author obua Fri, 16 Apr 2004 18:30:51 +0200 changeset 14593 90c88e7ef62d parent 14592 dd1a2905ea73 child 14594 3ff9cfc5c403
first version of matrices for HOL/Isabelle
 src/HOL/Matrix/LinProg.thy file | annotate | diff | comparison | revisions src/HOL/Matrix/Matrix.thy file | annotate | diff | comparison | revisions src/HOL/Matrix/MatrixGeneral.thy file | annotate | diff | comparison | revisions
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix/LinProg.thy	Fri Apr 16 18:30:51 2004 +0200
@@ -0,0 +1,84 @@
+(*  Title:      HOL/Matrix/LinProg.thy
+    ID:         $Id$
+    Author:     Steven Obua
+    License:    2004 Technische Universität München
+*)
+
+theory LinProg = Matrix:
+
+lemma linprog_by_duality_approx_general:
+  assumes
+  "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
+  "! c a b. 0 <= c & a <= b \<longrightarrow> fmul c a <= fmul c b"
+  "! a. fmul 0 a = 0"
+  "! a. fmul a 0 = 0"
+  "fadd 0 0 = 0"
+  "associative fmul"
+  and specificprops:
+  "mult_matrix fmul fadd (combine_matrix fadd A dA) x <= (b::('a::{order, zero}) matrix)"
+  "mult_matrix fmul fadd y A = c"
+  "0 <= y"
+  shows
+  <= mult_matrix fmul fadd y b"
+proof -
+  let ?mul = "mult_matrix fmul fadd"
+  let ?t1 = "?mul y (?mul (?add A dA) x)"
+  have a: "?t1 <= ?mul y b" by (rule le_left_mult, simp_all!)
+  have b: "?t1 = ?mul (?mul y (?add A dA)) x" by (simp! add: mult_matrix_assoc_simple[THEN sym])
+  have r_distr: "r_distributive ?mul ?add"
+    apply (rule r_distributive_matrix)
+  have l_distr: "l_distributive ?mul ?add"
+    apply (rule l_distributive_matrix)
+  have c:"?mul y (?add A dA) = ?add (?mul y A) (?mul y dA)"
+    by (insert r_distr, simp add: r_distributive_def)
+  have d:"?mul (?add (?mul y A) (?mul y dA)) x = ?add (?mul (?mul y A) x) (?mul (?mul y dA) x)"
+    by (insert l_distr, simp add: l_distributive_def)
+  have e:"\<dots> = ?add (?mul c x) (?mul (?mul y dA) x)" by (simp!)
+  from a b c d e show "?add (?mul c x) (?mul (?mul y dA) x) <= ?mul y b" by simp
+qed
+
+lemma linprog_by_duality_approx:
+  assumes
+  "(A + dA) * x <= (b::('a::pordered_matrix_element) matrix)"
+  "y * A = c"
+  "0 <= y"
+  shows
+  "c * x  + (y * dA) * x <= y * b"
+apply (rule linprog_by_duality_approx_general)
+apply (simp_all)
+apply (simp_all add: distributive_def l_distributive_def r_distributive_def matrix_left_distrib matrix_right_distrib)
+done
+
+lemma linprog_by_duality:
+  assumes
+  "A * x <= (b::('a::pordered_g_semiring) matrix)"
+  "y * A = c"
+  "0 <= y"
+  shows
+  "c * x  <= y * b"
+proof -
+  have a:"(A + 0) * x <= b" by (simp!)
+  have b:"c * x + (y*0)*x <= y * b" by (rule linprog_by_duality_approx, simp_all!)
+  show "c * x <= y*b" by (insert b, simp)
+qed
+
+end
+
+
+
+  
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix/Matrix.thy	Fri Apr 16 18:30:51 2004 +0200
@@ -0,0 +1,283 @@
+(*  Title:      HOL/Matrix/Matrix.thy
+    ID:         $Id$
+    Author:     Steven Obua
+    License:    2004 Technische Universität München
+*)
+
+theory Matrix=MatrixGeneral:
+
+axclass almost_matrix_element < zero, plus, times
+matrix_add_assoc: "(a+b)+c = a + (b+c)"
+
+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
+
+instance matrix :: (plus) plus
+by (intro_classes)
+
+instance matrix :: (times) times
+by (intro_classes)
+
+plus_matrix_def: "A + B == combine_matrix (op +) A B"
+times_matrix_def: "A * B == mult_matrix (op *) (op +) A B"
+
+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 (rule combine_matrix_assoc2)
+  }
+  note plus_assoc = this
+  {
+    fix A::"('a::matrix_element) matrix"
+    fix B
+    fix C
+    have "(A * B) * C = A * (B * C)"
+      apply (rule mult_matrix_assoc_simple)
+      apply (simp_all add: associative_def commutative_def distributive_def l_distributive_def r_distributive_def)
+      apply (auto)
+      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 (insert combine_matrix_commute2[of "op +"])
+      apply (rule combine_matrix_commute2)
+  }
+  note plus_commute = this
+  have plus_zero: "(0::('a::matrix_element) matrix) + 0 = 0"
+    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 (rule l_distributive_matrix2)
+      apply (simp_all add: associative_def commutative_def l_distributive_def)
+      apply (auto)
+  }
+  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 (rule r_distributive_matrix2)
+      apply (simp_all add: associative_def commutative_def r_distributive_def)
+      apply (auto)
+  }
+  note right_distrib = this
+  show "OFCLASS('a matrix, matrix_element_class)"
+    apply (intro_classes)
+    apply (simp_all add: mult_left_zero mult_right_zero)
+    by (simp_all add: left_distrib right_distrib)
+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"
+
+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 semiring < g_semiring
+apply (intro_classes)
+by (simp_all add: mult_assoc left_distrib right_distrib)
+
+instance matrix :: (g_almost_semiring) g_almost_semiring
+apply (intro_classes)
+by (simp add: plus_matrix_def combine_matrix_def combine_infmatrix_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"
+
+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"
+
+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)
+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)
+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])
+
+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
+
+instance matrix :: (pordered_matrix_element) pordered_matrix_element
+apply (intro_classes)
+apply (rule le_combine_matrix)
+apply (simp_all)
+apply (rule le_left_mult)
+apply (rule le_right_mult)
+
+axclass pordered_g_semiring < g_semiring, pordered_matrix_element
+
+instance almost_ordered_semiring < pordered_g_semiring
+apply (intro_classes)
+
+instance matrix :: (pordered_g_semiring) pordered_g_semiring
+by (intro_classes)
+
+lemma nrows_mult: "nrows ((A::('a::matrix_element) matrix) * B) <= nrows A"
+
+lemma ncols_mult: "ncols ((A::('a::matrix_element) matrix) * B) <= ncols B"
+
+constdefs
+  one_matrix :: "nat \<Rightarrow> ('a::semiring) 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)"
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ n], simp add: split_if)+
+
+lemma nrows_one_matrix[simp]: "nrows (one_matrix n) = n" (is "?r = _")
+proof -
+  have "?r <= n" by (simp add: nrows_le)
+  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 = _")
+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"
+apply (subst Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+apply (rule_tac j1="xa" in ssubst[OF foldseq_almostzero])
+apply (simp_all)
+
+lemma one_matrix_mult_left: "nrows A <= n \<Longrightarrow> (one_matrix n) * A = A"
+apply (subst Rep_matrix_inject[THEN sym])
+apply (rule ext)+
+apply (rule_tac j1="x" in ssubst[OF foldseq_almostzero])
+apply (simp_all)
+
+constdefs
+  right_inverse_matrix :: "('a::semiring) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
+  "right_inverse_matrix A X == (A * X = one_matrix (max (nrows A) (ncols X)))"
+  inverse_matrix :: "('a::semiring) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
+  "inverse_matrix A X == (right_inverse_matrix A X) \<and> (right_inverse_matrix X A)"
+
+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])
+
+(* to be continued \<dots> *)
+
+end
+
+
+
+
+
+
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+ 
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Matrix/MatrixGeneral.thy	Fri Apr 16 18:30:51 2004 +0200
@@ -0,0 +1,1292 @@
+(*  Title:      HOL/Matrix/MatrixGeneral.thy
+    ID:         $Id$
+    Author:     Steven Obua
+    License:    2004 Technische UniversitÃ¤t MÃ¼nchen
+*)
+
+theory MatrixGeneral = Main:
+
+types 'a infmatrix = "[nat, nat] \<Rightarrow> 'a"
+
+constdefs
+  nonzero_positions :: "('a::zero) infmatrix \<Rightarrow> (nat*nat) set"
+  "nonzero_positions A == {pos. A (fst pos) (snd pos) ~= 0}"
+
+typedef 'a matrix = "{(f::(('a::zero) infmatrix)). finite (nonzero_positions f)}"
+apply (rule_tac x="(% j i. 0)" in exI)
+
+declare Rep_matrix_inverse[simp]
+
+lemma finite_nonzero_positions : "finite (nonzero_positions (Rep_matrix A))"
+apply (rule Abs_matrix_induct)
+
+constdefs
+  nrows :: "('a::zero) matrix \<Rightarrow> nat"
+  "nrows A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image fst) (nonzero_positions (Rep_matrix A))))"
+  ncols :: "('a::zero) matrix \<Rightarrow> nat"
+  "ncols A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image snd) (nonzero_positions (Rep_matrix A))))"
+
+lemma nrows:
+  assumes hyp: "nrows A \<le> m"
+  shows "(Rep_matrix A m n) = 0" (is ?concl)
+proof cases
+  assume "nonzero_positions(Rep_matrix A) = {}"
+  then show "(Rep_matrix A m n) = 0" by (simp add: nonzero_positions_def)
+next
+  assume a: "nonzero_positions(Rep_matrix A) \<noteq> {}"
+  let ?S = "fst(nonzero_positions(Rep_matrix A))"
+  from a have b: "?S \<noteq> {}" by (simp)
+  have c: "finite (?S)" by (simp add: finite_nonzero_positions)
+  from hyp have d: "Max (?S) < m" by (simp add: a nrows_def)
+  have "m \<notin> ?S"
+    proof -
+      have "m \<in> ?S \<Longrightarrow> m <= Max(?S)" by (simp add: Max[OF c b])
+      moreover from d have "~(m <= Max ?S)" by (simp)
+      ultimately show "m \<notin> ?S" by (auto)
+    qed
+  thus "Rep_matrix A m n = 0" by (simp add: nonzero_positions_def image_Collect)
+qed
+text {* \matrix cool *}
+constdefs
+  transpose_infmatrix :: "'a infmatrix \<Rightarrow> 'a infmatrix"
+  "transpose_infmatrix A j i == A i j"
+  transpose_matrix :: "('a::zero) matrix \<Rightarrow> 'a matrix"
+  "transpose_matrix == Abs_matrix o transpose_infmatrix o Rep_matrix"
+
+declare transpose_infmatrix_def[simp]
+
+lemma transpose_infmatrix_twice[simp]: "transpose_infmatrix (transpose_infmatrix A) = A"
+by ((rule ext)+, simp)
+
+lemma transpose_infmatrix: "transpose_infmatrix (% j i. P j i) = (% j i. P i j)"
+  apply (rule ext)+
+
+lemma transpose_infmatrix_closed[simp]: "Rep_matrix (Abs_matrix (transpose_infmatrix (Rep_matrix x))) = transpose_infmatrix (Rep_matrix x)"
+apply (rule Abs_matrix_inverse)
+apply (simp add: matrix_def nonzero_positions_def image_def)
+proof -
+  let ?A = "{pos. Rep_matrix x (snd pos) (fst pos) \<noteq> 0}"
+  let ?swap = "% pos. (snd pos, fst pos)"
+  let ?B = "{pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0}"
+  have swap_image: "?swap?A = ?B"
+    apply (rule set_ext)
+    apply (simp)
+    proof
+      fix y
+      assume hyp: "\<exists>a b. Rep_matrix x b a \<noteq> 0 \<and> y = (b, a)"
+      thus "Rep_matrix x (fst y) (snd y) \<noteq> 0"
+	proof -
+	  from hyp obtain a b where "(Rep_matrix x b a \<noteq> 0 & y = (b,a))" by blast
+	  then show "Rep_matrix x (fst y) (snd y) \<noteq> 0" by (simp)
+	qed
+    next
+      fix y
+      assume hyp: "Rep_matrix x (fst y) (snd y) \<noteq> 0"
+      show "\<exists> a b. (Rep_matrix x b a \<noteq> 0 & y = (b,a))" by (rule exI[of _ "snd y"], rule exI[of _ "fst y"], simp)
+    qed
+  then have "finite (?swap?A)"
+    proof -
+      have "finite (nonzero_positions (Rep_matrix x))" by (simp add: finite_nonzero_positions)
+      then have "finite ?B" by (simp add: nonzero_positions_def)
+      with swap_image show "finite (?swap?A)" by (simp)
+    qed
+  moreover
+  have "inj_on ?swap ?A" by (simp add: inj_on_def)
+  ultimately show "finite ?A"by (rule finite_imageD[of ?swap ?A])
+qed
+
+lemma transpose_matrix[simp]: "Rep_matrix(transpose_matrix A) j i = Rep_matrix A i j"
+
+lemma transpose_transpose_id[simp]: "transpose_matrix (transpose_matrix A) = A"
+
+lemma nrows_transpose[simp]: "nrows (transpose_matrix A) = ncols A"
+by (simp add: nrows_def ncols_def nonzero_positions_def transpose_matrix_def image_def)
+
+lemma ncols_transpose[simp]: "ncols (transpose_matrix A) = nrows A"
+by (simp add: nrows_def ncols_def nonzero_positions_def transpose_matrix_def image_def)
+
+lemma ncols: "ncols A <= n \<Longrightarrow> Rep_matrix A m n = 0"
+proof -
+  assume "ncols A <= n"
+  then have "nrows (transpose_matrix A) <= n" by (simp)
+  then have "Rep_matrix (transpose_matrix A) n m = 0" by (rule nrows)
+  thus "Rep_matrix A m n = 0" by (simp add: transpose_matrix_def)
+qed
+
+lemma ncols_le: "(ncols A <= n) = (! j i. n <= i \<longrightarrow> (Rep_matrix A j i) = 0)" (is "_ = ?st")
+apply (auto)
+  let ?P = "nonzero_positions (Rep_matrix A)"
+  let ?p = "snd?P"
+  have a:"finite ?p" by (simp add: finite_nonzero_positions)
+  let ?m = "Max ?p"
+  assume "~(Suc (?m) <= n)"
+  then have b:"n <= ?m" by (simp)
+  fix a b
+  assume "(a,b) \<in> ?P"
+  then have "?p \<noteq> {}" by (auto)
+  with a have "?m \<in>  ?p" by (simp)
+  moreover have "!x. (x \<in> ?p \<longrightarrow> (? y. (Rep_matrix A y x) \<noteq> 0))" by (simp add: nonzero_positions_def image_def)
+  ultimately have "? y. (Rep_matrix A y ?m) \<noteq> 0" by (simp)
+  moreover assume ?st
+  ultimately show "False" using b by (simp)
+qed
+
+lemma less_ncols: "(n < ncols A) = (? j i. n <= i & (Rep_matrix A j i) \<noteq> 0)" (is ?concl)
+proof -
+  have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
+  show ?concl by (simp add: a ncols_le)
+qed
+
+lemma le_ncols: "(n <= ncols A) = (\<forall> m. (\<forall> j i. m <= i \<longrightarrow> (Rep_matrix A j i) = 0) \<longrightarrow> n <= m)" (is ?concl)
+apply (auto)
+apply (subgoal_tac "ncols A <= m")
+apply (simp)
+apply (drule_tac x="ncols A" in spec)
+
+lemma nrows_le: "(nrows A <= n) = (! j i. n <= j \<longrightarrow> (Rep_matrix A j i) = 0)" (is ?s)
+proof -
+  have "(nrows A <= n) = (ncols (transpose_matrix A) <= n)" by (simp)
+  also have "\<dots> = (! j i. n <= i \<longrightarrow> (Rep_matrix (transpose_matrix A) j i = 0))" by (rule ncols_le)
+  also have "\<dots> = (! j i. n <= i \<longrightarrow> (Rep_matrix A i j) = 0)" by (simp)
+  finally show "(nrows A <= n) = (! j i. n <= j \<longrightarrow> (Rep_matrix A j i) = 0)" by (auto)
+qed
+
+lemma less_nrows: "(m < nrows A) = (? j i. m <= j & (Rep_matrix A j i) \<noteq> 0)" (is ?concl)
+proof -
+  have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
+  show ?concl by (simp add: a nrows_le)
+qed
+
+lemma le_nrows: "(n <= nrows A) = (\<forall> m. (\<forall> j i. m <= j \<longrightarrow> (Rep_matrix A j i) = 0) \<longrightarrow> n <= m)" (is ?concl)
+apply (auto)
+apply (subgoal_tac "nrows A <= m")
+apply (simp)
+apply (drule_tac x="nrows A" in spec)
+
+lemma finite_natarray1: "finite {x. x < (n::nat)}"
+apply (induct n)
+apply (simp)
+proof -
+  fix n
+  have "{x. x < Suc n} = insert n {x. x < n}"  by (rule set_ext, simp, arith)
+  moreover assume "finite {x. x < n}"
+  ultimately show "finite {x. x < Suc n}" by (simp)
+qed
+
+lemma finite_natarray2: "finite {pos. (fst pos) < (m::nat) & (snd pos) < (n::nat)}"
+  apply (induct m)
+  apply (simp+)
+  proof -
+    fix m::nat
+    let ?s0 = "{pos. fst pos < m & snd pos < n}"
+    let ?s1 = "{pos. fst pos < (Suc m) & snd pos < n}"
+    let ?sd = "{pos. fst pos = m & snd pos < n}"
+    assume f0: "finite ?s0"
+    have f1: "finite ?sd"
+    proof -
+      let ?f = "% x. (m, x)"
+      have "{pos. fst pos = m & snd pos < n} = ?f  {x. x < n}" by (rule set_ext, simp add: image_def, auto)
+      moreover have "finite {x. x < n}" by (simp add: finite_natarray1)
+      ultimately show "finite {pos. fst pos = m & snd pos < n}" by (simp)
+    qed
+    have su: "?s0 \<union> ?sd = ?s1" by (rule set_ext, simp, arith)
+    from f0 f1 have "finite (?s0 \<union> ?sd)" by (rule finite_UnI)
+    with su show "finite ?s1" by (simp)
+qed
+
+lemma RepAbs_matrix:
+  assumes aem: "? m. ! j i. m <= j \<longrightarrow> x j i = 0" (is ?em) and aen:"? n. ! j i. (n <= i \<longrightarrow> x j i = 0)" (is ?en)
+  shows "(Rep_matrix (Abs_matrix x)) = x"
+apply (rule Abs_matrix_inverse)
+proof -
+  from aem obtain m where a: "! j i. m <= j \<longrightarrow> x j i = 0" by (blast)
+  from aen obtain n where b: "! j i. n <= i \<longrightarrow> x j i = 0" by (blast)
+  let ?u = "{pos. x (fst pos) (snd pos) \<noteq> 0}"
+  let ?v = "{pos. fst pos < m & snd pos < n}"
+  have c: "!! (m::nat) a. ~(m <= a) \<Longrightarrow> a < m" by (arith)
+  from a b have "(?u \<inter> (-?v)) = {}"
+    apply (simp)
+    apply (rule set_ext)
+    apply (simp)
+    apply auto
+    by (rule c, auto)+
+  then have d: "?u \<subseteq> ?v" by blast
+  moreover have "finite ?v" by (simp add: finite_natarray2)
+  ultimately show "finite ?u" by (rule finite_subset)
+qed
+
+constdefs
+  apply_infmatrix :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix"
+  "apply_infmatrix f == % A. (% j i. f (A j i))"
+  apply_matrix :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix"
+  "apply_matrix f == % A. Abs_matrix (apply_infmatrix f (Rep_matrix A))"
+  combine_infmatrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix \<Rightarrow> 'c infmatrix"
+  "combine_infmatrix f == % A B. (% j i. f (A j i) (B j i))"
+  combine_matrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix \<Rightarrow> ('c::zero) matrix"
+  "combine_matrix f == % A B. Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))"
+
+lemma expand_apply_infmatrix[simp]: "apply_infmatrix f A j i = f (A j i)"
+
+lemma expand_combine_infmatrix[simp]: "combine_infmatrix f A B j i = f (A j i) (B j i)"
+
+constdefs
+commutative :: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> bool"
+"commutative f == ! x y. f x y = f y x"
+associative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
+"associative f == ! x y z. f (f x y) z = f x (f y z)"
+
+text{*
+To reason about associativity and commutativity of operations on matrices,
+let's take a step back and look at the general situtation: Assume that we have
+sets $A$ and $B$ with $B \subset A$ and an abstraction $u: A \rightarrow B$. This abstraction has to fulfill $u(b) = b$ for all $b \in B$, but is arbitrary otherwise.
+Each function $f: A \times A \rightarrow A$ now induces a function $f': B \times B \rightarrow B$ by $f' = u \circ f$.
+It is obvious that commutativity of $f$ implies commutativity of $f'$: $f' x y = u (f x y) = u (f y x) = f' y x.$
+*}
+
+lemma combine_infmatrix_commute:
+  "commutative f \<Longrightarrow> commutative (combine_infmatrix f)"
+
+lemma combine_matrix_commute:
+"commutative f \<Longrightarrow> commutative (combine_matrix f)"
+by (simp add: combine_matrix_def commutative_def combine_infmatrix_def)
+
+text{*
+On the contrary, given an associative function $f$ we cannot expect $f'$ to be associative. A counterexample is given by $A=\ganz$, $B=\{-1, 0, 1\}$,
+as $f$ we take addition on $\ganz$, which is clearly associative. The abstraction is given by  $u(a) = 0$ for $a \notin B$. Then we have
+$f' (f' 1 1) -1 = u(f (u (f 1 1)) -1) = u(f (u 2) -1) = u (f 0 -1) = -1,$
+but on the other hand we have
+$f' 1 (f' 1 -1) = u (f 1 (u (f 1 -1))) = u (f 1 0) = 1.$
+A way out of this problem is to assume that $f(A\times A)\subset A$ holds, and this is what we are going to do:
+*}
+
+lemma nonzero_positions_combine_infmatrix[simp]: "f 0 0 = 0 \<Longrightarrow> nonzero_positions (combine_infmatrix f A B) \<subseteq> (nonzero_positions A) \<union> (nonzero_positions B)"
+by (rule subsetI, simp add: nonzero_positions_def combine_infmatrix_def, auto)
+
+lemma finite_nonzero_positions_Rep[simp]: "finite (nonzero_positions (Rep_matrix A))"
+by (insert Rep_matrix [of A], simp add: matrix_def)
+
+lemma combine_infmatrix_closed [simp]:
+  "f 0 0 = 0 \<Longrightarrow> Rep_matrix (Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) = combine_infmatrix f (Rep_matrix A) (Rep_matrix B)"
+apply (rule Abs_matrix_inverse)
+apply (rule finite_subset[of _ "(nonzero_positions (Rep_matrix A)) \<union> (nonzero_positions (Rep_matrix B))"])
+by (simp_all)
+
+text {* We need the next two lemmas only later, but it is analog to the above one, so we prove them now: *}
+lemma nonzero_positions_apply_infmatrix[simp]: "f 0 = 0 \<Longrightarrow> nonzero_positions (apply_infmatrix f A) \<subseteq> nonzero_positions A"
+by (rule subsetI, simp add: nonzero_positions_def apply_infmatrix_def, auto)
+
+lemma apply_infmatrix_closed [simp]:
+  "f 0 = 0 \<Longrightarrow> Rep_matrix (Abs_matrix (apply_infmatrix f (Rep_matrix A))) = apply_infmatrix f (Rep_matrix A)"
+apply (rule Abs_matrix_inverse)
+apply (rule finite_subset[of _ "nonzero_positions (Rep_matrix A)"])
+by (simp_all)
+
+lemma combine_infmatrix_assoc[simp]: "f 0 0 = 0 \<Longrightarrow> associative f \<Longrightarrow> associative (combine_infmatrix f)"
+
+lemma comb: "f = g \<Longrightarrow> x = y \<Longrightarrow> f x = g y"
+by (auto)
+
+lemma combine_matrix_assoc: "f 0 0 = 0 \<Longrightarrow> associative f \<Longrightarrow> associative (combine_matrix f)"
+apply (simp(no_asm) add: associative_def combine_matrix_def, auto)
+apply (rule comb [of Abs_matrix Abs_matrix])
+by (auto, insert combine_infmatrix_assoc[of f], simp add: associative_def)
+
+lemma Rep_apply_matrix[simp]: "f 0 = 0 \<Longrightarrow> Rep_matrix (apply_matrix f A) j i = f (Rep_matrix A j i)"
+
+lemma Rep_combine_matrix[simp]: "f 0 0 = 0 \<Longrightarrow> Rep_matrix (combine_matrix f A B) j i = f (Rep_matrix A j i) (Rep_matrix B j i)"
+
+lemma combine_nrows: "f 0 0 = 0  \<Longrightarrow> nrows (combine_matrix f A B) <= max (nrows A) (nrows B)"
+
+lemma combine_ncols: "f 0 0 = 0 \<Longrightarrow> ncols (combine_matrix f A B) <= max (ncols A) (ncols B)"
+
+lemma combine_nrows: "f 0 0 = 0 \<Longrightarrow> nrows A <= q \<Longrightarrow> nrows B <= q \<Longrightarrow> nrows(combine_matrix f A B) <= q"
+
+lemma combine_ncols: "f 0 0 = 0 \<Longrightarrow> ncols A <= q \<Longrightarrow> ncols B <= q \<Longrightarrow> ncols(combine_matrix f A B) <= q"
+
+constdefs
+  zero_r_neutral :: "('a \<Rightarrow> ('b::zero) \<Rightarrow> 'a) \<Rightarrow> bool"
+  "zero_r_neutral f == ! a. f a 0 = a"
+  zero_l_neutral :: "('a \<Rightarrow> ('b::zero) \<Rightarrow> 'a) \<Rightarrow> bool"
+  "zero_l_neutral f == ! a. f 0 a = a"
+  zero_closed :: "(('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> bool"
+  "zero_closed f == (!x. f x 0 = 0) & (!y. f 0 y = 0)"
+
+consts foldseq :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
+primrec
+  "foldseq f s 0 = s 0"
+  "foldseq f s (Suc n) = f (s 0) (foldseq f (% k. s(Suc k)) n)"
+
+consts foldseq_transposed ::  "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
+primrec
+  "foldseq_transposed f s 0 = s 0"
+  "foldseq_transposed f s (Suc n) = f (foldseq_transposed f s n) (s (Suc n))"
+
+lemma foldseq_assoc : "associative f \<Longrightarrow> foldseq f = foldseq_transposed f"
+proof -
+  assume a:"associative f"
+  then have sublemma: "!! n. ! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
+  proof -
+    fix n
+    show "!N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
+    proof (induct n)
+      show "!N s. N <= 0 \<longrightarrow> foldseq f s N = foldseq_transposed f s N" by simp
+    next
+      fix n
+      assume b:"! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
+      have c:"!!N s. N <= n \<Longrightarrow> foldseq f s N = foldseq_transposed f s N" by (simp add: b)
+      show "! N t. N <= Suc n \<longrightarrow> foldseq f t N = foldseq_transposed f t N"
+      proof (auto)
+	fix N t
+	assume Nsuc: "N <= Suc n"
+	show "foldseq f t N = foldseq_transposed f t N"
+	proof cases
+	  assume "N <= n"
+	  then show "foldseq f t N = foldseq_transposed f t N" by (simp add: b)
+	next
+	  assume "~(N <= n)"
+	  with Nsuc have Nsuceq: "N = Suc n" by simp
+	  have neqz: "n \<noteq> 0 \<Longrightarrow> ? m. n = Suc m & Suc m <= n" by arith
+	  have assocf: "!! x y z. f x (f y z) = f (f x y) z" by (insert a, simp add: associative_def)
+	  show "foldseq f t N = foldseq_transposed f t N"
+	    apply (subst c)
+	    apply (simp)
+	    apply (case_tac "n = 0")
+	    apply (simp)
+	    apply (drule neqz)
+	    apply (erule exE)
+	    apply (simp)
+	    apply (subst assocf)
+	    proof -
+	      fix m
+	      assume "n = Suc m & Suc m <= n"
+	      then have mless: "Suc m <= n" by arith
+	      then have step1: "foldseq_transposed f (% k. t (Suc k)) m = foldseq f (% k. t (Suc k)) m" (is "?T1 = ?T2")
+		apply (subst c)
+		by simp+
+	      have step2: "f (t 0) ?T2 = foldseq f t (Suc m)" (is "_ = ?T3") by simp
+	      have step3: "?T3 = foldseq_transposed f t (Suc m)" (is "_ = ?T4")
+		apply (subst c)
+	      have step4: "?T4 = f (foldseq_transposed f t m) (t (Suc m))" (is "_=?T5") by simp
+	      from step1 step2 step3 step4 show sowhat: "f (f (t 0) ?T1) (t (Suc (Suc m))) = f ?T5 (t (Suc (Suc m)))" by simp
+	    qed
+	  qed
+	qed
+      qed
+    qed
+    show "foldseq f = foldseq_transposed f" by ((rule ext)+, insert sublemma, auto)
+  qed
+
+lemma foldseq_distr: "\<lbrakk>associative f; commutative f\<rbrakk> \<Longrightarrow> foldseq f (% k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n)"
+proof -
+  assume assoc: "associative f"
+  assume comm: "commutative f"
+  from assoc have a:"!! x y z. f (f x y) z = f x (f y z)" by (simp add: associative_def)
+  from comm have b: "!! x y. f x y = f y x" by (simp add: commutative_def)
+  from assoc comm have c: "!! x y z. f x (f y z) = f y (f x z)" by (simp add: commutative_def associative_def)
+  have "!! n. (! u v. foldseq f (%k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n))"
+    apply (induct_tac n)
+    apply (simp+, auto)
+    by (simp add: a b c)
+  then show "foldseq f (% k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n)" by simp
+qed
+
+theorem "\<lbrakk>associative f; associative g; \<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d); ? x y. (f x) \<noteq> (f y); ? x y. (g x) \<noteq> (g y); f x x = x; g x x = x\<rbrakk> \<Longrightarrow> f=g | (! y. f y x = y) | (! y. g y x = y)"
+oops
+(* Model found
+
+Trying to find a model that refutes: \<lbrakk>associative f; associative g;
+ \<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d); \<exists>x y. f x \<noteq> f y;
+ \<exists>x y. g x \<noteq> g y; f x x = x; g x x = x\<rbrakk>
+\<Longrightarrow> f = g \<or> (\<forall>y. f y x = y) \<or> (\<forall>y. g y x = y)
+Searching for a model of size 1, translating term... invoking SAT solver... no model found.
+Searching for a model of size 2, translating term... invoking SAT solver... no model found.
+Searching for a model of size 3, translating term... invoking SAT solver...
+Model found:
+Size of types: 'a: 3
+x: a1
+g: (a0\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a0, a2\<mapsto>a1), a1\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a1, a2\<mapsto>a0), a2\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a0, a2\<mapsto>a1))
+f: (a0\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a0, a2\<mapsto>a0), a1\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a1, a2\<mapsto>a1), a2\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a0, a2\<mapsto>a0))
+*)
+
+lemma foldseq_zero:
+assumes fz: "f 0 0 = 0" and sz: "! i. i <= n \<longrightarrow> s i = 0"
+shows "foldseq f s n = 0"
+proof -
+  have "!! n. ! s. (! i. i <= n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s n = 0"
+    apply (induct_tac n)
+    apply (simp)
+  then show "foldseq f s n = 0" by (simp add: sz)
+qed
+
+lemma foldseq_significant_positions:
+  assumes p: "! i. i <= N \<longrightarrow> S i = T i"
+  shows "foldseq f S N = foldseq f T N" (is ?concl)
+proof -
+  have "!! m . ! s t. (! i. i<=m \<longrightarrow> s i = t i) \<longrightarrow> foldseq f s m = foldseq f t m"
+    apply (induct_tac m)
+    apply (simp)
+    apply (simp)
+    apply (auto)
+    proof -
+      fix n
+      fix s::"nat\<Rightarrow>'a"
+      fix t::"nat\<Rightarrow>'a"
+      assume a: "\<forall>s t. (\<forall>i\<le>n. s i = t i) \<longrightarrow> foldseq f s n = foldseq f t n"
+      assume b: "\<forall>i\<le>Suc n. s i = t i"
+      have c:"!! a b. a = b \<Longrightarrow> f (t 0) a = f (t 0) b" by blast
+      have d:"!! s t. (\<forall>i\<le>n. s i = t i) \<Longrightarrow> foldseq f s n = foldseq f t n" by (simp add: a)
+      show "f (t 0) (foldseq f (\<lambda>k. s (Suc k)) n) = f (t 0) (foldseq f (\<lambda>k. t (Suc k)) n)" by (rule c, simp add: d b)
+    qed
+  with p show ?concl by simp
+qed
+
+lemma foldseq_tail: "M <= N \<Longrightarrow> foldseq f S N = foldseq f (% k. (if k < M then (S k) else (foldseq f (% k. S(k+M)) (N-M)))) M" (is "?p \<Longrightarrow> ?concl")
+proof -
+  have suc: "!! a b. \<lbrakk>a <= Suc b; a \<noteq> Suc b\<rbrakk> \<Longrightarrow> a <= b" by arith
+  have suc2: "!! a b. \<lbrakk>a <= Suc b; ~ (a <= b)\<rbrakk> \<Longrightarrow> a = (Suc b)" by arith
+  have eq: "!! (a::nat) b. \<lbrakk>~(a < b); a <= b\<rbrakk> \<Longrightarrow> a = b" by arith
+  have a:"!! a b c . a = b \<Longrightarrow> f c a = f c b" by blast
+  have "!! n. ! m s. m <= n \<longrightarrow> foldseq f s n = foldseq f (% k. (if k < m then (s k) else (foldseq f (% k. s(k+m)) (n-m)))) m"
+    apply (induct_tac n)
+    apply (simp)
+    apply (simp)
+    apply (auto)
+    apply (case_tac "m = Suc na")
+    apply (simp)
+    apply (rule a)
+    apply (rule foldseq_significant_positions)
+    apply (simp, auto)
+    apply (drule eq, simp+)
+    apply (drule suc, simp+)
+    proof -
+      fix na m s
+      assume suba:"\<forall>m\<le>na. \<forall>s. foldseq f s na = foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (na - m))m"
+      assume subb:"m <= na"
+      from suba have subc:"!! m s. m <= na \<Longrightarrow>foldseq f s na = foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (na - m))m" by simp
+      have subd: "foldseq f (\<lambda>k. if k < m then s (Suc k) else foldseq f (\<lambda>k. s (Suc (k + m))) (na - m)) m =
+	foldseq f (% k. s(Suc k)) na"
+	by (rule subc[of m "% k. s(Suc k)", THEN sym], simp add: subb)
+      from subb have sube: "m \<noteq> 0 \<Longrightarrow> ? mm. m = Suc mm & mm <= na" by arith
+      show "f (s 0) (foldseq f (\<lambda>k. if k < m then s (Suc k) else foldseq f (\<lambda>k. s (Suc (k + m))) (na - m)) m) =
+        foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (Suc na - m)) m"
+	apply (case_tac "m=0")
+	apply (simp)
+	apply (drule sube)
+	apply (auto)
+	apply (rule a)
+	by (simp add: subc if_def)
+    qed
+  then show "?p \<Longrightarrow> ?concl" by simp
+qed
+
+lemma foldseq_zerotail:
+  assumes
+  fz: "f 0 0 = 0"
+  and sz: "! i.  n <= i \<longrightarrow> s i = 0"
+  and nm: "n <= m"
+  shows
+  "foldseq f s n = foldseq f s m"
+proof -
+  have a: "!! a b. ~(a < b) \<Longrightarrow> a <= b \<Longrightarrow> (a::nat) = b" by simp
+  show "foldseq f s n = foldseq f s m"
+    apply (simp add: foldseq_tail[OF nm, of f s])
+    apply (rule foldseq_significant_positions)
+    apply (auto)
+    apply (drule a)
+    apply (simp+)
+    apply (subst foldseq_zero)
+    by (simp add: fz sz)+
+qed
+
+lemma foldseq_zerotail2:
+  assumes "! x. f x 0 = x"
+  and "! i. n < i \<longrightarrow> s i = 0"
+  and nm: "n <= m"
+  shows
+  "foldseq f s n = foldseq f s m" (is ?concl)
+proof -
+  have "f 0 0 = 0" by (simp add: prems)
+  have a:"!! (i::nat) m. ~(i < m) \<Longrightarrow> i <= m \<Longrightarrow> i = m" by arith
+  have b:"!! m n. n <= m \<Longrightarrow> m \<noteq> n \<Longrightarrow> ? k. m-n = Suc k" by arith
+  have c: "0 <= m" by simp
+  have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
+  show ?concl
+    apply (subst foldseq_tail[OF nm])
+    apply (rule foldseq_significant_positions)
+    apply (auto)
+    apply (case_tac "m=n")
+    apply (drule a, simp+)
+    apply (drule b[OF nm])
+    apply (auto)
+    apply (drule a)
+    apply simp+
+    apply (case_tac "k=0")
+    apply (drule d)
+    apply (auto)
+    by (simp add: prems foldseq_zero)
+qed
+
+lemma foldseq_zerostart:
+  "! x. f 0 (f 0 x) = f 0 x \<Longrightarrow>  ! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
+proof -
+  assume f00x: "! x. f 0 (f 0 x) = f 0 x"
+  have "! s. (! i. i<=n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
+    apply (induct n)
+    apply (simp)
+    apply (rule allI, rule impI)
+    proof -
+      fix n
+      fix s
+      have a:"foldseq f s (Suc (Suc n)) = f (s 0) (foldseq f (% k. s(Suc k)) (Suc n))" by simp
+      assume b: "! s. ((\<forall>i\<le>n. s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n)))"
+      from b have c:"!! s. (\<forall>i\<le>n. s i = 0) \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
+      assume d: "! i. i <= Suc n \<longrightarrow> s i = 0"
+      show "foldseq f s (Suc (Suc n)) = f 0 (s (Suc (Suc n)))"
+	apply (subst a)
+	apply (subst c)
+	by (simp add: d f00x)+
+    qed
+  then show "! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
+qed
+
+lemma foldseq_zerostart2:
+  "! x. f 0 x = x \<Longrightarrow>  ! i. i < n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s n = s n"
+proof -
+  assume a:"! i. i<n \<longrightarrow> s i = 0"
+  assume x:"! x. f 0 x = x"
+  from x have f00x: "! x. f 0 (f 0 x) = f 0 x" by blast
+  have b: "!! i l. i < Suc l = (i <= l)" by arith
+  have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
+  show "foldseq f s n = s n"
+  apply (case_tac "n=0")
+  apply (simp)
+  apply (insert a)
+  apply (drule d)
+  apply (auto)
+  apply (insert f00x)
+  apply (drule foldseq_zerostart)
+qed
+
+lemma foldseq_almostzero:
+  assumes f0x:"! x. f 0 x = x" and fx0: "! x. f x 0 = x" and s0:"! i. i \<noteq> j \<longrightarrow> s i = 0"
+  shows "foldseq f s n = (if (j <= n) then (s j) else 0)" (is ?concl)
+proof -
+  from s0 have a: "! i. i < j \<longrightarrow> s i = 0" by simp
+  from s0 have b: "! i. j < i \<longrightarrow> s i = 0" by simp
+  show ?concl
+    apply auto
+    apply (subst foldseq_zerotail2[of f, OF fx0, of j, OF b, of n, THEN sym])
+    apply simp
+    apply (subst foldseq_zerostart2)
+    apply (simp add: f0x a)+
+    apply (subst foldseq_zero)
+    by (simp add: s0 f0x)+
+qed
+
+lemma foldseq_distr_unary:
+  assumes "!! a b. g (f a b) = f (g a) (g b)"
+  shows "g(foldseq f s n) = foldseq f (% x. g(s x)) n" (is ?concl)
+proof -
+  have "! s. g(foldseq f s n) = foldseq f (% x. g(s x)) n"
+    apply (induct_tac n)
+    apply (simp)
+    apply (simp)
+    apply (auto)
+    apply (drule_tac x="% k. s (Suc k)" in spec)
+  then show ?concl by simp
+qed
+
+constdefs
+  mult_matrix_n :: "nat \<Rightarrow> (('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'a matrix \<Rightarrow> 'b matrix \<Rightarrow> 'c matrix"
+  "mult_matrix_n n fmul fadd A B == Abs_matrix(% j i. foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) n)"
+  mult_matrix :: "(('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'a matrix \<Rightarrow> 'b matrix \<Rightarrow> 'c matrix"
+  "mult_matrix fmul fadd A B == mult_matrix_n (max (ncols A) (nrows B)) fmul fadd A B"
+
+lemma mult_matrix_n:
+  assumes prems: "ncols A \<le>  n" (is ?An) "nrows B \<le> n" (is ?Bn) "fadd 0 0 = 0" "fmul 0 0 = 0"
+  shows c:"mult_matrix fmul fadd A B = mult_matrix_n n fmul fadd A B" (is ?concl)
+proof -
+  show ?concl using prems
+    apply (simp add: mult_matrix_def mult_matrix_n_def)
+    apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
+    by (rule foldseq_zerotail, simp_all add: nrows_le ncols_le prems)
+qed
+
+lemma mult_matrix_nm:
+  assumes prems: "ncols A <= n" "nrows B <= n" "ncols A <= m" "nrows B <= m" "fadd 0 0 = 0" "fmul 0 0 = 0"
+  shows "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B"
+proof -
+  from prems have "mult_matrix_n n fmul fadd A B = mult_matrix fmul fadd A B" by (simp add: mult_matrix_n)
+  also from prems have "\<dots> = mult_matrix_n m fmul fadd A B" by (simp add: mult_matrix_n[THEN sym])
+  finally show "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B" by simp
+qed
+
+constdefs
+  r_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool"
+  "r_distributive fmul fadd == ! a u v. fmul a (fadd u v) = fadd (fmul a u) (fmul a v)"
+  l_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
+  "l_distributive fmul fadd == ! a u v. fmul (fadd u v) a = fadd (fmul u a) (fmul v a)"
+  distributive :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
+
+lemma max1: "!! a x y. (a::nat) <= x \<Longrightarrow> a <= max x y" by (arith)
+lemma max2: "!! b x y. (b::nat) <= y \<Longrightarrow> b <= max x y" by (arith)
+
+lemma r_distributive_matrix:
+ assumes prems:
+  "fadd 0 0 = 0"
+  "! a. fmul a 0 = 0"
+  "! a. fmul 0 a = 0"
+proof -
+  from prems show ?concl
+    apply (simp add: r_distributive_def mult_matrix_def, auto)
+    proof -
+      fix a::"'a matrix"
+      fix u::"'b matrix"
+      fix v::"'b matrix"
+      let ?mx = "max (ncols a) (max (nrows u) (nrows v))"
+      from prems show "mult_matrix_n (max (ncols a) (nrows (combine_matrix fadd u v))) fmul fadd a (combine_matrix fadd u v) =
+        combine_matrix fadd (mult_matrix_n (max (ncols a) (nrows u)) fmul fadd a u) (mult_matrix_n (max (ncols a) (nrows v)) fmul fadd a v)"
+	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+	apply (simp add: max1 max2 combine_nrows combine_ncols)+
+	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+	apply (simp add: max1 max2 combine_nrows combine_ncols)+
+	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+	apply (simp add: max1 max2 combine_nrows combine_ncols)+
+	apply (simp add: combine_matrix_def combine_infmatrix_def)
+	apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
+	apply (subst RepAbs_matrix)
+	apply (simp, auto)
+	apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
+	apply (rule exI[of _ "ncols v"], simp add: ncols_le foldseq_zero)
+	apply (subst RepAbs_matrix)
+	apply (simp, auto)
+	apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
+	apply (rule exI[of _ "ncols u"], simp add: ncols_le foldseq_zero)
+	done
+    qed
+qed
+
+lemma l_distributive_matrix:
+ assumes prems:
+  "fadd 0 0 = 0"
+  "! a. fmul a 0 = 0"
+  "! a. fmul 0 a = 0"
+proof -
+  from prems show ?concl
+    apply (simp add: l_distributive_def mult_matrix_def, auto)
+    proof -
+      fix a::"'b matrix"
+      fix u::"'a matrix"
+      fix v::"'a matrix"
+      let ?mx = "max (nrows a) (max (ncols u) (ncols v))"
+      from prems show "mult_matrix_n (max (ncols (combine_matrix fadd u v)) (nrows a)) fmul fadd (combine_matrix fadd u v) a =
+               combine_matrix fadd (mult_matrix_n (max (ncols u) (nrows a)) fmul fadd u a) (mult_matrix_n (max (ncols v) (nrows a)) fmul fadd v a)"
+	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+	apply (simp add: max1 max2 combine_nrows combine_ncols)+
+	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+	apply (simp add: max1 max2 combine_nrows combine_ncols)+
+	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+	apply (simp add: max1 max2 combine_nrows combine_ncols)+
+	apply (simp add: combine_matrix_def combine_infmatrix_def)
+	apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
+	apply (subst RepAbs_matrix)
+	apply (simp, auto)
+	apply (rule exI[of _ "nrows v"], simp add: nrows_le foldseq_zero)
+	apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
+	apply (subst RepAbs_matrix)
+	apply (simp, auto)
+	apply (rule exI[of _ "nrows u"], simp add: nrows_le foldseq_zero)
+	apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
+	done
+    qed
+qed
+
+
+instance matrix :: (zero)zero by intro_classes
+
+  zero_matrix_def: "(0::('a::zero) matrix) == Abs_matrix(% j i. 0)"
+
+lemma Rep_zero_matrix_def[simp]: "Rep_matrix 0 j i = 0"
+  apply (subst RepAbs_matrix)
+  by (auto)
+
+lemma zero_matrix_def_nrows[simp]: "nrows 0 = 0"
+proof -
+  have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
+  show "nrows 0 = 0" by (rule a, subst nrows_le, simp)
+qed
+
+lemma zero_matrix_def_ncols[simp]: "ncols 0 = 0"
+proof -
+  have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
+  show "ncols 0 = 0" by (rule a, subst ncols_le, simp)
+qed
+
+lemma combine_matrix_zero_r_neutral: "zero_r_neutral f \<Longrightarrow> zero_r_neutral (combine_matrix f)"
+  by (simp add: zero_r_neutral_def combine_matrix_def combine_infmatrix_def)
+
+lemma mult_matrix_zero_closed: "\<lbrakk>fadd 0 0 = 0; zero_closed fmul\<rbrakk> \<Longrightarrow> zero_closed (mult_matrix fmul fadd)"
+  apply (simp add: zero_closed_def mult_matrix_def mult_matrix_n_def)
+  apply (auto)
+  by (subst foldseq_zero, (simp add: zero_matrix_def)+)+
+
+lemma mult_matrix_n_zero_right[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul a 0 = 0\<rbrakk> \<Longrightarrow> mult_matrix_n n fmul fadd A 0 = 0"
+  apply (subst foldseq_zero)
+
+lemma mult_matrix_n_zero_left[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul 0 a = 0\<rbrakk> \<Longrightarrow> mult_matrix_n n fmul fadd 0 A = 0"
+  apply (subst foldseq_zero)
+
+lemma mult_matrix_zero_left[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul 0 a = 0\<rbrakk> \<Longrightarrow> mult_matrix fmul fadd 0 A = 0"
+
+lemma mult_matrix_zero_right[simp]: "\<lbrakk>fadd 0 0 = 0; !a. fmul a 0 = 0\<rbrakk> \<Longrightarrow> mult_matrix fmul fadd A 0 = 0"
+
+lemma apply_matrix_zero[simp]: "f 0 = 0 \<Longrightarrow> apply_matrix f 0 = 0"
+  apply (simp add: apply_matrix_def apply_infmatrix_def)
+
+lemma combine_matrix_zero: "f 0 0 = 0 \<Longrightarrow> combine_matrix f 0 0 = 0"
+  apply (simp add: combine_matrix_def combine_infmatrix_def)
+
+lemma apply_zero_matrix_def[simp]: "apply_matrix (% x. 0) A = 0"
+  apply (simp add: apply_matrix_def apply_infmatrix_def)
+
+constdefs
+  singleton_matrix :: "nat \<Rightarrow> nat \<Rightarrow> ('a::zero) \<Rightarrow> 'a matrix"
+  "singleton_matrix j i a == Abs_matrix(% m n. if j = m & i = n then a else 0)"
+  move_matrix :: "('a::zero) matrix \<Rightarrow> int \<Rightarrow> int \<Rightarrow> 'a matrix"
+  "move_matrix A y x == Abs_matrix(% j i. if (neg ((int j)-y)) | (neg ((int i)-x)) then 0 else Rep_matrix A (nat ((int j)-y)) (nat ((int i)-x)))"
+  take_rows :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
+  "take_rows A r == Abs_matrix(% j i. if (j < r) then (Rep_matrix A j i) else 0)"
+  take_columns :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
+  "take_columns A c == Abs_matrix(% j i. if (i < c) then (Rep_matrix A j i) else 0)"
+
+constdefs
+  column_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
+  "column_of_matrix A n == take_columns (move_matrix A 0 (- int n)) 1"
+  row_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
+  "row_of_matrix A m == take_rows (move_matrix A (- int m) 0) 1"
+
+lemma Rep_singleton_matrix[simp]: "Rep_matrix (singleton_matrix j i e) m n = (if j = m & i = n then e else 0)"
+apply (auto)
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ "Suc m"], simp)
+apply (rule exI[of _ "Suc n"], simp+)
+by (subst RepAbs_matrix, rule exI[of _ "Suc j"], simp, rule exI[of _ "Suc i"], simp+)+
+
+lemma singleton_matrix_zero[simp]: "singleton_matrix j i 0 = 0"
+  by (simp add: singleton_matrix_def zero_matrix_def)
+
+lemma nrows_singleton[simp]: "nrows(singleton_matrix j i e) = (if e = 0 then 0 else Suc j)"
+apply (auto)
+apply (rule le_anti_sym)
+apply (subst nrows_le)
+apply (subst RepAbs_matrix)
+apply (simp, arith)
+apply (simp, arith)
+apply (simp)
+apply (rule not_leE)
+apply (subst nrows_le)
+by simp
+
+lemma ncols_singleton[simp]: "ncols(singleton_matrix j i e) = (if e = 0 then 0 else Suc i)"
+apply (auto)
+apply (rule le_anti_sym)
+apply (subst ncols_le)
+apply (subst RepAbs_matrix)
+apply (simp, arith)
+apply (simp, arith)
+apply (simp)
+apply (rule not_leE)
+apply (subst ncols_le)
+by simp
+
+lemma combine_singleton: "f 0 0 = 0 \<Longrightarrow> combine_matrix f (singleton_matrix j i a) (singleton_matrix j i b) = singleton_matrix j i (f a b)"
+apply (simp add: singleton_matrix_def combine_matrix_def combine_infmatrix_def)
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ "Suc j"], simp)
+apply (rule exI[of _ "Suc i"], simp)
+apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ "Suc j"], simp)
+apply (rule exI[of _ "Suc i"], simp)
+by simp
+
+lemma Rep_move_matrix[simp]:
+  "Rep_matrix (move_matrix A y x) j i =
+  (if (neg ((int j)-y)) | (neg ((int i)-x)) then 0 else Rep_matrix A (nat((int j)-y)) (nat((int i)-x)))"
+apply (auto)
+by (subst RepAbs_matrix,
+  rule exI[of _ "(nrows A)+(nat (abs y))"], auto, rule nrows, arith,
+  rule exI[of _ "(ncols A)+(nat (abs x))"], auto, rule ncols, arith)+
+
+lemma Rep_take_columns[simp]:
+  "Rep_matrix (take_columns A c) j i =
+  (if i < c then (Rep_matrix A j i) else 0)"
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
+apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
+done
+
+lemma Rep_take_rows[simp]:
+  "Rep_matrix (take_rows A r) j i =
+  (if j < r then (Rep_matrix A j i) else 0)"
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
+apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
+done
+
+lemma Rep_column_of_matrix[simp]:
+  "Rep_matrix (column_of_matrix A c) j i = (if i = 0 then (Rep_matrix A j c) else 0)"
+
+lemma Rep_row_of_matrix[simp]:
+  "Rep_matrix (row_of_matrix A r) j i = (if j = 0 then (Rep_matrix A r i) else 0)"
+
+lemma mult_matrix_singleton_right[simp]:
+  assumes prems:
+  "! x. fmul x 0 = 0"
+  "! x. fmul 0 x = 0"
+  "! x. fadd 0 x = x"
+  "! x. fadd x 0 = x"
+  shows "(mult_matrix fmul fadd A (singleton_matrix j i e)) = apply_matrix (% x. fmul x e) (move_matrix (column_of_matrix A j) 0 (int i))"
+  apply (subst mult_matrix_nm[of _ _ _ "max (ncols A) (Suc j)"])
+  apply (auto)
+  apply (simp add: mult_matrix_n_def apply_matrix_def apply_infmatrix_def)
+  apply (rule comb[of "Abs_matrix" "Abs_matrix"], auto, (rule ext)+)
+  apply (subst foldseq_almostzero[of _ j])
+  apply (auto)
+  apply (insert ncols_le[of A j])
+  apply (arith)
+  proof -
+    fix k
+    fix l
+    assume a:"~neg(int l - int i)"
+    assume b:"nat (int l - int i) = 0"
+    from a b have a: "l = i" by(insert not_neg_nat[of "int l - int i"], simp add: a b)
+    assume c: "i \<noteq> l"
+    from c a show "fmul (Rep_matrix A k j) e = 0" by blast
+  qed
+
+lemma mult_matrix_ext:
+  assumes
+  eprem:
+  "? e. (! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)"
+  and fprems:
+  "! a. fmul 0 a = 0"
+  "! a. fmul a 0 = 0"
+  "! a. fadd a 0 = a"
+  "! a. fadd 0 a = a"
+  and contraprems:
+  shows
+  "A = B"
+proof(rule contrapos_np[of "False"], simp)
+  assume a: "A \<noteq> B"
+  have b: "!! f g. (! x y. f x y = g x y) \<Longrightarrow> f = g" by ((rule ext)+, auto)
+  have "? j i. (Rep_matrix A j i) \<noteq> (Rep_matrix B j i)"
+    apply (rule contrapos_np[of "False"], simp+)
+    apply (insert b[of "Rep_matrix A" "Rep_matrix B"], simp)
+    by (simp add: Rep_matrix_inject a)
+  then obtain J I where c:"(Rep_matrix A J I) \<noteq> (Rep_matrix B J I)" by blast
+  from eprem obtain e where eprops:"(! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)" by blast
+  let ?S = "singleton_matrix I 0 e"
+  let ?comp = "mult_matrix fmul fadd"
+  have d: "!!x f g. f = g \<Longrightarrow> f x = g x" by blast
+  have e: "(% x. fmul x e) 0 = 0" by (simp add: prems)
+  have "~(?comp A ?S = ?comp B ?S)"
+    apply (rule notI)
+    apply (simp add: fprems eprops)
+    apply (simp add: Rep_matrix_inject[THEN sym])
+    apply (drule d[of _ _ "J"], drule d[of _ _ "0"])
+    by (simp add: e c eprops)
+  with contraprems show "False" by simp
+qed
+
+constdefs
+  foldmatrix :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"
+  "foldmatrix f g A m n == foldseq_transposed g (% j. foldseq f (A j) n) m"
+  foldmatrix_transposed :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"
+  "foldmatrix_transposed f g A m n == foldseq g (% j. foldseq_transposed f (A j) n) m"
+
+lemma foldmatrix_transpose:
+  assumes
+  "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
+  shows
+  "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m" (is ?concl)
+proof -
+  have forall:"!! P x. (! x. P x) \<Longrightarrow> P x" by auto
+  have tworows:"! A. foldmatrix f g A 1 n = foldmatrix_transposed g f (transpose_infmatrix A) n 1"
+    apply (induct n)
+    apply (simp add: foldmatrix_def foldmatrix_transposed_def prems)+
+    apply (auto)
+    by (drule_tac x1="(% j i. A j (Suc i))" in forall, simp)
+  show "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m"
+    apply (simp add: foldmatrix_def foldmatrix_transposed_def)
+    apply (induct m, simp)
+    apply (simp)
+    apply (insert tworows)
+    apply (drule_tac x="% j i. (if j = 0 then (foldseq_transposed g (\<lambda>u. A u i) na) else (A (Suc na) i))" in spec)
+    by (simp add: foldmatrix_def foldmatrix_transposed_def)
+qed
+
+lemma foldseq_foldseq:
+assumes
+  "associative f"
+  "associative g"
+  "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
+shows
+  "foldseq g (% j. foldseq f (A j) n) m = foldseq f (% j. foldseq g ((transpose_infmatrix A) j) m) n"
+  apply (insert foldmatrix_transpose[of g f A m n])
+  by (simp add: foldmatrix_def foldmatrix_transposed_def foldseq_assoc[THEN sym] prems)
+
+lemma mult_n_nrows:
+assumes
+"! a. fmul 0 a = 0"
+"! a. fmul a 0 = 0"
+shows "nrows (mult_matrix_n n fmul fadd A B) \<le> nrows A"
+apply (subst nrows_le)
+apply (subst RepAbs_matrix)
+apply (rule_tac x="nrows A" in exI)
+apply (simp add: nrows prems foldseq_zero)
+apply (rule_tac x="ncols B" in exI)
+apply (simp add: ncols prems foldseq_zero)
+by (simp add: nrows prems foldseq_zero)
+
+lemma mult_n_ncols:
+assumes
+"! a. fmul 0 a = 0"
+"! a. fmul a 0 = 0"
+shows "ncols (mult_matrix_n n fmul fadd A B) \<le> ncols B"
+apply (subst ncols_le)
+apply (subst RepAbs_matrix)
+apply (rule_tac x="nrows A" in exI)
+apply (simp add: nrows prems foldseq_zero)
+apply (rule_tac x="ncols B" in exI)
+apply (simp add: ncols prems foldseq_zero)
+by (simp add: ncols prems foldseq_zero)
+
+lemma mult_nrows:
+assumes
+"! a. fmul 0 a = 0"
+"! a. fmul a 0 = 0"
+shows "nrows (mult_matrix fmul fadd A B) \<le> nrows A"
+by (simp add: mult_matrix_def mult_n_nrows prems)
+
+lemma mult_ncols:
+assumes
+"! a. fmul 0 a = 0"
+"! a. fmul a 0 = 0"
+shows "ncols (mult_matrix fmul fadd A B) \<le> ncols B"
+by (simp add: mult_matrix_def mult_n_ncols prems)
+
+lemma mult_matrix_assoc:
+  assumes prems:
+  "! a. fmul1 0 a = 0"
+  "! a. fmul1 a 0 = 0"
+  "! a. fmul2 0 a = 0"
+  "! a. fmul2 a 0 = 0"
+  "fadd1 0 0 = 0"
+  "fadd2 0 0 = 0"
+  "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
+  "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
+  "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
+  shows "mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B) C = mult_matrix fmul1 fadd1 A (mult_matrix fmul2 fadd2 B C)" (is ?concl)
+proof -
+  have comb_left:  "!! A B x y. A = B \<Longrightarrow> (Rep_matrix (Abs_matrix A)) x y = (Rep_matrix(Abs_matrix B)) x y" by blast
+  have fmul2fadd1fold: "!! x s n. fmul2 (foldseq fadd1 s n)  x = foldseq fadd1 (% k. fmul2 (s k) x) n"
+    by (rule_tac g1 = "% y. fmul2 y x" in ssubst [OF foldseq_distr_unary], simp_all!)
+  have fmul1fadd2fold: "!! x s n. fmul1 x (foldseq fadd2 s n) = foldseq fadd2 (% k. fmul1 x (s k)) n"
+      by (rule_tac g1 = "% y. fmul1 x y" in ssubst [OF foldseq_distr_unary], simp_all!)
+  let ?N = "max (ncols A) (max (ncols B) (max (nrows B) (nrows C)))"
+  show ?concl
+    apply (simp add: Rep_matrix_inject[THEN sym])
+    apply (rule ext)+
+    apply (subst mult_matrix_nm[of _ "max (ncols (mult_matrix_n (max (ncols A) (nrows B)) fmul1 fadd1 A B)) (nrows C)" _ "max (ncols B) (nrows C)"])
+    apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
+    apply (subst mult_matrix_nm[of _ "max (ncols A) (nrows (mult_matrix_n (max (ncols B) (nrows C)) fmul2 fadd2 B C))" _ "max (ncols A) (nrows B)"])     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
+    apply (subst mult_matrix_nm[of _ _ _ "?N"])
+    apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
+    apply (subst mult_matrix_nm[of _ _ _ "?N"])
+    apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
+    apply (subst mult_matrix_nm[of _ _ _ "?N"])
+    apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
+    apply (subst mult_matrix_nm[of _ _ _ "?N"])
+    apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
+    apply (rule comb_left)
+    apply ((rule ext)+, simp)
+    apply (subst RepAbs_matrix)
+    apply (rule exI[of _ "nrows B"])
+    apply (simp add: nrows prems foldseq_zero)
+    apply (rule exI[of _ "ncols C"])
+    apply (simp add: prems ncols foldseq_zero)
+    apply (subst RepAbs_matrix)
+    apply (rule exI[of _ "nrows A"])
+    apply (simp add: nrows prems foldseq_zero)
+    apply (rule exI[of _ "ncols B"])
+    apply (simp add: prems ncols foldseq_zero)
+    apply (subst foldseq_foldseq)
+qed
+
+lemma
+  assumes prems:
+  "! a. fmul1 0 a = 0"
+  "! a. fmul1 a 0 = 0"
+  "! a. fmul2 0 a = 0"
+  "! a. fmul2 a 0 = 0"
+  "fadd1 0 0 = 0"
+  "fadd2 0 0 = 0"
+  "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
+  "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
+  "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
+  shows
+apply (rule ext)+
+
+lemma mult_matrix_assoc_simple:
+  assumes prems:
+  "! a. fmul 0 a = 0"
+  "! a. fmul a 0 = 0"
+  "fadd 0 0 = 0"
+  "associative fmul"
+  shows "mult_matrix fmul fadd (mult_matrix fmul fadd A B) C = mult_matrix fmul fadd A (mult_matrix fmul fadd B C)" (is ?concl)
+proof -
+    by (simp! add: associative_def commutative_def)
+  then show ?concl
+    apply (subst mult_matrix_assoc)
+    apply (simp_all!)
+    by (simp add: associative_def distributive_def l_distributive_def r_distributive_def)+
+qed
+
+lemma transpose_apply_matrix: "f 0 = 0 \<Longrightarrow> transpose_matrix (apply_matrix f A) = apply_matrix f (transpose_matrix A)"
+apply (rule ext)+
+by simp
+
+lemma transpose_combine_matrix: "f 0 0 = 0 \<Longrightarrow> transpose_matrix (combine_matrix f A B) = combine_matrix f (transpose_matrix A) (transpose_matrix B)"
+apply (rule ext)+
+by simp
+
+lemma Rep_mult_matrix:
+  assumes
+  "! a. fmul 0 a = 0"
+  "! a. fmul a 0 = 0"
+  "fadd 0 0 = 0"
+  shows
+  "Rep_matrix(mult_matrix fmul fadd A B) j i =
+  foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) (max (ncols A) (nrows B))"
+apply (subst RepAbs_matrix)
+apply (rule exI[of _ "nrows A"], simp! add: nrows foldseq_zero)
+apply (rule exI[of _ "ncols B"], simp! add: ncols foldseq_zero)
+by simp
+
+lemma transpose_mult_matrix:
+  assumes
+  "! a. fmul 0 a = 0"
+  "! a. fmul a 0 = 0"
+  "fadd 0 0 = 0"
+  "! x y. fmul y x = fmul x y"
+  shows
+  "transpose_matrix (mult_matrix fmul fadd A B) = mult_matrix fmul fadd (transpose_matrix B) (transpose_matrix A)"
+  apply (simp add: Rep_matrix_inject[THEN sym])
+  apply (rule ext)+
+  by (simp! add: Rep_mult_matrix max_ac)
+
+instance matrix :: ("{ord,zero}") ord by intro_classes
+
+  le_matrix_def: "(A::('a::{ord,zero}) matrix) <= B == ! j i. (Rep_matrix A j i) <= (Rep_matrix B j i)"
+  less_def: "(A::('a::{ord,zero}) matrix) < B == (A <= B) & (A \<noteq> B)"
+
+instance matrix :: ("{order, zero}") order
+apply intro_classes
+apply (auto)
+apply (drule_tac x=j in spec, drule_tac x=j in spec)
+apply (drule_tac x=i in spec, drule_tac x=i in spec)
+apply (simp)
+apply (rule ext)+
+apply (drule_tac x=xa in spec, drule_tac x=xa in spec)
+apply (drule_tac x=xb in spec, drule_tac x=xb in spec)
+by simp
+
+lemma le_apply_matrix:
+  assumes
+  "f 0 = 0"
+  "! x y. x <= y \<longrightarrow> f x <= f y"
+  "(a::('a::{ord, zero}) matrix) <= b"
+  shows
+  "apply_matrix f a <= apply_matrix f b"
+
+lemma le_combine_matrix:
+  assumes
+  "f 0 0 = 0"
+  "! a b c d. a <= b & c <= d \<longrightarrow> f a c <= f b d"
+  "A <= B"
+  "C <= D"
+  shows
+  "combine_matrix f A C <= combine_matrix f B D"
+
+lemma le_left_combine_matrix:
+  assumes
+  "f 0 0 = 0"
+  "! a b c. 0 <= c & a <= b \<longrightarrow> f c a <= f c b"
+  "0 <= C"
+  "A <= B"
+  shows
+  "combine_matrix f C A <= combine_matrix f C B"
+
+lemma le_right_combine_matrix:
+  assumes
+  "f 0 0 = 0"
+  "! a b c. 0 <= c & a <= b \<longrightarrow> f a c <= f b c"
+  "0 <= C"
+  "A <= B"
+  shows
+  "combine_matrix f A C <= combine_matrix f B C"
+
+lemma le_transpose_matrix: "(A <= B) = (transpose_matrix A <= transpose_matrix B)"
+  by (simp add: le_matrix_def, auto)
+
+lemma le_foldseq:
+  assumes
+  "! a b c d . a <= b & c <= d \<longrightarrow> f a c <= f b d"
+  "! i. i <= n \<longrightarrow> s i <= t i"
+  shows
+  "foldseq f s n <= foldseq f t n"
+proof -
+  have "! s t. (! i. i<=n \<longrightarrow> s i <= t i) \<longrightarrow> foldseq f s n <= foldseq f t n" by (induct_tac n, simp_all!)
+  then show "foldseq f s n <= foldseq f t n" by (simp!)
+qed
+
+lemma le_left_mult:
+  assumes
+  "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
+  "! c a b. 0 <= c & a <= b \<longrightarrow> fmul c a <= fmul c b"
+  "! a. fmul 0 a = 0"
+  "! a. fmul a 0 = 0"
+  "fadd 0 0 = 0"
+  "0 <= C"
+  "A <= B"
+  shows
+  "mult_matrix fmul fadd C A <= mult_matrix fmul fadd C B"
+  apply (simp! add: le_matrix_def Rep_mult_matrix)
+  apply (auto)
+  apply (subst foldseq_zerotail[of _ _ _ "max (ncols C) (max (nrows A) (nrows B))"], simp_all add: nrows ncols max1 max2)+
+  apply (rule le_foldseq)
+  by (auto)
+
+lemma le_right_mult:
+  assumes
+  "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
+  "! c a b. 0 <= c & a <= b \<longrightarrow> fmul a c <= fmul b c"
+  "! a. fmul 0 a = 0"
+  "! a. fmul a 0 = 0"
+  "fadd 0 0 = 0"
+  "0 <= C"
+  "A <= B"
+  shows
+  "mult_matrix fmul fadd A C <= mult_matrix fmul fadd B C"
+  apply (simp! add: le_matrix_def Rep_mult_matrix)
+  apply (auto)
+  apply (subst foldseq_zerotail[of _ _ _ "max (nrows C) (max (ncols A) (ncols B))"], simp_all add: nrows ncols max1 max2)+
+  apply (rule le_foldseq)
+  by (auto)
+
+end