# HG changeset patch
# User huffman
# Date 1237451359 25200
# Node ID 638b088bb84076eced4df06a90fd4c5e4a2a4980
# Parent  ac50e7dedf6d2b5f32182f18a23f17839162d02d
imported patch euclidean

diff -r ac50e7dedf6d -r 638b088bb840 src/HOL/Library/Determinants.thy
--- a/src/HOL/Library/Determinants.thy	Wed Mar 18 22:17:23 2009 +0100
+++ b/src/HOL/Library/Determinants.thy	Thu Mar 19 01:29:19 2009 -0700
@@ -68,22 +68,22 @@
 subsection{* Trace *}
 
 definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a" where
-  "trace A = setsum (\<lambda>i. ((A$i)$i)) {1..dimindex(UNIV::'n set)}"
+  "trace A = setsum (\<lambda>i. ((A$i)$i)) (UNIV::'n set)"
 
 lemma trace_0: "trace(mat 0) = 0"
-  by (simp add: trace_def mat_def Cart_lambda_beta setsum_0)
+  by (simp add: trace_def mat_def)
 
-lemma trace_I: "trace(mat 1 :: 'a::semiring_1^'n^'n) = of_nat(dimindex(UNIV::'n set))"
-  by (simp add: trace_def mat_def Cart_lambda_beta)
+lemma trace_I: "trace(mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))"
+  by (simp add: trace_def mat_def)
 
 lemma trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
-  by (simp add: trace_def setsum_addf Cart_lambda_beta vector_component)
+  by (simp add: trace_def setsum_addf)
 
 lemma trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
-  by (simp add: trace_def setsum_subtractf Cart_lambda_beta vector_component)
+  by (simp add: trace_def setsum_subtractf)
 
 lemma trace_mul_sym:"trace ((A::'a::comm_semiring_1^'n^'n) ** B) = trace (B**A)"
-  apply (simp add: trace_def matrix_matrix_mult_def Cart_lambda_beta)
+  apply (simp add: trace_def matrix_matrix_mult_def)
   apply (subst setsum_commute)
   by (simp add: mult_commute)
 
@@ -92,18 +92,12 @@
 (* ------------------------------------------------------------------------- *)
 
 definition det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
-  "det A = setsum (\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) {1 .. dimindex(UNIV :: 'n set)}) {p. p permutes {1 .. dimindex(UNIV :: 'n set)}}"
+  "det A = setsum (\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)) {p. p permutes (UNIV :: 'n set)}"
 
 (* ------------------------------------------------------------------------- *)
 (* A few general lemmas we need below.                                       *)
 (* ------------------------------------------------------------------------- *)
 
-lemma Cart_lambda_beta_perm: assumes p: "p permutes {1..dimindex(UNIV::'n set)}"
-  and i: "i \<in> {1..dimindex(UNIV::'n set)}"
-  shows "Cart_nth (Cart_lambda g ::'a^'n) (p i) = g(p i)"
-  using permutes_in_image[OF p] i
-  by (simp add:  Cart_lambda_beta permutes_in_image[OF p])
-
 lemma setprod_permute:
   assumes p: "p permutes S"
   shows "setprod f S = setprod (f o p) S"
@@ -127,11 +121,11 @@
 (* Basic determinant properties.                                             *)
 (* ------------------------------------------------------------------------- *)
 
-lemma det_transp: "det (transp A) = det (A::'a::comm_ring_1 ^'n^'n)"
+lemma det_transp: "det (transp A) = det (A::'a::comm_ring_1 ^'n^'n::finite)"
 proof-
   let ?di = "\<lambda>A i j. A$i$j"
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
-  have fU: "finite ?U" by blast
+  let ?U = "(UNIV :: 'n set)"
+  have fU: "finite ?U" by simp
   {fix p assume p: "p \<in> {p. p permutes ?U}"
     from p have pU: "p permutes ?U" by blast
     have sth: "sign (inv p) = sign p"
@@ -147,7 +141,7 @@
       {fix i assume i: "i \<in> ?U"
 	from i permutes_inv_o[OF pU] permutes_in_image[OF pU]
 	have "((\<lambda>i. ?di (transp A) i (inv p i)) o p) i = ?di A i (p i)"
-	  unfolding transp_def by (simp add: Cart_lambda_beta expand_fun_eq)}
+	  unfolding transp_def by (simp add: expand_fun_eq)}
       then show "setprod ((\<lambda>i. ?di (transp A) i (inv p i)) o p) ?U = setprod (\<lambda>i. ?di A i (p i)) ?U" by (auto intro: setprod_cong)
     qed
     finally have "of_int (sign (inv p)) * (setprod (\<lambda>i. ?di (transp A) i (inv p i)) ?U) = of_int (sign p) * (setprod (\<lambda>i. ?di A i (p i)) ?U)" using sth
@@ -157,22 +151,21 @@
 qed
 
 lemma det_lowerdiagonal:
-  fixes A :: "'a::comm_ring_1^'n^'n"
-  assumes ld: "\<And>i j. i \<in> {1 .. dimindex (UNIV:: 'n set)} \<Longrightarrow> j \<in> {1 .. dimindex(UNIV:: 'n set)} \<Longrightarrow> i < j \<Longrightarrow> A$i$j = 0"
-  shows "det A = setprod (\<lambda>i. A$i$i) {1..dimindex(UNIV:: 'n set)}"
+  fixes A :: "'a::comm_ring_1^'n^'n::{finite,wellorder}"
+  assumes ld: "\<And>i j. i < j \<Longrightarrow> A$i$j = 0"
+  shows "det A = setprod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
 proof-
-  let ?U = "{1..dimindex(UNIV:: 'n set)}"
+  let ?U = "UNIV:: 'n set"
   let ?PU = "{p. p permutes ?U}"
-  let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) {1 .. dimindex(UNIV :: 'n set)}"
-  have fU: "finite ?U" by blast
+  let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
+  have fU: "finite ?U" by simp
   from finite_permutations[OF fU] have fPU: "finite ?PU" .
   have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
   {fix p assume p: "p \<in> ?PU -{id}"
     from p have pU: "p permutes ?U" and pid: "p \<noteq> id" by blast+
     from permutes_natset_le[OF pU] pid obtain i where
-      i: "i \<in> ?U" "p i > i" by (metis not_le)
-    from permutes_in_image[OF pU] i(1) have piU: "p i \<in> ?U" by blast
-    from ld[OF i(1) piU i(2)] i(1) have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
+      i: "p i > i" by (metis not_le)
+    from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
     from setprod_zero[OF fU ex] have "?pp p = 0" by simp}
   then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"  by blast
   from setsum_mono_zero_cong_left[OF fPU id0 p0] show ?thesis
@@ -180,32 +173,32 @@
 qed
 
 lemma det_upperdiagonal:
-  fixes A :: "'a::comm_ring_1^'n^'n"
-  assumes ld: "\<And>i j. i \<in> {1 .. dimindex (UNIV:: 'n set)} \<Longrightarrow> j \<in> {1 .. dimindex(UNIV:: 'n set)} \<Longrightarrow> i > j \<Longrightarrow> A$i$j = 0"
-  shows "det A = setprod (\<lambda>i. A$i$i) {1..dimindex(UNIV:: 'n set)}"
+  fixes A :: "'a::comm_ring_1^'n^'n::{finite,wellorder}"
+  assumes ld: "\<And>i j. i > j \<Longrightarrow> A$i$j = 0"
+  shows "det A = setprod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
 proof-
-  let ?U = "{1..dimindex(UNIV:: 'n set)}"
+  let ?U = "UNIV:: 'n set"
   let ?PU = "{p. p permutes ?U}"
-  let ?pp = "(\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) {1 .. dimindex(UNIV :: 'n set)})"
-  have fU: "finite ?U" by blast
+  let ?pp = "(\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set))"
+  have fU: "finite ?U" by simp
   from finite_permutations[OF fU] have fPU: "finite ?PU" .
   have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
   {fix p assume p: "p \<in> ?PU -{id}"
     from p have pU: "p permutes ?U" and pid: "p \<noteq> id" by blast+
     from permutes_natset_ge[OF pU] pid obtain i where
-      i: "i \<in> ?U" "p i < i" by (metis not_le)
-    from permutes_in_image[OF pU] i(1) have piU: "p i \<in> ?U" by blast
-    from ld[OF i(1) piU i(2)] i(1) have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
+      i: "p i < i" by (metis not_le)
+    from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
     from setprod_zero[OF fU ex] have "?pp p = 0" by simp}
   then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"  by blast
   from   setsum_mono_zero_cong_left[OF fPU id0 p0] show ?thesis
     unfolding det_def by (simp add: sign_id)
 qed
 
-lemma det_I: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
+(* FIXME: get rid of wellorder requirement *)
+lemma det_I: "det (mat 1 :: 'a::comm_ring_1^'n^'n::{finite,wellorder}) = 1"
 proof-
   let ?A = "mat 1 :: 'a::comm_ring_1^'n^'n"
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?U = "UNIV :: 'n set"
   let ?f = "\<lambda>i j. ?A$i$j"
   {fix i assume i: "i \<in> ?U"
     have "?f i i = 1" using i by (vector mat_def)}
@@ -219,60 +212,38 @@
   finally show ?thesis .
 qed
 
-lemma det_0: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
-proof-
-  let ?A = "mat 0 :: 'a::comm_ring_1^'n^'n"
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
-  let ?f = "\<lambda>i j. ?A$i$j"
-  have th:"setprod (\<lambda>i. ?f i i) ?U = 0"
-    apply (rule setprod_zero)
-    apply simp
-    apply (rule bexI[where x=1])
-    using dimindex_ge_1[of "UNIV :: 'n set"]
-    by (simp_all add: mat_def Cart_lambda_beta)
-  {fix i j assume i: "i \<in> ?U" and j: "j \<in> ?U" and ij: "i < j"
-    have "?f i j = 0" using i j ij by (vector mat_def) }
-  then have "det ?A = setprod (\<lambda>i. ?f i i) ?U" using det_lowerdiagonal
-    by blast
-  also have "\<dots> = 0" unfolding th  ..
-  finally show ?thesis .
-qed
+lemma det_0: "det (mat 0 :: 'a::comm_ring_1^'n^'n::finite) = 0"
+  by (simp add: det_def setprod_zero)
 
 lemma det_permute_rows:
-  fixes A :: "'a::comm_ring_1^'n^'n"
-  assumes p: "p permutes {1 .. dimindex (UNIV :: 'n set)}"
+  fixes A :: "'a::comm_ring_1^'n^'n::finite"
+  assumes p: "p permutes (UNIV :: 'n::finite set)"
   shows "det(\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
-  apply (simp add: det_def setsum_right_distrib mult_assoc[symmetric] del: One_nat_def)
+  apply (simp add: det_def setsum_right_distrib mult_assoc[symmetric])
   apply (subst sum_permutations_compose_right[OF p])
 proof(rule setsum_cong2)
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?U = "UNIV :: 'n set"
   let ?PU = "{p. p permutes ?U}"
-  let ?Ap = "(\<chi> i. A$p i :: 'a^'n^'n)"
   fix q assume qPU: "q \<in> ?PU"
-  have fU: "finite ?U" by blast
+  have fU: "finite ?U" by simp
   from qPU have q: "q permutes ?U" by blast
   from p q have pp: "permutation p" and qp: "permutation q"
     by (metis fU permutation_permutes)+
   from permutes_inv[OF p] have ip: "inv p permutes ?U" .
-    {fix i assume i: "i \<in> ?U"
-      from Cart_lambda_beta[rule_format, OF i, of "\<lambda>i. A$ p i"]
-      have "?Ap$i$ (q o p) i = A $ p i $ (q o p) i " by simp}
-    hence "setprod (\<lambda>i. ?Ap$i$ (q o p) i) ?U = setprod (\<lambda>i. A$p i$(q o p) i) ?U"
-      by (auto intro: setprod_cong)
-    also have "\<dots> = setprod ((\<lambda>i. A$p i$(q o p) i) o inv p) ?U"
+    have "setprod (\<lambda>i. A$p i$ (q o p) i) ?U = setprod ((\<lambda>i. A$p i$(q o p) i) o inv p) ?U"
       by (simp only: setprod_permute[OF ip, symmetric])
     also have "\<dots> = setprod (\<lambda>i. A $ (p o inv p) i $ (q o (p o inv p)) i) ?U"
       by (simp only: o_def)
     also have "\<dots> = setprod (\<lambda>i. A$i$q i) ?U" by (simp only: o_def permutes_inverses[OF p])
-    finally   have thp: "setprod (\<lambda>i. ?Ap$i$ (q o p) i) ?U = setprod (\<lambda>i. A$i$q i) ?U"
+    finally   have thp: "setprod (\<lambda>i. A$p i$ (q o p) i) ?U = setprod (\<lambda>i. A$i$q i) ?U"
       by blast
-  show "of_int (sign (q o p)) * setprod (\<lambda>i. ?Ap$i$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A$i$q i) ?U"
+  show "of_int (sign (q o p)) * setprod (\<lambda>i. A$ p i$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A$i$q i) ?U"
     by (simp only: thp sign_compose[OF qp pp] mult_commute of_int_mult)
 qed
 
 lemma det_permute_columns:
-  fixes A :: "'a::comm_ring_1^'n^'n"
-  assumes p: "p permutes {1 .. dimindex (UNIV :: 'n set)}"
+  fixes A :: "'a::comm_ring_1^'n^'n::finite"
+  assumes p: "p permutes (UNIV :: 'n set)"
   shows "det(\<chi> i j. A$i$ p j :: 'a^'n^'n) = of_int (sign p) * det A"
 proof-
   let ?Ap = "\<chi> i j. A$i$ p j :: 'a^'n^'n"
@@ -281,15 +252,13 @@
     unfolding det_permute_rows[OF p, of ?At] det_transp ..
   moreover
   have "?Ap = transp (\<chi> i. transp A $ p i)"
-    by (simp add: transp_def Cart_eq Cart_lambda_beta Cart_lambda_beta_perm[OF p])
+    by (simp add: transp_def Cart_eq)
   ultimately show ?thesis by simp
 qed
 
 lemma det_identical_rows:
-  fixes A :: "'a::ordered_idom^'n^'n"
-  assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
-  and j: "j\<in>{1 .. dimindex (UNIV :: 'n set)}"
-  and ij: "i \<noteq> j"
+  fixes A :: "'a::ordered_idom^'n^'n::finite"
+  assumes ij: "i \<noteq> j"
   and r: "row i A = row j A"
   shows	"det A = 0"
 proof-
@@ -298,94 +267,59 @@
   have th1: "of_int (-1) = - 1" by (metis of_int_1 of_int_minus number_of_Min)
   let ?p = "Fun.swap i j id"
   let ?A = "\<chi> i. A $ ?p i"
-  from r have "A = ?A" by (simp add: Cart_eq Cart_lambda_beta Cart_lambda_beta_perm[OF permutes_swap_id[OF i j]] row_def swap_def)
+  from r have "A = ?A" by (simp add: Cart_eq row_def swap_def)
   hence "det A = det ?A" by simp
   moreover have "det A = - det ?A"
-    by (simp add: det_permute_rows[OF permutes_swap_id[OF i j]] sign_swap_id ij th1)
+    by (simp add: det_permute_rows[OF permutes_swap_id] sign_swap_id ij th1)
   ultimately show "det A = 0" by (metis tha)
 qed
 
 lemma det_identical_columns:
-  fixes A :: "'a::ordered_idom^'n^'n"
-  assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
-  and j: "j\<in>{1 .. dimindex (UNIV :: 'n set)}"
-  and ij: "i \<noteq> j"
+  fixes A :: "'a::ordered_idom^'n^'n::finite"
+  assumes ij: "i \<noteq> j"
   and r: "column i A = column j A"
   shows	"det A = 0"
 apply (subst det_transp[symmetric])
-apply (rule det_identical_rows[OF i j ij])
-by (metis row_transp i j r)
+apply (rule det_identical_rows[OF ij])
+by (metis row_transp r)
 
 lemma det_zero_row:
-  fixes A :: "'a::{idom, ring_char_0}^'n^'n"
-  assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
-  and r: "row i A = 0"
+  fixes A :: "'a::{idom, ring_char_0}^'n^'n::finite"
+  assumes r: "row i A = 0"
   shows "det A = 0"
-using i r
-apply (simp add: row_def det_def Cart_lambda_beta Cart_eq vector_component del: One_nat_def)
+using r
+apply (simp add: row_def det_def Cart_eq)
 apply (rule setsum_0')
-apply (clarsimp simp add: sign_nz simp del: One_nat_def)
+apply (clarsimp simp add: sign_nz)
 apply (rule setprod_zero)
 apply simp
-apply (rule bexI[where x=i])
-apply (erule_tac x="a i" in ballE)
-apply (subgoal_tac "(0\<Colon>'a ^ 'n) $ a i = 0")
-apply simp
-apply (rule zero_index)
-apply (drule permutes_in_image[of _ _ i])
-apply simp
-apply (drule permutes_in_image[of _ _ i])
-apply simp
-apply simp
+apply auto
 done
 
 lemma det_zero_column:
-  fixes A :: "'a::{idom,ring_char_0}^'n^'n"
-  assumes i: "i\<in>{1 .. dimindex (UNIV :: 'n set)}"
-  and r: "column i A = 0"
+  fixes A :: "'a::{idom,ring_char_0}^'n^'n::finite"
+  assumes r: "column i A = 0"
   shows "det A = 0"
   apply (subst det_transp[symmetric])
-  apply (rule det_zero_row[OF i])
-  by (metis row_transp r i)
-
-lemma setsum_lambda_beta[simp]: "setsum (\<lambda>i. ((\<chi> i. g i) :: 'a::{comm_monoid_add}^'n) $ i ) {1 .. dimindex (UNIV :: 'n set)} = setsum g {1 .. dimindex (UNIV :: 'n set)}"
-  by (simp add: Cart_lambda_beta)
-
-lemma setprod_lambda_beta[simp]: "setprod (\<lambda>i. ((\<chi> i. g i) :: 'a::{comm_monoid_mult}^'n) $ i ) {1 .. dimindex (UNIV :: 'n set)} = setprod g {1 .. dimindex (UNIV :: 'n set)}"
-  apply (rule setprod_cong)
-  apply simp
-  apply (simp add: Cart_lambda_beta')
-  done
-
-lemma setprod_lambda_beta2[simp]: "setprod (\<lambda>i. ((\<chi> i. g i) :: 'a::{comm_monoid_mult}^'n^'n) $ i$ f i ) {1 .. dimindex (UNIV :: 'n set)} = setprod (\<lambda>i. g i $ f i) {1 .. dimindex (UNIV :: 'n set)}"
-proof(rule setprod_cong[OF refl])
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
-  fix i assume i: "i \<in> ?U"
-  from Cart_lambda_beta'[OF i, of g] have
-    "((\<chi> i. g i) :: 'a^'n^'n) $ i = g i" .
-  hence "((\<chi> i. g i) :: 'a^'n^'n) $ i $ f i = g i $ f i" by simp
-  then
-  show "((\<chi> i. g i):: 'a^'n^'n) $ i $ f i = g i $ f i"   .
-qed
+  apply (rule det_zero_row [of i])
+  by (metis row_transp r)
 
 lemma det_row_add:
-  assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
+  fixes a b c :: "'n::finite \<Rightarrow> _ ^ 'n"
   shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
              det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
              det((\<chi> i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)"
-unfolding det_def setprod_lambda_beta2 setsum_addf[symmetric]
+unfolding det_def Cart_lambda_beta setsum_addf[symmetric]
 proof (rule setsum_cong2)
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?U = "UNIV :: 'n set"
   let ?pU = "{p. p permutes ?U}"
-  let ?f = "(\<lambda>i. if i = k then a i + b i else c i)::nat \<Rightarrow> 'a::comm_ring_1^'n"
-  let ?g = "(\<lambda> i. if i = k then a i else c i)::nat \<Rightarrow> 'a::comm_ring_1^'n"
-  let ?h = "(\<lambda> i. if i = k then b i else c i)::nat \<Rightarrow> 'a::comm_ring_1^'n"
+  let ?f = "(\<lambda>i. if i = k then a i + b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
+  let ?g = "(\<lambda> i. if i = k then a i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
+  let ?h = "(\<lambda> i. if i = k then b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
   fix p assume p: "p \<in> ?pU"
   let ?Uk = "?U - {k}"
   from p have pU: "p permutes ?U" by blast
-  from k have pkU: "p k \<in> ?U" by (simp only: permutes_in_image[OF pU])
-  note pin[simp] = permutes_in_image[OF pU]
-  have kU: "?U = insert k ?Uk" using k by blast
+  have kU: "?U = insert k ?Uk" by blast
   {fix j assume j: "j \<in> ?Uk"
     from j have "?f j $ p j = ?g j $ p j" and "?f j $ p j= ?h j $ p j"
       by simp_all}
@@ -394,14 +328,14 @@
     apply -
     apply (rule setprod_cong, simp_all)+
     done
-  have th3: "finite ?Uk" "k \<notin> ?Uk" using k by auto
+  have th3: "finite ?Uk" "k \<notin> ?Uk" by auto
   have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?f i $ p i) (insert k ?Uk)"
     unfolding kU[symmetric] ..
   also have "\<dots> = ?f k $ p k  * setprod (\<lambda>i. ?f i $ p i) ?Uk"
     apply (rule setprod_insert)
     apply simp
-    using k by blast
-  also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk)" using pkU by (simp add: ring_simps vector_component)
+    by blast
+  also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk)" by (simp add: ring_simps)
   also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?g i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?h i $ p i) ?Uk)" by (metis th1 th2)
   also have "\<dots> = setprod (\<lambda>i. ?g i $ p i) (insert k ?Uk) + setprod (\<lambda>i. ?h i $ p i) (insert k ?Uk)"
     unfolding  setprod_insert[OF th3] by simp
@@ -411,38 +345,36 @@
 qed
 
 lemma det_row_mul:
-  assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
+  fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
   shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
              c* det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
 
-unfolding det_def setprod_lambda_beta2 setsum_right_distrib
+unfolding det_def Cart_lambda_beta setsum_right_distrib
 proof (rule setsum_cong2)
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?U = "UNIV :: 'n set"
   let ?pU = "{p. p permutes ?U}"
-  let ?f = "(\<lambda>i. if i = k then c*s a i else b i)::nat \<Rightarrow> 'a::comm_ring_1^'n"
-  let ?g = "(\<lambda> i. if i = k then a i else b i)::nat \<Rightarrow> 'a::comm_ring_1^'n"
+  let ?f = "(\<lambda>i. if i = k then c*s a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
+  let ?g = "(\<lambda> i. if i = k then a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
   fix p assume p: "p \<in> ?pU"
   let ?Uk = "?U - {k}"
   from p have pU: "p permutes ?U" by blast
-  from k have pkU: "p k \<in> ?U" by (simp only: permutes_in_image[OF pU])
-  note pin[simp] = permutes_in_image[OF pU]
-  have kU: "?U = insert k ?Uk" using k by blast
+  have kU: "?U = insert k ?Uk" by blast
   {fix j assume j: "j \<in> ?Uk"
     from j have "?f j $ p j = ?g j $ p j" by simp}
   then have th1: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?g i $ p i) ?Uk"
     apply -
     apply (rule setprod_cong, simp_all)
     done
-  have th3: "finite ?Uk" "k \<notin> ?Uk" using k by auto
+  have th3: "finite ?Uk" "k \<notin> ?Uk" by auto
   have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?f i $ p i) (insert k ?Uk)"
     unfolding kU[symmetric] ..
   also have "\<dots> = ?f k $ p k  * setprod (\<lambda>i. ?f i $ p i) ?Uk"
     apply (rule setprod_insert)
     apply simp
-    using k by blast
-  also have "\<dots> = (c*s a k) $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk" using pkU by (simp add: ring_simps vector_component)
+    by blast
+  also have "\<dots> = (c*s a k) $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk" by (simp add: ring_simps)
   also have "\<dots> = c* (a k $ p k * setprod (\<lambda>i. ?g i $ p i) ?Uk)"
-    unfolding th1 using pkU by (simp add: vector_component mult_ac)
+    unfolding th1 by (simp add: mult_ac)
   also have "\<dots> = c* (setprod (\<lambda>i. ?g i $ p i) (insert k ?Uk))"
     unfolding  setprod_insert[OF th3] by simp
   finally have "setprod (\<lambda>i. ?f i $ p i) ?U = c* (setprod (\<lambda>i. ?g i $ p i) ?U)" unfolding kU[symmetric] .
@@ -451,36 +383,33 @@
 qed
 
 lemma det_row_0:
-  assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
+  fixes b :: "'n::finite \<Rightarrow> _ ^ 'n"
   shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
-using det_row_mul[OF k, of 0 "\<lambda>i. 1" b]
+using det_row_mul[of k 0 "\<lambda>i. 1" b]
 apply (simp)
   unfolding vector_smult_lzero .
 
 lemma det_row_operation:
-  fixes A :: "'a::ordered_idom^'n^'n"
-  assumes i: "i \<in> {1 .. dimindex(UNIV :: 'n set)}"
-  and j: "j \<in> {1 .. dimindex(UNIV :: 'n set)}"
-  and ij: "i \<noteq> j"
+  fixes A :: "'a::ordered_idom^'n^'n::finite"
+  assumes ij: "i \<noteq> j"
   shows "det (\<chi> k. if k = i then row i A + c *s row j A else row k A) = det A"
 proof-
   let ?Z = "(\<chi> k. if k = i then row j A else row k A) :: 'a ^'n^'n"
-  have th: "row i ?Z = row j ?Z" using i j by (vector row_def)
+  have th: "row i ?Z = row j ?Z" by (vector row_def)
   have th2: "((\<chi> k. if k = i then row i A else row k A) :: 'a^'n^'n) = A"
-    using i j by (vector row_def)
+    by (vector row_def)
   show ?thesis
-    unfolding det_row_add [OF i] det_row_mul[OF i] det_identical_rows[OF i j ij th] th2
+    unfolding det_row_add [of i] det_row_mul[of i] det_identical_rows[OF ij th] th2
     by simp
 qed
 
 lemma det_row_span:
-  fixes A :: "'a:: ordered_idom^'n^'n"
-  assumes i: "i \<in> {1 .. dimindex(UNIV :: 'n set)}"
-  and x: "x \<in> span {row j A |j. j\<in> {1 .. dimindex(UNIV :: 'n set)} \<and> j\<noteq> i}"
+  fixes A :: "'a:: ordered_idom^'n^'n::finite"
+  assumes x: "x \<in> span {row j A |j. j \<noteq> i}"
   shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
 proof-
-  let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
-  let ?S = "{row j A |j. j\<in> ?U \<and> j\<noteq> i}"
+  let ?U = "UNIV :: 'n set"
+  let ?S = "{row j A |j. j \<noteq> i}"
   let ?d = "\<lambda>x. det (\<chi> k. if k = i then x else row k A)"
   let ?P = "\<lambda>x. ?d (row i A + x) = det A"
   {fix k
@@ -489,17 +418,17 @@
   then have P0: "?P 0"
     apply -
     apply (rule cong[of det, OF refl])
-    using i by (vector row_def)
+    by (vector row_def)
   moreover
   {fix c z y assume zS: "z \<in> ?S" and Py: "?P y"
-    from zS obtain j where j: "z = row j A" "j \<in> ?U" "i \<noteq> j" by blast
+    from zS obtain j where j: "z = row j A" "i \<noteq> j" by blast
     let ?w = "row i A + y"
     have th0: "row i A + (c*s z + y) = ?w + c*s z" by vector
     have thz: "?d z = 0"
-      apply (rule det_identical_rows[OF i j(2,3)])
-      using i j by (vector row_def)
+      apply (rule det_identical_rows[OF j(2)])
+      using j by (vector row_def)
     have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)" unfolding th0 ..
-    then have "?P (c*s z + y)" unfolding thz Py det_row_mul[OF i] det_row_add[OF i]
+    then have "?P (c*s z + y)" unfolding thz Py det_row_mul[of i] det_row_add[of i]
       by simp }
 
   ultimately show ?thesis
@@ -516,48 +445,47 @@
 (* ------------------------------------------------------------------------- *)
 
 lemma det_dependent_rows:
-  fixes A:: "'a::ordered_idom^'n^'n"
+  fixes A:: "'a::ordered_idom^'n^'n::finite"
   assumes d: "dependent (rows A)"
   shows "det A = 0"
 proof-
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
-  from d obtain i where i: "i \<in> ?U" "row i A \<in> span (rows A - {row i A})"
+  let ?U = "UNIV :: 'n set"
+  from d obtain i where i: "row i A \<in> span (rows A - {row i A})"
     unfolding dependent_def rows_def by blast
-  {fix j k assume j: "j \<in>?U" and k: "k \<in> ?U" and jk: "j \<noteq> k"
+  {fix j k assume jk: "j \<noteq> k"
     and c: "row j A = row k A"
-    from det_identical_rows[OF j k jk c] have ?thesis .}
+    from det_identical_rows[OF jk c] have ?thesis .}
   moreover
-  {assume H: "\<And> i j. i\<in> ?U \<Longrightarrow> j \<in> ?U \<Longrightarrow> i \<noteq> j \<Longrightarrow> row i A \<noteq> row j A"
-    have th0: "- row i A \<in> span {row j A|j. j \<in> ?U \<and> j \<noteq> i}"
+  {assume H: "\<And> i j. i \<noteq> j \<Longrightarrow> row i A \<noteq> row j A"
+    have th0: "- row i A \<in> span {row j A|j. j \<noteq> i}"
       apply (rule span_neg)
       apply (rule set_rev_mp)
-      apply (rule i(2))
+      apply (rule i)
       apply (rule span_mono)
       using H i by (auto simp add: rows_def)
-    from det_row_span[OF i(1) th0]
+    from det_row_span[OF th0]
     have "det A = det (\<chi> k. if k = i then 0 *s 1 else row k A)"
       unfolding right_minus vector_smult_lzero ..
-    with det_row_mul[OF i(1), of "0::'a" "\<lambda>i. 1"]
+    with det_row_mul[of i "0::'a" "\<lambda>i. 1"]
     have "det A = 0" by simp}
   ultimately show ?thesis by blast
 qed
 
-lemma det_dependent_columns: assumes d: "dependent(columns (A::'a::ordered_idom^'n^'n))" shows "det A = 0"
+lemma det_dependent_columns: assumes d: "dependent(columns (A::'a::ordered_idom^'n^'n::finite))" shows "det A = 0"
 by (metis d det_dependent_rows rows_transp det_transp)
 
 (* ------------------------------------------------------------------------- *)
 (* Multilinearity and the multiplication formula.                            *)
 (* ------------------------------------------------------------------------- *)
 
-lemma Cart_lambda_cong: "(\<And>x. x \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> f x = g x) \<Longrightarrow> (Cart_lambda f::'a^'n) = (Cart_lambda g :: 'a^'n)"
+lemma Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (Cart_lambda f::'a^'n) = (Cart_lambda g :: 'a^'n)"
   apply (rule iffD1[OF Cart_lambda_unique]) by vector
 
 lemma det_linear_row_setsum:
-  assumes fS: "finite S" and k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
-  shows "det ((\<chi> i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n) = setsum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
-  using k
+  assumes fS: "finite S"
+  shows "det ((\<chi> i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n::finite) = setsum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
 proof(induct rule: finite_induct[OF fS])
-  case 1 thus ?case apply simp  unfolding setsum_empty det_row_0[OF k] ..
+  case 1 thus ?case apply simp  unfolding setsum_empty det_row_0[of k] ..
 next
   case (2 x F)
   then  show ?case by (simp add: det_row_add cong del: if_weak_cong)
@@ -588,91 +516,89 @@
 lemma eq_id_iff[simp]: "(\<forall>x. f x = x) = (f = id)" by (auto intro: ext)
 
 lemma det_linear_rows_setsum_lemma:
-  assumes fS: "finite S" and k: "k \<le> dimindex (UNIV :: 'n set)"
-  shows "det((\<chi> i. if i <= k then setsum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
-             setsum (\<lambda>f. det((\<chi> i. if i <= k then a i (f i) else c i)::'a^'n^'n))
-                 {f. (\<forall>i \<in> {1 .. k}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1..k} \<longrightarrow> f i = i)}"
-using k
-proof(induct k arbitrary: a c)
-  case 0
-  have th0: "\<And>x y. (\<chi> i. if i <= 0 then x i else y i) = (\<chi> i. y i)" by vector
-  from "0.prems"  show ?case unfolding th0 by simp
+  assumes fS: "finite S" and fT: "finite T"
+  shows "det((\<chi> i. if i \<in> T then setsum (a i) S else c i):: 'a::comm_ring_1^'n^'n::finite) =
+             setsum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
+                 {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
+using fT
+proof(induct T arbitrary: a c set: finite)
+  case empty
+  have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)" by vector
+  from "empty.prems"  show ?case unfolding th0 by simp
 next
-  case (Suc k a c)
-  let ?F = "\<lambda>k. {f. (\<forall>i \<in> {1 .. k}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1..k} \<longrightarrow> f i = i)}"
-  let ?h = "\<lambda>(y::nat,g) i. if i = Suc k then y else g i"
-  let ?k = "\<lambda>h. (h(Suc k),(\<lambda>i. if i = Suc k then i else h i))"
-  let ?s = "\<lambda> k a c f. det((\<chi> i. if i <= k then a i (f i) else c i)::'a^'n^'n)"
-  let ?c = "\<lambda>i. if i = Suc k then a i j else c i"
-  from Suc.prems have Sk: "Suc k \<in> {1 .. dimindex (UNIV :: 'n set)}" by simp
-  from Suc.prems have k': "k \<le> dimindex (UNIV :: 'n set)" by arith
-  have thif: "\<And>a b c d. (if b \<or> a then c else d) = (if a then c else if b then c else d)" by simp
+  case (insert z T a c)
+  let ?F = "\<lambda>T. {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
+  let ?h = "\<lambda>(y,g) i. if i = z then y else g i"
+  let ?k = "\<lambda>h. (h(z),(\<lambda>i. if i = z then i else h i))"
+  let ?s = "\<lambda> k a c f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n)"
+  let ?c = "\<lambda>i. if i = z then a i j else c i"
+  have thif: "\<And>a b c d. (if a \<or> b then c else d) = (if a then c else if b then c else d)" by simp
   have thif2: "\<And>a b c d e. (if a then b else if c then d else e) =
      (if c then (if a then b else d) else (if a then b else e))" by simp
-  have "det (\<chi> i. if i \<le> Suc k then setsum (a i) S else c i) =
-        det (\<chi> i. if i = Suc k then setsum (a i) S
-                 else if i \<le> k then setsum (a i) S else c i)"
-    unfolding le_Suc_eq thif  ..
-  also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<le> k then setsum (a i) S
-                    else if i = Suc k then a i j else c i))"
-    unfolding det_linear_row_setsum[OF fS Sk]
+  from `z \<notin> T` have nz: "\<And>i. i \<in> T \<Longrightarrow> i = z \<longleftrightarrow> False" by auto
+  have "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
+        det (\<chi> i. if i = z then setsum (a i) S
+                 else if i \<in> T then setsum (a i) S else c i)"
+    unfolding insert_iff thif ..
+  also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<in> T then setsum (a i) S
+                    else if i = z then a i j else c i))"
+    unfolding det_linear_row_setsum[OF fS]
     apply (subst thif2)
-    by (simp cong del: if_weak_cong cong add: if_cong)
+    using nz by (simp cong del: if_weak_cong cong add: if_cong)
   finally have tha:
-    "det (\<chi> i. if i \<le> Suc k then setsum (a i) S else c i) =
-     (\<Sum>(j, f)\<in>S \<times> ?F k. det (\<chi> i. if i \<le> k then a i (f i)
-                                else if i = Suc k then a i j
+    "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
+     (\<Sum>(j, f)\<in>S \<times> ?F T. det (\<chi> i. if i \<in> T then a i (f i)
+                                else if i = z then a i j
                                 else c i))"
-    unfolding  Suc.hyps[OF k'] unfolding setsum_cartesian_product by blast
+    unfolding  insert.hyps unfolding setsum_cartesian_product by blast
   show ?case unfolding tha
     apply(rule setsum_eq_general_reverses[where h= "?h" and k= "?k"],
-      blast intro: finite_cartesian_product fS finite_bounded_functions[OF fS],
-      blast intro: finite_cartesian_product fS finite_bounded_functions[OF fS], auto intro: ext)
+      blast intro: finite_cartesian_product fS finite,
+      blast intro: finite_cartesian_product fS finite)
+    using `z \<notin> T`
+    apply (auto intro: ext)
     apply (rule cong[OF refl[of det]])
     by vector
 qed
 
 lemma det_linear_rows_setsum:
-  assumes fS: "finite S"
-  shows "det (\<chi> i. setsum (a i) S) = setsum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. (\<forall>i \<in> {1 .. dimindex (UNIV :: 'n set)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1.. dimindex (UNIV :: 'n set)} \<longrightarrow> f i = i)}"
+  assumes fS: "finite (S::'n::finite set)"
+  shows "det (\<chi> i. setsum (a i) S) = setsum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n::finite)) {f. \<forall>i. f i \<in> S}"
 proof-
-  have th0: "\<And>x y. ((\<chi> i. if i <= dimindex(UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\<chi> i. x i)" by vector
+  have th0: "\<And>x y. ((\<chi> i. if i \<in> (UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\<chi> i. x i)" by vector
 
-  from det_linear_rows_setsum_lemma[OF fS, of "dimindex (UNIV :: 'n set)" a, unfolded th0, OF order_refl] show ?thesis by blast
+  from det_linear_rows_setsum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite] show ?thesis by simp
 qed
 
 lemma matrix_mul_setsum_alt:
-  fixes A B :: "'a::comm_ring_1^'n^'n"
-  shows "A ** B = (\<chi> i. setsum (\<lambda>k. A$i$k *s B $ k) {1 .. dimindex (UNIV :: 'n set)})"
+  fixes A B :: "'a::comm_ring_1^'n^'n::finite"
+  shows "A ** B = (\<chi> i. setsum (\<lambda>k. A$i$k *s B $ k) (UNIV :: 'n set))"
   by (vector matrix_matrix_mult_def setsum_component)
 
 lemma det_rows_mul:
-  "det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
-  setprod (\<lambda>i. c i) {1..dimindex(UNIV:: 'n set)} * det((\<chi> i. a i)::'a^'n^'n)"
-proof (simp add: det_def Cart_lambda_beta' setsum_right_distrib vector_component cong add: setprod_cong del: One_nat_def, rule setsum_cong2)
-  let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
+  "det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n::finite) =
+  setprod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
+proof (simp add: det_def setsum_right_distrib cong add: setprod_cong, rule setsum_cong2)
+  let ?U = "UNIV :: 'n set"
   let ?PU = "{p. p permutes ?U}"
   fix p assume pU: "p \<in> ?PU"
   let ?s = "of_int (sign p)"
   from pU have p: "p permutes ?U" by blast
-  have "setprod (\<lambda>i. (c i *s a i) $ p i) ?U = setprod (\<lambda>i. c i * a i $ p i) ?U"
-    apply (rule setprod_cong, blast)
-    by (auto simp only: permutes_in_image[OF p] intro: vector_smult_component)
-  also have "\<dots> = setprod c ?U * setprod (\<lambda>i. a i $ p i) ?U"
+  have "setprod (\<lambda>i. c i * a i $ p i) ?U = setprod c ?U * setprod (\<lambda>i. a i $ p i) ?U"
     unfolding setprod_timesf ..
-  finally show "?s * (\<Prod>xa\<in>?U. (c xa *s a xa) $ p xa) =
+  then show "?s * (\<Prod>xa\<in>?U. c xa * a xa $ p xa) =
         setprod c ?U * (?s* (\<Prod>xa\<in>?U. a xa $ p xa))" by (simp add: ring_simps)
 qed
 
 lemma det_mul:
-  fixes A B :: "'a::ordered_idom^'n^'n"
+  fixes A B :: "'a::ordered_idom^'n^'n::finite"
   shows "det (A ** B) = det A * det B"
 proof-
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?U = "UNIV :: 'n set"
   let ?F = "{f. (\<forall>i\<in> ?U. f i \<in> ?U) \<and> (\<forall>i. i \<notin> ?U \<longrightarrow> f i = i)}"
   let ?PU = "{p. p permutes ?U}"
   have fU: "finite ?U" by simp
-  have fF: "finite ?F"  using finite_bounded_functions[OF fU] .
+  have fF: "finite ?F" by (rule finite)
   {fix p assume p: "p permutes ?U"
 
     have "p \<in> ?F" unfolding mem_Collect_eq permutes_in_image[OF p]
@@ -687,23 +613,21 @@
     let ?A = "(\<chi> i. A$i$f i *s B$f i) :: 'a^'n^'n"
     let ?B = "(\<chi> i. B$f i) :: 'a^'n^'n"
     {assume fni: "\<not> inj_on f ?U"
-      then obtain i j where ij: "i \<in> ?U" "j \<in> ?U" "f i = f j" "i \<noteq> j"
+      then obtain i j where ij: "f i = f j" "i \<noteq> j"
 	unfolding inj_on_def by blast
       from ij
       have rth: "row i ?B = row j ?B" by (vector row_def)
-      from det_identical_rows[OF ij(1,2,4) rth]
+      from det_identical_rows[OF ij(2) rth]
       have "det (\<chi> i. A$i$f i *s B$f i) = 0"
 	unfolding det_rows_mul by simp}
     moreover
     {assume fi: "inj_on f ?U"
       from f fi have fith: "\<And>i j. f i = f j \<Longrightarrow> i = j"
-	unfolding inj_on_def
-	apply (case_tac "i \<in> ?U")
-	apply (case_tac "j \<in> ?U") by metis+
+	unfolding inj_on_def by metis
       note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]]
 
       {fix y
-	from fs f have "\<exists>x. f x = y" by (cases "y \<in> ?U") blast+
+	from fs f have "\<exists>x. f x = y" by blast
 	then obtain x where x: "f x = y" by blast
 	{fix z assume z: "f z = y" from fith x z have "z = x" by metis}
 	with x have "\<exists>!x. f x = y" by blast}
@@ -747,7 +671,7 @@
     unfolding det_def setsum_product
     by (rule setsum_cong2)
   have "det (A**B) = setsum (\<lambda>f.  det (\<chi> i. A $ i $ f i *s B $ f i)) ?F"
-    unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU] ..
+    unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU] by simp
   also have "\<dots> = setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU"
     using setsum_mono_zero_cong_left[OF fF PUF zth, symmetric]
     unfolding det_rows_mul by auto
@@ -759,17 +683,18 @@
 (* ------------------------------------------------------------------------- *)
 
 lemma invertible_left_inverse:
-  fixes A :: "real^'n^'n"
+  fixes A :: "real^'n^'n::finite"
   shows "invertible A \<longleftrightarrow> (\<exists>(B::real^'n^'n). B** A = mat 1)"
   by (metis invertible_def matrix_left_right_inverse)
 
 lemma invertible_righ_inverse:
-  fixes A :: "real^'n^'n"
+  fixes A :: "real^'n^'n::finite"
   shows "invertible A \<longleftrightarrow> (\<exists>(B::real^'n^'n). A** B = mat 1)"
   by (metis invertible_def matrix_left_right_inverse)
 
+(* FIXME: get rid of wellorder requirement *)
 lemma invertible_det_nz:
-  fixes A::"real ^'n^'n"
+  fixes A::"real ^'n^'n::{finite,wellorder}"
   shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
 proof-
   {assume "invertible A"
@@ -780,7 +705,7 @@
       apply (simp add: det_mul det_I) by algebra }
   moreover
   {assume H: "\<not> invertible A"
-    let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
+    let ?U = "UNIV :: 'n set"
     have fU: "finite ?U" by simp
     from H obtain c i where c: "setsum (\<lambda>i. c i *s row i A) ?U = 0"
       and iU: "i \<in> ?U" and ci: "c i \<noteq> 0"
@@ -794,11 +719,11 @@
     from c ci
     have thr0: "- row i A = setsum (\<lambda>j. (1/ c i) *s c j *s row j A) (?U - {i})"
       unfolding setsum_diff1'[OF fU iU] setsum_cmul
-      apply (simp add: field_simps)
+      apply -
       apply (rule vector_mul_lcancel_imp[OF ci])
       apply (auto simp add: vector_smult_assoc vector_smult_rneg field_simps)
       unfolding stupid ..
-    have thr: "- row i A \<in> span {row j A| j. j\<in> ?U \<and> j \<noteq> i}"
+    have thr: "- row i A \<in> span {row j A| j. j \<noteq> i}"
       unfolding thr0
       apply (rule span_setsum)
       apply simp
@@ -810,8 +735,8 @@
     let ?B = "(\<chi> k. if k = i then 0 else row k A) :: real ^'n^'n"
     have thrb: "row i ?B = 0" using iU by (vector row_def)
     have "det A = 0"
-      unfolding det_row_span[OF iU thr, symmetric] right_minus
-      unfolding  det_zero_row[OF iU thrb]  ..}
+      unfolding det_row_span[OF thr, symmetric] right_minus
+      unfolding  det_zero_row[OF thrb]  ..}
   ultimately show ?thesis by blast
 qed
 
@@ -820,15 +745,14 @@
 (* ------------------------------------------------------------------------- *)
 
 lemma cramer_lemma_transp:
-  fixes A:: "'a::ordered_idom^'n^'n" and x :: "'a ^'n"
-  assumes k: "k \<in> {1 .. dimindex(UNIV ::'n set)}"
-  shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x$i *s row i A) {1 .. dimindex(UNIV::'n set)}
+  fixes A:: "'a::ordered_idom^'n^'n::finite" and x :: "'a ^'n::finite"
+  shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x$i *s row i A) (UNIV::'n set)
                            else row i A)::'a^'n^'n) = x$k * det A"
   (is "?lhs = ?rhs")
 proof-
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?U = "UNIV :: 'n set"
   let ?Uk = "?U - {k}"
-  have U: "?U = insert k ?Uk" using k by blast
+  have U: "?U = insert k ?Uk" by blast
   have fUk: "finite ?Uk" by simp
   have kUk: "k \<notin> ?Uk" by simp
   have th00: "\<And>k s. x$k *s row k A + s = (x$k - 1) *s row k A + row k A + s"
@@ -837,7 +761,7 @@
   have "(\<chi> i. row i A) = A" by (vector row_def)
   then have thd1: "det (\<chi> i. row i A) = det A"  by simp
   have thd0: "det (\<chi> i. if i = k then row k A + (\<Sum>i \<in> ?Uk. x $ i *s row i A) else row i A) = det A"
-    apply (rule det_row_span[OF k])
+    apply (rule det_row_span)
     apply (rule span_setsum[OF fUk])
     apply (rule ballI)
     apply (rule span_mul)
@@ -849,30 +773,31 @@
     unfolding setsum_insert[OF fUk kUk]
     apply (subst th00)
     unfolding add_assoc
-    apply (subst det_row_add[OF k])
+    apply (subst det_row_add)
     unfolding thd0
-    unfolding det_row_mul[OF k]
+    unfolding det_row_mul
     unfolding th001[of k "\<lambda>i. row i A"]
     unfolding thd1  by (simp add: ring_simps)
 qed
 
 lemma cramer_lemma:
-  fixes A :: "'a::ordered_idom ^'n^'n"
-  assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}" (is " _ \<in> ?U")
+  fixes A :: "'a::ordered_idom ^'n^'n::finite"
   shows "det((\<chi> i j. if j = k then (A *v x)$i else A$i$j):: 'a^'n^'n) = x$k * det A"
 proof-
+  let ?U = "UNIV :: 'n set"
   have stupid: "\<And>c. setsum (\<lambda>i. c i *s row i (transp A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
     by (auto simp add: row_transp intro: setsum_cong2)
   show ?thesis
   unfolding matrix_mult_vsum
-  unfolding cramer_lemma_transp[OF k, of x "transp A", unfolded det_transp, symmetric]
+  unfolding cramer_lemma_transp[of k x "transp A", unfolded det_transp, symmetric]
   unfolding stupid[of "\<lambda>i. x$i"]
   apply (subst det_transp[symmetric])
   apply (rule cong[OF refl[of det]]) by (vector transp_def column_def row_def)
 qed
 
+(* FIXME: get rid of wellorder requirement *)
 lemma cramer:
-  fixes A ::"real^'n^'n"
+  fixes A ::"real^'n^'n::{finite,wellorder}"
   assumes d0: "det A \<noteq> 0"
   shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)"
 proof-
@@ -884,7 +809,7 @@
   {fix x assume x: "A *v x = b"
   have "x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)"
     unfolding x[symmetric]
-    using d0 by (simp add: Cart_eq Cart_lambda_beta' cramer_lemma field_simps)}
+    using d0 by (simp add: Cart_eq cramer_lemma field_simps)}
   with xe show ?thesis by auto
 qed
 
@@ -894,7 +819,7 @@
 
 definition "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
 
-lemma orthogonal_transformation: "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>(v::real ^'n). norm (f v) = norm v)"
+lemma orthogonal_transformation: "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>(v::real ^_). norm (f v) = norm v)"
   unfolding orthogonal_transformation_def
   apply auto
   apply (erule_tac x=v in allE)+
@@ -903,14 +828,14 @@
 
 definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow> transp Q ** Q = mat 1 \<and> Q ** transp Q = mat 1"
 
-lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n)  \<longleftrightarrow> transp Q ** Q = mat 1"
+lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n::finite)  \<longleftrightarrow> transp Q ** Q = mat 1"
   by (metis matrix_left_right_inverse orthogonal_matrix_def)
 
-lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1)"
+lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n::finite)"
   by (simp add: orthogonal_matrix_def transp_mat matrix_mul_lid)
 
 lemma orthogonal_matrix_mul:
-  fixes A :: "real ^'n^'n"
+  fixes A :: "real ^'n^'n::finite"
   assumes oA : "orthogonal_matrix A"
   and oB: "orthogonal_matrix B"
   shows "orthogonal_matrix(A ** B)"
@@ -921,26 +846,26 @@
   by (simp add: matrix_mul_rid)
 
 lemma orthogonal_transformation_matrix:
-  fixes f:: "real^'n \<Rightarrow> real^'n"
+  fixes f:: "real^'n \<Rightarrow> real^'n::finite"
   shows "orthogonal_transformation f \<longleftrightarrow> linear f \<and> orthogonal_matrix(matrix f)"
   (is "?lhs \<longleftrightarrow> ?rhs")
 proof-
   let ?mf = "matrix f"
   let ?ot = "orthogonal_transformation f"
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?U = "UNIV :: 'n set"
   have fU: "finite ?U" by simp
   let ?m1 = "mat 1 :: real ^'n^'n"
   {assume ot: ?ot
     from ot have lf: "linear f" and fd: "\<forall>v w. f v \<bullet> f w = v \<bullet> w"
       unfolding  orthogonal_transformation_def orthogonal_matrix by blast+
-    {fix i j assume i: "i \<in> ?U" and j: "j \<in> ?U"
+    {fix i j
       let ?A = "transp ?mf ** ?mf"
       have th0: "\<And>b (x::'a::comm_ring_1). (if b then 1 else 0)*x = (if b then x else 0)"
 	"\<And>b (x::'a::comm_ring_1). x*(if b then 1 else 0) = (if b then x else 0)"
 	by simp_all
-      from fd[rule_format, of "basis i" "basis j", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul] i j
+      from fd[rule_format, of "basis i" "basis j", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul]
       have "?A$i$j = ?m1 $ i $ j"
-	by (simp add: Cart_lambda_beta' dot_def matrix_matrix_mult_def columnvector_def rowvector_def basis_def th0 setsum_delta[OF fU] mat_def del: One_nat_def)}
+	by (simp add: dot_def matrix_matrix_mult_def columnvector_def rowvector_def basis_def th0 setsum_delta[OF fU] mat_def)}
     hence "orthogonal_matrix ?mf" unfolding orthogonal_matrix by vector
     with lf have ?rhs by blast}
   moreover
@@ -949,12 +874,13 @@
       unfolding orthogonal_matrix_def norm_eq orthogonal_transformation
       unfolding matrix_works[OF lf, symmetric]
       apply (subst dot_matrix_vector_mul)
-      by (simp add: dot_matrix_product matrix_mul_lid del: One_nat_def)}
+      by (simp add: dot_matrix_product matrix_mul_lid)}
   ultimately show ?thesis by blast
 qed
 
+(* FIXME: get rid of wellorder requirement *)
 lemma det_orthogonal_matrix:
-  fixes Q:: "'a::ordered_idom^'n^'n"
+  fixes Q:: "'a::ordered_idom^'n^'n::{finite,wellorder}"
   assumes oQ: "orthogonal_matrix Q"
   shows "det Q = 1 \<or> det Q = - 1"
 proof-
@@ -979,7 +905,7 @@
 (* Linearity of scaling, and hence isometry, that preserves origin.          *)
 (* ------------------------------------------------------------------------- *)
 lemma scaling_linear:
-  fixes f :: "real ^'n \<Rightarrow> real ^'n"
+  fixes f :: "real ^'n \<Rightarrow> real ^'n::finite"
   assumes f0: "f 0 = 0" and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
   shows "linear f"
 proof-
@@ -995,7 +921,7 @@
 qed
 
 lemma isometry_linear:
-  "f (0:: real^'n) = (0:: real^'n) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y
+  "f (0:: real^'n) = (0:: real^'n::finite) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y
         \<Longrightarrow> linear f"
 by (rule scaling_linear[where c=1]) simp_all
 
@@ -1004,7 +930,7 @@
 (* ------------------------------------------------------------------------- *)
 
 lemma orthogonal_transformation_isometry:
-  "orthogonal_transformation f \<longleftrightarrow> f(0::real^'n) = (0::real^'n) \<and> (\<forall>x y. dist(f x) (f y) = dist x y)"
+  "orthogonal_transformation f \<longleftrightarrow> f(0::real^'n) = (0::real^'n::finite) \<and> (\<forall>x y. dist(f x) (f y) = dist x y)"
   unfolding orthogonal_transformation
   apply (rule iffI)
   apply clarify
@@ -1023,7 +949,7 @@
 (* ------------------------------------------------------------------------- *)
 
 lemma isometry_sphere_extend:
-  fixes f:: "real ^'n \<Rightarrow> real ^'n"
+  fixes f:: "real ^'n \<Rightarrow> real ^'n::finite"
   assumes f1: "\<forall>x. norm x = 1 \<longrightarrow> norm (f x) = 1"
   and fd1: "\<forall> x y. norm x = 1 \<longrightarrow> norm y = 1 \<longrightarrow> dist (f x) (f y) = dist x y"
   shows "\<exists>g. orthogonal_transformation g \<and> (\<forall>x. norm x = 1 \<longrightarrow> g x = f x)"
@@ -1094,8 +1020,9 @@
 definition "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
 definition "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
 
+(* FIXME: get rid of wellorder requirement *)
 lemma orthogonal_rotation_or_rotoinversion:
-  fixes Q :: "'a::ordered_idom^'n^'n"
+  fixes Q :: "'a::ordered_idom^'n^'n::{finite,wellorder}"
   shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
   by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
 (* ------------------------------------------------------------------------- *)
@@ -1110,17 +1037,16 @@
   by (simp add: nat_number setprod_numseg mult_commute)
 
 lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
-  by (simp add: det_def dimindex_def permutes_sing sign_id del: One_nat_def)
+  by (simp add: det_def permutes_sing sign_id UNIV_1)
 
 lemma det_2: "det (A::'a::comm_ring_1^2^2) = A$1$1 * A$2$2 - A$1$2 * A$2$1"
 proof-
-  have f12: "finite {2::nat}" "1 \<notin> {2::nat}" by auto
-  have th12: "{1 .. 2} = insert (1::nat) {2}" by auto
+  have f12: "finite {2::2}" "1 \<notin> {2::2}" by auto
   show ?thesis
-  apply (simp add: det_def dimindex_def th12 del: One_nat_def)
+  unfolding det_def UNIV_2
   unfolding setsum_over_permutations_insert[OF f12]
   unfolding permutes_sing
-  apply (simp add: sign_swap_id sign_id swap_id_eq del: One_nat_def)
+  apply (simp add: sign_swap_id sign_id swap_id_eq)
   by (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
 qed
 
@@ -1132,18 +1058,17 @@
   A$1$2 * A$2$1 * A$3$3 -
   A$1$3 * A$2$2 * A$3$1"
 proof-
-  have f123: "finite {(2::nat), 3}" "1 \<notin> {(2::nat), 3}" by auto
-  have f23: "finite {(3::nat)}" "2 \<notin> {(3::nat)}" by auto
-  have th12: "{1 .. 3} = insert (1::nat) (insert 2 {3})" by auto
+  have f123: "finite {2::3, 3}" "1 \<notin> {2::3, 3}" by auto
+  have f23: "finite {3::3}" "2 \<notin> {3::3}" by auto
 
   show ?thesis
-  apply (simp add: det_def dimindex_def th12 del: One_nat_def)
+  unfolding det_def UNIV_3
   unfolding setsum_over_permutations_insert[OF f123]
   unfolding setsum_over_permutations_insert[OF f23]
 
   unfolding permutes_sing
-  apply (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq del: One_nat_def)
-  apply (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31) One_nat_def)
+  apply (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
+  apply (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
   by (simp add: ring_simps)
 qed
 
diff -r ac50e7dedf6d -r 638b088bb840 src/HOL/Library/Euclidean_Space.thy
--- a/src/HOL/Library/Euclidean_Space.thy	Wed Mar 18 22:17:23 2009 +0100
+++ b/src/HOL/Library/Euclidean_Space.thy	Thu Mar 19 01:29:19 2009 -0700
@@ -13,32 +13,50 @@
 
 text{* Some common special cases.*}
 
-lemma forall_1: "(\<forall>(i::'a::{order,one}). 1 <= i \<and> i <= 1 --> P i) \<longleftrightarrow> P 1"
-  by (metis order_eq_iff)
-lemma forall_dimindex_1: "(\<forall>i \<in> {1..dimindex(UNIV:: 1 set)}. P i) \<longleftrightarrow> P 1"
-  by (simp add: dimindex_def)
-
-lemma forall_2: "(\<forall>(i::nat). 1 <= i \<and> i <= 2 --> P i) \<longleftrightarrow> P 1 \<and> P 2"
-proof-
-  have "\<And>i::nat. 1 <= i \<and> i <= 2 \<longleftrightarrow> i = 1 \<or> i = 2" by arith
-  thus ?thesis by metis
+lemma forall_1: "(\<forall>i::1. P i) \<longleftrightarrow> P 1"
+  by (metis num1_eq_iff)
+
+lemma exhaust_2:
+  fixes x :: 2 shows "x = 1 \<or> x = 2"
+proof (induct x)
+  case (of_int z)
+  then have "0 <= z" and "z < 2" by simp_all
+  then have "z = 0 | z = 1" by arith
+  then show ?case by auto
 qed
 
-lemma forall_3: "(\<forall>(i::nat). 1 <= i \<and> i <= 3 --> P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
-proof-
-  have "\<And>i::nat. 1 <= i \<and> i <= 3 \<longleftrightarrow> i = 1 \<or> i = 2 \<or> i = 3" by arith
-  thus ?thesis by metis
+lemma forall_2: "(\<forall>i::2. P i) \<longleftrightarrow> P 1 \<and> P 2"
+  by (metis exhaust_2)
+
+lemma exhaust_3:
+  fixes x :: 3 shows "x = 1 \<or> x = 2 \<or> x = 3"
+proof (induct x)
+  case (of_int z)
+  then have "0 <= z" and "z < 3" by simp_all
+  then have "z = 0 \<or> z = 1 \<or> z = 2" by arith
+  then show ?case by auto
 qed
 
-lemma setsum_singleton[simp]: "setsum f {x} = f x" by simp
-lemma setsum_1: "setsum f {(1::'a::{order,one})..1} = f 1"
-  by (simp add: atLeastAtMost_singleton)
-
-lemma setsum_2: "setsum f {1::nat..2} = f 1 + f 2"
-  by (simp add: nat_number  atLeastAtMostSuc_conv add_commute)
-
-lemma setsum_3: "setsum f {1::nat..3} = f 1 + f 2 + f 3"
-  by (simp add: nat_number  atLeastAtMostSuc_conv add_commute)
+lemma forall_3: "(\<forall>i::3. P i) \<longleftrightarrow> P 1 \<and> P 2 \<and> P 3"
+  by (metis exhaust_3)
+
+lemma UNIV_1: "UNIV = {1::1}"
+  by (auto simp add: num1_eq_iff)
+
+lemma UNIV_2: "UNIV = {1::2, 2::2}"
+  using exhaust_2 by auto
+
+lemma UNIV_3: "UNIV = {1::3, 2::3, 3::3}"
+  using exhaust_3 by auto
+
+lemma setsum_1: "setsum f (UNIV::1 set) = f 1"
+  unfolding UNIV_1 by simp
+
+lemma setsum_2: "setsum f (UNIV::2 set) = f 1 + f 2"
+  unfolding UNIV_2 by simp
+
+lemma setsum_3: "setsum f (UNIV::3 set) = f 1 + f 2 + f 3"
+  unfolding UNIV_3 by (simp add: add_ac)
 
 subsection{* Basic componentwise operations on vectors. *}
 
@@ -76,10 +94,8 @@
 instantiation "^" :: (ord,type) ord
  begin
 definition vector_less_eq_def:
-  "less_eq (x :: 'a ^'b) y = (ALL i : {1 .. dimindex (UNIV :: 'b set)}.
-  x$i <= y$i)"
-definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i : {1 ..
-  dimindex (UNIV :: 'b set)}. x$i < y$i)"
+  "less_eq (x :: 'a ^'b) y = (ALL i. x$i <= y$i)"
+definition vector_less_def: "less (x :: 'a ^'b) y = (ALL i. x$i < y$i)"
 
 instance by (intro_classes)
 end
@@ -102,19 +118,19 @@
 text{* Dot products. *}
 
 definition dot :: "'a::{comm_monoid_add, times} ^ 'n \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a" (infix "\<bullet>" 70) where
-  "x \<bullet> y = setsum (\<lambda>i. x$i * y$i) {1 .. dimindex (UNIV:: 'n set)}"
+  "x \<bullet> y = setsum (\<lambda>i. x$i * y$i) UNIV"
+
 lemma dot_1[simp]: "(x::'a::{comm_monoid_add, times}^1) \<bullet> y = (x$1) * (y$1)"
-  by (simp add: dot_def dimindex_def)
+  by (simp add: dot_def setsum_1)
 
 lemma dot_2[simp]: "(x::'a::{comm_monoid_add, times}^2) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2)"
-  by (simp add: dot_def dimindex_def nat_number)
+  by (simp add: dot_def setsum_2)
 
 lemma dot_3[simp]: "(x::'a::{comm_monoid_add, times}^3) \<bullet> y = (x$1) * (y$1) + (x$2) * (y$2) + (x$3) * (y$3)"
-  by (simp add: dot_def dimindex_def nat_number)
+  by (simp add: dot_def setsum_3)
 
 subsection {* A naive proof procedure to lift really trivial arithmetic stuff from the basis of the vector space. *}
 
-lemmas Cart_lambda_beta' = Cart_lambda_beta[rule_format]
 method_setup vector = {*
 let
   val ss1 = HOL_basic_ss addsimps [@{thm dot_def}, @{thm setsum_addf} RS sym,
@@ -125,7 +141,7 @@
               @{thm vector_minus_def}, @{thm vector_uminus_def},
               @{thm vector_one_def}, @{thm vector_zero_def}, @{thm vec_def},
               @{thm vector_scaleR_def},
-              @{thm Cart_lambda_beta'}, @{thm vector_scalar_mult_def}]
+              @{thm Cart_lambda_beta}, @{thm vector_scalar_mult_def}]
  fun vector_arith_tac ths =
    simp_tac ss1
    THEN' (fn i => rtac @{thm setsum_cong2} i
@@ -145,39 +161,38 @@
 
 text{* Obvious "component-pushing". *}
 
-lemma vec_component: " i \<in> {1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (vec x :: 'a ^ 'n)$i = x"
+lemma vec_component [simp]: "(vec x :: 'a ^ 'n)$i = x"
   by (vector vec_def)
 
-lemma vector_add_component:
-  fixes x y :: "'a::{plus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
+lemma vector_add_component [simp]:
+  fixes x y :: "'a::{plus} ^ 'n"
   shows "(x + y)$i = x$i + y$i"
-  using i by vector
-
-lemma vector_minus_component:
-  fixes x y :: "'a::{minus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
+  by vector
+
+lemma vector_minus_component [simp]:
+  fixes x y :: "'a::{minus} ^ 'n"
   shows "(x - y)$i = x$i - y$i"
-  using i  by vector
-
-lemma vector_mult_component:
-  fixes x y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
+  by vector
+
+lemma vector_mult_component [simp]:
+  fixes x y :: "'a::{times} ^ 'n"
   shows "(x * y)$i = x$i * y$i"
-  using i by vector
-
-lemma vector_smult_component:
-  fixes y :: "'a::{times} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
+  by vector
+
+lemma vector_smult_component [simp]:
+  fixes y :: "'a::{times} ^ 'n"
   shows "(c *s y)$i = c * (y$i)"
-  using i by vector
-
-lemma vector_uminus_component:
-  fixes x :: "'a::{uminus} ^ 'n"  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
+  by vector
+
+lemma vector_uminus_component [simp]:
+  fixes x :: "'a::{uminus} ^ 'n"
   shows "(- x)$i = - (x$i)"
-  using i by vector
-
-lemma vector_scaleR_component:
+  by vector
+
+lemma vector_scaleR_component [simp]:
   fixes x :: "'a::scaleR ^ 'n"
-  assumes i: "i \<in> {1 .. dimindex(UNIV :: 'n set)}"
   shows "(scaleR r x)$i = scaleR r (x$i)"
-  using i by vector
+  by vector
 
 lemma cond_component: "(if b then x else y)$i = (if b then x$i else y$i)" by vector
 
@@ -250,7 +265,7 @@
 instance "^" :: (semiring_0,type) semiring_0
   apply (intro_classes) by (vector ring_simps)+
 instance "^" :: (semiring_1,type) semiring_1
-  apply (intro_classes) apply vector using dimindex_ge_1 by auto
+  apply (intro_classes) by vector
 instance "^" :: (comm_semiring,type) comm_semiring
   apply (intro_classes) by (vector ring_simps)+
 
@@ -274,16 +289,16 @@
 instance "^" :: (real_algebra_1,type) real_algebra_1 ..
 
 lemma of_nat_index:
-  "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
+  "(of_nat n :: 'a::semiring_1 ^'n)$i = of_nat n"
   apply (induct n)
   apply vector
   apply vector
   done
 lemma zero_index[simp]:
-  "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (0 :: 'a::zero ^'n)$i = 0" by vector
+  "(0 :: 'a::zero ^'n)$i = 0" by vector
 
 lemma one_index[simp]:
-  "i\<in>{1 .. dimindex (UNIV :: 'n set)} \<Longrightarrow> (1 :: 'a::one ^'n)$i = 1" by vector
+  "(1 :: 'a::one ^'n)$i = 1" by vector
 
 lemma one_plus_of_nat_neq_0: "(1::'a::semiring_char_0) + of_nat n \<noteq> 0"
 proof-
@@ -296,28 +311,7 @@
 proof (intro_classes)
   fix m n ::nat
   show "(of_nat m :: 'a^'b) = of_nat n \<longleftrightarrow> m = n"
-  proof(induct m arbitrary: n)
-    case 0 thus ?case apply vector
-      apply (induct n,auto simp add: ring_simps)
-      using dimindex_ge_1 apply auto
-      apply vector
-      by (auto simp add: of_nat_index one_plus_of_nat_neq_0)
-  next
-    case (Suc n m)
-    thus ?case  apply vector
-      apply (induct m, auto simp add: ring_simps of_nat_index zero_index)
-      using dimindex_ge_1 apply simp apply blast
-      apply (simp add: one_plus_of_nat_neq_0)
-      using dimindex_ge_1 apply simp apply blast
-      apply (simp add: vector_component one_index of_nat_index)
-      apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
-      using  dimindex_ge_1 apply simp apply blast
-      apply (simp add: vector_component one_index of_nat_index)
-      apply (simp only: of_nat.simps(2)[where ?'a = 'a, symmetric] of_nat_eq_iff)
-      using dimindex_ge_1 apply simp apply blast
-      apply (simp add: vector_component one_index of_nat_index)
-      done
-  qed
+    by (simp add: Cart_eq of_nat_index)
 qed
 
 instance "^" :: (comm_ring_1,type) comm_ring_1 by intro_classes
@@ -341,8 +335,7 @@
   by (vector ring_simps)
 
 lemma vec_eq[simp]: "(vec m = vec n) \<longleftrightarrow> (m = n)"
-  apply (auto simp add: vec_def Cart_eq vec_component Cart_lambda_beta )
-  using dimindex_ge_1 apply auto done
+  by (simp add: Cart_eq)
 
 subsection {* Square root of sum of squares *}
 
@@ -513,11 +506,11 @@
 
 subsection {* Norms *}
 
-instantiation "^" :: (real_normed_vector, type) real_normed_vector
+instantiation "^" :: (real_normed_vector, finite) real_normed_vector
 begin
 
 definition vector_norm_def:
-  "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) {1 .. dimindex (UNIV:: 'b set)}"
+  "norm (x::'a^'b) = setL2 (\<lambda>i. norm (x$i)) UNIV"
 
 definition vector_sgn_def:
   "sgn (x::'a^'b) = scaleR (inverse (norm x)) x"
@@ -533,14 +526,11 @@
   show "norm (x + y) \<le> norm x + norm y"
     unfolding vector_norm_def
     apply (rule order_trans [OF _ setL2_triangle_ineq])
-    apply (rule setL2_mono)
-    apply (simp add: vector_component norm_triangle_ineq)
-    apply simp
+    apply (simp add: setL2_mono norm_triangle_ineq)
     done
   show "norm (scaleR a x) = \<bar>a\<bar> * norm x"
     unfolding vector_norm_def
-    by (simp add: vector_component norm_scaleR setL2_right_distrib
-             cong: strong_setL2_cong)
+    by (simp add: norm_scaleR setL2_right_distrib)
   show "sgn x = scaleR (inverse (norm x)) x"
     by (rule vector_sgn_def)
 qed
@@ -549,11 +539,11 @@
 
 subsection {* Inner products *}
 
-instantiation "^" :: (real_inner, type) real_inner
+instantiation "^" :: (real_inner, finite) real_inner
 begin
 
 definition vector_inner_def:
-  "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) {1 .. dimindex(UNIV::'b set)}"
+  "inner x y = setsum (\<lambda>i. inner (x$i) (y$i)) UNIV"
 
 instance proof
   fix r :: real and x y z :: "'a ^ 'b"
@@ -562,10 +552,10 @@
     by (simp add: inner_commute)
   show "inner (x + y) z = inner x z + inner y z"
     unfolding vector_inner_def
-    by (vector inner_left_distrib)
+    by (simp add: inner_left_distrib setsum_addf)
   show "inner (scaleR r x) y = r * inner x y"
     unfolding vector_inner_def
-    by (vector inner_scaleR_left)
+    by (simp add: inner_scaleR_left setsum_right_distrib)
   show "0 \<le> inner x x"
     unfolding vector_inner_def
     by (simp add: setsum_nonneg)
@@ -613,25 +603,16 @@
   show ?case by (simp add: h)
 qed
 
-lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) = 0"
-proof-
-  {assume f: "finite (UNIV :: 'n set)"
-    let ?S = "{Suc 0 .. card (UNIV :: 'n set)}"
-    have fS: "finite ?S" using f by simp
-    have fp: "\<forall> i\<in> ?S. x$i * x$i>= 0" by simp
-    have ?thesis by (vector dimindex_def f setsum_squares_eq_0_iff[OF fS fp])}
-  moreover
-  {assume "\<not> finite (UNIV :: 'n set)" then have ?thesis by (vector dimindex_def)}
-  ultimately show ?thesis by metis
-qed
-
-lemma dot_pos_lt[simp]: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]
+lemma dot_eq_0: "x \<bullet> x = 0 \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) = 0"
+  by (simp add: dot_def setsum_squares_eq_0_iff Cart_eq)
+
+lemma dot_pos_lt[simp]: "(0 < x \<bullet> x) \<longleftrightarrow> (x::'a::{ordered_ring_strict,ring_no_zero_divisors} ^ 'n::finite) \<noteq> 0" using dot_eq_0[of x] dot_pos_le[of x]
   by (auto simp add: le_less)
 
 subsection{* The collapse of the general concepts to dimension one. *}
 
 lemma vector_one: "(x::'a ^1) = (\<chi> i. (x$1))"
-  by (vector dimindex_def)
+  by (simp add: Cart_eq forall_1)
 
 lemma forall_one: "(\<forall>(x::'a ^1). P x) \<longleftrightarrow> (\<forall>x. P(\<chi> i. x))"
   apply auto
@@ -640,7 +621,7 @@
   done
 
 lemma norm_vector_1: "norm (x :: _^1) = norm (x$1)"
-  by (simp add: vector_norm_def dimindex_def)
+  by (simp add: vector_norm_def UNIV_1)
 
 lemma norm_real: "norm(x::real ^ 1) = abs(x$1)"
   by (simp add: norm_vector_1)
@@ -648,17 +629,16 @@
 text{* Metric *}
 
 text {* FIXME: generalize to arbitrary @{text real_normed_vector} types *}
-definition dist:: "real ^ 'n \<Rightarrow> real ^ 'n \<Rightarrow> real" where
+definition dist:: "real ^ 'n::finite \<Rightarrow> real ^ 'n \<Rightarrow> real" where
   "dist x y = norm (x - y)"
 
 lemma dist_real: "dist(x::real ^ 1) y = abs((x$1) - (y$1))"
-  using dimindex_ge_1[of "UNIV :: 1 set"]
-  by (auto simp add: norm_real dist_def vector_component Cart_lambda_beta[where ?'a = "1"] )
+  by (auto simp add: norm_real dist_def)
 
 subsection {* A connectedness or intermediate value lemma with several applications. *}
 
 lemma connected_real_lemma:
-  fixes f :: "real \<Rightarrow> real ^ 'n"
+  fixes f :: "real \<Rightarrow> real ^ 'n::finite"
   assumes ab: "a \<le> b" and fa: "f a \<in> e1" and fb: "f b \<in> e2"
   and dst: "\<And>e x. a <= x \<Longrightarrow> x <= b \<Longrightarrow> 0 < e ==> \<exists>d > 0. \<forall>y. abs(y - x) < d \<longrightarrow> dist(f y) (f x) < e"
   and e1: "\<forall>y \<in> e1. \<exists>e > 0. \<forall>y'. dist y' y < e \<longrightarrow> y' \<in> e1"
@@ -758,7 +738,11 @@
 
 text{* Hence derive more interesting properties of the norm. *}
 
-lemma norm_0[simp]: "norm (0::real ^ 'n) = 0"
+text {*
+  This type-specific version is only here
+  to make @{text normarith.ML} happy.
+*}
+lemma norm_0: "norm (0::real ^ _) = 0"
   by (rule norm_zero)
 
 lemma norm_mul[simp]: "norm(a *s x) = abs(a) * norm x"
@@ -770,7 +754,7 @@
   by (simp add: vector_norm_def setL2_def dot_def power2_eq_square)
 lemma norm_pow_2: "norm x ^ 2 = x \<bullet> x"
   by (simp add: real_vector_norm_def)
-lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n)" by (metis norm_eq_zero)
+lemma norm_eq_0_imp: "norm x = 0 ==> x = (0::real ^'n::finite)" by (metis norm_eq_zero)
 lemma vector_mul_eq_0[simp]: "(a *s x = 0) \<longleftrightarrow> a = (0::'a::idom) \<or> x = 0"
   by vector
 lemma vector_mul_lcancel[simp]: "a *s x = a *s y \<longleftrightarrow> a = (0::real) \<or> x = y"
@@ -781,7 +765,9 @@
   by (metis vector_mul_lcancel)
 lemma vector_mul_rcancel_imp: "x \<noteq> 0 \<Longrightarrow> (a::real) *s x = b *s x ==> a = b"
   by (metis vector_mul_rcancel)
-lemma norm_cauchy_schwarz: "x \<bullet> y <= norm x * norm y"
+lemma norm_cauchy_schwarz:
+  fixes x y :: "real ^ 'n::finite"
+  shows "x \<bullet> y <= norm x * norm y"
 proof-
   {assume "norm x = 0"
     hence ?thesis by (simp add: dot_lzero dot_rzero)}
@@ -802,50 +788,74 @@
   ultimately show ?thesis by metis
 qed
 
-lemma norm_cauchy_schwarz_abs: "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
+lemma norm_cauchy_schwarz_abs:
+  fixes x y :: "real ^ 'n::finite"
+  shows "\<bar>x \<bullet> y\<bar> \<le> norm x * norm y"
   using norm_cauchy_schwarz[of x y] norm_cauchy_schwarz[of x "-y"]
   by (simp add: real_abs_def dot_rneg)
 
-lemma norm_triangle_sub: "norm (x::real ^'n) <= norm(y) + norm(x - y)"
+lemma norm_triangle_sub: "norm (x::real ^'n::finite) <= norm(y) + norm(x - y)"
   using norm_triangle_ineq[of "y" "x - y"] by (simp add: ring_simps)
-lemma norm_triangle_le: "norm(x::real ^'n) + norm y <= e ==> norm(x + y) <= e"
+lemma norm_triangle_le: "norm(x::real ^'n::finite) + norm y <= e ==> norm(x + y) <= e"
   by (metis order_trans norm_triangle_ineq)
-lemma norm_triangle_lt: "norm(x::real ^'n) + norm(y) < e ==> norm(x + y) < e"
+lemma norm_triangle_lt: "norm(x::real ^'n::finite) + norm(y) < e ==> norm(x + y) < e"
   by (metis basic_trans_rules(21) norm_triangle_ineq)
 
-lemma component_le_norm: "i \<in> {1 .. dimindex(UNIV :: 'n set)} ==> \<bar>x$i\<bar> <= norm (x::real ^ 'n)"
+lemma setsum_delta:
+  assumes fS: "finite S"
+  shows "setsum (\<lambda>k. if k=a then b k else 0) S = (if a \<in> S then b a else 0)"
+proof-
+  let ?f = "(\<lambda>k. if k=a then b k else 0)"
+  {assume a: "a \<notin> S"
+    hence "\<forall> k\<in> S. ?f k = 0" by simp
+    hence ?thesis  using a by simp}
+  moreover
+  {assume a: "a \<in> S"
+    let ?A = "S - {a}"
+    let ?B = "{a}"
+    have eq: "S = ?A \<union> ?B" using a by blast
+    have dj: "?A \<inter> ?B = {}" by simp
+    from fS have fAB: "finite ?A" "finite ?B" by auto
+    have "setsum ?f S = setsum ?f ?A + setsum ?f ?B"
+      using setsum_Un_disjoint[OF fAB dj, of ?f, unfolded eq[symmetric]]
+      by simp
+    then have ?thesis  using a by simp}
+  ultimately show ?thesis by blast
+qed
+
+lemma component_le_norm: "\<bar>x$i\<bar> <= norm (x::real ^ 'n::finite)"
   apply (simp add: vector_norm_def)
   apply (rule member_le_setL2, simp_all)
   done
 
-lemma norm_bound_component_le: "norm(x::real ^ 'n) <= e
-                ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> <= e"
+lemma norm_bound_component_le: "norm(x::real ^ 'n::finite) <= e
+                ==> \<bar>x$i\<bar> <= e"
   by (metis component_le_norm order_trans)
 
-lemma norm_bound_component_lt: "norm(x::real ^ 'n) < e
-                ==> \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. \<bar>x$i\<bar> < e"
+lemma norm_bound_component_lt: "norm(x::real ^ 'n::finite) < e
+                ==> \<bar>x$i\<bar> < e"
   by (metis component_le_norm basic_trans_rules(21))
 
-lemma norm_le_l1: "norm (x:: real ^'n) <= setsum(\<lambda>i. \<bar>x$i\<bar>) {1..dimindex(UNIV::'n set)}"
+lemma norm_le_l1: "norm (x:: real ^'n::finite) <= setsum(\<lambda>i. \<bar>x$i\<bar>) UNIV"
   by (simp add: vector_norm_def setL2_le_setsum)
 
-lemma real_abs_norm[simp]: "\<bar> norm x\<bar> = norm (x :: real ^'n)"
+lemma real_abs_norm: "\<bar>norm x\<bar> = norm (x :: real ^ _)"
   by (rule abs_norm_cancel)
-lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n) - norm y\<bar> <= norm(x - y)"
+lemma real_abs_sub_norm: "\<bar>norm(x::real ^'n::finite) - norm y\<bar> <= norm(x - y)"
   by (rule norm_triangle_ineq3)
-lemma norm_le: "norm(x::real ^ 'n) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
+lemma norm_le: "norm(x::real ^ _) <= norm(y) \<longleftrightarrow> x \<bullet> x <= y \<bullet> y"
   by (simp add: real_vector_norm_def)
-lemma norm_lt: "norm(x::real ^'n) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
+lemma norm_lt: "norm(x::real ^ _) < norm(y) \<longleftrightarrow> x \<bullet> x < y \<bullet> y"
   by (simp add: real_vector_norm_def)
-lemma norm_eq: "norm (x::real ^'n) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
+lemma norm_eq: "norm (x::real ^ _) = norm y \<longleftrightarrow> x \<bullet> x = y \<bullet> y"
   by (simp add: order_eq_iff norm_le)
-lemma norm_eq_1: "norm(x::real ^ 'n) = 1 \<longleftrightarrow> x \<bullet> x = 1"
+lemma norm_eq_1: "norm(x::real ^ _) = 1 \<longleftrightarrow> x \<bullet> x = 1"
   by (simp add: real_vector_norm_def)
 
 text{* Squaring equations and inequalities involving norms.  *}
 
 lemma dot_square_norm: "x \<bullet> x = norm(x)^2"
-  by (simp add: real_vector_norm_def  dot_pos_le )
+  by (simp add: real_vector_norm_def)
 
 lemma norm_eq_square: "norm(x) = a \<longleftrightarrow> 0 <= a \<and> x \<bullet> x = a^2"
   by (auto simp add: real_vector_norm_def)
@@ -885,7 +895,7 @@
 
 text{* Equality of vectors in terms of @{term "op \<bullet>"} products.    *}
 
-lemma vector_eq: "(x:: real ^ 'n) = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y\<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
+lemma vector_eq: "(x:: real ^ 'n::finite) = y \<longleftrightarrow> x \<bullet> x = x \<bullet> y\<and> y \<bullet> y = x \<bullet> x" (is "?lhs \<longleftrightarrow> ?rhs")
 proof
   assume "?lhs" then show ?rhs by simp
 next
@@ -907,7 +917,7 @@
   done
 
   (* FIXME: Move all these theorems into the ML code using lemma antiquotation *)
-lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
+lemma norm_add_rule_thm: "b1 >= norm(x1 :: real ^'n::finite) \<Longrightarrow> b2 >= norm(x2) ==> b1 + b2 >= norm(x1 + x2)"
   apply (rule norm_triangle_le) by simp
 
 lemma ge_iff_diff_ge_0: "(a::'a::ordered_ring) \<ge> b == a - b \<ge> 0"
@@ -936,13 +946,13 @@
   "(c *s x + w) + (d *s y + z) == d *s y + ((c *s x + w) + z)" by ((atomize (full)), vector)+
 lemma pth_d: "x + (0::real ^'n) == x" by (atomize (full)) vector
 
-lemma norm_imp_pos_and_ge: "norm (x::real ^ 'n) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
+lemma norm_imp_pos_and_ge: "norm (x::real ^ _) == n \<Longrightarrow> norm x \<ge> 0 \<and> n \<ge> norm x"
   by (atomize) (auto simp add: norm_ge_zero)
 
 lemma real_eq_0_iff_le_ge_0: "(x::real) = 0 == x \<ge> 0 \<and> -x \<ge> 0" by arith
 
 lemma norm_pths:
-  "(x::real ^'n) = y \<longleftrightarrow> norm (x - y) \<le> 0"
+  "(x::real ^'n::finite) = y \<longleftrightarrow> norm (x - y) \<le> 0"
   "x \<noteq> y \<longleftrightarrow> \<not> (norm (x - y) \<le> 0)"
   using norm_ge_zero[of "x - y"] by auto
 
@@ -988,13 +998,13 @@
 
 lemma dist_le_0[simp]: "dist x y <= 0 \<longleftrightarrow> x = y" by norm
 
+lemma setsum_component [simp]:
+  fixes f:: " 'a \<Rightarrow> ('b::comm_monoid_add) ^'n"
+  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
+  by (cases "finite S", induct S set: finite, simp_all)
+
 lemma setsum_eq: "setsum f S = (\<chi> i. setsum (\<lambda>x. (f x)$i ) S)"
-  apply vector
-  apply auto
-  apply (cases "finite S")
-  apply (rule finite_induct[of S])
-  apply (auto simp add: vector_component zero_index)
-  done
+  by (simp add: Cart_eq)
 
 lemma setsum_clauses:
   shows "setsum f {} = 0"
@@ -1005,13 +1015,7 @@
 lemma setsum_cmul:
   fixes f:: "'c \<Rightarrow> ('a::semiring_1)^'n"
   shows "setsum (\<lambda>x. c *s f x) S = c *s setsum f S"
-  by (simp add: setsum_eq Cart_eq Cart_lambda_beta vector_component setsum_right_distrib)
-
-lemma setsum_component:
-  fixes f:: " 'a \<Rightarrow> ('b::semiring_1) ^'n"
-  assumes i: "i \<in> {1 .. dimindex(UNIV:: 'n set)}"
-  shows "(setsum f S)$i = setsum (\<lambda>x. (f x)$i) S"
-  using i by (simp add: setsum_eq Cart_lambda_beta)
+  by (simp add: Cart_eq setsum_right_distrib)
 
 lemma setsum_norm:
   fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
@@ -1028,7 +1032,7 @@
 qed
 
 lemma real_setsum_norm:
-  fixes f :: "'a \<Rightarrow> real ^'n"
+  fixes f :: "'a \<Rightarrow> real ^'n::finite"
   assumes fS: "finite S"
   shows "norm (setsum f S) <= setsum (\<lambda>x. norm(f x)) S"
 proof(induct rule: finite_induct[OF fS])
@@ -1054,7 +1058,7 @@
 qed
 
 lemma real_setsum_norm_le:
-  fixes f :: "'a \<Rightarrow> real ^ 'n"
+  fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
   assumes fS: "finite S"
   and fg: "\<forall>x \<in> S. norm (f x) \<le> g x"
   shows "norm (setsum f S) \<le> setsum g S"
@@ -1074,7 +1078,7 @@
   by simp
 
 lemma real_setsum_norm_bound:
-  fixes f :: "'a \<Rightarrow> real ^ 'n"
+  fixes f :: "'a \<Rightarrow> real ^ 'n::finite"
   assumes fS: "finite S"
   and K: "\<forall>x \<in> S. norm (f x) \<le> K"
   shows "norm (setsum f S) \<le> of_nat (card S) * K"
@@ -1155,13 +1159,13 @@
 by (auto intro: setsum_0')
 
 lemma vsum_norm_allsubsets_bound:
-  fixes f:: "'a \<Rightarrow> real ^'n"
+  fixes f:: "'a \<Rightarrow> real ^'n::finite"
   assumes fP: "finite P" and fPs: "\<And>Q. Q \<subseteq> P \<Longrightarrow> norm (setsum f Q) \<le> e"
-  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real (dimindex(UNIV :: 'n set)) *  e"
+  shows "setsum (\<lambda>x. norm (f x)) P \<le> 2 * real CARD('n) *  e"
 proof-
-  let ?d = "real (dimindex (UNIV ::'n set))"
+  let ?d = "real CARD('n)"
   let ?nf = "\<lambda>x. norm (f x)"
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?U = "UNIV :: 'n set"
   have th0: "setsum (\<lambda>x. setsum (\<lambda>i. \<bar>f x $ i\<bar>) ?U) P = setsum (\<lambda>i. setsum (\<lambda>x. \<bar>f x $ i\<bar>) P) ?U"
     by (rule setsum_commute)
   have th1: "2 * ?d * e = of_nat (card ?U) * (2 * e)" by (simp add: real_of_nat_def)
@@ -1178,11 +1182,11 @@
     have thp0: "?Pp \<inter> ?Pn ={}" by auto
     have PpP: "?Pp \<subseteq> P" and PnP: "?Pn \<subseteq> P" by blast+
     have Ppe:"setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp \<le> e"
-      using i component_le_norm[OF i, of "setsum (\<lambda>x. f x) ?Pp"]  fPs[OF PpP]
-      by (auto simp add: setsum_component intro: abs_le_D1)
+      using component_le_norm[of "setsum (\<lambda>x. f x) ?Pp" i]  fPs[OF PpP]
+      by (auto intro: abs_le_D1)
     have Pne: "setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn \<le> e"
-      using i component_le_norm[OF i, of "setsum (\<lambda>x. - f x) ?Pn"]  fPs[OF PnP]
-      by (auto simp add: setsum_negf setsum_component vector_component intro: abs_le_D1)
+      using component_le_norm[of "setsum (\<lambda>x. - f x) ?Pn" i]  fPs[OF PnP]
+      by (auto simp add: setsum_negf intro: abs_le_D1)
     have "setsum (\<lambda>x. \<bar>f x $ i\<bar>) P = setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pp + setsum (\<lambda>x. \<bar>f x $ i\<bar>) ?Pn"
       apply (subst thp)
       apply (rule setsum_Un_zero)
@@ -1204,32 +1208,29 @@
 
 definition "basis k = (\<chi> i. if i = k then 1 else 0)"
 
+lemma basis_component [simp]: "basis k $ i = (if k=i then 1 else 0)"
+  unfolding basis_def by simp
+
 lemma delta_mult_idempotent:
   "(if k=a then 1 else (0::'a::semiring_1)) * (if k=a then 1 else 0) = (if k=a then 1 else 0)" by (cases "k=a", auto)
 
 lemma norm_basis:
-  assumes k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
-  shows "norm (basis k :: real ^'n) = 1"
-  using k
+  shows "norm (basis k :: real ^'n::finite) = 1"
   apply (simp add: basis_def real_vector_norm_def dot_def)
   apply (vector delta_mult_idempotent)
-  using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "k" "\<lambda>k. 1::real"]
+  using setsum_delta[of "UNIV :: 'n set" "k" "\<lambda>k. 1::real"]
   apply auto
   done
 
-lemma norm_basis_1: "norm(basis 1 :: real ^'n) = 1"
-  apply (simp add: basis_def real_vector_norm_def dot_def)
-  apply (vector delta_mult_idempotent)
-  using setsum_delta[of "{1 .. dimindex (UNIV :: 'n set)}" "1" "\<lambda>k. 1::real"] dimindex_nonzero[of "UNIV :: 'n set"]
-  apply auto
-  done
-
-lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n). norm x = c"
-  apply (rule exI[where x="c *s basis 1"])
-  by (simp only: norm_mul norm_basis_1)
+lemma norm_basis_1: "norm(basis 1 :: real ^'n::{finite,one}) = 1"
+  by (rule norm_basis)
+
+lemma vector_choose_size: "0 <= c ==> \<exists>(x::real^'n::finite). norm x = c"
+  apply (rule exI[where x="c *s basis arbitrary"])
+  by (simp only: norm_mul norm_basis)
 
 lemma vector_choose_dist: assumes e: "0 <= e"
-  shows "\<exists>(y::real^'n). dist x y = e"
+  shows "\<exists>(y::real^'n::finite). dist x y = e"
 proof-
   from vector_choose_size[OF e] obtain c:: "real ^'n"  where "norm c = e"
     by blast
@@ -1237,56 +1238,50 @@
   then show ?thesis by blast
 qed
 
-lemma basis_inj: "inj_on (basis :: nat \<Rightarrow> real ^'n) {1 .. dimindex (UNIV :: 'n set)}"
-  by (auto simp add: inj_on_def basis_def Cart_eq Cart_lambda_beta)
-
-lemma basis_component: "i \<in> {1 .. dimindex(UNIV:: 'n set)} ==> (basis k ::('a::semiring_1)^'n)$i = (if k=i then 1 else 0)"
-  by (simp add: basis_def Cart_lambda_beta)
+lemma basis_inj: "inj (basis :: 'n \<Rightarrow> real ^'n::finite)"
+  by (simp add: inj_on_def Cart_eq)
 
 lemma cond_value_iff: "f (if b then x else y) = (if b then f x else f y)"
   by auto
 
 lemma basis_expansion:
-  "setsum (\<lambda>i. (x$i) *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::ring_1) ^'n)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
-  by (auto simp add: Cart_eq basis_component[where ?'n = "'n"] setsum_component vector_component cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
+  "setsum (\<lambda>i. (x$i) *s basis i) UNIV = (x::('a::ring_1) ^'n::finite)" (is "?lhs = ?rhs" is "setsum ?f ?S = _")
+  by (auto simp add: Cart_eq cond_value_iff setsum_delta[of "?S", where ?'b = "'a", simplified] cong del: if_weak_cong)
 
 lemma basis_expansion_unique:
-  "setsum (\<lambda>i. f i *s basis i) {1 .. dimindex (UNIV :: 'n set)} = (x::('a::comm_ring_1) ^'n) \<longleftrightarrow> (\<forall>i\<in>{1 .. dimindex(UNIV:: 'n set)}. f i = x$i)"
-  by (simp add: Cart_eq setsum_component vector_component basis_component setsum_delta cond_value_iff cong del: if_weak_cong)
+  "setsum (\<lambda>i. f i *s basis i) UNIV = (x::('a::comm_ring_1) ^'n::finite) \<longleftrightarrow> (\<forall>i. f i = x$i)"
+  by (simp add: Cart_eq setsum_delta cond_value_iff cong del: if_weak_cong)
 
 lemma cond_application_beta: "(if b then f else g) x = (if b then f x else g x)"
   by auto
 
 lemma dot_basis:
-  assumes i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
-  shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n) = (x$i :: 'a::semiring_1)"
-  using i
-  by (auto simp add: dot_def basis_def Cart_lambda_beta cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
-
-lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> i \<notin> {1..dimindex(UNIV ::'n set)}"
-  by (auto simp add: Cart_eq basis_component zero_index)
+  shows "basis i \<bullet> x = x$i" "x \<bullet> (basis i :: 'a^'n::finite) = (x$i :: 'a::semiring_1)"
+  by (auto simp add: dot_def basis_def cond_application_beta  cond_value_iff setsum_delta cong del: if_weak_cong)
+
+lemma basis_eq_0: "basis i = (0::'a::semiring_1^'n) \<longleftrightarrow> False"
+  by (auto simp add: Cart_eq)
 
 lemma basis_nonzero:
-  assumes k: "k \<in> {1 .. dimindex(UNIV ::'n set)}"
   shows "basis k \<noteq> (0:: 'a::semiring_1 ^'n)"
-  using k by (simp add: basis_eq_0)
-
-lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n)"
+  by (simp add: basis_eq_0)
+
+lemma vector_eq_ldot: "(\<forall>x. x \<bullet> y = x \<bullet> z) \<longleftrightarrow> y = (z::'a::semiring_1^'n::finite)"
   apply (auto simp add: Cart_eq dot_basis)
   apply (erule_tac x="basis i" in allE)
   apply (simp add: dot_basis)
   apply (subgoal_tac "y = z")
   apply simp
-  apply vector
+  apply (simp add: Cart_eq)
   done
 
-lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n)"
+lemma vector_eq_rdot: "(\<forall>z. x \<bullet> z = y \<bullet> z) \<longleftrightarrow> x = (y::'a::semiring_1^'n::finite)"
   apply (auto simp add: Cart_eq dot_basis)
   apply (erule_tac x="basis i" in allE)
   apply (simp add: dot_basis)
   apply (subgoal_tac "x = y")
   apply simp
-  apply vector
+  apply (simp add: Cart_eq)
   done
 
 subsection{* Orthogonality. *}
@@ -1294,16 +1289,12 @@
 definition "orthogonal x y \<longleftrightarrow> (x \<bullet> y = 0)"
 
 lemma orthogonal_basis:
-  assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}"
-  shows "orthogonal (basis i :: 'a^'n) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
-  using i
-  by (auto simp add: orthogonal_def dot_def basis_def Cart_lambda_beta cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
+  shows "orthogonal (basis i :: 'a^'n::finite) x \<longleftrightarrow> x$i = (0::'a::ring_1)"
+  by (auto simp add: orthogonal_def dot_def basis_def cond_value_iff cond_application_beta setsum_delta cong del: if_weak_cong)
 
 lemma orthogonal_basis_basis:
-  assumes i:"i \<in> {1 .. dimindex(UNIV ::'n set)}"
-  and j: "j \<in> {1 .. dimindex(UNIV ::'n set)}"
-  shows "orthogonal (basis i :: 'a::ring_1^'n) (basis j) \<longleftrightarrow> i \<noteq> j"
-  unfolding orthogonal_basis[OF i] basis_component[OF i] by simp
+  shows "orthogonal (basis i :: 'a::ring_1^'n::finite) (basis j) \<longleftrightarrow> i \<noteq> j"
+  unfolding orthogonal_basis[of i] basis_component[of j] by simp
 
   (* FIXME : Maybe some of these require less than comm_ring, but not all*)
 lemma orthogonal_clauses:
@@ -1326,51 +1317,43 @@
 
 subsection{* Explicit vector construction from lists. *}
 
-lemma Cart_lambda_beta_1[simp]: "(Cart_lambda g)$1 = g 1"
-  apply (rule Cart_lambda_beta[rule_format])
-  using dimindex_ge_1 apply auto done
-
-lemma Cart_lambda_beta_1'[simp]: "(Cart_lambda g)$(Suc 0) = g 1"
-  by (simp only: One_nat_def[symmetric] Cart_lambda_beta_1)
-
-definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"
+primrec from_nat :: "nat \<Rightarrow> 'a::{monoid_add,one}"
+where "from_nat 0 = 0" | "from_nat (Suc n) = 1 + from_nat n"
+
+lemma from_nat [simp]: "from_nat = of_nat"
+by (rule ext, induct_tac x, simp_all)
+
+primrec
+  list_fun :: "nat \<Rightarrow> _ list \<Rightarrow> _ \<Rightarrow> _"
+where
+  "list_fun n [] = (\<lambda>x. 0)"
+| "list_fun n (x # xs) = fun_upd (list_fun (Suc n) xs) (from_nat n) x"
+
+definition "vector l = (\<chi> i. list_fun 1 l i)"
+(*definition "vector l = (\<chi> i. if i <= length l then l ! (i - 1) else 0)"*)
 
 lemma vector_1: "(vector[x]) $1 = x"
-  using dimindex_ge_1
-  by (auto simp add: vector_def Cart_lambda_beta[rule_format])
-lemma dimindex_2[simp]: "2 \<in> {1 .. dimindex (UNIV :: 2 set)}"
-  by (auto simp add: dimindex_def)
-lemma dimindex_2'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 2 set)}"
-  by (auto simp add: dimindex_def)
-lemma dimindex_3[simp]: "2 \<in> {1 .. dimindex (UNIV :: 3 set)}" "3 \<in> {1 .. dimindex (UNIV :: 3 set)}"
-  by (auto simp add: dimindex_def)
-
-lemma dimindex_3'[simp]: "2 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}" "3 \<in> {Suc 0 .. dimindex (UNIV :: 3 set)}"
-  by (auto simp add: dimindex_def)
+  unfolding vector_def by simp
 
 lemma vector_2:
  "(vector[x,y]) $1 = x"
  "(vector[x,y] :: 'a^2)$2 = (y::'a::zero)"
-  apply (simp add: vector_def)
-  using Cart_lambda_beta[rule_format, OF dimindex_2, of "\<lambda>i. if i \<le> length [x,y] then [x,y] ! (i - 1) else (0::'a)"]
-  apply (simp only: vector_def )
-  apply auto
-  done
+  unfolding vector_def by simp_all
 
 lemma vector_3:
  "(vector [x,y,z] ::('a::zero)^3)$1 = x"
  "(vector [x,y,z] ::('a::zero)^3)$2 = y"
  "(vector [x,y,z] ::('a::zero)^3)$3 = z"
-apply (simp_all add: vector_def Cart_lambda_beta dimindex_3)
-  using Cart_lambda_beta[rule_format, OF dimindex_3(1), of "\<lambda>i. if i \<le> length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"]   using Cart_lambda_beta[rule_format, OF dimindex_3(2), of "\<lambda>i. if i \<le> length [x,y,z] then [x,y,z] ! (i - 1) else (0::'a)"]
-  by simp_all
+  unfolding vector_def by simp_all
 
 lemma forall_vector_1: "(\<forall>v::'a::zero^1. P v) \<longleftrightarrow> (\<forall>x. P(vector[x]))"
   apply auto
   apply (erule_tac x="v$1" in allE)
   apply (subgoal_tac "vector [v$1] = v")
   apply simp
-  by (vector vector_def dimindex_def)
+  apply (vector vector_def)
+  apply (simp add: forall_1)
+  done
 
 lemma forall_vector_2: "(\<forall>v::'a::zero^2. P v) \<longleftrightarrow> (\<forall>x y. P(vector[x, y]))"
   apply auto
@@ -1378,9 +1361,8 @@
   apply (erule_tac x="v$2" in allE)
   apply (subgoal_tac "vector [v$1, v$2] = v")
   apply simp
-  apply (vector vector_def dimindex_def)
-  apply auto
-  apply (subgoal_tac "i = 1 \<or> i =2", auto)
+  apply (vector vector_def)
+  apply (simp add: forall_2)
   done
 
 lemma forall_vector_3: "(\<forall>v::'a::zero^3. P v) \<longleftrightarrow> (\<forall>x y z. P(vector[x, y, z]))"
@@ -1390,9 +1372,8 @@
   apply (erule_tac x="v$3" in allE)
   apply (subgoal_tac "vector [v$1, v$2, v$3] = v")
   apply simp
-  apply (vector vector_def dimindex_def)
-  apply auto
-  apply (subgoal_tac "i = 1 \<or> i =2 \<or> i = 3", auto)
+  apply (vector vector_def)
+  apply (simp add: forall_3)
   done
 
 subsection{* Linear functions. *}
@@ -1400,7 +1381,7 @@
 definition "linear f \<longleftrightarrow> (\<forall>x y. f(x + y) = f x + f y) \<and> (\<forall>c x. f(c *s x) = c *s f x)"
 
 lemma linear_compose_cmul: "linear f ==> linear (\<lambda>x. (c::'a::comm_semiring) *s f x)"
-  by (vector linear_def Cart_eq Cart_lambda_beta[rule_format] ring_simps)
+  by (vector linear_def Cart_eq ring_simps)
 
 lemma linear_compose_neg: "linear (f :: 'a ^'n \<Rightarrow> 'a::comm_ring ^'m) ==> linear (\<lambda>x. -(f(x)))" by (vector linear_def Cart_eq)
 
@@ -1426,9 +1407,9 @@
 
 lemma linear_vmul_component:
   fixes f:: "'a::semiring_1^'m \<Rightarrow> 'a^'n"
-  assumes lf: "linear f" and k: "k \<in> {1 .. dimindex (UNIV :: 'n set)}"
+  assumes lf: "linear f"
   shows "linear (\<lambda>x. f x $ k *s v)"
-  using lf k
+  using lf
   apply (auto simp add: linear_def )
   by (vector ring_simps)+
 
@@ -1485,15 +1466,15 @@
 qed
 
 lemma linear_bounded:
-  fixes f:: "real ^'m \<Rightarrow> real ^'n"
+  fixes f:: "real ^'m::finite \<Rightarrow> real ^'n::finite"
   assumes lf: "linear f"
   shows "\<exists>B. \<forall>x. norm (f x) \<le> B * norm x"
 proof-
-  let ?S = "{1..dimindex(UNIV:: 'm set)}"
+  let ?S = "UNIV:: 'm set"
   let ?B = "setsum (\<lambda>i. norm(f(basis i))) ?S"
   have fS: "finite ?S" by simp
   {fix x:: "real ^ 'm"
-    let ?g = "(\<lambda>i::nat. (x$i) *s (basis i) :: real ^ 'm)"
+    let ?g = "(\<lambda>i. (x$i) *s (basis i) :: real ^ 'm)"
     have "norm (f x) = norm (f (setsum (\<lambda>i. (x$i) *s (basis i)) ?S))"
       by (simp only:  basis_expansion)
     also have "\<dots> = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)"
@@ -1501,7 +1482,7 @@
       by auto
     finally have th0: "norm (f x) = norm (setsum (\<lambda>i. (x$i) *s f (basis i))?S)" .
     {fix i assume i: "i \<in> ?S"
-      from component_le_norm[OF i, of x]
+      from component_le_norm[of x i]
       have "norm ((x$i) *s f (basis i :: real ^'m)) \<le> norm (f (basis i)) * norm x"
       unfolding norm_mul
       apply (simp only: mult_commute)
@@ -1514,7 +1495,7 @@
 qed
 
 lemma linear_bounded_pos:
-  fixes f:: "real ^'n \<Rightarrow> real ^ 'm"
+  fixes f:: "real ^'n::finite \<Rightarrow> real ^ 'm::finite"
   assumes lf: "linear f"
   shows "\<exists>B > 0. \<forall>x. norm (f x) \<le> B * norm x"
 proof-
@@ -1595,12 +1576,12 @@
 qed
 
 lemma bilinear_bounded:
-  fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
+  fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
   assumes bh: "bilinear h"
   shows "\<exists>B. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
 proof-
-  let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
-  let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?M = "UNIV :: 'm set"
+  let ?N = "UNIV :: 'n set"
   let ?B = "setsum (\<lambda>(i,j). norm (h (basis i) (basis j))) (?M \<times> ?N)"
   have fM: "finite ?M" and fN: "finite ?N" by simp_all
   {fix x:: "real ^ 'm" and  y :: "real^'n"
@@ -1622,7 +1603,7 @@
 qed
 
 lemma bilinear_bounded_pos:
-  fixes h:: "real ^'m \<Rightarrow> real^'n \<Rightarrow> real ^ 'k"
+  fixes h:: "real ^'m::finite \<Rightarrow> real^'n::finite \<Rightarrow> real ^ 'k::finite"
   assumes bh: "bilinear h"
   shows "\<exists>B > 0. \<forall>x y. norm (h x y) \<le> B * norm x * norm y"
 proof-
@@ -1649,12 +1630,12 @@
 lemma choice_iff: "(\<forall>x. \<exists>y. P x y) \<longleftrightarrow> (\<exists>f. \<forall>x. P x (f x))" by metis
 
 lemma adjoint_works_lemma:
-  fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^ 'm"
+  fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
   assumes lf: "linear f"
   shows "\<forall>x y. f x \<bullet> y = x \<bullet> adjoint f y"
 proof-
-  let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
-  let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
+  let ?N = "UNIV :: 'n set"
+  let ?M = "UNIV :: 'm set"
   have fN: "finite ?N" by simp
   have fM: "finite ?M" by simp
   {fix y:: "'a ^ 'm"
@@ -1667,7 +1648,7 @@
 	by (simp add: linear_cmul[OF lf])
       finally have "f x \<bullet> y = x \<bullet> ?w"
 	apply (simp only: )
-	apply (simp add: dot_def setsum_component Cart_lambda_beta setsum_left_distrib setsum_right_distrib vector_component setsum_commute[of _ ?M ?N] ring_simps del: One_nat_def)
+	apply (simp add: dot_def setsum_left_distrib setsum_right_distrib setsum_commute[of _ ?M ?N] ring_simps)
 	done}
   }
   then show ?thesis unfolding adjoint_def
@@ -1677,34 +1658,34 @@
 qed
 
 lemma adjoint_works:
-  fixes f:: "'a::ring_1 ^'n \<Rightarrow> 'a ^ 'm"
+  fixes f:: "'a::ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
   assumes lf: "linear f"
   shows "x \<bullet> adjoint f y = f x \<bullet> y"
   using adjoint_works_lemma[OF lf] by metis
 
 
 lemma adjoint_linear:
-  fixes f :: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^ 'm"
+  fixes f :: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
   assumes lf: "linear f"
   shows "linear (adjoint f)"
   by (simp add: linear_def vector_eq_ldot[symmetric] dot_radd dot_rmult adjoint_works[OF lf])
 
 lemma adjoint_clauses:
-  fixes f:: "'a::comm_ring_1 ^'n \<Rightarrow> 'a ^ 'm"
+  fixes f:: "'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a ^ 'm::finite"
   assumes lf: "linear f"
   shows "x \<bullet> adjoint f y = f x \<bullet> y"
   and "adjoint f y \<bullet> x = y \<bullet> f x"
   by (simp_all add: adjoint_works[OF lf] dot_sym )
 
 lemma adjoint_adjoint:
-  fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> _"
+  fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
   assumes lf: "linear f"
   shows "adjoint (adjoint f) = f"
   apply (rule ext)
   by (simp add: vector_eq_ldot[symmetric] adjoint_clauses[OF adjoint_linear[OF lf]] adjoint_clauses[OF lf])
 
 lemma adjoint_unique:
-  fixes f:: "'a::comm_ring_1 ^ 'n \<Rightarrow> 'a ^ 'm"
+  fixes f:: "'a::comm_ring_1 ^ 'n::finite \<Rightarrow> 'a ^ 'm::finite"
   assumes lf: "linear f" and u: "\<forall>x y. f' x \<bullet> y = x \<bullet> f y"
   shows "f' = adjoint f"
   apply (rule ext)
@@ -1716,14 +1697,14 @@
 consts generic_mult :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" (infixr "\<star>" 75)
 
 defs (overloaded)
-matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) \<star> (m' :: 'a ^'p^'n) \<equiv> (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) {1 .. dimindex (UNIV :: 'n set)}) ::'a ^ 'p ^'m"
+matrix_matrix_mult_def: "(m:: ('a::semiring_1) ^'n^'m) \<star> (m' :: 'a ^'p^'n) \<equiv> (\<chi> i j. setsum (\<lambda>k. ((m$i)$k) * ((m'$k)$j)) (UNIV :: 'n set)) ::'a ^ 'p ^'m"
 
 abbreviation
   matrix_matrix_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'p^'n \<Rightarrow> 'a ^ 'p ^'m"  (infixl "**" 70)
   where "m ** m' == m\<star> m'"
 
 defs (overloaded)
-  matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) {1..dimindex(UNIV ::'n set)}) :: 'a^'m"
+  matrix_vector_mult_def: "(m::('a::semiring_1) ^'n^'m) \<star> (x::'a ^'n) \<equiv> (\<chi> i. setsum (\<lambda>j. ((m$i)$j) * (x$j)) (UNIV ::'n set)) :: 'a^'m"
 
 abbreviation
   matrix_vector_mult' :: "('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n \<Rightarrow> 'a ^ 'm"  (infixl "*v" 70)
@@ -1731,19 +1712,19 @@
   "m *v v == m \<star> v"
 
 defs (overloaded)
-  vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) {1..dimindex(UNIV :: 'm set)}) :: 'a^'n"
+  vector_matrix_mult_def: "(x::'a^'m) \<star> (m::('a::semiring_1) ^'n^'m) \<equiv> (\<chi> j. setsum (\<lambda>i. ((m$i)$j) * (x$i)) (UNIV :: 'm set)) :: 'a^'n"
 
 abbreviation
   vactor_matrix_mult' :: "'a ^ 'm \<Rightarrow> ('a::semiring_1) ^'n^'m \<Rightarrow> 'a ^'n "  (infixl "v*" 70)
   where
   "v v* m == v \<star> m"
 
-definition "(mat::'a::zero => 'a ^'n^'m) k = (\<chi> i j. if i = j then k else 0)"
+definition "(mat::'a::zero => 'a ^'n^'n) k = (\<chi> i j. if i = j then k else 0)"
 definition "(transp::'a^'n^'m \<Rightarrow> 'a^'m^'n) A = (\<chi> i j. ((A$j)$i))"
-definition "(row::nat => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
-definition "(column::nat =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
-definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> {1 .. dimindex(UNIV :: 'm set)}}"
-definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> {1 .. dimindex(UNIV :: 'n set)}}"
+definition "(row::'m => 'a ^'n^'m \<Rightarrow> 'a ^'n) i A = (\<chi> j. ((A$i)$j))"
+definition "(column::'n =>'a^'n^'m =>'a^'m) j A = (\<chi> i. ((A$i)$j))"
+definition "rows(A::'a^'n^'m) = { row i A | i. i \<in> (UNIV :: 'm set)}"
+definition "columns(A::'a^'n^'m) = { column i A | i. i \<in> (UNIV :: 'n set)}"
 
 lemma mat_0[simp]: "mat 0 = 0" by (vector mat_def)
 lemma matrix_add_ldistrib: "(A ** (B + C)) = (A \<star> B) + (A \<star> C)"
@@ -1756,16 +1737,20 @@
   using setsum_delta[OF fS, of a b, symmetric]
   by (auto intro: setsum_cong)
 
-lemma matrix_mul_lid: "mat 1 ** A = A"
+lemma matrix_mul_lid:
+  fixes A :: "'a::semiring_1 ^ 'm ^ 'n::finite"
+  shows "mat 1 ** A = A"
   apply (simp add: matrix_matrix_mult_def mat_def)
   apply vector
-  by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite_atLeastAtMost]  mult_1_left mult_zero_left if_True)
-
-
-lemma matrix_mul_rid: "A ** mat 1 = A"
+  by (auto simp only: cond_value_iff cond_application_beta setsum_delta'[OF finite]  mult_1_left mult_zero_left if_True UNIV_I)
+
+
+lemma matrix_mul_rid:
+  fixes A :: "'a::semiring_1 ^ 'm::finite ^ 'n"
+  shows "A ** mat 1 = A"
   apply (simp add: matrix_matrix_mult_def mat_def)
   apply vector
-  by (auto simp only: cond_value_iff cond_application_beta setsum_delta[OF finite_atLeastAtMost]  mult_1_right mult_zero_right if_True cong: if_cong)
+  by (auto simp only: cond_value_iff cond_application_beta setsum_delta[OF finite]  mult_1_right mult_zero_right if_True UNIV_I cong: if_cong)
 
 lemma matrix_mul_assoc: "A ** (B ** C) = (A ** B) ** C"
   apply (vector matrix_matrix_mult_def setsum_right_distrib setsum_left_distrib mult_assoc)
@@ -1779,31 +1764,31 @@
   apply simp
   done
 
-lemma matrix_vector_mul_lid: "mat 1 *v x = x"
+lemma matrix_vector_mul_lid: "mat 1 *v x = (x::'a::semiring_1 ^ 'n::finite)"
   apply (vector matrix_vector_mult_def mat_def)
   by (simp add: cond_value_iff cond_application_beta
     setsum_delta' cong del: if_weak_cong)
 
 lemma matrix_transp_mul: "transp(A ** B) = transp B ** transp (A::'a::comm_semiring_1^'m^'n)"
-  by (simp add: matrix_matrix_mult_def transp_def Cart_eq Cart_lambda_beta mult_commute)
-
-lemma matrix_eq: "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
+  by (simp add: matrix_matrix_mult_def transp_def Cart_eq mult_commute)
+
+lemma matrix_eq:
+  fixes A B :: "'a::semiring_1 ^ 'n::finite ^ 'm"
+  shows "A = B \<longleftrightarrow>  (\<forall>x. A *v x = B *v x)" (is "?lhs \<longleftrightarrow> ?rhs")
   apply auto
   apply (subst Cart_eq)
   apply clarify
-  apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq Cart_lambda_beta cong del: if_weak_cong)
+  apply (clarsimp simp add: matrix_vector_mult_def basis_def cond_value_iff cond_application_beta Cart_eq cong del: if_weak_cong)
   apply (erule_tac x="basis ia" in allE)
-  apply (erule_tac x="i" in ballE)
-  by (auto simp add: basis_def cond_value_iff cond_application_beta Cart_lambda_beta setsum_delta[OF finite_atLeastAtMost] cong del: if_weak_cong)
+  apply (erule_tac x="i" in allE)
+  by (auto simp add: basis_def cond_value_iff cond_application_beta setsum_delta[OF finite] cong del: if_weak_cong)
 
 lemma matrix_vector_mul_component:
-  assumes k: "k \<in> {1.. dimindex (UNIV :: 'm set)}"
   shows "((A::'a::semiring_1^'n'^'m) *v x)$k = (A$k) \<bullet> x"
-  using k
-  by (simp add: matrix_vector_mult_def Cart_lambda_beta dot_def)
+  by (simp add: matrix_vector_mult_def dot_def)
 
 lemma dot_lmul_matrix: "((x::'a::comm_semiring_1 ^'n) v* A) \<bullet> y = x \<bullet> (A *v y)"
-  apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib Cart_lambda_beta mult_ac)
+  apply (simp add: dot_def matrix_vector_mult_def vector_matrix_mult_def setsum_left_distrib setsum_right_distrib mult_ac)
   apply (subst setsum_commute)
   by simp
 
@@ -1815,23 +1800,16 @@
 
 lemma row_transp:
   fixes A:: "'a::semiring_1^'n^'m"
-  assumes i: "i \<in> {1.. dimindex (UNIV :: 'n set)}"
   shows "row i (transp A) = column i A"
-  using i
-  by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
+  by (simp add: row_def column_def transp_def Cart_eq)
 
 lemma column_transp:
   fixes A:: "'a::semiring_1^'n^'m"
-  assumes i: "i \<in> {1.. dimindex (UNIV :: 'm set)}"
   shows "column i (transp A) = row i A"
-  using i
-  by (simp add: row_def column_def transp_def Cart_eq Cart_lambda_beta)
+  by (simp add: row_def column_def transp_def Cart_eq)
 
 lemma rows_transp: "rows(transp (A::'a::semiring_1^'n^'m)) = columns A"
-apply (auto simp add: rows_def columns_def row_transp intro: set_ext)
-apply (rule_tac x=i in exI)
-apply (auto simp add: row_transp)
-done
+by (auto simp add: rows_def columns_def row_transp intro: set_ext)
 
 lemma columns_transp: "columns(transp (A::'a::semiring_1^'n^'m)) = rows A" by (metis transp_transp rows_transp)
 
@@ -1840,25 +1818,25 @@
 lemma matrix_mult_dot: "A *v x = (\<chi> i. A$i \<bullet> x)"
   by (simp add: matrix_vector_mult_def dot_def)
 
-lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) {1 .. dimindex(UNIV:: 'n set)}"
-  by (simp add: matrix_vector_mult_def Cart_eq setsum_component Cart_lambda_beta vector_component column_def mult_commute)
+lemma matrix_mult_vsum: "(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s column i A) (UNIV:: 'n set)"
+  by (simp add: matrix_vector_mult_def Cart_eq column_def mult_commute)
 
 lemma vector_componentwise:
-  "(x::'a::ring_1^'n) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) {1..dimindex(UNIV :: 'n set)})"
+  "(x::'a::ring_1^'n::finite) = (\<chi> j. setsum (\<lambda>i. (x$i) * (basis i :: 'a^'n)$j) (UNIV :: 'n set))"
   apply (subst basis_expansion[symmetric])
-  by (vector Cart_eq Cart_lambda_beta setsum_component)
+  by (vector Cart_eq setsum_component)
 
 lemma linear_componentwise:
-  fixes f:: "'a::ring_1 ^ 'm \<Rightarrow> 'a ^ 'n"
-  assumes lf: "linear f" and j: "j \<in> {1 .. dimindex (UNIV :: 'n set)}"
-  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) {1 .. dimindex (UNIV :: 'm set)}" (is "?lhs = ?rhs")
+  fixes f:: "'a::ring_1 ^ 'm::finite \<Rightarrow> 'a ^ 'n"
+  assumes lf: "linear f"
+  shows "(f x)$j = setsum (\<lambda>i. (x$i) * (f (basis i)$j)) (UNIV :: 'm set)" (is "?lhs = ?rhs")
 proof-
-  let ?M = "{1 .. dimindex (UNIV :: 'm set)}"
-  let ?N = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?M = "(UNIV :: 'm set)"
+  let ?N = "(UNIV :: 'n set)"
   have fM: "finite ?M" by simp
   have "?rhs = (setsum (\<lambda>i.(x$i) *s f (basis i) ) ?M)$j"
-    unfolding vector_smult_component[OF j, symmetric]
-    unfolding setsum_component[OF j, of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
+    unfolding vector_smult_component[symmetric]
+    unfolding setsum_component[of "(\<lambda>i.(x$i) *s f (basis i :: 'a^'m))" ?M]
     ..
   then show ?thesis unfolding linear_setsum_mul[OF lf fM, symmetric] basis_expansion ..
 qed
@@ -1876,38 +1854,38 @@
 where "matrix f = (\<chi> i j. (f(basis j))$i)"
 
 lemma matrix_vector_mul_linear: "linear(\<lambda>x. A *v (x::'a::comm_semiring_1 ^ 'n))"
-  by (simp add: linear_def matrix_vector_mult_def Cart_eq Cart_lambda_beta vector_component ring_simps setsum_right_distrib setsum_addf)
-
-lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n)"
-apply (simp add: matrix_def matrix_vector_mult_def Cart_eq Cart_lambda_beta mult_commute del: One_nat_def)
+  by (simp add: linear_def matrix_vector_mult_def Cart_eq ring_simps setsum_right_distrib setsum_addf)
+
+lemma matrix_works: assumes lf: "linear f" shows "matrix f *v x = f (x::'a::comm_ring_1 ^ 'n::finite)"
+apply (simp add: matrix_def matrix_vector_mult_def Cart_eq mult_commute)
 apply clarify
 apply (rule linear_componentwise[OF lf, symmetric])
-apply simp
 done
 
-lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n))" by (simp add: ext matrix_works)
-
-lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n)) = A"
+lemma matrix_vector_mul: "linear f ==> f = (\<lambda>x. matrix f *v (x::'a::comm_ring_1 ^ 'n::finite))" by (simp add: ext matrix_works)
+
+lemma matrix_of_matrix_vector_mul: "matrix(\<lambda>x. A *v (x :: 'a:: comm_ring_1 ^ 'n::finite)) = A"
   by (simp add: matrix_eq matrix_vector_mul_linear matrix_works)
 
 lemma matrix_compose:
-  assumes lf: "linear (f::'a::comm_ring_1^'n \<Rightarrow> _)" and lg: "linear g"
+  assumes lf: "linear (f::'a::comm_ring_1^'n::finite \<Rightarrow> 'a^'m::finite)"
+  and lg: "linear (g::'a::comm_ring_1^'m::finite \<Rightarrow> 'a^'k)"
   shows "matrix (g o f) = matrix g ** matrix f"
   using lf lg linear_compose[OF lf lg] matrix_works[OF linear_compose[OF lf lg]]
   by (simp  add: matrix_eq matrix_works matrix_vector_mul_assoc[symmetric] o_def)
 
-lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s ((transp A)$i)) {1..dimindex(UNIV:: 'n set)}"
-  by (simp add: matrix_vector_mult_def transp_def Cart_eq Cart_lambda_beta setsum_component vector_component mult_commute)
-
-lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n^'m) *v x) = (\<lambda>x. transp A *v x)"
+lemma matrix_vector_column:"(A::'a::comm_semiring_1^'n^'m) *v x = setsum (\<lambda>i. (x$i) *s ((transp A)$i)) (UNIV:: 'n set)"
+  by (simp add: matrix_vector_mult_def transp_def Cart_eq mult_commute)
+
+lemma adjoint_matrix: "adjoint(\<lambda>x. (A::'a::comm_ring_1^'n::finite^'m::finite) *v x) = (\<lambda>x. transp A *v x)"
   apply (rule adjoint_unique[symmetric])
   apply (rule matrix_vector_mul_linear)
-  apply (simp add: transp_def dot_def Cart_lambda_beta matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
+  apply (simp add: transp_def dot_def matrix_vector_mult_def setsum_left_distrib setsum_right_distrib)
   apply (subst setsum_commute)
   apply (auto simp add: mult_ac)
   done
 
-lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n \<Rightarrow> 'a ^ 'm)"
+lemma matrix_adjoint: assumes lf: "linear (f :: 'a::comm_ring_1^'n::finite \<Rightarrow> 'a ^ 'm::finite)"
   shows "matrix(adjoint f) = transp(matrix f)"
   apply (subst matrix_vector_mul[OF lf])
   unfolding adjoint_matrix matrix_of_matrix_vector_mul ..
@@ -1980,21 +1958,21 @@
 qed
 
 
-lemma lambda_skolem: "(\<forall>i \<in> {1 .. dimindex(UNIV :: 'n set)}. \<exists>x. P i x) \<longleftrightarrow>
-   (\<exists>x::'a ^ 'n. \<forall>i \<in> {1 .. dimindex(UNIV:: 'n set)}. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
+lemma lambda_skolem: "(\<forall>i. \<exists>x. P i x) \<longleftrightarrow>
+   (\<exists>x::'a ^ 'n. \<forall>i. P i (x$i))" (is "?lhs \<longleftrightarrow> ?rhs")
 proof-
-  let ?S = "{1 .. dimindex(UNIV :: 'n set)}"
+  let ?S = "(UNIV :: 'n set)"
   {assume H: "?rhs"
     then have ?lhs by auto}
   moreover
   {assume H: "?lhs"
-    then obtain f where f:"\<forall>i\<in> ?S. P i (f i)" unfolding Ball_def choice_iff by metis
+    then obtain f where f:"\<forall>i. P i (f i)" unfolding choice_iff by metis
     let ?x = "(\<chi> i. (f i)) :: 'a ^ 'n"
-    {fix i assume i: "i \<in> ?S"
-      with f i have "P i (f i)" by metis
-      then have "P i (?x$i)" using Cart_lambda_beta[of f, rule_format, OF i] by auto
+    {fix i
+      from f have "P i (f i)" by metis
+      then have "P i (?x$i)" by auto
     }
-    hence "\<forall>i \<in> ?S. P i (?x$i)" by metis
+    hence "\<forall>i. P i (?x$i)" by metis
     hence ?rhs by metis }
   ultimately show ?thesis by metis
 qed
@@ -2237,7 +2215,7 @@
 definition "onorm f = rsup {norm (f x)| x. norm x = 1}"
 
 lemma norm_bound_generalize:
-  fixes f:: "real ^'n \<Rightarrow> real^'m"
+  fixes f:: "real ^'n::finite \<Rightarrow> real^'m::finite"
   assumes lf: "linear f"
   shows "(\<forall>x. norm x = 1 \<longrightarrow> norm (f x) \<le> b) \<longleftrightarrow> (\<forall>x. norm (f x) \<le> b * norm x)" (is "?lhs \<longleftrightarrow> ?rhs")
 proof-
@@ -2248,8 +2226,8 @@
 
   moreover
   {assume H: ?lhs
-    from H[rule_format, of "basis 1"]
-    have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis 1)"] dimindex_ge_1[of "UNIV:: 'n set"]
+    from H[rule_format, of "basis arbitrary"]
+    have bp: "b \<ge> 0" using norm_ge_zero[of "f (basis arbitrary)"]
       by (auto simp add: norm_basis elim: order_trans [OF norm_ge_zero])
     {fix x :: "real ^'n"
       {assume "x = 0"
@@ -2270,14 +2248,14 @@
 qed
 
 lemma onorm:
-  fixes f:: "real ^'n \<Rightarrow> real ^'m"
+  fixes f:: "real ^'n::finite \<Rightarrow> real ^'m::finite"
   assumes lf: "linear f"
   shows "norm (f x) <= onorm f * norm x"
   and "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
 proof-
   {
     let ?S = "{norm (f x) |x. norm x = 1}"
-    have Se: "?S \<noteq> {}" using  norm_basis_1 by auto
+    have Se: "?S \<noteq> {}" using  norm_basis by auto
     from linear_bounded[OF lf] have b: "\<exists> b. ?S *<= b"
       unfolding norm_bound_generalize[OF lf, symmetric] by (auto simp add: setle_def)
     {from rsup[OF Se b, unfolded onorm_def[symmetric]]
@@ -2294,10 +2272,10 @@
   }
 qed
 
-lemma onorm_pos_le: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" shows "0 <= onorm f"
-  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis 1"], unfolded norm_basis_1] by simp
-
-lemma onorm_eq_0: assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)"
+lemma onorm_pos_le: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" shows "0 <= onorm f"
+  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "basis arbitrary"], unfolded norm_basis] by simp
+
+lemma onorm_eq_0: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)"
   shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
   using onorm[OF lf]
   apply (auto simp add: onorm_pos_le)
@@ -2307,7 +2285,7 @@
   apply arith
   done
 
-lemma onorm_const: "onorm(\<lambda>x::real^'n. (y::real ^ 'm)) = norm y"
+lemma onorm_const: "onorm(\<lambda>x::real^'n::finite. (y::real ^ 'm::finite)) = norm y"
 proof-
   let ?f = "\<lambda>x::real^'n. (y::real ^ 'm)"
   have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
@@ -2317,13 +2295,14 @@
     apply (rule rsup_unique) by (simp_all  add: setle_def)
 qed
 
-lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n \<Rightarrow> real ^'m)"
+lemma onorm_pos_lt: assumes lf: "linear (f::real ^ 'n::finite \<Rightarrow> real ^'m::finite)"
   shows "0 < onorm f \<longleftrightarrow> ~(\<forall>x. f x = 0)"
   unfolding onorm_eq_0[OF lf, symmetric]
   using onorm_pos_le[OF lf] by arith
 
 lemma onorm_compose:
-  assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"
+  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)"
+  and lg: "linear (g::real^'k::finite \<Rightarrow> real^'n::finite)"
   shows "onorm (f o g) <= onorm f * onorm g"
   apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
   unfolding o_def
@@ -2335,18 +2314,18 @@
   apply (rule onorm_pos_le[OF lf])
   done
 
-lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
+lemma onorm_neg_lemma: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real^'m::finite)"
   shows "onorm (\<lambda>x. - f x) \<le> onorm f"
   using onorm[OF linear_compose_neg[OF lf]] onorm[OF lf]
   unfolding norm_minus_cancel by metis
 
-lemma onorm_neg: assumes lf: "linear (f::real ^'n \<Rightarrow> real^'m)"
+lemma onorm_neg: assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real^'m::finite)"
   shows "onorm (\<lambda>x. - f x) = onorm f"
   using onorm_neg_lemma[OF lf] onorm_neg_lemma[OF linear_compose_neg[OF lf]]
   by simp
 
 lemma onorm_triangle:
-  assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and lg: "linear g"
+  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" and lg: "linear g"
   shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
   apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
   apply (rule order_trans)
@@ -2357,14 +2336,14 @@
   apply (rule onorm(1)[OF lg])
   done
 
-lemma onorm_triangle_le: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
+lemma onorm_triangle_le: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
   \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
   apply (rule order_trans)
   apply (rule onorm_triangle)
   apply assumption+
   done
 
-lemma onorm_triangle_lt: "linear (f::real ^'n \<Rightarrow> real ^'m) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
+lemma onorm_triangle_lt: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
   ==> onorm(\<lambda>x. f x + g x) < e"
   apply (rule order_le_less_trans)
   apply (rule onorm_triangle)
@@ -2381,7 +2360,7 @@
   by (simp add: vec1_def)
 
 lemma vec1_dest_vec1[simp]: "vec1(dest_vec1 x) = x" "dest_vec1(vec1 y) = y"
-  by (simp_all add: vec1_def dest_vec1_def Cart_eq Cart_lambda_beta dimindex_def del: One_nat_def)
+  by (simp_all add: vec1_def dest_vec1_def Cart_eq forall_1)
 
 lemma forall_vec1: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (vec1 x))" by (metis vec1_dest_vec1)
 
@@ -2451,21 +2430,21 @@
   shows "linear f \<Longrightarrow> linear (\<lambda>x. dest_vec1(f x) *s v)"
   unfolding dest_vec1_def
   apply (rule linear_vmul_component)
-  by (auto simp add: dimindex_def)
+  by auto
 
 lemma linear_from_scalars:
   assumes lf: "linear (f::'a::comm_ring_1 ^1 \<Rightarrow> 'a^'n)"
   shows "f = (\<lambda>x. dest_vec1 x *s column 1 (matrix f))"
   apply (rule ext)
   apply (subst matrix_works[OF lf, symmetric])
-  apply (auto simp add: Cart_eq matrix_vector_mult_def dest_vec1_def column_def Cart_lambda_beta vector_component dimindex_def mult_commute del: One_nat_def )
+  apply (auto simp add: Cart_eq matrix_vector_mult_def dest_vec1_def column_def  mult_commute UNIV_1)
   done
 
-lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n \<Rightarrow> 'a^1)"
+lemma linear_to_scalars: assumes lf: "linear (f::'a::comm_ring_1 ^'n::finite \<Rightarrow> 'a^1)"
   shows "f = (\<lambda>x. vec1(row 1 (matrix f) \<bullet> x))"
   apply (rule ext)
   apply (subst matrix_works[OF lf, symmetric])
-  apply (auto simp add: Cart_eq matrix_vector_mult_def vec1_def row_def Cart_lambda_beta vector_component dimindex_def dot_def mult_commute)
+  apply (simp add: Cart_eq matrix_vector_mult_def vec1_def row_def dot_def mult_commute forall_1)
   done
 
 lemma dest_vec1_eq_0: "dest_vec1 x = 0 \<longleftrightarrow> x = 0"
@@ -2485,25 +2464,25 @@
 text{* Pasting vectors. *}
 
 lemma linear_fstcart: "linear fstcart"
-  by (auto simp add: linear_def fstcart_def Cart_eq Cart_lambda_beta vector_component dimindex_finite_sum)
+  by (auto simp add: linear_def Cart_eq)
 
 lemma linear_sndcart: "linear sndcart"
-  by (auto simp add: linear_def sndcart_def Cart_eq Cart_lambda_beta vector_component dimindex_finite_sum)
+  by (auto simp add: linear_def Cart_eq)
 
 lemma fstcart_vec[simp]: "fstcart(vec x) = vec x"
-  by (vector fstcart_def vec_def dimindex_finite_sum)
-
-lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b,'c) finite_sum) + fstcart y"
-  using linear_fstcart[unfolded linear_def] by blast
-
-lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b,'c) finite_sum)"
-  using linear_fstcart[unfolded linear_def] by blast
-
-lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^('b,'c) finite_sum)"
-unfolding vector_sneg_minus1 fstcart_cmul ..
-
-lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^('b,'c) finite_sum) - fstcart y"
-  unfolding diff_def fstcart_add fstcart_neg  ..
+  by (simp add: Cart_eq)
+
+lemma fstcart_add[simp]:"fstcart(x + y) = fstcart (x::'a::{plus,times}^('b + 'c)) + fstcart y"
+  by (simp add: Cart_eq)
+
+lemma fstcart_cmul[simp]:"fstcart(c*s x) = c*s fstcart (x::'a::{plus,times}^('b + 'c))"
+  by (simp add: Cart_eq)
+
+lemma fstcart_neg[simp]:"fstcart(- x) = - fstcart (x::'a::ring_1^('b + 'c))"
+  by (simp add: Cart_eq)
+
+lemma fstcart_sub[simp]:"fstcart(x - y) = fstcart (x::'a::ring_1^('b + 'c)) - fstcart y"
+  by (simp add: Cart_eq)
 
 lemma fstcart_setsum:
   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
@@ -2512,19 +2491,19 @@
   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
 
 lemma sndcart_vec[simp]: "sndcart(vec x) = vec x"
-  by (vector sndcart_def vec_def dimindex_finite_sum)
-
-lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^('b,'c) finite_sum) + sndcart y"
-  using linear_sndcart[unfolded linear_def] by blast
-
-lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^('b,'c) finite_sum)"
-  using linear_sndcart[unfolded linear_def] by blast
-
-lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^('b,'c) finite_sum)"
-unfolding vector_sneg_minus1 sndcart_cmul ..
-
-lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^('b,'c) finite_sum) - sndcart y"
-  unfolding diff_def sndcart_add sndcart_neg  ..
+  by (simp add: Cart_eq)
+
+lemma sndcart_add[simp]:"sndcart(x + y) = sndcart (x::'a::{plus,times}^('b + 'c)) + sndcart y"
+  by (simp add: Cart_eq)
+
+lemma sndcart_cmul[simp]:"sndcart(c*s x) = c*s sndcart (x::'a::{plus,times}^('b + 'c))"
+  by (simp add: Cart_eq)
+
+lemma sndcart_neg[simp]:"sndcart(- x) = - sndcart (x::'a::ring_1^('b + 'c))"
+  by (simp add: Cart_eq)
+
+lemma sndcart_sub[simp]:"sndcart(x - y) = sndcart (x::'a::ring_1^('b + 'c)) - sndcart y"
+  by (simp add: Cart_eq)
 
 lemma sndcart_setsum:
   fixes f:: "'d \<Rightarrow> 'a::semiring_1^_"
@@ -2533,10 +2512,10 @@
   by (induct rule: finite_induct[OF fS], simp_all add: vec_0[symmetric] del: vec_0)
 
 lemma pastecart_vec[simp]: "pastecart (vec x) (vec x) = vec x"
-  by (simp add: pastecart_eq fstcart_vec sndcart_vec fstcart_pastecart sndcart_pastecart)
+  by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
 
 lemma pastecart_add[simp]:"pastecart (x1::'a::{plus,times}^_) y1 + pastecart x2 y2 = pastecart (x1 + x2) (y1 + y2)"
-  by (simp add: pastecart_eq fstcart_add sndcart_add fstcart_pastecart sndcart_pastecart)
+  by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
 
 lemma pastecart_cmul[simp]: "pastecart (c *s (x1::'a::{plus,times}^_)) (c *s y1) = c *s pastecart x1 y1"
   by (simp add: pastecart_eq fstcart_pastecart sndcart_pastecart)
@@ -2553,109 +2532,53 @@
   shows "pastecart (setsum f S) (setsum g S) = setsum (\<lambda>i. pastecart (f i) (g i)) S"
   by (simp  add: pastecart_eq fstcart_setsum[OF fS] sndcart_setsum[OF fS] fstcart_pastecart sndcart_pastecart)
 
-lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n,'m) finite_sum)"
+lemma setsum_Plus:
+  "\<lbrakk>finite A; finite B\<rbrakk> \<Longrightarrow>
+    (\<Sum>x\<in>A <+> B. g x) = (\<Sum>x\<in>A. g (Inl x)) + (\<Sum>x\<in>B. g (Inr x))"
+  unfolding Plus_def
+  by (subst setsum_Un_disjoint, auto simp add: setsum_reindex)
+
+lemma setsum_UNIV_sum:
+  fixes g :: "'a::finite + 'b::finite \<Rightarrow> _"
+  shows "(\<Sum>x\<in>UNIV. g x) = (\<Sum>x\<in>UNIV. g (Inl x)) + (\<Sum>x\<in>UNIV. g (Inr x))"
+  apply (subst UNIV_Plus_UNIV [symmetric])
+  apply (rule setsum_Plus [OF finite finite])
+  done
+
+lemma norm_fstcart: "norm(fstcart x) <= norm (x::real ^('n::finite + 'm::finite))"
 proof-
-  let ?n = "dimindex (UNIV :: 'n set)"
-  let ?m = "dimindex (UNIV :: 'm set)"
-  let ?N = "{1 .. ?n}"
-  let ?M = "{1 .. ?m}"
-  let ?NM = "{1 .. dimindex (UNIV :: ('n,'m) finite_sum set)}"
-  have th_0: "1 \<le> ?n +1" by simp
   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
     by (simp add: pastecart_fst_snd)
   have th1: "fstcart x \<bullet> fstcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
-    by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square del: One_nat_def)
+    by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg)
   then show ?thesis
     unfolding th0
     unfolding real_vector_norm_def real_sqrt_le_iff id_def
-    by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
+    by (simp add: dot_def)
 qed
 
 lemma dist_fstcart: "dist(fstcart (x::real^_)) (fstcart y) <= dist x y"
   by (metis dist_def fstcart_sub[symmetric] norm_fstcart)
 
-lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n,'m) finite_sum)"
+lemma norm_sndcart: "norm(sndcart x) <= norm (x::real ^('n::finite + 'm::finite))"
 proof-
-  let ?n = "dimindex (UNIV :: 'n set)"
-  let ?m = "dimindex (UNIV :: 'm set)"
-  let ?N = "{1 .. ?n}"
-  let ?M = "{1 .. ?m}"
-  let ?nm = "dimindex (UNIV :: ('n,'m) finite_sum set)"
-  let ?NM = "{1 .. ?nm}"
-  have thnm[simp]: "?nm = ?n + ?m" by (simp add: dimindex_finite_sum)
-  have th_0: "1 \<le> ?n +1" by simp
   have th0: "norm x = norm (pastecart (fstcart x) (sndcart x))"
     by (simp add: pastecart_fst_snd)
-  let ?f = "\<lambda>n. n - ?n"
-  let ?S = "{?n+1 .. ?nm}"
-  have finj:"inj_on ?f ?S"
-    using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"]
-    apply (simp add: Ball_def atLeastAtMost_iff inj_on_def dimindex_finite_sum del: One_nat_def)
-    by arith
-  have fS: "?f ` ?S = ?M"
-    apply (rule set_ext)
-    apply (simp add: image_iff Bex_def) using dimindex_nonzero[of "UNIV :: 'n set"] dimindex_nonzero[of "UNIV :: 'm set"] by arith
   have th1: "sndcart x \<bullet> sndcart x \<le> pastecart (fstcart x) (sndcart x) \<bullet> pastecart (fstcart x) (sndcart x)"
-    by (simp add: dot_def setsum_add_split[OF th_0, of _ ?m] pastecart_def dimindex_finite_sum Cart_lambda_beta setsum_nonneg zero_le_square setsum_reindex[OF finj, unfolded fS] del: One_nat_def)
+    by (simp add: dot_def setsum_UNIV_sum pastecart_def setsum_nonneg)
   then show ?thesis
     unfolding th0
     unfolding real_vector_norm_def real_sqrt_le_iff id_def
-    by (simp add: dot_def dimindex_finite_sum Cart_lambda_beta)
+    by (simp add: dot_def)
 qed
 
 lemma dist_sndcart: "dist(sndcart (x::real^_)) (sndcart y) <= dist x y"
   by (metis dist_def sndcart_sub[symmetric] norm_sndcart)
 
-lemma dot_pastecart: "(pastecart (x1::'a::{times,comm_monoid_add}^'n) (x2::'a::{times,comm_monoid_add}^'m)) \<bullet> (pastecart y1 y2) =  x1 \<bullet> y1 + x2 \<bullet> y2"
-proof-
-  let ?n = "dimindex (UNIV :: 'n set)"
-  let ?m = "dimindex (UNIV :: 'm set)"
-  let ?N = "{1 .. ?n}"
-  let ?M = "{1 .. ?m}"
-  let ?nm = "dimindex (UNIV :: ('n,'m) finite_sum set)"
-  let ?NM = "{1 .. ?nm}"
-  have thnm: "?nm = ?n + ?m" by (simp add: dimindex_finite_sum)
-  have th_0: "1 \<le> ?n +1" by simp
-  have th_1: "\<And>i. i \<in> {?m + 1 .. ?nm} \<Longrightarrow> i - ?m \<in> ?N" apply (simp add: thnm) by arith
-  let ?f = "\<lambda>a b i. (a$i) * (b$i)"
-  let ?g = "?f (pastecart x1 x2) (pastecart y1 y2)"
-  let ?S = "{?n +1 .. ?nm}"
-  {fix i
-    assume i: "i \<in> ?N"
-    have "?g i = ?f x1 y1 i"
-      using i
-      apply (simp add: pastecart_def Cart_lambda_beta thnm) done
-  }
-  hence th2: "setsum ?g ?N = setsum (?f x1 y1) ?N"
-    apply -
-    apply (rule setsum_cong)
-    apply auto
-    done
-  {fix i
-    assume i: "i \<in> ?S"
-    have "?g i = ?f x2 y2 (i - ?n)"
-      using i
-      apply (simp add: pastecart_def Cart_lambda_beta thnm) done
-  }
-  hence th3: "setsum ?g ?S = setsum (\<lambda>i. ?f x2 y2 (i -?n)) ?S"
-    apply -
-    apply (rule setsum_cong)
-    apply auto
-    done
-  let ?r = "\<lambda>n. n - ?n"
-  have rinj: "inj_on ?r ?S" apply (simp add: inj_on_def Ball_def thnm) by arith
-  have rS: "?r ` ?S = ?M" apply (rule set_ext)
-    apply (simp add: thnm image_iff Bex_def) by arith
-  have "pastecart x1 x2 \<bullet> (pastecart y1 y2) = setsum ?g ?NM" by (simp add: dot_def)
-  also have "\<dots> = setsum ?g ?N + setsum ?g ?S"
-    by (simp add: dot_def thnm setsum_add_split[OF th_0, of _ ?m] del: One_nat_def)
-  also have "\<dots> = setsum (?f x1 y1) ?N + setsum (?f x2 y2) ?M"
-    unfolding setsum_reindex[OF rinj, unfolded rS o_def] th2 th3 ..
-  finally
-  show ?thesis by (simp add: dot_def)
-qed
-
-lemma norm_pastecart: "norm(pastecart x y) <= norm(x :: real ^ _) + norm(y)"
+lemma dot_pastecart: "(pastecart (x1::'a::{times,comm_monoid_add}^'n::finite) (x2::'a::{times,comm_monoid_add}^'m::finite)) \<bullet> (pastecart y1 y2) =  x1 \<bullet> y1 + x2 \<bullet> y2"
+  by (simp add: dot_def setsum_UNIV_sum pastecart_def)
+
+lemma norm_pastecart: "norm(pastecart x y) <= norm(x :: real ^ 'm::finite) + norm(y::real^'n::finite)"
   unfolding real_vector_norm_def dot_pastecart real_sqrt_le_iff id_def
   apply (rule power2_le_imp_le)
   apply (simp add: real_sqrt_pow2[OF add_nonneg_nonneg[OF dot_pos_le[of x] dot_pos_le[of y]]])
@@ -3419,7 +3342,7 @@
 
 (* Standard bases are a spanning set, and obviously finite.                  *)
 
-lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n | i. i \<in> {1 .. dimindex(UNIV :: 'n set)}} = UNIV"
+lemma span_stdbasis:"span {basis i :: 'a::ring_1^'n::finite | i. i \<in> (UNIV :: 'n set)} = UNIV"
 apply (rule set_ext)
 apply auto
 apply (subst basis_expansion[symmetric])
@@ -3431,47 +3354,43 @@
 apply (auto simp add: Collect_def mem_def)
 done
 
-
-lemma has_size_stdbasis: "{basis i ::real ^'n | i. i \<in> {1 .. dimindex (UNIV :: 'n set)}} hassize (dimindex(UNIV :: 'n set))" (is "?S hassize ?n")
+lemma has_size_stdbasis: "{basis i ::real ^'n::finite | i. i \<in> (UNIV :: 'n set)} hassize CARD('n)" (is "?S hassize ?n")
 proof-
-  have eq: "?S = basis ` {1 .. ?n}" by blast
+  have eq: "?S = basis ` UNIV" by blast
   show ?thesis unfolding eq
     apply (rule hassize_image_inj[OF basis_inj])
     by (simp add: hassize_def)
 qed
 
-lemma finite_stdbasis: "finite {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV:: 'n set)}}"
+lemma finite_stdbasis: "finite {basis i ::real^'n::finite |i. i\<in> (UNIV:: 'n set)}"
   using has_size_stdbasis[unfolded hassize_def]
   ..
 
-lemma card_stdbasis: "card {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} = dimindex(UNIV :: 'n set)"
+lemma card_stdbasis: "card {basis i ::real^'n::finite |i. i\<in> (UNIV :: 'n set)} = CARD('n)"
   using has_size_stdbasis[unfolded hassize_def]
   ..
 
 lemma independent_stdbasis_lemma:
   assumes x: "(x::'a::semiring_1 ^ 'n) \<in> span (basis ` S)"
-  and i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
   and iS: "i \<notin> S"
   shows "(x$i) = 0"
 proof-
-  let ?n = "dimindex (UNIV :: 'n set)"
-  let ?U = "{1 .. ?n}"
+  let ?U = "UNIV :: 'n set"
   let ?B = "basis ` S"
   let ?P = "\<lambda>(x::'a^'n). \<forall>i\<in> ?U. i \<notin> S \<longrightarrow> x$i =0"
  {fix x::"'a^'n" assume xS: "x\<in> ?B"
-   from xS have "?P x" by (auto simp add: basis_component)}
+   from xS have "?P x" by auto}
  moreover
  have "subspace ?P"
-   by (auto simp add: subspace_def Collect_def mem_def zero_index vector_component)
+   by (auto simp add: subspace_def Collect_def mem_def)
  ultimately show ?thesis
-   using x span_induct[of ?B ?P x] i iS by blast
+   using x span_induct[of ?B ?P x] iS by blast
 qed
 
-lemma independent_stdbasis: "independent {basis i ::real^'n |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
+lemma independent_stdbasis: "independent {basis i ::real^'n::finite |i. i\<in> (UNIV :: 'n set)}"
 proof-
-  let ?n = "dimindex (UNIV :: 'n set)"
-  let ?I = "{1 .. ?n}"
-  let ?b = "basis :: nat \<Rightarrow> real ^'n"
+  let ?I = "UNIV :: 'n set"
+  let ?b = "basis :: _ \<Rightarrow> real ^'n"
   let ?B = "?b ` ?I"
   have eq: "{?b i|i. i \<in> ?I} = ?B"
     by auto
@@ -3484,8 +3403,8 @@
       apply (rule inj_on_image_set_diff[symmetric])
       apply (rule basis_inj) using k(1) by auto
     from k(2) have th0: "?b k \<in> span (?b ` (?I - {k}))" unfolding eq2 .
-    from independent_stdbasis_lemma[OF th0 k(1), simplified]
-    have False by (simp add: basis_component[OF k(1), of k])}
+    from independent_stdbasis_lemma[OF th0, of k, simplified]
+    have False by simp}
   then show ?thesis unfolding eq dependent_def ..
 qed
 
@@ -3665,19 +3584,19 @@
     done
 qed
 
-lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> {(i::nat) .. j}}"
+lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x\<in> (UNIV::'a::finite set)}"
 proof-
-  have eq: "{f x |x. x\<in> {i .. j}} = f ` {i .. j}" by auto
+  have eq: "{f x |x. x\<in> UNIV} = f ` UNIV" by auto
   show ?thesis unfolding eq
     apply (rule finite_imageI)
-    apply (rule finite_atLeastAtMost)
+    apply (rule finite)
     done
 qed
 
 
 lemma independent_bound:
-  fixes S:: "(real^'n) set"
-  shows "independent S \<Longrightarrow> finite S \<and> card S <= dimindex(UNIV :: 'n set)"
+  fixes S:: "(real^'n::finite) set"
+  shows "independent S \<Longrightarrow> finite S \<and> card S <= CARD('n)"
   apply (subst card_stdbasis[symmetric])
   apply (rule independent_span_bound)
   apply (rule finite_Atleast_Atmost_nat)
@@ -3686,23 +3605,23 @@
   apply (rule subset_UNIV)
   done
 
-lemma dependent_biggerset: "(finite (S::(real ^'n) set) ==> card S > dimindex(UNIV:: 'n set)) ==> dependent S"
+lemma dependent_biggerset: "(finite (S::(real ^'n::finite) set) ==> card S > CARD('n)) ==> dependent S"
   by (metis independent_bound not_less)
 
 (* Hence we can create a maximal independent subset.                         *)
 
 lemma maximal_independent_subset_extend:
-  assumes sv: "(S::(real^'n) set) \<subseteq> V" and iS: "independent S"
+  assumes sv: "(S::(real^'n::finite) set) \<subseteq> V" and iS: "independent S"
   shows "\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
   using sv iS
-proof(induct d\<equiv> "dimindex (UNIV :: 'n set) - card S" arbitrary: S rule: nat_less_induct)
+proof(induct d\<equiv> "CARD('n) - card S" arbitrary: S rule: nat_less_induct)
   fix n and S:: "(real^'n) set"
-  assume H: "\<forall>m<n. \<forall>S \<subseteq> V. independent S \<longrightarrow> m = dimindex (UNIV::'n set) - card S \<longrightarrow>
+  assume H: "\<forall>m<n. \<forall>S \<subseteq> V. independent S \<longrightarrow> m = CARD('n) - card S \<longrightarrow>
               (\<exists>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B)"
-    and sv: "S \<subseteq> V" and i: "independent S" and n: "n = dimindex (UNIV :: 'n set) - card S"
+    and sv: "S \<subseteq> V" and i: "independent S" and n: "n = CARD('n) - card S"
   let ?P = "\<lambda>B. S \<subseteq> B \<and> B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
   let ?ths = "\<exists>x. ?P x"
-  let ?d = "dimindex (UNIV :: 'n set)"
+  let ?d = "CARD('n)"
   {assume "V \<subseteq> span S"
     then have ?ths  using sv i by blast }
   moreover
@@ -3713,7 +3632,7 @@
     from independent_insert[of a S]  i a
     have th1: "independent (insert a S)" by auto
     have mlt: "?d - card (insert a S) < n"
-      using aS a n independent_bound[OF th1] dimindex_ge_1[of "UNIV :: 'n set"]
+      using aS a n independent_bound[OF th1]
       by auto
 
     from H[rule_format, OF mlt th0 th1 refl]
@@ -3725,14 +3644,14 @@
 qed
 
 lemma maximal_independent_subset:
-  "\<exists>(B:: (real ^'n) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
+  "\<exists>(B:: (real ^'n::finite) set). B\<subseteq> V \<and> independent B \<and> V \<subseteq> span B"
   by (metis maximal_independent_subset_extend[of "{}:: (real ^'n) set"] empty_subsetI independent_empty)
 
 (* Notion of dimension.                                                      *)
 
 definition "dim V = (SOME n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n))"
 
-lemma basis_exists:  "\<exists>B. (B :: (real ^'n) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize dim V)"
+lemma basis_exists:  "\<exists>B. (B :: (real ^'n::finite) set) \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize dim V)"
 unfolding dim_def some_eq_ex[of "\<lambda>n. \<exists>B. B \<subseteq> V \<and> independent B \<and> V \<subseteq> span B \<and> (B hassize n)"]
 unfolding hassize_def
 using maximal_independent_subset[of V] independent_bound
@@ -3740,37 +3659,37 @@
 
 (* Consequences of independence or spanning for cardinality.                 *)
 
-lemma independent_card_le_dim: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B \<le> dim V"
+lemma independent_card_le_dim: "(B::(real ^'n::finite) set) \<subseteq> V \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B \<le> dim V"
 by (metis basis_exists[of V] independent_span_bound[where ?'a=real] hassize_def subset_trans)
 
-lemma span_card_ge_dim:  "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
+lemma span_card_ge_dim:  "(B::(real ^'n::finite) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> finite B \<Longrightarrow> dim V \<le> card B"
   by (metis basis_exists[of V] independent_span_bound hassize_def subset_trans)
 
 lemma basis_card_eq_dim:
-  "B \<subseteq> (V:: (real ^'n) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
+  "B \<subseteq> (V:: (real ^'n::finite) set) \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> finite B \<and> card B = dim V"
   by (metis order_eq_iff independent_card_le_dim span_card_ge_dim independent_mono)
 
-lemma dim_unique: "(B::(real ^'n) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> B hassize n \<Longrightarrow> dim V = n"
+lemma dim_unique: "(B::(real ^'n::finite) set) \<subseteq> V \<Longrightarrow> V \<subseteq> span B \<Longrightarrow> independent B \<Longrightarrow> B hassize n \<Longrightarrow> dim V = n"
   by (metis basis_card_eq_dim hassize_def)
 
 (* More lemmas about dimension.                                              *)
 
-lemma dim_univ: "dim (UNIV :: (real^'n) set) = dimindex (UNIV :: 'n set)"
-  apply (rule dim_unique[of "{basis i |i. i\<in> {1 .. dimindex (UNIV :: 'n set)}}"])
+lemma dim_univ: "dim (UNIV :: (real^'n::finite) set) = CARD('n)"
+  apply (rule dim_unique[of "{basis i |i. i\<in> (UNIV :: 'n set)}"])
   by (auto simp only: span_stdbasis has_size_stdbasis independent_stdbasis)
 
 lemma dim_subset:
-  "(S:: (real ^'n) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
+  "(S:: (real ^'n::finite) set) \<subseteq> T \<Longrightarrow> dim S \<le> dim T"
   using basis_exists[of T] basis_exists[of S]
   by (metis independent_span_bound[where ?'a = real and ?'n = 'n] subset_eq hassize_def)
 
-lemma dim_subset_univ: "dim (S:: (real^'n) set) \<le> dimindex (UNIV :: 'n set)"
+lemma dim_subset_univ: "dim (S:: (real^'n::finite) set) \<le> CARD('n)"
   by (metis dim_subset subset_UNIV dim_univ)
 
 (* Converses to those.                                                       *)
 
 lemma card_ge_dim_independent:
-  assumes BV:"(B::(real ^'n) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
+  assumes BV:"(B::(real ^'n::finite) set) \<subseteq> V" and iB:"independent B" and dVB:"dim V \<le> card B"
   shows "V \<subseteq> span B"
 proof-
   {fix a assume aV: "a \<in> V"
@@ -3784,7 +3703,7 @@
 qed
 
 lemma card_le_dim_spanning:
-  assumes BV: "(B:: (real ^'n) set) \<subseteq> V" and VB: "V \<subseteq> span B"
+  assumes BV: "(B:: (real ^'n::finite) set) \<subseteq> V" and VB: "V \<subseteq> span B"
   and fB: "finite B" and dVB: "dim V \<ge> card B"
   shows "independent B"
 proof-
@@ -3805,7 +3724,7 @@
   then show ?thesis unfolding dependent_def by blast
 qed
 
-lemma card_eq_dim: "(B:: (real ^'n) set) \<subseteq> V \<Longrightarrow> B hassize dim V \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
+lemma card_eq_dim: "(B:: (real ^'n::finite) set) \<subseteq> V \<Longrightarrow> B hassize dim V \<Longrightarrow> independent B \<longleftrightarrow> V \<subseteq> span B"
   by (metis hassize_def order_eq_iff card_le_dim_spanning
     card_ge_dim_independent)
 
@@ -3814,13 +3733,13 @@
 (* ------------------------------------------------------------------------- *)
 
 lemma independent_bound_general:
-  "independent (S:: (real^'n) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
+  "independent (S:: (real^'n::finite) set) \<Longrightarrow> finite S \<and> card S \<le> dim S"
   by (metis independent_card_le_dim independent_bound subset_refl)
 
-lemma dependent_biggerset_general: "(finite (S:: (real^'n) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
+lemma dependent_biggerset_general: "(finite (S:: (real^'n::finite) set) \<Longrightarrow> card S > dim S) \<Longrightarrow> dependent S"
   using independent_bound_general[of S] by (metis linorder_not_le)
 
-lemma dim_span: "dim (span (S:: (real ^'n) set)) = dim S"
+lemma dim_span: "dim (span (S:: (real ^'n::finite) set)) = dim S"
 proof-
   have th0: "dim S \<le> dim (span S)"
     by (auto simp add: subset_eq intro: dim_subset span_superset)
@@ -3833,10 +3752,10 @@
     using fB(2)  by arith
 qed
 
-lemma subset_le_dim: "(S:: (real ^'n) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
+lemma subset_le_dim: "(S:: (real ^'n::finite) set) \<subseteq> span T \<Longrightarrow> dim S \<le> dim T"
   by (metis dim_span dim_subset)
 
-lemma span_eq_dim: "span (S:: (real ^'n) set) = span T ==> dim S = dim T"
+lemma span_eq_dim: "span (S:: (real ^'n::finite) set) = span T ==> dim S = dim T"
   by (metis dim_span)
 
 lemma spans_image:
@@ -3845,7 +3764,9 @@
   unfolding span_linear_image[OF lf]
   by (metis VB image_mono)
 
-lemma dim_image_le: assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S:: (real ^'n) set)"
+lemma dim_image_le:
+  fixes f :: "real^'n::finite \<Rightarrow> real^'m::finite"
+  assumes lf: "linear f" shows "dim (f ` S) \<le> dim (S:: (real ^'n::finite) set)"
 proof-
   from basis_exists[of S] obtain B where
     B: "B \<subseteq> S" "independent B" "S \<subseteq> span B" "B hassize dim S" by blast
@@ -3889,14 +3810,14 @@
     (* FIXME : Move to some general theory ?*)
 definition "pairwise R S \<longleftrightarrow> (\<forall>x \<in> S. \<forall>y\<in> S. x\<noteq>y \<longrightarrow> R x y)"
 
-lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n) \<bullet> (x - ((b \<bullet> x) / (b\<bullet>b)) *s b) = 0"
+lemma vector_sub_project_orthogonal: "(b::'a::ordered_field^'n::finite) \<bullet> (x - ((b \<bullet> x) / (b\<bullet>b)) *s b) = 0"
   apply (cases "b = 0", simp)
   apply (simp add: dot_rsub dot_rmult)
   unfolding times_divide_eq_right[symmetric]
   by (simp add: field_simps dot_eq_0)
 
 lemma basis_orthogonal:
-  fixes B :: "(real ^'n) set"
+  fixes B :: "(real ^'n::finite) set"
   assumes fB: "finite B"
   shows "\<exists>C. finite C \<and> card C \<le> card B \<and> span C = span B \<and> pairwise orthogonal C"
   (is " \<exists>C. ?P B C")
@@ -3972,7 +3893,7 @@
 qed
 
 lemma orthogonal_basis_exists:
-  fixes V :: "(real ^'n) set"
+  fixes V :: "(real ^'n::finite) set"
   shows "\<exists>B. independent B \<and> B \<subseteq> span V \<and> V \<subseteq> span B \<and> (B hassize dim V) \<and> pairwise orthogonal B"
 proof-
   from basis_exists[of V] obtain B where B: "B \<subseteq> V" "independent B" "V \<subseteq> span B" "B hassize dim V" by blast
@@ -4000,7 +3921,7 @@
 
 lemma span_not_univ_orthogonal:
   assumes sU: "span S \<noteq> UNIV"
-  shows "\<exists>(a:: real ^'n). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
+  shows "\<exists>(a:: real ^'n::finite). a \<noteq>0 \<and> (\<forall>x \<in> span S. a \<bullet> x = 0)"
 proof-
   from sU obtain a where a: "a \<notin> span S" by blast
   from orthogonal_basis_exists obtain B where
@@ -4039,17 +3960,17 @@
 qed
 
 lemma span_not_univ_subset_hyperplane:
-  assumes SU: "span S \<noteq> (UNIV ::(real^'n) set)"
+  assumes SU: "span S \<noteq> (UNIV ::(real^'n::finite) set)"
   shows "\<exists> a. a \<noteq>0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
   using span_not_univ_orthogonal[OF SU] by auto
 
 lemma lowdim_subset_hyperplane:
-  assumes d: "dim S < dimindex (UNIV :: 'n set)"
-  shows "\<exists>(a::real ^'n). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
+  assumes d: "dim S < CARD('n::finite)"
+  shows "\<exists>(a::real ^'n::finite). a  \<noteq> 0 \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
 proof-
   {assume "span S = UNIV"
     hence "dim (span S) = dim (UNIV :: (real ^'n) set)" by simp
-    hence "dim S = dimindex (UNIV :: 'n set)" by (simp add: dim_span dim_univ)
+    hence "dim S = CARD('n)" by (simp add: dim_span dim_univ)
     with d have False by arith}
   hence th: "span S \<noteq> UNIV" by blast
   from span_not_univ_subset_hyperplane[OF th] show ?thesis .
@@ -4196,7 +4117,7 @@
 qed
 
 lemma linear_independent_extend:
-  assumes iB: "independent (B:: (real ^'n) set)"
+  assumes iB: "independent (B:: (real ^'n::finite) set)"
   shows "\<exists>g. linear g \<and> (\<forall>x\<in>B. g x = f x)"
 proof-
   from maximal_independent_subset_extend[of B UNIV] iB
@@ -4249,7 +4170,8 @@
 qed
 
 lemma subspace_isomorphism:
-  assumes s: "subspace (S:: (real ^'n) set)" and t: "subspace T"
+  assumes s: "subspace (S:: (real ^'n::finite) set)"
+  and t: "subspace (T :: (real ^ 'm::finite) set)"
   and d: "dim S = dim T"
   shows "\<exists>f. linear f \<and> f ` S = T \<and> inj_on f S"
 proof-
@@ -4324,12 +4246,12 @@
 qed
 
 lemma linear_eq_stdbasis:
-  assumes lf: "linear (f::'a::ring_1^'m \<Rightarrow> 'a^'n)" and lg: "linear g"
-  and fg: "\<forall>i \<in> {1 .. dimindex(UNIV :: 'm set)}. f (basis i) = g(basis i)"
+  assumes lf: "linear (f::'a::ring_1^'m::finite \<Rightarrow> 'a^'n::finite)" and lg: "linear g"
+  and fg: "\<forall>i. f (basis i) = g(basis i)"
   shows "f = g"
 proof-
   let ?U = "UNIV :: 'm set"
-  let ?I = "{basis i:: 'a^'m|i. i \<in> {1 .. dimindex ?U}}"
+  let ?I = "{basis i:: 'a^'m|i. i \<in> ?U}"
   {fix x assume x: "x \<in> (UNIV :: ('a^'m) set)"
     from equalityD2[OF span_stdbasis]
     have IU: " (UNIV :: ('a^'m) set) \<subseteq> span ?I" by blast
@@ -4369,12 +4291,12 @@
 qed
 
 lemma bilinear_eq_stdbasis:
-  assumes bf: "bilinear (f:: 'a::ring_1^'m \<Rightarrow> 'a^'n \<Rightarrow> 'a^'p)"
+  assumes bf: "bilinear (f:: 'a::ring_1^'m::finite \<Rightarrow> 'a^'n::finite \<Rightarrow> 'a^'p)"
   and bg: "bilinear g"
-  and fg: "\<forall>i\<in> {1 .. dimindex (UNIV :: 'm set)}. \<forall>j\<in>  {1 .. dimindex (UNIV :: 'n set)}. f (basis i) (basis j) = g (basis i) (basis j)"
+  and fg: "\<forall>i j. f (basis i) (basis j) = g (basis i) (basis j)"
   shows "f = g"
 proof-
-  from fg have th: "\<forall>x \<in> {basis i| i. i\<in> {1 .. dimindex (UNIV :: 'm set)}}. \<forall>y\<in>  {basis j |j. j \<in> {1 .. dimindex (UNIV :: 'n set)}}. f x y = g x y" by blast
+  from fg have th: "\<forall>x \<in> {basis i| i. i\<in> (UNIV :: 'm set)}. \<forall>y\<in>  {basis j |j. j \<in> (UNIV :: 'n set)}. f x y = g x y" by blast
   from bilinear_eq[OF bf bg equalityD2[OF span_stdbasis] equalityD2[OF span_stdbasis] th] show ?thesis by (blast intro: ext)
 qed
 
@@ -4389,16 +4311,14 @@
   by (metis matrix_transp_mul transp_mat transp_transp)
 
 lemma linear_injective_left_inverse:
-  assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'m)" and fi: "inj f"
+  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'m::finite)" and fi: "inj f"
   shows "\<exists>g. linear g \<and> g o f = id"
 proof-
   from linear_independent_extend[OF independent_injective_image, OF independent_stdbasis, OF lf fi]
-  obtain h:: "real ^'m \<Rightarrow> real ^'n" where h: "linear h" " \<forall>x \<in> f ` {basis i|i. i \<in> {1 .. dimindex (UNIV::'n set)}}. h x = inv f x" by blast
+  obtain h:: "real ^'m \<Rightarrow> real ^'n" where h: "linear h" " \<forall>x \<in> f ` {basis i|i. i \<in> (UNIV::'n set)}. h x = inv f x" by blast
   from h(2)
-  have th: "\<forall>i\<in>{1..dimindex (UNIV::'n set)}. (h \<circ> f) (basis i) = id (basis i)"
+  have th: "\<forall>i. (h \<circ> f) (basis i) = id (basis i)"
     using inv_o_cancel[OF fi, unfolded stupid_ext[symmetric] id_def o_def]
-    apply auto
-    apply (erule_tac x="basis i" in allE)
     by auto
 
   from linear_eq_stdbasis[OF linear_compose[OF lf h(1)] linear_id th]
@@ -4407,14 +4327,14 @@
 qed
 
 lemma linear_surjective_right_inverse:
-  assumes lf: "linear (f:: real ^'m \<Rightarrow> real ^'n)" and sf: "surj f"
+  assumes lf: "linear (f:: real ^'m::finite \<Rightarrow> real ^'n::finite)" and sf: "surj f"
   shows "\<exists>g. linear g \<and> f o g = id"
 proof-
   from linear_independent_extend[OF independent_stdbasis]
   obtain h:: "real ^'n \<Rightarrow> real ^'m" where
-    h: "linear h" "\<forall> x\<in> {basis i| i. i\<in> {1 .. dimindex (UNIV :: 'n set)}}. h x = inv f x" by blast
+    h: "linear h" "\<forall> x\<in> {basis i| i. i\<in> (UNIV :: 'n set)}. h x = inv f x" by blast
   from h(2)
-  have th: "\<forall>i\<in>{1..dimindex (UNIV::'n set)}. (f o h) (basis i) = id (basis i)"
+  have th: "\<forall>i. (f o h) (basis i) = id (basis i)"
     using sf
     apply (auto simp add: surj_iff o_def stupid_ext[symmetric])
     apply (erule_tac x="basis i" in allE)
@@ -4426,7 +4346,7 @@
 qed
 
 lemma matrix_left_invertible_injective:
-"(\<exists>B. (B::real^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
+"(\<exists>B. (B::real^'m^'n) ** (A::real^'n::finite^'m::finite) = mat 1) \<longleftrightarrow> (\<forall>x y. A *v x = A *v y \<longrightarrow> x = y)"
 proof-
   {fix B:: "real^'m^'n" and x y assume B: "B ** A = mat 1" and xy: "A *v x = A*v y"
     from xy have "B*v (A *v x) = B *v (A*v y)" by simp
@@ -4445,13 +4365,13 @@
 qed
 
 lemma matrix_left_invertible_ker:
-  "(\<exists>B. (B::real ^'m^'n) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
+  "(\<exists>B. (B::real ^'m::finite^'n::finite) ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> (\<forall>x. A *v x = 0 \<longrightarrow> x = 0)"
   unfolding matrix_left_invertible_injective
   using linear_injective_0[OF matrix_vector_mul_linear, of A]
   by (simp add: inj_on_def)
 
 lemma matrix_right_invertible_surjective:
-"(\<exists>B. (A::real^'n^'m) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
+"(\<exists>B. (A::real^'n::finite^'m::finite) ** (B::real^'m^'n) = mat 1) \<longleftrightarrow> surj (\<lambda>x. A *v x)"
 proof-
   {fix B :: "real ^'m^'n"  assume AB: "A ** B = mat 1"
     {fix x :: "real ^ 'm"
@@ -4475,11 +4395,11 @@
 qed
 
 lemma matrix_left_invertible_independent_columns:
-  fixes A :: "real^'n^'m"
-  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s column i A) {1 .. dimindex(UNIV :: 'n set)} = 0 \<longrightarrow> (\<forall>i\<in> {1 .. dimindex (UNIV :: 'n set)}. c i = 0))"
+  fixes A :: "real^'n::finite^'m::finite"
+  shows "(\<exists>(B::real ^'m^'n). B ** A = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s column i A) (UNIV :: 'n set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
    (is "?lhs \<longleftrightarrow> ?rhs")
 proof-
-  let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
+  let ?U = "UNIV :: 'n set"
   {assume k: "\<forall>x. A *v x = 0 \<longrightarrow> x = 0"
     {fix c i assume c: "setsum (\<lambda>i. c i *s column i A) ?U = 0"
       and i: "i \<in> ?U"
@@ -4487,7 +4407,7 @@
       have th0:"A *v ?x = 0"
 	using c
 	unfolding matrix_mult_vsum Cart_eq
-	by (auto simp add: vector_component zero_index setsum_component Cart_lambda_beta)
+	by auto
       from k[rule_format, OF th0] i
       have "c i = 0" by (vector Cart_eq)}
     hence ?rhs by blast}
@@ -4501,16 +4421,16 @@
 qed
 
 lemma matrix_right_invertible_independent_rows:
-  fixes A :: "real^'n^'m"
-  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s row i A) {1 .. dimindex(UNIV :: 'm set)} = 0 \<longrightarrow> (\<forall>i\<in> {1 .. dimindex (UNIV :: 'm set)}. c i = 0))"
+  fixes A :: "real^'n::finite^'m::finite"
+  shows "(\<exists>(B::real^'m^'n). A ** B = mat 1) \<longleftrightarrow> (\<forall>c. setsum (\<lambda>i. c i *s row i A) (UNIV :: 'm set) = 0 \<longrightarrow> (\<forall>i. c i = 0))"
   unfolding left_invertible_transp[symmetric]
     matrix_left_invertible_independent_columns
   by (simp add: column_transp)
 
 lemma matrix_right_invertible_span_columns:
-  "(\<exists>(B::real ^'n^'m). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
+  "(\<exists>(B::real ^'n::finite^'m::finite). (A::real ^'m^'n) ** B = mat 1) \<longleftrightarrow> span (columns A) = UNIV" (is "?lhs = ?rhs")
 proof-
-  let ?U = "{1 .. dimindex (UNIV :: 'm set)}"
+  let ?U = "UNIV :: 'm set"
   have fU: "finite ?U" by simp
   have lhseq: "?lhs \<longleftrightarrow> (\<forall>y. \<exists>(x::real^'m). setsum (\<lambda>i. (x$i) *s column i A) ?U = y)"
     unfolding matrix_right_invertible_surjective matrix_mult_vsum surj_def
@@ -4545,7 +4465,7 @@
 	  x: "setsum (\<lambda>i. (x$i) *s column i A) ?U = y2" by blast
 	let ?x = "(\<chi> j. if j = i then c + (x$i) else (x$j))::real^'m"
 	show "?P (c*s y1 + y2)"
-	  proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric]Cart_lambda_beta setsum_component cond_value_iff right_distrib cond_application_beta vector_component cong del: if_weak_cong, simp only: One_nat_def[symmetric])
+	  proof(rule exI[where x= "?x"], vector, auto simp add: i x[symmetric] cond_value_iff right_distrib cond_application_beta cong del: if_weak_cong)
 	    fix j
 	    have th: "\<forall>xa \<in> ?U. (if xa = i then (c + (x$i)) * ((column xa A)$j)
            else (x$xa) * ((column xa A$j))) = (if xa = i then c * ((column i A)$j) else 0) + ((x$xa) * ((column xa A)$j))" using i(1)
@@ -4570,7 +4490,7 @@
 qed
 
 lemma matrix_left_invertible_span_rows:
-  "(\<exists>(B::real^'m^'n). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
+  "(\<exists>(B::real^'m::finite^'n::finite). B ** (A::real^'n^'m) = mat 1) \<longleftrightarrow> span (rows A) = UNIV"
   unfolding right_invertible_transp[symmetric]
   unfolding columns_transp[symmetric]
   unfolding matrix_right_invertible_span_columns
@@ -4579,7 +4499,7 @@
 (* An injective map real^'n->real^'n is also surjective.                       *)
 
 lemma linear_injective_imp_surjective:
-  assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and fi: "inj f"
+  assumes lf: "linear (f:: real ^'n::finite \<Rightarrow> real ^'n)" and fi: "inj f"
   shows "surj f"
 proof-
   let ?U = "UNIV :: (real ^'n) set"
@@ -4641,7 +4561,7 @@
 qed
 
 lemma linear_surjective_imp_injective:
-  assumes lf: "linear (f::real ^'n => real ^'n)" and sf: "surj f"
+  assumes lf: "linear (f::real ^'n::finite => real ^'n)" and sf: "surj f"
   shows "inj f"
 proof-
   let ?U = "UNIV :: (real ^'n) set"
@@ -4701,14 +4621,14 @@
   by (simp add: expand_fun_eq o_def id_def)
 
 lemma linear_injective_isomorphism:
-  assumes lf: "linear (f :: real^'n \<Rightarrow> real ^'n)" and fi: "inj f"
+  assumes lf: "linear (f :: real^'n::finite \<Rightarrow> real ^'n)" and fi: "inj f"
   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
 unfolding isomorphism_expand[symmetric]
 using linear_surjective_right_inverse[OF lf linear_injective_imp_surjective[OF lf fi]] linear_injective_left_inverse[OF lf fi]
 by (metis left_right_inverse_eq)
 
 lemma linear_surjective_isomorphism:
-  assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and sf: "surj f"
+  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and sf: "surj f"
   shows "\<exists>f'. linear f' \<and> (\<forall>x. f' (f x) = x) \<and> (\<forall>x. f (f' x) = x)"
 unfolding isomorphism_expand[symmetric]
 using linear_surjective_right_inverse[OF lf sf] linear_injective_left_inverse[OF lf linear_surjective_imp_injective[OF lf sf]]
@@ -4717,7 +4637,7 @@
 (* Left and right inverses are the same for R^N->R^N.                        *)
 
 lemma linear_inverse_left:
-  assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and lf': "linear f'"
+  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and lf': "linear f'"
   shows "f o f' = id \<longleftrightarrow> f' o f = id"
 proof-
   {fix f f':: "real ^'n \<Rightarrow> real ^'n"
@@ -4735,7 +4655,7 @@
 (* Moreover, a one-sided inverse is automatically linear.                    *)
 
 lemma left_inverse_linear:
-  assumes lf: "linear (f::real ^'n \<Rightarrow> real ^'n)" and gf: "g o f = id"
+  assumes lf: "linear (f::real ^'n::finite \<Rightarrow> real ^'n)" and gf: "g o f = id"
   shows "linear g"
 proof-
   from gf have fi: "inj f" apply (auto simp add: inj_on_def o_def id_def stupid_ext[symmetric])
@@ -4750,7 +4670,7 @@
 qed
 
 lemma right_inverse_linear:
-  assumes lf: "linear (f:: real ^'n \<Rightarrow> real ^'n)" and gf: "f o g = id"
+  assumes lf: "linear (f:: real ^'n::finite \<Rightarrow> real ^'n)" and gf: "f o g = id"
   shows "linear g"
 proof-
   from gf have fi: "surj f" apply (auto simp add: surj_def o_def id_def stupid_ext[symmetric])
@@ -4767,7 +4687,7 @@
 (* The same result in terms of square matrices.                              *)
 
 lemma matrix_left_right_inverse:
-  fixes A A' :: "real ^'n^'n"
+  fixes A A' :: "real ^'n::finite^'n"
   shows "A ** A' = mat 1 \<longleftrightarrow> A' ** A = mat 1"
 proof-
   {fix A A' :: "real ^'n^'n" assume AA': "A ** A' = mat 1"
@@ -4796,21 +4716,20 @@
 
 lemma transp_columnvector:
  "transp(columnvector v) = rowvector v"
-  by (simp add: transp_def rowvector_def columnvector_def Cart_eq Cart_lambda_beta)
+  by (simp add: transp_def rowvector_def columnvector_def Cart_eq)
 
 lemma transp_rowvector: "transp(rowvector v) = columnvector v"
-  by (simp add: transp_def columnvector_def rowvector_def Cart_eq Cart_lambda_beta)
+  by (simp add: transp_def columnvector_def rowvector_def Cart_eq)
 
 lemma dot_rowvector_columnvector:
   "columnvector (A *v v) = A ** columnvector v"
   by (vector columnvector_def matrix_matrix_mult_def matrix_vector_mult_def)
 
-lemma dot_matrix_product: "(x::'a::semiring_1^'n) \<bullet> y = (((rowvector x ::'a^'n^1) ** (columnvector y :: 'a^1^'n))$1)$1"
-  apply (vector matrix_matrix_mult_def rowvector_def columnvector_def dot_def)
-  by (simp add: Cart_lambda_beta)
+lemma dot_matrix_product: "(x::'a::semiring_1^'n::finite) \<bullet> y = (((rowvector x ::'a^'n^1) ** (columnvector y :: 'a^1^'n))$1)$1"
+  by (vector matrix_matrix_mult_def rowvector_def columnvector_def dot_def)
 
 lemma dot_matrix_vector_mul:
-  fixes A B :: "real ^'n ^'n" and x y :: "real ^'n"
+  fixes A B :: "real ^'n::finite ^'n" and x y :: "real ^'n"
   shows "(A *v x) \<bullet> (B *v y) =
       (((rowvector x :: real^'n^1) ** ((transp A ** B) ** (columnvector y :: real ^1^'n)))$1)$1"
 unfolding dot_matrix_product transp_columnvector[symmetric]
@@ -4818,30 +4737,28 @@
 
 (* Infinity norm.                                                            *)
 
-definition "infnorm (x::real^'n) = rsup {abs(x$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
-
-lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> {1 .. dimindex (UNIV :: 'n set)}"
-  using dimindex_ge_1 by auto
+definition "infnorm (x::real^'n::finite) = rsup {abs(x$i) |i. i\<in> (UNIV :: 'n set)}"
+
+lemma numseg_dimindex_nonempty: "\<exists>i. i \<in> (UNIV :: 'n set)"
+  by auto
 
 lemma infnorm_set_image:
-  "{abs(x$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} =
-  (\<lambda>i. abs(x$i)) ` {1 .. dimindex(UNIV :: 'n set)}" by blast
+  "{abs(x$i) |i. i\<in> (UNIV :: 'n set)} =
+  (\<lambda>i. abs(x$i)) ` (UNIV :: 'n set)" by blast
 
 lemma infnorm_set_lemma:
-  shows "finite {abs((x::'a::abs ^'n)$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}}"
-  and "{abs(x$i) |i. i\<in> {1 .. dimindex(UNIV :: 'n set)}} \<noteq> {}"
+  shows "finite {abs((x::'a::abs ^'n::finite)$i) |i. i\<in> (UNIV :: 'n set)}"
+  and "{abs(x$i) |i. i\<in> (UNIV :: 'n::finite set)} \<noteq> {}"
   unfolding infnorm_set_image
-  using dimindex_ge_1[of "UNIV :: 'n set"]
   by (auto intro: finite_imageI)
 
-lemma infnorm_pos_le: "0 \<le> infnorm x"
+lemma infnorm_pos_le: "0 \<le> infnorm (x::real^'n::finite)"
   unfolding infnorm_def
   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
   unfolding infnorm_set_image
-  using dimindex_ge_1
   by auto
 
-lemma infnorm_triangle: "infnorm ((x::real^'n) + y) \<le> infnorm x + infnorm y"
+lemma infnorm_triangle: "infnorm ((x::real^'n::finite) + y) \<le> infnorm x + infnorm y"
 proof-
   have th: "\<And>x y (z::real). x - y <= z \<longleftrightarrow> x - z <= y" by arith
   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
@@ -4857,12 +4774,12 @@
   unfolding rsup_finite_ge_iff[ OF infnorm_set_lemma]
 
   unfolding infnorm_set_image ball_simps bex_simps
-  apply (simp add: vector_add_component)
-  apply (metis numseg_dimindex_nonempty th2)
+  apply simp
+  apply (metis th2)
   done
 qed
 
-lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::real ^'n) = 0"
+lemma infnorm_eq_0: "infnorm x = 0 \<longleftrightarrow> (x::real ^'n::finite) = 0"
 proof-
   have "infnorm x <= 0 \<longleftrightarrow> x = 0"
     unfolding infnorm_def
@@ -4880,9 +4797,7 @@
   apply (rule cong[of "rsup" "rsup"])
   apply blast
   apply (rule set_ext)
-  apply (auto simp add: vector_component abs_minus_cancel)
-  apply (rule_tac x="i" in exI)
-  apply (simp add: vector_component)
+  apply auto
   done
 
 lemma infnorm_sub: "infnorm (x - y) = infnorm (y - x)"
@@ -4905,16 +4820,16 @@
 lemma real_abs_infnorm: " \<bar>infnorm x\<bar> = infnorm x"
   using infnorm_pos_le[of x] by arith
 
-lemma component_le_infnorm: assumes i: "i \<in> {1 .. dimindex (UNIV :: 'n set)}"
-  shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n)"
+lemma component_le_infnorm:
+  shows "\<bar>x$i\<bar> \<le> infnorm (x::real^'n::finite)"
 proof-
-  let ?U = "{1 .. dimindex (UNIV :: 'n set)}"
+  let ?U = "UNIV :: 'n set"
   let ?S = "{\<bar>x$i\<bar> |i. i\<in> ?U}"
   have fS: "finite ?S" unfolding image_Collect[symmetric]
     apply (rule finite_imageI) unfolding Collect_def mem_def by simp
-  have S0: "?S \<noteq> {}" using numseg_dimindex_nonempty by blast
+  have S0: "?S \<noteq> {}" by blast
   have th1: "\<And>S f. f ` S = { f i| i. i \<in> S}" by blast
-  from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0] i
+  from rsup_finite_in[OF fS S0] rsup_finite_Ub[OF fS S0]
   show ?thesis unfolding infnorm_def isUb_def setle_def
     unfolding infnorm_set_image ball_simps by auto
 qed
@@ -4923,9 +4838,9 @@
   apply (subst infnorm_def)
   unfolding rsup_finite_le_iff[OF infnorm_set_lemma]
   unfolding infnorm_set_image ball_simps
-  apply (simp add: abs_mult vector_component del: One_nat_def)
-  apply (rule ballI)
-  apply (drule component_le_infnorm[of _ x])
+  apply (simp add: abs_mult)
+  apply (rule allI)
+  apply (cut_tac component_le_infnorm[of x])
   apply (rule mult_mono)
   apply auto
   done
@@ -4958,18 +4873,16 @@
   unfolding infnorm_set_image  ball_simps
   by (metis component_le_norm)
 lemma card_enum: "card {1 .. n} = n" by auto
-lemma norm_le_infnorm: "norm(x) <= sqrt(real (dimindex(UNIV ::'n set))) * infnorm(x::real ^'n)"
+lemma norm_le_infnorm: "norm(x) <= sqrt(real CARD('n)) * infnorm(x::real ^'n::finite)"
 proof-
-  let ?d = "dimindex(UNIV ::'n set)"
-  have d: "?d = card {1 .. ?d}" by auto
+  let ?d = "CARD('n)"
   have "real ?d \<ge> 0" by simp
   hence d2: "(sqrt (real ?d))^2 = real ?d"
     by (auto intro: real_sqrt_pow2)
   have th: "sqrt (real ?d) * infnorm x \<ge> 0"
-    by (simp add: dimindex_ge_1 zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
+    by (simp add: zero_le_mult_iff real_sqrt_ge_0_iff infnorm_pos_le)
   have th1: "x\<bullet>x \<le> (sqrt (real ?d) * infnorm x)^2"
     unfolding power_mult_distrib d2
-    apply (subst d)
     apply (subst power2_abs[symmetric])
     unfolding real_of_nat_def dot_def power2_eq_square[symmetric]
     apply (subst power2_abs[symmetric])
@@ -4986,7 +4899,7 @@
 
 (* Equality in Cauchy-Schwarz and triangle inequalities.                     *)
 
-lemma norm_cauchy_schwarz_eq: "(x::real ^'n) \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *s y = norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
+lemma norm_cauchy_schwarz_eq: "(x::real ^'n::finite) \<bullet> y = norm x * norm y \<longleftrightarrow> norm x *s y = norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
 proof-
   {assume h: "x = 0"
     hence ?thesis by simp}
@@ -5012,7 +4925,9 @@
   ultimately show ?thesis by blast
 qed
 
-lemma norm_cauchy_schwarz_abs_eq: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
+lemma norm_cauchy_schwarz_abs_eq:
+  fixes x y :: "real ^ 'n::finite"
+  shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow>
                 norm x *s y = norm y *s x \<or> norm(x) *s y = - norm y *s x" (is "?lhs \<longleftrightarrow> ?rhs")
 proof-
   have th: "\<And>(x::real) a. a \<ge> 0 \<Longrightarrow> abs x = a \<longleftrightarrow> x = a \<or> x = - a" by arith
@@ -5029,7 +4944,9 @@
   finally show ?thesis ..
 qed
 
-lemma norm_triangle_eq: "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
+lemma norm_triangle_eq:
+  fixes x y :: "real ^ 'n::finite"
+  shows "norm(x + y) = norm x + norm y \<longleftrightarrow> norm x *s y = norm y *s x"
 proof-
   {assume x: "x =0 \<or> y =0"
     hence ?thesis by (cases "x=0", simp_all)}
@@ -5107,7 +5024,9 @@
   ultimately show ?thesis by blast
 qed
 
-lemma norm_cauchy_schwarz_equal: "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
+lemma norm_cauchy_schwarz_equal:
+  fixes x y :: "real ^ 'n::finite"
+  shows "abs(x \<bullet> y) = norm x * norm y \<longleftrightarrow> collinear {(0::real^'n),x,y}"
 unfolding norm_cauchy_schwarz_abs_eq
 apply (cases "x=0", simp_all add: collinear_2)
 apply (cases "y=0", simp_all add: collinear_2 insert_commute)
diff -r ac50e7dedf6d -r 638b088bb840 src/HOL/Library/Finite_Cartesian_Product.thy
--- a/src/HOL/Library/Finite_Cartesian_Product.thy	Wed Mar 18 22:17:23 2009 +0100
+++ b/src/HOL/Library/Finite_Cartesian_Product.thy	Thu Mar 19 01:29:19 2009 -0700
@@ -10,202 +10,81 @@
 begin
 
   (* FIXME : ATP_Linkup is only needed for metis at a few places. We could dispense of that by changing the proofs*)
-subsection{* Dimention of sets *}
-
-definition "dimindex (S:: 'a set) = (if finite (UNIV::'a set) then card (UNIV:: 'a set) else 1)"
-
-syntax "_type_dimindex" :: "type => nat" ("(1DIM/(1'(_')))")
-translations "DIM(t)" => "CONST dimindex (CONST UNIV :: t set)"
-
-lemma dimindex_nonzero: "dimindex S \<noteq>  0"
-unfolding dimindex_def
-by (simp add: neq0_conv[symmetric] del: neq0_conv)
-
-lemma dimindex_ge_1: "dimindex S \<ge> 1"
-  using dimindex_nonzero[of S] by arith
-lemma dimindex_univ: "dimindex (S :: 'a set) = DIM('a)" by (simp add: dimindex_def)
 
 definition hassize (infixr "hassize" 12) where
   "(S hassize n) = (finite S \<and> card S = n)"
 
-lemma dimindex_unique: " (UNIV :: 'a set) hassize n ==> DIM('a) = n"
-by (simp add: dimindex_def hassize_def)
-
-
-subsection{* An indexing type parametrized by base type. *}
-
-typedef 'a finite_image = "{1 .. DIM('a)}"
-  using dimindex_ge_1 by auto
-
-lemma finite_image_image: "(UNIV :: 'a finite_image set) = Abs_finite_image ` {1 .. DIM('a)}"
-apply (auto simp add: Abs_finite_image_inverse image_def finite_image_def)
-apply (rule_tac x="Rep_finite_image x" in bexI)
-apply (simp_all add: Rep_finite_image_inverse Rep_finite_image)
-using Rep_finite_image[where ?'a = 'a]
-unfolding finite_image_def
-apply simp
-done
-
-text{* Dimension of such a type, and indexing over it. *}
-
-lemma inj_on_Abs_finite_image:
-  "inj_on (Abs_finite_image:: _ \<Rightarrow> 'a finite_image) {1 .. DIM('a)}"
-by (auto simp add: inj_on_def finite_image_def Abs_finite_image_inject[where ?'a='a])
-
-lemma has_size_finite_image: "(UNIV:: 'a finite_image set) hassize dimindex (S :: 'a set)"
-  unfolding hassize_def finite_image_image card_image[OF inj_on_Abs_finite_image[where ?'a='a]] by (auto simp add: dimindex_def)
-
 lemma hassize_image_inj: assumes f: "inj_on f S" and S: "S hassize n"
   shows "f ` S hassize n"
   using f S card_image[OF f]
     by (simp add: hassize_def inj_on_def)
 
-lemma card_finite_image: "card (UNIV:: 'a finite_image set) = dimindex(S:: 'a set)"
-using has_size_finite_image
-unfolding hassize_def by blast
-
-lemma finite_finite_image: "finite (UNIV:: 'a finite_image set)"
-using has_size_finite_image
-unfolding hassize_def by blast
-
-lemma dimindex_finite_image: "dimindex (S:: 'a finite_image set) = dimindex(T:: 'a set)"
-unfolding card_finite_image[of T, symmetric]
-by (auto simp add: dimindex_def finite_finite_image)
-
-lemma Abs_finite_image_works:
-  fixes i:: "'a finite_image"
-  shows " \<exists>!n \<in> {1 .. DIM('a)}. Abs_finite_image n = i"
-  unfolding Bex1_def Ex1_def
-  apply (rule_tac x="Rep_finite_image i" in exI)
-  using Rep_finite_image_inverse[where ?'a = 'a]
-    Rep_finite_image[where ?'a = 'a]
-  Abs_finite_image_inverse[where ?'a='a, symmetric]
-  by (auto simp add: finite_image_def)
-
-lemma Abs_finite_image_inj:
- "i \<in> {1 .. DIM('a)} \<Longrightarrow> j \<in> {1 .. DIM('a)}
-  \<Longrightarrow> (((Abs_finite_image i ::'a finite_image) = Abs_finite_image j) \<longleftrightarrow> (i = j))"
-  using Abs_finite_image_works[where ?'a = 'a]
-  by (auto simp add: atLeastAtMost_iff Bex1_def)
-
-lemma forall_Abs_finite_image:
-  "(\<forall>k:: 'a finite_image. P k) \<longleftrightarrow> (\<forall>i \<in> {1 .. DIM('a)}. P(Abs_finite_image i))"
-unfolding Ball_def atLeastAtMost_iff Ex1_def
-using Abs_finite_image_works[where ?'a = 'a, unfolded atLeastAtMost_iff Bex1_def]
-by metis
 
 subsection {* Finite Cartesian products, with indexing and lambdas. *}
 
-typedef (Cart)
+typedef (open Cart)
   ('a, 'b) "^" (infixl "^" 15)
-    = "{f:: 'b finite_image \<Rightarrow> 'a . True}" by simp
+    = "UNIV :: ('b \<Rightarrow> 'a) set"
+  morphisms Cart_nth Cart_lambda ..
 
-abbreviation dimset:: "('a ^ 'n) \<Rightarrow> nat set" where
-  "dimset a \<equiv> {1 .. DIM('n)}"
+notation Cart_nth (infixl "$" 90)
 
-definition Cart_nth :: "'a ^ 'b \<Rightarrow> nat \<Rightarrow> 'a" (infixl "$" 90) where
-  "x$i = Rep_Cart x (Abs_finite_image i)"
+notation (xsymbols) Cart_lambda (binder "\<chi>" 10)
 
 lemma stupid_ext: "(\<forall>x. f x = g x) \<longleftrightarrow> (f = g)"
   apply auto
   apply (rule ext)
   apply auto
   done
-lemma Cart_eq: "((x:: 'a ^ 'b) = y) \<longleftrightarrow> (\<forall>i\<in> dimset x. x$i = y$i)"
-  unfolding Cart_nth_def forall_Abs_finite_image[symmetric, where P = "\<lambda>i. Rep_Cart x i = Rep_Cart y i"] stupid_ext
-  using Rep_Cart_inject[of x y] ..
-
-consts Cart_lambda :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a ^ 'b"
-notation (xsymbols) Cart_lambda (binder "\<chi>" 10)
-
-defs Cart_lambda_def: "Cart_lambda g == (SOME (f:: 'a ^ 'b). \<forall>i \<in> {1 .. DIM('b)}. f$i = g i)"
 
-lemma  Cart_lambda_beta: " \<forall> i\<in> {1 .. DIM('b)}. (Cart_lambda g:: 'a ^ 'b)$i = g i"
-  unfolding Cart_lambda_def
-proof (rule someI_ex)
-  let ?p = "\<lambda>(i::nat) (k::'b finite_image). i \<in> {1 .. DIM('b)} \<and> (Abs_finite_image i = k)"
-  let ?f = "Abs_Cart (\<lambda>k. g (THE i. ?p i k)):: 'a ^ 'b"
-  let ?P = "\<lambda>f i. f$i = g i"
-  let ?Q = "\<lambda>(f::'a ^ 'b). \<forall> i \<in> {1 .. DIM('b)}. ?P f i"
-  {fix i
-    assume i: "i \<in> {1 .. DIM('b)}"
-    let ?j = "THE j. ?p j (Abs_finite_image i)"
-    from theI'[where P = "\<lambda>j. ?p (j::nat) (Abs_finite_image i :: 'b finite_image)", OF Abs_finite_image_works[of "Abs_finite_image i :: 'b finite_image", unfolded Bex1_def]]
-    have j: "?j \<in> {1 .. DIM('b)}" "(Abs_finite_image ?j :: 'b finite_image) = Abs_finite_image i" by blast+
-    from i j Abs_finite_image_inject[of i ?j, where ?'a = 'b]
-    have th: "?j = i" by (simp add: finite_image_def)
-    have "?P ?f i"
-      using th
-      by (simp add: Cart_nth_def Abs_Cart_inverse Rep_Cart_inverse Cart_def) }
-  hence th0: "?Q ?f" ..
-  with th0 show "\<exists>f. ?Q f" unfolding Ex1_def by auto
-qed
+lemma Cart_eq: "((x:: 'a ^ 'b) = y) \<longleftrightarrow> (\<forall>i. x$i = y$i)"
+  by (simp add: Cart_nth_inject [symmetric] expand_fun_eq)
 
-lemma  Cart_lambda_beta': "i\<in> {1 .. DIM('b)} \<Longrightarrow> (Cart_lambda g:: 'a ^ 'b)$i = g i"
-  using Cart_lambda_beta by blast
+lemma Cart_lambda_beta [simp]: "Cart_lambda g $ i = g i"
+  by (simp add: Cart_lambda_inverse)
 
 lemma Cart_lambda_unique:
   fixes f :: "'a ^ 'b"
-  shows "(\<forall>i\<in> {1 .. DIM('b)}. f$i = g i) \<longleftrightarrow> Cart_lambda g = f"
-  by (auto simp add: Cart_eq Cart_lambda_beta)
+  shows "(\<forall>i. f$i = g i) \<longleftrightarrow> Cart_lambda g = f"
+  by (auto simp add: Cart_eq)
 
-lemma Cart_lambda_eta: "(\<chi> i. (g$i)) = g" by (simp add: Cart_eq Cart_lambda_beta)
+lemma Cart_lambda_eta: "(\<chi> i. (g$i)) = g"
+  by (simp add: Cart_eq)
 
 text{* A non-standard sum to "paste" Cartesian products. *}
 
-typedef ('a,'b) finite_sum = "{1 .. DIM('a) + DIM('b)}"
-  apply (rule exI[where x="1"])
-  using dimindex_ge_1[of "UNIV :: 'a set"] dimindex_ge_1[of "UNIV :: 'b set"]
-  by auto
+definition pastecart :: "'a ^ 'm \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ ('m + 'n)" where
+  "pastecart f g = (\<chi> i. case i of Inl a \<Rightarrow> f$a | Inr b \<Rightarrow> g$b)"
+
+definition fstcart:: "'a ^('m + 'n) \<Rightarrow> 'a ^ 'm" where
+  "fstcart f = (\<chi> i. (f$(Inl i)))"
 
-definition pastecart :: "'a ^ 'm \<Rightarrow> 'a ^ 'n \<Rightarrow> 'a ^ ('m,'n) finite_sum" where
-  "pastecart f g = (\<chi> i. (if i <= DIM('m) then f$i else g$(i - DIM('m))))"
+definition sndcart:: "'a ^('m + 'n) \<Rightarrow> 'a ^ 'n" where
+  "sndcart f = (\<chi> i. (f$(Inr i)))"
 
-definition fstcart:: "'a ^('m, 'n) finite_sum \<Rightarrow> 'a ^ 'm" where
-  "fstcart f = (\<chi> i. (f$i))"
-
-definition sndcart:: "'a ^('m, 'n) finite_sum \<Rightarrow> 'a ^ 'n" where
-  "sndcart f = (\<chi> i. (f$(i + DIM('m))))"
+lemma nth_pastecart_Inl [simp]: "pastecart f g $ Inl a = f$a"
+  unfolding pastecart_def by simp
 
-lemma finite_sum_image: "(UNIV::('a,'b) finite_sum set) = Abs_finite_sum ` {1 .. DIM('a) + DIM('b)}"
-apply (auto  simp add: image_def)
-apply (rule_tac x="Rep_finite_sum x" in bexI)
-apply (simp add: Rep_finite_sum_inverse)
-using Rep_finite_sum[unfolded finite_sum_def, where ?'a = 'a and ?'b = 'b]
-apply (simp add: Rep_finite_sum)
-done
+lemma nth_pastecart_Inr [simp]: "pastecart f g $ Inr b = g$b"
+  unfolding pastecart_def by simp
+
+lemma nth_fstcart [simp]: "fstcart f $ i = f $ Inl i"
+  unfolding fstcart_def by simp
 
-lemma inj_on_Abs_finite_sum: "inj_on (Abs_finite_sum :: _ \<Rightarrow> ('a,'b) finite_sum) {1 .. DIM('a) + DIM('b)}"
-  using Abs_finite_sum_inject[where ?'a = 'a and ?'b = 'b]
-  by (auto simp add: inj_on_def finite_sum_def)
+lemma nth_sndtcart [simp]: "sndcart f $ i = f $ Inr i"
+  unfolding sndcart_def by simp
 
-lemma dimindex_has_size_finite_sum:
-  "(UNIV::('m,'n) finite_sum set) hassize (DIM('m) + DIM('n))"
-  by (simp add: finite_sum_image hassize_def card_image[OF inj_on_Abs_finite_sum[where ?'a = 'm and ?'b = 'n]] del: One_nat_def)
-
-lemma dimindex_finite_sum: "DIM(('m,'n) finite_sum) = DIM('m) + DIM('n)"
-  using dimindex_has_size_finite_sum[where ?'n = 'n and ?'m = 'm, unfolded hassize_def]
-  by (simp add: dimindex_def)
+lemma finite_sum_image: "(UNIV::('a + 'b) set) = range Inl \<union> range Inr"
+by (auto, case_tac x, auto)
 
 lemma fstcart_pastecart: "fstcart (pastecart (x::'a ^'m ) (y:: 'a ^ 'n)) = x"
-  by (simp add: pastecart_def fstcart_def Cart_eq Cart_lambda_beta dimindex_finite_sum)
+  by (simp add: Cart_eq)
 
 lemma sndcart_pastecart: "sndcart (pastecart (x::'a ^'m ) (y:: 'a ^ 'n)) = y"
-  by (simp add: pastecart_def sndcart_def Cart_eq Cart_lambda_beta dimindex_finite_sum)
+  by (simp add: Cart_eq)
 
 lemma pastecart_fst_snd: "pastecart (fstcart z) (sndcart z) = z"
-proof -
- {fix i
-  assume H: "i \<le> DIM('b) + DIM('c)"
-    "\<not> i \<le> DIM('b)"
-    from H have ith: "i - DIM('b) \<in> {1 .. DIM('c)}"
-      apply simp by arith
-    from H have th0: "i - DIM('b) + DIM('b) = i"
-      by simp
-  have "(\<chi> i. (z$(i + DIM('b))) :: 'a ^ 'c)$(i - DIM('b)) = z$i"
-    unfolding Cart_lambda_beta'[where g = "\<lambda> i. z$(i + DIM('b))", OF ith] th0 ..}
-thus ?thesis by (auto simp add: pastecart_def fstcart_def sndcart_def Cart_eq Cart_lambda_beta dimindex_finite_sum)
-qed
+  by (simp add: Cart_eq pastecart_def fstcart_def sndcart_def split: sum.split)
 
 lemma pastecart_eq: "(x = y) \<longleftrightarrow> (fstcart x = fstcart y) \<and> (sndcart x = sndcart y)"
   using pastecart_fst_snd[of x] pastecart_fst_snd[of y] by metis
@@ -216,53 +95,4 @@
 lemma exists_pastecart: "(\<exists>p. P p)  \<longleftrightarrow> (\<exists>x y. P (pastecart x y))"
   by (metis pastecart_fst_snd fstcart_pastecart sndcart_pastecart)
 
-text{* The finiteness lemma. *}
-
-lemma finite_cart:
- "\<forall>i \<in> {1 .. DIM('n)}. finite {x.  P i x}
-  \<Longrightarrow> finite {v::'a ^ 'n . (\<forall>i \<in> {1 .. DIM('n)}. P i (v$i))}"
-proof-
-  assume f: "\<forall>i \<in> {1 .. DIM('n)}. finite {x.  P i x}"
-  {fix n
-    assume n: "n \<le> DIM('n)"
-    have "finite {v:: 'a ^ 'n . (\<forall>i\<in> {1 .. DIM('n)}. i \<le> n \<longrightarrow> P i (v$i))
-                              \<and> (\<forall>i\<in> {1 .. DIM('n)}. n < i \<longrightarrow> v$i = (SOME x. False))}"
-      using n
-      proof(induct n)
-	case 0
-	have th0: "{v . (\<forall>i \<in> {1 .. DIM('n)}. v$i = (SOME x. False))} =
-      {(\<chi> i. (SOME x. False)::'a ^ 'n)}" by (auto simp add: Cart_lambda_beta Cart_eq)
-	with "0.prems" show ?case by auto
-      next
-	case (Suc n)
-	let ?h = "\<lambda>(x::'a,v:: 'a ^ 'n). (\<chi> i. if i = Suc n then x else v$i):: 'a ^ 'n"
-	let ?T = "{v\<Colon>'a ^ 'n.
-            (\<forall>i\<Colon>nat\<in>{1\<Colon>nat..DIM('n)}. i \<le> Suc n \<longrightarrow> P i (v$i)) \<and>
-            (\<forall>i\<Colon>nat\<in>{1\<Colon>nat..DIM('n)}.
-                Suc n < i \<longrightarrow> v$i = (SOME x\<Colon>'a. False))}"
-	let ?S = "{x::'a . P (Suc  n) x} \<times> {v:: 'a^'n. (\<forall>i \<in> {1 .. DIM('n)}. i <= n \<longrightarrow> P i (v$i)) \<and> (\<forall>i \<in> {1 .. DIM('n)}. n < i \<longrightarrow> v$i = (SOME x. False))}"
-	have th0: " ?T \<subseteq> (?h ` ?S)"
-	  using Suc.prems
-	  apply (auto simp add: image_def)
-	  apply (rule_tac x = "x$(Suc n)" in exI)
-	  apply (rule conjI)
-	  apply (rotate_tac)
-	  apply (erule ballE[where x="Suc n"])
-	  apply simp
-	  apply simp
-	  apply (rule_tac x= "\<chi> i. if i = Suc n then (SOME x:: 'a. False) else (x:: 'a ^ 'n)$i:: 'a ^ 'n" in exI)
-	  by (simp add: Cart_eq Cart_lambda_beta)
-	have th1: "finite ?S"
-	  apply (rule finite_cartesian_product)
-	  using f Suc.hyps Suc.prems by auto
-	from finite_imageI[OF th1] have th2: "finite (?h ` ?S)" .
-	from finite_subset[OF th0 th2] show ?case by blast
-      qed}
-
-  note th = this
-  from this[of "DIM('n)"] f
-  show ?thesis by auto
-qed
-
-
 end
diff -r ac50e7dedf6d -r 638b088bb840 src/HOL/Library/Topology_Euclidean_Space.thy
--- a/src/HOL/Library/Topology_Euclidean_Space.thy	Wed Mar 18 22:17:23 2009 +0100
+++ b/src/HOL/Library/Topology_Euclidean_Space.thy	Thu Mar 19 01:29:19 2009 -0700
@@ -488,7 +488,7 @@
 
 subsection{* Limit points *}
 
-definition islimpt:: "real ^'n \<Rightarrow> (real^'n) set \<Rightarrow> bool" (infixr "islimpt" 60) where
+definition islimpt:: "real ^'n::finite \<Rightarrow> (real^'n) set \<Rightarrow> bool" (infixr "islimpt" 60) where
   islimpt_def: "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
 
   (* FIXME: Sure this form is OK????*)
@@ -510,7 +510,7 @@
   using approachable_lt_le[where f="\<lambda>x'. dist x' x" and P="\<lambda>x'. \<not> (x'\<in>S \<and> x'\<noteq>x)"]
   by metis
 
-lemma islimpt_UNIV[simp, intro]: "(x:: real ^'n) islimpt UNIV"
+lemma islimpt_UNIV[simp, intro]: "(x:: real ^'n::finite) islimpt UNIV"
 proof-
   {
     fix e::real assume ep: "e>0"
@@ -532,20 +532,20 @@
 lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
   unfolding islimpt_approachable apply auto by ferrack
 
-lemma closed_positive_orthant: "closed {x::real^'n. \<forall>i\<in>{1.. dimindex(UNIV:: 'n set)}. 0 \<le>x$i}"
+lemma closed_positive_orthant: "closed {x::real^'n::finite. \<forall>i. 0 \<le>x$i}"
 proof-
-  let ?U = "{1 .. dimindex(UNIV :: 'n set)}"
-  let ?O = "{x::real^'n. \<forall>i\<in>?U. x$i\<ge>0}"
-  {fix x:: "real^'n" and i::nat assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e" and i: "i \<in> ?U"
+  let ?U = "UNIV :: 'n set"
+  let ?O = "{x::real^'n. \<forall>i. x$i\<ge>0}"
+  {fix x:: "real^'n" and i::'n assume H: "\<forall>e>0. \<exists>x'\<in>?O. x' \<noteq> x \<and> dist x' x < e"
     and xi: "x$i < 0"
     from xi have th0: "-x$i > 0" by arith
     from H[rule_format, OF th0] obtain x' where x': "x' \<in>?O" "x' \<noteq> x" "dist x' x < -x $ i" by blast
       have th:" \<And>b a (x::real). abs x <= b \<Longrightarrow> b <= a ==> ~(a + x < 0)" by arith
       have th': "\<And>x (y::real). x < 0 \<Longrightarrow> 0 <= y ==> abs x <= abs (y - x)" by arith
-      have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using i x'(1) xi
+      have th1: "\<bar>x$i\<bar> \<le> \<bar>(x' - x)$i\<bar>" using x'(1) xi
 	apply (simp only: vector_component)
 	by (rule th') auto
-      have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm[OF i, of "x'-x"]
+      have th2: "\<bar>dist x x'\<bar> \<ge> \<bar>(x' - x)$i\<bar>" using  component_le_norm[of "x'-x" i]
 	apply (simp add: dist_def) by norm
       from th[OF th1 th2] x'(3) have False by (simp add: dist_sym dist_pos_le) }
   then show ?thesis unfolding closed_limpt islimpt_approachable
@@ -965,7 +965,7 @@
 definition within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
   within_def: "net within S = mknet (\<lambda>x y. netord net x y \<and> x \<in> S)"
 
-definition indirection :: "real ^'n \<Rightarrow> real ^'n \<Rightarrow> (real ^'n) net" (infixr "indirection" 70) where
+definition indirection :: "real ^'n::finite \<Rightarrow> real ^'n \<Rightarrow> (real ^'n) net" (infixr "indirection" 70) where
   indirection_def: "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = c*s v}"
 
 text{* Prove That They are all nets. *}
@@ -1019,7 +1019,7 @@
   (\<forall>(a::'a) b. a = b) \<or> (\<exists>(a::'a) b. a \<noteq> b \<and> (\<forall>x. ~(netord (net) x a) \<and> ~(netord(net) x b)))"
 
 
-lemma trivial_limit_within: "trivial_limit (at (a::real^'n) within S) \<longleftrightarrow> ~(a islimpt S)"
+lemma trivial_limit_within: "trivial_limit (at (a::real^'n::finite) within S) \<longleftrightarrow> ~(a islimpt S)"
 proof-
   {assume "\<forall>(a::real^'n) b. a = b" hence "\<not> a islimpt S"
       apply (simp add: islimpt_approachable_le)
@@ -1104,7 +1104,7 @@
 apply (metis dlo_simps(7) dlo_simps(9) le_maxI2 min_max.le_iff_sup min_max.sup_absorb1 order_antisym_conv) done
 
 (* FIXME Declare this with P::'a::some_type \<Rightarrow> bool *)
-lemma eventually_at_infinity: "eventually (P::(real^'n \<Rightarrow> bool)) at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)" (is "?lhs = ?rhs")
+lemma eventually_at_infinity: "eventually (P::(real^'n::finite \<Rightarrow> bool)) at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)" (is "?lhs = ?rhs")
 proof
   assume "?lhs" thus "?rhs"
     unfolding eventually_def at_infinity
@@ -1145,7 +1145,7 @@
 
 subsection{* Limits, defined as vacuously true when the limit is trivial. *}
 
-definition tendsto:: "('a \<Rightarrow> real ^'n) \<Rightarrow> real ^'n \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "--->" 55) where
+definition tendsto:: "('a \<Rightarrow> real ^'n::finite) \<Rightarrow> real ^'n \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "--->" 55) where
   tendsto_def: "(f ---> l) net  \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
 
 lemma tendstoD: "(f ---> l) net \<Longrightarrow> e>0 \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
@@ -1177,7 +1177,7 @@
   by (auto simp add: tendsto_def eventually_at)
 
 lemma Lim_at_infinity:
-  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x::real^'n. norm x >= b \<longrightarrow> dist (f x) l < e)"
+  "(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x::real^'n::finite. norm x >= b \<longrightarrow> dist (f x) l < e)"
   by (auto simp add: tendsto_def eventually_at_infinity)
 
 lemma Lim_sequentially:
@@ -1210,7 +1210,7 @@
 qed
 
 lemma Lim_Un_univ:
- "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>  S \<union> T = (UNIV::(real^'n) set)
+ "(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>  S \<union> T = (UNIV::(real^'n::finite) set)
         ==> (f ---> l) (at x)"
   by (metis Lim_Un within_UNIV)
 
@@ -1275,7 +1275,7 @@
 
 text{* Basic arithmetical combining theorems for limits. *}
 
-lemma Lim_linear: fixes f :: "('a \<Rightarrow> real^'n)" and h :: "(real^'n \<Rightarrow> real^'m)"
+lemma Lim_linear: fixes f :: "('a \<Rightarrow> real^'n::finite)" and h :: "(real^'n \<Rightarrow> real^'m::finite)"
   assumes "(f ---> l) net" "linear h"
   shows "((\<lambda>x. h (f x)) ---> h l) net"
 proof (cases "trivial_limit net")
@@ -1315,7 +1315,7 @@
   apply (subst minus_diff_eq[symmetric])
   unfolding norm_minus_cancel by simp
 
-lemma Lim_add: fixes f :: "'a \<Rightarrow> real^'n" shows
+lemma Lim_add: fixes f :: "'a \<Rightarrow> real^'n::finite" shows
  "(f ---> l) net \<Longrightarrow> (g ---> m) net \<Longrightarrow> ((\<lambda>x. f(x) + g(x)) ---> l + m) net"
 proof-
   assume as:"(f ---> l) net" "(g ---> m) net"
@@ -1368,14 +1368,14 @@
     using assms `e>0` unfolding tendsto_def by auto
 qed
 
-lemma Lim_component: "(f ---> l) net \<Longrightarrow> i \<in> {1 .. dimindex(UNIV:: 'n set)}
-                      ==> ((\<lambda>a. vec1((f a :: real ^'n)$i)) ---> vec1(l$i)) net"
-  apply (simp add: Lim dist_def vec1_sub[symmetric] norm_vec1  vector_minus_component[symmetric] del: One_nat_def)
-  apply auto
+lemma Lim_component: "(f ---> l) net
+                      ==> ((\<lambda>a. vec1((f a :: real ^'n::finite)$i)) ---> vec1(l$i)) net"
+  apply (simp add: Lim dist_def vec1_sub[symmetric] norm_vec1  vector_minus_component[symmetric] del: vector_minus_component)
+  apply (auto simp del: vector_minus_component)
   apply (erule_tac x=e in allE)
   apply clarify
   apply (rule_tac x=y in exI)
-  apply auto
+  apply (auto simp del: vector_minus_component)
   apply (rule order_le_less_trans)
   apply (rule component_le_norm)
   by auto
@@ -1450,7 +1450,7 @@
 text{* Uniqueness of the limit, when nontrivial. *}
 
 lemma Lim_unique:
-  fixes l::"real^'a" and net::"'b::zero_neq_one net"
+  fixes l::"real^'a::finite" and net::"'b::zero_neq_one net"
   assumes "\<not>(trivial_limit net)"  "(f ---> l) net"  "(f ---> l') net"
   shows "l = l'"
 proof-
@@ -1472,7 +1472,7 @@
 text{* Limit under bilinear function (surprisingly tedious, but important) *}
 
 lemma norm_bound_lemma:
-  "0 < e \<Longrightarrow> \<exists>d>0. \<forall>(x'::real^'b) y'::real^'a. norm(x' - (x::real^'b)) < d \<and> norm(y' - y) < d \<longrightarrow> norm(x') * norm(y' - y) + norm(x' - x) * norm(y) < e"
+  "0 < e \<Longrightarrow> \<exists>d>0. \<forall>(x'::real^'b::finite) y'::real^'a::finite. norm(x' - (x::real^'b)) < d \<and> norm(y' - y) < d \<longrightarrow> norm(x') * norm(y' - y) + norm(x' - x) * norm(y) < e"
 proof-
   assume e: "0 < e"
   have th1: "(2 * norm x + 2 * norm y + 2) > 0" using norm_ge_zero[of x] norm_ge_zero[of y] by norm
@@ -1503,7 +1503,7 @@
 qed
 
 lemma Lim_bilinear:
-  fixes net :: "'a net" and h:: "real ^'m \<Rightarrow> real ^'n \<Rightarrow> real ^'p"
+  fixes net :: "'a net" and h:: "real ^'m::finite \<Rightarrow> real ^'n::finite \<Rightarrow> real ^'p::finite"
   assumes "(f ---> l) net" and "(g ---> m) net" and "bilinear h"
   shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
 proof(cases "trivial_limit net")
@@ -1541,7 +1541,7 @@
 lemma Lim_at_id: "(id ---> a) (at a)"
 apply (subst within_UNIV[symmetric]) by (simp add: Lim_within_id)
 
-lemma Lim_at_zero: "(f ---> l) (at (a::real^'a)) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
+lemma Lim_at_zero: "(f ---> l) (at (a::real^'a::finite)) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
 proof
   assume "?lhs"
   { fix e::real assume "e>0"
@@ -1619,7 +1619,7 @@
 text{* Common case assuming being away from some crucial point like 0. *}
 
 lemma Lim_transform_away_within:
-  fixes f:: "real ^'m \<Rightarrow> real ^'n"
+  fixes f:: "real ^'m::finite \<Rightarrow> real ^'n::finite"
   assumes "a\<noteq>b" "\<forall>x\<in> S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
   and "(f ---> l) (at a within S)"
   shows "(g ---> l) (at a within S)"
@@ -1630,7 +1630,7 @@
 qed
 
 lemma Lim_transform_away_at:
-  fixes f:: "real ^'m \<Rightarrow> real ^'n"
+  fixes f:: "real ^'m::finite \<Rightarrow> real ^'n::finite"
   assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
   and fl: "(f ---> l) (at a)"
   shows "(g ---> l) (at a)"
@@ -1640,7 +1640,7 @@
 text{* Alternatively, within an open set. *}
 
 lemma Lim_transform_within_open:
-  fixes f:: "real ^'m \<Rightarrow> real ^'n"
+  fixes f:: "real ^'m::finite \<Rightarrow> real ^'n::finite"
   assumes "open S"  "a \<in> S"  "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"  "(f ---> l) (at a)"
   shows "(g ---> l) (at a)"
 proof-
@@ -1917,7 +1917,7 @@
 subsection{* Boundedness. *}
 
   (* FIXME: This has to be unified with BSEQ!! *)
-definition "bounded S \<longleftrightarrow> (\<exists>a. \<forall>(x::real^'n) \<in> S. norm x <= a)"
+definition "bounded S \<longleftrightarrow> (\<exists>a. \<forall>(x::real^'n::finite) \<in> S. norm x <= a)"
 
 lemma bounded_empty[simp]: "bounded {}" by (simp add: bounded_def)
 lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
@@ -1978,7 +1978,7 @@
 lemma bounded_insert[intro]:"bounded(insert x S) \<longleftrightarrow> bounded S"
   by (metis Diff_cancel Un_empty_right Un_insert_right bounded_Un bounded_subset finite.emptyI finite_imp_bounded infinite_remove subset_insertI)
 
-lemma bot_bounded_UNIV[simp, intro]: "~(bounded (UNIV:: (real^'n) set))"
+lemma bot_bounded_UNIV[simp, intro]: "~(bounded (UNIV:: (real^'n::finite) set))"
 proof(auto simp add: bounded_pos not_le)
   fix b::real  assume b: "b >0"
   have b1: "b +1 \<ge> 0" using b by simp
@@ -1988,7 +1988,7 @@
 qed
 
 lemma bounded_linear_image:
-  fixes f :: "real^'m \<Rightarrow> real^'n"
+  fixes f :: "real^'m::finite \<Rightarrow> real^'n::finite"
   assumes "bounded S" "linear f"
   shows "bounded(f ` S)"
 proof-
@@ -2110,7 +2110,7 @@
 subsection{* Compactness (the definition is the one based on convegent subsequences). *}
 
 definition "compact S \<longleftrightarrow>
-   (\<forall>(f::nat \<Rightarrow> real^'n). (\<forall>n. f n \<in> S) \<longrightarrow>
+   (\<forall>(f::nat \<Rightarrow> real^'n::finite). (\<forall>n. f n \<in> S) \<longrightarrow>
        (\<exists>l\<in>S. \<exists>r. (\<forall>m n. m < n \<longrightarrow> r m < r n) \<and> ((f o r) ---> l) sequentially))"
 
 lemma monotone_bigger: fixes r::"nat\<Rightarrow>nat"
@@ -2178,81 +2178,69 @@
 qed
 
 lemma compact_lemma:
-  assumes "bounded s" and "\<forall>n. (x::nat \<Rightarrow>real^'a) n \<in> s"
-  shows "\<forall>d\<in>{1.. dimindex(UNIV::'a set)}.
-        \<exists>l::(real^'a). \<exists> r. (\<forall>n m::nat. m < n --> r m < r n) \<and>
-        (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>i\<in>{1..d}. \<bar>x (r n) $ i - l $ i\<bar> < e)"
+  assumes "bounded s" and "\<forall>n. (x::nat \<Rightarrow>real^'a::finite) n \<in> s"
+  shows "\<forall>d.
+        \<exists>l::(real^'a::finite). \<exists> r. (\<forall>n m::nat. m < n --> r m < r n) \<and>
+        (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>i\<in>d. \<bar>x (r n) $ i - l $ i\<bar> < e)"
 proof-
   obtain b where b:"\<forall>x\<in>s. norm x \<le> b" using assms(1)[unfolded bounded_def] by auto
-  { { fix i assume i:"i\<in>{1.. dimindex(UNIV::'a set)}"
+  { { fix i::'a
       { fix n::nat
-	have "\<bar>x n $ i\<bar> \<le> b" using b[THEN bspec[where x="x n"]] and component_le_norm[of i "x n"] and assms(2)[THEN spec[where x=n]] and i by auto }
+	have "\<bar>x n $ i\<bar> \<le> b" using b[THEN bspec[where x="x n"]] and component_le_norm[of "x n" i] and assms(2)[THEN spec[where x=n]] by auto }
       hence "\<forall>n. \<bar>x n $ i\<bar> \<le> b" by auto
     } note b' = this
 
-    fix d assume "d\<in>{1.. dimindex(UNIV::'a set)}"
+    fix d::"'a set" have "finite d" by simp
     hence "\<exists>l::(real^'a). \<exists> r. (\<forall>n m::nat. m < n --> r m < r n) \<and>
-        (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>i\<in>{1..d}. \<bar>x (r n) $ i - l $ i\<bar> < e)"
-    proof(induct d) case 0 thus ?case by auto
-      (* The induction really starts at Suc 0 *)
-    next case (Suc d)
-      show ?case proof(cases "d = 0")
-	case True hence "Suc d = Suc 0" by auto
-	obtain l r where r:"\<forall>m n::nat. m < n \<longrightarrow> r m < r n" and lr:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>x (r n) $ 1 - l\<bar> < e" using b' and dimindex_ge_1[of "UNIV::'a set"]
-	  using compact_real_lemma[of "\<lambda>i. (x i)$1" b] by auto
-	thus ?thesis apply(rule_tac x="vec l" in exI) apply(rule_tac x=r in exI)
-	  unfolding `Suc d = Suc 0` apply auto
-	  unfolding vec_component[OF Suc(2)[unfolded `Suc d = Suc 0`]] by auto
-      next
-	case False hence d:"d \<in>{1.. dimindex(UNIV::'a set)}" using Suc(2) by auto
-	obtain l1::"real^'a" and r1 where r1:"\<forall>n m::nat. m < n \<longrightarrow> r1 m < r1 n" and lr1:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>i\<in>{1..d}. \<bar>x (r1 n) $ i - l1 $ i\<bar> < e"
-	  using Suc(1)[OF d] by auto
-	obtain l2 r2 where r2:"\<forall>m n::nat. m < n \<longrightarrow> r2 m < r2 n" and lr2:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>(x \<circ> r1) (r2 n) $ (Suc d) - l2\<bar> < e"
-	  using b'[OF Suc(2)] and compact_real_lemma[of "\<lambda>i. ((x \<circ> r1) i)$(Suc d)" b] by auto
+        (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>i\<in>d. \<bar>x (r n) $ i - l $ i\<bar> < e)"
+    proof(induct d) case empty thus ?case by auto
+    next case (insert k d)
+	obtain l1::"real^'a" and r1 where r1:"\<forall>n m::nat. m < n \<longrightarrow> r1 m < r1 n" and lr1:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>i\<in>d. \<bar>x (r1 n) $ i - l1 $ i\<bar> < e"
+	  using insert(3) by auto
+	obtain l2 r2 where r2:"\<forall>m n::nat. m < n \<longrightarrow> r2 m < r2 n" and lr2:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<bar>(x \<circ> r1) (r2 n) $ k - l2\<bar> < e"
+	  using b'[of k] and compact_real_lemma[of "\<lambda>i. ((x \<circ> r1) i)$k" b] by auto
 	def r \<equiv> "r1 \<circ> r2" have r:"\<forall>m n. m < n \<longrightarrow> r m < r n" unfolding r_def o_def using r1 and r2 by auto
 	moreover
-	def l \<equiv> "(\<chi> i. if i = Suc d then l2 else l1$i)::real^'a"
+	def l \<equiv> "(\<chi> i. if i = k then l2 else l1$i)::real^'a"
 	{ fix e::real assume "e>0"
-	  from lr1 obtain N1 where N1:"\<forall>n\<ge>N1. \<forall>i\<in>{1..d}. \<bar>x (r1 n) $ i - l1 $ i\<bar> < e" using `e>0` by blast
-	  from lr2 obtain N2 where N2:"\<forall>n\<ge>N2. \<bar>(x \<circ> r1) (r2 n) $ (Suc d) - l2\<bar> < e" using `e>0` by blast
+	  from lr1 obtain N1 where N1:"\<forall>n\<ge>N1. \<forall>i\<in>d. \<bar>x (r1 n) $ i - l1 $ i\<bar> < e" using `e>0` by blast
+	  from lr2 obtain N2 where N2:"\<forall>n\<ge>N2. \<bar>(x \<circ> r1) (r2 n) $ k - l2\<bar> < e" using `e>0` by blast
 	  { fix n assume n:"n\<ge> N1 + N2"
-	    fix i assume i:"i\<in>{1..Suc d}" hence i':"i\<in>{1.. dimindex(UNIV::'a set)}" using Suc by auto
+	    fix i assume i:"i\<in>(insert k d)"
 	    hence "\<bar>x (r n) $ i - l $ i\<bar> < e"
 	      using N2[THEN spec[where x="n"]] and n
  	      using N1[THEN spec[where x="r2 n"]] and n
 	      using monotone_bigger[OF r] and i
-	      unfolding l_def and r_def and Cart_lambda_beta'[OF i']
+	      unfolding l_def and r_def
 	      using monotone_bigger[OF r2, of n] by auto  }
-	  hence "\<exists>N. \<forall>n\<ge>N. \<forall>i\<in>{1..Suc d}. \<bar>x (r n) $ i - l $ i\<bar> < e" by blast	}
-	ultimately show ?thesis by auto
-      qed
+	  hence "\<exists>N. \<forall>n\<ge>N. \<forall>i\<in>(insert k d). \<bar>x (r n) $ i - l $ i\<bar> < e" by blast	}
+	ultimately show ?case by auto
     qed  }
   thus ?thesis by auto
 qed
 
-lemma bounded_closed_imp_compact: fixes s::"(real^'a) set"
+lemma bounded_closed_imp_compact: fixes s::"(real^'a::finite) set"
   assumes "bounded s" and "closed s"
   shows "compact s"
 proof-
-  let ?d = "dimindex (UNIV::'a set)"
+  let ?d = "UNIV::'a set"
   { fix f assume as:"\<forall>n::nat. f n \<in> s"
     obtain l::"real^'a" and r where r:"\<forall>n m::nat. m < n \<longrightarrow> r m < r n"
-      and lr:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>i\<in>{1..?d}. \<bar>f (r n) $ i - l $ i\<bar> < e"
-      using compact_lemma[OF assms(1) as, THEN bspec[where x="?d"]] and dimindex_ge_1[of "UNIV::'a set"] by auto
+      and lr:"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>i\<in>?d. \<bar>f (r n) $ i - l $ i\<bar> < e"
+      using compact_lemma[OF assms(1) as, THEN spec[where x="?d"]] by auto
     { fix e::real assume "e>0"
-      hence "0 < e / (real_of_nat ?d)" using dimindex_nonzero[of "UNIV::'a set"] using divide_pos_pos[of e, of "real_of_nat ?d"] by auto
-      then obtain N::nat where N:"\<forall>n\<ge>N. \<forall>i\<in>{1..?d}. \<bar>f (r n) $ i - l $ i\<bar> < e / (real_of_nat ?d)" using lr[THEN spec[where x="e / (real_of_nat ?d)"]] by blast
+      hence "0 < e / (real_of_nat (card ?d))" using zero_less_card_finite using divide_pos_pos[of e, of "real_of_nat (card ?d)"] by auto
+      then obtain N::nat where N:"\<forall>n\<ge>N. \<forall>i\<in>?d. \<bar>f (r n) $ i - l $ i\<bar> < e / (real_of_nat (card ?d))" using lr[THEN spec[where x="e / (real_of_nat (card ?d))"]] by blast
       { fix n assume n:"n\<ge>N"
-	have "1 \<in> {1..?d}" using dimindex_nonzero[of "UNIV::'a set"] by auto
-	hence "finite {1..?d}"  "{1..?d} \<noteq> {}" by auto
+	hence "finite ?d"  "?d \<noteq> {}" by auto
 	moreover
-	{ fix i assume i:"i \<in> {1..?d}"
-	  hence "\<bar>((f \<circ> r) n - l) $ i\<bar> < e / real_of_nat ?d" using `n\<ge>N` using N[THEN spec[where x=n]]
-	    apply auto apply(erule_tac x=i in ballE) unfolding vector_minus_component[OF i] by auto  }
-	ultimately have "(\<Sum>i = 1..?d. \<bar>((f \<circ> r) n - l) $ i\<bar>)
-	  < (\<Sum>i = 1..?d. e / real_of_nat ?d)"
-	  using setsum_strict_mono[of "{1..?d}" "\<lambda>i. \<bar>((f \<circ> r) n - l) $ i\<bar>" "\<lambda>i. e / (real_of_nat ?d)"] by auto
-	hence "(\<Sum>i = 1..?d. \<bar>((f \<circ> r) n - l) $ i\<bar>) < e" unfolding setsum_constant using dimindex_nonzero[of "UNIV::'a set"] by auto
+	{ fix i assume i:"i \<in> ?d"
+	  hence "\<bar>((f \<circ> r) n - l) $ i\<bar> < e / real_of_nat (card ?d)" using `n\<ge>N` using N[THEN spec[where x=n]]
+	    by auto  }
+	ultimately have "(\<Sum>i \<in> ?d. \<bar>((f \<circ> r) n - l) $ i\<bar>)
+	  < (\<Sum>i \<in> ?d. e / real_of_nat (card ?d))"
+	  using setsum_strict_mono[of "?d" "\<lambda>i. \<bar>((f \<circ> r) n - l) $ i\<bar>" "\<lambda>i. e / (real_of_nat (card ?d))"] by auto
+	hence "(\<Sum>i \<in> ?d. \<bar>((f \<circ> r) n - l) $ i\<bar>) < e" unfolding setsum_constant by auto
 	hence "dist ((f \<circ> r) n) l < e" unfolding dist_def using norm_le_l1[of "(f \<circ> r) n - l"] by auto  }
       hence "\<exists>N. \<forall>n\<ge>N. dist ((f \<circ> r) n) l < e" by auto  }
     hence *:"((f \<circ> r) ---> l) sequentially" unfolding Lim_sequentially by auto
@@ -2268,7 +2256,7 @@
 
 definition cauchy_def:"cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
 
-definition complete_def:"complete s \<longleftrightarrow> (\<forall>f::(nat=>real^'a). (\<forall>n. f n \<in> s) \<and> cauchy f
+definition complete_def:"complete s \<longleftrightarrow> (\<forall>f::(nat=>real^'a::finite). (\<forall>n. f n \<in> s) \<and> cauchy f
                       --> (\<exists>l \<in> s. (f ---> l) sequentially))"
 
 lemma cauchy: "cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
@@ -2350,7 +2338,7 @@
 lemma complete_univ:
  "complete UNIV"
 proof(simp add: complete_def, rule, rule)
-  fix f::"nat \<Rightarrow> real^'n" assume "cauchy f"
+  fix f::"nat \<Rightarrow> real^'n::finite" assume "cauchy f"
   hence "bounded (f`UNIV)" using cauchy_imp_bounded[of f] unfolding image_def by auto
   hence "compact (closure (f`UNIV))"  using bounded_closed_imp_compact[of "closure (range f)"] by auto
   hence "complete (closure (range f))" using compact_imp_complete by auto
@@ -2389,7 +2377,7 @@
 
 subsection{* Total boundedness. *}
 
-fun helper_1::"((real^'n) set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real^'n" where
+fun helper_1::"((real^'n::finite) set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> real^'n" where
   "helper_1 s e n = (SOME y::real^'n. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
 declare helper_1.simps[simp del]
 
@@ -2422,7 +2410,7 @@
 
 subsection{* Heine-Borel theorem (following Burkill \& Burkill vol. 2) *}
 
-lemma heine_borel_lemma: fixes s::"(real^'n) set"
+lemma heine_borel_lemma: fixes s::"(real^'n::finite) set"
   assumes "compact s"  "s \<subseteq> (\<Union> t)"  "\<forall>b \<in> t. open b"
   shows "\<exists>e>0. \<forall>x \<in> s. \<exists>b \<in> t. ball x e \<subseteq> b"
 proof(rule ccontr)
@@ -2513,11 +2501,11 @@
 
 subsection{* Complete the chain of compactness variants. *}
 
-primrec helper_2::"(real \<Rightarrow> real^'n) \<Rightarrow> nat \<Rightarrow> real ^'n" where
+primrec helper_2::"(real \<Rightarrow> real^'n::finite) \<Rightarrow> nat \<Rightarrow> real ^'n" where
   "helper_2 beyond 0 = beyond 0" |
   "helper_2 beyond (Suc n) = beyond (norm (helper_2 beyond n) + 1 )"
 
-lemma bolzano_weierstrass_imp_bounded: fixes s::"(real^'n) set"
+lemma bolzano_weierstrass_imp_bounded: fixes s::"(real^'n::finite) set"
   assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
   shows "bounded s"
 proof(rule ccontr)
@@ -2576,7 +2564,7 @@
 
 lemma sequence_infinite_lemma:
   assumes "\<forall>n::nat. (f n  \<noteq> l)"  "(f ---> l) sequentially"
-  shows "infinite {y::real^'a. (\<exists> n. y = f n)}"
+  shows "infinite {y::real^'a::finite. (\<exists> n. y = f n)}"
 proof(rule ccontr)
   let ?A = "(\<lambda>x. dist x l) ` {y. \<exists>n. y = f n}"
   assume "\<not> infinite {y. \<exists>n. y = f n}"
@@ -2771,7 +2759,7 @@
 lemma bounded_closed_nest:
   assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
   "(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"
-  shows "\<exists> a::real^'a. \<forall>n::nat. a \<in> s(n)"
+  shows "\<exists> a::real^'a::finite. \<forall>n::nat. a \<in> s(n)"
 proof-
   from assms(2) obtain x where x:"\<forall>n::nat. x n \<in> s n" using choice[of "\<lambda>n x. x\<in> s n"] by auto
   from assms(4,1) have *:"compact (s 0)" using bounded_closed_imp_compact[of "s 0"] by auto
@@ -2803,7 +2791,7 @@
           "\<forall>n. (s n \<noteq> {})"
           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y \<in> (s n). dist x y < e"
-  shows "\<exists>a::real^'a. \<forall>n::nat. a \<in> s n"
+  shows "\<exists>a::real^'a::finite. \<forall>n::nat. a \<in> s n"
 proof-
   have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
   hence "\<exists>t. \<forall>n. t n \<in> s n" using choice[of "\<lambda> n x. x \<in> s n"] by auto
@@ -2836,7 +2824,7 @@
           "\<forall>n. s n \<noteq> {}"
           "\<forall>m n. m \<le> n --> s n \<subseteq> s m"
           "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
-  shows "\<exists>a::real^'a. \<Inter> {t. (\<exists>n::nat. t = s n)} = {a}"
+  shows "\<exists>a::real^'a::finite. \<Inter> {t. (\<exists>n::nat. t = s n)} = {a}"
 proof-
   obtain a where a:"\<forall>n. a \<in> s n" using decreasing_closed_nest[of s] using assms by auto
   { fix b assume b:"b \<in> \<Inter>{t. \<exists>n. t = s n}"
@@ -2851,7 +2839,7 @@
 
 text{* Cauchy-type criteria for uniform convergence. *}
 
-lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> real^'a" shows
+lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> real^'a::finite" shows
  "(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
   (\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs")
 proof(rule)
@@ -2960,7 +2948,7 @@
     apply (auto simp add: dist_sym) apply(erule_tac x=e in allE) by auto
 qed
 
-lemma continuous_at_ball: fixes f::"real^'a \<Rightarrow> real^'a"
+lemma continuous_at_ball: fixes f::"real^'a::finite \<Rightarrow> real^'a"
   shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f ` (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
 proof
   assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
@@ -3255,7 +3243,7 @@
 
 lemma uniformly_continuous_on_add:
   assumes "uniformly_continuous_on s f" "uniformly_continuous_on s g"
-  shows "uniformly_continuous_on s (\<lambda>x. f(x) + g(x) ::real^'n)"
+  shows "uniformly_continuous_on s (\<lambda>x. f(x) + g(x) ::real^'n::finite)"
 proof-
   have *:"\<And>fx fy gx gy::real^'n. fx - fy + (gx - gy) = fx + gx - (fy + gy)" by auto
   {  fix x y assume "((\<lambda>n. f (x n) - f (y n)) ---> 0) sequentially"
@@ -3729,7 +3717,7 @@
 
 subsection{* Topological properties of linear functions.                               *}
 
-lemma linear_lim_0: fixes f::"real^'a \<Rightarrow> real^'b"
+lemma linear_lim_0: fixes f::"real^'a::finite \<Rightarrow> real^'b::finite"
   assumes "linear f" shows "(f ---> 0) (at (0))"
 proof-
   obtain B where "B>0" and B:"\<forall>x. norm (f x) \<le> B * norm x" using linear_bounded_pos[OF assms] by auto
@@ -3813,19 +3801,18 @@
 unfolding continuous_on_vec1_range norm_vec1[THEN sym] by (metis norm_vec1 order_le_less_trans real_abs_sub_norm)
 
 lemma continuous_at_vec1_component:
-  assumes "1 \<le> i" "i \<le> dimindex(UNIV::('a set))"
-  shows "continuous (at (a::real^'a)) (\<lambda> x. vec1(x$i))"
+  shows "continuous (at (a::real^'a::finite)) (\<lambda> x. vec1(x$i))"
 proof-
   { fix e::real and x assume "0 < dist x a" "dist x a < e" "e>0"
-    hence "\<bar>x $ i - a $ i\<bar> < e" using component_le_norm[of i "x - a"] vector_minus_component[of i x a] assms unfolding dist_def by auto  }
+    hence "\<bar>x $ i - a $ i\<bar> < e" using component_le_norm[of "x - a" i] unfolding dist_def by auto  }
   thus ?thesis unfolding continuous_at tendsto_def eventually_at dist_vec1 by auto
 qed
 
 lemma continuous_on_vec1_component:
-  assumes "i \<in> {1..dimindex (UNIV::'a set)}"  shows "continuous_on s (\<lambda> x::real^'a. vec1(x$i))"
+  shows "continuous_on s (\<lambda> x::real^'a::finite. vec1(x$i))"
 proof-
   { fix e::real and x xa assume "x\<in>s" "e>0" "xa\<in>s" "0 < norm (xa - x) \<and> norm (xa - x) < e"
-    hence "\<bar>xa $ i - x $ i\<bar> < e" using component_le_norm[of i "xa - x"] vector_minus_component[of i xa x] assms by auto  }
+    hence "\<bar>xa $ i - x $ i\<bar> < e" using component_le_norm[of "xa - x" i] by auto  }
   thus ?thesis unfolding continuous_on Lim_within dist_vec1 unfolding dist_def by auto
 qed
 
@@ -4070,7 +4057,7 @@
 proof-
   have *:"{x + y | x y. x \<in> s \<and> y \<in> t} =(\<lambda>z. fstcart z + sndcart z) ` {pastecart x y | x y.  x \<in> s \<and> y \<in> t}"
     apply auto unfolding image_iff apply(rule_tac x="pastecart xa y" in bexI) unfolding fstcart_pastecart sndcart_pastecart by auto
-  have "linear (\<lambda>z::real^('a, 'a) finite_sum. fstcart z + sndcart z)" unfolding linear_def
+  have "linear (\<lambda>z::real^('a + 'a). fstcart z + sndcart z)" unfolding linear_def
     unfolding fstcart_add sndcart_add apply auto
     unfolding vector_add_ldistrib fstcart_cmul[THEN sym] sndcart_cmul[THEN sym] by auto
   hence "continuous_on {pastecart x y |x y. x \<in> s \<and> y \<in> t} (\<lambda>z. fstcart z + sndcart z)"
@@ -4306,90 +4293,86 @@
 
 (* A cute way of denoting open and closed intervals using overloading.       *)
 
-lemma interval: fixes a :: "'a::ord^'n" shows
-  "{a <..< b} = {x::'a^'n. \<forall>i \<in> dimset a. a$i < x$i \<and> x$i < b$i}" and
-  "{a .. b} = {x::'a^'n. \<forall>i \<in> dimset a. a$i \<le> x$i \<and> x$i \<le> b$i}"
+lemma interval: fixes a :: "'a::ord^'n::finite" shows
+  "{a <..< b} = {x::'a^'n. \<forall>i. a$i < x$i \<and> x$i < b$i}" and
+  "{a .. b} = {x::'a^'n. \<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i}"
   by (auto simp add: expand_set_eq vector_less_def vector_less_eq_def)
 
-lemma mem_interval:
-  "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i \<in> dimset a. a$i < x$i \<and> x$i < b$i)"
-  "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i \<in> dimset a. a$i \<le> x$i \<and> x$i \<le> b$i)"
+lemma mem_interval: fixes a :: "'a::ord^'n::finite" shows
+  "x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i. a$i < x$i \<and> x$i < b$i)"
+  "x \<in> {a .. b} \<longleftrightarrow> (\<forall>i. a$i \<le> x$i \<and> x$i \<le> b$i)"
   using interval[of a b]
   by(auto simp add: expand_set_eq vector_less_def vector_less_eq_def)
 
-lemma interval_eq_empty: fixes a :: "real^'n" shows
- "({a <..< b} = {} \<longleftrightarrow> (\<exists>i \<in> dimset a. b$i \<le> a$i))" (is ?th1) and
- "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i \<in> dimset a. b$i < a$i))" (is ?th2)
+lemma interval_eq_empty: fixes a :: "real^'n::finite" shows
+ "({a <..< b} = {} \<longleftrightarrow> (\<exists>i. b$i \<le> a$i))" (is ?th1) and
+ "({a  ..  b} = {} \<longleftrightarrow> (\<exists>i. b$i < a$i))" (is ?th2)
 proof-
-  { fix i x assume i:"i\<in>dimset a" and as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"
+  { fix i x assume as:"b$i \<le> a$i" and x:"x\<in>{a <..< b}"
     hence "a $ i < x $ i \<and> x $ i < b $ i" unfolding mem_interval by auto
     hence "a$i < b$i" by auto
     hence False using as by auto  }
   moreover
-  { assume as:"\<forall>i \<in> dimset a. \<not> (b$i \<le> a$i)"
+  { assume as:"\<forall>i. \<not> (b$i \<le> a$i)"
     let ?x = "(1/2) *s (a + b)"
-    { fix i assume i:"i\<in>dimset a"
-      hence "a$i < b$i" using as[THEN bspec[where x=i]] by auto
+    { fix i
+      have "a$i < b$i" using as[THEN spec[where x=i]] by auto
       hence "a$i < ((1/2) *s (a+b)) $ i" "((1/2) *s (a+b)) $ i < b$i"
-	unfolding vector_smult_component[OF i] and vector_add_component[OF i]
+	unfolding vector_smult_component and vector_add_component
 	by (auto simp add: Arith_Tools.less_divide_eq_number_of1)  }
     hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }
   ultimately show ?th1 by blast
 
-  { fix i x assume i:"i\<in>dimset a" and as:"b$i < a$i" and x:"x\<in>{a .. b}"
+  { fix i x assume as:"b$i < a$i" and x:"x\<in>{a .. b}"
     hence "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" unfolding mem_interval by auto
     hence "a$i \<le> b$i" by auto
     hence False using as by auto  }
   moreover
-  { assume as:"\<forall>i \<in> dimset a. \<not> (b$i < a$i)"
+  { assume as:"\<forall>i. \<not> (b$i < a$i)"
     let ?x = "(1/2) *s (a + b)"
-    { fix i assume i:"i\<in>dimset a"
-      hence "a$i \<le> b$i" using as[THEN bspec[where x=i]] by auto
+    { fix i
+      have "a$i \<le> b$i" using as[THEN spec[where x=i]] by auto
       hence "a$i \<le> ((1/2) *s (a+b)) $ i" "((1/2) *s (a+b)) $ i \<le> b$i"
-	unfolding vector_smult_component[OF i] and vector_add_component[OF i]
+	unfolding vector_smult_component and vector_add_component
 	by (auto simp add: Arith_Tools.less_divide_eq_number_of1)  }
     hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }
   ultimately show ?th2 by blast
 qed
 
-lemma interval_ne_empty: fixes a :: "real^'n" shows
-  "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i \<in> dimset a. a$i \<le> b$i)" and
-  "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i \<in> dimset a. a$i < b$i)"
-  unfolding interval_eq_empty[of a b] by auto
-
-lemma subset_interval_imp: fixes a :: "real^'n" shows
- "(\<forall>i \<in> dimset a. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
- "(\<forall>i \<in> dimset a. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
- "(\<forall>i \<in> dimset a. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
- "(\<forall>i \<in> dimset a. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
-  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval by(auto elim!: ballE)
-
-lemma interval_sing: fixes a :: "'a::linorder^'n" shows
+lemma interval_ne_empty: fixes a :: "real^'n::finite" shows
+  "{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i \<le> b$i)" and
+  "{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i. a$i < b$i)"
+  unfolding interval_eq_empty[of a b] by (auto simp add: not_less not_le) (* BH: Why doesn't just "auto" work here? *)
+
+lemma subset_interval_imp: fixes a :: "real^'n::finite" shows
+ "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
+ "(\<forall>i. a$i < c$i \<and> d$i < b$i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
+ "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
+ "(\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
+  unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
+  by (auto intro: order_trans less_le_trans le_less_trans less_imp_le) (* BH: Why doesn't just "auto" work here? *)
+
+lemma interval_sing: fixes a :: "'a::linorder^'n::finite" shows
  "{a .. a} = {a} \<and> {a<..<a} = {}"
 apply(auto simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
-apply (simp only: order_eq_iff)
-using dimindex_ge_1[of "UNIV :: 'n set"]
-apply (auto simp add: not_less )
-apply (erule_tac x= 1 in ballE)
-apply (rule bexI[where x=1])
-apply auto
+apply (simp add: order_eq_iff)
+apply (auto simp add: not_less less_imp_le)
 done
 
 
-lemma interval_open_subset_closed:  fixes a :: "'a::preorder^'n" shows
+lemma interval_open_subset_closed:  fixes a :: "'a::preorder^'n::finite" shows
  "{a<..<b} \<subseteq> {a .. b}"
 proof(simp add: subset_eq, rule)
   fix x
   assume x:"x \<in>{a<..<b}"
-  { fix i assume "i \<in> dimset a"
-    hence "a $ i \<le> x $ i"
+  { fix i
+    have "a $ i \<le> x $ i"
       using x order_less_imp_le[of "a$i" "x$i"]
       by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
   }
   moreover
-  { fix i assume "i \<in> dimset a"
-    hence "x $ i \<le> b $ i"
-      using x
+  { fix i
+    have "x $ i \<le> b $ i"
       using x order_less_imp_le[of "x$i" "b$i"]
       by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
   }
@@ -4398,76 +4381,76 @@
     by(simp add: expand_set_eq vector_less_def vector_less_eq_def Cart_eq)
 qed
 
-lemma subset_interval: fixes a :: "real^'n" shows
- "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i \<in> dimset a. c$i \<le> d$i) --> (\<forall>i \<in> dimset a. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1) and
- "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i \<in> dimset a. c$i \<le> d$i) --> (\<forall>i \<in> dimset a. a$i < c$i \<and> d$i < b$i)" (is ?th2) and
- "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i \<in> dimset a. c$i < d$i) --> (\<forall>i \<in> dimset a. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3) and
- "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i \<in> dimset a. c$i < d$i) --> (\<forall>i \<in> dimset a. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
+lemma subset_interval: fixes a :: "real^'n::finite" shows
+ "{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th1) and
+ "{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i \<le> d$i) --> (\<forall>i. a$i < c$i \<and> d$i < b$i)" (is ?th2) and
+ "{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th3) and
+ "{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i. c$i < d$i) --> (\<forall>i. a$i \<le> c$i \<and> d$i \<le> b$i)" (is ?th4)
 proof-
-  show ?th1 unfolding subset_eq and Ball_def and mem_interval apply auto by(erule_tac x=xa in allE, simp)+
-  show ?th2 unfolding subset_eq and Ball_def and mem_interval apply auto by(erule_tac x=xa in allE, simp)+
-  { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i \<in> dimset a. c$i < d$i"
-    hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
-    fix i assume i:"i \<in> dimset a"
+  show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
+  show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
+  { assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i. c$i < d$i"
+    hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by (auto, drule_tac x=i in spec, simp) (* BH: Why doesn't just "auto" work? *)
+    fix i
     (** TODO combine the following two parts as done in the HOL_light version. **)
     { let ?x = "(\<chi> j. (if j=i then ((min (a$j) (d$j))+c$j)/2 else (c$j+d$j)/2))::real^'n"
       assume as2: "a$i > c$i"
-      { fix j assume j:"j\<in>dimset a"
-	hence "c $ j < ?x $ j \<and> ?x $ j < d $ j" unfolding Cart_lambda_beta[THEN bspec[where x=j], OF j]
-	  apply(cases "j=i") using as(2)[THEN bspec[where x=j], OF j]
+      { fix j
+	have "c $ j < ?x $ j \<and> ?x $ j < d $ j" unfolding Cart_lambda_beta
+	  apply(cases "j=i") using as(2)[THEN spec[where x=j]]
 	  by (auto simp add: Arith_Tools.less_divide_eq_number_of1 as2)  }
       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
       moreover
       have "?x\<notin>{a .. b}"
-	unfolding mem_interval apply auto apply(rule_tac x=i in bexI)
-	unfolding Cart_lambda_beta[THEN bspec[where x=i], OF i]
-	using as(2)[THEN bspec[where x=i], OF i] and as2 and i
+	unfolding mem_interval apply auto apply(rule_tac x=i in exI)
+	using as(2)[THEN spec[where x=i]] and as2
 	by (auto simp add: Arith_Tools.less_divide_eq_number_of1)
       ultimately have False using as by auto  }
     hence "a$i \<le> c$i" by(rule ccontr)auto
     moreover
     { let ?x = "(\<chi> j. (if j=i then ((max (b$j) (c$j))+d$j)/2 else (c$j+d$j)/2))::real^'n"
       assume as2: "b$i < d$i"
-      { fix j assume j:"j\<in>dimset a"
-	hence "d $ j > ?x $ j \<and> ?x $ j > c $ j" unfolding Cart_lambda_beta[THEN bspec[where x=j], OF j]
-	  apply(cases "j=i") using as(2)[THEN bspec[where x=j], OF j]
+      { fix j
+	have "d $ j > ?x $ j \<and> ?x $ j > c $ j" unfolding Cart_lambda_beta
+	  apply(cases "j=i") using as(2)[THEN spec[where x=j]]
 	  by (auto simp add: Arith_Tools.less_divide_eq_number_of1 as2)  }
       hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
       moreover
       have "?x\<notin>{a .. b}"
-	unfolding mem_interval apply auto apply(rule_tac x=i in bexI)
-	unfolding Cart_lambda_beta[THEN bspec[where x=i], OF i]
-	using as(2)[THEN bspec[where x=i], OF i] and as2 and i
+	unfolding mem_interval apply auto apply(rule_tac x=i in exI)
+	using as(2)[THEN spec[where x=i]] and as2
 	by (auto simp add: Arith_Tools.less_divide_eq_number_of1)
       ultimately have False using as by auto  }
     hence "b$i \<ge> d$i" by(rule ccontr)auto
     ultimately
     have "a$i \<le> c$i \<and> d$i \<le> b$i" by auto
   } note part1 = this
-  thus ?th3 unfolding subset_eq and Ball_def and mem_interval apply auto by(erule_tac x=xa in allE, simp)+
-  { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i \<in> dimset a. c$i < d$i"
-    fix i assume i:"i \<in> dimset a"
+  thus ?th3 unfolding subset_eq and Ball_def and mem_interval apply auto apply (erule_tac x=ia in allE, simp)+ by (erule_tac x=i in allE, erule_tac x=i in allE, simp)+
+  { assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i. c$i < d$i"
+    fix i
     from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
-    hence "a$i \<le> c$i \<and> d$i \<le> b$i" using part1 and as(2) and i by auto  } note * = this
-  thus ?th4 unfolding subset_eq and Ball_def and mem_interval apply auto by(erule_tac x=xa in allE, simp)+
-qed
-
-lemma disjoint_interval: fixes a::"real^'n" shows
-  "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i \<in> dimset a. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1) and
-  "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i \<in> dimset a. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2) and
-  "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i \<in> dimset a. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3) and
-  "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i \<in> dimset a. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
+    hence "a$i \<le> c$i \<and> d$i \<le> b$i" using part1 and as(2) by auto  } note * = this
+  thus ?th4 unfolding subset_eq and Ball_def and mem_interval apply auto apply (erule_tac x=ia in allE, simp)+ by (erule_tac x=i in allE, erule_tac x=i in allE, simp)+
+qed
+
+lemma disjoint_interval: fixes a::"real^'n::finite" shows
+  "{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i < c$i \<or> b$i < c$i \<or> d$i < a$i))" (is ?th1) and
+  "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i < a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th2) and
+  "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i < c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th3) and
+  "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i. (b$i \<le> a$i \<or> d$i \<le> c$i \<or> b$i \<le> c$i \<or> d$i \<le> a$i))" (is ?th4)
 proof-
   let ?z = "(\<chi> i. ((max (a$i) (c$i)) + (min (b$i) (d$i))) / 2)::real^'n"
   show ?th1 ?th2 ?th3 ?th4
-  unfolding expand_set_eq and Int_iff and empty_iff and mem_interval and ball_conj_distrib[THEN sym] and eq_False
-  by (auto simp add: Cart_lambda_beta' Arith_Tools.less_divide_eq_number_of1 intro!: bexI elim!: allE[where x="?z"])
-qed
-
-lemma inter_interval: fixes a :: "'a::linorder^'n" shows
+  unfolding expand_set_eq and Int_iff and empty_iff and mem_interval and all_conj_distrib[THEN sym] and eq_False
+  apply (auto elim!: allE[where x="?z"])
+  apply ((rule_tac x=x in exI, force) | (rule_tac x=i in exI, force))+
+  done
+qed
+
+lemma inter_interval: fixes a :: "'a::linorder^'n::finite" shows
  "{a .. b} \<inter> {c .. d} =  {(\<chi> i. max (a$i) (c$i)) .. (\<chi> i. min (b$i) (d$i))}"
   unfolding expand_set_eq and Int_iff and mem_interval
-  by (auto simp add: Cart_lambda_beta' Arith_Tools.less_divide_eq_number_of1 intro!: bexI)
+  by (auto simp add: Arith_Tools.less_divide_eq_number_of1 intro!: bexI)
 
 (* Moved interval_open_subset_closed a bit upwards *)
 
@@ -4475,54 +4458,54 @@
  "a < x \<Longrightarrow> x < b ==> (\<exists>d>0. \<forall>x'. abs(x' - x) < d --> a < x' \<and> x' < b)"
   by(rule_tac x="min (x - a) (b - x)" in exI, auto)
 
-lemma open_interval: fixes a :: "real^'n" shows "open {a<..<b}"
+lemma open_interval: fixes a :: "real^'n::finite" shows "open {a<..<b}"
 proof-
   { fix x assume x:"x\<in>{a<..<b}"
-    { fix i assume "i\<in>dimset x"
-      hence "\<exists>d>0. \<forall>x'. abs (x' - (x$i)) < d \<longrightarrow> a$i < x' \<and> x' < b$i"
-	using x[unfolded mem_interval, THEN bspec[where x=i]]
+    { fix i
+      have "\<exists>d>0. \<forall>x'. abs (x' - (x$i)) < d \<longrightarrow> a$i < x' \<and> x' < b$i"
+	using x[unfolded mem_interval, THEN spec[where x=i]]
 	using open_interval_lemma[of "a$i" "x$i" "b$i"] by auto  }
 
-    hence "\<forall>i\<in>dimset x. \<exists>d>0. \<forall>x'. abs (x' - (x$i)) < d \<longrightarrow> a$i < x' \<and> x' < b$i" by auto
-    then obtain d where d:"\<forall>i\<in>dimset x. 0 < d i \<and> (\<forall>x'. \<bar>x' - x $ i\<bar> < d i \<longrightarrow> a $ i < x' \<and> x' < b $ i)"
-      using bchoice[of "dimset x" "\<lambda>i d. d>0 \<and> (\<forall>x'. \<bar>x' - x $ i\<bar> < d \<longrightarrow> a $ i < x' \<and> x' < b $ i)"] by auto
-
-    let ?d = "Min (d ` dimset x)"
-    have **:"finite (d ` dimset x)" "d ` dimset x \<noteq> {}" using dimindex_ge_1[of "UNIV::'n set"] by auto
+    hence "\<forall>i. \<exists>d>0. \<forall>x'. abs (x' - (x$i)) < d \<longrightarrow> a$i < x' \<and> x' < b$i" by auto
+    then obtain d where d:"\<forall>i. 0 < d i \<and> (\<forall>x'. \<bar>x' - x $ i\<bar> < d i \<longrightarrow> a $ i < x' \<and> x' < b $ i)"
+      using bchoice[of "UNIV" "\<lambda>i d. d>0 \<and> (\<forall>x'. \<bar>x' - x $ i\<bar> < d \<longrightarrow> a $ i < x' \<and> x' < b $ i)"] by auto
+
+    let ?d = "Min (range d)"
+    have **:"finite (range d)" "range d \<noteq> {}" by auto
     have "?d>0" unfolding Min_gr_iff[OF **] using d by auto
     moreover
     { fix x' assume as:"dist x' x < ?d"
-      { fix i assume i:"i \<in> dimset x"
+      { fix i
 	have "\<bar>x'$i - x $ i\<bar> < d i"
-	  using norm_bound_component_lt[OF as[unfolded dist_def], THEN bspec[where x=i], OF i]
-	  unfolding vector_minus_component[OF i] and Min_gr_iff[OF **] using i by auto
-	hence "a $ i < x' $ i" "x' $ i < b $ i" using d[THEN bspec[where x=i], OF i] by auto  }
+	  using norm_bound_component_lt[OF as[unfolded dist_def], of i]
+	  unfolding vector_minus_component and Min_gr_iff[OF **] by auto
+	hence "a $ i < x' $ i" "x' $ i < b $ i" using d[THEN spec[where x=i]] by auto  }
       hence "a < x' \<and> x' < b" unfolding vector_less_def by auto  }
-    ultimately have "\<exists>e>0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a<..<b}" by auto
+    ultimately have "\<exists>e>0. \<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a<..<b}" by (auto, rule_tac x="?d" in exI, simp)
   }
   thus ?thesis unfolding open_def using open_interval_lemma by auto
 qed
 
-lemma closed_interval: fixes a :: "real^'n" shows "closed {a .. b}"
+lemma closed_interval: fixes a :: "real^'n::finite" shows "closed {a .. b}"
 proof-
-  { fix x i assume i:"i\<in>dimset x" and as:"\<forall>e>0. \<exists>x'\<in>{a..b}. x' \<noteq> x \<and> dist x' x < e"(* and xab:"a$i > x$i \<or> b$i < x$i"*)
+  { fix x i assume as:"\<forall>e>0. \<exists>x'\<in>{a..b}. x' \<noteq> x \<and> dist x' x < e"(* and xab:"a$i > x$i \<or> b$i < x$i"*)
     { assume xa:"a$i > x$i"
       with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < a$i - x$i" by(erule_tac x="a$i - x$i" in allE)auto
       hence False unfolding mem_interval and dist_def
-	using component_le_norm[OF i, of "y-x", unfolded vector_minus_component[OF i]] and i and xa by(auto elim!: ballE[where x=i])
+	using component_le_norm[of "y-x" i, unfolded vector_minus_component] and xa by(auto elim!: allE[where x=i])
     } hence "a$i \<le> x$i" by(rule ccontr)auto
     moreover
     { assume xb:"b$i < x$i"
       with as obtain y where y:"y\<in>{a..b}" "y \<noteq> x" "dist y x < x$i - b$i" by(erule_tac x="x$i - b$i" in allE)auto
       hence False unfolding mem_interval and dist_def
-	using component_le_norm[OF i, of "y-x", unfolded vector_minus_component[OF i]] and i and xb by(auto elim!: ballE[where x=i])
+	using component_le_norm[of "y-x" i, unfolded vector_minus_component] and xb by(auto elim!: allE[where x=i])
     } hence "x$i \<le> b$i" by(rule ccontr)auto
     ultimately
     have "a $ i \<le> x $ i \<and> x $ i \<le> b $ i" by auto }
   thus ?thesis unfolding closed_limpt islimpt_approachable mem_interval by auto
 qed
 
-lemma interior_closed_interval: fixes a :: "real^'n" shows
+lemma interior_closed_interval: fixes a :: "real^'n::finite" shows
  "interior {a .. b} = {a<..<b}" (is "?L = ?R")
 proof(rule subset_antisym)
   show "?R \<subseteq> ?L" using interior_maximal[OF interval_open_subset_closed open_interval] by auto
@@ -4530,85 +4513,87 @@
   { fix x assume "\<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> {a..b}"
     then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" by auto
     then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_def and subset_eq by auto
-    { fix i assume i:"i\<in>dimset x"
+    { fix i
       have "dist (x - (e / 2) *s basis i) x < e"
 	   "dist (x + (e / 2) *s basis i) x < e"
 	unfolding dist_def apply auto
-	unfolding norm_minus_cancel and norm_mul using norm_basis[OF i] and `e>0` by auto
+	unfolding norm_minus_cancel and norm_mul using norm_basis[of i] and `e>0` by auto
       hence "a $ i \<le> (x - (e / 2) *s basis i) $ i"
                     "(x + (e / 2) *s basis i) $ i \<le> b $ i"
 	using e[THEN spec[where x="x - (e/2) *s basis i"]]
 	and   e[THEN spec[where x="x + (e/2) *s basis i"]]
-	unfolding mem_interval using i by auto
+	unfolding mem_interval by (auto elim!: allE[where x=i])
       hence "a $ i < x $ i" and "x $ i < b $ i"
-	unfolding vector_minus_component[OF i] and vector_add_component[OF i]
-	unfolding vector_smult_component[OF i] and basis_component[OF i] using `e>0` by auto   }
+	unfolding vector_minus_component and vector_add_component
+	unfolding vector_smult_component and basis_component using `e>0` by auto   }
     hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }
   thus "?L \<subseteq> ?R" unfolding interior_def and subset_eq by auto
 qed
 
-lemma bounded_closed_interval: fixes a :: "real^'n" shows
+lemma bounded_closed_interval: fixes a :: "real^'n::finite" shows
  "bounded {a .. b}"
 proof-
-  let ?b = "\<Sum>i\<in>dimset a. \<bar>a$i\<bar> + \<bar>b$i\<bar>"
-  { fix x::"real^'n" assume x:"\<forall>i\<in>dimset a. a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
-    { fix i assume "i\<in>dimset a"
-      hence "\<bar>x$i\<bar> \<le> \<bar>a$i\<bar> + \<bar>b$i\<bar>" using x[THEN bspec[where x=i]] by auto  }
-    hence "(\<Sum>i\<in>dimset a. \<bar>x $ i\<bar>) \<le> ?b" by(rule setsum_mono)auto
+  let ?b = "\<Sum>i\<in>UNIV. \<bar>a$i\<bar> + \<bar>b$i\<bar>"
+  { fix x::"real^'n" assume x:"\<forall>i. a $ i \<le> x $ i \<and> x $ i \<le> b $ i"
+    { fix i
+      have "\<bar>x$i\<bar> \<le> \<bar>a$i\<bar> + \<bar>b$i\<bar>" using x[THEN spec[where x=i]] by auto  }
+    hence "(\<Sum>i\<in>UNIV. \<bar>x $ i\<bar>) \<le> ?b" by(rule setsum_mono)
     hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }
   thus ?thesis unfolding interval and bounded_def by auto
 qed
 
-lemma bounded_interval: fixes a :: "real^'n" shows
+lemma bounded_interval: fixes a :: "real^'n::finite" shows
  "bounded {a .. b} \<and> bounded {a<..<b}"
   using bounded_closed_interval[of a b]
   using interval_open_subset_closed[of a b]
   using bounded_subset[of "{a..b}" "{a<..<b}"]
   by simp
 
-lemma not_interval_univ: fixes a :: "real^'n" shows
+lemma not_interval_univ: fixes a :: "real^'n::finite" shows
  "({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
   using bounded_interval[of a b]
   by auto
 
-lemma compact_interval: fixes a :: "real^'n" shows
+lemma compact_interval: fixes a :: "real^'n::finite" shows
  "compact {a .. b}"
   using bounded_closed_imp_compact using bounded_interval[of a b] using closed_interval[of a b] by auto
 
-lemma open_interval_midpoint: fixes a :: "real^'n"
+lemma open_interval_midpoint: fixes a :: "real^'n::finite"
   assumes "{a<..<b} \<noteq> {}" shows "((1/2) *s (a + b)) \<in> {a<..<b}"
 proof-
-  { fix i assume i:"i\<in>dimset a"
-    hence "a $ i < ((1 / 2) *s (a + b)) $ i \<and> ((1 / 2) *s (a + b)) $ i < b $ i"
-      using assms[unfolded interval_ne_empty, THEN bspec[where x=i]]
-      unfolding vector_smult_component[OF i] and vector_add_component[OF i]
+  { fix i
+    have "a $ i < ((1 / 2) *s (a + b)) $ i \<and> ((1 / 2) *s (a + b)) $ i < b $ i"
+      using assms[unfolded interval_ne_empty, THEN spec[where x=i]]
+      unfolding vector_smult_component and vector_add_component
       by(auto simp add: Arith_Tools.less_divide_eq_number_of1)  }
   thus ?thesis unfolding mem_interval by auto
 qed
 
-lemma open_closed_interval_convex: fixes x :: "real^'n"
+lemma open_closed_interval_convex: fixes x :: "real^'n::finite"
   assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
   shows "(e *s x + (1 - e) *s y) \<in> {a<..<b}"
 proof-
-  { fix i assume i:"i\<in>dimset a"
+  { fix i
     have "a $ i = e * a$i + (1 - e) * a$i" unfolding left_diff_distrib by simp
     also have "\<dots> < e * x $ i + (1 - e) * y $ i" apply(rule add_less_le_mono)
       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
-      using x i unfolding mem_interval  apply(erule_tac x=i in ballE) apply simp_all
-      using y i unfolding mem_interval  apply(erule_tac x=i in ballE) by simp_all
-    finally have "a $ i < (e *s x + (1 - e) *s y) $ i" using i by (auto simp add: vector_add_component vector_smult_component)
+      using x unfolding mem_interval  apply simp
+      using y unfolding mem_interval  apply simp
+      done
+    finally have "a $ i < (e *s x + (1 - e) *s y) $ i" by auto
     moreover {
     have "b $ i = e * b$i + (1 - e) * b$i" unfolding left_diff_distrib by simp
     also have "\<dots> > e * x $ i + (1 - e) * y $ i" apply(rule add_less_le_mono)
       using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
-      using x i unfolding mem_interval  apply(erule_tac x=i in ballE) apply simp_all
-      using y i unfolding mem_interval  apply(erule_tac x=i in ballE) by simp_all
-    finally have "(e *s x + (1 - e) *s y) $ i < b $ i" using i by (auto simp add: vector_add_component vector_smult_component)
+      using x unfolding mem_interval  apply simp
+      using y unfolding mem_interval  apply simp
+      done
+    finally have "(e *s x + (1 - e) *s y) $ i < b $ i" by auto
     } ultimately have "a $ i < (e *s x + (1 - e) *s y) $ i \<and> (e *s x + (1 - e) *s y) $ i < b $ i" by auto }
   thus ?thesis unfolding mem_interval by auto
 qed
 
-lemma closure_open_interval: fixes a :: "real^'n"
+lemma closure_open_interval: fixes a :: "real^'n::finite"
   assumes "{a<..<b} \<noteq> {}"
   shows "closure {a<..<b} = {a .. b}"
 proof-
@@ -4639,15 +4624,15 @@
   thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
 qed
 
-lemma bounded_subset_open_interval_symmetric: fixes s::"(real^'n) set"
+lemma bounded_subset_open_interval_symmetric: fixes s::"(real^'n::finite) set"
   assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"
 proof-
   obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
   def a \<equiv> "(\<chi> i. b+1)::real^'n"
   { fix x assume "x\<in>s"
-    fix i assume i:"i\<in>dimset a"
-    have "(-a)$i < x$i" and "x$i < a$i" using b[THEN bspec[where x=x], OF `x\<in>s`] and component_le_norm[OF i, of x]
-      unfolding vector_uminus_component[OF i] and a_def and Cart_lambda_beta'[OF i] by auto
+    fix i
+    have "(-a)$i < x$i" and "x$i < a$i" using b[THEN bspec[where x=x], OF `x\<in>s`] and component_le_norm[of x i]
+      unfolding vector_uminus_component and a_def and Cart_lambda_beta by auto
   }
   thus ?thesis by(auto intro: exI[where x=a] simp add: vector_less_def)
 qed
@@ -4679,30 +4664,32 @@
   case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
 qed
 
-lemma inter_interval_mixed_eq_empty: fixes a :: "real^'n"
+lemma inter_interval_mixed_eq_empty: fixes a :: "real^'n::finite"
   assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
   unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..
 
 
 (* Some special cases for intervals in R^1.                                  *)
 
-lemma dim1: "dimindex (UNIV::(1 set)) = 1"
-unfolding dimindex_def
-by simp
+lemma all_1: "(\<forall>x::1. P x) \<longleftrightarrow> P 1"
+  by (metis num1_eq_iff)
+
+lemma ex_1: "(\<exists>x::1. P x) \<longleftrightarrow> P 1"
+  by auto (metis num1_eq_iff)
 
 lemma interval_cases_1: fixes x :: "real^1" shows
  "x \<in> {a .. b} ==> x \<in> {a<..<b} \<or> (x = a) \<or> (x = b)"
-  by(simp add:  Cart_eq vector_less_def vector_less_eq_def dim1, auto)
+  by(simp add:  Cart_eq vector_less_def vector_less_eq_def all_1, auto)
 
 lemma in_interval_1: fixes x :: "real^1" shows
  "(x \<in> {a .. b} \<longleftrightarrow> dest_vec1 a \<le> dest_vec1 x \<and> dest_vec1 x \<le> dest_vec1 b) \<and>
   (x \<in> {a<..<b} \<longleftrightarrow> dest_vec1 a < dest_vec1 x \<and> dest_vec1 x < dest_vec1 b)"
-by(simp add: Cart_eq vector_less_def vector_less_eq_def dim1 dest_vec1_def)
+by(simp add: Cart_eq vector_less_def vector_less_eq_def all_1 dest_vec1_def)
 
 lemma interval_eq_empty_1: fixes a :: "real^1" shows
   "{a .. b} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a"
   "{a<..<b} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a"
-  unfolding interval_eq_empty and dim1 and dest_vec1_def by auto
+  unfolding interval_eq_empty and ex_1 and dest_vec1_def by auto
 
 lemma subset_interval_1: fixes a :: "real^1" shows
  "({a .. b} \<subseteq> {c .. d} \<longleftrightarrow>  dest_vec1 b < dest_vec1 a \<or>
@@ -4713,7 +4700,7 @@
                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
  "({a<..<b} \<subseteq> {c<..<d} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or>
                 dest_vec1 c \<le> dest_vec1 a \<and> dest_vec1 a < dest_vec1 b \<and> dest_vec1 b \<le> dest_vec1 d)"
-  unfolding subset_interval[of a b c d] unfolding forall_dimindex_1 and dest_vec1_def by auto
+  unfolding subset_interval[of a b c d] unfolding all_1 and dest_vec1_def by auto
 
 lemma eq_interval_1: fixes a :: "real^1" shows
  "{a .. b} = {c .. d} \<longleftrightarrow>
@@ -4729,37 +4716,37 @@
   "{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b < dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
   "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d < dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
   "{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> dest_vec1 b \<le> dest_vec1 a \<or> dest_vec1 d \<le> dest_vec1 c  \<or>  dest_vec1 b \<le> dest_vec1 c \<or> dest_vec1 d \<le> dest_vec1 a"
-  unfolding disjoint_interval and dest_vec1_def and dim1 by auto
+  unfolding disjoint_interval and dest_vec1_def ex_1 by auto
 
 lemma open_closed_interval_1: fixes a :: "real^1" shows
  "{a<..<b} = {a .. b} - {a, b}"
-  unfolding expand_set_eq apply simp unfolding vector_less_def and vector_less_eq_def and dim1 and dest_vec1_eq[THEN sym] and dest_vec1_def by auto
+  unfolding expand_set_eq apply simp unfolding vector_less_def and vector_less_eq_def and all_1 and dest_vec1_eq[THEN sym] and dest_vec1_def by auto
 
 lemma closed_open_interval_1: "dest_vec1 (a::real^1) \<le> dest_vec1 b ==> {a .. b} = {a<..<b} \<union> {a,b}"
-  unfolding expand_set_eq apply simp unfolding vector_less_def and vector_less_eq_def and dim1 and dest_vec1_eq[THEN sym] and dest_vec1_def by auto
+  unfolding expand_set_eq apply simp unfolding vector_less_def and vector_less_eq_def and all_1 and dest_vec1_eq[THEN sym] and dest_vec1_def by auto
 
 (* Some stuff for half-infinite intervals too; FIXME: notation?  *)
 
-lemma closed_interval_left: fixes b::"real^'n"
-  shows "closed {x::real^'n. \<forall>i \<in> dimset x. x$i \<le> b$i}"
+lemma closed_interval_left: fixes b::"real^'n::finite"
+  shows "closed {x::real^'n. \<forall>i. x$i \<le> b$i}"
 proof-
-  { fix i assume i:"i\<in>dimset b"
-    fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>dimset b. x $ i \<le> b $ i}. x' \<noteq> x \<and> dist x' x < e"
+  { fix i
+    fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. x $ i \<le> b $ i}. x' \<noteq> x \<and> dist x' x < e"
     { assume "x$i > b$i"
-      then obtain y where "y $ i \<le> b $ i"  "y \<noteq> x"  "dist y x < x$i - b$i" using x[THEN spec[where x="x$i - b$i"]] and i by (auto, erule_tac x=i in ballE)auto
-      hence False using component_le_norm[OF i, of "y - x"] unfolding dist_def and vector_minus_component[OF i] by auto   }
+      then obtain y where "y $ i \<le> b $ i"  "y \<noteq> x"  "dist y x < x$i - b$i" using x[THEN spec[where x="x$i - b$i"]] by auto
+      hence False using component_le_norm[of "y - x" i] unfolding dist_def and vector_minus_component by auto   }
     hence "x$i \<le> b$i" by(rule ccontr)auto  }
   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
 qed
 
-lemma closed_interval_right: fixes a::"real^'n"
-  shows "closed {x::real^'n. \<forall>i \<in> dimset x. a$i \<le> x$i}"
+lemma closed_interval_right: fixes a::"real^'n::finite"
+  shows "closed {x::real^'n. \<forall>i. a$i \<le> x$i}"
 proof-
-  { fix i assume i:"i\<in>dimset a"
-    fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>dimset a. a $ i \<le> x $ i}. x' \<noteq> x \<and> dist x' x < e"
+  { fix i
+    fix x::"real^'n" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i. a $ i \<le> x $ i}. x' \<noteq> x \<and> dist x' x < e"
     { assume "a$i > x$i"
-      then obtain y where "a $ i \<le> y $ i"  "y \<noteq> x"  "dist y x < a$i - x$i" using x[THEN spec[where x="a$i - x$i"]] and i by(auto, erule_tac x=i in ballE)auto
-      hence False using component_le_norm[OF i, of "y - x"] unfolding dist_def and vector_minus_component[OF i] by auto   }
+      then obtain y where "a $ i \<le> y $ i"  "y \<noteq> x"  "dist y x < a$i - x$i" using x[THEN spec[where x="a$i - x$i"]] by auto
+      hence False using component_le_norm[of "y - x" i] unfolding dist_def and vector_minus_component by auto   }
     hence "a$i \<le> x$i" by(rule ccontr)auto  }
   thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
 qed
@@ -4768,13 +4755,13 @@
 
 definition "is_interval s \<longleftrightarrow> (\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. a \<le> x \<and> x \<le> b \<longrightarrow> x \<in> s)"
 
-lemma is_interval_interval: fixes a::"real^'n" shows
+lemma is_interval_interval: fixes a::"real^'n::finite" shows
   "is_interval {a<..<b}" "is_interval {a .. b}"
   unfolding is_interval_def apply(auto simp add: vector_less_def vector_less_eq_def)
-  apply(erule_tac x=i in ballE)+ apply simp+
-  apply(erule_tac x=i in ballE)+ apply simp+
-  apply(erule_tac x=i in ballE)+ apply simp+
-  apply(erule_tac x=i in ballE)+ apply simp+
+  apply(erule_tac x=i in allE)+ apply simp
+  apply(erule_tac x=i in allE)+ apply simp
+  apply(erule_tac x=i in allE)+ apply simp
+  apply(erule_tac x=i in allE)+ apply simp
   done
 
 lemma is_interval_empty:
@@ -4789,7 +4776,7 @@
 
 subsection{* Closure of halfspaces and hyperplanes.                                    *}
 
-lemma Lim_vec1_dot: fixes f :: "real^'m \<Rightarrow> real^'n"
+lemma Lim_vec1_dot: fixes f :: "real^'m \<Rightarrow> real^'n::finite"
   assumes "(f ---> l) net"  shows "((vec1 o (\<lambda>y. a \<bullet> (f y))) ---> vec1(a \<bullet> l)) net"
 proof(cases "a = vec 0")
   case True thus ?thesis using dot_lzero and Lim_const[of 0 net] unfolding vec1_vec and o_def by auto
@@ -4821,14 +4808,14 @@
   using continuous_at_vec1_dot
   by auto
 
-lemma closed_halfspace_le: fixes a::"real^'n"
+lemma closed_halfspace_le: fixes a::"real^'n::finite"
   shows "closed {x. a \<bullet> x \<le> b}"
 proof-
   have *:"{x \<in> UNIV. (vec1 \<circ> op \<bullet> a) x \<in> vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b}} = {x. a \<bullet> x \<le> b}" by auto
-  let ?T = "{x::real^1. (\<forall>i\<in>dimset x. x$i \<le> (vec1 b)$i)}"
+  let ?T = "{x::real^1. (\<forall>i. x$i \<le> (vec1 b)$i)}"
   have "closed ?T" using closed_interval_left[of "vec1 b"] by simp
-  moreover have "vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b} = range (vec1 \<circ> op \<bullet> a) \<inter> ?T" unfolding dim1
-    unfolding image_def apply auto unfolding vec1_component[unfolded One_nat_def] by auto
+  moreover have "vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b} = range (vec1 \<circ> op \<bullet> a) \<inter> ?T" unfolding all_1
+    unfolding image_def by auto
   ultimately have "\<exists>T. closed T \<and> vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b} = range (vec1 \<circ> op \<bullet> a) \<inter> T" by auto
   hence "closedin euclidean {x \<in> UNIV. (vec1 \<circ> op \<bullet> a) x \<in> vec1 ` {r. \<exists>x. a \<bullet> x = r \<and> r \<le> b}}"
     using continuous_on_vec1_dot[of UNIV a, unfolded continuous_on_closed subtopology_UNIV] unfolding closedin_closed
@@ -4846,11 +4833,11 @@
 qed
 
 lemma closed_halfspace_component_le:
-  assumes "i \<in> {1 .. dimindex (UNIV::'n set)}" shows "closed {x::real^'n. x$i \<le> a}"
+  shows "closed {x::real^'n::finite. x$i \<le> a}"
   using closed_halfspace_le[of "(basis i)::real^'n" a] unfolding dot_basis[OF assms] by auto
 
 lemma closed_halfspace_component_ge:
-  assumes "i \<in> {1 .. dimindex (UNIV::'n set)}" shows "closed {x::real^'n. x$i \<ge> a}"
+  shows "closed {x::real^'n::finite. x$i \<ge> a}"
   using closed_halfspace_ge[of a "(basis i)::real^'n"] unfolding dot_basis[OF assms] by auto
 
 text{* Openness of halfspaces.                                                   *}
@@ -4868,48 +4855,45 @@
 qed
 
 lemma open_halfspace_component_lt:
-  assumes "i \<in> {1 .. dimindex(UNIV::'n set)}" shows "open {x::real^'n. x$i < a}"
+  shows "open {x::real^'n::finite. x$i < a}"
   using open_halfspace_lt[of "(basis i)::real^'n" a] unfolding dot_basis[OF assms] by auto
 
 lemma open_halfspace_component_gt:
-  assumes "i \<in> {1 .. dimindex(UNIV::'n set)}" shows "open {x::real^'n. x$i  > a}"
+  shows "open {x::real^'n::finite. x$i  > a}"
   using open_halfspace_gt[of a "(basis i)::real^'n"] unfolding dot_basis[OF assms] by auto
 
 text{* This gives a simple derivation of limit component bounds.                 *}
 
-lemma Lim_component_le: fixes f :: "'a \<Rightarrow> real^'n"
+lemma Lim_component_le: fixes f :: "'a \<Rightarrow> real^'n::finite"
   assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "eventually (\<lambda>x. f(x)$i \<le> b) net"
-  and i:"i\<in> {1 .. dimindex(UNIV::'n set)}"
   shows "l$i \<le> b"
 proof-
-  { fix x have "x \<in> {x::real^'n. basis i \<bullet> x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding dot_basis[OF i] by auto } note * = this
+  { fix x have "x \<in> {x::real^'n. basis i \<bullet> x \<le> b} \<longleftrightarrow> x$i \<le> b" unfolding dot_basis by auto } note * = this
   show ?thesis using Lim_in_closed_set[of "{x. basis i \<bullet> x \<le> b}" f net l] unfolding *
     using closed_halfspace_le[of "(basis i)::real^'n" b] and assms(1,2,3) by auto
 qed
 
-lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> real^'n"
+lemma Lim_component_ge: fixes f :: "'a \<Rightarrow> real^'n::finite"
   assumes "(f ---> l) net"  "\<not> (trivial_limit net)"  "eventually (\<lambda>x. b \<le> (f x)$i) net"
-  and i:"i\<in> {1 .. dimindex(UNIV::'n set)}"
   shows "b \<le> l$i"
 proof-
-  { fix x have "x \<in> {x::real^'n. basis i \<bullet> x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding dot_basis[OF i] by auto } note * = this
+  { fix x have "x \<in> {x::real^'n. basis i \<bullet> x \<ge> b} \<longleftrightarrow> x$i \<ge> b" unfolding dot_basis by auto } note * = this
   show ?thesis using Lim_in_closed_set[of "{x. basis i \<bullet> x \<ge> b}" f net l] unfolding *
     using closed_halfspace_ge[of b "(basis i)::real^'n"] and assms(1,2,3) by auto
 qed
 
-lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> real^'n"
+lemma Lim_component_eq: fixes f :: "'a \<Rightarrow> real^'n::finite"
   assumes net:"(f ---> l) net" "~(trivial_limit net)" and ev:"eventually (\<lambda>x. f(x)$i = b) net"
-  and i:"i\<in> {1 .. dimindex(UNIV::'n set)}"
   shows "l$i = b"
-  using ev[unfolded order_eq_iff eventually_and] using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] using i by auto
+  using ev[unfolded order_eq_iff eventually_and] using Lim_component_ge[OF net, of b i] and Lim_component_le[OF net, of i b] by auto
 
 lemma Lim_drop_le: fixes f :: "'a \<Rightarrow> real^1" shows
   "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. dest_vec1 (f x) \<le> b) net ==> dest_vec1 l \<le> b"
-  using Lim_component_le[of f l net 1 b] unfolding dest_vec1_def and dim1 by auto
+  using Lim_component_le[of f l net 1 b] unfolding dest_vec1_def by auto
 
 lemma Lim_drop_ge: fixes f :: "'a \<Rightarrow> real^1" shows
  "(f ---> l) net \<Longrightarrow> ~(trivial_limit net) \<Longrightarrow> eventually (\<lambda>x. b \<le> dest_vec1 (f x)) net ==> b \<le> dest_vec1 l"
-  using Lim_component_ge[of f l net b 1] unfolding dest_vec1_def and dim1 by auto
+  using Lim_component_ge[of f l net b 1] unfolding dest_vec1_def by auto
 
 text{* Limits relative to a union.                                               *}
 
@@ -4926,7 +4910,7 @@
   using assms unfolding continuous_on unfolding Lim_within_union
   unfolding Lim unfolding trivial_limit_within unfolding closed_limpt by auto
 
-lemma continuous_on_cases: fixes g :: "real^'m \<Rightarrow> real ^'n"
+lemma continuous_on_cases: fixes g :: "real^'m::finite \<Rightarrow> real ^'n::finite"
   assumes "closed s" "closed t" "continuous_on s f" "continuous_on t g"
           "\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x"
   shows "continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
@@ -4943,7 +4927,7 @@
 
 text{* Some more convenient intermediate-value theorem formulations.             *}
 
-lemma connected_ivt_hyperplane: fixes y :: "real^'n"
+lemma connected_ivt_hyperplane: fixes y :: "real^'n::finite"
   assumes "connected s" "x \<in> s" "y \<in> s" "a \<bullet> x \<le> b" "b \<le> a \<bullet> y"
   shows "\<exists>z \<in> s. a \<bullet> z = b"
 proof(rule ccontr)
@@ -4956,8 +4940,8 @@
   ultimately show False using assms(1)[unfolded connected_def not_ex, THEN spec[where x="?A"], THEN spec[where x="?B"]] and assms(2-5) by auto
 qed
 
-lemma connected_ivt_component: fixes x::"real^'n" shows
- "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> k \<in> dimset x \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"
+lemma connected_ivt_component: fixes x::"real^'n::finite" shows
+ "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x$k \<le> a \<Longrightarrow> a \<le> y$k \<Longrightarrow> (\<exists>z\<in>s.  z$k = a)"
   using connected_ivt_hyperplane[of s x y "(basis k)::real^'n" a] by (auto simp add: dot_basis)
 
 text{* Also more convenient formulations of monotone convergence.                *}
@@ -4982,7 +4966,7 @@
      (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
      (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
 
-definition homeomorphic :: "((real^'a) set) \<Rightarrow> ((real^'b) set) \<Rightarrow> bool" (infixr "homeomorphic" 60) where
+definition homeomorphic :: "((real^'a::finite) set) \<Rightarrow> ((real^'b::finite) set) \<Rightarrow> bool" (infixr "homeomorphic" 60) where
   homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
 
 lemma homeomorphic_refl: "s homeomorphic s"
@@ -5103,7 +5087,7 @@
     using homeomorphic_translation[of "(\<lambda>x. c *s x) ` s" a] unfolding * by auto
 qed
 
-lemma homeomorphic_balls: fixes a b ::"real^'a"
+lemma homeomorphic_balls: fixes a b ::"real^'a::finite"
   assumes "0 < d"  "0 < e"
   shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
         "(cball a d) homeomorphic (cball b e)" (is ?cth)
@@ -5173,7 +5157,7 @@
 
 lemma dist_0_norm:"dist 0 x = norm x" unfolding dist_def by(auto simp add: norm_minus_cancel)
 
-lemma injective_imp_isometric: fixes f::"real^'m \<Rightarrow> real^'n"
+lemma injective_imp_isometric: fixes f::"real^'m::finite \<Rightarrow> real^'n::finite"
   assumes s:"closed s"  "subspace s"  and f:"linear f" "\<forall>x\<in>s. (f x = 0) \<longrightarrow> (x = 0)"
   shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm(x)"
 proof(cases "s \<subseteq> {0::real^'m}")
@@ -5235,24 +5219,23 @@
 subsection{* Some properties of a canonical subspace.                                  *}
 
 lemma subspace_substandard:
- "subspace {x::real^'n. (\<forall>i \<in> dimset x. d < i \<longrightarrow> x$i = 0)}"
+ "subspace {x::real^'n. (\<forall>i. P i \<longrightarrow> x$i = 0)}"
   unfolding subspace_def by(auto simp add: vector_add_component vector_smult_component elim!: ballE)
 
 lemma closed_substandard:
- "closed {x::real^'n. \<forall>i \<in> dimset x. d < i --> x$i = 0}" (is "closed ?A")
+ "closed {x::real^'n::finite. \<forall>i. P i --> x$i = 0}" (is "closed ?A")
 proof-
-  let ?D = "{Suc d..dimindex(UNIV::('n set))}"
+  let ?D = "{i. P i}"
   let ?Bs = "{{x::real^'n. basis i \<bullet> x = 0}| i. i \<in> ?D}"
   { fix x
     { assume "x\<in>?A"
-      hence x:"\<forall>i\<in>?D. d < i \<longrightarrow> x $ i = 0" by auto
+      hence x:"\<forall>i\<in>?D. x $ i = 0" by auto
       hence "x\<in> \<Inter> ?Bs" by(auto simp add: dot_basis x) }
     moreover
     { assume x:"x\<in>\<Inter>?Bs"
-      { fix i assume i:"i\<in>dimset x" and "d < i"
-	hence "i \<in> ?D" by auto
+      { fix i assume i:"i \<in> ?D"
 	then obtain B where BB:"B \<in> ?Bs" and B:"B = {x::real^'n. basis i \<bullet> x = 0}" by auto
-	hence "x $ i = 0" unfolding B unfolding dot_basis[OF i] using x by auto  }
+	hence "x $ i = 0" unfolding B using x unfolding dot_basis by auto  }
       hence "x\<in>?A" by auto }
     ultimately have "x\<in>?A \<longleftrightarrow> x\<in> \<Inter>?Bs" by auto }
   hence "?A = \<Inter> ?Bs" by auto
@@ -5260,40 +5243,40 @@
 qed
 
 lemma dim_substandard:
-  assumes "d \<le> dimindex(UNIV::'n set)"
-  shows "dim {x::real^'n. \<forall>i \<in> dimset x. d < i --> x$i = 0} = d" (is "dim ?A = d")
+  shows "dim {x::real^'n::finite. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0} = card d" (is "dim ?A = _")
 proof-
-  let ?D = "{1..dimindex (UNIV::'n set)}"
-  let ?B = "(basis::nat\<Rightarrow>real^'n) ` {1..d}"
-
-    let ?bas = "basis::nat \<Rightarrow> real^'n"
-
-  have "?B \<subseteq> ?A" by (auto simp add: basis_component)
+  let ?D = "UNIV::'n set"
+  let ?B = "(basis::'n\<Rightarrow>real^'n) ` d"
+
+    let ?bas = "basis::'n \<Rightarrow> real^'n"
+
+  have "?B \<subseteq> ?A" by auto
 
   moreover
   { fix x::"real^'n" assume "x\<in>?A"
-    hence "x\<in> span ?B"
+    with finite[of d]
+    have "x\<in> span ?B"
     proof(induct d arbitrary: x)
-      case 0 hence "x=0" unfolding Cart_eq by auto
+      case empty hence "x=0" unfolding Cart_eq by auto
       thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
     next
-      case (Suc n)
-      hence *:"\<forall>i\<in>?D. Suc n < i \<longrightarrow> x $ i = 0" by auto
-      have **:"{1..n} \<subseteq> {1..Suc n}" by auto
-      def y \<equiv> "x - x$(Suc n) *s basis (Suc n)"
-      have y:"x = y + (x$Suc n) *s basis (Suc n)" unfolding y_def by auto
-      { fix i assume i:"i\<in>?D" and i':"n < i"
-	hence "y $ i = 0" unfolding y_def unfolding vector_minus_component[OF i]
-	  and vector_smult_component[OF i] and basis_component[OF i] using i'
-	  using *[THEN bspec[where x=i]] by auto }
-      hence "y \<in> span (basis ` {1..Suc n})" using Suc(1)[of y]
-	using span_mono[of "?bas ` {1..n}" "?bas ` {1..Suc n}"]
+      case (insert k F)
+      hence *:"\<forall>i. i \<notin> insert k F \<longrightarrow> x $ i = 0" by auto
+      have **:"F \<subseteq> insert k F" by auto
+      def y \<equiv> "x - x$k *s basis k"
+      have y:"x = y + (x$k) *s basis k" unfolding y_def by auto
+      { fix i assume i':"i \<notin> F"
+	hence "y $ i = 0" unfolding y_def unfolding vector_minus_component
+	  and vector_smult_component and basis_component
+	  using *[THEN spec[where x=i]] by auto }
+      hence "y \<in> span (basis ` (insert k F))" using insert(3)
+	using span_mono[of "?bas ` F" "?bas ` (insert k F)"]
 	using image_mono[OF **, of basis] by auto
       moreover
-      have "basis (Suc n) \<in> span (?bas ` {1..Suc n})" by(rule span_superset, auto)
-      hence "x$(Suc n) *s basis (Suc n) \<in> span (?bas ` {1..Suc n})" using span_mul by auto
+      have "basis k \<in> span (?bas ` (insert k F))" by(rule span_superset, auto)
+      hence "x$k *s basis k \<in> span (?bas ` (insert k F))" using span_mul by auto
       ultimately
-      have "y + x$(Suc n) *s basis (Suc n) \<in> span (?bas ` {1..Suc n})"
+      have "y + x$k *s basis k \<in> span (?bas ` (insert k F))"
 	using span_add by auto
       thus ?case using y by auto
     qed
@@ -5306,27 +5289,39 @@
   hence "independent ?B" using independent_mono[OF independent_stdbasis, of ?B] and assms by auto
 
   moreover
-  have "{1..d} \<subseteq> ?D" unfolding subset_eq using assms by auto
-  hence *:"inj_on (basis::nat\<Rightarrow>real^'n) {1..d}" using subset_inj_on[OF basis_inj, of "{1..d}"] using assms by auto
-  have "?B hassize d" unfolding hassize_def and card_image[OF *] by auto
-
-  ultimately show ?thesis using dim_unique[of "basis ` {1..d}" ?A] by auto
+  have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
+  hence *:"inj_on (basis::'n\<Rightarrow>real^'n) d" using subset_inj_on[OF basis_inj, of "d"] by auto
+  have "?B hassize (card d)" unfolding hassize_def and card_image[OF *] by auto
+
+  ultimately show ?thesis using dim_unique[of "basis ` d" ?A] by auto
 qed
 
 text{* Hence closure and completeness of all subspaces.                          *}
 
-lemma closed_subspace: fixes s::"(real^'n) set"
+lemma closed_subspace_lemma: "n \<le> card (UNIV::'n::finite set) \<Longrightarrow> \<exists>A::'n set. card A = n"
+apply (induct n)
+apply (rule_tac x="{}" in exI, simp)
+apply clarsimp
+apply (subgoal_tac "\<exists>x. x \<notin> A")
+apply (erule exE)
+apply (rule_tac x="insert x A" in exI, simp)
+apply (subgoal_tac "A \<noteq> UNIV", auto)
+done
+
+lemma closed_subspace: fixes s::"(real^'n::finite) set"
   assumes "subspace s" shows "closed s"
 proof-
-  let ?t = "{x::real^'n. \<forall>i\<in>{1..dimindex (UNIV :: 'n set)}. dim s<i \<longrightarrow> x$i = 0}"
-  have "dim s \<le> dimindex (UNIV :: 'n set)" using dim_subset_univ by auto
+  have "dim s \<le> card (UNIV :: 'n set)" using dim_subset_univ by auto
+  then obtain d::"'n set" where t: "card d = dim s"
+    using closed_subspace_lemma by auto
+  let ?t = "{x::real^'n. \<forall>i. i \<notin> d \<longrightarrow> x$i = 0}"
   obtain f where f:"linear f"  "f ` ?t = s" "inj_on f ?t"
-    using subspace_isomorphism[OF subspace_substandard[of "dim s"] assms]
-    using dim_substandard[OF  dim_subset_univ[of s]] by auto
+    using subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"] assms]
+    using dim_substandard[of d] and t by auto
   have "\<forall>x\<in>?t. f x = 0 \<longrightarrow> x = 0" using linear_0[OF f(1)] using f(3)[unfolded inj_on_def]
     by(erule_tac x=0 in ballE) auto
-  moreover have "closed ?t" using closed_substandard by auto
-  moreover have "subspace ?t" using subspace_substandard by auto
+  moreover have "closed ?t" using closed_substandard .
+  moreover have "subspace ?t" using subspace_substandard .
   ultimately show ?thesis using closed_injective_image_subspace[of ?t f]
     unfolding f(2) using f(1) by auto
 qed
@@ -5400,13 +5395,14 @@
   by metis
 
 lemma image_affinity_interval: fixes m::real
+  fixes a b c :: "real^'n::finite"
   shows "(\<lambda>x. m *s x + c) ` {a .. b} =
             (if {a .. b} = {} then {}
             else (if 0 \<le> m then {m *s a + c .. m *s b + c}
             else {m *s b + c .. m *s a + c}))"
 proof(cases "m=0")
   { fix x assume "x \<le> c" "c \<le> x"
-    hence "x=c" unfolding vector_less_eq_def and Cart_eq by(auto elim!: ballE)  }
+    hence "x=c" unfolding vector_less_eq_def and Cart_eq by (auto intro: order_antisym) }
   moreover case True
   moreover have "c \<in> {m *s a + c..m *s b + c}" unfolding True by(auto simp add: vector_less_eq_def)
   ultimately show ?thesis by auto
@@ -5425,14 +5421,14 @@
       unfolding image_iff Bex_def mem_interval vector_less_eq_def
       apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
 	intro!: exI[where x="(1 / m) *s (y - c)"])
-      by(auto elim!: ballE simp add: pos_le_divide_eq pos_divide_le_eq real_mult_commute)
+      by(auto simp add: pos_le_divide_eq pos_divide_le_eq real_mult_commute diff_le_iff)
   } moreover
   { fix y assume "m *s b + c \<le> y" "y \<le> m *s a + c" "m < 0"
     hence "y \<in> (\<lambda>x. m *s x + c) ` {a..b}"
       unfolding image_iff Bex_def mem_interval vector_less_eq_def
       apply(auto simp add: vector_smult_component vector_add_component vector_minus_component vector_smult_assoc pth_3[symmetric]
 	intro!: exI[where x="(1 / m) *s (y - c)"])
-      by(auto elim!: ballE simp add: neg_le_divide_eq neg_divide_le_eq real_mult_commute)
+      by(auto simp add: neg_le_divide_eq neg_divide_le_eq real_mult_commute diff_le_iff)
   }
   ultimately show ?thesis using False by auto
 qed
@@ -5569,7 +5565,7 @@
 lemma edelstein_fix:
   assumes s:"compact s" "s \<noteq> {}" and gs:"(g ` s) \<subseteq> s"
       and dist:"\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
-  shows "\<exists>! x::real^'a\<in>s. g x = x"
+  shows "\<exists>! x::real^'a::finite\<in>s. g x = x"
 proof(cases "\<exists>x\<in>s. g x \<noteq> x")
   obtain x where "x\<in>s" using s(2) by auto
   case False hence g:"\<forall>x\<in>s. g x = x" by auto