# Theory Determinants

theory Determinants
imports Cartesian_Euclidean_Space Permutations
```(*  Title:      HOL/Multivariate_Analysis/Determinants.thy
Author:     Amine Chaieb, University of Cambridge
*)

section ‹Traces, Determinant of square matrices and some properties›

theory Determinants
imports
Cartesian_Euclidean_Space
"~~/src/HOL/Library/Permutations"
begin

subsection ‹Trace›

definition trace :: "'a::semiring_1^'n^'n ⇒ 'a"
where "trace A = setsum (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 setsum.commute)
done

text ‹Definition of determinant.›

definition det:: "'a::comm_ring_1^'n^'n ⇒ 'a" where
"det A =
setsum (λp. of_int (sign p) * setprod (λi. A\$i\$p i) (UNIV :: 'n set))
{p. p permutes (UNIV :: 'n set)}"

text ‹A few general lemmas we need below.›

lemma setprod_permute:
assumes p: "p permutes S"
shows "setprod f S = setprod (f ∘ p) S"
using assms by (fact setprod.permute)

lemma setproduct_permute_nat_interval:
fixes m n :: nat
shows "p permutes {m..n} ⟹ setprod f {m..n} = setprod (f ∘ p) {m..n}"
by (blast intro!: setprod_permute)

text ‹Basic determinant properties.›

lemma det_transpose: "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 "setprod (λi. ?di (transpose A) i (inv p i)) ?U =
setprod (λi. ?di (transpose A) i (inv p i)) (p ` ?U)"
by simp
also have "… = setprod ((λi. ?di (transpose A) i (inv p i)) ∘ p) ?U"
unfolding setprod.reindex[OF pi] ..
also have "… = setprod (λi. ?di A i (p i)) ?U"
proof -
{
fix i
assume i: "i ∈ ?U"
from i permutes_inv_o[OF pU] permutes_in_image[OF pU]
have "((λi. ?di (transpose A) i (inv p i)) ∘ p) i = ?di A i (p i)"
unfolding transpose_def by (simp add: fun_eq_iff)
}
then show "setprod ((λi. ?di (transpose A) i (inv p i)) ∘ p) ?U =
setprod (λi. ?di A i (p i)) ?U"
by (auto intro: setprod.cong)
qed
finally have "of_int (sign (inv p)) * (setprod (λi. ?di (transpose A) i (inv p i)) ?U) =
of_int (sign p) * (setprod (λi. ?di A i (p i)) ?U)"
using sth by simp
}
then show ?thesis
unfolding det_def
apply (subst setsum_permutations_inverse)
apply (rule setsum.cong)
apply (rule refl)
apply blast
done
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 = setprod (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) * setprod (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"
{
fix p
assume p: "p ∈ ?PU - {id}"
from p have pU: "p permutes ?U" and pid: "p ≠ id"
by blast+
from permutes_natset_le[OF pU] pid obtain i where i: "p i > i"
by (metis not_le)
from ld[OF i] have ex:"∃i ∈ ?U. A\$i\$p i = 0"
by blast
from setprod_zero[OF fU ex] have "?pp p = 0"
by simp
}
then have p0: "∀p ∈ ?PU - {id}. ?pp p = 0"
by blast
from setsum.mono_neutral_cong_left[OF fPU 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 = setprod (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) * setprod (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"
{
fix p
assume p: "p ∈ ?PU - {id}"
from p have pU: "p permutes ?U" and pid: "p ≠ id"
by blast+
from permutes_natset_ge[OF pU] pid obtain i where i: "p i < i"
by (metis not_le)
from ld[OF i] have ex:"∃i ∈ ?U. A\$i\$p i = 0"
by blast
from setprod_zero[OF fU ex] have "?pp p = 0"
by simp
}
then have p0: "∀p ∈ ?PU -{id}. ?pp p = 0"
by blast
from setsum.mono_neutral_cong_left[OF fPU 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: "⋀i j. i ≠ j ⟹ A\$i\$j = 0"
shows "det A = setprod (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) * setprod (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"
{
fix p
assume p: "p ∈ ?PU - {id}"
then have "p ≠ id"
by simp
then obtain i where i: "p i ≠ i"
unfolding fun_eq_iff by auto
from ld [OF i [symmetric]] have ex:"∃i ∈ ?U. A\$i\$p i = 0"
by blast
from setprod_zero [OF fU ex] have "?pp p = 0"
by simp
}
then have p0: "∀p ∈ ?PU - {id}. ?pp p = 0"
by blast
from setsum.mono_neutral_cong_left[OF fPU id0 p0] show ?thesis
unfolding det_def by (simp add: sign_id)
qed

lemma det_I: "det (mat 1 :: 'a::comm_ring_1^'n^'n) = 1"
proof -
let ?A = "mat 1 :: 'a::comm_ring_1^'n^'n"
let ?U = "UNIV :: 'n set"
let ?f = "λi j. ?A\$i\$j"
{
fix i
assume i: "i ∈ ?U"
have "?f i i = 1"
using i by (vector mat_def)
}
then have th: "setprod (λi. ?f i i) ?U = setprod (λx. 1) ?U"
by (auto intro: setprod.cong)
{
fix i j
assume i: "i ∈ ?U" and j: "j ∈ ?U" and ij: "i ≠ j"
have "?f i j = 0" using i j ij
by (vector mat_def)
}
then have "det ?A = setprod (λi. ?f i i) ?U"
using det_diagonal by blast
also have "… = 1"
unfolding th setprod.neutral_const ..
finally show ?thesis .
qed

lemma det_0: "det (mat 0 :: 'a::comm_ring_1^'n^'n) = 0"

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"
apply (simp add: det_def setsum_right_distrib mult.assoc[symmetric])
apply (subst sum_permutations_compose_right[OF p])
proof (rule setsum.cong)
let ?U = "UNIV :: 'n set"
let ?PU = "{p. p permutes ?U}"
fix q
assume qPU: "q ∈ ?PU"
have fU: "finite ?U"
by simp
from qPU have q: "q permutes ?U"
by blast
from p q have pp: "permutation p" and qp: "permutation q"
by (metis fU permutation_permutes)+
from permutes_inv[OF p] have ip: "inv p permutes ?U" .
have "setprod (λi. A\$p i\$ (q ∘ p) i) ?U = setprod ((λi. A\$p i\$(q ∘ p) i) ∘ inv p) ?U"
by (simp only: setprod_permute[OF ip, symmetric])
also have "… = setprod (λi. A \$ (p ∘ inv p) i \$ (q ∘ (p ∘ inv p)) i) ?U"
by (simp only: o_def)
also have "… = setprod (λi. A\$i\$q i) ?U"
by (simp only: o_def permutes_inverses[OF p])
finally have thp: "setprod (λi. A\$p i\$ (q ∘ p) i) ?U = setprod (λi. A\$i\$q i) ?U"
by blast
show "of_int (sign (q ∘ p)) * setprod (λi. A\$ p i\$ (q ∘ p) i) ?U =
of_int (sign p) * of_int (sign q) * setprod (λi. A\$i\$q i) ?U"
by (simp only: thp sign_compose[OF qp pp] mult.commute of_int_mult)
qed rule

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_rows:
fixes A :: "'a::linordered_idom^'n^'n"
assumes ij: "i ≠ j"
and r: "row i A = row j A"
shows "det A = 0"
proof-
have tha: "⋀(a::'a) b. a = b ⟹ b = - a ⟹ a = 0"
by simp
have th1: "of_int (-1) = - 1" by simp
let ?p = "Fun.swap i j id"
let ?A = "χ i. A \$ ?p i"
from r have "A = ?A" by (simp add: vec_eq_iff row_def Fun.swap_def)
then have "det A = det ?A" by simp
moreover have "det A = - det ?A"
by (simp add: det_permute_rows[OF permutes_swap_id] sign_swap_id ij th1)
ultimately show "det A = 0" by (metis tha)
qed

lemma det_identical_columns:
fixes A :: "'a::linordered_idom^'n^'n"
assumes ij: "i ≠ j"
and r: "column i A = column j A"
shows "det A = 0"
apply (subst det_transpose[symmetric])
apply (rule det_identical_rows[OF ij])
apply (metis row_transpose r)
done

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

lemma det_zero_column:
fixes A :: "'a::{idom,ring_char_0}^'n^'n"
assumes r: "column i A = 0"
shows "det A = 0"
apply (subst det_transpose[symmetric])
apply (rule det_zero_row [of i])
apply (metis row_transpose r)
done

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 setsum.distrib[symmetric]
proof (rule setsum.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
{
fix j
assume j: "j ∈ ?Uk"
from j have "?f j \$ p j = ?g j \$ p j" and "?f j \$ p j= ?h j \$ p j"
by simp_all
}
then have th1: "setprod (λi. ?f i \$ p i) ?Uk = setprod (λi. ?g i \$ p i) ?Uk"
and th2: "setprod (λi. ?f i \$ p i) ?Uk = setprod (λi. ?h i \$ p i) ?Uk"
apply -
apply (rule setprod.cong, simp_all)+
done
have th3: "finite ?Uk" "k ∉ ?Uk"
by auto
have "setprod (λi. ?f i \$ p i) ?U = setprod (λi. ?f i \$ p i) (insert k ?Uk)"
unfolding kU[symmetric] ..
also have "… = ?f k \$ p k * setprod (λi. ?f i \$ p i) ?Uk"
apply (rule setprod.insert)
apply simp
apply blast
done
also have "… = (a k \$ p k * setprod (λi. ?f i \$ p i) ?Uk) + (b k\$ p k * setprod (λi. ?f i \$ p i) ?Uk)"
also have "… = (a k \$ p k * setprod (λi. ?g i \$ p i) ?Uk) + (b k\$ p k * setprod (λi. ?h i \$ p i) ?Uk)"
by (metis th1 th2)
also have "… = setprod (λi. ?g i \$ p i) (insert k ?Uk) + setprod (λi. ?h i \$ p i) (insert k ?Uk)"
unfolding  setprod.insert[OF th3] by simp
finally have "setprod (λi. ?f i \$ p i) ?U = setprod (λi. ?g i \$ p i) ?U + setprod (λi. ?h i \$ p i) ?U"
unfolding kU[symmetric] .
then show "of_int (sign p) * setprod (λi. ?f i \$ p i) ?U =
of_int (sign p) * setprod (λi. ?g i \$ p i) ?U + of_int (sign p) * setprod (λi. ?h i \$ p i) ?U"
qed rule

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 setsum_right_distrib
proof (rule setsum.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
{
fix j
assume j: "j ∈ ?Uk"
from j have "?f j \$ p j = ?g j \$ p j"
by simp
}
then have th1: "setprod (λi. ?f i \$ p i) ?Uk = setprod (λi. ?g i \$ p i) ?Uk"
apply -
apply (rule setprod.cong)
apply simp_all
done
have th3: "finite ?Uk" "k ∉ ?Uk"
by auto
have "setprod (λi. ?f i \$ p i) ?U = setprod (λi. ?f i \$ p i) (insert k ?Uk)"
unfolding kU[symmetric] ..
also have "… = ?f k \$ p k  * setprod (λi. ?f i \$ p i) ?Uk"
apply (rule setprod.insert)
apply simp
apply blast
done
also have "… = (c*s a k) \$ p k * setprod (λi. ?f i \$ p i) ?Uk"
also have "… = c* (a k \$ p k * setprod (λi. ?g i \$ p i) ?Uk)"
unfolding th1 by (simp add: ac_simps)
also have "… = c* (setprod (λi. ?g i \$ p i) (insert k ?Uk))"
unfolding setprod.insert[OF th3] by simp
finally have "setprod (λi. ?f i \$ p i) ?U = c* (setprod (λi. ?g i \$ p i) ?U)"
unfolding kU[symmetric] .
then show "of_int (sign p) * setprod (λi. ?f i \$ p i) ?U =
c * (of_int (sign p) * setprod (λi. ?g i \$ p i) ?U)"
qed rule

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::linordered_idom^'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 :: "real^'n^'n"
assumes x: "x ∈ 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"
proof -
let ?U = "UNIV :: 'n set"
let ?S = "{row j A |j. j ≠ i}"
let ?d = "λx. det (χ k. if k = i then x else row k A)"
let ?P = "λx. ?d (row i A + x) = det A"
{
fix k
have "(if k = i then row i A + 0 else row k A) = row k A"
by simp
}
then have P0: "?P 0"
apply -
apply (rule cong[of det, OF refl])
apply (vector row_def)
done
moreover
{
fix c z y
assume zS: "z ∈ ?S" and Py: "?P y"
from zS 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
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 "?P (c*s z + y)"
unfolding thz Py det_row_mul[of i] det_row_add[of i]
by simp
}
ultimately show ?thesis
apply -
apply (rule span_induct_alt[of ?P ?S, OF P0, folded scalar_mult_eq_scaleR])
apply blast
apply (rule x)
done
qed

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:: "real^'n^'n"
assumes d: "dependent (rows A)"
shows "det A = 0"
proof -
let ?U = "UNIV :: 'n set"
from d obtain i where i: "row i A ∈ span (rows A - {row i A})"
unfolding dependent_def rows_def by blast
{
fix j k
assume jk: "j ≠ k" and c: "row j A = row k A"
from det_identical_rows[OF jk c] have ?thesis .
}
moreover
{
assume H: "⋀ i j. i ≠ j ⟹ row i A ≠ row j A"
have th0: "- row i A ∈ span {row j A|j. j ≠ i}"
apply (rule span_neg)
apply (rule set_rev_mp)
apply (rule i)
apply (rule span_mono)
using H i
done
from det_row_span[OF th0]
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::real" "λi. 1"]
have "det A = 0" by simp
}
ultimately show ?thesis by blast
qed

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

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 (rule iffD1[OF vec_lambda_unique]) vector

lemma det_linear_row_setsum:
assumes fS: "finite S"
shows "det ((χ i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n) =
setsum (λj. det ((χ i. if i = k then a  i j else c i)::'a^'n^'n)) S"
proof (induct rule: finite_induct[OF fS])
case 1
then show ?case
apply simp
unfolding setsum.empty det_row_0[of k]
apply rule
done
next
case (2 x F)
then show ?case
qed

lemma finite_bounded_functions:
assumes fS: "finite S"
shows "finite {f. (∀i ∈ {1.. (k::nat)}. f i ∈ S) ∧ (∀i. i ∉ {1 .. k} ⟶ f i = i)}"
proof (induct k)
case 0
have th: "{f. ∀i. f i = i} = {id}"
by auto
show ?case
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 (rule_tac x="x (Suc k)" in bexI)
apply (rule_tac x = "λi. if i = Suc k then i else x i" in exI)
apply auto
done
with finite_imageI[OF finite_cartesian_product[OF fS Suc.hyps(1)], of ?f]
show ?case
by metis
qed

lemma eq_id_iff[simp]: "(∀x. f x = x) ⟷ f = id"
by auto

lemma det_linear_rows_setsum_lemma:
assumes fS: "finite S"
and fT: "finite T"
shows "det ((χ i. if i ∈ T then setsum (a i) S else c i):: 'a::comm_ring_1^'n^'n) =
setsum (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
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 ⟷ False"
by auto
have "det (χ i. if i ∈ insert z T then setsum (a i) S else c i) =
det (χ i. if i = z then setsum (a i) S else if i ∈ T then setsum (a i) S else c i)"
unfolding insert_iff thif ..
also have "… = (∑j∈S. det (χ i. if i ∈ T then setsum (a i) S else if i = z then a i j else c i))"
unfolding det_linear_row_setsum[OF fS]
apply (subst thif2)
using nz
apply (simp cong del: if_weak_cong cong add: if_cong)
done
finally have tha:
"det (χ i. if i ∈ insert z T then setsum (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 setsum.cartesian_product by blast
show ?case unfolding tha
using ‹z ∉ T›
by (intro setsum.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_setsum:
fixes S :: "'n::finite set"
assumes fS: "finite S"
shows "det (χ i. setsum (a i) S) =
setsum (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_setsum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite]
show ?thesis by simp
qed

lemma matrix_mul_setsum_alt:
fixes A B :: "'a::comm_ring_1^'n^'n"
shows "A ** B = (χ i. setsum (λk. A\$i\$k *s B \$ k) (UNIV :: 'n set))"
by (vector matrix_matrix_mult_def setsum_component)

lemma det_rows_mul:
"det((χ i. c i *s a i)::'a::comm_ring_1^'n^'n) =
setprod (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 "setprod (λi. c i * a i \$ p i) ?U = setprod c ?U * setprod (λi. a i \$ p i) ?U"
unfolding setprod.distrib ..
then show "?s * (∏xa∈?U. c xa * a xa \$ p xa) =
setprod c ?U * (?s* (∏xa∈?U. a xa \$ p xa))"
qed rule

lemma det_mul:
fixes A B :: "'a::linordered_idom^'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 fU: "finite ?U"
by simp
have fF: "finite ?F"
by (rule finite)
{
fix p
assume p: "p permutes ?U"
have "p ∈ ?F" unfolding mem_Collect_eq permutes_in_image[OF p]
using p[unfolded permutes_def] 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
from ij
have rth: "row i ?B = row j ?B"
by (vector row_def)
from det_identical_rows[OF ij(2) rth]
have "det (χ i. A\$i\$f i *s B\$f i) = 0"
unfolding det_rows_mul by simp
}
moreover
{
assume fi: "inj_on f ?U"
from f fi have fith: "⋀i j. f i = f j ⟹ i = j"
unfolding inj_on_def by metis
note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]]
{
fix y
from fs f have "∃x. f x = y"
by blast
then obtain x where x: "f x = y"
by blast
{
fix z
assume z: "f z = y"
from fith x z have "z = x"
by metis
}
with x have "∃!x. f x = y"
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 "(setsum (λq. ?s q *
(∏i∈ ?U. (χ i. A \$ i \$ p i *s B \$ p i :: 'a^'n^'n) \$ i \$ q i)) ?PU) =
(setsum (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 setsum.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: "setprod (λi. B\$i\$ q (inv p i)) ?U = setprod ((λi. B\$i\$ q (inv p i)) ∘ p) ?U"
by (rule setprod_permute[OF p])
have thp: "setprod (λi. (χ i. A\$i\$p i *s B\$p i :: 'a^'n^'n) \$i \$ q i) ?U =
setprod (λi. A\$i\$p i) ?U * setprod (λi. B\$i\$ q (inv p i)) ?U"
unfolding th001 setprod.distrib[symmetric] o_def permutes_inverses[OF p]
apply (rule setprod.cong[OF refl])
using permutes_in_image[OF q]
apply vector
done
show "?s q * setprod (λi. (((χ i. A\$i\$p i *s B\$p i) :: 'a^'n^'n)\$i\$q i)) ?U =
?s p * (setprod (λi. A\$i\$p i) ?U) * (?s (q ∘ inv p) * setprod (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: "setsum (λf. det (χ i. A\$i\$f i *s B\$f i)) ?PU = det A * det B"
unfolding det_def setsum_product
by (rule setsum.cong [OF refl])
have "det (A**B) = setsum (λf.  det (χ i. A \$ i \$ f i *s B \$ f i)) ?F"
unfolding matrix_mul_setsum_alt det_linear_rows_setsum[OF fU]
by simp
also have "… = setsum (λf. det (χ i. A\$i\$f i *s B\$f i)) ?PU"
using setsum.mono_neutral_cong_left[OF fF PUF zth, symmetric]
unfolding det_rows_mul by auto
finally show ?thesis unfolding th2 .
qed

text ‹Relation to invertibility.›

lemma invertible_left_inverse:
fixes A :: "real^'n^'n"
shows "invertible A ⟷ (∃(B::real^'n^'n). B** A = mat 1)"
by (metis invertible_def matrix_left_right_inverse)

lemma invertible_righ_inverse:
fixes A :: "real^'n^'n"
shows "invertible A ⟷ (∃(B::real^'n^'n). A** B = mat 1)"
by (metis invertible_def matrix_left_right_inverse)

lemma invertible_det_nz:
fixes A::"real ^'n^'n"
shows "invertible A ⟷ det A ≠ 0"
proof -
{
assume "invertible A"
then obtain B :: "real ^'n^'n" where B: "A ** B = mat 1"
unfolding invertible_righ_inverse by blast
then have "det (A ** B) = det (mat 1 :: real ^'n^'n)"
by simp
then have "det A ≠ 0"
by (simp add: det_mul det_I) algebra
}
moreover
{
assume H: "¬ invertible A"
let ?U = "UNIV :: 'n set"
have fU: "finite ?U"
by simp
from H obtain c i where c: "setsum (λi. c i *s row i A) ?U = 0"
and iU: "i ∈ ?U"
and ci: "c i ≠ 0"
unfolding invertible_righ_inverse
unfolding matrix_right_invertible_independent_rows
by blast
have *: "⋀(a::real^'n) b. a + b = 0 ⟹ -a = b"
apply (drule_tac f="op + (- a)" in cong[OF refl])
apply simp
done
from c ci
have thr0: "- row i A = setsum (λj. (1/ c i) *s (c j *s row j A)) (?U - {i})"
unfolding setsum.remove[OF fU iU] setsum_cmul
apply -
apply (rule vector_mul_lcancel_imp[OF ci])
unfolding *
apply rule
done
have thr: "- row i A ∈ span {row j A| j. j ≠ i}"
unfolding thr0
apply (rule span_setsum)
apply simp
apply (rule ballI)
apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+
apply (rule span_superset)
apply auto
done
let ?B = "(χ k. if k = i then 0 else row k A) :: real ^'n^'n"
have thrb: "row i ?B = 0" using iU by (vector row_def)
have "det A = 0"
unfolding det_row_span[OF thr, symmetric] right_minus
unfolding det_zero_row[OF thrb] ..
}
ultimately show ?thesis
by blast
qed

text ‹Cramer's rule.›

lemma cramer_lemma_transpose:
fixes A:: "real^'n^'n"
and x :: "real^'n"
shows "det ((χ i. if i = k then setsum (λi. x\$i *s row i A) (UNIV::'n set)
else row i A)::real^'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 fUk: "finite ?Uk"
by simp
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"
apply (rule det_row_span)
apply (rule span_setsum)
apply (rule ballI)
apply (rule span_mul [where 'a="real^'n", folded scalar_mult_eq_scaleR])+
apply (rule span_superset)
apply auto
done
show "?lhs = x\$k * det A"
apply (subst U)
unfolding setsum.insert[OF fUk kUk]
apply (subst th00)
unfolding thd0
unfolding det_row_mul
unfolding th001[of k "λi. row i A"]
unfolding thd1
done
qed

lemma cramer_lemma:
fixes A :: "real^'n^'n"
shows "det((χ i j. if j = k then (A *v x)\$i else A\$i\$j):: real^'n^'n) = x\$k * det A"
proof -
let ?U = "UNIV :: 'n set"
have *: "⋀c. setsum (λi. c i *s row i (transpose A)) ?U = setsum (λi. c i *s column i A) ?U"
by (auto simp add: row_transpose intro: setsum.cong)
show ?thesis
unfolding matrix_mult_vsum
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

lemma cramer:
fixes A ::"real^'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

text ‹Orthogonality of a transformation and matrix.›

definition "orthogonal_transformation f ⟷ linear f ∧ (∀v w. f v ∙ f w = v ∙ w)"

lemma orthogonal_transformation:
"orthogonal_transformation f ⟷ linear f ∧ (∀(v::real ^_). norm (f v) = norm v)"
unfolding orthogonal_transformation_def
apply auto
apply (erule_tac x=v in allE)+
done

definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) ⟷
transpose Q ** Q = mat 1 ∧ Q ** transpose Q = mat 1"

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

lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n)"
by (simp add: orthogonal_matrix_def transpose_mat matrix_mul_lid)

lemma orthogonal_matrix_mul:
fixes A :: "real ^'n^'n"
assumes oA : "orthogonal_matrix A"
and oB: "orthogonal_matrix B"
shows "orthogonal_matrix(A ** B)"
using oA oB
unfolding orthogonal_matrix matrix_transpose_mul
apply (subst matrix_mul_assoc)
apply (subst matrix_mul_assoc[symmetric])
done

lemma orthogonal_transformation_matrix:
fixes f:: "real^'n ⇒ real^'n"
shows "orthogonal_transformation f ⟷ linear f ∧ orthogonal_matrix(matrix f)"
(is "?lhs ⟷ ?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: "linear f" and fd: "∀v w. f v ∙ f w = v ∙ w"
unfolding  orthogonal_transformation_def orthogonal_matrix by blast+
{
fix i j
let ?A = "transpose ?mf ** ?mf"
have th0: "⋀b (x::'a::comm_ring_1). (if b then 1 else 0)*x = (if b then x else 0)"
"⋀b (x::'a::comm_ring_1). x*(if b then 1 else 0) = (if b then x else 0)"
by simp_all
from fd[rule_format, of "axis i 1" "axis j 1", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul]
have "?A\$i\$j = ?m1 \$ i \$ j"
by (simp add: inner_vec_def matrix_matrix_mult_def columnvector_def rowvector_def
th0 setsum.delta[OF fU] mat_def axis_def)
}
then have "orthogonal_matrix ?mf"
unfolding orthogonal_matrix
by vector
with lf have ?rhs
by blast
}
moreover
{
assume lf: "linear f" and om: "orthogonal_matrix ?mf"
from lf om have ?lhs
unfolding orthogonal_matrix_def norm_eq orthogonal_transformation
unfolding matrix_works[OF lf, symmetric]
apply (subst dot_matrix_vector_mul)
done
}
ultimately show ?thesis
by blast
qed

lemma det_orthogonal_matrix:
fixes Q:: "'a::linordered_idom^'n^'n"
assumes oQ: "orthogonal_matrix Q"
shows "det Q = 1 ∨ det Q = - 1"
proof -
have th: "⋀x::'a. x = 1 ∨ x = - 1 ⟷ x*x = 1" (is "⋀x::'a. ?ths x")
proof -
fix x:: 'a
have th0: "x * x - 1 = (x - 1) * (x + 1)"
have th1: "⋀(x::'a) y. x = - y ⟷ x + y = 0"
apply (subst eq_iff_diff_eq_0)
apply simp
done
have "x * x = 1 ⟷ x * x - 1 = 0"
by simp
also have "… ⟷ x = 1 ∨ x = - 1"
unfolding th0 th1 by simp
finally show "?ths x" ..
qed
from oQ have "Q ** transpose Q = mat 1"
by (metis orthogonal_matrix_def)
then have "det (Q ** transpose Q) = det (mat 1:: 'a^'n^'n)"
by simp
then have "det Q * det Q = 1"
by (simp add: det_mul det_I det_transpose)
then show ?thesis unfolding th .
qed

text ‹Linearity of scaling, and hence isometry, that preserves origin.›

lemma scaling_linear:
fixes f :: "real ^'n ⇒ real ^'n"
assumes f0: "f 0 = 0"
and fd: "∀x y. dist (f x) (f y) = c * dist x y"
shows "linear f"
proof -
{
fix v w
{
fix x
note fd[rule_format, of x 0, unfolded dist_norm f0 diff_0_right]
}
note th0 = this
have "f v ∙ f w = c⇧2 * (v ∙ w)"
unfolding dot_norm_neg dist_norm[symmetric]
unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)}
note fc = this
show ?thesis
unfolding linear_iff vector_eq[where 'a="real^'n"] scalar_mult_eq_scaleR
qed

lemma isometry_linear:
"f (0:: real^'n) = (0:: real^'n) ⟹ ∀x y. dist(f x) (f y) = dist x y ⟹ linear f"
by (rule scaling_linear[where c=1]) simp_all

text ‹Hence another formulation of orthogonal transformation.›

lemma orthogonal_transformation_isometry:
"orthogonal_transformation f ⟷ f(0::real^'n) = (0::real^'n) ∧ (∀x y. dist(f x) (f y) = dist x y)"
unfolding orthogonal_transformation
apply (rule iffI)
apply clarify
apply (clarsimp simp add: linear_0 linear_sub[symmetric] dist_norm)
apply (rule conjI)
apply (rule isometry_linear)
apply simp
apply simp
apply clarify
apply (erule_tac x=v in allE)
apply (erule_tac x=0 in allE)
done

text ‹Can extend an isometry from unit sphere.›

lemma isometry_sphere_extend:
fixes f:: "real ^'n ⇒ real ^'n"
assumes f1: "∀x. norm x = 1 ⟶ norm (f x) = 1"
and fd1: "∀ x y. norm x = 1 ⟶ norm y = 1 ⟶ dist (f x) (f y) = dist x y"
shows "∃g. orthogonal_transformation g ∧ (∀x. norm x = 1 ⟶ g x = f x)"
proof -
{
fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
assume H:
"x = norm x *⇩R x0"
"y = norm y *⇩R y0"
"x' = norm x *⇩R x0'" "y' = norm y *⇩R y0'"
"norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1"
"norm(x0' - y0') = norm(x0 - y0)"
then have *: "x0 ∙ y0 = x0' ∙ y0' + y0' ∙ x0' - y0 ∙ x0 "
have "norm(x' - y') = norm(x - y)"
apply (subst H(1))
apply (subst H(2))
apply (subst H(3))
apply (subst H(4))
using H(5-9)
unfolding *
done
}
note th0 = this
let ?g = "λx. if x = 0 then 0 else norm x *⇩R f (inverse (norm x) *⇩R x)"
{
fix x:: "real ^'n"
assume nx: "norm x = 1"
have "?g x = f x"
using nx by auto
}
then have thfg: "∀x. norm x = 1 ⟶ ?g x = f x"
by blast
have g0: "?g 0 = 0"
by simp
{
fix x y :: "real ^'n"
{
assume "x = 0" "y = 0"
then have "dist (?g x) (?g y) = dist x y"
by simp
}
moreover
{
assume "x = 0" "y ≠ 0"
then have "dist (?g x) (?g y) = dist x y"
apply (rule f1[rule_format])
done
}
moreover
{
assume "x ≠ 0" "y = 0"
then have "dist (?g x) (?g y) = dist x y"
apply (rule f1[rule_format])
done
}
moreover
{
assume z: "x ≠ 0" "y ≠ 0"
have th00:
"x = norm x *⇩R (inverse (norm x) *⇩R x)"
"y = norm y *⇩R (inverse (norm y) *⇩R y)"
"norm x *⇩R f ((inverse (norm x) *⇩R x)) = norm x *⇩R f (inverse (norm x) *⇩R x)"
"norm y *⇩R f (inverse (norm y) *⇩R y) = norm y *⇩R f (inverse (norm y) *⇩R y)"
"norm (inverse (norm x) *⇩R x) = 1"
"norm (f (inverse (norm x) *⇩R x)) = 1"
"norm (inverse (norm y) *⇩R y) = 1"
"norm (f (inverse (norm y) *⇩R y)) = 1"
"norm (f (inverse (norm x) *⇩R x) - f (inverse (norm y) *⇩R y)) =
norm (inverse (norm x) *⇩R x - inverse (norm y) *⇩R y)"
using z
by (auto simp add: field_simps intro: f1[rule_format] fd1[rule_format, unfolded dist_norm])
from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
}
ultimately have "dist (?g x) (?g y) = dist x y"
by blast
}
note thd = this
show ?thesis
apply (rule exI[where x= ?g])
unfolding orthogonal_transformation_isometry
using g0 thfg thd
apply metis
done
qed

text ‹Rotation, reflection, rotoinversion.›

definition "rotation_matrix Q ⟷ orthogonal_matrix Q ∧ det Q = 1"
definition "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 ‹Explicit formulas for low dimensions.›

lemma setprod_neutral_const: "setprod f {(1::nat)..1} = f 1"
by simp

lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2"
by (simp add: eval_nat_numeral atLeastAtMostSuc_conv mult.commute)

lemma setprod_3: "setprod 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"
by (simp add: det_def of_nat_Suc sign_id)

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 setsum_over_permutations_insert[OF f12]
unfolding permutes_sing
by (simp add: sign_swap_id sign_id swap_id_eq)
qed

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

end
```