author wenzelm Wed, 28 Aug 2013 23:41:21 +0200 changeset 53253 220f306f5c4e parent 53252 4766fbe322b5 child 53254 082d0972096b
tuned proofs;
```--- a/src/HOL/Multivariate_Analysis/Determinants.thy	Wed Aug 28 22:50:23 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Determinants.thy	Wed Aug 28 23:41:21 2013 +0200
@@ -11,14 +11,18 @@
begin

subsection{* First some facts about products*}
-lemma setprod_insert_eq: "finite A \<Longrightarrow> setprod f (insert a A) = (if a \<in> A then setprod f A else f a * setprod f A)"
-apply clarsimp
-by(subgoal_tac "insert a A = A", auto)
+
+lemma setprod_insert_eq:
+  "finite A \<Longrightarrow> setprod f (insert a A) = (if a \<in> A then setprod f A else f a * setprod f A)"
+  apply clarsimp
+  apply (subgoal_tac "insert a A = A")
+  apply auto
+  done

assumes mn: "(m::nat) <= n + 1"
shows "setprod f {m.. n+p} = setprod f {m .. n} * setprod f {n+1..n+p}"
-proof-
+proof -
let ?A = "{m .. n+p}"
let ?B = "{m .. n}"
let ?C = "{n+1..n+p}"
@@ -30,47 +34,56 @@

lemma setprod_offset: "setprod f {(m::nat) + p .. n + p} = setprod (\<lambda>i. f (i + p)) {m..n}"
-apply (rule setprod_reindex_cong[where f="op + p"])
-apply (auto simp add: image_iff Bex_def inj_on_def)
-apply arith
-apply (rule ext)
-done
+  apply (rule setprod_reindex_cong[where f="op + p"])
+  apply (auto simp add: image_iff Bex_def inj_on_def)
+  apply presburger
+  apply (rule ext)
+  done

-lemma setprod_singleton: "setprod f {x} = f x" by simp
-
-lemma setprod_singleton_nat_seg: "setprod f {n..n} = f (n::'a::order)" by simp
+lemma setprod_singleton: "setprod f {x} = f x"
+  by simp

-lemma setprod_numseg: "setprod f {m..0} = (if m=0 then f 0 else 1)"
-  "setprod f {m .. Suc n} = (if m \<le> Suc n then f (Suc n) * setprod f {m..n}
-                             else setprod f {m..n})"
+lemma setprod_singleton_nat_seg: "setprod f {n..n} = f (n::'a::order)"
+  by simp
+
+lemma setprod_numseg:
+  "setprod f {m..0} = (if m = 0 then f 0 else 1)"
+  "setprod f {m .. Suc n} =
+    (if m \<le> Suc n then f (Suc n) * setprod f {m..n} else setprod f {m..n})"

-lemma setprod_le: assumes fS: "finite S" and fg: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (g x :: 'a::linordered_idom)"
+lemma setprod_le:
+  assumes fS: "finite S"
+    and fg: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (g x :: 'a::linordered_idom)"
shows "setprod f S \<le> setprod g S"
-using fS fg
-apply(induct S)
-apply simp
-apply auto
-apply (rule mult_mono)
-apply (auto intro: setprod_nonneg)
-done
+  using fS fg
+  apply (induct S)
+  apply simp
+  apply auto
+  apply (rule mult_mono)
+  apply (auto intro: setprod_nonneg)
+  done

(* FIXME: In Finite_Set there is a useless further assumption *)
-lemma setprod_inversef: "finite A ==> setprod (inverse \<circ> f) A = (inverse (setprod f A) :: 'a:: field_inverse_zero)"
+lemma setprod_inversef:
+  "finite A \<Longrightarrow> setprod (inverse \<circ> f) A = (inverse (setprod f A) :: 'a:: field_inverse_zero)"
apply (erule finite_induct)
apply (simp)
apply simp
done

-lemma setprod_le_1: assumes fS: "finite S" and f: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (1::'a::linordered_idom)"
+lemma setprod_le_1:
+  assumes fS: "finite S"
+    and f: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (1::'a::linordered_idom)"
shows "setprod f S \<le> 1"
-using setprod_le[OF fS f] unfolding setprod_1 .
+  using setprod_le[OF fS f] unfolding setprod_1 .

-subsection{* Trace *}
+
+subsection {* Trace *}

-definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a" where
-  "trace A = setsum (\<lambda>i. ((A\$i)\$i)) (UNIV::'n set)"
+definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a"
+  where "trace A = setsum (\<lambda>i. ((A\$i)\$i)) (UNIV::'n set)"

lemma trace_0: "trace(mat 0) = 0"
@@ -87,14 +100,17 @@
lemma trace_mul_sym:"trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
apply (subst setsum_commute)
+  done

(* ------------------------------------------------------------------------- *)
(* Definition of determinant.                                                *)
(* ------------------------------------------------------------------------- *)

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) (UNIV :: 'n set)) {p. p permutes (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.                                       *)
@@ -105,7 +121,8 @@
shows "setprod f S = setprod (f o p) S"
using assms by (fact setprod.permute)

-lemma setproduct_permute_nat_interval: "p permutes {m::nat .. n} ==> setprod f {m..n} = setprod (f o p) {m..n}"
+lemma setproduct_permute_nat_interval:
+  "p permutes {m::nat .. n} ==> setprod f {m..n} = setprod (f o p) {m..n}"
by (blast intro!: setprod_permute)

(* ------------------------------------------------------------------------- *)
@@ -113,52 +130,71 @@
(* ------------------------------------------------------------------------- *)

lemma det_transpose: "det (transpose A) = det (A::'a::comm_ring_1 ^'n^'n)"
-proof-
+proof -
let ?di = "\<lambda>A i j. A\$i\$j"
let ?U = "(UNIV :: 'n set)"
have fU: "finite ?U" by simp
-  {fix p assume p: "p \<in> {p. p permutes ?U}"
+  {
+    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"
by (metis sign_inverse fU p mem_Collect_eq permutation_permutes)
from permutes_inj[OF pU]
have pi: "inj_on p ?U" by (blast intro: subset_inj_on)
from permutes_image[OF pU]
-    have "setprod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U = setprod (\<lambda>i. ?di (transpose A) i (inv p i)) (p ` ?U)" by simp
+    have "setprod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U =
+      setprod (\<lambda>i. ?di (transpose A) i (inv p i)) (p ` ?U)" by simp
also have "\<dots> = setprod ((\<lambda>i. ?di (transpose A) i (inv p i)) o p) ?U"
unfolding setprod_reindex[OF pi] ..
also have "\<dots> = setprod (\<lambda>i. ?di A i (p i)) ?U"
-    proof-
-      {fix i assume i: "i \<in> ?U"
+    proof -
+      {
+        fix i
+        assume i: "i \<in> ?U"
from i permutes_inv_o[OF pU] permutes_in_image[OF pU]
have "((\<lambda>i. ?di (transpose A) i (inv p i)) o p) i = ?di A i (p i)"
-          unfolding transpose_def by (simp add: fun_eq_iff)}
-      then show "setprod ((\<lambda>i. ?di (transpose A) i (inv p i)) o p) ?U = setprod (\<lambda>i. ?di A i (p i)) ?U" by (auto intro: setprod_cong)
+          unfolding transpose_def by (simp add: fun_eq_iff)
+      }
+      then show "setprod ((\<lambda>i. ?di (transpose 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 (transpose A) i (inv p i)) ?U) = of_int (sign p) * (setprod (\<lambda>i. ?di A i (p i)) ?U)" using sth
-      by simp}
-  then show ?thesis unfolding det_def apply (subst setsum_permutations_inverse)
-  apply (rule setsum_cong2) by blast
+    finally have "of_int (sign (inv p)) * (setprod (\<lambda>i. ?di (transpose A) i (inv p i)) ?U) =
+      of_int (sign p) * (setprod (\<lambda>i. ?di A i (p i)) ?U)" using sth by simp
+  }
+  then show ?thesis
+    unfolding det_def
+    apply (subst setsum_permutations_inverse)
+    apply (rule setsum_cong2)
+    apply blast
+    done
qed

lemma det_lowerdiagonal:
fixes A :: "'a::comm_ring_1^('n::{finite,wellorder})^('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-
+proof -
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) (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: "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
+  {
+    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: "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
@@ -167,21 +203,26 @@
fixes A :: "'a::comm_ring_1^'n::{finite,wellorder}^'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-
+proof -
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) (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: "p i < i" by (metis not_le)
+  {
+    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: "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
+    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

@@ -189,14 +230,16 @@
fixes A :: "'a::comm_ring_1^'n^'n"
assumes ld: "\<And>i j. i \<noteq> j \<Longrightarrow> A\$i\$j = 0"
shows "det A = setprod (\<lambda>i. A\$i\$i) (UNIV::'n set)"
-proof-
+proof -
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) (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}"
+  {
+    fix p
+    assume p: "p \<in> ?PU - {id}"
then have "p \<noteq> id" by simp
then obtain i where i: "p i \<noteq> i" unfolding fun_eq_iff by auto
from ld [OF i [symmetric]] have ex:"\<exists>i \<in> ?U. A\$i\$p i = 0" by blast
@@ -207,16 +250,22 @@
qed

lemma det_I: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
-proof-
+proof -
let ?A = "mat 1 :: 'a::comm_ring_1^'n^'n"
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)}
-  hence th: "setprod (\<lambda>i. ?f i i) ?U = setprod (\<lambda>x. 1) ?U"
+  {
+    fix i
+    assume i: "i \<in> ?U"
+    have "?f i i = 1" using i by (vector mat_def)
+  }
+  then have th: "setprod (\<lambda>i. ?f i i) ?U = setprod (\<lambda>x. 1) ?U"
by (auto intro: setprod_cong)
-  {fix i j assume i: "i \<in> ?U" and j: "j \<in> ?U" and ij: "i \<noteq> j"
-    have "?f i j = 0" using i j ij by (vector mat_def) }
+  {
+    fix i j
+    assume i: "i \<in> ?U" and j: "j \<in> ?U" and ij: "i \<noteq> 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_diagonal
by blast
also have "\<dots> = 1" unfolding th setprod_1 ..
@@ -232,23 +281,27 @@
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])
apply (subst sum_permutations_compose_right[OF p])
-proof(rule setsum_cong2)
+proof (rule setsum_cong2)
let ?U = "UNIV :: 'n set"
let ?PU = "{p. p permutes ?U}"
-  fix q assume qPU: "q \<in> ?PU"
+  fix q
+  assume qPU: "q \<in> ?PU"
have fU: "finite ?U" by simp
-  from qPU have q: "q permutes ?U" by blast
+  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" .
-    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. 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. A\$ p i\$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A\$i\$q i) ?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. 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. 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

@@ -256,7 +309,7 @@
fixes A :: "'a::comm_ring_1^'n^'n"
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-
+proof -
let ?Ap = "\<chi> i j. A\$i\$ p j :: 'a^'n^'n"
let ?At = "transpose A"
have "of_int (sign p) * det A = det (transpose (\<chi> i. transpose A \$ p i))"
@@ -270,16 +323,16 @@
lemma det_identical_rows:
fixes A :: "'a::linordered_idom^'n^'n"
assumes ij: "i \<noteq> j"
-  and r: "row i A = row j A"
+    and r: "row i A = row j A"
shows "det A = 0"
proof-
-  have tha: "\<And>(a::'a) b. a = b ==> b = - a ==> a = 0"
+  have tha: "\<And>(a::'a) b. a = b \<Longrightarrow> b = - a \<Longrightarrow> a = 0"
by simp
have th1: "of_int (-1) = - 1" by simp
let ?p = "Fun.swap i j id"
let ?A = "\<chi> i. A \$ ?p i"
from r have "A = ?A" by (simp add: vec_eq_iff row_def swap_def)
-  hence "det A = det ?A" by simp
+  then have "det A = det ?A" by simp
moreover have "det A = - det ?A"
by (simp add: det_permute_rows[OF permutes_swap_id] sign_swap_id ij th1)
ultimately show "det A = 0" by (metis tha)
@@ -288,21 +341,22 @@
lemma det_identical_columns:
fixes A :: "'a::linordered_idom^'n^'n"
assumes ij: "i \<noteq> j"
-  and r: "column i A = column j A"
+    and r: "column i A = column j A"
shows "det A = 0"
-apply (subst det_transpose[symmetric])
-apply (rule det_identical_rows[OF ij])
-by (metis row_transpose r)
+  apply (subst det_transpose[symmetric])
+  apply (rule det_identical_rows[OF ij])
+  apply (metis row_transpose r)
+  done

lemma det_zero_row:
fixes A :: "'a::{idom, ring_char_0}^'n^'n"
assumes r: "row i A = 0"
shows "det A = 0"
-using r
-apply (simp add: row_def det_def vec_eq_iff)
-apply (rule setsum_0')
-apply (auto simp: sign_nz)
-done
+  using r
+  apply (simp add: row_def det_def vec_eq_iff)
+  apply (rule setsum_0')
+  apply (auto simp: sign_nz)
+  done

lemma det_zero_column:
fixes A :: "'a::{idom,ring_char_0}^'n^'n"
@@ -310,27 +364,32 @@
shows "det A = 0"
apply (subst det_transpose[symmetric])
apply (rule det_zero_row [of i])
-  by (metis row_transpose r)
+  apply (metis row_transpose r)
+  done

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)"
+    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)"
proof (rule setsum_cong2)
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)::'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"
+  fix p
+  assume p: "p \<in> ?pU"
let ?Uk = "?U - {k}"
from p have pU: "p permutes ?U" by blast
have kU: "?U = insert k ?Uk" by blast
-  {fix j assume j: "j \<in> ?Uk"
+  {
+    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}
+      by simp_all
+  }
then have th1: "setprod (\<lambda>i. ?f i \$ p i) ?Uk = setprod (\<lambda>i. ?g i \$ p i) ?Uk"
and th2: "setprod (\<lambda>i. ?f i \$ p i) ?Uk = setprod (\<lambda>i. ?h i \$ p i) ?Uk"
apply -
@@ -342,36 +401,45 @@
also have "\<dots> = ?f k \$ p k  * setprod (\<lambda>i. ?f i \$ p i) ?Uk"
apply (rule setprod_insert)
apply simp
-    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: field_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)
+    apply blast
+    done
+  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)"
+  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
-  finally have "setprod (\<lambda>i. ?f i \$ p i) ?U = setprod (\<lambda>i. ?g i \$ p i) ?U + setprod (\<lambda>i. ?h i \$ p i) ?U" unfolding kU[symmetric] .
-  then show "of_int (sign p) * setprod (\<lambda>i. ?f i \$ p i) ?U = of_int (sign p) * setprod (\<lambda>i. ?g i \$ p i) ?U + of_int (sign p) * setprod (\<lambda>i. ?h i \$ p i) ?U"
+  finally have "setprod (\<lambda>i. ?f i \$ p i) ?U =
+    setprod (\<lambda>i. ?g i \$ p i) ?U + setprod (\<lambda>i. ?h i \$ p i) ?U" unfolding kU[symmetric] .
+  then show "of_int (sign p) * setprod (\<lambda>i. ?f i \$ p i) ?U =
+    of_int (sign p) * setprod (\<lambda>i. ?g i \$ p i) ?U + of_int (sign p) * setprod (\<lambda>i. ?h i \$ p i) ?U"
qed

lemma det_row_mul:
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 vec_lambda_beta setsum_right_distrib
+    c * det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
+  unfolding det_def vec_lambda_beta setsum_right_distrib
proof (rule setsum_cong2)
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)::'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"
+  fix p
+  assume p: "p \<in> ?pU"
let ?Uk = "?U - {k}"
from p have pU: "p permutes ?U" 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}
+  {
+    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)
+    apply (rule setprod_cong)
+    apply simp_all
done
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)"
@@ -379,29 +447,34 @@
also have "\<dots> = ?f k \$ p k  * setprod (\<lambda>i. ?f i \$ p i) ?Uk"
apply (rule setprod_insert)
apply simp
-    by blast
-  also have "\<dots> = (c*s a k) \$ p k * setprod (\<lambda>i. ?f i \$ p i) ?Uk" by (simp add: field_simps)
+    apply blast
+    done
+  also have "\<dots> = (c*s a k) \$ p k * setprod (\<lambda>i. ?f i \$ p i) ?Uk"
also have "\<dots> = c* (a k \$ p k * setprod (\<lambda>i. ?g i \$ p i) ?Uk)"
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] .
-  then show "of_int (sign p) * setprod (\<lambda>i. ?f i \$ p i) ?U = c * (of_int (sign p) * setprod (\<lambda>i. ?g i \$ p i) ?U)"
+    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] .
+  then show "of_int (sign p) * setprod (\<lambda>i. ?f i \$ p i) ?U =
+    c * (of_int (sign p) * setprod (\<lambda>i. ?g i \$ p i) ?U)"
qed

lemma det_row_0:
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 0 "\<lambda>i. 1" b]
-apply (simp)
-  unfolding vector_smult_lzero .
+  using det_row_mul[of k 0 "\<lambda>i. 1" b]
+  apply simp
+  apply (simp only: vector_smult_lzero)
+  done

lemma det_row_operation:
fixes A :: "'a::linordered_idom^'n^'n"
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-
+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" by (vector row_def)
have th2: "((\<chi> k. if k = i then row i A else row k A) :: 'a^'n^'n) = A"
@@ -415,30 +488,38 @@
fixes A :: "real^'n^'n"
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-
+proof -
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
-
-    have "(if k = i then row i A + 0 else row k A) = row k A" by simp}
+  {
+    fix k
+    have "(if k = i then row i A + 0 else row k A) = row k A" by simp
+  }
then have P0: "?P 0"
apply -
apply (rule cong[of det, OF refl])
-    by (vector row_def)
+    apply (vector row_def)
+    done
moreover
-  {fix c z y assume zS: "z \<in> ?S" and Py: "?P y"
+  {
+    fix c z y
+    assume zS: "z \<in> ?S" and Py: "?P y"
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 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]
-      by simp }
-
+      using j
+      apply (vector row_def)
+      done
+    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]
+      by simp
+  }
ultimately show ?thesis
apply -
apply (rule span_induct_alt[of ?P ?S, OF P0, folded scalar_mult_eq_scaleR])
@@ -456,53 +537,68 @@
fixes A:: "real^'n^'n"
assumes d: "dependent (rows A)"
shows "det A = 0"
-proof-
+proof -
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 jk: "j \<noteq> k"
-    and c: "row j A = row k A"
-    from det_identical_rows[OF jk c] have ?thesis .}
+  {
+    fix j k
+    assume jk: "j \<noteq> k" and c: "row j A = row k A"
+    from det_identical_rows[OF jk c] have ?thesis .
+  }
moreover
-  {assume H: "\<And> i j. i \<noteq> j \<Longrightarrow> row i A \<noteq> row j A"
+  {
+    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)
apply (rule span_mono)
-      using H i by (auto simp add: rows_def)
+      using H i
+      apply (auto simp add: rows_def)
+      done
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 "0::real" "\<lambda>i. 1"]
-    have "det A = 0" by simp}
+    have "det A = 0" by simp
+  }
ultimately show ?thesis by blast
qed

-lemma det_dependent_columns: assumes d: "dependent(columns (A::real^'n^'n))" shows "det A = 0"
-by (metis d det_dependent_rows rows_transpose det_transpose)
+lemma det_dependent_columns:
+  assumes d: "dependent (columns (A::real^'n^'n))"
+  shows "det A = 0"
+  by (metis d det_dependent_rows rows_transpose det_transpose)

(* ------------------------------------------------------------------------- *)
(* Multilinearity and the multiplication formula.                            *)
(* ------------------------------------------------------------------------- *)

lemma Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
-  apply (rule iffD1[OF vec_lambda_unique]) by vector
+  by (rule iffD1[OF vec_lambda_unique]) vector

lemma det_linear_row_setsum:
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) = 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] ..
+  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"
+proof (induct rule: finite_induct[OF fS])
+  case 1
+  then show ?case
+    apply simp
+    unfolding setsum_empty det_row_0[of k]
+    apply rule
+    done
next
case (2 x F)
+  then show ?case
qed

lemma finite_bounded_functions:
assumes fS: "finite S"
shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
-proof(induct k)
+proof (induct k)
case 0
have th: "{f. \<forall>i. f i = i} = {id}" by auto
show ?case by (auto simp add: th)
@@ -526,13 +622,14 @@
lemma det_linear_rows_setsum_lemma:
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) =
-             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)
+    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
+  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 (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)}"
@@ -540,42 +637,48 @@
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 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
-  from `z \<notin> T` have nz: "\<And>i. i \<in> T \<Longrightarrow> i = z \<longleftrightarrow> False" by auto
+     (if c then (if a then b else d) else (if a then b else e))"
+    by simp
+  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)"
+    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))"
+  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)
-    using nz by (simp cong del: if_weak_cong cong add: if_cong)
+    using nz
+    apply (simp cong del: if_weak_cong cong add: if_cong)
+    done
finally have tha:
"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  insert.hyps 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"],
+    apply (rule setsum_eq_general_reverses[where h= "?h" and k= "?k"],
blast intro: finite_cartesian_product fS finite,
blast intro: finite_cartesian_product fS finite)
using `z \<notin> T`
apply auto
apply (rule cong[OF refl[of det]])
-    by vector
+    apply vector
+    done
qed

lemma det_linear_rows_setsum:
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)) {f. \<forall>i. f i \<in> S}"
-proof-
-  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 "UNIV :: 'n set" a, unfolded th0, OF finite] show ?thesis by simp
+  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. f i \<in> S}"
+proof -
+  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 "UNIV :: 'n set" a, unfolded th0, OF finite]
+  show ?thesis by simp
qed

lemma matrix_mul_setsum_alt:
@@ -585,75 +688,93 @@

lemma det_rows_mul:
"det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
-  setprod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
+    setprod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
let ?U = "UNIV :: 'n set"
let ?PU = "{p. p permutes ?U}"
-  fix p assume pU: "p \<in> ?PU"
+  fix p
+  assume pU: "p \<in> ?PU"
let ?s = "of_int (sign p)"
-  from pU have p: "p permutes ?U" by blast
+  from pU have p: "p permutes ?U"
+    by blast
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 ..
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: field_simps)
+    setprod c ?U * (?s* (\<Prod>xa\<in>?U. a xa \$ p xa))" by (simp add: field_simps)
qed

lemma det_mul:
fixes A B :: "'a::linordered_idom^'n^'n"
shows "det (A ** B) = det A * det B"
-proof-
+proof -
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" by (rule finite)
-  {fix p assume p: "p permutes ?U"
-
+  {
+    fix p
+    assume p: "p permutes ?U"
have "p \<in> ?F" unfolding mem_Collect_eq permutes_in_image[OF p]
-      using p[unfolded permutes_def] by simp}
+      using p[unfolded permutes_def] by simp
+  }
then have PUF: "?PU \<subseteq> ?F"  by blast
-  {fix f assume fPU: "f \<in> ?F - ?PU"
+  {
+    fix f
+    assume fPU: "f \<in> ?F - ?PU"
have fUU: "f ` ?U \<subseteq> ?U" using fPU by auto
-    from fPU have f: "\<forall>i \<in> ?U. f i \<in> ?U"
-      "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)" unfolding permutes_def
-      by auto
+    from fPU have f: "\<forall>i \<in> ?U. f i \<in> ?U" "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)"
+      unfolding permutes_def by auto

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"
+    {
+      assume fni: "\<not> inj_on f ?U"
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(2) rth]
have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
-        unfolding det_rows_mul by simp}
+        unfolding det_rows_mul by simp
+    }
moreover
-    {assume fi: "inj_on f ?U"
+    {
+      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 by metis
note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]]

-      {fix y
+      {
+        fix y
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}
-      with f(3) have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0" by blast}
-    ultimately have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0" by blast}
-  hence zth: "\<forall> f\<in> ?F - ?PU. det (\<chi> i. A\$i\$f i *s B\$f i) = 0" by simp
-  {fix p assume pU: "p \<in> ?PU"
+        {
+          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
+      }
+      with f(3) have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0" by blast
+    }
+    ultimately have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0" by blast
+  }
+  hence zth: "\<forall> f\<in> ?F - ?PU. det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
+    by simp
+  {
+    fix p
+    assume pU: "p \<in> ?PU"
from pU have p: "p permutes ?U" by blast
let ?s = "\<lambda>p. of_int (sign p)"
-    let ?f = "\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A \$ i \$ p i) *
-               (?s q * (\<Prod>i\<in> ?U. B \$ i \$ q i))"
+    let ?f = "\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A \$ i \$ p i) * (?s q * (\<Prod>i\<in> ?U. B \$ i \$ q i))"
have "(setsum (\<lambda>q. ?s q *
-            (\<Prod>i\<in> ?U. (\<chi> i. A \$ i \$ p i *s B \$ p i :: 'a^'n^'n) \$ i \$ q i)) ?PU) =
-        (setsum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A \$ i \$ p i) *
-               (?s q * (\<Prod>i\<in> ?U. B \$ i \$ q i))) ?PU)"
+        (\<Prod>i\<in> ?U. (\<chi> i. A \$ i \$ p i *s B \$ p i :: 'a^'n^'n) \$ i \$ q i)) ?PU) =
+      (setsum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A \$ i \$ p i) * (?s q * (\<Prod>i\<in> ?U. B \$ i \$ q i))) ?PU)"
unfolding sum_permutations_compose_right[OF permutes_inv[OF p], of ?f]
proof(rule setsum_cong2)
-      fix q assume qU: "q \<in> ?PU"
+      fix q
+      assume qU: "q \<in> ?PU"
hence q: "q permutes ?U" by blast
from p q have pp: "permutation p" and pq: "permutation q"
unfolding permutation_permutes by auto
@@ -666,11 +787,15 @@
by (simp add:  th00 mult_ac sign_idempotent sign_compose)
have th001: "setprod (\<lambda>i. B\$i\$ q (inv p i)) ?U = setprod ((\<lambda>i. B\$i\$ q (inv p i)) o p) ?U"
by (rule setprod_permute[OF p])
-      have thp: "setprod (\<lambda>i. (\<chi> i. A\$i\$p i *s B\$p i :: 'a^'n^'n) \$i \$ q i) ?U = setprod (\<lambda>i. A\$i\$p i) ?U * setprod (\<lambda>i. B\$i\$ q (inv p i)) ?U"
+      have thp: "setprod (\<lambda>i. (\<chi> i. A\$i\$p i *s B\$p i :: 'a^'n^'n) \$i \$ q i) ?U =
+        setprod (\<lambda>i. A\$i\$p i) ?U * setprod (\<lambda>i. B\$i\$ q (inv p i)) ?U"
unfolding th001 setprod_timesf[symmetric] o_def permutes_inverses[OF p]
apply (rule setprod_cong[OF refl])
-        using permutes_in_image[OF q] by vector
-      show "?s q * setprod (\<lambda>i. (((\<chi> i. A\$i\$p i *s B\$p i) :: 'a^'n^'n)\$i\$q i)) ?U = ?s p * (setprod (\<lambda>i. A\$i\$p i) ?U) * (?s (q o inv p) * setprod (\<lambda>i. B\$i\$(q o inv p) i) ?U)"
+        using permutes_in_image[OF q]
+        apply vector
+        done
+      show "?s q * setprod (\<lambda>i. (((\<chi> i. A\$i\$p i *s B\$p i) :: 'a^'n^'n)\$i\$q i)) ?U =
+        ?s p * (setprod (\<lambda>i. A\$i\$p i) ?U) * (?s (q o inv p) * setprod (\<lambda>i. B\$i\$(q o inv p) i) ?U)"
using ths thp pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
by (simp add: sign_nz th00 field_simps sign_idempotent sign_compose)
qed
@@ -703,22 +828,24 @@
lemma invertible_det_nz:
fixes A::"real ^'n^'n"
shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
-proof-
-  {assume "invertible A"
+proof -
+  {
+    assume "invertible A"
then obtain B :: "real ^'n^'n" where B: "A ** B = mat 1"
unfolding invertible_righ_inverse by blast
hence "det (A ** B) = det (mat 1 :: real ^'n^'n)" by simp
-    hence "det A \<noteq> 0"
-      apply (simp add: det_mul det_I) by algebra }
+    hence "det A \<noteq> 0" by (simp add: det_mul det_I) algebra
+  }
moreover
-  {assume H: "\<not> invertible A"
+  {
+    assume H: "\<not> invertible A"
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"
unfolding invertible_righ_inverse
unfolding matrix_right_invertible_independent_rows by blast
-    have stupid: "\<And>(a::real^'n) b. a + b = 0 \<Longrightarrow> -a = b"
+    have *: "\<And>(a::real^'n) b. a + b = 0 \<Longrightarrow> -a = b"
apply (drule_tac f="op + (- a)" in cong[OF refl])
apply simp
@@ -729,7 +856,7 @@
apply -
apply (rule vector_mul_lcancel_imp[OF ci])
-      unfolding stupid ..
+      unfolding * ..
have thr: "- row i A \<in> span {row j A| j. j \<noteq> i}"
unfolding thr0
apply (rule span_setsum)
@@ -743,7 +870,8 @@
have thrb: "row i ?B = 0" using iU by (vector row_def)
have "det A = 0"
unfolding det_row_span[OF thr, symmetric] right_minus
-      unfolding  det_zero_row[OF thrb]  ..}
+      unfolding det_zero_row[OF thrb] ..
+  }
ultimately show ?thesis by blast
qed

@@ -756,7 +884,7 @@
shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x\$i *s row i A) (UNIV::'n set)
else row i A)::real^'n^'n) = x\$k * det A"
(is "?lhs = ?rhs")
-proof-
+proof -
let ?U = "UNIV :: 'n set"
let ?Uk = "?U - {k}"
have U: "?U = insert k ?Uk" by blast
@@ -766,7 +894,8 @@
by (vector field_simps)
have th001: "\<And>f k . (\<lambda>x. if x = k then f k else f x) = f" by auto
have "(\<chi> i. row i A) = A" by (vector row_def)
-  then have thd1: "det (\<chi> i. row i A) = det A"  by simp
+  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)
apply (rule span_setsum[OF fUk])
@@ -784,37 +913,44 @@
unfolding thd0
unfolding det_row_mul
unfolding th001[of k "\<lambda>i. row i A"]
-    unfolding thd1  by (simp add: field_simps)
+    unfolding thd1
+    done
qed

lemma cramer_lemma:
fixes A :: "real^'n^'n"
shows "det((\<chi> i j. if j = k then (A *v x)\$i else A\$i\$j):: real^'n^'n) = x\$k * det A"
-proof-
+proof -
let ?U = "UNIV :: 'n set"
-  have stupid: "\<And>c. setsum (\<lambda>i. c i *s row i (transpose A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
+  have *: "\<And>c. setsum (\<lambda>i. c i *s row i (transpose A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
by (auto simp add: row_transpose intro: setsum_cong2)
show ?thesis  unfolding matrix_mult_vsum
-  unfolding cramer_lemma_transpose[of k x "transpose A", unfolded det_transpose, symmetric]
-  unfolding stupid[of "\<lambda>i. x\$i"]
-  apply (subst det_transpose[symmetric])
-  apply (rule cong[OF refl[of det]]) by (vector transpose_def column_def row_def)
+    unfolding cramer_lemma_transpose[of k x "transpose A", unfolded det_transpose, symmetric]
+    unfolding *[of "\<lambda>i. x\$i"]
+    apply (subst det_transpose[symmetric])
+    apply (rule cong[OF refl[of det]])
+    apply (vector transpose_def column_def row_def)
+    done
qed

lemma cramer:
fixes A ::"real^'n^'n"
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) / det A)"
-proof-
+proof -
from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
unfolding invertible_det_nz[symmetric] invertible_def by blast
have "(A ** B) *v b = b" by (simp add: B matrix_vector_mul_lid)
-  hence "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc)
+  then have "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc)
then have xe: "\<exists>x. A*v x = b" by blast
-  {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) / det A)"
-    unfolding x[symmetric]
-    using d0 by (simp add: vec_eq_iff cramer_lemma field_simps)}
+  {
+    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) / det A)"
+      unfolding x[symmetric]
+      using d0 by (simp add: vec_eq_iff cramer_lemma field_simps)
+  }
with xe show ?thesis by auto
qed

@@ -824,16 +960,19 @@

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 ^_). 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)+
+  done

-definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1 \<and> Q ** transpose Q = mat 1"
+definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow>
+  transpose Q ** Q = mat 1 \<and> Q ** transpose Q = mat 1"

-lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n)  \<longleftrightarrow> transpose Q ** Q = mat 1"
+lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1"
by (metis matrix_left_right_inverse orthogonal_matrix_def)

lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"
@@ -842,28 +981,31 @@
lemma orthogonal_matrix_mul:
fixes A :: "real ^'n^'n"
assumes oA : "orthogonal_matrix A"
-  and oB: "orthogonal_matrix B"
+    and oB: "orthogonal_matrix B"
shows "orthogonal_matrix(A ** B)"
using oA oB
unfolding orthogonal_matrix matrix_transpose_mul
apply (subst matrix_mul_assoc)
apply (subst matrix_mul_assoc[symmetric])
+  done

lemma orthogonal_transformation_matrix:
fixes f:: "real^'n \<Rightarrow> real^'n"
shows "orthogonal_transformation f \<longleftrightarrow> linear f \<and> orthogonal_matrix(matrix f)"
(is "?lhs \<longleftrightarrow> ?rhs")
-proof-
+proof -
let ?mf = "matrix f"
let ?ot = "orthogonal_transformation f"
let ?U = "UNIV :: 'n set"
have fU: "finite ?U" by simp
let ?m1 = "mat 1 :: real ^'n^'n"
-  {assume ot: ?ot
+  {
+    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
+    {
+      fix i j
let ?A = "transpose ?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)"
@@ -871,16 +1013,22 @@
from fd[rule_format, of "axis i 1" "axis j 1", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul]
have "?A\$i\$j = ?m1 \$ i \$ j"
by (simp add: inner_vec_def matrix_matrix_mult_def columnvector_def rowvector_def
-            th0 setsum_delta[OF fU] mat_def axis_def) }
-    hence "orthogonal_matrix ?mf" unfolding orthogonal_matrix by vector
-    with lf have ?rhs by blast}
+            th0 setsum_delta[OF fU] mat_def axis_def)
+    }
+    then have "orthogonal_matrix ?mf" unfolding orthogonal_matrix
+      by vector
+    with lf have ?rhs by blast
+  }
moreover
-  {assume lf: "linear f" and om: "orthogonal_matrix ?mf"
+  {
+    assume lf: "linear f" and om: "orthogonal_matrix ?mf"
from lf om have ?lhs
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)}
+      apply (simp add: dot_matrix_product matrix_mul_lid)
+      done
+  }
ultimately show ?thesis by blast
qed

@@ -888,21 +1036,26 @@
fixes Q:: "'a::linordered_idom^'n^'n"
assumes oQ: "orthogonal_matrix Q"
shows "det Q = 1 \<or> det Q = - 1"
-proof-
-
+proof -
have th: "\<And>x::'a. x = 1 \<or> x = - 1 \<longleftrightarrow> x*x = 1" (is "\<And>x::'a. ?ths x")
-  proof-
+  proof -
fix x:: 'a
-    have th0: "x*x - 1 = (x - 1)*(x + 1)" by (simp add: field_simps)
+    have th0: "x*x - 1 = (x - 1)*(x + 1)"
have th1: "\<And>(x::'a) y. x = - y \<longleftrightarrow> x + y = 0"
-      apply (subst eq_iff_diff_eq_0) by simp
-    have "x*x = 1 \<longleftrightarrow> x*x - 1 = 0" by simp
+      apply (subst eq_iff_diff_eq_0)
+      apply simp
+      done
+    have "x * x = 1 \<longleftrightarrow> x*x - 1 = 0" by simp
also have "\<dots> \<longleftrightarrow> x = 1 \<or> x = - 1" unfolding th0 th1 by simp
finally show "?ths x" ..
qed
-  from oQ have "Q ** transpose Q = mat 1" by (metis orthogonal_matrix_def)
-  hence "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)" by simp
-  hence "det Q * det Q = 1" by (simp add: det_mul det_I det_transpose)
+  from oQ have "Q ** transpose Q = mat 1"
+    by (metis orthogonal_matrix_def)
+  then have "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)"
+    by simp
+  then have "det Q * det Q = 1"
+    by (simp add: det_mul det_I det_transpose)
then show ?thesis unfolding th .
qed

@@ -911,25 +1064,29 @@
(* ------------------------------------------------------------------------- *)
lemma scaling_linear:
fixes f :: "real ^'n \<Rightarrow> real ^'n"
-  assumes f0: "f 0 = 0" and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
+  assumes f0: "f 0 = 0"
+    and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
shows "linear f"
-proof-
-  {fix v w
-    {fix x note fd[rule_format, of x 0, unfolded dist_norm f0 diff_0_right] }
+proof -
+  {
+    fix v w
+    {
+      fix x
+      note fd[rule_format, of x 0, unfolded dist_norm f0 diff_0_right]
+    }
note th0 = this
have "f v \<bullet> f w = c\<^sup>2 * (v \<bullet> w)"
unfolding dot_norm_neg dist_norm[symmetric]
unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)}
note fc = this
show ?thesis
-    unfolding linear_def vector_eq[where 'a="real^'n"] scalar_mult_eq_scaleR
+    unfolding linear_def vector_eq[where 'a="real^'n"] scalar_mult_eq_scaleR
qed

lemma isometry_linear:
-  "f (0:: real^'n) = (0:: real^'n) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y
-        \<Longrightarrow> linear f"
-by (rule scaling_linear[where c=1]) simp_all
+  "f (0:: real^'n) = (0:: real^'n) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y \<Longrightarrow> linear f"
+  by (rule scaling_linear[where c=1]) simp_all

(* ------------------------------------------------------------------------- *)
(* Hence another formulation of orthogonal transformation.                   *)
@@ -948,7 +1105,8 @@
apply clarify
apply (erule_tac x=v in allE)
apply (erule_tac x=0 in allE)
+  done

(* ------------------------------------------------------------------------- *)
(* Can extend an isometry from unit sphere.                                  *)
@@ -957,15 +1115,19 @@
lemma isometry_sphere_extend:
fixes f:: "real ^'n \<Rightarrow> real ^'n"
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"
+    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)"
-proof-
-  {fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
-    assume H: "x = norm x *\<^sub>R x0" "y = norm y *\<^sub>R y0"
-    "x' = norm x *\<^sub>R x0'" "y' = norm y *\<^sub>R y0'"
-    "norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1"
-    "norm(x0' - y0') = norm(x0 - y0)"
-    hence *:"x0 \<bullet> y0 = x0' \<bullet> y0' + y0' \<bullet> x0' - y0 \<bullet> x0 " by(simp add: norm_eq norm_eq_1 inner_add inner_diff)
+proof -
+  {
+    fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
+    assume H:
+      "x = norm x *\<^sub>R x0"
+      "y = norm y *\<^sub>R y0"
+      "x' = norm x *\<^sub>R x0'" "y' = norm y *\<^sub>R y0'"
+      "norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1"
+      "norm(x0' - y0') = norm(x0 - y0)"
+    hence *: "x0 \<bullet> y0 = x0' \<bullet> y0' + y0' \<bullet> x0' - y0 \<bullet> x0 "
have "norm(x' - y') = norm(x - y)"
apply (subst H(1))
apply (subst H(2))
@@ -974,48 +1136,71 @@
using H(5-9)
apply (simp add: inner_diff scalar_mult_eq_scaleR) unfolding *
-      by (simp add: field_simps) }
+      done
+  }
note th0 = this
let ?g = "\<lambda>x. if x = 0 then 0 else norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)"
-  {fix x:: "real ^'n" assume nx: "norm x = 1"
-    have "?g x = f x" using nx by auto}
-  hence thfg: "\<forall>x. norm x = 1 \<longrightarrow> ?g x = f x" by blast
+  {
+    fix x:: "real ^'n"
+    assume nx: "norm x = 1"
+    have "?g x = f x" using nx by auto
+  }
+  then have thfg: "\<forall>x. norm x = 1 \<longrightarrow> ?g x = f x"
+    by blast
have g0: "?g 0 = 0" by simp
-  {fix x y :: "real ^'n"
-    {assume "x = 0" "y = 0"
-      then have "dist (?g x) (?g y) = dist x y" by simp }
+  {
+    fix x y :: "real ^'n"
+    {
+      assume "x = 0" "y = 0"
+      then have "dist (?g x) (?g y) = dist x y" by simp
+    }
moreover
-    {assume "x = 0" "y \<noteq> 0"
+    {
+      assume "x = 0" "y \<noteq> 0"
then have "dist (?g x) (?g y) = dist x y"
apply (rule f1[rule_format])
+        done
+    }
moreover
-    {assume "x \<noteq> 0" "y = 0"
+    {
+      assume "x \<noteq> 0" "y = 0"
then have "dist (?g x) (?g y) = dist x y"
apply (rule f1[rule_format])
+        done
+    }
moreover
-    {assume z: "x \<noteq> 0" "y \<noteq> 0"
-      have th00: "x = norm x *\<^sub>R (inverse (norm x) *\<^sub>R x)" "y = norm y *\<^sub>R (inverse (norm y) *\<^sub>R y)" "norm x *\<^sub>R f ((inverse (norm x) *\<^sub>R x)) = norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)"
+    {
+      assume z: "x \<noteq> 0" "y \<noteq> 0"
+      have th00:
+        "x = norm x *\<^sub>R (inverse (norm x) *\<^sub>R x)"
+        "y = norm y *\<^sub>R (inverse (norm y) *\<^sub>R y)"
+        "norm x *\<^sub>R f ((inverse (norm x) *\<^sub>R x)) = norm x *\<^sub>R f (inverse (norm x) *\<^sub>R x)"
"norm y *\<^sub>R f (inverse (norm y) *\<^sub>R y) = norm y *\<^sub>R f (inverse (norm y) *\<^sub>R y)"
"norm (inverse (norm x) *\<^sub>R x) = 1"
"norm (f (inverse (norm x) *\<^sub>R x)) = 1"
"norm (inverse (norm y) *\<^sub>R y) = 1"
"norm (f (inverse (norm y) *\<^sub>R y)) = 1"
"norm (f (inverse (norm x) *\<^sub>R x) - f (inverse (norm y) *\<^sub>R y)) =
-        norm (inverse (norm x) *\<^sub>R x - inverse (norm y) *\<^sub>R y)"
+          norm (inverse (norm x) *\<^sub>R x - inverse (norm y) *\<^sub>R y)"
using z
by (auto simp add: field_simps intro: f1[rule_format] fd1[rule_format, unfolded dist_norm])
from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
-    ultimately have "dist (?g x) (?g y) = dist x y" by blast}
+    }
+    ultimately have "dist (?g x) (?g y) = dist x y" by blast
+  }
note thd = this
show ?thesis
apply (rule exI[where x= ?g])
unfolding orthogonal_transformation_isometry
-      using  g0 thfg thd by metis
+    using g0 thfg thd
+    apply metis
+    done
qed

(* ------------------------------------------------------------------------- *)
@@ -1029,14 +1214,17 @@
fixes Q :: "'a::linordered_idom^'n^'n"
shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
+
(* ------------------------------------------------------------------------- *)
(* Explicit formulas for low dimensions.                                     *)
(* ------------------------------------------------------------------------- *)

-lemma setprod_1: "setprod f {(1::nat)..1} = f 1" by simp
+lemma setprod_1: "setprod f {(1::nat)..1} = f 1"
+  by simp

lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2"
by (simp add: eval_nat_numeral setprod_numseg mult_commute)
+
lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3"
by (simp add: eval_nat_numeral setprod_numseg mult_commute)

@@ -1044,33 +1232,33 @@

lemma det_2: "det (A::'a::comm_ring_1^2^2) = A\$1\$1 * A\$2\$2 - A\$1\$2 * A\$2\$1"
-proof-
+proof -
have f12: "finite {2::2}" "1 \<notin> {2::2}" by auto
show ?thesis
-  unfolding det_def UNIV_2
-  unfolding setsum_over_permutations_insert[OF f12]
-  unfolding permutes_sing
-  by (simp add: sign_swap_id sign_id swap_id_eq)
+    unfolding det_def UNIV_2
+    unfolding setsum_over_permutations_insert[OF f12]
+    unfolding permutes_sing
+    by (simp add: sign_swap_id sign_id swap_id_eq)
qed

-lemma det_3: "det (A::'a::comm_ring_1^3^3) =
-  A\$1\$1 * A\$2\$2 * A\$3\$3 +
-  A\$1\$2 * A\$2\$3 * A\$3\$1 +
-  A\$1\$3 * A\$2\$1 * A\$3\$2 -
-  A\$1\$1 * A\$2\$3 * A\$3\$2 -
-  A\$1\$2 * A\$2\$1 * A\$3\$3 -
-  A\$1\$3 * A\$2\$2 * A\$3\$1"
-proof-
+lemma det_3:
+  "det (A::'a::comm_ring_1^3^3) =
+    A\$1\$1 * A\$2\$2 * A\$3\$3 +
+    A\$1\$2 * A\$2\$3 * A\$3\$1 +
+    A\$1\$3 * A\$2\$1 * A\$3\$2 -
+    A\$1\$1 * A\$2\$3 * A\$3\$2 -
+    A\$1\$2 * A\$2\$1 * A\$3\$3 -
+    A\$1\$3 * A\$2\$2 * A\$3\$1"
+proof -
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
-  unfolding det_def UNIV_3
-  unfolding setsum_over_permutations_insert[OF f123]
-  unfolding setsum_over_permutations_insert[OF f23]
-
-  unfolding permutes_sing
-  by (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
+    unfolding det_def UNIV_3
+    unfolding setsum_over_permutations_insert[OF f123]
+    unfolding setsum_over_permutations_insert[OF f23]
+    unfolding permutes_sing
+    by (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
qed

end```
```--- a/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Wed Aug 28 22:50:23 2013 +0200
+++ b/src/HOL/Multivariate_Analysis/Operator_Norm.thy	Wed Aug 28 23:41:21 2013 +0200
@@ -11,72 +11,83 @@
definition "onorm f = Sup {norm (f x)| x. norm x = 1}"

lemma norm_bound_generalize:
-  fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
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-
-  {assume H: ?rhs
-    {fix x :: "'a" assume x: "norm x = 1"
-      from H[rule_format, of x] x have "norm (f x) \<le> b" by simp}
-    then have ?lhs by blast }
+  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
+  assume H: ?rhs
+  {
+    fix x :: "'a"
+    assume x: "norm x = 1"
+    from H[rule_format, of x] x have "norm (f x) \<le> b" by simp
+  }
+  then show ?lhs by blast
+next
+  assume H: ?lhs
+  have bp: "b \<ge> 0"
+    apply -
+    apply (rule order_trans [OF norm_ge_zero])
+    apply (rule H[rule_format, of "SOME x::'a. x \<in> Basis"])
+    apply (auto intro: SOME_Basis norm_Basis)
+    done
+  {
+    fix x :: "'a"
+    {
+      assume "x = 0"
+      then have "norm (f x) \<le> b * norm x"
+        by (simp add: linear_0[OF lf] bp)
+    }
+    moreover
+    {
+      assume x0: "x \<noteq> 0"
+      then have n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
+      let ?c = "1/ norm x"
+      have "norm (?c *\<^sub>R x) = 1" using x0 by (simp add: n0)
+      with H have "norm (f (?c *\<^sub>R x)) \<le> b" by blast
+      then have "?c * norm (f x) \<le> b"
+        by (simp add: linear_cmul[OF lf])
+      then have "norm (f x) \<le> b * norm x"
+        using n0 norm_ge_zero[of x] by (auto simp add: field_simps)
+    }
+    ultimately have "norm (f x) \<le> b * norm x" by blast
+  }
+  then show ?rhs by blast
+qed

-  moreover
-  {assume H: ?lhs
-    have bp: "b \<ge> 0"
-      apply -
-      apply(rule order_trans [OF norm_ge_zero])
-      apply(rule H[rule_format, of "SOME x::'a. x \<in> Basis"])
-      by (auto intro: SOME_Basis norm_Basis)
-    {fix x :: "'a"
-      {assume "x = 0"
-        then have "norm (f x) \<le> b * norm x" by (simp add: linear_0[OF lf] bp)}
-      moreover
-      {assume x0: "x \<noteq> 0"
-        hence n0: "norm x \<noteq> 0" by (metis norm_eq_zero)
-        let ?c = "1/ norm x"
-        have "norm (?c *\<^sub>R x) = 1" using x0 by (simp add: n0)
-        with H have "norm (f (?c *\<^sub>R x)) \<le> b" by blast
-        hence "?c * norm (f x) \<le> b"
-          by (simp add: linear_cmul[OF lf])
-        hence "norm (f x) \<le> b * norm x"
-          using n0 norm_ge_zero[of x] by (auto simp add: field_simps)}
-      ultimately have "norm (f x) \<le> b * norm x" by blast}
-    then have ?rhs by blast}
-  ultimately show ?thesis by blast
-qed
-
lemma onorm:
fixes f:: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
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 "norm (f (SOME i. i \<in> Basis)) \<in> ?S"
-      by (auto intro!: exI[of _ "SOME i. i \<in> Basis"] norm_Basis SOME_Basis)
-    hence Se: "?S \<noteq> {}" 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 isLub_cSup[OF Se b, unfolded onorm_def[symmetric]]
-      show "norm (f x) <= onorm f * norm x"
-        apply -
-        apply (rule spec[where x = x])
-        unfolding norm_bound_generalize[OF lf, symmetric]
-        by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
-    {
-      show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
-        using isLub_cSup[OF Se b, unfolded onorm_def[symmetric]]
-        unfolding norm_bound_generalize[OF lf, symmetric]
-        by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)}
-  }
+  shows "norm (f x) \<le> onorm f * norm x"
+    and "\<forall>x. norm (f x) \<le> b * norm x \<Longrightarrow> onorm f \<le> b"
+proof -
+  let ?S = "{norm (f x) |x. norm x = 1}"
+  have "norm (f (SOME i. i \<in> Basis)) \<in> ?S"
+    by (auto intro!: exI[of _ "SOME i. i \<in> Basis"] norm_Basis SOME_Basis)
+  then have Se: "?S \<noteq> {}" 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 isLub_cSup[OF Se b, unfolded onorm_def[symmetric]]
+  show "norm (f x) <= onorm f * norm x"
+    apply -
+    apply (rule spec[where x = x])
+    unfolding norm_bound_generalize[OF lf, symmetric]
+    apply (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)
+    done
+  show "\<forall>x. norm (f x) <= b * norm x \<Longrightarrow> onorm f <= b"
+    using isLub_cSup[OF Se b, unfolded onorm_def[symmetric]]
+    unfolding norm_bound_generalize[OF lf, symmetric]
+    by (auto simp add: isLub_def isUb_def leastP_def setge_def setle_def)
qed

-lemma onorm_pos_le: assumes lf: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)" shows "0 <= onorm f"
-  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "SOME i. i \<in> Basis"]]
+lemma onorm_pos_le:
+  assumes lf: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)"
+  shows "0 \<le> onorm f"
+  using order_trans[OF norm_ge_zero onorm(1)[OF lf, of "SOME i. i \<in> Basis"]]

-lemma onorm_eq_0: assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
+lemma onorm_eq_0:
+  assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
shows "onorm f = 0 \<longleftrightarrow> (\<forall>x. f x = 0)"
using onorm[OF lf]
@@ -87,47 +98,53 @@
done

lemma onorm_const: "onorm(\<lambda>x::'a::euclidean_space. (y::'b::euclidean_space)) = norm y"
-proof-
+proof -
let ?f = "\<lambda>x::'a. (y::'b)"
have th: "{norm (?f x)| x. norm x = 1} = {norm y}"
by (auto simp: SOME_Basis intro!: exI[of _ "SOME i. i \<in> Basis"])
show ?thesis
unfolding onorm_def th
-    apply (rule cSup_unique) by (simp_all  add: setle_def)
+    apply (rule cSup_unique)
+    done
qed

-lemma onorm_pos_lt: assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
+lemma onorm_pos_lt:
+  assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
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::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)"
-  and lg: "linear (g::'k::euclidean_space \<Rightarrow> 'n::euclidean_space)"
-  shows "onorm (f o g) <= onorm f * onorm g"
-  apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
-  unfolding o_def
-  apply (subst mult_assoc)
-  apply (rule order_trans)
-  apply (rule onorm(1)[OF lf])
-  apply (rule mult_left_mono)
-  apply (rule onorm(1)[OF lg])
-  apply (rule onorm_pos_le[OF lf])
-  done
+    and lg: "linear (g::'k::euclidean_space \<Rightarrow> 'n::euclidean_space)"
+  shows "onorm (f o g) \<le> onorm f * onorm g"
+    apply (rule onorm(2)[OF linear_compose[OF lg lf], rule_format])
+    unfolding o_def
+    apply (subst mult_assoc)
+    apply (rule order_trans)
+    apply (rule onorm(1)[OF lf])
+    apply (rule mult_left_mono)
+    apply (rule onorm(1)[OF lg])
+    apply (rule onorm_pos_le[OF lf])
+    done

-lemma onorm_neg_lemma: assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
+lemma onorm_neg_lemma:
+  assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
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::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
+lemma onorm_neg:
+  assumes lf: "linear (f::'a::euclidean_space \<Rightarrow> 'b::euclidean_space)"
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::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)" and lg: "linear g"
-  shows "onorm (\<lambda>x. f x + g x) <= onorm f + onorm g"
+  assumes lf: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space)"
+    and lg: "linear g"
+  shows "onorm (\<lambda>x. f x + g x) \<le> onorm f + onorm g"
apply(rule onorm(2)[OF linear_compose_add[OF lf lg], rule_format])
apply (rule order_trans)
apply (rule norm_triangle_ineq)
@@ -137,17 +154,20 @@
apply (rule onorm(1)[OF lg])
done

-lemma onorm_triangle_le: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) <= e
-  \<Longrightarrow> onorm(\<lambda>x. f x + g x) <= e"
+lemma onorm_triangle_le:
+  "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space) \<Longrightarrow>
+    linear g \<Longrightarrow> onorm f + onorm g \<le> e \<Longrightarrow> onorm (\<lambda>x. f x + g x) \<le> e"
apply (rule order_trans)
apply (rule onorm_triangle)
apply assumption+
done

-lemma onorm_triangle_lt: "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space) \<Longrightarrow> linear g \<Longrightarrow> onorm(f) + onorm(g) < e
-  ==> onorm(\<lambda>x. f x + g x) < e"
+lemma onorm_triangle_lt:
+  "linear (f::'n::euclidean_space \<Rightarrow> 'm::euclidean_space) \<Longrightarrow> linear g \<Longrightarrow>
+    onorm f + onorm g < e \<Longrightarrow> onorm(\<lambda>x. f x + g x) < e"
apply (rule order_le_less_trans)
apply (rule onorm_triangle)
-  by assumption+
+  apply assumption+
+  done

end```