(**************************************************************************)
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 ≝
/2 width=3 by ex2_intro/
qed-.
+(* Main inversion lemmas ****************************************************)
+
+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)
+| cases (sor_inv_npx … H … H1 H2)
+| cases (sor_inv_pnx … H … H1 H2)
+| cases (sor_inv_nnx … H … H1 H2)
+] -g1 -g2
+/3 width=5 by eq_push, eq_next/
+qed-.
+
(* Basic properties *********************************************************)
corec lemma sor_eq_repl_back1: ∀f2,f. eq_repl_back … (λf1. 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/
qed-.
+
+(* Properies on identity ****************************************************)
+
+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.
+#f2 * -f2
+#f2 #g2 #Hf2 #H2 #f1 cases (pn_split f1) *
+/3 width=7 by sor_pp, sor_np/
+qed.
+
+(* Properties on 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.
+#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
+[ /3 width=7 by fcla_push, sor_pp, ex4_2_intro/
+| /3 width=7 by fcla_next, sor_pn, max_S2_le_S, le_S_S, ex4_2_intro/
+| /3 width=7 by fcla_next, sor_np, max_S1_le_S, le_S_S, ex4_2_intro/
+| /4 width=7 by fcla_next, sor_nn, le_S, le_S_S, ex4_2_intro/
+]
+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.
+#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-.
+
+(* Properties on test for finite colength ***********************************)
+
+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⦄.
+#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-.