update in ground_2, static_2, basic_2, apps_2, alpha_1

[helm.git] / matita / matita / contribs / lambdadelta / ground_2 / relocation / rtmap_sor.ma

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 a461157d60256cccfc21128db5fa054b9575fe47..9191eb3212b6aa67185a8f444a3c7fc2fba45af8 100644 (file)
--- 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: â\88\80f1,f2,f,g1,g2,g. sor f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → sor g1 g2 g
-| sor_np: â\88\80f1,f2,f,g1,g2,g. sor f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 ⫯f = g → sor g1 g2 g
-| sor_pn: â\88\80f1,f2,f,g1,g2,g. sor f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → sor g1 g2 g
-| sor_nn: â\88\80f1,f2,f,g1,g2,g. sor f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → sor g1 g2 g
+| sor_pp: â\88\80f1,f2,f,g1,g2,g. sor f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → sor g1 g2 g
+| sor_np: â\88\80f1,f2,f,g1,g2,g. sor f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 â\86\91f = g → sor g1 g2 g
+| sor_pn: â\88\80f1,f2,f,g1,g2,g. sor f1 f2 f â\86\92 ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → sor g1 g2 g
+| sor_nn: â\88\80f1,f2,f,g1,g2,g. sor f1 f2 f â\86\92 â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → sor g1 g2 g
 .
 
 interpretation "union (rtmap)"
@@ -28,8 +29,8 @@ interpretation "union (rtmap)"
 
 (* Basic inversion lemmas ***************************************************)
 
-lemma sor_inv_ppx: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 →
-                   â\88\83â\88\83f. f1 â\8b\93 f2 â\89\98 f & â\86\91f = g.
+lemma sor_inv_ppx: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 ⫯f2 = g2 →
+                   â\88\83â\88\83f. f1 â\8b\93 f2 â\89\98 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: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 →
-                   â\88\83â\88\83f. f1 â\8b\93 f2 â\89\98 f & ⫯f = g.
+lemma sor_inv_npx: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 →
+                   â\88\83â\88\83f. f1 â\8b\93 f2 â\89\98 f & â\86\91f = 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: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 →
-                   â\88\83â\88\83f. f1 â\8b\93 f2 â\89\98 f & ⫯f = g.
+lemma sor_inv_pnx: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 →
+                   â\88\83â\88\83f. f1 â\8b\93 f2 â\89\98 f & â\86\91f = 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: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 ⫯f2 = g2 →
-                   â\88\83â\88\83f. f1 â\8b\93 f2 â\89\98 f & ⫯f = g.
+lemma sor_inv_nnx: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 →
+                   â\88\83â\88\83f. f1 â\8b\93 f2 â\89\98 f & â\86\91f = 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)
@@ -75,14 +76,14 @@ qed-.
 (* Advanced inversion lemmas ************************************************)
 
 lemma sor_inv_ppn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
-                   â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 ⫯f = g → ⊥.
+                   â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 â\86\91f = 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 →
-                   â\88\80f1,f. ⫯f1 = g1 â\86\92 â\86\91f = g → ⊥.
+                   â\88\80f1,f. â\86\91f1 = g1 â\86\92 ⫯f = g → ⊥.
 #g1 #g2 #g #H #f1 #f #H1 #H0
 elim (pn_split g2) * #f2 #H2
 [ elim (sor_inv_npx … H … H1 H2)
@@ -92,7 +93,7 @@ elim (pn_split g2) * #f2 #H2
 qed-.
 
 lemma sor_inv_xnp: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
-                   â\88\80f2,f. ⫯f2 = g2 â\86\92 â\86\91f = g → ⊥.
+                   â\88\80f2,f. â\86\91f2 = g2 â\86\92 ⫯f = g → ⊥.
 #g1 #g2 #g #H #f2 #f #H2 #H0
 elim (pn_split g1) * #f1 #H1
 [ elim (sor_inv_pnx … H … H1 H2)
@@ -102,36 +103,36 @@ elim (pn_split g1) * #f1 #H1
 qed-.
 
 lemma sor_inv_ppp: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
-                   â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → f1 ⋓ f2 ≘ f.
+                   â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯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 →
-                   â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 ⫯f = g → f1 ⋓ f2 ≘ f.
+                   â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 â\86\91f = 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 →
-                   â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → f1 ⋓ f2 ≘ f.
+                   â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = 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 →
-                   â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → f1 ⋓ f2 ≘ f.
+                   â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = 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 →
-                   â\88\80f1,f. â\86\91f1 = g1 â\86\92 â\86\91f = g →
-                   â\88\83â\88\83f2. f1 â\8b\93 f2 â\89\98 f & â\86\91f2 = g2.
+                   â\88\80f1,f. ⫯f1 = g1 â\86\92 ⫯f = g →
+                   â\88\83â\88\83f2. f1 â\8b\93 f2 â\89\98 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/
@@ -140,8 +141,8 @@ elim (pn_split g2) * #f2 #H2
 qed-.
 
 lemma sor_inv_xpp: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
-                   â\88\80f2,f. â\86\91f2 = g2 â\86\92 â\86\91f = g →
-                   â\88\83â\88\83f1. f1 â\8b\93 f2 â\89\98 f & â\86\91f1 = g1.
+                   â\88\80f2,f. ⫯f2 = g2 â\86\92 ⫯f = g →
+                   â\88\83â\88\83f1. f1 â\8b\93 f2 â\89\98 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/
@@ -150,8 +151,8 @@ elim (pn_split g1) * #f1 #H1
 qed-.
 
 lemma sor_inv_pxn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
-                   â\88\80f1,f. â\86\91f1 = g1 â\86\92 ⫯f = g →
-                   â\88\83â\88\83f2. f1 â\8b\93 f2 â\89\98 f & ⫯f2 = g2.
+                   â\88\80f1,f. ⫯f1 = g1 â\86\92 â\86\91f = g →
+                   â\88\83â\88\83f2. f1 â\8b\93 f2 â\89\98 f & â\86\91f2 = g2.
 #g1 #g2 #g #H #f1 #f #H1 #H0
 elim (pn_split g2) * #f2 #H2
 [ elim (sor_inv_ppn … H … H1 H2 H0)
@@ -160,8 +161,8 @@ elim (pn_split g2) * #f2 #H2
 qed-.
 
 lemma sor_inv_xpn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
-                   â\88\80f2,f. â\86\91f2 = g2 â\86\92 ⫯f = g →
-                   â\88\83â\88\83f1. f1 â\8b\93 f2 â\89\98 f & ⫯f1 = g1.
+                   â\88\80f2,f. ⫯f2 = g2 â\86\92 â\86\91f = g →
+                   â\88\83â\88\83f1. f1 â\8b\93 f2 â\89\98 f & â\86\91f1 = 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: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f. â\86\91f = g →
-                   â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & â\86\91f1 = g1 & â\86\91f2 = g2.
+lemma sor_inv_xxp: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f. ⫯f = g →
+                   â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 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/
@@ -179,25 +180,25 @@ elim (pn_split g1) * #f1 #H1
 qed-.
 
 lemma sor_inv_nxn: ∀g1,g2,g. g1 ⋓ g2 ≘ g →
-                   â\88\80f1,f. ⫯f1 = g1 â\86\92 ⫯f = g →
-                   (â\88\83â\88\83f2. f1 â\8b\93 f2 â\89\98 f & â\86\91f2 = g2) ∨
-                   â\88\83â\88\83f2. f1 â\8b\93 f2 â\89\98 f & ⫯f2 = g2.
+                   â\88\80f1,f. â\86\91f1 = g1 â\86\92 â\86\91f = g →
+                   (â\88\83â\88\83f2. f1 â\8b\93 f2 â\89\98 f & ⫯f2 = g2) ∨
+                   â\88\83â\88\83f2. f1 â\8b\93 f2 â\89\98 f & â\86\91f2 = 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 →
-                   â\88\80f2,f. ⫯f2 = g2 â\86\92 ⫯f = g →
-                   (â\88\83â\88\83f1. f1 â\8b\93 f2 â\89\98 f & â\86\91f1 = g1) ∨
-                   â\88\83â\88\83f1. f1 â\8b\93 f2 â\89\98 f & ⫯f1 = g1.
+                   â\88\80f2,f. â\86\91f2 = g2 â\86\92 â\86\91f = g →
+                   (â\88\83â\88\83f1. f1 â\8b\93 f2 â\89\98 f & ⫯f1 = g1) ∨
+                   â\88\83â\88\83f1. f1 â\8b\93 f2 â\89\98 f & â\86\91f1 = 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: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f. ⫯f = g →
-                   â\88¨â\88¨ â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & ⫯f1 = g1 & â\86\91f2 = g2
-                    | â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & â\86\91f1 = g1 & ⫯f2 = g2
-                    | â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & ⫯f1 = g1 & ⫯f2 = g2.
+lemma sor_inv_xxn: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f. â\86\91f = g →
+                   â\88¨â\88¨ â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & â\86\91f1 = g1 & ⫯f2 = g2
+                    | â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & ⫯f1 = g1 & â\86\91f2 = g2
+                    | â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & â\86\91f1 = g1 & â\86\91f2 = 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: â\88\80f1,f2,x,y. f1 â\8b\93 f2 â\89\98 x â\86\92 f1 â\8b\93 f2 â\89\98 y â\86\92 x â\89\97 y.
+corec theorem sor_mono: â\88\80f1,f2,x,y. f1 â\8b\93 f2 â\89\98 x â\86\92 f1 â\8b\93 f2 â\89\98 y â\86\92 x â\89¡ 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)
@@ -279,22 +280,22 @@ lemma sor_tl: ∀f1,f2,f. f1 ⋓ f2 ≘ f → ⫱f1 ⋓ ⫱f2 ≘ ⫱f.
 ] -Hf #g #Hg #H destruct //
 qed.
 
-lemma sor_xxn_tl: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f. ⫯f = g →
-                  (â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & ⫯f1 = g1 & ⫱g2 = f2) ∨
-                  (â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & ⫱g1 = f1 & ⫯f2 = g2).
+lemma sor_xxn_tl: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f. â\86\91f = g →
+                  (â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & â\86\91f1 = g1 & ⫱g2 = f2) ∨
+                  (â\88\83â\88\83f1,f2. f1 â\8b\93 f2 â\89\98 f & ⫱g1 = f1 & â\86\91f2 = 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: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f2. ⫯f2 = g2 →
-                  â\88\83â\88\83f1,f. f1 â\8b\93 f2 â\89\98 f & ⫱g1 = f1 & ⫯f = g.
+lemma sor_xnx_tl: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f2. â\86\91f2 = g2 →
+                  â\88\83â\88\83f1,f. f1 â\8b\93 f2 â\89\98 f & ⫱g1 = f1 & â\86\91f = 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: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f1. ⫯f1 = g1 →
-                  â\88\83â\88\83f2,f. f1 â\8b\93 f2 â\89\98 f & ⫱g2 = f2 & ⫯f = g.
+lemma sor_nxx_tl: â\88\80g1,g2,g. g1 â\8b\93 g2 â\89\98 g â\86\92 â\88\80f1. â\86\91f1 = g1 →
+                  â\88\83â\88\83f2,f. f1 â\8b\93 f2 â\89\98 f & ⫱g2 = f2 & â\86\91f = 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/
@@ -309,64 +310,64 @@ qed.
 
 (* Properies with test for identity *****************************************)
 
-corec lemma sor_isid_sn: â\88\80f1. ð\9d\90\88â¦\83f1â¦\84 → ∀f2. f1 ⋓ f2 ≘ f2.
+corec lemma sor_isid_sn: â\88\80f1. ð\9d\90\88â\9dªf1â\9d« → ∀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: â\88\80f2. ð\9d\90\88â¦\83f2â¦\84 → ∀f1. f1 ⋓ f2 ≘ f1.
+corec lemma sor_isid_dx: â\88\80f2. ð\9d\90\88â\9dªf2â\9d« → ∀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: â\88\80f1,f2,f. ð\9d\90\88â¦\83f1â¦\84 â\86\92 ð\9d\90\88â¦\83f2â¦\84 â\86\92 ð\9d\90\88â¦\83fâ¦\84 → f1 ⋓ f2 ≘ f.
+lemma sor_isid: â\88\80f1,f2,f. ð\9d\90\88â\9dªf1â\9d« â\86\92 ð\9d\90\88â\9dªf2â\9d« â\86\92 ð\9d\90\88â\9dªfâ\9d« → 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: â\88\80f1,f2,f. ⫱f1 â\8b\93 f2 â\89\98 f â\86\92 f1 â\8b\93 ⫯f2 â\89\98 ⫯f.
+lemma sor_inv_tl_sn: â\88\80f1,f2,f. ⫱f1 â\8b\93 f2 â\89\98 f â\86\92 f1 â\8b\93 â\86\91f2 â\89\98 â\86\91f.
 #f1 #f2 #f elim (pn_split f1) *
 #g1 #H destruct /2 width=7 by sor_pn, sor_nn/
 qed-.
 
-lemma sor_inv_tl_dx: â\88\80f1,f2,f. f1 â\8b\93 ⫱f2 â\89\98 f â\86\92 ⫯f1 â\8b\93 f2 â\89\98 ⫯f.
+lemma sor_inv_tl_dx: â\88\80f1,f2,f. f1 â\8b\93 ⫱f2 â\89\98 f â\86\92 â\86\91f1 â\8b\93 f2 â\89\98 â\86\91f.
 #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: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\88â¦\83f1â¦\84 â\86\92 f2 â\89\97 f.
+lemma sor_isid_inv_sn: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªf1â\9d« â\86\92 f2 â\89¡ f.
 /3 width=4 by sor_isid_sn, sor_mono/
 qed-.
 
-lemma sor_isid_inv_dx: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\88â¦\83f2â¦\84 â\86\92 f1 â\89\97 f.
+lemma sor_isid_inv_dx: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªf2â\9d« â\86\92 f1 â\89¡ f.
 /3 width=4 by sor_isid_dx, sor_mono/
 qed-.
 
-corec lemma sor_fwd_isid1: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\88â¦\83fâ¦\84 â\86\92 ð\9d\90\88â¦\83f1â¦\84.
+corec lemma sor_fwd_isid1: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªfâ\9d« â\86\92 ð\9d\90\88â\9dªf1â\9d«.
 #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: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\88â¦\83fâ¦\84 â\86\92 ð\9d\90\88â¦\83f2â¦\84.
+corec lemma sor_fwd_isid2: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªfâ\9d« â\86\92 ð\9d\90\88â\9dªf2â\9d«.
 #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: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\88â¦\83fâ¦\84 â\86\92 ð\9d\90\88â¦\83f1â¦\84 â\88§ ð\9d\90\88â¦\83f2â¦\84.
+lemma sor_inv_isid3: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªfâ\9d« â\86\92 ð\9d\90\88â\9dªf1â\9d« â\88§ ð\9d\90\88â\9dªf2â\9d«.
 /3 width=4 by sor_fwd_isid2, sor_fwd_isid1, conj/ qed-.
 
 (* Properties with finite colength assignment *******************************)
 
-lemma sor_fcla_ex: â\88\80f1,n1. ð\9d\90\82â¦\83f1â¦\84 â\89\98 n1 â\86\92 â\88\80f2,n2. ð\9d\90\82â¦\83f2â¦\84 ≘ n2 →
-                   â\88\83â\88\83f,n. f1 â\8b\93 f2 â\89\98 f & ð\9d\90\82â¦\83fâ¦\84 ≘ n & (n1 ∨ n2) ≤ n & n ≤ n1 + n2.
+lemma sor_fcla_ex: â\88\80f1,n1. ð\9d\90\82â\9dªf1â\9d« â\89\98 n1 â\86\92 â\88\80f2,n2. ð\9d\90\82â\9dªf2â\9d« ≘ n2 →
+                   â\88\83â\88\83f,n. f1 â\8b\93 f2 â\89\98 f & ð\9d\90\82â\9dªfâ\9d« ≘ 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 *)
@@ -377,16 +378,16 @@ lemma sor_fcla_ex: ∀f1,n1. 𝐂⦃f1⦄ ≘ n1 → ∀f2,n2. 𝐂⦃f2⦄ ≘
 ]
 qed-.
 
-lemma sor_fcla: â\88\80f1,n1. ð\9d\90\82â¦\83f1â¦\84 â\89\98 n1 â\86\92 â\88\80f2,n2. ð\9d\90\82â¦\83f2â¦\84 ≘ n2 → ∀f. f1 ⋓ f2 ≘ f →
-                â\88\83â\88\83n. ð\9d\90\82â¦\83fâ¦\84 ≘ n & (n1 ∨ n2) ≤ n & n ≤ n1 + n2.
+lemma sor_fcla: â\88\80f1,n1. ð\9d\90\82â\9dªf1â\9d« â\89\98 n1 â\86\92 â\88\80f2,n2. ð\9d\90\82â\9dªf2â\9d« ≘ n2 → ∀f. f1 ⋓ f2 ≘ f →
+                â\88\83â\88\83n. ð\9d\90\82â\9dªfâ\9d« ≘ 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: â\88\80f,n. ð\9d\90\82â¦\83fâ¦\84 ≘ n → ∀f1,f2. f1 ⋓ f2 ≘ f →
-                          â\88\83â\88\83n1.  ð\9d\90\82â¦\83f1â¦\84 ≘ n1 & n1 ≤ n.
+lemma sor_fwd_fcla_sn_ex: â\88\80f,n. ð\9d\90\82â\9dªfâ\9d« ≘ n → ∀f1,f2. f1 ⋓ f2 ≘ f →
+                          â\88\83â\88\83n1.  ð\9d\90\82â\9dªf1â\9d« ≘ 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
@@ -398,37 +399,37 @@ lemma sor_fwd_fcla_sn_ex: ∀f,n. 𝐂⦃f⦄ ≘ n → ∀f1,f2. f1 ⋓ f2 ≘
 ]
 qed-.
 
-lemma sor_fwd_fcla_dx_ex: â\88\80f,n. ð\9d\90\82â¦\83fâ¦\84 ≘ n → ∀f1,f2. f1 ⋓ f2 ≘ f →
-                          â\88\83â\88\83n2.  ð\9d\90\82â¦\83f2â¦\84 ≘ n2 & n2 ≤ n.
+lemma sor_fwd_fcla_dx_ex: â\88\80f,n. ð\9d\90\82â\9dªfâ\9d« ≘ n → ∀f1,f2. f1 ⋓ f2 ≘ f →
+                          â\88\83â\88\83n2.  ð\9d\90\82â\9dªf2â\9d« ≘ 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: â\88\80f1,f2. ð\9d\90\85â¦\83f1â¦\84 â\86\92 ð\9d\90\85â¦\83f2â¦\84 â\86\92 â\88\83â\88\83f. f1 â\8b\93 f2 â\89\98 f & ð\9d\90\85â¦\83fâ¦\84.
+lemma sor_isfin_ex: â\88\80f1,f2. ð\9d\90\85â\9dªf1â\9d« â\86\92 ð\9d\90\85â\9dªf2â\9d« â\86\92 â\88\83â\88\83f. f1 â\8b\93 f2 â\89\98 f & ð\9d\90\85â\9dªfâ\9d«.
 #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: â\88\80f1,f2. ð\9d\90\85â¦\83f1â¦\84 â\86\92 ð\9d\90\85â¦\83f2â¦\84 â\86\92 â\88\80f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\85â¦\83fâ¦\84.
+lemma sor_isfin: â\88\80f1,f2. ð\9d\90\85â\9dªf1â\9d« â\86\92 ð\9d\90\85â\9dªf2â\9d« â\86\92 â\88\80f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\85â\9dªfâ\9d«.
 #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: â\88\80f. ð\9d\90\85â¦\83fâ¦\84 â\86\92 â\88\80f1,f2. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\85â¦\83f1â¦\84.
+lemma sor_fwd_isfin_sn: â\88\80f. ð\9d\90\85â\9dªfâ\9d« â\86\92 â\88\80f1,f2. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\85â\9dªf1â\9d«.
 #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: â\88\80f. ð\9d\90\85â¦\83fâ¦\84 â\86\92 â\88\80f1,f2. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\85â¦\83f2â¦\84.
+lemma sor_fwd_isfin_dx: â\88\80f. ð\9d\90\85â\9dªfâ\9d« â\86\92 â\88\80f1,f2. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\85â\9dªf2â\9d«.
 #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: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\85â¦\83fâ¦\84 â\86\92 ð\9d\90\85â¦\83f1â¦\84 â\88§ ð\9d\90\85â¦\83f2â¦\84.
+lemma sor_inv_isfin3: â\88\80f1,f2,f. f1 â\8b\93 f2 â\89\98 f â\86\92 ð\9d\90\85â\9dªfâ\9d« â\86\92 ð\9d\90\85â\9dªf1â\9d« â\88§ ð\9d\90\85â\9dªf2â\9d«.
 /3 width=4 by sor_fwd_isfin_dx, sor_fwd_isfin_sn, conj/ qed-.
 
 (* Inversion lemmas with inclusion ******************************************)