(*  Title:      HOL/Matrix/SparseMatrix.thy

     ID:         $Id$

     Author:     Steven Obua

 *)

 theory SparseMatrix

 imports "./Matrix"

 begin

 types 

   'a spvec = "(nat * 'a) list"

   'a spmat = "('a spvec) spvec"

 definition sparse_row_vector :: "('a::ab_group_add) spvec \<Rightarrow> 'a matrix" where

   sparse_row_vector_def: "sparse_row_vector arr = foldl (% m x. m + (singleton_matrix 0 (fst x) (snd x))) 0 arr"

 definition sparse_row_matrix :: "('a::ab_group_add) spmat \<Rightarrow> 'a matrix" where

   sparse_row_matrix_def: "sparse_row_matrix arr = foldl (% m r. m + (move_matrix (sparse_row_vector (snd r)) (int (fst r)) 0)) 0 arr"

 code_datatype sparse_row_vector sparse_row_matrix

 lemma sparse_row_vector_empty [simp]: "sparse_row_vector [] = 0"

   by (simp add: sparse_row_vector_def)

 lemma sparse_row_matrix_empty [simp]: "sparse_row_matrix [] = 0"

   by (simp add: sparse_row_matrix_def)

 lemmas [code] = sparse_row_vector_empty [symmetric]

 lemma foldl_distrstart[rule_format]: "! a x y. (f (g x y) a = g x (f y a)) \<Longrightarrow> ! x y. (foldl f (g x y) l = g x (foldl f y l))"

   by (induct l, auto)

 lemma sparse_row_vector_cons[simp]:

   "sparse_row_vector (a # arr) = (singleton_matrix 0 (fst a) (snd a)) + (sparse_row_vector arr)"

   apply (induct arr)

   apply (auto simp add: sparse_row_vector_def)

   apply (simp add: foldl_distrstart [of "\<lambda>m x. m + singleton_matrix 0 (fst x) (snd x)" "\<lambda>x m. singleton_matrix 0 (fst x) (snd x) + m"])

   done

 lemma sparse_row_vector_append[simp]:

   "sparse_row_vector (a @ b) = (sparse_row_vector a) + (sparse_row_vector b)"

   by (induct a) auto

 lemma nrows_spvec[simp]: "nrows (sparse_row_vector x) <= (Suc 0)"

   apply (induct x)

   apply (simp_all add: add_nrows)

   done

 lemma sparse_row_matrix_cons: "sparse_row_matrix (a#arr) = ((move_matrix (sparse_row_vector (snd a)) (int (fst a)) 0)) + sparse_row_matrix arr"

   apply (induct arr)

   apply (auto simp add: sparse_row_matrix_def)

   apply (simp add: foldl_distrstart[of "\<lambda>m x. m + (move_matrix (sparse_row_vector (snd x)) (int (fst x)) 0)" 

     "% a m. (move_matrix (sparse_row_vector (snd a)) (int (fst a)) 0) + m"])

   done

 lemma sparse_row_matrix_append: "sparse_row_matrix (arr@brr) = (sparse_row_matrix arr) + (sparse_row_matrix brr)"

   apply (induct arr)

   apply (auto simp add: sparse_row_matrix_cons)

   done

 primrec sorted_spvec :: "'a spvec \<Rightarrow> bool" where

   "sorted_spvec [] = True"

   | sorted_spvec_step: "sorted_spvec (a#as) = (case as of [] \<Rightarrow> True | b#bs \<Rightarrow> ((fst a < fst b) & (sorted_spvec as)))" 

 primrec sorted_spmat :: "'a spmat \<Rightarrow> bool" where

   "sorted_spmat [] = True"

   | "sorted_spmat (a#as) = ((sorted_spvec (snd a)) & (sorted_spmat as))"

 declare sorted_spvec.simps [simp del]

 lemma sorted_spvec_empty[simp]: "sorted_spvec [] = True"

 by (simp add: sorted_spvec.simps)

 lemma sorted_spvec_cons1: "sorted_spvec (a#as) \<Longrightarrow> sorted_spvec as"

 apply (induct as)

 apply (auto simp add: sorted_spvec.simps)

 done

 lemma sorted_spvec_cons2: "sorted_spvec (a#b#t) \<Longrightarrow> sorted_spvec (a#t)"

 apply (induct t)

 apply (auto simp add: sorted_spvec.simps)

 done

 lemma sorted_spvec_cons3: "sorted_spvec(a#b#t) \<Longrightarrow> fst a < fst b"

 apply (auto simp add: sorted_spvec.simps)

 done

 lemma sorted_sparse_row_vector_zero[rule_format]: "m <= n \<longrightarrow> sorted_spvec ((n,a)#arr) \<longrightarrow> Rep_matrix (sparse_row_vector arr) j m = 0"

 apply (induct arr)

 apply (auto)

 apply (frule sorted_spvec_cons2,simp)+

 apply (frule sorted_spvec_cons3, simp)

 done

 lemma sorted_sparse_row_matrix_zero[rule_format]: "m <= n \<longrightarrow> sorted_spvec ((n,a)#arr) \<longrightarrow> Rep_matrix (sparse_row_matrix arr) m j = 0"

   apply (induct arr)

   apply (auto)

   apply (frule sorted_spvec_cons2, simp)

   apply (frule sorted_spvec_cons3, simp)

   apply (simp add: sparse_row_matrix_cons neg_def)

   done

 primrec minus_spvec :: "('a::ab_group_add) spvec \<Rightarrow> 'a spvec" where

   "minus_spvec [] = []"

   | "minus_spvec (a#as) = (fst a, -(snd a))#(minus_spvec as)"

 primrec abs_spvec ::  "('a::lordered_ab_group_add_abs) spvec \<Rightarrow> 'a spvec" where

   "abs_spvec [] = []"

   | "abs_spvec (a#as) = (fst a, abs (snd a))#(abs_spvec as)"

 lemma sparse_row_vector_minus: 

   "sparse_row_vector (minus_spvec v) = - (sparse_row_vector v)"

   apply (induct v)

   apply (simp_all add: sparse_row_vector_cons)

   apply (simp add: Rep_matrix_inject[symmetric])

   apply (rule ext)+

   apply simp

   done

 instance matrix :: (lordered_ab_group_add_abs) lordered_ab_group_add_abs

 apply default

 unfolding abs_matrix_def .. (*FIXME move*)

 lemma sparse_row_vector_abs:

   "sorted_spvec (v :: 'a::lordered_ring spvec) \<Longrightarrow> sparse_row_vector (abs_spvec v) = abs (sparse_row_vector v)"

   apply (induct v)

   apply simp_all

   apply (frule_tac sorted_spvec_cons1, simp)

   apply (simp only: Rep_matrix_inject[symmetric])

   apply (rule ext)+

   apply auto

   apply (subgoal_tac "Rep_matrix (sparse_row_vector v) 0 a = 0")

   apply (simp)

   apply (rule sorted_sparse_row_vector_zero)

   apply auto

   done

 lemma sorted_spvec_minus_spvec:

   "sorted_spvec v \<Longrightarrow> sorted_spvec (minus_spvec v)"

   apply (induct v)

   apply (simp)

   apply (frule sorted_spvec_cons1, simp)

   apply (simp add: sorted_spvec.simps split:list.split_asm)

   done

 lemma sorted_spvec_abs_spvec:

   "sorted_spvec v \<Longrightarrow> sorted_spvec (abs_spvec v)"

   apply (induct v)

   apply (simp)

   apply (frule sorted_spvec_cons1, simp)

   apply (simp add: sorted_spvec.simps split:list.split_asm)

   done

 definition

   "smult_spvec y = map (% a. (fst a, y * snd a))"  

 lemma smult_spvec_empty[simp]: "smult_spvec y [] = []"

   by (simp add: smult_spvec_def)

 lemma smult_spvec_cons: "smult_spvec y (a#arr) = (fst a, y * (snd a)) # (smult_spvec y arr)"

   by (simp add: smult_spvec_def)

 consts addmult_spvec :: "('a::ring) * 'a spvec * 'a spvec \<Rightarrow> 'a spvec"

 recdef addmult_spvec "measure (% (y, a, b). length a + (length b))"

   "addmult_spvec (y, arr, []) = arr"

   "addmult_spvec (y, [], brr) = smult_spvec y brr"

   "addmult_spvec (y, a#arr, b#brr) = (

     if (fst a) < (fst b) then (a#(addmult_spvec (y, arr, b#brr))) 

     else (if (fst b < fst a) then ((fst b, y * (snd b))#(addmult_spvec (y, a#arr, brr)))

     else ((fst a, (snd a)+ y*(snd b))#(addmult_spvec (y, arr,brr)))))"

 lemma addmult_spvec_empty1[simp]: "addmult_spvec (y, [], a) = smult_spvec y a"

   by (induct a) auto

 lemma addmult_spvec_empty2[simp]: "addmult_spvec (y, a, []) = a"

   by (induct a) auto

 lemma sparse_row_vector_map: "(! x y. f (x+y) = (f x) + (f y)) \<Longrightarrow> (f::'a\<Rightarrow>('a::lordered_ring)) 0 = 0 \<Longrightarrow> 

   sparse_row_vector (map (% x. (fst x, f (snd x))) a) = apply_matrix f (sparse_row_vector a)"

   apply (induct a)

   apply (simp_all add: apply_matrix_add)

   done

 lemma sparse_row_vector_smult: "sparse_row_vector (smult_spvec y a) = scalar_mult y (sparse_row_vector a)"

   apply (induct a)

   apply (simp_all add: smult_spvec_cons scalar_mult_add)

   done

 lemma sparse_row_vector_addmult_spvec: "sparse_row_vector (addmult_spvec (y::'a::lordered_ring, a, b)) = 

   (sparse_row_vector a) + (scalar_mult y (sparse_row_vector b))"

   apply (rule addmult_spvec.induct[of _ y])

   apply (simp add: scalar_mult_add smult_spvec_cons sparse_row_vector_smult singleton_matrix_add)+

   done

 lemma sorted_smult_spvec[rule_format]: "sorted_spvec a \<Longrightarrow> sorted_spvec (smult_spvec y a)"

   apply (auto simp add: smult_spvec_def)

   apply (induct a)

   apply (auto simp add: sorted_spvec.simps split:list.split_asm)

   done

 lemma sorted_spvec_addmult_spvec_helper: "\<lbrakk>sorted_spvec (addmult_spvec (y, (a, b) # arr, brr)); aa < a; sorted_spvec ((a, b) # arr); 

   sorted_spvec ((aa, ba) # brr)\<rbrakk> \<Longrightarrow> sorted_spvec ((aa, y * ba) # addmult_spvec (y, (a, b) # arr, brr))"  

   apply (induct brr)

   apply (auto simp add: sorted_spvec.simps)

   apply (simp split: list.split)

   apply (auto)

   apply (simp split: list.split)

   apply (auto)

   done

 lemma sorted_spvec_addmult_spvec_helper2: 

  "\<lbrakk>sorted_spvec (addmult_spvec (y, arr, (aa, ba) # brr)); a < aa; sorted_spvec ((a, b) # arr); sorted_spvec ((aa, ba) # brr)\<rbrakk>

        \<Longrightarrow> sorted_spvec ((a, b) # addmult_spvec (y, arr, (aa, ba) # brr))"

   apply (induct arr)

   apply (auto simp add: smult_spvec_def sorted_spvec.simps)

   apply (simp split: list.split)

   apply (auto)

   done

 lemma sorted_spvec_addmult_spvec_helper3[rule_format]:

   "sorted_spvec (addmult_spvec (y, arr, brr)) \<longrightarrow> sorted_spvec ((aa, b) # arr) \<longrightarrow> sorted_spvec ((aa, ba) # brr)

      \<longrightarrow> sorted_spvec ((aa, b + y * ba) # (addmult_spvec (y, arr, brr)))"

   apply (rule addmult_spvec.induct[of _ y arr brr])

   apply (simp_all add: sorted_spvec.simps smult_spvec_def)

   done

 lemma sorted_addmult_spvec[rule_format]: "sorted_spvec a \<longrightarrow> sorted_spvec b \<longrightarrow> sorted_spvec (addmult_spvec (y, a, b))"

   apply (rule addmult_spvec.induct[of _ y a b])

   apply (simp_all add: sorted_smult_spvec)

   apply (rule conjI, intro strip)

   apply (case_tac "~(a < aa)")

   apply (simp_all)

   apply (frule_tac as=brr in sorted_spvec_cons1)

   apply (simp add: sorted_spvec_addmult_spvec_helper)

   apply (intro strip | rule conjI)+

   apply (frule_tac as=arr in sorted_spvec_cons1)

   apply (simp add: sorted_spvec_addmult_spvec_helper2)

   apply (intro strip)

   apply (frule_tac as=arr in sorted_spvec_cons1)

   apply (frule_tac as=brr in sorted_spvec_cons1)

   apply (simp)

   apply (simp_all add: sorted_spvec_addmult_spvec_helper3)

   done

 consts 

   mult_spvec_spmat :: "('a::lordered_ring) spvec * 'a spvec * 'a spmat  \<Rightarrow> 'a spvec"

 recdef mult_spvec_spmat "measure (% (c, arr, brr). (length arr) + (length brr))"

   "mult_spvec_spmat (c, [], brr) = c"

   "mult_spvec_spmat (c, arr, []) = c"

   "mult_spvec_spmat (c, a#arr, b#brr) = (

      if ((fst a) < (fst b)) then (mult_spvec_spmat (c, arr, b#brr))

      else (if ((fst b) < (fst a)) then (mult_spvec_spmat (c, a#arr, brr)) 

      else (mult_spvec_spmat (addmult_spvec (snd a, c, snd b), arr, brr))))"

 lemma sparse_row_mult_spvec_spmat[rule_format]: "sorted_spvec (a::('a::lordered_ring) spvec) \<longrightarrow> sorted_spvec B \<longrightarrow> 

   sparse_row_vector (mult_spvec_spmat (c, a, B)) = (sparse_row_vector c) + (sparse_row_vector a) * (sparse_row_matrix B)"

 proof -

   have comp_1: "!! a b. a < b \<Longrightarrow> Suc 0 <= nat ((int b)-(int a))" by arith

   have not_iff: "!! a b. a = b \<Longrightarrow> (~ a) = (~ b)" by simp

   have max_helper: "!! a b. ~ (a <= max (Suc a) b) \<Longrightarrow> False"

     by arith

   {

     fix a 

     fix v

     assume a:"a < nrows(sparse_row_vector v)"

     have b:"nrows(sparse_row_vector v) <= 1" by simp

     note dummy = less_le_trans[of a "nrows (sparse_row_vector v)" 1, OF a b]   

     then have "a = 0" by simp

   }

   note nrows_helper = this

   show ?thesis

     apply (rule mult_spvec_spmat.induct)

     apply simp+

     apply (rule conjI)

     apply (intro strip)

     apply (frule_tac as=brr in sorted_spvec_cons1)

     apply (simp add: ring_simps sparse_row_matrix_cons)

     apply (simplesubst Rep_matrix_zero_imp_mult_zero) 

     apply (simp)

     apply (intro strip)

     apply (rule disjI2)

     apply (intro strip)

     apply (subst nrows)

     apply (rule  order_trans[of _ 1])

     apply (simp add: comp_1)+

     apply (subst Rep_matrix_zero_imp_mult_zero)

     apply (intro strip)

     apply (case_tac "k <= aa")

     apply (rule_tac m1 = k and n1 = a and a1 = b in ssubst[OF sorted_sparse_row_vector_zero])

     apply (simp_all)

     apply (rule impI)

     apply (rule disjI2)

     apply (rule nrows)

     apply (rule order_trans[of _ 1])

     apply (simp_all add: comp_1)

     apply (intro strip | rule conjI)+

     apply (frule_tac as=arr in sorted_spvec_cons1)

     apply (simp add: ring_simps)

     apply (subst Rep_matrix_zero_imp_mult_zero)

     apply (simp)

     apply (rule disjI2)

     apply (intro strip)

     apply (simp add: sparse_row_matrix_cons neg_def)

     apply (case_tac "a <= aa")  

     apply (erule sorted_sparse_row_matrix_zero)  

     apply (simp_all)

     apply (intro strip)

     apply (case_tac "a=aa")

     apply (simp_all)

     apply (frule_tac as=arr in sorted_spvec_cons1)

     apply (frule_tac as=brr in sorted_spvec_cons1)

     apply (simp add: sparse_row_matrix_cons ring_simps sparse_row_vector_addmult_spvec)

     apply (rule_tac B1 = "sparse_row_matrix brr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])

     apply (auto)

     apply (rule sorted_sparse_row_matrix_zero)

     apply (simp_all)

     apply (rule_tac A1 = "sparse_row_vector arr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])

     apply (auto)

     apply (rule_tac m=k and n = aa and a = b and arr=arr in sorted_sparse_row_vector_zero)

     apply (simp_all)

     apply (simp add: neg_def)

     apply (drule nrows_notzero)

     apply (drule nrows_helper)

     apply (arith)

     apply (subst Rep_matrix_inject[symmetric])

     apply (rule ext)+

     apply (simp)

     apply (subst Rep_matrix_mult)

     apply (rule_tac j1=aa in ssubst[OF foldseq_almostzero])

     apply (simp_all)

     apply (intro strip, rule conjI)

     apply (intro strip)

     apply (drule_tac max_helper)

     apply (simp)

     apply (auto)

     apply (rule zero_imp_mult_zero)

     apply (rule disjI2)

     apply (rule nrows)

     apply (rule order_trans[of _ 1])

     apply (simp)

     apply (simp)

     done

 qed

 lemma sorted_mult_spvec_spmat[rule_format]: 

   "sorted_spvec (c::('a::lordered_ring) spvec) \<longrightarrow> sorted_spmat B \<longrightarrow> sorted_spvec (mult_spvec_spmat (c, a, B))"

   apply (rule mult_spvec_spmat.induct[of _ c a B])

   apply (simp_all add: sorted_addmult_spvec)

   done

 consts 

   mult_spmat :: "('a::lordered_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"

 primrec 

   "mult_spmat [] A = []"

   "mult_spmat (a#as) A = (fst a, mult_spvec_spmat ([], snd a, A))#(mult_spmat as A)"

 lemma sparse_row_mult_spmat[rule_format]: 

   "sorted_spmat A \<longrightarrow> sorted_spvec B \<longrightarrow> sparse_row_matrix (mult_spmat A B) = (sparse_row_matrix A) * (sparse_row_matrix B)"

   apply (induct A)

   apply (auto simp add: sparse_row_matrix_cons sparse_row_mult_spvec_spmat ring_simps move_matrix_mult)

   done

 lemma sorted_spvec_mult_spmat[rule_format]:

   "sorted_spvec (A::('a::lordered_ring) spmat) \<longrightarrow> sorted_spvec (mult_spmat A B)"

   apply (induct A)

   apply (auto)

   apply (drule sorted_spvec_cons1, simp)

   apply (case_tac A)

   apply (auto simp add: sorted_spvec.simps)

   done

 lemma sorted_spmat_mult_spmat[rule_format]:

   "sorted_spmat (B::('a::lordered_ring) spmat) \<longrightarrow> sorted_spmat (mult_spmat A B)"

   apply (induct A)

   apply (auto simp add: sorted_mult_spvec_spmat) 

   done

 consts

   add_spvec :: "('a::lordered_ab_group_add) spvec * 'a spvec \<Rightarrow> 'a spvec"

   add_spmat :: "('a::lordered_ab_group_add) spmat * 'a spmat \<Rightarrow> 'a spmat"

 recdef add_spvec "measure (% (a, b). length a + (length b))"

   "add_spvec (arr, []) = arr"

   "add_spvec ([], brr) = brr"

   "add_spvec (a#arr, b#brr) = (

   if (fst a) < (fst b) then (a#(add_spvec (arr, b#brr))) 

      else (if (fst b < fst a) then (b#(add_spvec (a#arr, brr)))

      else ((fst a, (snd a)+(snd b))#(add_spvec (arr,brr)))))"

 lemma add_spvec_empty1[simp]: "add_spvec ([], a) = a"

   by (induct a, auto)

 lemma add_spvec_empty2[simp]: "add_spvec (a, []) = a"

   by (induct a, auto)

 lemma sparse_row_vector_add: "sparse_row_vector (add_spvec (a,b)) = (sparse_row_vector a) + (sparse_row_vector b)"

   apply (rule add_spvec.induct[of _ a b])

   apply (simp_all add: singleton_matrix_add)

   done

 recdef add_spmat "measure (% (A,B). (length A)+(length B))"

   "add_spmat ([], bs) = bs"

   "add_spmat (as, []) = as"

   "add_spmat (a#as, b#bs) = (

   if fst a < fst b then 

     (a#(add_spmat (as, b#bs)))

   else (if fst b < fst a then

     (b#(add_spmat (a#as, bs)))

   else

     ((fst a, add_spvec (snd a, snd b))#(add_spmat (as, bs)))))"

 lemma sparse_row_add_spmat: "sparse_row_matrix (add_spmat (A, B)) = (sparse_row_matrix A) + (sparse_row_matrix B)"

   apply (rule add_spmat.induct)

   apply (auto simp add: sparse_row_matrix_cons sparse_row_vector_add move_matrix_add)

   done

 lemmas [code] = sparse_row_add_spmat [symmetric]

 lemmas [code] = sparse_row_vector_add [symmetric]

 lemma sorted_add_spvec_helper1[rule_format]: "add_spvec ((a,b)#arr, brr) = (ab, bb) # list \<longrightarrow> (ab = a | (brr \<noteq> [] & ab = fst (hd brr)))"

   proof - 

     have "(! x ab a. x = (a,b)#arr \<longrightarrow> add_spvec (x, brr) = (ab, bb) # list \<longrightarrow> (ab = a | (ab = fst (hd brr))))"

       by (rule add_spvec.induct[of _ _ brr], auto)

     then show ?thesis

       by (case_tac brr, auto)

   qed

 lemma sorted_add_spmat_helper1[rule_format]: "add_spmat ((a,b)#arr, brr) = (ab, bb) # list \<longrightarrow> (ab = a | (brr \<noteq> [] & ab = fst (hd brr)))"

   proof - 

     have "(! x ab a. x = (a,b)#arr \<longrightarrow> add_spmat (x, brr) = (ab, bb) # list \<longrightarrow> (ab = a | (ab = fst (hd brr))))"

       by (rule add_spmat.induct[of _ _ brr], auto)

     then show ?thesis

       by (case_tac brr, auto)

   qed

 lemma sorted_add_spvec_helper[rule_format]: "add_spvec (arr, brr) = (ab, bb) # list \<longrightarrow> ((arr \<noteq> [] & ab = fst (hd arr)) | (brr \<noteq> [] & ab = fst (hd brr)))"

   apply (rule add_spvec.induct[of _ arr brr])

   apply (auto)

   done

 lemma sorted_add_spmat_helper[rule_format]: "add_spmat (arr, brr) = (ab, bb) # list \<longrightarrow> ((arr \<noteq> [] & ab = fst (hd arr)) | (brr \<noteq> [] & ab = fst (hd brr)))"

   apply (rule add_spmat.induct[of _ arr brr])

   apply (auto)

   done

 lemma add_spvec_commute: "add_spvec (a, b) = add_spvec (b, a)"

   by (rule add_spvec.induct[of _ a b], auto)

 lemma add_spmat_commute: "add_spmat (a, b) = add_spmat (b, a)"

   apply (rule add_spmat.induct[of _ a b])

   apply (simp_all add: add_spvec_commute)

   done

 lemma sorted_add_spvec_helper2: "add_spvec ((a,b)#arr, brr) = (ab, bb) # list \<Longrightarrow> aa < a \<Longrightarrow> sorted_spvec ((aa, ba) # brr) \<Longrightarrow> aa < ab"

   apply (drule sorted_add_spvec_helper1)

   apply (auto)

   apply (case_tac brr)

   apply (simp_all)

   apply (drule_tac sorted_spvec_cons3)

   apply (simp)

   done

 lemma sorted_add_spmat_helper2: "add_spmat ((a,b)#arr, brr) = (ab, bb) # list \<Longrightarrow> aa < a \<Longrightarrow> sorted_spvec ((aa, ba) # brr) \<Longrightarrow> aa < ab"

   apply (drule sorted_add_spmat_helper1)

   apply (auto)

   apply (case_tac brr)

   apply (simp_all)

   apply (drule_tac sorted_spvec_cons3)

   apply (simp)

   done

 lemma sorted_spvec_add_spvec[rule_format]: "sorted_spvec a \<longrightarrow> sorted_spvec b \<longrightarrow> sorted_spvec (add_spvec (a, b))"

   apply (rule add_spvec.induct[of _ a b])

   apply (simp_all)

   apply (rule conjI)

   apply (intro strip)

   apply (simp)

   apply (frule_tac as=brr in sorted_spvec_cons1)

   apply (simp)

   apply (subst sorted_spvec_step)

   apply (simp split: list.split)

   apply (clarify, simp)

   apply (simp add: sorted_add_spvec_helper2)

   apply (clarify)

   apply (rule conjI)

   apply (case_tac "a=aa")

   apply (simp)

   apply (clarify)

   apply (frule_tac as=arr in sorted_spvec_cons1, simp)

   apply (subst sorted_spvec_step)

   apply (simp split: list.split)

   apply (clarify, simp)

   apply (simp add: sorted_add_spvec_helper2 add_spvec_commute)

   apply (case_tac "a=aa")

   apply (simp_all)

   apply (clarify)

   apply (frule_tac as=arr in sorted_spvec_cons1)

   apply (frule_tac as=brr in sorted_spvec_cons1)

   apply (simp)

   apply (subst sorted_spvec_step)

   apply (simp split: list.split)

   apply (clarify, simp)

   apply (drule_tac sorted_add_spvec_helper)

   apply (auto)

   apply (case_tac arr)

   apply (simp_all)

   apply (drule sorted_spvec_cons3)

   apply (simp)

   apply (case_tac brr)

   apply (simp_all)

   apply (drule sorted_spvec_cons3)

   apply (simp)

   done

 lemma sorted_spvec_add_spmat[rule_format]: "sorted_spvec A \<longrightarrow> sorted_spvec B \<longrightarrow> sorted_spvec (add_spmat (A, B))"

   apply (rule add_spmat.induct[of _ A B])

   apply (simp_all)

   apply (rule conjI)

   apply (intro strip)

   apply (simp)

   apply (frule_tac as=bs in sorted_spvec_cons1)

   apply (simp)

   apply (subst sorted_spvec_step)

   apply (simp split: list.split)

   apply (clarify, simp)

   apply (simp add: sorted_add_spmat_helper2)

   apply (clarify)

   apply (rule conjI)

   apply (case_tac "a=aa")

   apply (simp)

   apply (clarify)

   apply (frule_tac as=as in sorted_spvec_cons1, simp)

   apply (subst sorted_spvec_step)

   apply (simp split: list.split)

   apply (clarify, simp)

   apply (simp add: sorted_add_spmat_helper2 add_spmat_commute)

   apply (case_tac "a=aa")

   apply (simp_all)

   apply (clarify)

   apply (frule_tac as=as in sorted_spvec_cons1)

   apply (frule_tac as=bs in sorted_spvec_cons1)

   apply (simp)

   apply (subst sorted_spvec_step)

   apply (simp split: list.split)

   apply (clarify, simp)

   apply (drule_tac sorted_add_spmat_helper)

   apply (auto)

   apply (case_tac as)

   apply (simp_all)

   apply (drule sorted_spvec_cons3)

   apply (simp)

   apply (case_tac bs)

   apply (simp_all)

   apply (drule sorted_spvec_cons3)

   apply (simp)

   done

 lemma sorted_spmat_add_spmat[rule_format]: "sorted_spmat A \<longrightarrow> sorted_spmat B \<longrightarrow> sorted_spmat (add_spmat (A, B))"

   apply (rule add_spmat.induct[of _ A B])

   apply (simp_all add: sorted_spvec_add_spvec)

   done

 consts

   le_spvec :: "('a::lordered_ab_group_add) spvec * 'a spvec \<Rightarrow> bool" 

   le_spmat :: "('a::lordered_ab_group_add) spmat * 'a spmat \<Rightarrow> bool" 

 recdef le_spvec "measure (% (a,b). (length a) + (length b))" 

   "le_spvec ([], []) = True"

   "le_spvec (a#as, []) = ((snd a <= 0) & (le_spvec (as, [])))"

   "le_spvec ([], b#bs) = ((0 <= snd b) & (le_spvec ([], bs)))"

   "le_spvec (a#as, b#bs) = (

   if (fst a < fst b) then 

     ((snd a <= 0) & (le_spvec (as, b#bs)))

   else (if (fst b < fst a) then

     ((0 <= snd b) & (le_spvec (a#as, bs)))

   else 

     ((snd a <= snd b) & (le_spvec (as, bs)))))"

 recdef le_spmat "measure (% (a,b). (length a) + (length b))"

   "le_spmat ([], []) = True"

   "le_spmat (a#as, []) = (le_spvec (snd a, []) & (le_spmat (as, [])))"

   "le_spmat ([], b#bs) = (le_spvec ([], snd b) & (le_spmat ([], bs)))"

   "le_spmat (a#as, b#bs) = (

   if fst a < fst b then

     (le_spvec(snd a,[]) & le_spmat(as, b#bs))

   else (if (fst b < fst a) then 

     (le_spvec([], snd b) & le_spmat(a#as, bs))

   else

     (le_spvec(snd a, snd b) & le_spmat (as, bs))))"

 constdefs

   disj_matrices :: "('a::zero) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool"

   "disj_matrices A B == (! j i. (Rep_matrix A j i \<noteq> 0) \<longrightarrow> (Rep_matrix B j i = 0)) & (! j i. (Rep_matrix B j i \<noteq> 0) \<longrightarrow> (Rep_matrix A j i = 0))"  

 declare [[simp_depth_limit = 6]]

 lemma disj_matrices_contr1: "disj_matrices A B \<Longrightarrow> Rep_matrix A j i \<noteq> 0 \<Longrightarrow> Rep_matrix B j i = 0"

    by (simp add: disj_matrices_def)

 lemma disj_matrices_contr2: "disj_matrices A B \<Longrightarrow> Rep_matrix B j i \<noteq> 0 \<Longrightarrow> Rep_matrix A j i = 0"

    by (simp add: disj_matrices_def)

 lemma disj_matrices_add: "disj_matrices A B \<Longrightarrow> disj_matrices C D \<Longrightarrow> disj_matrices A D \<Longrightarrow> disj_matrices B C \<Longrightarrow> 

   (A + B <= C + D) = (A <= C & B <= (D::('a::lordered_ab_group_add) matrix))"

   apply (auto)

   apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)

   apply (intro strip)

   apply (erule conjE)+

   apply (drule_tac j=j and i=i in spec2)+

   apply (case_tac "Rep_matrix B j i = 0")

   apply (case_tac "Rep_matrix D j i = 0")

   apply (simp_all)

   apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)

   apply (intro strip)

   apply (erule conjE)+

   apply (drule_tac j=j and i=i in spec2)+

   apply (case_tac "Rep_matrix A j i = 0")

   apply (case_tac "Rep_matrix C j i = 0")

   apply (simp_all)

   apply (erule add_mono)

   apply (assumption)

   done

 lemma disj_matrices_zero1[simp]: "disj_matrices 0 B"

 by (simp add: disj_matrices_def)

 lemma disj_matrices_zero2[simp]: "disj_matrices A 0"

 by (simp add: disj_matrices_def)

 lemma disj_matrices_commute: "disj_matrices A B = disj_matrices B A"

 by (auto simp add: disj_matrices_def)

 lemma disj_matrices_add_le_zero: "disj_matrices A B \<Longrightarrow>

   (A + B <= 0) = (A <= 0 & (B::('a::lordered_ab_group_add) matrix) <= 0)"

 by (rule disj_matrices_add[of A B 0 0, simplified])

 lemma disj_matrices_add_zero_le: "disj_matrices A B \<Longrightarrow>

   (0 <= A + B) = (0 <= A & 0 <= (B::('a::lordered_ab_group_add) matrix))"

 by (rule disj_matrices_add[of 0 0 A B, simplified])

 lemma disj_matrices_add_x_le: "disj_matrices A B \<Longrightarrow> disj_matrices B C \<Longrightarrow> 

   (A <= B + C) = (A <= C & 0 <= (B::('a::lordered_ab_group_add) matrix))"

 by (auto simp add: disj_matrices_add[of 0 A B C, simplified])

 lemma disj_matrices_add_le_x: "disj_matrices A B \<Longrightarrow> disj_matrices B C \<Longrightarrow> 

   (B + A <= C) = (A <= C &  (B::('a::lordered_ab_group_add) matrix) <= 0)"

 by (auto simp add: disj_matrices_add[of B A 0 C,simplified] disj_matrices_commute)

 lemma disj_sparse_row_singleton: "i <= j \<Longrightarrow> sorted_spvec((j,y)#v) \<Longrightarrow> disj_matrices (sparse_row_vector v) (singleton_matrix 0 i x)"

   apply (simp add: disj_matrices_def)

   apply (rule conjI)

   apply (rule neg_imp)

   apply (simp)

   apply (intro strip)

   apply (rule sorted_sparse_row_vector_zero)

   apply (simp_all)

   apply (intro strip)

   apply (rule sorted_sparse_row_vector_zero)

   apply (simp_all)

   done 

 lemma disj_matrices_x_add: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (A::('a::lordered_ab_group_add) matrix) (B+C)"

   apply (simp add: disj_matrices_def)

   apply (auto)

   apply (drule_tac j=j and i=i in spec2)+

   apply (case_tac "Rep_matrix B j i = 0")

   apply (case_tac "Rep_matrix C j i = 0")

   apply (simp_all)

   done

 lemma disj_matrices_add_x: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (B+C) (A::('a::lordered_ab_group_add) matrix)" 

   by (simp add: disj_matrices_x_add disj_matrices_commute)

 lemma disj_singleton_matrices[simp]: "disj_matrices (singleton_matrix j i x) (singleton_matrix u v y) = (j \<noteq> u | i \<noteq> v | x = 0 | y = 0)" 

   by (auto simp add: disj_matrices_def)

 lemma disj_move_sparse_vec_mat[simplified disj_matrices_commute]: 

   "j <= a \<Longrightarrow> sorted_spvec((a,c)#as) \<Longrightarrow> disj_matrices (move_matrix (sparse_row_vector b) (int j) i) (sparse_row_matrix as)"

   apply (auto simp add: neg_def disj_matrices_def)

   apply (drule nrows_notzero)

   apply (drule less_le_trans[OF _ nrows_spvec])

   apply (subgoal_tac "ja = j")

   apply (simp add: sorted_sparse_row_matrix_zero)

   apply (arith)

   apply (rule nrows)

   apply (rule order_trans[of _ 1 _])

   apply (simp)

   apply (case_tac "nat (int ja - int j) = 0")

   apply (case_tac "ja = j")

   apply (simp add: sorted_sparse_row_matrix_zero)

   apply arith+

   done

 lemma disj_move_sparse_row_vector_twice:

   "j \<noteq> u \<Longrightarrow> disj_matrices (move_matrix (sparse_row_vector a) j i) (move_matrix (sparse_row_vector b) u v)"

   apply (auto simp add: neg_def disj_matrices_def)

   apply (rule nrows, rule order_trans[of _ 1], simp, drule nrows_notzero, drule less_le_trans[OF _ nrows_spvec], arith)+

   done

 lemma le_spvec_iff_sparse_row_le[rule_format]: "(sorted_spvec a) \<longrightarrow> (sorted_spvec b) \<longrightarrow> (le_spvec (a,b)) = (sparse_row_vector a <= sparse_row_vector b)"

   apply (rule le_spvec.induct)

   apply (simp_all add: sorted_spvec_cons1 disj_matrices_add_le_zero disj_matrices_add_zero_le 

     disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)

   apply (rule conjI, intro strip)

   apply (simp add: sorted_spvec_cons1)

   apply (subst disj_matrices_add_x_le)

   apply (simp add: disj_sparse_row_singleton[OF less_imp_le] disj_matrices_x_add disj_matrices_commute)

   apply (simp add: disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)

   apply (simp, blast)

   apply (intro strip, rule conjI, intro strip)

   apply (simp add: sorted_spvec_cons1)

   apply (subst disj_matrices_add_le_x)

   apply (simp_all add: disj_sparse_row_singleton[OF order_refl] disj_sparse_row_singleton[OF less_imp_le] disj_matrices_commute disj_matrices_x_add)

   apply (blast)

   apply (intro strip)

   apply (simp add: sorted_spvec_cons1)

   apply (case_tac "a=aa", simp_all)

   apply (subst disj_matrices_add)

   apply (simp_all add: disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)

   done

 lemma le_spvec_empty2_sparse_row[rule_format]: "(sorted_spvec b) \<longrightarrow> (le_spvec (b,[]) = (sparse_row_vector b <= 0))"

   apply (induct b)

   apply (simp_all add: sorted_spvec_cons1)

   apply (intro strip)

   apply (subst disj_matrices_add_le_zero)

   apply (simp add: disj_matrices_commute disj_sparse_row_singleton sorted_spvec_cons1)

   apply (rule_tac y = "snd a" in disj_sparse_row_singleton[OF order_refl])

   apply (simp_all)

   done

 lemma le_spvec_empty1_sparse_row[rule_format]: "(sorted_spvec b) \<longrightarrow> (le_spvec ([],b) = (0 <= sparse_row_vector b))"

   apply (induct b)

   apply (simp_all add: sorted_spvec_cons1)

   apply (intro strip)

   apply (subst disj_matrices_add_zero_le)

   apply (simp add: disj_matrices_commute disj_sparse_row_singleton sorted_spvec_cons1)

   apply (rule_tac y = "snd a" in disj_sparse_row_singleton[OF order_refl])

   apply (simp_all)

   done

 lemma le_spmat_iff_sparse_row_le[rule_format]: "(sorted_spvec A) \<longrightarrow> (sorted_spmat A) \<longrightarrow> (sorted_spvec B) \<longrightarrow> (sorted_spmat B) \<longrightarrow> 

   le_spmat(A, B) = (sparse_row_matrix A <= sparse_row_matrix B)"

   apply (rule le_spmat.induct)

   apply (simp add: sparse_row_matrix_cons disj_matrices_add_le_zero disj_matrices_add_zero_le disj_move_sparse_vec_mat[OF order_refl] 

     disj_matrices_commute sorted_spvec_cons1 le_spvec_empty2_sparse_row le_spvec_empty1_sparse_row)+ 

   apply (rule conjI, intro strip)

   apply (simp add: sorted_spvec_cons1)

   apply (subst disj_matrices_add_x_le)

   apply (rule disj_matrices_add_x)

   apply (simp add: disj_move_sparse_row_vector_twice)

   apply (simp add: disj_move_sparse_vec_mat[OF less_imp_le] disj_matrices_commute)

   apply (simp add: disj_move_sparse_vec_mat[OF order_refl] disj_matrices_commute)

   apply (simp, blast)

   apply (intro strip, rule conjI, intro strip)

   apply (simp add: sorted_spvec_cons1)

   apply (subst disj_matrices_add_le_x)

   apply (simp add: disj_move_sparse_vec_mat[OF order_refl])

   apply (rule disj_matrices_x_add)

   apply (simp add: disj_move_sparse_row_vector_twice)

   apply (simp add: disj_move_sparse_vec_mat[OF less_imp_le] disj_matrices_commute)

   apply (simp, blast)

   apply (intro strip)

   apply (case_tac "a=aa")

   apply (simp_all)

   apply (subst disj_matrices_add)

   apply (simp_all add: disj_matrices_commute disj_move_sparse_vec_mat[OF order_refl])

   apply (simp add: sorted_spvec_cons1 le_spvec_iff_sparse_row_le)

   done

 declare [[simp_depth_limit = 999]]

 consts 

    abs_spmat :: "('a::lordered_ring) spmat \<Rightarrow> 'a spmat"

    minus_spmat :: "('a::lordered_ring) spmat \<Rightarrow> 'a spmat"

 primrec

   "abs_spmat [] = []"

   "abs_spmat (a#as) = (fst a, abs_spvec (snd a))#(abs_spmat as)"

 primrec

   "minus_spmat [] = []"

   "minus_spmat (a#as) = (fst a, minus_spvec (snd a))#(minus_spmat as)"

 lemma sparse_row_matrix_minus:

   "sparse_row_matrix (minus_spmat A) = - (sparse_row_matrix A)"

   apply (induct A)

   apply (simp_all add: sparse_row_vector_minus sparse_row_matrix_cons)

   apply (subst Rep_matrix_inject[symmetric])

   apply (rule ext)+

   apply simp

   done

 lemma Rep_sparse_row_vector_zero: "x \<noteq> 0 \<Longrightarrow> Rep_matrix (sparse_row_vector v) x y = 0"

 proof -

   assume x:"x \<noteq> 0"

   have r:"nrows (sparse_row_vector v) <= Suc 0" by (rule nrows_spvec)

   show ?thesis

     apply (rule nrows)

     apply (subgoal_tac "Suc 0 <= x")

     apply (insert r)

     apply (simp only:)

     apply (insert x)

     apply arith

     done

 qed

 lemma sparse_row_matrix_abs:

   "sorted_spvec A \<Longrightarrow> sorted_spmat A \<Longrightarrow> sparse_row_matrix (abs_spmat A) = abs (sparse_row_matrix A)"

   apply (induct A)

   apply (simp_all add: sparse_row_vector_abs sparse_row_matrix_cons)

   apply (frule_tac sorted_spvec_cons1, simp)

   apply (simplesubst Rep_matrix_inject[symmetric])

   apply (rule ext)+

   apply auto

   apply (case_tac "x=a")

   apply (simp)

   apply (simplesubst sorted_sparse_row_matrix_zero)

   apply auto

   apply (simplesubst Rep_sparse_row_vector_zero)

   apply (simp_all add: neg_def)

   done

 lemma sorted_spvec_minus_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec (minus_spmat A)"

   apply (induct A)

   apply (simp)

   apply (frule sorted_spvec_cons1, simp)

   apply (simp add: sorted_spvec.simps split:list.split_asm)

   done 

 lemma sorted_spvec_abs_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec (abs_spmat A)" 

   apply (induct A)

   apply (simp)

   apply (frule sorted_spvec_cons1, simp)

   apply (simp add: sorted_spvec.simps split:list.split_asm)

   done

 lemma sorted_spmat_minus_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (minus_spmat A)"

   apply (induct A)

   apply (simp_all add: sorted_spvec_minus_spvec)

   done

 lemma sorted_spmat_abs_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (abs_spmat A)"

   apply (induct A)

   apply (simp_all add: sorted_spvec_abs_spvec)

   done

 constdefs

   diff_spmat :: "('a::lordered_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"

   "diff_spmat A B == add_spmat (A, minus_spmat B)"

 lemma sorted_spmat_diff_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat B \<Longrightarrow> sorted_spmat (diff_spmat A B)"

   by (simp add: diff_spmat_def sorted_spmat_minus_spmat sorted_spmat_add_spmat)

 lemma sorted_spvec_diff_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec B \<Longrightarrow> sorted_spvec (diff_spmat A B)"

   by (simp add: diff_spmat_def sorted_spvec_minus_spmat sorted_spvec_add_spmat)

 lemma sparse_row_diff_spmat: "sparse_row_matrix (diff_spmat A B ) = (sparse_row_matrix A) - (sparse_row_matrix B)"

   by (simp add: diff_spmat_def sparse_row_add_spmat sparse_row_matrix_minus)

 constdefs

   sorted_sparse_matrix :: "'a spmat \<Rightarrow> bool"

   "sorted_sparse_matrix A == (sorted_spvec A) & (sorted_spmat A)"

 lemma sorted_sparse_matrix_imp_spvec: "sorted_sparse_matrix A \<Longrightarrow> sorted_spvec A"

   by (simp add: sorted_sparse_matrix_def)

 lemma sorted_sparse_matrix_imp_spmat: "sorted_sparse_matrix A \<Longrightarrow> sorted_spmat A"

   by (simp add: sorted_sparse_matrix_def)

 lemmas sorted_sp_simps = 

   sorted_spvec.simps

   sorted_spmat.simps

   sorted_sparse_matrix_def

 lemma bool1: "(\<not> True) = False"  by blast

 lemma bool2: "(\<not> False) = True"  by blast

 lemma bool3: "((P\<Colon>bool) \<and> True) = P" by blast

 lemma bool4: "(True \<and> (P\<Colon>bool)) = P" by blast

 lemma bool5: "((P\<Colon>bool) \<and> False) = False" by blast

 lemma bool6: "(False \<and> (P\<Colon>bool)) = False" by blast

 lemma bool7: "((P\<Colon>bool) \<or> True) = True" by blast

 lemma bool8: "(True \<or> (P\<Colon>bool)) = True" by blast

 lemma bool9: "((P\<Colon>bool) \<or> False) = P" by blast

 lemma bool10: "(False \<or> (P\<Colon>bool)) = P" by blast

 lemmas boolarith = bool1 bool2 bool3 bool4 bool5 bool6 bool7 bool8 bool9 bool10

 lemma if_case_eq: "(if b then x else y) = (case b of True => x | False => y)" by simp

 consts

   pprt_spvec :: "('a::{lordered_ab_group_add}) spvec \<Rightarrow> 'a spvec"

   nprt_spvec :: "('a::{lordered_ab_group_add}) spvec \<Rightarrow> 'a spvec"

   pprt_spmat :: "('a::{lordered_ab_group_add}) spmat \<Rightarrow> 'a spmat"

   nprt_spmat :: "('a::{lordered_ab_group_add}) spmat \<Rightarrow> 'a spmat"

 primrec

   "pprt_spvec [] = []"

   "pprt_spvec (a#as) = (fst a, pprt (snd a)) # (pprt_spvec as)"

 primrec

   "nprt_spvec [] = []"

   "nprt_spvec (a#as) = (fst a, nprt (snd a)) # (nprt_spvec as)"

 primrec 

   "pprt_spmat [] = []"

   "pprt_spmat (a#as) = (fst a, pprt_spvec (snd a))#(pprt_spmat as)"

   (*case (pprt_spvec (snd a)) of [] \<Rightarrow> (pprt_spmat as) | y#ys \<Rightarrow> (fst a, y#ys)#(pprt_spmat as))"*)

 primrec 

   "nprt_spmat [] = []"

   "nprt_spmat (a#as) = (fst a, nprt_spvec (snd a))#(nprt_spmat as)"

   (*case (nprt_spvec (snd a)) of [] \<Rightarrow> (nprt_spmat as) | y#ys \<Rightarrow> (fst a, y#ys)#(nprt_spmat as))"*)

 lemma pprt_add: "disj_matrices A (B::(_::lordered_ring) matrix) \<Longrightarrow> pprt (A+B) = pprt A + pprt B"

   apply (simp add: pprt_def sup_matrix_def)

   apply (simp add: Rep_matrix_inject[symmetric])

   apply (rule ext)+

   apply simp

   apply (case_tac "Rep_matrix A x xa \<noteq> 0")

   apply (simp_all add: disj_matrices_contr1)

   done

 lemma nprt_add: "disj_matrices A (B::(_::lordered_ring) matrix) \<Longrightarrow> nprt (A+B) = nprt A + nprt B"

   apply (simp add: nprt_def inf_matrix_def)

   apply (simp add: Rep_matrix_inject[symmetric])

   apply (rule ext)+

   apply simp

   apply (case_tac "Rep_matrix A x xa \<noteq> 0")

   apply (simp_all add: disj_matrices_contr1)

   done

 lemma pprt_singleton[simp]: "pprt (singleton_matrix j i (x::_::lordered_ring)) = singleton_matrix j i (pprt x)"

   apply (simp add: pprt_def sup_matrix_def)

   apply (simp add: Rep_matrix_inject[symmetric])

   apply (rule ext)+

   apply simp

   done

 lemma nprt_singleton[simp]: "nprt (singleton_matrix j i (x::_::lordered_ring)) = singleton_matrix j i (nprt x)"

   apply (simp add: nprt_def inf_matrix_def)

   apply (simp add: Rep_matrix_inject[symmetric])

   apply (rule ext)+

   apply simp

   done

 lemma less_imp_le: "a < b \<Longrightarrow> a <= (b::_::order)" by (simp add: less_def)

 lemma sparse_row_vector_pprt: "sorted_spvec (v :: 'a::lordered_ring spvec) \<Longrightarrow> sparse_row_vector (pprt_spvec v) = pprt (sparse_row_vector v)"

   apply (induct v)

   apply (simp_all)

   apply (frule sorted_spvec_cons1, auto)

   apply (subst pprt_add)

   apply (subst disj_matrices_commute)

   apply (rule disj_sparse_row_singleton)

   apply auto

   done

 lemma sparse_row_vector_nprt: "sorted_spvec (v :: 'a::lordered_ring spvec) \<Longrightarrow> sparse_row_vector (nprt_spvec v) = nprt (sparse_row_vector v)"

   apply (induct v)

   apply (simp_all)

   apply (frule sorted_spvec_cons1, auto)

   apply (subst nprt_add)

   apply (subst disj_matrices_commute)

   apply (rule disj_sparse_row_singleton)

   apply auto

   done

 lemma pprt_move_matrix: "pprt (move_matrix (A::('a::lordered_ring) matrix) j i) = move_matrix (pprt A) j i"

   apply (simp add: pprt_def)

   apply (simp add: sup_matrix_def)

   apply (simp add: Rep_matrix_inject[symmetric])

   apply (rule ext)+

   apply (simp)

   done

 lemma nprt_move_matrix: "nprt (move_matrix (A::('a::lordered_ring) matrix) j i) = move_matrix (nprt A) j i"

   apply (simp add: nprt_def)

   apply (simp add: inf_matrix_def)

   apply (simp add: Rep_matrix_inject[symmetric])

   apply (rule ext)+

   apply (simp)

   done

 lemma sparse_row_matrix_pprt: "sorted_spvec (m :: 'a::lordered_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (pprt_spmat m) = pprt (sparse_row_matrix m)"

   apply (induct m)

   apply simp

   apply simp

   apply (frule sorted_spvec_cons1)

   apply (simp add: sparse_row_matrix_cons sparse_row_vector_pprt)

   apply (subst pprt_add)

   apply (subst disj_matrices_commute)

   apply (rule disj_move_sparse_vec_mat)

   apply auto

   apply (simp add: sorted_spvec.simps)

   apply (simp split: list.split)

   apply auto

   apply (simp add: pprt_move_matrix)

   done

 lemma sparse_row_matrix_nprt: "sorted_spvec (m :: 'a::lordered_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (nprt_spmat m) = nprt (sparse_row_matrix m)"

   apply (induct m)

   apply simp

   apply simp

   apply (frule sorted_spvec_cons1)

   apply (simp add: sparse_row_matrix_cons sparse_row_vector_nprt)

   apply (subst nprt_add)

   apply (subst disj_matrices_commute)

   apply (rule disj_move_sparse_vec_mat)

   apply auto

   apply (simp add: sorted_spvec.simps)

   apply (simp split: list.split)

   apply auto

   apply (simp add: nprt_move_matrix)

   done

 lemma sorted_pprt_spvec: "sorted_spvec v \<Longrightarrow> sorted_spvec (pprt_spvec v)"

   apply (induct v)

   apply (simp)

   apply (frule sorted_spvec_cons1)

   apply simp

   apply (simp add: sorted_spvec.simps split:list.split_asm)

   done

 lemma sorted_nprt_spvec: "sorted_spvec v \<Longrightarrow> sorted_spvec (nprt_spvec v)"

   apply (induct v)

   apply (simp)

   apply (frule sorted_spvec_cons1)

   apply simp

   apply (simp add: sorted_spvec.simps split:list.split_asm)

   done

 lemma sorted_spvec_pprt_spmat: "sorted_spvec m \<Longrightarrow> sorted_spvec (pprt_spmat m)"

   apply (induct m)

   apply (simp)

   apply (frule sorted_spvec_cons1)

   apply simp

   apply (simp add: sorted_spvec.simps split:list.split_asm)

   done

 lemma sorted_spvec_nprt_spmat: "sorted_spvec m \<Longrightarrow> sorted_spvec (nprt_spmat m)"

   apply (induct m)

   apply (simp)

   apply (frule sorted_spvec_cons1)

   apply simp

   apply (simp add: sorted_spvec.simps split:list.split_asm)

   done

 lemma sorted_spmat_pprt_spmat: "sorted_spmat m \<Longrightarrow> sorted_spmat (pprt_spmat m)"

   apply (induct m)

   apply (simp_all add: sorted_pprt_spvec)

   done

 lemma sorted_spmat_nprt_spmat: "sorted_spmat m \<Longrightarrow> sorted_spmat (nprt_spmat m)"

   apply (induct m)

   apply (simp_all add: sorted_nprt_spvec)

   done

 constdefs

   mult_est_spmat :: "('a::lordered_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"

   "mult_est_spmat r1 r2 s1 s2 == 

   add_spmat (mult_spmat (pprt_spmat s2) (pprt_spmat r2), add_spmat (mult_spmat (pprt_spmat s1) (nprt_spmat r2), 

   add_spmat (mult_spmat (nprt_spmat s2) (pprt_spmat r1), mult_spmat (nprt_spmat s1) (nprt_spmat r1))))"  

 lemmas sparse_row_matrix_op_simps =

   sorted_sparse_matrix_imp_spmat sorted_sparse_matrix_imp_spvec

   sparse_row_add_spmat sorted_spvec_add_spmat sorted_spmat_add_spmat

   sparse_row_diff_spmat sorted_spvec_diff_spmat sorted_spmat_diff_spmat

   sparse_row_matrix_minus sorted_spvec_minus_spmat sorted_spmat_minus_spmat

   sparse_row_mult_spmat sorted_spvec_mult_spmat sorted_spmat_mult_spmat

   sparse_row_matrix_abs sorted_spvec_abs_spmat sorted_spmat_abs_spmat

   le_spmat_iff_sparse_row_le

   sparse_row_matrix_pprt sorted_spvec_pprt_spmat sorted_spmat_pprt_spmat

   sparse_row_matrix_nprt sorted_spvec_nprt_spmat sorted_spmat_nprt_spmat

 lemma zero_eq_Numeral0: "(0::_::number_ring) = Numeral0" by simp

 lemmas sparse_row_matrix_arith_simps[simplified zero_eq_Numeral0] = 

   mult_spmat.simps mult_spvec_spmat.simps 

   addmult_spvec.simps 

   smult_spvec_empty smult_spvec_cons

   add_spmat.simps add_spvec.simps

   minus_spmat.simps minus_spvec.simps

   abs_spmat.simps abs_spvec.simps

   diff_spmat_def

   le_spmat.simps le_spvec.simps

   pprt_spmat.simps pprt_spvec.simps

   nprt_spmat.simps nprt_spvec.simps

   mult_est_spmat_def

 (*lemma spm_linprog_dual_estimate_1:

   assumes  

   "sorted_sparse_matrix A1"

   "sorted_sparse_matrix A2"

   "sorted_sparse_matrix c1"

   "sorted_sparse_matrix c2"

   "sorted_sparse_matrix y"

   "sorted_spvec b"

   "sorted_spvec r"

   "le_spmat ([], y)"

   "A * x \<le> sparse_row_matrix (b::('a::lordered_ring) spmat)"

   "sparse_row_matrix A1 <= A"

   "A <= sparse_row_matrix A2"

   "sparse_row_matrix c1 <= c"

   "c <= sparse_row_matrix c2"

   "abs x \<le> sparse_row_matrix r"

   shows

   "c * x \<le> sparse_row_matrix (add_spmat (mult_spmat y b, mult_spmat (add_spmat (add_spmat (mult_spmat y (diff_spmat A2 A1), 

   abs_spmat (diff_spmat (mult_spmat y A1) c1)), diff_spmat c2 c1)) r))"

   by (insert prems, simp add: sparse_row_matrix_op_simps linprog_dual_estimate_1[where A=A])

 *)

 end