src/HOL/Matrix/Matrix.thy
 author haftmann Fri Nov 27 08:41:10 2009 +0100 (2009-11-27) changeset 33963 977b94b64905 parent 33657 a4179bf442d1 child 34872 6ca970cfa873 permissions -rw-r--r--
renamed former datatype.ML to datatype_data.ML; datatype.ML provides uniform view on datatype.ML and datatype_rep_proofs.ML
     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_antisym)

   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_antisym)

   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_antisym 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