wenzelm@16487
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(* Title: HOL/Matrix/SparseMatrix.thy
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wenzelm@16487
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Author: Steven Obua
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wenzelm@16487
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*)
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wenzelm@16487
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haftmann@27484
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theory SparseMatrix
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wenzelm@28637
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imports Matrix
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haftmann@27484
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begin
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obua@15009
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wenzelm@43334
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type_synonym 'a spvec = "(nat * 'a) list"
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wenzelm@43334
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type_synonym 'a spmat = "'a spvec spvec"
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obua@15009
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wenzelm@38571
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definition sparse_row_vector :: "('a::ab_group_add) spvec \<Rightarrow> 'a matrix"
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wenzelm@38571
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where "sparse_row_vector arr = foldl (% m x. m + (singleton_matrix 0 (fst x) (snd x))) 0 arr"
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obua@15009
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wenzelm@38571
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definition sparse_row_matrix :: "('a::ab_group_add) spmat \<Rightarrow> 'a matrix"
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wenzelm@38571
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where "sparse_row_matrix arr = foldl (% m r. m + (move_matrix (sparse_row_vector (snd r)) (int (fst r)) 0)) 0 arr"
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obua@15009
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haftmann@27484
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code_datatype sparse_row_vector sparse_row_matrix
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haftmann@27484
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lemma sparse_row_vector_empty [simp]: "sparse_row_vector [] = 0"
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obua@15009
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by (simp add: sparse_row_vector_def)
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obua@15009
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haftmann@27484
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lemma sparse_row_matrix_empty [simp]: "sparse_row_matrix [] = 0"
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obua@15009
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by (simp add: sparse_row_matrix_def)
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obua@15009
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haftmann@28562
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lemmas [code] = sparse_row_vector_empty [symmetric]
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haftmann@27484
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nipkow@31817
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lemma foldl_distrstart: "! a x y. (f (g x y) a = g x (f y a)) \<Longrightarrow> (foldl f (g x y) l = g x (foldl f y l))"
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nipkow@31817
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by (induct l arbitrary: x y, auto)
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obua@15009
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haftmann@27653
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lemma sparse_row_vector_cons[simp]:
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"sparse_row_vector (a # arr) = (singleton_matrix 0 (fst a) (snd a)) + (sparse_row_vector arr)"
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obua@15009
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apply (induct arr)
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obua@15009
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apply (auto simp add: sparse_row_vector_def)
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haftmann@27653
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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"])
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obua@15009
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done
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obua@15009
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haftmann@27653
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lemma sparse_row_vector_append[simp]:
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"sparse_row_vector (a @ b) = (sparse_row_vector a) + (sparse_row_vector b)"
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haftmann@27653
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by (induct a) auto
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obua@15009
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obua@15009
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lemma nrows_spvec[simp]: "nrows (sparse_row_vector x) <= (Suc 0)"
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obua@15009
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apply (induct x)
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obua@15009
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apply (simp_all add: add_nrows)
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obua@15009
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done
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obua@15009
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obua@15009
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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"
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obua@15009
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apply (induct arr)
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obua@15009
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apply (auto simp add: sparse_row_matrix_def)
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obua@15009
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apply (simp add: foldl_distrstart[of "\<lambda>m x. m + (move_matrix (sparse_row_vector (snd x)) (int (fst x)) 0)"
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obua@15009
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"% a m. (move_matrix (sparse_row_vector (snd a)) (int (fst a)) 0) + m"])
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done
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obua@15009
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obua@15009
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lemma sparse_row_matrix_append: "sparse_row_matrix (arr@brr) = (sparse_row_matrix arr) + (sparse_row_matrix brr)"
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obua@15009
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apply (induct arr)
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obua@15009
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apply (auto simp add: sparse_row_matrix_cons)
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done
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obua@15009
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primrec sorted_spvec :: "'a spvec \<Rightarrow> bool"
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where
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"sorted_spvec [] = True"
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| sorted_spvec_step: "sorted_spvec (a#as) = (case as of [] \<Rightarrow> True | b#bs \<Rightarrow> ((fst a < fst b) & (sorted_spvec as)))"
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obua@15009
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wenzelm@38571
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primrec sorted_spmat :: "'a spmat \<Rightarrow> bool"
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where
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"sorted_spmat [] = True"
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| "sorted_spmat (a#as) = ((sorted_spvec (snd a)) & (sorted_spmat as))"
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obua@15009
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obua@15009
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declare sorted_spvec.simps [simp del]
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obua@15009
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lemma sorted_spvec_empty[simp]: "sorted_spvec [] = True"
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obua@15009
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by (simp add: sorted_spvec.simps)
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obua@15009
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lemma sorted_spvec_cons1: "sorted_spvec (a#as) \<Longrightarrow> sorted_spvec as"
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obua@15009
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apply (induct as)
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obua@15009
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apply (auto simp add: sorted_spvec.simps)
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done
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lemma sorted_spvec_cons2: "sorted_spvec (a#b#t) \<Longrightarrow> sorted_spvec (a#t)"
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obua@15009
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apply (induct t)
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obua@15009
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apply (auto simp add: sorted_spvec.simps)
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done
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obua@15009
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obua@15009
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lemma sorted_spvec_cons3: "sorted_spvec(a#b#t) \<Longrightarrow> fst a < fst b"
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obua@15009
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apply (auto simp add: sorted_spvec.simps)
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obua@15009
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done
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obua@15009
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nipkow@31817
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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"
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obua@15009
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apply (induct arr)
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apply (auto)
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obua@15009
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apply (frule sorted_spvec_cons2,simp)+
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apply (frule sorted_spvec_cons3, simp)
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obua@15009
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done
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obua@15009
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nipkow@31817
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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"
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obua@15009
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apply (induct arr)
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obua@15009
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apply (auto)
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obua@15009
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apply (frule sorted_spvec_cons2, simp)
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obua@15009
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apply (frule sorted_spvec_cons3, simp)
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huffman@47573
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apply (simp add: sparse_row_matrix_cons)
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obua@15009
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done
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obua@15009
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wenzelm@38571
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primrec minus_spvec :: "('a::ab_group_add) spvec \<Rightarrow> 'a spvec"
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where
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obua@15178
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"minus_spvec [] = []"
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wenzelm@38571
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| "minus_spvec (a#as) = (fst a, -(snd a))#(minus_spvec as)"
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obua@15178
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wenzelm@38571
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primrec abs_spvec :: "('a::lattice_ab_group_add_abs) spvec \<Rightarrow> 'a spvec"
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where
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obua@15178
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"abs_spvec [] = []"
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wenzelm@38571
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| "abs_spvec (a#as) = (fst a, abs (snd a))#(abs_spvec as)"
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obua@15178
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obua@15178
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lemma sparse_row_vector_minus:
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obua@15178
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"sparse_row_vector (minus_spvec v) = - (sparse_row_vector v)"
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obua@15178
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apply (induct v)
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obua@15178
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apply (simp_all add: sparse_row_vector_cons)
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obua@15178
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apply (simp add: Rep_matrix_inject[symmetric])
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obua@15178
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apply (rule ext)+
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obua@15178
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apply simp
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obua@15178
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done
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obua@15178
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haftmann@35028
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instance matrix :: (lattice_ab_group_add_abs) lattice_ab_group_add_abs
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haftmann@27653
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apply default
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haftmann@27653
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unfolding abs_matrix_def .. (*FIXME move*)
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haftmann@27653
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obua@15178
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lemma sparse_row_vector_abs:
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haftmann@35028
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"sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (abs_spvec v) = abs (sparse_row_vector v)"
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obua@15178
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apply (induct v)
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haftmann@27653
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apply simp_all
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obua@15178
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apply (frule_tac sorted_spvec_cons1, simp)
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obua@15178
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apply (simp only: Rep_matrix_inject[symmetric])
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obua@15178
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apply (rule ext)+
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obua@15178
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apply auto
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nipkow@15236
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apply (subgoal_tac "Rep_matrix (sparse_row_vector v) 0 a = 0")
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obua@15178
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apply (simp)
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obua@15178
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apply (rule sorted_sparse_row_vector_zero)
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obua@15178
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apply auto
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obua@15178
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done
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obua@15178
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obua@15178
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lemma sorted_spvec_minus_spvec:
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obua@15178
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"sorted_spvec v \<Longrightarrow> sorted_spvec (minus_spvec v)"
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obua@15178
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apply (induct v)
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obua@15178
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apply (simp)
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obua@15178
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apply (frule sorted_spvec_cons1, simp)
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nipkow@15236
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apply (simp add: sorted_spvec.simps split:list.split_asm)
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obua@15178
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done
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obua@15178
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obua@15178
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lemma sorted_spvec_abs_spvec:
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obua@15178
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"sorted_spvec v \<Longrightarrow> sorted_spvec (abs_spvec v)"
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obua@15178
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apply (induct v)
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obua@15178
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apply (simp)
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obua@15178
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apply (frule sorted_spvec_cons1, simp)
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nipkow@15236
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apply (simp add: sorted_spvec.simps split:list.split_asm)
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obua@15178
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done
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obua@15178
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wenzelm@38571
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definition "smult_spvec y = map (% a. (fst a, y * snd a))"
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obua@15009
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obua@15009
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lemma smult_spvec_empty[simp]: "smult_spvec y [] = []"
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obua@15009
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by (simp add: smult_spvec_def)
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obua@15009
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obua@15009
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lemma smult_spvec_cons: "smult_spvec y (a#arr) = (fst a, y * (snd a)) # (smult_spvec y arr)"
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obua@15009
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by (simp add: smult_spvec_def)
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obua@15009
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wenzelm@38571
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fun addmult_spvec :: "('a::ring) \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec"
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wenzelm@38571
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where
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wenzelm@38571
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"addmult_spvec y arr [] = arr"
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wenzelm@38571
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| "addmult_spvec y [] brr = smult_spvec y brr"
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wenzelm@38571
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| "addmult_spvec y ((i,a)#arr) ((j,b)#brr) = (
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nipkow@31816
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if i < j then ((i,a)#(addmult_spvec y arr ((j,b)#brr)))
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nipkow@31816
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else (if (j < i) then ((j, y * b)#(addmult_spvec y ((i,a)#arr) brr))
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nipkow@31816
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else ((i, a + y*b)#(addmult_spvec y arr brr))))"
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nipkow@31816
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(* Steven used termination "measure (% (y, a, b). length a + (length b))" *)
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obua@15009
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nipkow@31816
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lemma addmult_spvec_empty1[simp]: "addmult_spvec y [] a = smult_spvec y a"
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haftmann@27484
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by (induct a) auto
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obua@15009
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nipkow@31816
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lemma addmult_spvec_empty2[simp]: "addmult_spvec y a [] = a"
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haftmann@27484
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by (induct a) auto
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obua@15009
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haftmann@35028
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lemma sparse_row_vector_map: "(! x y. f (x+y) = (f x) + (f y)) \<Longrightarrow> (f::'a\<Rightarrow>('a::lattice_ring)) 0 = 0 \<Longrightarrow>
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obua@15009
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sparse_row_vector (map (% x. (fst x, f (snd x))) a) = apply_matrix f (sparse_row_vector a)"
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obua@15009
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apply (induct a)
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obua@15009
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apply (simp_all add: apply_matrix_add)
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obua@15009
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done
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obua@15009
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obua@15009
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lemma sparse_row_vector_smult: "sparse_row_vector (smult_spvec y a) = scalar_mult y (sparse_row_vector a)"
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obua@15009
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apply (induct a)
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obua@15009
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apply (simp_all add: smult_spvec_cons scalar_mult_add)
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obua@15009
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done
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obua@15009
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haftmann@35028
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lemma sparse_row_vector_addmult_spvec: "sparse_row_vector (addmult_spvec (y::'a::lattice_ring) a b) =
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obua@15009
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(sparse_row_vector a) + (scalar_mult y (sparse_row_vector b))"
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nipkow@31817
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apply (induct y a b rule: addmult_spvec.induct)
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obua@15009
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apply (simp add: scalar_mult_add smult_spvec_cons sparse_row_vector_smult singleton_matrix_add)+
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obua@15009
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done
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obua@15009
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nipkow@31817
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lemma sorted_smult_spvec: "sorted_spvec a \<Longrightarrow> sorted_spvec (smult_spvec y a)"
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obua@15009
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apply (auto simp add: smult_spvec_def)
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obua@15009
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apply (induct a)
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nipkow@15236
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apply (auto simp add: sorted_spvec.simps split:list.split_asm)
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obua@15009
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done
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obua@15009
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nipkow@31816
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lemma sorted_spvec_addmult_spvec_helper: "\<lbrakk>sorted_spvec (addmult_spvec y ((a, b) # arr) brr); aa < a; sorted_spvec ((a, b) # arr);
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nipkow@31816
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sorted_spvec ((aa, ba) # brr)\<rbrakk> \<Longrightarrow> sorted_spvec ((aa, y * ba) # addmult_spvec y ((a, b) # arr) brr)"
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obua@15009
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apply (induct brr)
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obua@15009
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apply (auto simp add: sorted_spvec.simps)
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obua@15009
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done
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obua@15009
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obua@15009
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lemma sorted_spvec_addmult_spvec_helper2:
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nipkow@31816
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"\<lbrakk>sorted_spvec (addmult_spvec y arr ((aa, ba) # brr)); a < aa; sorted_spvec ((a, b) # arr); sorted_spvec ((aa, ba) # brr)\<rbrakk>
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nipkow@31816
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\<Longrightarrow> sorted_spvec ((a, b) # addmult_spvec y arr ((aa, ba) # brr))"
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obua@15009
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apply (induct arr)
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obua@15009
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apply (auto simp add: smult_spvec_def sorted_spvec.simps)
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obua@15009
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done
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obua@15009
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obua@15009
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lemma sorted_spvec_addmult_spvec_helper3[rule_format]:
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nipkow@31816
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"sorted_spvec (addmult_spvec y arr brr) \<longrightarrow> sorted_spvec ((aa, b) # arr) \<longrightarrow> sorted_spvec ((aa, ba) # brr)
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nipkow@31816
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\<longrightarrow> sorted_spvec ((aa, b + y * ba) # (addmult_spvec y arr brr))"
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nipkow@31816
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apply (induct y arr brr rule: addmult_spvec.induct)
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nipkow@31816
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apply (simp_all add: sorted_spvec.simps smult_spvec_def split:list.split)
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obua@15009
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done
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obua@15009
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nipkow@31817
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lemma sorted_addmult_spvec: "sorted_spvec a \<Longrightarrow> sorted_spvec b \<Longrightarrow> sorted_spvec (addmult_spvec y a b)"
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nipkow@31817
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apply (induct y a b rule: addmult_spvec.induct)
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obua@15009
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apply (simp_all add: sorted_smult_spvec)
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obua@15009
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apply (rule conjI, intro strip)
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nipkow@31816
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apply (case_tac "~(i < j)")
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obua@15009
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apply (simp_all)
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obua@15009
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apply (frule_tac as=brr in sorted_spvec_cons1)
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obua@15009
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apply (simp add: sorted_spvec_addmult_spvec_helper)
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obua@15009
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apply (intro strip | rule conjI)+
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obua@15009
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apply (frule_tac as=arr in sorted_spvec_cons1)
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obua@15009
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apply (simp add: sorted_spvec_addmult_spvec_helper2)
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obua@15009
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apply (intro strip)
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obua@15009
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apply (frule_tac as=arr in sorted_spvec_cons1)
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obua@15009
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apply (frule_tac as=brr in sorted_spvec_cons1)
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obua@15009
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apply (simp)
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obua@15009
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apply (simp_all add: sorted_spvec_addmult_spvec_helper3)
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obua@15009
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done
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obua@15009
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wenzelm@38571
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fun mult_spvec_spmat :: "('a::lattice_ring) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spmat \<Rightarrow> 'a spvec"
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wenzelm@38571
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where
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nipkow@31816
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(* recdef mult_spvec_spmat "measure (% (c, arr, brr). (length arr) + (length brr))" *)
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wenzelm@38571
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"mult_spvec_spmat c [] brr = c"
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wenzelm@38571
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| "mult_spvec_spmat c arr [] = c"
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wenzelm@38571
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| "mult_spvec_spmat c ((i,a)#arr) ((j,b)#brr) = (
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nipkow@31816
|
247 |
if (i < j) then mult_spvec_spmat c arr ((j,b)#brr)
|
nipkow@31816
|
248 |
else if (j < i) then mult_spvec_spmat c ((i,a)#arr) brr
|
nipkow@31816
|
249 |
else mult_spvec_spmat (addmult_spvec a c b) arr brr)"
|
obua@15009
|
250 |
|
haftmann@35028
|
251 |
lemma sparse_row_mult_spvec_spmat[rule_format]: "sorted_spvec (a::('a::lattice_ring) spvec) \<longrightarrow> sorted_spvec B \<longrightarrow>
|
nipkow@31816
|
252 |
sparse_row_vector (mult_spvec_spmat c a B) = (sparse_row_vector c) + (sparse_row_vector a) * (sparse_row_matrix B)"
|
obua@15009
|
253 |
proof -
|
obua@15009
|
254 |
have comp_1: "!! a b. a < b \<Longrightarrow> Suc 0 <= nat ((int b)-(int a))" by arith
|
obua@15009
|
255 |
have not_iff: "!! a b. a = b \<Longrightarrow> (~ a) = (~ b)" by simp
|
obua@15009
|
256 |
have max_helper: "!! a b. ~ (a <= max (Suc a) b) \<Longrightarrow> False"
|
obua@15009
|
257 |
by arith
|
obua@15009
|
258 |
{
|
obua@15009
|
259 |
fix a
|
obua@15009
|
260 |
fix v
|
obua@15009
|
261 |
assume a:"a < nrows(sparse_row_vector v)"
|
obua@15009
|
262 |
have b:"nrows(sparse_row_vector v) <= 1" by simp
|
obua@15009
|
263 |
note dummy = less_le_trans[of a "nrows (sparse_row_vector v)" 1, OF a b]
|
obua@15009
|
264 |
then have "a = 0" by simp
|
obua@15009
|
265 |
}
|
obua@15009
|
266 |
note nrows_helper = this
|
obua@15009
|
267 |
show ?thesis
|
nipkow@31817
|
268 |
apply (induct c a B rule: mult_spvec_spmat.induct)
|
obua@15009
|
269 |
apply simp+
|
obua@15009
|
270 |
apply (rule conjI)
|
obua@15009
|
271 |
apply (intro strip)
|
obua@15009
|
272 |
apply (frule_tac as=brr in sorted_spvec_cons1)
|
nipkow@29667
|
273 |
apply (simp add: algebra_simps sparse_row_matrix_cons)
|
paulson@15481
|
274 |
apply (simplesubst Rep_matrix_zero_imp_mult_zero)
|
obua@15009
|
275 |
apply (simp)
|
obua@15009
|
276 |
apply (rule disjI2)
|
obua@15009
|
277 |
apply (intro strip)
|
obua@15009
|
278 |
apply (subst nrows)
|
obua@15009
|
279 |
apply (rule order_trans[of _ 1])
|
obua@15009
|
280 |
apply (simp add: comp_1)+
|
obua@15009
|
281 |
apply (subst Rep_matrix_zero_imp_mult_zero)
|
obua@15009
|
282 |
apply (intro strip)
|
nipkow@31816
|
283 |
apply (case_tac "k <= j")
|
nipkow@31816
|
284 |
apply (rule_tac m1 = k and n1 = i and a1 = a in ssubst[OF sorted_sparse_row_vector_zero])
|
obua@15009
|
285 |
apply (simp_all)
|
obua@15009
|
286 |
apply (rule disjI2)
|
obua@15009
|
287 |
apply (rule nrows)
|
obua@15009
|
288 |
apply (rule order_trans[of _ 1])
|
obua@15009
|
289 |
apply (simp_all add: comp_1)
|
obua@15009
|
290 |
|
obua@15009
|
291 |
apply (intro strip | rule conjI)+
|
obua@15009
|
292 |
apply (frule_tac as=arr in sorted_spvec_cons1)
|
nipkow@29667
|
293 |
apply (simp add: algebra_simps)
|
obua@15009
|
294 |
apply (subst Rep_matrix_zero_imp_mult_zero)
|
obua@15009
|
295 |
apply (simp)
|
obua@15009
|
296 |
apply (rule disjI2)
|
obua@15009
|
297 |
apply (intro strip)
|
huffman@47573
|
298 |
apply (simp add: sparse_row_matrix_cons)
|
nipkow@31816
|
299 |
apply (case_tac "i <= j")
|
obua@15009
|
300 |
apply (erule sorted_sparse_row_matrix_zero)
|
obua@15009
|
301 |
apply (simp_all)
|
obua@15009
|
302 |
apply (intro strip)
|
nipkow@31816
|
303 |
apply (case_tac "i=j")
|
obua@15009
|
304 |
apply (simp_all)
|
obua@15009
|
305 |
apply (frule_tac as=arr in sorted_spvec_cons1)
|
obua@15009
|
306 |
apply (frule_tac as=brr in sorted_spvec_cons1)
|
nipkow@29667
|
307 |
apply (simp add: sparse_row_matrix_cons algebra_simps sparse_row_vector_addmult_spvec)
|
obua@15009
|
308 |
apply (rule_tac B1 = "sparse_row_matrix brr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])
|
obua@15009
|
309 |
apply (auto)
|
obua@15009
|
310 |
apply (rule sorted_sparse_row_matrix_zero)
|
obua@15009
|
311 |
apply (simp_all)
|
obua@15009
|
312 |
apply (rule_tac A1 = "sparse_row_vector arr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])
|
obua@15009
|
313 |
apply (auto)
|
nipkow@31816
|
314 |
apply (rule_tac m=k and n = j and a = a and arr=arr in sorted_sparse_row_vector_zero)
|
obua@15009
|
315 |
apply (simp_all)
|
obua@15009
|
316 |
apply (drule nrows_notzero)
|
obua@15009
|
317 |
apply (drule nrows_helper)
|
obua@15009
|
318 |
apply (arith)
|
obua@15009
|
319 |
|
obua@15009
|
320 |
apply (subst Rep_matrix_inject[symmetric])
|
obua@15009
|
321 |
apply (rule ext)+
|
obua@15009
|
322 |
apply (simp)
|
obua@15009
|
323 |
apply (subst Rep_matrix_mult)
|
nipkow@31816
|
324 |
apply (rule_tac j1=j in ssubst[OF foldseq_almostzero])
|
obua@15009
|
325 |
apply (simp_all)
|
webertj@20432
|
326 |
apply (intro strip, rule conjI)
|
obua@15009
|
327 |
apply (intro strip)
|
webertj@20432
|
328 |
apply (drule_tac max_helper)
|
webertj@20432
|
329 |
apply (simp)
|
webertj@20432
|
330 |
apply (auto)
|
obua@15009
|
331 |
apply (rule zero_imp_mult_zero)
|
obua@15009
|
332 |
apply (rule disjI2)
|
obua@15009
|
333 |
apply (rule nrows)
|
obua@15009
|
334 |
apply (rule order_trans[of _ 1])
|
webertj@20432
|
335 |
apply (simp)
|
webertj@20432
|
336 |
apply (simp)
|
obua@15009
|
337 |
done
|
obua@15009
|
338 |
qed
|
obua@15009
|
339 |
|
obua@15009
|
340 |
lemma sorted_mult_spvec_spmat[rule_format]:
|
haftmann@35028
|
341 |
"sorted_spvec (c::('a::lattice_ring) spvec) \<longrightarrow> sorted_spmat B \<longrightarrow> sorted_spvec (mult_spvec_spmat c a B)"
|
nipkow@31817
|
342 |
apply (induct c a B rule: mult_spvec_spmat.induct)
|
obua@15009
|
343 |
apply (simp_all add: sorted_addmult_spvec)
|
obua@15009
|
344 |
done
|
obua@15009
|
345 |
|
wenzelm@38571
|
346 |
primrec mult_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
|
wenzelm@38571
|
347 |
where
|
obua@15009
|
348 |
"mult_spmat [] A = []"
|
wenzelm@38571
|
349 |
| "mult_spmat (a#as) A = (fst a, mult_spvec_spmat [] (snd a) A)#(mult_spmat as A)"
|
obua@15009
|
350 |
|
nipkow@31817
|
351 |
lemma sparse_row_mult_spmat:
|
nipkow@31817
|
352 |
"sorted_spmat A \<Longrightarrow> sorted_spvec B \<Longrightarrow>
|
nipkow@31817
|
353 |
sparse_row_matrix (mult_spmat A B) = (sparse_row_matrix A) * (sparse_row_matrix B)"
|
obua@15009
|
354 |
apply (induct A)
|
nipkow@29667
|
355 |
apply (auto simp add: sparse_row_matrix_cons sparse_row_mult_spvec_spmat algebra_simps move_matrix_mult)
|
obua@15009
|
356 |
done
|
obua@15009
|
357 |
|
obua@15009
|
358 |
lemma sorted_spvec_mult_spmat[rule_format]:
|
haftmann@35028
|
359 |
"sorted_spvec (A::('a::lattice_ring) spmat) \<longrightarrow> sorted_spvec (mult_spmat A B)"
|
obua@15009
|
360 |
apply (induct A)
|
obua@15009
|
361 |
apply (auto)
|
obua@15009
|
362 |
apply (drule sorted_spvec_cons1, simp)
|
nipkow@15236
|
363 |
apply (case_tac A)
|
obua@15009
|
364 |
apply (auto simp add: sorted_spvec.simps)
|
obua@15009
|
365 |
done
|
obua@15009
|
366 |
|
nipkow@31817
|
367 |
lemma sorted_spmat_mult_spmat:
|
haftmann@35028
|
368 |
"sorted_spmat (B::('a::lattice_ring) spmat) \<Longrightarrow> sorted_spmat (mult_spmat A B)"
|
obua@15009
|
369 |
apply (induct A)
|
obua@15009
|
370 |
apply (auto simp add: sorted_mult_spvec_spmat)
|
obua@15009
|
371 |
done
|
obua@15009
|
372 |
|
obua@15009
|
373 |
|
wenzelm@38571
|
374 |
fun add_spvec :: "('a::lattice_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec"
|
wenzelm@38571
|
375 |
where
|
nipkow@31816
|
376 |
(* "measure (% (a, b). length a + (length b))" *)
|
wenzelm@38571
|
377 |
"add_spvec arr [] = arr"
|
wenzelm@38571
|
378 |
| "add_spvec [] brr = brr"
|
wenzelm@38571
|
379 |
| "add_spvec ((i,a)#arr) ((j,b)#brr) = (
|
wenzelm@38571
|
380 |
if i < j then (i,a)#(add_spvec arr ((j,b)#brr))
|
nipkow@31816
|
381 |
else if (j < i) then (j,b) # add_spvec ((i,a)#arr) brr
|
nipkow@31816
|
382 |
else (i, a+b) # add_spvec arr brr)"
|
obua@15009
|
383 |
|
nipkow@31816
|
384 |
lemma add_spvec_empty1[simp]: "add_spvec [] a = a"
|
nipkow@31816
|
385 |
by (cases a, auto)
|
obua@15009
|
386 |
|
nipkow@31816
|
387 |
lemma sparse_row_vector_add: "sparse_row_vector (add_spvec a b) = (sparse_row_vector a) + (sparse_row_vector b)"
|
nipkow@31817
|
388 |
apply (induct a b rule: add_spvec.induct)
|
obua@15009
|
389 |
apply (simp_all add: singleton_matrix_add)
|
obua@15009
|
390 |
done
|
obua@15009
|
391 |
|
wenzelm@38571
|
392 |
fun add_spmat :: "('a::lattice_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
|
wenzelm@38571
|
393 |
where
|
nipkow@31816
|
394 |
(* "measure (% (A,B). (length A)+(length B))" *)
|
wenzelm@38571
|
395 |
"add_spmat [] bs = bs"
|
wenzelm@38571
|
396 |
| "add_spmat as [] = as"
|
wenzelm@38571
|
397 |
| "add_spmat ((i,a)#as) ((j,b)#bs) = (
|
wenzelm@38571
|
398 |
if i < j then
|
wenzelm@38571
|
399 |
(i,a) # add_spmat as ((j,b)#bs)
|
wenzelm@38571
|
400 |
else if j < i then
|
wenzelm@38571
|
401 |
(j,b) # add_spmat ((i,a)#as) bs
|
wenzelm@38571
|
402 |
else
|
wenzelm@38571
|
403 |
(i, add_spvec a b) # add_spmat as bs)"
|
obua@15009
|
404 |
|
nipkow@31816
|
405 |
lemma add_spmat_Nil2[simp]: "add_spmat as [] = as"
|
nipkow@31816
|
406 |
by(cases as) auto
|
nipkow@31816
|
407 |
|
nipkow@31816
|
408 |
lemma sparse_row_add_spmat: "sparse_row_matrix (add_spmat A B) = (sparse_row_matrix A) + (sparse_row_matrix B)"
|
nipkow@31817
|
409 |
apply (induct A B rule: add_spmat.induct)
|
obua@15009
|
410 |
apply (auto simp add: sparse_row_matrix_cons sparse_row_vector_add move_matrix_add)
|
obua@15009
|
411 |
done
|
obua@15009
|
412 |
|
haftmann@28562
|
413 |
lemmas [code] = sparse_row_add_spmat [symmetric]
|
haftmann@28562
|
414 |
lemmas [code] = sparse_row_vector_add [symmetric]
|
haftmann@27484
|
415 |
|
nipkow@31816
|
416 |
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)))"
|
obua@15009
|
417 |
proof -
|
nipkow@31816
|
418 |
have "(! x ab a. x = (a,b)#arr \<longrightarrow> add_spvec x brr = (ab, bb) # list \<longrightarrow> (ab = a | (ab = fst (hd brr))))"
|
nipkow@31817
|
419 |
by (induct brr rule: add_spvec.induct) (auto split:if_splits)
|
obua@15009
|
420 |
then show ?thesis
|
obua@15009
|
421 |
by (case_tac brr, auto)
|
obua@15009
|
422 |
qed
|
obua@15009
|
423 |
|
nipkow@31816
|
424 |
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)))"
|
obua@15009
|
425 |
proof -
|
nipkow@31816
|
426 |
have "(! x ab a. x = (a,b)#arr \<longrightarrow> add_spmat x brr = (ab, bb) # list \<longrightarrow> (ab = a | (ab = fst (hd brr))))"
|
nipkow@31817
|
427 |
by (rule add_spmat.induct) (auto split:if_splits)
|
obua@15009
|
428 |
then show ?thesis
|
obua@15009
|
429 |
by (case_tac brr, auto)
|
obua@15009
|
430 |
qed
|
obua@15009
|
431 |
|
nipkow@31817
|
432 |
lemma sorted_add_spvec_helper: "add_spvec arr brr = (ab, bb) # list \<Longrightarrow> ((arr \<noteq> [] & ab = fst (hd arr)) | (brr \<noteq> [] & ab = fst (hd brr)))"
|
nipkow@31817
|
433 |
apply (induct arr brr rule: add_spvec.induct)
|
nipkow@31817
|
434 |
apply (auto split:if_splits)
|
obua@15009
|
435 |
done
|
obua@15009
|
436 |
|
nipkow@31817
|
437 |
lemma sorted_add_spmat_helper: "add_spmat arr brr = (ab, bb) # list \<Longrightarrow> ((arr \<noteq> [] & ab = fst (hd arr)) | (brr \<noteq> [] & ab = fst (hd brr)))"
|
nipkow@31817
|
438 |
apply (induct arr brr rule: add_spmat.induct)
|
nipkow@31817
|
439 |
apply (auto split:if_splits)
|
obua@15009
|
440 |
done
|
obua@15009
|
441 |
|
nipkow@31816
|
442 |
lemma add_spvec_commute: "add_spvec a b = add_spvec b a"
|
nipkow@31817
|
443 |
by (induct a b rule: add_spvec.induct) auto
|
obua@15009
|
444 |
|
nipkow@31816
|
445 |
lemma add_spmat_commute: "add_spmat a b = add_spmat b a"
|
nipkow@31817
|
446 |
apply (induct a b rule: add_spmat.induct)
|
obua@15009
|
447 |
apply (simp_all add: add_spvec_commute)
|
obua@15009
|
448 |
done
|
obua@15009
|
449 |
|
nipkow@31816
|
450 |
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"
|
obua@15009
|
451 |
apply (drule sorted_add_spvec_helper1)
|
obua@15009
|
452 |
apply (auto)
|
obua@15009
|
453 |
apply (case_tac brr)
|
obua@15009
|
454 |
apply (simp_all)
|
obua@15009
|
455 |
apply (drule_tac sorted_spvec_cons3)
|
obua@15009
|
456 |
apply (simp)
|
obua@15009
|
457 |
done
|
obua@15009
|
458 |
|
nipkow@31816
|
459 |
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"
|
obua@15009
|
460 |
apply (drule sorted_add_spmat_helper1)
|
obua@15009
|
461 |
apply (auto)
|
obua@15009
|
462 |
apply (case_tac brr)
|
obua@15009
|
463 |
apply (simp_all)
|
obua@15009
|
464 |
apply (drule_tac sorted_spvec_cons3)
|
obua@15009
|
465 |
apply (simp)
|
obua@15009
|
466 |
done
|
obua@15009
|
467 |
|
nipkow@31816
|
468 |
lemma sorted_spvec_add_spvec[rule_format]: "sorted_spvec a \<longrightarrow> sorted_spvec b \<longrightarrow> sorted_spvec (add_spvec a b)"
|
nipkow@31817
|
469 |
apply (induct a b rule: add_spvec.induct)
|
obua@15009
|
470 |
apply (simp_all)
|
obua@15009
|
471 |
apply (rule conjI)
|
nipkow@31816
|
472 |
apply (clarsimp)
|
obua@15009
|
473 |
apply (frule_tac as=brr in sorted_spvec_cons1)
|
obua@15009
|
474 |
apply (simp)
|
obua@15009
|
475 |
apply (subst sorted_spvec_step)
|
nipkow@31816
|
476 |
apply (clarsimp simp: sorted_add_spvec_helper2 split: list.split)
|
obua@15009
|
477 |
apply (clarify)
|
obua@15009
|
478 |
apply (rule conjI)
|
obua@15009
|
479 |
apply (clarify)
|
obua@15009
|
480 |
apply (frule_tac as=arr in sorted_spvec_cons1, simp)
|
obua@15009
|
481 |
apply (subst sorted_spvec_step)
|
nipkow@31816
|
482 |
apply (clarsimp simp: sorted_add_spvec_helper2 add_spvec_commute split: list.split)
|
obua@15009
|
483 |
apply (clarify)
|
obua@15009
|
484 |
apply (frule_tac as=arr in sorted_spvec_cons1)
|
obua@15009
|
485 |
apply (frule_tac as=brr in sorted_spvec_cons1)
|
obua@15009
|
486 |
apply (simp)
|
obua@15009
|
487 |
apply (subst sorted_spvec_step)
|
obua@15009
|
488 |
apply (simp split: list.split)
|
nipkow@31816
|
489 |
apply (clarsimp)
|
obua@15009
|
490 |
apply (drule_tac sorted_add_spvec_helper)
|
nipkow@31816
|
491 |
apply (auto simp: neq_Nil_conv)
|
obua@15009
|
492 |
apply (drule sorted_spvec_cons3)
|
obua@15009
|
493 |
apply (simp)
|
obua@15009
|
494 |
apply (drule sorted_spvec_cons3)
|
obua@15009
|
495 |
apply (simp)
|
obua@15009
|
496 |
done
|
obua@15009
|
497 |
|
nipkow@31816
|
498 |
lemma sorted_spvec_add_spmat[rule_format]: "sorted_spvec A \<longrightarrow> sorted_spvec B \<longrightarrow> sorted_spvec (add_spmat A B)"
|
nipkow@31817
|
499 |
apply (induct A B rule: add_spmat.induct)
|
obua@15009
|
500 |
apply (simp_all)
|
obua@15009
|
501 |
apply (rule conjI)
|
obua@15009
|
502 |
apply (intro strip)
|
obua@15009
|
503 |
apply (simp)
|
obua@15009
|
504 |
apply (frule_tac as=bs in sorted_spvec_cons1)
|
obua@15009
|
505 |
apply (simp)
|
obua@15009
|
506 |
apply (subst sorted_spvec_step)
|
obua@15009
|
507 |
apply (simp split: list.split)
|
obua@15009
|
508 |
apply (clarify, simp)
|
obua@15009
|
509 |
apply (simp add: sorted_add_spmat_helper2)
|
obua@15009
|
510 |
apply (clarify)
|
obua@15009
|
511 |
apply (rule conjI)
|
obua@15009
|
512 |
apply (clarify)
|
obua@15009
|
513 |
apply (frule_tac as=as in sorted_spvec_cons1, simp)
|
obua@15009
|
514 |
apply (subst sorted_spvec_step)
|
nipkow@31816
|
515 |
apply (clarsimp simp: sorted_add_spmat_helper2 add_spmat_commute split: list.split)
|
nipkow@31816
|
516 |
apply (clarsimp)
|
obua@15009
|
517 |
apply (frule_tac as=as in sorted_spvec_cons1)
|
obua@15009
|
518 |
apply (frule_tac as=bs in sorted_spvec_cons1)
|
obua@15009
|
519 |
apply (simp)
|
obua@15009
|
520 |
apply (subst sorted_spvec_step)
|
obua@15009
|
521 |
apply (simp split: list.split)
|
obua@15009
|
522 |
apply (clarify, simp)
|
obua@15009
|
523 |
apply (drule_tac sorted_add_spmat_helper)
|
nipkow@31816
|
524 |
apply (auto simp:neq_Nil_conv)
|
obua@15009
|
525 |
apply (drule sorted_spvec_cons3)
|
obua@15009
|
526 |
apply (simp)
|
obua@15009
|
527 |
apply (drule sorted_spvec_cons3)
|
obua@15009
|
528 |
apply (simp)
|
obua@15009
|
529 |
done
|
obua@15009
|
530 |
|
nipkow@31817
|
531 |
lemma sorted_spmat_add_spmat[rule_format]: "sorted_spmat A \<Longrightarrow> sorted_spmat B \<Longrightarrow> sorted_spmat (add_spmat A B)"
|
nipkow@31817
|
532 |
apply (induct A B rule: add_spmat.induct)
|
obua@15009
|
533 |
apply (simp_all add: sorted_spvec_add_spvec)
|
obua@15009
|
534 |
done
|
obua@15009
|
535 |
|
wenzelm@38571
|
536 |
fun le_spvec :: "('a::lattice_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> bool"
|
wenzelm@38571
|
537 |
where
|
nipkow@31816
|
538 |
(* "measure (% (a,b). (length a) + (length b))" *)
|
wenzelm@38571
|
539 |
"le_spvec [] [] = True"
|
wenzelm@38571
|
540 |
| "le_spvec ((_,a)#as) [] = (a <= 0 & le_spvec as [])"
|
wenzelm@38571
|
541 |
| "le_spvec [] ((_,b)#bs) = (0 <= b & le_spvec [] bs)"
|
wenzelm@38571
|
542 |
| "le_spvec ((i,a)#as) ((j,b)#bs) = (
|
wenzelm@38571
|
543 |
if (i < j) then a <= 0 & le_spvec as ((j,b)#bs)
|
wenzelm@38571
|
544 |
else if (j < i) then 0 <= b & le_spvec ((i,a)#as) bs
|
wenzelm@38571
|
545 |
else a <= b & le_spvec as bs)"
|
obua@15009
|
546 |
|
wenzelm@38571
|
547 |
fun le_spmat :: "('a::lattice_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> bool"
|
wenzelm@38571
|
548 |
where
|
nipkow@31816
|
549 |
(* "measure (% (a,b). (length a) + (length b))" *)
|
wenzelm@38571
|
550 |
"le_spmat [] [] = True"
|
wenzelm@38571
|
551 |
| "le_spmat ((i,a)#as) [] = (le_spvec a [] & le_spmat as [])"
|
wenzelm@38571
|
552 |
| "le_spmat [] ((j,b)#bs) = (le_spvec [] b & le_spmat [] bs)"
|
wenzelm@38571
|
553 |
| "le_spmat ((i,a)#as) ((j,b)#bs) = (
|
wenzelm@38571
|
554 |
if i < j then (le_spvec a [] & le_spmat as ((j,b)#bs))
|
wenzelm@38571
|
555 |
else if j < i then (le_spvec [] b & le_spmat ((i,a)#as) bs)
|
wenzelm@38571
|
556 |
else (le_spvec a b & le_spmat as bs))"
|
obua@15009
|
557 |
|
haftmann@35413
|
558 |
definition disj_matrices :: "('a::zero) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
|
wenzelm@38571
|
559 |
"disj_matrices A B \<longleftrightarrow>
|
wenzelm@38571
|
560 |
(! 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))"
|
obua@15009
|
561 |
|
wenzelm@24124
|
562 |
declare [[simp_depth_limit = 6]]
|
obua@15009
|
563 |
|
obua@15580
|
564 |
lemma disj_matrices_contr1: "disj_matrices A B \<Longrightarrow> Rep_matrix A j i \<noteq> 0 \<Longrightarrow> Rep_matrix B j i = 0"
|
obua@15580
|
565 |
by (simp add: disj_matrices_def)
|
obua@15580
|
566 |
|
obua@15580
|
567 |
lemma disj_matrices_contr2: "disj_matrices A B \<Longrightarrow> Rep_matrix B j i \<noteq> 0 \<Longrightarrow> Rep_matrix A j i = 0"
|
obua@15580
|
568 |
by (simp add: disj_matrices_def)
|
obua@15580
|
569 |
|
obua@15580
|
570 |
|
obua@15009
|
571 |
lemma disj_matrices_add: "disj_matrices A B \<Longrightarrow> disj_matrices C D \<Longrightarrow> disj_matrices A D \<Longrightarrow> disj_matrices B C \<Longrightarrow>
|
haftmann@35028
|
572 |
(A + B <= C + D) = (A <= C & B <= (D::('a::lattice_ab_group_add) matrix))"
|
obua@15009
|
573 |
apply (auto)
|
obua@15009
|
574 |
apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)
|
obua@15009
|
575 |
apply (intro strip)
|
obua@15009
|
576 |
apply (erule conjE)+
|
obua@15009
|
577 |
apply (drule_tac j=j and i=i in spec2)+
|
obua@15009
|
578 |
apply (case_tac "Rep_matrix B j i = 0")
|
obua@15009
|
579 |
apply (case_tac "Rep_matrix D j i = 0")
|
obua@15009
|
580 |
apply (simp_all)
|
obua@15009
|
581 |
apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)
|
obua@15009
|
582 |
apply (intro strip)
|
obua@15009
|
583 |
apply (erule conjE)+
|
obua@15009
|
584 |
apply (drule_tac j=j and i=i in spec2)+
|
obua@15009
|
585 |
apply (case_tac "Rep_matrix A j i = 0")
|
obua@15009
|
586 |
apply (case_tac "Rep_matrix C j i = 0")
|
obua@15009
|
587 |
apply (simp_all)
|
obua@15009
|
588 |
apply (erule add_mono)
|
obua@15009
|
589 |
apply (assumption)
|
obua@15009
|
590 |
done
|
obua@15009
|
591 |
|
obua@15009
|
592 |
lemma disj_matrices_zero1[simp]: "disj_matrices 0 B"
|
obua@15009
|
593 |
by (simp add: disj_matrices_def)
|
obua@15009
|
594 |
|
obua@15009
|
595 |
lemma disj_matrices_zero2[simp]: "disj_matrices A 0"
|
obua@15009
|
596 |
by (simp add: disj_matrices_def)
|
obua@15009
|
597 |
|
obua@15009
|
598 |
lemma disj_matrices_commute: "disj_matrices A B = disj_matrices B A"
|
obua@15009
|
599 |
by (auto simp add: disj_matrices_def)
|
obua@15009
|
600 |
|
obua@15009
|
601 |
lemma disj_matrices_add_le_zero: "disj_matrices A B \<Longrightarrow>
|
haftmann@35028
|
602 |
(A + B <= 0) = (A <= 0 & (B::('a::lattice_ab_group_add) matrix) <= 0)"
|
obua@15009
|
603 |
by (rule disj_matrices_add[of A B 0 0, simplified])
|
obua@15009
|
604 |
|
obua@15009
|
605 |
lemma disj_matrices_add_zero_le: "disj_matrices A B \<Longrightarrow>
|
haftmann@35028
|
606 |
(0 <= A + B) = (0 <= A & 0 <= (B::('a::lattice_ab_group_add) matrix))"
|
obua@15009
|
607 |
by (rule disj_matrices_add[of 0 0 A B, simplified])
|
obua@15009
|
608 |
|
obua@15009
|
609 |
lemma disj_matrices_add_x_le: "disj_matrices A B \<Longrightarrow> disj_matrices B C \<Longrightarrow>
|
haftmann@35028
|
610 |
(A <= B + C) = (A <= C & 0 <= (B::('a::lattice_ab_group_add) matrix))"
|
obua@15009
|
611 |
by (auto simp add: disj_matrices_add[of 0 A B C, simplified])
|
obua@15009
|
612 |
|
obua@15009
|
613 |
lemma disj_matrices_add_le_x: "disj_matrices A B \<Longrightarrow> disj_matrices B C \<Longrightarrow>
|
haftmann@35028
|
614 |
(B + A <= C) = (A <= C & (B::('a::lattice_ab_group_add) matrix) <= 0)"
|
obua@15009
|
615 |
by (auto simp add: disj_matrices_add[of B A 0 C,simplified] disj_matrices_commute)
|
obua@15009
|
616 |
|
obua@15009
|
617 |
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)"
|
obua@15009
|
618 |
apply (simp add: disj_matrices_def)
|
obua@15009
|
619 |
apply (rule conjI)
|
obua@15009
|
620 |
apply (rule neg_imp)
|
obua@15009
|
621 |
apply (simp)
|
obua@15009
|
622 |
apply (intro strip)
|
obua@15009
|
623 |
apply (rule sorted_sparse_row_vector_zero)
|
obua@15009
|
624 |
apply (simp_all)
|
obua@15009
|
625 |
apply (intro strip)
|
obua@15009
|
626 |
apply (rule sorted_sparse_row_vector_zero)
|
obua@15009
|
627 |
apply (simp_all)
|
obua@15009
|
628 |
done
|
obua@15009
|
629 |
|
haftmann@35028
|
630 |
lemma disj_matrices_x_add: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (A::('a::lattice_ab_group_add) matrix) (B+C)"
|
obua@15009
|
631 |
apply (simp add: disj_matrices_def)
|
obua@15009
|
632 |
apply (auto)
|
obua@15009
|
633 |
apply (drule_tac j=j and i=i in spec2)+
|
obua@15009
|
634 |
apply (case_tac "Rep_matrix B j i = 0")
|
obua@15009
|
635 |
apply (case_tac "Rep_matrix C j i = 0")
|
obua@15009
|
636 |
apply (simp_all)
|
obua@15009
|
637 |
done
|
obua@15009
|
638 |
|
haftmann@35028
|
639 |
lemma disj_matrices_add_x: "disj_matrices A B \<Longrightarrow> disj_matrices A C \<Longrightarrow> disj_matrices (B+C) (A::('a::lattice_ab_group_add) matrix)"
|
obua@15009
|
640 |
by (simp add: disj_matrices_x_add disj_matrices_commute)
|
obua@15009
|
641 |
|
obua@15009
|
642 |
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)"
|
obua@15009
|
643 |
by (auto simp add: disj_matrices_def)
|
obua@15009
|
644 |
|
obua@15009
|
645 |
lemma disj_move_sparse_vec_mat[simplified disj_matrices_commute]:
|
obua@15009
|
646 |
"j <= a \<Longrightarrow> sorted_spvec((a,c)#as) \<Longrightarrow> disj_matrices (move_matrix (sparse_row_vector b) (int j) i) (sparse_row_matrix as)"
|
huffman@47573
|
647 |
apply (auto simp add: disj_matrices_def)
|
obua@15009
|
648 |
apply (drule nrows_notzero)
|
obua@15009
|
649 |
apply (drule less_le_trans[OF _ nrows_spvec])
|
obua@15009
|
650 |
apply (subgoal_tac "ja = j")
|
obua@15009
|
651 |
apply (simp add: sorted_sparse_row_matrix_zero)
|
obua@15009
|
652 |
apply (arith)
|
obua@15009
|
653 |
apply (rule nrows)
|
obua@15009
|
654 |
apply (rule order_trans[of _ 1 _])
|
obua@15009
|
655 |
apply (simp)
|
obua@15009
|
656 |
apply (case_tac "nat (int ja - int j) = 0")
|
obua@15009
|
657 |
apply (case_tac "ja = j")
|
obua@15009
|
658 |
apply (simp add: sorted_sparse_row_matrix_zero)
|
obua@15009
|
659 |
apply arith+
|
obua@15009
|
660 |
done
|
obua@15009
|
661 |
|
obua@15009
|
662 |
lemma disj_move_sparse_row_vector_twice:
|
obua@15009
|
663 |
"j \<noteq> u \<Longrightarrow> disj_matrices (move_matrix (sparse_row_vector a) j i) (move_matrix (sparse_row_vector b) u v)"
|
huffman@47573
|
664 |
apply (auto simp add: disj_matrices_def)
|
obua@15009
|
665 |
apply (rule nrows, rule order_trans[of _ 1], simp, drule nrows_notzero, drule less_le_trans[OF _ nrows_spvec], arith)+
|
obua@15009
|
666 |
done
|
obua@15009
|
667 |
|
nipkow@31816
|
668 |
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)"
|
nipkow@31817
|
669 |
apply (induct a b rule: le_spvec.induct)
|
obua@15178
|
670 |
apply (simp_all add: sorted_spvec_cons1 disj_matrices_add_le_zero disj_matrices_add_zero_le
|
obua@15178
|
671 |
disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
|
obua@15178
|
672 |
apply (rule conjI, intro strip)
|
obua@15178
|
673 |
apply (simp add: sorted_spvec_cons1)
|
obua@15178
|
674 |
apply (subst disj_matrices_add_x_le)
|
obua@15178
|
675 |
apply (simp add: disj_sparse_row_singleton[OF less_imp_le] disj_matrices_x_add disj_matrices_commute)
|
obua@15178
|
676 |
apply (simp add: disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
|
obua@15178
|
677 |
apply (simp, blast)
|
obua@15178
|
678 |
apply (intro strip, rule conjI, intro strip)
|
obua@15178
|
679 |
apply (simp add: sorted_spvec_cons1)
|
obua@15178
|
680 |
apply (subst disj_matrices_add_le_x)
|
obua@15178
|
681 |
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)
|
obua@15178
|
682 |
apply (blast)
|
obua@15178
|
683 |
apply (intro strip)
|
obua@15178
|
684 |
apply (simp add: sorted_spvec_cons1)
|
nipkow@31816
|
685 |
apply (case_tac "a=b", simp_all)
|
obua@15178
|
686 |
apply (subst disj_matrices_add)
|
obua@15178
|
687 |
apply (simp_all add: disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
|
obua@15009
|
688 |
done
|
obua@15009
|
689 |
|
nipkow@31816
|
690 |
lemma le_spvec_empty2_sparse_row[rule_format]: "sorted_spvec b \<longrightarrow> le_spvec b [] = (sparse_row_vector b <= 0)"
|
obua@15009
|
691 |
apply (induct b)
|
obua@15009
|
692 |
apply (simp_all add: sorted_spvec_cons1)
|
obua@15009
|
693 |
apply (intro strip)
|
obua@15009
|
694 |
apply (subst disj_matrices_add_le_zero)
|
nipkow@31816
|
695 |
apply (auto simp add: disj_matrices_commute disj_sparse_row_singleton[OF order_refl] sorted_spvec_cons1)
|
obua@15009
|
696 |
done
|
obua@15009
|
697 |
|
nipkow@31816
|
698 |
lemma le_spvec_empty1_sparse_row[rule_format]: "(sorted_spvec b) \<longrightarrow> (le_spvec [] b = (0 <= sparse_row_vector b))"
|
obua@15009
|
699 |
apply (induct b)
|
obua@15009
|
700 |
apply (simp_all add: sorted_spvec_cons1)
|
obua@15009
|
701 |
apply (intro strip)
|
obua@15009
|
702 |
apply (subst disj_matrices_add_zero_le)
|
nipkow@31816
|
703 |
apply (auto simp add: disj_matrices_commute disj_sparse_row_singleton[OF order_refl] sorted_spvec_cons1)
|
obua@15009
|
704 |
done
|
obua@15009
|
705 |
|
obua@15009
|
706 |
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>
|
nipkow@31816
|
707 |
le_spmat A B = (sparse_row_matrix A <= sparse_row_matrix B)"
|
nipkow@31817
|
708 |
apply (induct A B rule: le_spmat.induct)
|
obua@15009
|
709 |
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]
|
obua@15009
|
710 |
disj_matrices_commute sorted_spvec_cons1 le_spvec_empty2_sparse_row le_spvec_empty1_sparse_row)+
|
obua@15009
|
711 |
apply (rule conjI, intro strip)
|
obua@15009
|
712 |
apply (simp add: sorted_spvec_cons1)
|
obua@15009
|
713 |
apply (subst disj_matrices_add_x_le)
|
obua@15009
|
714 |
apply (rule disj_matrices_add_x)
|
obua@15009
|
715 |
apply (simp add: disj_move_sparse_row_vector_twice)
|
obua@15009
|
716 |
apply (simp add: disj_move_sparse_vec_mat[OF less_imp_le] disj_matrices_commute)
|
obua@15009
|
717 |
apply (simp add: disj_move_sparse_vec_mat[OF order_refl] disj_matrices_commute)
|
obua@15009
|
718 |
apply (simp, blast)
|
obua@15009
|
719 |
apply (intro strip, rule conjI, intro strip)
|
obua@15009
|
720 |
apply (simp add: sorted_spvec_cons1)
|
obua@15009
|
721 |
apply (subst disj_matrices_add_le_x)
|
obua@15009
|
722 |
apply (simp add: disj_move_sparse_vec_mat[OF order_refl])
|
obua@15009
|
723 |
apply (rule disj_matrices_x_add)
|
obua@15009
|
724 |
apply (simp add: disj_move_sparse_row_vector_twice)
|
obua@15009
|
725 |
apply (simp add: disj_move_sparse_vec_mat[OF less_imp_le] disj_matrices_commute)
|
obua@15009
|
726 |
apply (simp, blast)
|
obua@15009
|
727 |
apply (intro strip)
|
nipkow@31816
|
728 |
apply (case_tac "i=j")
|
obua@15009
|
729 |
apply (simp_all)
|
obua@15009
|
730 |
apply (subst disj_matrices_add)
|
obua@15009
|
731 |
apply (simp_all add: disj_matrices_commute disj_move_sparse_vec_mat[OF order_refl])
|
obua@15009
|
732 |
apply (simp add: sorted_spvec_cons1 le_spvec_iff_sparse_row_le)
|
obua@15009
|
733 |
done
|
obua@15009
|
734 |
|
wenzelm@24124
|
735 |
declare [[simp_depth_limit = 999]]
|
obua@15009
|
736 |
|
wenzelm@38571
|
737 |
primrec abs_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat"
|
wenzelm@38571
|
738 |
where
|
wenzelm@38571
|
739 |
"abs_spmat [] = []"
|
wenzelm@38571
|
740 |
| "abs_spmat (a#as) = (fst a, abs_spvec (snd a))#(abs_spmat as)"
|
obua@15009
|
741 |
|
wenzelm@38571
|
742 |
primrec minus_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat"
|
wenzelm@38571
|
743 |
where
|
wenzelm@38571
|
744 |
"minus_spmat [] = []"
|
wenzelm@38571
|
745 |
| "minus_spmat (a#as) = (fst a, minus_spvec (snd a))#(minus_spmat as)"
|
obua@15178
|
746 |
|
obua@15178
|
747 |
lemma sparse_row_matrix_minus:
|
obua@15178
|
748 |
"sparse_row_matrix (minus_spmat A) = - (sparse_row_matrix A)"
|
obua@15178
|
749 |
apply (induct A)
|
obua@15178
|
750 |
apply (simp_all add: sparse_row_vector_minus sparse_row_matrix_cons)
|
obua@15178
|
751 |
apply (subst Rep_matrix_inject[symmetric])
|
obua@15178
|
752 |
apply (rule ext)+
|
obua@15178
|
753 |
apply simp
|
obua@15178
|
754 |
done
|
obua@15178
|
755 |
|
obua@15178
|
756 |
lemma Rep_sparse_row_vector_zero: "x \<noteq> 0 \<Longrightarrow> Rep_matrix (sparse_row_vector v) x y = 0"
|
obua@15178
|
757 |
proof -
|
obua@15178
|
758 |
assume x:"x \<noteq> 0"
|
obua@15178
|
759 |
have r:"nrows (sparse_row_vector v) <= Suc 0" by (rule nrows_spvec)
|
obua@15178
|
760 |
show ?thesis
|
obua@15178
|
761 |
apply (rule nrows)
|
obua@15178
|
762 |
apply (subgoal_tac "Suc 0 <= x")
|
obua@15178
|
763 |
apply (insert r)
|
obua@15178
|
764 |
apply (simp only:)
|
obua@15178
|
765 |
apply (insert x)
|
obua@15178
|
766 |
apply arith
|
obua@15178
|
767 |
done
|
obua@15178
|
768 |
qed
|
obua@15178
|
769 |
|
obua@15178
|
770 |
lemma sparse_row_matrix_abs:
|
obua@15178
|
771 |
"sorted_spvec A \<Longrightarrow> sorted_spmat A \<Longrightarrow> sparse_row_matrix (abs_spmat A) = abs (sparse_row_matrix A)"
|
obua@15178
|
772 |
apply (induct A)
|
obua@15178
|
773 |
apply (simp_all add: sparse_row_vector_abs sparse_row_matrix_cons)
|
obua@15178
|
774 |
apply (frule_tac sorted_spvec_cons1, simp)
|
obua@15580
|
775 |
apply (simplesubst Rep_matrix_inject[symmetric])
|
obua@15178
|
776 |
apply (rule ext)+
|
obua@15178
|
777 |
apply auto
|
obua@15178
|
778 |
apply (case_tac "x=a")
|
obua@15178
|
779 |
apply (simp)
|
paulson@15481
|
780 |
apply (simplesubst sorted_sparse_row_matrix_zero)
|
obua@15178
|
781 |
apply auto
|
paulson@15481
|
782 |
apply (simplesubst Rep_sparse_row_vector_zero)
|
huffman@47573
|
783 |
apply simp_all
|
obua@15178
|
784 |
done
|
obua@15178
|
785 |
|
obua@15178
|
786 |
lemma sorted_spvec_minus_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec (minus_spmat A)"
|
obua@15178
|
787 |
apply (induct A)
|
obua@15178
|
788 |
apply (simp)
|
obua@15178
|
789 |
apply (frule sorted_spvec_cons1, simp)
|
nipkow@15236
|
790 |
apply (simp add: sorted_spvec.simps split:list.split_asm)
|
obua@15178
|
791 |
done
|
obua@15178
|
792 |
|
obua@15178
|
793 |
lemma sorted_spvec_abs_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec (abs_spmat A)"
|
obua@15178
|
794 |
apply (induct A)
|
obua@15178
|
795 |
apply (simp)
|
obua@15178
|
796 |
apply (frule sorted_spvec_cons1, simp)
|
nipkow@15236
|
797 |
apply (simp add: sorted_spvec.simps split:list.split_asm)
|
obua@15178
|
798 |
done
|
obua@15178
|
799 |
|
obua@15178
|
800 |
lemma sorted_spmat_minus_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (minus_spmat A)"
|
obua@15178
|
801 |
apply (induct A)
|
obua@15178
|
802 |
apply (simp_all add: sorted_spvec_minus_spvec)
|
obua@15178
|
803 |
done
|
obua@15178
|
804 |
|
obua@15178
|
805 |
lemma sorted_spmat_abs_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (abs_spmat A)"
|
obua@15178
|
806 |
apply (induct A)
|
obua@15178
|
807 |
apply (simp_all add: sorted_spvec_abs_spvec)
|
obua@15178
|
808 |
done
|
obua@15178
|
809 |
|
wenzelm@38571
|
810 |
definition diff_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
|
wenzelm@38571
|
811 |
where "diff_spmat A B = add_spmat A (minus_spmat B)"
|
obua@15178
|
812 |
|
obua@15178
|
813 |
lemma sorted_spmat_diff_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat B \<Longrightarrow> sorted_spmat (diff_spmat A B)"
|
obua@15178
|
814 |
by (simp add: diff_spmat_def sorted_spmat_minus_spmat sorted_spmat_add_spmat)
|
obua@15178
|
815 |
|
obua@15178
|
816 |
lemma sorted_spvec_diff_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec B \<Longrightarrow> sorted_spvec (diff_spmat A B)"
|
obua@15178
|
817 |
by (simp add: diff_spmat_def sorted_spvec_minus_spmat sorted_spvec_add_spmat)
|
obua@15178
|
818 |
|
obua@15178
|
819 |
lemma sparse_row_diff_spmat: "sparse_row_matrix (diff_spmat A B ) = (sparse_row_matrix A) - (sparse_row_matrix B)"
|
obua@15178
|
820 |
by (simp add: diff_spmat_def sparse_row_add_spmat sparse_row_matrix_minus)
|
obua@15178
|
821 |
|
wenzelm@38571
|
822 |
definition sorted_sparse_matrix :: "'a spmat \<Rightarrow> bool"
|
wenzelm@38571
|
823 |
where "sorted_sparse_matrix A \<longleftrightarrow> sorted_spvec A & sorted_spmat A"
|
obua@15178
|
824 |
|
obua@15178
|
825 |
lemma sorted_sparse_matrix_imp_spvec: "sorted_sparse_matrix A \<Longrightarrow> sorted_spvec A"
|
obua@15178
|
826 |
by (simp add: sorted_sparse_matrix_def)
|
obua@15178
|
827 |
|
obua@15178
|
828 |
lemma sorted_sparse_matrix_imp_spmat: "sorted_sparse_matrix A \<Longrightarrow> sorted_spmat A"
|
obua@15178
|
829 |
by (simp add: sorted_sparse_matrix_def)
|
obua@15178
|
830 |
|
obua@15178
|
831 |
lemmas sorted_sp_simps =
|
obua@15178
|
832 |
sorted_spvec.simps
|
obua@15178
|
833 |
sorted_spmat.simps
|
obua@15178
|
834 |
sorted_sparse_matrix_def
|
obua@15178
|
835 |
|
obua@15178
|
836 |
lemma bool1: "(\<not> True) = False" by blast
|
obua@15178
|
837 |
lemma bool2: "(\<not> False) = True" by blast
|
obua@15178
|
838 |
lemma bool3: "((P\<Colon>bool) \<and> True) = P" by blast
|
obua@15178
|
839 |
lemma bool4: "(True \<and> (P\<Colon>bool)) = P" by blast
|
obua@15178
|
840 |
lemma bool5: "((P\<Colon>bool) \<and> False) = False" by blast
|
obua@15178
|
841 |
lemma bool6: "(False \<and> (P\<Colon>bool)) = False" by blast
|
obua@15178
|
842 |
lemma bool7: "((P\<Colon>bool) \<or> True) = True" by blast
|
obua@15178
|
843 |
lemma bool8: "(True \<or> (P\<Colon>bool)) = True" by blast
|
obua@15178
|
844 |
lemma bool9: "((P\<Colon>bool) \<or> False) = P" by blast
|
obua@15178
|
845 |
lemma bool10: "(False \<or> (P\<Colon>bool)) = P" by blast
|
obua@15178
|
846 |
lemmas boolarith = bool1 bool2 bool3 bool4 bool5 bool6 bool7 bool8 bool9 bool10
|
obua@15178
|
847 |
|
obua@15178
|
848 |
lemma if_case_eq: "(if b then x else y) = (case b of True => x | False => y)" by simp
|
obua@15178
|
849 |
|
wenzelm@38571
|
850 |
primrec pprt_spvec :: "('a::{lattice_ab_group_add}) spvec \<Rightarrow> 'a spvec"
|
wenzelm@38571
|
851 |
where
|
wenzelm@38571
|
852 |
"pprt_spvec [] = []"
|
wenzelm@38571
|
853 |
| "pprt_spvec (a#as) = (fst a, pprt (snd a)) # (pprt_spvec as)"
|
obua@15580
|
854 |
|
wenzelm@38571
|
855 |
primrec nprt_spvec :: "('a::{lattice_ab_group_add}) spvec \<Rightarrow> 'a spvec"
|
wenzelm@38571
|
856 |
where
|
wenzelm@38571
|
857 |
"nprt_spvec [] = []"
|
wenzelm@38571
|
858 |
| "nprt_spvec (a#as) = (fst a, nprt (snd a)) # (nprt_spvec as)"
|
obua@15580
|
859 |
|
wenzelm@38571
|
860 |
primrec pprt_spmat :: "('a::{lattice_ab_group_add}) spmat \<Rightarrow> 'a spmat"
|
wenzelm@38571
|
861 |
where
|
wenzelm@38571
|
862 |
"pprt_spmat [] = []"
|
wenzelm@38571
|
863 |
| "pprt_spmat (a#as) = (fst a, pprt_spvec (snd a))#(pprt_spmat as)"
|
obua@15580
|
864 |
|
wenzelm@38571
|
865 |
primrec nprt_spmat :: "('a::{lattice_ab_group_add}) spmat \<Rightarrow> 'a spmat"
|
wenzelm@38571
|
866 |
where
|
obua@15580
|
867 |
"nprt_spmat [] = []"
|
wenzelm@38571
|
868 |
| "nprt_spmat (a#as) = (fst a, nprt_spvec (snd a))#(nprt_spmat as)"
|
obua@15580
|
869 |
|
obua@15580
|
870 |
|
haftmann@35028
|
871 |
lemma pprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) \<Longrightarrow> pprt (A+B) = pprt A + pprt B"
|
haftmann@22452
|
872 |
apply (simp add: pprt_def sup_matrix_def)
|
obua@15580
|
873 |
apply (simp add: Rep_matrix_inject[symmetric])
|
obua@15580
|
874 |
apply (rule ext)+
|
obua@15580
|
875 |
apply simp
|
obua@15580
|
876 |
apply (case_tac "Rep_matrix A x xa \<noteq> 0")
|
obua@15580
|
877 |
apply (simp_all add: disj_matrices_contr1)
|
obua@15580
|
878 |
done
|
obua@15580
|
879 |
|
haftmann@35028
|
880 |
lemma nprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) \<Longrightarrow> nprt (A+B) = nprt A + nprt B"
|
haftmann@22452
|
881 |
apply (simp add: nprt_def inf_matrix_def)
|
obua@15580
|
882 |
apply (simp add: Rep_matrix_inject[symmetric])
|
obua@15580
|
883 |
apply (rule ext)+
|
obua@15580
|
884 |
apply simp
|
obua@15580
|
885 |
apply (case_tac "Rep_matrix A x xa \<noteq> 0")
|
obua@15580
|
886 |
apply (simp_all add: disj_matrices_contr1)
|
obua@15580
|
887 |
done
|
obua@15580
|
888 |
|
haftmann@35028
|
889 |
lemma pprt_singleton[simp]: "pprt (singleton_matrix j i (x::_::lattice_ring)) = singleton_matrix j i (pprt x)"
|
haftmann@22452
|
890 |
apply (simp add: pprt_def sup_matrix_def)
|
obua@15580
|
891 |
apply (simp add: Rep_matrix_inject[symmetric])
|
obua@15580
|
892 |
apply (rule ext)+
|
obua@15580
|
893 |
apply simp
|
obua@15580
|
894 |
done
|
obua@15580
|
895 |
|
haftmann@35028
|
896 |
lemma nprt_singleton[simp]: "nprt (singleton_matrix j i (x::_::lattice_ring)) = singleton_matrix j i (nprt x)"
|
haftmann@22452
|
897 |
apply (simp add: nprt_def inf_matrix_def)
|
obua@15580
|
898 |
apply (simp add: Rep_matrix_inject[symmetric])
|
obua@15580
|
899 |
apply (rule ext)+
|
obua@15580
|
900 |
apply simp
|
obua@15580
|
901 |
done
|
obua@15580
|
902 |
|
obua@15580
|
903 |
lemma less_imp_le: "a < b \<Longrightarrow> a <= (b::_::order)" by (simp add: less_def)
|
obua@15580
|
904 |
|
haftmann@35028
|
905 |
lemma sparse_row_vector_pprt: "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (pprt_spvec v) = pprt (sparse_row_vector v)"
|
obua@15580
|
906 |
apply (induct v)
|
obua@15580
|
907 |
apply (simp_all)
|
obua@15580
|
908 |
apply (frule sorted_spvec_cons1, auto)
|
obua@15580
|
909 |
apply (subst pprt_add)
|
obua@15580
|
910 |
apply (subst disj_matrices_commute)
|
obua@15580
|
911 |
apply (rule disj_sparse_row_singleton)
|
obua@15580
|
912 |
apply auto
|
obua@15580
|
913 |
done
|
obua@15580
|
914 |
|
haftmann@35028
|
915 |
lemma sparse_row_vector_nprt: "sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (nprt_spvec v) = nprt (sparse_row_vector v)"
|
obua@15580
|
916 |
apply (induct v)
|
obua@15580
|
917 |
apply (simp_all)
|
obua@15580
|
918 |
apply (frule sorted_spvec_cons1, auto)
|
obua@15580
|
919 |
apply (subst nprt_add)
|
obua@15580
|
920 |
apply (subst disj_matrices_commute)
|
obua@15580
|
921 |
apply (rule disj_sparse_row_singleton)
|
obua@15580
|
922 |
apply auto
|
obua@15580
|
923 |
done
|
obua@15580
|
924 |
|
obua@15580
|
925 |
|
haftmann@35028
|
926 |
lemma pprt_move_matrix: "pprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (pprt A) j i"
|
obua@15580
|
927 |
apply (simp add: pprt_def)
|
haftmann@22452
|
928 |
apply (simp add: sup_matrix_def)
|
obua@15580
|
929 |
apply (simp add: Rep_matrix_inject[symmetric])
|
obua@15580
|
930 |
apply (rule ext)+
|
obua@15580
|
931 |
apply (simp)
|
obua@15580
|
932 |
done
|
obua@15580
|
933 |
|
haftmann@35028
|
934 |
lemma nprt_move_matrix: "nprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (nprt A) j i"
|
obua@15580
|
935 |
apply (simp add: nprt_def)
|
haftmann@22452
|
936 |
apply (simp add: inf_matrix_def)
|
obua@15580
|
937 |
apply (simp add: Rep_matrix_inject[symmetric])
|
obua@15580
|
938 |
apply (rule ext)+
|
obua@15580
|
939 |
apply (simp)
|
obua@15580
|
940 |
done
|
obua@15580
|
941 |
|
haftmann@35028
|
942 |
lemma sparse_row_matrix_pprt: "sorted_spvec (m :: 'a::lattice_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (pprt_spmat m) = pprt (sparse_row_matrix m)"
|
obua@15580
|
943 |
apply (induct m)
|
obua@15580
|
944 |
apply simp
|
obua@15580
|
945 |
apply simp
|
obua@15580
|
946 |
apply (frule sorted_spvec_cons1)
|
obua@15580
|
947 |
apply (simp add: sparse_row_matrix_cons sparse_row_vector_pprt)
|
obua@15580
|
948 |
apply (subst pprt_add)
|
obua@15580
|
949 |
apply (subst disj_matrices_commute)
|
obua@15580
|
950 |
apply (rule disj_move_sparse_vec_mat)
|
obua@15580
|
951 |
apply auto
|
obua@15580
|
952 |
apply (simp add: sorted_spvec.simps)
|
obua@15580
|
953 |
apply (simp split: list.split)
|
obua@15580
|
954 |
apply auto
|
obua@15580
|
955 |
apply (simp add: pprt_move_matrix)
|
obua@15580
|
956 |
done
|
obua@15580
|
957 |
|
haftmann@35028
|
958 |
lemma sparse_row_matrix_nprt: "sorted_spvec (m :: 'a::lattice_ring spmat) \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (nprt_spmat m) = nprt (sparse_row_matrix m)"
|
obua@15580
|
959 |
apply (induct m)
|
obua@15580
|
960 |
apply simp
|
obua@15580
|
961 |
apply simp
|
obua@15580
|
962 |
apply (frule sorted_spvec_cons1)
|
obua@15580
|
963 |
apply (simp add: sparse_row_matrix_cons sparse_row_vector_nprt)
|
obua@15580
|
964 |
apply (subst nprt_add)
|
obua@15580
|
965 |
apply (subst disj_matrices_commute)
|
obua@15580
|
966 |
apply (rule disj_move_sparse_vec_mat)
|
obua@15580
|
967 |
apply auto
|
obua@15580
|
968 |
apply (simp add: sorted_spvec.simps)
|
obua@15580
|
969 |
apply (simp split: list.split)
|
obua@15580
|
970 |
apply auto
|
obua@15580
|
971 |
apply (simp add: nprt_move_matrix)
|
obua@15580
|
972 |
done
|
obua@15580
|
973 |
|
obua@15580
|
974 |
lemma sorted_pprt_spvec: "sorted_spvec v \<Longrightarrow> sorted_spvec (pprt_spvec v)"
|
obua@15580
|
975 |
apply (induct v)
|
obua@15580
|
976 |
apply (simp)
|
obua@15580
|
977 |
apply (frule sorted_spvec_cons1)
|
obua@15580
|
978 |
apply simp
|
obua@15580
|
979 |
apply (simp add: sorted_spvec.simps split:list.split_asm)
|
obua@15580
|
980 |
done
|
obua@15580
|
981 |
|
obua@15580
|
982 |
lemma sorted_nprt_spvec: "sorted_spvec v \<Longrightarrow> sorted_spvec (nprt_spvec v)"
|
obua@15580
|
983 |
apply (induct v)
|
obua@15580
|
984 |
apply (simp)
|
obua@15580
|
985 |
apply (frule sorted_spvec_cons1)
|
obua@15580
|
986 |
apply simp
|
obua@15580
|
987 |
apply (simp add: sorted_spvec.simps split:list.split_asm)
|
obua@15580
|
988 |
done
|
obua@15580
|
989 |
|
obua@15580
|
990 |
lemma sorted_spvec_pprt_spmat: "sorted_spvec m \<Longrightarrow> sorted_spvec (pprt_spmat m)"
|
obua@15580
|
991 |
apply (induct m)
|
obua@15580
|
992 |
apply (simp)
|
obua@15580
|
993 |
apply (frule sorted_spvec_cons1)
|
obua@15580
|
994 |
apply simp
|
obua@15580
|
995 |
apply (simp add: sorted_spvec.simps split:list.split_asm)
|
obua@15580
|
996 |
done
|
obua@15580
|
997 |
|
obua@15580
|
998 |
lemma sorted_spvec_nprt_spmat: "sorted_spvec m \<Longrightarrow> sorted_spvec (nprt_spmat m)"
|
obua@15580
|
999 |
apply (induct m)
|
obua@15580
|
1000 |
apply (simp)
|
obua@15580
|
1001 |
apply (frule sorted_spvec_cons1)
|
obua@15580
|
1002 |
apply simp
|
obua@15580
|
1003 |
apply (simp add: sorted_spvec.simps split:list.split_asm)
|
obua@15580
|
1004 |
done
|
obua@15580
|
1005 |
|
obua@15580
|
1006 |
lemma sorted_spmat_pprt_spmat: "sorted_spmat m \<Longrightarrow> sorted_spmat (pprt_spmat m)"
|
obua@15580
|
1007 |
apply (induct m)
|
obua@15580
|
1008 |
apply (simp_all add: sorted_pprt_spvec)
|
obua@15580
|
1009 |
done
|
obua@15580
|
1010 |
|
obua@15580
|
1011 |
lemma sorted_spmat_nprt_spmat: "sorted_spmat m \<Longrightarrow> sorted_spmat (nprt_spmat m)"
|
obua@15580
|
1012 |
apply (induct m)
|
obua@15580
|
1013 |
apply (simp_all add: sorted_nprt_spvec)
|
obua@15580
|
1014 |
done
|
obua@15580
|
1015 |
|
haftmann@35413
|
1016 |
definition mult_est_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat" where
|
wenzelm@38571
|
1017 |
"mult_est_spmat r1 r2 s1 s2 =
|
nipkow@31816
|
1018 |
add_spmat (mult_spmat (pprt_spmat s2) (pprt_spmat r2)) (add_spmat (mult_spmat (pprt_spmat s1) (nprt_spmat r2))
|
nipkow@31816
|
1019 |
(add_spmat (mult_spmat (nprt_spmat s2) (pprt_spmat r1)) (mult_spmat (nprt_spmat s1) (nprt_spmat r1))))"
|
obua@15580
|
1020 |
|
obua@15580
|
1021 |
lemmas sparse_row_matrix_op_simps =
|
obua@15580
|
1022 |
sorted_sparse_matrix_imp_spmat sorted_sparse_matrix_imp_spvec
|
obua@15580
|
1023 |
sparse_row_add_spmat sorted_spvec_add_spmat sorted_spmat_add_spmat
|
obua@15580
|
1024 |
sparse_row_diff_spmat sorted_spvec_diff_spmat sorted_spmat_diff_spmat
|
obua@15580
|
1025 |
sparse_row_matrix_minus sorted_spvec_minus_spmat sorted_spmat_minus_spmat
|
obua@15580
|
1026 |
sparse_row_mult_spmat sorted_spvec_mult_spmat sorted_spmat_mult_spmat
|
obua@15580
|
1027 |
sparse_row_matrix_abs sorted_spvec_abs_spmat sorted_spmat_abs_spmat
|
obua@15580
|
1028 |
le_spmat_iff_sparse_row_le
|
obua@15580
|
1029 |
sparse_row_matrix_pprt sorted_spvec_pprt_spmat sorted_spmat_pprt_spmat
|
obua@15580
|
1030 |
sparse_row_matrix_nprt sorted_spvec_nprt_spmat sorted_spmat_nprt_spmat
|
obua@15580
|
1031 |
|
obua@15580
|
1032 |
lemma zero_eq_Numeral0: "(0::_::number_ring) = Numeral0" by simp
|
obua@15580
|
1033 |
|
obua@15580
|
1034 |
lemmas sparse_row_matrix_arith_simps[simplified zero_eq_Numeral0] =
|
obua@15580
|
1035 |
mult_spmat.simps mult_spvec_spmat.simps
|
obua@15580
|
1036 |
addmult_spvec.simps
|
obua@15580
|
1037 |
smult_spvec_empty smult_spvec_cons
|
obua@15580
|
1038 |
add_spmat.simps add_spvec.simps
|
obua@15580
|
1039 |
minus_spmat.simps minus_spvec.simps
|
obua@15580
|
1040 |
abs_spmat.simps abs_spvec.simps
|
obua@15580
|
1041 |
diff_spmat_def
|
obua@15580
|
1042 |
le_spmat.simps le_spvec.simps
|
obua@15580
|
1043 |
pprt_spmat.simps pprt_spvec.simps
|
obua@15580
|
1044 |
nprt_spmat.simps nprt_spvec.simps
|
obua@15580
|
1045 |
mult_est_spmat_def
|
obua@15580
|
1046 |
|
obua@15580
|
1047 |
|
obua@15580
|
1048 |
(*lemma spm_linprog_dual_estimate_1:
|
obua@15178
|
1049 |
assumes
|
obua@15178
|
1050 |
"sorted_sparse_matrix A1"
|
obua@15178
|
1051 |
"sorted_sparse_matrix A2"
|
obua@15178
|
1052 |
"sorted_sparse_matrix c1"
|
obua@15178
|
1053 |
"sorted_sparse_matrix c2"
|
obua@15178
|
1054 |
"sorted_sparse_matrix y"
|
obua@15178
|
1055 |
"sorted_spvec b"
|
obua@15178
|
1056 |
"sorted_spvec r"
|
obua@15178
|
1057 |
"le_spmat ([], y)"
|
haftmann@35028
|
1058 |
"A * x \<le> sparse_row_matrix (b::('a::lattice_ring) spmat)"
|
obua@15178
|
1059 |
"sparse_row_matrix A1 <= A"
|
obua@15178
|
1060 |
"A <= sparse_row_matrix A2"
|
obua@15178
|
1061 |
"sparse_row_matrix c1 <= c"
|
obua@15178
|
1062 |
"c <= sparse_row_matrix c2"
|
obua@15178
|
1063 |
"abs x \<le> sparse_row_matrix r"
|
obua@15178
|
1064 |
shows
|
obua@15178
|
1065 |
"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),
|
obua@15178
|
1066 |
abs_spmat (diff_spmat (mult_spmat y A1) c1)), diff_spmat c2 c1)) r))"
|
obua@15178
|
1067 |
by (insert prems, simp add: sparse_row_matrix_op_simps linprog_dual_estimate_1[where A=A])
|
obua@15580
|
1068 |
*)
|
obua@15009
|
1069 |
|
obua@15009
|
1070 |
end
|