X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fground_2%2Frelocation%2Frtmap_sor.ma;h=9191eb3212b6aa67185a8f444a3c7fc2fba45af8;hb=bd53c4e895203eb049e75434f638f26b5a161a2b;hp=640f9e265b41726ace787548be926c48227700f9;hpb=325bc2fb36e8f8db99a152037d71332c9ac7eff9;p=helm.git

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 640f9e265..9191eb321 100644
--- a/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sor.ma
+++ b/matita/matita/contribs/lambdadelta/ground_2/relocation/rtmap_sor.ma
@@ -12,15 +12,16 @@
 (*                                                                        *)
 (**************************************************************************)
 
+include "ground_2/xoa/ex_4_2.ma".
 include "ground_2/notation/relations/runion_3.ma".
 include "ground_2/relocation/rtmap_isfin.ma".
 include "ground_2/relocation/rtmap_sle.ma".
 
 coinductive sor: relation3 rtmap rtmap rtmap ≝
-| sor_pp: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ↑f1 = g1 → ↑f2 = g2 → ↑f = g → sor g1 g2 g
-| sor_np: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ⫯f1 = g1 → ↑f2 = g2 → ⫯f = g → sor g1 g2 g
-| sor_pn: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → sor g1 g2 g
-| sor_nn: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → sor g1 g2 g
+| sor_pp: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → sor g1 g2 g
+| sor_np: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ↑f1 = g1 → ⫯f2 = g2 → ↑f = g → sor g1 g2 g
+| sor_pn: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ⫯f1 = g1 → ↑f2 = g2 → ↑f = g → sor g1 g2 g
+| sor_nn: ∀f1,f2,f,g1,g2,g. sor f1 f2 f → ↑f1 = g1 → ↑f2 = g2 → ↑f = g → sor g1 g2 g
 .
 
 interpretation "union (rtmap)"
@@ -28,8 +29,8 @@ interpretation "union (rtmap)"
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma sor_inv_ppx: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
-                   ∃∃f. f1 ⋓ f2 ≡ f & ↑f = g.
+lemma sor_inv_ppx: ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 →
+                   ∃∃f. f1 ⋓ f2 ≘ f & ⫯f = g.
 #g1 #g2 #g * -g1 -g2 -g
 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
 try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
@@ -39,8 +40,8 @@ try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
 /2 width=3 by ex2_intro/
 qed-.
 
-lemma sor_inv_npx: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 →
-                   ∃∃f. f1 ⋓ f2 ≡ f & ⫯f = g.
+lemma sor_inv_npx: ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 →
+                   ∃∃f. f1 ⋓ f2 ≘ f & ↑f = g.
 #g1 #g2 #g * -g1 -g2 -g
 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
 try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
@@ -50,8 +51,8 @@ try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
 /2 width=3 by ex2_intro/
 qed-.
 
-lemma sor_inv_pnx: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ⫯f2 = g2 →
-                   ∃∃f. f1 ⋓ f2 ≡ f & ⫯f = g.
+lemma sor_inv_pnx: ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f1,f2. ⫯f1 = g1 → ↑f2 = g2 →
+                   ∃∃f. f1 ⋓ f2 ≘ f & ↑f = g.
 #g1 #g2 #g * -g1 -g2 -g
 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
 try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
@@ -61,8 +62,8 @@ try elim (discr_push_next … Hx2) try elim (discr_next_push … Hx2)
 /2 width=3 by ex2_intro/
 qed-.
 
-lemma sor_inv_nnx: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f1,f2. ⫯f1 = g1 → ⫯f2 = g2 →
-                   ∃∃f. f1 ⋓ f2 ≡ f & ⫯f = g.
+lemma sor_inv_nnx: ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
+                   ∃∃f. f1 ⋓ f2 ≘ f & ↑f = g.
 #g1 #g2 #g * -g1 -g2 -g
 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x1 #x2 #Hx1 #Hx2 destruct
 try (>(injective_push … Hx1) -x1) try (>(injective_next … Hx1) -x1)
@@ -74,15 +75,15 @@ qed-.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma sor_inv_ppn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ⫯f = g → ⊥.
+lemma sor_inv_ppn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ↑f = g → ⊥.
 #g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
 elim (sor_inv_ppx … H … H1 H2) -g1 -g2 #x #_ #H destruct
 /2 width=3 by discr_push_next/
 qed-.
 
-lemma sor_inv_nxp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f1,f. ⫯f1 = g1 → ↑f = g → ⊥.
+lemma sor_inv_nxp: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f1,f. ↑f1 = g1 → ⫯f = g → ⊥.
 #g1 #g2 #g #H #f1 #f #H1 #H0
 elim (pn_split g2) * #f2 #H2
 [ elim (sor_inv_npx … H … H1 H2)
@@ -91,8 +92,8 @@ elim (pn_split g2) * #f2 #H2
 /2 width=3 by discr_next_push/
 qed-.
 
-lemma sor_inv_xnp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f2,f. ⫯f2 = g2 → ↑f = g → ⊥.
+lemma sor_inv_xnp: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f2,f. ↑f2 = g2 → ⫯f = g → ⊥.
 #g1 #g2 #g #H #f2 #f #H2 #H0
 elim (pn_split g1) * #f1 #H1
 [ elim (sor_inv_pnx … H … H1 H2)
@@ -101,37 +102,37 @@ elim (pn_split g1) * #f1 #H1
 /2 width=3 by discr_next_push/
 qed-.
 
-lemma sor_inv_ppp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ⋓ f2 ≡ f.
+lemma sor_inv_ppp: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⋓ f2 ≘ f.
 #g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
 elim (sor_inv_ppx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
 <(injective_push … H) -f //
 qed-.
 
-lemma sor_inv_npn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f1,f2,f. ⫯f1 = g1 → ↑f2 = g2 → ⫯f = g → f1 ⋓ f2 ≡ f.
+lemma sor_inv_npn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ↑f = g → f1 ⋓ f2 ≘ f.
 #g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
 elim (sor_inv_npx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
 <(injective_next … H) -f //
 qed-.
 
-lemma sor_inv_pnn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f1,f2,f. ↑f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⋓ f2 ≡ f.
+lemma sor_inv_pnn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f1,f2,f. ⫯f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ⋓ f2 ≘ f.
 #g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
 elim (sor_inv_pnx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
 <(injective_next … H) -f //
 qed-.
 
-lemma sor_inv_nnn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f1,f2,f. ⫯f1 = g1 → ⫯f2 = g2 → ⫯f = g → f1 ⋓ f2 ≡ f.
+lemma sor_inv_nnn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f1,f2,f. ↑f1 = g1 → ↑f2 = g2 → ↑f = g → f1 ⋓ f2 ≘ f.
 #g1 #g2 #g #H #f1 #f2 #f #H1 #H2 #H0
 elim (sor_inv_nnx … H … H1 H2) -g1 -g2 #x #Hx #H destruct
 <(injective_next … H) -f //
 qed-.
 
-lemma sor_inv_pxp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f1,f. ↑f1 = g1 → ↑f = g →
-                   ∃∃f2. f1 ⋓ f2 ≡ f & ↑f2 = g2.
+lemma sor_inv_pxp: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f1,f. ⫯f1 = g1 → ⫯f = g →
+                   ∃∃f2. f1 ⋓ f2 ≘ f & ⫯f2 = g2.
 #g1 #g2 #g #H #f1 #f #H1 #H0
 elim (pn_split g2) * #f2 #H2
 [ /3 width=7 by sor_inv_ppp, ex2_intro/
@@ -139,9 +140,9 @@ elim (pn_split g2) * #f2 #H2
 ]
 qed-.
 
-lemma sor_inv_xpp: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f2,f. ↑f2 = g2 → ↑f = g →
-                   ∃∃f1. f1 ⋓ f2 ≡ f & ↑f1 = g1.
+lemma sor_inv_xpp: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f2,f. ⫯f2 = g2 → ⫯f = g →
+                   ∃∃f1. f1 ⋓ f2 ≘ f & ⫯f1 = g1.
 #g1 #g2 #g #H #f2 #f #H2 #H0
 elim (pn_split g1) * #f1 #H1
 [ /3 width=7 by sor_inv_ppp, ex2_intro/
@@ -149,9 +150,9 @@ elim (pn_split g1) * #f1 #H1
 ]
 qed-.
 
-lemma sor_inv_pxn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f1,f. ↑f1 = g1 → ⫯f = g →
-                   ∃∃f2. f1 ⋓ f2 ≡ f & ⫯f2 = g2.
+lemma sor_inv_pxn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f1,f. ⫯f1 = g1 → ↑f = g →
+                   ∃∃f2. f1 ⋓ f2 ≘ f & ↑f2 = g2.
 #g1 #g2 #g #H #f1 #f #H1 #H0
 elim (pn_split g2) * #f2 #H2
 [ elim (sor_inv_ppn … H … H1 H2 H0)
@@ -159,9 +160,9 @@ elim (pn_split g2) * #f2 #H2
 ]
 qed-.
 
-lemma sor_inv_xpn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f2,f. ↑f2 = g2 → ⫯f = g →
-                   ∃∃f1. f1 ⋓ f2 ≡ f & ⫯f1 = g1.
+lemma sor_inv_xpn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f2,f. ⫯f2 = g2 → ↑f = g →
+                   ∃∃f1. f1 ⋓ f2 ≘ f & ↑f1 = g1.
 #g1 #g2 #g #H #f2 #f #H2 #H0
 elim (pn_split g1) * #f1 #H1
 [ elim (sor_inv_ppn … H … H1 H2 H0)
@@ -169,8 +170,8 @@ elim (pn_split g1) * #f1 #H1
 ]
 qed-.
 
-lemma sor_inv_xxp: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f. ↑f = g →
-                   ∃∃f1,f2. f1 ⋓ f2 ≡ f & ↑f1 = g1 & ↑f2 = g2.
+lemma sor_inv_xxp: ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f. ⫯f = g →
+                   ∃∃f1,f2. f1 ⋓ f2 ≘ f & ⫯f1 = g1 & ⫯f2 = g2.
 #g1 #g2 #g #H #f #H0
 elim (pn_split g1) * #f1 #H1
 [ elim (sor_inv_pxp … H … H1 H0) -g /2 width=5 by ex3_2_intro/
@@ -178,26 +179,26 @@ elim (pn_split g1) * #f1 #H1
 ]
 qed-.
 
-lemma sor_inv_nxn: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f1,f. ⫯f1 = g1 → ⫯f = g →
-                   (∃∃f2. f1 ⋓ f2 ≡ f & ↑f2 = g2) ∨
-                   ∃∃f2. f1 ⋓ f2 ≡ f & ⫯f2 = g2.
+lemma sor_inv_nxn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f1,f. ↑f1 = g1 → ↑f = g →
+                   (∃∃f2. f1 ⋓ f2 ≘ f & ⫯f2 = g2) ∨
+                   ∃∃f2. f1 ⋓ f2 ≘ f & ↑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: ∀g1,g2,g. g1 ⋓ g2 ≡ g →
-                   ∀f2,f. ⫯f2 = g2 → ⫯f = g →
-                   (∃∃f1. f1 ⋓ f2 ≡ f & ↑f1 = g1) ∨
-                   ∃∃f1. f1 ⋓ f2 ≡ f & ⫯f1 = g1.
+lemma sor_inv_xnn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                   ∀f2,f. ↑f2 = g2 → ↑f = g →
+                   (∃∃f1. f1 ⋓ f2 ≘ f & ⫯f1 = g1) ∨
+                   ∃∃f1. f1 ⋓ f2 ≘ f & ↑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: ∀g1,g2,g. g1 ⋓ g2 ≡ g → ∀f. ⫯f = g →
-                   ∨∨ ∃∃f1,f2. f1 ⋓ f2 ≡ f & ⫯f1 = g1 & ↑f2 = g2
-                    | ∃∃f1,f2. f1 ⋓ f2 ≡ f & ↑f1 = g1 & ⫯f2 = g2
-                    | ∃∃f1,f2. f1 ⋓ f2 ≡ f & ⫯f1 = g1 & ⫯f2 = g2.
+lemma sor_inv_xxn: ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f. ↑f = g →
+                   ∨∨ ∃∃f1,f2. f1 ⋓ f2 ≘ f & ↑f1 = g1 & ⫯f2 = g2
+                    | ∃∃f1,f2. f1 ⋓ f2 ≘ f & ⫯f1 = g1 & ↑f2 = g2
+                    | ∃∃f1,f2. f1 ⋓ f2 ≘ f & ↑f1 = g1 & ↑f2 = g2.
 #g1 #g2 #g #H #f #H0
 elim (pn_split g1) * #f1 #H1
 [ elim (sor_inv_pxn … H … H1 H0) -g
@@ -209,7 +210,7 @@ qed-.
 
 (* Main inversion lemmas ****************************************************)
 
-corec theorem sor_mono: ∀f1,f2,x,y. f1 ⋓ f2 ≡ x → f1 ⋓ f2 ≡ y → x ≗ y.
+corec theorem sor_mono: ∀f1,f2,x,y. f1 ⋓ f2 ≘ x → f1 ⋓ f2 ≘ y → x ≡ y.
 #f1 #f2 #x #y * -f1 -f2 -x
 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #H
 [ cases (sor_inv_ppx … H … H1 H2)
@@ -222,45 +223,45 @@ qed-.
 
 (* Basic properties *********************************************************)
 
-corec lemma sor_eq_repl_back1: ∀f2,f. eq_repl_back … (λf1. f1 ⋓ f2 ≡ f).
+corec lemma sor_eq_repl_back1: ∀f2,f. eq_repl_back … (λf1. f1 ⋓ f2 ≘ f).
 #f2 #f #f1 * -f1 -f2 -f
 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0 #x #Hx
 try cases (eq_inv_px … Hx … H1) try cases (eq_inv_nx … Hx … H1) -g1
 /3 width=7 by sor_pp, sor_np, sor_pn, sor_nn/
 qed-.
 
-lemma sor_eq_repl_fwd1: ∀f2,f. eq_repl_fwd … (λf1. f1 ⋓ f2 ≡ f).
+lemma sor_eq_repl_fwd1: ∀f2,f. eq_repl_fwd … (λf1. f1 ⋓ f2 ≘ f).
 #f2 #f @eq_repl_sym /2 width=3 by sor_eq_repl_back1/
 qed-.
 
-corec lemma sor_eq_repl_back2: ∀f1,f. eq_repl_back … (λf2. f1 ⋓ f2 ≡ f).
+corec lemma sor_eq_repl_back2: ∀f1,f. eq_repl_back … (λf2. f1 ⋓ f2 ≘ f).
 #f1 #f #f2 * -f1 -f2 -f
 #f1 #f2 #f #g1 #g2 #g #Hf #H #H2 #H0 #x #Hx
 try cases (eq_inv_px … Hx … H2) try cases (eq_inv_nx … Hx … H2) -g2
 /3 width=7 by sor_pp, sor_np, sor_pn, sor_nn/
 qed-.
 
-lemma sor_eq_repl_fwd2: ∀f1,f. eq_repl_fwd … (λf2. f1 ⋓ f2 ≡ f).
+lemma sor_eq_repl_fwd2: ∀f1,f. eq_repl_fwd … (λf2. f1 ⋓ f2 ≘ f).
 #f1 #f @eq_repl_sym /2 width=3 by sor_eq_repl_back2/
 qed-.
 
-corec lemma sor_eq_repl_back3: ∀f1,f2. eq_repl_back … (λf. f1 ⋓ f2 ≡ f).
+corec lemma sor_eq_repl_back3: ∀f1,f2. eq_repl_back … (λf. f1 ⋓ f2 ≘ f).
 #f1 #f2 #f * -f1 -f2 -f
 #f1 #f2 #f #g1 #g2 #g #Hf #H #H2 #H0 #x #Hx
 try cases (eq_inv_px … Hx … H0) try cases (eq_inv_nx … Hx … H0) -g
 /3 width=7 by sor_pp, sor_np, sor_pn, sor_nn/
 qed-.
 
-lemma sor_eq_repl_fwd3: ∀f1,f2. eq_repl_fwd … (λf. f1 ⋓ f2 ≡ f).
+lemma sor_eq_repl_fwd3: ∀f1,f2. eq_repl_fwd … (λf. f1 ⋓ f2 ≘ f).
 #f1 #f2 @eq_repl_sym /2 width=3 by sor_eq_repl_back3/
 qed-.
 
-corec lemma sor_refl: ∀f. f ⋓ f ≡ f.
+corec lemma sor_idem: ∀f. f ⋓ f ≘ f.
 #f cases (pn_split f) * #g #H
 [ @(sor_pp … H H H) | @(sor_nn … H H H) ] -H //
 qed.
 
-corec lemma sor_sym: ∀f1,f2,f. f1 ⋓ f2 ≡ f → f2 ⋓ f1 ≡ f.
+corec lemma sor_comm: ∀f1,f2,f. f1 ⋓ f2 ≘ f → f2 ⋓ f1 ≘ f.
 #f1 #f2 #f * -f1 -f2 -f
 #f1 #f2 #f #g1 #g2 #g #Hf * * * -g1 -g2 -g
 [ @sor_pp | @sor_pn | @sor_np | @sor_nn ] /2 width=7 by/
@@ -268,7 +269,7 @@ qed-.
 
 (* Properties with tail *****************************************************)
 
-lemma sor_tl: ∀f1,f2,f. f1 ⋓ f2 ≡ f → ⫱f1 ⋓ ⫱f2 ≡ ⫱f.
+lemma sor_tl: ∀f1,f2,f. f1 ⋓ f2 ≘ f → ⫱f1 ⋓ ⫱f2 ≘ ⫱f.
 #f1 cases (pn_split f1) * #g1 #H1
 #f2 cases (pn_split f2) * #g2 #H2
 #f #Hf
@@ -279,61 +280,94 @@ lemma sor_tl: ∀f1,f2,f. f1 ⋓ f2 ≡ f → ⫱f1 ⋓ ⫱f2 ≡ ⫱f.
 ] -Hf #g #Hg #H destruct //
 qed.
 
+lemma sor_xxn_tl: ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f. ↑f = g →
+                  (∃∃f1,f2. f1 ⋓ f2 ≘ f & ↑f1 = g1 & ⫱g2 = f2) ∨
+                  (∃∃f1,f2. f1 ⋓ f2 ≘ f & ⫱g1 = f1 & ↑f2 = g2).
+#g1 #g2 #g #H #f #H0 elim (sor_inv_xxn … H … H0) -H -H0 *
+/3 width=5 by ex3_2_intro, or_introl, or_intror/
+qed-.
+
+lemma sor_xnx_tl: ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f2. ↑f2 = g2 →
+                  ∃∃f1,f. f1 ⋓ f2 ≘ f & ⫱g1 = f1 & ↑f = g.
+#g1 elim (pn_split g1) * #f1 #H1 #g2 #g #H #f2 #H2
+[ elim (sor_inv_pnx … H … H1 H2) | elim (sor_inv_nnx … H … H1 H2) ] -g2
+/3 width=5 by ex3_2_intro/
+qed-.
+
+lemma sor_nxx_tl: ∀g1,g2,g. g1 ⋓ g2 ≘ g → ∀f1. ↑f1 = g1 →
+                  ∃∃f2,f. f1 ⋓ f2 ≘ f & ⫱g2 = f2 & ↑f = g.
+#g1 #g2 elim (pn_split g2) * #f2 #H2 #g #H #f1 #H1
+[ elim (sor_inv_npx … H … H1 H2) | elim (sor_inv_nnx … H … H1 H2) ] -g1
+/3 width=5 by ex3_2_intro/
+qed-.
+
 (* Properties with iterated tail ********************************************)
 
-lemma sor_tls: ∀f1,f2,f. f1 ⋓ f2 ≡ f →
-               ∀n. ⫱*[n]f1 ⋓ ⫱*[n]f2 ≡ ⫱*[n]f.
+lemma sor_tls: ∀f1,f2,f. f1 ⋓ f2 ≘ f →
+               ∀n. ⫱*[n]f1 ⋓ ⫱*[n]f2 ≘ ⫱*[n]f.
 #f1 #f2 #f #Hf #n elim n -n /2 width=1 by sor_tl/
 qed.
 
 (* Properies with test for identity *****************************************)
 
-corec lemma sor_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ⋓ f2 ≡ f2.
+corec lemma sor_isid_sn: ∀f1. 𝐈❪f1❫ → ∀f2. f1 ⋓ f2 ≘ f2.
 #f1 * -f1
 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) *
 /3 width=7 by sor_pp, sor_pn/
 qed.
 
-corec lemma sor_isid_dx: ∀f2. 𝐈⦃f2⦄ → ∀f1. f1 ⋓ f2 ≡ f1.
+corec lemma sor_isid_dx: ∀f2. 𝐈❪f2❫ → ∀f1. f1 ⋓ f2 ≘ f1.
 #f2 * -f2
 #f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) *
 /3 width=7 by sor_pp, sor_np/
 qed.
 
-lemma sor_isid: ∀f1,f2,f. 𝐈⦃f1⦄ → 𝐈⦃f2⦄ → 𝐈⦃f⦄ → f1 ⋓ f2 ≡ f.
+lemma sor_isid: ∀f1,f2,f. 𝐈❪f1❫ → 𝐈❪f2❫ → 𝐈❪f❫ → f1 ⋓ f2 ≘ f.
 /4 width=3 by sor_eq_repl_back2, sor_eq_repl_back1, isid_inv_eq_repl/ qed.
 
+(* Inversion lemmas with tail ***********************************************)
+
+lemma sor_inv_tl_sn: ∀f1,f2,f. ⫱f1 ⋓ f2 ≘ f → f1 ⋓ ↑f2 ≘ ↑f.
+#f1 #f2 #f elim (pn_split f1) *
+#g1 #H destruct /2 width=7 by sor_pn, sor_nn/
+qed-.
+
+lemma sor_inv_tl_dx: ∀f1,f2,f. f1 ⋓ ⫱f2 ≘ f → ↑f1 ⋓ f2 ≘ ↑f.
+#f1 #f2 #f elim (pn_split f2) *
+#g2 #H destruct /2 width=7 by sor_np, sor_nn/
+qed-.
+
 (* Inversion lemmas with test for identity **********************************)
 
-lemma sor_isid_inv_sn: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f1⦄ → f2 ≗ f.
+lemma sor_isid_inv_sn: ∀f1,f2,f. f1 ⋓ f2 ≘ f → 𝐈❪f1❫ → f2 ≡ f.
 /3 width=4 by sor_isid_sn, sor_mono/
 qed-.
 
-lemma sor_isid_inv_dx: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f2⦄ → f1 ≗ f.
+lemma sor_isid_inv_dx: ∀f1,f2,f. f1 ⋓ f2 ≘ f → 𝐈❪f2❫ → f1 ≡ f.
 /3 width=4 by sor_isid_dx, sor_mono/
 qed-.
 
-corec lemma sor_fwd_isid1: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄.
+corec lemma sor_fwd_isid1: ∀f1,f2,f. f1 ⋓ f2 ≘ f → 𝐈❪f❫ → 𝐈❪f1❫.
 #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 … Hg … H)
 qed-.
 
-corec lemma sor_fwd_isid2: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
+corec lemma sor_fwd_isid2: ∀f1,f2,f. f1 ⋓ f2 ≘ f → 𝐈❪f❫ → 𝐈❪f2❫.
 #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 … Hg … H)
 qed-.
 
-lemma sor_inv_isid3: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐈⦃f⦄ → 𝐈⦃f1⦄ ∧ 𝐈⦃f2⦄.
+lemma sor_inv_isid3: ∀f1,f2,f. f1 ⋓ f2 ≘ f → 𝐈❪f❫ → 𝐈❪f1❫ ∧ 𝐈❪f2❫.
 /3 width=4 by sor_fwd_isid2, sor_fwd_isid1, conj/ qed-.
 
 (* Properties with finite colength assignment *******************************)
 
-lemma sor_fcla_ex: ∀f1,n1. 𝐂⦃f1⦄ ≡ n1 → ∀f2,n2. 𝐂⦃f2⦄ ≡ n2 →
-                   ∃∃f,n. f1 ⋓ f2 ≡ f & 𝐂⦃f⦄ ≡ n & (n1 ∨ n2) ≤ n & n ≤ n1 + n2.
+lemma sor_fcla_ex: ∀f1,n1. 𝐂❪f1❫ ≘ n1 → ∀f2,n2. 𝐂❪f2❫ ≘ n2 →
+                   ∃∃f,n. f1 ⋓ f2 ≘ f & 𝐂❪f❫ ≘ n & (n1 ∨ n2) ≤ n & n ≤ n1 + n2.
 #f1 #n1 #Hf1 elim Hf1 -f1 -n1 /3 width=6 by sor_isid_sn, ex4_2_intro/
 #f1 #n1 #Hf1 #IH #f2 #n2 * -f2 -n2 /3 width=6 by fcla_push, fcla_next, ex4_2_intro, sor_isid_dx/
 #f2 #n2 #Hf2 elim (IH … Hf2) -IH -Hf2 -Hf1 [2,4: #f #n <plus_n_Sm ] (**) (* full auto fails *)
@@ -344,16 +378,16 @@ lemma sor_fcla_ex: ∀f1,n1. 𝐂⦃f1⦄ ≡ n1 → ∀f2,n2. 𝐂⦃f2⦄ ≡
 ]
 qed-.
 
-lemma sor_fcla: ∀f1,n1. 𝐂⦃f1⦄ ≡ n1 → ∀f2,n2. 𝐂⦃f2⦄ ≡ n2 → ∀f. f1 ⋓ f2 ≡ f →
-                ∃∃n. 𝐂⦃f⦄ ≡ n & (n1 ∨ n2) ≤ n & n ≤ n1 + n2.
+lemma sor_fcla: ∀f1,n1. 𝐂❪f1❫ ≘ n1 → ∀f2,n2. 𝐂❪f2❫ ≘ n2 → ∀f. f1 ⋓ f2 ≘ f →
+                ∃∃n. 𝐂❪f❫ ≘ n & (n1 ∨ n2) ≤ n & n ≤ n1 + n2.
 #f1 #n1 #Hf1 #f2 #n2 #Hf2 #f #Hf elim (sor_fcla_ex … Hf1 … Hf2) -Hf1 -Hf2
 /4 width=6 by sor_mono, fcla_eq_repl_back, ex3_intro/
 qed-.
 
 (* Forward lemmas with finite colength **************************************)
 
-lemma sor_fwd_fcla_sn_ex: ∀f,n. 𝐂⦃f⦄ ≡ n → ∀f1,f2. f1 ⋓ f2 ≡ f →
-                          ∃∃n1.  𝐂⦃f1⦄ ≡ n1 & n1 ≤ n.
+lemma sor_fwd_fcla_sn_ex: ∀f,n. 𝐂❪f❫ ≘ n → ∀f1,f2. f1 ⋓ f2 ≘ f →
+                          ∃∃n1.  𝐂❪f1❫ ≘ n1 & n1 ≤ n.
 #f #n #H elim H -f -n
 [ /4 width=4 by sor_fwd_isid1, fcla_isid, ex2_intro/
 | #f #n #_ #IH #f1 #f2 #H
@@ -365,57 +399,154 @@ lemma sor_fwd_fcla_sn_ex: ∀f,n. 𝐂⦃f⦄ ≡ n → ∀f1,f2. f1 ⋓ f2 ≡
 ]
 qed-.
 
-lemma sor_fwd_fcla_dx_ex: ∀f,n. 𝐂⦃f⦄ ≡ n → ∀f1,f2. f1 ⋓ f2 ≡ f →
-                          ∃∃n2.  𝐂⦃f2⦄ ≡ n2 & n2 ≤ n.
-/3 width=4 by sor_fwd_fcla_sn_ex, sor_sym/ qed-.
+lemma sor_fwd_fcla_dx_ex: ∀f,n. 𝐂❪f❫ ≘ n → ∀f1,f2. f1 ⋓ f2 ≘ f →
+                          ∃∃n2.  𝐂❪f2❫ ≘ n2 & n2 ≤ n.
+/3 width=4 by sor_fwd_fcla_sn_ex, sor_comm/ qed-.
 
 (* Properties with test for finite colength *********************************)
 
-lemma sor_isfin_ex: ∀f1,f2. 𝐅⦃f1⦄ → 𝐅⦃f2⦄ → ∃∃f. f1 ⋓ f2 ≡ f & 𝐅⦃f⦄.
+lemma sor_isfin_ex: ∀f1,f2. 𝐅❪f1❫ → 𝐅❪f2❫ → ∃∃f. f1 ⋓ f2 ≘ f & 𝐅❪f❫.
 #f1 #f2 * #n1 #H1 * #n2 #H2 elim (sor_fcla_ex … H1 … H2) -H1 -H2
 /3 width=4 by ex2_intro, ex_intro/
 qed-.
 
-lemma sor_isfin: ∀f1,f2. 𝐅⦃f1⦄ → 𝐅⦃f2⦄ → ∀f. f1 ⋓ f2 ≡ f → 𝐅⦃f⦄.
+lemma sor_isfin: ∀f1,f2. 𝐅❪f1❫ → 𝐅❪f2❫ → ∀f. f1 ⋓ f2 ≘ f → 𝐅❪f❫.
 #f1 #f2 #Hf1 #Hf2 #f #Hf elim (sor_isfin_ex … Hf1 … Hf2) -Hf1 -Hf2
 /3 width=6 by sor_mono, isfin_eq_repl_back/
 qed-.
 
 (* Forward lemmas with test for finite colength *****************************)
 
-lemma sor_fwd_isfin_sn: ∀f. 𝐅⦃f⦄ → ∀f1,f2. f1 ⋓ f2 ≡ f → 𝐅⦃f1⦄.
+lemma sor_fwd_isfin_sn: ∀f. 𝐅❪f❫ → ∀f1,f2. f1 ⋓ f2 ≘ f → 𝐅❪f1❫.
 #f * #n #Hf #f1 #f2 #H
 elim (sor_fwd_fcla_sn_ex … Hf … H) -f -f2 /2 width=2 by ex_intro/
 qed-.
 
-lemma sor_fwd_isfin_dx: ∀f. 𝐅⦃f⦄ → ∀f1,f2. f1 ⋓ f2 ≡ f → 𝐅⦃f2⦄.
+lemma sor_fwd_isfin_dx: ∀f. 𝐅❪f❫ → ∀f1,f2. f1 ⋓ f2 ≘ f → 𝐅❪f2❫.
 #f * #n #Hf #f1 #f2 #H
 elim (sor_fwd_fcla_dx_ex … Hf … H) -f -f1 /2 width=2 by ex_intro/
 qed-.
 
 (* Inversion lemmas with test for finite colength ***************************)
 
-lemma sor_inv_isfin3: ∀f1,f2,f. f1 ⋓ f2 ≡ f → 𝐅⦃f⦄ → 𝐅⦃f1⦄ ∧ 𝐅⦃f2⦄.
+lemma sor_inv_isfin3: ∀f1,f2,f. f1 ⋓ f2 ≘ f → 𝐅❪f❫ → 𝐅❪f1❫ ∧ 𝐅❪f2❫.
 /3 width=4 by sor_fwd_isfin_dx, sor_fwd_isfin_sn, conj/ qed-.
 
 (* Inversion lemmas with inclusion ******************************************)
 
-corec lemma sor_inv_sle_sn: ∀f1,f2,f. f1 ⋓ f2 ≡ f → f1 ⊆ f.
+corec lemma sor_inv_sle_sn: ∀f1,f2,f. f1 ⋓ f2 ≘ f → f1 ⊆ f.
 #f1 #f2 #f * -f1 -f2 -f
 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0
 /3 width=5 by sle_push, sle_next, sle_weak/
 qed-.
 
-corec lemma sor_inv_sle_dx: ∀f1,f2,f. f1 ⋓ f2 ≡ f → f2 ⊆ f.
+corec lemma sor_inv_sle_dx: ∀f1,f2,f. f1 ⋓ f2 ≘ f → f2 ⊆ f.
 #f1 #f2 #f * -f1 -f2 -f
 #f1 #f2 #f #g1 #g2 #g #Hf #H1 #H2 #H0
 /3 width=5 by sle_push, sle_next, sle_weak/
 qed-.
 
+lemma sor_inv_sle_sn_trans: ∀f1,f2,f. f1 ⋓ f2 ≘ f → ∀g. g ⊆ f1 → g ⊆ f.
+/3 width=4 by sor_inv_sle_sn, sle_trans/ qed-.
+
+lemma sor_inv_sle_dx_trans: ∀f1,f2,f. f1 ⋓ f2 ≘ f → ∀g. g ⊆ f2 → g ⊆ f.
+/3 width=4 by sor_inv_sle_dx, sle_trans/ qed-.
+
+axiom sor_inv_sle: ∀f1,f2,f. f1 ⋓ f2 ≘ f → ∀g. f1 ⊆ g → f2 ⊆ g → f ⊆ g.
+
 (* Properties with inclusion ************************************************)
 
-lemma sor_sle_sn: ∀f1,f2,f. f1 ⋓ f2 ≡ f → ∀g. g ⊆ f1 → g ⊆ f.
-/3 width=4 by sor_inv_sle_sn, sle_trans/ qed.
+corec lemma sor_sle_dx: ∀f1,f2. f1 ⊆ f2 → f1 ⋓ f2 ≘ f2.
+#f1 #f2 * -f1 -f2 /3 width=7 by sor_pp, sor_nn, sor_pn/
+qed.
 
-lemma sor_sle_dx: ∀f1,f2,f. f1 ⋓ f2 ≡ f → ∀g. g ⊆ f2 → g ⊆ f.
-/3 width=4 by sor_inv_sle_dx, sle_trans/ qed.
+corec lemma sor_sle_sn: ∀f1,f2. f1 ⊆ f2 → f2 ⋓ f1 ≘ f2.
+#f1 #f2 * -f1 -f2 /3 width=7 by sor_pp, sor_nn, sor_np/
+qed.
+
+(* Main properties **********************************************************)
+
+axiom monotonic_sle_sor: ∀f1,g1. f1 ⊆ g1 → ∀f2,g2. f2 ⊆ g2 →
+                         ∀f. f1 ⋓ f2 ≘ f → ∀g. g1 ⋓ g2 ≘ g → f ⊆ g.
+
+axiom sor_assoc_dx: ∀f0,f3,f4. f0 ⋓ f3 ≘ f4 →
+                    ∀f1,f2. f1 ⋓ f2 ≘ f0 →
+                    ∀f. f2 ⋓ f3 ≘ f → f1 ⋓ f ≘ f4.
+
+axiom sor_assoc_sn: ∀f1,f0,f4. f1 ⋓ f0 ≘ f4 →
+                    ∀f2, f3. f2 ⋓ f3 ≘ f0 →
+                    ∀f. f1 ⋓ f2 ≘ f → f ⋓ f3 ≘ f4.
+
+lemma sor_comm_23: ∀f0,f1,f2,f3,f4,f.
+                   f0⋓f4 ≘ f1 → f1⋓f2 ≘ f → f0⋓f2 ≘ f3 → f3⋓f4 ≘ f.
+/4 width=6 by sor_comm, sor_assoc_dx/ qed-.
+
+corec theorem sor_comm_23_idem: ∀f0,f1,f2. f0 ⋓ f1 ≘ f2 →
+                                ∀f. f1 ⋓ f2 ≘ f → f1 ⋓ f0 ≘ f.
+#f0 #f1 #f2 * -f0 -f1 -f2
+#f0 #f1 #f2 #g0 #g1 #g2 #Hf2 #H0 #H1 #H2 #g #Hg
+[ cases (sor_inv_ppx … Hg … H1 H2)
+| cases (sor_inv_pnx … Hg … H1 H2)
+| cases (sor_inv_nnx … Hg … H1 H2)
+| cases (sor_inv_nnx … Hg … H1 H2)
+] -g2 #f #Hf #H
+/3 width=7 by sor_nn, sor_np, sor_pn, sor_pp/
+qed-.
+
+corec theorem sor_coll_dx: ∀f1,f2,f. f1 ⋓ f2 ≘ f → ∀g1,g2,g. g1 ⋓ g2 ≘ g →
+                           ∀g0. g1 ⋓ g0 ≘ f1 → g2 ⋓ g0 ≘ f2 → g ⋓ g0 ≘ f.
+#f1 #f2 #f cases (pn_split f) * #x #Hx #Hf #g1 #g2 #g #Hg #g0 #Hf1 #Hf2
+[ cases (sor_inv_xxp … Hf … Hx) -Hf #x1 #x2 #Hf #Hx1 #Hx2
+  cases (sor_inv_xxp … Hf1 … Hx1) -f1 #y1 #y0 #Hf1 #Hy1 #Hy0
+  cases (sor_inv_xpp … Hf2 … Hy0 … Hx2) -f2 #y2 #Hf2 #Hy2
+  cases (sor_inv_ppx … Hg … Hy1 Hy2) -g1 -g2 #y #Hg #Hy
+  @(sor_pp … Hy Hy0 Hx) -g -g0 -f /2 width=8 by/
+| cases (pn_split g) * #y #Hy
+  [ cases (sor_inv_xxp … Hg … Hy) -Hg #y1 #y2 #Hg #Hy1 #Hy2
+    cases (sor_xxn_tl … Hf … Hx) * #x1 #x2 #_ #Hx1 #Hx2
+    [ cases (sor_inv_pxn … Hf1 … Hy1 Hx1) -g1 #y0 #Hf1 #Hy0
+      cases (sor_inv_pnx … Hf2 … Hy2 Hy0) -g2 -x2 #x2 #Hf2 #Hx2
+    | cases (sor_inv_pxn … Hf2 … Hy2 Hx2) -g2 #y0 #Hf2 #Hy0
+      cases (sor_inv_pnx … Hf1 … Hy1 Hy0) -g1 -x1 #x1 #Hf1 #Hx1
+    ]
+    lapply (sor_inv_nnn … Hf … Hx1 Hx2 Hx) -f1 -f2 #Hf
+    @(sor_pn … Hy Hy0 Hx) -g -g0 -f /2 width=8 by/
+  | lapply (sor_tl … Hf) -Hf #Hf
+    lapply (sor_tl … Hg) -Hg #Hg
+    lapply (sor_tl … Hf1) -Hf1 #Hf1
+    lapply (sor_tl … Hf2) -Hf2 #Hf2
+    cases (pn_split g0) * #y0 #Hy0
+    [ @(sor_np … Hy Hy0 Hx) /2 width=8 by/
+    | @(sor_nn … Hy Hy0 Hx) /2 width=8 by/
+    ]
+  ]
+]
+qed-.
+
+corec theorem sor_distr_dx: ∀g0,g1,g2,g. g1 ⋓ g2 ≘ g →
+                            ∀f1,f2,f. g1 ⋓ g0 ≘ f1 → g2 ⋓ g0 ≘ f2 → g ⋓ g0 ≘ f →
+                            f1 ⋓ f2 ≘ f.
+#g0 cases (pn_split g0) * #y0 #H0 #g1 #g2 #g
+[ * -g1 -g2 -g #y1 #y2 #y #g1 #g2 #g #Hy #Hy1 #Hy2 #Hy #f1 #f2 #f #Hf1 #Hf2 #Hf
+  [ cases (sor_inv_ppx … Hf1 … Hy1 H0) -g1
+    cases (sor_inv_ppx … Hf2 … Hy2 H0) -g2
+    cases (sor_inv_ppx … Hf … Hy H0) -g
+  | cases (sor_inv_npx … Hf1 … Hy1 H0) -g1
+    cases (sor_inv_ppx … Hf2 … Hy2 H0) -g2
+    cases (sor_inv_npx … Hf … Hy H0) -g
+  | cases (sor_inv_ppx … Hf1 … Hy1 H0) -g1
+    cases (sor_inv_npx … Hf2 … Hy2 H0) -g2
+    cases (sor_inv_npx … Hf … Hy H0) -g
+  | cases (sor_inv_npx … Hf1 … Hy1 H0) -g1
+    cases (sor_inv_npx … Hf2 … Hy2 H0) -g2
+    cases (sor_inv_npx … Hf … Hy H0) -g
+  ] -g0 #y #Hy #H #y2 #Hy2 #H2 #y1 #Hy1 #H1
+  /3 width=8 by sor_nn, sor_np, sor_pn, sor_pp/
+| #H #f1 #f2 #f #Hf1 #Hf2 #Hf
+  cases (sor_xnx_tl … Hf1 … H0) -Hf1
+  cases (sor_xnx_tl … Hf2 … H0) -Hf2
+  cases (sor_xnx_tl … Hf … H0) -Hf
+  -g0 #y #x #Hx #Hy #H #y2 #x2 #Hx2 #Hy2 #H2 #y1 #x1 #Hx1 #Hy1 #H1
+  /4 width=8 by sor_tl, sor_nn/
+]
+qed-.