induct: fixes P :: "'b tree234 \<Rightarrow> bool"   and a0 :: "'b tree234" assumes "\<And>a. P \<langle>\<langle>\<rangle>, a, \<langle>\<rangle>\<rangle>"   and "\<And>a b. P \<langle>\<langle>\<rangle>, a, \<langle>\<rangle>, b, \<langle>\<rangle>\<rangle>"   and "\<And>a b c. P \<langle>\<langle>\<rangle>, a, \<langle>\<rangle>, b, \<langle>\<rangle>, c, \<langle>\<rangle>\<rangle>"   and "\<And>v va vb a r. P \<langle>v, va, vb\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb\<rangle>, a, r\<rangle>"   and "\<And>v va vb vc vd a r. P \<langle>v, va, vb, vc, vd\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb, vc, vd\<rangle>, a, r\<rangle>"   and "\<And>v va vb vc vd ve vf a r. P \<langle>v, va, vb, vc, vd, ve, vf\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb, vc, vd, ve, vf\<rangle>, a, r\<rangle>"   and "\<And>l a v va vb. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb\<rangle>\<rangle>"   and "\<And>l a v va vb vc vd. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb, vc, vd\<rangle>\<rangle>"   and "\<And>l a v va vb vc vd ve vf. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb, vc, vd, ve, vf\<rangle>\<rangle>"   and "\<And>v va vb a m b r. P \<langle>v, va, vb\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb\<rangle>, a, m, b, r\<rangle>"   and "\<And>v va vb vc vd a m b r. P \<langle>v, va, vb, vc, vd\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb, vc, vd\<rangle>, a, m, b, r\<rangle>"   and "\<And>v va vb vc vd ve vf a m b r. P \<langle>v, va, vb, vc, vd, ve, vf\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb, vc, vd, ve, vf\<rangle>, a, m, b, r\<rangle>"   and "\<And>l a v va vb b r. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb\<rangle>, b, r\<rangle>"   and "\<And>l a v va vb vc vd b r. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb, vc, vd\<rangle>, b, r\<rangle>"   and "\<And>l a v va vb vc vd ve vf b r. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb, vc, vd, ve, vf\<rangle>, b, r\<rangle>"   and "\<And>l a m b v va vb. P l \<Longrightarrow> P \<langle>l, a, m, b, \<langle>v, va, vb\<rangle>\<rangle>"   and "\<And>l a m b v va vb vc vd. P l \<Longrightarrow> P \<langle>l, a, m, b, \<langle>v, va, vb, vc, vd\<rangle>\<rangle>"   and "\<And>l a m b v va vb vc vd ve vf. P l \<Longrightarrow> P \<langle>l, a, m, b, \<langle>v, va, vb, vc, vd, ve, vf\<rangle>\<rangle>"   and "\<And>v va vb a m b n c r. P \<langle>v, va, vb\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb\<rangle>, a, m, b, n, c, r\<rangle>"   and "\<And>v va vb vc vd a m b n c r. P \<langle>v, va, vb, vc, vd\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb, vc, vd\<rangle>, a, m, b, n, c, r\<rangle>"   and "\<And>v va vb vc vd ve vf a m b n c r. P \<langle>v, va, vb, vc, vd, ve, vf\<rangle> \<Longrightarrow> P \<langle>\<langle>v, va, vb, vc, vd, ve, vf\<rangle>, a, m, b, n, c, r\<rangle>"   and "\<And>l a v va vb b n c r. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb\<rangle>, b, n, c, r\<rangle>"   and "\<And>l a v va vb vc vd b n c r. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb, vc, vd\<rangle>, b, n, c, r\<rangle>"   and "\<And>l a v va vb vc vd ve vf b n c r. P l \<Longrightarrow> P \<langle>l, a, \<langle>v, va, vb, vc, vd, ve, vf\<rangle>, b, n, c, r\<rangle>"   and "\<And>l a m b v va vb c r. P l \<Longrightarrow> P \<langle>l, a, m, b, \<langle>v, va, vb\<rangle>, c, r\<rangle>"   and "\<And>l a m b v va vb vc vd c r. P l \<Longrightarrow> P \<langle>l, a, m, b, \<langle>v, va, vb, vc, vd\<rangle>, c, r\<rangle>"   and "\<And>l a m b v va vb vc vd ve vf c r. P l \<Longrightarrow> P \<langle>l, a, m, b, \<langle>v, va, vb, vc, vd, ve, vf\<rangle>, c, r\<rangle>"   and "\<And>l a m b n c v va vb. P l \<Longrightarrow> P \<langle>l, a, m, b, n, c, \<langle>v, va, vb\<rangle>\<rangle>"   and "\<And>l a m b n c v va vb vc vd. P l \<Longrightarrow> P \<langle>l, a, m, b, n, c, \<langle>v, va, vb, vc, vd\<rangle>\<rangle>"   and "\<And>l a m b n c v va vb vc vd ve vf. P l \<Longrightarrow> P \<langle>l, a, m, b, n, c, \<langle>v, va, vb, vc, vd, ve, vf\<rangle>\<rangle>"   and "P \<langle>\<rangle>" shows "P a0"