
diff -r f909f1a5cb22 -r b7b71952c194 src/HOL/Decision_Procs/ex/Dense_Linear_Order_Ex.thy
--- a/src/HOL/Decision_Procs/ex/Dense_Linear_Order_Ex.thy	Sat Jun 20 22:20:27 2015 +0200
+++ b/src/HOL/Decision_Procs/ex/Dense_Linear_Order_Ex.thy	Sat Jun 20 22:38:39 2015 +0200
@@ -6,148 +6,159 @@
 imports "../Dense_Linear_Order"
 begin
 
-lemma
-  "\<exists>(y::'a::{linordered_field}) <2. x + 3* y < 0 \<and> x - y >0"
+lemma "\<exists>(y::'a::linordered_field) < 2. x + 3* y < 0 \<and> x - y > 0"
   by ferrack
 
-lemma "~ (ALL x (y::'a::{linordered_field}). x < y --> 10*x < 11*y)"
+lemma "\<not> (\<forall>x (y::'a::linordered_field). x < y \<longrightarrow> 10 * x < 11 * y)"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y. x < y --> (10*(x + 5*y + -1) < 60*y)"
+lemma "\<forall>(x::'a::linordered_field) y. x < y \<longrightarrow> 10 * (x + 5 * y + -1) < 60 * y"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y. x ~= y --> x < y"
+lemma "\<exists>(x::'a::linordered_field) y. x \<noteq> y \<longrightarrow> x < y"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y. (x ~= y & 10*x ~= 9*y & 10*x < y) --> x < y"
+lemma "\<exists>(x::'a::linordered_field) y. x \<noteq> y \<and> 10 * x \<noteq> 9 * y \<and> 10 * x < y \<longrightarrow> x < y"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y. (x ~= y & 5*x <= y) --> 500*x <= 100*y"
+lemma "\<forall>(x::'a::linordered_field) y. x \<noteq> y \<and> 5 * x \<le> y \<longrightarrow> 500 * x \<le> 100 * y"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}). (EX (y::'a::{linordered_field}). 4*x + 3*y <= 0 & 4*x + 3*y >= -1)"
+lemma "\<forall>x::'a::linordered_field. \<exists>y::'a::linordered_field. 4 * x + 3 * y \<le> 0 \<and> 4 * x + 3 * y \<ge> -1"
+  by ferrack
+
+lemma "\<forall>(x::'a::linordered_field) < 0. \<exists>(y::'a::linordered_field) > 0. 7 * x + y > 0 \<and> x - y \<le> 9"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) < 0. (EX (y::'a::{linordered_field}) > 0. 7*x + y > 0 & x - y <= 9)"
+lemma "\<exists>x::'a::linordered_field. 0 < x \<and> x < 1 \<longrightarrow> (\<forall>y > 1. x + y \<noteq> 1)"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}). (0 < x & x < 1) --> (ALL y > 1. x + y ~= 1)"
+lemma "\<exists>x. \<forall>y::'a::linordered_field. y < 2 \<longrightarrow> 2 * (y - x) \<le> 0"
   by ferrack
 
-lemma "EX x. (ALL (y::'a::{linordered_field}). y < 2 -->  2*(y - x) \<le> 0 )"
+lemma "\<forall>x::'a::linordered_field. x < 10 \<or> x > 20 \<or> (\<exists>y. y \<ge> 0 \<and> y \<le> 10 \<and> x + y = 20)"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}). x < 10 | x > 20 | (EX y. y>= 0 & y <= 10 & x+y = 20)"
+lemma "\<forall>(x::'a::linordered_field) y z. x + y < z \<longrightarrow> y \<ge> z \<longrightarrow> x < 0"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y z. x + y < z --> y >= z --> x < 0"
+lemma "\<exists>(x::'a::linordered_field) y z. x + 7 * y < 5 * z \<and> 5 * y \<ge> 7 * z \<and> x < 0"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y z. x + 7*y < 5* z & 5*y >= 7*z & x < 0"
+lemma "\<forall>(x::'a::linordered_field) y z. \<bar>x + y\<bar> \<le> z \<longrightarrow> \<bar>z\<bar> = z"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y z. abs (x + y) <= z --> (abs z = z)"
+lemma "\<exists>(x::'a::linordered_field) y z. x + 7 * y - 5 * z < 0 \<and> 5 * y + 7 * z + 3 * x < 0"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y z. x + 7*y - 5* z < 0 & 5*y + 7*z + 3*x < 0"
+lemma "\<forall>(x::'a::linordered_field) y z.
+  (\<bar>5 * x + 3 * y + z\<bar> \<le> 5 * x + 3 * y + z \<and> \<bar>5 * x + 3 * y + z\<bar> \<ge> - (5 * x + 3 * y + z)) \<or>
+  (\<bar>5 * x + 3 * y + z\<bar> \<ge> 5 * x + 3 * y + z \<and> \<bar>5 * x + 3 * y + z\<bar> \<le> - (5 * x + 3 * y + z))"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y z. (abs (5*x+3*y+z) <= 5*x+3*y+z & abs (5*x+3*y+z) >= - (5*x+3*y+z)) | (abs (5*x+3*y+z) >= 5*x+3*y+z & abs (5*x+3*y+z) <= - (5*x+3*y+z))"
+lemma "\<forall>(x::'a::linordered_field) y. x < y \<longrightarrow> (\<exists>z>0. x + z = y)"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y. x < y --> (EX z>0. x+z = y)"
+lemma "\<forall>(x::'a::linordered_field) y. x < y \<longrightarrow> (\<exists>z>0. x + z = y)"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y. x < y --> (EX z>0. x+z = y)"
+lemma "\<forall>(x::'a::linordered_field) y. \<exists>z>0. \<bar>x - y\<bar> \<le> z"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y. (EX z>0. abs (x - y) <= z )"
+lemma "\<exists>(x::'a::linordered_field) y. \<forall>z<0. (z < x \<longrightarrow> z \<le> y) \<and> (z > y \<longrightarrow> z \<ge> x)"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y. (ALL z<0. (z < x --> z <= y) & (z > y --> z >= x))"
+lemma "\<exists>(x::'a::linordered_field) y. \<forall>z\<ge>0. \<bar>3 * x + 7 * y\<bar> \<le> 2 * z + 1"
+  by ferrack
+
+lemma "\<exists>(x::'a::linordered_field) y. \<forall>z<0. (z < x \<longrightarrow> z \<le> y) \<and> (z > y \<longrightarrow> z \<ge> x)"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y. (ALL z>=0. abs (3*x+7*y) <= 2*z + 1)"
+lemma "\<exists>(x::'a::linordered_field) > 0. \<forall>y. \<exists>z. 13 * \<bar>z\<bar> \<noteq> \<bar>12 * y - x\<bar> \<and> 5 * x - 3 * \<bar>y\<bar> \<le> 7 * z"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y. (ALL z<0. (z < x --> z <= y) & (z > y --> z >= x))"
-  by ferrack
-
-lemma "EX (x::'a::{linordered_field})>0. (ALL y. (EX z. 13* abs z \<noteq> abs (12*y - x) & 5*x - 3*(abs y) <= 7*z))"
+lemma "\<exists>x::'a::linordered_field.
+  \<bar>4 * x + 17\<bar> < 4 \<and> (\<forall>y. \<bar>x * 34 - 34 * y - 9\<bar> \<noteq> 0 \<longrightarrow> (\<exists>z. 5 * x - 3 * \<bar>y\<bar> \<le> 7 * z))"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}). abs (4*x + 17) < 4 & (ALL y . abs (x*34 - 34*y - 9) \<noteq> 0 \<longrightarrow> (EX z. 5*x - 3*abs y <= 7*z))"
+lemma "\<forall>x::'a::linordered_field. \<exists>y > \<bar>23 * x - 9\<bar>. \<forall>z > \<bar>3 * y - 19 * \<bar>x\<bar>\<bar>. x + z > 2 * y"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}). (EX y > abs (23*x - 9). (ALL z > abs (3*y - 19* abs x). x+z > 2*y))"
+lemma "\<forall>x::'a::linordered_field.
+  \<exists>y < \<bar>3 * x - 1\<bar>. \<forall>z \<ge> 3 * \<bar>x\<bar> - 1. \<bar>12 * x - 13 * y + 19 * z\<bar> > \<bar>23 * x\<bar>"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}). (EX y< abs (3*x - 1). (ALL z >= (3*abs x - 1). abs (12*x - 13*y + 19*z) > abs (23*x) ))"
+lemma "\<exists>x::'a::linordered_field. \<bar>x\<bar> < 100 \<and> (\<forall>y > x. (\<exists>z < 2 * y - x. 5 * x - 3 * y \<le> 7 * z))"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}). abs x < 100 & (ALL y > x. (EX z<2*y - x. 5*x - 3*y <= 7*z))"
+lemma "\<forall>(x::'a::linordered_field) y z w.
+  7 * x < 3 * y \<longrightarrow> 5 * y < 7 * z \<longrightarrow> z < 2 * w \<longrightarrow> 7 * (2 * w - x) > 2 * y"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y z w. 7*x<3*y --> 5*y < 7*z --> z < 2*w --> 7*(2*w-x) > 2*y"
+lemma "\<exists>(x::'a::linordered_field) y z w. 5 * x + 3 * z - 17 * w + \<bar>y - 8 * x + z\<bar> \<le> 89"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y z w. 5*x + 3*z - 17*w + abs (y - 8*x + z) <= 89"
+lemma "\<exists>(x::'a::linordered_field) y z w.
+  5 * x + 3 * z - 17 * w + 7 * (y - 8 * x + z) \<le> max y (7 * z - x + w)"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y z w. 5*x + 3*z - 17*w + 7* (y - 8*x + z) <= max y (7*z - x + w)"
-  by ferrack
-
-lemma "EX (x::'a::{linordered_field}) y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)"
+lemma "\<exists>(x::'a::linordered_field) y z w.
+  min (5 * x + 3 * z) (17 * w) + 5 * \<bar>y - 8 * x + z\<bar> \<le> max y (7 * z - x + w)"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y z. (EX w >= (x+y+z). w <= abs x + abs y + abs z)"
+lemma "\<forall>(x::'a::linordered_field) y z. \<exists>w \<ge> x + y + z. w \<le> \<bar>x\<bar> + \<bar>y\<bar> + \<bar>z\<bar>"
   by ferrack
 
-lemma "~(ALL (x::'a::{linordered_field}). (EX y z w. 3* x + z*4 = 3*y & x + y < z & x> w & 3*x < w + y))"
+lemma "\<not> (\<forall>x::'a::linordered_field. \<exists>y z w.
+  3 * x + z * 4 = 3 * y \<and> x + y < z \<and> x > w \<and> 3 * x < w + y)"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y. (EX z w. abs (x-y) = (z-w) & z*1234 < 233*x & w ~= y)"
+lemma "\<forall>(x::'a::linordered_field) y. \<exists>z w. \<bar>x - y\<bar> = z - w \<and> z * 1234 < 233 * x \<and> w \<noteq> y"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}). (EX y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w))"
+lemma "\<forall>x::'a::linordered_field. \<exists>y z w.
+  min (5 * x + 3 * z) (17 * w) + 5 * \<bar>y - 8 * x + z\<bar> \<le> max y (7 * z - x + w)"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y z. (ALL w >= abs (x+y+z). w >= abs x + abs y + abs z)"
+lemma "\<exists>(x::'a::linordered_field) y z. \<forall>w \<ge> \<bar>x + y + z\<bar>. w \<ge> \<bar>x\<bar> + \<bar>y\<bar> + \<bar>z\<bar>"
   by ferrack
 
-lemma "EX z. (ALL (x::'a::{linordered_field}) y. (EX w >= (x+y+z). w <= abs x + abs y + abs z))"
+lemma "\<exists>z. \<forall>(x::'a::linordered_field) y. \<exists>w \<ge> x + y + z. w \<le> \<bar>x\<bar> + \<bar>y\<bar> + \<bar>z\<bar>"
+  by ferrack
+
+lemma "\<exists>z. \<forall>(x::'a::linordered_field) < \<bar>z\<bar>. \<exists>y w. x < y \<and> x < z \<and> x > w \<and> 3 * x < w + y"
   by ferrack
 
-lemma "EX z. (ALL (x::'a::{linordered_field}) < abs z. (EX y w. x< y & x < z & x> w & 3*x < w + y))"
+lemma "\<forall>(x::'a::linordered_field) y. \<exists>z. \<forall>w. \<bar>x - y\<bar> = \<bar>z - w\<bar> \<longrightarrow> z < x \<and> w \<noteq> y"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}) y. (EX z. (ALL w. abs (x-y) = abs (z-w) --> z < x & w ~= y))"
+lemma "\<exists>y. \<forall>x::'a::linordered_field. \<exists>z w.
+  min (5 * x + 3 * z) (17 * w) + 5 * \<bar>y - 8 * x + z\<bar> \<le> max y (7 * z - x + w)"
   by ferrack
 
-lemma "EX y. (ALL (x::'a::{linordered_field}). (EX z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)))"
-  by ferrack
-
-lemma "EX (x::'a::{linordered_field}) z. (ALL w >= 13*x - 4*z. (EX y. w >= abs x + abs y + z))"
+lemma "\<exists>(x::'a::linordered_field) z. \<forall>w \<ge> 13 * x - 4 * z. \<exists>y. w \<ge> \<bar>x\<bar> + \<bar>y\<bar> + z"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}). (ALL y < x. (EX z > (x+y).
-  (ALL w. 5*w + 10*x - z >= y --> w + 7*x + 3*z >= 2*y)))"
+lemma "\<exists>x::'a::linordered_field. \<forall>y < x. \<exists>z > x + y.
+  \<forall>w. 5 * w + 10 * x - z \<ge> y \<longrightarrow> w + 7 * x + 3 * z \<ge> 2 * y"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}). (ALL y. (EX z > y.
-  (ALL w . w < 13 --> w + 10*x - z >= y --> 5*w + 7*x + 13*z >= 2*y)))"
+lemma "\<exists>x::'a::linordered_field. \<forall>y. \<exists>z > y.
+  \<forall>w. w < 13 \<longrightarrow> w + 10 * x - z \<ge> y \<longrightarrow> 5 * w + 7 * x + 13 * z \<ge> 2 * y"
   by ferrack
 
-lemma "EX (x::'a::{linordered_field}) y z w. min (5*x + 3*z) (17*w) + 5* abs (y - 8*x + z) <= max y (7*z - x + w)"
+lemma "\<exists>(x::'a::linordered_field) y z w.
+  min (5 * x + 3 * z) (17 * w) + 5 * \<bar>y - 8 * x + z\<bar> \<le> max y (7 * z - x + w)"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}). (EX y. (ALL z>19. y <= x + z & (EX w. abs (y - x) < w)))"
+lemma "\<forall>x::'a::linordered_field. \<exists>y. \<forall>z>19. y \<le> x + z \<and> (\<exists>w. \<bar>y - x\<bar> < w)"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}). (EX y. (ALL z>19. y <= x + z & (EX w. abs (x + z) < w - y)))"
+lemma "\<forall>x::'a::linordered_field. \<exists>y. \<forall>z>19. y \<le> x + z \<and> (\<exists>w. \<bar>x + z\<bar> < w - y)"
   by ferrack
 
-lemma "ALL (x::'a::{linordered_field}). (EX y. abs y ~= abs x & (ALL z> max x y. (EX w. w ~= y & w ~= z & 3*w - z >= x + y)))"
+lemma "\<forall>x::'a::linordered_field. \<exists>y.
+  \<bar>y\<bar> \<noteq> \<bar>x\<bar> \<and> (\<forall>z > max x y. \<exists>w. w \<noteq> y \<and> w \<noteq> z \<and> 3 * w - z \<ge> x + y)"
   by ferrack
 
 end




