author immler Wed, 16 Jan 2019 18:14:02 -0500 changeset 69675 880ab0f27ddf parent 69674 fc252acb7100 child 69676 56acd449da41
Reorg, in particular Determinants as well as some linear algebra from Starlike and Change_Of_Vars
 src/HOL/Analysis/Cartesian_Space.thy file | annotate | diff | comparison | revisions src/HOL/Analysis/Change_Of_Vars.thy file | annotate | diff | comparison | revisions src/HOL/Analysis/Convex.thy file | annotate | diff | comparison | revisions src/HOL/Analysis/Cross3.thy file | annotate | diff | comparison | revisions src/HOL/Analysis/Determinants.thy file | annotate | diff | comparison | revisions src/HOL/Analysis/Linear_Algebra.thy file | annotate | diff | comparison | revisions src/HOL/Analysis/Starlike.thy file | annotate | diff | comparison | revisions
```--- a/src/HOL/Analysis/Cartesian_Space.thy	Wed Jan 16 16:50:35 2019 -0500
+++ b/src/HOL/Analysis/Cartesian_Space.thy	Wed Jan 16 18:14:02 2019 -0500
@@ -930,4 +930,492 @@
lemma%unimportant const_vector_cart:"((\<chi> i. d)::real^'n) = (\<Sum>i\<in>Basis. d *\<^sub>R i)"
by (rule vector_cart)

+subsection%unimportant \<open>Explicit formulas for low dimensions\<close>
+
+lemma%unimportant  prod_neutral_const: "prod f {(1::nat)..1} = f 1"
+  by simp
+
+lemma%unimportant  prod_2: "prod f {(1::nat)..2} = f 1 * f 2"
+  by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)
+
+lemma%unimportant  prod_3: "prod f {(1::nat)..3} = f 1 * f 2 * f 3"
+  by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)
+
+
+subsection%important  \<open>Orthogonality of a matrix\<close>
+
+definition%important  "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow>
+  transpose Q ** Q = mat 1 \<and> Q ** transpose Q = mat 1"
+
+lemma%unimportant  orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1"
+  by (metis matrix_left_right_inverse orthogonal_matrix_def)
+
+lemma%unimportant  orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"
+
+lemma%unimportant  orthogonal_matrix_mul:
+  fixes A :: "real ^'n^'n"
+  assumes  "orthogonal_matrix A" "orthogonal_matrix B"
+  shows "orthogonal_matrix(A ** B)"
+  using assms
+  by (simp add: orthogonal_matrix matrix_transpose_mul matrix_left_right_inverse matrix_mul_assoc)
+
+lemma%important  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%unimportant -
+  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
+    from ot have lf: "Vector_Spaces.linear (*s) (*s) f" and fd: "\<And>v w. f v \<bullet> f w = v \<bullet> w"
+      unfolding orthogonal_transformation_def orthogonal_matrix linear_def scalar_mult_eq_scaleR
+      by blast+
+    {
+      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)"
+        by simp_all
+      from fd[of "axis i 1" "axis j 1",
+        simplified 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 sum.delta[OF fU] mat_def axis_def)
+    }
+    then have "orthogonal_matrix ?mf"
+      unfolding orthogonal_matrix
+      by vector
+    with lf have ?rhs
+      unfolding linear_def scalar_mult_eq_scaleR
+      by blast
+  }
+  moreover
+  {
+    assume lf: "Vector_Spaces.linear (*s) (*s) f" and om: "orthogonal_matrix ?mf"
+    from lf om have ?lhs
+      unfolding orthogonal_matrix_def norm_eq orthogonal_transformation
+      apply (simp only: matrix_works[OF lf, symmetric] dot_matrix_vector_mul)
+      apply (simp add: dot_matrix_product linear_def scalar_mult_eq_scaleR)
+      done
+  }
+  ultimately show ?thesis
+    by (auto simp: linear_def scalar_mult_eq_scaleR)
+qed
+
+
+subsection%important \<open> We can find an orthogonal matrix taking any unit vector to any other\<close>
+
+lemma%unimportant  orthogonal_matrix_transpose [simp]:
+     "orthogonal_matrix(transpose A) \<longleftrightarrow> orthogonal_matrix A"
+  by (auto simp: orthogonal_matrix_def)
+
+lemma%unimportant  orthogonal_matrix_orthonormal_columns:
+  fixes A :: "real^'n^'n"
+  shows "orthogonal_matrix A \<longleftrightarrow>
+          (\<forall>i. norm(column i A) = 1) \<and>
+          (\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (column i A) (column j A))"
+  by (auto simp: orthogonal_matrix matrix_mult_transpose_dot_column vec_eq_iff mat_def norm_eq_1 orthogonal_def)
+
+lemma%unimportant  orthogonal_matrix_orthonormal_rows:
+  fixes A :: "real^'n^'n"
+  shows "orthogonal_matrix A \<longleftrightarrow>
+          (\<forall>i. norm(row i A) = 1) \<and>
+          (\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (row i A) (row j A))"
+  using orthogonal_matrix_orthonormal_columns [of "transpose A"] by simp
+
+lemma%important  orthogonal_matrix_exists_basis:
+  fixes a :: "real^'n"
+  assumes "norm a = 1"
+  obtains A where "orthogonal_matrix A" "A *v (axis k 1) = a"
+proof%unimportant -
+  obtain S where "a \<in> S" "pairwise orthogonal S" and noS: "\<And>x. x \<in> S \<Longrightarrow> norm x = 1"
+   and "independent S" "card S = CARD('n)" "span S = UNIV"
+    using vector_in_orthonormal_basis assms by force
+  then obtain f0 where "bij_betw f0 (UNIV::'n set) S"
+    by (metis finite_class.finite_UNIV finite_same_card_bij finiteI_independent)
+  then obtain f where f: "bij_betw f (UNIV::'n set) S" and a: "a = f k"
+    using bij_swap_iff [of k "inv f0 a" f0]
+    by (metis UNIV_I \<open>a \<in> S\<close> bij_betw_inv_into_right bij_betw_swap_iff swap_apply(1))
+  show thesis
+  proof
+    have [simp]: "\<And>i. norm (f i) = 1"
+      using bij_betwE [OF \<open>bij_betw f UNIV S\<close>] by (blast intro: noS)
+    have [simp]: "\<And>i j. i \<noteq> j \<Longrightarrow> orthogonal (f i) (f j)"
+      using \<open>pairwise orthogonal S\<close> \<open>bij_betw f UNIV S\<close>
+      by (auto simp: pairwise_def bij_betw_def inj_on_def)
+    show "orthogonal_matrix (\<chi> i j. f j \$ i)"
+      by (simp add: orthogonal_matrix_orthonormal_columns column_def)
+    show "(\<chi> i j. f j \$ i) *v axis k 1 = a"
+      by (simp add: matrix_vector_mult_def axis_def a if_distrib cong: if_cong)
+  qed
+qed
+
+lemma%unimportant  orthogonal_transformation_exists_1:
+  fixes a b :: "real^'n"
+  assumes "norm a = 1" "norm b = 1"
+  obtains f where "orthogonal_transformation f" "f a = b"
+proof%unimportant -
+  obtain k::'n where True
+    by simp
+  obtain A B where AB: "orthogonal_matrix A" "orthogonal_matrix B" and eq: "A *v (axis k 1) = a" "B *v (axis k 1) = b"
+    using orthogonal_matrix_exists_basis assms by metis
+  let ?f = "\<lambda>x. (B ** transpose A) *v x"
+  show thesis
+  proof
+    show "orthogonal_transformation ?f"
+      by (subst orthogonal_transformation_matrix)
+        (auto simp: AB orthogonal_matrix_mul)
+  next
+    show "?f a = b"
+      using \<open>orthogonal_matrix A\<close> unfolding orthogonal_matrix_def
+      by (metis eq matrix_mul_rid matrix_vector_mul_assoc)
+  qed
+qed
+
+lemma%important  orthogonal_transformation_exists:
+  fixes a b :: "real^'n"
+  assumes "norm a = norm b"
+  obtains f where "orthogonal_transformation f" "f a = b"
+proof%unimportant (cases "a = 0 \<or> b = 0")
+  case True
+  with assms show ?thesis
+    using that by force
+next
+  case False
+  then obtain f where f: "orthogonal_transformation f" and eq: "f (a /\<^sub>R norm a) = (b /\<^sub>R norm b)"
+    by (auto intro: orthogonal_transformation_exists_1 [of "a /\<^sub>R norm a" "b /\<^sub>R norm b"])
+  show ?thesis
+  proof
+    interpret linear f
+      using f by (simp add: orthogonal_transformation_linear)
+    have "f a /\<^sub>R norm a = f (a /\<^sub>R norm a)"
+    also have "\<dots> = b /\<^sub>R norm a"
+      by (simp add: eq assms [symmetric])
+    finally show "f a = b"
+      using False by auto
+  qed (use f in auto)
+qed
+
+
+subsection%important  \<open>Linearity of scaling, and hence isometry, that preserves origin\<close>
+
+lemma%important  scaling_linear:
+  fixes f :: "'a::real_inner \<Rightarrow> 'a::real_inner"
+  assumes f0: "f 0 = 0"
+    and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
+  shows "linear f"
+proof%unimportant -
+  {
+    fix v w
+    have "norm (f x) = c * norm x" for x
+      by (metis dist_0_norm f0 fd)
+    then have "f v \<bullet> f w = c\<^sup>2 * (v \<bullet> w)"
+      unfolding dot_norm_neg dist_norm[symmetric]
+      by (simp add: fd power2_eq_square field_simps)
+  }
+  then show ?thesis
+    unfolding linear_iff vector_eq[where 'a="'a"] scalar_mult_eq_scaleR
+qed
+
+lemma%unimportant  isometry_linear:
+  "f (0::'a::real_inner) = (0::'a) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y \<Longrightarrow> linear f"
+  by (rule scaling_linear[where c=1]) simp_all
+
+text \<open>Hence another formulation of orthogonal transformation\<close>
+
+lemma%important  orthogonal_transformation_isometry:
+  "orthogonal_transformation f \<longleftrightarrow> f(0::'a::real_inner) = (0::'a) \<and> (\<forall>x y. dist(f x) (f y) = dist x y)"
+  unfolding orthogonal_transformation
+  apply (auto simp: linear_0 isometry_linear)
+   apply (metis (no_types, hide_lams) dist_norm linear_diff)
+  by (metis dist_0_norm)
+
+
+subsection%important  \<open>Can extend an isometry from unit sphere\<close>
+
+lemma%important  isometry_sphere_extend:
+  fixes f:: "'a::real_inner \<Rightarrow> 'a"
+  assumes f1: "\<And>x. norm x = 1 \<Longrightarrow> norm (f x) = 1"
+    and fd1: "\<And>x y. \<lbrakk>norm x = 1; norm y = 1\<rbrakk> \<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%unimportant -
+  {
+    fix x y x' y' u v u' v' :: "'a"
+    assume H: "x = norm x *\<^sub>R u" "y = norm y *\<^sub>R v"
+              "x' = norm x *\<^sub>R u'" "y' = norm y *\<^sub>R v'"
+      and J: "norm u = 1" "norm u' = 1" "norm v = 1" "norm v' = 1" "norm(u' - v') = norm(u - v)"
+    then have *: "u \<bullet> v = u' \<bullet> v' + v' \<bullet> u' - v \<bullet> u "
+    have "norm (norm x *\<^sub>R u' - norm y *\<^sub>R v') = norm (norm x *\<^sub>R u - norm y *\<^sub>R v)"
+      using J by (simp add: norm_eq norm_eq_1 inner_diff * field_simps)
+    then have "norm(x' - y') = norm(x - y)"
+      using H by metis
+  }
+  note norm_eq = this
+  let ?g = "\<lambda>x. if x = 0 then 0 else norm x *\<^sub>R f (x /\<^sub>R norm x)"
+  have thfg: "?g x = f x" if "norm x = 1" for x
+    using that by auto
+  have thd: "dist (?g x) (?g y) = dist x y" for x y
+  proof (cases "x=0 \<or> y=0")
+    case False
+    show "dist (?g x) (?g y) = dist x y"
+      unfolding dist_norm
+    proof (rule norm_eq)
+      show "x = norm x *\<^sub>R (x /\<^sub>R norm x)" "y = norm y *\<^sub>R (y /\<^sub>R norm y)"
+           "norm (f (x /\<^sub>R norm x)) = 1" "norm (f (y /\<^sub>R norm y)) = 1"
+        using False f1 by auto
+    qed (use False in \<open>auto simp: field_simps intro: f1 fd1[unfolded dist_norm]\<close>)
+  qed (auto simp: f1)
+  show ?thesis
+    unfolding orthogonal_transformation_isometry
+    by (rule exI[where x= ?g]) (metis thfg thd)
+qed
+
+subsection%important\<open>Induction on matrix row operations\<close>
+
+lemma%unimportant induct_matrix_row_operations:
+  fixes P :: "real^'n^'n \<Rightarrow> bool"
+  assumes zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
+    and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A\$i\$j = 0) \<Longrightarrow> P A"
+    and swap_cols: "\<And>A m n. \<lbrakk>P A; m \<noteq> n\<rbrakk> \<Longrightarrow> P(\<chi> i j. A \$ i \$ Fun.swap m n id j)"
+    and row_op: "\<And>A m n c. \<lbrakk>P A; m \<noteq> n\<rbrakk>
+                   \<Longrightarrow> P(\<chi> i. if i = m then row m A + c *\<^sub>R row n A else row i A)"
+  shows "P A"
+proof -
+  have "P A" if "(\<And>i j. \<lbrakk>j \<in> -K;  i \<noteq> j\<rbrakk> \<Longrightarrow> A\$i\$j = 0)" for A K
+  proof -
+    have "finite K"
+      by simp
+    then show ?thesis using that
+    proof (induction arbitrary: A rule: finite_induct)
+      case empty
+      with diagonal show ?case
+        by simp
+    next
+      case (insert k K)
+      note insertK = insert
+      have "P A" if kk: "A\$k\$k \<noteq> 0"
+        and 0: "\<And>i j. \<lbrakk>j \<in> - insert k K; i \<noteq> j\<rbrakk> \<Longrightarrow> A\$i\$j = 0"
+               "\<And>i. \<lbrakk>i \<in> -L; i \<noteq> k\<rbrakk> \<Longrightarrow> A\$i\$k = 0" for A L
+      proof -
+        have "finite L"
+          by simp
+        then show ?thesis using 0 kk
+        proof (induction arbitrary: A rule: finite_induct)
+          case (empty B)
+          show ?case
+          proof (rule insertK)
+            fix i j
+            assume "i \<in> - K" "j \<noteq> i"
+            show "B \$ j \$ i = 0"
+              using \<open>j \<noteq> i\<close> \<open>i \<in> - K\<close> empty
+              by (metis ComplD ComplI Compl_eq_Diff_UNIV Diff_empty UNIV_I insert_iff)
+          qed
+        next
+          case (insert l L B)
+          show ?case
+          proof (cases "k = l")
+            case True
+            with insert show ?thesis
+              by auto
+          next
+            case False
+            let ?C = "\<chi> i. if i = l then row l B - (B \$ l \$ k / B \$ k \$ k) *\<^sub>R row k B else row i B"
+            have 1: "\<lbrakk>j \<in> - insert k K; i \<noteq> j\<rbrakk> \<Longrightarrow> ?C \$ i \$ j = 0" for j i
+              by (auto simp: insert.prems(1) row_def)
+            have 2: "?C \$ i \$ k = 0"
+              if "i \<in> - L" "i \<noteq> k" for i
+            proof (cases "i=l")
+              case True
+              with that insert.prems show ?thesis
+            next
+              case False
+              with that show ?thesis
+                by (simp add: insert.prems(2) row_def)
+            qed
+            have 3: "?C \$ k \$ k \<noteq> 0"
+              by (auto simp: insert.prems row_def \<open>k \<noteq> l\<close>)
+            have PC: "P ?C"
+              using insert.IH [OF 1 2 3] by auto
+            have eqB: "(\<chi> i. if i = l then row l ?C + (B \$ l \$ k / B \$ k \$ k) *\<^sub>R row k ?C else row i ?C) = B"
+              using \<open>k \<noteq> l\<close> by (simp add: vec_eq_iff row_def)
+            show ?thesis
+              using row_op [OF PC, of l k, where c = "B\$l\$k / B\$k\$k"] eqB \<open>k \<noteq> l\<close>
+              by (simp add: cong: if_cong)
+          qed
+        qed
+      qed
+      then have nonzero_hyp: "P A"
+        if kk: "A\$k\$k \<noteq> 0" and zeroes: "\<And>i j. j \<in> - insert k K \<and> i\<noteq>j \<Longrightarrow> A\$i\$j = 0" for A
+        by (auto simp: intro!: kk zeroes)
+      show ?case
+      proof (cases "row k A = 0")
+        case True
+        with zero_row show ?thesis by auto
+      next
+        case False
+        then obtain l where l: "A\$k\$l \<noteq> 0"
+          by (auto simp: row_def zero_vec_def vec_eq_iff)
+        show ?thesis
+        proof (cases "k = l")
+          case True
+          with l nonzero_hyp insert.prems show ?thesis
+            by blast
+        next
+          case False
+          have *: "A \$ i \$ Fun.swap k l id j = 0" if "j \<noteq> k" "j \<notin> K" "i \<noteq> j" for i j
+            using False l insert.prems that
+            by (auto simp: swap_def insert split: if_split_asm)
+          have "P (\<chi> i j. (\<chi> i j. A \$ i \$ Fun.swap k l id j) \$ i \$ Fun.swap k l id j)"
+            by (rule swap_cols [OF nonzero_hyp False]) (auto simp: l *)
+          moreover
+          have "(\<chi> i j. (\<chi> i j. A \$ i \$ Fun.swap k l id j) \$ i \$ Fun.swap k l id j) = A"
+            by (vector Fun.swap_def)
+          ultimately show ?thesis
+            by simp
+        qed
+      qed
+    qed
+  qed
+  then show ?thesis
+    by blast
+qed
+
+lemma%unimportant induct_matrix_elementary:
+  fixes P :: "real^'n^'n \<Rightarrow> bool"
+  assumes mult: "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P(A ** B)"
+    and zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
+    and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A\$i\$j = 0) \<Longrightarrow> P A"
+    and swap1: "\<And>m n. m \<noteq> n \<Longrightarrow> P(\<chi> i j. mat 1 \$ i \$ Fun.swap m n id j)"
+    and idplus: "\<And>m n c. m \<noteq> n \<Longrightarrow> P(\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j))"
+  shows "P A"
+proof -
+  have swap: "P (\<chi> i j. A \$ i \$ Fun.swap m n id j)"  (is "P ?C")
+    if "P A" "m \<noteq> n" for A m n
+  proof -
+    have "A ** (\<chi> i j. mat 1 \$ i \$ Fun.swap m n id j) = ?C"
+      by (simp add: matrix_matrix_mult_def mat_def vec_eq_iff if_distrib sum.delta_remove)
+    then show ?thesis
+      using mult swap1 that by metis
+  qed
+  have row: "P (\<chi> i. if i = m then row m A + c *\<^sub>R row n A else row i A)"  (is "P ?C")
+    if "P A" "m \<noteq> n" for A m n c
+  proof -
+    let ?B = "\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j)"
+    have "?B ** A = ?C"
+      using \<open>m \<noteq> n\<close> unfolding matrix_matrix_mult_def row_def of_bool_def
+      by (auto simp: vec_eq_iff if_distrib [of "\<lambda>x. x * y" for y] sum.remove cong: if_cong)
+    then show ?thesis
+      by (rule subst) (auto simp: that mult idplus)
+  qed
+  show ?thesis
+    by (rule induct_matrix_row_operations [OF zero_row diagonal swap row])
+qed
+
+lemma%unimportant induct_matrix_elementary_alt:
+  fixes P :: "real^'n^'n \<Rightarrow> bool"
+  assumes mult: "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P(A ** B)"
+    and zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
+    and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A\$i\$j = 0) \<Longrightarrow> P A"
+    and swap1: "\<And>m n. m \<noteq> n \<Longrightarrow> P(\<chi> i j. mat 1 \$ i \$ Fun.swap m n id j)"
+    and idplus: "\<And>m n. m \<noteq> n \<Longrightarrow> P(\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j))"
+  shows "P A"
+proof -
+  have *: "P (\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j))"
+    if "m \<noteq> n" for m n c
+  proof (cases "c = 0")
+    case True
+    with diagonal show ?thesis by auto
+  next
+    case False
+    then have eq: "(\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j)) =
+                      (\<chi> i j. if i = j then (if j = n then inverse c else 1) else 0) **
+                      (\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j)) **
+                      (\<chi> i j. if i = j then if j = n then c else 1 else 0)"
+      using \<open>m \<noteq> n\<close>
+      apply (simp add: matrix_matrix_mult_def vec_eq_iff of_bool_def if_distrib [of "\<lambda>x. y * x" for y] cong: if_cong)
+      apply (simp add: if_if_eq_conj sum.neutral conj_commute cong: conj_cong)
+      done
+    show ?thesis
+      apply (subst eq)
+      apply (intro mult idplus that)
+       apply (auto intro: diagonal)
+      done
+  qed
+  show ?thesis
+    by (rule induct_matrix_elementary) (auto intro: assms *)
+qed
+
+lemma%unimportant matrix_vector_mult_matrix_matrix_mult_compose:
+  "(*v) (A ** B) = (*v) A \<circ> (*v) B"
+  by (auto simp: matrix_vector_mul_assoc)
+
+lemma%unimportant induct_linear_elementary:
+  fixes f :: "real^'n \<Rightarrow> real^'n"
+  assumes "linear f"
+    and comp: "\<And>f g. \<lbrakk>linear f; linear g; P f; P g\<rbrakk> \<Longrightarrow> P(f \<circ> g)"
+    and zeroes: "\<And>f i. \<lbrakk>linear f; \<And>x. (f x) \$ i = 0\<rbrakk> \<Longrightarrow> P f"
+    and const: "\<And>c. P(\<lambda>x. \<chi> i. c i * x\$i)"
+    and swap: "\<And>m n::'n. m \<noteq> n \<Longrightarrow> P(\<lambda>x. \<chi> i. x \$ Fun.swap m n id i)"
+    and idplus: "\<And>m n::'n. m \<noteq> n \<Longrightarrow> P(\<lambda>x. \<chi> i. if i = m then x\$m + x\$n else x\$i)"
+  shows "P f"
+proof -
+  have "P ((*v) A)" for A
+  proof (rule induct_matrix_elementary_alt)
+    fix A B
+    assume "P ((*v) A)" and "P ((*v) B)"
+    then show "P ((*v) (A ** B))"
+      by (auto simp add: matrix_vector_mult_matrix_matrix_mult_compose matrix_vector_mul_linear
+          intro!: comp)
+  next
+    fix A :: "real^'n^'n" and i
+    assume "row i A = 0"
+    show "P ((*v) A)"
+      using matrix_vector_mul_linear
+      by (rule zeroes[where i=i])
+        (metis \<open>row i A = 0\<close> inner_zero_left matrix_vector_mul_component row_def vec_lambda_eta)
+  next
+    fix A :: "real^'n^'n"
+    assume 0: "\<And>i j. i \<noteq> j \<Longrightarrow> A \$ i \$ j = 0"
+    have "A \$ i \$ i * x \$ i = (\<Sum>j\<in>UNIV. A \$ i \$ j * x \$ j)" for x and i :: "'n"
+    then have "(\<lambda>x. \<chi> i. A \$ i \$ i * x \$ i) = ((*v) A)"
+      by (auto simp: 0 matrix_vector_mult_def)
+    then show "P ((*v) A)"
+      using const [of "\<lambda>i. A \$ i \$ i"] by simp
+  next
+    fix m n :: "'n"
+    assume "m \<noteq> n"
+    have eq: "(\<Sum>j\<in>UNIV. if i = Fun.swap m n id j then x \$ j else 0) =
+              (\<Sum>j\<in>UNIV. if j = Fun.swap m n id i then x \$ j else 0)"
+      for i and x :: "real^'n"
+      unfolding swap_def by (rule sum.cong) auto
+    have "(\<lambda>x::real^'n. \<chi> i. x \$ Fun.swap m n id i) = ((*v) (\<chi> i j. if i = Fun.swap m n id j then 1 else 0))"
+      by (auto simp: mat_def matrix_vector_mult_def eq if_distrib [of "\<lambda>x. x * y" for y] cong: if_cong)
+    with swap [OF \<open>m \<noteq> n\<close>] show "P ((*v) (\<chi> i j. mat 1 \$ i \$ Fun.swap m n id j))"
+      by (simp add: mat_def matrix_vector_mult_def)
+  next
+    fix m n :: "'n"
+    assume "m \<noteq> n"
+    then have "x \$ m + x \$ n = (\<Sum>j\<in>UNIV. of_bool (j = n \<or> m = j) * x \$ j)" for x :: "real^'n"
+      by (auto simp: of_bool_def if_distrib [of "\<lambda>x. x * y" for y] sum.remove cong: if_cong)
+    then have "(\<lambda>x::real^'n. \<chi> i. if i = m then x \$ m + x \$ n else x \$ i) =
+               ((*v) (\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j)))"
+      unfolding matrix_vector_mult_def of_bool_def
+      by (auto simp: vec_eq_iff if_distrib [of "\<lambda>x. x * y" for y] cong: if_cong)
+    then show "P ((*v) (\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j)))"
+      using idplus [OF \<open>m \<noteq> n\<close>] by simp
+  qed
+  then show ?thesis
+    by (metis \<open>linear f\<close> matrix_vector_mul)
+qed
+
end
\ No newline at end of file```
```--- a/src/HOL/Analysis/Change_Of_Vars.thy	Wed Jan 16 16:50:35 2019 -0500
+++ b/src/HOL/Analysis/Change_Of_Vars.thy	Wed Jan 16 18:14:02 2019 -0500
@@ -9,246 +9,42 @@

begin

-subsection%important\<open>Induction on matrix row operations\<close>
+subsection%unimportant \<open>Orthogonal Transformation of Balls\<close>

-lemma%unimportant induct_matrix_row_operations:
-  fixes P :: "real^'n^'n \<Rightarrow> bool"
-  assumes zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
-    and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A\$i\$j = 0) \<Longrightarrow> P A"
-    and swap_cols: "\<And>A m n. \<lbrakk>P A; m \<noteq> n\<rbrakk> \<Longrightarrow> P(\<chi> i j. A \$ i \$ Fun.swap m n id j)"
-    and row_op: "\<And>A m n c. \<lbrakk>P A; m \<noteq> n\<rbrakk>
-                   \<Longrightarrow> P(\<chi> i. if i = m then row m A + c *\<^sub>R row n A else row i A)"
-  shows "P A"
-proof -
-  have "P A" if "(\<And>i j. \<lbrakk>j \<in> -K;  i \<noteq> j\<rbrakk> \<Longrightarrow> A\$i\$j = 0)" for A K
-  proof -
-    have "finite K"
-      by simp
-    then show ?thesis using that
-    proof (induction arbitrary: A rule: finite_induct)
-      case empty
-      with diagonal show ?case
-        by simp
-    next
-      case (insert k K)
-      note insertK = insert
-      have "P A" if kk: "A\$k\$k \<noteq> 0"
-        and 0: "\<And>i j. \<lbrakk>j \<in> - insert k K; i \<noteq> j\<rbrakk> \<Longrightarrow> A\$i\$j = 0"
-               "\<And>i. \<lbrakk>i \<in> -L; i \<noteq> k\<rbrakk> \<Longrightarrow> A\$i\$k = 0" for A L
-      proof -
-        have "finite L"
-          by simp
-        then show ?thesis using 0 kk
-        proof (induction arbitrary: A rule: finite_induct)
-          case (empty B)
-          show ?case
-          proof (rule insertK)
-            fix i j
-            assume "i \<in> - K" "j \<noteq> i"
-            show "B \$ j \$ i = 0"
-              using \<open>j \<noteq> i\<close> \<open>i \<in> - K\<close> empty
-              by (metis ComplD ComplI Compl_eq_Diff_UNIV Diff_empty UNIV_I insert_iff)
-          qed
-        next
-          case (insert l L B)
-          show ?case
-          proof (cases "k = l")
-            case True
-            with insert show ?thesis
-              by auto
-          next
-            case False
-            let ?C = "\<chi> i. if i = l then row l B - (B \$ l \$ k / B \$ k \$ k) *\<^sub>R row k B else row i B"
-            have 1: "\<lbrakk>j \<in> - insert k K; i \<noteq> j\<rbrakk> \<Longrightarrow> ?C \$ i \$ j = 0" for j i
-              by (auto simp: insert.prems(1) row_def)
-            have 2: "?C \$ i \$ k = 0"
-              if "i \<in> - L" "i \<noteq> k" for i
-            proof (cases "i=l")
-              case True
-              with that insert.prems show ?thesis
-            next
-              case False
-              with that show ?thesis
-                by (simp add: insert.prems(2) row_def)
-            qed
-            have 3: "?C \$ k \$ k \<noteq> 0"
-              by (auto simp: insert.prems row_def \<open>k \<noteq> l\<close>)
-            have PC: "P ?C"
-              using insert.IH [OF 1 2 3] by auto
-            have eqB: "(\<chi> i. if i = l then row l ?C + (B \$ l \$ k / B \$ k \$ k) *\<^sub>R row k ?C else row i ?C) = B"
-              using \<open>k \<noteq> l\<close> by (simp add: vec_eq_iff row_def)
-            show ?thesis
-              using row_op [OF PC, of l k, where c = "B\$l\$k / B\$k\$k"] eqB \<open>k \<noteq> l\<close>
-              by (simp add: cong: if_cong)
-          qed
-        qed
-      qed
-      then have nonzero_hyp: "P A"
-        if kk: "A\$k\$k \<noteq> 0" and zeroes: "\<And>i j. j \<in> - insert k K \<and> i\<noteq>j \<Longrightarrow> A\$i\$j = 0" for A
-        by (auto simp: intro!: kk zeroes)
-      show ?case
-      proof (cases "row k A = 0")
-        case True
-        with zero_row show ?thesis by auto
-      next
-        case False
-        then obtain l where l: "A\$k\$l \<noteq> 0"
-          by (auto simp: row_def zero_vec_def vec_eq_iff)
-        show ?thesis
-        proof (cases "k = l")
-          case True
-          with l nonzero_hyp insert.prems show ?thesis
-            by blast
-        next
-          case False
-          have *: "A \$ i \$ Fun.swap k l id j = 0" if "j \<noteq> k" "j \<notin> K" "i \<noteq> j" for i j
-            using False l insert.prems that
-            by (auto simp: swap_def insert split: if_split_asm)
-          have "P (\<chi> i j. (\<chi> i j. A \$ i \$ Fun.swap k l id j) \$ i \$ Fun.swap k l id j)"
-            by (rule swap_cols [OF nonzero_hyp False]) (auto simp: l *)
-          moreover
-          have "(\<chi> i j. (\<chi> i j. A \$ i \$ Fun.swap k l id j) \$ i \$ Fun.swap k l id j) = A"
-            by (metis (no_types, lifting) id_apply o_apply swap_id_idempotent vec_lambda_unique vec_lambda_unique)
-          ultimately show ?thesis
-            by simp
-        qed
-      qed
-    qed
-  qed
-  then show ?thesis
-    by blast
+lemma%unimportant image_orthogonal_transformation_ball:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
+  assumes "orthogonal_transformation f"
+  shows "f ` ball x r = ball (f x) r"
+proof (intro equalityI subsetI)
+  fix y assume "y \<in> f ` ball x r"
+  with assms show "y \<in> ball (f x) r"
+    by (auto simp: orthogonal_transformation_isometry)
+next
+  fix y assume y: "y \<in> ball (f x) r"
+  then obtain z where z: "y = f z"
+    using assms orthogonal_transformation_surj by blast
+  with y assms show "y \<in> f ` ball x r"
+    by (auto simp: orthogonal_transformation_isometry)
qed

-lemma%unimportant induct_matrix_elementary:
-  fixes P :: "real^'n^'n \<Rightarrow> bool"
-  assumes mult: "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P(A ** B)"
-    and zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
-    and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A\$i\$j = 0) \<Longrightarrow> P A"
-    and swap1: "\<And>m n. m \<noteq> n \<Longrightarrow> P(\<chi> i j. mat 1 \$ i \$ Fun.swap m n id j)"
-    and idplus: "\<And>m n c. m \<noteq> n \<Longrightarrow> P(\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j))"
-  shows "P A"
-proof -
-  have swap: "P (\<chi> i j. A \$ i \$ Fun.swap m n id j)"  (is "P ?C")
-    if "P A" "m \<noteq> n" for A m n
-  proof -
-    have "A ** (\<chi> i j. mat 1 \$ i \$ Fun.swap m n id j) = ?C"
-      by (simp add: matrix_matrix_mult_def mat_def vec_eq_iff if_distrib sum.delta_remove)
-    then show ?thesis
-      using mult swap1 that by metis
-  qed
-  have row: "P (\<chi> i. if i = m then row m A + c *\<^sub>R row n A else row i A)"  (is "P ?C")
-    if "P A" "m \<noteq> n" for A m n c
-  proof -
-    let ?B = "\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j)"
-    have "?B ** A = ?C"
-      using \<open>m \<noteq> n\<close> unfolding matrix_matrix_mult_def row_def of_bool_def
-      by (auto simp: vec_eq_iff if_distrib [of "\<lambda>x. x * y" for y] sum.remove cong: if_cong)
-    then show ?thesis
-      by (rule subst) (auto simp: that mult idplus)
-  qed
-  show ?thesis
-    by (rule induct_matrix_row_operations [OF zero_row diagonal swap row])
-qed
-
-lemma%unimportant induct_matrix_elementary_alt:
-  fixes P :: "real^'n^'n \<Rightarrow> bool"
-  assumes mult: "\<And>A B. \<lbrakk>P A; P B\<rbrakk> \<Longrightarrow> P(A ** B)"
-    and zero_row: "\<And>A i. row i A = 0 \<Longrightarrow> P A"
-    and diagonal: "\<And>A. (\<And>i j. i \<noteq> j \<Longrightarrow> A\$i\$j = 0) \<Longrightarrow> P A"
-    and swap1: "\<And>m n. m \<noteq> n \<Longrightarrow> P(\<chi> i j. mat 1 \$ i \$ Fun.swap m n id j)"
-    and idplus: "\<And>m n. m \<noteq> n \<Longrightarrow> P(\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j))"
-  shows "P A"
-proof -
-  have *: "P (\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j))"
-    if "m \<noteq> n" for m n c
-  proof (cases "c = 0")
-    case True
-    with diagonal show ?thesis by auto
-  next
-    case False
-    then have eq: "(\<chi> i j. if i = m \<and> j = n then c else of_bool (i = j)) =
-                      (\<chi> i j. if i = j then (if j = n then inverse c else 1) else 0) **
-                      (\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j)) **
-                      (\<chi> i j. if i = j then if j = n then c else 1 else 0)"
-      using \<open>m \<noteq> n\<close>
-      apply (simp add: matrix_matrix_mult_def vec_eq_iff of_bool_def if_distrib [of "\<lambda>x. y * x" for y] cong: if_cong)
-      apply (simp add: if_if_eq_conj sum.neutral conj_commute cong: conj_cong)
-      done
-    show ?thesis
-      apply (subst eq)
-      apply (intro mult idplus that)
-       apply (auto intro: diagonal)
-      done
-  qed
-  show ?thesis
-    by (rule induct_matrix_elementary) (auto intro: assms *)
+lemma%unimportant  image_orthogonal_transformation_cball:
+  fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
+  assumes "orthogonal_transformation f"
+  shows "f ` cball x r = cball (f x) r"
+proof (intro equalityI subsetI)
+  fix y assume "y \<in> f ` cball x r"
+  with assms show "y \<in> cball (f x) r"
+    by (auto simp: orthogonal_transformation_isometry)
+next
+  fix y assume y: "y \<in> cball (f x) r"
+  then obtain z where z: "y = f z"
+    using assms orthogonal_transformation_surj by blast
+  with y assms show "y \<in> f ` cball x r"
+    by (auto simp: orthogonal_transformation_isometry)
qed

-lemma%unimportant matrix_vector_mult_matrix_matrix_mult_compose:
-  "(*v) (A ** B) = (*v) A \<circ> (*v) B"
-  by (auto simp: matrix_vector_mul_assoc)

-lemma%unimportant induct_linear_elementary:
-  fixes f :: "real^'n \<Rightarrow> real^'n"
-  assumes "linear f"
-    and comp: "\<And>f g. \<lbrakk>linear f; linear g; P f; P g\<rbrakk> \<Longrightarrow> P(f \<circ> g)"
-    and zeroes: "\<And>f i. \<lbrakk>linear f; \<And>x. (f x) \$ i = 0\<rbrakk> \<Longrightarrow> P f"
-    and const: "\<And>c. P(\<lambda>x. \<chi> i. c i * x\$i)"
-    and swap: "\<And>m n::'n. m \<noteq> n \<Longrightarrow> P(\<lambda>x. \<chi> i. x \$ Fun.swap m n id i)"
-    and idplus: "\<And>m n::'n. m \<noteq> n \<Longrightarrow> P(\<lambda>x. \<chi> i. if i = m then x\$m + x\$n else x\$i)"
-  shows "P f"
-proof -
-  have "P ((*v) A)" for A
-  proof (rule induct_matrix_elementary_alt)
-    fix A B
-    assume "P ((*v) A)" and "P ((*v) B)"
-    then show "P ((*v) (A ** B))"
-      by (auto simp add: matrix_vector_mult_matrix_matrix_mult_compose matrix_vector_mul_linear
-          intro!: comp)
-  next
-    fix A :: "real^'n^'n" and i
-    assume "row i A = 0"
-    show "P ((*v) A)"
-      using matrix_vector_mul_linear
-      by (rule zeroes[where i=i])
-        (metis \<open>row i A = 0\<close> inner_zero_left matrix_vector_mul_component row_def vec_lambda_eta)
-  next
-    fix A :: "real^'n^'n"
-    assume 0: "\<And>i j. i \<noteq> j \<Longrightarrow> A \$ i \$ j = 0"
-    have "A \$ i \$ i * x \$ i = (\<Sum>j\<in>UNIV. A \$ i \$ j * x \$ j)" for x and i :: "'n"
-    then have "(\<lambda>x. \<chi> i. A \$ i \$ i * x \$ i) = ((*v) A)"
-      by (auto simp: 0 matrix_vector_mult_def)
-    then show "P ((*v) A)"
-      using const [of "\<lambda>i. A \$ i \$ i"] by simp
-  next
-    fix m n :: "'n"
-    assume "m \<noteq> n"
-    have eq: "(\<Sum>j\<in>UNIV. if i = Fun.swap m n id j then x \$ j else 0) =
-              (\<Sum>j\<in>UNIV. if j = Fun.swap m n id i then x \$ j else 0)"
-      for i and x :: "real^'n"
-      unfolding swap_def by (rule sum.cong) auto
-    have "(\<lambda>x::real^'n. \<chi> i. x \$ Fun.swap m n id i) = ((*v) (\<chi> i j. if i = Fun.swap m n id j then 1 else 0))"
-      by (auto simp: mat_def matrix_vector_mult_def eq if_distrib [of "\<lambda>x. x * y" for y] cong: if_cong)
-    with swap [OF \<open>m \<noteq> n\<close>] show "P ((*v) (\<chi> i j. mat 1 \$ i \$ Fun.swap m n id j))"
-      by (simp add: mat_def matrix_vector_mult_def)
-  next
-    fix m n :: "'n"
-    assume "m \<noteq> n"
-    then have "x \$ m + x \$ n = (\<Sum>j\<in>UNIV. of_bool (j = n \<or> m = j) * x \$ j)" for x :: "real^'n"
-      by (auto simp: of_bool_def if_distrib [of "\<lambda>x. x * y" for y] sum.remove cong: if_cong)
-    then have "(\<lambda>x::real^'n. \<chi> i. if i = m then x \$ m + x \$ n else x \$ i) =
-               ((*v) (\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j)))"
-      unfolding matrix_vector_mult_def of_bool_def
-      by (auto simp: vec_eq_iff if_distrib [of "\<lambda>x. x * y" for y] cong: if_cong)
-    then show "P ((*v) (\<chi> i j. of_bool (i = m \<and> j = n \<or> i = j)))"
-      using idplus [OF \<open>m \<noteq> n\<close>] by simp
-  qed
-  then show ?thesis
-    by (metis \<open>linear f\<close> matrix_vector_mul)
-qed
-
+subsection \<open>Measurable Shear and Stretch\<close>

proposition%important
fixes a :: "real^'n"```
```--- a/src/HOL/Analysis/Convex.thy	Wed Jan 16 16:50:35 2019 -0500
+++ b/src/HOL/Analysis/Convex.thy	Wed Jan 16 18:14:02 2019 -0500
@@ -14,79 +14,6 @@
"HOL-Library.Set_Algebras"
begin

-lemma substdbasis_expansion_unique:
-  assumes d: "d \<subseteq> Basis"
-  shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
-    (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
-proof -
-  have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
-    by auto
-  have **: "finite d"
-    by (auto intro: finite_subset[OF assms])
-  have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
-    using d
-    by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
-  show ?thesis
-    unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
-qed
-
-lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
-  by (rule independent_mono[OF independent_Basis])
-
-lemma sum_not_0: "sum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
-  by (rule ccontr) auto
-
-lemma subset_translation_eq [simp]:
-    fixes a :: "'a::real_vector" shows "(+) a ` s \<subseteq> (+) a ` t \<longleftrightarrow> s \<subseteq> t"
-  by auto
-
-lemma translate_inj_on:
-  fixes A :: "'a::ab_group_add set"
-  shows "inj_on (\<lambda>x. a + x) A"
-  unfolding inj_on_def by auto
-
-lemma translation_assoc:
-  fixes a b :: "'a::ab_group_add"
-  shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
-  by auto
-
-lemma translation_invert:
-  assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
-  shows "A = B"
-proof -
-  have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
-    using assms by auto
-  then show ?thesis
-    using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
-qed
-
-lemma translation_galois:
-  shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
-  using translation_assoc[of "-a" a S]
-  apply auto
-  using translation_assoc[of a "-a" T]
-  apply auto
-  done
-
-lemma translation_inverse_subset:
-  assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
-  shows "V \<le> ((\<lambda>x. a + x) ` S)"
-proof -
-  {
-    fix x
-    assume "x \<in> V"
-    then have "x-a \<in> S" using assms by auto
-    then have "x \<in> {a + v |v. v \<in> S}"
-      apply auto
-      apply (rule exI[of _ "x-a"], simp)
-      done
-    then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
-  }
-  then show ?thesis by auto
-qed
-
subsection \<open>Convexity\<close>

definition%important convex :: "'a::real_vector set \<Rightarrow> bool"```
```--- a/src/HOL/Analysis/Cross3.thy	Wed Jan 16 16:50:35 2019 -0500
+++ b/src/HOL/Analysis/Cross3.thy	Wed Jan 16 18:14:02 2019 -0500
@@ -7,7 +7,7 @@
section\<open>Vector Cross Products in 3 Dimensions\<close>

theory "Cross3"
-  imports Determinants
+  imports Determinants Cartesian_Euclidean_Space
begin

context includes no_Set_Product_syntax ```
```--- a/src/HOL/Analysis/Determinants.thy	Wed Jan 16 16:50:35 2019 -0500
+++ b/src/HOL/Analysis/Determinants.thy	Wed Jan 16 18:14:02 2019 -0500
@@ -6,7 +6,7 @@

theory Determinants
imports
-  Cartesian_Euclidean_Space
+  Cartesian_Space
"HOL-Library.Permutations"
begin

@@ -941,372 +941,6 @@
by auto
qed

-subsection%important  \<open>Orthogonality of a transformation and matrix\<close>
-
-definition%important  "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
-
-definition%important  "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow>
-  transpose Q ** Q = mat 1 \<and> Q ** transpose Q = mat 1"
-
-lemma%unimportant  orthogonal_transformation:
-  "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v. norm (f v) = norm v)"
-  unfolding orthogonal_transformation_def
-  apply auto
-  apply (erule_tac x=v in allE)+
-  done
-
-lemma%unimportant  orthogonal_transformation_id [simp]: "orthogonal_transformation (\<lambda>x. x)"
-  by (simp add: linear_iff orthogonal_transformation_def)
-
-lemma%unimportant  orthogonal_orthogonal_transformation:
-    "orthogonal_transformation f \<Longrightarrow> orthogonal (f x) (f y) \<longleftrightarrow> orthogonal x y"
-  by (simp add: orthogonal_def orthogonal_transformation_def)
-
-lemma%unimportant  orthogonal_transformation_compose:
-   "\<lbrakk>orthogonal_transformation f; orthogonal_transformation g\<rbrakk> \<Longrightarrow> orthogonal_transformation(f \<circ> g)"
-  by (auto simp: orthogonal_transformation_def linear_compose)
-
-lemma%unimportant  orthogonal_transformation_neg:
-  "orthogonal_transformation(\<lambda>x. -(f x)) \<longleftrightarrow> orthogonal_transformation f"
-  by (auto simp: orthogonal_transformation_def dest: linear_compose_neg)
-
-lemma%unimportant  orthogonal_transformation_scaleR: "orthogonal_transformation f \<Longrightarrow> f (c *\<^sub>R v) = c *\<^sub>R f v"
-  by (simp add: linear_iff orthogonal_transformation_def)
-
-lemma%unimportant  orthogonal_transformation_linear:
-   "orthogonal_transformation f \<Longrightarrow> linear f"
-
-lemma%unimportant  orthogonal_transformation_inj:
-  "orthogonal_transformation f \<Longrightarrow> inj f"
-  unfolding orthogonal_transformation_def inj_on_def
-  by (metis vector_eq)
-
-lemma%unimportant  orthogonal_transformation_surj:
-  "orthogonal_transformation f \<Longrightarrow> surj f"
-  for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
-  by (simp add: linear_injective_imp_surjective orthogonal_transformation_inj orthogonal_transformation_linear)
-
-lemma%unimportant  orthogonal_transformation_bij:
-  "orthogonal_transformation f \<Longrightarrow> bij f"
-  for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
-  by (simp add: bij_def orthogonal_transformation_inj orthogonal_transformation_surj)
-
-lemma%unimportant  orthogonal_transformation_inv:
-  "orthogonal_transformation f \<Longrightarrow> orthogonal_transformation (inv f)"
-  for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
-  by (metis (no_types, hide_lams) bijection.inv_right bijection_def inj_linear_imp_inv_linear orthogonal_transformation orthogonal_transformation_bij orthogonal_transformation_inj)
-
-lemma%unimportant  orthogonal_transformation_norm:
-  "orthogonal_transformation f \<Longrightarrow> norm (f x) = norm x"
-  by (metis orthogonal_transformation)
-
-lemma%unimportant  orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n) \<longleftrightarrow> transpose Q ** Q = mat 1"
-  by (metis matrix_left_right_inverse orthogonal_matrix_def)
-
-lemma%unimportant  orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"
-
-lemma%unimportant  orthogonal_matrix_mul:
-  fixes A :: "real ^'n^'n"
-  assumes  "orthogonal_matrix A" "orthogonal_matrix B"
-  shows "orthogonal_matrix(A ** B)"
-  using assms
-  by (simp add: orthogonal_matrix matrix_transpose_mul matrix_left_right_inverse matrix_mul_assoc)
-
-lemma%important  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%unimportant -
-  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
-    from ot have lf: "Vector_Spaces.linear (*s) (*s) f" and fd: "\<And>v w. f v \<bullet> f w = v \<bullet> w"
-      unfolding orthogonal_transformation_def orthogonal_matrix linear_def scalar_mult_eq_scaleR
-      by blast+
-    {
-      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)"
-        by simp_all
-      from fd[of "axis i 1" "axis j 1",
-        simplified 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 sum.delta[OF fU] mat_def axis_def)
-    }
-    then have "orthogonal_matrix ?mf"
-      unfolding orthogonal_matrix
-      by vector
-    with lf have ?rhs
-      unfolding linear_def scalar_mult_eq_scaleR
-      by blast
-  }
-  moreover
-  {
-    assume lf: "Vector_Spaces.linear (*s) (*s) f" and om: "orthogonal_matrix ?mf"
-    from lf om have ?lhs
-      unfolding orthogonal_matrix_def norm_eq orthogonal_transformation
-      apply (simp only: matrix_works[OF lf, symmetric] dot_matrix_vector_mul)
-      apply (simp add: dot_matrix_product linear_def scalar_mult_eq_scaleR)
-      done
-  }
-  ultimately show ?thesis
-    by (auto simp: linear_def scalar_mult_eq_scaleR)
-qed
-
-lemma%important  det_orthogonal_matrix:
-  fixes Q:: "'a::linordered_idom^'n^'n"
-  assumes oQ: "orthogonal_matrix Q"
-  shows "det Q = 1 \<or> det Q = - 1"
-proof%unimportant -
-  have "Q ** transpose Q = mat 1"
-    by (metis oQ orthogonal_matrix_def)
-  then have "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)"
-    by simp
-  then have "det Q * det Q = 1"
-  then show ?thesis
-qed
-
-lemma%important  orthogonal_transformation_det [simp]:
-  fixes f :: "real^'n \<Rightarrow> real^'n"
-  shows "orthogonal_transformation f \<Longrightarrow> \<bar>det (matrix f)\<bar> = 1"
-  using%unimportant det_orthogonal_matrix orthogonal_transformation_matrix by fastforce
-
-
-subsection%important  \<open>Linearity of scaling, and hence isometry, that preserves origin\<close>
-
-lemma%important  scaling_linear:
-  fixes f :: "'a::real_inner \<Rightarrow> 'a::real_inner"
-  assumes f0: "f 0 = 0"
-    and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
-  shows "linear f"
-proof%unimportant -
-  {
-    fix v w
-    have "norm (f x) = c * norm x" for x
-      by (metis dist_0_norm f0 fd)
-    then have "f v \<bullet> f w = c\<^sup>2 * (v \<bullet> w)"
-      unfolding dot_norm_neg dist_norm[symmetric]
-      by (simp add: fd power2_eq_square field_simps)
-  }
-  then show ?thesis
-    unfolding linear_iff vector_eq[where 'a="'a"] scalar_mult_eq_scaleR
-qed
-
-lemma%unimportant  isometry_linear:
-  "f (0::'a::real_inner) = (0::'a) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y \<Longrightarrow> linear f"
-  by (rule scaling_linear[where c=1]) simp_all
-
-text \<open>Hence another formulation of orthogonal transformation\<close>
-
-lemma%important  orthogonal_transformation_isometry:
-  "orthogonal_transformation f \<longleftrightarrow> f(0::'a::real_inner) = (0::'a) \<and> (\<forall>x y. dist(f x) (f y) = dist x y)"
-  unfolding orthogonal_transformation
-  apply (auto simp: linear_0 isometry_linear)
-   apply (metis (no_types, hide_lams) dist_norm linear_diff)
-  by (metis dist_0_norm)
-
-
-lemma%unimportant  image_orthogonal_transformation_ball:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
-  assumes "orthogonal_transformation f"
-  shows "f ` ball x r = ball (f x) r"
-proof (intro equalityI subsetI)
-  fix y assume "y \<in> f ` ball x r"
-  with assms show "y \<in> ball (f x) r"
-    by (auto simp: orthogonal_transformation_isometry)
-next
-  fix y assume y: "y \<in> ball (f x) r"
-  then obtain z where z: "y = f z"
-    using assms orthogonal_transformation_surj by blast
-  with y assms show "y \<in> f ` ball x r"
-    by (auto simp: orthogonal_transformation_isometry)
-qed
-
-lemma%unimportant  image_orthogonal_transformation_cball:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'a"
-  assumes "orthogonal_transformation f"
-  shows "f ` cball x r = cball (f x) r"
-proof (intro equalityI subsetI)
-  fix y assume "y \<in> f ` cball x r"
-  with assms show "y \<in> cball (f x) r"
-    by (auto simp: orthogonal_transformation_isometry)
-next
-  fix y assume y: "y \<in> cball (f x) r"
-  then obtain z where z: "y = f z"
-    using assms orthogonal_transformation_surj by blast
-  with y assms show "y \<in> f ` cball x r"
-    by (auto simp: orthogonal_transformation_isometry)
-qed
-
-subsection%important \<open> We can find an orthogonal matrix taking any unit vector to any other\<close>
-
-lemma%unimportant  orthogonal_matrix_transpose [simp]:
-     "orthogonal_matrix(transpose A) \<longleftrightarrow> orthogonal_matrix A"
-  by (auto simp: orthogonal_matrix_def)
-
-lemma%unimportant  orthogonal_matrix_orthonormal_columns:
-  fixes A :: "real^'n^'n"
-  shows "orthogonal_matrix A \<longleftrightarrow>
-          (\<forall>i. norm(column i A) = 1) \<and>
-          (\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (column i A) (column j A))"
-  by (auto simp: orthogonal_matrix matrix_mult_transpose_dot_column vec_eq_iff mat_def norm_eq_1 orthogonal_def)
-
-lemma%unimportant  orthogonal_matrix_orthonormal_rows:
-  fixes A :: "real^'n^'n"
-  shows "orthogonal_matrix A \<longleftrightarrow>
-          (\<forall>i. norm(row i A) = 1) \<and>
-          (\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (row i A) (row j A))"
-  using orthogonal_matrix_orthonormal_columns [of "transpose A"] by simp
-
-lemma%important  orthogonal_matrix_exists_basis:
-  fixes a :: "real^'n"
-  assumes "norm a = 1"
-  obtains A where "orthogonal_matrix A" "A *v (axis k 1) = a"
-proof%unimportant -
-  obtain S where "a \<in> S" "pairwise orthogonal S" and noS: "\<And>x. x \<in> S \<Longrightarrow> norm x = 1"
-   and "independent S" "card S = CARD('n)" "span S = UNIV"
-    using vector_in_orthonormal_basis assms by force
-  then obtain f0 where "bij_betw f0 (UNIV::'n set) S"
-    by (metis finite_class.finite_UNIV finite_same_card_bij finiteI_independent)
-  then obtain f where f: "bij_betw f (UNIV::'n set) S" and a: "a = f k"
-    using bij_swap_iff [of k "inv f0 a" f0]
-    by (metis UNIV_I \<open>a \<in> S\<close> bij_betw_inv_into_right bij_betw_swap_iff swap_apply1)
-  show thesis
-  proof
-    have [simp]: "\<And>i. norm (f i) = 1"
-      using bij_betwE [OF \<open>bij_betw f UNIV S\<close>] by (blast intro: noS)
-    have [simp]: "\<And>i j. i \<noteq> j \<Longrightarrow> orthogonal (f i) (f j)"
-      using \<open>pairwise orthogonal S\<close> \<open>bij_betw f UNIV S\<close>
-      by (auto simp: pairwise_def bij_betw_def inj_on_def)
-    show "orthogonal_matrix (\<chi> i j. f j \$ i)"
-      by (simp add: orthogonal_matrix_orthonormal_columns column_def)
-    show "(\<chi> i j. f j \$ i) *v axis k 1 = a"
-      by (simp add: matrix_vector_mult_def axis_def a if_distrib cong: if_cong)
-  qed
-qed
-
-lemma%unimportant  orthogonal_transformation_exists_1:
-  fixes a b :: "real^'n"
-  assumes "norm a = 1" "norm b = 1"
-  obtains f where "orthogonal_transformation f" "f a = b"
-proof%unimportant -
-  obtain k::'n where True
-    by simp
-  obtain A B where AB: "orthogonal_matrix A" "orthogonal_matrix B" and eq: "A *v (axis k 1) = a" "B *v (axis k 1) = b"
-    using orthogonal_matrix_exists_basis assms by metis
-  let ?f = "\<lambda>x. (B ** transpose A) *v x"
-  show thesis
-  proof
-    show "orthogonal_transformation ?f"
-      by (subst orthogonal_transformation_matrix)
-        (auto simp: AB orthogonal_matrix_mul)
-  next
-    show "?f a = b"
-      using \<open>orthogonal_matrix A\<close> unfolding orthogonal_matrix_def
-      by (metis eq matrix_mul_rid matrix_vector_mul_assoc)
-  qed
-qed
-
-lemma%important  orthogonal_transformation_exists:
-  fixes a b :: "real^'n"
-  assumes "norm a = norm b"
-  obtains f where "orthogonal_transformation f" "f a = b"
-proof%unimportant (cases "a = 0 \<or> b = 0")
-  case True
-  with assms show ?thesis
-    using that by force
-next
-  case False
-  then obtain f where f: "orthogonal_transformation f" and eq: "f (a /\<^sub>R norm a) = (b /\<^sub>R norm b)"
-    by (auto intro: orthogonal_transformation_exists_1 [of "a /\<^sub>R norm a" "b /\<^sub>R norm b"])
-  show ?thesis
-  proof
-    interpret linear f
-      using f by (simp add: orthogonal_transformation_linear)
-    have "f a /\<^sub>R norm a = f (a /\<^sub>R norm a)"
-    also have "\<dots> = b /\<^sub>R norm a"
-      by (simp add: eq assms [symmetric])
-    finally show "f a = b"
-      using False by auto
-  qed (use f in auto)
-qed
-
-
-subsection%important  \<open>Can extend an isometry from unit sphere\<close>
-
-lemma%important  isometry_sphere_extend:
-  fixes f:: "'a::real_inner \<Rightarrow> 'a"
-  assumes f1: "\<And>x. norm x = 1 \<Longrightarrow> norm (f x) = 1"
-    and fd1: "\<And>x y. \<lbrakk>norm x = 1; norm y = 1\<rbrakk> \<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%unimportant -
-  {
-    fix x y x' y' u v u' v' :: "'a"
-    assume H: "x = norm x *\<^sub>R u" "y = norm y *\<^sub>R v"
-              "x' = norm x *\<^sub>R u'" "y' = norm y *\<^sub>R v'"
-      and J: "norm u = 1" "norm u' = 1" "norm v = 1" "norm v' = 1" "norm(u' - v') = norm(u - v)"
-    then have *: "u \<bullet> v = u' \<bullet> v' + v' \<bullet> u' - v \<bullet> u "
-    have "norm (norm x *\<^sub>R u' - norm y *\<^sub>R v') = norm (norm x *\<^sub>R u - norm y *\<^sub>R v)"
-      using J by (simp add: norm_eq norm_eq_1 inner_diff * field_simps)
-    then have "norm(x' - y') = norm(x - y)"
-      using H by metis
-  }
-  note norm_eq = this
-  let ?g = "\<lambda>x. if x = 0 then 0 else norm x *\<^sub>R f (x /\<^sub>R norm x)"
-  have thfg: "?g x = f x" if "norm x = 1" for x
-    using that by auto
-  have thd: "dist (?g x) (?g y) = dist x y" for x y
-  proof (cases "x=0 \<or> y=0")
-    case False
-    show "dist (?g x) (?g y) = dist x y"
-      unfolding dist_norm
-    proof (rule norm_eq)
-      show "x = norm x *\<^sub>R (x /\<^sub>R norm x)" "y = norm y *\<^sub>R (y /\<^sub>R norm y)"
-           "norm (f (x /\<^sub>R norm x)) = 1" "norm (f (y /\<^sub>R norm y)) = 1"
-        using False f1 by auto
-    qed (use False in \<open>auto simp: field_simps intro: f1 fd1[unfolded dist_norm]\<close>)
-  qed (auto simp: f1)
-  show ?thesis
-    unfolding orthogonal_transformation_isometry
-    by (rule exI[where x= ?g]) (metis thfg thd)
-qed
-
-subsection%important  \<open>Rotation, reflection, rotoinversion\<close>
-
-definition%important  "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
-definition%important  "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
-
-lemma%important  orthogonal_rotation_or_rotoinversion:
-  fixes Q :: "'a::linordered_idom^'n^'n"
-  shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
-  by%unimportant (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
-
-text \<open>Explicit formulas for low dimensions\<close>
-
-lemma%unimportant  prod_neutral_const: "prod f {(1::nat)..1} = f 1"
-  by simp
-
-lemma%unimportant  prod_2: "prod f {(1::nat)..2} = f 1 * f 2"
-  by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)
-
-lemma%unimportant  prod_3: "prod f {(1::nat)..3} = f 1 * f 2 * f 3"
-  by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)
-
lemma%unimportant  det_1: "det (A::'a::comm_ring_1^1^1) = A\$1\$1"

@@ -1342,6 +976,36 @@
by (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
qed

+lemma%important  det_orthogonal_matrix:
+  fixes Q:: "'a::linordered_idom^'n^'n"
+  assumes oQ: "orthogonal_matrix Q"
+  shows "det Q = 1 \<or> det Q = - 1"
+proof%unimportant -
+  have "Q ** transpose Q = mat 1"
+    by (metis oQ orthogonal_matrix_def)
+  then have "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)"
+    by simp
+  then have "det Q * det Q = 1"
+  then show ?thesis
+qed
+
+lemma%important  orthogonal_transformation_det [simp]:
+  fixes f :: "real^'n \<Rightarrow> real^'n"
+  shows "orthogonal_transformation f \<Longrightarrow> \<bar>det (matrix f)\<bar> = 1"
+  using%unimportant det_orthogonal_matrix orthogonal_transformation_matrix by fastforce
+
+subsection%important  \<open>Rotation, reflection, rotoinversion\<close>
+
+definition%important  "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
+definition%important  "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"
+
+lemma%important  orthogonal_rotation_or_rotoinversion:
+  fixes Q :: "'a::linordered_idom^'n^'n"
+  shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
+  by%unimportant (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
+
text\<open> Slightly stronger results giving rotation, but only in two or more dimensions\<close>

lemma%unimportant  rotation_matrix_exists_basis:
@@ -1360,8 +1024,10 @@
then show ?thesis
using \<open>A *v axis k 1 = a\<close> that by auto
next
-    obtain j where "j \<noteq> k"
-      by (metis (full_types) 2 card_2_exists ex_card)
+    from ex_card[OF 2] obtain h i::'n where "h \<noteq> i"
+      by (auto simp add: eval_nat_numeral card_Suc_eq)
+    then obtain j where "j \<noteq> k"
+      by (metis (full_types))
let ?TA = "transpose A"
let ?A = "\<chi> i. if i = j then - 1 *\<^sub>R (?TA \$ i) else ?TA \$i"
assume "rotoinversion_matrix A"```
```--- a/src/HOL/Analysis/Linear_Algebra.thy	Wed Jan 16 16:50:35 2019 -0500
+++ b/src/HOL/Analysis/Linear_Algebra.thy	Wed Jan 16 18:14:02 2019 -0500
@@ -30,6 +30,79 @@
lemma finite_Atleast_Atmost_nat[simp]: "finite {f x |x. x \<in> (UNIV::'a::finite set)}"
using finite finite_image_set by blast

+lemma substdbasis_expansion_unique:
+  includes inner_syntax
+  assumes d: "d \<subseteq> Basis"
+  shows "(\<Sum>i\<in>d. f i *\<^sub>R i) = (x::'a::euclidean_space) \<longleftrightarrow>
+    (\<forall>i\<in>Basis. (i \<in> d \<longrightarrow> f i = x \<bullet> i) \<and> (i \<notin> d \<longrightarrow> x \<bullet> i = 0))"
+proof -
+  have *: "\<And>x a b P. x * (if P then a else b) = (if P then x * a else x * b)"
+    by auto
+  have **: "finite d"
+    by (auto intro: finite_subset[OF assms])
+  have ***: "\<And>i. i \<in> Basis \<Longrightarrow> (\<Sum>i\<in>d. f i *\<^sub>R i) \<bullet> i = (\<Sum>x\<in>d. if x = i then f x else 0)"
+    using d
+    by (auto intro!: sum.cong simp: inner_Basis inner_sum_left)
+  show ?thesis
+    unfolding euclidean_eq_iff[where 'a='a] by (auto simp: sum.delta[OF **] ***)
+qed
+
+lemma independent_substdbasis: "d \<subseteq> Basis \<Longrightarrow> independent d"
+  by (rule independent_mono[OF independent_Basis])
+
+lemma sum_not_0: "sum f A \<noteq> 0 \<Longrightarrow> \<exists>a \<in> A. f a \<noteq> 0"
+  by (rule ccontr) auto
+
+lemma subset_translation_eq [simp]:
+    fixes a :: "'a::real_vector" shows "(+) a ` s \<subseteq> (+) a ` t \<longleftrightarrow> s \<subseteq> t"
+  by auto
+
+lemma translate_inj_on:
+  fixes A :: "'a::ab_group_add set"
+  shows "inj_on (\<lambda>x. a + x) A"
+  unfolding inj_on_def by auto
+
+lemma translation_assoc:
+  fixes a b :: "'a::ab_group_add"
+  shows "(\<lambda>x. b + x) ` ((\<lambda>x. a + x) ` S) = (\<lambda>x. (a + b) + x) ` S"
+  by auto
+
+lemma translation_invert:
+  assumes "(\<lambda>x. a + x) ` A = (\<lambda>x. a + x) ` B"
+  shows "A = B"
+proof -
+  have "(\<lambda>x. -a + x) ` ((\<lambda>x. a + x) ` A) = (\<lambda>x. - a + x) ` ((\<lambda>x. a + x) ` B)"
+    using assms by auto
+  then show ?thesis
+    using translation_assoc[of "-a" a A] translation_assoc[of "-a" a B] by auto
+qed
+
+lemma translation_galois:
+  shows "T = ((\<lambda>x. a + x) ` S) \<longleftrightarrow> S = ((\<lambda>x. (- a) + x) ` T)"
+  using translation_assoc[of "-a" a S]
+  apply auto
+  using translation_assoc[of a "-a" T]
+  apply auto
+  done
+
+lemma translation_inverse_subset:
+  assumes "((\<lambda>x. - a + x) ` V) \<le> (S :: 'n::ab_group_add set)"
+  shows "V \<le> ((\<lambda>x. a + x) ` S)"
+proof -
+  {
+    fix x
+    assume "x \<in> V"
+    then have "x-a \<in> S" using assms by auto
+    then have "x \<in> {a + v |v. v \<in> S}"
+      apply auto
+      apply (rule exI[of _ "x-a"], simp)
+      done
+    then have "x \<in> ((\<lambda>x. a+x) ` S)" by auto
+  }
+  then show ?thesis by auto
+qed

subsection%unimportant \<open>More interesting properties of the norm\<close>

@@ -224,6 +297,66 @@
qed

+subsection%important  \<open>Orthogonality of a transformation\<close>
+
+definition%important  "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
+
+lemma%unimportant  orthogonal_transformation:
+  "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v. norm (f v) = norm v)"
+  unfolding orthogonal_transformation_def
+  apply auto
+  apply (erule_tac x=v in allE)+
+  done
+
+lemma%unimportant  orthogonal_transformation_id [simp]: "orthogonal_transformation (\<lambda>x. x)"
+  by (simp add: linear_iff orthogonal_transformation_def)
+
+lemma%unimportant  orthogonal_orthogonal_transformation:
+    "orthogonal_transformation f \<Longrightarrow> orthogonal (f x) (f y) \<longleftrightarrow> orthogonal x y"
+  by (simp add: orthogonal_def orthogonal_transformation_def)
+
+lemma%unimportant  orthogonal_transformation_compose:
+   "\<lbrakk>orthogonal_transformation f; orthogonal_transformation g\<rbrakk> \<Longrightarrow> orthogonal_transformation(f \<circ> g)"
+  by (auto simp: orthogonal_transformation_def linear_compose)
+
+lemma%unimportant  orthogonal_transformation_neg:
+  "orthogonal_transformation(\<lambda>x. -(f x)) \<longleftrightarrow> orthogonal_transformation f"
+  by (auto simp: orthogonal_transformation_def dest: linear_compose_neg)
+
+lemma%unimportant  orthogonal_transformation_scaleR: "orthogonal_transformation f \<Longrightarrow> f (c *\<^sub>R v) = c *\<^sub>R f v"
+  by (simp add: linear_iff orthogonal_transformation_def)
+
+lemma%unimportant  orthogonal_transformation_linear:
+   "orthogonal_transformation f \<Longrightarrow> linear f"
+
+lemma%unimportant  orthogonal_transformation_inj:
+  "orthogonal_transformation f \<Longrightarrow> inj f"
+  unfolding orthogonal_transformation_def inj_on_def
+  by (metis vector_eq)
+
+lemma%unimportant  orthogonal_transformation_surj:
+  "orthogonal_transformation f \<Longrightarrow> surj f"
+  for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  by (simp add: linear_injective_imp_surjective orthogonal_transformation_inj orthogonal_transformation_linear)
+
+lemma%unimportant  orthogonal_transformation_bij:
+  "orthogonal_transformation f \<Longrightarrow> bij f"
+  for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  by (simp add: bij_def orthogonal_transformation_inj orthogonal_transformation_surj)
+
+lemma%unimportant  orthogonal_transformation_inv:
+  "orthogonal_transformation f \<Longrightarrow> orthogonal_transformation (inv f)"
+  for f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
+  by (metis (no_types, hide_lams) bijection.inv_right bijection_def inj_linear_imp_inv_linear orthogonal_transformation orthogonal_transformation_bij orthogonal_transformation_inj)
+
+lemma%unimportant  orthogonal_transformation_norm:
+  "orthogonal_transformation f \<Longrightarrow> norm (f x) = norm x"
+  by (metis orthogonal_transformation)
+
+
subsection \<open>Bilinear functions\<close>

definition%important
@@ -1210,4 +1343,540 @@
qed
qed

+
+subsection\<open>Properties of special hyperplanes\<close>
+
+lemma subspace_hyperplane: "subspace {x. a \<bullet> x = 0}"
+  by (simp add: subspace_def inner_right_distrib)
+
+lemma subspace_hyperplane2: "subspace {x. x \<bullet> a = 0}"
+  by (simp add: inner_commute inner_right_distrib subspace_def)
+
+lemma special_hyperplane_span:
+  fixes S :: "'n::euclidean_space set"
+  assumes "k \<in> Basis"
+  shows "{x. k \<bullet> x = 0} = span (Basis - {k})"
+proof -
+  have *: "x \<in> span (Basis - {k})" if "k \<bullet> x = 0" for x
+  proof -
+    have "x = (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b)"
+    also have "... = (\<Sum>b \<in> Basis - {k}. (x \<bullet> b) *\<^sub>R b)"
+      by (auto simp: sum.remove [of _ k] inner_commute assms that)
+    finally have "x = (\<Sum>b\<in>Basis - {k}. (x \<bullet> b) *\<^sub>R b)" .
+    then show ?thesis
+  qed
+  show ?thesis
+    apply (rule span_subspace [symmetric])
+    using assms
+    apply (auto simp: inner_not_same_Basis intro: * subspace_hyperplane)
+    done
+qed
+
+lemma dim_special_hyperplane:
+  fixes k :: "'n::euclidean_space"
+  shows "k \<in> Basis \<Longrightarrow> dim {x. k \<bullet> x = 0} = DIM('n) - 1"
+apply (rule dim_unique [OF subset_refl])
+apply (auto simp: independent_substdbasis)
+apply (metis member_remove remove_def span_base)
+done
+
+proposition dim_hyperplane:
+  fixes a :: "'a::euclidean_space"
+  assumes "a \<noteq> 0"
+    shows "dim {x. a \<bullet> x = 0} = DIM('a) - 1"
+proof -
+  have span0: "span {x. a \<bullet> x = 0} = {x. a \<bullet> x = 0}"
+    by (rule span_unique) (auto simp: subspace_hyperplane)
+  then obtain B where "independent B"
+              and Bsub: "B \<subseteq> {x. a \<bullet> x = 0}"
+              and subspB: "{x. a \<bullet> x = 0} \<subseteq> span B"
+              and card0: "(card B = dim {x. a \<bullet> x = 0})"
+              and ortho: "pairwise orthogonal B"
+    using orthogonal_basis_exists by metis
+  with assms have "a \<notin> span B"
+    by (metis (mono_tags, lifting) span_eq inner_eq_zero_iff mem_Collect_eq span0)
+  then have ind: "independent (insert a B)"
+    by (simp add: \<open>independent B\<close> independent_insert)
+  have "finite B"
+    using \<open>independent B\<close> independent_bound by blast
+  have "UNIV \<subseteq> span (insert a B)"
+  proof fix y::'a
+    obtain r z where z: "y = r *\<^sub>R a + z" "a \<bullet> z = 0"
+      apply (rule_tac r="(a \<bullet> y) / (a \<bullet> a)" and z = "y - ((a \<bullet> y) / (a \<bullet> a)) *\<^sub>R a" in that)
+      using assms
+      by (auto simp: algebra_simps)
+    show "y \<in> span (insert a B)"
+      by (metis (mono_tags, lifting) z Bsub span_eq_iff
+         add_diff_cancel_left' mem_Collect_eq span0 span_breakdown_eq span_subspace subspB)
+  qed
+  then have dima: "DIM('a) = dim(insert a B)"
+    by (metis independent_Basis span_Basis dim_eq_card top.extremum_uniqueI)
+  then show ?thesis
+    by (metis (mono_tags, lifting) Bsub Diff_insert_absorb \<open>a \<notin> span B\<close> ind card0
+        card_Diff_singleton dim_span indep_card_eq_dim_span insertI1 subsetCE
+        subspB)
+qed
+
+lemma lowdim_eq_hyperplane:
+  fixes S :: "'a::euclidean_space set"
+  assumes "dim S = DIM('a) - 1"
+  obtains a where "a \<noteq> 0" and "span S = {x. a \<bullet> x = 0}"
+proof -
+  have dimS: "dim S < DIM('a)"
+  then obtain b where b: "b \<noteq> 0" "span S \<subseteq> {a. b \<bullet> a = 0}"
+    using lowdim_subset_hyperplane [of S] by fastforce
+  show ?thesis
+    apply (rule that[OF b(1)])
+    apply (rule subspace_dim_equal)
+    by (auto simp: assms b dim_hyperplane dim_span subspace_hyperplane
+        subspace_span)
+qed
+
+lemma dim_eq_hyperplane:
+  fixes S :: "'n::euclidean_space set"
+  shows "dim S = DIM('n) - 1 \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> span S = {x. a \<bullet> x = 0})"
+by (metis One_nat_def dim_hyperplane dim_span lowdim_eq_hyperplane)
+
+
+subsection\<open> Orthogonal bases, Gram-Schmidt process, and related theorems\<close>
+
+lemma pairwise_orthogonal_independent:
+  assumes "pairwise orthogonal S" and "0 \<notin> S"
+    shows "independent S"
+proof -
+  have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+    using assms by (simp add: pairwise_def orthogonal_def)
+  have "False" if "a \<in> S" and a: "a \<in> span (S - {a})" for a
+  proof -
+    obtain T U where "T \<subseteq> S - {a}" "a = (\<Sum>v\<in>T. U v *\<^sub>R v)"
+      using a by (force simp: span_explicit)
+    then have "a \<bullet> a = a \<bullet> (\<Sum>v\<in>T. U v *\<^sub>R v)"
+      by simp
+    also have "... = 0"
+      by (metis "0" DiffE \<open>T \<subseteq> S - {a}\<close> mult_not_zero singletonI subsetCE \<open>a \<in> S\<close>)
+    finally show ?thesis
+      using \<open>0 \<notin> S\<close> \<open>a \<in> S\<close> by auto
+  qed
+  then show ?thesis
+    by (force simp: dependent_def)
+qed
+
+lemma pairwise_orthogonal_imp_finite:
+  fixes S :: "'a::euclidean_space set"
+  assumes "pairwise orthogonal S"
+    shows "finite S"
+proof -
+  have "independent (S - {0})"
+    apply (rule pairwise_orthogonal_independent)
+     apply (metis Diff_iff assms pairwise_def)
+    by blast
+  then show ?thesis
+    by (meson independent_imp_finite infinite_remove)
+qed
+
+lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
+  by (simp add: subspace_def orthogonal_clauses)
+
+lemma subspace_orthogonal_to_vectors: "subspace {y. \<forall>x \<in> S. orthogonal x y}"
+  by (simp add: subspace_def orthogonal_clauses)
+
+lemma orthogonal_to_span:
+  assumes a: "a \<in> span S" and x: "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
+    shows "orthogonal x a"
+  by (metis a orthogonal_clauses(1,2,4)
+      span_induct_alt x)
+
+proposition Gram_Schmidt_step:
+  fixes S :: "'a::euclidean_space set"
+  assumes S: "pairwise orthogonal S" and x: "x \<in> span S"
+    shows "orthogonal x (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b))"
+proof -
+  have "finite S"
+    by (simp add: S pairwise_orthogonal_imp_finite)
+  have "orthogonal (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)) x"
+       if "x \<in> S" for x
+  proof -
+    have "a \<bullet> x = (\<Sum>y\<in>S. if y = x then y \<bullet> a else 0)"
+      by (simp add: \<open>finite S\<close> inner_commute sum.delta that)
+    also have "... =  (\<Sum>b\<in>S. b \<bullet> a * (b \<bullet> x) / (b \<bullet> b))"
+      apply (rule sum.cong [OF refl], simp)
+      by (meson S orthogonal_def pairwise_def that)
+   finally show ?thesis
+     by (simp add: orthogonal_def algebra_simps inner_sum_left)
+  qed
+  then show ?thesis
+    using orthogonal_to_span orthogonal_commute x by blast
+qed
+
+
+lemma orthogonal_extension_aux:
+  fixes S :: "'a::euclidean_space set"
+  assumes "finite T" "finite S" "pairwise orthogonal S"
+    shows "\<exists>U. pairwise orthogonal (S \<union> U) \<and> span (S \<union> U) = span (S \<union> T)"
+using assms
+proof (induction arbitrary: S)
+  case empty then show ?case
+    by simp (metis sup_bot_right)
+next
+  case (insert a T)
+  have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+    using insert by (simp add: pairwise_def orthogonal_def)
+  define a' where "a' = a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)"
+  obtain U where orthU: "pairwise orthogonal (S \<union> insert a' U)"
+             and spanU: "span (insert a' S \<union> U) = span (insert a' S \<union> T)"
+    by (rule exE [OF insert.IH [of "insert a' S"]])
+      (auto simp: Gram_Schmidt_step a'_def insert.prems orthogonal_commute
+        pairwise_orthogonal_insert span_clauses)
+  have orthS: "\<And>x. x \<in> S \<Longrightarrow> a' \<bullet> x = 0"
+    using Gram_Schmidt_step [OF \<open>pairwise orthogonal S\<close>]
+    apply (force simp: orthogonal_def inner_commute span_superset [THEN subsetD])
+    done
+  have "span (S \<union> insert a' U) = span (insert a' (S \<union> T))"
+    using spanU by simp
+  also have "... = span (insert a (S \<union> T))"
+    apply (rule eq_span_insert_eq)
+    apply (simp add: a'_def span_neg span_sum span_base span_mul)
+    done
+  also have "... = span (S \<union> insert a T)"
+    by simp
+  finally show ?case
+    by (rule_tac x="insert a' U" in exI) (use orthU in auto)
+qed
+
+
+proposition orthogonal_extension:
+  fixes S :: "'a::euclidean_space set"
+  assumes S: "pairwise orthogonal S"
+  obtains U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
+proof -
+  obtain B where "finite B" "span B = span T"
+    using basis_subspace_exists [of "span T"] subspace_span by metis
+  with orthogonal_extension_aux [of B S]
+  obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> B)"
+    using assms pairwise_orthogonal_imp_finite by auto
+  with \<open>span B = span T\<close> show ?thesis
+    by (rule_tac U=U in that) (auto simp: span_Un)
+qed
+
+corollary%unimportant orthogonal_extension_strong:
+  fixes S :: "'a::euclidean_space set"
+  assumes S: "pairwise orthogonal S"
+  obtains U where "U \<inter> (insert 0 S) = {}" "pairwise orthogonal (S \<union> U)"
+                  "span (S \<union> U) = span (S \<union> T)"
+proof -
+  obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
+    using orthogonal_extension assms by blast
+  then show ?thesis
+    apply (rule_tac U = "U - (insert 0 S)" in that)
+      apply blast
+     apply (force simp: pairwise_def)
+    apply (metis Un_Diff_cancel Un_insert_left span_redundant span_zero)
+    done
+qed
+
+subsection\<open>Decomposing a vector into parts in orthogonal subspaces\<close>
+
+text\<open>existence of orthonormal basis for a subspace.\<close>
+
+lemma orthogonal_spanningset_subspace:
+  fixes S :: "'a :: euclidean_space set"
+  assumes "subspace S"
+  obtains B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
+proof -
+  obtain B where "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
+    using basis_exists by blast
+  with orthogonal_extension [of "{}" B]
+  show ?thesis
+    by (metis Un_empty_left assms pairwise_empty span_superset span_subspace that)
+qed
+
+lemma orthogonal_basis_subspace:
+  fixes S :: "'a :: euclidean_space set"
+  assumes "subspace S"
+  obtains B where "0 \<notin> B" "B \<subseteq> S" "pairwise orthogonal B" "independent B"
+                  "card B = dim S" "span B = S"
+proof -
+  obtain B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
+    using assms orthogonal_spanningset_subspace by blast
+  then show ?thesis
+    apply (rule_tac B = "B - {0}" in that)
+    apply (auto simp: indep_card_eq_dim_span pairwise_subset pairwise_orthogonal_independent elim: pairwise_subset)
+    done
+qed
+
+proposition orthonormal_basis_subspace:
+  fixes S :: "'a :: euclidean_space set"
+  assumes "subspace S"
+  obtains B where "B \<subseteq> S" "pairwise orthogonal B"
+              and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
+              and "independent B" "card B = dim S" "span B = S"
+proof -
+  obtain B where "0 \<notin> B" "B \<subseteq> S"
+             and orth: "pairwise orthogonal B"
+             and "independent B" "card B = dim S" "span B = S"
+    by (blast intro: orthogonal_basis_subspace [OF assms])
+  have 1: "(\<lambda>x. x /\<^sub>R norm x) ` B \<subseteq> S"
+    using \<open>span B = S\<close> span_superset span_mul by fastforce
+  have 2: "pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` B)"
+    using orth by (force simp: pairwise_def orthogonal_clauses)
+  have 3: "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` B \<Longrightarrow> norm x = 1"
+    by (metis (no_types, lifting) \<open>0 \<notin> B\<close> image_iff norm_sgn sgn_div_norm)
+  have 4: "independent ((\<lambda>x. x /\<^sub>R norm x) ` B)"
+    by (metis "2" "3" norm_zero pairwise_orthogonal_independent zero_neq_one)
+  have "inj_on (\<lambda>x. x /\<^sub>R norm x) B"
+  proof
+    fix x y
+    assume "x \<in> B" "y \<in> B" "x /\<^sub>R norm x = y /\<^sub>R norm y"
+    moreover have "\<And>i. i \<in> B \<Longrightarrow> norm (i /\<^sub>R norm i) = 1"
+      using 3 by blast
+    ultimately show "x = y"
+      by (metis norm_eq_1 orth orthogonal_clauses(7) orthogonal_commute orthogonal_def pairwise_def zero_neq_one)
+  qed
+  then have 5: "card ((\<lambda>x. x /\<^sub>R norm x) ` B) = dim S"
+    by (metis \<open>card B = dim S\<close> card_image)
+  have 6: "span ((\<lambda>x. x /\<^sub>R norm x) ` B) = S"
+    by (metis "1" "4" "5" assms card_eq_dim independent_imp_finite span_subspace)
+  show ?thesis
+    by (rule that [OF 1 2 3 4 5 6])
+qed
+
+
+proposition%unimportant orthogonal_to_subspace_exists_gen:
+  fixes S :: "'a :: euclidean_space set"
+  assumes "span S \<subset> span T"
+  obtains x where "x \<noteq> 0" "x \<in> span T" "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
+proof -
+  obtain B where "B \<subseteq> span S" and orthB: "pairwise orthogonal B"
+             and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
+             and "independent B" "card B = dim S" "span B = span S"
+    by (rule orthonormal_basis_subspace [of "span S", OF subspace_span])
+      (auto simp: dim_span)
+  with assms obtain u where spanBT: "span B \<subseteq> span T" and "u \<notin> span B" "u \<in> span T"
+    by auto
+  obtain C where orthBC: "pairwise orthogonal (B \<union> C)" and spanBC: "span (B \<union> C) = span (B \<union> {u})"
+    by (blast intro: orthogonal_extension [OF orthB])
+  show thesis
+  proof (cases "C \<subseteq> insert 0 B")
+    case True
+    then have "C \<subseteq> span B"
+      using span_eq
+      by (metis span_insert_0 subset_trans)
+    moreover have "u \<in> span (B \<union> C)"
+      using \<open>span (B \<union> C) = span (B \<union> {u})\<close> span_superset by force
+    ultimately show ?thesis
+      using True \<open>u \<notin> span B\<close>
+      by (metis Un_insert_left span_insert_0 sup.orderE)
+  next
+    case False
+    then obtain x where "x \<in> C" "x \<noteq> 0" "x \<notin> B"
+      by blast
+    then have "x \<in> span T"
+      by (metis (no_types, lifting) Un_insert_right Un_upper2 \<open>u \<in> span T\<close> spanBT spanBC
+          \<open>u \<in> span T\<close> insert_subset span_superset span_mono
+          span_span subsetCE subset_trans sup_bot.comm_neutral)
+    moreover have "orthogonal x y" if "y \<in> span B" for y
+      using that
+    proof (rule span_induct)
+      show "subspace {a. orthogonal x a}"
+      show "\<And>b. b \<in> B \<Longrightarrow> orthogonal x b"
+        by (metis Un_iff \<open>x \<in> C\<close> \<open>x \<notin> B\<close> orthBC pairwise_def)
+    qed
+    ultimately show ?thesis
+      using \<open>x \<noteq> 0\<close> that \<open>span B = span S\<close> by auto
+  qed
+qed
+
+corollary%unimportant orthogonal_to_subspace_exists:
+  fixes S :: "'a :: euclidean_space set"
+  assumes "dim S < DIM('a)"
+  obtains x where "x \<noteq> 0" "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
+proof -
+have "span S \<subset> UNIV"
+  by (metis (mono_tags) UNIV_I assms inner_eq_zero_iff less_le lowdim_subset_hyperplane
+      mem_Collect_eq top.extremum_strict top.not_eq_extremum)
+  with orthogonal_to_subspace_exists_gen [of S UNIV] that show ?thesis
+    by (auto simp: span_UNIV)
+qed
+
+corollary%unimportant orthogonal_to_vector_exists:
+  fixes x :: "'a :: euclidean_space"
+  assumes "2 \<le> DIM('a)"
+  obtains y where "y \<noteq> 0" "orthogonal x y"
+proof -
+  have "dim {x} < DIM('a)"
+    using assms by auto
+  then show thesis
+    by (rule orthogonal_to_subspace_exists) (simp add: orthogonal_commute span_base that)
+qed
+
+proposition%unimportant orthogonal_subspace_decomp_exists:
+  fixes S :: "'a :: euclidean_space set"
+  obtains y z
+  where "y \<in> span S"
+    and "\<And>w. w \<in> span S \<Longrightarrow> orthogonal z w"
+    and "x = y + z"
+proof -
+  obtain T where "0 \<notin> T" "T \<subseteq> span S" "pairwise orthogonal T" "independent T"
+    "card T = dim (span S)" "span T = span S"
+    using orthogonal_basis_subspace subspace_span by blast
+  let ?a = "\<Sum>b\<in>T. (b \<bullet> x / (b \<bullet> b)) *\<^sub>R b"
+  have orth: "orthogonal (x - ?a) w" if "w \<in> span S" for w
+    by (simp add: Gram_Schmidt_step \<open>pairwise orthogonal T\<close> \<open>span T = span S\<close>
+        orthogonal_commute that)
+  show ?thesis
+    apply (rule_tac y = "?a" and z = "x - ?a" in that)
+      apply (meson \<open>T \<subseteq> span S\<close> span_scale span_sum subsetCE)
+     apply (fact orth, simp)
+    done
+qed
+
+lemma orthogonal_subspace_decomp_unique:
+  fixes S :: "'a :: euclidean_space set"
+  assumes "x + y = x' + y'"
+      and ST: "x \<in> span S" "x' \<in> span S" "y \<in> span T" "y' \<in> span T"
+      and orth: "\<And>a b. \<lbrakk>a \<in> S; b \<in> T\<rbrakk> \<Longrightarrow> orthogonal a b"
+  shows "x = x' \<and> y = y'"
+proof -
+  have "x + y - y' = x'"
+  moreover have "\<And>a b. \<lbrakk>a \<in> span S; b \<in> span T\<rbrakk> \<Longrightarrow> orthogonal a b"
+    by (meson orth orthogonal_commute orthogonal_to_span)
+  ultimately have "0 = x' - x"
+    by (metis (full_types) add_diff_cancel_left' ST diff_right_commute orthogonal_clauses(10) orthogonal_clauses(5) orthogonal_self)
+  with assms show ?thesis by auto
+qed
+
+lemma vector_in_orthogonal_spanningset:
+  fixes a :: "'a::euclidean_space"
+  obtains S where "a \<in> S" "pairwise orthogonal S" "span S = UNIV"
+  by (metis UNIV_I Un_iff empty_iff insert_subset orthogonal_extension pairwise_def
+      pairwise_orthogonal_insert span_UNIV subsetI subset_antisym)
+
+lemma vector_in_orthogonal_basis:
+  fixes a :: "'a::euclidean_space"
+  assumes "a \<noteq> 0"
+  obtains S where "a \<in> S" "0 \<notin> S" "pairwise orthogonal S" "independent S" "finite S"
+                  "span S = UNIV" "card S = DIM('a)"
+proof -
+  obtain S where S: "a \<in> S" "pairwise orthogonal S" "span S = UNIV"
+    using vector_in_orthogonal_spanningset .
+  show thesis
+  proof
+    show "pairwise orthogonal (S - {0})"
+      using pairwise_mono S(2) by blast
+    show "independent (S - {0})"
+      by (simp add: \<open>pairwise orthogonal (S - {0})\<close> pairwise_orthogonal_independent)
+    show "finite (S - {0})"
+      using \<open>independent (S - {0})\<close> independent_imp_finite by blast
+    show "card (S - {0}) = DIM('a)"
+      using span_delete_0 [of S] S
+      by (simp add: \<open>independent (S - {0})\<close> indep_card_eq_dim_span dim_UNIV)
+  qed (use S \<open>a \<noteq> 0\<close> in auto)
+qed
+
+lemma vector_in_orthonormal_basis:
+  fixes a :: "'a::euclidean_space"
+  assumes "norm a = 1"
+  obtains S where "a \<in> S" "pairwise orthogonal S" "\<And>x. x \<in> S \<Longrightarrow> norm x = 1"
+    "independent S" "card S = DIM('a)" "span S = UNIV"
+proof -
+  have "a \<noteq> 0"
+    using assms by auto
+  then obtain S where "a \<in> S" "0 \<notin> S" "finite S"
+          and S: "pairwise orthogonal S" "independent S" "span S = UNIV" "card S = DIM('a)"
+    by (metis vector_in_orthogonal_basis)
+  let ?S = "(\<lambda>x. x /\<^sub>R norm x) ` S"
+  show thesis
+  proof
+    show "a \<in> ?S"
+      using \<open>a \<in> S\<close> assms image_iff by fastforce
+  next
+    show "pairwise orthogonal ?S"
+      using \<open>pairwise orthogonal S\<close> by (auto simp: pairwise_def orthogonal_def)
+    show "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` S \<Longrightarrow> norm x = 1"
+      using \<open>0 \<notin> S\<close> by (auto simp: divide_simps)
+    then show "independent ?S"
+      by (metis \<open>pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` S)\<close> norm_zero pairwise_orthogonal_independent zero_neq_one)
+    have "inj_on (\<lambda>x. x /\<^sub>R norm x) S"
+      unfolding inj_on_def
+      by (metis (full_types) S(1) \<open>0 \<notin> S\<close> inverse_nonzero_iff_nonzero norm_eq_zero orthogonal_scaleR orthogonal_self pairwise_def)
+    then show "card ?S = DIM('a)"
+      by (simp add: card_image S)
+    show "span ?S = UNIV"
+      by (metis (no_types) \<open>0 \<notin> S\<close> \<open>finite S\<close> \<open>span S = UNIV\<close>
+          field_class.field_inverse_zero inverse_inverse_eq less_irrefl span_image_scale
+          zero_less_norm_iff)
+  qed
+qed
+
+proposition dim_orthogonal_sum:
+  fixes A :: "'a::euclidean_space set"
+  assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+    shows "dim(A \<union> B) = dim A + dim B"
+proof -
+  have 1: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+    by (erule span_induct [OF _ subspace_hyperplane2]; simp add: assms)
+  have "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+    using 1 by (simp add: span_induct [OF _ subspace_hyperplane])
+  then have 0: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
+    by simp
+  have "dim(A \<union> B) = dim (span (A \<union> B))"
+  also have "span (A \<union> B) = ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
+    by (auto simp add: span_Un image_def)
+  also have "dim \<dots> = dim {x + y |x y. x \<in> span A \<and> y \<in> span B}"
+    by (auto intro!: arg_cong [where f=dim])
+  also have "... = dim {x + y |x y. x \<in> span A \<and> y \<in> span B} + dim(span A \<inter> span B)"
+    by (auto simp: dest: 0)
+  also have "... = dim (span A) + dim (span B)"
+    by (rule dim_sums_Int) (auto simp: subspace_span)
+  also have "... = dim A + dim B"
+  finally show ?thesis .
+qed
+
+lemma dim_subspace_orthogonal_to_vectors:
+  fixes A :: "'a::euclidean_space set"
+  assumes "subspace A" "subspace B" "A \<subseteq> B"
+    shows "dim {y \<in> B. \<forall>x \<in> A. orthogonal x y} + dim A = dim B"
+proof -
+  have "dim (span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)) = dim (span B)"
+  proof (rule arg_cong [where f=dim, OF subset_antisym])
+    show "span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A) \<subseteq> span B"
+      by (simp add: \<open>A \<subseteq> B\<close> Collect_restrict span_mono)
+  next
+    have *: "x \<in> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
+         if "x \<in> B" for x
+    proof -
+      obtain y z where "x = y + z" "y \<in> span A" and orth: "\<And>w. w \<in> span A \<Longrightarrow> orthogonal z w"
+        using orthogonal_subspace_decomp_exists [of A x] that by auto
+      have "y \<in> span B"
+        using \<open>y \<in> span A\<close> assms(3) span_mono by blast
+      then have "z \<in> {a \<in> B. \<forall>x. x \<in> A \<longrightarrow> orthogonal x a}"
+        apply simp
+        using \<open>x = y + z\<close> assms(1) assms(2) orth orthogonal_commute span_add_eq
+          span_eq_iff that by blast
+      then have z: "z \<in> span {y \<in> B. \<forall>x\<in>A. orthogonal x y}"
+        by (meson span_superset subset_iff)
+      then show ?thesis
+        apply (auto simp: span_Un image_def  \<open>x = y + z\<close> \<open>y \<in> span A\<close>)
+        using \<open>y \<in> span A\<close> add.commute by blast
+    qed
+    show "span B \<subseteq> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
+      by (rule span_minimal)
+        (auto intro: * span_minimal simp: subspace_span)
+  qed
+  then show ?thesis
+    by (metis (no_types, lifting) dim_orthogonal_sum dim_span mem_Collect_eq
+        orthogonal_commute orthogonal_def)
+qed
+
end```
```--- a/src/HOL/Analysis/Starlike.thy	Wed Jan 16 16:50:35 2019 -0500
+++ b/src/HOL/Analysis/Starlike.thy	Wed Jan 16 18:14:02 2019 -0500
@@ -5469,103 +5469,6 @@
(if a = 0 \<and> r > 0 then -1 else DIM('a))"
using aff_dim_halfspace_le [of "-a" "-r"] by simp

-subsection\<open>Properties of special hyperplanes\<close>
-
-lemma subspace_hyperplane: "subspace {x. a \<bullet> x = 0}"
-  by (simp add: subspace_def inner_right_distrib)
-
-lemma subspace_hyperplane2: "subspace {x. x \<bullet> a = 0}"
-  by (simp add: inner_commute inner_right_distrib subspace_def)
-
-lemma special_hyperplane_span:
-  fixes S :: "'n::euclidean_space set"
-  assumes "k \<in> Basis"
-  shows "{x. k \<bullet> x = 0} = span (Basis - {k})"
-proof -
-  have *: "x \<in> span (Basis - {k})" if "k \<bullet> x = 0" for x
-  proof -
-    have "x = (\<Sum>b\<in>Basis. (x \<bullet> b) *\<^sub>R b)"
-    also have "... = (\<Sum>b \<in> Basis - {k}. (x \<bullet> b) *\<^sub>R b)"
-      by (auto simp: sum.remove [of _ k] inner_commute assms that)
-    finally have "x = (\<Sum>b\<in>Basis - {k}. (x \<bullet> b) *\<^sub>R b)" .
-    then show ?thesis
-  qed
-  show ?thesis
-    apply (rule span_subspace [symmetric])
-    using assms
-    apply (auto simp: inner_not_same_Basis intro: * subspace_hyperplane)
-    done
-qed
-
-lemma dim_special_hyperplane:
-  fixes k :: "'n::euclidean_space"
-  shows "k \<in> Basis \<Longrightarrow> dim {x. k \<bullet> x = 0} = DIM('n) - 1"
-apply (rule dim_unique [OF subset_refl])
-apply (auto simp: independent_substdbasis)
-apply (metis member_remove remove_def span_base)
-done
-
-proposition dim_hyperplane:
-  fixes a :: "'a::euclidean_space"
-  assumes "a \<noteq> 0"
-    shows "dim {x. a \<bullet> x = 0} = DIM('a) - 1"
-proof -
-  have span0: "span {x. a \<bullet> x = 0} = {x. a \<bullet> x = 0}"
-    by (rule span_unique) (auto simp: subspace_hyperplane)
-  then obtain B where "independent B"
-              and Bsub: "B \<subseteq> {x. a \<bullet> x = 0}"
-              and subspB: "{x. a \<bullet> x = 0} \<subseteq> span B"
-              and card0: "(card B = dim {x. a \<bullet> x = 0})"
-              and ortho: "pairwise orthogonal B"
-    using orthogonal_basis_exists by metis
-  with assms have "a \<notin> span B"
-    by (metis (mono_tags, lifting) span_eq inner_eq_zero_iff mem_Collect_eq span0)
-  then have ind: "independent (insert a B)"
-    by (simp add: \<open>independent B\<close> independent_insert)
-  have "finite B"
-    using \<open>independent B\<close> independent_bound by blast
-  have "UNIV \<subseteq> span (insert a B)"
-  proof fix y::'a
-    obtain r z where z: "y = r *\<^sub>R a + z" "a \<bullet> z = 0"
-      apply (rule_tac r="(a \<bullet> y) / (a \<bullet> a)" and z = "y - ((a \<bullet> y) / (a \<bullet> a)) *\<^sub>R a" in that)
-      using assms
-      by (auto simp: algebra_simps)
-    show "y \<in> span (insert a B)"
-      by (metis (mono_tags, lifting) z Bsub span_eq_iff
-         add_diff_cancel_left' mem_Collect_eq span0 span_breakdown_eq span_subspace subspB)
-  qed
-  then have dima: "DIM('a) = dim(insert a B)"
-    by (metis independent_Basis span_Basis dim_eq_card top.extremum_uniqueI)
-  then show ?thesis
-    by (metis (mono_tags, lifting) Bsub Diff_insert_absorb \<open>a \<notin> span B\<close> ind card0
-        card_Diff_singleton dim_span indep_card_eq_dim_span insertI1 subsetCE
-        subspB)
-qed
-
-lemma lowdim_eq_hyperplane:
-  fixes S :: "'a::euclidean_space set"
-  assumes "dim S = DIM('a) - 1"
-  obtains a where "a \<noteq> 0" and "span S = {x. a \<bullet> x = 0}"
-proof -
-  have dimS: "dim S < DIM('a)"
-  then obtain b where b: "b \<noteq> 0" "span S \<subseteq> {a. b \<bullet> a = 0}"
-    using lowdim_subset_hyperplane [of S] by fastforce
-  show ?thesis
-    apply (rule that[OF b(1)])
-    apply (rule subspace_dim_equal)
-    by (auto simp: assms b dim_hyperplane dim_span subspace_hyperplane
-        subspace_span)
-qed
-
-lemma dim_eq_hyperplane:
-  fixes S :: "'n::euclidean_space set"
-  shows "dim S = DIM('n) - 1 \<longleftrightarrow> (\<exists>a. a \<noteq> 0 \<and> span S = {x. a \<bullet> x = 0})"
-by (metis One_nat_def dim_hyperplane dim_span lowdim_eq_hyperplane)
-
proposition aff_dim_eq_hyperplane:
fixes S :: "'a::euclidean_space set"
shows "aff_dim S = DIM('a) - 1 \<longleftrightarrow> (\<exists>a b. a \<noteq> 0 \<and> affine hull S = {x. a \<bullet> x = b})"
@@ -6436,444 +6339,6 @@
shows "aff_dim S < DIM('a) \<longleftrightarrow> (affine hull S \<noteq> UNIV)"
by (metis (no_types) aff_dim_affine_hull aff_dim_le_DIM aff_dim_UNIV affine_hull_UNIV less_le)

-
-subsection\<open> Orthogonal bases, Gram-Schmidt process, and related theorems\<close>
-
-lemma pairwise_orthogonal_independent:
-  assumes "pairwise orthogonal S" and "0 \<notin> S"
-    shows "independent S"
-proof -
-  have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
-    using assms by (simp add: pairwise_def orthogonal_def)
-  have "False" if "a \<in> S" and a: "a \<in> span (S - {a})" for a
-  proof -
-    obtain T U where "T \<subseteq> S - {a}" "a = (\<Sum>v\<in>T. U v *\<^sub>R v)"
-      using a by (force simp: span_explicit)
-    then have "a \<bullet> a = a \<bullet> (\<Sum>v\<in>T. U v *\<^sub>R v)"
-      by simp
-    also have "... = 0"
-      by (metis "0" DiffE \<open>T \<subseteq> S - {a}\<close> mult_not_zero singletonI subsetCE \<open>a \<in> S\<close>)
-    finally show ?thesis
-      using \<open>0 \<notin> S\<close> \<open>a \<in> S\<close> by auto
-  qed
-  then show ?thesis
-    by (force simp: dependent_def)
-qed
-
-lemma pairwise_orthogonal_imp_finite:
-  fixes S :: "'a::euclidean_space set"
-  assumes "pairwise orthogonal S"
-    shows "finite S"
-proof -
-  have "independent (S - {0})"
-    apply (rule pairwise_orthogonal_independent)
-     apply (metis Diff_iff assms pairwise_def)
-    by blast
-  then show ?thesis
-    by (meson independent_imp_finite infinite_remove)
-qed
-
-lemma subspace_orthogonal_to_vector: "subspace {y. orthogonal x y}"
-  by (simp add: subspace_def orthogonal_clauses)
-
-lemma subspace_orthogonal_to_vectors: "subspace {y. \<forall>x \<in> S. orthogonal x y}"
-  by (simp add: subspace_def orthogonal_clauses)
-
-lemma orthogonal_to_span:
-  assumes a: "a \<in> span S" and x: "\<And>y. y \<in> S \<Longrightarrow> orthogonal x y"
-    shows "orthogonal x a"
-  by (metis a orthogonal_clauses(1,2,4)
-      span_induct_alt x)
-
-proposition Gram_Schmidt_step:
-  fixes S :: "'a::euclidean_space set"
-  assumes S: "pairwise orthogonal S" and x: "x \<in> span S"
-    shows "orthogonal x (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b))"
-proof -
-  have "finite S"
-    by (simp add: S pairwise_orthogonal_imp_finite)
-  have "orthogonal (a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)) x"
-       if "x \<in> S" for x
-  proof -
-    have "a \<bullet> x = (\<Sum>y\<in>S. if y = x then y \<bullet> a else 0)"
-      by (simp add: \<open>finite S\<close> inner_commute sum.delta that)
-    also have "... =  (\<Sum>b\<in>S. b \<bullet> a * (b \<bullet> x) / (b \<bullet> b))"
-      apply (rule sum.cong [OF refl], simp)
-      by (meson S orthogonal_def pairwise_def that)
-   finally show ?thesis
-     by (simp add: orthogonal_def algebra_simps inner_sum_left)
-  qed
-  then show ?thesis
-    using orthogonal_to_span orthogonal_commute x by blast
-qed
-
-
-lemma orthogonal_extension_aux:
-  fixes S :: "'a::euclidean_space set"
-  assumes "finite T" "finite S" "pairwise orthogonal S"
-    shows "\<exists>U. pairwise orthogonal (S \<union> U) \<and> span (S \<union> U) = span (S \<union> T)"
-using assms
-proof (induction arbitrary: S)
-  case empty then show ?case
-    by simp (metis sup_bot_right)
-next
-  case (insert a T)
-  have 0: "\<And>x y. \<lbrakk>x \<noteq> y; x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
-    using insert by (simp add: pairwise_def orthogonal_def)
-  define a' where "a' = a - (\<Sum>b\<in>S. (b \<bullet> a / (b \<bullet> b)) *\<^sub>R b)"
-  obtain U where orthU: "pairwise orthogonal (S \<union> insert a' U)"
-             and spanU: "span (insert a' S \<union> U) = span (insert a' S \<union> T)"
-    by (rule exE [OF insert.IH [of "insert a' S"]])
-      (auto simp: Gram_Schmidt_step a'_def insert.prems orthogonal_commute
-        pairwise_orthogonal_insert span_clauses)
-  have orthS: "\<And>x. x \<in> S \<Longrightarrow> a' \<bullet> x = 0"
-    using Gram_Schmidt_step [OF \<open>pairwise orthogonal S\<close>]
-    apply (force simp: orthogonal_def inner_commute span_superset [THEN subsetD])
-    done
-  have "span (S \<union> insert a' U) = span (insert a' (S \<union> T))"
-    using spanU by simp
-  also have "... = span (insert a (S \<union> T))"
-    apply (rule eq_span_insert_eq)
-    apply (simp add: a'_def span_neg span_sum span_base span_mul)
-    done
-  also have "... = span (S \<union> insert a T)"
-    by simp
-  finally show ?case
-    by (rule_tac x="insert a' U" in exI) (use orthU in auto)
-qed
-
-
-proposition orthogonal_extension:
-  fixes S :: "'a::euclidean_space set"
-  assumes S: "pairwise orthogonal S"
-  obtains U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
-proof -
-  obtain B where "finite B" "span B = span T"
-    using basis_subspace_exists [of "span T"] subspace_span by metis
-  with orthogonal_extension_aux [of B S]
-  obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> B)"
-    using assms pairwise_orthogonal_imp_finite by auto
-  with \<open>span B = span T\<close> show ?thesis
-    by (rule_tac U=U in that) (auto simp: span_Un)
-qed
-
-corollary%unimportant orthogonal_extension_strong:
-  fixes S :: "'a::euclidean_space set"
-  assumes S: "pairwise orthogonal S"
-  obtains U where "U \<inter> (insert 0 S) = {}" "pairwise orthogonal (S \<union> U)"
-                  "span (S \<union> U) = span (S \<union> T)"
-proof -
-  obtain U where "pairwise orthogonal (S \<union> U)" "span (S \<union> U) = span (S \<union> T)"
-    using orthogonal_extension assms by blast
-  then show ?thesis
-    apply (rule_tac U = "U - (insert 0 S)" in that)
-      apply blast
-     apply (force simp: pairwise_def)
-    apply (metis Un_Diff_cancel Un_insert_left span_redundant span_zero)
-    done
-qed
-
-subsection\<open>Decomposing a vector into parts in orthogonal subspaces\<close>
-
-text\<open>existence of orthonormal basis for a subspace.\<close>
-
-lemma orthogonal_spanningset_subspace:
-  fixes S :: "'a :: euclidean_space set"
-  assumes "subspace S"
-  obtains B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
-proof -
-  obtain B where "B \<subseteq> S" "independent B" "S \<subseteq> span B" "card B = dim S"
-    using basis_exists by blast
-  with orthogonal_extension [of "{}" B]
-  show ?thesis
-    by (metis Un_empty_left assms pairwise_empty span_superset span_subspace that)
-qed
-
-lemma orthogonal_basis_subspace:
-  fixes S :: "'a :: euclidean_space set"
-  assumes "subspace S"
-  obtains B where "0 \<notin> B" "B \<subseteq> S" "pairwise orthogonal B" "independent B"
-                  "card B = dim S" "span B = S"
-proof -
-  obtain B where "B \<subseteq> S" "pairwise orthogonal B" "span B = S"
-    using assms orthogonal_spanningset_subspace by blast
-  then show ?thesis
-    apply (rule_tac B = "B - {0}" in that)
-    apply (auto simp: indep_card_eq_dim_span pairwise_subset pairwise_orthogonal_independent elim: pairwise_subset)
-    done
-qed
-
-proposition orthonormal_basis_subspace:
-  fixes S :: "'a :: euclidean_space set"
-  assumes "subspace S"
-  obtains B where "B \<subseteq> S" "pairwise orthogonal B"
-              and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
-              and "independent B" "card B = dim S" "span B = S"
-proof -
-  obtain B where "0 \<notin> B" "B \<subseteq> S"
-             and orth: "pairwise orthogonal B"
-             and "independent B" "card B = dim S" "span B = S"
-    by (blast intro: orthogonal_basis_subspace [OF assms])
-  have 1: "(\<lambda>x. x /\<^sub>R norm x) ` B \<subseteq> S"
-    using \<open>span B = S\<close> span_superset span_mul by fastforce
-  have 2: "pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` B)"
-    using orth by (force simp: pairwise_def orthogonal_clauses)
-  have 3: "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` B \<Longrightarrow> norm x = 1"
-    by (metis (no_types, lifting) \<open>0 \<notin> B\<close> image_iff norm_sgn sgn_div_norm)
-  have 4: "independent ((\<lambda>x. x /\<^sub>R norm x) ` B)"
-    by (metis "2" "3" norm_zero pairwise_orthogonal_independent zero_neq_one)
-  have "inj_on (\<lambda>x. x /\<^sub>R norm x) B"
-  proof
-    fix x y
-    assume "x \<in> B" "y \<in> B" "x /\<^sub>R norm x = y /\<^sub>R norm y"
-    moreover have "\<And>i. i \<in> B \<Longrightarrow> norm (i /\<^sub>R norm i) = 1"
-      using 3 by blast
-    ultimately show "x = y"
-      by (metis norm_eq_1 orth orthogonal_clauses(7) orthogonal_commute orthogonal_def pairwise_def zero_neq_one)
-  qed
-  then have 5: "card ((\<lambda>x. x /\<^sub>R norm x) ` B) = dim S"
-    by (metis \<open>card B = dim S\<close> card_image)
-  have 6: "span ((\<lambda>x. x /\<^sub>R norm x) ` B) = S"
-    by (metis "1" "4" "5" assms card_eq_dim independent_finite span_subspace)
-  show ?thesis
-    by (rule that [OF 1 2 3 4 5 6])
-qed
-
-
-proposition%unimportant orthogonal_to_subspace_exists_gen:
-  fixes S :: "'a :: euclidean_space set"
-  assumes "span S \<subset> span T"
-  obtains x where "x \<noteq> 0" "x \<in> span T" "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
-proof -
-  obtain B where "B \<subseteq> span S" and orthB: "pairwise orthogonal B"
-             and "\<And>x. x \<in> B \<Longrightarrow> norm x = 1"
-             and "independent B" "card B = dim S" "span B = span S"
-    by (rule orthonormal_basis_subspace [of "span S", OF subspace_span])
-      (auto simp: dim_span)
-  with assms obtain u where spanBT: "span B \<subseteq> span T" and "u \<notin> span B" "u \<in> span T"
-    by auto
-  obtain C where orthBC: "pairwise orthogonal (B \<union> C)" and spanBC: "span (B \<union> C) = span (B \<union> {u})"
-    by (blast intro: orthogonal_extension [OF orthB])
-  show thesis
-  proof (cases "C \<subseteq> insert 0 B")
-    case True
-    then have "C \<subseteq> span B"
-      using span_eq
-      by (metis span_insert_0 subset_trans)
-    moreover have "u \<in> span (B \<union> C)"
-      using \<open>span (B \<union> C) = span (B \<union> {u})\<close> span_superset by force
-    ultimately show ?thesis
-      using True \<open>u \<notin> span B\<close>
-      by (metis Un_insert_left span_insert_0 sup.orderE)
-  next
-    case False
-    then obtain x where "x \<in> C" "x \<noteq> 0" "x \<notin> B"
-      by blast
-    then have "x \<in> span T"
-      by (metis (no_types, lifting) Un_insert_right Un_upper2 \<open>u \<in> span T\<close> spanBT spanBC
-          \<open>u \<in> span T\<close> insert_subset span_superset span_mono
-          span_span subsetCE subset_trans sup_bot.comm_neutral)
-    moreover have "orthogonal x y" if "y \<in> span B" for y
-      using that
-    proof (rule span_induct)
-      show "subspace {a. orthogonal x a}"
-      show "\<And>b. b \<in> B \<Longrightarrow> orthogonal x b"
-        by (metis Un_iff \<open>x \<in> C\<close> \<open>x \<notin> B\<close> orthBC pairwise_def)
-    qed
-    ultimately show ?thesis
-      using \<open>x \<noteq> 0\<close> that \<open>span B = span S\<close> by auto
-  qed
-qed
-
-corollary%unimportant orthogonal_to_subspace_exists:
-  fixes S :: "'a :: euclidean_space set"
-  assumes "dim S < DIM('a)"
-  obtains x where "x \<noteq> 0" "\<And>y. y \<in> span S \<Longrightarrow> orthogonal x y"
-proof -
-have "span S \<subset> UNIV"
-  by (metis (mono_tags) UNIV_I assms inner_eq_zero_iff less_le lowdim_subset_hyperplane
-      mem_Collect_eq top.extremum_strict top.not_eq_extremum)
-  with orthogonal_to_subspace_exists_gen [of S UNIV] that show ?thesis
-    by (auto simp: span_UNIV)
-qed
-
-corollary%unimportant orthogonal_to_vector_exists:
-  fixes x :: "'a :: euclidean_space"
-  assumes "2 \<le> DIM('a)"
-  obtains y where "y \<noteq> 0" "orthogonal x y"
-proof -
-  have "dim {x} < DIM('a)"
-    using assms by auto
-  then show thesis
-    by (rule orthogonal_to_subspace_exists) (simp add: orthogonal_commute span_base that)
-qed
-
-proposition%unimportant orthogonal_subspace_decomp_exists:
-  fixes S :: "'a :: euclidean_space set"
-  obtains y z
-  where "y \<in> span S"
-    and "\<And>w. w \<in> span S \<Longrightarrow> orthogonal z w"
-    and "x = y + z"
-proof -
-  obtain T where "0 \<notin> T" "T \<subseteq> span S" "pairwise orthogonal T" "independent T"
-    "card T = dim (span S)" "span T = span S"
-    using orthogonal_basis_subspace subspace_span by blast
-  let ?a = "\<Sum>b\<in>T. (b \<bullet> x / (b \<bullet> b)) *\<^sub>R b"
-  have orth: "orthogonal (x - ?a) w" if "w \<in> span S" for w
-    by (simp add: Gram_Schmidt_step \<open>pairwise orthogonal T\<close> \<open>span T = span S\<close>
-        orthogonal_commute that)
-  show ?thesis
-    apply (rule_tac y = "?a" and z = "x - ?a" in that)
-      apply (meson \<open>T \<subseteq> span S\<close> span_scale span_sum subsetCE)
-     apply (fact orth, simp)
-    done
-qed
-
-lemma orthogonal_subspace_decomp_unique:
-  fixes S :: "'a :: euclidean_space set"
-  assumes "x + y = x' + y'"
-      and ST: "x \<in> span S" "x' \<in> span S" "y \<in> span T" "y' \<in> span T"
-      and orth: "\<And>a b. \<lbrakk>a \<in> S; b \<in> T\<rbrakk> \<Longrightarrow> orthogonal a b"
-  shows "x = x' \<and> y = y'"
-proof -
-  have "x + y - y' = x'"
-  moreover have "\<And>a b. \<lbrakk>a \<in> span S; b \<in> span T\<rbrakk> \<Longrightarrow> orthogonal a b"
-    by (meson orth orthogonal_commute orthogonal_to_span)
-  ultimately have "0 = x' - x"
-    by (metis (full_types) add_diff_cancel_left' ST diff_right_commute orthogonal_clauses(10) orthogonal_clauses(5) orthogonal_self)
-  with assms show ?thesis by auto
-qed
-
-lemma vector_in_orthogonal_spanningset:
-  fixes a :: "'a::euclidean_space"
-  obtains S where "a \<in> S" "pairwise orthogonal S" "span S = UNIV"
-  by (metis UNIV_I Un_iff empty_iff insert_subset orthogonal_extension pairwise_def
-      pairwise_orthogonal_insert span_UNIV subsetI subset_antisym)
-
-lemma vector_in_orthogonal_basis:
-  fixes a :: "'a::euclidean_space"
-  assumes "a \<noteq> 0"
-  obtains S where "a \<in> S" "0 \<notin> S" "pairwise orthogonal S" "independent S" "finite S"
-                  "span S = UNIV" "card S = DIM('a)"
-proof -
-  obtain S where S: "a \<in> S" "pairwise orthogonal S" "span S = UNIV"
-    using vector_in_orthogonal_spanningset .
-  show thesis
-  proof
-    show "pairwise orthogonal (S - {0})"
-      using pairwise_mono S(2) by blast
-    show "independent (S - {0})"
-      by (simp add: \<open>pairwise orthogonal (S - {0})\<close> pairwise_orthogonal_independent)
-    show "finite (S - {0})"
-      using \<open>independent (S - {0})\<close> independent_finite by blast
-    show "card (S - {0}) = DIM('a)"
-      using span_delete_0 [of S] S
-      by (simp add: \<open>independent (S - {0})\<close> indep_card_eq_dim_span dim_UNIV)
-  qed (use S \<open>a \<noteq> 0\<close> in auto)
-qed
-
-lemma vector_in_orthonormal_basis:
-  fixes a :: "'a::euclidean_space"
-  assumes "norm a = 1"
-  obtains S where "a \<in> S" "pairwise orthogonal S" "\<And>x. x \<in> S \<Longrightarrow> norm x = 1"
-    "independent S" "card S = DIM('a)" "span S = UNIV"
-proof -
-  have "a \<noteq> 0"
-    using assms by auto
-  then obtain S where "a \<in> S" "0 \<notin> S" "finite S"
-          and S: "pairwise orthogonal S" "independent S" "span S = UNIV" "card S = DIM('a)"
-    by (metis vector_in_orthogonal_basis)
-  let ?S = "(\<lambda>x. x /\<^sub>R norm x) ` S"
-  show thesis
-  proof
-    show "a \<in> ?S"
-      using \<open>a \<in> S\<close> assms image_iff by fastforce
-  next
-    show "pairwise orthogonal ?S"
-      using \<open>pairwise orthogonal S\<close> by (auto simp: pairwise_def orthogonal_def)
-    show "\<And>x. x \<in> (\<lambda>x. x /\<^sub>R norm x) ` S \<Longrightarrow> norm x = 1"
-      using \<open>0 \<notin> S\<close> by (auto simp: divide_simps)
-    then show "independent ?S"
-      by (metis \<open>pairwise orthogonal ((\<lambda>x. x /\<^sub>R norm x) ` S)\<close> norm_zero pairwise_orthogonal_independent zero_neq_one)
-    have "inj_on (\<lambda>x. x /\<^sub>R norm x) S"
-      unfolding inj_on_def
-      by (metis (full_types) S(1) \<open>0 \<notin> S\<close> inverse_nonzero_iff_nonzero norm_eq_zero orthogonal_scaleR orthogonal_self pairwise_def)
-    then show "card ?S = DIM('a)"
-      by (simp add: card_image S)
-    show "span ?S = UNIV"
-      by (metis (no_types) \<open>0 \<notin> S\<close> \<open>finite S\<close> \<open>span S = UNIV\<close>
-          field_class.field_inverse_zero inverse_inverse_eq less_irrefl span_image_scale
-          zero_less_norm_iff)
-  qed
-qed
-
-proposition dim_orthogonal_sum:
-  fixes A :: "'a::euclidean_space set"
-  assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
-    shows "dim(A \<union> B) = dim A + dim B"
-proof -
-  have 1: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
-    by (erule span_induct [OF _ subspace_hyperplane2]; simp add: assms)
-  have "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
-    using 1 by (simp add: span_induct [OF _ subspace_hyperplane])
-  then have 0: "\<And>x y. \<lbrakk>x \<in> span A; y \<in> span B\<rbrakk> \<Longrightarrow> x \<bullet> y = 0"
-    by simp
-  have "dim(A \<union> B) = dim (span (A \<union> B))"
-  also have "span (A \<union> B) = ((\<lambda>(a, b). a + b) ` (span A \<times> span B))"
-    by (auto simp add: span_Un image_def)
-  also have "dim \<dots> = dim {x + y |x y. x \<in> span A \<and> y \<in> span B}"
-    by (auto intro!: arg_cong [where f=dim])
-  also have "... = dim {x + y |x y. x \<in> span A \<and> y \<in> span B} + dim(span A \<inter> span B)"
-    by (auto simp: dest: 0)
-  also have "... = dim (span A) + dim (span B)"
-    by (rule dim_sums_Int) (auto simp: subspace_span)
-  also have "... = dim A + dim B"
-  finally show ?thesis .
-qed
-
-lemma dim_subspace_orthogonal_to_vectors:
-  fixes A :: "'a::euclidean_space set"
-  assumes "subspace A" "subspace B" "A \<subseteq> B"
-    shows "dim {y \<in> B. \<forall>x \<in> A. orthogonal x y} + dim A = dim B"
-proof -
-  have "dim (span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)) = dim (span B)"
-  proof (rule arg_cong [where f=dim, OF subset_antisym])
-    show "span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A) \<subseteq> span B"
-      by (simp add: \<open>A \<subseteq> B\<close> Collect_restrict span_mono)
-  next
-    have *: "x \<in> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
-         if "x \<in> B" for x
-    proof -
-      obtain y z where "x = y + z" "y \<in> span A" and orth: "\<And>w. w \<in> span A \<Longrightarrow> orthogonal z w"
-        using orthogonal_subspace_decomp_exists [of A x] that by auto
-      have "y \<in> span B"
-        using \<open>y \<in> span A\<close> assms(3) span_mono by blast
-      then have "z \<in> {a \<in> B. \<forall>x. x \<in> A \<longrightarrow> orthogonal x a}"
-        apply simp
-        using \<open>x = y + z\<close> assms(1) assms(2) orth orthogonal_commute span_add_eq
-          span_eq_iff that by blast
-      then have z: "z \<in> span {y \<in> B. \<forall>x\<in>A. orthogonal x y}"
-        by (meson span_superset subset_iff)
-      then show ?thesis
-        apply (auto simp: span_Un image_def  \<open>x = y + z\<close> \<open>y \<in> span A\<close>)
-        using \<open>y \<in> span A\<close> add.commute by blast
-    qed
-    show "span B \<subseteq> span ({y \<in> B. \<forall>x\<in>A. orthogonal x y} \<union> A)"
-      by (rule span_minimal)
-        (auto intro: * span_minimal simp: subspace_span)
-  qed
-  then show ?thesis
-    by (metis (no_types, lifting) dim_orthogonal_sum dim_span mem_Collect_eq
-        orthogonal_commute orthogonal_def)
-qed
-
lemma aff_dim_openin:
fixes S :: "'a::euclidean_space set"
assumes ope: "openin (subtopology euclidean T) S" and "affine T" "S \<noteq> {}"```