diff --git a/thys/Relational_Disjoint_Set_Forests/Disjoint_Set_Forests.thy b/thys/Relational_Disjoint_Set_Forests/Disjoint_Set_Forests.thy
--- a/thys/Relational_Disjoint_Set_Forests/Disjoint_Set_Forests.thy
+++ b/thys/Relational_Disjoint_Set_Forests/Disjoint_Set_Forests.thy
@@ -1,2328 +1,2334 @@
 (* Title:      Disjoint-Set Forests
    Author:     Walter Guttmann
    Maintainer: Walter Guttmann <walter.guttmann at canterbury.ac.nz>
 *)
 
 theory Disjoint_Set_Forests
 
 imports
   "HOL-Hoare.Hoare_Logic"
   Stone_Kleene_Relation_Algebras.Kleene_Relation_Algebras
 begin
 
 no_notation
   minus (infixl "-" 65) and
   trancl ("(_\<^sup>+)" [1000] 999)
 
 context p_algebra
 begin
 
 abbreviation minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" (infixl "-" 65)
   where "x - y \<equiv> x \<sqinter> -y"
 
 end
 
 text \<open>
 An arc in a Stone relation algebra corresponds to an atom in a relation algebra and represents a single edge in a graph.
 A point represents a set of nodes.
 A rectangle represents the Cartesian product of two sets of nodes \cite{BerghammerStruth2010}.
 \<close>
 
 context times_top
 begin
 
 abbreviation rectangle :: "'a \<Rightarrow> bool"
   where "rectangle x \<equiv> x * top * x = x"
 
 end
 
 context stone_relation_algebra
 begin
 
 lemma arc_rectangle:
   "arc x \<Longrightarrow> rectangle x"
   using arc_top_arc by blast
 
 section \<open>Relation-Algebraic Semantics of Associative Array Access\<close>
 
 text \<open>
 The following two operations model updating array $x$ at index $y$ to value $z$, 
 and reading the content of array $x$ at index $y$, respectively.
 The read operation uses double brackets to avoid ambiguity with list syntax.
 The remainder of this section shows basic properties of these operations.
 \<close>
 
 abbreviation rel_update :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" ("(_[_\<longmapsto>_])" [70, 65, 65] 61)
   where "x[y\<longmapsto>z] \<equiv> (y \<sqinter> z\<^sup>T) \<squnion> (-y \<sqinter> x)"
 
 abbreviation rel_access :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" ("(2_[[_]])" [70, 65] 65)
   where "x[[y]] \<equiv> x\<^sup>T * y"
 
 text \<open>Theorem 1.1\<close>
 
 lemma update_univalent:
   assumes "univalent x"
     and "vector y"
     and "injective z"
   shows "univalent (x[y\<longmapsto>z])"
 proof -
   have 1: "univalent (y \<sqinter> z\<^sup>T)"
     using assms(3) inf_commute univalent_inf_closed by force
   have "(y \<sqinter> z\<^sup>T)\<^sup>T * (-y \<sqinter> x) = (y\<^sup>T \<sqinter> z) * (-y \<sqinter> x)"
     by (simp add: conv_dist_inf)
   also have "... = z * (y \<sqinter> -y \<sqinter> x)"
     by (metis assms(2) covector_inf_comp_3 inf.sup_monoid.add_assoc inf.sup_monoid.add_commute)
   finally have 2: "(y \<sqinter> z\<^sup>T)\<^sup>T * (-y \<sqinter> x) = bot"
     by simp
   have 3: "vector (-y)"
     using assms(2) vector_complement_closed by simp
   have "(-y \<sqinter> x)\<^sup>T * (y \<sqinter> z\<^sup>T) = (-y\<^sup>T \<sqinter> x\<^sup>T) * (y \<sqinter> z\<^sup>T)"
     by (simp add: conv_complement conv_dist_inf)
   also have "... = x\<^sup>T * (-y \<sqinter> y \<sqinter> z\<^sup>T)"
     using 3 by (metis (mono_tags, opaque_lifting) conv_complement covector_inf_comp_3 inf.sup_monoid.add_assoc inf.sup_monoid.add_commute)
   finally have 4: "(-y \<sqinter> x)\<^sup>T * (y \<sqinter> z\<^sup>T) = bot"
     by simp
   have 5: "univalent (-y \<sqinter> x)"
     using assms(1) inf_commute univalent_inf_closed by fastforce
   have "(x[y\<longmapsto>z])\<^sup>T * (x[y\<longmapsto>z]) = (y \<sqinter> z\<^sup>T)\<^sup>T * (x[y\<longmapsto>z]) \<squnion> (-y \<sqinter> x)\<^sup>T * (x[y\<longmapsto>z])"
     by (simp add: conv_dist_sup mult_right_dist_sup)
   also have "... = (y \<sqinter> z\<^sup>T)\<^sup>T * (y \<sqinter> z\<^sup>T) \<squnion> (y \<sqinter> z\<^sup>T)\<^sup>T * (-y \<sqinter> x) \<squnion> (-y \<sqinter> x)\<^sup>T * (y \<sqinter> z\<^sup>T) \<squnion> (-y \<sqinter> x)\<^sup>T * (-y \<sqinter> x)"
     by (simp add: mult_left_dist_sup sup_assoc)
   finally show ?thesis
     using 1 2 4 5 by simp
 qed
 
 text \<open>Theorem 1.2\<close>
 
 lemma update_total:
   assumes "total x"
     and "vector y"
     and "regular y"
     and "surjective z"
   shows "total (x[y\<longmapsto>z])"
 proof -
   have "(x[y\<longmapsto>z]) * top = x*top[y\<longmapsto>top*z]"
     by (simp add: assms(2) semiring.distrib_right vector_complement_closed vector_inf_comp conv_dist_comp)
   also have "... = top[y\<longmapsto>top]"
     using assms(1) assms(4) by simp
   also have "... = top"
     using assms(3) regular_complement_top by auto
   finally show ?thesis
     by simp
 qed
 
 text \<open>Theorem 1.3\<close>
 
 lemma update_mapping:
   assumes "mapping x"
     and "vector y"
     and "regular y"
     and "bijective z"
   shows "mapping (x[y\<longmapsto>z])"
   using assms update_univalent update_total by simp
 
 text \<open>Theorem 1.4\<close>
 
 lemma read_injective:
   assumes "injective y"
     and "univalent x"
   shows "injective (x[[y]])"
   using assms injective_mult_closed univalent_conv_injective by blast
 
 text \<open>Theorem 1.5\<close>
 
 lemma read_surjective:
   assumes "surjective y"
     and "total x"
   shows "surjective (x[[y]])"
   using assms surjective_mult_closed total_conv_surjective by blast
 
 text \<open>Theorem 1.6\<close>
 
 lemma read_bijective:
   assumes "bijective y"
     and "mapping x"
   shows "bijective (x[[y]])"
   by (simp add: assms read_injective read_surjective)
 
 text \<open>Theorem 1.7\<close>
 
 lemma read_point:
   assumes "point p"
     and "mapping x"
   shows "point (x[[p]])"
   using assms comp_associative read_injective read_surjective by auto
 
 text \<open>Theorem 1.8\<close>
 
 lemma update_postcondition:
   assumes "point x" "point y"
   shows "x \<sqinter> p = x * y\<^sup>T \<longleftrightarrow> p[[x]] = y"
   apply (rule iffI)
   subgoal by (metis assms comp_associative conv_dist_comp conv_involutive covector_inf_comp_3 equivalence_top_closed vector_covector)
   subgoal
     apply (rule order.antisym)
     subgoal by (metis assms conv_dist_comp conv_involutive inf.boundedI inf.cobounded1 vector_covector vector_restrict_comp_conv)
     subgoal by (smt assms comp_associative conv_dist_comp conv_involutive covector_restrict_comp_conv dense_conv_closed equivalence_top_closed inf.boundedI shunt_mapping vector_covector preorder_idempotent)
     done
   done
 
 text \<open>Back and von Wright's array independence requirements \cite{BackWright1998}, 
   later also lens laws \cite{FosterGreenwaldMoorePierceSchmitt2007}\<close>
 
 text \<open>Theorem 2.1\<close>
 
 lemma put_get:
   assumes "vector y" "surjective y" "vector z"
   shows "(x[y\<longmapsto>z])[[y]] = z"
 proof -
   have "(x[y\<longmapsto>z])[[y]] = (y\<^sup>T \<sqinter> z) * y \<squnion> (-y\<^sup>T \<sqinter> x\<^sup>T) * y"
     by (simp add: conv_complement conv_dist_inf conv_dist_sup mult_right_dist_sup)
   also have "... = z * y"
   proof -
     have "(-y\<^sup>T \<sqinter> x\<^sup>T) * y = bot"
       by (metis assms(1) covector_inf_comp_3 inf_commute conv_complement mult_right_zero p_inf vector_complement_closed)
     thus ?thesis
       by (simp add: assms covector_inf_comp_3 inf_commute)
   qed
   also have "... = z"
     by (metis assms(2,3) mult_assoc)
   finally show ?thesis
     .
 qed
 
 text \<open>Theorem 2.3\<close>
 
 lemma put_put:
   "(x[y\<longmapsto>z])[y\<longmapsto>w] = x[y\<longmapsto>w]"
   by (metis inf_absorb2 inf_commute inf_le1 inf_sup_distrib1 maddux_3_13 sup_inf_absorb)
 
 text \<open>Theorem 2.5\<close>
 
 lemma get_put:
   assumes "point y"
   shows "x[y\<longmapsto>x[[y]]] = x"
 proof -
   have "x[y\<longmapsto>x[[y]]] = (y \<sqinter> y\<^sup>T * x) \<squnion> (-y \<sqinter> x)"
     by (simp add: conv_dist_comp)
   also have "... = (y \<sqinter> x) \<squnion> (-y \<sqinter> x)"
   proof -
     have "y \<sqinter> y\<^sup>T * x = y \<sqinter> x"
     proof (rule order.antisym)
       have "y \<sqinter> y\<^sup>T * x = (y \<sqinter> y\<^sup>T) * x"
         by (simp add: assms vector_inf_comp)
       also have "(y \<sqinter> y\<^sup>T) * x = y * y\<^sup>T * x"
         by (simp add: assms vector_covector)
       also have "... \<le> x"
         using assms comp_isotone by fastforce
       finally show "y \<sqinter> y\<^sup>T * x \<le> y \<sqinter> x"
         by simp
       have "y \<sqinter> x \<le> y\<^sup>T * x"
         by (simp add: assms vector_restrict_comp_conv)
       thus "y \<sqinter> x \<le> y \<sqinter> y\<^sup>T * x"
         by simp
     qed
     thus ?thesis
       by simp
   qed
   also have "... = x"
   proof -
     have "regular y"
       using assms bijective_regular by blast
     thus ?thesis
       by (metis inf.sup_monoid.add_commute maddux_3_11_pp)
   qed
   finally show ?thesis
     .
 qed
 
 lemma update_inf:
   "u \<le> y \<Longrightarrow> (x[y\<longmapsto>z]) \<sqinter> u = z\<^sup>T \<sqinter> u"
   by (smt comp_inf.mult_right_dist_sup comp_inf.semiring.mult_zero_right inf.left_commute inf.sup_monoid.add_assoc inf_absorb2 p_inf sup_bot_right inf.sup_monoid.add_commute)
 
 lemma update_inf_same:
   "(x[y\<longmapsto>z]) \<sqinter> y = z\<^sup>T \<sqinter> y"
   by (simp add: update_inf)
 
 lemma update_inf_different:
   "u \<le> -y \<Longrightarrow> (x[y\<longmapsto>z]) \<sqinter> u = x \<sqinter> u"
   by (smt inf.right_idem inf.sup_monoid.add_commute inf.sup_relative_same_increasing inf_import_p maddux_3_13 sup.cobounded2 update_inf_same)
 
 end
 
 section \<open>Relation-Algebraic Semantics of Disjoint-Set Forests\<close>
 
 text \<open>
 A disjoint-set forest represents a partition of a set into equivalence classes.
 We take the represented equivalence relation as the semantics of a forest.
 It is obtained by operation \<open>fc\<close> below.
 Additionally, operation \<open>wcc\<close> giving the weakly connected components of a graph will be used for the semantics of the union of two disjoint sets.
 Finally, operation \<open>root\<close> yields the root of a component tree, that is, the representative of a set containing a given element.
 This section defines these operations and derives their properties.
 \<close>
 
 context stone_kleene_relation_algebra
 begin
 
 text \<open>Theorem 5.2\<close>
 
 lemma omit_redundant_points:
   assumes "point p"
   shows "p \<sqinter> x\<^sup>\<star> = (p \<sqinter> 1) \<squnion> (p \<sqinter> x) * (-p \<sqinter> x)\<^sup>\<star>"
 proof (rule order.antisym)
   let ?p = "p \<sqinter> 1"
   have "?p * x * (-p \<sqinter> x)\<^sup>\<star> * ?p \<le> ?p * top * ?p"
     by (metis comp_associative mult_left_isotone mult_right_isotone top.extremum)
   also have "... \<le> ?p"
     by (simp add: assms injective_codomain vector_inf_one_comp)
   finally have "?p * x * (-p \<sqinter> x)\<^sup>\<star> * ?p * x \<le> ?p * x"
     using mult_left_isotone by blast
   hence "?p * x * (-p \<sqinter> x)\<^sup>\<star> * (p \<sqinter> x) \<le> ?p * x"
     by (simp add: assms comp_associative vector_inf_one_comp)
   also have 1: "... \<le> ?p * x * (-p \<sqinter> x)\<^sup>\<star>"
     using mult_right_isotone star.circ_reflexive by fastforce
   finally have "?p * x * (-p \<sqinter> x)\<^sup>\<star> * (p \<sqinter> x) \<squnion> ?p * x * (-p \<sqinter> x)\<^sup>\<star> * (-p \<sqinter> x) \<le> ?p * x * (-p \<sqinter> x)\<^sup>\<star>"
     by (simp add: mult_right_isotone star.circ_plus_same star.left_plus_below_circ mult_assoc)
   hence "?p * x * (-p \<sqinter> x)\<^sup>\<star> * ((p \<squnion> -p) \<sqinter> x) \<le> ?p * x * (-p \<sqinter> x)\<^sup>\<star>"
     by (simp add: comp_inf.mult_right_dist_sup mult_left_dist_sup)
   hence "?p * x * (-p \<sqinter> x)\<^sup>\<star> * x \<le> ?p * x * (-p \<sqinter> x)\<^sup>\<star>"
     by (metis assms bijective_regular inf.absorb2 inf.cobounded1 inf.sup_monoid.add_commute shunting_p)
   hence "?p * x * (-p \<sqinter> x)\<^sup>\<star> * x \<squnion> ?p * x \<le> ?p * x * (-p \<sqinter> x)\<^sup>\<star>"
     using 1 by simp
   hence "?p * (1 \<squnion> x * (-p \<sqinter> x)\<^sup>\<star>) * x \<le> ?p * x * (-p \<sqinter> x)\<^sup>\<star>"
     by (simp add: comp_associative mult_left_dist_sup mult_right_dist_sup)
   also have "... \<le> ?p * (1 \<squnion> x * (-p \<sqinter> x)\<^sup>\<star>)"
     by (simp add: comp_associative mult_right_isotone)
   finally have "?p * x\<^sup>\<star> \<le> ?p * (1 \<squnion> x * (-p \<sqinter> x)\<^sup>\<star>)"
     using star_right_induct by (meson dual_order.trans le_supI mult_left_sub_dist_sup_left mult_sub_right_one)
   also have "... = ?p \<squnion> ?p * x * (-p \<sqinter> x)\<^sup>\<star>"
     by (simp add: comp_associative semiring.distrib_left)
   finally show "p \<sqinter> x\<^sup>\<star> \<le> ?p \<squnion> (p \<sqinter> x) * (-p \<sqinter> x)\<^sup>\<star>"
     by (simp add: assms vector_inf_one_comp)
   show "?p \<squnion> (p \<sqinter> x) * (-p \<sqinter> x)\<^sup>\<star> \<le> p \<sqinter> x\<^sup>\<star>"
     by (metis assms comp_isotone inf.boundedI inf.cobounded1 inf.coboundedI2 inf.sup_monoid.add_commute le_supI star.circ_increasing star.circ_transitive_equal star_isotone star_left_unfold_equal sup.cobounded1 vector_export_comp)
 qed
 
 text \<open>Weakly connected components\<close>
 
 abbreviation "wcc x \<equiv> (x \<squnion> x\<^sup>T)\<^sup>\<star>"
 
 text \<open>Theorem 7.1\<close>
 
 lemma wcc_equivalence:
   "equivalence (wcc x)"
   apply (intro conjI)
   subgoal by (simp add: star.circ_reflexive)
   subgoal by (simp add: star.circ_transitive_equal)
   subgoal by (simp add: conv_dist_sup conv_star_commute sup_commute)
   done
 
 text \<open>Theorem 7.2\<close>
 
 lemma wcc_increasing:
   "x \<le> wcc x"
   by (simp add: star.circ_sub_dist_1)
 
 lemma wcc_isotone:
   "x \<le> y \<Longrightarrow> wcc x \<le> wcc y"
   using conv_isotone star_isotone sup_mono by blast
 
 lemma wcc_idempotent:
   "wcc (wcc x) = wcc x"
   using star_involutive wcc_equivalence by auto
 
 text \<open>Theorem 7.3\<close>
 
 lemma wcc_below_wcc:
   "x \<le> wcc y \<Longrightarrow> wcc x \<le> wcc y"
   using wcc_idempotent wcc_isotone by fastforce
 
 lemma wcc_galois:
   "x \<le> wcc y \<longleftrightarrow> wcc x \<le> wcc y"
   using order_trans star.circ_sub_dist_1 wcc_below_wcc by blast
 
 text \<open>Theorem 7.4\<close>
 
 lemma wcc_bot:
   "wcc bot = 1"
   by (simp add: star.circ_zero)
 
 lemma wcc_one:
   "wcc 1 = 1"
   by (simp add: star_one)
 
 text \<open>Theorem 7.5\<close>
 
 lemma wcc_top:
   "wcc top = top"
   by (simp add: star.circ_top)
 
 text \<open>Theorem 7.6\<close>
 
 lemma wcc_with_loops:
   "wcc x = wcc (x \<squnion> 1)"
   by (metis conv_dist_sup star_decompose_1 star_sup_one sup_commute symmetric_one_closed)
 
 lemma wcc_without_loops:
   "wcc x = wcc (x - 1)"
   by (metis conv_star_commute star_sum reachable_without_loops)
 
 lemma forest_components_wcc:
   "injective x \<Longrightarrow> wcc x = forest_components x"
   by (simp add: cancel_separate_1)
 
 text \<open>Theorem 7.8\<close>
 
 lemma wcc_sup_wcc:
   "wcc (x \<squnion> y) = wcc (x \<squnion> wcc y)"
   by (smt (verit, ccfv_SIG) le_sup_iff order.antisym sup_right_divisibility wcc_below_wcc wcc_increasing)
 
 text \<open>Components of a forest, which is represented using edges directed towards the roots\<close>
 
 abbreviation "fc x \<equiv> x\<^sup>\<star> * x\<^sup>T\<^sup>\<star>"
 
 text \<open>Theorem 3.1\<close>
 
 lemma fc_equivalence:
   "univalent x \<Longrightarrow> equivalence (fc x)"
   apply (intro conjI)
   subgoal by (simp add: reflexive_mult_closed star.circ_reflexive)
   subgoal by (metis cancel_separate_1 order.eq_iff star.circ_transitive_equal)
   subgoal by (simp add: conv_dist_comp conv_star_commute)
   done
 
 text \<open>Theorem 3.2\<close>
 
 lemma fc_increasing:
   "x \<le> fc x"
   by (metis le_supE mult_left_isotone star.circ_back_loop_fixpoint star.circ_increasing)
 
 text \<open>Theorem 3.3\<close>
 
 lemma fc_isotone:
   "x \<le> y \<Longrightarrow> fc x \<le> fc y"
   by (simp add: comp_isotone conv_isotone star_isotone)
 
 text \<open>Theorem 3.4\<close>
 
 lemma fc_idempotent:
   "univalent x \<Longrightarrow> fc (fc x) = fc x"
   by (metis fc_equivalence cancel_separate_1 star.circ_transitive_equal star_involutive)
 
 text \<open>Theorem 3.5\<close>
 
 lemma fc_star:
   "univalent x \<Longrightarrow> (fc x)\<^sup>\<star> = fc x"
   using fc_equivalence fc_idempotent star.circ_transitive_equal by simp
 
 lemma fc_plus:
   "univalent x \<Longrightarrow> (fc x)\<^sup>+ = fc x"
   by (metis fc_star star.circ_decompose_9)
 
 text \<open>Theorem 3.6\<close>
 
 lemma fc_bot:
   "fc bot = 1"
   by (simp add: star.circ_zero)
 
 lemma fc_one:
   "fc 1 = 1"
   by (simp add: star_one)
 
 text \<open>Theorem 3.7\<close>
 
 lemma fc_top:
   "fc top = top"
   by (simp add: star.circ_top)
 
 text \<open>Theorem 7.7\<close>
 
 lemma fc_wcc:
   "univalent x \<Longrightarrow> wcc x = fc x"
   by (simp add: fc_star star_decompose_1)
 
 lemma fc_via_root:
   assumes "total (p\<^sup>\<star> * (p \<sqinter> 1))"
   shows "fc p = p\<^sup>\<star> * (p \<sqinter> 1) * p\<^sup>T\<^sup>\<star>"
 proof (rule order.antisym)
   have "1 \<le> p\<^sup>\<star> * (p \<sqinter> 1) * p\<^sup>T\<^sup>\<star>"
     by (smt assms comp_associative conv_dist_comp conv_star_commute coreflexive_idempotent coreflexive_symmetric inf.cobounded2 total_var)
   hence "fc p \<le> p\<^sup>\<star> * p\<^sup>\<star> * (p \<sqinter> 1) * p\<^sup>T\<^sup>\<star> * p\<^sup>T\<^sup>\<star>"
     by (metis comp_right_one mult_left_isotone mult_right_isotone mult_assoc)
   thus "fc p \<le> p\<^sup>\<star> * (p \<sqinter> 1) * p\<^sup>T\<^sup>\<star>"
     by (simp add: star.circ_transitive_equal mult_assoc)
   show "p\<^sup>\<star> * (p \<sqinter> 1) * p\<^sup>T\<^sup>\<star> \<le> fc p"
     by (metis comp_isotone inf.cobounded2 mult_1_right order.refl)
 qed
 
 text \<open>Theorem 5.1\<close>
 
 lemma update_acyclic_1:
   assumes "acyclic (p - 1)"
     and "point y"
     and "vector w"
     and "w \<le> p\<^sup>\<star> * y"
   shows "acyclic ((p[w\<longmapsto>y]) - 1)"
 proof -
   let ?p = "p[w\<longmapsto>y]"
   have "w * y\<^sup>T \<le> p\<^sup>\<star>"
     using assms(2,4) shunt_bijective by blast
   hence "w * y\<^sup>T \<le> (p - 1)\<^sup>\<star>"
     using reachable_without_loops by auto
   hence "w * y\<^sup>T - 1 \<le> (p - 1)\<^sup>\<star> - 1"
     by (simp add: inf.coboundedI2 inf.sup_monoid.add_commute)
   also have "... \<le> (p - 1)\<^sup>+"
     by (simp add: star_plus_without_loops)
   finally have 1: "w \<sqinter> y\<^sup>T \<sqinter> -1 \<le> (p - 1)\<^sup>+"
     using assms(2,3) vector_covector by auto
   have "?p - 1 = (w \<sqinter> y\<^sup>T \<sqinter> -1) \<squnion> (-w \<sqinter> p \<sqinter> -1)"
     by (simp add: inf_sup_distrib2)
   also have "... \<le> (p - 1)\<^sup>+ \<squnion> (-w \<sqinter> p \<sqinter> -1)"
     using 1 sup_left_isotone by blast
   also have "... \<le> (p - 1)\<^sup>+ \<squnion> (p - 1)"
     using comp_inf.mult_semi_associative sup_right_isotone by auto
   also have "... = (p - 1)\<^sup>+"
     by (metis star.circ_back_loop_fixpoint sup.right_idem)
   finally have "(?p - 1)\<^sup>+ \<le> (p - 1)\<^sup>+"
     by (metis comp_associative comp_isotone star.circ_transitive_equal star.left_plus_circ star_isotone)
   also have "... \<le> -1"
     using assms(1) by blast
   finally show ?thesis
     by simp
 qed
 
 lemma update_acyclic_2:
   assumes "acyclic (p - 1)"
     and "point y"
     and "point x"
     and "y \<le> p\<^sup>T\<^sup>\<star> * x"
     and "univalent p"
     and "p\<^sup>T * y \<le> y"
   shows "acyclic ((p[p\<^sup>T\<^sup>\<star>*x\<longmapsto>y]) - 1)"
 proof -
   have "p\<^sup>T * p\<^sup>\<star> * y = p\<^sup>T * p * p\<^sup>\<star> * y \<squnion> p\<^sup>T * y"
     by (metis comp_associative mult_left_dist_sup star.circ_loop_fixpoint)
   also have "... \<le> p\<^sup>\<star> * y"
     by (metis assms(5,6) comp_right_one le_supI le_supI2 mult_left_isotone star.circ_loop_fixpoint star.circ_transitive_equal)
   finally have "p\<^sup>T\<^sup>\<star> * x \<le> p\<^sup>\<star> * y"
     by (simp add: assms(2-4) bijective_reverse conv_star_commute comp_associative star_left_induct)
   thus ?thesis
     by (simp add: assms(1-3) vector_mult_closed update_acyclic_1)
 qed
 
 lemma update_acyclic_3:
   assumes "acyclic (p - 1)"
     and "point y"
     and "point w"
     and "y \<le> p\<^sup>T\<^sup>\<star> * w"
   shows "acyclic ((p[w\<longmapsto>y]) - 1)"
   by (simp add: assms bijective_reverse conv_star_commute update_acyclic_1)
 
 lemma rectangle_star_rectangle:
   "rectangle a \<Longrightarrow> a * x\<^sup>\<star> * a \<le> a"
   by (metis mult_left_isotone mult_right_isotone top.extremum)
 
 lemma arc_star_arc:
   "arc a \<Longrightarrow> a * x\<^sup>\<star> * a \<le> a"
   using arc_top_arc rectangle_star_rectangle by blast
 
 lemma star_rectangle_decompose:
   assumes "rectangle a"
   shows "(a \<squnion> x)\<^sup>\<star> = x\<^sup>\<star> \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>"
 proof (rule order.antisym)
   have 1: "1 \<le> x\<^sup>\<star> \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>"
     by (simp add: star.circ_reflexive sup.coboundedI1)
   have "(a \<squnion> x) * (x\<^sup>\<star> \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>) = a * x\<^sup>\<star> \<squnion> a * x\<^sup>\<star> * a * x\<^sup>\<star> \<squnion> x\<^sup>+ \<squnion> x\<^sup>+ * a * x\<^sup>\<star>"
     by (metis comp_associative semiring.combine_common_factor semiring.distrib_left sup_commute)
   also have "... = a * x\<^sup>\<star> \<squnion> x\<^sup>+ \<squnion> x\<^sup>+ * a * x\<^sup>\<star>"
     using assms rectangle_star_rectangle by (simp add: mult_left_isotone sup_absorb1)
   also have "... = x\<^sup>+ \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>"
     by (metis comp_associative star.circ_loop_fixpoint sup_assoc sup_commute)
   also have "... \<le> x\<^sup>\<star> \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>"
     using star.left_plus_below_circ sup_left_isotone by auto
   finally show "(a \<squnion> x)\<^sup>\<star> \<le> x\<^sup>\<star> \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>"
     using 1 by (metis comp_right_one le_supI star_left_induct)
 next
   show "x\<^sup>\<star> \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star> \<le> (a \<squnion> x)\<^sup>\<star>"
     by (metis comp_isotone le_supE le_supI star.circ_increasing star.circ_transitive_equal star_isotone sup_ge2)
 qed
 
 lemma star_arc_decompose:
   "arc a \<Longrightarrow> (a \<squnion> x)\<^sup>\<star> = x\<^sup>\<star> \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>"
   using arc_top_arc star_rectangle_decompose by blast
 
 lemma plus_rectangle_decompose:
   assumes "rectangle a"
   shows "(a \<squnion> x)\<^sup>+ = x\<^sup>+ \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>"
 proof -
   have "(a \<squnion> x)\<^sup>+ = (a \<squnion> x) * (x\<^sup>\<star> \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>)"
     by (simp add: assms star_rectangle_decompose)
   also have "... = a * x\<^sup>\<star> \<squnion> a * x\<^sup>\<star> * a * x\<^sup>\<star> \<squnion> x\<^sup>+ \<squnion> x\<^sup>+ * a * x\<^sup>\<star>"
     by (metis comp_associative semiring.combine_common_factor semiring.distrib_left sup_commute)
   also have "... = a * x\<^sup>\<star> \<squnion> x\<^sup>+ \<squnion> x\<^sup>+ * a * x\<^sup>\<star>"
     using assms rectangle_star_rectangle by (simp add: mult_left_isotone sup_absorb1)
   also have "... = x\<^sup>+ \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>"
     by (metis comp_associative star.circ_loop_fixpoint sup_assoc sup_commute)
   finally show ?thesis
     by simp
 qed
 
 text \<open>Theorem 8.1\<close>
 
 lemma plus_arc_decompose:
   "arc a \<Longrightarrow> (a \<squnion> x)\<^sup>+ = x\<^sup>+ \<squnion> x\<^sup>\<star> * a * x\<^sup>\<star>"
   using arc_top_arc plus_rectangle_decompose by blast
 
 text \<open>Theorem 8.2\<close>
 
 lemma update_acyclic_4:
   assumes "acyclic (p - 1)"
     and "point y"
     and "point w"
     and "y \<sqinter> p\<^sup>\<star> * w = bot"
   shows "acyclic ((p[w\<longmapsto>y]) - 1)"
 proof -
   let ?p = "p[w\<longmapsto>y]"
   have "y\<^sup>T * p\<^sup>\<star> * w \<le> -1"
     using assms(4) comp_associative pseudo_complement schroeder_3_p by auto
   hence 1: "p\<^sup>\<star> * w * y\<^sup>T * p\<^sup>\<star> \<le> -1"
     by (metis comp_associative comp_commute_below_diversity star.circ_transitive_equal)
   have "?p - 1 \<le> (w \<sqinter> y\<^sup>T) \<squnion> (p - 1)"
     by (metis comp_inf.mult_right_dist_sup dual_order.trans inf.cobounded1 inf.coboundedI2 inf.sup_monoid.add_assoc le_supI sup.cobounded1 sup_ge2)
   also have "... = w * y\<^sup>T \<squnion> (p - 1)"
     using assms(2,3) by (simp add: vector_covector)
   finally have "(?p - 1)\<^sup>+ \<le> (w * y\<^sup>T \<squnion> (p - 1))\<^sup>+"
     by (simp add: comp_isotone star_isotone)
   also have "... = (p - 1)\<^sup>+ \<squnion> (p - 1)\<^sup>\<star> * w * y\<^sup>T * (p - 1)\<^sup>\<star>"
     using assms(2,3) plus_arc_decompose points_arc by (simp add: comp_associative)
   also have "... \<le> (p - 1)\<^sup>+ \<squnion> p\<^sup>\<star> * w * y\<^sup>T * p\<^sup>\<star>"
     using reachable_without_loops by auto
   also have "... \<le> -1"
     using 1 assms(1) by simp
   finally show ?thesis
     by simp
 qed
 
 text \<open>Theorem 8.3\<close>
 
 lemma update_acyclic_5:
   assumes "acyclic (p - 1)"
     and "point w"
   shows "acyclic ((p[w\<longmapsto>w]) - 1)"
 proof -
   let ?p = "p[w\<longmapsto>w]"
   have "?p - 1 \<le> (w \<sqinter> w\<^sup>T \<sqinter> -1) \<squnion> (p - 1)"
     by (metis comp_inf.mult_right_dist_sup inf.cobounded2 inf.sup_monoid.add_assoc sup_right_isotone)
   also have "... = p - 1"
     using assms(2) by (metis comp_inf.covector_complement_closed equivalence_top_closed inf_top.right_neutral maddux_3_13 pseudo_complement regular_closed_top regular_one_closed vector_covector vector_top_closed)
   finally show ?thesis
     using assms(1) acyclic_down_closed by blast
 qed
 
 text \<open>Root of the tree containing point $x$ in the disjoint-set forest $p$\<close>
 
 abbreviation "roots p \<equiv> (p \<sqinter> 1) * top"
 abbreviation "root p x \<equiv> p\<^sup>T\<^sup>\<star> * x \<sqinter> roots p"
 
 text \<open>Theorem 4.1\<close>
 
 lemma root_var:
   "root p x = (p \<sqinter> 1) * p\<^sup>T\<^sup>\<star> * x"
   by (simp add: coreflexive_comp_top_inf inf_commute mult_assoc)
 
 text \<open>Theorem 4.2\<close>
 
 lemma root_successor_loop:
   "univalent p \<Longrightarrow> root p x = p[[root p x]]"
   by (metis root_var injective_codomain comp_associative conv_dist_inf coreflexive_symmetric equivalence_one_closed inf.cobounded2 univalent_conv_injective)
 
 lemma root_transitive_successor_loop:
   "univalent p \<Longrightarrow> root p x = p\<^sup>T\<^sup>\<star> * (root p x)"
   by (metis mult_1_right star_one star_simulation_right_equal root_successor_loop)
 
 lemma roots_successor_loop:
   "univalent p \<Longrightarrow> p[[roots p]] = roots p"
   by (metis conv_involutive inf_commute injective_codomain one_inf_conv mult_assoc)
 
 lemma roots_transitive_successor_loop:
   "univalent p \<Longrightarrow> p\<^sup>T\<^sup>\<star> * (roots p) = roots p"
   by (metis comp_associative star.circ_left_top star_simulation_right_equal roots_successor_loop)
 
 text \<open>The root of a tree of a node belongs to the same component as the node.\<close>
 
 lemma root_same_component:
   "injective x \<Longrightarrow> root p x * x\<^sup>T \<le> fc p"
   by (metis comp_associative coreflexive_comp_top_inf eq_refl inf.sup_left_divisibility inf.sup_monoid.add_commute mult_isotone star.circ_circ_mult star.circ_right_top star.circ_transitive_equal star_one star_outer_increasing test_preserves_equation top_greatest)
 
 lemma root_vector:
   "vector x \<Longrightarrow> vector (root p x)"
   by (simp add: vector_mult_closed root_var)
 
 lemma root_vector_inf:
   "vector x \<Longrightarrow> root p x * x\<^sup>T = root p x \<sqinter> x\<^sup>T"
   by (simp add: vector_covector root_vector)
 
 lemma root_same_component_vector:
   "injective x \<Longrightarrow> vector x \<Longrightarrow> root p x \<sqinter> x\<^sup>T \<le> fc p"
   using root_same_component root_vector_inf by fastforce
 
 lemma univalent_root_successors:
   assumes "univalent p"
   shows "(p \<sqinter> 1) * p\<^sup>\<star> = p \<sqinter> 1"
 proof (rule order.antisym)
   have "(p \<sqinter> 1) * p \<le> p \<sqinter> 1"
     by (smt assms(1) comp_inf.mult_semi_associative conv_dist_comp conv_dist_inf conv_order equivalence_one_closed inf.absorb1 inf.sup_monoid.add_assoc injective_codomain)
   thus "(p \<sqinter> 1) * p\<^sup>\<star> \<le> p \<sqinter> 1"
     using star_right_induct_mult by blast
   show "p \<sqinter> 1 \<le> (p \<sqinter> 1) * p\<^sup>\<star>"
     by (metis coreflexive_idempotent inf_le1 inf_le2 mult_right_isotone order_trans star.circ_increasing)
 qed
 
 lemma same_component_same_root_sub:
   assumes "univalent p"
     and "bijective y"
     and "x * y\<^sup>T \<le> fc p"
   shows "root p x \<le> root p y"
 proof -
   have "root p x * y\<^sup>T \<le> (p \<sqinter> 1) * p\<^sup>T\<^sup>\<star>"
     by (smt assms(1,3) mult_isotone mult_assoc root_var fc_plus fc_star order.eq_iff univalent_root_successors)
   thus ?thesis
     by (simp add: assms(2) shunt_bijective root_var)
 qed
 
 lemma same_component_same_root:
   assumes "univalent p"
     and "bijective x"
     and "bijective y"
     and "x * y\<^sup>T \<le> fc p"
   shows "root p x = root p y"
 proof (rule order.antisym)
   show "root p x \<le> root p y"
     using assms(1,3,4) same_component_same_root_sub by blast
   have "y * x\<^sup>T \<le> fc p"
     using assms(1,4) fc_equivalence conv_dist_comp conv_isotone by fastforce
   thus "root p y \<le> root p x"
     using assms(1,2) same_component_same_root_sub by blast
 qed
 
 lemma same_roots_sub:
   assumes "univalent q"
     and "p \<sqinter> 1 \<le> q \<sqinter> 1"
     and "fc p \<le> fc q"
   shows "p\<^sup>\<star> * (p \<sqinter> 1) \<le> q\<^sup>\<star> * (q \<sqinter> 1)"
 proof -
   have "p\<^sup>\<star> * (p \<sqinter> 1) \<le> p\<^sup>\<star> * (q \<sqinter> 1)"
     using assms(2) mult_right_isotone by auto
   also have "... \<le> fc p * (q \<sqinter> 1)"
     using mult_left_isotone mult_right_isotone star.circ_reflexive by fastforce
   also have "... \<le> fc q * (q \<sqinter> 1)"
     by (simp add: assms(3) mult_left_isotone)
   also have "... = q\<^sup>\<star> * (q \<sqinter> 1)"
     by (metis assms(1) conv_dist_comp conv_dist_inf conv_star_commute inf_commute one_inf_conv symmetric_one_closed mult_assoc univalent_root_successors)
   finally show ?thesis
     .
 qed
 
 lemma same_roots:
   assumes "univalent p"
     and "univalent q"
     and "p \<sqinter> 1 = q \<sqinter> 1"
     and "fc p = fc q"
   shows "p\<^sup>\<star> * (p \<sqinter> 1) = q\<^sup>\<star> * (q \<sqinter> 1)"
   by (smt assms conv_dist_comp conv_dist_inf conv_involutive conv_star_commute inf_commute one_inf_conv symmetric_one_closed root_var univalent_root_successors)
 
 lemma same_root:
   assumes "univalent p"
     and "univalent q"
     and "p \<sqinter> 1 = q \<sqinter> 1"
     and "fc p = fc q"
   shows "root p x = root q x"
   by (metis assms mult_assoc root_var univalent_root_successors)
 
 lemma loop_root:
   assumes "injective x"
     and "x = p[[x]]"
   shows "x = root p x"
 proof (rule order.antisym)
   have "x \<le> p * x"
     by (metis assms comp_associative comp_right_one conv_order equivalence_one_closed ex231c inf.orderE inf.sup_monoid.add_commute mult_left_isotone mult_right_isotone one_inf_conv)
   hence "x = (p \<sqinter> 1) * x"
     by (simp add: assms(1) inf_absorb2 injective_comp_right_dist_inf)
   thus "x \<le> root p x"
     by (metis assms(2) coreflexive_comp_top_inf inf.boundedI inf.cobounded1 inf.cobounded2 mult_isotone star.circ_increasing)
 next
   show "root p x \<le> x"
     using assms(2) le_infI1 star_left_induct_mult by auto
 qed
 
 lemma one_loop:
   assumes "acyclic (p - 1)"
     and "univalent p"
   shows "(p \<sqinter> 1) * (p\<^sup>T - 1)\<^sup>+ * (p \<sqinter> 1) = bot"
 proof -
   have "p\<^sup>T\<^sup>+ \<sqinter> (p \<sqinter> 1) * top * (p \<sqinter> 1) = (p \<sqinter> 1) * p\<^sup>T\<^sup>+ * (p \<sqinter> 1)"
     by (simp add: test_comp_test_top)
   also have "... \<le> p\<^sup>T\<^sup>\<star> * (p \<sqinter> 1)"
     by (simp add: inf.coboundedI2 mult_left_isotone star.circ_mult_upper_bound star.circ_reflexive star.left_plus_below_circ)
   also have "... = p \<sqinter> 1"
     by (metis assms(2) conv_dist_comp conv_dist_inf conv_star_commute inf_commute one_inf_conv symmetric_one_closed univalent_root_successors)
   also have "... \<le> 1"
     by simp
   finally have "(p \<sqinter> 1) * top * (p \<sqinter> 1) \<le> -(p\<^sup>T\<^sup>+ - 1)"
     using p_antitone p_antitone_iff p_shunting_swap by blast
   hence "(p \<sqinter> 1)\<^sup>T * (p\<^sup>T\<^sup>+ - 1) * (p \<sqinter> 1)\<^sup>T \<le> bot"
     using triple_schroeder_p p_top by blast
   hence "(p \<sqinter> 1) * (p\<^sup>T\<^sup>+ - 1) * (p \<sqinter> 1) = bot"
     by (simp add: coreflexive_symmetric le_bot)
   thus ?thesis
     by (smt assms(1) conv_complement conv_dist_comp conv_dist_inf conv_star_commute inf_absorb1 star.circ_plus_same symmetric_one_closed reachable_without_loops star_plus_without_loops)
 qed
 
 lemma root_root:
   "root p x = root p (root p x)"
   by (smt comp_associative comp_inf.mult_right_sub_dist_sup_right dual_order.eq_iff inf.cobounded1 inf.cobounded2 inf.orderE mult_right_isotone star.circ_loop_fixpoint star.circ_transitive_equal root_var)
 
 lemma loop_root_2:
   assumes "acyclic (p - 1)"
     and "univalent p"
     and "injective x"
     and "x \<le> p\<^sup>T\<^sup>+ * x"
   shows "x = root p x"
 proof (rule order.antisym)
   have 1: "x = x - (-1 * x)"
     by (metis assms(3) comp_injective_below_complement inf.orderE mult_1_left regular_one_closed)
   have "x \<le> (p\<^sup>T - 1)\<^sup>+ * x \<squnion> (p \<sqinter> 1) * x"
     by (metis assms(4) inf_commute mult_right_dist_sup one_inf_conv plus_reachable_without_loops)
   also have "... \<le> -1 * x \<squnion> (p \<sqinter> 1) * x"
     by (metis assms(1) conv_complement conv_dist_inf conv_isotone conv_plus_commute mult_left_isotone semiring.add_right_mono symmetric_one_closed)
   also have "... \<le> -1 * x \<squnion> root p x"
     using comp_isotone inf.coboundedI2 star.circ_reflexive sup_right_isotone by auto
   finally have "x \<le> (-1 * x \<squnion> root p x) - (-1 * x)"
     using 1 inf.boundedI inf.order_iff by blast
   also have "... \<le> root p x"
     using inf.sup_left_divisibility by auto
   finally show 2: "x \<le> root p x"
     .
   have "root p x = (p \<sqinter> 1) * x \<squnion> (p \<sqinter> 1) * (p\<^sup>T - 1)\<^sup>+ * x"
     by (metis comp_associative mult_left_dist_sup star.circ_loop_fixpoint sup_commute reachable_without_loops root_var)
   also have "... \<le> x \<squnion> (p \<sqinter> 1) * (p\<^sup>T - 1)\<^sup>+ * root p x"
     using 2 by (metis coreflexive_comp_top_inf inf.cobounded2 mult_right_isotone semiring.add_mono)
   also have "... = x"
     by (metis assms(1,2) one_loop root_var mult_assoc semiring.mult_not_zero sup_bot_right)
   finally show "root p x \<le> x"
     .
 qed
 
 lemma path_compression_invariant_simplify:
   assumes "point w"
       and "p\<^sup>T\<^sup>+ * w \<le> -w"
       and "w \<noteq> y"
     shows "p[[w]] \<noteq> w"
 proof
   assume "p[[w]] = w"
   hence "w \<le> p\<^sup>T\<^sup>+ * w"
     by (metis comp_isotone eq_refl star.circ_mult_increasing)
   also have "... \<le> -w"
     by (simp add: assms(2))
   finally have "w = bot"
     using inf.orderE by fastforce
   thus False
     using assms(1,3) le_bot by force
 qed
 
 end
 
 context stone_relation_algebra_tarski
 begin
 
 text \<open>Theorem 5.4 \<open>distinct_points\<close> has been moved to theory \<open>Relation_Algebras\<close> in entry \<open>Stone_Relation_Algebras\<close>\<close>
 
 text \<open>Back and von Wright's array independence requirements \cite{BackWright1998}\<close>
 
 text \<open>Theorem 2.2\<close>
 
 lemma put_get_different_vector:
   assumes "vector y" "w \<le> -y"
   shows "(x[y\<longmapsto>z])[[w]] = x[[w]]"
 proof -
   have "(x[y\<longmapsto>z])[[w]] = (y\<^sup>T \<sqinter> z) * w \<squnion> (-y\<^sup>T \<sqinter> x\<^sup>T) * w"
     by (simp add: conv_complement conv_dist_inf conv_dist_sup mult_right_dist_sup)
   also have "... = z * (w \<sqinter> y) \<squnion> x\<^sup>T * (w - y)"
     by (metis assms(1) conv_complement covector_inf_comp_3 inf_commute vector_complement_closed)
   also have "... = z * (w \<sqinter> y) \<squnion> x\<^sup>T * w"
     by (simp add: assms(2) inf.absorb1)
   also have "... = z * bot \<squnion> x\<^sup>T * w"
     by (metis assms(2) comp_inf.semiring.mult_zero_right inf.absorb1 inf.sup_monoid.add_assoc p_inf)
   also have "... = x\<^sup>T * w"
     by simp
   finally show ?thesis
     .
 qed
 
 lemma put_get_different:
   assumes "point y" "point w" "w \<noteq> y"
   shows "(x[y\<longmapsto>z])[[w]] = x[[w]]"
 proof -
   have "w \<sqinter> y = bot"
     using assms distinct_points by simp
   hence "w \<le> -y"
     using pseudo_complement by simp
   thus ?thesis
     by (simp add: assms(1) assms(2) put_get_different_vector)
 qed
 
 text \<open>Theorem 2.4\<close>
 
 lemma put_put_different_vector:
   assumes "vector y" "vector v" "v \<sqinter> y = bot"
   shows "(x[y\<longmapsto>z])[v\<longmapsto>w] = (x[v\<longmapsto>w])[y\<longmapsto>z]"
 proof -
   have "(x[y\<longmapsto>z])[v\<longmapsto>w] = (v \<sqinter> w\<^sup>T) \<squnion> (-v \<sqinter> y \<sqinter> z\<^sup>T) \<squnion> (-v \<sqinter> -y \<sqinter> x)"
     by (simp add: comp_inf.semiring.distrib_left inf_assoc sup_assoc)
   also have "... = (v \<sqinter> w\<^sup>T) \<squnion> (y \<sqinter> z\<^sup>T) \<squnion> (-v \<sqinter> -y \<sqinter> x)"
     by (metis assms(3) inf_commute inf_import_p p_inf selection_closed_id)
   also have "... = (y \<sqinter> z\<^sup>T) \<squnion> (v \<sqinter> w\<^sup>T) \<squnion> (-y \<sqinter> -v \<sqinter> x)"
     by (simp add: inf_commute sup_commute)
   also have "... = (y \<sqinter> z\<^sup>T) \<squnion> (-y \<sqinter> v \<sqinter> w\<^sup>T) \<squnion> (-y \<sqinter> -v \<sqinter> x)"
     using assms distinct_points pseudo_complement inf.absorb2 by simp
   also have "... = (x[v\<longmapsto>w])[y\<longmapsto>z]"
     by (simp add: comp_inf.semiring.distrib_left inf_assoc sup_assoc)
   finally show ?thesis
     .
 qed
 
 lemma put_put_different:
   assumes "point y" "point v" "v \<noteq> y"
   shows "(x[y\<longmapsto>z])[v\<longmapsto>w] = (x[v\<longmapsto>w])[y\<longmapsto>z]"
   using assms distinct_points put_put_different_vector by blast
 
 end
 
 section \<open>Verifying Operations on Disjoint-Set Forests\<close>
 
 text \<open>
 In this section we verify the make-set, find-set and union-sets operations of disjoint-set forests.
 We start by introducing syntax for updating arrays in programs.
 Updating the value at a given array index means updating the whole array.
 \<close>
 
 syntax
   "_rel_update" :: "idt \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b com" ("(2_[_] :=/ _)" [70, 65, 65] 61)
 
 translations
   "x[y] := z" => "(x := (y \<sqinter> z\<^sup>T) \<squnion> (CONST uminus y \<sqinter> x))"
 
 text \<open>
 The finiteness requirement in the following class is used for proving that the operations terminate.
 \<close>
 
 class finite_regular_p_algebra = p_algebra +
   assumes finite_regular: "finite { x . regular x }"
+begin
+
+abbreviation card_down_regular :: "'a \<Rightarrow> nat" ("_\<down>" [100] 100)
+  where "x\<down> \<equiv> card { z . regular z \<and> z \<le> x }"
+
+end
 
 class stone_kleene_relation_algebra_tarski_finite_regular = stone_kleene_relation_algebra_tarski + finite_regular_p_algebra
 begin
 
 subsection \<open>Make-Set\<close>
 
 text \<open>
 We prove two correctness results about make-set.
 The first shows that the forest changes only to the extent of making one node the root of a tree.
 The second result adds that only singleton sets are created.
 \<close>
 
 definition "make_set_postcondition p x p0 \<equiv> x \<sqinter> p = x * x\<^sup>T \<and> -x \<sqinter> p = -x \<sqinter> p0"
 
 theorem make_set:
   "VARS p
   [ point x \<and> p0 = p ]
   p[x] := x
   [ make_set_postcondition p x p0 ]"
   apply vcg_tc_simp
   by (simp add: make_set_postcondition_def inf_sup_distrib1 inf_assoc[THEN sym] vector_covector[THEN sym])
 
 theorem make_set_2:
   "VARS p
   [ point x \<and> p0 = p \<and> p \<le> 1 ]
   p[x] := x
   [ make_set_postcondition p x p0 \<and> p \<le> 1 ]"
 proof vcg_tc
   fix p
   assume 1: "point x \<and> p0 = p \<and> p \<le> 1"
   show "make_set_postcondition (p[x\<longmapsto>x]) x p0 \<and> p[x\<longmapsto>x] \<le> 1"
   proof (rule conjI)
     show "make_set_postcondition (p[x\<longmapsto>x]) x p0"
       using 1 by (simp add: make_set_postcondition_def inf_sup_distrib1 inf_assoc[THEN sym] vector_covector[THEN sym])
     show "p[x\<longmapsto>x] \<le> 1"
       using 1 by (metis coreflexive_sup_closed dual_order.trans inf.cobounded2 vector_covector)
   qed
 qed
 
 text \<open>
 The above total-correctness proof allows us to extract a function, which can be used in other implementations below.
 This is a technique of \cite{Guttmann2018c}.
 \<close>
 
 lemma make_set_exists:
   "point x \<Longrightarrow> \<exists>p' . make_set_postcondition p' x p"
   using tc_extract_function make_set by blast
 
 definition "make_set p x \<equiv> (SOME p' . make_set_postcondition p' x p)"
 
 lemma make_set_function:
   assumes "point x"
     and "p' = make_set p x"
   shows "make_set_postcondition p' x p"
 proof -
   let ?P = "\<lambda>p' . make_set_postcondition p' x p"
   have "?P (SOME z . ?P z)"
     using assms(1) make_set_exists by (meson someI)
   thus ?thesis
     using assms(2) make_set_def by auto
 qed
 
 end
 
 subsection \<open>Find-Set\<close>
 
 text \<open>
 Disjoint-set forests are represented by their parent mapping.
 It is a forest except each root of a component tree points to itself.
 
 We prove that find-set returns the root of the component tree of the given node.
 \<close>
 
 context pd_kleene_allegory
 begin
 
 abbreviation "disjoint_set_forest p \<equiv> mapping p \<and> acyclic (p - 1)"
 
 end
 
 context stone_kleene_relation_algebra_tarski_finite_regular
 begin
 
 definition "find_set_precondition p x \<equiv> disjoint_set_forest p \<and> point x"
 definition "find_set_invariant p x y \<equiv> find_set_precondition p x \<and> point y \<and> y \<le> p\<^sup>T\<^sup>\<star> * x"
 definition "find_set_postcondition p x y \<equiv> point y \<and> y = root p x"
 
 lemma find_set_1:
   "find_set_precondition p x \<Longrightarrow> find_set_invariant p x x"
   apply (unfold find_set_invariant_def)
   using mult_left_isotone star.circ_reflexive find_set_precondition_def by fastforce
 
 lemma find_set_2:
-  "find_set_invariant p x y \<and> y \<noteq> p[[y]] \<Longrightarrow> find_set_invariant p x (p[[y]]) \<and> card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * (p[[y]]) } < card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y }"
+  "find_set_invariant p x y \<and> y \<noteq> p[[y]] \<Longrightarrow> find_set_invariant p x (p[[y]]) \<and> (p\<^sup>T\<^sup>\<star> * (p[[y]]))\<down> < (p\<^sup>T\<^sup>\<star> * y)\<down>"
 proof -
   let ?s = "{ z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y }"
   let ?t = "{ z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * (p[[y]]) }"
   assume 1: "find_set_invariant p x y \<and> y \<noteq> p[[y]]"
   have 2: "point (p[[y]])"
     using 1 read_point find_set_invariant_def find_set_precondition_def by simp
   show "find_set_invariant p x (p[[y]]) \<and> card ?t < card ?s"
   proof (unfold find_set_invariant_def, intro conjI)
     show "find_set_precondition p x"
       using 1 find_set_invariant_def by simp
     show "vector (p[[y]])"
       using 2 by simp
     show "injective (p[[y]])"
       using 2 by simp
     show "surjective (p[[y]])"
       using 2 by simp
     show "p[[y]] \<le> p\<^sup>T\<^sup>\<star> * x"
       using 1 by (metis (opaque_lifting) find_set_invariant_def comp_associative comp_isotone star.circ_increasing star.circ_transitive_equal)
     show "card ?t < card ?s"
     proof -
       have "p[[y]] = (p\<^sup>T \<sqinter> 1) * y \<squnion> (p\<^sup>T - 1) * y"
         by (metis maddux_3_11_pp mult_right_dist_sup regular_one_closed)
       also have "... \<le> ((p[[y]]) \<sqinter> y) \<squnion> (p\<^sup>T - 1) * y"
         by (metis comp_left_subdist_inf mult_1_left semiring.add_right_mono)
       also have "... = (p\<^sup>T - 1) * y"
         using 1 2 find_set_invariant_def distinct_points by auto
       finally have 3: "(p\<^sup>T - 1)\<^sup>\<star> * (p[[y]]) \<le> (p\<^sup>T - 1)\<^sup>+ * y"
         by (simp add: mult_right_isotone star_simulation_right_equal mult_assoc)
       have "p\<^sup>T\<^sup>\<star> * (p[[y]]) \<le> p\<^sup>T\<^sup>\<star> * y"
         by (metis mult_left_isotone star.right_plus_below_circ mult_assoc)
       hence 4: "?t \<subseteq> ?s"
         using order_trans by auto
       have 5: "y \<in> ?s"
         using 1 find_set_invariant_def bijective_regular mult_left_isotone star.circ_reflexive by fastforce
       have 6: "\<not> y \<in> ?t"
       proof
         assume "y \<in> ?t"
         hence "y \<le> (p\<^sup>T - 1)\<^sup>+ * y"
           using 3 by (metis reachable_without_loops mem_Collect_eq order_trans)
         hence "y * y\<^sup>T \<le> (p\<^sup>T - 1)\<^sup>+"
           using 1 find_set_invariant_def shunt_bijective by simp
         also have "... \<le> -1"
           using 1 by (metis (mono_tags, lifting) find_set_invariant_def find_set_precondition_def conv_dist_comp conv_dist_inf conv_isotone conv_star_commute equivalence_one_closed star.circ_plus_same symmetric_complement_closed)
         finally have "y \<le> -y"
           using schroeder_4_p by auto
         thus False
           using 1 by (metis find_set_invariant_def comp_inf.coreflexive_idempotent conv_complement covector_vector_comp inf.absorb1 inf.sup_monoid.add_commute pseudo_complement surjective_conv_total top.extremum vector_top_closed regular_closed_top)
       qed
       show "card ?t < card ?s"
         apply (rule psubset_card_mono)
         subgoal using finite_regular by simp
         subgoal using 4 5 6 by auto
         done
     qed
   qed
 qed
 
 lemma find_set_3:
   "find_set_invariant p x y \<and> y = p[[y]] \<Longrightarrow> find_set_postcondition p x y"
 proof -
   assume 1: "find_set_invariant p x y \<and> y = p[[y]]"
   show "find_set_postcondition p x y"
   proof (unfold find_set_postcondition_def, rule conjI)
     show "point y"
       using 1 find_set_invariant_def by simp
     show "y = root p x"
     proof (rule order.antisym)
       have "y * y\<^sup>T \<le> p"
         using 1 by (metis find_set_invariant_def find_set_precondition_def shunt_bijective shunt_mapping top_right_mult_increasing)
       hence "y * y\<^sup>T \<le> p \<sqinter> 1"
         using 1 find_set_invariant_def le_infI by blast
       hence "y \<le> roots p"
         using 1 by (metis find_set_invariant_def order_lesseq_imp shunt_bijective top_right_mult_increasing mult_assoc)
       thus "y \<le> root p x"
         using 1 find_set_invariant_def by simp
     next
       have 2: "x \<le> p\<^sup>\<star> * y"
         using 1 find_set_invariant_def find_set_precondition_def bijective_reverse conv_star_commute by auto
       have "p\<^sup>T * p\<^sup>\<star> * y = p\<^sup>T * p * p\<^sup>\<star> * y \<squnion> (p[[y]])"
         by (metis comp_associative mult_left_dist_sup star.circ_loop_fixpoint)
       also have "... \<le> p\<^sup>\<star> * y \<squnion> y"
         using 1 by (metis find_set_invariant_def find_set_precondition_def comp_isotone mult_left_sub_dist_sup semiring.add_right_mono star.circ_back_loop_fixpoint star.circ_circ_mult star.circ_top star.circ_transitive_equal star_involutive star_one)
       also have "... = p\<^sup>\<star> * y"
         by (metis star.circ_loop_fixpoint sup.left_idem sup_commute)
       finally have 3: "p\<^sup>T\<^sup>\<star> * x \<le> p\<^sup>\<star> * y"
         using 2 by (simp add: comp_associative star_left_induct)
       have "p * y \<sqinter> roots p = (p \<sqinter> 1) * p * y"
         using comp_associative coreflexive_comp_top_inf inf_commute by auto
       also have "... \<le> p\<^sup>T * p * y"
         by (metis inf.cobounded2 inf.sup_monoid.add_commute mult_left_isotone one_inf_conv)
       also have "... \<le> y"
         using 1 find_set_invariant_def find_set_precondition_def mult_left_isotone by fastforce
       finally have 4: "p * y \<le> y \<squnion> -roots p"
         using 1 by (metis find_set_invariant_def shunting_p bijective_regular)
       have "p * -roots p \<le> -roots p"
         using 1 by (metis find_set_invariant_def find_set_precondition_def conv_complement_sub_leq conv_involutive roots_successor_loop)
       hence "p * y \<squnion> p * -roots p \<le> y \<squnion> -roots p"
         using 4 dual_order.trans le_supI sup_ge2 by blast
       hence "p * (y \<squnion> -roots p) \<le> y \<squnion> -roots p"
         by (simp add: mult_left_dist_sup)
       hence "p\<^sup>\<star> * y \<le> y \<squnion> -roots p"
         by (simp add: star_left_induct)
       hence "p\<^sup>T\<^sup>\<star> * x \<le> y \<squnion> -roots p"
         using 3 dual_order.trans by blast
       thus "root p x \<le> y"
         using 1 by (metis find_set_invariant_def shunting_p bijective_regular)
     qed
   qed
 qed
 
 theorem find_set:
   "VARS y
   [ find_set_precondition p x ]
   y := x;
   WHILE y \<noteq> p[[y]]
     INV { find_set_invariant p x y }
-    VAR { card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y } }
+    VAR { (p\<^sup>T\<^sup>\<star> * y)\<down> }
      DO y := p[[y]]
      OD
   [ find_set_postcondition p x y ]"
   apply vcg_tc_simp
     apply (fact find_set_1)
    apply (fact find_set_2)
   by (fact find_set_3)
 
 lemma find_set_exists:
   "find_set_precondition p x \<Longrightarrow> \<exists>y . find_set_postcondition p x y"
   using tc_extract_function find_set by blast
 
 text \<open>
 The root of a component tree is a point, that is, represents a singleton set of nodes.
 This could be proved from the definitions using Kleene-relation algebraic calculations.
 But they can be avoided because the property directly follows from the postcondition of the previous correctness proof.
 The corresponding algorithm shows how to obtain the root.
 We therefore have an essentially constructive proof of the following result.
 \<close>
 
 text \<open>Theorem 4.3\<close>
 
 lemma root_point:
   "disjoint_set_forest p \<Longrightarrow> point x \<Longrightarrow> point (root p x)"
   using find_set_exists find_set_precondition_def find_set_postcondition_def by simp
 
 definition "find_set p x \<equiv> (SOME y . find_set_postcondition p x y)"
 
 lemma find_set_function:
   assumes "find_set_precondition p x"
     and "y = find_set p x"
   shows "find_set_postcondition p x y"
   by (metis assms find_set_def find_set_exists someI)
 
 subsection \<open>Path Compression\<close>
 
 text \<open>
 The path-compression technique is frequently implemented in recursive implementations of find-set 
 modifying the tree on the way out from recursive calls. Here we implement it using a second while-loop, 
 which iterates over the same path to the root and changes edges to point to the root of the component, 
 which is known after the while-loop in find-set completes. We prove that path compression preserves 
 the equivalence-relational semantics of the disjoint-set forest and also preserves the roots of the 
 component trees. Additionally we prove the exact effect of path compression.
 \<close>
 
 definition "path_compression_precondition p x y \<equiv> disjoint_set_forest p \<and> point x \<and> point y \<and> y = root p x"
 definition "path_compression_invariant p x y p0 w \<equiv> 
   path_compression_precondition p x y \<and> point w \<and> y \<le> p\<^sup>T\<^sup>\<star> * w \<and> 
   (w \<noteq> x \<longrightarrow> p[[x]] = y \<and> y \<noteq> x \<and> p\<^sup>T\<^sup>+ * w \<le> -x) \<and> p \<sqinter> 1 = p0 \<sqinter> 1 \<and> fc p = fc p0 \<and>
   root p w = y \<and> (w \<noteq> y \<longrightarrow> p\<^sup>T\<^sup>+ * w \<le> -w) \<and> p[[w]] = p0[[w]] \<and> p0[p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w\<longmapsto>y] = p \<and>
   disjoint_set_forest p0 \<and> y = root p0 x \<and> w \<le> p0\<^sup>T\<^sup>\<star> * x"
 definition "path_compression_postcondition p x y p0 \<equiv> 
   path_compression_precondition p x y \<and> p \<sqinter> 1 = p0 \<sqinter> 1 \<and> fc p = fc p0 \<and>
   p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y] = p"
 
 text \<open>
 We first consider a variant that achieves the effect as a single update.
 The parents of all nodes reachable from x are simultaneously updated to the root of the component of x.
 \<close>
 
 lemma path_compression_exact:
   assumes "path_compression_precondition p0 x y"
     and "p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y] = p"
   shows "p \<sqinter> 1 = p0 \<sqinter> 1" "fc p = fc p0"
 proof -
   have a1: "disjoint_set_forest p0" and a2: "point x" and a3: "point y" and a4: "y = root p0 x"
     using path_compression_precondition_def assms(1) by auto
   have 1: "regular (p0\<^sup>T\<^sup>\<star> * x)"
     using a1 a2 bijective_regular mapping_regular regular_closed_star regular_conv_closed regular_mult_closed by auto
   have "p \<sqinter> 1 = (p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T \<sqinter> 1) \<squnion> (-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0 \<sqinter> 1)"
     using assms(2) inf_sup_distrib2 by auto
   also have "... = (p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0 \<sqinter> 1) \<squnion> (-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0 \<sqinter> 1)"
   proof -
     have "p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T \<sqinter> 1 = p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0 \<sqinter> 1"
     proof (rule order.antisym)
       have "(p0 \<sqinter> 1) * p0\<^sup>T\<^sup>\<star> * x \<sqinter> 1 \<le> p0"
         by (smt coreflexive_comp_top_inf_one inf.absorb_iff2 inf.cobounded2 inf.sup_monoid.add_assoc root_var)
       hence "p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T \<sqinter> 1 \<le> p0"
         by (metis inf_le1 a4 conv_dist_inf coreflexive_symmetric inf.absorb2 inf.cobounded2 inf.sup_monoid.add_assoc root_var symmetric_one_closed)
       thus "p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T \<sqinter> 1 \<le> p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0 \<sqinter> 1"
         by (meson inf.le_sup_iff order.refl)
       have "p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0 \<sqinter> 1 \<le> y"
         by (metis a4 coreflexive_comp_top_inf_one inf.cobounded1 inf_assoc inf_le2)
       thus "p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0 \<sqinter> 1 \<le> p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T \<sqinter> 1"
         by (smt conv_dist_inf coreflexive_symmetric inf.absorb_iff2 inf.cobounded2 inf.sup_monoid.add_assoc)
     qed
     thus ?thesis
       by simp
   qed
   also have "... = p0 \<sqinter> 1"
     using 1 by (metis inf.sup_monoid.add_commute inf_sup_distrib1 maddux_3_11_pp)
   finally show "p \<sqinter> 1 = p0 \<sqinter> 1"
     .
   show  "fc p = fc p0"
   proof (rule order.antisym)
     have 2: "univalent (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])"
       by (simp add: a1 a2 a3 update_univalent mult_assoc)
     have 3: "-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0 \<le> (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>\<star> * (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>T\<^sup>\<star>"
       using fc_increasing inf.order_trans sup.cobounded2 by blast
     have "p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0 \<le> (p0\<^sup>T\<^sup>\<star> \<sqinter> p0 * x\<^sup>T) * (x \<sqinter> p0\<^sup>\<star> * p0)"
       by (metis conv_involutive conv_star_commute dedekind)
     also have "... \<le> p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0 * x\<^sup>T * p0\<^sup>\<star> * p0"
       by (metis comp_associative inf.boundedI inf.cobounded2 inf_le1 mult_isotone)
     also have "... \<le> p0\<^sup>T\<^sup>\<star> * x \<sqinter> top * x\<^sup>T * p0\<^sup>\<star>"
       using comp_associative comp_inf.mult_right_isotone mult_isotone star.right_plus_below_circ by auto
     also have "... = p0\<^sup>T\<^sup>\<star> * x * x\<^sup>T * p0\<^sup>\<star>"
       by (metis a2 symmetric_top_closed vector_covector vector_inf_comp vector_mult_closed)
     also have "... \<le> (p0\<^sup>T\<^sup>\<star> * x * y\<^sup>T) * (y * x\<^sup>T * p0\<^sup>\<star>)"
       by (metis a3 order.antisym comp_inf.top_right_mult_increasing conv_involutive dedekind_1 inf.sup_left_divisibility inf.sup_monoid.add_commute mult_right_isotone surjective_conv_total mult_assoc)
     also have "... = (p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T) * (y \<sqinter> x\<^sup>T * p0\<^sup>\<star>)"
       by (metis a2 a3 vector_covector vector_inf_comp vector_mult_closed)
     also have "... = (p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T) * (p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T)\<^sup>T"
       by (simp add: conv_dist_comp conv_dist_inf conv_star_commute inf_commute)
     also have "... \<le> (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>\<star> * (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>T\<^sup>\<star>"
       by (meson conv_isotone dual_order.trans mult_isotone star.circ_increasing sup.cobounded1)
     finally have "p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0 \<le> (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>\<star> * (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>T\<^sup>\<star>"
       .
     hence "(p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0) \<squnion> (-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0) \<le> (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>\<star> * (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>T\<^sup>\<star>"
       using 3 le_supI by blast
     hence "p0 \<le> (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>\<star> * (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>T\<^sup>\<star>"
       using 1 by (metis inf_commute maddux_3_11_pp)
     hence "fc p0 \<le> (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>\<star> * (p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>y])\<^sup>T\<^sup>\<star>"
       using 2 fc_idempotent fc_isotone by fastforce
     thus "fc p0 \<le> fc p"
       by (simp add: assms(2))
     have "((p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T) \<squnion> (-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0))\<^sup>\<star> = (-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0)\<^sup>\<star> * ((p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T) \<squnion> 1)"
     proof (rule star_sup_2)
       have 4: "transitive (p0\<^sup>T\<^sup>\<star> * x)"
         using a2 comp_associative mult_right_isotone rectangle_star_rectangle by auto
       have "transitive (y\<^sup>T)"
         by (metis a3 conv_dist_comp inf.eq_refl mult_assoc)
       thus "transitive (p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T)"
         using 4 transitive_inf_closed by auto
       have 5: "p0\<^sup>T\<^sup>\<star> * x * (-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0) \<le> p0\<^sup>T\<^sup>\<star> * x"
         by (metis a2 mult_right_isotone top_greatest mult_assoc)
       have "(-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0)\<^sup>T * y \<le> p0\<^sup>T * y"
         by (simp add: conv_dist_inf mult_left_isotone)
       also have "... \<le> y"
         using a1 a4 root_successor_loop by auto
       finally have "y\<^sup>T * (-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0) \<le> y\<^sup>T"
         using conv_dist_comp conv_isotone by fastforce
       thus "(p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T) * (-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0) \<le> p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T"
         using 5 comp_left_subdist_inf inf_mono order_trans by blast
     qed
     hence "p\<^sup>\<star> = (-(p0\<^sup>T\<^sup>\<star> * x) \<sqinter> p0)\<^sup>\<star> * ((p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T) \<squnion> 1)"
       by (simp add: assms(2))
     also have "... \<le> p0\<^sup>\<star> * ((p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<^sup>T) \<squnion> 1)"
       by (simp add: mult_left_isotone star_isotone)
     also have "... = p0\<^sup>\<star> * (p0\<^sup>T\<^sup>\<star> * x * y\<^sup>T \<squnion> 1)"
       by (simp add: a2 a3 vector_covector vector_mult_closed)
     also have "... = p0\<^sup>\<star> * (p0\<^sup>T\<^sup>\<star> * (x * x\<^sup>T) * p0\<^sup>\<star> * (p0 \<sqinter> 1) \<squnion> 1)"
       by (metis a4 coreflexive_symmetric inf.cobounded2 root_var comp_associative conv_dist_comp conv_involutive conv_star_commute)
     also have "... \<le> p0\<^sup>\<star> * (p0\<^sup>T\<^sup>\<star> * 1 * p0\<^sup>\<star> * (p0 \<sqinter> 1) \<squnion> 1)"
       by (metis a2 mult_left_isotone mult_right_isotone semiring.add_left_mono sup_commute)
     also have "... = p0\<^sup>\<star> * (p0\<^sup>T\<^sup>\<star> * (p0 \<sqinter> 1) \<squnion> p0\<^sup>\<star> * (p0 \<sqinter> 1) \<squnion> 1)"
       by (simp add: a1 cancel_separate_eq mult_right_dist_sup)
     also have "... = p0\<^sup>\<star> * ((p0 \<sqinter> 1) \<squnion> p0\<^sup>\<star> * (p0 \<sqinter> 1) \<squnion> 1)"
       by (smt univalent_root_successors a1 conv_dist_comp conv_dist_inf coreflexive_idempotent coreflexive_symmetric inf.cobounded2 injective_codomain loop_root root_transitive_successor_loop symmetric_one_closed)
     also have "... = p0\<^sup>\<star> * (p0\<^sup>\<star> * (p0 \<sqinter> 1) \<squnion> 1)"
       by (metis inf.sup_left_divisibility inf_commute sup.left_idem sup_commute sup_relative_same_increasing)
     also have "... \<le> p0\<^sup>\<star> * p0\<^sup>\<star>"
       by (metis inf.cobounded2 inf_commute order.refl order_lesseq_imp star.circ_mult_upper_bound star.circ_reflexive star.circ_transitive_equal sup.boundedI sup_monoid.add_commute)
     also have "... = p0\<^sup>\<star>"
       by (simp add: star.circ_transitive_equal)
     finally show "fc p \<le> fc p0"
       by (metis conv_order conv_star_commute mult_isotone)
   qed
 qed
 
 lemma update_acyclic_6:
   assumes "disjoint_set_forest p"
     and "point x"
   shows "acyclic ((p[p\<^sup>T\<^sup>\<star>*x\<longmapsto>root p x]) - 1)"
   using assms root_point root_successor_loop update_acyclic_2 by auto
 
 theorem path_compression_assign:
   "VARS p
   [ path_compression_precondition p x y \<and> p0 = p ]
   p[p\<^sup>T\<^sup>\<star> * x] := y
   [ path_compression_postcondition p x y p0 ]"
   apply vcg_tc_simp
   apply (unfold path_compression_precondition_def path_compression_postcondition_def)
   apply (intro conjI)
   subgoal using update_univalent mult_assoc by auto
   subgoal using bijective_regular mapping_regular regular_closed_star regular_conv_closed regular_mult_closed update_mapping mult_assoc by auto
   subgoal using update_acyclic_6 by blast
   subgoal by blast
   subgoal by blast
   subgoal by blast
   subgoal by blast
   subgoal by blast
   subgoal by blast
   subgoal by (smt same_root path_compression_exact path_compression_precondition_def update_univalent vector_mult_closed)
   subgoal using path_compression_exact(1) path_compression_precondition_def by blast
   subgoal using path_compression_exact(2) path_compression_precondition_def by blast
   by blast
 
 text \<open>
 We next look at implementing these updates using a loop.
 \<close>
 
 lemma path_compression_1a:
   assumes "point x"
     and "disjoint_set_forest p"
     and "x \<noteq> root p x"
   shows "p\<^sup>T\<^sup>+ * x \<le> - x"
   by (meson assms bijective_regular mapping_regular regular_closed_star regular_conv_closed regular_mult_closed vector_mult_closed point_in_vector_or_complement_2 loop_root_2)
 
 lemma path_compression_1b:
   "x \<le> p\<^sup>T\<^sup>\<star> * x"
   using mult_left_isotone star.circ_reflexive by fastforce
 
 lemma path_compression_1:
   "path_compression_precondition p x y \<Longrightarrow> path_compression_invariant p x y p x"
   using path_compression_invariant_def path_compression_precondition_def loop_root path_compression_1a path_compression_1b by auto
 
 lemma path_compression_2:
-  "path_compression_invariant p x y p0 w \<and> y \<noteq> p[[w]] \<Longrightarrow> path_compression_invariant (p[w\<longmapsto>y]) x y p0 (p[[w]]) \<and> card { z . regular z \<and> z \<le> (p[w\<longmapsto>y])\<^sup>T\<^sup>\<star> * (p[[w]]) } < card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * w }"
+  "path_compression_invariant p x y p0 w \<and> y \<noteq> p[[w]] \<Longrightarrow> path_compression_invariant (p[w\<longmapsto>y]) x y p0 (p[[w]]) \<and> ((p[w\<longmapsto>y])\<^sup>T\<^sup>\<star> * (p[[w]]))\<down> < (p\<^sup>T\<^sup>\<star> * w)\<down>"
 proof -
   let ?p = "p[w\<longmapsto>y]"
   let ?s = "{ z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * w }"
   let ?t = "{ z . regular z \<and> z \<le> ?p\<^sup>T\<^sup>\<star> * (p[[w]]) }"
   assume 1: "path_compression_invariant p x y p0 w \<and> y \<noteq> p[[w]]"
   have i1: "disjoint_set_forest p" and i2: "point x" and i3: "point y" and i4: "y = root p x"
     using 1 path_compression_invariant_def path_compression_precondition_def by meson+
   have i5: "point w" and i6: "y \<le> p\<^sup>T\<^sup>\<star> * w"
     and i7: "w \<noteq> x \<longrightarrow> p[[x]] = y \<and> y \<noteq> x \<and> p\<^sup>T\<^sup>+ * w \<le> -x" and i8: "p \<sqinter> 1 = p0 \<sqinter> 1" and i9: "fc p = fc p0"
     and i10: "root p w = y" and i11: "p[[w]] = p0[[w]]" and i12: "p0[p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w\<longmapsto>y] = p"
     using 1 path_compression_invariant_def by blast+
   have i13: "disjoint_set_forest p0" and i14: "y = root p0 x" and i15: "w \<le> p0\<^sup>T\<^sup>\<star> * x"
     using 1 path_compression_invariant_def by auto
   have 2: "point (p[[w]])"
     using i1 i5 read_point by blast
   show "path_compression_invariant ?p x y p0 (p[[w]]) \<and> card ?t < card ?s"
   proof (unfold path_compression_invariant_def, intro conjI)
     have 3: "mapping ?p"
       by (simp add: i1 i3 i5 bijective_regular update_total update_univalent)
     have 4: "w \<noteq> y"
       using 1 i1 i4 root_successor_loop by blast
     hence 5: "w \<sqinter> y = bot"
       by (simp add: i3 i5 distinct_points)
     hence "y * w\<^sup>T \<le> -1"
       using pseudo_complement schroeder_4_p by auto
     hence "y * w\<^sup>T \<le> p\<^sup>T\<^sup>\<star> - 1"
       using i5 i6 shunt_bijective by auto
     also have "... \<le> p\<^sup>T\<^sup>+"
       by (simp add: star_plus_without_loops)
     finally have 6: "y \<le> p\<^sup>T\<^sup>+ * w"
       using i5 shunt_bijective by auto
     have 7: "w * w\<^sup>T \<le> -p\<^sup>T\<^sup>+"
     proof (rule ccontr)
       assume "\<not> w * w\<^sup>T \<le> -p\<^sup>T\<^sup>+"
       hence "w * w\<^sup>T \<le> --p\<^sup>T\<^sup>+"
         using i5 point_arc arc_in_partition by blast
       hence "w * w\<^sup>T \<le> p\<^sup>T\<^sup>+ \<sqinter> 1"
         using i1 i5 mapping_regular regular_conv_closed regular_closed_star regular_mult_closed by simp
       also have "... = ((p\<^sup>T \<sqinter> 1) * p\<^sup>T\<^sup>\<star> \<sqinter> 1) \<squnion> ((p\<^sup>T - 1) * p\<^sup>T\<^sup>\<star> \<sqinter> 1)"
         by (metis comp_inf.mult_right_dist_sup maddux_3_11_pp mult_right_dist_sup regular_one_closed)
       also have "... = ((p\<^sup>T \<sqinter> 1) * p\<^sup>T\<^sup>\<star> \<sqinter> 1) \<squnion> ((p - 1)\<^sup>+ \<sqinter> 1)\<^sup>T"
         by (metis conv_complement conv_dist_inf conv_plus_commute equivalence_one_closed reachable_without_loops)
       also have "... \<le> ((p\<^sup>T \<sqinter> 1) * p\<^sup>T\<^sup>\<star> \<sqinter> 1) \<squnion> (-1 \<sqinter> 1)\<^sup>T"
         by (metis (no_types, opaque_lifting) i1 sup_right_isotone inf.sup_left_isotone conv_isotone)
       also have "... = (p\<^sup>T \<sqinter> 1) * p\<^sup>T\<^sup>\<star> \<sqinter> 1"
         by simp
       also have "... \<le> (p\<^sup>T \<sqinter> 1) * top \<sqinter> 1"
         by (metis comp_inf.comp_isotone coreflexive_comp_top_inf equivalence_one_closed inf.cobounded1 inf.cobounded2)
       also have "... \<le> p\<^sup>T"
         by (simp add: coreflexive_comp_top_inf_one)
       finally have "w * w\<^sup>T \<le> p\<^sup>T"
         by simp
       hence "w \<le> p[[w]]"
         using i5 shunt_bijective by blast
       hence "w = p[[w]]"
         using 2 by (metis i5 epm_3 mult_semi_associative)
       thus False
         using 2 4 i10 loop_root by auto
     qed
     hence 8: "w \<sqinter> p\<^sup>T\<^sup>+ * w = bot"
       using p_antitone_iff pseudo_complement schroeder_4_p by blast
     show "y \<le> ?p\<^sup>T\<^sup>\<star> * (p[[w]])"
     proof -
       have "(w \<sqinter> y\<^sup>T)\<^sup>T * (-w \<sqinter> p)\<^sup>T\<^sup>\<star> * p\<^sup>T * w \<le> w\<^sup>T * (-w \<sqinter> p)\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (simp add: conv_isotone mult_left_isotone)
       also have "... \<le> w\<^sup>T * p\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (simp add: conv_isotone mult_left_isotone star_isotone mult_right_isotone)
       also have "... = w\<^sup>T * p\<^sup>T\<^sup>+ * w"
         by (simp add: star_plus mult_assoc)
       also have "... = bot"
         using 8 by (smt i5 covector_inf_comp_3 mult_assoc conv_dist_comp conv_star_commute covector_bot_closed equivalence_top_closed inf.le_iff_sup mult_left_isotone)
       finally have "((w \<sqinter> y\<^sup>T)\<^sup>T \<squnion> (-w \<sqinter> p)\<^sup>T) * (-w \<sqinter> p)\<^sup>T\<^sup>\<star> * p\<^sup>T * w \<le> (-w \<sqinter> p)\<^sup>T * (-w \<sqinter> p)\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (simp add: bot_unique mult_right_dist_sup)
       also have "... \<le> (-w \<sqinter> p)\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (simp add: mult_left_isotone star.left_plus_below_circ)
       finally have "?p\<^sup>T * (-w \<sqinter> p)\<^sup>T\<^sup>\<star> * p\<^sup>T * w \<le> (-w \<sqinter> p)\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (simp add: conv_dist_sup)
       hence "?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w \<le> (-w \<sqinter> p)\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (metis comp_associative star.circ_loop_fixpoint star_left_induct sup_commute sup_least sup_left_divisibility)
       hence "w \<sqinter> ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w \<le> w \<sqinter> (-w \<sqinter> p)\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         using inf.sup_right_isotone by blast
       also have "... \<le> w \<sqinter> p\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         using conv_isotone mult_left_isotone star_isotone inf.sup_right_isotone by simp
       also have "... = bot"
         using 8 by (simp add: star_plus)
       finally have 9: "w\<^sup>T * ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w = bot"
         by (smt i5 covector_inf_comp_3 mult_assoc conv_dist_comp covector_bot_closed equivalence_top_closed inf.le_iff_sup mult_left_isotone bot_least inf.absorb1)
       have "p\<^sup>T * ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w = ((w \<sqinter> p)\<^sup>T \<squnion> (-w \<sqinter> p)\<^sup>T) * ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (metis i5 bijective_regular conv_dist_sup inf.sup_monoid.add_commute maddux_3_11_pp)
       also have "... = (w \<sqinter> p)\<^sup>T * ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w \<squnion> (-w \<sqinter> p)\<^sup>T * ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (simp add: mult_right_dist_sup)
       also have "... \<le> w\<^sup>T * ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w \<squnion> (-w \<sqinter> p)\<^sup>T * ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         using semiring.add_right_mono comp_isotone conv_isotone by auto
       also have "... = (-w \<sqinter> p)\<^sup>T * ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         using 9 by simp
       also have "... \<le> ?p\<^sup>T\<^sup>+ * p\<^sup>T * w"
         by (simp add: conv_isotone mult_left_isotone)
       also have "... \<le> ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (simp add: comp_isotone star.left_plus_below_circ)
       finally have "p\<^sup>T\<^sup>\<star> * p\<^sup>T * w \<le> ?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (metis comp_associative star.circ_loop_fixpoint star_left_induct sup_commute sup_least sup_left_divisibility)
       thus "y \<le> ?p\<^sup>T\<^sup>\<star> * (p[[w]])"
         using 6 by (simp add: star_simulation_right_equal mult_assoc)
     qed
     have 10: "acyclic (?p - 1)"
       using i1 i10 i3 i5 inf_le1 update_acyclic_3 by blast
     have "?p[[p\<^sup>T\<^sup>+ * w]] \<le> p\<^sup>T\<^sup>+ * w"
     proof -
       have "(w\<^sup>T \<sqinter> y) * p\<^sup>T\<^sup>+ * w = y \<sqinter> w\<^sup>T * p\<^sup>T\<^sup>+ * w"
         by (metis i3 inf_vector_comp vector_inf_comp)
       hence "?p[[p\<^sup>T\<^sup>+ * w]] = (y \<sqinter> w\<^sup>T * p\<^sup>T\<^sup>+ * w) \<squnion> (-w\<^sup>T \<sqinter> p\<^sup>T) * p\<^sup>T\<^sup>+ * w"
         by (simp add: comp_associative conv_complement conv_dist_inf conv_dist_sup mult_right_dist_sup)
       also have "... \<le> y \<squnion> (-w\<^sup>T \<sqinter> p\<^sup>T) * p\<^sup>T\<^sup>+ * w"
         using sup_left_isotone by auto
       also have "... \<le> y \<squnion> p\<^sup>T * p\<^sup>T\<^sup>+ * w"
         using mult_left_isotone sup_right_isotone by auto
       also have "... \<le> y \<squnion> p\<^sup>T\<^sup>+ * w"
         using semiring.add_left_mono mult_left_isotone mult_right_isotone star.left_plus_below_circ by auto
       also have "... = p\<^sup>T\<^sup>+ * w"
         using 6 by (simp add: sup_absorb2)
       finally show ?thesis
         by simp
     qed
     hence 11: "?p\<^sup>T\<^sup>\<star> * (p[[w]]) \<le> p\<^sup>T\<^sup>+ * w"
       using star_left_induct by (simp add: mult_left_isotone star.circ_mult_increasing)
     hence 12: "?p\<^sup>T\<^sup>+ * (p[[w]]) \<le> p\<^sup>T\<^sup>+ * w"
       using dual_order.trans mult_left_isotone star.left_plus_below_circ by blast
     have 13: "?p[[x]] = y \<and> y \<noteq> x \<and> ?p\<^sup>T\<^sup>+ * (p[[w]]) \<le> -x"
     proof (cases "w = x")
       case True
       hence "?p[[x]] = (w\<^sup>T \<sqinter> y) * w \<squnion> (-w\<^sup>T \<sqinter> p\<^sup>T) * w"
         by (simp add: conv_complement conv_dist_inf conv_dist_sup mult_right_dist_sup)
       also have "... = (w\<^sup>T \<sqinter> y) * w \<squnion> p\<^sup>T * (-w \<sqinter> w)"
         by (metis i5 conv_complement covector_inf_comp_3 inf.sup_monoid.add_commute vector_complement_closed)
       also have "... = (w\<^sup>T \<sqinter> y) * w"
         by simp
       also have "... = y * w"
         by (simp add: i5 covector_inf_comp_3 inf.sup_monoid.add_commute)
       also have "... = y"
         by (metis i3 i5 comp_associative)
       finally show ?thesis
         using 4 8 12 True pseudo_complement inf.sup_monoid.add_commute order.trans by blast
     next
       case False
       have "?p[[x]] = (w\<^sup>T \<sqinter> y) * x \<squnion> (-w\<^sup>T \<sqinter> p\<^sup>T) * x"
         by (simp add: conv_complement conv_dist_inf conv_dist_sup mult_right_dist_sup)
       also have "... = y * (w \<sqinter> x) \<squnion> p\<^sup>T * (-w \<sqinter> x)"
         by (metis i5 conv_complement covector_inf_comp_3 inf_commute vector_complement_closed)
       also have "... = p\<^sup>T * (-w \<sqinter> x)"
         using i2 i5 False distinct_points by auto
       also have "... = y"
         using i2 i5 i7 False distinct_points inf.absorb2 pseudo_complement by auto
       finally show ?thesis
         using 12 False i7 dual_order.trans by blast
     qed
     thus "p[[w]] \<noteq> x \<longrightarrow> ?p[[x]] = y \<and> y \<noteq> x \<and> ?p\<^sup>T\<^sup>+ * (p[[w]]) \<le> -x"
       by simp
     have 14: "?p\<^sup>T\<^sup>\<star> * x = x \<squnion> y"
     proof (rule order.antisym)
       have "?p\<^sup>T * (x \<squnion> y) = y \<squnion> ?p\<^sup>T * y"
         using 13 by (simp add: mult_left_dist_sup)
       also have "... = y \<squnion> (w\<^sup>T \<sqinter> y) * y \<squnion> (-w\<^sup>T \<sqinter> p\<^sup>T) * y"
         by (simp add: conv_complement conv_dist_inf conv_dist_sup mult_right_dist_sup sup_assoc)
       also have "... \<le> y \<squnion> (w\<^sup>T \<sqinter> y) * y \<squnion> p\<^sup>T * y"
         using mult_left_isotone sup_right_isotone by auto
       also have "... = y \<squnion> (w\<^sup>T \<sqinter> y) * y"
         using i1 i10 root_successor_loop sup_commute by auto
       also have "... \<le> y \<squnion> y * y"
         using mult_left_isotone sup_right_isotone by auto
       also have "... = y"
         by (metis i3 comp_associative sup.idem)
       also have "... \<le> x \<squnion> y"
         by simp
       finally show "?p\<^sup>T\<^sup>\<star> * x \<le> x \<squnion> y"
         by (simp add: star_left_induct)
     next
       show "x \<squnion> y \<le> ?p\<^sup>T\<^sup>\<star> * x"
         using 13 by (metis mult_left_isotone star.circ_increasing star.circ_loop_fixpoint sup.boundedI sup_ge2)
     qed
     have 15: "y = root ?p x"
     proof -
       have "(p \<sqinter> 1) * y = (p \<sqinter> 1) * (p \<sqinter> 1) * p\<^sup>T\<^sup>\<star> * x"
         by (simp add: i4 comp_associative root_var)
       also have "... = (p \<sqinter> 1) * p\<^sup>T\<^sup>\<star> * x"
         using coreflexive_idempotent by auto
       finally have 16: "(p \<sqinter> 1) * y = y"
         by (simp add: i4 root_var)
       have 17: "(p \<sqinter> 1) * x \<le> y"
         by (metis (no_types, lifting) i4 comp_right_one mult_left_isotone mult_right_isotone star.circ_reflexive root_var)
       have "root ?p x = (?p \<sqinter> 1) * (x \<squnion> y)"
         using 14 by (metis mult_assoc root_var)
       also have "... = (w \<sqinter> y\<^sup>T \<sqinter> 1) * (x \<squnion> y) \<squnion> (-w \<sqinter> p \<sqinter> 1) * (x \<squnion> y)"
         by (simp add: inf_sup_distrib2 semiring.distrib_right)
       also have "... = (w \<sqinter> 1 \<sqinter> y\<^sup>T) * (x \<squnion> y) \<squnion> (-w \<sqinter> p \<sqinter> 1) * (x \<squnion> y)"
         by (simp add: inf.left_commute inf.sup_monoid.add_commute)
       also have "... = (w \<sqinter> 1) * (y \<sqinter> (x \<squnion> y)) \<squnion> (-w \<sqinter> p \<sqinter> 1) * (x \<squnion> y)"
         by (simp add: i3 covector_inf_comp_3)
       also have "... = (w \<sqinter> 1) * y \<squnion> (-w \<sqinter> p \<sqinter> 1) * (x \<squnion> y)"
         by (simp add: inf.absorb1)
       also have "... = (w \<sqinter> 1 * y) \<squnion> (-w \<sqinter> (p \<sqinter> 1) * (x \<squnion> y))"
         by (simp add: i5 inf_assoc vector_complement_closed vector_inf_comp)
       also have "... = (w \<sqinter> y) \<squnion> (-w \<sqinter> ((p \<sqinter> 1) * x \<squnion> y))"
         using 16 by (simp add: mult_left_dist_sup)
       also have "... = (w \<sqinter> y) \<squnion> (-w \<sqinter> y)"
         using 17 by (simp add: sup.absorb2)
       also have "... = y"
         using 5 inf.sup_monoid.add_commute le_iff_inf pseudo_complement sup_monoid.add_0_left by fastforce
       finally show ?thesis
         by simp
     qed
     show "path_compression_precondition ?p x y"
       using 3 10 15 i2 i3 path_compression_precondition_def by blast
     show "vector (p[[w]])"
       using 2 by simp
     show "injective (p[[w]])"
       using 2 by simp
     show "surjective (p[[w]])"
       using 2 by simp
     have "w \<sqinter> p \<sqinter> 1 \<le> w \<sqinter> w\<^sup>T \<sqinter> p"
       by (metis inf.boundedE inf.boundedI inf.cobounded1 inf.cobounded2 one_inf_conv)
     also have "... = w * w\<^sup>T \<sqinter> p"
       by (simp add: i5 vector_covector)
     also have "... \<le> -p\<^sup>T\<^sup>+ \<sqinter> p"
       using 7 by (simp add: inf.coboundedI2 inf.sup_monoid.add_commute)
     finally have "w \<sqinter> p \<sqinter> 1 = bot"
       by (metis (no_types, opaque_lifting) conv_dist_inf coreflexive_symmetric inf.absorb1 inf.boundedE inf.cobounded2 pseudo_complement star.circ_mult_increasing)
     also have "w \<sqinter> y\<^sup>T \<sqinter> 1 = bot"
       using 5 antisymmetric_bot_closed asymmetric_bot_closed comp_inf.schroeder_2 inf.absorb1 one_inf_conv by fastforce
     finally have "w \<sqinter> p \<sqinter> 1 = w \<sqinter> y\<^sup>T \<sqinter> 1"
       by simp
     thus 18: "?p \<sqinter> 1 = p0 \<sqinter> 1"
       by (metis i5 i8 bijective_regular inf.sup_monoid.add_commute inf_sup_distrib2 maddux_3_11_pp)
     show 19: "fc ?p = fc p0"
     proof -
       have "p[[w]] = p\<^sup>T * (w \<sqinter> p\<^sup>\<star> * y)"
         by (metis i3 i5 i6 bijective_reverse conv_star_commute inf.absorb1)
       also have "... = p\<^sup>T * (w \<sqinter> p\<^sup>\<star>) * y"
         by (simp add: i5 vector_inf_comp mult_assoc)
       also have "... = p\<^sup>T * ((w \<sqinter> 1) \<squnion> (w \<sqinter> p) * (-w \<sqinter> p)\<^sup>\<star>) * y"
         by (simp add: i5 omit_redundant_points)
       also have "... = p\<^sup>T * (w \<sqinter> 1) * y \<squnion> p\<^sup>T * (w \<sqinter> p) * (-w \<sqinter> p)\<^sup>\<star> * y"
         by (simp add: comp_associative mult_left_dist_sup mult_right_dist_sup)
       also have "... \<le> p\<^sup>T * y \<squnion> p\<^sup>T * (w \<sqinter> p) * (-w \<sqinter> p)\<^sup>\<star> * y"
         by (metis semiring.add_right_mono comp_isotone order.eq_iff inf.cobounded1 inf.sup_monoid.add_commute mult_1_right)
       also have "... = y \<squnion> p\<^sup>T * (w \<sqinter> p) * (-w \<sqinter> p)\<^sup>\<star> * y"
         using i1 i4 root_successor_loop by auto
       also have "... \<le> y \<squnion> p\<^sup>T * p * (-w \<sqinter> p)\<^sup>\<star> * y"
         using comp_isotone sup_right_isotone by auto
       also have "... \<le> y \<squnion> (-w \<sqinter> p)\<^sup>\<star> * y"
         by (metis i1 comp_associative eq_refl shunt_mapping sup_right_isotone)
       also have "... = (-w \<sqinter> p)\<^sup>\<star> * y"
         by (metis star.circ_loop_fixpoint sup.left_idem sup_commute)
       finally have 20: "p[[w]] \<le> (-w \<sqinter> p)\<^sup>\<star> * y"
         by simp
       have "p\<^sup>T * (-w \<sqinter> p)\<^sup>\<star> * y = p\<^sup>T * y \<squnion> p\<^sup>T * (-w \<sqinter> p) * (-w \<sqinter> p)\<^sup>\<star> * y"
         by (metis comp_associative mult_left_dist_sup star.circ_loop_fixpoint sup_commute)
       also have "... = y \<squnion> p\<^sup>T * (-w \<sqinter> p) * (-w \<sqinter> p)\<^sup>\<star> * y"
         using i1 i4 root_successor_loop by auto
       also have "... \<le> y \<squnion> p\<^sup>T * p * (-w \<sqinter> p)\<^sup>\<star> * y"
         using comp_isotone sup_right_isotone by auto
       also have "... \<le> y \<squnion> (-w \<sqinter> p)\<^sup>\<star> * y"
         by (metis i1 comp_associative eq_refl shunt_mapping sup_right_isotone)
       also have "... = (-w \<sqinter> p)\<^sup>\<star> * y"
         by (metis star.circ_loop_fixpoint sup.left_idem sup_commute)
       finally have 21: "p\<^sup>T\<^sup>\<star> * p\<^sup>T * w \<le> (-w \<sqinter> p)\<^sup>\<star> * y"
         using 20 by (simp add: comp_associative star_left_induct)
       have "w\<^sup>T \<sqinter> p\<^sup>T = p\<^sup>T * (w\<^sup>T \<sqinter> 1)"
         by (metis i5 comp_right_one covector_inf_comp_3 inf.sup_monoid.add_commute one_inf_conv)
       also have "... \<le> p[[w]]"
         by (metis comp_right_subdist_inf inf.boundedE inf.sup_monoid.add_commute one_inf_conv)
       also have "... \<le> p\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
         by (simp add: mult_left_isotone star.circ_mult_increasing_2)
       also have "... \<le> (-w \<sqinter> p)\<^sup>\<star> * y"
         using 21 by simp
       finally have "w \<sqinter> p \<le> y\<^sup>T * (-w \<sqinter> p)\<^sup>T\<^sup>\<star>"
         by (metis conv_dist_comp conv_dist_inf conv_involutive conv_isotone conv_star_commute)
       hence "w \<sqinter> p \<le> (w \<sqinter> y\<^sup>T) * (-w \<sqinter> p)\<^sup>T\<^sup>\<star>"
         by (simp add: i5 vector_inf_comp)
       also have "... \<le> (w \<sqinter> y\<^sup>T) * ?p\<^sup>T\<^sup>\<star>"
         by (simp add: conv_isotone mult_right_isotone star_isotone)
       also have "... \<le> ?p * ?p\<^sup>T\<^sup>\<star>"
         by (simp add: mult_left_isotone)
       also have "... \<le> fc ?p"
         by (simp add: mult_left_isotone star.circ_increasing)
       finally have 22: "w \<sqinter> p \<le> fc ?p"
         by simp
       have "-w \<sqinter> p \<le> ?p"
         by simp
       also have "... \<le> fc ?p"
         by (simp add: fc_increasing)
       finally have "(w \<squnion> -w) \<sqinter> p \<le> fc ?p"
         using 22 by (simp add: comp_inf.semiring.distrib_left inf.sup_monoid.add_commute)
       hence "p \<le> fc ?p"
         by (metis i5 bijective_regular inf.sup_monoid.add_commute inf_sup_distrib1 maddux_3_11_pp)
       hence 23: "fc p \<le> fc ?p"
         using 3 fc_idempotent fc_isotone by fastforce
       have "?p \<le> (w \<sqinter> y\<^sup>T) \<squnion> p"
         using sup_right_isotone by auto
       also have "... = w * y\<^sup>T \<squnion> p"
         by (simp add: i3 i5 vector_covector)
       also have "... \<le> p\<^sup>\<star> \<squnion> p"
         by (smt i5 i6 conv_dist_comp conv_involutive conv_isotone conv_star_commute le_supI shunt_bijective star.circ_increasing sup_absorb1)
       also have "... \<le> fc p"
         using fc_increasing star.circ_back_loop_prefixpoint by auto
       finally have "fc ?p \<le> fc p"
         using i1 fc_idempotent fc_isotone by fastforce
       thus ?thesis
         using 23 i9 by auto
     qed
     show "?p[[p[[w]]]] = p0[[p[[w]]]]"
     proof -
       have "?p[[p[[w]]]] = p[[p[[w]]]]"
         using 2 4 by (metis i5 i10 loop_root put_get_different)
       also have "... = p[[p0[[w]]]]"
         by (simp add: i11)
       also have "... = (p0[p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w\<longmapsto>y])[[p0[[w]]]]"
         using i12 by auto
       also have "... = p0[[p0[[w]]]]"
       proof -
         have "p0[[w]] \<le> -(p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w)"
           by (meson inf.coboundedI2 mult_left_isotone p_antitone p_antitone_iff star.circ_increasing)
         thus ?thesis
           by (meson i2 i5 put_get_different_vector vector_complement_closed vector_inf_closed vector_mult_closed)
       qed
       also have "... = p0[[p[[w]]]]"
         by (simp add: i11)
       finally show ?thesis
         .
     qed
     have 24: "root ?p (p[[w]]) = root p0 (p[[w]])"
       using 3 18 19 i13 same_root by blast
     also have "... = root p0 (p0[[w]])"
       by (simp add: i11)
     also have 25: "... = root p0 w"
       by (metis i5 i13 conv_involutive forest_components_increasing mult_left_isotone shunt_bijective injective_mult_closed read_surjective same_component_same_root)
     finally show 26: "root ?p (p[[w]]) = y"
       by (metis i1 i10 i13 i8 i9 same_root)
     thus "p[[w]] \<noteq> y \<longrightarrow> ?p\<^sup>T\<^sup>+ * (p[[w]]) \<le> -(p[[w]])"
       using 2 3 10 by (simp add: path_compression_1a)
     show "univalent p0" "total p0" "acyclic (p0 - 1)"
       by (simp_all add: i13)
     show "y = root p0 x"
       by (simp add: i14)
     show "p[[w]] \<le> p0\<^sup>T\<^sup>\<star> * x"
       by (metis i11 i15 mult_isotone star.circ_increasing star.circ_transitive_equal mult_assoc)
     let ?q = "p0[p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * (p[[w]])\<longmapsto>y]"
     show "?q = ?p"
     proof -
       have 27: "w \<squnion> p0\<^sup>T\<^sup>+ * w = p0\<^sup>T\<^sup>\<star> * w"
         using comp_associative star.circ_loop_fixpoint sup_commute by auto
       hence 28: "p0\<^sup>T\<^sup>+ * w = p0\<^sup>T\<^sup>\<star> * w - w"
         using 4 24 25 26 by (metis i11 i13 i5 inf.orderE maddux_3_13 path_compression_1a)
       hence "p0\<^sup>T\<^sup>\<star> * (p[[w]]) \<le> -w"
         by (metis i11 inf_le2 star_plus mult.assoc)
       hence "w \<le> -(p0\<^sup>T\<^sup>\<star> * (p[[w]]))"
         by (simp add: p_antitone_iff)
       hence "w \<le> p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * (p[[w]])"
         by (simp add: i15)
       hence 29: "?q \<sqinter> w = ?p \<sqinter> w"
         by (metis update_inf update_inf_same)
       have 30: "?q \<sqinter> p0\<^sup>T\<^sup>+ * w = ?p \<sqinter> p0\<^sup>T\<^sup>+ * w"
       proof -
         have "?q \<sqinter> p0\<^sup>T\<^sup>+ * w = p0 \<sqinter> p0\<^sup>T\<^sup>+ * w"
           by (metis i11 comp_associative inf.cobounded2 p_antitone_iff star.circ_plus_same update_inf_different)
         also have "... = p \<sqinter> p0\<^sup>T\<^sup>+ * w"
           using 28 by (metis i12 inf.cobounded2 inf.sup_monoid.add_assoc p_antitone_iff update_inf_different)
         also have "... = ?p \<sqinter> p0\<^sup>T\<^sup>+ * w"
           using 28 by (simp add: update_inf_different)
         finally show ?thesis
           .
       qed
       have 31: "?q \<sqinter> p0\<^sup>T\<^sup>\<star> * w = ?p \<sqinter> p0\<^sup>T\<^sup>\<star> * w"
         using 27 29 30 by (metis inf_sup_distrib1)
       have 32: "?q \<sqinter> (p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w) = ?p \<sqinter> (p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w)"
       proof -
         have "p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w \<le> p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * (p[[w]])"
           using 28 by (metis i11 inf.sup_right_isotone mult.semigroup_axioms p_antitone_inf star_plus semigroup.assoc)
         hence "?q \<sqinter> (p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w) = y\<^sup>T \<sqinter> p0\<^sup>T\<^sup>\<star> * x \<sqinter> -(p0\<^sup>T\<^sup>\<star> * w)"
           by (metis inf_assoc update_inf)
         also have "... = p \<sqinter> (p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w)"
           by (metis i12 inf_assoc update_inf_same)
         also have "... = ?p \<sqinter> (p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w)"
           by (simp add: inf.coboundedI2 p_antitone path_compression_1b inf_assoc update_inf_different)
         finally show ?thesis
           .
       qed
       have "p0\<^sup>T\<^sup>\<star> * w \<squnion> (p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w) = p0\<^sup>T\<^sup>\<star> * x"
       proof -
         have 33: "regular (p0\<^sup>T\<^sup>\<star> * w)"
           using i13 i5 bijective_regular mapping_regular regular_closed_star regular_conv_closed regular_mult_closed by auto
         have "p0\<^sup>T\<^sup>\<star> * w \<le> p0\<^sup>T\<^sup>\<star> * x"
           by (metis i15 comp_associative mult_right_isotone star.circ_transitive_equal)
         hence "p0\<^sup>T\<^sup>\<star> * w \<squnion> (p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w) = p0\<^sup>T\<^sup>\<star> * x \<sqinter> (p0\<^sup>T\<^sup>\<star> * w \<squnion> -(p0\<^sup>T\<^sup>\<star> * w))"
           by (simp add: comp_inf.semiring.distrib_left inf.absorb2)
         also have "... = p0\<^sup>T\<^sup>\<star> * x"
           using 33 by (metis inf_sup_distrib1 maddux_3_11_pp)
         finally show ?thesis
           .
       qed
       hence 34: "?q \<sqinter> p0\<^sup>T\<^sup>\<star> * x = ?p \<sqinter> p0\<^sup>T\<^sup>\<star> * x"
         using 31 32 by (metis inf_sup_distrib1)
       have 35: "regular (p0\<^sup>T\<^sup>\<star> * x)"
         using i13 i2 bijective_regular mapping_regular regular_closed_star regular_conv_closed regular_mult_closed by auto
       have "-(p0\<^sup>T\<^sup>\<star> * x) \<le> -w"
         by (simp add: i15 p_antitone)
       hence "?q - p0\<^sup>T\<^sup>\<star> * x = ?p - p0\<^sup>T\<^sup>\<star> * x"
         by (metis i12 p_antitone_inf update_inf_different)
       thus ?thesis
         using 34 35 by (metis maddux_3_11_pp)
     qed
     show "card ?t < card ?s"
     proof -
       have "?p\<^sup>T * p\<^sup>T\<^sup>\<star> * w = (w\<^sup>T \<sqinter> y) * p\<^sup>T\<^sup>\<star> * w \<squnion> (-w\<^sup>T \<sqinter> p\<^sup>T) * p\<^sup>T\<^sup>\<star> * w"
         by (simp add: conv_complement conv_dist_inf conv_dist_sup mult_right_dist_sup)
       also have "... \<le> (w\<^sup>T \<sqinter> y) * p\<^sup>T\<^sup>\<star> * w \<squnion> p\<^sup>T * p\<^sup>T\<^sup>\<star> * w"
         using mult_left_isotone sup_right_isotone by auto
       also have "... \<le> (w\<^sup>T \<sqinter> y) * p\<^sup>T\<^sup>\<star> * w \<squnion> p\<^sup>T\<^sup>\<star> * w"
         using mult_left_isotone star.left_plus_below_circ sup_right_isotone by blast
       also have "... \<le> y * p\<^sup>T\<^sup>\<star> * w \<squnion> p\<^sup>T\<^sup>\<star> * w"
         using semiring.add_right_mono mult_left_isotone by auto
       also have "... \<le> y * top \<squnion> p\<^sup>T\<^sup>\<star> * w"
         by (simp add: comp_associative le_supI1 mult_right_isotone)
       also have "... = p\<^sup>T\<^sup>\<star> * w"
         by (simp add: i3 i6 sup_absorb2)
       finally have "?p\<^sup>T\<^sup>\<star> * p\<^sup>T * w \<le> p\<^sup>T\<^sup>\<star> * w"
         using 11 by (metis dual_order.trans star.circ_loop_fixpoint sup_commute sup_ge2 mult_assoc)
       hence 36: "?t \<subseteq> ?s"
         using order_lesseq_imp mult_assoc by auto
       have 37: "w \<in> ?s"
         by (simp add: i5 bijective_regular path_compression_1b)
       have 38: "\<not> w \<in> ?t"
       proof
         assume "w \<in> ?t"
         hence 39: "w \<le> (?p\<^sup>T - 1)\<^sup>\<star> * (p[[w]])"
           using reachable_without_loops by auto
         hence "p[[w]] \<le> (?p - 1)\<^sup>\<star> * w"
           using 2 by (smt i5 bijective_reverse conv_star_commute reachable_without_loops)
         also have "... \<le> p\<^sup>\<star> * w"
         proof -
           have "p\<^sup>T\<^sup>\<star> * y = y"
             using i1 i4 root_transitive_successor_loop by auto
           hence "y\<^sup>T * p\<^sup>\<star> * w = y\<^sup>T * w"
             by (metis conv_dist_comp conv_involutive conv_star_commute)
           also have "... = bot"
             using 5 by (metis i5 inf.idem inf.sup_monoid.add_commute mult_left_zero schroeder_1 vector_inf_comp)
           finally have 40: "y\<^sup>T * p\<^sup>\<star> * w = bot"
             by simp
           have "(?p - 1) * p\<^sup>\<star> * w = (w \<sqinter> y\<^sup>T \<sqinter> -1) * p\<^sup>\<star> * w \<squnion> (-w \<sqinter> p \<sqinter> -1) * p\<^sup>\<star> * w"
             by (simp add: comp_inf.mult_right_dist_sup mult_right_dist_sup)
           also have "... \<le> (w \<sqinter> y\<^sup>T \<sqinter> -1) * p\<^sup>\<star> * w \<squnion> p * p\<^sup>\<star> * w"
             by (meson inf_le1 inf_le2 mult_left_isotone order_trans sup_right_isotone)
           also have "... \<le> (w \<sqinter> y\<^sup>T \<sqinter> -1) * p\<^sup>\<star> * w \<squnion> p\<^sup>\<star> * w"
             using mult_left_isotone star.left_plus_below_circ sup_right_isotone by blast
           also have "... \<le> y\<^sup>T * p\<^sup>\<star> * w \<squnion> p\<^sup>\<star> * w"
             by (meson inf_le1 inf_le2 mult_left_isotone order_trans sup_left_isotone)
           also have "... = p\<^sup>\<star> * w"
             using 40 by simp
           finally show ?thesis
             by (metis comp_associative le_supI star.circ_loop_fixpoint sup_ge2 star_left_induct)
         qed
         finally have "w \<le> p\<^sup>T\<^sup>\<star> * p\<^sup>T * w"
           using 11 39 reachable_without_loops star_plus by auto
         thus False
           using 4 i1 i10 i5 loop_root_2 star.circ_plus_same by auto
       qed
       show "card ?t < card ?s"
         apply (rule psubset_card_mono)
         subgoal using finite_regular by simp
         subgoal using 36 37 38 by auto
         done
     qed
   qed
 qed
 
 lemma path_compression_3a:
   assumes "path_compression_invariant p x (p[[w]]) p0 w"
   shows "p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>p[[w]]] = p"
 proof -
   let ?y = "p[[w]]"
   let ?p = "p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>?y]"
   have i1: "disjoint_set_forest p" and i2: "point x" and i3: "point ?y" and i4: "?y = root p x"
     using assms path_compression_invariant_def path_compression_precondition_def by meson+
   have i5: "point w" and i6: "?y \<le> p\<^sup>T\<^sup>\<star> * w"
     and i7: "w \<noteq> x \<longrightarrow> p[[x]] = ?y \<and> ?y \<noteq> x \<and> p\<^sup>T\<^sup>+ * w \<le> -x" and i8: "p \<sqinter> 1 = p0 \<sqinter> 1" and i9: "fc p = fc p0"
     and i10: "root p w = ?y" and i11: "p[[w]] = p0[[w]]" and i12: "p0[p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w\<longmapsto>?y] = p"
     and i13: "disjoint_set_forest p0" and i14: "?y = root p0 x" and i15: "w \<le> p0\<^sup>T\<^sup>\<star> * x"
     using assms path_compression_invariant_def by blast+
   have 1: "?p \<sqinter> ?y = p \<sqinter> ?y"
     by (metis i1 i14 i3 i4 get_put inf_le1 root_successor_loop update_inf update_inf_same)
   have 2: "?p \<sqinter> w = p \<sqinter> w"
     by (metis i5 i11 i15 get_put update_inf update_inf_same)
   have "?y = root p0 w"
     by (metis i1 i10 i13 i8 i9 same_root)
   hence "p0\<^sup>T\<^sup>\<star> * w = w \<squnion> ?y"
     by (metis i11 i13 root_transitive_successor_loop star.circ_loop_fixpoint star_plus sup_monoid.add_commute mult_assoc)
   hence 3: "?p \<sqinter> p0\<^sup>T\<^sup>\<star> * w = p \<sqinter> p0\<^sup>T\<^sup>\<star> * w"
     using 1 2 by (simp add: inf_sup_distrib1)
   have "p0\<^sup>T\<^sup>\<star> * w \<le> p0\<^sup>T\<^sup>\<star> * x"
     by (metis i15 comp_associative mult_right_isotone star.circ_transitive_equal)
   hence 4: "?p \<sqinter> (p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0\<^sup>T\<^sup>\<star> * w) = p \<sqinter> (p0\<^sup>T\<^sup>\<star> * x \<sqinter> p0\<^sup>T\<^sup>\<star> * w)"
     using 3 by (simp add: inf.absorb2)
   have 5: "?p \<sqinter> (p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w) = p \<sqinter> (p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * w)"
     by (metis i12 inf_le1 update_inf update_inf_same)
   have "regular (p0\<^sup>T\<^sup>\<star> * w)"
     using i13 i5 bijective_regular mapping_regular regular_closed_star regular_conv_closed regular_mult_closed by auto
   hence 6: "?p \<sqinter> p0\<^sup>T\<^sup>\<star> * x = p \<sqinter> p0\<^sup>T\<^sup>\<star> * x"
     using 4 5 by (smt inf_sup_distrib1 maddux_3_11_pp)
   have 7: "?p - p0\<^sup>T\<^sup>\<star> * x = p - p0\<^sup>T\<^sup>\<star> * x"
     by (smt i12 inf.sup_monoid.add_commute inf_import_p inf_sup_absorb le_iff_inf p_dist_inf update_inf_different inf.idem p_antitone_inf)
   have "regular (p0\<^sup>T\<^sup>\<star> * x)"
     using i13 i2 bijective_regular mapping_regular regular_closed_star regular_conv_closed regular_mult_closed by auto
   thus "?p = p"
     using 6 7 by (smt inf_sup_distrib1 maddux_3_11_pp)
 qed
 
 lemma path_compression_3:
   "path_compression_invariant p x (p[[w]]) p0 w \<Longrightarrow> path_compression_postcondition p x (p[[w]]) p0"
   using path_compression_invariant_def path_compression_postcondition_def path_compression_precondition_def path_compression_3a by blast
 
 theorem path_compression:
   "VARS p t w
   [ path_compression_precondition p x y \<and> p0 = p ]
   w := x;
   WHILE y \<noteq> p[[w]]
     INV { path_compression_invariant p x y p0 w }
-    VAR { card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * w } }
+    VAR { (p\<^sup>T\<^sup>\<star> * w)\<down> }
      DO t := w;
         w := p[[w]];
         p[t] := y
      OD
   [ path_compression_postcondition p x y p0 ]"
   apply vcg_tc_simp
     apply (fact path_compression_1)
    apply (fact path_compression_2)
   using path_compression_3 by auto
 
 lemma path_compression_exists:
   "path_compression_precondition p x y \<Longrightarrow> \<exists>p' . path_compression_postcondition p' x y p"
   using tc_extract_function path_compression by blast
 
 definition "path_compression p x y \<equiv> (SOME p' . path_compression_postcondition p' x y p)"
 
 lemma path_compression_function:
   assumes "path_compression_precondition p x y"
     and "p' = path_compression p x y"
   shows "path_compression_postcondition p' x y p"
   by (metis assms path_compression_def path_compression_exists someI)
 
 subsection \<open>Find-Set with Path Compression\<close>
 
 text \<open>
 We sequentially combine find-set and path compression.
 We consider implementations which use the previously derived functions and implementations which unfold their definitions.
 \<close>
 
 theorem find_set_path_compression:
   "VARS p y
   [ find_set_precondition p x \<and> p0 = p ]
   y := find_set p x;
   p := path_compression p x y
   [ path_compression_postcondition p x y p0 ]"
   apply vcg_tc_simp
   using find_set_function find_set_postcondition_def find_set_precondition_def path_compression_function path_compression_precondition_def by fastforce
 
 theorem find_set_path_compression_1:
   "VARS p t w y
   [ find_set_precondition p x \<and> p0 = p ]
   y := find_set p x;
   w := x;
   WHILE y \<noteq> p[[w]]
     INV { path_compression_invariant p x y p0 w }
-    VAR { card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * w } }
+    VAR { (p\<^sup>T\<^sup>\<star> * w)\<down> }
      DO t := w;
         w := p[[w]];
         p[t] := y
      OD
   [ path_compression_postcondition p x y p0 ]"
   apply vcg_tc_simp
     using find_set_function find_set_postcondition_def find_set_precondition_def path_compression_1 path_compression_precondition_def apply fastforce
    apply (fact path_compression_2)
   by (fact path_compression_3)
 
 theorem find_set_path_compression_2:
   "VARS p y
   [ find_set_precondition p x \<and> p0 = p ]
   y := x;
   WHILE y \<noteq> p[[y]]
     INV { find_set_invariant p x y \<and> p0 = p }
-    VAR { card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y } }
+    VAR { (p\<^sup>T\<^sup>\<star> * y)\<down> }
      DO y := p[[y]]
      OD;
   p := path_compression p x y
   [ path_compression_postcondition p x y p0 ]"
   apply vcg_tc_simp
     apply (fact find_set_1)
    apply (fact find_set_2)
   by (smt find_set_3 find_set_invariant_def find_set_postcondition_def find_set_precondition_def path_compression_function path_compression_precondition_def)
 
 theorem find_set_path_compression_3:
   "VARS p t w y
   [ find_set_precondition p x \<and> p0 = p ]
   y := x;
   WHILE y \<noteq> p[[y]]
     INV { find_set_invariant p x y \<and> p0 = p }
-    VAR { card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y } }
+    VAR { (p\<^sup>T\<^sup>\<star> * y)\<down> }
      DO y := p[[y]]
      OD;
   w := x;
   WHILE y \<noteq> p[[w]]
     INV { path_compression_invariant p x y p0 w }
-    VAR { card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * w } }
+    VAR { (p\<^sup>T\<^sup>\<star> * w)\<down> }
      DO t := w;
         w := p[[w]];
         p[t] := y
      OD
   [ path_compression_postcondition p x y p0 ]"
   apply vcg_tc_simp
       apply (simp add: find_set_1)
      apply (fact find_set_2)
     using find_set_3 find_set_invariant_def find_set_postcondition_def find_set_precondition_def path_compression_1 path_compression_precondition_def apply blast
    apply (fact path_compression_2)
   by (fact path_compression_3)
 
 text \<open>
 Find-set with path compression returns two results: the representative of the tree and the modified disjoint-set forest.
 \<close>
 
 lemma find_set_path_compression_exists:
   "find_set_precondition p x \<Longrightarrow> \<exists>p' y . path_compression_postcondition p' x y p"
   using tc_extract_function find_set_path_compression by blast
 
 definition "find_set_path_compression p x \<equiv> (SOME (p',y) . path_compression_postcondition p' x y p)"
 
 lemma find_set_path_compression_function:
   assumes "find_set_precondition p x"
     and "(p',y) = find_set_path_compression p x"
   shows "path_compression_postcondition p' x y p"
 proof -
   let ?P = "\<lambda>(p',y) . path_compression_postcondition p' x y p"
   have "?P (SOME z . ?P z)"
     apply (unfold some_eq_ex)
     using assms(1) find_set_path_compression_exists by simp
   thus ?thesis
     using assms(2) find_set_path_compression_def by auto
 qed
 
 text \<open>
 We prove that \<open>find_set_path_compression\<close> returns the same representative as \<open>find_set\<close>.
 \<close>
 
 lemma find_set_path_compression_find_set:
   assumes "find_set_precondition p x"
   shows "find_set p x = snd (find_set_path_compression p x)"
 proof -
   let ?r = "find_set p x"
   let ?p = "fst (find_set_path_compression p x)"
   let ?y = "snd (find_set_path_compression p x)"
   have 1: "find_set_postcondition p x ?r"
     by (simp add: assms find_set_function)
   have "path_compression_postcondition ?p x ?y p"
     using assms find_set_path_compression_function prod.collapse by blast
   thus "?r = ?y"
     using 1 by (smt assms same_root find_set_precondition_def find_set_postcondition_def path_compression_precondition_def path_compression_postcondition_def)
 qed
 
 text \<open>
 A weaker postcondition suffices to prove that the two forests have the same semantics; that is, they describe the same disjoint sets and have the same roots.
 \<close>
 
 lemma find_set_path_compression_path_compression_semantics:
   assumes "find_set_precondition p x"
   shows "fc (path_compression p x (find_set p x)) = fc (fst (find_set_path_compression p x))"
     and "path_compression p x (find_set p x) \<sqinter> 1 = fst (find_set_path_compression p x) \<sqinter> 1"
 proof -
   let ?r = "find_set p x"
   let ?q = "path_compression p x ?r"
   let ?p = "fst (find_set_path_compression p x)"
   let ?y = "snd (find_set_path_compression p x)"
   have 1: "path_compression_postcondition (path_compression p x ?r) x ?r p"
     using assms find_set_function find_set_postcondition_def find_set_precondition_def path_compression_function path_compression_precondition_def by auto
   have 2: "path_compression_postcondition ?p x ?y p"
     using assms find_set_path_compression_function prod.collapse by blast
   show "fc ?q = fc ?p"
     using 1 2 by (simp add: path_compression_postcondition_def)
   show "?q \<sqinter> 1 = ?p \<sqinter> 1"
     using 1 2 by (simp add: path_compression_postcondition_def)
 qed
 
 text \<open>
 With the current, stronger postcondition of path compression describing the precise effect of how links change, we can prove that the two forests are actually equal.
 \<close>
 
 lemma find_set_path_compression_find_set_pathcompression:
   assumes "find_set_precondition p x"
   shows "path_compression p x (find_set p x) = fst (find_set_path_compression p x)"
 proof -
   let ?r = "find_set p x"
   let ?q = "path_compression p x ?r"
   let ?p = "fst (find_set_path_compression p x)"
   let ?y = "snd (find_set_path_compression p x)"
   have 1: "path_compression_postcondition (path_compression p x ?r) x ?r p"
     using assms find_set_function find_set_postcondition_def find_set_precondition_def path_compression_function path_compression_precondition_def by auto
   have 2: "path_compression_postcondition ?p x ?y p"
     using assms find_set_path_compression_function prod.collapse by blast
   have "?r = ?y"
     by (simp add: assms find_set_path_compression_find_set)
   thus "?q = ?p"
     using 1 2 by (simp add: path_compression_postcondition_def)
 qed
 
 subsection \<open>Union-Sets\<close>
 
 text \<open>
 We only consider a naive union-sets operation (without ranks).
 The semantics is the equivalence closure obtained after adding the link between the two given nodes, 
 which requires those two elements to be in the same set.
 The implementation uses temporary variable \<open>t\<close> to store the two results returned by find-set with path compression.
 The disjoint-set forest, which keeps being updated, is threaded through the sequence of operations.
 \<close>
 
 definition "union_sets_precondition p x y \<equiv> disjoint_set_forest p \<and> point x \<and> point y"
 definition "union_sets_postcondition p x y p0 \<equiv> union_sets_precondition p x y \<and> fc p = wcc (p0 \<squnion> x * y\<^sup>T)"
 
 lemma union_sets_1:
   assumes "union_sets_precondition p0 x y"
     and "path_compression_postcondition p1 x r p0"
     and "path_compression_postcondition p2 y s p1"
   shows "union_sets_postcondition (p2[r\<longmapsto>s]) x y p0"
 proof (unfold union_sets_postcondition_def union_sets_precondition_def, intro conjI)
   let ?p = "p2[r\<longmapsto>s]"
   have 1: "disjoint_set_forest p1 \<and> point r \<and> r = root p1 x \<and> p1 \<sqinter> 1 = p0 \<sqinter> 1 \<and> fc p1 = fc p0"
     using assms(2) path_compression_precondition_def path_compression_postcondition_def by auto
   have 2: "disjoint_set_forest p2 \<and> point s \<and> s = root p2 y \<and> p2 \<sqinter> 1 = p1 \<sqinter> 1 \<and> fc p2 = fc p1"
     using assms(3) path_compression_precondition_def path_compression_postcondition_def by auto
   hence 3: "fc p2 = fc p0"
     using 1 by simp
   show 4: "univalent ?p"
     using 1 2 update_univalent by blast
   show "total ?p"
     using 1 2 bijective_regular update_total by blast
   show "acyclic (?p - 1)"
   proof (cases "r = s")
     case True
     thus ?thesis
       using 2 update_acyclic_5 by fastforce
   next
     case False
     hence "bot = r \<sqinter> s"
       using 1 2 distinct_points by blast
     also have "... = r \<sqinter> p2\<^sup>T\<^sup>\<star> * s"
       using 2 by (smt root_transitive_successor_loop)
     finally have "s \<sqinter> p2\<^sup>\<star> * r = bot"
       using schroeder_1 conv_star_commute inf.sup_monoid.add_commute by fastforce
     thus ?thesis
       using 1 2 update_acyclic_4 by blast
   qed
   show "vector x"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "injective x"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "surjective x"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "vector y"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "injective y"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "surjective y"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "fc ?p = wcc (p0 \<squnion> x * y\<^sup>T)"
   proof (rule order.antisym)
     have "r = p1[[r]]"
       using 1 by (metis root_successor_loop)
     hence "r * r\<^sup>T \<le> p1\<^sup>T"
       using 1 eq_refl shunt_bijective by blast
     hence "r * r\<^sup>T \<le> p1"
       using 1 conv_order coreflexive_symmetric by fastforce
     hence "r * r\<^sup>T \<le> p1 \<sqinter> 1"
       using 1 inf.boundedI by blast
     also have "... = p2 \<sqinter> 1"
       using 2 by simp
     finally have "r * r\<^sup>T \<le> p2"
       by simp
     hence "r \<le> p2 * r"
       using 1 shunt_bijective by blast
     hence 5: "p2[[r]] \<le> r"
       using 2 shunt_mapping by blast
     have "r \<sqinter> p2 \<le> r * (top \<sqinter> r\<^sup>T * p2)"
       using 1 by (metis dedekind_1)
     also have "... = r * r\<^sup>T * p2"
       by (simp add: mult_assoc)
     also have "... \<le> r * r\<^sup>T"
       using 5 by (metis comp_associative conv_dist_comp conv_involutive conv_order mult_right_isotone)
     also have "... \<le> 1"
       using 1 by blast
     finally have 6: "r \<sqinter> p2 \<le> 1"
       by simp
     have "p0 \<le> wcc p0"
       by (simp add: star.circ_sub_dist_1)
     also have "... = wcc p2"
       using 3 by (simp add: star_decompose_1)
     also have 7: "... \<le> wcc ?p"
     proof -
       have "wcc p2 = wcc ((-r \<sqinter> p2) \<squnion> (r \<sqinter> p2))"
         using 1 by (metis bijective_regular inf.sup_monoid.add_commute maddux_3_11_pp)
       also have "... \<le> wcc ((-r \<sqinter> p2) \<squnion> 1)"
         using 6 wcc_isotone sup_right_isotone by simp
       also have "... = wcc (-r \<sqinter> p2)"
         using wcc_with_loops by simp
       also have "... \<le> wcc ?p"
         using wcc_isotone sup_ge2 by blast
       finally show ?thesis
         by simp
     qed
     finally have 8: "p0 \<le> wcc ?p"
       by force
     have "r \<le> p1\<^sup>T\<^sup>\<star> * x"
       using 1 by (metis inf_le1)
     hence 9: "r * x\<^sup>T \<le> p1\<^sup>T\<^sup>\<star>"
       using assms(1) shunt_bijective union_sets_precondition_def by blast
     hence "x * r\<^sup>T \<le> p1\<^sup>\<star>"
       using conv_dist_comp conv_order conv_star_commute by force
     also have "... \<le> wcc p1"
       by (simp add: star.circ_sub_dist)
     also have "... = wcc p2"
       using 1 2 by (simp add: fc_wcc)
     also have "... \<le> wcc ?p"
       using 7 by simp
     finally have 10: "x * r\<^sup>T \<le> wcc ?p"
       by simp
     have 11: "r * s\<^sup>T \<le> wcc ?p"
       using 1 2 star.circ_sub_dist_1 sup_assoc vector_covector by auto
     have "s \<le> p2\<^sup>T\<^sup>\<star> * y"
       using 2 by (metis inf_le1)
     hence 12: "s * y\<^sup>T \<le> p2\<^sup>T\<^sup>\<star>"
       using assms(1) shunt_bijective union_sets_precondition_def by blast
     also have "... \<le> wcc p2"
       using star_isotone sup_ge2 by blast
     also have "... \<le> wcc ?p"
       using 7 by simp
     finally have 13: "s * y\<^sup>T \<le> wcc ?p"
       by simp
     have "x \<le> x * r\<^sup>T * r \<and> y \<le> y * s\<^sup>T * s"
       using 1 2 shunt_bijective by blast
     hence "x * y\<^sup>T \<le> x * r\<^sup>T * r * (y * s\<^sup>T * s)\<^sup>T"
       using comp_isotone conv_isotone by blast
     also have "... = x * r\<^sup>T * r * s\<^sup>T * s * y\<^sup>T"
       by (simp add: comp_associative conv_dist_comp)
     also have "... \<le> wcc ?p * (r * s\<^sup>T) * (s * y\<^sup>T)"
       using 10 by (metis mult_left_isotone mult_assoc)
     also have "... \<le> wcc ?p * wcc ?p * (s * y\<^sup>T)"
       using 11 by (metis mult_left_isotone mult_right_isotone)
     also have "... \<le> wcc ?p * wcc ?p * wcc ?p"
       using 13 by (metis mult_right_isotone)
     also have "... = wcc ?p"
       by (simp add: star.circ_transitive_equal)
     finally have "p0 \<squnion> x * y\<^sup>T \<le> wcc ?p"
       using 8 by simp
     hence "wcc (p0 \<squnion> x * y\<^sup>T) \<le> wcc ?p"
       using wcc_below_wcc by simp
     thus "wcc (p0 \<squnion> x * y\<^sup>T) \<le> fc ?p"
       using 4 fc_wcc by simp
     have "-r \<sqinter> p2 \<le> wcc p2"
       by (simp add: inf.coboundedI2 star.circ_sub_dist_1)
     also have "... = wcc p0"
       using 3 by (simp add: star_decompose_1)
     also have "... \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       by (simp add: wcc_isotone)
     finally have 14: "-r \<sqinter> p2 \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       by simp
     have "r * x\<^sup>T \<le> wcc p1"
       using 9 inf.order_trans star.circ_sub_dist sup_commute by fastforce
     also have "... = wcc p0"
       using 1 by (simp add: star_decompose_1)
     also have "... \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       by (simp add: wcc_isotone)
     finally have 15: "r * x\<^sup>T \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       by simp
     have 16: "x * y\<^sup>T \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       using le_supE star.circ_sub_dist_1 by blast
     have "y * s\<^sup>T \<le> p2\<^sup>\<star>"
       using 12 conv_dist_comp conv_order conv_star_commute by fastforce
     also have "... \<le> wcc p2"
       using star.circ_sub_dist sup_commute by fastforce
     also have "... = wcc p0"
       using 3 by (simp add: star_decompose_1)
     also have "... \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       by (simp add: wcc_isotone)
     finally have 17: "y * s\<^sup>T \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       by simp
     have "r \<le> r * x\<^sup>T * x \<and> s \<le> s * y\<^sup>T * y"
       using assms(1) shunt_bijective union_sets_precondition_def by blast
     hence "r * s\<^sup>T \<le> r * x\<^sup>T * x * (s * y\<^sup>T * y)\<^sup>T"
       using comp_isotone conv_isotone by blast
     also have "... = r * x\<^sup>T * x * y\<^sup>T * y * s\<^sup>T"
       by (simp add: comp_associative conv_dist_comp)
     also have "... \<le> wcc (p0 \<squnion> x * y\<^sup>T) * (x * y\<^sup>T) * (y * s\<^sup>T)"
       using 15 by (metis mult_left_isotone mult_assoc)
     also have "... \<le> wcc (p0 \<squnion> x * y\<^sup>T) * wcc (p0 \<squnion> x * y\<^sup>T) * (y * s\<^sup>T)"
       using 16 by (metis mult_left_isotone mult_right_isotone)
     also have "... \<le> wcc (p0 \<squnion> x * y\<^sup>T) * wcc (p0 \<squnion> x * y\<^sup>T) * wcc (p0 \<squnion> x * y\<^sup>T)"
       using 17 by (metis mult_right_isotone)
     also have "... = wcc (p0 \<squnion> x * y\<^sup>T)"
       by (simp add: star.circ_transitive_equal)
     finally have "?p \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       using 1 2 14 vector_covector by auto
     hence "wcc ?p \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       using wcc_below_wcc by blast
     thus "fc ?p \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       using 4 fc_wcc by simp
   qed
 qed
 
 theorem union_sets:
   "VARS p r s t
   [ union_sets_precondition p x y \<and> p0 = p ]
   t := find_set_path_compression p x;
   p := fst t;
   r := snd t;
   t := find_set_path_compression p y;
   p := fst t;
   s := snd t;
   p[r] := s
   [ union_sets_postcondition p x y p0 ]"
 proof vcg_tc_simp
   let ?t1 = "find_set_path_compression p0 x"
   let ?p1 = "fst ?t1"
   let ?r = "snd ?t1"
   let ?t2 = "find_set_path_compression ?p1 y"
   let ?p2 = "fst ?t2"
   let ?s = "snd ?t2"
   let ?p = "?p2[?r\<longmapsto>?s]"
   assume 1: "union_sets_precondition p0 x y"
   hence 2: "path_compression_postcondition ?p1 x ?r p0"
     by (simp add: find_set_precondition_def union_sets_precondition_def find_set_path_compression_function)
   hence "path_compression_postcondition ?p2 y ?s ?p1"
     using 1 by (meson find_set_precondition_def union_sets_precondition_def find_set_path_compression_function path_compression_postcondition_def path_compression_precondition_def prod.collapse)
   thus "union_sets_postcondition (?p2[?r\<longmapsto>?s]) x y p0"
     using 1 2 by (simp add: union_sets_1)
 qed
 
 lemma union_sets_exists:
   "union_sets_precondition p x y \<Longrightarrow> \<exists>p' . union_sets_postcondition p' x y p"
   using tc_extract_function union_sets by blast
 
 definition "union_sets p x y \<equiv> (SOME p' . union_sets_postcondition p' x y p)"
 
 lemma union_sets_function:
   assumes "union_sets_precondition p x y"
     and "p' = union_sets p x y"
   shows "union_sets_postcondition p' x y p"
   by (metis assms union_sets_def union_sets_exists someI)
 
 theorem union_sets_2:
   "VARS p r s
   [ union_sets_precondition p x y \<and> p0 = p ]
   r := find_set p x;
   p := path_compression p x r;
   s := find_set p y;
   p := path_compression p y s;
   p[r] := s
   [ union_sets_postcondition p x y p0 ]"
 proof vcg_tc_simp
   let ?r = "find_set p0 x"
   let ?p1 = "path_compression p0 x ?r"
   let ?s = "find_set ?p1 y"
   let ?p2 = "path_compression ?p1 y ?s"
   assume 1: "union_sets_precondition p0 x y"
   hence 2: "path_compression_postcondition ?p1 x ?r p0"
     using find_set_function find_set_postcondition_def find_set_precondition_def path_compression_function path_compression_precondition_def union_sets_precondition_def by auto
   hence "path_compression_postcondition ?p2 y ?s ?p1"
     using 1 find_set_function find_set_postcondition_def find_set_precondition_def path_compression_function path_compression_precondition_def union_sets_precondition_def path_compression_postcondition_def by meson
   thus "union_sets_postcondition (?p2[?r\<longmapsto>?s]) x y p0"
     using 1 2 by (simp add: union_sets_1)
 qed
 
 end
 
 end
 
diff --git a/thys/Relational_Disjoint_Set_Forests/More_Disjoint_Set_Forests.thy b/thys/Relational_Disjoint_Set_Forests/More_Disjoint_Set_Forests.thy
--- a/thys/Relational_Disjoint_Set_Forests/More_Disjoint_Set_Forests.thy
+++ b/thys/Relational_Disjoint_Set_Forests/More_Disjoint_Set_Forests.thy
@@ -1,3187 +1,3187 @@
 (* Title:      More on Disjoint-Set Forests
    Author:     Walter Guttmann
    Maintainer: Walter Guttmann <walter.guttmann at canterbury.ac.nz>
 *)
 
 theory More_Disjoint_Set_Forests
 
 imports Disjoint_Set_Forests
 
 begin
 
 section \<open>More on Array Access and Disjoint-Set Forests\<close>
 
 text \<open>
 This section contains further results about directed acyclic graphs and relational array operations.
 \<close>
 
 context stone_relation_algebra
 begin
 
 text \<open>Theorem 6.4\<close>
 
 lemma update_square:
   assumes "point y"
     shows "x[y\<longmapsto>x[[x[[y]]]]] \<le> x * x \<squnion> x"
 proof -
   have "x[y\<longmapsto>x[[x[[y]]]]] = (y \<sqinter> y\<^sup>T * x * x) \<squnion> (-y \<sqinter> x)"
     by (simp add: conv_dist_comp)
   also have "... \<le> (y \<sqinter> y\<^sup>T) * x * x \<squnion> x"
     by (smt assms inf.eq_refl inf.sup_monoid.add_commute inf_le1 sup_mono vector_inf_comp)
   also have "... \<le> x * x \<squnion> x"
     by (smt (z3) assms comp_associative conv_dist_comp coreflexive_comp_top_inf inf.cobounded2 sup_left_isotone symmetric_top_closed)
   finally show ?thesis
     .
 qed
 
 text \<open>Theorem 2.13\<close>
 
 lemma update_ub:
   "x[y\<longmapsto>z] \<le> x \<squnion> z\<^sup>T"
   by (meson dual_order.trans inf.cobounded2 le_supI sup.cobounded1 sup_ge2)
 
 text \<open>Theorem 6.7\<close>
 
 lemma update_square_ub:
   "x[y\<longmapsto>(x * x)\<^sup>T] \<le> x \<squnion> x * x"
   by (metis conv_involutive update_ub)
 
 text \<open>Theorem 2.14\<close>
 
 lemma update_same_sub:
   assumes "u \<sqinter> x = u \<sqinter> z"
       and "y \<le> u"
       and "regular y"
     shows "x[y\<longmapsto>z\<^sup>T] = x"
   by (smt (z3) assms conv_involutive inf.sup_monoid.add_commute inf.sup_relative_same_increasing maddux_3_11_pp)
 
 text \<open>Theorem 2.15\<close>
 
 lemma update_point_get:
   "point y \<Longrightarrow> x[y\<longmapsto>z[[y]]] = x[y\<longmapsto>z\<^sup>T]"
   by (metis conv_involutive get_put inf_commute update_inf_same)
 
 text \<open>Theorem 2.11\<close>
 
 lemma update_bot:
   "x[bot\<longmapsto>z] = x"
   by simp
 
 text \<open>Theorem 2.12\<close>
 
 lemma update_top:
   "x[top\<longmapsto>z] = z\<^sup>T"
   by simp
 
 text \<open>Theorem 2.6\<close>
 
 lemma update_same:
   assumes "regular u"
     shows "(x[y\<longmapsto>z])[u\<longmapsto>z] = x[y \<squnion> u\<longmapsto>z]"
 proof -
   have "(x[y\<longmapsto>z])[u\<longmapsto>z] = (u \<sqinter> z\<^sup>T) \<squnion> (-u \<sqinter> y \<sqinter> z\<^sup>T) \<squnion> (-u \<sqinter> -y \<sqinter> x)"
     using inf.sup_monoid.add_assoc inf_sup_distrib1 sup_assoc by force
   also have "... = (u \<sqinter> z\<^sup>T) \<squnion> (y \<sqinter> z\<^sup>T) \<squnion> (-(u \<squnion> y) \<sqinter> x)"
     by (metis assms inf_sup_distrib2 maddux_3_21_pp p_dist_sup)
   also have "... = x[y \<squnion> u\<longmapsto>z]"
     using comp_inf.mult_right_dist_sup sup_commute by auto
   finally show ?thesis
     .
 qed
 
 lemma update_same_3:
   assumes "regular u"
       and "regular v"
     shows "((x[y\<longmapsto>z])[u\<longmapsto>z])[v\<longmapsto>z] = x[y \<squnion> u \<squnion> v\<longmapsto>z]"
   by (metis assms update_same)
 
 text \<open>Theorem 2.7\<close>
 
 lemma update_split:
   assumes "regular w"
     shows "x[y\<longmapsto>z] = (x[y - w\<longmapsto>z])[y \<sqinter> w\<longmapsto>z]"
   by (smt (z3) assms comp_inf.semiring.distrib_left inf.left_commute inf.sup_monoid.add_commute inf_import_p maddux_3_11_pp maddux_3_12 p_dist_inf sup_assoc)
 
 text \<open>Theorem 2.8\<close>
 
 lemma update_injective_swap:
   assumes "injective x"
       and "point y"
       and "injective z"
       and "vector z"
     shows "injective ((x[y\<longmapsto>x[[z]]])[z\<longmapsto>x[[y]]])"
 proof -
   have 1: "(z \<sqinter> y\<^sup>T * x) * (z \<sqinter> y\<^sup>T * x)\<^sup>T \<le> 1"
     using assms(3) injective_inf_closed by auto
   have "(z \<sqinter> y\<^sup>T * x) * (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<le> (z \<sqinter> y\<^sup>T * x) * (y\<^sup>T \<sqinter> x\<^sup>T * z)"
     by (metis conv_dist_comp conv_involutive conv_order inf.boundedE inf.boundedI inf.cobounded1 inf.cobounded2 mult_right_isotone)
   also have "... = (z \<sqinter> z\<^sup>T * x) * (y\<^sup>T \<sqinter> x\<^sup>T * y)"
     by (smt (z3) assms(2,4) covector_inf_comp_3 inf.left_commute inf.sup_monoid.add_commute comp_associative conv_dist_comp conv_involutive)
   also have "... = (z \<sqinter> z\<^sup>T) * x * x\<^sup>T * (y \<sqinter> y\<^sup>T)"
     by (smt (z3) assms(2,4) comp_associative inf.sup_monoid.add_commute vector_covector vector_inf_comp)
   also have "... \<le> x * x\<^sup>T"
     by (metis assms(2-4) comp_associative comp_right_one coreflexive_comp_top_inf inf.coboundedI2 mult_right_isotone vector_covector)
   also have "... \<le> 1"
     by (simp add: assms(1))
   finally have 2: "(z \<sqinter> y\<^sup>T * x) * (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<le> 1"
     .
   have "(z \<sqinter> y\<^sup>T * x) * (-z \<sqinter> -y \<sqinter> x)\<^sup>T \<le> y\<^sup>T * x * (-y\<^sup>T \<sqinter> x\<^sup>T)"
     by (smt comp_isotone conv_complement conv_dist_inf inf.cobounded2 inf.sup_monoid.add_assoc)
   also have "... = y\<^sup>T * x * x\<^sup>T \<sqinter> -y\<^sup>T"
     by (simp add: inf.commute assms(2) covector_comp_inf vector_conv_compl)
   also have "... \<le> y\<^sup>T \<sqinter> -y\<^sup>T"
     by (metis assms(1) comp_associative comp_inf.mult_left_isotone comp_isotone comp_right_one mult_sub_right_one)
   finally have 3: "(z \<sqinter> y\<^sup>T * x) * (-z \<sqinter> -y \<sqinter> x)\<^sup>T \<le> 1"
     using pseudo_complement by fastforce
   have 4: "(-z \<sqinter> y \<sqinter> z\<^sup>T * x) * (z \<sqinter> y\<^sup>T * x)\<^sup>T \<le> 1"
     using 2 conv_dist_comp conv_order by force
   have 5: "(-z \<sqinter> y \<sqinter> z\<^sup>T * x) * (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<le> 1"
     by (simp add: assms(2) inf_assoc inf_left_commute injective_inf_closed)
   have "(-z \<sqinter> y \<sqinter> z\<^sup>T * x) * (-z \<sqinter> -y \<sqinter> x)\<^sup>T \<le> z\<^sup>T * x * (-z\<^sup>T \<sqinter> x\<^sup>T)"
     using comp_inf.mult_left_isotone comp_isotone conv_complement conv_dist_inf inf.cobounded1 inf.cobounded2 by auto
   also have "... = z\<^sup>T * x * x\<^sup>T \<sqinter> -z\<^sup>T"
     by (metis assms(4) covector_comp_inf inf.sup_monoid.add_commute vector_conv_compl)
   also have "... \<le> z\<^sup>T \<sqinter> -z\<^sup>T"
     by (metis assms(1) comp_associative comp_inf.mult_left_isotone comp_isotone comp_right_one mult_sub_right_one)
   finally have 6: "(-z \<sqinter> y \<sqinter> z\<^sup>T * x) * (-z \<sqinter> -y \<sqinter> x)\<^sup>T \<le> 1"
     using pseudo_complement by fastforce
   have 7: "(-z \<sqinter> -y \<sqinter> x) * (z \<sqinter> y\<^sup>T * x)\<^sup>T \<le> 1"
     using 3 conv_dist_comp coreflexive_symmetric by fastforce
   have 8: "(-z \<sqinter> -y \<sqinter> x) * (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<le> 1"
     using 6 conv_dist_comp coreflexive_symmetric by fastforce
   have 9: "(-z \<sqinter> -y \<sqinter> x) * (-z \<sqinter> -y \<sqinter> x)\<^sup>T \<le> 1"
     using assms(1) inf.sup_monoid.add_commute injective_inf_closed by auto
   have "(x[y\<longmapsto>x[[z]]])[z\<longmapsto>x[[y]]] = (z \<sqinter> y\<^sup>T * x) \<squnion> (-z \<sqinter> y \<sqinter> z\<^sup>T * x) \<squnion> (-z \<sqinter> -y \<sqinter> x)"
     by (simp add: comp_inf.comp_left_dist_sup conv_dist_comp inf_assoc sup_monoid.add_assoc)
   hence "((x[y\<longmapsto>x[[z]]])[z\<longmapsto>x[[y]]]) * ((x[y\<longmapsto>x[[z]]])[z\<longmapsto>x[[y]]])\<^sup>T = ((z \<sqinter> y\<^sup>T * x) \<squnion> (-z \<sqinter> y \<sqinter> z\<^sup>T * x) \<squnion> (-z \<sqinter> -y \<sqinter> x)) * ((z \<sqinter> y\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> -y \<sqinter> x)\<^sup>T)"
     by (simp add: conv_dist_sup)
   also have "... = (z \<sqinter> y\<^sup>T * x) * ((z \<sqinter> y\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> -y \<sqinter> x)\<^sup>T) \<squnion>
                    (-z \<sqinter> y \<sqinter> z\<^sup>T * x) * ((z \<sqinter> y\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> -y \<sqinter> x)\<^sup>T) \<squnion>
                    (-z \<sqinter> -y \<sqinter> x) * ((z \<sqinter> y\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> -y \<sqinter> x)\<^sup>T)"
     using mult_right_dist_sup by auto
   also have "... = (z \<sqinter> y\<^sup>T * x) * (z \<sqinter> y\<^sup>T * x)\<^sup>T \<squnion> (z \<sqinter> y\<^sup>T * x) * (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<squnion> (z \<sqinter> y\<^sup>T * x) * (-z \<sqinter> -y \<sqinter> x)\<^sup>T \<squnion>
                    (-z \<sqinter> y \<sqinter> z\<^sup>T * x) * (z \<sqinter> y\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> y \<sqinter> z\<^sup>T * x) * (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> y \<sqinter> z\<^sup>T * x) * (-z \<sqinter> -y \<sqinter> x)\<^sup>T \<squnion>
                    (-z \<sqinter> -y \<sqinter> x) * (z \<sqinter> y\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> -y \<sqinter> x) * (-z \<sqinter> y \<sqinter> z\<^sup>T * x)\<^sup>T \<squnion> (-z \<sqinter> -y \<sqinter> x) * (-z \<sqinter> -y \<sqinter> x)\<^sup>T"
     using mult_left_dist_sup sup.left_commute sup_commute by auto
   also have "... \<le> 1"
     using 1 2 3 4 5 6 7 8 9 by simp_all
   finally show ?thesis
     .
 qed
 
 lemma update_injective_swap_2:
   assumes "injective x"
     shows "injective ((x[y\<longmapsto>x[[bot]]])[bot\<longmapsto>x[[y]]])"
   by (simp add: assms inf.sup_monoid.add_commute injective_inf_closed)
 
 text \<open>Theorem 2.9\<close>
 
 lemma update_univalent_swap:
   assumes "univalent x"
       and "injective y"
       and "vector y"
       and "injective z"
       and "vector z"
     shows "univalent ((x[y\<longmapsto>x[[z]]])[z\<longmapsto>x[[y]]])"
   by (simp add: assms read_injective update_univalent)
 
 text \<open>Theorem 2.10\<close>
 
 lemma update_mapping_swap:
   assumes "mapping x"
       and "point y"
       and "point z"
     shows "mapping ((x[y\<longmapsto>x[[z]]])[z\<longmapsto>x[[y]]])"
   by (simp add: assms bijective_regular read_injective read_surjective update_total update_univalent)
 
 text \<open>Theorem 2.16 \<open>mapping_inf_point_arc\<close> has been moved to theory \<open>Relation_Algebras\<close> in entry \<open>Stone_Relation_Algebras\<close>\<close>
 
 end
 
 context stone_kleene_relation_algebra
 begin
 
 lemma omit_redundant_points_2:
   assumes "point p"
   shows "p \<sqinter> x\<^sup>\<star> = (p \<sqinter> 1) \<squnion> (p \<sqinter> x \<sqinter> -p\<^sup>T) * (x \<sqinter> -p\<^sup>T)\<^sup>\<star>"
 proof -
   let ?p = "p \<sqinter> 1"
   let ?np = "-p \<sqinter> 1"
   have 1: "p \<sqinter> x\<^sup>\<star> \<sqinter> 1 = p \<sqinter> 1"
     by (metis inf.le_iff_sup inf.left_commute inf.sup_monoid.add_commute star.circ_reflexive)
   have 2: "p \<sqinter> 1 \<sqinter> -p\<^sup>T = bot"
     by (smt (z3) inf_bot_right inf_commute inf_left_commute one_inf_conv p_inf)
   have "p \<sqinter> x\<^sup>\<star> \<sqinter> -1 = p \<sqinter> x\<^sup>\<star> \<sqinter> -p\<^sup>T"
     by (metis assms antisymmetric_inf_diversity inf.cobounded1 inf.sup_relative_same_increasing vector_covector)
   also have "... = (p \<sqinter> 1 \<sqinter> -p\<^sup>T) \<squnion> ((p \<sqinter> x) * (-p \<sqinter> x)\<^sup>\<star> \<sqinter> -p\<^sup>T)"
     by (simp add: assms omit_redundant_points comp_inf.semiring.distrib_right)
   also have "... = (p \<sqinter> x) * (-p \<sqinter> x)\<^sup>\<star> \<sqinter> -p\<^sup>T"
     using 2 by simp
   also have "... = ?p * x * (-p \<sqinter> x)\<^sup>\<star> \<sqinter> -p\<^sup>T"
     by (metis assms vector_export_comp_unit)
   also have "... = ?p * x * (?np * x)\<^sup>\<star> \<sqinter> -p\<^sup>T"
     by (metis assms vector_complement_closed vector_export_comp_unit)
   also have "... = ?p * x * (?np * x)\<^sup>\<star> * ?np"
     by (metis assms conv_complement covector_comp_inf inf.sup_monoid.add_commute mult_1_right one_inf_conv vector_conv_compl)
   also have "... = ?p * x * ?np * (x * ?np)\<^sup>\<star>"
     using star_slide mult_assoc by auto
   also have "... = (?p * x \<sqinter> -p\<^sup>T) * (x * ?np)\<^sup>\<star>"
     by (metis assms conv_complement covector_comp_inf inf.sup_monoid.add_commute mult_1_right one_inf_conv vector_conv_compl)
   also have "... = (?p * x \<sqinter> -p\<^sup>T) * (x \<sqinter> -p\<^sup>T)\<^sup>\<star>"
     by (metis assms conv_complement covector_comp_inf inf.sup_monoid.add_commute mult_1_right one_inf_conv vector_conv_compl)
   also have "... = (p \<sqinter> x \<sqinter> -p\<^sup>T) * (x \<sqinter> -p\<^sup>T)\<^sup>\<star>"
     by (metis assms vector_export_comp_unit)
   finally show ?thesis
     using 1 by (metis maddux_3_11_pp regular_one_closed)
 qed
 
 text \<open>Theorem 5.3\<close>
 
 lemma omit_redundant_points_3:
   assumes "point p"
   shows "p \<sqinter> x\<^sup>\<star> = (p \<sqinter> 1) \<squnion> (p \<sqinter> (x \<sqinter> -p\<^sup>T)\<^sup>+)"
   by (simp add: assms inf_assoc vector_inf_comp omit_redundant_points_2)
 
 text \<open>Theorem 6.1\<close>
 
 lemma even_odd_root:
   assumes "acyclic (x - 1)"
       and "regular x"
       and "univalent x"
     shows "(x * x)\<^sup>T\<^sup>\<star> \<sqinter> x\<^sup>T * (x * x)\<^sup>T\<^sup>\<star> = (1 \<sqinter> x) * ((x * x)\<^sup>T\<^sup>\<star> \<sqinter> x\<^sup>T * (x * x)\<^sup>T\<^sup>\<star>)"
 proof -
   have 1: "univalent (x * x)"
     by (simp add: assms(3) univalent_mult_closed)
   have "x \<sqinter> 1 \<le> top * (x \<sqinter> 1)"
     by (simp add: top_left_mult_increasing)
   hence "x \<sqinter> -(top * (x \<sqinter> 1)) \<le> x - 1"
     using assms(2) p_shunting_swap pp_dist_comp by auto
   hence "x\<^sup>\<star> * (x \<sqinter> -(top * (x \<sqinter> 1))) \<le> (x - 1)\<^sup>\<star> * (x - 1)"
     using mult_right_isotone reachable_without_loops by auto
   also have "... \<le> -1"
     by (simp add: assms(1) star_plus)
   finally have "(x \<sqinter> -(top * (x \<sqinter> 1)))\<^sup>T \<le> -x\<^sup>\<star>"
     using schroeder_4_p by force
   hence "x\<^sup>T \<sqinter> x\<^sup>\<star> \<le> (top * (x \<sqinter> 1))\<^sup>T"
     by (smt (z3) assms(2) conv_complement conv_dist_inf p_shunting_swap regular_closed_inf regular_closed_top regular_mult_closed regular_one_closed)
   also have "... = (1 \<sqinter> x) * top"
     by (metis conv_dist_comp conv_dist_inf inf_commute one_inf_conv symmetric_one_closed symmetric_top_closed)
   finally have 2: "(x\<^sup>T \<sqinter> x\<^sup>\<star>) * top \<le> (1 \<sqinter> x) * top"
     by (metis inf.orderE inf.orderI inf_commute inf_vector_comp)
   have "1 \<sqinter> x\<^sup>T\<^sup>+ \<le> (x\<^sup>T \<sqinter> 1 * x\<^sup>\<star>) * x\<^sup>T\<^sup>\<star>"
     by (metis conv_involutive conv_star_commute dedekind_2 inf_commute)
   also have "... \<le> (x\<^sup>T \<sqinter> x\<^sup>\<star>) * top"
     by (simp add: mult_right_isotone)
   also have "... \<le> (1 \<sqinter> x) * top"
     using 2 by simp
   finally have 3: "1 \<sqinter> x\<^sup>T\<^sup>+ \<le> (1 \<sqinter> x) * top"
     .
   have "x\<^sup>T \<sqinter> (x\<^sup>T * x\<^sup>T)\<^sup>+ = 1 * x\<^sup>T \<sqinter> (x\<^sup>T * x\<^sup>T)\<^sup>\<star> * x\<^sup>T * x\<^sup>T"
     using star_plus mult_assoc by auto
   also have "... = (1 \<sqinter> (x\<^sup>T * x\<^sup>T)\<^sup>\<star> * x\<^sup>T) * x\<^sup>T"
     using assms(3) injective_comp_right_dist_inf by force
   also have "... \<le> (1 \<sqinter> x\<^sup>T\<^sup>\<star> * x\<^sup>T) * x\<^sup>T"
     by (meson comp_inf.mult_right_isotone comp_isotone inf.eq_refl star.circ_square)
   also have "... \<le> (1 \<sqinter> x) * top * x\<^sup>T"
     using 3 by (simp add: mult_left_isotone star_plus)
   also have "... \<le> (1 \<sqinter> x) * top"
     by (simp add: comp_associative mult_right_isotone)
   finally have 4: "x\<^sup>T \<sqinter> (x\<^sup>T * x\<^sup>T)\<^sup>+ \<le> (1 \<sqinter> x) * top"
     .
   have "x\<^sup>T \<sqinter> (x\<^sup>T * x\<^sup>T)\<^sup>\<star> = (x\<^sup>T \<sqinter> 1) \<squnion> (x\<^sup>T \<sqinter> (x\<^sup>T * x\<^sup>T)\<^sup>+)"
     by (metis inf_sup_distrib1 star_left_unfold_equal)
   also have "... \<le> (1 \<sqinter> x) * top"
     using 4 by (metis inf.sup_monoid.add_commute le_supI one_inf_conv top_right_mult_increasing)
   finally have 4: "x\<^sup>T \<sqinter> (x\<^sup>T * x\<^sup>T)\<^sup>\<star> \<le> (1 \<sqinter> x) * top"
     .
   have "x\<^sup>T \<sqinter> (x * x)\<^sup>\<star> \<sqinter> -1 \<le> x\<^sup>T \<sqinter> x\<^sup>\<star> \<sqinter> -1"
     by (simp add: inf.coboundedI2 inf.sup_monoid.add_commute star.circ_square)
   also have "... = (x - 1)\<^sup>\<star> \<sqinter> (x - 1)\<^sup>T"
     using conv_complement conv_dist_inf inf_assoc inf_left_commute reachable_without_loops symmetric_one_closed by auto
   also have "... = bot"
     using assms(1) acyclic_star_below_complement_1 by auto
   finally have 5: "x\<^sup>T \<sqinter> (x * x)\<^sup>\<star> \<sqinter> -1 = bot"
     by (simp add: le_bot)
   have "x\<^sup>T \<sqinter> (x * x)\<^sup>\<star> = (x\<^sup>T \<sqinter> (x * x)\<^sup>\<star> \<sqinter> 1) \<squnion> (x\<^sup>T \<sqinter> (x * x)\<^sup>\<star> \<sqinter> -1)"
     by (metis maddux_3_11_pp regular_one_closed)
   also have "... = x\<^sup>T \<sqinter> (x * x)\<^sup>\<star> \<sqinter> 1"
     using 5 by simp
   also have "... = x\<^sup>T \<sqinter> 1"
     by (metis calculation comp_inf.semiring.distrib_left inf.sup_monoid.add_commute star.circ_transitive_equal star_involutive star_left_unfold_equal sup_inf_absorb)
   finally have "(x\<^sup>T \<sqinter> (x * x)\<^sup>\<star>) \<squnion> (x\<^sup>T \<sqinter> (x\<^sup>T * x\<^sup>T)\<^sup>\<star>) \<le> (1 \<sqinter> x) * top"
     using 4 inf.sup_monoid.add_commute one_inf_conv top_right_mult_increasing by auto
   hence "x\<^sup>T \<sqinter> ((x * x)\<^sup>\<star> \<squnion> (x * x)\<^sup>T\<^sup>\<star>) \<le> (1 \<sqinter> x) * top"
     by (simp add: comp_inf.semiring.distrib_left conv_dist_comp)
   hence 6: "x\<^sup>T \<sqinter> (x * x)\<^sup>T\<^sup>\<star> * (x * x)\<^sup>\<star> \<le> (1 \<sqinter> x) * top"
     using 1 by (simp add: cancel_separate_eq sup_commute)
   have "(x * x)\<^sup>T\<^sup>\<star> \<sqinter> x\<^sup>T * (x * x)\<^sup>T\<^sup>\<star> \<le> (x\<^sup>T \<sqinter> (x * x)\<^sup>T\<^sup>\<star> * (x * x)\<^sup>\<star>) * (x * x)\<^sup>T\<^sup>\<star>"
     by (metis conv_involutive conv_star_commute dedekind_2 inf_commute)
   also have "... \<le> (1 \<sqinter> x) * top * (x * x)\<^sup>T\<^sup>\<star>"
     using 6 by (simp add: mult_left_isotone)
   also have "... = (1 \<sqinter> x) * top"
     by (simp add: comp_associative star.circ_left_top)
   finally have "(x * x)\<^sup>T\<^sup>\<star> \<sqinter> x\<^sup>T * (x * x)\<^sup>T\<^sup>\<star> = (x * x)\<^sup>T\<^sup>\<star> \<sqinter> x\<^sup>T * (x * x)\<^sup>T\<^sup>\<star> \<sqinter> (1 \<sqinter> x) * top"
     using inf.order_iff by auto
   also have "... = (1 \<sqinter> x) * ((x * x)\<^sup>T\<^sup>\<star> \<sqinter> x\<^sup>T * (x * x)\<^sup>T\<^sup>\<star>)"
     by (metis coreflexive_comp_top_inf inf.cobounded1 inf.sup_monoid.add_commute)
   finally show ?thesis
     .
 qed
 
 lemma update_square_plus:
   "point y \<Longrightarrow> x[y\<longmapsto>x[[x[[y]]]]] \<le> x\<^sup>+"
   by (meson update_square comp_isotone dual_order.trans le_supI order_refl star.circ_increasing star.circ_mult_increasing)
 
 lemma update_square_ub_plus:
   "x[y\<longmapsto>(x * x)\<^sup>T] \<le> x\<^sup>+"
   by (simp add: comp_isotone inf.coboundedI2 star.circ_increasing star.circ_mult_increasing)
 
 text \<open>Theorem 6.2\<close>
 
 lemma acyclic_square:
   assumes "acyclic (x - 1)"
     shows "x * x \<sqinter> 1 = x \<sqinter> 1"
 proof (rule order.antisym)
   have "1 \<sqinter> x * x = 1 \<sqinter> ((x - 1) * x \<squnion> (x \<sqinter> 1) * x)"
     by (metis maddux_3_11_pp regular_one_closed semiring.distrib_right)
   also have "... \<le> 1 \<sqinter> ((x - 1) * x \<squnion> x)"
     by (metis inf.cobounded2 mult_1_left mult_left_isotone inf.sup_right_isotone semiring.add_left_mono)
   also have "... = 1 \<sqinter> ((x - 1) * (x - 1) \<squnion> (x - 1) * (x \<sqinter> 1) \<squnion> x)"
     by (metis maddux_3_11_pp mult_left_dist_sup regular_one_closed)
   also have "... \<le> (1 \<sqinter> (x - 1) * (x - 1)) \<squnion> (x - 1) * (x \<sqinter> 1) \<squnion> x"
     by (metis inf_le2 inf_sup_distrib1 semiring.add_left_mono sup_monoid.add_assoc)
   also have "... \<le> (1 \<sqinter> (x - 1)\<^sup>+) \<squnion> (x - 1) * (x \<sqinter> 1) \<squnion> x"
     by (metis comp_isotone inf.eq_refl inf.sup_right_isotone star.circ_increasing sup_monoid.add_commute sup_right_isotone)
   also have "... = (x - 1) * (x \<sqinter> 1) \<squnion> x"
     by (metis assms inf.le_iff_sup inf.sup_monoid.add_commute inf_import_p inf_p regular_one_closed sup_inf_absorb sup_monoid.add_commute)
   also have "... = x"
     by (metis comp_isotone inf.cobounded1 inf_le2 mult_1_right sup.absorb2)
   finally show "x * x \<sqinter> 1 \<le> x \<sqinter> 1"
     by (simp add: inf.sup_monoid.add_commute)
   show "x \<sqinter> 1 \<le> x * x \<sqinter> 1"
     by (metis coreflexive_idempotent inf_le1 inf_le2 le_infI mult_isotone)
 qed
 
 lemma diagonal_update_square_aux:
   assumes "acyclic (x - 1)"
       and "point y"
     shows "1 \<sqinter> y \<sqinter> y\<^sup>T * x * x = 1 \<sqinter> y \<sqinter> x"
 proof -
   have 1: "1 \<sqinter> y \<sqinter> x \<le> y\<^sup>T * x * x"
     by (metis comp_isotone coreflexive_idempotent inf.boundedE inf.cobounded1 inf.cobounded2 one_inf_conv)
   have "1 \<sqinter> y \<sqinter> y\<^sup>T * x * x = 1 \<sqinter> (y \<sqinter> y\<^sup>T) * x * x"
     by (simp add: assms(2) inf.sup_monoid.add_assoc vector_inf_comp)
   also have "... = 1 \<sqinter> (y \<sqinter> 1) * x * x"
     by (metis assms(2) inf.cobounded1 inf.sup_monoid.add_commute inf.sup_same_context one_inf_conv vector_covector)
   also have "... \<le> 1 \<sqinter> x * x"
     by (metis comp_left_subdist_inf inf.sup_right_isotone le_infE mult_left_isotone mult_left_one)
   also have "... \<le> x"
     using assms(1) acyclic_square inf.sup_monoid.add_commute by auto
   finally show ?thesis
     using 1 by (metis inf.absorb2 inf.left_commute inf.sup_monoid.add_commute)
 qed
 
 text \<open>Theorem 6.5\<close>
 
 lemma diagonal_update_square:
   assumes "acyclic (x - 1)"
       and "point y"
     shows "(x[y\<longmapsto>x[[x[[y]]]]]) \<sqinter> 1 = x \<sqinter> 1"
 proof -
   let ?xy = "x[[y]]"
   let ?xxy = "x[[?xy]]"
   let ?xyxxy = "x[y\<longmapsto>?xxy]"
   have "?xyxxy \<sqinter> 1 = ((y \<sqinter> y\<^sup>T * x * x) \<squnion> (-y \<sqinter> x)) \<sqinter> 1"
     by (simp add: conv_dist_comp)
   also have "... = (y \<sqinter> y\<^sup>T * x * x \<sqinter> 1) \<squnion> (-y \<sqinter> x \<sqinter> 1)"
     by (simp add: inf_sup_distrib2)
   also have "... = (y \<sqinter> x \<sqinter> 1) \<squnion> (-y \<sqinter> x \<sqinter> 1)"
     using assms by (smt (verit, ccfv_threshold) diagonal_update_square_aux find_set_precondition_def inf_assoc inf_commute)
   also have "... = x \<sqinter> 1"
     by (metis assms(2) bijective_regular comp_inf.mult_right_dist_sup inf.sup_monoid.add_commute maddux_3_11_pp)
   finally show ?thesis
     .
 qed
 
 text \<open>Theorem 6.6\<close>
 
 lemma fc_update_square:
   assumes "mapping x"
       and "point y"
     shows "fc (x[y\<longmapsto>x[[x[[y]]]]]) = fc x"
 proof (rule order.antisym)
   let ?xy = "x[[y]]"
   let ?xxy = "x[[?xy]]"
   let ?xyxxy = "x[y\<longmapsto>?xxy]"
   have 1: "y \<sqinter> y\<^sup>T * x * x \<le> x * x"
     by (smt (z3) assms(2) inf.cobounded2 inf.sup_monoid.add_commute inf.sup_same_context mult_1_left one_inf_conv vector_covector vector_inf_comp)
   have 2: "?xyxxy = (y \<sqinter> y\<^sup>T * x * x) \<squnion> (-y \<sqinter> x)"
     by (simp add: conv_dist_comp)
   also have "... \<le> x * x \<squnion> x"
     using 1 inf_le2 sup_mono by blast
   also have "... \<le> x\<^sup>\<star>"
     by (simp add: star.circ_increasing star.circ_mult_upper_bound)
   finally show "fc ?xyxxy \<le> fc x"
     by (metis comp_isotone conv_order conv_star_commute star_involutive star_isotone)
   have 3: "y \<sqinter> x \<sqinter> 1 \<le> fc ?xyxxy"
     using inf.coboundedI1 inf.sup_monoid.add_commute reflexive_mult_closed star.circ_reflexive by auto
   have 4: "y - 1 \<le> -y\<^sup>T"
     using assms(2) p_shunting_swap regular_one_closed vector_covector by auto
   have "y \<sqinter> x \<le> y\<^sup>T * x"
     by (simp add: assms(2) vector_restrict_comp_conv)
   also have "... \<le> y\<^sup>T * x * x * x\<^sup>T"
     by (metis assms(1) comp_associative mult_1_right mult_right_isotone total_var)
   finally have "y \<sqinter> x \<sqinter> -1 \<le> y \<sqinter> -y\<^sup>T \<sqinter> y\<^sup>T * x * x * x\<^sup>T"
     using 4 by (smt (z3) inf.cobounded1 inf.coboundedI2 inf.sup_monoid.add_assoc inf.sup_monoid.add_commute inf_greatest)
   also have "... = (y \<sqinter> y\<^sup>T * x * x) * x\<^sup>T \<sqinter> -y\<^sup>T"
     by (metis assms(2) inf.sup_monoid.add_assoc inf.sup_monoid.add_commute vector_inf_comp)
   also have "... = (y \<sqinter> y\<^sup>T * x * x) * (x\<^sup>T \<sqinter> -y\<^sup>T)"
     using assms(2) covector_comp_inf vector_conv_compl by auto
   also have "... = (y \<sqinter> y\<^sup>T * x * x) * (-y \<sqinter> x)\<^sup>T"
     by (simp add: conv_complement conv_dist_inf inf_commute)
   also have "... \<le> ?xyxxy * (-y \<sqinter> x)\<^sup>T"
     using 2 by (simp add: comp_left_increasing_sup)
   also have "... \<le> ?xyxxy * ?xyxxy\<^sup>T"
     by (simp add: conv_isotone mult_right_isotone)
   also have "... \<le> fc ?xyxxy"
     using comp_isotone star.circ_increasing by blast
   finally have 5: "y \<sqinter> x \<le> fc ?xyxxy"
     using 3 by (smt (z3) comp_inf.semiring.distrib_left inf.le_iff_sup maddux_3_11_pp regular_one_closed)
   have "x = (y \<sqinter> x) \<squnion> (-y \<sqinter> x)"
     by (metis assms(2) bijective_regular inf.sup_monoid.add_commute maddux_3_11_pp)
   also have "... \<le> fc ?xyxxy"
     using 5 dual_order.trans fc_increasing sup.cobounded2 sup_least by blast
   finally show "fc x \<le> fc ?xyxxy"
     by (smt (z3) assms fc_equivalence fc_isotone fc_wcc read_injective star.circ_decompose_9 star_decompose_1 update_univalent)
 qed
 
 text \<open>Theorem 6.2\<close>
 
 lemma acyclic_plus_loop:
   assumes "acyclic (x - 1)"
   shows "x\<^sup>+ \<sqinter> 1 = x \<sqinter> 1"
 proof -
   let ?r = "x \<sqinter> 1"
   let ?i = "x - 1"
   have "x\<^sup>+ \<sqinter> 1 = (?i \<squnion> ?r)\<^sup>+ \<sqinter> 1"
     by (metis maddux_3_11_pp regular_one_closed)
   also have "... = ((?i\<^sup>\<star> * ?r)\<^sup>\<star> * ?i\<^sup>+ \<squnion> (?i\<^sup>\<star> * ?r)\<^sup>+) \<sqinter> 1"
     using plus_sup by auto
   also have "... \<le> (?i\<^sup>+ \<squnion> (?i\<^sup>\<star> * ?r)\<^sup>+) \<sqinter> 1"
     by (metis comp_associative dual_order.eq_iff maddux_3_11_pp reachable_without_loops regular_one_closed star.circ_plus_same star.circ_sup_9)
   also have "... = (?i\<^sup>\<star> * ?r)\<^sup>+ \<sqinter> 1"
     by (smt (z3) assms comp_inf.mult_right_dist_sup inf.absorb2 inf.sup_monoid.add_commute inf_le2 maddux_3_11_pp pseudo_complement regular_one_closed)
   also have "... \<le> ?i\<^sup>\<star> * ?r \<sqinter> 1"
     by (metis comp_associative dual_order.eq_iff maddux_3_11_pp reachable_without_loops regular_one_closed star.circ_sup_9 star_slide)
   also have "... = (?r \<squnion> ?i\<^sup>+ * ?r) \<sqinter> 1"
     using comp_associative star.circ_loop_fixpoint sup_commute by force
   also have "... \<le> x \<squnion> (?i\<^sup>+ * ?r \<sqinter> 1)"
     by (metis comp_inf.mult_right_dist_sup inf.absorb1 inf.cobounded1 inf.cobounded2)
   also have "... \<le> x \<squnion> (-1 * ?r \<sqinter> 1)"
     by (meson assms comp_inf.comp_isotone mult_left_isotone order.refl semiring.add_left_mono)
   also have "... = x"
     by (metis comp_inf.semiring.mult_not_zero comp_right_one inf.cobounded2 inf_sup_absorb mult_right_isotone pseudo_complement sup.idem sup_inf_distrib1)
   finally show ?thesis
     by (meson inf.sup_same_context inf_le1 order_trans star.circ_mult_increasing)
 qed
 
 lemma star_irreflexive_part_eq:
   "x\<^sup>\<star> - 1 = (x - 1)\<^sup>+ - 1"
   by (metis reachable_without_loops star_plus_without_loops)
 
 text \<open>Theorem 6.3\<close>
 
 lemma star_irreflexive_part:
   "x\<^sup>\<star> - 1 \<le> (x - 1)\<^sup>+"
   using star_irreflexive_part_eq by auto
 
 lemma square_irreflexive_part:
   "x * x - 1 \<le> (x - 1)\<^sup>+"
 proof -
   have "x * x = (x \<sqinter> 1) * x \<squnion> (x - 1) * x"
     by (metis maddux_3_11_pp mult_right_dist_sup regular_one_closed)
   also have "... \<le> 1 * x \<squnion> (x - 1) * x"
     using comp_isotone inf.cobounded2 semiring.add_right_mono by blast
   also have "... \<le> 1 \<squnion> (x - 1) \<squnion> (x - 1) * x"
     by (metis inf.cobounded2 maddux_3_11_pp mult_1_left regular_one_closed sup_left_isotone)
   also have "... = (x - 1) * (x \<squnion> 1) \<squnion> 1"
     by (simp add: mult_left_dist_sup sup_assoc sup_commute)
   finally have "x * x - 1 \<le> (x - 1) * (x \<squnion> 1)"
     using shunting_var_p by auto
   also have "... = (x - 1) * (x - 1) \<squnion> (x - 1)"
     by (metis comp_right_one inf.sup_monoid.add_commute maddux_3_21_pp mult_left_dist_sup regular_one_closed sup_commute)
   also have "... \<le> (x - 1)\<^sup>+"
     by (metis mult_left_isotone star.circ_increasing star.circ_mult_increasing star.circ_plus_same sup.bounded_iff)
   finally show ?thesis
     .
 qed
 
 text \<open>Theorem 6.3\<close>
 
 lemma square_irreflexive_part_2:
   "x * x - 1 \<le> x\<^sup>\<star> - 1"
   using comp_inf.mult_left_isotone star.circ_increasing star.circ_mult_upper_bound by blast
 
 text \<open>Theorem 6.8\<close>
 
 lemma acyclic_update_square:
   assumes "acyclic (x - 1)"
   shows "acyclic ((x[y\<longmapsto>(x * x)\<^sup>T]) - 1)"
 proof -
   have "((x[y\<longmapsto>(x * x)\<^sup>T]) - 1)\<^sup>+ \<le> ((x \<squnion> x * x) - 1)\<^sup>+"
     by (metis comp_inf.mult_right_isotone comp_isotone inf.sup_monoid.add_commute star_isotone update_square_ub)
   also have "... = ((x - 1) \<squnion> (x * x - 1))\<^sup>+"
     using comp_inf.semiring.distrib_right by auto
   also have "... \<le> ((x - 1)\<^sup>+)\<^sup>+"
     by (smt (verit, del_insts) comp_isotone reachable_without_loops star.circ_mult_increasing star.circ_plus_same star.circ_right_slide star.circ_separate_5 star.circ_square star.circ_transitive_equal star.left_plus_circ sup.bounded_iff sup_ge1 square_irreflexive_part)
   also have "... \<le> -1"
     using assms by (simp add: acyclic_plus)
   finally show ?thesis
     .
 qed
 
 text \<open>Theorem 6.9\<close>
 
 lemma disjoint_set_forest_update_square:
   assumes "disjoint_set_forest x"
       and "vector y"
       and "regular y"
     shows "disjoint_set_forest (x[y\<longmapsto>(x * x)\<^sup>T])"
 proof (intro conjI)
   show "univalent (x[y\<longmapsto>(x * x)\<^sup>T])"
     using assms update_univalent mapping_mult_closed univalent_conv_injective by blast
   show "total (x[y\<longmapsto>(x * x)\<^sup>T])"
     using assms update_total total_conv_surjective total_mult_closed by blast
   show "acyclic ((x[y\<longmapsto>(x * x)\<^sup>T]) - 1)"
     using acyclic_update_square assms(1) by blast
 qed
 
 lemma disjoint_set_forest_update_square_point:
   assumes "disjoint_set_forest x"
       and "point y"
     shows "disjoint_set_forest (x[y\<longmapsto>(x * x)\<^sup>T])"
   using assms disjoint_set_forest_update_square bijective_regular by blast
 
 end
 
 section \<open>Verifying Further Operations on Disjoint-Set Forests\<close>
 
 text \<open>
 In this section we verify the init-sets, path-halving and path-splitting operations of disjoint-set forests.
 \<close>
 
 class choose_point =
   fixes choose_point :: "'a \<Rightarrow> 'a"
 
 text \<open>
 Using the \<open>choose_point\<close> operation we define a simple for-each-loop abstraction as syntactic sugar translated to a while-loop.
 Regular vector \<open>h\<close> describes the set of all elements that are yet to be processed.
 It is made explicit so that the invariant can refer to it.
 \<close>
 
 syntax
   "_Foreach" :: "idt \<Rightarrow> idt \<Rightarrow> 'assn \<Rightarrow> 'com \<Rightarrow> 'com"  ("(1FOREACH _/ USING _/ INV {_} //DO _ /OD)" [0,0,0,0] 61)
 translations "FOREACH x USING h INV { i } DO c OD" =>
   "h := CONST top;
    WHILE h \<noteq> CONST bot
      INV { CONST regular h \<and> CONST vector h \<and> i }
-     VAR { CONST card { x . CONST regular x \<and> x \<le> h } }
+     VAR { h\<down> }
       DO x := CONST choose_point h;
          c;
          h[x] := CONST bot
       OD"
 
 class stone_kleene_relation_algebra_choose_point_finite_regular = stone_kleene_relation_algebra + finite_regular_p_algebra + choose_point +
   assumes choose_point_point: "vector x \<Longrightarrow> x \<noteq> bot \<Longrightarrow> point (choose_point x)"
   assumes choose_point_decreasing: "choose_point x \<le> --x"
 begin
 
 subclass stone_kleene_relation_algebra_tarski_finite_regular
 proof unfold_locales
   fix x
   let ?p = "choose_point (x * top)"
   let ?q = "choose_point ((?p \<sqinter> x)\<^sup>T * top)"
   let ?y = "?p \<sqinter> ?q\<^sup>T"
   assume 1: "regular x" "x \<noteq> bot"
   hence 2: "x * top \<noteq> bot"
     using le_bot top_right_mult_increasing by auto
   hence 3: "point ?p"
     by (simp add: choose_point_point comp_associative)
   hence 4: "?p \<noteq> bot"
     using 2 mult_right_zero by force
   have "?p \<sqinter> x \<noteq> bot"
   proof
     assume "?p \<sqinter> x = bot"
     hence 5: "x \<le> -?p"
       using p_antitone_iff pseudo_complement by auto
     have "?p \<le> --(x * top)"
       by (simp add: choose_point_decreasing)
     also have "... \<le> --(-?p * top)"
       using 5 by (simp add: comp_isotone pp_isotone)
     also have "... = -?p * top"
       using regular_mult_closed by auto
     also have "... = -?p"
       using 3 vector_complement_closed by auto
     finally have "?p = bot"
       using inf_absorb2 by fastforce
     thus False
       using 4 by auto
   qed
   hence "(?p \<sqinter> x)\<^sup>T * top \<noteq> bot"
     by (metis comp_inf.semiring.mult_zero_left comp_right_one inf.sup_monoid.add_commute inf_top.left_neutral schroeder_1)
   hence "point ?q"
     using choose_point_point vector_top_closed mult_assoc by auto
   hence 6: "arc ?y"
     using 3 by (smt bijective_conv_mapping inf.sup_monoid.add_commute mapping_inf_point_arc)
   have "?q \<le> --((?p \<sqinter> x)\<^sup>T * top)"
     by (simp add: choose_point_decreasing)
   hence "?y \<le> ?p \<sqinter> --((?p \<sqinter> x)\<^sup>T * top)\<^sup>T"
     by (metis conv_complement conv_isotone inf.sup_right_isotone)
   also have "... = ?p \<sqinter> --(top * (?p \<sqinter> x))"
     by (simp add: conv_dist_comp)
   also have "... = ?p \<sqinter> top * (?p \<sqinter> x)"
     using 1 3 bijective_regular pp_dist_comp by auto
   also have "... = ?p \<sqinter> ?p\<^sup>T * x"
     using 3 by (metis comp_inf_vector conv_dist_comp inf.sup_monoid.add_commute inf_top_right symmetric_top_closed)
   also have "... = (?p \<sqinter> ?p\<^sup>T) * x"
     using 3 by (simp add: vector_inf_comp)
   also have "... \<le> 1 * x"
     using 3 point_antisymmetric mult_left_isotone by blast
   finally have "?y \<le> x"
     by simp
   thus "top * x * top = top"
     using 6 by (smt (verit, ccfv_SIG) mult_assoc le_iff_sup mult_left_isotone semiring.distrib_left sup.orderE top.extremum)
 qed
 
 subsection \<open>Init-Sets\<close>
 
 text \<open>
 A disjoint-set forest is initialised by applying \<open>make_set\<close> to each node.
 We prove that the resulting disjoint-set forest is the identity relation.
 \<close>
 
 theorem init_sets:
   "VARS h p x
   [ True ]
   FOREACH x
     USING h
       INV { p - h = 1 - h }
        DO p := make_set p x
        OD
   [ p = 1 \<and> disjoint_set_forest p \<and> h = bot ]"
 proof vcg_tc_simp
   fix h p
   let ?x = "choose_point h"
   let ?m = "make_set p ?x"
   assume 1: "regular h \<and> vector h \<and> p - h = 1 - h \<and> h \<noteq> bot"
   show "vector (-?x \<sqinter> h) \<and>
         ?m \<sqinter> (--?x \<squnion> -h) = 1 \<sqinter> (--?x \<squnion> -h) \<and>
-        card { x . regular x \<and> x \<le> -?x \<and> x \<le> h } < card { x . regular x \<and> x \<le> h }"
+        card { x . regular x \<and> x \<le> -?x \<and> x \<le> h } < h\<down>"
   proof (intro conjI)
     show "vector (-?x \<sqinter> h)"
       using 1 choose_point_point vector_complement_closed vector_inf_closed by blast
     have 2: "point ?x \<and> regular ?x"
       using 1 bijective_regular choose_point_point by blast
     have 3: "-h \<le> -?x"
       using choose_point_decreasing p_antitone_iff by auto
     have 4: "?x \<sqinter> ?m = ?x * ?x\<^sup>T \<and> -?x \<sqinter> ?m = -?x \<sqinter> p"
       using 1 choose_point_point make_set_function make_set_postcondition_def by auto
     have "?m \<sqinter> (--?x \<squnion> -h) = (?m \<sqinter> ?x) \<squnion> (?m - h)"
       using 2 comp_inf.comp_left_dist_sup by auto
     also have "... = ?x * ?x\<^sup>T \<squnion> (?m \<sqinter> -?x \<sqinter> -h)"
       using 3 4 by (smt (z3) inf_absorb2 inf_assoc inf_commute)
     also have "... = ?x * ?x\<^sup>T \<squnion> (1 - h)"
       using 1 3 4 inf.absorb2 inf.sup_monoid.add_assoc inf_commute by auto
     also have "... = (1 \<sqinter> ?x) \<squnion> (1 - h)"
       using 2 by (metis inf.cobounded2 inf.sup_same_context one_inf_conv vector_covector)
     also have "... = 1 \<sqinter> (--?x \<squnion> -h)"
       using 2 comp_inf.semiring.distrib_left by auto
     finally show "?m \<sqinter> (--?x \<squnion> -h) = 1 \<sqinter> (--?x \<squnion> -h)"
       .
     have 5: "\<not> ?x \<le> -?x"
       using 1 2 by (metis comp_commute_below_diversity conv_order inf.cobounded2 inf_absorb2 pseudo_complement strict_order_var top.extremum)
     have 6: "?x \<le> h"
       using 1 by (metis choose_point_decreasing)
-    show "card { x . regular x \<and> x \<le> -?x \<and> x \<le> h } < card { x . regular x \<and> x \<le> h }"
+    show "card { x . regular x \<and> x \<le> -?x \<and> x \<le> h } < h\<down>"
       apply (rule psubset_card_mono)
       using finite_regular apply simp
       using 2 5 6 by auto
   qed
 qed
 
 end
 
 subsection \<open>Path Halving\<close>
 
 text \<open>
 Path halving is a variant of the path compression technique.
 Similarly to path compression, we implement path halving independently of find-set, using a second while-loop which iterates over the same path to the root.
 We prove that path halving preserves the equivalence-relational semantics of the disjoint-set forest and also preserves the roots of the component trees.
 Additionally we prove the exact effect of path halving, which is to replace every other parent pointer with a pointer to the respective grandparent.
 \<close>
 
 context stone_kleene_relation_algebra_tarski_finite_regular
 begin
 
 definition "path_halving_invariant p x y p0 \<equiv> 
   find_set_precondition p x \<and> point y \<and> y \<le> p\<^sup>T\<^sup>\<star> * x \<and> y \<le> (p0 * p0)\<^sup>T\<^sup>\<star> * x \<and>
   p0[(p0 * p0)\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * y\<longmapsto>(p0 * p0)\<^sup>T] = p \<and>
   disjoint_set_forest p0"
 definition "path_halving_postcondition p x y p0 \<equiv> 
   path_compression_precondition p x y \<and> p \<sqinter> 1 = p0 \<sqinter> 1 \<and> fc p = fc p0 \<and>
   p0[(p0 * p0)\<^sup>T\<^sup>\<star> * x\<longmapsto>(p0 * p0)\<^sup>T] = p"
 
 lemma path_halving_invariant_aux_1:
   assumes "point x"
       and "point y"
       and "disjoint_set_forest p0"
   shows "p0 \<le> wcc (p0[(p0 * p0)\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * y\<longmapsto>(p0 * p0)\<^sup>T])"
 proof -
   let ?p2 = "p0 * p0"
   let ?p2t = "?p2\<^sup>T"
   let ?p2ts = "?p2t\<^sup>\<star>"
   let ?px = "?p2ts * x"
   let ?py = "-(p0\<^sup>T\<^sup>\<star> * y)"
   let ?pxy = "?px \<sqinter> ?py"
   let ?p = "p0[?pxy\<longmapsto>?p2t]"
   have 1: "regular ?pxy"
     using assms(1,3) bijective_regular find_set_precondition_def mapping_regular pp_dist_comp regular_closed_star regular_conv_closed path_halving_invariant_def by auto
   have 2: "vector x \<and> vector ?px \<and> vector ?py"
     using assms(1,2) find_set_precondition_def vector_complement_closed vector_mult_closed path_halving_invariant_def by auto
   have 3: "?pxy \<sqinter> p0 \<sqinter> -?p2 \<le> -?px\<^sup>T"
   proof -
     have 4: "injective x \<and> univalent ?p2 \<and> regular p0"
       using assms(1,3) find_set_precondition_def mapping_regular univalent_mult_closed path_halving_invariant_def by auto
     have "?p2\<^sup>\<star> * p0 \<sqinter> 1 \<le> p0\<^sup>+ \<sqinter> 1"
       using comp_inf.mult_left_isotone comp_isotone comp_right_one mult_sub_right_one star.circ_square star_slide by auto
     also have "... \<le> p0"
       using acyclic_plus_loop assms(3) path_halving_invariant_def by auto
     finally have 5: "?p2\<^sup>\<star> * p0 \<sqinter> 1 \<le> p0"
       .
     hence 6: "?p2ts * (1 - p0) \<le> -p0"
       by (smt (verit, ccfv_SIG) conv_star_commute dual_order.trans inf.sup_monoid.add_assoc order.refl p_antitone_iff pseudo_complement schroeder_4_p schroeder_6_p)
     have "?p2t\<^sup>+ * p0 \<sqinter> 1 = ?p2ts * p0\<^sup>T * (p0\<^sup>T * p0) \<sqinter> 1"
       by (metis conv_dist_comp star_plus mult_assoc)
     also have "... \<le> ?p2ts * p0\<^sup>T \<sqinter> 1"
       by (metis assms(3) comp_inf.mult_left_isotone comp_isotone comp_right_one mult_sub_right_one)
     also have "... \<le> p0"
       using 5 by (metis conv_dist_comp conv_star_commute inf_commute one_inf_conv star_slide)
     finally have "?p2t\<^sup>+ * p0 \<le> -1 \<squnion> p0"
       by (metis regular_one_closed shunting_var_p sup_commute)
     hence 7: "?p2\<^sup>+ * (1 - p0) \<le> -p0"
       by (smt (z3) conv_dist_comp conv_star_commute half_shunting inf.sup_monoid.add_assoc inf.sup_monoid.add_commute pseudo_complement schroeder_4_p schroeder_6_p star.circ_plus_same)
     have "(1 \<sqinter> ?px) * top * (1 \<sqinter> ?px \<sqinter> -p0) = ?px \<sqinter> top * (1 \<sqinter> ?px \<sqinter> -p0)"
       using 2 by (metis inf_commute vector_inf_one_comp mult_assoc)
     also have "... = ?px \<sqinter> ?px\<^sup>T * (1 - p0)"
       using 2 by (smt (verit, ccfv_threshold) covector_inf_comp_3 inf.sup_monoid.add_assoc inf.sup_monoid.add_commute inf_top.left_neutral)
     also have "... = ?px \<sqinter> x\<^sup>T * ?p2\<^sup>\<star> * (1 - p0)"
       by (simp add: conv_dist_comp conv_star_commute)
     also have "... = (?px \<sqinter> x\<^sup>T) * ?p2\<^sup>\<star> * (1 - p0)"
       using 2 vector_inf_comp by auto
     also have "... = ?p2ts * (x * x\<^sup>T) * ?p2\<^sup>\<star> * (1 - p0)"
       using 2 vector_covector mult_assoc by auto
     also have "... \<le> ?p2ts * ?p2\<^sup>\<star> * (1 - p0)"
       using 4 by (metis inf.order_lesseq_imp mult_left_isotone star.circ_mult_upper_bound star.circ_reflexive)
     also have "... = (?p2ts \<squnion> ?p2\<^sup>\<star>) * (1 - p0)"
       using 4 by (simp add: cancel_separate_eq)
     also have "... = (?p2ts \<squnion> ?p2\<^sup>+) * (1 - p0)"
       by (metis star.circ_plus_one star_plus_loops sup_assoc sup_commute)
     also have "... \<le> -p0"
       using 6 7 by (simp add: mult_right_dist_sup)
     finally have "(1 \<sqinter> ?px)\<^sup>T * p0 * (1 \<sqinter> ?px \<sqinter> -p0)\<^sup>T \<le> bot"
       by (smt (z3) inf.boundedI inf_p top.extremum triple_schroeder_p)
     hence 8: "(1 \<sqinter> ?px) * p0 * (1 \<sqinter> ?px \<sqinter> -p0) = bot"
       by (simp add: coreflexive_inf_closed coreflexive_symmetric le_bot)
     have "?px \<sqinter> p0 \<sqinter> ?px\<^sup>T = (1 \<sqinter> ?px) * p0 \<sqinter> ?px\<^sup>T"
       using 2 inf_commute vector_inf_one_comp by fastforce
     also have "... = (1 \<sqinter> ?px) * p0 * (1 \<sqinter> ?px)"
       using 2 by (metis comp_inf_vector mult_1_right vector_conv_covector)
     also have "... = (1 \<sqinter> ?px) * p0 * (1 \<sqinter> ?px \<sqinter> p0) \<squnion> (1 \<sqinter> ?px) * p0 * (1 \<sqinter> ?px \<sqinter> -p0)"
       using 4 by (metis maddux_3_11_pp mult_left_dist_sup)
     also have "... = (1 \<sqinter> ?px) * p0 * (1 \<sqinter> ?px \<sqinter> p0)"
       using 8 by simp
     also have "... \<le> ?p2"
       by (metis comp_isotone coreflexive_comp_top_inf inf.cobounded1 inf.cobounded2)
     finally have "?px \<sqinter> p0 \<sqinter> -?p2 \<le> -?px\<^sup>T"
       using 4 p_shunting_swap regular_mult_closed by fastforce
     thus ?thesis
       by (meson comp_inf.mult_left_isotone dual_order.trans inf.cobounded1)
   qed
   have "p0 \<le> ?p2 * p0\<^sup>T"
     by (metis assms(3) comp_associative comp_isotone comp_right_one eq_refl total_var)
   hence "?pxy \<sqinter> p0 \<sqinter> -?p2 \<le> ?p2 * p0\<^sup>T"
     by (metis inf.coboundedI1 inf.sup_monoid.add_commute)
   hence "?pxy \<sqinter> p0 \<sqinter> -?p2 \<le> ?pxy \<sqinter> ?p2 * p0\<^sup>T \<sqinter> -?px\<^sup>T"
     using 3 by (meson dual_order.trans inf.boundedI inf.cobounded1)
   also have "... = (?pxy \<sqinter> ?p2) * p0\<^sup>T \<sqinter> -?px\<^sup>T"
     using 2 vector_inf_comp by auto
   also have "... = (?pxy \<sqinter> ?p2) * (-?px \<sqinter> p0)\<^sup>T"
     using 2 by (simp add: covector_comp_inf inf.sup_monoid.add_commute vector_conv_compl conv_complement conv_dist_inf)
   also have "... \<le> ?p * (-?px \<sqinter> p0)\<^sup>T"
     using comp_left_increasing_sup by auto
   also have "... \<le> ?p * ?p\<^sup>T"
     by (metis comp_inf.mult_right_isotone comp_isotone conv_isotone inf.eq_refl inf.sup_monoid.add_commute le_supI1 p_antitone_inf sup_commute)
   also have "... \<le> wcc ?p"
     using star.circ_sub_dist_2 by auto
   finally have 9: "?pxy \<sqinter> p0 \<sqinter> -?p2 \<le> wcc ?p"
     .
   have "p0 = (?pxy \<sqinter> p0) \<squnion> (-?pxy \<sqinter> p0)"
     using 1 by (metis inf.sup_monoid.add_commute maddux_3_11_pp)
   also have "... \<le> (?pxy \<sqinter> p0) \<squnion> ?p"
     using sup_right_isotone by auto
   also have "... = (?pxy \<sqinter> p0 \<sqinter> -?p2) \<squnion> (?pxy \<sqinter> p0 \<sqinter> ?p2) \<squnion> ?p"
     by (smt (z3) assms(3) maddux_3_11_pp mapping_regular pp_dist_comp path_halving_invariant_def)
   also have "... \<le> (?pxy \<sqinter> p0 \<sqinter> -?p2) \<squnion> (?pxy \<sqinter> ?p2) \<squnion> ?p"
     by (meson comp_inf.comp_left_subdist_inf inf.boundedE semiring.add_left_mono semiring.add_right_mono)
   also have "... = (?pxy \<sqinter> p0 \<sqinter> -?p2) \<squnion> ?p"
     using sup_assoc by auto
   also have "... \<le> wcc ?p \<squnion> ?p"
     using 9 sup_left_isotone by blast
   also have "... \<le> wcc ?p"
     by (simp add: star.circ_sub_dist_1)
   finally show ?thesis
     .
 qed
 
 lemma path_halving_invariant_aux:
   assumes "path_halving_invariant p x y p0"
   shows "p[[y]] = p0[[y]]"
     and "p[[p[[y]]]] = p0[[p0[[y]]]]"
     and "p[[p[[p[[y]]]]]] = p0[[p0[[p0[[y]]]]]]"
     and "p \<sqinter> 1 = p0 \<sqinter> 1"
     and "fc p = fc p0"
 proof -
   let ?p2 = "p0 * p0"
   let ?p2t = "?p2\<^sup>T"
   let ?p2ts = "?p2t\<^sup>\<star>"
   let ?px = "?p2ts * x"
   let ?py = "-(p0\<^sup>T\<^sup>\<star> * y)"
   let ?pxy = "?px \<sqinter> ?py"
   let ?p = "p0[?pxy\<longmapsto>?p2t]"
   have "?p[[y]] = p0[[y]]"
     apply (rule put_get_different_vector)
     using assms find_set_precondition_def vector_complement_closed vector_inf_closed vector_mult_closed path_halving_invariant_def apply force
     by (meson inf.cobounded2 order_lesseq_imp p_antitone_iff path_compression_1b)
   thus 1: "p[[y]] = p0[[y]]"
     using assms path_halving_invariant_def by auto
   have "?p[[p0[[y]]]] = p0[[p0[[y]]]]"
     apply (rule put_get_different_vector)
     using assms find_set_precondition_def vector_complement_closed vector_inf_closed vector_mult_closed path_halving_invariant_def apply force
     by (metis comp_isotone inf.boundedE inf.coboundedI2 inf.eq_refl p_antitone_iff selection_closed_id star.circ_increasing)
   thus 2: "p[[p[[y]]]] = p0[[p0[[y]]]]"
     using 1 assms path_halving_invariant_def by auto
   have "?p[[p0[[p0[[y]]]]]] = p0[[p0[[p0[[y]]]]]]"
     apply (rule put_get_different_vector)
     using assms find_set_precondition_def vector_complement_closed vector_inf_closed vector_mult_closed path_halving_invariant_def apply force
     by (metis comp_associative comp_isotone conv_dist_comp conv_involutive conv_order inf.coboundedI2 inf.le_iff_sup mult_left_isotone p_antitone_iff p_antitone_inf star.circ_increasing star.circ_transitive_equal)
   thus "p[[p[[p[[y]]]]]] = p0[[p0[[p0[[y]]]]]]"
     using 2 assms path_halving_invariant_def by auto
   have 3: "regular ?pxy"
     using assms bijective_regular find_set_precondition_def mapping_regular pp_dist_comp regular_closed_star regular_conv_closed path_halving_invariant_def by auto
   have "p \<sqinter> 1 = ?p \<sqinter> 1"
     using assms path_halving_invariant_def by auto
   also have "... = (?pxy \<sqinter> ?p2 \<sqinter> 1) \<squnion> (-?pxy \<sqinter> p0 \<sqinter> 1)"
     using comp_inf.semiring.distrib_right conv_involutive by auto
   also have "... = (?pxy \<sqinter> p0 \<sqinter> 1) \<squnion> (-?pxy \<sqinter> p0 \<sqinter> 1)"
     using assms acyclic_square path_halving_invariant_def inf.sup_monoid.add_assoc by auto
   also have "... = (?pxy \<squnion> -?pxy) \<sqinter> p0 \<sqinter> 1"
     using inf_sup_distrib2 by auto
   also have "... = p0 \<sqinter> 1"
     using 3 by (metis inf.sup_monoid.add_commute inf_sup_distrib1 maddux_3_11_pp)
   finally show "p \<sqinter> 1 = p0 \<sqinter> 1"
     .
   have "p \<le> p0\<^sup>+"
     by (metis assms path_halving_invariant_def update_square_ub_plus)
   hence 4: "fc p \<le> fc p0"
     using conv_plus_commute fc_isotone star.left_plus_circ by fastforce
   have "wcc p0 \<le> wcc ?p"
     by (meson assms wcc_below_wcc path_halving_invariant_aux_1 path_halving_invariant_def find_set_precondition_def)
   hence "fc p0 \<le> fc ?p"
     using assms find_set_precondition_def path_halving_invariant_def fc_wcc by auto
   thus "fc p = fc p0"
     using 4 assms path_halving_invariant_def by auto
 qed
 
 lemma path_halving_1:
   "find_set_precondition p0 x \<Longrightarrow> path_halving_invariant p0 x x p0"
 proof -
   assume 1: "find_set_precondition p0 x"
   show "path_halving_invariant p0 x x p0"
   proof (unfold path_halving_invariant_def, intro conjI)
     show "find_set_precondition p0 x"
       using 1 by simp
     show "vector x" "injective x" "surjective x"
       using 1 find_set_precondition_def by auto
     show "x \<le> p0\<^sup>T\<^sup>\<star> * x"
       by (simp add: path_compression_1b)
     show "x \<le> (p0 * p0)\<^sup>T\<^sup>\<star> * x"
       by (simp add: path_compression_1b)
     have "(p0 * p0)\<^sup>T\<^sup>\<star> * x \<le> p0\<^sup>T\<^sup>\<star> * x"
       by (simp add: conv_dist_comp mult_left_isotone star.circ_square)
     thus "p0[(p0 * p0)\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * x\<longmapsto>(p0 * p0)\<^sup>T] = p0"
       by (smt (z3) inf.le_iff_sup inf_commute maddux_3_11_pp p_antitone_inf pseudo_complement)
     show "univalent p0" "total p0" "acyclic (p0 - 1)"
       using 1 find_set_precondition_def by auto
   qed
 qed
 
 lemma path_halving_2:
-  "path_halving_invariant p x y p0 \<and> y \<noteq> p[[y]] \<Longrightarrow> path_halving_invariant (p[y\<longmapsto>p[[p[[y]]]]]) x ((p[y\<longmapsto>p[[p[[y]]]]])[[y]]) p0 \<and> card { z . regular z \<and> z \<le> (p[y\<longmapsto>p[[p[[y]]]]])\<^sup>T\<^sup>\<star> * ((p[y\<longmapsto>p[[p[[y]]]]])[[y]]) } < card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y }"
+  "path_halving_invariant p x y p0 \<and> y \<noteq> p[[y]] \<Longrightarrow> path_halving_invariant (p[y\<longmapsto>p[[p[[y]]]]]) x ((p[y\<longmapsto>p[[p[[y]]]]])[[y]]) p0 \<and> ((p[y\<longmapsto>p[[p[[y]]]]])\<^sup>T\<^sup>\<star> * ((p[y\<longmapsto>p[[p[[y]]]]])[[y]]))\<down> < (p\<^sup>T\<^sup>\<star> * y)\<down>"
 proof -
   let ?py = "p[[y]]"
   let ?ppy = "p[[?py]]"
   let ?pyppy = "p[y\<longmapsto>?ppy]"
   let ?p2 = "p0 * p0"
   let ?p2t = "?p2\<^sup>T"
   let ?p2ts = "?p2t\<^sup>\<star>"
   let ?px = "?p2ts * x"
   let ?py2 = "-(p0\<^sup>T\<^sup>\<star> * y)"
   let ?pxy = "?px \<sqinter> ?py2"
   let ?p = "p0[?pxy\<longmapsto>?p2t]"
   let ?pty = "p0\<^sup>T * y"
   let ?pt2y = "p0\<^sup>T * p0\<^sup>T * y"
   let ?pt2sy = "p0\<^sup>T\<^sup>\<star> * p0\<^sup>T * p0\<^sup>T * y"
   assume 1: "path_halving_invariant p x y p0 \<and> y \<noteq> ?py"
   have 2: "point ?pty \<and> point ?pt2y"
     using 1 by (smt (verit) comp_associative read_injective read_surjective path_halving_invariant_def)
-  show "path_halving_invariant ?pyppy x (?pyppy[[y]]) p0 \<and> card { z . regular z \<and> z \<le> ?pyppy\<^sup>T\<^sup>\<star> * (?pyppy[[y]]) } < card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y }"
+  show "path_halving_invariant ?pyppy x (?pyppy[[y]]) p0 \<and> (?pyppy\<^sup>T\<^sup>\<star> * (?pyppy[[y]]))\<down> < (p\<^sup>T\<^sup>\<star> * y)\<down>"
   proof
     show "path_halving_invariant ?pyppy x (?pyppy[[y]]) p0"
     proof (unfold path_halving_invariant_def, intro conjI)
       show 3: "find_set_precondition ?pyppy x"
       proof (unfold find_set_precondition_def, intro conjI)
         show "univalent ?pyppy"
           using 1 find_set_precondition_def read_injective update_univalent path_halving_invariant_def by auto
         show "total ?pyppy"
           using 1 bijective_regular find_set_precondition_def read_surjective update_total path_halving_invariant_def by force
         show "acyclic (?pyppy - 1)"
           apply (rule update_acyclic_3)
           using 1 find_set_precondition_def path_halving_invariant_def apply blast
           using 1 2 comp_associative path_halving_invariant_aux(2) apply force
           using 1 path_halving_invariant_def apply blast
           by (metis inf.order_lesseq_imp mult_isotone star.circ_increasing star.circ_square mult_assoc)
         show "vector x" "injective x" "surjective x"
           using 1 find_set_precondition_def path_halving_invariant_def by auto
       qed
       show "vector (?pyppy[[y]])"
         using 1 comp_associative path_halving_invariant_def by auto
       show "injective (?pyppy[[y]])"
         using 1 3 read_injective path_halving_invariant_def find_set_precondition_def by auto
       show "surjective (?pyppy[[y]])"
         using 1 3 read_surjective path_halving_invariant_def find_set_precondition_def by auto
       show "?pyppy[[y]] \<le> ?pyppy\<^sup>T\<^sup>\<star> * x"
       proof -
         have "y = (y \<sqinter> p\<^sup>T\<^sup>\<star>) * x"
           using 1 le_iff_inf vector_inf_comp path_halving_invariant_def by auto
         also have "... = ((y \<sqinter> 1) \<squnion> (y \<sqinter> (p\<^sup>T \<sqinter> -y\<^sup>T)\<^sup>+)) * x"
           using 1 omit_redundant_points_3 path_halving_invariant_def by auto
         also have "... \<le> (1 \<squnion> (y \<sqinter> (p\<^sup>T \<sqinter> -y\<^sup>T)\<^sup>+)) * x"
           using 1 sup_inf_distrib2 vector_inf_comp path_halving_invariant_def by auto
         also have "... \<le> (1 \<squnion> (p\<^sup>T \<sqinter> -y\<^sup>T)\<^sup>+) * x"
           by (simp add: inf.coboundedI2 mult_left_isotone)
         also have "... = (p \<sqinter> -y)\<^sup>T\<^sup>\<star> * x"
           by (simp add: conv_complement conv_dist_inf star_left_unfold_equal)
         also have "... \<le> ?pyppy\<^sup>T\<^sup>\<star> * x"
           by (simp add: conv_isotone inf.sup_monoid.add_commute mult_left_isotone star_isotone)
         finally show ?thesis
           by (metis mult_isotone star.circ_increasing star.circ_transitive_equal mult_assoc)
       qed
       show "?pyppy[[y]] \<le> ?px"
       proof -
         have "?pyppy[[y]] = p[[?py]]"
           using 1 put_get vector_mult_closed path_halving_invariant_def by force
         also have "... = p0[[p0[[y]]]]"
           using 1 path_halving_invariant_aux(2) by blast
         also have "... = ?p2t * y"
           by (simp add: conv_dist_comp mult_assoc)
         also have "... \<le> ?p2t * ?px"
           using 1 path_halving_invariant_def comp_associative mult_right_isotone by force
         also have "... \<le> ?px"
           by (metis comp_associative mult_left_isotone star.left_plus_below_circ)
         finally show ?thesis
           .
       qed
       show "p0[?px - p0\<^sup>T\<^sup>\<star> * (?pyppy[[y]])\<longmapsto>?p2t] = ?pyppy"
       proof -
         have "?px \<sqinter> ?pty = ?px \<sqinter> p0\<^sup>T * ?px \<sqinter> ?pty"
           using 1 inf.absorb2 inf.sup_monoid.add_assoc mult_right_isotone path_halving_invariant_def by force
         also have "... = (?p2ts \<sqinter> p0\<^sup>T * ?p2ts) * x \<sqinter> ?pty"
           using 3 comp_associative find_set_precondition_def injective_comp_right_dist_inf by auto
         also have "... = (1 \<sqinter> p0) * (?p2ts \<sqinter> p0\<^sup>T * ?p2ts) * x \<sqinter> ?pty"
           using 1 even_odd_root mapping_regular path_halving_invariant_def by auto
         also have "... \<le> (1 \<sqinter> p0) * top \<sqinter> ?pty"
           by (metis comp_associative comp_inf.mult_left_isotone comp_inf.star.circ_sub_dist_2 comp_left_subdist_inf dual_order.trans mult_right_isotone)
         also have 4: "... = (1 \<sqinter> p0\<^sup>T) * ?pty"
           using coreflexive_comp_top_inf one_inf_conv by auto
         also have "... \<le> ?pt2y"
           by (simp add: mult_assoc mult_left_isotone)
         finally have 5: "?px \<sqinter> ?pty \<le> ?pt2y"
           .
         have 6: "p[?px \<sqinter> -?pt2sy \<sqinter> ?pty\<longmapsto>?p2t] = p"
         proof (cases "?pty \<le> ?px \<sqinter> -?pt2sy")
           case True
           hence "?pty \<le> ?pt2y"
             using 5 conv_dist_comp inf.absorb2 by auto
           hence 7: "?pty = ?pt2y"
             using 2 epm_3 by fastforce
           have "p[?px \<sqinter> -?pt2sy \<sqinter> ?pty\<longmapsto>?p2t] = p[?pty\<longmapsto>?p2t]"
             using True inf.absorb2 by auto
           also have "... = p[?pty\<longmapsto>?p2[[?pty]]]"
             using 2 update_point_get by auto
           also have "... = p[?pty\<longmapsto>p0\<^sup>T * p0\<^sup>T * p0\<^sup>T * y]"
             using comp_associative conv_dist_comp by auto
           also have "... = p[?pty\<longmapsto>?pt2y]"
             using 7 mult_assoc by simp
           also have "... = p[?pty\<longmapsto>p[[?pty]]]"
             using 1 path_halving_invariant_aux(1,2) mult_assoc by force
           also have "... = p"
             using 2 get_put by auto
           finally show ?thesis
             .
         next
           case False
           have "mapping ?p2"
             using 1 mapping_mult_closed path_halving_invariant_def by blast
           hence 8: "regular (?px \<sqinter> -?pt2sy)"
             using 1 bijective_regular find_set_precondition_def mapping_regular pp_dist_comp regular_closed_star regular_conv_closed path_halving_invariant_def by auto
           have "vector (?px \<sqinter> -?pt2sy)"
             using 1 find_set_precondition_def vector_complement_closed vector_inf_closed vector_mult_closed path_halving_invariant_def by force
           hence "?pty \<le> -(?px \<sqinter> -?pt2sy)"
             using 2 8 point_in_vector_or_complement False by blast
           hence "?px \<sqinter> -?pt2sy \<sqinter> ?pty = bot"
             by (simp add: p_antitone_iff pseudo_complement)
           thus ?thesis
             by simp
         qed
         have 9: "p[?px \<sqinter> -?pt2sy \<sqinter> y\<longmapsto>?p2t] = ?pyppy"
         proof (cases "y \<le> -?pt2sy")
           case True
           hence "p[?px \<sqinter> -?pt2sy \<sqinter> y\<longmapsto>?p2t] = p[y\<longmapsto>?p2t]"
             using 1 inf.absorb2 path_halving_invariant_def by auto
           also have "... = ?pyppy"
             using 1 by (metis comp_associative conv_dist_comp path_halving_invariant_aux(2) path_halving_invariant_def update_point_get)
           finally show ?thesis
             .
         next
           case False
           have "vector (-?pt2sy)"
             using 1 vector_complement_closed vector_mult_closed path_halving_invariant_def by blast
           hence 10: "y \<le> ?pt2sy"
             using 1 by (smt (verit, del_insts) False bijective_regular point_in_vector_or_complement regular_closed_star regular_mult_closed total_conv_surjective univalent_conv_injective path_halving_invariant_def)
           hence "?px \<sqinter> -?pt2sy \<sqinter> y = bot"
             by (simp add: inf.coboundedI2 p_antitone pseudo_complement)
           hence 11: "p[?px \<sqinter> -?pt2sy \<sqinter> y\<longmapsto>?p2t] = p"
             by simp
           have "y \<le> p0\<^sup>T\<^sup>+ * y"
             using 10 by (metis mult_left_isotone order_lesseq_imp star.circ_plus_same star.left_plus_below_circ)
           hence 12: "y = root p0 y"
             using 1 loop_root_2 path_halving_invariant_def by blast
           have "?pyppy = p[y\<longmapsto>p0[[p0[[y]]]]]"
             using 1 path_halving_invariant_aux(2) by force
           also have "... = p[y\<longmapsto>p0[[y]]]"
             using 1 12 by (metis root_successor_loop path_halving_invariant_def)
           also have "... = p[y\<longmapsto>?py]"
             using 1 path_halving_invariant_aux(1) by force
           also have "... = p"
             using 1 get_put path_halving_invariant_def by blast
           finally show ?thesis
             using 11 by simp
         qed
         have 13: "-?pt2sy = -(p0\<^sup>T\<^sup>\<star> * y) \<squnion> (-?pt2sy \<sqinter> ?pty) \<squnion> (-?pt2sy \<sqinter> y)"
         proof (rule order.antisym)
           have 14: "regular (p0\<^sup>T\<^sup>\<star> * y) \<and> regular ?pt2sy"
             using 1 by (metis order.antisym conv_complement conv_dist_comp conv_involutive conv_star_commute forest_components_increasing mapping_regular pp_dist_star regular_mult_closed top.extremum path_halving_invariant_def)
           have "p0\<^sup>T\<^sup>\<star> = p0\<^sup>T\<^sup>\<star> * p0\<^sup>T * p0\<^sup>T \<squnion> p0\<^sup>T \<squnion> 1"
             using star.circ_back_loop_fixpoint star.circ_plus_same star_left_unfold_equal sup_commute by auto
           hence "p0\<^sup>T\<^sup>\<star> * y \<le> ?pt2sy \<squnion> ?pty \<squnion> y"
             by (metis inf.eq_refl mult_1_left mult_right_dist_sup)
           also have "... = ?pt2sy \<squnion> (-?pt2sy \<sqinter> ?pty) \<squnion> y"
             using 14 by (metis maddux_3_21_pp)
           also have "... = ?pt2sy \<squnion> (-?pt2sy \<sqinter> ?pty) \<squnion> (-?pt2sy \<sqinter> y)"
             using 14 by (smt (z3) maddux_3_21_pp sup.left_commute sup_assoc)
           hence "p0\<^sup>T\<^sup>\<star> * y \<sqinter> -?pt2sy \<le> (-?pt2sy \<sqinter> ?pty) \<squnion> (-?pt2sy \<sqinter> y)"
             using calculation half_shunting sup_assoc sup_commute by auto
           thus "-?pt2sy \<le> -(p0\<^sup>T\<^sup>\<star> * y) \<squnion> (-?pt2sy \<sqinter> ?pty) \<squnion> (-?pt2sy \<sqinter> y)"
             using 14 by (smt (z3) inf.sup_monoid.add_commute shunting_var_p sup.left_commute sup_commute)
           have "-(p0\<^sup>T\<^sup>\<star> * y) \<le> -?pt2sy"
             by (meson mult_left_isotone order.trans p_antitone star.right_plus_below_circ)
           thus "-(p0\<^sup>T\<^sup>\<star> * y) \<squnion> (-?pt2sy \<sqinter> ?pty) \<squnion> (-?pt2sy \<sqinter> y) \<le> -?pt2sy"
             by simp
         qed
         have "regular ?px" "regular ?pty" "regular y"
           using 1 bijective_regular find_set_precondition_def mapping_regular pp_dist_comp regular_closed_star regular_conv_closed path_halving_invariant_def by auto
         hence 15: "regular (?px \<sqinter> -?pt2sy \<sqinter> ?pty)" "regular (?px \<sqinter> -?pt2sy \<sqinter> y)"
           by auto
         have "p0[?px - p0\<^sup>T\<^sup>\<star> * (?pyppy[[y]])\<longmapsto>?p2t] = p0[?px - p0\<^sup>T\<^sup>\<star> * (p[[?py]])\<longmapsto>?p2t]"
           using 1 put_get vector_mult_closed path_halving_invariant_def by auto
         also have "... = p0[?px - ?pt2sy\<longmapsto>?p2t]"
           using 1 comp_associative path_halving_invariant_aux(2) by force
         also have "... = p0[?pxy \<squnion> (?px \<sqinter> -?pt2sy \<sqinter> ?pty) \<squnion> (?px \<sqinter> -?pt2sy \<sqinter> y)\<longmapsto>?p2t]"
           using 13 by (metis comp_inf.semiring.distrib_left inf.sup_monoid.add_assoc)
         also have "... = (?p[?px \<sqinter> -?pt2sy \<sqinter> ?pty\<longmapsto>?p2t])[?px \<sqinter> -?pt2sy \<sqinter> y\<longmapsto>?p2t]"
           using 15 by (smt (z3) update_same_3 comp_inf.semiring.mult_not_zero inf.sup_monoid.add_assoc inf.sup_monoid.add_commute)
         also have "... = (p[?px \<sqinter> -?pt2sy \<sqinter> ?pty\<longmapsto>?p2t])[?px \<sqinter> -?pt2sy \<sqinter> y\<longmapsto>?p2t]"
           using 1 path_halving_invariant_def by auto
         also have "... = p[?px \<sqinter> -?pt2sy \<sqinter> y\<longmapsto>?p2t]"
           using 6 by simp
         also have "... = ?pyppy"
           using 9 by auto
         finally show ?thesis
           .
       qed
       show "univalent p0" "total p0" "acyclic (p0 - 1)"
         using 1 path_halving_invariant_def by auto
     qed
     let ?s = "{ z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y }"
     let ?t = "{ z . regular z \<and> z \<le> ?pyppy\<^sup>T\<^sup>\<star> * (?pyppy[[y]]) }"
     have "?pyppy\<^sup>T\<^sup>\<star> * (?pyppy[[y]]) = ?pyppy\<^sup>T\<^sup>\<star> * (p[[?py]])"
       using 1 put_get vector_mult_closed path_halving_invariant_def by force
     also have "... \<le> p\<^sup>+\<^sup>T\<^sup>\<star> * (p[[?py]])"
       using 1 path_halving_invariant_def update_square_plus conv_order mult_left_isotone star_isotone by force
     also have "... = p\<^sup>T\<^sup>\<star> * p\<^sup>T * p\<^sup>T * y"
       by (simp add: conv_plus_commute star.left_plus_circ mult_assoc)
     also have "... \<le> p\<^sup>T\<^sup>+ * y"
       by (metis mult_left_isotone star.left_plus_below_circ star_plus)
     finally have 16: "?pyppy\<^sup>T\<^sup>\<star> * (?pyppy[[y]]) \<le> p\<^sup>T\<^sup>+ * y"
       .
     hence "?pyppy\<^sup>T\<^sup>\<star> * (?pyppy[[y]]) \<le> p\<^sup>T\<^sup>\<star> * y"
       using mult_left_isotone order_lesseq_imp star.left_plus_below_circ by blast
     hence 17: "?t \<subseteq> ?s"
       using order_trans by auto
     have 18: "y \<in> ?s"
       using 1 bijective_regular path_compression_1b path_halving_invariant_def by force
     have 19: "\<not> y \<in> ?t"
     proof
       assume "y \<in> ?t"
       hence "y \<le> ?pyppy\<^sup>T\<^sup>\<star> * (?pyppy[[y]])"
         by simp
       hence "y \<le> p\<^sup>T\<^sup>+ * y"
         using 16 dual_order.trans by blast
       hence "y = root p y"
         using 1 find_set_precondition_def loop_root_2 path_halving_invariant_def by blast
       hence "y = ?py"
         using 1 by (metis find_set_precondition_def root_successor_loop path_halving_invariant_def)
       thus False
         using 1 by simp
     qed
     show "card ?t < card ?s"
       apply (rule psubset_card_mono)
       subgoal using finite_regular by simp
       subgoal using 17 18 19 by auto
       done
   qed
 qed
 
 lemma path_halving_3:
   "path_halving_invariant p x y p0 \<and> y = p[[y]] \<Longrightarrow> path_halving_postcondition p x y p0"
 proof -
   assume 1: "path_halving_invariant p x y p0 \<and> y = p[[y]]"
   show "path_halving_postcondition p x y p0"
   proof (unfold path_halving_postcondition_def path_compression_precondition_def, intro conjI)
     show "univalent p" "total p" "acyclic (p - 1)"
       using 1 find_set_precondition_def path_halving_invariant_def by blast+
     show "vector x" "injective x" "surjective x"
       using 1 find_set_precondition_def path_halving_invariant_def by blast+
     show 2: "vector y" "injective y" "surjective y"
       using 1 path_halving_invariant_def by blast+
     have "find_set_invariant p x y"
       using 1 find_set_invariant_def path_halving_invariant_def by blast
     thus "y = root p x"
       using 1 find_set_3 find_set_postcondition_def by blast
     show "p \<sqinter> 1 = p0 \<sqinter> 1"
       using 1 path_halving_invariant_aux(4) by blast
     show "fc p = fc p0"
       using 1 path_halving_invariant_aux(5) by blast
     have 3: "y = p0[[y]]"
       using 1 path_halving_invariant_aux(1) by auto
     hence "p0\<^sup>T\<^sup>\<star> * y = y"
       using order.antisym path_compression_1b star_left_induct_mult_equal by auto
     hence 4: "p0[(p0 * p0)\<^sup>T\<^sup>\<star> * x - y\<longmapsto>(p0 * p0)\<^sup>T] = p"
       using 1 path_halving_invariant_def by auto
     have "(p0 * p0)\<^sup>T * y = y"
       using 3 mult_assoc conv_dist_comp by auto
     hence "y \<sqinter> p0 * p0 = y \<sqinter> p0"
       using 2 3 by (metis update_postcondition)
     hence 5: "y \<sqinter> p = y \<sqinter> p0 * p0"
       using 1 2 3 by (smt update_postcondition)
     have "p0[(p0 * p0)\<^sup>T\<^sup>\<star> * x\<longmapsto>(p0 * p0)\<^sup>T] = (p0[(p0 * p0)\<^sup>T\<^sup>\<star> * x - y\<longmapsto>(p0 * p0)\<^sup>T])[(p0 * p0)\<^sup>T\<^sup>\<star> * x \<sqinter> y\<longmapsto>(p0 * p0)\<^sup>T]"
       using 1 bijective_regular path_halving_invariant_def update_split by blast
     also have "... = p[(p0 * p0)\<^sup>T\<^sup>\<star> * x \<sqinter> y\<longmapsto>(p0 * p0)\<^sup>T]"
       using 4 by simp
     also have "... = p"
       apply (rule update_same_sub)
       using 5 apply simp
       apply simp
       using 1 bijective_regular inf.absorb2 path_halving_invariant_def by auto
     finally show "p0[(p0 * p0)\<^sup>T\<^sup>\<star> * x\<longmapsto>(p0 * p0)\<^sup>T] = p"
       .
   qed
 qed
 
 theorem find_path_halving:
   "VARS p y
   [ find_set_precondition p x \<and> p0 = p ]
   y := x;
   WHILE y \<noteq> p[[y]]
     INV { path_halving_invariant p x y p0 }
-    VAR { card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y } }
+    VAR { (p\<^sup>T\<^sup>\<star> * y)\<down> }
      DO p[y] := p[[p[[y]]]];
         y := p[[y]]
      OD
   [ path_halving_postcondition p x y p0 ]"
   apply vcg_tc_simp
     apply (fact path_halving_1)
    apply (fact path_halving_2)
   by (fact path_halving_3)
 
 subsection \<open>Path Splitting\<close>
 
 text \<open>
 Path splitting is another variant of the path compression technique.
 We implement it again independently of find-set, using a second while-loop which iterates over the same path to the root.
 We prove that path splitting preserves the equivalence-relational semantics of the disjoint-set forest and also preserves the roots of the component trees.
 Additionally we prove the exact effect of path splitting, which is to replace every parent pointer with a pointer to the respective grandparent.
 \<close>
 
 definition "path_splitting_invariant p x y p0 \<equiv> 
   find_set_precondition p x \<and> point y \<and> y \<le> p0\<^sup>T\<^sup>\<star> * x \<and>
   p0[p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * y\<longmapsto>(p0 * p0)\<^sup>T] = p \<and>
   disjoint_set_forest p0"
 definition "path_splitting_postcondition p x y p0 \<equiv> 
   path_compression_precondition p x y \<and> p \<sqinter> 1 = p0 \<sqinter> 1 \<and> fc p = fc p0 \<and>
   p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>(p0 * p0)\<^sup>T] = p"
 
 lemma path_splitting_invariant_aux_1:
   assumes "point x"
       and "point y"
       and "disjoint_set_forest p0"
     shows "(p0[p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * y\<longmapsto>(p0 * p0)\<^sup>T]) \<sqinter> 1 = p0 \<sqinter> 1"
       and "fc (p0[p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * y\<longmapsto>(p0 * p0)\<^sup>T]) = fc p0"
       and "p0\<^sup>T\<^sup>\<star> * x \<le> p0\<^sup>\<star> * root p0 x"
 proof -
   let ?p2 = "p0 * p0"
   let ?p2t = "?p2\<^sup>T"
   let ?px = "p0\<^sup>T\<^sup>\<star> * x"
   let ?py = "-(p0\<^sup>T\<^sup>\<star> * y)"
   let ?pxy = "?px \<sqinter> ?py"
   let ?q1 = "?pxy \<sqinter> p0"
   let ?q2 = "-?pxy \<sqinter> p0"
   let ?q3 = "?pxy \<sqinter> ?p2"
   let ?q4 = "-?pxy \<sqinter> ?p2"
   let ?p = "p0[?pxy\<longmapsto>?p2t]"
   let ?r0 = "root p0 x"
   let ?rp = "root ?p x"
   have 1: "regular ?px \<and> regular (p0\<^sup>T\<^sup>\<star> * y) \<and> regular ?pxy"
     using assms bijective_regular find_set_precondition_def mapping_regular pp_dist_comp regular_closed_star regular_conv_closed path_halving_invariant_def regular_closed_inf by auto
   have 2: "vector x \<and> vector ?px \<and> vector ?py \<and> vector ?pxy"
     using assms(1,2) find_set_precondition_def vector_complement_closed vector_mult_closed path_halving_invariant_def vector_inf_closed by auto
   have 3: "?r0 \<le> p0 * ?r0"
     by (metis assms(3) dedekind_1 inf.le_iff_sup root_successor_loop top_greatest)
   hence "?pxy \<sqinter> p0 * ?r0 \<le> ?pxy \<sqinter> ?p2 * ?r0"
     by (metis comp_associative inf.eq_refl inf.sup_right_isotone mult_isotone)
   hence 4: "?q1 * ?r0 \<le> ?q3 * ?r0"
     using 2 by (simp add: vector_inf_comp)
   have 5: "?q1 * ?q2 \<le> ?q3"
     using 2 by (smt (z3) comp_isotone inf.cobounded1 inf.cobounded2 inf_greatest vector_export_comp)
   have "?q1 * ?q2\<^sup>\<star> * ?r0 = ?q1 * ?r0 \<squnion> ?q1 * ?q2 * ?q2\<^sup>\<star> * ?r0"
     by (metis comp_associative semiring.distrib_left star.circ_loop_fixpoint sup_commute)
   also have "... \<le> ?q1 * ?r0 \<squnion> ?q3 * ?q2\<^sup>\<star> * ?r0"
     using 5 by (meson mult_left_isotone sup_right_isotone)
   also have "... \<le> ?q3 * ?r0 \<squnion> ?q3 * ?q2\<^sup>\<star> * ?r0"
     using 4 sup_left_isotone by blast
   also have "... = ?q3 * ?q2\<^sup>\<star> * ?r0"
     by (smt (verit, del_insts) comp_associative semiring.distrib_left star.circ_loop_fixpoint star.circ_transitive_equal star_involutive sup_commute)
   finally have 6: "?q1 * ?q2\<^sup>\<star> * ?r0 \<le> ?q3 * ?q2\<^sup>\<star> * ?r0"
     .
   have "?q1 * (-?pxy \<sqinter> p0\<^sup>+) * ?pxy \<le> (?px \<sqinter> p0) * (-?pxy \<sqinter> p0\<^sup>+) * ?pxy"
     by (meson comp_inf.comp_left_subdist_inf inf.boundedE mult_left_isotone)
   also have "... \<le> (?px \<sqinter> p0) * (-?pxy \<sqinter> p0\<^sup>+) * ?py"
     by (simp add: mult_right_isotone)
   also have "... \<le> ?px\<^sup>T * (-?pxy \<sqinter> p0\<^sup>+) * ?py"
   proof -
     have "?px \<sqinter> p0 \<le> ?px\<^sup>T * p0"
       using 2 by (simp add: vector_restrict_comp_conv)
     also have "... \<le> ?px\<^sup>T"
       by (metis comp_associative conv_dist_comp conv_involutive conv_star_commute mult_right_isotone star.circ_increasing star.circ_transitive_equal)
     finally show ?thesis
       using mult_left_isotone by auto
   qed
   also have "... = top * (?px \<sqinter> -?pxy \<sqinter> p0\<^sup>+) * ?py"
     using 2 by (smt (z3) comp_inf.star_plus conv_dist_inf covector_inf_comp_3 inf_top.right_neutral vector_complement_closed vector_inf_closed)
   also have "... \<le> top * (-?py \<sqinter> p0\<^sup>+) * ?py"
     by (metis comp_inf.comp_isotone comp_isotone inf.cobounded2 inf.eq_refl inf_import_p)
   also have "... = top * (-?py \<sqinter> p0\<^sup>+ \<sqinter> ?py\<^sup>T) * top"
     using 2 by (simp add: comp_associative covector_inf_comp_3)
   also have "... = bot"
   proof -
     have "p0\<^sup>T\<^sup>\<star> * y - y\<^sup>T * p0\<^sup>\<star> = p0\<^sup>T\<^sup>\<star> * y * y\<^sup>T * -p0\<^sup>\<star>"
       using 2 by (metis assms(2) bijective_conv_mapping comp_mapping_complement vector_covector vector_export_comp vector_mult_closed)
     also have "... \<le> p0\<^sup>T\<^sup>\<star> * -p0\<^sup>\<star>"
       by (meson assms(2) mult_left_isotone order_refl shunt_bijective)
     also have "... \<le> -p0\<^sup>\<star>"
       by (simp add: conv_complement conv_star_commute pp_increasing schroeder_6_p star.circ_transitive_equal)
     also have "... \<le> -p0\<^sup>+"
       by (simp add: p_antitone star.left_plus_below_circ)
     finally have "-?py \<sqinter> p0\<^sup>+ \<sqinter> ?py\<^sup>T = bot"
       by (metis comp_inf.p_pp_comp conv_complement conv_dist_comp conv_involutive conv_star_commute p_shunting_swap pp_isotone pseudo_complement_pp regular_closed_p)
     thus ?thesis
       by simp
   qed
   finally have 7: "?q1 * (-?pxy \<sqinter> p0\<^sup>+) * ?pxy = bot"
     using le_bot by blast
   have "?q2\<^sup>+ \<le> -?pxy"
     using 2 by (smt (z3) comp_isotone complement_conv_sub inf.order_trans inf.sup_right_divisibility inf_commute symmetric_top_closed top_greatest)
   hence "?q2\<^sup>+ \<le> -?pxy \<sqinter> p0\<^sup>+"
     by (simp add: comp_isotone star_isotone)
   hence 8: "?q1 * ?q2\<^sup>+ * ?pxy = bot"
     using 7 mult_left_isotone mult_right_isotone le_bot by auto
   have "?q1 * ?q2\<^sup>+ * ?q3\<^sup>\<star> = ?q1 * ?q2\<^sup>+ \<squnion> ?q1 * ?q2\<^sup>+ * ?q3\<^sup>+"
     by (smt (z3) comp_associative star.circ_back_loop_fixpoint star.circ_plus_same sup_commute)
   also have "... \<le> ?q1 * ?q2\<^sup>+ \<squnion> ?q1 * ?q2\<^sup>+ * ?pxy"
     using 2 by (smt (z3) inf.cobounded1 mult_right_isotone sup_right_isotone vector_inf_comp)
   finally have 9: "?q1 * ?q2\<^sup>+ * ?q3\<^sup>\<star> \<le> ?q1 * ?q2\<^sup>+"
     using 8 by simp
   have 10: "?q1 * ?q4 * ?pxy = bot"
   proof -
     have "?p2 \<le> p0\<^sup>+"
       by (simp add: mult_right_isotone star.circ_increasing)
     thus ?thesis
       using 7 by (metis mult_left_isotone mult_right_isotone le_bot comp_inf.comp_isotone eq_refl)
   qed
   have 11: "?q1 * ?q2 * ?pxy = bot"
   proof -
     have "p0 \<le> p0\<^sup>+"
       by (simp add: star.circ_mult_increasing)
     thus ?thesis
       using 7 by (metis mult_left_isotone mult_right_isotone le_bot comp_inf.comp_isotone eq_refl)
   qed
   have 12: "?q2 \<le> p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     by (smt (verit, del_insts) conv_dist_comp conv_order conv_star_commute inf.coboundedI1 inf.orderE inf.sup_monoid.add_commute path_compression_1b)
   have "?q3 * p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> = ?q1 * p0 * p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     using 2 vector_inf_comp by auto
   also have "... = ?q1 * (?q3 \<squnion> ?q4) * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     using 1 by (smt (z3) comp_associative comp_inf.mult_right_dist_sup comp_inf.star_slide inf_top.right_neutral regular_complement_top)
   also have "... = ?q1 * ?q3 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q4 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     using mult_left_dist_sup mult_right_dist_sup by auto
   also have "... \<le> ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q4 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     by (smt (z3) mult_left_isotone mult_left_sub_dist_sup_right sup_left_isotone sup_right_divisibility mult_assoc star.left_plus_below_circ)
   also have "... = ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q4 * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q4 * ?q3\<^sup>+ * ?q2\<^sup>\<star>"
     by (smt (z3) semiring.combine_common_factor star.circ_back_loop_fixpoint star_plus sup_monoid.add_commute mult_assoc)
   also have "... \<le> ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q4 * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q4 * ?pxy * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     by (smt (verit, ccfv_threshold) comp_isotone inf.sup_right_divisibility inf_commute order.refl semiring.add_left_mono mult_assoc)
   also have "... = ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q4 * ?q2\<^sup>\<star>"
     using 10 by simp
   also have "... = ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q2 * p0 * ?q2\<^sup>\<star>"
     using 2 by (smt vector_complement_closed vector_inf_comp mult_assoc)
   also have "... = ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q2 * (?q2 \<squnion> ?q1) * ?q2\<^sup>\<star>"
     using 1 by (smt (z3) comp_associative comp_inf.mult_right_dist_sup comp_inf.star_slide inf_top.right_neutral regular_complement_top)
   also have "... = ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q2 * ?q2 * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q2 * ?q1 * ?q2\<^sup>\<star>"
     using mult_left_dist_sup mult_right_dist_sup sup_commute sup_left_commute by auto
   also have "... \<le> ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q2 * ?q2 * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q2 * ?pxy * ?q2\<^sup>\<star>"
     by (smt (verit, ccfv_threshold) comp_isotone inf.sup_right_divisibility inf_commute order.refl semiring.add_left_mono mult_assoc)
   also have "... = ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q2 * ?q2 * ?q2\<^sup>\<star>"
     using 11 by simp
   also have "... \<le> ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q1 * ?q2\<^sup>\<star>"
     by (smt comp_associative comp_isotone mult_right_isotone star.circ_increasing star.circ_transitive_equal star.left_plus_below_circ sup_right_isotone)
   also have "... = ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     by (smt (verit, best) comp_associative semiring.distrib_left star.circ_loop_fixpoint star.circ_transitive_equal star_involutive)
   finally have 13: "?q3 * p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<le> p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     by (meson inf.cobounded2 mult_left_isotone order_lesseq_imp)
   hence "?q3 * p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> \<squnion> ?q2 \<le> p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     using 12 by simp
   hence "?q3\<^sup>\<star> * ?q2 \<le> p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     by (simp add: star_left_induct mult_assoc)
   hence "?q1 * ?q3\<^sup>\<star> * ?q2 \<le> ?q1 * p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     by (simp add: comp_associative mult_right_isotone)
   hence "?q1 * ?q3\<^sup>\<star> * ?q2 \<le> ?q3\<^sup>+ * ?q2\<^sup>\<star>"
     using 2 by (simp add: vector_inf_comp)
   hence 14: "?q1 * ?q3\<^sup>\<star> * ?q2 \<le> ?q3\<^sup>\<star> * ?q2\<^sup>\<star>"
     using mult_left_isotone order_lesseq_imp star.left_plus_below_circ by blast
   have "p0 * ?r0 \<le> p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (metis comp_associative mult_1_right mult_left_isotone mult_right_isotone reflexive_mult_closed star.circ_reflexive)
   hence 15: "?r0 \<le> p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     using 3 dual_order.trans by blast
   have "?q3 * p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<le> p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     using 13 mult_left_isotone by blast
   hence "?q3 * p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?r0 \<le> p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     using 15 by simp
   hence "?q3\<^sup>\<star> * ?r0 \<le> p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (simp add: star_left_induct mult_assoc)
   hence "?q1 * ?q3\<^sup>\<star> * ?r0 \<le> ?q1 * p0 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (simp add: comp_associative mult_right_isotone)
   hence "?q1 * ?q3\<^sup>\<star> * ?r0 \<le> ?q3\<^sup>+ * ?q2\<^sup>\<star> * ?r0"
     using 2 by (simp add: vector_inf_comp)
   hence 16: "?q1 * ?q3\<^sup>\<star> * ?r0 \<le> ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     using mult_left_isotone order_lesseq_imp star.left_plus_below_circ by blast
   have "?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 = ?q1 * ?q3\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>+ * ?r0"
     by (smt (z3) comp_associative mult_right_dist_sup star.circ_back_loop_fixpoint star.circ_plus_same sup_commute)
   also have "... \<le> ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>+ * ?r0"
     using 16 sup_left_isotone by blast
   also have "... \<le> ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     using 14 by (smt (z3) inf.eq_refl semiring.distrib_right star.circ_transitive_equal sup.absorb2 sup_monoid.add_commute mult_assoc)
   also have "... = ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (simp add: comp_associative star.circ_transitive_equal)
   finally have 17: "?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<le> ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     .
   have "?r0 \<le> ?q2\<^sup>\<star> * ?r0"
     using star.circ_loop_fixpoint sup_right_divisibility by auto
   also have "... \<le> ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     using comp_associative star.circ_loop_fixpoint sup_right_divisibility by force
   also have "... \<le> ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     using comp_associative star.circ_loop_fixpoint sup_right_divisibility by force
   finally have 18: "?r0 \<le> ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     .
   have "p0 * ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 = (?q2 \<squnion> ?q1) * ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     using 1 by (smt (z3) comp_inf.mult_right_dist_sup comp_inf.star_plus inf_top.right_neutral regular_complement_top)
   also have "... = ?q2 * ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     using mult_right_dist_sup by auto
   also have "... \<le> ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (smt (z3) comp_left_increasing_sup star.circ_loop_fixpoint sup_left_isotone mult_assoc)
   also have "... = ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q2\<^sup>+ * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (smt (z3) mult_left_dist_sup semiring.combine_common_factor star.circ_loop_fixpoint sup_monoid.add_commute mult_assoc)
   also have "... \<le> ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q2\<^sup>+ * ?q2\<^sup>\<star> * ?r0"
     using 9 mult_left_isotone sup_right_isotone by auto
   also have "... \<le> ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q2\<^sup>\<star> * ?r0"
     by (smt (z3) comp_associative comp_isotone inf.eq_refl semiring.add_right_mono star.circ_transitive_equal star.left_plus_below_circ sup_commute)
   also have "... \<le> ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q1 * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q3 * ?q2\<^sup>\<star> * ?r0"
     using 6 sup_right_isotone by blast
   also have "... = ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q3 * ?q2\<^sup>\<star> * ?r0"
     using 17 by (smt (z3) le_iff_sup semiring.combine_common_factor semiring.distrib_right star.circ_loop_fixpoint sup_monoid.add_commute)
   also have "... \<le> ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (meson mult_left_isotone star.circ_increasing sup_right_isotone)
   also have "... = ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (smt (z3) comp_associative star.circ_loop_fixpoint star.circ_transitive_equal star_involutive)
   finally have "p0 * ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0 \<squnion> ?r0 \<le> ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     using 18 sup.boundedI by blast
   hence "p0\<^sup>\<star> * ?r0 \<le> ?q2\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (simp add: comp_associative star_left_induct)
   also have "... \<le> ?p\<^sup>\<star> * ?q3\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (metis mult_left_isotone star.circ_sub_dist sup_commute)
   also have "... \<le> ?p\<^sup>\<star> * ?p\<^sup>\<star> * ?q2\<^sup>\<star> * ?r0"
     by (simp add: mult_left_isotone mult_right_isotone star_isotone)
   also have "... \<le> ?p\<^sup>\<star> * ?p\<^sup>\<star> * ?p\<^sup>\<star> * ?r0"
     by (metis mult_isotone order.refl star.circ_sub_dist sup_commute)
   finally have 19: "p0\<^sup>\<star> * ?r0 \<le> ?p\<^sup>\<star> * ?r0"
     by (simp add: star.circ_transitive_equal)
   have 20: "?p\<^sup>\<star> \<le> p0\<^sup>\<star>"
     by (metis star.left_plus_circ star_isotone update_square_ub_plus)
   hence 21: "p0\<^sup>\<star> * ?r0 = ?p\<^sup>\<star> * ?r0"
     using 19 order.antisym mult_left_isotone by auto
   have "?p \<sqinter> 1 = (?q3 \<sqinter> 1) \<squnion> (?q2 \<sqinter> 1)"
     using comp_inf.semiring.distrib_right conv_involutive by auto
   also have "... = (?q1 \<sqinter> 1) \<squnion> (?q2 \<sqinter> 1)"
     using assms(3) acyclic_square path_splitting_invariant_def inf.sup_monoid.add_assoc by auto
   also have "... = (?pxy \<squnion> -?pxy) \<sqinter> p0 \<sqinter> 1"
     using inf_sup_distrib2 by auto
   also have "... = p0 \<sqinter> 1"
     using 1 by (metis inf.sup_monoid.add_commute inf_sup_distrib1 maddux_3_11_pp)
   finally show 22: "?p \<sqinter> 1 = p0 \<sqinter> 1"
     .
   have "?p\<^sup>T\<^sup>\<star> * x \<le> p0\<^sup>T\<^sup>\<star> * x"
     using 20 by (metis conv_isotone conv_star_commute mult_left_isotone)
   hence 23: "?rp \<le> ?r0"
     using 22 comp_inf.mult_left_isotone by auto
   have 24: "disjoint_set_forest ?p"
     using 1 2 assms(3) disjoint_set_forest_update_square by blast
   hence 25: "point ?rp"
     using root_point assms(1) by auto
   have "?r0 * ?rp\<^sup>T = ?r0 * x\<^sup>T * ?p\<^sup>\<star> * (?p \<sqinter> 1)"
     by (smt (z3) comp_associative conv_dist_comp conv_dist_inf conv_involutive conv_star_commute inf.sup_monoid.add_commute one_inf_conv root_var star_one star_sup_one wcc_one)
   also have "... \<le> (p0 \<sqinter> 1) * p0\<^sup>T\<^sup>\<star> * 1 * ?p\<^sup>\<star> * (?p \<sqinter> 1)"
     by (smt (z3) assms(1) comp_associative mult_left_isotone mult_right_isotone root_var)
   also have "... \<le> (p0 \<sqinter> 1) * p0\<^sup>T\<^sup>\<star> * p0\<^sup>\<star> * (p0 \<sqinter> 1)"
     using 20 22 comp_isotone by force
   also have "... = (p0 \<sqinter> 1) * p0\<^sup>\<star> * (p0 \<sqinter> 1) \<squnion> (p0 \<sqinter> 1) * p0\<^sup>T\<^sup>\<star> * (p0 \<sqinter> 1)"
     by (simp add: assms(3) cancel_separate_eq sup_monoid.add_commute mult_assoc mult_left_dist_sup semiring.distrib_right)
   also have "... = (p0 \<sqinter> 1) * (p0 \<sqinter> 1) \<squnion> (p0 \<sqinter> 1) * p0\<^sup>T\<^sup>\<star> * (p0 \<sqinter> 1)"
     using univalent_root_successors assms(3) by simp
   also have "... = (p0 \<sqinter> 1) * (p0 \<sqinter> 1) \<squnion> (p0 \<sqinter> 1) * ((p0 \<sqinter> 1) * p0\<^sup>\<star>)\<^sup>T"
     by (smt (z3) comp_associative conv_dist_comp conv_dist_inf conv_star_commute inf.sup_monoid.add_commute one_inf_conv star_one star_sup_one wcc_one)
   also have "... = (p0 \<sqinter> 1) * (p0 \<sqinter> 1)"
     by (metis univalent_root_successors assms(3) conv_dist_inf inf.sup_monoid.add_commute one_inf_conv sup_idem symmetric_one_closed)
   also have "... \<le> 1"
     by (simp add: coreflexive_mult_closed)
   finally have "?r0 * ?rp\<^sup>T \<le> 1"
     .
   hence "?r0 \<le> 1 * ?rp"
     using 25 shunt_bijective by blast
   hence 26: "?r0 = ?rp"
     using 23 order.antisym by simp
   have "?px * ?r0\<^sup>T = ?px * x\<^sup>T * p0\<^sup>\<star> * (p0 \<sqinter> 1)"
     by (smt (z3) comp_associative conv_dist_comp conv_dist_inf conv_involutive conv_star_commute inf.sup_monoid.add_commute one_inf_conv root_var star_one star_sup_one wcc_one)
   also have "... \<le> p0\<^sup>T\<^sup>\<star> * 1 * p0\<^sup>\<star> * (p0 \<sqinter> 1)"
     by (smt (z3) assms(1) comp_associative mult_left_isotone mult_right_isotone root_var)
   also have "... = p0\<^sup>\<star> * (p0 \<sqinter> 1) \<squnion> p0\<^sup>T\<^sup>\<star> * (p0 \<sqinter> 1)"
     by (simp add: assms(3) cancel_separate_eq sup_monoid.add_commute mult_right_dist_sup)
   also have "... = p0\<^sup>\<star> * (p0 \<sqinter> 1) \<squnion> ((p0 \<sqinter> 1) * p0\<^sup>\<star>)\<^sup>T"
     by (smt (z3) conv_dist_comp conv_dist_inf conv_star_commute inf.sup_monoid.add_commute one_inf_conv star_one star_sup_one wcc_one)
   also have "... = p0\<^sup>\<star> * (p0 \<sqinter> 1) \<squnion> (p0 \<sqinter> 1)"
     by (metis univalent_root_successors assms(3) conv_dist_inf inf.sup_monoid.add_commute one_inf_conv symmetric_one_closed)
   also have "... = p0\<^sup>\<star> * (p0 \<sqinter> 1)"
     by (metis conv_involutive path_compression_1b sup.absorb2 sup_commute)
   also have "... \<le> p0\<^sup>\<star>"
     by (simp add: inf.coboundedI1 star.circ_increasing star.circ_mult_upper_bound)
   finally have 27: "?px * ?r0\<^sup>T \<le> p0\<^sup>\<star>"
     .
   thus 28: "?px \<le> p0\<^sup>\<star> * ?r0"
     by (simp add: assms(1,3) root_point shunt_bijective)
   have 29: "point ?r0"
     using root_point assms(1,3) by auto
   hence 30: "mapping (?r0\<^sup>T)"
     using bijective_conv_mapping by blast
   have "?r0 * (?px \<sqinter> p0) = ?r0 * top * (?px \<sqinter> p0)"
     using 29 by force
   also have "... = ?r0 * ?px\<^sup>T * p0"
     using 29 by (metis assms(1) covector_inf_comp_3 vector_covector vector_mult_closed)
   also have "... = ?r0 * x\<^sup>T * p0\<^sup>\<star> * p0"
     using comp_associative conv_dist_comp conv_star_commute by auto
   also have "... \<le> ?r0 * x\<^sup>T * p0\<^sup>\<star>"
     by (simp add: comp_associative mult_right_isotone star.circ_plus_same star.left_plus_below_circ)
   also have "... = ?r0 * ?px\<^sup>T"
     by (simp add: comp_associative conv_dist_comp conv_star_commute)
   also have "... = (?px * ?r0\<^sup>T)\<^sup>T"
     by (simp add: conv_dist_comp)
   also have "... \<le> p0\<^sup>T\<^sup>\<star>"
     using 27 conv_isotone conv_star_commute by fastforce
   finally have "?r0 * (?px \<sqinter> p0) \<le> p0\<^sup>T\<^sup>\<star>"
     .
   hence "?px \<sqinter> p0 \<le> ?r0\<^sup>T * p0\<^sup>T\<^sup>\<star>"
     using 30 shunt_mapping by auto
   hence "?px \<sqinter> p0 \<le> p0\<^sup>\<star> * ?r0 \<sqinter> ?r0\<^sup>T * p0\<^sup>T\<^sup>\<star>"
     using 28 inf.coboundedI2 inf.sup_monoid.add_commute by fastforce
   also have "... = p0\<^sup>\<star> * ?r0 * ?r0\<^sup>T * p0\<^sup>T\<^sup>\<star>"
     using 29 by (smt (z3) vector_covector vector_inf_comp vector_mult_closed)
   also have "... = ?p\<^sup>\<star> * ?r0 * ?r0\<^sup>T * ?p\<^sup>T\<^sup>\<star>"
     using 21 by (smt comp_associative conv_dist_comp conv_star_commute)
   also have "... = ?p\<^sup>\<star> * ?rp * ?rp\<^sup>T * ?p\<^sup>T\<^sup>\<star>"
     using 26 by auto
   also have "... \<le> ?p\<^sup>\<star> * 1 * ?p\<^sup>T\<^sup>\<star>"
     using 25 by (smt (z3) comp_associative mult_left_isotone mult_right_isotone)
   finally have 31: "?px \<sqinter> p0 \<le> fc ?p"
     by auto
   have "-?px \<sqinter> p0 \<le> ?p"
     by (simp add: inf.sup_monoid.add_commute le_supI1 sup_commute)
   also have "... \<le> fc ?p"
     using fc_increasing by auto
   finally have "p0 \<le> fc ?p"
     using 1 31 by (smt (z3) inf.sup_monoid.add_commute maddux_3_11_pp semiring.add_left_mono sup.orderE sup_commute)
   also have "... \<le> wcc ?p"
     using star.circ_sub_dist_3 by auto
   finally have 32: "wcc p0 \<le> wcc ?p"
     using wcc_below_wcc by blast
   have "?p \<le> wcc p0"
     by (simp add: inf.coboundedI1 inf.sup_monoid.add_commute star.circ_mult_upper_bound star.circ_sub_dist_1)
   hence "wcc ?p \<le> wcc p0"
     using wcc_below_wcc by blast
   hence "wcc ?p = wcc p0"
     using 32 order.antisym by blast
   thus "fc ?p = fc p0"
     using 24 assms(3) fc_wcc by auto
 qed
 
 lemma path_splitting_invariant_aux:
   assumes "path_splitting_invariant p x y p0"
   shows "p[[y]] = p0[[y]]"
     and "p[[p[[y]]]] = p0[[p0[[y]]]]"
     and "p[[p[[p[[y]]]]]] = p0[[p0[[p0[[y]]]]]]"
     and "p \<sqinter> 1 = p0 \<sqinter> 1"
     and "fc p = fc p0"
 proof -
   let ?p2 = "p0 * p0"
   let ?p2t = "?p2\<^sup>T"
   let ?px = "p0\<^sup>T\<^sup>\<star> * x"
   let ?py = "-(p0\<^sup>T\<^sup>\<star> * y)"
   let ?pxy = "?px \<sqinter> ?py"
   let ?p = "p0[?pxy\<longmapsto>?p2t]"
   have "?p[[y]] = p0[[y]]"
     apply (rule put_get_different_vector)
     using assms find_set_precondition_def vector_complement_closed vector_inf_closed vector_mult_closed path_splitting_invariant_def apply force
     by (meson inf.cobounded2 order_lesseq_imp p_antitone_iff path_compression_1b)
   thus 1: "p[[y]] = p0[[y]]"
     using assms path_splitting_invariant_def by auto
   have "?p[[p0[[y]]]] = p0[[p0[[y]]]]"
     apply (rule put_get_different_vector)
     using assms find_set_precondition_def vector_complement_closed vector_inf_closed vector_mult_closed path_splitting_invariant_def apply force
     by (metis comp_isotone inf.boundedE inf.coboundedI2 inf.eq_refl p_antitone_iff selection_closed_id star.circ_increasing)
   thus 2: "p[[p[[y]]]] = p0[[p0[[y]]]]"
     using 1 assms path_splitting_invariant_def by auto
   have "?p[[p0[[p0[[y]]]]]] = p0[[p0[[p0[[y]]]]]]"
     apply (rule put_get_different_vector)
     using assms find_set_precondition_def vector_complement_closed vector_inf_closed vector_mult_closed path_splitting_invariant_def apply force
     by (metis comp_associative comp_isotone conv_dist_comp conv_involutive conv_order inf.coboundedI2 inf.le_iff_sup mult_left_isotone p_antitone_iff p_antitone_inf star.circ_increasing star.circ_transitive_equal)
   thus "p[[p[[p[[y]]]]]] = p0[[p0[[p0[[y]]]]]]"
     using 2 assms path_splitting_invariant_def by auto
   show "p \<sqinter> 1 = p0 \<sqinter> 1"
     using assms path_splitting_invariant_aux_1(1) path_splitting_invariant_def find_set_precondition_def by auto
   show "fc p = fc p0"
     using assms path_splitting_invariant_aux_1(2) path_splitting_invariant_def find_set_precondition_def by auto
 qed
 
 lemma path_splitting_1:
   "find_set_precondition p0 x \<Longrightarrow> path_splitting_invariant p0 x x p0"
 proof -
   assume 1: "find_set_precondition p0 x"
   show "path_splitting_invariant p0 x x p0"
   proof (unfold path_splitting_invariant_def, intro conjI)
     show "find_set_precondition p0 x"
       using 1 by simp
     show "vector x" "injective x" "surjective x"
       using 1 find_set_precondition_def by auto
     show "x \<le> p0\<^sup>T\<^sup>\<star> * x"
       by (simp add: path_compression_1b)
     have "(p0 * p0)\<^sup>T\<^sup>\<star> * x \<le> p0\<^sup>T\<^sup>\<star> * x"
       by (simp add: conv_dist_comp mult_left_isotone star.circ_square)
     thus "p0[p0\<^sup>T\<^sup>\<star> * x - p0\<^sup>T\<^sup>\<star> * x\<longmapsto>(p0 * p0)\<^sup>T] = p0"
       by (smt (z3) inf.le_iff_sup inf_commute maddux_3_11_pp p_antitone_inf pseudo_complement)
     show "univalent p0" "total p0" "acyclic (p0 - 1)"
       using 1 find_set_precondition_def by auto
   qed
 qed
 
 lemma path_splitting_2:
-  "path_splitting_invariant p x y p0 \<and> y \<noteq> p[[y]] \<Longrightarrow> path_splitting_invariant (p[y\<longmapsto>p[[p[[y]]]]]) x (p[[y]]) p0 \<and> card { z . regular z \<and> z \<le> (p[y\<longmapsto>p[[p[[y]]]]])\<^sup>T\<^sup>\<star> * (p[[y]]) } < card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y }"
+  "path_splitting_invariant p x y p0 \<and> y \<noteq> p[[y]] \<Longrightarrow> path_splitting_invariant (p[y\<longmapsto>p[[p[[y]]]]]) x (p[[y]]) p0 \<and> ((p[y\<longmapsto>p[[p[[y]]]]])\<^sup>T\<^sup>\<star> * (p[[y]]))\<down> < (p\<^sup>T\<^sup>\<star> * y)\<down>"
 proof -
   let ?py = "p[[y]]"
   let ?ppy = "p[[?py]]"
   let ?pyppy = "p[y\<longmapsto>?ppy]"
   let ?p2 = "p0 * p0"
   let ?p2t = "?p2\<^sup>T"
   let ?p2ts = "?p2t\<^sup>\<star>"
   let ?px = "p0\<^sup>T\<^sup>\<star> * x"
   let ?py2 = "-(p0\<^sup>T\<^sup>\<star> * y)"
   let ?pxy = "?px \<sqinter> ?py2"
   let ?p = "p0[?pxy\<longmapsto>?p2t]"
   let ?pty = "p0\<^sup>T * y"
   let ?pt2y = "p0\<^sup>T * p0\<^sup>T * y"
   let ?pt2sy = "p0\<^sup>T\<^sup>\<star> * p0\<^sup>T * p0\<^sup>T * y"
   let ?ptpy = "p0\<^sup>T\<^sup>+ * y"
   assume 1: "path_splitting_invariant p x y p0 \<and> y \<noteq> ?py"
   have 2: "point ?pty \<and> point ?pt2y"
     using 1 by (smt (verit) comp_associative read_injective read_surjective path_splitting_invariant_def)
-  show "path_splitting_invariant ?pyppy x (p[[y]]) p0 \<and> card { z . regular z \<and> z \<le> ?pyppy\<^sup>T\<^sup>\<star> * (p[[y]]) } < card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y }"
+  show "path_splitting_invariant ?pyppy x (p[[y]]) p0 \<and> (?pyppy\<^sup>T\<^sup>\<star> * (p[[y]]))\<down> < (p\<^sup>T\<^sup>\<star> * y)\<down>"
   proof
     show "path_splitting_invariant ?pyppy x (p[[y]]) p0"
     proof (unfold path_splitting_invariant_def, intro conjI)
       show 3: "find_set_precondition ?pyppy x"
       proof (unfold find_set_precondition_def, intro conjI)
         show "univalent ?pyppy"
           using 1 find_set_precondition_def read_injective update_univalent path_splitting_invariant_def by auto
         show "total ?pyppy"
           using 1 bijective_regular find_set_precondition_def read_surjective update_total path_splitting_invariant_def by force
         show "acyclic (?pyppy - 1)"
           apply (rule update_acyclic_3)
           using 1 find_set_precondition_def path_splitting_invariant_def apply blast
           using 1 2 comp_associative path_splitting_invariant_aux(2) apply force
           using 1 path_splitting_invariant_def apply blast
           by (metis inf.order_lesseq_imp mult_isotone star.circ_increasing star.circ_square mult_assoc)
         show "vector x" "injective x" "surjective x"
           using 1 find_set_precondition_def path_splitting_invariant_def by auto
       qed
       show "vector (p[[y]])"
         using 1 comp_associative path_splitting_invariant_def by auto
       show "injective (p[[y]])"
         using 1 3 read_injective path_splitting_invariant_def find_set_precondition_def by auto
       show "surjective (p[[y]])"
         using 1 3 read_surjective path_splitting_invariant_def find_set_precondition_def by auto
       show "p[[y]] \<le> ?px"
       proof -
         have "p[[y]] = p0[[y]]"
           using 1 path_splitting_invariant_aux(1) by blast
         also have "... \<le> p0\<^sup>T * ?px"
           using 1 path_splitting_invariant_def mult_right_isotone by force
         also have "... \<le> ?px"
           by (metis comp_associative mult_left_isotone star.left_plus_below_circ)
         finally show ?thesis
           .
       qed
       show "p0[?px - p0\<^sup>T\<^sup>\<star> * (p[[y]])\<longmapsto>?p2t] = ?pyppy"
       proof -
         have 4: "p[?px \<sqinter> -?ptpy \<sqinter> y\<longmapsto>?p2t] = ?pyppy"
         proof (cases "y \<le> -?ptpy")
           case True
           hence "p[?px \<sqinter> -?ptpy \<sqinter> y\<longmapsto>?p2t] = p[y\<longmapsto>?p2t]"
             using 1 inf.absorb2 path_splitting_invariant_def by auto
           also have "... = ?pyppy"
             using 1 by (metis comp_associative conv_dist_comp path_splitting_invariant_aux(2) path_splitting_invariant_def update_point_get)
           finally show ?thesis
             .
         next
           case False
           have "vector (-?ptpy)"
             using 1 vector_complement_closed vector_mult_closed path_splitting_invariant_def by blast
           hence 5: "y \<le> ?ptpy"
             using 1 by (smt (verit, del_insts) False bijective_regular point_in_vector_or_complement regular_closed_star regular_mult_closed total_conv_surjective univalent_conv_injective path_splitting_invariant_def)
           hence "?px \<sqinter> -?ptpy \<sqinter> y = bot"
             by (simp add: inf.coboundedI2 p_antitone pseudo_complement)
           hence 6: "p[?px \<sqinter> -?ptpy \<sqinter> y\<longmapsto>?p2t] = p"
             by simp
           have 7: "y = root p0 y"
             using 1 5 loop_root_2 path_splitting_invariant_def by blast
           have "?pyppy = p[y\<longmapsto>p0[[p0[[y]]]]]"
             using 1 path_splitting_invariant_aux(2) by force
           also have "... = p[y\<longmapsto>p0[[y]]]"
             using 1 7 by (metis root_successor_loop path_splitting_invariant_def)
           also have "... = p[y\<longmapsto>?py]"
             using 1 path_splitting_invariant_aux(1) by force
           also have "... = p"
             using 1 get_put path_splitting_invariant_def by blast
           finally show ?thesis
             using 6 by simp
         qed
         have 8: "-?ptpy = ?py2 \<squnion> (-?ptpy \<sqinter> y)"
         proof (rule order.antisym)
           have 9: "regular (p0\<^sup>T\<^sup>\<star> * y) \<and> regular ?ptpy"
             using 1 bijective_regular mapping_conv_bijective pp_dist_star regular_mult_closed path_splitting_invariant_def by auto
           have "p0\<^sup>T\<^sup>\<star> * y \<le> ?ptpy \<squnion> y"
             by (simp add: star.circ_loop_fixpoint mult_assoc)
           also have "... = ?ptpy \<squnion> (-?ptpy \<sqinter> y)"
             using 9 by (metis maddux_3_21_pp)
           hence "p0\<^sup>T\<^sup>\<star> * y \<sqinter> -?ptpy \<le> -?ptpy \<sqinter> y"
             using calculation half_shunting sup_commute by auto
           thus "-?ptpy \<le> ?py2 \<squnion> (-?ptpy \<sqinter> y)"
             using 9 by (smt (z3) inf.sup_monoid.add_commute shunting_var_p sup.left_commute sup_commute)
           have "-(p0\<^sup>T\<^sup>\<star> * y) \<le> -?ptpy"
             by (simp add: comp_isotone p_antitone star.left_plus_below_circ)
           thus "-(p0\<^sup>T\<^sup>\<star> * y) \<squnion> (-?ptpy \<sqinter> y) \<le> -?ptpy"
             by simp
         qed
         have "regular ?px" "regular y"
           using 1 bijective_regular find_set_precondition_def mapping_regular pp_dist_comp regular_closed_star regular_conv_closed path_splitting_invariant_def by auto
         hence 10: "regular (?px \<sqinter> -?ptpy \<sqinter> y)"
           by auto
         have "p0[?px \<sqinter> -(p0\<^sup>T\<^sup>\<star> * (p[[y]]))\<longmapsto>?p2t] = p0[?px \<sqinter> -?ptpy\<longmapsto>?p2t]"
           using 1 by (smt comp_associative path_splitting_invariant_aux(1) star_plus)
         also have "... = p0[?pxy \<squnion> (?px \<sqinter> -?ptpy \<sqinter> y)\<longmapsto>?p2t]"
           using 8 by (metis comp_inf.semiring.distrib_left inf.sup_monoid.add_assoc)
         also have "... = ?p[?px \<sqinter> -?ptpy \<sqinter> y\<longmapsto>?p2t]"
           using 10 by (smt (z3) update_same comp_inf.semiring.mult_not_zero inf.sup_monoid.add_assoc inf.sup_monoid.add_commute)
         also have "... = p[?px \<sqinter> -?ptpy \<sqinter> y\<longmapsto>?p2t]"
           using 1 path_splitting_invariant_def by auto
         also have "... = ?pyppy"
           using 4 by auto
         finally show ?thesis
           .
       qed
       show "univalent p0" "total p0" "acyclic (p0 - 1)"
         using 1 path_splitting_invariant_def by auto
     qed
     let ?s = "{ z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y }"
     let ?t = "{ z . regular z \<and> z \<le> ?pyppy\<^sup>T\<^sup>\<star> * (p[[y]]) }"
     have "?pyppy\<^sup>T\<^sup>\<star> * (p[[y]]) \<le> p\<^sup>+\<^sup>T\<^sup>\<star> * (p[[y]])"
       using 1 path_splitting_invariant_def update_square_plus conv_order mult_left_isotone star_isotone by force
     also have "... = p\<^sup>T\<^sup>\<star> * p\<^sup>T * y"
       by (simp add: conv_plus_commute star.left_plus_circ mult_assoc)
     also have "... = p\<^sup>T\<^sup>+ * y"
       by (simp add: star_plus)
     finally have 11: "?pyppy\<^sup>T\<^sup>\<star> * (p[[y]]) \<le> p\<^sup>T\<^sup>+ * y"
       .
     hence "?pyppy\<^sup>T\<^sup>\<star> * (p[[y]]) \<le> p\<^sup>T\<^sup>\<star> * y"
       using mult_left_isotone order_lesseq_imp star.left_plus_below_circ by blast
     hence 12: "?t \<subseteq> ?s"
       using order_trans by auto
     have 13: "y \<in> ?s"
       using 1 bijective_regular path_compression_1b path_splitting_invariant_def by force
     have 14: "\<not> y \<in> ?t"
     proof
       assume "y \<in> ?t"
       hence "y \<le> ?pyppy\<^sup>T\<^sup>\<star> * (p[[y]])"
         by simp
       hence "y \<le> p\<^sup>T\<^sup>+ * y"
         using 11 dual_order.trans by blast
       hence "y = root p y"
         using 1 find_set_precondition_def loop_root_2 path_splitting_invariant_def by blast
       hence "y = ?py"
         using 1 by (metis find_set_precondition_def root_successor_loop path_splitting_invariant_def)
       thus False
         using 1 by simp
     qed
     show "card ?t < card ?s"
       apply (rule psubset_card_mono)
       subgoal using finite_regular by simp
       subgoal using 12 13 14 by auto
       done
   qed
 qed
 
 lemma path_splitting_3:
   "path_splitting_invariant p x y p0 \<and> y = p[[y]] \<Longrightarrow> path_splitting_postcondition p x y p0"
 proof -
   assume 1: "path_splitting_invariant p x y p0 \<and> y = p[[y]]"
   show "path_splitting_postcondition p x y p0"
   proof (unfold path_splitting_postcondition_def path_compression_precondition_def, intro conjI)
     show "univalent p" "total p" "acyclic (p - 1)"
       using 1 find_set_precondition_def path_splitting_invariant_def by blast+
     show "vector x" "injective x" "surjective x"
       using 1 find_set_precondition_def path_splitting_invariant_def by blast+
     show 2: "vector y" "injective y" "surjective y"
       using 1 path_splitting_invariant_def by blast+
     show 3: "p \<sqinter> 1 = p0 \<sqinter> 1"
       using 1 path_splitting_invariant_aux(4) by blast
     show 4: "fc p = fc p0"
       using 1 path_splitting_invariant_aux(5) by blast
     have "y \<le> p0\<^sup>T\<^sup>\<star> * x"
       using 1 path_splitting_invariant_def by simp
     hence 5: "y * x\<^sup>T \<le> fc p0"
       using 1 by (metis dual_order.trans fc_wcc find_set_precondition_def shunt_bijective star.circ_decompose_11 star_decompose_1 star_outer_increasing path_splitting_invariant_def)
     have 6: "y = p0[[y]]"
       using 1 path_splitting_invariant_aux(1) by auto
     hence "y = root p0 y"
       using 2 loop_root by auto
     also have "... = root p0 x"
       using 1 2 5 find_set_precondition_def path_splitting_invariant_def same_component_same_root by auto
     also have "... = root p x"
       using 1 3 4 by (metis find_set_precondition_def path_splitting_invariant_def same_root)
     finally show "y = root p x"
       .
     have "p0\<^sup>T\<^sup>\<star> * y = y"
       using 6 order.antisym path_compression_1b star_left_induct_mult_equal by auto
     hence 7: "p0[p0\<^sup>T\<^sup>\<star> * x - y\<longmapsto>(p0 * p0)\<^sup>T] = p"
       using 1 path_splitting_invariant_def by auto
     have "(p0 * p0)\<^sup>T * y = y"
       using 6 mult_assoc conv_dist_comp by auto
     hence "y \<sqinter> p0 * p0 = y \<sqinter> p0"
       using 2 6 by (metis update_postcondition)
     hence 8: "y \<sqinter> p = y \<sqinter> p0 * p0"
       using 1 2 6 by (smt update_postcondition)
     have "p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>(p0 * p0)\<^sup>T] = (p0[p0\<^sup>T\<^sup>\<star> * x - y\<longmapsto>(p0 * p0)\<^sup>T])[p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<longmapsto>(p0 * p0)\<^sup>T]"
       using 1 bijective_regular path_splitting_invariant_def update_split by blast
     also have "... = p[p0\<^sup>T\<^sup>\<star> * x \<sqinter> y\<longmapsto>(p0 * p0)\<^sup>T]"
       using 7 by simp
     also have "... = p"
       apply (rule update_same_sub)
       using 8 apply simp
       apply simp
       using 1 bijective_regular inf.absorb2 path_splitting_invariant_def by auto
     finally show "p0[p0\<^sup>T\<^sup>\<star> * x\<longmapsto>(p0 * p0)\<^sup>T] = p"
       .
   qed
 qed
 
 theorem find_path_splitting:
   "VARS p t y
   [ find_set_precondition p x \<and> p0 = p ]
   y := x;
   WHILE y \<noteq> p[[y]]
     INV { path_splitting_invariant p x y p0 }
-    VAR { card { z . regular z \<and> z \<le> p\<^sup>T\<^sup>\<star> * y } }
+    VAR { (p\<^sup>T\<^sup>\<star> * y)\<down> }
      DO t := p[[y]];
         p[y] := p[[p[[y]]]];
         y := t
      OD
   [ path_splitting_postcondition p x y p0 ]"
   apply vcg_tc_simp
     apply (fact path_splitting_1)
    apply (fact path_splitting_2)
   by (fact path_splitting_3)
 
 end
 
 section \<open>Verifying Union by Rank\<close>
 
 text \<open>
 In this section we verify the union-by-rank operation of disjoint-set forests.
 The rank of a node is an upper bound of the height of the subtree rooted at that node.
 The rank array of a disjoint-set forest maps each node to its rank.
 This can be represented as a homogeneous relation since the possible rank values are $0, \dots, n-1$ where $n$ is the number of nodes of the disjoint-set forest.
 \<close>
 
 subsection \<open>Peano structures\<close>
 
 text \<open>
 Since ranks are natural numbers we start by introducing basic Peano arithmetic.
 Numbers are represented as (relational) points.
 Constant \<open>Z\<close> represents the number $0$.
 Constant \<open>S\<close> represents the successor function.
 The successor of a number $x$ is obtained by the relational composition \<open>S\<^sup>T * x\<close>.
 The composition \<open>S * x\<close> results in the predecessor of $x$.
 \<close>
 
 class peano_signature =
   fixes Z :: "'a"
   fixes S :: "'a"
 
 text \<open>
 The numbers will be used in arrays, which are represented by homogeneous finite relations.
 Such relations can only represent finitely many numbers.
 This means that we weaken the Peano axioms, which are usually used to obtain (infinitely many) natural numbers.
 Axiom \<open>Z_point\<close> specifies that $0$ is a number.
 Axiom \<open>S_univalent\<close> specifies that every number has at most one `successor'.
 Together with axiom \<open>S_total\<close>, which is added later, this means that every number has exactly one `successor'.
 Axiom \<open>S_injective\<close> specifies that numbers with the same successor are equal.
 Axiom \<open>S_star_Z_top\<close> specifies that every number can be obtained from $0$ by finitely many applications of the successor.
 We omit the Peano axiom \<open>S * Z = bot\<close> which would specify that $0$ is not the successor of any number.
 Since only finitely many numbers will be represented, the remaining axioms will model successor modulo $m$ for some $m$ depending on the carrier of the algebra.
 That is, the algebra will be able to represent numbers $0, \dots, m-1$ where the successor of $m-1$ is $0$.
 \<close>
 
 class skra_peano_1 = stone_kleene_relation_algebra + stone_relation_algebra_tarski_consistent + peano_signature +
   assumes Z_point: "point Z"
   assumes S_univalent: "univalent S"
   assumes S_injective: "injective S"
   assumes S_star_Z_top: "S\<^sup>T\<^sup>\<star> * Z = top"
 begin
 
 lemma conv_Z_Z:
   "Z\<^sup>T * Z = top"
   by (simp add: Z_point point_conv_comp)
 
 text \<open>Theorem 9.2\<close>
 
 lemma Z_below_S_star:
   "Z \<le> S\<^sup>\<star>"
 proof -
   have "top * Z\<^sup>T \<le> S\<^sup>T\<^sup>\<star>"
     using S_star_Z_top Z_point shunt_bijective by blast
   thus ?thesis
     using Z_point conv_order conv_star_commute vector_conv_covector by force
 qed
 
 text \<open>Theorem 9.3\<close>
 
 lemma S_connected:
   "S\<^sup>T\<^sup>\<star> * S\<^sup>\<star> = top"
   by (metis Z_below_S_star S_star_Z_top mult_left_dist_sup sup.orderE sup_commute top.extremum)
 
 text \<open>Theorem 9.4\<close>
 
 lemma S_star_connex:
   "S\<^sup>\<star> \<squnion> S\<^sup>T\<^sup>\<star> = top"
   using S_connected S_univalent cancel_separate_eq sup_commute by auto
 
 text \<open>Theorem 9.5\<close>
 
 lemma Z_sup_conv_S_top:
   "Z \<squnion> S\<^sup>T * top = top"
   using S_star_Z_top star.circ_loop_fixpoint sup_commute by auto
 
 lemma top_S_sup_conv_Z:
   "top * S \<squnion> Z\<^sup>T = top"
   by (metis S_star_Z_top conv_dist_comp conv_involutive conv_star_commute star.circ_back_loop_fixpoint symmetric_top_closed)
 
 text \<open>Theorem 9.1\<close>
 
 lemma S_inf_1_below_Z:
   "S \<sqinter> 1 \<le> Z"
 proof -
   have "(S \<sqinter> 1) * S\<^sup>T \<le> S \<sqinter> 1"
     by (metis S_injective conv_dist_comp coreflexive_symmetric inf.boundedI inf.cobounded1 inf.cobounded2 injective_codomain)
   hence "(S \<sqinter> 1) * S\<^sup>T\<^sup>\<star> \<le> S \<sqinter> 1"
     using star_right_induct_mult by blast
   hence "(S \<sqinter> 1) * S\<^sup>T\<^sup>\<star> * Z \<le> (S \<sqinter> 1) * Z"
     by (simp add: mult_left_isotone)
   also have "... \<le> Z"
     by (metis comp_left_subdist_inf inf.boundedE mult_1_left)
   finally show ?thesis
     using S_star_Z_top inf.order_trans top_right_mult_increasing mult_assoc by auto
 qed
 
 lemma S_inf_1_below_conv_Z:
   "S \<sqinter> 1 \<le> Z\<^sup>T"
   using S_inf_1_below_Z conv_order coreflexive_symmetric by fastforce
 
 text \<open>
 The successor operation provides a convenient way to compare two natural numbers.
 Namely, $k < m$ if $m$ can be reached from $k$ by finitely many applications of the successor, formally \<open>m \<le> S\<^sup>T\<^sup>\<star> * k\<close> or \<open>k \<le> S\<^sup>\<star> * m\<close>.
 This does not work for numbers modulo $m$ since comparison depends on the chosen representative.
 We therefore work with a modified successor relation \<open>S'\<close>, which is a partial function that computes the successor for all numbers except $m-1$.
 If $S$ is surjective, the point \<open>M\<close> representing the greatest number $m-1$ is the predecessor of $0$ under \<open>S\<close>.
 If $S$ is not surjective (like for the set of all natural numbers), \<open>M = bot\<close>.
 \<close>
 
 abbreviation "S' \<equiv> S - Z\<^sup>T"
 abbreviation "M \<equiv> S * Z"
 
 text \<open>Theorem 11.1\<close>
 
 lemma M_point_iff_S_surjective:
   "point M \<longleftrightarrow> surjective S"
 proof
   assume 1: "point M"
   hence "1 \<le> Z\<^sup>T * S\<^sup>T * S * Z"
     using comp_associative conv_dist_comp surjective_var by auto
   hence "Z \<le> S\<^sup>T * S * Z"
     using 1 Z_point bijective_reverse mult_assoc by auto
   also have "... \<le> S\<^sup>T * top"
     by (simp add: comp_isotone mult_assoc)
   finally have "S\<^sup>T * S\<^sup>T * top \<squnion> Z \<le> S\<^sup>T * top"
     using mult_isotone mult_assoc by force
   hence "S\<^sup>T\<^sup>\<star> * Z \<le> S\<^sup>T * top"
     by (simp add: star_left_induct mult_assoc)
   thus "surjective S"
     by (simp add: S_star_Z_top order.antisym surjective_conv_total)
 next
   assume "surjective S"
   thus "point M"
     by (metis S_injective Z_point comp_associative injective_mult_closed)
 qed
 
 text \<open>Theorem 10.1\<close>
 
 lemma S'_univalent:
   "univalent S'"
   by (simp add: S_univalent univalent_inf_closed)
 
 text \<open>Theorem 10.2\<close>
 
 lemma S'_injective:
   "injective S'"
   by (simp add: S_injective injective_inf_closed)
 
 text \<open>Theorem 10.9\<close>
 
 lemma S'_Z:
   "S' * Z = bot"
   by (simp add: Z_point covector_vector_comp injective_comp_right_dist_inf)
 
 text \<open>Theorem 10.4\<close>
 
 lemma S'_irreflexive:
   "irreflexive S'"
   using S_inf_1_below_conv_Z order_lesseq_imp p_shunting_swap pp_increasing by blast
 
 end
 
 class skra_peano_2 = skra_peano_1 +
   assumes S_total: "total S"
 begin
 
 lemma S_mapping:
   "mapping S"
   by (simp add: S_total S_univalent)
 
 text \<open>Theorem 11.2\<close>
 
 lemma M_bot_iff_S_not_surjective:
   "M \<noteq> bot \<longleftrightarrow> surjective S"
 proof
   assume "M \<noteq> bot"
   hence "top * S * Z = top"
     by (metis S_mapping Z_point bijective_regular comp_associative mapping_regular regular_mult_closed tarski)
   hence "Z\<^sup>T \<le> top * S"
     using M_point_iff_S_surjective S_injective Z_point comp_associative injective_mult_closed by auto
   thus "surjective S"
     using sup.orderE top_S_sup_conv_Z by fastforce
 next
   assume "surjective S"
   thus "M \<noteq> bot"
     using M_point_iff_S_surjective consistent covector_bot_closed by force
 qed
 
 text \<open>Theorem 11.3\<close>
 
 lemma M_point_or_bot:
   "point M \<or> M = bot"
   using M_bot_iff_S_not_surjective M_point_iff_S_surjective by blast
 
 text \<open>Alternative way to express \<open>S'\<close>\<close>
 
 text \<open>Theorem 12.1\<close>
 
 lemma S'_var:
   "S' = S - M"
 proof -
   have "S' = S * (1 - Z\<^sup>T)"
     by (simp add: Z_point covector_comp_inf vector_conv_compl)
   also have "... = S * (1 - Z)"
     by (metis conv_complement one_inf_conv)
   also have "... = S * 1 \<sqinter> S * -Z"
     by (simp add: S_mapping univalent_comp_left_dist_inf)
   also have "... = S - M"
     by (simp add: comp_mapping_complement S_mapping)
   finally show ?thesis
     .
 qed
 
 text \<open>Special case of just $1$ number\<close>
 
 text \<open>Theorem 12.2\<close>
 
 lemma M_is_Z_iff_1_is_top:
   "M = Z \<longleftrightarrow> 1 = top"
 proof
   assume "M = Z"
   hence "Z = S\<^sup>T * Z"
     by (metis S_mapping Z_point order.antisym bijective_reverse inf.eq_refl shunt_mapping)
   thus "1 = top"
     by (metis S_star_Z_top Z_point inf.eq_refl star_left_induct sup.absorb2 symmetric_top_closed top_le)
 next
   assume "1 = top"
   thus "M = Z"
     using S_mapping comp_right_one mult_1_left by auto
 qed
 
 text \<open>Theorem 12.3\<close>
 
 lemma S_irreflexive:
   assumes "M \<noteq> Z"
   shows "irreflexive S"
 proof -
   have "(S \<sqinter> 1) * S\<^sup>T \<le> S \<sqinter> 1"
     by (smt (z3) S_injective S_mapping coreflexive_comp_top_inf dual_order.eq_iff inf.cobounded1 inf.sup_monoid.add_commute inf.sup_same_context mult_left_isotone one_inf_conv top_right_mult_increasing total_var)
   hence "(S \<sqinter> 1) * S\<^sup>T\<^sup>\<star> \<le> S \<sqinter> 1"
     using star_right_induct_mult by blast
   hence "(S \<sqinter> 1) * S\<^sup>T\<^sup>\<star> * Z \<le> (S \<sqinter> 1) * Z"
     by (simp add: mult_left_isotone)
   also have "... = M \<sqinter> Z"
     by (simp add: Z_point injective_comp_right_dist_inf)
   also have "... = bot"
     by (smt (verit, ccfv_threshold) M_point_or_bot assms Z_point bijective_one_closed bijective_regular comp_associative conv_complement coreflexive_comp_top_inf epm_3 inf.sup_monoid.add_commute one_inf_conv regular_mult_closed star.circ_increasing star.circ_zero tarski vector_conv_covector vector_export_comp_unit)
   finally have "S \<sqinter> 1 \<le> bot"
     using S_star_Z_top comp_associative le_bot top_right_mult_increasing by fastforce
   thus ?thesis
     using le_bot pseudo_complement by blast
 qed
 
 text \<open>
 We show that \<open>S'\<close> satisfies most properties of \<open>S\<close>.
 \<close>
 
 lemma M_regular:
   "regular M"
   using S_mapping Z_point bijective_regular mapping_regular regular_mult_closed by blast
 
 lemma S'_regular:
   "regular S'"
   using S_mapping mapping_regular by auto
 
 text \<open>Theorem 10.3\<close>
 
 lemma S'_star_Z_top:
   "S'\<^sup>T\<^sup>\<star> * Z = top"
 proof -
   have "S\<^sup>T\<^sup>\<star> * Z = (S' \<squnion> (S \<sqinter> M))\<^sup>T\<^sup>\<star> * Z"
     by (metis M_regular maddux_3_11_pp S'_var)
   also have "... \<le> S'\<^sup>T\<^sup>\<star> * Z"
   proof (cases "M = bot")
     case True
     thus ?thesis
       by simp
   next
     case False
     hence "point M"
       using M_point_or_bot by auto
     hence "arc (S \<sqinter> M)"
       using S_mapping mapping_inf_point_arc by blast
     hence 1: "arc ((S \<sqinter> M)\<^sup>T)"
       using conv_involutive by auto
     have 2: "S \<sqinter> M \<le> Z\<^sup>T"
       by (metis S'_var Z_point bijective_regular conv_complement inf.cobounded2 p_shunting_swap)
     have "(S' \<squnion> (S \<sqinter> M))\<^sup>T\<^sup>\<star> * Z = (S'\<^sup>T \<squnion> (S \<sqinter> M)\<^sup>T)\<^sup>\<star> * Z"
       by (simp add: S'_var conv_dist_sup)
     also have "... = (S'\<^sup>T\<^sup>\<star> * (S \<sqinter> M)\<^sup>T * S'\<^sup>T\<^sup>\<star> \<squnion> S'\<^sup>T\<^sup>\<star>) * Z"
       using 1 star_arc_decompose sup_commute by auto
     also have "... = S'\<^sup>T\<^sup>\<star> * (S \<sqinter> M)\<^sup>T * S'\<^sup>T\<^sup>\<star> * Z \<squnion> S'\<^sup>T\<^sup>\<star> * Z"
       using mult_right_dist_sup by auto
     also have "... \<le> S'\<^sup>T\<^sup>\<star> * Z\<^sup>T\<^sup>T * S'\<^sup>T\<^sup>\<star> * Z \<squnion> S'\<^sup>T\<^sup>\<star> * Z"
       using 2 by (meson comp_isotone conv_isotone inf.eq_refl semiring.add_mono)
     also have "... \<le> S'\<^sup>T\<^sup>\<star> * Z"
       by (metis Z_point comp_associative conv_involutive le_supI mult_right_isotone top.extremum)
     finally show ?thesis
       .
   qed
   finally show ?thesis
     using S_star_Z_top top_le by auto
 qed
 
 text \<open>Theorem 10.5\<close>
 
 lemma Z_below_S'_star:
   "Z \<le> S'\<^sup>\<star>"
   by (metis S'_star_Z_top Z_point comp_associative comp_right_one conv_order conv_star_commute mult_right_isotone vector_conv_covector)
 
 text \<open>Theorem 10.6\<close>
 
 lemma S'_connected:
   "S'\<^sup>T\<^sup>\<star> * S'\<^sup>\<star> = top"
   by (metis Z_below_S'_star S'_star_Z_top mult_left_dist_sup sup.orderE sup_commute top.extremum)
 
 text \<open>Theorem 10.7\<close>
 
 lemma S'_star_connex:
   "S'\<^sup>\<star> \<squnion> S'\<^sup>T\<^sup>\<star> = top"
   using S'_connected S'_univalent cancel_separate_eq sup_commute by auto
 
 text \<open>Theorem 10.8\<close>
 
 lemma Z_sup_conv_S'_top:
   "Z \<squnion> S'\<^sup>T * top = top"
   using S'_star_Z_top star.circ_loop_fixpoint sup_commute by auto
 
 lemma top_S'_sup_conv_Z:
   "top * S' \<squnion> Z\<^sup>T = top"
   by (metis S'_star_Z_top conv_dist_comp conv_involutive conv_star_commute star.circ_back_loop_fixpoint symmetric_top_closed)
 
 end
 
 subsection \<open>Initialising Ranks\<close>
 
 text \<open>
 We show that the rank array satisfies three properties which are established/preserved by the union-find operations.
 First, every node has a rank, that is, the rank array is a mapping.
 Second, the rank of a node is strictly smaller than the rank of its parent, except if the node is a root.
 This implies that the rank of a node is an upper bound on the height of its subtree.
 Third, the number of roots in the disjoint-set forest (the number of disjoint sets) is not larger than $m-k$ where $m$ is the total number of nodes and $k$ is the maximum rank of any node.
 The third property is useful to show that ranks never overflow (exceed $m-1$).
 To compare the number of roots and $m-k$ we use the existence of an injective univalent relation between the set of roots and the set of $m-k$ largest numbers, both represented as vectors.
 The three properties are captured in \<open>rank_property\<close>.
 \<close>
 
 class skra_peano_3 = stone_kleene_relation_algebra_tarski_finite_regular + skra_peano_2
 begin
 
 definition "card_less_eq v w \<equiv> \<exists>i . injective i \<and> univalent i \<and> regular i \<and> v \<le> i * w"
 definition "rank_property p rank \<equiv> mapping rank \<and> (p - 1) * rank \<le> rank * S'\<^sup>+ \<and> card_less_eq (roots p) (-(S'\<^sup>+ * rank\<^sup>T * top))"
 
 end
 
 class skra_peano_4 = stone_kleene_relation_algebra_choose_point_finite_regular + skra_peano_2
 begin
 
 subclass skra_peano_3 ..
 
 text \<open>
 The initialisation loop is augmented by setting the rank of each node to $0$.
 The resulting rank array satisfies the desired properties explained above.
 \<close>
 
 theorem init_ranks:
   "VARS h p x rank
   [ True ]
   FOREACH x
     USING h
       INV { p - h = 1 - h \<and> rank - h = Z\<^sup>T - h }
        DO p := make_set p x;
           rank[x] := Z
        OD
   [ p = 1 \<and> disjoint_set_forest p \<and> rank = Z\<^sup>T \<and> rank_property p rank \<and> h = bot ]"
 proof vcg_tc_simp
   fix h p rank
   let ?x = "choose_point h"
   let ?m = "make_set p ?x"
   let ?rank = "rank[?x\<longmapsto>Z]"
   assume 1: "regular h \<and> vector h \<and> p - h = 1 - h \<and> rank - h = Z\<^sup>T - h \<and> h \<noteq> bot"
   show "vector (-?x \<sqinter> h) \<and>
         ?m \<sqinter> (--?x \<squnion> -h) = 1 \<sqinter> (--?x \<squnion> -h) \<and>
         ?rank \<sqinter> (--?x \<squnion> -h) = Z\<^sup>T \<sqinter> (--?x \<squnion> -h) \<and>
-        card { x . regular x \<and> x \<le> -?x \<and> x \<le> h } < card { x . regular x \<and> x \<le> h }"
+        card { x . regular x \<and> x \<le> -?x \<and> x \<le> h } < h\<down>"
   proof (intro conjI)
     show "vector (-?x \<sqinter> h)"
       using 1 choose_point_point vector_complement_closed vector_inf_closed by blast
     have 2: "point ?x \<and> regular ?x"
       using 1 bijective_regular choose_point_point by blast
     have 3: "-h \<le> -?x"
       using choose_point_decreasing p_antitone_iff by auto
     have 4: "?x \<sqinter> ?m = ?x * ?x\<^sup>T \<and> -?x \<sqinter> ?m = -?x \<sqinter> p"
       using 1 choose_point_point make_set_function make_set_postcondition_def by auto
     have "?m \<sqinter> (--?x \<squnion> -h) = (?m \<sqinter> ?x) \<squnion> (?m - h)"
       using 2 comp_inf.comp_left_dist_sup by auto
     also have "... = ?x * ?x\<^sup>T \<squnion> (?m \<sqinter> -?x \<sqinter> -h)"
       using 3 4 by (smt (z3) inf_absorb2 inf_assoc inf_commute)
     also have "... = ?x * ?x\<^sup>T \<squnion> (1 - h)"
       using 1 3 4 inf.absorb2 inf.sup_monoid.add_assoc inf_commute by auto
     also have "... = (1 \<sqinter> ?x) \<squnion> (1 - h)"
       using 2 by (metis inf.cobounded2 inf.sup_same_context one_inf_conv vector_covector)
     also have "... = 1 \<sqinter> (--?x \<squnion> -h)"
       using 2 comp_inf.semiring.distrib_left by auto
     finally show "?m \<sqinter> (--?x \<squnion> -h) = 1 \<sqinter> (--?x \<squnion> -h)"
       .
     have 5: "?x \<sqinter> ?rank = ?x \<sqinter> Z\<^sup>T \<and> -?x \<sqinter> ?rank = -?x \<sqinter> rank"
       by (smt (z3) inf_commute order_refl update_inf_different update_inf_same)
     have "?rank \<sqinter> (--?x \<squnion> -h) = (?rank \<sqinter> ?x) \<squnion> (?rank - h)"
       using 2 comp_inf.comp_left_dist_sup by auto
     also have "... = (?x \<sqinter> Z\<^sup>T) \<squnion> (?rank \<sqinter> -?x \<sqinter> -h)"
       using 3 5 by (smt (z3) inf_absorb2 inf_assoc inf_commute)
     also have "... = (Z\<^sup>T \<sqinter> ?x) \<squnion> (Z\<^sup>T - h)"
       using 1 3 5 inf.absorb2 inf.sup_monoid.add_assoc inf_commute by auto
     also have "... = Z\<^sup>T \<sqinter> (--?x \<squnion> -h)"
       using 2 comp_inf.semiring.distrib_left by auto
     finally show "?rank \<sqinter> (--?x \<squnion> -h) = Z\<^sup>T \<sqinter> (--?x \<squnion> -h)"
       .
     have 5: "\<not> ?x \<le> -?x"
       using 1 2 by (metis comp_commute_below_diversity conv_order inf.cobounded2 inf_absorb2 pseudo_complement strict_order_var top.extremum)
     have 6: "?x \<le> h"
       using 1 by (metis choose_point_decreasing)
-    show "card { x . regular x \<and> x \<le> -?x \<and> x \<le> h } < card { x . regular x \<and> x \<le> h }"
+    show "card { x . regular x \<and> x \<le> -?x \<and> x \<le> h } < h\<down>"
       apply (rule psubset_card_mono)
       using finite_regular apply simp
       using 2 5 6 by auto
   qed
 next
   show "rank_property 1 (Z\<^sup>T)"
   proof (unfold rank_property_def, intro conjI)
     show "univalent (Z\<^sup>T)" "total (Z\<^sup>T)"
       using Z_point surjective_conv_total by auto
     show "(1 - 1) * (Z\<^sup>T) \<le> (Z\<^sup>T) * S'\<^sup>+"
       by simp
     have "top \<le> 1 * -(S'\<^sup>+ * Z * top)"
       by (simp add: S'_Z comp_associative star_simulation_right_equal)
     thus "card_less_eq (roots 1) (-(S'\<^sup>+ * Z\<^sup>T\<^sup>T * top))"
       by (metis conv_involutive inf.idem mapping_one_closed regular_one_closed card_less_eq_def bijective_one_closed)
   qed
 qed
 
 end
 
 subsection \<open>Union by Rank\<close>
 
 text \<open>
 We show that path compression and union-by-rank preserve the rank property.
 \<close>
 
 context stone_kleene_relation_algebra_tarski_finite_regular
 begin
 
 lemma union_sets_1_swap:
   assumes "union_sets_precondition p0 x y"
     and "path_compression_postcondition p1 x r p0"
     and "path_compression_postcondition p2 y s p1"
   shows "union_sets_postcondition (p2[s\<longmapsto>r]) x y p0"
 proof (unfold union_sets_postcondition_def union_sets_precondition_def, intro conjI)
   let ?p = "p2[s\<longmapsto>r]"
   have 1: "disjoint_set_forest p1 \<and> point r \<and> r = root p1 x \<and> p1 \<sqinter> 1 = p0 \<sqinter> 1 \<and> fc p1 = fc p0"
     using assms(2) path_compression_precondition_def path_compression_postcondition_def by auto
   have 2: "disjoint_set_forest p2 \<and> point s \<and> s = root p2 y \<and> p2 \<sqinter> 1 = p1 \<sqinter> 1 \<and> fc p2 = fc p1"
     using assms(3) path_compression_precondition_def path_compression_postcondition_def by auto
   hence 3: "fc p2 = fc p0"
     using 1 by simp
   show 4: "univalent ?p"
     using 1 2 update_univalent by blast
   show "total ?p"
     using 1 2 bijective_regular update_total by blast
   show "acyclic (?p - 1)"
   proof (cases "r = s")
     case True
     thus ?thesis
       using 2 update_acyclic_5 by fastforce
   next
     case False
     hence "bot = s \<sqinter> r"
       using 1 2 distinct_points inf_commute by blast
     also have "... = s \<sqinter> p1\<^sup>T\<^sup>\<star> * r"
       using 1 by (smt root_transitive_successor_loop)
     also have "... = s \<sqinter> p2\<^sup>T\<^sup>\<star> * r"
       using 1 2 by (smt (z3) inf_assoc inf_commute same_root)
     finally have "r \<sqinter> p2\<^sup>\<star> * s = bot"
       using schroeder_1 conv_star_commute inf.sup_monoid.add_commute by fastforce
     thus ?thesis
       using 1 2 update_acyclic_4 by blast
   qed
   show "vector x"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "injective x"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "surjective x"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "vector y"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "injective y"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "surjective y"
     using assms(1) by (simp add: union_sets_precondition_def)
   show "fc ?p = wcc (p0 \<squnion> x * y\<^sup>T)"
   proof (rule order.antisym)
     have "s = p2[[s]]"
       using 2 by (metis root_successor_loop)
     hence "s * s\<^sup>T \<le> p2\<^sup>T"
       using 2 eq_refl shunt_bijective by blast
     hence "s * s\<^sup>T \<le> p2"
       using 2 conv_order coreflexive_symmetric by fastforce
     hence "s \<le> p2 * s"
       using 2 shunt_bijective by blast
     hence 5: "p2[[s]] \<le> s"
       using 2 shunt_mapping by blast
     have "s \<sqinter> p2 \<le> s * (top \<sqinter> s\<^sup>T * p2)"
       using 2 by (metis dedekind_1)
     also have "... = s * s\<^sup>T * p2"
       by (simp add: mult_assoc)
     also have "... \<le> s * s\<^sup>T"
       using 5 by (metis comp_associative conv_dist_comp conv_involutive conv_order mult_right_isotone)
     also have "... \<le> 1"
       using 2 by blast
     finally have 6: "s \<sqinter> p2 \<le> 1"
       by simp
     have "p0 \<le> wcc p0"
       by (simp add: star.circ_sub_dist_1)
     also have "... = wcc p2"
       using 3 by (simp add: star_decompose_1)
     also have 7: "... \<le> wcc ?p"
     proof -
       have "wcc p2 = wcc ((-s \<sqinter> p2) \<squnion> (s \<sqinter> p2))"
         using 2 by (metis bijective_regular inf.sup_monoid.add_commute maddux_3_11_pp)
       also have "... \<le> wcc ((-s \<sqinter> p2) \<squnion> 1)"
         using 6 wcc_isotone sup_right_isotone by simp
       also have "... = wcc (-s \<sqinter> p2)"
         using wcc_with_loops by simp
       also have "... \<le> wcc ?p"
         using wcc_isotone sup_ge2 by blast
       finally show ?thesis
         by simp
     qed
     finally have 8: "p0 \<le> wcc ?p"
       by force
     have "s \<le> p2\<^sup>T\<^sup>\<star> * y"
       using 2 by (metis inf_le1)
     hence 9: "s * y\<^sup>T \<le> p2\<^sup>T\<^sup>\<star>"
       using assms(1) shunt_bijective union_sets_precondition_def by blast
     hence "y * s\<^sup>T \<le> p2\<^sup>\<star>"
       using conv_dist_comp conv_order conv_star_commute by force
     also have "... \<le> wcc p2"
       by (simp add: star.circ_sub_dist)
     also have "... \<le> wcc ?p"
       using 7 by simp
     finally have 10: "y * s\<^sup>T \<le> wcc ?p"
       by simp
     have 11: "s * r\<^sup>T \<le> wcc ?p"
       using 1 2 star.circ_sub_dist_1 sup_assoc vector_covector by auto
     have "r \<le> p1\<^sup>T\<^sup>\<star> * x"
       using 1 by (metis inf_le1)
     hence 12: "r * x\<^sup>T \<le> p1\<^sup>T\<^sup>\<star>"
       using assms(1) shunt_bijective union_sets_precondition_def by blast
     also have "... \<le> wcc p1"
       using star_isotone sup_ge2 by blast
     also have "... = wcc p2"
       using 2 by (simp add: star_decompose_1)
     also have "... \<le> wcc ?p"
       using 7 by simp
     finally have 13: "r * x\<^sup>T \<le> wcc ?p"
       by simp
     have "x \<le> x * r\<^sup>T * r \<and> y \<le> y * s\<^sup>T * s"
       using 1 2 shunt_bijective by blast
     hence "y * x\<^sup>T \<le> y * s\<^sup>T * s * (x * r\<^sup>T * r)\<^sup>T"
       using comp_isotone conv_isotone by blast
     also have "... = y * s\<^sup>T * s * r\<^sup>T * r * x\<^sup>T"
       by (simp add: comp_associative conv_dist_comp)
     also have "... \<le> wcc ?p * (s * r\<^sup>T) * (r * x\<^sup>T)"
       using 10 by (metis mult_left_isotone mult_assoc)
     also have "... \<le> wcc ?p * wcc ?p * (r * x\<^sup>T)"
       using 11 by (metis mult_left_isotone mult_right_isotone)
     also have "... \<le> wcc ?p * wcc ?p * wcc ?p"
       using 13 by (metis mult_right_isotone)
     also have "... = wcc ?p"
       by (simp add: star.circ_transitive_equal)
     finally have "x * y\<^sup>T \<le> wcc ?p"
       by (metis conv_dist_comp conv_involutive conv_order wcc_equivalence)
     hence "p0 \<squnion> x * y\<^sup>T \<le> wcc ?p"
       using 8 by simp
     hence "wcc (p0 \<squnion> x * y\<^sup>T) \<le> wcc ?p"
       using wcc_below_wcc by simp
     thus "wcc (p0 \<squnion> x * y\<^sup>T) \<le> fc ?p"
       using 4 fc_wcc by simp
     have "-s \<sqinter> p2 \<le> wcc p2"
       by (simp add: inf.coboundedI2 star.circ_sub_dist_1)
     also have "... = wcc p0"
       using 3 by (simp add: star_decompose_1)
     also have "... \<le> wcc (p0 \<squnion> y * x\<^sup>T)"
       by (simp add: wcc_isotone)
     finally have 14: "-s \<sqinter> p2 \<le> wcc (p0 \<squnion> y * x\<^sup>T)"
       by simp
     have "s * y\<^sup>T \<le> wcc p2"
       using 9 inf.order_trans star.circ_sub_dist sup_commute by fastforce
     also have "... = wcc p0"
       using 1 2 by (simp add: star_decompose_1)
     also have "... \<le> wcc (p0 \<squnion> y * x\<^sup>T)"
       by (simp add: wcc_isotone)
     finally have 15: "s * y\<^sup>T \<le> wcc (p0 \<squnion> y * x\<^sup>T)"
       by simp
     have 16: "y * x\<^sup>T \<le> wcc (p0 \<squnion> y * x\<^sup>T)"
       using le_supE star.circ_sub_dist_1 by blast
     have "x * r\<^sup>T \<le> p1\<^sup>\<star>"
       using 12 conv_dist_comp conv_order conv_star_commute by fastforce
     also have "... \<le> wcc p1"
       using star.circ_sub_dist sup_commute by fastforce
     also have "... = wcc p0"
       using 1 by (simp add: star_decompose_1)
     also have "... \<le> wcc (p0 \<squnion> y * x\<^sup>T)"
       by (simp add: wcc_isotone)
     finally have 17: "x * r\<^sup>T \<le> wcc (p0 \<squnion> y * x\<^sup>T)"
       by simp
     have "r \<le> r * x\<^sup>T * x \<and> s \<le> s * y\<^sup>T * y"
       using assms(1) shunt_bijective union_sets_precondition_def by blast
     hence "s * r\<^sup>T \<le> s * y\<^sup>T * y * (r * x\<^sup>T * x)\<^sup>T"
       using comp_isotone conv_isotone by blast
     also have "... = s * y\<^sup>T * y * x\<^sup>T * x * r\<^sup>T"
       by (simp add: comp_associative conv_dist_comp)
     also have "... \<le> wcc (p0 \<squnion> y * x\<^sup>T) * (y * x\<^sup>T) * (x * r\<^sup>T)"
       using 15 by (metis mult_left_isotone mult_assoc)
     also have "... \<le> wcc (p0 \<squnion> y * x\<^sup>T) * wcc (p0 \<squnion> y * x\<^sup>T) * (x * r\<^sup>T)"
       using 16 by (metis mult_left_isotone mult_right_isotone)
     also have "... \<le> wcc (p0 \<squnion> y * x\<^sup>T) * wcc (p0 \<squnion> y * x\<^sup>T) * wcc (p0 \<squnion> y * x\<^sup>T)"
       using 17 by (metis mult_right_isotone)
     also have "... = wcc (p0 \<squnion> y * x\<^sup>T)"
       by (simp add: star.circ_transitive_equal)
     finally have "?p \<le> wcc (p0 \<squnion> y * x\<^sup>T)"
       using 1 2 14 vector_covector by auto
     hence "wcc ?p \<le> wcc (p0 \<squnion> y * x\<^sup>T)"
       using wcc_below_wcc by blast
     also have "... = wcc (p0 \<squnion> x * y\<^sup>T)"
       using conv_dist_comp conv_dist_sup sup_assoc sup_commute by auto
     finally show "fc ?p \<le> wcc (p0 \<squnion> x * y\<^sup>T)"
       using 4 fc_wcc by simp
   qed
 qed
 
 lemma union_sets_1_skip:
   assumes "union_sets_precondition p0 x y"
     and "path_compression_postcondition p1 x r p0"
     and "path_compression_postcondition p2 y r p1"
   shows "union_sets_postcondition p2 x y p0"
 proof (unfold union_sets_postcondition_def union_sets_precondition_def, intro conjI)
   have 1: "point r \<and> r = root p1 x \<and> fc p1 = fc p0 \<and> disjoint_set_forest p2 \<and> r = root p2 y \<and> fc p2 = fc p1"
     using assms(2,3) path_compression_precondition_def path_compression_postcondition_def by auto
   thus "univalent p2" "total p2" "acyclic (p2 - 1)"
     by auto
   show "vector x" "injective x" "surjective x" "vector y" "injective y" "surjective y"
     using assms(1) union_sets_precondition_def by auto
   have "r \<le> p1\<^sup>T\<^sup>\<star> * x"
     using 1 by (metis inf_le1)
   hence "r * x\<^sup>T \<le> p1\<^sup>T\<^sup>\<star>"
     using assms(1) shunt_bijective union_sets_precondition_def by blast
   hence 2: "x * r\<^sup>T \<le> p1\<^sup>\<star>"
     using conv_dist_comp conv_order conv_star_commute by force
   have "r \<le> p2\<^sup>T\<^sup>\<star> * y"
     using 1 by (metis inf_le1)
   hence 3: "r * y\<^sup>T \<le> p2\<^sup>T\<^sup>\<star>"
     using assms(1) shunt_bijective union_sets_precondition_def by blast
   have "x * y\<^sup>T \<le> x * r\<^sup>T * r * y\<^sup>T"
     using 1 mult_left_isotone shunt_bijective by blast
   also have "... \<le> p1\<^sup>\<star> * p2\<^sup>T\<^sup>\<star>"
     using 2 3 by (metis comp_associative comp_isotone)
   also have "... \<le> wcc p0"
     using 1 by (metis star.circ_mult_upper_bound star_decompose_1 star_isotone sup_ge2 star.circ_sub_dist)
   finally show "fc p2 = wcc (p0 \<squnion> x * y\<^sup>T)"
     using 1 by (smt (z3) fc_star star_decompose_1 sup_absorb1 wcc_sup_wcc star.circ_sub_dist_3 sup_commute wcc_equivalence)
 qed
 
 end
 
 syntax
   "_Cond1" :: "'bexp \<Rightarrow> 'com \<Rightarrow> 'com"  ("(1IF _/ THEN _/ FI)" [0,0] 61)
 translations "IF b THEN c FI" == "IF b THEN c ELSE SKIP FI"
 
 context skra_peano_3
 begin
 
 lemma path_compression_preserves_rank_property:
   assumes "path_compression_postcondition p x y p0"
       and "disjoint_set_forest p0"
       and "rank_property p0 rank"
     shows "rank_property p rank"
 proof (unfold rank_property_def, intro conjI)
   let ?px = "p0\<^sup>T\<^sup>\<star> * x"
   have 1: "point y"
     using assms(1,2) path_compression_postcondition_def path_compression_precondition_def root_point by auto
   have 2: "vector ?px"
     using assms(1) comp_associative path_compression_postcondition_def path_compression_precondition_def by auto
   have "root p0 x = root p x"
     by (smt (verit) assms(1,2) path_compression_postcondition_def path_compression_precondition_def same_root)
   hence "root p0 x = y"
     using assms(1) path_compression_postcondition_def path_compression_precondition_def by auto
   hence "?px \<le> p0\<^sup>\<star> * y"
     by (meson assms(1,2) path_splitting_invariant_aux_1(3) path_compression_precondition_def path_compression_postcondition_def)
   hence "?px * y\<^sup>T \<le> p0\<^sup>\<star>"
     using 1 shunt_bijective by blast
   hence "?px \<sqinter> y\<^sup>T \<le> p0\<^sup>\<star>"
     using 1 2 by (simp add: vector_covector)
   also have "... = (p0 - 1)\<^sup>+ \<squnion> 1"
     using reachable_without_loops star_left_unfold_equal sup_commute by fastforce
   finally have 3: "?px \<sqinter> y\<^sup>T \<sqinter> -1 \<le> (p0 - 1)\<^sup>+"
     using half_shunting by blast
   have "p0[?px\<longmapsto>y] = p"
     using assms(1) path_compression_postcondition_def by simp
   hence "(p - 1) * rank = (?px \<sqinter> y\<^sup>T \<sqinter> -1) * rank \<squnion> (-?px \<sqinter> p0 \<sqinter> -1) * rank"
     using inf_sup_distrib2 mult_right_dist_sup by force
   also have "... \<le> (?px \<sqinter> y\<^sup>T \<sqinter> -1) * rank \<squnion> (p0 - 1) * rank"
     by (meson comp_inf.mult_semi_associative le_infE mult_left_isotone sup_right_isotone)
   also have "... \<le> (?px \<sqinter> y\<^sup>T \<sqinter> -1) * rank \<squnion> rank * S'\<^sup>+"
     using assms(3) rank_property_def sup_right_isotone by auto
   also have "... \<le> (p0 - 1)\<^sup>+ * rank \<squnion> rank * S'\<^sup>+"
     using 3 mult_left_isotone sup_left_isotone by blast
   also have "... \<le> rank * S'\<^sup>+"
   proof -
     have "(p0 - 1)\<^sup>\<star> * rank \<le> rank * S'\<^sup>\<star>"
       using assms(3) rank_property_def star_simulation_left star.left_plus_circ by fastforce
     hence "(p0 - 1)\<^sup>+ * rank \<le> (p0 - 1) * rank * S'\<^sup>\<star>"
       by (simp add: comp_associative mult_right_isotone)
     also have "... \<le> rank * S'\<^sup>+"
       by (smt (z3) assms(3) rank_property_def comp_associative comp_left_subdist_inf inf.boundedE inf.sup_right_divisibility star.circ_transitive_equal)
     finally show ?thesis
       by simp
   qed
   finally show "(p - 1) * rank \<le> rank * S'\<^sup>+"
     .
   show "univalent rank" "total rank"
     using rank_property_def assms(3) by auto
   show "card_less_eq (roots p) (-(S'\<^sup>+ * rank\<^sup>T * top))"
     using assms(1,3) path_compression_postcondition_def rank_property_def by auto
 qed
 
 theorem union_sets_by_rank:
   "VARS p r s rank
   [ union_sets_precondition p x y \<and> rank_property p rank \<and> p0 = p ]
   r := find_set p x;
   p := path_compression p x r;
   s := find_set p y;
   p := path_compression p y s;
   IF r \<noteq> s THEN
     IF rank[[r]] \<le> S'\<^sup>+ * (rank[[s]]) THEN
       p[r] := s
     ELSE
       p[s] := r;
       IF rank[[r]] = rank[[s]] THEN
         rank[r] := S'\<^sup>T * (rank[[r]])
       FI
     FI
   FI
   [ union_sets_postcondition p x y p0 \<and> rank_property p rank ]"
 proof vcg_tc_simp
   fix rank
   let ?r = "find_set p0 x"
   let ?p1 = "path_compression p0 x ?r"
   let ?s = "find_set ?p1 y"
   let ?p2 = "path_compression ?p1 y ?s"
   let ?p5 = "path_compression ?p1 y ?r"
   let ?rr = "rank[[?r]]"
   let ?rs = "rank[[?s]]"
   let ?rank = "rank[?r\<longmapsto>S'\<^sup>T * ?rs]"
   let ?p3 = "?p2[?r\<longmapsto>?s]"
   let ?p4 = "?p2[?s\<longmapsto>?r]"
   assume 1: "union_sets_precondition p0 x y \<and> rank_property p0 rank"
   hence 2: "path_compression_postcondition ?p1 x ?r p0"
     using find_set_function find_set_postcondition_def find_set_precondition_def path_compression_function path_compression_precondition_def union_sets_precondition_def by auto
   hence 3: "path_compression_postcondition ?p2 y ?s ?p1"
     using 1 find_set_function find_set_postcondition_def find_set_precondition_def path_compression_function path_compression_precondition_def union_sets_precondition_def path_compression_postcondition_def by meson
   have "rank_property ?p1 rank"
     using 1 2 path_compression_preserves_rank_property union_sets_precondition_def by blast
   hence 4: "rank_property ?p2 rank"
     using 1 2 3 by (meson path_compression_preserves_rank_property path_compression_postcondition_def path_compression_precondition_def)
   have 5: "point ?r" "point ?s"
     using 2 3 path_compression_postcondition_def path_compression_precondition_def by auto
   hence 6: "point ?rr" "point ?rs"
     using 1 comp_associative read_injective read_surjective rank_property_def by auto
   have "top \<le> S'\<^sup>\<star> \<squnion> S'\<^sup>+\<^sup>T"
     by (metis S'_star_connex conv_dist_comp conv_star_commute eq_refl star.circ_reflexive star_left_unfold_equal star_simulation_right_equal sup.orderE sup_monoid.add_assoc)
   hence 7: "-S'\<^sup>+\<^sup>T \<le> S'\<^sup>\<star>"
     by (metis comp_inf.case_split_left comp_inf.star.circ_plus_one comp_inf.star.circ_sup_2 half_shunting)
   show "(?r \<noteq> ?s \<longrightarrow> (?rr \<le> S'\<^sup>+ * ?rs \<longrightarrow> union_sets_postcondition ?p3 x y p0 \<and> rank_property ?p3 rank) \<and>
                       (\<not> ?rr \<le> S'\<^sup>+ * ?rs \<longrightarrow> ((?rr = ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 ?rank) \<and>
                                               (?rr \<noteq> ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 rank)))) \<and>
         (?r = ?s \<longrightarrow> union_sets_postcondition ?p5 x y p0 \<and> rank_property ?p5 rank)"
   proof
     show "?r \<noteq> ?s \<longrightarrow> (?rr \<le> S'\<^sup>+ * ?rs \<longrightarrow> union_sets_postcondition ?p3 x y p0 \<and> rank_property ?p3 rank) \<and>
                       (\<not> ?rr \<le> S'\<^sup>+ * ?rs \<longrightarrow> ((?rr = ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 ?rank) \<and>
                                               (?rr \<noteq> ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 rank)))"
     proof
       assume 8: "?r \<noteq> ?s"
       show "(?rr \<le> S'\<^sup>+ * ?rs \<longrightarrow> union_sets_postcondition ?p3 x y p0 \<and> rank_property ?p3 rank) \<and>
             (\<not> ?rr \<le> S'\<^sup>+ * ?rs \<longrightarrow> ((?rr = ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 ?rank) \<and>
                                     (?rr \<noteq> ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 rank)))"
       proof
         show "?rr \<le> S'\<^sup>+ * ?rs \<longrightarrow> union_sets_postcondition ?p3 x y p0 \<and> rank_property ?p3 rank"
         proof
           assume 9: "?rr \<le> S'\<^sup>+ * ?rs"
           show "union_sets_postcondition ?p3 x y p0 \<and> rank_property ?p3 rank"
           proof
             show "union_sets_postcondition ?p3 x y p0"
               using 1 2 3 by (simp add: union_sets_1)
             show "rank_property ?p3 rank"
             proof (unfold rank_property_def, intro conjI)
               show "univalent rank" "total rank"
                 using 1 rank_property_def by auto
               have "?s \<le> -?r"
                 using 5 8 by (meson order.antisym bijective_regular point_in_vector_or_complement point_in_vector_or_complement_2)
               hence "?r \<sqinter> ?s\<^sup>T \<sqinter> 1 = bot"
                 by (metis (full_types) bot_least inf.left_commute inf.sup_monoid.add_commute one_inf_conv pseudo_complement)
               hence "?p3 \<sqinter> 1 \<le> ?p2"
                 by (smt half_shunting inf.cobounded2 pseudo_complement regular_one_closed semiring.add_mono sup_commute)
               hence "roots ?p3 \<le> roots ?p2"
                 by (simp add: mult_left_isotone)
               thus "card_less_eq (roots ?p3) (-(S'\<^sup>+ * rank\<^sup>T * top))"
                 using 4 by (meson rank_property_def card_less_eq_def order_trans)
               have "(?p3 - 1) * rank = (?r \<sqinter> ?s\<^sup>T \<sqinter> -1) * rank \<squnion> (-?r \<sqinter> ?p2 \<sqinter> -1) * rank"
                 using comp_inf.semiring.distrib_right mult_right_dist_sup by auto
               also have "... \<le> (?r \<sqinter> ?s\<^sup>T \<sqinter> -1) * rank \<squnion> (?p2 - 1) * rank"
                 using comp_inf.mult_semi_associative mult_left_isotone sup_right_isotone by auto
               also have "... \<le> (?r \<sqinter> ?s\<^sup>T \<sqinter> -1) * rank \<squnion> rank * S'\<^sup>+"
                 using 4 sup_right_isotone rank_property_def by blast
               also have "... \<le> (?r \<sqinter> ?s\<^sup>T) * rank \<squnion> rank * S'\<^sup>+"
                 using inf_le1 mult_left_isotone sup_left_isotone by blast
               also have "... = ?r * ?s\<^sup>T * rank \<squnion> rank * S'\<^sup>+"
                 using 5 by (simp add: vector_covector)
               also have "... = rank * S'\<^sup>+"
               proof -
                 have "rank\<^sup>T * ?r \<le> S'\<^sup>+ * rank\<^sup>T * ?s"
                   using 9 comp_associative by auto
                 hence "?r \<le> rank * S'\<^sup>+ * rank\<^sup>T * ?s"
                   using 4 shunt_mapping comp_associative rank_property_def by auto
                 hence "?r * ?s\<^sup>T \<le> rank * S'\<^sup>+ * rank\<^sup>T"
                   using 5 shunt_bijective by blast
                 hence "?r * ?s\<^sup>T * rank \<le> rank * S'\<^sup>+"
                   using 4 shunt_bijective rank_property_def mapping_conv_bijective by auto
                 thus ?thesis
                   using sup_absorb2 by blast
               qed
               finally show "(?p3 - 1) * rank \<le> rank * S'\<^sup>+"
                 .
             qed
           qed
         qed
         show "\<not> ?rr \<le> S'\<^sup>+ * ?rs \<longrightarrow> ((?rr = ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 ?rank) \<and>
                                      (?rr \<noteq> ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 rank))"
         proof
           assume "\<not> ?rr \<le> S'\<^sup>+ * ?rs"
           hence "?rr \<le> -(S'\<^sup>+ * ?rs)"
             using 6 by (meson point_in_vector_or_complement S'_regular bijective_regular regular_closed_star regular_mult_closed vector_mult_closed)
           also have "... = -S'\<^sup>+ * ?rs"
             using 6 comp_bijective_complement by simp
           finally have "?rs \<le> -S'\<^sup>+\<^sup>T * ?rr"
             using 6 by (metis bijective_reverse conv_complement)
           also have "... \<le> S'\<^sup>\<star> * ?rr"
             using 7 by (simp add: mult_left_isotone)
           also have "... = S'\<^sup>+ * ?rr \<squnion> ?rr"
             using star.circ_loop_fixpoint mult_assoc by auto
           finally have 10: "?rs - ?rr \<le> S'\<^sup>+ * ?rr"
             using half_shunting by blast
           show "((?rr = ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 ?rank) \<and>
                  (?rr \<noteq> ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 rank))"
           proof
             show "?rr = ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 ?rank"
             proof
               assume 11: "?rr = ?rs"
               show "union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 ?rank"
               proof
                 show "union_sets_postcondition ?p4 x y p0"
                   using 1 2 3 by (simp add: union_sets_1_swap)
                 show "rank_property ?p4 ?rank"
                 proof (unfold rank_property_def, intro conjI)
                   show "univalent ?rank"
                     using 4 5 6 by (meson S'_univalent read_injective update_univalent rank_property_def)
                   have "card_less_eq (roots ?p2) (-(S'\<^sup>+ * rank\<^sup>T * top))"
                     using 4 rank_property_def by blast
                   from this obtain i where 12: "injective i \<and> univalent i \<and> regular i \<and> roots ?p2 \<le> i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                     using card_less_eq_def by blast
                   let ?i = "(i[?s\<longmapsto>i[[i * ?rr]]])[i * ?rr\<longmapsto>i[[?s]]]"
                   have 13: "?i = (i * ?rr \<sqinter> ?s\<^sup>T * i) \<squnion> (-(i * ?rr) \<sqinter> ?s \<sqinter> ?rr\<^sup>T * i\<^sup>T * i) \<squnion> (-(i * ?rr) \<sqinter> -?s \<sqinter> i)"
                     by (smt (z3) conv_dist_comp conv_involutive inf.sup_monoid.add_assoc inf_sup_distrib1 sup_assoc)
                   have 14: "injective ?i"
                     apply (rule update_injective_swap)
                     subgoal using 12 by simp
                     subgoal using 5 by simp
                     subgoal using 6 12 injective_mult_closed by simp
                     subgoal using 5 comp_associative by simp
                     done
                   have 15: "univalent ?i"
                     apply (rule update_univalent_swap)
                     subgoal using 12 by simp
                     subgoal using 5 by simp
                     subgoal using 5 by simp
                     subgoal using 6 12 injective_mult_closed by simp
                     subgoal using 5 comp_associative by simp
                     done
                   have 16: "regular ?i"
                     using 5 6 12 by (smt (z3) bijective_regular p_dist_inf p_dist_sup pp_dist_comp regular_closed_inf regular_conv_closed)
                   have 17: "regular (i * ?rr)"
                     using 6 12 bijective_regular regular_mult_closed by blast
                   have 18: "find_set_precondition ?p1 y"
                     using 2 3 find_set_precondition_def path_compression_postcondition_def path_compression_precondition_def by blast
                   hence "?s = root ?p1 y"
                     by (meson find_set_function find_set_postcondition_def)
                   also have "... = root ?p2 y"
                     using 3 18 by (smt (z3) find_set_precondition_def path_compression_postcondition_def path_compression_precondition_def same_root)
                   also have "... \<le> roots ?p2"
                     by simp
                   also have "... \<le> i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                     using 12 by simp
                   finally have 19: "?s \<le> i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                     .
                   have "roots ?p4 \<le> ?i * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                   proof -
                     have "?r \<le> -?s"
                       using 5 8 by (meson order.antisym bijective_regular point_in_vector_or_complement point_in_vector_or_complement_2)
                     hence "?s \<sqinter> ?r\<^sup>T \<sqinter> 1 = bot"
                       by (metis (full_types) bot_least inf.left_commute inf.sup_monoid.add_commute one_inf_conv pseudo_complement)
                     hence "?p4 \<sqinter> 1 \<le> -?s \<sqinter> ?p2"
                       by (smt (z3) bot_least comp_inf.semiring.distrib_left inf.cobounded2 inf.sup_monoid.add_commute le_supI)
                     hence "roots ?p4 \<le> roots (-?s \<sqinter> ?p2)"
                       by (simp add: mult_left_isotone)
                     also have "... = -?s \<sqinter> roots ?p2"
                       using 5 inf_assoc vector_complement_closed vector_inf_comp by auto
                     also have "... = (i * ?rr \<sqinter> -?s \<sqinter> roots ?p2) \<squnion> (-(i * ?rr) \<sqinter> -?s \<sqinter> roots ?p2)"
                       using 17 by (smt (z3) comp_inf.star_plus inf_sup_distrib2 inf_top.right_neutral regular_complement_top)
                     also have "... \<le> ?i * (-(S'\<^sup>+ * ?rank\<^sup>T * top))"
                     proof (rule sup_least)
                       have "?rank\<^sup>T * top = (?r \<sqinter> (S'\<^sup>T * ?rs)\<^sup>T)\<^sup>T * top \<squnion> (-?r \<sqinter> rank)\<^sup>T * top"
                         using conv_dist_sup mult_right_dist_sup by auto
                       also have "... = (?r\<^sup>T \<sqinter> S'\<^sup>T * ?rs) * top \<squnion> (-?r\<^sup>T \<sqinter> rank\<^sup>T) * top"
                         using conv_complement conv_dist_inf conv_involutive by auto
                       also have "... = S'\<^sup>T * ?rs * (?r \<sqinter> top) \<squnion> (-?r\<^sup>T \<sqinter> rank\<^sup>T) * top"
                         using 5 by (smt (z3) covector_inf_comp_3 inf_commute)
                       also have "... = S'\<^sup>T * ?rs * (?r \<sqinter> top) \<squnion> rank\<^sup>T * (-?r \<sqinter> top)"
                         using 5 by (smt (z3) conv_complement vector_complement_closed covector_inf_comp_3 inf_commute)
                       also have "... = S'\<^sup>T * ?rs * ?r \<squnion> rank\<^sup>T * -?r"
                         by simp
                       also have "... \<le> S'\<^sup>T * ?rs * ?r \<squnion> rank\<^sup>T * top"
                         using mult_right_isotone sup_right_isotone by force
                       also have "... \<le> S'\<^sup>T * ?rs \<squnion> rank\<^sup>T * top"
                         using 5 6 by (metis inf.eq_refl mult_assoc)
                       finally have "S'\<^sup>+ * ?rank\<^sup>T * top \<le> S'\<^sup>+ * S'\<^sup>T * ?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top"
                         by (smt comp_associative mult_left_dist_sup mult_right_isotone)
                       also have "... = S'\<^sup>\<star> * (S' * S'\<^sup>T) * ?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top"
                         by (smt star_plus mult_assoc)
                       also have "... \<le> S'\<^sup>\<star> * ?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top"
                         by (metis S'_injective comp_right_one mult_left_isotone mult_right_isotone sup_left_isotone)
                       also have "... = ?rs \<squnion> S'\<^sup>+ * ?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top"
                         using comp_associative star.circ_loop_fixpoint sup_commute by fastforce
                       also have "... = ?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top"
                         by (smt (verit, del_insts) comp_associative mult_left_dist_sup sup.orderE sup_assoc sup_commute top.extremum)
                       finally have 20: "S'\<^sup>+ * ?rank\<^sup>T * top \<le> ?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top"
                         .
                       have "?s * ?s\<^sup>T = (?s \<sqinter> i * -(S'\<^sup>+ * rank\<^sup>T * top)) * ?s\<^sup>T"
                         using 19 inf.orderE by fastforce
                       also have "... = (?s \<sqinter> i * -(S'\<^sup>+ * rank\<^sup>T * top)) * top \<sqinter> ?s\<^sup>T"
                         using 5 by (smt (z3) covector_comp_inf vector_conv_covector vector_covector vector_top_closed)
                       also have "... = ?s \<sqinter> i * -(S'\<^sup>+ * rank\<^sup>T * top) * top \<sqinter> ?s\<^sup>T"
                         using 5 vector_inf_comp by auto
                       also have "... \<le> 1 \<sqinter> i * -(S'\<^sup>+ * rank\<^sup>T * top) * top"
                         using 5 by (smt (verit, ccfv_SIG) inf.cobounded1 inf.cobounded2 inf_greatest order_trans vector_covector)
                       also have "... = 1 \<sqinter> i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                         using comp_associative vector_complement_closed vector_top_closed by auto
                       also have "... \<le> 1 \<sqinter> i * -(S'\<^sup>+ * rank\<^sup>T)"
                         by (meson comp_inf.mult_right_isotone mult_right_isotone p_antitone top_right_mult_increasing)
                       also have "... \<le> 1 \<sqinter> i * S'\<^sup>\<star>\<^sup>T * rank\<^sup>T"
                       proof -
                         have "S'\<^sup>\<star>\<^sup>T * rank\<^sup>T \<squnion> S'\<^sup>+ * rank\<^sup>T = (S'\<^sup>T\<^sup>\<star> \<squnion> S'\<^sup>+) * rank\<^sup>T"
                           by (simp add: conv_star_commute mult_right_dist_sup)
                         also have "... = (S'\<^sup>T\<^sup>\<star> \<squnion> S'\<^sup>\<star>) * rank\<^sup>T"
                           by (smt (z3) comp_associative semiring.distrib_right star.circ_loop_fixpoint sup.left_commute sup_commute sup_idem)
                         also have "... = top * rank\<^sup>T"
                           by (simp add: S'_star_connex sup_commute)
                         also have "... = top"
                           using 4 rank_property_def total_conv_surjective by blast
                         finally have "-(S'\<^sup>+ * rank\<^sup>T) \<le> S'\<^sup>\<star>\<^sup>T * rank\<^sup>T"
                           by (metis half_shunting inf.idem top_greatest)
                         thus ?thesis
                           using comp_associative inf.sup_right_isotone mult_right_isotone by auto
                       qed
                       also have "... = 1 \<sqinter> rank * S'\<^sup>\<star> * i\<^sup>T"
                         by (metis comp_associative conv_dist_comp conv_involutive one_inf_conv)
                       also have "... \<le> rank * S'\<^sup>\<star> * i\<^sup>T"
                         by simp
                       finally have "?s \<le> rank * S'\<^sup>\<star> * i\<^sup>T * ?s"
                         using 5 shunt_bijective by auto
                       hence "?rs \<le> S'\<^sup>\<star> * i\<^sup>T * ?s"
                         using 4 shunt_mapping comp_associative rank_property_def by auto
                       hence "?s * (i * ?rr \<sqinter> -?s \<sqinter> roots ?p2) \<le> ?s * (i * S'\<^sup>\<star> * i\<^sup>T * ?s \<sqinter> -?s \<sqinter> roots ?p2)"
                         using 11 comp_associative comp_inf.mult_left_isotone comp_isotone inf.eq_refl by auto
                       also have "... = ?s * ((i * S'\<^sup>+ * i\<^sup>T * ?s \<squnion> i * i\<^sup>T * ?s) \<sqinter> -?s \<sqinter> roots ?p2)"
                         by (metis comp_associative mult_left_dist_sup star.circ_loop_fixpoint)
                       also have "... \<le> ?s * ((i * S'\<^sup>+ * i\<^sup>T * ?s \<squnion> 1 * ?s) \<sqinter> -?s \<sqinter> roots ?p2)"
                         using 12 by (metis mult_left_isotone sup_right_isotone comp_inf.mult_left_isotone mult_right_isotone)
                       also have "... = ?s * (i * S'\<^sup>+ * i\<^sup>T * ?s \<sqinter> -?s \<sqinter> roots ?p2)"
                         using comp_inf.comp_right_dist_sup by simp
                       also have "... \<le> ?s * (i * S'\<^sup>+ * i\<^sup>T * ?s \<sqinter> roots ?p2)"
                         using comp_inf.mult_left_isotone inf.cobounded1 mult_right_isotone by blast
                       also have "... \<le> ?s * (i * S'\<^sup>+ * i\<^sup>T * ?s \<sqinter> i * -(S'\<^sup>+ * rank\<^sup>T * top))"
                         using 12 comp_inf.mult_right_isotone mult_right_isotone by auto
                       also have "... = ?s * (i * S'\<^sup>+ * i\<^sup>T * ?s \<sqinter> i) * -(S'\<^sup>+ * rank\<^sup>T * top)"
                         using 5 by (simp add: comp_associative vector_inf_comp)
                       also have "... = (?s \<sqinter> (i * S'\<^sup>+ * i\<^sup>T * ?s)\<^sup>T) * i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                         using 5 covector_inf_comp_3 mult_assoc by auto
                       also have "... = (?s \<sqinter> ?s\<^sup>T * i * S'\<^sup>+\<^sup>T * i\<^sup>T) * i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                         using conv_dist_comp conv_involutive mult_assoc by auto
                       also have "... = (?s \<sqinter> ?s\<^sup>T) * i * S'\<^sup>+\<^sup>T * i\<^sup>T * i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                         using 5 vector_inf_comp by auto
                       also have "... \<le> i * S'\<^sup>+\<^sup>T * i\<^sup>T * i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                         using 5 by (metis point_antisymmetric mult_left_isotone mult_left_one)
                       also have "... \<le> i * S'\<^sup>+\<^sup>T * -(S'\<^sup>+ * rank\<^sup>T * top)"
                         using 12 by (smt mult_left_isotone mult_right_isotone mult_assoc comp_right_one)
                       also have "... \<le> i * -(S'\<^sup>\<star> * rank\<^sup>T * top)"
                       proof -
                         have "S'\<^sup>+ * S'\<^sup>\<star> * rank\<^sup>T * top \<le> S'\<^sup>+ * rank\<^sup>T * top"
                           by (simp add: comp_associative star.circ_transitive_equal)
                         hence "S'\<^sup>+\<^sup>T * -(S'\<^sup>+ * rank\<^sup>T * top) \<le> -(S'\<^sup>\<star> * rank\<^sup>T * top)"
                           by (smt (verit, ccfv_SIG) comp_associative conv_complement_sub_leq mult_right_isotone order.trans p_antitone)
                         thus ?thesis
                           by (simp add: comp_associative mult_right_isotone)
                       qed
                       also have "... \<le> i * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                       proof -
                         have "S'\<^sup>+ * ?rank\<^sup>T * top \<le> ?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top"
                           using 20 by simp
                         also have "... \<le> rank\<^sup>T * top \<squnion> S'\<^sup>+ * rank\<^sup>T * top"
                           using mult_right_isotone sup_left_isotone top.extremum by blast
                         also have "... = S'\<^sup>\<star> * rank\<^sup>T * top"
                           using comp_associative star.circ_loop_fixpoint sup_commute by auto
                         finally show ?thesis
                           using mult_right_isotone p_antitone by blast
                       qed
                       finally have "?s * (i * ?rr \<sqinter> -?s \<sqinter> roots ?p2) \<le> i * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                         .
                       hence "i * ?rr \<sqinter> -?s \<sqinter> roots ?p2 \<le> ?s\<^sup>T * i * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                         using 5 shunt_mapping bijective_conv_mapping mult_assoc by auto
                       hence "i * ?rr \<sqinter> -?s \<sqinter> roots ?p2 \<le> i * ?rr \<sqinter> ?s\<^sup>T * i * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                         by (simp add: inf.sup_monoid.add_assoc)
                       also have "... = (i * ?rr \<sqinter> ?s\<^sup>T * i) * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                         using 6 vector_inf_comp vector_mult_closed by simp
                       also have "... \<le> ?i * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                         using 13 comp_left_increasing_sup sup_assoc by auto
                       finally show "i * ?rr \<sqinter> -?s \<sqinter> roots ?p2 \<le> ?i * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                         .
                       have "-(i * ?rr) \<sqinter> roots ?p2 \<le> -(i * ?rr) \<sqinter> i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                         using 12 inf.sup_right_isotone by auto
                       also have "... \<le> -(i * ?rr) \<sqinter> i * -(?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top)"
                       proof -
                         have 21: "regular (?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top)"
                           using 4 6 rank_property_def mapping_regular S'_regular pp_dist_star regular_conv_closed regular_mult_closed bijective_regular regular_closed_sup by auto
                         have "?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top \<le> S'\<^sup>+ * rank\<^sup>T * top \<squnion> ?rr"
                           using 11 by simp
                         hence "(?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top) - S'\<^sup>+ * rank\<^sup>T * top \<le> ?rr"
                           using half_shunting sup_commute by auto
                         hence "-(S'\<^sup>+ * rank\<^sup>T * top) \<le> -(?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top) \<squnion> ?rr"
                           using 21 by (metis inf.sup_monoid.add_commute shunting_var_p sup_commute)
                         hence "i * -(S'\<^sup>+ * rank\<^sup>T * top) \<le> i * -(?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top) \<squnion> i * ?rr"
                           by (metis mult_left_dist_sup mult_right_isotone)
                         hence "-(i * ?rr) \<sqinter> i * -(S'\<^sup>+ * rank\<^sup>T * top) \<le> i * -(?rs \<squnion> S'\<^sup>+ * rank\<^sup>T * top)"
                           using half_shunting inf.sup_monoid.add_commute by fastforce
                         thus ?thesis
                           using inf.le_sup_iff by blast
                       qed
                       also have "... \<le> -(i * ?rr) \<sqinter> i * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                         using 20 by (meson comp_inf.mult_right_isotone mult_right_isotone p_antitone)
                       finally have "-(i * ?rr) \<sqinter> -?s \<sqinter> roots ?p2 \<le> -(i * ?rr) \<sqinter> -?s \<sqinter> i * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                         by (smt (z3) inf.boundedI inf.cobounded1 inf.coboundedI2 inf.sup_monoid.add_assoc inf.sup_monoid.add_commute)
                       also have "... \<le> ?i * (-(S'\<^sup>+ * ?rank\<^sup>T * top))"
                         using 5 6 13 by (smt (z3) sup_commute vector_complement_closed vector_inf_comp vector_mult_closed comp_left_increasing_sup)
                       finally show "-(i * ?rr) \<sqinter> -?s \<sqinter> roots ?p2 \<le> ?i * -(S'\<^sup>+ * ?rank\<^sup>T * top)"
                         .
                     qed
                     finally show ?thesis
                       .
                   qed
                   thus "card_less_eq (roots ?p4) (-(S'\<^sup>+ * ?rank\<^sup>T * top))"
                     using 14 15 16 card_less_eq_def by auto
                   have "?s \<le> i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                     using 19 by simp
                   also have "... \<le> i * -(S'\<^sup>+ * ?rr)"
                     using mult_right_isotone p_antitone top.extremum mult_assoc by auto
                   also have "... = i * -S'\<^sup>+ * ?rr"
                     using 6 comp_bijective_complement mult_assoc by fastforce
                   finally have "?rr \<le> -S'\<^sup>T\<^sup>+ * i\<^sup>T * ?s"
                     using 5 6 by (metis conv_complement conv_dist_comp conv_plus_commute bijective_reverse)
                   also have "... \<le> S'\<^sup>\<star> * i\<^sup>T * ?s"
                     using 7 conv_plus_commute mult_left_isotone by auto
                   finally have 22: "?rr \<le> S'\<^sup>\<star> * i\<^sup>T * ?s"
                     .
                   have "?r = root ?p1 x"
                     using 2 path_compression_postcondition_def path_compression_precondition_def by blast
                   also have "... = root ?p2 x"
                     using 3 18 by (smt (z3) find_set_precondition_def path_compression_postcondition_def path_compression_precondition_def same_root)
                   also have "... \<le> roots ?p2"
                     by simp
                   also have "... \<le> i * -(S'\<^sup>+ * rank\<^sup>T * top)"
                     using 12 by simp
                   also have "... \<le> i * -(S'\<^sup>+ * ?rr)"
                     using mult_right_isotone p_antitone top.extremum mult_assoc by auto
                   also have "... = i * -S'\<^sup>+ * ?rr"
                     using 6 comp_bijective_complement mult_assoc by fastforce
                   finally have "?rr \<le> -S'\<^sup>T\<^sup>+ * i\<^sup>T * ?r"
                     using 5 6 by (metis conv_complement conv_dist_comp conv_plus_commute bijective_reverse)
                   also have "... \<le> S'\<^sup>\<star> * i\<^sup>T * ?r"
                     using 7 conv_plus_commute mult_left_isotone by auto
                   finally have "?rr \<le> S'\<^sup>\<star> * i\<^sup>T * ?r"
                     .
                   hence "?rr \<le> S'\<^sup>\<star> * i\<^sup>T * ?r \<sqinter> S'\<^sup>\<star> * i\<^sup>T * ?s"
                     using 22 inf.boundedI by blast
                   also have "... = (S'\<^sup>+ * i\<^sup>T * ?r \<squnion> i\<^sup>T * ?r) \<sqinter> S'\<^sup>\<star> * i\<^sup>T * ?s"
                     by (simp add: star.circ_loop_fixpoint mult_assoc)
                   also have "... \<le> S'\<^sup>+ * i\<^sup>T * ?r \<squnion> (i\<^sup>T * ?r \<sqinter> S'\<^sup>\<star> * i\<^sup>T * ?s)"
                     by (metis comp_inf.mult_right_dist_sup eq_refl inf.cobounded1 semiring.add_mono)
                   also have "... \<le> S' * top \<squnion> (i\<^sup>T * ?r \<sqinter> S'\<^sup>\<star> * i\<^sup>T * ?s)"
                     using comp_associative mult_right_isotone sup_left_isotone top.extremum by auto
                   also have "... = S' * top \<squnion> (i\<^sup>T * ?r \<sqinter> (S'\<^sup>+ * i\<^sup>T * ?s \<squnion> i\<^sup>T * ?s))"
                     by (simp add: star.circ_loop_fixpoint mult_assoc)
                   also have "... \<le> S' * top \<squnion> S'\<^sup>+ * i\<^sup>T * ?s \<squnion> (i\<^sup>T * ?r \<sqinter> i\<^sup>T * ?s)"
                     by (smt (z3) comp_inf.semiring.distrib_left inf.sup_right_divisibility star.circ_loop_fixpoint sup_assoc sup_commute sup_inf_distrib1)
                   also have "... \<le> S' * top \<squnion> (i\<^sup>T * ?r \<sqinter> i\<^sup>T * ?s)"
                     by (metis comp_associative mult_right_isotone order.refl sup.orderE top.extremum)
                   also have "... = S' * top \<squnion> i\<^sup>T * (?r \<sqinter> ?s)"
                     using 12 conv_involutive univalent_comp_left_dist_inf by auto
                   also have "... = S' * top"
                     using 5 8 distinct_points by auto
                   finally have "top \<le> ?rr\<^sup>T * S' * top"
                     using 6 by (smt conv_involutive shunt_mapping bijective_conv_mapping mult_assoc)
                   hence "surjective (S'\<^sup>T * ?rs)"
                     using 6 11 by (smt conv_dist_comp conv_involutive point_conv_comp top_le)
                   thus "total ?rank"
                     using 4 5 bijective_regular update_total rank_property_def by blast
                   show "(?p4 - 1) * ?rank \<le> ?rank * S'\<^sup>+"
                   proof -
                     have 23: "univalent ?p2"
                       using 3 path_compression_postcondition_def path_compression_precondition_def by blast
                     have 24: "?r \<sqinter> (?p4 - 1) * ?rank \<le> ?s\<^sup>T * rank * S' * S'\<^sup>+"
                     proof -
                       have "?r \<sqinter> (?p4 - 1) * ?rank = (?r \<sqinter> ?p4 \<sqinter> -1) * ?rank"
                         using 5 vector_complement_closed vector_inf_comp inf_assoc by auto
                       also have "... = (?r \<sqinter> -?s \<sqinter> ?p4 \<sqinter> -1) * ?rank"
                           using 5 8 by (smt (z3) order.antisym bijective_regular point_in_vector_or_complement point_in_vector_or_complement_2 inf_absorb1)
                       also have "... = (?r \<sqinter> -?s \<sqinter> ?p2 \<sqinter> -1) * ?rank"
                         by (simp add: inf.left_commute inf.sup_monoid.add_commute inf_sup_distrib1 inf_assoc)
                       also have "... \<le> (?r \<sqinter> ?p2 \<sqinter> -1) * ?rank"
                         using inf.sup_left_isotone inf_le1 mult_left_isotone by blast
                       also have "... = bot"
                       proof -
                         have "?r = root ?p1 x"
                           using 2 path_compression_postcondition_def path_compression_precondition_def by blast
                         also have "... = root ?p2 x"
                           using 3 18 by (smt (z3) find_set_precondition_def path_compression_postcondition_def path_compression_precondition_def same_root)
                         also have "... \<le> roots ?p2"
                           by simp
                         finally have "?r \<sqinter> ?p2 \<le> roots ?p2 \<sqinter> ?p2"
                           using inf.sup_left_isotone by blast
                         also have "... \<le> (?p2 \<sqinter> 1) * (?p2 \<sqinter> 1)\<^sup>T * ?p2"
                           by (smt (z3) comp_associative comp_inf.star_plus dedekind_1 inf_top_right order_lesseq_imp)
                         also have "... = (?p2 \<sqinter> 1) * (?p2 \<sqinter> 1) * ?p2"
                           using coreflexive_symmetric by force
                         also have "... \<le> (?p2 \<sqinter> 1) * ?p2"
                           by (metis coreflexive_comp_top_inf inf.cobounded2 mult_left_isotone)
                         also have "... \<le> ?p2 \<sqinter> 1"
                           by (smt 23 comp_inf.mult_semi_associative conv_dist_comp conv_dist_inf conv_order equivalence_one_closed inf.absorb1 inf.sup_monoid.add_assoc injective_codomain)
                         also have "... \<le> 1"
                           by simp
                         finally have "?r \<sqinter> ?p2 \<le> 1"
                           .
                         thus ?thesis
                           by (metis pseudo_complement regular_one_closed semiring.mult_not_zero)
                       qed
                       finally show ?thesis
                         using bot_least le_bot by blast
                     qed
                     have 25: "-?r \<sqinter> (?p4 - 1) * ?rank \<le> rank * S'\<^sup>+"
                     proof -
                       have "-?r \<sqinter> (?p4 - 1) * ?rank = (-?r \<sqinter> ?p4 \<sqinter> -1) * ?rank"
                         using 5 vector_complement_closed vector_inf_comp inf_assoc by auto
                       also have "... = (-?r \<sqinter> (?s \<squnion> -?s) \<sqinter> ?p4 \<sqinter> -1) * ?rank"
                         using 5 bijective_regular inf_top_right regular_complement_top by auto
                       also have "... = (-?r \<sqinter> ?s \<sqinter> ?p4 \<sqinter> -1) * ?rank \<squnion> (-?r \<sqinter> -?s \<sqinter> ?p4 \<sqinter> -1) * ?rank"
                         by (smt (z3) inf_sup_distrib1 inf_sup_distrib2 mult_right_dist_sup)
                       also have "... = (-?r \<sqinter> ?s \<sqinter> ?r\<^sup>T \<sqinter> -1) * ?rank \<squnion> (-?r \<sqinter> -?s \<sqinter> ?p2 \<sqinter> -1) * ?rank"
                         using 5 by (smt (z3) bijective_regular comp_inf.comp_left_dist_sup inf_assoc inf_commute inf_top_right mult_right_dist_sup regular_complement_top)
                       also have "... \<le> (?s \<sqinter> ?r\<^sup>T \<sqinter> -1) * ?rank \<squnion> (-?s \<sqinter> ?p2 \<sqinter> -1) * ?rank"
                         by (smt (z3) comp_inf.semiring.distrib_left inf.cobounded2 inf.sup_monoid.add_assoc mult_left_isotone mult_right_dist_sup)
                       also have "... \<le> (?s \<sqinter> ?r\<^sup>T) * ?rank \<squnion> (?p2 - 1) * ?rank"
                         by (smt (z3) inf.cobounded1 inf.cobounded2 inf.sup_monoid.add_assoc mult_left_isotone semiring.add_mono)
                       also have "... = ?s * (?r \<sqinter> ?rank) \<squnion> (?p2 - 1) * ?rank"
                         using 5 by (simp add: covector_inf_comp_3)
                       also have "... = ?s * (?r \<sqinter> (S'\<^sup>T * ?rs)\<^sup>T) \<squnion> (?p2 - 1) * ?rank"
                         using inf_commute update_inf_same mult_assoc by force
                       also have "... = ?s * (?r \<sqinter> ?s\<^sup>T * rank * S') \<squnion> (?p2 - 1) * ?rank"
                         using comp_associative conv_dist_comp conv_involutive by auto
                       also have "... \<le> ?s * ?s\<^sup>T * rank * S' \<squnion> (?p2 - 1) * ?rank"
                         using comp_associative inf.cobounded2 mult_right_isotone semiring.add_right_mono by auto
                       also have "... \<le> 1 * rank * S' \<squnion> (?p2 - 1) * ?rank"
                         using 5 by (meson mult_left_isotone order.refl semiring.add_mono)
                       also have "... = rank * S' \<squnion> (?p2 - 1) * (?r \<sqinter> (S'\<^sup>T * ?rr)\<^sup>T) \<squnion> (?p2 - 1) * (-?r \<sqinter> rank)"
                         using 11 comp_associative mult_1_left mult_left_dist_sup sup_assoc by auto
                       also have "... \<le> rank * S' \<squnion> (?p2 - 1) * (?r \<sqinter> ?r\<^sup>T * rank * S') \<squnion> (?p2 - 1) * rank"
                         using comp_associative conv_dist_comp conv_involutive inf.cobounded1 inf.sup_monoid.add_commute mult_right_isotone semiring.add_left_mono by auto
                       also have "... = rank * S' \<squnion> (?p2 - 1) * (?r \<sqinter> ?r\<^sup>T) * rank * S' \<squnion> (?p2 - 1) * rank"
                         using 5 comp_associative vector_inf_comp by auto
                       also have "... \<le> rank * S' \<squnion> (?p2 - 1) * rank * S' \<squnion> (?p2 - 1) * rank"
                         using 5 by (metis point_antisymmetric mult_left_isotone mult_right_isotone sup_left_isotone sup_right_isotone comp_right_one)
                       also have "... \<le> rank * S' \<squnion> rank * S'\<^sup>+ * S' \<squnion> (?p2 - 1) * rank"
                         using 4 by (metis rank_property_def mult_left_isotone sup_left_isotone sup_right_isotone)
                       also have "... \<le> rank * S' \<squnion> rank * S'\<^sup>+ * S' \<squnion> rank * S'\<^sup>+"
                         using 4 by (metis rank_property_def sup_right_isotone)
                       also have "... \<le> rank * S'\<^sup>+"
                         using comp_associative eq_refl le_sup_iff mult_right_isotone star.circ_mult_increasing star.circ_plus_same star.left_plus_below_circ by auto
                       finally show ?thesis
                         .
                     qed
                     have "(?p4 - 1) * ?rank = (?r \<sqinter> (?p4 - 1) * ?rank) \<squnion> (-?r \<sqinter> (?p4 - 1) * ?rank)"
                       using 5 by (smt (verit, ccfv_threshold) bijective_regular inf_commute inf_sup_distrib2 inf_top_right regular_complement_top)
                     also have "... \<le> (?r \<sqinter> ?s\<^sup>T * rank * S' * S'\<^sup>+) \<squnion> (-?r \<sqinter> rank * S'\<^sup>+)"
                       using 24 25 by (meson inf.boundedI inf.cobounded1 semiring.add_mono)
                     also have "... = (?r \<sqinter> ?s\<^sup>T * rank * S') * S'\<^sup>+ \<squnion> (-?r \<sqinter> rank) * S'\<^sup>+"
                       using 5 vector_complement_closed vector_inf_comp by auto
                     also have "... = ?rank * S'\<^sup>+"
                       using conv_dist_comp mult_right_dist_sup by auto
                     finally show ?thesis
                       .
                   qed
                 qed
               qed
             qed
             show "?rr \<noteq> ?rs \<longrightarrow> union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 rank"
             proof
               assume "?rr \<noteq> ?rs"
               hence "?rs \<sqinter> ?rr = bot"
                 using 6 by (meson bijective_regular dual_order.eq_iff point_in_vector_or_complement point_in_vector_or_complement_2 pseudo_complement)
               hence 26: "?rs \<le> S'\<^sup>+ * ?rr"
                 using 10 le_iff_inf pseudo_complement by auto
               show "union_sets_postcondition ?p4 x y p0 \<and> rank_property ?p4 rank"
               proof
                 show "union_sets_postcondition ?p4 x y p0"
                   using 1 2 3 by (simp add: union_sets_1_swap)
                 show "rank_property ?p4 rank"
                 proof (unfold rank_property_def, intro conjI)
                   show "univalent rank" "total rank"
                     using 1 rank_property_def by auto
                   have "?r \<le> -?s"
                     using 5 8 by (meson order.antisym bijective_regular point_in_vector_or_complement point_in_vector_or_complement_2)
                   hence "?s \<sqinter> ?r\<^sup>T \<sqinter> 1 = bot"
                     by (metis (full_types) bot_least inf.left_commute inf.sup_monoid.add_commute one_inf_conv pseudo_complement)
                   hence "?p4 \<sqinter> 1 \<le> ?p2"
                     by (smt half_shunting inf.cobounded2 pseudo_complement regular_one_closed semiring.add_mono sup_commute)
                   hence "roots ?p4 \<le> roots ?p2"
                     by (simp add: mult_left_isotone)
                   thus "card_less_eq (roots ?p4) (-(S'\<^sup>+ * rank\<^sup>T * top))"
                     using 4 by (meson rank_property_def card_less_eq_def order_trans)
                   have "(?p4 - 1) * rank = (?s \<sqinter> ?r\<^sup>T \<sqinter> -1) * rank \<squnion> (-?s \<sqinter> ?p2 \<sqinter> -1) * rank"
                     using comp_inf.semiring.distrib_right mult_right_dist_sup by auto
                   also have "... \<le> (?s \<sqinter> ?r\<^sup>T \<sqinter> -1) * rank \<squnion> (?p2 - 1) * rank"
                     using comp_inf.mult_semi_associative mult_left_isotone sup_right_isotone by auto
                   also have "... \<le> (?s \<sqinter> ?r\<^sup>T \<sqinter> -1) * rank \<squnion> rank * S'\<^sup>+"
                     using 4 sup_right_isotone rank_property_def by blast
                   also have "... \<le> (?s \<sqinter> ?r\<^sup>T) * rank \<squnion> rank * S'\<^sup>+"
                     using inf_le1 mult_left_isotone sup_left_isotone by blast
                   also have "... = ?s * ?r\<^sup>T * rank \<squnion> rank * S'\<^sup>+"
                     using 5 by (simp add: vector_covector)
                   also have "... = rank * S'\<^sup>+"
                   proof -
                     have "rank\<^sup>T * ?s \<le> S'\<^sup>+ * rank\<^sup>T * ?r"
                       using 26 comp_associative by auto
                     hence "?s \<le> rank * S'\<^sup>+ * rank\<^sup>T * ?r"
                       using 4 shunt_mapping comp_associative rank_property_def by auto
                     hence "?s * ?r\<^sup>T \<le> rank * S'\<^sup>+ * rank\<^sup>T"
                       using 5 shunt_bijective by blast
                     hence "?s * ?r\<^sup>T * rank \<le> rank * S'\<^sup>+"
                       using 4 shunt_bijective rank_property_def mapping_conv_bijective by auto
                     thus ?thesis
                       using sup_absorb2 by blast
                   qed
                   finally show "(?p4 - 1) * rank \<le> rank * S'\<^sup>+"
                     .
                 qed
               qed
             qed
           qed
         qed
       qed
     qed
     show "?r = ?s \<longrightarrow> union_sets_postcondition ?p5 x y p0 \<and> rank_property ?p5 rank"
     proof
       assume 27: "?r = ?s"
       show "union_sets_postcondition ?p5 x y p0 \<and> rank_property ?p5 rank"
       proof
         show "union_sets_postcondition ?p5 x y p0"
           using 1 2 3 27 by (simp add: union_sets_1_skip)
         show "rank_property ?p5 rank"
           using 4 27 by simp
       qed
     qed
   qed
 qed
 
 end
 
 end