From: Ferruccio Guidi <ferruccio.guidi@unibo.it>
Date: Sun, 29 May 2016 18:13:10 +0000 (+0000)
Subject: more results on sor ...
X-Git-Tag: make_still_working~574
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=f215e6c18fbd22a049e6d34cf3bb52b0cabc4d58;p=helm.git

more results on sor ...
---

diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
index a00a10d16..fef085189 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_after.ma
@@ -13,6 +13,7 @@
 (**************************************************************************)
 
 include "ground_2/notation/relations/rafter_3.ma".
+include "ground_2/relocation/rtmap_sor.ma".
 include "ground_2/relocation/rtmap_istot.ma".
 
 (* RELOCATION MAP ***********************************************************)
@@ -161,7 +162,7 @@ qed-.
 
 (* Basic properties *********************************************************)
 
-corec lemma after_eq_repl_back_2: âf1,f. eq_repl_back (Î»f2. f2 â f1 â¡ f).
+corec lemma after_eq_repl_back2: âf1,f. eq_repl_back (Î»f2. f2 â f1 â¡ f).
 #f1 #f #f2 * -f2 -f1 -f
 #f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
 [ cases (eq_inv_px â¦  H0 â¦  H21) -g21 /3 width=7 by after_refl/
@@ -170,11 +171,11 @@ corec lemma after_eq_repl_back_2: âf1,f. eq_repl_back (Î»f2. f2 â f1 â¡ f).
 ]
 qed-.
 
-lemma after_eq_repl_fwd_2: âf1,f. eq_repl_fwd (Î»f2. f2 â f1 â¡ f).
-#f1 #f @eq_repl_sym /2 width=3 by after_eq_repl_back_2/
+lemma after_eq_repl_fwd2: âf1,f. eq_repl_fwd (Î»f2. f2 â f1 â¡ f).
+#f1 #f @eq_repl_sym /2 width=3 by after_eq_repl_back2/
 qed-.
 
-corec lemma after_eq_repl_back_1: âf2,f. eq_repl_back (Î»f1. f2 â f1 â¡ f).
+corec lemma after_eq_repl_back1: âf2,f. eq_repl_back (Î»f1. f2 â f1 â¡ f).
 #f2 #f #f1 * -f2 -f1 -f
 #f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
 [ cases (eq_inv_px â¦  H0 â¦  H11) -g11 /3 width=7 by after_refl/
@@ -183,21 +184,21 @@ corec lemma after_eq_repl_back_1: âf2,f. eq_repl_back (Î»f1. f2 â f1 â¡ f).
 ]
 qed-.
 
-lemma after_eq_repl_fwd_1: âf2,f. eq_repl_fwd (Î»f1. f2 â f1 â¡ f).
-#f2 #f @eq_repl_sym /2 width=3 by after_eq_repl_back_1/
+lemma after_eq_repl_fwd1: âf2,f. eq_repl_fwd (Î»f1. f2 â f1 â¡ f).
+#f2 #f @eq_repl_sym /2 width=3 by after_eq_repl_back1/
 qed-.
 
-corec lemma after_eq_repl_back_0: âf1,f2. eq_repl_back (Î»f. f2 â f1 â¡ f).
+corec lemma after_eq_repl_back0: âf1,f2. eq_repl_back (Î»f. f2 â f1 â¡ f).
 #f2 #f1 #f * -f2 -f1 -f
 #f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
 [ cases (eq_inv_px â¦  H0 â¦  H01) -g01 /3 width=7 by after_refl/
 | cases (eq_inv_nx â¦  H0 â¦  H01) -g01 /3 width=7 by after_push/
-| cases (eq_inv_nx â¦  H0 â¦  H01) -g01 /3 width=5 by after_next/ 
+| cases (eq_inv_nx â¦  H0 â¦  H01) -g01 /3 width=5 by after_next/
 ]
 qed-.
 
-lemma after_eq_repl_fwd_0: âf2,f1. eq_repl_fwd (Î»f. f2 â f1 â¡ f).
-#f2 #f1 @eq_repl_sym /2 width=3 by after_eq_repl_back_0/
+lemma after_eq_repl_fwd0: âf2,f1. eq_repl_fwd (Î»f. f2 â f1 â¡ f).
+#f2 #f1 @eq_repl_sym /2 width=3 by after_eq_repl_back0/
 qed-.
 
 (* Main properties **********************************************************)
@@ -263,7 +264,7 @@ qed-.
 
 lemma after_mono_eq: âf1,f2,f. f1 â f2 â¡ f â âg1,g2,g. g1 â g2 â¡ g â
                      f1 â g1 â f2 â g2 â f â g.
-/4 width=4 by after_mono, after_eq_repl_back_1, after_eq_repl_back_2/ qed-.
+/4 width=4 by after_mono, after_eq_repl_back1, after_eq_repl_back2/ qed-.
 
 (* Properties on tls ********************************************************)
 
@@ -274,27 +275,27 @@ lemma after_tls: ân,f1,f2,f. @â¦0, f1â¦ â¡ n â
 #g1 #Hg1 #H1 cases (after_inv_nxx â¦ Hf â¦ H1) -Hf /2 width=1 by/
 qed.
 
-(* Inversion lemmas on isid *************************************************)
+(* Properties on isid *******************************************************)
 
-corec lemma isid_after_sn: âf1. ðâ¦f1â¦ â âf2. f1 â f2 â¡ f2.
+corec lemma after_isid_sn: âf1. ðâ¦f1â¦ â âf2. f1 â f2 â¡ f2.
 #f1 * -f1 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) * #g2 #H2
 /3 width=7 by after_push, after_refl/
-qed-.
+qed.
 
-corec lemma isid_after_dx: âf2. ðâ¦f2â¦ â âf1. f1 â f2 â¡ f1.
+corec lemma after_isid_dx: âf2. ðâ¦f2â¦ â âf1. f1 â f2 â¡ f1.
 #f2 * -f2 #f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) * #g1 #H1
 [ /3 width=7 by after_refl/
 | @(after_next â¦ H1 H1) /3 width=3 by isid_push/
 ]
-qed-.
+qed.
 
-lemma after_isid_inv_sn: âf1,f2,f. f1 â f2 â¡ f â  ðâ¦f1â¦ â f2 â f.
-/3 width=6 by isid_after_sn, after_mono/
-qed-.
+(* Inversion lemmas on isid *************************************************)
 
-lemma after_isid_inv_dx: âf1,f2,f. f1 â f2 â¡ f â  ðâ¦f2â¦ â f1 â f.
-/3 width=6 by isid_after_dx, after_mono/
-qed-.
+lemma after_isid_inv_sn: âf1,f2,f. f1 â f2 â¡ f â ðâ¦f1â¦ â f2 â f.
+/3 width=6 by after_isid_sn, after_mono/ qed-.
+
+lemma after_isid_inv_dx: âf1,f2,f. f1 â f2 â¡ f â ðâ¦f2â¦ â f1 â f.
+/3 width=6 by after_isid_dx, after_mono/ qed-.
 
 corec lemma after_fwd_isid1: âf1,f2,f. f1 â f2 â¡ f â ðâ¦fâ¦ â ðâ¦f1â¦.
 #f1 #f2 #f * -f1 -f2 -f
@@ -317,14 +318,14 @@ lemma after_inv_isid3: âf1,f2,f. f1 â f2 â¡ f â ðâ¦fâ¦ â ðâ¦f1
 
 lemma after_isid_isuni: âf1,f2. ðâ¦f2â¦ â ðâ¦f1â¦ â f1 â â«¯f2 â¡ â«¯f1.
 #f1 #f2 #Hf2 #H elim H -H
-/5 width=7 by isid_after_dx, after_eq_repl_back_2, after_next, after_push, eq_push_inv_isid/
+/5 width=7 by after_isid_dx, after_eq_repl_back2, after_next, after_push, eq_push_inv_isid/
 qed.
 
 lemma after_uni_next2: âf2. ðâ¦f2â¦ â âf1,f. â«¯f2 â f1 â¡ f â f2 â â«¯f1 â¡ f.
 #f2 #H elim H -f2
 [ #f2 #Hf2 #f1 #f #Hf
   elim (after_inv_nxx â¦ Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
-  /4 width=7 by after_isid_inv_sn, isid_after_sn, after_eq_repl_back_0, eq_next/
+  /4 width=7 by after_isid_inv_sn, after_isid_sn, after_eq_repl_back0, eq_next/
 | #f2 #_ #g2 #H2 #IH #f1 #f #Hf
   elim (after_inv_nxx â¦ Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
   /3 width=5 by after_next/
@@ -335,9 +336,35 @@ qed.
 
 lemma after_uni: ân1,n2. ðâ´n1âµ â ðâ´n2âµ â¡ ðâ´n1+n2âµ.
 @nat_elim2
-/4 width=5 by after_uni_next2, isid_after_sn, isid_after_dx, after_next/
+/4 width=5 by after_uni_next2, after_isid_sn, after_isid_dx, after_next/
 qed.
 
+(* Inversion lemmas on sor **************************************************)
+
+lemma sor_isid: âf1,f2,f. ðâ¦f1â¦ â ðâ¦f2â¦ â ðâ¦fâ¦ â f1 â f2 â¡ f.
+/4 width=3 by sor_eq_repl_back2, sor_eq_repl_back1, isid_inv_eq_repl/ qed.
+(*
+lemma after_inv_sor: âf. ðâ¦fâ¦ â âf2,f1. f2 â f1 â¡ f â âfa,fb. fa â fb â¡ f â
+                     ââf1a,f1b. f2 â f1a â¡ fa & f2 â f1b â¡ fb & f1a â f1b â¡ f1.
+@isfin_ind
+[ #f #Hf #f2 #f1 #H1f #fa #fb #H2f
+  elim (after_inv_isid3 â¦ H1f) -H1f //
+  elim (sor_inv_isid3 â¦ H2f) -H2f //
+  /3 width=5 by ex3_2_intro, after_isid_sn, sor_isid/
+| #f #_ #IH #f2 #f1 #H1 #fa #fb #H2
+  elim (after_inv_xxp â¦ H1) -H1 [ |*: // ] #g2 #g1 #H1f
+  elim (sor_inv_xxp â¦ H2) -H2 [ |*: // ] #ga #gb #H2f
+  elim (IH â¦ H1f â¦ H2f) -f /3 width=11 by sor_pp, after_refl, ex3_2_intro/
+| #f #_ #IH #f2 #f1 #H1 #fa #fb #H2
+  elim (sor_inv_xxn â¦ H2) -H2 [1,3,4: * |*: // ] #ga #gb #H2f
+  elim (after_inv_xxn â¦ H1) -H1 [1,3,5,7,9,11: * |*: // ] #g2 [1,3,5: #g1 ] #H1f
+  elim (IH â¦ H1f â¦ H2f) -f
+  /3 width=11 by sor_np, sor_pn, sor_nn, after_refl, after_push, after_next, ex3_2_intro/
+  #x1a #x1b #H39 #H40 #H41 #H42 #H43 #H44
+  [ @ex3_2_intro
+    [3: /2 width=7 by after_next/ | skip
+    |5: @H41 | skip  
+*)
 (* Forward lemmas on at *****************************************************)
 
 lemma after_at_fwd: âi,i1,f. @â¦i1, fâ¦ â¡ i â âf2,f1. f2 â f1 â¡ f â
@@ -407,7 +434,7 @@ lemma after_uni_dx: âi2,i1,f2. @â¦i1, f2â¦ â¡ i2 â
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp â¦ Hf2) -Hf2 // #g2 #H1 #H2 destruct
   lapply (after_isid_inv_dx â¦ Hf ?) -Hf
-  /3 width=3 by isid_after_sn, after_eq_repl_back_0/
+  /3 width=3 by after_isid_sn, after_eq_repl_back0/
 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxn â¦ Hf2) -Hf2 [1,3: * |*: // ]
   [ #g2 #j1 #Hg2 #H1 #H2 destruct
@@ -426,7 +453,7 @@ lemma after_uni_sn: âi2,i1,f2. @â¦i1, f2â¦ â¡ i2 â
 [ #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxp â¦ Hf2) -Hf2 // #g2 #H1 #H2 destruct
   lapply (after_isid_inv_sn â¦ Hf ?) -Hf
-  /3 width=3 by isid_after_dx, after_eq_repl_back_0/
+  /3 width=3 by after_isid_dx, after_eq_repl_back0/
 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
   elim (after_inv_nxx â¦ Hf) -Hf [2,3: // ] #g #Hg #H destruct
   elim (at_inv_xxn â¦ Hf2) -Hf2 [1,3: * |*: // ]
@@ -443,7 +470,7 @@ lemma after_uni_succ_dx: âi2,i1,f2. @â¦i1, f2â¦ â¡ i2 â
   elim (at_inv_xxp â¦ Hf2) -Hf2 // #g2 #H1 #H2 destruct
   elim (after_inv_pnx â¦ Hf) -Hf [ |*: // ] #g #Hg #H
   lapply (after_isid_inv_dx â¦ Hg ?) -Hg
-  /4 width=5 by isid_after_sn, after_eq_repl_back_0, after_next/
+  /4 width=5 by after_isid_sn, after_eq_repl_back0, after_next/
 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
   elim (at_inv_xxn â¦ Hf2) -Hf2 [1,3: * |*: // ]
   [ #g2 #j1 #Hg2 #H1 #H2 destruct
@@ -463,7 +490,7 @@ lemma after_uni_succ_sn: âi2,i1,f2. @â¦i1, f2â¦ â¡ i2 â
   elim (at_inv_xxp â¦ Hf2) -Hf2 // #g2 #H1 #H2 destruct
   elim (after_inv_nxx â¦ Hf) -Hf [ |*: // ] #g #Hg #H destruct
   lapply (after_isid_inv_sn â¦ Hg ?) -Hg
-  /4 width=7 by isid_after_dx, after_eq_repl_back_0, after_push/
+  /4 width=7 by after_isid_dx, after_eq_repl_back0, after_push/
 | #i2 #IH #i1 #f2 #Hf2 #f #Hf
   elim (after_inv_nxx â¦ Hf) -Hf [2,3: // ] #g #Hg #H destruct
   elim (at_inv_xxn â¦ Hf2) -Hf2 [1,3: * |*: // ]
diff --git a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sor.ma b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sor.ma
index e76c1ca3c..8f38b8062 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sor.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sor.ma
@@ -72,6 +72,141 @@ try elim (discr_push_next â¦ Hx2) try elim (discr_next_push â¦ Hx2)
 /2 width=3 by ex2_intro/
 qed-.
 
+(* Advanced inversion lemmas ************************************************)
+
+lemma sor_inv_ppn: âg1,g2,g. g1 â g2 â¡ g â
+                   âf1,f2,f. âf1 = g1 â âf2 = g2 â â«¯f = g â â¥.
+#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
+elim (sor_inv_ppx â¦ H â¦ H1 H2) -g1 -g2 #x #_ #H destruct
+/2 width=3 by discr_push_next/
+qed-.
+
+lemma sor_inv_nxp: âg1,g2,g. g1 â g2 â¡ g â
+                   âf1,f. â«¯f1 = g1 â âf = g â â¥.
+#g1 #g2 #g #H #f1 #f #H1 #H0
+elim (pn_split g2) * #f2 #H2
+[ elim (sor_inv_npx â¦ H â¦ H1 H2)
+| elim (sor_inv_nnx â¦ H â¦ H1 H2)
+] -g1 -g2 #x #_ #H destruct
+/2 width=3 by discr_next_push/
+qed-.
+
+lemma sor_inv_xnp: âg1,g2,g. g1 â g2 â¡ g â
+                   âf2,f. â«¯f2 = g2 â âf = g â â¥.
+#g1 #g2 #g #H #f2 #f #H2 #H0
+elim (pn_split g1) * #f1 #H1
+[ elim (sor_inv_pnx â¦ H â¦ H1 H2)
+| elim (sor_inv_nnx â¦ H â¦ H1 H2)
+] -g1 -g2 #x #_ #H destruct
+/2 width=3 by discr_next_push/
+qed-.
+
+lemma sor_inv_ppp: âg1,g2,g. g1 â g2 â¡ g â
+                   âf1,f2,f. âf1 = g1 â âf2 = g2 â âf = g â f1 â f2 â¡ f.
+#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
+elim (sor_inv_ppx â¦ H â¦ H1 H2) -g1 -g2 #x #Hx #H destruct
+<(injective_push â¦ H) -f //
+qed-.
+
+lemma sor_inv_npn: âg1,g2,g. g1 â g2 â¡ g â
+                   âf1,f2,f. â«¯f1 = g1 â âf2 = g2 â â«¯f = g â f1 â f2 â¡ f.
+#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
+elim (sor_inv_npx â¦ H â¦ H1 H2) -g1 -g2 #x #Hx #H destruct
+<(injective_next â¦ H) -f //
+qed-.
+
+lemma sor_inv_pnn: âg1,g2,g. g1 â g2 â¡ g â
+                   âf1,f2,f. âf1 = g1 â â«¯f2 = g2 â â«¯f = g â f1 â f2 â¡ f.
+#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
+elim (sor_inv_pnx â¦ H â¦ H1 H2) -g1 -g2 #x #Hx #H destruct
+<(injective_next â¦ H) -f //
+qed-.
+
+lemma sor_inv_nnn: âg1,g2,g. g1 â g2 â¡ g â
+                   âf1,f2,f. â«¯f1 = g1 â â«¯f2 = g2 â â«¯f = g â f1 â f2 â¡ f.
+#g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
+elim (sor_inv_nnx â¦ H â¦ H1 H2) -g1 -g2 #x #Hx #H destruct
+<(injective_next â¦ H) -f //
+qed-.
+
+lemma sor_inv_pxp: âg1,g2,g. g1 â g2 â¡ g â
+                   âf1,f. âf1 = g1 â âf = g â
+                   ââf2. f1 â f2 â¡ f & âf2 = g2.
+#g1 #g2 #g #H #f1 #f #H1 #H0
+elim (pn_split g2) * #f2 #H2
+[ /3 width=7 by sor_inv_ppp, ex2_intro/
+| elim (sor_inv_xnp â¦ H â¦ H2 H0)
+]
+qed-.
+
+lemma sor_inv_xpp: âg1,g2,g. g1 â g2 â¡ g â
+                   âf2,f. âf2 = g2 â âf = g â
+                   ââf1. f1 â f2 â¡ f & âf1 = g1.
+#g1 #g2 #g #H #f2 #f #H2 #H0
+elim (pn_split g1) * #f1 #H1
+[ /3 width=7 by sor_inv_ppp, ex2_intro/
+| elim (sor_inv_nxp â¦ H â¦ H1 H0)
+]
+qed-.
+
+lemma sor_inv_pxn: âg1,g2,g. g1 â g2 â¡ g â
+                   âf1,f. âf1 = g1 â â«¯f = g â
+                   ââf2. f1 â f2 â¡ f & â«¯f2 = g2.
+#g1 #g2 #g #H #f1 #f #H1 #H0
+elim (pn_split g2) * #f2 #H2
+[ elim (sor_inv_ppn â¦ H â¦ H1 H2 H0)
+| /3 width=7 by sor_inv_pnn, ex2_intro/
+]
+qed-.
+
+lemma sor_inv_xpn: âg1,g2,g. g1 â g2 â¡ g â
+                   âf2,f. âf2 = g2 â â«¯f = g â
+                   ââf1. f1 â f2 â¡ f & â«¯f1 = g1.
+#g1 #g2 #g #H #f2 #f #H2 #H0
+elim (pn_split g1) * #f1 #H1
+[ elim (sor_inv_ppn â¦ H â¦ H1 H2 H0)
+| /3 width=7 by sor_inv_npn, ex2_intro/
+]
+qed-.
+
+lemma sor_inv_xxp: âg1,g2,g. g1 â g2 â¡ g â âf. âf = g â
+                   ââf1,f2. f1 â f2 â¡ f & âf1 = g1 & âf2 = g2.
+#g1 #g2 #g #H #f #H0
+elim (pn_split g1) * #f1 #H1
+[ elim (sor_inv_pxp â¦ H â¦ H1 H0) -g /2 width=5 by ex3_2_intro/
+| elim (sor_inv_nxp â¦ H â¦ H1 H0)
+]
+qed-.
+
+lemma sor_inv_nxn: âg1,g2,g. g1 â g2 â¡ g â
+                   âf1,f. â«¯f1 = g1 â â«¯f = g â
+                   (ââf2. f1 â f2 â¡ f & âf2 = g2) â¨
+                   ââf2. f1 â f2 â¡ f & â«¯f2 = g2.
+#g1 #g2 elim (pn_split g2) *
+/4 width=7 by sor_inv_npn, sor_inv_nnn, ex2_intro, or_intror, or_introl/
+qed-.
+
+lemma sor_inv_xnn: âg1,g2,g. g1 â g2 â¡ g â
+                   âf2,f. â«¯f2 = g2 â â«¯f = g â
+                   (ââf1. f1 â f2 â¡ f & âf1 = g1) â¨
+                   ââf1. f1 â f2 â¡ f & â«¯f1 = g1.
+#g1 elim (pn_split g1) *
+/4 width=7 by sor_inv_pnn, sor_inv_nnn, ex2_intro, or_intror, or_introl/
+qed-.
+
+lemma sor_inv_xxn: âg1,g2,g. g1 â g2 â¡ g â âf. â«¯f = g â
+                   â¨â¨ ââf1,f2. f1 â f2 â¡ f & â«¯f1 = g1 & âf2 = g2
+                    | ââf1,f2. f1 â f2 â¡ f & âf1 = g1 & â«¯f2 = g2
+                    | ââf1,f2. f1 â f2 â¡ f & â«¯f1 = g1 & â«¯f2 = g2.
+#g1 #g2 #g #H #f #H0
+elim (pn_split g1) * #f1 #H1
+[ elim (sor_inv_pxn â¦ H â¦ H1 H0) -g
+  /3 width=5 by or3_intro1, ex3_2_intro/
+| elim (sor_inv_nxn â¦ H â¦ H1 H0) -g *
+  /3 width=5 by or3_intro0, or3_intro2, ex3_2_intro/
+]
+qed-.
+
 (* Main inversion lemmas ****************************************************)
 
 corec theorem sor_mono: âf1,f2,x,y. f1 â f2 â¡ x â f1 â f2 â¡ y â x â y.
@@ -131,7 +266,7 @@ corec lemma sor_sym: âf1,f2,f. f1 â f2 â¡ f â f2 â f1 â¡ f.
 [ @sor_pp | @sor_pn | @sor_np | @sor_nn ] /2 width=7 by/
 qed-.
 
-(* Properies on identity ****************************************************)
+(* Properies on test for identity *******************************************)
 
 corec lemma sor_isid_sn: âf1. ðâ¦f1â¦ â âf2. f1 â f2 â¡ f2.
 #f1 * -f1
@@ -145,6 +280,33 @@ corec lemma sor_isid_dx: âf2. ðâ¦f2â¦ â âf1. f1 â f2 â¡ f1.
 /3 width=7 by sor_pp, sor_np/
 qed.
 
+(* Inversion lemmas on test for identity ************************************)
+
+lemma sor_isid_inv_sn: âf1,f2,f. f1 â f2 â¡ f â ðâ¦f1â¦ â f2 â f.
+/3 width=4 by sor_isid_sn, sor_mono/
+qed-.
+
+lemma sor_isid_inv_dx: âf1,f2,f. f1 â f2 â¡ f â ðâ¦f2â¦ â f1 â f.
+/3 width=4 by sor_isid_dx, sor_mono/
+qed-.
+
+corec lemma sor_fwd_isid1: âf1,f2,f. f1 â f2 â¡ f â ðâ¦fâ¦ â ðâ¦f1â¦.
+#f1 #f2 #f * -f1 -f2 -f
+#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H #Hg
+[ /4 width=6 by isid_inv_push, isid_push/ ]
+cases (isid_inv_next â¦ Hg â¦ H)
+qed-.
+
+corec lemma sor_fwd_isid2: âf1,f2,f. f1 â f2 â¡ f â ðâ¦fâ¦ â ðâ¦f2â¦.
+#f1 #f2 #f * -f1 -f2 -f
+#f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H #Hg
+[ /4 width=6 by isid_inv_push, isid_push/ ]
+cases (isid_inv_next â¦ Hg â¦ H)
+qed-.
+
+lemma sor_inv_isid3: âf1,f2,f. f1 â f2 â¡ f â ðâ¦fâ¦ â ðâ¦f1â¦ â§ ðâ¦f2â¦.
+/3 width=4 by sor_fwd_isid2, sor_fwd_isid1, conj/ qed-.
+
 (* Properties on finite colength assignment *********************************)
 
 lemma sor_fcla_ex: âf1,n1. ðâ¦f1â¦ â¡ n1 â âf2,n2. ðâ¦f2â¦ â¡ n2 â