src/HOL/Matrix/MatrixGeneral.thy

 (*  Title:      HOL/Matrix/MatrixGeneral.thy
     ID:         $Id$
     Author:     Steven Obua
-    License:    2004 Technische UniversitÃ\<currency>t MÃ\<onequarter>nchen
+    License:    2004 Technische Universitaet Muenchen
 *)
 
 theory MatrixGeneral = Main:
 
 types 'a infmatrix = "[nat, nat] \<Rightarrow> 'a"
 
 constdefs
   nonzero_positions :: "('a::zero) infmatrix \<Rightarrow> (nat*nat) set"
-  "nonzero_positions A == {pos. A (fst pos) (snd pos) ~= 0}"  
+  "nonzero_positions A == {pos. A (fst pos) (snd pos) ~= 0}"
 
 typedef 'a matrix = "{(f::(('a::zero) infmatrix)). finite (nonzero_positions f)}"
 apply (rule_tac x="(% j i. 0)" in exI)
 by (simp add: nonzero_positions_def)
 
 declare Rep_matrix_inverse[simp]
 
 lemma finite_nonzero_positions : "finite (nonzero_positions (Rep_matrix A))"
-apply (rule Abs_matrix_induct) 
+apply (rule Abs_matrix_induct)
 by (simp add: Abs_matrix_inverse matrix_def)
 
 constdefs
   nrows :: "('a::zero) matrix \<Rightarrow> nat"
   "nrows A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image fst) (nonzero_positions (Rep_matrix A))))"
   ncols :: "('a::zero) matrix \<Rightarrow> nat"
-  "ncols A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image snd) (nonzero_positions (Rep_matrix A))))" 
-
-lemma nrows: 
-  assumes hyp: "nrows A \<le> m" 
+  "ncols A == if nonzero_positions(Rep_matrix A) = {} then 0 else Suc(Max ((image snd) (nonzero_positions (Rep_matrix A))))"
+
+lemma nrows:
+  assumes hyp: "nrows A \<le> m"
   shows "(Rep_matrix A m n) = 0" (is ?concl)
 proof cases
-  assume "nonzero_positions(Rep_matrix A) = {}" 
+  assume "nonzero_positions(Rep_matrix A) = {}"
   then show "(Rep_matrix A m n) = 0" by (simp add: nonzero_positions_def)
 next
   assume a: "nonzero_positions(Rep_matrix A) \<noteq> {}"
-  let ?S = "fst`(nonzero_positions(Rep_matrix A))" 
+  let ?S = "fst`(nonzero_positions(Rep_matrix A))"
   from a have b: "?S \<noteq> {}" by (simp)
   have c: "finite (?S)" by (simp add: finite_nonzero_positions)
   from hyp have d: "Max (?S) < m" by (simp add: a nrows_def)
-  have "m \<notin> ?S" 
-    proof - 
-      have "m \<in> ?S \<Longrightarrow> m <= Max(?S)" by (simp add: Max[OF c b]) 
+  have "m \<notin> ?S"
+    proof -
+      have "m \<in> ?S \<Longrightarrow> m <= Max(?S)" by (simp add: Max[OF c b])
       moreover from d have "~(m <= Max ?S)" by (simp)
       ultimately show "m \<notin> ?S" by (auto)
     qed
   thus "Rep_matrix A m n = 0" by (simp add: nonzero_positions_def image_Collect)
 qed
-text {* \matrix cool *}
+
 constdefs
   transpose_infmatrix :: "'a infmatrix \<Rightarrow> 'a infmatrix"
   "transpose_infmatrix A j i == A i j"
   transpose_matrix :: "('a::zero) matrix \<Rightarrow> 'a matrix"
   "transpose_matrix == Abs_matrix o transpose_infmatrix o Rep_matrix"

 
 lemma transpose_infmatrix: "transpose_infmatrix (% j i. P j i) = (% j i. P i j)"
   apply (rule ext)+
   by (simp add: transpose_infmatrix_def)
 
-lemma transpose_infmatrix_closed[simp]: "Rep_matrix (Abs_matrix (transpose_infmatrix (Rep_matrix x))) = transpose_infmatrix (Rep_matrix x)" 
+lemma transpose_infmatrix_closed[simp]: "Rep_matrix (Abs_matrix (transpose_infmatrix (Rep_matrix x))) = transpose_infmatrix (Rep_matrix x)"
 apply (rule Abs_matrix_inverse)
 apply (simp add: matrix_def nonzero_positions_def image_def)
 proof -
   let ?A = "{pos. Rep_matrix x (snd pos) (fst pos) \<noteq> 0}"
   let ?swap = "% pos. (snd pos, fst pos)"
   let ?B = "{pos. Rep_matrix x (fst pos) (snd pos) \<noteq> 0}"
-  have swap_image: "?swap`?A = ?B" 
+  have swap_image: "?swap`?A = ?B"
     apply (simp add: image_def)
     apply (rule set_ext)
     apply (simp)
     proof
       fix y
-      assume hyp: "\<exists>a b. Rep_matrix x b a \<noteq> 0 \<and> y = (b, a)" 
-      thus "Rep_matrix x (fst y) (snd y) \<noteq> 0" 
-	proof -
-	  from hyp obtain a b where "(Rep_matrix x b a \<noteq> 0 & y = (b,a))" by blast
-	  then show "Rep_matrix x (fst y) (snd y) \<noteq> 0" by (simp)
-	qed 
+      assume hyp: "\<exists>a b. Rep_matrix x b a \<noteq> 0 \<and> y = (b, a)"
+      thus "Rep_matrix x (fst y) (snd y) \<noteq> 0"
+        proof -
+          from hyp obtain a b where "(Rep_matrix x b a \<noteq> 0 & y = (b,a))" by blast
+          then show "Rep_matrix x (fst y) (snd y) \<noteq> 0" by (simp)
+        qed
     next
       fix y
       assume hyp: "Rep_matrix x (fst y) (snd y) \<noteq> 0"
-      show "\<exists> a b. (Rep_matrix x b a \<noteq> 0 & y = (b,a))" by (rule exI[of _ "snd y"], rule exI[of _ "fst y"], simp) 
+      show "\<exists> a b. (Rep_matrix x b a \<noteq> 0 & y = (b,a))" by (rule exI[of _ "snd y"], rule exI[of _ "fst y"], simp)
     qed
-  then have "finite (?swap`?A)" 
+  then have "finite (?swap`?A)"
     proof -
       have "finite (nonzero_positions (Rep_matrix x))" by (simp add: finite_nonzero_positions)
       then have "finite ?B" by (simp add: nonzero_positions_def)
       with swap_image show "finite (?swap`?A)" by (simp)
-    qed    
+    qed
   moreover
   have "inj_on ?swap ?A" by (simp add: inj_on_def)
-  ultimately show "finite ?A"by (rule finite_imageD[of ?swap ?A]) 	
-qed 
+  ultimately show "finite ?A"by (rule finite_imageD[of ?swap ?A])
+qed
 
 lemma transpose_matrix[simp]: "Rep_matrix(transpose_matrix A) j i = Rep_matrix A i j"
 by (simp add: transpose_matrix_def)
 
 lemma transpose_transpose_id[simp]: "transpose_matrix (transpose_matrix A) = A"
 by (simp add: transpose_matrix_def)
 
-lemma nrows_transpose[simp]: "nrows (transpose_matrix A) = ncols A" 
+lemma nrows_transpose[simp]: "nrows (transpose_matrix A) = ncols A"
 by (simp add: nrows_def ncols_def nonzero_positions_def transpose_matrix_def image_def)
 
-lemma ncols_transpose[simp]: "ncols (transpose_matrix A) = nrows A" 
+lemma ncols_transpose[simp]: "ncols (transpose_matrix A) = nrows A"
 by (simp add: nrows_def ncols_def nonzero_positions_def transpose_matrix_def image_def)
 
-lemma ncols: "ncols A <= n \<Longrightarrow> Rep_matrix A m n = 0" 
+lemma ncols: "ncols A <= n \<Longrightarrow> Rep_matrix A m n = 0"
 proof -
   assume "ncols A <= n"
-  then have "nrows (transpose_matrix A) <= n" by (simp) 
+  then have "nrows (transpose_matrix A) <= n" by (simp)
   then have "Rep_matrix (transpose_matrix A) n m = 0" by (rule nrows)
   thus "Rep_matrix A m n = 0" by (simp add: transpose_matrix_def)
 qed
 
 lemma ncols_le: "(ncols A <= n) = (! j i. n <= i \<longrightarrow> (Rep_matrix A j i) = 0)" (is "_ = ?st")

 apply (simp add: ncols)
 proof (simp add: ncols_def, auto)
   let ?P = "nonzero_positions (Rep_matrix A)"
   let ?p = "snd`?P"
   have a:"finite ?p" by (simp add: finite_nonzero_positions)
-  let ?m = "Max ?p"  
+  let ?m = "Max ?p"
   assume "~(Suc (?m) <= n)"
   then have b:"n <= ?m" by (simp)
   fix a b
   assume "(a,b) \<in> ?P"
   then have "?p \<noteq> {}" by (auto)
-  with a have "?m \<in>  ?p" by (simp)   
+  with a have "?m \<in>  ?p" by (simp)
   moreover have "!x. (x \<in> ?p \<longrightarrow> (? y. (Rep_matrix A y x) \<noteq> 0))" by (simp add: nonzero_positions_def image_def)
   ultimately have "? y. (Rep_matrix A y ?m) \<noteq> 0" by (simp)
   moreover assume ?st
   ultimately show "False" using b by (simp)
 qed

     fix m::nat
     let ?s0 = "{pos. fst pos < m & snd pos < n}"
     let ?s1 = "{pos. fst pos < (Suc m) & snd pos < n}"
     let ?sd = "{pos. fst pos = m & snd pos < n}"
     assume f0: "finite ?s0"
-    have f1: "finite ?sd" 
+    have f1: "finite ?sd"
     proof -
-      let ?f = "% x. (m, x)"	
+      let ?f = "% x. (m, x)"
       have "{pos. fst pos = m & snd pos < n} = ?f ` {x. x < n}" by (rule set_ext, simp add: image_def, auto)
       moreover have "finite {x. x < n}" by (simp add: finite_natarray1)
       ultimately show "finite {pos. fst pos = m & snd pos < n}" by (simp)
     qed
     have su: "?s0 \<union> ?sd = ?s1" by (rule set_ext, simp, arith)
     from f0 f1 have "finite (?s0 \<union> ?sd)" by (rule finite_UnI)
     with su show "finite ?s1" by (simp)
 qed
-    
-lemma RepAbs_matrix: 
+
+lemma RepAbs_matrix:
   assumes aem: "? m. ! j i. m <= j \<longrightarrow> x j i = 0" (is ?em) and aen:"? n. ! j i. (n <= i \<longrightarrow> x j i = 0)" (is ?en)
-  shows "(Rep_matrix (Abs_matrix x)) = x"  
+  shows "(Rep_matrix (Abs_matrix x)) = x"
 apply (rule Abs_matrix_inverse)
 apply (simp add: matrix_def nonzero_positions_def)
-proof -  
+proof -
   from aem obtain m where a: "! j i. m <= j \<longrightarrow> x j i = 0" by (blast)
-  from aen obtain n where b: "! j i. n <= i \<longrightarrow> x j i = 0" by (blast)    
+  from aen obtain n where b: "! j i. n <= i \<longrightarrow> x j i = 0" by (blast)
   let ?u = "{pos. x (fst pos) (snd pos) \<noteq> 0}"
   let ?v = "{pos. fst pos < m & snd pos < n}"
-  have c: "!! (m::nat) a. ~(m <= a) \<Longrightarrow> a < m" by (arith)  
+  have c: "!! (m::nat) a. ~(m <= a) \<Longrightarrow> a < m" by (arith)
   from a b have "(?u \<inter> (-?v)) = {}"
     apply (simp)
     apply (rule set_ext)
     apply (simp)
     apply auto

   then have d: "?u \<subseteq> ?v" by blast
   moreover have "finite ?v" by (simp add: finite_natarray2)
   ultimately show "finite ?u" by (rule finite_subset)
 qed
 
-constdefs 
+constdefs
   apply_infmatrix :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix"
   "apply_infmatrix f == % A. (% j i. f (A j i))"
   apply_matrix :: "('a \<Rightarrow> 'b) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix"
-  "apply_matrix f == % A. Abs_matrix (apply_infmatrix f (Rep_matrix A))" 
+  "apply_matrix f == % A. Abs_matrix (apply_infmatrix f (Rep_matrix A))"
   combine_infmatrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'a infmatrix \<Rightarrow> 'b infmatrix \<Rightarrow> 'c infmatrix"
-  "combine_infmatrix f == % A B. (% j i. f (A j i) (B j i))"   
+  "combine_infmatrix f == % A B. (% j i. f (A j i) (B j i))"
   combine_matrix :: "('a \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> ('a::zero) matrix \<Rightarrow> ('b::zero) matrix \<Rightarrow> ('c::zero) matrix"
   "combine_matrix f == % A B. Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))"
 
 lemma expand_apply_infmatrix[simp]: "apply_infmatrix f A j i = f (A j i)"
 by (simp add: apply_infmatrix_def)
 
-lemma expand_combine_infmatrix[simp]: "combine_infmatrix f A B j i = f (A j i) (B j i)" 
-by (simp add: combine_infmatrix_def)   
+lemma expand_combine_infmatrix[simp]: "combine_infmatrix f A B j i = f (A j i) (B j i)"
+by (simp add: combine_infmatrix_def)
 
 constdefs
 commutative :: "('a \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> bool"
 "commutative f == ! x y. f x y = f y x"
 associative :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
 "associative f == ! x y z. f (f x y) z = f x (f y z)"
 
-text{* 
+text{*
 To reason about associativity and commutativity of operations on matrices,
 let's take a step back and look at the general situtation: Assume that we have
 sets $A$ and $B$ with $B \subset A$ and an abstraction $u: A \rightarrow B$. This abstraction has to fulfill $u(b) = b$ for all $b \in B$, but is arbitrary otherwise.
 Each function $f: A \times A \rightarrow A$ now induces a function $f': B \times B \rightarrow B$ by $f' = u \circ f$.
-It is obvious that commutativity of $f$ implies commutativity of $f'$: $f' x y = u (f x y) = u (f y x) = f' y x.$     
+It is obvious that commutativity of $f$ implies commutativity of $f'$: $f' x y = u (f x y) = u (f y x) = f' y x.$
 *}
 
-lemma combine_infmatrix_commute: 
+lemma combine_infmatrix_commute:
   "commutative f \<Longrightarrow> commutative (combine_infmatrix f)"
 by (simp add: commutative_def combine_infmatrix_def)
 
 lemma combine_matrix_commute:
 "commutative f \<Longrightarrow> commutative (combine_matrix f)"
 by (simp add: combine_matrix_def commutative_def combine_infmatrix_def)
 
 text{*
-On the contrary, given an associative function $f$ we cannot expect $f'$ to be associative. A counterexample is given by $A=\ganz$, $B=\{-1, 0, 1\}$, 
+On the contrary, given an associative function $f$ we cannot expect $f'$ to be associative. A counterexample is given by $A=\ganz$, $B=\{-1, 0, 1\}$,
 as $f$ we take addition on $\ganz$, which is clearly associative. The abstraction is given by  $u(a) = 0$ for $a \notin B$. Then we have
 \[ f' (f' 1 1) -1 = u(f (u (f 1 1)) -1) = u(f (u 2) -1) = u (f 0 -1) = -1, \]
 but on the other hand we have
 \[ f' 1 (f' 1 -1) = u (f 1 (u (f 1 -1))) = u (f 1 0) = 1.\]
 A way out of this problem is to assume that $f(A\times A)\subset A$ holds, and this is what we are going to do:

 by (rule subsetI, simp add: nonzero_positions_def combine_infmatrix_def, auto)
 
 lemma finite_nonzero_positions_Rep[simp]: "finite (nonzero_positions (Rep_matrix A))"
 by (insert Rep_matrix [of A], simp add: matrix_def)
 
-lemma combine_infmatrix_closed [simp]: 
+lemma combine_infmatrix_closed [simp]:
   "f 0 0 = 0 \<Longrightarrow> Rep_matrix (Abs_matrix (combine_infmatrix f (Rep_matrix A) (Rep_matrix B))) = combine_infmatrix f (Rep_matrix A) (Rep_matrix B)"
 apply (rule Abs_matrix_inverse)
 apply (simp add: matrix_def)
 apply (rule finite_subset[of _ "(nonzero_positions (Rep_matrix A)) \<union> (nonzero_positions (Rep_matrix B))"])
 by (simp_all)

 
 lemma Rep_apply_matrix[simp]: "f 0 = 0 \<Longrightarrow> Rep_matrix (apply_matrix f A) j i = f (Rep_matrix A j i)"
 by (simp add: apply_matrix_def)
 
 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)"
-  by(simp add: combine_matrix_def) 
+  by(simp add: combine_matrix_def)
 
 lemma combine_nrows: "f 0 0 = 0  \<Longrightarrow> nrows (combine_matrix f A B) <= max (nrows A) (nrows B)"
 by (simp add: nrows_le)
 
 lemma combine_ncols: "f 0 0 = 0 \<Longrightarrow> ncols (combine_matrix f A B) <= max (ncols A) (ncols B)"

   "zero_l_neutral f == ! a. f 0 a = a"
   zero_closed :: "(('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> bool"
   "zero_closed f == (!x. f x 0 = 0) & (!y. f 0 y = 0)"
 
 consts foldseq :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
-primrec 
+primrec
   "foldseq f s 0 = s 0"
   "foldseq f s (Suc n) = f (s 0) (foldseq f (% k. s(Suc k)) n)"
 
 consts foldseq_transposed ::  "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> (nat \<Rightarrow> 'a) \<Rightarrow> nat \<Rightarrow> 'a"
 primrec

   "foldseq_transposed f s (Suc n) = f (foldseq_transposed f s n) (s (Suc n))"
 
 lemma foldseq_assoc : "associative f \<Longrightarrow> foldseq f = foldseq_transposed f"
 proof -
   assume a:"associative f"
-  then have sublemma: "!! n. ! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N" 
+  then have sublemma: "!! n. ! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
   proof -
     fix n
-    show "!N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N" 
-    proof (induct n) 
+    show "!N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
+    proof (induct n)
       show "!N s. N <= 0 \<longrightarrow> foldseq f s N = foldseq_transposed f s N" by simp
     next
       fix n
       assume b:"! N s. N <= n \<longrightarrow> foldseq f s N = foldseq_transposed f s N"
       have c:"!!N s. N <= n \<Longrightarrow> foldseq f s N = foldseq_transposed f s N" by (simp add: b)
       show "! N t. N <= Suc n \<longrightarrow> foldseq f t N = foldseq_transposed f t N"
       proof (auto)
-	fix N t
-	assume Nsuc: "N <= Suc n"
-	show "foldseq f t N = foldseq_transposed f t N" 
-	proof cases 
-	  assume "N <= n"
-	  then show "foldseq f t N = foldseq_transposed f t N" by (simp add: b)
-	next
-	  assume "~(N <= n)"
-	  with Nsuc have Nsuceq: "N = Suc n" by simp
-	  have neqz: "n \<noteq> 0 \<Longrightarrow> ? m. n = Suc m & Suc m <= n" by arith
-	  have assocf: "!! x y z. f x (f y z) = f (f x y) z" by (insert a, simp add: associative_def)
-	  show "foldseq f t N = foldseq_transposed f t N"
-	    apply (simp add: Nsuceq)
-	    apply (subst c)
-	    apply (simp)
-	    apply (case_tac "n = 0")
-	    apply (simp)
-	    apply (drule neqz)
-	    apply (erule exE)
-	    apply (simp)
-	    apply (subst assocf)
-	    proof -
-	      fix m
-	      assume "n = Suc m & Suc m <= n"
-	      then have mless: "Suc m <= n" by arith
-	      then have step1: "foldseq_transposed f (% k. t (Suc k)) m = foldseq f (% k. t (Suc k)) m" (is "?T1 = ?T2")
-		apply (subst c)
-		by simp+
-	      have step2: "f (t 0) ?T2 = foldseq f t (Suc m)" (is "_ = ?T3") by simp
-	      have step3: "?T3 = foldseq_transposed f t (Suc m)" (is "_ = ?T4")
-		apply (subst c)
-		by (simp add: mless)+
-	      have step4: "?T4 = f (foldseq_transposed f t m) (t (Suc m))" (is "_=?T5") by simp
-	      from step1 step2 step3 step4 show sowhat: "f (f (t 0) ?T1) (t (Suc (Suc m))) = f ?T5 (t (Suc (Suc m)))" by simp
-	    qed
-	  qed
-	qed
+        fix N t
+        assume Nsuc: "N <= Suc n"
+        show "foldseq f t N = foldseq_transposed f t N"
+        proof cases
+          assume "N <= n"
+          then show "foldseq f t N = foldseq_transposed f t N" by (simp add: b)
+        next
+          assume "~(N <= n)"
+          with Nsuc have Nsuceq: "N = Suc n" by simp
+          have neqz: "n \<noteq> 0 \<Longrightarrow> ? m. n = Suc m & Suc m <= n" by arith
+          have assocf: "!! x y z. f x (f y z) = f (f x y) z" by (insert a, simp add: associative_def)
+          show "foldseq f t N = foldseq_transposed f t N"
+            apply (simp add: Nsuceq)
+            apply (subst c)
+            apply (simp)
+            apply (case_tac "n = 0")
+            apply (simp)
+            apply (drule neqz)
+            apply (erule exE)
+            apply (simp)
+            apply (subst assocf)
+            proof -
+              fix m
+              assume "n = Suc m & Suc m <= n"
+              then have mless: "Suc m <= n" by arith
+              then have step1: "foldseq_transposed f (% k. t (Suc k)) m = foldseq f (% k. t (Suc k)) m" (is "?T1 = ?T2")
+                apply (subst c)
+                by simp+
+              have step2: "f (t 0) ?T2 = foldseq f t (Suc m)" (is "_ = ?T3") by simp
+              have step3: "?T3 = foldseq_transposed f t (Suc m)" (is "_ = ?T4")
+                apply (subst c)
+                by (simp add: mless)+
+              have step4: "?T4 = f (foldseq_transposed f t m) (t (Suc m))" (is "_=?T5") by simp
+              from step1 step2 step3 step4 show sowhat: "f (f (t 0) ?T1) (t (Suc (Suc m))) = f ?T5 (t (Suc (Suc m)))" by simp
+            qed
+          qed
+        qed
       qed
     qed
     show "foldseq f = foldseq_transposed f" by ((rule ext)+, insert sublemma, auto)
   qed
 

   then show "foldseq f (% k. f (u k) (v k)) n = f (foldseq f u n) (foldseq f v n)" by simp
 qed
 
 theorem "\<lbrakk>associative f; associative g; \<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d); ? x y. (f x) \<noteq> (f y); ? x y. (g x) \<noteq> (g y); f x x = x; g x x = x\<rbrakk> \<Longrightarrow> f=g | (! y. f y x = y) | (! y. g y x = y)"
 oops
-(* Model found 
+(* Model found
 
 Trying to find a model that refutes: \<lbrakk>associative f; associative g;
  \<forall>a b c d. g (f a b) (f c d) = f (g a c) (g b d); \<exists>x y. f x \<noteq> f y;
  \<exists>x y. g x \<noteq> g y; f x x = x; g x x = x\<rbrakk>
 \<Longrightarrow> f = g \<or> (\<forall>y. f y x = y) \<or> (\<forall>y. g y x = y)

 x: a1
 g: (a0\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a0, a2\<mapsto>a1), a1\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a1, a2\<mapsto>a0), a2\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a0, a2\<mapsto>a1))
 f: (a0\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a0, a2\<mapsto>a0), a1\<mapsto>(a0\<mapsto>a1, a1\<mapsto>a1, a2\<mapsto>a1), a2\<mapsto>(a0\<mapsto>a0, a1\<mapsto>a0, a2\<mapsto>a0))
 *)
 
-lemma foldseq_zero: 
+lemma foldseq_zero:
 assumes fz: "f 0 0 = 0" and sz: "! i. i <= n \<longrightarrow> s i = 0"
 shows "foldseq f s n = 0"
 proof -
   have "!! n. ! s. (! i. i <= n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s n = 0"
     apply (induct_tac n)
     apply (simp)
     by (simp add: fz)
   then show "foldseq f s n = 0" by (simp add: sz)
 qed
 
-lemma foldseq_significant_positions: 
+lemma foldseq_significant_positions:
   assumes p: "! i. i <= N \<longrightarrow> S i = T i"
   shows "foldseq f S N = foldseq f T N" (is ?concl)
 proof -
   have "!! m . ! s t. (! i. i<=m \<longrightarrow> s i = t i) \<longrightarrow> foldseq f s m = foldseq f t m"
     apply (induct_tac m)
     apply (simp)
     apply (simp)
     apply (auto)
     proof -
-      fix n 
-      fix s::"nat\<Rightarrow>'a" 
+      fix n
+      fix s::"nat\<Rightarrow>'a"
       fix t::"nat\<Rightarrow>'a"
       assume a: "\<forall>s t. (\<forall>i\<le>n. s i = t i) \<longrightarrow> foldseq f s n = foldseq f t n"
       assume b: "\<forall>i\<le>Suc n. s i = t i"
       have c:"!! a b. a = b \<Longrightarrow> f (t 0) a = f (t 0) b" by blast
       have d:"!! s t. (\<forall>i\<le>n. s i = t i) \<Longrightarrow> foldseq f s n = foldseq f t n" by (simp add: a)
       show "f (t 0) (foldseq f (\<lambda>k. s (Suc k)) n) = f (t 0) (foldseq f (\<lambda>k. t (Suc k)) n)" by (rule c, simp add: d b)
     qed
   with p show ?concl by simp
 qed
-  
+
 lemma foldseq_tail: "M <= N \<Longrightarrow> foldseq f S N = foldseq f (% k. (if k < M then (S k) else (foldseq f (% k. S(k+M)) (N-M)))) M" (is "?p \<Longrightarrow> ?concl")
 proof -
   have suc: "!! a b. \<lbrakk>a <= Suc b; a \<noteq> Suc b\<rbrakk> \<Longrightarrow> a <= b" by arith
   have suc2: "!! a b. \<lbrakk>a <= Suc b; ~ (a <= b)\<rbrakk> \<Longrightarrow> a = (Suc b)" by arith
   have eq: "!! (a::nat) b. \<lbrakk>~(a < b); a <= b\<rbrakk> \<Longrightarrow> a = b" by arith
   have a:"!! a b c . a = b \<Longrightarrow> f c a = f c b" by blast
-  have "!! n. ! m s. m <= n \<longrightarrow> foldseq f s n = foldseq f (% k. (if k < m then (s k) else (foldseq f (% k. s(k+m)) (n-m)))) m" 
+  have "!! n. ! m s. m <= n \<longrightarrow> foldseq f s n = foldseq f (% k. (if k < m then (s k) else (foldseq f (% k. s(k+m)) (n-m)))) m"
     apply (induct_tac n)
     apply (simp)
     apply (simp)
     apply (auto)
     apply (case_tac "m = Suc na")
-    apply (simp) 
+    apply (simp)
     apply (rule a)
     apply (rule foldseq_significant_positions)
     apply (simp, auto)
     apply (drule eq, simp+)
     apply (drule suc, simp+)
     proof -
       fix na m s
       assume suba:"\<forall>m\<le>na. \<forall>s. foldseq f s na = foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (na - m))m"
       assume subb:"m <= na"
-      from suba have subc:"!! m s. m <= na \<Longrightarrow>foldseq f s na = foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (na - m))m" by simp  
-      have subd: "foldseq f (\<lambda>k. if k < m then s (Suc k) else foldseq f (\<lambda>k. s (Suc (k + m))) (na - m)) m = 
-	foldseq f (% k. s(Suc k)) na" 
-	by (rule subc[of m "% k. s(Suc k)", THEN sym], simp add: subb)
+      from suba have subc:"!! m s. m <= na \<Longrightarrow>foldseq f s na = foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (na - m))m" by simp
+      have subd: "foldseq f (\<lambda>k. if k < m then s (Suc k) else foldseq f (\<lambda>k. s (Suc (k + m))) (na - m)) m =
+        foldseq f (% k. s(Suc k)) na"
+        by (rule subc[of m "% k. s(Suc k)", THEN sym], simp add: subb)
       from subb have sube: "m \<noteq> 0 \<Longrightarrow> ? mm. m = Suc mm & mm <= na" by arith
       show "f (s 0) (foldseq f (\<lambda>k. if k < m then s (Suc k) else foldseq f (\<lambda>k. s (Suc (k + m))) (na - m)) m) =
         foldseq f (\<lambda>k. if k < m then s k else foldseq f (\<lambda>k. s (k + m)) (Suc na - m)) m"
-	apply (simp add: subd)
-	apply (case_tac "m=0")
-	apply (simp)
-	apply (drule sube)
-	apply (auto)
-	apply (rule a)
-	by (simp add: subc if_def)
+        apply (simp add: subd)
+        apply (case_tac "m=0")
+        apply (simp)
+        apply (drule sube)
+        apply (auto)
+        apply (rule a)
+        by (simp add: subc if_def)
     qed
-  then show "?p \<Longrightarrow> ?concl" by simp  
-qed
-	
+  then show "?p \<Longrightarrow> ?concl" by simp
+qed
+
 lemma foldseq_zerotail:
   assumes
-  fz: "f 0 0 = 0" 
-  and sz: "! i.  n <= i \<longrightarrow> s i = 0" 
-  and nm: "n <= m" 
-  shows
-  "foldseq f s n = foldseq f s m" 
+  fz: "f 0 0 = 0"
+  and sz: "! i.  n <= i \<longrightarrow> s i = 0"
+  and nm: "n <= m"
+  shows
+  "foldseq f s n = foldseq f s m"
 proof -
   have a: "!! a b. ~(a < b) \<Longrightarrow> a <= b \<Longrightarrow> (a::nat) = b" by simp
   show "foldseq f s n = foldseq f s m"
     apply (simp add: foldseq_tail[OF nm, of f s])
     apply (rule foldseq_significant_positions)

   have "f 0 0 = 0" by (simp add: prems)
   have a:"!! (i::nat) m. ~(i < m) \<Longrightarrow> i <= m \<Longrightarrow> i = m" by arith
   have b:"!! m n. n <= m \<Longrightarrow> m \<noteq> n \<Longrightarrow> ? k. m-n = Suc k" by arith
   have c: "0 <= m" by simp
   have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
-  show ?concl 
+  show ?concl
     apply (subst foldseq_tail[OF nm])
     apply (rule foldseq_significant_positions)
     apply (auto)
     apply (case_tac "m=n")
     apply (drule a, simp+)

     apply (simp add: prems)
     apply (drule d)
     apply (auto)
     by (simp add: prems foldseq_zero)
 qed
-    
+
 lemma foldseq_zerostart:
-  "! x. f 0 (f 0 x) = f 0 x \<Longrightarrow>  ! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" 
+  "! x. f 0 (f 0 x) = f 0 x \<Longrightarrow>  ! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
 proof -
   assume f00x: "! x. f 0 (f 0 x) = f 0 x"
   have "! s. (! i. i<=n \<longrightarrow> s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))"
     apply (induct n)
     apply (simp)

       have a:"foldseq f s (Suc (Suc n)) = f (s 0) (foldseq f (% k. s(Suc k)) (Suc n))" by simp
       assume b: "! s. ((\<forall>i\<le>n. s i = 0) \<longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n)))"
       from b have c:"!! s. (\<forall>i\<le>n. s i = 0) \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
       assume d: "! i. i <= Suc n \<longrightarrow> s i = 0"
       show "foldseq f s (Suc (Suc n)) = f 0 (s (Suc (Suc n)))"
-	apply (subst a)
-	apply (subst c)
-	by (simp add: d f00x)+
+        apply (subst a)
+        apply (subst c)
+        by (simp add: d f00x)+
     qed
   then show "! i. i <= n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s (Suc n) = f 0 (s (Suc n))" by simp
 qed
 
 lemma foldseq_zerostart2:
-  "! x. f 0 x = x \<Longrightarrow>  ! i. i < n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s n = s n" 
+  "! x. f 0 x = x \<Longrightarrow>  ! i. i < n \<longrightarrow> s i = 0 \<Longrightarrow> foldseq f s n = s n"
 proof -
   assume a:"! i. i<n \<longrightarrow> s i = 0"
   assume x:"! x. f 0 x = x"
-  from x have f00x: "! x. f 0 (f 0 x) = f 0 x" by blast  
+  from x have f00x: "! x. f 0 (f 0 x) = f 0 x" by blast
   have b: "!! i l. i < Suc l = (i <= l)" by arith
   have d: "!! k. k \<noteq> 0 \<Longrightarrow> ? l. k = Suc l" by arith
-  show "foldseq f s n = s n" 
+  show "foldseq f s n = s n"
   apply (case_tac "n=0")
   apply (simp)
   apply (insert a)
   apply (drule d)
   apply (auto)
   apply (simp add: b)
   apply (insert f00x)
   apply (drule foldseq_zerostart)
   by (simp add: x)+
 qed
-   
-lemma foldseq_almostzero: 
+
+lemma foldseq_almostzero:
   assumes f0x:"! x. f 0 x = x" and fx0: "! x. f x 0 = x" and s0:"! i. i \<noteq> j \<longrightarrow> s i = 0"
   shows "foldseq f s n = (if (j <= n) then (s j) else 0)" (is ?concl)
 proof -
   from s0 have a: "! i. i < j \<longrightarrow> s i = 0" by simp
-  from s0 have b: "! i. j < i \<longrightarrow> s i = 0" by simp 
+  from s0 have b: "! i. j < i \<longrightarrow> s i = 0" by simp
   show ?concl
     apply auto
     apply (subst foldseq_zerotail2[of f, OF fx0, of j, OF b, of n, THEN sym])
     apply simp
     apply (subst foldseq_zerostart2)

     apply (drule_tac x="% k. s (Suc k)" in spec)
     by (simp add: prems)
   then show ?concl by simp
 qed
 
-constdefs 
+constdefs
   mult_matrix_n :: "nat \<Rightarrow> (('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'a matrix \<Rightarrow> 'b matrix \<Rightarrow> 'c matrix"
-  "mult_matrix_n n fmul fadd A B == Abs_matrix(% j i. foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) n)"    
+  "mult_matrix_n n fmul fadd A B == Abs_matrix(% j i. foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) n)"
   mult_matrix :: "(('a::zero) \<Rightarrow> ('b::zero) \<Rightarrow> ('c::zero)) \<Rightarrow> ('c \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'a matrix \<Rightarrow> 'b matrix \<Rightarrow> 'c matrix"
   "mult_matrix fmul fadd A B == mult_matrix_n (max (ncols A) (nrows B)) fmul fadd A B"
 
-lemma mult_matrix_n: 
+lemma mult_matrix_n:
   assumes prems: "ncols A \<le>  n" (is ?An) "nrows B \<le> n" (is ?Bn) "fadd 0 0 = 0" "fmul 0 0 = 0"
   shows c:"mult_matrix fmul fadd A B = mult_matrix_n n fmul fadd A B" (is ?concl)
 proof -
   show ?concl using prems
     apply (simp add: mult_matrix_def mult_matrix_n_def)
     apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
-    by (rule foldseq_zerotail, simp_all add: nrows_le ncols_le prems) 
+    by (rule foldseq_zerotail, simp_all add: nrows_le ncols_le prems)
 qed
 
 lemma mult_matrix_nm:
   assumes prems: "ncols A <= n" "nrows B <= n" "ncols A <= m" "nrows B <= m" "fadd 0 0 = 0" "fmul 0 0 = 0"
   shows "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B"
 proof -
   from prems have "mult_matrix_n n fmul fadd A B = mult_matrix fmul fadd A B" by (simp add: mult_matrix_n)
   also from prems have "\<dots> = mult_matrix_n m fmul fadd A B" by (simp add: mult_matrix_n[THEN sym])
   finally show "mult_matrix_n n fmul fadd A B = mult_matrix_n m fmul fadd A B" by simp
 qed
-   
-constdefs 
+
+constdefs
   r_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> ('b \<Rightarrow> 'b \<Rightarrow> 'b) \<Rightarrow> bool"
   "r_distributive fmul fadd == ! a u v. fmul a (fadd u v) = fadd (fmul a u) (fmul a v)"
   l_distributive :: "('a \<Rightarrow> 'b \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
   "l_distributive fmul fadd == ! a u v. fmul (fadd u v) a = fadd (fmul u a) (fmul v a)"
   distributive :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> bool"
-  "distributive fmul fadd == l_distributive fmul fadd & r_distributive fmul fadd" 
+  "distributive fmul fadd == l_distributive fmul fadd & r_distributive fmul fadd"
 
 lemma max1: "!! a x y. (a::nat) <= x \<Longrightarrow> a <= max x y" by (arith)
 lemma max2: "!! b x y. (b::nat) <= y \<Longrightarrow> b <= max x y" by (arith)
 
 lemma r_distributive_matrix:
- assumes prems: 
-  "r_distributive fmul fadd" 
-  "associative fadd" 
-  "commutative fadd" 
-  "fadd 0 0 = 0" 
-  "! a. fmul a 0 = 0" 
+ assumes prems:
+  "r_distributive fmul fadd"
+  "associative fadd"
+  "commutative fadd"
+  "fadd 0 0 = 0"
+  "! a. fmul a 0 = 0"
   "! a. fmul 0 a = 0"
  shows "r_distributive (mult_matrix fmul fadd) (combine_matrix fadd)" (is ?concl)
 proof -
-  from prems show ?concl 
+  from prems show ?concl
     apply (simp add: r_distributive_def mult_matrix_def, auto)
     proof -
       fix a::"'a matrix"
       fix u::"'b matrix"
       fix v::"'b matrix"
       let ?mx = "max (ncols a) (max (nrows u) (nrows v))"
       from prems show "mult_matrix_n (max (ncols a) (nrows (combine_matrix fadd u v))) fmul fadd a (combine_matrix fadd u v) =
         combine_matrix fadd (mult_matrix_n (max (ncols a) (nrows u)) fmul fadd a u) (mult_matrix_n (max (ncols a) (nrows v)) fmul fadd a v)"
-	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
-	apply (simp add: max1 max2 combine_nrows combine_ncols)+
-	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
-	apply (simp add: max1 max2 combine_nrows combine_ncols)+
-	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
-	apply (simp add: max1 max2 combine_nrows combine_ncols)+
-	apply (simp add: mult_matrix_n_def r_distributive_def foldseq_distr[of fadd])
-	apply (simp add: combine_matrix_def combine_infmatrix_def)
-	apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
-	apply (subst RepAbs_matrix)
-	apply (simp, auto)
-	apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
-	apply (rule exI[of _ "ncols v"], simp add: ncols_le foldseq_zero)
-	apply (subst RepAbs_matrix)
-	apply (simp, auto)
-	apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
-	apply (rule exI[of _ "ncols u"], simp add: ncols_le foldseq_zero)
-	done
+        apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+        apply (simp add: max1 max2 combine_nrows combine_ncols)+
+        apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+        apply (simp add: max1 max2 combine_nrows combine_ncols)+
+        apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+        apply (simp add: max1 max2 combine_nrows combine_ncols)+
+        apply (simp add: mult_matrix_n_def r_distributive_def foldseq_distr[of fadd])
+        apply (simp add: combine_matrix_def combine_infmatrix_def)
+        apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
+        apply (subst RepAbs_matrix)
+        apply (simp, auto)
+        apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
+        apply (rule exI[of _ "ncols v"], simp add: ncols_le foldseq_zero)
+        apply (subst RepAbs_matrix)
+        apply (simp, auto)
+        apply (rule exI[of _ "nrows a"], simp add: nrows_le foldseq_zero)
+        apply (rule exI[of _ "ncols u"], simp add: ncols_le foldseq_zero)
+        done
     qed
 qed
 
 lemma l_distributive_matrix:
- assumes prems: 
-  "l_distributive fmul fadd" 
-  "associative fadd" 
-  "commutative fadd" 
-  "fadd 0 0 = 0" 
-  "! a. fmul a 0 = 0" 
+ assumes prems:
+  "l_distributive fmul fadd"
+  "associative fadd"
+  "commutative fadd"
+  "fadd 0 0 = 0"
+  "! a. fmul a 0 = 0"
   "! a. fmul 0 a = 0"
  shows "l_distributive (mult_matrix fmul fadd) (combine_matrix fadd)" (is ?concl)
 proof -
-  from prems show ?concl 
+  from prems show ?concl
     apply (simp add: l_distributive_def mult_matrix_def, auto)
     proof -
       fix a::"'b matrix"
       fix u::"'a matrix"
       fix v::"'a matrix"
       let ?mx = "max (nrows a) (max (ncols u) (ncols v))"
       from prems show "mult_matrix_n (max (ncols (combine_matrix fadd u v)) (nrows a)) fmul fadd (combine_matrix fadd u v) a =
                combine_matrix fadd (mult_matrix_n (max (ncols u) (nrows a)) fmul fadd u a) (mult_matrix_n (max (ncols v) (nrows a)) fmul fadd v a)"
-	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
-	apply (simp add: max1 max2 combine_nrows combine_ncols)+
-	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
-	apply (simp add: max1 max2 combine_nrows combine_ncols)+
-	apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
-	apply (simp add: max1 max2 combine_nrows combine_ncols)+
-	apply (simp add: mult_matrix_n_def l_distributive_def foldseq_distr[of fadd])
-	apply (simp add: combine_matrix_def combine_infmatrix_def)
-	apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
-	apply (subst RepAbs_matrix)
-	apply (simp, auto)
-	apply (rule exI[of _ "nrows v"], simp add: nrows_le foldseq_zero)
-	apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
-	apply (subst RepAbs_matrix)
-	apply (simp, auto)
-	apply (rule exI[of _ "nrows u"], simp add: nrows_le foldseq_zero)
-	apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
-	done
+        apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+        apply (simp add: max1 max2 combine_nrows combine_ncols)+
+        apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+        apply (simp add: max1 max2 combine_nrows combine_ncols)+
+        apply (subst mult_matrix_nm[of _ _ _ ?mx fadd fmul])
+        apply (simp add: max1 max2 combine_nrows combine_ncols)+
+        apply (simp add: mult_matrix_n_def l_distributive_def foldseq_distr[of fadd])
+        apply (simp add: combine_matrix_def combine_infmatrix_def)
+        apply (rule comb[of "Abs_matrix" "Abs_matrix"], simp, (rule ext)+)
+        apply (subst RepAbs_matrix)
+        apply (simp, auto)
+        apply (rule exI[of _ "nrows v"], simp add: nrows_le foldseq_zero)
+        apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
+        apply (subst RepAbs_matrix)
+        apply (simp, auto)
+        apply (rule exI[of _ "nrows u"], simp add: nrows_le foldseq_zero)
+        apply (rule exI[of _ "ncols a"], simp add: ncols_le foldseq_zero)
+        done
     qed
 qed
 
-    
+
 instance matrix :: (zero)zero by intro_classes
 
 defs(overloaded)
-  zero_matrix_def: "(0::('a::zero) matrix) == Abs_matrix(% j i. 0)" 
+  zero_matrix_def: "(0::('a::zero) matrix) == Abs_matrix(% j i. 0)"
 
 lemma Rep_zero_matrix_def[simp]: "Rep_matrix 0 j i = 0"
   apply (simp add: zero_matrix_def)
   apply (subst RepAbs_matrix)
   by (auto)

 lemma zero_matrix_def_ncols[simp]: "ncols 0 = 0"
 proof -
   have a:"!! (x::nat). x <= 0 \<Longrightarrow> x = 0" by (arith)
   show "ncols 0 = 0" by (rule a, subst ncols_le, simp)
 qed
-  
+
 lemma combine_matrix_zero_r_neutral: "zero_r_neutral f \<Longrightarrow> zero_r_neutral (combine_matrix f)"
   by (simp add: zero_r_neutral_def combine_matrix_def combine_infmatrix_def)
 
 lemma mult_matrix_zero_closed: "\<lbrakk>fadd 0 0 = 0; zero_closed fmul\<rbrakk> \<Longrightarrow> zero_closed (mult_matrix fmul fadd)"
   apply (simp add: zero_closed_def mult_matrix_def mult_matrix_n_def)

   take_columns :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
   "take_columns A c == Abs_matrix(% j i. if (i < c) then (Rep_matrix A j i) else 0)"
 
 constdefs
   column_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
-  "column_of_matrix A n == take_columns (move_matrix A 0 (- int n)) 1"  
+  "column_of_matrix A n == take_columns (move_matrix A 0 (- int n)) 1"
   row_of_matrix :: "('a::zero) matrix \<Rightarrow> nat \<Rightarrow> 'a matrix"
   "row_of_matrix A m == take_rows (move_matrix A (- int m) 0) 1"
 
 lemma Rep_singleton_matrix[simp]: "Rep_matrix (singleton_matrix j i e) m n = (if j = m & i = n then e else 0)"
 apply (simp add: singleton_matrix_def)

 apply (simp add: Suc_le_eq)
 apply (rule not_leE)
 apply (subst nrows_le)
 by simp
 
-lemma ncols_singleton[simp]: "ncols(singleton_matrix j i e) = (if e = 0 then 0 else Suc i)" 
+lemma ncols_singleton[simp]: "ncols(singleton_matrix j i e) = (if e = 0 then 0 else Suc i)"
 apply (auto)
 apply (rule le_anti_sym)
 apply (subst ncols_le)
 apply (simp add: singleton_matrix_def, auto)
 apply (subst RepAbs_matrix)

 apply (simp)
 apply (simp add: Suc_le_eq)
 apply (rule not_leE)
 apply (subst ncols_le)
 by simp
- 
+
 lemma combine_singleton: "f 0 0 = 0 \<Longrightarrow> combine_matrix f (singleton_matrix j i a) (singleton_matrix j i b) = singleton_matrix j i (f a b)"
 apply (simp add: singleton_matrix_def combine_matrix_def combine_infmatrix_def)
 apply (subst RepAbs_matrix)
 apply (rule exI[of _ "Suc j"], simp)
 apply (rule exI[of _ "Suc i"], simp)

 apply (subst RepAbs_matrix)
 apply (rule exI[of _ "Suc j"], simp)
 apply (rule exI[of _ "Suc i"], simp)
 by simp
 
-lemma Rep_move_matrix[simp]: 
-  "Rep_matrix (move_matrix A y x) j i = 
+lemma Rep_move_matrix[simp]:
+  "Rep_matrix (move_matrix A y x) j i =
   (if (neg ((int j)-y)) | (neg ((int i)-x)) then 0 else Rep_matrix A (nat((int j)-y)) (nat((int i)-x)))"
 apply (simp add: move_matrix_def)
 apply (auto)
 by (subst RepAbs_matrix,
   rule exI[of _ "(nrows A)+(nat (abs y))"], auto, rule nrows, arith,
   rule exI[of _ "(ncols A)+(nat (abs x))"], auto, rule ncols, arith)+
 
 lemma Rep_take_columns[simp]:
   "Rep_matrix (take_columns A c) j i =
-  (if i < c then (Rep_matrix A j i) else 0)" 
+  (if i < c then (Rep_matrix A j i) else 0)"
 apply (simp add: take_columns_def)
 apply (subst RepAbs_matrix)
 apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
 apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
 done
 
 lemma Rep_take_rows[simp]:
   "Rep_matrix (take_rows A r) j i =
-  (if j < r then (Rep_matrix A j i) else 0)" 
+  (if j < r then (Rep_matrix A j i) else 0)"
 apply (simp add: take_rows_def)
 apply (subst RepAbs_matrix)
 apply (rule exI[of _ "nrows A"], auto, simp add: nrows_le)
 apply (rule exI[of _ "ncols A"], auto, simp add: ncols_le)
 done

   eprem:
   "? e. (! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)"
   and fprems:
   "! a. fmul 0 a = 0"
   "! a. fmul a 0 = 0"
-  "! a. fadd a 0 = a" 
+  "! a. fadd a 0 = a"
   "! a. fadd 0 a = a"
   and contraprems:
   "mult_matrix fmul fadd A = mult_matrix fmul fadd B"
   shows
   "A = B"
 proof(rule contrapos_np[of "False"], simp)
   assume a: "A \<noteq> B"
   have b: "!! f g. (! x y. f x y = g x y) \<Longrightarrow> f = g" by ((rule ext)+, auto)
   have "? j i. (Rep_matrix A j i) \<noteq> (Rep_matrix B j i)"
-    apply (rule contrapos_np[of "False"], simp+)    
+    apply (rule contrapos_np[of "False"], simp+)
     apply (insert b[of "Rep_matrix A" "Rep_matrix B"], simp)
     by (simp add: Rep_matrix_inject a)
   then obtain J I where c:"(Rep_matrix A J I) \<noteq> (Rep_matrix B J I)" by blast
   from eprem obtain e where eprops:"(! a b. a \<noteq> b \<longrightarrow> fmul a e \<noteq> fmul b e)" by blast
   let ?S = "singleton_matrix I 0 e"

 
 constdefs
   foldmatrix :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"
   "foldmatrix f g A m n == foldseq_transposed g (% j. foldseq f (A j) n) m"
   foldmatrix_transposed :: "('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> ('a infmatrix) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"
-  "foldmatrix_transposed f g A m n == foldseq g (% j. foldseq_transposed f (A j) n) m" 
+  "foldmatrix_transposed f g A m n == foldseq g (% j. foldseq_transposed f (A j) n) m"
 
 lemma foldmatrix_transpose:
   assumes
   "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
   shows
   "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m" (is ?concl)
 proof -
   have forall:"!! P x. (! x. P x) \<Longrightarrow> P x" by auto
   have tworows:"! A. foldmatrix f g A 1 n = foldmatrix_transposed g f (transpose_infmatrix A) n 1"
-    apply (induct n) 
+    apply (induct n)
     apply (simp add: foldmatrix_def foldmatrix_transposed_def prems)+
     apply (auto)
     by (drule_tac x1="(% j i. A j (Suc i))" in forall, simp)
   show "foldmatrix f g A m n = foldmatrix_transposed g f (transpose_infmatrix A) n m"
     apply (simp add: foldmatrix_def foldmatrix_transposed_def)
     apply (induct m, simp)
-    apply (simp)    
+    apply (simp)
     apply (insert tworows)
     apply (drule_tac x="% j i. (if j = 0 then (foldseq_transposed g (\<lambda>u. A u i) na) else (A (Suc na) i))" in spec)
     by (simp add: foldmatrix_def foldmatrix_transposed_def)
 qed
 
 lemma foldseq_foldseq:
-assumes 
+assumes
   "associative f"
   "associative g"
-  "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)" 
-shows 
+  "! a b c d. g(f a b) (f c d) = f (g a c) (g b d)"
+shows
   "foldseq g (% j. foldseq f (A j) n) m = foldseq f (% j. foldseq g ((transpose_infmatrix A) j) m) n"
   apply (insert foldmatrix_transpose[of g f A m n])
   by (simp add: foldmatrix_def foldmatrix_transposed_def foldseq_assoc[THEN sym] prems)
-  
-lemma mult_n_nrows: 
-assumes 
+
+lemma mult_n_nrows:
+assumes
 "! a. fmul 0 a = 0"
 "! a. fmul a 0 = 0"
 "fadd 0 0 = 0"
 shows "nrows (mult_matrix_n n fmul fadd A B) \<le> nrows A"
 apply (subst nrows_le)

 apply (simp add: nrows prems foldseq_zero)
 apply (rule_tac x="ncols B" in exI)
 apply (simp add: ncols prems foldseq_zero)
 by (simp add: nrows prems foldseq_zero)
 
-lemma mult_n_ncols: 
-assumes 
+lemma mult_n_ncols:
+assumes
 "! a. fmul 0 a = 0"
 "! a. fmul a 0 = 0"
 "fadd 0 0 = 0"
 shows "ncols (mult_matrix_n n fmul fadd A B) \<le> ncols B"
 apply (subst ncols_le)

 apply (simp add: nrows prems foldseq_zero)
 apply (rule_tac x="ncols B" in exI)
 apply (simp add: ncols prems foldseq_zero)
 by (simp add: ncols prems foldseq_zero)
 
-lemma mult_nrows: 
-assumes 
+lemma mult_nrows:
+assumes
 "! a. fmul 0 a = 0"
 "! a. fmul a 0 = 0"
 "fadd 0 0 = 0"
 shows "nrows (mult_matrix fmul fadd A B) \<le> nrows A"
 by (simp add: mult_matrix_def mult_n_nrows prems)
 
 lemma mult_ncols:
-assumes 
+assumes
 "! a. fmul 0 a = 0"
 "! a. fmul a 0 = 0"
 "fadd 0 0 = 0"
 shows "ncols (mult_matrix fmul fadd A B) \<le> ncols B"
 by (simp add: mult_matrix_def mult_n_ncols prems)

   show ?concl
     apply (simp add: Rep_matrix_inject[THEN sym])
     apply (rule ext)+
     apply (simp add: mult_matrix_def)
     apply (subst mult_matrix_nm[of _ "max (ncols (mult_matrix_n (max (ncols A) (nrows B)) fmul1 fadd1 A B)) (nrows C)" _ "max (ncols B) (nrows C)"])
-    apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+    
+    apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
     apply (subst mult_matrix_nm[of _ "max (ncols A) (nrows (mult_matrix_n (max (ncols B) (nrows C)) fmul2 fadd2 B C))" _ "max (ncols A) (nrows B)"])     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
     apply (subst mult_matrix_nm[of _ _ _ "?N"])
     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+
     apply (subst mult_matrix_nm[of _ _ _ "?N"])
     apply (simp add: max1 max2 mult_n_ncols mult_n_nrows prems)+

     apply (simp add: fmul2fadd1fold fmul1fadd2fold prems)
     apply (subst foldseq_foldseq)
     apply (simp add: prems)+
     by (simp add: transpose_infmatrix)
 qed
- 
-lemma 
+
+lemma
   assumes prems:
   "! a. fmul1 0 a = 0"
   "! a. fmul1 a 0 = 0"
   "! a. fmul2 0 a = 0"
   "! a. fmul2 a 0 = 0"

   "associative fadd2"
   "! a b c. fmul2 (fmul1 a b) c = fmul1 a (fmul2 b c)"
   "! a b c. fmul2 (fadd1 a b) c = fadd1 (fmul2 a c) (fmul2 b c)"
   "! a b c. fmul1 c (fadd2 a b) = fadd2 (fmul1 c a) (fmul1 c b)"
   shows
-  "(mult_matrix fmul1 fadd1 A) o (mult_matrix fmul2 fadd2 B) = mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B)" 
+  "(mult_matrix fmul1 fadd1 A) o (mult_matrix fmul2 fadd2 B) = mult_matrix fmul2 fadd2 (mult_matrix fmul1 fadd1 A B)"
 apply (rule ext)+
 apply (simp add: comp_def )
 by (simp add: mult_matrix_assoc prems)
 
 lemma mult_matrix_assoc_simple:

   "distributive fmul fadd"
   shows "mult_matrix fmul fadd (mult_matrix fmul fadd A B) C = mult_matrix fmul fadd A (mult_matrix fmul fadd B C)" (is ?concl)
 proof -
   have "!! a b c d. fadd (fadd a b) (fadd c d) = fadd (fadd a c) (fadd b d)"
     by (simp! add: associative_def commutative_def)
-  then show ?concl 
+  then show ?concl
     apply (subst mult_matrix_assoc)
     apply (simp_all!)
     by (simp add: associative_def distributive_def l_distributive_def r_distributive_def)+
 qed
 

 lemma transpose_combine_matrix: "f 0 0 = 0 \<Longrightarrow> transpose_matrix (combine_matrix f A B) = combine_matrix f (transpose_matrix A) (transpose_matrix B)"
 apply (simp add: Rep_matrix_inject[THEN sym])
 apply (rule ext)+
 by simp
 
-lemma Rep_mult_matrix: 
-  assumes 
-  "! a. fmul 0 a = 0"  
+lemma Rep_mult_matrix:
+  assumes
+  "! a. fmul 0 a = 0"
   "! a. fmul a 0 = 0"
   "fadd 0 0 = 0"
   shows
-  "Rep_matrix(mult_matrix fmul fadd A B) j i = 
+  "Rep_matrix(mult_matrix fmul fadd A B) j i =
   foldseq fadd (% k. fmul (Rep_matrix A j k) (Rep_matrix B k i)) (max (ncols A) (nrows B))"
-apply (simp add: mult_matrix_def mult_matrix_n_def) 
+apply (simp add: mult_matrix_def mult_matrix_n_def)
 apply (subst RepAbs_matrix)
 apply (rule exI[of _ "nrows A"], simp! add: nrows foldseq_zero)
 apply (rule exI[of _ "ncols B"], simp! add: ncols foldseq_zero)
 by simp
 
 lemma transpose_mult_matrix:
-  assumes 
-  "! a. fmul 0 a = 0"  
+  assumes
+  "! a. fmul 0 a = 0"
   "! a. fmul a 0 = 0"
   "fadd 0 0 = 0"
-  "! x y. fmul y x = fmul x y" 
+  "! x y. fmul y x = fmul x y"
   shows
   "transpose_matrix (mult_matrix fmul fadd A B) = mult_matrix fmul fadd (transpose_matrix B) (transpose_matrix A)"
   apply (simp add: Rep_matrix_inject[THEN sym])
   apply (rule ext)+
   by (simp! add: Rep_mult_matrix max_ac)

 
 defs (overloaded)
   le_matrix_def: "(A::('a::{ord,zero}) matrix) <= B == ! j i. (Rep_matrix A j i) <= (Rep_matrix B j i)"
   less_def: "(A::('a::{ord,zero}) matrix) < B == (A <= B) & (A \<noteq> B)"
 
-instance matrix :: ("{order, zero}") order  
+instance matrix :: ("{order, zero}") order
 apply intro_classes
 apply (simp_all add: le_matrix_def less_def)
 apply (auto)
 apply (drule_tac x=j in spec, drule_tac x=j in spec)
 apply (drule_tac x=i in spec, drule_tac x=i in spec)

 apply (rule ext)+
 apply (drule_tac x=xa in spec, drule_tac x=xa in spec)
 apply (drule_tac x=xb in spec, drule_tac x=xb in spec)
 by simp
 
-lemma le_apply_matrix: 
+lemma le_apply_matrix:
   assumes
   "f 0 = 0"
   "! x y. x <= y \<longrightarrow> f x <= f y"
   "(a::('a::{ord, zero}) matrix) <= b"
   shows

 by (simp! add: le_matrix_def)
 
 lemma le_left_combine_matrix:
   assumes
   "f 0 0 = 0"
-  "! a b c. 0 <= c & a <= b \<longrightarrow> f c a <= f c b" 
+  "! a b c. 0 <= c & a <= b \<longrightarrow> f c a <= f c b"
   "0 <= C"
   "A <= B"
-  shows 
+  shows
   "combine_matrix f C A <= combine_matrix f C B"
   by (simp! add: le_matrix_def)
 
 lemma le_right_combine_matrix:
   assumes
   "f 0 0 = 0"
-  "! a b c. 0 <= c & a <= b \<longrightarrow> f a c <= f b c" 
+  "! a b c. 0 <= c & a <= b \<longrightarrow> f a c <= f b c"
   "0 <= C"
   "A <= B"
-  shows 
+  shows
   "combine_matrix f A C <= combine_matrix f B C"
   by (simp! add: le_matrix_def)
 
 lemma le_transpose_matrix: "(A <= B) = (transpose_matrix A <= transpose_matrix B)"
   by (simp add: le_matrix_def, auto)
 
 lemma le_foldseq:
   assumes
-  "! a b c d . a <= b & c <= d \<longrightarrow> f a c <= f b d" 
+  "! a b c d . a <= b & c <= d \<longrightarrow> f a c <= f b d"
   "! i. i <= n \<longrightarrow> s i <= t i"
   shows
   "foldseq f s n <= foldseq f t n"
 proof -
   have "! s t. (! i. i<=n \<longrightarrow> s i <= t i) \<longrightarrow> foldseq f s n <= foldseq f t n" by (induct_tac n, simp_all!)
   then show "foldseq f s n <= foldseq f t n" by (simp!)
-qed  
+qed
 
 lemma le_left_mult:
   assumes
   "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
   "! c a b. 0 <= c & a <= b \<longrightarrow> fmul c a <= fmul c b"
-  "! a. fmul 0 a = 0"  
+  "! a. fmul 0 a = 0"
   "! a. fmul a 0 = 0"
-  "fadd 0 0 = 0" 
+  "fadd 0 0 = 0"
   "0 <= C"
   "A <= B"
   shows
   "mult_matrix fmul fadd C A <= mult_matrix fmul fadd C B"
   apply (simp! add: le_matrix_def Rep_mult_matrix)

 
 lemma le_right_mult:
   assumes
   "! a b c d. a <= b & c <= d \<longrightarrow> fadd a c <= fadd b d"
   "! c a b. 0 <= c & a <= b \<longrightarrow> fmul a c <= fmul b c"
-  "! a. fmul 0 a = 0"  
+  "! a. fmul 0 a = 0"
   "! a. fmul a 0 = 0"
-  "fadd 0 0 = 0" 
+  "fadd 0 0 = 0"
   "0 <= C"
   "A <= B"
   shows
   "mult_matrix fmul fadd A C <= mult_matrix fmul fadd B C"
   apply (simp! add: le_matrix_def Rep_mult_matrix)