weakSimBangPres: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Rel'' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and R :: "('a, 'b, 'c) psi" assumes "(\<Psi>, P, Q) \<in> Rel"   and "eqvt Rel''"   and "guarded P"   and "guarded Q"   and "Rel' \<subseteq> Rel"   and "\<And>\<Psi>' \<Psi>\<^sub>U S T U A\<^sub>U. \<lbrakk>(\<Psi>' \<otimes> \<Psi>\<^sub>U, S, T) \<in> Rel; extractFrame U = \<langle>A\<^sub>U, \<Psi>\<^sub>U\<rangle>; A\<^sub>U \<sharp>* \<Psi>'; A\<^sub>U \<sharp>* S; A\<^sub>U \<sharp>* T\<rbrakk> \<Longrightarrow> (\<Psi>', U \<parallel> S, U \<parallel> T) \<in> Rel"   and "\<And>\<Psi>' S T U. \<lbrakk>(\<Psi>', S, T) \<in> Rel; guarded S; guarded T\<rbrakk> \<Longrightarrow> (\<Psi>', U \<parallel> !S, U \<parallel> !T) \<in> Rel''"   and "\<And>\<Psi>' S T xvec. \<lbrakk>(\<Psi>', S, T) \<in> Rel; xvec \<sharp>* \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>*xvec\<rparr>S, \<lparr>\<nu>*xvec\<rparr>T) \<in> Rel"   and "\<And>\<Psi>' S T xvec. \<lbrakk>(\<Psi>', S, T) \<in> Rel'; xvec \<sharp>* \<Psi>'\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>*xvec\<rparr>S, \<lparr>\<nu>*xvec\<rparr>T) \<in> Rel'"   and "\<And>\<Psi>' S T p. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> (p \<bullet> \<Psi>', p \<bullet> S, p \<bullet> T) \<in> Rel"   and "\<And>\<Psi>' S T p. (\<Psi>', S, T) \<in> Rel' \<Longrightarrow> (p \<bullet> \<Psi>', p \<bullet> S, p \<bullet> T) \<in> Rel'"   and "\<And>\<Psi>' S T \<Psi>''. \<lbrakk>(\<Psi>', S, T) \<in> Rel; \<Psi>' \<simeq> \<Psi>''\<rbrakk> \<Longrightarrow> (\<Psi>'', S, T) \<in> Rel"   and "\<And>\<Psi>' S T \<Psi>''. \<lbrakk>(\<Psi>', S, T) \<in> Rel'; \<Psi>' \<simeq> \<Psi>''\<rbrakk> \<Longrightarrow> (\<Psi>'', S, T) \<in> Rel'"   and "\<And>\<Psi>' S T U. \<lbrakk>(\<Psi>', S, T) \<in> Rel; (\<Psi>', T, U) \<in> Rel\<rbrakk> \<Longrightarrow> (\<Psi>', S, U) \<in> Rel"   and "\<And>\<Psi>' S T U. \<lbrakk>(\<Psi>', S, T) \<in> Rel'; (\<Psi>', T, U) \<in> Rel'\<rbrakk> \<Longrightarrow> (\<Psi>', S, U) \<in> Rel'"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> \<Psi>' \<rhd> S \<leadsto><Rel> T"   and "\<And>\<Psi>' S T \<Psi>''. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> (\<Psi>' \<otimes> \<Psi>'', S, T) \<in> Rel"   and "\<And>\<Psi>' S T \<Psi>''. (\<Psi>', S, T) \<in> Rel' \<Longrightarrow> (\<Psi>' \<otimes> \<Psi>'', S, T) \<in> Rel'"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> (\<Psi>', T, S) \<in> Rel"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel' \<Longrightarrow> (\<Psi>', T, S) \<in> Rel'"   and "\<And>\<Psi>' S T U. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> (\<Psi>', S \<parallel> U, T \<parallel> U) \<in> Rel"   and "\<And>\<Psi>' S T. (\<Psi>', S, T) \<in> Rel \<Longrightarrow> (\<Psi>', S \<parallel> S, T \<parallel> T) \<in> Rel"   and "\<And>\<Psi>' S T U. (\<Psi>', S, T) \<in> Rel' \<Longrightarrow> (\<Psi>', U \<parallel> S, U \<parallel> T) \<in> Rel'"   and "\<And>\<Psi>' S T U. (\<Psi>', S \<parallel> (T \<parallel> U), S \<parallel> T \<parallel> U) \<in> Rel"   and "\<And>\<Psi>' S T U. (\<Psi>', S \<parallel> (T \<parallel> U), S \<parallel> T \<parallel> U) \<in> Rel'"   and "\<And>xvec \<Psi>' T S. \<lbrakk>xvec \<sharp>* \<Psi>'; xvec \<sharp>* T\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>*xvec\<rparr>S \<parallel> T, (\<lparr>\<nu>*xvec\<rparr>S) \<parallel> T) \<in> Rel"   and "\<And>xvec \<Psi>' T S. \<lbrakk>xvec \<sharp>* \<Psi>'; xvec \<sharp>* T\<rbrakk> \<Longrightarrow> (\<Psi>', \<lparr>\<nu>*xvec\<rparr>S \<parallel> T, (\<lparr>\<nu>*xvec\<rparr>S) \<parallel> T) \<in> Rel'"   and "\<And>\<Psi>' S T U O. \<lbrakk>(\<Psi>', S, T) \<in> Rel; (\<Psi>', T, U) \<in> Rel''; (\<Psi>', U, O) \<in> Rel'\<rbrakk> \<Longrightarrow> (\<Psi>', S, O) \<in> Rel''"   and "\<And>\<Psi>' S \<alpha> S'. \<lbrakk>\<Psi>' \<rhd> !S \<longmapsto> \<alpha> \<prec> S'; guarded S; bn \<alpha> \<sharp>* S; \<alpha> \<noteq> \<tau>; bn \<alpha> \<sharp>* subject \<alpha>\<rbrakk> \<Longrightarrow> \<exists>T. \<Psi>' \<rhd> S \<longmapsto> \<alpha> \<prec> T \<and> (\<one>, S', T \<parallel> !S) \<in> Rel'"   and "\<And>\<Psi>' S S'. \<lbrakk>\<Psi>' \<rhd> !S \<longmapsto> \<tau> \<prec> S'; guarded S\<rbrakk> \<Longrightarrow> \<exists>T. \<Psi>' \<rhd> S \<parallel> S \<longmapsto> \<tau> \<prec> T \<and> (\<one>, S', T \<parallel> !S) \<in> Rel'"   and "\<And>\<Psi>' S S'. \<lbrakk>\<Psi>' \<rhd> S \<parallel> S \<Longrightarrow>\<^sup>^\<^sub>\<tau> S'; guarded S\<rbrakk> \<Longrightarrow> \<exists>T. \<Psi>' \<rhd> !S \<Longrightarrow>\<^sup>^\<^sub>\<tau> T \<and> (\<Psi>', T, S' \<parallel> !S) \<in> Rel'" shows "\<Psi> \<rhd> R \<parallel> !P \<leadsto><Rel''> R \<parallel> !Q"