK #HK
+elim (lveq_inj_length … H) -H normalize /3 width=1 by conj, eq_f/
+qed-.
+
+theorem lveq_inj: ∀L1,L2,n1,n2. L1 ≋ⓧ*[n1, n2] L2 →
+ ∀m1,m2. L1 ≋ⓧ*[m1, m2] L2 →
+ ∧∧ n1 = m1 & n2 = m2.
+#L1 #L2 #n1 #n2 #Hn #m1 #m2 #Hm
+elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct
+elim (lveq_fwd_length … Hm) -Hm #H1 #H2 destruct
+/2 width=1 by conj/
+qed-.
+
+theorem lveq_inj_void_sn_ge: ∀K1,K2. |K2| ≤ |K1| →
+ ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
+ ∀m1,m2. K1.ⓧ ≋ⓧ*[m1, m2] K2 →
+ ∧∧ ↑n1 = m1 & 0 = m2 & 0 = n2.
+#L1 #L2 #HL #n1 #n2 #Hn #m1 #m2 #Hm
+elim (lveq_fwd_length … Hn) -Hn #H1 #H2 destruct
+elim (lveq_fwd_length … Hm) -Hm #H1 #H2 destruct
+>length_bind >eq_minus_S_pred >(eq_minus_O … HL)
+/3 width=4 by plus_minus, and3_intro/
+qed-.
+
+theorem lveq_inj_void_dx_le: ∀K1,K2. |K1| ≤ |K2| →
+ ∀n1,n2. K1 ≋ⓧ*[n1, n2] K2 →
+ ∀m1,m2. K1 ≋ⓧ*[m1, m2] K2.ⓧ →
+ ∧∧ ↑n2 = m2 & 0 = m1 & 0 = n1.
+/3 width=5 by lveq_inj_void_sn_ge, lveq_sym/ qed-. (* auto: 2x lveq_sym *)
--- /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 "static_2/notation/relations/stareq_4.ma".
+include "static_2/syntax/item_sd.ma".
+include "static_2/syntax/term.ma".
+
+(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************)
+
+inductive tdeq (h) (o): relation term ≝
+| tdeq_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → tdeq h o (⋆s1) (⋆s2)
+| tdeq_lref: ∀i. tdeq h o (#i) (#i)
+| tdeq_gref: ∀l. tdeq h o (§l) (§l)
+| tdeq_pair: ∀I,V1,V2,T1,T2. tdeq h o V1 V2 → tdeq h o T1 T2 → tdeq h o (②{I}V1.T1) (②{I}V2.T2)
+.
+
+interpretation
+ "context-free degree-based equivalence (term)"
+ 'StarEq h o T1 T2 = (tdeq h o T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact tdeq_inv_sort1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀s1. X = ⋆s1 →
+ ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
+#h #o #X #Y * -X -Y
+[ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/
+| #i #s #H destruct
+| #l #s #H destruct
+| #I #V1 #V2 #T1 #T2 #_ #_ #s #H destruct
+]
+qed-.
+
+lemma tdeq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ≛[h, o] Y →
+ ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
+/2 width=3 by tdeq_inv_sort1_aux/ qed-.
+
+fact tdeq_inv_lref1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀i. X = #i → Y = #i.
+#h #o #X #Y * -X -Y //
+[ #s1 #s2 #d #_ #_ #j #H destruct
+| #I #V1 #V2 #T1 #T2 #_ #_ #j #H destruct
+]
+qed-.
+
+lemma tdeq_inv_lref1: ∀h,o,Y,i. #i ≛[h, o] Y → Y = #i.
+/2 width=5 by tdeq_inv_lref1_aux/ qed-.
+
+fact tdeq_inv_gref1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀l. X = §l → Y = §l.
+#h #o #X #Y * -X -Y //
+[ #s1 #s2 #d #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #_ #_ #k #H destruct
+]
+qed-.
+
+lemma tdeq_inv_gref1: ∀h,o,Y,l. §l ≛[h, o] Y → Y = §l.
+/2 width=5 by tdeq_inv_gref1_aux/ qed-.
+
+fact tdeq_inv_pair1_aux: ∀h,o,X,Y. X ≛[h, o] Y → ∀I,V1,T1. X = ②{I}V1.T1 →
+ ∃∃V2,T2. V1 ≛[h, o] V2 & T1 ≛[h, o] T2 & Y = ②{I}V2.T2.
+#h #o #X #Y * -X -Y
+[ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct
+| #i #J #W1 #U1 #H destruct
+| #l #J #W1 #U1 #H destruct
+| #I #V1 #V2 #T1 #T2 #HV #HT #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma tdeq_inv_pair1: ∀h,o,I,V1,T1,Y. ②{I}V1.T1 ≛[h, o] Y →
+ ∃∃V2,T2. V1 ≛[h, o] V2 & T1 ≛[h, o] T2 & Y = ②{I}V2.T2.
+/2 width=3 by tdeq_inv_pair1_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma tdeq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ≛[h, o] Y → ∀d. deg h o s1 d →
+ ∃∃s2. deg h o s2 d & Y = ⋆s2.
+#h #o #Y #s1 #H #d #Hs1 elim (tdeq_inv_sort1 … H) -H
+#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/
+qed-.
+
+lemma tdeq_inv_sort_deg: ∀h,o,s1,s2. ⋆s1 ≛[h, o] ⋆s2 →
+ ∀d1,d2. deg h o s1 d1 → deg h o s2 d2 →
+ d1 = d2.
+#h #o #s1 #y #H #d1 #d2 #Hs1 #Hy
+elim (tdeq_inv_sort1_deg … H … Hs1) -s1 #s2 #Hs2 #H destruct
+<(deg_mono h o … Hy … Hs2) -s2 -d1 //
+qed-.
+
+lemma tdeq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ≛[h, o] ②{I2}V2.T2 →
+ ∧∧ I1 = I2 & V1 ≛[h, o] V2 & T1 ≛[h, o] T2.
+#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H elim (tdeq_inv_pair1 … H) -H
+#V0 #T0 #HV #HT #H destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma tdeq_inv_pair_xy_x: ∀h,o,I,V,T. ②{I}V.T ≛[h, o] V → ⊥.
+#h #o #I #V elim V -V
+[ #J #T #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
+| #J #X #Y #IHX #_ #T #H elim (tdeq_inv_pair … H) -H #H #HY #_ destruct /2 width=2 by/
+]
+qed-.
+
+lemma tdeq_inv_pair_xy_y: ∀h,o,I,T,V. ②{I}V.T ≛[h, o] T → ⊥.
+#h #o #I #T elim T -T
+[ #J #V #H elim (tdeq_inv_pair1 … H) -H #X #Y #_ #_ #H destruct
+| #J #X #Y #_ #IHY #V #H elim (tdeq_inv_pair … H) -H #H #_ #HY destruct /2 width=2 by/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma tdeq_fwd_atom1: ∀h,o,I,Y. ⓪{I} ≛[h, o] Y → ∃J. Y = ⓪{J}.
+#h #o * #x #Y #H [ elim (tdeq_inv_sort1 … H) -H ]
+/3 width=4 by tdeq_inv_gref1, tdeq_inv_lref1, ex_intro/
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma tdeq_refl: ∀h,o. reflexive … (tdeq h o).
+#h #o #T elim T -T /2 width=1 by tdeq_pair/
+* /2 width=1 by tdeq_lref, tdeq_gref/
+#s elim (deg_total h o s) /2 width=3 by tdeq_sort/
+qed.
+
+lemma tdeq_sym: ∀h,o. symmetric … (tdeq h o).
+#h #o #T1 #T2 #H elim H -T1 -T2
+/2 width=3 by tdeq_sort, tdeq_lref, tdeq_gref, tdeq_pair/
+qed-.
+
+lemma tdeq_dec: ∀h,o,T1,T2. Decidable (T1 ≛[h, o] T2).
+#h #o #T1 elim T1 -T1 [ * #s1 | #I1 #V1 #T1 #IHV #IHT ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
+[ elim (deg_total h o s1) #d1 #H1
+ elim (deg_total h o s2) #d2 #H2
+ elim (eq_nat_dec d1 d2) #Hd12 destruct /3 width=3 by tdeq_sort, or_introl/
+ @or_intror #H
+ lapply (tdeq_inv_sort_deg … H … H1 H2) -H -H1 -H2 /2 width=1 by/
+|2,3,13:
+ @or_intror #H
+ elim (tdeq_inv_sort1 … H) -H #x1 #x2 #_ #_ #H destruct
+|4,6,14:
+ @or_intror #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct
+|5:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (tdeq_inv_lref1 … H) -H #H destruct /2 width=1 by/
+|7,8,15:
+ @or_intror #H
+ lapply (tdeq_inv_gref1 … H) -H #H destruct
+|9:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (tdeq_inv_gref1 … H) -H #H destruct /2 width=1 by/
+|10,11,12:
+ @or_intror #H
+ elim (tdeq_inv_pair1 … H) -H #X1 #X2 #_ #_ #H destruct
+|16:
+ elim (eq_item2_dec I1 I2) #HI12 destruct
+ [ elim (IHV V2) -IHV #HV12
+ elim (IHT T2) -IHT #HT12
+ [ /3 width=1 by tdeq_pair, or_introl/ ]
+ ]
+ @or_intror #H
+ elim (tdeq_inv_pair … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Negated inversion lemmas *************************************************)
+
+lemma tdneq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2.
+ (②{I1}V1.T1 ≛[h, o] ②{I2}V2.T2 → ⊥) →
+ ∨∨ I1 = I2 → ⊥
+ | (V1 ≛[h, o] V2 → ⊥)
+ | (T1 ≛[h, o] T2 → ⊥).
+#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H12
+elim (eq_item2_dec I1 I2) /3 width=1 by or3_intro0/ #H destruct
+elim (tdeq_dec h o V1 V2) /3 width=1 by or3_intro1/
+elim (tdeq_dec h o T1 T2) /4 width=1 by tdeq_pair, or3_intro2/
+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 "static_2/notation/relations/stareq_5.ma".
+include "static_2/syntax/cext2.ma".
+include "static_2/syntax/tdeq.ma".
+
+(* EXTENDED DEGREE-BASED EQUIVALENCE ****************************************)
+
+definition tdeq_ext: ∀h. sd h → relation bind ≝
+ λh,o. ext2 (tdeq h o).
+
+definition cdeq: ∀h. sd h → relation3 lenv term term ≝
+ λh,o,L. tdeq h o.
+
+definition cdeq_ext: ∀h. sd h → relation3 lenv bind bind ≝
+ λh,o. cext2 (cdeq h o).
+
+interpretation
+ "context-free degree-based equivalence (binder)"
+ 'StarEq h o I1 I2 = (tdeq_ext h o I1 I2).
+
+interpretation
+ "context-dependent degree-based equivalence (term)"
+ 'StarEq h o L T1 T2 = (cdeq h o L T1 T2).
+
+interpretation
+ "context-dependent degree-based equivalence (binder)"
+ 'StarEq h o L I1 I2 = (cdeq_ext h o L I1 I2).
--- /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 "static_2/syntax/tdeq.ma".
+
+(* DEGREE-BASED EQUIVALENCE ON TERMS ****************************************)
+
+(* Main properties **********************************************************)
+
+theorem tdeq_trans: ∀h,o. Transitive … (tdeq h o).
+#h #o #T1 #T #H elim H -T1 -T
+[ #s1 #s #d #Hs1 #Hs #X #H
+ elim (tdeq_inv_sort1_deg … H … Hs) -s /2 width=3 by tdeq_sort/
+| #i1 #i #H <(tdeq_inv_lref1 … H) -H //
+| #l1 #l #H <(tdeq_inv_gref1 … H) -H //
+| #I #V1 #V #T1 #T #_ #_ #IHV #IHT #X #H
+ elim (tdeq_inv_pair1 … H) -H /3 width=1 by tdeq_pair/
+]
+qed-.
+
+theorem tdeq_canc_sn: ∀h,o. left_cancellable … (tdeq h o).
+/3 width=3 by tdeq_trans, tdeq_sym/ qed-.
+
+theorem tdeq_canc_dx: ∀h,o. right_cancellable … (tdeq h o).
+/3 width=3 by tdeq_trans, tdeq_sym/ qed-.
+
+theorem tdeq_repl: ∀h,o,T1,T2. T1 ≛[h, o] T2 →
+ ∀U1. T1 ≛[h, o] U1 → ∀U2. T2 ≛[h, o] U2 → U1 ≛[h, o] U2.
+/3 width=3 by tdeq_canc_sn, tdeq_trans/ qed-.
+
+(* Negated main properies ***************************************************)
+
+theorem tdeq_tdneq_trans: ∀h,o,T1,T. T1 ≛[h, o] T → ∀T2. (T ≛[h, o] T2 → ⊥) →
+ T1 ≛[h, o] T2 → ⊥.
+/3 width=3 by tdeq_canc_sn/ qed-.
+
+theorem tdneq_tdeq_canc_dx: ∀h,o,T1,T. (T1 ≛[h, o] T → ⊥) → ∀T2. T2 ≛[h, o] T →
+ T1 ≛[h, o] T2 → ⊥.
+/3 width=3 by 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 "static_2/notation/functions/item0_1.ma".
+include "static_2/notation/functions/snitem2_3.ma".
+include "static_2/notation/functions/snbind2_4.ma".
+include "static_2/notation/functions/snbind2pos_3.ma".
+include "static_2/notation/functions/snbind2neg_3.ma".
+include "static_2/notation/functions/snflat2_3.ma".
+include "static_2/notation/functions/star_1.ma".
+include "static_2/notation/functions/lref_1.ma".
+include "static_2/notation/functions/gref_1.ma".
+include "static_2/notation/functions/snabbr_3.ma".
+include "static_2/notation/functions/snabbrpos_2.ma".
+include "static_2/notation/functions/snabbrneg_2.ma".
+include "static_2/notation/functions/snabst_3.ma".
+include "static_2/notation/functions/snabstpos_2.ma".
+include "static_2/notation/functions/snabstneg_2.ma".
+include "static_2/notation/functions/snappl_2.ma".
+include "static_2/notation/functions/sncast_2.ma".
+include "static_2/syntax/item.ma".
+
+(* TERMS ********************************************************************)
+
+(* terms *)
+inductive term: Type[0] ≝
+ | TAtom: item0 → term (* atomic item construction *)
+ | TPair: item2 → term → term → term (* binary item construction *)
+.
+
+interpretation "term construction (atomic)"
+ 'Item0 I = (TAtom I).
+
+interpretation "term construction (binary)"
+ 'SnItem2 I T1 T2 = (TPair I T1 T2).
+
+interpretation "term binding construction (binary)"
+ 'SnBind2 p I T1 T2 = (TPair (Bind2 p I) T1 T2).
+
+interpretation "term positive binding construction (binary)"
+ 'SnBind2Pos I T1 T2 = (TPair (Bind2 true I) T1 T2).
+
+interpretation "term negative binding construction (binary)"
+ 'SnBind2Neg I T1 T2 = (TPair (Bind2 false I) T1 T2).
+
+interpretation "term flat construction (binary)"
+ 'SnFlat2 I T1 T2 = (TPair (Flat2 I) T1 T2).
+
+interpretation "sort (term)"
+ 'Star s = (TAtom (Sort s)).
+
+interpretation "local reference (term)"
+ 'LRef i = (TAtom (LRef i)).
+
+interpretation "global reference (term)"
+ 'GRef l = (TAtom (GRef l)).
+
+interpretation "abbreviation (term)"
+ 'SnAbbr p T1 T2 = (TPair (Bind2 p Abbr) T1 T2).
+
+interpretation "positive abbreviation (term)"
+ 'SnAbbrPos T1 T2 = (TPair (Bind2 true Abbr) T1 T2).
+
+interpretation "negative abbreviation (term)"
+ 'SnAbbrNeg T1 T2 = (TPair (Bind2 false Abbr) T1 T2).
+
+interpretation "abstraction (term)"
+ 'SnAbst p T1 T2 = (TPair (Bind2 p Abst) T1 T2).
+
+interpretation "positive abstraction (term)"
+ 'SnAbstPos T1 T2 = (TPair (Bind2 true Abst) T1 T2).
+
+interpretation "negative abstraction (term)"
+ 'SnAbstNeg T1 T2 = (TPair (Bind2 false Abst) T1 T2).
+
+interpretation "application (term)"
+ 'SnAppl T1 T2 = (TPair (Flat2 Appl) T1 T2).
+
+interpretation "native type annotation (term)"
+ 'SnCast T1 T2 = (TPair (Flat2 Cast) T1 T2).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: term_dec *)
+lemma eq_term_dec: ∀T1,T2:term. Decidable (T1 = T2).
+#T1 elim T1 -T1 #I1 [| #V1 #T1 #IHV1 #IHT1 ] * #I2 [2,4: #V2 #T2 ]
+[1,4: @or_intror #H destruct
+| elim (eq_item2_dec I1 I2) #HI
+ [ elim (IHV1 V2) -IHV1 #HV
+ [ elim (IHT1 T2) -IHT1 /2 width=1 by or_introl/ #HT ]
+ ]
+ @or_intror #H destruct /2 width=1 by/
+| elim (eq_item0_dec I1 I2) /2 width=1 by or_introl/ #HI
+ @or_intror #H destruct /2 width=1 by/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact destruct_tatom_tatom_aux: ∀I1,I2. ⓪{I1} = ⓪{I2} → I1 = I2.
+#I1 #I2 #H destruct //
+qed-.
+
+fact destruct_tpair_tpair_aux: ∀I1,I2,T1,T2,V1,V2. ②{I1}T1.V1 = ②{I2}T2.V2 →
+ ∧∧T1 = T2 & I1 = I2 & V1 = V2.
+#I1 #I2 #T1 #T2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+lemma discr_tpair_xy_x: ∀I,T,V. ②{I}V.T = V → ⊥.
+#I #T #V elim V -V
+[ #J #H destruct
+| #J #W #U #IHW #_ #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+]
+qed-.
+
+(* Basic_1: was: thead_x_y_y *)
+lemma discr_tpair_xy_y: ∀I,V,T. ②{I}V.T = T → ⊥.
+#I #V #T elim T -T
+[ #J #H destruct
+| #J #W #U #_ #IHU #H elim (destruct_tpair_tpair_aux … H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+]
+qed-.
+
+lemma eq_false_inv_tpair_sn: ∀I,V1,T1,V2,T2.
+ (②{I}V1.T1 = ②{I}V2.T2 → ⊥) →
+ (V1 = V2 → ⊥) ∨ (V1 = V2 ∧ (T1 = T2 → ⊥)).
+#I #V1 #T1 #V2 #T2 #H
+elim (eq_term_dec V1 V2) /3 width=1 by or_introl/ #HV12 destruct
+@or_intror @conj // #HT12 destruct /2 width=1 by/
+qed-.
+
+lemma eq_false_inv_tpair_dx: ∀I,V1,T1,V2,T2.
+ (②{I} V1. T1 = ②{I}V2.T2 → ⊥) →
+ (T1 = T2 → ⊥) ∨ (T1 = T2 ∧ (V1 = V2 → ⊥)).
+#I #V1 #T1 #V2 #T2 #H
+elim (eq_term_dec T1 T2) /3 width=1 by or_introl/ #HT12 destruct
+@or_intror @conj // #HT12 destruct /2 width=1 by/
+qed-.
+
+(* Basic_1: removed theorems 3:
+ not_void_abst not_abbr_void not_abst_void
+*)
--- /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 "static_2/notation/relations/simple_1.ma".
+include "static_2/syntax/term.ma".
+
+(* SIMPLE (NEUTRAL) TERMS ***************************************************)
+
+inductive simple: predicate term ≝
+ | simple_atom: ∀I. simple (⓪{I})
+ | simple_flat: ∀I,V,T. simple (ⓕ{I}V.T)
+.
+
+interpretation "simple (term)" 'Simple T = (simple T).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact simple_inv_bind_aux: ∀T. 𝐒⦃T⦄ → ∀p,J,W,U. T = ⓑ{p,J}W.U → ⊥.
+#T * -T
+[ #I #p #J #W #U #H destruct
+| #I #V #T #a #J #W #U #H destruct
+]
+qed-.
+
+lemma simple_inv_bind: ∀p,I,V,T. 𝐒⦃ⓑ{p,I} V. T⦄ → ⊥.
+/2 width=7 by simple_inv_bind_aux/ qed-.
+
+lemma simple_inv_pair: ∀I,V,T. 𝐒⦃②{I}V.T⦄ → ∃J. I = Flat2 J.
+* /2 width=2 by ex_intro/
+#p #I #V #T #H elim (simple_inv_bind … 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 "ground_2/lib/list.ma".
+include "static_2/notation/functions/snapplvector_2.ma".
+include "static_2/syntax/term_simple.ma".
+
+(* TERMS ********************************************************************)
+
+rec definition applv Vs T on Vs ≝
+ match Vs with
+ [ nil ⇒ T
+ | cons hd tl ⇒ ⓐhd. (applv tl T)
+ ].
+
+interpretation "application to vector (term)"
+ 'SnApplVector Vs T = (applv Vs T).
+
+(* Basic properties *********************************************************)
+
+lemma applv_nil: ∀T. ⒶⒺ.T = T.
+// qed.
+
+lemma applv_cons: ∀V,Vs,T. ⒶV⨮Vs.T = ⓐV.ⒶVs.T.
+// qed.
+
+(* Properties with simple terms *********************************************)
+
+lemma applv_simple: ∀T,Vs. 𝐒⦃T⦄ → 𝐒⦃ⒶVs.T⦄.
+#T * //
+qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/notation/functions/weight_1.ma".
+include "static_2/syntax/term.ma".
+
+(* WEIGHT OF A TERM *********************************************************)
+
+rec definition tw T ≝ match T with
+[ TAtom _ ⇒ 1
+| TPair _ V T ⇒ ↑(tw V + tw T)
+].
+
+interpretation "weight (term)" 'Weight T = (tw T).
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: tweight_lt *)
+lemma tw_pos: ∀T. 1 ≤ ♯{T}.
+#T elim T -T //
+qed.
+
+(* Basic_1: removed theorems 11:
+ wadd_le wadd_lt wadd_O weight_le weight_eq weight_add_O
+ weight_add_S tlt_trans tlt_head_sx tlt_head_dx tlt_wf_ind
+*)
+(* Basic_1: removed local theorems 1: q_ind *)
--- /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 "static_2/notation/relations/topiso_4.ma".
+include "static_2/syntax/item_sd.ma".
+include "static_2/syntax/term.ma".
+
+(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
+
+(* Basic_2A1: includes: tsts_atom tsts_pair *)
+inductive theq (h) (o): relation term ≝
+| theq_sort: ∀s1,s2,d. deg h o s1 d → deg h o s2 d → theq h o (⋆s1) (⋆s2)
+| theq_lref: ∀i. theq h o (#i) (#i)
+| theq_gref: ∀l. theq h o (§l) (§l)
+| theq_pair: ∀I,V1,V2,T1,T2. theq h o (②{I}V1.T1) (②{I}V2.T2)
+.
+
+interpretation "head equivalence (term)" 'TopIso h o T1 T2 = (theq h o T1 T2).
+
+(* Basic inversion lemmas ***************************************************)
+
+fact theq_inv_sort1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀s1. X = ⋆s1 →
+ ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
+#h #o #X #Y * -X -Y
+[ #s1 #s2 #d #Hs1 #Hs2 #s #H destruct /2 width=5 by ex3_2_intro/
+| #i #s #H destruct
+| #l #s #H destruct
+| #I #V1 #V2 #T1 #T2 #s #H destruct
+]
+qed-.
+
+(* Basic_1: was just: iso_gen_sort *)
+lemma theq_inv_sort1: ∀h,o,Y,s1. ⋆s1 ⩳[h, o] Y →
+ ∃∃s2,d. deg h o s1 d & deg h o s2 d & Y = ⋆s2.
+/2 width=3 by theq_inv_sort1_aux/ qed-.
+
+fact theq_inv_lref1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀i. X = #i → Y = #i.
+#h #o #X #Y * -X -Y //
+[ #s1 #s2 #d #_ #_ #j #H destruct
+| #I #V1 #V2 #T1 #T2 #j #H destruct
+]
+qed-.
+
+(* Basic_1: was: iso_gen_lref *)
+lemma theq_inv_lref1: ∀h,o,Y,i. #i ⩳[h, o] Y → Y = #i.
+/2 width=5 by theq_inv_lref1_aux/ qed-.
+
+fact theq_inv_gref1_aux: ∀h,o,X,Y. X ⩳[h, o] Y → ∀l. X = §l → Y = §l.
+#h #o #X #Y * -X -Y //
+[ #s1 #s2 #d #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #k #H destruct
+]
+qed-.
+
+lemma theq_inv_gref1: ∀h,o,Y,l. §l ⩳[h, o] Y → Y = §l.
+/2 width=5 by theq_inv_gref1_aux/ qed-.
+
+fact theq_inv_pair1_aux: ∀h,o,T1,T2. T1 ⩳[h, o] T2 →
+ ∀J,W1,U1. T1 = ②{J}W1.U1 →
+ ∃∃W2,U2. T2 = ②{J}W2.U2.
+#h #o #T1 #T2 * -T1 -T2
+[ #s1 #s2 #d #_ #_ #J #W1 #U1 #H destruct
+| #i #J #W1 #U1 #H destruct
+| #l #J #W1 #U1 #H destruct
+| #I #V1 #V2 #T1 #T2 #J #W1 #U1 #H destruct /2 width=3 by ex1_2_intro/
+]
+qed-.
+
+(* Basic_1: was: iso_gen_head *)
+(* Basic_2A1: was: tsts_inv_pair1 *)
+lemma theq_inv_pair1: ∀h,o,J,W1,U1,T2. ②{J}W1.U1 ⩳[h, o] T2 →
+ ∃∃W2,U2. T2 = ②{J}W2. U2.
+/2 width=7 by theq_inv_pair1_aux/ qed-.
+
+fact theq_inv_pair2_aux: ∀h,o,T1,T2. T1 ⩳[h, o] T2 →
+ ∀J,W2,U2. T2 = ②{J}W2.U2 →
+ ∃∃W1,U1. T1 = ②{J}W1.U1.
+#h #o #T1 #T2 * -T1 -T2
+[ #s1 #s2 #d #_ #_ #J #W2 #U2 #H destruct
+| #i #J #W2 #U2 #H destruct
+| #l #J #W2 #U2 #H destruct
+| #I #V1 #V2 #T1 #T2 #J #W2 #U2 #H destruct /2 width=3 by ex1_2_intro/
+]
+qed-.
+
+(* Basic_2A1: was: tsts_inv_pair2 *)
+lemma theq_inv_pair2: ∀h,o,J,T1,W2,U2. T1 ⩳[h, o] ②{J}W2.U2 →
+ ∃∃W1,U1. T1 = ②{J}W1.U1.
+/2 width=7 by theq_inv_pair2_aux/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma theq_inv_sort1_deg: ∀h,o,Y,s1. ⋆s1 ⩳[h, o] Y → ∀d. deg h o s1 d →
+ ∃∃s2. deg h o s2 d & Y = ⋆s2.
+#h #o #Y #s1 #H #d #Hs1 elim (theq_inv_sort1 … H) -H
+#s2 #x #Hx <(deg_mono h o … Hx … Hs1) -s1 -d /2 width=3 by ex2_intro/
+qed-.
+
+lemma theq_inv_sort_deg: ∀h,o,s1,s2. ⋆s1 ⩳[h, o] ⋆s2 →
+ ∀d1,d2. deg h o s1 d1 → deg h o s2 d2 →
+ d1 = d2.
+#h #o #s1 #y #H #d1 #d2 #Hs1 #Hy
+elim (theq_inv_sort1_deg … H … Hs1) -s1 #s2 #Hs2 #H destruct
+<(deg_mono h o … Hy … Hs2) -s2 -d1 //
+qed-.
+
+lemma theq_inv_pair: ∀h,o,I1,I2,V1,V2,T1,T2. ②{I1}V1.T1 ⩳[h, o] ②{I2}V2.T2 →
+ I1 = I2.
+#h #o #I1 #I2 #V1 #V2 #T1 #T2 #H elim (theq_inv_pair1 … H) -H
+#V0 #T0 #H destruct //
+qed-.
+
+(* Basic properties *********************************************************)
+
+(* Basic_1: was: iso_refl *)
+(* Basic_2A1: was: tsts_refl *)
+lemma theq_refl: ∀h,o. reflexive … (theq h o).
+#h #o * //
+* /2 width=1 by theq_lref, theq_gref/
+#s elim (deg_total h o s) /2 width=3 by theq_sort/
+qed.
+
+(* Basic_2A1: was: tsts_sym *)
+lemma theq_sym: ∀h,o. symmetric … (theq h o).
+#h #o #T1 #T2 * -T1 -T2 /2 width=3 by theq_sort/
+qed-.
+
+(* Basic_2A1: was: tsts_dec *)
+lemma theq_dec: ∀h,o,T1,T2. Decidable (T1 ⩳[h, o] T2).
+#h #o * [ * #s1 | #I1 #V1 #T1 ] * [1,3,5,7: * #s2 |*: #I2 #V2 #T2 ]
+[ elim (deg_total h o s1) #d1 #H1
+ elim (deg_total h o s2) #d2 #H2
+ elim (eq_nat_dec d1 d2) #Hd12 destruct /3 width=3 by theq_sort, or_introl/
+ @or_intror #H
+ lapply (theq_inv_sort_deg … H … H1 H2) -H -H1 -H2 /2 width=1 by/
+|2,3,13:
+ @or_intror #H
+ elim (theq_inv_sort1 … H) -H #x1 #x2 #_ #_ #H destruct
+|4,6,14:
+ @or_intror #H
+ lapply (theq_inv_lref1 … H) -H #H destruct
+|5:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (theq_inv_lref1 … H) -H #H destruct /2 width=1 by/
+|7,8,15:
+ @or_intror #H
+ lapply (theq_inv_gref1 … H) -H #H destruct
+|9:
+ elim (eq_nat_dec s1 s2) #Hs12 destruct /2 width=1 by or_introl/
+ @or_intror #H
+ lapply (theq_inv_gref1 … H) -H #H destruct /2 width=1 by/
+|10,11,12:
+ @or_intror #H
+ elim (theq_inv_pair1 … H) -H #X1 #X2 #H destruct
+|16:
+ elim (eq_item2_dec I1 I2) #HI12 destruct
+ [ /3 width=1 by theq_pair, or_introl/ ]
+ @or_intror #H
+ lapply (theq_inv_pair … H) -H /2 width=1 by/
+]
+qed-.
+
+(* Basic_2A1: removed theorems 2:
+ tsts_inv_atom1 tsts_inv_atom2
+*)
--- /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 "static_2/syntax/term_simple.ma".
+include "static_2/syntax/theq.ma".
+
+(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
+
+(* Properies with simple (neutral) terms ************************************)
+
+(* Basic_2A1: was: simple_tsts_repl_dx *)
+lemma simple_theq_repl_dx: ∀h,o,T1,T2. T1 ⩳[h, o] T2 → 𝐒⦃T1⦄ → 𝐒⦃T2⦄.
+#h #o #T1 #T2 * -T1 -T2 //
+#I #V1 #V2 #T1 #T2 #H
+elim (simple_inv_pair … H) -H #J #H destruct //
+qed-.
+
+(* Basic_2A1: was: simple_tsts_repl_sn *)
+lemma simple_theq_repl_sn: ∀h,o,T1,T2. T1 ⩳[h, o] T2 → 𝐒⦃T2⦄ → 𝐒⦃T1⦄.
+/3 width=5 by simple_theq_repl_dx, theq_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 "static_2/syntax/term_vector.ma".
+include "static_2/syntax/theq_simple.ma".
+
+(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
+
+(* Advanced inversion lemmas with simple (neutral) terms ********************)
+
+(* Basic_1: was only: iso_flats_lref_bind_false iso_flats_flat_bind_false *)
+(* Basic_2A1: was: tsts_inv_bind_applv_simple *)
+lemma theq_inv_applv_bind_simple: ∀h,o,p,I,Vs,V2,T1,T2. ⒶVs.T1 ⩳[h, o] ⓑ{p,I}V2.T2 →
+ 𝐒⦃T1⦄ → ⊥.
+#h #o #p #I #Vs #V2 #T1 #T2 #H elim (theq_inv_pair2 … H) -H
+#V0 #T0 elim Vs -Vs normalize
+[ #H destruct #H /2 width=5 by simple_inv_bind/
+| #V #Vs #_ #H destruct
+]
+qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "static_2/syntax/tdeq.ma".
+include "static_2/syntax/theq.ma".
+
+(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
+
+(* Properties with degree-based equivalence for terms ***********************)
+
+lemma tdeq_theq: ∀h,o,T1,T2. T1 ≛[h, o] T2 → T1 ⩳[h, o] T2.
+#h #o #T1 #T2 * -T1 -T2 /2 width=3 by theq_sort, theq_pair/
+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 "static_2/syntax/theq.ma".
+
+(* HEAD EQUIVALENCE FOR TERMS ***********************************************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: iso_trans *)
+(* Basic_2A1: was: tsts_trans *)
+theorem theq_trans: ∀h,o. Transitive … (theq h o).
+#h #o #T1 #T * -T1 -T
+[ #s1 #s #d #Hs1 #Hs #X #H
+ elim (theq_inv_sort1_deg … H … Hs) -s /2 width=3 by theq_sort/
+| #i1 #i #H <(theq_inv_lref1 … H) -H //
+| #l1 #l #H <(theq_inv_gref1 … H) -H //
+| #I #V1 #V #T1 #T #X #H
+ elim (theq_inv_pair1 … H) -H #V2 #T2 #H destruct //
+]
+qed-.
+
+(* Basic_2A1: was: tsts_canc_sn *)
+theorem theq_canc_sn: ∀h,o. left_cancellable … (theq h o).
+/3 width=3 by theq_trans, theq_sym/ qed-.
+
+(* Basic_2A1: was: tsts_canc_dx *)
+theorem theq_canc_dx: ∀h,o. right_cancellable … (theq h o).
+/3 width=3 by theq_trans, theq_sym/ qed-.
--- /dev/null
+<?xml version="1.0" encoding="UTF-8"?>
+
+<page xmlns="http://lambdadelta.info/"
+ description = "\lambda\delta home page"
+ title = "\lambda\delta home page"
+ logo = "crux"
+ head = "cic:/matita/lambdadelta/static_2/ (syntactic components of λδ version 2)"
+>
+ <sitemap name="sitemap"/>
+
+ <section4 name="contents">Contents of the Specification</section4>
+ <body>This specification contains the syntactic components of λδ version 2
+ that every forthcoming system of the λδ family will share.
+ </body>
+
+ <section4 name="summary">Summary of the Specification</section4>
+ <body>Here is a numerical account of the specification's contents
+ and its timeline.
+ </body>
+ <table name="static_2_sum"/>
+
+ <news class="alpha" date="2018 June 30.">
+ The specification stands alone.
+ </news>
+
+ <section4 name="structure">Logical Structure of the Specification</section4>
+ <body>This table reports the specification's components and their planes.
+ </body>
+ <table name="static_2_src"/>
+
+ <footer/>
+</page>
--- /dev/null
+name "static_2_src"
+
+table {
+ class "gray"
+ [ { "component" * } {
+ [ { "section" * } {
+ [ [ "plane" ] "files" * ]
+ }
+ ]
+ }
+ ]
+ class "water"
+ [ { "iterated static typing" * } {
+ [ { "iterated generic extension of a context-sensitive relation" * } {
+ [ [ "for lenvs on referred entries" ] "rexs" + "( ? ⪤*[?,?] ? )" "rexs_length" + "rexs_lex" + "rexs_drops" + "rexs_fqup" + "rexs_rexs" * ]
+ }
+ ]
+ }
+ ]
+ class "green"
+ [ { "static typing" * } {
+ [ { "generic reducibility" * } {
+ [ [ "restricted refinement for lenvs" ] "lsubc" + "( ? ⊢ ? ⫃[?] ? )" "lsubc_drops" + "lsubc_lsubr" + "lsubc_lsuba" * ]
+ [ [ "candidates" ] "gcp_cr" + "( ⦃?,?,?⦄ ϵ[?] 〚?〛 )" "gcp_aaa" * ]
+ [ [ "computation properties" ] "gcp" *]
+ }
+ ]
+ [ { "atomic arity assignment" * } {
+ [ [ "restricted refinement for lenvs" ] "lsuba" + "( ? ⊢ ? ⫃⁝ ? )" "lsuba_drops" + "lsuba_lsubr" + "lsuba_aaa" + "lsuba_lsuba" * ]
+ [ [ "for terms" ] "aaa" + "( ⦃?,?⦄ ⊢ ? ⁝ ? )" "aaa_drops" + "aaa_fqus" + "aaa_rdeq" + "aaa_fdeq" + "aaa_aaa" * ]
+ }
+ ]
+ [ { "degree-based equivalence" * } {
+ [ [ "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" ] "req" + "( ? ≡[?] ? )" "req_fqup" + "req_fsle" * ]
+ }
+ ]
+ [ { "generic extension of a context-sensitive relation" * } {
+ [ [ "for lenvs on referred entries" ] "rex" + "( ? ⪤[?,?] ? )" "rex_length" + "rex_lex" + "rex_drops" + "rex_fqup" + "rex_fsle" + "rex_rex" * ]
+ }
+ ]
+ [ { "context-sensitive free variables" * } {
+ [ [ "inclusion for restricted closures" ] "fsle" + "( ⦃?,?⦄ ⊆ ⦃?,?⦄ )" "fsle_length" + "fsle_drops" + "fsle_fqup" + "fsle_fsle" * ]
+ [ [ "restricted refinement for lenvs" ] "lsubf" + "( ⦃?,?⦄ ⫃𝐅* ⦃?,?⦄ )" "lsubf_lsubr" + "lsubf_frees" + "lsubf_lsubf" * ]
+ [ [ "for terms" ] "frees" + "( ? ⊢ 𝐅*⦃?⦄ ≘ ? )" "frees_append" + "frees_drops" + "frees_fqup" + "frees_frees" * ]
+ }
+ ]
+ [ { "local environments" * } {
+ [ [ "restricted refinement" ] "lsubr" + "( ? ⫃ ? )" "lsubr_length" + "lsubr_drops" + "lsubr_lsubr" * ]
+ }
+ ]
+ }
+ ]
+ class "grass"
+ [ { "s-computation" * } {
+ [ { "iterated structural successor" * } {
+ [ [ "for closures" ] "fqus" + "( ⦃?,?,?⦄ ⊐*[?] ⦃?,?,?⦄ )" + "( ⦃?,?,?⦄ ⊐* ⦃?,?,?⦄ )" "fqus_weight" + "fqus_drops" + "fqus_fqup" + "fqus_fqus" * ]
+ [ [ "proper for closures" ] "fqup" + "( ⦃?,?,?⦄ ⊐+[?] ⦃?,?,?⦄ )" + "( ⦃?,?,?⦄ ⊐+ ⦃?,?,?⦄ )" "fqup_weight" + "fqup_drops" + "fqup_fqup" * ]
+ }
+ ]
+ }
+ ]
+ class "yellow"
+ [ { "s-transition" * } {
+ [ { "structural successor" * } {
+ [ [ "for closures" ] "fquq" + "( ⦃?,?,?⦄ ⊐⸮[?] ⦃?,?,?⦄ )" + "( ⦃?,?,?⦄ ⊐⸮ ⦃?,?,?⦄ )" "fquq_length" + "fquq_weight" * ]
+ [ [ "proper for closures" ] "fqu" + "( ⦃?,?,?⦄ ⊐[?] ⦃?,?,?⦄ )" + "( ⦃?,?,?⦄ ⊐ ⦃?,?,?⦄ )" "fqu_length" + "fqu_weight" + "fqu_tdeq" * ]
+ }
+ ]
+ }
+ ]
+ class "orange"
+ [ { "relocation" * } {
+ [ { "generic slicing" * } {
+ [ [ "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" * } {
+ [ [ "for binders" ] "lifts_bind" + "( ⬆*[?] ? ≘ ? )" "lifts_weight_bind" + "lifts_lifts_bind" * ]
+ [ [ "for term vectors" ] "lifts_vector" + "( ⬆*[?] ? ≘ ? )" "lifts_lifts_vector" * ]
+ [ [ "for terms" ] "lifts" + "( ⬆*[?] ? ≘ ? )" "lifts_simple" + "lifts_weight" + "lifts_tdeq" + "lifts_lifts" * ]
+ }
+ ]
+ [ { "syntactic equivalence" * } {
+ [ [ "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" ] "sex" + "( ? ⪤[?,?,?] ? )" "sex_tc" + "sex_length" + "sex_sex" * ]
+ }
+ ]
+ }
+ ]
+ class "red"
+ [ { "syntax" * } {
+ [ { "equivalence up to exclusion binders" * } {
+ [ [ "for lenvs" ] "lveq" + "( ? ≋ⓧ*[?,?] ? )" "lveq_length" + "lveq_lveq" * ]
+ }
+ ]
+ [ { "append" * } {
+ [ [ "for lenvs" ] "append" + "( ? + ? )" "append_length" * ]
+ }
+ ]
+ [ { "head equivalence" * } {
+ [ [ "for terms" ] "theq" + "( ? ⩳[?,?] ? )" "theq_simple" + "theq_tdeq" + "theq_theq" + "theq_simple_vector" * ]
+ }
+ ]
+ [ { "degree-based equivalence" * } {
+ [ [ "" ] "tdeq_ext" + "( ? ≛[?,?] ? )" + "( ? ⊢ ? ≛[?,?] ? )" * ]
+ [ [ "" ] "tdeq" + "( ? ≛[?,?] ? )" "tdeq_tdeq" * ]
+ }
+ ]
+ [ { "closures" * } {
+ [ [ "" ] "cl_weight" + "( ♯{?,?,?} )" * ]
+ [ [ "" ] "cl_restricted_weight" + "( ♯{?,?} )" * ]
+ }
+ ]
+ [ { "global environments" * } {
+ [ [ "" ] "genv_length" + "( |?| )" * ]
+ [ [ "" ] "genv_weight" + "( ♯{?} )" * ]
+ [ [ "" ] "genv" * ]
+ }
+ ]
+ [ { "local environments" * } {
+ [ [ "" ] "ceq_ext" "ceq_ext_ceq_ext" * ]
+ [ [ "" ] "cext2" * ]
+ [ [ "" ] "lenv_length" + "( |?| )" * ]
+ [ [ "" ] "lenv_weight" + "( ♯{?} )" * ]
+ [ [ "" ] "lenv" * ]
+ }
+ ]
+ [ { "binders for local environments" * } {
+ [ [ "" ] "ext2" "ext2_tc" + "ext2_ext2" * ]
+ [ [ "" ] "bind" "bind_weight" * ]
+ }
+ ]
+ [ { "terms" * } {
+ [ [ "" ] "term_vector" + "( Ⓐ?.? )" * ]
+ [ [ "" ] "term_simple" + "( 𝐒⦃?⦄ )" * ]
+ [ [ "" ] "term_weight" + "( ♯{?} )" * ]
+ [ [ "" ] "term" * ]
+ }
+ ]
+ [ { "items" * } {
+ [ [ "" ] "item_sd" * ]
+ [ [ "" ] "item_sh" * ]
+ [ [ "" ] "item" * ]
+ }
+ ]
+ [ { "atomic arities" * } {
+ [ [ "" ] "aarity" * ]
+ }
+ ]
+ }
+ ]
+}
+
+class "top" { * }
+
+class "capitalize italic" { 0 1 }
+
+class "italic" { 2 }
projects / helm.git / commitdiff