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zA#tx=8*wu2$6)170T9!BBXuhQOx(Yo3O9bvjwg!d;21=)3-mDMTUkXmQOR;PQQq=` z0AtghZ_aQDsK^Y3u_7~^$?!f77*sMyz?4u%utElT9@vNPQ%Xw|RZ46G3@H%XK&nfb z(F0GcUV>X%>J1G%lJ651goaxi+#k^N-04t=D>xkxp+1HxGo%>ACOM@{FL2gdZ{Mc5 zAO7#p(N z7?=aGhB-jMsyI$n2rBs@5OGe#@!(Tgynqzs*)uIFlPWcwd47n3I^DXwZo`)-h6ZKi zIRO+l>9FM|!7PgSaj>p$jh#6pyVb--{&!Y1}{!>FKykUtizTpq@OgwloqOgV#uZQ?Q`n*_BpPjqqd z&{5nHE^h|BDS`VG@B&4G4%*P%?C9b)*9E|oLv#nZYW++iF;Bp17pQ7Y3C9bSgE8Ey zRm6`k$qjgA3wH@;SY#R(7?i^mz>3;W{ttq|l`J;cp*o7epwztOk9tnAcbEveM|>=f zJ~@LFnSFLj=i)M5Q9;md!7sk@EC?`9SuqaFl(9H`K|6IsyWl7zqcz0*Dr*qSUCJ;r7_rDpFk@#c=IiIW)nQAfHD{bov8@Y1|-D zCb6+gOh1rM!Nyoq0zmm%Z9)9DdZGY4AC{iqoD>^I^VyyDo0bsaA%jx+hKl1Y&e4`V;Rsg920X0gUxeGI7AG zCE$hG+2#29zN2Xzh{abKeN{8USMhGzv;}Y1L0I%%e`jo5w+VkikK|wY^6JyxZ7w&L zFLjneU6u#9UxEV?&&6M!x7>a=?r1Em{QoEyRq$n9o42gKe9M)WuU*%SD}iXOH x$, e dunque $z > x$ per ipotesi induttiva, ovverosia esiste $z''$ non nullo tale per cui $z = x + z''$. Poiché - $z''$ è non nullo, esiste $m$ tale per cui $z'' = m + 1$. Quindi - $z = x + (m + 1) = (x + 1) + m$. Se fosse $m = 0$, si avrebbe - $(x + 1) + y' = y + z' = z = x + 1$, \Lightning. Dunque - $z > x + 1$. + $z''$ è non nullo, esiste $m$ tale per cui $z'' = S(m)$. Quindi + $z = x + S(m) = S(x) + m$. Se fosse $m = 0$, si avrebbe + $S(x) + y' = y + z' = z = S(x)$, \Lightning. Dunque + $z > S(x)$. \end{enumerate} \item[\fbox{Confrontabilità}] Per induzione su $x$ con la formula @@ -1262,12 +1297,12 @@ originale in una formula esplicita: \item[$\boxed{P(0)}$] Se $y \neq 0$, allora $0 < y$ dal momento che $y = 0 + y$. - \item[$\boxed{P(x + 1)}$] Per ipotesi induttiva, vale $y < x$, $y = x$ o - $y > x$. Se vale $y < x$, allora, per transitività, $y < x + 1$. Se - $y = x$, allora banalmente $y < x + 1$. Se invece $y > x$, esiste + \item[$\boxed{P(S(x))}$] Per ipotesi induttiva, vale $y < x$, $y = x$ o + $y > x$. Se vale $y < x$, allora, per transitività, $y < S(x)$. Se + $y = x$, allora banalmente $y < S(x)$. Se invece $y > x$, esiste $y'$ non nullo tale per cui $y = x + y'$. Poiché $y'$ non è nullo, allora - esiste $t$ tale per cui $y' = t + 1$. Dunque $y = x + (t + 1) = - (x + 1) + t$. Se $t = 0$, $y = x + 1$; altrimenti $y > x + 1$. + esiste $t$ tale per cui $y' = S(t)$. Dunque $y = x + S(t) = + S(x) + t$. Se $t = 0$, $y = S(x)$; altrimenti $y > S(x)$. \end{enumerate} \end{enumerate} @@ -1277,13 +1312,13 @@ originale in una formula esplicita: \item Se $y > x$, allora esiste $y'$ non nullo per cui $y = x + y''$. Allora: \[ y \cdot z = (x + y'') \cdot z = x \cdot z + y'' \cdot z, \] - e quindi, poiché $y'' \cdot z$ (sia $y''$, che $z$ sono non nulli), + e quindi, poiché $y'' \cdot z$ (sia $y''$, che $z$ sono non nulli, vd. \textit{Problema 31}), $y \cdot z > x \cdot z$. \item Mostriamo le due implicazioni. \begin{enumerate} - \item[$\boxed{n \in m \implies n < m}$] Se $n \in m$, allora $m = n \sqcup \underbrace{\{n, \ldots, m-1\}}_{=A}$. + \item[$\boxed{n \in m \implies n < m}$] Se $n \in m$, allora $m = n \sqcup \underbrace{\{n, \ldots\}}_{=\,A}$. Allora $m = n + \abs{A}$, e $\abs{A} \neq 0$ essendo $A$ non vuoto. Dunque $n < m$. \item[$\boxed{n \in m \impliedby n < m}$] Usiamo il fatto che $\in$ è un ordine totale su $\omega$.