# Theory Determinants

```(*  Title:      HOL/Analysis/Determinants.thy
Author:     Amine Chaieb, University of Cambridge; proofs reworked by LCP
*)

section ‹Traces and Determinants of Square Matrices›

theory Determinants
imports
"HOL-Combinatorics.Permutations"
Cartesian_Space
begin

subsection  ‹Trace›

definition✐‹tag important›  trace :: "'a::semiring_1^'n^'n ⇒ 'a"
where "trace A = sum (λi. ((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✐‹tag important›  ‹Definition of determinant›

definition✐‹tag important›  det:: "'a::comm_ring_1^'n^'n ⇒ 'a" where
"det A =
sum (λp. of_int (sign p) * prod (λi. A\$i\$p i) (UNIV :: 'n set))
{p. p permutes (UNIV :: 'n set)}"

text ‹Basic determinant properties›

lemma  det_transpose [simp]: "det (transpose A) = det (A::'a::comm_ring_1 ^'n^'n)"
proof -
let ?di = "λA i j. A\$i\$j"
let ?U = "(UNIV :: 'n set)"
have fU: "finite ?U" by simp
{
fix p
assume p: "p ∈ {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 (λi. ?di (transpose A) i (inv p i)) ?U =
prod (λi. ?di (transpose A) i (inv p i)) (p ` ?U)"
by simp
also have "… = prod ((λi. ?di (transpose A) i (inv p i)) ∘ p) ?U"
unfolding prod.reindex[OF pi] ..
also have "… = prod (λi. ?di A i (p i)) ?U"
proof -
have "((λi. ?di (transpose A) i (inv p i)) ∘ p) i = ?di A i (p i)" if "i ∈ ?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 ((λi. ?di (transpose A) i (inv p i)) ∘ p) ?U = prod (λi. ?di A i (p i)) ?U"
by (auto intro: prod.cong)
qed
finally have "of_int (sign (inv p)) * (prod (λi. ?di (transpose A) i (inv p i)) ?U) =
of_int (sign p) * (prod (λi. ?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: "⋀i j. i < j ⟹ A\$i\$j = 0"
shows "det A = prod (λi. A\$i\$i) (UNIV:: 'n set)"
proof -
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "λp. of_int (sign p) * prod (λi. A\$i\$p i) (UNIV :: 'n set)"
have fU: "finite ?U"
by simp
have id0: "{id} ⊆ ?PU"
by (auto simp: permutes_id)
have p0: "∀p ∈ ?PU - {id}. ?pp p = 0"
proof
fix p
assume "p ∈ ?PU - {id}"
then obtain i where i: "p i > i"
by clarify (meson leI permutes_natset_le)
from ld[OF i] have "∃i ∈ ?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: "⋀i j. i > j ⟹ A\$i\$j = 0"
shows "det A = prod (λi. A\$i\$i) (UNIV:: 'n set)"
proof -
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "(λp. of_int (sign p) * prod (λi. A\$i\$p i) (UNIV :: 'n set))"
have fU: "finite ?U"
by simp
have id0: "{id} ⊆ ?PU"
by (auto simp: permutes_id)
have p0: "∀p ∈ ?PU -{id}. ?pp p = 0"
proof
fix p
assume p: "p ∈ ?PU - {id}"
then obtain i where i: "p i < i"
by clarify (meson leI permutes_natset_ge)
from ld[OF i] have "∃i ∈ ?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

proposition  det_diagonal:
fixes A :: "'a::comm_ring_1^'n^'n"
assumes ld: "⋀i j. i ≠ j ⟹ A\$i\$j = 0"
shows "det A = prod (λi. A\$i\$i) (UNIV::'n set)"
proof -
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "λp. of_int (sign p) * prod (λi. 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} ⊆ ?PU"
by (auto simp: permutes_id)
have p0: "∀p ∈ ?PU - {id}. ?pp p = 0"
proof
fix p
assume p: "p ∈ ?PU - {id}"
then obtain i where i: "p i ≠ i"
by fastforce
with ld have "∃i ∈ ?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 (χ 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 *: "(∑q∈?PU. of_int (sign (q ∘ p)) * (∏i∈?U. A \$ p i \$ (q ∘ p) i)) =
(∑n∈?PU. of_int (sign p) * of_int (sign n) * (∏i∈?U. A \$ i \$ n i))"
proof (rule sum.cong)
fix q
assume qPU: "q ∈ ?PU"
have fU: "finite ?U"
by simp
from qPU have q: "q permutes ?U"
by blast
have "prod (λi. A\$p i\$ (q ∘ p) i) ?U = prod ((λi. A\$p i\$(q ∘ p) i) ∘ inv p) ?U"
by (simp only: prod.permute[OF permutes_inv[OF p], symmetric])
also have "… = prod (λi. A \$ (p ∘ inv p) i \$ (q ∘ (p ∘ inv p)) i) ?U"
by (simp only: o_def)
also have "… = prod (λi. A\$i\$q i) ?U"
by (simp only: o_def permutes_inverses[OF p])
finally have thp: "prod (λi. A\$p i\$ (q ∘ p) i) ?U = prod (λi. 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 ∘ p)) * prod (λi. A\$ p i\$ (q ∘ p) i) ?U =
of_int (sign p) * of_int (sign q) * prod (λi. 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(χ i j. A\$i\$ p j :: 'a^'n^'n) = of_int (sign p) * det A"
proof -
let ?Ap = "χ i j. A\$i\$ p j :: 'a^'n^'n"
let ?At = "transpose A"
have "of_int (sign p) * det A = det (transpose (χ i. transpose A \$ p i))"
unfolding det_permute_rows[OF p, of ?At] det_transpose ..
moreover
have "?Ap = transpose (χ 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 ≠ k"
and r: "column j A = column k A"
shows "det A = 0"
proof -
let ?U="UNIV::'n set"
let ?t_jk="Transposition.transpose j k"
let ?PU="{p. p permutes ?U}"
let ?S1="{p. p∈?PU ∧ evenperm p}"
let ?S2="{(?t_jk ∘ p) |p. p ∈?S1}"
let ?f="λp. of_int (sign p) * (∏i∈UNIV. A \$ i \$ p i)"
let ?g="λp. ?t_jk ∘ 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 ∘ x = ?t_jk ∘ y"
show "x = y" by (metis (opaque_lifting, no_types) comp_assoc eq id_comp swap_id_idempotent)
qed
have tjk_permutes: "?t_jk permutes ?U"
by (auto simp add: permutes_def dest: transpose_eq_imp_eq) (meson transpose_involutory)
have tjk_eq: "∀i l. A \$ i \$ ?t_jk l  =  A \$ i \$ l"
using r jk
unfolding column_def vec_eq_iff by (simp add: Transposition.transpose_def)
have sign_tjk: "sign ?t_jk = -1" using sign_swap_id[of j k] jk by auto
{fix x
assume x: "x∈ ?S1"
have "sign (?t_jk ∘ 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 "… = - sign x" using sign_tjk by simp
also have "… ≠ sign x" unfolding sign_def by simp
finally have "sign (?t_jk ∘ x) ≠ sign x" and "(?t_jk ∘ x) ∈ ?S2"
using x by force+
}
hence disjoint: "?S1 ∩ ?S2 = {}"
by (force simp: sign_def)
have PU_decomposition: "?PU = ?S1 ∪ ?S2"
proof (auto)
fix x
assume x: "x permutes ?U" and "∀p. p permutes ?U ⟶ x = Transposition.transpose j k ∘ p ⟶ ¬ evenperm p"
then obtain p where p: "p permutes UNIV" and x_eq: "x = Transposition.transpose j k ∘ p"
and odd_p: "¬ 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 "Transposition.transpose j k ∘ p permutes UNIV" by (metis p permutes_compose tjk_permutes)
qed
have "sum ?f ?S2 = sum ((λp. of_int (sign p) * (∏i∈UNIV. A \$ i \$ p i))
∘ (∘) (Transposition.transpose j k)) {p ∈ {p. p permutes UNIV}. evenperm p}"
unfolding g_S1 by (rule sum.reindex[OF inj_g])
also have "… = sum (λp. of_int (sign (?t_jk ∘ p)) * (∏i∈UNIV. A \$ i \$ p i)) ?S1"
unfolding o_def by (rule sum.cong, auto simp: tjk_eq)
also have "… = sum (λp. - ?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 ∘ x)) = - (sign x)"
unfolding sign_compose[OF perm_tjk perm_x] sign_tjk by auto
thus "of_int (sign (?t_jk ∘ x)) * (∏i∈UNIV. A \$ i \$ x i)
= - (of_int (sign x) * (∏i∈UNIV. A \$ i \$ x i))"
by auto
qed
also have "…= - sum ?f ?S1" unfolding sum_negf ..
finally have *: "sum ?f ?S2 = - sum ?f ?S1" .
have "det A = (∑p | p permutes UNIV. of_int (sign p) * (∏i∈UNIV. A \$ i \$ p i))"
unfolding det_def ..
also have "…= sum ?f ?S1 + sum ?f ?S2"
by (subst PU_decomposition, rule sum.union_disjoint[OF _ _ disjoint], auto)
also have "…= sum ?f ?S1 - sum ?f ?S1 " unfolding * by auto
also have "…= 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 ≠ 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 ⟹ det A = 0" and "row j F = 0 ⟹ 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 ⟹ det A = 0" and "column j F = 0 ⟹ det F = 0"
unfolding atomize_conj atomize_imp
by (metis det_transpose det_zero_row row_transpose)

fixes a b c :: "'n::finite ⇒ _ ^ 'n"
shows "det((χ i. if i = k then a i + b i else c i)::'a::comm_ring_1^'n^'n) =
det((χ i. if i = k then a i else c i)::'a::comm_ring_1^'n^'n) +
det((χ 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 = "(λi. if i = k then a i + b i else c i)::'n ⇒ 'a::comm_ring_1^'n"
let ?g = "(λ i. if i = k then a i else c i)::'n ⇒ 'a::comm_ring_1^'n"
let ?h = "(λ i. if i = k then b i else c i)::'n ⇒ 'a::comm_ring_1^'n"
fix p
assume p: "p ∈ ?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 (λi. ?f i \$ p i) ?Uk = prod (λi. ?g i \$ p i) ?Uk"
"prod (λi. ?f i \$ p i) ?Uk = prod (λi. ?h i \$ p i) ?Uk"
by auto
have Uk: "finite ?Uk" "k ∉ ?Uk"
by auto
have "prod (λi. ?f i \$ p i) ?U = prod (λi. ?f i \$ p i) (insert k ?Uk)"
unfolding kU[symmetric] ..
also have "… = ?f k \$ p k * prod (λi. ?f i \$ p i) ?Uk"
by (rule prod.insert) auto
also have "… = (a k \$ p k * prod (λi. ?f i \$ p i) ?Uk) + (b k\$ p k * prod (λi. ?f i \$ p i) ?Uk)"
also have "… = (a k \$ p k * prod (λi. ?g i \$ p i) ?Uk) + (b k\$ p k * prod (λi. ?h i \$ p i) ?Uk)"
by (metis eq)
also have "… = prod (λi. ?g i \$ p i) (insert k ?Uk) + prod (λi. ?h i \$ p i) (insert k ?Uk)"
unfolding  prod.insert[OF Uk] by simp
finally have "prod (λi. ?f i \$ p i) ?U = prod (λi. ?g i \$ p i) ?U + prod (λi. ?h i \$ p i) ?U"
unfolding kU[symmetric] .
then show "of_int (sign p) * prod (λi. ?f i \$ p i) ?U =
of_int (sign p) * prod (λi. ?g i \$ p i) ?U + of_int (sign p) * prod (λi. ?h i \$ p i) ?U"
qed auto

lemma  det_row_mul:
fixes a b :: "'n::finite ⇒ _ ^ 'n"
shows "det((χ i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
c * det((χ 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 = "(λi. if i = k then c*s a i else b i)::'n ⇒ 'a::comm_ring_1^'n"
let ?g = "(λ i. if i = k then a i else b i)::'n ⇒ 'a::comm_ring_1^'n"
fix p
assume p: "p ∈ ?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 (λi. ?f i \$ p i) ?Uk = prod (λi. ?g i \$ p i) ?Uk"
by auto
have Uk: "finite ?Uk" "k ∉ ?Uk"
by auto
have "prod (λi. ?f i \$ p i) ?U = prod (λi. ?f i \$ p i) (insert k ?Uk)"
unfolding kU[symmetric] ..
also have "… = ?f k \$ p k  * prod (λi. ?f i \$ p i) ?Uk"
by (rule prod.insert) auto
also have "… = (c*s a k) \$ p k * prod (λi. ?f i \$ p i) ?Uk"
also have "… = c* (a k \$ p k * prod (λi. ?g i \$ p i) ?Uk)"
unfolding eq by (simp add: ac_simps)
also have "… = c* (prod (λi. ?g i \$ p i) (insert k ?Uk))"
unfolding prod.insert[OF Uk] by simp
finally have "prod (λi. ?f i \$ p i) ?U = c* (prod (λi. ?g i \$ p i) ?U)"
unfolding kU[symmetric] .
then show "of_int (sign p) * prod (λi. ?f i \$ p i) ?U = c * (of_int (sign p) * prod (λi. ?g i \$ p i) ?U)"
qed auto

lemma  det_row_0:
fixes b :: "'n::finite ⇒ _ ^ 'n"
shows "det((χ i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
using det_row_mul[of k 0 "λi. 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 ≠ j"
shows "det (χ k. if k = i then row i A + c *s row j A else row k A) = det A"
proof -
let ?Z = "(χ 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: "((χ 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 ∈ vec.span {row j A |j. j ≠ i}"
shows "det (χ 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 ≠ 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 = "λx. det (χ 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"

proposition  det_matrix_scaleR [simp]: "det (matrix (((*⇩R) r)) :: real^'n^'n) = r ^ CARD('n::finite)"
apply (subst det_diagonal)
apply (auto simp: matrix_def mat_def)
done

text ‹
May as well do this, though it's a bit unsatisfactory since it ignores
exact duplicates by considering the rows/columns as a set.
›

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 ∈ vec.span (rows A - {row i A})"
unfolding vec.dependent_def rows_def by blast
show ?thesis
proof (cases "∀i j. i ≠ j ⟶ row i A ≠ row j A")
case True
with i have "vec.span (rows A - {row i A}) ⊆ vec.span {row j A |j. j ≠ i}"
by (auto simp: rows_def intro!: vec.span_mono)
then have "- row i A ∈ vec.span {row j A|j. j ≠ i}"
by (meson i subsetCE vec.span_neg)
from det_row_span[OF this]
have "det A = det (χ 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 "λi. 1"]
show ?thesis by simp
next
case False
then obtain j k where jk: "j ≠ 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 ‹Multilinearity and the multiplication formula›

lemma  Cart_lambda_cong: "(⋀x. f x = g x) ⟹ (vec_lambda f::'a^'n) = (vec_lambda g :: 'a^'n)"
by auto

lemma  det_linear_row_sum:
assumes fS: "finite S"
shows "det ((χ i. if i = k then sum (a i) S else c i)::'a::comm_ring_1^'n^'n) =
sum (λj. det ((χ 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. (∀i ∈ {1.. (k::nat)}. f i ∈ S) ∧ (∀i. i ∉ {1 .. k} ⟶ f i = i)}"
proof (induct k)
case 0
have *: "{f. ∀i. f i = i} = {id}"
by auto
show ?case
by (auto simp: *)
next
case (Suc k)
let ?f = "λ(y::nat,g) i. if i = Suc k then y else g i"
let ?S = "?f ` (S × {f. (∀i∈{1..k}. f i ∈ S) ∧ (∀i. i ∉ {1..k} ⟶ f i = i)})"
have "?S = {f. (∀i∈{1.. Suc k}. f i ∈ S) ∧ (∀i. i ∉ {1.. Suc k} ⟶ 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 = "λi. 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 ((χ i. if i ∈ T then sum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
sum (λf. det((χ i. if i ∈ T then a i (f i) else c i)::'a^'n^'n))
{f. (∀i ∈ T. f i ∈ S) ∧ (∀i. i ∉ T ⟶ f i = i)}"
using fT
proof (induct T arbitrary: a c set: finite)
case empty
have th0: "⋀x y. (χ i. if i ∈ {} then x i else y i) = (χ 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 = "λT. {f. (∀i ∈ T. f i ∈ S) ∧ (∀i. i ∉ T ⟶ f i = i)}"
let ?h = "λ(y,g) i. if i = z then y else g i"
let ?k = "λh. (h(z),(λi. if i = z then i else h i))"
let ?s = "λ k a c f. det((χ i. if i ∈ T then a i (f i) else c i)::'a^'n^'n)"
let ?c = "λj i. if i = z then a i j else c i"
have thif: "⋀a b c d. (if a ∨ b then c else d) = (if a then c else if b then c else d)"
by simp
have thif2: "⋀a b c d e. (if a then b else if c then d else e) =
(if c then (if a then b else d) else (if a then b else e))"
by simp
from ‹z ∉ T› have nz: "⋀i. i ∈ T ⟹ i ≠ z"
by auto
have "det (χ i. if i ∈ insert z T then sum (a i) S else c i) =
det (χ i. if i = z then sum (a i) S else if i ∈ T then sum (a i) S else c i)"
unfolding insert_iff thif ..
also have "… = (∑j∈S. det (χ i. if i ∈ 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 (χ i. if i ∈ insert z T then sum (a i) S else c i) =
(∑(j, f)∈S × ?F T. det (χ i. if i ∈ 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 ‹z ∉ T›
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 (χ i. sum (a i) S) =
sum (λf. det (χ i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n)) {f. ∀i. f i ∈ S}"
proof -
have th0: "⋀x y. ((χ i. if i ∈ (UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (χ 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 = (χ i. sum (λk. A\$i\$k *s B \$ k) (UNIV :: 'n set))"
by (vector matrix_matrix_mult_def sum_component)

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

proposition  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. (∀i ∈ ?U. f i ∈ ?U) ∧ (∀i. i ∉ ?U ⟶ f i = i)}"
let ?PU = "{p. p permutes ?U}"
have "p ∈ ?F" if "p permutes ?U" for p
by simp
then have PUF: "?PU ⊆ ?F" by blast
{
fix f
assume fPU: "f ∈ ?F - ?PU"
have fUU: "f ` ?U ⊆ ?U"
using fPU by auto
from fPU have f: "∀i ∈ ?U. f i ∈ ?U" "∀i. i ∉ ?U ⟶ f i = i" "¬(∀y. ∃!x. f x = y)"
unfolding permutes_def by auto

let ?A = "(χ i. A\$i\$f i *s B\$f i) :: 'a^'n^'n"
let ?B = "(χ i. B\$f i) :: 'a^'n^'n"
{
assume fni: "¬ inj_on f ?U"
then obtain i j where ij: "f i = f j" "i ≠ 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 (χ 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: "⋀i j. f i = f j ⟹ i = j"
unfolding inj_on_def by metis
note fs = fi[unfolded surjective_iff_injective_gen[OF finite finite refl fUU, symmetric]]
have "∃!x. f x = y" for y
using fith fs by blast
with f(3) have "det (χ i. A\$i\$f i *s B\$f i) = 0"
by blast
}
ultimately have "det (χ i. A\$i\$f i *s B\$f i) = 0"
by blast
}
then have zth: "∀ f∈ ?F - ?PU. det (χ i. A\$i\$f i *s B\$f i) = 0"
by simp
{
fix p
assume pU: "p ∈ ?PU"
from pU have p: "p permutes ?U"
by blast
let ?s = "λp. of_int (sign p)"
let ?f = "λq. ?s p * (∏i∈ ?U. A \$ i \$ p i) * (?s q * (∏i∈ ?U. B \$ i \$ q i))"
have "(sum (λq. ?s q *
(∏i∈ ?U. (χ i. A \$ i \$ p i *s B \$ p i :: 'a^'n^'n) \$ i \$ q i)) ?PU) =
(sum (λq. ?s p * (∏i∈ ?U. A \$ i \$ p i) * (?s q * (∏i∈ ?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 ∈ ?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)"
"⋀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 ∘ 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 (λi. B\$i\$ q (inv p i)) ?U = prod ((λi. B\$i\$ q (inv p i)) ∘ p) ?U"
by (rule prod.permute[OF p])
have thp: "prod (λi. (χ i. A\$i\$p i *s B\$p i :: 'a^'n^'n) \$i \$ q i) ?U =
prod (λi. A\$i\$p i) ?U * prod (λi. 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 (λi. (((χ i. A\$i\$p i *s B\$p i) :: 'a^'n^'n)\$i\$q i)) ?U =
?s p * (prod (λi. A\$i\$p i) ?U) * (?s (q ∘ inv p) * prod (λi. B\$i\$(q ∘ 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 (λf. det (χ 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 (λf.  det (χ 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 "… = sum (λf. det (χ 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 ‹Relation to invertibility›

proposition  invertible_det_nz:
fixes A::"'a::{field}^'n^'n"
shows "invertible A ⟷ det A ≠ 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 (λi. c i *s row i A) ?U = 0" and iU: "i ∈ ?U" and ci: "c i ≠ 0"
unfolding invertible_right_inverse matrix_right_invertible_independent_rows
by blast
have thr0: "- row i A = sum (λj. (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 ∈ vec.span {row j A| j. j ≠ i}"
unfolding thr0 by (auto intro: vec.span_base vec.span_scale vec.span_sum)
let ?B = "(χ 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 ⇒ 'a::field^'n"
assumes "Vector_Spaces.linear (*s) (*s) f"
shows "det (matrix f) ≠ 0 ⟷ inj f"
proof
assume "det (matrix f) ≠ 0"
then show "inj f"
using assms invertible_det_nz inj_matrix_vector_mult by force
next
assume "inj f"
show "det (matrix f) ≠ 0"
using vec.linear_injective_left_inverse [OF assms ‹inj f›]
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 ⇒ real^'n"
assumes "linear f"
shows "det (matrix f) ≠ 0 ⟷ 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 ⟷ rank A < CARD('n)"
using invertible_det_nz [of A]
by (auto simp: matrix_left_invertible_injective invertible_left_inverse less_rank_noninjective)

subsubsection✐‹tag important›  ‹Invertibility of matrices and corresponding linear functions›

lemma  matrix_left_invertible_gen:
fixes f :: "'a::field^'m ⇒ 'a::field^'n"
assumes "Vector_Spaces.linear (*s) (*s) f"
shows "((∃B. B ** matrix f = mat 1) ⟷ (∃g. Vector_Spaces.linear (*s) (*s) g ∧ g ∘ f = id))"
proof safe
fix B
assume 1: "B ** matrix f = mat 1"
show "∃g. Vector_Spaces.linear (*s) (*s) g ∧ g ∘ f = id"
proof (intro exI conjI)
show "Vector_Spaces.linear (*s) (*s) (λy. B *v y)"
by simp
show "((*v) B) ∘ 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 ∘ f = id"
then have "matrix g ** matrix f = mat 1"
by (metis assms matrix_compose_gen matrix_id_mat_1)
then show "∃B. B ** matrix f = mat 1" ..
qed

lemma  matrix_left_invertible:
"linear f ⟹ ((∃B. B ** matrix f = mat 1) ⟷ (∃g. linear g ∧ g ∘ f = id))" for f::"real^'m ⇒ 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 ⇒ 'a^'n"
assumes "Vector_Spaces.linear (*s) (*s) f"
shows "((∃B. matrix f ** B = mat 1) ⟷ (∃g. Vector_Spaces.linear (*s) (*s) g ∧ f ∘ g = id))"
proof safe
fix B
assume 1: "matrix f ** B = mat 1"
show "∃g. Vector_Spaces.linear (*s) (*s) g ∧ f ∘ g = id"
proof (intro exI conjI)
show "Vector_Spaces.linear (*s) (*s) ((*v) B)"
by simp
show "f ∘ (*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, opaque_lifting))
qed
next
fix g
assume "Vector_Spaces.linear (*s) (*s) g" and "f ∘ g = id"
then have "matrix f ** matrix g = mat 1"
by (metis assms matrix_compose_gen matrix_id_mat_1)
then show "∃B. matrix f ** B = mat 1" ..
qed

lemma  matrix_right_invertible:
"linear f ⟹ ((∃B. matrix f ** B = mat 1) ⟷ (∃g. linear g ∧ f ∘ g = id))" for f::"real^'m ⇒ 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 ⇒ 'a::field^'n"
assumes "Vector_Spaces.linear (*s) (*s) f"
shows  "invertible (matrix f) ⟷ (∃g. Vector_Spaces.linear (*s) (*s) g ∧ f ∘ g = id ∧ g ∘ 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 ⟹ invertible (matrix f) ⟷ (∃g. linear g ∧ f ∘ g = id ∧ g ∘ f = id)"
for f::"real^'m ⇒ 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 ⟷ 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 ‹Cramer's rule›

lemma  cramer_lemma_transpose:
fixes A:: "'a::{field}^'n^'n"
and x :: "'a::{field}^'n"
shows "det ((χ i. if i = k then sum (λi. 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 ∉ ?Uk"
by simp
have th00: "⋀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: "⋀f k . (λx. if x = k then f k else f x) = f"
by auto
have "(χ i. row i A) = A" by (vector row_def)
then have thd1: "det (χ i. row i A) = det A"
by simp
have thd0: "det (χ i. if i = k then row k A + (∑i ∈ ?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 "λi. row i A"]
unfolding thd1
done
qed

proposition  cramer_lemma:
fixes A :: "'a::{field}^'n^'n"
shows "det((χ 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 *: "⋀c. sum (λi. c i *s row i (transpose A)) ?U = sum (λi. 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 "λi. x\$i"]
apply (subst det_transpose[symmetric])
apply (rule cong[OF refl[of det]])
apply (vector transpose_def column_def row_def)
done
qed

proposition  cramer:
fixes A ::"'a::{field}^'n^'n"
assumes d0: "det A ≠ 0"
shows "A *v x = b ⟷ x = (χ k. det(χ 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: "∃x. A *v x = b"
by blast
{
fix x
assume x: "A *v x = b"
have "x = (χ k. det(χ 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

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 ∉ {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 ∉ {2::3, 3}"
by auto
have f23: "finite {3::3}" "2 ∉ {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

proposition  det_orthogonal_matrix:
fixes Q:: "'a::linordered_idom^'n^'n"
assumes oQ: "orthogonal_matrix Q"
shows "det Q = 1 ∨ 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

proposition  orthogonal_transformation_det [simp]:
fixes f :: "real^'n ⇒ real^'n"
shows "orthogonal_transformation f ⟹ ¦det (matrix f)¦ = 1"
using det_orthogonal_matrix orthogonal_transformation_matrix by fastforce

subsection  ‹Rotation, reflection, rotoinversion›

definition✐‹tag important›  "rotation_matrix Q ⟷ orthogonal_matrix Q ∧ det Q = 1"
definition✐‹tag important›  "rotoinversion_matrix Q ⟷ orthogonal_matrix Q ∧ det Q = - 1"

lemma  orthogonal_rotation_or_rotoinversion:
fixes Q :: "'a::linordered_idom^'n^'n"
shows " orthogonal_matrix Q ⟷ rotation_matrix Q ∨ rotoinversion_matrix Q"
by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)

text‹ Slightly stronger results giving rotation, but only in two or more dimensions›

lemma  rotation_matrix_exists_basis:
fixes a :: "real^'n"
assumes 2: "2 ≤ 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 ‹A *v axis k 1 = a› that by auto
next
from ex_card[OF 2] obtain h i::'n where "h ≠ i"
by (auto simp add: eval_nat_numeral card_Suc_eq)
then obtain j where "j ≠ k"
by (metis (full_types))
let ?TA = "transpose A"
let ?A = "χ i. if i = j then - 1 *⇩R (?TA \$ i) else ?TA \$i"
assume "rotoinversion_matrix A"
then have [simp]: "det A = -1"
show ?thesis
proof
have [simp]: "row i (χ i. if i = j then - 1 *⇩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 ‹orthogonal_matrix A› 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 _ "λi. ?TA \$ i", unfolded scalar_mult_eq_scaleR])
show "transpose ?A *v axis k 1 = a"
using ‹j ≠ k› A by (simp add: matrix_vector_column axis_def scalar_mult_eq_scaleR if_distrib [of "λz. z *⇩R c" for c] cong: if_cong)
qed
qed
qed

lemma  rotation_exists_1:
fixes a :: "real^'n"
assumes "2 ≤ 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 = "λx. (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 ≤ 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 ∨ 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 /⇩R norm a) = b /⇩R norm b"
using rotation_exists_1 [of "a /⇩R norm a" "b /⇩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 ≤ CARD('n) ⟹ det(matrix f) = 1"
"f(norm a *⇩R axis k 1) = a"
proof (cases "CARD('n) = 1")
case True
obtain f where "orthogonal_transformation f" "f (norm a *⇩R axis k (1::real)) = a"
proof (rule orthogonal_transformation_exists)
show "norm (norm a *⇩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 *⇩R axis k 1) = a"
proof (rule rotation_exists)
show "2 ≤ CARD('n)"
using False one_le_card_finite [where 'a='n] by linarith
show "norm (norm a *⇩R axis k (1::real)) = norm a"
by simp
qed auto
then show thesis
using that by blast
qed

end
```