src/HOL/Matrix/Matrix.thy
author wenzelm
Wed Dec 29 17:34:41 2010 +0100 (2010-12-29)
changeset 41413 64cd30d6b0b8
parent 39302 d7728f65b353
child 42463 f270e3e18be5
permissions -rw-r--r--
explicit file specifications -- avoid secondary load path;
     1 (*  Title:      HOL/Matrix/Matrix.thy
     2     Author:     Steven Obua
     3 *)
     4 
     5 theory Matrix
     6 imports Main "~~/src/HOL/Library/Lattice_Algebras"
     7 begin
     8 
     9 types 'a infmatrix = "nat \<Rightarrow> nat \<Rightarrow> 'a"
    10 
    11 definition nonzero_positions :: "(nat \<Rightarrow> nat \<Rightarrow> 'a::zero) \<Rightarrow> nat \<times> nat \<Rightarrow> bool" where
    12   "nonzero_positions A = {pos. A (fst pos) (snd pos) ~= 0}"
    13 
    14 typedef 'a matrix = "{(f::(nat \<Rightarrow> nat \<Rightarrow> 'a::zero)). finite (nonzero_positions f)}"
    15 proof -
    16   have "(\<lambda>j i. 0) \<in> {(f::(nat \<Rightarrow> nat \<Rightarrow> 'a::zero)). finite (nonzero_positions f)}"
    17     by (simp add: nonzero_positions_def)
    18   then show ?thesis by auto
    19 qed
    20 
    21 declare Rep_matrix_inverse[simp]
    22 
    23 lemma finite_nonzero_positions : "finite (nonzero_positions (Rep_matrix A))"
    24 apply (rule Abs_matrix_induct)
    25 by (simp add: Abs_matrix_inverse matrix_def)
    26 
    27 definition nrows :: "('a::zero) matrix \<Rightarrow> nat" where
    28   "nrows A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image fst) (nonzero_positions (Rep_matrix A))))"
    29 
    30 definition ncols :: "('a::zero) matrix \<Rightarrow> nat" where
    31   "ncols A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image snd) (nonzero_positions (Rep_matrix A))))"
    32 
    33 lemma nrows:
    34   assumes hyp: "nrows A \<le> m"
    35   shows "(Rep_matrix A m n) = 0"
    36 proof cases
    37   assume "nonzero_positions(Rep_matrix A) = {}"
    38   then show "(Rep_matrix A m n) = 0" by (simp add: nonzero_positions_def)
    39 next
    40   assume a: "nonzero_positions(Rep_matrix A) \<noteq> {}"
    41   let ?S = "fst`(nonzero_positions(Rep_matrix A))"
    42   have c: "finite (?S)" by (simp add: finite_nonzero_positions)
    43   from hyp have d: "Max (?S) < m" by (simp add: a nrows_def)
    44   have "m \<notin> ?S"
    45     proof -
    46       have "m \<in> ?S \<Longrightarrow> m <= Max(?S)" by (simp add: Max_ge [OF c])
    47       moreover from d have "~(m <= Max ?S)" by (simp)
    48       ultimately show "m \<notin> ?S" by (auto)
    49     qed
    50   thus "Rep_matrix A m n = 0" by (simp add: nonzero_positions_def image_Collect)
    51 qed
    52 
    53 definition transpose_infmatrix :: "'a infmatrix \<Rightarrow> 'a infmatrix" where
    54   "transpose_infmatrix A j i == A i j"
    55 
    56 definition transpose_matrix :: "('a::zero) matrix \<Rightarrow> 'a matrix" where
    57   "transpose_matrix == Abs_matrix o transpose_infmatrix o Rep_matrix"
    58 
    59 declare transpose_infmatrix_def[simp]
    60 
    61 lemma transpose_infmatrix_twice[simp]: "transpose_infmatrix (transpose_infmatrix A) = A"
    62 by ((rule ext)+, simp)
    63 
    64 lemma transpose_infmatrix: "transpose_infmatrix (% j i. P j i) = (% j i. P i j)"
    65   apply (rule ext)+
    66   by (simp add: transpose_infmatrix_def)
    67 
    68 lemma transpose_infmatrix_closed[simp]: "Rep_matrix (Abs_matrix (transpose_infmatrix (Rep_matrix x))) = transpose_infmatrix (Rep_matrix x)"
    69 apply (rule Abs_matrix_inverse)
    70 apply (simp add: matrix_def nonzero_positions_def image_def)
    71 proof -
    72   let ?A = "{pos. Rep_matrix x (snd pos) (fst pos) \<noteq> 0}"
    73   let ?swap = "% pos. (snd pos, fst pos)"
    74   let ?B = "{pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0}"
    75   have swap_image: "?swap`?A = ?B"
    76     apply (simp add: image_def)
    77     apply (rule set_eqI)
    78     apply (simp)
    79     proof
    80       fix y
    81       assume hyp: "\<exists>a b. Rep_matrix x b a \<noteq> 0 \<and> y = (b, a)"
    82       thus "Rep_matrix x (fst y) (snd y) \<noteq> 0"
    83         proof -
    84           from hyp obtain a b where "(Rep_matrix x b a \<noteq> 0 & y = (b,a))" by blast
    85           then show "Rep_matrix x (fst y) (snd y) \<noteq> 0" by (simp)
    86         qed
    87     next
    88       fix y
    89       assume hyp: "Rep_matrix x (fst y) (snd y) \<noteq> 0"
    90       show "\<exists> a b. (Rep_matrix x b a \<noteq> 0 & y = (b,a))"
    91         by (rule exI[of _ "snd y"], rule exI[of _ "fst y"]) (simp add: hyp)
    92     qed
    93   then have "finite (?swap`?A)"
    94     proof -
    95       have "finite (nonzero_positions (Rep_matrix x))" by (simp add: finite_nonzero_positions)
    96       then have "finite ?B" by (simp add: nonzero_positions_def)
    97       with swap_image show "finite (?swap`?A)" by (simp)
    98     qed
    99   moreover
   100   have "inj_on ?swap ?A" by (simp add: inj_on_def)
   101   ultimately show "finite ?A"by (rule finite_imageD[of ?swap ?A])
   102 qed
   103 
   104 lemma infmatrixforward: "(x::'a infmatrix) = y \<Longrightarrow> \<forall> a b. x a b = y a b" by auto
   105 
   106 lemma transpose_infmatrix_inject: "(transpose_infmatrix A = transpose_infmatrix B) = (A = B)"
   107 apply (auto)
   108 apply (rule ext)+
   109 apply (simp add: transpose_infmatrix)
   110 apply (drule infmatrixforward)
   111 apply (simp)
   112 done
   113 
   114 lemma transpose_matrix_inject: "(transpose_matrix A = transpose_matrix B) = (A = B)"
   115 apply (simp add: transpose_matrix_def)
   116 apply (subst Rep_matrix_inject[THEN sym])+
   117 apply (simp only: transpose_infmatrix_closed transpose_infmatrix_inject)
   118 done
   119 
   120 lemma transpose_matrix[simp]: "Rep_matrix(transpose_matrix A) j i = Rep_matrix A i j"
   121 by (simp add: transpose_matrix_def)
   122 
   123 lemma transpose_transpose_id[simp]: "transpose_matrix (transpose_matrix A) = A"
   124 by (simp add: transpose_matrix_def)
   125 
   126 lemma nrows_transpose[simp]: "nrows (transpose_matrix A) = ncols A"
   127 by (simp add: nrows_def ncols_def nonzero_positions_def transpose_matrix_def image_def)
   128 
   129 lemma ncols_transpose[simp]: "ncols (transpose_matrix A) = nrows A"
   130 by (simp add: nrows_def ncols_def nonzero_positions_def transpose_matrix_def image_def)
   131 
   132 lemma ncols: "ncols A <= n \<Longrightarrow> Rep_matrix A m n = 0"
   133 proof -
   134   assume "ncols A <= n"
   135   then have "nrows (transpose_matrix A) <= n" by (simp)
   136   then have "Rep_matrix (transpose_matrix A) n m = 0" by (rule nrows)
   137   thus "Rep_matrix A m n = 0" by (simp add: transpose_matrix_def)
   138 qed
   139 
   140 lemma ncols_le: "(ncols A <= n) = (! j i. n <= i \<longrightarrow> (Rep_matrix A j i) = 0)" (is "_ = ?st")
   141 apply (auto)
   142 apply (simp add: ncols)
   143 proof (simp add: ncols_def, auto)
   144   let ?P = "nonzero_positions (Rep_matrix A)"
   145   let ?p = "snd`?P"
   146   have a:"finite ?p" by (simp add: finite_nonzero_positions)
   147   let ?m = "Max ?p"
   148   assume "~(Suc (?m) <= n)"
   149   then have b:"n <= ?m" by (simp)
   150   fix a b
   151   assume "(a,b) \<in> ?P"
   152   then have "?p \<noteq> {}" by (auto)
   153   with a have "?m \<in>  ?p" by (simp)
   154   moreover have "!x. (x \<in> ?p \<longrightarrow> (? y. (Rep_matrix A y x) \<noteq> 0))" by (simp add: nonzero_positions_def image_def)
   155   ultimately have "? y. (Rep_matrix A y ?m) \<noteq> 0" by (simp)
   156   moreover assume ?st
   157   ultimately show "False" using b by (simp)
   158 qed
   159 
   160 lemma less_ncols: "(n < ncols A) = (? j i. n <= i & (Rep_matrix A j i) \<noteq> 0)"
   161 proof -
   162   have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
   163   show ?thesis by (simp add: a ncols_le)
   164 qed
   165 
   166 lemma le_ncols: "(n <= ncols A) = (\<forall> m. (\<forall> j i. m <= i \<longrightarrow> (Rep_matrix A j i) = 0) \<longrightarrow> n <= m)"
   167 apply (auto)
   168 apply (subgoal_tac "ncols A <= m")
   169 apply (simp)
   170 apply (simp add: ncols_le)
   171 apply (drule_tac x="ncols A" in spec)
   172 by (simp add: ncols)
   173 
   174 lemma nrows_le: "(nrows A <= n) = (! j i. n <= j \<longrightarrow> (Rep_matrix A j i) = 0)" (is ?s)
   175 proof -
   176   have "(nrows A <= n) = (ncols (transpose_matrix A) <= n)" by (simp)
   177   also have "\<dots> = (! j i. n <= i \<longrightarrow> (Rep_matrix (transpose_matrix A) j i = 0))" by (rule ncols_le)
   178   also have "\<dots> = (! j i. n <= i \<longrightarrow> (Rep_matrix A i j) = 0)" by (simp)
   179   finally show "(nrows A <= n) = (! j i. n <= j \<longrightarrow> (Rep_matrix A j i) = 0)" by (auto)
   180 qed
   181 
   182 lemma less_nrows: "(m < nrows A) = (? j i. m <= j & (Rep_matrix A j i) \<noteq> 0)"
   183 proof -
   184   have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
   185   show ?thesis by (simp add: a nrows_le)
   186 qed
   187 
   188 lemma le_nrows: "(n <= nrows A) = (\<forall> m. (\<forall> j i. m <= j \<longrightarrow> (Rep_matrix A j i) = 0) \<longrightarrow> n <= m)"
   189 apply (auto)
   190 apply (subgoal_tac "nrows A <= m")
   191 apply (simp)
   192 apply (simp add: nrows_le)
   193 apply (drule_tac x="nrows A" in spec)
   194 by (simp add: nrows)
   195 
   196 lemma nrows_notzero: "Rep_matrix A m n \<noteq> 0 \<Longrightarrow> m < nrows A"
   197 apply (case_tac "nrows A <= m")
   198 apply (simp_all add: nrows)
   199 done
   200 
   201 lemma ncols_notzero: "Rep_matrix A m n \<noteq> 0 \<Longrightarrow> n < ncols A"
   202 apply (case_tac "ncols A <= n")
   203 apply (simp_all add: ncols)
   204 done
   205 
   206 lemma finite_natarray1: "finite {x. x < (n::nat)}"
   207 apply (induct n)
   208 apply (simp)
   209 proof -
   210   fix n
   211   have "{x. x < Suc n} = insert n {x. x < n}"  by (rule set_eqI, simp, arith)
   212   moreover assume "finite {x. x < n}"
   213   ultimately show "finite {x. x < Suc n}" by (simp)
   214 qed
   215 
   216 lemma finite_natarray2: "finite {pos. (fst pos) < (m::nat) & (snd pos) < (n::nat)}"
   217   apply (induct m)
   218   apply (simp+)
   219   proof -
   220     fix m::nat
   221     let ?s0 = "{pos. fst pos < m & snd pos < n}"
   222     let ?s1 = "{pos. fst pos < (Suc m) & snd pos < n}"
   223     let ?sd = "{pos. fst pos = m & snd pos < n}"
   224     assume f0: "finite ?s0"
   225     have f1: "finite ?sd"
   226     proof -
   227       let ?f = "% x. (m, x)"
   228       have "{pos. fst pos = m & snd pos < n} = ?f ` {x. x < n}" by (rule set_eqI, simp add: image_def, auto)
   229       moreover have "finite {x. x < n}" by (simp add: finite_natarray1)
   230       ultimately show "finite {pos. fst pos = m & snd pos < n}" by (simp)
   231     qed
   232     have su: "?s0 \<union> ?sd = ?s1" by (rule set_eqI, simp, arith)
   233     from f0 f1 have "finite (?s0 \<union> ?sd)" by (rule finite_UnI)
   234     with su show "finite ?s1" by (simp)
   235 qed
   236 
   237 lemma RepAbs_matrix:
   238   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)
   239   shows "(Rep_matrix (Abs_matrix x)) = x"
   240 apply (rule Abs_matrix_inverse)
   241 apply (simp add: matrix_def nonzero_positions_def)
   242 proof -
   243   from aem obtain m where a: "! j i. m <= j \<longrightarrow> x j i = 0" by (blast)
   244   from aen obtain n where b: "! j i. n <= i \<longrightarrow> x j i = 0" by (blast)
   245   let ?u = "{pos. x (fst pos) (snd pos) \<noteq> 0}"
   246   let ?v = "{pos. fst pos < m & snd pos < n}"
   247   have c: "!! (m::nat) a. ~(m <= a) \<Longrightarrow> a < m" by (arith)
   248   from a b have "(?u \<inter> (-?v)) = {}"
   249     apply (simp)
   250     apply (rule set_eqI)
   251     apply (simp)
   252     apply auto
   253     by (rule c, auto)+
   254   then have d: "?u \<subseteq> ?v" by blast
   255   moreover have "finite ?v" by (simp add: finite_natarray2)
   256   ultimately show "finite ?u" by (rule finite_subset)
   257 qed
   258 
   259 definition apply_infmatrix :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix" where
   260   "apply_infmatrix f == % A. (% j i. f (A j i))"
   261 
   262 definition apply_matrix :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix" where
   263   "apply_matrix f == % A. Abs_matrix (apply_infmatrix f (Rep_matrix A))"
   264 
   265 definition combine_infmatrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix \<Rightarrow> 'c infmatrix" where
   266   "combine_infmatrix f == % A B. (% j i. f (A j i) (B j i))"
   267 
   268 definition combine_matrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix \<Rightarrow> ('c::zero) matrix" where
   269   "combine_matrix f == % A B. Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))"
   270 
   271 lemma expand_apply_infmatrix[simp]: "apply_infmatrix f A j i = f (A j i)"
   272 by (simp add: apply_infmatrix_def)
   273 
   274 lemma expand_combine_infmatrix[simp]: "combine_infmatrix f A B j i = f (A j i) (B j i)"
   275 by (simp add: combine_infmatrix_def)
   276 
   277 definition commutative :: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> bool" where
   278 "commutative f == ! x y. f x y = f y x"
   279 
   280 definition associative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool" where
   281 "associative f == ! x y z. f (f x y) z = f x (f y z)"
   282 
   283 text{*
   284 To reason about associativity and commutativity of operations on matrices,
   285 let's take a step back and look at the general situtation: Assume that we have
   286 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.
   287 Each function $f: A \times A \rightarrow A$ now induces a function $f': B \times B \rightarrow B$ by $f' = u \circ f$.
   288 It is obvious that commutativity of $f$ implies commutativity of $f'$: $f' x y = u (f x y) = u (f y x) = f' y x.$
   289 *}
   290 
   291 lemma combine_infmatrix_commute:
   292   "commutative f \<Longrightarrow> commutative (combine_infmatrix f)"
   293 by (simp add: commutative_def combine_infmatrix_def)
   294 
   295 lemma combine_matrix_commute:
   296 "commutative f \<Longrightarrow> commutative (combine_matrix f)"
   297 by (simp add: combine_matrix_def commutative_def combine_infmatrix_def)
   298 
   299 text{*
   300 On the contrary, given an associative function $f$ we cannot expect $f'$ to be associative. A counterexample is given by $A=\ganz$, $B=\{-1, 0, 1\}$,
   301 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
   302 \[ f' (f' 1 1) -1 = u(f (u (f 1 1)) -1) = u(f (u 2) -1) = u (f 0 -1) = -1, \]
   303 but on the other hand we have
   304 \[ f' 1 (f' 1 -1) = u (f 1 (u (f 1 -1))) = u (f 1 0) = 1.\]
   305 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:
   306 *}
   307 
   308 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)"
   309 by (rule subsetI, simp add: nonzero_positions_def combine_infmatrix_def, auto)
   310 
   311 lemma finite_nonzero_positions_Rep[simp]: "finite (nonzero_positions (Rep_matrix A))"
   312 by (insert Rep_matrix [of A], simp add: matrix_def)
   313 
   314 lemma combine_infmatrix_closed [simp]:
   315   "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)"
   316 apply (rule Abs_matrix_inverse)
   317 apply (simp add: matrix_def)
   318 apply (rule finite_subset[of _ "(nonzero_positions (Rep_matrix A)) \<union> (nonzero_positions (Rep_matrix B))"])
   319 by (simp_all)
   320 
   321 text {* We need the next two lemmas only later, but it is analog to the above one, so we prove them now: *}
   322 lemma nonzero_positions_apply_infmatrix[simp]: "f 0 = 0 \<Longrightarrow> nonzero_positions (apply_infmatrix f A) \<subseteq> nonzero_positions A"
   323 by (rule subsetI, simp add: nonzero_positions_def apply_infmatrix_def, auto)
   324 
   325 lemma apply_infmatrix_closed [simp]:
   326   "f 0 = 0 \<Longrightarrow> Rep_matrix (Abs_matrix (apply_infmatrix f (Rep_matrix A))) = apply_infmatrix f (Rep_matrix A)"
   327 apply (rule Abs_matrix_inverse)
   328 apply (simp add: matrix_def)
   329 apply (rule finite_subset[of _ "nonzero_positions (Rep_matrix A)"])
   330 by (simp_all)
   331 
   332 lemma combine_infmatrix_assoc[simp]: "f 0 0 = 0 \<Longrightarrow> associative f \<Longrightarrow> associative (combine_infmatrix f)"
   333 by (simp add: associative_def combine_infmatrix_def)
   334 
   335 lemma comb: "f = g \<Longrightarrow> x = y \<Longrightarrow> f x = g y"
   336 by (auto)
   337 
   338 lemma combine_matrix_assoc: "f 0 0 = 0 \<Longrightarrow> associative f \<Longrightarrow> associative (combine_matrix f)"
   339 apply (simp(no_asm) add: associative_def combine_matrix_def, auto)
   340 apply (rule comb [of Abs_matrix Abs_matrix])
   341 by (auto, insert combine_infmatrix_assoc[of f], simp add: associative_def)
   342 
   343 lemma Rep_apply_matrix[simp]: "f 0 = 0 \<Longrightarrow> Rep_matrix (apply_matrix f A) j i = f (Rep_matrix A j i)"
   344 by (simp add: apply_matrix_def)
   345 
   346 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)"
   347   by(simp add: combine_matrix_def)
   348 
   349 lemma combine_nrows_max: "f 0 0 = 0  \<Longrightarrow> nrows (combine_matrix f A B) <= max (nrows A) (nrows B)"
   350 by (simp add: nrows_le)
   351 
   352 lemma combine_ncols_max: "f 0 0 = 0 \<Longrightarrow> ncols (combine_matrix f A B) <= max (ncols A) (ncols B)"
   353 by (simp add: ncols_le)
   354 
   355 lemma combine_nrows: "f 0 0 = 0 \<Longrightarrow> nrows A <= q \<Longrightarrow> nrows B <= q \<Longrightarrow> nrows(combine_matrix f A B) <= q"
   356   by (simp add: nrows_le)
   357 
   358 lemma combine_ncols: "f 0 0 = 0 \<Longrightarrow> ncols A <= q \<Longrightarrow> ncols B <= q \<Longrightarrow> ncols(combine_matrix f A B) <= q"
   359   by (simp add: ncols_le)
   360 
   361 definition zero_r_neutral :: "('a \<Rightarrow> 'b::zero \<Rightarrow> 'a) \<Rightarrow> bool" where
   362   "zero_r_neutral f == ! a. f a 0 = a"
   363 
   364 definition zero_l_neutral :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool" where
   365   "zero_l_neutral f == ! a. f 0 a = a"
   366 
   367 definition zero_closed :: "(('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> bool" where
   368   "zero_closed f == (!x. f x 0 = 0) & (!y. f 0 y = 0)"
   369 
   370 primrec foldseq :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
   371 where
   372   "foldseq f s 0 = s 0"
   373 | "foldseq f s (Suc n) = f (s 0) (foldseq f (% k. s(Suc k)) n)"
   374 
   375 primrec foldseq_transposed ::  "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
   376 where
   377   "foldseq_transposed f s 0 = s 0"
   378 | "foldseq_transposed f s (Suc n) = f (foldseq_transposed f s n) (s (Suc n))"
   379 
   380 lemma foldseq_assoc : "associative f \<Longrightarrow> foldseq f = foldseq_transposed f"
   381 proof -
   382   assume a:"associative f"
   383   then have sublemma: "!! n. ! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
   384   proof -
   385     fix n
   386     show "!N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
   387     proof (induct n)
   388       show "!N s. N <= 0 \<longrightarrow> foldseq f s N = foldseq_transposed f s N" by simp
   389     next
   390       fix n
   391       assume b:"! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
   392       have c:"!!N s. N <= n \<Longrightarrow> foldseq f s N = foldseq_transposed f s N" by (simp add: b)
   393       show "! N t. N <= Suc n \<longrightarrow> foldseq f t N = foldseq_transposed f t N"
   394       proof (auto)
   395         fix N t
   396         assume Nsuc: "N <= Suc n"
   397         show "foldseq f t N = foldseq_transposed f t N"
   398         proof cases
   399           assume "N <= n"
   400           then show "foldseq f t N = foldseq_transposed f t N" by (simp add: b)
   401         next
   402           assume "~(N <= n)"
   403           with Nsuc have Nsuceq: "N = Suc n" by simp
   404           have neqz: "n \<noteq> 0 \<Longrightarrow> ? m. n = Suc m & Suc m <= n" by arith
   405           have assocf: "!! x y z. f x (f y z) = f (f x y) z" by (insert a, simp add: associative_def)
   406           show "foldseq f t N = foldseq_transposed f t N"
   407             apply (simp add: Nsuceq)
   408             apply (subst c)
   409             apply (simp)
   410             apply (case_tac "n = 0")
   411             apply (simp)
   412             apply (drule neqz)
   413             apply (erule exE)
   414             apply (simp)
   415             apply (subst assocf)
   416             proof -
   417               fix m
   418               assume "n = Suc m & Suc m <= n"
   419               then have mless: "Suc m <= n" by arith
   420               then have step1: "foldseq_transposed f (% k. t (Suc k)) m = foldseq f (% k. t (Suc k)) m" (is "?T1 = ?T2")
   421                 apply (subst c)
   422                 by simp+
   423               have step2: "f (t 0) ?T2 = foldseq f t (Suc m)" (is "_ = ?T3") by simp
   424               have step3: "?T3 = foldseq_transposed f t (Suc m)" (is "_ = ?T4")
   425                 apply (subst c)
   426                 by (simp add: mless)+
   427               have step4: "?T4 = f (foldseq_transposed f t m) (t (Suc m))" (is "_=?T5") by simp
   428               from step1 step2 step3 step4 show sowhat: "f (f (t 0) ?T1) (t (Suc (Suc m))) = f ?T5 (t (Suc (Suc m)))" by simp
   429             qed
   430           qed
   431         qed
   432       qed
   433     qed
   434     show "foldseq f = foldseq_transposed f" by ((rule ext)+, insert sublemma, auto)
   435   qed
   436 
   437 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)"
   438 proof -
   439   assume assoc: "associative f"
   440   assume comm: "commutative f"
   441   from assoc have a:"!! x y z. f (f x y) z = f x (f y z)" by (simp add: associative_def)
   442   from comm have b: "!! x y. f x y = f y x" by (simp add: commutative_def)
   443   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)
   444   have "!! n. (! u v. foldseq f (%k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n))"
   445     apply (induct_tac n)
   446     apply (simp+, auto)
   447     by (simp add: a b c)
   448   then show "foldseq f (% k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n)" by simp
   449 qed
   450 
   451 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)"
   452 oops
   453 (* Model found
   454 
   455 Trying to find a model that refutes: \<lbrakk>associative f; associative g;
   456  \<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;
   457  \<exists>x y. g x \<noteq> g y; f x x = x; g x x = x\<rbrakk>
   458 \<Longrightarrow> f = g \<or> (\<forall>y. f y x = y) \<or> (\<forall>y. g y x = y)
   459 Searching for a model of size 1, translating term... invoking SAT solver... no model found.
   460 Searching for a model of size 2, translating term... invoking SAT solver... no model found.
   461 Searching for a model of size 3, translating term... invoking SAT solver...
   462 Model found:
   463 Size of types: 'a: 3
   464 x: a1
   465 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))
   466 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))
   467 *)
   468 
   469 lemma foldseq_zero:
   470 assumes fz: "f 0 0 = 0" and sz: "! i. i <= n \<longrightarrow> s i = 0"
   471 shows "foldseq f s n = 0"
   472 proof -
   473   have "!! n. ! s. (! i. i <= n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s n = 0"
   474     apply (induct_tac n)
   475     apply (simp)
   476     by (simp add: fz)
   477   then show "foldseq f s n = 0" by (simp add: sz)
   478 qed
   479 
   480 lemma foldseq_significant_positions:
   481   assumes p: "! i. i <= N \<longrightarrow> S i = T i"
   482   shows "foldseq f S N = foldseq f T N"
   483 proof -
   484   have "!! m . ! s t. (! i. i<=m \<longrightarrow> s i = t i) \<longrightarrow> foldseq f s m = foldseq f t m"
   485     apply (induct_tac m)
   486     apply (simp)
   487     apply (simp)
   488     apply (auto)
   489     proof -
   490       fix n
   491       fix s::"nat\<Rightarrow>'a"
   492       fix t::"nat\<Rightarrow>'a"
   493       assume a: "\<forall>s t. (\<forall>i\<le>n. s i = t i) \<longrightarrow> foldseq f s n = foldseq f t n"
   494       assume b: "\<forall>i\<le>Suc n. s i = t i"
   495       have c:"!! a b. a = b \<Longrightarrow> f (t 0) a = f (t 0) b" by blast
   496       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)
   497       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)
   498     qed
   499   with p show ?thesis by simp
   500 qed
   501 
   502 lemma foldseq_tail:
   503   assumes "M <= N"
   504   shows "foldseq f S N = foldseq f (% k. (if k < M then (S k) else (foldseq f (% k. S(k+M)) (N-M)))) M"
   505 proof -
   506   have suc: "!! a b. \<lbrakk>a <= Suc b; a \<noteq> Suc b\<rbrakk> \<Longrightarrow> a <= b" by arith
   507   have a:"!! a b c . a = b \<Longrightarrow> f c a = f c b" by blast
   508   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"
   509     apply (induct_tac n)
   510     apply (simp)
   511     apply (simp)
   512     apply (auto)
   513     apply (case_tac "m = Suc na")
   514     apply (simp)
   515     apply (rule a)
   516     apply (rule foldseq_significant_positions)
   517     apply (auto)
   518     apply (drule suc, simp+)
   519     proof -
   520       fix na m s
   521       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"
   522       assume subb:"m <= na"
   523       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
   524       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 =
   525         foldseq f (% k. s(Suc k)) na"
   526         by (rule subc[of m "% k. s(Suc k)", THEN sym], simp add: subb)
   527       from subb have sube: "m \<noteq> 0 \<Longrightarrow> ? mm. m = Suc mm & mm <= na" by arith
   528       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) =
   529         foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (Suc na - m)) m"
   530         apply (simp add: subd)
   531         apply (cases "m = 0")
   532         apply (simp)
   533         apply (drule sube)
   534         apply (auto)
   535         apply (rule a)
   536         by (simp add: subc cong del: if_cong)
   537     qed
   538   then show ?thesis using assms by simp
   539 qed
   540 
   541 lemma foldseq_zerotail:
   542   assumes
   543   fz: "f 0 0 = 0"
   544   and sz: "! i.  n <= i \<longrightarrow> s i = 0"
   545   and nm: "n <= m"
   546   shows
   547   "foldseq f s n = foldseq f s m"
   548 proof -
   549   show "foldseq f s n = foldseq f s m"
   550     apply (simp add: foldseq_tail[OF nm, of f s])
   551     apply (rule foldseq_significant_positions)
   552     apply (auto)
   553     apply (subst foldseq_zero)
   554     by (simp add: fz sz)+
   555 qed
   556 
   557 lemma foldseq_zerotail2:
   558   assumes "! x. f x 0 = x"
   559   and "! i. n < i \<longrightarrow> s i = 0"
   560   and nm: "n <= m"
   561   shows "foldseq f s n = foldseq f s m"
   562 proof -
   563   have "f 0 0 = 0" by (simp add: assms)
   564   have b:"!! m n. n <= m \<Longrightarrow> m \<noteq> n \<Longrightarrow> ? k. m-n = Suc k" by arith
   565   have c: "0 <= m" by simp
   566   have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
   567   show ?thesis
   568     apply (subst foldseq_tail[OF nm])
   569     apply (rule foldseq_significant_positions)
   570     apply (auto)
   571     apply (case_tac "m=n")
   572     apply (simp+)
   573     apply (drule b[OF nm])
   574     apply (auto)
   575     apply (case_tac "k=0")
   576     apply (simp add: assms)
   577     apply (drule d)
   578     apply (auto)
   579     apply (simp add: assms foldseq_zero)
   580     done
   581 qed
   582 
   583 lemma foldseq_zerostart:
   584   "! 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))"
   585 proof -
   586   assume f00x: "! x. f 0 (f 0 x) = f 0 x"
   587   have "! s. (! i. i<=n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
   588     apply (induct n)
   589     apply (simp)
   590     apply (rule allI, rule impI)
   591     proof -
   592       fix n
   593       fix s
   594       have a:"foldseq f s (Suc (Suc n)) = f (s 0) (foldseq f (% k. s(Suc k)) (Suc n))" by simp
   595       assume b: "! s. ((\<forall>i\<le>n. s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n)))"
   596       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
   597       assume d: "! i. i <= Suc n \<longrightarrow> s i = 0"
   598       show "foldseq f s (Suc (Suc n)) = f 0 (s (Suc (Suc n)))"
   599         apply (subst a)
   600         apply (subst c)
   601         by (simp add: d f00x)+
   602     qed
   603   then show "! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
   604 qed
   605 
   606 lemma foldseq_zerostart2:
   607   "! x. f 0 x = x \<Longrightarrow>  ! i. i < n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s n = s n"
   608 proof -
   609   assume a:"! i. i<n \<longrightarrow> s i = 0"
   610   assume x:"! x. f 0 x = x"
   611   from x have f00x: "! x. f 0 (f 0 x) = f 0 x" by blast
   612   have b: "!! i l. i < Suc l = (i <= l)" by arith
   613   have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
   614   show "foldseq f s n = s n"
   615   apply (case_tac "n=0")
   616   apply (simp)
   617   apply (insert a)
   618   apply (drule d)
   619   apply (auto)
   620   apply (simp add: b)
   621   apply (insert f00x)
   622   apply (drule foldseq_zerostart)
   623   by (simp add: x)+
   624 qed
   625 
   626 lemma foldseq_almostzero:
   627   assumes f0x:"! x. f 0 x = x" and fx0: "! x. f x 0 = x" and s0:"! i. i \<noteq> j \<longrightarrow> s i = 0"
   628   shows "foldseq f s n = (if (j <= n) then (s j) else 0)"
   629 proof -
   630   from s0 have a: "! i. i < j \<longrightarrow> s i = 0" by simp
   631   from s0 have b: "! i. j < i \<longrightarrow> s i = 0" by simp
   632   show ?thesis
   633     apply auto
   634     apply (subst foldseq_zerotail2[of f, OF fx0, of j, OF b, of n, THEN sym])
   635     apply simp
   636     apply (subst foldseq_zerostart2)
   637     apply (simp add: f0x a)+
   638     apply (subst foldseq_zero)
   639     by (simp add: s0 f0x)+
   640 qed
   641 
   642 lemma foldseq_distr_unary:
   643   assumes "!! a b. g (f a b) = f (g a) (g b)"
   644   shows "g(foldseq f s n) = foldseq f (% x. g(s x)) n"
   645 proof -
   646   have "! s. g(foldseq f s n) = foldseq f (% x. g(s x)) n"
   647     apply (induct_tac n)
   648     apply (simp)
   649     apply (simp)
   650     apply (auto)
   651     apply (drule_tac x="% k. s (Suc k)" in spec)
   652     by (simp add: assms)
   653   then show ?thesis by simp
   654 qed
   655 
   656 definition 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" where
   657   "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)"
   658 
   659 definition 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" where
   660   "mult_matrix fmul fadd A B == mult_matrix_n (max (ncols A) (nrows B)) fmul fadd A B"
   661 
   662 lemma mult_matrix_n:
   663   assumes "ncols A \<le>  n" (is ?An) "nrows B \<le> n" (is ?Bn) "fadd 0 0 = 0" "fmul 0 0 = 0"
   664   shows c:"mult_matrix fmul fadd A B = mult_matrix_n n fmul fadd A B"
   665 proof -
   666   show ?thesis using assms
   667     apply (simp add: mult_matrix_def mult_matrix_n_def)
   668     apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
   669     apply (rule foldseq_zerotail, simp_all add: nrows_le ncols_le assms)
   670     done
   671 qed
   672 
   673 lemma mult_matrix_nm:
   674   assumes "ncols A <= n" "nrows B <= n" "ncols A <= m" "nrows B <= m" "fadd 0 0 = 0" "fmul 0 0 = 0"
   675   shows "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B"
   676 proof -
   677   from assms have "mult_matrix_n n fmul fadd A B = mult_matrix fmul fadd A B"
   678     by (simp add: mult_matrix_n)
   679   also from assms have "\<dots> = mult_matrix_n m fmul fadd A B"
   680     by (simp add: mult_matrix_n[THEN sym])
   681   finally show "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B" by simp
   682 qed
   683 
   684 definition r_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool" where
   685   "r_distributive fmul fadd == ! a u v. fmul a (fadd u v) = fadd (fmul a u) (fmul a v)"
   686 
   687 definition l_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool" where
   688   "l_distributive fmul fadd == ! a u v. fmul (fadd u v) a = fadd (fmul u a) (fmul v a)"
   689 
   690 definition distributive :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool" where
   691   "distributive fmul fadd == l_distributive fmul fadd & r_distributive fmul fadd"
   692 
   693 lemma max1: "!! a x y. (a::nat) <= x \<Longrightarrow> a <= max x y" by (arith)
   694 lemma max2: "!! b x y. (b::nat) <= y \<Longrightarrow> b <= max x y" by (arith)
   695 
   696 lemma r_distributive_matrix:
   697  assumes
   698   "r_distributive fmul fadd"
   699   "associative fadd"
   700   "commutative fadd"
   701   "fadd 0 0 = 0"
   702   "! a. fmul a 0 = 0"
   703   "! a. fmul 0 a = 0"
   704  shows "r_distributive (mult_matrix fmul fadd) (combine_matrix fadd)"
   705 proof -
   706   from assms show ?thesis
   707     apply (simp add: r_distributive_def mult_matrix_def, auto)
   708     proof -
   709       fix a::"'a matrix"
   710       fix u::"'b matrix"
   711       fix v::"'b matrix"
   712       let ?mx = "max (ncols a) (max (nrows u) (nrows v))"
   713       from assms show "mult_matrix_n (max (ncols a) (nrows (combine_matrix fadd u v))) fmul fadd a (combine_matrix fadd u v) =
   714         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)"
   715         apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
   716         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   717         apply (subst mult_matrix_nm[of _ _ v ?mx fadd fmul])
   718         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   719         apply (subst mult_matrix_nm[of _ _ u ?mx fadd fmul])
   720         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   721         apply (simp add: mult_matrix_n_def r_distributive_def foldseq_distr[of fadd])
   722         apply (simp add: combine_matrix_def combine_infmatrix_def)
   723         apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
   724         apply (simplesubst RepAbs_matrix)
   725         apply (simp, auto)
   726         apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
   727         apply (rule exI[of _ "ncols v"], simp add: ncols_le foldseq_zero)
   728         apply (subst RepAbs_matrix)
   729         apply (simp, auto)
   730         apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
   731         apply (rule exI[of _ "ncols u"], simp add: ncols_le foldseq_zero)
   732         done
   733     qed
   734 qed
   735 
   736 lemma l_distributive_matrix:
   737  assumes
   738   "l_distributive fmul fadd"
   739   "associative fadd"
   740   "commutative fadd"
   741   "fadd 0 0 = 0"
   742   "! a. fmul a 0 = 0"
   743   "! a. fmul 0 a = 0"
   744  shows "l_distributive (mult_matrix fmul fadd) (combine_matrix fadd)"
   745 proof -
   746   from assms show ?thesis
   747     apply (simp add: l_distributive_def mult_matrix_def, auto)
   748     proof -
   749       fix a::"'b matrix"
   750       fix u::"'a matrix"
   751       fix v::"'a matrix"
   752       let ?mx = "max (nrows a) (max (ncols u) (ncols v))"
   753       from assms show "mult_matrix_n (max (ncols (combine_matrix fadd u v)) (nrows a)) fmul fadd (combine_matrix fadd u v) a =
   754                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)"
   755         apply (subst mult_matrix_nm[of v _ _ ?mx fadd fmul])
   756         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   757         apply (subst mult_matrix_nm[of u _ _ ?mx fadd fmul])
   758         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   759         apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
   760         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   761         apply (simp add: mult_matrix_n_def l_distributive_def foldseq_distr[of fadd])
   762         apply (simp add: combine_matrix_def combine_infmatrix_def)
   763         apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
   764         apply (simplesubst RepAbs_matrix)
   765         apply (simp, auto)
   766         apply (rule exI[of _ "nrows v"], simp add: nrows_le foldseq_zero)
   767         apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
   768         apply (subst RepAbs_matrix)
   769         apply (simp, auto)
   770         apply (rule exI[of _ "nrows u"], simp add: nrows_le foldseq_zero)
   771         apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
   772         done
   773     qed
   774 qed
   775 
   776 instantiation matrix :: (zero) zero
   777 begin
   778 
   779 definition zero_matrix_def: "0 = Abs_matrix (\<lambda>j i. 0)"
   780 
   781 instance ..
   782 
   783 end
   784 
   785 lemma Rep_zero_matrix_def[simp]: "Rep_matrix 0 j i = 0"
   786   apply (simp add: zero_matrix_def)
   787   apply (subst RepAbs_matrix)
   788   by (auto)
   789 
   790 lemma zero_matrix_def_nrows[simp]: "nrows 0 = 0"
   791 proof -
   792   have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
   793   show "nrows 0 = 0" by (rule a, subst nrows_le, simp)
   794 qed
   795 
   796 lemma zero_matrix_def_ncols[simp]: "ncols 0 = 0"
   797 proof -
   798   have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
   799   show "ncols 0 = 0" by (rule a, subst ncols_le, simp)
   800 qed
   801 
   802 lemma combine_matrix_zero_l_neutral: "zero_l_neutral f \<Longrightarrow> zero_l_neutral (combine_matrix f)"
   803   by (simp add: zero_l_neutral_def combine_matrix_def combine_infmatrix_def)
   804 
   805 lemma combine_matrix_zero_r_neutral: "zero_r_neutral f \<Longrightarrow> zero_r_neutral (combine_matrix f)"
   806   by (simp add: zero_r_neutral_def combine_matrix_def combine_infmatrix_def)
   807 
   808 lemma mult_matrix_zero_closed: "\<lbrakk>fadd 0 0 = 0; zero_closed fmul\<rbrakk> \<Longrightarrow> zero_closed (mult_matrix fmul fadd)"
   809   apply (simp add: zero_closed_def mult_matrix_def mult_matrix_n_def)
   810   apply (auto)
   811   by (subst foldseq_zero, (simp add: zero_matrix_def)+)+
   812 
   813 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"
   814   apply (simp add: mult_matrix_n_def)
   815   apply (subst foldseq_zero)
   816   by (simp_all add: zero_matrix_def)
   817 
   818 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"
   819   apply (simp add: mult_matrix_n_def)
   820   apply (subst foldseq_zero)
   821   by (simp_all add: zero_matrix_def)
   822 
   823 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"
   824 by (simp add: mult_matrix_def)
   825 
   826 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"
   827 by (simp add: mult_matrix_def)
   828 
   829 lemma apply_matrix_zero[simp]: "f 0 = 0 \<Longrightarrow> apply_matrix f 0 = 0"
   830   apply (simp add: apply_matrix_def apply_infmatrix_def)
   831   by (simp add: zero_matrix_def)
   832 
   833 lemma combine_matrix_zero: "f 0 0 = 0 \<Longrightarrow> combine_matrix f 0 0 = 0"
   834   apply (simp add: combine_matrix_def combine_infmatrix_def)
   835   by (simp add: zero_matrix_def)
   836 
   837 lemma transpose_matrix_zero[simp]: "transpose_matrix 0 = 0"
   838 apply (simp add: transpose_matrix_def transpose_infmatrix_def zero_matrix_def RepAbs_matrix)
   839 apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
   840 apply (simp add: RepAbs_matrix)
   841 done
   842 
   843 lemma apply_zero_matrix_def[simp]: "apply_matrix (% x. 0) A = 0"
   844   apply (simp add: apply_matrix_def apply_infmatrix_def)
   845   by (simp add: zero_matrix_def)
   846 
   847 definition singleton_matrix :: "nat \<Rightarrow> nat \<Rightarrow> ('a::zero) \<Rightarrow> 'a matrix" where
   848   "singleton_matrix j i a == Abs_matrix(% m n. if j = m & i = n then a else 0)"
   849 
   850 definition move_matrix :: "('a::zero) matrix \<Rightarrow> int \<Rightarrow> int \<Rightarrow> 'a matrix" where
   851   "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)))"
   852 
   853 definition take_rows :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
   854   "take_rows A r == Abs_matrix(% j i. if (j < r) then (Rep_matrix A j i) else 0)"
   855 
   856 definition take_columns :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
   857   "take_columns A c == Abs_matrix(% j i. if (i < c) then (Rep_matrix A j i) else 0)"
   858 
   859 definition column_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
   860   "column_of_matrix A n == take_columns (move_matrix A 0 (- int n)) 1"
   861 
   862 definition row_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix" where
   863   "row_of_matrix A m == take_rows (move_matrix A (- int m) 0) 1"
   864 
   865 lemma Rep_singleton_matrix[simp]: "Rep_matrix (singleton_matrix j i e) m n = (if j = m & i = n then e else 0)"
   866 apply (simp add: singleton_matrix_def)
   867 apply (auto)
   868 apply (subst RepAbs_matrix)
   869 apply (rule exI[of _ "Suc m"], simp)
   870 apply (rule exI[of _ "Suc n"], simp+)
   871 by (subst RepAbs_matrix, rule exI[of _ "Suc j"], simp, rule exI[of _ "Suc i"], simp+)+
   872 
   873 lemma apply_singleton_matrix[simp]: "f 0 = 0 \<Longrightarrow> apply_matrix f (singleton_matrix j i x) = (singleton_matrix j i (f x))"
   874 apply (subst Rep_matrix_inject[symmetric])
   875 apply (rule ext)+
   876 apply (simp)
   877 done
   878 
   879 lemma singleton_matrix_zero[simp]: "singleton_matrix j i 0 = 0"
   880   by (simp add: singleton_matrix_def zero_matrix_def)
   881 
   882 lemma nrows_singleton[simp]: "nrows(singleton_matrix j i e) = (if e = 0 then 0 else Suc j)"
   883 proof-
   884 have th: "\<not> (\<forall>m. m \<le> j)" "\<exists>n. \<not> n \<le> i" by arith+
   885 from th show ?thesis 
   886 apply (auto)
   887 apply (rule le_antisym)
   888 apply (subst nrows_le)
   889 apply (simp add: singleton_matrix_def, auto)
   890 apply (subst RepAbs_matrix)
   891 apply auto
   892 apply (simp add: Suc_le_eq)
   893 apply (rule not_leE)
   894 apply (subst nrows_le)
   895 by simp
   896 qed
   897 
   898 lemma ncols_singleton[simp]: "ncols(singleton_matrix j i e) = (if e = 0 then 0 else Suc i)"
   899 proof-
   900 have th: "\<not> (\<forall>m. m \<le> j)" "\<exists>n. \<not> n \<le> i" by arith+
   901 from th show ?thesis 
   902 apply (auto)
   903 apply (rule le_antisym)
   904 apply (subst ncols_le)
   905 apply (simp add: singleton_matrix_def, auto)
   906 apply (subst RepAbs_matrix)
   907 apply auto
   908 apply (simp add: Suc_le_eq)
   909 apply (rule not_leE)
   910 apply (subst ncols_le)
   911 by simp
   912 qed
   913 
   914 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)"
   915 apply (simp add: singleton_matrix_def combine_matrix_def combine_infmatrix_def)
   916 apply (subst RepAbs_matrix)
   917 apply (rule exI[of _ "Suc j"], simp)
   918 apply (rule exI[of _ "Suc i"], simp)
   919 apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
   920 apply (subst RepAbs_matrix)
   921 apply (rule exI[of _ "Suc j"], simp)
   922 apply (rule exI[of _ "Suc i"], simp)
   923 by simp
   924 
   925 lemma transpose_singleton[simp]: "transpose_matrix (singleton_matrix j i a) = singleton_matrix i j a"
   926 apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
   927 apply (simp)
   928 done
   929 
   930 lemma Rep_move_matrix[simp]:
   931   "Rep_matrix (move_matrix A y x) j i =
   932   (if (neg ((int j)-y)) | (neg ((int i)-x)) then 0 else Rep_matrix A (nat((int j)-y)) (nat((int i)-x)))"
   933 apply (simp add: move_matrix_def)
   934 apply (auto)
   935 by (subst RepAbs_matrix,
   936   rule exI[of _ "(nrows A)+(nat (abs y))"], auto, rule nrows, arith,
   937   rule exI[of _ "(ncols A)+(nat (abs x))"], auto, rule ncols, arith)+
   938 
   939 lemma move_matrix_0_0[simp]: "move_matrix A 0 0 = A"
   940 by (simp add: move_matrix_def)
   941 
   942 lemma move_matrix_ortho: "move_matrix A j i = move_matrix (move_matrix A j 0) 0 i"
   943 apply (subst Rep_matrix_inject[symmetric])
   944 apply (rule ext)+
   945 apply (simp)
   946 done
   947 
   948 lemma transpose_move_matrix[simp]:
   949   "transpose_matrix (move_matrix A x y) = move_matrix (transpose_matrix A) y x"
   950 apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
   951 apply (simp)
   952 done
   953 
   954 lemma move_matrix_singleton[simp]: "move_matrix (singleton_matrix u v x) j i = 
   955   (if (j + int u < 0) | (i + int v < 0) then 0 else (singleton_matrix (nat (j + int u)) (nat (i + int v)) x))"
   956   apply (subst Rep_matrix_inject[symmetric])
   957   apply (rule ext)+
   958   apply (case_tac "j + int u < 0")
   959   apply (simp, arith)
   960   apply (case_tac "i + int v < 0")
   961   apply (simp add: neg_def, arith)
   962   apply (simp add: neg_def)
   963   apply arith
   964   done
   965 
   966 lemma Rep_take_columns[simp]:
   967   "Rep_matrix (take_columns A c) j i =
   968   (if i < c then (Rep_matrix A j i) else 0)"
   969 apply (simp add: take_columns_def)
   970 apply (simplesubst RepAbs_matrix)
   971 apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
   972 apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
   973 done
   974 
   975 lemma Rep_take_rows[simp]:
   976   "Rep_matrix (take_rows A r) j i =
   977   (if j < r then (Rep_matrix A j i) else 0)"
   978 apply (simp add: take_rows_def)
   979 apply (simplesubst RepAbs_matrix)
   980 apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
   981 apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
   982 done
   983 
   984 lemma Rep_column_of_matrix[simp]:
   985   "Rep_matrix (column_of_matrix A c) j i = (if i = 0 then (Rep_matrix A j c) else 0)"
   986   by (simp add: column_of_matrix_def)
   987 
   988 lemma Rep_row_of_matrix[simp]:
   989   "Rep_matrix (row_of_matrix A r) j i = (if j = 0 then (Rep_matrix A r i) else 0)"
   990   by (simp add: row_of_matrix_def)
   991 
   992 lemma column_of_matrix: "ncols A <= n \<Longrightarrow> column_of_matrix A n = 0"
   993 apply (subst Rep_matrix_inject[THEN sym])
   994 apply (rule ext)+
   995 by (simp add: ncols)
   996 
   997 lemma row_of_matrix: "nrows A <= n \<Longrightarrow> row_of_matrix A n = 0"
   998 apply (subst Rep_matrix_inject[THEN sym])
   999 apply (rule ext)+
  1000 by (simp add: nrows)
  1001 
  1002 lemma mult_matrix_singleton_right[simp]:
  1003   assumes
  1004   "! x. fmul x 0 = 0"
  1005   "! x. fmul 0 x = 0"
  1006   "! x. fadd 0 x = x"
  1007   "! x. fadd x 0 = x"
  1008   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))"
  1009   apply (simp add: mult_matrix_def)
  1010   apply (subst mult_matrix_nm[of _ _ _ "max (ncols A) (Suc j)"])
  1011   apply (auto)
  1012   apply (simp add: assms)+
  1013   apply (simp add: mult_matrix_n_def apply_matrix_def apply_infmatrix_def)
  1014   apply (rule comb[of "Abs_matrix" "Abs_matrix"], auto, (rule ext)+)
  1015   apply (subst foldseq_almostzero[of _ j])
  1016   apply (simp add: assms)+
  1017   apply (auto)
  1018   apply (metis add_0 le_antisym le_diff_eq not_neg_nat zero_le_imp_of_nat zle_int)
  1019   done
  1020 
  1021 lemma mult_matrix_ext:
  1022   assumes
  1023   eprem:
  1024   "? e. (! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)"
  1025   and fprems:
  1026   "! a. fmul 0 a = 0"
  1027   "! a. fmul a 0 = 0"
  1028   "! a. fadd a 0 = a"
  1029   "! a. fadd 0 a = a"
  1030   and contraprems:
  1031   "mult_matrix fmul fadd A = mult_matrix fmul fadd B"
  1032   shows
  1033   "A = B"
  1034 proof(rule contrapos_np[of "False"], simp)
  1035   assume a: "A \<noteq> B"
  1036   have b: "!! f g. (! x y. f x y = g x y) \<Longrightarrow> f = g" by ((rule ext)+, auto)
  1037   have "? j i. (Rep_matrix A j i) \<noteq> (Rep_matrix B j i)"
  1038     apply (rule contrapos_np[of "False"], simp+)
  1039     apply (insert b[of "Rep_matrix A" "Rep_matrix B"], simp)
  1040     by (simp add: Rep_matrix_inject a)
  1041   then obtain J I where c:"(Rep_matrix A J I) \<noteq> (Rep_matrix B J I)" by blast
  1042   from eprem obtain e where eprops:"(! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)" by blast
  1043   let ?S = "singleton_matrix I 0 e"
  1044   let ?comp = "mult_matrix fmul fadd"
  1045   have d: "!!x f g. f = g \<Longrightarrow> f x = g x" by blast
  1046   have e: "(% x. fmul x e) 0 = 0" by (simp add: assms)
  1047   have "~(?comp A ?S = ?comp B ?S)"
  1048     apply (rule notI)
  1049     apply (simp add: fprems eprops)
  1050     apply (simp add: Rep_matrix_inject[THEN sym])
  1051     apply (drule d[of _ _ "J"], drule d[of _ _ "0"])
  1052     by (simp add: e c eprops)
  1053   with contraprems show "False" by simp
  1054 qed
  1055 
  1056 definition foldmatrix :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a" where
  1057   "foldmatrix f g A m n == foldseq_transposed g (% j. foldseq f (A j) n) m"
  1058 
  1059 definition foldmatrix_transposed :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a" where
  1060   "foldmatrix_transposed f g A m n == foldseq g (% j. foldseq_transposed f (A j) n) m"
  1061 
  1062 lemma foldmatrix_transpose:
  1063   assumes
  1064   "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
  1065   shows
  1066   "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m"
  1067 proof -
  1068   have forall:"!! P x. (! x. P x) \<Longrightarrow> P x" by auto
  1069   have tworows:"! A. foldmatrix f g A 1 n = foldmatrix_transposed g f (transpose_infmatrix A) n 1"
  1070     apply (induct n)
  1071     apply (simp add: foldmatrix_def foldmatrix_transposed_def assms)+
  1072     apply (auto)
  1073     by (drule_tac x="(% j i. A j (Suc i))" in forall, simp)
  1074   show "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m"
  1075     apply (simp add: foldmatrix_def foldmatrix_transposed_def)
  1076     apply (induct m, simp)
  1077     apply (simp)
  1078     apply (insert tworows)
  1079     apply (drule_tac x="% j i. (if j = 0 then (foldseq_transposed g (\<lambda>u. A u i) m) else (A (Suc m) i))" in spec)
  1080     by (simp add: foldmatrix_def foldmatrix_transposed_def)
  1081 qed
  1082 
  1083 lemma foldseq_foldseq:
  1084 assumes
  1085   "associative f"
  1086   "associative g"
  1087   "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
  1088 shows
  1089   "foldseq g (% j. foldseq f (A j) n) m = foldseq f (% j. foldseq g ((transpose_infmatrix A) j) m) n"
  1090   apply (insert foldmatrix_transpose[of g f A m n])
  1091   by (simp add: foldmatrix_def foldmatrix_transposed_def foldseq_assoc[THEN sym] assms)
  1092 
  1093 lemma mult_n_nrows:
  1094 assumes
  1095 "! a. fmul 0 a = 0"
  1096 "! a. fmul a 0 = 0"
  1097 "fadd 0 0 = 0"
  1098 shows "nrows (mult_matrix_n n fmul fadd A B) \<le> nrows A"
  1099 apply (subst nrows_le)
  1100 apply (simp add: mult_matrix_n_def)
  1101 apply (subst RepAbs_matrix)
  1102 apply (rule_tac x="nrows A" in exI)
  1103 apply (simp add: nrows assms foldseq_zero)
  1104 apply (rule_tac x="ncols B" in exI)
  1105 apply (simp add: ncols assms foldseq_zero)
  1106 apply (simp add: nrows assms foldseq_zero)
  1107 done
  1108 
  1109 lemma mult_n_ncols:
  1110 assumes
  1111 "! a. fmul 0 a = 0"
  1112 "! a. fmul a 0 = 0"
  1113 "fadd 0 0 = 0"
  1114 shows "ncols (mult_matrix_n n fmul fadd A B) \<le> ncols B"
  1115 apply (subst ncols_le)
  1116 apply (simp add: mult_matrix_n_def)
  1117 apply (subst RepAbs_matrix)
  1118 apply (rule_tac x="nrows A" in exI)
  1119 apply (simp add: nrows assms foldseq_zero)
  1120 apply (rule_tac x="ncols B" in exI)
  1121 apply (simp add: ncols assms foldseq_zero)
  1122 apply (simp add: ncols assms foldseq_zero)
  1123 done
  1124 
  1125 lemma mult_nrows:
  1126 assumes
  1127 "! a. fmul 0 a = 0"
  1128 "! a. fmul a 0 = 0"
  1129 "fadd 0 0 = 0"
  1130 shows "nrows (mult_matrix fmul fadd A B) \<le> nrows A"
  1131 by (simp add: mult_matrix_def mult_n_nrows assms)
  1132 
  1133 lemma mult_ncols:
  1134 assumes
  1135 "! a. fmul 0 a = 0"
  1136 "! a. fmul a 0 = 0"
  1137 "fadd 0 0 = 0"
  1138 shows "ncols (mult_matrix fmul fadd A B) \<le> ncols B"
  1139 by (simp add: mult_matrix_def mult_n_ncols assms)
  1140 
  1141 lemma nrows_move_matrix_le: "nrows (move_matrix A j i) <= nat((int (nrows A)) + j)"
  1142   apply (auto simp add: nrows_le)
  1143   apply (rule nrows)
  1144   apply (arith)
  1145   done
  1146 
  1147 lemma ncols_move_matrix_le: "ncols (move_matrix A j i) <= nat((int (ncols A)) + i)"
  1148   apply (auto simp add: ncols_le)
  1149   apply (rule ncols)
  1150   apply (arith)
  1151   done
  1152 
  1153 lemma mult_matrix_assoc:
  1154   assumes
  1155   "! a. fmul1 0 a = 0"
  1156   "! a. fmul1 a 0 = 0"
  1157   "! a. fmul2 0 a = 0"
  1158   "! a. fmul2 a 0 = 0"
  1159   "fadd1 0 0 = 0"
  1160   "fadd2 0 0 = 0"
  1161   "! a b c d. fadd2 (fadd1 a b) (fadd1 c d) = fadd1 (fadd2 a c) (fadd2 b d)"
  1162   "associative fadd1"
  1163   "associative fadd2"
  1164   "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
  1165   "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
  1166   "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
  1167   shows "mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B) C = mult_matrix fmul1 fadd1 A (mult_matrix fmul2 fadd2 B C)"
  1168 proof -
  1169   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
  1170   have fmul2fadd1fold: "!! x s n. fmul2 (foldseq fadd1 s n)  x = foldseq fadd1 (% k. fmul2 (s k) x) n"
  1171     by (rule_tac g1 = "% y. fmul2 y x" in ssubst [OF foldseq_distr_unary], insert assms, simp_all)
  1172   have fmul1fadd2fold: "!! x s n. fmul1 x (foldseq fadd2 s n) = foldseq fadd2 (% k. fmul1 x (s k)) n"
  1173     using assms by (rule_tac g1 = "% y. fmul1 x y" in ssubst [OF foldseq_distr_unary], simp_all)
  1174   let ?N = "max (ncols A) (max (ncols B) (max (nrows B) (nrows C)))"
  1175   show ?thesis
  1176     apply (simp add: Rep_matrix_inject[THEN sym])
  1177     apply (rule ext)+
  1178     apply (simp add: mult_matrix_def)
  1179     apply (simplesubst 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)"])
  1180     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
  1181     apply (simplesubst 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)"])
  1182     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
  1183     apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
  1184     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
  1185     apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
  1186     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
  1187     apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
  1188     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
  1189     apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
  1190     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows assms)+
  1191     apply (simp add: mult_matrix_n_def)
  1192     apply (rule comb_left)
  1193     apply ((rule ext)+, simp)
  1194     apply (simplesubst RepAbs_matrix)
  1195     apply (rule exI[of _ "nrows B"])
  1196     apply (simp add: nrows assms foldseq_zero)
  1197     apply (rule exI[of _ "ncols C"])
  1198     apply (simp add: assms ncols foldseq_zero)
  1199     apply (subst RepAbs_matrix)
  1200     apply (rule exI[of _ "nrows A"])
  1201     apply (simp add: nrows assms foldseq_zero)
  1202     apply (rule exI[of _ "ncols B"])
  1203     apply (simp add: assms ncols foldseq_zero)
  1204     apply (simp add: fmul2fadd1fold fmul1fadd2fold assms)
  1205     apply (subst foldseq_foldseq)
  1206     apply (simp add: assms)+
  1207     apply (simp add: transpose_infmatrix)
  1208     done
  1209 qed
  1210 
  1211 lemma
  1212   assumes
  1213   "! a. fmul1 0 a = 0"
  1214   "! a. fmul1 a 0 = 0"
  1215   "! a. fmul2 0 a = 0"
  1216   "! a. fmul2 a 0 = 0"
  1217   "fadd1 0 0 = 0"
  1218   "fadd2 0 0 = 0"
  1219   "! a b c d. fadd2 (fadd1 a b) (fadd1 c d) = fadd1 (fadd2 a c) (fadd2 b d)"
  1220   "associative fadd1"
  1221   "associative fadd2"
  1222   "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
  1223   "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
  1224   "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
  1225   shows
  1226   "(mult_matrix fmul1 fadd1 A) o (mult_matrix fmul2 fadd2 B) = mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B)"
  1227 apply (rule ext)+
  1228 apply (simp add: comp_def )
  1229 apply (simp add: mult_matrix_assoc assms)
  1230 done
  1231 
  1232 lemma mult_matrix_assoc_simple:
  1233   assumes
  1234   "! a. fmul 0 a = 0"
  1235   "! a. fmul a 0 = 0"
  1236   "fadd 0 0 = 0"
  1237   "associative fadd"
  1238   "commutative fadd"
  1239   "associative fmul"
  1240   "distributive fmul fadd"
  1241   shows "mult_matrix fmul fadd (mult_matrix fmul fadd A B) C = mult_matrix fmul fadd A (mult_matrix fmul fadd B C)"
  1242 proof -
  1243   have "!! a b c d. fadd (fadd a b) (fadd c d) = fadd (fadd a c) (fadd b d)"
  1244     using assms by (simp add: associative_def commutative_def)
  1245   then show ?thesis
  1246     apply (subst mult_matrix_assoc)
  1247     using assms
  1248     apply simp_all
  1249     apply (simp_all add: associative_def distributive_def l_distributive_def r_distributive_def)
  1250     done
  1251 qed
  1252 
  1253 lemma transpose_apply_matrix: "f 0 = 0 \<Longrightarrow> transpose_matrix (apply_matrix f A) = apply_matrix f (transpose_matrix A)"
  1254 apply (simp add: Rep_matrix_inject[THEN sym])
  1255 apply (rule ext)+
  1256 by simp
  1257 
  1258 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)"
  1259 apply (simp add: Rep_matrix_inject[THEN sym])
  1260 apply (rule ext)+
  1261 by simp
  1262 
  1263 lemma Rep_mult_matrix:
  1264   assumes
  1265   "! a. fmul 0 a = 0"
  1266   "! a. fmul a 0 = 0"
  1267   "fadd 0 0 = 0"
  1268   shows
  1269   "Rep_matrix(mult_matrix fmul fadd A B) j i =
  1270   foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) (max (ncols A) (nrows B))"
  1271 apply (simp add: mult_matrix_def mult_matrix_n_def)
  1272 apply (subst RepAbs_matrix)
  1273 apply (rule exI[of _ "nrows A"], insert assms, simp add: nrows foldseq_zero)
  1274 apply (rule exI[of _ "ncols B"], insert assms, simp add: ncols foldseq_zero)
  1275 apply simp
  1276 done
  1277 
  1278 lemma transpose_mult_matrix:
  1279   assumes
  1280   "! a. fmul 0 a = 0"
  1281   "! a. fmul a 0 = 0"
  1282   "fadd 0 0 = 0"
  1283   "! x y. fmul y x = fmul x y"
  1284   shows
  1285   "transpose_matrix (mult_matrix fmul fadd A B) = mult_matrix fmul fadd (transpose_matrix B) (transpose_matrix A)"
  1286   apply (simp add: Rep_matrix_inject[THEN sym])
  1287   apply (rule ext)+
  1288   using assms
  1289   apply (simp add: Rep_mult_matrix max_ac)
  1290   done
  1291 
  1292 lemma column_transpose_matrix: "column_of_matrix (transpose_matrix A) n = transpose_matrix (row_of_matrix A n)"
  1293 apply (simp add:  Rep_matrix_inject[THEN sym])
  1294 apply (rule ext)+
  1295 by simp
  1296 
  1297 lemma take_columns_transpose_matrix: "take_columns (transpose_matrix A) n = transpose_matrix (take_rows A n)"
  1298 apply (simp add: Rep_matrix_inject[THEN sym])
  1299 apply (rule ext)+
  1300 by simp
  1301 
  1302 instantiation matrix :: ("{zero, ord}") ord
  1303 begin
  1304 
  1305 definition
  1306   le_matrix_def: "A \<le> B \<longleftrightarrow> (\<forall>j i. Rep_matrix A j i \<le> Rep_matrix B j i)"
  1307 
  1308 definition
  1309   less_def: "A < (B\<Colon>'a matrix) \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
  1310 
  1311 instance ..
  1312 
  1313 end
  1314 
  1315 instance matrix :: ("{zero, order}") order
  1316 apply intro_classes
  1317 apply (simp_all add: le_matrix_def less_def)
  1318 apply (auto)
  1319 apply (drule_tac x=j in spec, drule_tac x=j in spec)
  1320 apply (drule_tac x=i in spec, drule_tac x=i in spec)
  1321 apply (simp)
  1322 apply (simp add: Rep_matrix_inject[THEN sym])
  1323 apply (rule ext)+
  1324 apply (drule_tac x=xa in spec, drule_tac x=xa in spec)
  1325 apply (drule_tac x=xb in spec, drule_tac x=xb in spec)
  1326 apply simp
  1327 done
  1328 
  1329 lemma le_apply_matrix:
  1330   assumes
  1331   "f 0 = 0"
  1332   "! x y. x <= y \<longrightarrow> f x <= f y"
  1333   "(a::('a::{ord, zero}) matrix) <= b"
  1334   shows
  1335   "apply_matrix f a <= apply_matrix f b"
  1336   using assms by (simp add: le_matrix_def)
  1337 
  1338 lemma le_combine_matrix:
  1339   assumes
  1340   "f 0 0 = 0"
  1341   "! a b c d. a <= b & c <= d \<longrightarrow> f a c <= f b d"
  1342   "A <= B"
  1343   "C <= D"
  1344   shows
  1345   "combine_matrix f A C <= combine_matrix f B D"
  1346   using assms by (simp add: le_matrix_def)
  1347 
  1348 lemma le_left_combine_matrix:
  1349   assumes
  1350   "f 0 0 = 0"
  1351   "! a b c. a <= b \<longrightarrow> f c a <= f c b"
  1352   "A <= B"
  1353   shows
  1354   "combine_matrix f C A <= combine_matrix f C B"
  1355   using assms by (simp add: le_matrix_def)
  1356 
  1357 lemma le_right_combine_matrix:
  1358   assumes
  1359   "f 0 0 = 0"
  1360   "! a b c. a <= b \<longrightarrow> f a c <= f b c"
  1361   "A <= B"
  1362   shows
  1363   "combine_matrix f A C <= combine_matrix f B C"
  1364   using assms by (simp add: le_matrix_def)
  1365 
  1366 lemma le_transpose_matrix: "(A <= B) = (transpose_matrix A <= transpose_matrix B)"
  1367   by (simp add: le_matrix_def, auto)
  1368 
  1369 lemma le_foldseq:
  1370   assumes
  1371   "! a b c d . a <= b & c <= d \<longrightarrow> f a c <= f b d"
  1372   "! i. i <= n \<longrightarrow> s i <= t i"
  1373   shows
  1374   "foldseq f s n <= foldseq f t n"
  1375 proof -
  1376   have "! s t. (! i. i<=n \<longrightarrow> s i <= t i) \<longrightarrow> foldseq f s n <= foldseq f t n"
  1377     by (induct n) (simp_all add: assms)
  1378   then show "foldseq f s n <= foldseq f t n" using assms by simp
  1379 qed
  1380 
  1381 lemma le_left_mult:
  1382   assumes
  1383   "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
  1384   "! c a b.   0 <= c & a <= b \<longrightarrow> fmul c a <= fmul c b"
  1385   "! a. fmul 0 a = 0"
  1386   "! a. fmul a 0 = 0"
  1387   "fadd 0 0 = 0"
  1388   "0 <= C"
  1389   "A <= B"
  1390   shows
  1391   "mult_matrix fmul fadd C A <= mult_matrix fmul fadd C B"
  1392   using assms
  1393   apply (simp add: le_matrix_def Rep_mult_matrix)
  1394   apply (auto)
  1395   apply (simplesubst foldseq_zerotail[of _ _ _ "max (ncols C) (max (nrows A) (nrows B))"], simp_all add: nrows ncols max1 max2)+
  1396   apply (rule le_foldseq)
  1397   apply (auto)
  1398   done
  1399 
  1400 lemma le_right_mult:
  1401   assumes
  1402   "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
  1403   "! c a b. 0 <= c & a <= b \<longrightarrow> fmul a c <= fmul b c"
  1404   "! a. fmul 0 a = 0"
  1405   "! a. fmul a 0 = 0"
  1406   "fadd 0 0 = 0"
  1407   "0 <= C"
  1408   "A <= B"
  1409   shows
  1410   "mult_matrix fmul fadd A C <= mult_matrix fmul fadd B C"
  1411   using assms
  1412   apply (simp add: le_matrix_def Rep_mult_matrix)
  1413   apply (auto)
  1414   apply (simplesubst foldseq_zerotail[of _ _ _ "max (nrows C) (max (ncols A) (ncols B))"], simp_all add: nrows ncols max1 max2)+
  1415   apply (rule le_foldseq)
  1416   apply (auto)
  1417   done
  1418 
  1419 lemma spec2: "! j i. P j i \<Longrightarrow> P j i" by blast
  1420 lemma neg_imp: "(\<not> Q \<longrightarrow> \<not> P) \<Longrightarrow> P \<longrightarrow> Q" by blast
  1421 
  1422 lemma singleton_matrix_le[simp]: "(singleton_matrix j i a <= singleton_matrix j i b) = (a <= (b::_::order))"
  1423   by (auto simp add: le_matrix_def)
  1424 
  1425 lemma singleton_le_zero[simp]: "(singleton_matrix j i x <= 0) = (x <= (0::'a::{order,zero}))"
  1426   apply (auto)
  1427   apply (simp add: le_matrix_def)
  1428   apply (drule_tac j=j and i=i in spec2)
  1429   apply (simp)
  1430   apply (simp add: le_matrix_def)
  1431   done
  1432 
  1433 lemma singleton_ge_zero[simp]: "(0 <= singleton_matrix j i x) = ((0::'a::{order,zero}) <= x)"
  1434   apply (auto)
  1435   apply (simp add: le_matrix_def)
  1436   apply (drule_tac j=j and i=i in spec2)
  1437   apply (simp)
  1438   apply (simp add: le_matrix_def)
  1439   done
  1440 
  1441 lemma move_matrix_le_zero[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (move_matrix A j i <= 0) = (A <= (0::('a::{order,zero}) matrix))"
  1442   apply (auto simp add: le_matrix_def neg_def)
  1443   apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
  1444   apply (auto)
  1445   done
  1446 
  1447 lemma move_matrix_zero_le[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (0 <= move_matrix A j i) = ((0::('a::{order,zero}) matrix) <= A)"
  1448   apply (auto simp add: le_matrix_def neg_def)
  1449   apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
  1450   apply (auto)
  1451   done
  1452 
  1453 lemma move_matrix_le_move_matrix_iff[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (move_matrix A j i <= move_matrix B j i) = (A <= (B::('a::{order,zero}) matrix))"
  1454   apply (auto simp add: le_matrix_def neg_def)
  1455   apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
  1456   apply (auto)
  1457   done  
  1458 
  1459 instantiation matrix :: ("{lattice, zero}") lattice
  1460 begin
  1461 
  1462 definition "inf = combine_matrix inf"
  1463 
  1464 definition "sup = combine_matrix sup"
  1465 
  1466 instance
  1467   by default (auto simp add: inf_le1 inf_le2 le_infI le_matrix_def inf_matrix_def sup_matrix_def)
  1468 
  1469 end
  1470 
  1471 instantiation matrix :: ("{plus, zero}") plus
  1472 begin
  1473 
  1474 definition
  1475   plus_matrix_def: "A + B = combine_matrix (op +) A B"
  1476 
  1477 instance ..
  1478 
  1479 end
  1480 
  1481 instantiation matrix :: ("{uminus, zero}") uminus
  1482 begin
  1483 
  1484 definition
  1485   minus_matrix_def: "- A = apply_matrix uminus A"
  1486 
  1487 instance ..
  1488 
  1489 end
  1490 
  1491 instantiation matrix :: ("{minus, zero}") minus
  1492 begin
  1493 
  1494 definition
  1495   diff_matrix_def: "A - B = combine_matrix (op -) A B"
  1496 
  1497 instance ..
  1498 
  1499 end
  1500 
  1501 instantiation matrix :: ("{plus, times, zero}") times
  1502 begin
  1503 
  1504 definition
  1505   times_matrix_def: "A * B = mult_matrix (op *) (op +) A B"
  1506 
  1507 instance ..
  1508 
  1509 end
  1510 
  1511 instantiation matrix :: ("{lattice, uminus, zero}") abs
  1512 begin
  1513 
  1514 definition
  1515   abs_matrix_def: "abs (A \<Colon> 'a matrix) = sup A (- A)"
  1516 
  1517 instance ..
  1518 
  1519 end
  1520 
  1521 instance matrix :: (monoid_add) monoid_add
  1522 proof
  1523   fix A B C :: "'a matrix"
  1524   show "A + B + C = A + (B + C)"    
  1525     apply (simp add: plus_matrix_def)
  1526     apply (rule combine_matrix_assoc[simplified associative_def, THEN spec, THEN spec, THEN spec])
  1527     apply (simp_all add: add_assoc)
  1528     done
  1529   show "0 + A = A"
  1530     apply (simp add: plus_matrix_def)
  1531     apply (rule combine_matrix_zero_l_neutral[simplified zero_l_neutral_def, THEN spec])
  1532     apply (simp)
  1533     done
  1534   show "A + 0 = A"
  1535     apply (simp add: plus_matrix_def)
  1536     apply (rule combine_matrix_zero_r_neutral [simplified zero_r_neutral_def, THEN spec])
  1537     apply (simp)
  1538     done
  1539 qed
  1540 
  1541 instance matrix :: (comm_monoid_add) comm_monoid_add
  1542 proof
  1543   fix A B :: "'a matrix"
  1544   show "A + B = B + A"
  1545     apply (simp add: plus_matrix_def)
  1546     apply (rule combine_matrix_commute[simplified commutative_def, THEN spec, THEN spec])
  1547     apply (simp_all add: add_commute)
  1548     done
  1549   show "0 + A = A"
  1550     apply (simp add: plus_matrix_def)
  1551     apply (rule combine_matrix_zero_l_neutral[simplified zero_l_neutral_def, THEN spec])
  1552     apply (simp)
  1553     done
  1554 qed
  1555 
  1556 instance matrix :: (group_add) group_add
  1557 proof
  1558   fix A B :: "'a matrix"
  1559   show "- A + A = 0" 
  1560     by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
  1561   show "A - B = A + - B"
  1562     by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject [symmetric] ext diff_minus)
  1563 qed
  1564 
  1565 instance matrix :: (ab_group_add) ab_group_add
  1566 proof
  1567   fix A B :: "'a matrix"
  1568   show "- A + A = 0" 
  1569     by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
  1570   show "A - B = A + - B" 
  1571     by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
  1572 qed
  1573 
  1574 instance matrix :: (ordered_ab_group_add) ordered_ab_group_add
  1575 proof
  1576   fix A B C :: "'a matrix"
  1577   assume "A <= B"
  1578   then show "C + A <= C + B"
  1579     apply (simp add: plus_matrix_def)
  1580     apply (rule le_left_combine_matrix)
  1581     apply (simp_all)
  1582     done
  1583 qed
  1584   
  1585 instance matrix :: (lattice_ab_group_add) semilattice_inf_ab_group_add ..
  1586 instance matrix :: (lattice_ab_group_add) semilattice_sup_ab_group_add ..
  1587 
  1588 instance matrix :: (semiring_0) semiring_0
  1589 proof
  1590   fix A B C :: "'a matrix"
  1591   show "A * B * C = A * (B * C)"
  1592     apply (simp add: times_matrix_def)
  1593     apply (rule mult_matrix_assoc)
  1594     apply (simp_all add: associative_def algebra_simps)
  1595     done
  1596   show "(A + B) * C = A * C + B * C"
  1597     apply (simp add: times_matrix_def plus_matrix_def)
  1598     apply (rule l_distributive_matrix[simplified l_distributive_def, THEN spec, THEN spec, THEN spec])
  1599     apply (simp_all add: associative_def commutative_def algebra_simps)
  1600     done
  1601   show "A * (B + C) = A * B + A * C"
  1602     apply (simp add: times_matrix_def plus_matrix_def)
  1603     apply (rule r_distributive_matrix[simplified r_distributive_def, THEN spec, THEN spec, THEN spec])
  1604     apply (simp_all add: associative_def commutative_def algebra_simps)
  1605     done
  1606   show "0 * A = 0" by (simp add: times_matrix_def)
  1607   show "A * 0 = 0" by (simp add: times_matrix_def)
  1608 qed
  1609 
  1610 instance matrix :: (ring) ring ..
  1611 
  1612 instance matrix :: (ordered_ring) ordered_ring
  1613 proof
  1614   fix A B C :: "'a matrix"
  1615   assume a: "A \<le> B"
  1616   assume b: "0 \<le> C"
  1617   from a b show "C * A \<le> C * B"
  1618     apply (simp add: times_matrix_def)
  1619     apply (rule le_left_mult)
  1620     apply (simp_all add: add_mono mult_left_mono)
  1621     done
  1622   from a b show "A * C \<le> B * C"
  1623     apply (simp add: times_matrix_def)
  1624     apply (rule le_right_mult)
  1625     apply (simp_all add: add_mono mult_right_mono)
  1626     done
  1627 qed
  1628 
  1629 instance matrix :: (lattice_ring) lattice_ring
  1630 proof
  1631   fix A B C :: "('a :: lattice_ring) matrix"
  1632   show "abs A = sup A (-A)" 
  1633     by (simp add: abs_matrix_def)
  1634 qed
  1635 
  1636 lemma Rep_matrix_add[simp]:
  1637   "Rep_matrix ((a::('a::monoid_add)matrix)+b) j i  = (Rep_matrix a j i) + (Rep_matrix b j i)"
  1638   by (simp add: plus_matrix_def)
  1639 
  1640 lemma Rep_matrix_mult: "Rep_matrix ((a::('a::semiring_0) matrix) * b) j i = 
  1641   foldseq (op +) (% k.  (Rep_matrix a j k) * (Rep_matrix b k i)) (max (ncols a) (nrows b))"
  1642 apply (simp add: times_matrix_def)
  1643 apply (simp add: Rep_mult_matrix)
  1644 done
  1645 
  1646 lemma apply_matrix_add: "! x y. f (x+y) = (f x) + (f y) \<Longrightarrow> f 0 = (0::'a)
  1647   \<Longrightarrow> apply_matrix f ((a::('a::monoid_add) matrix) + b) = (apply_matrix f a) + (apply_matrix f b)"
  1648 apply (subst Rep_matrix_inject[symmetric])
  1649 apply (rule ext)+
  1650 apply (simp)
  1651 done
  1652 
  1653 lemma singleton_matrix_add: "singleton_matrix j i ((a::_::monoid_add)+b) = (singleton_matrix j i a) + (singleton_matrix j i b)"
  1654 apply (subst Rep_matrix_inject[symmetric])
  1655 apply (rule ext)+
  1656 apply (simp)
  1657 done
  1658 
  1659 lemma nrows_mult: "nrows ((A::('a::semiring_0) matrix) * B) <= nrows A"
  1660 by (simp add: times_matrix_def mult_nrows)
  1661 
  1662 lemma ncols_mult: "ncols ((A::('a::semiring_0) matrix) * B) <= ncols B"
  1663 by (simp add: times_matrix_def mult_ncols)
  1664 
  1665 definition
  1666   one_matrix :: "nat \<Rightarrow> ('a::{zero,one}) matrix" where
  1667   "one_matrix n = Abs_matrix (% j i. if j = i & j < n then 1 else 0)"
  1668 
  1669 lemma Rep_one_matrix[simp]: "Rep_matrix (one_matrix n) j i = (if (j = i & j < n) then 1 else 0)"
  1670 apply (simp add: one_matrix_def)
  1671 apply (simplesubst RepAbs_matrix)
  1672 apply (rule exI[of _ n], simp add: split_if)+
  1673 by (simp add: split_if)
  1674 
  1675 lemma nrows_one_matrix[simp]: "nrows ((one_matrix n) :: ('a::zero_neq_one)matrix) = n" (is "?r = _")
  1676 proof -
  1677   have "?r <= n" by (simp add: nrows_le)
  1678   moreover have "n <= ?r" by (simp add:le_nrows, arith)
  1679   ultimately show "?r = n" by simp
  1680 qed
  1681 
  1682 lemma ncols_one_matrix[simp]: "ncols ((one_matrix n) :: ('a::zero_neq_one)matrix) = n" (is "?r = _")
  1683 proof -
  1684   have "?r <= n" by (simp add: ncols_le)
  1685   moreover have "n <= ?r" by (simp add: le_ncols, arith)
  1686   ultimately show "?r = n" by simp
  1687 qed
  1688 
  1689 lemma one_matrix_mult_right[simp]: "ncols A <= n \<Longrightarrow> (A::('a::{semiring_1}) matrix) * (one_matrix n) = A"
  1690 apply (subst Rep_matrix_inject[THEN sym])
  1691 apply (rule ext)+
  1692 apply (simp add: times_matrix_def Rep_mult_matrix)
  1693 apply (rule_tac j1="xa" in ssubst[OF foldseq_almostzero])
  1694 apply (simp_all)
  1695 by (simp add: ncols)
  1696 
  1697 lemma one_matrix_mult_left[simp]: "nrows A <= n \<Longrightarrow> (one_matrix n) * A = (A::('a::semiring_1) matrix)"
  1698 apply (subst Rep_matrix_inject[THEN sym])
  1699 apply (rule ext)+
  1700 apply (simp add: times_matrix_def Rep_mult_matrix)
  1701 apply (rule_tac j1="x" in ssubst[OF foldseq_almostzero])
  1702 apply (simp_all)
  1703 by (simp add: nrows)
  1704 
  1705 lemma transpose_matrix_mult: "transpose_matrix ((A::('a::comm_ring) matrix)*B) = (transpose_matrix B) * (transpose_matrix A)"
  1706 apply (simp add: times_matrix_def)
  1707 apply (subst transpose_mult_matrix)
  1708 apply (simp_all add: mult_commute)
  1709 done
  1710 
  1711 lemma transpose_matrix_add: "transpose_matrix ((A::('a::monoid_add) matrix)+B) = transpose_matrix A + transpose_matrix B"
  1712 by (simp add: plus_matrix_def transpose_combine_matrix)
  1713 
  1714 lemma transpose_matrix_diff: "transpose_matrix ((A::('a::group_add) matrix)-B) = transpose_matrix A - transpose_matrix B"
  1715 by (simp add: diff_matrix_def transpose_combine_matrix)
  1716 
  1717 lemma transpose_matrix_minus: "transpose_matrix (-(A::('a::group_add) matrix)) = - transpose_matrix (A::'a matrix)"
  1718 by (simp add: minus_matrix_def transpose_apply_matrix)
  1719 
  1720 definition right_inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
  1721   "right_inverse_matrix A X == (A * X = one_matrix (max (nrows A) (ncols X))) \<and> nrows X \<le> ncols A" 
  1722 
  1723 definition left_inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
  1724   "left_inverse_matrix A X == (X * A = one_matrix (max(nrows X) (ncols A))) \<and> ncols X \<le> nrows A" 
  1725 
  1726 definition inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
  1727   "inverse_matrix A X == (right_inverse_matrix A X) \<and> (left_inverse_matrix A X)"
  1728 
  1729 lemma right_inverse_matrix_dim: "right_inverse_matrix A X \<Longrightarrow> nrows A = ncols X"
  1730 apply (insert ncols_mult[of A X], insert nrows_mult[of A X])
  1731 by (simp add: right_inverse_matrix_def)
  1732 
  1733 lemma left_inverse_matrix_dim: "left_inverse_matrix A Y \<Longrightarrow> ncols A = nrows Y"
  1734 apply (insert ncols_mult[of Y A], insert nrows_mult[of Y A]) 
  1735 by (simp add: left_inverse_matrix_def)
  1736 
  1737 lemma left_right_inverse_matrix_unique: 
  1738   assumes "left_inverse_matrix A Y" "right_inverse_matrix A X"
  1739   shows "X = Y"
  1740 proof -
  1741   have "Y = Y * one_matrix (nrows A)" 
  1742     apply (subst one_matrix_mult_right)
  1743     using assms
  1744     apply (simp_all add: left_inverse_matrix_def)
  1745     done
  1746   also have "\<dots> = Y * (A * X)" 
  1747     apply (insert assms)
  1748     apply (frule right_inverse_matrix_dim)
  1749     by (simp add: right_inverse_matrix_def)
  1750   also have "\<dots> = (Y * A) * X" by (simp add: mult_assoc)
  1751   also have "\<dots> = X" 
  1752     apply (insert assms)
  1753     apply (frule left_inverse_matrix_dim)
  1754     apply (simp_all add:  left_inverse_matrix_def right_inverse_matrix_def one_matrix_mult_left)
  1755     done
  1756   ultimately show "X = Y" by (simp)
  1757 qed
  1758 
  1759 lemma inverse_matrix_inject: "\<lbrakk> inverse_matrix A X; inverse_matrix A Y \<rbrakk> \<Longrightarrow> X = Y"
  1760   by (auto simp add: inverse_matrix_def left_right_inverse_matrix_unique)
  1761 
  1762 lemma one_matrix_inverse: "inverse_matrix (one_matrix n) (one_matrix n)"
  1763   by (simp add: inverse_matrix_def left_inverse_matrix_def right_inverse_matrix_def)
  1764 
  1765 lemma zero_imp_mult_zero: "(a::'a::semiring_0) = 0 | b = 0 \<Longrightarrow> a * b = 0"
  1766 by auto
  1767 
  1768 lemma Rep_matrix_zero_imp_mult_zero:
  1769   "! j i k. (Rep_matrix A j k = 0) | (Rep_matrix B k i) = 0  \<Longrightarrow> A * B = (0::('a::lattice_ring) matrix)"
  1770 apply (subst Rep_matrix_inject[symmetric])
  1771 apply (rule ext)+
  1772 apply (auto simp add: Rep_matrix_mult foldseq_zero zero_imp_mult_zero)
  1773 done
  1774 
  1775 lemma add_nrows: "nrows (A::('a::monoid_add) matrix) <= u \<Longrightarrow> nrows B <= u \<Longrightarrow> nrows (A + B) <= u"
  1776 apply (simp add: plus_matrix_def)
  1777 apply (rule combine_nrows)
  1778 apply (simp_all)
  1779 done
  1780 
  1781 lemma move_matrix_row_mult: "move_matrix ((A::('a::semiring_0) matrix) * B) j 0 = (move_matrix A j 0) * B"
  1782 apply (subst Rep_matrix_inject[symmetric])
  1783 apply (rule ext)+
  1784 apply (auto simp add: Rep_matrix_mult foldseq_zero)
  1785 apply (rule_tac foldseq_zerotail[symmetric])
  1786 apply (auto simp add: nrows zero_imp_mult_zero max2)
  1787 apply (rule order_trans)
  1788 apply (rule ncols_move_matrix_le)
  1789 apply (simp add: max1)
  1790 done
  1791 
  1792 lemma move_matrix_col_mult: "move_matrix ((A::('a::semiring_0) matrix) * B) 0 i = A * (move_matrix B 0 i)"
  1793 apply (subst Rep_matrix_inject[symmetric])
  1794 apply (rule ext)+
  1795 apply (auto simp add: Rep_matrix_mult foldseq_zero)
  1796 apply (rule_tac foldseq_zerotail[symmetric])
  1797 apply (auto simp add: ncols zero_imp_mult_zero max1)
  1798 apply (rule order_trans)
  1799 apply (rule nrows_move_matrix_le)
  1800 apply (simp add: max2)
  1801 done
  1802 
  1803 lemma move_matrix_add: "((move_matrix (A + B) j i)::(('a::monoid_add) matrix)) = (move_matrix A j i) + (move_matrix B j i)" 
  1804 apply (subst Rep_matrix_inject[symmetric])
  1805 apply (rule ext)+
  1806 apply (simp)
  1807 done
  1808 
  1809 lemma move_matrix_mult: "move_matrix ((A::('a::semiring_0) matrix)*B) j i = (move_matrix A j 0) * (move_matrix B 0 i)"
  1810 by (simp add: move_matrix_ortho[of "A*B"] move_matrix_col_mult move_matrix_row_mult)
  1811 
  1812 definition scalar_mult :: "('a::ring) \<Rightarrow> 'a matrix \<Rightarrow> 'a matrix" where
  1813   "scalar_mult a m == apply_matrix (op * a) m"
  1814 
  1815 lemma scalar_mult_zero[simp]: "scalar_mult y 0 = 0" 
  1816 by (simp add: scalar_mult_def)
  1817 
  1818 lemma scalar_mult_add: "scalar_mult y (a+b) = (scalar_mult y a) + (scalar_mult y b)"
  1819 by (simp add: scalar_mult_def apply_matrix_add algebra_simps)
  1820 
  1821 lemma Rep_scalar_mult[simp]: "Rep_matrix (scalar_mult y a) j i = y * (Rep_matrix a j i)" 
  1822 by (simp add: scalar_mult_def)
  1823 
  1824 lemma scalar_mult_singleton[simp]: "scalar_mult y (singleton_matrix j i x) = singleton_matrix j i (y * x)"
  1825 apply (subst Rep_matrix_inject[symmetric])
  1826 apply (rule ext)+
  1827 apply (auto)
  1828 done
  1829 
  1830 lemma Rep_minus[simp]: "Rep_matrix (-(A::_::group_add)) x y = - (Rep_matrix A x y)"
  1831 by (simp add: minus_matrix_def)
  1832 
  1833 lemma Rep_abs[simp]: "Rep_matrix (abs (A::_::lattice_ab_group_add)) x y = abs (Rep_matrix A x y)"
  1834 by (simp add: abs_lattice sup_matrix_def)
  1835 
  1836 end