# HG changeset patch
# User wenzelm
# Date 1083441905 -7200
# Node ID 4deda204e1d8f94e9fefe733acc850e9d4e6a9fd
# Parent  b8d6c395c9e2b677ab67a51de2a9ce746c9cfc54
improved syntax;

diff -r b8d6c395c9e2 -r 4deda204e1d8 src/HOL/Algebra/Group.thy
--- a/src/HOL/Algebra/Group.thy	Sat May 01 22:04:14 2004 +0200
+++ b/src/HOL/Algebra/Group.thy	Sat May 01 22:05:05 2004 +0200
@@ -42,10 +42,10 @@
   pow :: "[('a, 'm) monoid_scheme, 'a, 'b::number] => 'a" (infixr "'(^')\<index>" 75)
 
 defs (overloaded)
-  nat_pow_def: "pow G a n == nat_rec (one G) (%u b. mult G b a) n"
+  nat_pow_def: "pow G a n == nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a) n"
   int_pow_def: "pow G a z ==
-    let p = nat_rec (one G) (%u b. mult G b a)
-    in if neg z then m_inv G (p (nat (-z))) else p (nat z)"
+    let p = nat_rec \<one>\<^bsub>G\<^esub> (%u b. b \<otimes>\<^bsub>G\<^esub> a)
+    in if neg z then inv\<^bsub>G\<^esub> (p (nat (-z))) else p (nat z)"
 
 locale magma = struct G +
   assumes m_closed [intro, simp]:
@@ -62,14 +62,15 @@
     and r_one [simp]: "x \<in> carrier G ==> x \<otimes> \<one> = x"
 
 lemma monoidI:
+  includes struct G
   assumes m_closed:
-      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
-    and one_closed: "one G \<in> carrier G"
+      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
+    and one_closed: "\<one> \<in> carrier G"
     and m_assoc:
       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
-      mult G (mult G x y) z = mult G x (mult G y z)"
-    and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
-    and r_one: "!!x. x \<in> carrier G ==> mult G x (one G) = x"
+      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
+    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
+    and r_one: "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"
   shows "monoid G"
   by (fast intro!: monoid.intro magma.intro semigroup_axioms.intro
     semigroup.intro monoid_axioms.intro
@@ -80,8 +81,8 @@
   by (unfold Units_def) fast
 
 lemma (in monoid) inv_unique:
-  assumes eq: "y \<otimes> x = \<one>" "x \<otimes> y' = \<one>"
-    and G: "x \<in> carrier G" "y \<in> carrier G" "y' \<in> carrier G"
+  assumes eq: "y \<otimes> x = \<one>"  "x \<otimes> y' = \<one>"
+    and G: "x \<in> carrier G"  "y \<in> carrier G"  "y' \<in> carrier G"
   shows "y = y'"
 proof -
   from G eq have "y = y \<otimes> (x \<otimes> y')" by simp
@@ -129,13 +130,13 @@
    (x \<otimes> y = x \<otimes> z) = (y = z)"
 proof
   assume eq: "x \<otimes> y = x \<otimes> z"
-    and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G"
+    and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"
   then have "(inv x \<otimes> x) \<otimes> y = (inv x \<otimes> x) \<otimes> z"
     by (simp add: m_assoc Units_closed)
   with G show "y = z" by (simp add: Units_l_inv)
 next
   assume eq: "y = z"
-    and G: "x \<in> Units G" "y \<in> carrier G" "z \<in> carrier G"
+    and G: "x \<in> Units G"  "y \<in> carrier G"  "z \<in> carrier G"
   then show "x \<otimes> y = x \<otimes> z" by simp
 qed
 
@@ -152,14 +153,14 @@
   "inj_on (m_inv G) (Units G)"
 proof (rule inj_onI)
   fix x y
-  assume G: "x \<in> Units G" "y \<in> Units G" and eq: "inv x = inv y"
+  assume G: "x \<in> Units G"  "y \<in> Units G" and eq: "inv x = inv y"
   then have "inv (inv x) = inv (inv y)" by simp
   with G show "x = y" by simp
 qed
 
 lemma (in monoid) Units_inv_comm:
   assumes inv: "x \<otimes> y = \<one>"
-    and G: "x \<in> Units G" "y \<in> Units G"
+    and G: "x \<in> Units G"  "y \<in> Units G"
   shows "y \<otimes> x = \<one>"
 proof -
   from G have "x \<otimes> y \<otimes> x = x \<otimes> \<one>" by (auto simp add: inv Units_closed)
@@ -200,59 +201,58 @@
   assumes Units: "carrier G <= Units G"
 
 theorem groupI:
+  includes struct G
   assumes m_closed [simp]:
-      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
-    and one_closed [simp]: "one G \<in> carrier G"
+      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
+    and one_closed [simp]: "\<one> \<in> carrier G"
     and m_assoc:
       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
-      mult G (mult G x y) z = mult G x (mult G y z)"
-    and l_one [simp]: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
-    and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G"
+      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
+    and l_one [simp]: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
+    and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one>"
   shows "group G"
 proof -
   have l_cancel [simp]:
     "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
-    (mult G x y = mult G x z) = (y = z)"
+    (x \<otimes> y = x \<otimes> z) = (y = z)"
   proof
     fix x y z
-    assume eq: "mult G x y = mult G x z"
-      and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
+    assume eq: "x \<otimes> y = x \<otimes> z"
+      and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
-      and l_inv: "mult G x_inv x = one G" by fast
-    from G eq xG have "mult G (mult G x_inv x) y = mult G (mult G x_inv x) z"
+      and l_inv: "x_inv \<otimes> x = \<one>" by fast
+    from G eq xG have "(x_inv \<otimes> x) \<otimes> y = (x_inv \<otimes> x) \<otimes> z"
       by (simp add: m_assoc)
     with G show "y = z" by (simp add: l_inv)
   next
     fix x y z
     assume eq: "y = z"
-      and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
-    then show "mult G x y = mult G x z" by simp
+      and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
+    then show "x \<otimes> y = x \<otimes> z" by simp
   qed
   have r_one:
-    "!!x. x \<in> carrier G ==> mult G x (one G) = x"
+    "!!x. x \<in> carrier G ==> x \<otimes> \<one> = x"
   proof -
     fix x
     assume x: "x \<in> carrier G"
     with l_inv_ex obtain x_inv where xG: "x_inv \<in> carrier G"
-      and l_inv: "mult G x_inv x = one G" by fast
-    from x xG have "mult G x_inv (mult G x (one G)) = mult G x_inv x"
+      and l_inv: "x_inv \<otimes> x = \<one>" by fast
+    from x xG have "x_inv \<otimes> (x \<otimes> \<one>) = x_inv \<otimes> x"
       by (simp add: m_assoc [symmetric] l_inv)
-    with x xG show "mult G x (one G) = x" by simp 
+    with x xG show "x \<otimes> \<one> = x" by simp
   qed
   have inv_ex:
-    "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G &
-      mult G x y = one G"
+    "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"
   proof -
     fix x
     assume x: "x \<in> carrier G"
     with l_inv_ex obtain y where y: "y \<in> carrier G"
-      and l_inv: "mult G y x = one G" by fast
-    from x y have "mult G y (mult G x y) = mult G y (one G)"
+      and l_inv: "y \<otimes> x = \<one>" by fast
+    from x y have "y \<otimes> (x \<otimes> y) = y \<otimes> \<one>"
       by (simp add: m_assoc [symmetric] l_inv r_one)
-    with x y have r_inv: "mult G x y = one G"
+    with x y have r_inv: "x \<otimes> y = \<one>"
       by simp
-    from x y show "EX y : carrier G. mult G y x = one G &
-      mult G x y = one G"
+    from x y show "EX y : carrier G. y \<otimes> x = \<one> & x \<otimes> y = \<one>"
       by (fast intro: l_inv r_inv)
   qed
   then have carrier_subset_Units: "carrier G <= Units G"
@@ -265,7 +265,7 @@
 
 lemma (in monoid) monoid_groupI:
   assumes l_inv_ex:
-    "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G"
+    "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one>"
   shows "group G"
   by (rule groupI) (auto intro: m_assoc l_inv_ex)
 
@@ -306,13 +306,13 @@
    (y \<otimes> x = z \<otimes> x) = (y = z)"
 proof
   assume eq: "y \<otimes> x = z \<otimes> x"
-    and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
+    and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
   then have "y \<otimes> (x \<otimes> inv x) = z \<otimes> (x \<otimes> inv x)"
     by (simp add: m_assoc [symmetric])
   with G show "y = z" by (simp add: r_inv)
 next
   assume eq: "y = z"
-    and G: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
+    and G: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
   then show "y \<otimes> x = z \<otimes> x" by simp
 qed
 
@@ -335,7 +335,7 @@
 lemma (in group) inv_mult_group:
   "[| x \<in> carrier G; y \<in> carrier G |] ==> inv (x \<otimes> y) = inv y \<otimes> inv x"
 proof -
-  assume G: "x \<in> carrier G" "y \<in> carrier G"
+  assume G: "x \<in> carrier G"  "y \<in> carrier G"
   then have "inv (x \<otimes> y) \<otimes> (x \<otimes> y) = (inv y \<otimes> inv x) \<otimes> (x \<otimes> y)"
     by (simp add: m_assoc l_inv) (simp add: m_assoc [symmetric] l_inv)
   with G show ?thesis by simp
@@ -343,14 +343,14 @@
 
 lemma (in group) inv_comm:
   "[| x \<otimes> y = \<one>; x \<in> carrier G; y \<in> carrier G |] ==> y \<otimes> x = \<one>"
-  by (rule Units_inv_comm) auto                          
+  by (rule Units_inv_comm) auto
 
 lemma (in group) inv_equality:
      "[|y \<otimes> x = \<one>; x \<in> carrier G; y \<in> carrier G|] ==> inv x = y"
 apply (simp add: m_inv_def)
 apply (rule the_equality)
- apply (simp add: inv_comm [of y x]) 
-apply (rule r_cancel [THEN iffD1], auto) 
+ apply (simp add: inv_comm [of y x])
+apply (rule r_cancel [THEN iffD1], auto)
 done
 
 text {* Power *}
@@ -493,10 +493,10 @@
 constdefs (structure G and H)
   DirProdSemigroup :: "_ => _ => ('a \<times> 'b) semigroup"  (infixr "\<times>\<^sub>s" 80)
   "G \<times>\<^sub>s H == (| carrier = carrier G \<times> carrier H,
-    mult = (%(xg, xh) (yg, yh). (xg \<otimes> yg, xh \<otimes>\<^sub>2 yh)) |)"
+    mult = (%(g, h) (g', h'). (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')) |)"
 
   DirProdGroup :: "_ => _ => ('a \<times> 'b) monoid"  (infixr "\<times>\<^sub>g" 80)
-  "G \<times>\<^sub>g H == semigroup.extend (G \<times>\<^sub>s H) (| one = (\<one>, \<one>\<^sub>2) |)"
+  "G \<times>\<^sub>g H == semigroup.extend (G \<times>\<^sub>s H) (| one = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>) |)"
 
 lemma DirProdSemigroup_magma:
   includes magma G + magma H
@@ -550,18 +550,18 @@
   by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
 
 lemma one_DirProdGroup [simp]:
-     "one (G \<times>\<^sub>g H) = (one G, one H)"
+     "\<one>\<^bsub>(G \<times>\<^sub>g H)\<^esub> = (\<one>\<^bsub>G\<^esub>, \<one>\<^bsub>H\<^esub>)"
   by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
 
 lemma mult_DirProdGroup [simp]:
-     "mult (G \<times>\<^sub>g H) (g, h) (g', h') = (mult G g g', mult H h h')"
+     "(g, h) \<otimes>\<^bsub>(G \<times>\<^sub>g H)\<^esub> (g', h') = (g \<otimes>\<^bsub>G\<^esub> g', h \<otimes>\<^bsub>H\<^esub> h')"
   by (simp add: DirProdGroup_def DirProdSemigroup_def semigroup.defs)
 
 lemma inv_DirProdGroup [simp]:
   includes group G + group H
   assumes g: "g \<in> carrier G"
       and h: "h \<in> carrier H"
-  shows "m_inv (G \<times>\<^sub>g H) (g, h) = (m_inv G g, m_inv H h)"
+  shows "m_inv (G \<times>\<^sub>g H) (g, h) = (inv\<^bsub>G\<^esub> g, inv\<^bsub>H\<^esub> h)"
   apply (rule group.inv_equality [OF DirProdGroup_group])
   apply (simp_all add: prems group_def group.l_inv)
   done
@@ -572,7 +572,7 @@
   hom :: "_ => _ => ('a => 'b) set"
   "hom G H ==
     {h. h \<in> carrier G -> carrier H &
-      (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes> y) = h x \<otimes>\<^sub>2 h y)}"
+      (\<forall>x \<in> carrier G. \<forall>y \<in> carrier G. h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y)}"
 
 lemma (in semigroup) hom:
   includes semigroup G
@@ -586,9 +586,9 @@
 qed
 
 lemma hom_mult:
-  "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |] 
-   ==> h (mult G x y) = mult H (h x) (h y)"
-  by (simp add: hom_def) 
+  "[| h \<in> hom G H; x \<in> carrier G; y \<in> carrier G |]
+   ==> h (x \<otimes>\<^bsub>G\<^esub> y) = h x \<otimes>\<^bsub>H\<^esub> h y"
+  by (simp add: hom_def)
 
 lemma hom_closed:
   "[| h \<in> hom G H; x \<in> carrier G |] ==> h x \<in> carrier H"
@@ -598,9 +598,9 @@
      "[|group G; h \<in> hom G G; h' \<in> hom G G; h' \<in> carrier G -> carrier G|]
       ==> compose (carrier G) h h' \<in> hom G G"
 apply (simp (no_asm_simp) add: hom_def)
-apply (intro conjI) 
+apply (intro conjI)
  apply (force simp add: funcset_compose hom_def)
-apply (simp add: compose_def group.axioms hom_mult funcset_mem) 
+apply (simp add: compose_def group.axioms hom_mult funcset_mem)
 done
 
 locale group_hom = group G + group H + var h +
@@ -613,9 +613,9 @@
   by simp
 
 lemma (in group_hom) hom_one [simp]:
-  "h \<one> = \<one>\<^sub>2"
+  "h \<one> = \<one>\<^bsub>H\<^esub>"
 proof -
-  have "h \<one> \<otimes>\<^sub>2 \<one>\<^sub>2 = h \<one> \<otimes>\<^sub>2 h \<one>"
+  have "h \<one> \<otimes>\<^bsub>H\<^esub> \<one>\<^bsub>H\<^esub> = h \<one> \<otimes>\<^sub>2 h \<one>"
     by (simp add: hom_mult [symmetric] del: hom_mult)
   then show ?thesis by (simp del: r_one)
 qed
@@ -625,14 +625,14 @@
   by simp
 
 lemma (in group_hom) hom_inv [simp]:
-  "x \<in> carrier G ==> h (inv x) = inv\<^sub>2 (h x)"
+  "x \<in> carrier G ==> h (inv x) = inv\<^bsub>H\<^esub> (h x)"
 proof -
   assume x: "x \<in> carrier G"
-  then have "h x \<otimes>\<^sub>2 h (inv x) = \<one>\<^sub>2"
+  then have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = \<one>\<^bsub>H\<^esub>"
     by (simp add: hom_mult [symmetric] G.r_inv del: hom_mult)
-  also from x have "... = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)"
+  also from x have "... = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)"
     by (simp add: hom_mult [symmetric] H.r_inv del: hom_mult)
-  finally have "h x \<otimes>\<^sub>2 h (inv x) = h x \<otimes>\<^sub>2 inv\<^sub>2 (h x)" .
+  finally have "h x \<otimes>\<^bsub>H\<^esub> h (inv x) = h x \<otimes>\<^bsub>H\<^esub> inv\<^bsub>H\<^esub> (h x)" .
   with x show ?thesis by simp
 qed
 
@@ -653,7 +653,7 @@
   "[| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
    x \<otimes> (y \<otimes> z) = y \<otimes> (x \<otimes> z)"
 proof -
-  assume xyz: "x \<in> carrier G" "y \<in> carrier G" "z \<in> carrier G"
+  assume xyz: "x \<in> carrier G"  "y \<in> carrier G"  "z \<in> carrier G"
   from xyz have "x \<otimes> (y \<otimes> z) = (x \<otimes> y) \<otimes> z" by (simp add: m_assoc)
   also from xyz have "... = (y \<otimes> x) \<otimes> z" by (simp add: m_comm)
   also from xyz have "... = y \<otimes> (x \<otimes> z)" by (simp add: m_assoc)
@@ -665,15 +665,16 @@
 locale comm_monoid = comm_semigroup + monoid
 
 lemma comm_monoidI:
+  includes struct G
   assumes m_closed:
-      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
-    and one_closed: "one G \<in> carrier G"
+      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
+    and one_closed: "\<one> \<in> carrier G"
     and m_assoc:
       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
-      mult G (mult G x y) z = mult G x (mult G y z)"
-    and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
+      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
+    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
     and m_comm:
-      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
+      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
   shows "comm_monoid G"
   using l_one
   by (auto intro!: comm_monoid.intro magma.intro semigroup_axioms.intro
@@ -682,20 +683,17 @@
 
 lemma (in monoid) monoid_comm_monoidI:
   assumes m_comm:
-      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
+      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
   shows "comm_monoid G"
   by (rule comm_monoidI) (auto intro: m_assoc m_comm)
-(*
-lemma (in comm_monoid) r_one [simp]:
+(*lemma (in comm_monoid) r_one [simp]:
   "x \<in> carrier G ==> x \<otimes> \<one> = x"
 proof -
   assume G: "x \<in> carrier G"
   then have "x \<otimes> \<one> = \<one> \<otimes> x" by (simp add: m_comm)
   also from G have "... = x" by simp
   finally show ?thesis .
-qed
-*)
-
+qed*)
 lemma (in comm_monoid) nat_pow_distr:
   "[| x \<in> carrier G; y \<in> carrier G |] ==>
   (x \<otimes> y) (^) (n::nat) = x (^) n \<otimes> y (^) n"
@@ -705,22 +703,23 @@
 
 lemma (in group) group_comm_groupI:
   assumes m_comm: "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==>
-      mult G x y = mult G y x"
+      x \<otimes> y = y \<otimes> x"
   shows "comm_group G"
   by (fast intro: comm_group.intro comm_semigroup_axioms.intro
     group.axioms prems)
 
 lemma comm_groupI:
+  includes struct G
   assumes m_closed:
-      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y \<in> carrier G"
-    and one_closed: "one G \<in> carrier G"
+      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y \<in> carrier G"
+    and one_closed: "\<one> \<in> carrier G"
     and m_assoc:
       "!!x y z. [| x \<in> carrier G; y \<in> carrier G; z \<in> carrier G |] ==>
-      mult G (mult G x y) z = mult G x (mult G y z)"
+      (x \<otimes> y) \<otimes> z = x \<otimes> (y \<otimes> z)"
     and m_comm:
-      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> mult G x y = mult G y x"
-    and l_one: "!!x. x \<in> carrier G ==> mult G (one G) x = x"
-    and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. mult G y x = one G"
+      "!!x y. [| x \<in> carrier G; y \<in> carrier G |] ==> x \<otimes> y = y \<otimes> x"
+    and l_one: "!!x. x \<in> carrier G ==> \<one> \<otimes> x = x"
+    and l_inv_ex: "!!x. x \<in> carrier G ==> EX y : carrier G. y \<otimes> x = \<one>"
   shows "comm_group G"
   by (fast intro: group.group_comm_groupI groupI prems)
 
diff -r b8d6c395c9e2 -r 4deda204e1d8 src/HOL/Algebra/Lattice.thy
--- a/src/HOL/Algebra/Lattice.thy	Sat May 01 22:04:14 2004 +0200
+++ b/src/HOL/Algebra/Lattice.thy	Sat May 01 22:05:05 2004 +0200
@@ -14,9 +14,7 @@
 record 'a order = "'a partial_object" +
   le :: "['a, 'a] => bool" (infixl "\<sqsubseteq>\<index>" 50)
 
-locale order_syntax = struct L
-
-locale partial_order = order_syntax +
+locale partial_order = struct L +
   assumes refl [intro, simp]:
                   "x \<in> carrier L ==> x \<sqsubseteq> x"
     and anti_sym [intro]:
@@ -31,19 +29,19 @@
 
   -- {* Upper and lower bounds of a set. *}
   Upper :: "[_, 'a set] => 'a set"
-  "Upper L A == {u. (ALL x. x \<in> A \<inter> carrier L --> le L x u)} \<inter>
+  "Upper L A == {u. (ALL x. x \<in> A \<inter> carrier L --> x \<sqsubseteq> u)} \<inter>
                 carrier L"
 
   Lower :: "[_, 'a set] => 'a set"
-  "Lower L A == {l. (ALL x. x \<in> A \<inter> carrier L --> le L l x)} \<inter>
+  "Lower L A == {l. (ALL x. x \<in> A \<inter> carrier L --> l \<sqsubseteq> x)} \<inter>
                 carrier L"
 
   -- {* Least and greatest, as predicate. *}
   least :: "[_, 'a, 'a set] => bool"
-  "least L l A == A \<subseteq> carrier L & l \<in> A & (ALL x : A. le L l x)"
+  "least L l A == A \<subseteq> carrier L & l \<in> A & (ALL x : A. l \<sqsubseteq> x)"
 
   greatest :: "[_, 'a, 'a set] => bool"
-  "greatest L g A == A \<subseteq> carrier L & g \<in> A & (ALL x : A. le L x g)"
+  "greatest L g A == A \<subseteq> carrier L & g \<in> A & (ALL x : A. x \<sqsubseteq> g)"
 
   -- {* Supremum and infimum *}
   sup :: "[_, 'a set] => 'a" ("\<Squnion>\<index>_" [90] 90)
@@ -66,14 +64,14 @@
   by (unfold Upper_def) clarify
 
 lemma UpperD [dest]:
-  includes order_syntax
+  includes struct L
   shows "[| u \<in> Upper L A; x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> u"
-  by (unfold Upper_def) blast 
+  by (unfold Upper_def) blast
 
 lemma Upper_memI:
-  includes order_syntax
+  includes struct L
   shows "[| !! y. y \<in> A ==> y \<sqsubseteq> x; x \<in> carrier L |] ==> x \<in> Upper L A"
-  by (unfold Upper_def) blast 
+  by (unfold Upper_def) blast
 
 lemma Upper_antimono:
   "A \<subseteq> B ==> Upper L B \<subseteq> Upper L A"
@@ -87,14 +85,14 @@
   by (unfold Lower_def) clarify
 
 lemma LowerD [dest]:
-  includes order_syntax
+  includes struct L
   shows "[| l \<in> Lower L A; x \<in> A; A \<subseteq> carrier L |] ==> l \<sqsubseteq> x"
-  by (unfold Lower_def) blast 
+  by (unfold Lower_def) blast
 
 lemma Lower_memI:
-  includes order_syntax
+  includes struct L
   shows "[| !! y. y \<in> A ==> x \<sqsubseteq> y; x \<in> carrier L |] ==> x \<in> Lower L A"
-  by (unfold Lower_def) blast 
+  by (unfold Lower_def) blast
 
 lemma Lower_antimono:
   "A \<subseteq> B ==> Lower L B \<subseteq> Lower L A"
@@ -116,22 +114,21 @@
   by (unfold least_def) blast
 
 lemma least_le:
-  includes order_syntax
+  includes struct L
   shows "[| least L x A; a \<in> A |] ==> x \<sqsubseteq> a"
   by (unfold least_def) fast
 
 lemma least_UpperI:
-  includes order_syntax
+  includes struct L
   assumes above: "!! x. x \<in> A ==> x \<sqsubseteq> s"
     and below: "!! y. y \<in> Upper L A ==> s \<sqsubseteq> y"
-    and L: "A \<subseteq> carrier L" "s \<in> carrier L"
+    and L: "A \<subseteq> carrier L"  "s \<in> carrier L"
   shows "least L s (Upper L A)"
-proof (unfold least_def, intro conjI)
-  show "Upper L A \<subseteq> carrier L" by simp
-next
-  from above L show "s \<in> Upper L A" by (simp add: Upper_def)
-next
-  from below show "ALL x : Upper L A. s \<sqsubseteq> x" by fast
+proof -
+  have "Upper L A \<subseteq> carrier L" by simp
+  moreover from above L have "s \<in> Upper L A" by (simp add: Upper_def)
+  moreover from below have "ALL x : Upper L A. s \<sqsubseteq> x" by fast
+  ultimately show ?thesis by (simp add: least_def)
 qed
 
 
@@ -150,24 +147,24 @@
   by (unfold greatest_def) blast
 
 lemma greatest_le:
-  includes order_syntax
+  includes struct L
   shows "[| greatest L x A; a \<in> A |] ==> a \<sqsubseteq> x"
   by (unfold greatest_def) fast
 
 lemma greatest_LowerI:
-  includes order_syntax
+  includes struct L
   assumes below: "!! x. x \<in> A ==> i \<sqsubseteq> x"
     and above: "!! y. y \<in> Lower L A ==> y \<sqsubseteq> i"
-    and L: "A \<subseteq> carrier L" "i \<in> carrier L"
+    and L: "A \<subseteq> carrier L"  "i \<in> carrier L"
   shows "greatest L i (Lower L A)"
-proof (unfold greatest_def, intro conjI)
-  show "Lower L A \<subseteq> carrier L" by simp
-next
-  from below L show "i \<in> Lower L A" by (simp add: Lower_def)
-next
-  from above show "ALL x : Lower L A. x \<sqsubseteq> i" by fast
+proof -
+  have "Lower L A \<subseteq> carrier L" by simp
+  moreover from below L have "i \<in> Lower L A" by (simp add: Lower_def)
+  moreover from above have "ALL x : Lower L A. x \<sqsubseteq> i" by fast
+  ultimately show ?thesis by (simp add: greatest_def)
 qed
 
+
 subsection {* Lattices *}
 
 locale lattice = partial_order +
@@ -177,12 +174,12 @@
     "[| x \<in> carrier L; y \<in> carrier L |] ==> EX s. greatest L s (Lower L {x, y})"
 
 lemma least_Upper_above:
-  includes order_syntax
+  includes struct L
   shows "[| least L s (Upper L A); x \<in> A; A \<subseteq> carrier L |] ==> x \<sqsubseteq> s"
   by (unfold least_def) blast
 
 lemma greatest_Lower_above:
-  includes order_syntax
+  includes struct L
   shows "[| greatest L i (Lower L A); x \<in> A; A \<subseteq> carrier L |] ==> i \<sqsubseteq> x"
   by (unfold greatest_def) blast
 
@@ -193,11 +190,11 @@
   "[| !!l. least L l (Upper L {x, y}) ==> P l; x \<in> carrier L; y \<in> carrier L |]
   ==> P (x \<squnion> y)"
 proof (unfold join_def sup_def)
-  assume L: "x \<in> carrier L" "y \<in> carrier L"
+  assume L: "x \<in> carrier L"  "y \<in> carrier L"
     and P: "!!l. least L l (Upper L {x, y}) ==> P l"
   with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
   with L show "P (THE l. least L l (Upper L {x, y}))"
-  by (fast intro: theI2 least_unique P)
+    by (fast intro: theI2 least_unique P)
 qed
 
 lemma (in lattice) join_closed [simp]:
@@ -209,8 +206,8 @@
   by (rule least_UpperI) fast+
 
 lemma (in partial_order) sup_of_singleton [simp]:
-  includes order_syntax
-  shows "x \<in> carrier L ==> \<Squnion> {x} = x"
+  includes struct L
+  shows "x \<in> carrier L ==> \<Squnion>{x} = x"
   by (unfold sup_def) (blast intro: least_unique least_UpperI sup_of_singletonI)
 
 
@@ -219,129 +216,104 @@
 lemma (in lattice) sup_insertI:
   "[| !!s. least L s (Upper L (insert x A)) ==> P s;
   least L a (Upper L A); x \<in> carrier L; A \<subseteq> carrier L |]
-  ==> P (\<Squnion> (insert x A))"
+  ==> P (\<Squnion>(insert x A))"
 proof (unfold sup_def)
-  assume L: "x \<in> carrier L" "A \<subseteq> carrier L"
+  assume L: "x \<in> carrier L"  "A \<subseteq> carrier L"
     and P: "!!l. least L l (Upper L (insert x A)) ==> P l"
     and least_a: "least L a (Upper L A)"
   from L least_a have La: "a \<in> carrier L" by simp
   from L sup_of_two_exists least_a
   obtain s where least_s: "least L s (Upper L {a, x})" by blast
   show "P (THE l. least L l (Upper L (insert x A)))"
-  proof (rule theI2 [where a = s])
+  proof (rule theI2)
     show "least L s (Upper L (insert x A))"
     proof (rule least_UpperI)
       fix z
-      assume xA: "z \<in> insert x A"
-      show "z \<sqsubseteq> s"
-      proof -
-	{
-	  assume "z = x" then have ?thesis
-	    by (simp add: least_Upper_above [OF least_s] L La)
-        }
-	moreover
-        {
-	  assume "z \<in> A"
-          with L least_s least_a have ?thesis
-	    by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
-        }
-      moreover note xA
-      ultimately show ?thesis by blast
+      assume "z \<in> insert x A"
+      then show "z \<sqsubseteq> s"
+      proof
+        assume "z = x" then show ?thesis
+          by (simp add: least_Upper_above [OF least_s] L La)
+      next
+        assume "z \<in> A"
+        with L least_s least_a show ?thesis
+          by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
+      qed
+    next
+      fix y
+      assume y: "y \<in> Upper L (insert x A)"
+      show "s \<sqsubseteq> y"
+      proof (rule least_le [OF least_s], rule Upper_memI)
+	fix z
+	assume z: "z \<in> {a, x}"
+	then show "z \<sqsubseteq> y"
+	proof
+          have y': "y \<in> Upper L A"
+            apply (rule subsetD [where A = "Upper L (insert x A)"])
+            apply (rule Upper_antimono) apply clarify apply assumption
+            done
+          assume "z = a"
+          with y' least_a show ?thesis by (fast dest: least_le)
+	next
+	  assume "z \<in> {x}"  (* FIXME "z = x"; declare specific elim rule for "insert x {}" (!?) *)
+          with y L show ?thesis by blast
+	qed
+      qed (rule Upper_closed [THEN subsetD])
+    next
+      from L show "insert x A \<subseteq> carrier L" by simp
+      from least_s show "s \<in> carrier L" by simp
     qed
   next
-    fix y
-    assume y: "y \<in> Upper L (insert x A)"
-    show "s \<sqsubseteq> y"
-    proof (rule least_le [OF least_s], rule Upper_memI)
-      fix z
-      assume z: "z \<in> {a, x}"
-      show "z \<sqsubseteq> y"
-      proof -
-	{
-          have y': "y \<in> Upper L A"
-	    apply (rule subsetD [where A = "Upper L (insert x A)"])
-	    apply (rule Upper_antimono) apply clarify apply assumption
-	    done
-	  assume "z = a"
-	  with y' least_a have ?thesis by (fast dest: least_le)
-        }
-	moreover
-	{
-           assume "z = x"
-           with y L have ?thesis by blast
-        }
-        moreover note z
-        ultimately show ?thesis by blast
-      qed
-    qed (rule Upper_closed [THEN subsetD])
-  next
-    from L show "insert x A \<subseteq> carrier L" by simp
-  next
-    from least_s show "s \<in> carrier L" by simp
-  qed
-next
     fix l
     assume least_l: "least L l (Upper L (insert x A))"
     show "l = s"
     proof (rule least_unique)
       show "least L s (Upper L (insert x A))"
       proof (rule least_UpperI)
-	fix z
-	assume xA: "z \<in> insert x A"
-	show "z \<sqsubseteq> s"
-      proof -
-	{
-	  assume "z = x" then have ?thesis
-	    by (simp add: least_Upper_above [OF least_s] L La)
-        }
-	moreover
-        {
-	  assume "z \<in> A"
-          with L least_s least_a have ?thesis
-	    by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
-        }
-	  moreover note xA
-	  ultimately show ?thesis by blast
+        fix z
+        assume "z \<in> insert x A"
+        then show "z \<sqsubseteq> s"
+	proof
+          assume "z = x" then show ?thesis
+            by (simp add: least_Upper_above [OF least_s] L La)
+	next
+          assume "z \<in> A"
+          with L least_s least_a show ?thesis
+            by (rule_tac trans [where y = a]) (auto dest: least_Upper_above)
 	qed
       next
-	fix y
-	assume y: "y \<in> Upper L (insert x A)"
-	show "s \<sqsubseteq> y"
-	proof (rule least_le [OF least_s], rule Upper_memI)
-	  fix z
-	  assume z: "z \<in> {a, x}"
-	  show "z \<sqsubseteq> y"
-	  proof -
-	    {
-          have y': "y \<in> Upper L A"
-	    apply (rule subsetD [where A = "Upper L (insert x A)"])
-	    apply (rule Upper_antimono) apply clarify apply assumption
-	    done
-	  assume "z = a"
-	  with y' least_a have ?thesis by (fast dest: least_le)
-        }
-	moreover
-	{
-           assume "z = x"
-           with y L have ?thesis by blast
-            }
-            moreover note z
-            ultimately show ?thesis by blast
-	  qed
-	qed (rule Upper_closed [THEN subsetD])
+        fix y
+        assume y: "y \<in> Upper L (insert x A)"
+        show "s \<sqsubseteq> y"
+        proof (rule least_le [OF least_s], rule Upper_memI)
+          fix z
+          assume z: "z \<in> {a, x}"
+          then show "z \<sqsubseteq> y"
+          proof
+            have y': "y \<in> Upper L A"
+	      apply (rule subsetD [where A = "Upper L (insert x A)"])
+	      apply (rule Upper_antimono) apply clarify apply assumption
+	      done
+            assume "z = a"
+            with y' least_a show ?thesis by (fast dest: least_le)
+	  next
+            assume "z \<in> {x}"
+            with y L show ?thesis by blast
+          qed
+        qed (rule Upper_closed [THEN subsetD])
       next
-	from L show "insert x A \<subseteq> carrier L" by simp
-      next
-	from least_s show "s \<in> carrier L" by simp
+        from L show "insert x A \<subseteq> carrier L" by simp
+        from least_s show "s \<in> carrier L" by simp
       qed
     qed
   qed
 qed
 
 lemma (in lattice) finite_sup_least:
-  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion> A) (Upper L A)"
+  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> least L (\<Squnion>A) (Upper L A)"
 proof (induct set: Finites)
-  case empty then show ?case by simp
+  case empty
+  then show ?case by simp
 next
   case (insert A x)
   show ?case
@@ -350,16 +322,15 @@
     with insert show ?thesis by (simp add: sup_of_singletonI)
   next
     case False
-    from insert show ?thesis
-    proof (rule_tac sup_insertI)
-      from False insert show "least L (\<Squnion> A) (Upper L A)" by simp
-    qed simp_all
+    with insert have "least L (\<Squnion>A) (Upper L A)" by simp
+    with _ show ?thesis
+      by (rule sup_insertI) (simp_all add: insert [simplified])
   qed
 qed
 
 lemma (in lattice) finite_sup_insertI:
   assumes P: "!!l. least L l (Upper L (insert x A)) ==> P l"
-    and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L"
+    and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
   shows "P (\<Squnion> (insert x A))"
 proof (cases "A = {}")
   case True with P and xA show ?thesis
@@ -370,44 +341,44 @@
 qed
 
 lemma (in lattice) finite_sup_closed:
-  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion> A \<in> carrier L"
+  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Squnion>A \<in> carrier L"
 proof (induct set: Finites)
   case empty then show ?case by simp
 next
   case (insert A x) then show ?case
-    by (rule_tac finite_sup_insertI) (simp_all)
+    by - (rule finite_sup_insertI, simp_all)
 qed
 
 lemma (in lattice) join_left:
   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqsubseteq> x \<squnion> y"
-  by (rule joinI [folded join_def]) (blast dest: least_mem )
+  by (rule joinI [folded join_def]) (blast dest: least_mem)
 
 lemma (in lattice) join_right:
   "[| x \<in> carrier L; y \<in> carrier L |] ==> y \<sqsubseteq> x \<squnion> y"
-  by (rule joinI [folded join_def]) (blast dest: least_mem )
+  by (rule joinI [folded join_def]) (blast dest: least_mem)
 
 lemma (in lattice) sup_of_two_least:
-  "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion> {x, y}) (Upper L {x, y})"
+  "[| x \<in> carrier L; y \<in> carrier L |] ==> least L (\<Squnion>{x, y}) (Upper L {x, y})"
 proof (unfold sup_def)
-  assume L: "x \<in> carrier L" "y \<in> carrier L"
+  assume L: "x \<in> carrier L"  "y \<in> carrier L"
   with sup_of_two_exists obtain s where "least L s (Upper L {x, y})" by fast
   with L show "least L (THE xa. least L xa (Upper L {x, y})) (Upper L {x, y})"
   by (fast intro: theI2 least_unique)  (* blast fails *)
 qed
 
 lemma (in lattice) join_le:
-  assumes sub: "x \<sqsubseteq> z" "y \<sqsubseteq> z"
-    and L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+  assumes sub: "x \<sqsubseteq> z"  "y \<sqsubseteq> z"
+    and L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
   shows "x \<squnion> y \<sqsubseteq> z"
 proof (rule joinI)
   fix s
   assume "least L s (Upper L {x, y})"
   with sub L show "s \<sqsubseteq> z" by (fast elim: least_le intro: Upper_memI)
 qed
-  
+
 lemma (in lattice) join_assoc_lemma:
-  assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
-  shows "x \<squnion> (y \<squnion> z) = \<Squnion> {x, y, z}"
+  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
+  shows "x \<squnion> (y \<squnion> z) = \<Squnion>{x, y, z}"
 proof (rule finite_sup_insertI)
   -- {* The textbook argument in Jacobson I, p 457 *}
   fix s
@@ -424,21 +395,22 @@
 qed (simp_all add: L)
 
 lemma join_comm:
-  includes order_syntax
+  includes struct L
   shows "x \<squnion> y = y \<squnion> x"
   by (unfold join_def) (simp add: insert_commute)
 
 lemma (in lattice) join_assoc:
-  assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
   shows "(x \<squnion> y) \<squnion> z = x \<squnion> (y \<squnion> z)"
 proof -
   have "(x \<squnion> y) \<squnion> z = z \<squnion> (x \<squnion> y)" by (simp only: join_comm)
-  also from L have "... = \<Squnion> {z, x, y}" by (simp add: join_assoc_lemma)
-  also from L have "... = \<Squnion> {x, y, z}" by (simp add: insert_commute)
+  also from L have "... = \<Squnion>{z, x, y}" by (simp add: join_assoc_lemma)
+  also from L have "... = \<Squnion>{x, y, z}" by (simp add: insert_commute)
   also from L have "... = x \<squnion> (y \<squnion> z)" by (simp add: join_assoc_lemma)
   finally show ?thesis .
 qed
 
+
 subsubsection {* Infimum *}
 
 lemma (in lattice) meetI:
@@ -446,7 +418,7 @@
   x \<in> carrier L; y \<in> carrier L |]
   ==> P (x \<sqinter> y)"
 proof (unfold meet_def inf_def)
-  assume L: "x \<in> carrier L" "y \<in> carrier L"
+  assume L: "x \<in> carrier L"  "y \<in> carrier L"
     and P: "!!g. greatest L g (Lower L {x, y}) ==> P g"
   with inf_of_two_exists obtain i where "greatest L i (Lower L {x, y})" by fast
   with L show "P (THE g. greatest L g (Lower L {x, y}))"
@@ -462,7 +434,7 @@
   by (rule greatest_LowerI) fast+
 
 lemma (in partial_order) inf_of_singleton [simp]:
-  includes order_syntax
+  includes struct L
   shows "x \<in> carrier L ==> \<Sqinter> {x} = x"
   by (unfold inf_def) (blast intro: greatest_unique greatest_LowerI inf_of_singletonI)
 
@@ -471,127 +443,101 @@
 lemma (in lattice) inf_insertI:
   "[| !!i. greatest L i (Lower L (insert x A)) ==> P i;
   greatest L a (Lower L A); x \<in> carrier L; A \<subseteq> carrier L |]
-  ==> P (\<Sqinter> (insert x A))"
+  ==> P (\<Sqinter>(insert x A))"
 proof (unfold inf_def)
-  assume L: "x \<in> carrier L" "A \<subseteq> carrier L"
+  assume L: "x \<in> carrier L"  "A \<subseteq> carrier L"
     and P: "!!g. greatest L g (Lower L (insert x A)) ==> P g"
     and greatest_a: "greatest L a (Lower L A)"
   from L greatest_a have La: "a \<in> carrier L" by simp
   from L inf_of_two_exists greatest_a
   obtain i where greatest_i: "greatest L i (Lower L {a, x})" by blast
   show "P (THE g. greatest L g (Lower L (insert x A)))"
-  proof (rule theI2 [where a = i])
+  proof (rule theI2)
     show "greatest L i (Lower L (insert x A))"
     proof (rule greatest_LowerI)
       fix z
-      assume xA: "z \<in> insert x A"
-      show "i \<sqsubseteq> z"
-      proof -
-	{
-	  assume "z = x" then have ?thesis
-	    by (simp add: greatest_Lower_above [OF greatest_i] L La)
-        }
-	moreover
-        {
-	  assume "z \<in> A"
-          with L greatest_i greatest_a have ?thesis
-	    by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
-        }
-      moreover note xA
-      ultimately show ?thesis by blast
+      assume "z \<in> insert x A"
+      then show "i \<sqsubseteq> z"
+      proof
+        assume "z = x" then show ?thesis
+          by (simp add: greatest_Lower_above [OF greatest_i] L La)
+      next
+        assume "z \<in> A"
+        with L greatest_i greatest_a show ?thesis
+          by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
+      qed
+    next
+      fix y
+      assume y: "y \<in> Lower L (insert x A)"
+      show "y \<sqsubseteq> i"
+      proof (rule greatest_le [OF greatest_i], rule Lower_memI)
+	fix z
+	assume z: "z \<in> {a, x}"
+	then show "y \<sqsubseteq> z"
+	proof
+          have y': "y \<in> Lower L A"
+            apply (rule subsetD [where A = "Lower L (insert x A)"])
+            apply (rule Lower_antimono) apply clarify apply assumption
+            done
+          assume "z = a"
+          with y' greatest_a show ?thesis by (fast dest: greatest_le)
+	next
+          assume "z \<in> {x}"
+          with y L show ?thesis by blast
+	qed
+      qed (rule Lower_closed [THEN subsetD])
+    next
+      from L show "insert x A \<subseteq> carrier L" by simp
+      from greatest_i show "i \<in> carrier L" by simp
     qed
   next
-    fix y
-    assume y: "y \<in> Lower L (insert x A)"
-    show "y \<sqsubseteq> i"
-    proof (rule greatest_le [OF greatest_i], rule Lower_memI)
-      fix z
-      assume z: "z \<in> {a, x}"
-      show "y \<sqsubseteq> z"
-      proof -
-	{
-          have y': "y \<in> Lower L A"
-	    apply (rule subsetD [where A = "Lower L (insert x A)"])
-	    apply (rule Lower_antimono) apply clarify apply assumption
-	    done
-	  assume "z = a"
-	  with y' greatest_a have ?thesis by (fast dest: greatest_le)
-        }
-	moreover
-	{
-           assume "z = x"
-           with y L have ?thesis by blast
-        }
-        moreover note z
-        ultimately show ?thesis by blast
-      qed
-    qed (rule Lower_closed [THEN subsetD])
-  next
-    from L show "insert x A \<subseteq> carrier L" by simp
-  next
-    from greatest_i show "i \<in> carrier L" by simp
-  qed
-next
     fix g
     assume greatest_g: "greatest L g (Lower L (insert x A))"
     show "g = i"
     proof (rule greatest_unique)
       show "greatest L i (Lower L (insert x A))"
       proof (rule greatest_LowerI)
-	fix z
-	assume xA: "z \<in> insert x A"
-	show "i \<sqsubseteq> z"
-      proof -
-	{
-	  assume "z = x" then have ?thesis
-	    by (simp add: greatest_Lower_above [OF greatest_i] L La)
-        }
-	moreover
-        {
-	  assume "z \<in> A"
-          with L greatest_i greatest_a have ?thesis
-	    by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
-        }
-	  moreover note xA
-	  ultimately show ?thesis by blast
-	qed
+        fix z
+        assume "z \<in> insert x A"
+        then show "i \<sqsubseteq> z"
+	proof
+          assume "z = x" then show ?thesis
+            by (simp add: greatest_Lower_above [OF greatest_i] L La)
+	next
+          assume "z \<in> A"
+          with L greatest_i greatest_a show ?thesis
+            by (rule_tac trans [where y = a]) (auto dest: greatest_Lower_above)
+        qed
       next
-	fix y
-	assume y: "y \<in> Lower L (insert x A)"
-	show "y \<sqsubseteq> i"
-	proof (rule greatest_le [OF greatest_i], rule Lower_memI)
-	  fix z
-	  assume z: "z \<in> {a, x}"
-	  show "y \<sqsubseteq> z"
-	  proof -
-	    {
-          have y': "y \<in> Lower L A"
-	    apply (rule subsetD [where A = "Lower L (insert x A)"])
-	    apply (rule Lower_antimono) apply clarify apply assumption
-	    done
-	  assume "z = a"
-	  with y' greatest_a have ?thesis by (fast dest: greatest_le)
-        }
-	moreover
-	{
-           assume "z = x"
-           with y L have ?thesis by blast
-            }
-            moreover note z
-            ultimately show ?thesis by blast
+        fix y
+        assume y: "y \<in> Lower L (insert x A)"
+        show "y \<sqsubseteq> i"
+        proof (rule greatest_le [OF greatest_i], rule Lower_memI)
+          fix z
+          assume z: "z \<in> {a, x}"
+          then show "y \<sqsubseteq> z"
+          proof
+            have y': "y \<in> Lower L A"
+	      apply (rule subsetD [where A = "Lower L (insert x A)"])
+	      apply (rule Lower_antimono) apply clarify apply assumption
+	      done
+            assume "z = a"
+            with y' greatest_a show ?thesis by (fast dest: greatest_le)
+	  next
+            assume "z \<in> {x}"
+            with y L show ?thesis by blast
 	  qed
-	qed (rule Lower_closed [THEN subsetD])
+        qed (rule Lower_closed [THEN subsetD])
       next
-	from L show "insert x A \<subseteq> carrier L" by simp
-      next
-	from greatest_i show "i \<in> carrier L" by simp
+        from L show "insert x A \<subseteq> carrier L" by simp
+        from greatest_i show "i \<in> carrier L" by simp
       qed
     qed
   qed
 qed
 
 lemma (in lattice) finite_inf_greatest:
-  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter> A) (Lower L A)"
+  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> greatest L (\<Sqinter>A) (Lower L A)"
 proof (induct set: Finites)
   case empty then show ?case by simp
 next
@@ -604,14 +550,14 @@
     case False
     from insert show ?thesis
     proof (rule_tac inf_insertI)
-      from False insert show "greatest L (\<Sqinter> A) (Lower L A)" by simp
+      from False insert show "greatest L (\<Sqinter>A) (Lower L A)" by simp
     qed simp_all
   qed
 qed
 
 lemma (in lattice) finite_inf_insertI:
   assumes P: "!!i. greatest L i (Lower L (insert x A)) ==> P i"
-    and xA: "finite A" "x \<in> carrier L" "A \<subseteq> carrier L"
+    and xA: "finite A"  "x \<in> carrier L"  "A \<subseteq> carrier L"
   shows "P (\<Sqinter> (insert x A))"
 proof (cases "A = {}")
   case True with P and xA show ?thesis
@@ -622,7 +568,7 @@
 qed
 
 lemma (in lattice) finite_inf_closed:
-  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter> A \<in> carrier L"
+  "[| finite A; A \<subseteq> carrier L; A ~= {} |] ==> \<Sqinter>A \<in> carrier L"
 proof (induct set: Finites)
   case empty then show ?case by simp
 next
@@ -632,17 +578,17 @@
 
 lemma (in lattice) meet_left:
   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> x"
-  by (rule meetI [folded meet_def]) (blast dest: greatest_mem )
+  by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
 
 lemma (in lattice) meet_right:
   "[| x \<in> carrier L; y \<in> carrier L |] ==> x \<sqinter> y \<sqsubseteq> y"
-  by (rule meetI [folded meet_def]) (blast dest: greatest_mem )
+  by (rule meetI [folded meet_def]) (blast dest: greatest_mem)
 
 lemma (in lattice) inf_of_two_greatest:
   "[| x \<in> carrier L; y \<in> carrier L |] ==>
   greatest L (\<Sqinter> {x, y}) (Lower L {x, y})"
 proof (unfold inf_def)
-  assume L: "x \<in> carrier L" "y \<in> carrier L"
+  assume L: "x \<in> carrier L"  "y \<in> carrier L"
   with inf_of_two_exists obtain s where "greatest L s (Lower L {x, y})" by fast
   with L
   show "greatest L (THE xa. greatest L xa (Lower L {x, y})) (Lower L {x, y})"
@@ -650,18 +596,18 @@
 qed
 
 lemma (in lattice) meet_le:
-  assumes sub: "z \<sqsubseteq> x" "z \<sqsubseteq> y"
-    and L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+  assumes sub: "z \<sqsubseteq> x"  "z \<sqsubseteq> y"
+    and L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
   shows "z \<sqsubseteq> x \<sqinter> y"
 proof (rule meetI)
   fix i
   assume "greatest L i (Lower L {x, y})"
   with sub L show "z \<sqsubseteq> i" by (fast elim: greatest_le intro: Lower_memI)
 qed
-  
+
 lemma (in lattice) meet_assoc_lemma:
-  assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
-  shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter> {x, y, z}"
+  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
+  shows "x \<sqinter> (y \<sqinter> z) = \<Sqinter>{x, y, z}"
 proof (rule finite_inf_insertI)
   txt {* The textbook argument in Jacobson I, p 457 *}
   fix i
@@ -678,12 +624,12 @@
 qed (simp_all add: L)
 
 lemma meet_comm:
-  includes order_syntax
+  includes struct L
   shows "x \<sqinter> y = y \<sqinter> x"
   by (unfold meet_def) (simp add: insert_commute)
 
 lemma (in lattice) meet_assoc:
-  assumes L: "x \<in> carrier L" "y \<in> carrier L" "z \<in> carrier L"
+  assumes L: "x \<in> carrier L"  "y \<in> carrier L"  "z \<in> carrier L"
   shows "(x \<sqinter> y) \<sqinter> z = x \<sqinter> (y \<sqinter> z)"
 proof -
   have "(x \<sqinter> y) \<sqinter> z = z \<sqinter> (x \<sqinter> y)" by (simp only: meet_comm)
@@ -693,6 +639,7 @@
   finally show ?thesis .
 qed
 
+
 subsection {* Total Orders *}
 
 locale total_order = lattice +
@@ -707,47 +654,48 @@
   show "lattice_axioms L"
   proof (rule lattice_axioms.intro)
     fix x y
-    assume L: "x \<in> carrier L" "y \<in> carrier L"
+    assume L: "x \<in> carrier L"  "y \<in> carrier L"
     show "EX s. least L s (Upper L {x, y})"
     proof -
       note total L
       moreover
       {
-	assume "x \<sqsubseteq> y"
+        assume "x \<sqsubseteq> y"
         with L have "least L y (Upper L {x, y})"
-	  by (rule_tac least_UpperI) auto
+          by (rule_tac least_UpperI) auto
       }
       moreover
       {
-	assume "y \<sqsubseteq> x"
+        assume "y \<sqsubseteq> x"
         with L have "least L x (Upper L {x, y})"
-	  by (rule_tac least_UpperI) auto
+          by (rule_tac least_UpperI) auto
       }
       ultimately show ?thesis by blast
     qed
   next
     fix x y
-    assume L: "x \<in> carrier L" "y \<in> carrier L"
+    assume L: "x \<in> carrier L"  "y \<in> carrier L"
     show "EX i. greatest L i (Lower L {x, y})"
     proof -
       note total L
       moreover
       {
-	assume "y \<sqsubseteq> x"
+        assume "y \<sqsubseteq> x"
         with L have "greatest L y (Lower L {x, y})"
-	  by (rule_tac greatest_LowerI) auto
+          by (rule_tac greatest_LowerI) auto
       }
       moreover
       {
-	assume "x \<sqsubseteq> y"
+        assume "x \<sqsubseteq> y"
         with L have "greatest L x (Lower L {x, y})"
-	  by (rule_tac greatest_LowerI) auto
+          by (rule_tac greatest_LowerI) auto
       }
       ultimately show ?thesis by blast
     qed
   qed
 qed (assumption | rule total_order_axioms.intro)+
 
+
 subsection {* Complete lattices *}
 
 locale complete_lattice = lattice +
@@ -766,7 +714,7 @@
   shows "complete_lattice L"
 proof (rule complete_lattice.intro)
   show "lattice_axioms L"
-  by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
+    by (rule lattice_axioms.intro) (blast intro: sup_exists inf_exists)+
 qed (assumption | rule complete_lattice_axioms.intro)+
 
 constdefs (structure L)
@@ -789,7 +737,7 @@
 qed
 
 lemma (in complete_lattice) sup_closed [simp]:
-  "A \<subseteq> carrier L ==> \<Squnion> A \<in> carrier L"
+  "A \<subseteq> carrier L ==> \<Squnion>A \<in> carrier L"
   by (rule supI) simp_all
 
 lemma (in complete_lattice) top_closed [simp, intro]:
@@ -798,7 +746,7 @@
 
 lemma (in complete_lattice) infI:
   "[| !!i. greatest L i (Lower L A) ==> P i; A \<subseteq> carrier L |]
-  ==> P (\<Sqinter> A)"
+  ==> P (\<Sqinter>A)"
 proof (unfold inf_def)
   assume L: "A \<subseteq> carrier L"
     and P: "!!l. greatest L l (Lower L A) ==> P l"
@@ -808,7 +756,7 @@
 qed
 
 lemma (in complete_lattice) inf_closed [simp]:
-  "A \<subseteq> carrier L ==> \<Sqinter> A \<in> carrier L"
+  "A \<subseteq> carrier L ==> \<Sqinter>A \<in> carrier L"
   by (rule infI) simp_all
 
 lemma (in complete_lattice) bottom_closed [simp, intro]:
@@ -842,7 +790,7 @@
   obtain b where b_inf_B: "greatest L b (Lower L ?B)" ..
   have "least L b (Upper L A)"
 apply (rule least_UpperI)
-   apply (rule greatest_le [where A = "Lower L ?B"]) 
+   apply (rule greatest_le [where A = "Lower L ?B"])
     apply (rule b_inf_B)
    apply (rule Lower_memI)
     apply (erule UpperD)
@@ -942,7 +890,7 @@
   with subgr [THEN subgroup.m_inv_closed]
   show "inv x \<in> \<Inter>A" by blast
 next
-  fix x y assume "x \<in> \<Inter>A" "y \<in> \<Inter>A"
+  fix x y assume "x \<in> \<Inter>A"  "y \<in> \<Inter>A"
   with subgr [THEN subgroup.m_closed]
   show "x \<otimes> y \<in> \<Inter>A" by blast
 qed
@@ -986,4 +934,4 @@
   then show "EX I. greatest ?L I (Lower ?L A)" ..
 qed
 
-end
\ No newline at end of file
+end