--- a/src/HOL/Matrix/Matrix.thy Sun Jun 13 17:57:35 2004 +0200
+++ b/src/HOL/Matrix/Matrix.thy Mon Jun 14 14:20:55 2004 +0200
@@ -1,201 +1,127 @@
(* Title: HOL/Matrix/Matrix.thy
ID: $Id$
Author: Steven Obua
- License: 2004 Technische Universität München
*)
-theory Matrix = MatrixGeneral:
+theory Matrix=MatrixGeneral:
+
+instance matrix :: (minus) minus
+by intro_classes
+
+instance matrix :: (plus) plus
+by (intro_classes)
-axclass almost_matrix_element < zero, plus, times
-matrix_add_assoc: "(a+b)+c = a + (b+c)"
-matrix_add_commute: "a+b = b+a"
+instance matrix :: ("{plus,times}") times
+by (intro_classes)
+
+defs (overloaded)
+ plus_matrix_def: "A + B == combine_matrix (op +) A B"
+ diff_matrix_def: "A - B == combine_matrix (op -) A B"
+ minus_matrix_def: "- A == apply_matrix uminus A"
+ times_matrix_def: "A * B == mult_matrix (op *) (op +) A B"
+
+lemma is_meet_combine_matrix_meet: "is_meet (combine_matrix meet)"
+by (simp_all add: is_meet_def le_matrix_def meet_left_le meet_right_le meet_imp_le)
-matrix_mult_assoc: "(a*b)*c = a*(b*c)"
-matrix_mult_left_0[simp]: "0 * a = 0"
-matrix_mult_right_0[simp]: "a * 0 = 0"
-
-matrix_left_distrib: "(a+b)*c = a*c+b*c"
-matrix_right_distrib: "a*(b+c) = a*b+a*c"
-
-axclass matrix_element < almost_matrix_element
-matrix_add_0[simp]: "0+0 = 0"
-
-instance matrix :: (plus) plus ..
-instance matrix :: (times) times ..
+instance matrix :: (lordered_ab_group) lordered_ab_group_meet
+proof
+ fix A B C :: "('a::lordered_ab_group) matrix"
+ show "A + B + C = A + (B + C)"
+ apply (simp add: plus_matrix_def)
+ apply (rule combine_matrix_assoc[simplified associative_def, THEN spec, THEN spec, THEN spec])
+ apply (simp_all add: add_assoc)
+ done
+ show "A + B = B + A"
+ apply (simp add: plus_matrix_def)
+ apply (rule combine_matrix_commute[simplified commutative_def, THEN spec, THEN spec])
+ apply (simp_all add: add_commute)
+ done
+ show "0 + A = A"
+ apply (simp add: plus_matrix_def)
+ apply (rule combine_matrix_zero_l_neutral[simplified zero_l_neutral_def, THEN spec])
+ apply (simp)
+ done
+ show "- A + A = 0"
+ by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
+ show "A - B = A + - B"
+ by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
+ show "\<exists>m\<Colon>'a matrix \<Rightarrow> 'a matrix \<Rightarrow> 'a matrix. is_meet m"
+ by (auto intro: is_meet_combine_matrix_meet)
+ assume "A <= B"
+ then show "C + A <= C + B"
+ apply (simp add: plus_matrix_def)
+ apply (rule le_left_combine_matrix)
+ apply (simp_all)
+ done
+qed
defs (overloaded)
-plus_matrix_def: "A + B == combine_matrix (op +) A B"
-times_matrix_def: "A * B == mult_matrix (op *) (op +) A B"
+ abs_matrix_def: "abs (A::('a::lordered_ab_group) matrix) == join A (- A)"
-instance matrix :: (matrix_element) matrix_element
-proof -
- note combine_matrix_assoc2 = combine_matrix_assoc[simplified associative_def, THEN spec, THEN spec, THEN spec]
- {
- fix A::"('a::matrix_element) matrix"
- fix B
- fix C
- have "(A + B) + C = A + (B + C)"
- apply (simp add: plus_matrix_def)
- apply (rule combine_matrix_assoc2)
- by (simp_all add: matrix_add_assoc)
- }
- note plus_assoc = this
- {
- fix A::"('a::matrix_element) matrix"
- fix B
- fix C
- have "(A * B) * C = A * (B * C)"
- apply (simp add: times_matrix_def)
- apply (rule mult_matrix_assoc_simple)
- apply (simp_all add: associative_def commutative_def distributive_def l_distributive_def r_distributive_def)
- apply (auto)
- apply (simp_all add: matrix_add_assoc)
- apply (simp_all add: matrix_add_commute)
- apply (simp_all add: matrix_mult_assoc)
- by (simp_all add: matrix_left_distrib matrix_right_distrib)
- }
- note mult_assoc = this
- note combine_matrix_commute2 = combine_matrix_commute[simplified commutative_def, THEN spec, THEN spec]
- {
- fix A::"('a::matrix_element) matrix"
- fix B
- have "A + B = B + A"
- apply (simp add: plus_matrix_def)
- apply (insert combine_matrix_commute2[of "op +"])
- apply (rule combine_matrix_commute2)
- by (simp add: matrix_add_commute)
- }
- note plus_commute = this
- have plus_zero: "(0::('a::matrix_element) matrix) + 0 = 0"
- apply (simp add: plus_matrix_def)
- apply (rule combine_matrix_zero)
- by (simp)
- have mult_left_zero: "!! A. (0::('a::matrix_element) matrix) * A = 0" by(simp add: times_matrix_def)
- have mult_right_zero: "!! A. A * (0::('a::matrix_element) matrix) = 0" by (simp add: times_matrix_def)
- note l_distributive_matrix2 = l_distributive_matrix[simplified l_distributive_def matrix_left_distrib, THEN spec, THEN spec, THEN spec]
- {
- fix A::"('a::matrix_element) matrix"
- fix B
- fix C
- have "(A + B) * C = A * C + B * C"
- apply (simp add: plus_matrix_def)
- apply (simp add: times_matrix_def)
- apply (rule l_distributive_matrix2)
- apply (simp_all add: associative_def commutative_def l_distributive_def)
- apply (auto)
- apply (simp_all add: matrix_add_assoc)
- apply (simp_all add: matrix_add_commute)
- by (simp_all add: matrix_left_distrib)
- }
- note left_distrib = this
- note r_distributive_matrix2 = r_distributive_matrix[simplified r_distributive_def matrix_right_distrib, THEN spec, THEN spec, THEN spec]
- {
- fix A::"('a::matrix_element) matrix"
- fix B
- fix C
- have "C * (A + B) = C * A + C * B"
- apply (simp add: plus_matrix_def)
- apply (simp add: times_matrix_def)
- apply (rule r_distributive_matrix2)
- apply (simp_all add: associative_def commutative_def r_distributive_def)
- apply (auto)
- apply (simp_all add: matrix_add_assoc)
- apply (simp_all add: matrix_add_commute)
- by (simp_all add: matrix_right_distrib)
- }
- note right_distrib = this
- show "OFCLASS('a matrix, matrix_element_class)"
- apply (intro_classes)
- apply (simp_all add: plus_assoc)
- apply (simp_all add: plus_commute)
- apply (simp_all add: plus_zero)
- apply (simp_all add: mult_assoc)
- apply (simp_all add: mult_left_zero mult_right_zero)
- by (simp_all add: left_distrib right_distrib)
+instance matrix :: (lordered_ring) lordered_ring
+proof
+ fix A B C :: "('a :: lordered_ring) matrix"
+ show "A * B * C = A * (B * C)"
+ apply (simp add: times_matrix_def)
+ apply (rule mult_matrix_assoc)
+ apply (simp_all add: associative_def ring_eq_simps)
+ done
+ show "(A + B) * C = A * C + B * C"
+ apply (simp add: times_matrix_def plus_matrix_def)
+ apply (rule l_distributive_matrix[simplified l_distributive_def, THEN spec, THEN spec, THEN spec])
+ apply (simp_all add: associative_def commutative_def ring_eq_simps)
+ done
+ show "A * (B + C) = A * B + A * C"
+ apply (simp add: times_matrix_def plus_matrix_def)
+ apply (rule r_distributive_matrix[simplified r_distributive_def, THEN spec, THEN spec, THEN spec])
+ apply (simp_all add: associative_def commutative_def ring_eq_simps)
+ done
+ show "abs A = join A (-A)"
+ by (simp add: abs_matrix_def)
+ assume a: "A \<le> B"
+ assume b: "0 \<le> C"
+ from a b show "C * A \<le> C * B"
+ apply (simp add: times_matrix_def)
+ apply (rule le_left_mult)
+ apply (simp_all add: add_mono mult_left_mono)
+ done
+ from a b show "A * C \<le> B * C"
+ apply (simp add: times_matrix_def)
+ apply (rule le_right_mult)
+ apply (simp_all add: add_mono mult_right_mono)
+ done
qed
-axclass g_almost_semiring < almost_matrix_element
-g_add_left_0[simp]: "0 + a = a"
-
-lemma g_add_right_0[simp]: "(a::'a::g_almost_semiring) + 0 = a"
-by (simp add: matrix_add_commute)
-
-axclass g_semiring < g_almost_semiring
-g_add_leftimp_eq: "a+b = a+c \<Longrightarrow> b = c"
-
-instance g_almost_semiring < matrix_element
- by intro_classes simp
-
-instance matrix :: (g_almost_semiring) g_almost_semiring
-apply (intro_classes)
-by (simp add: plus_matrix_def combine_matrix_def combine_infmatrix_def)
+lemma Rep_matrix_add[simp]: "Rep_matrix ((a::('a::lordered_ab_group)matrix)+b) j i = (Rep_matrix a j i) + (Rep_matrix b j i)"
+by (simp add: plus_matrix_def)
-lemma RepAbs_matrix_eq_left: " Rep_matrix(Abs_matrix f) = g \<Longrightarrow> \<exists>m. \<forall>j i. m \<le> j \<longrightarrow> f j i = 0 \<Longrightarrow> \<exists>n. \<forall>j i. n \<le> i \<longrightarrow> f j i = 0 \<Longrightarrow> f = g"
-by (simp add: RepAbs_matrix)
-
-lemma RepAbs_matrix_eq_right: "g = Rep_matrix(Abs_matrix f) \<Longrightarrow> \<exists>m. \<forall>j i. m \<le> j \<longrightarrow> f j i = 0 \<Longrightarrow> \<exists>n. \<forall>j i. n \<le> i \<longrightarrow> f j i = 0 \<Longrightarrow> g = f"
-by (simp add: RepAbs_matrix)
+lemma Rep_matrix_mult: "Rep_matrix ((a::('a::lordered_ring) matrix) * b) j i =
+ foldseq (op +) (% k. (Rep_matrix a j k) * (Rep_matrix b k i)) (max (ncols a) (nrows b))"
+apply (simp add: times_matrix_def)
+apply (simp add: Rep_mult_matrix)
+done
+
-instance matrix :: (g_semiring) g_semiring
-apply (intro_classes)
-apply (simp add: plus_matrix_def combine_matrix_def combine_infmatrix_def)
-apply (subst Rep_matrix_inject[THEN sym])
-apply (drule ssubst[OF Rep_matrix_inject, of "% x. x"])
-apply (drule RepAbs_matrix_eq_left)
-apply (auto)
-apply (rule_tac x="max (nrows a) (nrows b)" in exI, simp add: nrows_le)
-apply (rule_tac x="max (ncols a) (ncols b)" in exI, simp add: ncols_le)
-apply (drule RepAbs_matrix_eq_right)
-apply (rule_tac x="max (nrows a) (nrows c)" in exI, simp add: nrows_le)
-apply (rule_tac x="max (ncols a) (ncols c)" in exI, simp add: ncols_le)
+lemma apply_matrix_add: "! x y. f (x+y) = (f x) + (f y) \<Longrightarrow> f 0 = (0::'a) \<Longrightarrow> apply_matrix f ((a::('a::lordered_ab_group) matrix) + b) = (apply_matrix f a) + (apply_matrix f b)"
+apply (subst Rep_matrix_inject[symmetric])
apply (rule ext)+
-apply (drule_tac x="x" and y="x" in comb, simp)
-apply (drule_tac x="xa" and y="xa" in comb, simp)
-apply (drule g_add_leftimp_eq)
-by simp
-
-axclass pordered_matrix_element < matrix_element, order, zero
-pordered_add_right_mono: "a <= b \<Longrightarrow> a + c <= b + c"
-pordered_mult_left: "0 <= c \<Longrightarrow> a <= b \<Longrightarrow> c*a <= c*b"
-pordered_mult_right: "0 <= c \<Longrightarrow> a <= b \<Longrightarrow> a*c <= b*c"
-
-lemma pordered_add_left_mono: "(a::'a::pordered_matrix_element) <= b \<Longrightarrow> c + a <= c + b"
-apply (insert pordered_add_right_mono[of a b c])
-by (simp add: matrix_add_commute)
+apply (simp)
+done
-lemma pordered_add: "(a::'a::pordered_matrix_element) <= b \<Longrightarrow> (c::'a::pordered_matrix_element) <= d \<Longrightarrow> a+c <= b+d"
-proof -
- assume p1:"a <= b"
- assume p2:"c <= d"
- have "a+c <= b+c" by (rule pordered_add_right_mono)
- also have "\<dots> <= b+d" by (rule pordered_add_left_mono)
- ultimately show "a+c <= b+d" by simp
-qed
+lemma singleton_matrix_add: "singleton_matrix j i ((a::_::lordered_ab_group)+b) = (singleton_matrix j i a) + (singleton_matrix j i b)"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (simp)
+done
-instance matrix :: (pordered_matrix_element) pordered_matrix_element
-apply (intro_classes)
-apply (simp_all add: plus_matrix_def times_matrix_def)
-apply (rule le_combine_matrix)
-apply (simp_all)
-apply (simp_all add: pordered_add)
-apply (rule le_left_mult)
-apply (simp_all add: matrix_add_0 g_add_left_0 pordered_add pordered_mult_left matrix_mult_left_0 matrix_mult_right_0)
-apply (rule le_right_mult)
-by (simp_all add: pordered_add pordered_mult_right)
-
-axclass pordered_g_semiring < g_semiring, pordered_matrix_element
-
-instance matrix :: (pordered_g_semiring) pordered_g_semiring ..
-
-lemma nrows_mult: "nrows ((A::('a::matrix_element) matrix) * B) <= nrows A"
+lemma nrows_mult: "nrows ((A::('a::lordered_ring) matrix) * B) <= nrows A"
by (simp add: times_matrix_def mult_nrows)
-lemma ncols_mult: "ncols ((A::('a::matrix_element) matrix) * B) <= ncols B"
+lemma ncols_mult: "ncols ((A::('a::lordered_ring) matrix) * B) <= ncols B"
by (simp add: times_matrix_def mult_ncols)
-(*
constdefs
- one_matrix :: "nat \<Rightarrow> ('a::comm_semiring_1_cancel) matrix"
+ one_matrix :: "nat \<Rightarrow> ('a::{zero,one}) matrix"
"one_matrix n == Abs_matrix (% j i. if j = i & j < n then 1 else 0)"
lemma Rep_one_matrix[simp]: "Rep_matrix (one_matrix n) j i = (if (j = i & j < n) then 1 else 0)"
@@ -204,21 +130,21 @@
apply (rule exI[of _ n], simp add: split_if)+
by (simp add: split_if, arith)
-lemma nrows_one_matrix[simp]: "nrows (one_matrix n) = n" (is "?r = _")
+lemma nrows_one_matrix[simp]: "nrows ((one_matrix n) :: ('a::axclass_0_neq_1)matrix) = n" (is "?r = _")
proof -
have "?r <= n" by (simp add: nrows_le)
- moreover have "n <= ?r" by (simp add: le_nrows, arith)
+ moreover have "n <= ?r" by (simp add:le_nrows, arith)
ultimately show "?r = n" by simp
qed
-lemma ncols_one_matrix[simp]: "ncols (one_matrix n) = n" (is "?r = _")
+lemma ncols_one_matrix[simp]: "ncols ((one_matrix n) :: ('a::axclass_0_neq_1)matrix) = n" (is "?r = _")
proof -
have "?r <= n" by (simp add: ncols_le)
moreover have "n <= ?r" by (simp add: le_ncols, arith)
ultimately show "?r = n" by simp
qed
-lemma one_matrix_mult_right: "ncols A <= n \<Longrightarrow> A * (one_matrix n) = A"
+lemma one_matrix_mult_right[simp]: "ncols A <= n \<Longrightarrow> (A::('a::{lordered_ring,ring_1}) matrix) * (one_matrix n) = A"
apply (subst Rep_matrix_inject[THEN sym])
apply (rule ext)+
apply (simp add: times_matrix_def Rep_mult_matrix)
@@ -226,7 +152,7 @@
apply (simp_all)
by (simp add: max_def ncols)
-lemma one_matrix_mult_left: "nrows A <= n \<Longrightarrow> (one_matrix n) * A = A"
+lemma one_matrix_mult_left[simp]: "nrows A <= n \<Longrightarrow> (one_matrix n) * A = (A::('a::{lordered_ring, ring_1}) matrix)"
apply (subst Rep_matrix_inject[THEN sym])
apply (rule ext)+
apply (simp add: times_matrix_def Rep_mult_matrix)
@@ -234,16 +160,131 @@
apply (simp_all)
by (simp add: max_def nrows)
-constdefs
- right_inverse_matrix :: "('a::comm_semiring_1_cancel) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
- "right_inverse_matrix A X == (A * X = one_matrix (max (nrows A) (ncols X)))"
- inverse_matrix :: "('a::comm_semiring_1_cancel) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
- "inverse_matrix A X == (right_inverse_matrix A X) \<and> (right_inverse_matrix X A)"
+lemma transpose_matrix_mult: "transpose_matrix ((A::('a::{lordered_ring,comm_ring}) matrix)*B) = (transpose_matrix B) * (transpose_matrix A)"
+apply (simp add: times_matrix_def)
+apply (subst transpose_mult_matrix)
+apply (simp_all add: mult_commute)
+done
+
+lemma transpose_matrix_add: "transpose_matrix ((A::('a::lordered_ab_group) matrix)+B) = transpose_matrix A + transpose_matrix B"
+by (simp add: plus_matrix_def transpose_combine_matrix)
+
+lemma transpose_matrix_diff: "transpose_matrix ((A::('a::lordered_ab_group) matrix)-B) = transpose_matrix A - transpose_matrix B"
+by (simp add: diff_matrix_def transpose_combine_matrix)
+
+lemma transpose_matrix_minus: "transpose_matrix (-(A::('a::lordered_ring) matrix)) = - transpose_matrix (A::('a::lordered_ring) matrix)"
+by (simp add: minus_matrix_def transpose_apply_matrix)
+
+constdefs
+ right_inverse_matrix :: "('a::{lordered_ring, ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
+ "right_inverse_matrix A X == (A * X = one_matrix (max (nrows A) (ncols X))) \<and> nrows X \<le> ncols A"
+ left_inverse_matrix :: "('a::{lordered_ring, ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
+ "left_inverse_matrix A X == (X * A = one_matrix (max(nrows X) (ncols A))) \<and> ncols X \<le> nrows A"
+ inverse_matrix :: "('a::{lordered_ring, ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
+ "inverse_matrix A X == (right_inverse_matrix A X) \<and> (left_inverse_matrix A X)"
lemma right_inverse_matrix_dim: "right_inverse_matrix A X \<Longrightarrow> nrows A = ncols X"
apply (insert ncols_mult[of A X], insert nrows_mult[of A X])
by (simp add: right_inverse_matrix_def)
-text {* to be continued \dots *}
-*)
+lemma left_inverse_matrix_dim: "left_inverse_matrix A Y \<Longrightarrow> ncols A = nrows Y"
+apply (insert ncols_mult[of Y A], insert nrows_mult[of Y A])
+by (simp add: left_inverse_matrix_def)
+
+lemma left_right_inverse_matrix_unique:
+ assumes "left_inverse_matrix A Y" "right_inverse_matrix A X"
+ shows "X = Y"
+proof -
+ have "Y = Y * one_matrix (nrows A)"
+ apply (subst one_matrix_mult_right)
+ apply (insert prems)
+ by (simp_all add: left_inverse_matrix_def)
+ also have "\<dots> = Y * (A * X)"
+ apply (insert prems)
+ apply (frule right_inverse_matrix_dim)
+ by (simp add: right_inverse_matrix_def)
+ also have "\<dots> = (Y * A) * X" by (simp add: mult_assoc)
+ also have "\<dots> = X"
+ apply (insert prems)
+ apply (frule left_inverse_matrix_dim)
+ apply (simp_all add: left_inverse_matrix_def right_inverse_matrix_def one_matrix_mult_left)
+ done
+ ultimately show "X = Y" by (simp)
+qed
+
+lemma inverse_matrix_inject: "\<lbrakk> inverse_matrix A X; inverse_matrix A Y \<rbrakk> \<Longrightarrow> X = Y"
+ by (auto simp add: inverse_matrix_def left_right_inverse_matrix_unique)
+
+lemma one_matrix_inverse: "inverse_matrix (one_matrix n) (one_matrix n)"
+ by (simp add: inverse_matrix_def left_inverse_matrix_def right_inverse_matrix_def)
+
+lemma zero_imp_mult_zero: "(a::'a::ring) = 0 | b = 0 \<Longrightarrow> a * b = 0"
+by auto
+
+lemma Rep_matrix_zero_imp_mult_zero:
+ "! j i k. (Rep_matrix A j k = 0) | (Rep_matrix B k i) = 0 \<Longrightarrow> A * B = (0::('a::lordered_ring) matrix)"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (auto simp add: Rep_matrix_mult foldseq_zero zero_imp_mult_zero)
+done
+
+lemma add_nrows: "nrows (A::('a::comm_monoid_add) matrix) <= u \<Longrightarrow> nrows B <= u \<Longrightarrow> nrows (A + B) <= u"
+apply (simp add: plus_matrix_def)
+apply (rule combine_nrows)
+apply (simp_all)
+done
+
+lemma move_matrix_row_mult: "move_matrix ((A::('a::lordered_ring) matrix) * B) j 0 = (move_matrix A j 0) * B"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (auto simp add: Rep_matrix_mult foldseq_zero)
+apply (rule_tac foldseq_zerotail[symmetric])
+apply (auto simp add: nrows zero_imp_mult_zero max2)
+apply (rule order_trans)
+apply (rule ncols_move_matrix_le)
+apply (simp add: max1)
+done
+
+lemma move_matrix_col_mult: "move_matrix ((A::('a::lordered_ring) matrix) * B) 0 i = A * (move_matrix B 0 i)"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (auto simp add: Rep_matrix_mult foldseq_zero)
+apply (rule_tac foldseq_zerotail[symmetric])
+apply (auto simp add: ncols zero_imp_mult_zero max1)
+apply (rule order_trans)
+apply (rule nrows_move_matrix_le)
+apply (simp add: max2)
+done
+
+lemma move_matrix_add: "((move_matrix (A + B) j i)::(('a::lordered_ab_group) matrix)) = (move_matrix A j i) + (move_matrix B j i)"
+apply (subst Rep_matrix_inject[symmetric])
+apply (rule ext)+
+apply (simp)
+done
+
+lemma move_matrix_mult: "move_matrix ((A::('a::lordered_ring) matrix)*B) j i = (move_matrix A j 0) * (move_matrix B 0 i)"
+by (simp add: move_matrix_ortho[of "A*B"] move_matrix_col_mult move_matrix_row_mult)
+
+constdefs
+ scalar_mult :: "('a::lordered_ring) \<Rightarrow> 'a matrix \<Rightarrow> 'a matrix"
+ "scalar_mult a m == apply_matrix (op * a) m"
+
+lemma scalar_mult_zero[simp]: "scalar_mult y 0 = 0"
+ by (simp add: scalar_mult_def)
+
+lemma scalar_mult_add: "scalar_mult y (a+b) = (scalar_mult y a) + (scalar_mult y b)"
+ by (simp add: scalar_mult_def apply_matrix_add ring_eq_simps)
+
+lemma Rep_scalar_mult[simp]: "Rep_matrix (scalar_mult y a) j i = y * (Rep_matrix a j i)"
+ by (simp add: scalar_mult_def)
+
+lemma scalar_mult_singleton[simp]: "scalar_mult y (singleton_matrix j i x) = singleton_matrix j i (y * x)"
+ apply (subst Rep_matrix_inject[symmetric])
+ apply (rule ext)+
+ apply (auto)
+ done
+
+
+
+
end