src/HOL/Real/HahnBanach/HahnBanachExtLemmas.thy

 \begin{matharray}{l} \All
 {u\in F}{\All {v\in F}{a\ap u \leq b \ap v}} 
 \end{matharray} *};
 
 lemma ex_xi: 
-  "[| is_vectorspace F; !! u v. [| u:F; v:F |] ==> a u <= b v |]
-  ==> EX (xi::real). ALL y:F. a y <= xi & xi <= b y"; 
+  "[| is_vectorspace F; !! u v. [| u \<in> F; v \<in> F |] ==> a u <= b v |]
+  ==> \<exists>xi::real. \<forall>y \<in> F. a y <= xi \<and> xi <= b y"; 
 proof -;
   assume vs: "is_vectorspace F";
-  assume r: "(!! u v. [| u:F; v:F |] ==> a u <= (b v::real))";
+  assume r: "(!! u v. [| u \<in> F; v \<in> F |] ==> a u <= (b v::real))";
 
   txt {* From the completeness of the reals follows:
   The set $S = \{a\: u\dt\: u\in F\}$ has a supremum, if
   it is non-empty and has an upper bound. *};
 
-  let ?S = "{a u :: real | u. u:F}";
-
-  have "EX xi. isLub UNIV ?S xi";  
+  let ?S = "{a u :: real | u. u \<in> F}";
+
+  have "\<exists>xi. isLub UNIV ?S xi";  
   proof (rule reals_complete);
   
     txt {* The set $S$ is non-empty, since $a\ap\zero \in S$: *};
 
-    from vs; have "a \<zero> : ?S"; by force;
-    thus "EX X. X : ?S"; ..;
+    from vs; have "a 0 \<in> ?S"; by force;
+    thus "\<exists>X. X \<in> ?S"; ..;
 
     txt {* $b\ap \zero$ is an upper bound of $S$: *};
 
-    show "EX Y. isUb UNIV ?S Y"; 
+    show "\<exists>Y. isUb UNIV ?S Y"; 
     proof; 
-      show "isUb UNIV ?S (b \<zero>)";
+      show "isUb UNIV ?S (b 0)";
       proof (intro isUbI setleI ballI);
-        show "b \<zero> : UNIV"; ..;
+        show "b 0 \<in> UNIV"; ..;
       next;
 
         txt {* Every element $y\in S$ is less than $b\ap \zero$: *};
 
-        fix y; assume y: "y : ?S"; 
-        from y; have "EX u:F. y = a u"; by fast;
-        thus "y <= b \<zero>"; 
+        fix y; assume y: "y \<in> ?S"; 
+        from y; have "\<exists>u \<in> F. y = a u"; by fast;
+        thus "y <= b 0"; 
         proof;
-          fix u; assume "u:F"; 
+          fix u; assume "u \<in> F"; 
           assume "y = a u";
-          also; have "a u <= b \<zero>"; by (rule r) (simp!)+;
+          also; have "a u <= b 0"; by (rule r) (simp!)+;
           finally; show ?thesis; .;
         qed;
       qed;
     qed;
   qed;
 
-  thus "EX xi. ALL y:F. a y <= xi & xi <= b y"; 
+  thus "\<exists>xi. \<forall>y \<in> F. a y <= xi \<and> xi <= b y"; 
   proof (elim exE);
     fix xi; assume "isLub UNIV ?S xi"; 
     show ?thesis;
     proof (intro exI conjI ballI); 
    
       txt {* For all $y\in F$ holds $a\ap y \leq \xi$: *};
      
-      fix y; assume y: "y:F";
+      fix y; assume y: "y \<in> F";
       show "a y <= xi";    
       proof (rule isUbD);  
         show "isUb UNIV ?S xi"; ..;
       qed (force!);
     next;
 
       txt {* For all $y\in F$ holds $\xi\leq b\ap y$: *};
 
-      fix y; assume "y:F";
+      fix y; assume "y \<in> F";
       show "xi <= b y";  
       proof (intro isLub_le_isUb isUbI setleI);
-        show "b y : UNIV"; ..;
-        show "ALL ya : ?S. ya <= b y"; 
+        show "b y \<in> UNIV"; ..;
+        show "\<forall>ya \<in> ?S. ya <= b y"; 
         proof;
-          fix au; assume au: "au : ?S ";
-          hence "EX u:F. au = a u"; by fast;
+          fix au; assume au: "au \<in> ?S ";
+          hence "\<exists>u \<in> F. au = a u"; by fast;
           thus "au <= b y";
           proof;
-            fix u; assume "u:F"; assume "au = a u";  
+            fix u; assume "u \<in> F"; assume "au = a u";  
             also; have "... <= b y"; by (rule r);
             finally; show ?thesis; .;
           qed;
         qed;
       qed; 

 
 text{* \medskip The function $h_0$ is defined as a
 $h_0\ap x = h\ap y + a\cdot \xi$ where $x = y + a\mult \xi$
 is a linear extension of $h$ to $H_0$. *};
 
-lemma h0_lf: 
-  "[| h0 == (\\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<prod> x0 & y:H 
+lemma h'_lf: 
+  "[| h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H 
                 in h y + a * xi);
-  H0 == H + lin x0; is_subspace H E; is_linearform H h; x0 ~: H; 
-  x0 : E; x0 ~= \<zero>; is_vectorspace E |]
-  ==> is_linearform H0 h0";
+  H' == H + lin x0; is_subspace H E; is_linearform H h; x0 \<notin> H; 
+  x0 \<in> E; x0 \<noteq> 0; is_vectorspace E |]
+  ==> is_linearform H' h'";
 proof -;
-  assume h0_def: 
-    "h0 == (\\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<prod> x0 & y:H 
+  assume h'_def: 
+    "h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H 
                in h y + a * xi)"
-    and H0_def: "H0 == H + lin x0" 
-    and vs: "is_subspace H E" "is_linearform H h" "x0 ~: H"
-      "x0 ~= \<zero>" "x0 : E" "is_vectorspace E";
-
-  have h0: "is_vectorspace H0"; 
-  proof (unfold H0_def, rule vs_sum_vs);
+    and H'_def: "H' == H + lin x0" 
+    and vs: "is_subspace H E" "is_linearform H h" "x0 \<notin> H"
+      "x0 \<noteq> 0" "x0 \<in> E" "is_vectorspace E";
+
+  have h': "is_vectorspace H'"; 
+  proof (unfold H'_def, rule vs_sum_vs);
     show "is_subspace (lin x0) E"; ..;
   qed; 
 
   show ?thesis;
   proof;
-    fix x1 x2; assume x1: "x1 : H0" and x2: "x2 : H0"; 
+    fix x1 x2; assume x1: "x1 \<in> H'" and x2: "x2 \<in> H'"; 
 
     txt{* We now have to show that $h_0$ is additive, i.~e.\
     $h_0 \ap (x_1\plus x_2) = h_0\ap x_1 + h_0\ap x_2$
     for $x_1, x_2\in H$. *}; 
 
-    have x1x2: "x1 + x2 : H0"; 
-      by (rule vs_add_closed, rule h0); 
+    have x1x2: "x1 + x2 \<in> H'"; 
+      by (rule vs_add_closed, rule h'); 
     from x1; 
-    have ex_x1: "EX y1 a1. x1 = y1 + a1 \<prod> x0  & y1 : H"; 
-      by (unfold H0_def vs_sum_def lin_def) fast;
+    have ex_x1: "\<exists> y1 a1. x1 = y1 + a1 \<cdot> x0  \<and> y1 \<in> H"; 
+      by (unfold H'_def vs_sum_def lin_def) fast;
     from x2; 
-    have ex_x2: "EX y2 a2. x2 = y2 + a2 \<prod> x0 & y2 : H"; 
-      by (unfold H0_def vs_sum_def lin_def) fast;
+    have ex_x2: "\<exists> y2 a2. x2 = y2 + a2 \<cdot> x0 \<and> y2 \<in> H"; 
+      by (unfold H'_def vs_sum_def lin_def) fast;
     from x1x2; 
-    have ex_x1x2: "EX y a. x1 + x2 = y + a \<prod> x0 & y : H";
-      by (unfold H0_def vs_sum_def lin_def) fast;
+    have ex_x1x2: "\<exists> y a. x1 + x2 = y + a \<cdot> x0 \<and> y \<in> H";
+      by (unfold H'_def vs_sum_def lin_def) fast;
 
     from ex_x1 ex_x2 ex_x1x2;
-    show "h0 (x1 + x2) = h0 x1 + h0 x2";
+    show "h' (x1 + x2) = h' x1 + h' x2";
     proof (elim exE conjE);
       fix y1 y2 y a1 a2 a;
-      assume y1: "x1 = y1 + a1 \<prod> x0"     and y1': "y1 : H"
-         and y2: "x2 = y2 + a2 \<prod> x0"     and y2': "y2 : H" 
-         and y: "x1 + x2 = y + a \<prod> x0"   and y':  "y  : H"; 
-
-      have ya: "y1 + y2 = y & a1 + a2 = a"; 
-      proof (rule decomp_H0);;
-	txt_raw {* \label{decomp-H0-use} *};;
-        show "y1 + y2 + (a1 + a2) \<prod> x0 = y + a \<prod> x0"; 
+      assume y1: "x1 = y1 + a1 \<cdot> x0"     and y1': "y1 \<in> H"
+         and y2: "x2 = y2 + a2 \<cdot> x0"     and y2': "y2 \<in> H" 
+         and y: "x1 + x2 = y + a \<cdot> x0"   and y':  "y  \<in> H"; 
+
+      have ya: "y1 + y2 = y \<and> a1 + a2 = a"; 
+      proof (rule decomp_H');;
+	txt_raw {* \label{decomp-H-use} *};;
+        show "y1 + y2 + (a1 + a2) \<cdot> x0 = y + a \<cdot> x0"; 
           by (simp! add: vs_add_mult_distrib2 [of E]);
-        show "y1 + y2 : H"; ..;
+        show "y1 + y2 \<in> H"; ..;
       qed;
 
-      have "h0 (x1 + x2) = h y + a * xi";
-	by (rule h0_definite);
+      have "h' (x1 + x2) = h y + a * xi";
+	by (rule h'_definite);
       also; have "... = h (y1 + y2) + (a1 + a2) * xi"; 
         by (simp add: ya);
       also; from vs y1' y2'; 
       have "... = h y1 + h y2 + a1 * xi + a2 * xi"; 
 	by (simp add: linearform_add [of H] 
                       real_add_mult_distrib);
       also; have "... = (h y1 + a1 * xi) + (h y2 + a2 * xi)"; 
         by simp;
-      also; have "h y1 + a1 * xi = h0 x1"; 
-        by (rule h0_definite [RS sym]);
-      also; have "h y2 + a2 * xi = h0 x2"; 
-        by (rule h0_definite [RS sym]);
+      also; have "h y1 + a1 * xi = h' x1"; 
+        by (rule h'_definite [RS sym]);
+      also; have "h y2 + a2 * xi = h' x2"; 
+        by (rule h'_definite [RS sym]);
       finally; show ?thesis; .;
     qed;
  
     txt{* We further have to show that $h_0$ is multiplicative, 
     i.~e.\ $h_0\ap (c \mult x_1) = c \cdot h_0\ap x_1$
     for $x\in H$ and $c\in \bbbR$. 
     *}; 
 
   next;  
-    fix c x1; assume x1: "x1 : H0";    
-    have ax1: "c \<prod> x1 : H0";
-      by (rule vs_mult_closed, rule h0);
-    from x1; have ex_x: "!! x. x: H0 
-                        ==> EX y a. x = y + a \<prod> x0 & y : H";
-      by (unfold H0_def vs_sum_def lin_def) fast;
-
-    from x1; have ex_x1: "EX y1 a1. x1 = y1 + a1 \<prod> x0 & y1 : H";
-      by (unfold H0_def vs_sum_def lin_def) fast;
-    with ex_x [of "c \<prod> x1", OF ax1];
-    show "h0 (c \<prod> x1) = c * (h0 x1)";  
+    fix c x1; assume x1: "x1 \<in> H'";    
+    have ax1: "c \<cdot> x1 \<in> H'";
+      by (rule vs_mult_closed, rule h');
+    from x1; have ex_x: "!! x. x\<in> H' 
+                        ==> \<exists> y a. x = y + a \<cdot> x0 \<and> y \<in> H";
+      by (unfold H'_def vs_sum_def lin_def) fast;
+
+    from x1; have ex_x1: "\<exists> y1 a1. x1 = y1 + a1 \<cdot> x0 \<and> y1 \<in> H";
+      by (unfold H'_def vs_sum_def lin_def) fast;
+    with ex_x [of "c \<cdot> x1", OF ax1];
+    show "h' (c \<cdot> x1) = c * (h' x1)";  
     proof (elim exE conjE);
       fix y1 y a1 a; 
-      assume y1: "x1 = y1 + a1 \<prod> x0"       and y1': "y1 : H"
-        and y: "c \<prod> x1 = y  + a  \<prod> x0"   and y':  "y  : H"; 
-
-      have ya: "c \<prod> y1 = y & c * a1 = a"; 
-      proof (rule decomp_H0); 
-	show "c \<prod> y1 + (c * a1) \<prod> x0 = y + a \<prod> x0"; 
-          by (simp! add: add: vs_add_mult_distrib1);
-        show "c \<prod> y1 : H"; ..;
+      assume y1: "x1 = y1 + a1 \<cdot> x0"     and y1': "y1 \<in> H"
+        and y: "c \<cdot> x1 = y  + a \<cdot> x0"    and y': "y \<in> H"; 
+
+      have ya: "c \<cdot> y1 = y \<and> c * a1 = a"; 
+      proof (rule decomp_H'); 
+	show "c \<cdot> y1 + (c * a1) \<cdot> x0 = y + a \<cdot> x0"; 
+          by (simp! add: vs_add_mult_distrib1);
+        show "c \<cdot> y1 \<in> H"; ..;
       qed;
 
-      have "h0 (c \<prod> x1) = h y + a * xi"; 
-	by (rule h0_definite);
-      also; have "... = h (c \<prod> y1) + (c * a1) * xi";
+      have "h' (c \<cdot> x1) = h y + a * xi"; 
+	by (rule h'_definite);
+      also; have "... = h (c \<cdot> y1) + (c * a1) * xi";
         by (simp add: ya);
       also; from vs y1'; have "... = c * h y1 + c * a1 * xi"; 
 	by (simp add: linearform_mult [of H]);
       also; from vs y1'; have "... = c * (h y1 + a1 * xi)"; 
 	by (simp add: real_add_mult_distrib2 real_mult_assoc);
-      also; have "h y1 + a1 * xi = h0 x1"; 
-        by (rule h0_definite [RS sym]);
+      also; have "h y1 + a1 * xi = h' x1"; 
+        by (rule h'_definite [RS sym]);
       finally; show ?thesis; .;
     qed;
   qed;
 qed;
 
 text{* \medskip The linear extension $h_0$ of $h$
 is bounded by the seminorm $p$. *};
 
-lemma h0_norm_pres:
-  "[| h0 == (\\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<prod> x0 & y:H 
+lemma h'_norm_pres:
+  "[| h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H 
                 in h y + a * xi);
-  H0 == H + lin x0; x0 ~: H; x0 : E; x0 ~= \<zero>; is_vectorspace E; 
-  is_subspace H E; is_seminorm E p; is_linearform H h; ALL y:H. h y <= p y; 
-  ALL y:H. - p (y + x0) - h y <= xi & xi <= p (y + x0) - h y |]
-   ==> ALL x:H0. h0 x <= p x"; 
+  H' == H + lin x0; x0 \<notin> H; x0 \<in> E; x0 \<noteq> 0; is_vectorspace E; 
+  is_subspace H E; is_seminorm E p; is_linearform H h; \<forall>y \<in> H. h y <= p y; 
+  \<forall>y \<in> H. - p (y + x0) - h y <= xi \<and> xi <= p (y + x0) - h y |]
+   ==> \<forall> x \<in> H'. h' x <= p x"; 
 proof; 
-  assume h0_def: 
-    "h0 == (\\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<prod> x0 & y:H 
+  assume h'_def: 
+    "h' == (\<lambda>x. let (y, a) = SOME (y, a). x = y + a \<cdot> x0 \<and> y \<in> H 
                in (h y) + a * xi)"
-    and H0_def: "H0 == H + lin x0" 
-    and vs: "x0 ~: H" "x0 : E" "x0 ~= \<zero>" "is_vectorspace E" 
+    and H'_def: "H' == H + lin x0" 
+    and vs: "x0 \<notin> H" "x0 \<in> E" "x0 \<noteq> 0" "is_vectorspace E" 
             "is_subspace H E" "is_seminorm E p" "is_linearform H h" 
-    and a: "ALL y:H. h y <= p y";
-  presume a1: "ALL ya:H. - p (ya + x0) - h ya <= xi";
-  presume a2: "ALL ya:H. xi <= p (ya + x0) - h ya";
-  fix x; assume "x : H0"; 
+    and a: "\<forall>y \<in> H. h y <= p y";
+  presume a1: "\<forall>ya \<in> H. - p (ya + x0) - h ya <= xi";
+  presume a2: "\<forall>ya \<in> H. xi <= p (ya + x0) - h ya";
+  fix x; assume "x \<in> H'"; 
   have ex_x: 
-    "!! x. x : H0 ==> EX y a. x = y + a \<prod> x0 & y : H";
-    by (unfold H0_def vs_sum_def lin_def) fast;
-  have "EX y a. x = y + a \<prod> x0 & y : H";
+    "!! x. x \<in> H' ==> \<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H";
+    by (unfold H'_def vs_sum_def lin_def) fast;
+  have "\<exists>y a. x = y + a \<cdot> x0 \<and> y \<in> H";
     by (rule ex_x);
-  thus "h0 x <= p x";
+  thus "h' x <= p x";
   proof (elim exE conjE);
-    fix y a; assume x: "x = y + a \<prod> x0" and y: "y : H";
-    have "h0 x = h y + a * xi";
-      by (rule h0_definite);
+    fix y a; assume x: "x = y + a \<cdot> x0" and y: "y \<in> H";
+    have "h' x = h y + a * xi";
+      by (rule h'_definite);
 
     txt{* Now we show  
     $h\ap y + a \cdot \xi\leq  p\ap (y\plus a \mult x_0)$ 
     by case analysis on $a$. \label{linorder_linear_split}*};
 
-    also; have "... <= p (y + a \<prod> x0)";
+    also; have "... <= p (y + a \<cdot> x0)";
     proof (rule linorder_linear_split); 
 
       assume z: "a = #0"; 
       with vs y a; show ?thesis; by simp;
 
     txt {* In the case $a < 0$, we use $a_1$ with $\idt{ya}$ 
     taken as $y/a$: *};
 
     next;
-      assume lz: "a < #0"; hence nz: "a ~= #0"; by simp;
+      assume lz: "a < #0"; hence nz: "a \<noteq> #0"; by simp;
       from a1; 
-      have "- p (rinv a \<prod> y + x0) - h (rinv a \<prod> y) <= xi";
+      have "- p (rinv a \<cdot> y + x0) - h (rinv a \<cdot> y) <= xi";
         by (rule bspec) (simp!);
 
       txt {* The thesis for this case now follows by a short  
       calculation. *};      
       hence "a * xi 
-            <= a * (- p (rinv a \<prod> y + x0) - h (rinv a \<prod> y))";
+            <= a * (- p (rinv a \<cdot> y + x0) - h (rinv a \<cdot> y))";
         by (rule real_mult_less_le_anti [OF lz]);
-      also; have "... = - a * (p (rinv a \<prod> y + x0)) 
-                        - a * (h (rinv a \<prod> y))";
+      also; have "... = - a * (p (rinv a \<cdot> y + x0)) 
+                        - a * (h (rinv a \<cdot> y))";
         by (rule real_mult_diff_distrib);
-      also; from lz vs y; have "- a * (p (rinv a \<prod> y + x0)) 
-                               = p (a \<prod> (rinv a \<prod> y + x0))";
+      also; from lz vs y; have "- a * (p (rinv a \<cdot> y + x0)) 
+                               = p (a \<cdot> (rinv a \<cdot> y + x0))";
         by (simp add: seminorm_abs_homogenous abs_minus_eqI2);
-      also; from nz vs y; have "... = p (y + a \<prod> x0)";
+      also; from nz vs y; have "... = p (y + a \<cdot> x0)";
         by (simp add: vs_add_mult_distrib1);
-      also; from nz vs y; have "a * (h (rinv a \<prod> y)) =  h y";
+      also; from nz vs y; have "a * (h (rinv a \<cdot> y)) =  h y";
         by (simp add: linearform_mult [RS sym]);
-      finally; have "a * xi <= p (y + a \<prod> x0) - h y"; .;
-
-      hence "h y + a * xi <= h y + p (y + a \<prod> x0) - h y";
+      finally; have "a * xi <= p (y + a \<cdot> x0) - h y"; .;
+
+      hence "h y + a * xi <= h y + p (y + a \<cdot> x0) - h y";
         by (simp add: real_add_left_cancel_le);
       thus ?thesis; by simp;
 
       txt {* In the case $a > 0$, we use $a_2$ with $\idt{ya}$ 
       taken as $y/a$: *};
 
     next; 
-      assume gz: "#0 < a"; hence nz: "a ~= #0"; by simp;
+      assume gz: "#0 < a"; hence nz: "a \<noteq> #0"; by simp;
       from a2;
-      have "xi <= p (rinv a \<prod> y + x0) - h (rinv a \<prod> y)";
+      have "xi <= p (rinv a \<cdot> y + x0) - h (rinv a \<cdot> y)";
         by (rule bspec) (simp!);
 
       txt {* The thesis for this case follows by a short
       calculation: *};
 
       with gz; have "a * xi 
-            <= a * (p (rinv a \<prod> y + x0) - h (rinv a \<prod> y))";
+            <= a * (p (rinv a \<cdot> y + x0) - h (rinv a \<cdot> y))";
         by (rule real_mult_less_le_mono);
-      also; have "... = a * p (rinv a \<prod> y + x0) 
-                        - a * h (rinv a \<prod> y)";
+      also; have "... = a * p (rinv a \<cdot> y + x0) 
+                        - a * h (rinv a \<cdot> y)";
         by (rule real_mult_diff_distrib2); 
       also; from gz vs y; 
-      have "a * p (rinv a \<prod> y + x0) 
-           = p (a \<prod> (rinv a \<prod> y + x0))";
+      have "a * p (rinv a \<cdot> y + x0) 
+           = p (a \<cdot> (rinv a \<cdot> y + x0))";
         by (simp add: seminorm_abs_homogenous abs_eqI2);
       also; from nz vs y; 
-      have "... = p (y + a \<prod> x0)";
+      have "... = p (y + a \<cdot> x0)";
         by (simp add: vs_add_mult_distrib1);
-      also; from nz vs y; have "a * h (rinv a \<prod> y) = h y";
+      also; from nz vs y; have "a * h (rinv a \<cdot> y) = h y";
         by (simp add: linearform_mult [RS sym]); 
-      finally; have "a * xi <= p (y + a \<prod> x0) - h y"; .;
+      finally; have "a * xi <= p (y + a \<cdot> x0) - h y"; .;
  
-      hence "h y + a * xi <= h y + (p (y + a \<prod> x0) - h y)";
+      hence "h y + a * xi <= h y + (p (y + a \<cdot> x0) - h y)";
         by (simp add: real_add_left_cancel_le);
       thus ?thesis; by simp;
     qed;
     also; from x; have "... = p x"; by simp;
     finally; show ?thesis; .;