diff --git a/thys/Relational_Minimum_Spanning_Trees/Boruvka.thy b/thys/Relational_Minimum_Spanning_Trees/Boruvka.thy
--- a/thys/Relational_Minimum_Spanning_Trees/Boruvka.thy
+++ b/thys/Relational_Minimum_Spanning_Trees/Boruvka.thy
@@ -1,3619 +1,3819 @@
 (* Title:      Borůvka's Minimum Spanning Tree Algorithm
    Author:     Nicolas Robinson-O'Brien
    Maintainer: Walter Guttmann <walter.guttmann at canterbury.ac.nz>
 *)
 
 section \<open>Bor\r{u}vka's Minimum Spanning Tree Algorithm\<close>
 
 text \<open>
 In this theory we prove partial correctness of Bor\r{u}vka's minimum spanning tree algorithm.
 \<close>
 
 theory Boruvka
 
 imports
   Relational_Disjoint_Set_Forests.Disjoint_Set_Forests
   Kruskal
 
 begin
 
 subsection \<open>General results\<close>
 
 text \<open>
 The proof is carried out in $m$-$k$-Stone-Kleene relation algebras.
 In this section we give results that hold more generally.
 \<close>
 
 context stone_kleene_relation_algebra
 begin
 
-definition "big_forest H d \<equiv>
-    equivalence H
-  \<and> d \<le> -H
-  \<and> univalent (H * d)
-  \<and> H \<sqinter> d * d\<^sup>T \<le> 1
-  \<and> (H * d)\<^sup>+ \<le> - H"
-
-definition "bf_between_points p q H d \<equiv> point p \<and> point q \<and> p \<le> (H * d)\<^sup>\<star> * H * d"
-
-definition "bf_between_arcs a b H d \<equiv> arc a \<and> arc b \<and> a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * b * top"
-
-text \<open>Theorem 3\<close>
-
 lemma He_eq_He_THe_star:
   assumes "arc e"
     and "equivalence H"
   shows "H * e = H * e * (top * H * e)\<^sup>\<star>"
 proof -
   let ?x = "H * e"
   have 1: "H * e \<le> H * e * (top * H * e)\<^sup>\<star>"
     using comp_isotone star.circ_reflexive by fastforce
   have "H * e * (top * H * e)\<^sup>\<star> = H * e * (top * e)\<^sup>\<star>"
     by (metis assms(2) preorder_idempotent surjective_var)
   also have "... \<le> H * e * (1 \<squnion> top * (e * top)\<^sup>\<star> * e)"
     by (metis eq_refl star.circ_mult_1)
   also have "... \<le> H * e * (1 \<squnion> top * top * e)"
     by (simp add: star.circ_left_top)
   also have "... = H * e \<squnion> H * e * top * e"
     by (simp add: mult.semigroup_axioms semiring.distrib_left semigroup.assoc)
   finally have 2: "H * e * (top * H * e)\<^sup>\<star> \<le> H * e"
     using assms arc_top_arc mult_assoc by auto
   thus ?thesis
     using 1 2 by simp
 qed
 
 lemma path_through_components:
   assumes "equivalence H"
     and "arc e"
   shows "(H * (d \<squnion> e))\<^sup>\<star> = (H * d)\<^sup>\<star> \<squnion> (H * d)\<^sup>\<star> * H * e * (H * d)\<^sup>\<star>"
 proof -
   have "H * e * (H * d)\<^sup>\<star> * H * e \<le> H * e * top * H * e"
     by (simp add: comp_isotone)
   also have "... = H * e * top * e"
     by (metis assms(1) preorder_idempotent surjective_var mult_assoc)
   also have "... = H * e"
     using assms(2) arc_top_arc mult_assoc by auto
   finally have 1: "H * e * (H * d)\<^sup>\<star> * H * e \<le> H * e"
     by simp
   have "(H * (d \<squnion> e))\<^sup>\<star> = (H * d \<squnion> H * e)\<^sup>\<star>"
     by (simp add: comp_left_dist_sup)
   also have "... = (H * d)\<^sup>\<star> \<squnion> (H * d)\<^sup>\<star> * H * e * (H * d)\<^sup>\<star>"
     using 1 star_separate_3 by (simp add: mult_assoc)
   finally show ?thesis
     by simp
 qed
 
 lemma simplify_f:
   assumes "regular p"
     and "regular e"
   shows "(f \<sqinter> - e\<^sup>T \<sqinter> - p) \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p) \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> - p)\<^sup>T \<squnion> e\<^sup>T \<squnion> e = f \<squnion> f\<^sup>T \<squnion> e \<squnion> e\<^sup>T"
 proof -
   have "(f \<sqinter> - e\<^sup>T \<sqinter> - p) \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p) \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> - p)\<^sup>T \<squnion> e\<^sup>T \<squnion> e
     = (f \<sqinter> - e\<^sup>T \<sqinter> - p) \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p) \<squnion> (f\<^sup>T \<sqinter> - e \<sqinter> p\<^sup>T) \<squnion> (f\<^sup>T \<sqinter> - e \<sqinter> - p\<^sup>T) \<squnion> e\<^sup>T \<squnion> e"
     by (simp add: conv_complement conv_dist_inf)
   also have "... =
       ((f \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)) \<sqinter> (- e\<^sup>T \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)) \<sqinter> (- p \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)))
     \<squnion> ((f\<^sup>T \<squnion> (f\<^sup>T \<sqinter> - e \<sqinter> - p\<^sup>T)) \<sqinter> (- e \<squnion> (f\<^sup>T \<sqinter> - e \<sqinter> - p\<^sup>T)) \<sqinter> (p\<^sup>T \<squnion> (f\<^sup>T \<sqinter> - e \<sqinter> - p\<^sup>T)))
     \<squnion> e\<^sup>T \<squnion> e"
     by (metis sup_inf_distrib2 sup_assoc)
   also have "... =
       ((f \<squnion> f) \<sqinter> (f \<squnion> - e\<^sup>T) \<sqinter> (f \<squnion> p) \<sqinter> (- e\<^sup>T \<squnion> f) \<sqinter> (- e\<^sup>T \<squnion> - e\<^sup>T) \<sqinter> (- e\<^sup>T \<squnion> p) \<sqinter> (- p \<squnion> f) \<sqinter> (- p \<squnion> - e\<^sup>T) \<sqinter> (- p \<squnion> p))
     \<squnion> ((f\<^sup>T \<squnion> f\<^sup>T) \<sqinter> (f\<^sup>T \<squnion> - e) \<sqinter> (f\<^sup>T \<squnion> - p\<^sup>T) \<sqinter> (- e \<squnion> f\<^sup>T) \<sqinter> (- e \<squnion> - e) \<sqinter> (- e \<squnion> - p\<^sup>T) \<sqinter> (p\<^sup>T \<squnion> f\<^sup>T) \<sqinter> (p\<^sup>T \<squnion> - e) \<sqinter> (p\<^sup>T \<squnion> - p\<^sup>T))
     \<squnion> e\<^sup>T \<squnion> e"
     using sup_inf_distrib1 sup_assoc inf_assoc sup_inf_distrib1 by simp
   also have "... =
       ((f \<squnion> f) \<sqinter> (f \<squnion> - e\<^sup>T) \<sqinter> (f \<squnion> p) \<sqinter> (f \<squnion> - p) \<sqinter> (- e\<^sup>T \<squnion> f) \<sqinter> (- e\<^sup>T \<squnion> - e\<^sup>T) \<sqinter> (- e\<^sup>T \<squnion> p) \<sqinter> (- e\<^sup>T \<squnion> - p) \<sqinter> (- p \<squnion> p))
     \<squnion> ((f\<^sup>T \<squnion> f\<^sup>T) \<sqinter> (f\<^sup>T \<squnion> - e) \<sqinter> (f\<^sup>T \<squnion> - p\<^sup>T) \<sqinter> (- e \<squnion> f\<^sup>T) \<sqinter> (f\<^sup>T \<squnion> p\<^sup>T) \<sqinter> (- e \<squnion> - e) \<sqinter> (- e \<squnion> - p\<^sup>T) \<sqinter> (- e \<squnion> p\<^sup>T) \<sqinter> (p\<^sup>T \<squnion> - p\<^sup>T))
     \<squnion> e\<^sup>T \<squnion> e"
     by (smt abel_semigroup.commute inf.abel_semigroup_axioms inf.left_commute sup.abel_semigroup_axioms)
   also have "... = (f \<sqinter> - e\<^sup>T \<sqinter> (- p \<squnion> p)) \<squnion> (f\<^sup>T \<sqinter> - e \<sqinter> (p\<^sup>T \<squnion> - p\<^sup>T)) \<squnion> e\<^sup>T \<squnion> e"
     by (smt inf.sup_monoid.add_assoc inf.sup_monoid.add_commute inf_sup_absorb sup.idem)
   also have "... = (f \<sqinter> - e\<^sup>T) \<squnion> (f\<^sup>T \<sqinter> - e) \<squnion> e\<^sup>T \<squnion> e"
     by (metis assms(1) conv_complement inf_top_right stone)
   also have "... = (f \<squnion> e\<^sup>T) \<sqinter> (- e\<^sup>T \<squnion> e\<^sup>T) \<squnion> (f\<^sup>T \<squnion> e) \<sqinter> (- e \<squnion> e)"
     by (metis sup.left_commute sup_assoc sup_inf_distrib2)
   finally show ?thesis
     by (metis abel_semigroup.commute assms(2) conv_complement inf_top_right stone sup.abel_semigroup_axioms sup_assoc)
 qed
 
 lemma simplify_forest_components_f:
   assumes "regular p"
     and "regular e"
     and "injective (f \<sqinter> - e\<^sup>T \<sqinter> - p \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e)"
     and "injective f"
   shows "forest_components ((f \<sqinter> - e\<^sup>T \<sqinter> - p) \<squnion> (f \<sqinter> -e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e) = (f \<squnion> f\<^sup>T \<squnion> e \<squnion> e\<^sup>T)\<^sup>\<star>"
 proof -
   have "forest_components ((f \<sqinter> - e\<^sup>T \<sqinter> - p) \<squnion> (f \<sqinter> -e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e) = wcc ((f \<sqinter> - e\<^sup>T \<sqinter> - p) \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e)"
     by (simp add: assms(3) forest_components_wcc)
   also have "... = ((f \<sqinter> - e\<^sup>T \<sqinter> - p) \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> - p)\<^sup>T \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p) \<squnion> e\<^sup>T)\<^sup>\<star>"
     using conv_dist_sup sup_assoc by auto
   also have "... = ((f \<sqinter> - e\<^sup>T \<sqinter> - p) \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p) \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> - p)\<^sup>T \<squnion> e\<^sup>T \<squnion> e)\<^sup>\<star>"
     using sup_assoc sup_commute by auto
   also have "... = (f \<squnion> f\<^sup>T \<squnion> e \<squnion> e\<^sup>T)\<^sup>\<star>"
     using assms(1, 2, 3, 4) simplify_f by auto
   finally show ?thesis
     by simp
 qed
 
 lemma components_disj_increasing:
   assumes "regular p"
     and "regular e"
     and "injective (f \<sqinter> - e\<^sup>T \<sqinter> - p \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e)"
     and "injective f"
   shows "forest_components f \<le> forest_components (f \<sqinter> - e\<^sup>T \<sqinter> - p \<squnion> (f \<sqinter> - e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e)"
 proof -
   have 1: "forest_components ((f \<sqinter> - e\<^sup>T \<sqinter> - p) \<squnion> (f \<sqinter> -e\<^sup>T \<sqinter> p)\<^sup>T \<squnion> e) = (f \<squnion> f\<^sup>T \<squnion> e \<squnion> e\<^sup>T)\<^sup>\<star>"
     using simplify_forest_components_f assms(1, 2, 3, 4) by blast
   have "forest_components f = wcc f"
     by (simp add: assms(4) forest_components_wcc)
   also have "... \<le> (f \<squnion> f\<^sup>T \<squnion> e\<^sup>T \<squnion> e)\<^sup>\<star>"
     by (simp add: le_supI2 star_isotone sup_commute)
   finally show ?thesis
     using 1 sup.left_commute sup_commute by simp
 qed
 
 lemma fch_equivalence:
   assumes "forest h"
   shows "equivalence (forest_components h)"
   by (simp add: assms forest_components_equivalence)
 
-lemma big_forest_path_split_1:
+lemma forest_modulo_equivalence_path_split_1:
   assumes "arc a"
     and "equivalence H"
   shows "(H * d)\<^sup>\<star> * H * a * top = (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
 proof -
   let ?H = "H"
   let ?x = "?H * (d \<sqinter> -a)"
   let ?y = "?H * a"
   let ?a = "?H * a * top"
   let ?d = "?H * d"
   have 1: "?d\<^sup>\<star> * ?a \<le> ?x\<^sup>\<star> * ?a"
   proof -
     have "?x\<^sup>\<star> *?y * ?x\<^sup>\<star> * ?a \<le> ?x\<^sup>\<star> * ?a * ?a"
       by (smt mult_left_isotone star.circ_right_top top_right_mult_increasing mult_assoc)
     also have "... = ?x\<^sup>\<star> * ?a * a * top"
       by (metis ex231e mult_assoc)
     also have "... = ?x\<^sup>\<star> * ?a"
       by (simp add: assms(1) mult_assoc)
     finally have 11: "?x\<^sup>\<star> *?y * ?x\<^sup>\<star> * ?a \<le> ?x\<^sup>\<star> * ?a"
       by simp
     have "?d\<^sup>\<star> * ?a = (?H * (d \<sqinter> a) \<squnion> ?H * (d \<sqinter> -a))\<^sup>\<star> * ?a"
     proof -
       have 12: "regular a"
         using assms(1) arc_regular by simp
       have "?H * ((d \<sqinter> a) \<squnion> (d \<sqinter> - a)) = ?H * (d \<sqinter> top)"
         using 12 by (metis inf_top_right maddux_3_11_pp)
       thus ?thesis
         using mult_left_dist_sup by auto
     qed
     also have "... \<le> (?y \<squnion> ?x)\<^sup>\<star> * ?a"
       by (metis comp_inf.coreflexive_idempotent comp_isotone inf.cobounded1 inf.sup_monoid.add_commute semiring.add_mono star_isotone top.extremum)
     also have "... = (?x \<squnion> ?y)\<^sup>\<star> * ?a"
       by (simp add: sup_commute mult_assoc)
     also have "... = ?x\<^sup>\<star> * ?a \<squnion> (?x\<^sup>\<star> * ?y * (?x\<^sup>\<star> * ?y)\<^sup>\<star> * ?x\<^sup>\<star>) * ?a"
       by (smt mult_right_dist_sup star.circ_sup_9 star.circ_unfold_sum mult_assoc)
     also have "... \<le> ?x\<^sup>\<star> * ?a \<squnion> (?x\<^sup>\<star> * ?y * (top * ?y)\<^sup>\<star> * ?x\<^sup>\<star>) * ?a"
     proof -
       have "(?x\<^sup>\<star> * ?y)\<^sup>\<star> \<le> (top * ?y)\<^sup>\<star>"
         by (simp add: mult_left_isotone star_isotone)
       thus ?thesis
         by (metis comp_inf.coreflexive_idempotent comp_inf.transitive_star eq_refl mult_left_dist_sup top.extremum mult_assoc)
     qed
     also have "... = ?x\<^sup>\<star> * ?a \<squnion> (?x\<^sup>\<star> * ?y * ?x\<^sup>\<star>) * ?a"
       using assms(1, 2) He_eq_He_THe_star arc_regular mult_assoc by auto
     finally have 13: "(?H * d)\<^sup>\<star> * ?a \<le> ?x\<^sup>\<star> * ?a \<squnion> ?x\<^sup>\<star> * ?y * ?x\<^sup>\<star> * ?a"
       by (simp add: mult_assoc)
     have 14: "?x\<^sup>\<star> * ?y * ?x\<^sup>\<star> * ?a \<le> ?x\<^sup>\<star> * ?a"
       using 11 mult_assoc by auto
     thus ?thesis
       using 13 14 sup.absorb1 by auto
   qed
   have 2: "?d\<^sup>\<star> * ?a \<ge> ?x\<^sup>\<star> *?a"
     by (simp add: comp_isotone star_isotone)
   thus ?thesis
-    using 1 2 antisym mult_assoc by simp
+    using 1 2 order.antisym mult_assoc by simp
 qed
 
 lemma dTransHd_le_1:
   assumes "equivalence H"
     and "univalent (H * d)"
   shows "d\<^sup>T * H * d \<le> 1"
 proof -
   have "d\<^sup>T * H\<^sup>T * H * d \<le> 1"
     using assms(2) conv_dist_comp mult_assoc by auto
   thus ?thesis
     using assms(1) mult_assoc by (simp add: preorder_idempotent)
 qed
 
 lemma HcompaT_le_compHaT:
   assumes "equivalence H"
     and "injective (a * top)"
   shows "-H * a * top \<le> - (H * a * top)"
 proof -
   have "a * top * a\<^sup>T \<le> 1"
     by (metis assms(2) conv_dist_comp symmetric_top_closed vector_top_closed mult_assoc)
   hence "a * top * a\<^sup>T * H \<le> H"
     using assms(1) comp_isotone order_trans by blast
   hence "a * top * top * a\<^sup>T * H \<le> H"
     by (simp add: vector_mult_closed)
   hence "a * top * (H * a * top)\<^sup>T \<le> H"
     by (metis assms(1) conv_dist_comp symmetric_top_closed vector_top_closed mult_assoc)
   thus ?thesis
     using assms(2) comp_injective_below_complement mult_assoc by auto
 qed
 
-text \<open>Theorem 4\<close>
+subsection \<open>Forests modulo an equivalence\<close>
 
-lemma expand_big_forest:
-  assumes "big_forest H d"
+text \<open>
+In the graphical interpretation, the arcs of d are directed towards the root(s) of the \<open>forest_modulo_equivalence\<close>.
+\<close>
+
+definition "forest_modulo_equivalence x d \<equiv> equivalence x \<and> univalent (x * d) \<and> x \<sqinter> d * d\<^sup>T \<le> 1 \<and> (x * d)\<^sup>+ \<sqinter> x \<le> bot"
+
+definition "forest_modulo_equivalence_path a b H d \<equiv> arc a \<and> arc b \<and> a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * b * top"
+
+lemma d_separates_forest_modulo_equivalence_1:
+  assumes "forest_modulo_equivalence x d"
+  shows "x * d \<le> -x"
+proof -
+  have "x * d \<le> (x * d)\<^sup>+"
+    using star.circ_mult_increasing by simp
+  also have "... \<le> -x"
+    using assms(1) forest_modulo_equivalence_def le_bot pseudo_complement by blast
+  finally show ?thesis
+    by simp
+qed
+
+lemma d_separates_forest_modulo_equivalence_2:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> d * x \<le> -x"
+  using forest_modulo_equivalence_def schroeder_6_p d_separates_forest_modulo_equivalence_1 by metis
+
+lemma d_separates_forest_modulo_equivalence_3:
+  assumes "forest_modulo_equivalence x d"
+  shows "d \<le> -x"
+proof -
+  have "1 \<le> x"
+    using assms(1) forest_modulo_equivalence_def by auto
+  then have "d \<le> x * d"
+    using mult_left_isotone by fastforce
+  also have "... \<le> (x * d)\<^sup>+"
+    using star.circ_mult_increasing by simp
+  also have "... \<le> -x"
+    using assms(1) forest_modulo_equivalence_def le_bot pseudo_complement by blast
+  finally show ?thesis
+    by simp
+qed
+
+lemma d_separates_forest_modulo_equivalence_4:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> d\<^sup>T \<le> -x"
+  using d_separates_forest_modulo_equivalence_3 forest_modulo_equivalence_def conv_isotone symmetric_complement_closed by metis
+
+lemma d_separates_forest_modulo_equivalence_5:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> d \<squnion> d\<^sup>T \<le> -x"
+  using d_separates_forest_modulo_equivalence_3 d_separates_forest_modulo_equivalence_4 sup_least by blast
+
+lemma d_separates_forest_modulo_equivalence_6:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> d * x \<squnion> x * d \<le> -x"
+  using d_separates_forest_modulo_equivalence_1 d_separates_forest_modulo_equivalence_2 sup_least by blast
+
+lemma d_separates_forest_modulo_equivalence_7:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> x * (d \<squnion> d\<^sup>T) * x \<le> -x"
+  using d_separates_forest_modulo_equivalence_5 forest_modulo_equivalence_def inf.sup_monoid.add_commute preorder_idempotent pseudo_complement triple_schroeder_p by metis
+
+lemma d_separates_forest_modulo_equivalence_8:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> (d * x)\<^sup>T \<le> -x"
+  using d_separates_forest_modulo_equivalence_2 forest_modulo_equivalence_def conv_isotone symmetric_complement_closed by metis
+
+lemma d_separates_forest_modulo_equivalence_9:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> (x * d)\<^sup>T \<le> -x"
+  by (metis d_separates_forest_modulo_equivalence_1 forest_modulo_equivalence_def conv_isotone symmetric_complement_closed)
+
+lemma d_separates_forest_modulo_equivalence_10:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> (d * x)\<^sup>+ \<le> -x"
+  using forest_modulo_equivalence_def le_bot pseudo_complement schroeder_5_p star_slide mult_assoc by metis
+
+lemma d_separates_forest_modulo_equivalence_11:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> (x * d)\<^sup>+ \<le> -x"
+  using forest_modulo_equivalence_def le_bot pseudo_complement by blast
+
+lemma d_separates_forest_modulo_equivalence_12:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> (d * x)\<^sup>T\<^sup>+ \<le> -x"
+  using d_separates_forest_modulo_equivalence_10 forest_modulo_equivalence_def conv_isotone conv_plus_commute symmetric_complement_closed by metis
+
+lemma d_separates_x_in_forest_13:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> (x * d)\<^sup>T\<^sup>+ \<le> -x"
+  using d_separates_forest_modulo_equivalence_11 forest_modulo_equivalence_def conv_isotone conv_plus_commute symmetric_complement_closed by metis
+
+lemma irreflexive_d_in_forest_modulo_equivalence:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> irreflexive d"
+  by (metis d_separates_forest_modulo_equivalence_3 forest_modulo_equivalence_def inf.cobounded2 inf.left_commute inf.orderE pseudo_complement)
+
+lemma univalent_d_in_forest_modulo_equivalence:
+  assumes "forest_modulo_equivalence x d"
+  shows "univalent d"
+proof -
+  have "d\<^sup>T * d \<le> d\<^sup>T * x\<^sup>T * x * d"
+    using assms(1) forest_modulo_equivalence_def comp_isotone comp_right_one mult_sub_right_one by metis
+  also have "... \<le> 1"
+    using assms(1) forest_modulo_equivalence_def comp_associative conv_dist_comp by auto
+  finally show ?thesis
+    by simp
+qed
+
+lemma acyclic_d_in_forest_modulo_equivalence:
+  assumes "forest_modulo_equivalence x d"
+  shows "acyclic d"
+proof -
+  have "d\<^sup>\<star> \<le> (x * d)\<^sup>\<star>"
+    using assms(1) forest_modulo_equivalence_def mult_left_isotone star.circ_circ_mult star.circ_circ_mult_1 star.circ_extra_circ star.left_plus_circ star_involutive star_isotone star_one star_slide mult_assoc by metis
+  then have "d * d\<^sup>\<star> \<le> d * (x * d)\<^sup>\<star>"
+    using mult_right_isotone by blast
+  also have "... \<le> x * d * (x * d)\<^sup>\<star>"
+    using assms(1) forest_modulo_equivalence_def eq_refl inf.order_trans mult_isotone star.circ_circ_mult_1 star_involutive star_one star_outer_increasing mult_assoc by metis
+  also have "... \<le> -x"
+    using assms d_separates_forest_modulo_equivalence_11 by blast
+  also have "... \<le> -1"
+    using assms(1) forest_modulo_equivalence_def p_antitone by blast
+  finally show ?thesis
+    by simp
+qed
+
+lemma acyclic_dt_d_in_forest_modulo_equivalence:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> acyclic (d\<^sup>T)"
+  using acyclic_d_in_forest_modulo_equivalence conv_plus_commute irreflexive_conv_closed by fastforce
+
+lemma dt_forest_modulo_equivalence_forest:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> forest (d\<^sup>T)"
+  using acyclic_dt_d_in_forest_modulo_equivalence univalent_d_in_forest_modulo_equivalence by simp
+
+lemma var_forest_modulo_equivalence_axiom:
+  shows "forest_modulo_equivalence x d \<Longrightarrow> d\<^sup>T * x * d \<le> 1"
+  using forest_modulo_equivalence_def comp_associative conv_dist_comp preorder_idempotent by metis
+
+lemma forest_modulo_equivalence_wcc:
+  assumes "forest_modulo_equivalence x d"
+  shows "(x * d)\<^sup>\<star> * (x * d)\<^sup>T\<^sup>\<star> = ((x * d) \<squnion> (x * d)\<^sup>T)\<^sup>\<star>"
+  using assms(1) forest_modulo_equivalence_def fc_wcc by force
+
+lemma forest_modulo_equivalence_direction_1:
+  assumes "forest_modulo_equivalence x d"
+  shows "(x * d)\<^sup>\<star> \<sqinter> (x * d)\<^sup>T = bot"
+  using assms(1) d_separates_forest_modulo_equivalence_11 forest_modulo_equivalence_def acyclic_star_below_complement_1 order_lesseq_imp p_antitone_iff by meson
+
+lemma forest_modulo_equivalence_direction_2:
+  assumes "forest_modulo_equivalence x d"
+  shows "(x * d)\<^sup>T\<^sup>\<star> \<sqinter> (x * d) \<le> bot"
+  using assms(1) forest_modulo_equivalence_direction_1 comp_inf.idempotent_bot_closed conv_inf_bot_iff conv_star_commute inf.sup_left_divisibility by metis
+
+lemma forest_modulo_equivalence_separate:
+  assumes "forest_modulo_equivalence x d"
+  shows "(x * d)\<^sup>\<star> * (x * d)\<^sup>T\<^sup>\<star> \<sqinter> (x * d)\<^sup>T * (x * d) \<le> 1"
+proof -
+  have "(x * d)\<^sup>\<star> \<sqinter> (x * d)\<^sup>T * (x * d) = (1 \<squnion> (x * d)\<^sup>+) \<sqinter> (x * d)\<^sup>T * (x * d)"
+    using star_left_unfold_equal by simp
+  also have "... = (1 \<sqinter> (x * d)\<^sup>T * (x * d)) \<squnion> ((x * d)\<^sup>+ \<sqinter> (x * d)\<^sup>T * (x * d))"
+    using comp_inf.semiring.distrib_right by simp
+  also have "... \<le> 1 \<squnion> ((x * d)\<^sup>+ \<sqinter> (x * d)\<^sup>T * (x * d))"
+    using inf.cobounded1 semiring.add_right_mono by blast
+  also have "... = 1 \<squnion> ((x * d)\<^sup>\<star> \<sqinter> (x * d)\<^sup>T) * (x * d)"
+    using assms(1) forest_modulo_equivalence_def forest_modulo_equivalence_direction_1 comp_inf.semiring.mult_zero_right inf.sup_left_divisibility le_infI2 semiring.mult_not_zero sup.orderE by metis
+  also have "... \<le> 1 \<squnion> bot"
+    using assms(1) forest_modulo_equivalence_direction_1 by simp
+  finally have 2: "(x * d)\<^sup>\<star> \<sqinter> (x * d)\<^sup>T * (x * d) \<le> 1"
+    by simp
+  then have 3: "(x * d)\<^sup>T\<^sup>\<star> \<sqinter> (x * d)\<^sup>T * (x * d) \<le> 1"
+    using assms(1) forest_modulo_equivalence_def conv_dist_comp conv_dist_inf conv_involutive conv_star_commute coreflexive_symmetric by metis
+  have "((x * d)\<^sup>\<star> \<squnion> (x * d)\<^sup>T\<^sup>\<star>) \<sqinter> ((x * d)\<^sup>T * (x * d)) \<le> 1"
+    using 2 3 inf_sup_distrib2 by simp
+  thus ?thesis
+    using assms(1) le_infI2 forest_modulo_equivalence_def by blast
+qed
+
+lemma forest_modulo_equivalence_path_trans_closure:
+  assumes "forest_modulo_equivalence x d"
+  shows "(x * d\<^sup>T)\<^sup>+ * x * d * x * d\<^sup>T \<le> (x * d\<^sup>T)\<^sup>+"
+proof -
+  have "(x * d\<^sup>T)\<^sup>+ * x * d * x * d\<^sup>T = (x * d\<^sup>T)\<^sup>\<star> * x * d\<^sup>T * x * d * x * d\<^sup>T"
+    using comp_associative star.circ_plus_same by metis
+  also have "... \<le> (x * d\<^sup>T)\<^sup>\<star> * x * 1 * x * d\<^sup>T"
+    using assms(1) forest_modulo_equivalence_def var_forest_modulo_equivalence_axiom comp_associative mult_left_isotone mult_right_isotone by metis
+  also have "... \<le> (x * d\<^sup>T)\<^sup>\<star> * x * d\<^sup>T"
+    using assms(1) forest_modulo_equivalence_def by (simp add: preorder_idempotent mult_assoc)
+  finally show ?thesis
+    using star.circ_plus_same mult_assoc by simp
+qed
+
+text \<open>
+The \<open>forest_modulo_equivalence\<close> structure is preserved if d is decreased.
+\<close>
+
+lemma forest_modulo_equivalence_decrease_d:
+  assumes "forest_modulo_equivalence x d"
+  shows "forest_modulo_equivalence x (d \<sqinter> c)"
+proof (unfold forest_modulo_equivalence_def, intro conjI)
+  show "reflexive x"
+    using assms(1) forest_modulo_equivalence_def by blast
+next
+  show "transitive x"
+    using assms(1) forest_modulo_equivalence_def by blast
+next
+  show "symmetric x"
+    using assms(1) forest_modulo_equivalence_def by blast
+next
+  show "univalent (x * (d \<sqinter> c))"
+  proof -
+    have "(x * (d \<sqinter> c))\<^sup>T * x * (d \<sqinter> c) \<le> (x * d)\<^sup>T * x * d"
+      using conv_order mult_isotone by auto
+    also have "... \<le> 1"
+      using assms(1) forest_modulo_equivalence_def mult_assoc by auto
+    finally show ?thesis
+      using mult_assoc by auto
+  qed
+next
+  show "coreflexive (x \<sqinter> ((d \<sqinter> c) * (d \<sqinter> c)\<^sup>T))"
+  proof -
+    have "x \<sqinter> (d \<sqinter> c) * (d \<sqinter> c)\<^sup>T \<le> x \<sqinter> d * d\<^sup>T"
+      using conv_dist_inf inf.sup_right_isotone mult_isotone by auto
+    thus ?thesis
+      using assms(1) forest_modulo_equivalence_def order_lesseq_imp by blast
+  qed
+next
+  show "(x * (d \<sqinter> c))\<^sup>+ \<sqinter> x \<le> bot"
+  proof -
+    have "(x * (d \<sqinter> c))\<^sup>+ \<le> (x * d)\<^sup>+"
+      using comp_isotone star_isotone by simp
+    thus ?thesis
+      using assms d_separates_forest_modulo_equivalence_11 dual_order.eq_iff dual_order.trans pseudo_complement by blast
+  qed
+qed
+
+lemma expand_forest_modulo_equivalence:
+  assumes "forest_modulo_equivalence H d"
   shows "(d\<^sup>T * H)\<^sup>\<star> * (H * d)\<^sup>\<star> = (d\<^sup>T * H)\<^sup>\<star> \<squnion> (H * d)\<^sup>\<star>"
 proof -
   have "(H * d)\<^sup>T * H * d \<le> 1"
-    using assms big_forest_def mult_assoc by auto
+    using assms forest_modulo_equivalence_def mult_assoc by auto
   hence "d\<^sup>T * H * H * d \<le> 1"
-    using assms big_forest_def conv_dist_comp by auto
+    using assms forest_modulo_equivalence_def conv_dist_comp by auto
   thus ?thesis
     by (simp add: cancel_separate_eq comp_associative)
 qed
 
-
-lemma big_forest_path_bot:
+lemma forest_modulo_equivalence_path_bot:
   assumes "arc a"
     and "a \<le> d"
-    and "big_forest H d"
+    and "forest_modulo_equivalence H d"
   shows "(d \<sqinter> - a)\<^sup>T * (H * a * top) \<le> bot"
 proof -
   have 1: "d\<^sup>T * H * d \<le> 1"
-    using assms(3) big_forest_def dTransHd_le_1 by blast
+    using assms(3) forest_modulo_equivalence_def dTransHd_le_1 by blast
   hence "d * - 1 * d\<^sup>T \<le> - H"
     using triple_schroeder_p by force
   hence "d * - 1 * d\<^sup>T \<le> 1 \<squnion> - H"
     by (simp add: le_supI2)
   hence "d * d\<^sup>T \<squnion> d * - 1 * d\<^sup>T \<le> 1 \<squnion> - H"
-    by (metis assms(3) big_forest_def inf_commute regular_one_closed shunting_p le_supI)
+    by (metis assms(3) forest_modulo_equivalence_def inf_commute regular_one_closed shunting_p le_supI)
   hence "d * 1 * d\<^sup>T \<squnion> d * - 1 * d\<^sup>T \<le> 1 \<squnion> - H"
     by simp
   hence "d * (1 \<squnion> - 1) * d\<^sup>T \<le> 1 \<squnion> - H"
     using comp_associative mult_right_dist_sup by (simp add: mult_left_dist_sup)
   hence "d * top * d\<^sup>T \<le> 1 \<squnion> - H"
     using regular_complement_top by auto
   hence "d * top * a\<^sup>T \<le> 1 \<squnion> - H"
     using assms(2) conv_isotone dual_order.trans mult_right_isotone by blast
   hence "d * (a * top)\<^sup>T \<le> 1 \<squnion> - H"
     by (simp add: comp_associative conv_dist_comp)
   hence "d \<le> (1 \<squnion> - H) * (a * top)"
     by (simp add: assms(1) shunt_bijective)
   hence "d \<le> a * top \<squnion> - H * a * top"
     by (simp add: comp_associative mult_right_dist_sup)
   also have "... \<le> a * top \<squnion> - (H * a * top)"
-    using assms(1, 3) HcompaT_le_compHaT big_forest_def sup_right_isotone by auto
+    using assms(1, 3) HcompaT_le_compHaT forest_modulo_equivalence_def sup_right_isotone by auto
   finally have "d \<le> a * top \<squnion> - (H * a * top)"
     by simp
   hence "d \<sqinter> --( H * a * top) \<le> a * top"
     using shunting_var_p by auto
   hence 2:"d \<sqinter> H * a * top \<le> a * top"
     using inf.sup_right_isotone order.trans pp_increasing by blast
   have 3:"d \<sqinter> H * a * top \<le> top * a"
   proof -
     have "d \<sqinter> H * a * top \<le> (H * a \<sqinter> d * top\<^sup>T) * (top \<sqinter> (H * a)\<^sup>T * d)"
       by (metis dedekind inf_commute)
     also have "... = d * top \<sqinter> H * a * a\<^sup>T * H\<^sup>T * d"
       by (simp add: conv_dist_comp inf_vector_comp mult_assoc)
     also have "... \<le> d * top \<sqinter> H * a * d\<^sup>T * H\<^sup>T * d"
       using assms(2) mult_right_isotone mult_left_isotone conv_isotone inf.sup_right_isotone by auto
     also have "... = d * top \<sqinter> H * a * d\<^sup>T * H * d"
-      using assms(3) big_forest_def by auto
+      using assms(3) forest_modulo_equivalence_def by auto
     also have "... \<le> d * top \<sqinter> H * a * 1"
       using 1 by (metis inf.sup_right_isotone mult_right_isotone mult_assoc)
     also have "... \<le> H * a"
       by simp
     also have "... \<le> top * a"
       by (simp add: mult_left_isotone)
     finally have "d \<sqinter> H * a * top \<le> top * a"
       by simp
     thus ?thesis
       by simp
   qed
   have "d \<sqinter> H * a * top \<le> a * top \<sqinter> top * a"
     using 2 3 by simp
   also have "... = a * top * top * a"
     by (metis comp_associative comp_inf.star.circ_decompose_9 comp_inf.star_star_absorb comp_inf_covector vector_inf_comp vector_top_closed)
   also have "... = a * top * a"
     by (simp add: vector_mult_closed)
   finally have 4:"d \<sqinter> H * a * top \<le> a"
     by (simp add: assms(1) arc_top_arc)
   hence "d \<sqinter> - a \<le> -(H * a * top)"
     using assms(1) arc_regular p_shunting_swap by fastforce
   hence "(d \<sqinter> - a) * top \<le> -(H * a * top)"
     using mult.semigroup_axioms p_antitone_iff schroeder_4_p semigroup.assoc by fastforce
   thus ?thesis
     by (simp add: schroeder_3_p)
 qed
 
-lemma big_forest_path_split_2:
+lemma forest_modulo_equivalence_path_split_2:
   assumes "arc a"
     and "a \<le> d"
-    and "big_forest H d"
+    and "forest_modulo_equivalence H d"
   shows "(H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top = (H * ((d \<sqinter> - a) \<squnion> (d \<sqinter> - a)\<^sup>T))\<^sup>\<star> * H * a * top"
 proof -
   let ?lhs = "(H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
   have 1: "d\<^sup>T * H * d \<le> 1"
-    using assms(3) big_forest_def dTransHd_le_1 by blast
+    using assms(3) forest_modulo_equivalence_def dTransHd_le_1 by blast
   have 2: "H * a * top \<le> ?lhs"
     by (metis le_iff_sup star.circ_loop_fixpoint star.circ_transitive_equal star_involutive sup_commute mult_assoc)
   have "(d \<sqinter> - a)\<^sup>T * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top) = (d \<sqinter> - a)\<^sup>T * H * (d \<sqinter> - a) * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top)"
   proof -
     have "(d \<sqinter> - a)\<^sup>T * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top) = (d \<sqinter> - a)\<^sup>T * (1 \<squnion> H * (d \<sqinter> - a) * (H * (d \<sqinter> - a))\<^sup>\<star>) * (H * a * top)"
       by (simp add: star_left_unfold_equal)
     also have "... = (d \<sqinter> - a)\<^sup>T * H * a * top \<squnion> (d \<sqinter> - a)\<^sup>T * H * (d \<sqinter> - a) * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top)"
       by (smt mult_left_dist_sup star.circ_loop_fixpoint star.circ_mult_1 star_slide sup_commute mult_assoc)
     also have "... = bot \<squnion> (d \<sqinter> - a)\<^sup>T * H * (d \<sqinter> - a) * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top)"
-      by (metis assms(1, 2, 3) big_forest_path_bot mult_assoc le_bot)
+      by (metis assms(1, 2, 3) forest_modulo_equivalence_path_bot mult_assoc le_bot)
     thus ?thesis
       by (simp add: calculation)
   qed
   also have "... \<le> d\<^sup>T * H * d * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top)"
     using conv_isotone inf.cobounded1 mult_isotone by auto
   also have "... \<le> 1 * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top)"
     using 1 by (metis le_iff_sup mult_right_dist_sup)
   finally have 3: "(d \<sqinter> - a)\<^sup>T * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top) \<le> ?lhs"
     using mult_assoc by auto
   hence 4: "H * (d \<sqinter> - a)\<^sup>T * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top) \<le> ?lhs"
   proof -
     have "H * (d \<sqinter> - a)\<^sup>T * (H * (d \<sqinter> - a))\<^sup>\<star> * (H * a * top) \<le> H * (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
       using 3 mult_right_isotone mult_assoc by auto
     also have "... = H * H * ((d \<sqinter> - a) * H)\<^sup>\<star> * H * a * top"
-      by (metis assms(3) big_forest_def star_slide mult_assoc preorder_idempotent)
+      by (metis assms(3) forest_modulo_equivalence_def star_slide mult_assoc preorder_idempotent)
     also have "... = H * ((d \<sqinter> - a) * H)\<^sup>\<star> * H * a * top"
-      using assms(3) big_forest_def preorder_idempotent by fastforce
+      using assms(3) forest_modulo_equivalence_def preorder_idempotent by fastforce
     finally show ?thesis
-      by (metis assms(3) big_forest_def preorder_idempotent star_slide mult_assoc)
+      by (metis assms(3) forest_modulo_equivalence_def preorder_idempotent star_slide mult_assoc)
   qed
   have 5: "(H * (d \<sqinter> - a) \<squnion> H * (d \<sqinter> - a)\<^sup>T) * (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top \<le> ?lhs"
   proof -
     have 51: "H * (d \<sqinter> - a) * (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top \<le> (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
       using star.left_plus_below_circ mult_left_isotone by simp
     have 52: "(H * (d \<sqinter> - a) \<squnion> H * (d \<sqinter> - a)\<^sup>T) * (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top = H * (d \<sqinter> - a) * (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top \<squnion> H * (d \<sqinter> - a)\<^sup>T * (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
       using mult_right_dist_sup by auto
     hence "... \<le> (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top \<squnion> H * (d \<sqinter> - a)\<^sup>T * (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top"
       using star.left_plus_below_circ mult_left_isotone sup_left_isotone by auto
     thus ?thesis
       using 4 51 52 mult_assoc by auto
   qed
   hence "(H * (d \<sqinter> - a) \<squnion> H * (d \<sqinter> - a)\<^sup>T)\<^sup>\<star> * H * a * top \<le> ?lhs"
   proof -
     have "(H * (d \<sqinter> - a) \<squnion> H * (d \<sqinter> - a)\<^sup>T)\<^sup>\<star> * (H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top \<le> ?lhs"
       using 5 star_left_induct_mult_iff mult_assoc by auto
     thus ?thesis
       using star.circ_decompose_11 star_decompose_1 by auto
   qed
   hence 6: "(H * ((d \<sqinter> - a) \<squnion> (d \<sqinter> - a)\<^sup>T))\<^sup>\<star> * H * a * top \<le> ?lhs"
     using mult_left_dist_sup by auto
   have 7: "(H * (d \<sqinter> - a))\<^sup>\<star> * H * a * top \<le> (H * ((d \<sqinter> - a) \<squnion> (d \<sqinter> - a)\<^sup>T))\<^sup>\<star> * H * a * top"
     by (simp add: mult_left_isotone semiring.distrib_left star_isotone)
   thus ?thesis
     using 6 7 by (simp add: mult_assoc)
 qed
 
 end
 
+context stone_relation_algebra
+begin
+
+text \<open>
+A \<open>vector_classes\<close> corresponds to one or more equivalence classes and a \<open>unique_vector_class\<close> corresponds to a single equivalence class.
+\<close>
+
+definition vector_classes        :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "vector_classes x v \<equiv> regular x \<and> regular v \<and> equivalence x \<and> vector v \<and> x * v \<le> v \<and> v \<noteq> bot"
+definition unique_vector_class   :: "'a \<Rightarrow> 'a \<Rightarrow> bool" where "unique_vector_class x v \<equiv> vector_classes x v \<and> v * v\<^sup>T \<le> x"
+
+end
+
 subsection \<open>An operation to select components\<close>
 
 text \<open>
 We introduce the operation \<open>choose_component\<close>.
 \begin{itemize}
-\item Axiom \<open>component_in_v\<close> expresses that the result of \<open>choose_component\<close> is contained in the set of vertices, $v$, we are selecting from, ignoring the weights.
-\item Axiom \<open>component_is_vector\<close> states that the result of \<open>choose_component\<close> is a vector.
-\item Axiom \<open>component_is_regular\<close> states that the result of \<open>choose_component\<close> is regular.
-\item Axiom \<open>component_is_connected\<close> states that any two vertices from the result of \<open>choose_component\<close> are connected in $e$.
-\item Axiom \<open>component_single\<close> states that the result of \<open>choose_component\<close> is closed under being connected in $e$.
-\item Finally, axiom \<open>component_not_bot_when_v_bot_bot\<close> expresses that the operation \<open>choose_component\<close> returns a non-empty component if the input satisfies the given criteria.
+  \item Axiom \<open>component_in_v\<close> expresses that the result of \<open>choose_component\<close> is contained in the set of vertices, $v$, we are selecting from, ignoring the weights.
+  \item Axiom \<open>component_is_vector\<close> states that the result of \<open>choose_component\<close> is a vector.
+  \item Axiom \<open>component_is_regular\<close> states that the result of \<open>choose_component\<close> is regular.
+  \item Axiom \<open>component_is_connected\<close> states that any two vertices from the result of \<open>choose_component\<close> are connected in $e$.
+  \item Axiom \<open>component_single\<close> states that the result of \<open>choose_component\<close> is closed under being connected in $e$.
+  \item Finally, axiom \<open>component_not_bot_when_v_bot_bot\<close> expresses that the operation \<open>choose_component\<close> returns a non-empty component if the input satisfies the given criteria.
 \end{itemize}
 \<close>
 
 class choose_component =
   fixes choose_component :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"
 
 class choose_component_algebra = choose_component + stone_relation_algebra +
-  assumes component_in_v: "choose_component e v \<le> --v"
-  assumes component_is_vector: "vector (choose_component e v)"
-  assumes component_is_regular: "regular (choose_component e v)"
-  assumes component_is_connected: "choose_component e v * (choose_component e v)\<^sup>T \<le> e"
-  assumes component_single: "choose_component e v = e * choose_component e v"
-  assumes component_not_bot_when_v_bot_bot: "
-      regular e
-    \<and> equivalence e
-    \<and> vector v
-    \<and> regular v
-    \<and> e * v = v
-    \<and> v \<noteq> bot \<longrightarrow> choose_component e v \<noteq> bot"
-
-text \<open>Theorem 1\<close>
+  assumes component_is_vector:              "vector (choose_component e v)"
+  assumes component_is_regular:             "regular (choose_component e v)"
+  assumes component_in_v:                   "choose_component e v \<le> --v"
+  assumes component_is_connected:           "choose_component e v * (choose_component e v)\<^sup>T \<le> e"
+  assumes component_single:                 "e * choose_component e v \<le> choose_component e v"
+  assumes component_not_bot_when_v_bot_bot: "vector_classes e v \<longrightarrow> choose_component e v \<noteq> bot"
 
 text \<open>
 Every \<open>m_kleene_algebra\<close> is an instance of \<open>choose_component_algebra\<close> when the \<open>choose_component\<close> operation is defined as follows:
 \<close>
 
 context m_kleene_algebra
 begin
 
-definition "choose_component_input_condition e v \<equiv>
-    regular e
-  \<and> equivalence e
-  \<and> vector v
-  \<and> regular v
-  \<and> e * v = v"
-
 definition "m_choose_component e v \<equiv>
-  if choose_component_input_condition e v then
+  if vector_classes e v then
     e * minarc(v) * top
   else
     bot"
 
 sublocale m_choose_component_algebra: choose_component_algebra where choose_component = m_choose_component
 proof
   fix e v
   show "m_choose_component e v \<le> -- v"
-  proof (cases "choose_component_input_condition e v")
+  proof (cases "vector_classes e v")
     case True
     hence "m_choose_component e v = e * minarc(v) * top"
       by (simp add: m_choose_component_def)
     also have "... \<le> e * --v * top"
       by (simp add: comp_isotone minarc_below)
     also have "... = e * v * top"
-      using True choose_component_input_condition_def by auto
-    also have "... = v * top"
-      using True choose_component_input_condition_def by auto
+      using True vector_classes_def by auto
+    also have "... \<le> v * top"
+      using True vector_classes_def mult_assoc by auto
     finally show ?thesis
-      using True choose_component_input_condition_def by auto
+      using True vector_classes_def by auto
   next
     case False
     hence "m_choose_component e v = bot"
       using False m_choose_component_def by auto
     thus ?thesis
       by simp
   qed
 next
   fix e v
   show "vector (m_choose_component e v)"
-  proof (cases "choose_component_input_condition e v")
+  proof (cases "vector_classes e v")
     case True
     thus ?thesis
       by (simp add: mult_assoc m_choose_component_def)
   next
     case False
     thus ?thesis
       by (simp add: m_choose_component_def)
   qed
 next
   fix e v
   show "regular (m_choose_component e v)"
-    using choose_component_input_condition_def minarc_regular regular_closed_star regular_mult_closed m_choose_component_def by auto
+    using minarc_regular regular_mult_closed vector_classes_def m_choose_component_def by auto
 next
   fix e v
   show "m_choose_component e v * (m_choose_component e v)\<^sup>T \<le> e"
-  proof (cases "choose_component_input_condition e v")
+  proof (cases "vector_classes e v")
     case True
-    assume 1: "choose_component_input_condition e v"
+    assume 1: "vector_classes e v"
     hence "m_choose_component e v * (m_choose_component e v)\<^sup>T = e * minarc(v) * top * (e * minarc(v) * top)\<^sup>T"
       by (simp add: m_choose_component_def)
     also have "... = e * minarc(v) * top * top\<^sup>T * minarc(v)\<^sup>T * e\<^sup>T"
       by (metis comp_associative conv_dist_comp)
     also have "... = e * minarc(v) * top * top * minarc(v)\<^sup>T * e"
-      using 1 choose_component_input_condition_def by auto
+      using True vector_classes_def by auto
     also have "... = e * minarc(v) * top * minarc(v)\<^sup>T * e"
       by (simp add: comp_associative)
     also have "... \<le> e"
     proof (cases "v = bot")
       case True
       thus ?thesis
         by (simp add: True minarc_bot)
     next
       case False
       assume 3: "v \<noteq> bot"
       hence "e * minarc(v) * top * minarc(v)\<^sup>T \<le> e * 1"
         using 3 minarc_arc arc_expanded comp_associative mult_right_isotone by fastforce
       hence "e * minarc(v) * top * minarc(v)\<^sup>T * e \<le> e * 1 * e"
         using mult_left_isotone by auto
       also have "... = e"
-        using 1 choose_component_input_condition_def preorder_idempotent by auto
+        using True preorder_idempotent vector_classes_def by auto
       thus ?thesis
         using calculation by auto
     qed
     thus ?thesis
       by (simp add: calculation)
   next
     case False
     thus ?thesis
       by (simp add: m_choose_component_def)
   qed
 next
   fix e v
-  show "m_choose_component e v = e * m_choose_component e v"
-  proof (cases "choose_component_input_condition e v")
+  show "e * m_choose_component e v \<le> m_choose_component e v"
+  proof (cases "vector_classes e v")
     case True
     thus ?thesis
-      by (metis choose_component_input_condition_def preorder_idempotent m_choose_component_def mult_assoc)
+      using comp_right_one dual_order.eq_iff mult_isotone vector_classes_def m_choose_component_def mult_assoc by metis
   next
     case False
     thus ?thesis
       by (simp add: m_choose_component_def)
   qed
 next
   fix e v
-  show "regular e \<and> equivalence e \<and> vector v \<and> regular v \<and> e * v = v \<and> v \<noteq> bot \<longrightarrow> m_choose_component e v \<noteq> bot"
-  proof (cases "choose_component_input_condition e v")
+  show "vector_classes e v \<longrightarrow> m_choose_component e v \<noteq> bot"
+  proof (cases "vector_classes e v")
     case True
     hence "m_choose_component e v \<ge> minarc(v) * top"
-      by (metis choose_component_input_condition_def mult_1_left mult_left_isotone m_choose_component_def)
+      using vector_classes_def m_choose_component_def comp_associative minarc_arc shunt_bijective by fastforce
     also have "... \<ge> minarc(v)"
       using calculation dual_order.trans top_right_mult_increasing by blast
     thus ?thesis
-      using True bot_unique minarc_bot_iff by fastforce
+      using le_bot minarc_bot_iff vector_classes_def by fastforce
   next
     case False
     thus ?thesis
-      using choose_component_input_condition_def by blast
+      by blast
   qed
 qed
 
 end
 
 subsection \<open>m-k-Stone-Kleene relation algebras\<close>
 
 text \<open>
 $m$-$k$-Stone-Kleene relation algebras are an extension of $m$-Kleene algebras where the \<open>choose_component\<close> operation has been added.
 \<close>
 
 class m_kleene_algebra_choose_component =
   m_kleene_algebra
   + choose_component_algebra
 begin
 
 text \<open>
 A \<open>selected_edge\<close> is a minimum-weight edge whose source is in a component, with respect to $h$, $j$ and $g$, and whose target is not in that component.
 \<close>
 
 abbreviation "selected_edge h j g \<equiv> minarc (choose_component (forest_components h) j * - choose_component (forest_components h) j\<^sup>T \<sqinter> g)"
 
 text \<open>
 A \<open>path\<close> is any sequence of edges in the forest, $f$, of the graph, $g$, backwards from the target of the \<open>selected_edge\<close> to a root in $f$.
 \<close>
 
 abbreviation "path f h j g \<equiv> top * selected_edge h j g * (f \<sqinter> - selected_edge h j g\<^sup>T)\<^sup>T\<^sup>\<star>"
 
 definition "boruvka_outer_invariant f g \<equiv>
     symmetric g
   \<and> forest f
   \<and> f \<le> --g
   \<and> regular f
   \<and> (\<exists>w . minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T)"
 
 definition "boruvka_inner_invariant j f h g d \<equiv>
     boruvka_outer_invariant f g
   \<and> g \<noteq> bot
-  \<and> vector j
-  \<and> regular j
-  \<and> boruvka_outer_invariant h g
-  \<and> forest h
-  \<and> forest_components h \<le> forest_components f
-  \<and> big_forest (forest_components h) d
+  \<and> regular d
+  \<and> regular j \<and> vector j
+  \<and> regular h \<and> forest h
+  \<and> forest_components h * j = j
+  \<and> forest_modulo_equivalence (forest_components h) d
   \<and> d * top \<le> - j
-  \<and> forest_components h * j = j
-  \<and> forest_components f = (forest_components h * (d \<squnion> d\<^sup>T))\<^sup>\<star> * forest_components h
   \<and> f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T
-  \<and> (\<forall> a b . bf_between_arcs a b (forest_components h) d \<and> a \<le> -(forest_components h) \<sqinter> -- g \<and> b \<le> d
-    \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g))
-  \<and> regular d"
+  \<and> (\<forall> a b . forest_modulo_equivalence_path a b (forest_components h) d \<and> a \<le> -(forest_components h) \<sqinter> -- g \<and> b \<le> d \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g))"
+
+lemma F_is_H_and_d:
+  assumes "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
+    and "injective f"
+    and "injective h"
+  shows "forest_components f = (forest_components h * (d \<squnion> d\<^sup>T))\<^sup>\<star> * forest_components h"
+proof -
+  have "forest_components f = (f \<squnion> f\<^sup>T)\<^sup>\<star>"
+    using assms(2) cancel_separate_1 by simp
+  also have "... = (h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T)\<^sup>\<star>"
+    using assms(1) by auto
+  also have "... = ((h \<squnion> h\<^sup>T)\<^sup>\<star> * (d \<squnion> d\<^sup>T))\<^sup>\<star> * (h \<squnion> h\<^sup>T)\<^sup>\<star>"
+    using star.circ_sup_9 sup_assoc by metis
+  also have "... = (forest_components h * (d \<squnion> d\<^sup>T))\<^sup>\<star> * forest_components h"
+    using assms(3) forest_components_wcc by simp
+  finally show ?thesis
+    by simp
+qed
+
+lemma H_below_F:
+  assumes "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
+    and "injective f"
+    and "injective h"
+  shows "forest_components h \<le> forest_components f"
+  using assms(1, 2, 3) cancel_separate_1 dual_order.trans star.circ_sub_dist by metis
+
+lemma H_below_regular_g:
+  assumes "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
+    and "f \<le> --g"
+    and "symmetric g"
+  shows "h \<le> --g"
+proof -
+  have "h \<le> f \<squnion> f\<^sup>T"
+    using assms(1) sup_assoc by simp
+  also have "... \<le> --g"
+    using assms(2, 3) conv_complement conv_isotone by fastforce
+  finally show ?thesis
+    using order_trans by blast
+qed
 
 lemma expression_equivalent_without_e_complement:
   assumes "selected_edge h j g \<le> - forest_components f"
   shows "f \<sqinter> - (selected_edge h j g)\<^sup>T \<sqinter> - (path f h j g) \<squnion> (f \<sqinter> - (selected_edge h j g)\<^sup>T \<sqinter> (path f h j g))\<^sup>T \<squnion> (selected_edge h j g)
          = f \<sqinter> - (path f h j g) \<squnion> (f \<sqinter> (path f h j g))\<^sup>T \<squnion> (selected_edge h j g)"
 proof -
   let ?p = "path f h j g"
   let ?e = "selected_edge h j g"
   let ?F = "forest_components f"
   have 1: "?e \<le> - ?F"
     by (simp add: assms)
   have "f\<^sup>T \<le> ?F"
     by (metis conv_dist_comp conv_involutive conv_order conv_star_commute forest_components_increasing)
   hence "- ?F \<le> - f\<^sup>T"
     using p_antitone by auto
   hence "?e \<le> - f\<^sup>T"
     using 1 dual_order.trans by blast
   hence "f\<^sup>T \<le> - ?e"
     by (simp add: p_antitone_iff)
   hence "f\<^sup>T\<^sup>T \<le> - ?e\<^sup>T"
     by (metis conv_complement conv_dist_inf inf.orderE inf.orderI)
   hence "f \<le> - ?e\<^sup>T"
     by auto
   hence "f = f \<sqinter> - ?e\<^sup>T"
     using inf.orderE by blast
   hence "f \<sqinter> - ?e\<^sup>T \<sqinter> - ?p \<squnion> (f \<sqinter> - ?e\<^sup>T \<sqinter> ?p)\<^sup>T \<squnion> ?e = f \<sqinter> - ?p \<squnion> (f \<sqinter> ?p)\<^sup>T \<squnion> ?e"
     by auto
   thus ?thesis by auto
 qed
 
-text \<open>Theorem 2\<close>
-
 text \<open>
 The source of the \<open>selected_edge\<close> is contained in $j$, the vector describing those vertices still to be processed in the inner loop of Bor\r{u}vka's algorithm.
 \<close>
 
 lemma et_below_j:
   assumes "vector j"
     and "regular j"
     and "j \<noteq> bot"
   shows "selected_edge h j g * top \<le> j"
 proof -
   let ?e = "selected_edge h j g"
   let ?c = "choose_component (forest_components h) j"
   have "?e * top \<le> --(?c * -?c\<^sup>T \<sqinter> g) * top"
     using comp_isotone minarc_below by blast
   also have "... = (--(?c * -?c\<^sup>T) \<sqinter> --g) * top"
     by simp
   also have "... = (?c * -?c\<^sup>T \<sqinter> --g) * top"
     using component_is_regular regular_mult_closed by auto
   also have "... = (?c \<sqinter> -?c\<^sup>T \<sqinter> --g) * top"
     by (metis component_is_vector conv_complement vector_complement_closed vector_covector)
   also have "... \<le> ?c * top"
     using inf.cobounded1 mult_left_isotone order_trans by blast
   also have "... \<le> j * top"
     by (metis assms(2) comp_inf.star.circ_sup_2 comp_isotone component_in_v)
   also have "... = j"
     by (simp add: assms(1))
   finally show ?thesis
     by simp
 qed
 
-subsubsection \<open>Components of forests and big forests\<close>
+subsubsection \<open>Components of forests and forests modulo an equivalence\<close>
 
 text \<open>
-We prove a number of properties about \<open>big_forest\<close> and \<open>forest_components\<close>.
+We prove a number of properties about \<open>forest_modulo_equivalence\<close> and \<open>forest_components\<close>.
 \<close>
 
+lemma component_single_eq:
+  assumes "equivalence x"
+  shows "choose_component x v = x * choose_component x v"
+proof -
+  have 1: "choose_component x v \<le> x * choose_component x v"
+    by (meson component_is_connected ex231c mult_isotone order_lesseq_imp)
+  thus ?thesis
+    by (simp add: component_single order.antisym)
+qed
+
 lemma fc_j_eq_j_inv:
   assumes "forest h"
     and "forest_components h * j = j"
   shows "forest_components h * (j \<sqinter> - choose_component (forest_components h) j) = j \<sqinter> - choose_component (forest_components h) j"
 proof -
   let ?c = "choose_component (forest_components h) j"
   let ?H = "forest_components h"
   have 1:"equivalence ?H"
     by (simp add: assms(1) forest_components_equivalence)
   have "?H * (j \<sqinter> - ?c) = ?H * j \<sqinter> ?H * - ?c"
     using 1 by (metis assms(2) equivalence_comp_dist_inf inf.sup_monoid.add_commute)
   hence 2: "?H * (j \<sqinter> - ?c) = j \<sqinter> ?H * - ?c"
     by (simp add: assms(2))
   have 3: "j \<sqinter> - ?c \<le> ?H * - ?c"
     using 1 by (metis assms(2) dedekind_1 dual_order.trans equivalence_comp_dist_inf inf.cobounded2)
   have "?H * ?c \<le> ?c"
     using component_single by auto
   hence "?H\<^sup>T * ?c \<le> ?c"
     using 1 by simp
   hence "?H * - ?c \<le> - ?c"
     using component_is_regular schroeder_3_p by force
   hence "j \<sqinter> ?H * - ?c \<le> j \<sqinter> - ?c"
     using inf.sup_right_isotone by auto
   thus ?thesis
     using 2 3 order.antisym by simp
 qed
 
-text \<open>Theorem 5\<close>
-
 text \<open>
-There is a path in the \<open>big_forest\<close> between edges $a$ and $b$ if and only if there is either a path in the \<open>big_forest\<close> from $a$ to $b$ or one from $a$ to $c$ and one from $c$ to $b$.
+There is a path in the \<open>forest_modulo_equivalence\<close> between edges $a$ and $b$ if and only if there is either a path in the \<open>forest_modulo_equivalence\<close> from $a$ to $b$ or one from $a$ to $c$ and one from $c$ to $b$.
 \<close>
 
-lemma big_forest_path_split_disj:
+lemma forest_modulo_equivalence_path_split_disj:
   assumes "equivalence H"
     and "arc c"
     and "regular a \<and> regular b \<and> regular c \<and> regular d \<and> regular H"
-  shows "bf_between_arcs a b H (d \<squnion> c) \<longleftrightarrow> bf_between_arcs a b H d \<or> (bf_between_arcs a c H d \<and> bf_between_arcs c b H d)"
+  shows "forest_modulo_equivalence_path a b H (d \<squnion> c) \<longleftrightarrow> forest_modulo_equivalence_path a b H d \<or> (forest_modulo_equivalence_path a c H d \<and> forest_modulo_equivalence_path c b H d)"
 proof -
-  have 1: "bf_between_arcs a b H (d \<squnion> c) \<longrightarrow> bf_between_arcs a b H d \<or> (bf_between_arcs a c H d \<and> bf_between_arcs c b H d)"
+  have 1: "forest_modulo_equivalence_path a b H (d \<squnion> c) \<longrightarrow> forest_modulo_equivalence_path a b H d \<or> (forest_modulo_equivalence_path a c H d \<and> forest_modulo_equivalence_path c b H d)"
   proof (rule impI)
-    assume 11: "bf_between_arcs a b H (d \<squnion> c)"
+    assume 11: "forest_modulo_equivalence_path a b H (d \<squnion> c)"
     hence "a\<^sup>T * top \<le> (H * (d \<squnion> c))\<^sup>\<star> * H * b * top"
-      by (simp add: bf_between_arcs_def)
+      by (simp add: forest_modulo_equivalence_path_def)
     also have "... = ((H * d)\<^sup>\<star> \<squnion> (H * d)\<^sup>\<star> * H * c * (H * d)\<^sup>\<star>) * H * b * top"
       using assms(1, 2) path_through_components by simp
     also have "... = (H * d)\<^sup>\<star> * H * b * top \<squnion> (H * d)\<^sup>\<star> * H * c * (H * d)\<^sup>\<star> * H * b * top"
       by (simp add: mult_right_dist_sup)
     finally have 12:"a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * b * top \<squnion> (H * d)\<^sup>\<star> * H * c * (H * d)\<^sup>\<star> * H * b * top"
       by simp
     have 13: "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * b * top \<or> a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * c * (H * d)\<^sup>\<star> * H * b * top"
     proof (rule point_in_vector_sup)
       show "point (a\<^sup>T * top)"
-        using 11 bf_between_arcs_def mult_assoc by auto
+        using 11 forest_modulo_equivalence_path_def mult_assoc by auto
     next
       show "vector ((H * d)\<^sup>\<star> * H * b * top)"
         using vector_mult_closed by simp
     next
       show "regular ((H * d)\<^sup>\<star> * H * b * top)"
         using assms(3) pp_dist_comp pp_dist_star by auto
     next
       show "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * b * top \<squnion> (H * d)\<^sup>\<star> * H * c * (H * d)\<^sup>\<star> * H * b * top"
         using 12 by simp
     qed
-    thus "bf_between_arcs a b H d \<or> (bf_between_arcs a c H d \<and> bf_between_arcs c b H d)"
+    thus "forest_modulo_equivalence_path a b H d \<or> (forest_modulo_equivalence_path a c H d \<and> forest_modulo_equivalence_path c b H d)"
     proof (cases "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * b * top")
       case True
       assume "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * b * top"
-      hence "bf_between_arcs a b H d"
-        using 11 bf_between_arcs_def by auto
+      hence "forest_modulo_equivalence_path a b H d"
+        using 11 forest_modulo_equivalence_path_def by auto
       thus ?thesis
         by simp
     next
       case False
       have 14: "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * c * (H * d)\<^sup>\<star> * H * b * top"
         using 13 by (simp add: False)
       hence 15: "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * c * top"
         by (metis mult_right_isotone order_lesseq_imp top_greatest mult_assoc)
       have "c\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * b * top"
       proof (rule ccontr)
         assume "\<not> c\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * b * top"
         hence "c\<^sup>T * top \<le> -((H * d)\<^sup>\<star> * H * b * top)"
           by (meson assms(2, 3) point_in_vector_or_complement regular_closed_star regular_closed_top regular_mult_closed vector_mult_closed vector_top_closed)
         hence "c * (H * d)\<^sup>\<star> * H * b * top \<le> bot"
           using schroeder_3_p mult_assoc by auto
         thus "False"
           using 13 False sup.absorb_iff1 mult_assoc by auto
       qed
-      hence "bf_between_arcs a c H d \<and> bf_between_arcs c b H d"
-        using 11 15 assms(2) bf_between_arcs_def by auto
+      hence "forest_modulo_equivalence_path a c H d \<and> forest_modulo_equivalence_path c b H d"
+        using 11 15 assms(2) forest_modulo_equivalence_path_def by auto
       thus ?thesis
         by simp
     qed
   qed
-  have 2: "bf_between_arcs a b H d \<or> (bf_between_arcs a c H d \<and> bf_between_arcs c b H d) \<longrightarrow> bf_between_arcs a b H (d \<squnion> c)"
+  have 2: "forest_modulo_equivalence_path a b H d \<or> (forest_modulo_equivalence_path a c H d \<and> forest_modulo_equivalence_path c b H d) \<longrightarrow> forest_modulo_equivalence_path a b H (d \<squnion> c)"
   proof -
-    have 21: "bf_between_arcs a b H d \<longrightarrow> bf_between_arcs a b H (d \<squnion> c)"
+    have 21: "forest_modulo_equivalence_path a b H d \<longrightarrow> forest_modulo_equivalence_path a b H (d \<squnion> c)"
     proof (rule impI)
-      assume 22:"bf_between_arcs a b H d"
+      assume 22:"forest_modulo_equivalence_path a b H d"
       hence "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * b * top"
-        using bf_between_arcs_def by blast
+        using forest_modulo_equivalence_path_def by blast
       hence "a\<^sup>T * top \<le> (H * (d \<squnion> c))\<^sup>\<star> * H * b * top"
         by (simp add: mult_left_isotone mult_right_dist_sup mult_right_isotone order.trans star_isotone star_slide)
-      thus "bf_between_arcs a b H (d \<squnion> c)"
-        using 22 bf_between_arcs_def by blast
+      thus "forest_modulo_equivalence_path a b H (d \<squnion> c)"
+        using 22 forest_modulo_equivalence_path_def by blast
     qed
-    have "bf_between_arcs a c H d \<and> bf_between_arcs c b H d \<longrightarrow> bf_between_arcs a b H (d \<squnion> c)"
+    have "forest_modulo_equivalence_path a c H d \<and> forest_modulo_equivalence_path c b H d \<longrightarrow> forest_modulo_equivalence_path a b H (d \<squnion> c)"
     proof (rule impI)
-      assume 23: "bf_between_arcs a c H d \<and> bf_between_arcs c b H d"
+      assume 23: "forest_modulo_equivalence_path a c H d \<and> forest_modulo_equivalence_path c b H d"
       hence "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * c * top"
-        using bf_between_arcs_def by blast
+        using forest_modulo_equivalence_path_def by blast
       also have "... \<le> (H * d)\<^sup>\<star> * H * c * c\<^sup>T * c * top"
         by (metis ex231c comp_inf.star.circ_sup_2 mult_isotone mult_right_isotone mult_assoc)
       also have "... \<le> (H * d)\<^sup>\<star> * H * c * c\<^sup>T * top"
         by (simp add: mult_right_isotone mult_assoc)
       also have "... \<le> (H * d)\<^sup>\<star> * H * c * (H * d)\<^sup>\<star> * H * b * top"
-        using 23 mult_right_isotone mult_assoc by (simp add: bf_between_arcs_def)
+        using 23 mult_right_isotone mult_assoc by (simp add: forest_modulo_equivalence_path_def)
       also have "... \<le> (H * d)\<^sup>\<star> * H * b * top \<squnion> (H * d)\<^sup>\<star> * H * c * (H * d)\<^sup>\<star> * H * b * top"
-        by (simp add: bf_between_arcs_def)
+        by (simp add: forest_modulo_equivalence_path_def)
       finally have "a\<^sup>T * top \<le> (H * (d \<squnion> c))\<^sup>\<star> * H * b * top"
         using assms(1, 2) path_through_components mult_right_dist_sup by simp
-      thus "bf_between_arcs a b H (d \<squnion> c)"
-        using 23 bf_between_arcs_def by blast
+      thus "forest_modulo_equivalence_path a b H (d \<squnion> c)"
+        using 23 forest_modulo_equivalence_path_def by blast
     qed
     thus ?thesis
       using 21 by auto
   qed
   thus ?thesis
     using 1 2 by blast
 qed
 
 lemma dT_He_eq_bot:
   assumes "vector j"
     and "regular j"
     and "d * top \<le> - j"
     and "forest_components h * j = j"
     and "j \<noteq> bot"
   shows "d\<^sup>T * forest_components h * selected_edge h j g \<le> bot"
 proof -
   let ?e = "selected_edge h j g"
   let ?H = "forest_components h"
   have 1: "?e * top \<le> j"
     using assms(1, 2, 5) et_below_j by auto
   have "d\<^sup>T * ?H * ?e \<le> (d * top)\<^sup>T * ?H * (?e * top)"
     by (simp add: comp_isotone conv_isotone top_right_mult_increasing)
   also have "... \<le> (d * top)\<^sup>T * ?H * j"
     using 1 mult_right_isotone by auto
   also have "... \<le> (- j)\<^sup>T * ?H * j"
     by (simp add: assms(3) conv_isotone mult_left_isotone)
   also have "... = (- j)\<^sup>T * j"
     using assms(4) comp_associative by auto
   also have "... = bot"
     by (simp add: assms(1) conv_complement covector_vector_comp)
   finally show ?thesis
     using coreflexive_bot_closed le_bot by blast
 qed
 
-lemma big_forest_d_U_e:
+lemma forest_modulo_equivalence_d_U_e:
   assumes "forest f"
     and "vector j"
     and "regular j"
     and "forest h"
-    and "forest_components h \<le> forest_components f"
-    and "big_forest (forest_components h) d"
+    and "forest_modulo_equivalence (forest_components h) d"
     and "d * top \<le> - j"
     and "forest_components h * j = j"
     and "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
     and "selected_edge h j g \<le> - forest_components f"
-    and "selected_edge h j g \<noteq> bot"
     and "j \<noteq> bot"
-  shows "big_forest (forest_components h) (d \<squnion> selected_edge h j g)"
-proof (unfold big_forest_def, intro conjI)
+  shows "forest_modulo_equivalence (forest_components h) (d \<squnion> selected_edge h j g)"
+proof (cases "selected_edge h j g = bot")
+  let ?e = "selected_edge h j g"
+  case True
+  assume "?e = bot"
+  thus ?thesis
+    by (simp add: True assms(5))
+next
   let ?H = "forest_components h"
   let ?F = "forest_components f"
   let ?e = "selected_edge h j g"
   let ?d' = "d \<squnion> ?e"
-  show 01: "reflexive ?H"
-    by (simp add: assms(4) forest_components_equivalence)
-  show 02: "transitive ?H"
-    by (simp add: assms(4) forest_components_equivalence)
-  show 03: "symmetric ?H"
-    by (simp add: assms(4) forest_components_equivalence)
-  have 04: "equivalence ?H"
-    by (simp add: 01 02 03)
-  show 1: "?d' \<le> - ?H"
-  proof -
-    have "?H \<le> ?F"
-      by (simp add: assms(5))
-    hence 11: "?e \<le> - ?H"
-      using assms(10) order_lesseq_imp p_antitone by blast
-    have "d \<le> - ?H"
-      using assms(6) big_forest_def by auto
-    thus ?thesis
-      by (simp add: 11)
-  qed
-  show "univalent (?H * ?d')"
-  proof -
-    have "(?H * ?d')\<^sup>T * (?H * ?d') = ?d'\<^sup>T * ?H\<^sup>T * ?H * ?d'"
-      using conv_dist_comp mult_assoc by auto
-    also have "... = ?d'\<^sup>T * ?H * ?H * ?d'"
-      by (simp add: conv_dist_comp conv_star_commute)
-    also have "... = ?d'\<^sup>T * ?H * ?d'"
-      using 01 02 by (metis preorder_idempotent mult_assoc)
-    finally have 21: "univalent (?H * ?d') \<longleftrightarrow> ?e\<^sup>T * ?H * d \<squnion> d\<^sup>T * ?H * ?e \<squnion> ?e\<^sup>T * ?H * ?e \<squnion> d\<^sup>T * ?H * d \<le> 1"
-      using conv_dist_sup semiring.distrib_left semiring.distrib_right by auto
-    have 22: "?e\<^sup>T * ?H * ?e \<le> 1"
+  case False
+  assume e_not_bot: "?e \<noteq> bot"
+  have "forest_modulo_equivalence (forest_components h) (d \<squnion> selected_edge h j g)"
+  proof (unfold forest_modulo_equivalence_def, intro conjI)
+    show 01: "reflexive ?H"
+      by (simp add: assms(4) forest_components_equivalence)
+    show 02: "transitive ?H"
+      by (simp add: assms(4) forest_components_equivalence)
+    show 03: "symmetric ?H"
+      by (simp add: assms(4) forest_components_equivalence)
+    have 04: "equivalence ?H"
+      by (simp add: 01 02 03)
+    show "univalent (?H * ?d')"
     proof -
-      have 221: "?e\<^sup>T * ?H * ?e \<le> ?e\<^sup>T * top * ?e"
-        by (simp add: mult_left_isotone mult_right_isotone)
-      have "arc ?e"
-        using assms(11) minarc_arc minarc_bot_iff by blast
-      hence "?e\<^sup>T * top * ?e \<le> 1"
-        using arc_expanded by blast
-      thus ?thesis
-        using 221 dual_order.trans by blast
-    qed
-    have 24: "d\<^sup>T * ?H * ?e \<le> 1"
-      by (metis assms(2, 3, 7, 8, 12) dT_He_eq_bot coreflexive_bot_closed le_bot)
-    hence "(d\<^sup>T * ?H * ?e)\<^sup>T \<le> 1\<^sup>T"
-      using conv_isotone by blast
-    hence "?e\<^sup>T * ?H\<^sup>T * d\<^sup>T\<^sup>T \<le> 1"
-      by (simp add: conv_dist_comp mult_assoc)
-    hence 25: "?e\<^sup>T * ?H * d \<le> 1"
-      using assms(4) fch_equivalence by auto
-    have 8: "d\<^sup>T * ?H * d \<le> 1"
-      using 04 assms(6) dTransHd_le_1 big_forest_def by blast
-    thus ?thesis
-      using 21 22 24 25 by simp
-  qed
-  show "coreflexive (?H \<sqinter> ?d' * ?d'\<^sup>T)"
-  proof -
-    have "coreflexive (?H \<sqinter> ?d' * ?d'\<^sup>T) \<longleftrightarrow> ?H \<sqinter> (d \<squnion> ?e) * (d\<^sup>T \<squnion> ?e\<^sup>T) \<le> 1"
-      by (simp add: conv_dist_sup)
-    also have "... \<longleftrightarrow> ?H \<sqinter> (d * d\<^sup>T \<squnion> d * ?e\<^sup>T \<squnion> ?e * d\<^sup>T \<squnion> ?e * ?e\<^sup>T) \<le> 1"
-      by (metis mult_left_dist_sup mult_right_dist_sup sup.left_commute sup_commute)
-    finally have 1: "coreflexive (?H \<sqinter> ?d' * ?d'\<^sup>T) \<longleftrightarrow> ?H \<sqinter> d * d\<^sup>T \<squnion> ?H \<sqinter> d * ?e\<^sup>T \<squnion> ?H \<sqinter> ?e * d\<^sup>T \<squnion> ?H \<sqinter> ?e * ?e\<^sup>T \<le> 1"
-      by (simp add: inf_sup_distrib1)
-    have 31: "?H \<sqinter> d * d\<^sup>T \<le> 1"
-      using assms(6) big_forest_def by blast
-    have 32: "?H \<sqinter> ?e * d\<^sup>T \<le> 1"
-    proof -
-      have "?e * d\<^sup>T \<le> ?e * top * (d * top)\<^sup>T"
-        by (simp add: conv_isotone mult_isotone top_right_mult_increasing)
-      also have "... \<le> ?e * top * - j\<^sup>T"
-        by (metis assms(7) conv_complement conv_isotone mult_right_isotone)
-      also have "... \<le> j * - j\<^sup>T"
-        using assms(2, 3, 12) et_below_j mult_left_isotone by auto
-      also have "... \<le> - ?H"
-        using 03 by (metis assms(2, 3, 8) conv_complement conv_dist_comp equivalence_top_closed mult_left_isotone schroeder_3_p vector_top_closed)
-      finally have "?e * d\<^sup>T \<le> - ?H"
-        by simp
-      thus ?thesis
-        by (metis inf.coboundedI1 p_antitone_iff p_shunting_swap regular_one_closed)
-    qed
-    have 33: "?H \<sqinter> d * ?e\<^sup>T \<le> 1"
-    proof -
-      have 331: "injective h"
-        by (simp add: assms(4))
-      have "(?H \<sqinter> ?e * d\<^sup>T)\<^sup>T \<le> 1"
-        using 32 coreflexive_conv_closed by auto
-      hence "?H \<sqinter> (?e * d\<^sup>T)\<^sup>T \<le> 1"
-        using 331 conv_dist_inf forest_components_equivalence by auto
-      thus ?thesis
-        using conv_dist_comp by auto
-    qed
-    have 34: "?H \<sqinter> ?e * ?e\<^sup>T \<le> 1"
-    proof -
-      have 341:"arc ?e \<and> arc (?e\<^sup>T)"
-        using assms(11) minarc_arc minarc_bot_iff by auto
-      have "?H \<sqinter> ?e * ?e\<^sup>T \<le> ?e * ?e\<^sup>T"
-        by auto
+      have "(?H * ?d')\<^sup>T * (?H * ?d') = ?d'\<^sup>T * ?H\<^sup>T * ?H * ?d'"
+        using conv_dist_comp mult_assoc by auto
+      also have "... = ?d'\<^sup>T * ?H * ?H * ?d'"
+        by (simp add: conv_dist_comp conv_star_commute)
+      also have "... = ?d'\<^sup>T * ?H * ?d'"
+        using 01 02 by (metis preorder_idempotent mult_assoc)
+      finally have 21: "univalent (?H * ?d') \<longleftrightarrow> ?e\<^sup>T * ?H * d \<squnion> d\<^sup>T * ?H * ?e \<squnion> ?e\<^sup>T * ?H * ?e \<squnion> d\<^sup>T * ?H * d \<le> 1"
+        using conv_dist_sup semiring.distrib_left semiring.distrib_right by auto
+      have 22: "?e\<^sup>T * ?H * ?e \<le> 1"
+      proof -
+        have 221: "?e\<^sup>T * ?H * ?e \<le> ?e\<^sup>T * top * ?e"
+          by (simp add: mult_left_isotone mult_right_isotone)
+        have "arc ?e"
+          using e_not_bot minarc_arc minarc_bot_iff by blast
+        hence "?e\<^sup>T * top * ?e \<le> 1"
+          using arc_expanded by blast
+        thus ?thesis
+          using 221 dual_order.trans by blast
+      qed
+      have 24: "d\<^sup>T * ?H * ?e \<le> 1"
+        by (metis assms(2, 3, 6, 7, 10) dT_He_eq_bot coreflexive_bot_closed le_bot)
+      hence "(d\<^sup>T * ?H * ?e)\<^sup>T \<le> 1\<^sup>T"
+        using conv_isotone by blast
+      hence "?e\<^sup>T * ?H\<^sup>T * d\<^sup>T\<^sup>T \<le> 1"
+        by (simp add: conv_dist_comp mult_assoc)
+      hence 25: "?e\<^sup>T * ?H * d \<le> 1"
+        using assms(4) fch_equivalence by auto
+      have 8: "d\<^sup>T * ?H * d \<le> 1"
+        using 04 assms(5) dTransHd_le_1 forest_modulo_equivalence_def by blast
       thus ?thesis
-        using 341 arc_injective le_infI2 by blast
+        using 21 22 24 25 by simp
     qed
-    thus ?thesis
-      using 1 31 32 33 34 by simp
-  qed
-  show 4:"(?H * (d \<squnion> ?e))\<^sup>+ \<le> - ?H"
-  proof -
-    have "?e \<le> - ?F"
-      by (simp add: assms(10))
-    hence "?F \<le> - ?e"
-      by (simp add: p_antitone_iff)
-    hence "?F\<^sup>T * ?F \<le> - ?e"
-      using assms(1) fch_equivalence by fastforce
-    hence "?F\<^sup>T * ?F * ?F\<^sup>T \<le> - ?e"
-      by (metis assms(1) fch_equivalence forest_components_star star.circ_decompose_9)
-    hence 41: "?F * ?e * ?F \<le> - ?F"
-      using triple_schroeder_p by blast
-    hence 42:"(?F * ?F)\<^sup>\<star> * ?F * ?e * (?F * ?F)\<^sup>\<star> \<le> - ?F"
-    proof -
-      have 43: "?F * ?F = ?F"
-        using assms(1) forest_components_equivalence preorder_idempotent by auto
-      hence "?F * ?e * ?F = ?F * ?F * ?e * ?F"
-        by simp
-      also have "... = (?F)\<^sup>\<star> * ?F * ?e * (?F)\<^sup>\<star>"
-        by (simp add: assms(1) forest_components_star)
-      also have "... = (?F * ?F)\<^sup>\<star> * ?F * ?e * (?F * ?F)\<^sup>\<star>"
-        using 43 by simp
-      finally show ?thesis
-        using 41 by simp
-    qed
-    hence 44: "(?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> \<le> - ?H"
+    show "coreflexive (?H \<sqinter> ?d' * ?d'\<^sup>T)"
     proof -
-      have 45: "?H \<le> ?F"
-        by (simp add: assms(5))
-      hence 46:"?H * ?e \<le> ?F * ?e"
-        by (simp add: mult_left_isotone)
-      have "d \<le> f \<squnion> f\<^sup>T"
-        using assms(9) sup.left_commute sup_commute by auto
-      also have "... \<le> ?F"
-        by (metis forest_components_increasing le_supI2 star.circ_back_loop_fixpoint star.circ_increasing sup.bounded_iff)
-      finally have "d \<le> ?F"
-        by simp
-      hence "?H * d \<le> ?F * ?F"
-        using 45 mult_isotone by auto
-      hence 47: "(?H * d)\<^sup>\<star> \<le> (?F * ?F)\<^sup>\<star>"
-        by (simp add: star_isotone)
-      hence "(?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> \<le> (?H * d)\<^sup>\<star> * ?F * ?e * (?H * d)\<^sup>\<star>"
-        using 46 by (metis mult_left_isotone mult_right_isotone mult_assoc)
-      also have "... \<le> (?F * ?F)\<^sup>\<star> * ?F * ?e * (?F * ?F)\<^sup>\<star>"
-        using 47 mult_left_isotone mult_right_isotone by (simp add: comp_isotone)
-      also have "... \<le> - ?F"
-        using 42 by simp
-      also have "... \<le> - ?H"
-        using 45 by (simp add: p_antitone)
-      finally show ?thesis
-        by simp
+      have "coreflexive (?H \<sqinter> ?d' * ?d'\<^sup>T) \<longleftrightarrow> ?H \<sqinter> (d \<squnion> ?e) * (d\<^sup>T \<squnion> ?e\<^sup>T) \<le> 1"
+        by (simp add: conv_dist_sup)
+      also have "... \<longleftrightarrow> ?H \<sqinter> (d * d\<^sup>T \<squnion> d * ?e\<^sup>T \<squnion> ?e * d\<^sup>T \<squnion> ?e * ?e\<^sup>T) \<le> 1"
+        by (metis mult_left_dist_sup mult_right_dist_sup sup.left_commute sup_commute)
+      finally have 1: "coreflexive (?H \<sqinter> ?d' * ?d'\<^sup>T) \<longleftrightarrow> ?H \<sqinter> d * d\<^sup>T \<squnion> ?H \<sqinter> d * ?e\<^sup>T \<squnion> ?H \<sqinter> ?e * d\<^sup>T \<squnion> ?H \<sqinter> ?e * ?e\<^sup>T \<le> 1"
+        by (simp add: inf_sup_distrib1)
+      have 31: "?H \<sqinter> d * d\<^sup>T \<le> 1"
+        using assms(5) forest_modulo_equivalence_def by blast
+      have 32: "?H \<sqinter> ?e * d\<^sup>T \<le> 1"
+      proof -
+        have "?e * d\<^sup>T \<le> ?e * top * (d * top)\<^sup>T"
+          by (simp add: conv_isotone mult_isotone top_right_mult_increasing)
+        also have "... \<le> ?e * top * - j\<^sup>T"
+          by (metis assms(6) conv_complement conv_isotone mult_right_isotone)
+        also have "... \<le> j * - j\<^sup>T"
+          using assms(2, 3, 10) et_below_j mult_left_isotone by auto
+        also have "... \<le> - ?H"
+          using 03 by (metis assms(2, 3, 7) conv_complement conv_dist_comp equivalence_top_closed mult_left_isotone schroeder_3_p vector_top_closed)
+        finally have "?e * d\<^sup>T \<le> - ?H"
+          by simp
+        thus ?thesis
+          by (metis inf.coboundedI1 p_antitone_iff p_shunting_swap regular_one_closed)
+      qed
+      have 33: "?H \<sqinter> d * ?e\<^sup>T \<le> 1"
+      proof -
+        have 331: "injective h"
+          by (simp add: assms(4))
+        have "(?H \<sqinter> ?e * d\<^sup>T)\<^sup>T \<le> 1"
+          using 32 coreflexive_conv_closed by auto
+        hence "?H \<sqinter> (?e * d\<^sup>T)\<^sup>T \<le> 1"
+          using 331 conv_dist_inf forest_components_equivalence by auto
+        thus ?thesis
+          using conv_dist_comp by auto
+      qed
+      have 34: "?H \<sqinter> ?e * ?e\<^sup>T \<le> 1"
+      proof -
+        have 341:"arc ?e \<and> arc (?e\<^sup>T)"
+          using e_not_bot minarc_arc minarc_bot_iff by auto
+        have "?H \<sqinter> ?e * ?e\<^sup>T \<le> ?e * ?e\<^sup>T"
+          by auto
+        thus ?thesis
+          using 341 arc_injective le_infI2 by blast
+      qed
+      thus ?thesis
+        using 1 31 32 33 34 by simp
     qed
-    have "(?H * (d \<squnion> ?e))\<^sup>+ = (?H * (d \<squnion> ?e))\<^sup>\<star> * (?H * (d \<squnion> ?e))"
-      using star.circ_plus_same by auto
-    also have "... = ((?H * d)\<^sup>\<star> \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star>) * (?H * (d \<squnion> ?e))"
-      using assms(4, 11) forest_components_equivalence minarc_arc minarc_bot_iff path_through_components by auto
-    also have "... = (?H * d)\<^sup>\<star> * (?H * (d \<squnion> ?e)) \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * (?H * (d \<squnion> ?e))"
-      using mult_right_dist_sup by auto
-    also have "... = (?H * d)\<^sup>\<star> * (?H * d \<squnion> ?H * ?e) \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * (?H * d \<squnion> ?H * ?e)"
-      by (simp add: mult_left_dist_sup)
-    also have "... = (?H * d)\<^sup>\<star> * ?H * d \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * (?H * d \<squnion> ?H * ?e)"
-      using mult_left_dist_sup mult_assoc by auto
-    also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * (?H * d \<squnion> ?H * ?e)"
-      by (simp add: star.circ_plus_same mult_assoc)
-    also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * ?H * d \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * ?H * ?e"
-      by (simp add: mult.semigroup_axioms semiring.distrib_left sup.semigroup_axioms semigroup.assoc)
-    also have "... \<le> (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * ?H * d \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e"
+    show 4:"(?H * (d \<squnion> ?e))\<^sup>+ \<sqinter> ?H \<le> bot"
     proof -
-      have "?e * (?H * d)\<^sup>\<star> * ?H * ?e \<le> ?e * top * ?e"
-        by (metis comp_associative comp_inf.coreflexive_idempotent comp_inf.coreflexive_transitive comp_isotone top.extremum)
-      also have "... \<le> ?e"
-        using assms(11) arc_top_arc minarc_arc minarc_bot_iff by auto
-      finally have "?e * (?H * d)\<^sup>\<star> * ?H * ?e \<le> ?e"
-        by simp
-      hence "(?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
-        by (simp add: comp_associative comp_isotone)
-      thus ?thesis
-        using sup_right_isotone by blast
+      have 40: "(?H * d)\<^sup>+ \<le> -?H"
+        using assms(5) forest_modulo_equivalence_def bot_unique pseudo_complement by blast
+      have "?e \<le> - ?F"
+        by (simp add: assms(9))
+      hence "?F \<le> - ?e"
+        by (simp add: p_antitone_iff)
+      hence "?F\<^sup>T * ?F \<le> - ?e"
+        using assms(1) fch_equivalence by fastforce
+      hence "?F\<^sup>T * ?F * ?F\<^sup>T \<le> - ?e"
+        by (metis assms(1) fch_equivalence forest_components_star star.circ_decompose_9)
+      hence 41: "?F * ?e * ?F \<le> - ?F"
+        using triple_schroeder_p by blast
+      hence 42:"(?F * ?F)\<^sup>\<star> * ?F * ?e * (?F * ?F)\<^sup>\<star> \<le> - ?F"
+      proof -
+        have 43: "?F * ?F = ?F"
+          using assms(1) forest_components_equivalence preorder_idempotent by auto
+        hence "?F * ?e * ?F = ?F * ?F * ?e * ?F"
+          by simp
+        also have "... = (?F)\<^sup>\<star> * ?F * ?e * (?F)\<^sup>\<star>"
+          by (simp add: assms(1) forest_components_star)
+        also have "... = (?F * ?F)\<^sup>\<star> * ?F * ?e * (?F * ?F)\<^sup>\<star>"
+          using 43 by simp
+        finally show ?thesis
+          using 41 by simp
+      qed
+      hence 44: "(?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> \<le> - ?H"
+      proof -
+        have 45: "?H \<le> ?F"
+          using assms(1, 4, 8) H_below_F by blast
+        hence 46:"?H * ?e \<le> ?F * ?e"
+          by (simp add: mult_left_isotone)
+        have "d \<le> f \<squnion> f\<^sup>T"
+          using assms(8) sup.left_commute sup_commute by auto
+        also have "... \<le> ?F"
+          by (metis forest_components_increasing le_supI2 star.circ_back_loop_fixpoint star.circ_increasing sup.bounded_iff)
+        finally have "d \<le> ?F"
+          by simp
+        hence "?H * d \<le> ?F * ?F"
+          using 45 mult_isotone by auto
+        hence 47: "(?H * d)\<^sup>\<star> \<le> (?F * ?F)\<^sup>\<star>"
+          by (simp add: star_isotone)
+        hence "(?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> \<le> (?H * d)\<^sup>\<star> * ?F * ?e * (?H * d)\<^sup>\<star>"
+          using 46 by (metis mult_left_isotone mult_right_isotone mult_assoc)
+        also have "... \<le> (?F * ?F)\<^sup>\<star> * ?F * ?e * (?F * ?F)\<^sup>\<star>"
+          using 47 mult_left_isotone mult_right_isotone by (simp add: comp_isotone)
+        also have "... \<le> - ?F"
+          using 42 by simp
+        also have "... \<le> - ?H"
+          using 45 by (simp add: p_antitone)
+        finally show ?thesis
+          by simp
+      qed
+      have "(?H * (d \<squnion> ?e))\<^sup>+ = (?H * (d \<squnion> ?e))\<^sup>\<star> * (?H * (d \<squnion> ?e))"
+        using star.circ_plus_same by auto
+      also have "... = ((?H * d)\<^sup>\<star> \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star>) * (?H * (d \<squnion> ?e))"
+        using assms(4) e_not_bot forest_components_equivalence minarc_arc minarc_bot_iff path_through_components by auto
+      also have "... = (?H * d)\<^sup>\<star> * (?H * (d \<squnion> ?e)) \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * (?H * (d \<squnion> ?e))"
+        using mult_right_dist_sup by auto
+      also have "... = (?H * d)\<^sup>\<star> * (?H * d \<squnion> ?H * ?e) \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * (?H * d \<squnion> ?H * ?e)"
+        by (simp add: mult_left_dist_sup)
+      also have "... = (?H * d)\<^sup>\<star> * ?H * d \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * (?H * d \<squnion> ?H * ?e)"
+        using mult_left_dist_sup mult_assoc by auto
+      also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * (?H * d \<squnion> ?H * ?e)"
+        by (simp add: star.circ_plus_same mult_assoc)
+      also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * ?H * d \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * ?H * ?e"
+        by (simp add: mult.semigroup_axioms semiring.distrib_left sup.semigroup_axioms semigroup.assoc)
+      also have "... \<le> (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * ?H * d \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e"
+      proof -
+        have "?e * (?H * d)\<^sup>\<star> * ?H * ?e \<le> ?e * top * ?e"
+          by (metis comp_associative comp_inf.coreflexive_idempotent comp_inf.coreflexive_transitive comp_isotone top.extremum)
+        also have "... \<le> ?e"
+          using e_not_bot arc_top_arc minarc_arc minarc_bot_iff by auto
+        finally have "?e * (?H * d)\<^sup>\<star> * ?H * ?e \<le> ?e"
+          by simp
+        hence "(?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
+          by (simp add: comp_associative comp_isotone)
+        thus ?thesis
+          using sup_right_isotone by blast
+      qed
+      also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * ?H * d"
+        by (simp add: order.eq_iff ac_simps)
+      also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>+"
+        using star.circ_plus_same mult_assoc by auto
+      also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (1 \<squnion> (?H * d)\<^sup>+)"
+        by (simp add: mult_left_dist_sup sup_assoc)
+      also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star>"
+        by (simp add: star_left_unfold_equal)
+      also have "... \<le> - ?H"
+        using 40 44 assms(5) sup.boundedI by blast
+      finally show ?thesis
+        using pseudo_complement by force
     qed
-    also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star> * ?H * d"
-      by (simp add: order.eq_iff ac_simps)
-    also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>+"
-      using star.circ_plus_same mult_assoc by auto
-    also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (1 \<squnion> (?H * d)\<^sup>+)"
-      by (simp add: mult_left_dist_sup sup_assoc)
-    also have "... = (?H * d)\<^sup>+ \<squnion> (?H * d)\<^sup>\<star> * ?H * ?e * (?H * d)\<^sup>\<star>"
-      by (simp add: star_left_unfold_equal)
-    also have "... \<le> - ?H"
-      using 44 assms(6) big_forest_def by auto
-    finally show ?thesis
-      by simp
   qed
+  thus ?thesis
+    by blast
 qed
 
 subsubsection \<open>Identifying arcs\<close>
 
 text \<open>
 The expression $d \sqcap \top e^\top H \sqcap (H d^\top)^* H a^\top \top$ identifies the edge incoming to the component that the \<open>selected_edge\<close>, $e$, is outgoing from and which is on the path from edge $a$ to $e$.
 Here, we prove this expression is an \<open>arc\<close>.
 \<close>
 
 lemma shows_arc_x:
-  assumes "big_forest H d"
-    and "bf_between_arcs a e H d"
+  assumes "forest_modulo_equivalence H d"
+    and "forest_modulo_equivalence_path a e H d"
     and "H * d * (H * d)\<^sup>\<star> \<le> - H"
     and "\<not> a\<^sup>T * top \<le> H * e * top"
     and "regular a"
     and "regular e"
     and "regular H"
     and "regular d"
   shows "arc (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)"
 proof -
   let ?x = "d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top"
   have 1:"regular ?x"
     using assms(5, 6, 7, 8) regular_closed_star regular_conv_closed regular_mult_closed by auto
   have 2: "a\<^sup>T * top * a \<le> 1"
-    using arc_expanded assms(2) bf_between_arcs_def by auto
+    using arc_expanded assms(2) forest_modulo_equivalence_path_def by auto
   have 3: "e * top * e\<^sup>T \<le> 1"
-    using assms(2) bf_between_arcs_def arc_expanded by blast
+    using assms(2) forest_modulo_equivalence_path_def arc_expanded by blast
   have 4: "top * ?x * top = top"
   proof -
     have "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * e * top"
-      using assms(2) bf_between_arcs_def by blast
+      using assms(2) forest_modulo_equivalence_path_def by blast
     also have "... = H * e * top \<squnion> (H * d)\<^sup>\<star> * H * d * H * e * top"
       by (metis star.circ_loop_fixpoint star.circ_plus_same sup_commute mult_assoc)
     finally have "a\<^sup>T * top \<le> H * e * top \<squnion> (H * d)\<^sup>\<star> * H * d * H * e * top"
       by simp
     hence "a\<^sup>T * top \<le> H * e * top \<or> a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * d * H * e * top"
-      using assms(2, 6, 7) point_in_vector_sup bf_between_arcs_def regular_mult_closed vector_mult_closed by auto
+      using assms(2, 6, 7) point_in_vector_sup forest_modulo_equivalence_path_def regular_mult_closed vector_mult_closed by auto
     hence "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * d * H * e * top"
       using assms(4) by blast
     also have "... = (H * d)\<^sup>\<star> * H * d * (H * e * top \<sqinter> H * e * top)"
       by (simp add: mult_assoc)
     also have "... = (H * d)\<^sup>\<star> * H * (d \<sqinter> (H * e * top)\<^sup>T) * H * e * top"
       by (metis comp_associative covector_inf_comp_3 star.circ_left_top star.circ_top)
     also have "... = (H * d)\<^sup>\<star> * H * (d \<sqinter> top\<^sup>T * e\<^sup>T * H\<^sup>T) * H * e * top"
       using conv_dist_comp mult_assoc by auto
     also have "... = (H * d)\<^sup>\<star> * H * (d \<sqinter> top * e\<^sup>T * H) * H * e * top"
-      using assms(1) by (simp add: big_forest_def)
+      using assms(1) by (simp add: forest_modulo_equivalence_def)
     finally have 2: "a\<^sup>T * top \<le> (H * d)\<^sup>\<star> * H * (d \<sqinter> top * e\<^sup>T * H) * H * e * top"
       by simp
     hence "e * top \<le> ((H * d)\<^sup>\<star> * H * (d \<sqinter> top * e\<^sup>T * H) * H)\<^sup>T * a\<^sup>T * top"
     proof -
       have "bijective (e * top) \<and> bijective (a\<^sup>T * top)"
-        using assms(2) bf_between_arcs_def by auto
+        using assms(2) forest_modulo_equivalence_path_def by auto
       thus ?thesis
         using 2 by (metis bijective_reverse mult_assoc)
     qed
     also have "... = H\<^sup>T * (d \<sqinter> top * e\<^sup>T * H)\<^sup>T * H\<^sup>T * (H * d)\<^sup>\<star>\<^sup>T * a\<^sup>T * top"
       by (simp add: conv_dist_comp mult_assoc)
     also have "... = H * (d \<sqinter> top * e\<^sup>T * H)\<^sup>T * H * (H * d)\<^sup>\<star>\<^sup>T * a\<^sup>T * top"
-      using assms(1) big_forest_def by auto
+      using assms(1) forest_modulo_equivalence_def by auto
     also have "... = H * (d \<sqinter> top * e\<^sup>T * H)\<^sup>T * H * (d\<^sup>T * H)\<^sup>\<star> * a\<^sup>T * top"
-      using assms(1) big_forest_def conv_dist_comp conv_star_commute by auto
+      using assms(1) forest_modulo_equivalence_def conv_dist_comp conv_star_commute by auto
     also have "... = H * (d\<^sup>T \<sqinter> H * e * top) * H * (d\<^sup>T * H)\<^sup>\<star> * a\<^sup>T * top"
-      using assms(1) conv_dist_comp big_forest_def comp_associative conv_dist_inf by auto
+      using assms(1) conv_dist_comp forest_modulo_equivalence_def comp_associative conv_dist_inf by auto
     also have "... = H * (d\<^sup>T \<sqinter> H * e * top) * (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top"
       by (simp add: comp_associative star_slide)
     also have "... = H * (d\<^sup>T \<sqinter> H * e * top) * ((H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)"
       using mult_assoc by auto
     also have "... = H * (d\<^sup>T \<sqinter> H * e * top \<sqinter> ((H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)\<^sup>T) * (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top"
       by (smt comp_inf_vector covector_comp_inf vector_conv_covector vector_top_closed mult_assoc)
     also have "... = H * (d\<^sup>T \<sqinter> (top * e\<^sup>T * H)\<^sup>T \<sqinter> ((H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)\<^sup>T) * (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top"
-      using assms(1) big_forest_def conv_dist_comp mult_assoc by auto
+      using assms(1) forest_modulo_equivalence_def conv_dist_comp mult_assoc by auto
     also have "... = H * (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)\<^sup>T * (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top"
       by (simp add: conv_dist_inf)
     finally have 3: "e * top \<le> H * ?x\<^sup>T * (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top"
       by auto
     have "?x \<noteq> bot"
     proof (rule ccontr)
       assume "\<not> ?x \<noteq> bot"
       hence "e * top = bot"
         using 3 le_bot by auto
       thus "False"
-        using assms(2, 4) bf_between_arcs_def mult_assoc semiring.mult_zero_right by auto
+        using assms(2, 4) forest_modulo_equivalence_path_def mult_assoc semiring.mult_zero_right by auto
     qed
     thus ?thesis
       using 1 using tarski by blast
   qed
   have 5: "?x * top * ?x\<^sup>T \<le> 1"
   proof -
     have 51: "H * (d * H)\<^sup>\<star> \<sqinter> d * H * d\<^sup>T \<le> 1"
     proof -
       have 511: "d * (H * d)\<^sup>\<star> \<le> - H"
-        using assms(1, 3) big_forest_def preorder_idempotent schroeder_4_p triple_schroeder_p by fastforce
+        using assms(1, 3) forest_modulo_equivalence_def preorder_idempotent schroeder_4_p triple_schroeder_p by fastforce
       hence "(d * H)\<^sup>\<star> * d \<le> - H"
         using star_slide by auto
       hence "H * (d\<^sup>T * H)\<^sup>\<star> \<le> - d"
-        by (smt assms(1) big_forest_def conv_dist_comp conv_star_commute schroeder_4_p star_slide)
+        by (smt assms(1) forest_modulo_equivalence_def conv_dist_comp conv_star_commute schroeder_4_p star_slide)
       hence "H * (d * H)\<^sup>\<star> \<le> - d\<^sup>T"
-        using 511 by (metis assms(1) big_forest_def schroeder_5_p star_slide)
+        using 511 by (metis assms(1) forest_modulo_equivalence_def schroeder_5_p star_slide)
       hence "H * (d * H)\<^sup>\<star> \<le> - (H * d\<^sup>T)"
         by (metis assms(3) p_antitone_iff schroeder_4_p star_slide mult_assoc)
       hence "H * (d * H)\<^sup>\<star> \<sqinter> H * d\<^sup>T \<le> bot"
         by (simp add: bot_unique pseudo_complement)
       hence "H * d * (H * (d * H)\<^sup>\<star> \<sqinter> H * d\<^sup>T) \<le> 1"
         by (simp add: bot_unique)
       hence 512: "H * d * H * (d * H)\<^sup>\<star> \<sqinter> H * d * H * d\<^sup>T \<le> 1"
-        using univalent_comp_left_dist_inf assms(1) big_forest_def mult_assoc by fastforce
+        using univalent_comp_left_dist_inf assms(1) forest_modulo_equivalence_def mult_assoc by fastforce
       hence 513: "H * d * H * (d * H)\<^sup>\<star> \<sqinter> d * H * d\<^sup>T \<le> 1"
       proof -
         have "d * H * d\<^sup>T \<le> H * d * H * d\<^sup>T"
-          by (metis assms(1) big_forest_def conv_dist_comp conv_involutive mult_1_right mult_left_isotone)
+          by (metis assms(1) forest_modulo_equivalence_def conv_dist_comp conv_involutive mult_1_right mult_left_isotone)
         thus ?thesis
           using 512 by (smt dual_order.trans p_antitone p_shunting_swap regular_one_closed)
       qed
       have "d\<^sup>T * H * d \<le> 1 \<squnion> - H"
-        using assms(1) big_forest_def dTransHd_le_1 le_supI1 by blast
+        using assms(1) forest_modulo_equivalence_def dTransHd_le_1 le_supI1 by blast
       hence "(- 1 \<sqinter> H) * d\<^sup>T * H \<le> - d\<^sup>T"
-        by (metis assms(1) big_forest_def dTransHd_le_1 inf.sup_monoid.add_commute le_infI2 p_antitone_iff regular_one_closed schroeder_4_p mult_assoc)
+        by (metis assms(1) forest_modulo_equivalence_def dTransHd_le_1 inf.sup_monoid.add_commute le_infI2 p_antitone_iff regular_one_closed schroeder_4_p mult_assoc)
       hence "d * (- 1 \<sqinter> H) * d\<^sup>T \<le> - H"
-        by (metis assms(1) big_forest_def conv_dist_comp schroeder_3_p triple_schroeder_p)
+        by (metis assms(1) forest_modulo_equivalence_def conv_dist_comp schroeder_3_p triple_schroeder_p)
       hence "H \<sqinter> d * (- 1 \<sqinter> H) * d\<^sup>T \<le> 1"
         by (metis inf.coboundedI1 p_antitone_iff p_shunting_swap regular_one_closed)
       hence "H \<sqinter> d * d\<^sup>T \<squnion> H \<sqinter> d * (- 1 \<sqinter> H) * d\<^sup>T \<le> 1"
-        using assms(1) big_forest_def le_supI by blast
+        using assms(1) forest_modulo_equivalence_def le_supI by blast
       hence "H \<sqinter> (d * 1 * d\<^sup>T \<squnion> d * (- 1 \<sqinter> H) * d\<^sup>T) \<le> 1"
         using comp_inf.semiring.distrib_left by auto
       hence "H \<sqinter> (d * (1 \<squnion> (- 1 \<sqinter> H)) * d\<^sup>T) \<le> 1"
         by (simp add: mult_left_dist_sup mult_right_dist_sup)
       hence 514: "H \<sqinter> d * H * d\<^sup>T \<le> 1"
-        by (metis assms(1) big_forest_def comp_inf.semiring.distrib_left inf.le_iff_sup inf.sup_monoid.add_commute inf_top_right regular_one_closed stone)
+        by (metis assms(1) forest_modulo_equivalence_def comp_inf.semiring.distrib_left inf.le_iff_sup inf.sup_monoid.add_commute inf_top_right regular_one_closed stone)
       thus ?thesis
       proof -
         have "H \<sqinter> d * H * d\<^sup>T \<squnion> H * d * H * (d * H)\<^sup>\<star> \<sqinter> d * H * d\<^sup>T \<le> 1"
           using 513 514 by simp
         hence "d * H * d\<^sup>T \<sqinter> (H \<squnion> H * d * H * (d * H)\<^sup>\<star>) \<le> 1"
           by (simp add: comp_inf.semiring.distrib_left inf.sup_monoid.add_commute)
         hence "d * H * d\<^sup>T \<sqinter> H * (1 \<squnion> d * H * (d * H)\<^sup>\<star>) \<le> 1"
           by (simp add: mult_left_dist_sup mult_assoc)
         thus ?thesis
           by (simp add: inf.sup_monoid.add_commute star_left_unfold_equal)
       qed
     qed
     have "?x * top * ?x\<^sup>T = (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top) * top * (d\<^sup>T \<sqinter> H\<^sup>T * e\<^sup>T\<^sup>T * top\<^sup>T \<sqinter> top\<^sup>T * a\<^sup>T\<^sup>T * H\<^sup>T * (d\<^sup>T\<^sup>T * H\<^sup>T)\<^sup>\<star>)"
       by (simp add: conv_dist_comp conv_dist_inf conv_star_commute mult_assoc)
     also have "... = (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top) * top * (d\<^sup>T \<sqinter> H * e * top \<sqinter> top * a * H * (d * H)\<^sup>\<star>)"
-      using assms(1) big_forest_def by auto
+      using assms(1) forest_modulo_equivalence_def by auto
     also have "... = (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top \<sqinter> (d \<sqinter> top * e\<^sup>T * H) * top * (d\<^sup>T \<sqinter> H * e * top \<sqinter> top * a * H * (d * H)\<^sup>\<star>)"
       by (metis inf_vector_comp vector_export_comp)
     also have "... = (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top \<sqinter> (d \<sqinter> top * e\<^sup>T * H) * top * top * (d\<^sup>T \<sqinter> H * e * top \<sqinter> top * a * H * (d * H)\<^sup>\<star>)"
       by (simp add: vector_mult_closed)
     also have "... = (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top \<sqinter> d * ((top * e\<^sup>T * H)\<^sup>T \<sqinter> top) * top * (d\<^sup>T \<sqinter> H * e * top \<sqinter> top * a * H * (d * H)\<^sup>\<star>)"
       by (simp add: covector_comp_inf_1 covector_mult_closed)
     also have "... = (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top \<sqinter> d * ((top * e\<^sup>T * H)\<^sup>T \<sqinter> (H * e * top)\<^sup>T) * d\<^sup>T \<sqinter> top * a * H * (d * H)\<^sup>\<star>"
       by (smt comp_associative comp_inf.star_star_absorb comp_inf_vector conv_star_commute covector_comp_inf covector_conv_vector fc_top star.circ_top total_conv_surjective vector_conv_covector vector_inf_comp)
     also have "... = (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top \<sqinter> top * a * H * (d * H)\<^sup>\<star> \<sqinter> d * ((top * e\<^sup>T * H)\<^sup>T \<sqinter> (H * e * top)\<^sup>T) * d\<^sup>T"
       using inf.sup_monoid.add_assoc inf.sup_monoid.add_commute by auto
     also have "... = (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top * top * a * H * (d * H)\<^sup>\<star> \<sqinter> d * ((top * e\<^sup>T * H)\<^sup>T \<sqinter> (H * e * top)\<^sup>T) * d\<^sup>T"
       by (smt comp_inf.star.circ_decompose_9 comp_inf.star_star_absorb comp_inf_covector fc_top star.circ_decompose_11 star.circ_top vector_export_comp)
     also have "... = (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top * a * H * (d * H)\<^sup>\<star> \<sqinter> d * (H * e * top \<sqinter> top * e\<^sup>T * H) * d\<^sup>T"
-      using assms(1) big_forest_def conv_dist_comp mult_assoc by auto
+      using assms(1) forest_modulo_equivalence_def conv_dist_comp mult_assoc by auto
     also have "... = (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top * a * H * (d * H)\<^sup>\<star> \<sqinter> d * H * e * top * e\<^sup>T * H * d\<^sup>T"
       by (metis comp_inf_covector inf_top.left_neutral mult_assoc)
     also have "... \<le> (H * d\<^sup>T)\<^sup>\<star> * (H * d)\<^sup>\<star> * H \<sqinter> d * H * e * top * e\<^sup>T * H * d\<^sup>T"
     proof -
       have "(H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top * a * H * (d * H)\<^sup>\<star> \<le> (H * d\<^sup>T)\<^sup>\<star> * H * 1 * H * (d * H)\<^sup>\<star>"
         using 2 by (metis comp_associative comp_isotone mult_left_isotone mult_semi_associative star.circ_transitive_equal)
       also have "... = (H * d\<^sup>T)\<^sup>\<star> * H * (d * H)\<^sup>\<star>"
-        using assms(1) big_forest_def mult.semigroup_axioms preorder_idempotent semigroup.assoc by fastforce
+        using assms(1) forest_modulo_equivalence_def mult.semigroup_axioms preorder_idempotent semigroup.assoc by fastforce
       also have "... = (H * d\<^sup>T)\<^sup>\<star> * (H * d)\<^sup>\<star> * H"
         by (metis star_slide mult_assoc)
       finally show ?thesis
         using inf.sup_left_isotone by auto
     qed
     also have "... \<le> (H * d\<^sup>T)\<^sup>\<star> * (H * d)\<^sup>\<star> * H \<sqinter> d * H * d\<^sup>T"
     proof -
       have "d * H * e * top * e\<^sup>T * H * d\<^sup>T \<le> d * H * 1 * H * d\<^sup>T"
         using 3 by (metis comp_isotone idempotent_one_closed mult_left_isotone mult_sub_right_one mult_assoc)
       also have "... \<le> d * H * d\<^sup>T"
-        by (metis assms(1) big_forest_def mult_left_isotone mult_one_associative mult_semi_associative preorder_idempotent)
+        by (metis assms(1) forest_modulo_equivalence_def mult_left_isotone mult_one_associative mult_semi_associative preorder_idempotent)
       finally show ?thesis
         using inf.sup_right_isotone by auto
     qed
     also have "... = H * (d\<^sup>T * H)\<^sup>\<star> * (H * d)\<^sup>\<star> * H \<sqinter> d * H * d\<^sup>T"
-      by (metis assms(1) big_forest_def comp_associative preorder_idempotent star_slide)
+      by (metis assms(1) forest_modulo_equivalence_def comp_associative preorder_idempotent star_slide)
     also have "... = H * ((d\<^sup>T * H)\<^sup>\<star> \<squnion> (H * d)\<^sup>\<star>) * H \<sqinter> d * H * d\<^sup>T"
-      by (simp add: assms(1) expand_big_forest mult.semigroup_axioms semigroup.assoc)
+      by (simp add: assms(1) expand_forest_modulo_equivalence mult.semigroup_axioms semigroup.assoc)
     also have "... = (H * (d\<^sup>T * H)\<^sup>\<star> * H \<squnion> H * (H * d)\<^sup>\<star> * H) \<sqinter> d * H * d\<^sup>T"
       by (simp add: mult_left_dist_sup mult_right_dist_sup)
     also have "... = (H * d\<^sup>T)\<^sup>\<star> * H \<sqinter> d * H * d\<^sup>T \<squnion> H * (d * H)\<^sup>\<star> \<sqinter> d * H * d\<^sup>T"
-      by (smt assms(1) big_forest_def inf_sup_distrib2 mult.semigroup_axioms preorder_idempotent star_slide semigroup.assoc)
+      by (smt assms(1) forest_modulo_equivalence_def inf_sup_distrib2 mult.semigroup_axioms preorder_idempotent star_slide semigroup.assoc)
     also have "... \<le> (H * d\<^sup>T)\<^sup>\<star> * H \<sqinter> d * H * d\<^sup>T \<squnion> 1"
       using 51 comp_inf.semiring.add_left_mono by blast
     finally have "?x * top * ?x\<^sup>T \<le> 1"
-      using 51 by (smt assms(1) big_forest_def conv_dist_comp conv_dist_inf conv_dist_sup conv_involutive conv_star_commute equivalence_one_closed mult.semigroup_axioms sup.absorb2 semigroup.assoc conv_isotone conv_order)
+      using 51 by (smt assms(1) forest_modulo_equivalence_def conv_dist_comp conv_dist_inf conv_dist_sup conv_involutive conv_star_commute equivalence_one_closed mult.semigroup_axioms sup.absorb2 semigroup.assoc conv_isotone conv_order)
     thus ?thesis
       by simp
   qed
   have 6: "?x\<^sup>T * top * ?x \<le> 1"
   proof -
     have "?x\<^sup>T * top * ?x = (d\<^sup>T \<sqinter> H\<^sup>T * e\<^sup>T\<^sup>T * top\<^sup>T \<sqinter> top\<^sup>T * a\<^sup>T\<^sup>T * H\<^sup>T * (d\<^sup>T\<^sup>T * H\<^sup>T)\<^sup>\<star>) * top * (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)"
       by (simp add: conv_dist_comp conv_dist_inf conv_star_commute mult_assoc)
     also have "... = (d\<^sup>T \<sqinter> H * e * top \<sqinter> top * a * H * (d * H)\<^sup>\<star>) * top * (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)"
-      using assms(1) big_forest_def by auto
+      using assms(1) forest_modulo_equivalence_def by auto
     also have "... = H * e * top \<sqinter> (d\<^sup>T \<sqinter> top * a * H * (d * H)\<^sup>\<star>) * top * (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)"
       by (smt comp_associative inf.sup_monoid.add_assoc inf.sup_monoid.add_commute star.circ_left_top star.circ_top vector_inf_comp)
     also have "... = H * e * top \<sqinter> d\<^sup>T * ((top * a * H * (d * H)\<^sup>\<star>)\<^sup>T \<sqinter> top) * (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)"
       by (simp add: covector_comp_inf_1 covector_mult_closed)
     also have "... = H * e * top \<sqinter> d\<^sup>T * (d * H)\<^sup>\<star>\<^sup>T * H * a\<^sup>T * top * (d \<sqinter> top * e\<^sup>T * H \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)"
-      using assms(1) big_forest_def comp_associative conv_dist_comp by auto
+      using assms(1) forest_modulo_equivalence_def comp_associative conv_dist_comp by auto
     also have "... = H * e * top \<sqinter> d\<^sup>T * (d * H)\<^sup>\<star>\<^sup>T * H * a\<^sup>T * top * (d \<sqinter> (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top) \<sqinter> top * e\<^sup>T * H"
       by (smt comp_associative comp_inf_covector inf.sup_monoid.add_assoc inf.sup_monoid.add_commute)
     also have "... = H * e * top \<sqinter> d\<^sup>T * (d * H)\<^sup>\<star>\<^sup>T * H * a\<^sup>T * (top \<sqinter> ((H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)\<^sup>T) * d \<sqinter> top * e\<^sup>T * H"
       by (metis comp_associative comp_inf_vector vector_conv_covector vector_top_closed)
     also have "... = H * e * top \<sqinter> (H * e * top)\<^sup>T \<sqinter> d\<^sup>T * (d * H)\<^sup>\<star>\<^sup>T * H * a\<^sup>T * ((H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)\<^sup>T * d"
-      by (smt assms(1) big_forest_def conv_dist_comp inf.left_commute inf.sup_monoid.add_commute symmetric_top_closed mult_assoc inf_top.left_neutral)
+      by (smt assms(1) forest_modulo_equivalence_def conv_dist_comp inf.left_commute inf.sup_monoid.add_commute symmetric_top_closed mult_assoc inf_top.left_neutral)
     also have "... = H * e * top * (H * e * top)\<^sup>T \<sqinter> d\<^sup>T * (d * H)\<^sup>\<star>\<^sup>T * H * a\<^sup>T * ((H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top)\<^sup>T * d"
       using vector_covector vector_mult_closed by auto
     also have "... = H * e * top * top\<^sup>T * e\<^sup>T * H\<^sup>T \<sqinter> d\<^sup>T * (d * H)\<^sup>\<star>\<^sup>T * H * a\<^sup>T * top\<^sup>T * a\<^sup>T\<^sup>T * H\<^sup>T * (H * d\<^sup>T)\<^sup>\<star>\<^sup>T * d"
       by (smt conv_dist_comp mult.semigroup_axioms symmetric_top_closed semigroup.assoc)
     also have "... = H * e * top * top * e\<^sup>T * H \<sqinter> d\<^sup>T * (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top * a * H * (d * H)\<^sup>\<star> * d"
-      using assms(1) big_forest_def conv_dist_comp conv_star_commute by auto
+      using assms(1) forest_modulo_equivalence_def conv_dist_comp conv_star_commute by auto
     also have "... = H * e * top * e\<^sup>T * H \<sqinter> d\<^sup>T * (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top * a * H * (d * H)\<^sup>\<star> * d"
       using vector_top_closed mult_assoc by auto
     also have "... \<le> H \<sqinter> d\<^sup>T * (H * d\<^sup>T)\<^sup>\<star> * H * (d * H)\<^sup>\<star> * d"
     proof -
       have "H * e * top * e\<^sup>T * H \<le> H * 1 * H"
         using 3 by (metis comp_associative mult_left_isotone mult_right_isotone)
       also have "... = H"
-        using assms(1) big_forest_def preorder_idempotent by auto
+        using assms(1) forest_modulo_equivalence_def preorder_idempotent by auto
       finally have 611: "H * e * top * e\<^sup>T * H \<le> H"
         by simp
       have "d\<^sup>T * (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top * a * H * (d * H)\<^sup>\<star> * d \<le> d\<^sup>T * (H * d\<^sup>T)\<^sup>\<star> * H * 1 * H * (d * H)\<^sup>\<star> * d"
         using 2 by (metis comp_associative mult_left_isotone mult_right_isotone)
       also have "... = d\<^sup>T * (H * d\<^sup>T)\<^sup>\<star> * H * (d * H)\<^sup>\<star> * d"
-        using assms(1) big_forest_def mult.semigroup_axioms preorder_idempotent semigroup.assoc by fastforce
+        using assms(1) forest_modulo_equivalence_def mult.semigroup_axioms preorder_idempotent semigroup.assoc by fastforce
       finally have "d\<^sup>T * (H * d\<^sup>T)\<^sup>\<star> * H * a\<^sup>T * top * a * H * (d * H)\<^sup>\<star> * d \<le> d\<^sup>T * (H * d\<^sup>T)\<^sup>\<star> * H * (d * H)\<^sup>\<star> * d"
         by simp
       thus ?thesis
         using 611 comp_inf.comp_isotone by blast
     qed
     also have "... = H \<sqinter> (d\<^sup>T * H)\<^sup>\<star> * d\<^sup>T * H * d * (H * d)\<^sup>\<star>"
       using star_slide mult_assoc by auto
     also have "... \<le> H \<sqinter> (d\<^sup>T * H)\<^sup>\<star> * (H * d)\<^sup>\<star>"
     proof -
       have "(d\<^sup>T * H)\<^sup>\<star> * d\<^sup>T * H * d * (H * d)\<^sup>\<star> \<le> (d\<^sup>T * H)\<^sup>\<star> * 1 * (H * d)\<^sup>\<star>"
-        by (smt assms(1) big_forest_def conv_dist_comp mult_left_isotone mult_right_isotone preorder_idempotent mult_assoc)
+        by (smt assms(1) forest_modulo_equivalence_def conv_dist_comp mult_left_isotone mult_right_isotone preorder_idempotent mult_assoc)
       also have "... = (d\<^sup>T * H)\<^sup>\<star> * (H * d)\<^sup>\<star>"
         by simp
       finally show ?thesis
         using inf.sup_right_isotone by blast
     qed
     also have "... = H \<sqinter> ((d\<^sup>T * H)\<^sup>\<star> \<squnion> (H * d)\<^sup>\<star>)"
-      by (simp add: assms(1) expand_big_forest)
+      by (simp add: assms(1) expand_forest_modulo_equivalence)
     also have "... = H \<sqinter> (d\<^sup>T * H)\<^sup>\<star> \<squnion> H \<sqinter> (H * d)\<^sup>\<star>"
       by (simp add: comp_inf.semiring.distrib_left)
     also have "... = 1 \<squnion> H \<sqinter> (d\<^sup>T * H)\<^sup>+ \<squnion> H \<sqinter> (H * d)\<^sup>+"
     proof -
       have 612: "H \<sqinter> (H * d)\<^sup>\<star> = 1 \<squnion> H \<sqinter> (H * d)\<^sup>+"
-        using assms(1) big_forest_def reflexive_inf_star by blast
+        using assms(1) forest_modulo_equivalence_def reflexive_inf_star by blast
       have "H \<sqinter> (d\<^sup>T * H)\<^sup>\<star> = 1 \<squnion> H \<sqinter> (d\<^sup>T * H)\<^sup>+"
-        using assms(1) big_forest_def reflexive_inf_star by auto
+        using assms(1) forest_modulo_equivalence_def reflexive_inf_star by auto
       thus ?thesis
         using 612 sup_assoc sup_commute by auto
     qed
     also have "... \<le> 1"
     proof -
       have 613: "H \<sqinter> (H * d)\<^sup>+ \<le> 1"
         by (metis assms(3) inf.coboundedI1 p_antitone_iff p_shunting_swap regular_one_closed)
       hence "H \<sqinter> (d\<^sup>T * H)\<^sup>+ \<le> 1"
-        by (metis assms(1) big_forest_def conv_dist_comp conv_dist_inf conv_plus_commute coreflexive_symmetric)
+        by (metis assms(1) forest_modulo_equivalence_def conv_dist_comp conv_dist_inf conv_plus_commute coreflexive_symmetric)
       thus ?thesis
         by (simp add: 613)
     qed
     finally show ?thesis
       by simp
   qed
   have 7:"bijective (?x * top)"
     using 4 5 6 arc_expanded by blast
   have "bijective (?x\<^sup>T * top)"
     using 4 5 6 arc_expanded by blast
   thus ?thesis
     using 7 by simp
 qed
 
 text \<open>
 To maintain that $f$ can be extended to a minimum spanning forest we identify an edge, $i = v \sqcap \overline{F}e\top \sqcap \top e^\top F$, that may be exchanged with the \<open>selected_edge\<close>, $e$.
 Here, we show that $i$ is an \<open>arc\<close>.
 \<close>
 
 lemma boruvka_edge_arc:
   assumes "equivalence F"
     and "forest v"
     and "arc e"
     and "regular F"
     and "F \<le> forest_components (F \<sqinter> v)"
     and "regular v"
     and "v * e\<^sup>T = bot"
     and "e * F * e = bot"
     and "e\<^sup>T \<le> v\<^sup>\<star>"
     and "e \<noteq> bot"
   shows "arc (v \<sqinter> -F * e * top \<sqinter> top * e\<^sup>T * F)"
 proof -
   let ?i = "v \<sqinter> -F * e * top \<sqinter> top * e\<^sup>T * F"
   have 1: "?i\<^sup>T * top * ?i \<le> 1"
   proof -
     have "?i\<^sup>T * top * ?i = (v\<^sup>T \<sqinter> top * e\<^sup>T * -F \<sqinter> F * e * top) * top * (v \<sqinter> -F * e * top \<sqinter> top * e\<^sup>T * F)"
       using assms(1) conv_complement conv_dist_comp conv_dist_inf mult.semigroup_axioms semigroup.assoc by fastforce
     also have "... = F * e * top \<sqinter> (v\<^sup>T \<sqinter> top * e\<^sup>T * -F) * top * (v \<sqinter> -F * e * top) \<sqinter> top * e\<^sup>T * F"
       by (smt covector_comp_inf covector_mult_closed inf_vector_comp vector_export_comp vector_top_closed)
     also have "... = F * e * top \<sqinter> (v\<^sup>T \<sqinter> top * e\<^sup>T * -F) * top * top * (v \<sqinter> -F * e * top) \<sqinter> top * e\<^sup>T * F"
       by (simp add: comp_associative)
     also have "... = F * e * top \<sqinter> v\<^sup>T * (top \<sqinter> (top * e\<^sup>T * -F)\<^sup>T) * top * (v \<sqinter> -F * e * top) \<sqinter> top * e\<^sup>T * F"
       using comp_associative comp_inf_vector_1 by auto
     also have "... = F * e * top \<sqinter> v\<^sup>T * (top \<sqinter> (top * e\<^sup>T * -F)\<^sup>T) * (top \<sqinter> (-F * e * top)\<^sup>T) * v \<sqinter> top * e\<^sup>T * F"
       by (smt comp_inf_vector conv_dist_comp mult.semigroup_axioms symmetric_top_closed semigroup.assoc)
     also have "... = F * e * top \<sqinter> v\<^sup>T * (top * e\<^sup>T * -F)\<^sup>T * (-F * e * top)\<^sup>T * v \<sqinter> top * e\<^sup>T * F"
       by simp
     also have "... = F * e * top \<sqinter> v\<^sup>T * -F\<^sup>T * e\<^sup>T\<^sup>T * top\<^sup>T * top\<^sup>T * e\<^sup>T * -F\<^sup>T * v \<sqinter> top * e\<^sup>T * F"
       by (metis comp_associative conv_complement conv_dist_comp)
     also have "... = F * e * top \<sqinter> v\<^sup>T * -F * e * top * top * e\<^sup>T * -F * v \<sqinter> top * e\<^sup>T * F"
       by (simp add: assms(1))
     also have "... = F * e * top \<sqinter> v\<^sup>T * -F * e * top \<sqinter> top * e\<^sup>T * -F * v \<sqinter> top * e\<^sup>T * F"
       by (metis comp_associative comp_inf_covector inf.sup_monoid.add_assoc inf_top.left_neutral vector_top_closed)
     also have "... = (F \<sqinter> v\<^sup>T * -F) * e * top \<sqinter> top * e\<^sup>T * -F * v \<sqinter> top * e\<^sup>T * F"
       using assms(3) injective_comp_right_dist_inf mult_assoc by auto
     also have "... = (F \<sqinter> v\<^sup>T * -F) * e * top \<sqinter> top * e\<^sup>T * (F \<sqinter> -F * v)"
       using assms(3) conv_dist_comp inf.sup_monoid.add_assoc inf.sup_monoid.add_commute mult.semigroup_axioms univalent_comp_left_dist_inf semigroup.assoc by fastforce
     also have "... = (F \<sqinter> v\<^sup>T * -F) * e * top * top * e\<^sup>T * (F \<sqinter> -F * v)"
       by (metis comp_associative comp_inf_covector inf_top.left_neutral vector_top_closed)
     also have "... = (F \<sqinter> v\<^sup>T * -F) * e * top * e\<^sup>T * (F \<sqinter> -F * v)"
       by (simp add: comp_associative)
     also have "... \<le> (F \<sqinter> v\<^sup>T * -F) * (F \<sqinter> -F * v)"
       by (smt assms(3) conv_dist_comp mult_left_isotone shunt_bijective symmetric_top_closed top_right_mult_increasing mult_assoc)
     also have "... \<le> (F \<sqinter> v\<^sup>T * -F) * (F \<sqinter> -F * v) \<sqinter> F"
       by (metis assms(1) inf.absorb1 inf.cobounded1 mult_isotone preorder_idempotent)
     also have "... \<le> (F \<sqinter> v\<^sup>T * -F) * (F \<sqinter> -F * v) \<sqinter> (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star>"
       using assms(5) comp_inf.mult_right_isotone by auto
     also have "... \<le> (-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star>"
     proof -
       have "F \<sqinter> v\<^sup>T * -F \<le> (v\<^sup>T \<sqinter> F * -F\<^sup>T) * -F"
         by (metis conv_complement dedekind_2 inf_commute)
       also have "... = (v\<^sup>T \<sqinter> -F\<^sup>T) * -F"
         using assms(1) equivalence_comp_left_complement by simp
       finally have "F \<sqinter> v\<^sup>T * -F \<le> F \<sqinter> (v\<^sup>T \<sqinter> -F) * -F"
         using assms(1) by auto
       hence 11: "F \<sqinter> v\<^sup>T * -F = F \<sqinter> (-F \<sqinter> v\<^sup>T) * -F"
         by (metis inf.antisym_conv inf.sup_monoid.add_commute comp_left_subdist_inf inf.boundedE inf.sup_right_isotone)
       hence "F\<^sup>T \<sqinter> -F\<^sup>T * v\<^sup>T\<^sup>T = F\<^sup>T \<sqinter> -F\<^sup>T * (-F\<^sup>T \<sqinter> v\<^sup>T\<^sup>T)"
         by (metis (full_types) assms(1) conv_complement conv_dist_comp conv_dist_inf)
       hence 12: "F \<sqinter> -F * v = F \<sqinter> -F * (-F \<sqinter> v)"
         using assms(1) by (simp add: abel_semigroup.commute inf.abel_semigroup_axioms)
       have "(F \<sqinter> v\<^sup>T * -F) * (F \<sqinter> -F * v) = (F \<sqinter> (-F \<sqinter> v\<^sup>T) * -F) * (F \<sqinter> -F * (-F \<sqinter> v))"
         using 11 12 by auto
       also have "... \<le> (-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v)"
         by (metis comp_associative comp_isotone inf.cobounded2)
       finally show ?thesis
         using comp_inf.mult_left_isotone by blast
     qed
     also have "... = ((-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> (F \<sqinter> v)\<^sup>T * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star>) \<squnion> ((-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> (F \<sqinter> v)\<^sup>\<star>)"
       by (metis comp_associative inf_sup_distrib1 star.circ_loop_fixpoint)
     also have "... = ((-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> (F \<sqinter> v\<^sup>T) * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star>) \<squnion> ((-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> (F \<sqinter> v)\<^sup>\<star>)"
       using assms(1) conv_dist_inf by auto
     also have "... = bot \<squnion> ((-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> (F \<sqinter> v)\<^sup>\<star>)"
     proof -
       have "(-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> (F \<sqinter> v\<^sup>T) * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star> \<le> bot"
         using assms(1, 2) forests_bot_2 by (simp add: comp_associative)
       thus ?thesis
         using le_bot by blast
     qed
     also have "... = (-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> (1 \<squnion> (F \<sqinter> v)\<^sup>\<star> * (F \<sqinter> v))"
       by (simp add: star.circ_plus_same star_left_unfold_equal)
     also have "... = ((-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> 1) \<squnion> ((-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> (F \<sqinter> v)\<^sup>\<star> * (F \<sqinter> v))"
       by (simp add: comp_inf.semiring.distrib_left)
     also have "... \<le> 1 \<squnion> ((-F \<sqinter> v\<^sup>T) * -F * -F * (-F \<sqinter> v) \<sqinter> (F \<sqinter> v)\<^sup>\<star> * (F \<sqinter> v))"
       using sup_left_isotone by auto
     also have "... \<le> 1 \<squnion> bot"
       using assms(1, 2) forests_bot_3 comp_inf.semiring.add_left_mono by simp
     finally show ?thesis
       by simp
   qed
   have 2: "?i * top * ?i\<^sup>T \<le> 1"
   proof -
     have "?i * top * ?i\<^sup>T = (v \<sqinter> -F * e * top \<sqinter> top * e\<^sup>T * F) * top * (v\<^sup>T \<sqinter> (-F * e * top)\<^sup>T \<sqinter> (top * e\<^sup>T * F)\<^sup>T)"
       by (simp add: conv_dist_inf)
     also have "... = (v \<sqinter> -F * e * top \<sqinter> top * e\<^sup>T * F) * top * (v\<^sup>T \<sqinter> top\<^sup>T * e\<^sup>T * -F\<^sup>T \<sqinter> F\<^sup>T * e\<^sup>T\<^sup>T * top\<^sup>T)"
       by (simp add: conv_complement conv_dist_comp mult_assoc)
     also have "... = (v \<sqinter> -F * e * top \<sqinter> top * e\<^sup>T * F) * top * (v\<^sup>T \<sqinter> top * e\<^sup>T * -F \<sqinter> F * e * top)"
       by (simp add: assms(1))
     also have "... = -F * e * top \<sqinter> (v \<sqinter> top * e\<^sup>T * F) * top * (v\<^sup>T \<sqinter> top * e\<^sup>T * -F \<sqinter> F * e * top)"
       by (smt inf.left_commute inf.sup_monoid.add_assoc vector_export_comp)
     also have "... = -F * e * top \<sqinter> (v \<sqinter> top * e\<^sup>T * F) * top * (v\<^sup>T \<sqinter> F * e * top) \<sqinter> top * e\<^sup>T * -F"
       by (smt comp_inf_covector inf.sup_monoid.add_assoc inf.sup_monoid.add_commute mult_assoc)
     also have "... = -F * e * top \<sqinter> (v \<sqinter> top * e\<^sup>T * F) * top * top * (v\<^sup>T \<sqinter> F * e * top) \<sqinter> top * e\<^sup>T * -F"
       by (simp add: mult_assoc)
     also have "... = -F * e * top \<sqinter> v * ((top * e\<^sup>T * F)\<^sup>T \<sqinter> top) * top * (v\<^sup>T \<sqinter> F * e * top) \<sqinter> top * e\<^sup>T * -F"
       by (simp add: comp_inf_vector_1 mult.semigroup_axioms semigroup.assoc)
     also have "... = -F * e * top \<sqinter> v * ((top * e\<^sup>T * F)\<^sup>T \<sqinter> top) * (top \<sqinter> (F * e * top)\<^sup>T) * v\<^sup>T \<sqinter> top * e\<^sup>T * -F"
       by (smt comp_inf_vector covector_comp_inf vector_conv_covector vector_mult_closed vector_top_closed)
     also have "... = -F * e * top \<sqinter> v * (top * e\<^sup>T * F)\<^sup>T * (F * e * top)\<^sup>T * v\<^sup>T \<sqinter> top * e\<^sup>T * -F"
       by simp
     also have "... = -F * e * top \<sqinter> v * F\<^sup>T * e\<^sup>T\<^sup>T * top\<^sup>T * top\<^sup>T * e\<^sup>T * F\<^sup>T * v\<^sup>T \<sqinter> top * e\<^sup>T * -F"
       by (metis comp_associative conv_dist_comp)
     also have "... = -F * e * top \<sqinter> v * F * e * top * top * e\<^sup>T * F * v\<^sup>T \<sqinter> top * e\<^sup>T * -F"
       using assms(1) by auto
     also have "... = -F * e * top \<sqinter> v * F * e * top \<sqinter> top * e\<^sup>T * F * v\<^sup>T \<sqinter> top * e\<^sup>T * -F"
       by (smt comp_associative comp_inf_covector inf.sup_monoid.add_assoc inf_top.left_neutral vector_top_closed)
     also have "... = (-F \<sqinter> v * F) * e * top \<sqinter> top * e\<^sup>T * F * v\<^sup>T \<sqinter> top * e\<^sup>T * -F"
       using injective_comp_right_dist_inf assms(3) mult.semigroup_axioms semigroup.assoc by fastforce
     also have "... = (-F \<sqinter> v * F) * e * top \<sqinter> top * e\<^sup>T * (F * v\<^sup>T \<sqinter> -F)"
       using injective_comp_right_dist_inf assms(3) conv_dist_comp inf.sup_monoid.add_assoc mult.semigroup_axioms univalent_comp_left_dist_inf semigroup.assoc by fastforce
     also have "... = (-F \<sqinter> v * F) * e * top * top * e\<^sup>T * (F * v\<^sup>T \<sqinter> -F)"
       by (metis inf_top_right vector_export_comp vector_top_closed)
     also have "... = (-F \<sqinter> v * F) * e * top * e\<^sup>T * (F * v\<^sup>T \<sqinter> -F)"
       by (simp add: comp_associative)
     also have "... \<le> (-F \<sqinter> v * F) * (F * v\<^sup>T \<sqinter> -F)"
       by (smt assms(3) conv_dist_comp mult.semigroup_axioms mult_left_isotone shunt_bijective symmetric_top_closed top_right_mult_increasing semigroup.assoc)
     also have "... = (-F \<sqinter> v * F) * ((v * F)\<^sup>T \<sqinter> -F)"
       by (simp add: assms(1) conv_dist_comp)
     also have "... = (-F \<sqinter> v * F) * (-F \<sqinter> v * F)\<^sup>T"
       using assms(1) conv_complement conv_dist_inf by (simp add: inf.sup_monoid.add_commute)
     also have "... \<le> (-F \<sqinter> v) * (F \<sqinter> v)\<^sup>\<star> * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (-F \<sqinter> v)\<^sup>T"
     proof -
       let ?Fv = "F \<sqinter> v"
       have "-F \<sqinter> v * F \<le> -F \<sqinter> v * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star>"
         using assms(5) inf.sup_right_isotone mult_right_isotone comp_associative by auto
       also have "... \<le> -F \<sqinter> v * (F \<sqinter> v)\<^sup>\<star>"
       proof -
         have "v * v\<^sup>T \<le> 1"
           by (simp add: assms(2))
         hence "v * v\<^sup>T * F \<le> F"
           using assms(1) dual_order.trans mult_left_isotone by blast
         hence "v * v\<^sup>T * F\<^sup>T\<^sup>\<star> * F\<^sup>\<star> \<le> F"
           by (metis assms(1) mult_1_right preorder_idempotent star.circ_sup_one_right_unfold star.circ_transitive_equal star_one star_simulation_right_equal mult_assoc)
         hence "v * (F \<sqinter> v)\<^sup>T * F\<^sup>T\<^sup>\<star> * F\<^sup>\<star> \<le> F"
           by (meson conv_isotone dual_order.trans inf.cobounded2 inf.sup_monoid.add_commute mult_left_isotone mult_right_isotone)
         hence "v * (F \<sqinter> v)\<^sup>T * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star> \<le> F"
           by (meson conv_isotone dual_order.trans inf.cobounded2 inf.sup_monoid.add_commute mult_left_isotone mult_right_isotone comp_isotone conv_dist_inf inf.cobounded1 star_isotone)
         hence "-F \<sqinter> v * (F \<sqinter> v)\<^sup>T * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star> \<le> bot"
           using order.eq_iff p_antitone pseudo_complement by auto
         hence "(-F \<sqinter> v * (F \<sqinter> v)\<^sup>T * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star>) \<squnion> v * (v \<sqinter> F)\<^sup>\<star> \<le> v * (v \<sqinter> F)\<^sup>\<star>"
           using bot_least le_bot by fastforce
         hence "(-F \<squnion> v * (v \<sqinter> F)\<^sup>\<star>) \<sqinter> (v * (F \<sqinter> v)\<^sup>T * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star> \<squnion> v * (v \<sqinter> F)\<^sup>\<star>) \<le> v * (v \<sqinter> F)\<^sup>\<star>"
           by (simp add: sup_inf_distrib2)
         hence "(-F \<squnion> v * (v \<sqinter> F)\<^sup>\<star>) \<sqinter> v * ((F \<sqinter> v)\<^sup>T * (F \<sqinter> v)\<^sup>T\<^sup>\<star> \<squnion> 1) * (v \<sqinter> F)\<^sup>\<star> \<le> v * (v \<sqinter> F)\<^sup>\<star>"
           by (simp add: inf.sup_monoid.add_commute mult.semigroup_axioms mult_left_dist_sup mult_right_dist_sup semigroup.assoc)
         hence "(-F \<squnion> v * (v \<sqinter> F)\<^sup>\<star>) \<sqinter> v * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (v \<sqinter> F)\<^sup>\<star> \<le> v * (v \<sqinter> F)\<^sup>\<star>"
           by (simp add: star_left_unfold_equal sup_commute)
         hence "-F \<sqinter> v * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (v \<sqinter> F)\<^sup>\<star> \<le> v * (v \<sqinter> F)\<^sup>\<star>"
           using comp_inf.mult_right_sub_dist_sup_left inf.order_lesseq_imp by blast
         thus ?thesis
           by (simp add: inf.sup_monoid.add_commute)
       qed
       also have "... \<le> (v \<sqinter> -F * (F \<sqinter> v)\<^sup>T\<^sup>\<star>) * (F \<sqinter> v)\<^sup>\<star>"
         by (metis dedekind_2 conv_star_commute inf.sup_monoid.add_commute)
       also have "... \<le> (v \<sqinter> -F * F\<^sup>T\<^sup>\<star>) * (F \<sqinter> v)\<^sup>\<star>"
         using conv_isotone inf.sup_right_isotone mult_left_isotone mult_right_isotone star_isotone by auto
       also have "... = (v \<sqinter> -F * F) * (F \<sqinter> v)\<^sup>\<star>"
         by (metis assms(1) equivalence_comp_right_complement mult_left_one star_one star_simulation_right_equal)
       also have "... = (-F \<sqinter> v) * (F \<sqinter> v)\<^sup>\<star>"
         using assms(1) equivalence_comp_right_complement inf.sup_monoid.add_commute by auto
       finally have "-F \<sqinter> v * F \<le> (-F \<sqinter> v) * (F \<sqinter> v)\<^sup>\<star>"
         by simp
       hence "(-F \<sqinter> v * F) * (-F \<sqinter> v * F)\<^sup>T \<le> (-F \<sqinter> v) * (F \<sqinter> v)\<^sup>\<star> * ((-F \<sqinter> v) * (F \<sqinter> v)\<^sup>\<star>)\<^sup>T"
         by (simp add: comp_isotone conv_isotone)
       also have "... = (-F \<sqinter> v) * (F \<sqinter> v)\<^sup>\<star> * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (-F \<sqinter> v)\<^sup>T"
         by (simp add: comp_associative conv_dist_comp conv_star_commute)
       finally show ?thesis
         by simp
     qed
     also have "... \<le> (-F \<sqinter> v) * ((F \<sqinter> v\<^sup>\<star>) \<squnion> (F \<sqinter> v\<^sup>T\<^sup>\<star>)) * (-F \<sqinter> v)\<^sup>T"
     proof -
       have "(F \<sqinter> v)\<^sup>\<star> * (F \<sqinter> v)\<^sup>T\<^sup>\<star> \<le> F\<^sup>\<star> * F\<^sup>T\<^sup>\<star>"
         using fc_isotone by auto
       also have "... \<le> F * F"
         by (metis assms(1) preorder_idempotent star.circ_sup_one_left_unfold star.circ_transitive_equal star_right_induct_mult)
       finally have 21: "(F \<sqinter> v)\<^sup>\<star> * (F \<sqinter> v)\<^sup>T\<^sup>\<star> \<le> F"
         using assms(1) dual_order.trans by blast
       have "(F \<sqinter> v)\<^sup>\<star> * (F \<sqinter> v)\<^sup>T\<^sup>\<star> \<le> v\<^sup>\<star> * v\<^sup>T\<^sup>\<star>"
         by (simp add: fc_isotone)
       hence "(F \<sqinter> v)\<^sup>\<star> * (F \<sqinter> v)\<^sup>T\<^sup>\<star> \<le> F \<sqinter> v\<^sup>\<star> * v\<^sup>T\<^sup>\<star>"
         using 21 by simp
       also have "... = F \<sqinter> (v\<^sup>\<star> \<squnion> v\<^sup>T\<^sup>\<star>)"
         by (simp add: assms(2) cancel_separate_eq)
       finally show ?thesis
         by (metis assms(4, 6) comp_associative comp_inf.semiring.distrib_left comp_isotone inf_pp_semi_commute mult_left_isotone regular_closed_inf)
     qed
     also have "... \<le> (-F \<sqinter> v) * (F \<sqinter> v\<^sup>T\<^sup>\<star>) * (-F \<sqinter> v)\<^sup>T \<squnion> (-F \<sqinter> v) * (F \<sqinter> v\<^sup>\<star>) * (-F \<sqinter> v)\<^sup>T"
       by (simp add: mult_left_dist_sup mult_right_dist_sup)
     also have "... \<le> (-F \<sqinter> v) * (-F \<sqinter> v)\<^sup>T \<squnion> (-F \<sqinter> v) * (-F \<sqinter> v)\<^sup>T"
     proof -
       have "(-F \<sqinter> v) * (F \<sqinter> v\<^sup>T\<^sup>\<star>) \<le> (-F \<sqinter> v) * ((F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star> \<sqinter> v\<^sup>T\<^sup>\<star>)"
         by (simp add: assms(5) inf.coboundedI1 mult_right_isotone)
       also have "... = (-F \<sqinter> v) * ((F \<sqinter> v)\<^sup>T * (F \<sqinter> v)\<^sup>T\<^sup>\<star> * (F \<sqinter> v)\<^sup>\<star> \<sqinter> v\<^sup>T\<^sup>\<star>) \<squnion> (-F \<sqinter> v) * ((F \<sqinter> v)\<^sup>\<star> \<sqinter> v\<^sup>T\<^sup>\<star>)"
         by (metis comp_associative comp_inf.mult_right_dist_sup mult_left_dist_sup star.circ_loop_fixpoint)
       also have "... \<le> (-F \<sqinter> v) * (F \<sqinter> v)\<^sup>T * top \<squnion> (-F \<sqinter> v) * ((F \<sqinter> v)\<^sup>\<star> \<sqinter> v\<^sup>T\<^sup>\<star>)"
         by (simp add: comp_associative comp_isotone inf.coboundedI2 inf.sup_monoid.add_commute le_supI1)
       also have "... \<le> (-F \<sqinter> v) * (F \<sqinter> v)\<^sup>T * top \<squnion> (-F \<sqinter> v) * (v\<^sup>\<star> \<sqinter> v\<^sup>T\<^sup>\<star>)"
         by (smt comp_inf.mult_right_isotone comp_inf.semiring.add_mono order.eq_iff inf.cobounded2 inf.sup_monoid.add_commute mult_right_isotone star_isotone)
       also have "... \<le> bot \<squnion> (-F \<sqinter> v) * (v\<^sup>\<star> \<sqinter> v\<^sup>T\<^sup>\<star>)"
         by (metis assms(1, 2) forests_bot_1 comp_associative comp_inf.semiring.add_right_mono mult_semi_associative vector_bot_closed)
       also have "... \<le> -F \<sqinter> v"
         by (simp add: assms(2) acyclic_star_inf_conv)
       finally have 22: "(-F \<sqinter> v) * (F \<sqinter> v\<^sup>T\<^sup>\<star>) \<le> -F \<sqinter> v"
         by simp
       have "((-F \<sqinter> v) * (F \<sqinter> v\<^sup>T\<^sup>\<star>))\<^sup>T = (F \<sqinter> v\<^sup>\<star>) * (-F \<sqinter> v)\<^sup>T"
         by (simp add: assms(1) conv_dist_inf conv_star_commute conv_dist_comp)
       hence "(F \<sqinter> v\<^sup>\<star>) * (-F \<sqinter> v)\<^sup>T \<le> (-F \<sqinter> v)\<^sup>T"
         using 22 conv_isotone by fastforce
       thus ?thesis
         using 22 by (metis assms(4, 6) comp_associative comp_inf.pp_comp_semi_commute comp_inf.semiring.add_mono comp_isotone inf_pp_commute mult_left_isotone)
     qed
     also have "... = (-F \<sqinter> v) * (-F \<sqinter> v)\<^sup>T"
       by simp
     also have "... \<le> v * v\<^sup>T"
       by (simp add: comp_isotone conv_isotone)
     also have "... \<le> 1"
       by (simp add: assms(2))
     thus ?thesis
       using calculation dual_order.trans by blast
   qed
   have 3: "top * ?i * top = top"
   proof -
     have 31: "regular (e\<^sup>T * -F * v * F * e)"
       using assms(3, 4, 6) arc_regular regular_mult_closed by auto
     have 32: "bijective ((top * e\<^sup>T)\<^sup>T)"
       using assms(3) by (simp add: conv_dist_comp)
     have "top * ?i * top = top * (v \<sqinter> -F * e * top) * ((top * e\<^sup>T * F)\<^sup>T \<sqinter> top)"
       by (simp add: comp_associative comp_inf_vector_1)
     also have "... = (top \<sqinter> (-F * e * top)\<^sup>T) * v * ((top * e\<^sup>T * F)\<^sup>T \<sqinter> top)"
       using comp_inf_vector conv_dist_comp by auto
     also have "... = (-F * e * top)\<^sup>T * v * (top * e\<^sup>T * F)\<^sup>T"
       by simp
     also have "... = top\<^sup>T * e\<^sup>T * -F\<^sup>T * v * F\<^sup>T * e\<^sup>T\<^sup>T * top\<^sup>T"
       by (simp add: comp_associative conv_complement conv_dist_comp)
     finally have 33: "top * ?i * top = top * e\<^sup>T * -F * v * F * e * top"
       by (simp add: assms(1))
     have "top * ?i * top \<noteq> bot"
     proof (rule ccontr)
       assume "\<not> top * (v \<sqinter> - F * e * top \<sqinter> top * e\<^sup>T * F) * top \<noteq> bot"
       hence "top * e\<^sup>T * -F * v * F * e * top = bot"
         using 33 by auto
       hence "e\<^sup>T * -F * v * F * e = bot"
         using 31 tarski comp_associative le_bot by fastforce
       hence "top * (-F * v * F * e)\<^sup>T \<le> -(e\<^sup>T)"
         by (metis comp_associative conv_complement_sub_leq conv_involutive p_bot schroeder_5_p)
       hence "top * e\<^sup>T * F\<^sup>T * v\<^sup>T * -F\<^sup>T \<le> -(e\<^sup>T)"
         by (simp add: comp_associative conv_complement conv_dist_comp)
       hence "v * F * e * top * e\<^sup>T \<le> F"
         by (metis assms(1, 4) comp_associative conv_dist_comp schroeder_3_p symmetric_top_closed)
       hence "v * F * e * top * top * e\<^sup>T \<le> F"
         by (simp add: comp_associative)
       hence "v * F * e * top \<le> F * (top * e\<^sup>T)\<^sup>T"
         using 32 by (metis shunt_bijective comp_associative conv_involutive)
       hence "v * F * e * top \<le> F * e * top"
         using comp_associative conv_dist_comp by auto
       hence "v\<^sup>\<star> * F * e * top \<le> F * e * top"
         using comp_associative star_left_induct_mult_iff by auto
       hence "e\<^sup>T * F * e * top \<le> F * e * top"
         by (meson assms(9) mult_left_isotone order_trans)
       hence "e\<^sup>T * F * e * top * (e * top)\<^sup>T \<le> F"
         using 32 shunt_bijective assms(3) mult_assoc by auto
       hence 34: "e\<^sup>T * F * e * top * top * e\<^sup>T \<le> F"
         by (metis conv_dist_comp mult.semigroup_axioms symmetric_top_closed semigroup.assoc)
       hence "e\<^sup>T \<le> F"
       proof -
         have "e\<^sup>T \<le> e\<^sup>T * e * e\<^sup>T"
           by (metis conv_involutive ex231c)
         also have "... \<le> e\<^sup>T * F * e * e\<^sup>T"
           using assms(1) comp_associative mult_left_isotone mult_right_isotone by fastforce
         also have "... \<le> e\<^sup>T * F * e * top * top * e\<^sup>T"
           by (simp add: mult_left_isotone top_right_mult_increasing vector_mult_closed)
         finally show ?thesis
           using 34 by simp
       qed
       hence 35: "e \<le> F"
         using assms(1) conv_order by fastforce
       have "top * (F * e)\<^sup>T \<le> - e"
         using assms(8) comp_associative schroeder_4_p by auto
       hence "top * e\<^sup>T * F \<le> - e"
         by (simp add: assms(1) comp_associative conv_dist_comp)
       hence "(top * e\<^sup>T)\<^sup>T * e \<le> - F"
         using schroeder_3_p by auto
       hence "e * top * e \<le> - F"
         by (simp add: conv_dist_comp)
       hence "e \<le> - F"
         by (simp add: assms(3) arc_top_arc)
       hence "e \<le> F \<sqinter> - F"
         using 35 inf.boundedI by blast
       hence "e = bot"
         using bot_unique by auto
       thus "False"
         using assms(10) by auto
     qed
     thus ?thesis
       by (metis assms(3, 4, 6) arc_regular regular_closed_inf regular_closed_top regular_conv_closed regular_mult_closed semiring.mult_not_zero tarski)
   qed
   have "bijective (?i * top) \<and> bijective (?i\<^sup>T * top)"
     using 1 2 3 arc_expanded by blast
   thus ?thesis
     by blast
 qed
 
 subsubsection \<open>Comparison of edge weights\<close>
 
 text \<open>
 In this section we compare the weight of the \<open>selected_edge\<close> with other edges of interest.
-Theorems 8, 9, 10 and 11 are supporting lemmas.
-For example, Theorem 8 is used to show that the \<open>selected_edge\<close> has its source inside and its target outside the component it is chosen for.
+For example, Theorem \<open>e_leq_c_c_complement_transpose_general\<close> is used to show that the \<open>selected_edge\<close> has its source inside and its target outside the component it is chosen for.
 \<close>
 
-text \<open>Theorem 8\<close>
-
 lemma e_leq_c_c_complement_transpose_general:
-  assumes "e = minarc (c * -(c)\<^sup>T \<sqinter> g)"
-    and "regular c"
-  shows "e \<le> c * -(c)\<^sup>T"
+  assumes "e = minarc (v * -(v)\<^sup>T \<sqinter> g)"
+    and "regular v"
+  shows "e \<le> v * -(v)\<^sup>T"
 proof -
-  have "e \<le> -- (c * - c\<^sup>T \<sqinter> g)"
+  have "e \<le> -- (v * - v\<^sup>T \<sqinter> g)"
     using assms(1) minarc_below order_trans by blast
-  also have "... \<le> -- (c * - c\<^sup>T)"
+  also have "... \<le> -- (v * - v\<^sup>T)"
     using order_lesseq_imp pp_isotone_inf by blast
-  also have "... = c * - c\<^sup>T"
+  also have "... = v * - v\<^sup>T"
     using assms(2) regular_mult_closed by auto
   finally show ?thesis
     by simp
 qed
 
-text \<open>Theorem 9\<close>
-
 lemma x_leq_c_transpose_general:
-  assumes "forest h"
-    and "vector c"
-    and "x\<^sup>T * top \<le> forest_components(h) * e * top"
-    and "e \<le> c * -c\<^sup>T"
-    and "c = forest_components(h) * c"
-  shows "x \<le> c\<^sup>T"
+  assumes "vector_classes x v"
+    and "a\<^sup>T * top \<le> x * e * top"
+    and "e \<le> v * -v\<^sup>T"
+  shows "a \<le> v\<^sup>T"
 proof -
-  let ?H = "forest_components h"
-  have "x \<le> top * x"
+  have 1: "equivalence x"
+    using assms(1) using vector_classes_def by blast
+  have "a \<le> top * a"
     using top_left_mult_increasing by blast
-  also have "... \<le> (?H * e * top)\<^sup>T"
-    using assms(3) conv_dist_comp conv_order by force
-  also have "... = top * e\<^sup>T * ?H"
-    using assms(1) comp_associative conv_dist_comp forest_components_equivalence by auto
-  also have "... \<le> top * (c * - c\<^sup>T)\<^sup>T * ?H"
-    by (simp add: assms(4) conv_isotone mult_left_isotone mult_right_isotone)
-  also have "... = top * (- c * c\<^sup>T) * ?H"
+  also have "... \<le> (x * e * top)\<^sup>T"
+    using assms(2) conv_dist_comp conv_isotone by fastforce
+  also have "... = top * e\<^sup>T * x"
+    using 1 by (simp add: conv_dist_comp mult_assoc)
+  also have "... \<le> top * (v * - v\<^sup>T)\<^sup>T * x"
+    by (metis assms(3) conv_dist_comp conv_isotone mult_left_isotone symmetric_top_closed)
+  also have "... = top * (- v * v\<^sup>T) * x"
     by (simp add: conv_complement conv_dist_comp)
-  also have "... \<le> top * c\<^sup>T * ?H"
+  also have "... \<le> top * v\<^sup>T * x"
     by (metis mult_left_isotone top.extremum mult_assoc)
-  also have "... = c\<^sup>T * ?H"
-    using assms(1, 2) component_is_vector vector_conv_covector by auto
-  also have "... = c\<^sup>T"
-    by (metis assms(1, 5) fch_equivalence conv_dist_comp)
+  also have "... = v\<^sup>T * x"
+    using assms(1) vector_classes_def vector_conv_covector by auto
+  also have "... = v\<^sup>T"
+    by (metis assms(1) order.antisym conv_dist_comp conv_order dual_order.trans mult_right_isotone mult_sub_right_one vector_classes_def)
   finally show ?thesis
     by simp
 qed
 
-text \<open>Theorem 10\<close>
-
 lemma x_leq_c_complement_general:
-  assumes "vector c"
-    and "c * c\<^sup>T \<le> forest_components h"
-    and "x \<le> c\<^sup>T"
-    and "x \<le> -forest_components h"
-  shows "x \<le> -c"
+  assumes "vector v"
+    and "v * v\<^sup>T \<le> x"
+    and "a \<le> v\<^sup>T"
+    and "a \<le> -x"
+  shows "a \<le> -v"
 proof -
-  let ?H = "forest_components h"
-  have "x \<le> - ?H \<sqinter> c\<^sup>T"
+  have "a \<le> -x \<sqinter> v\<^sup>T"
     using assms(3, 4) by auto
-  also have "... \<le> - c"
+  also have "... \<le> - v"
   proof -
-    have "c \<sqinter> c\<^sup>T \<le> ?H"
+    have "v \<sqinter> v\<^sup>T \<le> x"
       using assms(1, 2) vector_covector by auto
-    hence "-?H \<sqinter> c \<sqinter> c\<^sup>T \<le> bot"
+    hence "-x \<sqinter> v \<sqinter> v\<^sup>T \<le> bot"
       using inf.sup_monoid.add_assoc p_antitone pseudo_complement by fastforce
     thus ?thesis
       using le_bot p_shunting_swap pseudo_complement by blast
   qed
   finally show ?thesis
     by simp
 qed
 
-text \<open>Theorem 11\<close>
-
-lemma sum_e_below_sum_x_when_outgoing_same_component_general:
-  assumes "e = minarc (c * -(c)\<^sup>T \<sqinter> g)"
-    and "regular c"
-    and "forest h"
-    and "vector c"
-    and "x\<^sup>T * top \<le> (forest_components h) * e * top"
-    and "c = (forest_components h) * c"
-    and "c * c\<^sup>T \<le> forest_components h"
-    and "x \<le> - forest_components h \<sqinter> -- g"
+lemma sum_e_below_sum_a_when_outgoing_same_component_general:
+  assumes "e = minarc (v * -(v)\<^sup>T \<sqinter> g)"
     and "symmetric g"
-    and "arc x"
-    and "c \<noteq> bot"
-  shows "sum (e \<sqinter> g) \<le> sum (x \<sqinter> g)"
+    and "arc a"
+    and "a \<le> -x \<sqinter> -- g"
+    and "a\<^sup>T * top \<le> x * e * top"
+    and "unique_vector_class x v"
+  shows "sum (e \<sqinter> g) \<le> sum (a \<sqinter> g)"
 proof -
-  let ?H = "forest_components h"
-  have 1:"e \<le> c * - c\<^sup>T"
-    using assms(1, 2) e_leq_c_c_complement_transpose_general by auto
-  have 2: "x \<le> c\<^sup>T"
-    using 1 assms(3, 4, 5, 6) x_leq_c_transpose_general by auto
-  hence "x \<le> -c"
-    using assms(4, 7, 8) x_leq_c_complement_general inf.boundedE by blast
-  hence "x \<le> - c \<sqinter> c\<^sup>T"
+  have 1:"e \<le> v * - v\<^sup>T"
+    using assms(1, 6) e_leq_c_c_complement_transpose_general unique_vector_class_def vector_classes_def by auto
+  have 2: "a \<le> v\<^sup>T"
+    using 1 assms(5) assms(6) x_leq_c_transpose_general unique_vector_class_def by blast
+  hence "a \<le> -v"
+    using assms(4, 6) inf.boundedE unique_vector_class_def vector_classes_def x_leq_c_complement_general by meson
+  hence "a \<le> - v \<sqinter> v\<^sup>T"
     using 2 by simp
-  hence "x \<le> - c * c\<^sup>T"
-    using assms(4) by (simp add: vector_complement_closed vector_covector)
-  hence "x\<^sup>T \<le> c\<^sup>T\<^sup>T * - c\<^sup>T"
-    by (metis conv_complement conv_dist_comp conv_isotone)
-  hence 3: "x\<^sup>T \<le> c * - c\<^sup>T"
+  hence "a \<le> - v * v\<^sup>T"
+    using assms(6) vector_complement_closed vector_covector unique_vector_class_def vector_classes_def by metis
+  hence "a\<^sup>T \<le> v\<^sup>T\<^sup>T * - v\<^sup>T"
+    using conv_complement conv_dist_comp conv_isotone by metis
+  hence 3: "a\<^sup>T \<le> v * - v\<^sup>T"
     by simp
-  hence "x \<le> -- g"
-    using assms(8) by auto
-  hence "x\<^sup>T \<le> -- g"
-    using assms(9) conv_complement conv_isotone by fastforce
-  hence "x\<^sup>T \<sqinter> c * - c\<^sup>T \<sqinter> -- g \<noteq> bot"
-    using 3 by (metis assms(10, 11) comp_inf.semiring.mult_not_zero conv_dist_comp
-          conv_involutive inf.orderE mult_right_zero top.extremum)
-  hence "x\<^sup>T \<sqinter> c * - c\<^sup>T \<sqinter> g \<noteq> bot"
+  hence "a \<le> -- g"
+    using assms(4) by auto
+  hence "a\<^sup>T \<le> -- g"
+    using assms(2) conv_complement conv_isotone by fastforce
+  hence "a\<^sup>T \<sqinter> v * - v\<^sup>T \<sqinter> -- g \<noteq> bot"
+    using 3 assms(3, 6) inf.orderE semiring.mult_not_zero unique_vector_class_def vector_classes_def by metis
+  hence "a\<^sup>T \<sqinter> v * - v\<^sup>T \<sqinter> g \<noteq> bot"
     using inf.sup_monoid.add_commute pp_inf_bot_iff by auto
-  hence "sum (minarc (c * - c\<^sup>T \<sqinter> g) \<sqinter> (c * - c\<^sup>T \<sqinter> g)) \<le> sum (x\<^sup>T \<sqinter> c * - c\<^sup>T \<sqinter> g)"
-    using assms(10) minarc_min inf.sup_monoid.add_assoc by auto
-  hence "sum (e \<sqinter> c * - c\<^sup>T \<sqinter> g) \<le> sum (x\<^sup>T \<sqinter> c * - c\<^sup>T \<sqinter> g)"
-    using assms(1) inf.sup_monoid.add_assoc by auto
-  hence "sum (e \<sqinter> g) \<le> sum (x\<^sup>T \<sqinter> g)"
+  hence "sum (minarc (v * - v\<^sup>T \<sqinter> g) \<sqinter> (v * - v\<^sup>T \<sqinter> g)) \<le> sum (a\<^sup>T \<sqinter> v * - v\<^sup>T \<sqinter> g)"
+    using assms(3) minarc_min inf.sup_monoid.add_assoc by simp
+  hence "sum (e \<sqinter> v * - v\<^sup>T \<sqinter> g) \<le> sum (a\<^sup>T \<sqinter> v * - v\<^sup>T \<sqinter> g)"
+    using assms(1, 6) inf.sup_monoid.add_assoc by simp
+  hence "sum (e \<sqinter> g) \<le> sum (a\<^sup>T \<sqinter> g)"
     using 1 3 by (metis inf.orderE)
-  hence "sum (e \<sqinter> g) \<le> sum (x \<sqinter> g)"
-    using assms(9) sum_symmetric by auto
+  hence "sum (e \<sqinter> g) \<le> sum (a \<sqinter> g)"
+    by (simp add: assms(2) sum_symmetric)
   thus ?thesis
     by simp
 qed
 
 lemma sum_e_below_sum_x_when_outgoing_same_component:
   assumes "symmetric g"
     and "vector j"
     and "forest h"
+    and "regular h"
     and "x \<le> - forest_components h \<sqinter> -- g"
     and "x\<^sup>T * top \<le> forest_components h * selected_edge h j g * top"
     and "j \<noteq> bot"
     and "arc x"
   shows "sum (selected_edge h j g \<sqinter> g) \<le> sum (x \<sqinter> g)"
 proof -
   let ?e = "selected_edge h j g"
   let ?c = "choose_component (forest_components h) j"
   let ?H = "forest_components h"
   show ?thesis
-  proof (rule sum_e_below_sum_x_when_outgoing_same_component_general)
+  proof (rule sum_e_below_sum_a_when_outgoing_same_component_general)
   next
     show "?e = minarc (?c * - ?c\<^sup>T \<sqinter> g)"
       by simp
   next
-    show "regular ?c"
-      using component_is_regular by auto
-  next
-    show "forest h"
-      by (simp add: assms(3))
-  next
-    show "vector ?c"
-      by (simp add: assms(2, 6) component_is_vector)
-  next
-    show "x\<^sup>T * top \<le> ?H * ?e * top"
-      by (simp add: assms(5))
-  next
-    show "?c = ?H * ?c"
-      using component_single by auto
-  next
-    show "?c * ?c\<^sup>T \<le> ?H"
-      by (simp add: component_is_connected)
-  next
-    show "x \<le> -?H \<sqinter> -- g"
-      using assms(4) by auto
-  next
     show "symmetric g"
       by (simp add: assms(1))
   next
     show "arc x"
-      by (simp add: assms(7))
+      by (simp add: assms(8))
   next
-    show "?c \<noteq> bot"
-      using assms(2, 5 , 6, 7) inf_bot_left le_bot minarc_bot mult_left_zero mult_right_zero by fastforce
+    show "x \<le> -?H \<sqinter> -- g"
+      using assms(5) by auto
+  next
+    show "x\<^sup>T * top \<le> ?H * ?e * top"
+      by (simp add: assms(6))
+  next
+    show "unique_vector_class ?H ?c"
+    proof (unfold unique_vector_class_def, unfold vector_classes_def, intro conjI)
+    next
+      show "regular ?H"
+        by (metis assms(4) conv_complement pp_dist_star regular_mult_closed)
+    next
+      show "regular ?c"
+        using component_is_regular by auto
+    next
+      show "reflexive ?H"
+        using assms(3) forest_components_equivalence by blast
+    next
+      show "transitive ?H"
+        using assms(3) fch_equivalence by blast
+    next
+      show "symmetric ?H"
+        by (simp add: assms(3) fch_equivalence)
+    next
+      show "vector ?c"
+        by (simp add: assms(2, 6) component_is_vector)
+    next
+      show "?H * ?c \<le> ?c"
+        using component_single by auto
+    next
+      show "?c \<noteq> bot"
+        using assms(2, 6 , 7, 8) inf_bot_left le_bot minarc_bot mult_left_zero mult_right_zero by fastforce
+    next
+      show "?c * ?c\<^sup>T \<le> ?H"
+        by (simp add: component_is_connected)
+    qed
   qed
 qed
 
 text \<open>
-If there is a path in the \<open>big_forest\<close> from an edge between components, $a$, to the \<open>selected_edge\<close>, $e$, then the weight of $e$ is no greater than the weight of $a$.
+If there is a path in the \<open>forest_modulo_equivalence\<close> from an edge between components, $a$, to the \<open>selected_edge\<close>, $e$, then the weight of $e$ is no greater than the weight of $a$.
 This is because either,
 \begin{itemize}
 \item the edges $a$ and $e$ are adjacent the same component so that we can use \<open>sum_e_below_sum_x_when_outgoing_same_component\<close>, or
 \item there is at least one edge between $a$ and $e$, namely $x$, the edge incoming to the component that $e$ is outgoing from.
-      The path from $a$ to $e$ is split on $x$ using \<open>big_forest_path_split_disj\<close>.
+      The path from $a$ to $e$ is split on $x$ using \<open>forest_modulo_equivalence_path_split_disj\<close>.
       We show that the weight of $e$ is no greater than the weight of $x$ by making use of lemma \<open>sum_e_below_sum_x_when_outgoing_same_component\<close>.
       We define $x$ in a way that we can show that the weight of $x$ is no greater than the weight of $a$ using the invariant.
       Then, it follows that the weight of $e$ is no greater than the weight of $a$ owing to transitivity.
 \end{itemize}
 \<close>
 
-lemma a_to_e_in_bigforest:
+lemma a_to_e_in_forest_modulo_equivalence:
   assumes "symmetric g"
     and "f \<le> --g"
     and "vector j"
     and "forest h"
-    and "big_forest (forest_components h) d"
+    and "forest_modulo_equivalence (forest_components h) d"
     and "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
-    and "(\<forall> a b . bf_between_arcs a b (forest_components h) d \<and> a \<le> -(forest_components h) \<sqinter> -- g \<and> b \<le> d \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g))"
+    and "(\<forall> a b . forest_modulo_equivalence_path a b (forest_components h) d \<and> a \<le> -(forest_components h) \<sqinter> -- g \<and> b \<le> d \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g))"
     and "regular d"
     and "j \<noteq> bot"
     and "b = selected_edge h j g"
     and "arc a"
-    and "bf_between_arcs a b (forest_components h) (d \<squnion> selected_edge h j g)"
+    and "forest_modulo_equivalence_path a b (forest_components h) (d \<squnion> selected_edge h j g)"
     and "a \<le> - forest_components h \<sqinter> -- g"
     and "regular h"
   shows "sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
 proof -
   let ?p = "path f h j g"
   let ?e = "selected_edge h j g"
   let ?F = "forest_components f"
   let ?H = "forest_components h"
   have "sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
   proof (cases "a\<^sup>T * top \<le> ?H * ?e * top")
     case True
     show "a\<^sup>T * top \<le> ?H * ?e * top \<Longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
     proof-
       have "sum (?e \<sqinter> g) \<le> sum (a \<sqinter> g)"
       proof (rule sum_e_below_sum_x_when_outgoing_same_component)
         show "symmetric g"
           using assms(1) by auto
       next
         show "vector j"
           using assms(3) by blast
       next
         show "forest h"
           by (simp add: assms(4))
       next
         show "a \<le> - ?H \<sqinter> -- g"
           using assms(13) by auto
       next
         show "a\<^sup>T * top \<le> ?H * ?e * top"
           using True by auto
       next
         show "j \<noteq> bot"
           by (simp add: assms(9))
       next
         show "arc a"
           by (simp add: assms(11))
+      next
+        show "regular h"
+          using assms(14) by auto
       qed
       thus ?thesis
         using assms(10) by auto
     qed
   next
     case False
     show "\<not> a\<^sup>T * top \<le> ?H * ?e * top \<Longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
     proof -
       let ?d' = "d \<squnion> ?e"
       let ?x = "d \<sqinter> top * ?e\<^sup>T * ?H \<sqinter> (?H * d\<^sup>T)\<^sup>\<star> * ?H * a\<^sup>T * top"
       have 61: "arc (?x)"
       proof (rule shows_arc_x)
-        show "big_forest ?H d"
+        show "forest_modulo_equivalence ?H d"
           by (simp add: assms(5))
       next
-        show "bf_between_arcs a ?e ?H d"
+        show "forest_modulo_equivalence_path a ?e ?H d"
         proof -
-          have 611: "bf_between_arcs a b ?H (d \<squnion> b)"
+          have 611: "forest_modulo_equivalence_path a b ?H (d \<squnion> b)"
             using assms(10, 12) by auto
           have 616: "regular h"
             using assms(14) by auto
           have "regular a"
-            using 611 bf_between_arcs_def arc_regular by fastforce
+            using 611 forest_modulo_equivalence_path_def arc_regular by fastforce
           thus ?thesis
-            using 616 by (smt big_forest_path_split_disj assms(4, 8, 10, 12) bf_between_arcs_def fch_equivalence minarc_regular regular_closed_star regular_conv_closed regular_mult_closed)
+            using 616 by (smt forest_modulo_equivalence_path_split_disj assms(4, 8, 10, 12) forest_modulo_equivalence_path_def fch_equivalence minarc_regular regular_closed_star regular_conv_closed regular_mult_closed)
         qed
       next
         show "(?H * d)\<^sup>+ \<le> - ?H"
-          using assms(5) big_forest_def by blast
+          using assms(5) forest_modulo_equivalence_def le_bot pseudo_complement by blast
       next
         show "\<not> a\<^sup>T * top \<le> ?H * ?e * top"
           by (simp add: False)
       next
         show "regular a"
-          using assms(12) bf_between_arcs_def arc_regular by auto
+          using assms(12) forest_modulo_equivalence_path_def arc_regular by auto
       next
         show "regular ?e"
           using minarc_regular by auto
       next
         show "regular ?H"
           using assms(14) pp_dist_star regular_conv_closed regular_mult_closed by auto
       next
         show "regular d"
           using assms(8) by auto
       qed
       have 62: "bijective (a\<^sup>T * top)"
         by (simp add: assms(11))
       have 63: "bijective (?x * top)"
         using 61 by simp
       have 64: "?x \<le> (?H * d\<^sup>T)\<^sup>\<star> * ?H * a\<^sup>T * top"
         by simp
       hence "?x * top \<le> (?H * d\<^sup>T)\<^sup>\<star> * ?H * a\<^sup>T * top"
         using mult_left_isotone inf_vector_comp by auto
       hence "a\<^sup>T * top \<le> ((?H * d\<^sup>T)\<^sup>\<star> * ?H)\<^sup>T * ?x * top"
         using 62 63 64 by (smt bijective_reverse mult_assoc)
       also have "... = ?H * (d * ?H)\<^sup>\<star> * ?x * top"
         using conv_dist_comp conv_star_commute by auto
       also have "... = (?H * d)\<^sup>\<star> * ?H * ?x * top"
         by (simp add: star_slide)
       finally have "a\<^sup>T * top \<le> (?H * d)\<^sup>\<star> * ?H * ?x * top"
         by simp
-      hence 65: "bf_between_arcs a ?x ?H d"
-        using 61 assms(12) bf_between_arcs_def by blast
+      hence 65: "forest_modulo_equivalence_path a ?x ?H d"
+        using 61 assms(12) forest_modulo_equivalence_path_def by blast
       have 66: "?x \<le> d"
         by (simp add: inf.sup_monoid.add_assoc)
       hence x_below_a: "sum (?x \<sqinter> g) \<le> sum (a \<sqinter> g)"
-        using 65 bf_between_arcs_def assms(7, 13) by blast
+        using 65 forest_modulo_equivalence_path_def assms(7, 13) by blast
       have "sum (?e \<sqinter> g) \<le> sum (?x \<sqinter> g)"
       proof (rule sum_e_below_sum_x_when_outgoing_same_component)
         show "symmetric g"
           using assms(1) by auto
       next
         show "vector j"
           using assms(3) by blast
       next
         show "forest h"
           by (simp add: assms(4))
       next
         show "?x \<le> - ?H \<sqinter> -- g"
         proof -
           have 67: "?x \<le> - ?H"
-            using 66 assms(5) big_forest_def order_lesseq_imp by blast
+          proof -
+            have "?x \<le> d"
+              using 66 by blast
+            also have "... \<le> ?H * d"
+              using dual_order.trans star.circ_loop_fixpoint sup.cobounded2 mult_assoc by metis
+            also have "... \<le> (?H * d)\<^sup>+"
+              using star.circ_mult_increasing by blast
+            also have "... \<le> - ?H"
+              using assms(5) bot_unique pseudo_complement forest_modulo_equivalence_def by blast
+            thus ?thesis
+              using calculation inf.order_trans by blast
+          qed
           have "?x \<le> d"
             by (simp add: conv_isotone inf.sup_monoid.add_assoc)
           also have "... \<le> f \<squnion> f\<^sup>T"
           proof -
             have "h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T = f \<squnion> f\<^sup>T"
               by (simp add: assms(6))
             thus ?thesis
               by (metis (no_types) le_supE sup.absorb_iff2 sup.idem)
           qed
           also have "... \<le> -- g"
             using assms(1, 2) conv_complement conv_isotone by fastforce
           finally have "?x \<le> -- g"
             by simp
           thus ?thesis
             by (simp add: 67)
         qed
       next
         show "?x\<^sup>T * top \<le> ?H * ?e * top"
         proof -
           have "?x \<le> top * ?e\<^sup>T * ?H"
             using inf.coboundedI1 by auto
           hence "?x\<^sup>T \<le> ?H * ?e * top"
             using conv_dist_comp conv_dist_inf conv_star_commute inf.orderI inf.sup_monoid.add_assoc inf.sup_monoid.add_commute mult_assoc by auto
           hence "?x\<^sup>T * top \<le> ?H * ?e * top * top"
             by (simp add: mult_left_isotone)
           thus ?thesis
             by (simp add: mult_assoc)
         qed
       next
         show "j \<noteq> bot"
           by (simp add: assms(9))
       next
         show "arc (?x)"
           using 61 by blast
+      next
+        show "regular h"
+          using assms(14) by auto
       qed
       hence "sum (?e \<sqinter> g) \<le> sum (a \<sqinter> g)"
         using x_below_a order.trans by blast
       thus ?thesis
         by (simp add: assms(10))
     qed
   qed
   thus ?thesis
     by simp
 qed
 
 subsubsection \<open>Maintenance of algorithm invariants\<close>
 
 text \<open>
 In this section, most of the work is done to maintain the invariants of the inner and outer loops of the algorithm.
 In particular, we use \<open>exists_a_w\<close> to maintain that $f$ can be extended to a minimum spanning forest.
 \<close>
 
 lemma boruvka_exchange_spanning_inv:
   assumes "forest v"
     and "v\<^sup>\<star> * e\<^sup>T = e\<^sup>T"
     and "i \<le> v \<sqinter> top * e\<^sup>T * w\<^sup>T\<^sup>\<star>"
     and "arc i"
     and "arc e"
     and "v \<le> --g"
     and "w \<le> --g"
     and "e \<le> --g"
     and "components g \<le> forest_components v"
   shows "i \<le> (v \<sqinter> -i)\<^sup>T\<^sup>\<star> * e\<^sup>T * top"
 proof -
   have 1: "(v \<sqinter> -i \<sqinter> -i\<^sup>T) * (v\<^sup>T \<sqinter> -i \<sqinter> -i\<^sup>T) \<le> 1"
     using assms(1) comp_isotone order.trans inf.cobounded1 by blast
   have 2: "bijective (i * top) \<and> bijective (e\<^sup>T * top)"
     using assms(4, 5) mult_assoc by auto
   have "i \<le> v * (top * e\<^sup>T * w\<^sup>T\<^sup>\<star>)\<^sup>T"
     using assms(3) covector_mult_closed covector_restrict_comp_conv order_lesseq_imp vector_top_closed by blast
   also have "... \<le> v * w\<^sup>T\<^sup>\<star>\<^sup>T * e\<^sup>T\<^sup>T * top\<^sup>T"
     by (simp add: comp_associative conv_dist_comp)
   also have "... \<le> v * w\<^sup>\<star> * e * top"
     by (simp add: conv_star_commute)
   also have "... = v * w\<^sup>\<star> * e * e\<^sup>T * e * top"
     using assms(5) arc_eq_1 by (simp add: comp_associative)
   also have "... \<le> v * w\<^sup>\<star> * e * e\<^sup>T * top"
     by (simp add: comp_associative mult_right_isotone)
   also have "... \<le> (--g) * (--g)\<^sup>\<star> * (--g) * e\<^sup>T * top"
     using assms(6, 7, 8) by (simp add: comp_isotone star_isotone)
   also have "... \<le> (--g)\<^sup>\<star> * e\<^sup>T * top"
     by (metis comp_isotone mult_left_isotone star.circ_increasing star.circ_transitive_equal)
   also have "... \<le> v\<^sup>T\<^sup>\<star> * v\<^sup>\<star> * e\<^sup>T * top"
     by (simp add: assms(9) mult_left_isotone)
   also have "... \<le> v\<^sup>T\<^sup>\<star> * e\<^sup>T * top"
     by (simp add: assms(2) comp_associative)
   finally have "i \<le> v\<^sup>T\<^sup>\<star> * e\<^sup>T * top"
     by simp
   hence "i * top \<le> v\<^sup>T\<^sup>\<star> * e\<^sup>T * top"
     by (metis comp_associative mult_left_isotone vector_top_closed)
   hence "e\<^sup>T * top \<le> v\<^sup>T\<^sup>\<star>\<^sup>T * i * top"
     using 2 by (metis bijective_reverse mult_assoc)
   also have "... = v\<^sup>\<star> * i * top"
     by (simp add: conv_star_commute)
   also have "... \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
   proof -
     have 3: "i * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
       using star.circ_loop_fixpoint sup_right_divisibility mult_assoc by auto
     have "(v \<sqinter> i) * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<le> i * top * i * top"
       by (metis comp_isotone inf.cobounded1 inf.sup_monoid.add_commute mult_left_isotone top.extremum)
     also have "... \<le> i * top"
       by simp
     finally have 4: "(v \<sqinter> i) * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
       using 3 dual_order.trans by blast
     have 5: "(v \<sqinter> -i \<sqinter> -i\<^sup>T) * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
       by (metis mult_left_isotone star.circ_increasing star.left_plus_circ)
     have "v\<^sup>+ \<le> -1"
       by (simp add: assms(1))
     hence "v * v \<le> -1"
       by (metis mult_left_isotone order_trans star.circ_increasing star.circ_plus_same)
     hence "v * 1 \<le> -v\<^sup>T"
       by (simp add: schroeder_5_p)
     hence "v \<le> -v\<^sup>T"
       by simp
     hence "v \<sqinter> v\<^sup>T \<le> bot"
       by (simp add: bot_unique pseudo_complement)
     hence 7: "v \<sqinter> i\<^sup>T \<le> bot"
       by (metis assms(3) comp_inf.mult_right_isotone conv_dist_inf inf.boundedE inf.le_iff_sup le_bot)
     hence "(v \<sqinter> i\<^sup>T) * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<le> bot"
       using le_bot semiring.mult_zero_left by fastforce
     hence 6: "(v \<sqinter> i\<^sup>T) * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
       using bot_least le_bot by blast
     have 8: "v = (v \<sqinter> i) \<squnion> (v \<sqinter> i\<^sup>T) \<squnion> (v \<sqinter> -i \<sqinter> -i\<^sup>T)"
     proof -
       have 81: "regular i"
         by (simp add: assms(4) arc_regular)
       have "(v \<sqinter> i\<^sup>T) \<squnion> (v \<sqinter> -i \<sqinter> -i\<^sup>T) = (v \<sqinter> -i)"
         using 7 by (metis comp_inf.coreflexive_comp_inf_complement inf_import_p inf_p le_bot maddux_3_11_pp top.extremum)
       hence "(v \<sqinter> i) \<squnion> (v \<sqinter> i\<^sup>T) \<squnion> (v \<sqinter> -i \<sqinter> -i\<^sup>T) = (v \<sqinter> i) \<squnion> (v \<sqinter> -i)"
         by (simp add: sup.semigroup_axioms semigroup.assoc)
       also have "... = v"
         using 81 by (metis maddux_3_11_pp)
       finally show ?thesis
         by simp
     qed
     have "(v \<sqinter> i) * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<squnion> (v \<sqinter> i\<^sup>T) * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<squnion> (v \<sqinter> -i \<sqinter> -i\<^sup>T) * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
       using 4 5 6 by simp
     hence "((v \<sqinter> i) \<squnion> (v \<sqinter> i\<^sup>T) \<squnion> (v \<sqinter> -i \<sqinter> -i\<^sup>T)) * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
       by (simp add: mult_right_dist_sup)
     hence "v * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
       using 8 by auto
     hence "i * top \<squnion> v * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
       using 3 by auto
     hence 9:"v\<^sup>\<star> * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
       by (simp add: star_left_induct_mult mult_assoc)
     have "v\<^sup>\<star> * i * top \<le> v\<^sup>\<star> * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
       using 3 mult_right_isotone mult_assoc by auto
     thus ?thesis
       using 9 order.trans by blast
   qed
   finally have "e\<^sup>T * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * i * top"
     by simp
   hence "i * top \<le> (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star>\<^sup>T * e\<^sup>T * top"
     using 2 by (metis bijective_reverse mult_assoc)
   also have "... = (v\<^sup>T \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * e\<^sup>T * top"
     using comp_inf.inf_vector_comp conv_complement conv_dist_inf conv_star_commute inf.sup_monoid.add_commute by auto
   also have "... \<le> ((v \<sqinter> -i \<sqinter> -i\<^sup>T) \<squnion> (v\<^sup>T \<sqinter> -i \<sqinter> -i\<^sup>T))\<^sup>\<star> * e\<^sup>T * top"
     by (simp add: mult_left_isotone star_isotone)
   finally have "i \<le> ((v\<^sup>T \<sqinter> -i \<sqinter> -i\<^sup>T) \<squnion> (v \<sqinter> -i \<sqinter> -i\<^sup>T))\<^sup>\<star> * e\<^sup>T * top"
     using dual_order.trans top_right_mult_increasing sup_commute by auto
   also have "... = (v\<^sup>T \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * (v \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * e\<^sup>T * top"
     using 1 cancel_separate_1 by (simp add: sup_commute)
   also have "... \<le> (v\<^sup>T \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * v\<^sup>\<star> * e\<^sup>T * top"
     by (simp add: inf_assoc mult_left_isotone mult_right_isotone star_isotone)
   also have "... = (v\<^sup>T \<sqinter> -i \<sqinter> -i\<^sup>T)\<^sup>\<star> * e\<^sup>T * top"
     using assms(2) mult_assoc by simp
   also have "... \<le> (v\<^sup>T \<sqinter> -i\<^sup>T)\<^sup>\<star> * e\<^sup>T * top"
     by (metis mult_left_isotone star_isotone inf.cobounded2 inf.left_commute inf.sup_monoid.add_commute)
   also have "... = (v \<sqinter> -i)\<^sup>T\<^sup>\<star> * e\<^sup>T * top"
     using conv_complement conv_dist_inf by auto
   finally show ?thesis
     by simp
 qed
 
 lemma exists_a_w:
   assumes "symmetric g"
     and "forest f"
     and "f \<le> --g"
     and "regular f"
     and "(\<exists>w . minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T)"
     and "vector j"
     and "regular j"
     and "forest h"
-    and "forest_components h \<le> forest_components f"
-    and "big_forest (forest_components h) d"
+    and "forest_modulo_equivalence (forest_components h) d"
     and "d * top \<le> - j"
     and "forest_components h * j = j"
-    and "forest_components f = (forest_components h * (d \<squnion> d\<^sup>T))\<^sup>\<star> * forest_components h"
     and "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
-    and "(\<forall> a b . bf_between_arcs a b (forest_components h) d \<and> a \<le> -(forest_components h) \<sqinter> -- g \<and> b \<le> d \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g))"
+    and "(\<forall> a b . forest_modulo_equivalence_path a b (forest_components h) d \<and> a \<le> -(forest_components h) \<sqinter> -- g \<and> b \<le> d \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g))"
     and "regular d"
     and "selected_edge h j g \<le> - forest_components f"
     and "selected_edge h j g \<noteq> bot"
     and "j \<noteq> bot"
     and "regular h"
     and "h \<le> --g"
   shows "\<exists>w. minimum_spanning_forest w g \<and>
     f \<sqinter> - (selected_edge h j g)\<^sup>T \<sqinter> - (path f h j g) \<squnion> (f \<sqinter> - (selected_edge h j g)\<^sup>T \<sqinter> (path f h j g))\<^sup>T \<squnion> (selected_edge h j g) \<le> w \<squnion> w\<^sup>T"
 proof -
   let ?p = "path f h j g"
   let ?e = "selected_edge h j g"
   let ?f = "(f \<sqinter> -?e\<^sup>T \<sqinter> -?p) \<squnion> (f \<sqinter> -?e\<^sup>T \<sqinter> ?p)\<^sup>T \<squnion> ?e"
   let ?F = "forest_components f"
   let ?H = "forest_components h"
   let ?ec = "choose_component (forest_components h) j * - choose_component (forest_components h) j\<^sup>T \<sqinter> g"
   from assms(4) obtain w where 2: "minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T"
     using assms(5) by blast
   hence 3: "regular w \<and> regular f \<and> regular ?e"
     by (metis assms(4) minarc_regular minimum_spanning_forest_def spanning_forest_def)
   have 5: "equivalence ?F"
     using assms(2) forest_components_equivalence by auto
   have "?e\<^sup>T * top * ?e\<^sup>T = ?e\<^sup>T"
     by (metis arc_conv_closed arc_top_arc coreflexive_bot_closed coreflexive_symmetric minarc_arc minarc_bot_iff semiring.mult_not_zero)
   hence "?e\<^sup>T * top * ?e\<^sup>T \<le> -?F"
-    using 5 assms(17) conv_complement conv_isotone by fastforce
+    using 5 assms(15) conv_complement conv_isotone by fastforce
   hence 6: "?e * ?F * ?e = bot"
     using assms(2) le_bot triple_schroeder_p by simp
   let ?q = "w \<sqinter> top * ?e * w\<^sup>T\<^sup>\<star>"
   let ?v = "(w \<sqinter> -(top * ?e * w\<^sup>T\<^sup>\<star>)) \<squnion> ?q\<^sup>T"
   have 7: "regular ?q"
     using 3 regular_closed_star regular_conv_closed regular_mult_closed by auto
   have 8: "injective ?v"
   proof (rule kruskal_exchange_injective_inv_1)
     show "injective w"
       using 2 minimum_spanning_forest_def spanning_forest_def by blast
   next
     show "covector (top * ?e * w\<^sup>T\<^sup>\<star>)"
       by (simp add: covector_mult_closed)
   next
     show "top * ?e * w\<^sup>T\<^sup>\<star> * w\<^sup>T \<le> top * ?e * w\<^sup>T\<^sup>\<star>"
       by (simp add: mult_right_isotone star.right_plus_below_circ mult_assoc)
   next
     show "coreflexive ((top * ?e * w\<^sup>T\<^sup>\<star>)\<^sup>T * (top * ?e * w\<^sup>T\<^sup>\<star>) \<sqinter> w\<^sup>T * w)"
       using 2 by (metis comp_inf.semiring.mult_not_zero forest_bot kruskal_injective_inv_3 minarc_arc minarc_bot_iff minimum_spanning_forest_def semiring.mult_not_zero spanning_forest_def)
   qed
   have 9: "components g \<le> forest_components ?v"
   proof (rule kruskal_exchange_spanning_inv_1)
     show "injective (w \<sqinter> - (top *?e * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * ?e * w\<^sup>T\<^sup>\<star>)\<^sup>T)"
       using 8 by simp
   next
     show "regular (w \<sqinter> top * ?e * w\<^sup>T\<^sup>\<star>)"
       using 7 by simp
   next
     show "components g \<le> forest_components w"
       using 2 minimum_spanning_forest_def spanning_forest_def by blast
   qed
   have 10: "spanning_forest ?v g"
   proof (unfold spanning_forest_def, intro conjI)
     show "injective ?v"
       using 8 by auto
   next
     show "acyclic ?v"
     proof (rule kruskal_exchange_acyclic_inv_1)
       show "pd_kleene_allegory_class.acyclic w"
         using 2 minimum_spanning_forest_def spanning_forest_def by blast
     next
       show "covector (top * ?e * w\<^sup>T\<^sup>\<star>)"
         by (simp add: covector_mult_closed)
     qed
   next
     show "?v \<le> --g"
     proof (rule sup_least)
       show "w \<sqinter> - (top * ?e * w\<^sup>T\<^sup>\<star>) \<le> - - g"
         using 7 inf.coboundedI1 minimum_spanning_forest_def spanning_forest_def 2 by blast
     next
       show "(w \<sqinter> top * ?e * w\<^sup>T\<^sup>\<star>)\<^sup>T \<le> - - g"
         using 2 by (metis assms(1) conv_complement conv_isotone inf.coboundedI1 minimum_spanning_forest_def spanning_forest_def)
     qed
   next
     show "components g \<le> forest_components ?v"
       using 9 by simp
   next
     show "regular ?v"
       using 3 regular_closed_star regular_conv_closed regular_mult_closed by auto
   qed
   have 11: "sum (?v \<sqinter> g) = sum (w \<sqinter> g)"
   proof -
     have "sum (?v \<sqinter> g) = sum (w \<sqinter> -(top * ?e * w\<^sup>T\<^sup>\<star>) \<sqinter> g) + sum (?q\<^sup>T \<sqinter> g)"
       using 2 by (smt conv_complement conv_top epm_8 inf_import_p inf_top_right regular_closed_top vector_top_closed minimum_spanning_forest_def spanning_forest_def sum_disjoint)
     also have "... = sum (w \<sqinter> -(top * ?e * w\<^sup>T\<^sup>\<star>) \<sqinter> g) + sum (?q \<sqinter> g)"
       by (simp add: assms(1) sum_symmetric)
     also have "... = sum (((w \<sqinter> -(top * ?e * w\<^sup>T\<^sup>\<star>)) \<squnion> ?q) \<sqinter> g)"
       using inf_commute inf_left_commute sum_disjoint by simp
     also have "... = sum (w \<sqinter> g)"
       using 3 7 8 maddux_3_11_pp by auto
     finally show ?thesis
       by simp
   qed
   have 12: "?v \<squnion> ?v\<^sup>T = w \<squnion> w\<^sup>T"
   proof -
     have "?v \<squnion> ?v\<^sup>T = (w \<sqinter> -?q) \<squnion> ?q\<^sup>T \<squnion> (w\<^sup>T \<sqinter> -?q\<^sup>T) \<squnion> ?q"
       using conv_complement conv_dist_inf conv_dist_sup inf_import_p sup_assoc by simp
     also have "... = w \<squnion> w\<^sup>T"
       using 3 7 conv_complement conv_dist_inf inf_import_p maddux_3_11_pp sup_monoid.add_assoc sup_monoid.add_commute by auto
     finally show ?thesis
       by simp
   qed
   have 13: "?v * ?e\<^sup>T = bot"
   proof (rule kruskal_reroot_edge)
     show "injective (?e\<^sup>T * top)"
-      using assms(18) minarc_arc minarc_bot_iff by blast
+      using assms(16) minarc_arc minarc_bot_iff by blast
   next
     show "pd_kleene_allegory_class.acyclic w"
       using 2 minimum_spanning_forest_def spanning_forest_def by simp
   qed
   have "?v \<sqinter> ?e \<le> ?v \<sqinter> top * ?e"
     using inf.sup_right_isotone top_left_mult_increasing by simp
   also have "... \<le> ?v * (top * ?e)\<^sup>T"
     using covector_restrict_comp_conv covector_mult_closed vector_top_closed by simp
   finally have 14: "?v \<sqinter> ?e = bot"
     using 13 by (metis conv_dist_comp mult_assoc le_bot mult_left_zero)
   let ?i = "?v \<sqinter> (- ?F) * ?e * top \<sqinter> top * ?e\<^sup>T * ?F"
   let ?w = "(?v \<sqinter> -?i) \<squnion> ?e"
   have 15: "regular ?i"
     using 3 regular_closed_star regular_conv_closed regular_mult_closed by simp
   have 16: "?F \<le> -?i"
   proof -
     have 161: "bijective (?e * top)"
-      using assms(18) minarc_arc minarc_bot_iff by auto
+      using assms(16) minarc_arc minarc_bot_iff by auto
     have "?i \<le> - ?F * ?e * top"
       using inf.cobounded2 inf.coboundedI1 by blast
     also have "... = - (?F * ?e * top)"
       using 161 comp_bijective_complement by (simp add: mult_assoc)
     finally have "?i \<le> - (?F * ?e * top)"
       by blast
     hence 162: "?i \<sqinter> ?F \<le> - (?F * ?e * top)"
       using inf.coboundedI1 by blast
     have "?i \<sqinter> ?F \<le> ?F \<sqinter> (top * ?e\<^sup>T * ?F)"
       by (meson inf_le1 inf_le2 le_infI order_trans)
     also have "... \<le> ?F * (top * ?e\<^sup>T * ?F)\<^sup>T"
       by (simp add: covector_mult_closed covector_restrict_comp_conv)
     also have "... = ?F * ?F\<^sup>T * ?e\<^sup>T\<^sup>T * top\<^sup>T"
       by (simp add: conv_dist_comp mult_assoc)
     also have "... = ?F * ?F * ?e * top"
       by (simp add: conv_dist_comp conv_star_commute)
     also have "... = ?F * ?e * top"
       by (simp add: 5 preorder_idempotent)
     finally have "?i \<sqinter> ?F \<le> ?F * ?e * top"
       by simp
     hence "?i \<sqinter> ?F \<le> ?F * ?e * top \<sqinter> - (?F * ?e * top)"
       using 162 inf.bounded_iff by blast
     also have "... = bot"
       by simp
     finally show ?thesis
       using le_bot p_antitone_iff pseudo_complement by blast
   qed
   have 17: "?i \<le> top * ?e\<^sup>T * (?F \<sqinter> ?v \<sqinter> -?i)\<^sup>T\<^sup>\<star>"
   proof -
     have "?i \<le> ?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> * (?F \<sqinter> ?v)\<^sup>\<star>"
       using 2 8 12 by (smt inf.sup_right_isotone kruskal_forest_components_inf mult_right_isotone mult_assoc)
     also have "... = ?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> * (1 \<squnion> (?F \<sqinter> ?v)\<^sup>\<star> * (?F \<sqinter> ?v))"
       using star_left_unfold_equal star.circ_right_unfold_1 by auto
     also have "... = ?v \<sqinter> - ?F * ?e * top \<sqinter> (top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> \<squnion> top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> * (?F \<sqinter> ?v)\<^sup>\<star> * (?F \<sqinter> ?v))"
       by (simp add: mult_left_dist_sup mult_assoc)
     also have "... = (?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star>) \<squnion> (?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> * (?F \<sqinter> ?v)\<^sup>\<star> * (?F \<sqinter> ?v))"
       using comp_inf.semiring.distrib_left by blast
     also have "... \<le> top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> \<squnion> (?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> * (?F \<sqinter> ?v)\<^sup>\<star> * (?F \<sqinter> ?v))"
       using comp_inf.semiring.add_right_mono inf_le2 by blast
     also have "... \<le> top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> \<squnion> (?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * (?F\<^sup>T \<sqinter> ?v\<^sup>T)\<^sup>\<star> * (?F \<sqinter> ?v)\<^sup>\<star> * (?F \<sqinter> ?v))"
       by (simp add: conv_dist_inf)
     also have "... \<le> top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> \<squnion> (?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * ?F\<^sup>T\<^sup>\<star> * ?F\<^sup>\<star> * (?F \<sqinter> ?v))"
     proof -
       have "top * ?e\<^sup>T * (?F\<^sup>T \<sqinter> ?v\<^sup>T)\<^sup>\<star> * (?F \<sqinter> ?v)\<^sup>\<star> * (?F \<sqinter> ?v) \<le> top * ?e\<^sup>T * ?F\<^sup>T\<^sup>\<star> * ?F\<^sup>\<star> * (?F \<sqinter> ?v)"
         using star_isotone by (simp add: comp_isotone)
       hence "?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * (?F\<^sup>T \<sqinter> ?v\<^sup>T)\<^sup>\<star> * (?F \<sqinter> ?v)\<^sup>\<star> * (?F \<sqinter> ?v) \<le> ?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * ?F\<^sup>T\<^sup>\<star> * ?F\<^sup>\<star> * (?F \<sqinter> ?v)"
         using inf.sup_right_isotone by blast
       thus ?thesis
         using sup_right_isotone by blast
     qed
     also have "... = top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> \<squnion> (?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * ?F\<^sup>\<star> * ?F\<^sup>\<star> * (?F \<sqinter> ?v))"
       using 5 by auto
     also have "... = top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> \<squnion> (?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * ?F * ?F * (?F \<sqinter> ?v))"
       by (simp add: assms(2) forest_components_star)
     also have "... = top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star> \<squnion> (?v \<sqinter> - ?F * ?e * top \<sqinter> top * ?e\<^sup>T * ?F * (?F \<sqinter> ?v))"
       using 5 mult.semigroup_axioms preorder_idempotent semigroup.assoc by fastforce
     also have "... = top * ?e\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>\<star>"
     proof -
       have "?e * top * ?e\<^sup>T \<le> 1"
-        using assms(18) arc_expanded minarc_arc minarc_bot_iff by auto
+        using assms(16) arc_expanded minarc_arc minarc_bot_iff by auto
       hence "?F * ?e * top * ?e\<^sup>T \<le> ?F * 1"
         by (metis comp_associative comp_isotone mult_semi_associative star.circ_transitive_equal)
       hence "?v * ?v\<^sup>T * ?F * ?e * top * ?e\<^sup>T \<le> 1 * ?F * 1"
         using 8 by (smt comp_isotone mult_assoc)
       hence 171: "?v * ?v\<^sup>T * ?F * ?e * top * ?e\<^sup>T \<le> ?F"
         by simp
       hence "?v * (?F \<sqinter> ?v)\<^sup>T * ?F * ?e * top * ?e\<^sup>T \<le> ?F"
       proof -
         have "?v * (?F \<sqinter> ?v)\<^sup>T * ?F * ?e * top * ?e\<^sup>T \<le> ?v * ?v\<^sup>T * ?F * ?e * top * ?e\<^sup>T"
           by (simp add: conv_dist_inf mult_left_isotone mult_right_isotone)
         thus ?thesis
           using 171 order_trans by blast
       qed
       hence 172: "-?F * ((?F \<sqinter> ?v)\<^sup>T * ?F * ?e * top * ?e\<^sup>T)\<^sup>T \<le> -?v"
         by (smt schroeder_4_p comp_associative order_lesseq_imp pp_increasing)
       have "-?F * ((?F \<sqinter> ?v)\<^sup>T * ?F * ?e * top * ?e\<^sup>T)\<^sup>T = -?F * ?e\<^sup>T\<^sup>T * top\<^sup>T * ?e\<^sup>T * ?F\<^sup>T * (?F \<sqinter> ?v)\<^sup>T\<^sup>T"
         by (simp add: comp_associative conv_dist_comp)
       also have "... = -?F * ?e * top * ?e\<^sup>T * ?F * (?F \<sqinter> ?v)"
         using 5 by auto
       also have "... = -?F * ?e * top * top * ?e\<^sup>T * ?F * (?F \<sqinter> ?v)"
         using comp_associative by auto
       also have "... = -?F * ?e * top \<sqinter> top * ?e\<^sup>T * ?F * (?F \<sqinter> ?v)"
         by (smt comp_associative comp_inf.star.circ_decompose_9 comp_inf.star_star_absorb comp_inf_covector inf_vector_comp vector_top_closed)
       finally have "-?F * ((?F \<sqinter> ?v)\<^sup>T * ?F * ?e * top * ?e\<^sup>T)\<^sup>T = -?F * ?e * top \<sqinter> top * ?e\<^sup>T * ?F * (?F \<sqinter> ?v)"
         by simp
       hence "-?F * ?e * top \<sqinter> top * ?e\<^sup>T * ?F * (?F \<sqinter> ?v) \<le> -?v"
         using 172 by auto
       hence "?v \<sqinter> -?F * ?e * top \<sqinter> top * ?e\<^sup>T * ?F * (?F \<sqinter> ?v) \<le> bot"
         by (smt bot_unique inf.sup_monoid.add_commute p_shunting_swap pseudo_complement)
       thus ?thesis
         using le_bot sup_monoid.add_0_right by blast
     qed
     also have "... = top * ?e\<^sup>T * (?F \<sqinter> ?v \<sqinter> -?i)\<^sup>T\<^sup>\<star>"
       using 16 by (smt comp_inf.coreflexive_comp_inf_complement inf_top_right p_bot pseudo_complement top.extremum)
     finally show ?thesis
       by blast
   qed
   have 18: "?i \<le> top * ?e\<^sup>T * ?w\<^sup>T\<^sup>\<star>"
   proof -
     have "?i \<le> top * ?e\<^sup>T * (?F \<sqinter> ?v \<sqinter> -?i)\<^sup>T\<^sup>\<star>"
       using 17 by simp
     also have "... \<le> top * ?e\<^sup>T * (?v \<sqinter> -?i)\<^sup>T\<^sup>\<star>"
       using mult_right_isotone conv_isotone star_isotone inf.cobounded2 inf.sup_monoid.add_assoc by (simp add: inf.sup_monoid.add_assoc order.eq_iff inf.sup_monoid.add_commute)
     also have "... \<le> top * ?e\<^sup>T * ((?v \<sqinter> -?i) \<squnion> ?e)\<^sup>T\<^sup>\<star>"
       using mult_right_isotone conv_isotone star_isotone sup_ge1 by simp
     finally show ?thesis
       by blast
   qed
   have 19: "?i \<le> top * ?e\<^sup>T * ?v\<^sup>T\<^sup>\<star>"
   proof -
     have "?i \<le> top * ?e\<^sup>T * (?F \<sqinter> ?v \<sqinter> -?i)\<^sup>T\<^sup>\<star>"
       using 17 by simp
     also have "... \<le> top * ?e\<^sup>T * (?v \<sqinter> -?i)\<^sup>T\<^sup>\<star>"
       using mult_right_isotone conv_isotone star_isotone inf.cobounded2 inf.sup_monoid.add_assoc by (simp add: inf.sup_monoid.add_assoc order.eq_iff inf.sup_monoid.add_commute)
     also have "... \<le> top * ?e\<^sup>T * (?v)\<^sup>T\<^sup>\<star>"
       using mult_right_isotone conv_isotone star_isotone by auto
     finally show ?thesis
       by blast
   qed
   have 20: "f \<squnion> f\<^sup>T \<le> (?v \<sqinter> -?i \<sqinter> -?i\<^sup>T) \<squnion> (?v\<^sup>T \<sqinter> -?i \<sqinter> -?i\<^sup>T)"
   proof (rule kruskal_edge_between_components_2)
     show "?F \<le> - ?i"
       using 16 by simp
   next
     show "injective f"
       by (simp add: assms(2))
   next
     show "f \<squnion> f\<^sup>T \<le> w \<sqinter> - (top * ?e * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * ?e * w\<^sup>T\<^sup>\<star>)\<^sup>T \<squnion> (w \<sqinter> - (top * ?e * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * ?e * w\<^sup>T\<^sup>\<star>)\<^sup>T)\<^sup>T"
       using 2 12 by (metis conv_dist_sup conv_involutive conv_isotone le_supI sup_commute)
   qed
   have "minimum_spanning_forest ?w g \<and> ?f \<le> ?w \<squnion> ?w\<^sup>T"
   proof (intro conjI)
     have 211: "?e\<^sup>T \<le> ?v\<^sup>\<star>"
     proof (rule kruskal_edge_arc_1[where g=g and h="?ec"])
       show "?e \<le> -- ?ec"
         using minarc_below by blast
     next
       show "?ec \<le> g"
         using assms(4) inf.cobounded2 by (simp add: boruvka_inner_invariant_def boruvka_outer_invariant_def conv_dist_inf)
     next
       show "symmetric g"
         by (meson assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def)
     next
       show "components g \<le> forest_components (w \<sqinter> - (top * ?e * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * ?e * w\<^sup>T\<^sup>\<star>)\<^sup>T)"
         using 9 by simp
     next
       show "(w \<sqinter> - (top * ?e * w\<^sup>T\<^sup>\<star>) \<squnion> (w \<sqinter> top * ?e * w\<^sup>T\<^sup>\<star>)\<^sup>T) * ?e\<^sup>T = bot"
         using 13 by blast
     qed
     have 212: "arc ?i"
     proof (rule boruvka_edge_arc)
       show "equivalence ?F"
         by (simp add: 5)
     next
       show "forest ?v"
         using 10 spanning_forest_def by blast
     next
       show "arc ?e"
-        using assms(18) minarc_arc minarc_bot_iff by blast
+        using assms(16) minarc_arc minarc_bot_iff by blast
     next
       show "regular ?F"
         using 3 regular_closed_star regular_conv_closed regular_mult_closed by auto
     next
       show "?F \<le> forest_components (?F \<sqinter> ?v)"
         by (simp add: 12 2 8 kruskal_forest_components_inf)
     next
       show "regular ?v"
         using 10 spanning_forest_def by blast
     next
       show "?v * ?e\<^sup>T = bot"
         using 13 by auto
     next
       show "?e * ?F * ?e = bot"
         by (simp add: 6)
     next
       show "?e\<^sup>T \<le> ?v\<^sup>\<star>"
         using 211 by auto
 
     next
       show "?e \<noteq> bot"
-        by (simp add: assms(18))
+        by (simp add: assms(16))
     qed
     show "minimum_spanning_forest ?w g"
     proof (unfold minimum_spanning_forest_def, intro conjI)
       have "(?v \<sqinter> -?i) * ?e\<^sup>T \<le> ?v * ?e\<^sup>T"
         using inf_le1 mult_left_isotone by simp
       hence "(?v \<sqinter> -?i) * ?e\<^sup>T = bot"
         using 13 le_bot by simp
       hence 221: "?e * (?v \<sqinter> -?i)\<^sup>T = bot"
         using conv_dist_comp conv_involutive conv_bot by force
       have 222: "injective ?w"
       proof (rule injective_sup)
         show "injective (?v \<sqinter> -?i)"
           using 8 by (simp add: injective_inf_closed)
       next
         show "coreflexive (?e * (?v \<sqinter> -?i)\<^sup>T)"
           using 221 by simp
       next
         show "injective ?e"
           by (metis arc_injective minarc_arc coreflexive_bot_closed coreflexive_injective minarc_bot_iff)
       qed
       show "spanning_forest ?w g"
       proof (unfold spanning_forest_def, intro conjI)
         show "injective ?w"
           using 222 by simp
       next
         show "acyclic ?w"
         proof (rule kruskal_exchange_acyclic_inv_2)
           show "acyclic ?v"
             using 10 spanning_forest_def by blast
         next
           show "injective ?v"
             using 8 by simp
         next
           show "?i \<le>?v"
             using inf.coboundedI1 by simp
         next
           show "bijective (?i\<^sup>T * top)"
             using 212 by simp
         next
           show "bijective (?e * top)"
             using 14 212 by (smt assms(4) comp_inf.idempotent_bot_closed conv_complement minarc_arc minarc_bot_iff p_bot regular_closed_bot semiring.mult_not_zero symmetric_top_closed)
         next
           show "?i \<le> top * ?e\<^sup>T *?v\<^sup>T\<^sup>\<star>"
             using 19 by simp
         next
           show "?v * ?e\<^sup>T * top = bot"
             using 13 by simp
         qed
       next
         have "?w \<le> ?v \<squnion> ?e"
           using inf_le1 sup_left_isotone by simp
         also have "... \<le> --g \<squnion> ?e"
           using 10 sup_left_isotone spanning_forest_def by blast
         also have "... \<le> --g \<squnion> --h"
         proof -
           have 1: "--g \<le> --g \<squnion> --h"
             by simp
           have 2: "?e \<le> --g \<squnion> --h"
             by (metis inf.coboundedI1 inf.sup_monoid.add_commute minarc_below order.trans p_dist_inf p_dist_sup sup.cobounded1)
           thus ?thesis
             using 1 2 by simp
         qed
         also have "... \<le> --g"
-            using assms(20, 21) by auto
+            using assms(18, 19) by auto
         finally show "?w \<le> --g"
           by simp
       next
         have 223: "?i \<le> (?v \<sqinter> -?i)\<^sup>T\<^sup>\<star> * ?e\<^sup>T * top"
         proof (rule boruvka_exchange_spanning_inv)
           show "forest ?v"
             using 10 spanning_forest_def by blast
         next
           show "?v\<^sup>\<star> * ?e\<^sup>T = ?e\<^sup>T"
             using 13 by (smt conv_complement conv_dist_comp conv_involutive conv_star_commute dense_pp fc_top regular_closed_top star_absorb)
         next
           show "?i \<le> ?v \<sqinter> top * ?e\<^sup>T * ?w\<^sup>T\<^sup>\<star>"
             using 18 inf.sup_monoid.add_assoc by auto
         next
           show "arc ?i"
             using 212 by blast
         next
           show "arc ?e"
-            using assms(18) minarc_arc minarc_bot_iff by auto
+            using assms(16) minarc_arc minarc_bot_iff by auto
         next
           show "?v \<le> --g"
             using 10 spanning_forest_def by blast
         next
           show "?w \<le> --g"
           proof -
             have 2231: "?e \<le> --g"
               by (metis inf.boundedE minarc_below pp_dist_inf)
             have "?w \<le> ?v \<squnion> ?e"
               using inf_le1 sup_left_isotone by simp
             also have "... \<le> --g"
               using 2231 10 spanning_forest_def sup_least by blast
             finally show ?thesis
               by blast
           qed
         next
           show "?e \<le> --g"
             by (metis inf.boundedE minarc_below pp_dist_inf)
         next
           show "components g \<le> forest_components ?v"
             by (simp add: 9)
         qed
         have "components g \<le> forest_components ?v"
           using 10 spanning_forest_def by auto
         also have "... \<le> forest_components ?w"
         proof (rule kruskal_exchange_forest_components_inv)
         next
           show "injective ((?v \<sqinter> -?i) \<squnion> ?e)"
             using 222 by simp
         next
           show "regular ?i"
             using 15 by simp
         next
           show "?e * top * ?e = ?e"
             by (metis arc_top_arc minarc_arc minarc_bot_iff semiring.mult_not_zero)
         next
           show "?i \<le> top * ?e\<^sup>T * ?v\<^sup>T\<^sup>\<star>"
             using 19 by blast
         next
           show "?v * ?e\<^sup>T * top = bot"
             using 13 by simp
         next
           show "injective ?v"
             using 8 by simp
         next
           show "?i \<le> ?v"
             by (simp add: le_infI1)
         next
           show "?i \<le> (?v \<sqinter> -?i)\<^sup>T\<^sup>\<star> * ?e\<^sup>T * top"
             using 223 by blast
         qed
         finally show "components g \<le> forest_components ?w"
           by simp
       next
         show "regular ?w"
           using 3 7 regular_conv_closed by simp
       qed
     next
       have 224: "?e \<sqinter> g \<noteq> bot"
-        using assms(18) inf.left_commute inf_bot_right minarc_meet_bot by fastforce
+        using assms(16) inf.left_commute inf_bot_right minarc_meet_bot by fastforce
       have 225: "sum (?e \<sqinter> g) \<le> sum (?i \<sqinter> g)"
-      proof (rule a_to_e_in_bigforest)
+      proof (rule a_to_e_in_forest_modulo_equivalence)
         show "symmetric g"
           using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
       next
         show "j \<noteq> bot"
-          by (simp add: assms(19))
+          by (simp add: assms(17))
       next
         show "f \<le> -- g"
           by (simp add: assms(3))
       next
         show "vector j"
           using assms(6) boruvka_inner_invariant_def by blast
       next
         show "forest h"
           by (simp add: assms(8))
       next
-        show " big_forest (forest_components h) d"
-          by (simp add: assms(10))
+        show " forest_modulo_equivalence (forest_components h) d"
+          by (simp add: assms(9))
       next
         show "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
-          by (simp add: assms(14))
+          by (simp add: assms(12))
       next
-        show "\<forall>a b. bf_between_arcs a b (?H) d \<and> a \<le> - ?H \<sqinter> - - g \<and> b \<le> d \<longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
-          by (simp add: assms(15))
+        show "\<forall>a b. forest_modulo_equivalence_path a b (?H) d \<and> a \<le> - ?H \<sqinter> - - g \<and> b \<le> d \<longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
+          by (simp add: assms(13))
       next
         show "regular d"
-          using assms(16) by auto
+          using assms(14) by auto
       next
         show "?e = ?e"
           by simp
       next
         show "arc ?i"
           using 212 by blast
       next
-        show "bf_between_arcs ?i ?e ?H (d \<squnion> ?e)"
+        show "forest_modulo_equivalence_path ?i ?e ?H (d \<squnion> ?e)"
         proof -
           have "d\<^sup>T * ?H * ?e = bot"
-            using assms(6, 7, 11, 12, 19) dT_He_eq_bot le_bot by blast
+            using assms(6, 7, 10 ,11, 17) dT_He_eq_bot le_bot by blast
           hence 251: "d\<^sup>T * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
             by simp
           hence "d\<^sup>T * ?H * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
             by (metis assms(8) forest_components_star star.circ_decompose_9 mult_assoc)
           hence "d\<^sup>T * (?H * d)\<^sup>\<star> * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
           proof -
             have "d\<^sup>T * ?H * d \<le> 1"
-              using assms(10) big_forest_def dTransHd_le_1 by blast
+              using assms(9) forest_modulo_equivalence_def dTransHd_le_1 by blast
             hence "d\<^sup>T * ?H * d * (?H * d)\<^sup>\<star> * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
               by (metis mult_left_isotone star.circ_circ_mult star_involutive star_one)
             hence "d\<^sup>T * ?H * ?e \<squnion> d\<^sup>T * ?H * d * (?H * d)\<^sup>\<star> * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
               using 251 by simp
             hence "d\<^sup>T * (1 \<squnion> ?H * d * (?H * d)\<^sup>\<star>) * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
               by (simp add: comp_associative comp_left_dist_sup semiring.distrib_right)
             thus ?thesis
               by (simp add: star_left_unfold_equal)
           qed
           hence "?H * d\<^sup>T * (?H * d)\<^sup>\<star> * ?H * ?e \<le> ?H * (?H * d)\<^sup>\<star> * ?H * ?e"
             by (simp add: mult_right_isotone mult_assoc)
           hence "?H * d\<^sup>T * (?H * d)\<^sup>\<star> * ?H * ?e \<le> ?H * ?H * (d * ?H)\<^sup>\<star> * ?e"
             by (smt star_slide mult_assoc)
           hence "?H * d\<^sup>T * (?H * d)\<^sup>\<star> * ?H * ?e \<le> ?H * (d * ?H)\<^sup>\<star> * ?e"
             by (metis assms(8) forest_components_star star.circ_decompose_9)
           hence "?H * d\<^sup>T * (?H * d)\<^sup>\<star> * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
             using star_slide by auto
           hence "?H * d * (?H * d)\<^sup>\<star> * ?H * ?e \<squnion> ?H * d\<^sup>T * (?H * d)\<^sup>\<star> * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
             by (smt le_supI star.circ_loop_fixpoint sup.cobounded2 sup_commute mult_assoc)
           hence "(?H * (d \<squnion> d\<^sup>T)) * (?H * d)\<^sup>\<star> * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
             by (simp add: semiring.distrib_left semiring.distrib_right)
           hence "(?H * (d \<squnion> d\<^sup>T))\<^sup>\<star> * (?H * d)\<^sup>\<star> * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
             by (simp add: star_left_induct_mult mult_assoc)
           hence 252: "(?H * (d \<squnion> d\<^sup>T))\<^sup>\<star> * ?H * ?e \<le> (?H * d)\<^sup>\<star> * ?H * ?e"
             by (smt mult_left_dist_sup star.circ_transitive_equal star_slide star_sup_1 mult_assoc)
           have "?i \<le> top * ?e\<^sup>T * ?F"
             by auto
           hence "?i\<^sup>T \<le> ?F\<^sup>T * ?e\<^sup>T\<^sup>T * top\<^sup>T"
             by (simp add: conv_dist_comp conv_dist_inf mult_assoc)
           hence "?i\<^sup>T * top \<le> ?F\<^sup>T * ?e\<^sup>T\<^sup>T * top\<^sup>T * top"
             using comp_isotone by blast
           also have "... = ?F\<^sup>T * ?e\<^sup>T\<^sup>T * top\<^sup>T"
             by (simp add: vector_mult_closed)
           also have "... = ?F * ?e\<^sup>T\<^sup>T * top\<^sup>T"
             by (simp add: conv_dist_comp conv_star_commute)
           also have "... = ?F * ?e * top"
             by simp
           also have "... = (?H * (d \<squnion> d\<^sup>T))\<^sup>\<star> * ?H * ?e * top"
-            by (simp add: assms(13))
+            using assms(2, 8, 12) F_is_H_and_d by simp
           also have "... \<le> (?H * d)\<^sup>\<star> * ?H * ?e * top"
             by (simp add: 252 comp_isotone)
           also have "... \<le> (?H * (d \<squnion> ?e))\<^sup>\<star> * ?H * ?e * top"
             by (simp add: comp_isotone star_isotone)
           finally have "?i\<^sup>T * top \<le> (?H * (d \<squnion> ?e))\<^sup>\<star> * ?H * ?e * top"
             by blast
           thus ?thesis
-            using 212 assms(18) bf_between_arcs_def minarc_arc minarc_bot_iff by blast
+            using 212 assms(16) forest_modulo_equivalence_path_def minarc_arc minarc_bot_iff by blast
         qed
       next
         show "?i \<le> - ?H \<sqinter> -- g"
         proof -
-          have 241: "?i \<le> -?H"
+          have "forest_components h \<le> forest_components f"
+            using assms(2, 8, 12) H_below_F by blast
+          then have 241: "?i \<le> -?H"
             using 16 assms(9) inf.order_lesseq_imp p_antitone_iff by blast
           have "?i \<le> -- g"
             using 10 inf.coboundedI1 spanning_forest_def by blast
           thus ?thesis
             using 241 inf_greatest by blast
         qed
       next
         show "regular h"
-          using assms(20) by auto
+          using assms(18) by auto
       qed
       have "?v \<sqinter> ?e \<sqinter> -?i = bot"
         using 14 by simp
       hence "sum (?w \<sqinter> g) = sum (?v \<sqinter> -?i \<sqinter> g) + sum (?e \<sqinter> g)"
         using sum_disjoint inf_commute inf_assoc by simp
       also have "... \<le> sum (?v \<sqinter> -?i \<sqinter> g) + sum (?i \<sqinter> g)"
         using 224 225 sum_plus_right_isotone by simp
       also have "... = sum (((?v \<sqinter> -?i) \<squnion> ?i) \<sqinter> g)"
         using sum_disjoint inf_le2 pseudo_complement by simp
       also have "... = sum ((?v \<squnion> ?i) \<sqinter> (-?i \<squnion> ?i) \<sqinter> g)"
         by (simp add: sup_inf_distrib2)
       also have "... = sum ((?v \<squnion> ?i) \<sqinter> g)"
         using 15 by (metis inf_top_right stone)
       also have "... = sum (?v \<sqinter> g)"
         by (simp add: inf.sup_monoid.add_assoc)
       finally have "sum (?w \<sqinter> g) \<le> sum (?v \<sqinter> g)"
         by simp
       thus "\<forall>u . spanning_forest u g \<longrightarrow> sum (?w \<sqinter> g) \<le> sum (u \<sqinter> g)"
         using 2 11 minimum_spanning_forest_def by auto
     qed
   next
     have "?f \<le> f \<squnion> f\<^sup>T \<squnion> ?e"
       by (smt conv_dist_inf inf_le1 sup_left_isotone sup_mono inf.order_lesseq_imp)
     also have "... \<le> (?v \<sqinter> -?i \<sqinter> -?i\<^sup>T) \<squnion> (?v\<^sup>T \<sqinter> -?i \<sqinter> -?i\<^sup>T) \<squnion> ?e"
       using 20 sup_left_isotone by simp
     also have "... \<le> (?v \<sqinter> -?i) \<squnion> (?v\<^sup>T \<sqinter> -?i \<sqinter> -?i\<^sup>T) \<squnion> ?e"
       by (metis inf.cobounded1 sup_inf_distrib2)
     also have "... = ?w \<squnion> (?v\<^sup>T \<sqinter> -?i \<sqinter> -?i\<^sup>T)"
       by (simp add: sup_assoc sup_commute)
     also have "... \<le> ?w \<squnion> (?v\<^sup>T \<sqinter> -?i\<^sup>T)"
       using inf.sup_right_isotone inf_assoc sup_right_isotone by simp
     also have "... \<le> ?w \<squnion> ?w\<^sup>T"
       using conv_complement conv_dist_inf conv_dist_sup sup_right_isotone by simp
     finally show "?f \<le> ?w \<squnion> ?w\<^sup>T"
       by simp
   qed
   thus ?thesis by auto
 qed
 
 lemma boruvka_outer_invariant_when_e_not_bot:
   assumes "boruvka_inner_invariant j f h g d"
     and "j \<noteq> bot"
     and "selected_edge h j g \<le> - forest_components f"
     and "selected_edge h j g \<noteq> bot"
   shows "boruvka_outer_invariant (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> - path f h j g \<squnion> (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> path f h j g)\<^sup>T \<squnion> selected_edge h j g) g"
 proof -
   let ?c = "choose_component (forest_components h) j"
   let ?p = "path f h j g"
   let ?F = "forest_components f"
   let ?H = "forest_components h"
   let ?e = "selected_edge h j g"
   let ?f' = "f \<sqinter> -?e\<^sup>T \<sqinter> -?p \<squnion> (f \<sqinter> -?e\<^sup>T \<sqinter> ?p)\<^sup>T \<squnion> ?e"
   let ?d' = "d \<squnion> ?e"
   let ?j' = "j \<sqinter> -?c"
   show "boruvka_outer_invariant ?f' g"
   proof (unfold boruvka_outer_invariant_def, intro conjI)
     show "symmetric g"
       by (meson assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def)
   next
     show "injective ?f'"
     proof (rule kruskal_injective_inv)
       show "injective (f \<sqinter> - ?e\<^sup>T)"
         by (meson assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def injective_inf_closed)
       show "covector (?p)"
         using covector_mult_closed by simp
       show "?p * (f \<sqinter> - ?e\<^sup>T)\<^sup>T \<le> ?p"
         by (simp add: mult_right_isotone star.left_plus_below_circ star_plus mult_assoc)
       show "?e \<le> ?p"
         by (meson mult_left_isotone order.trans star_outer_increasing top.extremum)
       show "?p * (f \<sqinter> - ?e\<^sup>T)\<^sup>T \<le> - ?e"
       proof -
         have "?p * (f \<sqinter> - ?e\<^sup>T)\<^sup>T \<le> ?p * f\<^sup>T"
           by (simp add: conv_dist_inf mult_right_isotone)
         also have "... \<le> top * ?e * (f)\<^sup>T\<^sup>\<star> * f\<^sup>T"
           using conv_dist_inf star_isotone comp_isotone by simp
         also have "... \<le> - ?e"
           using assms(1, 4) boruvka_inner_invariant_def boruvka_outer_invariant_def kruskal_injective_inv_2 minarc_arc minarc_bot_iff by auto
         finally show ?thesis .
       qed
       show "injective (?e)"
         by (metis arc_injective coreflexive_bot_closed minarc_arc minarc_bot_iff semiring.mult_not_zero)
       show "coreflexive (?p\<^sup>T * ?p \<sqinter> (f \<sqinter> - ?e\<^sup>T)\<^sup>T * (f \<sqinter> - ?e\<^sup>T))"
       proof -
         have "(?p\<^sup>T * ?p \<sqinter> (f \<sqinter> - ?e\<^sup>T)\<^sup>T * (f \<sqinter> - ?e\<^sup>T)) \<le> ?p\<^sup>T * ?p \<sqinter> f\<^sup>T * f"
           using conv_dist_inf inf.sup_right_isotone mult_isotone by simp
         also have "... \<le> (top * ?e * f\<^sup>T\<^sup>\<star>)\<^sup>T * (top * ?e * f\<^sup>T\<^sup>\<star>) \<sqinter> f\<^sup>T * f"
           by (metis comp_associative comp_inf.coreflexive_transitive comp_inf.mult_right_isotone comp_isotone conv_isotone inf.cobounded1 inf.idem inf.sup_monoid.add_commute star_isotone top.extremum)
         also have "... \<le> 1"
           using assms(1, 4) boruvka_inner_invariant_def boruvka_outer_invariant_def kruskal_injective_inv_3 minarc_arc minarc_bot_iff by auto
         finally show ?thesis
           by simp
       qed
     qed
   next
     show "acyclic ?f'"
     proof (rule kruskal_acyclic_inv)
       show "acyclic (f \<sqinter> - ?e\<^sup>T)"
       proof -
         have f_intersect_below: "(f \<sqinter> - ?e\<^sup>T) \<le> f" by simp
         have "acyclic f"
           by (meson assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def)
         thus ?thesis
           using comp_isotone dual_order.trans star_isotone f_intersect_below by blast
       qed
     next
       show "covector ?p"
         by (metis comp_associative vector_top_closed)
     next
       show "(f \<sqinter> - ?e\<^sup>T \<sqinter> ?p)\<^sup>T * (f \<sqinter> - ?e\<^sup>T)\<^sup>\<star> * ?e = bot"
       proof -
         have "?e \<le> - (f\<^sup>T\<^sup>\<star> * f\<^sup>\<star>)"
           by (simp add: assms(3))
         hence "?e * top * ?e \<le> - (f\<^sup>T\<^sup>\<star> * f\<^sup>\<star>)"
           by (metis arc_top_arc minarc_arc minarc_bot_iff semiring.mult_not_zero)
         hence "?e\<^sup>T * top * ?e\<^sup>T \<le> - (f\<^sup>T\<^sup>\<star> * f\<^sup>\<star>)\<^sup>T"
           by (metis comp_associative conv_complement conv_dist_comp conv_isotone symmetric_top_closed)
         hence "?e\<^sup>T * top * ?e\<^sup>T \<le> - (f\<^sup>T\<^sup>\<star> * f\<^sup>\<star>)"
           by (simp add: conv_dist_comp conv_star_commute)
         hence "?e * (f\<^sup>T\<^sup>\<star> * f\<^sup>\<star>) * ?e \<le> bot"
           using triple_schroeder_p by auto
         hence 1: "?e * f\<^sup>T\<^sup>\<star> * f\<^sup>\<star> * ?e \<le> bot"
           using mult_assoc by auto
         have 2: "(f \<sqinter> - ?e\<^sup>T)\<^sup>T\<^sup>\<star> \<le> f\<^sup>T\<^sup>\<star>"
           by (simp add: conv_dist_inf star_isotone)
         have "(f \<sqinter> - ?e\<^sup>T \<sqinter> ?p)\<^sup>T * (f \<sqinter> - ?e\<^sup>T)\<^sup>\<star> * ?e \<le> (f \<sqinter> ?p)\<^sup>T * (f \<sqinter> - ?e\<^sup>T)\<^sup>\<star> * ?e"
           by (simp add: comp_isotone conv_dist_inf inf.orderI inf.sup_monoid.add_assoc)
         also have "... \<le> (f \<sqinter> ?p)\<^sup>T * f\<^sup>\<star> * ?e"
           by (simp add: comp_isotone star_isotone)
         also have "... \<le> (f \<sqinter> top * ?e * (f)\<^sup>T\<^sup>\<star>)\<^sup>T * f\<^sup>\<star> * ?e"
           using 2 by (metis comp_inf.comp_isotone comp_inf.coreflexive_transitive comp_isotone conv_isotone inf.idem top.extremum)
         also have "... = (f\<^sup>T \<sqinter> (top * ?e * f\<^sup>T\<^sup>\<star>)\<^sup>T) * f\<^sup>\<star> * ?e"
           by (simp add: conv_dist_inf)
         also have "... \<le> top * (f\<^sup>T \<sqinter> (top * ?e * f\<^sup>T\<^sup>\<star>)\<^sup>T) * f\<^sup>\<star> * ?e"
           using top_left_mult_increasing mult_assoc by auto
         also have "... = (top \<sqinter> top * ?e * f\<^sup>T\<^sup>\<star>) * f\<^sup>T * f\<^sup>\<star> * ?e"
           by (smt covector_comp_inf_1 covector_mult_closed order.eq_iff inf.sup_monoid.add_commute vector_top_closed)
         also have "... = top * ?e * f\<^sup>T\<^sup>\<star> * f\<^sup>T * f\<^sup>\<star> * ?e"
           by simp
         also have "... \<le> top * ?e * f\<^sup>T\<^sup>\<star> * f\<^sup>\<star> * ?e"
           by (smt conv_dist_comp conv_isotone conv_star_commute mult_left_isotone mult_right_isotone star.left_plus_below_circ mult_assoc)
         also have "... \<le> bot"
           using 1 covector_bot_closed le_bot mult_assoc by fastforce
         finally show ?thesis
           using le_bot by auto
       qed
     next
       show "?e * (f \<sqinter> - ?e\<^sup>T)\<^sup>\<star> * ?e = bot"
       proof -
         have 1: "?e \<le> - ?F"
           by (simp add: assms(3))
         have 2: "injective f"
           by (meson assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def)
         have 3: "equivalence ?F"
           using 2 forest_components_equivalence by simp
         hence 4: "?e\<^sup>T = ?e\<^sup>T * top * ?e\<^sup>T"
           by (smt arc_conv_closed arc_top_arc covector_complement_closed covector_conv_vector ex231e minarc_arc minarc_bot_iff pp_surjective regular_closed_top vector_mult_closed vector_top_closed)
         also have "... \<le> - ?F" using 1 3 conv_isotone conv_complement calculation by fastforce
         finally have 5: "?e * ?F * ?e = bot"
           using 4 by (smt triple_schroeder_p le_bot pp_total regular_closed_top vector_top_closed)
         have "(f \<sqinter> - ?e\<^sup>T)\<^sup>\<star> \<le> f\<^sup>\<star>"
           by (simp add: star_isotone)
         hence "?e * (f \<sqinter> - ?e\<^sup>T)\<^sup>\<star> * ?e \<le> ?e * f\<^sup>\<star> * ?e"
           using mult_left_isotone mult_right_isotone by blast
         also have "... \<le> ?e * ?F * ?e"
           by (metis conv_star_commute forest_components_increasing mult_left_isotone mult_right_isotone star_involutive)
         also have 6: "... = bot"
           using 5 by simp
         finally show ?thesis using 6 le_bot by blast
       qed
     next
       show "forest_components (f \<sqinter> -?e\<^sup>T) \<le> - ?e"
       proof -
         have 1: "?e \<le> - ?F"
           by (simp add: assms(3))
         have "f \<sqinter> - ?e\<^sup>T \<le> f"
           by simp
         hence "forest_components (f \<sqinter> - ?e\<^sup>T) \<le> ?F"
           using forest_components_isotone by blast
         thus ?thesis
           using 1 order_lesseq_imp p_antitone_iff by blast
       qed
     qed
   next
     show "?f' \<le> --g"
     proof -
       have 1: "(f \<sqinter> - ?e\<^sup>T \<sqinter> - ?p) \<le> --g"
         by (meson assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def inf.coboundedI1)
       have 2: "(f \<sqinter> - ?e\<^sup>T \<sqinter> ?p)\<^sup>T \<le> --g"
       proof -
         have "(f \<sqinter> - ?e\<^sup>T \<sqinter> ?p)\<^sup>T \<le> f\<^sup>T"
           by (simp add: conv_isotone inf.sup_monoid.add_assoc)
         also have "... \<le> --g"
           by (metis assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def conv_complement conv_isotone)
         finally show ?thesis
           by simp
       qed
       have 3: "?e \<le> --g"
         by (metis inf.boundedE minarc_below pp_dist_inf)
       show ?thesis using 1 2 3
         by simp
     qed
   next
     show "regular ?f'"
       using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def minarc_regular regular_closed_star regular_conv_closed regular_mult_closed by auto
   next
     show "\<exists>w. minimum_spanning_forest w g \<and> ?f' \<le> w \<squnion> w\<^sup>T"
     proof (rule exists_a_w)
       show "symmetric g"
         using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
     next
       show "forest f"
         using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
     next
       show "f \<le> --g"
         using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
     next
       show "regular f"
         using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
     next
       show "(\<exists>w . minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T)"
         using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
     next
       show "vector j"
         using assms(1) boruvka_inner_invariant_def by blast
     next
       show "regular j"
         using assms(1) boruvka_inner_invariant_def by blast
     next
       show "forest h"
         using assms(1) boruvka_inner_invariant_def by blast
     next
-      show "forest_components h \<le> forest_components f"
-        using assms(1) boruvka_inner_invariant_def by blast
-    next
-      show "big_forest (forest_components h) d"
+      show "forest_modulo_equivalence (forest_components h) d"
         using assms(1) boruvka_inner_invariant_def by blast
     next
       show "d * top \<le> - j"
         using assms(1) boruvka_inner_invariant_def by blast
     next
       show "forest_components h * j = j"
         using assms(1) boruvka_inner_invariant_def by blast
     next
-      show "forest_components f = (forest_components h * (d \<squnion> d\<^sup>T))\<^sup>\<star> * forest_components h"
-        using assms(1) boruvka_inner_invariant_def by blast
-    next
       show "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
         using assms(1) boruvka_inner_invariant_def by blast
     next
-      show "(\<forall> a b . bf_between_arcs a b (forest_components h) d \<and> a \<le> -(forest_components h) \<sqinter> -- g \<and> b \<le> d \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g))"
+      show "(\<forall> a b . forest_modulo_equivalence_path a b (forest_components h) d \<and> a \<le> -(forest_components h) \<sqinter> -- g \<and> b \<le> d \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g))"
         using assms(1) boruvka_inner_invariant_def by blast
     next
       show "regular d"
         using assms(1) boruvka_inner_invariant_def by blast
     next
       show "selected_edge h j g \<le> - forest_components f"
         by (simp add: assms(3))
     next
       show "selected_edge h j g \<noteq> bot"
         by (simp add: assms(4))
     next
       show "j \<noteq> bot"
         by (simp add: assms(2))
     next
       show "regular h"
         using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
     next
       show "h \<le> --g"
-        using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
+        using H_below_regular_g assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
     qed
   qed
 qed
 
 lemma second_inner_invariant_when_e_not_bot:
   assumes "boruvka_inner_invariant j f h g d"
     and "j \<noteq> bot"
     and "selected_edge h j g \<le> - forest_components f"
     and "selected_edge h j g \<noteq> bot"
   shows "boruvka_inner_invariant
          (j \<sqinter> - choose_component (forest_components h) j)
          (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> - path f h j g \<squnion>
           (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> path f h j g)\<^sup>T \<squnion>
           selected_edge h j g)
          h g (d \<squnion> selected_edge h j g)"
 proof -
   let ?c = "choose_component (forest_components h) j"
   let ?p = "path f h j g"
   let ?F = "forest_components f"
   let ?H = "forest_components h"
   let ?e = "selected_edge h j g"
   let ?f' = "f \<sqinter> -?e\<^sup>T \<sqinter> -?p \<squnion> (f \<sqinter> -?e\<^sup>T \<sqinter> ?p)\<^sup>T \<squnion> ?e"
   let ?d' = "d \<squnion> ?e"
   let ?j' = "j \<sqinter> -?c"
   show "boruvka_inner_invariant ?j' ?f' h g ?d'"
   proof (unfold boruvka_inner_invariant_def, intro conjI)
-    have 1: "boruvka_outer_invariant ?f' g"
+    show 1: "boruvka_outer_invariant ?f' g"
       using assms(1, 2, 3, 4) boruvka_outer_invariant_when_e_not_bot by blast
-    show "boruvka_outer_invariant ?f' g"
-      using assms(1, 2, 3, 4) boruvka_outer_invariant_when_e_not_bot by blast
+  next
     show "g \<noteq> bot"
       using assms(1) boruvka_inner_invariant_def by force
-    show "vector ?j'"
-      using assms(1, 2) boruvka_inner_invariant_def component_is_vector vector_complement_closed vector_inf_closed by simp
+  next
+    show "regular ?d'"
+      using assms(1) boruvka_inner_invariant_def minarc_regular by auto
+  next
     show "regular ?j'"
       using assms(1) boruvka_inner_invariant_def by auto
-    show "boruvka_outer_invariant h g"
+  next
+    show "vector ?j'"
+      using assms(1, 2) boruvka_inner_invariant_def component_is_vector vector_complement_closed vector_inf_closed by simp
+  next
+    show "regular h"
       by (meson assms(1) boruvka_inner_invariant_def)
+  next
     show "injective h"
       by (meson assms(1) boruvka_inner_invariant_def)
+  next
     show "pd_kleene_allegory_class.acyclic h"
       by (meson assms(1) boruvka_inner_invariant_def)
-    show "?H \<le> forest_components ?f'"
-    proof -
-      have 2: "?F \<le> forest_components ?f'"
-      proof (rule components_disj_increasing)
-        show "regular ?p"
-          using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def minarc_regular regular_closed_star regular_conv_closed regular_mult_closed by auto[1]
-      next
-        show "regular ?e"
-          using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def minarc_regular regular_closed_star regular_conv_closed regular_mult_closed by auto[1]
-      next
-        show "injective ?f'"
-          using 1 boruvka_outer_invariant_def by blast
-      next
-        show "injective f"
-          using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by blast
-      qed
-      thus ?thesis
-        using assms(1) boruvka_inner_invariant_def dual_order.trans by blast
-    qed
-    show "big_forest ?H ?d'"
-      using assms(1, 2, 3, 4) big_forest_d_U_e boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
+  next
+    show "?H * ?j' = ?j'"
+      using fc_j_eq_j_inv assms(1) boruvka_inner_invariant_def by blast
+  next
+    show "forest_modulo_equivalence ?H ?d'"
+      using assms(1, 2, 3) forest_modulo_equivalence_d_U_e boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
   next
     show "?d' * top \<le> -?j'"
     proof -
       have 31: "?d' * top = d * top \<squnion> ?e * top"
         by (simp add: mult_right_dist_sup)
       have 32: "d * top \<le> -?j'"
         by (meson assms(1) boruvka_inner_invariant_def inf.coboundedI1 p_antitone_iff)
       have "regular (?c * - ?c\<^sup>T)"
-        using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def component_is_regular regular_conv_closed regular_mult_closed by auto
-      hence "minarc(?c * - ?c\<^sup>T \<sqinter> g) = minarc(?c \<sqinter> - ?c\<^sup>T \<sqinter> g)"
+        using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def component_is_regular regular_conv_closed regular_mult_closed by presburger
+      then have "minarc(?c * - ?c\<^sup>T \<sqinter> g) = minarc(?c \<sqinter> - ?c\<^sup>T \<sqinter> g)"
         by (metis component_is_vector covector_comp_inf inf_top.left_neutral vector_conv_compl)
       also have "... \<le> -- (?c \<sqinter> - ?c\<^sup>T \<sqinter> g)"
         using minarc_below by blast
       also have "... \<le> -- ?c"
         by (simp add: inf.sup_monoid.add_assoc)
       also have "... = ?c"
         using component_is_regular by auto
       finally have "?e \<le> ?c"
         by simp
-      hence "?e * top \<le> ?c"
+      then have "?e * top \<le> ?c"
         by (metis component_is_vector mult_left_isotone)
       also have "... \<le> -j \<squnion> ?c"
         by simp
       also have "... = - (j \<sqinter> - ?c)"
         using component_is_regular by auto
       finally have 33: "?e * top \<le> - (j \<sqinter> - ?c)"
         by simp
       show ?thesis
         using 31 32 33 by auto
     qed
   next
-    show "?H * ?j' = ?j'"
-      using fc_j_eq_j_inv assms(1) boruvka_inner_invariant_def by blast
-  next
-    show "forest_components ?f' = (?H * (?d' \<squnion> ?d'\<^sup>T))\<^sup>\<star> * ?H"
-    proof -
-      have "forest_components ?f' = (f \<squnion> f\<^sup>T \<squnion> ?e \<squnion> ?e\<^sup>T)\<^sup>\<star>"
-      proof (rule simplify_forest_components_f)
-        show "regular ?p"
-          using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def minarc_regular regular_closed_star regular_conv_closed regular_mult_closed by auto
-      next
-        show "regular ?e"
-          using minarc_regular by auto
-      next
-        show "injective ?f'"
-          using assms(1, 2, 3, 4) boruvka_outer_invariant_def boruvka_outer_invariant_when_e_not_bot by blast
-      next
-        show "injective f"
-          using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by blast
-      qed
-      also have "... = (h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T \<squnion> ?e \<squnion> ?e\<^sup>T)\<^sup>\<star>"
-        using assms(1) boruvka_inner_invariant_def by simp
-      also have "... = (h \<squnion> h\<^sup>T \<squnion> ?d' \<squnion> ?d'\<^sup>T)\<^sup>\<star>"
-        by (smt conv_dist_sup sup_monoid.add_assoc sup_monoid.add_commute)
-      also have "... = ((h \<squnion> h\<^sup>T)\<^sup>\<star> * (?d' \<squnion> ?d'\<^sup>T))\<^sup>\<star> * (h \<squnion> h\<^sup>T)\<^sup>\<star>"
-        by (metis star.circ_sup_9 sup_assoc)
-      finally show ?thesis
-        using assms(1) boruvka_inner_invariant_def forest_components_wcc by simp
-    qed
-  next
     show "?f' \<squnion> ?f'\<^sup>T = h \<squnion> h\<^sup>T \<squnion> ?d' \<squnion> ?d'\<^sup>T"
     proof -
       have "?f' \<squnion> ?f'\<^sup>T = f \<sqinter> - ?e\<^sup>T \<sqinter> - ?p \<squnion> (f \<sqinter> - ?e\<^sup>T \<sqinter> ?p)\<^sup>T \<squnion> ?e \<squnion> (f \<sqinter> - ?e\<^sup>T \<sqinter> - ?p)\<^sup>T \<squnion> (f \<sqinter> - ?e\<^sup>T \<sqinter> ?p) \<squnion> ?e\<^sup>T"
         by (simp add: conv_dist_sup sup_monoid.add_assoc)
       also have "... = (f \<sqinter> - ?e\<^sup>T \<sqinter> - ?p) \<squnion> (f \<sqinter> - ?e\<^sup>T \<sqinter> ?p) \<squnion> (f \<sqinter> - ?e\<^sup>T \<sqinter> ?p)\<^sup>T \<squnion> (f \<sqinter> - ?e\<^sup>T \<sqinter> - ?p)\<^sup>T \<squnion> ?e\<^sup>T \<squnion> ?e"
         by (simp add: sup.left_commute sup_commute)
       also have "... = f \<squnion> f\<^sup>T \<squnion> ?e \<squnion> ?e\<^sup>T"
       proof (rule simplify_f)
         show "regular ?p"
           using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def minarc_regular regular_closed_star regular_conv_closed regular_mult_closed by auto
       next
         show "regular ?e"
           using minarc_regular by blast
       qed
       also have "... = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T \<squnion> ?e \<squnion> ?e\<^sup>T"
         using assms(1) boruvka_inner_invariant_def by auto
       finally show ?thesis
         by (smt conv_dist_sup sup.left_commute sup_commute)
     qed
   next
-    show "\<forall> a b . bf_between_arcs a b ?H ?d' \<and> a \<le> - ?H \<sqinter> -- g \<and> b \<le> ?d' \<longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
-    proof (intro allI, rule impI, unfold bf_between_arcs_def)
+    show "\<forall> a b . forest_modulo_equivalence_path a b ?H ?d' \<and> a \<le> - ?H \<sqinter> -- g \<and> b \<le> ?d' \<longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
+    proof (intro allI, rule impI, unfold forest_modulo_equivalence_path_def)
       fix a b
       assume 1: "(arc a \<and> arc b \<and> a\<^sup>T * top \<le> (?H * ?d')\<^sup>\<star> * ?H * b * top) \<and> a \<le> - ?H \<sqinter> -- g \<and> b \<le> ?d'"
       thus "sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
       proof (cases "b = ?e")
         case b_equals_e: True
         thus ?thesis
         proof (cases "a = ?e")
           case True
           thus ?thesis
             using b_equals_e by auto
         next
           case a_ne_e: False
           have "sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
-          proof (rule a_to_e_in_bigforest)
+          proof (rule a_to_e_in_forest_modulo_equivalence)
             show "symmetric g"
               using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
           next
             show "j \<noteq> bot"
               by (simp add: assms(2))
           next
             show "f \<le> -- g"
               using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
           next
             show "vector j"
               using assms(1) boruvka_inner_invariant_def by blast
           next
             show "forest h"
               using assms(1) boruvka_inner_invariant_def by blast
           next
-            show " big_forest (forest_components h) d"
+            show "forest_modulo_equivalence (forest_components h) d"
               using assms(1) boruvka_inner_invariant_def by blast
           next
             show "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
               using assms(1) boruvka_inner_invariant_def by blast
           next
-            show "\<forall>a b. bf_between_arcs a b (?H) d \<and> a \<le> - ?H \<sqinter> - - g \<and> b \<le> d \<longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
+            show "\<forall>a b. forest_modulo_equivalence_path a b (?H) d \<and> a \<le> - ?H \<sqinter> - - g \<and> b \<le> d \<longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
               using assms(1) boruvka_inner_invariant_def by blast
           next
             show "regular d"
               using assms(1) boruvka_inner_invariant_def by blast
           next
             show "b = ?e"
               using b_equals_e by simp
           next
             show "arc a"
               using 1 by simp
           next
-            show "bf_between_arcs a b ?H ?d'"
-              using 1 bf_between_arcs_def by simp
+            show "forest_modulo_equivalence_path a b ?H ?d'"
+              using 1 forest_modulo_equivalence_path_def by simp
           next
             show "a \<le> - ?H \<sqinter> -- g"
               using 1 by simp
           next
             show "regular h"
               using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
           qed
           thus ?thesis
             by simp
         qed
       next
         case b_not_equal_e: False
-        hence b_below_d: "b \<le> d"
-          using 1 by (metis assms(4) different_arc_in_sup_arc minarc_arc minarc_bot_iff)
+        then have b_below_d: "b \<le> d"
+          using 1 assms(4) different_arc_in_sup_arc minarc_arc minarc_bot_iff by metis
         thus ?thesis
         proof (cases "?e \<le> d")
           case True
-          hence "bf_between_arcs a b ?H d \<and> b \<le> d"
-            using 1 bf_between_arcs_def sup.absorb1 by auto
+          then have "forest_modulo_equivalence_path a b ?H d \<and> b \<le> d"
+            using 1 forest_modulo_equivalence_path_def sup.absorb1 by auto
           thus ?thesis
             using 1 assms(1) boruvka_inner_invariant_def by blast
         next
           case e_not_less_than_d: False
           have 71:"equivalence ?H"
             using assms(1) fch_equivalence boruvka_inner_invariant_def by auto
-          hence 72: "bf_between_arcs a b ?H ?d' \<longleftrightarrow> bf_between_arcs a b ?H d \<or> (bf_between_arcs a ?e ?H d \<and> bf_between_arcs ?e b ?H d)"
-          proof (rule big_forest_path_split_disj)
+          then have 72: "forest_modulo_equivalence_path a b ?H ?d' \<longleftrightarrow> forest_modulo_equivalence_path a b ?H d \<or> (forest_modulo_equivalence_path a ?e ?H d \<and> forest_modulo_equivalence_path ?e b ?H d)"
+          proof (rule forest_modulo_equivalence_path_split_disj)
             show "arc ?e"
               using assms(4) minarc_arc minarc_bot_iff by blast
           next
             show "regular a \<and> regular b \<and> regular ?e \<and> regular d \<and> regular ?H"
               using assms(1) 1 boruvka_inner_invariant_def boruvka_outer_invariant_def arc_regular minarc_regular regular_closed_star regular_conv_closed regular_mult_closed by auto
           qed
           thus ?thesis
-          proof (cases "bf_between_arcs a b ?H d")
+          proof (cases "forest_modulo_equivalence_path a b ?H d")
             case True
-            have "bf_between_arcs a b ?H d \<and> b \<le> d"
-              using 1 by (metis assms(4) True b_not_equal_e minarc_arc minarc_bot_iff different_arc_in_sup_arc)
+            have "forest_modulo_equivalence_path a b ?H d \<and> b \<le> d"
+              using 1 True forest_modulo_equivalence_path_def sup.absorb1 by (metis assms(4) b_not_equal_e minarc_arc minarc_bot_iff different_arc_in_sup_arc)
             thus ?thesis
               using 1 assms(1) b_below_d boruvka_inner_invariant_def by auto
           next
             case False
-            have 73:"bf_between_arcs a ?e ?H d \<and> bf_between_arcs ?e b ?H d"
-              using 1 72 False bf_between_arcs_def by blast
+            have 73:"forest_modulo_equivalence_path a ?e ?H d \<and> forest_modulo_equivalence_path ?e b ?H d"
+              using 1 72 False forest_modulo_equivalence_path_def by blast
             have 74: "?e \<le> --g"
               by (metis inf.boundedE minarc_below pp_dist_inf)
-            have "?e \<le> - ?H"
-              by (meson assms(1, 3) boruvka_inner_invariant_def dual_order.trans p_antitone_iff)
-            hence "?e \<le> - ?H \<sqinter> --g"
+            have "?H \<le> ?F"
+                using assms(1) H_below_F boruvka_inner_invariant_def boruvka_outer_invariant_def by blast
+            then have "?e \<le> - ?H"
+              using assms(3) order_trans p_antitone by blast
+            then have "?e \<le> - ?H \<sqinter> --g"
               using 74 by simp
-            hence 75: "sum (b \<sqinter> g) \<le> sum (?e \<sqinter> g)"
+            then have 75: "sum (b \<sqinter> g) \<le> sum (?e \<sqinter> g)"
               using assms(1) b_below_d 73 boruvka_inner_invariant_def by blast
-            have 76: "bf_between_arcs a ?e ?H ?d'"
-              using 73 by (meson big_forest_path_split_disj assms(1) bf_between_arcs_def boruvka_inner_invariant_def boruvka_outer_invariant_def fch_equivalence arc_regular regular_closed_star regular_conv_closed regular_mult_closed)
+            have 76: "forest_modulo_equivalence_path a ?e ?H ?d'"
+              by (meson 73 forest_modulo_equivalence_path_split_disj assms(1) forest_modulo_equivalence_path_def boruvka_inner_invariant_def boruvka_outer_invariant_def fch_equivalence arc_regular regular_closed_star regular_conv_closed regular_mult_closed)
             have 77: "sum (?e \<sqinter> g) \<le> sum (a \<sqinter> g)"
-            proof (rule a_to_e_in_bigforest)
+            proof (rule a_to_e_in_forest_modulo_equivalence)
               show "symmetric g"
                 using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
             next
               show "j \<noteq> bot"
                 by (simp add: assms(2))
             next
               show "f \<le> -- g"
                 using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
             next
               show "vector j"
                 using assms(1) boruvka_inner_invariant_def by blast
             next
               show "forest h"
                 using assms(1) boruvka_inner_invariant_def by blast
             next
-              show " big_forest (forest_components h) d"
+              show "forest_modulo_equivalence (forest_components h) d"
                 using assms(1) boruvka_inner_invariant_def by blast
             next
               show "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
                 using assms(1) boruvka_inner_invariant_def by blast
             next
-              show "\<forall>a b. bf_between_arcs a b (?H) d \<and> a \<le> - ?H \<sqinter> - - g \<and> b \<le> d \<longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
+              show "\<forall>a b. forest_modulo_equivalence_path a b (?H) d \<and> a \<le> - ?H \<sqinter> - - g \<and> b \<le> d \<longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
                 using assms(1) boruvka_inner_invariant_def by blast
             next
               show "regular d"
                 using assms(1) boruvka_inner_invariant_def by blast
             next
               show "?e = ?e"
                 by simp
             next
               show "arc a"
                 using 1 by simp
             next
-              show "bf_between_arcs a ?e ?H ?d'"
-                by (simp add: 76)
+              show "forest_modulo_equivalence_path a ?e ?H ?d'"
+                by (simp add: "76")
             next
               show "a \<le> - ?H \<sqinter> --g"
                 using 1 by simp
             next
               show "regular h"
                 using assms(1) boruvka_inner_invariant_def boruvka_outer_invariant_def by auto
             qed
             thus ?thesis
               using 75 order.trans by blast
           qed
         qed
       qed
     qed
-  next
-    show "regular ?d'"
-      using assms(1) boruvka_inner_invariant_def minarc_regular by auto
   qed
 qed
 
 lemma second_inner_invariant_when_e_bot:
   assumes "selected_edge h j g = bot"
     and "selected_edge h j g \<le> - forest_components f"
     and "boruvka_inner_invariant j f h g d"
   shows "boruvka_inner_invariant
      (j \<sqinter> - choose_component (forest_components h) j)
      (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> - path f h j g \<squnion>
       (f \<sqinter> - selected_edge h j g\<^sup>T \<sqinter> path f h j g)\<^sup>T \<squnion>
       selected_edge h j g)
      h g (d \<squnion> selected_edge h j g)"
 proof -
   let ?c = "choose_component (forest_components h) j"
   let ?p = "path f h j g"
   let ?F = "forest_components f"
   let ?H = "forest_components h"
   let ?e = "selected_edge h j g"
   let ?f' = "f \<sqinter> -?e\<^sup>T \<sqinter> -?p \<squnion> (f \<sqinter> -?e\<^sup>T \<sqinter> ?p)\<^sup>T \<squnion> ?e"
   let ?d' = "d \<squnion> ?e"
   let ?j' = "j \<sqinter> -?c"
   show "boruvka_inner_invariant ?j' ?f' h g ?d'"
   proof (unfold boruvka_inner_invariant_def, intro conjI)
   next
     show "boruvka_outer_invariant ?f' g"
-      using assms(1, 3) boruvka_inner_invariant_def by auto
+      using assms(1) assms(3) boruvka_inner_invariant_def by auto
   next
     show "g \<noteq> bot"
       using assms(3) boruvka_inner_invariant_def by blast
   next
-    show "vector ?j'"
-      by (metis assms(3) boruvka_inner_invariant_def component_is_vector vector_complement_closed vector_inf_closed)
+    show "regular ?d'"
+      using assms(1) assms(3) boruvka_inner_invariant_def by auto
   next
     show "regular ?j'"
       using assms(3) boruvka_inner_invariant_def by auto
   next
-    show "boruvka_outer_invariant h g"
+    show "vector ?j'"
+      by (metis assms(3) boruvka_inner_invariant_def component_is_vector vector_complement_closed vector_inf_closed)
+  next
+    show "regular h"
       using assms(3) boruvka_inner_invariant_def by blast
   next
     show "injective h"
       using assms(3) boruvka_inner_invariant_def by blast
   next
     show "pd_kleene_allegory_class.acyclic h"
       using assms(3) boruvka_inner_invariant_def by blast
   next
-    show "?H \<le> forest_components ?f'"
-      using assms(1, 3) boruvka_inner_invariant_def by auto
-  next
-    show " big_forest ?H ?d'"
-      using assms(1, 3) boruvka_inner_invariant_def by auto
-  next
-    show "?d' * top \<le> -?j'"
-      by (metis assms(1, 3) boruvka_inner_invariant_def order.trans p_antitone_inf sup_monoid.add_0_right)
-  next
     show "?H * ?j' = ?j'"
       using assms(3) fc_j_eq_j_inv boruvka_inner_invariant_def by blast
   next
-    show "forest_components ?f' = (?H * (?d' \<squnion> ?d'\<^sup>T))\<^sup>\<star> *?H"
-      using assms(1, 3) boruvka_inner_invariant_def by auto
+    show "forest_modulo_equivalence ?H ?d'"
+      using assms(1) assms(3) boruvka_inner_invariant_def by auto
+  next
+    show "?d' * top \<le> -?j'"
+      using assms(1) assms(3) boruvka_inner_invariant_def by (metis order.trans p_antitone_inf sup_monoid.add_0_right)
   next
     show "?f' \<squnion> ?f'\<^sup>T = h \<squnion> h\<^sup>T \<squnion> ?d' \<squnion> ?d'\<^sup>T"
       using assms(1, 3) boruvka_inner_invariant_def by auto
   next
-    show "\<forall>a b. bf_between_arcs a b ?H ?d' \<and> a \<le> -?H \<sqinter> --g \<and> b \<le> ?d' \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g)"
-      using assms(1, 3) boruvka_inner_invariant_def by auto
-  next
-    show "regular ?d'"
+    show "\<forall>a b. forest_modulo_equivalence_path a b ?H ?d' \<and> a \<le> -?H \<sqinter> --g \<and> b \<le> ?d' \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g)"
       using assms(1, 3) boruvka_inner_invariant_def by auto
   qed
 qed
 
 subsection \<open>Formalization and correctness proof\<close>
 
 text \<open>
 The following result shows that Bor\r{u}vka's algorithm constructs a minimum spanning forest.
 We have the same postcondition as the proof of Kruskal's minimum spanning tree algorithm.
 We show only partial correctness.
 \<close>
 
 theorem boruvka_mst:
   "VARS f j h c e d
   { symmetric g }
   f := bot;
   WHILE -(forest_components f) \<sqinter> g \<noteq> bot
     INV { boruvka_outer_invariant f g }
     DO
       j := top;
       h := f;
       d := bot;
       WHILE j \<noteq> bot
         INV { boruvka_inner_invariant j f h g d }
         DO
           c := choose_component (forest_components h) j;
           e := minarc(c * -c\<^sup>T \<sqinter> g);
           IF e \<le> -(forest_components f) THEN
             f := f \<sqinter> -e\<^sup>T;
             f := (f \<sqinter> -(top * e * f\<^sup>T\<^sup>\<star>)) \<squnion> (f \<sqinter> top * e * f\<^sup>T\<^sup>\<star>)\<^sup>T \<squnion> e;
             d := d \<squnion> e
           ELSE
             SKIP
           FI;
           j := j \<sqinter> -c
         OD
     OD
   { minimum_spanning_forest f g }"
 proof vcg_simp
   assume 1: "symmetric g"
   show "boruvka_outer_invariant bot g"
     using 1 boruvka_outer_invariant_def kruskal_exists_minimal_spanning by auto
 next
   fix f
   let ?F = "forest_components f"
   assume 1: "boruvka_outer_invariant f g \<and> - ?F \<sqinter> g \<noteq> bot"
   have 2: "equivalence ?F"
     using 1 boruvka_outer_invariant_def forest_components_equivalence by auto
   show "boruvka_inner_invariant top f f g bot"
   proof (unfold boruvka_inner_invariant_def, intro conjI)
     show "boruvka_outer_invariant f g"
       by (simp add: 1)
   next
     show "g \<noteq> bot"
       using 1 by auto
   next
     show "surjective top"
       by simp
   next
     show "regular top"
       by simp
   next
-    show "boruvka_outer_invariant f g"
-      using 1 by auto
+    show "regular bot"
+      by auto
+  next
+    show "regular f"
+      using 1 boruvka_outer_invariant_def by blast
   next
     show "injective f"
       using 1 boruvka_outer_invariant_def by blast
   next
     show "pd_kleene_allegory_class.acyclic f"
       using 1 boruvka_outer_invariant_def by blast
   next
-    show "?F \<le> ?F"
-      by simp
-  next
-    show "big_forest ?F bot"
-      by (simp add: 2 big_forest_def)
+    show "forest_modulo_equivalence ?F bot"
+      by (simp add: 2 forest_modulo_equivalence_def)
   next
     show "bot * top \<le> - top"
       by simp
   next
     show "times_top_class.total (?F)"
       by (simp add: star.circ_right_top mult_assoc)
   next
-    show "?F = (?F * (bot \<squnion> bot\<^sup>T))\<^sup>\<star> * ?F"
-      by (metis mult_right_zero semiring.mult_zero_left star.circ_loop_fixpoint sup_commute sup_monoid.add_0_right symmetric_bot_closed)
-  next
     show "f \<squnion> f\<^sup>T = f \<squnion> f\<^sup>T \<squnion> bot \<squnion> bot\<^sup>T"
       by simp
   next
-    show "\<forall> a b. bf_between_arcs a b ?F bot \<and> a \<le> - ?F \<sqinter> -- g \<and> b \<le> bot \<longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
-      by (metis (full_types) bf_between_arcs_def bot_unique mult_left_zero mult_right_zero top.extremum)
-  next
-    show "regular bot"
-      by auto
+    show "\<forall> a b. forest_modulo_equivalence_path a b ?F bot \<and> a \<le> - ?F \<sqinter> -- g \<and> b \<le> bot \<longrightarrow> sum (b \<sqinter> g) \<le> sum (a \<sqinter> g)"
+      by (metis (full_types) forest_modulo_equivalence_path_def bot_unique mult_left_zero mult_right_zero top.extremum)
   qed
 next
   fix f j h d
   let ?c = "choose_component (forest_components h) j"
   let ?p = "path f h j g"
   let ?F = "forest_components f"
   let ?H = "forest_components h"
   let ?e = "selected_edge h j g"
   let ?f' = "f \<sqinter> -?e\<^sup>T \<sqinter> -?p \<squnion> (f \<sqinter> -?e\<^sup>T \<sqinter> ?p)\<^sup>T \<squnion> ?e"
   let ?d' = "d \<squnion> ?e"
   let ?j' = "j \<sqinter> -?c"
   assume 1: "boruvka_inner_invariant j f h g d \<and> j \<noteq> bot"
   show "(?e \<le> -?F \<longrightarrow> boruvka_inner_invariant ?j' ?f' h g ?d') \<and> (\<not> ?e \<le> -?F \<longrightarrow> boruvka_inner_invariant ?j' f h g d)"
   proof (intro conjI)
     show "?e \<le> -?F \<longrightarrow> boruvka_inner_invariant ?j' ?f' h g ?d'"
     proof (cases "?e = bot")
       case True
       thus ?thesis
         using 1 second_inner_invariant_when_e_bot by simp
     next
       case False
       thus ?thesis
         using 1 second_inner_invariant_when_e_not_bot by simp
     qed
   next
     show "\<not> ?e \<le> -?F \<longrightarrow> boruvka_inner_invariant ?j' f h g d"
     proof (rule impI, unfold boruvka_inner_invariant_def, intro conjI)
       show "boruvka_outer_invariant f g"
         using 1 boruvka_inner_invariant_def by blast
     next
       show "g \<noteq> bot"
         using 1 boruvka_inner_invariant_def by blast
     next
       show "vector ?j'"
         using 1 boruvka_inner_invariant_def component_is_vector vector_complement_closed vector_inf_closed by auto
     next
       show "regular ?j'"
         using 1 boruvka_inner_invariant_def by auto
     next
-      show "boruvka_outer_invariant h g"
-        using 1 boruvka_inner_invariant_def by auto
+      show "regular d"
+        using 1 boruvka_inner_invariant_def by blast
+    next
+      show "regular h"
+        using 1 boruvka_inner_invariant_def by blast
     next
       show "injective h"
         using 1 boruvka_inner_invariant_def by blast
     next
       show "pd_kleene_allegory_class.acyclic h"
         using 1 boruvka_inner_invariant_def by blast
     next
-      show "?H \<le> ?F"
-        using 1 boruvka_inner_invariant_def by blast
-    next
-      show "big_forest ?H d"
+      show "forest_modulo_equivalence ?H d"
         using 1 boruvka_inner_invariant_def by blast
     next
       show "d * top \<le> -?j'"
         using 1 by (meson boruvka_inner_invariant_def dual_order.trans p_antitone_inf)
     next
       show "?H * ?j' = ?j'"
         using 1 fc_j_eq_j_inv boruvka_inner_invariant_def by blast
     next
-      show "?F = (?H * (d \<squnion> d\<^sup>T))\<^sup>\<star> * ?H"
-        using 1 boruvka_inner_invariant_def by blast
-    next
       show "f \<squnion> f\<^sup>T = h \<squnion> h\<^sup>T \<squnion> d \<squnion> d\<^sup>T"
         using 1 boruvka_inner_invariant_def by blast
     next
-      show "\<not> ?e \<le> -?F \<Longrightarrow> \<forall>a b. bf_between_arcs a b ?H d \<and> a \<le> -?H \<sqinter> --g \<and> b \<le> d \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g)"
-        using 1 boruvka_inner_invariant_def by blast
-    next
-      show "\<not> ?e \<le> -?F \<Longrightarrow> regular d"
+      show "\<not> ?e \<le> -?F \<Longrightarrow> \<forall>a b. forest_modulo_equivalence_path a b ?H d \<and> a \<le> -?H \<sqinter> --g \<and> b \<le> d \<longrightarrow> sum(b \<sqinter> g) \<le> sum(a \<sqinter> g)"
         using 1 boruvka_inner_invariant_def by blast
     qed
   qed
 next
   fix f h d
   assume "boruvka_inner_invariant bot f h g d"
   thus "boruvka_outer_invariant f g"
     by (meson boruvka_inner_invariant_def)
 next
   fix f
   assume 1: "boruvka_outer_invariant f g \<and> - forest_components f \<sqinter> g = bot"
   hence 2:"spanning_forest f g"
   proof (unfold spanning_forest_def, intro conjI)
     show "injective f"
       using 1 boruvka_outer_invariant_def by blast
   next
     show "acyclic f"
       using 1 boruvka_outer_invariant_def by blast
   next
     show "f \<le> --g"
       using 1 boruvka_outer_invariant_def by blast
   next
     show "components g \<le> forest_components f"
     proof -
       let ?F = "forest_components f"
       have "-?F \<sqinter> g \<le> bot"
         by (simp add: 1)
       hence "--g \<le> bot \<squnion> --?F"
         using 1 shunting_p p_antitone pseudo_complement by auto
       hence "--g \<le> ?F"
         using 1 boruvka_outer_invariant_def pp_dist_comp pp_dist_star regular_conv_closed by auto
       hence "(--g)\<^sup>\<star> \<le> ?F\<^sup>\<star>"
         by (simp add: star_isotone)
       thus ?thesis
         using 1 boruvka_outer_invariant_def forest_components_star by auto
     qed
   next
     show "regular f"
       using 1 boruvka_outer_invariant_def by auto
   qed
   from 1 obtain w where 3: "minimum_spanning_forest w g \<and> f \<le> w \<squnion> w\<^sup>T"
     using boruvka_outer_invariant_def by blast
   hence "w = w \<sqinter> --g"
     by (simp add: inf.absorb1 minimum_spanning_forest_def spanning_forest_def)
   also have "... \<le> w \<sqinter> components g"
     by (metis inf.sup_right_isotone star.circ_increasing)
   also have "... \<le> w \<sqinter> f\<^sup>T\<^sup>\<star> * f\<^sup>\<star>"
     using 2 spanning_forest_def inf.sup_right_isotone by simp
   also have "... \<le> f \<squnion> f\<^sup>T"
   proof (rule cancel_separate_6[where z=w and y="w\<^sup>T"])
     show "injective w"
       using 3 minimum_spanning_forest_def spanning_forest_def by simp
   next
     show "f\<^sup>T \<le> w\<^sup>T \<squnion> w"
       using 3 by (metis conv_dist_inf conv_dist_sup conv_involutive inf.cobounded2 inf.orderE)
   next
     show "f \<le> w\<^sup>T \<squnion> w"
       using 3 by (simp add: sup_commute)
   next
     show "injective w"
       using 3 minimum_spanning_forest_def spanning_forest_def by simp
   next
     show "w \<sqinter> w\<^sup>T\<^sup>\<star> = bot"
       using 3 by (metis acyclic_star_below_complement comp_inf.mult_right_isotone inf_p le_bot minimum_spanning_forest_def spanning_forest_def)
   qed
   finally have 4: "w \<le> f \<squnion> f\<^sup>T"
     by simp
   have "sum (f \<sqinter> g) = sum ((w \<squnion> w\<^sup>T) \<sqinter> (f \<sqinter> g))"
     using 3 by (metis inf_absorb2 inf.assoc)
   also have "... = sum (w \<sqinter> (f \<sqinter> g)) + sum (w\<^sup>T \<sqinter> (f \<sqinter> g))"
     using 3 inf.commute acyclic_asymmetric sum_disjoint minimum_spanning_forest_def spanning_forest_def by simp
   also have "... = sum (w \<sqinter> (f \<sqinter> g)) + sum (w \<sqinter> (f\<^sup>T \<sqinter> g\<^sup>T))"
     by (metis conv_dist_inf conv_involutive sum_conv)
   also have "... = sum (f \<sqinter> (w \<sqinter> g)) + sum (f\<^sup>T \<sqinter> (w \<sqinter> g))"
   proof -
     have 51:"f\<^sup>T \<sqinter> (w \<sqinter> g) = f\<^sup>T \<sqinter> (w \<sqinter> g\<^sup>T)"
       using 1 boruvka_outer_invariant_def by auto
     have 52:"f \<sqinter> (w \<sqinter> g) = w \<sqinter> (f \<sqinter> g)"
       by (simp add: inf.left_commute)
     thus ?thesis
       using 51 52 abel_semigroup.left_commute inf.abel_semigroup_axioms by fastforce
   qed
   also have "... = sum ((f \<squnion> f\<^sup>T) \<sqinter> (w \<sqinter> g))"
     using 2 acyclic_asymmetric inf.sup_monoid.add_commute sum_disjoint spanning_forest_def by simp
   also have "... = sum (w \<sqinter> g)"
     using 4 by (metis inf_absorb2 inf.assoc)
   finally show "minimum_spanning_forest f g"
     using 2 3 minimum_spanning_forest_def by simp
 qed
 
 end
 
 end
+
diff --git a/thys/Relational_Minimum_Spanning_Trees/document/root.tex b/thys/Relational_Minimum_Spanning_Trees/document/root.tex
--- a/thys/Relational_Minimum_Spanning_Trees/document/root.tex
+++ b/thys/Relational_Minimum_Spanning_Trees/document/root.tex
@@ -1,58 +1,57 @@
 \documentclass[11pt,a4paper]{article}
 \usepackage[T1]{fontenc}
 \usepackage{isabelle,isabellesym}
 \usepackage{amssymb,ragged2e}
 \usepackage{pdfsetup}
 
 \isabellestyle{it}
 \renewenvironment{isamarkuptext}{\par\isastyletext\begin{isapar}\justifying\color{blue}}{\end{isapar}}
 \renewcommand\labelitemi{$*$}
 \urlstyle{rm}
 
 \begin{document}
 
 \title{Relational Minimum Spanning Tree Algorithms}
 \author{Walter Guttmann and Nicolas Robinson-O'Brien}
 \maketitle
 
 \begin{abstract}
   We verify the correctness of Prim's, Kruskal's and Bor\r{u}vka's minimum spanning tree algorithms based on algebras for aggregation and minimisation.
 \end{abstract}
 
 \tableofcontents
 
 \section{Overview}
 
 The theories described in this document prove the correctness of Prim's, Kruskal's and Bor\r{u}vka's minimum spanning tree algorithms.
 Specifications and algorithms work in Stone-Kleene relation algebras extended by operations for aggregation and minimisation.
 The algorithms are implemented in a simple imperative language and their proof uses Hoare logic.
 The correctness proofs are discussed in \cite{Guttmann2016c,Guttmann2018b,Guttmann2018c,RobinsonOBrien2020}.
 
 \subsection{Prim's and Kruskal's minimum spanning tree algorithms}
 
 A framework based on Stone relation algebras and Kleene algebras and extended by operations for aggregation and minimisation was presented by the first author in \cite{Guttmann2016c,Guttmann2018b} and used to formally verify the correctness of Prim's minimum spanning tree algorithm.
 It was extended in \cite{Guttmann2018c} and applied to prove the correctness of Kruskal's minimum spanning tree algorithm.
 
 Two theories, one each for Prim's and Kruskal's algorithms, prove total correctness of these algorithms.
 As case studies for the algebraic framework, these two theories combined were originally part of another AFP entry \cite{Guttmann2018a}.
 
 \subsection{Bor\r{u}vka's minimum spanning tree algorithm}
 
 Otakar Bor\r{u}vka formalised the minimum spanning tree problem and proposed a solution to it \cite{Boruvka1926}.
 Bor\r{u}vka's original paper is written in Czech; translations of varying completeness can be found in \cite{GrahamHell1985,NesetrilMilkovaNesetrilova2001}.
 
 The theory for Bor\r{u}vka's minimum spanning tree algorithm proves partial correctness of this algorithm.
 This work is based on the same algebraic framework as the proof of Kruskal's algorithm; in particular it uses many theories from the hierarchy underlying \cite{Guttmann2018a}.
 
 The theory for Bor\r{u}vka's algorithm formally verifies results from the second author's Master's thesis \cite{RobinsonOBrien2020}.
-Certain lemmas in this theory are numbered for easy correlation to theorems from the thesis.
 
 \begin{flushleft}
 \input{session}
 \end{flushleft}
 
 \bibliographystyle{abbrv}
 \bibliography{root}
 
 \end{document}