From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Tue, 11 Mar 2014 17:57:05 +0000 (+0000)
Subject: reaxiomatized lleq fixes a bug in it and allows to park substitution in etc
X-Git-Tag: make_still_working~949
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=8a5a354c9ac3ef20ca01dbeb61f6b99902f172a7;p=helm.git
reaxiomatized lleq fixes a bug in it and allows to park substitution in etc
---
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpys.ma
deleted file mode 100644
index 881030572..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpys.ma
+++ /dev/null
@@ -1,24 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/reduction/cpx_cpys.ma".
-include "basic_2/computation/cpxs.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
-
-(* Properties on local environment refinement for extended substitution *****)
-
-lemma lsuby_cpxs_trans: âh,g,G. lsub_trans ⦠(cpxs h g G) (lsuby 0 (â)).
-/3 width=5 by lsuby_cpx_trans, LTC_lsub_trans/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_leq.ma
new file mode 100644
index 000000000..1d35deb53
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_leq.ma
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/cpx_leq.ma".
+include "basic_2/computation/cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma leq_cpxs_trans: âh,g,G. lsub_trans ⦠(cpxs h g G) (leq 0 (â)).
+/3 width=5 by leq_cpx_trans, LTC_lsub_trans/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbu.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbu.ma
index 7165feaee..93e837f9e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbu.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbu.ma
@@ -13,7 +13,7 @@
(**************************************************************************)
include "basic_2/notation/relations/btpredproper_8.ma".
-include "basic_2/substitution/lleq.ma".
+include "basic_2/relocation/lleq.ma".
include "basic_2/computation/fpbs.ma".
(* UNITARY "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **************)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbu_fpns.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbu_fpns.ma
index 9f5111ad2..863b08196 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbu_fpns.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbu_fpns.ma
@@ -12,6 +12,7 @@
(* *)
(**************************************************************************)
+include "basic_2/relocation/lleq_lleq.ma".
include "basic_2/computation/cpxs_lleq.ma".
include "basic_2/computation/lpxs_lleq.ma".
include "basic_2/computation/lpxs_lpxs.ma".
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpns.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpns.ma
index 6e68fef14..dce46b065 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpns.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpns.ma
@@ -13,7 +13,7 @@
(**************************************************************************)
include "basic_2/notation/relations/btpredsnstar_8.ma".
-include "basic_2/substitution/lleq.ma".
+include "basic_2/relocation/lleq.ma".
include "basic_2/computation/lpxs.ma".
(* PARALLEL COMPUTATION FOR "BIG TREE" NORMAL FORMS *************************)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpns_fpns.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpns_fpns.ma
index 728f48df9..deaa7a967 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpns_fpns.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpns_fpns.ma
@@ -12,7 +12,7 @@
(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_lleq.ma".
+include "basic_2/relocation/lleq_lleq.ma".
include "basic_2/computation/lpxs_lpxs.ma".
include "basic_2/computation/fpns.ma".
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpxs.ma
index 3ed17d258..aef5014e7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpxs.ma
@@ -12,6 +12,7 @@
(* *)
(**************************************************************************)
+include "ground_2/ynat/ynat_max.ma".
include "basic_2/computation/lsx_ldrop.ma".
include "basic_2/computation/lsx_lpxs.ma".
include "basic_2/computation/lcosx.ma".
@@ -36,7 +37,7 @@ lemma lsx_cpx_trans_lcosx: âh,g,G,L,T1,T2. â¦G, L⦠⢠T1 â¡[h, g] T2 â
| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #HL #H
elim (lsx_inv_bind ⦠H) -H #HV1 #HT1
@lsx_bind /2 width=2 by/ (**) (* explicit constructor *)
- @(lsx_leqy_conf ⦠(L.â{I}V1)) /3 width=1 by lcosx_pair, lsuby_succ/
+ @(lsx_leq_conf ⦠(L.â{I}V1)) /3 width=1 by lcosx_pair, leq_succ/
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #HL #H
elim (lsx_inv_flat ⦠H) -H /3 width=1 by lsx_flat/
| #G #L #V #U1 #U2 #T2 #_ #HTU2 #IHU12 #d #HL #H
@@ -50,13 +51,13 @@ lemma lsx_cpx_trans_lcosx: âh,g,G,L,T1,T2. â¦G, L⦠⢠T1 â¡[h, g] T2 â
elim (lsx_inv_flat ⦠H) -H #HV1 #H
elim (lsx_inv_bind ⦠H) -H #HW1 #HT1
@lsx_bind /3 width=1 by lsx_flat/ (**) (* explicit constructor *)
- @(lsx_leqy_conf ⦠(L.âW1)) /3 width=1 by lcosx_pair, lsuby_succ/
+ @(lsx_leq_conf ⦠(L.âW1)) /3 width=1 by lcosx_pair, leq_succ/
| #a #G #L #V1 #V2 #U2 #W1 #W2 #T1 #T2 #_ #HVU2 #_ #_ #IHV12 #IHW12 #IHT12 #d #HL #H
elim (lsx_inv_flat ⦠H) -H #HV1 #H
elim (lsx_inv_bind ⦠H) -H #HW1 #HT1
@lsx_bind /2 width=1 by/ (**) (* explicit constructor *)
@lsx_flat [ /3 width=7 by lsx_lift_ge, ldrop_drop/ ]
- @(lsx_leqy_conf ⦠(L.âW1)) /3 width=1 by lcosx_pair, lsuby_succ/
+ @(lsx_leq_conf ⦠(L.âW1)) /3 width=1 by lcosx_pair, leq_succ/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
index 2ca1e193d..e21c79912 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
@@ -12,8 +12,9 @@
(* *)
(**************************************************************************)
+include "basic_2/relocation/lleq_leq.ma".
include "basic_2/reduction/lpx_lleq.ma".
-include "basic_2/computation/cpxs_cpys.ma".
+include "basic_2/computation/cpxs_leq.ma".
include "basic_2/computation/lpxs_ldrop.ma".
include "basic_2/computation/lpxs_cpxs.ma".
@@ -37,7 +38,7 @@ lemma lpxs_lleq_fqu_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2,
[ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpxs_inv_pair2 ⦠H1) -H1
#K0 #V0 #H1KL1 #_ #H destruct
elim (lleq_inv_lref_ge_dx ⦠H2 ? I L1 V1) -H2 //
- #I1 #K1 #H #H2KL1 lapply (ldrop_inv_O2 ⦠H) -H #H destruct
+ #K1 #H #H2KL1 lapply (ldrop_inv_O2 ⦠H) -H #H destruct
/2 width=4 by fqu_lref_O, ex3_intro/
| * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
[ elim (lleq_inv_bind ⦠H)
@@ -93,36 +94,32 @@ elim (fqus_inv_gen ⦠H) -H
]
qed-.
-fact lsuby_lpxs_trans_lleq_aux: âh,g,G,L1,L0,d,e. L1 âÃ[d, e] L0 â e = â â
- âL2. â¦G, L0⦠⢠â¡*[h, g] L2 â
- ââL. L âÃ[d, e] L2 & â¦G, L1⦠⢠â¡*[h, g] L &
- (âT. |L1| = |L0| â |L| = |L2| â L0 â[T, d] L2 â L1 â[T, d] L).
+fact leq_lpxs_trans_lleq_aux: âh,g,G,L1,L0,d,e. L1 â[d, e] L0 â e = â â
+ âL2. â¦G, L0⦠⢠â¡*[h, g] L2 â
+ ââL. L â[d, e] L2 & â¦G, L1⦠⢠â¡*[h, g] L &
+ (âT. L0 â[T, d] L2 â L1 â[T, d] L).
#h #g #G #L1 #L0 #d #e #H elim H -L1 -L0 -d -e
-[ #L1 #d #e #_ #L2 #H >(lpxs_inv_atom1 ⦠H) -H
+[ #d #e #_ #L2 #H >(lpxs_inv_atom1 ⦠H) -H
/3 width=5 by ex3_intro, conj/
| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
-| #I1 #I0 #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
+| #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
elim (lpxs_inv_pair1 ⦠H) -H #L2 #V2 #HL02 #HV02 #H destruct
lapply (ysucc_inv_Y_dx ⦠He) -He #He
elim (IHL10 ⦠HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
- @(ex3_intro ⦠(L.â{I1}V2)) /3 width=3 by lpxs_pair, lsuby_cpxs_trans, lsuby_pair/
- #T #H1 #H2 lapply (injective_plus_l ⦠H1) lapply (injective_plus_l ⦠H2) -H1 -H2
- #H1 #H2 elim (IH T) // #HL0dx #HL0sn
- @conj #H @(lleq_lsuby_repl ⦠H) -H normalize
- /3 width=1 by lsuby_sym, lsuby_pair_O_Y/
+ @(ex3_intro ⦠(L.â{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/
+ #T elim (IH T) #HL0dx #HL0sn
+ @conj #H @(lleq_leq_repl ⦠H) -H /3 width=1 by leq_sym, leq_pair_O_Y/
| #I1 #I0 #L1 #L0 #V1 #V0 #d #e #HL10 #IHL10 #He #Y #H
elim (lpxs_inv_pair1 ⦠H) -H #L2 #V2 #HL02 #HV02 #H destruct
elim (IHL10 ⦠HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
- @(ex3_intro ⦠(L.â{I1}V1)) /3 width=1 by lpxs_pair, lsuby_succ/
- #T #H1 #H2 lapply (injective_plus_l ⦠H1) lapply (injective_plus_l ⦠H2) -H1 -H2
- #H1 #H2 elim (IH T) // #HL0dx #HL0sn
- @conj #H @(lleq_lsuby_repl ⦠H) -H
- /3 width=1 by lsuby_sym, lsuby_succ/ normalize //
+ @(ex3_intro ⦠(L.â{I1}V1)) /3 width=1 by lpxs_pair, leq_succ/
+ #T elim (IH T) #HL0dx #HL0sn
+ @conj #H @(lleq_leq_repl ⦠H) -H /3 width=1 by leq_sym, leq_succ/
]
qed-.
-lemma lsuby_lpxs_trans_lleq: âh,g,G,L1,L0,d. L1 âÃ[d, â] L0 â
- âL2. â¦G, L0⦠⢠â¡*[h, g] L2 â
- ââL. L âÃ[d, â] L2 & â¦G, L1⦠⢠â¡*[h, g] L &
- (âT. |L1| = |L0| â |L| = |L2| â L0 â[T, d] L2 â L1 â[T, d] L).
-/2 width=1 by lsuby_lpxs_trans_lleq_aux/ qed-.
+lemma leq_lpxs_trans_lleq: âh,g,G,L1,L0,d. L1 â[d, â] L0 â
+ âL2. â¦G, L0⦠⢠â¡*[h, g] L2 â
+ ââL. L â[d, â] L2 & â¦G, L1⦠⢠â¡*[h, g] L &
+ (âT. L0 â[T, d] L2 â L1 â[T, d] L).
+/2 width=1 by leq_lpxs_trans_lleq_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
index 5307af43e..59dda4e82 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
@@ -13,7 +13,7 @@
(**************************************************************************)
include "basic_2/notation/relations/lazysn_6.ma".
-include "basic_2/substitution/lleq.ma".
+include "basic_2/relocation/lleq.ma".
include "basic_2/computation/lpxs.ma".
(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
@@ -62,10 +62,10 @@ lemma lsx_gref: âh,g,G,L,d,p. G ⢠ââ¬*[h, g, §p, d] L.
/3 width=4 by lpxs_fwd_length, lleq_gref/
qed.
-lemma lsx_be: âh,g,G,L,T,U,dt,d,e. yinj d ⤠dt â dt ⤠d + e â
- â§[d, e] T â¡ U â G ⢠ââ¬*[h, g, U, dt] L â G ⢠ââ¬*[h, g, U, d] L.
-#h #g #G #L #T #U #dt #d #e #Hddt #Hdtde #HTU #H @(lsx_ind ⦠H) -L
-/5 width=7 by lsx_intro, lleq_be/
+lemma lsx_ge_up: âh,g,G,L,T,U,dt,d,e. dt ⤠yinj d + yinj e â
+ â§[d, e] T â¡ U â G ⢠ââ¬*[h, g, U, dt] L â G ⢠ââ¬*[h, g, U, d] L.
+#h #g #G #L #T #U #dt #d #e #Hdtde #HTU #H @(lsx_ind ⦠H) -L
+/5 width=7 by lsx_intro, lleq_ge_up/
qed-.
(* Basic forward lemmas *****************************************************)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_ldrop.ma
index 61466ca35..9bd29a5f1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_ldrop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_ldrop.ma
@@ -12,7 +12,7 @@
(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_ldrop.ma".
+include "basic_2/relocation/lleq_ldrop.ma".
include "basic_2/computation/lpxs_ldrop.ma".
include "basic_2/computation/lsx.ma".
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
index b3cb36781..9762de28b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
@@ -12,6 +12,7 @@
(* *)
(**************************************************************************)
+include "basic_2/relocation/lleq_lleq.ma".
include "basic_2/computation/lpxs_lleq.ma".
include "basic_2/computation/lpxs_lpxs.ma".
include "basic_2/computation/lsx.ma".
@@ -20,15 +21,12 @@ include "basic_2/computation/lsx.ma".
(* Advanced properties ******************************************************)
-lemma lsx_leqy_conf: âh,g,G,L1,T,d. G ⢠ââ¬*[h, g, T, d] L1 â
- âL2. L1 âÃ[d, â] L2 â |L1| = |L2| â G ⢠ââ¬*[h, g, T, d] L2.
+lemma lsx_leq_conf: âh,g,G,L1,T,d. G ⢠ââ¬*[h, g, T, d] L1 â
+ âL2. L1 â[d, â] L2 â G ⢠ââ¬*[h, g, T, d] L2.
#h #g #G #L1 #T #d #H @(lsx_ind ⦠H) -L1
-#L1 #_ #IHL1 #L2 #H1L12 #H2L12 @lsx_intro
-#L3 #H1L23 #HnL23 lapply (lpxs_fwd_length ⦠H1L23)
-#H2L23 elim (lsuby_lpxs_trans_lleq ⦠H1L12 ⦠H1L23) -H1L12 -H1L23
-#L0 #H1L03 #H1L10 #H lapply (lpxs_fwd_length ⦠H1L10)
-#H2L10 elim (H T) -H //
-#_ #H @(IHL1 ⦠H1L10) -IHL1 -H1L10 /3 width=1 by/
+#L1 #_ #IHL1 #L2 #HL12 @lsx_intro
+#L3 #HL23 #HnL23 elim (leq_lpxs_trans_lleq ⦠HL12 ⦠HL23) -HL12 -HL23
+#L0 #HL03 #HL10 #H elim (H T) -H /4 width=4 by/
qed-.
lemma lsx_ge: âh,g,G,L,T,d1,d2. d1 ⤠d2 â
@@ -113,10 +111,10 @@ lemma lsx_fwd_bind_dx: âh,g,a,I,G,L,V,T,d. G ⢠ââ¬*[h, g, â{a,I}V.T, d]
#L1 #_ #IHL1 @lsx_intro
#Y #H #HT elim (lpxs_inv_pair1 ⦠H) -H
#L2 #V2 #HL12 #_ #H destruct
-@(lsx_leqy_conf ⦠(L2.â{I}V1)) /2 width=1 by lsuby_succ/
+@(lsx_leq_conf ⦠(L2.â{I}V1)) /2 width=1 by leq_succ/
@IHL1 // #H @HT -IHL1 -HL12 -HT
-@(lleq_lsuby_trans ⦠(L2.â{I}V1))
-/2 width=2 by lleq_fwd_bind_dx, lsuby_succ/
+@(lleq_leq_trans ⦠(L2.â{I}V1))
+/2 width=2 by lleq_fwd_bind_dx, leq_succ/
qed-.
(* Advanced inversion lemmas ************************************************)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpx_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpx_cpys.etc
new file mode 100644
index 000000000..3b38ee6ab
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpx_cpys.etc
@@ -0,0 +1,42 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpys_alt.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on local environment refinement for extended substitution *****)
+
+lemma lsuby_cpx_trans: âh,g,G. lsub_trans ⦠(cpx h g G) (lsuby 0 (â)).
+#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+[ //
+| /2 width=2 by cpx_sort/
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (lsuby_ldrop_trans_be ⦠HL12 ⦠HLK1) // -HL12 -HLK1 /3 width=7 by cpx_delta/
+|4,9: /4 width=1 by cpx_bind, cpx_beta, lsuby_pair_O_Y/
+|5,7,8: /3 width=1 by cpx_flat, cpx_tau, cpx_ti/
+|6,10: /4 width=3 by cpx_zeta, cpx_theta, lsuby_pair_O_Y/
+]
+qed-.
+
+(* Properties on context-sensitive extended multiple substitution for terms *)
+
+lemma cpys_cpx: âh,g,G,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¡[h, g] T2.
+#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt ⦠H) -G -L -T1 -T2 -d -e
+/2 width=7 by cpx_delta, cpx_bind, cpx_flat/
+qed.
+
+lemma cpy_cpx: âh,g,G,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â â¦G, L⦠⢠T1 â¡[h, g] T2.
+/3 width=3 by cpy_cpys, cpys_cpx/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpxs_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpxs_cpys.etc
new file mode 100644
index 000000000..881030572
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpxs_cpys.etc
@@ -0,0 +1,24 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/reduction/cpx_cpys.ma".
+include "basic_2/computation/cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Properties on local environment refinement for extended substitution *****)
+
+lemma lsuby_cpxs_trans: âh,g,G. lsub_trans ⦠(cpxs h g G) (lsuby 0 (â)).
+/3 width=5 by lsuby_cpx_trans, LTC_lsub_trans/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpy.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpy.etc
new file mode 100644
index 000000000..a1e538ae0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpy.etc
@@ -0,0 +1,296 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/ynat/ynat_max.ma".
+include "basic_2/notation/relations/psubst_6.ma".
+include "basic_2/grammar/genv.ma".
+include "basic_2/relocation/lsuby.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* activate genv *)
+inductive cpy: ynat â ynat â relation4 genv lenv term term â
+| cpy_atom : âI,G,L,d,e. cpy d e G L (âª{I}) (âª{I})
+| cpy_subst: âI,G,L,K,V,W,i,d,e. d ⤠yinj i â i < d+e â
+ â©[i] L â¡ K.â{I}V â â§[0, i+1] V â¡ W â cpy d e G L (#i) W
+| cpy_bind : âa,I,G,L,V1,V2,T1,T2,d,e.
+ cpy d e G L V1 V2 â cpy (⫯d) e G (L.â{I}V1) T1 T2 â
+ cpy d e G L (â{a,I}V1.T1) (â{a,I}V2.T2)
+| cpy_flat : âI,G,L,V1,V2,T1,T2,d,e.
+ cpy d e G L V1 V2 â cpy d e G L T1 T2 â
+ cpy d e G L (â{I}V1.T1) (â{I}V2.T2)
+.
+
+interpretation "context-sensitive extended ordinary substritution (term)"
+ 'PSubst G L T1 d e T2 = (cpy d e G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cpy_trans: âG,d,e. lsub_trans ⦠(cpy d e G) (lsuby d e).
+#G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
+[ //
+| #I #G #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
+ elim (lsuby_ldrop_trans_be ⦠HL12 ⦠HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
+| /4 width=1 by lsuby_succ, cpy_bind/
+| /3 width=1 by cpy_flat/
+]
+qed-.
+
+lemma cpy_refl: âG,T,L,d,e. â¦G, L⦠⢠T â¶[d, e] T.
+#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
+qed.
+
+(* Basic_1: was: subst1_ex *)
+lemma cpy_full: âI,G,K,V,T1,L,d. â©[d] L â¡ K.â{I}V â
+ ââT2,T. â¦G, L⦠⢠T1 â¶[d, 1] T2 & â§[d, 1] T â¡ T2.
+#I #G #K #V #T1 elim T1 -T1
+[ * #i #L #d #HLK
+ /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
+ elim (lt_or_eq_or_gt i d) #Hid
+ /3 width=4 by lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/
+ destruct
+ elim (lift_total V 0 (i+1)) #W #HVW
+ elim (lift_split ⦠HVW i i)
+ /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
+| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+ elim (IHW1 ⦠HLK) -IHW1 #W2 #W #HW12 #HW2
+ [ elim (IHU1 (L.â{J}W1) (d+1)) -IHU1
+ /3 width=9 by cpy_bind, ldrop_drop, lift_bind, ex2_2_intro/
+ | elim (IHU1 ⦠HLK) -IHU1 -HLK
+ /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
+ ]
+]
+qed-.
+
+lemma cpy_weak: âG,L,T1,T2,d1,e1. â¦G, L⦠⢠T1 â¶[d1, e1] T2 â
+ âd2,e2. d2 ⤠d1 â d1 + e1 ⤠d2 + e2 â
+ â¦G, L⦠⢠T1 â¶[d2, e2] T2.
+#G #L #T1 #T2 #d1 #e1 #H elim H -G -L -T1 -T2 -d1 -e1 //
+[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
+| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
+| /3 width=1 by cpy_flat/
+]
+qed-.
+
+lemma cpy_weak_top: âG,L,T1,T2,d,e.
+ â¦G, L⦠⢠T1 â¶[d, e] T2 â â¦G, L⦠⢠T1 â¶[d, |L| - d] T2.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e //
+[ #I #G #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
+ lapply (ldrop_fwd_length_lt2 ⦠HLK)
+ /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
+| #a #I #G #L #V1 #V2 normalize in match (|L.â{I}V2|); (**) (* |?| does not work *)
+ /2 width=1 by cpy_bind/
+| /2 width=1 by cpy_flat/
+]
+qed-.
+
+lemma cpy_weak_full: âG,L,T1,T2,d,e.
+ â¦G, L⦠⢠T1 â¶[d, e] T2 â â¦G, L⦠⢠T1 â¶[0, |L|] T2.
+#G #L #T1 #T2 #d #e #HT12
+lapply (cpy_weak ⦠HT12 0 (d + e) ? ?) -HT12
+/2 width=2 by cpy_weak_top/
+qed-.
+
+lemma cpy_split_up: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â âi. i ⤠d + e â
+ ââT. â¦G, L⦠⢠T1 â¶[d, i-d] T & â¦G, L⦠⢠T â¶[i, d+e-i] T2.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
+ elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+ /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // #V
+ elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+ >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans ⦠HT2 (L.â{I}V) ?) -HT2
+ /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
+ /3 width=5 by ex2_intro, cpy_flat/
+]
+qed-.
+
+lemma cpy_split_down: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â âi. i ⤠d + e â
+ ââT. â¦G, L⦠⢠T1 â¶[i, d+e-i] T & â¦G, L⦠⢠T â¶[d, i-d] T2.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
+ elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+ /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // #V
+ elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+ >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans ⦠HT2 (L.â{I}V) ?) -HT2
+ /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
+ /3 width=5 by ex2_intro, cpy_flat/
+]
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpy_fwd_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
+ âT1,d,e. â§[d, e] T1 â¡ U1 â
+ d ⤠dt â d + e ⤠dt + et â
+ ââT2. â¦G, L⦠⢠U1 â¶[d+e, dt+et-(d+e)] U2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
+[ * #i #G #L #dt #et #T1 #d #e #H #_
+ [ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #K #V #W #i #dt #et #Hdti #Hidet #HLK #HVW #T1 #d #e #H #Hddt #Hdedet
+ elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct [ -V -Hidet -Hdedet | -Hdti -Hddt ]
+ [ elim (ylt_yle_false ⦠Hddt) -Hddt /3 width=3 by yle_ylt_trans, ylt_inj/
+ | elim (le_inv_plus_l ⦠Hid) #Hdie #Hei
+ elim (lift_split ⦠HVW d (i-e+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdie
+ #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hei
+ @(ex2_intro ⦠H) -H @(cpy_subst ⦠HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
+ elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 ⦠HVW1) -V1 -IHW12 //
+ elim (IHU12 ⦠HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
+ <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
+ /3 width=2 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
+ elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 ⦠HVW1) -V1 -IHW12 // elim (IHU12 ⦠HTU1) -T1 -IHU12
+ /3 width=2 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+lemma cpy_fwd_tw: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â â¯{T1} ⤠â¯{T2}.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e normalize
+/3 width=1 by monotonic_le_plus_l, le_plus/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact cpy_inv_atom1_aux: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â âJ. T1 = âª{J} â
+ T2 = âª{J} â¨
+ ââI,K,V,i. d ⤠yinj i & i < d + e &
+ â©[i] L â¡ K.â{I}V &
+ â§[O, i+1] V â¡ T2 &
+ J = LRef i.
+#G #L #T1 #T2 #d #e * -G -L -T1 -T2 -d -e
+[ #I #G #L #d #e #J #H destruct /2 width=1 by or_introl/
+| #I #G #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
+]
+qed-.
+
+lemma cpy_inv_atom1: âI,G,L,T2,d,e. â¦G, L⦠⢠âª{I} â¶[d, e] T2 â
+ T2 = âª{I} â¨
+ ââJ,K,V,i. d ⤠yinj i & i < d + e &
+ â©[i] L â¡ K.â{J}V &
+ â§[O, i+1] V â¡ T2 &
+ I = LRef i.
+/2 width=4 by cpy_inv_atom1_aux/ qed-.
+
+(* Basic_1: was: subst1_gen_sort *)
+lemma cpy_inv_sort1: âG,L,T2,k,d,e. â¦G, L⦠⢠âk â¶[d, e] T2 â T2 = âk.
+#G #L #T2 #k #d #e #H
+elim (cpy_inv_atom1 ⦠H) -H //
+* #I #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+(* Basic_1: was: subst1_gen_lref *)
+lemma cpy_inv_lref1: âG,L,T2,i,d,e. â¦G, L⦠⢠#i â¶[d, e] T2 â
+ T2 = #i â¨
+ ââI,K,V. d ⤠i & i < d + e &
+ â©[i] L â¡ K.â{I}V &
+ â§[O, i+1] V â¡ T2.
+#G #L #T2 #i #d #e #H
+elim (cpy_inv_atom1 ⦠H) -H /2 width=1 by or_introl/
+* #I #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
+qed-.
+
+lemma cpy_inv_gref1: âG,L,T2,p,d,e. â¦G, L⦠⢠§p â¶[d, e] T2 â T2 = §p.
+#G #L #T2 #p #d #e #H
+elim (cpy_inv_atom1 ⦠H) -H //
+* #I #K #V #i #_ #_ #_ #_ #H destruct
+qed-.
+
+fact cpy_inv_bind1_aux: âG,L,U1,U2,d,e. â¦G, L⦠⢠U1 â¶[d, e] U2 â
+ âa,I,V1,T1. U1 = â{a,I}V1.T1 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¶[d, e] V2 &
+ â¦G, L. â{I}V1⦠⢠T1 â¶[⫯d, e] T2 &
+ U2 = â{a,I}V2.T2.
+#G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
+[ #I #G #L #d #e #b #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #b #J #W1 #U1 #H destruct
+]
+qed-.
+
+lemma cpy_inv_bind1: âa,I,G,L,V1,T1,U2,d,e. â¦G, L⦠⢠â{a,I} V1. T1 â¶[d, e] U2 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¶[d, e] V2 &
+ â¦G, L.â{I}V1⦠⢠T1 â¶[⫯d, e] T2 &
+ U2 = â{a,I}V2.T2.
+/2 width=3 by cpy_inv_bind1_aux/ qed-.
+
+fact cpy_inv_flat1_aux: âG,L,U1,U2,d,e. â¦G, L⦠⢠U1 â¶[d, e] U2 â
+ âI,V1,T1. U1 = â{I}V1.T1 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¶[d, e] V2 &
+ â¦G, L⦠⢠T1 â¶[d, e] T2 &
+ U2 = â{I}V2.T2.
+#G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
+[ #I #G #L #d #e #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #W1 #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma cpy_inv_flat1: âI,G,L,V1,T1,U2,d,e. â¦G, L⦠⢠â{I} V1. T1 â¶[d, e] U2 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¶[d, e] V2 &
+ â¦G, L⦠⢠T1 â¶[d, e] T2 &
+ U2 = â{I}V2.T2.
+/2 width=3 by cpy_inv_flat1_aux/ qed-.
+
+
+fact cpy_inv_refl_O2_aux: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â e = 0 â T1 = T2.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
+[ //
+| #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
+ elim (ylt_yle_false ⦠Hdi) -Hdi //
+| /3 width=1 by eq_f2/
+| /3 width=1 by eq_f2/
+]
+qed-.
+
+lemma cpy_inv_refl_O2: âG,L,T1,T2,d. â¦G, L⦠⢠T1 â¶[d, 0] T2 â T1 = T2.
+/2 width=6 by cpy_inv_refl_O2_aux/ qed-.
+
+(* Basic_1: was: subst1_gen_lift_eq *)
+lemma cpy_inv_lift1_eq: âG,T1,U1,d,e. â§[d, e] T1 â¡ U1 â
+ âL,U2. â¦G, L⦠⢠U1 â¶[d, e] U2 â U1 = U2.
+#G #T1 #U1 #d #e #HTU1 #L #U2 #HU12 elim (cpy_fwd_up ⦠HU12 ⦠HTU1) -HU12 -HTU1
+/2 width=4 by cpy_inv_refl_O2/
+qed-.
+
+(* Basic_1: removed theorems 25:
+ subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
+ subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
+ subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
+ subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
+ subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
+ subst0_confluence_lift subst0_tlt
+ subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
+*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpy_cpy.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpy_cpy.etc
new file mode 100644
index 000000000..66b0487db
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpy_cpy.etc
@@ -0,0 +1,122 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/cpy_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Main properties **********************************************************)
+
+(* Basic_1: was: subst1_confluence_eq *)
+theorem cpy_conf_eq: âG,L,T0,T1,d1,e1. â¦G, L⦠⢠T0 â¶[d1, e1] T1 â
+ âT2,d2,e2. â¦G, L⦠⢠T0 â¶[d2, e2] T2 â
+ ââT. â¦G, L⦠⢠T1 â¶[d2, e2] T & â¦G, L⦠⢠T2 â¶[d1, e1] T.
+#G #L #T0 #T1 #d1 #e1 #H elim H -G -L -T0 -T1 -d1 -e1
+[ /2 width=3 by ex2_intro/
+| #I1 #G #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
+ elim (cpy_inv_lref1 ⦠H) -H
+ [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
+ | -Hd1 -Hde1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
+ lapply (ldrop_mono ⦠HLK1 ⦠HLK2) -HLK1 -HLK2 #H destruct
+ >(lift_mono ⦠HVT1 ⦠HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
+ ]
+| #a #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+ elim (cpy_inv_bind1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 ⦠HV02) -IHV01 -HV02 #V #HV1 #HV2
+ elim (IHT01 ⦠HT02) -T0 #T #HT1 #HT2
+ lapply (lsuby_cpy_trans ⦠HT1 (L.â{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+ lapply (lsuby_cpy_trans ⦠HT2 (L.â{I}V2) ?) -HT2
+ /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+| #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+ elim (cpy_inv_flat1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 ⦠HV02) -V0
+ elim (IHT01 ⦠HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: subst1_confluence_neq *)
+theorem cpy_conf_neq: âG,L1,T0,T1,d1,e1. â¦G, L1⦠⢠T0 â¶[d1, e1] T1 â
+ âL2,T2,d2,e2. â¦G, L2⦠⢠T0 â¶[d2, e2] T2 â
+ (d1 + e1 ⤠d2 ⨠d2 + e2 ⤠d1) â
+ ââT. â¦G, L2⦠⢠T1 â¶[d2, e2] T & â¦G, L1⦠⢠T2 â¶[d1, e1] T.
+#G #L1 #T0 #T1 #d1 #e1 #H elim H -G -L1 -T0 -T1 -d1 -e1
+[ /2 width=3 by ex2_intro/
+| #I1 #G #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
+ elim (cpy_inv_lref1 ⦠H1) -H1
+ [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
+ | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded [ -Hd1 -Hde2 | -Hd2 -Hde1 ]
+ [ elim (ylt_yle_false ⦠Hde1) -Hde1 /2 width=3 by yle_trans/
+ | elim (ylt_yle_false ⦠Hde2) -Hde2 /2 width=3 by yle_trans/
+ ]
+ ]
+| #a #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+ elim (cpy_inv_bind1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 ⦠HV02 H) -IHV01 -HV02 #V #HV1 #HV2
+ elim (IHT01 ⦠HT02) -T0
+ [ -H #T #HT1 #HT2
+ lapply (lsuby_cpy_trans ⦠HT1 (L2.â{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
+ lapply (lsuby_cpy_trans ⦠HT2 (L1.â{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
+ | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
+ ]
+| #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+ elim (cpy_inv_flat1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV01 ⦠HV02 H) -V0
+ elim (IHT01 ⦠HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
+]
+qed-.
+
+(* Note: the constant 1 comes from cpy_subst *)
+(* Basic_1: was: subst1_trans *)
+theorem cpy_trans_ge: âG,L,T1,T0,d,e. â¦G, L⦠⢠T1 â¶[d, e] T0 â
+ âT2. â¦G, L⦠⢠T0 â¶[d, 1] T2 â 1 ⤠e â â¦G, L⦠⢠T1 â¶[d, e] T2.
+#G #L #T1 #T0 #d #e #H elim H -G -L -T1 -T0 -d -e
+[ #I #G #L #d #e #T2 #H #He
+ elim (cpy_inv_atom1 ⦠H) -H
+ [ #H destruct //
+ | * #J #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
+ lapply (ylt_yle_trans ⦠(d+e) ⦠Hide2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
+ ]
+| #I #G #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
+ lapply (cpy_weak ⦠HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
+ >yplus_inj #HVT2 <(cpy_inv_lift1_eq ⦠HVW ⦠HVT2) -HVT2 /2 width=5 by cpy_subst/
+| #a #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+ elim (cpy_inv_bind1 ⦠H) -H #V2 #T2 #HV02 #HT02 #H destruct
+ lapply (lsuby_cpy_trans ⦠HT02 (L.â{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+ lapply (IHT10 ⦠HT02 He) -T0 /3 width=1 by cpy_bind/
+| #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+ elim (cpy_inv_flat1 ⦠H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
+]
+qed-.
+
+theorem cpy_trans_down: âG,L,T1,T0,d1,e1. â¦G, L⦠⢠T1 â¶[d1, e1] T0 â
+ âT2,d2,e2. â¦G, L⦠⢠T0 â¶[d2, e2] T2 â d2 + e2 ⤠d1 â
+ ââT. â¦G, L⦠⢠T1 â¶[d2, e2] T & â¦G, L⦠⢠T â¶[d1, e1] T2.
+#G #L #T1 #T0 #d1 #e1 #H elim H -G -L -T1 -T0 -d1 -e1
+[ /2 width=3 by ex2_intro/
+| #I #G #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
+ lapply (yle_trans ⦠Hde2d1 ⦠Hdi1) -Hde2d1 #Hde2i1
+ lapply (cpy_weak ⦠HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hde2i1
+ >yplus_inj #HWT2 <(cpy_inv_lift1_eq ⦠HVW ⦠HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
+| #a #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (cpy_inv_bind1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ lapply (lsuby_cpy_trans ⦠HT02 (L.â{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
+ elim (IHV10 ⦠HV02) -IHV10 -HV02 // #V
+ elim (IHT10 ⦠HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
+ lapply (lsuby_cpy_trans ⦠HT2 (L.â{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
+| #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+ elim (cpy_inv_flat1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
+ elim (IHV10 ⦠HV02) -V0 //
+ elim (IHT10 ⦠HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpy_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpy_lift.etc
new file mode 100644
index 000000000..aa0b2416a
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpy_lift.etc
@@ -0,0 +1,249 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/ldrop_ldrop.ma".
+include "basic_2/relocation/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
+
+(* Properties on relocation *************************************************)
+
+(* Basic_1: was: subst1_lift_lt *)
+lemma cpy_lift_le: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶[dt, et] T2 â
+ âL,U1,U2,s,d,e. â©[s, d, e] L â¡ K â
+ â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â
+ dt + et ⤠d â â¦G, L⦠⢠U1 â¶[dt, et] U2.
+#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
+[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
+ >(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdetd
+ lapply (ylt_yle_trans ⦠Hdetd ⦠Hidet) -Hdetd #Hid
+ lapply (ylt_inv_inj ⦠Hid) -Hid #Hid
+ lapply (lift_inv_lref1_lt ⦠H ⦠Hid) -H #H destruct
+ elim (lift_trans_ge ⦠HVW ⦠HWU2) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le ⦠HLK ⦠HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+ elim (ldrop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
+ >(lift_mono ⦠HVY ⦠HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
+| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
+| #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=7 by cpy_flat/
+]
+qed-.
+
+lemma cpy_lift_be: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶[dt, et] T2 â
+ âL,U1,U2,s,d,e. â©[s, d, e] L â¡ K â
+ â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â
+ dt ⤠d â d ⤠dt + et â â¦G, L⦠⢠U1 â¶[dt, et + e] U2.
+#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
+[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_ #_
+ >(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdtd #_
+ elim (lift_inv_lref1 ⦠H) -H * #Hid #H destruct
+ [ -Hdtd
+ lapply (ylt_yle_trans ⦠(dt+et+e) ⦠Hidet) // -Hidet #Hidete
+ elim (lift_trans_ge ⦠HVW ⦠HWU2) -W // <minus_plus #W #HVW #HWU2
+ elim (ldrop_trans_le ⦠HLK ⦠HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
+ elim (ldrop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
+ >(lift_mono ⦠HVY ⦠HVW) -V #H destruct /2 width=5 by cpy_subst/
+ | -Hdti
+ elim (yle_inv_inj2 ⦠Hdtd) -Hdtd #dtt #Hdtd #H destruct
+ lapply (transitive_le ⦠Hdtd Hid) -Hdtd #Hdti
+ lapply (lift_trans_be ⦠HVW ⦠HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm ⦠HLK ⦠HKV ?) -K // -Hid
+ /4 width=5 by cpy_subst, ldrop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
+ ]
+| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdtd #Hddet
+ elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+ elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=7 by cpy_flat/
+]
+qed-.
+
+(* Basic_1: was: subst1_lift_ge *)
+lemma cpy_lift_ge: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶[dt, et] T2 â
+ âL,U1,U2,s,d,e. â©[s, d, e] L â¡ K â
+ â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â
+ d ⤠dt â â¦G, L⦠⢠U1 â¶[dt+e, et] U2.
+#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
+[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
+ >(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
+| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hddt
+ lapply (yle_trans ⦠Hddt ⦠Hdti) -Hddt #Hid
+ elim (yle_inv_inj2 ⦠Hid) -Hid #dd #Hddi #H0 destruct
+ lapply (lift_inv_lref1_ge ⦠H ⦠Hddi) -H #H destruct
+ lapply (lift_trans_be ⦠HVW ⦠HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
+ lapply (ldrop_trans_ge_comm ⦠HLK ⦠HKV ?) -K // -Hddi
+ /3 width=5 by cpy_subst, ldrop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
+| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
+ elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /4 width=6 by cpy_bind, ldrop_skip, yle_succ/
+| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
+ elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
+ elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
+ /3 width=6 by cpy_flat/
+]
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+(* Basic_1: was: subst1_gen_lift_lt *)
+lemma cpy_inv_lift1_le: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ dt + et ⤠d â
+ ââT2. â¦G, K⦠⢠T1 â¶[dt, et] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
+[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdetd
+ lapply (ylt_yle_trans ⦠Hdetd ⦠Hidet) -Hdetd #Hid
+ lapply (ylt_inv_inj ⦠Hid) -Hid #Hid
+ lapply (lift_inv_lref2_lt ⦠H ⦠Hid) -H #H destruct
+ elim (ldrop_conf_lt ⦠HLK ⦠HLKV) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le ⦠HUV ⦠HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 ⦠HLK ⦠HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 ⦠HTU1) -IHU12 -HTU1
+ /3 width=6 by cpy_bind, yle_succ, ldrop_skip, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 ⦠HLK ⦠HVW1) -W1 //
+ elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK
+ /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+lemma cpy_inv_lift1_be: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ dt ⤠d â yinj d + e ⤠dt + et â
+ ââT2. â¦G, K⦠⢠T1 â¶[dt, et-e] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
+[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_ #_
+ [ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdtd #Hdedet
+ lapply (yle_fwd_plus_ge_inj ⦠Hdtd Hdedet) #Heet
+ elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct [ -Hdtd -Hidet | -Hdti -Hdedet ]
+ [ lapply (ylt_yle_trans i d (dt+(et-e)) ? ?) /2 width=1 by ylt_inj/
+ [ >yplus_minus_assoc_inj /2 width=1 by yle_plus_to_minus_inj2/ ] -Hdedet #Hidete
+ elim (ldrop_conf_lt ⦠HLK ⦠HLKV) -L // #L #U #HKL #_ #HUV
+ elim (lift_trans_le ⦠HUV ⦠HVW) -V // >minus_plus <plus_minus_m_m // -Hid
+ /3 width=5 by cpy_subst, ex2_intro/
+ | elim (le_inv_plus_l ⦠Hid) #Hdie #Hei
+ lapply (yle_trans ⦠Hdtd (i-e) ?) /2 width=1 by yle_inj/ -Hdtd #Hdtie
+ lapply (ldrop_conf_ge ⦠HLK ⦠HLKV ?) -L // #HKV
+ elim (lift_split ⦠HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hid -Hdie
+ #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
+ @(ex2_intro ⦠H) @(cpy_subst ⦠HKV HV1) // (**) (* explicit constructor *)
+ >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 ⦠HLK ⦠HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 ⦠HTU1) -U1
+ /3 width=6 by cpy_bind, ldrop_skip, lift_bind, yle_succ, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
+ elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 ⦠HLK ⦠HVW1) -W1 //
+ elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK //
+ /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Basic_1: was: subst1_gen_lift_ge *)
+lemma cpy_inv_lift1_ge: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ yinj d + e ⤠dt â
+ ââT2. â¦G, K⦠⢠T1 â¶[dt-e, et] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
+[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_
+ [ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
+ | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
+ ]
+| #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdedt
+ lapply (yle_trans ⦠Hdedt ⦠Hdti) #Hdei
+ elim (yle_inv_plus_inj2 ⦠Hdedt) -Hdedt #_ #Hedt
+ elim (yle_inv_plus_inj2 ⦠Hdei) #Hdie #Hei
+ lapply (lift_inv_lref2_ge ⦠H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
+ lapply (ldrop_conf_ge ⦠HLK ⦠HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
+ elim (lift_split ⦠HVW d (i-e+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hdei -Hdie
+ #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
+ @(ex2_intro ⦠H) @(cpy_subst ⦠HKV HV10) (**) (* explicit constructor *)
+ [ /2 width=1 by monotonic_yle_minus_dx/
+ | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
+ ]
+| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (yle_inv_plus_inj2 ⦠Hdetd) #_ #Hedt
+ elim (IHW12 ⦠HLK ⦠HVW1) -IHW12 // #V2 #HV12 #HVW2
+ elim (IHU12 ⦠HTU1) -U1 [4: @ldrop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
+ >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
+| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+ elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
+ elim (IHW12 ⦠HLK ⦠HVW1) -W1 //
+ elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
+]
+qed-.
+
+(* Advancd inversion lemmas on relocation ***********************************)
+
+lemma cpy_inv_lift1_ge_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ d ⤠dt â dt ⤠yinj d + e â yinj d + e ⤠dt + et â
+ ââT2. â¦G, K⦠⢠T1 â¶[d, dt + et - (yinj d + e)] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (cpy_split_up ⦠HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (cpy_weak ⦠HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
+lapply (cpy_inv_lift1_eq ⦠HTU1 ⦠HU1) -HU1 #HU1 destruct
+elim (cpy_inv_lift1_ge ⦠HU2 ⦠HLK ⦠HTU1) -U -L /2 width=3 by ex2_intro/
+qed-.
+
+lemma cpy_inv_lift1_be_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ dt ⤠d â dt + et ⤠yinj d + e â
+ ââT2. â¦G, K⦠⢠T1 â¶[dt, d-dt] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
+lapply (cpy_weak ⦠HU12 dt (d+e-dt) ? ?) -HU12 //
+[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hdetde #HU12
+elim (cpy_inv_lift1_be ⦠HU12 ⦠HLK ⦠HTU1) -U1 -L /2 width=3 by ex2_intro/
+qed-.
+
+lemma cpy_inv_lift1_le_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ dt ⤠d â d ⤠dt + et â dt + et ⤠yinj d + e â
+ ââT2. â¦G, K⦠⢠T1 â¶[dt, d - dt] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
+elim (cpy_split_up ⦠HU12 d) -HU12 // #U #HU1 #HU2
+elim (cpy_inv_lift1_le ⦠HU1 ⦠HLK ⦠HTU1) -U1
+[2: >ymax_pre_sn_comm // ] -Hdtd #T #HT1 #HTU
+lapply (cpy_weak ⦠HU2 d e ? ?) -HU2 //
+[ >ymax_pre_sn_comm // ] -Hddet -Hdetde #HU2
+lapply (cpy_inv_lift1_eq ⦠HTU ⦠HU2) -L #H destruct /2 width=3 by ex2_intro/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys.etc
new file mode 100644
index 000000000..10c59120d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys.etc
@@ -0,0 +1,166 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/psubststar_6.ma".
+include "basic_2/relocation/cpy.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+definition cpys: ynat â ynat â relation4 genv lenv term term â
+ λd,e,G. LTC ⦠(cpy d e G).
+
+interpretation "context-sensitive extended multiple substritution (term)"
+ 'PSubstStar G L T1 d e T2 = (cpys d e G L T1 T2).
+
+(* Basic eliminators ********************************************************)
+
+lemma cpys_ind: âG,L,T1,d,e. âR:predicate term. R T1 â
+ (âT,T2. â¦G, L⦠⢠T1 â¶*[d, e] T â â¦G, L⦠⢠T â¶[d, e] T2 â R T â R T2) â
+ âT2. â¦G, L⦠⢠T1 â¶*[d, e] T2 â R T2.
+#G #L #T1 #d #e #R #HT1 #IHT1 #T2 #HT12
+@(TC_star_ind ⦠HT1 IHT1 ⦠HT12) //
+qed-.
+
+lemma cpys_ind_dx: âG,L,T2,d,e. âR:predicate term. R T2 â
+ (âT1,T. â¦G, L⦠⢠T1 â¶[d, e] T â â¦G, L⦠⢠T â¶*[d, e] T2 â R T â R T1) â
+ âT1. â¦G, L⦠⢠T1 â¶*[d, e] T2 â R T1.
+#G #L #T2 #d #e #R #HT2 #IHT2 #T1 #HT12
+@(TC_star_ind_dx ⦠HT2 IHT2 ⦠HT12) //
+qed-.
+
+(* Basic properties *********************************************************)
+
+lemma cpy_cpys: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â â¦G, L⦠⢠T1 â¶*[d, e] T2.
+/2 width=1 by inj/ qed.
+
+lemma cpys_strap1: âG,L,T1,T,T2,d,e.
+ â¦G, L⦠⢠T1 â¶*[d, e] T â â¦G, L⦠⢠T â¶[d, e] T2 â â¦G, L⦠⢠T1 â¶*[d, e] T2.
+normalize /2 width=3 by step/ qed-.
+
+lemma cpys_strap2: âG,L,T1,T,T2,d,e.
+ â¦G, L⦠⢠T1 â¶[d, e] T â â¦G, L⦠⢠T â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¶*[d, e] T2.
+normalize /2 width=3 by TC_strap/ qed-.
+
+lemma lsuby_cpys_trans: âG,d,e. lsub_trans ⦠(cpys d e G) (lsuby d e).
+/3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/
+qed-.
+
+lemma cpys_refl: âG,L,d,e. reflexive ⦠(cpys d e G L).
+/2 width=1 by cpy_cpys/ qed.
+
+lemma cpys_bind: âG,L,V1,V2,d,e. â¦G, L⦠⢠V1 â¶*[d, e] V2 â
+ âI,T1,T2. â¦G, L.â{I}V1⦠⢠T1 â¶*[⫯d, e] T2 â
+ âa. â¦G, L⦠⢠â{a,I}V1.T1 â¶*[d, e] â{a,I}V2.T2.
+#G #L #V1 #V2 #d #e #HV12 @(cpys_ind ⦠HV12) -V2
+[ #I #T1 #T2 #HT12 @(cpys_ind ⦠HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/
+| /3 width=5 by cpys_strap1, cpy_bind/
+]
+qed.
+
+lemma cpys_flat: âG,L,V1,V2,d,e. â¦G, L⦠⢠V1 â¶*[d, e] V2 â
+ âT1,T2. â¦G, L⦠⢠T1 â¶*[d, e] T2 â
+ âI. â¦G, L⦠⢠â{I}V1.T1 â¶*[d, e] â{I}V2.T2.
+#G #L #V1 #V2 #d #e #HV12 @(cpys_ind ⦠HV12) -V2
+[ #T1 #T2 #HT12 @(cpys_ind ⦠HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/
+| /3 width=5 by cpys_strap1, cpy_flat/
+qed.
+
+lemma cpys_weak: âG,L,T1,T2,d1,e1. â¦G, L⦠⢠T1 â¶*[d1, e1] T2 â
+ âd2,e2. d2 ⤠d1 â d1 + e1 ⤠d2 + e2 â
+ â¦G, L⦠⢠T1 â¶*[d2, e2] T2.
+#G #L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(cpys_ind ⦠H) -T2
+/3 width=7 by cpys_strap1, cpy_weak/
+qed-.
+
+lemma cpys_weak_top: âG,L,T1,T2,d,e.
+ â¦G, L⦠⢠T1 â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¶*[d, |L| - d] T2.
+#G #L #T1 #T2 #d #e #H @(cpys_ind ⦠H) -T2
+/3 width=4 by cpys_strap1, cpy_weak_top/
+qed-.
+
+lemma cpys_weak_full: âG,L,T1,T2,d,e.
+ â¦G, L⦠⢠T1 â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¶*[0, |L|] T2.
+#G #L #T1 #T2 #d #e #H @(cpys_ind ⦠H) -T2
+/3 width=5 by cpys_strap1, cpy_weak_full/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma cpys_fwd_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
+ âT1,d,e. â§[d, e] T1 â¡ U1 â
+ d ⤠dt â d + e ⤠dt + et â
+ ââT2. â¦G, L⦠⢠U1 â¶*[d+e, dt+et-(d+e)] U2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H #T1 #d #e #HTU1 #Hddt #Hdedet @(cpys_ind ⦠H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU
+ elim (cpy_fwd_up ⦠HU2 ⦠HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_fwd_tw: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶*[d, e] T2 â â¯{T1} ⤠â¯{T2}.
+#G #L #T1 #T2 #d #e #H @(cpys_ind ⦠H) -T2 //
+#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw ⦠HT2) -HT2
+/2 width=3 by transitive_le/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+(* Note: this can be derived from cpys_inv_atom1 *)
+lemma cpys_inv_sort1: âG,L,T2,k,d,e. â¦G, L⦠⢠âk â¶*[d, e] T2 â T2 = âk.
+#G #L #T2 #k #d #e #H @(cpys_ind ⦠H) -T2 //
+#T #T2 #_ #HT2 #IHT1 destruct
+>(cpy_inv_sort1 ⦠HT2) -HT2 //
+qed-.
+
+(* Note: this can be derived from cpys_inv_atom1 *)
+lemma cpys_inv_gref1: âG,L,T2,p,d,e. â¦G, L⦠⢠§p â¶*[d, e] T2 â T2 = §p.
+#G #L #T2 #p #d #e #H @(cpys_ind ⦠H) -T2 //
+#T #T2 #_ #HT2 #IHT1 destruct
+>(cpy_inv_gref1 ⦠HT2) -HT2 //
+qed-.
+
+lemma cpys_inv_bind1: âa,I,G,L,V1,T1,U2,d,e. â¦G, L⦠⢠â{a,I}V1.T1 â¶*[d, e] U2 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¶*[d, e] V2 &
+ â¦G, L.â{I}V1⦠⢠T1 â¶*[⫯d, e] T2 &
+ U2 = â{a,I}V2.T2.
+#a #I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind ⦠H) -U2
+[ /2 width=5 by ex3_2_intro/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (cpy_inv_bind1 ⦠HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
+ lapply (lsuby_cpy_trans ⦠HT2 (L.â{I}V1) ?) -HT2
+ /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/
+]
+qed-.
+
+lemma cpys_inv_flat1: âI,G,L,V1,T1,U2,d,e. â¦G, L⦠⢠â{I}V1.T1 â¶*[d, e] U2 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¶*[d, e] V2 & â¦G, L⦠⢠T1 â¶*[d, e] T2 &
+ U2 = â{I}V2.T2.
+#I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind ⦠H) -U2
+[ /2 width=5 by ex3_2_intro/
+| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
+ elim (cpy_inv_flat1 ⦠HU2) -HU2
+ /3 width=5 by cpys_strap1, ex3_2_intro/
+]
+qed-.
+
+lemma cpys_inv_refl_O2: âG,L,T1,T2,d. â¦G, L⦠⢠T1 â¶*[d, 0] T2 â T1 = T2.
+#G #L #T1 #T2 #d #H @(cpys_ind ⦠H) -T2 //
+#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 ⦠HT2) -HT2 //
+qed-.
+
+lemma cpys_inv_lift1_eq: âG,L,U1,U2. âd,e:nat.
+ â¦G, L⦠⢠U1 â¶*[d, e] U2 â âT1. â§[d, e] T1 â¡ U1 â U1 = U2.
+#G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind ⦠H) -U2
+/2 width=7 by cpy_inv_lift1_eq/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys_alt.etc
new file mode 100644
index 000000000..e297be698
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys_alt.etc
@@ -0,0 +1,102 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/psubststaralt_6.ma".
+include "basic_2/substitution/cpys_lift.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* alternative definition of cpys *)
+inductive cpysa: ynat â ynat â relation4 genv lenv term term â
+| cpysa_atom : âI,G,L,d,e. cpysa d e G L (âª{I}) (âª{I})
+| cpysa_subst: âI,G,L,K,V1,V2,W2,i,d,e. d ⤠yinj i â i < d+e â
+ â©[i] L â¡ K.â{I}V1 â cpysa 0 (â«°(d+e-i)) G K V1 V2 â
+ â§[0, i+1] V2 â¡ W2 â cpysa d e G L (#i) W2
+| cpysa_bind : âa,I,G,L,V1,V2,T1,T2,d,e.
+ cpysa d e G L V1 V2 â cpysa (⫯d) e G (L.â{I}V1) T1 T2 â
+ cpysa d e G L (â{a,I}V1.T1) (â{a,I}V2.T2)
+| cpysa_flat : âI,G,L,V1,V2,T1,T2,d,e.
+ cpysa d e G L V1 V2 â cpysa d e G L T1 T2 â
+ cpysa d e G L (â{I}V1.T1) (â{I}V2.T2)
+.
+
+interpretation
+ "context-sensitive extended multiple substritution (term) alternative"
+ 'PSubstStarAlt G L T1 d e T2 = (cpysa d e G L T1 T2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_cpysa_trans: âG,d,e. lsub_trans ⦠(cpysa d e G) (lsuby d e).
+#G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
+[ //
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (lsuby_ldrop_trans_be ⦠HL12 ⦠HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/
+| /4 width=1 by lsuby_succ, cpysa_bind/
+| /3 width=1 by cpysa_flat/
+]
+qed-.
+
+lemma cpysa_refl: âG,T,L,d,e. â¦G, L⦠⢠T â¶â¶*[d, e] T.
+#G #T elim T -T //
+#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/
+qed.
+
+lemma cpysa_cpy_trans: âG,L,T1,T,d,e. â¦G, L⦠⢠T1 â¶â¶*[d, e] T â
+ âT2. â¦G, L⦠⢠T â¶[d, e] T2 â â¦G, L⦠⢠T1 â¶â¶*[d, e] T2.
+#G #L #T1 #T #d #e #H elim H -G -L -T1 -T -d -e
+[ #I #G #L #d #e #X #H
+ elim (cpy_inv_atom1 ⦠H) -H // * /2 width=7 by cpysa_subst/
+| #I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
+ lapply (ldrop_fwd_drop2 ⦠HLK) #H0LK
+ lapply (cpy_weak ⦠H 0 (d+e) ? ?) -H // #H
+ elim (cpy_inv_lift1_be ⦠H ⦠H0LK ⦠HVW2) -H -H0LK -HVW2
+ /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
+| #a #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
+ elim (cpy_inv_bind1 ⦠H) -H #V2 #T2 #HV2 #HT2 #H destruct
+ /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/
+| #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
+ elim (cpy_inv_flat1 ⦠H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/
+]
+qed-.
+
+lemma cpys_cpysa: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¶â¶*[d, e] T2.
+/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma cpysa_inv_cpys: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¶*[d, e] T2.
+#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
+/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
+qed-.
+
+(* Advanced eliminators *****************************************************)
+
+lemma cpys_ind_alt: âR:ynatâynatârelation4 genv lenv term term.
+ (âI,G,L,d,e. R d e G L (âª{I}) (âª{I})) â
+ (âI,G,L,K,V1,V2,W2,i,d,e. d ⤠yinj i â i < d + e â
+ â©[i] L â¡ K.â{I}V1 â â¦G, K⦠⢠V1 â¶*[O, â«°(d+e-i)] V2 â
+ â§[O, i+1] V2 â¡ W2 â R O (â«°(d+e-i)) G K V1 V2 â R d e G L (#i) W2
+ ) â
+ (âa,I,G,L,V1,V2,T1,T2,d,e. â¦G, L⦠⢠V1 â¶*[d, e] V2 â
+ â¦G, L.â{I}V1⦠⢠T1 â¶*[⫯d, e] T2 â R d e G L V1 V2 â
+ R (⫯d) e G (L.â{I}V1) T1 T2 â R d e G L (â{a,I}V1.T1) (â{a,I}V2.T2)
+ ) â
+ (âI,G,L,V1,V2,T1,T2,d,e. â¦G, L⦠⢠V1 â¶*[d, e] V2 â
+ â¦G, L⦠⢠T1 â¶*[d, e] T2 â R d e G L V1 V2 â
+ R d e G L T1 T2 â R d e G L (â{I}V1.T1) (â{I}V2.T2)
+ ) â
+ âd,e,G,L,T1,T2. â¦G, L⦠⢠T1 â¶*[d, e] T2 â R d e G L T1 T2.
+#R #H1 #H2 #H3 #H4 #d #e #G #L #T1 #T2 #H elim (cpys_cpysa ⦠H) -G -L -T1 -T2 -d -e
+/3 width=8 by cpysa_inv_cpys/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys_cpys.etc
new file mode 100644
index 000000000..52759ca7f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys_cpys.etc
@@ -0,0 +1,117 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/cpy_cpy.ma".
+include "basic_2/substitution/cpys_alt.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma cpys_inv_SO2: âG,L,T1,T2,d. â¦G, L⦠⢠T1 â¶*[d, 1] T2 â â¦G, L⦠⢠T1 â¶[d, 1] T2.
+#G #L #T1 #T2 #d #H @(cpys_ind ⦠H) -T2 /2 width=3 by cpy_trans_ge/
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma cpys_strip_eq: âG,L,T0,T1,d1,e1. â¦G, L⦠⢠T0 â¶*[d1, e1] T1 â
+ âT2,d2,e2. â¦G, L⦠⢠T0 â¶[d2, e2] T2 â
+ ââT. â¦G, L⦠⢠T1 â¶[d2, e2] T & â¦G, L⦠⢠T2 â¶*[d1, e1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-.
+
+lemma cpys_strip_neq: âG,L1,T0,T1,d1,e1. â¦G, L1⦠⢠T0 â¶*[d1, e1] T1 â
+ âL2,T2,d2,e2. â¦G, L2⦠⢠T0 â¶[d2, e2] T2 â
+ (d1 + e1 ⤠d2 ⨠d2 + e2 ⤠d1) â
+ ââT. â¦G, L2⦠⢠T1 â¶[d2, e2] T & â¦G, L1⦠⢠T2 â¶*[d1, e1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-.
+
+lemma cpys_strap1_down: âG,L,T1,T0,d1,e1. â¦G, L⦠⢠T1 â¶*[d1, e1] T0 â
+ âT2,d2,e2. â¦G, L⦠⢠T0 â¶[d2, e2] T2 â d2 + e2 ⤠d1 â
+ ââT. â¦G, L⦠⢠T1 â¶[d2, e2] T & â¦G, L⦠⢠T â¶*[d1, e1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed.
+
+lemma cpys_strap2_down: âG,L,T1,T0,d1,e1. â¦G, L⦠⢠T1 â¶[d1, e1] T0 â
+ âT2,d2,e2. â¦G, L⦠⢠T0 â¶*[d2, e2] T2 â d2 + e2 ⤠d1 â
+ ââT. â¦G, L⦠⢠T1 â¶*[d2, e2] T & â¦G, L⦠⢠T â¶[d1, e1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-.
+
+lemma cpys_split_up: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶*[d, e] T2 â
+ âi. d ⤠i â i ⤠d + e â
+ ââT. â¦G, L⦠⢠T1 â¶*[d, i - d] T & â¦G, L⦠⢠T â¶*[i, d + e - i] T2.
+#G #L #T1 #T2 #d #e #H #i #Hdi #Hide @(cpys_ind ⦠H) -T2
+[ /2 width=3 by ex2_intro/
+| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
+ elim (cpy_split_up ⦠HT12 ⦠Hide) -HT12 -Hide #T0 #HT0 #HT02
+ elim (cpys_strap1_down ⦠HT3 ⦠HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
+ >ymax_pre_sn_comm //
+]
+qed-.
+
+lemma cpys_inv_lift1_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ d ⤠dt â dt ⤠yinj d + e â yinj d + e ⤠dt + et â
+ ââT2. â¦G, K⦠⢠T1 â¶*[d, dt + et - (yinj d + e)] T2 &
+ â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
+elim (cpys_split_up ⦠HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
+lapply (cpys_weak ⦠HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
+lapply (cpys_inv_lift1_eq ⦠HU1 ⦠HTU1) -HU1 #HU1 destruct
+elim (cpys_inv_lift1_ge ⦠HU2 ⦠HLK ⦠HTU1) -HU2 -HLK -HTU1 //
+>yplus_minus_inj /2 width=3 by ex2_intro/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem cpys_conf_eq: âG,L,T0,T1,d1,e1. â¦G, L⦠⢠T0 â¶*[d1, e1] T1 â
+ âT2,d2,e2. â¦G, L⦠⢠T0 â¶*[d2, e2] T2 â
+ ââT. â¦G, L⦠⢠T1 â¶*[d2, e2] T & â¦G, L⦠⢠T2 â¶*[d1, e1] T.
+normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-.
+
+theorem cpys_conf_neq: âG,L1,T0,T1,d1,e1. â¦G, L1⦠⢠T0 â¶*[d1, e1] T1 â
+ âL2,T2,d2,e2. â¦G, L2⦠⢠T0 â¶*[d2, e2] T2 â
+ (d1 + e1 ⤠d2 ⨠d2 + e2 ⤠d1) â
+ ââT. â¦G, L2⦠⢠T1 â¶*[d2, e2] T & â¦G, L1⦠⢠T2 â¶*[d1, e1] T.
+normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-.
+
+theorem cpys_trans_eq: âG,L,T1,T,T2,d,e.
+ â¦G, L⦠⢠T1 â¶*[d, e] T â â¦G, L⦠⢠T â¶*[d, e] T2 â
+ â¦G, L⦠⢠T1 â¶*[d, e] T2.
+normalize /2 width=3 by trans_TC/ qed-.
+
+theorem cpys_trans_down: âG,L,T1,T0,d1,e1. â¦G, L⦠⢠T1 â¶*[d1, e1] T0 â
+ âT2,d2,e2. â¦G, L⦠⢠T0 â¶*[d2, e2] T2 â d2 + e2 ⤠d1 â
+ ââT. â¦G, L⦠⢠T1 â¶*[d2, e2] T & â¦G, L⦠⢠T â¶*[d1, e1] T2.
+normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-.
+
+theorem cpys_antisym_eq: âG,L1,T1,T2,d,e. â¦G, L1⦠⢠T1 â¶*[d, e] T2 â
+ âL2. â¦G, L2⦠⢠T2 â¶*[d, e] T1 â T1 = T2.
+#G #L1 #T1 #T2 #d #e #H @(cpys_ind_alt ⦠H) -G -L1 -T1 -T2 //
+[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #_ #_ #HVW2 #_ #L2 #HW2
+ elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hdi -Hide | ]
+ [ lapply (cpys_weak_full ⦠HW2) -HW2 #HW2
+ lapply (cpys_weak ⦠HW2 0 (i+1) ? ?) -HW2 //
+ [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi
+ #HW2 >(cpys_inv_lift1_eq ⦠HW2) -HW2 //
+ | elim (ldrop_O1_le ⦠Hi) -Hi #K2 #HLK2
+ elim (cpys_inv_lift1_ge_up ⦠HW2 ⦠HLK2 ⦠HVW2 ? ? ?) -HW2 -HLK2 -HVW2
+ /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ -Hdi -Hide
+ #X #_ #H elim (lift_inv_lref2_be ⦠H) -H //
+ ]
+| #a #I #G #L1 #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 ⦠H) -H
+ #V #T #HV2 #HT2 #H destruct
+ lapply (IHV12 ⦠HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/
+| #I #G #L1 #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 ⦠H) -H
+ #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys_lift.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys_lift.etc
new file mode 100644
index 000000000..b6d9e45a1
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/cpys_lift.etc
@@ -0,0 +1,218 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/cpy_lift.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
+
+(* Advanced properties ******************************************************)
+
+lemma cpys_subst: âI,G,L,K,V,U1,i,d,e.
+ d ⤠yinj i â i < d + e â
+ â©[i] L â¡ K.â{I}V â â¦G, K⦠⢠V â¶*[0, â«°(d+e-i)] U1 â
+ âU2. â§[0, i+1] U1 â¡ U2 â â¦G, L⦠⢠#i â¶*[d, e] U2.
+#I #G #L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(cpys_ind ⦠H) -U1
+[ /3 width=5 by cpy_cpys, cpy_subst/
+| #U #U1 #_ #HU1 #IHU #U2 #HU12
+ elim (lift_total U 0 (i+1)) #U0 #HU0
+ lapply (IHU ⦠HU0) -IHU #H
+ lapply (ldrop_fwd_drop2 ⦠HLK) -HLK #HLK
+ lapply (cpy_lift_ge ⦠HU1 ⦠HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
+ lapply (cpy_weak ⦠HU02 d e ? ?) -HU02
+ [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
+ >yplus_O1 <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
+]
+qed.
+
+lemma cpys_subst_Y2: âI,G,L,K,V,U1,i,d.
+ d ⤠yinj i â
+ â©[i] L â¡ K.â{I}V â â¦G, K⦠⢠V â¶*[0, â] U1 â
+ âU2. â§[0, i+1] U1 â¡ U2 â â¦G, L⦠⢠#i â¶*[d, â] U2.
+#I #G #L #K #V #U1 #i #d #Hdi #HLK #HVU1 #U2 #HU12
+@(cpys_subst ⦠HLK ⦠HU12) >yminus_Y_inj //
+qed.
+
+(* Advanced inverion lemmas *************************************************)
+
+lemma cpys_inv_atom1: âI,G,L,T2,d,e. â¦G, L⦠⢠âª{I} â¶*[d, e] T2 â
+ T2 = âª{I} â¨
+ ââJ,K,V1,V2,i. d ⤠yinj i & i < d + e &
+ â©[i] L â¡ K.â{J}V1 &
+ â¦G, K⦠⢠V1 â¶*[0, â«°(d+e-i)] V2 &
+ â§[O, i+1] V2 â¡ T2 &
+ I = LRef i.
+#I #G #L #T2 #d #e #H @(cpys_ind ⦠H) -T2
+[ /2 width=1 by or_introl/
+| #T #T2 #_ #HT2 *
+ [ #H destruct
+ elim (cpy_inv_atom1 ⦠HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
+ | * #J #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
+ lapply (ldrop_fwd_drop2 ⦠HLK) #H
+ elim (cpy_inv_lift1_ge_up ⦠HT2 ⦠H ⦠HVT) -HT2 -H -HVT
+ [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
+ /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
+ ]
+]
+qed-.
+
+lemma cpys_inv_lref1: âG,L,T2,i,d,e. â¦G, L⦠⢠#i â¶*[d, e] T2 â
+ T2 = #i â¨
+ ââI,K,V1,V2. d ⤠i & i < d + e &
+ â©[i] L â¡ K.â{I}V1 &
+ â¦G, K⦠⢠V1 â¶*[0, â«°(d+e-i)] V2 &
+ â§[O, i+1] V2 â¡ T2.
+#G #L #T2 #i #d #e #H elim (cpys_inv_atom1 ⦠H) -H /2 width=1 by or_introl/
+* #I #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
+qed-.
+
+lemma cpys_inv_lref1_ldrop: âG,L,T2,i,d,e. â¦G, L⦠⢠#i â¶*[d, e] T2 â
+ âI,K,V1. â©[i] L â¡ K.â{I}V1 â
+ âV2. â§[O, i+1] V2 â¡ T2 â
+ â§â§ â¦G, K⦠⢠V1 â¶*[0, â«°(d+e-i)] V2
+ & d ⤠i
+ & i < d + e.
+#G #L #T2 #i #d #e #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 ⦠H) -H
+[ #H destruct elim (lift_inv_lref2_be ⦠HVT2) -HVT2 -HLK //
+| * #Z #Y #X1 #X2 #Hdi #Hide #HLY #HX12 #HXT2
+ lapply (lift_inj ⦠HXT2 ⦠HVT2) -T2 #H destruct
+ lapply (ldrop_mono ⦠HLY ⦠HLK) -L #H destruct
+ /2 width=1 by and3_intro/
+]
+qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma cpys_lift_le: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶*[dt, et] T2 â
+ âL,U1,s,d,e. dt + et ⤠yinj d â â©[s, d, e] L â¡ K â
+ â§[d, e] T1 â¡ U1 â âU2. â§[d, e] T2 â¡ U2 â
+ â¦G, L⦠⢠U1 â¶*[dt, et] U2.
+#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdetd #HLK #HTU1 @(cpys_ind ⦠H) -T2
+[ #U2 #H >(lift_mono ⦠HTU1 ⦠H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (IHT ⦠HTU) -IHT #HU1
+ lapply (cpy_lift_le ⦠HT2 ⦠HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+lemma cpys_lift_be: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶*[dt, et] T2 â
+ âL,U1,s,d,e. dt ⤠yinj d â d ⤠dt + et â
+ â©[s, d, e] L â¡ K â â§[d, e] T1 â¡ U1 â
+ âU2. â§[d, e] T2 â¡ U2 â â¦G, L⦠⢠U1 â¶*[dt, et + e] U2.
+#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind ⦠H) -T2
+[ #U2 #H >(lift_mono ⦠HTU1 ⦠H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (IHT ⦠HTU) -IHT #HU1
+ lapply (cpy_lift_be ⦠HT2 ⦠HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+lemma cpys_lift_ge: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶*[dt, et] T2 â
+ âL,U1,s,d,e. yinj d ⤠dt â â©[s, d, e] L â¡ K â
+ â§[d, e] T1 â¡ U1 â âU2. â§[d, e] T2 â¡ U2 â
+ â¦G, L⦠⢠U1 â¶*[dt+e, et] U2.
+#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hddt #HLK #HTU1 @(cpys_ind ⦠H) -T2
+[ #U2 #H >(lift_mono ⦠HTU1 ⦠H) -H //
+| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
+ elim (lift_total T d e) #U #HTU
+ lapply (IHT ⦠HTU) -IHT #HU1
+ lapply (cpy_lift_ge ⦠HT2 ⦠HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
+]
+qed-.
+
+(* Inversion lemmas for relocation ******************************************)
+
+lemma cpys_inv_lift1_le: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ dt + et ⤠d â
+ ââT2. â¦G, K⦠⢠T1 â¶*[dt, et] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdetd @(cpys_ind ⦠H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_le ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_be: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ dt ⤠d â yinj d + e ⤠dt + et â
+ ââT2. â¦G, K⦠⢠T1 â¶*[dt, et - e] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(cpys_ind ⦠H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_be ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_ge: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ yinj d + e ⤠dt â
+ ââT2. â¦G, K⦠⢠T1 â¶*[dt - e, et] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind ⦠H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_ge ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+(* Advanced inversion lemmas on relocation **********************************)
+
+lemma cpys_inv_lift1_ge_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ d ⤠dt â dt ⤠yinj d + e â yinj d + e ⤠dt + et â
+ ââT2. â¦G, K⦠⢠T1 â¶*[d, dt + et - (yinj d + e)] T2 &
+ â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(cpys_ind ⦠H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_ge_up ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_be_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ dt ⤠d â dt + et ⤠yinj d + e â
+ ââT2. â¦G, K⦠⢠T1 â¶*[dt, d - dt] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(cpys_ind ⦠H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_be_up ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_le_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
+ âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
+ dt ⤠d â d ⤠dt + et â dt + et ⤠yinj d + e â
+ ââT2. â¦G, K⦠⢠T1 â¶*[dt, d - dt] T2 & â§[d, e] T2 â¡ U2.
+#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(cpys_ind ⦠H) -U2
+[ /2 width=3 by ex2_intro/
+| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
+ elim (cpy_inv_lift1_le_up ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
+]
+qed-.
+
+lemma cpys_inv_lift1_subst: âG,L,W1,W2,d,e. â¦G, L⦠⢠W1 â¶*[d, e] W2 â
+ âK,V1,i. â©[i+1] L â¡ K â â§[O, i+1] V1 â¡ W1 â
+ d ⤠yinj i â i < d + e â
+ ââV2. â¦G, K⦠⢠V1 â¶*[O, â«°(d+e-i)] V2 & â§[O, i+1] V2 â¡ W2.
+#G #L #W1 #W2 #d #e #HW12 #K #V1 #i #HLK #HVW1 #Hdi #Hide
+elim (cpys_inv_lift1_ge_up ⦠HW12 ⦠HLK ⦠HVW1 ? ? ?) //
+>yplus_O1 <yplus_inj >yplus_SO2
+[ >yminus_succ2 /2 width=3 by ex2_intro/
+| /2 width=1 by ylt_fwd_le_succ1/
+| /2 width=3 by yle_trans/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/extlrsubeq_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/extlrsubeq_4.etc
new file mode 100644
index 000000000..8a9714dd8
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/extlrsubeq_4.etc
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The 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 â Ã [ term 46 d , break term 46 e ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'ExtLRSubEq $L1 $d $e $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lazyeqalt_4.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lazyeqalt_4.etc
new file mode 100644
index 000000000..b89d7844c
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lazyeqalt_4.etc
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The 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 â â break [ term 46 T , break term 46 d ] break term 46 L2 )"
+ non associative with precedence 45
+ for @{ 'LazyEqAlt $T $d $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq.etc
new file mode 100644
index 000000000..72a5346ce
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq.etc
@@ -0,0 +1,160 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/lazyeq_4.ma".
+include "basic_2/substitution/cpys.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+definition lleq: relation4 ynat term lenv lenv â
+ λd,T,L1,L2. |L1| = |L2| â§
+ (âU. â¦â, L1⦠⢠T â¶*[d, â] U â â¦â, L2⦠⢠T â¶*[d, â] U).
+
+interpretation
+ "lazy equivalence (local environment)"
+ 'LazyEq T d L1 L2 = (lleq d T L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lleq_refl: âd,T. reflexive ⦠(lleq d T).
+/3 width=1 by conj/ qed.
+
+lemma lleq_sym: âd,T. symmetric ⦠(lleq d T).
+#d #T #L1 #L2 * /3 width=1 by iff_sym, conj/
+qed-.
+
+lemma lleq_sort: âL1,L2,d,k. |L1| = |L2| â L1 â[âk, d] L2.
+#L1 #L2 #d #k #HL12 @conj // -HL12
+#U @conj #H >(cpys_inv_sort1 ⦠H) -H //
+qed.
+
+lemma lleq_gref: âL1,L2,d,p. |L1| = |L2| â L1 â[§p, d] L2.
+#L1 #L2 #d #k #HL12 @conj // -HL12
+#U @conj #H >(cpys_inv_gref1 ⦠H) -H //
+qed.
+
+lemma lleq_bind: âa,I,L1,L2,V,T,d.
+ L1 â[V, d] L2 â L1.â{I}V â[T, ⫯d] L2.â{I}V â
+ L1 â[â{a,I}V.T, d] L2.
+#a #I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12
+#X @conj #H elim (cpys_inv_bind1 ⦠H) -H
+#W #U #HVW #HTU #H destruct
+elim (IHV W) -IHV elim (IHT U) -IHT /3 width=1 by cpys_bind/
+qed.
+
+lemma lleq_flat: âI,L1,L2,V,T,d.
+ L1 â[V, d] L2 â L1 â[T, d] L2 â L1 â[â{I}V.T, d] L2.
+#I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12
+#X @conj #H elim (cpys_inv_flat1 ⦠H) -H
+#W #U #HVW #HTU #H destruct
+elim (IHV W) -IHV elim (IHT U) -IHT
+/3 width=1 by cpys_flat/
+qed.
+
+lemma lleq_be: âL1,L2,U,dt. L1 â[U, dt] L2 â âT,d,e. â§[d, e] T â¡ U â
+ d ⤠dt â dt ⤠d + e â L1 â[U, d] L2.
+#L1 #L2 #U #dt * #HL12 #IH #T #d #e #HTU #Hddt #Hdtde @conj // -HL12
+#U0 elim (IH U0) -IH #H12 #H21 @conj
+#HU0 elim (cpys_fwd_up ⦠HU0 ⦠HTU) // -HU0 /4 width=5 by cpys_weak/
+qed-.
+
+lemma lsuby_lleq_trans: âL2,L,T,d. L2 â[T, d] L â
+ âL1. L1 âÃ[d, â] L2 â |L1| = |L2| â L1 â[T, d] L.
+#L2 #L #T #d * #HL2 #IH #L1 #HL12 #H @conj // -HL2
+#U elim (IH U) -IH #Hdx #Hsn @conj #HTU
+[ @Hdx -Hdx -Hsn @(lsuby_cpys_trans ⦠HTU) -HTU
+ /2 width=1 by lsuby_sym/ (**) (* full auto does not work *)
+| -H -Hdx /3 width=3 by lsuby_cpys_trans/
+]
+qed-.
+
+lemma lleq_lsuby_trans: âL,L1,T,d. L â[T, d] L1 â
+ âL2. L1 âÃ[d, â] L2 â |L1| = |L2| â L â[T, d] L2.
+/5 width=4 by lsuby_lleq_trans, lleq_sym, lsuby_sym/ qed-.
+
+lemma lleq_lsuby_repl: âL1,L2,T,d. L1 â[T, d] L2 â
+ âK1. K1 âÃ[d, â] L1 â |K1| = |L1| â
+ âK2. L2 âÃ[d, â] K2 â |L2| = |K2| â
+ K1 â[T, d] K2.
+/3 width=4 by lleq_lsuby_trans, lsuby_lleq_trans/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lleq_fwd_length: âL1,L2,T,d. L1 â[T, d] L2 â |L1| = |L2|.
+#L1 #L2 #T #d * //
+qed-.
+
+lemma lleq_fwd_ldrop_sn: âL1,L2,T,d. L1 â[d, T] L2 â âK1,i. â©[i] L1 â¡ K1 â
+ âK2. â©[i] L2 â¡ K2.
+#L1 #L2 #T #d #H #K1 #i #HLK1 lapply (lleq_fwd_length ⦠H) -H
+#HL12 lapply (ldrop_fwd_length_le2 ⦠HLK1) -HLK1 /2 width=1 by ldrop_O1_le/
+qed-.
+
+lemma lleq_fwd_ldrop_dx: âL1,L2,T,d. L1 â[d, T] L2 â âK2,i. â©[i] L2 â¡ K2 â
+ âK1. â©[i] L1 â¡ K1.
+/3 width=6 by lleq_fwd_ldrop_sn, lleq_sym/ qed-.
+
+lemma lleq_fwd_bind_sn: âa,I,L1,L2,V,T,d.
+ L1 â[â{a,I}V.T, d] L2 â L1 â[V, d] L2.
+#a #I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
+#U elim (H (â{a,I}U.T)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_bind/ -H
+#H elim (cpys_inv_bind1 ⦠H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+lemma lleq_fwd_bind_dx: âa,I,L1,L2,V,T,d.
+ L1 â[â{a,I}V.T, d] L2 â L1.â{I}V â[T, ⫯d] L2.â{I}V.
+#a #I #L1 #L2 #V #T #d * #HL12 #H @conj [ normalize // ] -HL12
+#U elim (H (â{a,I}V.U)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_bind/ -H
+#H elim (cpys_inv_bind1 ⦠H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+lemma lleq_fwd_flat_sn: âI,L1,L2,V,T,d.
+ L1 â[â{I}V.T, d] L2 â L1 â[V, d] L2.
+#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
+#U elim (H (â{I}U.T)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_flat/ -H
+#H elim (cpys_inv_flat1 ⦠H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+lemma lleq_fwd_flat_dx: âI,L1,L2,V,T,d.
+ L1 â[â{I}V.T, d] L2 â L1 â[T, d] L2.
+#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
+#U elim (H (â{I}V.U)) -H
+#H1 #H2 @conj
+#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
+/2 width=1 by cpys_flat/ -H
+#H elim (cpys_inv_flat1 ⦠H) -H
+#X #Y #H1 #H2 #H destruct //
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+lemma lleq_inv_bind: âa,I,L1,L2,V,T,d. L1 â[â{a,I}V.T, d] L2 â
+ L1 â[V, d] L2 ⧠L1.â{I}V â[T, ⫯d] L2.â{I}V.
+/3 width=4 by lleq_fwd_bind_sn, lleq_fwd_bind_dx, conj/ qed-.
+
+lemma lleq_inv_flat: âI,L1,L2,V,T,d. L1 â[â{I}V.T, d] L2 â
+ L1 â[V, d] L2 ⧠L1 â[T, d] L2.
+/3 width=3 by lleq_fwd_flat_sn, lleq_fwd_flat_dx, conj/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_alt.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_alt.etc
new file mode 100644
index 000000000..a8fb526e5
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_alt.etc
@@ -0,0 +1,85 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/lazyeqalt_4.ma".
+include "basic_2/substitution/lleq_lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Note: alternative definition of lleq *)
+inductive lleqa: relation4 ynat term lenv lenv â
+| lleqa_sort: âL1,L2,d,k. |L1| = |L2| â lleqa d (âk) L1 L2
+| lleqa_skip: âL1,L2,d,i. |L1| = |L2| â yinj i < d â lleqa d (#i) L1 L2
+| lleqa_lref: âI1,I2,L1,L2,K1,K2,V,d,i. d ⤠yinj i â
+ â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â
+ lleqa (yinj 0) V K1 K2 â lleqa d (#i) L1 L2
+| lleqa_free: âL1,L2,d,i. |L1| = |L2| â |L1| ⤠i â |L2| ⤠i â lleqa d (#i) L1 L2
+| lleqa_gref: âL1,L2,d,p. |L1| = |L2| â lleqa d (§p) L1 L2
+| lleqa_bind: âa,I,L1,L2,V,T,d.
+ lleqa d V L1 L2 â lleqa (⫯d) T (L1.â{I}V) (L2.â{I}V) â
+ lleqa d (â{a,I}V.T) L1 L2
+| lleqa_flat: âI,L1,L2,V,T,d.
+ lleqa d V L1 L2 â lleqa d T L1 L2 â lleqa d (â{I}V.T) L1 L2
+.
+
+interpretation
+ "lazy equivalence (local environment) alternative"
+ 'LazyEqAlt T d L1 L2 = (lleqa d T L1 L2).
+
+(* Main inversion lemmas ****************************************************)
+
+theorem lleqa_inv_lleq: âL1,L2,T,d. L1 ââ[T, d] L2 â L1 â[T, d] L2.
+#L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+/2 width=8 by lleq_flat, lleq_bind, lleq_gref, lleq_free, lleq_lref, lleq_skip, lleq_sort/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lleq_lleqa: âL1,T,L2,d. L1 â[T, d] L2 â L1 ââ[T, d] L2.
+#L1 #T @(f2_ind ⦠rfw ⦠L1 T) -L1 -T
+#n #IH #L1 * * /3 width=3 by lleqa_gref, lleqa_sort, lleq_fwd_length/
+[ #i #Hn #L2 #d #H elim (lleq_fwd_lref ⦠H) [ * || * ]
+ /4 width=9 by lleqa_free, lleqa_lref, lleqa_skip, lleq_fwd_length, ldrop_fwd_rfw/
+| #a #I #V #T #Hn #L2 #d #H elim (lleq_inv_bind ⦠H) -H /3 width=1 by lleqa_bind/
+| #I #V #T #Hn #L2 #d #H elim (lleq_inv_flat ⦠H) -H /3 width=1 by lleqa_flat/
+]
+qed.
+
+(* Advanced eliminators *****************************************************)
+
+lemma lleq_ind_alt: âR:relation4 ynat term lenv lenv. (
+ âL1,L2,d,k. |L1| = |L2| â R d (âk) L1 L2
+ ) â (
+ âL1,L2,d,i. |L1| = |L2| â yinj i < d â R d (#i) L1 L2
+ ) â (
+ âI1,I2,L1,L2,K1,K2,V,d,i. d ⤠yinj i â
+ â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â
+ K1 â[V, yinj O] K2 â R (yinj O) V K1 K2 â R d (#i) L1 L2
+ ) â (
+ âL1,L2,d,i. |L1| = |L2| â |L1| ⤠i â |L2| ⤠i â R d (#i) L1 L2
+ ) â (
+ âL1,L2,d,p. |L1| = |L2| â R d (§p) L1 L2
+ ) â (
+ âa,I,L1,L2,V,T,d.
+ L1 â[V, d]L2 â L1.â{I}V â[T, ⫯d] L2.â{I}V â
+ R d V L1 L2 â R (⫯d) T (L1.â{I}V) (L2.â{I}V) â R d (â{a,I}V.T) L1 L2
+ ) â (
+ âI,L1,L2,V,T,d.
+ L1 â[V, d]L2 â L1 â[T, d] L2 â
+ R d V L1 L2 â R d T L1 L2 â R d (â{I}V.T) L1 L2
+ ) â
+ âd,T,L1,L2. L1 â[T, d] L2 â R d T L1 L2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim (lleq_lleqa ⦠H) -H
+/3 width=9 by lleqa_inv_lleq/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_ext.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_ext.etc
new file mode 100644
index 000000000..d6bb03fa7
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_ext.etc
@@ -0,0 +1,91 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/lleq_alt.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Advanced inversion lemmas ************************************************)
+
+fact lleq_inv_S_aux: âL1,L2,T,d0. L1 â[T, d0] L2 â âd. d0 = d + 1 â
+ âK1,K2,I,V. â©[d] L1 â¡ K1.â{I}V â â©[d] L2 â¡ K2.â{I}V â
+ K1 â[V, 0] K2 â L1 â[T, d] L2.
+#L1 #L2 #T #d0 #H @(lleq_ind_alt ⦠H) -L1 -L2 -T -d0
+/2 width=1 by lleq_gref, lleq_free, lleq_sort/
+[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V #HLK1 #HLK2 #HV destruct
+ elim (yle_split_eq i d) /2 width=1 by lleq_skip, ylt_fwd_succ2/ -HL12 -Hid
+ #H destruct /2 width=8 by lleq_lref/
+| #I1 #I2 #L1 #L2 #K11 #K22 #V #d0 #i #Hd0i #HLK11 #HLK22 #HV #_ #d #H #K1 #K2 #J #W #_ #_ #_ destruct
+ /3 width=8 by lleq_lref, yle_pred_sn/
+| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
+ /4 width=7 by lleq_bind, ldrop_drop/
+| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
+ /3 width=7 by lleq_flat/
+]
+qed-.
+
+lemma lleq_inv_S: âT,L1,L2,d. L1 â[T, d+1] L2 â
+ âK1,K2,I,V. â©[d] L1 â¡ K1.â{I}V â â©[d] L2 â¡ K2.â{I}V â
+ K1 â[V, 0] K2 â L1 â[T, d] L2.
+/2 width=7 by lleq_inv_S_aux/ qed-.
+
+lemma lleq_inv_bind_O: âa,I,L1,L2,V,T. L1 â[â{a,I}V.T, 0] L2 â
+ L1 â[V, 0] L2 ⧠L1.â{I}V â[T, 0] L2.â{I}V.
+#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind ⦠H) -H
+/3 width=7 by ldrop_pair, conj, lleq_inv_S/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lleq_fwd_bind_O: âa,I,L1,L2,V,T. L1 â[â{a,I}V.T, 0] L2 â
+ L1.â{I}V â[T, 0] L2.â{I}V.
+#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind_O ⦠H) -H //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lleq_ge: âL1,L2,T,d1. L1 â[T, d1] L2 â âd2. d1 ⤠d2 â L1 â[T, d2] L2.
+#L1 #L2 #T #d1 #H @(lleq_ind_alt ⦠H) -L1 -L2 -T -d1
+/4 width=1 by lleq_sort, lleq_free, lleq_gref, lleq_bind, lleq_flat, yle_succ/
+[ /3 width=3 by lleq_skip, ylt_yle_trans/
+| #I1 #I2 #L1 #L2 #K1 #K2 #V #d1 #i #Hi #HLK1 #HLK2 #HV #IHV #d2 #Hd12 elim (ylt_split i d2)
+ [ lapply (lleq_fwd_length ⦠HV) #HK12 #Hid2
+ lapply (ldrop_fwd_length ⦠HLK1) lapply (ldrop_fwd_length ⦠HLK2)
+ normalize in ⢠(%â%â?); -I1 -I2 -V -d1 /2 width=1 by lleq_skip/
+ | /3 width=8 by lleq_lref, yle_trans/
+ ]
+]
+qed-.
+
+lemma lleq_bind_O: âa,I,L1,L2,V,T. L1 â[V, 0] L2 â L1.â{I}V â[T, 0] L2.â{I}V â
+ L1 â[â{a,I}V.T, 0] L2.
+/3 width=3 by lleq_ge, lleq_bind/ qed.
+
+lemma lleq_bind_repl_SO: âI1,I2,L1,L2,V1,V2,T. L1.â{I1}V1 â[T, 0] L2.â{I2}V2 â
+ âJ1,J2,W1,W2. L1.â{J1}W1 â[T, 1] L2.â{J2}W2.
+#I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (lleq_ge ⦠HT 1 ?) // -HT
+#HT @(lleq_lsuby_repl ⦠HT) /2 width=1 by lsuby_succ/ (**) (* full auto fails *)
+qed-.
+
+lemma lleq_bind_repl_O: âI,L1,L2,V,T. L1.â{I}V â[T, 0] L2.â{I}V â
+ âJ,W. L1 â[W, 0] L2 â L1.â{J}W â[T, 0] L2.â{J}W.
+/3 width=7 by lleq_bind_repl_SO, lleq_inv_S/ qed-.
+
+(* Inversion lemmas on negated lazy quivalence for local environments *******)
+
+lemma nlleq_inv_bind_O: âa,I,L1,L2,V,T. (L1 â[â{a,I}V.T, 0] L2 â â¥) â
+ (L1 â[V, 0] L2 â â¥) ⨠(L1.â{I}V â[T, 0] L2.â{I}V â â¥).
+#a #I #L1 #L2 #V #T #H elim (lleq_dec V L1 L2 0)
+/4 width=1 by lleq_bind_O, or_intror, or_introl/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_fqus.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_fqus.etc
new file mode 100644
index 000000000..18744a539
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_fqus.etc
@@ -0,0 +1,75 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/fqus_alt.ma".
+include "basic_2/substitution/lleq_ext.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on supclosure and derivatives *********************************)
+
+lemma lleq_fqu_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â â¦G2, K2, U⦠â
+ âL1. L1 â[T, 0] L2 â
+ ââK1. â¦G1, L1, T⦠â â¦G2, K1, U⦠& K1 â[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
+[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx ⦠H ⦠I L2 V) -H //
+ #I1 #K1 #H1 #H2 lapply (ldrop_inv_O2 ⦠H1) -H1
+ #H destruct /2 width=3 by fqu_lref_O, ex2_intro/
+| * [ #a ] #I #G #L2 #V #T #L1 #H
+ [ elim (lleq_inv_bind ⦠H)
+ | elim (lleq_inv_flat ⦠H)
+ ] -H
+ /2 width=3 by fqu_pair_sn, ex2_intro/
+| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O ⦠H) -H
+ #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/
+| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat ⦠H) -H
+ /2 width=3 by fqu_flat_dx, ex2_intro/
+| #G #L2 #K2 #T #U #e #HLK2 #HTU #L1 #HL12
+ elim (ldrop_O1_le (e+1) L1)
+ [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
+ | lapply (ldrop_fwd_length_le2 ⦠HLK2) -K2
+ lapply (lleq_fwd_length ⦠HL12) -T -U //
+ ]
+]
+qed-.
+
+lemma lleq_fquq_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â⸮ â¦G2, K2, U⦠â
+ âL1. L1 â[T, 0] L2 â
+ ââK1. â¦G1, L1, T⦠â⸮ â¦G2, K1, U⦠& K1 â[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen ⦠H) -H
+[ #H elim (lleq_fqu_trans ⦠H ⦠HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma lleq_fqup_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â+ â¦G2, K2, U⦠â
+ âL1. L1 â[T, 0] L2 â
+ ââK1. â¦G1, L1, T⦠â+ â¦G2, K1, U⦠& K1 â[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind ⦠H) -G2 -K2 -U
+[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans ⦠HTU ⦠HL12) -L2
+ /3 width=3 by fqu_fqup, ex2_intro/
+| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU ⦠HL12) -L2
+ #K1 #HTU #HK1 elim (lleq_fqu_trans ⦠HU2 ⦠HK1) -K
+ /3 width=5 by fqup_strap1, ex2_intro/
+]
+qed-.
+
+lemma lleq_fqus_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â* â¦G2, K2, U⦠â
+ âL1. L1 â[T, 0] L2 â
+ ââK1. â¦G1, L1, T⦠â* â¦G2, K1, U⦠& K1 â[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen ⦠H) -H
+[ #H elim (lleq_fqup_trans ⦠H ⦠HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_ldrop.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_ldrop.etc
new file mode 100644
index 000000000..9cb597c6f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_ldrop.etc
@@ -0,0 +1,123 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpys_lift.ma".
+include "basic_2/substitution/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lleq_skip: âL1,L2,d,i. yinj i < d â |L1| = |L2| â L1 â[#i, d] L2.
+#L1 #L2 #d #i #Hid #HL12 @conj // -HL12
+#U @conj #H elim (cpys_inv_lref1 ⦠H) -H // *
+#I #Z #Y #X #H elim (ylt_yle_false ⦠Hid ⦠H)
+qed.
+
+lemma lleq_lref: âI1,I2,L1,L2,K1,K2,V,d,i. d ⤠yinj i â
+ â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â
+ K1 â[V, 0] K2 â L1 â[#i, d] L2.
+#I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 * #HK12 #IH @conj [ -IH | -HK12 ]
+[ lapply (ldrop_fwd_length ⦠HLK1) -HLK1 #H1
+ lapply (ldrop_fwd_length ⦠HLK2) -HLK2 #H2
+ >H1 >H2 -H1 -H2 normalize //
+| #U @conj #H elim (cpys_inv_lref1 ⦠H) -H // *
+ >yminus_Y_inj #I #K #X #W #_ #_ #H #HVW #HWU
+ [ letin HLK â HLK1 | letin HLK â HLK2 ]
+ lapply (ldrop_mono ⦠H ⦠HLK) -H #H destruct elim (IH W)
+ /3 width=7 by cpys_subst_Y2/
+]
+qed.
+
+lemma lleq_free: âL1,L2,d,i. |L1| ⤠i â |L2| ⤠i â |L1| = |L2| â L1 â[#i, d] L2.
+#L1 #L2 #d #i #HL1 #HL2 #HL12 @conj // -HL12
+#U @conj #H elim (cpys_inv_lref1 ⦠H) -H // *
+#I #Z #Y #X #_ #_ #H lapply (ldrop_fwd_length_lt2 ⦠H) -H
+#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
+qed.
+
+(* Properties on relocation *************************************************)
+
+lemma lleq_lift_le: âK1,K2,T,dt. K1 â[T, dt] K2 â
+ âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âU. â§[d, e] T â¡ U â dt ⤠d â L1 â[U, dt] L2.
+#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hdtd
+lapply (ldrop_fwd_length ⦠HLK1) lapply (ldrop_fwd_length ⦠HLK2)
+#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0
+[ letin HLKA â HLK1 letin HLKB â HLK2 | letin HLKA â HLK2 letin HLKB â HLK1 ]
+elim (cpys_inv_lift1_be ⦠HU0 ⦠HLKA ⦠HTU) // -HU0 >yminus_Y_inj #T0 #HT0 #HTU0
+elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=12 by cpys_lift_be/
+qed-.
+
+lemma lleq_lift_ge: âK1,K2,T,dt. K1 â[T, dt] K2 â
+ âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âU. â§[d, e] T â¡ U â d ⤠dt â L1 â[U, dt+e] L2.
+#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hddt
+lapply (ldrop_fwd_length ⦠HLK1) lapply (ldrop_fwd_length ⦠HLK2)
+#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0
+[ letin HLKA â HLK1 letin HLKB â HLK2 | letin HLKA â HLK2 letin HLKB â HLK1 ]
+elim (cpys_inv_lift1_ge ⦠HU0 ⦠HLKA ⦠HTU) /2 width=1 by monotonic_yle_plus_dx/ -HU0 >yplus_minus_inj #T0 #HT0 #HTU0
+elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=10 by cpys_lift_ge/
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma lleq_inv_lift_le: âL1,L2,U,dt. L1 â[U, dt] L2 â
+ âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âT. â§[d, e] T â¡ U â dt ⤠d â K1 â[T, dt] K2.
+#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdtd
+lapply (ldrop_fwd_length_minus2 ⦠HLK1) lapply (ldrop_fwd_length_minus2 ⦠HLK2)
+#H2 #H1 @conj // -HL12 -H1 -H2
+#T0 elim (lift_total T0 d e)
+#U0 #HTU0 elim (IH U0) -IH
+#H12 #H21 @conj #HT0
+[ letin HLKA â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ]
+lapply (cpys_lift_be ⦠HT0 ⦠HLKA ⦠HTU ⦠HTU0) // -HT0
+>yplus_Y1 #HU0 elim (cpys_inv_lift1_be ⦠(H0 HU0) ⦠HLKB ⦠HTU) // -L1 -L2 -U -Hdtd
+#X #HT0 #HX lapply (lift_inj ⦠HX ⦠HTU0) -U0 //
+qed-.
+
+lemma lleq_inv_lift_ge: âL1,L2,U,dt. L1 â[U, dt] L2 â
+ âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âT. â§[d, e] T â¡ U â yinj d + e ⤠dt â K1 â[T, dt-e] K2.
+#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdedt
+lapply (ldrop_fwd_length_minus2 ⦠HLK1) lapply (ldrop_fwd_length_minus2 ⦠HLK2)
+#H2 #H1 @conj // -HL12 -H1 -H2
+elim (yle_inv_plus_inj2 ⦠Hdedt) #Hddt #Hedt
+#T0 elim (lift_total T0 d e)
+#U0 #HTU0 elim (IH U0) -IH
+#H12 #H21 @conj #HT0
+[ letin HLKA â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ]
+lapply (cpys_lift_ge ⦠HT0 ⦠HLKA ⦠HTU ⦠HTU0) // -HT0 -Hddt
+>ymax_pre_sn // #HU0 elim (cpys_inv_lift1_ge ⦠(H0 HU0) ⦠HLKB ⦠HTU) // -L1 -L2 -U -Hdedt -Hedt
+#X #HT0 #HX lapply (lift_inj ⦠HX ⦠HTU0) -U0 //
+qed-.
+
+lemma lleq_inv_lift_be: âL1,L2,U,dt. L1 â[U, dt] L2 â
+ âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âT. â§[d, e] T â¡ U â d ⤠dt â dt ⤠yinj d + e â K1 â[T, d] K2.
+#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hddt #Hdtde
+lapply (ldrop_fwd_length_minus2 ⦠HLK1) lapply (ldrop_fwd_length_minus2 ⦠HLK2)
+#H2 #H1 @conj // -HL12 -H1 -H2
+#T0 elim (lift_total T0 d e)
+#U0 #HTU0 elim (IH U0) -IH
+#H12 #H21 @conj #HT0
+[ letin HLKA â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ]
+lapply (cpys_lift_ge ⦠HT0 ⦠HLKA ⦠HTU ⦠HTU0) // -HT0
+#HU0 lapply (cpys_weak ⦠HU0 dt (â) ? ?) // -HU0
+#HU0 lapply (H0 HU0)
+#HU0 lapply (cpys_weak ⦠HU0 d (â) ? ?) // -HU0
+#HU0 elim (cpys_inv_lift1_ge_up ⦠HU0 ⦠HLKB ⦠HTU) // -L1 -L2 -U -Hddt -Hdtde
+#X #HT0 #HX lapply (lift_inj ⦠HX ⦠HTU0) -U0 //
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_lleq.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_lleq.etc
new file mode 100644
index 000000000..2a8873d9b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lleq_lleq.etc
@@ -0,0 +1,175 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/cpys_cpys.ma".
+include "basic_2/substitution/lleq_ldrop.ma".
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lleq_fwd_lref: âL1,L2. âd:ynat. âi:nat. L1 â[#i, d] L2 â
+ â¨â¨ |L1| ⤠i ⧠|L2| ⤠i
+ | yinj i < d
+ | ââI1,I2,K1,K2,V. â©[i] L1 â¡ K1.â{I1}V &
+ â©[i] L2 â¡ K2.â{I2}V &
+ K1 â[V, yinj 0] K2 & d ⤠yinj i.
+#L1 #L2 #d #i * #HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=3 by or3_intro0, conj/
+elim (ylt_split i d) /2 width=1 by or3_intro1/ #Hdi #Hi
+elim (ldrop_O1_lt ⦠Hi) #I1 #K1 #V1 #HLK1
+elim (ldrop_O1_lt L2 i) // -Hi #I2 #K2 #V2 #HLK2
+lapply (ldrop_fwd_length_minus2 ⦠HLK2) #H
+lapply (ldrop_fwd_length_minus2 ⦠HLK1) >HL12 <H -HL12 -H
+#H lapply (injective_plus_l ⦠H) -H #HK12
+elim (lift_total V1 0 (i+1)) #W1 #HVW1
+elim (lift_total V2 0 (i+1)) #W2 #HVW2
+elim (IH W1) elim (IH W2) #_ #H2 #H1 #_
+elim (cpys_inv_lref1_ldrop ⦠(H1 ?) ⦠HLK2 ⦠HVW1) -H1
+[ elim (cpys_inv_lref1_ldrop ⦠(H2 ?) ⦠HLK1 ⦠HVW2) -H2 ]
+/3 width=7 by cpys_subst, yle_inj/ -W1 -W2 #H12 #_ #_ #H21 #_ #_
+lapply (cpys_antisym_eq ⦠H12 ⦠H21) -H12 -H21 #H destruct
+@or3_intro2 @(ex4_5_intro ⦠HLK1 HLK2) // @conj // -HK12
+#V elim (lift_total V 0 (i+1))
+#W #HVW elim (IH W) -IH #H12 #H21 @conj #H
+[ elim (cpys_inv_lref1_ldrop ⦠(H12 ?) ⦠HLK2 ⦠HVW) -H12 -H21
+| elim (cpys_inv_lref1_ldrop ⦠(H21 ?) ⦠HLK1 ⦠HVW) -H21 -H12
+] [1,3: >yminus_Y_inj ] /3 width=7 by cpys_subst_Y2, yle_inj/
+qed-.
+
+lemma lleq_fwd_lref_dx: âL1,L2,d,i. L1 â[#i, d] L2 â
+ âI2,K2,V. â©[i] L2 â¡ K2.â{I2}V â
+ i < d â¨
+ ââI1,K1. â©[i] L1 â¡ K1.â{I1}V & K1 â[V, 0] K2 & d ⤠i.
+#L1 #L2 #d #i #H #I2 #K2 #V #HLK2 elim (lleq_fwd_lref ⦠H) -H [ * || * ]
+[ #_ #H elim (lt_refl_false i)
+ lapply (ldrop_fwd_length_lt2 ⦠HLK2) -HLK2
+ /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+| /2 width=1 by or_introl/
+| #I1 #I2 #K11 #K22 #V0 #HLK11 #HLK22 #HV0 #Hdi lapply (ldrop_mono ⦠HLK22 ⦠HLK2) -L2
+ #H destruct /3 width=5 by ex3_2_intro, or_intror/
+]
+qed-.
+
+lemma lleq_fwd_lref_sn: âL1,L2,d,i. L1 â[#i, d] L2 â
+ âI1,K1,V. â©[i] L1 â¡ K1.â{I1}V â
+ i < d â¨
+ ââI2,K2. â©[i] L2 â¡ K2.â{I2}V & K1 â[V, 0] K2 & d ⤠i.
+#L1 #L2 #d #i #HL12 #I1 #K1 #V #HLK1 elim (lleq_fwd_lref_dx L2 ⦠d ⦠HLK1) -HLK1
+[2: * ] /4 width=6 by lleq_sym, ex3_2_intro, or_introl, or_intror/
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lleq_inv_lref_ge_dx: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
+ âI2,K2,V. â©[i] L2 â¡ K2.â{I2}V â
+ ââI1,K1. â©[i] L1 â¡ K1.â{I1}V & K1 â[V, 0] K2.
+#L1 #L2 #d #i #H #Hdi #I2 #K2 #V #HLK2 elim (lleq_fwd_lref_dx ⦠H ⦠HLK2) -L2
+[ #H elim (ylt_yle_false ⦠H Hdi)
+| * /2 width=4 by ex2_2_intro/
+]
+qed-.
+
+lemma lleq_inv_lref_ge_sn: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
+ âI1,K1,V. â©[i] L1 â¡ K1.â{I1}V â
+ ââI2,K2. â©[i] L2 â¡ K2.â{I2}V & K1 â[V, 0] K2.
+#L1 #L2 #d #i #HL12 #Hdi #I1 #K1 #V #HLK1 elim (lleq_inv_lref_ge_dx L2 ⦠Hdi ⦠HLK1) -Hdi -HLK1
+/3 width=4 by lleq_sym, ex2_2_intro/
+qed-.
+
+lemma lleq_inv_lref_ge_gen: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
+ âI1,I2,K1,K2,V1,V2.
+ â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â
+ V1 = V2 ⧠K1 â[V2, 0] K2.
+#L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
+elim (lleq_inv_lref_ge_sn ⦠HL12 ⦠HLK1) // -L1 -d
+#J #Y #HY lapply (ldrop_mono ⦠HY ⦠HLK2) -L2 -i #H destruct /2 width=1 by conj/
+qed-.
+
+lemma lleq_inv_lref_ge: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
+ âI,K1,K2,V. â©[i] L1 â¡ K1.â{I}V â â©[i] L2 â¡ K2.â{I}V â
+ K1 â[V, 0] K2.
+#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2
+elim (lleq_inv_lref_ge_gen ⦠HL12 ⦠HLK1 HLK2) //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma lleq_dec: âT,L1,L2,d. Decidable (L1 â[T, d] L2).
+#T #L1 @(f2_ind ⦠rfw ⦠L1 T) -L1 -T
+#n #IH #L1 * *
+[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_sort/
+| #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|))
+ [ #HL12 #d elim (ylt_split i d) /3 width=1 by lleq_skip, or_introl/
+ #Hdi elim (lt_or_ge i (|L1|)) #HiL1
+ elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, lleq_free/
+ elim (ldrop_O1_lt ⦠HiL2) #I2 #K2 #V2 #HLK2
+ elim (ldrop_O1_lt ⦠HiL1) #I1 #K1 #V1 #HLK1
+ elim (eq_term_dec V2 V1)
+ [ #H3 elim (IH K1 V1 ⦠K2 0) destruct
+ /3 width=8 by lleq_lref, ldrop_fwd_rfw, or_introl/
+ ]
+ -IH #H3 @or_intror
+ #H elim (lleq_fwd_lref ⦠H) -H [1,3,4,6: * ]
+ [1,3: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
+ |5,6: /2 width=4 by ylt_yle_false/
+ ]
+ #Z1 #Z2 #Y1 #Y2 #X #HLY1 #HLY2 #HX #_
+ lapply (ldrop_mono ⦠HLY1 ⦠HLK1) -HLY1 -HLK1
+ lapply (ldrop_mono ⦠HLY2 ⦠HLK2) -HLY2 -HLK2
+ #H2 #H1 destruct /2 width=1 by/
+ ]
+| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_gref/
+| #a #I #V #T #Hn #L2 #d destruct
+ elim (IH L1 V ⦠L2 d) /2 width=1 by/
+ elim (IH (L1.â{I}V) T ⦠(L2.â{I}V) (d+1)) -IH /3 width=1 by or_introl, lleq_bind/
+ #H1 #H2 @or_intror
+ #H elim (lleq_inv_bind ⦠H) -H /2 width=1 by/
+| #I #V #T #Hn #L2 #d destruct
+ elim (IH L1 V ⦠L2 d) /2 width=1 by/
+ elim (IH L1 T ⦠L2 d) -IH /3 width=1 by or_introl, lleq_flat/
+ #H1 #H2 @or_intror
+ #H elim (lleq_inv_flat ⦠H) -H /2 width=1 by/
+]
+-n /4 width=3 by lleq_fwd_length, or_intror/
+qed-.
+
+(* Main properties **********************************************************)
+
+theorem lleq_trans: âd,T. Transitive ⦠(lleq d T).
+#d #T #L1 #L * #HL1 #IH1 #L2 * #HL2 #IH2 /3 width=3 by conj, iff_trans/
+qed-.
+
+theorem lleq_canc_sn: âL,L1,L2,T,d. L â[d, T] L1â L â[d, T] L2 â L1 â[d, T] L2.
+/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+theorem lleq_canc_dx: âL1,L2,L,T,d. L1 â[d, T] L â L2 â[d, T] L â L1 â[d, T] L2.
+/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+(* Inversion lemmas on negated lazy quivalence for local environments *******)
+
+lemma nlleq_inv_bind: âa,I,L1,L2,V,T,d. (L1 â[â{a,I}V.T, d] L2 â â¥) â
+ (L1 â[V, d] L2 â â¥) ⨠(L1.â{I}V â[T, ⫯d] L2.â{I}V â â¥).
+#a #I #L1 #L2 #V #T #d #H elim (lleq_dec V L1 L2 d)
+/4 width=1 by lleq_bind, or_intror, or_introl/
+qed-.
+
+lemma nlleq_inv_flat: âI,L1,L2,V,T,d. (L1 â[â{I}V.T, d] L2 â â¥) â
+ (L1 â[V, d] L2 â â¥) ⨠(L1 â[T, d] L2 â â¥).
+#I #L1 #L2 #V #T #d #H elim (lleq_dec V L1 L2 d)
+/4 width=1 by lleq_flat, or_intror, or_introl/
+qed-.
+
+(* Note: lleq_nlleq_trans: âd,T,L1,L. L1â[T, d] L â
+ âL2. (L â[T, d] L2 â â¥) â (L1 â[T, d] L2 â â¥).
+/3 width=3 by lleq_canc_sn/ qed-.
+works with /4 width=8/ so lleq_canc_sn is more convenient
+*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lpx_cpys.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lpx_cpys.etc
new file mode 100644
index 000000000..d3c490a89
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lpx_cpys.etc
@@ -0,0 +1,69 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/cpy_lift.ma".
+include "basic_2/substitution/cpys.ma".
+include "basic_2/reduction/lpx_ldrop.ma".
+
+(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
+
+(* Properties on context-sensitive extended substitution for terms **********)
+
+lemma cpx_cpy_trans_lpx: âh,g,G,L1,T1,T. â¦G, L1⦠⢠T1 â¡[h, g] T â
+ âL2. â¦G, L1⦠⢠â¡[h, g] L2 â
+ âT2,d,e. â¦G, L2⦠⢠T â¶[d, e] T2 â â¦G, L1⦠⢠T1 â¡[h, g] T2.
+#h #g #G #L1 #T1 #T #H elim H -G -L1 -T1 -T
+[ #J #G #L1 #L2 #HL12 #T2 #d #e #H elim (cpy_inv_atom1 ⦠H) -H //
+ * #I #K2 #V2 #i #_ #_ #HLK2 #HVT2 #H destruct
+ elim (lpx_ldrop_trans_O1 ⦠HL12 ⦠HLK2) -L2 #X #HLK1 #H
+ elim (lpx_inv_pair2 ⦠H) -H #K1 #V1 #_ #HV12 #H destruct
+ /2 width=7 by cpx_delta/
+| #G #L1 #k #l #Hkl #L2 #_ #X #d #e #H >(cpy_inv_sort1 ⦠H) -X /2 width=2 by cpx_sort/
+| #I #G #L1 #K1 #V1 #V #T #i #HLK1 #_ #HVT #IHV1 #L2 #HL12 #T2 #d #e #HT2
+ elim (lpx_ldrop_conf ⦠HLK1 ⦠HL12) -HL12 #X #H #HLK2
+ elim (lpx_inv_pair1 ⦠H) -H #K2 #V0 #HK12 #_ #H destruct
+ lapply (ldrop_fwd_drop2 ⦠HLK2) -V0 #HLK2
+ lapply (cpy_weak ⦠HT2 0 (â) ? ?) -HT2 // #HT2
+ elim (cpy_inv_lift1_be ⦠HT2 ⦠HLK2 ⦠HVT) -HT2 -HLK2 -HVT
+ /3 width=7 by cpx_delta/
+| #a #I #G #L1 #V1 #V #T1 #T #HV1 #_ #IHV1 #IHT1 #L2 #HL12 #X #d #e #H elim (cpy_inv_bind1 ⦠H) -H
+ #V2 #T2 #HV2 #HT2 #H destruct /4 width=5 by lpx_pair, cpx_bind/
+| #I #G #L1 #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #L2 #HL12 #X #d #e #H elim (cpy_inv_flat1 ⦠H) -H
+ #V2 #T2 #HV2 #HT2 #H destruct /3 width=5 by cpx_flat/
+| #G #L1 #V1 #U1 #U #T #_ #HTU #IHU1 #L2 #HL12 #T2 #d #e #HT2
+ lapply (cpy_weak ⦠HT2 0 (â) ? ?) -HT2 // #HT2
+ elim (lift_total T2 0 1) #U2 #HTU2
+ lapply (cpy_lift_be ⦠HT2 (L2.âV1) ⦠(â») ⦠HTU ⦠HTU2 ? ?) -T
+ /4 width=5 by cpx_zeta, lpx_pair, ldrop_drop/
+| /3 width=5 by cpx_tau/
+| /3 width=5 by cpx_ti/
+| #a #G #L1 #V1 #V #W1 #W #T1 #T #HV1 #HW1 #_ #IHV1 #IHW1 #IHT1 #L2 #HL12 #X #d #e #HX
+ elim (cpy_inv_bind1 ⦠HX) -HX #Y #T2 #HY #HT2 #H destruct
+ elim (cpy_inv_flat1 ⦠HY) -HY #W2 #V2 #HW2 #HV2 #H destruct
+ /5 width=11 by lpx_pair, cpx_beta, lsuby_cpy_trans, lsuby_succ/
+| #a #G #L1 #V1 #V #U #W1 #W #T1 #T #_ #HVU #HW1 #_ #IHV1 #IHW1 #IHT1 #L2 #HL12 #X #d #e #HX
+ elim (cpy_inv_bind1 ⦠HX) -HX #W2 #Y #HW2 #HY #H destruct
+ elim (cpy_inv_flat1 ⦠HY) -HY #U2 #T2 #HU2 #HT2 #H destruct
+ lapply (cpy_weak ⦠HU2 0 (â) ? ?) -HU2 // #HU2
+ elim (cpy_inv_lift1_be ⦠HU2 L2 ⦠HVU) -U
+ /4 width=7 by lpx_pair, cpx_theta, ldrop_drop/
+]
+qed-.
+
+lemma cpx_cpys_trans_lpx: âh,g,G,L1,T1,T. â¦G, L1⦠⢠T1 â¡[h, g] T â
+ âL2. â¦G, L1⦠⢠â¡[h, g] L2 â
+ âT2,d,e. â¦G, L2⦠⢠T â¶*[d, e] T2 â â¦G, L1⦠⢠T1 â¡[h, g] T2.
+#h #g #G #L1 #T1 #T #HT1 #L2 #HL12 #T2 #d #e #H @(cpys_ind ⦠H) -T2
+/2 width=7 by cpx_cpy_trans_lpx/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby.etc
new file mode 100644
index 000000000..6c90b202b
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby.etc
@@ -0,0 +1,237 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/ynat/ynat_plus.ma".
+include "basic_2/notation/relations/extlrsubeq_4.ma".
+include "basic_2/relocation/ldrop.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+inductive lsuby: relation4 ynat ynat lenv lenv â
+| lsuby_atom: âL,d,e. lsuby d e L (â)
+| lsuby_zero: âI1,I2,L1,L2,V1,V2.
+ lsuby 0 0 L1 L2 â lsuby 0 0 (L1.â{I1}V1) (L2.â{I2}V2)
+| lsuby_pair: âI1,I2,L1,L2,V,e. lsuby 0 e L1 L2 â
+ lsuby 0 (⫯e) (L1.â{I1}V) (L2.â{I2}V)
+| lsuby_succ: âI1,I2,L1,L2,V1,V2,d,e.
+ lsuby d e L1 L2 â lsuby (⫯d) e (L1. â{I1}V1) (L2. â{I2} V2)
+.
+
+interpretation
+ "local environment refinement (extended substitution)"
+ 'ExtLRSubEq L1 d e L2 = (lsuby d e L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lsuby_pair_lt: âI1,I2,L1,L2,V,e. L1 âÃ[0, â«°e] L2 â 0 < e â
+ L1.â{I1}V âÃ[0, e] L2.â{I2}V.
+#I1 #I2 #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 ⦠He) /2 width=1 by lsuby_pair/
+qed.
+
+lemma lsuby_succ_lt: âI1,I2,L1,L2,V1,V2,d,e. L1 âÃ[â«°d, e] L2 â 0 < d â
+ L1.â{I1}V1 âÃ[d, e] L2. â{I2}V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd <(ylt_inv_O1 ⦠Hd) /2 width=1 by lsuby_succ/
+qed.
+
+lemma lsuby_pair_O_Y: âL1,L2. L1 âÃ[0, â] L2 â
+ âI1,I2,V. L1.â{I1}V âÃ[0,â] L2.â{I2}V.
+#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 ⦠V ⦠HL12) -HL12 //
+qed.
+
+lemma lsuby_refl: âL,d,e. L âÃ[d, e] L.
+#L elim L -L //
+#L #I #V #IHL #d elim (ynat_cases ⦠d) [| * #x ]
+#Hd destruct /2 width=1 by lsuby_succ/
+#e elim (ynat_cases ⦠e) [| * #x ]
+#He destruct /2 width=1 by lsuby_zero, lsuby_pair/
+qed.
+
+lemma lsuby_O2: âL2,L1,d. |L2| ⤠|L1| â L1 âÃ[d, yinj 0] L2.
+#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
+[ #d #H lapply (le_n_O_to_eq ⦠H) -H <plus_n_Sm #H destruct
+| #L1 #I1 #V1 #d #H lapply (le_plus_to_le_r ⦠H) -H #HL12
+ elim (ynat_cases d) /3 width=1 by lsuby_zero/
+ * /3 width=1 by lsuby_succ/
+]
+qed.
+
+lemma lsuby_sym: âd,e,L1,L2. L1 âÃ[d, e] L2 â |L1| = |L2| â L2 âÃ[d, e] L1.
+#d #e #L1 #L2 #H elim H -d -e -L1 -L2
+[ #L1 #d #e #H >(length_inv_zero_dx ⦠H) -L1 //
+| /2 width=1 by lsuby_O2/
+| #I1 #I2 #L1 #L2 #V #e #_ #IHL12 #H lapply (injective_plus_l ⦠H)
+ /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #H lapply (injective_plus_l ⦠H)
+ /3 width=1 by lsuby_succ/
+]
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lsuby_inv_atom1_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â L1 = â â L2 = â.
+#L1 #L2 #d #e * -L1 -L2 -d -e //
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
+| #I1 #I2 #L1 #L2 #V #e #_ #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #H destruct
+]
+qed-.
+
+lemma lsuby_inv_atom1: âL2,d,e. â âÃ[d, e] L2 â L2 = â.
+/2 width=5 by lsuby_inv_atom1_aux/ qed-.
+
+fact lsuby_inv_zero1_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
+ âJ1,K1,W1. L1 = K1.â{J1}W1 â d = 0 â e = 0 â
+ L2 = â â¨
+ ââJ2,K2,W2. K1 âÃ[0, 0] K2 & L2 = K2.â{J2}W2.
+#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
+ /3 width=5 by ex2_3_intro, or_intror/
+| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_ #H
+ elim (ysucc_inv_O_dx ⦠H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_ #H
+ elim (ysucc_inv_O_dx ⦠H)
+]
+qed-.
+
+lemma lsuby_inv_zero1: âI1,K1,L2,V1. K1.â{I1}V1 âÃ[0, 0] L2 â
+ L2 = â â¨
+ ââI2,K2,V2. K1 âÃ[0, 0] K2 & L2 = K2.â{I2}V2.
+/2 width=9 by lsuby_inv_zero1_aux/ qed-.
+
+fact lsuby_inv_pair1_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
+ âJ1,K1,W. L1 = K1.â{J1}W â d = 0 â 0 < e â
+ L2 = â â¨
+ ââJ2,K2. K1 âÃ[0, â«°e] K2 & L2 = K2.â{J2}W.
+#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I1 #I2 #L1 #L2 #V #e #HL12 #J1 #K1 #W #H #_ #_ destruct
+ /3 width=4 by ex2_2_intro, or_intror/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_ #H
+ elim (ysucc_inv_O_dx ⦠H)
+]
+qed-.
+
+lemma lsuby_inv_pair1: âI1,K1,L2,V,e. K1.â{I1}V âÃ[0, e] L2 â 0 < e â
+ L2 = â â¨
+ ââI2,K2. K1 âÃ[0, â«°e] K2 & L2 = K2.â{I2}V.
+/2 width=6 by lsuby_inv_pair1_aux/ qed-.
+
+fact lsuby_inv_succ1_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
+ âJ1,K1,W1. L1 = K1.â{J1}W1 â 0 < d â
+ L2 = â â¨
+ ââJ2,K2,W2. K1 âÃ[â«°d, e] K2 & L2 = K2.â{J2}W2.
+#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
+ /3 width=5 by ex2_3_intro, or_intror/
+]
+qed-.
+
+lemma lsuby_inv_succ1: âI1,K1,L2,V1,d,e. K1.â{I1}V1 âÃ[d, e] L2 â 0 < d â
+ L2 = â â¨
+ ââI2,K2,V2. K1 âÃ[â«°d, e] K2 & L2 = K2.â{I2}V2.
+/2 width=5 by lsuby_inv_succ1_aux/ qed-.
+
+fact lsuby_inv_zero2_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
+ âJ2,K2,W2. L2 = K2.â{J2}W2 â d = 0 â e = 0 â
+ ââJ1,K1,W1. K1 âÃ[0, 0] K2 & L1 = K1.â{J1}W1.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #L1 #d #e #J2 #K2 #W1 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
+ /2 width=5 by ex2_3_intro/
+| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_ #H
+ elim (ysucc_inv_O_dx ⦠H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_ #H
+ elim (ysucc_inv_O_dx ⦠H)
+]
+qed-.
+
+lemma lsuby_inv_zero2: âI2,K2,L1,V2. L1 âÃ[0, 0] K2.â{I2}V2 â
+ ââI1,K1,V1. K1 âÃ[0, 0] K2 & L1 = K1.â{I1}V1.
+/2 width=9 by lsuby_inv_zero2_aux/ qed-.
+
+fact lsuby_inv_pair2_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
+ âJ2,K2,W. L2 = K2.â{J2}W â d = 0 â 0 < e â
+ ââJ1,K1. K1 âÃ[0, â«°e] K2 & L1 = K1.â{J1}W.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #L1 #d #e #J2 #K2 #W #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I1 #I2 #L1 #L2 #V #e #HL12 #J2 #K2 #W #H #_ #_ destruct
+ /2 width=4 by ex2_2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_ #H
+ elim (ysucc_inv_O_dx ⦠H)
+]
+qed-.
+
+lemma lsuby_inv_pair2: âI2,K2,L1,V,e. L1 âÃ[0, e] K2.â{I2}V â 0 < e â
+ ââI1,K1. K1 âÃ[0, â«°e] K2 & L1 = K1.â{I1}V.
+/2 width=6 by lsuby_inv_pair2_aux/ qed-.
+
+fact lsuby_inv_succ2_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
+ âJ2,K2,W2. L2 = K2.â{J2}W2 â 0 < d â
+ ââJ1,K1,W1. K1 âÃ[â«°d, e] K2 & L1 = K1.â{J1}W1.
+#L1 #L2 #d #e * -L1 -L2 -d -e
+[ #L1 #d #e #J2 #K2 #W2 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K1 #W2 #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J2 #K2 #W2 #H #_ destruct
+ /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+lemma lsuby_inv_succ2: âI2,K2,L1,V2,d,e. L1 âÃ[d, e] K2.â{I2}V2 â 0 < d â
+ ââI1,K1,V1. K1 âÃ[â«°d, e] K2 & L1 = K1.â{I1}V1.
+/2 width=5 by lsuby_inv_succ2_aux/ qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lsuby_fwd_length: âL1,L2,d,e. L1 âÃ[d, e] L2 â |L2| ⤠|L1|.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize /2 width=1 by le_S_S/
+qed-.
+
+(* Properties on basic slicing **********************************************)
+
+lemma lsuby_ldrop_trans_be: âL1,L2,d,e. L1 âÃ[d, e] L2 â
+ âI2,K2,W,s,i. â©[s, 0, i] L2 â¡ K2.â{I2}W â
+ d ⤠i â i < d + e â
+ ââI1,K1. K1 âÃ[0, â«°(d+e-i)] K2 & â©[s, 0, i] L1 â¡ K1.â{I1}W.
+#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+[ #L1 #d #e #J2 #K2 #W #s #i #H
+ elim (ldrop_inv_atom1 ⦠H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
+ elim (ldrop_inv_O1_pair1 ⦠H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+ [ #_ destruct -I2 >ypred_succ
+ /2 width=4 by ldrop_pair, ex2_2_intro/
+ | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
+ #H <H -H #H lapply (ylt_inv_succ ⦠H) -H
+ #Hie elim (IHL12 ⦠HLK1) -IHL12 -HLK1 // -Hie
+ >yminus_succ <yminus_inj /3 width=4 by ldrop_drop_lt, ex2_2_intro/
+ ]
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hdi
+ elim (yle_inv_succ1 ⦠Hdi) -Hdi
+ #Hdi #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ ⦠H) -H
+ #Hide lapply (ldrop_inv_drop1_lt ⦠HLK2 ?) -HLK2 /2 width=1 by ylt_O/
+ #HLK1 elim (IHL12 ⦠HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
+ /4 width=4 by ylt_O, ldrop_drop_lt, ex2_2_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby_lsuby.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby_lsuby.etc
new file mode 100644
index 000000000..24361d3c0
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/lsuby_lsuby.etc
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/lsuby.ma".
+
+(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
+
+(* Main properties **********************************************************)
+
+theorem lsuby_trans: âd,e. Transitive ⦠(lsuby d e).
+#d #e #L1 #L2 #H elim H -L1 -L2 -d -e
+[ #L1 #d #e #X #H lapply (lsuby_inv_atom1 ⦠H) -H
+ #H destruct //
+| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 ⦠H) -H //
+ * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_zero/
+| #I1 #I2 #L1 #L2 #V #e #_ #IHL1 #X #H elim (lsuby_inv_pair1 ⦠H) -H //
+ * #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL1 #X #H elim (lsuby_inv_succ1 ⦠H) -H //
+ * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/psubst_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/psubst_6.etc
new file mode 100644
index 000000000..31fe21410
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/psubst_6.etc
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The 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 T1 break ⶠ[ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubst $G $L $T1 $d $e $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/psubststar_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/psubststar_6.etc
new file mode 100644
index 000000000..f2bb4bd78
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/psubststar_6.etc
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The 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 T1 break ⶠ* [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStar $G $L $T1 $d $e $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/psubststaralt_6.etc b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/psubststaralt_6.etc
new file mode 100644
index 000000000..e727d4945
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/etc/lsuby/psubststaralt_6.etc
@@ -0,0 +1,19 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The 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 T1 break ⶠⶠ* [ term 46 d , break term 46 e ] break term 46 T2 )"
+ non associative with precedence 45
+ for @{ 'PSubstStarAlt $G $L $T1 $d $e $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/extlrsubeq_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/extlrsubeq_4.ma
deleted file mode 100644
index 8a9714dd8..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/extlrsubeq_4.ma
+++ /dev/null
@@ -1,19 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The 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 â Ã [ term 46 d , break term 46 e ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'ExtLRSubEq $L1 $d $e $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeqalt_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeqalt_4.ma
deleted file mode 100644
index b89d7844c..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeqalt_4.ma
+++ /dev/null
@@ -1,19 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The 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 â â break [ term 46 T , break term 46 d ] break term 46 L2 )"
- non associative with precedence 45
- for @{ 'LazyEqAlt $T $d $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubst_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubst_6.ma
deleted file mode 100644
index 31fe21410..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubst_6.ma
+++ /dev/null
@@ -1,19 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The 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 T1 break ⶠ[ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubst $G $L $T1 $d $e $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubststar_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubststar_6.ma
deleted file mode 100644
index f2bb4bd78..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubststar_6.ma
+++ /dev/null
@@ -1,19 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The 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 T1 break ⶠ* [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStar $G $L $T1 $d $e $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubststaralt_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubststaralt_6.ma
deleted file mode 100644
index e727d4945..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubststaralt_6.ma
+++ /dev/null
@@ -1,19 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The 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 T1 break ⶠⶠ* [ term 46 d , break term 46 e ] break term 46 T2 )"
- non associative with precedence 45
- for @{ 'PSubstStarAlt $G $L $T1 $d $e $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cpys.ma
deleted file mode 100644
index 3b38ee6ab..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cpys.ma
+++ /dev/null
@@ -1,42 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpys_alt.ma".
-include "basic_2/reduction/cpx.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
-
-(* Properties on local environment refinement for extended substitution *****)
-
-lemma lsuby_cpx_trans: âh,g,G. lsub_trans ⦠(cpx h g G) (lsuby 0 (â)).
-#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
-[ //
-| /2 width=2 by cpx_sort/
-| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsuby_ldrop_trans_be ⦠HL12 ⦠HLK1) // -HL12 -HLK1 /3 width=7 by cpx_delta/
-|4,9: /4 width=1 by cpx_bind, cpx_beta, lsuby_pair_O_Y/
-|5,7,8: /3 width=1 by cpx_flat, cpx_tau, cpx_ti/
-|6,10: /4 width=3 by cpx_zeta, cpx_theta, lsuby_pair_O_Y/
-]
-qed-.
-
-(* Properties on context-sensitive extended multiple substitution for terms *)
-
-lemma cpys_cpx: âh,g,G,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¡[h, g] T2.
-#h #g #G #L #T1 #T2 #d #e #H @(cpys_ind_alt ⦠H) -G -L -T1 -T2 -d -e
-/2 width=7 by cpx_delta, cpx_bind, cpx_flat/
-qed.
-
-lemma cpy_cpx: âh,g,G,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â â¦G, L⦠⢠T1 â¡[h, g] T2.
-/3 width=3 by cpy_cpys, cpys_cpx/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_leq.ma
new file mode 100644
index 000000000..7d86e027d
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_leq.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/ldrop_leq.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma leq_cpx_trans: âh,g,G. lsub_trans ⦠(cpx h g G) (leq 0 (â)).
+#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+[ //
+| /2 width=2 by cpx_sort/
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (leq_ldrop_trans_be ⦠HL12 ⦠HLK1) // -HL12 -HLK1 /3 width=7 by cpx_delta/
+|4,9: /4 width=1 by cpx_bind, cpx_beta, leq_pair_O_Y/
+|5,7,8: /3 width=1 by cpx_flat, cpx_tau, cpx_ti/
+|6,10: /4 width=3 by cpx_zeta, cpx_theta, leq_pair_O_Y/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lleq.ma
index 6ff6c39d0..ae6ddac78 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lleq.ma
@@ -12,7 +12,8 @@
(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_ext.ma".
+include "basic_2/relocation/lleq_leq.ma".
+include "basic_2/relocation/lleq_ldrop.ma".
include "basic_2/reduction/cpx.ma".
(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
@@ -49,8 +50,8 @@ lemma lleq_cpx_conf_dx: âh,g,G,L2,T1,T2. â¦G, L2⦠⢠T1 â¡[h, g] T2 â
#h #g #G #L2 #T1 #T2 #H elim H -G -L2 -T1 -T2
[ //
| /3 width=3 by lleq_fwd_length, lleq_sort/
-| #I2 #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #H elim (lleq_inv_lref_ge_dx ⦠H ⦠HLK2) // -H
- #I1 #K1 #HLK1 #HV1
+| #I #G #L2 #K2 #V1 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV12 #L1 #H elim (lleq_inv_lref_ge_dx ⦠H ⦠HLK2) // -H
+ #K1 #HLK1 #HV1
lapply (ldrop_fwd_drop2 ⦠HLK1) -HLK1 #HLK1
lapply (ldrop_fwd_drop2 ⦠HLK2) -HLK2 #HLK2
@(lleq_lift_le ⦠HLK1 HLK2 ⦠HVW2) -HLK1 -HLK2 -HVW2 /2 width=1 by/ (**) (* full auto too slow *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_cpys.ma
deleted file mode 100644
index d3c490a89..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_cpys.ma
+++ /dev/null
@@ -1,69 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| 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/cpy_lift.ma".
-include "basic_2/substitution/cpys.ma".
-include "basic_2/reduction/lpx_ldrop.ma".
-
-(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
-
-(* Properties on context-sensitive extended substitution for terms **********)
-
-lemma cpx_cpy_trans_lpx: âh,g,G,L1,T1,T. â¦G, L1⦠⢠T1 â¡[h, g] T â
- âL2. â¦G, L1⦠⢠â¡[h, g] L2 â
- âT2,d,e. â¦G, L2⦠⢠T â¶[d, e] T2 â â¦G, L1⦠⢠T1 â¡[h, g] T2.
-#h #g #G #L1 #T1 #T #H elim H -G -L1 -T1 -T
-[ #J #G #L1 #L2 #HL12 #T2 #d #e #H elim (cpy_inv_atom1 ⦠H) -H //
- * #I #K2 #V2 #i #_ #_ #HLK2 #HVT2 #H destruct
- elim (lpx_ldrop_trans_O1 ⦠HL12 ⦠HLK2) -L2 #X #HLK1 #H
- elim (lpx_inv_pair2 ⦠H) -H #K1 #V1 #_ #HV12 #H destruct
- /2 width=7 by cpx_delta/
-| #G #L1 #k #l #Hkl #L2 #_ #X #d #e #H >(cpy_inv_sort1 ⦠H) -X /2 width=2 by cpx_sort/
-| #I #G #L1 #K1 #V1 #V #T #i #HLK1 #_ #HVT #IHV1 #L2 #HL12 #T2 #d #e #HT2
- elim (lpx_ldrop_conf ⦠HLK1 ⦠HL12) -HL12 #X #H #HLK2
- elim (lpx_inv_pair1 ⦠H) -H #K2 #V0 #HK12 #_ #H destruct
- lapply (ldrop_fwd_drop2 ⦠HLK2) -V0 #HLK2
- lapply (cpy_weak ⦠HT2 0 (â) ? ?) -HT2 // #HT2
- elim (cpy_inv_lift1_be ⦠HT2 ⦠HLK2 ⦠HVT) -HT2 -HLK2 -HVT
- /3 width=7 by cpx_delta/
-| #a #I #G #L1 #V1 #V #T1 #T #HV1 #_ #IHV1 #IHT1 #L2 #HL12 #X #d #e #H elim (cpy_inv_bind1 ⦠H) -H
- #V2 #T2 #HV2 #HT2 #H destruct /4 width=5 by lpx_pair, cpx_bind/
-| #I #G #L1 #V1 #V #T1 #T #_ #_ #IHV1 #IHT1 #L2 #HL12 #X #d #e #H elim (cpy_inv_flat1 ⦠H) -H
- #V2 #T2 #HV2 #HT2 #H destruct /3 width=5 by cpx_flat/
-| #G #L1 #V1 #U1 #U #T #_ #HTU #IHU1 #L2 #HL12 #T2 #d #e #HT2
- lapply (cpy_weak ⦠HT2 0 (â) ? ?) -HT2 // #HT2
- elim (lift_total T2 0 1) #U2 #HTU2
- lapply (cpy_lift_be ⦠HT2 (L2.âV1) ⦠(â») ⦠HTU ⦠HTU2 ? ?) -T
- /4 width=5 by cpx_zeta, lpx_pair, ldrop_drop/
-| /3 width=5 by cpx_tau/
-| /3 width=5 by cpx_ti/
-| #a #G #L1 #V1 #V #W1 #W #T1 #T #HV1 #HW1 #_ #IHV1 #IHW1 #IHT1 #L2 #HL12 #X #d #e #HX
- elim (cpy_inv_bind1 ⦠HX) -HX #Y #T2 #HY #HT2 #H destruct
- elim (cpy_inv_flat1 ⦠HY) -HY #W2 #V2 #HW2 #HV2 #H destruct
- /5 width=11 by lpx_pair, cpx_beta, lsuby_cpy_trans, lsuby_succ/
-| #a #G #L1 #V1 #V #U #W1 #W #T1 #T #_ #HVU #HW1 #_ #IHV1 #IHW1 #IHT1 #L2 #HL12 #X #d #e #HX
- elim (cpy_inv_bind1 ⦠HX) -HX #W2 #Y #HW2 #HY #H destruct
- elim (cpy_inv_flat1 ⦠HY) -HY #U2 #T2 #HU2 #HT2 #H destruct
- lapply (cpy_weak ⦠HU2 0 (â) ? ?) -HU2 // #HU2
- elim (cpy_inv_lift1_be ⦠HU2 L2 ⦠HVU) -U
- /4 width=7 by lpx_pair, cpx_theta, ldrop_drop/
-]
-qed-.
-
-lemma cpx_cpys_trans_lpx: âh,g,G,L1,T1,T. â¦G, L1⦠⢠T1 â¡[h, g] T â
- âL2. â¦G, L1⦠⢠â¡[h, g] L2 â
- âT2,d,e. â¦G, L2⦠⢠T â¶*[d, e] T2 â â¦G, L1⦠⢠T1 â¡[h, g] T2.
-#h #g #G #L1 #T1 #T #HT1 #L2 #HL12 #T2 #d #e #H @(cpys_ind ⦠H) -T2
-/2 width=7 by cpx_cpy_trans_lpx/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma
index 9daa44656..353f810c3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma
@@ -12,7 +12,7 @@
(* *)
(**************************************************************************)
-include "basic_2/substitution/lleq_ext.ma".
+include "basic_2/relocation/lleq_ldrop.ma".
include "basic_2/reduction/lpx_ldrop.ma".
(* SN EXTENDED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS ********************)
@@ -51,7 +51,7 @@ lemma lpx_lleq_fqu_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2,
[ #I #G1 #L1 #V1 #X #H1 #H2 elim (lpx_inv_pair2 ⦠H1) -H1
#K0 #V0 #H1KL1 #_ #H destruct
elim (lleq_inv_lref_ge_dx ⦠H2 ? I L1 V1) -H2 //
- #I1 #K1 #H #H2KL1 lapply (ldrop_inv_O2 ⦠H) -H #H destruct
+ #K1 #H #H2KL1 lapply (ldrop_inv_O2 ⦠H) -H #H destruct
/2 width=4 by fqu_lref_O, ex3_intro/
| * [ #a ] #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H
[ elim (lleq_inv_bind ⦠H)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
deleted file mode 100644
index a1e538ae0..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy.ma
+++ /dev/null
@@ -1,296 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_max.ma".
-include "basic_2/notation/relations/psubst_6.ma".
-include "basic_2/grammar/genv.ma".
-include "basic_2/relocation/lsuby.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* activate genv *)
-inductive cpy: ynat â ynat â relation4 genv lenv term term â
-| cpy_atom : âI,G,L,d,e. cpy d e G L (âª{I}) (âª{I})
-| cpy_subst: âI,G,L,K,V,W,i,d,e. d ⤠yinj i â i < d+e â
- â©[i] L â¡ K.â{I}V â â§[0, i+1] V â¡ W â cpy d e G L (#i) W
-| cpy_bind : âa,I,G,L,V1,V2,T1,T2,d,e.
- cpy d e G L V1 V2 â cpy (⫯d) e G (L.â{I}V1) T1 T2 â
- cpy d e G L (â{a,I}V1.T1) (â{a,I}V2.T2)
-| cpy_flat : âI,G,L,V1,V2,T1,T2,d,e.
- cpy d e G L V1 V2 â cpy d e G L T1 T2 â
- cpy d e G L (â{I}V1.T1) (â{I}V2.T2)
-.
-
-interpretation "context-sensitive extended ordinary substritution (term)"
- 'PSubst G L T1 d e T2 = (cpy d e G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_cpy_trans: âG,d,e. lsub_trans ⦠(cpy d e G) (lsuby d e).
-#G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
-[ //
-| #I #G #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
- elim (lsuby_ldrop_trans_be ⦠HL12 ⦠HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
-| /4 width=1 by lsuby_succ, cpy_bind/
-| /3 width=1 by cpy_flat/
-]
-qed-.
-
-lemma cpy_refl: âG,T,L,d,e. â¦G, L⦠⢠T â¶[d, e] T.
-#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
-qed.
-
-(* Basic_1: was: subst1_ex *)
-lemma cpy_full: âI,G,K,V,T1,L,d. â©[d] L â¡ K.â{I}V â
- ââT2,T. â¦G, L⦠⢠T1 â¶[d, 1] T2 & â§[d, 1] T â¡ T2.
-#I #G #K #V #T1 elim T1 -T1
-[ * #i #L #d #HLK
- /2 width=4 by lift_sort, lift_gref, ex2_2_intro/
- elim (lt_or_eq_or_gt i d) #Hid
- /3 width=4 by lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/
- destruct
- elim (lift_total V 0 (i+1)) #W #HVW
- elim (lift_split ⦠HVW i i)
- /4 width=5 by cpy_subst, ylt_inj, ex2_2_intro/
-| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #d #HLK
- elim (IHW1 ⦠HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L.â{J}W1) (d+1)) -IHU1
- /3 width=9 by cpy_bind, ldrop_drop, lift_bind, ex2_2_intro/
- | elim (IHU1 ⦠HLK) -IHU1 -HLK
- /3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
- ]
-]
-qed-.
-
-lemma cpy_weak: âG,L,T1,T2,d1,e1. â¦G, L⦠⢠T1 â¶[d1, e1] T2 â
- âd2,e2. d2 ⤠d1 â d1 + e1 ⤠d2 + e2 â
- â¦G, L⦠⢠T1 â¶[d2, e2] T2.
-#G #L #T1 #T2 #d1 #e1 #H elim H -G -L -T1 -T2 -d1 -e1 //
-[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
-| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
-| /3 width=1 by cpy_flat/
-]
-qed-.
-
-lemma cpy_weak_top: âG,L,T1,T2,d,e.
- â¦G, L⦠⢠T1 â¶[d, e] T2 â â¦G, L⦠⢠T1 â¶[d, |L| - d] T2.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e //
-[ #I #G #L #K #V #W #i #d #e #Hdi #_ #HLK #HVW
- lapply (ldrop_fwd_length_lt2 ⦠HLK)
- /4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
-| #a #I #G #L #V1 #V2 normalize in match (|L.â{I}V2|); (**) (* |?| does not work *)
- /2 width=1 by cpy_bind/
-| /2 width=1 by cpy_flat/
-]
-qed-.
-
-lemma cpy_weak_full: âG,L,T1,T2,d,e.
- â¦G, L⦠⢠T1 â¶[d, e] T2 â â¦G, L⦠⢠T1 â¶[0, |L|] T2.
-#G #L #T1 #T2 #d #e #HT12
-lapply (cpy_weak ⦠HT12 0 (d + e) ? ?) -HT12
-/2 width=2 by cpy_weak_top/
-qed-.
-
-lemma cpy_split_up: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â âi. i ⤠d + e â
- ââT. â¦G, L⦠⢠T1 â¶[d, i-d] T & â¦G, L⦠⢠T â¶[i, d+e-i] T2.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
-[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
- elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
- /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
- elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
- >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
- lapply (lsuby_cpy_trans ⦠HT2 (L.â{I}V) ?) -HT2
- /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
- elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
- /3 width=5 by ex2_intro, cpy_flat/
-]
-qed-.
-
-lemma cpy_split_down: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â âi. i ⤠d + e â
- ââT. â¦G, L⦠⢠T1 â¶[i, d+e-i] T & â¦G, L⦠⢠T â¶[d, i-d] T2.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
-[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
- elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
- /4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
- elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
- >yplus_SO2 >yplus_succ1 #T #HT1 #HT2
- lapply (lsuby_cpy_trans ⦠HT2 (L.â{I}V) ?) -HT2
- /3 width=5 by lsuby_succ, ex2_intro, cpy_bind/
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
- elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hide
- /3 width=5 by ex2_intro, cpy_flat/
-]
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpy_fwd_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
- âT1,d,e. â§[d, e] T1 â¡ U1 â
- d ⤠dt â d + e ⤠dt + et â
- ââT2. â¦G, L⦠⢠U1 â¶[d+e, dt+et-(d+e)] U2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
-[ * #i #G #L #dt #et #T1 #d #e #H #_
- [ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #K #V #W #i #dt #et #Hdti #Hidet #HLK #HVW #T1 #d #e #H #Hddt #Hdedet
- elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct [ -V -Hidet -Hdedet | -Hdti -Hddt ]
- [ elim (ylt_yle_false ⦠Hddt) -Hddt /3 width=3 by yle_ylt_trans, ylt_inj/
- | elim (le_inv_plus_l ⦠Hid) #Hdie #Hei
- elim (lift_split ⦠HVW d (i-e+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hdie
- #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hei
- @(ex2_intro ⦠H) -H @(cpy_subst ⦠HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
- elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 ⦠HVW1) -V1 -IHW12 //
- elim (IHU12 ⦠HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
- <yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
- /3 width=2 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
- elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 ⦠HVW1) -V1 -IHW12 // elim (IHU12 ⦠HTU1) -T1 -IHU12
- /3 width=2 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-lemma cpy_fwd_tw: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â â¯{T1} ⤠â¯{T2}.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e normalize
-/3 width=1 by monotonic_le_plus_l, le_plus/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact cpy_inv_atom1_aux: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â âJ. T1 = âª{J} â
- T2 = âª{J} â¨
- ââI,K,V,i. d ⤠yinj i & i < d + e &
- â©[i] L â¡ K.â{I}V &
- â§[O, i+1] V â¡ T2 &
- J = LRef i.
-#G #L #T1 #T2 #d #e * -G -L -T1 -T2 -d -e
-[ #I #G #L #d #e #J #H destruct /2 width=1 by or_introl/
-| #I #G #L #K #V #T2 #i #d #e #Hdi #Hide #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #H destruct
-]
-qed-.
-
-lemma cpy_inv_atom1: âI,G,L,T2,d,e. â¦G, L⦠⢠âª{I} â¶[d, e] T2 â
- T2 = âª{I} â¨
- ââJ,K,V,i. d ⤠yinj i & i < d + e &
- â©[i] L â¡ K.â{J}V &
- â§[O, i+1] V â¡ T2 &
- I = LRef i.
-/2 width=4 by cpy_inv_atom1_aux/ qed-.
-
-(* Basic_1: was: subst1_gen_sort *)
-lemma cpy_inv_sort1: âG,L,T2,k,d,e. â¦G, L⦠⢠âk â¶[d, e] T2 â T2 = âk.
-#G #L #T2 #k #d #e #H
-elim (cpy_inv_atom1 ⦠H) -H //
-* #I #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-(* Basic_1: was: subst1_gen_lref *)
-lemma cpy_inv_lref1: âG,L,T2,i,d,e. â¦G, L⦠⢠#i â¶[d, e] T2 â
- T2 = #i â¨
- ââI,K,V. d ⤠i & i < d + e &
- â©[i] L â¡ K.â{I}V &
- â§[O, i+1] V â¡ T2.
-#G #L #T2 #i #d #e #H
-elim (cpy_inv_atom1 ⦠H) -H /2 width=1 by or_introl/
-* #I #K #V #j #Hdj #Hjde #HLK #HVT2 #H destruct /3 width=5 by ex4_3_intro, or_intror/
-qed-.
-
-lemma cpy_inv_gref1: âG,L,T2,p,d,e. â¦G, L⦠⢠§p â¶[d, e] T2 â T2 = §p.
-#G #L #T2 #p #d #e #H
-elim (cpy_inv_atom1 ⦠H) -H //
-* #I #K #V #i #_ #_ #_ #_ #H destruct
-qed-.
-
-fact cpy_inv_bind1_aux: âG,L,U1,U2,d,e. â¦G, L⦠⢠U1 â¶[d, e] U2 â
- âa,I,V1,T1. U1 = â{a,I}V1.T1 â
- ââV2,T2. â¦G, L⦠⢠V1 â¶[d, e] V2 &
- â¦G, L. â{I}V1⦠⢠T1 â¶[⫯d, e] T2 &
- U2 = â{a,I}V2.T2.
-#G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
-[ #I #G #L #d #e #b #J #W1 #U1 #H destruct
-| #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #b #J #W1 #U1 #H destruct
-]
-qed-.
-
-lemma cpy_inv_bind1: âa,I,G,L,V1,T1,U2,d,e. â¦G, L⦠⢠â{a,I} V1. T1 â¶[d, e] U2 â
- ââV2,T2. â¦G, L⦠⢠V1 â¶[d, e] V2 &
- â¦G, L.â{I}V1⦠⢠T1 â¶[⫯d, e] T2 &
- U2 = â{a,I}V2.T2.
-/2 width=3 by cpy_inv_bind1_aux/ qed-.
-
-fact cpy_inv_flat1_aux: âG,L,U1,U2,d,e. â¦G, L⦠⢠U1 â¶[d, e] U2 â
- âI,V1,T1. U1 = â{I}V1.T1 â
- ââV2,T2. â¦G, L⦠⢠V1 â¶[d, e] V2 &
- â¦G, L⦠⢠T1 â¶[d, e] T2 &
- U2 = â{I}V2.T2.
-#G #L #U1 #U2 #d #e * -G -L -U1 -U2 -d -e
-[ #I #G #L #d #e #J #W1 #U1 #H destruct
-| #I #G #L #K #V #W #i #d #e #_ #_ #_ #_ #J #W1 #U1 #H destruct
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #J #W1 #U1 #H destruct
-| #I #G #L #V1 #V2 #T1 #T2 #d #e #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-]
-qed-.
-
-lemma cpy_inv_flat1: âI,G,L,V1,T1,U2,d,e. â¦G, L⦠⢠â{I} V1. T1 â¶[d, e] U2 â
- ââV2,T2. â¦G, L⦠⢠V1 â¶[d, e] V2 &
- â¦G, L⦠⢠T1 â¶[d, e] T2 &
- U2 = â{I}V2.T2.
-/2 width=3 by cpy_inv_flat1_aux/ qed-.
-
-
-fact cpy_inv_refl_O2_aux: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â e = 0 â T1 = T2.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
-[ //
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
- elim (ylt_yle_false ⦠Hdi) -Hdi //
-| /3 width=1 by eq_f2/
-| /3 width=1 by eq_f2/
-]
-qed-.
-
-lemma cpy_inv_refl_O2: âG,L,T1,T2,d. â¦G, L⦠⢠T1 â¶[d, 0] T2 â T1 = T2.
-/2 width=6 by cpy_inv_refl_O2_aux/ qed-.
-
-(* Basic_1: was: subst1_gen_lift_eq *)
-lemma cpy_inv_lift1_eq: âG,T1,U1,d,e. â§[d, e] T1 â¡ U1 â
- âL,U2. â¦G, L⦠⢠U1 â¶[d, e] U2 â U1 = U2.
-#G #T1 #U1 #d #e #HTU1 #L #U2 #HU12 elim (cpy_fwd_up ⦠HU12 ⦠HTU1) -HU12 -HTU1
-/2 width=4 by cpy_inv_refl_O2/
-qed-.
-
-(* Basic_1: removed theorems 25:
- subst0_gen_sort subst0_gen_lref subst0_gen_head subst0_gen_lift_lt
- subst0_gen_lift_false subst0_gen_lift_ge subst0_refl subst0_trans
- subst0_lift_lt subst0_lift_ge subst0_lift_ge_S subst0_lift_ge_s
- subst0_subst0 subst0_subst0_back subst0_weight_le subst0_weight_lt
- subst0_confluence_neq subst0_confluence_eq subst0_tlt_head
- subst0_confluence_lift subst0_tlt
- subst1_head subst1_gen_head subst1_lift_S subst1_confluence_lift
-*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma
deleted file mode 100644
index 66b0487db..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_cpy.ma
+++ /dev/null
@@ -1,122 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| 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/cpy_lift.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Main properties **********************************************************)
-
-(* Basic_1: was: subst1_confluence_eq *)
-theorem cpy_conf_eq: âG,L,T0,T1,d1,e1. â¦G, L⦠⢠T0 â¶[d1, e1] T1 â
- âT2,d2,e2. â¦G, L⦠⢠T0 â¶[d2, e2] T2 â
- ââT. â¦G, L⦠⢠T1 â¶[d2, e2] T & â¦G, L⦠⢠T2 â¶[d1, e1] T.
-#G #L #T0 #T1 #d1 #e1 #H elim H -G -L -T0 -T1 -d1 -e1
-[ /2 width=3 by ex2_intro/
-| #I1 #G #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
- elim (cpy_inv_lref1 ⦠H) -H
- [ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
- | -Hd1 -Hde1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
- lapply (ldrop_mono ⦠HLK1 ⦠HLK2) -HLK1 -HLK2 #H destruct
- >(lift_mono ⦠HVT1 ⦠HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
- ]
-| #a #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
- elim (cpy_inv_bind1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 ⦠HV02) -IHV01 -HV02 #V #HV1 #HV2
- elim (IHT01 ⦠HT02) -T0 #T #HT1 #HT2
- lapply (lsuby_cpy_trans ⦠HT1 (L.â{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans ⦠HT2 (L.â{I}V2) ?) -HT2
- /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
-| #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
- elim (cpy_inv_flat1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 ⦠HV02) -V0
- elim (IHT01 ⦠HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
-]
-qed-.
-
-(* Basic_1: was: subst1_confluence_neq *)
-theorem cpy_conf_neq: âG,L1,T0,T1,d1,e1. â¦G, L1⦠⢠T0 â¶[d1, e1] T1 â
- âL2,T2,d2,e2. â¦G, L2⦠⢠T0 â¶[d2, e2] T2 â
- (d1 + e1 ⤠d2 ⨠d2 + e2 ⤠d1) â
- ââT. â¦G, L2⦠⢠T1 â¶[d2, e2] T & â¦G, L1⦠⢠T2 â¶[d1, e1] T.
-#G #L1 #T0 #T1 #d1 #e1 #H elim H -G -L1 -T0 -T1 -d1 -e1
-[ /2 width=3 by ex2_intro/
-| #I1 #G #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
- elim (cpy_inv_lref1 ⦠H1) -H1
- [ #H destruct /3 width=7 by cpy_subst, ex2_intro/
- | -HLK1 -HVT1 * #I2 #K2 #V2 #Hd2 #Hde2 #_ #_ elim H2 -H2 #Hded [ -Hd1 -Hde2 | -Hd2 -Hde1 ]
- [ elim (ylt_yle_false ⦠Hde1) -Hde1 /2 width=3 by yle_trans/
- | elim (ylt_yle_false ⦠Hde2) -Hde2 /2 width=3 by yle_trans/
- ]
- ]
-| #a #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
- elim (cpy_inv_bind1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 ⦠HV02 H) -IHV01 -HV02 #V #HV1 #HV2
- elim (IHT01 ⦠HT02) -T0
- [ -H #T #HT1 #HT2
- lapply (lsuby_cpy_trans ⦠HT1 (L2.â{I}V1) ?) -HT1 /2 width=1 by lsuby_succ/
- lapply (lsuby_cpy_trans ⦠HT2 (L1.â{I}V2) ?) -HT2 /3 width=5 by cpy_bind, lsuby_succ, ex2_intro/
- | -HV1 -HV2 elim H -H /3 width=1 by yle_succ, or_introl, or_intror/
- ]
-| #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
- elim (cpy_inv_flat1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV01 ⦠HV02 H) -V0
- elim (IHT01 ⦠HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
-]
-qed-.
-
-(* Note: the constant 1 comes from cpy_subst *)
-(* Basic_1: was: subst1_trans *)
-theorem cpy_trans_ge: âG,L,T1,T0,d,e. â¦G, L⦠⢠T1 â¶[d, e] T0 â
- âT2. â¦G, L⦠⢠T0 â¶[d, 1] T2 â 1 ⤠e â â¦G, L⦠⢠T1 â¶[d, e] T2.
-#G #L #T1 #T0 #d #e #H elim H -G -L -T1 -T0 -d -e
-[ #I #G #L #d #e #T2 #H #He
- elim (cpy_inv_atom1 ⦠H) -H
- [ #H destruct //
- | * #J #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
- lapply (ylt_yle_trans ⦠(d+e) ⦠Hide2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
- ]
-| #I #G #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
- lapply (cpy_weak ⦠HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
- >yplus_inj #HVT2 <(cpy_inv_lift1_eq ⦠HVW ⦠HVT2) -HVT2 /2 width=5 by cpy_subst/
-| #a #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
- elim (cpy_inv_bind1 ⦠H) -H #V2 #T2 #HV02 #HT02 #H destruct
- lapply (lsuby_cpy_trans ⦠HT02 (L.â{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- lapply (IHT10 ⦠HT02 He) -T0 /3 width=1 by cpy_bind/
-| #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
- elim (cpy_inv_flat1 ⦠H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
-]
-qed-.
-
-theorem cpy_trans_down: âG,L,T1,T0,d1,e1. â¦G, L⦠⢠T1 â¶[d1, e1] T0 â
- âT2,d2,e2. â¦G, L⦠⢠T0 â¶[d2, e2] T2 â d2 + e2 ⤠d1 â
- ââT. â¦G, L⦠⢠T1 â¶[d2, e2] T & â¦G, L⦠⢠T â¶[d1, e1] T2.
-#G #L #T1 #T0 #d1 #e1 #H elim H -G -L -T1 -T0 -d1 -e1
-[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i1 #d1 #e1 #Hdi1 #Hide1 #HLK #HVW #T2 #d2 #e2 #HWT2 #Hde2d1
- lapply (yle_trans ⦠Hde2d1 ⦠Hdi1) -Hde2d1 #Hde2i1
- lapply (cpy_weak ⦠HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hde2i1
- >yplus_inj #HWT2 <(cpy_inv_lift1_eq ⦠HVW ⦠HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
-| #a #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
- elim (cpy_inv_bind1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- lapply (lsuby_cpy_trans ⦠HT02 (L.â{I}V1) ?) -HT02 /2 width=1 by lsuby_succ/ #HT02
- elim (IHV10 ⦠HV02) -IHV10 -HV02 // #V
- elim (IHT10 ⦠HT02) -T0 /2 width=1 by yle_succ/ #T #HT1 #HT2
- lapply (lsuby_cpy_trans ⦠HT2 (L.â{I}V) ?) -HT2 /3 width=6 by cpy_bind, lsuby_succ, ex2_intro/
-| #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
- elim (cpy_inv_flat1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
- elim (IHV10 ⦠HV02) -V0 //
- elim (IHT10 ⦠HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
deleted file mode 100644
index aa0b2416a..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/cpy_lift.ma
+++ /dev/null
@@ -1,249 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| 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/ldrop_ldrop.ma".
-include "basic_2/relocation/cpy.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED ORDINARY SUBSTITUTION FOR TERMS ***************)
-
-(* Properties on relocation *************************************************)
-
-(* Basic_1: was: subst1_lift_lt *)
-lemma cpy_lift_le: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶[dt, et] T2 â
- âL,U1,U2,s,d,e. â©[s, d, e] L â¡ K â
- â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â
- dt + et ⤠d â â¦G, L⦠⢠U1 â¶[dt, et] U2.
-#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
-[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
- >(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdetd
- lapply (ylt_yle_trans ⦠Hdetd ⦠Hidet) -Hdetd #Hid
- lapply (ylt_inv_inj ⦠Hid) -Hid #Hid
- lapply (lift_inv_lref1_lt ⦠H ⦠Hid) -H #H destruct
- elim (lift_trans_ge ⦠HVW ⦠HWU2) -W // <minus_plus #W #HVW #HWU2
- elim (ldrop_trans_le ⦠HLK ⦠HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (ldrop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
- >(lift_mono ⦠HVY ⦠HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
-| #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=7 by cpy_flat/
-]
-qed-.
-
-lemma cpy_lift_be: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶[dt, et] T2 â
- âL,U1,U2,s,d,e. â©[s, d, e] L â¡ K â
- â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â
- dt ⤠d â d ⤠dt + et â â¦G, L⦠⢠U1 â¶[dt, et + e] U2.
-#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
-[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_ #_
- >(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hdtd #_
- elim (lift_inv_lref1 ⦠H) -H * #Hid #H destruct
- [ -Hdtd
- lapply (ylt_yle_trans ⦠(dt+et+e) ⦠Hidet) // -Hidet #Hidete
- elim (lift_trans_ge ⦠HVW ⦠HWU2) -W // <minus_plus #W #HVW #HWU2
- elim (ldrop_trans_le ⦠HLK ⦠HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (ldrop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
- >(lift_mono ⦠HVY ⦠HVW) -V #H destruct /2 width=5 by cpy_subst/
- | -Hdti
- elim (yle_inv_inj2 ⦠Hdtd) -Hdtd #dtt #Hdtd #H destruct
- lapply (transitive_le ⦠Hdtd Hid) -Hdtd #Hdti
- lapply (lift_trans_be ⦠HVW ⦠HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm ⦠HLK ⦠HKV ?) -K // -Hid
- /4 width=5 by cpy_subst, ldrop_inv_gen, monotonic_ylt_plus_dx, yle_plus_dx1_trans, yle_inj/
- ]
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdtd #Hddet
- elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=7 by cpy_bind, ldrop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
- elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=7 by cpy_flat/
-]
-qed-.
-
-(* Basic_1: was: subst1_lift_ge *)
-lemma cpy_lift_ge: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶[dt, et] T2 â
- âL,U1,U2,s,d,e. â©[s, d, e] L â¡ K â
- â§[d, e] T1 â¡ U1 â â§[d, e] T2 â¡ U2 â
- d ⤠dt â â¦G, L⦠⢠U1 â¶[dt+e, et] U2.
-#G #K #T1 #T2 #dt #et #H elim H -G -K -T1 -T2 -dt -et
-[ #I #G #K #dt #et #L #U1 #U2 #s #d #e #_ #H1 #H2 #_
- >(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #dt #et #Hdti #Hidet #HKV #HVW #L #U1 #U2 #s #d #e #HLK #H #HWU2 #Hddt
- lapply (yle_trans ⦠Hddt ⦠Hdti) -Hddt #Hid
- elim (yle_inv_inj2 ⦠Hid) -Hid #dd #Hddi #H0 destruct
- lapply (lift_inv_lref1_ge ⦠H ⦠Hddi) -H #H destruct
- lapply (lift_trans_be ⦠HVW ⦠HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (ldrop_trans_ge_comm ⦠HLK ⦠HKV ?) -K // -Hddi
- /3 width=5 by cpy_subst, ldrop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /4 width=6 by cpy_bind, ldrop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
- elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
- elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
- /3 width=6 by cpy_flat/
-]
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-(* Basic_1: was: subst1_gen_lift_lt *)
-lemma cpy_inv_lift1_le: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- dt + et ⤠d â
- ââT2. â¦G, K⦠⢠T1 â¶[dt, et] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
-[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdetd
- lapply (ylt_yle_trans ⦠Hdetd ⦠Hidet) -Hdetd #Hid
- lapply (ylt_inv_inj ⦠Hid) -Hid #Hid
- lapply (lift_inv_lref2_lt ⦠H ⦠Hid) -H #H destruct
- elim (ldrop_conf_lt ⦠HLK ⦠HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le ⦠HUV ⦠HVW) -V // >minus_plus <plus_minus_m_m // -Hid /3 width=5 by cpy_subst, ex2_intro/
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
- elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 ⦠HLK ⦠HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 ⦠HTU1) -IHU12 -HTU1
- /3 width=6 by cpy_bind, yle_succ, ldrop_skip, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
- elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 ⦠HLK ⦠HVW1) -W1 //
- elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK
- /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-lemma cpy_inv_lift1_be: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- dt ⤠d â yinj d + e ⤠dt + et â
- ââT2. â¦G, K⦠⢠T1 â¶[dt, et-e] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
-[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_ #_
- [ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdtd #Hdedet
- lapply (yle_fwd_plus_ge_inj ⦠Hdtd Hdedet) #Heet
- elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct [ -Hdtd -Hidet | -Hdti -Hdedet ]
- [ lapply (ylt_yle_trans i d (dt+(et-e)) ? ?) /2 width=1 by ylt_inj/
- [ >yplus_minus_assoc_inj /2 width=1 by yle_plus_to_minus_inj2/ ] -Hdedet #Hidete
- elim (ldrop_conf_lt ⦠HLK ⦠HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le ⦠HUV ⦠HVW) -V // >minus_plus <plus_minus_m_m // -Hid
- /3 width=5 by cpy_subst, ex2_intro/
- | elim (le_inv_plus_l ⦠Hid) #Hdie #Hei
- lapply (yle_trans ⦠Hdtd (i-e) ?) /2 width=1 by yle_inj/ -Hdtd #Hdtie
- lapply (ldrop_conf_ge ⦠HLK ⦠HLKV ?) -L // #HKV
- elim (lift_split ⦠HVW d (i-e+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hid -Hdie
- #V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
- @(ex2_intro ⦠H) @(cpy_subst ⦠HKV HV1) // (**) (* explicit constructor *)
- >yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
- elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 ⦠HLK ⦠HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 ⦠HTU1) -U1
- /3 width=6 by cpy_bind, ldrop_skip, lift_bind, yle_succ, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
- elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 ⦠HLK ⦠HVW1) -W1 //
- elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK //
- /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-(* Basic_1: was: subst1_gen_lift_ge *)
-lemma cpy_inv_lift1_ge: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- yinj d + e ⤠dt â
- ââT2. â¦G, K⦠⢠T1 â¶[dt-e, et] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H elim H -G -L -U1 -U2 -dt -et
-[ * #i #G #L #dt #et #K #s #d #e #_ #T1 #H #_
- [ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
- | lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- ]
-| #I #G #L #KV #V #W #i #dt #et #Hdti #Hidet #HLKV #HVW #K #s #d #e #HLK #T1 #H #Hdedt
- lapply (yle_trans ⦠Hdedt ⦠Hdti) #Hdei
- elim (yle_inv_plus_inj2 ⦠Hdedt) -Hdedt #_ #Hedt
- elim (yle_inv_plus_inj2 ⦠Hdei) #Hdie #Hei
- lapply (lift_inv_lref2_ge ⦠H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
- lapply (ldrop_conf_ge ⦠HLK ⦠HLKV ?) -L /2 width=1 by yle_inv_inj/ #HKV
- elim (lift_split ⦠HVW d (i-e+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hdei -Hdie
- #V0 #HV10 >plus_minus /2 width=1 by yle_inv_inj/ <minus_minus /3 width=1 by yle_inv_inj, le_S/ <minus_n_n <plus_n_O #H
- @(ex2_intro ⦠H) @(cpy_subst ⦠HKV HV10) (**) (* explicit constructor *)
- [ /2 width=1 by monotonic_yle_minus_dx/
- | <yplus_minus_comm_inj /2 width=1 by monotonic_ylt_minus_dx/
- ]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
- elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (yle_inv_plus_inj2 ⦠Hdetd) #_ #Hedt
- elim (IHW12 ⦠HLK ⦠HVW1) -IHW12 // #V2 #HV12 #HVW2
- elim (IHU12 ⦠HTU1) -U1 [4: @ldrop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
- >yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
- elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (IHW12 ⦠HLK ⦠HVW1) -W1 //
- elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
-]
-qed-.
-
-(* Advancd inversion lemmas on relocation ***********************************)
-
-lemma cpy_inv_lift1_ge_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- d ⤠dt â dt ⤠yinj d + e â yinj d + e ⤠dt + et â
- ââT2. â¦G, K⦠⢠T1 â¶[d, dt + et - (yinj d + e)] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (cpy_split_up ⦠HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (cpy_weak ⦠HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
-lapply (cpy_inv_lift1_eq ⦠HTU1 ⦠HU1) -HU1 #HU1 destruct
-elim (cpy_inv_lift1_ge ⦠HU2 ⦠HLK ⦠HTU1) -U -L /2 width=3 by ex2_intro/
-qed-.
-
-lemma cpy_inv_lift1_be_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- dt ⤠d â dt + et ⤠yinj d + e â
- ââT2. â¦G, K⦠⢠T1 â¶[dt, d-dt] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde
-lapply (cpy_weak ⦠HU12 dt (d+e-dt) ? ?) -HU12 //
-[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hdetde #HU12
-elim (cpy_inv_lift1_be ⦠HU12 ⦠HLK ⦠HTU1) -U1 -L /2 width=3 by ex2_intro/
-qed-.
-
-lemma cpy_inv_lift1_le_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- dt ⤠d â d ⤠dt + et â dt + et ⤠yinj d + e â
- ââT2. â¦G, K⦠⢠T1 â¶[dt, d - dt] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde
-elim (cpy_split_up ⦠HU12 d) -HU12 // #U #HU1 #HU2
-elim (cpy_inv_lift1_le ⦠HU1 ⦠HLK ⦠HTU1) -U1
-[2: >ymax_pre_sn_comm // ] -Hdtd #T #HT1 #HTU
-lapply (cpy_weak ⦠HU2 d e ? ?) -HU2 //
-[ >ymax_pre_sn_comm // ] -Hddet -Hdetde #HU2
-lapply (cpy_inv_lift1_eq ⦠HTU ⦠HU2) -L #H destruct /2 width=3 by ex2_intro/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fqu_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqu_lleq.ma
new file mode 100644
index 000000000..b4235ba93
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fqu_lleq.ma
@@ -0,0 +1,45 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/lleq_ldrop.ma".
+include "basic_2/relocation/fqu.ma".
+
+(* SUPCLOSURE ***************************************************************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_fqu_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â â¦G2, K2, U⦠â
+ âL1. L1 â[T, 0] L2 â
+ ââK1. â¦G1, L1, T⦠â â¦G2, K1, U⦠& K1 â[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
+[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx ⦠H ⦠I L2 V) -H //
+ #K1 #H1 #H2 lapply (ldrop_inv_O2 ⦠H1) -H1
+ #H destruct /2 width=3 by fqu_lref_O, ex2_intro/
+| * [ #a ] #I #G #L2 #V #T #L1 #H
+ [ elim (lleq_inv_bind ⦠H)
+ | elim (lleq_inv_flat ⦠H)
+ ] -H
+ /2 width=3 by fqu_pair_sn, ex2_intro/
+| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O ⦠H) -H
+ #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/
+| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat ⦠H) -H
+ /2 width=3 by fqu_flat_dx, ex2_intro/
+| #G #L2 #K2 #T #U #e #HLK2 #HTU #L1 #HL12
+ elim (ldrop_O1_le (e+1) L1)
+ [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
+ | lapply (ldrop_fwd_length_le2 ⦠HLK2) -K2
+ lapply (lleq_fwd_length ⦠HL12) -T -U //
+ ]
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/fquq_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/fquq_lleq.ma
new file mode 100644
index 000000000..880c387e4
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/fquq_lleq.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/fqu_lleq.ma".
+include "basic_2/relocation/fquq_alt.ma".
+
+(* OPTIONAL SUPCLOSURE ******************************************************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_fquq_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â⸮ â¦G2, K2, U⦠â
+ âL1. L1 â[T, 0] L2 â
+ ââK1. â¦G1, L1, T⦠â⸮ â¦G2, K1, U⦠& K1 â[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen ⦠H) -H
+[ #H elim (lleq_fqu_trans ⦠H ⦠HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_leq.ma
index 1355827f7..cdca16bb5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_leq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/ldrop_leq.ma
@@ -12,7 +12,6 @@
(* *)
(**************************************************************************)
-include "ground_2/ynat/ynat_plus.ma".
include "basic_2/grammar/leq_leq.ma".
include "basic_2/relocation/ldrop.ma".
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq.ma
index 7a9ec530e..b431e0d18 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq.ma
@@ -25,6 +25,9 @@ interpretation
"lazy equivalence (local environment)"
'LazyEq T d L1 L2 = (lleq d T L1 L2).
+definition lleq_transitive: predicate (relation3 lenv term term) â
+ λR. âL2,T1,T2. R L2 T1 T2 â âL1. L1 â[T1, 0] L2 â R L1 T1 T2.
+
(* Basic inversion lemmas ***************************************************)
lemma lleq_ind: âR:relation4 ynat term lenv lenv. (
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_ldrop.ma
new file mode 100644
index 000000000..299412217
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_ldrop.ma
@@ -0,0 +1,150 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/llpx_sn_ldrop.ma".
+include "basic_2/relocation/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Advanced properties ******************************************************)
+
+lemma lleq_bind_repl_O: âI,L1,L2,V,T. L1.â{I}V â[T, 0] L2.â{I}V â
+ âJ,W. L1 â[W, 0] L2 â L1.â{J}W â[T, 0] L2.â{J}W.
+/2 width=7 by llpx_sn_bind_repl_O/ qed-.
+
+lemma lleq_dec: âT,L1,L2,d. Decidable (L1 â[T, d] L2).
+/3 width=1 by llpx_sn_dec, eq_term_dec/ qed-.
+
+lemma lleq_llpx_sn_trans: âR. lleq_transitive R â
+ âL1,L2,T,d. L1 â[T, d] L2 â
+ âL. llpx_sn R d T L2 L â llpx_sn R d T L1 L.
+#R #HR #L1 #L2 #T #d #H @(lleq_ind ⦠H) -L1 -L2 -T -d
+[1,2,5: /4 width=6 by llpx_sn_fwd_length, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, trans_eq/
+|4: /4 width=6 by llpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux, trans_eq/
+| #I #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 #HK12 #IHK12 #L #H elim (llpx_sn_inv_lref_ge_sn ⦠H ⦠HLK2) -H -HLK2
+ /3 width=11 by llpx_sn_lref/
+| #a #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_bind ⦠H) -H
+ /3 width=1 by llpx_sn_bind/
+| #I #L1 #L2 #V #T #d #_ #_ #IHV #IHT #L #H elim (llpx_sn_inv_flat ⦠H) -H
+ /3 width=1 by llpx_sn_flat/
+]
+qed-.
+
+lemma lleq_llpx_sn_conf: âR. lleq_transitive R â
+ âL1,L2,T,d. L1 â[T, d] L2 â
+ âL. llpx_sn R d T L1 L â llpx_sn R d T L2 L.
+/3 width=3 by lleq_llpx_sn_trans, lleq_sym/ qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma lleq_inv_lref_ge_dx: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
+ âI,K2,V. â©[i] L2 â¡ K2.â{I}V â
+ ââK1. â©[i] L1 â¡ K1.â{I}V & K1 â[V, 0] K2.
+#L1 #L2 #d #i #H #Hdi #I #K2 #V #HLK2 elim (llpx_sn_inv_lref_ge_dx ⦠H ⦠HLK2) -L2
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lleq_inv_lref_ge_sn: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
+ âI,K1,V. â©[i] L1 â¡ K1.â{I}V â
+ ââK2. â©[i] L2 â¡ K2.â{I}V & K1 â[V, 0] K2.
+#L1 #L2 #d #i #H #Hdi #I1 #K1 #V #HLK1 elim (llpx_sn_inv_lref_ge_sn ⦠H ⦠HLK1) -L1
+/2 width=3 by ex2_intro/
+qed-.
+
+lemma lleq_inv_lref_ge_bi: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
+ âI1,I2,K1,K2,V1,V2.
+ â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â
+ â§â§ I1 = I2 & K1 â[V1, 0] K2 & V1 = V2.
+/2 width=8 by llpx_sn_inv_lref_ge_bi/ qed-.
+
+lemma lleq_inv_lref_ge: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
+ âI,K1,K2,V. â©[i] L1 â¡ K1.â{I}V â â©[i] L2 â¡ K2.â{I}V â
+ K1 â[V, 0] K2.
+#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2
+elim (lleq_inv_lref_ge_bi ⦠HL12 ⦠HLK1 HLK2) //
+qed-.
+
+lemma lleq_inv_S: âL1,L2,T,d. L1 â[T, d+1] L2 â
+ âI,K1,K2,V. â©[d] L1 â¡ K1.â{I}V â â©[d] L2 â¡ K2.â{I}V â
+ K1 â[V, 0] K2 â L1 â[T, d] L2.
+/2 width=9 by llpx_sn_inv_S/ qed-.
+
+lemma lleq_inv_bind_O: âa,I,L1,L2,V,T. L1 â[â{a,I}V.T, 0] L2 â
+ L1 â[V, 0] L2 ⧠L1.â{I}V â[T, 0] L2.â{I}V.
+/2 width=2 by llpx_sn_inv_bind_O/ qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma lleq_fwd_lref_dx: âL1,L2,d,i. L1 â[#i, d] L2 â
+ âI,K2,V. â©[i] L2 â¡ K2.â{I}V â
+ i < d â¨
+ ââK1. â©[i] L1 â¡ K1.â{I}V & K1 â[V, 0] K2 & d ⤠i.
+#L1 #L2 #d #i #H #I #K2 #V #HLK2 elim (llpx_sn_fwd_lref_dx ⦠H ⦠HLK2) -L2
+[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
+qed-.
+
+lemma lleq_fwd_lref_sn: âL1,L2,d,i. L1 â[#i, d] L2 â
+ âI,K1,V. â©[i] L1 â¡ K1.â{I}V â
+ i < d â¨
+ ââK2. â©[i] L2 â¡ K2.â{I}V & K1 â[V, 0] K2 & d ⤠i.
+#L1 #L2 #d #i #H #I #K1 #V #HLK1 elim (llpx_sn_fwd_lref_sn ⦠H ⦠HLK1) -L1
+[ | * ] /3 width=3 by ex3_intro, or_intror, or_introl/
+qed-.
+
+lemma lleq_fwd_bind_O_dx: âa,I,L1,L2,V,T. L1 â[â{a,I}V.T, 0] L2 â
+ L1.â{I}V â[T, 0] L2.â{I}V.
+/2 width=2 by llpx_sn_fwd_bind_O_dx/ qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma lleq_lift_le: âK1,K2,T,dt. K1 â[T, dt] K2 â
+ âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âU. â§[d, e] T â¡ U â dt ⤠d â L1 â[U, dt] L2.
+/3 width=10 by llpx_sn_lift_le, lift_mono/ qed-.
+
+lemma lleq_lift_ge: âK1,K2,T,dt. K1 â[T, dt] K2 â
+ âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âU. â§[d, e] T â¡ U â d ⤠dt â L1 â[U, dt+e] L2.
+/2 width=9 by llpx_sn_lift_ge/ qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma lleq_inv_lift_le: âL1,L2,U,dt. L1 â[U, dt] L2 â
+ âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âT. â§[d, e] T â¡ U â dt ⤠d â K1 â[T, dt] K2.
+/3 width=10 by llpx_sn_inv_lift_le, ex2_intro/ qed-.
+
+lemma lleq_inv_lift_be: âL1,L2,U,dt. L1 â[U, dt] L2 â
+ âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âT. â§[d, e] T â¡ U â d ⤠dt â dt ⤠yinj d + e â K1 â[T, d] K2.
+/2 width=11 by llpx_sn_inv_lift_be/ qed-.
+
+lemma lleq_inv_lift_ge: âL1,L2,U,dt. L1 â[U, dt] L2 â
+ âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âT. â§[d, e] T â¡ U â yinj d + e ⤠dt â K1 â[T, dt-e] K2.
+/2 width=9 by llpx_sn_inv_lift_ge/ qed-.
+
+(* Inversion lemmas on negated lazy quivalence for local environments *******)
+
+lemma nlleq_inv_bind: âa,I,L1,L2,V,T,d. (L1 â[â{a,I}V.T, d] L2 â â¥) â
+ (L1 â[V, d] L2 â â¥) ⨠(L1.â{I}V â[T, ⫯d] L2.â{I}V â â¥).
+/3 width=2 by nllpx_sn_inv_bind, eq_term_dec/ qed-.
+
+lemma nlleq_inv_flat: âI,L1,L2,V,T,d. (L1 â[â{I}V.T, d] L2 â â¥) â
+ (L1 â[V, d] L2 â â¥) ⨠(L1 â[T, d] L2 â â¥).
+/3 width=2 by nllpx_sn_inv_flat, eq_term_dec/ qed-.
+
+lemma nlleq_inv_bind_O: âa,I,L1,L2,V,T. (L1 â[â{a,I}V.T, 0] L2 â â¥) â
+ (L1 â[V, 0] L2 â â¥) ⨠(L1.â{I}V â[T, 0] L2.â{I}V â â¥).
+/3 width=2 by nllpx_sn_inv_bind_O, eq_term_dec/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_lleq.ma
new file mode 100644
index 000000000..a99b1e2f3
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/lleq_lleq.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/lleq_ldrop.ma".
+
+(* Main properties **********************************************************)
+
+theorem lleq_trans: âd,T. Transitive ⦠(lleq d T).
+/2 width=3 by lleq_llpx_sn_trans/ qed-.
+
+theorem lleq_canc_sn: âL,L1,L2,T,d. L â[d, T] L1â L â[d, T] L2 â L1 â[d, T] L2.
+/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+theorem lleq_canc_dx: âL1,L2,L,T,d. L1 â[d, T] L â L2 â[d, T] L â L1 â[d, T] L2.
+/3 width=3 by lleq_trans, lleq_sym/ qed-.
+
+(* Note: lleq_nlleq_trans: âd,T,L1,L. L1â[T, d] L â
+ âL2. (L â[T, d] L2 â â¥) â (L1 â[T, d] L2 â â¥).
+/3 width=3 by lleq_canc_sn/ qed-.
+works with /4 width=8/ so lleq_canc_sn is more convenient
+*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_ldrop.ma
new file mode 100644
index 000000000..4d90e2332
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/relocation/llpx_sn_ldrop.ma
@@ -0,0 +1,410 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/ldrop_ldrop.ma".
+include "basic_2/relocation/llpx_sn_leq.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Advanced forward lemmas **************************************************)
+
+lemma llpx_sn_fwd_lref_dx: âR,L1,L2,d,i. llpx_sn R d (#i) L1 L2 â
+ âI,K2,V2. â©[i] L2 â¡ K2.â{I}V2 â
+ i < d â¨
+ ââK1,V1. â©[i] L1 â¡ K1.â{I}V1 & llpx_sn R 0 V1 K1 K2 &
+ R K1 V1 V2 & d ⤠i.
+#R #L1 #L2 #d #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref ⦠H) -H [ * || * ]
+[ #_ #H elim (lt_refl_false i)
+ lapply (ldrop_fwd_length_lt2 ⦠HLK2) -HLK2
+ /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+| /2 width=1 by or_introl/
+| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi
+ lapply (ldrop_mono ⦠HLK22 ⦠HLK2) -L2 #H destruct
+ /3 width=5 by ex4_2_intro, or_intror/
+]
+qed-.
+
+lemma llpx_sn_fwd_lref_sn: âR,L1,L2,d,i. llpx_sn R d (#i) L1 L2 â
+ âI,K1,V1. â©[i] L1 â¡ K1.â{I}V1 â
+ i < d â¨
+ ââK2,V2. â©[i] L2 â¡ K2.â{I}V2 & llpx_sn R 0 V1 K1 K2 &
+ R K1 V1 V2 & d ⤠i.
+#R #L1 #L2 #d #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref ⦠H) -H [ * || * ]
+[ #H #_ elim (lt_refl_false i)
+ lapply (ldrop_fwd_length_lt2 ⦠HLK1) -HLK1
+ /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
+| /2 width=1 by or_introl/
+| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hdi
+ lapply (ldrop_mono ⦠HLK11 ⦠HLK1) -L1 #H destruct
+ /3 width=5 by ex4_2_intro, or_intror/
+]
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+lemma llpx_sn_inv_lref_ge_dx: âR,L1,L2,d,i. llpx_sn R d (#i) L1 L2 â d ⤠i â
+ âI,K2,V2. â©[i] L2 â¡ K2.â{I}V2 â
+ ââK1,V1. â©[i] L1 â¡ K1.â{I}V1 &
+ llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #d #i #H #Hdi #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx ⦠H ⦠HLK2) -L2
+[ #H elim (ylt_yle_false ⦠H Hdi)
+| * /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma llpx_sn_inv_lref_ge_sn: âR,L1,L2,d,i. llpx_sn R d (#i) L1 L2 â d ⤠i â
+ âI,K1,V1. â©[i] L1 â¡ K1.â{I}V1 â
+ ââK2,V2. â©[i] L2 â¡ K2.â{I}V2 &
+ llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #d #i #H #Hdi #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn ⦠H ⦠HLK1) -L1
+[ #H elim (ylt_yle_false ⦠H Hdi)
+| * /2 width=5 by ex3_2_intro/
+]
+qed-.
+
+lemma llpx_sn_inv_lref_ge_bi: âR,L1,L2,d,i. llpx_sn R d (#i) L1 L2 â d ⤠i â
+ âI1,I2,K1,K2,V1,V2.
+ â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â
+ â§â§ I1 = I2 & llpx_sn R 0 V1 K1 K2 & R K1 V1 V2.
+#R #L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
+elim (llpx_sn_inv_lref_ge_sn ⦠HL12 ⦠HLK1) // -L1 -d
+#J #Y #HY lapply (ldrop_mono ⦠HY ⦠HLK2) -L2 -i #H destruct /2 width=1 by and3_intro/
+qed-.
+
+fact llpx_sn_inv_S_aux: âR,L1,L2,T,d0. llpx_sn R d0 T L1 L2 â âd. d0 = d + 1 â
+ âK1,K2,I,V1,V2. â©[d] L1 â¡ K1.â{I}V1 â â©[d] L2 â¡ K2.â{I}V2 â
+ llpx_sn R 0 V1 K1 K2 â R K1 V1 V2 â llpx_sn R d T L1 L2.
+#R #L1 #L2 #T #d0 #H elim H -L1 -L2 -T -d0
+/2 width=1 by llpx_sn_gref, llpx_sn_free, llpx_sn_sort/
+[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct
+ elim (yle_split_eq i d) /2 width=1 by llpx_sn_skip, ylt_fwd_succ2/ -HL12 -Hid
+ #H destruct /2 width=9 by llpx_sn_lref/
+| #I #L1 #L2 #K11 #K22 #V1 #V2 #d0 #i #Hd0i #HLK11 #HLK22 #HK12 #HV12 #_ #d #H #K1 #K2 #J #W1 #W2 #_ #_ #_ #_ destruct
+ /3 width=9 by llpx_sn_lref, yle_pred_sn/
+| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
+ /4 width=9 by llpx_sn_bind, ldrop_drop/
+| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
+ /3 width=9 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_S: âR,L1,L2,T,d. llpx_sn R (d + 1) T L1 L2 â
+ âK1,K2,I,V1,V2. â©[d] L1 â¡ K1.â{I}V1 â â©[d] L2 â¡ K2.â{I}V2 â
+ llpx_sn R 0 V1 K1 K2 â R K1 V1 V2 â llpx_sn R d T L1 L2.
+/2 width=9 by llpx_sn_inv_S_aux/ qed-.
+
+lemma llpx_sn_inv_bind_O: âR. (âL. reflexive ⦠(R L)) â
+ âa,I,L1,L2,V,T. llpx_sn R 0 (â{a,I}V.T) L1 L2 â
+ llpx_sn R 0 V L1 L2 ⧠llpx_sn R 0 T (L1.â{I}V) (L2.â{I}V).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind ⦠H) -H
+/3 width=9 by ldrop_pair, conj, llpx_sn_inv_S/
+qed-.
+
+(* More advanced forward lemmas *********************************************)
+
+lemma llpx_sn_fwd_bind_O_dx: âR. (âL. reflexive ⦠(R L)) â
+ âa,I,L1,L2,V,T. llpx_sn R 0 (â{a,I}V.T) L1 L2 â
+ llpx_sn R 0 T (L1.â{I}V) (L2.â{I}V).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_inv_bind_O ⦠H) -H //
+qed-.
+
+(* Advanced properties ******************************************************)
+
+lemma llpx_sn_bind_repl_O: âR,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.â{I}V1) (L2.â{I}V2) â
+ âJ,W1,W2. llpx_sn R 0 W1 L1 L2 â R L1 W1 W2 â llpx_sn R 0 T (L1.â{J}W1) (L2.â{J}W2).
+/3 width=9 by llpx_sn_bind_repl_SO, llpx_sn_inv_S/ qed-.
+
+lemma llpx_sn_dec: âR. (âL,T1,T2. Decidable (R L T1 T2)) â
+ âT,L1,L2,d. Decidable (llpx_sn R d T L1 L2).
+#R #HR #T #L1 @(f2_ind ⦠rfw ⦠L1 T) -L1 -T
+#n #IH #L1 * *
+[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_sort/
+| #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|))
+ [ #HL12 #d elim (ylt_split i d) /3 width=1 by llpx_sn_skip, or_introl/
+ #Hdi elim (lt_or_ge i (|L1|)) #HiL1
+ elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, llpx_sn_free/
+ elim (ldrop_O1_lt ⦠HiL2) #I2 #K2 #V2 #HLK2
+ elim (ldrop_O1_lt ⦠HiL1) #I1 #K1 #V1 #HLK1
+ elim (eq_bind2_dec I2 I1)
+ [ #H2 elim (HR K1 V1 V2) -HR
+ [ #H3 elim (IH K1 V1 ⦠K2 0) destruct
+ /3 width=9 by llpx_sn_lref, ldrop_fwd_rfw, or_introl/
+ ]
+ ]
+ -IH #H3 @or_intror
+ #H elim (llpx_sn_fwd_lref ⦠H) -H [1,3,4,6,7,9: * ]
+ [1,3,5: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
+ |7,8,9: /2 width=4 by ylt_yle_false/
+ ]
+ #Z #Y1 #Y2 #X1 #X2 #HLY1 #HLY2 #HY12 #HX12
+ lapply (ldrop_mono ⦠HLY1 ⦠HLK1) -HLY1 -HLK1
+ lapply (ldrop_mono ⦠HLY2 ⦠HLK2) -HLY2 -HLK2
+ #H #H0 destruct /2 width=1 by/
+ ]
+| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, llpx_sn_gref/
+| #a #I #V #T #Hn #L2 #d destruct
+ elim (IH L1 V ⦠L2 d) /2 width=1 by/
+ elim (IH (L1.â{I}V) T ⦠(L2.â{I}V) (⫯d)) -IH /3 width=1 by or_introl, llpx_sn_bind/
+ #H1 #H2 @or_intror
+ #H elim (llpx_sn_inv_bind ⦠H) -H /2 width=1 by/
+| #I #V #T #Hn #L2 #d destruct
+ elim (IH L1 V ⦠L2 d) /2 width=1 by/
+ elim (IH L1 T ⦠L2 d) -IH /3 width=1 by or_introl, llpx_sn_flat/
+ #H1 #H2 @or_intror
+ #H elim (llpx_sn_inv_flat ⦠H) -H /2 width=1 by/
+]
+-n /4 width=4 by llpx_sn_fwd_length, or_intror/
+qed-.
+
+(* Properties on relocation *************************************************)
+
+lemma llpx_sn_lift_le: âR. l_liftable R â
+ âK1,K2,T,d0. llpx_sn R d0 T K1 K2 â
+ âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âU. â§[d, e] T â¡ U â d0 ⤠d â llpx_sn R d0 U L1 L2.
+#R #HR #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0
+[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 ⦠H) -X
+ lapply (ldrop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_sort/
+| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 ⦠H) -H
+ * #Hdi #H destruct
+ [ lapply (ldrop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_skip/
+ | elim (ylt_yle_false ⦠Hid0) -L1 -L2 -K1 -K2 -e -Hid0
+ /3 width=3 by yle_trans, yle_inj/
+ ]
+| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 ⦠H) -H
+ * #Hdi #H destruct [ -HK12 | -IHK12 ]
+ [ elim (ldrop_trans_lt ⦠HLK1 ⦠HK11) // -K1
+ elim (ldrop_trans_lt ⦠HLK2 ⦠HK22) // -Hdi -K2
+ /3 width=18 by llpx_sn_lref/
+ | lapply (ldrop_trans_ge_comm ⦠HLK1 ⦠HK11 ?) // -K1
+ lapply (ldrop_trans_ge_comm ⦠HLK2 ⦠HK22 ?) // -Hdi -Hd0 -K2
+ /3 width=9 by llpx_sn_lref, yle_plus_dx1_trans/
+ ]
+| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 ⦠H) -H
+ * #Hid #H destruct
+ lapply (ldrop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -HK12
+ [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/
+ | lapply (ldrop_fwd_length ⦠HLK1) -HLK1 #HLK1
+ lapply (ldrop_fwd_length ⦠HLK2) -HLK2 #HLK2
+ @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
+ ]
+| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 ⦠H) -X
+ lapply (ldrop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -d -e
+ /2 width=1 by llpx_sn_gref/
+| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 ⦠H) -H
+ #W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, ldrop_skip, yle_succ/
+| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 ⦠H) -H
+ #W #U #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_lift_ge: âR,K1,K2,T,d0. llpx_sn R d0 T K1 K2 â
+ âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âU. â§[d, e] T â¡ U â d ⤠d0 â llpx_sn R (d0+e) U L1 L2.
+#R #K1 #K2 #T #d0 #H elim H -K1 -K2 -T -d0
+[ #K1 #K2 #d0 #k #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 ⦠H) -X
+ lapply (ldrop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_sort/
+| #K1 #K2 #d0 #i #HK12 #Hid0 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 ⦠H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2
+ [ /3 width=3 by llpx_sn_skip, ylt_plus_dx2_trans/
+ | /3 width=3 by llpx_sn_skip, monotonic_ylt_plus_dx/
+ ]
+| #I #K1 #K2 #K11 #K22 #V1 #V2 #d0 #i #Hid0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 ⦠H) -H
+ * #Hid #H destruct
+ [ elim (ylt_yle_false ⦠Hid0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -e -Hid0
+ /3 width=3 by ylt_yle_trans, ylt_inj/
+ | lapply (ldrop_trans_ge_comm ⦠HLK1 ⦠HK11 ?) // -K1
+ lapply (ldrop_trans_ge_comm ⦠HLK2 ⦠HK22 ?) // -Hid -Hd0 -K2
+ /3 width=9 by llpx_sn_lref, monotonic_yle_plus_dx/
+ ]
+| #K1 #K2 #d0 #i #HK1 #HK2 #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref1 ⦠H) -H
+ * #Hid #H destruct
+ lapply (ldrop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -HK12
+ [ /3 width=7 by llpx_sn_free, ldrop_fwd_be/
+ | lapply (ldrop_fwd_length ⦠HLK1) -HLK1 #HLK1
+ lapply (ldrop_fwd_length ⦠HLK2) -HLK2 #HLK2
+ @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
+ ]
+| #K1 #K2 #d0 #p #HK12 #L1 #L2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 ⦠H) -X
+ lapply (ldrop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -d
+ /2 width=1 by llpx_sn_gref/
+| #a #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind1 ⦠H) -H
+ #W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, ldrop_skip, yle_succ/
+| #I #K1 #K2 #V #T #d0 #_ #_ #IHV #IHT #L1 #L2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat1 ⦠H) -H
+ #W #U #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
+]
+qed-.
+
+(* Inversion lemmas on relocation *******************************************)
+
+lemma llpx_sn_inv_lift_le: âR. l_deliftable_sn R â
+ âL1,L2,U,d0. llpx_sn R d0 U L1 L2 â
+ âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âT. â§[d, e] T â¡ U â d0 ⤠d â llpx_sn R d0 T K1 K2.
+#R #HR #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
+[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 ⦠H) -X
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d -e
+ /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 ⦠H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2
+ [ /2 width=1 by llpx_sn_skip/
+ | /3 width=3 by llpx_sn_skip, yle_ylt_trans/
+ ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 ⦠H) -H
+ * #Hid #H destruct [ -HK12 | -IHK12 ]
+ [ elim (ldrop_conf_lt ⦠HLK1 ⦠HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1
+ elim (ldrop_conf_lt ⦠HLK2 ⦠HLK22) // -Hid -L2 #L2 #V2 #HKL2 #HKL22 #HVW2
+ elim (HR ⦠HW12 ⦠HKL11 ⦠HVW1) -HR #V0 #HV0 #HV12
+ lapply (lift_inj ⦠HV0 ⦠HVW2) -HV0 -HVW2 #H destruct
+ /3 width=10 by llpx_sn_lref/
+ | lapply (ldrop_conf_ge ⦠HLK1 ⦠HLK11 ?) // -L1
+ lapply (ldrop_conf_ge ⦠HLK2 ⦠HLK22 ?) // -L2 -Hid0
+ elim (le_inv_plus_l ⦠Hid) -Hid /4 width=9 by llpx_sn_lref, yle_trans, yle_inj/ (**) (* slow *)
+ ]
+| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_lref2 ⦠H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12)
+ [ lapply (ldrop_fwd_length_le4 ⦠HLK1) -HLK1
+ lapply (ldrop_fwd_length_le4 ⦠HLK2) -HLK2
+ #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+ | lapply (ldrop_fwd_length ⦠HLK1) -HLK1 #H >H in HL1; -H
+ lapply (ldrop_fwd_length ⦠HLK2) -HLK2 #H >H in HL2; -H
+ /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+ ]
+| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 ⦠H) -X
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d -e
+ /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_bind2 ⦠H) -H
+ #V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, ldrop_skip, yle_succ/
+| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 elim (lift_inv_flat2 ⦠H) -H
+ #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_lift_be: âR,L1,L2,U,d0. llpx_sn R d0 U L1 L2 â
+ âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âT. â§[d, e] T â¡ U â d ⤠d0 â d0 ⤠yinj d + e â llpx_sn R d T K1 K2.
+#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
+[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 ⦠H) -X
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d0 -e
+ /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 ⦠H) -H
+ * #Hid #H destruct
+ [ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2
+ -Hid0 /3 width=1 by llpx_sn_skip, ylt_inj/
+ | elim (ylt_yle_false ⦠Hid0) -L1 -L2 -Hd0 -Hid0
+ /3 width=3 by yle_trans, yle_inj/ (**) (* slow *)
+ ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 ⦠H) -H
+ * #Hid #H destruct
+ [ elim (ylt_yle_false ⦠Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hd0e -Hid0
+ /3 width=3 by ylt_yle_trans, ylt_inj/
+ | lapply (ldrop_conf_ge ⦠HLK1 ⦠HLK11 ?) // -L1
+ lapply (ldrop_conf_ge ⦠HLK2 ⦠HLK22 ?) // -L2 -Hid0 -Hd0 -Hd0e
+ elim (le_inv_plus_l ⦠Hid) -Hid /3 width=9 by llpx_sn_lref, yle_inj/
+ ]
+| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_lref2 ⦠H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12)
+ [ lapply (ldrop_fwd_length_le4 ⦠HLK1) -HLK1
+ lapply (ldrop_fwd_length_le4 ⦠HLK2) -HLK2
+ #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+ | lapply (ldrop_fwd_length ⦠HLK1) -HLK1 #H >H in HL1; -H
+ lapply (ldrop_fwd_length ⦠HLK2) -HLK2 #H >H in HL2; -H
+ /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+ ]
+| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 ⦠H) -X
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d0 -e
+ /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_bind2 ⦠H) -H
+ >commutative_plus #V #T #HVW #HTU #H destruct
+ @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
+ @(IHU ⦠HTU) -IHU -HTU /2 width=1 by ldrop_skip, yle_succ/
+| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hd0 #Hd0e elim (lift_inv_flat2 ⦠H) -H
+ #V #T #HVW #HTU #H destruct /3 width=6 by llpx_sn_flat/
+]
+qed-.
+
+lemma llpx_sn_inv_lift_ge: âR,L1,L2,U,d0. llpx_sn R d0 U L1 L2 â
+ âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
+ âT. â§[d, e] T â¡ U â yinj d + e ⤠d0 â llpx_sn R (d0-e) T K1 K2.
+#R #L1 #L2 #U #d0 #H elim H -L1 -L2 -U -d0
+[ #L1 #L2 #d0 #k #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 ⦠H) -X
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d
+ /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #d0 #i #HL12 #Hid0 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 ⦠H) -H
+ * #Hid #H destruct [ -Hid0 | -Hded0 ]
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2
+ [ /4 width=3 by llpx_sn_skip, yle_plus_to_minus_inj2, ylt_yle_trans, ylt_inj/
+ | elim (le_inv_plus_l ⦠Hid) -Hid #_
+ /4 width=1 by llpx_sn_skip, monotonic_ylt_minus_dx, yle_inj/
+ ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #d0 #i #Hid0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 ⦠H) -H
+ * #Hid #H destruct
+ [ elim (ylt_yle_false ⦠Hid0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hid0
+ /3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/
+ | lapply (ldrop_conf_ge ⦠HLK1 ⦠HLK11 ?) // -L1
+ lapply (ldrop_conf_ge ⦠HLK2 ⦠HLK22 ?) // -L2 -Hded0 -Hid
+ /3 width=9 by llpx_sn_lref, monotonic_yle_minus_dx/
+ ]
+| #L1 #L2 #d0 #i #HL1 #HL2 #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_lref2 ⦠H) -H
+ * #_ #H destruct
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12)
+ [ lapply (ldrop_fwd_length_le4 ⦠HLK1) -HLK1
+ lapply (ldrop_fwd_length_le4 ⦠HLK2) -HLK2
+ #HKL2 #HKL1 #HK12 @llpx_sn_free // /2 width=3 by transitive_le/ (**) (* full auto too slow *)
+ | lapply (ldrop_fwd_length ⦠HLK1) -HLK1 #H >H in HL1; -H
+ lapply (ldrop_fwd_length ⦠HLK2) -HLK2 #H >H in HL2; -H
+ /3 width=1 by llpx_sn_free, le_plus_to_minus_r/
+ ]
+| #L1 #L2 #d0 #p #HL12 #K1 #K2 #d #e #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 ⦠H) -X
+ lapply (ldrop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d
+ /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_bind2 ⦠H) -H
+ #V #T #HVW #HTU #H destruct
+ @llpx_sn_bind [ /2 width=5 by/ ] -IHW (**) (* explicit constructor *)
+ <yminus_succ1_inj /2 width=2 by yle_fwd_plus_sn2/
+ @(IHU ⦠HTU) -IHU -HTU /2 width=1 by ldrop_skip, yle_succ/
+| #I #L1 #L2 #W #U #d0 #_ #_ #IHW #IHU #K1 #K2 #d #e #HLK1 #HLK2 #X #H #Hded0 elim (lift_inv_flat2 ⦠H) -H
+ #V #T #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
+]
+qed-.
+
+(* Inversion lemmas on negated lazy pointwise extension *********************)
+
+lemma nllpx_sn_inv_bind: âR. (âL,T1,T2. Decidable (R L T1 T2)) â
+ âa,I,L1,L2,V,T,d. (llpx_sn R d (â{a,I}V.T) L1 L2 â â¥) â
+ (llpx_sn R d V L1 L2 â â¥) ⨠(llpx_sn R (⫯d) T (L1.â{I}V) (L2.â{I}V) â â¥).
+#R #HR #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec ⦠HR V L1 L2 d)
+/4 width=1 by llpx_sn_bind, or_intror, or_introl/
+qed-.
+
+lemma nllpx_sn_inv_flat: âR. (âL,T1,T2. Decidable (R L T1 T2)) â
+ âI,L1,L2,V,T,d. (llpx_sn R d (â{I}V.T) L1 L2 â â¥) â
+ (llpx_sn R d V L1 L2 â â¥) ⨠(llpx_sn R d T L1 L2 â â¥).
+#R #HR #I #L1 #L2 #V #T #d #H elim (llpx_sn_dec ⦠HR V L1 L2 d)
+/4 width=1 by llpx_sn_flat, or_intror, or_introl/
+qed-.
+
+lemma nllpx_sn_inv_bind_O: âR. (âL,T1,T2. Decidable (R L T1 T2)) â
+ âa,I,L1,L2,V,T. (llpx_sn R 0 (â{a,I}V.T) L1 L2 â â¥) â
+ (llpx_sn R 0 V L1 L2 â â¥) ⨠(llpx_sn R 0 T (L1.â{I}V) (L2.â{I}V) â â¥).
+#R #HR #a #I #L1 #L2 #V #T #H elim (llpx_sn_dec ⦠HR V L1 L2 0)
+/4 width=1 by llpx_sn_bind_O, or_intror, or_introl/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma
deleted file mode 100644
index 6c90b202b..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby.ma
+++ /dev/null
@@ -1,237 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "ground_2/ynat/ynat_plus.ma".
-include "basic_2/notation/relations/extlrsubeq_4.ma".
-include "basic_2/relocation/ldrop.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-
-inductive lsuby: relation4 ynat ynat lenv lenv â
-| lsuby_atom: âL,d,e. lsuby d e L (â)
-| lsuby_zero: âI1,I2,L1,L2,V1,V2.
- lsuby 0 0 L1 L2 â lsuby 0 0 (L1.â{I1}V1) (L2.â{I2}V2)
-| lsuby_pair: âI1,I2,L1,L2,V,e. lsuby 0 e L1 L2 â
- lsuby 0 (⫯e) (L1.â{I1}V) (L2.â{I2}V)
-| lsuby_succ: âI1,I2,L1,L2,V1,V2,d,e.
- lsuby d e L1 L2 â lsuby (⫯d) e (L1. â{I1}V1) (L2. â{I2} V2)
-.
-
-interpretation
- "local environment refinement (extended substitution)"
- 'ExtLRSubEq L1 d e L2 = (lsuby d e L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_pair_lt: âI1,I2,L1,L2,V,e. L1 âÃ[0, â«°e] L2 â 0 < e â
- L1.â{I1}V âÃ[0, e] L2.â{I2}V.
-#I1 #I2 #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 ⦠He) /2 width=1 by lsuby_pair/
-qed.
-
-lemma lsuby_succ_lt: âI1,I2,L1,L2,V1,V2,d,e. L1 âÃ[â«°d, e] L2 â 0 < d â
- L1.â{I1}V1 âÃ[d, e] L2. â{I2}V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd <(ylt_inv_O1 ⦠Hd) /2 width=1 by lsuby_succ/
-qed.
-
-lemma lsuby_pair_O_Y: âL1,L2. L1 âÃ[0, â] L2 â
- âI1,I2,V. L1.â{I1}V âÃ[0,â] L2.â{I2}V.
-#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 ⦠V ⦠HL12) -HL12 //
-qed.
-
-lemma lsuby_refl: âL,d,e. L âÃ[d, e] L.
-#L elim L -L //
-#L #I #V #IHL #d elim (ynat_cases ⦠d) [| * #x ]
-#Hd destruct /2 width=1 by lsuby_succ/
-#e elim (ynat_cases ⦠e) [| * #x ]
-#He destruct /2 width=1 by lsuby_zero, lsuby_pair/
-qed.
-
-lemma lsuby_O2: âL2,L1,d. |L2| ⤠|L1| â L1 âÃ[d, yinj 0] L2.
-#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
-[ #d #H lapply (le_n_O_to_eq ⦠H) -H <plus_n_Sm #H destruct
-| #L1 #I1 #V1 #d #H lapply (le_plus_to_le_r ⦠H) -H #HL12
- elim (ynat_cases d) /3 width=1 by lsuby_zero/
- * /3 width=1 by lsuby_succ/
-]
-qed.
-
-lemma lsuby_sym: âd,e,L1,L2. L1 âÃ[d, e] L2 â |L1| = |L2| â L2 âÃ[d, e] L1.
-#d #e #L1 #L2 #H elim H -d -e -L1 -L2
-[ #L1 #d #e #H >(length_inv_zero_dx ⦠H) -L1 //
-| /2 width=1 by lsuby_O2/
-| #I1 #I2 #L1 #L2 #V #e #_ #IHL12 #H lapply (injective_plus_l ⦠H)
- /3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #H lapply (injective_plus_l ⦠H)
- /3 width=1 by lsuby_succ/
-]
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact lsuby_inv_atom1_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â L1 = â â L2 = â.
-#L1 #L2 #d #e * -L1 -L2 -d -e //
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
-| #I1 #I2 #L1 #L2 #V #e #_ #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #H destruct
-]
-qed-.
-
-lemma lsuby_inv_atom1: âL2,d,e. â âÃ[d, e] L2 â L2 = â.
-/2 width=5 by lsuby_inv_atom1_aux/ qed-.
-
-fact lsuby_inv_zero1_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
- âJ1,K1,W1. L1 = K1.â{J1}W1 â d = 0 â e = 0 â
- L2 = â â¨
- ââJ2,K2,W2. K1 âÃ[0, 0] K2 & L2 = K2.â{J2}W2.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
- /3 width=5 by ex2_3_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_ #H
- elim (ysucc_inv_O_dx ⦠H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_ #H
- elim (ysucc_inv_O_dx ⦠H)
-]
-qed-.
-
-lemma lsuby_inv_zero1: âI1,K1,L2,V1. K1.â{I1}V1 âÃ[0, 0] L2 â
- L2 = â â¨
- ââI2,K2,V2. K1 âÃ[0, 0] K2 & L2 = K2.â{I2}V2.
-/2 width=9 by lsuby_inv_zero1_aux/ qed-.
-
-fact lsuby_inv_pair1_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
- âJ1,K1,W. L1 = K1.â{J1}W â d = 0 â 0 < e â
- L2 = â â¨
- ââJ2,K2. K1 âÃ[0, â«°e] K2 & L2 = K2.â{J2}W.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
- elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #J1 #K1 #W #H #_ #_ destruct
- /3 width=4 by ex2_2_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_ #H
- elim (ysucc_inv_O_dx ⦠H)
-]
-qed-.
-
-lemma lsuby_inv_pair1: âI1,K1,L2,V,e. K1.â{I1}V âÃ[0, e] L2 â 0 < e â
- L2 = â â¨
- ââI2,K2. K1 âÃ[0, â«°e] K2 & L2 = K2.â{I2}V.
-/2 width=6 by lsuby_inv_pair1_aux/ qed-.
-
-fact lsuby_inv_succ1_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
- âJ1,K1,W1. L1 = K1.â{J1}W1 â 0 < d â
- L2 = â â¨
- ââJ2,K2,W2. K1 âÃ[â«°d, e] K2 & L2 = K2.â{J2}W2.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
- /3 width=5 by ex2_3_intro, or_intror/
-]
-qed-.
-
-lemma lsuby_inv_succ1: âI1,K1,L2,V1,d,e. K1.â{I1}V1 âÃ[d, e] L2 â 0 < d â
- L2 = â â¨
- ââI2,K2,V2. K1 âÃ[â«°d, e] K2 & L2 = K2.â{I2}V2.
-/2 width=5 by lsuby_inv_succ1_aux/ qed-.
-
-fact lsuby_inv_zero2_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
- âJ2,K2,W2. L2 = K2.â{J2}W2 â d = 0 â e = 0 â
- ââJ1,K1,W1. K1 âÃ[0, 0] K2 & L1 = K1.â{J1}W1.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W1 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
- /2 width=5 by ex2_3_intro/
-| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_ #H
- elim (ysucc_inv_O_dx ⦠H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_ #H
- elim (ysucc_inv_O_dx ⦠H)
-]
-qed-.
-
-lemma lsuby_inv_zero2: âI2,K2,L1,V2. L1 âÃ[0, 0] K2.â{I2}V2 â
- ââI1,K1,V1. K1 âÃ[0, 0] K2 & L1 = K1.â{I1}V1.
-/2 width=9 by lsuby_inv_zero2_aux/ qed-.
-
-fact lsuby_inv_pair2_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
- âJ2,K2,W. L2 = K2.â{J2}W â d = 0 â 0 < e â
- ââJ1,K1. K1 âÃ[0, â«°e] K2 & L1 = K1.â{J1}W.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
- elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #J2 #K2 #W #H #_ #_ destruct
- /2 width=4 by ex2_2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_ #H
- elim (ysucc_inv_O_dx ⦠H)
-]
-qed-.
-
-lemma lsuby_inv_pair2: âI2,K2,L1,V,e. L1 âÃ[0, e] K2.â{I2}V â 0 < e â
- ââI1,K1. K1 âÃ[0, â«°e] K2 & L1 = K1.â{I1}V.
-/2 width=6 by lsuby_inv_pair2_aux/ qed-.
-
-fact lsuby_inv_succ2_aux: âL1,L2,d,e. L1 âÃ[d, e] L2 â
- âJ2,K2,W2. L2 = K2.â{J2}W2 â 0 < d â
- ââJ1,K1,W1. K1 âÃ[â«°d, e] K2 & L1 = K1.â{J1}W1.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W2 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
- elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K1 #W2 #_ #H
- elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J2 #K2 #W2 #H #_ destruct
- /2 width=5 by ex2_3_intro/
-]
-qed-.
-
-lemma lsuby_inv_succ2: âI2,K2,L1,V2,d,e. L1 âÃ[d, e] K2.â{I2}V2 â 0 < d â
- ââI1,K1,V1. K1 âÃ[â«°d, e] K2 & L1 = K1.â{I1}V1.
-/2 width=5 by lsuby_inv_succ2_aux/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lsuby_fwd_length: âL1,L2,d,e. L1 âÃ[d, e] L2 â |L2| ⤠|L1|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize /2 width=1 by le_S_S/
-qed-.
-
-(* Properties on basic slicing **********************************************)
-
-lemma lsuby_ldrop_trans_be: âL1,L2,d,e. L1 âÃ[d, e] L2 â
- âI2,K2,W,s,i. â©[s, 0, i] L2 â¡ K2.â{I2}W â
- d ⤠i â i < d + e â
- ââI1,K1. K1 âÃ[0, â«°(d+e-i)] K2 & â©[s, 0, i] L1 â¡ K1.â{I1}W.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W #s #i #H
- elim (ldrop_inv_atom1 ⦠H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
- elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
- elim (ldrop_inv_O1_pair1 ⦠H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
- [ #_ destruct -I2 >ypred_succ
- /2 width=4 by ldrop_pair, ex2_2_intro/
- | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
- #H <H -H #H lapply (ylt_inv_succ ⦠H) -H
- #Hie elim (IHL12 ⦠HLK1) -IHL12 -HLK1 // -Hie
- >yminus_succ <yminus_inj /3 width=4 by ldrop_drop_lt, ex2_2_intro/
- ]
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hdi
- elim (yle_inv_succ1 ⦠Hdi) -Hdi
- #Hdi #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ ⦠H) -H
- #Hide lapply (ldrop_inv_drop1_lt ⦠HLK2 ?) -HLK2 /2 width=1 by ylt_O/
- #HLK1 elim (IHL12 ⦠HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
- /4 width=4 by ylt_O, ldrop_drop_lt, ex2_2_intro/
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby_lsuby.ma b/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby_lsuby.ma
deleted file mode 100644
index 24361d3c0..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/relocation/lsuby_lsuby.ma
+++ /dev/null
@@ -1,32 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| 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/lsuby.ma".
-
-(* LOCAL ENVIRONMENT REFINEMENT FOR EXTENDED SUBSTITUTION *******************)
-
-(* Main properties **********************************************************)
-
-theorem lsuby_trans: âd,e. Transitive ⦠(lsuby d e).
-#d #e #L1 #L2 #H elim H -L1 -L2 -d -e
-[ #L1 #d #e #X #H lapply (lsuby_inv_atom1 ⦠H) -H
- #H destruct //
-| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 ⦠H) -H //
- * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_zero/
-| #I1 #I2 #L1 #L2 #V #e #_ #IHL1 #X #H elim (lsuby_inv_pair1 ⦠H) -H //
- * #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL1 #X #H elim (lsuby_inv_succ1 ⦠H) -H //
- * #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma
deleted file mode 100644
index 10c59120d..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys.ma
+++ /dev/null
@@ -1,166 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| 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/psubststar_6.ma".
-include "basic_2/relocation/cpy.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-definition cpys: ynat â ynat â relation4 genv lenv term term â
- λd,e,G. LTC ⦠(cpy d e G).
-
-interpretation "context-sensitive extended multiple substritution (term)"
- 'PSubstStar G L T1 d e T2 = (cpys d e G L T1 T2).
-
-(* Basic eliminators ********************************************************)
-
-lemma cpys_ind: âG,L,T1,d,e. âR:predicate term. R T1 â
- (âT,T2. â¦G, L⦠⢠T1 â¶*[d, e] T â â¦G, L⦠⢠T â¶[d, e] T2 â R T â R T2) â
- âT2. â¦G, L⦠⢠T1 â¶*[d, e] T2 â R T2.
-#G #L #T1 #d #e #R #HT1 #IHT1 #T2 #HT12
-@(TC_star_ind ⦠HT1 IHT1 ⦠HT12) //
-qed-.
-
-lemma cpys_ind_dx: âG,L,T2,d,e. âR:predicate term. R T2 â
- (âT1,T. â¦G, L⦠⢠T1 â¶[d, e] T â â¦G, L⦠⢠T â¶*[d, e] T2 â R T â R T1) â
- âT1. â¦G, L⦠⢠T1 â¶*[d, e] T2 â R T1.
-#G #L #T2 #d #e #R #HT2 #IHT2 #T1 #HT12
-@(TC_star_ind_dx ⦠HT2 IHT2 ⦠HT12) //
-qed-.
-
-(* Basic properties *********************************************************)
-
-lemma cpy_cpys: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â â¦G, L⦠⢠T1 â¶*[d, e] T2.
-/2 width=1 by inj/ qed.
-
-lemma cpys_strap1: âG,L,T1,T,T2,d,e.
- â¦G, L⦠⢠T1 â¶*[d, e] T â â¦G, L⦠⢠T â¶[d, e] T2 â â¦G, L⦠⢠T1 â¶*[d, e] T2.
-normalize /2 width=3 by step/ qed-.
-
-lemma cpys_strap2: âG,L,T1,T,T2,d,e.
- â¦G, L⦠⢠T1 â¶[d, e] T â â¦G, L⦠⢠T â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¶*[d, e] T2.
-normalize /2 width=3 by TC_strap/ qed-.
-
-lemma lsuby_cpys_trans: âG,d,e. lsub_trans ⦠(cpys d e G) (lsuby d e).
-/3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/
-qed-.
-
-lemma cpys_refl: âG,L,d,e. reflexive ⦠(cpys d e G L).
-/2 width=1 by cpy_cpys/ qed.
-
-lemma cpys_bind: âG,L,V1,V2,d,e. â¦G, L⦠⢠V1 â¶*[d, e] V2 â
- âI,T1,T2. â¦G, L.â{I}V1⦠⢠T1 â¶*[⫯d, e] T2 â
- âa. â¦G, L⦠⢠â{a,I}V1.T1 â¶*[d, e] â{a,I}V2.T2.
-#G #L #V1 #V2 #d #e #HV12 @(cpys_ind ⦠HV12) -V2
-[ #I #T1 #T2 #HT12 @(cpys_ind ⦠HT12) -T2 /3 width=5 by cpys_strap1, cpy_bind/
-| /3 width=5 by cpys_strap1, cpy_bind/
-]
-qed.
-
-lemma cpys_flat: âG,L,V1,V2,d,e. â¦G, L⦠⢠V1 â¶*[d, e] V2 â
- âT1,T2. â¦G, L⦠⢠T1 â¶*[d, e] T2 â
- âI. â¦G, L⦠⢠â{I}V1.T1 â¶*[d, e] â{I}V2.T2.
-#G #L #V1 #V2 #d #e #HV12 @(cpys_ind ⦠HV12) -V2
-[ #T1 #T2 #HT12 @(cpys_ind ⦠HT12) -T2 /3 width=5 by cpys_strap1, cpy_flat/
-| /3 width=5 by cpys_strap1, cpy_flat/
-qed.
-
-lemma cpys_weak: âG,L,T1,T2,d1,e1. â¦G, L⦠⢠T1 â¶*[d1, e1] T2 â
- âd2,e2. d2 ⤠d1 â d1 + e1 ⤠d2 + e2 â
- â¦G, L⦠⢠T1 â¶*[d2, e2] T2.
-#G #L #T1 #T2 #d1 #e1 #H #d1 #d2 #Hd21 #Hde12 @(cpys_ind ⦠H) -T2
-/3 width=7 by cpys_strap1, cpy_weak/
-qed-.
-
-lemma cpys_weak_top: âG,L,T1,T2,d,e.
- â¦G, L⦠⢠T1 â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¶*[d, |L| - d] T2.
-#G #L #T1 #T2 #d #e #H @(cpys_ind ⦠H) -T2
-/3 width=4 by cpys_strap1, cpy_weak_top/
-qed-.
-
-lemma cpys_weak_full: âG,L,T1,T2,d,e.
- â¦G, L⦠⢠T1 â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¶*[0, |L|] T2.
-#G #L #T1 #T2 #d #e #H @(cpys_ind ⦠H) -T2
-/3 width=5 by cpys_strap1, cpy_weak_full/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma cpys_fwd_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
- âT1,d,e. â§[d, e] T1 â¡ U1 â
- d ⤠dt â d + e ⤠dt + et â
- ââT2. â¦G, L⦠⢠U1 â¶*[d+e, dt+et-(d+e)] U2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H #T1 #d #e #HTU1 #Hddt #Hdedet @(cpys_ind ⦠H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HU1 #HTU
- elim (cpy_fwd_up ⦠HU2 ⦠HTU) -HU2 -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_fwd_tw: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶*[d, e] T2 â â¯{T1} ⤠â¯{T2}.
-#G #L #T1 #T2 #d #e #H @(cpys_ind ⦠H) -T2 //
-#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw ⦠HT2) -HT2
-/2 width=3 by transitive_le/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-(* Note: this can be derived from cpys_inv_atom1 *)
-lemma cpys_inv_sort1: âG,L,T2,k,d,e. â¦G, L⦠⢠âk â¶*[d, e] T2 â T2 = âk.
-#G #L #T2 #k #d #e #H @(cpys_ind ⦠H) -T2 //
-#T #T2 #_ #HT2 #IHT1 destruct
->(cpy_inv_sort1 ⦠HT2) -HT2 //
-qed-.
-
-(* Note: this can be derived from cpys_inv_atom1 *)
-lemma cpys_inv_gref1: âG,L,T2,p,d,e. â¦G, L⦠⢠§p â¶*[d, e] T2 â T2 = §p.
-#G #L #T2 #p #d #e #H @(cpys_ind ⦠H) -T2 //
-#T #T2 #_ #HT2 #IHT1 destruct
->(cpy_inv_gref1 ⦠HT2) -HT2 //
-qed-.
-
-lemma cpys_inv_bind1: âa,I,G,L,V1,T1,U2,d,e. â¦G, L⦠⢠â{a,I}V1.T1 â¶*[d, e] U2 â
- ââV2,T2. â¦G, L⦠⢠V1 â¶*[d, e] V2 &
- â¦G, L.â{I}V1⦠⢠T1 â¶*[⫯d, e] T2 &
- U2 = â{a,I}V2.T2.
-#a #I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind ⦠H) -U2
-[ /2 width=5 by ex3_2_intro/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (cpy_inv_bind1 ⦠HU2) -HU2 #V2 #T2 #HV2 #HT2 #H
- lapply (lsuby_cpy_trans ⦠HT2 (L.â{I}V1) ?) -HT2
- /3 width=5 by cpys_strap1, lsuby_succ, ex3_2_intro/
-]
-qed-.
-
-lemma cpys_inv_flat1: âI,G,L,V1,T1,U2,d,e. â¦G, L⦠⢠â{I}V1.T1 â¶*[d, e] U2 â
- ââV2,T2. â¦G, L⦠⢠V1 â¶*[d, e] V2 & â¦G, L⦠⢠T1 â¶*[d, e] T2 &
- U2 = â{I}V2.T2.
-#I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind ⦠H) -U2
-[ /2 width=5 by ex3_2_intro/
-| #U #U2 #_ #HU2 * #V #T #HV1 #HT1 #H destruct
- elim (cpy_inv_flat1 ⦠HU2) -HU2
- /3 width=5 by cpys_strap1, ex3_2_intro/
-]
-qed-.
-
-lemma cpys_inv_refl_O2: âG,L,T1,T2,d. â¦G, L⦠⢠T1 â¶*[d, 0] T2 â T1 = T2.
-#G #L #T1 #T2 #d #H @(cpys_ind ⦠H) -T2 //
-#T #T2 #_ #HT2 #IHT1 <(cpy_inv_refl_O2 ⦠HT2) -HT2 //
-qed-.
-
-lemma cpys_inv_lift1_eq: âG,L,U1,U2. âd,e:nat.
- â¦G, L⦠⢠U1 â¶*[d, e] U2 â âT1. â§[d, e] T1 â¡ U1 â U1 = U2.
-#G #L #U1 #U2 #d #e #H #T1 #HTU1 @(cpys_ind ⦠H) -U2
-/2 width=7 by cpy_inv_lift1_eq/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
deleted file mode 100644
index e297be698..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_alt.ma
+++ /dev/null
@@ -1,102 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| 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/psubststaralt_6.ma".
-include "basic_2/substitution/cpys_lift.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* alternative definition of cpys *)
-inductive cpysa: ynat â ynat â relation4 genv lenv term term â
-| cpysa_atom : âI,G,L,d,e. cpysa d e G L (âª{I}) (âª{I})
-| cpysa_subst: âI,G,L,K,V1,V2,W2,i,d,e. d ⤠yinj i â i < d+e â
- â©[i] L â¡ K.â{I}V1 â cpysa 0 (â«°(d+e-i)) G K V1 V2 â
- â§[0, i+1] V2 â¡ W2 â cpysa d e G L (#i) W2
-| cpysa_bind : âa,I,G,L,V1,V2,T1,T2,d,e.
- cpysa d e G L V1 V2 â cpysa (⫯d) e G (L.â{I}V1) T1 T2 â
- cpysa d e G L (â{a,I}V1.T1) (â{a,I}V2.T2)
-| cpysa_flat : âI,G,L,V1,V2,T1,T2,d,e.
- cpysa d e G L V1 V2 â cpysa d e G L T1 T2 â
- cpysa d e G L (â{I}V1.T1) (â{I}V2.T2)
-.
-
-interpretation
- "context-sensitive extended multiple substritution (term) alternative"
- 'PSubstStarAlt G L T1 d e T2 = (cpysa d e G L T1 T2).
-
-(* Basic properties *********************************************************)
-
-lemma lsuby_cpysa_trans: âG,d,e. lsub_trans ⦠(cpysa d e G) (lsuby d e).
-#G #d #e #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -d -e
-[ //
-| #I #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (lsuby_ldrop_trans_be ⦠HL12 ⦠HLK1) -HL12 -HLK1 /3 width=7 by cpysa_subst/
-| /4 width=1 by lsuby_succ, cpysa_bind/
-| /3 width=1 by cpysa_flat/
-]
-qed-.
-
-lemma cpysa_refl: âG,T,L,d,e. â¦G, L⦠⢠T â¶â¶*[d, e] T.
-#G #T elim T -T //
-#I elim I -I /2 width=1 by cpysa_bind, cpysa_flat/
-qed.
-
-lemma cpysa_cpy_trans: âG,L,T1,T,d,e. â¦G, L⦠⢠T1 â¶â¶*[d, e] T â
- âT2. â¦G, L⦠⢠T â¶[d, e] T2 â â¦G, L⦠⢠T1 â¶â¶*[d, e] T2.
-#G #L #T1 #T #d #e #H elim H -G -L -T1 -T -d -e
-[ #I #G #L #d #e #X #H
- elim (cpy_inv_atom1 ⦠H) -H // * /2 width=7 by cpysa_subst/
-| #I #G #L #K #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK #_ #HVW2 #IHV12 #T2 #H
- lapply (ldrop_fwd_drop2 ⦠HLK) #H0LK
- lapply (cpy_weak ⦠H 0 (d+e) ? ?) -H // #H
- elim (cpy_inv_lift1_be ⦠H ⦠H0LK ⦠HVW2) -H -H0LK -HVW2
- /3 width=7 by cpysa_subst, ylt_fwd_le_succ/
-| #a #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
- elim (cpy_inv_bind1 ⦠H) -H #V2 #T2 #HV2 #HT2 #H destruct
- /5 width=5 by cpysa_bind, lsuby_cpy_trans, lsuby_succ/
-| #I #G #L #V1 #V #T1 #T #d #e #_ #_ #IHV1 #IHT1 #X #H
- elim (cpy_inv_flat1 ⦠H) -H #V2 #T2 #HV2 #HT2 #H destruct /3 width=1 by cpysa_flat/
-]
-qed-.
-
-lemma cpys_cpysa: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¶â¶*[d, e] T2.
-/3 width=8 by cpysa_cpy_trans, cpys_ind/ qed.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma cpysa_inv_cpys: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶â¶*[d, e] T2 â â¦G, L⦠⢠T1 â¶*[d, e] T2.
-#G #L #T1 #T2 #d #e #H elim H -G -L -T1 -T2 -d -e
-/2 width=7 by cpys_subst, cpys_flat, cpys_bind, cpy_cpys/
-qed-.
-
-(* Advanced eliminators *****************************************************)
-
-lemma cpys_ind_alt: âR:ynatâynatârelation4 genv lenv term term.
- (âI,G,L,d,e. R d e G L (âª{I}) (âª{I})) â
- (âI,G,L,K,V1,V2,W2,i,d,e. d ⤠yinj i â i < d + e â
- â©[i] L â¡ K.â{I}V1 â â¦G, K⦠⢠V1 â¶*[O, â«°(d+e-i)] V2 â
- â§[O, i+1] V2 â¡ W2 â R O (â«°(d+e-i)) G K V1 V2 â R d e G L (#i) W2
- ) â
- (âa,I,G,L,V1,V2,T1,T2,d,e. â¦G, L⦠⢠V1 â¶*[d, e] V2 â
- â¦G, L.â{I}V1⦠⢠T1 â¶*[⫯d, e] T2 â R d e G L V1 V2 â
- R (⫯d) e G (L.â{I}V1) T1 T2 â R d e G L (â{a,I}V1.T1) (â{a,I}V2.T2)
- ) â
- (âI,G,L,V1,V2,T1,T2,d,e. â¦G, L⦠⢠V1 â¶*[d, e] V2 â
- â¦G, L⦠⢠T1 â¶*[d, e] T2 â R d e G L V1 V2 â
- R d e G L T1 T2 â R d e G L (â{I}V1.T1) (â{I}V2.T2)
- ) â
- âd,e,G,L,T1,T2. â¦G, L⦠⢠T1 â¶*[d, e] T2 â R d e G L T1 T2.
-#R #H1 #H2 #H3 #H4 #d #e #G #L #T1 #T2 #H elim (cpys_cpysa ⦠H) -G -L -T1 -T2 -d -e
-/3 width=8 by cpysa_inv_cpys/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_cpys.ma
deleted file mode 100644
index 52759ca7f..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_cpys.ma
+++ /dev/null
@@ -1,117 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| 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/cpy_cpy.ma".
-include "basic_2/substitution/cpys_alt.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma cpys_inv_SO2: âG,L,T1,T2,d. â¦G, L⦠⢠T1 â¶*[d, 1] T2 â â¦G, L⦠⢠T1 â¶[d, 1] T2.
-#G #L #T1 #T2 #d #H @(cpys_ind ⦠H) -T2 /2 width=3 by cpy_trans_ge/
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma cpys_strip_eq: âG,L,T0,T1,d1,e1. â¦G, L⦠⢠T0 â¶*[d1, e1] T1 â
- âT2,d2,e2. â¦G, L⦠⢠T0 â¶[d2, e2] T2 â
- ââT. â¦G, L⦠⢠T1 â¶[d2, e2] T & â¦G, L⦠⢠T2 â¶*[d1, e1] T.
-normalize /3 width=3 by cpy_conf_eq, TC_strip1/ qed-.
-
-lemma cpys_strip_neq: âG,L1,T0,T1,d1,e1. â¦G, L1⦠⢠T0 â¶*[d1, e1] T1 â
- âL2,T2,d2,e2. â¦G, L2⦠⢠T0 â¶[d2, e2] T2 â
- (d1 + e1 ⤠d2 ⨠d2 + e2 ⤠d1) â
- ââT. â¦G, L2⦠⢠T1 â¶[d2, e2] T & â¦G, L1⦠⢠T2 â¶*[d1, e1] T.
-normalize /3 width=3 by cpy_conf_neq, TC_strip1/ qed-.
-
-lemma cpys_strap1_down: âG,L,T1,T0,d1,e1. â¦G, L⦠⢠T1 â¶*[d1, e1] T0 â
- âT2,d2,e2. â¦G, L⦠⢠T0 â¶[d2, e2] T2 â d2 + e2 ⤠d1 â
- ââT. â¦G, L⦠⢠T1 â¶[d2, e2] T & â¦G, L⦠⢠T â¶*[d1, e1] T2.
-normalize /3 width=3 by cpy_trans_down, TC_strap1/ qed.
-
-lemma cpys_strap2_down: âG,L,T1,T0,d1,e1. â¦G, L⦠⢠T1 â¶[d1, e1] T0 â
- âT2,d2,e2. â¦G, L⦠⢠T0 â¶*[d2, e2] T2 â d2 + e2 ⤠d1 â
- ââT. â¦G, L⦠⢠T1 â¶*[d2, e2] T & â¦G, L⦠⢠T â¶[d1, e1] T2.
-normalize /3 width=3 by cpy_trans_down, TC_strap2/ qed-.
-
-lemma cpys_split_up: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶*[d, e] T2 â
- âi. d ⤠i â i ⤠d + e â
- ââT. â¦G, L⦠⢠T1 â¶*[d, i - d] T & â¦G, L⦠⢠T â¶*[i, d + e - i] T2.
-#G #L #T1 #T2 #d #e #H #i #Hdi #Hide @(cpys_ind ⦠H) -T2
-[ /2 width=3 by ex2_intro/
-| #T #T2 #_ #HT12 * #T3 #HT13 #HT3
- elim (cpy_split_up ⦠HT12 ⦠Hide) -HT12 -Hide #T0 #HT0 #HT02
- elim (cpys_strap1_down ⦠HT3 ⦠HT0) -T /3 width=5 by cpys_strap1, ex2_intro/
- >ymax_pre_sn_comm //
-]
-qed-.
-
-lemma cpys_inv_lift1_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- d ⤠dt â dt ⤠yinj d + e â yinj d + e ⤠dt + et â
- ââT2. â¦G, K⦠⢠T1 â¶*[d, dt + et - (yinj d + e)] T2 &
- â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #HU12 #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet
-elim (cpys_split_up ⦠HU12 (d + e)) -HU12 // -Hdedet #U #HU1 #HU2
-lapply (cpys_weak ⦠HU1 d e ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hddt -Hdtde #HU1
-lapply (cpys_inv_lift1_eq ⦠HU1 ⦠HTU1) -HU1 #HU1 destruct
-elim (cpys_inv_lift1_ge ⦠HU2 ⦠HLK ⦠HTU1) -HU2 -HLK -HTU1 //
->yplus_minus_inj /2 width=3 by ex2_intro/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem cpys_conf_eq: âG,L,T0,T1,d1,e1. â¦G, L⦠⢠T0 â¶*[d1, e1] T1 â
- âT2,d2,e2. â¦G, L⦠⢠T0 â¶*[d2, e2] T2 â
- ââT. â¦G, L⦠⢠T1 â¶*[d2, e2] T & â¦G, L⦠⢠T2 â¶*[d1, e1] T.
-normalize /3 width=3 by cpy_conf_eq, TC_confluent2/ qed-.
-
-theorem cpys_conf_neq: âG,L1,T0,T1,d1,e1. â¦G, L1⦠⢠T0 â¶*[d1, e1] T1 â
- âL2,T2,d2,e2. â¦G, L2⦠⢠T0 â¶*[d2, e2] T2 â
- (d1 + e1 ⤠d2 ⨠d2 + e2 ⤠d1) â
- ââT. â¦G, L2⦠⢠T1 â¶*[d2, e2] T & â¦G, L1⦠⢠T2 â¶*[d1, e1] T.
-normalize /3 width=3 by cpy_conf_neq, TC_confluent2/ qed-.
-
-theorem cpys_trans_eq: âG,L,T1,T,T2,d,e.
- â¦G, L⦠⢠T1 â¶*[d, e] T â â¦G, L⦠⢠T â¶*[d, e] T2 â
- â¦G, L⦠⢠T1 â¶*[d, e] T2.
-normalize /2 width=3 by trans_TC/ qed-.
-
-theorem cpys_trans_down: âG,L,T1,T0,d1,e1. â¦G, L⦠⢠T1 â¶*[d1, e1] T0 â
- âT2,d2,e2. â¦G, L⦠⢠T0 â¶*[d2, e2] T2 â d2 + e2 ⤠d1 â
- ââT. â¦G, L⦠⢠T1 â¶*[d2, e2] T & â¦G, L⦠⢠T â¶*[d1, e1] T2.
-normalize /3 width=3 by cpy_trans_down, TC_transitive2/ qed-.
-
-theorem cpys_antisym_eq: âG,L1,T1,T2,d,e. â¦G, L1⦠⢠T1 â¶*[d, e] T2 â
- âL2. â¦G, L2⦠⢠T2 â¶*[d, e] T1 â T1 = T2.
-#G #L1 #T1 #T2 #d #e #H @(cpys_ind_alt ⦠H) -G -L1 -T1 -T2 //
-[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #_ #_ #HVW2 #_ #L2 #HW2
- elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hdi -Hide | ]
- [ lapply (cpys_weak_full ⦠HW2) -HW2 #HW2
- lapply (cpys_weak ⦠HW2 0 (i+1) ? ?) -HW2 //
- [ >yplus_O1 >yplus_O1 /3 width=1 by ylt_fwd_le, ylt_inj/ ] -Hi
- #HW2 >(cpys_inv_lift1_eq ⦠HW2) -HW2 //
- | elim (ldrop_O1_le ⦠Hi) -Hi #K2 #HLK2
- elim (cpys_inv_lift1_ge_up ⦠HW2 ⦠HLK2 ⦠HVW2 ? ? ?) -HW2 -HLK2 -HVW2
- /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ -Hdi -Hide
- #X #_ #H elim (lift_inv_lref2_be ⦠H) -H //
- ]
-| #a #I #G #L1 #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_bind1 ⦠H) -H
- #V #T #HV2 #HT2 #H destruct
- lapply (IHV12 ⦠HV2) #H destruct -IHV12 -HV2 /3 width=2 by eq_f2/
-| #I #G #L1 #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #L2 #H elim (cpys_inv_flat1 ⦠H) -H
- #V #T #HV2 #HT2 #H destruct /3 width=2 by eq_f2/
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
deleted file mode 100644
index b6d9e45a1..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpys_lift.ma
+++ /dev/null
@@ -1,218 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| 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/cpy_lift.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* CONTEXT-SENSITIVE EXTENDED MULTIPLE SUBSTITUTION FOR TERMS ***************)
-
-(* Advanced properties ******************************************************)
-
-lemma cpys_subst: âI,G,L,K,V,U1,i,d,e.
- d ⤠yinj i â i < d + e â
- â©[i] L â¡ K.â{I}V â â¦G, K⦠⢠V â¶*[0, â«°(d+e-i)] U1 â
- âU2. â§[0, i+1] U1 â¡ U2 â â¦G, L⦠⢠#i â¶*[d, e] U2.
-#I #G #L #K #V #U1 #i #d #e #Hdi #Hide #HLK #H @(cpys_ind ⦠H) -U1
-[ /3 width=5 by cpy_cpys, cpy_subst/
-| #U #U1 #_ #HU1 #IHU #U2 #HU12
- elim (lift_total U 0 (i+1)) #U0 #HU0
- lapply (IHU ⦠HU0) -IHU #H
- lapply (ldrop_fwd_drop2 ⦠HLK) -HLK #HLK
- lapply (cpy_lift_ge ⦠HU1 ⦠HLK HU0 HU12 ?) -HU1 -HLK -HU0 -HU12 // #HU02
- lapply (cpy_weak ⦠HU02 d e ? ?) -HU02
- [2,3: /2 width=3 by cpys_strap1, yle_succ_dx/ ]
- >yplus_O1 <yplus_inj >ymax_pre_sn_comm /2 width=1 by ylt_fwd_le_succ/
-]
-qed.
-
-lemma cpys_subst_Y2: âI,G,L,K,V,U1,i,d.
- d ⤠yinj i â
- â©[i] L â¡ K.â{I}V â â¦G, K⦠⢠V â¶*[0, â] U1 â
- âU2. â§[0, i+1] U1 â¡ U2 â â¦G, L⦠⢠#i â¶*[d, â] U2.
-#I #G #L #K #V #U1 #i #d #Hdi #HLK #HVU1 #U2 #HU12
-@(cpys_subst ⦠HLK ⦠HU12) >yminus_Y_inj //
-qed.
-
-(* Advanced inverion lemmas *************************************************)
-
-lemma cpys_inv_atom1: âI,G,L,T2,d,e. â¦G, L⦠⢠âª{I} â¶*[d, e] T2 â
- T2 = âª{I} â¨
- ââJ,K,V1,V2,i. d ⤠yinj i & i < d + e &
- â©[i] L â¡ K.â{J}V1 &
- â¦G, K⦠⢠V1 â¶*[0, â«°(d+e-i)] V2 &
- â§[O, i+1] V2 â¡ T2 &
- I = LRef i.
-#I #G #L #T2 #d #e #H @(cpys_ind ⦠H) -T2
-[ /2 width=1 by or_introl/
-| #T #T2 #_ #HT2 *
- [ #H destruct
- elim (cpy_inv_atom1 ⦠HT2) -HT2 [ /2 width=1 by or_introl/ | * /3 width=11 by ex6_5_intro, or_intror/ ]
- | * #J #K #V1 #V #i #Hdi #Hide #HLK #HV1 #HVT #HI
- lapply (ldrop_fwd_drop2 ⦠HLK) #H
- elim (cpy_inv_lift1_ge_up ⦠HT2 ⦠H ⦠HVT) -HT2 -H -HVT
- [2,3,4: /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ ]
- /4 width=11 by cpys_strap1, ex6_5_intro, or_intror/
- ]
-]
-qed-.
-
-lemma cpys_inv_lref1: âG,L,T2,i,d,e. â¦G, L⦠⢠#i â¶*[d, e] T2 â
- T2 = #i â¨
- ââI,K,V1,V2. d ⤠i & i < d + e &
- â©[i] L â¡ K.â{I}V1 &
- â¦G, K⦠⢠V1 â¶*[0, â«°(d+e-i)] V2 &
- â§[O, i+1] V2 â¡ T2.
-#G #L #T2 #i #d #e #H elim (cpys_inv_atom1 ⦠H) -H /2 width=1 by or_introl/
-* #I #K #V1 #V2 #j #Hdj #Hjde #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
-qed-.
-
-lemma cpys_inv_lref1_ldrop: âG,L,T2,i,d,e. â¦G, L⦠⢠#i â¶*[d, e] T2 â
- âI,K,V1. â©[i] L â¡ K.â{I}V1 â
- âV2. â§[O, i+1] V2 â¡ T2 â
- â§â§ â¦G, K⦠⢠V1 â¶*[0, â«°(d+e-i)] V2
- & d ⤠i
- & i < d + e.
-#G #L #T2 #i #d #e #H #I #K #V1 #HLK #V2 #HVT2 elim (cpys_inv_lref1 ⦠H) -H
-[ #H destruct elim (lift_inv_lref2_be ⦠HVT2) -HVT2 -HLK //
-| * #Z #Y #X1 #X2 #Hdi #Hide #HLY #HX12 #HXT2
- lapply (lift_inj ⦠HXT2 ⦠HVT2) -T2 #H destruct
- lapply (ldrop_mono ⦠HLY ⦠HLK) -L #H destruct
- /2 width=1 by and3_intro/
-]
-qed-.
-
-(* Properties on relocation *************************************************)
-
-lemma cpys_lift_le: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶*[dt, et] T2 â
- âL,U1,s,d,e. dt + et ⤠yinj d â â©[s, d, e] L â¡ K â
- â§[d, e] T1 â¡ U1 â âU2. â§[d, e] T2 â¡ U2 â
- â¦G, L⦠⢠U1 â¶*[dt, et] U2.
-#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdetd #HLK #HTU1 @(cpys_ind ⦠H) -T2
-[ #U2 #H >(lift_mono ⦠HTU1 ⦠H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (IHT ⦠HTU) -IHT #HU1
- lapply (cpy_lift_le ⦠HT2 ⦠HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
-]
-qed-.
-
-lemma cpys_lift_be: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶*[dt, et] T2 â
- âL,U1,s,d,e. dt ⤠yinj d â d ⤠dt + et â
- â©[s, d, e] L â¡ K â â§[d, e] T1 â¡ U1 â
- âU2. â§[d, e] T2 â¡ U2 â â¦G, L⦠⢠U1 â¶*[dt, et + e] U2.
-#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hdtd #Hddet #HLK #HTU1 @(cpys_ind ⦠H) -T2
-[ #U2 #H >(lift_mono ⦠HTU1 ⦠H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (IHT ⦠HTU) -IHT #HU1
- lapply (cpy_lift_be ⦠HT2 ⦠HLK HTU HTU2 ? ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
-]
-qed-.
-
-lemma cpys_lift_ge: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶*[dt, et] T2 â
- âL,U1,s,d,e. yinj d ⤠dt â â©[s, d, e] L â¡ K â
- â§[d, e] T1 â¡ U1 â âU2. â§[d, e] T2 â¡ U2 â
- â¦G, L⦠⢠U1 â¶*[dt+e, et] U2.
-#G #K #T1 #T2 #dt #et #H #L #U1 #s #d #e #Hddt #HLK #HTU1 @(cpys_ind ⦠H) -T2
-[ #U2 #H >(lift_mono ⦠HTU1 ⦠H) -H //
-| -HTU1 #T #T2 #_ #HT2 #IHT #U2 #HTU2
- elim (lift_total T d e) #U #HTU
- lapply (IHT ⦠HTU) -IHT #HU1
- lapply (cpy_lift_ge ⦠HT2 ⦠HLK HTU HTU2 ?) -HT2 -HLK -HTU -HTU2 /2 width=3 by cpys_strap1/
-]
-qed-.
-
-(* Inversion lemmas for relocation ******************************************)
-
-lemma cpys_inv_lift1_le: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- dt + et ⤠d â
- ââT2. â¦G, K⦠⢠T1 â¶*[dt, et] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdetd @(cpys_ind ⦠H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_le ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_be: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- dt ⤠d â yinj d + e ⤠dt + et â
- ââT2. â¦G, K⦠⢠T1 â¶*[dt, et - e] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdedet @(cpys_ind ⦠H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_be ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_ge: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- yinj d + e ⤠dt â
- ââT2. â¦G, K⦠⢠T1 â¶*[dt - e, et] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdedt @(cpys_ind ⦠H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_ge ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-(* Advanced inversion lemmas on relocation **********************************)
-
-lemma cpys_inv_lift1_ge_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- d ⤠dt â dt ⤠yinj d + e â yinj d + e ⤠dt + et â
- ââT2. â¦G, K⦠⢠T1 â¶*[d, dt + et - (yinj d + e)] T2 &
- â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hddt #Hdtde #Hdedet @(cpys_ind ⦠H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_ge_up ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_be_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- dt ⤠d â dt + et ⤠yinj d + e â
- ââT2. â¦G, K⦠⢠T1 â¶*[dt, d - dt] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hdetde @(cpys_ind ⦠H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_be_up ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_le_up: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶*[dt, et] U2 â
- âK,s,d,e. â©[s, d, e] L â¡ K â âT1. â§[d, e] T1 â¡ U1 â
- dt ⤠d â d ⤠dt + et â dt + et ⤠yinj d + e â
- ââT2. â¦G, K⦠⢠T1 â¶*[dt, d - dt] T2 & â§[d, e] T2 â¡ U2.
-#G #L #U1 #U2 #dt #et #H #K #s #d #e #HLK #T1 #HTU1 #Hdtd #Hddet #Hdetde @(cpys_ind ⦠H) -U2
-[ /2 width=3 by ex2_intro/
-| -HTU1 #U #U2 #_ #HU2 * #T #HT1 #HTU
- elim (cpy_inv_lift1_le_up ⦠HU2 ⦠HLK ⦠HTU) -HU2 -HLK -HTU /3 width=3 by cpys_strap1, ex2_intro/
-]
-qed-.
-
-lemma cpys_inv_lift1_subst: âG,L,W1,W2,d,e. â¦G, L⦠⢠W1 â¶*[d, e] W2 â
- âK,V1,i. â©[i+1] L â¡ K â â§[O, i+1] V1 â¡ W1 â
- d ⤠yinj i â i < d + e â
- ââV2. â¦G, K⦠⢠V1 â¶*[O, â«°(d+e-i)] V2 & â§[O, i+1] V2 â¡ W2.
-#G #L #W1 #W2 #d #e #HW12 #K #V1 #i #HLK #HVW1 #Hdi #Hide
-elim (cpys_inv_lift1_ge_up ⦠HW12 ⦠HLK ⦠HVW1 ? ? ?) //
->yplus_O1 <yplus_inj >yplus_SO2
-[ >yminus_succ2 /2 width=3 by ex2_intro/
-| /2 width=1 by ylt_fwd_le_succ1/
-| /2 width=3 by yle_trans/
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/fqup_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/fqup_lleq.ma
new file mode 100644
index 000000000..c0bfa41fe
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/fqup_lleq.ma
@@ -0,0 +1,32 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| 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/fqu_lleq.ma".
+include "basic_2/substitution/fqup.ma".
+
+(* PLUS-ITERATED SUPCLOSURE *************************************************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_fqup_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â+ â¦G2, K2, U⦠â
+ âL1. L1 â[T, 0] L2 â
+ ââK1. â¦G1, L1, T⦠â+ â¦G2, K1, U⦠& K1 â[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind ⦠H) -G2 -K2 -U
+[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans ⦠HTU ⦠HL12) -L2
+ /3 width=3 by fqu_fqup, ex2_intro/
+| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU ⦠HL12) -L2
+ #K1 #HTU #HK1 elim (lleq_fqu_trans ⦠HU2 ⦠HK1) -K
+ /3 width=5 by fqup_strap1, ex2_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/fqus_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/fqus_lleq.ma
new file mode 100644
index 000000000..ae209399f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/fqus_lleq.ma
@@ -0,0 +1,29 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/fqup_lleq.ma".
+include "basic_2/substitution/fqus_alt.ma".
+
+(* STAR-ITERATED SUPCLOSURE *************************************************)
+
+(* Properties on lazy equivalence for local environments ********************)
+
+lemma lleq_fqus_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â* â¦G2, K2, U⦠â
+ âL1. L1 â[T, 0] L2 â
+ ââK1. â¦G1, L1, T⦠â* â¦G2, K1, U⦠& K1 â[U, 0] K2.
+#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen ⦠H) -H
+[ #H elim (lleq_fqup_trans ⦠H ⦠HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/
+| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq.ma
deleted file mode 100644
index 72a5346ce..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq.ma
+++ /dev/null
@@ -1,160 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| 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/lazyeq_4.ma".
-include "basic_2/substitution/cpys.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-definition lleq: relation4 ynat term lenv lenv â
- λd,T,L1,L2. |L1| = |L2| â§
- (âU. â¦â, L1⦠⢠T â¶*[d, â] U â â¦â, L2⦠⢠T â¶*[d, â] U).
-
-interpretation
- "lazy equivalence (local environment)"
- 'LazyEq T d L1 L2 = (lleq d T L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma lleq_refl: âd,T. reflexive ⦠(lleq d T).
-/3 width=1 by conj/ qed.
-
-lemma lleq_sym: âd,T. symmetric ⦠(lleq d T).
-#d #T #L1 #L2 * /3 width=1 by iff_sym, conj/
-qed-.
-
-lemma lleq_sort: âL1,L2,d,k. |L1| = |L2| â L1 â[âk, d] L2.
-#L1 #L2 #d #k #HL12 @conj // -HL12
-#U @conj #H >(cpys_inv_sort1 ⦠H) -H //
-qed.
-
-lemma lleq_gref: âL1,L2,d,p. |L1| = |L2| â L1 â[§p, d] L2.
-#L1 #L2 #d #k #HL12 @conj // -HL12
-#U @conj #H >(cpys_inv_gref1 ⦠H) -H //
-qed.
-
-lemma lleq_bind: âa,I,L1,L2,V,T,d.
- L1 â[V, d] L2 â L1.â{I}V â[T, ⫯d] L2.â{I}V â
- L1 â[â{a,I}V.T, d] L2.
-#a #I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12
-#X @conj #H elim (cpys_inv_bind1 ⦠H) -H
-#W #U #HVW #HTU #H destruct
-elim (IHV W) -IHV elim (IHT U) -IHT /3 width=1 by cpys_bind/
-qed.
-
-lemma lleq_flat: âI,L1,L2,V,T,d.
- L1 â[V, d] L2 â L1 â[T, d] L2 â L1 â[â{I}V.T, d] L2.
-#I #L1 #L2 #V #T #d * #HL12 #IHV * #_ #IHT @conj // -HL12
-#X @conj #H elim (cpys_inv_flat1 ⦠H) -H
-#W #U #HVW #HTU #H destruct
-elim (IHV W) -IHV elim (IHT U) -IHT
-/3 width=1 by cpys_flat/
-qed.
-
-lemma lleq_be: âL1,L2,U,dt. L1 â[U, dt] L2 â âT,d,e. â§[d, e] T â¡ U â
- d ⤠dt â dt ⤠d + e â L1 â[U, d] L2.
-#L1 #L2 #U #dt * #HL12 #IH #T #d #e #HTU #Hddt #Hdtde @conj // -HL12
-#U0 elim (IH U0) -IH #H12 #H21 @conj
-#HU0 elim (cpys_fwd_up ⦠HU0 ⦠HTU) // -HU0 /4 width=5 by cpys_weak/
-qed-.
-
-lemma lsuby_lleq_trans: âL2,L,T,d. L2 â[T, d] L â
- âL1. L1 âÃ[d, â] L2 â |L1| = |L2| â L1 â[T, d] L.
-#L2 #L #T #d * #HL2 #IH #L1 #HL12 #H @conj // -HL2
-#U elim (IH U) -IH #Hdx #Hsn @conj #HTU
-[ @Hdx -Hdx -Hsn @(lsuby_cpys_trans ⦠HTU) -HTU
- /2 width=1 by lsuby_sym/ (**) (* full auto does not work *)
-| -H -Hdx /3 width=3 by lsuby_cpys_trans/
-]
-qed-.
-
-lemma lleq_lsuby_trans: âL,L1,T,d. L â[T, d] L1 â
- âL2. L1 âÃ[d, â] L2 â |L1| = |L2| â L â[T, d] L2.
-/5 width=4 by lsuby_lleq_trans, lleq_sym, lsuby_sym/ qed-.
-
-lemma lleq_lsuby_repl: âL1,L2,T,d. L1 â[T, d] L2 â
- âK1. K1 âÃ[d, â] L1 â |K1| = |L1| â
- âK2. L2 âÃ[d, â] K2 â |L2| = |K2| â
- K1 â[T, d] K2.
-/3 width=4 by lleq_lsuby_trans, lsuby_lleq_trans/ qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma lleq_fwd_length: âL1,L2,T,d. L1 â[T, d] L2 â |L1| = |L2|.
-#L1 #L2 #T #d * //
-qed-.
-
-lemma lleq_fwd_ldrop_sn: âL1,L2,T,d. L1 â[d, T] L2 â âK1,i. â©[i] L1 â¡ K1 â
- âK2. â©[i] L2 â¡ K2.
-#L1 #L2 #T #d #H #K1 #i #HLK1 lapply (lleq_fwd_length ⦠H) -H
-#HL12 lapply (ldrop_fwd_length_le2 ⦠HLK1) -HLK1 /2 width=1 by ldrop_O1_le/
-qed-.
-
-lemma lleq_fwd_ldrop_dx: âL1,L2,T,d. L1 â[d, T] L2 â âK2,i. â©[i] L2 â¡ K2 â
- âK1. â©[i] L1 â¡ K1.
-/3 width=6 by lleq_fwd_ldrop_sn, lleq_sym/ qed-.
-
-lemma lleq_fwd_bind_sn: âa,I,L1,L2,V,T,d.
- L1 â[â{a,I}V.T, d] L2 â L1 â[V, d] L2.
-#a #I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
-#U elim (H (â{a,I}U.T)) -H
-#H1 #H2 @conj
-#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
-/2 width=1 by cpys_bind/ -H
-#H elim (cpys_inv_bind1 ⦠H) -H
-#X #Y #H1 #H2 #H destruct //
-qed-.
-
-lemma lleq_fwd_bind_dx: âa,I,L1,L2,V,T,d.
- L1 â[â{a,I}V.T, d] L2 â L1.â{I}V â[T, ⫯d] L2.â{I}V.
-#a #I #L1 #L2 #V #T #d * #HL12 #H @conj [ normalize // ] -HL12
-#U elim (H (â{a,I}V.U)) -H
-#H1 #H2 @conj
-#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
-/2 width=1 by cpys_bind/ -H
-#H elim (cpys_inv_bind1 ⦠H) -H
-#X #Y #H1 #H2 #H destruct //
-qed-.
-
-lemma lleq_fwd_flat_sn: âI,L1,L2,V,T,d.
- L1 â[â{I}V.T, d] L2 â L1 â[V, d] L2.
-#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
-#U elim (H (â{I}U.T)) -H
-#H1 #H2 @conj
-#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
-/2 width=1 by cpys_flat/ -H
-#H elim (cpys_inv_flat1 ⦠H) -H
-#X #Y #H1 #H2 #H destruct //
-qed-.
-
-lemma lleq_fwd_flat_dx: âI,L1,L2,V,T,d.
- L1 â[â{I}V.T, d] L2 â L1 â[T, d] L2.
-#I #L1 #L2 #V #T #d * #HL12 #H @conj // -HL12
-#U elim (H (â{I}V.U)) -H
-#H1 #H2 @conj
-#H [ lapply (H1 ?) | lapply (H2 ?) ] -H1 -H2
-/2 width=1 by cpys_flat/ -H
-#H elim (cpys_inv_flat1 ⦠H) -H
-#X #Y #H1 #H2 #H destruct //
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-lemma lleq_inv_bind: âa,I,L1,L2,V,T,d. L1 â[â{a,I}V.T, d] L2 â
- L1 â[V, d] L2 ⧠L1.â{I}V â[T, ⫯d] L2.â{I}V.
-/3 width=4 by lleq_fwd_bind_sn, lleq_fwd_bind_dx, conj/ qed-.
-
-lemma lleq_inv_flat: âI,L1,L2,V,T,d. L1 â[â{I}V.T, d] L2 â
- L1 â[V, d] L2 ⧠L1 â[T, d] L2.
-/3 width=3 by lleq_fwd_flat_sn, lleq_fwd_flat_dx, conj/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
deleted file mode 100644
index a8fb526e5..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_alt.ma
+++ /dev/null
@@ -1,85 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| 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/lazyeqalt_4.ma".
-include "basic_2/substitution/lleq_lleq.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Note: alternative definition of lleq *)
-inductive lleqa: relation4 ynat term lenv lenv â
-| lleqa_sort: âL1,L2,d,k. |L1| = |L2| â lleqa d (âk) L1 L2
-| lleqa_skip: âL1,L2,d,i. |L1| = |L2| â yinj i < d â lleqa d (#i) L1 L2
-| lleqa_lref: âI1,I2,L1,L2,K1,K2,V,d,i. d ⤠yinj i â
- â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â
- lleqa (yinj 0) V K1 K2 â lleqa d (#i) L1 L2
-| lleqa_free: âL1,L2,d,i. |L1| = |L2| â |L1| ⤠i â |L2| ⤠i â lleqa d (#i) L1 L2
-| lleqa_gref: âL1,L2,d,p. |L1| = |L2| â lleqa d (§p) L1 L2
-| lleqa_bind: âa,I,L1,L2,V,T,d.
- lleqa d V L1 L2 â lleqa (⫯d) T (L1.â{I}V) (L2.â{I}V) â
- lleqa d (â{a,I}V.T) L1 L2
-| lleqa_flat: âI,L1,L2,V,T,d.
- lleqa d V L1 L2 â lleqa d T L1 L2 â lleqa d (â{I}V.T) L1 L2
-.
-
-interpretation
- "lazy equivalence (local environment) alternative"
- 'LazyEqAlt T d L1 L2 = (lleqa d T L1 L2).
-
-(* Main inversion lemmas ****************************************************)
-
-theorem lleqa_inv_lleq: âL1,L2,T,d. L1 ââ[T, d] L2 â L1 â[T, d] L2.
-#L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-/2 width=8 by lleq_flat, lleq_bind, lleq_gref, lleq_free, lleq_lref, lleq_skip, lleq_sort/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem lleq_lleqa: âL1,T,L2,d. L1 â[T, d] L2 â L1 ââ[T, d] L2.
-#L1 #T @(f2_ind ⦠rfw ⦠L1 T) -L1 -T
-#n #IH #L1 * * /3 width=3 by lleqa_gref, lleqa_sort, lleq_fwd_length/
-[ #i #Hn #L2 #d #H elim (lleq_fwd_lref ⦠H) [ * || * ]
- /4 width=9 by lleqa_free, lleqa_lref, lleqa_skip, lleq_fwd_length, ldrop_fwd_rfw/
-| #a #I #V #T #Hn #L2 #d #H elim (lleq_inv_bind ⦠H) -H /3 width=1 by lleqa_bind/
-| #I #V #T #Hn #L2 #d #H elim (lleq_inv_flat ⦠H) -H /3 width=1 by lleqa_flat/
-]
-qed.
-
-(* Advanced eliminators *****************************************************)
-
-lemma lleq_ind_alt: âR:relation4 ynat term lenv lenv. (
- âL1,L2,d,k. |L1| = |L2| â R d (âk) L1 L2
- ) â (
- âL1,L2,d,i. |L1| = |L2| â yinj i < d â R d (#i) L1 L2
- ) â (
- âI1,I2,L1,L2,K1,K2,V,d,i. d ⤠yinj i â
- â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â
- K1 â[V, yinj O] K2 â R (yinj O) V K1 K2 â R d (#i) L1 L2
- ) â (
- âL1,L2,d,i. |L1| = |L2| â |L1| ⤠i â |L2| ⤠i â R d (#i) L1 L2
- ) â (
- âL1,L2,d,p. |L1| = |L2| â R d (§p) L1 L2
- ) â (
- âa,I,L1,L2,V,T,d.
- L1 â[V, d]L2 â L1.â{I}V â[T, ⫯d] L2.â{I}V â
- R d V L1 L2 â R (⫯d) T (L1.â{I}V) (L2.â{I}V) â R d (â{a,I}V.T) L1 L2
- ) â (
- âI,L1,L2,V,T,d.
- L1 â[V, d]L2 â L1 â[T, d] L2 â
- R d V L1 L2 â R d T L1 L2 â R d (â{I}V.T) L1 L2
- ) â
- âd,T,L1,L2. L1 â[T, d] L2 â R d T L1 L2.
-#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #d #T #L1 #L2 #H elim (lleq_lleqa ⦠H) -H
-/3 width=9 by lleqa_inv_lleq/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_ext.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_ext.ma
deleted file mode 100644
index d6bb03fa7..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_ext.ma
+++ /dev/null
@@ -1,91 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/lleq_alt.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Advanced inversion lemmas ************************************************)
-
-fact lleq_inv_S_aux: âL1,L2,T,d0. L1 â[T, d0] L2 â âd. d0 = d + 1 â
- âK1,K2,I,V. â©[d] L1 â¡ K1.â{I}V â â©[d] L2 â¡ K2.â{I}V â
- K1 â[V, 0] K2 â L1 â[T, d] L2.
-#L1 #L2 #T #d0 #H @(lleq_ind_alt ⦠H) -L1 -L2 -T -d0
-/2 width=1 by lleq_gref, lleq_free, lleq_sort/
-[ #L1 #L2 #d0 #i #HL12 #Hid #d #H #K1 #K2 #I #V #HLK1 #HLK2 #HV destruct
- elim (yle_split_eq i d) /2 width=1 by lleq_skip, ylt_fwd_succ2/ -HL12 -Hid
- #H destruct /2 width=8 by lleq_lref/
-| #I1 #I2 #L1 #L2 #K11 #K22 #V #d0 #i #Hd0i #HLK11 #HLK22 #HV #_ #d #H #K1 #K2 #J #W #_ #_ #_ destruct
- /3 width=8 by lleq_lref, yle_pred_sn/
-| #a #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
- /4 width=7 by lleq_bind, ldrop_drop/
-| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W #HLK1 #HLK2 destruct
- /3 width=7 by lleq_flat/
-]
-qed-.
-
-lemma lleq_inv_S: âT,L1,L2,d. L1 â[T, d+1] L2 â
- âK1,K2,I,V. â©[d] L1 â¡ K1.â{I}V â â©[d] L2 â¡ K2.â{I}V â
- K1 â[V, 0] K2 â L1 â[T, d] L2.
-/2 width=7 by lleq_inv_S_aux/ qed-.
-
-lemma lleq_inv_bind_O: âa,I,L1,L2,V,T. L1 â[â{a,I}V.T, 0] L2 â
- L1 â[V, 0] L2 ⧠L1.â{I}V â[T, 0] L2.â{I}V.
-#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind ⦠H) -H
-/3 width=7 by ldrop_pair, conj, lleq_inv_S/
-qed-.
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lleq_fwd_bind_O: âa,I,L1,L2,V,T. L1 â[â{a,I}V.T, 0] L2 â
- L1.â{I}V â[T, 0] L2.â{I}V.
-#a #I #L1 #L2 #V #T #H elim (lleq_inv_bind_O ⦠H) -H //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_ge: âL1,L2,T,d1. L1 â[T, d1] L2 â âd2. d1 ⤠d2 â L1 â[T, d2] L2.
-#L1 #L2 #T #d1 #H @(lleq_ind_alt ⦠H) -L1 -L2 -T -d1
-/4 width=1 by lleq_sort, lleq_free, lleq_gref, lleq_bind, lleq_flat, yle_succ/
-[ /3 width=3 by lleq_skip, ylt_yle_trans/
-| #I1 #I2 #L1 #L2 #K1 #K2 #V #d1 #i #Hi #HLK1 #HLK2 #HV #IHV #d2 #Hd12 elim (ylt_split i d2)
- [ lapply (lleq_fwd_length ⦠HV) #HK12 #Hid2
- lapply (ldrop_fwd_length ⦠HLK1) lapply (ldrop_fwd_length ⦠HLK2)
- normalize in ⢠(%â%â?); -I1 -I2 -V -d1 /2 width=1 by lleq_skip/
- | /3 width=8 by lleq_lref, yle_trans/
- ]
-]
-qed-.
-
-lemma lleq_bind_O: âa,I,L1,L2,V,T. L1 â[V, 0] L2 â L1.â{I}V â[T, 0] L2.â{I}V â
- L1 â[â{a,I}V.T, 0] L2.
-/3 width=3 by lleq_ge, lleq_bind/ qed.
-
-lemma lleq_bind_repl_SO: âI1,I2,L1,L2,V1,V2,T. L1.â{I1}V1 â[T, 0] L2.â{I2}V2 â
- âJ1,J2,W1,W2. L1.â{J1}W1 â[T, 1] L2.â{J2}W2.
-#I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (lleq_ge ⦠HT 1 ?) // -HT
-#HT @(lleq_lsuby_repl ⦠HT) /2 width=1 by lsuby_succ/ (**) (* full auto fails *)
-qed-.
-
-lemma lleq_bind_repl_O: âI,L1,L2,V,T. L1.â{I}V â[T, 0] L2.â{I}V â
- âJ,W. L1 â[W, 0] L2 â L1.â{J}W â[T, 0] L2.â{J}W.
-/3 width=7 by lleq_bind_repl_SO, lleq_inv_S/ qed-.
-
-(* Inversion lemmas on negated lazy quivalence for local environments *******)
-
-lemma nlleq_inv_bind_O: âa,I,L1,L2,V,T. (L1 â[â{a,I}V.T, 0] L2 â â¥) â
- (L1 â[V, 0] L2 â â¥) ⨠(L1.â{I}V â[T, 0] L2.â{I}V â â¥).
-#a #I #L1 #L2 #V #T #H elim (lleq_dec V L1 L2 0)
-/4 width=1 by lleq_bind_O, or_intror, or_introl/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_fqus.ma
deleted file mode 100644
index 18744a539..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_fqus.ma
+++ /dev/null
@@ -1,75 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/fqus_alt.ma".
-include "basic_2/substitution/lleq_ext.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Properties on supclosure and derivatives *********************************)
-
-lemma lleq_fqu_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â â¦G2, K2, U⦠â
- âL1. L1 â[T, 0] L2 â
- ââK1. â¦G1, L1, T⦠â â¦G2, K1, U⦠& K1 â[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H elim H -G1 -G2 -L2 -K2 -T -U
-[ #I #G #L2 #V #L1 #H elim (lleq_inv_lref_ge_dx ⦠H ⦠I L2 V) -H //
- #I1 #K1 #H1 #H2 lapply (ldrop_inv_O2 ⦠H1) -H1
- #H destruct /2 width=3 by fqu_lref_O, ex2_intro/
-| * [ #a ] #I #G #L2 #V #T #L1 #H
- [ elim (lleq_inv_bind ⦠H)
- | elim (lleq_inv_flat ⦠H)
- ] -H
- /2 width=3 by fqu_pair_sn, ex2_intro/
-| #a #I #G #L2 #V #T #L1 #H elim (lleq_inv_bind_O ⦠H) -H
- #H3 #H4 /2 width=3 by fqu_bind_dx, ex2_intro/
-| #I #G #L2 #V #T #L1 #H elim (lleq_inv_flat ⦠H) -H
- /2 width=3 by fqu_flat_dx, ex2_intro/
-| #G #L2 #K2 #T #U #e #HLK2 #HTU #L1 #HL12
- elim (ldrop_O1_le (e+1) L1)
- [ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
- | lapply (ldrop_fwd_length_le2 ⦠HLK2) -K2
- lapply (lleq_fwd_length ⦠HL12) -T -U //
- ]
-]
-qed-.
-
-lemma lleq_fquq_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â⸮ â¦G2, K2, U⦠â
- âL1. L1 â[T, 0] L2 â
- ââK1. â¦G1, L1, T⦠â⸮ â¦G2, K1, U⦠& K1 â[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fquq_inv_gen ⦠H) -H
-[ #H elim (lleq_fqu_trans ⦠H ⦠HL12) -L2 /3 width=3 by fqu_fquq, ex2_intro/
-| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
-]
-qed-.
-
-lemma lleq_fqup_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â+ â¦G2, K2, U⦠â
- âL1. L1 â[T, 0] L2 â
- ââK1. â¦G1, L1, T⦠â+ â¦G2, K1, U⦠& K1 â[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H @(fqup_ind ⦠H) -G2 -K2 -U
-[ #G2 #K2 #U #HTU #L1 #HL12 elim (lleq_fqu_trans ⦠HTU ⦠HL12) -L2
- /3 width=3 by fqu_fqup, ex2_intro/
-| #G #G2 #K #K2 #U #U2 #_ #HU2 #IHTU #L1 #HL12 elim (IHTU ⦠HL12) -L2
- #K1 #HTU #HK1 elim (lleq_fqu_trans ⦠HU2 ⦠HK1) -K
- /3 width=5 by fqup_strap1, ex2_intro/
-]
-qed-.
-
-lemma lleq_fqus_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â* â¦G2, K2, U⦠â
- âL1. L1 â[T, 0] L2 â
- ââK1. â¦G1, L1, T⦠â* â¦G2, K1, U⦠& K1 â[U, 0] K2.
-#G1 #G2 #L2 #K2 #T #U #H #L1 #HL12 elim(fqus_inv_gen ⦠H) -H
-[ #H elim (lleq_fqup_trans ⦠H ⦠HL12) -L2 /3 width=3 by fqup_fqus, ex2_intro/
-| * #HG #HL #HT destruct /2 width=3 by ex2_intro/
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_ldrop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_ldrop.ma
deleted file mode 100644
index 9cb597c6f..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_ldrop.ma
+++ /dev/null
@@ -1,123 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpys_lift.ma".
-include "basic_2/substitution/lleq.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_skip: âL1,L2,d,i. yinj i < d â |L1| = |L2| â L1 â[#i, d] L2.
-#L1 #L2 #d #i #Hid #HL12 @conj // -HL12
-#U @conj #H elim (cpys_inv_lref1 ⦠H) -H // *
-#I #Z #Y #X #H elim (ylt_yle_false ⦠Hid ⦠H)
-qed.
-
-lemma lleq_lref: âI1,I2,L1,L2,K1,K2,V,d,i. d ⤠yinj i â
- â©[i] L1 â¡ K1.â{I1}V â â©[i] L2 â¡ K2.â{I2}V â
- K1 â[V, 0] K2 â L1 â[#i, d] L2.
-#I1 #I2 #L1 #L2 #K1 #K2 #V #d #i #Hdi #HLK1 #HLK2 * #HK12 #IH @conj [ -IH | -HK12 ]
-[ lapply (ldrop_fwd_length ⦠HLK1) -HLK1 #H1
- lapply (ldrop_fwd_length ⦠HLK2) -HLK2 #H2
- >H1 >H2 -H1 -H2 normalize //
-| #U @conj #H elim (cpys_inv_lref1 ⦠H) -H // *
- >yminus_Y_inj #I #K #X #W #_ #_ #H #HVW #HWU
- [ letin HLK â HLK1 | letin HLK â HLK2 ]
- lapply (ldrop_mono ⦠H ⦠HLK) -H #H destruct elim (IH W)
- /3 width=7 by cpys_subst_Y2/
-]
-qed.
-
-lemma lleq_free: âL1,L2,d,i. |L1| ⤠i â |L2| ⤠i â |L1| = |L2| â L1 â[#i, d] L2.
-#L1 #L2 #d #i #HL1 #HL2 #HL12 @conj // -HL12
-#U @conj #H elim (cpys_inv_lref1 ⦠H) -H // *
-#I #Z #Y #X #_ #_ #H lapply (ldrop_fwd_length_lt2 ⦠H) -H
-#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
-qed.
-
-(* Properties on relocation *************************************************)
-
-lemma lleq_lift_le: âK1,K2,T,dt. K1 â[T, dt] K2 â
- âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
- âU. â§[d, e] T â¡ U â dt ⤠d â L1 â[U, dt] L2.
-#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hdtd
-lapply (ldrop_fwd_length ⦠HLK1) lapply (ldrop_fwd_length ⦠HLK2)
-#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0
-[ letin HLKA â HLK1 letin HLKB â HLK2 | letin HLKA â HLK2 letin HLKB â HLK1 ]
-elim (cpys_inv_lift1_be ⦠HU0 ⦠HLKA ⦠HTU) // -HU0 >yminus_Y_inj #T0 #HT0 #HTU0
-elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=12 by cpys_lift_be/
-qed-.
-
-lemma lleq_lift_ge: âK1,K2,T,dt. K1 â[T, dt] K2 â
- âL1,L2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
- âU. â§[d, e] T â¡ U â d ⤠dt â L1 â[U, dt+e] L2.
-#K1 #K2 #T #dt * #HK12 #IHT #L1 #L2 #d #e #HLK1 #HLK2 #U #HTU #Hddt
-lapply (ldrop_fwd_length ⦠HLK1) lapply (ldrop_fwd_length ⦠HLK2)
-#H2 #H1 @conj // -HK12 -H1 -H2 #U0 @conj #HU0
-[ letin HLKA â HLK1 letin HLKB â HLK2 | letin HLKA â HLK2 letin HLKB â HLK1 ]
-elim (cpys_inv_lift1_ge ⦠HU0 ⦠HLKA ⦠HTU) /2 width=1 by monotonic_yle_plus_dx/ -HU0 >yplus_minus_inj #T0 #HT0 #HTU0
-elim (IHT T0) [ #H #_ | #_ #H ] -IHT /3 width=10 by cpys_lift_ge/
-qed-.
-
-(* Inversion lemmas on relocation *******************************************)
-
-lemma lleq_inv_lift_le: âL1,L2,U,dt. L1 â[U, dt] L2 â
- âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
- âT. â§[d, e] T â¡ U â dt ⤠d â K1 â[T, dt] K2.
-#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdtd
-lapply (ldrop_fwd_length_minus2 ⦠HLK1) lapply (ldrop_fwd_length_minus2 ⦠HLK2)
-#H2 #H1 @conj // -HL12 -H1 -H2
-#T0 elim (lift_total T0 d e)
-#U0 #HTU0 elim (IH U0) -IH
-#H12 #H21 @conj #HT0
-[ letin HLKA â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ]
-lapply (cpys_lift_be ⦠HT0 ⦠HLKA ⦠HTU ⦠HTU0) // -HT0
->yplus_Y1 #HU0 elim (cpys_inv_lift1_be ⦠(H0 HU0) ⦠HLKB ⦠HTU) // -L1 -L2 -U -Hdtd
-#X #HT0 #HX lapply (lift_inj ⦠HX ⦠HTU0) -U0 //
-qed-.
-
-lemma lleq_inv_lift_ge: âL1,L2,U,dt. L1 â[U, dt] L2 â
- âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
- âT. â§[d, e] T â¡ U â yinj d + e ⤠dt â K1 â[T, dt-e] K2.
-#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hdedt
-lapply (ldrop_fwd_length_minus2 ⦠HLK1) lapply (ldrop_fwd_length_minus2 ⦠HLK2)
-#H2 #H1 @conj // -HL12 -H1 -H2
-elim (yle_inv_plus_inj2 ⦠Hdedt) #Hddt #Hedt
-#T0 elim (lift_total T0 d e)
-#U0 #HTU0 elim (IH U0) -IH
-#H12 #H21 @conj #HT0
-[ letin HLKA â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ]
-lapply (cpys_lift_ge ⦠HT0 ⦠HLKA ⦠HTU ⦠HTU0) // -HT0 -Hddt
->ymax_pre_sn // #HU0 elim (cpys_inv_lift1_ge ⦠(H0 HU0) ⦠HLKB ⦠HTU) // -L1 -L2 -U -Hdedt -Hedt
-#X #HT0 #HX lapply (lift_inj ⦠HX ⦠HTU0) -U0 //
-qed-.
-
-lemma lleq_inv_lift_be: âL1,L2,U,dt. L1 â[U, dt] L2 â
- âK1,K2,d,e. â©[â», d, e] L1 â¡ K1 â â©[â», d, e] L2 â¡ K2 â
- âT. â§[d, e] T â¡ U â d ⤠dt â dt ⤠yinj d + e â K1 â[T, d] K2.
-#L1 #L2 #U #dt * #HL12 #IH #K1 #K2 #d #e #HLK1 #HLK2 #T #HTU #Hddt #Hdtde
-lapply (ldrop_fwd_length_minus2 ⦠HLK1) lapply (ldrop_fwd_length_minus2 ⦠HLK2)
-#H2 #H1 @conj // -HL12 -H1 -H2
-#T0 elim (lift_total T0 d e)
-#U0 #HTU0 elim (IH U0) -IH
-#H12 #H21 @conj #HT0
-[ letin HLKA â HLK1 letin HLKB â HLK2 letin H0 â H12 | letin HLKA â HLK2 letin HLKB â HLK1 letin H0 â H21 ]
-lapply (cpys_lift_ge ⦠HT0 ⦠HLKA ⦠HTU ⦠HTU0) // -HT0
-#HU0 lapply (cpys_weak ⦠HU0 dt (â) ? ?) // -HU0
-#HU0 lapply (H0 HU0)
-#HU0 lapply (cpys_weak ⦠HU0 d (â) ? ?) // -HU0
-#HU0 elim (cpys_inv_lift1_ge_up ⦠HU0 ⦠HLKB ⦠HTU) // -L1 -L2 -U -Hddt -Hdtde
-#X #HT0 #HX lapply (lift_inj ⦠HX ⦠HTU0) -U0 //
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_lleq.ma
deleted file mode 100644
index 2a8873d9b..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lleq_lleq.ma
+++ /dev/null
@@ -1,175 +0,0 @@
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basic_2/substitution/cpys_cpys.ma".
-include "basic_2/substitution/lleq_ldrop.ma".
-
-(* Advanced forward lemmas **************************************************)
-
-lemma lleq_fwd_lref: âL1,L2. âd:ynat. âi:nat. L1 â[#i, d] L2 â
- â¨â¨ |L1| ⤠i ⧠|L2| ⤠i
- | yinj i < d
- | ââI1,I2,K1,K2,V. â©[i] L1 â¡ K1.â{I1}V &
- â©[i] L2 â¡ K2.â{I2}V &
- K1 â[V, yinj 0] K2 & d ⤠yinj i.
-#L1 #L2 #d #i * #HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=3 by or3_intro0, conj/
-elim (ylt_split i d) /2 width=1 by or3_intro1/ #Hdi #Hi
-elim (ldrop_O1_lt ⦠Hi) #I1 #K1 #V1 #HLK1
-elim (ldrop_O1_lt L2 i) // -Hi #I2 #K2 #V2 #HLK2
-lapply (ldrop_fwd_length_minus2 ⦠HLK2) #H
-lapply (ldrop_fwd_length_minus2 ⦠HLK1) >HL12 <H -HL12 -H
-#H lapply (injective_plus_l ⦠H) -H #HK12
-elim (lift_total V1 0 (i+1)) #W1 #HVW1
-elim (lift_total V2 0 (i+1)) #W2 #HVW2
-elim (IH W1) elim (IH W2) #_ #H2 #H1 #_
-elim (cpys_inv_lref1_ldrop ⦠(H1 ?) ⦠HLK2 ⦠HVW1) -H1
-[ elim (cpys_inv_lref1_ldrop ⦠(H2 ?) ⦠HLK1 ⦠HVW2) -H2 ]
-/3 width=7 by cpys_subst, yle_inj/ -W1 -W2 #H12 #_ #_ #H21 #_ #_
-lapply (cpys_antisym_eq ⦠H12 ⦠H21) -H12 -H21 #H destruct
-@or3_intro2 @(ex4_5_intro ⦠HLK1 HLK2) // @conj // -HK12
-#V elim (lift_total V 0 (i+1))
-#W #HVW elim (IH W) -IH #H12 #H21 @conj #H
-[ elim (cpys_inv_lref1_ldrop ⦠(H12 ?) ⦠HLK2 ⦠HVW) -H12 -H21
-| elim (cpys_inv_lref1_ldrop ⦠(H21 ?) ⦠HLK1 ⦠HVW) -H21 -H12
-] [1,3: >yminus_Y_inj ] /3 width=7 by cpys_subst_Y2, yle_inj/
-qed-.
-
-lemma lleq_fwd_lref_dx: âL1,L2,d,i. L1 â[#i, d] L2 â
- âI2,K2,V. â©[i] L2 â¡ K2.â{I2}V â
- i < d â¨
- ââI1,K1. â©[i] L1 â¡ K1.â{I1}V & K1 â[V, 0] K2 & d ⤠i.
-#L1 #L2 #d #i #H #I2 #K2 #V #HLK2 elim (lleq_fwd_lref ⦠H) -H [ * || * ]
-[ #_ #H elim (lt_refl_false i)
- lapply (ldrop_fwd_length_lt2 ⦠HLK2) -HLK2
- /2 width=3 by lt_to_le_to_lt/ (**) (* full auto too slow *)
-| /2 width=1 by or_introl/
-| #I1 #I2 #K11 #K22 #V0 #HLK11 #HLK22 #HV0 #Hdi lapply (ldrop_mono ⦠HLK22 ⦠HLK2) -L2
- #H destruct /3 width=5 by ex3_2_intro, or_intror/
-]
-qed-.
-
-lemma lleq_fwd_lref_sn: âL1,L2,d,i. L1 â[#i, d] L2 â
- âI1,K1,V. â©[i] L1 â¡ K1.â{I1}V â
- i < d â¨
- ââI2,K2. â©[i] L2 â¡ K2.â{I2}V & K1 â[V, 0] K2 & d ⤠i.
-#L1 #L2 #d #i #HL12 #I1 #K1 #V #HLK1 elim (lleq_fwd_lref_dx L2 ⦠d ⦠HLK1) -HLK1
-[2: * ] /4 width=6 by lleq_sym, ex3_2_intro, or_introl, or_intror/
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lemma lleq_inv_lref_ge_dx: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
- âI2,K2,V. â©[i] L2 â¡ K2.â{I2}V â
- ââI1,K1. â©[i] L1 â¡ K1.â{I1}V & K1 â[V, 0] K2.
-#L1 #L2 #d #i #H #Hdi #I2 #K2 #V #HLK2 elim (lleq_fwd_lref_dx ⦠H ⦠HLK2) -L2
-[ #H elim (ylt_yle_false ⦠H Hdi)
-| * /2 width=4 by ex2_2_intro/
-]
-qed-.
-
-lemma lleq_inv_lref_ge_sn: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
- âI1,K1,V. â©[i] L1 â¡ K1.â{I1}V â
- ââI2,K2. â©[i] L2 â¡ K2.â{I2}V & K1 â[V, 0] K2.
-#L1 #L2 #d #i #HL12 #Hdi #I1 #K1 #V #HLK1 elim (lleq_inv_lref_ge_dx L2 ⦠Hdi ⦠HLK1) -Hdi -HLK1
-/3 width=4 by lleq_sym, ex2_2_intro/
-qed-.
-
-lemma lleq_inv_lref_ge_gen: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
- âI1,I2,K1,K2,V1,V2.
- â©[i] L1 â¡ K1.â{I1}V1 â â©[i] L2 â¡ K2.â{I2}V2 â
- V1 = V2 ⧠K1 â[V2, 0] K2.
-#L1 #L2 #d #i #HL12 #Hdi #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
-elim (lleq_inv_lref_ge_sn ⦠HL12 ⦠HLK1) // -L1 -d
-#J #Y #HY lapply (ldrop_mono ⦠HY ⦠HLK2) -L2 -i #H destruct /2 width=1 by conj/
-qed-.
-
-lemma lleq_inv_lref_ge: âL1,L2,d,i. L1 â[#i, d] L2 â d ⤠i â
- âI,K1,K2,V. â©[i] L1 â¡ K1.â{I}V â â©[i] L2 â¡ K2.â{I}V â
- K1 â[V, 0] K2.
-#L1 #L2 #d #i #HL12 #Hdi #I #K1 #K2 #V #HLK1 #HLK2
-elim (lleq_inv_lref_ge_gen ⦠HL12 ⦠HLK1 HLK2) //
-qed-.
-
-(* Advanced properties ******************************************************)
-
-lemma lleq_dec: âT,L1,L2,d. Decidable (L1 â[T, d] L2).
-#T #L1 @(f2_ind ⦠rfw ⦠L1 T) -L1 -T
-#n #IH #L1 * *
-[ #k #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_sort/
-| #i #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|))
- [ #HL12 #d elim (ylt_split i d) /3 width=1 by lleq_skip, or_introl/
- #Hdi elim (lt_or_ge i (|L1|)) #HiL1
- elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, lleq_free/
- elim (ldrop_O1_lt ⦠HiL2) #I2 #K2 #V2 #HLK2
- elim (ldrop_O1_lt ⦠HiL1) #I1 #K1 #V1 #HLK1
- elim (eq_term_dec V2 V1)
- [ #H3 elim (IH K1 V1 ⦠K2 0) destruct
- /3 width=8 by lleq_lref, ldrop_fwd_rfw, or_introl/
- ]
- -IH #H3 @or_intror
- #H elim (lleq_fwd_lref ⦠H) -H [1,3,4,6: * ]
- [1,3: /3 width=4 by lt_to_le_to_lt, lt_refl_false/
- |5,6: /2 width=4 by ylt_yle_false/
- ]
- #Z1 #Z2 #Y1 #Y2 #X #HLY1 #HLY2 #HX #_
- lapply (ldrop_mono ⦠HLY1 ⦠HLK1) -HLY1 -HLK1
- lapply (ldrop_mono ⦠HLY2 ⦠HLK2) -HLY2 -HLK2
- #H2 #H1 destruct /2 width=1 by/
- ]
-| #p #Hn #L2 elim (eq_nat_dec (|L1|) (|L2|)) /3 width=1 by or_introl, lleq_gref/
-| #a #I #V #T #Hn #L2 #d destruct
- elim (IH L1 V ⦠L2 d) /2 width=1 by/
- elim (IH (L1.â{I}V) T ⦠(L2.â{I}V) (d+1)) -IH /3 width=1 by or_introl, lleq_bind/
- #H1 #H2 @or_intror
- #H elim (lleq_inv_bind ⦠H) -H /2 width=1 by/
-| #I #V #T #Hn #L2 #d destruct
- elim (IH L1 V ⦠L2 d) /2 width=1 by/
- elim (IH L1 T ⦠L2 d) -IH /3 width=1 by or_introl, lleq_flat/
- #H1 #H2 @or_intror
- #H elim (lleq_inv_flat ⦠H) -H /2 width=1 by/
-]
--n /4 width=3 by lleq_fwd_length, or_intror/
-qed-.
-
-(* Main properties **********************************************************)
-
-theorem lleq_trans: âd,T. Transitive ⦠(lleq d T).
-#d #T #L1 #L * #HL1 #IH1 #L2 * #HL2 #IH2 /3 width=3 by conj, iff_trans/
-qed-.
-
-theorem lleq_canc_sn: âL,L1,L2,T,d. L â[d, T] L1â L â[d, T] L2 â L1 â[d, T] L2.
-/3 width=3 by lleq_trans, lleq_sym/ qed-.
-
-theorem lleq_canc_dx: âL1,L2,L,T,d. L1 â[d, T] L â L2 â[d, T] L â L1 â[d, T] L2.
-/3 width=3 by lleq_trans, lleq_sym/ qed-.
-
-(* Inversion lemmas on negated lazy quivalence for local environments *******)
-
-lemma nlleq_inv_bind: âa,I,L1,L2,V,T,d. (L1 â[â{a,I}V.T, d] L2 â â¥) â
- (L1 â[V, d] L2 â â¥) ⨠(L1.â{I}V â[T, ⫯d] L2.â{I}V â â¥).
-#a #I #L1 #L2 #V #T #d #H elim (lleq_dec V L1 L2 d)
-/4 width=1 by lleq_bind, or_intror, or_introl/
-qed-.
-
-lemma nlleq_inv_flat: âI,L1,L2,V,T,d. (L1 â[â{I}V.T, d] L2 â â¥) â
- (L1 â[V, d] L2 â â¥) ⨠(L1 â[T, d] L2 â â¥).
-#I #L1 #L2 #V #T #d #H elim (lleq_dec V L1 L2 d)
-/4 width=1 by lleq_flat, or_intror, or_introl/
-qed-.
-
-(* Note: lleq_nlleq_trans: âd,T,L1,L. L1â[T, d] L â
- âL2. (L â[T, d] L2 â â¥) â (L1 â[T, d] L2 â â¥).
-/3 width=3 by lleq_canc_sn/ qed-.
-works with /4 width=8/ so lleq_canc_sn is more convenient
-*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
index 4bb4189a0..a80f8d3b5 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2_src.tbl
@@ -107,7 +107,7 @@ table {
]
[ { "context-sensitive extended computation" * } {
[ "lpxs ( â¦?,?⦠⢠â¡*[?,?] ? )" "lpxs_alt ( â¦?,?⦠⢠â¡â¡*[?,?] ? )" "lpxs_ldrop" + "lpxs_lleq" + "lpxs_aaa" + "lpxs_cpxs" + "lpxs_lpxs" * ]
- [ "cpxs ( â¦?,?⦠⢠? â¡*[?,?] ? )" "cpxs_tstc" + "cpxs_tstc_vector" + "cpxs_lift" + "cpxs_cpys" + "cpxs_lleq" + "cpxs_aaa" + "cpxs_cpxs" * ]
+ [ "cpxs ( â¦?,?⦠⢠? â¡*[?,?] ? )" "cpxs_tstc" + "cpxs_tstc_vector" + "cpxs_leq" + "cpxs_lift" + "cpxs_lleq" + "cpxs_aaa" + "cpxs_cpxs" * ]
}
]
[ { "context-sensitive computation" * } {
@@ -136,8 +136,8 @@ table {
}
]
[ { "context-sensitive extended reduction" * } {
- [ "lpx ( â¦?,?⦠⢠â¡[?,?] ? )" "lpx_ldrop" + "lpx_cpys" + "lpx_lleq" + "lpx_aaa" * ]
- [ "cpx ( â¦?,?⦠⢠? â¡[?,?] ? )" "cpx_lift" + "cpx_cpys" + "cpx_lleq" + "cpx_cix" * ]
+ [ "lpx ( â¦?,?⦠⢠â¡[?,?] ? )" "lpx_ldrop" + "lpx_lleq" + "lpx_aaa" * ]
+ [ "cpx ( â¦?,?⦠⢠? â¡[?,?] ? )" "cpx_leq" + "cpx_lift" + "cpx_lleq" + "cpx_cix" * ]
}
]
[ { "irreducible forms for context-sensitive extended reduction" * } {
@@ -213,17 +213,9 @@ table {
]
class "yellow"
[ { "substitution" * } {
- [ { "lazy equivalence for local environments" * } {
- [ "lleq ( ? â[?,?] ? )" "lleq_alt ( ? ââ[?,?] ? )" "lleq_ldrop" + "lleq_fqus" + "lleq_lleq" + "lleq_ext" * ]
- }
- ]
- [ { "contxt-sensitive extended multiple substitution" * } {
- [ "cpys ( â¦?,?⦠⢠? â¶*[?,?] ? )" "cpys_alt ( â¦?,?⦠⢠? â¶â¶*[?,?] ? )" "cpys_lift" + "cpys_cpys" * ]
- }
- ]
[ { "iterated structural successor for closures" * } {
- [ "fqus ( â¦?,?,?⦠â* â¦?,?,?⦠)" "fqus_alt" + "fqus_fqus" * ]
- [ "fqup ( â¦?,?,?⦠â+ â¦?,?,?⦠)" "fqup_fqup" * ]
+ [ "fqus ( â¦?,?,?⦠â* â¦?,?,?⦠)" "fqus_alt" + "fqus_lleq" + "fqus_fqus" * ]
+ [ "fqup ( â¦?,?,?⦠â+ â¦?,?,?⦠)" "fqup_lleq" + "fqup_fqup" * ]
}
]
[ { "iterated local env. slicing" * } {
@@ -243,17 +235,9 @@ table {
]
class "orange"
[ { "relocation" * } {
- [ { "contxt-sensitive extended ordinary substitution" * } {
- [ "cpy ( â¦?,?⦠⢠? â¶[?,?] ? )" "cpy_lift" + "cpy_cpy" * ]
- }
- ]
- [ { "local env. ref. for extended substitution" * } {
- [ "lsuby ( ? âÃ[?,?] ? )" "lsuby_lsuby" * ]
- }
- ]
[ { "structural successor for closures" * } {
- [ "fquq ( â¦?,?,?⦠â⸮ â¦?,?,?⦠)" "fquq_alt ( â¦?,?,?⦠ââ⸮ â¦?,?,?⦠)" * ]
- [ "fqu ( â¦?,?,?⦠â â¦?,?,?⦠)" * ]
+ [ "fquq ( â¦?,?,?⦠â⸮ â¦?,?,?⦠)" "fquq_alt ( â¦?,?,?⦠ââ⸮ â¦?,?,?⦠)" "fquq_lleq" * ]
+ [ "fqu ( â¦?,?,?⦠â â¦?,?,?⦠)" "fqu_lleq" * ]
}
]
[ { "global env. slicing" * } {
@@ -261,11 +245,11 @@ table {
}
]
[ { "lazy equivalence for local environments" * } {
- [ "lleq ( ? â[?,?] ? )" "lleq_leq" * ]
+ [ "lleq ( ? â[?,?] ? )" "lleq_leq" + "lleq_ldrop" + "lleq_lleq" * ]
}
]
[ { "lazy pointwise extension of a relation" * } {
- [ "llpx_sn" "llpx_sn_leq" * ]
+ [ "llpx_sn" "llpx_sn_leq" + "llpx_sn_ldrop" * ]
}
]
[ { "basic local env. slicing" * } {
diff --git a/matita/matita/predefined_virtuals.ml b/matita/matita/predefined_virtuals.ml
index 14fff0a3c..6311d5933 100644
--- a/matita/matita/predefined_virtuals.ml
+++ b/matita/matita/predefined_virtuals.ml
@@ -1531,6 +1531,7 @@ let predefined_classes = [
["â"; "â¢"; "⧫"; "â¦"; "â"; "â "; ] ;
[">"; "â"; "â§"; "âª"; "»"; "â"; "â¯"; "â±"; "â¸"; "âº"; "â¶"; ] ;
["â¥"; "âª"; "â½"; "⪴"; "⥸"; "â"; ];
+ ["â¨"; "â©"; "â"; ] ;
["a"; "α"; "ð"; "ð"; "ð"; "â"; ] ;
["A"; "âµ"; "ð¸"; "ð"; "â¶"; ] ;
["b"; "β"; "Ã"; "ð"; "ð"; "ð"; "â"; ] ;