(* Title: Determinants
Author: Amine Chaieb, University of Cambridge
*)
header {* Traces, Determinant of square matrices and some properties *}
theory Determinants
imports Euclidean_Space Permutations
begin
subsection{* First some facts about products*}
lemma setprod_insert_eq: "finite A \<Longrightarrow> setprod f (insert a A) = (if a \<in> A then setprod f A else f a * setprod f A)"
apply clarsimp
by(subgoal_tac "insert a A = A", auto)
lemma setprod_add_split:
assumes mn: "(m::nat) <= n + 1"
shows "setprod f {m.. n+p} = setprod f {m .. n} * setprod f {n+1..n+p}"
proof-
let ?A = "{m .. n+p}"
let ?B = "{m .. n}"
let ?C = "{n+1..n+p}"
from mn have un: "?B \<union> ?C = ?A" by auto
from mn have dj: "?B \<inter> ?C = {}" by auto
have f: "finite ?B" "finite ?C" by simp_all
from setprod_Un_disjoint[OF f dj, of f, unfolded un] show ?thesis .
qed
lemma setprod_offset: "setprod f {(m::nat) + p .. n + p} = setprod (\<lambda>i. f (i + p)) {m..n}"
apply (rule setprod_reindex_cong[where f="op + p"])
apply (auto simp add: image_iff Bex_def inj_on_def)
apply arith
apply (rule ext)
apply (simp add: add_commute)
done
lemma setprod_singleton: "setprod f {x} = f x" by simp
lemma setprod_singleton_nat_seg: "setprod f {n..n} = f (n::'a::order)" by simp
lemma setprod_numseg: "setprod f {m..0} = (if m=0 then f 0 else 1)"
"setprod f {m .. Suc n} = (if m \<le> Suc n then f (Suc n) * setprod f {m..n}
else setprod f {m..n})"
by (auto simp add: atLeastAtMostSuc_conv)
lemma setprod_le: assumes fS: "finite S" and fg: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (g x :: 'a::ordered_idom)"
shows "setprod f S \<le> setprod g S"
using fS fg
apply(induct S)
apply simp
apply auto
apply (rule mult_mono)
apply (auto intro: setprod_nonneg)
done
(* FIXME: In Finite_Set there is a useless further assumption *)
lemma setprod_inversef: "finite A ==> setprod (inverse \<circ> f) A = (inverse (setprod f A) :: 'a:: {division_by_zero, field})"
apply (erule finite_induct)
apply (simp)
apply simp
done
lemma setprod_le_1: assumes fS: "finite S" and f: "\<forall>x\<in>S. f x \<ge> 0 \<and> f x \<le> (1::'a::ordered_idom)"
shows "setprod f S \<le> 1"
using setprod_le[OF fS f] unfolding setprod_1 .
subsection{* Trace *}
definition trace :: "'a::semiring_1^'n^'n \<Rightarrow> 'a" where
"trace A = setsum (\<lambda>i. ((A$i)$i)) (UNIV::'n set)"
lemma trace_0: "trace(mat 0) = 0"
by (simp add: trace_def mat_def)
lemma trace_I: "trace(mat 1 :: 'a::semiring_1^'n^'n) = of_nat(CARD('n))"
by (simp add: trace_def mat_def)
lemma trace_add: "trace ((A::'a::comm_semiring_1^'n^'n) + B) = trace A + trace B"
by (simp add: trace_def setsum_addf)
lemma trace_sub: "trace ((A::'a::comm_ring_1^'n^'n) - B) = trace A - trace B"
by (simp add: trace_def setsum_subtractf)
lemma trace_mul_sym:"trace ((A::'a::comm_semiring_1^'n^'n) ** B) = trace (B**A)"
apply (simp add: trace_def matrix_matrix_mult_def)
apply (subst setsum_commute)
by (simp add: mult_commute)
(* ------------------------------------------------------------------------- *)
(* Definition of determinant. *)
(* ------------------------------------------------------------------------- *)
definition det:: "'a::comm_ring_1^'n^'n \<Rightarrow> 'a" where
"det A = setsum (\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)) {p. p permutes (UNIV :: 'n set)}"
(* ------------------------------------------------------------------------- *)
(* A few general lemmas we need below. *)
(* ------------------------------------------------------------------------- *)
lemma setprod_permute:
assumes p: "p permutes S"
shows "setprod f S = setprod (f o p) S"
proof-
{assume "\<not> finite S" hence ?thesis by simp}
moreover
{assume fS: "finite S"
then have ?thesis
apply (simp add: setprod_def cong del:strong_setprod_cong)
apply (rule ab_semigroup_mult.fold_image_permute)
apply (auto simp add: p)
apply unfold_locales
done}
ultimately show ?thesis by blast
qed
lemma setproduct_permute_nat_interval: "p permutes {m::nat .. n} ==> setprod f {m..n} = setprod (f o p) {m..n}"
by (blast intro!: setprod_permute)
(* ------------------------------------------------------------------------- *)
(* Basic determinant properties. *)
(* ------------------------------------------------------------------------- *)
lemma det_transp: "det (transp A) = det (A::'a::comm_ring_1 ^'n^'n::finite)"
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_def Collect_def permutation_permutes)
from permutes_inj[OF pU]
have pi: "inj_on p ?U" by (blast intro: subset_inj_on)
from permutes_image[OF pU]
have "setprod (\<lambda>i. ?di (transp A) i (inv p i)) ?U = setprod (\<lambda>i. ?di (transp A) i (inv p i)) (p ` ?U)" by simp
also have "\<dots> = setprod ((\<lambda>i. ?di (transp A) i (inv p i)) o p) ?U"
unfolding setprod_reindex[OF pi] ..
also have "\<dots> = setprod (\<lambda>i. ?di A i (p i)) ?U"
proof-
{fix i assume i: "i \<in> ?U"
from i permutes_inv_o[OF pU] permutes_in_image[OF pU]
have "((\<lambda>i. ?di (transp A) i (inv p i)) o p) i = ?di A i (p i)"
unfolding transp_def by (simp add: expand_fun_eq)}
then show "setprod ((\<lambda>i. ?di (transp A) i (inv p i)) o p) ?U = setprod (\<lambda>i. ?di A i (p i)) ?U" by (auto intro: setprod_cong)
qed
finally have "of_int (sign (inv p)) * (setprod (\<lambda>i. ?di (transp A) i (inv p i)) ?U) = of_int (sign p) * (setprod (\<lambda>i. ?di A i (p i)) ?U)" using sth
by simp}
then show ?thesis unfolding det_def apply (subst setsum_permutations_inverse)
apply (rule setsum_cong2) by blast
qed
lemma det_lowerdiagonal:
fixes A :: "'a::comm_ring_1^'n^'n::{finite,wellorder}"
assumes ld: "\<And>i j. i < j \<Longrightarrow> A$i$j = 0"
shows "det A = setprod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
proof-
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
have fU: "finite ?U" by simp
from finite_permutations[OF fU] have fPU: "finite ?PU" .
have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
{fix p assume p: "p \<in> ?PU -{id}"
from p have pU: "p permutes ?U" and pid: "p \<noteq> id" by blast+
from permutes_natset_le[OF pU] pid obtain i where
i: "p i > i" by (metis not_le)
from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
from setprod_zero[OF fU ex] have "?pp p = 0" by simp}
then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0" by blast
from setsum_mono_zero_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^'n::{finite,wellorder}"
assumes ld: "\<And>i j. i > j \<Longrightarrow> A$i$j = 0"
shows "det A = setprod (\<lambda>i. A$i$i) (UNIV:: 'n set)"
proof-
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "(\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set))"
have fU: "finite ?U" by simp
from finite_permutations[OF fU] have fPU: "finite ?PU" .
have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
{fix p assume p: "p \<in> ?PU -{id}"
from p have pU: "p permutes ?U" and pid: "p \<noteq> id" by blast+
from permutes_natset_ge[OF pU] pid obtain i where
i: "p i < i" by (metis not_le)
from ld[OF i] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
from setprod_zero[OF fU ex] have "?pp p = 0" by simp}
then have p0: "\<forall>p \<in> ?PU -{id}. ?pp p = 0" by blast
from setsum_mono_zero_cong_left[OF fPU id0 p0] show ?thesis
unfolding det_def by (simp add: sign_id)
qed
lemma det_diagonal:
fixes A :: "'a::comm_ring_1^'n^'n::finite"
assumes ld: "\<And>i j. i \<noteq> j \<Longrightarrow> A$i$j = 0"
shows "det A = setprod (\<lambda>i. A$i$i) (UNIV::'n set)"
proof-
let ?U = "UNIV:: 'n set"
let ?PU = "{p. p permutes ?U}"
let ?pp = "\<lambda>p. of_int (sign p) * setprod (\<lambda>i. A$i$p i) (UNIV :: 'n set)"
have fU: "finite ?U" by simp
from finite_permutations[OF fU] have fPU: "finite ?PU" .
have id0: "{id} \<subseteq> ?PU" by (auto simp add: permutes_id)
{fix p assume p: "p \<in> ?PU - {id}"
then have "p \<noteq> id" by simp
then obtain i where i: "p i \<noteq> i" unfolding expand_fun_eq by auto
from ld [OF i [symmetric]] have ex:"\<exists>i \<in> ?U. A$i$p i = 0" by blast
from setprod_zero [OF fU ex] have "?pp p = 0" by simp}
then have p0: "\<forall>p \<in> ?PU - {id}. ?pp p = 0" by blast
from setsum_mono_zero_cong_left[OF fPU id0 p0] show ?thesis
unfolding det_def by (simp add: sign_id)
qed
lemma det_I: "det (mat 1 :: 'a::comm_ring_1^'n^'n::finite) = 1"
proof-
let ?A = "mat 1 :: 'a::comm_ring_1^'n^'n"
let ?U = "UNIV :: 'n set"
let ?f = "\<lambda>i j. ?A$i$j"
{fix i assume i: "i \<in> ?U"
have "?f i i = 1" using i by (vector mat_def)}
hence th: "setprod (\<lambda>i. ?f i i) ?U = setprod (\<lambda>x. 1) ?U"
by (auto intro: setprod_cong)
{fix i j assume i: "i \<in> ?U" and j: "j \<in> ?U" and ij: "i \<noteq> j"
have "?f i j = 0" using i j ij by (vector mat_def) }
then have "det ?A = setprod (\<lambda>i. ?f i i) ?U" using det_diagonal
by blast
also have "\<dots> = 1" unfolding th setprod_1 ..
finally show ?thesis .
qed
lemma det_0: "det (mat 0 :: 'a::comm_ring_1^'n^'n::finite) = 0"
by (simp add: det_def setprod_zero)
lemma det_permute_rows:
fixes A :: "'a::comm_ring_1^'n^'n::finite"
assumes p: "p permutes (UNIV :: 'n::finite set)"
shows "det(\<chi> i. A$p i :: 'a^'n^'n) = of_int (sign p) * det A"
apply (simp add: det_def setsum_right_distrib mult_assoc[symmetric])
apply (subst sum_permutations_compose_right[OF p])
proof(rule setsum_cong2)
let ?U = "UNIV :: 'n set"
let ?PU = "{p. p permutes ?U}"
fix q assume qPU: "q \<in> ?PU"
have fU: "finite ?U" by simp
from qPU have q: "q permutes ?U" by blast
from p q have pp: "permutation p" and qp: "permutation q"
by (metis fU permutation_permutes)+
from permutes_inv[OF p] have ip: "inv p permutes ?U" .
have "setprod (\<lambda>i. A$p i$ (q o p) i) ?U = setprod ((\<lambda>i. A$p i$(q o p) i) o inv p) ?U"
by (simp only: setprod_permute[OF ip, symmetric])
also have "\<dots> = setprod (\<lambda>i. A $ (p o inv p) i $ (q o (p o inv p)) i) ?U"
by (simp only: o_def)
also have "\<dots> = setprod (\<lambda>i. A$i$q i) ?U" by (simp only: o_def permutes_inverses[OF p])
finally have thp: "setprod (\<lambda>i. A$p i$ (q o p) i) ?U = setprod (\<lambda>i. A$i$q i) ?U"
by blast
show "of_int (sign (q o p)) * setprod (\<lambda>i. A$ p i$ (q o p) i) ?U = of_int (sign p) * of_int (sign q) * setprod (\<lambda>i. A$i$q i) ?U"
by (simp only: thp sign_compose[OF qp pp] mult_commute of_int_mult)
qed
lemma det_permute_columns:
fixes A :: "'a::comm_ring_1^'n^'n::finite"
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 = "transp A"
have "of_int (sign p) * det A = det (transp (\<chi> i. transp A $ p i))"
unfolding det_permute_rows[OF p, of ?At] det_transp ..
moreover
have "?Ap = transp (\<chi> i. transp A $ p i)"
by (simp add: transp_def Cart_eq)
ultimately show ?thesis by simp
qed
lemma det_identical_rows:
fixes A :: "'a::ordered_idom^'n^'n::finite"
assumes ij: "i \<noteq> j"
and r: "row i A = row j A"
shows "det A = 0"
proof-
have tha: "\<And>(a::'a) b. a = b ==> b = - a ==> a = 0"
by simp
have th1: "of_int (-1) = - 1" by (metis of_int_1 of_int_minus number_of_Min)
let ?p = "Fun.swap i j id"
let ?A = "\<chi> i. A $ ?p i"
from r have "A = ?A" by (simp add: Cart_eq row_def swap_def)
hence "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::ordered_idom^'n^'n::finite"
assumes ij: "i \<noteq> j"
and r: "column i A = column j A"
shows "det A = 0"
apply (subst det_transp[symmetric])
apply (rule det_identical_rows[OF ij])
by (metis row_transp r)
lemma det_zero_row:
fixes A :: "'a::{idom, ring_char_0}^'n^'n::finite"
assumes r: "row i A = 0"
shows "det A = 0"
using r
apply (simp add: row_def det_def Cart_eq)
apply (rule setsum_0')
apply (auto simp: sign_nz)
done
lemma det_zero_column:
fixes A :: "'a::{idom,ring_char_0}^'n^'n::finite"
assumes r: "column i A = 0"
shows "det A = 0"
apply (subst det_transp[symmetric])
apply (rule det_zero_row [of i])
by (metis row_transp r)
lemma det_row_add:
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 Cart_lambda_beta setsum_addf[symmetric]
proof (rule setsum_cong2)
let ?U = "UNIV :: 'n set"
let ?pU = "{p. p permutes ?U}"
let ?f = "(\<lambda>i. if i = k then a i + b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
let ?g = "(\<lambda> i. if i = k then a i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
let ?h = "(\<lambda> i. if i = k then b i else c i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
fix p assume p: "p \<in> ?pU"
let ?Uk = "?U - {k}"
from p have pU: "p permutes ?U" by blast
have kU: "?U = insert k ?Uk" by blast
{fix j assume j: "j \<in> ?Uk"
from j have "?f j $ p j = ?g j $ p j" and "?f j $ p j= ?h j $ p j"
by simp_all}
then have th1: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?g i $ p i) ?Uk"
and th2: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?h i $ p i) ?Uk"
apply -
apply (rule setprod_cong, simp_all)+
done
have th3: "finite ?Uk" "k \<notin> ?Uk" by auto
have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?f i $ p i) (insert k ?Uk)"
unfolding kU[symmetric] ..
also have "\<dots> = ?f k $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk"
apply (rule setprod_insert)
apply simp
by blast
also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk)" by (simp add: ring_simps)
also have "\<dots> = (a k $ p k * setprod (\<lambda>i. ?g i $ p i) ?Uk) + (b k$ p k * setprod (\<lambda>i. ?h i $ p i) ?Uk)" by (metis th1 th2)
also have "\<dots> = setprod (\<lambda>i. ?g i $ p i) (insert k ?Uk) + setprod (\<lambda>i. ?h i $ p i) (insert k ?Uk)"
unfolding setprod_insert[OF th3] by simp
finally have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?g i $ p i) ?U + setprod (\<lambda>i. ?h i $ p i) ?U" unfolding kU[symmetric] .
then show "of_int (sign p) * setprod (\<lambda>i. ?f i $ p i) ?U = of_int (sign p) * setprod (\<lambda>i. ?g i $ p i) ?U + of_int (sign p) * setprod (\<lambda>i. ?h i $ p i) ?U"
by (simp add: ring_simps)
qed
lemma det_row_mul:
fixes a b :: "'n::finite \<Rightarrow> _ ^ 'n"
shows "det((\<chi> i. if i = k then c *s a i else b i)::'a::comm_ring_1^'n^'n) =
c* det((\<chi> i. if i = k then a i else b i)::'a::comm_ring_1^'n^'n)"
unfolding det_def Cart_lambda_beta setsum_right_distrib
proof (rule setsum_cong2)
let ?U = "UNIV :: 'n set"
let ?pU = "{p. p permutes ?U}"
let ?f = "(\<lambda>i. if i = k then c*s a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
let ?g = "(\<lambda> i. if i = k then a i else b i)::'n \<Rightarrow> 'a::comm_ring_1^'n"
fix p assume p: "p \<in> ?pU"
let ?Uk = "?U - {k}"
from p have pU: "p permutes ?U" by blast
have kU: "?U = insert k ?Uk" by blast
{fix j assume j: "j \<in> ?Uk"
from j have "?f j $ p j = ?g j $ p j" by simp}
then have th1: "setprod (\<lambda>i. ?f i $ p i) ?Uk = setprod (\<lambda>i. ?g i $ p i) ?Uk"
apply -
apply (rule setprod_cong, simp_all)
done
have th3: "finite ?Uk" "k \<notin> ?Uk" by auto
have "setprod (\<lambda>i. ?f i $ p i) ?U = setprod (\<lambda>i. ?f i $ p i) (insert k ?Uk)"
unfolding kU[symmetric] ..
also have "\<dots> = ?f k $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk"
apply (rule setprod_insert)
apply simp
by blast
also have "\<dots> = (c*s a k) $ p k * setprod (\<lambda>i. ?f i $ p i) ?Uk" by (simp add: ring_simps)
also have "\<dots> = c* (a k $ p k * setprod (\<lambda>i. ?g i $ p i) ?Uk)"
unfolding th1 by (simp add: mult_ac)
also have "\<dots> = c* (setprod (\<lambda>i. ?g i $ p i) (insert k ?Uk))"
unfolding setprod_insert[OF th3] by simp
finally have "setprod (\<lambda>i. ?f i $ p i) ?U = c* (setprod (\<lambda>i. ?g i $ p i) ?U)" unfolding kU[symmetric] .
then show "of_int (sign p) * setprod (\<lambda>i. ?f i $ p i) ?U = c * (of_int (sign p) * setprod (\<lambda>i. ?g i $ p i) ?U)"
by (simp add: ring_simps)
qed
lemma det_row_0:
fixes b :: "'n::finite \<Rightarrow> _ ^ 'n"
shows "det((\<chi> i. if i = k then 0 else b i)::'a::comm_ring_1^'n^'n) = 0"
using det_row_mul[of k 0 "\<lambda>i. 1" b]
apply (simp)
unfolding vector_smult_lzero .
lemma det_row_operation:
fixes A :: "'a::ordered_idom^'n^'n::finite"
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:: ordered_idom^'n^'n::finite"
assumes x: "x \<in> span {row j A |j. j \<noteq> i}"
shows "det (\<chi> k. if k = i then row i A + x else row k A) = det A"
proof-
let ?U = "UNIV :: 'n set"
let ?S = "{row j A |j. j \<noteq> i}"
let ?d = "\<lambda>x. det (\<chi> k. if k = i then x else row k A)"
let ?P = "\<lambda>x. ?d (row i A + x) = det A"
{fix k
have "(if k = i then row i A + 0 else row k A) = row k A" by simp}
then have P0: "?P 0"
apply -
apply (rule cong[of det, OF refl])
by (vector row_def)
moreover
{fix c z y assume zS: "z \<in> ?S" and Py: "?P y"
from zS obtain j where j: "z = row j A" "i \<noteq> j" by blast
let ?w = "row i A + y"
have th0: "row i A + (c*s z + y) = ?w + c*s z" by vector
have thz: "?d z = 0"
apply (rule det_identical_rows[OF j(2)])
using j by (vector row_def)
have "?d (row i A + (c*s z + y)) = ?d (?w + c*s z)" unfolding th0 ..
then have "?P (c*s z + y)" unfolding thz Py det_row_mul[of i] det_row_add[of i]
by simp }
ultimately show ?thesis
apply -
apply (rule span_induct_alt[of ?P ?S, OF P0])
apply blast
apply (rule x)
done
qed
(* ------------------------------------------------------------------------- *)
(* 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::ordered_idom^'n^'n::finite"
assumes d: "dependent (rows A)"
shows "det A = 0"
proof-
let ?U = "UNIV :: 'n set"
from d obtain i where i: "row i A \<in> span (rows A - {row i A})"
unfolding dependent_def rows_def by blast
{fix j k assume jk: "j \<noteq> k"
and c: "row j A = row k A"
from det_identical_rows[OF jk c] have ?thesis .}
moreover
{assume H: "\<And> i j. i \<noteq> j \<Longrightarrow> row i A \<noteq> row j A"
have th0: "- row i A \<in> span {row j A|j. j \<noteq> i}"
apply (rule span_neg)
apply (rule set_rev_mp)
apply (rule i)
apply (rule span_mono)
using H i by (auto simp add: rows_def)
from det_row_span[OF th0]
have "det A = det (\<chi> k. if k = i then 0 *s 1 else row k A)"
unfolding right_minus vector_smult_lzero ..
with det_row_mul[of i "0::'a" "\<lambda>i. 1"]
have "det A = 0" by simp}
ultimately show ?thesis by blast
qed
lemma det_dependent_columns: assumes d: "dependent(columns (A::'a::ordered_idom^'n^'n::finite))" shows "det A = 0"
by (metis d det_dependent_rows rows_transp det_transp)
(* ------------------------------------------------------------------------- *)
(* Multilinearity and the multiplication formula. *)
(* ------------------------------------------------------------------------- *)
lemma Cart_lambda_cong: "(\<And>x. f x = g x) \<Longrightarrow> (Cart_lambda f::'a^'n) = (Cart_lambda g :: 'a^'n)"
apply (rule iffD1[OF Cart_lambda_unique]) by vector
lemma det_linear_row_setsum:
assumes fS: "finite S"
shows "det ((\<chi> i. if i = k then setsum (a i) S else c i)::'a::comm_ring_1^'n^'n::finite) = setsum (\<lambda>j. det ((\<chi> i. if i = k then a i j else c i)::'a^'n^'n)) S"
proof(induct rule: finite_induct[OF fS])
case 1 thus ?case apply simp unfolding setsum_empty det_row_0[of k] ..
next
case (2 x F)
then show ?case by (simp add: det_row_add cong del: if_weak_cong)
qed
lemma finite_bounded_functions:
assumes fS: "finite S"
shows "finite {f. (\<forall>i \<in> {1.. (k::nat)}. f i \<in> S) \<and> (\<forall>i. i \<notin> {1 .. k} \<longrightarrow> f i = i)}"
proof(induct k)
case 0
have th: "{f. \<forall>i. f i = i} = {id}" by (auto intro: ext)
show ?case by (auto simp add: th)
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 add: image_iff)
apply (rule_tac x="x (Suc k)" in bexI)
apply (rule_tac x = "\<lambda>i. if i = Suc k then i else x i" in exI)
apply (auto intro: ext)
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]: "(\<forall>x. f x = x) = (f = id)" by (auto intro: ext)
lemma det_linear_rows_setsum_lemma:
assumes fS: "finite S" and fT: "finite T"
shows "det((\<chi> i. if i \<in> T then setsum (a i) S else c i):: 'a::comm_ring_1^'n^'n::finite) =
setsum (\<lambda>f. det((\<chi> i. if i \<in> T then a i (f i) else c i)::'a^'n^'n))
{f. (\<forall>i \<in> T. f i \<in> S) \<and> (\<forall>i. i \<notin> T \<longrightarrow> f i = i)}"
using fT
proof(induct T arbitrary: a c set: finite)
case empty
have th0: "\<And>x y. (\<chi> i. if i \<in> {} then x i else y i) = (\<chi> i. y i)" by vector
from "empty.prems" show ?case unfolding th0 by simp
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>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 `z \<notin> T` have nz: "\<And>i. i \<in> T \<Longrightarrow> i = z \<longleftrightarrow> False" by auto
have "det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
det (\<chi> i. if i = z then setsum (a i) S
else if i \<in> T then setsum (a i) S else c i)"
unfolding insert_iff thif ..
also have "\<dots> = (\<Sum>j\<in>S. det (\<chi> i. if i \<in> T then setsum (a i) S
else if i = z then a i j else c i))"
unfolding det_linear_row_setsum[OF fS]
apply (subst thif2)
using nz by (simp cong del: if_weak_cong cong add: if_cong)
finally have tha:
"det (\<chi> i. if i \<in> insert z T then setsum (a i) S else c i) =
(\<Sum>(j, f)\<in>S \<times> ?F T. det (\<chi> i. if i \<in> T then a i (f i)
else if i = z then a i j
else c i))"
unfolding insert.hyps unfolding setsum_cartesian_product by blast
show ?case unfolding tha
apply(rule setsum_eq_general_reverses[where h= "?h" and k= "?k"],
blast intro: finite_cartesian_product fS finite,
blast intro: finite_cartesian_product fS finite)
using `z \<notin> T`
apply (auto intro: ext)
apply (rule cong[OF refl[of det]])
by vector
qed
lemma det_linear_rows_setsum:
assumes fS: "finite (S::'n::finite set)"
shows "det (\<chi> i. setsum (a i) S) = setsum (\<lambda>f. det (\<chi> i. a i (f i) :: 'a::comm_ring_1 ^ 'n^'n::finite)) {f. \<forall>i. f i \<in> S}"
proof-
have th0: "\<And>x y. ((\<chi> i. if i \<in> (UNIV:: 'n set) then x i else y i) :: 'a^'n^'n) = (\<chi> i. x i)" by vector
from det_linear_rows_setsum_lemma[OF fS, of "UNIV :: 'n set" a, unfolded th0, OF finite] show ?thesis by simp
qed
lemma matrix_mul_setsum_alt:
fixes A B :: "'a::comm_ring_1^'n^'n::finite"
shows "A ** B = (\<chi> i. setsum (\<lambda>k. A$i$k *s B $ k) (UNIV :: 'n set))"
by (vector matrix_matrix_mult_def setsum_component)
lemma det_rows_mul:
"det((\<chi> i. c i *s a i)::'a::comm_ring_1^'n^'n::finite) =
setprod (\<lambda>i. c i) (UNIV:: 'n set) * det((\<chi> i. a i)::'a^'n^'n)"
proof (simp add: det_def setsum_right_distrib cong add: setprod_cong, rule setsum_cong2)
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 "setprod (\<lambda>i. c i * a i $ p i) ?U = setprod c ?U * setprod (\<lambda>i. a i $ p i) ?U"
unfolding setprod_timesf ..
then show "?s * (\<Prod>xa\<in>?U. c xa * a xa $ p xa) =
setprod c ?U * (?s* (\<Prod>xa\<in>?U. a xa $ p xa))" by (simp add: ring_simps)
qed
lemma det_mul:
fixes A B :: "'a::ordered_idom^'n^'n::finite"
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 fU: "finite ?U" by simp
have fF: "finite ?F" by (rule finite)
{fix p assume p: "p permutes ?U"
have "p \<in> ?F" unfolding mem_Collect_eq permutes_in_image[OF p]
using p[unfolded permutes_def] by simp}
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
from ij
have rth: "row i ?B = row j ?B" by (vector row_def)
from det_identical_rows[OF ij(2) rth]
have "det (\<chi> i. A$i$f i *s B$f i) = 0"
unfolding det_rows_mul by simp}
moreover
{assume fi: "inj_on f ?U"
from f fi have fith: "\<And>i j. f i = f j \<Longrightarrow> i = j"
unfolding inj_on_def by metis
note fs = fi[unfolded surjective_iff_injective_gen[OF fU fU refl fUU, symmetric]]
{fix y
from fs f have "\<exists>x. f x = y" by blast
then obtain x where x: "f x = y" by blast
{fix z assume z: "f z = y" from fith x z have "z = x" by metis}
with x have "\<exists>!x. f x = y" by blast}
with f(3) have "det (\<chi> i. A$i$f i *s B$f i) = 0" by blast}
ultimately have "det (\<chi> i. A$i$f i *s B$f i) = 0" by blast}
hence zth: "\<forall> f\<in> ?F - ?PU. det (\<chi> i. A$i$f i *s B$f i) = 0" by simp
{fix p assume pU: "p \<in> ?PU"
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 "(setsum (\<lambda>q. ?s q *
(\<Prod>i\<in> ?U. (\<chi> i. A $ i $ p i *s B $ p i :: 'a^'n^'n) $ i $ q i)) ?PU) =
(setsum (\<lambda>q. ?s p * (\<Prod>i\<in> ?U. A $ i $ p i) *
(?s q * (\<Prod>i\<in> ?U. B $ i $ q i))) ?PU)"
unfolding sum_permutations_compose_right[OF permutes_inv[OF p], of ?f]
proof(rule setsum_cong2)
fix q assume qU: "q \<in> ?PU"
hence q: "q permutes ?U" by blast
from p q have pp: "permutation p" and pq: "permutation q"
unfolding permutation_permutes by auto
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]
by (simp_all add: sign_idempotent)
have ths: "?s q = ?s p * ?s (q o inv p)"
using pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
by (simp add: th00 mult_ac sign_idempotent sign_compose)
have th001: "setprod (\<lambda>i. B$i$ q (inv p i)) ?U = setprod ((\<lambda>i. B$i$ q (inv p i)) o p) ?U"
by (rule setprod_permute[OF p])
have thp: "setprod (\<lambda>i. (\<chi> i. A$i$p i *s B$p i :: 'a^'n^'n) $i $ q i) ?U = setprod (\<lambda>i. A$i$p i) ?U * setprod (\<lambda>i. B$i$ q (inv p i)) ?U"
unfolding th001 setprod_timesf[symmetric] o_def permutes_inverses[OF p]
apply (rule setprod_cong[OF refl])
using permutes_in_image[OF q] by vector
show "?s q * setprod (\<lambda>i. (((\<chi> i. A$i$p i *s B$p i) :: 'a^'n^'n)$i$q i)) ?U = ?s p * (setprod (\<lambda>i. A$i$p i) ?U) * (?s (q o inv p) * setprod (\<lambda>i. B$i$(q o inv p) i) ?U)"
using ths thp pp pq permutation_inverse[OF pp] sign_inverse[OF pp]
by (simp add: sign_nz th00 ring_simps sign_idempotent sign_compose)
qed
}
then have th2: "setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU = det A * det B"
unfolding det_def setsum_product
by (rule setsum_cong2)
have "det (A**B) = setsum (\<lambda>f. det (\<chi> 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 "\<dots> = setsum (\<lambda>f. det (\<chi> i. A$i$f i *s B$f i)) ?PU"
using setsum_mono_zero_cong_left[OF fF PUF zth, symmetric]
unfolding det_rows_mul by auto
finally show ?thesis unfolding th2 .
qed
(* ------------------------------------------------------------------------- *)
(* Relation to invertibility. *)
(* ------------------------------------------------------------------------- *)
lemma invertible_left_inverse:
fixes A :: "real^'n^'n::finite"
shows "invertible A \<longleftrightarrow> (\<exists>(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::finite"
shows "invertible A \<longleftrightarrow> (\<exists>(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::finite"
shows "invertible A \<longleftrightarrow> det A \<noteq> 0"
proof-
{assume "invertible A"
then obtain B :: "real ^'n^'n" where B: "A ** B = mat 1"
unfolding invertible_righ_inverse by blast
hence "det (A ** B) = det (mat 1 :: real ^'n^'n)" by simp
hence "det A \<noteq> 0"
apply (simp add: det_mul det_I) by algebra }
moreover
{assume H: "\<not> invertible A"
let ?U = "UNIV :: 'n set"
have fU: "finite ?U" by simp
from H obtain c i where c: "setsum (\<lambda>i. c i *s row i A) ?U = 0"
and iU: "i \<in> ?U" and ci: "c i \<noteq> 0"
unfolding invertible_righ_inverse
unfolding matrix_right_invertible_independent_rows by blast
have stupid: "\<And>(a::real^'n) b. a + b = 0 \<Longrightarrow> -a = b"
apply (drule_tac f="op + (- a)" in cong[OF refl])
apply (simp only: ab_left_minus add_assoc[symmetric])
apply simp
done
from c ci
have thr0: "- row i A = setsum (\<lambda>j. (1/ c i) *s (c j *s row j A)) (?U - {i})"
unfolding setsum_diff1'[OF fU iU] setsum_cmul
apply -
apply (rule vector_mul_lcancel_imp[OF ci])
apply (auto simp add: vector_smult_assoc vector_smult_rneg field_simps)
unfolding stupid ..
have thr: "- row i A \<in> span {row j A| j. j \<noteq> i}"
unfolding thr0
apply (rule span_setsum)
apply simp
apply (rule ballI)
apply (rule span_mul)+
apply (rule span_superset)
apply auto
done
let ?B = "(\<chi> 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
(* ------------------------------------------------------------------------- *)
(* Cramer's rule. *)
(* ------------------------------------------------------------------------- *)
lemma cramer_lemma_transp:
fixes A:: "'a::ordered_idom^'n^'n::finite" and x :: "'a ^'n::finite"
shows "det ((\<chi> i. if i = k then setsum (\<lambda>i. x$i *s row i A) (UNIV::'n set)
else row i A)::'a^'n^'n) = x$k * det A"
(is "?lhs = ?rhs")
proof-
let ?U = "UNIV :: 'n set"
let ?Uk = "?U - {k}"
have U: "?U = insert k ?Uk" by blast
have fUk: "finite ?Uk" by simp
have kUk: "k \<notin> ?Uk" by simp
have th00: "\<And>k s. x$k *s row k A + s = (x$k - 1) *s row k A + row k A + s"
by (vector ring_simps)
have th001: "\<And>f k . (\<lambda>x. if x = k then f k else f x) = f" by (auto intro: ext)
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"
apply (rule det_row_span)
apply (rule span_setsum[OF fUk])
apply (rule ballI)
apply (rule span_mul)
apply (rule span_superset)
apply auto
done
show "?lhs = x$k * det A"
apply (subst U)
unfolding setsum_insert[OF fUk kUk]
apply (subst th00)
unfolding add_assoc
apply (subst det_row_add)
unfolding thd0
unfolding det_row_mul
unfolding th001[of k "\<lambda>i. row i A"]
unfolding thd1 by (simp add: ring_simps)
qed
lemma cramer_lemma:
fixes A :: "'a::ordered_idom ^'n^'n::finite"
shows "det((\<chi> i j. if j = k then (A *v x)$i else A$i$j):: 'a^'n^'n) = x$k * det A"
proof-
let ?U = "UNIV :: 'n set"
have stupid: "\<And>c. setsum (\<lambda>i. c i *s row i (transp A)) ?U = setsum (\<lambda>i. c i *s column i A) ?U"
by (auto simp add: row_transp intro: setsum_cong2)
show ?thesis unfolding matrix_mult_vsum
unfolding cramer_lemma_transp[of k x "transp A", unfolded det_transp, symmetric]
unfolding stupid[of "\<lambda>i. x$i"]
apply (subst det_transp[symmetric])
apply (rule cong[OF refl[of det]]) by (vector transp_def column_def row_def)
qed
lemma cramer:
fixes A ::"real^'n^'n::finite"
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$j :: real^'n^'n) / det A)"
proof-
from d0 obtain B where B: "A ** B = mat 1" "B ** A = mat 1"
unfolding invertible_det_nz[symmetric] invertible_def by blast
have "(A ** B) *v b = b" by (simp add: B matrix_vector_mul_lid)
hence "A *v (B *v b) = b" by (simp add: matrix_vector_mul_assoc)
then have xe: "\<exists>x. A*v x = b" by blast
{fix x assume x: "A *v x = b"
have "x = (\<chi> k. det(\<chi> i j. if j=k then b$i else A$i$j :: real^'n^'n) / det A)"
unfolding x[symmetric]
using d0 by (simp add: Cart_eq cramer_lemma field_simps)}
with xe show ?thesis by auto
qed
(* ------------------------------------------------------------------------- *)
(* Orthogonality of a transformation and matrix. *)
(* ------------------------------------------------------------------------- *)
definition "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>v w. f v \<bullet> f w = v \<bullet> w)"
lemma orthogonal_transformation: "orthogonal_transformation f \<longleftrightarrow> linear f \<and> (\<forall>(v::real ^_). norm (f v) = norm v)"
unfolding orthogonal_transformation_def
apply auto
apply (erule_tac x=v in allE)+
apply (simp add: real_vector_norm_def)
by (simp add: dot_norm linear_add[symmetric])
definition "orthogonal_matrix (Q::'a::semiring_1^'n^'n) \<longleftrightarrow> transp Q ** Q = mat 1 \<and> Q ** transp Q = mat 1"
lemma orthogonal_matrix: "orthogonal_matrix (Q:: real ^'n^'n::finite) \<longleftrightarrow> transp Q ** Q = mat 1"
by (metis matrix_left_right_inverse orthogonal_matrix_def)
lemma orthogonal_matrix_id: "orthogonal_matrix (mat 1 :: _^'n^'n::finite)"
by (simp add: orthogonal_matrix_def transp_mat matrix_mul_lid)
lemma orthogonal_matrix_mul:
fixes A :: "real ^'n^'n::finite"
assumes oA : "orthogonal_matrix A"
and oB: "orthogonal_matrix B"
shows "orthogonal_matrix(A ** B)"
using oA oB
unfolding orthogonal_matrix matrix_transp_mul
apply (subst matrix_mul_assoc)
apply (subst matrix_mul_assoc[symmetric])
by (simp add: matrix_mul_rid)
lemma orthogonal_transformation_matrix:
fixes f:: "real^'n \<Rightarrow> real^'n::finite"
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: "linear f" and fd: "\<forall>v w. f v \<bullet> f w = v \<bullet> w"
unfolding orthogonal_transformation_def orthogonal_matrix by blast+
{fix i j
let ?A = "transp ?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[rule_format, of "basis i" "basis j", unfolded matrix_works[OF lf, symmetric] dot_matrix_vector_mul]
have "?A$i$j = ?m1 $ i $ j"
by (simp add: dot_def matrix_matrix_mult_def columnvector_def rowvector_def basis_def th0 setsum_delta[OF fU] mat_def)}
hence "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)
by (simp add: dot_matrix_product matrix_mul_lid)}
ultimately show ?thesis by blast
qed
lemma det_orthogonal_matrix:
fixes Q:: "'a::ordered_idom^'n^'n::finite"
assumes oQ: "orthogonal_matrix Q"
shows "det Q = 1 \<or> det Q = - 1"
proof-
have th: "\<And>x::'a. x = 1 \<or> x = - 1 \<longleftrightarrow> x*x = 1" (is "\<And>x::'a. ?ths x")
proof-
fix x:: 'a
have th0: "x*x - 1 = (x - 1)*(x + 1)" by (simp add: ring_simps)
have th1: "\<And>(x::'a) y. x = - y \<longleftrightarrow> x + y = 0"
apply (subst eq_iff_diff_eq_0) by simp
have "x*x = 1 \<longleftrightarrow> x*x - 1 = 0" by simp
also have "\<dots> \<longleftrightarrow> x = 1 \<or> x = - 1" unfolding th0 th1 by simp
finally show "?ths x" ..
qed
from oQ have "Q ** transp Q = mat 1" by (metis orthogonal_matrix_def)
hence "det (Q ** transp Q) = det (mat 1:: 'a^'n^'n)" by simp
hence "det Q * det Q = 1" by (simp add: det_mul det_I det_transp)
then show ?thesis unfolding th .
qed
(* ------------------------------------------------------------------------- *)
(* Linearity of scaling, and hence isometry, that preserves origin. *)
(* ------------------------------------------------------------------------- *)
lemma scaling_linear:
fixes f :: "real ^'n \<Rightarrow> real ^'n::finite"
assumes f0: "f 0 = 0" and fd: "\<forall>x y. dist (f x) (f y) = c * dist x y"
shows "linear f"
proof-
{fix v w
{fix x note fd[rule_format, of x 0, unfolded dist_def f0 diff_0_right] }
note th0 = this
have "f v \<bullet> f w = c^2 * (v \<bullet> w)"
unfolding dot_norm_neg dist_def[symmetric]
unfolding th0 fd[rule_format] by (simp add: power2_eq_square field_simps)}
note fc = this
show ?thesis unfolding linear_def vector_eq
by (simp add: dot_lmult dot_ladd dot_rmult dot_radd fc ring_simps)
qed
lemma isometry_linear:
"f (0:: real^'n) = (0:: real^'n::finite) \<Longrightarrow> \<forall>x y. dist(f x) (f y) = dist x y
\<Longrightarrow> linear f"
by (rule scaling_linear[where c=1]) simp_all
(* ------------------------------------------------------------------------- *)
(* Hence another formulation of orthogonal transformation. *)
(* ------------------------------------------------------------------------- *)
lemma orthogonal_transformation_isometry:
"orthogonal_transformation f \<longleftrightarrow> f(0::real^'n) = (0::real^'n::finite) \<and> (\<forall>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_def)
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)
by (simp add: dist_def)
(* ------------------------------------------------------------------------- *)
(* Can extend an isometry from unit sphere. *)
(* ------------------------------------------------------------------------- *)
lemma isometry_sphere_extend:
fixes f:: "real ^'n \<Rightarrow> real ^'n::finite"
assumes f1: "\<forall>x. norm x = 1 \<longrightarrow> norm (f x) = 1"
and fd1: "\<forall> x y. norm x = 1 \<longrightarrow> norm y = 1 \<longrightarrow> dist (f x) (f y) = dist x y"
shows "\<exists>g. orthogonal_transformation g \<and> (\<forall>x. norm x = 1 \<longrightarrow> g x = f x)"
proof-
{fix x y x' y' x0 y0 x0' y0' :: "real ^'n"
assume H: "x = norm x *s x0" "y = norm y *s y0"
"x' = norm x *s x0'" "y' = norm y *s y0'"
"norm x0 = 1" "norm x0' = 1" "norm y0 = 1" "norm y0' = 1"
"norm(x0' - y0') = norm(x0 - y0)"
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)
apply (simp add: norm_eq norm_eq_1)
apply (simp add: dot_lsub dot_rsub dot_lmult dot_rmult)
apply (simp add: ring_simps)
by (simp only: right_distrib[symmetric])}
note th0 = this
let ?g = "\<lambda>x. if x = 0 then 0 else norm x *s f (inverse (norm x) *s x)"
{fix x:: "real ^'n" assume nx: "norm x = 1"
have "?g x = f x" using nx by auto}
hence thfg: "\<forall>x. norm x = 1 \<longrightarrow> ?g x = f x" by blast
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 \<noteq> 0"
then have "dist (?g x) (?g y) = dist x y"
apply (simp add: dist_def norm_mul)
apply (rule f1[rule_format])
by(simp add: norm_mul field_simps)}
moreover
{assume "x \<noteq> 0" "y = 0"
then have "dist (?g x) (?g y) = dist x y"
apply (simp add: dist_def norm_mul)
apply (rule f1[rule_format])
by(simp add: norm_mul field_simps)}
moreover
{assume z: "x \<noteq> 0" "y \<noteq> 0"
have th00: "x = norm x *s (inverse (norm x) *s x)" "y = norm y *s (inverse (norm y) *s y)" "norm x *s f ((inverse (norm x) *s x)) = norm x *s f (inverse (norm x) *s x)"
"norm y *s f (inverse (norm y) *s y) = norm y *s f (inverse (norm y) *s y)"
"norm (inverse (norm x) *s x) = 1"
"norm (f (inverse (norm x) *s x)) = 1"
"norm (inverse (norm y) *s y) = 1"
"norm (f (inverse (norm y) *s y)) = 1"
"norm (f (inverse (norm x) *s x) - f (inverse (norm y) *s y)) =
norm (inverse (norm x) *s x - inverse (norm y) *s y)"
using z
by (auto simp add: vector_smult_assoc field_simps norm_mul intro: f1[rule_format] fd1[rule_format, unfolded dist_def])
from z th0[OF th00] have "dist (?g x) (?g y) = dist x y"
by (simp add: dist_def)}
ultimately have "dist (?g x) (?g y) = dist x y" by blast}
note thd = this
show ?thesis
apply (rule exI[where x= ?g])
unfolding orthogonal_transformation_isometry
using g0 thfg thd by metis
qed
(* ------------------------------------------------------------------------- *)
(* Rotation, reflection, rotoinversion. *)
(* ------------------------------------------------------------------------- *)
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::ordered_idom^'n^'n::finite"
shows " orthogonal_matrix Q \<longleftrightarrow> rotation_matrix Q \<or> rotoinversion_matrix Q"
by (metis rotoinversion_matrix_def rotation_matrix_def det_orthogonal_matrix)
(* ------------------------------------------------------------------------- *)
(* Explicit formulas for low dimensions. *)
(* ------------------------------------------------------------------------- *)
lemma setprod_1: "setprod f {(1::nat)..1} = f 1" by simp
lemma setprod_2: "setprod f {(1::nat)..2} = f 1 * f 2"
by (simp add: nat_number setprod_numseg mult_commute)
lemma setprod_3: "setprod f {(1::nat)..3} = f 1 * f 2 * f 3"
by (simp add: nat_number setprod_numseg mult_commute)
lemma det_1: "det (A::'a::comm_ring_1^1^1) = A$1$1"
by (simp add: det_def permutes_sing sign_id UNIV_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 setsum_over_permutations_insert[OF f12]
unfolding permutes_sing
apply (simp add: sign_swap_id sign_id swap_id_eq)
by (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
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 setsum_over_permutations_insert[OF f123]
unfolding setsum_over_permutations_insert[OF f23]
unfolding permutes_sing
apply (simp add: sign_swap_id permutation_swap_id sign_compose sign_id swap_id_eq)
apply (simp add: arith_simps(31)[symmetric] of_int_minus of_int_1 del: arith_simps(31))
by (simp add: ring_simps)
qed
end