src/HOL/Analysis/Determinants.thy
 author immler Thu, 24 May 2018 17:06:39 +0200 changeset 68263 e4e536a71e3d parent 68143 58c9231c2937 child 68833 fde093888c16 permissions -rw-r--r--
generalized Cramer's rule
```
(*  Title:      HOL/Analysis/Determinants.thy
Author:     Amine Chaieb, University of Cambridge; proofs reworked by LCP
*)

section \<open>Traces, Determinant of square matrices and some properties\<close>

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

subsection \<open>Trace\<close>

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

lemma trace_0: "trace (mat 0) = 0"

lemma trace_I: "trace (mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))"

lemma trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"

lemma trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"

lemma trace_mul_sym: "trace ((A::'a::comm_semiring_1^'n^'m) ** B) = trace (B**A)"
apply (subst sum.swap)
done

subsubsection \<open>Definition of determinant\<close>

definition det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
"det A =
sum (\<lambda>p. of_int (sign p) * prod (\<lambda>i. A\$i\$p i) (UNIV :: 'n set))
{p. p permutes (UNIV :: 'n set)}"

text \<open>Basic determinant properties\<close>

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

lemma det_lowerdiagonal:
fixes A :: "'a::comm_ring_1^('n::{finite,wellorder})^('n::{finite,wellorder})"
assumes ld: "\<And>i j. i < j \<Longrightarrow> A\$i\$j = 0"
shows "det A = prod (\<lambda>i. A\$i\$i) (UNIV:: 'n set)"
proof -
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "\<lambda>p. of_int (sign p) * prod (\<lambda>i. A\$i\$p i) (UNIV :: 'n set)"
have fU: "finite ?U"
by simp
have id0: "{id} \<subseteq> ?PU"
by (auto simp: permutes_id)
have p0: "\<forall>p \<in> ?PU - {id}. ?pp p = 0"
proof
fix p
assume "p \<in> ?PU - {id}"
then obtain i where i: "p i > i"
by clarify (meson leI permutes_natset_le)
from ld[OF i] have "\<exists>i \<in> ?U. A\$i\$p i = 0"
by blast
with prod_zero[OF fU] show "?pp p = 0"
by force
qed
from sum.mono_neutral_cong_left[OF finite_permutations[OF fU] id0 p0] show ?thesis
unfolding det_def by (simp add: sign_id)
qed

lemma det_upperdiagonal:
fixes A :: "'a::comm_ring_1^'n::{finite,wellorder}^'n::{finite,wellorder}"
assumes ld: "\<And>i j. i > j \<Longrightarrow> A\$i\$j = 0"
shows "det A = prod (\<lambda>i. A\$i\$i) (UNIV:: 'n set)"
proof -
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "(\<lambda>p. of_int (sign p) * prod (\<lambda>i. A\$i\$p i) (UNIV :: 'n set))"
have fU: "finite ?U"
by simp
have id0: "{id} \<subseteq> ?PU"
by (auto simp: permutes_id)
have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0"
proof
fix p
assume p: "p \<in> ?PU - {id}"
then obtain i where i: "p i < i"
by clarify (meson leI permutes_natset_ge)
from ld[OF i] have "\<exists>i \<in> ?U. A\$i\$p i = 0"
by blast
with prod_zero[OF fU]  show "?pp p = 0"
by force
qed
from sum.mono_neutral_cong_left[OF finite_permutations[OF fU] id0 p0] show ?thesis
unfolding det_def by (simp add: sign_id)
qed

lemma det_diagonal:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes ld: "\<And>i j. i \<noteq> j \<Longrightarrow> A\$i\$j = 0"
shows "det A = prod (\<lambda>i. A\$i\$i) (UNIV::'n set)"
proof -
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "\<lambda>p. of_int (sign p) * prod (\<lambda>i. A\$i\$p i) (UNIV :: 'n set)"
have fU: "finite ?U" by simp
from finite_permutations[OF fU] have fPU: "finite ?PU" .
have id0: "{id} \<subseteq> ?PU"
by (auto simp: permutes_id)
have p0: "\<forall>p \<in> ?PU - {id}. ?pp p = 0"
proof
fix p
assume p: "p \<in> ?PU - {id}"
then obtain i where i: "p i \<noteq> i"
by fastforce
with ld have "\<exists>i \<in> ?U. A\$i\$p i = 0"
by (metis UNIV_I)
with prod_zero [OF fU] show "?pp p = 0"
by force
qed
from sum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
unfolding det_def by (simp add: sign_id)
qed

lemma det_I [simp]: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"

lemma det_0 [simp]: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"
by (simp add: det_def prod_zero power_0_left)

lemma det_permute_rows:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes p: "p permutes (UNIV :: 'n::finite set)"
shows "det (\<chi> i. A\$p i :: 'a^'n^'n) = of_int (sign p) * det A"
proof -
let ?U = "UNIV :: 'n set"
let ?PU = "{p. p permutes ?U}"
have *: "(\<Sum>q\<in>?PU. of_int (sign (q \<circ> p)) * (\<Prod>i\<in>?U. A \$ p i \$ (q \<circ> p) i)) =
(\<Sum>n\<in>?PU. of_int (sign p) * of_int (sign n) * (\<Prod>i\<in>?U. A \$ i \$ n i))"
proof (rule sum.cong)
fix q
assume qPU: "q \<in> ?PU"
have fU: "finite ?U"
by simp
from qPU have q: "q permutes ?U"
by blast
have "prod (\<lambda>i. A\$p i\$ (q \<circ> p) i) ?U = prod ((\<lambda>i. A\$p i\$(q \<circ> p) i) \<circ> inv p) ?U"
by (simp only: prod.permute[OF permutes_inv[OF p], symmetric])
also have "\<dots> = prod (\<lambda>i. A \$ (p \<circ> inv p) i \$ (q \<circ> (p \<circ> inv p)) i) ?U"
by (simp only: o_def)
also have "\<dots> = prod (\<lambda>i. A\$i\$q i) ?U"
by (simp only: o_def permutes_inverses[OF p])
finally have thp: "prod (\<lambda>i. A\$p i\$ (q \<circ> p) i) ?U = prod (\<lambda>i. A\$i\$q i) ?U"
by blast
from p q have pp: "permutation p" and qp: "permutation q"
by (metis fU permutation_permutes)+
show "of_int (sign (q \<circ> p)) * prod (\<lambda>i. A\$ p i\$ (q \<circ> p) i) ?U =
of_int (sign p) * of_int (sign q) * prod (\<lambda>i. A\$i\$q i) ?U"
by (simp only: thp sign_compose[OF qp pp] mult.commute of_int_mult)
qed auto
show ?thesis
apply (simp add: det_def sum_distrib_left mult.assoc[symmetric])
apply (subst sum_permutations_compose_right[OF p])
apply (rule *)
done
qed

lemma det_permute_columns:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes p: "p permutes (UNIV :: 'n set)"
shows "det(\<chi> i j. A\$i\$ p j :: 'a^'n^'n) = of_int (sign p) * det A"
proof -
let ?Ap = "\<chi> i j. A\$i\$ p j :: 'a^'n^'n"
let ?At = "transpose A"
have "of_int (sign p) * det A = det (transpose (\<chi> i. transpose A \$ p i))"
unfolding det_permute_rows[OF p, of ?At] det_transpose ..
moreover
have "?Ap = transpose (\<chi> i. transpose A \$ p i)"
ultimately show ?thesis
by simp
qed

lemma det_identical_columns:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes jk: "j \<noteq> k"
and r: "column j A = column k A"
shows "det A = 0"
proof -
let ?U="UNIV::'n set"
let ?t_jk="Fun.swap j k id"
let ?PU="{p. p permutes ?U}"
let ?S1="{p. p\<in>?PU \<and> evenperm p}"
let ?S2="{(?t_jk \<circ> p) |p. p \<in>?S1}"
let ?f="\<lambda>p. of_int (sign p) * (\<Prod>i\<in>UNIV. A \$ i \$ p i)"
let ?g="\<lambda>p. ?t_jk \<circ> p"
have g_S1: "?S2 = ?g` ?S1" by auto
have inj_g: "inj_on ?g ?S1"
proof (unfold inj_on_def, auto)
fix x y assume x: "x permutes ?U" and even_x: "evenperm x"
and y: "y permutes ?U" and even_y: "evenperm y" and eq: "?t_jk \<circ> x = ?t_jk \<circ> y"
show "x = y" by (metis (hide_lams, no_types) comp_assoc eq id_comp swap_id_idempotent)
qed
have tjk_permutes: "?t_jk permutes ?U" unfolding permutes_def swap_id_eq by (auto,metis)
have tjk_eq: "\<forall>i l. A \$ i \$ ?t_jk l  =  A \$ i \$ l"
using r jk
unfolding column_def vec_eq_iff swap_id_eq by fastforce
have sign_tjk: "sign ?t_jk = -1" using sign_swap_id[of j k] jk by auto
{fix x
assume x: "x\<in> ?S1"
have "sign (?t_jk \<circ> x) = sign (?t_jk) * sign x"
by (metis (lifting) finite_class.finite_UNIV mem_Collect_eq
permutation_permutes permutation_swap_id sign_compose x)
also have "\<dots> = - sign x" using sign_tjk by simp
also have "\<dots> \<noteq> sign x" unfolding sign_def by simp
finally have "sign (?t_jk \<circ> x) \<noteq> sign x" and "(?t_jk \<circ> x) \<in> ?S2"
using x by force+
}
hence disjoint: "?S1 \<inter> ?S2 = {}"
by (force simp: sign_def)
have PU_decomposition: "?PU = ?S1 \<union> ?S2"
proof (auto)
fix x
assume x: "x permutes ?U" and "\<forall>p. p permutes ?U \<longrightarrow> x = Fun.swap j k id \<circ> p \<longrightarrow> \<not> evenperm p"
then obtain p where p: "p permutes UNIV" and x_eq: "x = Fun.swap j k id \<circ> p"
and odd_p: "\<not> evenperm p"
by (metis (mono_tags) id_o o_assoc permutes_compose swap_id_idempotent tjk_permutes)
thus "evenperm x"
by (meson evenperm_comp evenperm_swap finite_class.finite_UNIV
jk permutation_permutes permutation_swap_id)
next
fix p assume p: "p permutes ?U"
show "Fun.swap j k id \<circ> p permutes UNIV" by (metis p permutes_compose tjk_permutes)
qed
have "sum ?f ?S2 = sum ((\<lambda>p. of_int (sign p) * (\<Prod>i\<in>UNIV. A \$ i \$ p i))
\<circ> (\<circ>) (Fun.swap j k id)) {p \<in> {p. p permutes UNIV}. evenperm p}"
unfolding g_S1 by (rule sum.reindex[OF inj_g])
also have "\<dots> = sum (\<lambda>p. of_int (sign (?t_jk \<circ> p)) * (\<Prod>i\<in>UNIV. A \$ i \$ p i)) ?S1"
unfolding o_def by (rule sum.cong, auto simp: tjk_eq)
also have "\<dots> = sum (\<lambda>p. - ?f p) ?S1"
proof (rule sum.cong, auto)
fix x assume x: "x permutes ?U"
and even_x: "evenperm x"
hence perm_x: "permutation x" and perm_tjk: "permutation ?t_jk"
using permutation_permutes[of x] permutation_permutes[of ?t_jk] permutation_swap_id
by (metis finite_code)+
have "(sign (?t_jk \<circ> x)) = - (sign x)"
unfolding sign_compose[OF perm_tjk perm_x] sign_tjk by auto
thus "of_int (sign (?t_jk \<circ> x)) * (\<Prod>i\<in>UNIV. A \$ i \$ x i)
= - (of_int (sign x) * (\<Prod>i\<in>UNIV. A \$ i \$ x i))"
by auto
qed
also have "\<dots>= - sum ?f ?S1" unfolding sum_negf ..
finally have *: "sum ?f ?S2 = - sum ?f ?S1" .
have "det A = (\<Sum>p | p permutes UNIV. of_int (sign p) * (\<Prod>i\<in>UNIV. A \$ i \$ p i))"
unfolding det_def ..
also have "\<dots>= sum ?f ?S1 + sum ?f ?S2"
by (subst PU_decomposition, rule sum.union_disjoint[OF _ _ disjoint], auto)
also have "\<dots>= sum ?f ?S1 - sum ?f ?S1 " unfolding * by auto
also have "\<dots>= 0" by simp
finally show "det A = 0" by simp
qed

lemma det_identical_rows:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes ij: "i \<noteq> j" and r: "row i A = row j A"
shows "det A = 0"
by (metis column_transpose det_identical_columns det_transpose ij r)

lemma det_zero_row:
fixes A :: "'a::{idom, ring_char_0}^'n^'n" and F :: "'b::{field}^'m^'m"
shows "row i A = 0 \<Longrightarrow> det A = 0" and "row j F = 0 \<Longrightarrow> det F = 0"
by (force simp: row_def det_def vec_eq_iff sign_nz intro!: sum.neutral)+

lemma det_zero_column:
fixes A :: "'a::{idom, ring_char_0}^'n^'n" and F :: "'b::{field}^'m^'m"
shows "column i A = 0 \<Longrightarrow> det A = 0" and "column j F = 0 \<Longrightarrow> det F = 0"
unfolding atomize_conj atomize_imp
by (metis det_transpose det_zero_row row_transpose)

fixes a b c :: "'n::finite \<Rightarrow> _ ^ 'n"
shows "det((\<chi> i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
det((\<chi> i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
det((\<chi> i. if i = k then b i else c i)::'a::comm_ring_1^'n^'n)"
unfolding det_def vec_lambda_beta sum.distrib[symmetric]
proof (rule sum.cong)
let ?U = "UNIV :: 'n set"
let ?pU = "{p. p permutes ?U}"
let ?f = "(\<lambda>i. if i = k then a i + b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
let ?g = "(\<lambda> i. if i = k then a i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
let ?h = "(\<lambda> i. if i = k then b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
fix p
assume p: "p \<in> ?pU"
let ?Uk = "?U - {k}"
from p have pU: "p permutes ?U"
by blast
have kU: "?U = insert k ?Uk"
by blast
have eq: "prod (\<lambda>i. ?f i \$ p i) ?Uk = prod (\<lambda>i. ?g i \$ p i) ?Uk"
"prod (\<lambda>i. ?f i \$ p i) ?Uk = prod (\<lambda>i. ?h i \$ p i) ?Uk"
by auto
have Uk: "finite ?Uk" "k \<notin> ?Uk"
by auto
have "prod (\<lambda>i. ?f i \$ p i) ?U = prod (\<lambda>i. ?f i \$ p i) (insert k ?Uk)"
unfolding kU[symmetric] ..
also have "\<dots> = ?f k \$ p k * prod (\<lambda>i. ?f i \$ p i) ?Uk"
by (rule prod.insert) auto
also have "\<dots> = (a k \$ p k * prod (\<lambda>i. ?f i \$ p i) ?Uk) + (b k\$ p k * prod (\<lambda>i. ?f i \$ p i) ?Uk)"
also have "\<dots> = (a k \$ p k * prod (\<lambda>i. ?g i \$ p i) ?Uk) + (b k\$ p k * prod (\<lambda>i. ?h i \$ p i) ?Uk)"
by (metis eq)
also have "\<dots> = prod (\<lambda>i. ?g i \$ p i) (insert k ?Uk) + prod (\<lambda>i. ?h i \$ p i) (insert k ?Uk)"
unfolding  prod.insert[OF Uk] by simp
finally have "prod (\<lambda>i. ?f i \$ p i) ?U = prod (\<lambda>i. ?g i \$ p i) ?U + prod (\<lambda>i. ?h i \$ p i) ?U"
unfolding kU[symmetric] .
then show "of_int (sign p) * prod (\<lambda>i. ?f i \$ p i) ?U =
of_int (sign p) * prod (\<lambda>i. ?g i \$ p i) ?U + of_int (sign p) * prod (\<lambda>i. ?h i \$ p i) ?U"
qed auto

lemma det_row_mul:
fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
c * det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
unfolding det_def vec_lambda_beta sum_distrib_left
proof (rule sum.cong)
let ?U = "UNIV :: 'n set"
let ?pU = "{p. p permutes ?U}"
let ?f = "(\<lambda>i. if i = k then c*s a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
let ?g = "(\<lambda> i. if i = k then a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
fix p
assume p: "p \<in> ?pU"
let ?Uk = "?U - {k}"
from p have pU: "p permutes ?U"
by blast
have kU: "?U = insert k ?Uk"
by blast
have eq: "prod (\<lambda>i. ?f i \$ p i) ?Uk = prod (\<lambda>i. ?g i \$ p i) ?Uk"
by auto
have Uk: "finite ?Uk" "k \<notin> ?Uk"
by auto
have "prod (\<lambda>i. ?f i \$ p i) ?U = prod (\<lambda>i. ?f i \$ p i) (insert k ?Uk)"
unfolding kU[symmetric] ..
also have "\<dots> = ?f k \$ p k  * prod (\<lambda>i. ?f i \$ p i) ?Uk"
by (rule prod.insert) auto
also have "\<dots> = (c*s a k) \$ p k * prod (\<lambda>i. ?f i \$ p i) ?Uk"
also have "\<dots> = c* (a k \$ p k * prod (\<lambda>i. ?g i \$ p i) ?Uk)"
unfolding eq by (simp add: ac_simps)
also have "\<dots> = c* (prod (\<lambda>i. ?g i \$ p i) (insert k ?Uk))"
unfolding prod.insert[OF Uk] by simp
finally have "prod (\<lambda>i. ?f i \$ p i) ?U = c* (prod (\<lambda>i. ?g i \$ p i) ?U)"
unfolding kU[symmetric] .
then show "of_int (sign p) * prod (\<lambda>i. ?f i \$ p i) ?U = c * (of_int (sign p) * prod (\<lambda>i. ?g i \$ p i) ?U)"
qed auto

lemma det_row_0:
fixes b :: "'n::finite \<Rightarrow> _ ^ 'n"
shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
using det_row_mul[of k 0 "\<lambda>i. 1" b]
apply simp
apply (simp only: vector_smult_lzero)
done

lemma det_row_operation:
fixes A :: "'a::{comm_ring_1}^'n^'n"
assumes ij: "i \<noteq> j"
shows "det (\<chi> k. if k = i then row i A + c *s row j A else row k A) = det A"
proof -
let ?Z = "(\<chi> k. if k = i then row j A else row k A) :: 'a ^'n^'n"
have th: "row i ?Z = row j ?Z" by (vector row_def)
have th2: "((\<chi> k. if k = i then row i A else row k A) :: 'a^'n^'n) = A"
by (vector row_def)
show ?thesis
unfolding det_row_add [of i] det_row_mul[of i] det_identical_rows[OF ij th] th2
by simp
qed

lemma det_row_span:
fixes A :: "'a::{field}^'n^'n"
assumes x: "x \<in> vec.span {row j A |j. j \<noteq> i}"
shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
using x
proof (induction rule: vec.span_induct_alt)
case base
have "(if k = i then row i A + 0 else row k A) = row k A" for k
by simp
then show ?case
next
case (step c z y)
then obtain j where j: "z = row j A" "i \<noteq> j"
by blast
let ?w = "row i A + y"
have th0: "row i A + (c*s z + y) = ?w + c*s z"
by vector
let ?d = "\<lambda>x. det (\<chi> k. if k = i then x else row k A)"
have thz: "?d z = 0"
apply (rule det_identical_rows[OF j(2)])
using j
apply (vector row_def)
done
have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)"
unfolding th0 ..
then have "?d (row i A + (c*s z + y)) = det A"
unfolding thz step.IH det_row_mul[of i] det_row_add[of i] by simp
then show ?case
unfolding scalar_mult_eq_scaleR .
qed

lemma matrix_id [simp]: "det (matrix id) = 1"

lemma det_matrix_scaleR [simp]: "det (matrix ((( *\<^sub>R) r)) :: real^'n^'n) = r ^ CARD('n::finite)"
apply (subst det_diagonal)
apply (auto simp: matrix_def mat_def)
done

text \<open>
May as well do this, though it's a bit unsatisfactory since it ignores
exact duplicates by considering the rows/columns as a set.
\<close>

lemma det_dependent_rows:
fixes A:: "'a::{field}^'n^'n"
assumes d: "vec.dependent (rows A)"
shows "det A = 0"
proof -
let ?U = "UNIV :: 'n set"
from d obtain i where i: "row i A \<in> vec.span (rows A - {row i A})"
unfolding vec.dependent_def rows_def by blast
show ?thesis
proof (cases "\<forall>i j. i \<noteq> j \<longrightarrow> row i A \<noteq> row j A")
case True
with i have "vec.span (rows A - {row i A}) \<subseteq> vec.span {row j A |j. j \<noteq> i}"
by (auto simp: rows_def intro!: vec.span_mono)
then have "- row i A \<in> vec.span {row j A|j. j \<noteq> i}"
by (meson i subsetCE vec.span_neg)
from det_row_span[OF this]
have "det A = det (\<chi> k. if k = i then 0 *s 1 else row k A)"
unfolding right_minus vector_smult_lzero ..
with det_row_mul[of i 0 "\<lambda>i. 1"]
show ?thesis by simp
next
case False
then obtain j k where jk: "j \<noteq> k" "row j A = row k A"
by auto
from det_identical_rows[OF jk] show ?thesis .
qed
qed

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

text \<open>Multilinearity and the multiplication formula\<close>

lemma Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
by auto

lemma det_linear_row_sum:
assumes fS: "finite S"
shows "det ((\<chi> i. if i = k then sum (a i) S else c i)::'a::comm_ring_1^'n^'n) =
sum (\<lambda>j. det ((\<chi> i. if i = k then a  i j else c i)::'a^'n^'n)) S"
using fS  by (induct rule: finite_induct; simp add: det_row_0 det_row_add cong: if_cong)

lemma finite_bounded_functions:
assumes fS: "finite S"
shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
proof (induct k)
case 0
have *: "{f. \<forall>i. f i = i} = {id}"
by auto
show ?case
by (auto simp: *)
next
case (Suc k)
let ?f = "\<lambda>(y::nat,g) i. if i = Suc k then y else g i"
let ?S = "?f ` (S \<times> {f. (\<forall>i\<in>{1..k}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1..k} \<longrightarrow> f i = i)})"
have "?S = {f. (\<forall>i\<in>{1.. Suc k}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1.. Suc k} \<longrightarrow> f i = i)}"
apply (auto simp: image_iff)
apply (rename_tac f)
apply (rule_tac x="f (Suc k)" in bexI)
apply (rule_tac x = "\<lambda>i. if i = Suc k then i else f i" in exI, auto)
done
with finite_imageI[OF finite_cartesian_product[OF fS Suc.hyps(1)], of ?f]
show ?case
by metis
qed

lemma det_linear_rows_sum_lemma:
assumes fS: "finite S"
and fT: "finite T"
shows "det ((\<chi> i. if i \<in> T then sum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
sum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
{f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
using fT
proof (induct T arbitrary: a c set: finite)
case empty
have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)"
by vector
from empty.prems show ?case
unfolding th0 by (simp add: eq_id_iff)
next
case (insert z T a c)
let ?F = "\<lambda>T. {f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
let ?h = "\<lambda>(y,g) i. if i = z then y else g i"
let ?k = "\<lambda>h. (h(z),(\<lambda>i. if i = z then i else h i))"
let ?s = "\<lambda> k a c f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n)"
let ?c = "\<lambda>j i. if i = z then a i j else c i"
have thif: "\<And>a b c d. (if a \<or> b then c else d) = (if a then c else if b then c else d)"
by simp
have thif2: "\<And>a b c d e. (if a then b else if c then d else e) =
(if c then (if a then b else d) else (if a then b else e))"
by simp
from \<open>z \<notin> T\<close> have nz: "\<And>i. i \<in> T \<Longrightarrow> i \<noteq> z"
by auto
have "det (\<chi> i. if i \<in> insert z T then sum (a i) S else c i) =
det (\<chi> i. if i = z then sum (a i) S else if i \<in> T then sum (a i) S else c i)"
unfolding insert_iff thif ..
also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<in> T then sum (a i) S else if i = z then a i j else c i))"
unfolding det_linear_row_sum[OF fS]
by (subst thif2) (simp add: nz cong: if_cong)
finally have tha:
"det (\<chi> i. if i \<in> insert z T then sum (a i) S else c i) =
(\<Sum>(j, f)\<in>S \<times> ?F T. det (\<chi> i. if i \<in> T then a i (f i)
else if i = z then a i j
else c i))"
unfolding insert.hyps unfolding sum.cartesian_product by blast
show ?case unfolding tha
using \<open>z \<notin> T\<close>
by (intro sum.reindex_bij_witness[where i="?k" and j="?h"])
(auto intro!: cong[OF refl[of det]] simp: vec_eq_iff)
qed

lemma det_linear_rows_sum:
fixes S :: "'n::finite set"
assumes fS: "finite S"
shows "det (\<chi> i. sum (a i) S) =
sum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. \<forall>i. f i \<in> S}"
proof -
have th0: "\<And>x y. ((\<chi> i. if i \<in> (UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\<chi> i. x i)"
by vector
from det_linear_rows_sum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite]
show ?thesis by simp
qed

lemma matrix_mul_sum_alt:
fixes A B :: "'a::comm_ring_1^'n^'n"
shows "A ** B = (\<chi> i. sum (\<lambda>k. A\$i\$k *s B \$ k) (UNIV :: 'n set))"
by (vector matrix_matrix_mult_def sum_component)

lemma det_rows_mul:
"det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n) =
prod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
let ?U = "UNIV :: 'n set"
let ?PU = "{p. p permutes ?U}"
fix p
assume pU: "p \<in> ?PU"
let ?s = "of_int (sign p)"
from pU have p: "p permutes ?U"
by blast
have "prod (\<lambda>i. c i * a i \$ p i) ?U = prod c ?U * prod (\<lambda>i. a i \$ p i) ?U"
unfolding prod.distrib ..
then show "?s * (\<Prod>xa\<in>?U. c xa * a xa \$ p xa) =
prod c ?U * (?s* (\<Prod>xa\<in>?U. a xa \$ p xa))"
qed rule

lemma det_mul:
fixes A B :: "'a::comm_ring_1^'n^'n"
shows "det (A ** B) = det A * det B"
proof -
let ?U = "UNIV :: 'n set"
let ?F = "{f. (\<forall>i \<in> ?U. f i \<in> ?U) \<and> (\<forall>i. i \<notin> ?U \<longrightarrow> f i = i)}"
let ?PU = "{p. p permutes ?U}"
have "p \<in> ?F" if "p permutes ?U" for p
by simp
then have PUF: "?PU \<subseteq> ?F" by blast
{
fix f
assume fPU: "f \<in> ?F - ?PU"
have fUU: "f ` ?U \<subseteq> ?U"
using fPU by auto
from fPU have f: "\<forall>i \<in> ?U. f i \<in> ?U" "\<forall>i. i \<notin> ?U \<longrightarrow> f i = i" "\<not>(\<forall>y. \<exists>!x. f x = y)"
unfolding permutes_def by auto

let ?A = "(\<chi> i. A\$i\$f i *s B\$f i) :: 'a^'n^'n"
let ?B = "(\<chi> i. B\$f i) :: 'a^'n^'n"
{
assume fni: "\<not> inj_on f ?U"
then obtain i j where ij: "f i = f j" "i \<noteq> j"
unfolding inj_on_def by blast
then have "row i ?B = row j ?B"
by (vector row_def)
with det_identical_rows[OF ij(2)]
have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
unfolding det_rows_mul by force
}
moreover
{
assume fi: "inj_on f ?U"
from f fi have fith: "\<And>i j. f i = f j \<Longrightarrow> i = j"
unfolding inj_on_def by metis
note fs = fi[unfolded surjective_iff_injective_gen[OF finite finite refl fUU, symmetric]]
have "\<exists>!x. f x = y" for y
using fith fs by blast
with f(3) have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
by blast
}
ultimately have "det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
by blast
}
then have zth: "\<forall> f\<in> ?F - ?PU. det (\<chi> i. A\$i\$f i *s B\$f i) = 0"
by simp
{
fix p
assume pU: "p \<in> ?PU"
from pU have p: "p permutes ?U"
by blast
let ?s = "\<lambda>p. of_int (sign p)"
let ?f = "\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A \$ i \$ p i) * (?s q * (\<Prod>i\<in> ?U. B \$ i \$ q i))"
have "(sum (\<lambda>q. ?s q *
(\<Prod>i\<in> ?U. (\<chi> i. A \$ i \$ p i *s B \$ p i :: 'a^'n^'n) \$ i \$ q i)) ?PU) =
(sum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A \$ i \$ p i) * (?s q * (\<Prod>i\<in> ?U. B \$ i \$ q i))) ?PU)"
unfolding sum_permutations_compose_right[OF permutes_inv[OF p], of ?f]
proof (rule sum.cong)
fix q
assume qU: "q \<in> ?PU"
then have q: "q permutes ?U"
by blast
from p q have pp: "permutation p" and pq: "permutation q"
unfolding permutation_permutes by auto
have th00: "of_int (sign p) * of_int (sign p) = (1::'a)"
"\<And>a. of_int (sign p) * (of_int (sign p) * a) = a"
unfolding mult.assoc[symmetric]
unfolding of_int_mult[symmetric]
have ths: "?s q = ?s p * ?s (q \<circ> inv p)"
using pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
by (simp add: th00 ac_simps sign_idempotent sign_compose)
have th001: "prod (\<lambda>i. B\$i\$ q (inv p i)) ?U = prod ((\<lambda>i. B\$i\$ q (inv p i)) \<circ> p) ?U"
by (rule prod.permute[OF p])
have thp: "prod (\<lambda>i. (\<chi> i. A\$i\$p i *s B\$p i :: 'a^'n^'n) \$i \$ q i) ?U =
prod (\<lambda>i. A\$i\$p i) ?U * prod (\<lambda>i. B\$i\$ q (inv p i)) ?U"
unfolding th001 prod.distrib[symmetric] o_def permutes_inverses[OF p]
apply (rule prod.cong[OF refl])
using permutes_in_image[OF q]
apply vector
done
show "?s q * prod (\<lambda>i. (((\<chi> i. A\$i\$p i *s B\$p i) :: 'a^'n^'n)\$i\$q i)) ?U =
?s p * (prod (\<lambda>i. A\$i\$p i) ?U) * (?s (q \<circ> inv p) * prod (\<lambda>i. B\$i\$(q \<circ> inv p) i) ?U)"
using ths thp pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
by (simp add: sign_nz th00 field_simps sign_idempotent sign_compose)
qed rule
}
then have th2: "sum (\<lambda>f. det (\<chi> i. A\$i\$f i *s B\$f i)) ?PU = det A * det B"
unfolding det_def sum_product
by (rule sum.cong [OF refl])
have "det (A**B) = sum (\<lambda>f.  det (\<chi> i. A \$ i \$ f i *s B \$ f i)) ?F"
unfolding matrix_mul_sum_alt det_linear_rows_sum[OF finite]
by simp
also have "\<dots> = sum (\<lambda>f. det (\<chi> i. A\$i\$f i *s B\$f i)) ?PU"
using sum.mono_neutral_cong_left[OF finite PUF zth, symmetric]
unfolding det_rows_mul by auto
finally show ?thesis unfolding th2 .
qed

subsection \<open>Relation to invertibility\<close>

lemma invertible_det_nz:
fixes A::"'a::{field}^'n^'n"
shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
proof (cases "invertible A")
case True
then obtain B :: "'a^'n^'n" where B: "A ** B = mat 1"
unfolding invertible_right_inverse by blast
then have "det (A ** B) = det (mat 1 :: 'a^'n^'n)"
by simp
then show ?thesis
by (metis True det_I det_mul mult_zero_left one_neq_zero)
next
case False
let ?U = "UNIV :: 'n set"
have fU: "finite ?U"
by simp
from False obtain c i where c: "sum (\<lambda>i. c i *s row i A) ?U = 0" and iU: "i \<in> ?U" and ci: "c i \<noteq> 0"
unfolding invertible_right_inverse matrix_right_invertible_independent_rows
by blast
have thr0: "- row i A = sum (\<lambda>j. (1/ c i) *s (c j *s row j A)) (?U - {i})"
unfolding sum_cmul  using c ci
by (auto simp: sum.remove[OF fU iU] eq_vector_fraction_iff add_eq_0_iff)
have thr: "- row i A \<in> vec.span {row j A| j. j \<noteq> i}"
unfolding thr0 by (auto intro: vec.span_base vec.span_scale vec.span_sum)
let ?B = "(\<chi> k. if k = i then 0 else row k A) :: 'a^'n^'n"
have thrb: "row i ?B = 0" using iU by (vector row_def)
have "det A = 0"
unfolding det_row_span[OF thr, symmetric] right_minus
unfolding det_zero_row(2)[OF thrb] ..
then show ?thesis
qed

lemma det_nz_iff_inj_gen:
fixes f :: "'a::field^'n \<Rightarrow> 'a::field^'n"
assumes "Vector_Spaces.linear ( *s) ( *s) f"
shows "det (matrix f) \<noteq> 0 \<longleftrightarrow> inj f"
proof
assume "det (matrix f) \<noteq> 0"
then show "inj f"
using assms invertible_det_nz inj_matrix_vector_mult by force
next
assume "inj f"
show "det (matrix f) \<noteq> 0"
using vec.linear_injective_left_inverse [OF assms \<open>inj f\<close>]
by (metis assms invertible_det_nz invertible_left_inverse matrix_compose_gen matrix_id_mat_1)
qed

lemma det_nz_iff_inj:
fixes f :: "real^'n \<Rightarrow> real^'n"
assumes "linear f"
shows "det (matrix f) \<noteq> 0 \<longleftrightarrow> inj f"
using det_nz_iff_inj_gen[of f] assms
unfolding linear_matrix_vector_mul_eq .

lemma det_eq_0_rank:
fixes A :: "real^'n^'n"
shows "det A = 0 \<longleftrightarrow> rank A < CARD('n)"
using invertible_det_nz [of A]
by (auto simp: matrix_left_invertible_injective invertible_left_inverse less_rank_noninjective)

subsubsection\<open>Invertibility of matrices and corresponding linear functions\<close>

lemma matrix_left_invertible_gen:
fixes f :: "'a::field^'m \<Rightarrow> 'a::field^'n"
assumes "Vector_Spaces.linear ( *s) ( *s) f"
shows "((\<exists>B. B ** matrix f = mat 1) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear ( *s) ( *s) g \<and> g \<circ> f = id))"
proof safe
fix B
assume 1: "B ** matrix f = mat 1"
show "\<exists>g. Vector_Spaces.linear ( *s) ( *s) g \<and> g \<circ> f = id"
proof (intro exI conjI)
show "Vector_Spaces.linear ( *s) ( *s) (\<lambda>y. B *v y)"
by simp
show "(( *v) B) \<circ> f = id"
unfolding o_def
by (metis assms 1 eq_id_iff matrix_vector_mul(1) matrix_vector_mul_assoc matrix_vector_mul_lid)
qed
next
fix g
assume "Vector_Spaces.linear ( *s) ( *s) g" "g \<circ> f = id"
then have "matrix g ** matrix f = mat 1"
by (metis assms matrix_compose_gen matrix_id_mat_1)
then show "\<exists>B. B ** matrix f = mat 1" ..
qed

lemma matrix_left_invertible:
"linear f \<Longrightarrow> ((\<exists>B. B ** matrix f = mat 1) \<longleftrightarrow> (\<exists>g. linear g \<and> g \<circ> f = id))" for f::"real^'m \<Rightarrow> real^'n"
using matrix_left_invertible_gen[of f]
by (auto simp: linear_matrix_vector_mul_eq)

lemma matrix_right_invertible_gen:
fixes f :: "'a::field^'m \<Rightarrow> 'a^'n"
assumes "Vector_Spaces.linear ( *s) ( *s) f"
shows "((\<exists>B. matrix f ** B = mat 1) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear ( *s) ( *s) g \<and> f \<circ> g = id))"
proof safe
fix B
assume 1: "matrix f ** B = mat 1"
show "\<exists>g. Vector_Spaces.linear ( *s) ( *s) g \<and> f \<circ> g = id"
proof (intro exI conjI)
show "Vector_Spaces.linear ( *s) ( *s) (( *v) B)"
by simp
show "f \<circ> ( *v) B = id"
using 1 assms comp_apply eq_id_iff vec.linear_id matrix_id_mat_1 matrix_vector_mul_assoc matrix_works
by (metis (no_types, hide_lams))
qed
next
fix g
assume "Vector_Spaces.linear ( *s) ( *s) g" and "f \<circ> g = id"
then have "matrix f ** matrix g = mat 1"
by (metis assms matrix_compose_gen matrix_id_mat_1)
then show "\<exists>B. matrix f ** B = mat 1" ..
qed

lemma matrix_right_invertible:
"linear f \<Longrightarrow> ((\<exists>B. matrix f ** B = mat 1) \<longleftrightarrow> (\<exists>g. linear g \<and> f \<circ> g = id))" for f::"real^'m \<Rightarrow> real^'n"
using matrix_right_invertible_gen[of f]
by (auto simp: linear_matrix_vector_mul_eq)

lemma matrix_invertible_gen:
fixes f :: "'a::field^'m \<Rightarrow> 'a::field^'n"
assumes "Vector_Spaces.linear ( *s) ( *s) f"
shows  "invertible (matrix f) \<longleftrightarrow> (\<exists>g. Vector_Spaces.linear ( *s) ( *s) g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
(is "?lhs = ?rhs")
proof
assume ?lhs then show ?rhs
by (metis assms invertible_def left_right_inverse_eq matrix_left_invertible_gen matrix_right_invertible_gen)
next
assume ?rhs then show ?lhs
by (metis assms invertible_def matrix_compose_gen matrix_id_mat_1)
qed

lemma matrix_invertible:
"linear f \<Longrightarrow> invertible (matrix f) \<longleftrightarrow> (\<exists>g. linear g \<and> f \<circ> g = id \<and> g \<circ> f = id)"
for f::"real^'m \<Rightarrow> real^'n"
using matrix_invertible_gen[of f]
by (auto simp: linear_matrix_vector_mul_eq)

lemma invertible_eq_bij:
fixes m :: "'a::field^'m^'n"
shows "invertible m \<longleftrightarrow> bij (( *v) m)"
using matrix_invertible_gen[OF matrix_vector_mul_linear_gen, of m, simplified matrix_of_matrix_vector_mul]
by (metis bij_betw_def left_right_inverse_eq matrix_vector_mul_linear_gen o_bij
vec.linear_injective_left_inverse vec.linear_surjective_right_inverse)

subsection \<open>Cramer's rule\<close>

lemma cramer_lemma_transpose:
fixes A:: "'a::{field}^'n^'n"
and x :: "'a::{field}^'n"
shows "det ((\<chi> i. if i = k then sum (\<lambda>i. x\$i *s row i A) (UNIV::'n set)
else row i A)::'a::{field}^'n^'n) = x\$k * det A"
(is "?lhs = ?rhs")
proof -
let ?U = "UNIV :: 'n set"
let ?Uk = "?U - {k}"
have U: "?U = insert k ?Uk"
by blast
have kUk: "k \<notin> ?Uk"
by simp
have th00: "\<And>k s. x\$k *s row k A + s = (x\$k - 1) *s row k A + row k A + s"
by (vector field_simps)
have th001: "\<And>f k . (\<lambda>x. if x = k then f k else f x) = f"
by auto
have "(\<chi> i. row i A) = A" by (vector row_def)
then have thd1: "det (\<chi> i. row i A) = det A"
by simp
have thd0: "det (\<chi> i. if i = k then row k A + (\<Sum>i \<in> ?Uk. x \$ i *s row i A) else row i A) = det A"
by (force intro: det_row_span vec.span_sum vec.span_scale vec.span_base)
show "?lhs = x\$k * det A"
apply (subst U)
unfolding sum.insert[OF finite kUk]
apply (subst th00)
unfolding thd0
unfolding det_row_mul
unfolding th001[of k "\<lambda>i. row i A"]
unfolding thd1
done
qed

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

lemma cramer:
fixes A ::"'a::{field}^'n^'n"
assumes d0: "det A \<noteq> 0"
shows "A *v x = b \<longleftrightarrow> x = (\<chi> k. det(\<chi> i j. if j=k then b\$i else A\$i\$j) / det A)"
proof -
from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
unfolding invertible_det_nz[symmetric] invertible_def
by blast
have "(A ** B) *v b = b"
then have "A *v (B *v b) = b"
then have xe: "\<exists>x. A *v x = b"
by blast
{
fix x
assume x: "A *v x = b"
have "x = (\<chi> k. det(\<chi> i j. if j=k then b\$i else A\$i\$j) / det A)"
unfolding x[symmetric]
using d0 by (simp add: vec_eq_iff cramer_lemma field_simps)
}
with xe show ?thesis
by auto
qed

subsection \<open>Orthogonality of a transformation and matrix\<close>

definition "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"

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

lemma 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 orthogonal_transformation_id [simp]: "orthogonal_transformation (\<lambda>x. x)"

lemma orthogonal_orthogonal_transformation:
"orthogonal_transformation f \<Longrightarrow> orthogonal (f x) (f y) \<longleftrightarrow> orthogonal x y"

lemma 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 orthogonal_transformation_neg:
"orthogonal_transformation(\<lambda>x. -(f x)) \<longleftrightarrow> orthogonal_transformation f"
by (auto simp: orthogonal_transformation_def dest: linear_compose_neg)

lemma orthogonal_transformation_scaleR: "orthogonal_transformation f \<Longrightarrow> f (c *\<^sub>R v) = c *\<^sub>R f v"

lemma orthogonal_transformation_linear:
"orthogonal_transformation f \<Longrightarrow> linear f"

lemma orthogonal_transformation_inj:
"orthogonal_transformation f \<Longrightarrow> inj f"
unfolding orthogonal_transformation_def inj_on_def
by (metis vector_eq)

lemma 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 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 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 orthogonal_transformation_norm:
"orthogonal_transformation f \<Longrightarrow> norm (f x) = norm x"
by (metis orthogonal_transformation)

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

lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"

lemma 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 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 -
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 det_orthogonal_matrix:
fixes Q:: "'a::linordered_idom^'n^'n"
assumes oQ: "orthogonal_matrix Q"
shows "det Q = 1 \<or> det Q = - 1"
proof -
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 orthogonal_transformation_det [simp]:
fixes f :: "real^'n \<Rightarrow> real^'n"
shows "orthogonal_transformation f \<Longrightarrow> \<bar>det (matrix f)\<bar> = 1"
using det_orthogonal_matrix orthogonal_transformation_matrix by fastforce

subsection \<open>Linearity of scaling, and hence isometry, that preserves origin\<close>

lemma 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 -
{
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 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 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 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 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\<open> We can find an orthogonal matrix taking any unit vector to any other\<close>

lemma orthogonal_matrix_transpose [simp]:
"orthogonal_matrix(transpose A) \<longleftrightarrow> orthogonal_matrix A"
by (auto simp: orthogonal_matrix_def)

lemma 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 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 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 -
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)"
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 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 -
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 orthogonal_transformation_exists:
fixes a b :: "real^'n"
assumes "norm a = norm b"
obtains f where "orthogonal_transformation f" "f a = b"
proof (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 \<open>Can extend an isometry from unit sphere\<close>

lemma 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 -
{
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 \<open>Rotation, reflection, rotoinversion\<close>

definition "rotation_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = 1"
definition "rotoinversion_matrix Q \<longleftrightarrow> orthogonal_matrix Q \<and> det Q = - 1"

lemma orthogonal_rotation_or_rotoinversion:
fixes Q :: "'a::linordered_idom^'n^'n"
shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)

text \<open>Explicit formulas for low dimensions\<close>

lemma prod_neutral_const: "prod f {(1::nat)..1} = f 1"
by simp

lemma prod_2: "prod f {(1::nat)..2} = f 1 * f 2"
by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)

lemma prod_3: "prod f {(1::nat)..3} = f 1 * f 2 * f 3"
by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)

lemma det_1: "det (A::'a::comm_ring_1^1^1) = A\$1\$1"

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

lemma det_3:
"det (A::'a::comm_ring_1^3^3) =
A\$1\$1 * A\$2\$2 * A\$3\$3 +
A\$1\$2 * A\$2\$3 * A\$3\$1 +
A\$1\$3 * A\$2\$1 * A\$3\$2 -
A\$1\$1 * A\$2\$3 * A\$3\$2 -
A\$1\$2 * A\$2\$1 * A\$3\$3 -
A\$1\$3 * A\$2\$2 * A\$3\$1"
proof -
have f123: "finite {2::3, 3}" "1 \<notin> {2::3, 3}"
by auto
have f23: "finite {3::3}" "2 \<notin> {3::3}"
by auto

show ?thesis
unfolding det_def UNIV_3
unfolding sum_over_permutations_insert[OF f123]
unfolding sum_over_permutations_insert[OF f23]
unfolding permutes_sing
by (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
qed

text\<open> Slightly stronger results giving rotation, but only in two or more dimensions\<close>

lemma rotation_matrix_exists_basis:
fixes a :: "real^'n"
assumes 2: "2 \<le> CARD('n)" and "norm a = 1"
obtains A where "rotation_matrix A" "A *v (axis k 1) = a"
proof -
obtain A where "orthogonal_matrix A" and A: "A *v (axis k 1) = a"
using orthogonal_matrix_exists_basis assms by metis
with orthogonal_rotation_or_rotoinversion
consider "rotation_matrix A" | "rotoinversion_matrix A"
by metis
then show thesis
proof cases
assume "rotation_matrix A"
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)
let ?TA = "transpose A"
let ?A = "\<chi> i. if i = j then - 1 *\<^sub>R (?TA \$ i) else ?TA \$i"
assume "rotoinversion_matrix A"
then have [simp]: "det A = -1"
show ?thesis
proof
have [simp]: "row i (\<chi> i. if i = j then - 1 *\<^sub>R ?TA \$ i else ?TA \$ i) = (if i = j then - row i ?TA else row i ?TA)" for i
by (auto simp: row_def)
have "orthogonal_matrix ?A"
unfolding orthogonal_matrix_orthonormal_rows
using \<open>orthogonal_matrix A\<close> by (auto simp: orthogonal_matrix_orthonormal_columns orthogonal_clauses)
then show "rotation_matrix (transpose ?A)"
unfolding rotation_matrix_def
by (simp add: det_row_mul[of j _ "\<lambda>i. ?TA \$ i", unfolded scalar_mult_eq_scaleR])
show "transpose ?A *v axis k 1 = a"
using \<open>j \<noteq> k\<close> A by (simp add: matrix_vector_column axis_def scalar_mult_eq_scaleR if_distrib [of "\<lambda>z. z *\<^sub>R c" for c] cong: if_cong)
qed
qed
qed

lemma rotation_exists_1:
fixes a :: "real^'n"
assumes "2 \<le> CARD('n)" "norm a = 1" "norm b = 1"
obtains f where "orthogonal_transformation f" "det(matrix f) = 1" "f a = b"
proof -
obtain k::'n where True
by simp
obtain A B where AB: "rotation_matrix A" "rotation_matrix B"
and eq: "A *v (axis k 1) = a" "B *v (axis k 1) = b"
using rotation_matrix_exists_basis assms by metis
let ?f = "\<lambda>x. (B ** transpose A) *v x"
show thesis
proof
show "orthogonal_transformation ?f"
using AB orthogonal_matrix_mul orthogonal_transformation_matrix rotation_matrix_def matrix_vector_mul_linear by force
show "det (matrix ?f) = 1"
using AB by (auto simp: det_mul rotation_matrix_def)
show "?f a = b"
using AB unfolding orthogonal_matrix_def rotation_matrix_def
by (metis eq matrix_mul_rid matrix_vector_mul_assoc)
qed
qed

lemma rotation_exists:
fixes a :: "real^'n"
assumes 2: "2 \<le> CARD('n)" and eq: "norm a = norm b"
obtains f where "orthogonal_transformation f" "det(matrix f) = 1" "f a = b"
proof (cases "a = 0 \<or> b = 0")
case True
with assms have "a = 0" "b = 0"
by auto
then show ?thesis
by (metis eq_id_iff matrix_id orthogonal_transformation_id that)
next
case False
then obtain f where f: "orthogonal_transformation f" "det (matrix f) = 1"
and f': "f (a /\<^sub>R norm a) = b /\<^sub>R norm b"
using rotation_exists_1 [of "a /\<^sub>R norm a" "b /\<^sub>R norm b", OF 2] by auto
then interpret linear f by (simp add: orthogonal_transformation)
have "f a = b"
using f' False
with f show thesis ..
qed

lemma rotation_rightward_line:
fixes a :: "real^'n"
obtains f where "orthogonal_transformation f" "2 \<le> CARD('n) \<Longrightarrow> det(matrix f) = 1"
"f(norm a *\<^sub>R axis k 1) = a"
proof (cases "CARD('n) = 1")
case True
obtain f where "orthogonal_transformation f" "f (norm a *\<^sub>R axis k (1::real)) = a"
proof (rule orthogonal_transformation_exists)
show "norm (norm a *\<^sub>R axis k (1::real)) = norm a"
by simp
qed auto
then show thesis
using True that by auto
next
case False
obtain f where "orthogonal_transformation f" "det(matrix f) = 1" "f (norm a *\<^sub>R axis k 1) = a"
proof (rule rotation_exists)
show "2 \<le> CARD('n)"
using False one_le_card_finite [where 'a='n] by linarith
show "norm (norm a *\<^sub>R axis k (1::real)) = norm a"
by simp
qed auto
then show thesis
using that by blast
qed

end
```