src/HOL/Matrix/Matrix.thy
author wenzelm
Wed Sep 02 16:25:44 2009 +0200 (2009-09-02 ago)
changeset 32491 d5d8bea0cd94
parent 32440 153965be0f4b
child 32960 69916a850301
permissions -rw-r--r--
reorganized Compute theories for HOL-Matrix -- avoiding theory files within main HOL/Tools;
     1 (*  Title:      HOL/Matrix/Matrix.thy
     2     Author:     Steven Obua
     3 *)
     4 
     5 theory Matrix
     6 imports Main
     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 constdefs
    28   nrows :: "('a::zero) matrix \<Rightarrow> nat"
    29   "nrows A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image fst) (nonzero_positions (Rep_matrix A))))"
    30   ncols :: "('a::zero) matrix \<Rightarrow> nat"
    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" (is ?concl)
    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 constdefs
    54   transpose_infmatrix :: "'a infmatrix \<Rightarrow> 'a infmatrix"
    55   "transpose_infmatrix A j i == A i j"
    56   transpose_matrix :: "('a::zero) matrix \<Rightarrow> 'a matrix"
    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_ext)
    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)" (is ?concl)
   161 proof -
   162   have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
   163   show ?concl 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)" (is ?concl)
   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)" (is ?concl)
   183 proof -
   184   have a: "!! (a::nat) b. (a < b) = (~(b <= a))" by arith
   185   show ?concl 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)" (is ?concl)
   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_ext, 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_ext, 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_ext, 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_ext)
   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 constdefs
   260   apply_infmatrix :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix"
   261   "apply_infmatrix f == % A. (% j i. f (A j i))"
   262   apply_matrix :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix"
   263   "apply_matrix f == % A. Abs_matrix (apply_infmatrix f (Rep_matrix A))"
   264   combine_infmatrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix \<Rightarrow> 'c infmatrix"
   265   "combine_infmatrix f == % A B. (% j i. f (A j i) (B j i))"
   266   combine_matrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix \<Rightarrow> ('c::zero) matrix"
   267   "combine_matrix f == % A B. Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))"
   268 
   269 lemma expand_apply_infmatrix[simp]: "apply_infmatrix f A j i = f (A j i)"
   270 by (simp add: apply_infmatrix_def)
   271 
   272 lemma expand_combine_infmatrix[simp]: "combine_infmatrix f A B j i = f (A j i) (B j i)"
   273 by (simp add: combine_infmatrix_def)
   274 
   275 constdefs
   276 commutative :: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> bool"
   277 "commutative f == ! x y. f x y = f y x"
   278 associative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
   279 "associative f == ! x y z. f (f x y) z = f x (f y z)"
   280 
   281 text{*
   282 To reason about associativity and commutativity of operations on matrices,
   283 let's take a step back and look at the general situtation: Assume that we have
   284 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.
   285 Each function $f: A \times A \rightarrow A$ now induces a function $f': B \times B \rightarrow B$ by $f' = u \circ f$.
   286 It is obvious that commutativity of $f$ implies commutativity of $f'$: $f' x y = u (f x y) = u (f y x) = f' y x.$
   287 *}
   288 
   289 lemma combine_infmatrix_commute:
   290   "commutative f \<Longrightarrow> commutative (combine_infmatrix f)"
   291 by (simp add: commutative_def combine_infmatrix_def)
   292 
   293 lemma combine_matrix_commute:
   294 "commutative f \<Longrightarrow> commutative (combine_matrix f)"
   295 by (simp add: combine_matrix_def commutative_def combine_infmatrix_def)
   296 
   297 text{*
   298 On the contrary, given an associative function $f$ we cannot expect $f'$ to be associative. A counterexample is given by $A=\ganz$, $B=\{-1, 0, 1\}$,
   299 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
   300 \[ f' (f' 1 1) -1 = u(f (u (f 1 1)) -1) = u(f (u 2) -1) = u (f 0 -1) = -1, \]
   301 but on the other hand we have
   302 \[ f' 1 (f' 1 -1) = u (f 1 (u (f 1 -1))) = u (f 1 0) = 1.\]
   303 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:
   304 *}
   305 
   306 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)"
   307 by (rule subsetI, simp add: nonzero_positions_def combine_infmatrix_def, auto)
   308 
   309 lemma finite_nonzero_positions_Rep[simp]: "finite (nonzero_positions (Rep_matrix A))"
   310 by (insert Rep_matrix [of A], simp add: matrix_def)
   311 
   312 lemma combine_infmatrix_closed [simp]:
   313   "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)"
   314 apply (rule Abs_matrix_inverse)
   315 apply (simp add: matrix_def)
   316 apply (rule finite_subset[of _ "(nonzero_positions (Rep_matrix A)) \<union> (nonzero_positions (Rep_matrix B))"])
   317 by (simp_all)
   318 
   319 text {* We need the next two lemmas only later, but it is analog to the above one, so we prove them now: *}
   320 lemma nonzero_positions_apply_infmatrix[simp]: "f 0 = 0 \<Longrightarrow> nonzero_positions (apply_infmatrix f A) \<subseteq> nonzero_positions A"
   321 by (rule subsetI, simp add: nonzero_positions_def apply_infmatrix_def, auto)
   322 
   323 lemma apply_infmatrix_closed [simp]:
   324   "f 0 = 0 \<Longrightarrow> Rep_matrix (Abs_matrix (apply_infmatrix f (Rep_matrix A))) = apply_infmatrix f (Rep_matrix A)"
   325 apply (rule Abs_matrix_inverse)
   326 apply (simp add: matrix_def)
   327 apply (rule finite_subset[of _ "nonzero_positions (Rep_matrix A)"])
   328 by (simp_all)
   329 
   330 lemma combine_infmatrix_assoc[simp]: "f 0 0 = 0 \<Longrightarrow> associative f \<Longrightarrow> associative (combine_infmatrix f)"
   331 by (simp add: associative_def combine_infmatrix_def)
   332 
   333 lemma comb: "f = g \<Longrightarrow> x = y \<Longrightarrow> f x = g y"
   334 by (auto)
   335 
   336 lemma combine_matrix_assoc: "f 0 0 = 0 \<Longrightarrow> associative f \<Longrightarrow> associative (combine_matrix f)"
   337 apply (simp(no_asm) add: associative_def combine_matrix_def, auto)
   338 apply (rule comb [of Abs_matrix Abs_matrix])
   339 by (auto, insert combine_infmatrix_assoc[of f], simp add: associative_def)
   340 
   341 lemma Rep_apply_matrix[simp]: "f 0 = 0 \<Longrightarrow> Rep_matrix (apply_matrix f A) j i = f (Rep_matrix A j i)"
   342 by (simp add: apply_matrix_def)
   343 
   344 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)"
   345   by(simp add: combine_matrix_def)
   346 
   347 lemma combine_nrows_max: "f 0 0 = 0  \<Longrightarrow> nrows (combine_matrix f A B) <= max (nrows A) (nrows B)"
   348 by (simp add: nrows_le)
   349 
   350 lemma combine_ncols_max: "f 0 0 = 0 \<Longrightarrow> ncols (combine_matrix f A B) <= max (ncols A) (ncols B)"
   351 by (simp add: ncols_le)
   352 
   353 lemma combine_nrows: "f 0 0 = 0 \<Longrightarrow> nrows A <= q \<Longrightarrow> nrows B <= q \<Longrightarrow> nrows(combine_matrix f A B) <= q"
   354   by (simp add: nrows_le)
   355 
   356 lemma combine_ncols: "f 0 0 = 0 \<Longrightarrow> ncols A <= q \<Longrightarrow> ncols B <= q \<Longrightarrow> ncols(combine_matrix f A B) <= q"
   357   by (simp add: ncols_le)
   358 
   359 constdefs
   360   zero_r_neutral :: "('a \<Rightarrow> 'b::zero \<Rightarrow> 'a) \<Rightarrow> bool"
   361   "zero_r_neutral f == ! a. f a 0 = a"
   362   zero_l_neutral :: "('a::zero \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool"
   363   "zero_l_neutral f == ! a. f 0 a = a"
   364   zero_closed :: "(('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> bool"
   365   "zero_closed f == (!x. f x 0 = 0) & (!y. f 0 y = 0)"
   366 
   367 consts foldseq :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
   368 primrec
   369   "foldseq f s 0 = s 0"
   370   "foldseq f s (Suc n) = f (s 0) (foldseq f (% k. s(Suc k)) n)"
   371 
   372 consts foldseq_transposed ::  "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
   373 primrec
   374   "foldseq_transposed f s 0 = s 0"
   375   "foldseq_transposed f s (Suc n) = f (foldseq_transposed f s n) (s (Suc n))"
   376 
   377 lemma foldseq_assoc : "associative f \<Longrightarrow> foldseq f = foldseq_transposed f"
   378 proof -
   379   assume a:"associative f"
   380   then have sublemma: "!! n. ! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
   381   proof -
   382     fix n
   383     show "!N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
   384     proof (induct n)
   385       show "!N s. N <= 0 \<longrightarrow> foldseq f s N = foldseq_transposed f s N" by simp
   386     next
   387       fix n
   388       assume b:"! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
   389       have c:"!!N s. N <= n \<Longrightarrow> foldseq f s N = foldseq_transposed f s N" by (simp add: b)
   390       show "! N t. N <= Suc n \<longrightarrow> foldseq f t N = foldseq_transposed f t N"
   391       proof (auto)
   392         fix N t
   393         assume Nsuc: "N <= Suc n"
   394         show "foldseq f t N = foldseq_transposed f t N"
   395         proof cases
   396           assume "N <= n"
   397           then show "foldseq f t N = foldseq_transposed f t N" by (simp add: b)
   398         next
   399           assume "~(N <= n)"
   400           with Nsuc have Nsuceq: "N = Suc n" by simp
   401           have neqz: "n \<noteq> 0 \<Longrightarrow> ? m. n = Suc m & Suc m <= n" by arith
   402           have assocf: "!! x y z. f x (f y z) = f (f x y) z" by (insert a, simp add: associative_def)
   403           show "foldseq f t N = foldseq_transposed f t N"
   404             apply (simp add: Nsuceq)
   405             apply (subst c)
   406             apply (simp)
   407             apply (case_tac "n = 0")
   408             apply (simp)
   409             apply (drule neqz)
   410             apply (erule exE)
   411             apply (simp)
   412             apply (subst assocf)
   413             proof -
   414               fix m
   415               assume "n = Suc m & Suc m <= n"
   416               then have mless: "Suc m <= n" by arith
   417               then have step1: "foldseq_transposed f (% k. t (Suc k)) m = foldseq f (% k. t (Suc k)) m" (is "?T1 = ?T2")
   418                 apply (subst c)
   419                 by simp+
   420               have step2: "f (t 0) ?T2 = foldseq f t (Suc m)" (is "_ = ?T3") by simp
   421               have step3: "?T3 = foldseq_transposed f t (Suc m)" (is "_ = ?T4")
   422                 apply (subst c)
   423                 by (simp add: mless)+
   424               have step4: "?T4 = f (foldseq_transposed f t m) (t (Suc m))" (is "_=?T5") by simp
   425               from step1 step2 step3 step4 show sowhat: "f (f (t 0) ?T1) (t (Suc (Suc m))) = f ?T5 (t (Suc (Suc m)))" by simp
   426             qed
   427           qed
   428         qed
   429       qed
   430     qed
   431     show "foldseq f = foldseq_transposed f" by ((rule ext)+, insert sublemma, auto)
   432   qed
   433 
   434 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)"
   435 proof -
   436   assume assoc: "associative f"
   437   assume comm: "commutative f"
   438   from assoc have a:"!! x y z. f (f x y) z = f x (f y z)" by (simp add: associative_def)
   439   from comm have b: "!! x y. f x y = f y x" by (simp add: commutative_def)
   440   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)
   441   have "!! n. (! u v. foldseq f (%k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n))"
   442     apply (induct_tac n)
   443     apply (simp+, auto)
   444     by (simp add: a b c)
   445   then show "foldseq f (% k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n)" by simp
   446 qed
   447 
   448 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)"
   449 oops
   450 (* Model found
   451 
   452 Trying to find a model that refutes: \<lbrakk>associative f; associative g;
   453  \<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;
   454  \<exists>x y. g x \<noteq> g y; f x x = x; g x x = x\<rbrakk>
   455 \<Longrightarrow> f = g \<or> (\<forall>y. f y x = y) \<or> (\<forall>y. g y x = y)
   456 Searching for a model of size 1, translating term... invoking SAT solver... no model found.
   457 Searching for a model of size 2, translating term... invoking SAT solver... no model found.
   458 Searching for a model of size 3, translating term... invoking SAT solver...
   459 Model found:
   460 Size of types: 'a: 3
   461 x: a1
   462 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))
   463 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))
   464 *)
   465 
   466 lemma foldseq_zero:
   467 assumes fz: "f 0 0 = 0" and sz: "! i. i <= n \<longrightarrow> s i = 0"
   468 shows "foldseq f s n = 0"
   469 proof -
   470   have "!! n. ! s. (! i. i <= n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s n = 0"
   471     apply (induct_tac n)
   472     apply (simp)
   473     by (simp add: fz)
   474   then show "foldseq f s n = 0" by (simp add: sz)
   475 qed
   476 
   477 lemma foldseq_significant_positions:
   478   assumes p: "! i. i <= N \<longrightarrow> S i = T i"
   479   shows "foldseq f S N = foldseq f T N" (is ?concl)
   480 proof -
   481   have "!! m . ! s t. (! i. i<=m \<longrightarrow> s i = t i) \<longrightarrow> foldseq f s m = foldseq f t m"
   482     apply (induct_tac m)
   483     apply (simp)
   484     apply (simp)
   485     apply (auto)
   486     proof -
   487       fix n
   488       fix s::"nat\<Rightarrow>'a"
   489       fix t::"nat\<Rightarrow>'a"
   490       assume a: "\<forall>s t. (\<forall>i\<le>n. s i = t i) \<longrightarrow> foldseq f s n = foldseq f t n"
   491       assume b: "\<forall>i\<le>Suc n. s i = t i"
   492       have c:"!! a b. a = b \<Longrightarrow> f (t 0) a = f (t 0) b" by blast
   493       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)
   494       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)
   495     qed
   496   with p show ?concl by simp
   497 qed
   498 
   499 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")
   500 proof -
   501   have suc: "!! a b. \<lbrakk>a <= Suc b; a \<noteq> Suc b\<rbrakk> \<Longrightarrow> a <= b" by arith
   502   have a:"!! a b c . a = b \<Longrightarrow> f c a = f c b" by blast
   503   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"
   504     apply (induct_tac n)
   505     apply (simp)
   506     apply (simp)
   507     apply (auto)
   508     apply (case_tac "m = Suc na")
   509     apply (simp)
   510     apply (rule a)
   511     apply (rule foldseq_significant_positions)
   512     apply (auto)
   513     apply (drule suc, simp+)
   514     proof -
   515       fix na m s
   516       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"
   517       assume subb:"m <= na"
   518       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
   519       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 =
   520         foldseq f (% k. s(Suc k)) na"
   521         by (rule subc[of m "% k. s(Suc k)", THEN sym], simp add: subb)
   522       from subb have sube: "m \<noteq> 0 \<Longrightarrow> ? mm. m = Suc mm & mm <= na" by arith
   523       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) =
   524         foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (Suc na - m)) m"
   525         apply (simp add: subd)
   526         apply (case_tac "m=0")
   527         apply (simp)
   528         apply (drule sube)
   529         apply (auto)
   530         apply (rule a)
   531         by (simp add: subc if_def)
   532     qed
   533   then show "?p \<Longrightarrow> ?concl" by simp
   534 qed
   535 
   536 lemma foldseq_zerotail:
   537   assumes
   538   fz: "f 0 0 = 0"
   539   and sz: "! i.  n <= i \<longrightarrow> s i = 0"
   540   and nm: "n <= m"
   541   shows
   542   "foldseq f s n = foldseq f s m"
   543 proof -
   544   show "foldseq f s n = foldseq f s m"
   545     apply (simp add: foldseq_tail[OF nm, of f s])
   546     apply (rule foldseq_significant_positions)
   547     apply (auto)
   548     apply (subst foldseq_zero)
   549     by (simp add: fz sz)+
   550 qed
   551 
   552 lemma foldseq_zerotail2:
   553   assumes "! x. f x 0 = x"
   554   and "! i. n < i \<longrightarrow> s i = 0"
   555   and nm: "n <= m"
   556   shows
   557   "foldseq f s n = foldseq f s m" (is ?concl)
   558 proof -
   559   have "f 0 0 = 0" by (simp add: prems)
   560   have b:"!! m n. n <= m \<Longrightarrow> m \<noteq> n \<Longrightarrow> ? k. m-n = Suc k" by arith
   561   have c: "0 <= m" by simp
   562   have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
   563   show ?concl
   564     apply (subst foldseq_tail[OF nm])
   565     apply (rule foldseq_significant_positions)
   566     apply (auto)
   567     apply (case_tac "m=n")
   568     apply (simp+)
   569     apply (drule b[OF nm])
   570     apply (auto)
   571     apply (case_tac "k=0")
   572     apply (simp add: prems)
   573     apply (drule d)
   574     apply (auto)
   575     by (simp add: prems foldseq_zero)
   576 qed
   577 
   578 lemma foldseq_zerostart:
   579   "! 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))"
   580 proof -
   581   assume f00x: "! x. f 0 (f 0 x) = f 0 x"
   582   have "! s. (! i. i<=n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
   583     apply (induct n)
   584     apply (simp)
   585     apply (rule allI, rule impI)
   586     proof -
   587       fix n
   588       fix s
   589       have a:"foldseq f s (Suc (Suc n)) = f (s 0) (foldseq f (% k. s(Suc k)) (Suc n))" by simp
   590       assume b: "! s. ((\<forall>i\<le>n. s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n)))"
   591       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
   592       assume d: "! i. i <= Suc n \<longrightarrow> s i = 0"
   593       show "foldseq f s (Suc (Suc n)) = f 0 (s (Suc (Suc n)))"
   594         apply (subst a)
   595         apply (subst c)
   596         by (simp add: d f00x)+
   597     qed
   598   then show "! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
   599 qed
   600 
   601 lemma foldseq_zerostart2:
   602   "! x. f 0 x = x \<Longrightarrow>  ! i. i < n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s n = s n"
   603 proof -
   604   assume a:"! i. i<n \<longrightarrow> s i = 0"
   605   assume x:"! x. f 0 x = x"
   606   from x have f00x: "! x. f 0 (f 0 x) = f 0 x" by blast
   607   have b: "!! i l. i < Suc l = (i <= l)" by arith
   608   have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
   609   show "foldseq f s n = s n"
   610   apply (case_tac "n=0")
   611   apply (simp)
   612   apply (insert a)
   613   apply (drule d)
   614   apply (auto)
   615   apply (simp add: b)
   616   apply (insert f00x)
   617   apply (drule foldseq_zerostart)
   618   by (simp add: x)+
   619 qed
   620 
   621 lemma foldseq_almostzero:
   622   assumes f0x:"! x. f 0 x = x" and fx0: "! x. f x 0 = x" and s0:"! i. i \<noteq> j \<longrightarrow> s i = 0"
   623   shows "foldseq f s n = (if (j <= n) then (s j) else 0)" (is ?concl)
   624 proof -
   625   from s0 have a: "! i. i < j \<longrightarrow> s i = 0" by simp
   626   from s0 have b: "! i. j < i \<longrightarrow> s i = 0" by simp
   627   show ?concl
   628     apply auto
   629     apply (subst foldseq_zerotail2[of f, OF fx0, of j, OF b, of n, THEN sym])
   630     apply simp
   631     apply (subst foldseq_zerostart2)
   632     apply (simp add: f0x a)+
   633     apply (subst foldseq_zero)
   634     by (simp add: s0 f0x)+
   635 qed
   636 
   637 lemma foldseq_distr_unary:
   638   assumes "!! a b. g (f a b) = f (g a) (g b)"
   639   shows "g(foldseq f s n) = foldseq f (% x. g(s x)) n" (is ?concl)
   640 proof -
   641   have "! s. g(foldseq f s n) = foldseq f (% x. g(s x)) n"
   642     apply (induct_tac n)
   643     apply (simp)
   644     apply (simp)
   645     apply (auto)
   646     apply (drule_tac x="% k. s (Suc k)" in spec)
   647     by (simp add: prems)
   648   then show ?concl by simp
   649 qed
   650 
   651 constdefs
   652   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"
   653   "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)"
   654   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"
   655   "mult_matrix fmul fadd A B == mult_matrix_n (max (ncols A) (nrows B)) fmul fadd A B"
   656 
   657 lemma mult_matrix_n:
   658   assumes prems: "ncols A \<le>  n" (is ?An) "nrows B \<le> n" (is ?Bn) "fadd 0 0 = 0" "fmul 0 0 = 0"
   659   shows c:"mult_matrix fmul fadd A B = mult_matrix_n n fmul fadd A B" (is ?concl)
   660 proof -
   661   show ?concl using prems
   662     apply (simp add: mult_matrix_def mult_matrix_n_def)
   663     apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
   664     by (rule foldseq_zerotail, simp_all add: nrows_le ncols_le prems)
   665 qed
   666 
   667 lemma mult_matrix_nm:
   668   assumes prems: "ncols A <= n" "nrows B <= n" "ncols A <= m" "nrows B <= m" "fadd 0 0 = 0" "fmul 0 0 = 0"
   669   shows "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B"
   670 proof -
   671   from prems have "mult_matrix_n n fmul fadd A B = mult_matrix fmul fadd A B" by (simp add: mult_matrix_n)
   672   also from prems have "\<dots> = mult_matrix_n m fmul fadd A B" by (simp add: mult_matrix_n[THEN sym])
   673   finally show "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B" by simp
   674 qed
   675 
   676 constdefs
   677   r_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool"
   678   "r_distributive fmul fadd == ! a u v. fmul a (fadd u v) = fadd (fmul a u) (fmul a v)"
   679   l_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
   680   "l_distributive fmul fadd == ! a u v. fmul (fadd u v) a = fadd (fmul u a) (fmul v a)"
   681   distributive :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
   682   "distributive fmul fadd == l_distributive fmul fadd & r_distributive fmul fadd"
   683 
   684 lemma max1: "!! a x y. (a::nat) <= x \<Longrightarrow> a <= max x y" by (arith)
   685 lemma max2: "!! b x y. (b::nat) <= y \<Longrightarrow> b <= max x y" by (arith)
   686 
   687 lemma r_distributive_matrix:
   688  assumes prems:
   689   "r_distributive fmul fadd"
   690   "associative fadd"
   691   "commutative fadd"
   692   "fadd 0 0 = 0"
   693   "! a. fmul a 0 = 0"
   694   "! a. fmul 0 a = 0"
   695  shows "r_distributive (mult_matrix fmul fadd) (combine_matrix fadd)" (is ?concl)
   696 proof -
   697   from prems show ?concl
   698     apply (simp add: r_distributive_def mult_matrix_def, auto)
   699     proof -
   700       fix a::"'a matrix"
   701       fix u::"'b matrix"
   702       fix v::"'b matrix"
   703       let ?mx = "max (ncols a) (max (nrows u) (nrows v))"
   704       from prems show "mult_matrix_n (max (ncols a) (nrows (combine_matrix fadd u v))) fmul fadd a (combine_matrix fadd u v) =
   705         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)"
   706         apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
   707         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   708         apply (subst mult_matrix_nm[of _ _ v ?mx fadd fmul])
   709         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   710         apply (subst mult_matrix_nm[of _ _ u ?mx fadd fmul])
   711         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   712         apply (simp add: mult_matrix_n_def r_distributive_def foldseq_distr[of fadd])
   713         apply (simp add: combine_matrix_def combine_infmatrix_def)
   714         apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
   715         apply (simplesubst RepAbs_matrix)
   716         apply (simp, auto)
   717         apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
   718         apply (rule exI[of _ "ncols v"], simp add: ncols_le foldseq_zero)
   719         apply (subst RepAbs_matrix)
   720         apply (simp, auto)
   721         apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
   722         apply (rule exI[of _ "ncols u"], simp add: ncols_le foldseq_zero)
   723         done
   724     qed
   725 qed
   726 
   727 lemma l_distributive_matrix:
   728  assumes prems:
   729   "l_distributive fmul fadd"
   730   "associative fadd"
   731   "commutative fadd"
   732   "fadd 0 0 = 0"
   733   "! a. fmul a 0 = 0"
   734   "! a. fmul 0 a = 0"
   735  shows "l_distributive (mult_matrix fmul fadd) (combine_matrix fadd)" (is ?concl)
   736 proof -
   737   from prems show ?concl
   738     apply (simp add: l_distributive_def mult_matrix_def, auto)
   739     proof -
   740       fix a::"'b matrix"
   741       fix u::"'a matrix"
   742       fix v::"'a matrix"
   743       let ?mx = "max (nrows a) (max (ncols u) (ncols v))"
   744       from prems show "mult_matrix_n (max (ncols (combine_matrix fadd u v)) (nrows a)) fmul fadd (combine_matrix fadd u v) a =
   745                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)"
   746         apply (subst mult_matrix_nm[of v _ _ ?mx fadd fmul])
   747         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   748         apply (subst mult_matrix_nm[of u _ _ ?mx fadd fmul])
   749         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   750         apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
   751         apply (simp add: max1 max2 combine_nrows combine_ncols)+
   752         apply (simp add: mult_matrix_n_def l_distributive_def foldseq_distr[of fadd])
   753         apply (simp add: combine_matrix_def combine_infmatrix_def)
   754         apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
   755         apply (simplesubst RepAbs_matrix)
   756         apply (simp, auto)
   757         apply (rule exI[of _ "nrows v"], simp add: nrows_le foldseq_zero)
   758         apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
   759         apply (subst RepAbs_matrix)
   760         apply (simp, auto)
   761         apply (rule exI[of _ "nrows u"], simp add: nrows_le foldseq_zero)
   762         apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
   763         done
   764     qed
   765 qed
   766 
   767 instantiation matrix :: (zero) zero
   768 begin
   769 
   770 definition zero_matrix_def [code del]: "0 = Abs_matrix (\<lambda>j i. 0)"
   771 
   772 instance ..
   773 
   774 end
   775 
   776 lemma Rep_zero_matrix_def[simp]: "Rep_matrix 0 j i = 0"
   777   apply (simp add: zero_matrix_def)
   778   apply (subst RepAbs_matrix)
   779   by (auto)
   780 
   781 lemma zero_matrix_def_nrows[simp]: "nrows 0 = 0"
   782 proof -
   783   have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
   784   show "nrows 0 = 0" by (rule a, subst nrows_le, simp)
   785 qed
   786 
   787 lemma zero_matrix_def_ncols[simp]: "ncols 0 = 0"
   788 proof -
   789   have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
   790   show "ncols 0 = 0" by (rule a, subst ncols_le, simp)
   791 qed
   792 
   793 lemma combine_matrix_zero_l_neutral: "zero_l_neutral f \<Longrightarrow> zero_l_neutral (combine_matrix f)"
   794   by (simp add: zero_l_neutral_def combine_matrix_def combine_infmatrix_def)
   795 
   796 lemma combine_matrix_zero_r_neutral: "zero_r_neutral f \<Longrightarrow> zero_r_neutral (combine_matrix f)"
   797   by (simp add: zero_r_neutral_def combine_matrix_def combine_infmatrix_def)
   798 
   799 lemma mult_matrix_zero_closed: "\<lbrakk>fadd 0 0 = 0; zero_closed fmul\<rbrakk> \<Longrightarrow> zero_closed (mult_matrix fmul fadd)"
   800   apply (simp add: zero_closed_def mult_matrix_def mult_matrix_n_def)
   801   apply (auto)
   802   by (subst foldseq_zero, (simp add: zero_matrix_def)+)+
   803 
   804 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"
   805   apply (simp add: mult_matrix_n_def)
   806   apply (subst foldseq_zero)
   807   by (simp_all add: zero_matrix_def)
   808 
   809 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"
   810   apply (simp add: mult_matrix_n_def)
   811   apply (subst foldseq_zero)
   812   by (simp_all add: zero_matrix_def)
   813 
   814 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"
   815 by (simp add: mult_matrix_def)
   816 
   817 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"
   818 by (simp add: mult_matrix_def)
   819 
   820 lemma apply_matrix_zero[simp]: "f 0 = 0 \<Longrightarrow> apply_matrix f 0 = 0"
   821   apply (simp add: apply_matrix_def apply_infmatrix_def)
   822   by (simp add: zero_matrix_def)
   823 
   824 lemma combine_matrix_zero: "f 0 0 = 0 \<Longrightarrow> combine_matrix f 0 0 = 0"
   825   apply (simp add: combine_matrix_def combine_infmatrix_def)
   826   by (simp add: zero_matrix_def)
   827 
   828 lemma transpose_matrix_zero[simp]: "transpose_matrix 0 = 0"
   829 apply (simp add: transpose_matrix_def transpose_infmatrix_def zero_matrix_def RepAbs_matrix)
   830 apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
   831 apply (simp add: RepAbs_matrix)
   832 done
   833 
   834 lemma apply_zero_matrix_def[simp]: "apply_matrix (% x. 0) A = 0"
   835   apply (simp add: apply_matrix_def apply_infmatrix_def)
   836   by (simp add: zero_matrix_def)
   837 
   838 constdefs
   839   singleton_matrix :: "nat \<Rightarrow> nat \<Rightarrow> ('a::zero) \<Rightarrow> 'a matrix"
   840   "singleton_matrix j i a == Abs_matrix(% m n. if j = m & i = n then a else 0)"
   841   move_matrix :: "('a::zero) matrix \<Rightarrow> int \<Rightarrow> int \<Rightarrow> 'a matrix"
   842   "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)))"
   843   take_rows :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
   844   "take_rows A r == Abs_matrix(% j i. if (j < r) then (Rep_matrix A j i) else 0)"
   845   take_columns :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
   846   "take_columns A c == Abs_matrix(% j i. if (i < c) then (Rep_matrix A j i) else 0)"
   847 
   848 constdefs
   849   column_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
   850   "column_of_matrix A n == take_columns (move_matrix A 0 (- int n)) 1"
   851   row_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
   852   "row_of_matrix A m == take_rows (move_matrix A (- int m) 0) 1"
   853 
   854 lemma Rep_singleton_matrix[simp]: "Rep_matrix (singleton_matrix j i e) m n = (if j = m & i = n then e else 0)"
   855 apply (simp add: singleton_matrix_def)
   856 apply (auto)
   857 apply (subst RepAbs_matrix)
   858 apply (rule exI[of _ "Suc m"], simp)
   859 apply (rule exI[of _ "Suc n"], simp+)
   860 by (subst RepAbs_matrix, rule exI[of _ "Suc j"], simp, rule exI[of _ "Suc i"], simp+)+
   861 
   862 lemma apply_singleton_matrix[simp]: "f 0 = 0 \<Longrightarrow> apply_matrix f (singleton_matrix j i x) = (singleton_matrix j i (f x))"
   863 apply (subst Rep_matrix_inject[symmetric])
   864 apply (rule ext)+
   865 apply (simp)
   866 done
   867 
   868 lemma singleton_matrix_zero[simp]: "singleton_matrix j i 0 = 0"
   869   by (simp add: singleton_matrix_def zero_matrix_def)
   870 
   871 lemma nrows_singleton[simp]: "nrows(singleton_matrix j i e) = (if e = 0 then 0 else Suc j)"
   872 proof-
   873 have th: "\<not> (\<forall>m. m \<le> j)" "\<exists>n. \<not> n \<le> i" by arith+
   874 from th show ?thesis 
   875 apply (auto)
   876 apply (rule le_anti_sym)
   877 apply (subst nrows_le)
   878 apply (simp add: singleton_matrix_def, auto)
   879 apply (subst RepAbs_matrix)
   880 apply auto
   881 apply (simp add: Suc_le_eq)
   882 apply (rule not_leE)
   883 apply (subst nrows_le)
   884 by simp
   885 qed
   886 
   887 lemma ncols_singleton[simp]: "ncols(singleton_matrix j i e) = (if e = 0 then 0 else Suc i)"
   888 proof-
   889 have th: "\<not> (\<forall>m. m \<le> j)" "\<exists>n. \<not> n \<le> i" by arith+
   890 from th show ?thesis 
   891 apply (auto)
   892 apply (rule le_anti_sym)
   893 apply (subst ncols_le)
   894 apply (simp add: singleton_matrix_def, auto)
   895 apply (subst RepAbs_matrix)
   896 apply auto
   897 apply (simp add: Suc_le_eq)
   898 apply (rule not_leE)
   899 apply (subst ncols_le)
   900 by simp
   901 qed
   902 
   903 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)"
   904 apply (simp add: singleton_matrix_def combine_matrix_def combine_infmatrix_def)
   905 apply (subst RepAbs_matrix)
   906 apply (rule exI[of _ "Suc j"], simp)
   907 apply (rule exI[of _ "Suc i"], simp)
   908 apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
   909 apply (subst RepAbs_matrix)
   910 apply (rule exI[of _ "Suc j"], simp)
   911 apply (rule exI[of _ "Suc i"], simp)
   912 by simp
   913 
   914 lemma transpose_singleton[simp]: "transpose_matrix (singleton_matrix j i a) = singleton_matrix i j a"
   915 apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
   916 apply (simp)
   917 done
   918 
   919 lemma Rep_move_matrix[simp]:
   920   "Rep_matrix (move_matrix A y x) j i =
   921   (if (neg ((int j)-y)) | (neg ((int i)-x)) then 0 else Rep_matrix A (nat((int j)-y)) (nat((int i)-x)))"
   922 apply (simp add: move_matrix_def)
   923 apply (auto)
   924 by (subst RepAbs_matrix,
   925   rule exI[of _ "(nrows A)+(nat (abs y))"], auto, rule nrows, arith,
   926   rule exI[of _ "(ncols A)+(nat (abs x))"], auto, rule ncols, arith)+
   927 
   928 lemma move_matrix_0_0[simp]: "move_matrix A 0 0 = A"
   929 by (simp add: move_matrix_def)
   930 
   931 lemma move_matrix_ortho: "move_matrix A j i = move_matrix (move_matrix A j 0) 0 i"
   932 apply (subst Rep_matrix_inject[symmetric])
   933 apply (rule ext)+
   934 apply (simp)
   935 done
   936 
   937 lemma transpose_move_matrix[simp]:
   938   "transpose_matrix (move_matrix A x y) = move_matrix (transpose_matrix A) y x"
   939 apply (subst Rep_matrix_inject[symmetric], (rule ext)+)
   940 apply (simp)
   941 done
   942 
   943 lemma move_matrix_singleton[simp]: "move_matrix (singleton_matrix u v x) j i = 
   944   (if (j + int u < 0) | (i + int v < 0) then 0 else (singleton_matrix (nat (j + int u)) (nat (i + int v)) x))"
   945   apply (subst Rep_matrix_inject[symmetric])
   946   apply (rule ext)+
   947   apply (case_tac "j + int u < 0")
   948   apply (simp, arith)
   949   apply (case_tac "i + int v < 0")
   950   apply (simp add: neg_def, arith)
   951   apply (simp add: neg_def)
   952   apply arith
   953   done
   954 
   955 lemma Rep_take_columns[simp]:
   956   "Rep_matrix (take_columns A c) j i =
   957   (if i < c then (Rep_matrix A j i) else 0)"
   958 apply (simp add: take_columns_def)
   959 apply (simplesubst RepAbs_matrix)
   960 apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
   961 apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
   962 done
   963 
   964 lemma Rep_take_rows[simp]:
   965   "Rep_matrix (take_rows A r) j i =
   966   (if j < r then (Rep_matrix A j i) else 0)"
   967 apply (simp add: take_rows_def)
   968 apply (simplesubst RepAbs_matrix)
   969 apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
   970 apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
   971 done
   972 
   973 lemma Rep_column_of_matrix[simp]:
   974   "Rep_matrix (column_of_matrix A c) j i = (if i = 0 then (Rep_matrix A j c) else 0)"
   975   by (simp add: column_of_matrix_def)
   976 
   977 lemma Rep_row_of_matrix[simp]:
   978   "Rep_matrix (row_of_matrix A r) j i = (if j = 0 then (Rep_matrix A r i) else 0)"
   979   by (simp add: row_of_matrix_def)
   980 
   981 lemma column_of_matrix: "ncols A <= n \<Longrightarrow> column_of_matrix A n = 0"
   982 apply (subst Rep_matrix_inject[THEN sym])
   983 apply (rule ext)+
   984 by (simp add: ncols)
   985 
   986 lemma row_of_matrix: "nrows A <= n \<Longrightarrow> row_of_matrix A n = 0"
   987 apply (subst Rep_matrix_inject[THEN sym])
   988 apply (rule ext)+
   989 by (simp add: nrows)
   990 
   991 lemma mult_matrix_singleton_right[simp]:
   992   assumes prems:
   993   "! x. fmul x 0 = 0"
   994   "! x. fmul 0 x = 0"
   995   "! x. fadd 0 x = x"
   996   "! x. fadd x 0 = x"
   997   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))"
   998   apply (simp add: mult_matrix_def)
   999   apply (subst mult_matrix_nm[of _ _ _ "max (ncols A) (Suc j)"])
  1000   apply (auto)
  1001   apply (simp add: prems)+
  1002   apply (simp add: mult_matrix_n_def apply_matrix_def apply_infmatrix_def)
  1003   apply (rule comb[of "Abs_matrix" "Abs_matrix"], auto, (rule ext)+)
  1004   apply (subst foldseq_almostzero[of _ j])
  1005   apply (simp add: prems)+
  1006   apply (auto)
  1007   apply (metis comm_monoid_add.mult_1 le_anti_sym le_diff_eq not_neg_nat zero_le_imp_of_nat zle_int)
  1008   done
  1009 
  1010 lemma mult_matrix_ext:
  1011   assumes
  1012   eprem:
  1013   "? e. (! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)"
  1014   and fprems:
  1015   "! a. fmul 0 a = 0"
  1016   "! a. fmul a 0 = 0"
  1017   "! a. fadd a 0 = a"
  1018   "! a. fadd 0 a = a"
  1019   and contraprems:
  1020   "mult_matrix fmul fadd A = mult_matrix fmul fadd B"
  1021   shows
  1022   "A = B"
  1023 proof(rule contrapos_np[of "False"], simp)
  1024   assume a: "A \<noteq> B"
  1025   have b: "!! f g. (! x y. f x y = g x y) \<Longrightarrow> f = g" by ((rule ext)+, auto)
  1026   have "? j i. (Rep_matrix A j i) \<noteq> (Rep_matrix B j i)"
  1027     apply (rule contrapos_np[of "False"], simp+)
  1028     apply (insert b[of "Rep_matrix A" "Rep_matrix B"], simp)
  1029     by (simp add: Rep_matrix_inject a)
  1030   then obtain J I where c:"(Rep_matrix A J I) \<noteq> (Rep_matrix B J I)" by blast
  1031   from eprem obtain e where eprops:"(! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)" by blast
  1032   let ?S = "singleton_matrix I 0 e"
  1033   let ?comp = "mult_matrix fmul fadd"
  1034   have d: "!!x f g. f = g \<Longrightarrow> f x = g x" by blast
  1035   have e: "(% x. fmul x e) 0 = 0" by (simp add: prems)
  1036   have "~(?comp A ?S = ?comp B ?S)"
  1037     apply (rule notI)
  1038     apply (simp add: fprems eprops)
  1039     apply (simp add: Rep_matrix_inject[THEN sym])
  1040     apply (drule d[of _ _ "J"], drule d[of _ _ "0"])
  1041     by (simp add: e c eprops)
  1042   with contraprems show "False" by simp
  1043 qed
  1044 
  1045 constdefs
  1046   foldmatrix :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"
  1047   "foldmatrix f g A m n == foldseq_transposed g (% j. foldseq f (A j) n) m"
  1048   foldmatrix_transposed :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"
  1049   "foldmatrix_transposed f g A m n == foldseq g (% j. foldseq_transposed f (A j) n) m"
  1050 
  1051 lemma foldmatrix_transpose:
  1052   assumes
  1053   "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
  1054   shows
  1055   "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m" (is ?concl)
  1056 proof -
  1057   have forall:"!! P x. (! x. P x) \<Longrightarrow> P x" by auto
  1058   have tworows:"! A. foldmatrix f g A 1 n = foldmatrix_transposed g f (transpose_infmatrix A) n 1"
  1059     apply (induct n)
  1060     apply (simp add: foldmatrix_def foldmatrix_transposed_def prems)+
  1061     apply (auto)
  1062     by (drule_tac x="(% j i. A j (Suc i))" in forall, simp)
  1063   show "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m"
  1064     apply (simp add: foldmatrix_def foldmatrix_transposed_def)
  1065     apply (induct m, simp)
  1066     apply (simp)
  1067     apply (insert tworows)
  1068     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)
  1069     by (simp add: foldmatrix_def foldmatrix_transposed_def)
  1070 qed
  1071 
  1072 lemma foldseq_foldseq:
  1073 assumes
  1074   "associative f"
  1075   "associative g"
  1076   "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
  1077 shows
  1078   "foldseq g (% j. foldseq f (A j) n) m = foldseq f (% j. foldseq g ((transpose_infmatrix A) j) m) n"
  1079   apply (insert foldmatrix_transpose[of g f A m n])
  1080   by (simp add: foldmatrix_def foldmatrix_transposed_def foldseq_assoc[THEN sym] prems)
  1081 
  1082 lemma mult_n_nrows:
  1083 assumes
  1084 "! a. fmul 0 a = 0"
  1085 "! a. fmul a 0 = 0"
  1086 "fadd 0 0 = 0"
  1087 shows "nrows (mult_matrix_n n fmul fadd A B) \<le> nrows A"
  1088 apply (subst nrows_le)
  1089 apply (simp add: mult_matrix_n_def)
  1090 apply (subst RepAbs_matrix)
  1091 apply (rule_tac x="nrows A" in exI)
  1092 apply (simp add: nrows prems foldseq_zero)
  1093 apply (rule_tac x="ncols B" in exI)
  1094 apply (simp add: ncols prems foldseq_zero)
  1095 by (simp add: nrows prems foldseq_zero)
  1096 
  1097 lemma mult_n_ncols:
  1098 assumes
  1099 "! a. fmul 0 a = 0"
  1100 "! a. fmul a 0 = 0"
  1101 "fadd 0 0 = 0"
  1102 shows "ncols (mult_matrix_n n fmul fadd A B) \<le> ncols B"
  1103 apply (subst ncols_le)
  1104 apply (simp add: mult_matrix_n_def)
  1105 apply (subst RepAbs_matrix)
  1106 apply (rule_tac x="nrows A" in exI)
  1107 apply (simp add: nrows prems foldseq_zero)
  1108 apply (rule_tac x="ncols B" in exI)
  1109 apply (simp add: ncols prems foldseq_zero)
  1110 by (simp add: ncols prems foldseq_zero)
  1111 
  1112 lemma mult_nrows:
  1113 assumes
  1114 "! a. fmul 0 a = 0"
  1115 "! a. fmul a 0 = 0"
  1116 "fadd 0 0 = 0"
  1117 shows "nrows (mult_matrix fmul fadd A B) \<le> nrows A"
  1118 by (simp add: mult_matrix_def mult_n_nrows prems)
  1119 
  1120 lemma mult_ncols:
  1121 assumes
  1122 "! a. fmul 0 a = 0"
  1123 "! a. fmul a 0 = 0"
  1124 "fadd 0 0 = 0"
  1125 shows "ncols (mult_matrix fmul fadd A B) \<le> ncols B"
  1126 by (simp add: mult_matrix_def mult_n_ncols prems)
  1127 
  1128 lemma nrows_move_matrix_le: "nrows (move_matrix A j i) <= nat((int (nrows A)) + j)"
  1129   apply (auto simp add: nrows_le)
  1130   apply (rule nrows)
  1131   apply (arith)
  1132   done
  1133 
  1134 lemma ncols_move_matrix_le: "ncols (move_matrix A j i) <= nat((int (ncols A)) + i)"
  1135   apply (auto simp add: ncols_le)
  1136   apply (rule ncols)
  1137   apply (arith)
  1138   done
  1139 
  1140 lemma mult_matrix_assoc:
  1141   assumes prems:
  1142   "! a. fmul1 0 a = 0"
  1143   "! a. fmul1 a 0 = 0"
  1144   "! a. fmul2 0 a = 0"
  1145   "! a. fmul2 a 0 = 0"
  1146   "fadd1 0 0 = 0"
  1147   "fadd2 0 0 = 0"
  1148   "! a b c d. fadd2 (fadd1 a b) (fadd1 c d) = fadd1 (fadd2 a c) (fadd2 b d)"
  1149   "associative fadd1"
  1150   "associative fadd2"
  1151   "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
  1152   "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
  1153   "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
  1154   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)
  1155 proof -
  1156   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
  1157   have fmul2fadd1fold: "!! x s n. fmul2 (foldseq fadd1 s n)  x = foldseq fadd1 (% k. fmul2 (s k) x) n"
  1158     by (rule_tac g1 = "% y. fmul2 y x" in ssubst [OF foldseq_distr_unary], simp_all!)
  1159   have fmul1fadd2fold: "!! x s n. fmul1 x (foldseq fadd2 s n) = foldseq fadd2 (% k. fmul1 x (s k)) n"
  1160       by (rule_tac g1 = "% y. fmul1 x y" in ssubst [OF foldseq_distr_unary], simp_all!)
  1161   let ?N = "max (ncols A) (max (ncols B) (max (nrows B) (nrows C)))"
  1162   show ?concl
  1163     apply (simp add: Rep_matrix_inject[THEN sym])
  1164     apply (rule ext)+
  1165     apply (simp add: mult_matrix_def)
  1166     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)"])
  1167     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
  1168     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)"])     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
  1169     apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
  1170     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
  1171     apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
  1172     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
  1173     apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
  1174     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
  1175     apply (simplesubst mult_matrix_nm[of _ _ _ "?N"])
  1176     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
  1177     apply (simp add: mult_matrix_n_def)
  1178     apply (rule comb_left)
  1179     apply ((rule ext)+, simp)
  1180     apply (simplesubst RepAbs_matrix)
  1181     apply (rule exI[of _ "nrows B"])
  1182     apply (simp add: nrows prems foldseq_zero)
  1183     apply (rule exI[of _ "ncols C"])
  1184     apply (simp add: prems ncols foldseq_zero)
  1185     apply (subst RepAbs_matrix)
  1186     apply (rule exI[of _ "nrows A"])
  1187     apply (simp add: nrows prems foldseq_zero)
  1188     apply (rule exI[of _ "ncols B"])
  1189     apply (simp add: prems ncols foldseq_zero)
  1190     apply (simp add: fmul2fadd1fold fmul1fadd2fold prems)
  1191     apply (subst foldseq_foldseq)
  1192     apply (simp add: prems)+
  1193     by (simp add: transpose_infmatrix)
  1194 qed
  1195 
  1196 lemma
  1197   assumes prems:
  1198   "! a. fmul1 0 a = 0"
  1199   "! a. fmul1 a 0 = 0"
  1200   "! a. fmul2 0 a = 0"
  1201   "! a. fmul2 a 0 = 0"
  1202   "fadd1 0 0 = 0"
  1203   "fadd2 0 0 = 0"
  1204   "! a b c d. fadd2 (fadd1 a b) (fadd1 c d) = fadd1 (fadd2 a c) (fadd2 b d)"
  1205   "associative fadd1"
  1206   "associative fadd2"
  1207   "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
  1208   "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
  1209   "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
  1210   shows
  1211   "(mult_matrix fmul1 fadd1 A) o (mult_matrix fmul2 fadd2 B) = mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B)"
  1212 apply (rule ext)+
  1213 apply (simp add: comp_def )
  1214 by (simp add: mult_matrix_assoc prems)
  1215 
  1216 lemma mult_matrix_assoc_simple:
  1217   assumes prems:
  1218   "! a. fmul 0 a = 0"
  1219   "! a. fmul a 0 = 0"
  1220   "fadd 0 0 = 0"
  1221   "associative fadd"
  1222   "commutative fadd"
  1223   "associative fmul"
  1224   "distributive fmul fadd"
  1225   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)
  1226 proof -
  1227   have "!! a b c d. fadd (fadd a b) (fadd c d) = fadd (fadd a c) (fadd b d)"
  1228     by (simp! add: associative_def commutative_def)
  1229   then show ?concl
  1230     apply (subst mult_matrix_assoc)
  1231     apply (simp_all!)
  1232     by (simp add: associative_def distributive_def l_distributive_def r_distributive_def)+
  1233 qed
  1234 
  1235 lemma transpose_apply_matrix: "f 0 = 0 \<Longrightarrow> transpose_matrix (apply_matrix f A) = apply_matrix f (transpose_matrix A)"
  1236 apply (simp add: Rep_matrix_inject[THEN sym])
  1237 apply (rule ext)+
  1238 by simp
  1239 
  1240 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)"
  1241 apply (simp add: Rep_matrix_inject[THEN sym])
  1242 apply (rule ext)+
  1243 by simp
  1244 
  1245 lemma Rep_mult_matrix:
  1246   assumes
  1247   "! a. fmul 0 a = 0"
  1248   "! a. fmul a 0 = 0"
  1249   "fadd 0 0 = 0"
  1250   shows
  1251   "Rep_matrix(mult_matrix fmul fadd A B) j i =
  1252   foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) (max (ncols A) (nrows B))"
  1253 apply (simp add: mult_matrix_def mult_matrix_n_def)
  1254 apply (subst RepAbs_matrix)
  1255 apply (rule exI[of _ "nrows A"], simp! add: nrows foldseq_zero)
  1256 apply (rule exI[of _ "ncols B"], simp! add: ncols foldseq_zero)
  1257 by simp
  1258 
  1259 lemma transpose_mult_matrix:
  1260   assumes
  1261   "! a. fmul 0 a = 0"
  1262   "! a. fmul a 0 = 0"
  1263   "fadd 0 0 = 0"
  1264   "! x y. fmul y x = fmul x y"
  1265   shows
  1266   "transpose_matrix (mult_matrix fmul fadd A B) = mult_matrix fmul fadd (transpose_matrix B) (transpose_matrix A)"
  1267   apply (simp add: Rep_matrix_inject[THEN sym])
  1268   apply (rule ext)+
  1269   by (simp! add: Rep_mult_matrix max_ac)
  1270 
  1271 lemma column_transpose_matrix: "column_of_matrix (transpose_matrix A) n = transpose_matrix (row_of_matrix A n)"
  1272 apply (simp add:  Rep_matrix_inject[THEN sym])
  1273 apply (rule ext)+
  1274 by simp
  1275 
  1276 lemma take_columns_transpose_matrix: "take_columns (transpose_matrix A) n = transpose_matrix (take_rows A n)"
  1277 apply (simp add: Rep_matrix_inject[THEN sym])
  1278 apply (rule ext)+
  1279 by simp
  1280 
  1281 instantiation matrix :: ("{zero, ord}") ord
  1282 begin
  1283 
  1284 definition
  1285   le_matrix_def: "A \<le> B \<longleftrightarrow> (\<forall>j i. Rep_matrix A j i \<le> Rep_matrix B j i)"
  1286 
  1287 definition
  1288   less_def: "A < (B\<Colon>'a matrix) \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
  1289 
  1290 instance ..
  1291 
  1292 end
  1293 
  1294 instance matrix :: ("{zero, order}") order
  1295 apply intro_classes
  1296 apply (simp_all add: le_matrix_def less_def)
  1297 apply (auto)
  1298 apply (drule_tac x=j in spec, drule_tac x=j in spec)
  1299 apply (drule_tac x=i in spec, drule_tac x=i in spec)
  1300 apply (simp)
  1301 apply (simp add: Rep_matrix_inject[THEN sym])
  1302 apply (rule ext)+
  1303 apply (drule_tac x=xa in spec, drule_tac x=xa in spec)
  1304 apply (drule_tac x=xb in spec, drule_tac x=xb in spec)
  1305 apply simp
  1306 done
  1307 
  1308 lemma le_apply_matrix:
  1309   assumes
  1310   "f 0 = 0"
  1311   "! x y. x <= y \<longrightarrow> f x <= f y"
  1312   "(a::('a::{ord, zero}) matrix) <= b"
  1313   shows
  1314   "apply_matrix f a <= apply_matrix f b"
  1315   by (simp! add: le_matrix_def)
  1316 
  1317 lemma le_combine_matrix:
  1318   assumes
  1319   "f 0 0 = 0"
  1320   "! a b c d. a <= b & c <= d \<longrightarrow> f a c <= f b d"
  1321   "A <= B"
  1322   "C <= D"
  1323   shows
  1324   "combine_matrix f A C <= combine_matrix f B D"
  1325 by (simp! add: le_matrix_def)
  1326 
  1327 lemma le_left_combine_matrix:
  1328   assumes
  1329   "f 0 0 = 0"
  1330   "! a b c. a <= b \<longrightarrow> f c a <= f c b"
  1331   "A <= B"
  1332   shows
  1333   "combine_matrix f C A <= combine_matrix f C B"
  1334   by (simp! add: le_matrix_def)
  1335 
  1336 lemma le_right_combine_matrix:
  1337   assumes
  1338   "f 0 0 = 0"
  1339   "! a b c. a <= b \<longrightarrow> f a c <= f b c"
  1340   "A <= B"
  1341   shows
  1342   "combine_matrix f A C <= combine_matrix f B C"
  1343   by (simp! add: le_matrix_def)
  1344 
  1345 lemma le_transpose_matrix: "(A <= B) = (transpose_matrix A <= transpose_matrix B)"
  1346   by (simp add: le_matrix_def, auto)
  1347 
  1348 lemma le_foldseq:
  1349   assumes
  1350   "! a b c d . a <= b & c <= d \<longrightarrow> f a c <= f b d"
  1351   "! i. i <= n \<longrightarrow> s i <= t i"
  1352   shows
  1353   "foldseq f s n <= foldseq f t n"
  1354 proof -
  1355   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!)
  1356   then show "foldseq f s n <= foldseq f t n" by (simp!)
  1357 qed
  1358 
  1359 lemma le_left_mult:
  1360   assumes
  1361   "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
  1362   "! c a b.   0 <= c & a <= b \<longrightarrow> fmul c a <= fmul c b"
  1363   "! a. fmul 0 a = 0"
  1364   "! a. fmul a 0 = 0"
  1365   "fadd 0 0 = 0"
  1366   "0 <= C"
  1367   "A <= B"
  1368   shows
  1369   "mult_matrix fmul fadd C A <= mult_matrix fmul fadd C B"
  1370   apply (simp! add: le_matrix_def Rep_mult_matrix)
  1371   apply (auto)
  1372   apply (simplesubst foldseq_zerotail[of _ _ _ "max (ncols C) (max (nrows A) (nrows B))"], simp_all add: nrows ncols max1 max2)+
  1373   apply (rule le_foldseq)
  1374   by (auto)
  1375 
  1376 lemma le_right_mult:
  1377   assumes
  1378   "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
  1379   "! c a b. 0 <= c & a <= b \<longrightarrow> fmul a c <= fmul b c"
  1380   "! a. fmul 0 a = 0"
  1381   "! a. fmul a 0 = 0"
  1382   "fadd 0 0 = 0"
  1383   "0 <= C"
  1384   "A <= B"
  1385   shows
  1386   "mult_matrix fmul fadd A C <= mult_matrix fmul fadd B C"
  1387   apply (simp! add: le_matrix_def Rep_mult_matrix)
  1388   apply (auto)
  1389   apply (simplesubst foldseq_zerotail[of _ _ _ "max (nrows C) (max (ncols A) (ncols B))"], simp_all add: nrows ncols max1 max2)+
  1390   apply (rule le_foldseq)
  1391   by (auto)
  1392 
  1393 lemma spec2: "! j i. P j i \<Longrightarrow> P j i" by blast
  1394 lemma neg_imp: "(\<not> Q \<longrightarrow> \<not> P) \<Longrightarrow> P \<longrightarrow> Q" by blast
  1395 
  1396 lemma singleton_matrix_le[simp]: "(singleton_matrix j i a <= singleton_matrix j i b) = (a <= (b::_::order))"
  1397   by (auto simp add: le_matrix_def)
  1398 
  1399 lemma singleton_le_zero[simp]: "(singleton_matrix j i x <= 0) = (x <= (0::'a::{order,zero}))"
  1400   apply (auto)
  1401   apply (simp add: le_matrix_def)
  1402   apply (drule_tac j=j and i=i in spec2)
  1403   apply (simp)
  1404   apply (simp add: le_matrix_def)
  1405   done
  1406 
  1407 lemma singleton_ge_zero[simp]: "(0 <= singleton_matrix j i x) = ((0::'a::{order,zero}) <= x)"
  1408   apply (auto)
  1409   apply (simp add: le_matrix_def)
  1410   apply (drule_tac j=j and i=i in spec2)
  1411   apply (simp)
  1412   apply (simp add: le_matrix_def)
  1413   done
  1414 
  1415 lemma move_matrix_le_zero[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (move_matrix A j i <= 0) = (A <= (0::('a::{order,zero}) matrix))"
  1416   apply (auto simp add: le_matrix_def neg_def)
  1417   apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
  1418   apply (auto)
  1419   done
  1420 
  1421 lemma move_matrix_zero_le[simp]: "0 <= j \<Longrightarrow> 0 <= i \<Longrightarrow> (0 <= move_matrix A j i) = ((0::('a::{order,zero}) matrix) <= A)"
  1422   apply (auto simp add: le_matrix_def neg_def)
  1423   apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
  1424   apply (auto)
  1425   done
  1426 
  1427 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))"
  1428   apply (auto simp add: le_matrix_def neg_def)
  1429   apply (drule_tac j="ja+(nat j)" and i="ia+(nat i)" in spec2)
  1430   apply (auto)
  1431   done  
  1432 
  1433 instantiation matrix :: ("{lattice, zero}") lattice
  1434 begin
  1435 
  1436 definition [code del]: "inf = combine_matrix inf"
  1437 
  1438 definition [code del]: "sup = combine_matrix sup"
  1439 
  1440 instance
  1441   by default (auto simp add: inf_le1 inf_le2 le_infI le_matrix_def inf_matrix_def sup_matrix_def)
  1442 
  1443 end
  1444 
  1445 instantiation matrix :: ("{plus, zero}") plus
  1446 begin
  1447 
  1448 definition
  1449   plus_matrix_def [code del]: "A + B = combine_matrix (op +) A B"
  1450 
  1451 instance ..
  1452 
  1453 end
  1454 
  1455 instantiation matrix :: ("{uminus, zero}") uminus
  1456 begin
  1457 
  1458 definition
  1459   minus_matrix_def [code del]: "- A = apply_matrix uminus A"
  1460 
  1461 instance ..
  1462 
  1463 end
  1464 
  1465 instantiation matrix :: ("{minus, zero}") minus
  1466 begin
  1467 
  1468 definition
  1469   diff_matrix_def [code del]: "A - B = combine_matrix (op -) A B"
  1470 
  1471 instance ..
  1472 
  1473 end
  1474 
  1475 instantiation matrix :: ("{plus, times, zero}") times
  1476 begin
  1477 
  1478 definition
  1479   times_matrix_def [code del]: "A * B = mult_matrix (op *) (op +) A B"
  1480 
  1481 instance ..
  1482 
  1483 end
  1484 
  1485 instantiation matrix :: ("{lattice, uminus, zero}") abs
  1486 begin
  1487 
  1488 definition
  1489   abs_matrix_def [code del]: "abs (A \<Colon> 'a matrix) = sup A (- A)"
  1490 
  1491 instance ..
  1492 
  1493 end
  1494 
  1495 instance matrix :: (monoid_add) monoid_add
  1496 proof
  1497   fix A B C :: "'a matrix"
  1498   show "A + B + C = A + (B + C)"    
  1499     apply (simp add: plus_matrix_def)
  1500     apply (rule combine_matrix_assoc[simplified associative_def, THEN spec, THEN spec, THEN spec])
  1501     apply (simp_all add: add_assoc)
  1502     done
  1503   show "0 + A = A"
  1504     apply (simp add: plus_matrix_def)
  1505     apply (rule combine_matrix_zero_l_neutral[simplified zero_l_neutral_def, THEN spec])
  1506     apply (simp)
  1507     done
  1508   show "A + 0 = A"
  1509     apply (simp add: plus_matrix_def)
  1510     apply (rule combine_matrix_zero_r_neutral [simplified zero_r_neutral_def, THEN spec])
  1511     apply (simp)
  1512     done
  1513 qed
  1514 
  1515 instance matrix :: (comm_monoid_add) comm_monoid_add
  1516 proof
  1517   fix A B :: "'a matrix"
  1518   show "A + B = B + A"
  1519     apply (simp add: plus_matrix_def)
  1520     apply (rule combine_matrix_commute[simplified commutative_def, THEN spec, THEN spec])
  1521     apply (simp_all add: add_commute)
  1522     done
  1523   show "0 + A = A"
  1524     apply (simp add: plus_matrix_def)
  1525     apply (rule combine_matrix_zero_l_neutral[simplified zero_l_neutral_def, THEN spec])
  1526     apply (simp)
  1527     done
  1528 qed
  1529 
  1530 instance matrix :: (group_add) group_add
  1531 proof
  1532   fix A B :: "'a matrix"
  1533   show "- A + A = 0" 
  1534     by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
  1535   show "A - B = A + - B"
  1536     by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject [symmetric] ext diff_minus)
  1537 qed
  1538 
  1539 instance matrix :: (ab_group_add) ab_group_add
  1540 proof
  1541   fix A B :: "'a matrix"
  1542   show "- A + A = 0" 
  1543     by (simp add: plus_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
  1544   show "A - B = A + - B" 
  1545     by (simp add: plus_matrix_def diff_matrix_def minus_matrix_def Rep_matrix_inject[symmetric] ext)
  1546 qed
  1547 
  1548 instance matrix :: (pordered_ab_group_add) pordered_ab_group_add
  1549 proof
  1550   fix A B C :: "'a matrix"
  1551   assume "A <= B"
  1552   then show "C + A <= C + B"
  1553     apply (simp add: plus_matrix_def)
  1554     apply (rule le_left_combine_matrix)
  1555     apply (simp_all)
  1556     done
  1557 qed
  1558   
  1559 instance matrix :: (lordered_ab_group_add) lordered_ab_group_add_meet ..
  1560 instance matrix :: (lordered_ab_group_add) lordered_ab_group_add_join ..
  1561 
  1562 instance matrix :: (ring) ring
  1563 proof
  1564   fix A B C :: "'a matrix"
  1565   show "A * B * C = A * (B * C)"
  1566     apply (simp add: times_matrix_def)
  1567     apply (rule mult_matrix_assoc)
  1568     apply (simp_all add: associative_def algebra_simps)
  1569     done
  1570   show "(A + B) * C = A * C + B * C"
  1571     apply (simp add: times_matrix_def plus_matrix_def)
  1572     apply (rule l_distributive_matrix[simplified l_distributive_def, THEN spec, THEN spec, THEN spec])
  1573     apply (simp_all add: associative_def commutative_def algebra_simps)
  1574     done
  1575   show "A * (B + C) = A * B + A * C"
  1576     apply (simp add: times_matrix_def plus_matrix_def)
  1577     apply (rule r_distributive_matrix[simplified r_distributive_def, THEN spec, THEN spec, THEN spec])
  1578     apply (simp_all add: associative_def commutative_def algebra_simps)
  1579     done
  1580 qed  
  1581 
  1582 instance matrix :: (pordered_ring) pordered_ring
  1583 proof
  1584   fix A B C :: "'a matrix"
  1585   assume a: "A \<le> B"
  1586   assume b: "0 \<le> C"
  1587   from a b show "C * A \<le> C * B"
  1588     apply (simp add: times_matrix_def)
  1589     apply (rule le_left_mult)
  1590     apply (simp_all add: add_mono mult_left_mono)
  1591     done
  1592   from a b show "A * C \<le> B * C"
  1593     apply (simp add: times_matrix_def)
  1594     apply (rule le_right_mult)
  1595     apply (simp_all add: add_mono mult_right_mono)
  1596     done
  1597 qed
  1598 
  1599 instance matrix :: (lordered_ring) lordered_ring
  1600 proof
  1601   fix A B C :: "('a :: lordered_ring) matrix"
  1602   show "abs A = sup A (-A)" 
  1603     by (simp add: abs_matrix_def)
  1604 qed
  1605 
  1606 lemma Rep_matrix_add[simp]:
  1607   "Rep_matrix ((a::('a::monoid_add)matrix)+b) j i  = (Rep_matrix a j i) + (Rep_matrix b j i)"
  1608   by (simp add: plus_matrix_def)
  1609 
  1610 lemma Rep_matrix_mult: "Rep_matrix ((a::('a::ring) matrix) * b) j i = 
  1611   foldseq (op +) (% k.  (Rep_matrix a j k) * (Rep_matrix b k i)) (max (ncols a) (nrows b))"
  1612 apply (simp add: times_matrix_def)
  1613 apply (simp add: Rep_mult_matrix)
  1614 done
  1615 
  1616 lemma apply_matrix_add: "! x y. f (x+y) = (f x) + (f y) \<Longrightarrow> f 0 = (0::'a)
  1617   \<Longrightarrow> apply_matrix f ((a::('a::monoid_add) matrix) + b) = (apply_matrix f a) + (apply_matrix f b)"
  1618 apply (subst Rep_matrix_inject[symmetric])
  1619 apply (rule ext)+
  1620 apply (simp)
  1621 done
  1622 
  1623 lemma singleton_matrix_add: "singleton_matrix j i ((a::_::monoid_add)+b) = (singleton_matrix j i a) + (singleton_matrix j i b)"
  1624 apply (subst Rep_matrix_inject[symmetric])
  1625 apply (rule ext)+
  1626 apply (simp)
  1627 done
  1628 
  1629 lemma nrows_mult: "nrows ((A::('a::ring) matrix) * B) <= nrows A"
  1630 by (simp add: times_matrix_def mult_nrows)
  1631 
  1632 lemma ncols_mult: "ncols ((A::('a::ring) matrix) * B) <= ncols B"
  1633 by (simp add: times_matrix_def mult_ncols)
  1634 
  1635 definition
  1636   one_matrix :: "nat \<Rightarrow> ('a::{zero,one}) matrix" where
  1637   "one_matrix n = Abs_matrix (% j i. if j = i & j < n then 1 else 0)"
  1638 
  1639 lemma Rep_one_matrix[simp]: "Rep_matrix (one_matrix n) j i = (if (j = i & j < n) then 1 else 0)"
  1640 apply (simp add: one_matrix_def)
  1641 apply (simplesubst RepAbs_matrix)
  1642 apply (rule exI[of _ n], simp add: split_if)+
  1643 by (simp add: split_if)
  1644 
  1645 lemma nrows_one_matrix[simp]: "nrows ((one_matrix n) :: ('a::zero_neq_one)matrix) = n" (is "?r = _")
  1646 proof -
  1647   have "?r <= n" by (simp add: nrows_le)
  1648   moreover have "n <= ?r" by (simp add:le_nrows, arith)
  1649   ultimately show "?r = n" by simp
  1650 qed
  1651 
  1652 lemma ncols_one_matrix[simp]: "ncols ((one_matrix n) :: ('a::zero_neq_one)matrix) = n" (is "?r = _")
  1653 proof -
  1654   have "?r <= n" by (simp add: ncols_le)
  1655   moreover have "n <= ?r" by (simp add: le_ncols, arith)
  1656   ultimately show "?r = n" by simp
  1657 qed
  1658 
  1659 lemma one_matrix_mult_right[simp]: "ncols A <= n \<Longrightarrow> (A::('a::{ring_1}) matrix) * (one_matrix n) = A"
  1660 apply (subst Rep_matrix_inject[THEN sym])
  1661 apply (rule ext)+
  1662 apply (simp add: times_matrix_def Rep_mult_matrix)
  1663 apply (rule_tac j1="xa" in ssubst[OF foldseq_almostzero])
  1664 apply (simp_all)
  1665 by (simp add: ncols)
  1666 
  1667 lemma one_matrix_mult_left[simp]: "nrows A <= n \<Longrightarrow> (one_matrix n) * A = (A::('a::ring_1) matrix)"
  1668 apply (subst Rep_matrix_inject[THEN sym])
  1669 apply (rule ext)+
  1670 apply (simp add: times_matrix_def Rep_mult_matrix)
  1671 apply (rule_tac j1="x" in ssubst[OF foldseq_almostzero])
  1672 apply (simp_all)
  1673 by (simp add: nrows)
  1674 
  1675 lemma transpose_matrix_mult: "transpose_matrix ((A::('a::comm_ring) matrix)*B) = (transpose_matrix B) * (transpose_matrix A)"
  1676 apply (simp add: times_matrix_def)
  1677 apply (subst transpose_mult_matrix)
  1678 apply (simp_all add: mult_commute)
  1679 done
  1680 
  1681 lemma transpose_matrix_add: "transpose_matrix ((A::('a::monoid_add) matrix)+B) = transpose_matrix A + transpose_matrix B"
  1682 by (simp add: plus_matrix_def transpose_combine_matrix)
  1683 
  1684 lemma transpose_matrix_diff: "transpose_matrix ((A::('a::group_add) matrix)-B) = transpose_matrix A - transpose_matrix B"
  1685 by (simp add: diff_matrix_def transpose_combine_matrix)
  1686 
  1687 lemma transpose_matrix_minus: "transpose_matrix (-(A::('a::group_add) matrix)) = - transpose_matrix (A::'a matrix)"
  1688 by (simp add: minus_matrix_def transpose_apply_matrix)
  1689 
  1690 constdefs 
  1691   right_inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
  1692   "right_inverse_matrix A X == (A * X = one_matrix (max (nrows A) (ncols X))) \<and> nrows X \<le> ncols A" 
  1693   left_inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
  1694   "left_inverse_matrix A X == (X * A = one_matrix (max(nrows X) (ncols A))) \<and> ncols X \<le> nrows A" 
  1695   inverse_matrix :: "('a::{ring_1}) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"
  1696   "inverse_matrix A X == (right_inverse_matrix A X) \<and> (left_inverse_matrix A X)"
  1697 
  1698 lemma right_inverse_matrix_dim: "right_inverse_matrix A X \<Longrightarrow> nrows A = ncols X"
  1699 apply (insert ncols_mult[of A X], insert nrows_mult[of A X])
  1700 by (simp add: right_inverse_matrix_def)
  1701 
  1702 lemma left_inverse_matrix_dim: "left_inverse_matrix A Y \<Longrightarrow> ncols A = nrows Y"
  1703 apply (insert ncols_mult[of Y A], insert nrows_mult[of Y A]) 
  1704 by (simp add: left_inverse_matrix_def)
  1705 
  1706 lemma left_right_inverse_matrix_unique: 
  1707   assumes "left_inverse_matrix A Y" "right_inverse_matrix A X"
  1708   shows "X = Y"
  1709 proof -
  1710   have "Y = Y * one_matrix (nrows A)" 
  1711     apply (subst one_matrix_mult_right)
  1712     apply (insert prems)
  1713     by (simp_all add: left_inverse_matrix_def)
  1714   also have "\<dots> = Y * (A * X)" 
  1715     apply (insert prems)
  1716     apply (frule right_inverse_matrix_dim)
  1717     by (simp add: right_inverse_matrix_def)
  1718   also have "\<dots> = (Y * A) * X" by (simp add: mult_assoc)
  1719   also have "\<dots> = X" 
  1720     apply (insert prems)
  1721     apply (frule left_inverse_matrix_dim)
  1722     apply (simp_all add:  left_inverse_matrix_def right_inverse_matrix_def one_matrix_mult_left)
  1723     done
  1724   ultimately show "X = Y" by (simp)
  1725 qed
  1726 
  1727 lemma inverse_matrix_inject: "\<lbrakk> inverse_matrix A X; inverse_matrix A Y \<rbrakk> \<Longrightarrow> X = Y"
  1728   by (auto simp add: inverse_matrix_def left_right_inverse_matrix_unique)
  1729 
  1730 lemma one_matrix_inverse: "inverse_matrix (one_matrix n) (one_matrix n)"
  1731   by (simp add: inverse_matrix_def left_inverse_matrix_def right_inverse_matrix_def)
  1732 
  1733 lemma zero_imp_mult_zero: "(a::'a::ring) = 0 | b = 0 \<Longrightarrow> a * b = 0"
  1734 by auto
  1735 
  1736 lemma Rep_matrix_zero_imp_mult_zero:
  1737   "! j i k. (Rep_matrix A j k = 0) | (Rep_matrix B k i) = 0  \<Longrightarrow> A * B = (0::('a::lordered_ring) matrix)"
  1738 apply (subst Rep_matrix_inject[symmetric])
  1739 apply (rule ext)+
  1740 apply (auto simp add: Rep_matrix_mult foldseq_zero zero_imp_mult_zero)
  1741 done
  1742 
  1743 lemma add_nrows: "nrows (A::('a::monoid_add) matrix) <= u \<Longrightarrow> nrows B <= u \<Longrightarrow> nrows (A + B) <= u"
  1744 apply (simp add: plus_matrix_def)
  1745 apply (rule combine_nrows)
  1746 apply (simp_all)
  1747 done
  1748 
  1749 lemma move_matrix_row_mult: "move_matrix ((A::('a::ring) matrix) * B) j 0 = (move_matrix A j 0) * B"
  1750 apply (subst Rep_matrix_inject[symmetric])
  1751 apply (rule ext)+
  1752 apply (auto simp add: Rep_matrix_mult foldseq_zero)
  1753 apply (rule_tac foldseq_zerotail[symmetric])
  1754 apply (auto simp add: nrows zero_imp_mult_zero max2)
  1755 apply (rule order_trans)
  1756 apply (rule ncols_move_matrix_le)
  1757 apply (simp add: max1)
  1758 done
  1759 
  1760 lemma move_matrix_col_mult: "move_matrix ((A::('a::ring) matrix) * B) 0 i = A * (move_matrix B 0 i)"
  1761 apply (subst Rep_matrix_inject[symmetric])
  1762 apply (rule ext)+
  1763 apply (auto simp add: Rep_matrix_mult foldseq_zero)
  1764 apply (rule_tac foldseq_zerotail[symmetric])
  1765 apply (auto simp add: ncols zero_imp_mult_zero max1)
  1766 apply (rule order_trans)
  1767 apply (rule nrows_move_matrix_le)
  1768 apply (simp add: max2)
  1769 done
  1770 
  1771 lemma move_matrix_add: "((move_matrix (A + B) j i)::(('a::monoid_add) matrix)) = (move_matrix A j i) + (move_matrix B j i)" 
  1772 apply (subst Rep_matrix_inject[symmetric])
  1773 apply (rule ext)+
  1774 apply (simp)
  1775 done
  1776 
  1777 lemma move_matrix_mult: "move_matrix ((A::('a::ring) matrix)*B) j i = (move_matrix A j 0) * (move_matrix B 0 i)"
  1778 by (simp add: move_matrix_ortho[of "A*B"] move_matrix_col_mult move_matrix_row_mult)
  1779 
  1780 constdefs
  1781   scalar_mult :: "('a::ring) \<Rightarrow> 'a matrix \<Rightarrow> 'a matrix"
  1782   "scalar_mult a m == apply_matrix (op * a) m"
  1783 
  1784 lemma scalar_mult_zero[simp]: "scalar_mult y 0 = 0" 
  1785 by (simp add: scalar_mult_def)
  1786 
  1787 lemma scalar_mult_add: "scalar_mult y (a+b) = (scalar_mult y a) + (scalar_mult y b)"
  1788 by (simp add: scalar_mult_def apply_matrix_add algebra_simps)
  1789 
  1790 lemma Rep_scalar_mult[simp]: "Rep_matrix (scalar_mult y a) j i = y * (Rep_matrix a j i)" 
  1791 by (simp add: scalar_mult_def)
  1792 
  1793 lemma scalar_mult_singleton[simp]: "scalar_mult y (singleton_matrix j i x) = singleton_matrix j i (y * x)"
  1794 apply (subst Rep_matrix_inject[symmetric])
  1795 apply (rule ext)+
  1796 apply (auto)
  1797 done
  1798 
  1799 lemma Rep_minus[simp]: "Rep_matrix (-(A::_::group_add)) x y = - (Rep_matrix A x y)"
  1800 by (simp add: minus_matrix_def)
  1801 
  1802 lemma Rep_abs[simp]: "Rep_matrix (abs (A::_::lordered_ab_group_add)) x y = abs (Rep_matrix A x y)"
  1803 by (simp add: abs_lattice sup_matrix_def)
  1804 
  1805 end