From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sun, 26 Oct 2014 19:07:45 +0000 (+0000)
Subject: - some renaming according to the written version of basic_2
X-Git-Tag: make_still_working~810
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=c60524dec7ace912c416a90d6b926bee8553250b;p=helm.git
- some renaming according to the written version of basic_2
- more destructing lemmas invoked in place of buggy destructs
---
diff --git a/matita/matita/contribs/lambdadelta/apps_2/web/apps_2.ldw.xml b/matita/matita/contribs/lambdadelta/apps_2/web/apps_2.ldw.xml
index f13d943f8..e4499a294 100644
--- a/matita/matita/contribs/lambdadelta/apps_2/web/apps_2.ldw.xml
+++ b/matita/matita/contribs/lambdadelta/apps_2/web/apps_2.ldw.xml
@@ -10,32 +10,30 @@
applications of λδ version 2.
In particular it contains the components below.
</body>
- <news date="MLTT1.">
- Martin-Löf's Type Theory with one universe
- using λδ as theory of expressions.
- </news>
- <news date="Functional.">
- The validation algorithm for λδ as implemented in
- <rlink to="implementation.html#helena">Helena 0.8</rlink>.
- </news>
+ <topitem name="MLTT1">
+ <notice class="alpha" notice="MLTT1."/>
+ Martin-Löf's Type Theory with one universe
+ using λδ as theory of expressions.
+ </topitem>
+ <topitem name="functional">
+ <notice class="alpha" notice="Functional."/>
+ The validation algorithm for λδ as implemented in
+ <rlink to="implementation.html#helena">Helena 0.8</rlink>.
+ </topitem>
<section>Summary of the Specification</section>
<body>Here is a numerical acount of the specification's contents
and its timeline.
- Nodes are counted according to the "intrinsinc complexity measure"
- [F. Guidi: "Procedural Representation of CIC Proof Terms"
- Journal of Automated Reasoning 44(1-2), Springer (February 2010),
- pp. 53-78].
</body>
<table name="apps_2_sum"/>
- <news date="2012 February 24.">
+ <news class="alpha" date="2012 February 24.">
The Applications directory is started.
</news>
- <news date="2011 December 20.">
+ <news class="alpha" date="2011 December 20.">
The Functional component is started
inside the specification of λδ version 2.
</news>
- <news date="2011 December 12.">
+ <news class="alpha" date="2011 December 12.">
The MLTT1 component is started.
</news>
@@ -48,9 +46,5 @@
</body>
<table name="apps_2_src"/>
- <section>Physical Structure of the Specification</section>
- <body>The source files are grouped in directories,
- one for each component.
- </body>
<footer/>
</page>
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma
index c36bf3cc7..1f12f5557 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cprs_lift.ma
@@ -51,10 +51,10 @@ qed-.
(* Relocation properties ****************************************************)
(* Basic_1: was: pr3_lift *)
-lemma cprs_lift: âG. l_liftable (cprs G).
-/3 width=10 by l_liftable_LTC, cpr_lift/ qed.
+lemma cprs_lift: âG. d_liftable (cprs G).
+/3 width=10 by d_liftable_LTC, cpr_lift/ qed.
(* Basic_1: was: pr3_gen_lift *)
-lemma cprs_inv_lift1: âG. l_deliftable_sn (cprs G).
-/3 width=6 by l_deliftable_sn_LTC, cpr_inv_lift1/
+lemma cprs_inv_lift1: âG. d_deliftable_sn (cprs G).
+/3 width=6 by d_deliftable_sn_LTC, cpr_inv_lift1/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
index 2f316a737..75c56d8a4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs.ma
@@ -64,13 +64,13 @@ lemma cprs_cpxs: âh,g,G,L,T1,T2. â¦G, L⦠⢠T1 â¡* T2 â â¦G, L⦠â¢
#h #g #G #L #T1 #T2 #H @(cprs_ind ⦠H) -T2 /3 width=3 by cpxs_strap1, cpr_cpx/
qed.
-lemma cpxs_sort: âh,g,G,L,k,l1. deg h g k l1 â
- âl2. l2 ⤠l1 â â¦G, L⦠⢠âk â¡*[h, g] â((next h)^l2 k).
-#h #g #G #L #k #l1 #Hkl1 #l2 @(nat_ind_plus ⦠l2) -l2 /2 width=1 by cpx_cpxs/
-#l2 #IHl2 #Hl21 >iter_SO
-@(cpxs_strap1 ⦠(â(iter l2 â (next h) k)))
+lemma cpxs_sort: âh,g,G,L,k,d1. deg h g k d1 â
+ âd2. d2 ⤠d1 â â¦G, L⦠⢠âk â¡*[h, g] â((next h)^d2 k).
+#h #g #G #L #k #d1 #Hkd1 #d2 @(nat_ind_plus ⦠d2) -d2 /2 width=1 by cpx_cpxs/
+#d2 #IHd2 #Hd21 >iter_SO
+@(cpxs_strap1 ⦠(â(iter d2 â (next h) k)))
[ /3 width=3 by lt_to_le/
-| @(cpx_st ⦠(l1-l2-1)) <plus_minus_m_m
+| @(cpx_st ⦠(d1-d2-1)) <plus_minus_m_m
/2 width=1 by deg_iter, monotonic_le_minus_r/
]
qed.
@@ -137,15 +137,15 @@ qed.
(* Basic inversion lemmas ***************************************************)
lemma cpxs_inv_sort1: âh,g,G,L,U2,k. â¦G, L⦠⢠âk â¡*[h, g] U2 â
- âân,l. deg h g k (n+l) & U2 = â((next h)^n k).
+ âân,d. deg h g k (n+d) & U2 = â((next h)^n k).
#h #g #G #L #U2 #k #H @(cpxs_ind ⦠H) -U2
-[ elim (deg_total h g k) #l #Hkl
- @(ex2_2_intro ⦠0 ⦠Hkl) -Hkl //
-| #U #U2 #_ #HU2 * #n #l #Hknl #H destruct
+[ elim (deg_total h g k) #d #Hkd
+ @(ex2_2_intro ⦠0 ⦠Hkd) -Hkd //
+| #U #U2 #_ #HU2 * #n #d #Hknd #H destruct
elim (cpx_inv_sort1 ⦠HU2) -HU2
[ #H destruct /2 width=4 by ex2_2_intro/
- | * #l0 #Hkl0 #H destruct -l
- @(ex2_2_intro ⦠(n+1) l0) /2 width=1 by deg_inv_prec/ >iter_SO //
+ | * #d0 #Hkd0 #H destruct -d
+ @(ex2_2_intro ⦠(n+1) d0) /2 width=1 by deg_inv_prec/ >iter_SO //
]
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma
index 6f8e0ffcb..c02d0cd06 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_cpxs.ma
@@ -147,7 +147,7 @@ lemma fqu_cpxs_trans_neq: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2,
[1,3: /2 width=4 by fqu_flat_dx, cpxs_flat/
| #H0 destruct /2 width=1 by/
]
-| #G #L #K #T1 #U1 #e #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (e+1))
+| #G #L #K #T1 #U1 #m #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (m+1))
#U2 #HTU2 @(ex3_intro ⦠U2)
[1,3: /2 width=10 by cpxs_lift, fqu_drop/
| #H0 destruct /3 width=5 by lift_inj/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_leq.ma
deleted file mode 100644
index 1d35deb53..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_leq.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_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/cpxs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lift.ma
index de80cace1..494eee301 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lift.ma
@@ -30,17 +30,17 @@ lemma cpxs_delta: âh,g,I,G,L,K,V,V2,i.
]
qed.
-lemma lstas_cpxs: âh,g,G,L,T1,T2,l2. â¦G, L⦠⢠T1 â¢*[h, l2] T2 â
- âl1. â¦G, L⦠⢠T1 âª[h, g] l1 â l2 ⤠l1 â â¦G, L⦠⢠T1 â¡*[h, g] T2.
-#h #g #G #L #T1 #T2 #l2 #H elim H -G -L -T1 -T2 -l2 //
+lemma lstas_cpxs: âh,g,G,L,T1,T2,d2. â¦G, L⦠⢠T1 â¢*[h, d2] T2 â
+ âd1. â¦G, L⦠⢠T1 âª[h, g] d1 â d2 ⤠d1 â â¦G, L⦠⢠T1 â¡*[h, g] T2.
+#h #g #G #L #T1 #T2 #d2 #H elim H -G -L -T1 -T2 -d2 //
[ /3 width=3 by cpxs_sort, da_inv_sort/
-| #G #L #K #V1 #V2 #W2 #i #l2 #HLK #_ #HVW2 #IHV12 #l1 #H #Hl21
- elim (da_inv_lref ⦠H) -H * #K0 #V0 [| #l0 ] #HLK0
+| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
+ elim (da_inv_lref ⦠H) -H * #K0 #V0 [| #d0 ] #HLK0
lapply (drop_mono ⦠HLK0 ⦠HLK) -HLK0 #H destruct /3 width=7 by cpxs_delta/
-| #G #L #K #V1 #V2 #W2 #i #l2 #HLK #_ #HVW2 #IHV12 #l1 #H #Hl21
- elim (da_inv_lref ⦠H) -H * #K0 #V0 [| #l0 ] #HLK0
+| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #d1 #H #Hd21
+ elim (da_inv_lref ⦠H) -H * #K0 #V0 [| #d0 ] #HLK0
lapply (drop_mono ⦠HLK0 ⦠HLK) -HLK0 #H destruct
- #HV1 #H destruct lapply (le_plus_to_le_r ⦠Hl21) -Hl21
+ #HV1 #H destruct lapply (le_plus_to_le_r ⦠Hd21) -Hd21
/3 width=7 by cpxs_delta/
| /4 width=3 by cpxs_bind_dx, da_inv_bind/
| /4 width=3 by cpxs_flat_dx, da_inv_flat/
@@ -68,11 +68,11 @@ qed-.
(* Relocation properties ****************************************************)
-lemma cpxs_lift: âh,g,G. l_liftable (cpxs h g G).
-/3 width=10 by cpx_lift, cpxs_strap1, l_liftable_LTC/ qed.
+lemma cpxs_lift: âh,g,G. d_liftable (cpxs h g G).
+/3 width=10 by cpx_lift, cpxs_strap1, d_liftable_LTC/ qed.
-lemma cpxs_inv_lift1: âh,g,G. l_deliftable_sn (cpxs h g G).
-/3 width=6 by l_deliftable_sn_LTC, cpx_inv_lift1/
+lemma cpxs_inv_lift1: âh,g,G. d_deliftable_sn (cpxs h g G).
+/3 width=6 by d_deliftable_sn_LTC, cpx_inv_lift1/
qed-.
(* Properties on supclosure *************************************************)
@@ -95,8 +95,8 @@ lemma fquq_cpxs_trans: âh,g,G1,G2,L1,L2,T2,U2. â¦G2, L2⦠⢠T2 â¡*[h, g]
qed-.
lemma fquq_lstas_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â⸮ â¦G2, L2, T2⦠â
- âU2,l1. â¦G2, L2⦠⢠T2 â¢*[h, l1] U2 â
- âl2. â¦G2, L2⦠⢠T2 âª[h, g] l2 â l1 ⤠l2 â
+ âU2,d1. â¦G2, L2⦠⢠T2 â¢*[h, d1] U2 â
+ âd2. â¦G2, L2⦠⢠T2 âª[h, g] d2 â d1 ⤠d2 â
ââU1. â¦G1, L1⦠⢠T1 â¡*[h, g] U1 & â¦G1, L1, U1⦠â⸮ â¦G2, L2, U2â¦.
/3 width=5 by fquq_cpxs_trans, lstas_cpxs/ qed-.
@@ -118,7 +118,7 @@ lemma fqus_cpxs_trans: âh,g,G1,G2,L1,L2,T2,U2. â¦G2, L2⦠⢠T2 â¡*[h, g]
qed-.
lemma fqus_lstas_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â* â¦G2, L2, T2⦠â
- âU2,l1. â¦G2, L2⦠⢠T2 â¢*[h, l1] U2 â
- âl2. â¦G2, L2⦠⢠T2 âª[h, g] l2 â l1 ⤠l2 â
+ âU2,d1. â¦G2, L2⦠⢠T2 â¢*[h, d1] U2 â
+ âd2. â¦G2, L2⦠⢠T2 âª[h, g] d2 â d1 ⤠d2 â
ââU1. â¦G1, L1⦠⢠T1 â¡*[h, g] U1 & â¦G1, L1, U1⦠â* â¦G2, L2, U2â¦.
/3 width=6 by fqus_cpxs_trans, lstas_cpxs/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lreq.ma
new file mode 100644
index 000000000..4abb0b91e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_lreq.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_lreq.ma".
+include "basic_2/computation/cpxs.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL COMPUTATION ON TERMS *****************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_cpxs_trans: âh,g,G. lsub_trans ⦠(cpxs h g G) (lreq 0 (â)).
+/3 width=5 by lreq_cpx_trans, LTC_lsub_trans/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts.ma
index 2991481d9..dbed5a334 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/cpxs_tsts.ma
@@ -27,13 +27,13 @@ qed-.
lemma cpxs_fwd_sort: âh,g,G,L,U,k. â¦G, L⦠⢠âk â¡*[h, g] U â
âk â U ⨠â¦G, L⦠⢠â(next h k) â¡*[h, g] U.
#h #g #G #L #U #k #H
-elim (cpxs_inv_sort1 ⦠H) -H #n #l generalize in match k; -k @(nat_ind_plus ⦠n) -n
-[ #k #_ #H -l destruct /2 width=1 by or_introl/
-| #n #IHn #k >plus_plus_comm_23 #Hnl #H destruct
- lapply (deg_next_SO ⦠Hnl) -Hnl #Hnl
- elim (IHn ⦠Hnl) -IHn
+elim (cpxs_inv_sort1 ⦠H) -H #n #d generalize in match k; -k @(nat_ind_plus ⦠n) -n
+[ #k #_ #H -d destruct /2 width=1 by or_introl/
+| #n #IHn #k >plus_plus_comm_23 #Hnd #H destruct
+ lapply (deg_next_SO ⦠Hnd) -Hnd #Hnd
+ elim (IHn ⦠Hnd) -IHn
[ #H lapply (tsts_inv_atom1 ⦠H) -H #H >H -H /2 width=1 by or_intror/
- | generalize in match Hnl; -Hnl @(nat_ind_plus ⦠n) -n
+ | generalize in match Hnd; -Hnd @(nat_ind_plus ⦠n) -n
/4 width=3 by cpxs_strap2, cpx_st, or_intror/
| >iter_SO >iter_n_Sm //
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma
index cd28a524b..9b83b6084 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx.ma
@@ -56,11 +56,11 @@ lemma cnx_csx: âh,g,G,L,T. â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â â¦G, Lâ¦
lemma csx_sort: âh,g,G,L,k. â¦G, L⦠⢠â¬*[h, g] âk.
#h #g #G #L #k elim (deg_total h g k)
-#l generalize in match k; -k @(nat_ind_plus ⦠l) -l /3 width=6 by cnx_csx, cnx_sort/
-#l #IHl #k #Hkl lapply (deg_next_SO ⦠Hkl) -Hkl
-#Hkl @csx_intro #X #H #HX elim (cpx_inv_sort1 ⦠H) -H
+#d generalize in match k; -k @(nat_ind_plus ⦠d) -d /3 width=6 by cnx_csx, cnx_sort/
+#d #IHd #k #Hkd lapply (deg_next_SO ⦠Hkd) -Hkd
+#Hkd @csx_intro #X #H #HX elim (cpx_inv_sort1 ⦠H) -H
[ #H destruct elim HX //
-| -HX * #l0 #_ #H destruct -l0 /2 width=1 by/
+| -HX * #d0 #_ #H destruct -d0 /2 width=1 by/
]
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lift.ma
index 6b2ddff83..fabd9b1b9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_lift.ma
@@ -21,9 +21,9 @@ include "basic_2/computation/csx.ma".
(* Relocation properties ****************************************************)
(* Basic_1: was just: sn3_lift *)
-lemma csx_lift: âh,g,G,L2,L1,T1,s,d,e. â¦G, L1⦠⢠â¬*[h, g] T1 â
- âT2. â¬[s, d, e] L2 â¡ L1 â â¬[d, e] T1 â¡ T2 â â¦G, L2⦠⢠â¬*[h, g] T2.
-#h #g #G #L2 #L1 #T1 #s #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
+lemma csx_lift: âh,g,G,L2,L1,T1,s,l,m. â¦G, L1⦠⢠â¬*[h, g] T1 â
+ âT2. â¬[s, l, m] L2 â¡ L1 â â¬[l, m] T1 â¡ T2 â â¦G, L2⦠⢠â¬*[h, g] T2.
+#h #g #G #L2 #L1 #T1 #s #l #m #H elim H -T1 #T1 #_ #IHT1 #T2 #HL21 #HT12
@csx_intro #T #HLT2 #HT2
elim (cpx_inv_lift1 ⦠HLT2 ⦠HL21 ⦠HT12) -HLT2 #T0 #HT0 #HLT10
@(IHT1 ⦠HLT10) // -L1 -L2 #H destruct
@@ -31,11 +31,11 @@ elim (cpx_inv_lift1 ⦠HLT2 ⦠HL21 ⦠HT12) -HLT2 #T0 #HT0 #HLT10
qed.
(* Basic_1: was just: sn3_gen_lift *)
-lemma csx_inv_lift: âh,g,G,L2,L1,T1,s,d,e. â¦G, L1⦠⢠â¬*[h, g] T1 â
- âT2. â¬[s, d, e] L1 â¡ L2 â â¬[d, e] T2 â¡ T1 â â¦G, L2⦠⢠â¬*[h, g] T2.
-#h #g #G #L2 #L1 #T1 #s #d #e #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
+lemma csx_inv_lift: âh,g,G,L2,L1,T1,s,l,m. â¦G, L1⦠⢠â¬*[h, g] T1 â
+ âT2. â¬[s, l, m] L1 â¡ L2 â â¬[l, m] T2 â¡ T1 â â¦G, L2⦠⢠â¬*[h, g] T2.
+#h #g #G #L2 #L1 #T1 #s #l #m #H elim H -T1 #T1 #_ #IHT1 #T2 #HL12 #HT21
@csx_intro #T #HLT2 #HT2
-elim (lift_total T d e) #T0 #HT0
+elim (lift_total T l m) #T0 #HT0
lapply (cpx_lift ⦠HLT2 ⦠HL12 ⦠HT21 ⦠HT0) -HLT2 #HLT10
@(IHT1 ⦠HLT10) // -L1 -L2 #H destruct
>(lift_inj ⦠HT0 ⦠HT21) in HT2; -T1 /2 width=1 by/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_tsts_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_tsts_vector.ma
index 5c11070ed..d420563d7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/csx_tsts_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/csx_tsts_vector.ma
@@ -35,9 +35,9 @@ qed.
lemma csx_applv_sort: âh,g,G,L,k,Vs. â¦G, L⦠⢠â¬*[h, g] Vs â â¦G, L⦠⢠â¬*[h, g] â¶Vs.âk.
#h #g #G #L #k elim (deg_total h g k)
-#l generalize in match k; -k @(nat_ind_plus ⦠l) -l [ /3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ ]
-#l #IHl #k #Hkl lapply (deg_next_SO ⦠Hkl) -Hkl
-#Hkl #Vs elim Vs -Vs /2 width=1 by/
+#d generalize in match k; -k @(nat_ind_plus ⦠d) -d [ /3 width=6 by csx_applv_cnx, cnx_sort, simple_atom/ ]
+#d #IHd #k #Hkd lapply (deg_next_SO ⦠Hkd) -Hkd
+#Hkd #Vs elim Vs -Vs /2 width=1 by/
#V #Vs #IHVs #HVVs
elim (csxv_inv_cons ⦠HVVs) #HV #HVs
@csx_appl_simple_tsts /2 width=1 by applv_simple, simple_atom/ -IHVs -HV -HVs
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma
index b5a303df6..e39f3a5a3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_fpbs.ma
@@ -57,8 +57,8 @@ lemma cpxs_fpbg: âh,g,G,L,T1,T2. â¦G, L⦠⢠T1 â¡*[h, g] T2 â
/4 width=5 by cpxs_fpbs, fpb_cpx, ex2_3_intro/
qed.
-lemma lstas_fpbg: âh,g,G,L,T1,T2,l2. â¦G, L⦠⢠T1 â¢*[h, l2] T2 â (T1 = T2 â â¥) â
- âl1. l2 ⤠l1 â â¦G, L⦠⢠T1 âª[h, g] l1 â â¦G, L, T1⦠>â¡[h, g] â¦G, L, T2â¦.
+lemma lstas_fpbg: âh,g,G,L,T1,T2,d2. â¦G, L⦠⢠T1 â¢*[h, d2] T2 â (T1 = T2 â â¥) â
+ âd1. d2 ⤠d1 â â¦G, L⦠⢠T1 âª[h, g] d1 â â¦G, L, T1⦠>â¡[h, g] â¦G, L, T2â¦.
/3 width=5 by lstas_cpxs, cpxs_fpbg/ qed.
lemma lpxs_fpbg: âh,g,G,L1,L2,T. â¦G, L1⦠⢠â¡*[h, g] L2 â
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_lift.ma
index 1a6da7e17..d34676e48 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbg_lift.ma
@@ -19,6 +19,6 @@ include "basic_2/computation/fpbg.ma".
(* Advanced properties ******************************************************)
-lemma sta_fpbg: âh,g,G,L,T1,T2,l. â¦G, L⦠⢠T1 âª[h, g] l+1 â
+lemma sta_fpbg: âh,g,G,L,T1,T2,d. â¦G, L⦠⢠T1 âª[h, g] d+1 â
â¦G, L⦠⢠T1 â¢*[h, 1] T2 â â¦G, L, T1⦠>â¡[h, g] â¦G, L, T2â¦.
/4 width=2 by fpb_fpbg, sta_fpb/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_lift.ma
index 57c53b371..a1edecf0c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fpbs_lift.ma
@@ -19,18 +19,18 @@ include "basic_2/computation/fpbs.ma".
(* Advanced properties ******************************************************)
-lemma lstas_fpbs: âh,g,G,L,T1,T2,l2. â¦G, L⦠⢠T1 â¢*[h, l2] T2 â
- âl1. l2 ⤠l1 â â¦G, L⦠⢠T1 âª[h, g] l1 â â¦G, L, T1⦠â¥[h, g] â¦G, L, T2â¦.
+lemma lstas_fpbs: âh,g,G,L,T1,T2,d2. â¦G, L⦠⢠T1 â¢*[h, d2] T2 â
+ âd1. d2 ⤠d1 â â¦G, L⦠⢠T1 âª[h, g] d1 â â¦G, L, T1⦠â¥[h, g] â¦G, L, T2â¦.
/3 width=5 by cpxs_fpbs, lstas_cpxs/ qed.
-lemma sta_fpbs: âh,g,G,L,T,U,l.
- â¦G, L⦠⢠T âª[h, g] l+1 â â¦G, L⦠⢠T â¢*[h, 1] U â
+lemma sta_fpbs: âh,g,G,L,T,U,d.
+ â¦G, L⦠⢠T âª[h, g] d+1 â â¦G, L⦠⢠T â¢*[h, 1] U â
â¦G, L, T⦠â¥[h, g] â¦G, L, Uâ¦.
/2 width=5 by lstas_fpbs/ qed.
(* Note: this is used in the closure proof *)
-lemma cpr_lpr_sta_fpbs: âh,g,G,L1,L2,T1,T2,U2,l2.
+lemma cpr_lpr_sta_fpbs: âh,g,G,L1,L2,T1,T2,U2,d2.
â¦G, L1⦠⢠T1 â¡ T2 â â¦G, L1⦠⢠⡠L2 â
- â¦G, L2⦠⢠T2 âª[h, g] l2+1 â â¦G, L2⦠⢠T2 â¢*[h, 1] U2 â
+ â¦G, L2⦠⢠T2 âª[h, g] d2+1 â â¦G, L2⦠⢠T2 â¢*[h, 1] U2 â
â¦G, L1, T1⦠â¥[h, g] â¦G, L2, U2â¦.
/4 width=5 by fpbs_strap1, cpr_lpr_fpbs, sta_cpx, fpbq_cpx/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma
index d424374a6..403226304 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/fsb_csx.ma
@@ -29,12 +29,12 @@ lemma csx_fsb_fpbs: âh,g,G1,L1,T1. â¦G1, L1⦠⢠â¬*[h, g] T1 â
#G0 #L0 #T0 #IHu #H10 lapply (csx_fpbs_conf ⦠H10) // -HT1
#HT0 generalize in match IHu; -IHu generalize in match H10; -H10
@(lsx_ind ⦠(csx_lsx ⦠HT0 0)) -L0
-#L0 #_ #IHl #H10 #IHu @fsb_intro
-#G2 #L2 #T2 * -G2 -L2 -T2 [ -IHl -IHc | -IHu -IHl | ]
+#L0 #_ #IHd #H10 #IHu @fsb_intro
+#G2 #L2 #T2 * -G2 -L2 -T2 [ -IHd -IHc | -IHu -IHd | ]
[ /4 width=5 by fpbs_fqup_trans, fqu_fqup/
| #T2 #HT02 #HnT02 elim (fpbs_cpx_trans_neq ⦠H10 ⦠HT02 HnT02) -T0
/3 width=4 by/
-| #L2 #HL02 #HnL02 @(IHl ⦠HL02 HnL02) -IHl -HnL02 [ -IHu -IHc | ]
+| #L2 #HL02 #HnL02 @(IHd ⦠HL02 HnL02) -IHd -HnL02 [ -IHu -IHc | ]
[ /3 width=3 by fpbs_lpxs_trans, lpx_lpxs/
| #G3 #L3 #T3 #H03 #_ elim (lpx_fqup_trans ⦠H03 ⦠HL02) -L2
#L4 #T4 elim (eq_term_dec T0 T4) [ -IHc | -IHu ]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp.ma
index 3cab7cbaf..4266d9275 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp.ma
@@ -23,12 +23,12 @@ definition nf â λRR:relation4 genv lenv term term. λRS:relation term.
definition candidate: Type[0] â relation3 genv lenv term.
definition CP0 â λRR:relation4 genv lenv term term. λRS:relation term.
- âG. l_liftable1 (nf RR RS G) (â»).
+ âG. d_liftable1 (nf RR RS G) (â»).
definition CP1 â λRR:relation4 genv lenv term term. λRS:relation term.
âG,L. âk. NF ⦠(RR G L) RS (âk).
-definition CP2 â λRP:candidate. âG. l_liftable1 (RP G) (â»).
+definition CP2 â λRP:candidate. âG. d_liftable1 (RP G) (â»).
definition CP3 â λRP:candidate.
âG,L,T,k. RP G L (ââk.T) â RP G L T.
@@ -44,15 +44,15 @@ record gcp (RR:relation4 genv lenv term term) (RS:relation term) (RP:candidate)
(* Basic properties *********************************************************)
(* Basic_1: was: nf2_lift1 *)
-lemma gcp0_lifts: âRR,RS,RP. gcp RR RS RP â âG. l_liftables1 (nf RR RS G) (â»).
-#RR #RS #RP #H #G @l1_liftable_liftables @(cp0 ⦠H)
+lemma gcp0_lifts: âRR,RS,RP. gcp RR RS RP â âG. d_liftables1 (nf RR RS G) (â»).
+#RR #RS #RP #H #G @d1_liftable_liftables @(cp0 ⦠H)
qed.
-lemma gcp2_lifts: âRR,RS,RP. gcp RR RS RP â âG. l_liftables1 (RP G) (â»).
-#RR #RS #RP #H #G @l1_liftable_liftables @(cp2 ⦠H)
+lemma gcp2_lifts: âRR,RS,RP. gcp RR RS RP â âG. d_liftables1 (RP G) (â»).
+#RR #RS #RP #H #G @d1_liftable_liftables @(cp2 ⦠H)
qed.
(* Basic_1: was only: sns3_lifts1 *)
-lemma gcp2_lifts_all: âRR,RS,RP. gcp RR RS RP â âG. l_liftables1_all (RP G) (â»).
-#RR #RS #RP #H #G @l1_liftables_liftables_all /2 width=7 by gcp2_lifts/
+lemma gcp2_lifts_all: âRR,RS,RP. gcp RR RS RP â âG. d_liftables1_all (RP G) (â»).
+#RR #RS #RP #H #G @d1_liftables_liftables_all /2 width=7 by gcp2_lifts/
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_aaa.ma
index 58a128bd8..6ff7ee804 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_aaa.ma
@@ -37,16 +37,16 @@ theorem acr_aaa_csubc_lifts: âRR,RS,RP.
lapply (acr_gcr ⦠H1RP H2RP B) #HB
elim (lifts_inv_lref1 ⦠H) -H #i1 #Hi1 #H destruct
lapply (drop_fwd_drop2 ⦠HLK1) #HK1b
- elim (drops_drop_trans ⦠HL01 ⦠HLK1) #X #des1 #i0 #HL0 #H #Hi0 #Hdes1
+ elim (drops_drop_trans ⦠HL01 ⦠HLK1) #X #des1 #i0 #HL0 #H #Hi0 #Hcs1
>(at_mono ⦠Hi1 ⦠Hi0) -i1
- elim (drops_inv_skip2 ⦠Hdes1 ⦠H) -des1 #K0 #V0 #des0 #Hdes0 #HK01 #HV10 #H destruct
+ elim (drops_inv_skip2 ⦠Hcs1 ⦠H) -des1 #K0 #V0 #des0 #Hcs0 #HK01 #HV10 #H destruct
elim (lsubc_drop_O1_trans ⦠HL20 ⦠HL0) -HL0 #X #HLK2 #H
elim (lsubc_inv_pair2 ⦠H) -H *
[ #K2 #HK20 #H destruct
elim (lift_total V0 0 (i0 +1)) #V #HV0
- elim (lifts_lift_trans ⦠Hi0 ⦠Hdes0 ⦠HV10 ⦠HV0) -HV10 #V2 #HV12 #HV2
+ elim (lifts_lift_trans ⦠Hi0 ⦠Hcs0 ⦠HV10 ⦠HV0) -HV10 #V2 #HV12 #HV2
lapply (s5 ⦠HB ? G ? ? (â) ⦠HV0 HLK2) /3 width=7 by drops_cons, lifts_cons/ (* Note: uses IHB HL20 V2 HV0 *)
- | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hdes0
+ | -HLK1 -IHB -HL01 -HL20 -HK1b -Hi0 -Hcs0
#K2 #V2 #A2 #HKV2A #H1KV0A #H2KV0A #_ #H1 #H2 destruct
lapply (drop_fwd_drop2 ⦠HLK2) #HLK2b
lapply (aaa_lifts ⦠HK01 ⦠HV10 HKV1B) -HKV1B -HK01 -HV10 #HKV0B
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_cr.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_cr.ma
index fa040b915..1da427f5b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_cr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/gcp_cr.ma
@@ -79,14 +79,14 @@ interpretation
(* Basic properties *********************************************************)
(* Basic 1: was: sc3_lift *)
-lemma gcr_lift: âRR,RS,RP. gcp RR RS RP â âA,G. l_liftable1 (acr RP A G) (â»).
+lemma gcr_lift: âRR,RS,RP. gcp RR RS RP â âA,G. d_liftable1 (acr RP A G) (â»).
#RR #RS #RP #H #A elim A -A
/3 width=8 by cp2, drops_cons, lifts_cons/
qed.
(* Basic_1: was: sc3_lift1 *)
-lemma gcr_lifts: âRR,RS,RP. gcp RR RS RP â âA,G. l_liftables1 (acr RP A G) (â»).
-#RR #RS #RP #H #A #G @l1_liftable_liftables /2 width=7 by gcr_lift/
+lemma gcr_lifts: âRR,RS,RP. gcp RR RS RP â âA,G. d_liftables1 (acr RP A G) (â»).
+#RR #RS #RP #H #A #G @d1_liftable_liftables /2 width=7 by gcr_lift/
qed.
(* Basic_1: was:
@@ -120,11 +120,11 @@ lemma acr_gcr: âRR,RS,RP. gcp RR RS RP â gcr RR RS RP RP â
| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #des #HL0 #H #HB
elim (lifts_inv_applv1 ⦠H) -H #V0s #Y #HV0s #HY #H destruct
elim (lifts_inv_lref1 ⦠HY) -HY #i0 #Hi0 #H destruct
- elim (drops_drop_trans ⦠HL0 ⦠HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hdes0
+ elim (drops_drop_trans ⦠HL0 ⦠HLK) #X #des0 #i1 #HL02 #H #Hi1 #Hcs0
>(at_mono ⦠Hi1 ⦠Hi0) in HL02; -i1 #HL02
- elim (drops_inv_skip2 ⦠Hdes0 ⦠H) -H -des0 #L2 #W1 #des0 #Hdes0 #HLK #HVW1 #H destruct
+ elim (drops_inv_skip2 ⦠Hcs0 ⦠H) -H -des0 #L2 #W1 #des0 #Hcs0 #HLK #HVW1 #H destruct
elim (lift_total W1 0 (i0 + 1)) #W2 #HW12
- elim (lifts_lift_trans ⦠Hdes0 ⦠HVW1 ⦠HW12) // -Hdes0 -Hi0 #V3 #HV13 #HVW2
+ elim (lifts_lift_trans ⦠Hcs0 ⦠HVW1 ⦠HW12) // -Hcs0 -Hi0 #V3 #HV13 #HVW2
>(lift_mono ⦠HV13 ⦠HV12) in HVW2; -V3 #HVW2
@(s5 ⦠IHA ⦠(V0 @ V0s) ⦠HW12 HL02) /3 width=5 by lifts_applv/
| #G #L #V1s #V2s #HV12s #a #V #T #HA #HV #L0 #V10 #X #des #HL0 #H #HB
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx.ma
index 03187e34c..8997c6ec9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx.ma
@@ -18,15 +18,15 @@ include "basic_2/computation/lsx.ma".
(* SN EXTENDED STRONGLY CONORMALIZING LOCAL ENVIRONMENTS ********************)
inductive lcosx (h) (g) (G): relation2 ynat lenv â
-| lcosx_sort: âd. lcosx h g G d (â)
+| lcosx_sort: âl. lcosx h g G l (â)
| lcosx_skip: âI,L,T. lcosx h g G 0 L â lcosx h g G 0 (L.â{I}T)
-| lcosx_pair: âI,L,T,d. G ⢠â¬*[h, g, T, d] L â
- lcosx h g G d L â lcosx h g G (⫯d) (L.â{I}T)
+| lcosx_pair: âI,L,T,l. G ⢠â¬*[h, g, T, l] L â
+ lcosx h g G l L â lcosx h g G (⫯l) (L.â{I}T)
.
interpretation
"sn extended strong conormalization (local environment)"
- 'CoSN h g d G L = (lcosx h g G d L).
+ 'CoSN h g l G L = (lcosx h g G l L).
(* Basic properties *********************************************************)
@@ -34,43 +34,43 @@ lemma lcosx_O: âh,g,G,L. G ⢠~â¬*[h, g, 0] L.
#h #g #G #L elim L /2 width=1 by lcosx_skip/
qed.
-lemma lcosx_drop_trans_lt: âh,g,G,L,d. G ⢠~â¬*[h, g, d] L â
- âI,K,V,i. â¬[i] L â¡ K.â{I}V â i < d â
- G ⢠~â¬*[h, g, â«°(d-i)] K ⧠G ⢠â¬*[h, g, V, â«°(d-i)] K.
-#h #g #G #L #d #H elim H -L -d
-[ #d #J #K #V #i #H elim (drop_inv_atom1 ⦠H) -H #H destruct
+lemma lcosx_drop_trans_lt: âh,g,G,L,l. G ⢠~â¬*[h, g, l] L â
+ âI,K,V,i. â¬[i] L â¡ K.â{I}V â i < l â
+ G ⢠~â¬*[h, g, â«°(l-i)] K ⧠G ⢠â¬*[h, g, V, â«°(l-i)] K.
+#h #g #G #L #l #H elim H -L -l
+[ #l #J #K #V #i #H elim (drop_inv_atom1 ⦠H) -H #H destruct
| #I #L #T #_ #_ #J #K #V #i #_ #H elim (ylt_yle_false ⦠H) -H //
-| #I #L #T #d #HT #HL #IHL #J #K #V #i #H #Hid
+| #I #L #T #l #HT #HL #IHL #J #K #V #i #H #Hil
elim (drop_inv_O1_pair1 ⦠H) -H * #Hi #HLK destruct
[ >ypred_succ /2 width=1 by conj/
- | lapply (ylt_pred ⦠Hid ?) -Hid /2 width=1 by ylt_inj/ >ypred_succ #Hid
+ | lapply (ylt_pred ⦠Hil ?) -Hil /2 width=1 by ylt_inj/ >ypred_succ #Hil
elim (IHL ⦠HLK ?) -IHL -HLK <yminus_inj >yminus_SO2 //
- <(ypred_succ d) in ⢠(%â%â?); >yminus_pred /2 width=1 by ylt_inj, conj/
+ <(ypred_succ l) in ⢠(%â%â?); >yminus_pred /2 width=1 by ylt_inj, conj/
]
]
qed-.
(* Basic inversion lemmas ***************************************************)
-fact lcosx_inv_succ_aux: âh,g,G,L,x. G ⢠~â¬*[h, g, x] L â âd. x = ⫯d â
+fact lcosx_inv_succ_aux: âh,g,G,L,x. G ⢠~â¬*[h, g, x] L â âl. x = ⫯l â
L = â â¨
- ââI,K,V. L = K.â{I}V & G ⢠~â¬*[h, g, d] K &
- G ⢠â¬*[h, g, V, d] K.
-#h #g #G #L #d * -L -d /2 width=1 by or_introl/
+ ââI,K,V. L = K.â{I}V & G ⢠~â¬*[h, g, l] K &
+ G ⢠â¬*[h, g, V, l] K.
+#h #g #G #L #l * -L -l /2 width=1 by or_introl/
[ #I #L #T #_ #x #H elim (ysucc_inv_O_sn ⦠H)
-| #I #L #T #d #HT #HL #x #H <(ysucc_inj ⦠H) -x
+| #I #L #T #l #HT #HL #x #H <(ysucc_inj ⦠H) -x
/3 width=6 by ex3_3_intro, or_intror/
]
qed-.
-lemma lcosx_inv_succ: âh,g,G,L,d. G ⢠~â¬*[h, g, ⫯d] L â L = â â¨
- ââI,K,V. L = K.â{I}V & G ⢠~â¬*[h, g, d] K &
- G ⢠â¬*[h, g, V, d] K.
+lemma lcosx_inv_succ: âh,g,G,L,l. G ⢠~â¬*[h, g, ⫯l] L â L = â â¨
+ ââI,K,V. L = K.â{I}V & G ⢠~â¬*[h, g, l] K &
+ G ⢠â¬*[h, g, V, l] K.
/2 width=3 by lcosx_inv_succ_aux/ qed-.
-lemma lcosx_inv_pair: âh,g,I,G,L,T,d. G ⢠~â¬*[h, g, ⫯d] L.â{I}T â
- G ⢠~â¬*[h, g, d] L ⧠G ⢠â¬*[h, g, T, d] L.
-#h #g #I #G #L #T #d #H elim (lcosx_inv_succ ⦠H) -H
+lemma lcosx_inv_pair: âh,g,I,G,L,T,l. G ⢠~â¬*[h, g, ⫯l] L.â{I}T â
+ G ⢠~â¬*[h, g, l] L ⧠G ⢠â¬*[h, g, T, l] L.
+#h #g #I #G #L #T #l #H elim (lcosx_inv_succ ⦠H) -H
[ #H destruct
| * #Z #Y #X #H destruct /2 width=1 by conj/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpx.ma
index 8583b9dea..12b9a05d2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lcosx_cpx.ma
@@ -23,42 +23,42 @@ include "basic_2/computation/lcosx.ma".
(* Properties on extended context-sensitive parallel reduction for term *****)
lemma lsx_cpx_trans_lcosx: âh,g,G,L,T1,T2. â¦G, L⦠⢠T1 â¡[h, g] T2 â
- âd. G ⢠~â¬*[h, g, d] L â
- G ⢠â¬*[h, g, T1, d] L â G ⢠â¬*[h, g, T2, d] L.
+ âl. G ⢠~â¬*[h, g, l] L â
+ G ⢠â¬*[h, g, T1, l] L â G ⢠â¬*[h, g, T2, l] L.
#h #g #G #L #T1 #T2 #H elim H -G -L -T1 -T2 //
-[ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #d #HL #H
- elim (ylt_split i d) #Hdi [ -H | -HL ]
- [ <(ymax_pre_sn d (⫯i)) /2 width=1 by ylt_fwd_le_succ/
- elim (lcosx_drop_trans_lt ⦠HL ⦠HLK) // -HL -Hdi
+[ #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #l #HL #H
+ elim (ylt_split i l) #Hli [ -H | -HL ]
+ [ <(ymax_pre_sn l (⫯i)) /2 width=1 by ylt_fwd_le_succ/
+ elim (lcosx_drop_trans_lt ⦠HL ⦠HLK) // -HL -Hli
lapply (drop_fwd_drop2 ⦠HLK) -HLK /3 width=7 by lsx_lift_ge/
- | lapply (lsx_fwd_lref_be ⦠H ⦠HLK) // -H -Hdi
+ | lapply (lsx_fwd_lref_be ⦠H ⦠HLK) // -H -Hli
lapply (drop_fwd_drop2 ⦠HLK) -HLK
/4 width=10 by lsx_ge, lsx_lift_le/
]
-| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #d #HL #H
+| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #l #HL #H
elim (lsx_inv_bind ⦠H) -H #HV1 #HT1
@lsx_bind /2 width=2 by/ (**) (* explicit constructor *)
- @(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
+ @(lsx_lreq_conf ⦠(L.â{I}V1)) /3 width=1 by lcosx_pair, lreq_succ/
+| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #l #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
+| #G #L #V #U1 #U2 #T2 #_ #HTU2 #IHU12 #l #HL #H
elim (lsx_inv_bind ⦠H) -H
/4 width=9 by lcosx_pair, lsx_inv_lift_ge, drop_drop/
-| #G #L #V #T1 #T2 #_ #IHT12 #d #HL #H
+| #G #L #V #T1 #T2 #_ #IHT12 #l #HL #H
elim (lsx_inv_flat ⦠H) -H /2 width=1 by/
-| #G #L #V1 #V2 #T #_ #IHV12 #d #HL #H
+| #G #L #V1 #V2 #T #_ #IHV12 #l #HL #H
elim (lsx_inv_flat ⦠H) -H /2 width=1 by/
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #d #HL #H
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #l #HL #H
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_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
+ @(lsx_lreq_conf ⦠(L.âW1)) /3 width=1 by lcosx_pair, lreq_succ/
+| #a #G #L #V1 #V2 #U2 #W1 #W2 #T1 #T2 #_ #HVU2 #_ #_ #IHV12 #IHW12 #IHT12 #l #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, drop_drop/ ]
- @(lsx_leq_conf ⦠(L.âW1)) /3 width=1 by lcosx_pair, leq_succ/
+ @(lsx_lreq_conf ⦠(L.âW1)) /3 width=1 by lcosx_pair, lreq_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 a3ca7a1ac..6f6045dab 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lpxs_lleq.ma
@@ -14,7 +14,7 @@
include "basic_2/multiple/lleq_lleq.ma".
include "basic_2/reduction/lpx_lleq.ma".
-include "basic_2/computation/cpxs_leq.ma".
+include "basic_2/computation/cpxs_lreq.ma".
include "basic_2/computation/lpxs_drop.ma".
include "basic_2/computation/lpxs_cpxs.ma".
@@ -23,19 +23,19 @@ include "basic_2/computation/lpxs_cpxs.ma".
(* Properties on lazy equivalence for local environments ********************)
lemma lleq_lpxs_trans: âh,g,G,L2,K2. â¦G, L2⦠⢠â¡*[h, g] K2 â
- âL1,T,d. L1 â¡[T, d] L2 â
- ââK1. â¦G, L1⦠⢠â¡*[h, g] K1 & K1 â¡[T, d] K2.
+ âL1,T,l. L1 â¡[T, l] L2 â
+ ââK1. â¦G, L1⦠⢠â¡*[h, g] K1 & K1 â¡[T, l] K2.
#h #g #G #L2 #K2 #H @(lpxs_ind ⦠H) -K2 /2 width=3 by ex2_intro/
-#K #K2 #_ #HK2 #IH #L1 #T #d #HT elim (IH ⦠HT) -L2
+#K #K2 #_ #HK2 #IH #L1 #T #l #HT elim (IH ⦠HT) -L2
#L #HL1 #HT elim (lleq_lpx_trans ⦠HK2 ⦠HT) -K
/3 width=3 by lpxs_strap1, ex2_intro/
qed-.
-lemma lpxs_nlleq_inv_step_sn: âh,g,G,L1,L2,T,d. â¦G, L1⦠⢠â¡*[h, g] L2 â (L1 â¡[T, d] L2 â â¥) â
- ââL,L0. â¦G, L1⦠⢠â¡[h, g] L & L1 â¡[T, d] L â ⥠& â¦G, L⦠⢠â¡*[h, g] L0 & L0 â¡[T, d] L2.
-#h #g #G #L1 #L2 #T #d #H @(lpxs_ind_dx ⦠H) -L1
+lemma lpxs_nlleq_inv_step_sn: âh,g,G,L1,L2,T,l. â¦G, L1⦠⢠â¡*[h, g] L2 â (L1 â¡[T, l] L2 â â¥) â
+ ââL,L0. â¦G, L1⦠⢠â¡[h, g] L & L1 â¡[T, l] L â ⥠& â¦G, L⦠⢠â¡*[h, g] L0 & L0 â¡[T, l] L2.
+#h #g #G #L1 #L2 #T #l #H @(lpxs_ind_dx ⦠H) -L1
[ #H elim H -H //
-| #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L d) #H
+| #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L l) #H
[ -H1 -H2 elim IH2 -IH2 /3 width=3 by lleq_trans/ -H12
#L0 #L3 #H1 #H2 #H3 #H4 lapply (lleq_nlleq_trans ⦠H ⦠H2) -H2
#H2 elim (lleq_lpx_trans ⦠H1 ⦠H) -L
@@ -64,8 +64,8 @@ lemma lpxs_lleq_fqu_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2,
/3 width=4 by lpxs_pair, fqu_bind_dx, ex3_intro/
| #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat ⦠H) -H
/2 width=4 by fqu_flat_dx, ex3_intro/
-| #G1 #L1 #L #T1 #U1 #e #HL1 #HTU1 #K1 #H1KL1 #H2KL1
- elim (drop_O1_le (â») (e+1) K1)
+| #G1 #L1 #L #T1 #U1 #m #HL1 #HTU1 #K1 #H1KL1 #H2KL1
+ elim (drop_O1_le (â») (m+1) K1)
[ #K #HK1 lapply (lleq_inv_lift_le ⦠H2KL1 ⦠HK1 HL1 ⦠HTU1 ?) -H2KL1 //
#H2KL elim (lpxs_drop_trans_O1 ⦠H1KL1 ⦠HL1) -L1
#K0 #HK10 #H1KL lapply (drop_mono ⦠HK10 ⦠HK1) -HK10 #H destruct
@@ -110,32 +110,32 @@ elim (fqus_inv_gen ⦠H) -H
]
qed-.
-fact leq_lpxs_trans_lleq_aux: âh,g,G,L1,L0,d,e. L1 ⩬[d, e] L0 â e = â â
- âL2. â¦G, L0⦠⢠â¡*[h, g] L2 â
- ââL. L ⩬[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
-[ #d #e #_ #L2 #H >(lpxs_inv_atom1 ⦠H) -H
+fact lreq_lpxs_trans_lleq_aux: âh,g,G,L1,L0,l,m. L1 ⩬[l, m] L0 â m = â â
+ âL2. â¦G, L0⦠⢠â¡*[h, g] L2 â
+ ââL. L ⩬[l, m] L2 & â¦G, L1⦠⢠â¡*[h, g] L &
+ (âT. L0 â¡[T, l] L2 â L1 â¡[T, l] L).
+#h #g #G #L1 #L0 #l #m #H elim H -L1 -L0 -l -m
+[ #l #m #_ #L2 #H >(lpxs_inv_atom1 ⦠H) -H
/3 width=5 by ex3_intro, conj/
-| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
-| #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
+| #I #L1 #L0 #V1 #m #HL10 #IHL10 #Hm #Y #H
elim (lpxs_inv_pair1 ⦠H) -H #L2 #V2 #HL02 #HV02 #H destruct
- lapply (ysucc_inv_Y_dx ⦠He) -He #He
+ lapply (ysucc_inv_Y_dx ⦠Hm) -Hm #Hm
elim (IHL10 ⦠HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
- @(ex3_intro ⦠(L.â{I}V2)) /3 width=3 by lpxs_pair, leq_cpxs_trans, leq_pair/
+ @(ex3_intro ⦠(L.â{I}V2)) /3 width=3 by lpxs_pair, lreq_cpxs_trans, lreq_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
+ @conj #H @(lleq_lreq_repl ⦠H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #l #m #HL10 #IHL10 #Hm #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, leq_succ/
+ @(ex3_intro ⦠(L.â{I1}V1)) /3 width=1 by lpxs_pair, lreq_succ/
#T elim (IH T) #HL0dx #HL0sn
- @conj #H @(lleq_leq_repl ⦠H) -H /3 width=1 by leq_sym, leq_succ/
+ @conj #H @(lleq_lreq_repl ⦠H) -H /3 width=1 by lreq_sym, lreq_succ/
]
qed-.
-lemma leq_lpxs_trans_lleq: âh,g,G,L1,L0,d. L1 ⩬[d, â] L0 â
- âL2. â¦G, L0⦠⢠â¡*[h, g] L2 â
- ââL. L ⩬[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-.
+lemma lreq_lpxs_trans_lleq: âh,g,G,L1,L0,l. L1 ⩬[l, â] L0 â
+ âL2. â¦G, L0⦠⢠â¡*[h, g] L2 â
+ ââL. L ⩬[l, â] L2 & â¦G, L1⦠⢠â¡*[h, g] L &
+ (âT. L0 â¡[T, l] L2 â L1 â¡[T, l] L).
+/2 width=1 by lreq_lpxs_trans_lleq_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drop.ma
index 404f7f582..64b236cb0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drop.ma
@@ -21,18 +21,18 @@ include "basic_2/computation/lsubc.ma".
(* Basic_1: was: csubc_drop_conf_O *)
(* Note: the constant 0 can not be generalized *)
-lemma lsubc_drop_O1_trans: âRP,G,L1,L2. G ⢠L1 â«[RP] L2 â âK2,s,e. â¬[s, 0, e] L2 â¡ K2 â
- ââK1. â¬[s, 0, e] L1 â¡ K1 & G ⢠K1 â«[RP] K2.
+lemma lsubc_drop_O1_trans: âRP,G,L1,L2. G ⢠L1 â«[RP] L2 â âK2,s,m. â¬[s, 0, m] L2 â¡ K2 â
+ ââK1. â¬[s, 0, m] L1 â¡ K1 & G ⢠K1 â«[RP] K2.
#RP #G #L1 #L2 #H elim H -L1 -L2
-[ #X #s #e #H elim (drop_inv_atom1 ⦠H) -H /4 width=3 by drop_atom, ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #X #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #H destruct
+[ #X #s #m #H elim (drop_inv_atom1 ⦠H) -H /4 width=3 by drop_atom, ex2_intro/
+| #I #L1 #L2 #V #_ #IHL12 #X #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #H destruct
[ elim (IHL12 L2 s 0) -IHL12 // #X #H <(drop_inv_O2 ⦠H) -H
/3 width=3 by lsubc_pair, drop_pair, ex2_intro/
| elim (IHL12 ⦠H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #V #W #A #HV #H1W #H2W #_ #IHL12 #X #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #H destruct
+| #L1 #L2 #V #W #A #HV #H1W #H2W #_ #IHL12 #X #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #H destruct
[ elim (IHL12 L2 s 0) -IHL12 // #X #H <(drop_inv_O2 ⦠H) -H
/3 width=8 by lsubc_beta, drop_pair, ex2_intro/
| elim (IHL12 ⦠H) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
@@ -42,20 +42,20 @@ qed-.
(* Basic_1: was: csubc_drop_conf_rev *)
lemma drop_lsubc_trans: âRR,RS,RP. gcp RR RS RP â
- âG,L1,K1,d,e. â¬[â», d, e] L1 â¡ K1 â âK2. G ⢠K1 â«[RP] K2 â
- ââL2. G ⢠L1 â«[RP] L2 & â¬[â», d, e] L2 â¡ K2.
-#RR #RS #RP #Hgcp #G #L1 #K1 #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H elim (lsubc_inv_atom1 ⦠H) -H
- >He /2 width=3 by ex2_intro/
+ âG,L1,K1,l,m. â¬[â», l, m] L1 â¡ K1 â âK2. G ⢠K1 â«[RP] K2 â
+ ââL2. G ⢠L1 â«[RP] L2 & â¬[â», l, m] L2 â¡ K2.
+#RR #RS #RP #Hgcp #G #L1 #K1 #l #m #H elim H -L1 -K1 -l -m
+[ #l #m #Hm #X #H elim (lsubc_inv_atom1 ⦠H) -H
+ >Hm /2 width=3 by ex2_intro/
| #L1 #I #V1 #X #H
elim (lsubc_inv_pair1 ⦠H) -H *
[ #K1 #HLK1 #H destruct /3 width=3 by lsubc_pair, drop_pair, ex2_intro/
| #K1 #V #W1 #A #HV1 #H1W1 #H2W1 #HLK1 #H1 #H2 #H3 destruct
/3 width=4 by lsubc_beta, drop_pair, ex2_intro/
]
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #K2 #HK12
+| #I #L1 #K1 #V1 #m #_ #IHLK1 #K2 #HK12
elim (IHLK1 ⦠HK12) -K1 /3 width=5 by lsubc_pair, drop_drop, ex2_intro/
-| #I #L1 #K1 #V1 #V2 #d #e #HLK1 #HV21 #IHLK1 #X #H
+| #I #L1 #K1 #V1 #V2 #l #m #HLK1 #HV21 #IHLK1 #X #H
elim (lsubc_inv_pair1 ⦠H) -H *
[ #K2 #HK12 #H destruct
elim (IHLK1 ⦠HK12) -K1 /3 width=5 by lsubc_pair, drop_skip, ex2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drops.ma
index 088cd210d..f76065b80 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsubc_drops.ma
@@ -24,7 +24,7 @@ lemma drops_lsubc_trans: âRR,RS,RP. gcp RR RS RP â
ââL2. G ⢠L1 â«[RP] L2 & â¬*[â», des] L2 â¡ K2.
#RR #RS #RP #Hgcp #G #L1 #K1 #des #H elim H -L1 -K1 -des
[ /2 width=3 by drops_nil, ex2_intro/
-| #L1 #L #K1 #des #d #e #_ #HLK1 #IHL #K2 #HK12
+| #L1 #L #K1 #des #l #m #_ #HLK1 #IHL #K2 #HK12
elim (drop_lsubc_trans ⦠Hgcp ⦠HLK1 ⦠HK12) -Hgcp -K1 #K #HLK #HK2
elim (IHL ⦠HLK) -IHL -HLK /3 width=5 by drops_cons, ex2_intro/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
index 3c2b01edd..1cc24d432 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx.ma
@@ -19,91 +19,91 @@ include "basic_2/reduction/lpx.ma".
(* SN EXTENDED STRONGLY NORMALIZING LOCAL ENVIRONMENTS **********************)
definition lsx: âh. sd h â relation4 ynat term genv lenv â
- λh,g,d,T,G. SN ⦠(lpx h g G) (lleq d T).
+ λh,g,l,T,G. SN ⦠(lpx h g G) (lleq l T).
interpretation
"extended strong normalization (local environment)"
- 'SN h g d T G L = (lsx h g T d G L).
+ 'SN h g l T G L = (lsx h g T l G L).
(* Basic eliminators ********************************************************)
-lemma lsx_ind: âh,g,G,T,d. âR:predicate lenv.
- (âL1. G ⢠â¬*[h, g, T, d] L1 â
- (âL2. â¦G, L1⦠⢠â¡[h, g] L2 â (L1 â¡[T, d] L2 â â¥) â R L2) â
+lemma lsx_ind: âh,g,G,T,l. âR:predicate lenv.
+ (âL1. G ⢠â¬*[h, g, T, l] L1 â
+ (âL2. â¦G, L1⦠⢠â¡[h, g] L2 â (L1 â¡[T, l] L2 â â¥) â R L2) â
R L1
) â
- âL. G ⢠â¬*[h, g, T, d] L â R L.
-#h #g #G #T #d #R #H0 #L1 #H elim H -L1
+ âL. G ⢠â¬*[h, g, T, l] L â R L.
+#h #g #G #T #l #R #H0 #L1 #H elim H -L1
/5 width=1 by lleq_sym, SN_intro/
qed-.
(* Basic properties *********************************************************)
-lemma lsx_intro: âh,g,G,L1,T,d.
- (âL2. â¦G, L1⦠⢠â¡[h, g] L2 â (L1 â¡[T, d] L2 â â¥) â G ⢠â¬*[h, g, T, d] L2) â
- G ⢠â¬*[h, g, T, d] L1.
+lemma lsx_intro: âh,g,G,L1,T,l.
+ (âL2. â¦G, L1⦠⢠â¡[h, g] L2 â (L1 â¡[T, l] L2 â â¥) â G ⢠â¬*[h, g, T, l] L2) â
+ G ⢠â¬*[h, g, T, l] L1.
/5 width=1 by lleq_sym, SN_intro/ qed.
-lemma lsx_atom: âh,g,G,T,d. G ⢠â¬*[h, g, T, d] â.
-#h #g #G #T #d @lsx_intro
+lemma lsx_atom: âh,g,G,T,l. G ⢠â¬*[h, g, T, l] â.
+#h #g #G #T #l @lsx_intro
#X #H #HT lapply (lpx_inv_atom1 ⦠H) -H
#H destruct elim HT -HT //
qed.
-lemma lsx_sort: âh,g,G,L,d,k. G ⢠â¬*[h, g, âk, d] L.
-#h #g #G #L1 #d #k @lsx_intro
+lemma lsx_sort: âh,g,G,L,l,k. G ⢠â¬*[h, g, âk, l] L.
+#h #g #G #L1 #l #k @lsx_intro
#L2 #HL12 #H elim H -H
/3 width=4 by lpx_fwd_length, lleq_sort/
qed.
-lemma lsx_gref: âh,g,G,L,d,p. G ⢠â¬*[h, g, §p, d] L.
-#h #g #G #L1 #d #p @lsx_intro
+lemma lsx_gref: âh,g,G,L,l,p. G ⢠â¬*[h, g, §p, l] L.
+#h #g #G #L1 #l #p @lsx_intro
#L2 #HL12 #H elim H -H
/3 width=4 by lpx_fwd_length, lleq_gref/
qed.
-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 ⢠â¬*[h, g, U, dt] L â G ⢠â¬*[h, g, U, d] L.
-#h #g #G #L #T #U #dt #d #e #Hdtde #HTU #H @(lsx_ind ⦠H) -L
+lemma lsx_ge_up: âh,g,G,L,T,U,lt,l,m. lt ⤠yinj l + yinj m â
+ â¬[l, m] T â¡ U â G ⢠â¬*[h, g, U, lt] L â G ⢠â¬*[h, g, U, l] L.
+#h #g #G #L #T #U #lt #l #m #Hltlm #HTU #H @(lsx_ind ⦠H) -L
/5 width=7 by lsx_intro, lleq_ge_up/
qed-.
-lemma lsx_ge: âh,g,G,L,T,d1,d2. d1 ⤠d2 â
- G ⢠â¬*[h, g, T, d1] L â G ⢠â¬*[h, g, T, d2] L.
-#h #g #G #L #T #d1 #d2 #Hd12 #H @(lsx_ind ⦠H) -L
+lemma lsx_ge: âh,g,G,L,T,l1,l2. l1 ⤠l2 â
+ G ⢠â¬*[h, g, T, l1] L â G ⢠â¬*[h, g, T, l2] L.
+#h #g #G #L #T #l1 #l2 #Hl12 #H @(lsx_ind ⦠H) -L
/5 width=7 by lsx_intro, lleq_ge/
qed-.
(* Basic forward lemmas *****************************************************)
-lemma lsx_fwd_bind_sn: âh,g,a,I,G,L,V,T,d. G ⢠â¬*[h, g, â{a,I}V.T, d] L â
- G ⢠â¬*[h, g, V, d] L.
-#h #g #a #I #G #L #V #T #d #H @(lsx_ind ⦠H) -L
+lemma lsx_fwd_bind_sn: âh,g,a,I,G,L,V,T,l. G ⢠â¬*[h, g, â{a,I}V.T, l] L â
+ G ⢠â¬*[h, g, V, l] L.
+#h #g #a #I #G #L #V #T #l #H @(lsx_ind ⦠H) -L
#L1 #_ #IHL1 @lsx_intro
#L2 #HL12 #HV @IHL1 /3 width=4 by lleq_fwd_bind_sn/
qed-.
-lemma lsx_fwd_flat_sn: âh,g,I,G,L,V,T,d. G ⢠â¬*[h, g, â{I}V.T, d] L â
- G ⢠â¬*[h, g, V, d] L.
-#h #g #I #G #L #V #T #d #H @(lsx_ind ⦠H) -L
+lemma lsx_fwd_flat_sn: âh,g,I,G,L,V,T,l. G ⢠â¬*[h, g, â{I}V.T, l] L â
+ G ⢠â¬*[h, g, V, l] L.
+#h #g #I #G #L #V #T #l #H @(lsx_ind ⦠H) -L
#L1 #_ #IHL1 @lsx_intro
#L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_sn/
qed-.
-lemma lsx_fwd_flat_dx: âh,g,I,G,L,V,T,d. G ⢠â¬*[h, g, â{I}V.T, d] L â
- G ⢠â¬*[h, g, T, d] L.
-#h #g #I #G #L #V #T #d #H @(lsx_ind ⦠H) -L
+lemma lsx_fwd_flat_dx: âh,g,I,G,L,V,T,l. G ⢠â¬*[h, g, â{I}V.T, l] L â
+ G ⢠â¬*[h, g, T, l] L.
+#h #g #I #G #L #V #T #l #H @(lsx_ind ⦠H) -L
#L1 #_ #IHL1 @lsx_intro
#L2 #HL12 #HV @IHL1 /3 width=3 by lleq_fwd_flat_dx/
qed-.
-lemma lsx_fwd_pair_sn: âh,g,I,G,L,V,T,d. G ⢠â¬*[h, g, â¡{I}V.T, d] L â
- G ⢠â¬*[h, g, V, d] L.
+lemma lsx_fwd_pair_sn: âh,g,I,G,L,V,T,l. G ⢠â¬*[h, g, â¡{I}V.T, l] L â
+ G ⢠â¬*[h, g, V, l] L.
#h #g * /2 width=4 by lsx_fwd_bind_sn, lsx_fwd_flat_sn/
qed-.
(* Basic inversion lemmas ***************************************************)
-lemma lsx_inv_flat: âh,g,I,G,L,V,T,d. G ⢠â¬*[h, g, â{I}V.T, d] L â
- G ⢠â¬*[h, g, V, d] L ⧠G ⢠â¬*[h, g, T, d] L.
+lemma lsx_inv_flat: âh,g,I,G,L,V,T,l. G ⢠â¬*[h, g, â{I}V.T, l] L â
+ G ⢠â¬*[h, g, V, l] L ⧠G ⢠â¬*[h, g, T, l] L.
/3 width=3 by lsx_fwd_flat_sn, lsx_fwd_flat_dx, conj/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma
index 48a58442b..05637c0e7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_alt.ma
@@ -20,57 +20,57 @@ include "basic_2/computation/lsx.ma".
(* alternative definition of lsx *)
definition lsxa: âh. sd h â relation4 ynat term genv lenv â
- λh,g,d,T,G. SN ⦠(lpxs h g G) (lleq d T).
+ λh,g,l,T,G. SN ⦠(lpxs h g G) (lleq l T).
interpretation
"extended strong normalization (local environment) alternative"
- 'SNAlt h g d T G L = (lsxa h g T d G L).
+ 'SNAlt h g l T G L = (lsxa h g T l G L).
(* Basic eliminators ********************************************************)
-lemma lsxa_ind: âh,g,G,T,d. âR:predicate lenv.
- (âL1. G ⢠â¬â¬*[h, g, T, d] L1 â
- (âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â (L1 â¡[T, d] L2 â â¥) â R L2) â
+lemma lsxa_ind: âh,g,G,T,l. âR:predicate lenv.
+ (âL1. G ⢠â¬â¬*[h, g, T, l] L1 â
+ (âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â (L1 â¡[T, l] L2 â â¥) â R L2) â
R L1
) â
- âL. G ⢠â¬â¬*[h, g, T, d] L â R L.
-#h #g #G #T #d #R #H0 #L1 #H elim H -L1
+ âL. G ⢠â¬â¬*[h, g, T, l] L â R L.
+#h #g #G #T #l #R #H0 #L1 #H elim H -L1
/5 width=1 by lleq_sym, SN_intro/
qed-.
(* Basic properties *********************************************************)
-lemma lsxa_intro: âh,g,G,L1,T,d.
- (âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â (L1 â¡[T, d] L2 â â¥) â G ⢠â¬â¬*[h, g, T, d] L2) â
- G ⢠â¬â¬*[h, g, T, d] L1.
+lemma lsxa_intro: âh,g,G,L1,T,l.
+ (âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â (L1 â¡[T, l] L2 â â¥) â G ⢠â¬â¬*[h, g, T, l] L2) â
+ G ⢠â¬â¬*[h, g, T, l] L1.
/5 width=1 by lleq_sym, SN_intro/ qed.
-fact lsxa_intro_aux: âh,g,G,L1,T,d.
- (âL,L2. â¦G, L⦠⢠â¡*[h, g] L2 â L1 â¡[T, d] L â (L1 â¡[T, d] L2 â â¥) â G ⢠â¬â¬*[h, g, T, d] L2) â
- G ⢠â¬â¬*[h, g, T, d] L1.
+fact lsxa_intro_aux: âh,g,G,L1,T,l.
+ (âL,L2. â¦G, L⦠⢠â¡*[h, g] L2 â L1 â¡[T, l] L â (L1 â¡[T, l] L2 â â¥) â G ⢠â¬â¬*[h, g, T, l] L2) â
+ G ⢠â¬â¬*[h, g, T, l] L1.
/4 width=3 by lsxa_intro/ qed-.
-lemma lsxa_lleq_trans: âh,g,T,G,L1,d. G ⢠â¬â¬*[h, g, T, d] L1 â
- âL2. L1 â¡[T, d] L2 â G ⢠â¬â¬*[h, g, T, d] L2.
-#h #g #T #G #L1 #d #H @(lsxa_ind ⦠H) -L1
+lemma lsxa_lleq_trans: âh,g,T,G,L1,l. G ⢠â¬â¬*[h, g, T, l] L1 â
+ âL2. L1 â¡[T, l] L2 â G ⢠â¬â¬*[h, g, T, l] L2.
+#h #g #T #G #L1 #l #H @(lsxa_ind ⦠H) -L1
#L1 #_ #IHL1 #L2 #HL12 @lsxa_intro
#K2 #HLK2 #HnLK2 elim (lleq_lpxs_trans ⦠HLK2 ⦠HL12) -HLK2
/5 width=4 by lleq_canc_sn, lleq_trans/
qed-.
-lemma lsxa_lpxs_trans: âh,g,T,G,L1,d. G ⢠â¬â¬*[h, g, T, d] L1 â
- âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â G ⢠â¬â¬*[h, g, T, d] L2.
-#h #g #T #G #L1 #d #H @(lsxa_ind ⦠H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
-elim (lleq_dec T L1 L2 d) /3 width=4 by lsxa_lleq_trans/
+lemma lsxa_lpxs_trans: âh,g,T,G,L1,l. G ⢠â¬â¬*[h, g, T, l] L1 â
+ âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â G ⢠â¬â¬*[h, g, T, l] L2.
+#h #g #T #G #L1 #l #H @(lsxa_ind ⦠H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
+elim (lleq_dec T L1 L2 l) /3 width=4 by lsxa_lleq_trans/
qed-.
-lemma lsxa_intro_lpx: âh,g,G,L1,T,d.
- (âL2. â¦G, L1⦠⢠â¡[h, g] L2 â (L1 â¡[T, d] L2 â â¥) â G ⢠â¬â¬*[h, g, T, d] L2) â
- G ⢠â¬â¬*[h, g, T, d] L1.
-#h #g #G #L1 #T #d #IH @lsxa_intro_aux
+lemma lsxa_intro_lpx: âh,g,G,L1,T,l.
+ (âL2. â¦G, L1⦠⢠â¡[h, g] L2 â (L1 â¡[T, l] L2 â â¥) â G ⢠â¬â¬*[h, g, T, l] L2) â
+ G ⢠â¬â¬*[h, g, T, l] L1.
+#h #g #G #L1 #T #l #IH @lsxa_intro_aux
#L #L2 #H @(lpxs_ind_dx ⦠H) -L
[ #H destruct #H elim H //
-| #L0 #L elim (lleq_dec T L1 L d) /3 width=1 by/
+| #L0 #L elim (lleq_dec T L1 L l) /3 width=1 by/
#HnT #HL0 #HL2 #_ #HT #_ elim (lleq_lpx_trans ⦠HL0 ⦠HT) -L0
#L0 #HL10 #HL0 @(lsxa_lpxs_trans ⦠HL2) -HL2
/5 width=3 by lsxa_lleq_trans, lleq_trans/
@@ -79,37 +79,37 @@ qed-.
(* Main properties **********************************************************)
-theorem lsx_lsxa: âh,g,G,L,T,d. G ⢠â¬*[h, g, T, d] L â G ⢠â¬â¬*[h, g, T, d] L.
-#h #g #G #L #T #d #H @(lsx_ind ⦠H) -L
+theorem lsx_lsxa: âh,g,G,L,T,l. G ⢠â¬*[h, g, T, l] L â G ⢠â¬â¬*[h, g, T, l] L.
+#h #g #G #L #T #l #H @(lsx_ind ⦠H) -L
/4 width=1 by lsxa_intro_lpx/
qed.
(* Main inversion lemmas ****************************************************)
-theorem lsxa_inv_lsx: âh,g,G,L,T,d. G ⢠â¬â¬*[h, g, T, d] L â G ⢠â¬*[h, g, T, d] L.
-#h #g #G #L #T #d #H @(lsxa_ind ⦠H) -L
+theorem lsxa_inv_lsx: âh,g,G,L,T,l. G ⢠â¬â¬*[h, g, T, l] L â G ⢠â¬*[h, g, T, l] L.
+#h #g #G #L #T #l #H @(lsxa_ind ⦠H) -L
/4 width=1 by lsx_intro, lpx_lpxs/
qed-.
(* Advanced properties ******************************************************)
-lemma lsx_intro_alt: âh,g,G,L1,T,d.
- (âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â (L1 â¡[T, d] L2 â â¥) â G ⢠â¬*[h, g, T, d] L2) â
- G ⢠â¬*[h, g, T, d] L1.
+lemma lsx_intro_alt: âh,g,G,L1,T,l.
+ (âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â (L1 â¡[T, l] L2 â â¥) â G ⢠â¬*[h, g, T, l] L2) â
+ G ⢠â¬*[h, g, T, l] L1.
/6 width=1 by lsxa_inv_lsx, lsx_lsxa, lsxa_intro/ qed.
-lemma lsx_lpxs_trans: âh,g,G,L1,T,d. G ⢠â¬*[h, g, T, d] L1 â
- âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â G ⢠â¬*[h, g, T, d] L2.
+lemma lsx_lpxs_trans: âh,g,G,L1,T,l. G ⢠â¬*[h, g, T, l] L1 â
+ âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â G ⢠â¬*[h, g, T, l] L2.
/4 width=3 by lsxa_inv_lsx, lsx_lsxa, lsxa_lpxs_trans/ qed-.
(* Advanced eliminators *****************************************************)
-lemma lsx_ind_alt: âh,g,G,T,d. âR:predicate lenv.
- (âL1. G ⢠â¬*[h, g, T, d] L1 â
- (âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â (L1 â¡[T, d] L2 â â¥) â R L2) â
+lemma lsx_ind_alt: âh,g,G,T,l. âR:predicate lenv.
+ (âL1. G ⢠â¬*[h, g, T, l] L1 â
+ (âL2. â¦G, L1⦠⢠â¡*[h, g] L2 â (L1 â¡[T, l] L2 â â¥) â R L2) â
R L1
) â
- âL. G ⢠â¬*[h, g, T, d] L â R L.
-#h #g #G #T #d #R #IH #L #H @(lsxa_ind h g G T d ⦠L)
+ âL. G ⢠â¬*[h, g, T, l] L â R L.
+#h #g #G #T #l #R #IH #L #H @(lsxa_ind h g G T l ⦠L)
/4 width=1 by lsxa_inv_lsx, lsx_lsxa/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma
index a49378955..67f754b7b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_csx.ma
@@ -19,10 +19,10 @@ include "basic_2/computation/lcosx_cpx.ma".
(* Advanced properties ******************************************************)
-lemma lsx_lref_be_lpxs: âh,g,I,G,K1,V,i,d. d ⤠yinj i â â¦G, K1⦠⢠â¬*[h, g] V â
+lemma lsx_lref_be_lpxs: âh,g,I,G,K1,V,i,l. l ⤠yinj i â â¦G, K1⦠⢠â¬*[h, g] V â
âK2. G ⢠â¬*[h, g, V, 0] K2 â â¦G, K1⦠⢠â¡*[h, g] K2 â
- âL2. â¬[i] L2 â¡ K2.â{I}V â G ⢠â¬*[h, g, #i, d] L2.
-#h #g #I #G #K1 #V #i #d #Hdi #H @(csx_ind_alt ⦠H) -V
+ âL2. â¬[i] L2 â¡ K2.â{I}V â G ⢠â¬*[h, g, #i, l] L2.
+#h #g #I #G #K1 #V #i #l #Hli #H @(csx_ind_alt ⦠H) -V
#V0 #_ #IHV0 #K2 #H @(lsx_ind ⦠H) -K2
#K0 #HK0 #IHK0 #HK10 #L0 #HLK0 @lsx_intro
#L2 #HL02 #HnL02 elim (lpx_drop_conf ⦠HLK0 ⦠HL02) -HL02
@@ -35,25 +35,25 @@ elim (eq_term_dec V0 V2) #HnV02 destruct [ -IHV0 -HV02 -HK0 | -IHK0 -HnL02 -HLK0
]
qed.
-lemma lsx_lref_be: âh,g,I,G,K,V,i,d. d ⤠yinj i â â¦G, K⦠⢠â¬*[h, g] V â
+lemma lsx_lref_be: âh,g,I,G,K,V,i,l. l ⤠yinj i â â¦G, K⦠⢠â¬*[h, g] V â
G ⢠â¬*[h, g, V, 0] K â
- âL. â¬[i] L â¡ K.â{I}V â G ⢠â¬*[h, g, #i, d] L.
+ âL. â¬[i] L â¡ K.â{I}V â G ⢠â¬*[h, g, #i, l] L.
/2 width=8 by lsx_lref_be_lpxs/ qed.
(* Main properties **********************************************************)
-theorem csx_lsx: âh,g,G,L,T. â¦G, L⦠⢠â¬*[h, g] T â âd. G ⢠â¬*[h, g, T, d] L.
+theorem csx_lsx: âh,g,G,L,T. â¦G, L⦠⢠â¬*[h, g] T â âl. G ⢠â¬*[h, g, T, l] L.
#h #g #G #L #T @(fqup_wf_ind_eq ⦠G L T) -G -L -T
#Z #Y #X #IH #G #L * * //
-[ #i #HG #HL #HT #H #d destruct
+[ #i #HG #HL #HT #H #l destruct
elim (lt_or_ge i (|L|)) /2 width=1 by lsx_lref_free/
- elim (ylt_split i d) /2 width=1 by lsx_lref_skip/
- #Hdi #Hi elim (drop_O1_lt (â») ⦠Hi) -Hi
+ elim (ylt_split i l) /2 width=1 by lsx_lref_skip/
+ #Hli #Hi elim (drop_O1_lt (â») ⦠Hi) -Hi
#I #K #V #HLK lapply (csx_inv_lref_bind ⦠HLK ⦠H) -H
/4 width=6 by lsx_lref_be, fqup_lref/
-| #a #I #V #T #HG #HL #HT #H #d destruct
+| #a #I #V #T #HG #HL #HT #H #l destruct
elim (csx_fwd_bind ⦠H) -H /3 width=1 by lsx_bind/
-| #I #V #T #HG #HL #HT #H #d destruct
+| #I #V #T #HG #HL #HT #H #l destruct
elim (csx_fwd_flat ⦠H) -H /3 width=1 by lsx_flat/
]
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_drop.ma
index 664ff62a8..f1a4c7b64 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_drop.ma
@@ -20,27 +20,27 @@ include "basic_2/computation/lsx.ma".
(* Advanced properties ******************************************************)
-lemma lsx_lref_free: âh,g,G,L,d,i. |L| ⤠i â G ⢠â¬*[h, g, #i, d] L.
-#h #g #G #L1 #d #i #HL1 @lsx_intro
+lemma lsx_lref_free: âh,g,G,L,l,i. |L| ⤠i â G ⢠â¬*[h, g, #i, l] L.
+#h #g #G #L1 #l #i #HL1 @lsx_intro
#L2 #HL12 #H elim H -H
/4 width=6 by lpx_fwd_length, lleq_free, le_repl_sn_conf_aux/
qed.
-lemma lsx_lref_skip: âh,g,G,L,d,i. yinj i < d â G ⢠â¬*[h, g, #i, d] L.
-#h #g #G #L1 #d #i #HL1 @lsx_intro
+lemma lsx_lref_skip: âh,g,G,L,l,i. yinj i < l â G ⢠â¬*[h, g, #i, l] L.
+#h #g #G #L1 #l #i #HL1 @lsx_intro
#L2 #HL12 #H elim H -H
/3 width=4 by lpx_fwd_length, lleq_skip/
qed.
(* Advanced forward lemmas **************************************************)
-lemma lsx_fwd_lref_be: âh,g,I,G,L,d,i. d ⤠yinj i â G ⢠â¬*[h, g, #i, d] L â
+lemma lsx_fwd_lref_be: âh,g,I,G,L,l,i. l ⤠yinj i â G ⢠â¬*[h, g, #i, l] L â
âK,V. â¬[i] L â¡ K.â{I}V â G ⢠â¬*[h, g, V, 0] K.
-#h #g #I #G #L #d #i #Hdi #H @(lsx_ind ⦠H) -L
+#h #g #I #G #L #l #i #Hli #H @(lsx_ind ⦠H) -L
#L1 #_ #IHL1 #K1 #V #HLK1 @lsx_intro
#K2 #HK12 #HnK12 lapply (drop_fwd_drop2 ⦠HLK1)
#H2LK1 elim (drop_lpx_trans ⦠H2LK1 ⦠HK12) -H2LK1 -HK12
-#L2 #HL12 #H2LK2 #H elim (leq_drop_conf_be ⦠H ⦠HLK1) -H /2 width=1 by ylt_inj/
+#L2 #HL12 #H2LK2 #H elim (lreq_drop_conf_be ⦠H ⦠HLK1) -H /2 width=1 by ylt_inj/
#Y #_ #HLK2 lapply (drop_fwd_drop2 ⦠HLK2)
#HY lapply (drop_mono ⦠HY ⦠H2LK2) -HY -H2LK2 #H destruct
/4 width=10 by lleq_inv_lref_ge/
@@ -48,19 +48,19 @@ qed-.
(* Properties on relocation *************************************************)
-lemma lsx_lift_le: âh,g,G,K,T,U,dt,d,e. dt ⤠yinj d â
- â¬[d, e] T â¡ U â G ⢠â¬*[h, g, T, dt] K â
- âL. â¬[â», d, e] L â¡ K â G ⢠â¬*[h, g, U, dt] L.
-#h #g #G #K #T #U #dt #d #e #Hdtd #HTU #H @(lsx_ind ⦠H) -K
+lemma lsx_lift_le: âh,g,G,K,T,U,lt,l,m. lt ⤠yinj l â
+ â¬[l, m] T â¡ U â G ⢠â¬*[h, g, T, lt] K â
+ âL. â¬[â», l, m] L â¡ K â G ⢠â¬*[h, g, U, lt] L.
+#h #g #G #K #T #U #lt #l #m #Hltl #HTU #H @(lsx_ind ⦠H) -K
#K1 #_ #IHK1 #L1 #HLK1 @lsx_intro
#L2 #HL12 #HnU elim (lpx_drop_conf ⦠HLK1 ⦠HL12) -HL12
/4 width=10 by lleq_lift_le/
qed-.
-lemma lsx_lift_ge: âh,g,G,K,T,U,dt,d,e. yinj d ⤠dt â
- â¬[d, e] T â¡ U â G ⢠â¬*[h, g, T, dt] K â
- âL. â¬[â», d, e] L â¡ K â G ⢠â¬*[h, g, U, dt + e] L.
-#h #g #G #K #T #U #dt #d #e #Hddt #HTU #H @(lsx_ind ⦠H) -K
+lemma lsx_lift_ge: âh,g,G,K,T,U,lt,l,m. yinj l ⤠lt â
+ â¬[l, m] T â¡ U â G ⢠â¬*[h, g, T, lt] K â
+ âL. â¬[â», l, m] L â¡ K â G ⢠â¬*[h, g, U, lt + m] L.
+#h #g #G #K #T #U #lt #l #m #Hllt #HTU #H @(lsx_ind ⦠H) -K
#K1 #_ #IHK1 #L1 #HLK1 @lsx_intro
#L2 #HL12 #HnU elim (lpx_drop_conf ⦠HLK1 ⦠HL12) -HL12
/4 width=9 by lleq_lift_ge/
@@ -68,28 +68,28 @@ qed-.
(* Inversion lemmas on relocation *******************************************)
-lemma lsx_inv_lift_le: âh,g,G,L,T,U,dt,d,e. dt ⤠yinj d â
- â¬[d, e] T â¡ U â G ⢠â¬*[h, g, U, dt] L â
- âK. â¬[â», d, e] L â¡ K â G ⢠â¬*[h, g, T, dt] K.
-#h #g #G #L #T #U #dt #d #e #Hdtd #HTU #H @(lsx_ind ⦠H) -L
+lemma lsx_inv_lift_le: âh,g,G,L,T,U,lt,l,m. lt ⤠yinj l â
+ â¬[l, m] T â¡ U â G ⢠â¬*[h, g, U, lt] L â
+ âK. â¬[â», l, m] L â¡ K â G ⢠â¬*[h, g, T, lt] K.
+#h #g #G #L #T #U #lt #l #m #Hltl #HTU #H @(lsx_ind ⦠H) -L
#L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
#K2 #HK12 #HnT elim (drop_lpx_trans ⦠HLK1 ⦠HK12) -HK12
/4 width=10 by lleq_inv_lift_le/
qed-.
-lemma lsx_inv_lift_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 ⢠â¬*[h, g, U, dt] L â
- âK. â¬[â», d, e] L â¡ K â G ⢠â¬*[h, g, T, d] K.
-#h #g #G #L #T #U #dt #d #e #Hddt #Hdtde #HTU #H @(lsx_ind ⦠H) -L
+lemma lsx_inv_lift_be: âh,g,G,L,T,U,lt,l,m. yinj l ⤠lt â lt ⤠l + m â
+ â¬[l, m] T â¡ U â G ⢠â¬*[h, g, U, lt] L â
+ âK. â¬[â», l, m] L â¡ K â G ⢠â¬*[h, g, T, l] K.
+#h #g #G #L #T #U #lt #l #m #Hllt #Hltlm #HTU #H @(lsx_ind ⦠H) -L
#L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
#K2 #HK12 #HnT elim (drop_lpx_trans ⦠HLK1 ⦠HK12) -HK12
/4 width=11 by lleq_inv_lift_be/
qed-.
-lemma lsx_inv_lift_ge: âh,g,G,L,T,U,dt,d,e. yinj d + yinj e ⤠dt â
- â¬[d, e] T â¡ U â G ⢠â¬*[h, g, U, dt] L â
- âK. â¬[â», d, e] L â¡ K â G ⢠â¬*[h, g, T, dt-e] K.
-#h #g #G #L #T #U #dt #d #e #Hdedt #HTU #H @(lsx_ind ⦠H) -L
+lemma lsx_inv_lift_ge: âh,g,G,L,T,U,lt,l,m. yinj l + yinj m ⤠lt â
+ â¬[l, m] T â¡ U â G ⢠â¬*[h, g, U, lt] L â
+ âK. â¬[â», l, m] L â¡ K â G ⢠â¬*[h, g, T, lt-m] K.
+#h #g #G #L #T #U #lt #l #m #Hlmlt #HTU #H @(lsx_ind ⦠H) -L
#L1 #_ #IHL1 #K1 #HLK1 @lsx_intro
#K2 #HK12 #HnT elim (drop_lpx_trans ⦠HLK1 ⦠HK12) -HK12
/4 width=9 by lleq_inv_lift_ge/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpx.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpx.ma
index 034024b72..05fc77f90 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpx.ma
@@ -20,44 +20,44 @@ include "basic_2/computation/lsx.ma".
(* Advanced properties ******************************************************)
-lemma lsx_lleq_trans: âh,g,T,G,L1,d. G ⢠â¬*[h, g, T, d] L1 â
- âL2. L1 â¡[T, d] L2 â G ⢠â¬*[h, g, T, d] L2.
-#h #g #T #G #L1 #d #H @(lsx_ind ⦠H) -L1
+lemma lsx_lleq_trans: âh,g,T,G,L1,l. G ⢠â¬*[h, g, T, l] L1 â
+ âL2. L1 â¡[T, l] L2 â G ⢠â¬*[h, g, T, l] L2.
+#h #g #T #G #L1 #l #H @(lsx_ind ⦠H) -L1
#L1 #_ #IHL1 #L2 #HL12 @lsx_intro
#K2 #HLK2 #HnLK2 elim (lleq_lpx_trans ⦠HLK2 ⦠HL12) -HLK2
/5 width=4 by lleq_canc_sn, lleq_trans/
qed-.
-lemma lsx_lpx_trans: âh,g,T,G,L1,d. G ⢠â¬*[h, g, T, d] L1 â
- âL2. â¦G, L1⦠⢠â¡[h, g] L2 â G ⢠â¬*[h, g, T, d] L2.
-#h #g #T #G #L1 #d #H @(lsx_ind ⦠H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
-elim (lleq_dec T L1 L2 d) /3 width=4 by lsx_lleq_trans/
+lemma lsx_lpx_trans: âh,g,T,G,L1,l. G ⢠â¬*[h, g, T, l] L1 â
+ âL2. â¦G, L1⦠⢠â¡[h, g] L2 â G ⢠â¬*[h, g, T, l] L2.
+#h #g #T #G #L1 #l #H @(lsx_ind ⦠H) -L1 #L1 #HL1 #IHL1 #L2 #HL12
+elim (lleq_dec T L1 L2 l) /3 width=4 by lsx_lleq_trans/
qed-.
-lemma lsx_leq_conf: âh,g,G,L1,T,d. G ⢠â¬*[h, g, T, d] L1 â
- âL2. L1 ⩬[d, â] L2 â G ⢠â¬*[h, g, T, d] L2.
-#h #g #G #L1 #T #d #H @(lsx_ind ⦠H) -L1
+lemma lsx_lreq_conf: âh,g,G,L1,T,l. G ⢠â¬*[h, g, T, l] L1 â
+ âL2. L1 ⩬[l, â] L2 â G ⢠â¬*[h, g, T, l] L2.
+#h #g #G #L1 #T #l #H @(lsx_ind ⦠H) -L1
#L1 #_ #IHL1 #L2 #HL12 @lsx_intro
-#L3 #HL23 #HnL23 elim (leq_lpx_trans_lleq ⦠HL12 ⦠HL23) -HL12 -HL23
+#L3 #HL23 #HnL23 elim (lreq_lpx_trans_lleq ⦠HL12 ⦠HL23) -HL12 -HL23
#L0 #HL03 #HL10 #H elim (H T) -H /4 width=4 by/
qed-.
(* Advanced forward lemmas **************************************************)
-lemma lsx_fwd_bind_dx: âh,g,a,I,G,L,V,T,d. G ⢠â¬*[h, g, â{a,I}V.T, d] L â
- G ⢠â¬*[h, g, T, ⫯d] L.â{I}V.
-#h #g #a #I #G #L #V1 #T #d #H @(lsx_ind ⦠H) -L
+lemma lsx_fwd_bind_dx: âh,g,a,I,G,L,V,T,l. G ⢠â¬*[h, g, â{a,I}V.T, l] L â
+ G ⢠â¬*[h, g, T, ⫯l] L.â{I}V.
+#h #g #a #I #G #L #V1 #T #l #H @(lsx_ind ⦠H) -L
#L1 #_ #IHL1 @lsx_intro
#Y #H #HT elim (lpx_inv_pair1 ⦠H) -H
#L2 #V2 #HL12 #_ #H destruct
-@(lsx_leq_conf ⦠(L2.â{I}V1)) /2 width=1 by leq_succ/
+@(lsx_lreq_conf ⦠(L2.â{I}V1)) /2 width=1 by lreq_succ/
@IHL1 // #H @HT -IHL1 -HL12 -HT
-@(lleq_leq_trans ⦠(L2.â{I}V1))
-/2 width=2 by lleq_fwd_bind_dx, leq_succ/
+@(lleq_lreq_trans ⦠(L2.â{I}V1))
+/2 width=2 by lleq_fwd_bind_dx, lreq_succ/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lsx_inv_bind: âh,g,a,I,G,L,V,T,d. G ⢠â¬*[h, g, â{a, I}V.T, d] L â
- G ⢠â¬*[h, g, V, d] L ⧠G ⢠â¬*[h, g, T, ⫯d] L.â{I}V.
+lemma lsx_inv_bind: âh,g,a,I,G,L,V,T,l. G ⢠â¬*[h, g, â{a, I}V.T, l] L â
+ G ⢠â¬*[h, g, V, l] L ⧠G ⢠â¬*[h, g, T, ⫯l] L.â{I}V.
/3 width=4 by lsx_fwd_bind_sn, lsx_fwd_bind_dx, conj/ qed-.
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 0accd974f..d84202bbc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lsx_lpxs.ma
@@ -19,16 +19,16 @@ include "basic_2/computation/lsx_alt.ma".
(* Advanced properties ******************************************************)
-fact lsx_bind_lpxs_aux: âh,g,a,I,G,L1,V,d. G ⢠â¬*[h, g, V, d] L1 â
- âY,T. G ⢠â¬*[h, g, T, ⫯d] Y â
+fact lsx_bind_lpxs_aux: âh,g,a,I,G,L1,V,l. G ⢠â¬*[h, g, V, l] L1 â
+ âY,T. G ⢠â¬*[h, g, T, ⫯l] Y â
âL2. Y = L2.â{I}V â â¦G, L1⦠⢠â¡*[h, g] L2 â
- G ⢠â¬*[h, g, â{a,I}V.T, d] L2.
-#h #g #a #I #G #L1 #V #d #H @(lsx_ind_alt ⦠H) -L1
+ G ⢠â¬*[h, g, â{a,I}V.T, l] L2.
+#h #g #a #I #G #L1 #V #l #H @(lsx_ind_alt ⦠H) -L1
#L1 #HL1 #IHL1 #Y #T #H @(lsx_ind_alt ⦠H) -Y
#Y #HY #IHY #L2 #H #HL12 destruct @lsx_intro_alt
#L0 #HL20 lapply (lpxs_trans ⦠HL12 ⦠HL20)
#HL10 #H elim (nlleq_inv_bind ⦠H) -H [ -HL1 -IHY | -HY -IHL1 ]
-[ #HnV elim (lleq_dec V L1 L2 d)
+[ #HnV elim (lleq_dec V L1 L2 l)
[ #HV @(IHL1 ⦠L0) /3 width=5 by lsx_lpxs_trans, lpxs_pair, lleq_canc_sn/ (**) (* full auto too slow *)
| -HnV -HL10 /4 width=5 by lsx_lpxs_trans, lpxs_pair/
]
@@ -36,20 +36,20 @@ fact lsx_bind_lpxs_aux: âh,g,a,I,G,L1,V,d. G ⢠â¬*[h, g, V, d] L1 â
]
qed-.
-lemma lsx_bind: âh,g,a,I,G,L,V,d. G ⢠â¬*[h, g, V, d] L â
- âT. G ⢠â¬*[h, g, T, ⫯d] L.â{I}V â
- G ⢠â¬*[h, g, â{a,I}V.T, d] L.
+lemma lsx_bind: âh,g,a,I,G,L,V,l. G ⢠â¬*[h, g, V, l] L â
+ âT. G ⢠â¬*[h, g, T, ⫯l] L.â{I}V â
+ G ⢠â¬*[h, g, â{a,I}V.T, l] L.
/2 width=3 by lsx_bind_lpxs_aux/ qed.
-lemma lsx_flat_lpxs: âh,g,I,G,L1,V,d. G ⢠â¬*[h, g, V, d] L1 â
- âL2,T. G ⢠â¬*[h, g, T, d] L2 â â¦G, L1⦠⢠â¡*[h, g] L2 â
- G ⢠â¬*[h, g, â{I}V.T, d] L2.
-#h #g #I #G #L1 #V #d #H @(lsx_ind_alt ⦠H) -L1
+lemma lsx_flat_lpxs: âh,g,I,G,L1,V,l. G ⢠â¬*[h, g, V, l] L1 â
+ âL2,T. G ⢠â¬*[h, g, T, l] L2 â â¦G, L1⦠⢠â¡*[h, g] L2 â
+ G ⢠â¬*[h, g, â{I}V.T, l] L2.
+#h #g #I #G #L1 #V #l #H @(lsx_ind_alt ⦠H) -L1
#L1 #HL1 #IHL1 #L2 #T #H @(lsx_ind_alt ⦠H) -L2
#L2 #HL2 #IHL2 #HL12 @lsx_intro_alt
#L0 #HL20 lapply (lpxs_trans ⦠HL12 ⦠HL20)
#HL10 #H elim (nlleq_inv_flat ⦠H) -H [ -HL1 -IHL2 | -HL2 -IHL1 ]
-[ #HnV elim (lleq_dec V L1 L2 d)
+[ #HnV elim (lleq_dec V L1 L2 l)
[ #HV @(IHL1 ⦠L0) /3 width=3 by lsx_lpxs_trans, lleq_canc_sn/ (**) (* full auto too slow: 47s *)
| -HnV -HL10 /3 width=4 by lsx_lpxs_trans/
]
@@ -57,6 +57,6 @@ lemma lsx_flat_lpxs: âh,g,I,G,L1,V,d. G ⢠â¬*[h, g, V, d] L1 â
]
qed-.
-lemma lsx_flat: âh,g,I,G,L,V,d. G ⢠â¬*[h, g, V, d] L â
- âT. G ⢠â¬*[h, g, T, d] L â G ⢠â¬*[h, g, â{I}V.T, d] L.
+lemma lsx_flat: âh,g,I,G,L,V,l. G ⢠â¬*[h, g, V, l] L â
+ âT. G ⢠â¬*[h, g, T, l] L â G ⢠â¬*[h, g, â{I}V.T, l] L.
/2 width=3 by lsx_flat_lpxs/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds.ma
index 87fe99592..abc22cc37 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds.ma
@@ -19,25 +19,25 @@ include "basic_2/computation/cprs.ma".
(* STRATIFIED DECOMPOSED PARALLEL COMPUTATION ON TERMS **********************)
definition scpds: âh. sd h â nat â relation4 genv lenv term term â
- λh,g,l2,G,L,T1,T2.
- ââT,l1. l2 ⤠l1 & â¦G, L⦠⢠T1 âª[h, g] l1 & â¦G, L⦠⢠T1 â¢*[h, l2] T & â¦G, L⦠⢠T â¡* T2.
+ λh,g,d2,G,L,T1,T2.
+ ââT,d1. d2 ⤠d1 & â¦G, L⦠⢠T1 âª[h, g] d1 & â¦G, L⦠⢠T1 â¢*[h, d2] T & â¦G, L⦠⢠T â¡* T2.
interpretation "stratified decomposed parallel computation (term)"
- 'DPRedStar h g l G L T1 T2 = (scpds h g l G L T1 T2).
+ 'DPRedStar h g d G L T1 T2 = (scpds h g d G L T1 T2).
(* Basic properties *********************************************************)
-lemma sta_cprs_scpds: âh,g,G,L,T1,T,T2,l. â¦G, L⦠⢠T1 âª[h, g] l+1 â â¦G, L⦠⢠T1 â¢*[h, 1] T â
+lemma sta_cprs_scpds: âh,g,G,L,T1,T,T2,d. â¦G, L⦠⢠T1 âª[h, g] d+1 â â¦G, L⦠⢠T1 â¢*[h, 1] T â
â¦G, L⦠⢠T â¡* T2 â â¦G, L⦠⢠T1 â¢*â¡*[h, g, 1] T2.
/2 width=6 by ex4_2_intro/ qed.
-lemma lstas_scpds: âh,g,G,L,T1,T2,l1. â¦G, L⦠⢠T1 âª[h, g] l1 â
- âl2. l2 ⤠l1 â â¦G, L⦠⢠T1 â¢*[h, l2] T2 â â¦G, L⦠⢠T1 â¢*â¡*[h, g, l2] T2.
+lemma lstas_scpds: âh,g,G,L,T1,T2,d1. â¦G, L⦠⢠T1 âª[h, g] d1 â
+ âd2. d2 ⤠d1 â â¦G, L⦠⢠T1 â¢*[h, d2] T2 â â¦G, L⦠⢠T1 â¢*â¡*[h, g, d2] T2.
/2 width=6 by ex4_2_intro/ qed.
-lemma scpds_strap1: âh,g,G,L,T1,T,T2,l.
- â¦G, L⦠⢠T1 â¢*â¡*[h, g, l] T â â¦G, L⦠⢠T â¡ T2 â â¦G, L⦠⢠T1 â¢*â¡*[h, g, l] T2.
-#h #g #G #L #T1 #T #T2 #l * /3 width=8 by cprs_strap1, ex4_2_intro/
+lemma scpds_strap1: âh,g,G,L,T1,T,T2,d.
+ â¦G, L⦠⢠T1 â¢*â¡*[h, g, d] T â â¦G, L⦠⢠T â¡ T2 â â¦G, L⦠⢠T1 â¢*â¡*[h, g, d] T2.
+#h #g #G #L #T1 #T #T2 #d * /3 width=8 by cprs_strap1, ex4_2_intro/
qed.
(* Basic forward lemmas *****************************************************)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_aaa.ma
index ad534204f..f14723fdc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_aaa.ma
@@ -20,6 +20,6 @@ include "basic_2/computation/scpds.ma".
(* Properties on atomic arity assignment for terms **************************)
-lemma scpds_aaa_conf: âh,g,G,L,l. Conf3 ⦠(aaa G L) (scpds h g l G L).
-#h #g #G #L #l #A #T #HT #U * /3 width=6 by lstas_aaa_conf, cprs_aaa_conf/
+lemma scpds_aaa_conf: âh,g,G,L,d. Conf3 ⦠(aaa G L) (scpds h g d G L).
+#h #g #G #L #d #A #T #HT #U * /3 width=6 by lstas_aaa_conf, cprs_aaa_conf/
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_lift.ma
index 794e5c60e..ee212e99c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_lift.ma
@@ -21,15 +21,15 @@ include "basic_2/computation/scpds.ma".
(* Relocation properties ****************************************************)
-lemma scpds_lift: âh,g,G,l. l_liftable (scpds h g l G).
-#h #g #G #l2 #K #T1 #T2 * #T #l1 #Hl21 #Hl1 #HT1 #HT2 #L #s #d #e
-elim (lift_total T d e)
+lemma scpds_lift: âh,g,G,d. d_liftable (scpds h g d G).
+#h #g #G #d2 #K #T1 #T2 * #T #d1 #Hd21 #Hd1 #HT1 #HT2 #L #s #l #m
+elim (lift_total T l m)
/3 width=15 by cprs_lift, da_lift, lstas_lift, ex4_2_intro/
qed.
-lemma scpds_inv_lift1: âh,g,G,l. l_deliftable_sn (scpds h g l G).
-#h #g #G #l2 #L #U1 #U2 * #U #l1 #Hl21 #Hl1 #HU1 #HU2 #K #s #d #e #HLK #T1 #HTU1
-lapply (da_inv_lift ⦠Hl1 ⦠HLK ⦠HTU1) -Hl1 #Hl1
+lemma scpds_inv_lift1: âh,g,G,d. d_deliftable_sn (scpds h g d G).
+#h #g #G #d2 #L #U1 #U2 * #U #d1 #Hd21 #Hd1 #HU1 #HU2 #K #s #l #m #HLK #T1 #HTU1
+lapply (da_inv_lift ⦠Hd1 ⦠HLK ⦠HTU1) -Hd1 #Hd1
elim (lstas_inv_lift1 ⦠HU1 ⦠HLK ⦠HTU1) -U1 #T #HTU #HT1
elim (cprs_inv_lift1 ⦠HU2 ⦠HLK ⦠HTU) -U -L
/3 width=8 by ex4_2_intro, ex2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_scpds.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_scpds.ma
index a7213b0d2..e9afcf4ef 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_scpds.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/scpds_scpds.ma
@@ -21,48 +21,48 @@ include "basic_2/computation/scpds.ma".
(* Advanced properties ******************************************************)
-lemma scpds_strap2: âh,g,G,L,T1,T,T2,l1,l. â¦G, L⦠⢠T1 âª[h, g] l1+1 â
- â¦G, L⦠⢠T1 â¢*[h, 1] T â â¦G, L⦠⢠T â¢*â¡*[h, g, l] T2 â
- â¦G, L⦠⢠T1 â¢*â¡*[h, g, l+1] T2.
-#h #g #G #L #T1 #T #T2 #l1 #l #Hl1 #HT1 *
-#T0 #l0 #Hl0 #HTl0 #HT0 #HT02
-lapply (lstas_da_conf ⦠HT1 ⦠Hl1) <minus_plus_m_m #HTl1
-lapply (da_mono ⦠HTl0 ⦠HTl1) -HTl0 -HTl1 #H destruct
+lemma scpds_strap2: âh,g,G,L,T1,T,T2,d1,d. â¦G, L⦠⢠T1 âª[h, g] d1+1 â
+ â¦G, L⦠⢠T1 â¢*[h, 1] T â â¦G, L⦠⢠T â¢*â¡*[h, g, d] T2 â
+ â¦G, L⦠⢠T1 â¢*â¡*[h, g, d+1] T2.
+#h #g #G #L #T1 #T #T2 #d1 #d #Hd1 #HT1 *
+#T0 #d0 #Hd0 #HTd0 #HT0 #HT02
+lapply (lstas_da_conf ⦠HT1 ⦠Hd1) <minus_plus_m_m #HTd1
+lapply (da_mono ⦠HTd0 ⦠HTd1) -HTd0 -HTd1 #H destruct
lapply (lstas_trans ⦠HT1 ⦠HT0) -T >commutative_plus
/3 width=6 by le_S_S, ex4_2_intro/
qed.
-lemma scpds_cprs_trans: âh,g,G,L,T1,T,T2,l.
- â¦G, L⦠⢠T1 â¢*â¡*[h, g, l] T â â¦G, L⦠⢠T â¡* T2 â â¦G, L⦠⢠T1 â¢*â¡*[h, g, l] T2.
-#h #g #G #L #T1 #T #T2 #l * /3 width=8 by cprs_trans, ex4_2_intro/
+lemma scpds_cprs_trans: âh,g,G,L,T1,T,T2,d.
+ â¦G, L⦠⢠T1 â¢*â¡*[h, g, d] T â â¦G, L⦠⢠T â¡* T2 â â¦G, L⦠⢠T1 â¢*â¡*[h, g, d] T2.
+#h #g #G #L #T1 #T #T2 #d * /3 width=8 by cprs_trans, ex4_2_intro/
qed-.
-lemma lstas_scpds_trans: âh,g,G,L,T1,T,T2,l1,l2,l.
- l2 ⤠l1 â â¦G, L⦠⢠T1 âª[h, g] l1 â
- â¦G, L⦠⢠T1 â¢*[h, l2] T â â¦G, L⦠⢠T â¢*â¡*[h, g, l] T2 â â¦G, L⦠⢠T1 â¢*â¡*[h, g, l2+l] T2.
-#h #g #G #L #T1 #T #T2 #l1 #l2 #l #Hl21 #HTl1 #HT1 * #T0 #l0 #Hl0 #HTl0 #HT0 #HT02
-lapply (lstas_da_conf ⦠HT1 ⦠HTl1) #HTl12
-lapply (da_mono ⦠HTl12 ⦠HTl0) -HTl12 -HTl0 #H destruct
-lapply (le_minus_to_plus_r ⦠Hl21 Hl0) -Hl21 -Hl0
+lemma lstas_scpds_trans: âh,g,G,L,T1,T,T2,d1,d2,d.
+ d2 ⤠d1 â â¦G, L⦠⢠T1 âª[h, g] d1 â
+ â¦G, L⦠⢠T1 â¢*[h, d2] T â â¦G, L⦠⢠T â¢*â¡*[h, g, d] T2 â â¦G, L⦠⢠T1 â¢*â¡*[h, g, d2+d] T2.
+#h #g #G #L #T1 #T #T2 #d1 #d2 #d #Hd21 #HTd1 #HT1 * #T0 #d0 #Hd0 #HTd0 #HT0 #HT02
+lapply (lstas_da_conf ⦠HT1 ⦠HTd1) #HTd12
+lapply (da_mono ⦠HTd12 ⦠HTd0) -HTd12 -HTd0 #H destruct
+lapply (le_minus_to_plus_r ⦠Hd21 Hd0) -Hd21 -Hd0
/3 width=7 by lstas_trans, ex4_2_intro/
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma scpds_inv_abst1: âh,g,a,G,L,V1,T1,U2,l. â¦G, L⦠⢠â{a}V1.T1 â¢*â¡*[h, g, l] U2 â
- ââV2,T2. â¦G, L⦠⢠V1 â¡* V2 & â¦G, L.âV1⦠⢠T1 â¢*â¡*[h, g, l] T2 &
+lemma scpds_inv_abst1: âh,g,a,G,L,V1,T1,U2,d. â¦G, L⦠⢠â{a}V1.T1 â¢*â¡*[h, g, d] U2 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¡* V2 & â¦G, L.âV1⦠⢠T1 â¢*â¡*[h, g, d] T2 &
U2 = â{a}V2.T2.
-#h #g #a #G #L #V1 #T1 #U2 #l2 * #X #l1 #Hl21 #Hl1 #H1 #H2
-lapply (da_inv_bind ⦠Hl1) -Hl1 #Hl1
+#h #g #a #G #L #V1 #T1 #U2 #d2 * #X #d1 #Hd21 #Hd1 #H1 #H2
+lapply (da_inv_bind ⦠Hd1) -Hd1 #Hd1
elim (lstas_inv_bind1 ⦠H1) -H1 #U #HTU1 #H destruct
elim (cprs_inv_abst1 ⦠H2) -H2 #V2 #T2 #HV12 #HUT2 #H destruct
/3 width=6 by ex4_2_intro, ex3_2_intro/
qed-.
-lemma scpds_inv_abbr_abst: âh,g,a1,a2,G,L,V1,W2,T1,T2,l. â¦G, L⦠⢠â{a1}V1.T1 â¢*â¡*[h, g, l] â{a2}W2.T2 â
- ââT. â¦G, L.âV1⦠⢠T1 â¢*â¡*[h, g, l] T & â¬[0, 1] â{a2}W2.T2 â¡ T & a1 = true.
-#h #g #a1 #a2 #G #L #V1 #W2 #T1 #T2 #l2 * #X #l1 #Hl21 #Hl1 #H1 #H2
-lapply (da_inv_bind ⦠Hl1) -Hl1 #Hl1
+lemma scpds_inv_abbr_abst: âh,g,a1,a2,G,L,V1,W2,T1,T2,d. â¦G, L⦠⢠â{a1}V1.T1 â¢*â¡*[h, g, d] â{a2}W2.T2 â
+ ââT. â¦G, L.âV1⦠⢠T1 â¢*â¡*[h, g, d] T & â¬[0, 1] â{a2}W2.T2 â¡ T & a1 = true.
+#h #g #a1 #a2 #G #L #V1 #W2 #T1 #T2 #d2 * #X #d1 #Hd21 #Hd1 #H1 #H2
+lapply (da_inv_bind ⦠Hd1) -Hd1 #Hd1
elim (lstas_inv_bind1 ⦠H1) -H1 #U1 #HTU1 #H destruct
elim (cprs_inv_abbr1 ⦠H2) -H2 *
[ #V2 #U2 #HV12 #HU12 #H destruct
@@ -70,24 +70,24 @@ elim (cprs_inv_abbr1 ⦠H2) -H2 *
]
qed-.
-lemma scpds_inv_lstas_eq: âh,g,G,L,T1,T2,l. â¦G, L⦠⢠T1 â¢*â¡*[h, g, l] T2 â
- âT. â¦G, L⦠⢠T1 â¢*[h, l] T â â¦G, L⦠⢠T â¡* T2.
-#h #g #G #L #T1 #T2 #l2 *
-#T0 #l1 #_ #_ #HT10 #HT02 #T #HT1
+lemma scpds_inv_lstas_eq: âh,g,G,L,T1,T2,d. â¦G, L⦠⢠T1 â¢*â¡*[h, g, d] T2 â
+ âT. â¦G, L⦠⢠T1 â¢*[h, d] T â â¦G, L⦠⢠T â¡* T2.
+#h #g #G #L #T1 #T2 #d2 *
+#T0 #d1 #_ #_ #HT10 #HT02 #T #HT1
lapply (lstas_mono ⦠HT10 ⦠HT1) #H destruct //
qed-.
(* Advanced forward lemmas **************************************************)
-lemma scpds_fwd_cpxs: âh,g,G,L,T1,T2,l. â¦G, L⦠⢠T1 â¢*â¡*[h, g, l] T2 â â¦G, L⦠⢠T1 â¡*[h, g] T2.
-#h #g #G #L #T1 #T2 #l * /3 width=5 by cpxs_trans, lstas_cpxs, cprs_cpxs/
+lemma scpds_fwd_cpxs: âh,g,G,L,T1,T2,d. â¦G, L⦠⢠T1 â¢*â¡*[h, g, d] T2 â â¦G, L⦠⢠T1 â¡*[h, g] T2.
+#h #g #G #L #T1 #T2 #d * /3 width=5 by cpxs_trans, lstas_cpxs, cprs_cpxs/
qed-.
(* Main properties **********************************************************)
-theorem scpds_conf_eq: âh,g,G,L,T0,T1,l. â¦G, L⦠⢠T0 â¢*â¡*[h, g, l] T1 â
- âT2. â¦G, L⦠⢠T0 â¢*â¡*[h, g, l] T2 â
+theorem scpds_conf_eq: âh,g,G,L,T0,T1,d. â¦G, L⦠⢠T0 â¢*â¡*[h, g, d] T1 â
+ âT2. â¦G, L⦠⢠T0 â¢*â¡*[h, g, d] T2 â
ââT. â¦G, L⦠⢠T1 â¡* T & â¦G, L⦠⢠T2 â¡* T.
-#h #g #G #L #T0 #T1 #l0 * #U1 #l1 #_ #_ #H1 #HUT1 #T2 * #U2 #l2 #_ #_ #H2 #HUT2 -l1 -l2
-lapply (lstas_mono ⦠H1 ⦠H2) #H destruct -h -l0 /2 width=3 by cprs_conf/
+#h #g #G #L #T0 #T1 #d0 * #U1 #d1 #_ #_ #H1 #HUT1 #T2 * #U2 #d2 #_ #_ #H2 #HUT2 -d1 -d2
+lapply (lstas_mono ⦠H1 ⦠H2) #H destruct -h -d0 /2 width=3 by cprs_conf/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
index 18a49d6bf..61f62965f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv.ma
@@ -22,8 +22,8 @@ inductive lsubsv (h) (g) (G): relation lenv â
| lsubsv_atom: lsubsv h g G (â) (â)
| lsubsv_pair: âI,L1,L2,V. lsubsv h g G L1 L2 â
lsubsv h g G (L1.â{I}V) (L2.â{I}V)
-| lsubsv_beta: âL1,L2,W,V,l1. â¦G, L1⦠⢠âW.V ¡[h, g, l1] â â¦G, L2⦠⢠W ¡[h, g] â
- â¦G, L1⦠⢠V âª[h, g] l1+1 â â¦G, L2⦠⢠W âª[h, g] l1 â
+| lsubsv_beta: âL1,L2,W,V,d1. â¦G, L1⦠⢠âW.V ¡[h, g, d1] â â¦G, L2⦠⢠W ¡[h, g] â
+ â¦G, L1⦠⢠V âª[h, g] d1+1 â â¦G, L2⦠⢠W âª[h, g] d1 â
lsubsv h g G L1 L2 â lsubsv h g G (L1.ââW.V) (L2.âW)
.
@@ -37,7 +37,7 @@ fact lsubsv_inv_atom1_aux: âh,g,G,L1,L2. G ⢠L1 â«Â¡[h, g] L2 â L1 = â
#h #g #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #l1 #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
]
qed-.
@@ -47,21 +47,21 @@ lemma lsubsv_inv_atom1: âh,g,G,L2. G ⢠â â«Â¡[h, g] L2 â L2 = â.
fact lsubsv_inv_pair1_aux: âh,g,G,L1,L2. G ⢠L1 â«Â¡[h, g] L2 â
âI,K1,X. L1 = K1.â{I}X â
(ââK2. G ⢠K1 â«Â¡[h, g] K2 & L2 = K2.â{I}X) â¨
- ââK2,W,V,l1. â¦G, K1⦠⢠âW.V ¡[h, g, l1] & â¦G, K2⦠⢠W ¡[h, g] &
- â¦G, K1⦠⢠V âª[h, g] l1+1 & â¦G, K2⦠⢠W âª[h, g] l1 &
+ ââK2,W,V,d1. â¦G, K1⦠⢠âW.V ¡[h, g, d1] & â¦G, K2⦠⢠W ¡[h, g] &
+ â¦G, K1⦠⢠V âª[h, g] d1+1 & â¦G, K2⦠⢠W âª[h, g] d1 &
G ⢠K1 â«Â¡[h, g] K2 &
I = Abbr & L2 = K2.âW & X = âW.V.
#h #g #G #L1 #L2 * -L1 -L2
[ #J #K1 #X #H destruct
| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #W #V #l1 #HWV #HW #HVl1 #HWl1 #HL12 #J #K1 #X #H destruct /3 width=11 by or_intror, ex8_4_intro/
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #HL12 #J #K1 #X #H destruct /3 width=11 by or_intror, ex8_4_intro/
]
qed-.
lemma lsubsv_inv_pair1: âh,g,I,G,K1,L2,X. G ⢠K1.â{I}X â«Â¡[h, g] L2 â
(ââK2. G ⢠K1 â«Â¡[h, g] K2 & L2 = K2.â{I}X) â¨
- ââK2,W,V,l1. â¦G, K1⦠⢠âW.V ¡[h, g, l1] & â¦G, K2⦠⢠W ¡[h, g] &
- â¦G, K1⦠⢠V âª[h, g] l1+1 & â¦G, K2⦠⢠W âª[h, g] l1 &
+ ââK2,W,V,d1. â¦G, K1⦠⢠âW.V ¡[h, g, d1] & â¦G, K2⦠⢠W ¡[h, g] &
+ â¦G, K1⦠⢠V âª[h, g] d1+1 & â¦G, K2⦠⢠W âª[h, g] d1 &
G ⢠K1 â«Â¡[h, g] K2 &
I = Abbr & L2 = K2.âW & X = âW.V.
/2 width=3 by lsubsv_inv_pair1_aux/ qed-.
@@ -70,7 +70,7 @@ fact lsubsv_inv_atom2_aux: âh,g,G,L1,L2. G ⢠L1 â«Â¡[h, g] L2 â L2 = â
#h #g #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #l1 #_ #_ #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #d1 #_ #_ #_ #_ #_ #H destruct
]
qed-.
@@ -80,20 +80,20 @@ lemma lsubsv_inv_atom2: âh,g,G,L1. G ⢠L1 â«Â¡[h, g] â â L1 = â.
fact lsubsv_inv_pair2_aux: âh,g,G,L1,L2. G ⢠L1 â«Â¡[h, g] L2 â
âI,K2,W. L2 = K2.â{I}W â
(ââK1. G ⢠K1 â«Â¡[h, g] K2 & L1 = K1.â{I}W) â¨
- ââK1,V,l1. â¦G, K1⦠⢠âW.V ¡[h, g, l1] & â¦G, K2⦠⢠W ¡[h, g] &
- â¦G, K1⦠⢠V âª[h, g] l1+1 & â¦G, K2⦠⢠W âª[h, g] l1 &
+ ââK1,V,d1. â¦G, K1⦠⢠âW.V ¡[h, g, d1] & â¦G, K2⦠⢠W ¡[h, g] &
+ â¦G, K1⦠⢠V âª[h, g] d1+1 & â¦G, K2⦠⢠W âª[h, g] d1 &
G ⢠K1 â«Â¡[h, g] K2 & I = Abst & L1 = K1.ââW.V.
#h #g #G #L1 #L2 * -L1 -L2
[ #J #K2 #U #H destruct
| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #W #V #l1 #HWV #HW #HVl1 #HWl1 #HL12 #J #K2 #U #H destruct /3 width=8 by or_intror, ex7_3_intro/
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #HL12 #J #K2 #U #H destruct /3 width=8 by or_intror, ex7_3_intro/
]
qed-.
lemma lsubsv_inv_pair2: âh,g,I,G,L1,K2,W. G ⢠L1 â«Â¡[h, g] K2.â{I}W â
(ââK1. G ⢠K1 â«Â¡[h, g] K2 & L1 = K1.â{I}W) â¨
- ââK1,V,l1. â¦G, K1⦠⢠âW.V ¡[h, g, l1] & â¦G, K2⦠⢠W ¡[h, g] &
- â¦G, K1⦠⢠V âª[h, g] l1+1 & â¦G, K2⦠⢠W âª[h, g] l1 &
+ ââK1,V,d1. â¦G, K1⦠⢠âW.V ¡[h, g, d1] & â¦G, K2⦠⢠W ¡[h, g] &
+ â¦G, K1⦠⢠V âª[h, g] d1+1 & â¦G, K2⦠⢠W âª[h, g] d1 &
G ⢠K1 â«Â¡[h, g] K2 & I = Abst & L1 = K1.ââW.V.
/2 width=3 by lsubsv_inv_pair2_aux/ qed-.
@@ -116,19 +116,19 @@ qed-.
(* Note: the constant 0 cannot be generalized *)
lemma lsubsv_drop_O1_conf: âh,g,G,L1,L2. G ⢠L1 â«Â¡[h, g] L2 â
- âK1,s,e. â¬[s, 0, e] L1 â¡ K1 â
- ââK2. G ⢠K1 â«Â¡[h, g] K2 & â¬[s, 0, e] L2 â¡ K2.
+ âK1,s,m. â¬[s, 0, m] L1 â¡ K1 â
+ ââK2. G ⢠K1 â«Â¡[h, g] K2 & â¬[s, 0, m] L2 â¡ K2.
#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK1
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
| elim (IHL12 ⦠HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #l1 #HWV #HW #HVl1 #HWl1 #_ #IHL12 #K1 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK1
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K1 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
@@ -139,19 +139,19 @@ qed-.
(* Note: the constant 0 cannot be generalized *)
lemma lsubsv_drop_O1_trans: âh,g,G,L1,L2. G ⢠L1 â«Â¡[h, g] L2 â
- âK2,s, e. â¬[s, 0, e] L2 â¡ K2 â
- ââK1. G ⢠K1 â«Â¡[h, g] K2 & â¬[s, 0, e] L1 â¡ K1.
+ âK2,s, m. â¬[s, 0, m] L2 â¡ K2 â
+ ââK1. G ⢠K1 â«Â¡[h, g] K2 & â¬[s, 0, m] L1 â¡ K1.
#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK2
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=3 by lsubsv_pair, drop_pair, ex2_intro/
| elim (IHL12 ⦠HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #l1 #HWV #HW #HVl1 #HWl1 #_ #IHL12 #K2 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK2
+| #L1 #L2 #W #V #d1 #HWV #HW #HVd1 #HWd1 #_ #IHL12 #K2 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=4 by lsubsv_beta, drop_pair, ex2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lstas.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lstas.ma
index 979eebb8f..ee2dea7fa 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lstas.ma
@@ -19,32 +19,32 @@ include "basic_2/dynamic/lsubsv.ma".
(* Properties on nat-iterated static type assignment ************************)
-lemma lsubsv_lstas_trans: âh,g,G,L2,T,U2,l2. â¦G, L2⦠⢠T â¢*[h, l2] U2 â
- âl1. l2 ⤠l1 â â¦G, L2⦠⢠T âª[h, g] l1 â
+lemma lsubsv_lstas_trans: âh,g,G,L2,T,U2,d2. â¦G, L2⦠⢠T â¢*[h, d2] U2 â
+ âd1. d2 ⤠d1 â â¦G, L2⦠⢠T âª[h, g] d1 â
âL1. G ⢠L1 â«Â¡[h, g] L2 â
- ââU1. â¦G, L1⦠⢠T â¢*[h, l2] U1 & â¦G, L1⦠⢠U1 â¬* U2.
-#h #g #G #L2 #T #U #l2 #H elim H -G -L2 -T -U -l2
+ ââU1. â¦G, L1⦠⢠T â¢*[h, d2] U1 & â¦G, L1⦠⢠U1 â¬* U2.
+#h #g #G #L2 #T #U #d2 #H elim H -G -L2 -T -U -d2
[ /2 width=3 by ex2_intro/
-| #G #L2 #K2 #V #W #U #i #l2 #HLK2 #_ #HWU #IHVW #l1 #Hl21 #Hl1 #L1 #HL12
- elim (da_inv_lref ⦠Hl1) -Hl1 * #K0 #V0 [| #l0 ] #HK0 #HV0
+| #G #L2 #K2 #V #W #U #i #d2 #HLK2 #_ #HWU #IHVW #d1 #Hd21 #Hd1 #L1 #HL12
+ elim (da_inv_lref ⦠Hd1) -Hd1 * #K0 #V0 [| #d0 ] #HK0 #HV0
lapply (drop_mono ⦠HK0 ⦠HLK2) -HK0 #H destruct
elim (lsubsv_drop_O1_trans ⦠HL12 ⦠HLK2) -L2 #Y #H #HLK1
elim (lsubsv_inv_pair2 ⦠H) -H * #K1 [ | -HWU -IHVW -HLK1 ]
[ #HK12 #H destruct
- elim (IHVW ⦠Hl21 HV0 ⦠HK12) -K2 -l1 #T #HVT #HTW
+ elim (IHVW ⦠Hd21 HV0 ⦠HK12) -K2 -d1 #T #HVT #HTW
lapply (drop_fwd_drop2 ⦠HLK1) #H
elim (lift_total T 0 (i+1))
/3 width=12 by lstas_ldef, cpcs_lift, ex2_intro/
- | #V0 #l0 #_ #_ #_ #_ #_ #H destruct
+ | #V0 #d0 #_ #_ #_ #_ #_ #H destruct
]
-| #G #L2 #K2 #V #W #i #HLK2 #_ #IHVW #l1 #_ #Hl1 #L1 #HL12
- elim (da_inv_lref ⦠Hl1) -Hl1 * #K0 #V0 [| #l0 ] #HK0 #HV0 [| #H1 ]
+| #G #L2 #K2 #V #W #i #HLK2 #_ #IHVW #d1 #_ #Hd1 #L1 #HL12
+ elim (da_inv_lref ⦠Hd1) -Hd1 * #K0 #V0 [| #d0 ] #HK0 #HV0 [| #H1 ]
lapply (drop_mono ⦠HK0 ⦠HLK2) -HK0 #H2 destruct
elim (lsubsv_drop_O1_trans ⦠HL12 ⦠HLK2) -L2 #Y #H #HLK1
elim (lsubsv_inv_pair2 ⦠H) -H * #K1
[ #HK12 #H destruct
elim (IHVW ⦠HV0 ⦠HK12) -K2 /3 width=5 by lstas_zero, ex2_intro/
- | #V1 #l1 #_ #_ #HV1 #HV #HK12 #_ #H destruct
+ | #V1 #d1 #_ #_ #HV1 #HV #HK12 #_ #H destruct
lapply (da_mono ⦠HV0 ⦠HV) -HV #H destruct
elim (da_lstas ⦠HV1 0) -HV1 #W1 #HVW1 #_
elim (lift_total W1 0 (i+1)) #U1 #HWU1
@@ -53,46 +53,46 @@ lemma lsubsv_lstas_trans: âh,g,G,L2,T,U2,l2. â¦G, L2⦠⢠T â¢*[h, l2] U2
@cpcs_cprs_sn @(cprs_delta ⦠HLK1 ⦠HWU1)
/4 width=2 by cprs_strap1, cpr_cprs, lstas_cpr, cpr_eps/
]
-| #G #L2 #K2 #V #W #U #i #l2 #HLK2 #_ #HWU #IHVW #l1 #Hl21 #Hl1 #L1 #HL12
- elim (da_inv_lref ⦠Hl1) -Hl1 * #K0 #V0 [| #l0 ] #HK0 #HV0 [| #H1 ]
+| #G #L2 #K2 #V #W #U #i #d2 #HLK2 #_ #HWU #IHVW #d1 #Hd21 #Hd1 #L1 #HL12
+ elim (da_inv_lref ⦠Hd1) -Hd1 * #K0 #V0 [| #d0 ] #HK0 #HV0 [| #H1 ]
lapply (drop_mono ⦠HK0 ⦠HLK2) -HK0 #H2 destruct
- lapply (le_plus_to_le_r ⦠Hl21) -Hl21 #Hl21
+ lapply (le_plus_to_le_r ⦠Hd21) -Hd21 #Hd21
elim (lsubsv_drop_O1_trans ⦠HL12 ⦠HLK2) -L2 #Y #H #HLK1
elim (lsubsv_inv_pair2 ⦠H) -H * #K1
[ #HK12 #H destruct
- elim (IHVW ⦠Hl21 HV0 ⦠HK12) -K2 -Hl21 #X
+ elim (IHVW ⦠Hd21 HV0 ⦠HK12) -K2 -Hd21 #X
lapply (drop_fwd_drop2 ⦠HLK1)
elim (lift_total X 0 (i+1))
/3 width=12 by lstas_succ, cpcs_lift, ex2_intro/
- | #V1 #l1 #H0 #_ #HV1 #HV #HK12 #_ #H destruct
+ | #V1 #d1 #H0 #_ #HV1 #HV #HK12 #_ #H destruct
lapply (da_mono ⦠HV0 ⦠HV) -HV #H destruct
elim (shnv_inv_cast ⦠H0) -H0 #_ #_ #H
- lapply (H ⦠Hl21) -H #HVV1
- elim (IHVW ⦠Hl21 HV0 ⦠HK12) -K2 -Hl21 #X #HVX #HXW
- elim (da_lstas ⦠HV1 (l2+1)) -HV1 #X1 #HVX1 #_
+ lapply (H ⦠Hd21) -H #HVV1
+ elim (IHVW ⦠Hd21 HV0 ⦠HK12) -K2 -Hd21 #X #HVX #HXW
+ elim (da_lstas ⦠HV1 (d2+1)) -HV1 #X1 #HVX1 #_
lapply (scpes_inv_lstas_eq ⦠HVV1 ⦠HVX ⦠HVX1) -HVV1 -HVX #HXX1
lapply (cpcs_canc_sn ⦠HXX1 ⦠HXW) -X
elim (lift_total X1 0 (i+1))
lapply (drop_fwd_drop2 ⦠HLK1)
/4 width=12 by cpcs_lift, lstas_cast, lstas_ldef, ex2_intro/
]
-| #a #I #G #L2 #V2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
- lapply (da_inv_bind ⦠Hl2) -Hl2 #Hl2
- elim (IHTU2 ⦠Hl2 (L1.â{I}V2) â¦)
+| #a #I #G #L2 #V2 #T2 #U2 #d1 #_ #IHTU2 #d2 #Hd12 #Hd2 #L1 #HL12
+ lapply (da_inv_bind ⦠Hd2) -Hd2 #Hd2
+ elim (IHTU2 ⦠Hd2 (L1.â{I}V2) â¦)
/3 width=3 by lsubsv_pair, lstas_bind, cpcs_bind_dx, ex2_intro/
-| #G #L2 #V2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
- lapply (da_inv_flat ⦠Hl2) -Hl2 #Hl2
- elim (IHTU2 ⦠Hl2 ⦠HL12) -L2
+| #G #L2 #V2 #T2 #U2 #d1 #_ #IHTU2 #d2 #Hd12 #Hd2 #L1 #HL12
+ lapply (da_inv_flat ⦠Hd2) -Hd2 #Hd2
+ elim (IHTU2 ⦠Hd2 ⦠HL12) -L2
/3 width=5 by lstas_appl, cpcs_flat, ex2_intro/
-| #G #L2 #W2 #T2 #U2 #l1 #_ #IHTU2 #l2 #Hl12 #Hl2 #L1 #HL12
- lapply (da_inv_flat ⦠Hl2) -Hl2 #Hl2
- elim (IHTU2 ⦠Hl2 ⦠HL12) -L2
+| #G #L2 #W2 #T2 #U2 #d1 #_ #IHTU2 #d2 #Hd12 #Hd2 #L1 #HL12
+ lapply (da_inv_flat ⦠Hd2) -Hd2 #Hd2
+ elim (IHTU2 ⦠Hd2 ⦠HL12) -L2
/3 width=3 by lstas_cast, ex2_intro/
]
qed-.
lemma lsubsv_sta_trans: âh,g,G,L2,T,U2. â¦G, L2⦠⢠T â¢*[h, 1] U2 â
- âl. â¦G, L2⦠⢠T âª[h, g] l+1 â
+ âd. â¦G, L2⦠⢠T âª[h, g] d+1 â
âL1. G ⢠L1 â«Â¡[h, g] L2 â
ââU1. â¦G, L1⦠⢠T â¢*[h, 1] U1 & â¦G, L1⦠⢠U1 â¬* U2.
/2 width=7 by lsubsv_lstas_trans/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lsuba.ma
index 54be766bc..b76c12da8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lsuba.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_lsuba.ma
@@ -22,9 +22,9 @@ include "basic_2/dynamic/lsubsv.ma".
lemma lsubsv_fwd_lsuba: âh,g,G,L1,L2. G ⢠L1 â«Â¡[h, g] L2 â G ⢠L1 â«â L2.
#h #g #G #L1 #L2 #H elim H -L1 -L2 /2 width=1 by lsuba_pair/
-#L1 #L2 #W #V #l1 #H #_ #_ #_ #_ #IHL12
+#L1 #L2 #W #V #d1 #H #_ #_ #_ #_ #IHL12
elim (shnv_inv_cast ⦠H) -H #HW #HV #H
-lapply (H 0 ?) // -l1 #HWV
+lapply (H 0 ?) // -d1 #HWV
elim (snv_fwd_aaa ⦠HW) -HW #B #HW
elim (snv_fwd_aaa ⦠HV) -HV #A #HV
lapply (scpes_aaa_mono ⦠HWV ⦠HW ⦠HV) #H destruct
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_scpds.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_scpds.ma
index 3e9772958..e310d8186 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_scpds.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_scpds.ma
@@ -20,12 +20,12 @@ include "basic_2/dynamic/lsubsv_lstas.ma".
(* Properties on decomposed extended parallel computation on terms **********)
-lemma lsubsv_scpds_trans: âh,g,G,L2,T1,T2,l. â¦G, L2⦠⢠T1 â¢*â¡*[h, g, l] T2 â
+lemma lsubsv_scpds_trans: âh,g,G,L2,T1,T2,d. â¦G, L2⦠⢠T1 â¢*â¡*[h, g, d] T2 â
âL1. G ⢠L1 â«Â¡[h, g] L2 â
- ââT. â¦G, L1⦠⢠T1 â¢*â¡*[h, g, l] T & â¦G, L1⦠⢠T2 â¡* T.
-#h #g #G #L2 #T1 #T2 #l2 * #T #l1 #Hl21 #Hl1 #HT1 #HT2 #L1 #HL12
+ ââT. â¦G, L1⦠⢠T1 â¢*â¡*[h, g, d] T & â¦G, L1⦠⢠T2 â¡* T.
+#h #g #G #L2 #T1 #T2 #d2 * #T #d1 #Hd21 #Hd1 #HT1 #HT2 #L1 #HL12
lapply (lsubsv_cprs_trans ⦠HL12 ⦠HT2) -HT2 #HT2
-elim (lsubsv_lstas_trans ⦠HT1 ⦠Hl1 ⦠HL12) // #T0 #HT10 #HT0
+elim (lsubsv_lstas_trans ⦠HT1 ⦠Hd1 ⦠HL12) // #T0 #HT10 #HT0
lapply (cpcs_cprs_strap1 ⦠HT0 ⦠HT2) -T #HT02
elim (cpcs_inv_cprs ⦠HT02) -HT02
/5 width=5 by lsubsv_fwd_lsubd, lsubd_da_trans, ex4_2_intro, ex2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_snv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_snv.ma
index 9dbb862c7..5dc36d782 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_snv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/lsubsv_snv.ma
@@ -25,12 +25,12 @@ lemma lsubsv_snv_trans: âh,g,G,L2,T. â¦G, L2⦠⢠T ¡[h, g] â
elim (lsubsv_drop_O1_trans ⦠HL12 ⦠HLK2) -L2 #X #H #HLK1
elim (lsubsv_inv_pair2 ⦠H) -H * #K1
[ #HK12 #H destruct /3 width=5 by snv_lref/
- | #W #l #HVW #_ #_ #_ #_ #H1 #H2 destruct -IHV
+ | #W #d #HVW #_ #_ #_ #_ #H1 #H2 destruct -IHV
/3 width=6 by shnv_inv_snv, snv_lref/
]
| #a #I #G #L2 #V #T #_ #_ #IHV #IHT #L1 #HL12 destruct
/4 width=1 by snv_bind, lsubsv_pair/
-| #a #G #L2 #V #W0 #T #U0 #l #_ #_ #HVW0 #HTU0 #IHV #IHT #L1 #HL12
+| #a #G #L2 #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 #IHV #IHT #L1 #HL12
elim (lsubsv_scpds_trans ⦠HVW0 ⦠HL12) -HVW0 #V0 #HV0 #HWV0
elim (lsubsv_scpds_trans ⦠HTU0 ⦠HL12) -HTU0 #X #HT0 #H
elim (cprs_inv_abst1 ⦠H) -H #W #T0 #HW0 #_ #H destruct
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/shnv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/shnv.ma
index 24bbe1e15..deacf88dd 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/shnv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/shnv.ma
@@ -18,31 +18,31 @@ include "basic_2/dynamic/snv.ma".
(* STRATIFIED HIGHER NATIVE VALIDITY FOR TERMS ******************************)
-inductive shnv (h) (g) (l1) (G) (L): predicate term â
+inductive shnv (h) (g) (d1) (G) (L): predicate term â
| shnv_cast: âU,T. â¦G, L⦠⢠U ¡[h, g] â â¦G, L⦠⢠T ¡[h, g] â
- (âl2. l2 ⤠l1 â â¦G, L⦠⢠U â¢*â¬*[h, g, l2, l2+1] T) â
- shnv h g l1 G L (âU.T)
+ (âd2. d2 ⤠d1 â â¦G, L⦠⢠U â¢*â¬*[h, g, d2, d2+1] T) â
+ shnv h g d1 G L (âU.T)
.
interpretation "stratified higher native validity (term)"
- 'NativeValid h g l G L T = (shnv h g l G L T).
+ 'NativeValid h g d G L T = (shnv h g d G L T).
(* Basic inversion lemmas ***************************************************)
-fact shnv_inv_cast_aux: âh,g,G,L,X,l1. â¦G, L⦠⢠X ¡[h, g, l1] â âU,T. X = âU.T â
+fact shnv_inv_cast_aux: âh,g,G,L,X,d1. â¦G, L⦠⢠X ¡[h, g, d1] â âU,T. X = âU.T â
â§â§ â¦G, L⦠⢠U ¡[h, g] & â¦G, L⦠⢠T ¡[h, g]
- & (âl2. l2 ⤠l1 â â¦G, L⦠⢠U â¢*â¬*[h, g, l2, l2+1] T).
-#h #g #G #L #X #l1 * -X
+ & (âd2. d2 ⤠d1 â â¦G, L⦠⢠U â¢*â¬*[h, g, d2, d2+1] T).
+#h #g #G #L #X #d1 * -X
#U #T #HU #HT #HUT #U1 #T1 #H destruct /3 width=1 by and3_intro/
qed-.
-lemma shnv_inv_cast: âh,g,G,L,U,T,l1. â¦G, L⦠⢠âU.T ¡[h, g, l1] â
+lemma shnv_inv_cast: âh,g,G,L,U,T,d1. â¦G, L⦠⢠âU.T ¡[h, g, d1] â
â§â§ â¦G, L⦠⢠U ¡[h, g] & â¦G, L⦠⢠T ¡[h, g]
- & (âl2. l2 ⤠l1 â â¦G, L⦠⢠U â¢*â¬*[h, g, l2, l2+1] T).
+ & (âd2. d2 ⤠d1 â â¦G, L⦠⢠U â¢*â¬*[h, g, d2, d2+1] T).
/2 width=3 by shnv_inv_cast_aux/ qed-.
-lemma shnv_inv_snv: âh,g,G,L,T,l. â¦G, L⦠⢠T ¡[h, g, l] â â¦G, L⦠⢠T ¡[h, g].
-#h #g #G #L #T #l * -T
+lemma shnv_inv_snv: âh,g,G,L,T,d. â¦G, L⦠⢠T ¡[h, g, d] â â¦G, L⦠⢠T ¡[h, g].
+#h #g #G #L #T #d * -T
#U #T #HU #HT #HUT elim (HUT 0) -HUT /2 width=3 by snv_cast/
qed-.
@@ -51,5 +51,5 @@ qed-.
lemma snv_shnv_cast: âh,g,G,L,U,T. â¦G, L⦠⢠âU.T ¡[h, g] â â¦G, L⦠⢠âU.T ¡[h, g, 0].
#h #g #G #L #U #T #H elim (snv_inv_cast ⦠H) -H
#U0 #HU #HT #HU0 #HTU0 @shnv_cast // -HU -HT
-#l #H <(le_n_O_to_eq ⦠H) -l /2 width=3 by scpds_div/
+#d #H <(le_n_O_to_eq ⦠H) -d /2 width=3 by scpds_div/
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
index 88d4db421..6eb3dff24 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv.ma
@@ -22,8 +22,8 @@ inductive snv (h) (g): relation3 genv lenv term â
| snv_sort: âG,L,k. snv h g G L (âk)
| snv_lref: âI,G,L,K,V,i. â¬[i] L â¡ K.â{I}V â snv h g G K V â snv h g G L (#i)
| snv_bind: âa,I,G,L,V,T. snv h g G L V â snv h g G (L.â{I}V) T â snv h g G L (â{a,I}V.T)
-| snv_appl: âa,G,L,V,W0,T,U0,l. snv h g G L V â snv h g G L T â
- â¦G, L⦠⢠V â¢*â¡*[h, g, 1] W0 â â¦G, L⦠⢠T â¢*â¡*[h, g, l] â{a}W0.U0 â snv h g G L (âV.T)
+| snv_appl: âa,G,L,V,W0,T,U0,d. snv h g G L V â snv h g G L T â
+ â¦G, L⦠⢠V â¢*â¡*[h, g, 1] W0 â â¦G, L⦠⢠T â¢*â¡*[h, g, d] â{a}W0.U0 â snv h g G L (âV.T)
| snv_cast: âG,L,U,T,U0. snv h g G L U â snv h g G L T â
â¦G, L⦠⢠U â¢*â¡*[h, g, 0] U0 â â¦G, L⦠⢠T â¢*â¡*[h, g, 1] U0 â snv h g G L (âU.T)
.
@@ -39,7 +39,7 @@ fact snv_inv_lref_aux: âh,g,G,L,X. â¦G, L⦠⢠X ¡[h, g] â âi. X = #i
[ #G #L #k #i #H destruct
| #I #G #L #K #V #i0 #HLK #HV #i #H destruct /2 width=5 by ex2_3_intro/
| #a #I #G #L #V #T #_ #_ #i #H destruct
-| #a #G #L #V #W0 #T #U0 #l #_ #_ #_ #_ #i #H destruct
+| #a #G #L #V #W0 #T #U0 #d #_ #_ #_ #_ #i #H destruct
| #G #L #U #T #U0 #_ #_ #_ #_ #i #H destruct
]
qed-.
@@ -53,7 +53,7 @@ fact snv_inv_gref_aux: âh,g,G,L,X. â¦G, L⦠⢠X ¡[h, g] â âp. X = §p
[ #G #L #k #p #H destruct
| #I #G #L #K #V #i #_ #_ #p #H destruct
| #a #I #G #L #V #T #_ #_ #p #H destruct
-| #a #G #L #V #W0 #T #U0 #l #_ #_ #_ #_ #p #H destruct
+| #a #G #L #V #W0 #T #U0 #d #_ #_ #_ #_ #p #H destruct
| #G #L #U #T #U0 #_ #_ #_ #_ #p #H destruct
]
qed-.
@@ -67,7 +67,7 @@ fact snv_inv_bind_aux: âh,g,G,L,X. â¦G, L⦠⢠X ¡[h, g] â âa,I,V,T. X
[ #G #L #k #b #Z #X1 #X2 #H destruct
| #I #G #L #K #V #i #_ #_ #b #Z #X1 #X2 #H destruct
| #a #I #G #L #V #T #HV #HT #b #Z #X1 #X2 #H destruct /2 width=1 by conj/
-| #a #G #L #V #W0 #T #U0 #l #_ #_ #_ #_ #b #Z #X1 #X2 #H destruct
+| #a #G #L #V #W0 #T #U0 #d #_ #_ #_ #_ #b #Z #X1 #X2 #H destruct
| #G #L #U #T #U0 #_ #_ #_ #_ #b #Z #X1 #X2 #H destruct
]
qed-.
@@ -77,20 +77,20 @@ lemma snv_inv_bind: âh,g,a,I,G,L,V,T. â¦G, L⦠⢠â{a,I}V.T ¡[h, g] â
/2 width=4 by snv_inv_bind_aux/ qed-.
fact snv_inv_appl_aux: âh,g,G,L,X. â¦G, L⦠⢠X ¡[h, g] â âV,T. X = âV.T â
- ââa,W0,U0,l. â¦G, L⦠⢠V ¡[h, g] & â¦G, L⦠⢠T ¡[h, g] &
- â¦G, L⦠⢠V â¢*â¡*[h, g, 1] W0 & â¦G, L⦠⢠T â¢*â¡*[h, g, l] â{a}W0.U0.
+ ââa,W0,U0,d. â¦G, L⦠⢠V ¡[h, g] & â¦G, L⦠⢠T ¡[h, g] &
+ â¦G, L⦠⢠V â¢*â¡*[h, g, 1] W0 & â¦G, L⦠⢠T â¢*â¡*[h, g, d] â{a}W0.U0.
#h #g #G #L #X * -L -X
[ #G #L #k #X1 #X2 #H destruct
| #I #G #L #K #V #i #_ #_ #X1 #X2 #H destruct
| #a #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
-| #a #G #L #V #W0 #T #U0 #l #HV #HT #HVW0 #HTU0 #X1 #X2 #H destruct /2 width=6 by ex4_4_intro/
+| #a #G #L #V #W0 #T #U0 #d #HV #HT #HVW0 #HTU0 #X1 #X2 #H destruct /2 width=6 by ex4_4_intro/
| #G #L #U #T #U0 #_ #_ #_ #_ #X1 #X2 #H destruct
]
qed-.
lemma snv_inv_appl: âh,g,G,L,V,T. â¦G, L⦠⢠âV.T ¡[h, g] â
- ââa,W0,U0,l. â¦G, L⦠⢠V ¡[h, g] & â¦G, L⦠⢠T ¡[h, g] &
- â¦G, L⦠⢠V â¢*â¡*[h, g, 1] W0 & â¦G, L⦠⢠T â¢*â¡*[h, g, l] â{a}W0.U0.
+ ââa,W0,U0,d. â¦G, L⦠⢠V ¡[h, g] & â¦G, L⦠⢠T ¡[h, g] &
+ â¦G, L⦠⢠V â¢*â¡*[h, g, 1] W0 & â¦G, L⦠⢠T â¢*â¡*[h, g, d] â{a}W0.U0.
/2 width=3 by snv_inv_appl_aux/ qed-.
fact snv_inv_cast_aux: âh,g,G,L,X. â¦G, L⦠⢠X ¡[h, g] â âU,T. X = âU.T â
@@ -100,7 +100,7 @@ fact snv_inv_cast_aux: âh,g,G,L,X. â¦G, L⦠⢠X ¡[h, g] â âU,T. X =
[ #G #L #k #X1 #X2 #H destruct
| #I #G #L #K #V #i #_ #_ #X1 #X2 #H destruct
| #a #I #G #L #V #T #_ #_ #X1 #X2 #H destruct
-| #a #G #L #V #W0 #T #U0 #l #_ #_ #_ #_ #X1 #X2 #H destruct
+| #a #G #L #V #W0 #T #U0 #d #_ #_ #_ #_ #X1 #X2 #H destruct
| #G #L #U #T #U0 #HV #HT #HU0 #HTU0 #X1 #X2 #H destruct /2 width=3 by ex4_intro/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma
index d663b3009..c404ee6f3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_aaa.ma
@@ -25,7 +25,7 @@ lemma snv_fwd_aaa: âh,g,G,L,T. â¦G, L⦠⢠T ¡[h, g] â âA. â¦G, Lâ¦
[ /2 width=2 by aaa_sort, ex_intro/
| #I #G #L #K #V #i #HLK #_ * /3 width=6 by aaa_lref, ex_intro/
| #a * #G #L #V #T #_ #_ * #B #HV * #A #HA /3 width=2 by aaa_abbr, aaa_abst, ex_intro/
-| #a #G #L #V #W0 #T #U0 #l #_ #_ #HVW0 #HTU0 * #B #HV * #X #HT
+| #a #G #L #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 * #B #HV * #X #HT
lapply (scpds_aaa_conf ⦠HV ⦠HVW0) -HVW0 #HW0
lapply (scpds_aaa_conf ⦠HT ⦠HTU0) -HTU0 #H
elim (aaa_inv_abst ⦠H) -H #B0 #A #H1 #HU #H2 destruct
@@ -39,12 +39,12 @@ qed-.
(* Advanced forward lemmas **************************************************)
-lemma snv_fwd_da: âh,g,G,L,T. â¦G, L⦠⢠T ¡[h, g] â âl. â¦G, L⦠⢠T âª[h, g] l.
+lemma snv_fwd_da: âh,g,G,L,T. â¦G, L⦠⢠T ¡[h, g] â âd. â¦G, L⦠⢠T âª[h, g] d.
#h #g #G #L #T #H elim (snv_fwd_aaa ⦠H) -H /2 width=2 by aaa_da/
qed-.
lemma snv_fwd_lstas: âh,g,G,L,T. â¦G, L⦠⢠T ¡[h, g] â
- âl. âU. â¦G, L⦠⢠T â¢*[h, l] U.
-#h #g #G #L #T #H #l elim (snv_fwd_aaa ⦠H) -H
-#A #HT elim (aaa_lstas h ⦠HT l) -HT /2 width=2 by ex_intro/
+ âd. âU. â¦G, L⦠⢠T â¢*[h, d] U.
+#h #g #G #L #T #H #d elim (snv_fwd_aaa ⦠H) -H
+#A #HT elim (aaa_lstas h ⦠HT d) -HT /2 width=2 by ex_intro/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_da_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_da_lpr.ma
index 7f38d6474..091f51048 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_da_lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_da_lpr.ma
@@ -26,12 +26,12 @@ fact da_cpr_lpr_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_da_cpr_lpr h g G1 L1 T1) â
âG1,L1,T1. G0 = G1 â L0 = L1 â T0 = T1 â IH_da_cpr_lpr h g G1 L1 T1.
#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #k #_ #_ #_ #_ #l #H2 #X3 #H3 #L2 #_ -IH3 -IH2 -IH1
+[ #k #_ #_ #_ #_ #d #H2 #X3 #H3 #L2 #_ -IH3 -IH2 -IH1
lapply (da_inv_sort ⦠H2) -H2
lapply (cpr_inv_sort1 ⦠H3) -H3 #H destruct /2 width=1 by da_sort/
-| #i #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+| #i #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
elim (snv_inv_lref ⦠H1) -H1 #I0 #K0 #X0 #H #HX0
- elim (da_inv_lref ⦠H2) -H2 * #K1 [ #V1 | #W1 #l1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct
+ elim (da_inv_lref ⦠H2) -H2 * #K1 [ #V1 | #W1 #d1 ] #HLK1 [ #HV1 | #HW1 #H ] destruct
lapply (drop_mono ⦠H ⦠HLK1) -H #H destruct
elim (cpr_inv_lref1 ⦠H3) -H3
[1,3: #H destruct
@@ -49,7 +49,7 @@ fact da_cpr_lpr_aux: âh,g,G0,L0,T0.
]
| #p #_ #_ #HT0 #H1 destruct -IH3 -IH2 -IH1
elim (snv_inv_gref ⦠H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH2
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH2
elim (snv_inv_bind ⦠H1) -H1 #_ #HT1
lapply (da_inv_bind ⦠H2) -H2
elim (cpr_inv_bind1 ⦠H3) -H3 *
@@ -58,32 +58,32 @@ fact da_cpr_lpr_aux: âh,g,G0,L0,T0.
| #T2 #HT12 #HT2 #H1 #H2 destruct
/4 width=11 by da_inv_lift, fqup_fpbg, lpr_pair, drop_drop/
]
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct
- elim (snv_inv_appl ⦠H1) -H1 #b1 #W1 #U1 #l1 #HV1 #HT1 #HVW1 #HTU1
- lapply (da_inv_flat ⦠H2) -H2 #Hl
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct
+ elim (snv_inv_appl ⦠H1) -H1 #b1 #W1 #U1 #d1 #HV1 #HT1 #HVW1 #HTU1
+ lapply (da_inv_flat ⦠H2) -H2 #Hd
elim (cpr_inv_appl1 ⦠H3) -H3 *
[ #V2 #T2 #HV12 #HT12 #H destruct -IH3 -IH2 /4 width=7 by da_flat, fqup_fpbg/
| #b #V2 #W #W2 #U #U2 #HV12 #HW2 #HU2 #H1 #H2 destruct
elim (snv_inv_bind ⦠HT1) -HT1 #HW #HU
- lapply (da_inv_bind ⦠Hl) -Hl #Hl
- elim (scpds_inv_abst1 ⦠HTU1) -HTU1 #W3 #U3 #HW3 #_ #H destruct -U3 -l1
- elim (snv_fwd_da ⦠HV1) #l1 #Hl1
- elim (snv_fwd_da ⦠HW) #l0 #Hl0
- lapply (cprs_scpds_div ⦠HW3 ⦠Hl0 ⦠1 HVW1) -W3 #H
- elim (da_scpes_aux ⦠IH3 IH2 IH1 ⦠Hl0 ⦠Hl1 ⦠H) -IH3 -IH2 -H /2 width=1 by fqup_fpbg/ #_ #H1
- <minus_n_O #H destruct >(plus_minus_m_m l1 1) in Hl1; // -H1 #Hl1
- lapply (IH1 ⦠HV1 ⦠Hl1 ⦠HV12 ⦠HL12) -HV1 -Hl1 -HV12 [ /2 by fqup_fpbg/ ]
- lapply (IH1 ⦠Hl0 ⦠HW2 ⦠HL12) -Hl0 /2 width=1 by fqup_fpbg/ -HW
- lapply (IH1 ⦠HU ⦠Hl ⦠HU2 (L2.âW2) ?) -IH1 -HU -Hl -HU2 [1,2: /2 by fqup_fpbg, lpr_pair/ ] -HL12 -HW2
+ lapply (da_inv_bind ⦠Hd) -Hd #Hd
+ elim (scpds_inv_abst1 ⦠HTU1) -HTU1 #W3 #U3 #HW3 #_ #H destruct -U3 -d1
+ elim (snv_fwd_da ⦠HV1) #d1 #Hd1
+ elim (snv_fwd_da ⦠HW) #d0 #Hd0
+ lapply (cprs_scpds_div ⦠HW3 ⦠Hd0 ⦠1 HVW1) -W3 #H
+ elim (da_scpes_aux ⦠IH3 IH2 IH1 ⦠Hd0 ⦠Hd1 ⦠H) -IH3 -IH2 -H /2 width=1 by fqup_fpbg/ #_ #H1
+ <minus_n_O #H destruct >(plus_minus_m_m d1 1) in Hd1; // -H1 #Hd1
+ lapply (IH1 ⦠HV1 ⦠Hd1 ⦠HV12 ⦠HL12) -HV1 -Hd1 -HV12 [ /2 by fqup_fpbg/ ]
+ lapply (IH1 ⦠Hd0 ⦠HW2 ⦠HL12) -Hd0 /2 width=1 by fqup_fpbg/ -HW
+ lapply (IH1 ⦠HU ⦠Hd ⦠HU2 (L2.âW2) ?) -IH1 -HU -Hd -HU2 [1,2: /2 by fqup_fpbg, lpr_pair/ ] -HL12 -HW2
/4 width=6 by da_bind, lsubd_da_trans, lsubd_beta/
- | #b #V0 #V2 #W #W2 #U #U2 #HV10 #HV02 #HW2 #HU2 #H1 #H2 destruct -IH3 -IH2 -b1 -V0 -W1 -U1 -l1 -HV1
+ | #b #V0 #V2 #W #W2 #U #U2 #HV10 #HV02 #HW2 #HU2 #H1 #H2 destruct -IH3 -IH2 -b1 -V0 -W1 -U1 -d1 -HV1
elim (snv_inv_bind ⦠HT1) -HT1 #_
- lapply (da_inv_bind ⦠Hl) -Hl
+ lapply (da_inv_bind ⦠Hd) -Hd
/5 width=9 by da_bind, da_flat, fqup_fpbg, lpr_pair/
]
-| #W1 #T1 #HG0 #HL0 #HT0 #H1 #l #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #d #H2 #X3 #H3 #L2 #HL12 destruct -IH3 -IH2
elim (snv_inv_cast ⦠H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
- lapply (da_inv_flat ⦠H2) -H2 #Hl
+ lapply (da_inv_flat ⦠H2) -H2 #Hd
elim (cpr_inv_cast1 ⦠H3) -H3
[ * #W2 #T2 #HW12 #HT12 #H destruct /4 width=7 by da_flat, fqup_fpbg/
| /3 width=7 by fqup_fpbg/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma
index b1b995c2b..4d9e1d0cc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lift.ma
@@ -20,57 +20,57 @@ include "basic_2/dynamic/snv.ma".
(* Relocation properties ****************************************************)
-lemma snv_lift: âh,g,G,K,T. â¦G, K⦠⢠T ¡[h, g] â âL,s,d,e. â¬[s, d, e] L â¡ K â
- âU. â¬[d, e] T â¡ U â â¦G, L⦠⢠U ¡[h, g].
+lemma snv_lift: âh,g,G,K,T. â¦G, K⦠⢠T ¡[h, g] â âL,s,l,m. â¬[s, l, m] L â¡ K â
+ âU. â¬[l, m] T â¡ U â â¦G, L⦠⢠U ¡[h, g].
#h #g #G #K #T #H elim H -G -K -T
-[ #G #K #k #L #s #d #e #_ #X #H
- >(lift_inv_sort1 ⦠H) -X -K -d -e //
-| #I #G #K #K0 #V #i #HK0 #_ #IHV #L #s #d #e #HLK #X #H
- elim (lift_inv_lref1 ⦠H) * #Hid #H destruct
+[ #G #K #k #L #s #l #m #_ #X #H
+ >(lift_inv_sort1 ⦠H) -X -K -l -m //
+| #I #G #K #K0 #V #i #HK0 #_ #IHV #L #s #l #m #HLK #X #H
+ elim (lift_inv_lref1 ⦠H) * #Hil #H destruct
[ elim (drop_trans_le ⦠HLK ⦠HK0) -K /2 width=2 by lt_to_le/ #X #HL0 #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #L0 #W #HLK0 #HVW #H destruct
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #L0 #W #HLK0 #HVW #H destruct
/3 width=9 by snv_lref/
| lapply (drop_trans_ge ⦠HLK ⦠HK0 ?) -K
/3 width=9 by snv_lref, drop_inv_gen/
]
-| #a #I #G #K #V #T #_ #_ #IHV #IHT #L #s #d #e #HLK #X #H
+| #a #I #G #K #V #T #_ #_ #IHV #IHT #L #s #l #m #HLK #X #H
elim (lift_inv_bind1 ⦠H) -H #W #U #HVW #HTU #H destruct
/4 width=5 by snv_bind, drop_skip/
-| #a #G #K #V #W0 #T #U0 #l #_ #_ #HVW0 #HTU0 #IHV #IHT #L #s #d #e #HLK #X #H
+| #a #G #K #V #W0 #T #U0 #d #_ #_ #HVW0 #HTU0 #IHV #IHT #L #s #l #m #HLK #X #H
elim (lift_inv_flat1 ⦠H) -H #W #U #HVW #HTU #H destruct
- elim (lift_total W0 d e)
- elim (lift_total U0 (d+1) e)
+ elim (lift_total W0 l m)
+ elim (lift_total U0 (l+1) m)
/4 width=17 by snv_appl, scpds_lift, lift_bind/
-| #G #K #V #T #U0 #_ #_ #HVU0 #HTU0 #IHV #IHT #L #s #d #e #HLK #X #H
+| #G #K #V #T #U0 #_ #_ #HVU0 #HTU0 #IHV #IHT #L #s #l #m #HLK #X #H
elim (lift_inv_flat1 ⦠H) -H #W #U #HVW #HTU #H destruct
- elim (lift_total U0 d e)
+ elim (lift_total U0 l m)
/3 width=12 by snv_cast, scpds_lift/
]
qed.
-lemma snv_inv_lift: âh,g,G,L,U. â¦G, L⦠⢠U ¡[h, g] â âK,s,d,e. â¬[s, d, e] L â¡ K â
- âT. â¬[d, e] T â¡ U â â¦G, K⦠⢠T ¡[h, g].
+lemma snv_inv_lift: âh,g,G,L,U. â¦G, L⦠⢠U ¡[h, g] â âK,s,l,m. â¬[s, l, m] L â¡ K â
+ âT. â¬[l, m] T â¡ U â â¦G, K⦠⢠T ¡[h, g].
#h #g #G #L #U #H elim H -G -L -U
-[ #G #L #k #K #s #d #e #_ #X #H
- >(lift_inv_sort2 ⦠H) -X -L -d -e //
-| #I #G #L #L0 #W #i #HL0 #_ #IHW #K #s #d #e #HLK #X #H
- elim (lift_inv_lref2 ⦠H) * #Hid #H destruct
+[ #G #L #k #K #s #l #m #_ #X #H
+ >(lift_inv_sort2 ⦠H) -X -L -l -m //
+| #I #G #L #L0 #W #i #HL0 #_ #IHW #K #s #l #m #HLK #X #H
+ elim (lift_inv_lref2 ⦠H) * #Hil #H destruct
[ elim (drop_conf_le ⦠HLK ⦠HL0) -L /2 width=2 by lt_to_le/ #X #HK0 #H
- elim (drop_inv_skip1 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K0 #V #HLK0 #HVW #H destruct
+ elim (drop_inv_skip1 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K0 #V #HLK0 #HVW #H destruct
/3 width=12 by snv_lref/
| lapply (drop_conf_ge ⦠HLK ⦠HL0 ?) -L /3 width=9 by snv_lref/
]
-| #a #I #G #L #W #U #_ #_ #IHW #IHU #K #s #d #e #HLK #X #H
+| #a #I #G #L #W #U #_ #_ #IHW #IHU #K #s #l #m #HLK #X #H
elim (lift_inv_bind2 ⦠H) -H #V #T #HVW #HTU #H destruct
/4 width=5 by snv_bind, drop_skip/
-| #a #G #L #W #W1 #U #U1 #l #_ #_ #HW1 #HU1 #IHW #IHU #K #s #d #e #HLK #X #H
+| #a #G #L #W #W1 #U #U1 #d #_ #_ #HW1 #HU1 #IHW #IHU #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #V #T #HVW #HTU #H destruct
elim (scpds_inv_lift1 ⦠HW1 ⦠HLK ⦠HVW) -HW1 #W0 #HW01 #HVW0
elim (scpds_inv_lift1 ⦠HU1 ⦠HLK ⦠HTU) -HU1 #X #H #HTU0
elim (lift_inv_bind2 ⦠H) -H #Y #U0 #HY #HU01 #H destruct
lapply (lift_inj ⦠HY ⦠HW01) -HY #H destruct
/3 width=6 by snv_appl/
-| #G #L #W #U #U1 #_ #_ #HWU1 #HU1 #IHW #IHU #K #s #d #e #HLK #X #H
+| #G #L #W #U #U1 #_ #_ #HWU1 #HU1 #IHW #IHU #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #V #T #HVW #HTU #H destruct
elim (scpds_inv_lift1 ⦠HWU1 ⦠HLK ⦠HVW) -HWU1 #U0 #HU01 #HVU0
elim (scpds_inv_lift1 ⦠HU1 ⦠HLK ⦠HTU) -HU1 #X #HX #HTU0
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lpr.ma
index f6ea17c6c..8c1350a82 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lpr.ma
@@ -51,7 +51,7 @@ fact snv_cpr_lpr_aux: âh,g,G0,L0,T0.
/4 width=10 by fqup_fpbg, snv_inv_lift, lpr_pair, drop_drop/
]
| #V1 #T1 #HG0 #HL0 #HT0 #H1 #X #H2 #L2 #HL12 destruct
- elim (snv_inv_appl ⦠H1) -H1 #a #W1 #U1 #l0 #HV1 #HT1 #HVW1 #HTU1
+ elim (snv_inv_appl ⦠H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
elim (cpr_inv_appl1 ⦠H2) -H2 *
[ #V2 #T2 #HV12 #HT12 #H destruct -IH4
lapply (IH1 ⦠HV12 ⦠HL12) /2 width=1 by fqup_fpbg/ #HV2
@@ -65,23 +65,23 @@ fact snv_cpr_lpr_aux: âh,g,G0,L0,T0.
/2 width=7 by snv_appl, cprs_bind/
| #b #V2 #W10 #W20 #T10 #T20 #HV12 #HW120 #HT120 #H1 #H2 destruct
elim (snv_inv_bind ⦠HT1) -HT1 #HW10 #HT10
- elim (scpds_inv_abst1 ⦠HTU1) -HTU1 #W30 #T30 #HW130 #_ #H destruct -T30 -l0
- elim (snv_fwd_da ⦠HV1) #l #HV1l
- elim (snv_fwd_da ⦠HW10) #l0 #HW10l
- lapply (cprs_scpds_div ⦠HW130 ⦠HW10l ⦠1 HVW1) -W30 #HVW10
- elim (da_scpes_aux ⦠IH4 IH1 IH2 ⦠HW10l ⦠HV1l ⦠HVW10) /2 width=1 by fqup_fpbg/
- #_ #Hl <minus_n_O #H destruct >(plus_minus_m_m l 1) in HV1l; // -Hl #HV1l
+ elim (scpds_inv_abst1 ⦠HTU1) -HTU1 #W30 #T30 #HW130 #_ #H destruct -T30 -d0
+ elim (snv_fwd_da ⦠HV1) #d #HV1d
+ elim (snv_fwd_da ⦠HW10) #d0 #HW10d
+ lapply (cprs_scpds_div ⦠HW130 ⦠HW10d ⦠1 HVW1) -W30 #HVW10
+ elim (da_scpes_aux ⦠IH4 IH1 IH2 ⦠HW10d ⦠HV1d ⦠HVW10) /2 width=1 by fqup_fpbg/
+ #_ #Hd <minus_n_O #H destruct >(plus_minus_m_m d 1) in HV1d; // -Hd #HV1d
lapply (scpes_cpr_lpr_aux ⦠IH2 IH3 ⦠HVW10 ⦠HW120 ⦠HV12 ⦠HL12) /2 width=1 by fqup_fpbg/ -HVW10 #HVW20
- lapply (IH2 ⦠HV1l ⦠HV12 ⦠HL12) /2 width=1 by fqup_fpbg/ -HV1l #HV2l
- lapply (IH2 ⦠HW10l ⦠HW120 ⦠HL12) /2 width=1 by fqup_fpbg/ -HW10l #HW20l
+ lapply (IH2 ⦠HV1d ⦠HV12 ⦠HL12) /2 width=1 by fqup_fpbg/ -HV1d #HV2d
+ lapply (IH2 ⦠HW10d ⦠HW120 ⦠HL12) /2 width=1 by fqup_fpbg/ -HW10d #HW20d
lapply (IH1 ⦠HV12 ⦠HL12) /2 width=1 by fqup_fpbg/ #HV2
lapply (IH1 ⦠HW120 ⦠HL12) /2 width=1 by fqup_fpbg/ -HW10 #HW20
lapply (IH1 ⦠HT10 ⦠HT120 ⦠(L2.âW20) ?) /2 width=1 by fqup_fpbg, lpr_pair/ -HT10 #HT20
@snv_bind /2 width=1 by snv_cast_scpes/
@(lsubsv_snv_trans ⦠HT20) -HT20
- @(lsubsv_beta ⦠(l-1)) //
- @shnv_cast [1,2: // ] #l0 #Hl0
- lapply (scpes_le_aux ⦠IH4 IH1 IH2 IH3 ⦠HW20l ⦠HV2l ⦠l0 ⦠HVW20) -IH4 -IH3 -IH2 -IH1 -HW20l -HV2l -HVW20
+ @(lsubsv_beta ⦠(d-1)) //
+ @shnv_cast [1,2: // ] #d0 #Hd0
+ lapply (scpes_le_aux ⦠IH4 IH1 IH2 IH3 ⦠HW20d ⦠HV2d ⦠d0 ⦠HVW20) -IH4 -IH3 -IH2 -IH1 -HW20d -HV2d -HVW20
/3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs, le_S_S/
| #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -IH4
elim (snv_inv_bind ⦠HT1) -HT1 #HW0 #HT0
@@ -106,8 +106,8 @@ fact snv_cpr_lpr_aux: âh,g,G0,L0,T0.
elim (snv_inv_cast ⦠H1) -H1 #U1 #HW1 #HT1 #HWU1 #HTU1
elim (cpr_inv_cast1 ⦠H2) -H2
[ * #W2 #T2 #HW12 #HT12 #H destruct
- elim (snv_fwd_da ⦠HW1) #l #HW1l
- lapply (scpds_div ⦠HWU1 ⦠HTU1) -U1 -l #HWT1
+ elim (snv_fwd_da ⦠HW1) #d #HW1d
+ lapply (scpds_div ⦠HWU1 ⦠HTU1) -U1 -d #HWT1
lapply (scpes_cpr_lpr_aux ⦠IH2 IH3 ⦠HWT1 ⦠HW12 ⦠HT12 ⦠HL12) /2 width=1 by fqup_fpbg/
lapply (IH1 ⦠HW12 ⦠HL12) /2 width=1 by fqup_fpbg/
lapply (IH1 ⦠HT12 ⦠HL12) /2 width=1 by fqup_fpbg/ -L1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lstas.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lstas.ma
index 3d4d1f35a..e8f7ff0b4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lstas.ma
@@ -26,33 +26,33 @@ fact snv_lstas_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_snv_lstas h g G1 L1 T1) â
âG1,L1,T1. G0 = G1 â L0 = L1 â T0 = T1 â IH_snv_lstas h g G1 L1 T1.
#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #k #HG0 #HL0 #HT0 #_ #l1 #l2 #Hl21 #Hl1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
+[ #k #HG0 #HL0 #HT0 #_ #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
>(lstas_inv_sort1 ⦠H2) -X //
-| #i #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #T #H2 destruct -IH4 -IH3 -IH2
+| #i #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #T #H2 destruct -IH4 -IH3 -IH2
elim (snv_inv_lref ⦠H1) -H1 #I0 #K0 #X0 #HLK0 #HX0
- elim (da_inv_lref ⦠Hl1) -Hl1 * #K1 [ #V1 | #W1 #l0 ] #HLK1 [ #Hl1 | #Hl0 #H ]
+ elim (da_inv_lref ⦠Hd1) -Hd1 * #K1 [ #V1 | #W1 #d0 ] #HLK1 [ #Hd1 | #Hd0 #H ]
lapply (drop_mono ⦠HLK0 ⦠HLK1) -HLK0 #H0 destruct
- elim (lstas_inv_lref1 ⦠H2) -H2 * #K #Y #X [3,6: #l ] #HLK #HYX [1,2: #HXT #H0 |3,5: #HXT |4,6: #H1 #H2 ]
+ elim (lstas_inv_lref1 ⦠H2) -H2 * #K #Y #X [3,6: #d ] #HLK #HYX [1,2: #HXT #H0 |3,5: #HXT |4,6: #H1 #H2 ]
lapply (drop_mono ⦠HLK ⦠HLK1) -HLK #H destruct
- [ lapply (le_plus_to_le_r ⦠Hl21) -Hl21 #Hl21 |3: -Hl21 ]
+ [ lapply (le_plus_to_le_r ⦠Hd21) -Hd21 #Hd21 |3: -Hd21 ]
lapply (fqup_lref ⦠G1 ⦠HLK1) #H
lapply (drop_fwd_drop2 ⦠HLK1) /4 width=8 by snv_lift, snv_lref, fqup_fpbg/
-| #p #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
+| #p #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2 -IH1
elim (snv_inv_gref ⦠H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X #H2 destruct -IH4 -IH3 -IH2
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2
elim (snv_inv_bind ⦠H1) -H1 #HV1 #HT1
- lapply (da_inv_bind ⦠Hl1) -Hl1 #Hl1
+ lapply (da_inv_bind ⦠Hd1) -Hd1 #Hd1
elim (lstas_inv_bind1 ⦠H2) -H2 #U1 #HTU1 #H destruct /4 width=8 by fqup_fpbg, snv_bind/
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X #H2 destruct
- elim (snv_inv_appl ⦠H1) -H1 #a #W1 #U1 #l0 #HV1 #HT1 #HVW1 #HTU1
- lapply (da_inv_flat ⦠Hl1) -Hl1 #Hl1
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct
+ elim (snv_inv_appl ⦠H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
+ lapply (da_inv_flat ⦠Hd1) -Hd1 #Hd1
elim (lstas_inv_appl1 ⦠H2) -H2 #T0 #HT10 #H destruct
- lapply (IH1 ⦠HT1 ⦠Hl1 ⦠HT10) /2 width=1 by fqup_fpbg/ #HT0
- lapply (lstas_scpds_aux ⦠IH1 IH4 IH3 IH2 ⦠Hl1 ⦠HT10 ⦠HTU1) -IH4 -IH3 -IH2 -IH1 /2 width=1 by fqup_fpbg/ -T1 -l1 #H
+ lapply (IH1 ⦠HT1 ⦠Hd1 ⦠HT10) /2 width=1 by fqup_fpbg/ #HT0
+ lapply (lstas_scpds_aux ⦠IH1 IH4 IH3 IH2 ⦠Hd1 ⦠HT10 ⦠HTU1) -IH4 -IH3 -IH2 -IH1 /2 width=1 by fqup_fpbg/ -T1 -d1 #H
elim (scpes_inv_abst2 ⦠H) -H /3 width=6 by snv_appl, scpds_cprs_trans/
-| #U1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X #H2 destruct -IH4 -IH3 -IH2
+| #U1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X #H2 destruct -IH4 -IH3 -IH2
elim (snv_inv_cast ⦠H1) -H1
- lapply (da_inv_flat ⦠Hl1) -Hl1
+ lapply (da_inv_flat ⦠Hd1) -Hd1
lapply (lstas_inv_cast1 ⦠H2) -H2 /3 width=8 by fqup_fpbg/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lstas_lpr.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lstas_lpr.ma
index 71d0aa836..dff71f336 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lstas_lpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_lstas_lpr.ma
@@ -27,16 +27,16 @@ fact lstas_cpr_lpr_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_lstas_cpr_lpr h g G1 L1 T1) â
âG1,L1,T1. G0 = G1 â L0 = L1 â T0 = T1 â IH_lstas_cpr_lpr h g G1 L1 T1.
#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G1 #L1 * * [|||| * ]
-[ #k #_ #_ #_ #_ #l1 #l2 #_ #_ #X2 #H2 #X3 #H3 #L2 #_ -IH4 -IH3 -IH2 -IH1
+[ #k #_ #_ #_ #_ #d1 #d2 #_ #_ #X2 #H2 #X3 #H3 #L2 #_ -IH4 -IH3 -IH2 -IH1
>(lstas_inv_sort1 ⦠H2) -X2
>(cpr_inv_sort1 ⦠H3) -X3 /2 width=3 by ex2_intro/
-| #i #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3
+| #i #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3
elim (snv_inv_lref ⦠H1) -H1 #I0 #K0 #X0 #HK0 #HX0
- elim (da_inv_lref ⦠Hl1) -Hl1 * #K1 [ #V1 | #W1 #l ] #HLK1 [ #HVl1 | #HWl1 #H destruct ]
+ elim (da_inv_lref ⦠Hd1) -Hd1 * #K1 [ #V1 | #W1 #d ] #HLK1 [ #HVd1 | #HWd1 #H destruct ]
lapply (drop_mono ⦠HK0 ⦠HLK1) -HK0 #H destruct
- elim (lstas_inv_lref1 ⦠H2) -H2 * #K0 #V0 #X0 [3,6: #l0 ] #HK0 #HVX0 [1,2: #HX02 #H |3,5: #HX02 |4,6: #H1 #H2 ] destruct
+ elim (lstas_inv_lref1 ⦠H2) -H2 * #K0 #V0 #X0 [3,6: #d0 ] #HK0 #HVX0 [1,2: #HX02 #H |3,5: #HX02 |4,6: #H1 #H2 ] destruct
lapply (drop_mono ⦠HK0 ⦠HLK1) -HK0 #H destruct
- [ lapply (le_plus_to_le_r ⦠Hl21) -Hl21 #Hl21 |3: -Hl21 ]
+ [ lapply (le_plus_to_le_r ⦠Hd21) -Hd21 #Hd21 |3: -Hd21 ]
lapply (fqup_lref ⦠G1 ⦠HLK1) #HKV1
elim (lpr_drop_conf ⦠HLK1 ⦠HL12) -HL12 #X #H #HLK2
elim (lpr_inv_pair1 ⦠H) -H #K2 [ #W2 | #W2 | #V2 ] #HK12 [ #HW12 | #HW12 | #HV12 ] #H destruct
@@ -46,77 +46,77 @@ fact lstas_cpr_lpr_aux: âh,g,G0,L0,T0.
|2,4,6: * #K #V #X #H #HVX #HX3
lapply (drop_mono ⦠H ⦠HLK1) -H -HLK1 #H destruct
]
- [ lapply (IH2 ⦠HWl1 ⦠HW12 ⦠HK12) /2 width=1 by fqup_fpbg/ -IH2 #H
- elim (da_lstas ⦠H l0) -H
- elim (IH1 ⦠HWl1 ⦠HVX0 ⦠HW12 ⦠HK12) -IH1 -HVX0 /2 width=1 by fqup_fpbg/ #V2 #HWV2 #HV2
+ [ lapply (IH2 ⦠HWd1 ⦠HW12 ⦠HK12) /2 width=1 by fqup_fpbg/ -IH2 #H
+ elim (da_lstas ⦠H d0) -H
+ elim (IH1 ⦠HWd1 ⦠HVX0 ⦠HW12 ⦠HK12) -IH1 -HVX0 /2 width=1 by fqup_fpbg/ #V2 #HWV2 #HV2
elim (lift_total V2 0 (i+1))
/3 width=12 by cpcs_lift, lstas_succ, ex2_intro/
- | elim (IH1 ⦠HWl1 ⦠HVX0 ⦠HW12 ⦠HK12) -IH1 -HVX0
+ | elim (IH1 ⦠HWd1 ⦠HVX0 ⦠HW12 ⦠HK12) -IH1 -HVX0
/3 width=5 by fqup_fpbg, lstas_zero, ex2_intro/
- | elim (IH1 ⦠HVl1 ⦠HVX0 ⦠HV12 ⦠HK12) -IH1 -HVl1 -HVX0 -HV12 -HK12 -IH2 /2 width=1 by fqup_fpbg/ #W2 #HVW2 #HW02
+ | elim (IH1 ⦠HVd1 ⦠HVX0 ⦠HV12 ⦠HK12) -IH1 -HVd1 -HVX0 -HV12 -HK12 -IH2 /2 width=1 by fqup_fpbg/ #W2 #HVW2 #HW02
elim (lift_total W2 0 (i+1))
/4 width=12 by cpcs_lift, lstas_ldef, ex2_intro/
- | elim (IH1 ⦠HVl1 ⦠HVX0 ⦠HVX ⦠HK12) -IH1 -HVl1 -HVX0 -HVX -HK12 -IH2 -V2 /2 width=1 by fqup_fpbg/ -l1 #W2 #HXW2 #HW02
+ | elim (IH1 ⦠HVd1 ⦠HVX0 ⦠HVX ⦠HK12) -IH1 -HVd1 -HVX0 -HVX -HK12 -IH2 -V2 /2 width=1 by fqup_fpbg/ -d1 #W2 #HXW2 #HW02
elim (lift_total W2 0 (i+1))
/3 width=12 by cpcs_lift, lstas_lift, ex2_intro/
]
| #p #_ #_ #HT0 #H1 destruct -IH4 -IH3 -IH2 -IH1
elim (snv_inv_gref ⦠H1)
-| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
+| #a #I #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
elim (snv_inv_bind ⦠H1) -H1 #_ #HT1
- lapply (da_inv_bind ⦠Hl1) -Hl1 #Hl1
+ lapply (da_inv_bind ⦠Hd1) -Hd1 #Hd1
elim (lstas_inv_bind1 ⦠H2) -H2 #U1 #HTU1 #H destruct
elim (cpr_inv_bind1 ⦠H3) -H3 *
[ #V2 #T2 #HV12 #HT12 #H destruct
- elim (IH1 ⦠Hl1 ⦠HTU1 ⦠HT12 (L2.â{I}V2)) -IH1 -Hl1 -HTU1 -HT12 /2 width=1 by fqup_fpbg, lpr_pair/ -T1
+ elim (IH1 ⦠Hd1 ⦠HTU1 ⦠HT12 (L2.â{I}V2)) -IH1 -Hd1 -HTU1 -HT12 /2 width=1 by fqup_fpbg, lpr_pair/ -T1
/4 width=5 by cpcs_bind2, lpr_cpr_conf, lstas_bind, ex2_intro/
| #T3 #HT13 #HXT3 #H1 #H2 destruct
- elim (IH1 ⦠Hl1 ⦠HTU1 ⦠HT13 (L2.âV1)) -IH1 -Hl1 -HTU1 -HT13 /2 width=1 by fqup_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
+ elim (IH1 ⦠Hd1 ⦠HTU1 ⦠HT13 (L2.âV1)) -IH1 -Hd1 -HTU1 -HT13 /2 width=1 by fqup_fpbg, lpr_pair/ -T1 -HL12 #U3 #HTU3 #HU13
elim (lstas_inv_lift1 ⦠HTU3 L2 ⦠HXT3) -T3
/5 width=8 by cpcs_cpr_strap1, cpcs_bind1, cpr_zeta, drop_drop, ex2_intro/
]
-| #V1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct
- elim (snv_inv_appl ⦠H1) -H1 #a #W1 #U1 #l0 #HV1 #HT1 #HVW1 #HTU1
- lapply (da_inv_flat ⦠Hl1) -Hl1 #Hl1
+| #V1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct
+ elim (snv_inv_appl ⦠H1) -H1 #a #W1 #U1 #d0 #HV1 #HT1 #HVW1 #HTU1
+ lapply (da_inv_flat ⦠Hd1) -Hd1 #Hd1
elim (lstas_inv_appl1 ⦠H2) -H2 #X #HT1U #H destruct
elim (cpr_inv_appl1 ⦠H3) -H3 *
- [ #V2 #T2 #HV12 #HT12 #H destruct -a -l0 -W1 -U1 -HV1 -IH4 -IH3 -IH2
- elim (IH1 ⦠Hl1 ⦠HT1U ⦠HT12 ⦠HL12) -IH1 -Hl1 -HT1U
+ [ #V2 #T2 #HV12 #HT12 #H destruct -a -d0 -W1 -U1 -HV1 -IH4 -IH3 -IH2
+ elim (IH1 ⦠Hd1 ⦠HT1U ⦠HT12 ⦠HL12) -IH1 -Hd1 -HT1U
/4 width=5 by fqup_fpbg, cpcs_flat, lpr_cpr_conf, lstas_appl, ex2_intro/
| #b #V2 #W2 #W3 #T2 #T3 #HV12 #HW23 #HT23 #H1 #H2 destruct
elim (snv_inv_bind ⦠HT1) -HT1 #HW2 #HT2
- lapply (da_inv_bind ⦠Hl1) -Hl1 #Hl1
+ lapply (da_inv_bind ⦠Hd1) -Hd1 #Hd1
elim (lstas_inv_bind1 ⦠HT1U) -HT1U #U #HT2U #H destruct
- elim (scpds_inv_abst1 ⦠HTU1) -HTU1 #W0 #U0 #HW20 #_ #H destruct -U0 -l0
- elim (snv_fwd_da ⦠HW2) #l0 #HW2l
- lapply (cprs_scpds_div ⦠HW20 ⦠HW2l ⦠HVW1) -W0 #H21
- elim (snv_fwd_da ⦠HV1) #l #HV1l
- elim (da_scpes_aux ⦠IH4 IH3 IH2 ⦠HW2l ⦠HV1l ⦠H21) /2 width=1 by fqup_fpbg/ #_ #H
- <minus_n_O #H0 destruct >(plus_minus_m_m l 1) in HV1l; // -H #HV1l
+ elim (scpds_inv_abst1 ⦠HTU1) -HTU1 #W0 #U0 #HW20 #_ #H destruct -U0 -d0
+ elim (snv_fwd_da ⦠HW2) #d0 #HW2d
+ lapply (cprs_scpds_div ⦠HW20 ⦠HW2d ⦠HVW1) -W0 #H21
+ elim (snv_fwd_da ⦠HV1) #d #HV1d
+ elim (da_scpes_aux ⦠IH4 IH3 IH2 ⦠HW2d ⦠HV1d ⦠H21) /2 width=1 by fqup_fpbg/ #_ #H
+ <minus_n_O #H0 destruct >(plus_minus_m_m d 1) in HV1d; // -H #HV1d
lapply (scpes_cpr_lpr_aux ⦠IH2 IH1 ⦠H21 ⦠HW23 ⦠HV12 ⦠HL12) -H21 /2 width=1 by fqup_fpbg/ #H32
lapply (IH3 ⦠HW23 ⦠HL12) /2 width=1 by fqup_fpbg/ #HW3
lapply (IH3 ⦠HV12 ⦠HL12) /2 width=1 by fqup_fpbg/ #HV2
- lapply (IH2 ⦠HW2l ⦠HW23 ⦠HL12) /2 width=1 by fqup_fpbg/ -HW2 -HW2l #HW3l
- lapply (IH2 ⦠HV1l ⦠HV12 ⦠HL12) /2 width=1 by fqup_fpbg/ -HV1 -HV1l #HV2l
- elim (IH1 ⦠Hl1 ⦠HT2U ⦠HT23 (L2.âW3)) -HT2U /2 width=1 by fqup_fpbg, lpr_pair/ #U3 #HTU3 #HU23
- elim (lsubsv_lstas_trans ⦠g ⦠HTU3 ⦠Hl21 ⦠(L2.ââW3.V2)) -HTU3
- [ #U4 #HT3U4 #HU43 -IH1 -IH2 -IH3 -IH4 -l -l1 -HW3 -HV2 -HT2
+ lapply (IH2 ⦠HW2d ⦠HW23 ⦠HL12) /2 width=1 by fqup_fpbg/ -HW2 -HW2d #HW3d
+ lapply (IH2 ⦠HV1d ⦠HV12 ⦠HL12) /2 width=1 by fqup_fpbg/ -HV1 -HV1d #HV2d
+ elim (IH1 ⦠Hd1 ⦠HT2U ⦠HT23 (L2.âW3)) -HT2U /2 width=1 by fqup_fpbg, lpr_pair/ #U3 #HTU3 #HU23
+ elim (lsubsv_lstas_trans ⦠g ⦠HTU3 ⦠Hd21 ⦠(L2.ââW3.V2)) -HTU3
+ [ #U4 #HT3U4 #HU43 -IH1 -IH2 -IH3 -IH4 -d -d1 -HW3 -HV2 -HT2
@(ex2_intro ⦠(â{b}âW3.V2.U4)) /2 width=1 by lstas_bind/ -HT3U4
@(cpcs_canc_dx ⦠(â{b}âW3.V2.U3)) /2 width=1 by cpcs_bind_dx/ -HU43
@(cpcs_cpr_strap1 ⦠(âV2.â{b}W3.U3)) /2 width=1 by cpr_beta/
/4 width=3 by cpcs_flat, cpcs_bind2, lpr_cpr_conf/
| -U3
- @(lsubsv_beta ⦠(l-1)) /3 width=7 by fqup_fpbg/
- @shnv_cast [1,2: // ] #l0 #Hl0
- lapply (scpes_le_aux ⦠IH4 IH3 IH2 IH1 ⦠HW3l ⦠HV2l ⦠l0 ⦠H32) -IH4 -IH3 -IH2 -IH1 -HW3l -HV2l -H32
+ @(lsubsv_beta ⦠(d-1)) /3 width=7 by fqup_fpbg/
+ @shnv_cast [1,2: // ] #d0 #Hd0
+ lapply (scpes_le_aux ⦠IH4 IH3 IH2 IH1 ⦠HW3d ⦠HV2d ⦠d0 ⦠H32) -IH4 -IH3 -IH2 -IH1 -HW3d -HV2d -H32
/3 width=5 by fpbg_fpbs_trans, fqup_fpbg, cpr_lpr_fpbs, le_S_S/
| -IH1 -IH3 -IH4 /3 width=9 by fqup_fpbg, lpr_pair/
]
- | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -l0 -W1 -HV1 -IH4 -IH3 -IH2
+ | #b #V0 #V2 #W0 #W2 #T0 #T2 #HV10 #HV02 #HW02 #HT02 #H1 #H2 destruct -a -d0 -W1 -HV1 -IH4 -IH3 -IH2
elim (snv_inv_bind ⦠HT1) -HT1 #_ #HT0
- lapply (da_inv_bind ⦠Hl1) -Hl1 #Hl1
+ lapply (da_inv_bind ⦠Hd1) -Hd1 #Hd1
elim (lstas_inv_bind1 ⦠HT1U) -HT1U #U0 #HTU0 #H destruct
- elim (IH1 ⦠Hl1 ⦠HTU0 ⦠HT02 (L2.âW2)) -IH1 -Hl1 -HTU0 /2 width=1 by fqup_fpbg, lpr_pair/ -T0 #U2 #HTU2 #HU02
+ elim (IH1 ⦠Hd1 ⦠HTU0 ⦠HT02 (L2.âW2)) -IH1 -Hd1 -HTU0 /2 width=1 by fqup_fpbg, lpr_pair/ -T0 #U2 #HTU2 #HU02
lapply (lpr_cpr_conf ⦠HL12 ⦠HV10) -HV10 #HV10
lapply (lpr_cpr_conf ⦠HL12 ⦠HW02) -L1 #HW02
lapply (cpcs_bind2 b ⦠HW02 ⦠HU02) -HW02 -HU02 #HU02
@@ -124,15 +124,15 @@ fact lstas_cpr_lpr_aux: âh,g,G0,L0,T0.
lapply (cpcs_cpr_strap1 ⦠HU02 (â{b}W2.âV2.U2) ?)
/4 width=3 by lstas_appl, lstas_bind, cpr_theta, ex2_intro/
]
-| #W1 #T1 #HG0 #HL0 #HT0 #H1 #l1 #l2 #Hl21 #Hl1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
+| #W1 #T1 #HG0 #HL0 #HT0 #H1 #d1 #d2 #Hd21 #Hd1 #X2 #H2 #X3 #H3 #L2 #HL12 destruct -IH4 -IH3 -IH2
elim (snv_inv_cast ⦠H1) -H1 #U1 #_ #HT1 #_ #_ -U1
- lapply (da_inv_flat ⦠Hl1) -Hl1 #Hl1
+ lapply (da_inv_flat ⦠Hd1) -Hd1 #Hd1
lapply (lstas_inv_cast1 ⦠H2) -H2 #HTU1
elim (cpr_inv_cast1 ⦠H3) -H3
[ * #U2 #T2 #_ #HT12 #H destruct
- elim (IH1 ⦠Hl1 ⦠HTU1 ⦠HT12 ⦠HL12) -IH1 -Hl1 -HTU1 -HL12
+ elim (IH1 ⦠Hd1 ⦠HTU1 ⦠HT12 ⦠HL12) -IH1 -Hd1 -HTU1 -HL12
/3 width=3 by fqup_fpbg, lstas_cast, ex2_intro/
- | #HT1X3 elim (IH1 ⦠Hl1 ⦠HTU1 ⦠HT1X3 ⦠HL12) -IH1 -Hl1 -HTU1 -HL12
+ | #HT1X3 elim (IH1 ⦠Hd1 ⦠HTU1 ⦠HT1X3 ⦠HL12) -IH1 -Hd1 -HTU1 -HL12
/2 width=3 by fqup_fpbg, ex2_intro/
]
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_preserve.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_preserve.ma
index 8376ac4c1..23914794d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_preserve.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_preserve.ma
@@ -61,34 +61,34 @@ lemma snv_cprs_lpr: âh,g,G,L1,T1. â¦G, L1⦠⢠T1 ¡[h, g] â
qed-.
lemma da_cprs_lpr: âh,g,G,L1,T1. â¦G, L1⦠⢠T1 ¡[h, g] â
- âl. â¦G, L1⦠⢠T1 âª[h, g] l â
- âT2. â¦G, L1⦠⢠T1 â¡* T2 â âL2. â¦G, L1⦠⢠⡠L2 â â¦G, L2⦠⢠T2 âª[h, g] l.
-#h #g #G #L1 #T1 #HT1 #l #Hl #T2 #H
+ âd. â¦G, L1⦠⢠T1 âª[h, g] d â
+ âT2. â¦G, L1⦠⢠T1 â¡* T2 â âL2. â¦G, L1⦠⢠⡠L2 â â¦G, L2⦠⢠T2 âª[h, g] d.
+#h #g #G #L1 #T1 #HT1 #d #Hd #T2 #H
@(cprs_ind ⦠H) -T2 /3 width=6 by snv_cprs_lpr, da_cpr_lpr/
qed-.
lemma lstas_cprs_lpr: âh,g,G,L1,T1. â¦G, L1⦠⢠T1 ¡[h, g] â
- âl1,l2. l2 ⤠l1 â â¦G, L1⦠⢠T1 âª[h, g] l1 â
- âU1. â¦G, L1⦠⢠T1 â¢*[h, l2] U1 â
+ âd1,d2. d2 ⤠d1 â â¦G, L1⦠⢠T1 âª[h, g] d1 â
+ âU1. â¦G, L1⦠⢠T1 â¢*[h, d2] U1 â
âT2. â¦G, L1⦠⢠T1 â¡* T2 â âL2. â¦G, L1⦠⢠⡠L2 â
- ââU2. â¦G, L2⦠⢠T2 â¢*[h, l2] U2 & â¦G, L2⦠⢠U1 â¬* U2.
-#h #g #G #L1 #T1 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 #H
+ ââU2. â¦G, L2⦠⢠T2 â¢*[h, d2] U2 & â¦G, L2⦠⢠U1 â¬* U2.
+#h #g #G #L1 #T1 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H
@(cprs_ind ⦠H) -T2 [ /2 width=10 by lstas_cpr_lpr/ ]
#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
elim (IHT1 L1) // -IHT1 #U #HTU #HU1
-elim (lstas_cpr_lpr ⦠g ⦠Hl21 ⦠HTU ⦠HTT2 ⦠HL12) -HTU -HTT2
-[2,3: /2 width=7 by snv_cprs_lpr, da_cprs_lpr/ ] -T1 -T -l1
+elim (lstas_cpr_lpr ⦠g ⦠Hd21 ⦠HTU ⦠HTT2 ⦠HL12) -HTU -HTT2
+[2,3: /2 width=7 by snv_cprs_lpr, da_cprs_lpr/ ] -T1 -T -d1
/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
qed-.
lemma lstas_cpcs_lpr: âh,g,G,L1,T1. â¦G, L1⦠⢠T1 ¡[h, g] â
- âl,l1. l ⤠l1 â â¦G, L1⦠⢠T1 âª[h, g] l1 â âU1. â¦G, L1⦠⢠T1 â¢*[h, l] U1 â
+ âd,d1. d ⤠d1 â â¦G, L1⦠⢠T1 âª[h, g] d1 â âU1. â¦G, L1⦠⢠T1 â¢*[h, d] U1 â
âT2. â¦G, L1⦠⢠T2 ¡[h, g] â
- âl2. l ⤠l2 â â¦G, L1⦠⢠T2 âª[h, g] l2 â âU2. â¦G, L1⦠⢠T2 â¢*[h, l] U2 â
+ âd2. d ⤠d2 â â¦G, L1⦠⢠T2 âª[h, g] d2 â âU2. â¦G, L1⦠⢠T2 â¢*[h, d] U2 â
â¦G, L1⦠⢠T1 â¬* T2 â âL2. â¦G, L1⦠⢠⡠L2 â â¦G, L2⦠⢠U1 â¬* U2.
-#h #g #G #L1 #T1 #HT1 #l #l1 #Hl1 #HTl1 #U1 #HTU1 #T2 #HT2 #l2 #Hl2 #HTl2 #U2 #HTU2 #H #L2 #HL12
+#h #g #G #L1 #T1 #HT1 #d #d1 #Hd1 #HTd1 #U1 #HTU1 #T2 #HT2 #d2 #Hd2 #HTd2 #U2 #HTU2 #H #L2 #HL12
elim (cpcs_inv_cprs ⦠H) -H #T #H1 #H2
-elim (lstas_cprs_lpr ⦠HT1 ⦠Hl1 HTl1 ⦠HTU1 ⦠H1 ⦠HL12) -T1 #W1 #H1 #HUW1
-elim (lstas_cprs_lpr ⦠HT2 ⦠Hl2 HTl2 ⦠HTU2 ⦠H2 ⦠HL12) -T2 #W2 #H2 #HUW2
-lapply (lstas_mono ⦠H1 ⦠H2) -h -T -l #H destruct /2 width=3 by cpcs_canc_dx/
+elim (lstas_cprs_lpr ⦠HT1 ⦠Hd1 HTd1 ⦠HTU1 ⦠H1 ⦠HL12) -T1 #W1 #H1 #HUW1
+elim (lstas_cprs_lpr ⦠HT2 ⦠Hd2 HTd2 ⦠HTU2 ⦠H2 ⦠HL12) -T2 #W2 #H2 #HUW2
+lapply (lstas_mono ⦠H1 ⦠H2) -h -T -d #H destruct /2 width=3 by cpcs_canc_dx/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_scpes.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_scpes.ma
index 9ed7e06cb..185f0d52c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_scpes.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_scpes.ma
@@ -27,21 +27,21 @@ definition IH_snv_cpr_lpr: âh:sh. sd h â relation3 genv lenv term â
definition IH_da_cpr_lpr: âh:sh. sd h â relation3 genv lenv term â
λh,g,G,L1,T1. â¦G, L1⦠⢠T1 ¡[h, g] â
- âl. â¦G, L1⦠⢠T1 âª[h, g] l â
+ âd. â¦G, L1⦠⢠T1 âª[h, g] d â
âT2. â¦G, L1⦠⢠T1 â¡ T2 â âL2. â¦G, L1⦠⢠⡠L2 â
- â¦G, L2⦠⢠T2 âª[h, g] l.
+ â¦G, L2⦠⢠T2 âª[h, g] d.
definition IH_lstas_cpr_lpr: âh:sh. sd h â relation3 genv lenv term â
λh,g,G,L1,T1. â¦G, L1⦠⢠T1 ¡[h, g] â
- âl1,l2. l2 ⤠l1 â â¦G, L1⦠⢠T1 âª[h, g] l1 â
- âU1. â¦G, L1⦠⢠T1 â¢*[h, l2] U1 â
+ âd1,d2. d2 ⤠d1 â â¦G, L1⦠⢠T1 âª[h, g] d1 â
+ âU1. â¦G, L1⦠⢠T1 â¢*[h, d2] U1 â
âT2. â¦G, L1⦠⢠T1 â¡ T2 â âL2. â¦G, L1⦠⢠⡠L2 â
- ââU2. â¦G, L2⦠⢠T2 â¢*[h, l2] U2 & â¦G, L2⦠⢠U1 â¬* U2.
+ ââU2. â¦G, L2⦠⢠T2 â¢*[h, d2] U2 & â¦G, L2⦠⢠U1 â¬* U2.
definition IH_snv_lstas: âh:sh. sd h â relation3 genv lenv term â
λh,g,G,L,T. â¦G, L⦠⢠T ¡[h, g] â
- âl1,l2. l2 ⤠l1 â â¦G, L⦠⢠T âª[h, g] l1 â
- âU. â¦G, L⦠⢠T â¢*[h, l2] U â â¦G, L⦠⢠U ¡[h, g].
+ âd1,d2. d2 ⤠d1 â â¦G, L⦠⢠T âª[h, g] d1 â
+ âU. â¦G, L⦠⢠T â¢*[h, d2] U â â¦G, L⦠⢠U ¡[h, g].
(* Properties for the preservation results **********************************)
@@ -57,9 +57,9 @@ fact da_cprs_lpr_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_snv_cpr_lpr h g G1 L1 T1) â
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_da_cpr_lpr h g G1 L1 T1) â
âG,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L1, T1⦠â â¦G, L1⦠⢠T1 ¡[h, g] â
- âl. â¦G, L1⦠⢠T1 âª[h, g] l â
- âT2. â¦G, L1⦠⢠T1 â¡* T2 â âL2. â¦G, L1⦠⢠⡠L2 â â¦G, L2⦠⢠T2 âª[h, g] l.
-#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #l #Hl #T2 #H
+ âd. â¦G, L1⦠⢠T1 âª[h, g] d â
+ âT2. â¦G, L1⦠⢠T1 â¡* T2 â âL2. â¦G, L1⦠⢠⡠L2 â â¦G, L2⦠⢠T2 âª[h, g] d.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d #Hd #T2 #H
@(cprs_ind ⦠H) -T2 /4 width=10 by snv_cprs_lpr_aux, fpbg_fpbs_trans, cprs_fpbs/
qed-.
@@ -68,13 +68,13 @@ fact da_scpds_lpr_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_snv_cpr_lpr h g G1 L1 T1) â
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_da_cpr_lpr h g G1 L1 T1) â
âG,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L1, T1⦠â â¦G, L1⦠⢠T1 ¡[h, g] â
- âl1. â¦G, L1⦠⢠T1 âª[h, g] l1 â
- âT2,l2. â¦G, L1⦠⢠T1 â¢*â¡*[h, g, l2] T2 â âL2. â¦G, L1⦠⢠⡠L2 â
- l2 ⤠l1 ⧠â¦G, L2⦠⢠T2 âª[h, g] l1-l2.
-#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #l1 #Hl1 #T2 #l2 * #T #l0 #Hl20 #H #HT1 #HT2 #L2 #HL12
-lapply (da_mono ⦠H ⦠Hl1) -H #H destruct
-lapply (lstas_da_conf ⦠HT1 ⦠Hl1) #Hl12
-lapply (da_cprs_lpr_aux ⦠IH2 IH1 ⦠Hl12 ⦠HT2 ⦠HL12) -IH2 -IH1 -HT2 -HL12
+ âd1. â¦G, L1⦠⢠T1 âª[h, g] d1 â
+ âT2,d2. â¦G, L1⦠⢠T1 â¢*â¡*[h, g, d2] T2 â âL2. â¦G, L1⦠⢠⡠L2 â
+ d2 ⤠d1 ⧠â¦G, L2⦠⢠T2 âª[h, g] d1-d2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #HLT0 #HT1 #d1 #Hd1 #T2 #d2 * #T #d0 #Hd20 #H #HT1 #HT2 #L2 #HL12
+lapply (da_mono ⦠H ⦠Hd1) -H #H destruct
+lapply (lstas_da_conf ⦠HT1 ⦠Hd1) #Hd12
+lapply (da_cprs_lpr_aux ⦠IH2 IH1 ⦠Hd12 ⦠HT2 ⦠HL12) -IH2 -IH1 -HT2 -HL12
/3 width=8 by fpbg_fpbs_trans, lstas_fpbs, conj/
qed-.
@@ -84,12 +84,12 @@ fact da_scpes_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_da_cpr_lpr h g G1 L1 T1) â
âG,L,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L, T1⦠â â¦G, L⦠⢠T1 ¡[h, g] â
âT2. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L, T2⦠â â¦G, L⦠⢠T2 ¡[h, g] â
- âl11. â¦G, L⦠⢠T1 âª[h, g] l11 â âl12. â¦G, L⦠⢠T2 âª[h, g] l12 â
- âl21,l22. â¦G, L⦠⢠T1 â¢*â¬*[h, g, l21, l22] T2 â
- â§â§ l21 ⤠l11 & l22 ⤠l12 & l11 - l21 = l12 - l22.
-#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #l11 #Hl11 #l12 #Hl12 #l21 #l22 * #T #HT1 #HT2
-elim (da_scpds_lpr_aux ⦠IH3 IH2 IH1 ⦠Hl11 ⦠HT1 ⦠L) -Hl11 -HT1 //
-elim (da_scpds_lpr_aux ⦠IH3 IH2 IH1 ⦠Hl12 ⦠HT2 ⦠L) -Hl12 -HT2 //
+ âd11. â¦G, L⦠⢠T1 âª[h, g] d11 â âd12. â¦G, L⦠⢠T2 âª[h, g] d12 â
+ âd21,d22. â¦G, L⦠⢠T1 â¢*â¬*[h, g, d21, d22] T2 â
+ â§â§ d21 ⤠d11 & d22 ⤠d12 & d11 - d21 = d12 - d22.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L #T1 #HLT01 #HT1 #T2 #HLT02 #HT2 #d11 #Hd11 #d12 #Hd12 #d21 #d22 * #T #HT1 #HT2
+elim (da_scpds_lpr_aux ⦠IH3 IH2 IH1 ⦠Hd11 ⦠HT1 ⦠L) -Hd11 -HT1 //
+elim (da_scpds_lpr_aux ⦠IH3 IH2 IH1 ⦠Hd12 ⦠HT2 ⦠L) -Hd12 -HT2 //
/3 width=7 by da_mono, and3_intro/
qed-.
@@ -98,19 +98,19 @@ fact lstas_cprs_lpr_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_da_cpr_lpr h g G1 L1 T1) â
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_lstas_cpr_lpr h g G1 L1 T1) â
âG,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L1, T1⦠â â¦G, L1⦠⢠T1 ¡[h, g] â
- âl1,l2. l2 ⤠l1 â â¦G, L1⦠⢠T1 âª[h, g] l1 â
- âU1. â¦G, L1⦠⢠T1 â¢*[h, l2] U1 â
+ âd1,d2. d2 ⤠d1 â â¦G, L1⦠⢠T1 âª[h, g] d1 â
+ âU1. â¦G, L1⦠⢠T1 â¢*[h, d2] U1 â
âT2. â¦G, L1⦠⢠T1 â¡* T2 â âL2. â¦G, L1⦠⢠⡠L2 â
- ââU2. â¦G, L2⦠⢠T2 â¢*[h, l2] U2 & â¦G, L2⦠⢠U1 â¬* U2.
-#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #l1 #l2 #Hl21 #Hl1 #U1 #HTU1 #T2 #H
+ ââU2. â¦G, L2⦠⢠T2 â¢*[h, d2] U2 & â¦G, L2⦠⢠U1 â¬* U2.
+#h #g #G0 #L0 #T0 #IH3 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #d1 #d2 #Hd21 #Hd1 #U1 #HTU1 #T2 #H
@(cprs_ind ⦠H) -T2 [ /2 width=10 by/ ]
#T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
elim (IHT1 L1) // -IHT1 #U #HTU #HU1
-elim (IH1 ⦠Hl21 ⦠HTU ⦠HTT2 ⦠HL12) -IH1 -HTU -HTT2
+elim (IH1 ⦠Hd21 ⦠HTU ⦠HTT2 ⦠HL12) -IH1 -HTU -HTT2
[2: /3 width=12 by da_cprs_lpr_aux/
|3: /3 width=10 by snv_cprs_lpr_aux/
|4: /3 width=5 by fpbg_fpbs_trans, cprs_fpbs/
-] -G0 -L0 -T0 -T1 -T -l1
+] -G0 -L0 -T0 -T1 -T -d1
/4 width=5 by lpr_cpcs_conf, cpcs_trans, ex2_intro/
qed-.
@@ -118,12 +118,12 @@ fact scpds_cpr_lpr_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_da_cpr_lpr h g G1 L1 T1) â
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_lstas_cpr_lpr h g G1 L1 T1) â
âG,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L1, T1⦠â â¦G, L1⦠⢠T1 ¡[h, g] â
- âU1,l. â¦G, L1⦠⢠T1 â¢*â¡*[h, g, l] U1 â
+ âU1,d. â¦G, L1⦠⢠T1 â¢*â¡*[h, g, d] U1 â
âT2. â¦G, L1⦠⢠T1 â¡ T2 â âL2. â¦G, L1⦠⢠⡠L2 â
- ââU2. â¦G, L2⦠⢠T2 â¢*â¡*[h, g, l] U2 & â¦G, L2⦠⢠U1 â¡* U2.
-#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 #l2 * #W1 #l1 #Hl21 #HTl1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
+ ââU2. â¦G, L2⦠⢠T2 â¢*â¡*[h, g, d] U2 & â¦G, L2⦠⢠U1 â¡* U2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #U1 #d2 * #W1 #d1 #Hd21 #HTd1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
elim (IH1 ⦠H01 ⦠HTW1 ⦠HT12 ⦠HL12) -IH1 // #W2 #HTW2 #HW12
-lapply (IH2 ⦠H01 ⦠HTl1 ⦠HT12 ⦠HL12) -L0 -T0 // -T1
+lapply (IH2 ⦠H01 ⦠HTd1 ⦠HT12 ⦠HL12) -L0 -T0 // -T1
lapply (lpr_cprs_conf ⦠HL12 ⦠HWU1) -L1 #HWU1
lapply (cpcs_canc_sn ⦠HW12 HWU1) -W1 #H
elim (cpcs_inv_cprs ⦠H) -H /3 width=6 by ex4_2_intro, ex2_intro/
@@ -134,10 +134,10 @@ fact scpes_cpr_lpr_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_lstas_cpr_lpr h g G1 L1 T1) â
âG,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L1, T1⦠â â¦G, L1⦠⢠T1 ¡[h, g] â
âT2. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L1, T2⦠â â¦G, L1⦠⢠T2 ¡[h, g] â
- âl1,l2. â¦G, L1⦠⢠T1 â¢*â¬*[h, g, l1, l2] T2 â
+ âd1,d2. â¦G, L1⦠⢠T1 â¢*â¬*[h, g, d1, d2] T2 â
âU1. â¦G, L1⦠⢠T1 â¡ U1 â âU2. â¦G, L1⦠⢠T2 â¡ U2 â âL2. â¦G, L1⦠⢠⡠L2 â
- â¦G, L2⦠⢠U1 â¢*â¬*[h, g, l1, l2] U2.
-#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #T2 #HT02 #HT2 #l1 #l2 * #T0 #HT10 #HT20 #U1 #HTU1 #U2 #HTU2 #L2 #HL12
+ â¦G, L2⦠⢠U1 â¢*â¬*[h, g, d1, d2] U2.
+#h #g #G0 #L0 #T0 #IH2 #IH1 #G #L1 #T1 #H01 #HT1 #T2 #HT02 #HT2 #d1 #d2 * #T0 #HT10 #HT20 #U1 #HTU1 #U2 #HTU2 #L2 #HL12
elim (scpds_cpr_lpr_aux ⦠IH2 IH1 ⦠HT10 ⦠HTU1 ⦠HL12) -HT10 -HTU1 // #X1 #HUX1 #H1
elim (scpds_cpr_lpr_aux ⦠IH2 IH1 ⦠HT20 ⦠HTU2 ⦠HL12) -HT20 -HTU2 // #X2 #HUX2 #H2
elim (cprs_conf ⦠H1 ⦠H2) -T0
@@ -150,20 +150,20 @@ fact lstas_scpds_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_da_cpr_lpr h g G1 L1 T1) â
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_lstas_cpr_lpr h g G1 L1 T1) â
âG,L,T. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L, T⦠â â¦G, L⦠⢠T ¡[h, g] â
- âl,l1. l1 ⤠l â â¦G, L⦠⢠T âª[h, g] l â âT1. â¦G, L⦠⢠T â¢*[h, l1] T1 â
- âT2,l2. â¦G, L⦠⢠T â¢*â¡*[h, g, l2] T2 â â¦G, L⦠⢠T1 â¢*â¬*[h, g, l2-l1, l1-l2] T2.
-#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T #H0 #HT #l #l1 #Hl1 #HTl #T1 #HT1 #T2 #l2 * #X #l0 #Hl20 #H #HTX #HXT2
-lapply (da_mono ⦠H ⦠HTl) -H #H destruct
-lapply (lstas_da_conf ⦠HT1 ⦠HTl) #HTl1
-lapply (lstas_da_conf ⦠HTX ⦠HTl) #HXl
-lapply (da_cprs_lpr_aux ⦠IH3 IH2 ⦠HXl ⦠HXT2 L ?)
-[1,2,3: /3 width=8 by fpbg_fpbs_trans, lstas_fpbs/ ] #HTl2
-elim (le_or_ge l1 l2) #Hl12 >(eq_minus_O ⦠Hl12)
-[ elim (da_lstas ⦠HTl2 0) #X2 #HTX2 #_ -IH4 -IH3 -IH2 -IH1 -H0 -HT -HTl -HXl
+ âd,d1. d1 ⤠d â â¦G, L⦠⢠T âª[h, g] d â âT1. â¦G, L⦠⢠T â¢*[h, d1] T1 â
+ âT2,d2. â¦G, L⦠⢠T â¢*â¡*[h, g, d2] T2 â â¦G, L⦠⢠T1 â¢*â¬*[h, g, d2-d1, d1-d2] T2.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T #H0 #HT #d #d1 #Hd1 #HTd #T1 #HT1 #T2 #d2 * #X #d0 #Hd20 #H #HTX #HXT2
+lapply (da_mono ⦠H ⦠HTd) -H #H destruct
+lapply (lstas_da_conf ⦠HT1 ⦠HTd) #HTd1
+lapply (lstas_da_conf ⦠HTX ⦠HTd) #HXd
+lapply (da_cprs_lpr_aux ⦠IH3 IH2 ⦠HXd ⦠HXT2 L ?)
+[1,2,3: /3 width=8 by fpbg_fpbs_trans, lstas_fpbs/ ] #HTd2
+elim (le_or_ge d1 d2) #Hd12 >(eq_minus_O ⦠Hd12)
+[ elim (da_lstas ⦠HTd2 0) #X2 #HTX2 #_ -IH4 -IH3 -IH2 -IH1 -H0 -HT -HTd -HXd
/5 width=6 by lstas_scpds, scpds_div, cprs_strap1, lstas_cpr, lstas_conf_le, monotonic_le_minus_l, ex4_2_intro/
-| elim (da_lstas ⦠HTl1 0) #X1 #HTX1 #_
+| elim (da_lstas ⦠HTd1 0) #X1 #HTX1 #_
lapply (lstas_conf_le ⦠HTX ⦠HT1) // #HXT1 -HT1
- elim (lstas_cprs_lpr_aux ⦠IH3 IH2 IH1 ⦠HXl ⦠HXT1 ⦠HXT2 L) -IH3 -IH2 -IH1 -HXl -HXT1 -HXT2
+ elim (lstas_cprs_lpr_aux ⦠IH3 IH2 IH1 ⦠HXd ⦠HXT1 ⦠HXT2 L) -IH3 -IH2 -IH1 -HXd -HXT1 -HXT2
/4 width=8 by cpcs_scpes, cpcs_cpr_conf, fpbg_fpbs_trans, lstas_fpbs, lstas_cpr, monotonic_le_minus_l/
]
qed-.
@@ -175,18 +175,18 @@ fact scpes_le_aux: âh,g,G0,L0,T0.
(âG1,L1,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G1, L1, T1⦠â IH_lstas_cpr_lpr h g G1 L1 T1) â
âG,L,T1. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L, T1⦠â â¦G, L⦠⢠T1 ¡[h, g] â
âT2. â¦G0, L0, T0⦠>â¡[h, g] â¦G, L, T2⦠â â¦G, L⦠⢠T2 ¡[h, g] â
- âl11. â¦G, L⦠⢠T1 âª[h, g] l11 â âl12. â¦G, L⦠⢠T2 âª[h, g] l12 â
- âl21,l22,l. l21 + l ⤠l11 â l22 + l ⤠l12 â
- â¦G, L⦠⢠T1 â¢*â¬*[h, g, l21, l22] T2 â â¦G, L⦠⢠T1 â¢*â¬*[h, g, l21+l, l22+l] T2.
-#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #T2 #H02 #HT2 #l11 #Hl11 #Hl12 #Hl12 #l21 #l22 #l #H1 #H2 * #T0 #HT10 #HT20
-elim (da_lstas ⦠Hl11 (l21+l)) #X1 #HTX1 #_
-elim (da_lstas ⦠Hl12 (l22+l)) #X2 #HTX2 #_
-lapply (lstas_scpds_aux ⦠IH4 IH3 IH2 IH1 ⦠Hl11 ⦠HTX1 ⦠HT10) -HT10
+ âd11. â¦G, L⦠⢠T1 âª[h, g] d11 â âd12. â¦G, L⦠⢠T2 âª[h, g] d12 â
+ âd21,d22,d. d21 + d ⤠d11 â d22 + d ⤠d12 â
+ â¦G, L⦠⢠T1 â¢*â¬*[h, g, d21, d22] T2 â â¦G, L⦠⢠T1 â¢*â¬*[h, g, d21+d, d22+d] T2.
+#h #g #G0 #L0 #T0 #IH4 #IH3 #IH2 #IH1 #G #L #T1 #H01 #HT1 #T2 #H02 #HT2 #d11 #Hd11 #Hd12 #Hd12 #d21 #d22 #d #H1 #H2 * #T0 #HT10 #HT20
+elim (da_lstas ⦠Hd11 (d21+d)) #X1 #HTX1 #_
+elim (da_lstas ⦠Hd12 (d22+d)) #X2 #HTX2 #_
+lapply (lstas_scpds_aux ⦠IH4 IH3 IH2 IH1 ⦠Hd11 ⦠HTX1 ⦠HT10) -HT10
[1,2,3: // | >eq_minus_O [2: // ] <minus_plus_m_m_commutative #HX1 ]
-lapply (lstas_scpds_aux ⦠IH4 IH3 IH2 IH1 ⦠Hl12 ⦠HTX2 ⦠HT20) -HT20
+lapply (lstas_scpds_aux ⦠IH4 IH3 IH2 IH1 ⦠Hd12 ⦠HTX2 ⦠HT20) -HT20
[1,2,3: // | >eq_minus_O [2: // ] <minus_plus_m_m_commutative #HX2 ]
-lapply (lstas_scpes_trans ⦠Hl11 ⦠HTX1 ⦠HX1) [ // ] -Hl11 -HTX1 -HX1 -H1 #H1
-lapply (lstas_scpes_trans ⦠Hl12 ⦠HTX2 ⦠HX2) [ // ] -Hl12 -HTX2 -HX2 -H2 #H2
+lapply (lstas_scpes_trans ⦠Hd11 ⦠HTX1 ⦠HX1) [ // ] -Hd11 -HTX1 -HX1 -H1 #H1
+lapply (lstas_scpes_trans ⦠Hd12 ⦠HTX2 ⦠HX2) [ // ] -Hd12 -HTX2 -HX2 -H2 #H2
/2 width=4 by scpes_canc_dx/ (**) (* full auto fails *)
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
index 083194053..0675804cc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/cpcs_cpcs.ma
@@ -59,14 +59,14 @@ elim (cprs_inv_abst1 ⦠H2) -H2 #W0 #T0 #_ #_ #H destruct
qed-.
(* Basic_1: was: pc3_gen_lift *)
-lemma cpcs_inv_lift: âG,L,K,s,d,e. â¬[s, d, e] L â¡ K â
- âT1,U1. â¬[d, e] T1 â¡ U1 â âT2,U2. â¬[d, e] T2 â¡ U2 â
+lemma cpcs_inv_lift: âG,L,K,s,l,m. â¬[s, l, m] L â¡ K â
+ âT1,U1. â¬[l, m] T1 â¡ U1 â âT2,U2. â¬[l, m] T2 â¡ U2 â
â¦G, L⦠⢠U1 â¬* U2 â â¦G, K⦠⢠T1 â¬* T2.
-#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
+#G #L #K #s #l #m #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HU12
elim (cpcs_inv_cprs ⦠HU12) -HU12 #U #HU1 #HU2
elim (cprs_inv_lift1 ⦠HU1 ⦠HLK ⦠HTU1) -U1 #T #HTU #HT1
elim (cprs_inv_lift1 ⦠HU2 ⦠HLK ⦠HTU2) -L -U2 #X #HXU
->(lift_inj ⦠HXU ⦠HTU) -X -U -d -e /2 width=3 by cprs_div/
+>(lift_inj ⦠HXU ⦠HTU) -X -U -l -m /2 width=3 by cprs_div/
qed-.
(* Advanced properties ******************************************************)
@@ -150,12 +150,12 @@ lemma lsubr_cpcs_trans: âG,L1,T1,T2. â¦G, L1⦠⢠T1 â¬* T2 â
qed-.
(* Basic_1: was: pc3_lift *)
-lemma cpcs_lift: âG,L,K,s,d,e. â¬[s, d, e] L â¡ K â
- âT1,U1. â¬[d, e] T1 â¡ U1 â âT2,U2. â¬[d, e] T2 â¡ U2 â
+lemma cpcs_lift: âG,L,K,s,l,m. â¬[s, l, m] L â¡ K â
+ âT1,U1. â¬[l, m] T1 â¡ U1 â âT2,U2. â¬[l, m] T2 â¡ U2 â
â¦G, K⦠⢠T1 â¬* T2 â â¦G, L⦠⢠U1 â¬* U2.
-#G #L #K #s #d #e #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
+#G #L #K #s #l #m #HLK #T1 #U1 #HTU1 #T2 #U2 #HTU2 #HT12
elim (cpcs_inv_cprs ⦠HT12) -HT12 #T #HT1 #HT2
-elim (lift_total T d e) /3 width=12 by cprs_div, cprs_lift/
+elim (lift_total T l m) /3 width=12 by cprs_div, cprs_lift/
qed.
lemma cpcs_strip: âG,L,T1,T. â¦G, L⦠⢠T â¬* T1 â âT2. â¦G, L⦠⢠T ⬠T2 â
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes.ma
index fcc2ea971..c9fa6d0f6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes.ma
@@ -18,20 +18,20 @@ include "basic_2/computation/scpds.ma".
(* STRATIFIED DECOMPOSED PARALLEL EQUIVALENCE FOR TERMS *********************)
definition scpes: âh. sd h â nat â nat â relation4 genv lenv term term â
- λh,g,l1,l2,G,L,T1,T2.
- ââT. â¦G, L⦠⢠T1 â¢*â¡*[h, g, l1] T & â¦G, L⦠⢠T2 â¢*â¡*[h, g, l2] T.
+ λh,g,d1,d2,G,L,T1,T2.
+ ââT. â¦G, L⦠⢠T1 â¢*â¡*[h, g, d1] T & â¦G, L⦠⢠T2 â¢*â¡*[h, g, d2] T.
interpretation "stratified decomposed parallel equivalence (term)"
- 'DPConvStar h g l1 l2 G L T1 T2 = (scpes h g l1 l2 G L T1 T2).
+ 'DPConvStar h g d1 d2 G L T1 T2 = (scpes h g d1 d2 G L T1 T2).
(* Basic properties *********************************************************)
-lemma scpds_div: âh,g,G,L,T1,T2,T,l1,l2.
- â¦G, L⦠⢠T1 â¢*â¡*[h, g, l1] T â â¦G, L⦠⢠T2 â¢*â¡*[h, g, l2] T â
- â¦G, L⦠⢠T1 â¢*â¬*[h, g, l1, l2] T2.
+lemma scpds_div: âh,g,G,L,T1,T2,T,d1,d2.
+ â¦G, L⦠⢠T1 â¢*â¡*[h, g, d1] T â â¦G, L⦠⢠T2 â¢*â¡*[h, g, d2] T â
+ â¦G, L⦠⢠T1 â¢*â¬*[h, g, d1, d2] T2.
/2 width=3 by ex2_intro/ qed.
-lemma scpes_sym: âh,g,G,L,T1,T2,l1,l2. â¦G, L⦠⢠T1 â¢*â¬*[h, g, l1, l2] T2 â
- â¦G, L⦠⢠T2 â¢*â¬*[h, g, l2, l1] T1.
-#h #g #G #L #T1 #T2 #L1 #l2 * /2 width=3 by scpds_div/
+lemma scpes_sym: âh,g,G,L,T1,T2,d1,d2. â¦G, L⦠⢠T1 â¢*â¬*[h, g, d1, d2] T2 â
+ â¦G, L⦠⢠T2 â¢*â¬*[h, g, d2, d1] T1.
+#h #g #G #L #T1 #T2 #L1 #d2 * /2 width=3 by scpds_div/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_aaa.ma
index 39bc00547..761ead9be 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_aaa.ma
@@ -19,10 +19,10 @@ include "basic_2/equivalence/scpes.ma".
(* Main inversion lemmas about atomic arity assignment on terms *************)
-theorem scpes_aaa_mono: âh,g,G,L,T1,T2,l1,l2. â¦G, L⦠⢠T1 â¢*â¬*[h, g, l1, l2] T2 â
+theorem scpes_aaa_mono: âh,g,G,L,T1,T2,d1,d2. â¦G, L⦠⢠T1 â¢*â¬*[h, g, d1, d2] T2 â
âA1. â¦G, L⦠⢠T1 â A1 â âA2. â¦G, L⦠⢠T2 â A2 â
A1 = A2.
-#h #g #G #L #T1 #T2 #l1 #l2 * #T #HT1 #HT2 #A1 #HA1 #A2 #HA2
+#h #g #G #L #T1 #T2 #d1 #d2 * #T #HT1 #HT2 #A1 #HA1 #A2 #HA2
lapply (scpds_aaa_conf ⦠HA1 ⦠HT1) -T1 #HA1
lapply (scpds_aaa_conf ⦠HA2 ⦠HT2) -T2 #HA2
lapply (aaa_mono ⦠HA1 ⦠HA2) -L -T //
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_cpcs.ma
index aa22846fd..20b688472 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_cpcs.ma
@@ -20,20 +20,20 @@ include "basic_2/equivalence/scpes.ma".
(* Inversion lemmas on parallel equivalence for terms ***********************)
-lemma scpes_inv_lstas_eq: âh,g,G,L,T1,T2,l1,l2. â¦G, L⦠⢠T1 â¢*â¬*[h, g, l1, l2] T2 â
- âU1. â¦G, L⦠⢠T1 â¢*[h, l1] U1 â
- âU2. â¦G, L⦠⢠T2 â¢*[h, l2] U2 â â¦G, L⦠⢠U1 â¬* U2.
-#h #g #G #L #T1 #T2 #l1 #l2 * #T #HT1 #HT2 #U1 #HTU1 #U2 #HTU2
+lemma scpes_inv_lstas_eq: âh,g,G,L,T1,T2,d1,d2. â¦G, L⦠⢠T1 â¢*â¬*[h, g, d1, d2] T2 â
+ âU1. â¦G, L⦠⢠T1 â¢*[h, d1] U1 â
+ âU2. â¦G, L⦠⢠T2 â¢*[h, d2] U2 â â¦G, L⦠⢠U1 â¬* U2.
+#h #g #G #L #T1 #T2 #d1 #d2 * #T #HT1 #HT2 #U1 #HTU1 #U2 #HTU2
/3 width=8 by scpds_inv_lstas_eq, cprs_div/
qed-.
(* Properties on parallel equivalence for terms *****************************)
-lemma cpcs_scpes: âh,g,G,L,T1,l11. â¦G, L⦠⢠T1 âª[h, g] l11 â
- âU1,l12. l12 ⤠l11 â â¦G, L⦠⢠T1 â¢*[h, l12] U1 â
- âT2,l21. â¦G, L⦠⢠T2 âª[h, g] l21 â
- âU2,l22. l22 ⤠l21 â â¦G, L⦠⢠T2 â¢*[h, l22] U2 â
- â¦G, L⦠⢠U1 â¬* U2 â â¦G, L⦠⢠T1 â¢*â¬*[h, g, l12, l22] T2.
-#h #g #G #L #T1 #l11 #HT1 #U1 #l12 #Hl121 #HTU1 #T2 #l21 #HT2 #U2 #l22 #Hl221 #HTU2 #HU12
+lemma cpcs_scpes: âh,g,G,L,T1,d11. â¦G, L⦠⢠T1 âª[h, g] d11 â
+ âU1,d12. d12 ⤠d11 â â¦G, L⦠⢠T1 â¢*[h, d12] U1 â
+ âT2,d21. â¦G, L⦠⢠T2 âª[h, g] d21 â
+ âU2,d22. d22 ⤠d21 â â¦G, L⦠⢠T2 â¢*[h, d22] U2 â
+ â¦G, L⦠⢠U1 â¬* U2 â â¦G, L⦠⢠T1 â¢*â¬*[h, g, d12, d22] T2.
+#h #g #G #L #T1 #d11 #HT1 #U1 #d12 #Hd121 #HTU1 #T2 #d21 #HT2 #U2 #d22 #Hd221 #HTU2 #HU12
elim (cpcs_inv_cprs ⦠HU12) -HU12 /3 width=6 by scpds_div, ex4_2_intro/
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_scpes.ma b/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_scpes.ma
index e0f79b1c0..e51027994 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_scpes.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/equivalence/scpes_scpes.ma
@@ -19,51 +19,51 @@ include "basic_2/equivalence/scpes.ma".
(* Advanced inversion lemmas ************************************************)
-lemma scpes_inv_abst2: âh,g,a,G,L,T1,T2,W2,l1,l2. â¦G, L⦠⢠T1 â¢*â¬*[h, g, l1, l2] â{a}W2.T2 â
- ââW,T. â¦G, L⦠⢠T1 â¢*â¡*[h, g, l1] â{a}W.T & â¦G, L⦠⢠W2 â¡* W &
- â¦G, L.âW2⦠⢠T2 â¢*â¡*[h, g, l2] T.
-#h #g #a #G #L #T1 #T2 #W2 #l1 #l2 * #T0 #HT10 #H
+lemma scpes_inv_abst2: âh,g,a,G,L,T1,T2,W2,d1,d2. â¦G, L⦠⢠T1 â¢*â¬*[h, g, d1, d2] â{a}W2.T2 â
+ ââW,T. â¦G, L⦠⢠T1 â¢*â¡*[h, g, d1] â{a}W.T & â¦G, L⦠⢠W2 â¡* W &
+ â¦G, L.âW2⦠⢠T2 â¢*â¡*[h, g, d2] T.
+#h #g #a #G #L #T1 #T2 #W2 #d1 #d2 * #T0 #HT10 #H
elim (scpds_inv_abst1 ⦠H) -H #W #T #HW2 #HT2 #H destruct /2 width=5 by ex3_2_intro/
qed-.
(* Advanced properties ******************************************************)
-lemma scpes_refl: âh,g,G,L,T,l1,l2. l2 ⤠l1 â â¦G, L⦠⢠T âª[h, g] l1 â
- â¦G, L⦠⢠T â¢*â¬*[h, g, l2, l2] T.
-#h #g #G #L #T #l1 #l2 #Hl21 #Hl1
-elim (da_lstas ⦠Hl1 ⦠l2) #U #HTU #_
+lemma scpes_refl: âh,g,G,L,T,d1,d2. d2 ⤠d1 â â¦G, L⦠⢠T âª[h, g] d1 â
+ â¦G, L⦠⢠T â¢*â¬*[h, g, d2, d2] T.
+#h #g #G #L #T #d1 #d2 #Hd21 #Hd1
+elim (da_lstas ⦠Hd1 ⦠d2) #U #HTU #_
/3 width=3 by scpds_div, lstas_scpds/
qed.
-lemma lstas_scpes_trans: âh,g,G,L,T1,l0,l1. â¦G, L⦠⢠T1 âª[h, g] l0 â l1 ⤠l0 â
- âT. â¦G, L⦠⢠T1 â¢*[h, l1] T â
- âT2,l,l2. â¦G, L⦠⢠T â¢*â¬*[h,g,l,l2] T2 â â¦G, L⦠⢠T1 â¢*â¬*[h,g,l1+l,l2] T2.
-#h #g #G #L #T1 #l0 #l1 #Hl0 #Hl10 #T #HT1 #T2 #l #l2 *
+lemma lstas_scpes_trans: âh,g,G,L,T1,d0,d1. â¦G, L⦠⢠T1 âª[h, g] d0 â d1 ⤠d0 â
+ âT. â¦G, L⦠⢠T1 â¢*[h, d1] T â
+ âT2,d,d2. â¦G, L⦠⢠T â¢*â¬*[h,g,d,d2] T2 â â¦G, L⦠⢠T1 â¢*â¬*[h,g,d1+d,d2] T2.
+#h #g #G #L #T1 #d0 #d1 #Hd0 #Hd10 #T #HT1 #T2 #d #d2 *
/3 width=3 by scpds_div, lstas_scpds_trans/ qed-.
(* Properties on parallel computation for terms *****************************)
lemma cprs_scpds_div: âh,g,G,L,T1,T. â¦G, L⦠⢠T1 â¡* T â
- âl. â¦G, L⦠⢠T1 âª[h, g] l â
- âT2,l2. â¦G, L⦠⢠T2 â¢*â¡*[h, g, l2] T â
- â¦G, Lâ¦â¢ T1 â¢*â¬*[h, g, 0, l2] T2.
-#h #g #G #L #T1 #T #HT1 #l #Hl elim (da_lstas ⦠Hl 0)
+ âd. â¦G, L⦠⢠T1 âª[h, g] d â
+ âT2,d2. â¦G, L⦠⢠T2 â¢*â¡*[h, g, d2] T â
+ â¦G, Lâ¦â¢ T1 â¢*â¬*[h, g, 0, d2] T2.
+#h #g #G #L #T1 #T #HT1 #d #Hd elim (da_lstas ⦠Hd 0)
#X1 #HTX1 #_ elim (cprs_strip ⦠HT1 X1) -HT1
/3 width=5 by scpds_strap1, scpds_div, lstas_cpr, ex4_2_intro/
qed.
(* Main properties **********************************************************)
-theorem scpes_trans: âh,g,G,L,T1,T,l1,l. â¦G, L⦠⢠T1 â¢*â¬*[h, g, l1, l] T â
- âT2,l2. â¦G, L⦠⢠T â¢*â¬*[h, g, l, l2] T2 â â¦G, L⦠⢠T1 â¢*â¬*[h, g, l1, l2] T2.
-#h #g #G #L #T1 #T #l1 #l * #X1 #HT1X1 #HTX1 #T2 #l2 * #X2 #HTX2 #HT2X2
-elim (scpds_conf_eq ⦠HTX1 ⦠HTX2) -T -l /3 width=5 by scpds_cprs_trans, scpds_div/
+theorem scpes_trans: âh,g,G,L,T1,T,d1,d. â¦G, L⦠⢠T1 â¢*â¬*[h, g, d1, d] T â
+ âT2,d2. â¦G, L⦠⢠T â¢*â¬*[h, g, d, d2] T2 â â¦G, L⦠⢠T1 â¢*â¬*[h, g, d1, d2] T2.
+#h #g #G #L #T1 #T #d1 #d * #X1 #HT1X1 #HTX1 #T2 #d2 * #X2 #HTX2 #HT2X2
+elim (scpds_conf_eq ⦠HTX1 ⦠HTX2) -T -d /3 width=5 by scpds_cprs_trans, scpds_div/
qed-.
-theorem scpes_canc_sn: âh,g,G,L,T,T1,l,l1. â¦G, L⦠⢠T â¢*â¬*[h, g, l, l1] T1 â
- âT2,l2. â¦G, L⦠⢠T â¢*â¬*[h, g, l, l2] T2 â â¦G, L⦠⢠T1 â¢*â¬*[h, g, l1, l2] T2.
+theorem scpes_canc_sn: âh,g,G,L,T,T1,d,d1. â¦G, L⦠⢠T â¢*â¬*[h, g, d, d1] T1 â
+ âT2,d2. â¦G, L⦠⢠T â¢*â¬*[h, g, d, d2] T2 â â¦G, L⦠⢠T1 â¢*â¬*[h, g, d1, d2] T2.
/3 width=4 by scpes_trans, scpes_sym/ qed-.
-theorem scpes_canc_dx: âh,g,G,L,T1,T,l1,l. â¦G, L⦠⢠T1 â¢*â¬*[h, g, l1, l] T â
- âT2,l2. â¦G, L⦠⢠T2 â¢*â¬*[h, g, l2, l] T â â¦G, L⦠⢠T1 â¢*â¬*[h, g, l1, l2] T2.
+theorem scpes_canc_dx: âh,g,G,L,T1,T,d1,d. â¦G, L⦠⢠T1 â¢*â¬*[h, g, d1, d] T â
+ âT2,d2. â¦G, L⦠⢠T2 â¢*â¬*[h, g, d2, d] T â â¦G, L⦠⢠T1 â¢*â¬*[h, g, d1, d2] T2.
/3 width=4 by scpes_trans, scpes_sym/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/examples/ex_cpr_omega.ma b/matita/matita/contribs/lambdadelta/basic_2/examples/ex_cpr_omega.ma
index 72869603d..68166c915 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/examples/ex_cpr_omega.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/examples/ex_cpr_omega.ma
@@ -26,8 +26,8 @@ definition Omega2: term â term â λW. +ââW.(Delta W).â#0.#0.
(* Basic properties *********************************************************)
-lemma Delta_lift: âW1,W2,d,e. â¬[d, e] W1 â¡ W2 â
- â¬[d, e] (Delta W1) â¡ (Delta W2).
+lemma Delta_lift: âW1,W2,l,m. â¬[l, m] W1 â¡ W2 â
+ â¬[l, m] (Delta W1) â¡ (Delta W2).
/4 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
(* Main properties **********************************************************)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/examples/ex_fpbg_refl.ma b/matita/matita/contribs/lambdadelta/basic_2/examples/ex_fpbg_refl.ma
index a0f03d62b..fbc2bd7cd 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/examples/ex_fpbg_refl.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/examples/ex_fpbg_refl.ma
@@ -28,8 +28,8 @@ definition ApplOmega3: term â nat â term â λW,k. ââk.(ApplOmega1 W k)
(* Basic properties *********************************************************)
-lemma ApplDelta_lift: âW1,W2,k,d,e. â¬[d, e] W1 â¡ W2 â
- â¬[d, e] (ApplDelta W1 k) â¡ (ApplDelta W2 k).
+lemma ApplDelta_lift: âW1,W2,k,l,m. â¬[l, m] W1 â¡ W2 â
+ â¬[l, m] (ApplDelta W1 k) â¡ (ApplDelta W2 k).
/5 width=1 by lift_flat, lift_bind, lift_lref_lt/ qed.
lemma cpr_ApplOmega_12: âG,L,W,k. â¦G, L⦠⢠ApplOmega1 W k â¡ ApplOmega2 W k.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/examples/ex_snv_eta.ma b/matita/matita/contribs/lambdadelta/basic_2/examples/ex_snv_eta.ma
index 2320cb5bd..3a6d5a753 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/examples/ex_snv_eta.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/examples/ex_snv_eta.ma
@@ -21,11 +21,11 @@ include "basic_2/dynamic/snv.ma".
(* extended validity of a closure, last arg of snv_appl > 1 *)
lemma snv_extended: âh,g,a,G,L,k. â¦G, L.ââk.ââ{a}âk.âk.â#0⦠⢠â#2.#0 ¡[h, g].
#h #g #a #G #L #k elim (deg_total h g k)
-#l #Hl @(snv_appl ⦠a ⦠(âk) ⦠(âk) (0+1+1))
+#d #Hd @(snv_appl ⦠a ⦠(âk) ⦠(âk) (0+1+1))
[ /4 width=5 by snv_lref, drop_drop_lt/
| /4 width=13 by snv_bind, snv_lref/
| /5 width=6 by lstas_scpds, lstas_succ, da_ldec, da_sort, drop_drop_lt/
-| @(lstas_scpds ⦠(l+1+1))
+| @(lstas_scpds ⦠(d+1+1))
/5 width=11 by lstas_bind, lstas_succ, da_bind, da_ldec, da_sort, lift_bind/
]
qed.
@@ -33,19 +33,19 @@ qed.
(* restricted validity of the η-expanded closure, last arg of snv_appl = 1 **)
lemma snv_restricted: âh,g,a,G,L,k. â¦G, L.ââk.ââ{a}âk.âk.ââ{a}âk.â#0.#1⦠⢠â#2.#0 ¡[h, g].
#h #g #a #G #L #k elim (deg_total h g k)
-#l #Hl @(snv_appl ⦠a ⦠(âk) ⦠(â#0.#2) (0+1))
+#d #Hd @(snv_appl ⦠a ⦠(âk) ⦠(â#0.#2) (0+1))
[ /4 width=5 by snv_lref, drop_drop_lt/
| @snv_lref [4: // |1,2,3: skip ]
@snv_bind //
@(snv_appl ⦠a ⦠(âk) ⦠(âk) (0+1))
[ @snv_lref [4: // |1,2,3: skip ] //
| @snv_lref [4: /2 width=1 by drop_drop_lt/ |1,2,3: skip ] @snv_bind //
- | @(lstas_scpds ⦠(l+1)) /3 width=6 by da_sort, da_ldec, lstas_succ/
- | @(lstas_scpds ⦠(l+1)) /3 width=8 by lstas_succ, lstas_bind, drop_drop, lift_bind/
+ | @(lstas_scpds ⦠(d+1)) /3 width=6 by da_sort, da_ldec, lstas_succ/
+ | @(lstas_scpds ⦠(d+1)) /3 width=8 by lstas_succ, lstas_bind, drop_drop, lift_bind/
@da_ldec [3: /2 width=1 by drop_drop_lt/ |1,2: skip ] /3 width=1 by da_sort, da_bind/
]
| /5 width=6 by lstas_scpds, lstas_succ, da_ldec, da_sort, drop_drop_lt/
-| @(lstas_scpds ⦠(l+1+1)) //
+| @(lstas_scpds ⦠(d+1+1)) //
[ @da_ldec [3: // |1,2: skip ]
@da_bind @da_flat @da_ldec [3: /2 width=1 by drop_drop_lt/ |1,2: skip ]
/3 width=1 by da_sort, da_bind/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/aarity.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/aarity.ma
index 27c775019..96dec98ba 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/aarity.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/aarity.ma
@@ -35,21 +35,21 @@ interpretation "atomic arity construction (binary)"
(* Basic inversion lemmas ***************************************************)
+fact destruct_apair_apair_aux: âA1,A2,B1,B2. â¡B1.A1 = â¡B2.A2 â B1 = B2 ⧠A1 = A2.
+#A1 #A2 #B1 #B2 #H destruct /2 width=1 by conj/
+qed-.
+
lemma discr_apair_xy_x: âA,B. â¡B. A = B â â¥.
#A #B elim B -B
[ #H destruct
-| #Y #X #IHY #_ #H destruct
- -H >e0 in e1; normalize (**) (* destruct: one quality is not simplified, the destructed equality is not erased *)
- /2 width=1/
+| #Y #X #IHY #_ #H elim (destruct_apair_apair_aux ⦠H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
]
qed-.
lemma discr_tpair_xy_y: âB,A. â¡B. A = A â â¥.
#B #A elim A -A
[ #H destruct
-| #Y #X #_ #IHX #H destruct
- -H (**) (* destruct: the destructed equality is not erased *)
- /2 width=1/
+| #Y #X #_ #IHX #H elim (destruct_apair_apair_aux ⦠H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
]
qed-.
@@ -57,16 +57,16 @@ qed-.
lemma eq_aarity_dec: âA1,A2:aarity. Decidable (A1 = A2).
#A1 elim A1 -A1
-[ #A2 elim A2 -A2 /2 width=1/
+[ #A2 elim A2 -A2 /2 width=1 by or_introl/
#B2 #A2 #_ #_ @or_intror #H destruct
| #B1 #A1 #IHB1 #IHA1 #A2 elim A2 -A2
[ -IHB1 -IHA1 @or_intror #H destruct
| #B2 #A2 #_ #_ elim (IHB1 B2) -IHB1
[ #H destruct elim (IHA1 A2) -IHA1
- [ #H destruct /2 width=1/
- | #HA12 @or_intror #H destruct /2 width=1/
+ [ #H destruct /2 width=1 by or_introl/
+ | #HA12 @or_intror #H destruct /2 width=1 by/
]
- | -IHA1 #HB12 @or_intror #H destruct /2 width=1/
+ | -IHA1 #HB12 @or_intror #H destruct /2 width=1 by/
]
]
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/item.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/item.ma
index 99260ba79..97207c56c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/item.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/item.ma
@@ -42,6 +42,12 @@ inductive item2: Type[0] â
| Flat2: flat2 â item2 (* non-binding item *)
.
+(* Basic inversion lemmas ***************************************************)
+
+fact destruct_sort_sort_aux: âk1,k2. Sort k1 = Sort k2 â k1 = k2.
+#k1 #k2 #H destruct //
+qed-.
+
(* Basic properties *********************************************************)
lemma eq_item0_dec: âI1,I2:item0. Decidable (I1 = I2).
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv.ma
index 49d6cf5b8..ff183d177 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv.ma
@@ -54,8 +54,8 @@ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma destruct_lpair_lpair: âI1,I2,L1,L2,V1,V2. L1.â{I1}V1 = L2.â{I2}V2 â
- â§â§L1 = L2 & I1 = I2 & V1 = V2.
+fact destruct_lpair_lpair_aux: âI1,I2,L1,L2,V1,V2. L1.â{I1}V1 = L2.â{I2}V2 â
+ â§â§L1 = L2 & I1 = I2 & V1 = V2.
#I1 #I2 #L1 #L2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
qed-.
@@ -63,6 +63,6 @@ lemma discr_lpair_x_xy: âI,V,L. L = L.â{I}V â â¥.
#I #V #L elim L -L
[ #H destruct
| #L #J #W #IHL #H
- elim (destruct_lpair_lpair ⦠H) -H #H1 #H2 #H3 destruct /2 width=1 by/ (**) (* destruct lemma needed *)
+ elim (destruct_lpair_lpair_aux ⦠H) -H #H1 #H2 #H3 destruct /2 width=1 by/ (**) (* destruct lemma needed *)
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_append.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_append.ma
index 68b7a39fe..cec15c818 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_append.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_append.ma
@@ -36,7 +36,7 @@ interpretation "tail abbreviation (local environment)"
interpretation "tail abstraction (local environment)"
'SnAbst L T = (append (LPair LAtom Abst T) L).
-definition l_appendable_sn: predicate (lenvârelation term) â λR.
+definition d_appendable_sn: predicate (lenvârelation term) â λR.
âK,T1,T2. R K T1 T2 â âL. R (L @@ K) T1 T2.
(* Basic properties *********************************************************)
@@ -75,7 +75,7 @@ lemma append_inj_sn: âK1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 â |K1| = |K2| â
| #K1 #I1 #V1 #IH * normalize
[ #L1 #L2 #_ <plus_n_Sm #H destruct
| #K2 #I2 #V2 #L1 #L2 #H1 #H2
- elim (destruct_lpair_lpair ⦠H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+ elim (destruct_lpair_lpair_aux ⦠H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
elim (IH ⦠H1) -IH -H1 /2 width=1 by conj/
]
]
@@ -94,7 +94,7 @@ lemma append_inj_dx: âK1,K2,L1,L2. L1 @@ K1 = L2 @@ K2 â |L1| = |L2| â
normalize in H2; >append_length in H2; #H
elim (plus_xySz_x_false ⦠(sym_eq ⦠H))
| #K2 #I2 #V2 #L1 #L2 #H1 #H2
- elim (destruct_lpair_lpair ⦠H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
+ elim (destruct_lpair_lpair_aux ⦠H1) -H1 #H1 #H3 #H4 destruct (**) (* destruct lemma needed *)
elim (IH ⦠H1) -IH -H1 /2 width=1 by conj/
]
]
@@ -108,15 +108,15 @@ lemma append_inv_pair_dx: âI,L,K,V. L @@ K = L.â{I}V â K = â.â{I}V.
#I #L #K #V #H elim (append_inj_dx ⦠(â.â{I}V) ⦠H) //
qed-.
-lemma length_inv_pos_dx_ltail: âL,d. |L| = d + 1 â
- ââI,K,V. |K| = d & L = â{I}V.K.
-#Y #d #H elim (length_inv_pos_dx ⦠H) -H #I #L #V #Hd #HLK destruct
+lemma length_inv_pos_dx_ltail: âL,l. |L| = l + 1 â
+ ââI,K,V. |K| = l & L = â{I}V.K.
+#Y #l #H elim (length_inv_pos_dx ⦠H) -H #I #L #V #Hl #HLK destruct
elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
qed-.
-lemma length_inv_pos_sn_ltail: âL,d. d + 1 = |L| â
- ââI,K,V. d = |K| & L = â{I}V.K.
-#Y #d #H elim (length_inv_pos_sn ⦠H) -H #I #L #V #Hd #HLK destruct
+lemma length_inv_pos_sn_ltail: âL,l. l + 1 = |L| â
+ ââI,K,V. l = |K| & L = â{I}V.K.
+#Y #l #H elim (length_inv_pos_sn ⦠H) -H #I #L #V #Hl #HLK destruct
elim (lpair_ltail L I V) /2 width=5 by ex2_3_intro/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_length.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_length.ma
index faf6de02d..7c31b59e7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_length.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/lenv_length.ma
@@ -33,20 +33,20 @@ lemma length_inv_zero_sn: âL. 0 = |L| â L = â.
* // #L #I #V normalize <plus_n_Sm #H destruct
qed-.
-lemma length_inv_pos_dx: âd,L. |L| = d + 1 â
- ââI,K,V. |K| = d & L = K. â{I}V.
-#d *
+lemma length_inv_pos_dx: âl,L. |L| = l + 1 â
+ ââI,K,V. |K| = l & L = K. â{I}V.
+#l *
[ normalize <plus_n_Sm #H destruct
| #K #I #V #H
- lapply (injective_plus_l ⦠H) -H /2 width=5/
+ lapply (injective_plus_l ⦠H) -H /2 width=5 by ex2_3_intro/
]
qed-.
-lemma length_inv_pos_sn: âd,L. d + 1 = |L| â
- ââI,K,V. d = |K| & L = K. â{I}V.
-#d *
+lemma length_inv_pos_sn: âl,L. l + 1 = |L| â
+ ââI,K,V. l = |K| & L = K. â{I}V.
+#l *
[ normalize <plus_n_Sm #H destruct
| #K #I #V #H
- lapply (injective_plus_l ⦠H) -H /2 width=5/
+ lapply (injective_plus_l ⦠H) -H /2 width=5 by ex2_3_intro/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma
deleted file mode 100644
index a898abdb5..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/leq.ma
+++ /dev/null
@@ -1,192 +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_lt.ma".
-include "basic_2/notation/relations/midiso_4.ma".
-include "basic_2/grammar/lenv_length.ma".
-
-(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
-
-inductive leq: relation4 ynat ynat lenv lenv â
-| leq_atom: âd,e. leq d e (â) (â)
-| leq_zero: âI1,I2,L1,L2,V1,V2.
- leq 0 0 L1 L2 â leq 0 0 (L1.â{I1}V1) (L2.â{I2}V2)
-| leq_pair: âI,L1,L2,V,e. leq 0 e L1 L2 â
- leq 0 (⫯e) (L1.â{I}V) (L2.â{I}V)
-| leq_succ: âI1,I2,L1,L2,V1,V2,d,e.
- leq d e L1 L2 â leq (⫯d) e (L1.â{I1}V1) (L2.â{I2}V2)
-.
-
-interpretation
- "equivalence (local environment)"
- 'MidIso d e L1 L2 = (leq d e L1 L2).
-
-(* Basic properties *********************************************************)
-
-lemma leq_pair_lt: âI,L1,L2,V,e. L1 ⩬[0, â«°e] L2 â 0 < e â
- L1.â{I}V ⩬[0, e] L2.â{I}V.
-#I #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 ⦠He) /2 width=1 by leq_pair/
-qed.
-
-lemma leq_succ_lt: âI1,I2,L1,L2,V1,V2,d,e. L1 ⩬[â«°d, e] L2 â 0 < d â
- L1.â{I1}V1 ⩬[d, e] L2. â{I2}V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #Hd <(ylt_inv_O1 ⦠Hd) /2 width=1 by leq_succ/
-qed.
-
-lemma leq_pair_O_Y: âL1,L2. L1 ⩬[0, â] L2 â
- âI,V. L1.â{I}V ⩬[0, â] L2.â{I}V.
-#L1 #L2 #HL12 #I #V lapply (leq_pair I ⦠V ⦠HL12) -HL12 //
-qed.
-
-lemma leq_refl: âL,d,e. L ⩬[d, e] L.
-#L elim L -L //
-#L #I #V #IHL #d elim (ynat_cases ⦠d) [| * #x ]
-#Hd destruct /2 width=1 by leq_succ/
-#e elim (ynat_cases ⦠e) [| * #x ]
-#He destruct /2 width=1 by leq_zero, leq_pair/
-qed.
-
-lemma leq_O2: âL1,L2,d. |L1| = |L2| â L1 ⩬[d, yinj 0] L2.
-#L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ]
-* // [1,3: #L2 #I2 #V2 ] #d normalize
-[1,3: <plus_n_Sm #H destruct ]
-#H lapply (injective_plus_l ⦠H) -H #HL12
-elim (ynat_cases d) /3 width=1 by leq_zero/
-* /3 width=1 by leq_succ/
-qed.
-
-lemma leq_sym: âd,e. symmetric ⦠(leq d e).
-#d #e #L1 #L2 #H elim H -L1 -L2 -d -e
-/2 width=1 by leq_zero, leq_pair, leq_succ/
-qed-.
-
-(* Basic inversion lemmas ***************************************************)
-
-fact leq_inv_atom1_aux: âL1,L2,d,e. L1 ⩬[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
-| #I #L1 #L2 #V #e #_ #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #H destruct
-]
-qed-.
-
-lemma leq_inv_atom1: âL2,d,e. â ⩬[d, e] L2 â L2 = â.
-/2 width=5 by leq_inv_atom1_aux/ qed-.
-
-fact leq_inv_zero1_aux: âL1,L2,d,e. L1 ⩬[d, e] L2 â
- âJ1,K1,W1. L1 = K1.â{J1}W1 â d = 0 â e = 0 â
- ââJ2,K2,W2. K1 ⩬[0, 0] K2 & L2 = K2.â{J2}W2.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #J1 #K1 #W1 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
- /2 width=5 by ex2_3_intro/
-| #I #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 leq_inv_zero1: âI1,K1,L2,V1. K1.â{I1}V1 ⩬[0, 0] L2 â
- ââI2,K2,V2. K1 ⩬[0, 0] K2 & L2 = K2.â{I2}V2.
-/2 width=9 by leq_inv_zero1_aux/ qed-.
-
-fact leq_inv_pair1_aux: âL1,L2,d,e. L1 ⩬[d, e] L2 â
- âJ,K1,W. L1 = K1.â{J}W â d = 0 â 0 < e â
- ââK2. K1 ⩬[0, â«°e] K2 & L2 = K2.â{J}W.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #J #K1 #W #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J #K1 #W #_ #_ #H
- elim (ylt_yle_false ⦠H) //
-| #I #L1 #L2 #V #e #HL12 #J #K1 #W #H #_ #_ destruct
- /2 width=3 by ex2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J #K1 #W #_ #H
- elim (ysucc_inv_O_dx ⦠H)
-]
-qed-.
-
-lemma leq_inv_pair1: âI,K1,L2,V,e. K1.â{I}V ⩬[0, e] L2 â 0 < e â
- ââK2. K1 ⩬[0, â«°e] K2 & L2 = K2.â{I}V.
-/2 width=6 by leq_inv_pair1_aux/ qed-.
-
-lemma leq_inv_pair: âI1,I2,L1,L2,V1,V2,e. L1.â{I1}V1 ⩬[0, e] L2.â{I2}V2 â 0 < e â
- â§â§ L1 ⩬[0, â«°e] L2 & I1 = I2 & V1 = V2.
-#I1 #I2 #L1 #L2 #V1 #V2 #e #H #He elim (leq_inv_pair1 ⦠H) -H //
-#Y #HL12 #H destruct /2 width=1 by and3_intro/
-qed-.
-
-fact leq_inv_succ1_aux: âL1,L2,d,e. L1 ⩬[d, e] L2 â
- âJ1,K1,W1. L1 = K1.â{J1}W1 â 0 < d â
- ââJ2,K2,W2. K1 ⩬[â«°d, e] K2 & L2 = K2.â{J2}W2.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #d #e #J1 #K1 #W1 #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
- elim (ylt_yle_false ⦠H) //
-| #I #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
- /2 width=5 by ex2_3_intro/
-]
-qed-.
-
-lemma leq_inv_succ1: âI1,K1,L2,V1,d,e. K1.â{I1}V1 ⩬[d, e] L2 â 0 < d â
- ââI2,K2,V2. K1 ⩬[â«°d, e] K2 & L2 = K2.â{I2}V2.
-/2 width=5 by leq_inv_succ1_aux/ qed-.
-
-lemma leq_inv_atom2: âL1,d,e. L1 ⩬[d, e] â â L1 = â.
-/3 width=3 by leq_inv_atom1, leq_sym/
-qed-.
-
-lemma leq_inv_succ: âI1,I2,L1,L2,V1,V2,d,e. L1.â{I1}V1 ⩬[d, e] L2.â{I2}V2 â 0 < d â
- L1 ⩬[⫰d, e] L2.
-#I1 #I2 #L1 #L2 #V1 #V2 #d #e #H #Hd elim (leq_inv_succ1 ⦠H) -H //
-#Z #Y #X #HL12 #H destruct //
-qed-.
-
-lemma leq_inv_zero2: âI2,K2,L1,V2. L1 ⩬[0, 0] K2.â{I2}V2 â
- ââI1,K1,V1. K1 ⩬[0, 0] K2 & L1 = K1.â{I1}V1.
-#I2 #K2 #L1 #V2 #H elim (leq_inv_zero1 ⦠(leq_sym ⦠H)) -H
-/3 width=5 by leq_sym, ex2_3_intro/
-qed-.
-
-lemma leq_inv_pair2: âI,K2,L1,V,e. L1 ⩬[0, e] K2.â{I}V â 0 < e â
- ââK1. K1 ⩬[0, â«°e] K2 & L1 = K1.â{I}V.
-#I #K2 #L1 #V #e #H #He elim (leq_inv_pair1 ⦠(leq_sym ⦠H)) -H
-/3 width=3 by leq_sym, ex2_intro/
-qed-.
-
-lemma leq_inv_succ2: âI2,K2,L1,V2,d,e. L1 ⩬[d, e] K2.â{I2}V2 â 0 < d â
- ââI1,K1,V1. K1 ⩬[â«°d, e] K2 & L1 = K1.â{I1}V1.
-#I2 #K2 #L1 #V2 #d #e #H #Hd elim (leq_inv_succ1 ⦠(leq_sym ⦠H)) -H
-/3 width=5 by leq_sym, ex2_3_intro/
-qed-.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma leq_fwd_length: âL1,L2,d,e. L1 ⩬[d, e] L2 â |L1| = |L2|.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e normalize //
-qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-fact leq_inv_O_Y_aux: âL1,L2,d,e. L1 ⩬[d, e] L2 â d = 0 â e = â â L1 = L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e //
-[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #_ #H destruct
-| /4 width=1 by eq_f3, ysucc_inv_Y_dx/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #_ #H elim (ysucc_inv_O_dx ⦠H)
-]
-qed-.
-
-lemma leq_inv_O_Y: âL1,L2. L1 ⩬[0, â] L2 â L1 = L2.
-/2 width=5 by leq_inv_O_Y_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/leq_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/leq_leq.ma
deleted file mode 100644
index 6c0059a6f..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/leq_leq.ma
+++ /dev/null
@@ -1,49 +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/grammar/leq.ma".
-
-(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
-
-(* Main properties **********************************************************)
-
-theorem leq_trans: âd,e. Transitive ⦠(leq d e).
-#d #e #L1 #L2 #H elim H -L1 -L2 -d -e
-[ #d #e #X #H lapply (leq_inv_atom1 ⦠H) -H
- #H destruct //
-| #I1 #I #L1 #L #V1 #V #_ #IHL1 #X #H elim (leq_inv_zero1 ⦠H) -H
- #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by leq_zero/
-| #I #L1 #L #V #e #_ #IHL1 #X #H elim (leq_inv_pair1 ⦠H) -H //
- #L2 #HL2 #H destruct /3 width=1 by leq_pair/
-| #I1 #I #L1 #L #V1 #V #d #e #_ #IHL1 #X #H elim (leq_inv_succ1 ⦠H) -H //
- #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by leq_succ/
-]
-qed-.
-
-theorem leq_canc_sn: âd,e,L,L1,L2. L ⩬[d, e] L1 â L ⩬[d, e] L2 â L1 ⩬[d, e] L2.
-/3 width=3 by leq_trans, leq_sym/ qed-.
-
-theorem leq_canc_dx: âd,e,L,L1,L2. L1 ⩬[d, e] L â L2 ⩬[d, e] L â L1 ⩬[d, e] L2.
-/3 width=3 by leq_trans, leq_sym/ qed-.
-
-theorem leq_join: âL1,L2,d,i. L1 ⩬[d, i] L2 â
- âe. L1 ⩬[i+d, e] L2 â L1 ⩬[d, i+e] L2.
-#L1 #L2 #d #i #H elim H -L1 -L2 -d -i //
-[ #I #L1 #L2 #V #e #_ #IHL12 #e #H
- lapply (leq_inv_succ ⦠H ?) -H // >ypred_succ /3 width=1 by leq_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #e #H
- lapply (leq_inv_succ ⦠H ?) -H // >yplus_succ2 >ypred_succ /3 width=1 by leq_succ/
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/lreq.ma
new file mode 100644
index 000000000..b68a25659
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/lreq.ma
@@ -0,0 +1,192 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM 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_lt.ma".
+include "basic_2/notation/relations/midiso_4.ma".
+include "basic_2/grammar/lenv_length.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
+
+inductive lreq: relation4 ynat ynat lenv lenv â
+| lreq_atom: âl,m. lreq l m (â) (â)
+| lreq_zero: âI1,I2,L1,L2,V1,V2.
+ lreq 0 0 L1 L2 â lreq 0 0 (L1.â{I1}V1) (L2.â{I2}V2)
+| lreq_pair: âI,L1,L2,V,m. lreq 0 m L1 L2 â
+ lreq 0 (⫯m) (L1.â{I}V) (L2.â{I}V)
+| lreq_succ: âI1,I2,L1,L2,V1,V2,l,m.
+ lreq l m L1 L2 â lreq (⫯l) m (L1.â{I1}V1) (L2.â{I2}V2)
+.
+
+interpretation
+ "equivalence (local environment)"
+ 'MidIso l m L1 L2 = (lreq l m L1 L2).
+
+(* Basic properties *********************************************************)
+
+lemma lreq_pair_lt: âI,L1,L2,V,m. L1 ⩬[0, â«°m] L2 â 0 < m â
+ L1.â{I}V ⩬[0, m] L2.â{I}V.
+#I #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 ⦠Hm) /2 width=1 by lreq_pair/
+qed.
+
+lemma lreq_succ_lt: âI1,I2,L1,L2,V1,V2,l,m. L1 ⩬[â«°l, m] L2 â 0 < l â
+ L1.â{I1}V1 ⩬[l, m] L2. â{I2}V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 ⦠Hl) /2 width=1 by lreq_succ/
+qed.
+
+lemma lreq_pair_O_Y: âL1,L2. L1 ⩬[0, â] L2 â
+ âI,V. L1.â{I}V ⩬[0, â] L2.â{I}V.
+#L1 #L2 #HL12 #I #V lapply (lreq_pair I ⦠V ⦠HL12) -HL12 //
+qed.
+
+lemma lreq_refl: âL,l,m. L ⩬[l, m] L.
+#L elim L -L //
+#L #I #V #IHL #l elim (ynat_cases ⦠l) [| * #x ]
+#Hl destruct /2 width=1 by lreq_succ/
+#m elim (ynat_cases ⦠m) [| * #x ]
+#Hm destruct /2 width=1 by lreq_zero, lreq_pair/
+qed.
+
+lemma lreq_O2: âL1,L2,l. |L1| = |L2| â L1 ⩬[l, yinj 0] L2.
+#L1 elim L1 -L1 [| #L1 #I1 #V1 #IHL1 ]
+* // [1,3: #L2 #I2 #V2 ] #l normalize
+[1,3: <plus_n_Sm #H destruct ]
+#H lapply (injective_plus_l ⦠H) -H #HL12
+elim (ynat_cases l) /3 width=1 by lreq_zero/
+* /3 width=1 by lreq_succ/
+qed.
+
+lemma lreq_sym: âl,m. symmetric ⦠(lreq l m).
+#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
+/2 width=1 by lreq_zero, lreq_pair, lreq_succ/
+qed-.
+
+(* Basic inversion lemmas ***************************************************)
+
+fact lreq_inv_atom1_aux: âL1,L2,l,m. L1 ⩬[l, m] L2 â L1 = â â L2 = â.
+#L1 #L2 #l #m * -L1 -L2 -l -m //
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #H destruct
+| #I #L1 #L2 #V #m #_ #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
+]
+qed-.
+
+lemma lreq_inv_atom1: âL2,l,m. â ⩬[l, m] L2 â L2 = â.
+/2 width=5 by lreq_inv_atom1_aux/ qed-.
+
+fact lreq_inv_zero1_aux: âL1,L2,l,m. L1 ⩬[l, m] L2 â
+ âJ1,K1,W1. L1 = K1.â{J1}W1 â l = 0 â m = 0 â
+ ââJ2,K2,W2. K1 ⩬[0, 0] K2 & L2 = K2.â{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #l #m #J1 #K1 #W1 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
+ /2 width=5 by ex2_3_intro/
+| #I #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
+ elim (ysucc_inv_O_dx ⦠H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
+ elim (ysucc_inv_O_dx ⦠H)
+]
+qed-.
+
+lemma lreq_inv_zero1: âI1,K1,L2,V1. K1.â{I1}V1 ⩬[0, 0] L2 â
+ ââI2,K2,V2. K1 ⩬[0, 0] K2 & L2 = K2.â{I2}V2.
+/2 width=9 by lreq_inv_zero1_aux/ qed-.
+
+fact lreq_inv_pair1_aux: âL1,L2,l,m. L1 ⩬[l, m] L2 â
+ âJ,K1,W. L1 = K1.â{J}W â l = 0 â 0 < m â
+ ââK2. K1 ⩬[0, â«°m] K2 & L2 = K2.â{J}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #l #m #J #K1 #W #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J #K1 #W #_ #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I #L1 #L2 #V #m #HL12 #J #K1 #W #H #_ #_ destruct
+ /2 width=3 by ex2_intro/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J #K1 #W #_ #H
+ elim (ysucc_inv_O_dx ⦠H)
+]
+qed-.
+
+lemma lreq_inv_pair1: âI,K1,L2,V,m. K1.â{I}V ⩬[0, m] L2 â 0 < m â
+ ââK2. K1 ⩬[0, â«°m] K2 & L2 = K2.â{I}V.
+/2 width=6 by lreq_inv_pair1_aux/ qed-.
+
+lemma lreq_inv_pair: âI1,I2,L1,L2,V1,V2,m. L1.â{I1}V1 ⩬[0, m] L2.â{I2}V2 â 0 < m â
+ â§â§ L1 ⩬[0, â«°m] L2 & I1 = I2 & V1 = V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #m #H #Hm elim (lreq_inv_pair1 ⦠H) -H //
+#Y #HL12 #H destruct /2 width=1 by and3_intro/
+qed-.
+
+fact lreq_inv_succ1_aux: âL1,L2,l,m. L1 ⩬[l, m] L2 â
+ âJ1,K1,W1. L1 = K1.â{J1}W1 â 0 < l â
+ ââJ2,K2,W2. K1 ⩬[â«°l, m] K2 & L2 = K2.â{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #l #m #J1 #K1 #W1 #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
+ /2 width=5 by ex2_3_intro/
+]
+qed-.
+
+lemma lreq_inv_succ1: âI1,K1,L2,V1,l,m. K1.â{I1}V1 ⩬[l, m] L2 â 0 < l â
+ ââI2,K2,V2. K1 ⩬[â«°l, m] K2 & L2 = K2.â{I2}V2.
+/2 width=5 by lreq_inv_succ1_aux/ qed-.
+
+lemma lreq_inv_atom2: âL1,l,m. L1 ⩬[l, m] â â L1 = â.
+/3 width=3 by lreq_inv_atom1, lreq_sym/
+qed-.
+
+lemma lreq_inv_succ: âI1,I2,L1,L2,V1,V2,l,m. L1.â{I1}V1 ⩬[l, m] L2.â{I2}V2 â 0 < l â
+ L1 ⩬[⫰l, m] L2.
+#I1 #I2 #L1 #L2 #V1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 ⦠H) -H //
+#Z #Y #X #HL12 #H destruct //
+qed-.
+
+lemma lreq_inv_zero2: âI2,K2,L1,V2. L1 ⩬[0, 0] K2.â{I2}V2 â
+ ââI1,K1,V1. K1 ⩬[0, 0] K2 & L1 = K1.â{I1}V1.
+#I2 #K2 #L1 #V2 #H elim (lreq_inv_zero1 ⦠(lreq_sym ⦠H)) -H
+/3 width=5 by lreq_sym, ex2_3_intro/
+qed-.
+
+lemma lreq_inv_pair2: âI,K2,L1,V,m. L1 ⩬[0, m] K2.â{I}V â 0 < m â
+ ââK1. K1 ⩬[0, â«°m] K2 & L1 = K1.â{I}V.
+#I #K2 #L1 #V #m #H #Hm elim (lreq_inv_pair1 ⦠(lreq_sym ⦠H)) -H
+/3 width=3 by lreq_sym, ex2_intro/
+qed-.
+
+lemma lreq_inv_succ2: âI2,K2,L1,V2,l,m. L1 ⩬[l, m] K2.â{I2}V2 â 0 < l â
+ ââI1,K1,V1. K1 ⩬[â«°l, m] K2 & L1 = K1.â{I1}V1.
+#I2 #K2 #L1 #V2 #l #m #H #Hl elim (lreq_inv_succ1 ⦠(lreq_sym ⦠H)) -H
+/3 width=5 by lreq_sym, ex2_3_intro/
+qed-.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma lreq_fwd_length: âL1,L2,l,m. L1 ⩬[l, m] L2 â |L1| = |L2|.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m normalize //
+qed-.
+
+(* Advanced inversion lemmas ************************************************)
+
+fact lreq_inv_O_Y_aux: âL1,L2,l,m. L1 ⩬[l, m] L2 â l = 0 â m = â â L1 = L2.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m //
+[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #_ #H destruct
+| /4 width=1 by eq_f3, ysucc_inv_Y_dx/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #_ #H elim (ysucc_inv_O_dx ⦠H)
+]
+qed-.
+
+lemma lreq_inv_O_Y: âL1,L2. L1 ⩬[0, â] L2 â L1 = L2.
+/2 width=5 by lreq_inv_O_Y_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/lreq_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/lreq_lreq.ma
new file mode 100644
index 000000000..20d6c0002
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/lreq_lreq.ma
@@ -0,0 +1,49 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM 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/grammar/lreq.ma".
+
+(* EQUIVALENCE FOR LOCAL ENVIRONMENTS ***************************************)
+
+(* Main properties **********************************************************)
+
+theorem lreq_trans: âl,m. Transitive ⦠(lreq l m).
+#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
+[ #l #m #X #H lapply (lreq_inv_atom1 ⦠H) -H
+ #H destruct //
+| #I1 #I #L1 #L #V1 #V #_ #IHL1 #X #H elim (lreq_inv_zero1 ⦠H) -H
+ #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lreq_zero/
+| #I #L1 #L #V #m #_ #IHL1 #X #H elim (lreq_inv_pair1 ⦠H) -H //
+ #L2 #HL2 #H destruct /3 width=1 by lreq_pair/
+| #I1 #I #L1 #L #V1 #V #l #m #_ #IHL1 #X #H elim (lreq_inv_succ1 ⦠H) -H //
+ #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lreq_succ/
+]
+qed-.
+
+theorem lreq_canc_sn: âl,m,L,L1,L2. L ⩬[l, m] L1 â L ⩬[l, m] L2 â L1 ⩬[l, m] L2.
+/3 width=3 by lreq_trans, lreq_sym/ qed-.
+
+theorem lreq_canc_dx: âl,m,L,L1,L2. L1 ⩬[l, m] L â L2 ⩬[l, m] L â L1 ⩬[l, m] L2.
+/3 width=3 by lreq_trans, lreq_sym/ qed-.
+
+theorem lreq_join: âL1,L2,l,i. L1 ⩬[l, i] L2 â
+ âm. L1 ⩬[i+l, m] L2 â L1 ⩬[l, i+m] L2.
+#L1 #L2 #l #i #H elim H -L1 -L2 -l -i //
+[ #I #L1 #L2 #V #m #_ #IHL12 #m #H
+ lapply (lreq_inv_succ ⦠H ?) -H // >ypred_succ /3 width=1 by lreq_pair/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #m #H
+ lapply (lreq_inv_succ ⦠H ?) -H // >yplus_succ2 >ypred_succ /3 width=1 by lreq_succ/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma b/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma
index fc039863e..e70058dc9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/grammar/term.ma
@@ -108,15 +108,19 @@ qed-.
(* Basic inversion lemmas ***************************************************)
-lemma destruct_tpair_tpair: âI1,I2,T1,T2,V1,V2. â¡{I1}T1.V1 = â¡{I2}T2.V2 â
- â§â§T1 = T2 & I1 = I2 & V1 = V2.
+fact destruct_tatom_tatom_aux: âI1,I2. âª{I1} = âª{I2} â I1 = I2.
+#I1 #I2 #H destruct //
+qed-.
+
+fact destruct_tpair_tpair_aux: âI1,I2,T1,T2,V1,V2. â¡{I1}T1.V1 = â¡{I2}T2.V2 â
+ â§â§T1 = T2 & I1 = I2 & V1 = V2.
#I1 #I2 #T1 #T2 #V1 #V2 #H destruct /2 width=1 by and3_intro/
qed-.
lemma discr_tpair_xy_x: âI,T,V. â¡{I} V. T = V â â¥.
#I #T #V elim V -V
[ #J #H destruct
-| #J #W #U #IHW #_ #H elim (destruct_tpair_tpair ⦠H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+| #J #W #U #IHW #_ #H elim (destruct_tpair_tpair_aux ⦠H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
]
qed-.
@@ -124,7 +128,7 @@ qed-.
lemma discr_tpair_xy_y: âI,V,T. â¡{I} V. T = T â â¥.
#I #V #T elim T -T
[ #J #H destruct
-| #J #W #U #_ #IHU #H elim (destruct_tpair_tpair ⦠H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
+| #J #W #U #_ #IHU #H elim (destruct_tpair_tpair_aux ⦠H) -H /2 width=1 by/ (**) (* destruct lemma needed *)
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys.ma
index eda8ee1e1..2ca575e2a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys.ma
@@ -18,98 +18,98 @@ include "basic_2/substitution/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).
+ λl,m,G. LTC ⦠(cpy l m G).
interpretation "context-sensitive extended multiple substritution (term)"
- 'PSubstStar G L T1 d e T2 = (cpys d e G L T1 T2).
+ 'PSubstStar G L T1 l m T2 = (cpys l m 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
+lemma cpys_ind: âG,L,T1,l,m. âR:predicate term. R T1 â
+ (âT,T2. â¦G, L⦠⢠T1 â¶*[l, m] T â â¦G, L⦠⢠T â¶[l, m] T2 â R T â R T2) â
+ âT2. â¦G, L⦠⢠T1 â¶*[l, m] T2 â R T2.
+#G #L #T1 #l #m #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
+lemma cpys_ind_dx: âG,L,T2,l,m. âR:predicate term. R T2 â
+ (âT1,T. â¦G, L⦠⢠T1 â¶[l, m] T â â¦G, L⦠⢠T â¶*[l, m] T2 â R T â R T1) â
+ âT1. â¦G, L⦠⢠T1 â¶*[l, m] T2 â R T1.
+#G #L #T2 #l #m #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.
+lemma cpy_cpys: âG,L,T1,T2,l,m. â¦G, L⦠⢠T1 â¶[l, m] T2 â â¦G, L⦠⢠T1 â¶*[l, m] 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.
+lemma cpys_strap1: âG,L,T1,T,T2,l,m.
+ â¦G, L⦠⢠T1 â¶*[l, m] T â â¦G, L⦠⢠T â¶[l, m] T2 â â¦G, L⦠⢠T1 â¶*[l, m] 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.
+lemma cpys_strap2: âG,L,T1,T,T2,l,m.
+ â¦G, L⦠⢠T1 â¶[l, m] T â â¦G, L⦠⢠T â¶*[l, m] T2 â â¦G, L⦠⢠T1 â¶*[l, m] 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).
+lemma lsuby_cpys_trans: âG,l,m. lsub_trans ⦠(cpys l m G) (lsuby l m).
/3 width=5 by lsuby_cpy_trans, LTC_lsub_trans/
qed-.
-lemma cpys_refl: âG,L,d,e. reflexive ⦠(cpys d e G L).
+lemma cpys_refl: âG,L,l,m. reflexive ⦠(cpys l m 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
+lemma cpys_bind: âG,L,V1,V2,l,m. â¦G, L⦠⢠V1 â¶*[l, m] V2 â
+ âI,T1,T2. â¦G, L.â{I}V1⦠⢠T1 â¶*[⫯l, m] T2 â
+ âa. â¦G, L⦠⢠â{a,I}V1.T1 â¶*[l, m] â{a,I}V2.T2.
+#G #L #V1 #V2 #l #m #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
+lemma cpys_flat: âG,L,V1,V2,l,m. â¦G, L⦠⢠V1 â¶*[l, m] V2 â
+ âT1,T2. â¦G, L⦠⢠T1 â¶*[l, m] T2 â
+ âI. â¦G, L⦠⢠â{I}V1.T1 â¶*[l, m] â{I}V2.T2.
+#G #L #V1 #V2 #l #m #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
+lemma cpys_weak: âG,L,T1,T2,l1,m1. â¦G, L⦠⢠T1 â¶*[l1, m1] T2 â
+ âl2,m2. l2 ⤠l1 â l1 + m1 ⤠l2 + m2 â
+ â¦G, L⦠⢠T1 â¶*[l2, m2] T2.
+#G #L #T1 #T2 #l1 #m1 #H #l1 #l2 #Hl21 #Hlm12 @(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
+lemma cpys_weak_top: âG,L,T1,T2,l,m.
+ â¦G, L⦠⢠T1 â¶*[l, m] T2 â â¦G, L⦠⢠T1 â¶*[l, |L| - l] T2.
+#G #L #T1 #T2 #l #m #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
+lemma cpys_weak_full: âG,L,T1,T2,l,m.
+ â¦G, L⦠⢠T1 â¶*[l, m] T2 â â¦G, L⦠⢠T1 â¶*[0, |L|] T2.
+#G #L #T1 #T2 #l #m #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
+lemma cpys_fwd_up: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶*[lt, mt] U2 â
+ âT1,l,m. â¬[l, m] T1 â¡ U1 â
+ l ⤠lt â l + m ⤠lt + mt â
+ ââT2. â¦G, L⦠⢠U1 â¶*[l+m, lt+mt-(l+m)] U2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H #T1 #l #m #HTU1 #Hllt #Hlmlmt @(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 //
+lemma cpys_fwd_tw: âG,L,T1,T2,l,m. â¦G, L⦠⢠T1 â¶*[l, m] T2 â â¯{T1} ⤠â¯{T2}.
+#G #L #T1 #T2 #l #m #H @(cpys_ind ⦠H) -T2 //
#T #T2 #_ #HT2 #IHT1 lapply (cpy_fwd_tw ⦠HT2) -HT2
/2 width=3 by transitive_le/
qed-.
@@ -117,24 +117,24 @@ 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 //
+lemma cpys_inv_sort1: âG,L,T2,k,l,m. â¦G, L⦠⢠âk â¶*[l, m] T2 â T2 = âk.
+#G #L #T2 #k #l #m #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 //
+lemma cpys_inv_gref1: âG,L,T2,p,l,m. â¦G, L⦠⢠§p â¶*[l, m] T2 â T2 = §p.
+#G #L #T2 #p #l #m #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 &
+lemma cpys_inv_bind1: âa,I,G,L,V1,T1,U2,l,m. â¦G, L⦠⢠â{a,I}V1.T1 â¶*[l, m] U2 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¶*[l, m] V2 &
+ â¦G, L.â{I}V1⦠⢠T1 â¶*[⫯l, m] T2 &
U2 = â{a,I}V2.T2.
-#a #I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind ⦠H) -U2
+#a #I #G #L #V1 #T1 #U2 #l #m #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
@@ -143,10 +143,10 @@ lemma cpys_inv_bind1: âa,I,G,L,V1,T1,U2,d,e. â¦G, L⦠⢠â{a,I}V1.T1 â¶*
]
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 &
+lemma cpys_inv_flat1: âI,G,L,V1,T1,U2,l,m. â¦G, L⦠⢠â{I}V1.T1 â¶*[l, m] U2 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¶*[l, m] V2 & â¦G, L⦠⢠T1 â¶*[l, m] T2 &
U2 = â{I}V2.T2.
-#I #G #L #V1 #T1 #U2 #d #e #H @(cpys_ind ⦠H) -U2
+#I #G #L #V1 #T1 #U2 #l #m #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
@@ -154,13 +154,13 @@ lemma cpys_inv_flat1: âI,G,L,V1,T1,U2,d,e. â¦G, L⦠⢠â{I}V1.T1 â¶*[d,
]
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 //
+lemma cpys_inv_refl_O2: âG,L,T1,T2,l. â¦G, L⦠⢠T1 â¶*[l, 0] T2 â T1 = T2.
+#G #L #T1 #T2 #l #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
+lemma cpys_inv_lift1_eq: âG,L,U1,U2. âl,m:nat.
+ â¦G, L⦠⢠U1 â¶*[l, m] U2 â âT1. â¬[l, m] T1 â¡ U1 â U1 = U2.
+#G #L #U1 #U2 #l #m #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/multiple/cpys_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_alt.ma
index 2d550cdbc..d868ef407 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_alt.ma
@@ -19,84 +19,84 @@ include "basic_2/multiple/cpys_lift.ma".
(* 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)
+| cpysa_atom : âI,G,L,l,m. cpysa l m G L (âª{I}) (âª{I})
+| cpysa_subst: âI,G,L,K,V1,V2,W2,i,l,m. l ⤠yinj i â i < l+m â
+ â¬[i] L â¡ K.â{I}V1 â cpysa 0 (â«°(l+m-i)) G K V1 V2 â
+ â¬[0, i+1] V2 â¡ W2 â cpysa l m G L (#i) W2
+| cpysa_bind : âa,I,G,L,V1,V2,T1,T2,l,m.
+ cpysa l m G L V1 V2 â cpysa (⫯l) m G (L.â{I}V1) T1 T2 â
+ cpysa l m G L (â{a,I}V1.T1) (â{a,I}V2.T2)
+| cpysa_flat : âI,G,L,V1,V2,T1,T2,l,m.
+ cpysa l m G L V1 V2 â cpysa l m G L T1 T2 â
+ cpysa l m 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).
+ 'PSubstStarAlt G L T1 l m T2 = (cpysa l m 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
+lemma lsuby_cpysa_trans: âG,l,m. lsub_trans ⦠(cpysa l m G) (lsuby l m).
+#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
[ //
-| #I #G #L1 #K1 #V1 #V2 #W2 #i #d #e #Hdi #Hide #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
elim (lsuby_drop_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.
+lemma cpysa_refl: âG,T,L,l,m. â¦G, L⦠⢠T â¶â¶*[l, m] 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
+lemma cpysa_cpy_trans: âG,L,T1,T,l,m. â¦G, L⦠⢠T1 â¶â¶*[l, m] T â
+ âT2. â¦G, L⦠⢠T â¶[l, m] T2 â â¦G, L⦠⢠T1 â¶â¶*[l, m] T2.
+#G #L #T1 #T #l #m #H elim H -G -L -T1 -T -l -m
+[ #I #G #L #l #m #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
+| #I #G #L #K #V1 #V2 #W2 #i #l #m #Hli #Hilm #HLK #_ #HVW2 #IHV12 #T2 #H
lapply (drop_fwd_drop2 ⦠HLK) #H0LK
- lapply (cpy_weak ⦠H 0 (d+e) ? ?) -H // #H
+ lapply (cpy_weak ⦠H 0 (l+m) ? ?) -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
+| #a #I #G #L #V1 #V #T1 #T #l #m #_ #_ #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
+| #I #G #L #V1 #V #T1 #T #l #m #_ #_ #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.
+lemma cpys_cpysa: âG,L,T1,T2,l,m. â¦G, L⦠⢠T1 â¶*[l, m] T2 â â¦G, L⦠⢠T1 â¶â¶*[l, m] 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
+lemma cpysa_inv_cpys: âG,L,T1,T2,l,m. â¦G, L⦠⢠T1 â¶â¶*[l, m] T2 â â¦G, L⦠⢠T1 â¶*[l, m] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
/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
+ (âI,G,L,l,m. R l m G L (âª{I}) (âª{I})) â
+ (âI,G,L,K,V1,V2,W2,i,l,m. l ⤠yinj i â i < l + m â
+ â¬[i] L â¡ K.â{I}V1 â â¦G, K⦠⢠V1 â¶*[O, â«°(l+m-i)] V2 â
+ â¬[O, i+1] V2 â¡ W2 â R O (â«°(l+m-i)) G K V1 V2 â R l m 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)
+ (âa,I,G,L,V1,V2,T1,T2,l,m. â¦G, L⦠⢠V1 â¶*[l, m] V2 â
+ â¦G, L.â{I}V1⦠⢠T1 â¶*[⫯l, m] T2 â R l m G L V1 V2 â
+ R (⫯l) m G (L.â{I}V1) T1 T2 â R l m 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)
+ (âI,G,L,V1,V2,T1,T2,l,m. â¦G, L⦠⢠V1 â¶*[l, m] V2 â
+ â¦G, L⦠⢠T1 â¶*[l, m] T2 â R l m G L V1 V2 â
+ R l m G L T1 T2 â R l m 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
+ âl,m,G,L,T1,T2. â¦G, L⦠⢠T1 â¶*[l, m] T2 â R l m G L T1 T2.
+#R #H1 #H2 #H3 #H4 #l #m #G #L #T1 #T2 #H elim (cpys_cpysa ⦠H) -G -L -T1 -T2 -l -m
/3 width=8 by cpysa_inv_cpys/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_cpys.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_cpys.ma
index 9c21cb39d..60557d094 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_cpys.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_cpys.ma
@@ -19,53 +19,53 @@ include "basic_2/multiple/cpys_alt.ma".
(* 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/
+lemma cpys_inv_SO2: âG,L,T1,T2,l. â¦G, L⦠⢠T1 â¶*[l, 1] T2 â â¦G, L⦠⢠T1 â¶[l, 1] T2.
+#G #L #T1 #T2 #l #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.
+lemma cpys_strip_eq: âG,L,T0,T1,l1,m1. â¦G, L⦠⢠T0 â¶*[l1, m1] T1 â
+ âT2,l2,m2. â¦G, L⦠⢠T0 â¶[l2, m2] T2 â
+ ââT. â¦G, L⦠⢠T1 â¶[l2, m2] T & â¦G, L⦠⢠T2 â¶*[l1, m1] 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.
+lemma cpys_strip_neq: âG,L1,T0,T1,l1,m1. â¦G, L1⦠⢠T0 â¶*[l1, m1] T1 â
+ âL2,T2,l2,m2. â¦G, L2⦠⢠T0 â¶[l2, m2] T2 â
+ (l1 + m1 ⤠l2 ⨠l2 + m2 ⤠l1) â
+ ââT. â¦G, L2⦠⢠T1 â¶[l2, m2] T & â¦G, L1⦠⢠T2 â¶*[l1, m1] 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.
+lemma cpys_strap1_down: âG,L,T1,T0,l1,m1. â¦G, L⦠⢠T1 â¶*[l1, m1] T0 â
+ âT2,l2,m2. â¦G, L⦠⢠T0 â¶[l2, m2] T2 â l2 + m2 ⤠l1 â
+ ââT. â¦G, L⦠⢠T1 â¶[l2, m2] T & â¦G, L⦠⢠T â¶*[l1, m1] 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.
+lemma cpys_strap2_down: âG,L,T1,T0,l1,m1. â¦G, L⦠⢠T1 â¶[l1, m1] T0 â
+ âT2,l2,m2. â¦G, L⦠⢠T0 â¶*[l2, m2] T2 â l2 + m2 ⤠l1 â
+ ââT. â¦G, L⦠⢠T1 â¶*[l2, m2] T & â¦G, L⦠⢠T â¶[l1, m1] 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
+lemma cpys_split_up: âG,L,T1,T2,l,m. â¦G, L⦠⢠T1 â¶*[l, m] T2 â
+ âi. l ⤠i â i ⤠l + m â
+ ââT. â¦G, L⦠⢠T1 â¶*[l, i - l] T & â¦G, L⦠⢠T â¶*[i, l + m - i] T2.
+#G #L #T1 #T2 #l #m #H #i #Hli #Hilm @(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 (cpy_split_up ⦠HT12 ⦠Hilm) -HT12 -Hilm #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
+lemma cpys_inv_lift1_up: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶*[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ l ⤠lt â lt ⤠yinj l + m â yinj l + m ⤠lt + mt â
+ ââT2. â¦G, K⦠⢠T1 â¶*[l, lt + mt - (yinj l + m)] T2 &
+ â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt
+elim (cpys_split_up ⦠HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
+lapply (cpys_weak ⦠HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #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/
@@ -73,45 +73,45 @@ 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.
+theorem cpys_conf_eq: âG,L,T0,T1,l1,m1. â¦G, L⦠⢠T0 â¶*[l1, m1] T1 â
+ âT2,l2,m2. â¦G, L⦠⢠T0 â¶*[l2, m2] T2 â
+ ââT. â¦G, L⦠⢠T1 â¶*[l2, m2] T & â¦G, L⦠⢠T2 â¶*[l1, m1] 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.
+theorem cpys_conf_neq: âG,L1,T0,T1,l1,m1. â¦G, L1⦠⢠T0 â¶*[l1, m1] T1 â
+ âL2,T2,l2,m2. â¦G, L2⦠⢠T0 â¶*[l2, m2] T2 â
+ (l1 + m1 ⤠l2 ⨠l2 + m2 ⤠l1) â
+ ââT. â¦G, L2⦠⢠T1 â¶*[l2, m2] T & â¦G, L1⦠⢠T2 â¶*[l1, m1] 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.
+theorem cpys_trans_eq: âG,L,T1,T,T2,l,m.
+ â¦G, L⦠⢠T1 â¶*[l, m] T â â¦G, L⦠⢠T â¶*[l, m] T2 â
+ â¦G, L⦠⢠T1 â¶*[l, m] 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.
+theorem cpys_trans_down: âG,L,T1,T0,l1,m1. â¦G, L⦠⢠T1 â¶*[l1, m1] T0 â
+ âT2,l2,m2. â¦G, L⦠⢠T0 â¶*[l2, m2] T2 â l2 + m2 ⤠l1 â
+ ââT. â¦G, L⦠⢠T1 â¶*[l2, m2] T & â¦G, L⦠⢠T â¶*[l1, m1] 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 | ]
+theorem cpys_antisym_eq: âG,L1,T1,T2,l,m. â¦G, L1⦠⢠T1 â¶*[l, m] T2 â
+ âL2. â¦G, L2⦠⢠T2 â¶*[l, m] T1 â T1 = T2.
+#G #L1 #T1 #T2 #l #m #H @(cpys_ind_alt ⦠H) -G -L1 -T1 -T2 //
+[ #I1 #G #L1 #K1 #V1 #V2 #W2 #i #l #m #Hli #Hilm #_ #_ #HVW2 #_ #L2 #HW2
+ elim (lt_or_ge (|L2|) (i+1)) #Hi [ -Hli -Hilm | ]
[ 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 (drop_O1_le (â») ⦠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
+ /2 width=1 by ylt_fwd_le_succ, yle_succ_dx/ -Hli -Hilm
#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
+| #a #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #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
+| #I #G #L1 #V1 #V2 #T1 #T2 #l #m #_ #_ #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/multiple/cpys_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_lift.ma
index d21e9cce0..848284a25 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/cpys_lift.ma
@@ -19,46 +19,46 @@ include "basic_2/multiple/cpys.ma".
(* 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
+lemma cpys_subst: âI,G,L,K,V,U1,i,l,m.
+ l ⤠yinj i â i < l + m â
+ â¬[i] L â¡ K.â{I}V â â¦G, K⦠⢠V â¶*[0, â«°(l+m-i)] U1 â
+ âU2. â¬[0, i+1] U1 â¡ U2 â â¦G, L⦠⢠#i â¶*[l, m] U2.
+#I #G #L #K #V #U1 #i #l #m #Hli #Hilm #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 (drop_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
+ lapply (cpy_weak ⦠HU02 l m ? ?) -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
+lemma cpys_subst_Y2: âI,G,L,K,V,U1,i,l.
+ l ⤠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
+ âU2. â¬[0, i+1] U1 â¡ U2 â â¦G, L⦠⢠#i â¶*[l, â] U2.
+#I #G #L #K #V #U1 #i #l #Hli #HLK #HVU1 #U2 #HU12
@(cpys_subst ⦠HLK ⦠HU12) >yminus_Y_inj //
qed.
(* Advanced inversion lemmas *************************************************)
-lemma cpys_inv_atom1: âI,G,L,T2,d,e. â¦G, L⦠⢠âª{I} â¶*[d, e] T2 â
+lemma cpys_inv_atom1: âI,G,L,T2,l,m. â¦G, L⦠⢠âª{I} â¶*[l, m] T2 â
T2 = âª{I} â¨
- ââJ,K,V1,V2,i. d ⤠yinj i & i < d + e &
+ ââJ,K,V1,V2,i. l ⤠yinj i & i < l + m &
â¬[i] L â¡ K.â{J}V1 &
- â¦G, K⦠⢠V1 â¶*[0, â«°(d+e-i)] V2 &
+ â¦G, K⦠⢠V1 â¶*[0, â«°(l+m-i)] V2 &
â¬[O, i+1] V2 â¡ T2 &
I = LRef i.
-#I #G #L #T2 #d #e #H @(cpys_ind ⦠H) -T2
+#I #G #L #T2 #l #m #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
+ | * #J #K #V1 #V #i #Hli #Hilm #HLK #HV1 #HVT #HI
lapply (drop_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/ ]
@@ -67,33 +67,33 @@ lemma cpys_inv_atom1: âI,G,L,T2,d,e. â¦G, L⦠⢠âª{I} â¶*[d, e] T2 â
]
qed-.
-lemma cpys_inv_lref1: âG,L,T2,i,d,e. â¦G, L⦠⢠#i â¶*[d, e] T2 â
+lemma cpys_inv_lref1: âG,L,T2,i,l,m. â¦G, L⦠⢠#i â¶*[l, m] T2 â
T2 = #i â¨
- ââI,K,V1,V2. d ⤠i & i < d + e &
+ ââI,K,V1,V2. l ⤠i & i < l + m &
â¬[i] L â¡ K.â{I}V1 &
- â¦G, K⦠⢠V1 â¶*[0, â«°(d+e-i)] V2 &
+ â¦G, K⦠⢠V1 â¶*[0, â«°(l+m-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/
+#G #L #T2 #i #l #m #H elim (cpys_inv_atom1 ⦠H) -H /2 width=1 by or_introl/
+* #I #K #V1 #V2 #j #Hlj #Hjlm #HLK #HV12 #HVT2 #H destruct /3 width=7 by ex5_4_intro, or_intror/
qed-.
-lemma cpys_inv_lref1_Y2: âG,L,T2,i,d. â¦G, L⦠⢠#i â¶*[d, â] T2 â
+lemma cpys_inv_lref1_Y2: âG,L,T2,i,l. â¦G, L⦠⢠#i â¶*[l, â] T2 â
T2 = #i â¨
- ââI,K,V1,V2. d ⤠i & â¬[i] L â¡ K.â{I}V1 &
+ ââI,K,V1,V2. l ⤠i & â¬[i] L â¡ K.â{I}V1 &
â¦G, K⦠⢠V1 â¶*[0, â] V2 & â¬[O, i+1] V2 â¡ T2.
-#G #L #T2 #i #d #H elim (cpys_inv_lref1 ⦠H) -H /2 width=1 by or_introl/
+#G #L #T2 #i #l #H elim (cpys_inv_lref1 ⦠H) -H /2 width=1 by or_introl/
* >yminus_Y_inj /3 width=7 by or_intror, ex4_4_intro/
qed-.
-lemma cpys_inv_lref1_drop: âG,L,T2,i,d,e. â¦G, L⦠⢠#i â¶*[d, e] T2 â
+lemma cpys_inv_lref1_drop: âG,L,T2,i,l,m. â¦G, L⦠⢠#i â¶*[l, m] 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
+ â§â§ â¦G, K⦠⢠V1 â¶*[0, â«°(l+m-i)] V2
+ & l ⤠i
+ & i < l + m.
+#G #L #T2 #i #l #m #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
+| * #Z #Y #X1 #X2 #Hli #Hilm #HLY #HX12 #HXT2
lapply (lift_inj ⦠HXT2 ⦠HVT2) -T2 #H destruct
lapply (drop_mono ⦠HLY ⦠HLK) -L #H destruct
/2 width=1 by and3_intro/
@@ -102,40 +102,40 @@ 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
+lemma cpys_lift_le: âG,K,T1,T2,lt,mt. â¦G, K⦠⢠T1 â¶*[lt, mt] T2 â
+ âL,U1,s,l,m. lt + mt ⤠yinj l â â¬[s, l, m] L â¡ K â
+ â¬[l, m] T1 â¡ U1 â âU2. â¬[l, m] T2 â¡ U2 â
+ â¦G, L⦠⢠U1 â¶*[lt, mt] U2.
+#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hlmtl #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
+ elim (lift_total T l m) #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
+lemma cpys_lift_be: âG,K,T1,T2,lt,mt. â¦G, K⦠⢠T1 â¶*[lt, mt] T2 â
+ âL,U1,s,l,m. lt ⤠yinj l â l ⤠lt + mt â
+ â¬[s, l, m] L â¡ K â â¬[l, m] T1 â¡ U1 â
+ âU2. â¬[l, m] T2 â¡ U2 â â¦G, L⦠⢠U1 â¶*[lt, mt + m] U2.
+#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hltl #Hllmt #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
+ elim (lift_total T l m) #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
+lemma cpys_lift_ge: âG,K,T1,T2,lt,mt. â¦G, K⦠⢠T1 â¶*[lt, mt] T2 â
+ âL,U1,s,l,m. yinj l ⤠lt â â¬[s, l, m] L â¡ K â
+ â¬[l, m] T1 â¡ U1 â âU2. â¬[l, m] T2 â¡ U2 â
+ â¦G, L⦠⢠U1 â¶*[lt+m, mt] U2.
+#G #K #T1 #T2 #lt #mt #H #L #U1 #s #l #m #Hllt #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
+ elim (lift_total T l m) #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/
]
@@ -143,33 +143,33 @@ 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
+lemma cpys_inv_lift1_le: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶*[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ lt + mt ⤠l â
+ ââT2. â¦G, K⦠⢠T1 â¶*[lt, mt] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmtl @(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
+lemma cpys_inv_lift1_be: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶*[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ lt ⤠l â yinj l + m ⤠lt + mt â
+ ââT2. â¦G, K⦠⢠T1 â¶*[lt, mt - m] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmlmt @(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
+lemma cpys_inv_lift1_ge: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶*[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ yinj l + m ⤠lt â
+ ââT2. â¦G, K⦠⢠T1 â¶*[lt - m, mt] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hlmlt @(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/
@@ -178,45 +178,45 @@ 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
+lemma cpys_inv_lift1_ge_up: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶*[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ l ⤠lt â lt ⤠yinj l + m â yinj l + m ⤠lt + mt â
+ ââT2. â¦G, K⦠⢠T1 â¶*[l, lt + mt - (yinj l + m)] T2 &
+ â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt @(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
+lemma cpys_inv_lift1_be_up: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶*[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ lt ⤠l â lt + mt ⤠yinj l + m â
+ ââT2. â¦G, K⦠⢠T1 â¶*[lt, l - lt] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm @(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
+lemma cpys_inv_lift1_le_up: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶*[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ lt ⤠l â l ⤠lt + mt â lt + mt ⤠yinj l + m â
+ ââT2. â¦G, K⦠⢠T1 â¶*[lt, l - lt] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm @(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
+lemma cpys_inv_lift1_subst: âG,L,W1,W2,l,m. â¦G, L⦠⢠W1 â¶*[l, m] 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
+ l ⤠yinj i â i < l + m â
+ ââV2. â¦G, K⦠⢠V1 â¶*[O, â«°(l+m-i)] V2 & â¬[O, i+1] V2 â¡ W2.
+#G #L #W1 #W2 #l #m #HW12 #K #V1 #i #HLK #HVW1 #Hli #Hilm
elim (cpys_inv_lift1_ge_up ⦠HW12 ⦠HLK ⦠HVW1 ? ? ?) //
>yplus_O1 <yplus_inj >yplus_SO2
[ >yminus_succ2 /2 width=3 by ex2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/drops.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/drops.ma
index 3a2546564..c6edbf3ed 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/drops.ma
@@ -22,8 +22,8 @@ include "basic_2/multiple/lifts_vector.ma".
inductive drops (s:bool): list2 nat nat â relation lenv â
| drops_nil : âL. drops s (â) L L
-| drops_cons: âL1,L,L2,des,d,e.
- drops s des L1 L â â¬[s, d, e] L â¡ L2 â drops s ({d, e} @ des) L1 L2
+| drops_cons: âL1,L,L2,des,l,m.
+ drops s des L1 L â â¬[s, l, m] L â¡ L2 â drops s ({l, m} @ des) L1 L2
.
interpretation "iterated slicing (local environment) abstract"
@@ -33,15 +33,15 @@ interpretation "iterated slicing (local environment) general"
'RDropStar des T1 T2 = (drops true des T1 T2).
*)
-definition l_liftable1: relation2 lenv term â predicate bool â
- λR,s. âK,T. R K T â âL,d,e. â¬[s, d, e] L â¡ K â
- âU. â¬[d, e] T â¡ U â R L U.
+definition d_liftable1: relation2 lenv term â predicate bool â
+ λR,s. âK,T. R K T â âL,l,m. â¬[s, l, m] L â¡ K â
+ âU. â¬[l, m] T â¡ U â R L U.
-definition l_liftables1: relation2 lenv term â predicate bool â
+definition d_liftables1: relation2 lenv term â predicate bool â
λR,s. âL,K,des. â¬*[s, des] L â¡ K â
âT,U. â¬*[des] T â¡ U â R K T â R L U.
-definition l_liftables1_all: relation2 lenv term â predicate bool â
+definition d_liftables1_all: relation2 lenv term â predicate bool â
λR,s. âL,K,des. â¬*[s, des] L â¡ K â
âTs,Us. â¬*[des] Ts â¡ Us â
all ⦠(R K) Ts â all ⦠(R L) Us.
@@ -50,7 +50,7 @@ definition l_liftables1_all: relation2 lenv term â predicate bool â
fact drops_inv_nil_aux: âL1,L2,s,des. â¬*[s, des] L1 â¡ L2 â des = â â L1 = L2.
#L1 #L2 #s #des * -L1 -L2 -des //
-#L1 #L #L2 #d #e #des #_ #_ #H destruct
+#L1 #L #L2 #l #m #des #_ #_ #H destruct
qed-.
(* Basic_1: was: drop1_gen_pnil *)
@@ -58,18 +58,18 @@ lemma drops_inv_nil: âL1,L2,s. â¬*[s, â] L1 â¡ L2 â L1 = L2.
/2 width=4 by drops_inv_nil_aux/ qed-.
fact drops_inv_cons_aux: âL1,L2,s,des. â¬*[s, des] L1 â¡ L2 â
- âd,e,tl. des = {d, e} @ tl â
- ââL. â¬*[s, tl] L1 â¡ L & â¬[s, d, e] L â¡ L2.
+ âl,m,tl. des = {l, m} @ tl â
+ ââL. â¬*[s, tl] L1 â¡ L & â¬[s, l, m] L â¡ L2.
#L1 #L2 #s #des * -L1 -L2 -des
-[ #L #d #e #tl #H destruct
-| #L1 #L #L2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
+[ #L #l #m #tl #H destruct
+| #L1 #L #L2 #des #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct
/2 width=3 by ex2_intro/
]
qed-.
(* Basic_1: was: drop1_gen_pcons *)
-lemma drops_inv_cons: âL1,L2,s,d,e,des. â¬*[s, {d, e} @ des] L1 â¡ L2 â
- ââL. â¬*[s, des] L1 â¡ L & â¬[s, d, e] L â¡ L2.
+lemma drops_inv_cons: âL1,L2,s,l,m,des. â¬*[s, {l, m} @ des] L1 â¡ L2 â
+ ââL. â¬*[s, des] L1 â¡ L & â¬[s, l, m] L â¡ L2.
/2 width=3 by drops_inv_cons_aux/ qed-.
lemma drops_inv_skip2: âI,s,des,des2,i. des â i â¡ des2 â
@@ -81,14 +81,14 @@ lemma drops_inv_skip2: âI,s,des,des2,i. des â i â¡ des2 â
#I #s #des #des2 #i #H elim H -des -des2 -i
[ #i #L1 #K2 #V2 #H
>(drops_inv_nil ⦠H) -L1 /2 width=7 by lifts_nil, minuss_nil, ex4_3_intro, drops_nil/
-| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
+| #des #des2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
elim (drops_inv_cons ⦠H) -H #L #HL1 #H
elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ #K #V >minus_plus #HK2 #HV2 #H destruct
- elim (IHdes2 ⦠HL1) -IHdes2 -HL1 #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
+ elim (IHcs2 ⦠HL1) -IHcs2 -HL1 #K1 #V1 #des1 #Hcs1 #HK1 #HV1 #X destruct
@(ex4_3_intro ⦠K1 V1 ⦠) // [3,4: /2 width=7 by lifts_cons, drops_cons/ | skip ]
normalize >plus_minus /3 width=1 by minuss_lt, lt_minus_to_plus/ (**) (* explicit constructors *)
-| #des #des2 #d #e #i #Hid #_ #IHdes2 #L1 #K2 #V2 #H
- elim (IHdes2 ⦠H) -IHdes2 -H #K1 #V1 #des1 #Hdes1 #HK1 #HV1 #X destruct
+| #des #des2 #l #m #i #Hil #_ #IHcs2 #L1 #K2 #V2 #H
+ elim (IHcs2 ⦠H) -IHcs2 -H #K1 #V1 #des1 #Hcs1 #HK1 #HV1 #X destruct
/4 width=7 by minuss_ge, ex4_3_intro, le_S_S/
]
qed-.
@@ -101,20 +101,20 @@ lemma drops_skip: âL1,L2,s,des. â¬*[s, des] L1 â¡ L2 â âV1,V2. â¬*[des]
#L1 #L2 #s #des #H elim H -L1 -L2 -des
[ #L #V1 #V2 #HV12 #I
>(lifts_inv_nil ⦠HV12) -HV12 //
-| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #V1 #V2 #H #I
+| #L1 #L #L2 #des #l #m #_ #HL2 #IHL #V1 #V2 #H #I
elim (lifts_inv_cons ⦠H) -H /3 width=5 by drop_skip, drops_cons/
].
qed.
-lemma l1_liftable_liftables: âR,s. l_liftable1 R s â l_liftables1 R s.
+lemma d1_liftable_liftables: âR,s. d_liftable1 R s â d_liftables1 R s.
#R #s #HR #L #K #des #H elim H -L -K -des
[ #L #T #U #H #HT <(lifts_inv_nil ⦠H) -H //
-| #L1 #L #L2 #des #d #e #_ #HL2 #IHL #T2 #T1 #H #HLT2
+| #L1 #L #L2 #des #l #m #_ #HL2 #IHL #T2 #T1 #H #HLT2
elim (lifts_inv_cons ⦠H) -H /3 width=10 by/
]
qed.
-lemma l1_liftables_liftables_all: âR,s. l_liftables1 R s â l_liftables1_all R s.
+lemma d1_liftables_liftables_all: âR,s. d_liftables1 R s â d_liftables1_all R s.
#R #s #HR #L #K #des #HLK #Ts #Us #H elim H -Ts -Us normalize //
#Ts #Us #T #U #HTU #_ #IHTUs * /3 width=7 by conj/
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/drops_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/drops_drop.ma
index 99b69bdc7..e835b2e18 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/drops_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/drops_drop.ma
@@ -24,8 +24,8 @@ lemma drops_drop_trans: âL1,L,des. â¬*[â», des] L1 â¡ L â âL2,i. â¬[i]
@â¦i, des⦠⡠i0 & des â i â¡ des0.
#L1 #L #des #H elim H -L1 -L -des
[ /2 width=7 by drops_nil, minuss_nil, at_nil, ex4_3_intro/
-| #L1 #L0 #L #des #d #e #_ #HL0 #IHL0 #L2 #i #HL2
- elim (lt_or_ge i d) #Hid
+| #L1 #L0 #L #des #l #m #_ #HL0 #IHL0 #L2 #i #HL2
+ elim (lt_or_ge i l) #Hil
[ elim (drop_trans_le ⦠HL0 ⦠HL2) -L /2 width=2 by lt_to_le/
#L #HL0 #HL2 elim (IHL0 ⦠HL0) -L0 /3 width=7 by drops_cons, minuss_lt, at_lt, ex4_3_intro/
| lapply (drop_trans_ge ⦠HL0 ⦠HL2 ?) -L // #HL02
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq.ma
index 910dca14c..1ca5528a6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq.ma
@@ -18,26 +18,26 @@ include "basic_2/multiple/lleq.ma".
(* LAZY EQUIVALENCE FOR CLOSURES ********************************************)
-inductive fleq (d) (G) (L1) (T): relation3 genv lenv term â
-| fleq_intro: âL2. L1 â¡[T, d] L2 â fleq d G L1 T G L2 T
+inductive fleq (l) (G) (L1) (T): relation3 genv lenv term â
+| fleq_intro: âL2. L1 â¡[T, l] L2 â fleq l G L1 T G L2 T
.
interpretation
"lazy equivalence (closure)"
- 'LazyEq d G1 L1 T1 G2 L2 T2 = (fleq d G1 L1 T1 G2 L2 T2).
+ 'LazyEq l G1 L1 T1 G2 L2 T2 = (fleq l G1 L1 T1 G2 L2 T2).
(* Basic properties *********************************************************)
-lemma fleq_refl: âd. tri_reflexive ⦠(fleq d).
+lemma fleq_refl: âl. tri_reflexive ⦠(fleq l).
/2 width=1 by fleq_intro/ qed.
-lemma fleq_sym: âd. tri_symmetric ⦠(fleq d).
-#d #G1 #L1 #T1 #G2 #L2 #T2 * /3 width=1 by fleq_intro, lleq_sym/
+lemma fleq_sym: âl. tri_symmetric ⦠(fleq l).
+#l #G1 #L1 #T1 #G2 #L2 #T2 * /3 width=1 by fleq_intro, lleq_sym/
qed-.
(* Basic inversion lemmas ***************************************************)
-lemma fleq_inv_gen: âG1,G2,L1,L2,T1,T2,d. â¦G1, L1, T1⦠â¡[d] â¦G2, L2, T2⦠â
- â§â§ G1 = G2 & L1 â¡[T1, d] L2 & T1 = T2.
-#G1 #G2 #L1 #L2 #T1 #T2 #d * -G2 -L2 -T2 /2 width=1 by and3_intro/
+lemma fleq_inv_gen: âG1,G2,L1,L2,T1,T2,l. â¦G1, L1, T1⦠â¡[l] â¦G2, L2, T2⦠â
+ â§â§ G1 = G2 & L1 â¡[T1, l] L2 & T1 = T2.
+#G1 #G2 #L1 #L2 #T1 #T2 #l * -G2 -L2 -T2 /2 width=1 by and3_intro/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq_fleq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq_fleq.ma
index 781b1f1e5..915be044e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq_fleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/fleq_fleq.ma
@@ -19,16 +19,16 @@ include "basic_2/multiple/fleq.ma".
(* Main properties **********************************************************)
-theorem fleq_trans: âd. tri_transitive ⦠(fleq d).
-#d #G1 #G #L1 #L #T1 #T * -G -L -T
+theorem fleq_trans: âl. tri_transitive ⦠(fleq l).
+#l #G1 #G #L1 #L #T1 #T * -G -L -T
#L #HT1 #G2 #L2 #T2 * -G2 -L2 -T2
/3 width=3 by lleq_trans, fleq_intro/
qed-.
-theorem fleq_canc_sn: âG,G1,G2,L,L1,L2,T,T1,T2,d.
- â¦G, L, T⦠â¡[d] â¦G1, L1, T1â¦â â¦G, L, T⦠â¡[d] â¦G2, L2, T2⦠â â¦G1, L1, T1⦠â¡[d] â¦G2, L2, T2â¦.
+theorem fleq_canc_sn: âG,G1,G2,L,L1,L2,T,T1,T2,l.
+ â¦G, L, T⦠â¡[l] â¦G1, L1, T1â¦â â¦G, L, T⦠â¡[l] â¦G2, L2, T2⦠â â¦G1, L1, T1⦠â¡[l] â¦G2, L2, T2â¦.
/3 width=5 by fleq_trans, fleq_sym/ qed-.
-theorem fleq_canc_dx: âG1,G2,G,L1,L2,L,T1,T2,T,d.
- â¦G1, L1, T1⦠â¡[d] â¦G, L, T⦠â â¦G2, L2, T2⦠â¡[d] â¦G, L, T⦠â â¦G1, L1, T1⦠â¡[d] â¦G2, L2, T2â¦.
+theorem fleq_canc_dx: âG1,G2,G,L1,L2,L,T1,T2,T,l.
+ â¦G1, L1, T1⦠â¡[l] â¦G, L, T⦠â â¦G2, L2, T2⦠â¡[l] â¦G, L, T⦠â â¦G1, L1, T1⦠â¡[l] â¦G2, L2, T2â¦.
/3 width=5 by fleq_trans, fleq_sym/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqup.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fqup.ma
index aa253f700..6c1e0f127 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqup.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/fqup.ma
@@ -37,9 +37,9 @@ lemma fqup_strap2: âG1,G,G2,L1,L,L2,T1,T,T2.
â¦G1, L1, T1⦠â+ â¦G2, L2, T2â¦.
/2 width=5 by tri_TC_strap/ qed.
-lemma fqup_drop: âG1,G2,L1,K1,K2,T1,T2,U1,e. â¬[e] L1 â¡ K1 â â¬[0, e] T1 â¡ U1 â
+lemma fqup_drop: âG1,G2,L1,K1,K2,T1,T2,U1,m. â¬[m] L1 â¡ K1 â â¬[0, m] T1 â¡ U1 â
â¦G1, K1, T1⦠â+ â¦G2, K2, T2⦠â â¦G1, L1, U1⦠â+ â¦G2, K2, T2â¦.
-#G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #e #HLK1 #HTU1 #HT12 elim (eq_or_gt ⦠e) #H destruct
+#G1 #G2 #L1 #K1 #K2 #T1 #T2 #U1 #m #HLK1 #HTU1 #HT12 elim (eq_or_gt ⦠m) #H destruct
[ >(drop_inv_O2 ⦠HLK1) -L1 <(lift_inv_O2 ⦠HTU1) -U1 //
| /3 width=5 by fqup_strap2, fqu_drop_lt/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus.ma
index 048dbf5ad..92679770e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/fqus.ma
@@ -56,7 +56,7 @@ lemma fqus_strap2: âG1,G,G2,L1,L,L2,T1,T,T2. â¦G1, L1, T1⦠â⸮ â¦G, L,
/2 width=5 by tri_TC_strap/ qed-.
lemma fqus_drop: âG1,G2,K1,K2,T1,T2. â¦G1, K1, T1⦠â* â¦G2, K2, T2⦠â
- âL1,U1,e. â¬[e] L1 â¡ K1 â â¬[0, e] T1 â¡ U1 â
+ âL1,U1,m. â¬[m] L1 â¡ K1 â â¬[0, m] T1 â¡ U1 â
â¦G1, L1, U1⦠â* â¦G2, K2, T2â¦.
#G1 #G2 #K1 #K2 #T1 #T2 #H @(fqus_ind ⦠H) -G2 -K2 -T2
/3 width=5 by fqus_strap1, fquq_fqus, fquq_drop/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees.ma
index f7269bc09..fe9292e9d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees.ma
@@ -20,75 +20,75 @@ include "basic_2/substitution/drop.ma".
(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
inductive frees: relation4 ynat lenv term nat â
-| frees_eq: âL,U,d,i. (âT. â¬[i, 1] T â¡ U â â¥) â frees d L U i
-| frees_be: âI,L,K,U,W,d,i,j. d ⤠yinj j â j < i â
+| frees_eq: âL,U,l,i. (âT. â¬[i, 1] T â¡ U â â¥) â frees l L U i
+| frees_be: âI,L,K,U,W,l,i,j. l ⤠yinj j â j < i â
(âT. â¬[j, 1] T â¡ U â â¥) â â¬[j]L â¡ K.â{I}W â
- frees 0 K W (i-j-1) â frees d L U i.
+ frees 0 K W (i-j-1) â frees l L U i.
interpretation
"context-sensitive free variables (term)"
- 'FreeStar L i d U = (frees d L U i).
+ 'FreeStar L i l U = (frees l L U i).
definition frees_trans: predicate (relation3 lenv term term) â
λR. âL,U1,U2,i. R L U1 U2 â L ⢠i ϵ ð
*[0]â¦U2⦠â L ⢠i ϵ ð
*[0]â¦U1â¦.
(* Basic inversion lemmas ***************************************************)
-lemma frees_inv: âL,U,d,i. L ⢠i ϵ ð
*[d]â¦U⦠â
+lemma frees_inv: âL,U,l,i. L ⢠i ϵ ð
*[l]â¦U⦠â
(âT. â¬[i, 1] T â¡ U â â¥) â¨
- ââI,K,W,j. d ⤠yinj j & j < i & (âT. â¬[j, 1] T â¡ U â â¥) &
+ ââI,K,W,j. l ⤠yinj j & j < i & (âT. â¬[j, 1] T â¡ U â â¥) &
â¬[j]L â¡ K.â{I}W & K ⢠(i-j-1) ϵ ð
*[yinj 0]â¦Wâ¦.
-#L #U #d #i * -L -U -d -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
+#L #U #l #i * -L -U -l -i /4 width=9 by ex5_4_intro, or_intror, or_introl/
qed-.
-lemma frees_inv_sort: âL,d,i,k. L ⢠i ϵ ð
*[d]â¦âk⦠â â¥.
-#L #d #i #k #H elim (frees_inv ⦠H) -H [|*] /2 width=2 by/
+lemma frees_inv_sort: âL,l,i,k. L ⢠i ϵ ð
*[l]â¦âk⦠â â¥.
+#L #l #i #k #H elim (frees_inv ⦠H) -H [|*] /2 width=2 by/
qed-.
-lemma frees_inv_gref: âL,d,i,p. L ⢠i ϵ ð
*[d]â¦Â§p⦠â â¥.
-#L #d #i #p #H elim (frees_inv ⦠H) -H [|*] /2 width=2 by/
+lemma frees_inv_gref: âL,l,i,p. L ⢠i ϵ ð
*[l]â¦Â§p⦠â â¥.
+#L #l #i #p #H elim (frees_inv ⦠H) -H [|*] /2 width=2 by/
qed-.
-lemma frees_inv_lref: âL,d,j,i. L ⢠i ϵ ð
*[d]â¦#j⦠â
+lemma frees_inv_lref: âL,l,j,i. L ⢠i ϵ ð
*[l]â¦#j⦠â
j = i â¨
- ââI,K,W. d ⤠yinj j & j < i & â¬[j] L â¡ K.â{I}W & K ⢠(i-j-1) ϵ ð
*[yinj 0]â¦Wâ¦.
-#L #d #x #i #H elim (frees_inv ⦠H) -H
+ ââI,K,W. l ⤠yinj j & j < i & â¬[j] L â¡ K.â{I}W & K ⢠(i-j-1) ϵ ð
*[yinj 0]â¦Wâ¦.
+#L #l #x #i #H elim (frees_inv ⦠H) -H
[ /4 width=2 by nlift_inv_lref_be_SO, or_introl/
-| * #I #K #W #j #Hdj #Hji #Hnx #HLK #HW
+| * #I #K #W #j #Hlj #Hji #Hnx #HLK #HW
>(nlift_inv_lref_be_SO ⦠Hnx) -x /3 width=5 by ex4_3_intro, or_intror/
]
qed-.
-lemma frees_inv_lref_free: âL,d,j,i. L ⢠i ϵ ð
*[d]â¦#j⦠â |L| ⤠j â j = i.
-#L #d #j #i #H #Hj elim (frees_inv_lref ⦠H) -H //
+lemma frees_inv_lref_free: âL,l,j,i. L ⢠i ϵ ð
*[l]â¦#j⦠â |L| ⤠j â j = i.
+#L #l #j #i #H #Hj elim (frees_inv_lref ⦠H) -H //
* #I #K #W #_ #_ #HLK lapply (drop_fwd_length_lt2 ⦠HLK) -I
#H elim (lt_refl_false j) /2 width=3 by lt_to_le_to_lt/
qed-.
-lemma frees_inv_lref_skip: âL,d,j,i. L ⢠i ϵ ð
*[d]â¦#j⦠â yinj j < d â j = i.
-#L #d #j #i #H #Hjd elim (frees_inv_lref ⦠H) -H //
-* #I #K #W #Hdj elim (ylt_yle_false ⦠Hdj) -Hdj //
+lemma frees_inv_lref_skip: âL,l,j,i. L ⢠i ϵ ð
*[l]â¦#j⦠â yinj j < l â j = i.
+#L #l #j #i #H #Hjl elim (frees_inv_lref ⦠H) -H //
+* #I #K #W #Hlj elim (ylt_yle_false ⦠Hlj) -Hlj //
qed-.
-lemma frees_inv_lref_ge: âL,d,j,i. L ⢠i ϵ ð
*[d]â¦#j⦠â i ⤠j â j = i.
-#L #d #j #i #H #Hij elim (frees_inv_lref ⦠H) -H //
-* #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -d /2 width=3 by lt_to_le_to_lt/
+lemma frees_inv_lref_ge: âL,l,j,i. L ⢠i ϵ ð
*[l]â¦#j⦠â i ⤠j â j = i.
+#L #l #j #i #H #Hij elim (frees_inv_lref ⦠H) -H //
+* #I #K #W #_ #Hji elim (lt_refl_false j) -I -L -K -W -l /2 width=3 by lt_to_le_to_lt/
qed-.
-lemma frees_inv_lref_lt: âL,d,j,i.L ⢠i ϵ ð
*[d]â¦#j⦠â j < i â
- ââI,K,W. d ⤠yinj j & â¬[j] L â¡ K.â{I}W & K ⢠(i-j-1) ϵ ð
*[yinj 0]â¦Wâ¦.
-#L #d #j #i #H #Hji elim (frees_inv_lref ⦠H) -H
+lemma frees_inv_lref_lt: âL,l,j,i.L ⢠i ϵ ð
*[l]â¦#j⦠â j < i â
+ ââI,K,W. l ⤠yinj j & â¬[j] L â¡ K.â{I}W & K ⢠(i-j-1) ϵ ð
*[yinj 0]â¦Wâ¦.
+#L #l #j #i #H #Hji elim (frees_inv_lref ⦠H) -H
[ #H elim (lt_refl_false j) //
| * /2 width=5 by ex3_3_intro/
]
qed-.
-lemma frees_inv_bind: âa,I,L,W,U,d,i. L ⢠i ϵ ð
*[d]â¦â{a,I}W.U⦠â
- L ⢠i ϵ ð
*[d]â¦W⦠⨠L.â{I}W ⢠i+1 ϵ ð
*[⫯d]â¦U⦠.
-#a #J #L #V #U #d #i #H elim (frees_inv ⦠H) -H
+lemma frees_inv_bind: âa,I,L,W,U,l,i. L ⢠i ϵ ð
*[l]â¦â{a,I}W.U⦠â
+ L ⢠i ϵ ð
*[l]â¦W⦠⨠L.â{I}W ⢠i+1 ϵ ð
*[⫯l]â¦U⦠.
+#a #J #L #V #U #l #i #H elim (frees_inv ⦠H) -H
[ #HnX elim (nlift_inv_bind ⦠HnX) -HnX
/4 width=2 by frees_eq, or_intror, or_introl/
-| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_bind ⦠HnX) -HnX
+| * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_bind ⦠HnX) -HnX
[ /4 width=9 by frees_be, or_introl/
| #HnT @or_intror @(frees_be ⦠HnT) -HnT
[4,5,6: /2 width=1 by drop_drop, yle_succ, lt_minus_to_plus/
@@ -99,58 +99,58 @@ lemma frees_inv_bind: âa,I,L,W,U,d,i. L ⢠i ϵ ð
*[d]â¦â{a,I}W.U⦠â
]
qed-.
-lemma frees_inv_flat: âI,L,W,U,d,i. L ⢠i ϵ ð
*[d]â¦â{I}W.U⦠â
- L ⢠i ϵ ð
*[d]â¦W⦠⨠L ⢠i ϵ ð
*[d]â¦U⦠.
-#J #L #V #U #d #i #H elim (frees_inv ⦠H) -H
+lemma frees_inv_flat: âI,L,W,U,l,i. L ⢠i ϵ ð
*[l]â¦â{I}W.U⦠â
+ L ⢠i ϵ ð
*[l]â¦W⦠⨠L ⢠i ϵ ð
*[l]â¦U⦠.
+#J #L #V #U #l #i #H elim (frees_inv ⦠H) -H
[ #HnX elim (nlift_inv_flat ⦠HnX) -HnX
/4 width=2 by frees_eq, or_intror, or_introl/
-| * #I #K #W #j #Hdj #Hji #HnX #HLK #HW elim (nlift_inv_flat ⦠HnX) -HnX
+| * #I #K #W #j #Hlj #Hji #HnX #HLK #HW elim (nlift_inv_flat ⦠HnX) -HnX
/4 width=9 by frees_be, or_intror, or_introl/
]
qed-.
(* Basic properties *********************************************************)
-lemma frees_lref_eq: âL,d,i. L ⢠i ϵ ð
*[d]â¦#iâ¦.
+lemma frees_lref_eq: âL,l,i. L ⢠i ϵ ð
*[l]â¦#iâ¦.
/3 width=7 by frees_eq, lift_inv_lref2_be/ qed.
-lemma frees_lref_be: âI,L,K,W,d,i,j. d ⤠yinj j â j < i â â¬[j]L â¡ K.â{I}W â
- K ⢠i-j-1 ϵ ð
*[0]â¦W⦠â L ⢠i ϵ ð
*[d]â¦#jâ¦.
+lemma frees_lref_be: âI,L,K,W,l,i,j. l ⤠yinj j â j < i â â¬[j]L â¡ K.â{I}W â
+ K ⢠i-j-1 ϵ ð
*[0]â¦W⦠â L ⢠i ϵ ð
*[l]â¦#jâ¦.
/3 width=9 by frees_be, lift_inv_lref2_be/ qed.
-lemma frees_bind_sn: âa,I,L,W,U,d,i. L ⢠i ϵ ð
*[d]â¦W⦠â
- L ⢠i ϵ ð
*[d]â¦â{a,I}W.Uâ¦.
-#a #I #L #W #U #d #i #H elim (frees_inv ⦠H) -H [|*]
+lemma frees_bind_sn: âa,I,L,W,U,l,i. L ⢠i ϵ ð
*[l]â¦W⦠â
+ L ⢠i ϵ ð
*[l]â¦â{a,I}W.Uâ¦.
+#a #I #L #W #U #l #i #H elim (frees_inv ⦠H) -H [|*]
/4 width=9 by frees_be, frees_eq, nlift_bind_sn/
qed.
-lemma frees_bind_dx: âa,I,L,W,U,d,i. L.â{I}W ⢠i+1 ϵ ð
*[⫯d]â¦U⦠â
- L ⢠i ϵ ð
*[d]â¦â{a,I}W.Uâ¦.
-#a #J #L #V #U #d #i #H elim (frees_inv ⦠H) -H
+lemma frees_bind_dx: âa,I,L,W,U,l,i. L.â{I}W ⢠i+1 ϵ ð
*[⫯l]â¦U⦠â
+ L ⢠i ϵ ð
*[l]â¦â{a,I}W.Uâ¦.
+#a #J #L #V #U #l #i #H elim (frees_inv ⦠H) -H
[ /4 width=9 by frees_eq, nlift_bind_dx/
-| * #I #K #W #j #Hdj #Hji #HnU #HLK #HW
- elim (yle_inv_succ1 ⦠Hdj) -Hdj <yminus_SO2 #Hyj #H
+| * #I #K #W #j #Hlj #Hji #HnU #HLK #HW
+ elim (yle_inv_succ1 ⦠Hlj) -Hlj <yminus_SO2 #Hyj #H
lapply (ylt_O ⦠H) -H #Hj
>(plus_minus_m_m j 1) in HnU; // <minus_le_minus_minus_comm in HW;
/4 width=9 by frees_be, nlift_bind_dx, drop_inv_drop1_lt, lt_plus_to_minus/
]
qed.
-lemma frees_flat_sn: âI,L,W,U,d,i. L ⢠i ϵ ð
*[d]â¦W⦠â
- L ⢠i ϵ ð
*[d]â¦â{I}W.Uâ¦.
-#I #L #W #U #d #i #H elim (frees_inv ⦠H) -H [|*]
+lemma frees_flat_sn: âI,L,W,U,l,i. L ⢠i ϵ ð
*[l]â¦W⦠â
+ L ⢠i ϵ ð
*[l]â¦â{I}W.Uâ¦.
+#I #L #W #U #l #i #H elim (frees_inv ⦠H) -H [|*]
/4 width=9 by frees_be, frees_eq, nlift_flat_sn/
qed.
-lemma frees_flat_dx: âI,L,W,U,d,i. L ⢠i ϵ ð
*[d]â¦U⦠â
- L ⢠i ϵ ð
*[d]â¦â{I}W.Uâ¦.
-#I #L #W #U #d #i #H elim (frees_inv ⦠H) -H [|*]
+lemma frees_flat_dx: âI,L,W,U,l,i. L ⢠i ϵ ð
*[l]â¦U⦠â
+ L ⢠i ϵ ð
*[l]â¦â{I}W.Uâ¦.
+#I #L #W #U #l #i #H elim (frees_inv ⦠H) -H [|*]
/4 width=9 by frees_be, frees_eq, nlift_flat_dx/
qed.
-lemma frees_weak: âL,U,d1,i. L ⢠i ϵ ð
*[d1]â¦U⦠â
- âd2. d2 ⤠d1 â L ⢠i ϵ ð
*[d2]â¦Uâ¦.
-#L #U #d1 #i #H elim H -L -U -d1 -i
+lemma frees_weak: âL,U,l1,i. L ⢠i ϵ ð
*[l1]â¦U⦠â
+ âl2. l2 ⤠l1 â L ⢠i ϵ ð
*[l2]â¦Uâ¦.
+#L #U #l1 #i #H elim H -L -U -l1 -i
/3 width=9 by frees_be, frees_eq, yle_trans/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_append.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_append.ma
index e43517ff9..8f6835955 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_append.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_append.ma
@@ -19,13 +19,13 @@ include "basic_2/multiple/frees.ma".
(* Properties on append for local environments ******************************)
-lemma frees_append: âL2,U,d,i. L2 ⢠i ϵ ð
*[d]â¦U⦠â i ⤠|L2| â
- âL1. L1 @@ L2 ⢠i ϵ ð
*[d]â¦Uâ¦.
-#L2 #U #d #i #H elim H -L2 -U -d -i /3 width=2 by frees_eq/
-#I #L2 #K2 #U #W #d #i #j #Hdj #Hji #HnU #HLK2 #_ #IHW #Hi #L1
+lemma frees_append: âL2,U,l,i. L2 ⢠i ϵ ð
*[l]â¦U⦠â i ⤠|L2| â
+ âL1. L1 @@ L2 ⢠i ϵ ð
*[l]â¦Uâ¦.
+#L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
+#I #L2 #K2 #U #W #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW #Hi #L1
lapply (drop_fwd_length_minus2 ⦠HLK2) normalize #H0
lapply (drop_O1_append_sn_le ⦠HLK2 ⦠L1) -HLK2
-[ -I -L1 -K2 -U -W -d /3 width=3 by lt_to_le, lt_to_le_to_lt/
+[ -I -L1 -K2 -U -W -l /3 width=3 by lt_to_le, lt_to_le_to_lt/
| #HLK2 @(frees_be ⦠HnU HLK2) // -HnU -HLK2 @IHW -IHW
>(minus_plus_m_m (|K2|) 1) >H0 -H0 /2 width=1 by monotonic_le_minus_l2/
]
@@ -33,20 +33,20 @@ qed.
(* Inversion lemmas on append for local environments ************************)
-fact frees_inv_append_aux: âL,U,d,i. L ⢠i ϵ ð
*[d]â¦U⦠â âL1,L2. L = L1 @@ L2 â
- i ⤠|L2| â L2 ⢠i ϵ ð
*[d]â¦Uâ¦.
-#L #U #d #i #H elim H -L -U -d -i /3 width=2 by frees_eq/
-#Z #L #Y #U #X #d #i #j #Hdj #Hji #HnU #HLY #_ #IHW #L1 #L2 #H #Hi destruct
-elim (drop_O1_lt (â») L2 j) [2: -Z -Y -L1 -X -U -d /2 width=3 by lt_to_le_to_lt/ ]
+fact frees_inv_append_aux: âL,U,l,i. L ⢠i ϵ ð
*[l]â¦U⦠â âL1,L2. L = L1 @@ L2 â
+ i ⤠|L2| â L2 ⢠i ϵ ð
*[l]â¦Uâ¦.
+#L #U #l #i #H elim H -L -U -l -i /3 width=2 by frees_eq/
+#Z #L #Y #U #X #l #i #j #Hlj #Hji #HnU #HLY #_ #IHW #L1 #L2 #H #Hi destruct
+elim (drop_O1_lt (â») L2 j) [2: -Z -Y -L1 -X -U -l /2 width=3 by lt_to_le_to_lt/ ]
#I #K2 #W #HLK2 lapply (drop_fwd_length_minus2 ⦠HLK2) normalize #H0
lapply (drop_O1_inv_append1_le ⦠HLY ⦠HLK2) -HLY
-[ -Z -I -Y -K2 -L1 -X -U -W -d /3 width=3 by lt_to_le, lt_to_le_to_lt/
+[ -Z -I -Y -K2 -L1 -X -U -W -l /3 width=3 by lt_to_le, lt_to_le_to_lt/
| normalize #H destruct
@(frees_be ⦠HnU HLK2) -HnU -HLK2 // @IHW -IHW //
>(minus_plus_m_m (|K2|) 1) >H0 -H0 /2 width=1 by monotonic_le_minus_l2/
]
qed-.
-lemma frees_inv_append: âL1,L2,U,d,i. L1 @@ L2 ⢠i ϵ ð
*[d]â¦U⦠â
- i ⤠|L2| â L2 ⢠i ϵ ð
*[d]â¦Uâ¦.
+lemma frees_inv_append: âL1,L2,U,l,i. L1 @@ L2 ⢠i ϵ ð
*[l]â¦U⦠â
+ i ⤠|L2| â L2 ⢠i ϵ ð
*[l]â¦Uâ¦.
/2 width=4 by frees_inv_append_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_leq.ma
deleted file mode 100644
index c8cf8f0c2..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_leq.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/substitution/drop_leq.ma".
-include "basic_2/multiple/frees.ma".
-
-(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
-
-(* Properties on equivalence for local environments *************************)
-
-lemma leq_frees_trans: âL2,U,d,i. L2 ⢠i ϵ ð
*[d]â¦U⦠â
- âL1. L1 ⩬[d, â] L2 â L1 ⢠i ϵ ð
*[d]â¦Uâ¦.
-#L2 #U #d #i #H elim H -L2 -U -d -i /3 width=2 by frees_eq/
-#I2 #L2 #K2 #U #W2 #d #i #j #Hdj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
-elim (leq_drop_trans_be ⦠HL12 ⦠HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
-lapply (leq_inv_O_Y ⦠HK12) -HK12 #H destruct /3 width=9 by frees_be/
-qed-.
-
-lemma frees_leq_conf: âL1,U,d,i. L1 ⢠i ϵ ð
*[d]â¦U⦠â
- âL2. L1 ⩬[d, â] L2 â L2 ⢠i ϵ ð
*[d]â¦Uâ¦.
-/3 width=3 by leq_sym, leq_frees_trans/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_lift.ma
index 14b5eff6c..0b60f7af7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_lift.ma
@@ -19,45 +19,45 @@ include "basic_2/multiple/frees.ma".
(* Advanced properties ******************************************************)
-lemma frees_dec: âL,U,d,i. Decidable (frees d L U i).
+lemma frees_dec: âL,U,l,i. Decidable (frees l L U i).
#L #U @(f2_ind ⦠rfw ⦠L U) -L -U
#n #IH #L * *
[ -IH /3 width=5 by frees_inv_sort, or_intror/
-| #j #Hn #d #i elim (lt_or_eq_or_gt i j) #Hji
+| #j #Hn #l #i elim (lt_or_eq_or_gt i j) #Hji
[ -n @or_intror #H elim (lt_refl_false i)
- lapply (frees_inv_lref_ge ⦠H ?) -L -d /2 width=1 by lt_to_le/
+ lapply (frees_inv_lref_ge ⦠H ?) -L -l /2 width=1 by lt_to_le/
| -n /2 width=1 by or_introl/
- | elim (ylt_split j d) #Hdi
+ | elim (ylt_split j l) #Hli
[ -n @or_intror #H elim (lt_refl_false i)
lapply (frees_inv_lref_skip ⦠H ?) -L //
| elim (lt_or_ge j (|L|)) #Hj
[ elim (drop_O1_lt (â») L j) // -Hj #I #K #W #HLK destruct
elim (IH K W ⦠0 (i-j-1)) -IH [1,3: /3 width=5 by frees_lref_be, drop_fwd_rfw, or_introl/ ] #HnW
- @or_intror #H elim (frees_inv_lref_lt ⦠H) // #Z #Y #X #_ #HLY -d
+ @or_intror #H elim (frees_inv_lref_lt ⦠H) // #Z #Y #X #_ #HLY -l
lapply (drop_mono ⦠HLY ⦠HLK) -L #H destruct /2 width=1 by/
| -n @or_intror #H elim (lt_refl_false i)
- lapply (frees_inv_lref_free ⦠H ?) -d //
+ lapply (frees_inv_lref_free ⦠H ?) -l //
]
]
]
| -IH /3 width=5 by frees_inv_gref, or_intror/
-| #a #I #W #U #Hn #d #i destruct
- elim (IH L W ⦠d i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
- elim (IH (L.â{I}W) U ⦠(⫯d) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
+| #a #I #W #U #Hn #l #i destruct
+ elim (IH L W ⦠l i) [1,3: /3 width=1 by frees_bind_sn, or_introl/ ] #HnW
+ elim (IH (L.â{I}W) U ⦠(⫯l) (i+1)) -IH [1,3: /3 width=1 by frees_bind_dx, or_introl/ ] #HnU
@or_intror #H elim (frees_inv_bind ⦠H) -H /2 width=1 by/
-| #I #W #U #Hn #d #i destruct
- elim (IH L W ⦠d i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
- elim (IH L U ⦠d i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
+| #I #W #U #Hn #l #i destruct
+ elim (IH L W ⦠l i) [1,3: /3 width=1 by frees_flat_sn, or_introl/ ] #HnW
+ elim (IH L U ⦠l i) -IH [1,3: /3 width=1 by frees_flat_dx, or_introl/ ] #HnU
@or_intror #H elim (frees_inv_flat ⦠H) -H /2 width=1 by/
]
qed-.
-lemma frees_S: âL,U,d,i. L ⢠i ϵ ð
*[yinj d]â¦U⦠â âI,K,W. â¬[d] L â¡ K.â{I}W â
- (K ⢠i-d-1 ϵ ð
*[0]â¦W⦠â â¥) â L ⢠i ϵ ð
*[⫯d]â¦Uâ¦.
-#L #U #d #i #H elim (frees_inv ⦠H) -H /3 width=2 by frees_eq/
-* #I #K #W #j #Hdj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
-lapply (yle_inv_inj ⦠Hdj) -Hdj #Hdj
-elim (le_to_or_lt_eq ⦠Hdj) -Hdj
+lemma frees_S: âL,U,l,i. L ⢠i ϵ ð
*[yinj l]â¦U⦠â âI,K,W. â¬[l] L â¡ K.â{I}W â
+ (K ⢠i-l-1 ϵ ð
*[0]â¦W⦠â â¥) â L ⢠i ϵ ð
*[⫯l]â¦Uâ¦.
+#L #U #l #i #H elim (frees_inv ⦠H) -H /3 width=2 by frees_eq/
+* #I #K #W #j #Hlj #Hji #HnU #HLK #HW #I0 #K0 #W0 #HLK0 #HnW0
+lapply (yle_inv_inj ⦠Hlj) -Hlj #Hlj
+elim (le_to_or_lt_eq ⦠Hlj) -Hlj
[ -I0 -K0 -W0 /3 width=9 by frees_be, yle_inj/
| -Hji -HnU #H destruct
lapply (drop_mono ⦠HLK0 ⦠HLK) #H destruct -I
@@ -74,19 +74,19 @@ qed.
(* Properties on relocation *************************************************)
-lemma frees_lift_ge: âK,T,d,i. K ⢠i ϵð
*[d]â¦T⦠â
- âL,s,d0,e0. â¬[s, d0, e0] L â¡ K â
- âU. â¬[d0, e0] T â¡ U â d0 ⤠i â
- L ⢠i+e0 ϵ ð
*[d]â¦Uâ¦.
-#K #T #d #i #H elim H -K -T -d -i
-[ #K #T #d #i #HnT #L #s #d0 #e0 #_ #U #HTU #Hd0i -K -s
+lemma frees_lift_ge: âK,T,l,i. K ⢠i ϵð
*[l]â¦T⦠â
+ âL,s,l0,m0. â¬[s, l0, m0] L â¡ K â
+ âU. â¬[l0, m0] T â¡ U â l0 ⤠i â
+ L ⢠i+m0 ϵ ð
*[l]â¦Uâ¦.
+#K #T #l #i #H elim H -K -T -l -i
+[ #K #T #l #i #HnT #L #s #l0 #m0 #_ #U #HTU #Hl0i -K -s
@frees_eq #X #HXU elim (lift_div_le ⦠HTU ⦠HXU) -U /2 width=2 by/
-| #I #K #K0 #T #V #d #i #j #Hdj #Hji #HnT #HK0 #HV #IHV #L #s #d0 #e0 #HLK #U #HTU #Hd0i
- elim (lt_or_ge j d0) #H1
+| #I #K #K0 #T #V #l #i #j #Hlj #Hji #HnT #HK0 #HV #IHV #L #s #l0 #m0 #HLK #U #HTU #Hl0i
+ elim (lt_or_ge j l0) #H1
[ elim (drop_trans_lt ⦠HLK ⦠HK0) // -K #L0 #W #HL0 #HLK0 #HVW
@(frees_be ⦠HL0) -HL0 -HV
/3 width=3 by lt_plus_to_minus_r, lt_to_le_to_lt/
- [ #X #HXU >(plus_minus_m_m d0 1) in HTU; /2 width=2 by ltn_to_ltO/ #HTU
+ [ #X #HXU >(plus_minus_m_m l0 1) in HTU; /2 width=2 by ltn_to_ltO/ #HTU
elim (lift_div_le ⦠HXU ⦠HTU ?) -U /2 width=2 by monotonic_pred/
| >minus_plus <plus_minus // <minus_plus
/3 width=5 by monotonic_le_minus_l2/
@@ -102,36 +102,36 @@ qed.
(* Inversion lemmas on relocation *******************************************)
-lemma frees_inv_lift_be: âL,U,d,i. L ⢠i ϵ ð
*[d]â¦U⦠â
- âK,s,d0,e0. â¬[s, d0, e0+1] L â¡ K â
- âT. â¬[d0, e0+1] T â¡ U â d0 ⤠i â i ⤠d0 + e0 â â¥.
-#L #U #d #i #H elim H -L -U -d -i
-[ #L #U #d #i #HnU #K #s #d0 #e0 #_ #T #HTU #Hd0i #Hide0
- elim (lift_split ⦠HTU i e0) -HTU /2 width=2 by/
-| #I #L #K0 #U #W #d #i #j #Hdi #Hij #HnU #HLK0 #_ #IHW #K #s #d0 #e0 #HLK #T #HTU #Hd0i #Hide0
- elim (lt_or_ge j d0) #H1
+lemma frees_inv_lift_be: âL,U,l,i. L ⢠i ϵ ð
*[l]â¦U⦠â
+ âK,s,l0,m0. â¬[s, l0, m0+1] L â¡ K â
+ âT. â¬[l0, m0+1] T â¡ U â l0 ⤠i â i ⤠l0 + m0 â â¥.
+#L #U #l #i #H elim H -L -U -l -i
+[ #L #U #l #i #HnU #K #s #l0 #m0 #_ #T #HTU #Hl0i #Hilm0
+ elim (lift_split ⦠HTU i m0) -HTU /2 width=2 by/
+| #I #L #K0 #U #W #l #i #j #Hli #Hij #HnU #HLK0 #_ #IHW #K #s #l0 #m0 #HLK #T #HTU #Hl0i #Hilm0
+ elim (lt_or_ge j l0) #H1
[ elim (drop_conf_lt ⦠HLK ⦠HLK0) -L // #L0 #V #H #HKL0 #HVW
@(IHW ⦠HKL0 ⦠HVW)
[ /2 width=1 by monotonic_le_minus_l2/
| >minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
]
- | elim (lift_split ⦠HTU j e0) -HTU /3 width=3 by lt_to_le_to_lt, lt_to_le/
+ | elim (lift_split ⦠HTU j m0) -HTU /3 width=3 by lt_to_le_to_lt, lt_to_le/
]
]
qed-.
-lemma frees_inv_lift_ge: âL,U,d,i. L ⢠i ϵ ð
*[d]â¦U⦠â
- âK,s,d0,e0. â¬[s, d0, e0] L â¡ K â
- âT. â¬[d0, e0] T â¡ U â d0 + e0 ⤠i â
- K ⢠i-e0 ϵð
*[d-yinj e0]â¦Tâ¦.
-#L #U #d #i #H elim H -L -U -d -i
-[ #L #U #d #i #HnU #K #s #d0 #e0 #HLK #T #HTU #Hde0i -L -s
- elim (le_inv_plus_l ⦠Hde0i) -Hde0i #Hd0ie0 #He0i @frees_eq #X #HXT -K
+lemma frees_inv_lift_ge: âL,U,l,i. L ⢠i ϵ ð
*[l]â¦U⦠â
+ âK,s,l0,m0. â¬[s, l0, m0] L â¡ K â
+ âT. â¬[l0, m0] T â¡ U â l0 + m0 ⤠i â
+ K ⢠i-m0 ϵð
*[l-yinj m0]â¦Tâ¦.
+#L #U #l #i #H elim H -L -U -l -i
+[ #L #U #l #i #HnU #K #s #l0 #m0 #HLK #T #HTU #Hlm0i -L -s
+ elim (le_inv_plus_l ⦠Hlm0i) -Hlm0i #Hl0im0 #Hm0i @frees_eq #X #HXT -K
elim (lift_trans_le ⦠HXT ⦠HTU) -T // <plus_minus_m_m /2 width=2 by/
-| #I #L #K0 #U #W #d #i #j #Hdi #Hij #HnU #HLK0 #_ #IHW #K #s #d0 #e0 #HLK #T #HTU #Hde0i
- elim (lt_or_ge j d0) #H1
+| #I #L #K0 #U #W #l #i #j #Hli #Hij #HnU #HLK0 #_ #IHW #K #s #l0 #m0 #HLK #T #HTU #Hlm0i
+ elim (lt_or_ge j l0) #H1
[ elim (drop_conf_lt ⦠HLK ⦠HLK0) -L // #L0 #V #H #HKL0 #HVW
- elim (le_inv_plus_l ⦠Hde0i) #H0 #He0i
+ elim (le_inv_plus_l ⦠Hlm0i) #H0 #Hm0i
@(frees_be ⦠H) -H
[ /3 width=1 by yle_plus_dx1_trans, monotonic_yle_minus_dx/
| /2 width=3 by lt_to_le_to_lt/
@@ -139,15 +139,15 @@ lemma frees_inv_lift_ge: âL,U,d,i. L ⢠i ϵ ð
*[d]â¦U⦠â
| lapply (IHW ⦠HKL0 ⦠HVW ?) // -I -K -K0 -L0 -V -W -T -U -s
>minus_plus >minus_plus >plus_minus /2 width=1 by monotonic_le_minus_l/
]
- | elim (lt_or_ge j (d0+e0)) [ >commutative_plus |] #H2
+ | elim (lt_or_ge j (l0+m0)) [ >commutative_plus |] #H2
[ -L -I -W lapply (lt_plus_to_minus ???? H2) // -H2 #H2
- elim (lift_split ⦠HTU j (e0-1)) -HTU //
+ elim (lift_split ⦠HTU j (m0-1)) -HTU //
[ >minus_minus_associative /2 width=2 by ltn_to_ltO/ <minus_n_n
#X #_ #H elim (HnU ⦠H)
| >commutative_plus /3 width=1 by le_minus_to_plus, monotonic_pred/
]
| lapply (drop_conf_ge ⦠HLK ⦠HLK0 ?) // -L #HK0
- elim (le_inv_plus_l ⦠H2) -H2 #H2 #He0j
+ elim (le_inv_plus_l ⦠H2) -H2 #H2 #Hm0j
@(frees_be ⦠HK0)
[ /2 width=1 by monotonic_yle_minus_dx/
| /2 width=1 by monotonic_lt_minus_l/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_lreq.ma
new file mode 100644
index 000000000..606c95cfb
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/frees_lreq.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/substitution/drop_lreq.ma".
+include "basic_2/multiple/frees.ma".
+
+(* CONTEXT-SENSITIVE FREE VARIABLES *****************************************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_frees_trans: âL2,U,l,i. L2 ⢠i ϵ ð
*[l]â¦U⦠â
+ âL1. L1 ⩬[l, â] L2 â L1 ⢠i ϵ ð
*[l]â¦Uâ¦.
+#L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
+#I2 #L2 #K2 #U #W2 #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
+elim (lreq_drop_trans_be ⦠HL12 ⦠HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
+lapply (lreq_inv_O_Y ⦠HK12) -HK12 #H destruct /3 width=9 by frees_be/
+qed-.
+
+lemma frees_lreq_conf: âL1,U,l,i. L1 ⢠i ϵ ð
*[l]â¦U⦠â
+ âL2. L1 ⩬[l, â] L2 â L2 ⢠i ϵ ð
*[l]â¦Uâ¦.
+/3 width=3 by lreq_sym, lreq_frees_trans/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts.ma
index 8d3d9ccd5..d4a5a61cc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts.ma
@@ -20,8 +20,8 @@ include "basic_2/multiple/mr2_plus.ma".
inductive lifts: list2 nat nat â relation term â
| lifts_nil : âT. lifts (â) T T
-| lifts_cons: âT1,T,T2,des,d,e.
- â¬[d,e] T1 â¡ T â lifts des T T2 â lifts ({d, e} @ des) T1 T2
+| lifts_cons: âT1,T,T2,des,l,m.
+ â¬[l,m] T1 â¡ T â lifts des T T2 â lifts ({l, m} @ des) T1 T2
.
interpretation "generic relocation (term)"
@@ -31,30 +31,30 @@ interpretation "generic relocation (term)"
fact lifts_inv_nil_aux: âT1,T2,des. â¬*[des] T1 â¡ T2 â des = â â T1 = T2.
#T1 #T2 #des * -T1 -T2 -des //
-#T1 #T #T2 #d #e #des #_ #_ #H destruct
+#T1 #T #T2 #l #m #des #_ #_ #H destruct
qed-.
lemma lifts_inv_nil: âT1,T2. â¬*[â] T1 â¡ T2 â T1 = T2.
/2 width=3 by lifts_inv_nil_aux/ qed-.
fact lifts_inv_cons_aux: âT1,T2,des. â¬*[des] T1 â¡ T2 â
- âd,e,tl. des = {d, e} @ tl â
- ââT. â¬[d, e] T1 â¡ T & â¬*[tl] T â¡ T2.
+ âl,m,tl. des = {l, m} @ tl â
+ ââT. â¬[l, m] T1 â¡ T & â¬*[tl] T â¡ T2.
#T1 #T2 #des * -T1 -T2 -des
-[ #T #d #e #tl #H destruct
-| #T1 #T #T2 #des #d #e #HT1 #HT2 #hd #he #tl #H destruct
+[ #T #l #m #tl #H destruct
+| #T1 #T #T2 #des #l #m #HT1 #HT2 #l0 #m0 #tl #H destruct
/2 width=3 by ex2_intro/
qed-.
-lemma lifts_inv_cons: âT1,T2,d,e,des. â¬*[{d, e} @ des] T1 â¡ T2 â
- ââT. â¬[d, e] T1 â¡ T & â¬*[des] T â¡ T2.
+lemma lifts_inv_cons: âT1,T2,l,m,des. â¬*[{l, m} @ des] T1 â¡ T2 â
+ ââT. â¬[l, m] T1 â¡ T & â¬*[des] T â¡ T2.
/2 width=3 by lifts_inv_cons_aux/ qed-.
(* Basic_1: was: lift1_sort *)
lemma lifts_inv_sort1: âT2,k,des. â¬*[des] âk â¡ T2 â T2 = âk.
#T2 #k #des elim des -des
[ #H <(lifts_inv_nil ⦠H) -H //
-| #d #e #des #IH #H
+| #l #m #des #IH #H
elim (lifts_inv_cons ⦠H) -H #X #H
>(lift_inv_sort1 ⦠H) -H /2 width=1 by/
]
@@ -65,9 +65,9 @@ lemma lifts_inv_lref1: âT2,des,i1. â¬*[des] #i1 â¡ T2 â
ââi2. @â¦i1, des⦠⡠i2 & T2 = #i2.
#T2 #des elim des -des
[ #i1 #H <(lifts_inv_nil ⦠H) -H /2 width=3 by at_nil, ex2_intro/
-| #d #e #des #IH #i1 #H
+| #l #m #des #IH #i1 #H
elim (lifts_inv_cons ⦠H) -H #X #H1 #H2
- elim (lift_inv_lref1 ⦠H1) -H1 * #Hdi1 #H destruct
+ elim (lift_inv_lref1 ⦠H1) -H1 * #Hli1 #H destruct
elim (IH ⦠H2) -IH -H2 /3 width=3 by at_lt, at_ge, ex2_intro/
]
qed-.
@@ -75,7 +75,7 @@ qed-.
lemma lifts_inv_gref1: âT2,p,des. â¬*[des] §p â¡ T2 â T2 = §p.
#T2 #p #des elim des -des
[ #H <(lifts_inv_nil ⦠H) -H //
-| #d #e #des #IH #H
+| #l #m #des #IH #H
elim (lifts_inv_cons ⦠H) -H #X #H
>(lift_inv_gref1 ⦠H) -H /2 width=1 by/
]
@@ -88,10 +88,10 @@ lemma lifts_inv_bind1: âa,I,T2,des,V1,U1. â¬*[des] â{a,I} V1. U1 â¡ T2 â
#a #I #T2 #des elim des -des
[ #V1 #U1 #H
<(lifts_inv_nil ⦠H) -H /2 width=5 by ex3_2_intro, lifts_nil/
-| #d #e #des #IHdes #V1 #U1 #H
+| #l #m #des #IHcs #V1 #U1 #H
elim (lifts_inv_cons ⦠H) -H #X #H #HT2
elim (lift_inv_bind1 ⦠H) -H #V #U #HV1 #HU1 #H destruct
- elim (IHdes ⦠HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+ elim (IHcs ⦠HT2) -IHcs -HT2 #V2 #U2 #HV2 #HU2 #H destruct
/3 width=5 by ex3_2_intro, lifts_cons/
]
qed-.
@@ -103,10 +103,10 @@ lemma lifts_inv_flat1: âI,T2,des,V1,U1. â¬*[des] â{I} V1. U1 â¡ T2 â
#I #T2 #des elim des -des
[ #V1 #U1 #H
<(lifts_inv_nil ⦠H) -H /2 width=5 by ex3_2_intro, lifts_nil/
-| #d #e #des #IHdes #V1 #U1 #H
+| #l #m #des #IHcs #V1 #U1 #H
elim (lifts_inv_cons ⦠H) -H #X #H #HT2
elim (lift_inv_flat1 ⦠H) -H #V #U #HV1 #HU1 #H destruct
- elim (IHdes ⦠HT2) -IHdes -HT2 #V2 #U2 #HV2 #HU2 #H destruct
+ elim (IHcs ⦠HT2) -IHcs -HT2 #V2 #U2 #HV2 #HU2 #H destruct
/3 width=5 by ex3_2_intro, lifts_cons/
]
qed-.
@@ -128,7 +128,7 @@ lemma lifts_bind: âa,I,T2,V1,V2,des. â¬*[des] V1 â¡ V2 â
â¬*[des] â{a,I} V1. T1 â¡ â{a,I} V2. T2.
#a #I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
[ #V #T1 #H >(lifts_inv_nil ⦠H) -H //
-| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
+| #V1 #V #V2 #des #l #m #HV1 #_ #IHV #T1 #H
elim (lifts_inv_cons ⦠H) -H /3 width=3 by lift_bind, lifts_cons/
]
qed.
@@ -138,13 +138,13 @@ lemma lifts_flat: âI,T2,V1,V2,des. â¬*[des] V1 â¡ V2 â
â¬*[des] â{I} V1. T1 â¡ â{I} V2. T2.
#I #T2 #V1 #V2 #des #H elim H -V1 -V2 -des
[ #V #T1 #H >(lifts_inv_nil ⦠H) -H //
-| #V1 #V #V2 #des #d #e #HV1 #_ #IHV #T1 #H
+| #V1 #V #V2 #des #l #m #HV1 #_ #IHV #T1 #H
elim (lifts_inv_cons ⦠H) -H /3 width=3 by lift_flat, lifts_cons/
]
qed.
lemma lifts_total: âdes,T1. âT2. â¬*[des] T1 â¡ T2.
#des elim des -des /2 width=2 by lifts_nil, ex_intro/
-#d #e #des #IH #T1 elim (lift_total T1 d e)
+#l #m #des #IH #T1 elim (lift_total T1 l m)
#T #HT1 elim (IH T) -IH /3 width=4 by lifts_cons, ex_intro/
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lift.ma
index ae4801a5e..990878bee 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/lifts_lift.ma
@@ -25,7 +25,7 @@ lemma lifts_lift_trans_le: âT1,T,des. â¬*[des] T1 â¡ T â âT2. â¬[0, 1]
ââT0. â¬[0, 1] T1 â¡ T0 & â¬*[des + 1] T0 â¡ T2.
#T1 #T #des #H elim H -T1 -T -des
[ /2 width=3/
-| #T1 #T3 #T #des #d #e #HT13 #_ #IHT13 #T2 #HT2
+| #T1 #T3 #T #des #l #m #HT13 #_ #IHT13 #T2 #HT2
elim (IHT13 ⦠HT2) -T #T #HT3 #HT2
elim (lift_trans_le ⦠HT13 ⦠HT3) -T3 // /3 width=5/
]
@@ -43,17 +43,17 @@ lemma lifts_lift_trans: âdes,i,i0. @â¦i, des⦠⡠i0 â
lapply (minuss_inv_nil1 ⦠H2) -H2 #H
>(pluss_inv_nil2 ⦠H) in HT10; -des0 #H
>(lifts_inv_nil ⦠H) -T1 /2 width=3/
-| #d #e #des #IHdes #i #i0 #H1 #des0 #H2 #T1 #T0 #HT10 #T2 #HT02
- elim (at_inv_cons ⦠H1) -H1 * #Hid #Hi0
- [ elim (minuss_inv_cons1_lt ⦠H2) -H2 [2: /2 width=1/ ] #des1 #Hdes1 <minus_le_minus_minus_comm // <minus_plus_m_m #H
+| #l #m #des #IHcs #i #i0 #H1 #des0 #H2 #T1 #T0 #HT10 #T2 #HT02
+ elim (at_inv_cons ⦠H1) -H1 * #Hil #Hi0
+ [ elim (minuss_inv_cons1_lt ⦠H2) -H2 [2: /2 width=1/ ] #des1 #Hcs1 <minus_le_minus_minus_comm // <minus_plus_m_m #H
elim (pluss_inv_cons2 ⦠H) -H #des2 #H1 #H2 destruct
elim (lifts_inv_cons ⦠HT10) -HT10 #T >minus_plus #HT1 #HT0
- elim (IHdes ⦠Hi0 ⦠Hdes1 ⦠HT0 ⦠HT02) -IHdes -Hi0 -Hdes1 -T0 #T0 #HT0 #HT02
+ elim (IHcs ⦠Hi0 ⦠Hcs1 ⦠HT0 ⦠HT02) -IHcs -Hi0 -Hcs1 -T0 #T0 #HT0 #HT02
elim (lift_trans_le ⦠HT1 ⦠HT0) -T /2 width=1/ #T #HT1 <plus_minus_m_m /2 width=1/ /3 width=5/
| >commutative_plus in Hi0; #Hi0
- lapply (minuss_inv_cons1_ge ⦠H2 ?) -H2 [ /2 width=1/ ] <associative_plus #Hdes0
- elim (IHdes ⦠Hi0 ⦠Hdes0 ⦠HT10 ⦠HT02) -IHdes -Hi0 -Hdes0 -T0 #T0 #HT0 #HT02
- elim (lift_split ⦠HT0 d (i+1)) -HT0 /2 width=1/ /3 width=5/
+ lapply (minuss_inv_cons1_ge ⦠H2 ?) -H2 [ /2 width=1/ ] <associative_plus #Hcs0
+ elim (IHcs ⦠Hi0 ⦠Hcs0 ⦠HT10 ⦠HT02) -IHcs -Hi0 -Hcs0 -T0 #T0 #HT0 #HT02
+ elim (lift_split ⦠HT0 l (i+1)) -HT0 /2 width=1/ /3 width=5/
]
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq.ma
index 666d22eec..c9cd4c54c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq.ma
@@ -23,7 +23,7 @@ definition lleq: relation4 ynat term lenv lenv â llpx_sn ceq.
interpretation
"lazy equivalence (local environment)"
- 'LazyEq T d L1 L2 = (lleq d T L1 L2).
+ 'LazyEq T l L1 L2 = (lleq l 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.
@@ -31,122 +31,122 @@ definition lleq_transitive: predicate (relation3 lenv term term) â
(* Basic inversion lemmas ***************************************************)
lemma lleq_ind: âR:relation4 ynat term lenv lenv. (
- âL1,L2,d,k. |L1| = |L2| â R d (âk) L1 L2
+ âL1,L2,l,k. |L1| = |L2| â R l (âk) L1 L2
) â (
- âL1,L2,d,i. |L1| = |L2| â yinj i < d â R d (#i) L1 L2
+ âL1,L2,l,i. |L1| = |L2| â yinj i < l â R l (#i) L1 L2
) â (
- âI,L1,L2,K1,K2,V,d,i. d ⤠yinj i â
+ âI,L1,L2,K1,K2,V,l,i. l ⤠yinj i â
â¬[i] L1 â¡ K1.â{I}V â â¬[i] L2 â¡ K2.â{I}V â
- K1 â¡[V, yinj O] K2 â R (yinj O) V K1 K2 â R d (#i) L1 L2
+ K1 â¡[V, yinj O] K2 â R (yinj O) V K1 K2 â R l (#i) L1 L2
) â (
- âL1,L2,d,i. |L1| = |L2| â |L1| ⤠i â |L2| ⤠i â R d (#i) L1 L2
+ âL1,L2,l,i. |L1| = |L2| â |L1| ⤠i â |L2| ⤠i â R l (#i) L1 L2
) â (
- âL1,L2,d,p. |L1| = |L2| â R d (§p) L1 L2
+ âL1,L2,l,p. |L1| = |L2| â R l (§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
+ âa,I,L1,L2,V,T,l.
+ L1 â¡[V, l]L2 â L1.â{I}V â¡[T, ⫯l] L2.â{I}V â
+ R l V L1 L2 â R (⫯l) T (L1.â{I}V) (L2.â{I}V) â R l (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
+ âI,L1,L2,V,T,l.
+ L1 â¡[V, l]L2 â L1 â¡[T, l] L2 â
+ R l V L1 L2 â R l T L1 L2 â R l (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 H -L1 -L2 -T -d /2 width=8 by/
+ âl,T,L1,L2. L1 â¡[T, l] L2 â R l T L1 L2.
+#R #H1 #H2 #H3 #H4 #H5 #H6 #H7 #l #T #L1 #L2 #H elim H -L1 -L2 -T -l /2 width=8 by/
qed-.
-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.
+lemma lleq_inv_bind: âa,I,L1,L2,V,T,l. L1 â¡[â{a,I}V.T, l] L2 â
+ L1 â¡[V, l] L2 ⧠L1.â{I}V â¡[T, ⫯l] L2.â{I}V.
/2 width=2 by llpx_sn_inv_bind/ 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.
+lemma lleq_inv_flat: âI,L1,L2,V,T,l. L1 â¡[â{I}V.T, l] L2 â
+ L1 â¡[V, l] L2 ⧠L1 â¡[T, l] L2.
/2 width=2 by llpx_sn_inv_flat/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma lleq_fwd_length: âL1,L2,T,d. L1 â¡[T, d] L2 â |L1| = |L2|.
+lemma lleq_fwd_length: âL1,L2,T,l. L1 â¡[T, l] L2 â |L1| = |L2|.
/2 width=4 by llpx_sn_fwd_length/ qed-.
-lemma lleq_fwd_lref: âL1,L2,d,i. L1 â¡[#i, d] L2 â
+lemma lleq_fwd_lref: âL1,L2,l,i. L1 â¡[#i, l] L2 â
â¨â¨ |L1| ⤠i ⧠|L2| ⤠i
- | yinj i < d
+ | yinj i < l
| ââI,K1,K2,V. â¬[i] L1 â¡ K1.â{I}V &
â¬[i] L2 â¡ K2.â{I}V &
- K1 â¡[V, yinj 0] K2 & d ⤠yinj i.
-#L1 #L2 #d #i #H elim (llpx_sn_fwd_lref ⦠H) /2 width=1/
+ K1 â¡[V, yinj 0] K2 & l ⤠yinj i.
+#L1 #L2 #l #i #H elim (llpx_sn_fwd_lref ⦠H) /2 width=1/
* /3 width=7 by or3_intro2, ex4_4_intro/
qed-.
-lemma lleq_fwd_drop_sn: âL1,L2,T,d. L1 â¡[d, T] L2 â âK1,i. â¬[i] L1 â¡ K1 â
+lemma lleq_fwd_drop_sn: âL1,L2,T,l. L1 â¡[l, T] L2 â âK1,i. â¬[i] L1 â¡ K1 â
âK2. â¬[i] L2 â¡ K2.
/2 width=7 by llpx_sn_fwd_drop_sn/ qed-.
-lemma lleq_fwd_drop_dx: âL1,L2,T,d. L1 â¡[d, T] L2 â âK2,i. â¬[i] L2 â¡ K2 â
+lemma lleq_fwd_drop_dx: âL1,L2,T,l. L1 â¡[l, T] L2 â âK2,i. â¬[i] L2 â¡ K2 â
âK1. â¬[i] L1 â¡ K1.
/2 width=7 by llpx_sn_fwd_drop_dx/ 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.
+lemma lleq_fwd_bind_sn: âa,I,L1,L2,V,T,l.
+ L1 â¡[â{a,I}V.T, l] L2 â L1 â¡[V, l] L2.
/2 width=4 by llpx_sn_fwd_bind_sn/ 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.
+lemma lleq_fwd_bind_dx: âa,I,L1,L2,V,T,l.
+ L1 â¡[â{a,I}V.T, l] L2 â L1.â{I}V â¡[T, ⫯l] L2.â{I}V.
/2 width=2 by llpx_sn_fwd_bind_dx/ 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.
+lemma lleq_fwd_flat_sn: âI,L1,L2,V,T,l.
+ L1 â¡[â{I}V.T, l] L2 â L1 â¡[V, l] L2.
/2 width=3 by llpx_sn_fwd_flat_sn/ 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.
+lemma lleq_fwd_flat_dx: âI,L1,L2,V,T,l.
+ L1 â¡[â{I}V.T, l] L2 â L1 â¡[T, l] L2.
/2 width=3 by llpx_sn_fwd_flat_dx/ qed-.
(* Basic properties *********************************************************)
-lemma lleq_sort: âL1,L2,d,k. |L1| = |L2| â L1 â¡[âk, d] L2.
+lemma lleq_sort: âL1,L2,l,k. |L1| = |L2| â L1 â¡[âk, l] L2.
/2 width=1 by llpx_sn_sort/ qed.
-lemma lleq_skip: âL1,L2,d,i. yinj i < d â |L1| = |L2| â L1 â¡[#i, d] L2.
+lemma lleq_skip: âL1,L2,l,i. yinj i < l â |L1| = |L2| â L1 â¡[#i, l] L2.
/2 width=1 by llpx_sn_skip/ qed.
-lemma lleq_lref: âI,L1,L2,K1,K2,V,d,i. d ⤠yinj i â
+lemma lleq_lref: âI,L1,L2,K1,K2,V,l,i. l ⤠yinj i â
â¬[i] L1 â¡ K1.â{I}V â â¬[i] L2 â¡ K2.â{I}V â
- K1 â¡[V, 0] K2 â L1 â¡[#i, d] L2.
+ K1 â¡[V, 0] K2 â L1 â¡[#i, l] L2.
/2 width=9 by llpx_sn_lref/ qed.
-lemma lleq_free: âL1,L2,d,i. |L1| ⤠i â |L2| ⤠i â |L1| = |L2| â L1 â¡[#i, d] L2.
+lemma lleq_free: âL1,L2,l,i. |L1| ⤠i â |L2| ⤠i â |L1| = |L2| â L1 â¡[#i, l] L2.
/2 width=1 by llpx_sn_free/ qed.
-lemma lleq_gref: âL1,L2,d,p. |L1| = |L2| â L1 â¡[§p, d] L2.
+lemma lleq_gref: âL1,L2,l,p. |L1| = |L2| â L1 â¡[§p, l] L2.
/2 width=1 by llpx_sn_gref/ 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.
+lemma lleq_bind: âa,I,L1,L2,V,T,l.
+ L1 â¡[V, l] L2 â L1.â{I}V â¡[T, ⫯l] L2.â{I}V â
+ L1 â¡[â{a,I}V.T, l] L2.
/2 width=1 by llpx_sn_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.
+lemma lleq_flat: âI,L1,L2,V,T,l.
+ L1 â¡[V, l] L2 â L1 â¡[T, l] L2 â L1 â¡[â{I}V.T, l] L2.
/2 width=1 by llpx_sn_flat/ qed.
-lemma lleq_refl: âd,T. reflexive ⦠(lleq d T).
+lemma lleq_refl: âl,T. reflexive ⦠(lleq l T).
/2 width=1 by llpx_sn_refl/ qed.
lemma lleq_Y: âL1,L2,T. |L1| = |L2| â L1 â¡[T, â] L2.
/2 width=1 by llpx_sn_Y/ qed.
-lemma lleq_sym: âd,T. symmetric ⦠(lleq d T).
-#d #T #L1 #L2 #H @(lleq_ind ⦠H) -d -T -L1 -L2
+lemma lleq_sym: âl,T. symmetric ⦠(lleq l T).
+#l #T #L1 #L2 #H @(lleq_ind ⦠H) -l -T -L1 -L2
/2 width=7 by lleq_sort, lleq_skip, lleq_lref, lleq_free, lleq_gref, lleq_bind, lleq_flat/
qed-.
-lemma lleq_ge_up: âL1,L2,U,dt. L1 â¡[U, dt] L2 â
- âT,d,e. â¬[d, e] T â¡ U â
- dt ⤠d + e â L1 â¡[U, d] L2.
+lemma lleq_ge_up: âL1,L2,U,lt. L1 â¡[U, lt] L2 â
+ âT,l,m. â¬[l, m] T â¡ U â
+ lt ⤠l + m â L1 â¡[U, l] L2.
/2 width=6 by llpx_sn_ge_up/ qed-.
-lemma lleq_ge: âL1,L2,T,d1. L1 â¡[T, d1] L2 â âd2. d1 ⤠d2 â L1 â¡[T, d2] L2.
+lemma lleq_ge: âL1,L2,T,l1. L1 â¡[T, l1] L2 â âl2. l1 ⤠l2 â L1 â¡[T, l2] L2.
/2 width=3 by llpx_sn_ge/ 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
@@ -156,5 +156,5 @@ lemma lleq_bind_O: âa,I,L1,L2,V,T. L1 â¡[V, 0] L2 â L1.â{I}V â¡[T, 0] L2
(* Advanceded properties on lazy pointwise extensions ************************)
lemma llpx_sn_lrefl: âR. (âL. reflexive ⦠(R L)) â
- âL1,L2,T,d. L1 â¡[T, d] L2 â llpx_sn R d T L1 L2.
+ âL1,L2,T,l. L1 â¡[T, l] L2 â llpx_sn R l T L1 L2.
/2 width=3 by llpx_sn_co/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt.ma
index 2c6b8feb6..d2919f5b6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt.ma
@@ -19,23 +19,23 @@ include "basic_2/multiple/lleq.ma".
(* Alternative definition (not recursive) ***********************************)
-theorem lleq_intro_alt: âL1,L2,T,d. |L1| = |L2| â
- (âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â L1 ⢠i ϵ ð
*[d]â¦T⦠â
+theorem lleq_intro_alt: âL1,L2,T,l. |L1| = |L2| â
+ (âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â L1 ⢠i ϵ ð
*[l]â¦T⦠â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
I1 = I2 ⧠V1 = V2
- ) â L1 â¡[T, d] L2.
-#L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+ ) â L1 â¡[T, l] L2.
+#L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_inv_llpx_sn @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
@(IH ⦠HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
qed.
-theorem lleq_inv_alt: âL1,L2,T,d. L1 â¡[T, d] L2 â
+theorem lleq_inv_alt: âL1,L2,T,l. L1 â¡[T, l] L2 â
|L1| = |L2| â§
- âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â L1 ⢠i ϵ ð
*[d]â¦T⦠â
+ âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â L1 ⢠i ϵ ð
*[l]â¦T⦠â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
I1 = I2 ⧠V1 = V2.
-#L1 #L2 #T #d #H elim (llpx_sn_llpx_sn_alt ⦠H) -H
+#L1 #L2 #T #l #H elim (llpx_sn_llpx_sn_alt ⦠H) -H
#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
@(IH ⦠HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 //
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt_rec.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt_rec.ma
index 692ece149..440e0f510 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt_rec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_alt_rec.ma
@@ -19,36 +19,36 @@ include "basic_2/multiple/lleq.ma".
(* Alternative definition (recursive) ***************************************)
-theorem lleq_intro_alt_r: âL1,L2,T,d. |L1| = |L2| â
- (âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+theorem lleq_intro_alt_r: âL1,L2,T,l. |L1| = |L2| â
+ (âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
â§â§ I1 = I2 & V1 = V2 & K1 â¡[V1, 0] K2
- ) â L1 â¡[T, d] L2.
-#L1 #L2 #T #d #HL12 #IH @llpx_sn_intro_alt_r // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+ ) â L1 â¡[T, l] L2.
+#L1 #L2 #T #l #HL12 #IH @llpx_sn_intro_alt_r // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
elim (IH ⦠HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
qed.
theorem lleq_ind_alt_r: âS:relation4 ynat term lenv lenv.
- (âL1,L2,T,d. |L1| = |L2| â (
- âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+ (âL1,L2,T,l. |L1| = |L2| â (
+ âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
â§â§ I1 = I2 & V1 = V2 & K1 â¡[V1, 0] K2 & S 0 V1 K1 K2
- ) â S d T L1 L2) â
- âL1,L2,T,d. L1 â¡[T, d] L2 â S d T L1 L2.
-#S #IH1 #L1 #L2 #T #d #H @(llpx_sn_ind_alt_r ⦠H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+ ) â S l T L1 L2) â
+ âL1,L2,T,l. L1 â¡[T, l] L2 â S l T L1 L2.
+#S #IH1 #L1 #L2 #T #l #H @(llpx_sn_ind_alt_r ⦠H) -L1 -L2 -T -l
+#L1 #L2 #T #l #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
elim (IH2 ⦠HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /2 width=1 by and4_intro/
qed-.
-theorem lleq_inv_alt_r: âL1,L2,T,d. L1 â¡[T, d] L2 â
+theorem lleq_inv_alt_r: âL1,L2,T,l. L1 â¡[T, l] L2 â
|L1| = |L2| â§
- âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+ âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
â§â§ I1 = I2 & V1 = V2 & K1 â¡[V1, 0] K2.
-#L1 #L2 #T #d #H elim (llpx_sn_inv_alt_r ⦠H) -H
+#L1 #L2 #T #l #H elim (llpx_sn_inv_alt_r ⦠H) -H
#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
elim (IH ⦠HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_drop.ma
index 2a31e44dc..a2df46620 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_drop.ma
@@ -23,61 +23,61 @@ 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).
+lemma lleq_dec: âT,L1,L2,l. Decidable (L1 â¡[T, l] 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
+ âL1,L2,T,l. L1 â¡[T, l] L2 â
+ âL. llpx_sn R l T L2 L â llpx_sn R l T L1 L.
+#R #HR #L1 #L2 #T #l #H @(lleq_ind ⦠H) -L1 -L2 -T -l
[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
+| #I #L1 #L2 #K1 #K2 #V #l #i #Hli #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
+| #a #I #L1 #L2 #V #T #l #_ #_ #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
+| #I #L1 #L2 #V #T #l #_ #_ #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.
+ âL1,L2,T,l. L1 â¡[T, l] L2 â
+ âL. llpx_sn R l T L1 L â llpx_sn R l 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
+lemma lleq_inv_lref_ge_dx: âL1,L2,l,i. L1 â¡[#i, l] L2 â l ⤠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
+#L1 #L2 #l #i #H #Hli #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
+lemma lleq_inv_lref_ge_sn: âL1,L2,l,i. L1 â¡[#i, l] L2 â l ⤠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
+#L1 #L2 #l #i #H #Hli #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
+lemma lleq_inv_lref_ge_bi: âL1,L2,l,i. L1 â¡[#i, l] L2 â l ⤠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
+lemma lleq_inv_lref_ge: âL1,L2,l,i. L1 â¡[#i, l] L2 â l ⤠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
+#L1 #L2 #l #i #HL12 #Hli #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.
+lemma lleq_inv_S: âL1,L2,T,l. L1 â¡[T, l+1] L2 â
+ âI,K1,K2,V. â¬[l] L1 â¡ K1.â{I}V â â¬[l] L2 â¡ K2.â{I}V â
+ K1 â¡[V, 0] K2 â L1 â¡[T, l] 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
@@ -86,19 +86,19 @@ lemma lleq_inv_bind_O: âa,I,L1,L2,V,T. L1 â¡[â{a,I}V.T, 0] L2 â
(* Advanced forward lemmas **************************************************)
-lemma lleq_fwd_lref_dx: âL1,L2,d,i. L1 â¡[#i, d] L2 â
+lemma lleq_fwd_lref_dx: âL1,L2,l,i. L1 â¡[#i, l] 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
+ i < l â¨
+ ââK1. â¬[i] L1 â¡ K1.â{I}V & K1 â¡[V, 0] K2 & l ⤠i.
+#L1 #L2 #l #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
+lemma lleq_fwd_lref_sn: âL1,L2,l,i. L1 â¡[#i, l] 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
+ i < l â¨
+ ââK2. â¬[i] L2 â¡ K2.â{I}V & K1 â¡[V, 0] K2 & l ⤠i.
+#L1 #L2 #l #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-.
@@ -108,41 +108,41 @@ lemma lleq_fwd_bind_O_dx: âa,I,L1,L2,V,T. L1 â¡[â{a,I}V.T, 0] L2 â
(* 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.
+lemma lleq_lift_le: âK1,K2,T,lt. K1 â¡[T, lt] K2 â
+ âL1,L2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
+ âU. â¬[l, m] T â¡ U â lt ⤠l â L1 â¡[U, lt] 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.
+lemma lleq_lift_ge: âK1,K2,T,lt. K1 â¡[T, lt] K2 â
+ âL1,L2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
+ âU. â¬[l, m] T â¡ U â l ⤠lt â L1 â¡[U, lt+m] 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.
+lemma lleq_inv_lift_le: âL1,L2,U,lt. L1 â¡[U, lt] L2 â
+ âK1,K2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
+ âT. â¬[l, m] T â¡ U â lt ⤠l â K1 â¡[T, lt] 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.
+lemma lleq_inv_lift_be: âL1,L2,U,lt. L1 â¡[U, lt] L2 â
+ âK1,K2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
+ âT. â¬[l, m] T â¡ U â l ⤠lt â lt ⤠yinj l + m â K1 â¡[T, l] 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.
+lemma lleq_inv_lift_ge: âL1,L2,U,lt. L1 â¡[U, lt] L2 â
+ âK1,K2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
+ âT. â¬[l, m] T â¡ U â yinj l + m ⤠lt â K1 â¡[T, lt-m] 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¥).
+lemma nlleq_inv_bind: âa,I,L1,L2,V,T,l. (L1 â¡[â{a,I}V.T, l] L2 â â¥) â
+ (L1 â¡[V, l] L2 â â¥) ⨠(L1.â{I}V â¡[T, ⫯l] 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¥).
+lemma nlleq_inv_flat: âI,L1,L2,V,T,l. (L1 â¡[â{I}V.T, l] L2 â â¥) â
+ (L1 â¡[V, l] L2 â â¥) ⨠(L1 â¡[T, l] 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
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_fqus.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_fqus.ma
index eea2d3486..28bb175ed 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_fqus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_fqus.ma
@@ -35,8 +35,8 @@ lemma lleq_fqu_trans: âG1,G2,L2,K2,T,U. â¦G1, L2, T⦠â â¦G2, K2, Uâ¦
#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 (drop_O1_le (â») (e+1) L1)
+| #G #L2 #K2 #T #U #m #HLK2 #HTU #L1 #HL12
+ elim (drop_O1_le (â») (m+1) L1)
[ /3 width=12 by fqu_drop, lleq_inv_lift_le, ex2_intro/
| lapply (drop_fwd_length_le2 ⦠HLK2) -K2
lapply (lleq_fwd_length ⦠HL12) -T -U //
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_leq.ma
deleted file mode 100644
index e23a1521f..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_leq.ma
+++ /dev/null
@@ -1,36 +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/multiple/llpx_sn_leq.ma".
-include "basic_2/multiple/lleq.ma".
-
-(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
-
-(* Properties on equivalence for local environments *************************)
-
-lemma leq_lleq_trans: âL2,L,T,d. L2 â¡[T, d] L â
- âL1. L1 ⩬[d, â] L2 â L1 â¡[T, d] L.
-/2 width=3 by leq_llpx_sn_trans/ qed-.
-
-lemma lleq_leq_trans: âL,L1,T,d. L â¡[T, d] L1 â
- âL2. L1 ⩬[d, â] L2 â L â¡[T, d] L2.
-/2 width=3 by llpx_sn_leq_trans/ qed-.
-
-lemma lleq_leq_repl: âL1,L2,T,d. L1 â¡[T, d] L2 â âK1. K1 ⩬[d, â] L1 â
- âK2. L2 ⩬[d, â] K2 â K1 â¡[T, d] K2.
-/2 width=5 by llpx_sn_leq_repl/ 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.
-/2 width=5 by llpx_sn_bind_repl_SO/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lleq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lleq.ma
index d07e13fce..630b41a32 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lleq.ma
@@ -18,22 +18,22 @@ include "basic_2/multiple/lleq_drop.ma".
(* Main properties **********************************************************)
-theorem lleq_trans: âd,T. Transitive ⦠(lleq d T).
+theorem lleq_trans: âl,T. Transitive ⦠(lleq l 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.
+theorem lleq_canc_sn: âL,L1,L2,T,l. L â¡[l, T] L1â L â¡[l, T] L2 â L1 â¡[l, 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.
+theorem lleq_canc_dx: âL1,L2,L,T,l. L1 â¡[l, T] L â L2 â¡[l, T] L â L1 â¡[l, T] L2.
/3 width=3 by lleq_trans, lleq_sym/ qed-.
(* Advanced properies on negated lazy equivalence *****************************)
(* Note: for use in auto, works with /4 width=8/ so lleq_canc_sn is preferred *)
-lemma lleq_nlleq_trans: âd,T,L1,L. L1 â¡[T, d] L â
- âL2. (L â¡[T, d] L2 â â¥) â (L1 â¡[T, d] L2 â â¥).
+lemma lleq_nlleq_trans: âl,T,L1,L. L1 â¡[T, l] L â
+ âL2. (L â¡[T, l] L2 â â¥) â (L1 â¡[T, l] L2 â â¥).
/3 width=3 by lleq_canc_sn/ qed-.
-lemma nlleq_lleq_div: âd,T,L2,L. L2 â¡[T, d] L â
- âL1. (L1 â¡[T, d] L â â¥) â (L1 â¡[T, d] L2 â â¥).
+lemma nlleq_lleq_div: âl,T,L2,L. L2 â¡[T, l] L â
+ âL1. (L1 â¡[T, l] L â â¥) â (L1 â¡[T, l] L2 â â¥).
/3 width=3 by lleq_trans/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_llor.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_llor.ma
index 1cfaa4013..06893aad6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_llor.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_llor.ma
@@ -21,8 +21,8 @@ include "basic_2/multiple/lleq_alt.ma".
(* Properties on pointwise union for local environments **********************)
lemma llpx_sn_llor_dx: âR. (s_r_confluent1 ⦠R (llpx_sn R 0)) â (frees_trans R) â
- âL1,L2,T,d. llpx_sn R d T L1 L2 â âL. L1 â[T, d] L2 â¡ L â L2 â¡[T, d] L.
-#R #H1R #H2R #L1 #L2 #T #d #H1 #L #H2
+ âL1,L2,T,l. llpx_sn R l T L1 L2 â âL. L1 â[T, l] L2 â¡ L â L2 â¡[T, l] L.
+#R #H1R #H2R #L1 #L2 #T #l #H1 #L #H2
lapply (llpx_sn_frees_trans ⦠H1R H2R ⦠H1) -H1R -H2R #HR
elim (llpx_sn_llpx_sn_alt ⦠H1) -H1 #HL12 #IH1
elim H2 -H2 #_ #HL1 #IH2
@@ -37,5 +37,5 @@ elim (IH2 ⦠HLK1 HLK2 HLK) -IH2 -HLK1 -HLK2 -HLK * /2 width=1 by conj/ #H
qed.
lemma llpx_sn_llor_dx_sym: âR. (s_r_confluent1 ⦠R (llpx_sn R 0)) â (frees_trans R) â
- âL1,L2,T,d. llpx_sn R d T L1 L2 â âL. L1 â[T, d] L2 â¡ L â L â¡[T, d] L2.
+ âL1,L2,T,l. llpx_sn R l T L1 L2 â âL. L1 â[T, l] L2 â¡ L â L â¡[T, l] L2.
/3 width=6 by llpx_sn_llor_dx, lleq_sym/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lreq.ma
new file mode 100644
index 000000000..8fa21a5da
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/lleq_lreq.ma
@@ -0,0 +1,36 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/multiple/llpx_sn_lreq.ma".
+include "basic_2/multiple/lleq.ma".
+
+(* LAZY EQUIVALENCE FOR LOCAL ENVIRONMENTS **********************************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_lleq_trans: âL2,L,T,l. L2 â¡[T, l] L â
+ âL1. L1 ⩬[l, â] L2 â L1 â¡[T, l] L.
+/2 width=3 by lreq_llpx_sn_trans/ qed-.
+
+lemma lleq_lreq_trans: âL,L1,T,l. L â¡[T, l] L1 â
+ âL2. L1 ⩬[l, â] L2 â L â¡[T, l] L2.
+/2 width=3 by llpx_sn_lreq_trans/ qed-.
+
+lemma lleq_lreq_repl: âL1,L2,T,l. L1 â¡[T, l] L2 â âK1. K1 ⩬[l, â] L1 â
+ âK2. L2 ⩬[l, â] K2 â K1 â¡[T, l] K2.
+/2 width=5 by llpx_sn_lreq_repl/ 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.
+/2 width=5 by llpx_sn_bind_repl_SO/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llor.ma
index 29d4c1d51..a2d1e61a7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llor.ma
@@ -17,24 +17,24 @@ include "basic_2/multiple/frees.ma".
(* POINTWISE UNION FOR LOCAL ENVIRONMENTS ***********************************)
-definition llor: ynat â relation4 term lenv lenv lenv â λd,T,L2,L1,L.
+definition llor: ynat â relation4 term lenv lenv lenv â λl,T,L2,L1,L.
â§â§ |L1| = |L2| & |L1| = |L|
& (âI1,I2,I,K1,K2,K,V1,V2,V,i.
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â â¬[i] L â¡ K.â{I}V â â¨â¨
- (â§â§ yinj i < d & I1 = I & V1 = V) |
- (â§â§ (L1 ⢠i ϵ ð
*[d]â¦T⦠â â¥) & I1 = I & V1 = V) |
- (â§â§ d ⤠yinj i & L1 ⢠i ϵ ð
*[d]â¦T⦠& I2 = I & V2 = V)
+ (â§â§ yinj i < l & I1 = I & V1 = V) |
+ (â§â§ (L1 ⢠i ϵ ð
*[l]â¦T⦠â â¥) & I1 = I & V1 = V) |
+ (â§â§ l ⤠yinj i & L1 ⢠i ϵ ð
*[l]â¦T⦠& I2 = I & V2 = V)
).
interpretation
"lazy union (local environment)"
- 'LazyOr L1 T d L2 L = (llor d T L2 L1 L).
+ 'LazyOr L1 T l L2 L = (llor l T L2 L1 L).
(* Basic properties *********************************************************)
(* Note: this can be proved by llor_skip *)
-lemma llor_atom: âT,d. â â[T, d] â â¡ â.
-#T #d @and3_intro //
+lemma llor_atom: âT,l. â â[T, l] â â¡ â.
+#T #l @and3_intro //
#I1 #I2 #I #K1 #K2 #K #V1 #V2 #V #i #HLK1
elim (drop_inv_atom1 ⦠HLK1) -HLK1 #H destruct
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_alt.ma
index 721d18c94..dd27e6a05 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_alt.ma
@@ -19,10 +19,10 @@ include "basic_2/multiple/llor.ma".
(* Alternative definition ***************************************************)
-lemma llor_tail_frees: âL1,L2,L,U,d. L1 â[U, d] L2 â¡ L â d ⤠yinj (|L1|) â
- âI1,W1. â{I1}W1.L1 ⢠|L1| ϵ ð
*[d]â¦U⦠â
- âI2,W2. â{I1}W1.L1 â[U, d] â{I2}W2.L2 â¡ â{I2}W2.L.
-#L1 #L2 #L #U #d * #HL12 #HL1 #IH #Hd #I1 #W1 #HU #I2 #W2
+lemma llor_tail_frees: âL1,L2,L,U,l. L1 â[U, l] L2 â¡ L â l ⤠yinj (|L1|) â
+ âI1,W1. â{I1}W1.L1 ⢠|L1| ϵ ð
*[l]â¦U⦠â
+ âI2,W2. â{I1}W1.L1 â[U, l] â{I2}W2.L2 â¡ â{I2}W2.L.
+#L1 #L2 #L #U #l * #HL12 #HL1 #IH #Hl #I1 #W1 #HU #I2 #W2
@and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ]
#J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK
lapply (drop_fwd_length_lt2 ⦠HLK1) >ltail_length #H
@@ -46,10 +46,10 @@ elim (le_to_or_lt_eq ⦠H) -H #H
]
qed.
-lemma llor_tail_cofrees: âL1,L2,L,U,d. L1 â[U, d] L2 â¡ L â
- âI1,W1. (â{I1}W1.L1 ⢠|L1| ϵ ð
*[d]â¦U⦠â â¥) â
- âI2,W2. â{I1}W1.L1 â[U, d] â{I2}W2.L2 â¡ â{I1}W1.L.
-#L1 #L2 #L #U #d * #HL12 #HL1 #IH #I1 #W1 #HU #I2 #W2
+lemma llor_tail_cofrees: âL1,L2,L,U,l. L1 â[U, l] L2 â¡ L â
+ âI1,W1. (â{I1}W1.L1 ⢠|L1| ϵ ð
*[l]â¦U⦠â â¥) â
+ âI2,W2. â{I1}W1.L1 â[U, l] â{I2}W2.L2 â¡ â{I1}W1.L.
+#L1 #L2 #L #U #l * #HL12 #HL1 #IH #I1 #W1 #HU #I2 #W2
@and3_intro [1,2: >ltail_length /2 width=1 by le_S_S/ ]
#J1 #J2 #J #K1 #K2 #K #V1 #V2 #V #i #HLK1 #HLK2 #HLK
lapply (drop_fwd_length_lt2 ⦠HLK1) >ltail_length #H
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_drop.ma
index 8692a91f1..cbeab4a59 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llor_drop.ma
@@ -19,8 +19,8 @@ include "basic_2/multiple/llor_alt.ma".
(* Advanced properties ******************************************************)
-lemma llor_skip: âL1,L2,U,d. |L1| = |L2| â yinj (|L1|) ⤠d â L1 â[U, d] L2 â¡ L1.
-#L1 #L2 #U #d #HL12 #Hd @and3_intro // -HL12
+lemma llor_skip: âL1,L2,U,l. |L1| = |L2| â yinj (|L1|) ⤠l â L1 â[U, l] L2 â¡ L1.
+#L1 #L2 #U #l #HL12 #Hl @and3_intro // -HL12
#I1 #I2 #I #K1 #K2 #K #W1 #W2 #W #i #HLK1 #_ #HLK -L2 -K2
lapply (drop_mono ⦠HLK ⦠HLK1) -HLK #H destruct
lapply (drop_fwd_length_lt2 ⦠HLK1) -K1
@@ -28,14 +28,14 @@ lapply (drop_fwd_length_lt2 ⦠HLK1) -K1
qed.
(* Note: lemma 1400 concludes the "big tree" theorem *)
-lemma llor_total: âL1,L2,T,d. |L1| = |L2| â âL. L1 â[T, d] L2 â¡ L.
+lemma llor_total: âL1,L2,T,l. |L1| = |L2| â âL. L1 â[T, l] L2 â¡ L.
#L1 @(lenv_ind_alt ⦠L1) -L1
-[ #L2 #T #d #H >(length_inv_zero_sn ⦠H) -L2 /2 width=2 by ex_intro/
-| #I1 #L1 #V1 #IHL1 #Y #T #d >ltail_length #H
+[ #L2 #T #l #H >(length_inv_zero_sn ⦠H) -L2 /2 width=2 by ex_intro/
+| #I1 #L1 #V1 #IHL1 #Y #T #l >ltail_length #H
elim (length_inv_pos_sn_ltail ⦠H) -H #I2 #L2 #V2 #HL12 #H destruct
- elim (ylt_split d (|â{I1}V1.L1|))
- [ elim (frees_dec (â{I1}V1.L1) T d (|L1|)) #HnU
- elim (IHL1 L2 T d) // -IHL1 -HL12
+ elim (ylt_split l (|â{I1}V1.L1|))
+ [ elim (frees_dec (â{I1}V1.L1) T l (|L1|)) #HnU
+ elim (IHL1 L2 T l) // -IHL1 -HL12
[ #L #HL12 >ltail_length /4 width=2 by llor_tail_frees, ylt_fwd_succ2, ex_intro/
| /4 width=2 by llor_tail_cofrees, ex_intro/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn.ma
index c78445bef..f75d1a4a9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn.ma
@@ -18,140 +18,140 @@ include "basic_2/substitution/drop.ma".
(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
inductive llpx_sn (R:relation3 lenv term term): relation4 ynat term lenv lenv â
-| llpx_sn_sort: âL1,L2,d,k. |L1| = |L2| â llpx_sn R d (âk) L1 L2
-| llpx_sn_skip: âL1,L2,d,i. |L1| = |L2| â yinj i < d â llpx_sn R d (#i) L1 L2
-| llpx_sn_lref: âI,L1,L2,K1,K2,V1,V2,d,i. d ⤠yinj i â
+| llpx_sn_sort: âL1,L2,l,k. |L1| = |L2| â llpx_sn R l (âk) L1 L2
+| llpx_sn_skip: âL1,L2,l,i. |L1| = |L2| â yinj i < l â llpx_sn R l (#i) L1 L2
+| llpx_sn_lref: âI,L1,L2,K1,K2,V1,V2,l,i. l ⤠yinj i â
â¬[i] L1 â¡ K1.â{I}V1 â â¬[i] L2 â¡ K2.â{I}V2 â
- llpx_sn R (yinj 0) V1 K1 K2 â R K1 V1 V2 â llpx_sn R d (#i) L1 L2
-| llpx_sn_free: âL1,L2,d,i. |L1| ⤠i â |L2| ⤠i â |L1| = |L2| â llpx_sn R d (#i) L1 L2
-| llpx_sn_gref: âL1,L2,d,p. |L1| = |L2| â llpx_sn R d (§p) L1 L2
-| llpx_sn_bind: âa,I,L1,L2,V,T,d.
- llpx_sn R d V L1 L2 â llpx_sn R (⫯d) T (L1.â{I}V) (L2.â{I}V) â
- llpx_sn R d (â{a,I}V.T) L1 L2
-| llpx_sn_flat: âI,L1,L2,V,T,d.
- llpx_sn R d V L1 L2 â llpx_sn R d T L1 L2 â llpx_sn R d (â{I}V.T) L1 L2
+ llpx_sn R (yinj 0) V1 K1 K2 â R K1 V1 V2 â llpx_sn R l (#i) L1 L2
+| llpx_sn_free: âL1,L2,l,i. |L1| ⤠i â |L2| ⤠i â |L1| = |L2| â llpx_sn R l (#i) L1 L2
+| llpx_sn_gref: âL1,L2,l,p. |L1| = |L2| â llpx_sn R l (§p) L1 L2
+| llpx_sn_bind: âa,I,L1,L2,V,T,l.
+ llpx_sn R l V L1 L2 â llpx_sn R (⫯l) T (L1.â{I}V) (L2.â{I}V) â
+ llpx_sn R l (â{a,I}V.T) L1 L2
+| llpx_sn_flat: âI,L1,L2,V,T,l.
+ llpx_sn R l V L1 L2 â llpx_sn R l T L1 L2 â llpx_sn R l (â{I}V.T) L1 L2
.
(* Basic inversion lemmas ***************************************************)
-fact llpx_sn_inv_bind_aux: âR,L1,L2,X,d. llpx_sn R d X L1 L2 â
+fact llpx_sn_inv_bind_aux: âR,L1,L2,X,l. llpx_sn R l X L1 L2 â
âa,I,V,T. X = â{a,I}V.T â
- llpx_sn R d V L1 L2 ⧠llpx_sn R (⫯d) T (L1.â{I}V) (L2.â{I}V).
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-[ #L1 #L2 #d #k #_ #b #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #b #J #W #U #H destruct
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #_ #_ #_ #_ #b #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #_ #b #J #W #U #H destruct
-| #L1 #L2 #d #p #_ #b #J #W #U #H destruct
-| #a #I #L1 #L2 #V #T #d #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/
-| #I #L1 #L2 #V #T #d #_ #_ #b #J #W #U #H destruct
+ llpx_sn R l V L1 L2 ⧠llpx_sn R (⫯l) T (L1.â{I}V) (L2.â{I}V).
+#R #L1 #L2 #X #l * -L1 -L2 -X -l
+[ #L1 #L2 #l #k #_ #b #J #W #U #H destruct
+| #L1 #L2 #l #i #_ #_ #b #J #W #U #H destruct
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #_ #_ #_ #_ #b #J #W #U #H destruct
+| #L1 #L2 #l #i #_ #_ #_ #b #J #W #U #H destruct
+| #L1 #L2 #l #p #_ #b #J #W #U #H destruct
+| #a #I #L1 #L2 #V #T #l #HV #HT #b #J #W #U #H destruct /2 width=1 by conj/
+| #I #L1 #L2 #V #T #l #_ #_ #b #J #W #U #H destruct
]
qed-.
-lemma llpx_sn_inv_bind: âR,a,I,L1,L2,V,T,d. llpx_sn R d (â{a,I}V.T) L1 L2 â
- llpx_sn R d V L1 L2 ⧠llpx_sn R (⫯d) T (L1.â{I}V) (L2.â{I}V).
+lemma llpx_sn_inv_bind: âR,a,I,L1,L2,V,T,l. llpx_sn R l (â{a,I}V.T) L1 L2 â
+ llpx_sn R l V L1 L2 ⧠llpx_sn R (⫯l) T (L1.â{I}V) (L2.â{I}V).
/2 width=4 by llpx_sn_inv_bind_aux/ qed-.
-fact llpx_sn_inv_flat_aux: âR,L1,L2,X,d. llpx_sn R d X L1 L2 â
+fact llpx_sn_inv_flat_aux: âR,L1,L2,X,l. llpx_sn R l X L1 L2 â
âI,V,T. X = â{I}V.T â
- llpx_sn R d V L1 L2 ⧠llpx_sn R d T L1 L2.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-[ #L1 #L2 #d #k #_ #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #J #W #U #H destruct
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #_ #_ #_ #_ #J #W #U #H destruct
-| #L1 #L2 #d #i #_ #_ #_ #J #W #U #H destruct
-| #L1 #L2 #d #p #_ #J #W #U #H destruct
-| #a #I #L1 #L2 #V #T #d #_ #_ #J #W #U #H destruct
-| #I #L1 #L2 #V #T #d #HV #HT #J #W #U #H destruct /2 width=1 by conj/
+ llpx_sn R l V L1 L2 ⧠llpx_sn R l T L1 L2.
+#R #L1 #L2 #X #l * -L1 -L2 -X -l
+[ #L1 #L2 #l #k #_ #J #W #U #H destruct
+| #L1 #L2 #l #i #_ #_ #J #W #U #H destruct
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #_ #_ #_ #_ #J #W #U #H destruct
+| #L1 #L2 #l #i #_ #_ #_ #J #W #U #H destruct
+| #L1 #L2 #l #p #_ #J #W #U #H destruct
+| #a #I #L1 #L2 #V #T #l #_ #_ #J #W #U #H destruct
+| #I #L1 #L2 #V #T #l #HV #HT #J #W #U #H destruct /2 width=1 by conj/
]
qed-.
-lemma llpx_sn_inv_flat: âR,I,L1,L2,V,T,d. llpx_sn R d (â{I}V.T) L1 L2 â
- llpx_sn R d V L1 L2 ⧠llpx_sn R d T L1 L2.
+lemma llpx_sn_inv_flat: âR,I,L1,L2,V,T,l. llpx_sn R l (â{I}V.T) L1 L2 â
+ llpx_sn R l V L1 L2 ⧠llpx_sn R l T L1 L2.
/2 width=4 by llpx_sn_inv_flat_aux/ qed-.
(* Basic forward lemmas *****************************************************)
-lemma llpx_sn_fwd_length: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â |L1| = |L2|.
-#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d //
-#I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #_ #HLK1 #HLK2 #_ #_ #HK12
+lemma llpx_sn_fwd_length: âR,L1,L2,T,l. llpx_sn R l T L1 L2 â |L1| = |L2|.
+#R #L1 #L2 #T #l #H elim H -L1 -L2 -T -l //
+#I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #_ #HLK1 #HLK2 #_ #_ #HK12
lapply (drop_fwd_length ⦠HLK1) -HLK1
lapply (drop_fwd_length ⦠HLK2) -HLK2
normalize //
qed-.
-lemma llpx_sn_fwd_drop_sn: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â
+lemma llpx_sn_fwd_drop_sn: âR,L1,L2,T,l. llpx_sn R l T L1 L2 â
âK1,i. â¬[i] L1 â¡ K1 â âK2. â¬[i] L2 â¡ K2.
-#R #L1 #L2 #T #d #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length ⦠H) -H
+#R #L1 #L2 #T #l #H #K1 #i #HLK1 lapply (llpx_sn_fwd_length ⦠H) -H
#HL12 lapply (drop_fwd_length_le2 ⦠HLK1) -HLK1 /2 width=1 by drop_O1_le/
qed-.
-lemma llpx_sn_fwd_drop_dx: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â
+lemma llpx_sn_fwd_drop_dx: âR,L1,L2,T,l. llpx_sn R l T L1 L2 â
âK2,i. â¬[i] L2 â¡ K2 â âK1. â¬[i] L1 â¡ K1.
-#R #L1 #L2 #T #d #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length ⦠H) -H
+#R #L1 #L2 #T #l #H #K2 #i #HLK2 lapply (llpx_sn_fwd_length ⦠H) -H
#HL12 lapply (drop_fwd_length_le2 ⦠HLK2) -HLK2 /2 width=1 by drop_O1_le/
qed-.
-fact llpx_sn_fwd_lref_aux: âR,L1,L2,X,d. llpx_sn R d X L1 L2 â âi. X = #i â
+fact llpx_sn_fwd_lref_aux: âR,L1,L2,X,l. llpx_sn R l X L1 L2 â âi. X = #i â
â¨â¨ |L1| ⤠i ⧠|L2| ⤠i
- | yinj i < d
+ | yinj i < l
| ââI,K1,K2,V1,V2. â¬[i] L1 â¡ K1.â{I}V1 &
â¬[i] L2 â¡ K2.â{I}V2 &
llpx_sn R (yinj 0) V1 K1 K2 &
- R K1 V1 V2 & d ⤠yinj i.
-#R #L1 #L2 #X #d * -L1 -L2 -X -d
-[ #L1 #L2 #d #k #_ #j #H destruct
-| #L1 #L2 #d #i #_ #Hid #j #H destruct /2 width=1 by or3_intro1/
-| #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 #j #H destruct
+ R K1 V1 V2 & l ⤠yinj i.
+#R #L1 #L2 #X #l * -L1 -L2 -X -l
+[ #L1 #L2 #l #k #_ #j #H destruct
+| #L1 #L2 #l #i #_ #Hil #j #H destruct /2 width=1 by or3_intro1/
+| #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #Hli #HLK1 #HLK2 #HK12 #HV12 #j #H destruct
/3 width=9 by or3_intro2, ex5_5_intro/
-| #L1 #L2 #d #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or3_intro0, conj/
-| #L1 #L2 #d #p #_ #j #H destruct
-| #a #I #L1 #L2 #V #T #d #_ #_ #j #H destruct
-| #I #L1 #L2 #V #T #d #_ #_ #j #H destruct
+| #L1 #L2 #l #i #HL1 #HL2 #_ #j #H destruct /3 width=1 by or3_intro0, conj/
+| #L1 #L2 #l #p #_ #j #H destruct
+| #a #I #L1 #L2 #V #T #l #_ #_ #j #H destruct
+| #I #L1 #L2 #V #T #l #_ #_ #j #H destruct
]
qed-.
-lemma llpx_sn_fwd_lref: âR,L1,L2,d,i. llpx_sn R d (#i) L1 L2 â
+lemma llpx_sn_fwd_lref: âR,L1,L2,l,i. llpx_sn R l (#i) L1 L2 â
â¨â¨ |L1| ⤠i ⧠|L2| ⤠i
- | yinj i < d
+ | yinj i < l
| ââI,K1,K2,V1,V2. â¬[i] L1 â¡ K1.â{I}V1 &
â¬[i] L2 â¡ K2.â{I}V2 &
llpx_sn R (yinj 0) V1 K1 K2 &
- R K1 V1 V2 & d ⤠yinj i.
+ R K1 V1 V2 & l ⤠yinj i.
/2 width=3 by llpx_sn_fwd_lref_aux/ qed-.
-lemma llpx_sn_fwd_bind_sn: âR,a,I,L1,L2,V,T,d. llpx_sn R d (â{a,I}V.T) L1 L2 â
- llpx_sn R d V L1 L2.
-#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind ⦠H) -H //
+lemma llpx_sn_fwd_bind_sn: âR,a,I,L1,L2,V,T,l. llpx_sn R l (â{a,I}V.T) L1 L2 â
+ llpx_sn R l V L1 L2.
+#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind ⦠H) -H //
qed-.
-lemma llpx_sn_fwd_bind_dx: âR,a,I,L1,L2,V,T,d. llpx_sn R d (â{a,I}V.T) L1 L2 â
- llpx_sn R (⫯d) T (L1.â{I}V) (L2.â{I}V).
-#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_bind ⦠H) -H //
+lemma llpx_sn_fwd_bind_dx: âR,a,I,L1,L2,V,T,l. llpx_sn R l (â{a,I}V.T) L1 L2 â
+ llpx_sn R (⫯l) T (L1.â{I}V) (L2.â{I}V).
+#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_bind ⦠H) -H //
qed-.
-lemma llpx_sn_fwd_flat_sn: âR,I,L1,L2,V,T,d. llpx_sn R d (â{I}V.T) L1 L2 â
- llpx_sn R d V L1 L2.
-#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat ⦠H) -H //
+lemma llpx_sn_fwd_flat_sn: âR,I,L1,L2,V,T,l. llpx_sn R l (â{I}V.T) L1 L2 â
+ llpx_sn R l V L1 L2.
+#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_flat ⦠H) -H //
qed-.
-lemma llpx_sn_fwd_flat_dx: âR,I,L1,L2,V,T,d. llpx_sn R d (â{I}V.T) L1 L2 â
- llpx_sn R d T L1 L2.
-#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_inv_flat ⦠H) -H //
+lemma llpx_sn_fwd_flat_dx: âR,I,L1,L2,V,T,l. llpx_sn R l (â{I}V.T) L1 L2 â
+ llpx_sn R l T L1 L2.
+#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_inv_flat ⦠H) -H //
qed-.
-lemma llpx_sn_fwd_pair_sn: âR,I,L1,L2,V,T,d. llpx_sn R d (â¡{I}V.T) L1 L2 â
- llpx_sn R d V L1 L2.
+lemma llpx_sn_fwd_pair_sn: âR,I,L1,L2,V,T,l. llpx_sn R l (â¡{I}V.T) L1 L2 â
+ llpx_sn R l V L1 L2.
#R * /2 width=4 by llpx_sn_fwd_flat_sn, llpx_sn_fwd_bind_sn/
qed-.
(* Basic properties *********************************************************)
-lemma llpx_sn_refl: âR. (âL. reflexive ⦠(R L)) â âT,L,d. llpx_sn R d T L L.
+lemma llpx_sn_refl: âR. (âL. reflexive ⦠(R L)) â âT,L,l. llpx_sn R l T L L.
#R #HR #T #L @(f2_ind ⦠rfw ⦠L T) -L -T
#n #IH #L * * /3 width=1 by llpx_sn_sort, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
#i #Hn elim (lt_or_ge i (|L|)) /2 width=1 by llpx_sn_free/
-#HiL #d elim (ylt_split i d) /2 width=1 by llpx_sn_skip/
+#HiL #l elim (ylt_split i l) /2 width=1 by llpx_sn_skip/
elim (drop_O1_lt ⦠HiL) -HiL destruct /4 width=9 by llpx_sn_lref, drop_fwd_rfw/
qed-.
@@ -163,37 +163,37 @@ lemma llpx_sn_Y: âR,T,L1,L2. |L1| = |L2| â llpx_sn R (â) T L1 L2.
@IH -IH // normalize /2 width=1 by eq_f2/
qed-.
-lemma llpx_sn_ge_up: âR,L1,L2,U,dt. llpx_sn R dt U L1 L2 â âT,d,e. â¬[d, e] T â¡ U â
- dt ⤠d + e â llpx_sn R d U L1 L2.
-#R #L1 #L2 #U #dt #H elim H -L1 -L2 -U -dt
-[ #L1 #L2 #dt #k #HL12 #X #d #e #H #_ >(lift_inv_sort2 ⦠H) -H /2 width=1 by llpx_sn_sort/
-| #L1 #L2 #dt #i #HL12 #Hidt #X #d #e #H #Hdtde
- elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12
- elim (ylt_yle_false ⦠Hidt) -Hidt
- @(yle_trans ⦠Hdtde) /2 width=1 by yle_inj/ (**) (* full auto too slow 11s *)
-| #I #L1 #L2 #K1 #K2 #W1 #W2 #dt #i #Hdti #HLK1 #HLK2 #HW1 #HW12 #_ #X #d #e #H #_
- elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct
+lemma llpx_sn_ge_up: âR,L1,L2,U,lt. llpx_sn R lt U L1 L2 â âT,l,m. â¬[l, m] T â¡ U â
+ lt ⤠l + m â llpx_sn R l U L1 L2.
+#R #L1 #L2 #U #lt #H elim H -L1 -L2 -U -lt
+[ #L1 #L2 #lt #k #HL12 #X #l #m #H #_ >(lift_inv_sort2 ⦠H) -H /2 width=1 by llpx_sn_sort/
+| #L1 #L2 #lt #i #HL12 #Hilt #X #l #m #H #Hltlm
+ elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct /3 width=1 by llpx_sn_skip, ylt_inj/ -HL12
+ elim (ylt_yle_false ⦠Hilt) -Hilt
+ @(yle_trans ⦠Hltlm) /2 width=1 by yle_inj/ (**) (* full auto too slow 11s *)
+| #I #L1 #L2 #K1 #K2 #W1 #W2 #lt #i #Hlti #HLK1 #HLK2 #HW1 #HW12 #_ #X #l #m #H #_
+ elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct
[ lapply (llpx_sn_fwd_length ⦠HW1) -HW1 #HK12
lapply (drop_fwd_length ⦠HLK1) lapply (drop_fwd_length ⦠HLK2)
- normalize in ⢠(%â%â?); -I -W1 -W2 -dt /3 width=1 by llpx_sn_skip, ylt_inj/
+ normalize in ⢠(%â%â?); -I -W1 -W2 -lt /3 width=1 by llpx_sn_skip, ylt_inj/
| /4 width=9 by llpx_sn_lref, yle_inj, le_plus_b/
]
| /2 width=1 by llpx_sn_free/
-| #L1 #L2 #dt #p #HL12 #X #d #e #H #_ >(lift_inv_gref2 ⦠H) -H /2 width=1 by llpx_sn_gref/
-| #a #I #L1 #L2 #W #U #dt #_ #_ #IHV #IHT #X #d #e #H #Hdtde destruct
+| #L1 #L2 #lt #p #HL12 #X #l #m #H #_ >(lift_inv_gref2 ⦠H) -H /2 width=1 by llpx_sn_gref/
+| #a #I #L1 #L2 #W #U #lt #_ #_ #IHV #IHT #X #l #m #H #Hltlm destruct
elim (lift_inv_bind2 ⦠H) -H #V #T #HVW >commutative_plus #HTU #H destruct
@(llpx_sn_bind) /2 width=4 by/ (**) (* full auto fails *)
@(IHT ⦠HTU) /2 width=1 by yle_succ/
-| #I #L1 #L2 #W #U #dt #_ #_ #IHV #IHT #X #d #e #H #Hdtde destruct
+| #I #L1 #L2 #W #U #lt #_ #_ #IHV #IHT #X #l #m #H #Hltlm destruct
elim (lift_inv_flat2 ⦠H) -H #HVW #HTU #H destruct
/3 width=4 by llpx_sn_flat/
]
qed-.
(**) (* the minor premise comes first *)
-lemma llpx_sn_ge: âR,L1,L2,T,d1,d2. d1 ⤠d2 â
- llpx_sn R d1 T L1 L2 â llpx_sn R d2 T L1 L2.
-#R #L1 #L2 #T #d1 #d2 * -d1 -d2 (**) (* destructed yle *)
+lemma llpx_sn_ge: âR,L1,L2,T,l1,l2. l1 ⤠l2 â
+ llpx_sn R l1 T L1 L2 â llpx_sn R l2 T L1 L2.
+#R #L1 #L2 #T #l1 #l2 * -l1 -l2 (**) (* destructed yle *)
/3 width=6 by llpx_sn_ge_up, llpx_sn_Y, llpx_sn_fwd_length, yle_inj/
qed-.
@@ -203,7 +203,7 @@ lemma llpx_sn_bind_O: âR,a,I,L1,L2,V,T. llpx_sn R 0 V L1 L2 â
/3 width=3 by llpx_sn_ge, llpx_sn_bind/ qed-.
lemma llpx_sn_co: âR1,R2. (âL,T1,T2. R1 L T1 T2 â R2 L T1 T2) â
- âL1,L2,T,d. llpx_sn R1 d T L1 L2 â llpx_sn R2 d T L1 L2.
-#R1 #R2 #HR12 #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+ âL1,L2,T,l. llpx_sn R1 l T L1 L2 â llpx_sn R2 l T L1 L2.
+#R1 #R2 #HR12 #L1 #L2 #T #l #H elim H -L1 -L2 -T -l
/3 width=9 by llpx_sn_sort, llpx_sn_skip, llpx_sn_lref, llpx_sn_free, llpx_sn_gref, llpx_sn_bind, llpx_sn_flat/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt.ma
index 956ae4699..ee49dc20c 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt.ma
@@ -19,21 +19,21 @@ include "basic_2/multiple/llpx_sn_alt_rec.ma".
(* alternative definition of llpx_sn (not recursive) *)
definition llpx_sn_alt: relation3 lenv term term â relation4 ynat term lenv lenv â
- λR,d,T,L1,L2. |L1| = |L2| â§
- (âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â L1 ⢠i ϵ ð
*[d]â¦T⦠â
+ λR,l,T,L1,L2. |L1| = |L2| â§
+ (âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â L1 ⢠i ϵ ð
*[l]â¦T⦠â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
I1 = I2 ⧠R K1 V1 V2
).
(* Main properties **********************************************************)
-theorem llpx_sn_llpx_sn_alt: âR,T,L1,L2,d. llpx_sn R d T L1 L2 â llpx_sn_alt R d T L1 L2.
+theorem llpx_sn_llpx_sn_alt: âR,T,L1,L2,l. llpx_sn R l T L1 L2 â llpx_sn_alt R l T L1 L2.
#R #U #L1 @(f2_ind ⦠rfw ⦠L1 U) -L1 -U
-#n #IHn #L1 #U #Hn #L2 #d #H elim (llpx_sn_inv_alt_r ⦠H) -H
+#n #IHn #L1 #U #Hn #L2 #l #H elim (llpx_sn_inv_alt_r ⦠H) -H
#HL12 #IHU @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2 elim (frees_inv ⦠H) -H
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2 elim (frees_inv ⦠H) -H
[ -n #HnU elim (IHU ⦠HnU HLK1 HLK2) -IHU -HnU -HLK1 -HLK2 /2 width=1 by conj/
-| * #J1 #K10 #W10 #j #Hdj #Hji #HnU #HLK10 #HnW10 destruct
+| * #J1 #K10 #W10 #j #Hlj #Hji #HnU #HLK10 #HnW10 destruct
lapply (drop_fwd_drop2 ⦠HLK10) #H
lapply (drop_conf_ge ⦠H ⦠HLK1 ?) -H /2 width=1 by lt_to_le/ <minus_plus #HK10
elim (drop_O1_lt (â») L2 j) [2: <HL12 /2 width=5 by drop_fwd_length_lt2/ ] #J2 #K20 #W20 #HLK20
@@ -44,10 +44,10 @@ theorem llpx_sn_llpx_sn_alt: âR,T,L1,L2,d. llpx_sn R d T L1 L2 â llpx_sn_alt
]
qed.
-theorem llpx_sn_alt_inv_llpx_sn: âR,T,L1,L2,d. llpx_sn_alt R d T L1 L2 â llpx_sn R d T L1 L2.
+theorem llpx_sn_alt_inv_llpx_sn: âR,T,L1,L2,l. llpx_sn_alt R l T L1 L2 â llpx_sn R l T L1 L2.
#R #U #L1 @(f2_ind ⦠rfw ⦠L1 U) -L1 -U
-#n #IHn #L1 #U #Hn #L2 #d * #HL12 #IHU @llpx_sn_intro_alt_r //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnU #HLK1 #HLK2 destruct
+#n #IHn #L1 #U #Hn #L2 #l * #HL12 #IHU @llpx_sn_intro_alt_r //
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #HnU #HLK1 #HLK2 destruct
elim (IHU ⦠HLK1 HLK2) /3 width=2 by frees_eq/
#H #HV12 @and3_intro // @IHn -IHn /3 width=6 by drop_fwd_rfw/
lapply (drop_fwd_drop2 ⦠HLK1) #H1
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt_rec.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt_rec.ma
index d428e5900..0f38faaa8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt_rec.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_alt_rec.ma
@@ -20,70 +20,70 @@ include "basic_2/multiple/llpx_sn.ma".
(* alternative definition of llpx_sn (recursive) *)
inductive llpx_sn_alt_r (R:relation3 lenv term term): relation4 ynat term lenv lenv â
-| llpx_sn_alt_r_intro: âL1,L2,T,d.
- (âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+| llpx_sn_alt_r_intro: âL1,L2,T,l.
+ (âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â I1 = I2 ⧠R K1 V1 V2
) â
- (âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+ (âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â llpx_sn_alt_r R 0 V1 K1 K2
- ) â |L1| = |L2| â llpx_sn_alt_r R d T L1 L2
+ ) â |L1| = |L2| â llpx_sn_alt_r R l T L1 L2
.
(* Compact definition of llpx_sn_alt_r **************************************)
-lemma llpx_sn_alt_r_intro_alt: âR,L1,L2,T,d. |L1| = |L2| â
- (âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+lemma llpx_sn_alt_r_intro_alt: âR,L1,L2,T,l. |L1| = |L2| â
+ (âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
â§â§ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2
- ) â llpx_sn_alt_r R d T L1 L2.
-#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_intro // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+ ) â llpx_sn_alt_r R l T L1 L2.
+#R #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_r_intro // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
elim (IH ⦠HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by conj/
qed.
lemma llpx_sn_alt_r_ind_alt: âR. âS:relation4 ynat term lenv lenv.
- (âL1,L2,T,d. |L1| = |L2| â (
- âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+ (âL1,L2,T,l. |L1| = |L2| â (
+ âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
â§â§ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2 & S 0 V1 K1 K2
- ) â S d T L1 L2) â
- âL1,L2,T,d. llpx_sn_alt_r R d T L1 L2 â S d T L1 L2.
-#R #S #IH #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
-#L1 #L2 #T #d #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+ ) â S l T L1 L2) â
+ âL1,L2,T,l. llpx_sn_alt_r R l T L1 L2 â S l T L1 L2.
+#R #S #IH #L1 #L2 #T #l #H elim H -L1 -L2 -T -l
+#L1 #L2 #T #l #H1 #H2 #HL12 #IH2 @IH -IH // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
elim (H1 ⦠HnT HLK1 HLK2) -H1 /4 width=8 by and4_intro/
qed-.
-lemma llpx_sn_alt_r_inv_alt: âR,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 â
+lemma llpx_sn_alt_r_inv_alt: âR,L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 â
|L1| = |L2| â§
- âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+ âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
â§â§ I1 = I2 & R K1 V1 V2 & llpx_sn_alt_r R 0 V1 K1 K2.
-#R #L1 #L2 #T #d #H @(llpx_sn_alt_r_ind_alt ⦠H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH @conj // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+#R #L1 #L2 #T #l #H @(llpx_sn_alt_r_ind_alt ⦠H) -L1 -L2 -T -l
+#L1 #L2 #T #l #HL12 #IH @conj // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
elim (IH ⦠HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /2 width=1 by and3_intro/
qed-.
(* Basic inversion lemmas ***************************************************)
-lemma llpx_sn_alt_r_inv_flat: âR,I,L1,L2,V,T,d. llpx_sn_alt_r R d (â{I}V.T) L1 L2 â
- llpx_sn_alt_r R d V L1 L2 ⧠llpx_sn_alt_r R d T L1 L2.
-#R #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt ⦠H) -H
+lemma llpx_sn_alt_r_inv_flat: âR,I,L1,L2,V,T,l. llpx_sn_alt_r R l (â{I}V.T) L1 L2 â
+ llpx_sn_alt_r R l V L1 L2 ⧠llpx_sn_alt_r R l T L1 L2.
+#R #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt ⦠H) -H
#HL12 #IH @conj @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2
elim (IH ⦠HLK1 HLK2) -IH -HLK1 -HLK2 //
/3 width=8 by nlift_flat_sn, nlift_flat_dx, and3_intro/
qed-.
-lemma llpx_sn_alt_r_inv_bind: âR,a,I,L1,L2,V,T,d. llpx_sn_alt_r R d (â{a,I}V.T) L1 L2 â
- llpx_sn_alt_r R d V L1 L2 ⧠llpx_sn_alt_r R (⫯d) T (L1.â{I}V) (L2.â{I}V).
-#R #a #I #L1 #L2 #V #T #d #H elim (llpx_sn_alt_r_inv_alt ⦠H) -H
+lemma llpx_sn_alt_r_inv_bind: âR,a,I,L1,L2,V,T,l. llpx_sn_alt_r R l (â{a,I}V.T) L1 L2 â
+ llpx_sn_alt_r R l V L1 L2 ⧠llpx_sn_alt_r R (⫯l) T (L1.â{I}V) (L2.â{I}V).
+#R #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_alt_r_inv_alt ⦠H) -H
#HL12 #IH @conj @llpx_sn_alt_r_intro_alt [1,3: normalize // ] -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #H #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #H #HLK1 #HLK2
[ elim (IH ⦠HLK1 HLK2) -IH -HLK1 -HLK2
/3 width=9 by nlift_bind_sn, and3_intro/
-| lapply (yle_inv_succ1 ⦠Hdi) -Hdi * #Hdi #Hi
+| lapply (yle_inv_succ1 ⦠Hli) -Hli * #Hli #Hi
lapply (drop_inv_drop1_lt ⦠HLK1 ?) -HLK1 /2 width=1 by ylt_O/ #HLK1
lapply (drop_inv_drop1_lt ⦠HLK2 ?) -HLK2 /2 width=1 by ylt_O/ #HLK2
elim (IH ⦠HLK1 HLK2) -IH -HLK1 -HLK2 /2 width=1 by and3_intro/
@@ -93,21 +93,21 @@ qed-.
(* Basic forward lemmas *****************************************************)
-lemma llpx_sn_alt_r_fwd_length: âR,L1,L2,T,d. llpx_sn_alt_r R d T L1 L2 â |L1| = |L2|.
-#R #L1 #L2 #T #d #H elim (llpx_sn_alt_r_inv_alt ⦠H) -H //
+lemma llpx_sn_alt_r_fwd_length: âR,L1,L2,T,l. llpx_sn_alt_r R l T L1 L2 â |L1| = |L2|.
+#R #L1 #L2 #T #l #H elim (llpx_sn_alt_r_inv_alt ⦠H) -H //
qed-.
-lemma llpx_sn_alt_r_fwd_lref: âR,L1,L2,d,i. llpx_sn_alt_r R d (#i) L1 L2 â
+lemma llpx_sn_alt_r_fwd_lref: âR,L1,L2,l,i. llpx_sn_alt_r R l (#i) L1 L2 â
â¨â¨ |L1| ⤠i ⧠|L2| ⤠i
- | yinj i < d
+ | yinj i < l
| ââI,K1,K2,V1,V2. â¬[i] L1 â¡ K1.â{I}V1 &
â¬[i] L2 â¡ K2.â{I}V2 &
llpx_sn_alt_r R (yinj 0) V1 K1 K2 &
- R K1 V1 V2 & d ⤠yinj i.
-#R #L1 #L2 #d #i #H elim (llpx_sn_alt_r_inv_alt ⦠H) -H
+ R K1 V1 V2 & l ⤠yinj i.
+#R #L1 #L2 #l #i #H elim (llpx_sn_alt_r_inv_alt ⦠H) -H
#HL12 #IH elim (lt_or_ge i (|L1|)) /3 width=1 by or3_intro0, conj/
-elim (ylt_split i d) /3 width=1 by or3_intro1/
-#Hdi #HL1 elim (drop_O1_lt (â») ⦠HL1)
+elim (ylt_split i l) /3 width=1 by or3_intro1/
+#Hli #HL1 elim (drop_O1_lt (â») ⦠HL1)
#I1 #K1 #V1 #HLK1 elim (drop_O1_lt (â») L2 i) //
#I2 #K2 #V2 #HLK2 elim (IH ⦠HLK1 HLK2) -IH
/3 width=9 by nlift_lref_be_SO, or3_intro2, ex5_5_intro/
@@ -115,38 +115,38 @@ qed-.
(* Basic properties *********************************************************)
-lemma llpx_sn_alt_r_sort: âR,L1,L2,d,k. |L1| = |L2| â llpx_sn_alt_r R d (âk) L1 L2.
-#R #L1 #L2 #d #k #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+lemma llpx_sn_alt_r_sort: âR,L1,L2,l,k. |L1| = |L2| â llpx_sn_alt_r R l (âk) L1 L2.
+#R #L1 #L2 #l #k #HL12 @llpx_sn_alt_r_intro_alt // -HL12
#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (âk)) //
qed.
-lemma llpx_sn_alt_r_gref: âR,L1,L2,d,p. |L1| = |L2| â llpx_sn_alt_r R d (§p) L1 L2.
-#R #L1 #L2 #d #p #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+lemma llpx_sn_alt_r_gref: âR,L1,L2,l,p. |L1| = |L2| â llpx_sn_alt_r R l (§p) L1 L2.
+#R #L1 #L2 #l #p #HL12 @llpx_sn_alt_r_intro_alt // -HL12
#I1 #I2 #K1 #K2 #V1 #V2 #i #_ #H elim (H (§p)) //
qed.
-lemma llpx_sn_alt_r_skip: âR,L1,L2,d,i. |L1| = |L2| â yinj i < d â llpx_sn_alt_r R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL12 #Hid @llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #j #Hdj #H elim (H (#i)) -H
+lemma llpx_sn_alt_r_skip: âR,L1,L2,l,i. |L1| = |L2| â yinj i < l â llpx_sn_alt_r R l (#i) L1 L2.
+#R #L1 #L2 #l #i #HL12 #Hil @llpx_sn_alt_r_intro_alt // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #j #Hlj #H elim (H (#i)) -H
/4 width=3 by lift_lref_lt, ylt_yle_trans, ylt_inv_inj/
qed.
-lemma llpx_sn_alt_r_free: âR,L1,L2,d,i. |L1| ⤠i â |L2| ⤠i â |L1| = |L2| â
- llpx_sn_alt_r R d (#i) L1 L2.
-#R #L1 #L2 #d #i #HL1 #_ #HL12 @llpx_sn_alt_r_intro_alt // -HL12
+lemma llpx_sn_alt_r_free: âR,L1,L2,l,i. |L1| ⤠i â |L2| ⤠i â |L1| = |L2| â
+ llpx_sn_alt_r R l (#i) L1 L2.
+#R #L1 #L2 #l #i #HL1 #_ #HL12 @llpx_sn_alt_r_intro_alt // -HL12
#I1 #I2 #K1 #K2 #V1 #V2 #j #_ #H #HLK1 elim (H (#(i-1))) -H
lapply (drop_fwd_length_lt2 ⦠HLK1) -HLK1
/3 width=3 by lift_lref_ge_minus, lt_to_le_to_lt/
qed.
-lemma llpx_sn_alt_r_lref: âR,I,L1,L2,K1,K2,V1,V2,d,i. d ⤠yinj i â
+lemma llpx_sn_alt_r_lref: âR,I,L1,L2,K1,K2,V1,V2,l,i. l ⤠yinj i â
â¬[i] L1 â¡ K1.â{I}V1 â â¬[i] L2 â¡ K2.â{I}V2 â
llpx_sn_alt_r R 0 V1 K1 K2 â R K1 V1 V2 â
- llpx_sn_alt_r R d (#i) L1 L2.
-#R #I #L1 #L2 #K1 #K2 #V1 #V2 #d #i #Hdi #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_r_intro_alt
+ llpx_sn_alt_r R l (#i) L1 L2.
+#R #I #L1 #L2 #K1 #K2 #V1 #V2 #l #i #Hli #HLK1 #HLK2 #HK12 #HV12 @llpx_sn_alt_r_intro_alt
[ lapply (llpx_sn_alt_r_fwd_length ⦠HK12) -HK12 #HK12
@(drop_fwd_length_eq2 ⦠HLK1 HLK2) normalize //
-| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hdj #H #HLY1 #HLY2
+| #Z1 #Z2 #Y1 #Y2 #X1 #X2 #j #Hlj #H #HLY1 #HLY2
elim (lt_or_eq_or_gt i j) #Hij destruct
[ elim (H (#i)) -H /2 width=1 by lift_lref_lt/
| lapply (drop_mono ⦠HLY1 ⦠HLK1) -HLY1 -HLK1 #H destruct
@@ -156,29 +156,29 @@ lemma llpx_sn_alt_r_lref: âR,I,L1,L2,K1,K2,V1,V2,d,i. d ⤠yinj i â
]
qed.
-lemma llpx_sn_alt_r_flat: âR,I,L1,L2,V,T,d.
- llpx_sn_alt_r R d V L1 L2 â llpx_sn_alt_r R d T L1 L2 â
- llpx_sn_alt_r R d (â{I}V.T) L1 L2.
-#R #I #L1 #L2 #V #T #d #HV #HT
+lemma llpx_sn_alt_r_flat: âR,I,L1,L2,V,T,l.
+ llpx_sn_alt_r R l V L1 L2 â llpx_sn_alt_r R l T L1 L2 â
+ llpx_sn_alt_r R l (â{I}V.T) L1 L2.
+#R #I #L1 #L2 #V #T #l #HV #HT
elim (llpx_sn_alt_r_inv_alt ⦠HV) -HV #HL12 #IHV
elim (llpx_sn_alt_r_inv_alt ⦠HT) -HT #_ #IHT
@llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #HnVT #HLK1 #HLK2
elim (nlift_inv_flat ⦠HnVT) -HnVT #H
[ elim (IHV ⦠HLK1 ⦠HLK2) -IHV /2 width=2 by and3_intro/
| elim (IHT ⦠HLK1 ⦠HLK2) -IHT /3 width=2 by and3_intro/
]
qed.
-lemma llpx_sn_alt_r_bind: âR,a,I,L1,L2,V,T,d.
- llpx_sn_alt_r R d V L1 L2 â
- llpx_sn_alt_r R (⫯d) T (L1.â{I}V) (L2.â{I}V) â
- llpx_sn_alt_r R d (â{a,I}V.T) L1 L2.
-#R #a #I #L1 #L2 #V #T #d #HV #HT
+lemma llpx_sn_alt_r_bind: âR,a,I,L1,L2,V,T,l.
+ llpx_sn_alt_r R l V L1 L2 â
+ llpx_sn_alt_r R (⫯l) T (L1.â{I}V) (L2.â{I}V) â
+ llpx_sn_alt_r R l (â{a,I}V.T) L1 L2.
+#R #a #I #L1 #L2 #V #T #l #HV #HT
elim (llpx_sn_alt_r_inv_alt ⦠HV) -HV #HL12 #IHV
elim (llpx_sn_alt_r_inv_alt ⦠HT) -HT #_ #IHT
@llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hdi #HnVT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hli #HnVT #HLK1 #HLK2
elim (nlift_inv_bind ⦠HnVT) -HnVT #H
[ elim (IHV ⦠HLK1 ⦠HLK2) -IHV /2 width=2 by and3_intro/
| elim IHT -IHT /2 width=12 by drop_drop, yle_succ, and3_intro/
@@ -187,64 +187,64 @@ qed.
(* Main properties **********************************************************)
-theorem llpx_sn_lpx_sn_alt_r: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â llpx_sn_alt_r R d T L1 L2.
-#R #L1 #L2 #T #d #H elim H -L1 -L2 -T -d
+theorem llpx_sn_lpx_sn_alt_r: âR,L1,L2,T,l. llpx_sn R l T L1 L2 â llpx_sn_alt_r R l T L1 L2.
+#R #L1 #L2 #T #l #H elim H -L1 -L2 -T -l
/2 width=9 by llpx_sn_alt_r_sort, llpx_sn_alt_r_gref, llpx_sn_alt_r_skip, llpx_sn_alt_r_free, llpx_sn_alt_r_lref, llpx_sn_alt_r_flat, llpx_sn_alt_r_bind/
qed.
(* Main inversion lemmas ****************************************************)
-theorem llpx_sn_alt_r_inv_lpx_sn: âR,T,L1,L2,d. llpx_sn_alt_r R d T L1 L2 â llpx_sn R d T L1 L2.
+theorem llpx_sn_alt_r_inv_lpx_sn: âR,T,L1,L2,l. llpx_sn_alt_r R l T L1 L2 â llpx_sn R l T L1 L2.
#R #T #L1 @(f2_ind ⦠rfw ⦠L1 T) -L1 -T #n #IH #L1 * *
[1,3: /3 width=4 by llpx_sn_alt_r_fwd_length, llpx_sn_gref, llpx_sn_sort/
-| #i #Hn #L2 #d #H lapply (llpx_sn_alt_r_fwd_length ⦠H)
+| #i #Hn #L2 #l #H lapply (llpx_sn_alt_r_fwd_length ⦠H)
#HL12 elim (llpx_sn_alt_r_fwd_lref ⦠H) -H
[ * /2 width=1 by llpx_sn_free/
| /2 width=1 by llpx_sn_skip/
| * /4 width=9 by llpx_sn_lref, drop_fwd_rfw/
]
-| #a #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_bind ⦠H) -H
+| #a #I #V #T #Hn #L2 #l #H elim (llpx_sn_alt_r_inv_bind ⦠H) -H
/3 width=1 by llpx_sn_bind/
-| #I #V #T #Hn #L2 #d #H elim (llpx_sn_alt_r_inv_flat ⦠H) -H
+| #I #V #T #Hn #L2 #l #H elim (llpx_sn_alt_r_inv_flat ⦠H) -H
/3 width=1 by llpx_sn_flat/
]
qed-.
(* Alternative definition of llpx_sn (recursive) ****************************)
-lemma llpx_sn_intro_alt_r: âR,L1,L2,T,d. |L1| = |L2| â
- (âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+lemma llpx_sn_intro_alt_r: âR,L1,L2,T,l. |L1| = |L2| â
+ (âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
â§â§ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2
- ) â llpx_sn R d T L1 L2.
-#R #L1 #L2 #T #d #HL12 #IH @llpx_sn_alt_r_inv_lpx_sn
+ ) â llpx_sn R l T L1 L2.
+#R #L1 #L2 #T #l #HL12 #IH @llpx_sn_alt_r_inv_lpx_sn
@llpx_sn_alt_r_intro_alt // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
elim (IH ⦠HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_lpx_sn_alt_r, and3_intro/
qed.
lemma llpx_sn_ind_alt_r: âR. âS:relation4 ynat term lenv lenv.
- (âL1,L2,T,d. |L1| = |L2| â (
- âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+ (âL1,L2,T,l. |L1| = |L2| â (
+ âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
â§â§ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2 & S 0 V1 K1 K2
- ) â S d T L1 L2) â
- âL1,L2,T,d. llpx_sn R d T L1 L2 â S d T L1 L2.
-#R #S #IH1 #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r ⦠H) -H
-#H @(llpx_sn_alt_r_ind_alt ⦠H) -L1 -L2 -T -d
-#L1 #L2 #T #d #HL12 #IH2 @IH1 -IH1 // -HL12
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+ ) â S l T L1 L2) â
+ âL1,L2,T,l. llpx_sn R l T L1 L2 â S l T L1 L2.
+#R #S #IH1 #L1 #L2 #T #l #H lapply (llpx_sn_lpx_sn_alt_r ⦠H) -H
+#H @(llpx_sn_alt_r_ind_alt ⦠H) -L1 -L2 -T -l
+#L1 #L2 #T #l #HL12 #IH2 @IH1 -IH1 // -HL12
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
elim (IH2 ⦠HnT HLK1 HLK2) -IH2 -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and4_intro/
qed-.
-lemma llpx_sn_inv_alt_r: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â
+lemma llpx_sn_inv_alt_r: âR,L1,L2,T,l. llpx_sn R l T L1 L2 â
|L1| = |L2| â§
- âI1,I2,K1,K2,V1,V2,i. d ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
+ âI1,I2,K1,K2,V1,V2,i. l ⤠yinj i â (âU. â¬[i, 1] U â¡ T â â¥) â
â¬[i] L1 â¡ K1.â{I1}V1 â â¬[i] L2 â¡ K2.â{I2}V2 â
â§â§ I1 = I2 & R K1 V1 V2 & llpx_sn R 0 V1 K1 K2.
-#R #L1 #L2 #T #d #H lapply (llpx_sn_lpx_sn_alt_r ⦠H) -H
+#R #L1 #L2 #T #l #H lapply (llpx_sn_lpx_sn_alt_r ⦠H) -H
#H elim (llpx_sn_alt_r_inv_alt ⦠H) -H
#HL12 #IH @conj //
-#I1 #I2 #K1 #K2 #V1 #V2 #i #Hid #HnT #HLK1 #HLK2
+#I1 #I2 #K1 #K2 #V1 #V2 #i #Hil #HnT #HLK1 #HLK2
elim (IH ⦠HnT HLK1 HLK2) -IH -HnT -HLK1 -HLK2 /3 width=1 by llpx_sn_alt_r_inv_lpx_sn, and3_intro/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_drop.ma
index cc21b8b2e..de36874c9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_drop.ma
@@ -13,39 +13,39 @@
(**************************************************************************)
include "basic_2/substitution/drop_drop.ma".
-include "basic_2/multiple/llpx_sn_leq.ma".
+include "basic_2/multiple/llpx_sn_lreq.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
+lemma llpx_sn_fwd_lref_dx: âR,L1,L2,l,i. llpx_sn R l (#i) L1 L2 â
âI,K2,V2. â¬[i] L2 â¡ K2.â{I}V2 â
- i < d â¨
+ i < l 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 [ * || * ]
+ R K1 V1 V2 & l ⤠i.
+#R #L1 #L2 #l #i #H #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref ⦠H) -H [ * || * ]
[ #_ #H elim (lt_refl_false i)
lapply (drop_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
+| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hli
lapply (drop_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
+lemma llpx_sn_fwd_lref_sn: âR,L1,L2,l,i. llpx_sn R l (#i) L1 L2 â
âI,K1,V1. â¬[i] L1 â¡ K1.â{I}V1 â
- i < d â¨
+ i < l 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 [ * || * ]
+ R K1 V1 V2 & l ⤠i.
+#R #L1 #L2 #l #i #H #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref ⦠H) -H [ * || * ]
[ #H #_ elim (lt_refl_false i)
lapply (drop_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
+| #I #K11 #K22 #V11 #V22 #HLK11 #HLK22 #HK12 #HV12 #Hli
lapply (drop_mono ⦠HLK11 ⦠HLK1) -L1 #H destruct
/3 width=5 by ex4_2_intro, or_intror/
]
@@ -53,55 +53,55 @@ 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
+lemma llpx_sn_inv_lref_ge_dx: âR,L1,L2,l,i. llpx_sn R l (#i) L1 L2 â l ⤠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)
+#R #L1 #L2 #l #i #H #Hli #I #K2 #V2 #HLK2 elim (llpx_sn_fwd_lref_dx ⦠H ⦠HLK2) -L2
+[ #H elim (ylt_yle_false ⦠H Hli)
| * /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
+lemma llpx_sn_inv_lref_ge_sn: âR,L1,L2,l,i. llpx_sn R l (#i) L1 L2 â l ⤠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)
+#R #L1 #L2 #l #i #H #Hli #I #K1 #V1 #HLK1 elim (llpx_sn_fwd_lref_sn ⦠H ⦠HLK1) -L1
+[ #H elim (ylt_yle_false ⦠H Hli)
| * /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
+lemma llpx_sn_inv_lref_ge_bi: âR,L1,L2,l,i. llpx_sn R l (#i) L1 L2 â l ⤠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
+#R #L1 #L2 #l #i #HL12 #Hli #I1 #I2 #K1 #K2 #V1 #V2 #HLK1 #HLK2
+elim (llpx_sn_inv_lref_ge_sn ⦠HL12 ⦠HLK1) // -L1 -l
#J #Y #HY lapply (drop_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
+fact llpx_sn_inv_S_aux: âR,L1,L2,T,l0. llpx_sn R l0 T L1 L2 â âl. l0 = l + 1 â
+ âK1,K2,I,V1,V2. â¬[l] L1 â¡ K1.â{I}V1 â â¬[l] L2 â¡ K2.â{I}V2 â
+ llpx_sn R 0 V1 K1 K2 â R K1 V1 V2 â llpx_sn R l T L1 L2.
+#R #L1 #L2 #T #l0 #H elim H -L1 -L2 -T -l0
/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
+[ #L1 #L2 #l0 #i #HL12 #Hil #l #H #K1 #K2 #I #V1 #V2 #HLK1 #HLK2 #HK12 #HV12 destruct
+ elim (yle_split_eq i l) /2 width=1 by llpx_sn_skip, ylt_fwd_succ2/ -HL12 -Hil
#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
+| #I #L1 #L2 #K11 #K22 #V1 #V2 #l0 #i #Hl0i #HLK11 #HLK22 #HK12 #HV12 #_ #l #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
+| #a #I #L1 #L2 #V #T #l0 #_ #_ #IHV #IHT #l #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
/4 width=9 by llpx_sn_bind, drop_drop/
-| #I #L1 #L2 #V #T #d0 #_ #_ #IHV #IHT #d #H #K1 #K2 #J #W1 #W2 #HLK1 #HLK2 #HK12 #HW12 destruct
+| #I #L1 #L2 #V #T #l0 #_ #_ #IHV #IHT #l #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.
+lemma llpx_sn_inv_S: âR,L1,L2,T,l. llpx_sn R (l + 1) T L1 L2 â
+ âK1,K2,I,V1,V2. â¬[l] L1 â¡ K1.â{I}V1 â â¬[l] L2 â¡ K2.â{I}V2 â
+ llpx_sn R 0 V1 K1 K2 â R K1 V1 V2 â llpx_sn R l 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
@@ -126,13 +126,13 @@ lemma llpx_sn_bind_repl_O: âR,I,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.â{I}V1) (L2
/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).
+ âT,L1,L2,l. Decidable (llpx_sn R l 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
+ [ #HL12 #l elim (ylt_split i l) /3 width=1 by llpx_sn_skip, or_introl/
+ #Hli elim (lt_or_ge i (|L1|)) #HiL1
elim (lt_or_ge i (|L2|)) #HiL2 /3 width=1 by or_introl, llpx_sn_free/
elim (drop_O1_lt (â») ⦠HiL2) #I2 #K2 #V2 #HLK2
elim (drop_O1_lt (â») ⦠HiL1) #I1 #K1 #V1 #HLK1
@@ -153,14 +153,14 @@ lemma llpx_sn_dec: âR. (âL,T1,T2. Decidable (R L T1 T2)) â
#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/
+| #a #I #V #T #Hn #L2 #l destruct
+ elim (IH L1 V ⦠L2 l) /2 width=1 by/
+ elim (IH (L1.â{I}V) T ⦠(L2.â{I}V) (⫯l)) -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/
+| #I #V #T #Hn #L2 #l destruct
+ elim (IH L1 V ⦠L2 l) /2 width=1 by/
+ elim (IH L1 T ⦠L2 l) -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/
]
@@ -169,115 +169,115 @@ 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 (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -d
+lemma llpx_sn_lift_le: âR. d_liftable R â
+ âK1,K2,T,l0. llpx_sn R l0 T K1 K2 â
+ âL1,L2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
+ âU. â¬[l, m] T â¡ U â l0 ⤠l â llpx_sn R l0 U L1 L2.
+#R #HR #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0
+[ #K1 #K2 #l0 #k #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 ⦠H) -X
+ lapply (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -l
/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 (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -d
+| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 ⦠H) -H
+ * #Hli #H destruct
+ [ lapply (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -l
/2 width=1 by llpx_sn_skip/
- | elim (ylt_yle_false ⦠Hid0) -L1 -L2 -K1 -K2 -e -Hid0
+ | elim (ylt_yle_false ⦠Hil0) -L1 -L2 -K1 -K2 -m -Hil0
/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 ]
+| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #IHK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 ⦠H) -H
+ * #Hli #H destruct [ -HK12 | -IHK12 ]
[ elim (drop_trans_lt ⦠HLK1 ⦠HK11) // -K1
- elim (drop_trans_lt ⦠HLK2 ⦠HK22) // -Hdi -K2
+ elim (drop_trans_lt ⦠HLK2 ⦠HK22) // -Hli -K2
/3 width=18 by llpx_sn_lref/
| lapply (drop_trans_ge_comm ⦠HLK1 ⦠HK11 ?) // -K1
- lapply (drop_trans_ge_comm ⦠HLK2 ⦠HK22 ?) // -Hdi -Hd0 -K2
+ lapply (drop_trans_ge_comm ⦠HLK2 ⦠HK22 ?) // -Hli -Hl0 -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
+| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 ⦠H) -H
+ * #Hil #H destruct
lapply (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -HK12
[ /3 width=7 by llpx_sn_free, drop_fwd_be/
| lapply (drop_fwd_length ⦠HLK1) -HLK1 #HLK1
lapply (drop_fwd_length ⦠HLK2) -HLK2 #HLK2
- @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
+ @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -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 (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -d -e
+| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 ⦠H) -X
+ lapply (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -l -m
/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
+| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 ⦠H) -H
#W #U #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_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
+| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 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 (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -d
+lemma llpx_sn_lift_ge: âR,K1,K2,T,l0. llpx_sn R l0 T K1 K2 â
+ âL1,L2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
+ âU. â¬[l, m] T â¡ U â l ⤠l0 â llpx_sn R (l0+m) U L1 L2.
+#R #K1 #K2 #T #l0 #H elim H -K1 -K2 -T -l0
+[ #K1 #K2 #l0 #k #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort1 ⦠H) -X
+ lapply (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -l
/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
+| #K1 #K2 #l0 #i #HK12 #Hil0 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref1 ⦠H) -H
* #_ #H destruct
lapply (drop_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
+| #I #K1 #K2 #K11 #K22 #V1 #V2 #l0 #i #Hil0 #HK11 #HK22 #HK12 #HV12 #_ #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 ⦠H) -H
+ * #Hil #H destruct
+ [ elim (ylt_yle_false ⦠Hil0) -I -L1 -L2 -K1 -K2 -K11 -K22 -V1 -V2 -m -Hil0
/3 width=3 by ylt_yle_trans, ylt_inj/
| lapply (drop_trans_ge_comm ⦠HLK1 ⦠HK11 ?) // -K1
- lapply (drop_trans_ge_comm ⦠HLK2 ⦠HK22 ?) // -Hid -Hd0 -K2
+ lapply (drop_trans_ge_comm ⦠HLK2 ⦠HK22 ?) // -Hil -Hl0 -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
+| #K1 #K2 #l0 #i #HK1 #HK2 #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref1 ⦠H) -H
+ * #Hil #H destruct
lapply (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -HK12
[ /3 width=7 by llpx_sn_free, drop_fwd_be/
| lapply (drop_fwd_length ⦠HLK1) -HLK1 #HLK1
lapply (drop_fwd_length ⦠HLK2) -HLK2 #HLK2
- @llpx_sn_free [ >HLK1 | >HLK2 ] -Hid -HLK1 -HLK2 /2 width=1 by monotonic_le_plus_r/ (**) (* explicit constructor *)
+ @llpx_sn_free [ >HLK1 | >HLK2 ] -Hil -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 (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -d
+| #K1 #K2 #l0 #p #HK12 #L1 #L2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref1 ⦠H) -X
+ lapply (drop_fwd_length_eq2 ⦠HLK1 HLK2 HK12) -K1 -K2 -l
/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
+| #a #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind1 ⦠H) -H
#W #U #HVW #HTU #H destruct /4 width=5 by llpx_sn_bind, drop_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
+| #I #K1 #K2 #V #T #l0 #_ #_ #IHV #IHT #L1 #L2 #l #m #HLK1 #HLK2 #X #H #Hl0 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 (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d -e
+lemma llpx_sn_inv_lift_le: âR. d_deliftable_sn R â
+ âL1,L2,U,l0. llpx_sn R l0 U L1 L2 â
+ âK1,K2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
+ âT. â¬[l, m] T â¡ U â l0 ⤠l â llpx_sn R l0 T K1 K2.
+#R #HR #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
+[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 ⦠H) -X
+ lapply (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -l -m
/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
+| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ elim (lift_inv_lref2 ⦠H) -H
* #_ #H destruct
lapply (drop_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 ]
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #IHK12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 ⦠H) -H
+ * #Hil #H destruct [ -HK12 | -IHK12 ]
[ elim (drop_conf_lt ⦠HLK1 ⦠HLK11) // -L1 #L1 #V1 #HKL1 #HKL11 #HVW1
- elim (drop_conf_lt ⦠HLK2 ⦠HLK22) // -Hid -L2 #L2 #V2 #HKL2 #HKL22 #HVW2
+ elim (drop_conf_lt ⦠HLK2 ⦠HLK22) // -Hil -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 (drop_conf_ge ⦠HLK1 ⦠HLK11 ?) // -L1
- lapply (drop_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 *)
+ lapply (drop_conf_ge ⦠HLK2 ⦠HLK22 ?) // -L2 -Hil0
+ elim (le_inv_plus_l ⦠Hil) -Hil /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
+| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_lref2 ⦠H) -H
* #_ #H destruct
lapply (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12)
[ lapply (drop_fwd_length_le4 ⦠HLK1) -HLK1
@@ -287,39 +287,39 @@ lemma llpx_sn_inv_lift_le: âR. l_deliftable_sn R â
lapply (drop_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 (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d -e
+| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 ⦠H) -X
+ lapply (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -l -m
/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
+| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 elim (lift_inv_bind2 ⦠H) -H
#V #T #HVW #HTU #H destruct /4 width=6 by llpx_sn_bind, drop_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
+| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 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 (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d0 -e
+lemma llpx_sn_inv_lift_be: âR,L1,L2,U,l0. llpx_sn R l0 U L1 L2 â
+ âK1,K2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
+ âT. â¬[l, m] T â¡ U â l ⤠l0 â l0 ⤠yinj l + m â llpx_sn R l T K1 K2.
+#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
+[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_sort2 ⦠H) -X
+ lapply (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -l0 -m
/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
+| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 ⦠H) -H
+ * #Hil #H destruct
[ lapply (drop_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
+ -Hil0 /3 width=1 by llpx_sn_skip, ylt_inj/
+ | elim (ylt_yle_false ⦠Hil0) -L1 -L2 -Hl0 -Hil0
/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
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 ⦠H) -H
+ * #Hil #H destruct
+ [ elim (ylt_yle_false ⦠Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hl0m -Hil0
/3 width=3 by ylt_yle_trans, ylt_inj/
| lapply (drop_conf_ge ⦠HLK1 ⦠HLK11 ?) // -L1
- lapply (drop_conf_ge ⦠HLK2 ⦠HLK22 ?) // -L2 -Hid0 -Hd0 -Hd0e
- elim (le_inv_plus_l ⦠Hid) -Hid /3 width=9 by llpx_sn_lref, yle_inj/
+ lapply (drop_conf_ge ⦠HLK2 ⦠HLK22 ?) // -L2 -Hil0 -Hl0 -Hl0m
+ elim (le_inv_plus_l ⦠Hil) -Hil /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
+| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m elim (lift_inv_lref2 ⦠H) -H
* #_ #H destruct
lapply (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12)
[ lapply (drop_fwd_length_le4 ⦠HLK1) -HLK1
@@ -329,41 +329,41 @@ lemma llpx_sn_inv_lift_be: âR,L1,L2,U,d0. llpx_sn R d0 U L1 L2 â
lapply (drop_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 (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d0 -e
+| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ #_ >(lift_inv_gref2 ⦠H) -X
+ lapply (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -l0 -m
/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
+| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m 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 drop_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
+| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hl0 #Hl0m 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 (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d
+lemma llpx_sn_inv_lift_ge: âR,L1,L2,U,l0. llpx_sn R l0 U L1 L2 â
+ âK1,K2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
+ âT. â¬[l, m] T â¡ U â yinj l + m ⤠l0 â llpx_sn R (l0-m) T K1 K2.
+#R #L1 #L2 #U #l0 #H elim H -L1 -L2 -U -l0
+[ #L1 #L2 #l0 #k #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_sort2 ⦠H) -X
+ lapply (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -l
/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 ]
+| #L1 #L2 #l0 #i #HL12 #Hil0 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 ⦠H) -H
+ * #Hil #H destruct [ -Hil0 | -Hlml0 ]
lapply (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2
[ /4 width=3 by llpx_sn_skip, yle_plus1_to_minus_inj2, ylt_yle_trans, ylt_inj/
- | elim (le_inv_plus_l ⦠Hid) -Hid #_
+ | elim (le_inv_plus_l ⦠Hil) -Hil #_
/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
+| #I #L1 #L2 #K11 #K22 #W1 #W2 #l0 #i #Hil0 #HLK11 #HLK22 #HK12 #HW12 #_ #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 ⦠H) -H
+ * #Hil #H destruct
+ [ elim (ylt_yle_false ⦠Hil0) -I -L1 -L2 -K11 -K22 -W1 -W2 -Hil0
/3 width=3 by yle_fwd_plus_sn1, ylt_yle_trans, ylt_inj/
| lapply (drop_conf_ge ⦠HLK1 ⦠HLK11 ?) // -L1
- lapply (drop_conf_ge ⦠HLK2 ⦠HLK22 ?) // -L2 -Hded0 -Hid
+ lapply (drop_conf_ge ⦠HLK2 ⦠HLK22 ?) // -L2 -Hlml0 -Hil
/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
+| #L1 #L2 #l0 #i #HL1 #HL2 #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_lref2 ⦠H) -H
* #_ #H destruct
lapply (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12)
[ lapply (drop_fwd_length_le4 ⦠HLK1) -HLK1
@@ -373,15 +373,15 @@ lemma llpx_sn_inv_lift_ge: âR,L1,L2,U,d0. llpx_sn R d0 U L1 L2 â
lapply (drop_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 (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -d
+| #L1 #L2 #l0 #p #HL12 #K1 #K2 #l #m #HLK1 #HLK2 #X #H #_ >(lift_inv_gref2 ⦠H) -X
+ lapply (drop_fwd_length_eq1 ⦠HLK1 HLK2 HL12) -L1 -L2 -l
/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
+| #a #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 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 drop_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
+| #I #L1 #L2 #W #U #l0 #_ #_ #IHW #IHU #K1 #K2 #l #m #HLK1 #HLK2 #X #H #Hlml0 elim (lift_inv_flat2 ⦠H) -H
#V #T #HVW #HTU #H destruct /3 width=5 by llpx_sn_flat/
]
qed-.
@@ -389,37 +389,37 @@ qed-.
(* Advanced inversion lemmas on relocation **********************************)
lemma llpx_sn_inv_lift_O: âR,L1,L2,U. llpx_sn R 0 U L1 L2 â
- âK1,K2,e. â¬[e] L1 â¡ K1 â â¬[e] L2 â¡ K2 â
- âT. â¬[0, e] T â¡ U â llpx_sn R 0 T K1 K2.
+ âK1,K2,m. â¬[m] L1 â¡ K1 â â¬[m] L2 â¡ K2 â
+ âT. â¬[0, m] T â¡ U â llpx_sn R 0 T K1 K2.
/2 width=11 by llpx_sn_inv_lift_be/ qed-.
lemma llpx_sn_drop_conf_O: âR,L1,L2,U. llpx_sn R 0 U L1 L2 â
- âK1,e. â¬[e] L1 â¡ K1 â âT. â¬[0, e] T â¡ U â
- ââK2. â¬[e] L2 â¡ K2 & llpx_sn R 0 T K1 K2.
-#R #L1 #L2 #U #HU #K1 #e #HLK1 #T #HTU elim (llpx_sn_fwd_drop_sn ⦠HU ⦠HLK1)
+ âK1,m. â¬[m] L1 â¡ K1 â âT. â¬[0, m] T â¡ U â
+ ââK2. â¬[m] L2 â¡ K2 & llpx_sn R 0 T K1 K2.
+#R #L1 #L2 #U #HU #K1 #m #HLK1 #T #HTU elim (llpx_sn_fwd_drop_sn ⦠HU ⦠HLK1)
/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
qed-.
lemma llpx_sn_drop_trans_O: âR,L1,L2,U. llpx_sn R 0 U L1 L2 â
- âK2,e. â¬[e] L2 â¡ K2 â âT. â¬[0, e] T â¡ U â
- ââK1. â¬[e] L1 â¡ K1 & llpx_sn R 0 T K1 K2.
-#R #L1 #L2 #U #HU #K2 #e #HLK2 #T #HTU elim (llpx_sn_fwd_drop_dx ⦠HU ⦠HLK2)
+ âK2,m. â¬[m] L2 â¡ K2 â âT. â¬[0, m] T â¡ U â
+ ââK1. â¬[m] L1 â¡ K1 & llpx_sn R 0 T K1 K2.
+#R #L1 #L2 #U #HU #K2 #m #HLK2 #T #HTU elim (llpx_sn_fwd_drop_dx ⦠HU ⦠HLK2)
/3 width=10 by llpx_sn_inv_lift_O, ex2_intro/
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)
+ âa,I,L1,L2,V,T,l. (llpx_sn R l (â{a,I}V.T) L1 L2 â â¥) â
+ (llpx_sn R l V L1 L2 â â¥) ⨠(llpx_sn R (⫯l) T (L1.â{I}V) (L2.â{I}V) â â¥).
+#R #HR #a #I #L1 #L2 #V #T #l #H elim (llpx_sn_dec ⦠HR V L1 L2 l)
/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)
+ âI,L1,L2,V,T,l. (llpx_sn R l (â{I}V.T) L1 L2 â â¥) â
+ (llpx_sn R l V L1 L2 â â¥) ⨠(llpx_sn R l T L1 L2 â â¥).
+#R #HR #I #L1 #L2 #V #T #l #H elim (llpx_sn_dec ⦠HR V L1 L2 l)
/4 width=1 by llpx_sn_flat, or_intror, or_introl/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_frees.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_frees.ma
index fdb31ffc5..81eff3bda 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_frees.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_frees.ma
@@ -20,10 +20,10 @@ include "basic_2/multiple/llpx_sn_alt_rec.ma".
(* Properties on context-sensitive free variables ***************************)
fact llpx_sn_frees_trans_aux: âR. (s_r_confluent1 ⦠R (llpx_sn R 0)) â (frees_trans R) â
- âL2,U,d,i. L2 ⢠i ϵ ð
*[d]â¦U⦠â
- âL1. llpx_sn R d U L1 L2 â L1 ⢠i ϵ ð
*[d]â¦Uâ¦.
-#R #H1R #H2R #L2 #U #d #i #H elim H -L2 -U -d -i /3 width=2 by frees_eq/
-#I2 #L2 #K2 #U #W2 #d #i #j #Hdj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
+ âL2,U,l,i. L2 ⢠i ϵ ð
*[l]â¦U⦠â
+ âL1. llpx_sn R l U L1 L2 â L1 ⢠i ϵ ð
*[l]â¦Uâ¦.
+#R #H1R #H2R #L2 #U #l #i #H elim H -L2 -U -l -i /3 width=2 by frees_eq/
+#I2 #L2 #K2 #U #W2 #l #i #j #Hlj #Hji #HnU #HLK2 #_ #IHW2 #L1 #HL12
elim (llpx_sn_inv_alt_r ⦠HL12) -HL12 #HL12 #IH
lapply (drop_fwd_length_lt2 ⦠HLK2) #Hj
elim (drop_O1_lt (â») L1 j) // -Hj -HL12 #I1 #K1 #W1 #HLK1
@@ -31,6 +31,6 @@ elim (IH ⦠HnU HLK1 HLK2) // -IH -HLK2 /5 width=11 by frees_be/
qed-.
lemma llpx_sn_frees_trans: âR. (s_r_confluent1 ⦠R (llpx_sn R 0)) â (frees_trans R) â
- âL1,L2,U,d. llpx_sn R d U L1 L2 â
- âi. L2 ⢠i ϵ ð
*[d]â¦U⦠â L1 ⢠i ϵ ð
*[d]â¦Uâ¦.
+ âL1,L2,U,l. llpx_sn R l U L1 L2 â
+ âi. L2 ⢠i ϵ ð
*[l]â¦U⦠â L1 ⢠i ϵ ð
*[l]â¦Uâ¦.
/2 width=6 by llpx_sn_frees_trans_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_leq.ma
deleted file mode 100644
index 5075a332e..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_leq.ma
+++ /dev/null
@@ -1,54 +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/drop_leq.ma".
-include "basic_2/multiple/llpx_sn.ma".
-
-(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
-
-(* Properties on equivalence for local environments *************************)
-
-lemma leq_llpx_sn_trans: âR,L2,L,T,d. llpx_sn R d T L2 L â
- âL1. L1 ⩬[d, â] L2 â llpx_sn R d T L1 L.
-#R #L2 #L #T #d #H elim H -L2 -L -T -d
-/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, leq_fwd_length, trans_eq/
-[ #I #L2 #L #K2 #K #V2 #V #d #i #Hdi #HLK2 #HLK #HK2 #HV2 #_ #L1 #HL12
- elim (leq_drop_trans_be ⦠HL12 ⦠HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
- lapply (leq_inv_O_Y ⦠HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
-| /4 width=5 by llpx_sn_free, leq_fwd_length, le_repl_sn_trans_aux, trans_eq/
-| /4 width=1 by llpx_sn_bind, leq_succ/
-]
-qed-.
-
-lemma llpx_sn_leq_trans: âR,L,L1,T,d. llpx_sn R d T L L1 â
- âL2. L1 ⩬[d, â] L2 â llpx_sn R d T L L2.
-#R #L #L1 #T #d #H elim H -L -L1 -T -d
-/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, leq_fwd_length, trans_eq/
-[ #I #L #L1 #K #K1 #V #V1 #d #i #Hdi #HLK #HLK1 #HK1 #HV1 #_ #L2 #HL12
- elim (leq_drop_conf_be ⦠HL12 ⦠HLK1) -L1 // >yminus_Y_inj #K2 #HK12 #HLK2
- lapply (leq_inv_O_Y ⦠HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
-| /4 width=5 by llpx_sn_free, leq_fwd_length, le_repl_sn_conf_aux, trans_eq/
-| /4 width=1 by llpx_sn_bind, leq_succ/
-]
-qed-.
-
-lemma llpx_sn_leq_repl: âR,L1,L2,T,d. llpx_sn R d T L1 L2 â âK1. K1 ⩬[d, â] L1 â
- âK2. L2 ⩬[d, â] K2 â llpx_sn R d T K1 K2.
-/3 width=4 by llpx_sn_leq_trans, leq_llpx_sn_trans/ qed-.
-
-lemma llpx_sn_bind_repl_SO: âR,I1,I2,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.â{I1}V1) (L2.â{I2}V2) â
- âJ1,J2,W1,W2. llpx_sn R 1 T (L1.â{J1}W1) (L2.â{J2}W2).
-#R #I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (llpx_sn_ge R ⦠1 ⦠HT) -HT
-/3 width=7 by llpx_sn_leq_repl, leq_succ/
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_llor.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_llor.ma
index 91d9e649e..235eedbda 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_llor.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_llor.ma
@@ -21,9 +21,9 @@ include "basic_2/multiple/lleq_alt.ma".
(* Inversion lemmas on pointwise union for local environments ****************)
lemma llpx_sn_llor_fwd_sn: âR. (âL. reflexive ⦠(R L)) â
- âL1,L2,T,d. llpx_sn R d T L1 L2 â
- âL. L1 â[T, d] L2 â¡ L â lpx_sn R L1 L.
-#R #HR #L1 #L2 #T #d #H1 #L #H2
+ âL1,L2,T,l. llpx_sn R l T L1 L2 â
+ âL. L1 â[T, l] L2 â¡ L â lpx_sn R L1 L.
+#R #HR #L1 #L2 #T #l #H1 #L #H2
elim (llpx_sn_llpx_sn_alt ⦠H1) -H1 #HL12 #IH1
elim H2 -H2 #_ #HL1 #IH2
@lpx_sn_intro_alt // #I1 #I #K1 #K #V1 #V #i #HLK1 #HLK
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_lpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_lpx_sn.ma
index 9ce6a75d9..b4b7f7ebe 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_lpx_sn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_lpx_sn.ma
@@ -20,15 +20,15 @@ include "basic_2/multiple/llpx_sn.ma".
(* Properties on pointwise extensions ***************************************)
lemma lpx_sn_llpx_sn: âR. (âL. reflexive ⦠(R L)) â
- âT,L1,L2,d. lpx_sn R L1 L2 â llpx_sn R d T L1 L2.
+ âT,L1,L2,l. lpx_sn R L1 L2 â llpx_sn R l T L1 L2.
#R #HR #T #L1 @(f2_ind ⦠rfw ⦠L1 T) -L1 -T
#n #IH #L1 * *
[ -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_sort/
| -HR #i elim (lt_or_ge i (|L1|))
[2: -IH /4 width=4 by lpx_sn_fwd_length, llpx_sn_free, le_repl_sn_conf_aux/ ]
- #Hi #Hn #L2 #d elim (ylt_split i d)
+ #Hi #Hn #L2 #l elim (ylt_split i l)
[ -n /3 width=2 by llpx_sn_skip, lpx_sn_fwd_length/ ]
- #Hdi #HL12 elim (drop_O1_lt (â») L1 i) //
+ #Hli #HL12 elim (drop_O1_lt (â») L1 i) //
#I #K1 #V1 #HLK1 elim (lpx_sn_drop_conf ⦠HL12 ⦠HLK1) -HL12
/4 width=9 by llpx_sn_lref, drop_fwd_rfw/
| -HR -IH /4 width=2 by lpx_sn_fwd_length, llpx_sn_gref/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_lreq.ma
new file mode 100644
index 000000000..3351b326e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/llpx_sn_lreq.ma
@@ -0,0 +1,54 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/substitution/drop_lreq.ma".
+include "basic_2/multiple/llpx_sn.ma".
+
+(* LAZY SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS ****)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_llpx_sn_trans: âR,L2,L,T,l. llpx_sn R l T L2 L â
+ âL1. L1 ⩬[l, â] L2 â llpx_sn R l T L1 L.
+#R #L2 #L #T #l #H elim H -L2 -L -T -l
+/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, lreq_fwd_length, trans_eq/
+[ #I #L2 #L #K2 #K #V2 #V #l #i #Hli #HLK2 #HLK #HK2 #HV2 #_ #L1 #HL12
+ elim (lreq_drop_trans_be ⦠HL12 ⦠HLK2) -L2 // >yminus_Y_inj #K1 #HK12 #HLK1
+ lapply (lreq_inv_O_Y ⦠HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
+| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_trans_aux, trans_eq/
+| /4 width=1 by llpx_sn_bind, lreq_succ/
+]
+qed-.
+
+lemma llpx_sn_lreq_trans: âR,L,L1,T,l. llpx_sn R l T L L1 â
+ âL2. L1 ⩬[l, â] L2 â llpx_sn R l T L L2.
+#R #L #L1 #T #l #H elim H -L -L1 -T -l
+/4 width=5 by llpx_sn_flat, llpx_sn_gref, llpx_sn_skip, llpx_sn_sort, lreq_fwd_length, trans_eq/
+[ #I #L #L1 #K #K1 #V #V1 #l #i #Hli #HLK #HLK1 #HK1 #HV1 #_ #L2 #HL12
+ elim (lreq_drop_conf_be ⦠HL12 ⦠HLK1) -L1 // >yminus_Y_inj #K2 #HK12 #HLK2
+ lapply (lreq_inv_O_Y ⦠HK12) -HK12 #H destruct /2 width=9 by llpx_sn_lref/
+| /4 width=5 by llpx_sn_free, lreq_fwd_length, le_repl_sn_conf_aux, trans_eq/
+| /4 width=1 by llpx_sn_bind, lreq_succ/
+]
+qed-.
+
+lemma llpx_sn_lreq_repl: âR,L1,L2,T,l. llpx_sn R l T L1 L2 â âK1. K1 ⩬[l, â] L1 â
+ âK2. L2 ⩬[l, â] K2 â llpx_sn R l T K1 K2.
+/3 width=4 by llpx_sn_lreq_trans, lreq_llpx_sn_trans/ qed-.
+
+lemma llpx_sn_bind_repl_SO: âR,I1,I2,L1,L2,V1,V2,T. llpx_sn R 0 T (L1.â{I1}V1) (L2.â{I2}V2) â
+ âJ1,J2,W1,W2. llpx_sn R 1 T (L1.â{J1}W1) (L2.â{J2}W2).
+#R #I1 #I2 #L1 #L2 #V1 #V2 #T #HT #J1 #J2 #W1 #W2 lapply (llpx_sn_ge R ⦠1 ⦠HT) -HT
+/3 width=7 by llpx_sn_lreq_repl, lreq_succ/
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2.ma
index 137c179fa..9691de8a2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2.ma
@@ -19,10 +19,10 @@ include "basic_2/grammar/term_vector.ma".
inductive at: list2 nat nat â relation nat â
| at_nil: âi. at (â) i i
-| at_lt : âdes,d,e,i1,i2. i1 < d â
- at des i1 i2 â at ({d, e} @ des) i1 i2
-| at_ge : âdes,d,e,i1,i2. d ⤠i1 â
- at des (i1 + e) i2 â at ({d, e} @ des) i1 i2
+| at_lt : âdes,l,m,i1,i2. i1 < l â
+ at des i1 i2 â at ({l, m} @ des) i1 i2
+| at_ge : âdes,l,m,i1,i2. l ⤠i1 â
+ at des (i1 + m) i2 â at ({l, m} @ des) i1 i2
.
interpretation "application (multiple relocation with pairs)"
@@ -33,8 +33,8 @@ interpretation "application (multiple relocation with pairs)"
fact at_inv_nil_aux: âdes,i1,i2. @â¦i1, des⦠⡠i2 â des = â â i1 = i2.
#des #i1 #i2 * -des -i1 -i2
[ //
-| #des #d #e #i1 #i2 #_ #_ #H destruct
-| #des #d #e #i1 #i2 #_ #_ #H destruct
+| #des #l #m #i1 #i2 #_ #_ #H destruct
+| #des #l #m #i1 #i2 #_ #_ #H destruct
]
qed-.
@@ -42,33 +42,33 @@ lemma at_inv_nil: âi1,i2. @â¦i1, â⦠⡠i2 â i1 = i2.
/2 width=3 by at_inv_nil_aux/ qed-.
fact at_inv_cons_aux: âdes,i1,i2. @â¦i1, des⦠⡠i2 â
- âd,e,des0. des = {d, e} @ des0 â
- i1 < d ⧠@â¦i1, des0⦠⡠i2 â¨
- d ⤠i1 ⧠@â¦i1 + e, des0⦠⡠i2.
+ âl,m,des0. des = {l, m} @ des0 â
+ i1 < l ⧠@â¦i1, des0⦠⡠i2 â¨
+ l ⤠i1 ⧠@â¦i1 + m, des0⦠⡠i2.
#des #i1 #i2 * -des -i1 -i2
-[ #i #d #e #des #H destruct
-| #des1 #d1 #e1 #i1 #i2 #Hid1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1 by or_introl, conj/
-| #des1 #d1 #e1 #i1 #i2 #Hdi1 #Hi12 #d2 #e2 #des2 #H destruct /3 width=1 by or_intror, conj/
+[ #i #l #m #des #H destruct
+| #des1 #l1 #m1 #i1 #i2 #Hil1 #Hi12 #l2 #m2 #des2 #H destruct /3 width=1 by or_introl, conj/
+| #des1 #l1 #m1 #i1 #i2 #Hli1 #Hi12 #l2 #m2 #des2 #H destruct /3 width=1 by or_intror, conj/
]
qed-.
-lemma at_inv_cons: âdes,d,e,i1,i2. @â¦i1, {d, e} @ des⦠⡠i2 â
- i1 < d ⧠@â¦i1, des⦠⡠i2 â¨
- d ⤠i1 ⧠@â¦i1 + e, des⦠⡠i2.
+lemma at_inv_cons: âdes,l,m,i1,i2. @â¦i1, {l, m} @ des⦠⡠i2 â
+ i1 < l ⧠@â¦i1, des⦠⡠i2 â¨
+ l ⤠i1 ⧠@â¦i1 + m, des⦠⡠i2.
/2 width=3 by at_inv_cons_aux/ qed-.
-lemma at_inv_cons_lt: âdes,d,e,i1,i2. @â¦i1, {d, e} @ des⦠⡠i2 â
- i1 < d â @â¦i1, des⦠⡠i2.
-#des #d #e #i1 #e2 #H
-elim (at_inv_cons ⦠H) -H * // #Hdi1 #_ #Hi1d
-lapply (le_to_lt_to_lt ⦠Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
-elim (lt_refl_false ⦠Hd)
+lemma at_inv_cons_lt: âdes,l,m,i1,i2. @â¦i1, {l, m} @ des⦠⡠i2 â
+ i1 < l â @â¦i1, des⦠⡠i2.
+#des #l #m #i1 #m2 #H
+elim (at_inv_cons ⦠H) -H * // #Hli1 #_ #Hi1l
+lapply (le_to_lt_to_lt ⦠Hli1 Hi1l) -Hli1 -Hi1l #Hl
+elim (lt_refl_false ⦠Hl)
qed-.
-lemma at_inv_cons_ge: âdes,d,e,i1,i2. @â¦i1, {d, e} @ des⦠⡠i2 â
- d ⤠i1 â @â¦i1 + e, des⦠⡠i2.
-#des #d #e #i1 #e2 #H
-elim (at_inv_cons ⦠H) -H * // #Hi1d #_ #Hdi1
-lapply (le_to_lt_to_lt ⦠Hdi1 Hi1d) -Hdi1 -Hi1d #Hd
-elim (lt_refl_false ⦠Hd)
+lemma at_inv_cons_ge: âdes,l,m,i1,i2. @â¦i1, {l, m} @ des⦠⡠i2 â
+ l ⤠i1 â @â¦i1 + m, des⦠⡠i2.
+#des #l #m #i1 #m2 #H
+elim (at_inv_cons ⦠H) -H * // #Hi1l #_ #Hli1
+lapply (le_to_lt_to_lt ⦠Hli1 Hi1l) -Hli1 -Hi1l #Hl
+elim (lt_refl_false ⦠Hl)
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_minus.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_minus.ma
index a4f7bb851..ce2918fdd 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_minus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_minus.ma
@@ -19,10 +19,10 @@ include "basic_2/multiple/mr2.ma".
inductive minuss: nat â relation (list2 nat nat) â
| minuss_nil: âi. minuss i (â) (â)
-| minuss_lt : âdes1,des2,d,e,i. i < d â minuss i des1 des2 â
- minuss i ({d, e} @ des1) ({d - i, e} @ des2)
-| minuss_ge : âdes1,des2,d,e,i. d ⤠i â minuss (e + i) des1 des2 â
- minuss i ({d, e} @ des1) des2
+| minuss_lt : âdes1,des2,l,m,i. i < l â minuss i des1 des2 â
+ minuss i ({l, m} @ des1) ({l - i, m} @ des2)
+| minuss_ge : âdes1,des2,l,m,i. l ⤠i â minuss (m + i) des1 des2 â
+ minuss i ({l, m} @ des1) des2
.
interpretation "minus (multiple relocation with pairs)"
@@ -33,8 +33,8 @@ interpretation "minus (multiple relocation with pairs)"
fact minuss_inv_nil1_aux: âdes1,des2,i. des1 â i â¡ des2 â des1 = â â des2 = â.
#des1 #des2 #i * -des1 -des2 -i
[ //
-| #des1 #des2 #d #e #i #_ #_ #H destruct
-| #des1 #des2 #d #e #i #_ #_ #H destruct
+| #des1 #des2 #l #m #i #_ #_ #H destruct
+| #des1 #des2 #l #m #i #_ #_ #H destruct
]
qed-.
@@ -42,35 +42,35 @@ lemma minuss_inv_nil1: âdes2,i. â â i â¡ des2 â des2 = â.
/2 width=4 by minuss_inv_nil1_aux/ qed-.
fact minuss_inv_cons1_aux: âdes1,des2,i. des1 â i â¡ des2 â
- âd,e,des. des1 = {d, e} @ des â
- d ⤠i ⧠des â e + i â¡ des2 â¨
- ââdes0. i < d & des â i â¡ des0 &
- des2 = {d - i, e} @ des0.
+ âl,m,des. des1 = {l, m} @ des â
+ l ⤠i ⧠des â m + i â¡ des2 â¨
+ ââdes0. i < l & des â i â¡ des0 &
+ des2 = {l - i, m} @ des0.
#des1 #des2 #i * -des1 -des2 -i
-[ #i #d #e #des #H destruct
-| #des1 #des #d1 #e1 #i1 #Hid1 #Hdes #d2 #e2 #des2 #H destruct /3 width=3 by ex3_intro, or_intror/
-| #des1 #des #d1 #e1 #i1 #Hdi1 #Hdes #d2 #e2 #des2 #H destruct /3 width=1 by or_introl, conj/
+[ #i #l #m #des #H destruct
+| #des1 #des #l1 #m1 #i1 #Hil1 #Hcs #l2 #m2 #des2 #H destruct /3 width=3 by ex3_intro, or_intror/
+| #des1 #des #l1 #m1 #i1 #Hli1 #Hcs #l2 #m2 #des2 #H destruct /3 width=1 by or_introl, conj/
]
qed-.
-lemma minuss_inv_cons1: âdes1,des2,d,e,i. {d, e} @ des1 â i â¡ des2 â
- d ⤠i ⧠des1 â e + i â¡ des2 â¨
- ââdes. i < d & des1 â i â¡ des &
- des2 = {d - i, e} @ des.
+lemma minuss_inv_cons1: âdes1,des2,l,m,i. {l, m} @ des1 â i â¡ des2 â
+ l ⤠i ⧠des1 â m + i â¡ des2 â¨
+ ââdes. i < l & des1 â i â¡ des &
+ des2 = {l - i, m} @ des.
/2 width=3 by minuss_inv_cons1_aux/ qed-.
-lemma minuss_inv_cons1_ge: âdes1,des2,d,e,i. {d, e} @ des1 â i â¡ des2 â
- d ⤠i â des1 â e + i â¡ des2.
-#des1 #des2 #d #e #i #H
-elim (minuss_inv_cons1 ⦠H) -H * // #des #Hid #_ #_ #Hdi
-lapply (lt_to_le_to_lt ⦠Hid Hdi) -Hid -Hdi #Hi
+lemma minuss_inv_cons1_ge: âdes1,des2,l,m,i. {l, m} @ des1 â i â¡ des2 â
+ l ⤠i â des1 â m + i â¡ des2.
+#des1 #des2 #l #m #i #H
+elim (minuss_inv_cons1 ⦠H) -H * // #des #Hil #_ #_ #Hli
+lapply (lt_to_le_to_lt ⦠Hil Hli) -Hil -Hli #Hi
elim (lt_refl_false ⦠Hi)
qed-.
-lemma minuss_inv_cons1_lt: âdes1,des2,d,e,i. {d, e} @ des1 â i â¡ des2 â
- i < d â
- ââdes. des1 â i â¡ des & des2 = {d - i, e} @ des.
-#des1 #des2 #d #e #i #H elim (minuss_inv_cons1 ⦠H) -H * /2 width=3 by ex2_intro/
-#Hdi #_ #Hid lapply (lt_to_le_to_lt ⦠Hid Hdi) -Hid -Hdi
+lemma minuss_inv_cons1_lt: âdes1,des2,l,m,i. {l, m} @ des1 â i â¡ des2 â
+ i < l â
+ ââdes. des1 â i â¡ des & des2 = {l - i, m} @ des.
+#des1 #des2 #l #m #i #H elim (minuss_inv_cons1 ⦠H) -H * /2 width=3 by ex2_intro/
+#Hli #_ #Hil lapply (lt_to_le_to_lt ⦠Hil Hli) -Hil -Hli
#Hi elim (lt_refl_false ⦠Hi)
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_mr2.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_mr2.ma
index 03c660fe0..2126666e3 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_mr2.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_mr2.ma
@@ -21,9 +21,9 @@ include "basic_2/multiple/mr2.ma".
theorem at_mono: âdes,i,i1. @â¦i, des⦠⡠i1 â âi2. @â¦i, des⦠⡠i2 â i1 = i2.
#des #i #i1 #H elim H -des -i -i1
[ #i #x #H <(at_inv_nil ⦠H) -x //
-| #des #d #e #i #i1 #Hid #_ #IHi1 #x #H
- lapply (at_inv_cons_lt ⦠H Hid) -H -Hid /2 width=1 by/
-| #des #d #e #i #i1 #Hdi #_ #IHi1 #x #H
- lapply (at_inv_cons_ge ⦠H Hdi) -H -Hdi /2 width=1 by/
+| #des #l #m #i #i1 #Hil #_ #IHi1 #x #H
+ lapply (at_inv_cons_lt ⦠H Hil) -H -Hil /2 width=1 by/
+| #des #l #m #i #i1 #Hli #_ #IHi1 #x #H
+ lapply (at_inv_cons_ge ⦠H Hli) -H -Hli /2 width=1 by/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_plus.ma b/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_plus.ma
index 6528fb827..bedc3dae2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_plus.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/multiple/mr2_plus.ma
@@ -18,7 +18,7 @@ include "basic_2/multiple/mr2.ma".
let rec pluss (des:list2 nat nat) (i:nat) on des â match des with
[ nil2 â â
-| cons2 d e des â {d + i, e} @ pluss des i
+| cons2 l m des â {l + i, m} @ pluss des i
].
interpretation "plus (multiple relocation with pairs)"
@@ -28,13 +28,13 @@ interpretation "plus (multiple relocation with pairs)"
lemma pluss_inv_nil2: âi,des. des + i = â â des = â.
#i * // normalize
-#d #e #des #H destruct
+#l #m #des #H destruct
qed.
-lemma pluss_inv_cons2: âi,d,e,des2,des. des + i = {d, e} @ des2 â
- ââdes1. des1 + i = des2 & des = {d - i, e} @ des1.
-#i #d #e #des2 * normalize
+lemma pluss_inv_cons2: âi,l,m,des2,des. des + i = {l, m} @ des2 â
+ ââdes1. des1 + i = des2 & des = {l - i, m} @ des1.
+#i #l #m #des2 * normalize
[ #H destruct
-| #d1 #e1 #des1 #H destruct /2 width=3/
+| #l1 #m1 #des1 #H destruct /2 width=3/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/names.txt b/matita/matita/contribs/lambdadelta/basic_2/names.txt
index 9a3319f5c..ed799bf44 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/names.txt
+++ b/matita/matita/contribs/lambdadelta/basic_2/names.txt
@@ -16,16 +16,17 @@ T,U,V,W: term
X,Y,Z : reserved: transient objet denoted by a capital letter
a,b : binder polarity
-c : reserved: future use (\lambda\delta 3)
-d : relocation depth
-e : relocation height
+c : relocation
+d : term degree
+e : reserved: future use (\lambda\delta 3)
f :
g : sort degree parameter
h : sort hierarchy parameter
i,j : local reference position index (de Bruijn's)
k : sort index
-l : term degree
-m,n : reserved: future use
+l : relocation depth
+m : relocation height
+n : type iterations
o :
p,q : global reference position
r : reduction kind parameter (true = ordinary, false = extended)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/cosn_5.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/cosn_5.ma
index 01a9040a6..be4f9a4a9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/cosn_5.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/cosn_5.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( G ⢠~ ⬠* break [ term 46 h , break term 46 g , break term 46 d ] break term 46 L )"
+notation "hvbox( G ⢠~ ⬠* break [ term 46 h , break term 46 g , break term 46 l ] break term 46 L )"
non associative with precedence 45
- for @{ 'CoSN $h $g $d $G $L }.
+ for @{ 'CoSN $h $g $l $G $L }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/degree_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/degree_6.ma
index 4c78c8126..7602a6544 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/degree_6.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/degree_6.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T ⪠break [ term 46 h , break term 46 g ] break term 46 l )"
+notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T ⪠break [ term 46 h , break term 46 g ] break term 46 d )"
non associative with precedence 45
- for @{ 'Degree $h $g $G $L $T $l }.
+ for @{ 'Degree $h $g $G $L $T $d }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/dpconvstar_8.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/dpconvstar_8.ma
index cb60c21e6..9aa3568af 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/dpconvstar_8.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/dpconvstar_8.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T1 ⢠* ⬠* break [ term 46 h , break term 46 g , break term 46 l1 , break term 46 l2 ] break term 46 T2 )"
+notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T1 ⢠* ⬠* break [ term 46 h , break term 46 g , break term 46 n1 , break term 46 n2 ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'DPConvStar $h $g $l1 $l2 $G $L $T1 $T2 }.
+ for @{ 'DPConvStar $h $g $n1 $n2 $G $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/dpredstar_7.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/dpredstar_7.ma
index cbe3d09e2..c5f55aed8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/dpredstar_7.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/dpredstar_7.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T1 ⢠* ⡠* break [ term 46 h , break term 46 g , break term 46 l ] break term 46 T2 )"
+notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T1 ⢠* ⡠* break [ term 46 h , break term 46 g , break term 46 n ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'DPRedStar $h $g $l $G $L $T1 $T2 }.
+ for @{ 'DPRedStar $h $g $n $G $L $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/freestar_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/freestar_4.ma
index 4ce6d2dc7..e46480639 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/freestar_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/freestar_4.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L ⢠break term 46 i ϵ ð
* [ break term 46 d ] ⦠break term 46 T ⦠)"
+notation "hvbox( L ⢠break term 46 i ϵ ð
* [ break term 46 l ] ⦠break term 46 T ⦠)"
non associative with precedence 45
- for @{ 'FreeStar $L $i $d $T }.
+ for @{ 'FreeStar $L $i $l $T }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_4.ma
index 0d9afe1ca..a6b02b3cc 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_4.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L1 â¡ break [ term 46 T , break term 46 d ] break term 46 L2 )"
+notation "hvbox( L1 â¡ break [ term 46 T , break term 46 l ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'LazyEq $T $d $L1 $L2 }.
+ for @{ 'LazyEq $T $l $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_7.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_7.ma
index 4bfa2c754..79a79e2c0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_7.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyeq_7.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦠term 46 G1, break term 46 L1, break term 46 T1 ⦠⡠break [ term 46 d ] break ⦠term 46 G2, break term 46 L2 , break term 46 T2 ⦠)"
+notation "hvbox( ⦠term 46 G1, break term 46 L1, break term 46 T1 ⦠⡠break [ term 46 l ] break ⦠term 46 G2, break term 46 L2 , break term 46 T2 ⦠)"
non associative with precedence 45
- for @{ 'LazyEq $d $G1 $L1 $T1 $G2 $L2 $T2 }.
+ for @{ 'LazyEq $l $G1 $L1 $T1 $G2 $L2 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyor_5.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyor_5.ma
index 905905214..6479c1f71 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyor_5.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lazyor_5.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L1 â break [ term 46 T , break term 46 d ] break term 46 L2 â¡ break term 46 L )"
+notation "hvbox( L1 â break [ term 46 T , break term 46 l ] break term 46 L2 â¡ break term 46 L )"
non associative with precedence 45
- for @{ 'LazyOr $L1 $T $d $L2 $L }.
+ for @{ 'LazyOr $L1 $T $l $L2 $L }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lrsubeq_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lrsubeq_4.ma
index f38c1f86a..994e6b22b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lrsubeq_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/lrsubeq_4.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L1 break â [ term 46 d , break term 46 e ] break term 46 L2 )"
+notation "hvbox( L1 break â [ term 46 l , break term 46 m ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'LRSubEq $L1 $d $e $L2 }.
+ for @{ 'LRSubEq $L1 $l $m $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/midiso_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/midiso_4.ma
index ea603e8b2..dc380d849 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/midiso_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/midiso_4.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( L1 ⩬ break [ term 46 d , break term 46 e ] break term 46 L2 )"
+notation "hvbox( L1 ⩬ break [ term 46 l , break term 46 m ] break term 46 L2 )"
non associative with precedence 45
- for @{ 'MidIso $d $e $L1 $L2 }.
+ for @{ 'MidIso $l $m $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/nativevalid_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/nativevalid_6.ma
index a192a58cd..449fe1452 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/nativevalid_6.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/nativevalid_6.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T ¡ break [ term 46 h , break term 46 g , break term 46 l ] )"
+notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T ¡ break [ term 46 h , break term 46 g , break term 46 d ] )"
non associative with precedence 45
- for @{ 'NativeValid $h $g $l $G $L $T }.
+ for @{ 'NativeValid $h $g $d $G $L $T }.
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
index 31fe21410..e35f7ca23 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubst_6.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubst_6.ma
@@ -14,6 +14,6 @@
(* 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 )"
+notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T1 break ⶠ[ term 46 l , break term 46 m ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'PSubst $G $L $T1 $d $e $T2 }.
+ for @{ 'PSubst $G $L $T1 $l $m $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
index f2bb4bd78..2dd1f2d6e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubststar_6.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubststar_6.ma
@@ -14,6 +14,6 @@
(* 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 )"
+notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T1 break ⶠ* [ term 46 l , break term 46 m ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'PSubstStar $G $L $T1 $d $e $T2 }.
+ for @{ 'PSubstStar $G $L $T1 $l $m $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
index e727d4945..97227dccf 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubststaralt_6.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/psubststaralt_6.ma
@@ -14,6 +14,6 @@
(* 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 )"
+notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T1 break ⶠⶠ* [ term 46 l , break term 46 m ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'PSubstStarAlt $G $L $T1 $d $e $T2 }.
+ for @{ 'PSubstStarAlt $G $L $T1 $l $m $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_3.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_3.ma
index 2d43a13fb..a761d47c6 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_3.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⬠[ term 46 e ] break term 46 L1 ⡠break term 46 L2 )"
+notation "hvbox( ⬠[ term 46 m ] break term 46 L1 ⡠break term 46 L2 )"
non associative with precedence 45
- for @{ 'RDrop $e $L1 $L2 }.
+ for @{ 'RDrop $m $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_4.ma
index 1728856fd..2345f0684 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_4.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⬠[ term 46 d , break term 46 e ] break term 46 L1 ⡠break term 46 L2 )"
+notation "hvbox( ⬠[ term 46 l , break term 46 m ] break term 46 L1 ⡠break term 46 L2 )"
non associative with precedence 45
- for @{ 'RDrop $d $e $L1 $L2 }.
+ for @{ 'RDrop $l $m $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_5.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_5.ma
index 3fbc10c64..d8c06726e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_5.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdrop_5.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⬠[ term 46 s , break term 46 d , break term 46 e ] break term 46 L1 ⡠break term 46 L2 )"
+notation "hvbox( ⬠[ term 46 s , break term 46 l , break term 46 m ] break term 46 L1 ⡠break term 46 L2 )"
non associative with precedence 45
- for @{ 'RDrop $s $d $e $L1 $L2 }.
+ for @{ 'RDrop $s $l $m $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdropstar_3.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdropstar_3.ma
index 85ce21e3d..8a6f2f4c1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdropstar_3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdropstar_3.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⬠* [ term 46 e ] break term 46 L1 ⡠break term 46 L2 )"
+notation "hvbox( ⬠* [ term 46 m ] break term 46 L1 ⡠break term 46 L2 )"
non associative with precedence 45
- for @{ 'RDropStar $e $L1 $L2 }.
+ for @{ 'RDropStar $m $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdropstar_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdropstar_4.ma
index e7a9c77e6..291de4c0e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdropstar_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rdropstar_4.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⬠* [ term 46 s , break term 46 e ] break term 46 L1 ⡠break term 46 L2 )"
+notation "hvbox( ⬠* [ term 46 s , break term 46 m ] break term 46 L1 ⡠break term 46 L2 )"
non associative with precedence 45
- for @{ 'RDropStar $s $e $L1 $L2 }.
+ for @{ 'RDropStar $s $m $L1 $L2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rlift_4.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rlift_4.ma
index 77c922073..e2a03284e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rlift_4.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rlift_4.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⬠[ term 46 d , break term 46 e ] break term 46 T1 ⡠break term 46 T2 )"
+notation "hvbox( ⬠[ term 46 l , break term 46 m ] break term 46 T1 ⡠break term 46 T2 )"
non associative with precedence 45
- for @{ 'RLift $d $e $T1 $T2 }.
+ for @{ 'RLift $l $m $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rliftstar_3.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rliftstar_3.ma
index ce6541c52..af4de47ba 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rliftstar_3.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/rliftstar_3.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⬠* [ term 46 e ] break term 46 T1 ⡠break term 46 T2 )"
+notation "hvbox( ⬠* [ term 46 m ] break term 46 T1 ⡠break term 46 T2 )"
non associative with precedence 45
- for @{ 'RLiftStar $e $T1 $T2 }.
+ for @{ 'RLiftStar $m $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/sn_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/sn_6.ma
index 95bec1559..c402cdc8a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/sn_6.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/sn_6.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( G ⢠⬠* break [ term 46 h , break term 46 g , break term 46 T , break term 46 d ] break term 46 L )"
+notation "hvbox( G ⢠⬠* break [ term 46 h , break term 46 g , break term 46 T , break term 46 l ] break term 46 L )"
non associative with precedence 45
- for @{ 'SN $h $g $T $d $G $L }.
+ for @{ 'SN $h $g $T $l $G $L }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/snalt_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/snalt_6.ma
index f1baa6805..4beb9fdcf 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/snalt_6.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/snalt_6.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( G ⢠⬠⬠* break [ term 46 h , break term 46 g , break term 46 T , break term 46 d ] break term 46 L )"
+notation "hvbox( G ⢠⬠⬠* break [ term 46 h , break term 46 g , break term 46 T , break term 46 l ] break term 46 L )"
non associative with precedence 45
- for @{ 'SNAlt $h $g $T $d $G $L }.
+ for @{ 'SNAlt $h $g $T $l $G $L }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/statictypestar_6.ma b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/statictypestar_6.ma
index a6c09b0a0..b7d6a71e0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/notation/relations/statictypestar_6.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/notation/relations/statictypestar_6.ma
@@ -14,6 +14,6 @@
(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T1 â¢* break [ term 46 h , break term 46 l ] break term 46 T2 )"
+notation "hvbox( ⦠term 46 G , break term 46 L ⦠⢠break term 46 T1 â¢* break [ term 46 h , break term 46 n ] break term 46 T2 )"
non associative with precedence 45
- for @{ 'StaticTypeStar $h $G $L $l $T1 $T2 }.
+ for @{ 'StaticTypeStar $h $G $L $n $T1 $T2 }.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_lift.ma
index b9f4102a6..68578e5f0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cir_lift.ma
@@ -19,10 +19,10 @@ include "basic_2/reduction/cir.ma".
(* Properties on relocation *************************************************)
-lemma cir_lift: âG,K,T. â¦G, K⦠⢠⡠ðâ¦T⦠â âL,s,d,e. â¬[s, d, e] L â¡ K â
- âU. â¬[d, e] T â¡ U â â¦G, L⦠⢠⡠ðâ¦Uâ¦.
+lemma cir_lift: âG,K,T. â¦G, K⦠⢠⡠ðâ¦T⦠â âL,s,l,m. â¬[s, l, m] L â¡ K â
+ âU. â¬[l, m] T â¡ U â â¦G, L⦠⢠⡠ðâ¦Uâ¦.
/3 width=8 by crr_inv_lift/ qed.
-lemma cir_inv_lift: âG,L,U. â¦G, L⦠⢠⡠ðâ¦U⦠â âK,s,d,e. â¬[s, d, e] L â¡ K â
- âT. â¬[d, e] T â¡ U â â¦G, K⦠⢠⡠ðâ¦Tâ¦.
+lemma cir_inv_lift: âG,L,U. â¦G, L⦠⢠⡠ðâ¦U⦠â âK,s,l,m. â¬[s, l, m] L â¡ K â
+ âT. â¬[l, m] T â¡ U â â¦G, K⦠⢠⡠ðâ¦Tâ¦.
/3 width=8 by crr_lift/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma
index 6683871ad..9532052b8 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix.ma
@@ -26,7 +26,7 @@ interpretation "irreducibility for context-sensitive extended reduction (term)"
(* Basic inversion lemmas ***************************************************)
-lemma cix_inv_sort: âh,g,G,L,k,l. deg h g k (l+1) â â¦G, L⦠⢠â¡[h, g] ðâ¦âk⦠â â¥.
+lemma cix_inv_sort: âh,g,G,L,k,d. deg h g k (d+1) â â¦G, L⦠⢠â¡[h, g] ðâ¦âk⦠â â¥.
/3 width=2 by crx_sort/ qed-.
lemma cix_inv_delta: âh,g,I,G,L,K,V,i. â¬[i] L â¡ K.â{I}V â â¦G, L⦠⢠â¡[h, g] ðâ¦#i⦠â â¥.
@@ -69,12 +69,12 @@ lemma cix_inv_cir: âh,g,G,L,T. â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â â¦G,
(* Basic properties *********************************************************)
lemma cix_sort: âh,g,G,L,k. deg h g k 0 â â¦G, L⦠⢠â¡[h, g] ðâ¦âkâ¦.
-#h #g #G #L #k #Hk #H elim (crx_inv_sort ⦠H) -L #l #Hkl
-lapply (deg_mono ⦠Hk Hkl) -h -k <plus_n_Sm #H destruct
+#h #g #G #L #k #Hk #H elim (crx_inv_sort ⦠H) -L #d #Hkd
+lapply (deg_mono ⦠Hk Hkd) -h -k <plus_n_Sm #H destruct
qed.
lemma tix_lref: âh,g,G,i. â¦G, â⦠⢠â¡[h, g] ðâ¦#iâ¦.
-#h #g #G #i #H elim (trx_inv_atom ⦠H) -H #k #l #_ #H destruct
+#h #g #G #i #H elim (trx_inv_atom ⦠H) -H #k #d #_ #H destruct
qed.
lemma cix_gref: âh,g,G,L,p. â¦G, L⦠⢠â¡[h, g] ðâ¦Â§pâ¦.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_lift.ma
index 37835b0f1..e27040b5f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cix_lift.ma
@@ -26,10 +26,10 @@ qed.
(* Properties on relocation *************************************************)
-lemma cix_lift: âh,g,G,K,T. â¦G, K⦠⢠â¡[h, g] ðâ¦T⦠â âL,s,d,e. â¬[s, d, e] L â¡ K â
- âU. â¬[d, e] T â¡ U â â¦G, L⦠⢠â¡[h, g] ðâ¦Uâ¦.
+lemma cix_lift: âh,g,G,K,T. â¦G, K⦠⢠â¡[h, g] ðâ¦T⦠â âL,s,l,m. â¬[s, l, m] L â¡ K â
+ âU. â¬[l, m] T â¡ U â â¦G, L⦠⢠â¡[h, g] ðâ¦Uâ¦.
/3 width=8 by crx_inv_lift/ qed.
-lemma cix_inv_lift: âh,g,G,L,U. â¦G, L⦠⢠â¡[h, g] ðâ¦U⦠â âK,s,d,e. â¬[s, d, e] L â¡ K â
- âT. â¬[d, e] T â¡ U â â¦G, K⦠⢠â¡[h, g] ðâ¦Tâ¦.
+lemma cix_inv_lift: âh,g,G,L,U. â¦G, L⦠⢠â¡[h, g] ðâ¦U⦠â âK,s,l,m. â¬[s, l, m] L â¡ K â
+ âT. â¬[l, m] T â¡ U â â¦G, K⦠⢠â¡[h, g] ðâ¦Tâ¦.
/3 width=8 by crx_lift/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_lift.ma
index 96ba32505..d280b1e83 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnr_lift.ma
@@ -30,20 +30,20 @@ qed.
(* Relocation properties ****************************************************)
(* Basic_1: was: nf2_lift *)
-lemma cnr_lift: âG,L0,L,T,T0,s,d,e. â¦G, L⦠⢠⡠ðâ¦T⦠â
- â¬[s, d, e] L0 â¡ L â â¬[d, e] T â¡ T0 â â¦G, L0⦠⢠⡠ðâ¦T0â¦.
-#G #L0 #L #T #T0 #s #d #e #HLT #HL0 #HT0 #X #H
+lemma cnr_lift: âG,L0,L,T,T0,s,l,m. â¦G, L⦠⢠⡠ðâ¦T⦠â
+ â¬[s, l, m] L0 â¡ L â â¬[l, m] T â¡ T0 â â¦G, L0⦠⢠⡠ðâ¦T0â¦.
+#G #L0 #L #T #T0 #s #l #m #HLT #HL0 #HT0 #X #H
elim (cpr_inv_lift1 ⦠H ⦠HL0 ⦠HT0) -L0 #T1 #HT10 #HT1
<(HLT ⦠HT1) in HT0; -L #HT0
>(lift_mono ⦠HT10 ⦠HT0) -T1 -X //
qed.
(* Note: this was missing in basic_1 *)
-lemma cnr_inv_lift: âG,L0,L,T,T0,s,d,e. â¦G, L0⦠⢠⡠ðâ¦T0⦠â
- â¬[s, d, e] L0 â¡ L â â¬[d, e] T â¡ T0 â â¦G, L⦠⢠⡠ðâ¦Tâ¦.
-#G #L0 #L #T #T0 #s #d #e #HLT0 #HL0 #HT0 #X #H
-elim (lift_total X d e) #X0 #HX0
+lemma cnr_inv_lift: âG,L0,L,T,T0,s,l,m. â¦G, L0⦠⢠⡠ðâ¦T0⦠â
+ â¬[s, l, m] L0 â¡ L â â¬[l, m] T â¡ T0 â â¦G, L⦠⢠⡠ðâ¦Tâ¦.
+#G #L0 #L #T #T0 #s #l #m #HLT0 #HL0 #HT0 #X #H
+elim (lift_total X l m) #X0 #HX0
lapply (cpr_lift ⦠H ⦠HL0 ⦠HT0 ⦠HX0) -L #HTX0
>(HLT0 ⦠HTX0) in HX0; -L0 -X0 #H
->(lift_inj ⦠H ⦠HT0) -T0 -X -s -d -e //
+>(lift_inj ⦠H ⦠HT0) -T0 -X -s -l -m //
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx.ma
index 0a23bcf03..e6d1fcf4b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx.ma
@@ -29,9 +29,11 @@ interpretation
lemma cnx_inv_sort: âh,g,G,L,k. â¦G, L⦠⢠â¡[h, g] ðâ¦âk⦠â deg h g k 0.
#h #g #G #L #k #H elim (deg_total h g k)
-#l @(nat_ind_plus ⦠l) -l // #l #_ #Hkl
-lapply (H (â(next h k)) ?) -H /2 width=2 by cpx_st/ -L -l #H destruct -H -e0 (**) (* destruct does not remove some premises *)
-lapply (next_lt h k) >e1 -e1 #H elim (lt_refl_false ⦠H)
+#d @(nat_ind_plus ⦠d) -d // #d #_ #Hkd
+lapply (H (â(next h k)) ?) -H /2 width=2 by cpx_st/ -L -d #H
+lapply (destruct_tatom_tatom_aux ⦠H) -H #H (**) (* destruct lemma needed *)
+lapply (destruct_sort_sort_aux ⦠H) -H #H (**) (* destruct lemma needed *)
+lapply (next_lt h k) >H -H #H elim (lt_refl_false ⦠H)
qed-.
lemma cnx_inv_delta: âh,g,I,G,L,K,V,i. â¬[i] L â¡ K.â{I}V â â¦G, L⦠⢠â¡[h, g] ðâ¦#i⦠â â¥.
@@ -97,13 +99,13 @@ qed-.
(* Basic properties *********************************************************)
lemma cnx_sort: âh,g,G,L,k. deg h g k 0 â â¦G, L⦠⢠â¡[h, g] ðâ¦âkâ¦.
-#h #g #G #L #k #Hk #X #H elim (cpx_inv_sort1 ⦠H) -H // * #l #Hkl #_
-lapply (deg_mono ⦠Hkl Hk) -h -L <plus_n_Sm #H destruct
+#h #g #G #L #k #Hk #X #H elim (cpx_inv_sort1 ⦠H) -H // * #d #Hkd #_
+lapply (deg_mono ⦠Hkd Hk) -h -L <plus_n_Sm #H destruct
qed.
-lemma cnx_sort_iter: âh,g,G,L,k,l. deg h g k l â â¦G, L⦠⢠â¡[h, g] ðâ¦â((next h)^l k)â¦.
-#h #g #G #L #k #l #Hkl
-lapply (deg_iter ⦠l Hkl) -Hkl <minus_n_n /2 width=6 by cnx_sort/
+lemma cnx_sort_iter: âh,g,G,L,k,d. deg h g k d â â¦G, L⦠⢠â¡[h, g] ðâ¦â((next h)^d k)â¦.
+#h #g #G #L #k #d #Hkd
+lapply (deg_iter ⦠d Hkd) -Hkd <minus_n_n /2 width=6 by cnx_sort/
qed.
lemma cnx_lref_free: âh,g,G,L,i. |L| ⤠i â â¦G, L⦠⢠â¡[h, g] ðâ¦#iâ¦.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_crx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_crx.ma
index c4948b776..ca66d9271 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_crx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_crx.ma
@@ -22,9 +22,9 @@ include "basic_2/reduction/cnx.ma".
(* Note: this property is unusual *)
lemma cnx_inv_crx: âh,g,G,L,T. â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â â¥.
#h #g #G #L #T #H elim H -L -T
-[ #L #k #l #Hkl #H
+[ #L #k #d #Hkd #H
lapply (cnx_inv_sort ⦠H) -H #H
- lapply (deg_mono ⦠H Hkl) -h -L -k <plus_n_Sm #H destruct
+ lapply (deg_mono ⦠H Hkd) -h -L -k <plus_n_Sm #H destruct
| #I #L #K #V #i #HLK #H
elim (cnx_inv_delta ⦠HLK H)
| #L #V #T #_ #IHV #H
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_lift.ma
index f313ee8bf..e66223cac 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cnx_lift.ma
@@ -19,19 +19,19 @@ include "basic_2/reduction/cnx.ma".
(* Relocation properties ****************************************************)
-lemma cnx_lift: âh,g,G,L0,L,T,T0,s,d,e. â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â â¬[s, d, e] L0 â¡ L â
- â¬[d, e] T â¡ T0 â â¦G, L0⦠⢠â¡[h, g] ðâ¦T0â¦.
-#h #g #G #L0 #L #T #T0 #s #d #e #HLT #HL0 #HT0 #X #H
+lemma cnx_lift: âh,g,G,L0,L,T,T0,s,l,m. â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â â¬[s, l, m] L0 â¡ L â
+ â¬[l, m] T â¡ T0 â â¦G, L0⦠⢠â¡[h, g] ðâ¦T0â¦.
+#h #g #G #L0 #L #T #T0 #s #l #m #HLT #HL0 #HT0 #X #H
elim (cpx_inv_lift1 ⦠H ⦠HL0 ⦠HT0) -L0 #T1 #HT10 #HT1
<(HLT ⦠HT1) in HT0; -L #HT0
>(lift_mono ⦠HT10 ⦠HT0) -T1 -X //
qed.
-lemma cnx_inv_lift: âh,g,G,L0,L,T,T0,s,d,e. â¦G, L0⦠⢠â¡[h, g] ðâ¦T0⦠â â¬[s, d, e] L0 â¡ L â
- â¬[d, e] T â¡ T0 â â¦G, L⦠⢠â¡[h, g] ðâ¦Tâ¦.
-#h #g #G #L0 #L #T #T0 #s #d #e #HLT0 #HL0 #HT0 #X #H
-elim (lift_total X d e) #X0 #HX0
+lemma cnx_inv_lift: âh,g,G,L0,L,T,T0,s,l,m. â¦G, L0⦠⢠â¡[h, g] ðâ¦T0⦠â â¬[s, l, m] L0 â¡ L â
+ â¬[l, m] T â¡ T0 â â¦G, L⦠⢠â¡[h, g] ðâ¦Tâ¦.
+#h #g #G #L0 #L #T #T0 #s #l #m #HLT0 #HL0 #HT0 #X #H
+elim (lift_total X l m) #X0 #HX0
lapply (cpx_lift ⦠H ⦠HL0 ⦠HT0 ⦠HX0) -L #HTX0
>(HLT0 ⦠HTX0) in HX0; -L0 -X0 #H
->(lift_inj ⦠H ⦠HT0) -T0 -X -d -e //
+>(lift_inj ⦠H ⦠HT0) -T0 -X -l -m //
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma
index dc9203525..61d8fd931 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr.ma
@@ -76,31 +76,31 @@ lemma cpr_pair_sn: âI,G,L,V1,V2. â¦G, L⦠⢠V1 â¡ V2 â
âT. â¦G, L⦠⢠â¡{I}V1.T â¡ â¡{I}V2.T.
* /2 width=1 by cpr_bind, cpr_flat/ qed.
-lemma cpr_delift: âG,K,V,T1,L,d. â¬[d] L â¡ (K.âV) â
- ââT2,T. â¦G, L⦠⢠T1 â¡ T2 & â¬[d, 1] T â¡ T2.
+lemma cpr_delift: âG,K,V,T1,L,l. â¬[l] L â¡ (K.âV) â
+ ââT2,T. â¦G, L⦠⢠T1 â¡ T2 & â¬[l, 1] T â¡ T2.
#G #K #V #T1 elim T1 -T1
[ * /2 width=4 by cpr_atom, lift_sort, lift_gref, ex2_2_intro/
- #i #L #d #HLK elim (lt_or_eq_or_gt i d)
- #Hid [1,3: /3 width=4 by cpr_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
+ #i #L #l #HLK elim (lt_or_eq_or_gt i l)
+ #Hil [1,3: /3 width=4 by cpr_atom, 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) /3 width=6 by cpr_delta, ex2_2_intro/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
elim (IHW1 ⦠HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. â{I}W1) (d+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/
+ [ elim (IHU1 (L. â{I}W1) (l+1)) -IHU1 /3 width=9 by drop_drop, cpr_bind, lift_bind, ex2_2_intro/
| elim (IHU1 ⦠HLK) -IHU1 -HLK /3 width=8 by cpr_flat, lift_flat, ex2_2_intro/
]
]
qed-.
-fact lstas_cpr_aux: âh,G,L,T1,T2,l. â¦G, L⦠⢠T1 â¢*[h, l] T2 â
- l = 0 â â¦G, L⦠⢠T1 â¡ T2.
-#h #G #L #T1 #T2 #l #H elim H -G -L -T1 -T2 -l
+fact lstas_cpr_aux: âh,G,L,T1,T2,d. â¦G, L⦠⢠T1 â¢*[h, d] T2 â
+ d = 0 â â¦G, L⦠⢠T1 â¡ T2.
+#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d
/3 width=1 by cpr_eps, cpr_flat, cpr_bind/
-[ #G #L #l #k #H0 destruct normalize //
-| #G #L #K #V1 #V2 #W2 #i #l #HLK #_ #HVW2 #IHV12 #H destruct
+[ #G #L #d #k #H0 destruct normalize //
+| #G #L #K #V1 #V2 #W2 #i #d #HLK #_ #HVW2 #IHV12 #H destruct
/3 width=6 by cpr_delta/
-| #G #L #K #V1 #V2 #W2 #i #l #_ #_ #_ #_ <plus_n_Sm #H destruct
+| #G #L #K #V1 #V2 #W2 #i #d #_ #_ #_ #_ <plus_n_Sm #H destruct
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_lift.ma
index 33e019d2d..85d034bd9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_lift.ma
@@ -20,36 +20,36 @@ include "basic_2/reduction/cpr.ma".
(* Relocation properties ****************************************************)
(* Basic_1: includes: pr0_lift pr2_lift *)
-lemma cpr_lift: âG. l_liftable (cpr G).
+lemma cpr_lift: âG. d_liftable (cpr G).
#G #K #T1 #T2 #H elim H -G -K -T1 -T2
-[ #I #G #K #L #s #d #e #_ #U1 #H1 #U2 #H2
+[ #I #G #K #L #s #l #m #_ #U1 #H1 #U2 #H2
>(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
-| #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #d #e #HLK #U1 #H #U2 #HWU2
- elim (lift_inv_lref1 ⦠H) * #Hid #H destruct
+| #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #l #m #HLK #U1 #H #U2 #HWU2
+ elim (lift_inv_lref1 ⦠H) * #Hil #H destruct
[ elim (lift_trans_ge ⦠HVW2 ⦠HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
elim (drop_trans_le ⦠HLK ⦠HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil
#K #Y #HKV #HVY #H destruct /3 width=9 by cpr_delta/
| lapply (lift_trans_be ⦠HVW2 ⦠HWU2 ? ?) -W2 /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (drop_trans_ge_comm ⦠HLK ⦠HKV ?) -K // -Hid /3 width=6 by cpr_delta, drop_inv_gen/
+ lapply (drop_trans_ge_comm ⦠HLK ⦠HKV ?) -K // -Hil /3 width=6 by cpr_delta, drop_inv_gen/
]
-| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #d #e #HLK #U1 #H1 #U2 #H2
+| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=6 by cpr_bind, drop_skip/
-| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #d #e #HLK #U1 #H1 #U2 #H2
+| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpr_flat/
-| #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #s #d #e #HLK #U1 #H #U2 #HTU2
+| #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #H #U2 #HTU2
elim (lift_inv_bind1 ⦠H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
elim (lift_conf_O1 ⦠HTU2 ⦠HT2) -T2 /4 width=6 by cpr_zeta, drop_skip/
-| #G #K #V #T1 #T2 #_ #IHT12 #L #s #d #e #HLK #U1 #H #U2 #HTU2
+| #G #K #V #T1 #T2 #_ #IHT12 #L #s #l #m #HLK #U1 #H #U2 #HTU2
elim (lift_inv_flat1 ⦠H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpr_eps/
-| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #d #e #HLK #X1 #HX1 #X2 #HX2
+| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 ⦠HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 ⦠HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_inv_bind1 ⦠HX2) -HX2 #X #T3 #HX #HT23 #HX2 destruct
elim (lift_inv_flat1 ⦠HX) -HX #W3 #V3 #HW23 #HV23 #HX destruct /4 width=6 by cpr_beta, drop_skip/
-| #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #s #d #e #HLK #X1 #HX1 #X2 #HX2
+| #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 ⦠HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 ⦠HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_inv_bind1 ⦠HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
@@ -59,46 +59,46 @@ lemma cpr_lift: âG. l_liftable (cpr G).
qed.
(* Basic_1: includes: pr0_gen_lift pr2_gen_lift *)
-lemma cpr_inv_lift1: âG. l_deliftable_sn (cpr G).
+lemma cpr_inv_lift1: âG. d_deliftable_sn (cpr G).
#G #L #U1 #U2 #H elim H -G -L -U1 -U2
-[ * #i #G #L #K #s #d #e #_ #T1 #H
+[ * #i #G #L #K #s #l #m #_ #T1 #H
[ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
| lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
]
-| #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #d #e #HLK #T1 #H
- elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct
+| #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #l #m #HLK #T1 #H
+ elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct
[ elim (drop_conf_lt ⦠HLK ⦠HLV) -L // #L #U #HKL #HLV #HUV
elim (IHV2 ⦠HLV ⦠HUV) -V #U2 #HUV2 #HU2
elim (lift_trans_le ⦠HUV2 ⦠HVW2) -V2 // >minus_plus <plus_minus_m_m /3 width=8 by cpr_delta, ex2_intro/
- | elim (le_inv_plus_l ⦠Hid) #Hdie #Hei
+ | elim (le_inv_plus_l ⦠Hil) #Hlim #Hmi
lapply (drop_conf_ge ⦠HLK ⦠HLV ?) -L // #HKLV
- elim (lift_split ⦠HVW2 d (i - e + 1)) -HVW2 [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hid -Hdie
+ elim (lift_split ⦠HVW2 l (i - m + 1)) -HVW2 [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hil -Hlim
#V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O
/3 width=8 by cpr_delta, ex2_intro/
]
-| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
+| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
elim (lift_inv_bind2 ⦠H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 ⦠HLK ⦠HWV1) -IHV12 #W2 #HW12 #HWV2
elim (IHU12 ⦠HTU1) -IHU12 -HTU1 /3 width=6 by cpr_bind, drop_skip, lift_bind, ex2_intro/
-| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
+| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 ⦠HLK ⦠HWV1) -V1
elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK /3 width=6 by cpr_flat, lift_flat, ex2_intro/
-| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #X #H
+| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #X #H
elim (lift_inv_bind2 ⦠H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHU1 (K.âW1) s ⦠HTU1) /2 width=1 by drop_skip/ -L -U1 #T #HTU #HT1
elim (lift_div_le ⦠HU2 ⦠HTU) -U /3 width=6 by cpr_zeta, ex2_intro/
-| #G #L #V #U1 #U2 #_ #IHU12 #K #s #d #e #HLK #X #H
+| #G #L #V #U1 #U2 #_ #IHU12 #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHU12 ⦠HLK ⦠HTU1) -L -U1 /3 width=3 by cpr_eps, ex2_intro/
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
elim (lift_inv_flat2 ⦠HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 ⦠HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
elim (IHV12 ⦠HLK ⦠HV01) -V1
elim (IHT12 (K.âW0) s ⦠HT01) -T1 /2 width=1 by drop_skip/
elim (IHW12 ⦠HLK ⦠HW01) -W1 /4 width=7 by cpr_beta, lift_flat, lift_bind, ex2_intro/
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
elim (lift_inv_flat2 ⦠HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 ⦠HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
elim (IHV1 ⦠HLK ⦠HV01) -V1 #V3 #HV3 #HV03
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_llpx_sn.ma
index 814cba15d..d8d4df555 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_llpx_sn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpr_llpx_sn.ma
@@ -19,12 +19,12 @@ include "basic_2/reduction/cpr.ma".
(* Properties on lazy sn pointwise extensions *******************************)
-lemma cpr_llpx_sn_conf: âR. (âL. reflexive ⦠(R L)) â l_liftable R â l_deliftable_sn R â
+lemma cpr_llpx_sn_conf: âR. (âL. reflexive ⦠(R L)) â d_liftable R â d_deliftable_sn R â
âG. s_r_confluent1 ⦠(cpr G) (llpx_sn R 0).
#R #H1R #H2R #H3R #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
[ //
| #G #Ls #Ks #V1s #V2s #W2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn ⦠H ⦠HLKs) // -H
- #Kd #V1d #HLKd #HV1s #HV1sd
+ #Kd #V1l #HLKd #HV1s #HV1sd
lapply (drop_fwd_drop2 ⦠HLKs) -HLKs #HLKs
lapply (drop_fwd_drop2 ⦠HLKd) -HLKd #HLKd
@(llpx_sn_lift_le ⦠HLKs HLKd ⦠HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
index dd8fcdb85..6de953b28 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx.ma
@@ -21,7 +21,7 @@ include "basic_2/reduction/cpr.ma".
(* avtivate genv *)
inductive cpx (h) (g): relation4 genv lenv term term â
| cpx_atom : âI,G,L. cpx h g G L (âª{I}) (âª{I})
-| cpx_st : âG,L,k,l. deg h g k (l+1) â cpx h g G L (âk) (â(next h k))
+| cpx_st : âG,L,k,d. deg h g k (d+1) â cpx h g G L (âk) (â(next h k))
| cpx_delta: âI,G,L,K,V,V2,W2,i.
â¬[i] L â¡ K.â{I}V â cpx h g G K V V2 â
â¬[0, i+1] V2 â¡ W2 â cpx h g G L (#i) W2
@@ -78,17 +78,17 @@ lemma cpx_pair_sn: âh,g,I,G,L,V1,V2. â¦G, L⦠⢠V1 â¡[h, g] V2 â
#h #g * /2 width=1 by cpx_bind, cpx_flat/
qed.
-lemma cpx_delift: âh,g,I,G,K,V,T1,L,d. â¬[d] L â¡ (K.â{I}V) â
- ââT2,T. â¦G, L⦠⢠T1 â¡[h, g] T2 & â¬[d, 1] T â¡ T2.
+lemma cpx_delift: âh,g,I,G,K,V,T1,L,l. â¬[l] L â¡ (K.â{I}V) â
+ ââT2,T. â¦G, L⦠⢠T1 â¡[h, g] T2 & â¬[l, 1] T â¡ T2.
#h #g #I #G #K #V #T1 elim T1 -T1
-[ * #i #L #d /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
- elim (lt_or_eq_or_gt i d) #Hid [1,3: /3 width=4 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_2_intro/ ]
+[ * #i #L #l /2 width=4 by cpx_atom, lift_sort, lift_gref, ex2_2_intro/
+ elim (lt_or_eq_or_gt i l) #Hil [1,3: /3 width=4 by cpx_atom, 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) /3 width=7 by cpx_delta, ex2_2_intro/
-| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #d #HLK
+| * [ #a ] #I #W1 #U1 #IHW1 #IHU1 #L #l #HLK
elim (IHW1 ⦠HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L. â{I} W1) (d+1)) -IHU1 /3 width=9 by cpx_bind, drop_drop, lift_bind, ex2_2_intro/
+ [ elim (IHU1 (L. â{I} W1) (l+1)) -IHU1 /3 width=9 by cpx_bind, drop_drop, lift_bind, ex2_2_intro/
| elim (IHU1 ⦠HLK) -IHU1 -HLK /3 width=8 by cpx_flat, lift_flat, ex2_2_intro/
]
]
@@ -98,12 +98,12 @@ qed-.
fact cpx_inv_atom1_aux: âh,g,G,L,T1,T2. â¦G, L⦠⢠T1 â¡[h, g] T2 â âJ. T1 = âª{J} â
â¨â¨ T2 = âª{J}
- | ââk,l. deg h g k (l+1) & T2 = â(next h k) & J = Sort k
+ | ââk,d. deg h g k (d+1) & T2 = â(next h k) & J = Sort k
| ââI,K,V,V2,i. â¬[i] L â¡ K.â{I}V & â¦G, K⦠⢠V â¡[h, g] V2 &
â¬[O, i+1] V2 â¡ T2 & J = LRef i.
#G #h #g #L #T1 #T2 * -L -T1 -T2
[ #I #G #L #J #H destruct /2 width=1 by or3_intro0/
-| #G #L #k #l #Hkl #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
+| #G #L #k #d #Hkd #J #H destruct /3 width=5 by or3_intro1, ex3_2_intro/
| #I #G #L #K #V #V2 #T2 #i #HLK #HV2 #HVT2 #J #H destruct /3 width=9 by or3_intro2, ex4_5_intro/
| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #H destruct
@@ -117,16 +117,16 @@ qed-.
lemma cpx_inv_atom1: âh,g,J,G,L,T2. â¦G, L⦠⢠âª{J} â¡[h, g] T2 â
â¨â¨ T2 = âª{J}
- | ââk,l. deg h g k (l+1) & T2 = â(next h k) & J = Sort k
+ | ââk,d. deg h g k (d+1) & T2 = â(next h k) & J = Sort k
| ââI,K,V,V2,i. â¬[i] L â¡ K.â{I}V & â¦G, K⦠⢠V â¡[h, g] V2 &
â¬[O, i+1] V2 â¡ T2 & J = LRef i.
/2 width=3 by cpx_inv_atom1_aux/ qed-.
lemma cpx_inv_sort1: âh,g,G,L,T2,k. â¦G, L⦠⢠âk â¡[h, g] T2 â T2 = âk â¨
- ââl. deg h g k (l+1) & T2 = â(next h k).
+ ââd. deg h g k (d+1) & T2 = â(next h k).
#h #g #G #L #T2 #k #H
elim (cpx_inv_atom1 ⦠H) -H /2 width=1 by or_introl/ *
-[ #k0 #l0 #Hkl0 #H1 #H2 destruct /3 width=4 by ex2_intro, or_intror/
+[ #k0 #d0 #Hkd0 #H1 #H2 destruct /3 width=4 by ex2_intro, or_intror/
| #I #K #V #V2 #i #_ #_ #_ #H destruct
]
qed-.
@@ -137,7 +137,7 @@ lemma cpx_inv_lref1: âh,g,G,L,T2,i. â¦G, L⦠⢠#i â¡[h, g] T2 â
â¬[O, i+1] V2 â¡ T2.
#h #g #G #L #T2 #i #H
elim (cpx_inv_atom1 ⦠H) -H /2 width=1 by or_introl/ *
-[ #k #l #_ #_ #H destruct
+[ #k #d #_ #_ #H destruct
| #I #K #V #V2 #j #HLK #HV2 #HVT2 #H destruct /3 width=7 by ex3_4_intro, or_intror/
]
qed-.
@@ -151,7 +151,7 @@ qed-.
lemma cpx_inv_gref1: âh,g,G,L,T2,p. â¦G, L⦠⢠§p â¡[h, g] T2 â T2 = §p.
#h #g #G #L #T2 #p #H
elim (cpx_inv_atom1 ⦠H) -H // *
-[ #k #l #_ #_ #H destruct
+[ #k #d #_ #_ #H destruct
| #I #K #V #V2 #i #_ #_ #_ #H destruct
]
qed-.
@@ -165,7 +165,7 @@ fact cpx_inv_bind1_aux: âh,g,G,L,U1,U2. â¦G, L⦠⢠U1 â¡[h, g] U2 â
a = true & J = Abbr.
#h #g #G #L #U1 #U2 * -L -U1 -U2
[ #I #G #L #b #J #W #U1 #H destruct
-| #G #L #k #l #_ #b #J #W #U1 #H destruct
+| #G #L #k #d #_ #b #J #W #U1 #H destruct
| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #b #J #W #U1 #H destruct
| #a #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #b #J #W #U1 #H destruct /3 width=5 by ex3_2_intro, or_introl/
| #I #G #L #V1 #V2 #T1 #T2 #_ #_ #b #J #W #U1 #H destruct
@@ -220,7 +220,7 @@ fact cpx_inv_flat1_aux: âh,g,G,L,U,U2. â¦G, L⦠⢠U â¡[h, g] U2 â
U2 = â{a}W2.âV2.T2 & J = Appl.
#h #g #G #L #U #U2 * -L -U -U2
[ #I #G #L #J #W #U1 #H destruct
-| #G #L #k #l #_ #J #W #U1 #H destruct
+| #G #L #k #d #_ #J #W #U1 #H destruct
| #I #G #L #K #V #V2 #W2 #i #_ #_ #_ #J #W #U1 #H destruct
| #a #I #G #L #V1 #V2 #T1 #T2 #_ #_ #J #W #U1 #H destruct
| #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #J #W #U1 #H destruct /3 width=5 by or5_intro0, ex3_2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cix.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cix.ma
index b6e057702..ecd2a5317 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cix.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_cix.ma
@@ -22,7 +22,7 @@ include "basic_2/reduction/cpx.ma".
lemma cpx_fwd_cix: âh,g,G,L,T1,T2. â¦G, L⦠⢠T1 â¡[h, g] T2 â â¦G, L⦠⢠â¡[h, g] ðâ¦T1⦠â T2 = T1.
#h #g #G #L #T1 #T2 #H elim H -G -L -T1 -T2
[ //
-| #G #L #k #l #Hkl #H elim (cix_inv_sort ⦠Hkl H)
+| #G #L #k #d #Hkd #H elim (cix_inv_sort ⦠Hkd H)
| #I #G #L #K #V1 #V2 #W2 #i #HLK #_ #HVW2 #IHV12 #H
elim (cix_inv_delta ⦠HLK) //
| #a * #G #L #V1 #V2 #T1 #T2 #_ #_ #IHV1 #IHT1 #H
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_leq.ma
deleted file mode 100644
index 85b3bec55..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_leq.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/substitution/drop_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_st/
-| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
- elim (leq_drop_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_eps, cpx_ct/
-|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_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma
index f6df750d1..d16ff6b2e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lift.ma
@@ -21,18 +21,18 @@ include "basic_2/reduction/cpx.ma".
(* Advanced properties ******************************************************)
-fact sta_cpx_aux: âh,g,G,L,T1,T2,l2,l1. â¦G, L⦠⢠T1 â¢*[h, l2] T2 â l2 = 1 â
- â¦G, L⦠⢠T1 âª[h, g] l1+1 â â¦G, L⦠⢠T1 â¡[h, g] T2.
-#h #g #G #L #T1 #T2 #l2 #l1 #H elim H -G -L -T1 -T2 -l2
-[ #G #L #l2 #k #H0 destruct normalize
+fact sta_cpx_aux: âh,g,G,L,T1,T2,d2,d1. â¦G, L⦠⢠T1 â¢*[h, d2] T2 â d2 = 1 â
+ â¦G, L⦠⢠T1 âª[h, g] d1+1 â â¦G, L⦠⢠T1 â¡[h, g] T2.
+#h #g #G #L #T1 #T2 #d2 #d1 #H elim H -G -L -T1 -T2 -d2
+[ #G #L #d2 #k #H0 destruct normalize
/3 width=4 by cpx_st, da_inv_sort/
-| #G #L #K #V1 #V2 #W2 #i #l2 #HLK #_ #HVW2 #IHV12 #H0 #H destruct
- elim (da_inv_lref ⦠H) -H * #K0 #V0 [| #l0 ] #HLK0
+| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #_ #HVW2 #IHV12 #H0 #H destruct
+ elim (da_inv_lref ⦠H) -H * #K0 #V0 [| #d0 ] #HLK0
lapply (drop_mono ⦠HLK0 ⦠HLK) -HLK0 #H destruct /3 width=7 by cpx_delta/
| #G #L #K #V1 #V2 #i #_ #_ #_ #H destruct
-| #G #L #K #V1 #V2 #W2 #i #l2 #HLK #HV12 #HVW2 #_ #H0 #H
+| #G #L #K #V1 #V2 #W2 #i #d2 #HLK #HV12 #HVW2 #_ #H0 #H
lapply (discr_plus_xy_y ⦠H0) -H0 #H0 destruct
- elim (da_inv_lref ⦠H) -H * #K0 #V0 [| #l0 ] #HLK0
+ elim (da_inv_lref ⦠H) -H * #K0 #V0 [| #d0 ] #HLK0
lapply (drop_mono ⦠HLK0 ⦠HLK) -HLK0 #H destruct
/4 width=7 by cpx_delta, cpr_cpx, lstas_cpr/
| /4 width=2 by cpx_bind, da_inv_bind/
@@ -41,47 +41,47 @@ fact sta_cpx_aux: âh,g,G,L,T1,T2,l2,l1. â¦G, L⦠⢠T1 â¢*[h, l2] T2 â l
]
qed-.
-lemma sta_cpx: âh,g,G,L,T1,T2,l. â¦G, L⦠⢠T1 â¢*[h, 1] T2 â
- â¦G, L⦠⢠T1 âª[h, g] l+1 â â¦G, L⦠⢠T1 â¡[h, g] T2.
+lemma sta_cpx: âh,g,G,L,T1,T2,d. â¦G, L⦠⢠T1 â¢*[h, 1] T2 â
+ â¦G, L⦠⢠T1 âª[h, g] d+1 â â¦G, L⦠⢠T1 â¡[h, g] T2.
/2 width=3 by sta_cpx_aux/ qed.
(* Relocation properties ****************************************************)
-lemma cpx_lift: âh,g,G. l_liftable (cpx h g G).
+lemma cpx_lift: âh,g,G. d_liftable (cpx h g G).
#h #g #G #K #T1 #T2 #H elim H -G -K -T1 -T2
-[ #I #G #K #L #s #d #e #_ #U1 #H1 #U2 #H2
+[ #I #G #K #L #s #l #m #_ #U1 #H1 #U2 #H2
>(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
-| #G #K #k #l #Hkl #L #s #d #e #_ #U1 #H1 #U2 #H2
+| #G #K #k #d #Hkd #L #s #l #m #_ #U1 #H1 #U2 #H2
>(lift_inv_sort1 ⦠H1) -U1
>(lift_inv_sort1 ⦠H2) -U2 /2 width=2 by cpx_st/
-| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #d #e #HLK #U1 #H #U2 #HWU2
- elim (lift_inv_lref1 ⦠H) * #Hid #H destruct
+| #I #G #K #KV #V #V2 #W2 #i #HKV #HV2 #HVW2 #IHV2 #L #s #l #m #HLK #U1 #H #U2 #HWU2
+ elim (lift_inv_lref1 ⦠H) * #Hil #H destruct
[ elim (lift_trans_ge ⦠HVW2 ⦠HWU2) -W2 // <minus_plus #W2 #HVW2 #HWU2
elim (drop_trans_le ⦠HLK ⦠HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil
#K #Y #HKV #HVY #H destruct /3 width=10 by cpx_delta/
| lapply (lift_trans_be ⦠HVW2 ⦠HWU2 ? ?) -W2 /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
lapply (drop_trans_ge_comm ⦠HLK ⦠HKV ?) -K /3 width=7 by cpx_delta, drop_inv_gen/
]
-| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #d #e #HLK #U1 #H1 #U2 #H2
+| #a #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /4 width=6 by cpx_bind, drop_skip/
-| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #d #e #HLK #U1 #H1 #U2 #H2
+| #I #G #K #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L #s #l #m #HLK #U1 #H1 #U2 #H2
elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1 destruct
elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct /3 width=6 by cpx_flat/
-| #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #s #d #e #HLK #U1 #H #U2 #HTU2
+| #G #K #V #T1 #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #H #U2 #HTU2
elim (lift_inv_bind1 ⦠H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct
elim (lift_conf_O1 ⦠HTU2 ⦠HT2) -T2 /4 width=6 by cpx_zeta, drop_skip/
-| #G #K #V #T1 #T2 #_ #IHT12 #L #s #d #e #HLK #U1 #H #U2 #HTU2
+| #G #K #V #T1 #T2 #_ #IHT12 #L #s #l #m #HLK #U1 #H #U2 #HTU2
elim (lift_inv_flat1 ⦠H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_eps/
-| #G #K #V1 #V2 #T #_ #IHV12 #L #s #d #e #HLK #U1 #H #U2 #HVU2
+| #G #K #V1 #V2 #T #_ #IHV12 #L #s #l #m #HLK #U1 #H #U2 #HVU2
elim (lift_inv_flat1 ⦠H) -H #VV1 #TT1 #HVV1 #HTT1 #H destruct /3 width=6 by cpx_ct/
-| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #d #e #HLK #X1 #HX1 #X2 #HX2
+| #a #G #K #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 ⦠HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 ⦠HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_inv_bind1 ⦠HX2) -HX2 #X #T3 #HX #HT23 #HX2 destruct
elim (lift_inv_flat1 ⦠HX) -HX #W3 #V3 #HW23 #HV23 #HX destruct /4 width=6 by cpx_beta, drop_skip/
-| #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #s #d #e #HLK #X1 #HX1 #X2 #HX2
+| #a #G #K #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L #s #l #m #HLK #X1 #HX1 #X2 #HX2
elim (lift_inv_flat1 ⦠HX1) -HX1 #V0 #X #HV10 #HX #HX1 destruct
elim (lift_inv_bind1 ⦠HX) -HX #W0 #T0 #HW0 #HT10 #HX destruct
elim (lift_inv_bind1 ⦠HX2) -HX2 #W3 #X #HW23 #HX #HX2 destruct
@@ -90,51 +90,51 @@ lemma cpx_lift: âh,g,G. l_liftable (cpx h g G).
]
qed.
-lemma cpx_inv_lift1: âh,g,G. l_deliftable_sn (cpx h g G).
+lemma cpx_inv_lift1: âh,g,G. d_deliftable_sn (cpx h g G).
#h #g #G #L #U1 #U2 #H elim H -G -L -U1 -U2
-[ * #i #G #L #K #s #d #e #_ #T1 #H
+[ * #i #G #L #K #s #l #m #_ #T1 #H
[ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by cpx_atom, lift_sort, ex2_intro/
- | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct /3 width=3 by cpx_atom, lift_lref_ge_minus, lift_lref_lt, ex2_intro/
| lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by cpx_atom, lift_gref, ex2_intro/
]
-| #G #L #k #l #Hkl #K #s #d #e #_ #T1 #H
+| #G #L #k #d #Hkd #K #s #l #m #_ #T1 #H
lapply (lift_inv_sort2 ⦠H) -H #H destruct /3 width=3 by cpx_st, lift_sort, ex2_intro/
-| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #d #e #HLK #T1 #H
- elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct
+| #I #G #L #LV #V #V2 #W2 #i #HLV #HV2 #HVW2 #IHV2 #K #s #l #m #HLK #T1 #H
+ elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct
[ elim (drop_conf_lt ⦠HLK ⦠HLV) -L // #L #U #HKL #HLV #HUV
elim (IHV2 ⦠HLV ⦠HUV) -V #U2 #HUV2 #HU2
elim (lift_trans_le ⦠HUV2 ⦠HVW2) -V2 // >minus_plus <plus_minus_m_m /3 width=9 by cpx_delta, ex2_intro/
- | elim (le_inv_plus_l ⦠Hid) #Hdie #Hei
+ | elim (le_inv_plus_l ⦠Hil) #Hlim #Hmi
lapply (drop_conf_ge ⦠HLK ⦠HLV ?) -L // #HKLV
- elim (lift_split ⦠HVW2 d (i - e + 1)) -HVW2 /3 width=1 by le_S, le_S_S/ -Hid -Hdie
+ elim (lift_split ⦠HVW2 l (i - m + 1)) -HVW2 /3 width=1 by le_S, le_S_S/ -Hil -Hlim
#V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O /3 width=9 by cpx_delta, ex2_intro/
]
-| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
+| #a #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
elim (lift_inv_bind2 ⦠H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 ⦠HLK ⦠HWV1) -IHV12 #W2 #HW12 #HWV2
elim (IHU12 ⦠HTU1) -IHU12 -HTU1 /3 width=6 by cpx_bind, drop_skip, lift_bind, ex2_intro/
-| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #d #e #HLK #X #H
+| #I #G #L #V1 #V2 #U1 #U2 #_ #_ #IHV12 #IHU12 #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 ⦠HLK ⦠HWV1) -V1
elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK /3 width=5 by cpx_flat, lift_flat, ex2_intro/
-| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #X #H
+| #G #L #V #U1 #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #X #H
elim (lift_inv_bind2 ⦠H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHU1 (K.âW1) s ⦠HTU1) /2 width=1/ -L -U1 #T #HTU #HT1
elim (lift_div_le ⦠HU2 ⦠HTU) -U /3 width=5 by cpx_zeta, ex2_intro/
-| #G #L #V #U1 #U2 #_ #IHU12 #K #s #d #e #HLK #X #H
+| #G #L #V #U1 #U2 #_ #IHU12 #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHU12 ⦠HLK ⦠HTU1) -L -U1 /3 width=3 by cpx_eps, ex2_intro/
-| #G #L #V1 #V2 #U1 #_ #IHV12 #K #s #d #e #HLK #X #H
+| #G #L #V1 #V2 #U1 #_ #IHV12 #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #W1 #T1 #HWV1 #HTU1 #H destruct
elim (IHV12 ⦠HLK ⦠HWV1) -L -V1 /3 width=3 by cpx_ct, ex2_intro/
-| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
+| #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
elim (lift_inv_flat2 ⦠HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 ⦠HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
elim (IHV12 ⦠HLK ⦠HV01) -V1 #V3 #HV32 #HV03
elim (IHT12 (K.âW0) s ⦠HT01) -T1 /2 width=1 by drop_skip/ #T3 #HT32 #HT03
elim (IHW12 ⦠HLK ⦠HW01) -W1
/4 width=7 by cpx_beta, lift_bind, lift_flat, ex2_intro/
-| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #s #d #e #HLK #X #HX
+| #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #K #s #l #m #HLK #X #HX
elim (lift_inv_flat2 ⦠HX) -HX #V0 #Y #HV01 #HY #HX destruct
elim (lift_inv_bind2 ⦠HY) -HY #W0 #T0 #HW01 #HT01 #HY destruct
elim (IHV1 ⦠HLK ⦠HV01) -V1 #V3 #HV3 #HV03
@@ -155,15 +155,15 @@ lemma fqu_cpx_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2, L2, T
[ #I #G #L #V2 #U2 #HVU2
elim (lift_total U2 0 1)
/4 width=7 by fqu_drop, cpx_delta, drop_pair, drop_drop, ex2_intro/
-| #G #L #K #T1 #U1 #e #HLK1 #HTU1 #T2 #HTU2
- elim (lift_total T2 0 (e+1))
+| #G #L #K #T1 #U1 #m #HLK1 #HTU1 #T2 #HTU2
+ elim (lift_total T2 0 (m+1))
/3 width=11 by cpx_lift, fqu_drop, ex2_intro/
]
qed-.
lemma fqu_sta_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2, L2, T2⦠â
âU2. â¦G2, L2⦠⢠T2 â¢*[h, 1] U2 â
- âl. â¦G2, L2⦠⢠T2 âª[h, g] l+1 â
+ âd. â¦G2, L2⦠⢠T2 âª[h, g] d+1 â
ââU1. â¦G1, L1⦠⢠T1 â¡[h, g] U1 & â¦G1, L1, U1⦠â â¦G2, L2, U2â¦.
/3 width=5 by fqu_cpx_trans, sta_cpx/ qed-.
@@ -178,7 +178,7 @@ qed-.
lemma fquq_sta_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â⸮ â¦G2, L2, T2⦠â
âU2. â¦G2, L2⦠⢠T2 â¢*[h, 1] U2 â
- âl. â¦G2, L2⦠⢠T2 âª[h, g] l+1 â
+ âd. â¦G2, L2⦠⢠T2 âª[h, g] d+1 â
ââU1. â¦G1, L1⦠⢠T1 â¡[h, g] U1 & â¦G1, L1, U1⦠â⸮ â¦G2, L2, U2â¦.
/3 width=5 by fquq_cpx_trans, sta_cpx/ qed-.
@@ -224,7 +224,7 @@ lemma fqu_cpx_trans_neq: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2, L
[1,3: /2 width=4 by fqu_flat_dx, cpx_flat/
| #H0 destruct /2 width=1 by/
]
-| #G #L #K #T1 #U1 #e #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (e+1))
+| #G #L #K #T1 #U1 #m #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (m+1))
#U2 #HTU2 @(ex3_intro ⦠U2)
[1,3: /2 width=10 by cpx_lift, fqu_drop/
| #H0 destruct /3 width=5 by lift_inj/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_llpx_sn.ma
index 3a12659f3..13fe1ae17 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_llpx_sn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_llpx_sn.ma
@@ -20,13 +20,13 @@ include "basic_2/reduction/cpx.ma".
(* Properties on lazy sn pointwise extensions *******************************)
(* Note: lemma 1000 *)
-lemma cpx_llpx_sn_conf: âR. (âL. reflexive ⦠(R L)) â l_liftable R â l_deliftable_sn R â
+lemma cpx_llpx_sn_conf: âR. (âL. reflexive ⦠(R L)) â d_liftable R â d_deliftable_sn R â
âh,g,G. s_r_confluent1 ⦠(cpx h g G) (llpx_sn R 0).
#R #H1R #H2R #H3R #h #g #G #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2
[ //
| /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
| #I #G #Ls #Ks #V1s #V2s #W2s #i #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn ⦠H ⦠HLKs) // -H
- #Kd #V1d #HLKd #HV1s #HV1sd
+ #Kd #V1l #HLKd #HV1s #HV1sd
lapply (drop_fwd_drop2 ⦠HLKs) -HLKs #HLKs
lapply (drop_fwd_drop2 ⦠HLKd) -HLKd #HLKd
@(llpx_sn_lift_le ⦠HLKs HLKd ⦠HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lreq.ma
new file mode 100644
index 000000000..4ab54737f
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/cpx_lreq.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/substitution/drop_lreq.ma".
+include "basic_2/reduction/cpx.ma".
+
+(* CONTEXT-SENSITIVE EXTENDED PARALLEL REDUCTION FOR TERMS ******************)
+
+(* Properties on equivalence for local environments *************************)
+
+lemma lreq_cpx_trans: âh,g,G. lsub_trans ⦠(cpx h g G) (lreq 0 (â)).
+#h #g #G #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2
+[ //
+| /2 width=2 by cpx_st/
+| #I #G #L1 #K1 #V1 #V2 #W2 #i #HLK1 #_ #HVW2 #IHV12 #L2 #HL12
+ elim (lreq_drop_trans_be ⦠HL12 ⦠HLK1) // -HL12 -HLK1 /3 width=7 by cpx_delta/
+|4,9: /4 width=1 by cpx_bind, cpx_beta, lreq_pair_O_Y/
+|5,7,8: /3 width=1 by cpx_flat, cpx_eps, cpx_ct/
+|6,10: /4 width=3 by cpx_zeta, cpx_theta, lreq_pair_O_Y/
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_lift.ma
index ee142ffe3..cc4370f45 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/crr_lift.ma
@@ -19,55 +19,55 @@ include "basic_2/reduction/crr.ma".
(* Properties on relocation *************************************************)
-lemma crr_lift: âG,K,T. â¦G, K⦠⢠⡠ðâ¦T⦠â âL,s,d,e. â¬[s, d, e] L â¡ K â
- âU. â¬[d, e] T â¡ U â â¦G, L⦠⢠⡠ðâ¦Uâ¦.
+lemma crr_lift: âG,K,T. â¦G, K⦠⢠⡠ðâ¦T⦠â âL,s,l,m. â¬[s, l, m] L â¡ K â
+ âU. â¬[l, m] T â¡ U â â¦G, L⦠⢠⡠ðâ¦Uâ¦.
#G #K #T #H elim H -K -T
-[ #K #K0 #V #i #HK0 #L #s #d #e #HLK #X #H
- elim (lift_inv_lref1 ⦠H) -H * #Hid #H destruct
+[ #K #K0 #V #i #HK0 #L #s #l #m #HLK #X #H
+ elim (lift_inv_lref1 ⦠H) -H * #Hil #H destruct
[ elim (drop_trans_lt ⦠HLK ⦠HK0) -K /2 width=4 by crr_delta/
| lapply (drop_trans_ge ⦠HLK ⦠HK0 ?) -K /3 width=4 by crr_delta, drop_inv_gen/
]
-| #K #V #T #_ #IHV #L #s #d #e #HLK #X #H
+| #K #V #T #_ #IHV #L #s #l #m #HLK #X #H
elim (lift_inv_flat1 ⦠H) -H #W #U #HVW #_ #H destruct /3 width=5 by crr_appl_sn/
-| #K #V #T #_ #IHT #L #s #d #e #HLK #X #H
+| #K #V #T #_ #IHT #L #s #l #m #HLK #X #H
elim (lift_inv_flat1 ⦠H) -H #W #U #_ #HTU #H destruct /3 width=5 by crr_appl_dx/
-| #I #K #V #T #HI #L #s #d #e #_ #X #H
+| #I #K #V #T #HI #L #s #l #m #_ #X #H
elim (lift_fwd_pair1 ⦠H) -H #W #U #_ #H destruct /2 width=1 by crr_ri2/
-| #a #I #K #V #T #HI #_ #IHV #L #s #d #e #HLK #X #H
+| #a #I #K #V #T #HI #_ #IHV #L #s #l #m #HLK #X #H
elim (lift_inv_bind1 ⦠H) -H #W #U #HVW #_ #H destruct /3 width=5 by crr_ib2_sn/
-| #a #I #K #V #T #HI #_ #IHT #L #s #d #e #HLK #X #H
+| #a #I #K #V #T #HI #_ #IHT #L #s #l #m #HLK #X #H
elim (lift_inv_bind1 ⦠H) -H #W #U #HVW #HTU #H destruct /4 width=5 by crr_ib2_dx, drop_skip/
-| #a #K #V #V0 #T #L #s #d #e #_ #X #H
+| #a #K #V #V0 #T #L #s #l #m #_ #X #H
elim (lift_inv_flat1 ⦠H) -H #W #X0 #_ #H0 #H destruct
elim (lift_inv_bind1 ⦠H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crr_beta/
-| #a #K #V #V0 #T #L #s #d #e #_ #X #H
+| #a #K #V #V0 #T #L #s #l #m #_ #X #H
elim (lift_inv_flat1 ⦠H) -H #W #X0 #_ #H0 #H destruct
elim (lift_inv_bind1 ⦠H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crr_theta/
]
qed.
-lemma crr_inv_lift: âG,L,U. â¦G, L⦠⢠⡠ðâ¦U⦠â âK,s,d,e. â¬[s, d, e] L â¡ K â
- âT. â¬[d, e] T â¡ U â â¦G, K⦠⢠⡠ðâ¦Tâ¦.
+lemma crr_inv_lift: âG,L,U. â¦G, L⦠⢠⡠ðâ¦U⦠â âK,s,l,m. â¬[s, l, m] L â¡ K â
+ âT. â¬[l, m] T â¡ U â â¦G, K⦠⢠⡠ðâ¦Tâ¦.
#G #L #U #H elim H -L -U
-[ #L #L0 #W #i #HK0 #K #s #d #e #HLK #X #H
- elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct
+[ #L #L0 #W #i #HK0 #K #s #l #m #HLK #X #H
+ elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct
[ elim (drop_conf_lt ⦠HLK ⦠HK0) -L /2 width=4 by crr_delta/
| lapply (drop_conf_ge ⦠HLK ⦠HK0 ?) -L /2 width=4 by crr_delta/
]
-| #L #W #U #_ #IHW #K #s #d #e #HLK #X #H
+| #L #W #U #_ #IHW #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #V #T #HVW #_ #H destruct /3 width=5 by crr_appl_sn/
-| #L #W #U #_ #IHU #K #s #d #e #HLK #X #H
+| #L #W #U #_ #IHU #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #V #T #_ #HTU #H destruct /3 width=5 by crr_appl_dx/
-| #I #L #W #U #HI #K #s #d #e #_ #X #H
+| #I #L #W #U #HI #K #s #l #m #_ #X #H
elim (lift_fwd_pair2 ⦠H) -H #V #T #_ #H destruct /2 width=1 by crr_ri2/
-| #a #I #L #W #U #HI #_ #IHW #K #s #d #e #HLK #X #H
+| #a #I #L #W #U #HI #_ #IHW #K #s #l #m #HLK #X #H
elim (lift_inv_bind2 ⦠H) -H #V #T #HVW #_ #H destruct /3 width=5 by crr_ib2_sn/
-| #a #I #L #W #U #HI #_ #IHU #K #s #d #e #HLK #X #H
+| #a #I #L #W #U #HI #_ #IHU #K #s #l #m #HLK #X #H
elim (lift_inv_bind2 ⦠H) -H #V #T #HVW #HTU #H destruct /4 width=5 by crr_ib2_dx, drop_skip/
-| #a #L #W #W0 #U #K #s #d #e #_ #X #H
+| #a #L #W #W0 #U #K #s #l #m #_ #X #H
elim (lift_inv_flat2 ⦠H) -H #V #X0 #_ #H0 #H destruct
elim (lift_inv_bind2 ⦠H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crr_beta/
-| #a #L #W #W0 #U #K #s #d #e #_ #X #H
+| #a #L #W #W0 #U #K #s #l #m #_ #X #H
elim (lift_inv_flat2 ⦠H) -H #V #X0 #_ #H0 #H destruct
elim (lift_inv_bind2 ⦠H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crr_theta/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma
index fe2ff03f9..a4509ec79 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx.ma
@@ -21,7 +21,7 @@ include "basic_2/reduction/crr.ma".
(* activate genv *)
(* extended reducible terms *)
inductive crx (h) (g) (G:genv): relation2 lenv term â
-| crx_sort : âL,k,l. deg h g k (l+1) â crx h g G L (âk)
+| crx_sort : âL,k,d. deg h g k (d+1) â crx h g G L (âk)
| crx_delta : âI,L,K,V,i. â¬[i] L â¡ K.â{I}V â crx h g G L (#i)
| crx_appl_sn: âL,V,T. crx h g G L V â crx h g G L (âV.T)
| crx_appl_dx: âL,V,T. crx h g G L T â crx h g G L (âV.T)
@@ -46,9 +46,9 @@ qed.
(* Basic inversion lemmas ***************************************************)
fact crx_inv_sort_aux: âh,g,G,L,T,k. â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â T = âk â
- âl. deg h g k (l+1).
+ âd. deg h g k (d+1).
#h #g #G #L #T #k0 * -L -T
-[ #L #k #l #Hkl #H destruct /2 width=2 by ex_intro/
+[ #L #k #d #Hkd #H destruct /2 width=2 by ex_intro/
| #I #L #K #V #i #HLK #H destruct
| #L #V #T #_ #H destruct
| #L #V #T #_ #H destruct
@@ -60,13 +60,13 @@ fact crx_inv_sort_aux: âh,g,G,L,T,k. â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â
]
qed-.
-lemma crx_inv_sort: âh,g,G,L,k. â¦G, L⦠⢠â¡[h, g] ðâ¦âk⦠â âl. deg h g k (l+1).
+lemma crx_inv_sort: âh,g,G,L,k. â¦G, L⦠⢠â¡[h, g] ðâ¦âk⦠â âd. deg h g k (d+1).
/2 width=5 by crx_inv_sort_aux/ qed-.
fact crx_inv_lref_aux: âh,g,G,L,T,i. â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â T = #i â
ââI,K,V. â¬[i] L â¡ K.â{I}V.
#h #g #G #L #T #j * -L -T
-[ #L #k #l #_ #H destruct
+[ #L #k #d #_ #H destruct
| #I #L #K #V #i #HLK #H destruct /2 width=4 by ex1_3_intro/
| #L #V #T #_ #H destruct
| #L #V #T #_ #H destruct
@@ -83,7 +83,7 @@ lemma crx_inv_lref: âh,g,G,L,i. â¦G, L⦠⢠â¡[h, g] ðâ¦#i⦠â â
fact crx_inv_gref_aux: âh,g,G,L,T,p. â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â T = §p â â¥.
#h #g #G #L #T #q * -L -T
-[ #L #k #l #_ #H destruct
+[ #L #k #d #_ #H destruct
| #I #L #K #V #i #HLK #H destruct
| #L #V #T #_ #H destruct
| #L #V #T #_ #H destruct
@@ -99,7 +99,7 @@ lemma crx_inv_gref: âh,g,G,L,p. â¦G, L⦠⢠â¡[h, g] ðâ¦Â§p⦠â â¥
/2 width=8 by crx_inv_gref_aux/ qed-.
lemma trx_inv_atom: âh,g,I,G. â¦G, â⦠⢠â¡[h, g] ðâ¦âª{I}⦠â
- ââk,l. deg h g k (l+1) & I = Sort k.
+ ââk,d. deg h g k (d+1) & I = Sort k.
#h #g * #i #G #H
[ elim (crx_inv_sort ⦠H) -H /2 width=4 by ex2_2_intro/
| elim (crx_inv_lref ⦠H) -H #I #L #V #H
@@ -111,7 +111,7 @@ qed-.
fact crx_inv_ib2_aux: âh,g,a,I,G,L,W,U,T. ib2 a I â â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â
T = â{a,I}W.U â â¦G, L⦠⢠â¡[h, g] ðâ¦W⦠⨠â¦G, L.â{I}W⦠⢠â¡[h, g] ðâ¦Uâ¦.
#h #g #b #J #G #L #W0 #U #T #HI * -L -T
-[ #L #k #l #_ #H destruct
+[ #L #k #d #_ #H destruct
| #I #L #K #V #i #_ #H destruct
| #L #V #T #_ #H destruct
| #L #V #T #_ #H destruct
@@ -132,7 +132,7 @@ lemma crx_inv_ib2: âh,g,a,I,G,L,W,T. ib2 a I â â¦G, L⦠⢠â¡[h, g] ð
fact crx_inv_appl_aux: âh,g,G,L,W,U,T. â¦G, L⦠⢠â¡[h, g] ðâ¦T⦠â T = âW.U â
â¨â¨ â¦G, L⦠⢠â¡[h, g] ðâ¦W⦠| â¦G, L⦠⢠â¡[h, g] ðâ¦U⦠| (ðâ¦U⦠â â¥).
#h #g #G #L #W0 #U #T * -L -T
-[ #L #k #l #_ #H destruct
+[ #L #k #d #_ #H destruct
| #I #L #K #V #i #_ #H destruct
| #L #V #T #HV #H destruct /2 width=1 by or3_intro0/
| #L #V #T #HT #H destruct /2 width=1 by or3_intro1/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_lift.ma
index 3298022ae..017f089fe 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/crx_lift.ma
@@ -19,59 +19,59 @@ include "basic_2/reduction/crx.ma".
(* Properties on relocation *************************************************)
-lemma crx_lift: âh,g,G,K,T. â¦G, K⦠⢠â¡[h, g] ðâ¦T⦠â âL,s,d,e. â¬[s, d, e] L â¡ K â
- âU. â¬[d, e] T â¡ U â â¦G, L⦠⢠â¡[h, g] ðâ¦Uâ¦.
+lemma crx_lift: âh,g,G,K,T. â¦G, K⦠⢠â¡[h, g] ðâ¦T⦠â âL,s,l,m. â¬[s, l, m] L â¡ K â
+ âU. â¬[l, m] T â¡ U â â¦G, L⦠⢠â¡[h, g] ðâ¦Uâ¦.
#h #g #G #K #T #H elim H -K -T
-[ #K #k #l #Hkl #L #s #d #e #_ #X #H
+[ #K #k #d #Hkd #L #s #l #m #_ #X #H
>(lift_inv_sort1 ⦠H) -X /2 width=2 by crx_sort/
-| #I #K #K0 #V #i #HK0 #L #s #d #e #HLK #X #H
- elim (lift_inv_lref1 ⦠H) -H * #Hid #H destruct
+| #I #K #K0 #V #i #HK0 #L #s #l #m #HLK #X #H
+ elim (lift_inv_lref1 ⦠H) -H * #Hil #H destruct
[ elim (drop_trans_lt ⦠HLK ⦠HK0) -K /2 width=4 by crx_delta/
| lapply (drop_trans_ge ⦠HLK ⦠HK0 ?) -K /3 width=5 by crx_delta, drop_inv_gen/
]
-| #K #V #T #_ #IHV #L #s #d #e #HLK #X #H
+| #K #V #T #_ #IHV #L #s #l #m #HLK #X #H
elim (lift_inv_flat1 ⦠H) -H #W #U #HVW #_ #H destruct /3 width=5 by crx_appl_sn/
-| #K #V #T #_ #IHT #L #s #d #e #HLK #X #H
+| #K #V #T #_ #IHT #L #s #l #m #HLK #X #H
elim (lift_inv_flat1 ⦠H) -H #W #U #_ #HTU #H destruct /3 width=5 by crx_appl_dx/
-| #I #K #V #T #HI #L #s #d #e #_ #X #H
+| #I #K #V #T #HI #L #s #l #m #_ #X #H
elim (lift_fwd_pair1 ⦠H) -H #W #U #_ #H destruct /2 width=1 by crx_ri2/
-| #a #I #K #V #T #HI #_ #IHV #L #s #d #e #HLK #X #H
+| #a #I #K #V #T #HI #_ #IHV #L #s #l #m #HLK #X #H
elim (lift_inv_bind1 ⦠H) -H #W #U #HVW #_ #H destruct /3 width=5 by crx_ib2_sn/
-| #a #I #K #V #T #HI #_ #IHT #L #s #d #e #HLK #X #H
+| #a #I #K #V #T #HI #_ #IHT #L #s #l #m #HLK #X #H
elim (lift_inv_bind1 ⦠H) -H #W #U #HVW #HTU #H destruct /4 width=5 by crx_ib2_dx, drop_skip/
-| #a #K #V #V0 #T #L #s #d #e #_ #X #H
+| #a #K #V #V0 #T #L #s #l #m #_ #X #H
elim (lift_inv_flat1 ⦠H) -H #W #X0 #_ #H0 #H destruct
elim (lift_inv_bind1 ⦠H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crx_beta/
-| #a #K #V #V0 #T #L #s #d #e #_ #X #H
+| #a #K #V #V0 #T #L #s #l #m #_ #X #H
elim (lift_inv_flat1 ⦠H) -H #W #X0 #_ #H0 #H destruct
elim (lift_inv_bind1 ⦠H0) -H0 #W0 #U #_ #_ #H0 destruct /2 width=1 by crx_theta/
]
qed.
-lemma crx_inv_lift: âh,g,G,L,U. â¦G, L⦠⢠â¡[h, g] ðâ¦U⦠â âK,s,d,e. â¬[s, d, e] L â¡ K â
- âT. â¬[d, e] T â¡ U â â¦G, K⦠⢠â¡[h, g] ðâ¦Tâ¦.
+lemma crx_inv_lift: âh,g,G,L,U. â¦G, L⦠⢠â¡[h, g] ðâ¦U⦠â âK,s,l,m. â¬[s, l, m] L â¡ K â
+ âT. â¬[l, m] T â¡ U â â¦G, K⦠⢠â¡[h, g] ðâ¦Tâ¦.
#h #g #G #L #U #H elim H -L -U
-[ #L #k #l #Hkl #K #s #d #e #_ #X #H
+[ #L #k #d #Hkd #K #s #l #m #_ #X #H
>(lift_inv_sort2 ⦠H) -X /2 width=2 by crx_sort/
-| #I #L #L0 #W #i #HK0 #K #s #d #e #HLK #X #H
- elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct
+| #I #L #L0 #W #i #HK0 #K #s #l #m #HLK #X #H
+ elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct
[ elim (drop_conf_lt ⦠HLK ⦠HK0) -L /2 width=4 by crx_delta/
| lapply (drop_conf_ge ⦠HLK ⦠HK0 ?) -L /2 width=4 by crx_delta/
]
-| #L #W #U #_ #IHW #K #s #d #e #HLK #X #H
+| #L #W #U #_ #IHW #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #V #T #HVW #_ #H destruct /3 width=5 by crx_appl_sn/
-| #L #W #U #_ #IHU #K #s #d #e #HLK #X #H
+| #L #W #U #_ #IHU #K #s #l #m #HLK #X #H
elim (lift_inv_flat2 ⦠H) -H #V #T #_ #HTU #H destruct /3 width=5 by crx_appl_dx/
-| #I #L #W #U #HI #K #s #d #e #_ #X #H
+| #I #L #W #U #HI #K #s #l #m #_ #X #H
elim (lift_fwd_pair2 ⦠H) -H #V #T #_ #H destruct /2 width=1 by crx_ri2/
-| #a #I #L #W #U #HI #_ #IHW #K #s #d #e #HLK #X #H
+| #a #I #L #W #U #HI #_ #IHW #K #s #l #m #HLK #X #H
elim (lift_inv_bind2 ⦠H) -H #V #T #HVW #_ #H destruct /3 width=5 by crx_ib2_sn/
-| #a #I #L #W #U #HI #_ #IHU #K #s #d #e #HLK #X #H
+| #a #I #L #W #U #HI #_ #IHU #K #s #l #m #HLK #X #H
elim (lift_inv_bind2 ⦠H) -H #V #T #HVW #HTU #H destruct /4 width=5 by crx_ib2_dx, drop_skip/
-| #a #L #W #W0 #U #K #s #d #e #_ #X #H
+| #a #L #W #W0 #U #K #s #l #m #_ #X #H
elim (lift_inv_flat2 ⦠H) -H #V #X0 #_ #H0 #H destruct
elim (lift_inv_bind2 ⦠H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crx_beta/
-| #a #L #W #W0 #U #K #s #d #e #_ #X #H
+| #a #L #W #W0 #U #K #s #l #m #_ #X #H
elim (lift_inv_flat2 ⦠H) -H #V #X0 #_ #H0 #H destruct
elim (lift_inv_bind2 ⦠H0) -H0 #V0 #T #_ #_ #H0 destruct /2 width=1 by crx_theta/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/fpb_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/fpb_lift.ma
index 5adf85e51..8efea75f0 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/fpb_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/fpb_lift.ma
@@ -20,9 +20,9 @@ include "basic_2/reduction/fpb.ma".
(* Advanced properties ******************************************************)
-lemma sta_fpb: âh,g,G,L,T1,T2,l. â¦G, L⦠⢠T1 âª[h, g] l+1 â
+lemma sta_fpb: âh,g,G,L,T1,T2,d. â¦G, L⦠⢠T1 âª[h, g] d+1 â
â¦G, L⦠⢠T1 â¢*[h, 1] T2 â â¦G, L, T1⦠â»[h, g] â¦G, L, T2â¦.
-#h #g #G #L #T1 #T2 #l #HT1 #HT12 elim (eq_term_dec T1 T2)
+#h #g #G #L #T1 #T2 #d #HT1 #HT12 elim (eq_term_dec T1 T2)
/3 width=2 by fpb_cpx, sta_cpx/ #H destruct
elim (lstas_inv_refl_pos h G L T2 0) //
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/fpbq_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/fpbq_lift.ma
index d432fae27..9d2846331 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/fpbq_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/fpbq_lift.ma
@@ -19,7 +19,7 @@ include "basic_2/reduction/fpbq.ma".
(* Advanced properties ******************************************************)
-lemma sta_fpbq: âh,g,G,L,T1,T2,l.
- â¦G, L⦠⢠T1 âª[h, g] l+1 â â¦G, L⦠⢠T1 â¢*[h, 1] T2 â
+lemma sta_fpbq: âh,g,G,L,T1,T2,d.
+ â¦G, L⦠⢠T1 âª[h, g] d+1 â â¦G, L⦠⢠T1 â¢*[h, 1] T2 â
â¦G, L, T1⦠â½[h, g] â¦G, L, T2â¦.
/3 width=4 by fpbq_cpx, sta_cpx/ qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_drop.ma
index f9701a1ab..8254c0bde 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpr_drop.ma
@@ -39,8 +39,8 @@ lemma fqu_cpr_trans_dx: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2, L2, T2
ââL,U1. â¦G1, L1⦠⢠⡠L & â¦G1, L⦠⢠T1 â¡ U1 & â¦G1, L, U1⦠â â¦G2, L2, U2â¦.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
-#G #L #K #U #T #e #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (e+1)) #T2 #HUT2
+#G #L #K #U #T #m #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (m+1)) #T2 #HUT2
lapply (cpr_lift ⦠HU2 ⦠HLK ⦠HUT ⦠HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
qed-.
@@ -58,8 +58,8 @@ lemma fqu_cpr_trans_sn: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2, L2, T2
ââL,U1. â¦G1, L1⦠⢠⡠L & â¦G1, L1⦠⢠T1 â¡ U1 & â¦G1, L, U1⦠â â¦G2, L2, U2â¦.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
/3 width=5 by fqu_lref_O, fqu_pair_sn, fqu_bind_dx, fqu_flat_dx, lpr_pair, cpr_pair_sn, cpr_atom, cpr_bind, cpr_flat, ex3_2_intro/
-#G #L #K #U #T #e #HLK #HUT #U2 #HU2
-elim (lift_total U2 0 (e+1)) #T2 #HUT2
+#G #L #K #U #T #m #HLK #HUT #U2 #HU2
+elim (lift_total U2 0 (m+1)) #T2 #HUT2
lapply (cpr_lift ⦠HU2 ⦠HLK ⦠HUT ⦠HUT2) -HU2 -HUT /3 width=9 by fqu_drop, ex3_2_intro/
qed-.
@@ -80,7 +80,7 @@ lemma fqu_lpr_trans: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2, L2, T2â¦
[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpr_inv_pair1 ⦠H) -H
#K2 #W2 #HLK2 #HVW2 #H destruct
/3 width=5 by fqu_fquq, cpr_pair_sn, fqu_bind_dx, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #K2 #HK12
+| #G #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #K2 #HK12
elim (drop_lpr_trans ⦠HLK1 ⦠HK12) -HK12
/3 width=7 by fqu_drop, ex3_2_intro/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_drop.ma
index 4ad9ae29e..1944e9dd2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_drop.ma
@@ -39,7 +39,7 @@ lemma fqu_lpx_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2, L2, T
[ #a #I #G2 #L2 #V2 #T2 #X #H elim (lpx_inv_pair1 ⦠H) -H
#K2 #W2 #HLK2 #HVW2 #H destruct
/3 width=5 by cpx_pair_sn, fqu_bind_dx, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #K2 #HK12
+| #G #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #K2 #HK12
elim (drop_lpx_trans ⦠HLK1 ⦠HK12) -HK12
/3 width=7 by fqu_drop, ex3_2_intro/
]
@@ -62,7 +62,7 @@ lemma lpx_fqu_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2, L2, T
[ #I #G1 #L1 #V1 #X #H elim (lpx_inv_pair2 ⦠H) -H
#K1 #W1 #HKL1 #HWV1 #H destruct elim (lift_total V1 0 1)
/4 width=7 by cpx_delta, fqu_drop, drop_drop, ex3_2_intro/
-| #G #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #L0 #HL01
+| #G #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #L0 #HL01
elim (lpx_drop_trans_O1 ⦠HL01 ⦠HLK1) -L1
/3 width=5 by fqu_drop, ex3_2_intro/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_frees.ma b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_frees.ma
index 6f4aac874..a4a7e5f2a 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_frees.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_frees.ma
@@ -12,7 +12,7 @@
(* *)
(**************************************************************************)
-include "basic_2/multiple/frees_leq.ma".
+include "basic_2/multiple/frees_lreq.ma".
include "basic_2/multiple/frees_lift.ma".
include "basic_2/reduction/lpx_drop.ma".
@@ -26,7 +26,7 @@ lemma lpx_cpx_frees_trans: âh,g,G,L1,U1,U2. â¦G, L1⦠⢠U1 â¡[h, g] U2
#h #g #G #L1 #U1 @(fqup_wf_ind_eq ⦠G L1 U1) -G -L1 -U1
#G0 #L0 #U0 #IH #G #L1 * *
[ -IH #k #HG #HL #HU #U2 #H1 #L2 #_ #i #H2 elim (cpx_inv_sort1 ⦠H1) -H1
- [| * #l #_ ] #H destruct elim (frees_inv_sort ⦠H2)
+ [| * #d #_ ] #H destruct elim (frees_inv_sort ⦠H2)
| #j #HG #HL #HU #U2 #H1 #L2 #HL12 #i #H2 elim (cpx_inv_lref1 ⦠H1) -H1
[ #H destruct elim (frees_inv_lref ⦠H2) -H2 //
* #I #K2 #W2 #Hj #Hji #HLK2 #HW2
@@ -63,7 +63,7 @@ lemma lpx_cpx_frees_trans: âh,g,G,L1,U1,U2. â¦G, L1⦠⢠U1 â¡[h, g] U2
elim (frees_inv_bind ⦠Hi) -Hi #Hi
[ elim (frees_inv_flat ⦠Hi) -Hi
/4 width=7 by frees_flat_dx, frees_flat_sn, frees_bind_sn/
- | lapply (leq_frees_trans ⦠Hi (L2.âV2) ?) /2 width=1 by leq_succ/ -Hi #HU2
+ | lapply (lreq_frees_trans ⦠Hi (L2.âV2) ?) /2 width=1 by lreq_succ/ -Hi #HU2
lapply (frees_weak ⦠HU2 0 ?) -HU2
/5 width=7 by frees_bind_dx_O, frees_flat_dx, lpx_pair/
]
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 885687055..9e731c5a9 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/reduction/lpx_lleq.ma
@@ -15,9 +15,9 @@
include "basic_2/multiple/llor_drop.ma".
include "basic_2/multiple/llpx_sn_llor.ma".
include "basic_2/multiple/llpx_sn_lpx_sn.ma".
-include "basic_2/multiple/lleq_leq.ma".
+include "basic_2/multiple/lleq_lreq.ma".
include "basic_2/multiple/lleq_llor.ma".
-include "basic_2/reduction/cpx_leq.ma".
+include "basic_2/reduction/cpx_lreq.ma".
include "basic_2/reduction/cpx_lleq.ma".
include "basic_2/reduction/lpx_frees.ma".
@@ -27,14 +27,14 @@ include "basic_2/reduction/lpx_frees.ma".
(* Note: contains a proof of llpx_cpx_conf *)
lemma lleq_lpx_trans: âh,g,G,L2,K2. â¦G, L2⦠⢠â¡[h, g] K2 â
- âL1,T,d. L1 â¡[T, d] L2 â
- ââK1. â¦G, L1⦠⢠â¡[h, g] K1 & K1 â¡[T, d] K2.
-#h #g #G #L2 #K2 #HLK2 #L1 #T #d #HL12
+ âL1,T,l. L1 â¡[T, l] L2 â
+ ââK1. â¦G, L1⦠⢠â¡[h, g] K1 & K1 â¡[T, l] K2.
+#h #g #G #L2 #K2 #HLK2 #L1 #T #l #HL12
lapply (lpx_fwd_length ⦠HLK2) #H1
lapply (lleq_fwd_length ⦠HL12) #H2
-lapply (lpx_sn_llpx_sn ⦠T ⦠d HLK2) // -HLK2 #H
+lapply (lpx_sn_llpx_sn ⦠T ⦠l HLK2) // -HLK2 #H
lapply (lleq_llpx_sn_trans ⦠HL12 ⦠H) /2 width=3 by lleq_cpx_trans/ -HL12 -H #H
-elim (llor_total L1 K2 T d) // -H1 -H2 #K1 #HLK1
+elim (llor_total L1 K2 T l) // -H1 -H2 #K1 #HLK1
lapply (llpx_sn_llor_dx_sym ⦠H ⦠HLK1)
[ /2 width=6 by cpx_frees_trans/
| /3 width=10 by cpx_llpx_sn_conf, cpx_inv_lift1, cpx_lift/
@@ -59,8 +59,8 @@ lemma lpx_lleq_fqu_trans: âh,g,G1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2,
/3 width=4 by lpx_pair, fqu_bind_dx, ex3_intro/
| #I #G1 #L1 #V1 #T1 #K1 #HLK1 #H elim (lleq_inv_flat ⦠H) -H
/2 width=4 by fqu_flat_dx, ex3_intro/
-| #G1 #L1 #L #T1 #U1 #e #HL1 #HTU1 #K1 #H1KL1 #H2KL1
- elim (drop_O1_le (â») (e+1) K1)
+| #G1 #L1 #L #T1 #U1 #m #HL1 #HTU1 #K1 #H1KL1 #H2KL1
+ elim (drop_O1_le (â») (m+1) K1)
[ #K #HK1 lapply (lleq_inv_lift_le ⦠H2KL1 ⦠HK1 HL1 ⦠HTU1 ?) -H2KL1 //
#H2KL elim (lpx_drop_trans_O1 ⦠H1KL1 ⦠HL1) -L1
#K0 #HK10 #H1KL lapply (drop_mono ⦠HK10 ⦠HK1) -HK10 #H destruct
@@ -105,32 +105,32 @@ elim (fqus_inv_gen ⦠H) -H
]
qed-.
-fact leq_lpx_trans_lleq_aux: âh,g,G,L1,L0,d,e. L1 ⩬[d, e] L0 â e = â â
- âL2. â¦G, L0⦠⢠â¡[h, g] L2 â
- ââL. L ⩬[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
-[ #d #e #_ #L2 #H >(lpx_inv_atom1 ⦠H) -H
+fact lreq_lpx_trans_lleq_aux: âh,g,G,L1,L0,l,m. L1 ⩬[l, m] L0 â m = â â
+ âL2. â¦G, L0⦠⢠â¡[h, g] L2 â
+ ââL. L ⩬[l, m] L2 & â¦G, L1⦠⢠â¡[h, g] L &
+ (âT. L0 â¡[T, l] L2 â L1 â¡[T, l] L).
+#h #g #G #L1 #L0 #l #m #H elim H -L1 -L0 -l -m
+[ #l #m #_ #L2 #H >(lpx_inv_atom1 ⦠H) -H
/3 width=5 by ex3_intro, conj/
-| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #He destruct
-| #I #L1 #L0 #V1 #e #HL10 #IHL10 #He #Y #H
+| #I1 #I0 #L1 #L0 #V1 #V0 #_ #_ #Hm destruct
+| #I #L1 #L0 #V1 #m #HL10 #IHL10 #Hm #Y #H
elim (lpx_inv_pair1 ⦠H) -H #L2 #V2 #HL02 #HV02 #H destruct
- lapply (ysucc_inv_Y_dx ⦠He) -He #He
+ lapply (ysucc_inv_Y_dx ⦠Hm) -Hm #Hm
elim (IHL10 ⦠HL02) // -IHL10 -HL02 #L #HL2 #HL1 #IH
- @(ex3_intro ⦠(L.â{I}V2)) /3 width=3 by lpx_pair, leq_cpx_trans, leq_pair/
+ @(ex3_intro ⦠(L.â{I}V2)) /3 width=3 by lpx_pair, lreq_cpx_trans, lreq_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
+ @conj #H @(lleq_lreq_repl ⦠H) -H /3 width=1 by lreq_sym, lreq_pair_O_Y/
+| #I1 #I0 #L1 #L0 #V1 #V0 #l #m #HL10 #IHL10 #Hm #Y #H
elim (lpx_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 lpx_pair, leq_succ/
+ @(ex3_intro ⦠(L.â{I1}V1)) /3 width=1 by lpx_pair, lreq_succ/
#T elim (IH T) #HL0dx #HL0sn
- @conj #H @(lleq_leq_repl ⦠H) -H /3 width=1 by leq_sym, leq_succ/
+ @conj #H @(lleq_lreq_repl ⦠H) -H /3 width=1 by lreq_sym, lreq_succ/
]
qed-.
-lemma leq_lpx_trans_lleq: âh,g,G,L1,L0,d. L1 ⩬[d, â] L0 â
- âL2. â¦G, L0⦠⢠â¡[h, g] L2 â
- ââL. L ⩬[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_lpx_trans_lleq_aux/ qed-.
+lemma lreq_lpx_trans_lleq: âh,g,G,L1,L0,l. L1 ⩬[l, â] L0 â
+ âL2. â¦G, L0⦠⢠â¡[h, g] L2 â
+ ââL. L ⩬[l, â] L2 & â¦G, L1⦠⢠â¡[h, g] L &
+ (âT. L0 â¡[T, l] L2 â L1 â¡[T, l] L).
+/2 width=1 by lreq_lpx_trans_lleq_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma
index 736f45d98..91723374f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lift.ma
@@ -19,54 +19,54 @@ include "basic_2/static/aaa.ma".
(* Properties on basic relocation *******************************************)
-lemma aaa_lift: âG,L1,T1,A. â¦G, L1⦠⢠T1 â A â âL2,s,d,e. â¬[s, d, e] L2 â¡ L1 â
- âT2. â¬[d, e] T1 â¡ T2 â â¦G, L2⦠⢠T2 â A.
+lemma aaa_lift: âG,L1,T1,A. â¦G, L1⦠⢠T1 â A â âL2,s,l,m. â¬[s, l, m] L2 â¡ L1 â
+ âT2. â¬[l, m] T1 â¡ T2 â â¦G, L2⦠⢠T2 â A.
#G #L1 #T1 #A #H elim H -G -L1 -T1 -A
-[ #G #L1 #k #L2 #s #d #e #_ #T2 #H
+[ #G #L1 #k #L2 #s #l #m #_ #T2 #H
>(lift_inv_sort1 ⦠H) -H //
-| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #s #d #e #HL21 #T2 #H
- elim (lift_inv_lref1 ⦠H) -H * #Hid #H destruct
+| #I #G #L1 #K1 #V1 #B #i #HLK1 #_ #IHB #L2 #s #l #m #HL21 #T2 #H
+ elim (lift_inv_lref1 ⦠H) -H * #Hil #H destruct
[ elim (drop_trans_le ⦠HL21 ⦠HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
/3 width=9 by aaa_lref/
| lapply (drop_trans_ge ⦠HL21 ⦠HLK1 ?) -L1
/3 width=9 by aaa_lref, drop_inv_gen/
]
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #d #e #HL21 #X #H
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #l #m #HL21 #X #H
elim (lift_inv_bind1 ⦠H) -H #V2 #T2 #HV12 #HT12 #H destruct
/4 width=5 by aaa_abbr, drop_skip/
-| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #d #e #HL21 #X #H
+| #a #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #l #m #HL21 #X #H
elim (lift_inv_bind1 ⦠H) -H #V2 #T2 #HV12 #HT12 #H destruct
/4 width=5 by aaa_abst, drop_skip/
-| #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #d #e #HL21 #X #H
+| #G #L1 #V1 #T1 #B #A #_ #_ #IHB #IHA #L2 #s #l #m #HL21 #X #H
elim (lift_inv_flat1 ⦠H) -H #V2 #T2 #HV12 #HT12 #H destruct
/3 width=5 by aaa_appl/
-| #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #s #d #e #HL21 #X #H
+| #G #L1 #V1 #T1 #A #_ #_ #IH1 #IH2 #L2 #s #l #m #HL21 #X #H
elim (lift_inv_flat1 ⦠H) -H #V2 #T2 #HV12 #HT12 #H destruct
/3 width=5 by aaa_cast/
]
qed.
-lemma aaa_inv_lift: âG,L2,T2,A. â¦G, L2⦠⢠T2 â A â âL1,s,d,e. â¬[s, d, e] L2 â¡ L1 â
- âT1. â¬[d, e] T1 â¡ T2 â â¦G, L1⦠⢠T1 â A.
+lemma aaa_inv_lift: âG,L2,T2,A. â¦G, L2⦠⢠T2 â A â âL1,s,l,m. â¬[s, l, m] L2 â¡ L1 â
+ âT1. â¬[l, m] T1 â¡ T2 â â¦G, L1⦠⢠T1 â A.
#G #L2 #T2 #A #H elim H -G -L2 -T2 -A
-[ #G #L2 #k #L1 #s #d #e #_ #T1 #H
+[ #G #L2 #k #L1 #s #l #m #_ #T1 #H
>(lift_inv_sort2 ⦠H) -H //
-| #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #s #d #e #HL21 #T1 #H
- elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct
+| #I #G #L2 #K2 #V2 #B #i #HLK2 #_ #IHB #L1 #s #l #m #HL21 #T1 #H
+ elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct
[ elim (drop_conf_lt ⦠HL21 ⦠HLK2) -L2 /3 width=9 by aaa_lref/
| lapply (drop_conf_ge ⦠HL21 ⦠HLK2 ?) -L2 /3 width=9 by aaa_lref/
]
-| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #d #e #HL21 #X #H
+| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #l #m #HL21 #X #H
elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HV12 #HT12 #H destruct
/4 width=5 by aaa_abbr, drop_skip/
-| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #d #e #HL21 #X #H
+| #a #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #l #m #HL21 #X #H
elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HV12 #HT12 #H destruct
/4 width=5 by aaa_abst, drop_skip/
-| #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #d #e #HL21 #X #H
+| #G #L2 #V2 #T2 #B #A #_ #_ #IHB #IHA #L1 #s #l #m #HL21 #X #H
elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HV12 #HT12 #H destruct
/3 width=5 by aaa_appl/
-| #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #s #d #e #HL21 #X #H
+| #G #L2 #V2 #T2 #A #_ #_ #IH1 #IH2 #L1 #s #l #m #HL21 #X #H
elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HV12 #HT12 #H destruct
/3 width=5 by aaa_cast/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma
index d3018298d..051ab4d29 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/aaa_lifts.ma
@@ -24,7 +24,7 @@ lemma aaa_lifts: âG,L1,L2,T2,A,s,des. â¬*[s, des] L2 â¡ L1 â âT1. â¬*[d
#G #L1 #L2 #T2 #A #s #des #H elim H -L1 -L2 -des
[ #L #T1 #H #HT1
<(lifts_inv_nil ⦠H) -H //
-| #L1 #L #L2 #des #d #e #_ #HL2 #IHL1 #T1 #H #HT1
+| #L1 #L #L2 #des #l #m #_ #HL2 #IHL1 #T1 #H #HT1
elim (lifts_inv_cons ⦠H) -H /3 width=10 by aaa_lift/
]
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/da.ma b/matita/matita/contribs/lambdadelta/basic_2/static/da.ma
index 81201f52a..9b1331820 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/da.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/da.ma
@@ -21,88 +21,88 @@ include "basic_2/static/sd.ma".
(* activate genv *)
inductive da (h:sh) (g:sd h): relation4 genv lenv term nat â
-| da_sort: âG,L,k,l. deg h g k l â da h g G L (âk) l
-| da_ldef: âG,L,K,V,i,l. â¬[i] L â¡ K.âV â da h g G K V l â da h g G L (#i) l
-| da_ldec: âG,L,K,W,i,l. â¬[i] L â¡ K.âW â da h g G K W l â da h g G L (#i) (l+1)
-| da_bind: âa,I,G,L,V,T,l. da h g G (L.â{I}V) T l â da h g G L (â{a,I}V.T) l
-| da_flat: âI,G,L,V,T,l. da h g G L T l â da h g G L (â{I}V.T) l
+| da_sort: âG,L,k,d. deg h g k d â da h g G L (âk) d
+| da_ldef: âG,L,K,V,i,d. â¬[i] L â¡ K.âV â da h g G K V d â da h g G L (#i) d
+| da_ldec: âG,L,K,W,i,d. â¬[i] L â¡ K.âW â da h g G K W d â da h g G L (#i) (d+1)
+| da_bind: âa,I,G,L,V,T,d. da h g G (L.â{I}V) T d â da h g G L (â{a,I}V.T) d
+| da_flat: âI,G,L,V,T,d. da h g G L T d â da h g G L (â{I}V.T) d
.
interpretation "degree assignment (term)"
- 'Degree h g G L T l = (da h g G L T l).
+ 'Degree h g G L T d = (da h g G L T d).
(* Basic inversion lemmas ***************************************************)
-fact da_inv_sort_aux: âh,g,G,L,T,l. â¦G, L⦠⢠T âª[h, g] l â
- âk0. T = âk0 â deg h g k0 l.
-#h #g #G #L #T #l * -G -L -T -l
-[ #G #L #k #l #Hkl #k0 #H destruct //
-| #G #L #K #V #i #l #_ #_ #k0 #H destruct
-| #G #L #K #W #i #l #_ #_ #k0 #H destruct
-| #a #I #G #L #V #T #l #_ #k0 #H destruct
-| #I #G #L #V #T #l #_ #k0 #H destruct
+fact da_inv_sort_aux: âh,g,G,L,T,d. â¦G, L⦠⢠T âª[h, g] d â
+ âk0. T = âk0 â deg h g k0 d.
+#h #g #G #L #T #d * -G -L -T -d
+[ #G #L #k #d #Hkd #k0 #H destruct //
+| #G #L #K #V #i #d #_ #_ #k0 #H destruct
+| #G #L #K #W #i #d #_ #_ #k0 #H destruct
+| #a #I #G #L #V #T #d #_ #k0 #H destruct
+| #I #G #L #V #T #d #_ #k0 #H destruct
]
qed-.
-lemma da_inv_sort: âh,g,G,L,k,l. â¦G, L⦠⢠âk âª[h, g] l â deg h g k l.
+lemma da_inv_sort: âh,g,G,L,k,d. â¦G, L⦠⢠âk âª[h, g] d â deg h g k d.
/2 width=5 by da_inv_sort_aux/ qed-.
-fact da_inv_lref_aux: âh,g,G,L,T,l. â¦G, L⦠⢠T âª[h, g] l â âj. T = #j â
- (ââK,V. â¬[j] L â¡ K.âV & â¦G, K⦠⢠V âª[h, g] l) â¨
- (ââK,W,l0. â¬[j] L â¡ K.âW & â¦G, K⦠⢠W âª[h, g] l0 &
- l = l0 + 1
+fact da_inv_lref_aux: âh,g,G,L,T,d. â¦G, L⦠⢠T âª[h, g] d â âj. T = #j â
+ (ââK,V. â¬[j] L â¡ K.âV & â¦G, K⦠⢠V âª[h, g] d) â¨
+ (ââK,W,d0. â¬[j] L â¡ K.âW & â¦G, K⦠⢠W âª[h, g] d0 &
+ d = d0 + 1
).
-#h #g #G #L #T #l * -G -L -T -l
-[ #G #L #k #l #_ #j #H destruct
-| #G #L #K #V #i #l #HLK #HV #j #H destruct /3 width=4 by ex2_2_intro, or_introl/
-| #G #L #K #W #i #l #HLK #HW #j #H destruct /3 width=6 by ex3_3_intro, or_intror/
-| #a #I #G #L #V #T #l #_ #j #H destruct
-| #I #G #L #V #T #l #_ #j #H destruct
+#h #g #G #L #T #d * -G -L -T -d
+[ #G #L #k #d #_ #j #H destruct
+| #G #L #K #V #i #d #HLK #HV #j #H destruct /3 width=4 by ex2_2_intro, or_introl/
+| #G #L #K #W #i #d #HLK #HW #j #H destruct /3 width=6 by ex3_3_intro, or_intror/
+| #a #I #G #L #V #T #d #_ #j #H destruct
+| #I #G #L #V #T #d #_ #j #H destruct
]
qed-.
-lemma da_inv_lref: âh,g,G,L,j,l. â¦G, L⦠⢠#j âª[h, g] l â
- (ââK,V. â¬[j] L â¡ K.âV & â¦G, K⦠⢠V âª[h, g] l) â¨
- (ââK,W,l0. â¬[j] L â¡ K.âW & â¦G, K⦠⢠W âª[h, g] l0 & l = l0+1).
+lemma da_inv_lref: âh,g,G,L,j,d. â¦G, L⦠⢠#j âª[h, g] d â
+ (ââK,V. â¬[j] L â¡ K.âV & â¦G, K⦠⢠V âª[h, g] d) â¨
+ (ââK,W,d0. â¬[j] L â¡ K.âW & â¦G, K⦠⢠W âª[h, g] d0 & d = d0+1).
/2 width=3 by da_inv_lref_aux/ qed-.
-fact da_inv_gref_aux: âh,g,G,L,T,l. â¦G, L⦠⢠T âª[h, g] l â âp0. T = §p0 â â¥.
-#h #g #G #L #T #l * -G -L -T -l
-[ #G #L #k #l #_ #p0 #H destruct
-| #G #L #K #V #i #l #_ #_ #p0 #H destruct
-| #G #L #K #W #i #l #_ #_ #p0 #H destruct
-| #a #I #G #L #V #T #l #_ #p0 #H destruct
-| #I #G #L #V #T #l #_ #p0 #H destruct
+fact da_inv_gref_aux: âh,g,G,L,T,d. â¦G, L⦠⢠T âª[h, g] d â âp0. T = §p0 â â¥.
+#h #g #G #L #T #d * -G -L -T -d
+[ #G #L #k #d #_ #p0 #H destruct
+| #G #L #K #V #i #d #_ #_ #p0 #H destruct
+| #G #L #K #W #i #d #_ #_ #p0 #H destruct
+| #a #I #G #L #V #T #d #_ #p0 #H destruct
+| #I #G #L #V #T #d #_ #p0 #H destruct
]
qed-.
-lemma da_inv_gref: âh,g,G,L,p,l. â¦G, L⦠⢠§p âª[h, g] l â â¥.
+lemma da_inv_gref: âh,g,G,L,p,d. â¦G, L⦠⢠§p âª[h, g] d â â¥.
/2 width=9 by da_inv_gref_aux/ qed-.
-fact da_inv_bind_aux: âh,g,G,L,T,l. â¦G, L⦠⢠T âª[h, g] l â
- âb,J,X,Y. T = â{b,J}Y.X â â¦G, L.â{J}Y⦠⢠X âª[h, g] l.
-#h #g #G #L #T #l * -G -L -T -l
-[ #G #L #k #l #_ #b #J #X #Y #H destruct
-| #G #L #K #V #i #l #_ #_ #b #J #X #Y #H destruct
-| #G #L #K #W #i #l #_ #_ #b #J #X #Y #H destruct
-| #a #I #G #L #V #T #l #HT #b #J #X #Y #H destruct //
-| #I #G #L #V #T #l #_ #b #J #X #Y #H destruct
+fact da_inv_bind_aux: âh,g,G,L,T,d. â¦G, L⦠⢠T âª[h, g] d â
+ âb,J,X,Y. T = â{b,J}Y.X â â¦G, L.â{J}Y⦠⢠X âª[h, g] d.
+#h #g #G #L #T #d * -G -L -T -d
+[ #G #L #k #d #_ #b #J #X #Y #H destruct
+| #G #L #K #V #i #d #_ #_ #b #J #X #Y #H destruct
+| #G #L #K #W #i #d #_ #_ #b #J #X #Y #H destruct
+| #a #I #G #L #V #T #d #HT #b #J #X #Y #H destruct //
+| #I #G #L #V #T #d #_ #b #J #X #Y #H destruct
]
qed-.
-lemma da_inv_bind: âh,g,b,J,G,L,Y,X,l. â¦G, L⦠⢠â{b,J}Y.X âª[h, g] l â â¦G, L.â{J}Y⦠⢠X âª[h, g] l.
+lemma da_inv_bind: âh,g,b,J,G,L,Y,X,d. â¦G, L⦠⢠â{b,J}Y.X âª[h, g] d â â¦G, L.â{J}Y⦠⢠X âª[h, g] d.
/2 width=4 by da_inv_bind_aux/ qed-.
-fact da_inv_flat_aux: âh,g,G,L,T,l. â¦G, L⦠⢠T âª[h, g] l â
- âJ,X,Y. T = â{J}Y.X â â¦G, L⦠⢠X âª[h, g] l.
-#h #g #G #L #T #l * -G -L -T -l
-[ #G #L #k #l #_ #J #X #Y #H destruct
-| #G #L #K #V #i #l #_ #_ #J #X #Y #H destruct
-| #G #L #K #W #i #l #_ #_ #J #X #Y #H destruct
-| #a #I #G #L #V #T #l #_ #J #X #Y #H destruct
-| #I #G #L #V #T #l #HT #J #X #Y #H destruct //
+fact da_inv_flat_aux: âh,g,G,L,T,d. â¦G, L⦠⢠T âª[h, g] d â
+ âJ,X,Y. T = â{J}Y.X â â¦G, L⦠⢠X âª[h, g] d.
+#h #g #G #L #T #d * -G -L -T -d
+[ #G #L #k #d #_ #J #X #Y #H destruct
+| #G #L #K #V #i #d #_ #_ #J #X #Y #H destruct
+| #G #L #K #W #i #d #_ #_ #J #X #Y #H destruct
+| #a #I #G #L #V #T #d #_ #J #X #Y #H destruct
+| #I #G #L #V #T #d #HT #J #X #Y #H destruct //
]
qed-.
-lemma da_inv_flat: âh,g,J,G,L,Y,X,l. â¦G, L⦠⢠â{J}Y.X âª[h, g] l â â¦G, L⦠⢠X âª[h, g] l.
+lemma da_inv_flat: âh,g,J,G,L,Y,X,d. â¦G, L⦠⢠â{J}Y.X âª[h, g] d â â¦G, L⦠⢠X âª[h, g] d.
/2 width=5 by da_inv_flat_aux/ qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/da_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/static/da_aaa.ma
index 7e9e4cc88..c7903c24e 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/da_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/da_aaa.ma
@@ -19,7 +19,7 @@ include "basic_2/static/da.ma".
(* Properties on atomic arity assignment for terms **************************)
-lemma aaa_da: âh,g,G,L,T,A. â¦G, L⦠⢠T â A â âl. â¦G, L⦠⢠T âª[h, g] l.
+lemma aaa_da: âh,g,G,L,T,A. â¦G, L⦠⢠T â A â âd. â¦G, L⦠⢠T âª[h, g] d.
#h #g #G #L #T #A #H elim H -G -L -T -A
[ #G #L #k elim (deg_total h g k) /3 width=2 by da_sort, ex_intro/
| * #G #L #K #V #B #i #HLK #_ * /3 width=5 by da_ldef, da_ldec, ex_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/da_da.ma b/matita/matita/contribs/lambdadelta/basic_2/static/da_da.ma
index 09d8eceec..64b7b7d84 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/da_da.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/da_da.ma
@@ -18,21 +18,21 @@ include "basic_2/static/da_lift.ma".
(* Main properties **********************************************************)
-theorem da_mono: âh,g,G,L,T,l1. â¦G, L⦠⢠T âª[h, g] l1 â
- âl2. â¦G, L⦠⢠T âª[h, g] l2 â l1 = l2.
-#h #g #G #L #T #l1 #H elim H -G -L -T -l1
-[ #G #L #k #l1 #Hkl1 #l2 #H
- lapply (da_inv_sort ⦠H) -G -L #Hkl2
- >(deg_mono ⦠Hkl2 ⦠Hkl1) -h -k -l2 //
-| #G #L #K #V #i #l1 #HLK #_ #IHV #l2 #H
- elim (da_inv_lref ⦠H) -H * #K0 #V0 [| #l0 ] #HLK0 #HV0 [| #Hl0 ]
+theorem da_mono: âh,g,G,L,T,d1. â¦G, L⦠⢠T âª[h, g] d1 â
+ âd2. â¦G, L⦠⢠T âª[h, g] d2 â d1 = d2.
+#h #g #G #L #T #d1 #H elim H -G -L -T -d1
+[ #G #L #k #d1 #Hkd1 #d2 #H
+ lapply (da_inv_sort ⦠H) -G -L #Hkd2
+ >(deg_mono ⦠Hkd2 ⦠Hkd1) -h -k -d2 //
+| #G #L #K #V #i #d1 #HLK #_ #IHV #d2 #H
+ elim (da_inv_lref ⦠H) -H * #K0 #V0 [| #d0 ] #HLK0 #HV0 [| #Hd0 ]
lapply (drop_mono ⦠HLK0 ⦠HLK) -HLK -HLK0 #H destruct /2 width=1/
-| #G #L #K #W #i #l1 #HLK #_ #IHW #l2 #H
- elim (da_inv_lref ⦠H) -H * #K0 #W0 [| #l0 ] #HLK0 #HW0 [| #Hl0 ]
+| #G #L #K #W #i #d1 #HLK #_ #IHW #d2 #H
+ elim (da_inv_lref ⦠H) -H * #K0 #W0 [| #d0 ] #HLK0 #HW0 [| #Hd0 ]
lapply (drop_mono ⦠HLK0 ⦠HLK) -HLK -HLK0 #H destruct /3 width=1/
-| #a #I #G #L #V #T #l1 #_ #IHT #l2 #H
+| #a #I #G #L #V #T #d1 #_ #IHT #d2 #H
lapply (da_inv_bind ⦠H) -H /2 width=1/
-| #I #G #L #V #T #l1 #_ #IHT #l2 #H
+| #I #G #L #V #T #d1 #_ #IHT #d2 #H
lapply (da_inv_flat ⦠H) -H /2 width=1/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/da_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/static/da_lift.ma
index 0e7428fa3..3f3926c17 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/da_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/da_lift.ma
@@ -19,32 +19,32 @@ include "basic_2/static/da.ma".
(* Properties on relocation *************************************************)
-lemma da_lift: âh,g,G,L1,T1,l. â¦G, L1⦠⢠T1 âª[h, g] l â
- âL2,s,d,e. â¬[s, d, e] L2 â¡ L1 â âT2. â¬[d, e] T1 â¡ T2 â
- â¦G, L2⦠⢠T2 âª[h, g] l.
-#h #g #G #L1 #T1 #l #H elim H -G -L1 -T1 -l
-[ #G #L1 #k #l #Hkl #L2 #s #d #e #_ #X #H
+lemma da_lift: âh,g,G,L1,T1,d. â¦G, L1⦠⢠T1 âª[h, g] d â
+ âL2,s,l,m. â¬[s, l, m] L2 â¡ L1 â âT2. â¬[l, m] T1 â¡ T2 â
+ â¦G, L2⦠⢠T2 âª[h, g] d.
+#h #g #G #L1 #T1 #d #H elim H -G -L1 -T1 -d
+[ #G #L1 #k #d #Hkd #L2 #s #l #m #_ #X #H
>(lift_inv_sort1 ⦠H) -X /2 width=1 by da_sort/
-| #G #L1 #K1 #V1 #i #l #HLK1 #_ #IHV1 #L2 #s #d #e #HL21 #X #H
- elim (lift_inv_lref1 ⦠H) * #Hid #H destruct
+| #G #L1 #K1 #V1 #i #d #HLK1 #_ #IHV1 #L2 #s #l #m #HL21 #X #H
+ elim (lift_inv_lref1 ⦠H) * #Hil #H destruct
[ elim (drop_trans_le ⦠HL21 ⦠HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
/3 width=9 by da_ldef/
| lapply (drop_trans_ge ⦠HL21 ⦠HLK1 ?) -L1
/3 width=8 by da_ldef, drop_inv_gen/
]
-| #G #L1 #K1 #W1 #i #l #HLK1 #_ #IHW1 #L2 #s #d #e #HL21 #X #H
- elim (lift_inv_lref1 ⦠H) * #Hid #H destruct
+| #G #L1 #K1 #W1 #i #d #HLK1 #_ #IHW1 #L2 #s #l #m #HL21 #X #H
+ elim (lift_inv_lref1 ⦠H) * #Hil #H destruct
[ elim (drop_trans_le ⦠HL21 ⦠HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #W2 #HK21 #HW12 #H destruct
/3 width=8 by da_ldec/
| lapply (drop_trans_ge ⦠HL21 ⦠HLK1 ?) -L1
/3 width=8 by da_ldec, drop_inv_gen/
]
-| #a #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #s #d #e #HL21 #X #H
+| #a #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #s #l #m #HL21 #X #H
elim (lift_inv_bind1 ⦠H) -H #V2 #T2 #HV12 #HU12 #H destruct
/4 width=5 by da_bind, drop_skip/
-| #I #G #L1 #V1 #T1 #l #_ #IHT1 #L2 #s #d #e #HL21 #X #H
+| #I #G #L1 #V1 #T1 #d #_ #IHT1 #L2 #s #l #m #HL21 #X #H
elim (lift_inv_flat1 ⦠H) -H #V2 #T2 #HV12 #HU12 #H destruct
/3 width=5 by da_flat/
]
@@ -52,26 +52,26 @@ qed.
(* Inversion lemmas on relocation *******************************************)
-lemma da_inv_lift: âh,g,G,L2,T2,l. â¦G, L2⦠⢠T2 âª[h, g] l â
- âL1,s,d,e. â¬[s, d, e] L2 â¡ L1 â âT1. â¬[d, e] T1 â¡ T2 â
- â¦G, L1⦠⢠T1 âª[h, g] l.
-#h #g #G #L2 #T2 #l #H elim H -G -L2 -T2 -l
-[ #G #L2 #k #l #Hkl #L1 #s #d #e #_ #X #H
+lemma da_inv_lift: âh,g,G,L2,T2,d. â¦G, L2⦠⢠T2 âª[h, g] d â
+ âL1,s,l,m. â¬[s, l, m] L2 â¡ L1 â âT1. â¬[l, m] T1 â¡ T2 â
+ â¦G, L1⦠⢠T1 âª[h, g] d.
+#h #g #G #L2 #T2 #d #H elim H -G -L2 -T2 -d
+[ #G #L2 #k #d #Hkd #L1 #s #l #m #_ #X #H
>(lift_inv_sort2 ⦠H) -X /2 width=1 by da_sort/
-| #G #L2 #K2 #V2 #i #l #HLK2 #HV2 #IHV2 #L1 #s #d #e #HL21 #X #H
- elim (lift_inv_lref2 ⦠H) * #Hid #H destruct [ -HV2 | -IHV2 ]
+| #G #L2 #K2 #V2 #i #d #HLK2 #HV2 #IHV2 #L1 #s #l #m #HL21 #X #H
+ elim (lift_inv_lref2 ⦠H) * #Hil #H destruct [ -HV2 | -IHV2 ]
[ elim (drop_conf_lt ⦠HL21 ⦠HLK2) -L2 /3 width=8 by da_ldef/
| lapply (drop_conf_ge ⦠HL21 ⦠HLK2 ?) -L2 /2 width=4 by da_ldef/
]
-| #G #L2 #K2 #W2 #i #l #HLK2 #HW2 #IHW2 #L1 #s #d #e #HL21 #X #H
- elim (lift_inv_lref2 ⦠H) * #Hid #H destruct [ -HW2 | -IHW2 ]
+| #G #L2 #K2 #W2 #i #d #HLK2 #HW2 #IHW2 #L1 #s #l #m #HL21 #X #H
+ elim (lift_inv_lref2 ⦠H) * #Hil #H destruct [ -HW2 | -IHW2 ]
[ elim (drop_conf_lt ⦠HL21 ⦠HLK2) -L2 /3 width=8 by da_ldec/
| lapply (drop_conf_ge ⦠HL21 ⦠HLK2 ?) -L2 /2 width=4 by da_ldec/
]
-| #a #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #s #d #e #HL21 #X #H
+| #a #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #s #l #m #HL21 #X #H
elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HV12 #HT12 #H destruct
/4 width=5 by da_bind, drop_skip/
-| #I #G #L2 #V2 #T2 #l #_ #IHT2 #L1 #s #d #e #HL21 #X #H
+| #I #G #L2 #V2 #T2 #d #_ #IHT2 #L1 #s #l #m #HL21 #X #H
elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HV12 #HT12 #H destruct
/3 width=5 by da_flat/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma
index be272d27a..e1869f384 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsuba.ma
@@ -100,19 +100,19 @@ lemma lsuba_refl: âG,L. G ⢠L â«â L.
qed.
(* Note: the constant 0 cannot be generalized *)
-lemma lsuba_drop_O1_conf: âG,L1,L2. G ⢠L1 â«â L2 â âK1,s,e. â¬[s, 0, e] L1 â¡ K1 â
- ââK2. G ⢠K1 â«â K2 & â¬[s, 0, e] L2 â¡ K2.
+lemma lsuba_drop_O1_conf: âG,L1,L2. G ⢠L1 â«â L2 â âK1,s,m. â¬[s, 0, m] L1 â¡ K1 â
+ ââK2. G ⢠K1 â«â K2 & â¬[s, 0, m] L2 â¡ K2.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK1
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=3 by lsuba_pair, drop_pair, ex2_intro/
| elim (IHL12 ⦠HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK1
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K1 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=3 by lsuba_beta, drop_pair, ex2_intro/
@@ -122,19 +122,19 @@ lemma lsuba_drop_O1_conf: âG,L1,L2. G ⢠L1 â«â L2 â âK1,s,e. â¬[s, 0
qed-.
(* Note: the constant 0 cannot be generalized *)
-lemma lsuba_drop_O1_trans: âG,L1,L2. G ⢠L1 â«â L2 â âK2,s,e. â¬[s, 0, e] L2 â¡ K2 â
- ââK1. G ⢠K1 â«â K2 & â¬[s, 0, e] L1 â¡ K1.
+lemma lsuba_drop_O1_trans: âG,L1,L2. G ⢠L1 â«â L2 â âK2,s,m. â¬[s, 0, m] L2 â¡ K2 â
+ ââK1. G ⢠K1 â«â K2 & â¬[s, 0, m] L1 â¡ K1.
#G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK2
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=3 by lsuba_pair, drop_pair, ex2_intro/
| elim (IHL12 ⦠HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK2
+| #L1 #L2 #W #V #A #HV #HW #_ #IHL12 #K2 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=3 by lsuba_beta, drop_pair, ex2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsubd.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsubd.ma
index 14abf913d..29da4dad7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lsubd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsubd.ma
@@ -22,7 +22,7 @@ inductive lsubd (h) (g) (G): relation lenv â
| lsubd_atom: lsubd h g G (â) (â)
| lsubd_pair: âI,L1,L2,V. lsubd h g G L1 L2 â
lsubd h g G (L1.â{I}V) (L2.â{I}V)
-| lsubd_beta: âL1,L2,W,V,l. â¦G, L1⦠⢠V âª[h, g] l+1 â â¦G, L2⦠⢠W âª[h, g] l â
+| lsubd_beta: âL1,L2,W,V,d. â¦G, L1⦠⢠V âª[h, g] d+1 â â¦G, L2⦠⢠W âª[h, g] d â
lsubd h g G L1 L2 â lsubd h g G (L1.ââW.V) (L2.âW)
.
@@ -42,7 +42,7 @@ fact lsubd_inv_atom1_aux: âh,g,G,L1,L2. G ⢠L1 â«âª[h, g] L2 â L1 = â
#h #g #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #l #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #d #_ #_ #_ #H destruct
]
qed-.
@@ -52,19 +52,19 @@ lemma lsubd_inv_atom1: âh,g,G,L2. G ⢠â â«âª[h, g] L2 â L2 = â.
fact lsubd_inv_pair1_aux: âh,g,G,L1,L2. G ⢠L1 â«âª[h, g] L2 â
âI,K1,X. L1 = K1.â{I}X â
(ââK2. G ⢠K1 â«âª[h, g] K2 & L2 = K2.â{I}X) â¨
- ââK2,W,V,l. â¦G, K1⦠⢠V âª[h, g] l+1 & â¦G, K2⦠⢠W âª[h, g] l &
+ ââK2,W,V,d. â¦G, K1⦠⢠V âª[h, g] d+1 & â¦G, K2⦠⢠W âª[h, g] d &
G ⢠K1 â«âª[h, g] K2 &
I = Abbr & L2 = K2.âW & X = âW.V.
#h #g #G #L1 #L2 * -L1 -L2
[ #J #K1 #X #H destruct
| #I #L1 #L2 #V #HL12 #J #K1 #X #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #W #V #l #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by ex6_4_intro, or_intror/
+| #L1 #L2 #W #V #d #HV #HW #HL12 #J #K1 #X #H destruct /3 width=9 by ex6_4_intro, or_intror/
]
qed-.
lemma lsubd_inv_pair1: âh,g,I,G,K1,L2,X. G ⢠K1.â{I}X â«âª[h, g] L2 â
(ââK2. G ⢠K1 â«âª[h, g] K2 & L2 = K2.â{I}X) â¨
- ââK2,W,V,l. â¦G, K1⦠⢠V âª[h, g] l+1 & â¦G, K2⦠⢠W âª[h, g] l &
+ ââK2,W,V,d. â¦G, K1⦠⢠V âª[h, g] d+1 & â¦G, K2⦠⢠W âª[h, g] d &
G ⢠K1 â«âª[h, g] K2 &
I = Abbr & L2 = K2.âW & X = âW.V.
/2 width=3 by lsubd_inv_pair1_aux/ qed-.
@@ -73,7 +73,7 @@ fact lsubd_inv_atom2_aux: âh,g,G,L1,L2. G ⢠L1 â«âª[h, g] L2 â L2 = â
#h #g #G #L1 #L2 * -L1 -L2
[ //
| #I #L1 #L2 #V #_ #H destruct
-| #L1 #L2 #W #V #l #_ #_ #_ #H destruct
+| #L1 #L2 #W #V #d #_ #_ #_ #H destruct
]
qed-.
@@ -83,18 +83,18 @@ lemma lsubd_inv_atom2: âh,g,G,L1. G ⢠L1 â«âª[h, g] â â L1 = â.
fact lsubd_inv_pair2_aux: âh,g,G,L1,L2. G ⢠L1 â«âª[h, g] L2 â
âI,K2,W. L2 = K2.â{I}W â
(ââK1. G ⢠K1 â«âª[h, g] K2 & L1 = K1.â{I}W) â¨
- ââK1,V,l. â¦G, K1⦠⢠V âª[h, g] l+1 & â¦G, K2⦠⢠W âª[h, g] l &
+ ââK1,V,d. â¦G, K1⦠⢠V âª[h, g] d+1 & â¦G, K2⦠⢠W âª[h, g] d &
G ⢠K1 â«âª[h, g] K2 & I = Abst & L1 = K1. ââW.V.
#h #g #G #L1 #L2 * -L1 -L2
[ #J #K2 #U #H destruct
| #I #L1 #L2 #V #HL12 #J #K2 #U #H destruct /3 width=3 by ex2_intro, or_introl/
-| #L1 #L2 #W #V #l #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by ex5_3_intro, or_intror/
+| #L1 #L2 #W #V #d #HV #HW #HL12 #J #K2 #U #H destruct /3 width=7 by ex5_3_intro, or_intror/
]
qed-.
lemma lsubd_inv_pair2: âh,g,I,G,L1,K2,W. G ⢠L1 â«âª[h, g] K2.â{I}W â
(ââK1. G ⢠K1 â«âª[h, g] K2 & L1 = K1.â{I}W) â¨
- ââK1,V,l. â¦G, K1⦠⢠V âª[h, g] l+1 & â¦G, K2⦠⢠W âª[h, g] l &
+ ââK1,V,d. â¦G, K1⦠⢠V âª[h, g] d+1 & â¦G, K2⦠⢠W âª[h, g] d &
G ⢠K1 â«âª[h, g] K2 & I = Abst & L1 = K1. ââW.V.
/2 width=3 by lsubd_inv_pair2_aux/ qed-.
@@ -106,19 +106,19 @@ qed.
(* Note: the constant 0 cannot be generalized *)
lemma lsubd_drop_O1_conf: âh,g,G,L1,L2. G ⢠L1 â«âª[h, g] L2 â
- âK1,s,e. â¬[s, 0, e] L1 â¡ K1 â
- ââK2. G ⢠K1 â«âª[h, g] K2 & â¬[s, 0, e] L2 â¡ K2.
+ âK1,s,m. â¬[s, 0, m] L1 â¡ K1 â
+ ââK2. G ⢠K1 â«âª[h, g] K2 & â¬[s, 0, m] L2 â¡ K2.
#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K1 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK1
+| #I #L1 #L2 #V #_ #IHL12 #K1 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
| elim (IHL12 ⦠HLK1) -L1 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K1 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK1
+| #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K1 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK1
[ destruct
elim (IHL12 L1 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
@@ -129,19 +129,19 @@ qed-.
(* Note: the constant 0 cannot be generalized *)
lemma lsubd_drop_O1_trans: âh,g,G,L1,L2. G ⢠L1 â«âª[h, g] L2 â
- âK2,s,e. â¬[s, 0, e] L2 â¡ K2 â
- ââK1. G ⢠K1 â«âª[h, g] K2 & â¬[s, 0, e] L1 â¡ K1.
+ âK2,s,m. â¬[s, 0, m] L2 â¡ K2 â
+ ââK1. G ⢠K1 â«âª[h, g] K2 & â¬[s, 0, m] L1 â¡ K1.
#h #g #G #L1 #L2 #H elim H -L1 -L2
[ /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #_ #IHL12 #K2 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK2
+| #I #L1 #L2 #V #_ #IHL12 #K2 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=3 by lsubd_pair, drop_pair, ex2_intro/
| elim (IHL12 ⦠HLK2) -L2 /3 width=3 by drop_drop_lt, ex2_intro/
]
-| #L1 #L2 #W #V #l #HV #HW #_ #IHL12 #K2 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK2
+| #L1 #L2 #W #V #d #HV #HW #_ #IHL12 #K2 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK2
[ destruct
elim (IHL12 L2 s 0) -IHL12 // #X #HL12 #H
<(drop_inv_O2 ⦠H) in HL12; -H /3 width=3 by lsubd_beta, drop_pair, ex2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsubd_da.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsubd_da.ma
index 2ccf11bad..457b64e77 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lsubd_da.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsubd_da.ma
@@ -19,21 +19,21 @@ include "basic_2/static/lsubd.ma".
(* Properties on degree assignment ******************************************)
-lemma lsubd_da_trans: âh,g,G,L2,T,l. â¦G, L2⦠⢠T âª[h, g] l â
- âL1. G ⢠L1 â«âª[h, g] L2 â â¦G, L1⦠⢠T âª[h, g] l.
-#h #g #G #L2 #T #l #H elim H -G -L2 -T -l
+lemma lsubd_da_trans: âh,g,G,L2,T,d. â¦G, L2⦠⢠T âª[h, g] d â
+ âL1. G ⢠L1 â«âª[h, g] L2 â â¦G, L1⦠⢠T âª[h, g] d.
+#h #g #G #L2 #T #d #H elim H -G -L2 -T -d
[ /2 width=1/
-| #G #L2 #K2 #V #i #l #HLK2 #_ #IHV #L1 #HL12
+| #G #L2 #K2 #V #i #d #HLK2 #_ #IHV #L1 #HL12
elim (lsubd_drop_O1_trans ⦠HL12 ⦠HLK2) -L2 #X #H #HLK1
elim (lsubd_inv_pair2 ⦠H) -H * #K1 [ | -IHV -HLK1 ]
[ #HK12 #H destruct /3 width=4/
- | #W #l0 #_ #_ #_ #H destruct
+ | #W #d0 #_ #_ #_ #H destruct
]
-| #G #L2 #K2 #W #i #l #HLK2 #HW #IHW #L1 #HL12
+| #G #L2 #K2 #W #i #d #HLK2 #HW #IHW #L1 #HL12
elim (lsubd_drop_O1_trans ⦠HL12 ⦠HLK2) -L2 #X #H #HLK1
elim (lsubd_inv_pair2 ⦠H) -H * #K1 [ -HW | -IHW ]
[ #HK12 #H destruct /3 width=4/
- | #V #l0 #HV #H0W #_ #_ #H destruct
+ | #V #d0 #HV #H0W #_ #_ #H destruct
lapply (da_mono ⦠H0W ⦠HW) -H0W -HW #H destruct /3 width=7/
]
| /4 width=1/
@@ -41,23 +41,23 @@ lemma lsubd_da_trans: âh,g,G,L2,T,l. â¦G, L2⦠⢠T âª[h, g] l â
]
qed-.
-lemma lsubd_da_conf: âh,g,G,L1,T,l. â¦G, L1⦠⢠T âª[h, g] l â
- âL2. G ⢠L1 â«âª[h, g] L2 â â¦G, L2⦠⢠T âª[h, g] l.
-#h #g #G #L1 #T #l #H elim H -G -L1 -T -l
+lemma lsubd_da_conf: âh,g,G,L1,T,d. â¦G, L1⦠⢠T âª[h, g] d â
+ âL2. G ⢠L1 â«âª[h, g] L2 â â¦G, L2⦠⢠T âª[h, g] d.
+#h #g #G #L1 #T #d #H elim H -G -L1 -T -d
[ /2 width=1/
-| #G #L1 #K1 #V #i #l #HLK1 #HV #IHV #L2 #HL12
+| #G #L1 #K1 #V #i #d #HLK1 #HV #IHV #L2 #HL12
elim (lsubd_drop_O1_conf ⦠HL12 ⦠HLK1) -L1 #X #H #HLK2
elim (lsubd_inv_pair1 ⦠H) -H * #K2 [ -HV | -IHV ]
[ #HK12 #H destruct /3 width=4/
- | #W0 #V0 #l0 #HV0 #HW0 #_ #_ #H1 #H2 destruct
+ | #W0 #V0 #d0 #HV0 #HW0 #_ #_ #H1 #H2 destruct
lapply (da_inv_flat ⦠HV) -HV #H0V0
lapply (da_mono ⦠H0V0 ⦠HV0) -H0V0 -HV0 #H destruct /2 width=4/
]
-| #G #L1 #K1 #W #i #l #HLK1 #HW #IHW #L2 #HL12
+| #G #L1 #K1 #W #i #d #HLK1 #HW #IHW #L2 #HL12
elim (lsubd_drop_O1_conf ⦠HL12 ⦠HLK1) -L1 #X #H #HLK2
elim (lsubd_inv_pair1 ⦠H) -H * #K2 [ -HW | -IHW ]
[ #HK12 #H destruct /3 width=4/
- | #W0 #V0 #l0 #HV0 #HW0 #_ #H destruct
+ | #W0 #V0 #d0 #HV0 #HW0 #_ #H destruct
]
| /4 width=1/
| /3 width=1/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lsubd_lsubd.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lsubd_lsubd.ma
index 15704de44..18670b4f1 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lsubd_lsubd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lsubd_lsubd.ma
@@ -24,13 +24,13 @@ theorem lsubd_trans: âh,g,G. Transitive ⦠(lsubd h g G).
| #I #L1 #L #Y #HL1 #IHL1 #X #H
elim (lsubd_inv_pair1 ⦠H) -H * #L2
[ #HL2 #H destruct /3 width=1/
- | #W #V #l #HV #HW #HL2 #H1 #H2 #H3 destruct
+ | #W #V #d #HV #HW #HL2 #H1 #H2 #H3 destruct
/3 width=3 by lsubd_beta, lsubd_da_trans/
]
-| #L1 #L #W #V #l #HV #HW #HL1 #IHL1 #X #H
+| #L1 #L #W #V #d #HV #HW #HL1 #IHL1 #X #H
elim (lsubd_inv_pair1 ⦠H) -H * #L2
[ #HL2 #H destruct /3 width=5 by lsubd_beta, lsubd_da_conf/
- | #W0 #V0 #l0 #_ #_ #_ #H destruct
+ | #W0 #V0 #d0 #_ #_ #_ #H destruct
]
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/sd.ma b/matita/matita/contribs/lambdadelta/basic_2/static/sd.ma
index 5e31749e1..a16d19042 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/sd.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/sd.ma
@@ -19,33 +19,33 @@ include "basic_2/static/sh.ma".
(* sort degree specification *)
record sd (h:sh): Type[0] â {
deg : relation nat; (* degree of the sort *)
- deg_total: âk. âl. deg k l; (* functional relation axioms *)
- deg_mono : âk,l1,l2. deg k l1 â deg k l2 â l1 = l2;
- deg_next : âk,l. deg k l â deg (next h k) (l - 1) (* compatibility condition *)
+ deg_total: âk. âd. deg k d; (* functional relation axioms *)
+ deg_mono : âk,d1,d2. deg k d1 â deg k d2 â d1 = d2;
+ deg_next : âk,d. deg k d â deg (next h k) (d - 1) (* compatibility condition *)
}.
(* Notable specifications ***************************************************)
-definition deg_O: relation nat â λk,l. l = 0.
+definition deg_O: relation nat â λk,d. d = 0.
definition sd_O: âh. sd h â λh. mk_sd h deg_O â¦.
/2 width=2 by le_n_O_to_eq, le_n, ex_intro/ defined.
inductive deg_SO (h:sh) (k:nat) (k0:nat): predicate nat â
-| deg_SO_pos : âl0. (next h)^l0 k0 = k â deg_SO h k k0 (l0 + 1)
-| deg_SO_zero: ((âl0. (next h)^l0 k0 = k) â â¥) â deg_SO h k k0 0
+| deg_SO_pos : âd0. (next h)^d0 k0 = k â deg_SO h k k0 (d0 + 1)
+| deg_SO_zero: ((âd0. (next h)^d0 k0 = k) â â¥) â deg_SO h k k0 0
.
-fact deg_SO_inv_pos_aux: âh,k,k0,l0. deg_SO h k k0 l0 â âl. l0 = l + 1 â
- (next h)^l k0 = k.
-#h #k #k0 #l0 * -l0
-[ #l0 #Hl0 #l #H
+fact deg_SO_inv_pos_aux: âh,k,k0,d0. deg_SO h k k0 d0 â âd. d0 = d + 1 â
+ (next h)^d k0 = k.
+#h #k #k0 #d0 * -d0
+[ #d0 #Hd0 #d #H
lapply (injective_plus_l ⦠H) -H #H destruct //
-| #_ #l0 <plus_n_Sm #H destruct
+| #_ #d0 <plus_n_Sm #H destruct
]
qed.
-lemma deg_SO_inv_pos: âh,k,k0,l0. deg_SO h k k0 (l0 + 1) â (next h)^l0 k0 = k.
+lemma deg_SO_inv_pos: âh,k,k0,d0. deg_SO h k k0 (d0 + 1) â (next h)^d0 k0 = k.
/2 width=3 by deg_SO_inv_pos_aux/ qed-.
lemma deg_SO_refl: âh,k. deg_SO h k k 1.
@@ -53,14 +53,14 @@ lemma deg_SO_refl: âh,k. deg_SO h k k 1.
qed.
lemma deg_SO_gt: âh,k1,k2. k1 < k2 â deg_SO h k1 k2 0.
-#h #k1 #k2 #HK12 @deg_SO_zero * #l elim l -l normalize
+#h #k1 #k2 #HK12 @deg_SO_zero * #d elim d -d normalize
[ #H destruct
elim (lt_refl_false ⦠HK12)
-| #l #_ #H
- lapply (next_lt h ((next h)^l k2)) >H -H #H
+| #d #_ #H
+ lapply (next_lt h ((next h)^d k2)) >H -H #H
lapply (transitive_lt ⦠H HK12) -k1 #H1
- lapply (nexts_le h k2 l) #H2
- lapply (le_to_lt_to_lt ⦠H2 H1) -h -l #H
+ lapply (nexts_le h k2 d) #H2
+ lapply (le_to_lt_to_lt ⦠H2 H1) -h -d #H
elim (lt_refl_false ⦠H)
]
qed.
@@ -69,63 +69,63 @@ definition sd_SO: âh. nat â sd h â λh,k. mk_sd h (deg_SO h k) â¦.
[ #k0
lapply (nexts_dec h k0 k) *
[ * /3 width=2 by deg_SO_pos, ex_intro/ | /4 width=2 by deg_SO_zero, ex_intro/ ]
-| #K0 #l1 #l2 * [ #l01 ] #H1 * [1,3: #l02 ] #H2 //
+| #K0 #d1 #d2 * [ #d01 ] #H1 * [1,3: #d02 ] #H2 //
[ < H2 in H1; -H2 #H
lapply (nexts_inj ⦠H) -H #H destruct //
| elim H1 /2 width=2 by ex_intro/
| elim H2 /2 width=2 by ex_intro/
]
-| #k0 #l0 *
- [ #l #H destruct elim l -l normalize
+| #k0 #d0 *
+ [ #d #H destruct elim d -d normalize
/2 width=1 by deg_SO_gt, deg_SO_pos, next_lt/
- | #H1 @deg_SO_zero * #l #H2 destruct
- @H1 -H1 @(ex_intro ⦠(S l)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
+ | #H1 @deg_SO_zero * #d #H2 destruct
+ @H1 -H1 @(ex_intro ⦠(S d)) /2 width=1 by sym_eq/ (**) (* explicit constructor *)
]
]
defined.
-let rec sd_l (h:sh) (k:nat) (l:nat) on l : sd h â
- match l with
+let rec sd_d (h:sh) (k:nat) (d:nat) on d : sd h â
+ match d with
[ O â sd_O h
- | S l â match l with
+ | S d â match d with
[ O â sd_SO h k
- | _ â sd_l h (next h k) l
+ | _ â sd_d h (next h k) d
]
].
(* Basic inversion lemmas ***************************************************)
-lemma deg_inv_pred: âh,g,k,l. deg h g (next h k) (l+1) â deg h g k (l+2).
-#h #g #k #l #H1
-elim (deg_total h g k) #l0 #H0
+lemma deg_inv_pred: âh,g,k,d. deg h g (next h k) (d+1) â deg h g k (d+2).
+#h #g #k #d #H1
+elim (deg_total h g k) #d0 #H0
lapply (deg_next ⦠H0) #H2
lapply (deg_mono ⦠H1 H2) -H1 -H2 #H
-<(associative_plus l 1 1) >H <plus_minus_m_m /2 width=3 by transitive_le/
+<(associative_plus d 1 1) >H <plus_minus_m_m /2 width=3 by transitive_le/
qed-.
-lemma deg_inv_prec: âh,g,k,l,l0. deg h g ((next h)^l k) (l0+1) â deg h g k (l+l0+1).
-#h #g #k #l @(nat_ind_plus ⦠l) -l //
-#l #IHl #l0 >iter_SO #H
-lapply (deg_inv_pred ⦠H) -H <(associative_plus l0 1 1) #H
-lapply (IHl ⦠H) -IHl -H //
+lemma deg_inv_prec: âh,g,k,d,d0. deg h g ((next h)^d k) (d0+1) â deg h g k (d+d0+1).
+#h #g #k #d @(nat_ind_plus ⦠d) -d //
+#d #IHd #d0 >iter_SO #H
+lapply (deg_inv_pred ⦠H) -H <(associative_plus d0 1 1) #H
+lapply (IHd ⦠H) -IHd -H //
qed-.
(* Basic properties *********************************************************)
-lemma deg_iter: âh,g,k,l1,l2. deg h g k l1 â deg h g ((next h)^l2 k) (l1-l2).
-#h #g #k #l1 #l2 @(nat_ind_plus ⦠l2) -l2 [ <minus_n_O // ]
-#l2 #IHl2 #Hkl1 >iter_SO <minus_plus /3 width=1 by deg_next/
+lemma deg_iter: âh,g,k,d1,d2. deg h g k d1 â deg h g ((next h)^d2 k) (d1-d2).
+#h #g #k #d1 #d2 @(nat_ind_plus ⦠d2) -d2 [ <minus_n_O // ]
+#d2 #IHd2 #Hkd1 >iter_SO <minus_plus /3 width=1 by deg_next/
qed.
-lemma deg_next_SO: âh,g,k,l. deg h g k (l+1) â deg h g (next h k) l.
-#h #g #k #l #Hkl
-lapply (deg_next ⦠Hkl) -Hkl <minus_plus_m_m //
+lemma deg_next_SO: âh,g,k,d. deg h g k (d+1) â deg h g (next h k) d.
+#h #g #k #d #Hkd
+lapply (deg_next ⦠Hkd) -Hkd <minus_plus_m_m //
qed-.
-lemma sd_l_SS: âh,k,l. sd_l h k (l + 2) = sd_l h (next h k) (l + 1).
-#h #k #l <plus_n_Sm <plus_n_Sm //
+lemma sd_d_SS: âh,k,d. sd_d h k (d + 2) = sd_d h (next h k) (d + 1).
+#h #k #d <plus_n_Sm <plus_n_Sm //
qed.
-lemma sd_l_correct: âh,l,k. deg h (sd_l h k l) k l.
-#h #l @(nat_ind_plus ⦠l) -l // #l @(nat_ind_plus ⦠l) -l /3 width=1 by deg_inv_pred/
+lemma sd_d_correct: âh,d,k. deg h (sd_d h k d) k d.
+#h #d @(nat_ind_plus ⦠d) -d // #d @(nat_ind_plus ⦠d) -d /3 width=1 by deg_inv_pred/
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/sh.ma b/matita/matita/contribs/lambdadelta/basic_2/static/sh.ma
index 0d09320cc..824e37e1f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/sh.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/sh.ma
@@ -27,18 +27,18 @@ definition sh_N: sh â mk_sh S â¦.
(* Basic properties *********************************************************)
-lemma nexts_le: âh,k,l. k ⤠(next h)^l k.
-#h #k #l elim l -l // normalize #l #IHl
-lapply (next_lt h ((next h)^l k)) #H
-lapply (le_to_lt_to_lt ⦠IHl H) -IHl -H /2 width=2 by lt_to_le/
+lemma nexts_le: âh,k,d. k ⤠(next h)^d k.
+#h #k #d elim d -d // normalize #d #IHd
+lapply (next_lt h ((next h)^d k)) #H
+lapply (le_to_lt_to_lt ⦠IHd H) -IHd -H /2 width=2 by lt_to_le/
qed.
-lemma nexts_lt: âh,k,l. k < (next h)^(l+1) k.
-#h #k #l >iter_SO
-lapply (nexts_le h k l) #H
+lemma nexts_lt: âh,k,d. k < (next h)^(d+1) k.
+#h #k #d >iter_SO
+lapply (nexts_le h k d) #H
@(le_to_lt_to_lt ⦠H) //
qed.
-axiom nexts_dec: âh,k1,k2. Decidable (âl. (next h)^l k1 = k2).
+axiom nexts_dec: âh,k1,k2. Decidable (âd. (next h)^d k1 = k2).
-axiom nexts_inj: âh,k,l1,l2. (next h)^l1 k = (next h)^l2 k â l1 = l2.
+axiom nexts_inj: âh,k,d1,d2. (next h)^d1 k = (next h)^d2 k â d1 = d2.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy.ma
index 439a2cba2..8fe2e8d26 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy.ma
@@ -21,51 +21,51 @@ include "basic_2/substitution/lsuby.ma".
(* 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)
+| cpy_atom : âI,G,L,l,m. cpy l m G L (âª{I}) (âª{I})
+| cpy_subst: âI,G,L,K,V,W,i,l,m. l ⤠yinj i â i < l+m â
+ â¬[i] L â¡ K.â{I}V â â¬[0, i+1] V â¡ W â cpy l m G L (#i) W
+| cpy_bind : âa,I,G,L,V1,V2,T1,T2,l,m.
+ cpy l m G L V1 V2 â cpy (⫯l) m G (L.â{I}V1) T1 T2 â
+ cpy l m G L (â{a,I}V1.T1) (â{a,I}V2.T2)
+| cpy_flat : âI,G,L,V1,V2,T1,T2,l,m.
+ cpy l m G L V1 V2 â cpy l m G L T1 T2 â
+ cpy l m 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).
+ 'PSubst G L T1 l m T2 = (cpy l m 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
+lemma lsuby_cpy_trans: âG,l,m. lsub_trans ⦠(cpy l m G) (lsuby l m).
+#G #l #m #L1 #T1 #T2 #H elim H -G -L1 -T1 -T2 -l -m
[ //
-| #I #G #L1 #K1 #V #W #i #d #e #Hdi #Hide #HLK1 #HVW #L2 #HL12
+| #I #G #L1 #K1 #V #W #i #l #m #Hli #Hilm #HLK1 #HVW #L2 #HL12
elim (lsuby_drop_trans_be ⦠HL12 ⦠HLK1) -HL12 -HLK1 /2 width=5 by cpy_subst/
| /4 width=1 by lsuby_succ, cpy_bind/
| /3 width=1 by cpy_flat/
]
qed-.
-lemma cpy_refl: âG,T,L,d,e. â¦G, L⦠⢠T â¶[d, e] T.
+lemma cpy_refl: âG,T,L,l,m. â¦G, L⦠⢠T â¶[l, m] T.
#G #T elim T -T // * /2 width=1 by cpy_bind, cpy_flat/
qed.
(* Basic_1: was: subst1_ex *)
-lemma cpy_full: 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.
+lemma cpy_full: âI,G,K,V,T1,L,l. â¬[l] L â¡ K.â{I}V â
+ ââT2,T. â¦G, L⦠⢠T1 â¶[l, 1] T2 & â¬[l, 1] T â¡ T2.
#I #G #K #V #T1 elim T1 -T1
-[ * #i #L #d #HLK
+[ * #i #L #l #HLK
/2 width=4 by lift_sort, lift_gref, ex2_2_intro/
- elim (lt_or_eq_or_gt i d) #Hid
+ elim (lt_or_eq_or_gt i l) #Hil
/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
+| * [ #a ] #J #W1 #U1 #IHW1 #IHU1 #L #l #HLK
elim (IHW1 ⦠HLK) -IHW1 #W2 #W #HW12 #HW2
- [ elim (IHU1 (L.â{J}W1) (d+1)) -IHU1
+ [ elim (IHU1 (L.â{J}W1) (l+1)) -IHU1
/3 width=9 by cpy_bind, drop_drop, lift_bind, ex2_2_intro/
| elim (IHU1 ⦠HLK) -IHU1 -HLK
/3 width=8 by cpy_flat, lift_flat, ex2_2_intro/
@@ -73,20 +73,20 @@ lemma cpy_full: âI,G,K,V,T1,L,d. â¬[d] L â¡ K.â{I}V â
]
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 //
+lemma cpy_weak: âG,L,T1,T2,l1,m1. â¦G, L⦠⢠T1 â¶[l1, m1] T2 â
+ âl2,m2. l2 ⤠l1 â l1 + m1 ⤠l2 + m2 â
+ â¦G, L⦠⢠T1 â¶[l2, m2] T2.
+#G #L #T1 #T2 #l1 #m1 #H elim H -G -L -T1 -T2 -l1 -m1 //
[ /3 width=5 by cpy_subst, ylt_yle_trans, yle_trans/
| /4 width=3 by cpy_bind, ylt_yle_trans, yle_succ/
| /3 width=1 by cpy_flat/
]
qed-.
-lemma cpy_weak_top: 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
+lemma cpy_weak_top: âG,L,T1,T2,l,m.
+ â¦G, L⦠⢠T1 â¶[l, m] T2 â â¦G, L⦠⢠T1 â¶[l, |L| - l] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m //
+[ #I #G #L #K #V #W #i #l #m #Hli #_ #HLK #HVW
lapply (drop_fwd_length_lt2 ⦠HLK)
/4 width=5 by cpy_subst, ylt_yle_trans, ylt_inj/
| #a #I #G #L #V1 #V2 normalize in match (|L.â{I}V2|); (**) (* |?| does not work *)
@@ -95,193 +95,193 @@ lemma cpy_weak_top: âG,L,T1,T2,d,e.
]
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
+lemma cpy_weak_full: âG,L,T1,T2,l,m.
+ â¦G, L⦠⢠T1 â¶[l, m] T2 â â¦G, L⦠⢠T1 â¶[0, |L|] T2.
+#G #L #T1 #T2 #l #m #HT12
+lapply (cpy_weak ⦠HT12 0 (l + m) ? ?) -HT12
/2 width=2 by cpy_weak_top/
qed-.
-lemma cpy_split_up: 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
+lemma cpy_split_up: âG,L,T1,T2,l,m. â¦G, L⦠⢠T1 â¶[l, m] T2 â âi. i ⤠l + m â
+ ââT. â¦G, L⦠⢠T1 â¶[l, i-l] T & â¦G, L⦠⢠T â¶[i, l+m-i] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
- elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
+ elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
/4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+ elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
>yplus_SO2 >yplus_succ1 #T #HT1 #HT2
lapply (lsuby_cpy_trans ⦠HT2 (L.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
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
/3 width=5 by ex2_intro, cpy_flat/
]
qed-.
-lemma cpy_split_down: 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
+lemma cpy_split_down: âG,L,T1,T2,l,m. â¦G, L⦠⢠T1 â¶[l, m] T2 â âi. i ⤠l + m â
+ ââT. â¦G, L⦠⢠T1 â¶[i, l+m-i] T & â¦G, L⦠⢠T â¶[l, i-l] T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #HLK #HVW #j #Hjde
- elim (ylt_split i j) [ -Hide -Hjde | -Hdi ]
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #HLK #HVW #j #Hjlm
+ elim (ylt_split i j) [ -Hilm -Hjlm | -Hli ]
/4 width=9 by cpy_subst, ylt_yle_trans, ex2_intro/
-| #a #I #G #L #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #i #Hide
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
elim (IHV12 i) -IHV12 // #V
- elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hide
+ elim (IHT12 (i+1)) -IHT12 /2 width=1 by yle_succ/ -Hilm
>yplus_SO2 >yplus_succ1 #T #HT1 #HT2
lapply (lsuby_cpy_trans ⦠HT2 (L.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
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #i #Hilm
+ elim (IHV12 i) -IHV12 // elim (IHT12 i) -IHT12 // -Hilm
/3 width=5 by ex2_intro, cpy_flat/
]
qed-.
(* Basic forward lemmas *****************************************************)
-lemma cpy_fwd_up: 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 #_
+lemma cpy_fwd_up: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶[lt, mt] U2 â
+ âT1,l,m. â¬[l, m] T1 â¡ U1 â
+ l ⤠lt â l + m ⤠lt + mt â
+ ââT2. â¦G, L⦠⢠U1 â¶[l+m, lt+mt-(l+m)] U2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #T1 #l #m #H #_
[ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
| lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
]
-| #I #G #L #K #V #W #i #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
+| #I #G #L #K #V #W #i #lt #mt #Hlti #Hilmt #HLK #HVW #T1 #l #m #H #Hllt #Hlmlmt
+ elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct [ -V -Hilmt -Hlmlmt | -Hlti -Hllt ]
+ [ elim (ylt_yle_false ⦠Hllt) -Hllt /3 width=3 by yle_ylt_trans, ylt_inj/
+ | elim (le_inv_plus_l ⦠Hil) #Hlim #Hmi
+ elim (lift_split ⦠HVW l (i-m+1) ? ? ?) [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hlim
+ #T2 #_ >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H -Hmi
@(ex2_intro ⦠H) -H @(cpy_subst ⦠HLK HVW) /2 width=1 by yle_inj/ >ymax_pre_sn_comm // (**) (* explicit constructor *)
]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
elim (IHW12 ⦠HVW1) -V1 -IHW12 //
elim (IHU12 ⦠HTU1) -T1 -IHU12 /2 width=1 by yle_succ/
<yplus_inj >yplus_SO2 >yplus_succ1 >yplus_succ1
/3 width=2 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #X #d #e #H #Hddt #Hdedet
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #X #l #m #H #Hllt #Hlmlmt
elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
elim (IHW12 ⦠HVW1) -V1 -IHW12 // elim (IHU12 ⦠HTU1) -T1 -IHU12
/3 width=2 by cpy_flat, lift_flat, ex2_intro/
]
qed-.
-lemma cpy_fwd_tw: 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
+lemma cpy_fwd_tw: âG,L,T1,T2,l,m. â¦G, L⦠⢠T1 â¶[l, m] T2 â â¯{T1} ⤠â¯{T2}.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m normalize
/3 width=1 by monotonic_le_plus_l, le_plus/
qed-.
(* Basic inversion lemmas ***************************************************)
-fact cpy_inv_atom1_aux: âG,L,T1,T2,d,e. â¦G, L⦠⢠T1 â¶[d, e] T2 â âJ. T1 = âª{J} â
+fact cpy_inv_atom1_aux: âG,L,T1,T2,l,m. â¦G, L⦠⢠T1 â¶[l, m] T2 â âJ. T1 = âª{J} â
T2 = âª{J} â¨
- ââI,K,V,i. d ⤠yinj i & i < d + e &
+ ââI,K,V,i. l ⤠yinj i & i < l + m &
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
+#G #L #T1 #T2 #l #m * -G -L -T1 -T2 -l -m
+[ #I #G #L #l #m #J #H destruct /2 width=1 by or_introl/
+| #I #G #L #K #V #T2 #i #l #m #Hli #Hilm #HLK #HVT2 #J #H destruct /3 width=9 by ex5_4_intro, or_intror/
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #H destruct
]
qed-.
-lemma cpy_inv_atom1: âI,G,L,T2,d,e. â¦G, L⦠⢠âª{I} â¶[d, e] T2 â
+lemma cpy_inv_atom1: âI,G,L,T2,l,m. â¦G, L⦠⢠âª{I} â¶[l, m] T2 â
T2 = âª{I} â¨
- ââJ,K,V,i. d ⤠yinj i & i < d + e &
+ ââJ,K,V,i. l ⤠yinj i & i < l + m &
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
+lemma cpy_inv_sort1: âG,L,T2,k,l,m. â¦G, L⦠⢠âk â¶[l, m] T2 â T2 = âk.
+#G #L #T2 #k #l #m #H
elim (cpy_inv_atom1 ⦠H) -H //
* #I #K #V #i #_ #_ #_ #_ #H destruct
qed-.
(* Basic_1: was: subst1_gen_lref *)
-lemma cpy_inv_lref1: âG,L,T2,i,d,e. â¦G, L⦠⢠#i â¶[d, e] T2 â
+lemma cpy_inv_lref1: âG,L,T2,i,l,m. â¦G, L⦠⢠#i â¶[l, m] T2 â
T2 = #i â¨
- ââI,K,V. d ⤠i & i < d + e &
+ ââI,K,V. l ⤠i & i < l + m &
â¬[i] L â¡ K.â{I}V &
â¬[O, i+1] V â¡ T2.
-#G #L #T2 #i #d #e #H
+#G #L #T2 #i #l #m #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/
+* #I #K #V #j #Hlj #Hjlm #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
+lemma cpy_inv_gref1: âG,L,T2,p,l,m. â¦G, L⦠⢠§p â¶[l, m] T2 â T2 = §p.
+#G #L #T2 #p #l #m #H
elim (cpy_inv_atom1 ⦠H) -H //
* #I #K #V #i #_ #_ #_ #_ #H destruct
qed-.
-fact cpy_inv_bind1_aux: âG,L,U1,U2,d,e. â¦G, L⦠⢠U1 â¶[d, e] U2 â
+fact cpy_inv_bind1_aux: âG,L,U1,U2,l,m. â¦G, L⦠⢠U1 â¶[l, m] 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 &
+ ââV2,T2. â¦G, L⦠⢠V1 â¶[l, m] V2 &
+ â¦G, L. â{I}V1⦠⢠T1 â¶[⫯l, m] 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
+#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
+[ #I #G #L #l #m #b #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #b #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #b #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #b #J #W1 #U1 #H destruct
]
qed-.
-lemma cpy_inv_bind1: 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 &
+lemma cpy_inv_bind1: âa,I,G,L,V1,T1,U2,l,m. â¦G, L⦠⢠â{a,I} V1. T1 â¶[l, m] U2 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¶[l, m] V2 &
+ â¦G, L.â{I}V1⦠⢠T1 â¶[⫯l, m] 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
+fact cpy_inv_flat1_aux: âG,L,U1,U2,l,m. â¦G, L⦠⢠U1 â¶[l, m] 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 &
+ ââV2,T2. â¦G, L⦠⢠V1 â¶[l, m] V2 &
+ â¦G, L⦠⢠T1 â¶[l, m] 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/
+#G #L #U1 #U2 #l #m * -G -L -U1 -U2 -l -m
+[ #I #G #L #l #m #J #W1 #U1 #H destruct
+| #I #G #L #K #V #W #i #l #m #_ #_ #_ #_ #J #W1 #U1 #H destruct
+| #a #I #G #L #V1 #V2 #T1 #T2 #l #m #_ #_ #J #W1 #U1 #H destruct
+| #I #G #L #V1 #V2 #T1 #T2 #l #m #HV12 #HT12 #J #W1 #U1 #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma cpy_inv_flat1: 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 &
+lemma cpy_inv_flat1: âI,G,L,V1,T1,U2,l,m. â¦G, L⦠⢠â{I} V1. T1 â¶[l, m] U2 â
+ ââV2,T2. â¦G, L⦠⢠V1 â¶[l, m] V2 &
+ â¦G, L⦠⢠T1 â¶[l, m] 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
+fact cpy_inv_refl_O2_aux: âG,L,T1,T2,l,m. â¦G, L⦠⢠T1 â¶[l, m] T2 â m = 0 â T1 = T2.
+#G #L #T1 #T2 #l #m #H elim H -G -L -T1 -T2 -l -m
[ //
-| #I #G #L #K #V #W #i #d #e #Hdi #Hide #_ #_ #H destruct
- elim (ylt_yle_false ⦠Hdi) -Hdi //
+| #I #G #L #K #V #W #i #l #m #Hli #Hilm #_ #_ #H destruct
+ elim (ylt_yle_false ⦠Hli) -Hli //
| /3 width=1 by eq_f2/
| /3 width=1 by eq_f2/
]
qed-.
-lemma cpy_inv_refl_O2: âG,L,T1,T2,d. â¦G, L⦠⢠T1 â¶[d, 0] T2 â T1 = T2.
+lemma cpy_inv_refl_O2: âG,L,T1,T2,l. â¦G, L⦠⢠T1 â¶[l, 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
+lemma cpy_inv_lift1_eq: âG,T1,U1,l,m. â¬[l, m] T1 â¡ U1 â
+ âL,U2. â¦G, L⦠⢠U1 â¶[l, m] U2 â U1 = U2.
+#G #T1 #U1 #l #m #HTU1 #L #U2 #HU12 elim (cpy_fwd_up ⦠HU12 ⦠HTU1) -HU12 -HTU1
/2 width=4 by cpy_inv_refl_O2/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_cpy.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_cpy.ma
index 28b5982c4..ef48b05af 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_cpy.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_cpy.ma
@@ -19,26 +19,26 @@ include "basic_2/substitution/cpy_lift.ma".
(* 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
+theorem cpy_conf_eq: âG,L,T0,T1,l1,m1. â¦G, L⦠⢠T0 â¶[l1, m1] T1 â
+ âT2,l2,m2. â¦G, L⦠⢠T0 â¶[l2, m2] T2 â
+ ââT. â¦G, L⦠⢠T1 â¶[l2, m2] T & â¦G, L⦠⢠T2 â¶[l1, m1] T.
+#G #L #T0 #T1 #l1 #m1 #H elim H -G -L -T0 -T1 -l1 -m1
[ /2 width=3 by ex2_intro/
-| #I1 #G #L #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #T2 #d2 #e2 #H
+| #I1 #G #L #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #T2 #l2 #m2 #H
elim (cpy_inv_lref1 ⦠H) -H
[ #HX destruct /3 width=7 by cpy_subst, ex2_intro/
- | -Hd1 -Hde1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
+ | -Hl1 -Hlm1 * #I2 #K2 #V2 #_ #_ #HLK2 #HVT2
lapply (drop_mono ⦠HLK1 ⦠HLK2) -HLK1 -HLK2 #H destruct
>(lift_mono ⦠HVT1 ⦠HVT2) -HVT1 -HVT2 /2 width=3 by ex2_intro/
]
-| #a #I #G #L #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #X #d2 #e2 #HX
+| #a #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
elim (cpy_inv_bind1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
elim (IHV01 ⦠HV02) -IHV01 -HV02 #V #HV1 #HV2
elim (IHT01 ⦠HT02) -T0 #T #HT1 #HT2
lapply (lsuby_cpy_trans ⦠HT1 (L.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
+| #I #G #L #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #X #l2 #m2 #HX
elim (cpy_inv_flat1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
elim (IHV01 ⦠HV02) -V0
elim (IHT01 ⦠HT02) -T0 /3 width=5 by cpy_flat, ex2_intro/
@@ -46,21 +46,21 @@ theorem cpy_conf_eq: âG,L,T0,T1,d1,e1. â¦G, L⦠⢠T0 â¶[d1, e1] T1 â
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
+theorem cpy_conf_neq: âG,L1,T0,T1,l1,m1. â¦G, L1⦠⢠T0 â¶[l1, m1] T1 â
+ âL2,T2,l2,m2. â¦G, L2⦠⢠T0 â¶[l2, m2] T2 â
+ (l1 + m1 ⤠l2 ⨠l2 + m2 ⤠l1) â
+ ââT. â¦G, L2⦠⢠T1 â¶[l2, m2] T & â¦G, L1⦠⢠T2 â¶[l1, m1] T.
+#G #L1 #T0 #T1 #l1 #m1 #H elim H -G -L1 -T0 -T1 -l1 -m1
[ /2 width=3 by ex2_intro/
-| #I1 #G #L1 #K1 #V1 #T1 #i0 #d1 #e1 #Hd1 #Hde1 #HLK1 #HVT1 #L2 #T2 #d2 #e2 #H1 #H2
+| #I1 #G #L1 #K1 #V1 #T1 #i0 #l1 #m1 #Hl1 #Hlm1 #HLK1 #HVT1 #L2 #T2 #l2 #m2 #H1 #H2
elim (cpy_inv_lref1 ⦠H1) -H1
[ #H destruct /3 width=7 by cpy_subst, ex2_intro/
- | -HLK1 -HVT1 * #I2 #K2 #V2 #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/
+ | -HLK1 -HVT1 * #I2 #K2 #V2 #Hl2 #Hlm2 #_ #_ elim H2 -H2 #Hlml [ -Hl1 -Hlm2 | -Hl2 -Hlm1 ]
+ [ elim (ylt_yle_false ⦠Hlm1) -Hlm1 /2 width=3 by yle_trans/
+ | elim (ylt_yle_false ⦠Hlm2) -Hlm2 /2 width=3 by yle_trans/
]
]
-| #a #I #G #L1 #V0 #V1 #T0 #T1 #d1 #e1 #_ #_ #IHV01 #IHT01 #L2 #X #d2 #e2 #HX #H
+| #a #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
elim (cpy_inv_bind1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
elim (IHV01 ⦠HV02 H) -IHV01 -HV02 #V #HV1 #HV2
elim (IHT01 ⦠HT02) -T0
@@ -69,7 +69,7 @@ theorem cpy_conf_neq: âG,L1,T0,T1,d1,e1. â¦G, L1⦠⢠T0 â¶[d1, e1] T1 â
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
+| #I #G #L1 #V0 #V1 #T0 #T1 #l1 #m1 #_ #_ #IHV01 #IHT01 #L2 #X #l2 #m2 #HX #H
elim (cpy_inv_flat1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
elim (IHV01 ⦠HV02 H) -V0
elim (IHT01 ⦠HT02 H) -T0 -H /3 width=5 by cpy_flat, ex2_intro/
@@ -78,43 +78,43 @@ 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
+theorem cpy_trans_ge: âG,L,T1,T0,l,m. â¦G, L⦠⢠T1 â¶[l, m] T0 â
+ âT2. â¦G, L⦠⢠T0 â¶[l, 1] T2 â 1 ⤠m â â¦G, L⦠⢠T1 â¶[l, m] T2.
+#G #L #T1 #T0 #l #m #H elim H -G -L -T1 -T0 -l -m
+[ #I #G #L #l #m #T2 #H #Hm
elim (cpy_inv_atom1 ⦠H) -H
[ #H destruct //
- | * #J #K #V #i #Hd2i #Hide2 #HLK #HVT2 #H destruct
- lapply (ylt_yle_trans ⦠(d+e) ⦠Hide2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
+ | * #J #K #V #i #Hl2i #Hilm2 #HLK #HVT2 #H destruct
+ lapply (ylt_yle_trans ⦠(l+m) ⦠Hilm2) /2 width=5 by cpy_subst, monotonic_yle_plus_dx/
]
-| #I #G #L #K #V #V2 #i #d #e #Hdi #Hide #HLK #HVW #T2 #HVT2 #He
+| #I #G #L #K #V #V2 #i #l #m #Hli #Hilm #HLK #HVW #T2 #HVT2 #Hm
lapply (cpy_weak ⦠HVT2 0 (i+1) ? ?) -HVT2 /3 width=1 by yle_plus_dx2_trans, yle_succ/
>yplus_inj #HVT2 <(cpy_inv_lift1_eq ⦠HVW ⦠HVT2) -HVT2 /2 width=5 by cpy_subst/
-| #a #I #G #L #V1 #V0 #T1 #T0 #d #e #_ #_ #IHV10 #IHT10 #X #H #He
+| #a #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
elim (cpy_inv_bind1 ⦠H) -H #V2 #T2 #HV02 #HT02 #H destruct
lapply (lsuby_cpy_trans ⦠HT02 (L.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
+ lapply (IHT10 ⦠HT02 Hm) -T0 /3 width=1 by cpy_bind/
+| #I #G #L #V1 #V0 #T1 #T0 #l #m #_ #_ #IHV10 #IHT10 #X #H #Hm
elim (cpy_inv_flat1 ⦠H) -H #V2 #T2 #HV02 #HT02 #H destruct /3 width=1 by cpy_flat/
]
qed-.
-theorem cpy_trans_down: 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
+theorem cpy_trans_down: âG,L,T1,T0,l1,m1. â¦G, L⦠⢠T1 â¶[l1, m1] T0 â
+ âT2,l2,m2. â¦G, L⦠⢠T0 â¶[l2, m2] T2 â l2 + m2 ⤠l1 â
+ ââT. â¦G, L⦠⢠T1 â¶[l2, m2] T & â¦G, L⦠⢠T â¶[l1, m1] T2.
+#G #L #T1 #T0 #l1 #m1 #H elim H -G -L -T1 -T0 -l1 -m1
[ /2 width=3 by ex2_intro/
-| #I #G #L #K #V #W #i1 #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
+| #I #G #L #K #V #W #i1 #l1 #m1 #Hli1 #Hilm1 #HLK #HVW #T2 #l2 #m2 #HWT2 #Hlm2l1
+ lapply (yle_trans ⦠Hlm2l1 ⦠Hli1) -Hlm2l1 #Hlm2i1
+ lapply (cpy_weak ⦠HWT2 0 (i1+1) ? ?) -HWT2 /3 width=1 by yle_succ, yle_pred_sn/ -Hlm2i1
>yplus_inj #HWT2 <(cpy_inv_lift1_eq ⦠HVW ⦠HWT2) -HWT2 /3 width=9 by cpy_subst, ex2_intro/
-| #a #I #G #L #V1 #V0 #T1 #T0 #d1 #e1 #_ #_ #IHV10 #IHT10 #X #d2 #e2 #HX #de2d1
+| #a #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
elim (cpy_inv_bind1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
lapply (lsuby_cpy_trans ⦠HT02 (L.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
+| #I #G #L #V1 #V0 #T1 #T0 #l1 #m1 #_ #_ #IHV10 #IHT10 #X #l2 #m2 #HX #lm2l1
elim (cpy_inv_flat1 ⦠HX) -HX #V2 #T2 #HV02 #HT02 #HX destruct
elim (IHV10 ⦠HV02) -V0 //
elim (IHT10 ⦠HT02) -T0 /3 width=6 by cpy_flat, ex2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_lift.ma
index 991d4db29..1d42acd6f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_lift.ma
@@ -20,59 +20,59 @@ include "basic_2/substitution/cpy.ma".
(* 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 #_
+lemma cpy_lift_le: âG,K,T1,T2,lt,mt. â¦G, K⦠⢠T1 â¶[lt, mt] T2 â
+ âL,U1,U2,s,l,m. â¬[s, l, m] L â¡ K â
+ â¬[l, m] T1 â¡ U1 â â¬[l, m] T2 â¡ U2 â
+ lt + mt ⤠l â â¦G, L⦠⢠U1 â¶[lt, mt] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
>(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #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
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hlmtl
+ lapply (ylt_yle_trans ⦠Hlmtl ⦠Hilmt) -Hlmtl #Hil
+ lapply (ylt_inv_inj ⦠Hil) -Hil #Hil
+ lapply (lift_inv_lref1_lt ⦠H ⦠Hil) -H #H destruct
elim (lift_trans_ge ⦠HVW ⦠HWU2) -W // <minus_plus #W #HVW #HWU2
elim (drop_trans_le ⦠HLK ⦠HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K #Y #_ #HVY
>(lift_mono ⦠HVY ⦠HVW) -Y -HVW #H destruct /2 width=5 by cpy_subst/
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
/4 width=7 by cpy_bind, drop_skip, yle_succ/
-| #G #I #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+| #G #I #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
/3 width=7 by cpy_flat/
]
qed-.
-lemma cpy_lift_be: 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 #_ #_
+lemma cpy_lift_be: âG,K,T1,T2,lt,mt. â¦G, K⦠⢠T1 â¶[lt, mt] T2 â
+ âL,U1,U2,s,l,m. â¬[s, l, m] L â¡ K â
+ â¬[l, m] T1 â¡ U1 â â¬[l, m] T2 â¡ U2 â
+ lt ⤠l â l ⤠lt + mt â â¦G, L⦠⢠U1 â¶[lt, mt + m] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_ #_
>(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #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
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hltl #_
+ elim (lift_inv_lref1 ⦠H) -H * #Hil #H destruct
+ [ -Hltl
+ lapply (ylt_yle_trans ⦠(lt+mt+m) ⦠Hilmt) // -Hilmt #Hilmtm
elim (lift_trans_ge ⦠HVW ⦠HWU2) -W // <minus_plus #W #HVW #HWU2
elim (drop_trans_le ⦠HLK ⦠HKV) -K /2 width=2 by lt_to_le/ #X #HLK #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K #Y #_ #HVY
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #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
+ | -Hlti
+ elim (yle_inv_inj2 ⦠Hltl) -Hltl #ltt #Hltl #H destruct
+ lapply (transitive_le ⦠Hltl Hil) -Hltl #Hlti
lapply (lift_trans_be ⦠HVW ⦠HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (drop_trans_ge_comm ⦠HLK ⦠HKV ?) -K // -Hid
+ lapply (drop_trans_ge_comm ⦠HLK ⦠HKV ?) -K // -Hil
/4 width=5 by cpy_subst, drop_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
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hltl #Hllmt
elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
/4 width=7 by cpy_bind, drop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hdetd
+| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hlmtl
elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
/3 width=7 by cpy_flat/
@@ -80,25 +80,25 @@ lemma cpy_lift_be: âG,K,T1,T2,dt,et. â¦G, K⦠⢠T1 â¶[dt, et] T2 â
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 #_
+lemma cpy_lift_ge: âG,K,T1,T2,lt,mt. â¦G, K⦠⢠T1 â¶[lt, mt] T2 â
+ âL,U1,U2,s,l,m. â¬[s, l, m] L â¡ K â
+ â¬[l, m] T1 â¡ U1 â â¬[l, m] T2 â¡ U2 â
+ l ⤠lt â â¦G, L⦠⢠U1 â¶[lt+m, mt] U2.
+#G #K #T1 #T2 #lt #mt #H elim H -G -K -T1 -T2 -lt -mt
+[ #I #G #K #lt #mt #L #U1 #U2 #s #l #m #_ #H1 #H2 #_
>(lift_mono ⦠H1 ⦠H2) -H1 -H2 //
-| #I #G #K #KV #V #W #i #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
+| #I #G #K #KV #V #W #i #lt #mt #Hlti #Hilmt #HKV #HVW #L #U1 #U2 #s #l #m #HLK #H #HWU2 #Hllt
+ lapply (yle_trans ⦠Hllt ⦠Hlti) -Hllt #Hil
+ elim (yle_inv_inj2 ⦠Hil) -Hil #ll #Hlli #H0 destruct
+ lapply (lift_inv_lref1_ge ⦠H ⦠Hlli) -H #H destruct
lapply (lift_trans_be ⦠HVW ⦠HWU2 ? ?) -W /2 width=1 by le_S/ >plus_plus_comm_23 #HVU2
- lapply (drop_trans_ge_comm ⦠HLK ⦠HKV ?) -K // -Hddi
+ lapply (drop_trans_ge_comm ⦠HLK ⦠HKV ?) -K // -Hlli
/3 width=5 by cpy_subst, drop_inv_gen, monotonic_ylt_plus_dx, monotonic_yle_plus_dx/
-| #a #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
+| #a #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
elim (lift_inv_bind1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_bind1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
/4 width=6 by cpy_bind, drop_skip, yle_succ/
-| #I #G #K #V1 #V2 #T1 #T2 #dt #et #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #d #e #HLK #H1 #H2 #Hddt
+| #I #G #K #V1 #V2 #T1 #T2 #lt #mt #_ #_ #IHV12 #IHT12 #L #U1 #U2 #s #l #m #HLK #H1 #H2 #Hllt
elim (lift_inv_flat1 ⦠H1) -H1 #VV1 #TT1 #HVV1 #HTT1 #H1
elim (lift_inv_flat1 ⦠H2) -H2 #VV2 #TT2 #HVV2 #HTT2 #H2 destruct
/3 width=6 by cpy_flat/
@@ -108,28 +108,28 @@ 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 #_
+lemma cpy_inv_lift1_le: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ lt + mt ⤠l â
+ ââT2. â¦G, K⦠⢠T1 â¶[lt, mt] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
[ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
| lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
]
-| #I #G #L #KV #V #W #i #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
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmtl
+ lapply (ylt_yle_trans ⦠Hlmtl ⦠Hilmt) -Hlmtl #Hil
+ lapply (ylt_inv_inj ⦠Hil) -Hil #Hil
+ lapply (lift_inv_lref2_lt ⦠H ⦠Hil) -H #H destruct
elim (drop_conf_lt ⦠HLK ⦠HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le ⦠HUV ⦠HVW) -V // >minus_plus <plus_minus_m_m // -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_trans_le ⦠HUV ⦠HVW) -V // >minus_plus <plus_minus_m_m // -Hil /3 width=5 by cpy_subst, ex2_intro/
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
elim (IHW12 ⦠HLK ⦠HVW1) -IHW12 // #V2 #HV12 #HVW2
elim (IHU12 ⦠HTU1) -IHU12 -HTU1
/3 width=6 by cpy_bind, yle_succ, drop_skip, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
elim (IHW12 ⦠HLK ⦠HVW1) -W1 //
elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK
@@ -137,38 +137,38 @@ lemma cpy_inv_lift1_le: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
]
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 #_ #_
+lemma cpy_inv_lift1_be: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ lt ⤠l â yinj l + m ⤠lt + mt â
+ ââT2. â¦G, K⦠⢠T1 â¶[lt, mt-m] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_ #_
[ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
| lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
]
-| #I #G #L #KV #V #W #i #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_plus1_to_minus_inj2/ ] -Hdedet #Hidete
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hltl #Hlmlmt
+ lapply (yle_fwd_plus_ge_inj ⦠Hltl Hlmlmt) #Hmmt
+ elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct [ -Hltl -Hilmt | -Hlti -Hlmlmt ]
+ [ lapply (ylt_yle_trans i l (lt+(mt-m)) ? ?) /2 width=1 by ylt_inj/
+ [ >yplus_minus_assoc_inj /2 width=1 by yle_plus1_to_minus_inj2/ ] -Hlmlmt #Hilmtm
elim (drop_conf_lt ⦠HLK ⦠HLKV) -L // #L #U #HKL #_ #HUV
- elim (lift_trans_le ⦠HUV ⦠HVW) -V // >minus_plus <plus_minus_m_m // -Hid
+ elim (lift_trans_le ⦠HUV ⦠HVW) -V // >minus_plus <plus_minus_m_m // -Hil
/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
+ | elim (le_inv_plus_l ⦠Hil) #Hlim #Hmi
+ lapply (yle_trans ⦠Hltl (i-m) ?) /2 width=1 by yle_inj/ -Hltl #Hltim
lapply (drop_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
+ elim (lift_split ⦠HVW l (i-m+1)) -HVW [2,3,4: /2 width=1 by le_S_S, le_S/ ] -Hil -Hlim
#V1 #HV1 >plus_minus // <minus_minus /2 width=1 by le_S/ <minus_n_n <plus_n_O #H
@(ex2_intro ⦠H) @(cpy_subst ⦠HKV HV1) // (**) (* explicit constructor *)
>yplus_minus_assoc_inj /3 width=1 by monotonic_ylt_minus_dx, yle_inj/
]
-| #a #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
elim (IHW12 ⦠HLK ⦠HVW1) -IHW12 // #V2 #HV12 #HVW2
elim (IHU12 ⦠HTU1) -U1
/3 width=6 by cpy_bind, drop_skip, lift_bind, yle_succ, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdtd #Hdedet
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hltl #Hlmlmt
elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
elim (IHW12 ⦠HLK ⦠HVW1) -W1 //
elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK //
@@ -177,35 +177,35 @@ lemma cpy_inv_lift1_be: âG,L,U1,U2,dt,et. â¦G, L⦠⢠U1 â¶[dt, et] U2 â
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 #_
+lemma cpy_inv_lift1_ge: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ yinj l + m ⤠lt â
+ ââT2. â¦G, K⦠⢠T1 â¶[lt-m, mt] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #H elim H -G -L -U1 -U2 -lt -mt
+[ * #i #G #L #lt #mt #K #s #l #m #_ #T1 #H #_
[ lapply (lift_inv_sort2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
- | elim (lift_inv_lref2 ⦠H) -H * #Hid #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
+ | elim (lift_inv_lref2 ⦠H) -H * #Hil #H destruct /3 width=3 by lift_lref_ge_minus, lift_lref_lt, ex2_intro/
| lapply (lift_inv_gref2 ⦠H) -H #H destruct /2 width=3 by ex2_intro/
]
-| #I #G #L #KV #V #W #i #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
+| #I #G #L #KV #V #W #i #lt #mt #Hlti #Hilmt #HLKV #HVW #K #s #l #m #HLK #T1 #H #Hlmlt
+ lapply (yle_trans ⦠Hlmlt ⦠Hlti) #Hlmi
+ elim (yle_inv_plus_inj2 ⦠Hlmlt) -Hlmlt #_ #Hmlt
+ elim (yle_inv_plus_inj2 ⦠Hlmi) #Hlim #Hmi
lapply (lift_inv_lref2_ge ⦠H ?) -H /2 width=1 by yle_inv_inj/ #H destruct
lapply (drop_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
+ elim (lift_split ⦠HVW l (i-m+1)) -HVW [2,3,4: /3 width=1 by yle_inv_inj, le_S_S, le_S/ ] -Hlmi -Hlim
#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
+| #a #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
- elim (yle_inv_plus_inj2 ⦠Hdetd) #_ #Hedt
+ elim (yle_inv_plus_inj2 ⦠Hlmtl) #_ #Hmlt
elim (IHW12 ⦠HLK ⦠HVW1) -IHW12 // #V2 #HV12 #HVW2
elim (IHU12 ⦠HTU1) -U1 [4: @drop_skip // |2,5: skip |3: /2 width=1 by yle_succ/ ]
>yminus_succ1_inj /3 width=5 by cpy_bind, lift_bind, ex2_intro/
-| #I #G #L #W1 #W2 #U1 #U2 #dt #et #_ #_ #IHW12 #IHU12 #K #s #d #e #HLK #X #H #Hdetd
+| #I #G #L #W1 #W2 #U1 #U2 #lt #mt #_ #_ #IHW12 #IHU12 #K #s #l #m #HLK #X #H #Hlmtl
elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HVW1 #HTU1 #H destruct
elim (IHW12 ⦠HLK ⦠HVW1) -W1 //
elim (IHU12 ⦠HLK ⦠HTU1) -U1 -HLK /3 width=5 by cpy_flat, lift_flat, ex2_intro/
@@ -214,36 +214,36 @@ qed-.
(* Advanced 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
+lemma cpy_inv_lift1_ge_up: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ l ⤠lt â lt ⤠yinj l + m â yinj l + m ⤠lt + mt â
+ ââT2. â¦G, K⦠⢠T1 â¶[l, lt + mt - (yinj l + m)] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hllt #Hltlm #Hlmlmt
+elim (cpy_split_up ⦠HU12 (l + m)) -HU12 // -Hlmlmt #U #HU1 #HU2
+lapply (cpy_weak ⦠HU1 l m ? ?) -HU1 // [ >ymax_pre_sn_comm // ] -Hllt -Hltlm #HU1
lapply (cpy_inv_lift1_eq ⦠HTU1 ⦠HU1) -HU1 #HU1 destruct
elim (cpy_inv_lift1_ge ⦠HU2 ⦠HLK ⦠HTU1) -U -L /2 width=3 by ex2_intro/
qed-.
-lemma cpy_inv_lift1_be_up: 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
+lemma cpy_inv_lift1_be_up: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ lt ⤠l â lt + mt ⤠yinj l + m â
+ ââT2. â¦G, K⦠⢠T1 â¶[lt, l-lt] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hlmtlm
+lapply (cpy_weak ⦠HU12 lt (l+m-lt) ? ?) -HU12 //
+[ >ymax_pre_sn_comm /2 width=1 by yle_plus_dx1_trans/ ] -Hlmtlm #HU12
elim (cpy_inv_lift1_be ⦠HU12 ⦠HLK ⦠HTU1) -U1 -L /2 width=3 by ex2_intro/
qed-.
-lemma cpy_inv_lift1_le_up: 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
+lemma cpy_inv_lift1_le_up: âG,L,U1,U2,lt,mt. â¦G, L⦠⢠U1 â¶[lt, mt] U2 â
+ âK,s,l,m. â¬[s, l, m] L â¡ K â âT1. â¬[l, m] T1 â¡ U1 â
+ lt ⤠l â l ⤠lt + mt â lt + mt ⤠yinj l + m â
+ ââT2. â¦G, K⦠⢠T1 â¶[lt, l - lt] T2 & â¬[l, m] T2 â¡ U2.
+#G #L #U1 #U2 #lt #mt #HU12 #K #s #l #m #HLK #T1 #HTU1 #Hltl #Hllmt #Hlmtlm
+elim (cpy_split_up ⦠HU12 l) -HU12 // #U #HU1 #HU2
elim (cpy_inv_lift1_le ⦠HU1 ⦠HLK ⦠HTU1) -U1
-[2: >ymax_pre_sn_comm // ] -Hdtd #T #HT1 #HTU
-lapply (cpy_weak ⦠HU2 d e ? ?) -HU2 //
-[ >ymax_pre_sn_comm // ] -Hddet -Hdetde #HU2
+[2: >ymax_pre_sn_comm // ] -Hltl #T #HT1 #HTU
+lapply (cpy_weak ⦠HU2 l m ? ?) -HU2 //
+[ >ymax_pre_sn_comm // ] -Hllmt -Hlmtlm #HU2
lapply (cpy_inv_lift1_eq ⦠HTU ⦠HU2) -L #H destruct /2 width=3 by ex2_intro/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_nlift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_nlift.ma
index b48aca5c1..bcf52015d 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_nlift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/cpy_nlift.ma
@@ -20,14 +20,14 @@ include "basic_2/substitution/cpy.ma".
(* Inversion lemmas on negated relocation ***********************************)
-lemma cpy_fwd_nlift2_ge: âG,L,U1,U2,d,e. â¦G, L⦠⢠U1 â¶[d, e] U2 â
- âi. d ⤠yinj i â (âT2. â¬[i, 1] T2 â¡ U2 â â¥) â
+lemma cpy_fwd_nlift2_ge: âG,L,U1,U2,l,m. â¦G, L⦠⢠U1 â¶[l, m] U2 â
+ âi. l ⤠yinj i â (âT2. â¬[i, 1] T2 â¡ U2 â â¥) â
(âT1. â¬[i, 1] T1 â¡ U1 â â¥) â¨
- ââI,K,W,j. d ⤠yinj j & j < i & â¬[j]L â¡ K.â{I}W &
+ ââI,K,W,j. l ⤠yinj j & j < i & â¬[j]L â¡ K.â{I}W &
(âV. â¬[i-j-1, 1] V â¡ W â â¥) & (âT1. â¬[j, 1] T1 â¡ U1 â â¥).
-#G #L #U1 #U2 #d #e #H elim H -G -L -U1 -U2 -d -e
+#G #L #U1 #U2 #l #m #H elim H -G -L -U1 -U2 -l -m
[ /3 width=2 by or_introl/
-| #I #G #L #K #V #W #j #d #e #Hdj #Hjde #HLK #HVW #i #Hdi #HnW
+| #I #G #L #K #V #W #j #l #m #Hlj #Hjlm #HLK #HVW #i #Hli #HnW
elim (lt_or_ge j i) #Hij
[ @or_intror @(ex5_4_intro ⦠HLK) // -HLK
[ #X #HXV elim (lift_trans_le ⦠HXV ⦠HVW ?) -V //
@@ -37,15 +37,15 @@ lemma cpy_fwd_nlift2_ge: âG,L,U1,U2,d,e. â¦G, L⦠⢠U1 â¶[d, e] U2 â
| elim (lift_split ⦠HVW i j) -HVW //
#X #_ #H elim HnW -HnW //
]
-| #a #I #G #L #W1 #W2 #U1 #U2 #d #e #_ #_ #IHW12 #IHU12 #i #Hdi #H elim (nlift_inv_bind ⦠H) -H
+| #a #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_bind ⦠H) -H
[ #HnW2 elim (IHW12 ⦠HnW2) -IHW12 -HnW2 -IHU12 //
[ /4 width=9 by nlift_bind_sn, or_introl/
| * /5 width=9 by nlift_bind_sn, ex5_4_intro, or_intror/
]
| #HnU2 elim (IHU12 ⦠HnU2) -IHU12 -HnU2 -IHW12 /2 width=1 by yle_succ/
[ /4 width=9 by nlift_bind_dx, or_introl/
- | * #J #K #W #j #Hdj #Hji #HLK #HnW
- elim (yle_inv_succ1 ⦠Hdj) -Hdj #Hdj #Hj
+ | * #J #K #W #j #Hlj #Hji #HLK #HnW
+ elim (yle_inv_succ1 ⦠Hlj) -Hlj #Hlj #Hj
lapply (ylt_O ⦠Hj) -Hj #Hj
lapply (drop_inv_drop1_lt ⦠HLK ?) // -HLK #HLK
>(plus_minus_m_m j 1) in ⢠(%â?); [2: /3 width=3 by yle_trans, yle_inv_inj/ ]
@@ -53,7 +53,7 @@ lemma cpy_fwd_nlift2_ge: âG,L,U1,U2,d,e. â¦G, L⦠⢠U1 â¶[d, e] U2 â
/5 width=9 by nlift_bind_dx, monotonic_lt_pred, lt_plus_to_minus_r, ex5_4_intro, or_intror/
]
]
-| #I #G #L #W1 #W2 #U1 #U2 #d #e #_ #_ #IHW12 #IHU12 #i #Hdi #H elim (nlift_inv_flat ⦠H) -H
+| #I #G #L #W1 #W2 #U1 #U2 #l #m #_ #_ #IHW12 #IHU12 #i #Hli #H elim (nlift_inv_flat ⦠H) -H
[ #HnW2 elim (IHW12 ⦠HnW2) -IHW12 -HnW2 -IHU12 //
[ /4 width=9 by nlift_flat_sn, or_introl/
| * /5 width=9 by nlift_flat_sn, ex5_4_intro, or_intror/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/drop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/drop.ma
index 475110434..cc6ffe84b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/drop.ma
@@ -24,17 +24,17 @@ include "basic_2/substitution/lift.ma".
(* Basic_1: includes: drop_skip_bind *)
inductive drop (s:bool): relation4 nat nat lenv lenv â
-| drop_atom: âd,e. (s = â» â e = 0) â drop s d e (â) (â)
+| drop_atom: âl,m. (s = â» â m = 0) â drop s l m (â) (â)
| drop_pair: âI,L,V. drop s 0 0 (L.â{I}V) (L.â{I}V)
-| drop_drop: âI,L1,L2,V,e. drop s 0 e L1 L2 â drop s 0 (e+1) (L1.â{I}V) L2
-| drop_skip: âI,L1,L2,V1,V2,d,e.
- drop s d e L1 L2 â â¬[d, e] V2 â¡ V1 â
- drop s (d+1) e (L1.â{I}V1) (L2.â{I}V2)
+| drop_drop: âI,L1,L2,V,m. drop s 0 m L1 L2 â drop s 0 (m+1) (L1.â{I}V) L2
+| drop_skip: âI,L1,L2,V1,V2,l,m.
+ drop s l m L1 L2 â â¬[l, m] V2 â¡ V1 â
+ drop s (l+1) m (L1.â{I}V1) (L2.â{I}V2)
.
interpretation
"basic slicing (local environment) abstract"
- 'RDrop s d e L1 L2 = (drop s d e L1 L2).
+ 'RDrop s l m L1 L2 = (drop s l m L1 L2).
(*
interpretation
"basic slicing (local environment) general"
@@ -42,56 +42,56 @@ interpretation
*)
interpretation
"basic slicing (local environment) lget"
- 'RDrop e L1 L2 = (drop false O e L1 L2).
+ 'RDrop m L1 L2 = (drop false O m L1 L2).
-definition l_liftable: predicate (lenv â relation term) â
- λR. âK,T1,T2. R K T1 T2 â âL,s,d,e. â¬[s, d, e] L â¡ K â
- âU1. â¬[d, e] T1 â¡ U1 â âU2. â¬[d, e] T2 â¡ U2 â R L U1 U2.
+definition d_liftable: predicate (lenv â relation term) â
+ λR. âK,T1,T2. R K T1 T2 â âL,s,l,m. â¬[s, l, m] L â¡ K â
+ âU1. â¬[l, m] T1 â¡ U1 â âU2. â¬[l, m] T2 â¡ U2 â R L U1 U2.
-definition l_deliftable_sn: predicate (lenv â relation term) â
- λR. âL,U1,U2. R L U1 U2 â âK,s,d,e. â¬[s, d, e] L â¡ K â
- âT1. â¬[d, e] T1 â¡ U1 â
- ââT2. â¬[d, e] T2 â¡ U2 & R K T1 T2.
+definition d_deliftable_sn: predicate (lenv â relation term) â
+ λR. âL,U1,U2. R L U1 U2 â âK,s,l,m. â¬[s, l, m] L â¡ K â
+ âT1. â¬[l, m] T1 â¡ U1 â
+ ââT2. â¬[l, m] T2 â¡ U2 & R K T1 T2.
definition dropable_sn: predicate (relation lenv) â
- λR. âL1,K1,s,d,e. â¬[s, d, e] L1 â¡ K1 â âL2. R L1 L2 â
- ââK2. R K1 K2 & â¬[s, d, e] L2 â¡ K2.
+ λR. âL1,K1,s,l,m. â¬[s, l, m] L1 â¡ K1 â âL2. R L1 L2 â
+ ââK2. R K1 K2 & â¬[s, l, m] L2 â¡ K2.
definition dropable_dx: predicate (relation lenv) â
- λR. âL1,L2. R L1 L2 â âK2,s,e. â¬[s, 0, e] L2 â¡ K2 â
- ââK1. â¬[s, 0, e] L1 â¡ K1 & R K1 K2.
+ λR. âL1,L2. R L1 L2 â âK2,s,m. â¬[s, 0, m] L2 â¡ K2 â
+ ââK1. â¬[s, 0, m] L1 â¡ K1 & R K1 K2.
(* Basic inversion lemmas ***************************************************)
-fact drop_inv_atom1_aux: âL1,L2,s,d,e. â¬[s, d, e] L1 â¡ L2 â L1 = â â
- L2 = â ⧠(s = â» â e = 0).
-#L1 #L2 #s #d #e * -L1 -L2 -d -e
+fact drop_inv_atom1_aux: âL1,L2,s,l,m. â¬[s, l, m] L1 â¡ L2 â L1 = â â
+ L2 = â ⧠(s = â» â m = 0).
+#L1 #L2 #s #l #m * -L1 -L2 -l -m
[ /3 width=1 by conj/
| #I #L #V #H destruct
-| #I #L1 #L2 #V #e #_ #H destruct
-| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #H destruct
+| #I #L1 #L2 #V #m #_ #H destruct
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #H destruct
]
qed-.
(* Basic_1: was: drop_gen_sort *)
-lemma drop_inv_atom1: âL2,s,d,e. â¬[s, d, e] â â¡ L2 â L2 = â ⧠(s = â» â e = 0).
+lemma drop_inv_atom1: âL2,s,l,m. â¬[s, l, m] â â¡ L2 â L2 = â ⧠(s = â» â m = 0).
/2 width=4 by drop_inv_atom1_aux/ qed-.
-fact drop_inv_O1_pair1_aux: âL1,L2,s,d,e. â¬[s, d, e] L1 â¡ L2 â d = 0 â
+fact drop_inv_O1_pair1_aux: âL1,L2,s,l,m. â¬[s, l, m] L1 â¡ L2 â l = 0 â
âK,I,V. L1 = K.â{I}V â
- (e = 0 ⧠L2 = K.â{I}V) â¨
- (0 < e ⧠â¬[s, d, e-1] K â¡ L2).
-#L1 #L2 #s #d #e * -L1 -L2 -d -e
-[ #d #e #_ #_ #K #J #W #H destruct
+ (m = 0 ⧠L2 = K.â{I}V) â¨
+ (0 < m ⧠â¬[s, l, m-1] K â¡ L2).
+#L1 #L2 #s #l #m * -L1 -L2 -l -m
+[ #l #m #_ #_ #K #J #W #H destruct
| #I #L #V #_ #K #J #W #HX destruct /3 width=1 by or_introl, conj/
-| #I #L1 #L2 #V #e #HL12 #_ #K #J #W #H destruct /3 width=1 by or_intror, conj/
-| #I #L1 #L2 #V1 #V2 #d #e #_ #_ >commutative_plus normalize #H destruct
+| #I #L1 #L2 #V #m #HL12 #_ #K #J #W #H destruct /3 width=1 by or_intror, conj/
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ >commutative_plus normalize #H destruct
]
qed-.
-lemma drop_inv_O1_pair1: âI,K,L2,V,s,e. â¬[s, 0, e] K. â{I} V â¡ L2 â
- (e = 0 ⧠L2 = K.â{I}V) â¨
- (0 < e ⧠â¬[s, 0, e-1] K â¡ L2).
+lemma drop_inv_O1_pair1: âI,K,L2,V,s,m. â¬[s, 0, m] K. â{I} V â¡ L2 â
+ (m = 0 ⧠L2 = K.â{I}V) â¨
+ (0 < m ⧠â¬[s, 0, m-1] K â¡ L2).
/2 width=3 by drop_inv_O1_pair1_aux/ qed-.
lemma drop_inv_pair1: âI,K,L2,V,s. â¬[s, 0, 0] K.â{I}V â¡ L2 â L2 = K.â{I}V.
@@ -101,42 +101,42 @@ elim (lt_refl_false ⦠H)
qed-.
(* Basic_1: was: drop_gen_drop *)
-lemma drop_inv_drop1_lt: âI,K,L2,V,s,e.
- â¬[s, 0, e] K.â{I}V â¡ L2 â 0 < e â â¬[s, 0, e-1] K â¡ L2.
-#I #K #L2 #V #s #e #H #He
+lemma drop_inv_drop1_lt: âI,K,L2,V,s,m.
+ â¬[s, 0, m] K.â{I}V â¡ L2 â 0 < m â â¬[s, 0, m-1] K â¡ L2.
+#I #K #L2 #V #s #m #H #Hm
elim (drop_inv_O1_pair1 ⦠H) -H * // #H destruct
-elim (lt_refl_false ⦠He)
+elim (lt_refl_false ⦠Hm)
qed-.
-lemma drop_inv_drop1: âI,K,L2,V,s,e.
- â¬[s, 0, e+1] K.â{I}V â¡ L2 â â¬[s, 0, e] K â¡ L2.
-#I #K #L2 #V #s #e #H lapply (drop_inv_drop1_lt ⦠H ?) -H //
+lemma drop_inv_drop1: âI,K,L2,V,s,m.
+ â¬[s, 0, m+1] K.â{I}V â¡ L2 â â¬[s, 0, m] K â¡ L2.
+#I #K #L2 #V #s #m #H lapply (drop_inv_drop1_lt ⦠H ?) -H //
qed-.
-fact drop_inv_skip1_aux: âL1,L2,s,d,e. â¬[s, d, e] L1 â¡ L2 â 0 < d â
+fact drop_inv_skip1_aux: âL1,L2,s,l,m. â¬[s, l, m] L1 â¡ L2 â 0 < l â
âI,K1,V1. L1 = K1.â{I}V1 â
- ââK2,V2. â¬[s, d-1, e] K1 â¡ K2 &
- â¬[d-1, e] V2 â¡ V1 &
+ ââK2,V2. â¬[s, l-1, m] K1 â¡ K2 &
+ â¬[l-1, m] V2 â¡ V1 &
L2 = K2.â{I}V2.
-#L1 #L2 #s #d #e * -L1 -L2 -d -e
-[ #d #e #_ #_ #J #K1 #W1 #H destruct
+#L1 #L2 #s #l #m * -L1 -L2 -l -m
+[ #l #m #_ #_ #J #K1 #W1 #H destruct
| #I #L #V #H elim (lt_refl_false ⦠H)
-| #I #L1 #L2 #V #e #_ #H elim (lt_refl_false ⦠H)
-| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV21 #_ #J #K1 #W1 #H destruct /2 width=5 by ex3_2_intro/
+| #I #L1 #L2 #V #m #_ #H elim (lt_refl_false ⦠H)
+| #I #L1 #L2 #V1 #V2 #l #m #HL12 #HV21 #_ #J #K1 #W1 #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_1: was: drop_gen_skip_l *)
-lemma drop_inv_skip1: âI,K1,V1,L2,s,d,e. â¬[s, d, e] K1.â{I}V1 â¡ L2 â 0 < d â
- ââK2,V2. â¬[s, d-1, e] K1 â¡ K2 &
- â¬[d-1, e] V2 â¡ V1 &
+lemma drop_inv_skip1: âI,K1,V1,L2,s,l,m. â¬[s, l, m] K1.â{I}V1 â¡ L2 â 0 < l â
+ ââK2,V2. â¬[s, l-1, m] K1 â¡ K2 &
+ â¬[l-1, m] V2 â¡ V1 &
L2 = K2.â{I}V2.
/2 width=3 by drop_inv_skip1_aux/ qed-.
-lemma drop_inv_O1_pair2: âI,K,V,s,e,L1. â¬[s, 0, e] L1 â¡ K.â{I}V â
- (e = 0 ⧠L1 = K.â{I}V) â¨
- ââI1,K1,V1. â¬[s, 0, e-1] K1 â¡ K.â{I}V & L1 = K1.â{I1}V1 & 0 < e.
-#I #K #V #s #e *
+lemma drop_inv_O1_pair2: âI,K,V,s,m,L1. â¬[s, 0, m] L1 â¡ K.â{I}V â
+ (m = 0 ⧠L1 = K.â{I}V) â¨
+ ââI1,K1,V1. â¬[s, 0, m-1] K1 â¡ K.â{I}V & L1 = K1.â{I1}V1 & 0 < m.
+#I #K #V #s #m *
[ #H elim (drop_inv_atom1 ⦠H) -H #H destruct
| #L1 #I1 #V1 #H
elim (drop_inv_O1_pair1 ⦠H) -H *
@@ -146,32 +146,32 @@ lemma drop_inv_O1_pair2: âI,K,V,s,e,L1. â¬[s, 0, e] L1 â¡ K.â{I}V â
]
qed-.
-fact drop_inv_skip2_aux: âL1,L2,s,d,e. â¬[s, d, e] L1 â¡ L2 â 0 < d â
+fact drop_inv_skip2_aux: âL1,L2,s,l,m. â¬[s, l, m] L1 â¡ L2 â 0 < l â
âI,K2,V2. L2 = K2.â{I}V2 â
- ââK1,V1. â¬[s, d-1, e] K1 â¡ K2 &
- â¬[d-1, e] V2 â¡ V1 &
+ ââK1,V1. â¬[s, l-1, m] K1 â¡ K2 &
+ â¬[l-1, m] V2 â¡ V1 &
L1 = K1.â{I}V1.
-#L1 #L2 #s #d #e * -L1 -L2 -d -e
-[ #d #e #_ #_ #J #K2 #W2 #H destruct
+#L1 #L2 #s #l #m * -L1 -L2 -l -m
+[ #l #m #_ #_ #J #K2 #W2 #H destruct
| #I #L #V #H elim (lt_refl_false ⦠H)
-| #I #L1 #L2 #V #e #_ #H elim (lt_refl_false ⦠H)
-| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV21 #_ #J #K2 #W2 #H destruct /2 width=5 by ex3_2_intro/
+| #I #L1 #L2 #V #m #_ #H elim (lt_refl_false ⦠H)
+| #I #L1 #L2 #V1 #V2 #l #m #HL12 #HV21 #_ #J #K2 #W2 #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_1: was: drop_gen_skip_r *)
-lemma drop_inv_skip2: âI,L1,K2,V2,s,d,e. â¬[s, d, e] L1 â¡ K2.â{I}V2 â 0 < d â
- ââK1,V1. â¬[s, d-1, e] K1 â¡ K2 & â¬[d-1, e] V2 â¡ V1 &
+lemma drop_inv_skip2: âI,L1,K2,V2,s,l,m. â¬[s, l, m] L1 â¡ K2.â{I}V2 â 0 < l â
+ ââK1,V1. â¬[s, l-1, m] K1 â¡ K2 & â¬[l-1, m] V2 â¡ V1 &
L1 = K1.â{I}V1.
/2 width=3 by drop_inv_skip2_aux/ qed-.
-lemma drop_inv_O1_gt: âL,K,e,s. â¬[s, 0, e] L â¡ K â |L| < e â
+lemma drop_inv_O1_gt: âL,K,m,s. â¬[s, 0, m] L â¡ K â |L| < m â
s = â ⧠K = â.
-#L elim L -L [| #L #Z #X #IHL ] #K #e #s #H normalize in ⢠(?%?â?); #H1e
+#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H normalize in ⢠(?%?â?); #H1m
[ elim (drop_inv_atom1 ⦠H) -H elim s -s /2 width=1 by conj/
- #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (lt_zero_false ⦠H1e)
-| elim (drop_inv_O1_pair1 ⦠H) -H * #H2e #HLK destruct
- [ elim (lt_zero_false ⦠H1e)
+ #_ #Hs lapply (Hs ?) // -Hs #H destruct elim (lt_zero_false ⦠H1m)
+| elim (drop_inv_O1_pair1 ⦠H) -H * #H2m #HLK destruct
+ [ elim (lt_zero_false ⦠H1m)
| elim (IHL ⦠HLK) -IHL -HLK /2 width=1 by lt_plus_to_minus_r, conj/
]
]
@@ -179,56 +179,56 @@ qed-.
(* Basic properties *********************************************************)
-lemma drop_refl_atom_O2: âs,d. â¬[s, d, O] â â¡ â.
+lemma drop_refl_atom_O2: âs,l. â¬[s, l, O] â â¡ â.
/2 width=1 by drop_atom/ qed.
(* Basic_1: was by definition: drop_refl *)
-lemma drop_refl: âL,d,s. â¬[s, d, 0] L â¡ L.
+lemma drop_refl: âL,l,s. â¬[s, l, 0] L â¡ L.
#L elim L -L //
-#L #I #V #IHL #d #s @(nat_ind_plus ⦠d) -d /2 width=1 by drop_pair, drop_skip/
+#L #I #V #IHL #l #s @(nat_ind_plus ⦠l) -l /2 width=1 by drop_pair, drop_skip/
qed.
-lemma drop_drop_lt: âI,L1,L2,V,s,e.
- â¬[s, 0, e-1] L1 â¡ L2 â 0 < e â â¬[s, 0, e] L1.â{I}V â¡ L2.
-#I #L1 #L2 #V #s #e #HL12 #He >(plus_minus_m_m e 1) /2 width=1 by drop_drop/
+lemma drop_drop_lt: âI,L1,L2,V,s,m.
+ â¬[s, 0, m-1] L1 â¡ L2 â 0 < m â â¬[s, 0, m] L1.â{I}V â¡ L2.
+#I #L1 #L2 #V #s #m #HL12 #Hm >(plus_minus_m_m m 1) /2 width=1 by drop_drop/
qed.
-lemma drop_skip_lt: âI,L1,L2,V1,V2,s,d,e.
- â¬[s, d-1, e] L1 â¡ L2 â â¬[d-1, e] V2 â¡ V1 â 0 < d â
- â¬[s, d, e] L1. â{I} V1 â¡ L2.â{I}V2.
-#I #L1 #L2 #V1 #V2 #s #d #e #HL12 #HV21 #Hd >(plus_minus_m_m d 1) /2 width=1 by drop_skip/
+lemma drop_skip_lt: âI,L1,L2,V1,V2,s,l,m.
+ â¬[s, l-1, m] L1 â¡ L2 â â¬[l-1, m] V2 â¡ V1 â 0 < l â
+ â¬[s, l, m] L1. â{I} V1 â¡ L2.â{I}V2.
+#I #L1 #L2 #V1 #V2 #s #l #m #HL12 #HV21 #Hl >(plus_minus_m_m l 1) /2 width=1 by drop_skip/
qed.
-lemma drop_O1_le: âs,e,L. e ⤠|L| â âK. â¬[s, 0, e] L â¡ K.
-#s #e @(nat_ind_plus ⦠e) -e /2 width=2 by ex_intro/
-#e #IHe *
+lemma drop_O1_le: âs,m,L. m ⤠|L| â âK. â¬[s, 0, m] L â¡ K.
+#s #m @(nat_ind_plus ⦠m) -m /2 width=2 by ex_intro/
+#m #IHm *
[ #H elim (le_plus_xSy_O_false ⦠H)
-| #L #I #V normalize #H elim (IHe L) -IHe /3 width=2 by drop_drop, monotonic_pred, ex_intro/
+| #L #I #V normalize #H elim (IHm L) -IHm /3 width=2 by drop_drop, monotonic_pred, ex_intro/
]
qed-.
-lemma drop_O1_lt: âs,L,e. e < |L| â ââI,K,V. â¬[s, 0, e] L â¡ K.â{I}V.
+lemma drop_O1_lt: âs,L,m. m < |L| â ââI,K,V. â¬[s, 0, m] L â¡ K.â{I}V.
#s #L elim L -L
-[ #e #H elim (lt_zero_false ⦠H)
-| #L #I #V #IHL #e @(nat_ind_plus ⦠e) -e /2 width=4 by drop_pair, ex1_3_intro/
- #e #_ normalize #H elim (IHL e) -IHL /3 width=4 by drop_drop, lt_plus_to_minus_r, lt_plus_to_lt_l, ex1_3_intro/
+[ #m #H elim (lt_zero_false ⦠H)
+| #L #I #V #IHL #m @(nat_ind_plus ⦠m) -m /2 width=4 by drop_pair, ex1_3_intro/
+ #m #_ normalize #H elim (IHL m) -IHL /3 width=4 by drop_drop, lt_plus_to_minus_r, lt_plus_to_lt_l, ex1_3_intro/
]
qed-.
-lemma drop_O1_pair: âL,K,e,s. â¬[s, 0, e] L â¡ K â e ⤠|L| â âI,V.
- ââJ,W. â¬[s, 0, e] L.â{I}V â¡ K.â{J}W.
-#L elim L -L [| #L #Z #X #IHL ] #K #e #s #H normalize #He #I #V
-[ elim (drop_inv_atom1 ⦠H) -H #H <(le_n_O_to_eq ⦠He) -e
+lemma drop_O1_pair: âL,K,m,s. â¬[s, 0, m] L â¡ K â m ⤠|L| â âI,V.
+ ââJ,W. â¬[s, 0, m] L.â{I}V â¡ K.â{J}W.
+#L elim L -L [| #L #Z #X #IHL ] #K #m #s #H normalize #Hm #I #V
+[ elim (drop_inv_atom1 ⦠H) -H #H <(le_n_O_to_eq ⦠Hm) -m
#Hs destruct /2 width=3 by ex1_2_intro/
-| elim (drop_inv_O1_pair1 ⦠H) -H * #He #HLK destruct /2 width=3 by ex1_2_intro/
+| elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #HLK destruct /2 width=3 by ex1_2_intro/
elim (IHL ⦠HLK ⦠Z X) -IHL -HLK
/3 width=3 by drop_drop_lt, le_plus_to_minus, ex1_2_intro/
]
qed-.
-lemma drop_O1_ge: âL,e. |L| ⤠e â â¬[â, 0, e] L â¡ â.
-#L elim L -L [ #e #_ @drop_atom #H destruct ]
-#L #I #V #IHL #e @(nat_ind_plus ⦠e) -e [ #H elim (le_plus_xSy_O_false ⦠H) ]
+lemma drop_O1_ge: âL,m. |L| ⤠m â â¬[â, 0, m] L â¡ â.
+#L elim L -L [ #m #_ @drop_atom #H destruct ]
+#L #I #V #IHL #m @(nat_ind_plus ⦠m) -m [ #H elim (le_plus_xSy_O_false ⦠H) ]
normalize /4 width=1 by drop_drop, monotonic_pred/
qed.
@@ -236,18 +236,18 @@ lemma drop_O1_eq: âL,s. â¬[s, 0, |L|] L â¡ â.
#L elim L -L /2 width=1 by drop_drop, drop_atom/
qed.
-lemma drop_split: âL1,L2,d,e2,s. â¬[s, d, e2] L1 â¡ L2 â âe1. e1 ⤠e2 â
- ââL. â¬[s, d, e2 - e1] L1 â¡ L & â¬[s, d, e1] L â¡ L2.
-#L1 #L2 #d #e2 #s #H elim H -L1 -L2 -d -e2
-[ #d #e2 #Hs #e1 #He12 @(ex2_intro ⦠(â))
+lemma drop_split: âL1,L2,l,m2,s. â¬[s, l, m2] L1 â¡ L2 â âm1. m1 ⤠m2 â
+ ââL. â¬[s, l, m2 - m1] L1 â¡ L & â¬[s, l, m1] L â¡ L2.
+#L1 #L2 #l #m2 #s #H elim H -L1 -L2 -l -m2
+[ #l #m2 #Hs #m1 #Hm12 @(ex2_intro ⦠(â))
@drop_atom #H lapply (Hs H) -s #H destruct /2 width=1 by le_n_O_to_eq/
-| #I #L1 #V #e1 #He1 lapply (le_n_O_to_eq ⦠He1) -He1
+| #I #L1 #V #m1 #Hm1 lapply (le_n_O_to_eq ⦠Hm1) -Hm1
#H destruct /2 width=3 by ex2_intro/
-| #I #L1 #L2 #V #e2 #HL12 #IHL12 #e1 @(nat_ind_plus ⦠e1) -e1
+| #I #L1 #L2 #V #m2 #HL12 #IHL12 #m1 @(nat_ind_plus ⦠m1) -m1
[ /3 width=3 by drop_drop, ex2_intro/
- | -HL12 #e1 #_ #He12 lapply (le_plus_to_le_r ⦠He12) -He12
- #He12 elim (IHL12 ⦠He12) -IHL12 >minus_plus_plus_l
- #L #HL1 #HL2 elim (lt_or_ge (|L1|) (e2-e1)) #H0
+ | -HL12 #m1 #_ #Hm12 lapply (le_plus_to_le_r ⦠Hm12) -Hm12
+ #Hm12 elim (IHL12 ⦠Hm12) -IHL12 >minus_plus_plus_l
+ #L #HL1 #HL2 elim (lt_or_ge (|L1|) (m2-m1)) #H0
[ elim (drop_inv_O1_gt ⦠HL1 H0) -HL1 #H1 #H2 destruct
elim (drop_inv_atom1 ⦠HL2) -HL2 #H #_ destruct
@(ex2_intro ⦠(â)) [ @drop_O1_ge normalize // ]
@@ -255,44 +255,44 @@ lemma drop_split: âL1,L2,d,e2,s. â¬[s, d, e2] L1 â¡ L2 â âe1. e1 ⤠e2
| elim (drop_O1_pair ⦠HL1 H0 I V) -HL1 -H0 /3 width=5 by drop_drop, ex2_intro/
]
]
-| #I #L1 #L2 #V1 #V2 #d #e2 #_ #HV21 #IHL12 #e1 #He12 elim (IHL12 ⦠He12) -IHL12
- #L #HL1 #HL2 elim (lift_split ⦠HV21 d e1) -HV21 /3 width=5 by drop_skip, ex2_intro/
+| #I #L1 #L2 #V1 #V2 #l #m2 #_ #HV21 #IHL12 #m1 #Hm12 elim (IHL12 ⦠Hm12) -IHL12
+ #L #HL1 #HL2 elim (lift_split ⦠HV21 l m1) -HV21 /3 width=5 by drop_skip, ex2_intro/
]
qed-.
-lemma drop_FT: âL1,L2,d,e. â¬[â», d, e] L1 â¡ L2 â â¬[â, d, e] L1 â¡ L2.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
+lemma drop_FT: âL1,L2,l,m. â¬[â», l, m] L1 â¡ L2 â â¬[â, l, m] L1 â¡ L2.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
/3 width=1 by drop_atom, drop_drop, drop_skip/
qed.
-lemma drop_gen: âL1,L2,s,d,e. â¬[â», d, e] L1 â¡ L2 â â¬[s, d, e] L1 â¡ L2.
+lemma drop_gen: âL1,L2,s,l,m. â¬[â», l, m] L1 â¡ L2 â â¬[s, l, m] L1 â¡ L2.
#L1 #L2 * /2 width=1 by drop_FT/
qed-.
-lemma drop_T: âL1,L2,s,d,e. â¬[s, d, e] L1 â¡ L2 â â¬[â, d, e] L1 â¡ L2.
+lemma drop_T: âL1,L2,s,l,m. â¬[s, l, m] L1 â¡ L2 â â¬[â, l, m] L1 â¡ L2.
#L1 #L2 * /2 width=1 by drop_FT/
qed-.
-lemma l_liftable_LTC: âR. l_liftable R â l_liftable (LTC ⦠R).
+lemma d_liftable_LTC: âR. d_liftable R â d_liftable (LTC ⦠R).
#R #HR #K #T1 #T2 #H elim H -T2
[ /3 width=10 by inj/
-| #T #T2 #_ #HT2 #IHT1 #L #s #d #e #HLK #U1 #HTU1 #U2 #HTU2
- elim (lift_total T d e) /4 width=12 by step/
+| #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #HTU1 #U2 #HTU2
+ elim (lift_total T l m) /4 width=12 by step/
]
qed-.
-lemma l_deliftable_sn_LTC: âR. l_deliftable_sn R â l_deliftable_sn (LTC ⦠R).
+lemma d_deliftable_sn_LTC: âR. d_deliftable_sn R â d_deliftable_sn (LTC ⦠R).
#R #HR #L #U1 #U2 #H elim H -U2
-[ #U2 #HU12 #K #s #d #e #HLK #T1 #HTU1
+[ #U2 #HU12 #K #s #l #m #HLK #T1 #HTU1
elim (HR ⦠HU12 ⦠HLK ⦠HTU1) -HR -L -U1 /3 width=3 by inj, ex2_intro/
-| #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #T1 #HTU1
+| #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #T1 #HTU1
elim (IHU1 ⦠HLK ⦠HTU1) -IHU1 -U1 #T #HTU #HT1
elim (HR ⦠HU2 ⦠HLK ⦠HTU) -HR -L -U /3 width=5 by step, ex2_intro/
]
qed-.
lemma dropable_sn_TC: âR. dropable_sn R â dropable_sn (TC ⦠R).
-#R #HR #L1 #K1 #s #d #e #HLK1 #L2 #H elim H -L2
+#R #HR #L1 #K1 #s #l #m #HLK1 #L2 #H elim H -L2
[ #L2 #HL12 elim (HR ⦠HLK1 ⦠HL12) -HR -L1
/3 width=3 by inj, ex2_intro/
| #L #L2 #_ #HL2 * #K #HK1 #HLK elim (HR ⦠HLK ⦠HL2) -HR -L
@@ -302,19 +302,19 @@ qed-.
lemma dropable_dx_TC: âR. dropable_dx R â dropable_dx (TC ⦠R).
#R #HR #L1 #L2 #H elim H -L2
-[ #L2 #HL12 #K2 #s #e #HLK2 elim (HR ⦠HL12 ⦠HLK2) -HR -L2
+[ #L2 #HL12 #K2 #s #m #HLK2 elim (HR ⦠HL12 ⦠HLK2) -HR -L2
/3 width=3 by inj, ex2_intro/
-| #L #L2 #_ #HL2 #IHL1 #K2 #s #e #HLK2 elim (HR ⦠HL2 ⦠HLK2) -HR -L2
+| #L #L2 #_ #HL2 #IHL1 #K2 #s #m #HLK2 elim (HR ⦠HL2 ⦠HLK2) -HR -L2
#K #HLK #HK2 elim (IHL1 ⦠HLK) -L
/3 width=5 by step, ex2_intro/
]
qed-.
-lemma l_deliftable_sn_llstar: âR. l_deliftable_sn R â
- âl. l_deliftable_sn (llstar ⦠R l).
-#R #HR #l #L #U1 #U2 #H @(lstar_ind_r ⦠l U2 H) -l -U2
+lemma d_deliftable_sn_llstar: âR. d_deliftable_sn R â
+ âd. d_deliftable_sn (llstar ⦠R d).
+#R #HR #d #L #U1 #U2 #H @(lstar_ind_r ⦠d U2 H) -d -U2
[ /2 width=3 by lstar_O, ex2_intro/
-| #l #U #U2 #_ #HU2 #IHU1 #K #s #d #e #HLK #T1 #HTU1
+| #d #U #U2 #_ #HU2 #IHU1 #K #s #l #m #HLK #T1 #HTU1
elim (IHU1 ⦠HLK ⦠HTU1) -IHU1 -U1 #T #HTU #HT1
elim (HR ⦠HU2 ⦠HLK ⦠HTU) -HR -L -U /3 width=5 by lstar_dx, ex2_intro/
]
@@ -323,36 +323,36 @@ qed-.
(* Basic forward lemmas *****************************************************)
(* Basic_1: was: drop_S *)
-lemma drop_fwd_drop2: âL1,I2,K2,V2,s,e. â¬[s, O, e] L1 â¡ K2. â{I2} V2 â
- â¬[s, O, e + 1] L1 â¡ K2.
+lemma drop_fwd_drop2: âL1,I2,K2,V2,s,m. â¬[s, O, m] L1 â¡ K2. â{I2} V2 â
+ â¬[s, O, m + 1] L1 â¡ K2.
#L1 elim L1 -L1
-[ #I2 #K2 #V2 #s #e #H lapply (drop_inv_atom1 ⦠H) -H * #H destruct
-| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #s #e #H
- elim (drop_inv_O1_pair1 ⦠H) -H * #He #H
+[ #I2 #K2 #V2 #s #m #H lapply (drop_inv_atom1 ⦠H) -H * #H destruct
+| #K1 #I1 #V1 #IHL1 #I2 #K2 #V2 #s #m #H
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hm #H
[ -IHL1 destruct /2 width=1 by drop_drop/
- | @drop_drop >(plus_minus_m_m e 1) /2 width=3 by/
+ | @drop_drop >(plus_minus_m_m m 1) /2 width=3 by/
]
]
qed-.
-lemma drop_fwd_length_ge: âL1,L2,d,e,s. â¬[s, d, e] L1 â¡ L2 â |L1| ⤠d â |L2| = |L1|.
-#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e // normalize
-[ #I #L1 #L2 #V #e #_ #_ #H elim (le_plus_xSy_O_false ⦠H)
+lemma drop_fwd_length_ge: âL1,L2,l,m,s. â¬[s, l, m] L1 â¡ L2 â |L1| ⤠l â |L2| = |L1|.
+#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m // normalize
+[ #I #L1 #L2 #V #m #_ #_ #H elim (le_plus_xSy_O_false ⦠H)
| /4 width=2 by le_plus_to_le_r, eq_f/
]
qed-.
-lemma drop_fwd_length_le_le: âL1,L2,d,e,s. â¬[s, d, e] L1 â¡ L2 â d ⤠|L1| â e ⤠|L1| - d â |L2| = |L1| - e.
-#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e // normalize
+lemma drop_fwd_length_le_le: âL1,L2,l,m,s. â¬[s, l, m] L1 â¡ L2 â l ⤠|L1| â m ⤠|L1| - l â |L2| = |L1| - m.
+#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m // normalize
[ /3 width=2 by le_plus_to_le_r/
-| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #IHL12 >minus_plus_plus_l
- #Hd #He lapply (le_plus_to_le_r ⦠Hd) -Hd
- #Hd >IHL12 // -L2 >plus_minus /2 width=3 by transitive_le/
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IHL12 >minus_plus_plus_l
+ #Hl #Hm lapply (le_plus_to_le_r ⦠Hl) -Hl
+ #Hl >IHL12 // -L2 >plus_minus /2 width=3 by transitive_le/
]
qed-.
-lemma drop_fwd_length_le_ge: âL1,L2,d,e,s. â¬[s, d, e] L1 â¡ L2 â d ⤠|L1| â |L1| - d ⤠e â |L2| = d.
-#L1 #L2 #d #e #s #H elim H -L1 -L2 -d -e normalize
+lemma drop_fwd_length_le_ge: âL1,L2,l,m,s. â¬[s, l, m] L1 â¡ L2 â l ⤠|L1| â |L1| - l ⤠m â |L2| = l.
+#L1 #L2 #l #m #s #H elim H -L1 -L2 -l -m normalize
[ /2 width=1 by le_n_O_to_eq/
| #I #L #V #_ <minus_n_O #H elim (le_plus_xSy_O_false ⦠H)
| /3 width=2 by le_plus_to_le_r/
@@ -360,67 +360,67 @@ lemma drop_fwd_length_le_ge: âL1,L2,d,e,s. â¬[s, d, e] L1 â¡ L2 â d ⤠|L
]
qed-.
-lemma drop_fwd_length: âL1,L2,d,e. â¬[â», d, e] L1 â¡ L2 â |L1| = |L2| + e.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e // normalize /2 width=1 by/
+lemma drop_fwd_length: âL1,L2,l,m. â¬[â», l, m] L1 â¡ L2 â |L1| = |L2| + m.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m // normalize /2 width=1 by/
qed-.
-lemma drop_fwd_length_minus2: âL1,L2,d,e. â¬[â», d, e] L1 â¡ L2 â |L2| = |L1| - e.
-#L1 #L2 #d #e #H lapply (drop_fwd_length ⦠H) -H /2 width=1 by plus_minus, le_n/
+lemma drop_fwd_length_minus2: âL1,L2,l,m. â¬[â», l, m] L1 â¡ L2 â |L2| = |L1| - m.
+#L1 #L2 #l #m #H lapply (drop_fwd_length ⦠H) -H /2 width=1 by plus_minus, le_n/
qed-.
-lemma drop_fwd_length_minus4: âL1,L2,d,e. â¬[â», d, e] L1 â¡ L2 â e = |L1| - |L2|.
-#L1 #L2 #d #e #H lapply (drop_fwd_length ⦠H) -H //
+lemma drop_fwd_length_minus4: âL1,L2,l,m. â¬[â», l, m] L1 â¡ L2 â m = |L1| - |L2|.
+#L1 #L2 #l #m #H lapply (drop_fwd_length ⦠H) -H //
qed-.
-lemma drop_fwd_length_le2: âL1,L2,d,e. â¬[â», d, e] L1 â¡ L2 â e ⤠|L1|.
-#L1 #L2 #d #e #H lapply (drop_fwd_length ⦠H) -H //
+lemma drop_fwd_length_le2: âL1,L2,l,m. â¬[â», l, m] L1 â¡ L2 â m ⤠|L1|.
+#L1 #L2 #l #m #H lapply (drop_fwd_length ⦠H) -H //
qed-.
-lemma drop_fwd_length_le4: âL1,L2,d,e. â¬[â», d, e] L1 â¡ L2 â |L2| ⤠|L1|.
-#L1 #L2 #d #e #H lapply (drop_fwd_length ⦠H) -H //
+lemma drop_fwd_length_le4: âL1,L2,l,m. â¬[â», l, m] L1 â¡ L2 â |L2| ⤠|L1|.
+#L1 #L2 #l #m #H lapply (drop_fwd_length ⦠H) -H //
qed-.
-lemma drop_fwd_length_lt2: âL1,I2,K2,V2,d,e.
- â¬[â», d, e] L1 â¡ K2. â{I2} V2 â e < |L1|.
-#L1 #I2 #K2 #V2 #d #e #H
+lemma drop_fwd_length_lt2: âL1,I2,K2,V2,l,m.
+ â¬[â», l, m] L1 â¡ K2. â{I2} V2 â m < |L1|.
+#L1 #I2 #K2 #V2 #l #m #H
lapply (drop_fwd_length ⦠H) normalize in ⢠(%â?); -I2 -V2 //
qed-.
-lemma drop_fwd_length_lt4: âL1,L2,d,e. â¬[â», d, e] L1 â¡ L2 â 0 < e â |L2| < |L1|.
-#L1 #L2 #d #e #H lapply (drop_fwd_length ⦠H) -H /2 width=1 by lt_minus_to_plus_r/
+lemma drop_fwd_length_lt4: âL1,L2,l,m. â¬[â», l, m] L1 â¡ L2 â 0 < m â |L2| < |L1|.
+#L1 #L2 #l #m #H lapply (drop_fwd_length ⦠H) -H /2 width=1 by lt_minus_to_plus_r/
qed-.
-lemma drop_fwd_length_eq1: âL1,L2,K1,K2,d,e. â¬[â», d, e] L1 â¡ K1 â â¬[â», d, e] L2 â¡ K2 â
+lemma drop_fwd_length_eq1: âL1,L2,K1,K2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
|L1| = |L2| â |K1| = |K2|.
-#L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
+#L1 #L2 #K1 #K2 #l #m #HLK1 #HLK2 #HL12
lapply (drop_fwd_length ⦠HLK1) -HLK1
lapply (drop_fwd_length ⦠HLK2) -HLK2
/2 width=2 by injective_plus_r/
qed-.
-lemma drop_fwd_length_eq2: âL1,L2,K1,K2,d,e. â¬[â», d, e] L1 â¡ K1 â â¬[â», d, e] L2 â¡ K2 â
+lemma drop_fwd_length_eq2: âL1,L2,K1,K2,l,m. â¬[â», l, m] L1 â¡ K1 â â¬[â», l, m] L2 â¡ K2 â
|K1| = |K2| â |L1| = |L2|.
-#L1 #L2 #K1 #K2 #d #e #HLK1 #HLK2 #HL12
+#L1 #L2 #K1 #K2 #l #m #HLK1 #HLK2 #HL12
lapply (drop_fwd_length ⦠HLK1) -HLK1
lapply (drop_fwd_length ⦠HLK2) -HLK2 //
qed-.
-lemma drop_fwd_lw: âL1,L2,s,d,e. â¬[s, d, e] L1 â¡ L2 â â¯{L2} ⤠â¯{L1}.
-#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e // normalize
+lemma drop_fwd_lw: âL1,L2,s,l,m. â¬[s, l, m] L1 â¡ L2 â â¯{L2} ⤠â¯{L1}.
+#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m // normalize
[ /2 width=3 by transitive_le/
-| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12
+| #I #L1 #L2 #V1 #V2 #l #m #_ #HV21 #IHL12
>(lift_fwd_tw ⦠HV21) -HV21 /2 width=1 by monotonic_le_plus_l/
]
qed-.
-lemma drop_fwd_lw_lt: âL1,L2,d,e. â¬[â», d, e] L1 â¡ L2 â 0 < e â â¯{L2} < â¯{L1}.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-[ #d #e #H >H -H //
+lemma drop_fwd_lw_lt: âL1,L2,l,m. â¬[â», l, m] L1 â¡ L2 â 0 < m â â¯{L2} < â¯{L1}.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+[ #l #m #H >H -H //
| #I #L #V #H elim (lt_refl_false ⦠H)
-| #I #L1 #L2 #V #e #HL12 #_ #_
+| #I #L1 #L2 #V #m #HL12 #_ #_
lapply (drop_fwd_lw ⦠HL12) -HL12 #HL12
@(le_to_lt_to_lt ⦠HL12) -HL12 //
-| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12 #H normalize in ⢠(?%%); -I
+| #I #L1 #L2 #V1 #V2 #l #m #_ #HV21 #IHL12 #H normalize in ⢠(?%%); -I
>(lift_fwd_tw ⦠HV21) -V2 /3 by lt_minus_to_plus/
]
qed-.
@@ -432,48 +432,48 @@ qed-.
(* Advanced inversion lemmas ************************************************)
-fact drop_inv_O2_aux: âL1,L2,s,d,e. â¬[s, d, e] L1 â¡ L2 â e = 0 â L1 = L2.
-#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e
+fact drop_inv_O2_aux: âL1,L2,s,l,m. â¬[s, l, m] L1 â¡ L2 â m = 0 â L1 = L2.
+#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m
[ //
| //
-| #I #L1 #L2 #V #e #_ #_ >commutative_plus normalize #H destruct
-| #I #L1 #L2 #V1 #V2 #d #e #_ #HV21 #IHL12 #H
- >(IHL12 H) -L1 >(lift_inv_O2_aux ⦠HV21 ⦠H) -V2 -d -e //
+| #I #L1 #L2 #V #m #_ #_ >commutative_plus normalize #H destruct
+| #I #L1 #L2 #V1 #V2 #l #m #_ #HV21 #IHL12 #H
+ >(IHL12 H) -L1 >(lift_inv_O2_aux ⦠HV21 ⦠H) -V2 -l -m //
]
qed-.
(* Basic_1: was: drop_gen_refl *)
-lemma drop_inv_O2: âL1,L2,s,d. â¬[s, d, 0] L1 â¡ L2 â L1 = L2.
+lemma drop_inv_O2: âL1,L2,s,l. â¬[s, l, 0] L1 â¡ L2 â L1 = L2.
/2 width=5 by drop_inv_O2_aux/ qed-.
-lemma drop_inv_length_eq: âL1,L2,d,e. â¬[â», d, e] L1 â¡ L2 â |L1| = |L2| â e = 0.
-#L1 #L2 #d #e #H #HL12 lapply (drop_fwd_length_minus4 ⦠H) //
+lemma drop_inv_length_eq: âL1,L2,l,m. â¬[â», l, m] L1 â¡ L2 â |L1| = |L2| â m = 0.
+#L1 #L2 #l #m #H #HL12 lapply (drop_fwd_length_minus4 ⦠H) //
qed-.
-lemma drop_inv_refl: âL,d,e. â¬[â», d, e] L â¡ L â e = 0.
+lemma drop_inv_refl: âL,l,m. â¬[â», l, m] L â¡ L â m = 0.
/2 width=5 by drop_inv_length_eq/ qed-.
-fact drop_inv_FT_aux: âL1,L2,s,d,e. â¬[s, d, e] L1 â¡ L2 â
- âI,K,V. L2 = K.â{I}V â s = â â d = 0 â
- â¬[â», d, e] L1 â¡ K.â{I}V.
-#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e
-[ #d #e #_ #J #K #W #H destruct
+fact drop_inv_FT_aux: âL1,L2,s,l,m. â¬[s, l, m] L1 â¡ L2 â
+ âI,K,V. L2 = K.â{I}V â s = â â l = 0 â
+ â¬[â», l, m] L1 â¡ K.â{I}V.
+#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m
+[ #l #m #_ #J #K #W #H destruct
| #I #L #V #J #K #W #H destruct //
-| #I #L1 #L2 #V #e #_ #IHL12 #J #K #W #H1 #H2 destruct
+| #I #L1 #L2 #V #m #_ #IHL12 #J #K #W #H1 #H2 destruct
/3 width=1 by drop_drop/
-| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #_ #J #K #W #_ #_
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #_ #J #K #W #_ #_
<plus_n_Sm #H destruct
]
qed-.
-lemma drop_inv_FT: âI,L,K,V,e. â¬[â, 0, e] L â¡ K.â{I}V â â¬[e] L â¡ K.â{I}V.
+lemma drop_inv_FT: âI,L,K,V,m. â¬[â, 0, m] L â¡ K.â{I}V â â¬[m] L â¡ K.â{I}V.
/2 width=5 by drop_inv_FT_aux/ qed.
-lemma drop_inv_gen: âI,L,K,V,s,e. â¬[s, 0, e] L â¡ K.â{I}V â â¬[e] L â¡ K.â{I}V.
+lemma drop_inv_gen: âI,L,K,V,s,m. â¬[s, 0, m] L â¡ K.â{I}V â â¬[m] L â¡ K.â{I}V.
#I #L #K #V * /2 width=1 by drop_inv_FT/
qed-.
-lemma drop_inv_T: âI,L,K,V,s,e. â¬[â, 0, e] L â¡ K.â{I}V â â¬[s, 0, e] L â¡ K.â{I}V.
+lemma drop_inv_T: âI,L,K,V,s,m. â¬[â, 0, m] L â¡ K.â{I}V â â¬[s, 0, m] L â¡ K.â{I}V.
#I #L #K #V * /2 width=1 by drop_inv_FT/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_append.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_append.ma
index 2cb36a8c7..ee2e49245 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_append.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_append.ma
@@ -19,38 +19,38 @@ include "basic_2/substitution/drop.ma".
(* Properties on append for local environments ******************************)
-fact drop_O1_append_sn_le_aux: âL1,L2,s,d,e. â¬[s, d, e] L1 â¡ L2 â
- d = 0 â e ⤠|L1| â
- âL. â¬[s, 0, e] L @@ L1 â¡ L @@ L2.
-#L1 #L2 #s #d #e #H elim H -L1 -L2 -d -e normalize
+fact drop_O1_append_sn_le_aux: âL1,L2,s,l,m. â¬[s, l, m] L1 â¡ L2 â
+ l = 0 â m ⤠|L1| â
+ âL. â¬[s, 0, m] L @@ L1 â¡ L @@ L2.
+#L1 #L2 #s #l #m #H elim H -L1 -L2 -l -m normalize
[2,3,4: /4 width=1 by drop_skip_lt, drop_drop, arith_b1, lt_minus_to_plus_r, monotonic_pred/ ]
-#d #e #_ #_ #H <(le_n_O_to_eq ⦠H) -H //
+#l #m #_ #_ #H <(le_n_O_to_eq ⦠H) -H //
qed-.
-lemma drop_O1_append_sn_le: âL1,L2,s,e. â¬[s, 0, e] L1 â¡ L2 â e ⤠|L1| â
- âL. â¬[s, 0, e] L @@ L1 â¡ L @@ L2.
+lemma drop_O1_append_sn_le: âL1,L2,s,m. â¬[s, 0, m] L1 â¡ L2 â m ⤠|L1| â
+ âL. â¬[s, 0, m] L @@ L1 â¡ L @@ L2.
/2 width=3 by drop_O1_append_sn_le_aux/ qed.
(* Inversion lemmas on append for local environments ************************)
-lemma drop_O1_inv_append1_ge: âK,L1,L2,s,e. â¬[s, 0, e] L1 @@ L2 â¡ K â
- |L2| ⤠e â â¬[s, 0, e - |L2|] L1 â¡ K.
+lemma drop_O1_inv_append1_ge: âK,L1,L2,s,m. â¬[s, 0, m] L1 @@ L2 â¡ K â
+ |L2| ⤠m â â¬[s, 0, m - |L2|] L1 â¡ K.
#K #L1 #L2 elim L2 -L2 normalize //
-#L2 #I #V #IHL2 #s #e #H #H1e
-elim (drop_inv_O1_pair1 ⦠H) -H * #H2e #HL12 destruct
-[ lapply (le_n_O_to_eq ⦠H1e) -H1e -IHL2
+#L2 #I #V #IHL2 #s #m #H #H1m
+elim (drop_inv_O1_pair1 ⦠H) -H * #H2m #HL12 destruct
+[ lapply (le_n_O_to_eq ⦠H1m) -H1m -IHL2
>commutative_plus normalize #H destruct
| <minus_plus >minus_minus_comm /3 width=1 by monotonic_pred/
]
qed-.
-lemma drop_O1_inv_append1_le: âK,L1,L2,s,e. â¬[s, 0, e] L1 @@ L2 â¡ K â e ⤠|L2| â
- âK2. â¬[s, 0, e] L2 â¡ K2 â K = L1 @@ K2.
+lemma drop_O1_inv_append1_le: âK,L1,L2,s,m. â¬[s, 0, m] L1 @@ L2 â¡ K â m ⤠|L2| â
+ âK2. â¬[s, 0, m] L2 â¡ K2 â K = L1 @@ K2.
#K #L1 #L2 elim L2 -L2 normalize
-[ #s #e #H1 #H2 #K2 #H3 lapply (le_n_O_to_eq ⦠H2) -H2
+[ #s #m #H1 #H2 #K2 #H3 lapply (le_n_O_to_eq ⦠H2) -H2
#H2 elim (drop_inv_atom1 ⦠H3) -H3 #H3 #_ destruct
>(drop_inv_O2 ⦠H1) -H1 //
-| #L2 #I #V #IHL2 #s #e @(nat_ind_plus ⦠e) -e [ -IHL2 ]
+| #L2 #I #V #IHL2 #s #m @(nat_ind_plus ⦠m) -m [ -IHL2 ]
[ #H1 #_ #K2 #H2
lapply (drop_inv_O2 ⦠H1) -H1 #H1
lapply (drop_inv_O2 ⦠H2) -H2 #H2 destruct //
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_drop.ma
index a2c8b19ef..83b347d39 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_drop.ma
@@ -20,14 +20,14 @@ include "basic_2/substitution/drop.ma".
(* Main properties **********************************************************)
(* Basic_1: was: drop_mono *)
-theorem drop_mono: âL,L1,s1,d,e. â¬[s1, d, e] L â¡ L1 â
- âL2,s2. â¬[s2, d, e] L â¡ L2 â L1 = L2.
-#L #L1 #s1 #d #e #H elim H -L -L1 -d -e
-[ #d #e #He #L2 #s2 #H elim (drop_inv_atom1 ⦠H) -H //
+theorem drop_mono: âL,L1,s1,l,m. â¬[s1, l, m] L â¡ L1 â
+ âL2,s2. â¬[s2, l, m] L â¡ L2 â L1 = L2.
+#L #L1 #s1 #l #m #H elim H -L -L1 -l -m
+[ #l #m #Hm #L2 #s2 #H elim (drop_inv_atom1 ⦠H) -H //
| #I #K #V #L2 #s2 #HL12 <(drop_inv_O2 ⦠HL12) -L2 //
-| #I #L #K #V #e #_ #IHLK #L2 #s2 #H
+| #I #L #K #V #m #_ #IHLK #L2 #s2 #H
lapply (drop_inv_drop1 ⦠H) -H /2 width=2 by/
-| #I #L #K1 #T #V1 #d #e #_ #HVT1 #IHLK1 #X #s2 #H
+| #I #L #K1 #T #V1 #l #m #_ #HVT1 #IHLK1 #X #s2 #H
elim (drop_inv_skip1 ⦠H) -H // <minus_plus_m_m #K2 #V2 #HLK2 #HVT2 #H destruct
>(lift_inj ⦠HVT1 ⦠HVT2) -HVT1 -HVT2
>(IHLK1 ⦠HLK2) -IHLK1 -HLK2 //
@@ -35,43 +35,43 @@ theorem drop_mono: âL,L1,s1,d,e. â¬[s1, d, e] L â¡ L1 â
qed-.
(* Basic_1: was: drop_conf_ge *)
-theorem drop_conf_ge: âL,L1,s1,d1,e1. â¬[s1, d1, e1] L â¡ L1 â
- âL2,s2,e2. â¬[s2, 0, e2] L â¡ L2 â d1 + e1 ⤠e2 â
- â¬[s2, 0, e2 - e1] L1 â¡ L2.
-#L #L1 #s1 #d1 #e1 #H elim H -L -L1 -d1 -e1 //
-[ #d #e #_ #L2 #s2 #e2 #H #_ elim (drop_inv_atom1 ⦠H) -H
- #H #He destruct
- @drop_atom #H >He // (**) (* explicit constructor *)
-| #I #L #K #V #e #_ #IHLK #L2 #s2 #e2 #H #He2
+theorem drop_conf_ge: âL,L1,s1,l1,m1. â¬[s1, l1, m1] L â¡ L1 â
+ âL2,s2,m2. â¬[s2, 0, m2] L â¡ L2 â l1 + m1 ⤠m2 â
+ â¬[s2, 0, m2 - m1] L1 â¡ L2.
+#L #L1 #s1 #l1 #m1 #H elim H -L -L1 -l1 -m1 //
+[ #l #m #_ #L2 #s2 #m2 #H #_ elim (drop_inv_atom1 ⦠H) -H
+ #H #Hm destruct
+ @drop_atom #H >Hm // (**) (* explicit constructor *)
+| #I #L #K #V #m #_ #IHLK #L2 #s2 #m2 #H #Hm2
lapply (drop_inv_drop1_lt ⦠H ?) -H /2 width=2 by ltn_to_ltO/ #HL2
<minus_plus >minus_minus_comm /3 width=1 by monotonic_pred/
-| #I #L #K #V1 #V2 #d #e #_ #_ #IHLK #L2 #s2 #e2 #H #Hdee2
- lapply (transitive_le 1 ⦠Hdee2) // #He2
- lapply (drop_inv_drop1_lt ⦠H ?) -H // -He2 #HL2
- lapply (transitive_le (1+e) ⦠Hdee2) // #Hee2
+| #I #L #K #V1 #V2 #l #m #_ #_ #IHLK #L2 #s2 #m2 #H #Hlmm2
+ lapply (transitive_le 1 ⦠Hlmm2) // #Hm2
+ lapply (drop_inv_drop1_lt ⦠H ?) -H // -Hm2 #HL2
+ lapply (transitive_le (1+m) ⦠Hlmm2) // #Hmm2
@drop_drop_lt >minus_minus_comm /3 width=1 by lt_minus_to_plus_r, monotonic_le_minus_r, monotonic_pred/ (**) (* explicit constructor *)
]
qed.
(* Note: apparently this was missing in basic_1 *)
-theorem drop_conf_be: âL0,L1,s1,d1,e1. â¬[s1, d1, e1] L0 â¡ L1 â
- âL2,e2. â¬[e2] L0 â¡ L2 â d1 ⤠e2 â e2 ⤠d1 + e1 â
- ââL. â¬[s1, 0, d1 + e1 - e2] L2 â¡ L & â¬[d1] L1 â¡ L.
-#L0 #L1 #s1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #He1 #L2 #e2 #H #Hd1 #_ elim (drop_inv_atom1 ⦠H) -H #H #He2 destruct
- >(He2 ?) in Hd1; // -He2 #Hd1 <(le_n_O_to_eq ⦠Hd1) -d1
+theorem drop_conf_be: âL0,L1,s1,l1,m1. â¬[s1, l1, m1] L0 â¡ L1 â
+ âL2,m2. â¬[m2] L0 â¡ L2 â l1 ⤠m2 â m2 ⤠l1 + m1 â
+ ââL. â¬[s1, 0, l1 + m1 - m2] L2 â¡ L & â¬[l1] L1 â¡ L.
+#L0 #L1 #s1 #l1 #m1 #H elim H -L0 -L1 -l1 -m1
+[ #l1 #m1 #Hm1 #L2 #m2 #H #Hl1 #_ elim (drop_inv_atom1 ⦠H) -H #H #Hm2 destruct
+ >(Hm2 ?) in Hl1; // -Hm2 #Hl1 <(le_n_O_to_eq ⦠Hl1) -l1
/4 width=3 by drop_atom, ex2_intro/
-| normalize #I #L #V #L2 #e2 #HL2 #_ #He2
- lapply (le_n_O_to_eq ⦠He2) -He2 #H destruct
+| normalize #I #L #V #L2 #m2 #HL2 #_ #Hm2
+ lapply (le_n_O_to_eq ⦠Hm2) -Hm2 #H destruct
lapply (drop_inv_O2 ⦠HL2) -HL2 #H destruct /2 width=3 by drop_pair, ex2_intro/
-| normalize #I #L0 #K0 #V1 #e1 #HLK0 #IHLK0 #L2 #e2 #H #_ #He21
- lapply (drop_inv_O1_pair1 ⦠H) -H * * #He2 #HL20
- [ -IHLK0 -He21 destruct <minus_n_O /3 width=3 by drop_drop, ex2_intro/
+| normalize #I #L0 #K0 #V1 #m1 #HLK0 #IHLK0 #L2 #m2 #H #_ #Hm21
+ lapply (drop_inv_O1_pair1 ⦠H) -H * * #Hm2 #HL20
+ [ -IHLK0 -Hm21 destruct <minus_n_O /3 width=3 by drop_drop, ex2_intro/
| -HLK0 <minus_le_minus_minus_comm //
elim (IHLK0 ⦠HL20) -L0 /2 width=3 by monotonic_pred, ex2_intro/
]
-| #I #L0 #K0 #V0 #V1 #d1 #e1 >plus_plus_comm_23 #_ #_ #IHLK0 #L2 #e2 #H #Hd1e2 #He2de1
- elim (le_inv_plus_l ⦠Hd1e2) #_ #He2
+| #I #L0 #K0 #V0 #V1 #l1 #m1 >plus_plus_comm_23 #_ #_ #IHLK0 #L2 #m2 #H #Hl1m2 #Hm2lm1
+ elim (le_inv_plus_l ⦠Hl1m2) #_ #Hm2
<minus_le_minus_minus_comm //
lapply (drop_inv_drop1_lt ⦠H ?) -H // #HL02
elim (IHLK0 ⦠HL02) -L0 /3 width=3 by drop_drop, monotonic_pred, ex2_intro/
@@ -79,22 +79,22 @@ theorem drop_conf_be: âL0,L1,s1,d1,e1. â¬[s1, d1, e1] L0 â¡ L1 â
qed-.
(* Note: apparently this was missing in basic_1 *)
-theorem drop_conf_le: âL0,L1,s1,d1,e1. â¬[s1, d1, e1] L0 â¡ L1 â
- âL2,s2,e2. â¬[s2, 0, e2] L0 â¡ L2 â e2 ⤠d1 â
- ââL. â¬[s2, 0, e2] L1 â¡ L & â¬[s1, d1 - e2, e1] L2 â¡ L.
-#L0 #L1 #s1 #d1 #e1 #H elim H -L0 -L1 -d1 -e1
-[ #d1 #e1 #He1 #L2 #s2 #e2 #H elim (drop_inv_atom1 ⦠H) -H
- #H #He2 #_ destruct /4 width=3 by drop_atom, ex2_intro/
-| #I #K0 #V0 #L2 #s2 #e2 #H1 #H2
+theorem drop_conf_le: âL0,L1,s1,l1,m1. â¬[s1, l1, m1] L0 â¡ L1 â
+ âL2,s2,m2. â¬[s2, 0, m2] L0 â¡ L2 â m2 ⤠l1 â
+ ââL. â¬[s2, 0, m2] L1 â¡ L & â¬[s1, l1 - m2, m1] L2 â¡ L.
+#L0 #L1 #s1 #l1 #m1 #H elim H -L0 -L1 -l1 -m1
+[ #l1 #m1 #Hm1 #L2 #s2 #m2 #H elim (drop_inv_atom1 ⦠H) -H
+ #H #Hm2 #_ destruct /4 width=3 by drop_atom, ex2_intro/
+| #I #K0 #V0 #L2 #s2 #m2 #H1 #H2
lapply (le_n_O_to_eq ⦠H2) -H2 #H destruct
lapply (drop_inv_pair1 ⦠H1) -H1 #H destruct /2 width=3 by drop_pair, ex2_intro/
-| #I #K0 #K1 #V0 #e1 #HK01 #_ #L2 #s2 #e2 #H1 #H2
+| #I #K0 #K1 #V0 #m1 #HK01 #_ #L2 #s2 #m2 #H1 #H2
lapply (le_n_O_to_eq ⦠H2) -H2 #H destruct
lapply (drop_inv_pair1 ⦠H1) -H1 #H destruct /3 width=3 by drop_drop, ex2_intro/
-| #I #K0 #K1 #V0 #V1 #d1 #e1 #HK01 #HV10 #IHK01 #L2 #s2 #e2 #H #He2d1
+| #I #K0 #K1 #V0 #V1 #l1 #m1 #HK01 #HV10 #IHK01 #L2 #s2 #m2 #H #Hm2l1
elim (drop_inv_O1_pair1 ⦠H) -H *
- [ -IHK01 -He2d1 #H1 #H2 destruct /3 width=5 by drop_pair, drop_skip, ex2_intro/
- | -HK01 -HV10 #He2 #HK0L2
+ [ -IHK01 -Hm2l1 #H1 #H2 destruct /3 width=5 by drop_pair, drop_skip, ex2_intro/
+ | -HK01 -HV10 #Hm2 #HK0L2
elim (IHK01 ⦠HK0L2) -IHK01 -HK0L2 /2 width=1 by monotonic_pred/
>minus_le_minus_minus_comm /3 width=3 by drop_drop_lt, ex2_intro/
]
@@ -103,38 +103,38 @@ qed-.
(* Note: with "s2", the conclusion parameter is "s1 ⨠s2" *)
(* Basic_1: was: drop_trans_ge *)
-theorem drop_trans_ge: âL1,L,s1,d1,e1. â¬[s1, d1, e1] L1 â¡ L â
- âL2,e2. â¬[e2] L â¡ L2 â d1 ⤠e2 â â¬[s1, 0, e1 + e2] L1 â¡ L2.
-#L1 #L #s1 #d1 #e1 #H elim H -L1 -L -d1 -e1
-[ #d1 #e1 #He1 #L2 #e2 #H #_ elim (drop_inv_atom1 ⦠H) -H
- #H #He2 destruct /4 width=1 by drop_atom, eq_f2/
+theorem drop_trans_ge: âL1,L,s1,l1,m1. â¬[s1, l1, m1] L1 â¡ L â
+ âL2,m2. â¬[m2] L â¡ L2 â l1 ⤠m2 â â¬[s1, 0, m1 + m2] L1 â¡ L2.
+#L1 #L #s1 #l1 #m1 #H elim H -L1 -L -l1 -m1
+[ #l1 #m1 #Hm1 #L2 #m2 #H #_ elim (drop_inv_atom1 ⦠H) -H
+ #H #Hm2 destruct /4 width=1 by drop_atom, eq_f2/
| /2 width=1 by drop_gen/
| /3 width=1 by drop_drop/
-| #I #L1 #L2 #V1 #V2 #d #e #_ #_ #IHL12 #L #e2 #H #Hde2
- lapply (lt_to_le_to_lt 0 ⦠Hde2) // #He2
- lapply (lt_to_le_to_lt ⦠(e + e2) He2 ?) // #Hee2
+| #I #L1 #L2 #V1 #V2 #l #m #_ #_ #IHL12 #L #m2 #H #Hlm2
+ lapply (lt_to_le_to_lt 0 ⦠Hlm2) // #Hm2
+ lapply (lt_to_le_to_lt ⦠(m + m2) Hm2 ?) // #Hmm2
lapply (drop_inv_drop1_lt ⦠H ?) -H // #HL2
@drop_drop_lt // >le_plus_minus /3 width=1 by monotonic_pred/
]
qed.
(* Basic_1: was: drop_trans_le *)
-theorem drop_trans_le: âL1,L,s1,d1,e1. â¬[s1, d1, e1] L1 â¡ L â
- âL2,s2,e2. â¬[s2, 0, e2] L â¡ L2 â e2 ⤠d1 â
- ââL0. â¬[s2, 0, e2] L1 â¡ L0 & â¬[s1, d1 - e2, e1] L0 â¡ L2.
-#L1 #L #s1 #d1 #e1 #H elim H -L1 -L -d1 -e1
-[ #d1 #e1 #He1 #L2 #s2 #e2 #H #_ elim (drop_inv_atom1 ⦠H) -H
- #H #He2 destruct /4 width=3 by drop_atom, ex2_intro/
-| #I #K #V #L2 #s2 #e2 #HL2 #H lapply (le_n_O_to_eq ⦠H) -H
+theorem drop_trans_le: âL1,L,s1,l1,m1. â¬[s1, l1, m1] L1 â¡ L â
+ âL2,s2,m2. â¬[s2, 0, m2] L â¡ L2 â m2 ⤠l1 â
+ ââL0. â¬[s2, 0, m2] L1 â¡ L0 & â¬[s1, l1 - m2, m1] L0 â¡ L2.
+#L1 #L #s1 #l1 #m1 #H elim H -L1 -L -l1 -m1
+[ #l1 #m1 #Hm1 #L2 #s2 #m2 #H #_ elim (drop_inv_atom1 ⦠H) -H
+ #H #Hm2 destruct /4 width=3 by drop_atom, ex2_intro/
+| #I #K #V #L2 #s2 #m2 #HL2 #H lapply (le_n_O_to_eq ⦠H) -H
#H destruct /2 width=3 by drop_pair, ex2_intro/
-| #I #L1 #L2 #V #e #_ #IHL12 #L #s2 #e2 #HL2 #H lapply (le_n_O_to_eq ⦠H) -H
+| #I #L1 #L2 #V #m #_ #IHL12 #L #s2 #m2 #HL2 #H lapply (le_n_O_to_eq ⦠H) -H
#H destruct elim (IHL12 ⦠HL2) -IHL12 -HL2 //
#L0 #H #HL0 lapply (drop_inv_O2 ⦠H) -H #H destruct
/3 width=5 by drop_pair, drop_drop, ex2_intro/
-| #I #L1 #L2 #V1 #V2 #d #e #HL12 #HV12 #IHL12 #L #s2 #e2 #H #He2d
+| #I #L1 #L2 #V1 #V2 #l #m #HL12 #HV12 #IHL12 #L #s2 #m2 #H #Hm2l
elim (drop_inv_O1_pair1 ⦠H) -H *
- [ -He2d -IHL12 #H1 #H2 destruct /3 width=5 by drop_pair, drop_skip, ex2_intro/
- | -HL12 -HV12 #He2 #HL2
+ [ -Hm2l -IHL12 #H1 #H2 destruct /3 width=5 by drop_pair, drop_skip, ex2_intro/
+ | -HL12 -HV12 #Hm2 #HL2
elim (IHL12 ⦠HL2) -L2 [ >minus_le_minus_minus_comm // /3 width=3 by drop_drop_lt, ex2_intro/ | /2 width=1 by monotonic_pred/ ]
]
]
@@ -142,50 +142,50 @@ qed-.
(* Advanced properties ******************************************************)
-lemma l_liftable_llstar: âR. l_liftable R â âl. l_liftable (llstar ⦠R l).
-#R #HR #l #K #T1 #T2 #H @(lstar_ind_r ⦠l T2 H) -l -T2
-[ #L #s #d #e #_ #U1 #HTU1 #U2 #HTU2 -HR -K
- >(lift_mono ⦠HTU2 ⦠HTU1) -T1 -U2 -d -e //
-| #l #T #T2 #_ #HT2 #IHT1 #L #s #d #e #HLK #U1 #HTU1 #U2 #HTU2
- elim (lift_total T d e) /3 width=12 by lstar_dx/
+lemma d_liftable_llstar: âR. d_liftable R â âd. d_liftable (llstar ⦠R d).
+#R #HR #d #K #T1 #T2 #H @(lstar_ind_r ⦠d T2 H) -d -T2
+[ #L #s #l #m #_ #U1 #HTU1 #U2 #HTU2 -HR -K
+ >(lift_mono ⦠HTU2 ⦠HTU1) -T1 -U2 -l -m //
+| #d #T #T2 #_ #HT2 #IHT1 #L #s #l #m #HLK #U1 #HTU1 #U2 #HTU2
+ elim (lift_total T l m) /3 width=12 by lstar_dx/
]
qed-.
(* Basic_1: was: drop_conf_lt *)
-lemma drop_conf_lt: âL,L1,s1,d1,e1. â¬[s1, d1, e1] L â¡ L1 â
- âI,K2,V2,s2,e2. â¬[s2, 0, e2] L â¡ K2.â{I}V2 â
- e2 < d1 â let d â d1 - e2 - 1 in
- ââK1,V1. â¬[s2, 0, e2] L1 â¡ K1.â{I}V1 &
- â¬[s1, d, e1] K2 â¡ K1 & â¬[d, e1] V1 â¡ V2.
-#L #L1 #s1 #d1 #e1 #H1 #I #K2 #V2 #s2 #e2 #H2 #He2d1
+lemma drop_conf_lt: âL,L1,s1,l1,m1. â¬[s1, l1, m1] L â¡ L1 â
+ âI,K2,V2,s2,m2. â¬[s2, 0, m2] L â¡ K2.â{I}V2 â
+ m2 < l1 â let l â l1 - m2 - 1 in
+ ââK1,V1. â¬[s2, 0, m2] L1 â¡ K1.â{I}V1 &
+ â¬[s1, l, m1] K2 â¡ K1 & â¬[l, m1] V1 â¡ V2.
+#L #L1 #s1 #l1 #m1 #H1 #I #K2 #V2 #s2 #m2 #H2 #Hm2l1
elim (drop_conf_le ⦠H1 ⦠H2) -L /2 width=2 by lt_to_le/ #K #HL1K #HK2
elim (drop_inv_skip1 ⦠HK2) -HK2 /2 width=1 by lt_plus_to_minus_r/
#K1 #V1 #HK21 #HV12 #H destruct /2 width=5 by ex3_2_intro/
qed-.
(* Note: apparently this was missing in basic_1 *)
-lemma drop_trans_lt: âL1,L,s1,d1,e1. â¬[s1, d1, e1] L1 â¡ L â
- âI,L2,V2,s2,e2. â¬[s2, 0, e2] L â¡ L2.â{I}V2 â
- e2 < d1 â let d â d1 - e2 - 1 in
- ââL0,V0. â¬[s2, 0, e2] L1 â¡ L0.â{I}V0 &
- â¬[s1, d, e1] L0 â¡ L2 & â¬[d, e1] V2 â¡ V0.
-#L1 #L #s1 #d1 #e1 #HL1 #I #L2 #V2 #s2 #e2 #HL2 #Hd21
+lemma drop_trans_lt: âL1,L,s1,l1,m1. â¬[s1, l1, m1] L1 â¡ L â
+ âI,L2,V2,s2,m2. â¬[s2, 0, m2] L â¡ L2.â{I}V2 â
+ m2 < l1 â let l â l1 - m2 - 1 in
+ ââL0,V0. â¬[s2, 0, m2] L1 â¡ L0.â{I}V0 &
+ â¬[s1, l, m1] L0 â¡ L2 & â¬[l, m1] V2 â¡ V0.
+#L1 #L #s1 #l1 #m1 #HL1 #I #L2 #V2 #s2 #m2 #HL2 #Hl21
elim (drop_trans_le ⦠HL1 ⦠HL2) -L /2 width=1 by lt_to_le/ #L0 #HL10 #HL02
elim (drop_inv_skip2 ⦠HL02) -HL02 /2 width=1 by lt_plus_to_minus_r/ #L #V1 #HL2 #HV21 #H destruct /2 width=5 by ex3_2_intro/
qed-.
-lemma drop_trans_ge_comm: âL1,L,L2,s1,d1,e1,e2.
- â¬[s1, d1, e1] L1 â¡ L â â¬[e2] L â¡ L2 â d1 ⤠e2 â
- â¬[s1, 0, e2 + e1] L1 â¡ L2.
-#L1 #L #L2 #s1 #d1 #e1 #e2
+lemma drop_trans_ge_comm: âL1,L,L2,s1,l1,m1,m2.
+ â¬[s1, l1, m1] L1 â¡ L â â¬[m2] L â¡ L2 â l1 ⤠m2 â
+ â¬[s1, 0, m2 + m1] L1 â¡ L2.
+#L1 #L #L2 #s1 #l1 #m1 #m2
>commutative_plus /2 width=5 by drop_trans_ge/
qed.
-lemma drop_conf_div: âI1,L,K,V1,e1. â¬[e1] L â¡ K.â{I1}V1 â
- âI2,V2,e2. â¬[e2] L â¡ K.â{I2}V2 â
- â§â§ e1 = e2 & I1 = I2 & V1 = V2.
-#I1 #L #K #V1 #e1 #HLK1 #I2 #V2 #e2 #HLK2
-elim (le_or_ge e1 e2) #He
+lemma drop_conf_div: âI1,L,K,V1,m1. â¬[m1] L â¡ K.â{I1}V1 â
+ âI2,V2,m2. â¬[m2] L â¡ K.â{I2}V2 â
+ â§â§ m1 = m2 & I1 = I2 & V1 = V2.
+#I1 #L #K #V1 #m1 #HLK1 #I2 #V2 #m2 #HLK2
+elim (le_or_ge m1 m2) #Hm
[ lapply (drop_conf_ge ⦠HLK1 ⦠HLK2 ?)
| lapply (drop_conf_ge ⦠HLK2 ⦠HLK1 ?)
] -HLK1 -HLK2 // #HK
@@ -199,10 +199,10 @@ qed-.
(* Advanced forward lemmas **************************************************)
-lemma drop_fwd_be: âL,K,s,d,e,i. â¬[s, d, e] L â¡ K â |K| ⤠i â i < d â |L| ⤠i.
-#L #K #s #d #e #i #HLK #HK #Hd elim (lt_or_ge i (|L|)) //
+lemma drop_fwd_be: âL,K,s,l,m,i. â¬[s, l, m] L â¡ K â |K| ⤠i â i < l â |L| ⤠i.
+#L #K #s #l #m #i #HLK #HK #Hl elim (lt_or_ge i (|L|)) //
#HL elim (drop_O1_lt (â») ⦠HL) #I #K0 #V #HLK0 -HL
-elim (drop_conf_lt ⦠HLK ⦠HLK0) // -HLK -HLK0 -Hd
+elim (drop_conf_lt ⦠HLK ⦠HLK0) // -HLK -HLK0 -Hl
#K1 #V1 #HK1 #_ #_ lapply (drop_fwd_length_lt2 ⦠HK1) -I -K1 -V1
#H elim (lt_refl_false i) /2 width=3 by lt_to_le_to_lt/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_leq.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_leq.ma
deleted file mode 100644
index d04c13b55..000000000
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_leq.ma
+++ /dev/null
@@ -1,92 +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/grammar/leq_leq.ma".
-include "basic_2/substitution/drop.ma".
-
-(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
-
-definition dedropable_sn: predicate (relation lenv) â
- λR. âL1,K1,s,d,e. â¬[s, d, e] L1 â¡ K1 â âK2. R K1 K2 â
- ââL2. R L1 L2 & â¬[s, d, e] L2 â¡ K2 & L1 ⩬[d, e] L2.
-
-(* Properties on equivalence ************************************************)
-
-lemma leq_drop_trans_be: âL1,L2,d,e. L1 ⩬[d, e] L2 â
- âI,K2,W,s,i. â¬[s, 0, i] L2 â¡ K2.â{I}W â
- d ⤠i â i < d + e â
- ââK1. K1 ⩬[0, â«°(d+e-i)] K2 & â¬[s, 0, i] L1 â¡ K1.â{I}W.
-#L1 #L2 #d #e #H elim H -L1 -L2 -d -e
-[ #d #e #J #K2 #W #s #i #H
- elim (drop_inv_atom1 ⦠H) -H #H destruct
-| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J #K2 #W #s #i #_ #_ #H
- elim (ylt_yle_false ⦠H) //
-| #I #L1 #L2 #V #e #HL12 #IHL12 #J #K2 #W #s #i #H #_ >yplus_O1
- elim (drop_inv_O1_pair1 ⦠H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
- [ #_ destruct >ypred_succ
- /2 width=3 by drop_pair, ex2_intro/
- | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
- #H <H -H #H lapply (ylt_inv_succ ⦠H) -H
- #Hie elim (IHL12 ⦠HLK1) -IHL12 -HLK1 // -Hie
- >yminus_succ <yminus_inj /3 width=3 by drop_drop_lt, ex2_intro/
- ]
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #J #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 (drop_inv_drop1_lt ⦠HLK2 ?) -HLK2 /2 width=1 by ylt_O/
- #HLK1 elim (IHL12 ⦠HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
- /4 width=3 by ylt_O, drop_drop_lt, ex2_intro/
-]
-qed-.
-
-lemma leq_drop_conf_be: âL1,L2,d,e. L1 ⩬[d, e] L2 â
- âI,K1,W,s,i. â¬[s, 0, i] L1 â¡ K1.â{I}W â
- d ⤠i â i < d + e â
- ââK2. K1 ⩬[0, â«°(d+e-i)] K2 & â¬[s, 0, i] L2 â¡ K2.â{I}W.
-#L1 #L2 #d #e #HL12 #I #K1 #W #s #i #HLK1 #Hdi #Hide
-elim (leq_drop_trans_be ⦠(leq_sym ⦠HL12) ⦠HLK1) // -L1 -Hdi -Hide
-/3 width=3 by leq_sym, ex2_intro/
-qed-.
-
-lemma drop_O1_ex: âK2,i,L1. |L1| = |K2| + i â
- ââL2. L1 ⩬[0, i] L2 & â¬[i] L2 â¡ K2.
-#K2 #i @(nat_ind_plus ⦠i) -i
-[ /3 width=3 by leq_O2, ex2_intro/
-| #i #IHi #Y #Hi elim (drop_O1_lt (â») Y 0) //
- #I #L1 #V #H lapply (drop_inv_O2 ⦠H) -H #H destruct
- normalize in Hi; elim (IHi L1) -IHi
- /3 width=5 by drop_drop, leq_pair, injective_plus_l, ex2_intro/
-]
-qed-.
-
-lemma dedropable_sn_TC: âR. dedropable_sn R â dedropable_sn (TC ⦠R).
-#R #HR #L1 #K1 #s #d #e #HLK1 #K2 #H elim H -K2
-[ #K2 #HK12 elim (HR ⦠HLK1 ⦠HK12) -HR -K1
- /3 width=4 by inj, ex3_intro/
-| #K #K2 #_ #HK2 * #L #H1L1 #HLK #H2L1 elim (HR ⦠HLK ⦠HK2) -HR -K
- /3 width=6 by leq_trans, step, ex3_intro/
-]
-qed-.
-
-(* Inversion lemmas on equivalence ******************************************)
-
-lemma drop_O1_inj: âi,L1,L2,K. â¬[i] L1 â¡ K â â¬[i] L2 â¡ K â L1 ⩬[i, â] L2.
-#i @(nat_ind_plus ⦠i) -i
-[ #L1 #L2 #K #H <(drop_inv_O2 ⦠H) -K #H <(drop_inv_O2 ⦠H) -L1 //
-| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 //
- lapply (drop_fwd_length ⦠HLK1)
- <(drop_fwd_length ⦠HLK2) [ /4 width=5 by drop_inv_drop1, leq_succ/ ]
- normalize <plus_n_Sm #H destruct
-]
-qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_lreq.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_lreq.ma
new file mode 100644
index 000000000..0b777278e
--- /dev/null
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/drop_lreq.ma
@@ -0,0 +1,92 @@
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/grammar/lreq_lreq.ma".
+include "basic_2/substitution/drop.ma".
+
+(* BASIC SLICING FOR LOCAL ENVIRONMENTS *************************************)
+
+definition dedropable_sn: predicate (relation lenv) â
+ λR. âL1,K1,s,l,m. â¬[s, l, m] L1 â¡ K1 â âK2. R K1 K2 â
+ ââL2. R L1 L2 & â¬[s, l, m] L2 â¡ K2 & L1 ⩬[l, m] L2.
+
+(* Properties on equivalence ************************************************)
+
+lemma lreq_drop_trans_be: âL1,L2,l,m. L1 ⩬[l, m] L2 â
+ âI,K2,W,s,i. â¬[s, 0, i] L2 â¡ K2.â{I}W â
+ l ⤠i â i < l + m â
+ ââK1. K1 ⩬[0, â«°(l+m-i)] K2 & â¬[s, 0, i] L1 â¡ K1.â{I}W.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+[ #l #m #J #K2 #W #s #i #H
+ elim (drop_inv_atom1 ⦠H) -H #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J #K2 #W #s #i #_ #_ #H
+ elim (ylt_yle_false ⦠H) //
+| #I #L1 #L2 #V #m #HL12 #IHL12 #J #K2 #W #s #i #H #_ >yplus_O1
+ elim (drop_inv_O1_pair1 ⦠H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
+ [ #_ destruct >ypred_succ
+ /2 width=3 by drop_pair, ex2_intro/
+ | lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
+ #H <H -H #H lapply (ylt_inv_succ ⦠H) -H
+ #Him elim (IHL12 ⦠HLK1) -IHL12 -HLK1 // -Him
+ >yminus_succ <yminus_inj /3 width=3 by drop_drop_lt, ex2_intro/
+ ]
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J #K2 #W #s #i #HLK2 #Hli
+ elim (yle_inv_succ1 ⦠Hli) -Hli
+ #Hli #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ ⦠H) -H
+ #Hilm lapply (drop_inv_drop1_lt ⦠HLK2 ?) -HLK2 /2 width=1 by ylt_O/
+ #HLK1 elim (IHL12 ⦠HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
+ /4 width=3 by ylt_O, drop_drop_lt, ex2_intro/
+]
+qed-.
+
+lemma lreq_drop_conf_be: âL1,L2,l,m. L1 ⩬[l, m] L2 â
+ âI,K1,W,s,i. â¬[s, 0, i] L1 â¡ K1.â{I}W â
+ l ⤠i â i < l + m â
+ ââK2. K1 ⩬[0, â«°(l+m-i)] K2 & â¬[s, 0, i] L2 â¡ K2.â{I}W.
+#L1 #L2 #l #m #HL12 #I #K1 #W #s #i #HLK1 #Hli #Hilm
+elim (lreq_drop_trans_be ⦠(lreq_sym ⦠HL12) ⦠HLK1) // -L1 -Hli -Hilm
+/3 width=3 by lreq_sym, ex2_intro/
+qed-.
+
+lemma drop_O1_ex: âK2,i,L1. |L1| = |K2| + i â
+ ââL2. L1 ⩬[0, i] L2 & â¬[i] L2 â¡ K2.
+#K2 #i @(nat_ind_plus ⦠i) -i
+[ /3 width=3 by lreq_O2, ex2_intro/
+| #i #IHi #Y #Hi elim (drop_O1_lt (â») Y 0) //
+ #I #L1 #V #H lapply (drop_inv_O2 ⦠H) -H #H destruct
+ normalize in Hi; elim (IHi L1) -IHi
+ /3 width=5 by drop_drop, lreq_pair, injective_plus_l, ex2_intro/
+]
+qed-.
+
+lemma dedropable_sn_TC: âR. dedropable_sn R â dedropable_sn (TC ⦠R).
+#R #HR #L1 #K1 #s #l #m #HLK1 #K2 #H elim H -K2
+[ #K2 #HK12 elim (HR ⦠HLK1 ⦠HK12) -HR -K1
+ /3 width=4 by inj, ex3_intro/
+| #K #K2 #_ #HK2 * #L #H1L1 #HLK #H2L1 elim (HR ⦠HLK ⦠HK2) -HR -K
+ /3 width=6 by lreq_trans, step, ex3_intro/
+]
+qed-.
+
+(* Inversion lemmas on equivalence ******************************************)
+
+lemma drop_O1_inj: âi,L1,L2,K. â¬[i] L1 â¡ K â â¬[i] L2 â¡ K â L1 ⩬[i, â] L2.
+#i @(nat_ind_plus ⦠i) -i
+[ #L1 #L2 #K #H <(drop_inv_O2 ⦠H) -K #H <(drop_inv_O2 ⦠H) -L1 //
+| #i #IHi * [2: #L1 #I1 #V1 ] * [2,4: #L2 #I2 #V2 ] #K #HLK1 #HLK2 //
+ lapply (drop_fwd_length ⦠HLK1)
+ <(drop_fwd_length ⦠HLK2) [ /4 width=5 by drop_inv_drop1, lreq_succ/ ]
+ normalize <plus_n_Sm #H destruct
+]
+qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/fqu.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/fqu.ma
index ffc2d1965..fa388daca 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/fqu.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/fqu.ma
@@ -24,8 +24,8 @@ inductive fqu: tri_relation genv lenv term â
| fqu_pair_sn: âI,G,L,V,T. fqu G L (â¡{I}V.T) G L V
| fqu_bind_dx: âa,I,G,L,V,T. fqu G L (â{a,I}V.T) G (L.â{I}V) T
| fqu_flat_dx: âI,G,L,V,T. fqu G L (â{I}V.T) G L T
-| fqu_drop : âG,L,K,T,U,e.
- â¬[e+1] L â¡ K â â¬[0, e+1] T â¡ U â fqu G L U G K T
+| fqu_drop : âG,L,K,T,U,m.
+ â¬[m+1] L â¡ K â â¬[0, m+1] T â¡ U â fqu G L U G K T
.
interpretation
@@ -34,9 +34,9 @@ interpretation
(* Basic properties *********************************************************)
-lemma fqu_drop_lt: âG,L,K,T,U,e. 0 < e â
- â¬[e] L â¡ K â â¬[0, e] T â¡ U â â¦G, L, U⦠â â¦G, K, Tâ¦.
-#G #L #K #T #U #e #He >(plus_minus_m_m e 1) /2 width=3 by fqu_drop/
+lemma fqu_drop_lt: âG,L,K,T,U,m. 0 < m â
+ â¬[m] L â¡ K â â¬[0, m] T â¡ U â â¦G, L, U⦠â â¦G, K, Tâ¦.
+#G #L #K #T #U #m #Hm >(plus_minus_m_m m 1) /2 width=3 by fqu_drop/
qed.
lemma fqu_lref_S_lt: âI,G,L,V,i. 0 < i â â¦G, L.â{I}V, #i⦠â â¦G, L, #(i-1)â¦.
@@ -47,9 +47,9 @@ qed.
lemma fqu_fwd_fw: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2, L2, T2⦠â â¯{G2, L2, T2} < â¯{G1, L1, T1}.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
-#G #L #K #T #U #e #HLK #HTU
+#G #L #K #T #U #m #HLK #HTU
lapply (drop_fwd_lw_lt ⦠HLK ?) -HLK // #HKL
-lapply (lift_fwd_tw ⦠HTU) -e #H
+lapply (lift_fwd_tw ⦠HTU) -m #H
normalize in ⢠(?%%); /2 width=1 by lt_minus_to_plus/
qed-.
@@ -72,7 +72,7 @@ fact fqu_inv_eq_aux: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â â¦G2, L2, T2â¦
G1 = G2 â |L1| = |L2| â T1 = T2 â â¥.
#G1 #G2 #L1 #L2 #T1 #T2 * -G1 -G2 -L1 -L2 -T1 -T2 normalize
/2 width=4 by discr_tpair_xy_y, discr_tpair_xy_x, plus_xSy_x_false/
-#G #L #K #T #U #e #HLK #_ #_ #H #_ -G -T -U >(drop_fwd_length ⦠HLK) in H; -L
+#G #L #K #T #U #m #HLK #_ #_ #H #_ -G -T -U >(drop_fwd_length ⦠HLK) in H; -L
/2 width=4 by plus_xySz_x_false/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/fquq.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/fquq.ma
index 48503a312..914af3702 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/fquq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/fquq.ma
@@ -23,8 +23,8 @@ inductive fquq: tri_relation genv lenv term â
| fquq_pair_sn: âI,G,L,V,T. fquq G L (â¡{I}V.T) G L V
| fquq_bind_dx: âa,I,G,L,V,T. fquq G L (â{a,I}V.T) G (L.â{I}V) T
| fquq_flat_dx: âI,G, L,V,T. fquq G L (â{I}V.T) G L T
-| fquq_drop : âG,L,K,T,U,e.
- â¬[e] L â¡ K â â¬[0, e] T â¡ U â fquq G L U G K T
+| fquq_drop : âG,L,K,T,U,m.
+ â¬[m] L â¡ K â â¬[0, m] T â¡ U â fquq G L U G K T
.
interpretation
@@ -44,7 +44,7 @@ qed.
lemma fquq_fwd_fw: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â⸮ â¦G2, L2, T2⦠â â¯{G2, L2, T2} ⤠â¯{G1, L1, T1}.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 /2 width=1 by lt_to_le/
-#G1 #L1 #K1 #T1 #U1 #e #HLK1 #HTU1
+#G1 #L1 #K1 #T1 #U1 #m #HLK1 #HTU1
lapply (drop_fwd_lw ⦠HLK1) -HLK1
lapply (lift_fwd_tw ⦠HTU1) -HTU1
/2 width=1 by le_plus, le_n/
@@ -54,7 +54,7 @@ fact fquq_fwd_length_lref1_aux: âG1,G2,L1,L2,T1,T2. â¦G1, L1, T1⦠â⸮
âi. T1 = #i â |L2| ⤠|L1|.
#G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2 //
[ #a #I #G #L #V #T #j #H destruct
-| #G1 #L1 #K1 #T1 #U1 #e #HLK1 #HTU1 #i #H destruct
+| #G1 #L1 #K1 #T1 #U1 #m #HLK1 #HTU1 #i #H destruct
/2 width=3 by drop_fwd_length_le4/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/fquq_alt.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/fquq_alt.ma
index 619bb0b4a..08a755b55 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/fquq_alt.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/fquq_alt.ma
@@ -29,9 +29,9 @@ interpretation
lemma fquqa_refl: tri_reflexive ⦠fquqa.
// qed.
-lemma fquqa_drop: âG,L,K,T,U,e.
- â¬[e] L â¡ K â â¬[0, e] T â¡ U â â¦G, L, U⦠ââ⸮ â¦G, K, Tâ¦.
-#G #L #K #T #U #e #HLK #HTU elim (eq_or_gt e)
+lemma fquqa_drop: âG,L,K,T,U,m.
+ â¬[m] L â¡ K â â¬[0, m] T â¡ U â â¦G, L, U⦠ââ⸮ â¦G, K, Tâ¦.
+#G #L #K #T #U #m #HLK #HTU elim (eq_or_gt m)
/3 width=5 by fqu_drop_lt, or_introl/ #H destruct
>(drop_inv_O2 ⦠HLK) -L >(lift_inv_O2 ⦠HTU) -T //
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/gget.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/gget.ma
index acf9602be..eb5c8fdab 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/gget.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/gget.ma
@@ -17,19 +17,19 @@ include "basic_2/grammar/genv.ma".
(* GLOBAL ENVIRONMENT READING ***********************************************)
-inductive gget (e:nat): relation genv â
-| gget_gt: âG. |G| ⤠e â gget e G (â)
-| gget_eq: âG. |G| = e + 1 â gget e G G
-| gget_lt: âI,G1,G2,V. e < |G1| â gget e G1 G2 â gget e (G1. â{I} V) G2
+inductive gget (m:nat): relation genv â
+| gget_gt: âG. |G| ⤠m â gget m G (â)
+| gget_eq: âG. |G| = m + 1 â gget m G G
+| gget_lt: âI,G1,G2,V. m < |G1| â gget m G1 G2 â gget m (G1. â{I} V) G2
.
interpretation "global reading"
- 'RDrop e G1 G2 = (gget e G1 G2).
+ 'RDrop m G1 G2 = (gget m G1 G2).
(* basic inversion lemmas ***************************************************)
-lemma gget_inv_gt: âG1,G2,e. â¬[e] G1 â¡ G2 â |G1| ⤠e â G2 = â.
-#G1 #G2 #e * -G1 -G2 //
+lemma gget_inv_gt: âG1,G2,m. â¬[m] G1 â¡ G2 â |G1| ⤠m â G2 = â.
+#G1 #G2 #m * -G1 -G2 //
[ #G #H >H -H >commutative_plus #H (**) (* lemma needed here *)
lapply (le_plus_to_le_r ⦠0 H) -H #H
lapply (le_n_O_to_eq ⦠H) -H #H destruct
@@ -40,8 +40,8 @@ lemma gget_inv_gt: âG1,G2,e. â¬[e] G1 â¡ G2 â |G1| ⤠e â G2 = â.
]
qed-.
-lemma gget_inv_eq: âG1,G2,e. â¬[e] G1 â¡ G2 â |G1| = e + 1 â G1 = G2.
-#G1 #G2 #e * -G1 -G2 //
+lemma gget_inv_eq: âG1,G2,m. â¬[m] G1 â¡ G2 â |G1| = m + 1 â G1 = G2.
+#G1 #G2 #m * -G1 -G2 //
[ #G #H1 #H2 >H2 in H1; -H2 >commutative_plus #H (**) (* lemma needed here *)
lapply (le_plus_to_le_r ⦠0 H) -H #H
lapply (le_n_O_to_eq ⦠H) -H #H destruct
@@ -51,9 +51,9 @@ lemma gget_inv_eq: âG1,G2,e. â¬[e] G1 â¡ G2 â |G1| = e + 1 â G1 = G2.
]
qed-.
-fact gget_inv_lt_aux: âI,G,G1,G2,V,e. â¬[e] G â¡ G2 â G = G1. â{I} V â
- e < |G1| â â¬[e] G1 â¡ G2.
-#I #G #G1 #G2 #V #e * -G -G2
+fact gget_inv_lt_aux: âI,G,G1,G2,V,m. â¬[m] G â¡ G2 â G = G1. â{I} V â
+ m < |G1| â â¬[m] G1 â¡ G2.
+#I #G #G1 #G2 #V #m * -G -G2
[ #G #H1 #H destruct #H2
lapply (le_to_lt_to_lt ⦠H1 H2) -H1 -H2 normalize in ⢠(? % ? â ?); >commutative_plus #H
lapply (lt_plus_to_lt_l ⦠0 H) -H #H
@@ -64,18 +64,18 @@ fact gget_inv_lt_aux: âI,G,G1,G2,V,e. â¬[e] G â¡ G2 â G = G1. â{I} V â
]
qed-.
-lemma gget_inv_lt: âI,G1,G2,V,e.
- â¬[e] G1. â{I} V â¡ G2 â e < |G1| â â¬[e] G1 â¡ G2.
+lemma gget_inv_lt: âI,G1,G2,V,m.
+ â¬[m] G1. â{I} V â¡ G2 â m < |G1| â â¬[m] G1 â¡ G2.
/2 width=5 by gget_inv_lt_aux/ qed-.
(* Basic properties *********************************************************)
-lemma gget_total: âe,G1. âG2. â¬[e] G1 â¡ G2.
-#e #G1 elim G1 -G1 /3 width=2/
+lemma gget_total: âm,G1. âG2. â¬[m] G1 â¡ G2.
+#m #G1 elim G1 -G1 /3 width=2 by gget_gt, ex_intro/
#I #V #G1 * #G2 #HG12
-elim (lt_or_eq_or_gt e (|G1|)) #He
+elim (lt_or_eq_or_gt m (|G1|)) #Hm
[ /3 width=2/
-| destruct /3 width=2/
-| @ex_intro [2: @gget_gt normalize /2 width=1/ | skip ] (**) (* explicit constructor *)
+| destruct /3 width=2 by gget_eq, ex_intro/
+| @ex_intro [2: @gget_gt normalize /2 width=1 by/ | skip ] (**) (* explicit constructor *)
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/gget_gget.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/gget_gget.ma
index bf14f9e44..c08ab3960 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/gget_gget.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/gget_gget.ma
@@ -18,20 +18,20 @@ include "basic_2/substitution/gget.ma".
(* Main properties **********************************************************)
-theorem gget_mono: âG,G1,e. â¬[e] G â¡ G1 â âG2. â¬[e] G â¡ G2 â G1 = G2.
-#G #G1 #e #H elim H -G -G1
-[ #G #He #G2 #H
- >(gget_inv_gt ⦠H He) -H -He //
-| #G #He #G2 #H
- >(gget_inv_eq ⦠H He) -H -He //
-| #I #G #G1 #V #He #_ #IHG1 #G2 #H
- lapply (gget_inv_lt ⦠H He) -H -He /2 width=1/
+theorem gget_mono: âG,G1,m. â¬[m] G â¡ G1 â âG2. â¬[m] G â¡ G2 â G1 = G2.
+#G #G1 #m #H elim H -G -G1
+[ #G #Hm #G2 #H
+ >(gget_inv_gt ⦠H Hm) -H -Hm //
+| #G #Hm #G2 #H
+ >(gget_inv_eq ⦠H Hm) -H -Hm //
+| #I #G #G1 #V #Hm #_ #IHG1 #G2 #H
+ lapply (gget_inv_lt ⦠H Hm) -H -Hm /2 width=1/
]
qed-.
-lemma gget_dec: âG1,G2,e. Decidable (â¬[e] G1 â¡ G2).
-#G1 #G2 #e
-elim (gget_total e G1) #G #HG1
+lemma gget_dec: âG1,G2,m. Decidable (â¬[m] G1 â¡ G2).
+#G1 #G2 #m
+elim (gget_total m G1) #G #HG1
elim (eq_genv_dec G G2) #HG2
[ destruct /2 width=1/
| @or_intror #HG12
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma
index d6537046c..fdb8b6a72 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift.ma
@@ -22,224 +22,224 @@ include "basic_2/grammar/term_simple.ma".
lift_sort lift_lref_lt lift_lref_ge lift_bind lift_flat
*)
inductive lift: relation4 nat nat term term â
-| lift_sort : âk,d,e. lift d e (âk) (âk)
-| lift_lref_lt: âi,d,e. i < d â lift d e (#i) (#i)
-| lift_lref_ge: âi,d,e. d ⤠i â lift d e (#i) (#(i + e))
-| lift_gref : âp,d,e. lift d e (§p) (§p)
-| lift_bind : âa,I,V1,V2,T1,T2,d,e.
- lift d e V1 V2 â lift (d + 1) e T1 T2 â
- lift d e (â{a,I} V1. T1) (â{a,I} V2. T2)
-| lift_flat : âI,V1,V2,T1,T2,d,e.
- lift d e V1 V2 â lift d e T1 T2 â
- lift d e (â{I} V1. T1) (â{I} V2. T2)
+| lift_sort : âk,l,m. lift l m (âk) (âk)
+| lift_lref_lt: âi,l,m. i < l â lift l m (#i) (#i)
+| lift_lref_ge: âi,l,m. l ⤠i â lift l m (#i) (#(i + m))
+| lift_gref : âp,l,m. lift l m (§p) (§p)
+| lift_bind : âa,I,V1,V2,T1,T2,l,m.
+ lift l m V1 V2 â lift (l + 1) m T1 T2 â
+ lift l m (â{a,I} V1. T1) (â{a,I} V2. T2)
+| lift_flat : âI,V1,V2,T1,T2,l,m.
+ lift l m V1 V2 â lift l m T1 T2 â
+ lift l m (â{I} V1. T1) (â{I} V2. T2)
.
-interpretation "relocation" 'RLift d e T1 T2 = (lift d e T1 T2).
+interpretation "relocation" 'RLift l m T1 T2 = (lift l m T1 T2).
(* Basic inversion lemmas ***************************************************)
-fact lift_inv_O2_aux: âd,e,T1,T2. â¬[d, e] T1 â¡ T2 â e = 0 â T1 = T2.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 /3 width=1 by eq_f2/
+fact lift_inv_O2_aux: âl,m,T1,T2. â¬[l, m] T1 â¡ T2 â m = 0 â T1 = T2.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 /3 width=1 by eq_f2/
qed-.
-lemma lift_inv_O2: âd,T1,T2. â¬[d, 0] T1 â¡ T2 â T1 = T2.
+lemma lift_inv_O2: âl,T1,T2. â¬[l, 0] T1 â¡ T2 â T1 = T2.
/2 width=4 by lift_inv_O2_aux/ qed-.
-fact lift_inv_sort1_aux: âd,e,T1,T2. â¬[d,e] T1 â¡ T2 â âk. T1 = âk â T2 = âk.
-#d #e #T1 #T2 * -d -e -T1 -T2 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+fact lift_inv_sort1_aux: âl,m,T1,T2. â¬[l,m] T1 â¡ T2 â âk. T1 = âk â T2 = âk.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
]
qed-.
-lemma lift_inv_sort1: âd,e,T2,k. â¬[d,e] âk â¡ T2 â T2 = âk.
+lemma lift_inv_sort1: âl,m,T2,k. â¬[l,m] âk â¡ T2 â T2 = âk.
/2 width=5 by lift_inv_sort1_aux/ qed-.
-fact lift_inv_lref1_aux: âd,e,T1,T2. â¬[d,e] T1 â¡ T2 â âi. T1 = #i â
- (i < d ⧠T2 = #i) ⨠(d ⤠i ⧠T2 = #(i + e)).
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
-| #j #d #e #Hj #i #Hi destruct /3 width=1 by or_intror, conj/
-| #p #d #e #i #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+fact lift_inv_lref1_aux: âl,m,T1,T2. â¬[l,m] T1 â¡ T2 â âi. T1 = #i â
+ (i < l ⧠T2 = #i) ⨠(l ⤠i ⧠T2 = #(i + m)).
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #i #H destruct
+| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
+| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_intror, conj/
+| #p #l #m #i #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
]
qed-.
-lemma lift_inv_lref1: âd,e,T2,i. â¬[d,e] #i â¡ T2 â
- (i < d ⧠T2 = #i) ⨠(d ⤠i ⧠T2 = #(i + e)).
+lemma lift_inv_lref1: âl,m,T2,i. â¬[l,m] #i â¡ T2 â
+ (i < l ⧠T2 = #i) ⨠(l ⤠i ⧠T2 = #(i + m)).
/2 width=3 by lift_inv_lref1_aux/ qed-.
-lemma lift_inv_lref1_lt: âd,e,T2,i. â¬[d,e] #i â¡ T2 â i < d â T2 = #i.
-#d #e #T2 #i #H elim (lift_inv_lref1 ⦠H) -H * //
-#Hdi #_ #Hid lapply (le_to_lt_to_lt ⦠Hdi Hid) -Hdi -Hid #Hdd
-elim (lt_refl_false ⦠Hdd)
+lemma lift_inv_lref1_lt: âl,m,T2,i. â¬[l,m] #i â¡ T2 â i < l â T2 = #i.
+#l #m #T2 #i #H elim (lift_inv_lref1 ⦠H) -H * //
+#Hli #_ #Hil lapply (le_to_lt_to_lt ⦠Hli Hil) -Hli -Hil #Hll
+elim (lt_refl_false ⦠Hll)
qed-.
-lemma lift_inv_lref1_ge: âd,e,T2,i. â¬[d,e] #i â¡ T2 â d ⤠i â T2 = #(i + e).
-#d #e #T2 #i #H elim (lift_inv_lref1 ⦠H) -H * //
-#Hid #_ #Hdi lapply (le_to_lt_to_lt ⦠Hdi Hid) -Hdi -Hid #Hdd
-elim (lt_refl_false ⦠Hdd)
+lemma lift_inv_lref1_ge: âl,m,T2,i. â¬[l,m] #i â¡ T2 â l ⤠i â T2 = #(i + m).
+#l #m #T2 #i #H elim (lift_inv_lref1 ⦠H) -H * //
+#Hil #_ #Hli lapply (le_to_lt_to_lt ⦠Hli Hil) -Hli -Hil #Hll
+elim (lt_refl_false ⦠Hll)
qed-.
-fact lift_inv_gref1_aux: âd,e,T1,T2. â¬[d,e] T1 â¡ T2 â âp. T1 = §p â T2 = §p.
-#d #e #T1 #T2 * -d -e -T1 -T2 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+fact lift_inv_gref1_aux: âl,m,T1,T2. â¬[l,m] T1 â¡ T2 â âp. T1 = §p â T2 = §p.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
]
qed-.
-lemma lift_inv_gref1: âd,e,T2,p. â¬[d,e] §p â¡ T2 â T2 = §p.
+lemma lift_inv_gref1: âl,m,T2,p. â¬[l,m] §p â¡ T2 â T2 = §p.
/2 width=5 by lift_inv_gref1_aux/ qed-.
-fact lift_inv_bind1_aux: âd,e,T1,T2. â¬[d,e] T1 â¡ T2 â
+fact lift_inv_bind1_aux: âl,m,T1,T2. â¬[l,m] T1 â¡ T2 â
âa,I,V1,U1. T1 = â{a,I} V1.U1 â
- ââV2,U2. â¬[d,e] V1 â¡ V2 & â¬[d+1,e] U1 â¡ U2 &
+ ââV2,U2. â¬[l,m] V1 â¡ V2 & â¬[l+1,m] U1 â¡ U2 &
T2 = â{a,I} V2. U2.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #a #I #V1 #U1 #H destruct
-| #i #d #e #_ #a #I #V1 #U1 #H destruct
-| #i #d #e #_ #a #I #V1 #U1 #H destruct
-| #p #d #e #a #I #V1 #U1 #H destruct
-| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
-| #J #W1 #W2 #T1 #T2 #d #e #_ #HT #a #I #V1 #U1 #H destruct
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #a #I #V1 #U1 #H destruct
+| #i #l #m #_ #a #I #V1 #U1 #H destruct
+| #i #l #m #_ #a #I #V1 #U1 #H destruct
+| #p #l #m #a #I #V1 #U1 #H destruct
+| #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+| #J #W1 #W2 #T1 #T2 #l #m #_ #HT #a #I #V1 #U1 #H destruct
]
qed-.
-lemma lift_inv_bind1: âd,e,T2,a,I,V1,U1. â¬[d,e] â{a,I} V1. U1 â¡ T2 â
- ââV2,U2. â¬[d,e] V1 â¡ V2 & â¬[d+1,e] U1 â¡ U2 &
+lemma lift_inv_bind1: âl,m,T2,a,I,V1,U1. â¬[l,m] â{a,I} V1. U1 â¡ T2 â
+ ââV2,U2. â¬[l,m] V1 â¡ V2 & â¬[l+1,m] U1 â¡ U2 &
T2 = â{a,I} V2. U2.
/2 width=3 by lift_inv_bind1_aux/ qed-.
-fact lift_inv_flat1_aux: âd,e,T1,T2. â¬[d,e] T1 â¡ T2 â
+fact lift_inv_flat1_aux: âl,m,T1,T2. â¬[l,m] T1 â¡ T2 â
âI,V1,U1. T1 = â{I} V1.U1 â
- ââV2,U2. â¬[d,e] V1 â¡ V2 & â¬[d,e] U1 â¡ U2 &
+ ââV2,U2. â¬[l,m] V1 â¡ V2 & â¬[l,m] U1 â¡ U2 &
T2 = â{I} V2. U2.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #I #V1 #U1 #H destruct
-| #i #d #e #_ #I #V1 #U1 #H destruct
-| #i #d #e #_ #I #V1 #U1 #H destruct
-| #p #d #e #I #V1 #U1 #H destruct
-| #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V1 #U1 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #I #V1 #U1 #H destruct
+| #i #l #m #_ #I #V1 #U1 #H destruct
+| #i #l #m #_ #I #V1 #U1 #H destruct
+| #p #l #m #I #V1 #U1 #H destruct
+| #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V1 #U1 #H destruct
+| #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V1 #U1 #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma lift_inv_flat1: âd,e,T2,I,V1,U1. â¬[d,e] â{I} V1. U1 â¡ T2 â
- ââV2,U2. â¬[d,e] V1 â¡ V2 & â¬[d,e] U1 â¡ U2 &
+lemma lift_inv_flat1: âl,m,T2,I,V1,U1. â¬[l,m] â{I} V1. U1 â¡ T2 â
+ ââV2,U2. â¬[l,m] V1 â¡ V2 & â¬[l,m] U1 â¡ U2 &
T2 = â{I} V2. U2.
/2 width=3 by lift_inv_flat1_aux/ qed-.
-fact lift_inv_sort2_aux: âd,e,T1,T2. â¬[d,e] T1 â¡ T2 â âk. T2 = âk â T1 = âk.
-#d #e #T1 #T2 * -d -e -T1 -T2 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+fact lift_inv_sort2_aux: âl,m,T1,T2. â¬[l,m] T1 â¡ T2 â âk. T2 = âk â T1 = âk.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
]
qed-.
(* Basic_1: was: lift_gen_sort *)
-lemma lift_inv_sort2: âd,e,T1,k. â¬[d,e] T1 â¡ âk â T1 = âk.
+lemma lift_inv_sort2: âl,m,T1,k. â¬[l,m] T1 â¡ âk â T1 = âk.
/2 width=5 by lift_inv_sort2_aux/ qed-.
-fact lift_inv_lref2_aux: âd,e,T1,T2. â¬[d,e] T1 â¡ T2 â âi. T2 = #i â
- (i < d ⧠T1 = #i) ⨠(d + e ⤠i ⧠T1 = #(i - e)).
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #i #H destruct
-| #j #d #e #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
-| #j #d #e #Hj #i #Hi destruct <minus_plus_m_m /4 width=1 by monotonic_le_plus_l, or_intror, conj/
-| #p #d #e #i #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #i #H destruct
+fact lift_inv_lref2_aux: âl,m,T1,T2. â¬[l,m] T1 â¡ T2 â âi. T2 = #i â
+ (i < l ⧠T1 = #i) ⨠(l + m ⤠i ⧠T1 = #(i - m)).
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #i #H destruct
+| #j #l #m #Hj #i #Hi destruct /3 width=1 by or_introl, conj/
+| #j #l #m #Hj #i #Hi destruct <minus_plus_m_m /4 width=1 by monotonic_le_plus_l, or_intror, conj/
+| #p #l #m #i #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #i #H destruct
]
qed-.
(* Basic_1: was: lift_gen_lref *)
-lemma lift_inv_lref2: âd,e,T1,i. â¬[d,e] T1 â¡ #i â
- (i < d ⧠T1 = #i) ⨠(d + e ⤠i ⧠T1 = #(i - e)).
+lemma lift_inv_lref2: âl,m,T1,i. â¬[l,m] T1 â¡ #i â
+ (i < l ⧠T1 = #i) ⨠(l + m ⤠i ⧠T1 = #(i - m)).
/2 width=3 by lift_inv_lref2_aux/ qed-.
(* Basic_1: was: lift_gen_lref_lt *)
-lemma lift_inv_lref2_lt: âd,e,T1,i. â¬[d,e] T1 â¡ #i â i < d â T1 = #i.
-#d #e #T1 #i #H elim (lift_inv_lref2 ⦠H) -H * //
-#Hdi #_ #Hid lapply (le_to_lt_to_lt ⦠Hdi Hid) -Hdi -Hid #Hdd
-elim (lt_inv_plus_l ⦠Hdd) -Hdd #Hdd
-elim (lt_refl_false ⦠Hdd)
+lemma lift_inv_lref2_lt: âl,m,T1,i. â¬[l,m] T1 â¡ #i â i < l â T1 = #i.
+#l #m #T1 #i #H elim (lift_inv_lref2 ⦠H) -H * //
+#Hli #_ #Hil lapply (le_to_lt_to_lt ⦠Hli Hil) -Hli -Hil #Hll
+elim (lt_inv_plus_l ⦠Hll) -Hll #Hll
+elim (lt_refl_false ⦠Hll)
qed-.
(* Basic_1: was: lift_gen_lref_false *)
-lemma lift_inv_lref2_be: âd,e,T1,i. â¬[d,e] T1 â¡ #i â
- d ⤠i â i < d + e â â¥.
-#d #e #T1 #i #H elim (lift_inv_lref2 ⦠H) -H *
+lemma lift_inv_lref2_be: âl,m,T1,i. â¬[l,m] T1 â¡ #i â
+ l ⤠i â i < l + m â â¥.
+#l #m #T1 #i #H elim (lift_inv_lref2 ⦠H) -H *
[ #H1 #_ #H2 #_ | #H2 #_ #_ #H1 ]
lapply (le_to_lt_to_lt ⦠H2 H1) -H2 -H1 #H
elim (lt_refl_false ⦠H)
qed-.
(* Basic_1: was: lift_gen_lref_ge *)
-lemma lift_inv_lref2_ge: âd,e,T1,i. â¬[d,e] T1 â¡ #i â d + e ⤠i â T1 = #(i - e).
-#d #e #T1 #i #H elim (lift_inv_lref2 ⦠H) -H * //
-#Hid #_ #Hdi lapply (le_to_lt_to_lt ⦠Hdi Hid) -Hdi -Hid #Hdd
-elim (lt_inv_plus_l ⦠Hdd) -Hdd #Hdd
-elim (lt_refl_false ⦠Hdd)
+lemma lift_inv_lref2_ge: âl,m,T1,i. â¬[l,m] T1 â¡ #i â l + m ⤠i â T1 = #(i - m).
+#l #m #T1 #i #H elim (lift_inv_lref2 ⦠H) -H * //
+#Hil #_ #Hli lapply (le_to_lt_to_lt ⦠Hli Hil) -Hli -Hil #Hll
+elim (lt_inv_plus_l ⦠Hll) -Hll #Hll
+elim (lt_refl_false ⦠Hll)
qed-.
-fact lift_inv_gref2_aux: âd,e,T1,T2. â¬[d,e] T1 â¡ T2 â âp. T2 = §p â T1 = §p.
-#d #e #T1 #T2 * -d -e -T1 -T2 //
-[ #i #d #e #_ #k #H destruct
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #k #H destruct
+fact lift_inv_gref2_aux: âl,m,T1,T2. â¬[l,m] T1 â¡ T2 â âp. T2 = §p â T1 = §p.
+#l #m #T1 #T2 * -l -m -T1 -T2 //
+[ #i #l #m #_ #k #H destruct
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #k #H destruct
]
qed-.
-lemma lift_inv_gref2: âd,e,T1,p. â¬[d,e] T1 ⡠§p â T1 = §p.
+lemma lift_inv_gref2: âl,m,T1,p. â¬[l,m] T1 ⡠§p â T1 = §p.
/2 width=5 by lift_inv_gref2_aux/ qed-.
-fact lift_inv_bind2_aux: âd,e,T1,T2. â¬[d,e] T1 â¡ T2 â
+fact lift_inv_bind2_aux: âl,m,T1,T2. â¬[l,m] T1 â¡ T2 â
âa,I,V2,U2. T2 = â{a,I} V2.U2 â
- ââV1,U1. â¬[d,e] V1 â¡ V2 & â¬[d+1,e] U1 â¡ U2 &
+ ââV1,U1. â¬[l,m] V1 â¡ V2 & â¬[l+1,m] U1 â¡ U2 &
T1 = â{a,I} V1. U1.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #a #I #V2 #U2 #H destruct
-| #i #d #e #_ #a #I #V2 #U2 #H destruct
-| #i #d #e #_ #a #I #V2 #U2 #H destruct
-| #p #d #e #a #I #V2 #U2 #H destruct
-| #b #J #W1 #W2 #T1 #T2 #d #e #HW #HT #a #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
-| #J #W1 #W2 #T1 #T2 #d #e #_ #_ #a #I #V2 #U2 #H destruct
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #a #I #V2 #U2 #H destruct
+| #i #l #m #_ #a #I #V2 #U2 #H destruct
+| #i #l #m #_ #a #I #V2 #U2 #H destruct
+| #p #l #m #a #I #V2 #U2 #H destruct
+| #b #J #W1 #W2 #T1 #T2 #l #m #HW #HT #a #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
+| #J #W1 #W2 #T1 #T2 #l #m #_ #_ #a #I #V2 #U2 #H destruct
]
qed-.
(* Basic_1: was: lift_gen_bind *)
-lemma lift_inv_bind2: âd,e,T1,a,I,V2,U2. â¬[d,e] T1 â¡ â{a,I} V2. U2 â
- ââV1,U1. â¬[d,e] V1 â¡ V2 & â¬[d+1,e] U1 â¡ U2 &
+lemma lift_inv_bind2: âl,m,T1,a,I,V2,U2. â¬[l,m] T1 â¡ â{a,I} V2. U2 â
+ ââV1,U1. â¬[l,m] V1 â¡ V2 & â¬[l+1,m] U1 â¡ U2 &
T1 = â{a,I} V1. U1.
/2 width=3 by lift_inv_bind2_aux/ qed-.
-fact lift_inv_flat2_aux: âd,e,T1,T2. â¬[d,e] T1 â¡ T2 â
+fact lift_inv_flat2_aux: âl,m,T1,T2. â¬[l,m] T1 â¡ T2 â
âI,V2,U2. T2 = â{I} V2.U2 â
- ââV1,U1. â¬[d,e] V1 â¡ V2 & â¬[d,e] U1 â¡ U2 &
+ ââV1,U1. â¬[l,m] V1 â¡ V2 & â¬[l,m] U1 â¡ U2 &
T1 = â{I} V1. U1.
-#d #e #T1 #T2 * -d -e -T1 -T2
-[ #k #d #e #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #i #d #e #_ #I #V2 #U2 #H destruct
-| #p #d #e #I #V2 #U2 #H destruct
-| #a #J #W1 #W2 #T1 #T2 #d #e #_ #_ #I #V2 #U2 #H destruct
-| #J #W1 #W2 #T1 #T2 #d #e #HW #HT #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
+#l #m #T1 #T2 * -l -m -T1 -T2
+[ #k #l #m #I #V2 #U2 #H destruct
+| #i #l #m #_ #I #V2 #U2 #H destruct
+| #i #l #m #_ #I #V2 #U2 #H destruct
+| #p #l #m #I #V2 #U2 #H destruct
+| #a #J #W1 #W2 #T1 #T2 #l #m #_ #_ #I #V2 #U2 #H destruct
+| #J #W1 #W2 #T1 #T2 #l #m #HW #HT #I #V2 #U2 #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
(* Basic_1: was: lift_gen_flat *)
-lemma lift_inv_flat2: âd,e,T1,I,V2,U2. â¬[d,e] T1 â¡ â{I} V2. U2 â
- ââV1,U1. â¬[d,e] V1 â¡ V2 & â¬[d,e] U1 â¡ U2 &
+lemma lift_inv_flat2: âl,m,T1,I,V2,U2. â¬[l,m] T1 â¡ â{I} V2. U2 â
+ ââV1,U1. â¬[l,m] V1 â¡ V2 & â¬[l,m] U1 â¡ U2 &
T1 = â{I} V1. U1.
/2 width=3 by lift_inv_flat2_aux/ qed-.
-lemma lift_inv_pair_xy_x: âd,e,I,V,T. â¬[d, e] â¡{I} V. T â¡ V â â¥.
-#d #e #J #V elim V -V
+lemma lift_inv_pair_xy_x: âl,m,I,V,T. â¬[l, m] â¡{I} V. T â¡ V â â¥.
+#l #m #J #V elim V -V
[ * #i #T #H
[ lapply (lift_inv_sort2 ⦠H) -H #H destruct
| elim (lift_inv_lref2 ⦠H) -H * #_ #H destruct
@@ -253,14 +253,14 @@ lemma lift_inv_pair_xy_x: âd,e,I,V,T. â¬[d, e] â¡{I} V. T â¡ V â â¥.
qed-.
(* Basic_1: was: thead_x_lift_y_y *)
-lemma lift_inv_pair_xy_y: âI,T,V,d,e. â¬[d, e] â¡{I} V. T â¡ T â â¥.
+lemma lift_inv_pair_xy_y: âI,T,V,l,m. â¬[l, m] â¡{I} V. T â¡ T â â¥.
#J #T elim T -T
-[ * #i #V #d #e #H
+[ * #i #V #l #m #H
[ lapply (lift_inv_sort2 ⦠H) -H #H destruct
| elim (lift_inv_lref2 ⦠H) -H * #_ #H destruct
| lapply (lift_inv_gref2 ⦠H) -H #H destruct
]
-| * [ #a ] #I #W2 #U2 #_ #IHU2 #V #d #e #H
+| * [ #a ] #I #W2 #U2 #_ #IHU2 #V #l #m #H
[ elim (lift_inv_bind2 ⦠H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
| elim (lift_inv_flat2 ⦠H) -H #W1 #U1 #_ #HU12 #H destruct /2 width=4 by/
]
@@ -269,102 +269,102 @@ qed-.
(* Basic forward lemmas *****************************************************)
-lemma lift_fwd_pair1: âI,T2,V1,U1,d,e. â¬[d,e] â¡{I}V1.U1 â¡ T2 â
- ââV2,U2. â¬[d,e] V1 â¡ V2 & T2 = â¡{I}V2.U2.
-* [ #a ] #I #T2 #V1 #U1 #d #e #H
+lemma lift_fwd_pair1: âI,T2,V1,U1,l,m. â¬[l,m] â¡{I}V1.U1 â¡ T2 â
+ ââV2,U2. â¬[l,m] V1 â¡ V2 & T2 = â¡{I}V2.U2.
+* [ #a ] #I #T2 #V1 #U1 #l #m #H
[ elim (lift_inv_bind1 ⦠H) -H /2 width=4 by ex2_2_intro/
| elim (lift_inv_flat1 ⦠H) -H /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lift_fwd_pair2: âI,T1,V2,U2,d,e. â¬[d,e] T1 â¡ â¡{I}V2.U2 â
- ââV1,U1. â¬[d,e] V1 â¡ V2 & T1 = â¡{I}V1.U1.
-* [ #a ] #I #T1 #V2 #U2 #d #e #H
+lemma lift_fwd_pair2: âI,T1,V2,U2,l,m. â¬[l,m] T1 â¡ â¡{I}V2.U2 â
+ ââV1,U1. â¬[l,m] V1 â¡ V2 & T1 = â¡{I}V1.U1.
+* [ #a ] #I #T1 #V2 #U2 #l #m #H
[ elim (lift_inv_bind2 ⦠H) -H /2 width=4 by ex2_2_intro/
| elim (lift_inv_flat2 ⦠H) -H /2 width=4 by ex2_2_intro/
]
qed-.
-lemma lift_fwd_tw: âd,e,T1,T2. â¬[d, e] T1 â¡ T2 â â¯{T1} = â¯{T2}.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 normalize //
+lemma lift_fwd_tw: âl,m,T1,T2. â¬[l, m] T1 â¡ T2 â â¯{T1} = â¯{T2}.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 normalize //
qed-.
-lemma lift_simple_dx: âd,e,T1,T2. â¬[d, e] T1 â¡ T2 â ðâ¦T1⦠â ðâ¦T2â¦.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
-#a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
+lemma lift_simple_dx: âl,m,T1,T2. â¬[l, m] T1 â¡ T2 â ðâ¦T1⦠â ðâ¦T2â¦.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
+#a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
elim (simple_inv_bind ⦠H)
qed-.
-lemma lift_simple_sn: âd,e,T1,T2. â¬[d, e] T1 â¡ T2 â ðâ¦T2⦠â ðâ¦T1â¦.
-#d #e #T1 #T2 #H elim H -d -e -T1 -T2 //
-#a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #_ #_ #H
+lemma lift_simple_sn: âl,m,T1,T2. â¬[l, m] T1 â¡ T2 â ðâ¦T2⦠â ðâ¦T1â¦.
+#l #m #T1 #T2 #H elim H -l -m -T1 -T2 //
+#a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #_ #_ #H
elim (simple_inv_bind ⦠H)
qed-.
(* Basic properties *********************************************************)
(* Basic_1: was: lift_lref_gt *)
-lemma lift_lref_ge_minus: âd,e,i. d + e ⤠i â â¬[d, e] #(i - e) â¡ #i.
-#d #e #i #H >(plus_minus_m_m i e) in ⢠(? ? ? ? %); /3 width=2 by lift_lref_ge, le_plus_to_minus_r, le_plus_b/
+lemma lift_lref_ge_minus: âl,m,i. l + m ⤠i â â¬[l, m] #(i - m) â¡ #i.
+#l #m #i #H >(plus_minus_m_m i m) in ⢠(? ? ? ? %); /3 width=2 by lift_lref_ge, le_plus_to_minus_r, le_plus_b/
qed.
-lemma lift_lref_ge_minus_eq: âd,e,i,j. d + e ⤠i â j = i - e â â¬[d, e] #j â¡ #i.
+lemma lift_lref_ge_minus_eq: âl,m,i,j. l + m ⤠i â j = i - m â â¬[l, m] #j â¡ #i.
/2 width=1/ qed-.
(* Basic_1: was: lift_r *)
-lemma lift_refl: âT,d. â¬[d, 0] T â¡ T.
+lemma lift_refl: âT,l. â¬[l, 0] T â¡ T.
#T elim T -T
-[ * #i // #d elim (lt_or_ge i d) /2 width=1 by lift_lref_lt, lift_lref_ge/
+[ * #i // #l elim (lt_or_ge i l) /2 width=1 by lift_lref_lt, lift_lref_ge/
| * /2 width=1 by lift_bind, lift_flat/
]
qed.
-lemma lift_total: âT1,d,e. âT2. â¬[d,e] T1 â¡ T2.
+lemma lift_total: âT1,l,m. âT2. â¬[l,m] T1 â¡ T2.
#T1 elim T1 -T1
-[ * #i /2 width=2/ #d #e elim (lt_or_ge i d) /3 width=2 by lift_lref_lt, lift_lref_ge, ex_intro/
-| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #d #e
- elim (IHV1 d e) -IHV1 #V2 #HV12
- [ elim (IHT1 (d+1) e) -IHT1 /3 width=2 by lift_bind, ex_intro/
- | elim (IHT1 d e) -IHT1 /3 width=2 by lift_flat, ex_intro/
+[ * #i /2 width=2/ #l #m elim (lt_or_ge i l) /3 width=2 by lift_lref_lt, lift_lref_ge, ex_intro/
+| * [ #a ] #I #V1 #T1 #IHV1 #IHT1 #l #m
+ elim (IHV1 l m) -IHV1 #V2 #HV12
+ [ elim (IHT1 (l+1) m) -IHT1 /3 width=2 by lift_bind, ex_intro/
+ | elim (IHT1 l m) -IHT1 /3 width=2 by lift_flat, ex_intro/
]
]
qed.
(* Basic_1: was: lift_free (right to left) *)
-lemma lift_split: âd1,e2,T1,T2. â¬[d1, e2] T1 â¡ T2 â
- âd2,e1. d1 ⤠d2 â d2 ⤠d1 + e1 â e1 ⤠e2 â
- ââT. â¬[d1, e1] T1 â¡ T & â¬[d2, e2 - e1] T â¡ T2.
-#d1 #e2 #T1 #T2 #H elim H -d1 -e2 -T1 -T2
+lemma lift_split: âl1,m2,T1,T2. â¬[l1, m2] T1 â¡ T2 â
+ âl2,m1. l1 ⤠l2 â l2 ⤠l1 + m1 â m1 ⤠m2 â
+ ââT. â¬[l1, m1] T1 â¡ T & â¬[l2, m2 - m1] T â¡ T2.
+#l1 #m2 #T1 #T2 #H elim H -l1 -m2 -T1 -T2
[ /3 width=3/
-| #i #d1 #e2 #Hid1 #d2 #e1 #Hd12 #_ #_
- lapply (lt_to_le_to_lt ⦠Hid1 Hd12) -Hd12 #Hid2 /4 width=3 by lift_lref_lt, ex2_intro/
-| #i #d1 #e2 #Hid1 #d2 #e1 #_ #Hd21 #He12
- lapply (transitive_le ⦠(i+e1) Hd21 ?) /2 width=1 by monotonic_le_plus_l/ -Hd21 #Hd21
- >(plus_minus_m_m e2 e1 ?) /3 width=3 by lift_lref_ge, ex2_intro/
+| #i #l1 #m2 #Hil1 #l2 #m1 #Hl12 #_ #_
+ lapply (lt_to_le_to_lt ⦠Hil1 Hl12) -Hl12 #Hil2 /4 width=3 by lift_lref_lt, ex2_intro/
+| #i #l1 #m2 #Hil1 #l2 #m1 #_ #Hl21 #Hm12
+ lapply (transitive_le ⦠(i+m1) Hl21 ?) /2 width=1 by monotonic_le_plus_l/ -Hl21 #Hl21
+ >(plus_minus_m_m m2 m1 ?) /3 width=3 by lift_lref_ge, ex2_intro/
| /3 width=3/
-| #a #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
- elim (IHV ⦠Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
- elim (IHT (d2+1) ⦠? ? He12) /3 width=5 by lift_bind, le_S_S, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #d1 #e2 #_ #_ #IHV #IHT #d2 #e1 #Hd12 #Hd21 #He12
- elim (IHV ⦠Hd12 Hd21 He12) -IHV #V0 #HV0a #HV0b
- elim (IHT d2 ⦠? ? He12) /3 width=5 by lift_flat, ex2_intro/
+| #a #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
+ elim (IHV ⦠Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
+ elim (IHT (l2+1) ⦠? ? Hm12) /3 width=5 by lift_bind, le_S_S, ex2_intro/
+| #I #V1 #V2 #T1 #T2 #l1 #m2 #_ #_ #IHV #IHT #l2 #m1 #Hl12 #Hl21 #Hm12
+ elim (IHV ⦠Hl12 Hl21 Hm12) -IHV #V0 #HV0a #HV0b
+ elim (IHT l2 ⦠? ? Hm12) /3 width=5 by lift_flat, ex2_intro/
]
qed.
(* Basic_1: was only: dnf_dec2 dnf_dec *)
-lemma is_lift_dec: âT2,d,e. Decidable (âT1. â¬[d,e] T1 â¡ T2).
+lemma is_lift_dec: âT2,l,m. Decidable (âT1. â¬[l,m] T1 â¡ T2).
#T1 elim T1 -T1
-[ * [1,3: /3 width=2 by lift_sort, lift_gref, ex_intro, or_introl/ ] #i #d #e
- elim (lt_or_ge i d) #Hdi
+[ * [1,3: /3 width=2 by lift_sort, lift_gref, ex_intro, or_introl/ ] #i #l #m
+ elim (lt_or_ge i l) #Hli
[ /4 width=3 by lift_lref_lt, ex_intro, or_introl/
- | elim (lt_or_ge i (d + e)) #Hide
- [ @or_intror * #T1 #H elim (lift_inv_lref2_be ⦠H Hdi Hide)
- | -Hdi /4 width=2 by lift_lref_ge_minus, ex_intro, or_introl/
+ | elim (lt_or_ge i (l + m)) #Hilm
+ [ @or_intror * #T1 #H elim (lift_inv_lref2_be ⦠H Hli Hilm)
+ | -Hli /4 width=2 by lift_lref_ge_minus, ex_intro, or_introl/
]
]
-| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #d #e
- [ elim (IHV2 d e) -IHV2
- [ * #V1 #HV12 elim (IHT2 (d+1) e) -IHT2
+| * [ #a ] #I #V2 #T2 #IHV2 #IHT2 #l #m
+ [ elim (IHV2 l m) -IHV2
+ [ * #V1 #HV12 elim (IHT2 (l+1) m) -IHT2
[ * #T1 #HT12 @or_introl /3 width=2 by lift_bind, ex_intro/
| -V1 #HT2 @or_intror * #X #H
elim (lift_inv_bind2 ⦠H) -H /3 width=2 by ex_intro/
@@ -372,8 +372,8 @@ lemma is_lift_dec: âT2,d,e. Decidable (âT1. â¬[d,e] T1 â¡ T2).
| -IHT2 #HV2 @or_intror * #X #H
elim (lift_inv_bind2 ⦠H) -H /3 width=2 by ex_intro/
]
- | elim (IHV2 d e) -IHV2
- [ * #V1 #HV12 elim (IHT2 d e) -IHT2
+ | elim (IHV2 l m) -IHV2
+ [ * #V1 #HV12 elim (IHT2 l m) -IHT2
[ * #T1 #HT12 /4 width=2/
| -V1 #HT2 @or_intror * #X #H
elim (lift_inv_flat2 ⦠H) -H /3 width=2 by ex_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma
index 06452afe4..9ab57806f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift.ma
@@ -19,50 +19,50 @@ include "basic_2/substitution/lift.ma".
(* Main properties ***********************************************************)
(* Basic_1: was: lift_inj *)
-theorem lift_inj: âd,e,T1,U. â¬[d,e] T1 â¡ U â âT2. â¬[d,e] T2 â¡ U â T1 = T2.
-#d #e #T1 #U #H elim H -d -e -T1 -U
-[ #k #d #e #X #HX
+theorem lift_inj: âl,m,T1,U. â¬[l,m] T1 â¡ U â âT2. â¬[l,m] T2 â¡ U â T1 = T2.
+#l #m #T1 #U #H elim H -l -m -T1 -U
+[ #k #l #m #X #HX
lapply (lift_inv_sort2 ⦠HX) -HX //
-| #i #d #e #Hid #X #HX
+| #i #l #m #Hil #X #HX
lapply (lift_inv_lref2_lt ⦠HX ?) -HX //
-| #i #d #e #Hdi #X #HX
+| #i #l #m #Hli #X #HX
lapply (lift_inv_lref2_ge ⦠HX ?) -HX /2 width=1 by monotonic_le_plus_l/
-| #p #d #e #X #HX
+| #p #l #m #X #HX
lapply (lift_inv_gref2 ⦠HX) -HX //
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
elim (lift_inv_bind2 ⦠HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
elim (lift_inv_flat2 ⦠HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
]
qed-.
(* Basic_1: was: lift_gen_lift *)
-theorem lift_div_le: âd1,e1,T1,T. â¬[d1, e1] T1 â¡ T â
- âd2,e2,T2. â¬[d2 + e1, e2] T2 â¡ T â
- d1 ⤠d2 â
- ââT0. â¬[d1, e1] T0 â¡ T2 & â¬[d2, e2] T0 â¡ T1.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+theorem lift_div_le: âl1,m1,T1,T. â¬[l1, m1] T1 â¡ T â
+ âl2,m2,T2. â¬[l2 + m1, m2] T2 â¡ T â
+ l1 ⤠l2 â
+ ââT0. â¬[l1, m1] T0 â¡ T2 & â¬[l2, m2] T0 â¡ T1.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
lapply (lift_inv_sort2 ⦠Hk) -Hk #Hk destruct /3 width=3 by lift_sort, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
- lapply (lt_to_le_to_lt ⦠Hid1 Hd12) -Hd12 #Hid2
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
+ lapply (lt_to_le_to_lt ⦠Hil1 Hl12) -Hl12 #Hil2
lapply (lift_inv_lref2_lt ⦠Hi ?) -Hi /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #Hi #Hd12
- elim (lift_inv_lref2 ⦠Hi) -Hi * #Hid2 #H destruct
- [ -Hd12 lapply (lt_plus_to_lt_l ⦠Hid2) -Hid2 #Hid2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
- | -Hid1 >plus_plus_comm_23 in Hid2; #H lapply (le_plus_to_le_r ⦠H) -H #H
- elim (le_inv_plus_l ⦠H) -H #Hide2 #He2i
- lapply (transitive_le ⦠Hd12 Hide2) -Hd12 #Hd12
- >le_plus_minus_comm // >(plus_minus_m_m i e2) in ⢠(? ? ? %);
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #Hi #Hl12
+ elim (lift_inv_lref2 ⦠Hi) -Hi * #Hil2 #H destruct
+ [ -Hl12 lapply (lt_plus_to_lt_l ⦠Hil2) -Hil2 #Hil2 /3 width=3 by lift_lref_lt, lift_lref_ge, ex2_intro/
+ | -Hil1 >plus_plus_comm_23 in Hil2; #H lapply (le_plus_to_le_r ⦠H) -H #H
+ elim (le_inv_plus_l ⦠H) -H #Hilm2 #Hm2i
+ lapply (transitive_le ⦠Hl12 Hilm2) -Hl12 #Hl12
+ >le_plus_minus_comm // >(plus_minus_m_m i m2) in ⢠(? ? ? %);
/4 width=3 by lift_lref_ge, ex2_intro/
]
-| #p #d1 #e1 #d2 #e2 #T2 #Hk #Hd12
+| #p #l1 #m1 #l2 #m2 #T2 #Hk #Hl12
lapply (lift_inv_gref2 ⦠Hk) -Hk #Hk destruct /3 width=3 by lift_gref, ex2_intro/
-| #a #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+| #a #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
lapply (lift_inv_bind2 ⦠H) -H * #W2 #U2 #HW2 #HU2 #H destruct
elim (IHW ⦠HW2) // -IHW -HW2 #W0 #HW2 #HW1
>plus_plus_comm_23 in HU2; #HU2 elim (IHU ⦠HU2) /3 width=5 by lift_bind, le_S_S, ex2_intro/
-| #I #W1 #W #U1 #U #d1 #e1 #_ #_ #IHW #IHU #d2 #e2 #T2 #H #Hd12
+| #I #W1 #W #U1 #U #l1 #m1 #_ #_ #IHW #IHU #l2 #m2 #T2 #H #Hl12
lapply (lift_inv_flat2 ⦠H) -H * #W2 #U2 #HW2 #HU2 #H destruct
elim (IHW ⦠HW2) // -IHW -HW2 #W0 #HW2 #HW1
elim (IHU ⦠HU2) /3 width=5 by lift_flat, ex2_intro/
@@ -70,74 +70,74 @@ theorem lift_div_le: âd1,e1,T1,T. â¬[d1, e1] T1 â¡ T â
qed.
(* Note: apparently this was missing in basic_1 *)
-theorem lift_div_be: âd1,e1,T1,T. â¬[d1, e1] T1 â¡ T â
- âe,e2,T2. â¬[d1 + e, e2] T2 â¡ T â
- e ⤠e1 â e1 ⤠e + e2 â
- ââT0. â¬[d1, e] T0 â¡ T2 & â¬[d1, e + e2 - e1] T0 â¡ T1.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #e #e2 #T2 #H >(lift_inv_sort2 ⦠H) -H /2 width=3 by lift_sort, ex2_intro/
-| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
+theorem lift_div_be: âl1,m1,T1,T. â¬[l1, m1] T1 â¡ T â
+ âm,m2,T2. â¬[l1 + m, m2] T2 â¡ T â
+ m ⤠m1 â m1 ⤠m + m2 â
+ ââT0. â¬[l1, m] T0 â¡ T2 & â¬[l1, m + m2 - m1] T0 â¡ T1.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #m #m2 #T2 #H >(lift_inv_sort2 ⦠H) -H /2 width=3 by lift_sort, ex2_intro/
+| #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
>(lift_inv_lref2_lt ⦠H) -H /3 width=3 by lift_lref_lt, lt_plus_to_minus_r, lt_to_le_to_lt, ex2_intro/
-| #i #d1 #e1 #Hid1 #e #e2 #T2 #H #He1 #He1e2
- elim (lt_or_ge (i+e1) (d1+e+e2)) #Hie1d1e2
+| #i #l1 #m1 #Hil1 #m #m2 #T2 #H #Hm1 #Hm1m2
+ elim (lt_or_ge (i+m1) (l1+m+m2)) #Him1l1m2
[ elim (lift_inv_lref2_be ⦠H) -H /2 width=1 by le_plus/
| >(lift_inv_lref2_ge ⦠H ?) -H //
- lapply (le_plus_to_minus ⦠Hie1d1e2) #Hd1e21i
- elim (le_inv_plus_l ⦠Hie1d1e2) -Hie1d1e2 #Hd1e12 #He2ie1
- @ex2_intro [2: /2 width=1/ | skip ] -Hd1e12
+ lapply (le_plus_to_minus ⦠Him1l1m2) #Hl1m21i
+ elim (le_inv_plus_l ⦠Him1l1m2) -Him1l1m2 #Hl1m12 #Hm2im1
+ @ex2_intro [2: /2 width=1 by lift_lref_ge_minus/ | skip ] -Hl1m12
@lift_lref_ge_minus_eq [ >plus_minus_associative // | /2 width=1 by minus_le_minus_minus_comm/ ]
]
-| #p #d1 #e1 #e #e2 #T2 #H >(lift_inv_gref2 ⦠H) -H /2 width=3 by lift_gref, ex2_intro/
-| #a #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+| #p #l1 #m1 #m #m2 #T2 #H >(lift_inv_gref2 ⦠H) -H /2 width=3 by lift_gref, ex2_intro/
+| #a #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
elim (lift_inv_bind2 ⦠H) -H #V2 #T2 #HV2 #HT2 #H destruct
elim (IHV1 ⦠HV2) -V // >plus_plus_comm_23 in HT2; #HT2
elim (IHT1 ⦠HT2) -T /3 width=5 by lift_bind, ex2_intro/
-| #I #V1 #V #T1 #T #d1 #e1 #_ #_ #IHV1 #IHT1 #e #e2 #X #H #He1 #He1e2
+| #I #V1 #V #T1 #T #l1 #m1 #_ #_ #IHV1 #IHT1 #m #m2 #X #H #Hm1 #Hm1m2
elim (lift_inv_flat2 ⦠H) -H #V2 #T2 #HV2 #HT2 #H destruct
elim (IHV1 ⦠HV2) -V //
elim (IHT1 ⦠HT2) -T /3 width=5 by lift_flat, ex2_intro/
]
qed.
-theorem lift_mono: âd,e,T,U1. â¬[d,e] T â¡ U1 â âU2. â¬[d,e] T â¡ U2 â U1 = U2.
-#d #e #T #U1 #H elim H -d -e -T -U1
-[ #k #d #e #X #HX
+theorem lift_mono: âl,m,T,U1. â¬[l,m] T â¡ U1 â âU2. â¬[l,m] T â¡ U2 â U1 = U2.
+#l #m #T #U1 #H elim H -l -m -T -U1
+[ #k #l #m #X #HX
lapply (lift_inv_sort1 ⦠HX) -HX //
-| #i #d #e #Hid #X #HX
+| #i #l #m #Hil #X #HX
lapply (lift_inv_lref1_lt ⦠HX ?) -HX //
-| #i #d #e #Hdi #X #HX
+| #i #l #m #Hli #X #HX
lapply (lift_inv_lref1_ge ⦠HX ?) -HX //
-| #p #d #e #X #HX
+| #p #l #m #X #HX
lapply (lift_inv_gref1 ⦠HX) -HX //
-| #a #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #a #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
elim (lift_inv_bind1 ⦠HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
-| #I #V1 #V2 #T1 #T2 #d #e #_ #_ #IHV12 #IHT12 #X #HX
+| #I #V1 #V2 #T1 #T2 #l #m #_ #_ #IHV12 #IHT12 #X #HX
elim (lift_inv_flat1 ⦠HX) -HX #V #T #HV1 #HT1 #HX destruct /3 width=1 by eq_f2/
]
qed-.
(* Basic_1: was: lift_free (left to right) *)
-theorem lift_trans_be: âd1,e1,T1,T. â¬[d1, e1] T1 â¡ T â
- âd2,e2,T2. â¬[d2, e2] T â¡ T2 â
- d1 ⤠d2 â d2 ⤠d1 + e1 â â¬[d1, e1 + e2] T1 â¡ T2.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
+theorem lift_trans_be: âl1,m1,T1,T. â¬[l1, m1] T1 â¡ T â
+ âl2,m2,T2. â¬[l2, m2] T â¡ T2 â
+ l1 ⤠l2 â l2 ⤠l1 + m1 â â¬[l1, m1 + m2] T1 â¡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
>(lift_inv_sort1 ⦠HT2) -HT2 //
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #Hd12 #_
- lapply (lt_to_le_to_lt ⦠Hid1 Hd12) -Hd12 #Hid2
- lapply (lift_inv_lref1_lt ⦠HT2 Hid2) /2 width=1 by lift_lref_lt/
-| #i #d1 #e1 #Hid1 #d2 #e2 #T2 #HT2 #_ #Hd21
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #Hl12 #_
+ lapply (lt_to_le_to_lt ⦠Hil1 Hl12) -Hl12 #Hil2
+ lapply (lift_inv_lref1_lt ⦠HT2 Hil2) /2 width=1 by lift_lref_lt/
+| #i #l1 #m1 #Hil1 #l2 #m2 #T2 #HT2 #_ #Hl21
lapply (lift_inv_lref1_ge ⦠HT2 ?) -HT2
- [ @(transitive_le ⦠Hd21 ?) -Hd21 /2 width=1 by monotonic_le_plus_l/
- | -Hd21 /2 width=1 by lift_lref_ge/
+ [ @(transitive_le ⦠Hl21 ?) -Hl21 /2 width=1 by monotonic_le_plus_l/
+ | -Hl21 /2 width=1 by lift_lref_ge/
]
-| #p #d1 #e1 #d2 #e2 #T2 #HT2 #_ #_
+| #p #l1 #m1 #l2 #m2 #T2 #HT2 #_ #_
>(lift_inv_gref1 ⦠HT2) -HT2 //
-| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
elim (lift_inv_bind1 ⦠HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
lapply (IHV12 ⦠HV20 ? ?) // -IHV12 -HV20 #HV10
lapply (IHT12 ⦠HT20 ? ?) /2 width=1 by lift_bind, le_S_S/ (**) (* full auto a bit slow *)
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd12 #Hd21
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl12 #Hl21
elim (lift_inv_flat1 ⦠HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
lapply (IHV12 ⦠HV20 ? ?) // -IHV12 -HV20 #HV10
lapply (IHT12 ⦠HT20 ? ?) /2 width=1 by lift_flat/ (**) (* full auto a bit slow *)
@@ -145,26 +145,26 @@ theorem lift_trans_be: âd1,e1,T1,T. â¬[d1, e1] T1 â¡ T â
qed.
(* Basic_1: was: lift_d (right to left) *)
-theorem lift_trans_le: âd1,e1,T1,T. â¬[d1, e1] T1 â¡ T â
- âd2,e2,T2. â¬[d2, e2] T â¡ T2 â d2 ⤠d1 â
- ââT0. â¬[d2, e2] T1 â¡ T0 & â¬[d1 + e2, e1] T0 â¡ T2.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #X #HX #_
+theorem lift_trans_le: âl1,m1,T1,T. â¬[l1, m1] T1 â¡ T â
+ âl2,m2,T2. â¬[l2, m2] T â¡ T2 â l2 ⤠l1 â
+ ââT0. â¬[l2, m2] T1 â¡ T0 & â¬[l1 + m2, m1] T0 â¡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #X #HX #_
>(lift_inv_sort1 ⦠HX) -HX /2 width=3 by lift_sort, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
- lapply (lt_to_le_to_lt ⦠(d1+e2) Hid1 ?) // #Hie2
- elim (lift_inv_lref1 ⦠HX) -HX * #Hid2 #HX destruct /4 width=3 by lift_lref_ge_minus, lift_lref_lt, lt_minus_to_plus, monotonic_le_plus_l, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hd21
- lapply (transitive_le ⦠Hd21 Hid1) -Hd21 #Hid2
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
+ lapply (lt_to_le_to_lt ⦠(l1+m2) Hil1 ?) // #Him2
+ elim (lift_inv_lref1 ⦠HX) -HX * #Hil2 #HX destruct /4 width=3 by lift_lref_ge_minus, lift_lref_lt, lt_minus_to_plus, monotonic_le_plus_l, ex2_intro/
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hl21
+ lapply (transitive_le ⦠Hl21 Hil1) -Hl21 #Hil2
lapply (lift_inv_lref1_ge ⦠HX ?) -HX /2 width=3 by transitive_le/ #HX destruct
>plus_plus_comm_23 /4 width=3 by lift_lref_ge_minus, lift_lref_ge, monotonic_le_plus_l, ex2_intro/
-| #p #d1 #e1 #d2 #e2 #X #HX #_
+| #p #l1 #m1 #l2 #m2 #X #HX #_
>(lift_inv_gref1 ⦠HX) -HX /2 width=3 by lift_gref, ex2_intro/
-| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
elim (lift_inv_bind1 ⦠HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
elim (IHV12 ⦠HV20) -IHV12 -HV20 //
elim (IHT12 ⦠HT20) -IHT12 -HT20 /3 width=5 by lift_bind, le_S_S, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hd21
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hl21
elim (lift_inv_flat1 ⦠HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
elim (IHV12 ⦠HV20) -IHV12 -HV20 //
elim (IHT12 ⦠HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
@@ -172,27 +172,27 @@ theorem lift_trans_le: âd1,e1,T1,T. â¬[d1, e1] T1 â¡ T â
qed.
(* Basic_1: was: lift_d (left to right) *)
-theorem lift_trans_ge: âd1,e1,T1,T. â¬[d1, e1] T1 â¡ T â
- âd2,e2,T2. â¬[d2, e2] T â¡ T2 â d1 + e1 ⤠d2 â
- ââT0. â¬[d2 - e1, e2] T1 â¡ T0 & â¬[d1, e1] T0 â¡ T2.
-#d1 #e1 #T1 #T #H elim H -d1 -e1 -T1 -T
-[ #k #d1 #e1 #d2 #e2 #X #HX #_
+theorem lift_trans_ge: âl1,m1,T1,T. â¬[l1, m1] T1 â¡ T â
+ âl2,m2,T2. â¬[l2, m2] T â¡ T2 â l1 + m1 ⤠l2 â
+ ââT0. â¬[l2 - m1, m2] T1 â¡ T0 & â¬[l1, m1] T0 â¡ T2.
+#l1 #m1 #T1 #T #H elim H -l1 -m1 -T1 -T
+[ #k #l1 #m1 #l2 #m2 #X #HX #_
>(lift_inv_sort1 ⦠HX) -HX /2 width=3 by lift_sort, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #Hded
- lapply (lt_to_le_to_lt ⦠(d1+e1) Hid1 ?) // #Hid1e
- lapply (lt_to_le_to_lt ⦠(d2-e1) Hid1 ?) /2 width=1 by le_plus_to_minus_r/ #Hid2e
- lapply (lt_to_le_to_lt ⦠Hid1e Hded) -Hid1e -Hded #Hid2
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #Hlml
+ lapply (lt_to_le_to_lt ⦠(l1+m1) Hil1 ?) // #Hil1m
+ lapply (lt_to_le_to_lt ⦠(l2-m1) Hil1 ?) /2 width=1 by le_plus_to_minus_r/ #Hil2m
+ lapply (lt_to_le_to_lt ⦠Hil1m Hlml) -Hil1m -Hlml #Hil2
lapply (lift_inv_lref1_lt ⦠HX ?) -HX // #HX destruct /3 width=3 by lift_lref_lt, ex2_intro/
-| #i #d1 #e1 #Hid1 #d2 #e2 #X #HX #_
- elim (lift_inv_lref1 ⦠HX) -HX * #Hied #HX destruct /4 width=3 by lift_lref_lt, lift_lref_ge, monotonic_le_minus_l, lt_plus_to_minus_r, transitive_le, ex2_intro/
-| #p #d1 #e1 #d2 #e2 #X #HX #_
+| #i #l1 #m1 #Hil1 #l2 #m2 #X #HX #_
+ elim (lift_inv_lref1 ⦠HX) -HX * #Himl #HX destruct /4 width=3 by lift_lref_lt, lift_lref_ge, monotonic_le_minus_l, lt_plus_to_minus_r, transitive_le, ex2_intro/
+| #p #l1 #m1 #l2 #m2 #X #HX #_
>(lift_inv_gref1 ⦠HX) -HX /2 width=3 by lift_gref, ex2_intro/
-| #a #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+| #a #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
elim (lift_inv_bind1 ⦠HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
elim (IHV12 ⦠HV20) -IHV12 -HV20 //
elim (IHT12 ⦠HT20) -IHT12 -HT20 /2 width=1 by le_S_S/ #T
<plus_minus /3 width=5 by lift_bind, le_plus_to_minus_r, le_plus_b, ex2_intro/
-| #I #V1 #V2 #T1 #T2 #d1 #e1 #_ #_ #IHV12 #IHT12 #d2 #e2 #X #HX #Hded
+| #I #V1 #V2 #T1 #T2 #l1 #m1 #_ #_ #IHV12 #IHT12 #l2 #m2 #X #HX #Hlml
elim (lift_inv_flat1 ⦠HX) -HX #V0 #T0 #HV20 #HT20 #HX destruct
elim (IHV12 ⦠HV20) -IHV12 -HV20 //
elim (IHT12 ⦠HT20) -IHT12 -HT20 /3 width=5 by lift_flat, ex2_intro/
@@ -201,17 +201,17 @@ qed.
(* Advanced properties ******************************************************)
-lemma lift_conf_O1: âT,T1,d1,e1. â¬[d1, e1] T â¡ T1 â âT2,e2. â¬[0, e2] T â¡ T2 â
- ââT0. â¬[0, e2] T1 â¡ T0 & â¬[d1 + e2, e1] T2 â¡ T0.
-#T #T1 #d1 #e1 #HT1 #T2 #e2 #HT2
-elim (lift_total T1 0 e2) #T0 #HT10
+lemma lift_conf_O1: âT,T1,l1,m1. â¬[l1, m1] T â¡ T1 â âT2,m2. â¬[0, m2] T â¡ T2 â
+ ââT0. â¬[0, m2] T1 â¡ T0 & â¬[l1 + m2, m1] T2 â¡ T0.
+#T #T1 #l1 #m1 #HT1 #T2 #m2 #HT2
+elim (lift_total T1 0 m2) #T0 #HT10
elim (lift_trans_le ⦠HT1 ⦠HT10) -HT1 // #X #HTX #HT20
lapply (lift_mono ⦠HTX ⦠HT2) -T #H destruct /2 width=3 by ex2_intro/
qed.
-lemma lift_conf_be: âT,T1,d,e1. â¬[d, e1] T â¡ T1 â âT2,e2. â¬[d, e2] T â¡ T2 â
- e1 ⤠e2 â â¬[d + e1, e2 - e1] T1 â¡ T2.
-#T #T1 #d #e1 #HT1 #T2 #e2 #HT2 #He12
-elim (lift_split ⦠HT2 (d+e1) e1) -HT2 // #X #H
+lemma lift_conf_be: âT,T1,l,m1. â¬[l, m1] T â¡ T1 â âT2,m2. â¬[l, m2] T â¡ T2 â
+ m1 ⤠m2 â â¬[l + m1, m2 - m1] T1 â¡ T2.
+#T #T1 #l #m1 #HT1 #T2 #m2 #HT2 #Hm12
+elim (lift_split ⦠HT2 (l+m1) m1) -HT2 // #X #H
>(lift_mono ⦠H ⦠HT1) -T //
qed.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift_vector.ma
index c0c83975e..44ed6a7b4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_lift_vector.ma
@@ -19,9 +19,9 @@ include "basic_2/substitution/lift_vector.ma".
(* Main properties ***********************************************************)
-theorem liftv_mono: âTs,U1s,d,e. â¬[d,e] Ts â¡ U1s â
- âU2s:list term. â¬[d,e] Ts â¡ U2s â U1s = U2s.
-#Ts #U1s #d #e #H elim H -Ts -U1s
+theorem liftv_mono: âTs,U1s,l,m. â¬[l,m] Ts â¡ U1s â
+ âU2s:list term. â¬[l,m] Ts â¡ U2s â U1s = U2s.
+#Ts #U1s #l #m #H elim H -Ts -U1s
[ #U2s #H >(liftv_inv_nil1 ⦠H) -H //
| #Ts #U1s #T #U1 #HTU1 #_ #IHTU1s #X #H destruct
elim (liftv_inv_cons1 ⦠H) -H #U2 #U2s #HTU2 #HTU2s #H destruct
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_neg.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_neg.ma
index a2d7cd7e1..434ad02e4 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_neg.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_neg.ma
@@ -21,24 +21,24 @@ include "basic_2/substitution/lift.ma".
lemma nlift_lref_be_SO: âX,i. â¬[i, 1] X â¡ #i â â¥.
/2 width=7 by lift_inv_lref2_be/ qed-.
-lemma nlift_bind_sn: âW,d,e. (âV. â¬[d, e] V â¡ W â â¥) â
- âa,I,U. (âX. â¬[d, e] X â¡ â{a,I}W.U â â¥).
-#W #d #e #HW #a #I #U #X #H elim (lift_inv_bind2 ⦠H) -H /2 width=2 by/
+lemma nlift_bind_sn: âW,l,m. (âV. â¬[l, m] V â¡ W â â¥) â
+ âa,I,U. (âX. â¬[l, m] X â¡ â{a,I}W.U â â¥).
+#W #l #m #HW #a #I #U #X #H elim (lift_inv_bind2 ⦠H) -H /2 width=2 by/
qed-.
-lemma nlift_bind_dx: âU,d,e. (âT. â¬[d+1, e] T â¡ U â â¥) â
- âa,I,W. (âX. â¬[d, e] X â¡ â{a,I}W.U â â¥).
-#U #d #e #HU #a #I #W #X #H elim (lift_inv_bind2 ⦠H) -H /2 width=2 by/
+lemma nlift_bind_dx: âU,l,m. (âT. â¬[l+1, m] T â¡ U â â¥) â
+ âa,I,W. (âX. â¬[l, m] X â¡ â{a,I}W.U â â¥).
+#U #l #m #HU #a #I #W #X #H elim (lift_inv_bind2 ⦠H) -H /2 width=2 by/
qed-.
-lemma nlift_flat_sn: âW,d,e. (âV. â¬[d, e] V â¡ W â â¥) â
- âI,U. (âX. â¬[d, e] X â¡ â{I}W.U â â¥).
-#W #d #e #HW #I #U #X #H elim (lift_inv_flat2 ⦠H) -H /2 width=2 by/
+lemma nlift_flat_sn: âW,l,m. (âV. â¬[l, m] V â¡ W â â¥) â
+ âI,U. (âX. â¬[l, m] X â¡ â{I}W.U â â¥).
+#W #l #m #HW #I #U #X #H elim (lift_inv_flat2 ⦠H) -H /2 width=2 by/
qed-.
-lemma nlift_flat_dx: âU,d,e. (âT. â¬[d, e] T â¡ U â â¥) â
- âI,W. (âX. â¬[d, e] X â¡ â{I}W.U â â¥).
-#U #d #e #HU #I #W #X #H elim (lift_inv_flat2 ⦠H) -H /2 width=2 by/
+lemma nlift_flat_dx: âU,l,m. (âT. â¬[l, m] T â¡ U â â¥) â
+ âI,W. (âX. â¬[l, m] X â¡ â{I}W.U â â¥).
+#U #l #m #HU #I #W #X #H elim (lift_inv_flat2 ⦠H) -H /2 width=2 by/
qed-.
(* Inversion lemmas on negated basic relocation *****************************)
@@ -50,17 +50,17 @@ lemma nlift_inv_lref_be_SO: âi,j. (âX. â¬[i, 1] X â¡ #j â â¥) â j = i
]
qed-.
-lemma nlift_inv_bind: âa,I,W,U,d,e. (âX. â¬[d, e] X â¡ â{a,I}W.U â â¥) â
- (âV. â¬[d, e] V â¡ W â â¥) ⨠(âT. â¬[d+1, e] T â¡ U â â¥).
-#a #I #W #U #d #e #H elim (is_lift_dec W d e)
+lemma nlift_inv_bind: âa,I,W,U,l,m. (âX. â¬[l, m] X â¡ â{a,I}W.U â â¥) â
+ (âV. â¬[l, m] V â¡ W â â¥) ⨠(âT. â¬[l+1, m] T â¡ U â â¥).
+#a #I #W #U #l #m #H elim (is_lift_dec W l m)
[ * /4 width=2 by lift_bind, or_intror/
| /4 width=2 by ex_intro, or_introl/
]
qed-.
-lemma nlift_inv_flat: âI,W,U,d,e. (âX. â¬[d, e] X â¡ â{I}W.U â â¥) â
- (âV. â¬[d, e] V â¡ W â â¥) ⨠(âT. â¬[d, e] T â¡ U â â¥).
-#I #W #U #d #e #H elim (is_lift_dec W d e)
+lemma nlift_inv_flat: âI,W,U,l,m. (âX. â¬[l, m] X â¡ â{I}W.U â â¥) â
+ (âV. â¬[l, m] V â¡ W â â¥) ⨠(âT. â¬[l, m] T â¡ U â â¥).
+#I #W #U #l #m #H elim (is_lift_dec W l m)
[ * /4 width=2 by lift_flat, or_intror/
| /4 width=2 by ex_intro, or_introl/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_vector.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_vector.ma
index 1aa6978f6..42cef005b 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_vector.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lift_vector.ma
@@ -17,46 +17,46 @@ include "basic_2/substitution/lift.ma".
(* BASIC TERM VECTOR RELOCATION *********************************************)
-inductive liftv (d,e:nat) : relation (list term) â
-| liftv_nil : liftv d e (â) (â)
+inductive liftv (l,m:nat) : relation (list term) â
+| liftv_nil : liftv l m (â) (â)
| liftv_cons: âT1s,T2s,T1,T2.
- â¬[d, e] T1 â¡ T2 â liftv d e T1s T2s â
- liftv d e (T1 @ T1s) (T2 @ T2s)
+ â¬[l, m] T1 â¡ T2 â liftv l m T1s T2s â
+ liftv l m (T1 @ T1s) (T2 @ T2s)
.
-interpretation "relocation (vector)" 'RLift d e T1s T2s = (liftv d e T1s T2s).
+interpretation "relocation (vector)" 'RLift l m T1s T2s = (liftv l m T1s T2s).
(* Basic inversion lemmas ***************************************************)
-fact liftv_inv_nil1_aux: âT1s,T2s,d,e. â¬[d, e] T1s â¡ T2s â T1s = â â T2s = â.
-#T1s #T2s #d #e * -T1s -T2s //
+fact liftv_inv_nil1_aux: âT1s,T2s,l,m. â¬[l, m] T1s â¡ T2s â T1s = â â T2s = â.
+#T1s #T2s #l #m * -T1s -T2s //
#T1s #T2s #T1 #T2 #_ #_ #H destruct
qed-.
-lemma liftv_inv_nil1: âT2s,d,e. â¬[d, e] â â¡ T2s â T2s = â.
+lemma liftv_inv_nil1: âT2s,l,m. â¬[l, m] â â¡ T2s â T2s = â.
/2 width=5 by liftv_inv_nil1_aux/ qed-.
-fact liftv_inv_cons1_aux: âT1s,T2s,d,e. â¬[d, e] T1s â¡ T2s â
+fact liftv_inv_cons1_aux: âT1s,T2s,l,m. â¬[l, m] T1s â¡ T2s â
âU1,U1s. T1s = U1 @ U1s â
- ââU2,U2s. â¬[d, e] U1 â¡ U2 & â¬[d, e] U1s â¡ U2s &
+ ââU2,U2s. â¬[l, m] U1 â¡ U2 & â¬[l, m] U1s â¡ U2s &
T2s = U2 @ U2s.
-#T1s #T2s #d #e * -T1s -T2s
+#T1s #T2s #l #m * -T1s -T2s
[ #U1 #U1s #H destruct
| #T1s #T2s #T1 #T2 #HT12 #HT12s #U1 #U1s #H destruct /2 width=5 by ex3_2_intro/
]
qed-.
-lemma liftv_inv_cons1: âU1,U1s,T2s,d,e. â¬[d, e] U1 @ U1s â¡ T2s â
- ââU2,U2s. â¬[d, e] U1 â¡ U2 & â¬[d, e] U1s â¡ U2s &
+lemma liftv_inv_cons1: âU1,U1s,T2s,l,m. â¬[l, m] U1 @ U1s â¡ T2s â
+ ââU2,U2s. â¬[l, m] U1 â¡ U2 & â¬[l, m] U1s â¡ U2s &
T2s = U2 @ U2s.
/2 width=3 by liftv_inv_cons1_aux/ qed-.
(* Basic properties *********************************************************)
-lemma liftv_total: âd,e. âT1s:list term. âT2s. â¬[d, e] T1s â¡ T2s.
-#d #e #T1s elim T1s -T1s
+lemma liftv_total: âl,m. âT1s:list term. âT2s. â¬[l, m] T1s â¡ T2s.
+#l #m #T1s elim T1s -T1s
[ /2 width=2 by liftv_nil, ex_intro/
| #T1 #T1s * #T2s #HT12s
- elim (lift_total T1 d e) /3 width=2 by liftv_cons, ex_intro/
+ elim (lift_total T1 l m) /3 width=2 by liftv_cons, ex_intro/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lpx_sn_drop.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lpx_sn_drop.ma
index b5ff6d204..3b7b6ebab 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lpx_sn_drop.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lpx_sn_drop.ma
@@ -12,7 +12,7 @@
(* *)
(**************************************************************************)
-include "basic_2/substitution/drop_leq.ma".
+include "basic_2/substitution/drop_lreq.ma".
include "basic_2/substitution/lpx_sn.ma".
(* SN POINTWISE EXTENSION OF A CONTEXT-SENSITIVE REALTION FOR TERMS *********)
@@ -45,17 +45,17 @@ lemma lpx_sn_drop_trans: âR,L1,L2. lpx_sn R L1 L2 â
]
qed-.
-lemma lpx_sn_deliftable_dropable: âR. l_deliftable_sn R â dropable_sn (lpx_sn R).
-#R #HR #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom1 ⦠H) -H
+lemma lpx_sn_deliftable_dropable: âR. d_deliftable_sn R â dropable_sn (lpx_sn R).
+#R #HR #L1 #K1 #s #l #m #H elim H -L1 -K1 -l -m
+[ #l #m #Hm #X #H >(lpx_sn_inv_atom1 ⦠H) -H
/4 width=3 by drop_atom, lpx_sn_atom, ex2_intro/
| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 ⦠H) -H
#L2 #V2 #HL12 #HV12 #H destruct
/3 width=5 by drop_pair, lpx_sn_pair, ex2_intro/
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 ⦠H) -H
+| #I #L1 #K1 #V1 #m #_ #IHLK1 #X #H elim (lpx_sn_inv_pair1 ⦠H) -H
#L2 #V2 #HL12 #HV12 #H destruct
elim (IHLK1 ⦠HL12) -L1 /3 width=3 by drop_drop, ex2_intro/
-| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
+| #I #L1 #K1 #V1 #W1 #l #m #HLK1 #HWV1 #IHLK1 #X #H
elim (lpx_sn_inv_pair1 ⦠H) -H #L2 #V2 #HL12 #HV12 #H destruct
elim (HR ⦠HV12 ⦠HLK1 ⦠HWV1) -V1
elim (IHLK1 ⦠HL12) -L1 /3 width=5 by drop_skip, lpx_sn_pair, ex2_intro/
@@ -63,39 +63,39 @@ lemma lpx_sn_deliftable_dropable: âR. l_deliftable_sn R â dropable_sn (lpx_s
qed-.
lemma lpx_sn_liftable_dedropable: âR. (âL. reflexive ? (R L)) â
- l_liftable R â dedropable_sn (lpx_sn R).
-#R #H1R #H2R #L1 #K1 #s #d #e #H elim H -L1 -K1 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom1 ⦠H) -H
+ d_liftable R â dedropable_sn (lpx_sn R).
+#R #H1R #H2R #L1 #K1 #s #l #m #H elim H -L1 -K1 -l -m
+[ #l #m #Hm #X #H >(lpx_sn_inv_atom1 ⦠H) -H
/4 width=4 by drop_atom, lpx_sn_atom, ex3_intro/
| #I #K1 #V1 #X #H elim (lpx_sn_inv_pair1 ⦠H) -H
#K2 #V2 #HK12 #HV12 #H destruct
lapply (lpx_sn_fwd_length ⦠HK12)
#H @(ex3_intro ⦠(K2.â{I}V2)) (**) (* explicit constructor *)
/3 width=1 by lpx_sn_pair, monotonic_le_plus_l/
- @leq_O2 normalize //
-| #I #L1 #K1 #V1 #e #_ #IHLK1 #K2 #HK12 elim (IHLK1 ⦠HK12) -K1
- /3 width=5 by drop_drop, leq_pair, lpx_sn_pair, ex3_intro/
-| #I #L1 #K1 #V1 #W1 #d #e #HLK1 #HWV1 #IHLK1 #X #H
+ @lreq_O2 normalize //
+| #I #L1 #K1 #V1 #m #_ #IHLK1 #K2 #HK12 elim (IHLK1 ⦠HK12) -K1
+ /3 width=5 by drop_drop, lreq_pair, lpx_sn_pair, ex3_intro/
+| #I #L1 #K1 #V1 #W1 #l #m #HLK1 #HWV1 #IHLK1 #X #H
elim (lpx_sn_inv_pair1 ⦠H) -H #K2 #W2 #HK12 #HW12 #H destruct
- elim (lift_total W2 d e) #V2 #HWV2
+ elim (lift_total W2 l m) #V2 #HWV2
lapply (H2R ⦠HW12 ⦠HLK1 ⦠HWV1 ⦠HWV2) -W1
elim (IHLK1 ⦠HK12) -K1
- /3 width=6 by drop_skip, leq_succ, lpx_sn_pair, ex3_intro/
+ /3 width=6 by drop_skip, lreq_succ, lpx_sn_pair, ex3_intro/
]
qed-.
-fact lpx_sn_dropable_aux: âR,L2,K2,s,d,e. â¬[s, d, e] L2 â¡ K2 â âL1. lpx_sn R L1 L2 â
- d = 0 â ââK1. â¬[s, 0, e] L1 â¡ K1 & lpx_sn R K1 K2.
-#R #L2 #K2 #s #d #e #H elim H -L2 -K2 -d -e
-[ #d #e #He #X #H >(lpx_sn_inv_atom2 ⦠H) -H
+fact lpx_sn_dropable_aux: âR,L2,K2,s,l,m. â¬[s, l, m] L2 â¡ K2 â âL1. lpx_sn R L1 L2 â
+ l = 0 â ââK1. â¬[s, 0, m] L1 â¡ K1 & lpx_sn R K1 K2.
+#R #L2 #K2 #s #l #m #H elim H -L2 -K2 -l -m
+[ #l #m #Hm #X #H >(lpx_sn_inv_atom2 ⦠H) -H
/4 width=3 by drop_atom, lpx_sn_atom, ex2_intro/
| #I #K2 #V2 #X #H elim (lpx_sn_inv_pair2 ⦠H) -H
#K1 #V1 #HK12 #HV12 #H destruct
/3 width=5 by drop_pair, lpx_sn_pair, ex2_intro/
-| #I #L2 #K2 #V2 #e #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 ⦠H) -H
+| #I #L2 #K2 #V2 #m #_ #IHLK2 #X #H #_ elim (lpx_sn_inv_pair2 ⦠H) -H
#L1 #V1 #HL12 #HV12 #H destruct
elim (IHLK2 ⦠HL12) -L2 /3 width=3 by drop_drop, ex2_intro/
-| #I #L2 #K2 #V2 #W2 #d #e #_ #_ #_ #L1 #_
+| #I #L2 #K2 #V2 #W2 #l #m #_ #_ #_ #L1 #_
<plus_n_Sm #H destruct
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lsuby.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lsuby.ma
index 578612a71..623f8c6aa 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lsuby.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lsuby.ma
@@ -19,29 +19,29 @@ include "basic_2/substitution/drop.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_atom: âL,l,m. lsuby l m 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)
+| lsuby_pair: âI1,I2,L1,L2,V,m. lsuby 0 m L1 L2 â
+ lsuby 0 (⫯m) (L1.â{I1}V) (L2.â{I2}V)
+| lsuby_succ: âI1,I2,L1,L2,V1,V2,l,m.
+ lsuby l m L1 L2 â lsuby (⫯l) m (L1.â{I1}V1) (L2.â{I2}V2)
.
interpretation
"local environment refinement (extended substitution)"
- 'LRSubEq L1 d e L2 = (lsuby d e L1 L2).
+ 'LRSubEq L1 l m L2 = (lsuby l m L1 L2).
(* Basic properties *********************************************************)
-lemma lsuby_pair_lt: âI1,I2,L1,L2,V,e. L1 â[0, â«°e] L2 â 0 < e â
- L1.â{I1}V â[0, e] L2.â{I2}V.
-#I1 #I2 #L1 #L2 #V #e #HL12 #He <(ylt_inv_O1 ⦠He) /2 width=1 by lsuby_pair/
+lemma lsuby_pair_lt: âI1,I2,L1,L2,V,m. L1 â[0, â«°m] L2 â 0 < m â
+ L1.â{I1}V â[0, m] L2.â{I2}V.
+#I1 #I2 #L1 #L2 #V #m #HL12 #Hm <(ylt_inv_O1 ⦠Hm) /2 width=1 by lsuby_pair/
qed.
-lemma lsuby_succ_lt: âI1,I2,L1,L2,V1,V2,d,e. L1 â[â«°d, e] L2 â 0 < d â
- L1.â{I1}V1 â[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/
+lemma lsuby_succ_lt: âI1,I2,L1,L2,V1,V2,l,m. L1 â[â«°l, m] L2 â 0 < l â
+ L1.â{I1}V1 â[l, m] L2. â{I2}V2.
+#I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #Hl <(ylt_inv_O1 ⦠Hl) /2 width=1 by lsuby_succ/
qed.
lemma lsuby_pair_O_Y: âL1,L2. L1 â[0, â] L2 â
@@ -49,57 +49,57 @@ lemma lsuby_pair_O_Y: âL1,L2. L1 â[0, â] L2 â
#L1 #L2 #HL12 #I1 #I2 #V lapply (lsuby_pair I1 I2 ⦠V ⦠HL12) -HL12 //
qed.
-lemma lsuby_refl: âL,d,e. L â[d, e] L.
+lemma lsuby_refl: âL,l,m. L â[l, m] 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/
+#L #I #V #IHL #l elim (ynat_cases ⦠l) [| * #x ]
+#Hl destruct /2 width=1 by lsuby_succ/
+#m elim (ynat_cases ⦠m) [| * #x ]
+#Hm destruct /2 width=1 by lsuby_zero, lsuby_pair/
qed.
-lemma lsuby_O2: âL2,L1,d. |L2| ⤠|L1| â L1 â[d, yinj 0] L2.
+lemma lsuby_O2: âL2,L1,l. |L2| ⤠|L1| â L1 â[l, yinj 0] L2.
#L2 elim L2 -L2 // #L2 #I2 #V2 #IHL2 * normalize
-[ #d #H elim (le_plus_xSy_O_false ⦠H)
-| #L1 #I1 #V1 #d #H lapply (le_plus_to_le_r ⦠H) -H #HL12
- elim (ynat_cases d) /3 width=1 by lsuby_zero/
+[ #l #H elim (le_plus_xSy_O_false ⦠H)
+| #L1 #I1 #V1 #l #H lapply (le_plus_to_le_r ⦠H) -H #HL12
+ elim (ynat_cases l) /3 width=1 by lsuby_zero/
* /3 width=1 by lsuby_succ/
]
qed.
-lemma lsuby_sym: âd,e,L1,L2. L1 â[d, e] L2 â |L1| = |L2| â L2 â[d, e] L1.
-#d #e #L1 #L2 #H elim H -d -e -L1 -L2
-[ #L1 #d #e #H >(length_inv_zero_dx ⦠H) -L1 //
+lemma lsuby_sym: âl,m,L1,L2. L1 â[l, m] L2 â |L1| = |L2| â L2 â[l, m] L1.
+#l #m #L1 #L2 #H elim H -l -m -L1 -L2
+[ #L1 #l #m #H >(length_inv_zero_dx ⦠H) -L1 //
| /2 width=1 by lsuby_O2/
-| #I1 #I2 #L1 #L2 #V #e #_ #IHL12 #H lapply (injective_plus_l ⦠H)
+| #I1 #I2 #L1 #L2 #V #m #_ #IHL12 #H lapply (injective_plus_l ⦠H)
/3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL12 #H lapply (injective_plus_l ⦠H)
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #H lapply (injective_plus_l ⦠H)
/3 width=1 by lsuby_succ/
]
qed-.
(* Basic inversion lemmas ***************************************************)
-fact lsuby_inv_atom1_aux: âL1,L2,d,e. L1 â[d, e] L2 â L1 = â â L2 = â.
-#L1 #L2 #d #e * -L1 -L2 -d -e //
+fact lsuby_inv_atom1_aux: âL1,L2,l,m. L1 â[l, m] L2 â L1 = â â L2 = â.
+#L1 #L2 #l #m * -L1 -L2 -l -m //
[ #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
+| #I1 #I2 #L1 #L2 #V #m #_ #H destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #H destruct
]
qed-.
-lemma lsuby_inv_atom1: âL2,d,e. â â[d, e] L2 â L2 = â.
+lemma lsuby_inv_atom1: âL2,l,m. â â[l, m] L2 â L2 = â.
/2 width=5 by lsuby_inv_atom1_aux/ qed-.
-fact lsuby_inv_zero1_aux: âL1,L2,d,e. L1 â[d, e] L2 â
- âJ1,K1,W1. L1 = K1.â{J1}W1 â d = 0 â e = 0 â
+fact lsuby_inv_zero1_aux: âL1,L2,l,m. L1 â[l, m] L2 â
+ âJ1,K1,W1. L1 = K1.â{J1}W1 â l = 0 â m = 0 â
L2 = â â¨
ââJ2,K2,W2. K1 â[0, 0] K2 & L2 = K2.â{J2}W2.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
[ #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J1 #K1 #W1 #H #_ #_ destruct
/3 width=5 by ex2_3_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #_ #H
+| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #_ #H
elim (ysucc_inv_O_dx ⦠H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W1 #_ #H
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W1 #_ #H
elim (ysucc_inv_O_dx ⦠H)
]
qed-.
@@ -109,54 +109,54 @@ lemma lsuby_inv_zero1: âI1,K1,L2,V1. K1.â{I1}V1 â[0, 0] L2 â
ââI2,K2,V2. K1 â[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 â[d, e] L2 â
- âJ1,K1,W. L1 = K1.â{J1}W â d = 0 â 0 < e â
+fact lsuby_inv_pair1_aux: âL1,L2,l,m. L1 â[l, m] L2 â
+ âJ1,K1,W. L1 = K1.â{J1}W â l = 0 â 0 < m â
L2 = â â¨
- ââJ2,K2. K1 â[0, â«°e] K2 & L2 = K2.â{J2}W.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+ ââJ2,K2. K1 â[0, â«°m] K2 & L2 = K2.â{J2}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W #_ #_ #H
elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #J1 #K1 #W #H #_ #_ destruct
+| #I1 #I2 #L1 #L2 #V #m #HL12 #J1 #K1 #W #H #_ #_ destruct
/3 width=4 by ex2_2_intro, or_intror/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J1 #K1 #W #_ #H
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J1 #K1 #W #_ #H
elim (ysucc_inv_O_dx ⦠H)
]
qed-.
-lemma lsuby_inv_pair1: âI1,K1,L2,V,e. K1.â{I1}V â[0, e] L2 â 0 < e â
+lemma lsuby_inv_pair1: âI1,K1,L2,V,m. K1.â{I1}V â[0, m] L2 â 0 < m â
L2 = â â¨
- ââI2,K2. K1 â[0, â«°e] K2 & L2 = K2.â{I2}V.
+ ââI2,K2. K1 â[0, â«°m] 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 â[d, e] L2 â
- âJ1,K1,W1. L1 = K1.â{J1}W1 â 0 < d â
+fact lsuby_inv_succ1_aux: âL1,L2,l,m. L1 â[l, m] L2 â
+ âJ1,K1,W1. L1 = K1.â{J1}W1 â 0 < l â
L2 = â â¨
- ââJ2,K2,W2. K1 â[â«°d, e] K2 & L2 = K2.â{J2}W2.
-#L1 #L2 #d #e * -L1 -L2 -d -e /2 width=1 by or_introl/
+ ââJ2,K2,W2. K1 â[â«°l, m] K2 & L2 = K2.â{J2}W2.
+#L1 #L2 #l #m * -L1 -L2 -l -m /2 width=1 by or_introl/
[ #I1 #I2 #L1 #L2 #V1 #V2 #_ #J1 #K1 #W1 #_ #H
elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V #e #_ #J1 #K1 #W1 #_ #H
+| #I1 #I2 #L1 #L2 #V #m #_ #J1 #K1 #W1 #_ #H
elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J1 #K1 #W1 #H #_ destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J1 #K1 #W1 #H #_ destruct
/3 width=5 by ex2_3_intro, or_intror/
]
qed-.
-lemma lsuby_inv_succ1: âI1,K1,L2,V1,d,e. K1.â{I1}V1 â[d, e] L2 â 0 < d â
+lemma lsuby_inv_succ1: âI1,K1,L2,V1,l,m. K1.â{I1}V1 â[l, m] L2 â 0 < l â
L2 = â â¨
- ââI2,K2,V2. K1 â[â«°d, e] K2 & L2 = K2.â{I2}V2.
+ ââI2,K2,V2. K1 â[â«°l, m] 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 â[d, e] L2 â
- âJ2,K2,W2. L2 = K2.â{J2}W2 â d = 0 â e = 0 â
+fact lsuby_inv_zero2_aux: âL1,L2,l,m. L1 â[l, m] L2 â
+ âJ2,K2,W2. L2 = K2.â{J2}W2 â l = 0 â m = 0 â
ââJ1,K1,W1. K1 â[0, 0] K2 & L1 = K1.â{J1}W1.
-#L1 #L2 #d #e * -L1 -L2 -d -e
-[ #L1 #d #e #J2 #K2 #W1 #H destruct
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W1 #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #HL12 #J2 #K2 #W2 #H #_ #_ destruct
/2 width=5 by ex2_3_intro/
-| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K2 #W2 #_ #_ #H
+| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K2 #W2 #_ #_ #H
elim (ysucc_inv_O_dx ⦠H)
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W2 #_ #H
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W2 #_ #H
elim (ysucc_inv_O_dx ⦠H)
]
qed-.
@@ -165,72 +165,72 @@ lemma lsuby_inv_zero2: âI2,K2,L1,V2. L1 â[0, 0] K2.â{I2}V2 â
ââI1,K1,V1. K1 â[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 â[d, e] L2 â
- âJ2,K2,W. L2 = K2.â{J2}W â d = 0 â 0 < e â
- ââJ1,K1. K1 â[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
+fact lsuby_inv_pair2_aux: âL1,L2,l,m. L1 â[l, m] L2 â
+ âJ2,K2,W. L2 = K2.â{J2}W â l = 0 â 0 < m â
+ ââJ1,K1. K1 â[0, â«°m] K2 & L1 = K1.â{J1}W.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W #_ #_ #H
elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #J2 #K2 #W #H #_ #_ destruct
+| #I1 #I2 #L1 #L2 #V #m #HL12 #J2 #K2 #W #H #_ #_ destruct
/2 width=4 by ex2_2_intro/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #J2 #K2 #W #_ #H
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #J2 #K2 #W #_ #H
elim (ysucc_inv_O_dx ⦠H)
]
qed-.
-lemma lsuby_inv_pair2: âI2,K2,L1,V,e. L1 â[0, e] K2.â{I2}V â 0 < e â
- ââI1,K1. K1 â[0, â«°e] K2 & L1 = K1.â{I1}V.
+lemma lsuby_inv_pair2: âI2,K2,L1,V,m. L1 â[0, m] K2.â{I2}V â 0 < m â
+ ââI1,K1. K1 â[0, â«°m] 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 â[d, e] L2 â
- âJ2,K2,W2. L2 = K2.â{J2}W2 â 0 < d â
- ââJ1,K1,W1. K1 â[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
+fact lsuby_inv_succ2_aux: âL1,L2,l,m. L1 â[l, m] L2 â
+ âJ2,K2,W2. L2 = K2.â{J2}W2 â 0 < l â
+ ââJ1,K1,W1. K1 â[â«°l, m] K2 & L1 = K1.â{J1}W1.
+#L1 #L2 #l #m * -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W2 #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #J2 #K2 #W2 #_ #H
elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V #e #_ #J2 #K1 #W2 #_ #H
+| #I1 #I2 #L1 #L2 #V #m #_ #J2 #K1 #W2 #_ #H
elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #HL12 #J2 #K2 #W2 #H #_ destruct
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #HL12 #J2 #K2 #W2 #H #_ destruct
/2 width=5 by ex2_3_intro/
]
qed-.
-lemma lsuby_inv_succ2: âI2,K2,L1,V2,d,e. L1 â[d, e] K2.â{I2}V2 â 0 < d â
- ââI1,K1,V1. K1 â[â«°d, e] K2 & L1 = K1.â{I1}V1.
+lemma lsuby_inv_succ2: âI2,K2,L1,V2,l,m. L1 â[l, m] K2.â{I2}V2 â 0 < l â
+ ââI1,K1,V1. K1 â[â«°l, m] 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 â[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/
+lemma lsuby_fwd_length: âL1,L2,l,m. L1 â[l, m] L2 â |L2| ⤠|L1|.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m normalize /2 width=1 by le_S_S/
qed-.
(* Properties on basic slicing **********************************************)
-lemma lsuby_drop_trans_be: âL1,L2,d,e. L1 â[d, e] L2 â
+lemma lsuby_drop_trans_be: âL1,L2,l,m. L1 â[l, m] 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 â[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
+ l ⤠i â i < l + m â
+ ââI1,K1. K1 â[0, â«°(l+m-i)] K2 & â¬[s, 0, i] L1 â¡ K1.â{I1}W.
+#L1 #L2 #l #m #H elim H -L1 -L2 -l -m
+[ #L1 #l #m #J2 #K2 #W #s #i #H
elim (drop_inv_atom1 ⦠H) -H #H destruct
| #I1 #I2 #L1 #L2 #V1 #V2 #_ #_ #J2 #K2 #W #s #i #_ #_ #H
elim (ylt_yle_false ⦠H) //
-| #I1 #I2 #L1 #L2 #V #e #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
+| #I1 #I2 #L1 #L2 #V #m #HL12 #IHL12 #J2 #K2 #W #s #i #H #_ >yplus_O1
elim (drop_inv_O1_pair1 ⦠H) -H * #Hi #HLK1 [ -IHL12 | -HL12 ]
[ #_ destruct -I2 >ypred_succ
/2 width=4 by drop_pair, ex2_2_intro/
| lapply (ylt_inv_O1 i ?) /2 width=1 by ylt_inj/
#H <H -H #H lapply (ylt_inv_succ ⦠H) -H
- #Hie elim (IHL12 ⦠HLK1) -IHL12 -HLK1 // -Hie
+ #Him elim (IHL12 ⦠HLK1) -IHL12 -HLK1 // -Him
>yminus_succ <yminus_inj /3 width=4 by drop_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 (drop_inv_drop1_lt ⦠HLK2 ?) -HLK2 /2 width=1 by ylt_O/
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL12 #J2 #K2 #W #s #i #HLK2 #Hli
+ elim (yle_inv_succ1 ⦠Hli) -Hli
+ #Hli #Hi <Hi >yplus_succ1 #H lapply (ylt_inv_succ ⦠H) -H
+ #Hilm lapply (drop_inv_drop1_lt ⦠HLK2 ?) -HLK2 /2 width=1 by ylt_O/
#HLK1 elim (IHL12 ⦠HLK1) -IHL12 -HLK1 <yminus_inj >yminus_SO2
/4 width=4 by ylt_O, drop_drop_lt, ex2_2_intro/
]
diff --git a/matita/matita/contribs/lambdadelta/basic_2/substitution/lsuby_lsuby.ma b/matita/matita/contribs/lambdadelta/basic_2/substitution/lsuby_lsuby.ma
index c3b792c42..ff914eca7 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/substitution/lsuby_lsuby.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/substitution/lsuby_lsuby.ma
@@ -18,15 +18,15 @@ include "basic_2/substitution/lsuby.ma".
(* 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
+theorem lsuby_trans: âl,m. Transitive ⦠(lsuby l m).
+#l #m #L1 #L2 #H elim H -L1 -L2 -l -m
+[ #L1 #l #m #X #H lapply (lsuby_inv_atom1 ⦠H) -H
#H destruct //
| #I1 #I2 #L1 #L #V1 #V #_ #IHL1 #X #H elim (lsuby_inv_zero1 ⦠H) -H //
* #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_zero/
-| #I1 #I2 #L1 #L2 #V #e #_ #IHL1 #X #H elim (lsuby_inv_pair1 ⦠H) -H //
+| #I1 #I2 #L1 #L2 #V #m #_ #IHL1 #X #H elim (lsuby_inv_pair1 ⦠H) -H //
* #I2 #L2 #HL2 #H destruct /3 width=1 by lsuby_pair/
-| #I1 #I2 #L1 #L2 #V1 #V2 #d #e #_ #IHL1 #X #H elim (lsuby_inv_succ1 ⦠H) -H //
+| #I1 #I2 #L1 #L2 #V1 #V2 #l #m #_ #IHL1 #X #H elim (lsuby_inv_succ1 ⦠H) -H //
* #I2 #L2 #V2 #HL2 #H destruct /3 width=1 by lsuby_succ/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas.ma
index 39a972c3f..289992d5f 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas.ma
@@ -21,71 +21,71 @@ include "basic_2/static/sh.ma".
(* activate genv *)
inductive lstas (h): nat â relation4 genv lenv term term â
-| lstas_sort: âG,L,l,k. lstas h l G L (âk) (â((next h)^l k))
-| lstas_ldef: âG,L,K,V,W,U,i,l. â¬[i] L â¡ K.âV â lstas h l G K V W â
- â¬[0, i+1] W â¡ U â lstas h l G L (#i) U
+| lstas_sort: âG,L,d,k. lstas h d G L (âk) (â((next h)^d k))
+| lstas_ldef: âG,L,K,V,W,U,i,d. â¬[i] L â¡ K.âV â lstas h d G K V W â
+ â¬[0, i+1] W â¡ U â lstas h d G L (#i) U
| lstas_zero: âG,L,K,W,V,i. â¬[i] L â¡ K.âW â lstas h 0 G K W V â
lstas h 0 G L (#i) (#i)
-| lstas_succ: âG,L,K,W,V,U,i,l. â¬[i] L â¡ K.âW â lstas h l G K W V â
- â¬[0, i+1] V â¡ U â lstas h (l+1) G L (#i) U
-| lstas_bind: âa,I,G,L,V,T,U,l. lstas h l G (L.â{I}V) T U â
- lstas h l G L (â{a,I}V.T) (â{a,I}V.U)
-| lstas_appl: âG,L,V,T,U,l. lstas h l G L T U â lstas h l G L (âV.T) (âV.U)
-| lstas_cast: âG,L,W,T,U,l. lstas h l G L T U â lstas h l G L (âW.T) U
+| lstas_succ: âG,L,K,W,V,U,i,d. â¬[i] L â¡ K.âW â lstas h d G K W V â
+ â¬[0, i+1] V â¡ U â lstas h (d+1) G L (#i) U
+| lstas_bind: âa,I,G,L,V,T,U,d. lstas h d G (L.â{I}V) T U â
+ lstas h d G L (â{a,I}V.T) (â{a,I}V.U)
+| lstas_appl: âG,L,V,T,U,d. lstas h d G L T U â lstas h d G L (âV.T) (âV.U)
+| lstas_cast: âG,L,W,T,U,d. lstas h d G L T U â lstas h d G L (âW.T) U
.
interpretation "nat-iterated static type assignment (term)"
- 'StaticTypeStar h G L l T U = (lstas h l G L T U).
+ 'StaticTypeStar h G L d T U = (lstas h d G L T U).
(* Basic inversion lemmas ***************************************************)
-fact lstas_inv_sort1_aux: âh,G,L,T,U,l. â¦G, L⦠⢠T â¢*[h, l] U â âk0. T = âk0 â
- U = â((next h)^l k0).
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #k0 #H destruct //
-| #G #L #K #V #W #U #i #l #_ #_ #_ #k0 #H destruct
+fact lstas_inv_sort1_aux: âh,G,L,T,U,d. â¦G, L⦠⢠T â¢*[h, d] U â âk0. T = âk0 â
+ U = â((next h)^d k0).
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #k0 #H destruct //
+| #G #L #K #V #W #U #i #d #_ #_ #_ #k0 #H destruct
| #G #L #K #W #V #i #_ #_ #k0 #H destruct
-| #G #L #K #W #V #U #i #l #_ #_ #_ #k0 #H destruct
-| #a #I #G #L #V #T #U #l #_ #k0 #H destruct
-| #G #L #V #T #U #l #_ #k0 #H destruct
-| #G #L #W #T #U #l #_ #k0 #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #k0 #H destruct
+| #a #I #G #L #V #T #U #d #_ #k0 #H destruct
+| #G #L #V #T #U #d #_ #k0 #H destruct
+| #G #L #W #T #U #d #_ #k0 #H destruct
qed-.
(* Basic_1: was just: sty0_gen_sort *)
-lemma lstas_inv_sort1: âh,G,L,X,k,l. â¦G, L⦠⢠âk â¢*[h, l] X â X = â((next h)^l k).
+lemma lstas_inv_sort1: âh,G,L,X,k,d. â¦G, L⦠⢠âk â¢*[h, d] X â X = â((next h)^d k).
/2 width=5 by lstas_inv_sort1_aux/
qed-.
-fact lstas_inv_lref1_aux: âh,G,L,T,U,l. â¦G, L⦠⢠T â¢*[h, l] U â âj. T = #j â â¨â¨
- (ââK,V,W. â¬[j] L â¡ K.âV & â¦G, K⦠⢠V â¢*[h, l] W &
+fact lstas_inv_lref1_aux: âh,G,L,T,U,d. â¦G, L⦠⢠T â¢*[h, d] U â âj. T = #j â â¨â¨
+ (ââK,V,W. â¬[j] L â¡ K.âV & â¦G, K⦠⢠V â¢*[h, d] W &
â¬[0, j+1] W â¡ U
) |
(ââK,W,V. â¬[j] L â¡ K.âW & â¦G, K⦠⢠W â¢*[h, 0] V &
- U = #j & l = 0
+ U = #j & d = 0
) |
- (ââK,W,V,l0. â¬[j] L â¡ K.âW & â¦G, K⦠⢠W â¢*[h, l0] V &
- â¬[0, j+1] V â¡ U & l = l0+1
+ (ââK,W,V,d0. â¬[j] L â¡ K.âW & â¦G, K⦠⢠W â¢*[h, d0] V &
+ â¬[0, j+1] V â¡ U & d = d0+1
).
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #j #H destruct
-| #G #L #K #V #W #U #i #l #HLK #HVW #HWU #j #H destruct /3 width=6 by or3_intro0, ex3_3_intro/
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #j #H destruct
+| #G #L #K #V #W #U #i #d #HLK #HVW #HWU #j #H destruct /3 width=6 by or3_intro0, ex3_3_intro/
| #G #L #K #W #V #i #HLK #HWV #j #H destruct /3 width=5 by or3_intro1, ex4_3_intro/
-| #G #L #K #W #V #U #i #l #HLK #HWV #HWU #j #H destruct /3 width=8 by or3_intro2, ex4_4_intro/
-| #a #I #G #L #V #T #U #l #_ #j #H destruct
-| #G #L #V #T #U #l #_ #j #H destruct
-| #G #L #W #T #U #l #_ #j #H destruct
+| #G #L #K #W #V #U #i #d #HLK #HWV #HWU #j #H destruct /3 width=8 by or3_intro2, ex4_4_intro/
+| #a #I #G #L #V #T #U #d #_ #j #H destruct
+| #G #L #V #T #U #d #_ #j #H destruct
+| #G #L #W #T #U #d #_ #j #H destruct
]
qed-.
-lemma lstas_inv_lref1: âh,G,L,X,i,l. â¦G, L⦠⢠#i â¢*[h, l] X â â¨â¨
- (ââK,V,W. â¬[i] L â¡ K.âV & â¦G, K⦠⢠V â¢*[h, l] W &
+lemma lstas_inv_lref1: âh,G,L,X,i,d. â¦G, L⦠⢠#i â¢*[h, d] X â â¨â¨
+ (ââK,V,W. â¬[i] L â¡ K.âV & â¦G, K⦠⢠V â¢*[h, d] W &
â¬[0, i+1] W â¡ X
) |
(ââK,W,V. â¬[i] L â¡ K.âW & â¦G, K⦠⢠W â¢*[h, 0] V &
- X = #i & l = 0
+ X = #i & d = 0
) |
- (ââK,W,V,l0. â¬[i] L â¡ K.âW & â¦G, K⦠⢠W â¢*[h, l0] V &
- â¬[0, i+1] V â¡ X & l = l0+1
+ (ââK,W,V,d0. â¬[i] L â¡ K.âW & â¦G, K⦠⢠W â¢*[h, d0] V &
+ â¬[0, i+1] V â¡ X & d = d0+1
).
/2 width=3 by lstas_inv_lref1_aux/
qed-.
@@ -98,89 +98,89 @@ lemma lstas_inv_lref1_O: âh,G,L,X,i. â¦G, L⦠⢠#i â¢*[h, 0] X â
X = #i
).
#h #G #L #X #i #H elim (lstas_inv_lref1 ⦠H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/
-#K #W #V #l #_ #_ #_ <plus_n_Sm #H destruct
+#K #W #V #d #_ #_ #_ <plus_n_Sm #H destruct
qed-.
(* Basic_1: was just: sty0_gen_lref *)
-lemma lstas_inv_lref1_S: âh,G,L,X,i,l. â¦G, L⦠⢠#i â¢*[h, l+1] X â
- (ââK,V,W. â¬[i] L â¡ K.âV & â¦G, K⦠⢠V â¢*[h, l+1] W &
+lemma lstas_inv_lref1_S: âh,G,L,X,i,d. â¦G, L⦠⢠#i â¢*[h, d+1] X â
+ (ââK,V,W. â¬[i] L â¡ K.âV & â¦G, K⦠⢠V â¢*[h, d+1] W &
â¬[0, i+1] W â¡ X
) â¨
- (ââK,W,V. â¬[i] L â¡ K.âW & â¦G, K⦠⢠W â¢*[h, l] V &
+ (ââK,W,V. â¬[i] L â¡ K.âW & â¦G, K⦠⢠W â¢*[h, d] V &
â¬[0, i+1] V â¡ X
).
-#h #G #L #X #i #l #H elim (lstas_inv_lref1 ⦠H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/
+#h #G #L #X #i #d #H elim (lstas_inv_lref1 ⦠H) -H * /3 width=6 by ex3_3_intro, or_introl, or_intror/
#K #W #V #_ #_ #_ <plus_n_Sm #H destruct
qed-.
-fact lstas_inv_gref1_aux: âh,G,L,T,U,l. â¦G, L⦠⢠T â¢*[h, l] U â âp0. T = §p0 â â¥.
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #p0 #H destruct
-| #G #L #K #V #W #U #i #l #_ #_ #_ #p0 #H destruct
+fact lstas_inv_gref1_aux: âh,G,L,T,U,d. â¦G, L⦠⢠T â¢*[h, d] U â âp0. T = §p0 â â¥.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #p0 #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #p0 #H destruct
| #G #L #K #W #V #i #_ #_ #p0 #H destruct
-| #G #L #K #W #V #U #i #l #_ #_ #_ #p0 #H destruct
-| #a #I #G #L #V #T #U #l #_ #p0 #H destruct
-| #G #L #V #T #U #l #_ #p0 #H destruct
-| #G #L #W #T #U #l #_ #p0 #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #p0 #H destruct
+| #a #I #G #L #V #T #U #d #_ #p0 #H destruct
+| #G #L #V #T #U #d #_ #p0 #H destruct
+| #G #L #W #T #U #d #_ #p0 #H destruct
qed-.
-lemma lstas_inv_gref1: âh,G,L,X,p,l. â¦G, L⦠⢠§p â¢*[h, l] X â â¥.
+lemma lstas_inv_gref1: âh,G,L,X,p,d. â¦G, L⦠⢠§p â¢*[h, d] X â â¥.
/2 width=9 by lstas_inv_gref1_aux/
qed-.
-fact lstas_inv_bind1_aux: âh,G,L,T,U,l. â¦G, L⦠⢠T â¢*[h, l] U â âb,J,X,Y. T = â{b,J}Y.X â
- ââZ. â¦G, L.â{J}Y⦠⢠X â¢*[h, l] Z & U = â{b,J}Y.Z.
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #b #J #X #Y #H destruct
-| #G #L #K #V #W #U #i #l #_ #_ #_ #b #J #X #Y #H destruct
+fact lstas_inv_bind1_aux: âh,G,L,T,U,d. â¦G, L⦠⢠T â¢*[h, d] U â âb,J,X,Y. T = â{b,J}Y.X â
+ ââZ. â¦G, L.â{J}Y⦠⢠X â¢*[h, d] Z & U = â{b,J}Y.Z.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #b #J #X #Y #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #b #J #X #Y #H destruct
| #G #L #K #W #V #i #_ #_ #b #J #X #Y #H destruct
-| #G #L #K #W #V #U #i #l #_ #_ #_ #b #J #X #Y #H destruct
-| #a #I #G #L #V #T #U #l #HTU #b #J #X #Y #H destruct /2 width=3 by ex2_intro/
-| #G #L #V #T #U #l #_ #b #J #X #Y #H destruct
-| #G #L #W #T #U #l #_ #b #J #X #Y #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #b #J #X #Y #H destruct
+| #a #I #G #L #V #T #U #d #HTU #b #J #X #Y #H destruct /2 width=3 by ex2_intro/
+| #G #L #V #T #U #d #_ #b #J #X #Y #H destruct
+| #G #L #W #T #U #d #_ #b #J #X #Y #H destruct
]
qed-.
(* Basic_1: was just: sty0_gen_bind *)
-lemma lstas_inv_bind1: âh,a,I,G,L,V,T,X,l. â¦G, L⦠⢠â{a,I}V.T â¢*[h, l] X â
- ââU. â¦G, L.â{I}V⦠⢠T â¢*[h, l] U & X = â{a,I}V.U.
+lemma lstas_inv_bind1: âh,a,I,G,L,V,T,X,d. â¦G, L⦠⢠â{a,I}V.T â¢*[h, d] X â
+ ââU. â¦G, L.â{I}V⦠⢠T â¢*[h, d] U & X = â{a,I}V.U.
/2 width=3 by lstas_inv_bind1_aux/
qed-.
-fact lstas_inv_appl1_aux: âh,G,L,T,U,l. â¦G, L⦠⢠T â¢*[h, l] U â âX,Y. T = âY.X â
- ââZ. â¦G, L⦠⢠X â¢*[h, l] Z & U = âY.Z.
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #X #Y #H destruct
-| #G #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
+fact lstas_inv_appl1_aux: âh,G,L,T,U,d. â¦G, L⦠⢠T â¢*[h, d] U â âX,Y. T = âY.X â
+ ââZ. â¦G, L⦠⢠X â¢*[h, d] Z & U = âY.Z.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #X #Y #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #X #Y #H destruct
| #G #L #K #W #V #i #_ #_ #X #Y #H destruct
-| #G #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
-| #a #I #G #L #V #T #U #l #_ #X #Y #H destruct
-| #G #L #V #T #U #l #HTU #X #Y #H destruct /2 width=3 by ex2_intro/
-| #G #L #W #T #U #l #_ #X #Y #H destruct
+| #G #L #K #W #V #U #i #d #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #d #_ #X #Y #H destruct
+| #G #L #V #T #U #d #HTU #X #Y #H destruct /2 width=3 by ex2_intro/
+| #G #L #W #T #U #d #_ #X #Y #H destruct
]
qed-.
(* Basic_1: was just: sty0_gen_appl *)
-lemma lstas_inv_appl1: âh,G,L,V,T,X,l. â¦G, L⦠⢠âV.T â¢*[h, l] X â
- ââU. â¦G, L⦠⢠T â¢*[h, l] U & X = âV.U.
+lemma lstas_inv_appl1: âh,G,L,V,T,X,d. â¦G, L⦠⢠âV.T â¢*[h, d] X â
+ ââU. â¦G, L⦠⢠T â¢*[h, d] U & X = âV.U.
/2 width=3 by lstas_inv_appl1_aux/
qed-.
-fact lstas_inv_cast1_aux: âh,G,L,T,U,l. â¦G, L⦠⢠T â¢*[h, l] U â âX,Y. T = âY.X â
- â¦G, L⦠⢠X â¢*[h, l] U.
-#h #G #L #T #U #l * -G -L -T -U -l
-[ #G #L #l #k #X #Y #H destruct
-| #G #L #K #V #W #U #i #l #_ #_ #_ #X #Y #H destruct
+fact lstas_inv_cast1_aux: âh,G,L,T,U,d. â¦G, L⦠⢠T â¢*[h, d] U â âX,Y. T = âY.X â
+ â¦G, L⦠⢠X â¢*[h, d] U.
+#h #G #L #T #U #d * -G -L -T -U -d
+[ #G #L #d #k #X #Y #H destruct
+| #G #L #K #V #W #U #i #d #_ #_ #_ #X #Y #H destruct
| #G #L #K #W #V #i #_ #_ #X #Y #H destruct
-| #G #L #K #W #V #U #i #l #_ #_ #_ #X #Y #H destruct
-| #a #I #G #L #V #T #U #l #_ #X #Y #H destruct
-| #G #L #V #T #U #l #_ #X #Y #H destruct
-| #G #L #W #T #U #l #HTU #X #Y #H destruct //
+| #G #L #K #W #V #U #i #d #_ #_ #_ #X #Y #H destruct
+| #a #I #G #L #V #T #U #d #_ #X #Y #H destruct
+| #G #L #V #T #U #d #_ #X #Y #H destruct
+| #G #L #W #T #U #d #HTU #X #Y #H destruct //
]
qed-.
(* Basic_1: was just: sty0_gen_cast *)
-lemma lstas_inv_cast1: âh,G,L,W,T,U,l. â¦G, L⦠⢠âW.T â¢*[h, l] U â â¦G, L⦠⢠T â¢*[h, l] U.
+lemma lstas_inv_cast1: âh,G,L,W,T,U,d. â¦G, L⦠⢠âW.T â¢*[h, d] U â â¦G, L⦠⢠T â¢*[h, d] U.
/2 width=4 by lstas_inv_cast1_aux/
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_aaa.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_aaa.ma
index 434c6e991..405f62ba2 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_aaa.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_aaa.ma
@@ -19,36 +19,36 @@ include "basic_2/unfold/lstas_lstas.ma".
(* Properties on atomic arity assignment for terms **************************)
-lemma aaa_lstas: âh,G,L,T,A. â¦G, L⦠⢠T â A â âl.
- ââU. â¦G, L⦠⢠T â¢*[h, l] U & â¦G, L⦠⢠U â A.
+lemma aaa_lstas: âh,G,L,T,A. â¦G, L⦠⢠T â A â âd.
+ ââU. â¦G, L⦠⢠T â¢*[h, d] U & â¦G, L⦠⢠U â A.
#h #G #L #T #A #H elim H -G -L -T -A
[ /2 width=3 by ex2_intro/
-| * #G #L #K #V #B #i #HLK #HV #IHV #l
- [ elim (IHV l) -IHV #W
+| * #G #L #K #V #B #i #HLK #HV #IHV #d
+ [ elim (IHV d) -IHV #W
elim (lift_total W 0 (i+1))
lapply (drop_fwd_drop2 ⦠HLK)
/3 width=10 by lstas_ldef, aaa_lift, ex2_intro/
- | @(nat_ind_plus ⦠l) -l
+ | @(nat_ind_plus ⦠d) -d
[ elim (IHV 0) -IHV /3 width=7 by lstas_zero, aaa_lref, ex2_intro/
- | #l #_ elim (IHV l) -IHV #W
+ | #d #_ elim (IHV d) -IHV #W
elim (lift_total W 0 (i+1))
lapply (drop_fwd_drop2 ⦠HLK)
/3 width=10 by lstas_succ, aaa_lift, ex2_intro/
]
]
-| #a #G #L #V #T #B #A #HV #_ #_ #IHT #l elim (IHT l) -IHT
+| #a #G #L #V #T #B #A #HV #_ #_ #IHT #d elim (IHT d) -IHT
/3 width=7 by lstas_bind, aaa_abbr, ex2_intro/
-| #a #G #L #V #T #B #A #HV #_ #_ #IHT #l elim (IHT l) -IHT
+| #a #G #L #V #T #B #A #HV #_ #_ #IHT #d elim (IHT d) -IHT
/3 width=6 by lstas_bind, aaa_abst, ex2_intro/
-| #G #L #V #T #B #A #HV #_ #_ #IHT #l elim (IHT l) -IHT
+| #G #L #V #T #B #A #HV #_ #_ #IHT #d elim (IHT d) -IHT
/3 width=6 by lstas_appl, aaa_appl, ex2_intro/
-| #G #L #W #T #A #HW #_ #_ #IHT #l elim (IHT l) -IHT
+| #G #L #W #T #A #HW #_ #_ #IHT #d elim (IHT d) -IHT
/3 width=3 by lstas_cast, aaa_cast, ex2_intro/
]
qed-.
-lemma lstas_aaa_conf: âh,G,L,l. Conf3 ⦠(aaa G L) (lstas h l G L).
-#h #G #L #l #A #T #HT #U #HTU
-elim (aaa_lstas h ⦠HT l) -HT #X #HTX
+lemma lstas_aaa_conf: âh,G,L,d. Conf3 ⦠(aaa G L) (lstas h d G L).
+#h #G #L #d #A #T #HT #U #HTU
+elim (aaa_lstas h ⦠HT d) -HT #X #HTX
lapply (lstas_mono ⦠HTX ⦠HTU) -T //
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_da.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_da.ma
index 8d1feca9f..6ecadc335 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_da.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_da.ma
@@ -19,77 +19,77 @@ include "basic_2/unfold/lstas_lstas.ma".
(* Properties on degree assignment for terms ********************************)
-lemma da_lstas: âh,g,G,L,T,l1. â¦G, L⦠⢠T âª[h, g] l1 â âl2.
- ââU. â¦G, L⦠⢠T â¢*[h, l2] U & â¦G, L⦠⢠U âª[h, g] l1-l2.
-#h #g #G #L #T #l1 #H elim H -G -L -T -l1
+lemma da_lstas: âh,g,G,L,T,d1. â¦G, L⦠⢠T âª[h, g] d1 â âd2.
+ ââU. â¦G, L⦠⢠T â¢*[h, d2] U & â¦G, L⦠⢠U âª[h, g] d1-d2.
+#h #g #G #L #T #d1 #H elim H -G -L -T -d1
[ /4 width=3 by da_sort, deg_iter, ex2_intro/
-| #G #L #K #V #i #l1 #HLK #_ #IHV #l2
- elim (IHV l2) -IHV #W
+| #G #L #K #V #i #d1 #HLK #_ #IHV #d2
+ elim (IHV d2) -IHV #W
elim (lift_total W 0 (i+1))
lapply (drop_fwd_drop2 ⦠HLK)
/3 width=10 by lstas_ldef, da_lift, ex2_intro/
-| #G #L #K #W #i #l1 #HLK #HW #IHW #l2 @(nat_ind_plus ⦠l2) -l2
+| #G #L #K #W #i #d1 #HLK #HW #IHW #d2 @(nat_ind_plus ⦠d2) -d2
[ elim (IHW 0) -IHW /3 width=6 by lstas_zero, da_ldec, ex2_intro/
- | #l #_ elim (IHW l) -IHW #V
+ | #d #_ elim (IHW d) -IHW #V
elim (lift_total V 0 (i+1))
lapply (drop_fwd_drop2 ⦠HLK)
/3 width=10 by lstas_succ, da_lift, ex2_intro/
]
-| #a #I #G #L #V #T #l1 #_ #IHT #l2 elim (IHT ⦠l2) -IHT
+| #a #I #G #L #V #T #d1 #_ #IHT #d2 elim (IHT ⦠d2) -IHT
/3 width=6 by lstas_bind, da_bind, ex2_intro/
-| * #G #L #V #T #l1 #_ #IHT #l2 elim (IHT ⦠l2) -IHT
+| * #G #L #V #T #d1 #_ #IHT #d2 elim (IHT ⦠d2) -IHT
/3 width=5 by lstas_appl, lstas_cast, da_flat, ex2_intro/
]
qed-.
-lemma lstas_da_conf: âh,g,G,L,T,U,l2. â¦G, L⦠⢠T â¢*[h, l2] U â
- âl1. â¦G, L⦠⢠T âª[h, g] l1 â â¦G, L⦠⢠U âª[h, g] l1-l2.
-#h #g #G #L #T #U #l2 #HTU #l1 #HT
-elim (da_lstas ⦠HT l2) -HT #X #HTX
+lemma lstas_da_conf: âh,g,G,L,T,U,d2. â¦G, L⦠⢠T â¢*[h, d2] U â
+ âd1. â¦G, L⦠⢠T âª[h, g] d1 â â¦G, L⦠⢠U âª[h, g] d1-d2.
+#h #g #G #L #T #U #d2 #HTU #d1 #HT
+elim (da_lstas ⦠HT d2) -HT #X #HTX
lapply (lstas_mono ⦠HTX ⦠HTU) -T //
qed-.
(* inversion lemmas on degree assignment for terms **************************)
-lemma lstas_inv_da: âh,g,G,L,T,U,l2. â¦G, L⦠⢠T â¢*[h, l2] U â
- ââl1. â¦G, L⦠⢠T âª[h, g] l1 & â¦G, L⦠⢠U âª[h, g] l1-l2.
-#h #g #G #L #T #U #l2 #H elim H -G -L -T -U -l2
-[ #G #L #l2 #k elim (deg_total h g k) /4 width=3 by da_sort, deg_iter, ex2_intro/
-| #G #L #K #V #W #U #i #l2 #HLK #_ #HWU *
+lemma lstas_inv_da: âh,g,G,L,T,U,d2. â¦G, L⦠⢠T â¢*[h, d2] U â
+ ââd1. â¦G, L⦠⢠T âª[h, g] d1 & â¦G, L⦠⢠U âª[h, g] d1-d2.
+#h #g #G #L #T #U #d2 #H elim H -G -L -T -U -d2
+[ #G #L #d2 #k elim (deg_total h g k) /4 width=3 by da_sort, deg_iter, ex2_intro/
+| #G #L #K #V #W #U #i #d2 #HLK #_ #HWU *
lapply (drop_fwd_drop2 ⦠HLK) /3 width=10 by da_lift, da_ldef, ex2_intro/
| #G #L #K #W #V #i #HLK #_ * /3 width=6 by da_ldec, ex2_intro/
-| #G #L #K #W #V #U #i #l2 #HLK #_ #HVU *
+| #G #L #K #W #V #U #i #d2 #HLK #_ #HVU *
lapply (drop_fwd_drop2 ⦠HLK) /3 width=10 by da_lift, da_ldec, ex2_intro/
-| #a #I #G #L #V #T #U #l2 #_ * /3 width=3 by da_bind, ex2_intro/
-| #G #L #V #T #U #l2 #_ * /3 width=3 by da_flat, ex2_intro/
-| #G #L #W #T #U #l2 #_ * /3 width=3 by da_flat, ex2_intro/
+| #a #I #G #L #V #T #U #d2 #_ * /3 width=3 by da_bind, ex2_intro/
+| #G #L #V #T #U #d2 #_ * /3 width=3 by da_flat, ex2_intro/
+| #G #L #W #T #U #d2 #_ * /3 width=3 by da_flat, ex2_intro/
]
qed-.
-lemma lstas_inv_da_ge: âh,G,L,T,U,l2,l. â¦G, L⦠⢠T â¢*[h, l2] U â
- ââg,l1. â¦G, L⦠⢠T âª[h, g] l1 & â¦G, L⦠⢠U âª[h, g] l1-l2 & l ⤠l1.
-#h #G #L #T #U #l2 #l #H elim H -G -L -T -U -l2
+lemma lstas_inv_da_ge: âh,G,L,T,U,d2,d. â¦G, L⦠⢠T â¢*[h, d2] U â
+ ââg,d1. â¦G, L⦠⢠T âª[h, g] d1 & â¦G, L⦠⢠U âª[h, g] d1-d2 & d ⤠d1.
+#h #G #L #T #U #d2 #d #H elim H -G -L -T -U -d2
[ /4 width=5 by da_sort, deg_iter, ex3_2_intro/
-| #G #L #K #V #W #U #i #l2 #HLK #_ #HWU *
+| #G #L #K #V #W #U #i #d2 #HLK #_ #HWU *
lapply (drop_fwd_drop2 ⦠HLK) /3 width=10 by da_lift, da_ldef, ex3_2_intro/
| #G #L #K #W #V #i #HLK #_ *
- #g #l1 #HW #HV #Hl1 /4 width=6 by da_ldec, lt_to_le, le_S_S, ex3_2_intro/
-| #G #L #K #W #V #U #i #l2 #HLK #_ #HVU *
+ #g #d1 #HW #HV #Hd1 /4 width=6 by da_ldec, lt_to_le, le_S_S, ex3_2_intro/
+| #G #L #K #W #V #U #i #d2 #HLK #_ #HVU *
lapply (drop_fwd_drop2 ⦠HLK)
/4 width=10 by da_lift, da_ldec, lt_to_le, le_S_S, ex3_2_intro/
-| #a #I #G #L #V #T #U #l2 #_ * /3 width=5 by da_bind, ex3_2_intro/
-| #G #L #V #T #U #l2 #_ * /3 width=5 by da_flat, ex3_2_intro/
-| #G #L #W #T #U #l2 #_ * /3 width=5 by da_flat, ex3_2_intro/
+| #a #I #G #L #V #T #U #d2 #_ * /3 width=5 by da_bind, ex3_2_intro/
+| #G #L #V #T #U #d2 #_ * /3 width=5 by da_flat, ex3_2_intro/
+| #G #L #W #T #U #d2 #_ * /3 width=5 by da_flat, ex3_2_intro/
]
qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lstas_inv_refl_pos: âh,G,L,T,l. â¦G, L⦠⢠T â¢*[h, l+1] T â â¥.
-#h #G #L #T #l2 #H elim (lstas_inv_da_ge ⦠(l2+1) H) -H
-#g #l1 #HT1 #HT12 #Hl21 lapply (da_mono ⦠HT1 ⦠HT12) -h -G -L -T
+lemma lstas_inv_refl_pos: âh,G,L,T,d. â¦G, L⦠⢠T â¢*[h, d+1] T â â¥.
+#h #G #L #T #d2 #H elim (lstas_inv_da_ge ⦠(d2+1) H) -H
+#g #d1 #HT1 #HT12 #Hd21 lapply (da_mono ⦠HT1 ⦠HT12) -h -G -L -T
#H elim (discr_x_minus_xy ⦠H) -H
[ #H destruct /2 width=3 by le_plus_xSy_O_false/
-| -l1 <plus_n_Sm #H destruct
+| -d1 <plus_n_Sm #H destruct
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lift.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lift.ma
index aea1d9f85..770d96725 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lift.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lift.ma
@@ -20,48 +20,48 @@ include "basic_2/unfold/lstas.ma".
(* Properties on relocation *************************************************)
(* Basic_1: was just: sty0_lift *)
-lemma lstas_lift: âh,G,l. l_liftable (lstas h G l).
-#h #G #l #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1 -l
-[ #G #L1 #l #k #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
+lemma lstas_lift: âh,G,d. d_liftable (lstas h G d).
+#h #G #d #L1 #T1 #U1 #H elim H -G -L1 -T1 -U1 -d
+[ #G #L1 #d #k #L2 #s #l #m #HL21 #X1 #H1 #X2 #H2
>(lift_inv_sort1 ⦠H1) -X1
>(lift_inv_sort1 ⦠H2) -X2 //
-| #G #L1 #K1 #V1 #W1 #W #i #l #HLK1 #_ #HW1 #IHVW1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
- elim (lift_inv_lref1 ⦠H) * #Hid #H destruct
+| #G #L1 #K1 #V1 #W1 #W #i #d #HLK1 #_ #HW1 #IHVW1 #L2 #s #l #m #HL21 #X #H #U2 #HWU2
+ elim (lift_inv_lref1 ⦠H) * #Hil #H destruct
[ elim (lift_trans_ge ⦠HW1 ⦠HWU2) -W // #W2 #HW12 #HWU2
elim (drop_trans_le ⦠HL21 ⦠HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
/3 width=9 by lstas_ldef/
| lapply (lift_trans_be ⦠HW1 ⦠HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
lapply (drop_trans_ge ⦠HL21 ⦠HLK1 ?) -L1 /3 width=9 by lstas_ldef, drop_inv_gen/
]
-| #G #L1 #K1 #V1 #W1 #i #HLK1 #_ #IHVW1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
+| #G #L1 #K1 #V1 #W1 #i #HLK1 #_ #IHVW1 #L2 #s #l #m #HL21 #X #H #U2 #HWU2
>(lift_mono ⦠HWU2 ⦠H) -U2
- elim (lift_inv_lref1 ⦠H) * #Hid #H destruct
- [ elim (lift_total W1 (d-i-1) e) #W2 #HW12
+ elim (lift_inv_lref1 ⦠H) * #Hil #H destruct
+ [ elim (lift_total W1 (l-i-1) m) #W2 #HW12
elim (drop_trans_le ⦠HL21 ⦠HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #V2 #HK21 #HV12 #H destruct
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #V2 #HK21 #HV12 #H destruct
/3 width=10 by lstas_zero/
| lapply (drop_trans_ge ⦠HL21 ⦠HLK1 ?) -L1
/3 width=10 by lstas_zero, drop_inv_gen/
]
-| #G #L1 #K1 #W1 #V1 #W #i #l #HLK1 #_ #HW1 #IHWV1 #L2 #s #d #e #HL21 #X #H #U2 #HWU2
- elim (lift_inv_lref1 ⦠H) * #Hid #H destruct
+| #G #L1 #K1 #W1 #V1 #W #i #d #HLK1 #_ #HW1 #IHWV1 #L2 #s #l #m #HL21 #X #H #U2 #HWU2
+ elim (lift_inv_lref1 ⦠H) * #Hil #H destruct
[ elim (lift_trans_ge ⦠HW1 ⦠HWU2) -W // <minus_plus #W #HW1 #HWU2
elim (drop_trans_le ⦠HL21 ⦠HLK1) -L1 /2 width=2 by lt_to_le/ #X #HLK2 #H
- elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hid #K2 #W2 #HK21 #HW12 #H destruct
+ elim (drop_inv_skip2 ⦠H) -H /2 width=1 by lt_plus_to_minus_r/ -Hil #K2 #W2 #HK21 #HW12 #H destruct
/3 width=9 by lstas_succ/
| lapply (lift_trans_be ⦠HW1 ⦠HWU2 ? ?) -W /2 width=1 by le_S/ #HW1U2
lapply (drop_trans_ge ⦠HL21 ⦠HLK1 ?) -L1 /3 width=9 by lstas_succ, drop_inv_gen/
]
-| #a #I #G #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
+| #a #I #G #L1 #V1 #T1 #U1 #d #_ #IHTU1 #L2 #s #l #m #HL21 #X1 #H1 #X2 #H2
elim (lift_inv_bind1 ⦠H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
elim (lift_inv_bind1 ⦠H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
lapply (lift_mono ⦠H1 ⦠HV12) -H1 #H destruct /4 width=6 by lstas_bind, drop_skip/
-| #G #L1 #V1 #T1 #U1 #l #_ #IHTU1 #L2 #s #d #e #HL21 #X1 #H1 #X2 #H2
+| #G #L1 #V1 #T1 #U1 #d #_ #IHTU1 #L2 #s #l #m #HL21 #X1 #H1 #X2 #H2
elim (lift_inv_flat1 ⦠H1) -H1 #V2 #T2 #HV12 #HT12 #H destruct
elim (lift_inv_flat1 ⦠H2) -H2 #X #U2 #H1 #HU12 #H2 destruct
lapply (lift_mono ⦠H1 ⦠HV12) -H1 #H destruct /4 width=6 by lstas_appl/
-| #G #L1 #W1 #T1 #U1 #l #_ #IHTU1 #L2 #s #d #e #HL21 #X #H #U2 #HU12
+| #G #L1 #W1 #T1 #U1 #d #_ #IHTU1 #L2 #s #l #m #HL21 #X #H #U2 #HU12
elim (lift_inv_flat1 ⦠H) -H #W2 #T2 #_ #HT12 #H destruct /3 width=6 by lstas_cast/
]
qed.
@@ -69,45 +69,45 @@ qed.
(* Inversion lemmas on relocation *******************************************)
(* Note: apparently this was missing in basic_1 *)
-lemma lstas_inv_lift1: âh,G,l. l_deliftable_sn (lstas h G l).
-#h #G #l #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2 -l
-[ #G #L2 #l #k #L1 #s #d #e #_ #X #H
+lemma lstas_inv_lift1: âh,G,d. d_deliftable_sn (lstas h G d).
+#h #G #d #L2 #T2 #U2 #H elim H -G -L2 -T2 -U2 -d
+[ #G #L2 #d #k #L1 #s #l #m #_ #X #H
>(lift_inv_sort2 ⦠H) -X /2 width=3 by lstas_sort, lift_sort, ex2_intro/
-| #G #L2 #K2 #V2 #W2 #W #i #l #HLK2 #HVW2 #HW2 #IHVW2 #L1 #s #d #e #HL21 #X #H
- elim (lift_inv_lref2 ⦠H) * #Hid #H destruct [ -HVW2 | -IHVW2 ]
+| #G #L2 #K2 #V2 #W2 #W #i #d #HLK2 #HVW2 #HW2 #IHVW2 #L1 #s #l #m #HL21 #X #H
+ elim (lift_inv_lref2 ⦠H) * #Hil #H destruct [ -HVW2 | -IHVW2 ]
[ elim (drop_conf_lt ⦠HL21 ⦠HLK2) -L2 // #K1 #V1 #HLK1 #HK21 #HV12
elim (IHVW2 ⦠HK21 ⦠HV12) -K2 -V2 #W1 #HW12 #HVW1
elim (lift_trans_le ⦠HW12 ⦠HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by lstas_ldef, ex2_intro/
| lapply (drop_conf_ge ⦠HL21 ⦠HLK2 ?) -L2 // #HL1K2
- elim (le_inv_plus_l ⦠Hid) -Hid #Hdie #ei
- elim (lift_split ⦠HW2 d (i-e+1)) -HW2 /2 width=1 by le_S_S, le_S/
+ elim (le_inv_plus_l ⦠Hil) -Hil #Hlim #mi
+ elim (lift_split ⦠HW2 l (i-m+1)) -HW2 /2 width=1 by le_S_S, le_S/
#W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by lstas_ldef, le_S, ex2_intro/
]
-| #G #L2 #K2 #W2 #V2 #i #HLK2 #HWV2 #IHWV2 #L1 #s #d #e #HL21 #X #H
- elim (lift_inv_lref2 ⦠H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
+| #G #L2 #K2 #W2 #V2 #i #HLK2 #HWV2 #IHWV2 #L1 #s #l #m #HL21 #X #H
+ elim (lift_inv_lref2 ⦠H) * #Hil #H destruct [ -HWV2 | -IHWV2 ]
[ elim (drop_conf_lt ⦠HL21 ⦠HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
elim (IHWV2 ⦠HK21 ⦠HW12) -K2
/3 width=5 by lstas_zero, lift_lref_lt, ex2_intro/
| lapply (drop_conf_ge ⦠HL21 ⦠HLK2 ?) -L2
/3 width=5 by lstas_zero, lift_lref_ge_minus, ex2_intro/
]
-| #G #L2 #K2 #W2 #V2 #W #i #l #HLK2 #HWV2 #HW2 #IHWV2 #L1 #s #d #e #HL21 #X #H
- elim (lift_inv_lref2 ⦠H) * #Hid #H destruct [ -HWV2 | -IHWV2 ]
+| #G #L2 #K2 #W2 #V2 #W #i #d #HLK2 #HWV2 #HW2 #IHWV2 #L1 #s #l #m #HL21 #X #H
+ elim (lift_inv_lref2 ⦠H) * #Hil #H destruct [ -HWV2 | -IHWV2 ]
[ elim (drop_conf_lt ⦠HL21 ⦠HLK2) -L2 // #K1 #W1 #HLK1 #HK21 #HW12
elim (IHWV2 ⦠HK21 ⦠HW12) -K2 #V1 #HV12 #HWV1
elim (lift_trans_le ⦠HV12 ⦠HW2) -W2 // >minus_plus <plus_minus_m_m /3 width=8 by lstas_succ, ex2_intro/
| lapply (drop_conf_ge ⦠HL21 ⦠HLK2 ?) -L2 // #HL1K2
- elim (le_inv_plus_l ⦠Hid) -Hid #Hdie #ei
- elim (lift_split ⦠HW2 d (i-e+1)) -HW2 /2 width=1 by le_S_S, le_S/
+ elim (le_inv_plus_l ⦠Hil) -Hil #Hlim #mi
+ elim (lift_split ⦠HW2 l (i-m+1)) -HW2 /2 width=1 by le_S_S, le_S/
#W0 #HW20 <le_plus_minus_comm // >minus_minus_m_m /3 width=8 by lstas_succ, le_S, ex2_intro/
]
-| #a #I #G #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
+| #a #I #G #L2 #V2 #T2 #U2 #d #_ #IHTU2 #L1 #s #l #m #HL21 #X #H
elim (lift_inv_bind2 ⦠H) -H #V1 #T1 #HV12 #HT12 #H destruct
elim (IHTU2 (L1.â{I}V1) ⦠HT12) -IHTU2 -HT12 /3 width=5 by lstas_bind, drop_skip, lift_bind, ex2_intro/
-| #G #L2 #V2 #T2 #U2 #l #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
+| #G #L2 #V2 #T2 #U2 #d #_ #IHTU2 #L1 #s #l #m #HL21 #X #H
elim (lift_inv_flat2 ⦠H) -H #V1 #T1 #HV12 #HT12 #H destruct
elim (IHTU2 ⦠HL21 ⦠HT12) -L2 -HT12 /3 width=5 by lstas_appl, lift_flat, ex2_intro/
-| #G #L2 #W2 #T2 #U2 #l #_ #IHTU2 #L1 #s #d #e #HL21 #X #H
+| #G #L2 #W2 #T2 #U2 #d #_ #IHTU2 #L1 #s #l #m #HL21 #X #H
elim (lift_inv_flat2 ⦠H) -H #W1 #T1 #_ #HT12 #H destruct
elim (IHTU2 ⦠HL21 ⦠HT12) -L2 -HT12 /3 width=3 by lstas_cast, ex2_intro/
]
@@ -115,69 +115,69 @@ qed-.
(* Advanced inversion lemmas ************************************************)
-lemma lstas_split_aux: âh,G,L,T1,T2,l. â¦G, L⦠⢠T1 â¢*[h, l] T2 â âl1,l2. l = l1 + l2 â
- ââT. â¦G, L⦠⢠T1 â¢*[h, l1] T & â¦G, L⦠⢠T â¢*[h, l2] T2.
-#h #G #L #T1 #T2 #l #H elim H -G -L -T1 -T2 -l
-[ #G #L #l #k #l1 #l2 #H destruct
+lemma lstas_split_aux: âh,G,L,T1,T2,d. â¦G, L⦠⢠T1 â¢*[h, d] T2 â âd1,d2. d = d1 + d2 â
+ ââT. â¦G, L⦠⢠T1 â¢*[h, d1] T & â¦G, L⦠⢠T â¢*[h, d2] T2.
+#h #G #L #T1 #T2 #d #H elim H -G -L -T1 -T2 -d
+[ #G #L #d #k #d1 #d2 #H destruct
>commutative_plus >iter_plus /2 width=3 by lstas_sort, ex2_intro/
-| #G #L #K #V1 #V2 #U2 #i #l #HLK #_ #VU2 #IHV12 #l1 #l2 #H destruct
- elim (IHV12 l1 l2) -IHV12 // #V
+| #G #L #K #V1 #V2 #U2 #i #d #HLK #_ #VU2 #IHV12 #d1 #d2 #H destruct
+ elim (IHV12 d1 d2) -IHV12 // #V
elim (lift_total V 0 (i+1))
lapply (drop_fwd_drop2 ⦠HLK)
/3 width=12 by lstas_lift, lstas_ldef, ex2_intro/
-| #G #L #K #W1 #W2 #i #HLK #HW12 #_ #l1 #l2 #H
+| #G #L #K #W1 #W2 #i #HLK #HW12 #_ #d1 #d2 #H
elim (zero_eq_plus ⦠H) -H #H1 #H2 destruct
/3 width=5 by lstas_zero, ex2_intro/
-| #G #L #K #W1 #W2 #U2 #i #l #HLK #HW12 #HWU2 #IHW12 #l1 @(nat_ind_plus ⦠l1) -l1
- [ #l2 normalize #H destruct
- elim (IHW12 0 l) -IHW12 //
+| #G #L #K #W1 #W2 #U2 #i #d #HLK #HW12 #HWU2 #IHW12 #d1 @(nat_ind_plus ⦠d1) -d1
+ [ #d2 normalize #H destruct
+ elim (IHW12 0 d) -IHW12 //
lapply (drop_fwd_drop2 ⦠HLK)
/3 width=8 by lstas_succ, lstas_zero, ex2_intro/
- | #l1 #_ #l2 <plus_plus_comm_23 #H lapply (injective_plus_l ⦠H) -H #H
- elim (IHW12 ⦠H) -l #W
+ | #d1 #_ #d2 <plus_plus_comm_23 #H lapply (injective_plus_l ⦠H) -H #H
+ elim (IHW12 ⦠H) -d #W
elim (lift_total W 0 (i+1))
lapply (drop_fwd_drop2 ⦠HLK)
/3 width=12 by lstas_lift, lstas_succ, ex2_intro/
]
-| #a #I #G #L #V #T #U #l #_ #IHTU #l1 #l2 #H
- elim (IHTU ⦠H) -l /3 width=3 by lstas_bind, ex2_intro/
-| #G #L #V #T #U #l #_ #IHTU #l1 #l2 #H
- elim (IHTU ⦠H) -l /3 width=3 by lstas_appl, ex2_intro/
-| #G #L #W #T #U #l #_ #IHTU #l1 #l2 #H
- elim (IHTU ⦠H) -l /3 width=3 by lstas_cast, ex2_intro/
+| #a #I #G #L #V #T #U #d #_ #IHTU #d1 #d2 #H
+ elim (IHTU ⦠H) -d /3 width=3 by lstas_bind, ex2_intro/
+| #G #L #V #T #U #d #_ #IHTU #d1 #d2 #H
+ elim (IHTU ⦠H) -d /3 width=3 by lstas_appl, ex2_intro/
+| #G #L #W #T #U #d #_ #IHTU #d1 #d2 #H
+ elim (IHTU ⦠H) -d /3 width=3 by lstas_cast, ex2_intro/
]
qed-.
-lemma lstas_split: âh,G,L,T1,T2,l1,l2. â¦G, L⦠⢠T1 â¢*[h, l1 + l2] T2 â
- ââT. â¦G, L⦠⢠T1 â¢*[h, l1] T & â¦G, L⦠⢠T â¢*[h, l2] T2.
+lemma lstas_split: âh,G,L,T1,T2,d1,d2. â¦G, L⦠⢠T1 â¢*[h, d1 + d2] T2 â
+ ââT. â¦G, L⦠⢠T1 â¢*[h, d1] T & â¦G, L⦠⢠T â¢*[h, d2] T2.
/2 width=3 by lstas_split_aux/ qed-.
(* Advanced properties ******************************************************)
-lemma lstas_lstas: âh,G,L,T,T1,l1. â¦G, L⦠⢠T â¢*[h, l1] T1 â
- âl2. âT2. â¦G, L⦠⢠T â¢*[h, l2] T2.
-#h #G #L #T #T1 #l1 #H elim H -G -L -T -T1 -l1
+lemma lstas_lstas: âh,G,L,T,T1,d1. â¦G, L⦠⢠T â¢*[h, d1] T1 â
+ âd2. âT2. â¦G, L⦠⢠T â¢*[h, d2] T2.
+#h #G #L #T #T1 #d1 #H elim H -G -L -T -T1 -d1
[ /2 width=2 by lstas_sort, ex_intro/
-| #G #L #K #V #V1 #U1 #i #l1 #HLK #_ #HVU1 #IHV1 #l2
- elim (IHV1 l2) -IHV1 #V2
+| #G #L #K #V #V1 #U1 #i #d1 #HLK #_ #HVU1 #IHV1 #d2
+ elim (IHV1 d2) -IHV1 #V2
elim (lift_total V2 0 (i+1))
/3 width=7 by ex_intro, lstas_ldef/
-| #G #L #K #W #W1 #i #HLK #HW1 #IHW1 #l2
- @(nat_ind_plus ⦠l2) -l2 /3 width=5 by lstas_zero, ex_intro/
- #l2 #_ elim (IHW1 l2) -IHW1 #W2
+| #G #L #K #W #W1 #i #HLK #HW1 #IHW1 #d2
+ @(nat_ind_plus ⦠d2) -d2 /3 width=5 by lstas_zero, ex_intro/
+ #d2 #_ elim (IHW1 d2) -IHW1 #W2
elim (lift_total W2 0 (i+1))
/3 width=7 by lstas_succ, ex_intro/
-| #G #L #K #W #W1 #U1 #i #l #HLK #_ #_ #IHW1 #l2
- @(nat_ind_plus ⦠l2) -l2
+| #G #L #K #W #W1 #U1 #i #d #HLK #_ #_ #IHW1 #d2
+ @(nat_ind_plus ⦠d2) -d2
[ elim (IHW1 0) -IHW1 /3 width=5 by lstas_zero, ex_intro/
- | #l2 #_ elim (IHW1 l2) -IHW1
+ | #d2 #_ elim (IHW1 d2) -IHW1
#W2 elim (lift_total W2 0 (i+1)) /3 width=7 by ex_intro, lstas_succ/
]
-| #a #I #G #L #V #T #U #l #_ #IHTU #l2
- elim (IHTU l2) -IHTU /3 width=2 by lstas_bind, ex_intro/
-| #G #L #V #T #U #l #_ #IHTU #l2
- elim (IHTU l2) -IHTU /3 width=2 by lstas_appl, ex_intro/
-| #G #L #W #T #U #l #_ #IHTU #l2
- elim (IHTU l2) -IHTU /3 width=2 by lstas_cast, ex_intro/
+| #a #I #G #L #V #T #U #d #_ #IHTU #d2
+ elim (IHTU d2) -IHTU /3 width=2 by lstas_bind, ex_intro/
+| #G #L #V #T #U #d #_ #IHTU #d2
+ elim (IHTU d2) -IHTU /3 width=2 by lstas_appl, ex_intro/
+| #G #L #W #T #U #d #_ #IHTU #d2
+ elim (IHTU d2) -IHTU /3 width=2 by lstas_cast, ex_intro/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_llpx_sn.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_llpx_sn.ma
index afa9d46d3..db9cada18 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_llpx_sn.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_llpx_sn.ma
@@ -19,26 +19,26 @@ include "basic_2/unfold/lstas.ma".
(* Properties on lazy sn pointwise extensions *******************************)
-lemma lstas_llpx_sn_conf: âR. (âL. reflexive ⦠(R L)) â l_liftable R â
- âh,G,l. s_r_confluent1 ⦠(lstas h l G) (llpx_sn R 0).
-#R #H1R #H2R #h #G #l #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2 -l
+lemma lstas_llpx_sn_conf: âR. (âL. reflexive ⦠(R L)) â d_liftable R â
+ âh,G,d. s_r_confluent1 ⦠(lstas h d G) (llpx_sn R 0).
+#R #H1R #H2R #h #G #d #Ls #T1 #T2 #H elim H -G -Ls -T1 -T2 -d
[ /3 width=4 by llpx_sn_fwd_length, llpx_sn_sort/
-| #G #Ls #Ks #V1s #V2s #W2s #i #l #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn ⦠H ⦠HLKs) // -H
- #Kd #V1d #HLKd #HV1s #HV1sd
+| #G #Ls #Ks #V1s #V2s #W2s #i #d #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn ⦠H ⦠HLKs) // -H
+ #Kd #V1l #HLKd #HV1s #HV1sd
lapply (drop_fwd_drop2 ⦠HLKs) -HLKs #HLKs
lapply (drop_fwd_drop2 ⦠HLKd) -HLKd #HLKd
@(llpx_sn_lift_le ⦠HLKs HLKd ⦠HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
| //
-| #G #Ls #Ks #V1s #V2s #W2s #i #l #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn ⦠H ⦠HLKs) // -H
- #Kd #V1d #HLKd #HV1s #HV1sd
+| #G #Ls #Ks #V1s #V2s #W2s #i #d #HLKs #_ #HVW2s #IHV12s #Ld #H elim (llpx_sn_inv_lref_ge_sn ⦠H ⦠HLKs) // -H
+ #Kd #V1l #HLKd #HV1s #HV1sd
lapply (drop_fwd_drop2 ⦠HLKs) -HLKs #HLKs
lapply (drop_fwd_drop2 ⦠HLKd) -HLKd #HLKd
@(llpx_sn_lift_le ⦠HLKs HLKd ⦠HVW2s) -HLKs -HLKd -HVW2s /2 width=1 by/ (**) (* full auto too slow *)
-| #a #I #G #Ls #V #T1 #T2 #l #_ #IHT12 #Ld #H elim (llpx_sn_inv_bind_O ⦠H) -H
+| #a #I #G #Ls #V #T1 #T2 #d #_ #IHT12 #Ld #H elim (llpx_sn_inv_bind_O ⦠H) -H
/4 width=5 by llpx_sn_bind_repl_SO, llpx_sn_bind/
-| #G #Ls #V #T1 #T2 #l #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat ⦠H) -H
+| #G #Ls #V #T1 #T2 #d #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat ⦠H) -H
/3 width=1 by llpx_sn_flat/
-| #G #Ls #V #T1 #T2 #l #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat ⦠H) -H
+| #G #Ls #V #T1 #T2 #d #_ #IHT12 #Ld #H elim (llpx_sn_inv_flat ⦠H) -H
/3 width=1 by llpx_sn_flat/
]
qed-.
diff --git a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma
index 1b5843657..0f1c9cb31 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/unfold/lstas_lstas.ma
@@ -18,24 +18,24 @@ include "basic_2/unfold/lstas_lift.ma".
(* Main properties **********************************************************)
-theorem lstas_trans: âh,G,L,T1,T,l1. â¦G, L⦠⢠T1 â¢*[h, l1] T â
- âT2,l2. â¦G, L⦠⢠T â¢*[h, l2] T2 â â¦G, L⦠⢠T1 â¢*[h, l1+l2] T2.
-#h #G #L #T1 #T #l1 #H elim H -G -L -T1 -T -l1
-[ #G #L #l1 #k #X #l2 #H >(lstas_inv_sort1 ⦠H) -X
+theorem lstas_trans: âh,G,L,T1,T,d1. â¦G, L⦠⢠T1 â¢*[h, d1] T â
+ âT2,d2. â¦G, L⦠⢠T â¢*[h, d2] T2 â â¦G, L⦠⢠T1 â¢*[h, d1+d2] T2.
+#h #G #L #T1 #T #d1 #H elim H -G -L -T1 -T -d1
+[ #G #L #d1 #k #X #d2 #H >(lstas_inv_sort1 ⦠H) -X
<iter_plus /2 width=1 by lstas_sort/
-| #G #L #K #V1 #V #U #i #l1 #HLK #_ #HVU #IHV1 #U2 #l2 #HU2
+| #G #L #K #V1 #V #U #i #d1 #HLK #_ #HVU #IHV1 #U2 #d2 #HU2
lapply (drop_fwd_drop2 ⦠HLK) #H0
elim (lstas_inv_lift1 ⦠HU2 ⦠H0 ⦠HVU)
/3 width=6 by lstas_ldef/
| //
-| #G #L #K #W1 #W #U #i #l1 #HLK #_ #HWU #IHW1 #U2 #l2 #HU2
+| #G #L #K #W1 #W #U #i #d1 #HLK #_ #HWU #IHW1 #U2 #d2 #HU2
lapply (drop_fwd_drop2 ⦠HLK) #H0
elim (lstas_inv_lift1 ⦠HU2 ⦠H0 ⦠HWU)
/3 width=6 by lstas_succ/
-| #a #I #G #L #V #T1 #T #l1 #_ #IHT1 #X #l2 #H
+| #a #I #G #L #V #T1 #T #d1 #_ #IHT1 #X #d2 #H
elim (lstas_inv_bind1 ⦠H) -H #T2 #HT2 #H destruct
/3 width=1 by lstas_bind/
-| #G #L #V #T1 #T #l1 #_ #IHT1 #X #l2 #H
+| #G #L #V #T1 #T #d1 #_ #IHT1 #X #d2 #H
elim (lstas_inv_appl1 ⦠H) -H #T2 #HT2 #H destruct
/3 width=1 by lstas_appl/
| /3 width=1 by lstas_cast/
@@ -43,12 +43,12 @@ theorem lstas_trans: âh,G,L,T1,T,l1. â¦G, L⦠⢠T1 â¢*[h, l1] T â
qed-.
(* Note: apparently this was missing in basic_1 *)
-theorem lstas_mono: âh,G,L,l. singlevalued ⦠(lstas h l G L).
-#h #G #L #l #T #T1 #H elim H -G -L -T -T1 -l
-[ #G #L #l #k #X #H >(lstas_inv_sort1 ⦠H) -X //
-| #G #L #K #V #V1 #U1 #i #l #HLK #_ #HVU1 #IHV1 #X #H
+theorem lstas_mono: âh,G,L,d. singlevalued ⦠(lstas h d G L).
+#h #G #L #d #T #T1 #H elim H -G -L -T -T1 -d
+[ #G #L #d #k #X #H >(lstas_inv_sort1 ⦠H) -X //
+| #G #L #K #V #V1 #U1 #i #d #HLK #_ #HVU1 #IHV1 #X #H
elim (lstas_inv_lref1 ⦠H) -H *
- #K0 #V0 #W0 [3: #l0 ] #HLK0
+ #K0 #V0 #W0 [3: #d0 ] #HLK0
lapply (drop_mono ⦠HLK0 ⦠HLK) -HLK -HLK0 #H destruct
#HVW0 #HX lapply (IHV1 ⦠HVW0) -IHV1 -HVW0 #H destruct
/2 width=5 by lift_mono/
@@ -56,16 +56,16 @@ theorem lstas_mono: âh,G,L,l. singlevalued ⦠(lstas h l G L).
elim (lstas_inv_lref1_O ⦠H) -H *
#K0 #V0 #W0 #HLK0
lapply (drop_mono ⦠HLK0 ⦠HLK) -HLK -HLK0 #H destruct //
-| #G #L #K #W #W1 #U1 #i #l #HLK #_ #HWU1 #IHWV #X #H
+| #G #L #K #W #W1 #U1 #i #d #HLK #_ #HWU1 #IHWV #X #H
elim (lstas_inv_lref1_S ⦠H) -H * #K0 #W0 #V0 #HLK0
lapply (drop_mono ⦠HLK0 ⦠HLK) -HLK -HLK0 #H destruct
#HW0 #HX lapply (IHWV ⦠HW0) -IHWV -HW0 #H destruct
/2 width=5 by lift_mono/
-| #a #I #G #L #V #T #U1 #l #_ #IHTU1 #X #H
+| #a #I #G #L #V #T #U1 #d #_ #IHTU1 #X #H
elim (lstas_inv_bind1 ⦠H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
-| #G #L #V #T #U1 #l #_ #IHTU1 #X #H
+| #G #L #V #T #U1 #d #_ #IHTU1 #X #H
elim (lstas_inv_appl1 ⦠H) -H #U2 #HTU2 #H destruct /3 width=1 by eq_f/
-| #G #L #W #T #U1 #l #_ #IHTU1 #U2 #H
+| #G #L #W #T #U1 #d #_ #IHTU1 #U2 #H
lapply (lstas_inv_cast1 ⦠H) -H /2 width=1 by/
]
qed-.
@@ -73,51 +73,51 @@ qed-.
(* Advanced inversion lemmas ************************************************)
(* Basic_1: was just: sty0_correct *)
-lemma lstas_correct: âh,G,L,T1,T,l1. â¦G, L⦠⢠T1 â¢*[h, l1] T â
- âl2. âT2. â¦G, L⦠⢠T â¢*[h, l2] T2.
-#h #G #L #T1 #T #l1 #H elim H -G -L -T1 -T -l1
+lemma lstas_correct: âh,G,L,T1,T,d1. â¦G, L⦠⢠T1 â¢*[h, d1] T â
+ âd2. âT2. â¦G, L⦠⢠T â¢*[h, d2] T2.
+#h #G #L #T1 #T #d1 #H elim H -G -L -T1 -T -d1
[ /2 width=2 by lstas_sort, ex_intro/
-| #G #L #K #V1 #V #U #i #l #HLK #_ #HVU #IHV1 #l2
- elim (IHV1 l2) -IHV1 #V2
+| #G #L #K #V1 #V #U #i #d #HLK #_ #HVU #IHV1 #d2
+ elim (IHV1 d2) -IHV1 #V2
elim (lift_total V2 0 (i+1))
lapply (drop_fwd_drop2 ⦠HLK) -HLK
/3 width=11 by ex_intro, lstas_lift/
-| #G #L #K #W1 #W #i #HLK #HW1 #IHW1 #l2
- @(nat_ind_plus ⦠l2) -l2 /3 width=5 by lstas_zero, ex_intro/
- #l2 #_ elim (IHW1 l2) -IHW1 #W2 #HW2
+| #G #L #K #W1 #W #i #HLK #HW1 #IHW1 #d2
+ @(nat_ind_plus ⦠d2) -d2 /3 width=5 by lstas_zero, ex_intro/
+ #d2 #_ elim (IHW1 d2) -IHW1 #W2 #HW2
lapply (lstas_trans ⦠HW1 ⦠HW2) -W
elim (lift_total W2 0 (i+1))
/3 width=7 by lstas_succ, ex_intro/
-| #G #L #K #W1 #W #U #i #l #HLK #_ #HWU #IHW1 #l2
- elim (IHW1 l2) -IHW1 #W2
+| #G #L #K #W1 #W #U #i #d #HLK #_ #HWU #IHW1 #d2
+ elim (IHW1 d2) -IHW1 #W2
elim (lift_total W2 0 (i+1))
lapply (drop_fwd_drop2 ⦠HLK) -HLK
/3 width=11 by ex_intro, lstas_lift/
-| #a #I #G #L #V #T #U #l #_ #IHTU #l2
- elim (IHTU l2) -IHTU /3 width=2 by lstas_bind, ex_intro/
-| #G #L #V #T #U #l #_ #IHTU #l2
- elim (IHTU l2) -IHTU /3 width=2 by lstas_appl, ex_intro/
-| #G #L #W #T #U #l #_ #IHTU #l2
- elim (IHTU l2) -IHTU /2 width=2 by ex_intro/
+| #a #I #G #L #V #T #U #d #_ #IHTU #d2
+ elim (IHTU d2) -IHTU /3 width=2 by lstas_bind, ex_intro/
+| #G #L #V #T #U #d #_ #IHTU #d2
+ elim (IHTU d2) -IHTU /3 width=2 by lstas_appl, ex_intro/
+| #G #L #W #T #U #d #_ #IHTU #d2
+ elim (IHTU d2) -IHTU /2 width=2 by ex_intro/
]
qed-.
(* more main properties *****************************************************)
-theorem lstas_conf_le: âh,G,L,T,U1,l1. â¦G, L⦠⢠T â¢*[h, l1] U1 â
- âU2,l2. l1 ⤠l2 â â¦G, L⦠⢠T â¢*[h, l2] U2 â
- â¦G, L⦠⢠U1 â¢*[h, l2-l1] U2.
-#h #G #L #T #U1 #l1 #HTU1 #U2 #l2 #Hl12
->(plus_minus_m_m ⦠Hl12) in ⢠(%â?); -Hl12 >commutative_plus #H
+theorem lstas_conf_le: âh,G,L,T,U1,d1. â¦G, L⦠⢠T â¢*[h, d1] U1 â
+ âU2,d2. d1 ⤠d2 â â¦G, L⦠⢠T â¢*[h, d2] U2 â
+ â¦G, L⦠⢠U1 â¢*[h, d2-d1] U2.
+#h #G #L #T #U1 #d1 #HTU1 #U2 #d2 #Hd12
+>(plus_minus_m_m ⦠Hd12) in ⢠(%â?); -Hd12 >commutative_plus #H
elim (lstas_split ⦠H) -H #U #HTU
>(lstas_mono ⦠HTU ⦠HTU1) -T //
qed-.
-theorem lstas_conf: âh,G,L,T0,T1,l1. â¦G, L⦠⢠T0 â¢*[h, l1] T1 â
- âT2,l2. â¦G, L⦠⢠T0 â¢*[h, l2] T2 â
- ââT. â¦G, L⦠⢠T1 â¢*[h, l2] T & â¦G, L⦠⢠T2 â¢*[h, l1] T.
-#h #G #L #T0 #T1 #l1 #HT01 #T2 #l2 #HT02
-elim (lstas_lstas ⦠HT01 (l1+l2)) #T #HT0
+theorem lstas_conf: âh,G,L,T0,T1,d1. â¦G, L⦠⢠T0 â¢*[h, d1] T1 â
+ âT2,d2. â¦G, L⦠⢠T0 â¢*[h, d2] T2 â
+ ââT. â¦G, L⦠⢠T1 â¢*[h, d2] T & â¦G, L⦠⢠T2 â¢*[h, d1] T.
+#h #G #L #T0 #T1 #d1 #HT01 #T2 #d2 #HT02
+elim (lstas_lstas ⦠HT01 (d1+d2)) #T #HT0
lapply (lstas_conf_le ⦠HT01 ⦠HT0) // -HT01 <minus_plus_m_m_commutative
lapply (lstas_conf_le ⦠HT02 ⦠HT0) // -T0 <minus_plus_m_m
/2 width=3 by ex2_intro/
diff --git a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
index 9f2c61a64..f69061eaf 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
+++ b/matita/matita/contribs/lambdadelta/basic_2/web/basic_2.ldw.xml
@@ -21,86 +21,82 @@
<section>Summary of the Specification</section>
<body>Here is a numerical acount of the specification's contents
and its timeline.
- Nodes are counted according to the "intrinsinc complexity measure"
- [F. Guidi: "Procedural Representation of CIC Proof Terms"
- Journal of Automated Reasoning 44(1-2), Springer (February 2010),
- pp. 53-78].
</body>
<table name="basic_2_sum"/>
<subsection>Stage "B"</subsection>
- <news date="In progress.">
+ <news class="alpha" date="Ongoing.">
Context-sensitive subject equivalence
for native type assignment.
</news>
<subsection>Stage "A": "Weakening the Applicability Condition"</subsection>
- <news date="2014 September 9.">
+ <news class="beta" date="2014 September 9.">
Iterated static type assignment defined (more elegantly)
as a primitive notion.
</news>
- <news date="2014 June 18.">
+ <news class="beta" date="2014 June 18.">
Preservation of stratified native validity
for context-sensitive computation on terms.
</news>
- <news date="2014 June 9.">
+ <news class="alpha" date="2014 June 9.">
Strong qrst-normalization
for simply typed terms.
</news>
- <news date="2014 April 16.">
+ <news class="alpha" date="2014 April 16.">
Lazy equivalence on local environments
addded as q-step to rst-computation on closures
(anniversary milestone).
</news>
- <news date="2014 January 20.">
+ <news class="alpha" date="2014 January 20.">
Parametrized slicing of local environments
comprises both versions of this operation
(one from basic_1, the other used in basic_2 till now).
</news>
- <news date="2013 August 7.">
+ <news class="alpha" date="2013 August 7.">
Passive support for global environments.
</news>
- <news date="2013 July 27.">
+ <news class="alpha" date="2013 July 27.">
Reaxiomatized β-reductum as in rt-reduction.
</news>
- <news date="2013 July 20.">
+ <news class="alpha" date="2013 July 20.">
Context-sensitive strong rt-normalization
for simply typed terms.
</news>
- <news date="2013 April 16.">
+ <news class="alpha" date="2013 April 16.">
Reaxiomatized substitution and reduction
commute with respect to subclosure
(anniversary milestone).
</news>
- <news date="2013 March 16.">
+ <news class="alpha" date="2013 March 16.">
Mutual recursive preservation of stratified native validity
for rst-computation on closures.
</news>
- <news date="2012 October 16.">
+ <news class="alpha" date="2012 October 16.">
Confluence for context-free parallel reduction on closures.
</news>
- <news date="2012 July 26.">
+ <news class="alpha" date="2012 July 26.">
Term binders polarized to control ζ-reduction (not released).
</news>
- <news date="2012 April 16.">
+ <news class="alpha" date="2012 April 16.">
Context-sensitive subject equivalence
for atomic arity assignment
(anniversary milestone).
</news>
- <news date="2012 March 15.">
+ <news class="alpha" date="2012 March 15.">
Context-sensitive strong normalization
for simply typed terms.
</news>
- <news date="2012 January 27.">
+ <news class="alpha" date="2012 January 27.">
Support for abstract candidates of reducibility.
</news>
- <news date="2011 September 21.">
+ <news class="alpha" date="2011 September 21.">
Confluence for context-sensitive parallel reduction on terms.
</news>
- <news date="2011 September 6.">
+ <news class="alpha" date="2011 September 6.">
Confluence for context-free parallel reduction on terms.
</news>
- <news date="2011 April 17.">
+ <news class="alpha" date="2011 April 17.">
Specification starts.
</news>
@@ -113,9 +109,5 @@
</body>
<table name="basic_2_src"/>
- <section>Physical Structure of the Specification</section>
- <body>The source files are grouped in directories,
- one for each component.
- </body>
<footer/>
</page>
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 c31b9bf06..0714b94ec 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
@@ -109,7 +109,7 @@ table {
]
[ { "context-sensitive rt-computation" * } {
[ "lpxs ( â¦?,?⦠⢠â¡*[?,?] ? )" "lpxs_drop" + "lpxs_lleq" + "lpxs_aaa" + "lpxs_cpxs" + "lpxs_lpxs" * ]
- [ "cpxs ( â¦?,?⦠⢠? â¡*[?,?] ? )" "cpxs_tsts" + "cpxs_tsts_vector" + "cpxs_leq" + "cpxs_lift" + "cpxs_lleq" + "cpxs_aaa" + "cpxs_cpxs" * ]
+ [ "cpxs ( â¦?,?⦠⢠? â¡*[?,?] ? )" "cpxs_tsts" + "cpxs_tsts_vector" + "cpxs_lreq" + "cpxs_lift" + "cpxs_lleq" + "cpxs_aaa" + "cpxs_cpxs" * ]
}
]
[ { "context-sensitive computation" * } {
@@ -140,7 +140,7 @@ table {
]
[ { "context-sensitive rt-reduction" * } {
[ "lpx ( â¦?,?⦠⢠â¡[?,?] ? )" "lpx_drop" + "lpx_frees" + "lpx_lleq" + "lpx_aaa" * ]
- [ "cpx ( â¦?,?⦠⢠? â¡[?,?] ? )" "cpx_leq" + "cpx_lift" + "cpx_llpx_sn" + "cpx_lleq" + "cpx_cix" * ]
+ [ "cpx ( â¦?,?⦠⢠? â¡[?,?] ? )" "cpx_lreq" + "cpx_lift" + "cpx_llpx_sn" + "cpx_lleq" + "cpx_cix" * ]
}
]
[ { "irreducible forms for context-sensitive rt-reduction" * } {
@@ -214,11 +214,11 @@ table {
[ { "multiple substitution" * } {
[ { "lazy equivalence" * } {
[ "fleq ( â¦?,?,?⦠â¡[?] â¦?,?,?⦠)" "fleq_fleq" * ]
- [ "lleq ( ? â¡[?,?] ? )" "lleq_alt" + "lleq_alt_rec" + "lleq_leq" + "lleq_drop" + "lleq_fqus" + "lleq_llor" + "lleq_lleq" * ]
+ [ "lleq ( ? â¡[?,?] ? )" "lleq_alt" + "lleq_alt_rec" + "lleq_lreq" + "lleq_drop" + "lleq_fqus" + "lleq_llor" + "lleq_lleq" * ]
}
]
[ { "lazy pointwise extension of a relation" * } {
- [ "llpx_sn" "llpx_sn_alt" + "llpx_sn_alt_rec" + "llpx_sn_tc" + "llpx_sn_leq" + "llpx_sn_drop" + "llpx_sn_lpx_sn" + "llpx_sn_frees" + "llpx_sn_llor" * ]
+ [ "llpx_sn" "llpx_sn_alt" + "llpx_sn_alt_rec" + "llpx_sn_tc" + "llpx_sn_lreq" + "llpx_sn_drop" + "llpx_sn_lpx_sn" + "llpx_sn_frees" + "llpx_sn_llor" * ]
}
]
[ { "pointwise union for local environments" * } {
@@ -226,7 +226,7 @@ table {
}
]
[ { "context-sensitive exclusion from free variables" * } {
- [ "frees ( ? ⢠? ϵ ð
*[?]�⦠)" "frees_append" + "frees_leq" + "frees_lift" * ]
+ [ "frees ( ? ⢠? ϵ ð
*[?]�⦠)" "frees_append" + "frees_lreq" + "frees_lift" * ]
}
]
[ { "contxt-sensitive multiple rt-substitution" * } {
@@ -277,7 +277,7 @@ table {
}
]
[ { "basic local env. slicing" * } {
- [ "drop ( â¬[?,?,?] ? â¡ ? )" "drop_append" + "drop_leq" + "drop_drop" * ]
+ [ "drop ( â¬[?,?,?] ? â¡ ? )" "drop_append" + "drop_lreq" + "drop_drop" * ]
}
]
[ { "basic term relocation" * } {
@@ -290,7 +290,7 @@ table {
class "red"
[ { "grammar" * } {
[ { "equivalence for local environments" * } {
- [ "leq ( ? ⩬[?,?] ? )" "leq_leq" * ]
+ [ "lreq ( ? ⩬[?,?] ? )" "lreq_lreq" * ]
}
]
[ { "same top term structure" * } {
diff --git a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml
index 3123e6807..522ac9312 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml
+++ b/matita/matita/contribs/lambdadelta/ground_2/web/ground_2.ldw.xml
@@ -14,10 +14,10 @@
pp. 53-78].
</body>
<table name="ground_2_sum"/>
- <news date="2013 November 27.">
+ <news class="alpha" date="2013 November 27.">
Natural numbers with infinity.
</news>
- <news date="2011 August 10.">
+ <news class="alpha" date="2011 August 10.">
Specification starts.
</news>
@@ -30,9 +30,5 @@
</body>
<table name="ground_2_src"/>
- <section>Physical Structure of the Specification</section>
- <body>The source files are grouped in directories,
- one for each plane.
- </body>
<footer/>
</page>