parCases: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and T :: "('a, 'b, 'c) psi"   and C :: "'d"   and Prop :: "'a action \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> P \<parallel> Q \<longmapsto> \<alpha> \<prec> T"   and "bn \<alpha> \<sharp>* \<Psi>"   and "bn \<alpha> \<sharp>* P"   and "bn \<alpha> \<sharp>* Q"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "\<And>P' A\<^sub>Q \<Psi>\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> \<alpha> \<prec> P'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* \<alpha>; A\<^sub>Q \<sharp>* P'; A\<^sub>Q \<sharp>* C\<rbrakk> \<Longrightarrow> Prop \<alpha> (P' \<parallel> Q)"   and "\<And>Q' A\<^sub>P \<Psi>\<^sub>P. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> \<alpha> \<prec> Q'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* \<alpha>; A\<^sub>P \<sharp>* Q'; A\<^sub>P \<sharp>* C\<rbrakk> \<Longrightarrow> Prop \<alpha> (P \<parallel> Q')"   and "\<And>\<Psi>\<^sub>Q M N P' A\<^sub>P \<Psi>\<^sub>P K xvec Q' A\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; \<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; \<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K; distinct xvec; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* \<Psi>\<^sub>Q; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* P'; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* Q'; A\<^sub>P \<sharp>* A\<^sub>Q; A\<^sub>P \<sharp>* C; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* \<Psi>\<^sub>P; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* K; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* P'; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* Q'; A\<^sub>Q \<sharp>* C; xvec \<sharp>* \<Psi>; xvec \<sharp>* \<Psi>\<^sub>P; xvec \<sharp>* P; xvec \<sharp>* M; xvec \<sharp>* K; xvec \<sharp>* Q; xvec \<sharp>* \<Psi>\<^sub>Q; xvec \<sharp>* C\<rbrakk> \<Longrightarrow> Prop (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')"   and "\<And>\<Psi>\<^sub>Q M xvec N P' A\<^sub>P \<Psi>\<^sub>P K Q' A\<^sub>Q. \<lbrakk>\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> P'; extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>; distinct A\<^sub>P; \<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>N\<rparr> \<prec> Q'; extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>; distinct A\<^sub>Q; \<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K; distinct xvec; A\<^sub>P \<sharp>* \<Psi>; A\<^sub>P \<sharp>* \<Psi>\<^sub>Q; A\<^sub>P \<sharp>* P; A\<^sub>P \<sharp>* M; A\<^sub>P \<sharp>* N; A\<^sub>P \<sharp>* P'; A\<^sub>P \<sharp>* Q; A\<^sub>P \<sharp>* xvec; A\<^sub>P \<sharp>* Q'; A\<^sub>P \<sharp>* A\<^sub>Q; A\<^sub>P \<sharp>* C; A\<^sub>Q \<sharp>* \<Psi>; A\<^sub>Q \<sharp>* \<Psi>\<^sub>P; A\<^sub>Q \<sharp>* P; A\<^sub>Q \<sharp>* K; A\<^sub>Q \<sharp>* N; A\<^sub>Q \<sharp>* P'; A\<^sub>Q \<sharp>* Q; A\<^sub>Q \<sharp>* xvec; A\<^sub>Q \<sharp>* Q'; A\<^sub>Q \<sharp>* C; xvec \<sharp>* \<Psi>; xvec \<sharp>* \<Psi>\<^sub>P; xvec \<sharp>* P; xvec \<sharp>* M; xvec \<sharp>* K; xvec \<sharp>* Q; xvec \<sharp>* \<Psi>\<^sub>Q; xvec \<sharp>* C\<rbrakk> \<Longrightarrow> Prop (\<tau>) (\<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q')" shows "Prop \<alpha> T"