(* Title: HOL/Matrix_LP/SparseMatrix.thy
Author: Steven Obua; updated to Isar by LCP
*)
theory SparseMatrix
imports Matrix
begin
type_synonym 'a spvec = "(nat * 'a) list"
type_synonym 'a spmat = "'a spvec spvec"
definition sparse_row_vector :: "('a::ab_group_add) spvec \<Rightarrow> 'a matrix"
where "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 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: "\<forall>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))"
by (induct l arbitrary: x y, auto)
lemma sparse_row_vector_cons[simp]:
"sparse_row_vector (a # arr) = (singleton_matrix 0 (fst a) (snd a)) + (sparse_row_vector arr)"
by (induct arr) (auto simp: foldl_distrstart sparse_row_vector_def)
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) \<le> (Suc 0)"
by (induct x) (auto simp: add_nrows)
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"
by (induct arr) (auto simp: foldl_distrstart sparse_row_matrix_def)
lemma sparse_row_matrix_append: "sparse_row_matrix (arr@brr) = (sparse_row_matrix arr) + (sparse_row_matrix brr)"
by (induct arr) (auto simp: sparse_row_matrix_cons)
fun sorted_spvec :: "'a spvec \<Rightarrow> bool"
where
"sorted_spvec [] = True"
| sorted_spvec_step1: "sorted_spvec [a] = True"
| sorted_spvec_step: "sorted_spvec ((m,x)#(n,y)#bs) = ((m < n) \<and> (sorted_spvec ((n,y)#bs)))"
primrec sorted_spmat :: "'a spmat \<Rightarrow> bool"
where
"sorted_spmat [] = True"
| "sorted_spmat (a#as) = ((sorted_spvec (snd a)) \<and> (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"
using sorted_spvec.elims(2) sorted_spvec_empty by blast
lemma sorted_spvec_cons2: "sorted_spvec (a#b#t) \<Longrightarrow> sorted_spvec (a#t)"
by (smt (verit, del_insts) sorted_spvec_step order.strict_trans list.inject sorted_spvec.elims(3) surj_pair)
lemma sorted_spvec_cons3: "sorted_spvec(a#b#t) \<Longrightarrow> fst a < fst b"
by (metis sorted_spvec_step prod.collapse)
lemma sorted_sparse_row_vector_zero:
assumes "m \<le> n"
shows "sorted_spvec ((n,a)#arr) \<Longrightarrow> Rep_matrix (sparse_row_vector arr) j m = 0"
proof (induct arr)
case Nil
then show ?case by auto
next
case (Cons a arr)
with assms show ?case
by (auto dest: sorted_spvec_cons2 sorted_spvec_cons3)
qed
lemma sorted_sparse_row_matrix_zero[rule_format]:
assumes "m \<le> n"
shows "sorted_spvec ((n,a)#arr) \<Longrightarrow> Rep_matrix (sparse_row_matrix arr) m j = 0"
proof (induct arr)
case Nil
then show ?case by auto
next
case (Cons a arr)
with assms show ?case
unfolding sparse_row_matrix_cons
by (auto dest: sorted_spvec_cons2 sorted_spvec_cons3)
qed
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::lattice_ab_group_add_abs) spvec \<Rightarrow> 'a spvec"
where
"abs_spvec [] = []"
| "abs_spvec (a#as) = (fst a, \<bar>snd a\<bar>)#(abs_spvec as)"
lemma sparse_row_vector_minus:
"sparse_row_vector (minus_spvec v) = - (sparse_row_vector v)"
proof (induct v)
case Nil
then show ?case
by auto
next
case (Cons a v)
then have "singleton_matrix 0 (fst a) (- snd a) = - singleton_matrix 0 (fst a) (snd a)"
by (simp add: Rep_matrix_inject minus_matrix_def)
then show ?case
by (simp add: local.Cons)
qed
lemma sparse_row_vector_abs:
"sorted_spvec (v :: 'a::lattice_ring spvec) \<Longrightarrow> sparse_row_vector (abs_spvec v) = \<bar>sparse_row_vector v\<bar>"
proof (induct v)
case Nil
then show ?case
by simp
next
case (Cons ab v)
then have v: "sorted_spvec v"
using sorted_spvec_cons1 by blast
show ?case
proof (cases ab)
case (Pair a b)
then have 0: "Rep_matrix (sparse_row_vector v) 0 a = 0"
using Cons.prems sorted_sparse_row_vector_zero by blast
with v Cons show ?thesis
by (fastforce simp: Pair simp flip: Rep_matrix_inject)
qed
qed
lemma sorted_spvec_minus_spvec:
"sorted_spvec v \<Longrightarrow> sorted_spvec (minus_spvec v)"
by (induct v rule: sorted_spvec.induct) (auto simp: sorted_spvec_step1 sorted_spvec_step)
lemma sorted_spvec_abs_spvec:
"sorted_spvec v \<Longrightarrow> sorted_spvec (abs_spvec v)"
by (induct v rule: sorted_spvec.induct) (auto simp: sorted_spvec_step1 sorted_spvec_step)
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)
fun addmult_spvec :: "('a::ring) \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec"
where
"addmult_spvec y arr [] = arr"
| "addmult_spvec y [] brr = smult_spvec y brr"
| "addmult_spvec y ((i,a)#arr) ((j,b)#brr) = (
if i < j then ((i,a)#(addmult_spvec y arr ((j,b)#brr)))
else (if (j < i) then ((j, y * b)#(addmult_spvec y ((i,a)#arr) brr))
else ((i, a + y*b)#(addmult_spvec y arr brr))))"
(* Steven used termination "measure (% (y, a, b). length a + (length b))" *)
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 simp
lemma sparse_row_vector_map: "(\<forall>x y. f (x+y) = (f x) + (f y)) \<Longrightarrow> (f::'a\<Rightarrow>('a::lattice_ring)) 0 = 0 \<Longrightarrow>
sparse_row_vector (map (% x. (fst x, f (snd x))) a) = apply_matrix f (sparse_row_vector a)"
by (induct a) (simp_all add: apply_matrix_add)
lemma sparse_row_vector_smult: "sparse_row_vector (smult_spvec y a) = scalar_mult y (sparse_row_vector a)"
by (induct a) (simp_all add: smult_spvec_cons scalar_mult_add)
lemma sparse_row_vector_addmult_spvec: "sparse_row_vector (addmult_spvec (y::'a::lattice_ring) a b) =
(sparse_row_vector a) + (scalar_mult y (sparse_row_vector b))"
by (induct y a b rule: addmult_spvec.induct)
(simp_all add: scalar_mult_add smult_spvec_cons sparse_row_vector_smult singleton_matrix_add)
lemma sorted_smult_spvec: "sorted_spvec a \<Longrightarrow> sorted_spvec (smult_spvec y a)"
by (induct a rule: sorted_spvec.induct) (auto simp: smult_spvec_def sorted_spvec_step1 sorted_spvec_step)
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)"
by (induct brr) (auto simp: sorted_spvec.simps)
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))"
by (induct arr) (auto simp: smult_spvec_def sorted_spvec.simps)
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))"
by (smt (verit, ccfv_threshold) sorted_spvec_step addmult_spvec.simps(1) list.distinct(1) list.sel(3) sorted_spvec.elims(1) sorted_spvec_addmult_spvec_helper2)
lemma sorted_addmult_spvec: "sorted_spvec a \<Longrightarrow> sorted_spvec b \<Longrightarrow> sorted_spvec (addmult_spvec y a b)"
proof (induct y a b rule: addmult_spvec.induct)
case (1 y arr)
then show ?case
by simp
next
case (2 y v va)
then show ?case
by (simp add: sorted_smult_spvec)
next
case (3 y i a arr j b brr)
show ?case
proof (cases i j rule: linorder_cases)
case less
with 3 show ?thesis
by (simp add: sorted_spvec_addmult_spvec_helper2 sorted_spvec_cons1)
next
case equal
with 3 show ?thesis
by (simp add: sorted_spvec_addmult_spvec_helper3 sorted_spvec_cons1)
next
case greater
with 3 show ?thesis
by (simp add: sorted_spvec_addmult_spvec_helper sorted_spvec_cons1)
qed
qed
fun mult_spvec_spmat :: "('a::lattice_ring) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spmat \<Rightarrow> 'a spvec"
where
"mult_spvec_spmat c [] brr = c"
| "mult_spvec_spmat c arr [] = c"
| "mult_spvec_spmat c ((i,a)#arr) ((j,b)#brr) = (
if (i < j) then mult_spvec_spmat c arr ((j,b)#brr)
else if (j < i) then mult_spvec_spmat c ((i,a)#arr) brr
else mult_spvec_spmat (addmult_spvec a c b) arr brr)"
lemma sparse_row_mult_spvec_spmat:
assumes "sorted_spvec (a::('a::lattice_ring) spvec)" "sorted_spvec B"
shows "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 \<le> nat ((int b)-(int a))" by arith
have not_iff: "!! a b. a = b \<Longrightarrow> (~ a) = (~ b)" by simp
{
fix a
fix v :: "(nat \<times> 'a) list"
assume a: "a < nrows(sparse_row_vector v)"
have "nrows(sparse_row_vector v) \<le> 1" by simp
then have "a = 0"
using a dual_order.strict_trans1 by blast
}
note nrows_helper = this
show ?thesis
using assms
proof (induct c a B rule: mult_spvec_spmat.induct)
case (1 c brr)
then show ?case
by simp
next
case (2 c v va)
then show ?case
by simp
next
case (3 c i a arr j b brr)
then have abrr: "sorted_spvec arr" "sorted_spvec brr"
using sorted_spvec_cons1 by blast+
have "\<And>m n. \<lbrakk>a \<noteq> 0; 0 < m\<rbrakk>
\<Longrightarrow> a * Rep_matrix (sparse_row_vector b) m n = 0"
by (metis mult_zero_right neq0_conv nrows_helper nrows_notzero)
then have \<dagger>: "scalar_mult a (sparse_row_vector b) =
singleton_matrix 0 j a * move_matrix (sparse_row_vector b) (int j) 0"
apply (intro matrix_eqI)
apply (simp)
apply (subst Rep_matrix_mult)
apply (subst foldseq_almostzero, auto)
done
show ?case
proof (cases i j rule: linorder_cases)
case less
with 3 abrr \<dagger> show ?thesis
apply (simp add: algebra_simps sparse_row_matrix_cons Rep_matrix_zero_imp_mult_zero)
by (metis Rep_matrix_zero_imp_mult_zero Rep_singleton_matrix less_imp_le_nat sorted_sparse_row_matrix_zero)
next
case equal
with 3 abrr \<dagger> show ?thesis
apply (simp add: sparse_row_matrix_cons algebra_simps sparse_row_vector_addmult_spvec)
apply (subst Rep_matrix_zero_imp_mult_zero)
using sorted_sparse_row_matrix_zero apply fastforce
apply (subst Rep_matrix_zero_imp_mult_zero)
apply (metis Rep_move_matrix comp_1 nrows_le nrows_spvec sorted_sparse_row_vector_zero verit_comp_simplify1(3))
apply simp
done
next
case greater
have "Rep_matrix (sparse_row_vector arr) j' k = 0 \<or>
Rep_matrix (move_matrix (sparse_row_vector b) (int j) 0) k
i' = 0"
if "sorted_spvec ((i, a) # arr)" for j' i' k
proof (cases "k \<le> j")
case True
with greater that show ?thesis
by (meson order.trans nat_less_le sorted_sparse_row_vector_zero)
qed (use nrows_helper nrows_notzero in force)
then have "sparse_row_vector arr * move_matrix (sparse_row_vector b) (int j) 0 = 0"
using greater 3
by (simp add: Rep_matrix_zero_imp_mult_zero)
with greater 3 abrr show ?thesis
apply (simp add: algebra_simps sparse_row_matrix_cons)
by (metis Rep_matrix_zero_imp_mult_zero Rep_move_matrix Rep_singleton_matrix comp_1 nrows_le nrows_spvec)
qed
qed
qed
lemma sorted_mult_spvec_spmat:
"sorted_spvec (c::('a::lattice_ring) spvec) \<Longrightarrow> sorted_spmat B \<Longrightarrow> sorted_spvec (mult_spvec_spmat c a B)"
by (induct c a B rule: mult_spvec_spmat.induct) (simp_all add: sorted_addmult_spvec)
primrec mult_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
where
"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:
"sorted_spmat A \<Longrightarrow> sorted_spvec B \<Longrightarrow>
sparse_row_matrix (mult_spmat A B) = (sparse_row_matrix A) * (sparse_row_matrix B)"
by (induct A) (auto simp: sparse_row_matrix_cons sparse_row_mult_spvec_spmat algebra_simps move_matrix_mult)
lemma sorted_spvec_mult_spmat:
fixes A :: "('a::lattice_ring) spmat"
shows "sorted_spvec A \<Longrightarrow> sorted_spvec (mult_spmat A B)"
by (induct A rule: sorted_spvec.induct) (auto simp: sorted_spvec.simps)
lemma sorted_spmat_mult_spmat:
"sorted_spmat (B::('a::lattice_ring) spmat) \<Longrightarrow> sorted_spmat (mult_spmat A B)"
by (induct A) (auto simp: sorted_mult_spvec_spmat)
fun add_spvec :: "('a::lattice_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> 'a spvec"
where
(* "measure (% (a, b). length a + (length b))" *)
"add_spvec arr [] = arr"
| "add_spvec [] brr = brr"
| "add_spvec ((i,a)#arr) ((j,b)#brr) = (
if i < j then (i,a)#(add_spvec arr ((j,b)#brr))
else if (j < i) then (j,b) # add_spvec ((i,a)#arr) brr
else (i, a+b) # add_spvec arr brr)"
lemma add_spvec_empty1[simp]: "add_spvec [] a = a"
by (cases a, auto)
lemma sparse_row_vector_add: "sparse_row_vector (add_spvec a b) = (sparse_row_vector a) + (sparse_row_vector b)"
by (induct a b rule: add_spvec.induct) (simp_all add: singleton_matrix_add)
fun add_spmat :: "('a::lattice_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
where
(* "measure (% (A,B). (length A)+(length B))" *)
"add_spmat [] bs = bs"
| "add_spmat as [] = as"
| "add_spmat ((i,a)#as) ((j,b)#bs) = (
if i < j then
(i,a) # add_spmat as ((j,b)#bs)
else if j < i then
(j,b) # add_spmat ((i,a)#as) bs
else
(i, add_spvec a b) # add_spmat as bs)"
lemma add_spmat_Nil2[simp]: "add_spmat as [] = as"
by(cases as) auto
lemma sparse_row_add_spmat: "sparse_row_matrix (add_spmat A B) = (sparse_row_matrix A) + (sparse_row_matrix B)"
by (induct A B rule: add_spmat.induct) (auto simp: sparse_row_matrix_cons sparse_row_vector_add move_matrix_add)
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 "(\<forall>x ab a. x = (a,b)#arr \<longrightarrow> add_spvec x brr = (ab, bb) # list \<longrightarrow> (ab = a | (ab = fst (hd brr))))"
by (induct brr rule: add_spvec.induct) (auto split:if_splits)
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)))"
by (smt (verit) add_spmat.elims fst_conv list.distinct(1) list.sel(1))
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)))"
by (induct arr brr rule: add_spvec.induct) (auto split:if_splits)
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)))"
by (induct arr brr rule: add_spmat.induct) (auto split:if_splits)
lemma add_spvec_commute: "add_spvec a b = add_spvec b a"
by (induct a b rule: add_spvec.induct) auto
lemma add_spmat_commute: "add_spmat a b = add_spmat b a"
by (induct a b rule: add_spmat.induct) (simp_all add: add_spvec_commute)
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"
by (smt (verit, best) add_spvec.elims fst_conv list.sel(1) sorted_spvec_cons3)
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"
by (metis (no_types, opaque_lifting) add_spmat.simps(1) list.sel(1) neq_Nil_conv sorted_add_spmat_helper sorted_spvec_cons3)
lemma sorted_spvec_add_spvec: "sorted_spvec a \<Longrightarrow> sorted_spvec b \<Longrightarrow> sorted_spvec (add_spvec a b)"
proof (induct a b rule: add_spvec.induct)
case (3 i a arr j b brr)
then have "sorted_spvec arr" "sorted_spvec brr"
using sorted_spvec_cons1 by blast+
with 3 show ?case
apply simp
by (smt (verit, ccfv_SIG) add_spvec.simps(2) list.sel(3) sorted_add_spvec_helper sorted_spvec.elims(1))
qed auto
lemma sorted_spvec_add_spmat:
"sorted_spvec A \<Longrightarrow> sorted_spvec B \<Longrightarrow> sorted_spvec (add_spmat A B)"
proof (induct A B rule: add_spmat.induct)
case (1 bs)
then show ?case by auto
next
case (2 v va)
then show ?case by auto
next
case (3 i a as j b bs)
then have "sorted_spvec as" "sorted_spvec bs"
using sorted_spvec_cons1 by blast+
with 3 show ?case
apply simp
by (smt (verit) Pair_inject add_spmat.elims list.discI list.inject sorted_spvec.elims(1))
qed
lemma sorted_spmat_add_spmat[rule_format]: "sorted_spmat A \<Longrightarrow> sorted_spmat B \<Longrightarrow> sorted_spmat (add_spmat A B)"
by (induct A B rule: add_spmat.induct) (simp_all add: sorted_spvec_add_spvec)
fun le_spvec :: "('a::lattice_ab_group_add) spvec \<Rightarrow> 'a spvec \<Rightarrow> bool"
where
(* "measure (% (a,b). (length a) + (length b))" *)
"le_spvec [] [] = True"
| "le_spvec ((_,a)#as) [] = (a \<le> 0 & le_spvec as [])"
| "le_spvec [] ((_,b)#bs) = (0 \<le> b & le_spvec [] bs)"
| "le_spvec ((i,a)#as) ((j,b)#bs) = (
if (i < j) then a \<le> 0 & le_spvec as ((j,b)#bs)
else if (j < i) then 0 \<le> b & le_spvec ((i,a)#as) bs
else a \<le> b & le_spvec as bs)"
fun le_spmat :: "('a::lattice_ab_group_add) spmat \<Rightarrow> 'a spmat \<Rightarrow> bool"
where
(* "measure (% (a,b). (length a) + (length b))" *)
"le_spmat [] [] = True"
| "le_spmat ((i,a)#as) [] = (le_spvec a [] & le_spmat as [])"
| "le_spmat [] ((j,b)#bs) = (le_spvec [] b & le_spmat [] bs)"
| "le_spmat ((i,a)#as) ((j,b)#bs) = (
if i < j then (le_spvec a [] & le_spmat as ((j,b)#bs))
else if j < i then (le_spvec [] b & le_spmat ((i,a)#as) bs)
else (le_spvec a b & le_spmat as bs))"
definition disj_matrices :: "('a::zero) matrix \<Rightarrow> 'a matrix \<Rightarrow> bool" where
"disj_matrices A B \<longleftrightarrow>
(\<forall>j i. (Rep_matrix A j i \<noteq> 0) \<longrightarrow> (Rep_matrix B j i = 0)) & (\<forall>j i. (Rep_matrix B j i \<noteq> 0) \<longrightarrow> (Rep_matrix A j i = 0))"
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:
fixes A :: "('a::lattice_ab_group_add) matrix"
shows "disj_matrices A B \<Longrightarrow> disj_matrices C D \<Longrightarrow> disj_matrices A D
\<Longrightarrow> disj_matrices B C \<Longrightarrow> (A + B \<le> C + D) = (A \<le> C \<and> B \<le> D)"
apply (intro iffI conjI)
unfolding le_matrix_def disj_matrices_def
apply (metis Rep_matrix_add group_cancel.rule0 order_refl)
apply (metis (no_types, lifting) Rep_matrix_add add_cancel_right_left dual_order.refl)
by (meson add_mono le_matrix_def)
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: disj_matrices_def)
lemma disj_matrices_add_le_zero: "disj_matrices A B \<Longrightarrow>
(A + B \<le> 0) = (A \<le> 0 & (B::('a::lattice_ab_group_add) matrix) \<le> 0)"
by (rule disj_matrices_add[of A B 0 0, simplified])
lemma disj_matrices_add_zero_le: "disj_matrices A B \<Longrightarrow>
(0 \<le> A + B) = (0 \<le> A & 0 \<le> (B::('a::lattice_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 \<le> B + C) = (A \<le> C & 0 \<le> (B::('a::lattice_ab_group_add) matrix))"
by (auto simp: 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 \<le> C) = (A \<le> C & (B::('a::lattice_ab_group_add) matrix) \<le> 0)"
by (auto simp: disj_matrices_add[of B A 0 C,simplified] disj_matrices_commute)
lemma disj_sparse_row_singleton: "i \<le> 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)
using sorted_sparse_row_vector_zero by blast
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)"
by (smt (verit, ccfv_SIG) Rep_matrix_add add_0 disj_matrices_def)
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)"
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: disj_matrices_def)
lemma disj_move_sparse_vec_mat:
assumes "j \<le> a" and "sorted_spvec ((a, c) # as)"
shows "disj_matrices (sparse_row_matrix as) (move_matrix (sparse_row_vector b) (int j) i)"
proof -
have "Rep_matrix (sparse_row_vector b) (n-j) (nat (int m - i)) = 0"
if "\<not> n<j" and nz: "Rep_matrix (sparse_row_matrix as) n m \<noteq> 0"
for n m
proof -
have "n \<noteq> j"
using assms sorted_sparse_row_matrix_zero nz by blast
with that have "j < n" by auto
then show ?thesis
by (metis One_nat_def Suc_diff_Suc nrows nrows_spvec plus_1_eq_Suc trans_le_add1)
qed
then show ?thesis
by (auto simp: disj_matrices_def nat_minus_as_int)
qed
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)"
unfolding disj_matrices_def
by (smt (verit, ccfv_SIG) One_nat_def Rep_move_matrix of_nat_1 le_nat_iff nrows nrows_spvec of_nat_le_iff)
lemma le_spvec_iff_sparse_row_le:
"sorted_spvec a \<Longrightarrow> sorted_spvec b \<Longrightarrow> (le_spvec a b) \<longleftrightarrow> (sparse_row_vector a \<le> sparse_row_vector b)"
proof (induct a b rule: le_spvec.induct)
case 1
then show ?case
by auto
next
case (2 uu a as)
then have "sorted_spvec as"
by (metis sorted_spvec_cons1)
with 2 show ?case
apply (simp add: add.commute)
by (metis disj_matrices_add_le_zero disj_sparse_row_singleton le_refl singleton_le_zero)
next
case (3 uv b bs)
then have "sorted_spvec bs"
by (metis sorted_spvec_cons1)
with 3 show ?case
apply (simp add: add.commute)
by (metis disj_matrices_add_zero_le disj_sparse_row_singleton le_refl singleton_ge_zero)
next
case (4 i a as j b bs)
then obtain \<section>: "sorted_spvec as" "sorted_spvec bs"
by (metis sorted_spvec_cons1)
show ?case
proof (cases i j rule: linorder_cases)
case less
with 4 \<section> show ?thesis
apply (simp add: )
by (metis disj_matrices_add_le_x disj_matrices_add_x disj_matrices_commute disj_singleton_matrices disj_sparse_row_singleton less_imp_le_nat singleton_le_zero not_le)
next
case equal
with 4 \<section> show ?thesis
apply (simp add: )
by (metis disj_matrices_add disj_matrices_commute disj_sparse_row_singleton order_refl singleton_matrix_le)
next
case greater
with 4 \<section> show ?thesis
apply (simp add: )
by (metis disj_matrices_add_x disj_matrices_add_x_le disj_matrices_commute disj_singleton_matrices disj_sparse_row_singleton le_refl order_less_le singleton_ge_zero)
qed
qed
lemma le_spvec_empty2_sparse_row:
"sorted_spvec b \<Longrightarrow> le_spvec b [] = (sparse_row_vector b \<le> 0)"
by (simp add: le_spvec_iff_sparse_row_le)
lemma le_spvec_empty1_sparse_row:
"(sorted_spvec b) \<Longrightarrow> (le_spvec [] b = (0 \<le> sparse_row_vector b))"
by (simp add: le_spvec_iff_sparse_row_le)
lemma le_spmat_iff_sparse_row_le:
"\<lbrakk>sorted_spvec A; sorted_spmat A; sorted_spvec B; sorted_spmat B\<rbrakk> \<Longrightarrow>
le_spmat A B = (sparse_row_matrix A \<le> sparse_row_matrix B)"
proof (induct A B rule: le_spmat.induct)
case (4 i a as j b bs)
then obtain \<section>: "sorted_spvec as" "sorted_spvec bs"
by (metis sorted_spvec_cons1)
show ?case
proof (cases i j rule: linorder_cases)
case less
with 4 \<section> show ?thesis
apply (simp add: sparse_row_matrix_cons le_spvec_empty2_sparse_row)
by (metis disj_matrices_add_le_x disj_matrices_add_x disj_matrices_commute disj_move_sparse_row_vector_twice disj_move_sparse_vec_mat int_eq_iff less_not_refl move_matrix_le_zero order_le_less)
next
case equal
with 4 \<section> show ?thesis
by (simp add: sparse_row_matrix_cons le_spvec_iff_sparse_row_le disj_matrices_commute disj_move_sparse_vec_mat[OF order_refl] disj_matrices_add)
next
case greater
with 4 \<section> show ?thesis
apply (simp add: sparse_row_matrix_cons le_spvec_empty1_sparse_row)
by (metis disj_matrices_add_x disj_matrices_add_x_le disj_matrices_commute disj_move_sparse_row_vector_twice disj_move_sparse_vec_mat move_matrix_zero_le nat_int nat_less_le of_nat_0_le_iff order_refl)
qed
qed (auto 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)
primrec abs_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat"
where
"abs_spmat [] = []"
| "abs_spmat (a#as) = (fst a, abs_spvec (snd a))#(abs_spmat as)"
primrec minus_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat"
where
"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)"
proof (induct A)
case Nil
then show ?case by auto
next
case (Cons a A)
then show ?case
by (simp add: sparse_row_vector_minus sparse_row_matrix_cons matrix_eqI)
qed
lemma Rep_sparse_row_vector_zero:
assumes "x \<noteq> 0"
shows "Rep_matrix (sparse_row_vector v) x y = 0"
by (metis Suc_leI assms le0 le_eq_less_or_eq nrows_le nrows_spvec)
lemma sparse_row_matrix_abs:
"sorted_spvec A \<Longrightarrow> sorted_spmat A \<Longrightarrow> sparse_row_matrix (abs_spmat A) = \<bar>sparse_row_matrix A\<bar>"
proof (induct A)
case Nil
then show ?case by auto
next
case (Cons ab A)
then have A: "sorted_spvec A"
using sorted_spvec_cons1 by blast
show ?case
proof (cases ab)
case (Pair a b)
show ?thesis
unfolding Pair
proof (intro matrix_eqI)
fix m n
show "Rep_matrix (sparse_row_matrix (abs_spmat ((a,b) # A))) m n
= Rep_matrix \<bar>sparse_row_matrix ((a,b) # A)\<bar> m n"
using Cons Pair A
apply (simp add: sparse_row_vector_abs sparse_row_matrix_cons)
apply (cases "m=a")
using sorted_sparse_row_matrix_zero apply fastforce
by (simp add: Rep_sparse_row_vector_zero)
qed
qed
qed
lemma sorted_spvec_minus_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec (minus_spmat A)"
by (induct A rule: sorted_spvec.induct) (auto simp: sorted_spvec.simps)
lemma sorted_spvec_abs_spmat: "sorted_spvec A \<Longrightarrow> sorted_spvec (abs_spmat A)"
by (induct A rule: sorted_spvec.induct) (auto simp: sorted_spvec.simps)
lemma sorted_spmat_minus_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (minus_spmat A)"
by (induct A) (simp_all add: sorted_spvec_minus_spvec)
lemma sorted_spmat_abs_spmat: "sorted_spmat A \<Longrightarrow> sorted_spmat (abs_spmat A)"
by (induct A) (simp_all add: sorted_spvec_abs_spvec)
definition diff_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat"
where "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)
definition sorted_sparse_matrix :: "'a spmat \<Rightarrow> bool"
where "sorted_sparse_matrix A \<longleftrightarrow> 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::bool) \<and> True) = P" by blast
lemma bool4: "(True \<and> (P::bool)) = P" by blast
lemma bool5: "((P::bool) \<and> False) = False" by blast
lemma bool6: "(False \<and> (P::bool)) = False" by blast
lemma bool7: "((P::bool) \<or> True) = True" by blast
lemma bool8: "(True \<or> (P::bool)) = True" by blast
lemma bool9: "((P::bool) \<or> False) = P" by blast
lemma bool10: "(False \<or> (P::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
primrec pprt_spvec :: "('a::{lattice_ab_group_add}) spvec \<Rightarrow> 'a spvec"
where
"pprt_spvec [] = []"
| "pprt_spvec (a#as) = (fst a, pprt (snd a)) # (pprt_spvec as)"
primrec nprt_spvec :: "('a::{lattice_ab_group_add}) spvec \<Rightarrow> 'a spvec"
where
"nprt_spvec [] = []"
| "nprt_spvec (a#as) = (fst a, nprt (snd a)) # (nprt_spvec as)"
primrec pprt_spmat :: "('a::{lattice_ab_group_add}) spmat \<Rightarrow> 'a spmat"
where
"pprt_spmat [] = []"
| "pprt_spmat (a#as) = (fst a, pprt_spvec (snd a))#(pprt_spmat as)"
primrec nprt_spmat :: "('a::{lattice_ab_group_add}) spmat \<Rightarrow> 'a spmat"
where
"nprt_spmat [] = []"
| "nprt_spmat (a#as) = (fst a, nprt_spvec (snd a))#(nprt_spmat as)"
lemma pprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) \<Longrightarrow> pprt (A+B) = pprt A + pprt B"
apply (simp add: pprt_def sup_matrix_def)
apply (intro matrix_eqI)
by (smt (verit, del_insts) Rep_combine_matrix Rep_zero_matrix_def add.commute comm_monoid_add_class.add_0 disj_matrices_def plus_matrix_def sup.idem)
lemma nprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) \<Longrightarrow> nprt (A+B) = nprt A + nprt B"
unfolding nprt_def inf_matrix_def
apply (intro matrix_eqI)
by (smt (verit, ccfv_threshold) Rep_combine_matrix Rep_matrix_add add.commute add_cancel_right_right add_eq_inf_sup disj_matrices_contr2 sup.idem)
lemma pprt_singleton[simp]:
fixes x:: "_::lattice_ring"
shows "pprt (singleton_matrix j i x) = singleton_matrix j i (pprt x)"
unfolding pprt_def sup_matrix_def
by (simp add: matrix_eqI)
lemma nprt_singleton[simp]:
fixes x:: "_::lattice_ring"
shows "nprt (singleton_matrix j i x) = singleton_matrix j i (nprt x)"
by (metis add_left_imp_eq pprt_singleton prts singleton_matrix_add)
lemma sparse_row_vector_pprt:
fixes v:: "_::lattice_ring spvec"
shows "sorted_spvec v \<Longrightarrow> sparse_row_vector (pprt_spvec v) = pprt (sparse_row_vector v)"
proof (induct v rule: sorted_spvec.induct)
case (3 m x n y bs)
then show ?case
apply simp
apply (subst pprt_add)
apply (metis disj_matrices_commute disj_sparse_row_singleton order.refl fst_conv prod.sel(2) sparse_row_vector_cons)
by (metis pprt_singleton sorted_spvec_cons1)
qed auto
lemma sparse_row_vector_nprt:
fixes v:: "_::lattice_ring spvec"
shows "sorted_spvec v \<Longrightarrow> sparse_row_vector (nprt_spvec v) = nprt (sparse_row_vector v)"
proof (induct v rule: sorted_spvec.induct)
case (3 m x n y bs)
then show ?case
apply simp
apply (subst nprt_add)
apply (metis disj_matrices_commute disj_sparse_row_singleton dual_order.refl fst_conv prod.sel(2) sparse_row_vector_cons)
using sorted_spvec_cons1 by force
qed auto
lemma pprt_move_matrix: "pprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (pprt A) j i"
by (simp add: pprt_def sup_matrix_def matrix_eqI)
lemma nprt_move_matrix: "nprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (nprt A) j i"
by (simp add: nprt_def inf_matrix_def matrix_eqI)
lemma sparse_row_matrix_pprt:
fixes m:: "'a::lattice_ring spmat"
shows "sorted_spvec m \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (pprt_spmat m) = pprt (sparse_row_matrix m)"
proof (induct m rule: sorted_spvec.induct)
case (2 a)
then show ?case
by (simp add: pprt_move_matrix sparse_row_matrix_cons sparse_row_vector_pprt)
next
case (3 m x n y bs)
then show ?case
apply (simp add: sparse_row_matrix_cons sparse_row_vector_pprt)
apply (subst pprt_add)
apply (subst disj_matrices_commute)
apply (metis disj_move_sparse_vec_mat eq_imp_le fst_conv prod.sel(2) sparse_row_matrix_cons)
apply (simp add: sorted_spvec.simps pprt_move_matrix)
done
qed auto
lemma sparse_row_matrix_nprt:
fixes m:: "'a::lattice_ring spmat"
shows "sorted_spvec m \<Longrightarrow> sorted_spmat m \<Longrightarrow> sorted_spmat m \<Longrightarrow> sparse_row_matrix (nprt_spmat m) = nprt (sparse_row_matrix m)"
proof (induct m rule: sorted_spvec.induct)
case (2 a)
then show ?case
by (simp add: nprt_move_matrix sparse_row_matrix_cons sparse_row_vector_nprt)
next
case (3 m x n y bs)
then show ?case
apply (simp add: sparse_row_matrix_cons sparse_row_vector_nprt)
apply (subst nprt_add)
apply (subst disj_matrices_commute)
apply (metis disj_move_sparse_vec_mat fst_conv nle_le prod.sel(2) sparse_row_matrix_cons)
apply (simp add: sorted_spvec.simps nprt_move_matrix)
done
qed auto
lemma sorted_pprt_spvec: "sorted_spvec v \<Longrightarrow> sorted_spvec (pprt_spvec v)"
proof (induct v rule: sorted_spvec.induct)
case 1
then show ?case by auto
next
case (2 a)
then show ?case
by (simp add: sorted_spvec_step1)
next
case (3 m x n y bs)
then show ?case
by (simp add: sorted_spvec_step)
qed
lemma sorted_nprt_spvec: "sorted_spvec v \<Longrightarrow> sorted_spvec (nprt_spvec v)"
by (induct v rule: sorted_spvec.induct) (simp_all add: sorted_spvec.simps split:list.split_asm)
lemma sorted_spvec_pprt_spmat: "sorted_spvec m \<Longrightarrow> sorted_spvec (pprt_spmat m)"
by (induct m rule: sorted_spvec.induct) (simp_all add: sorted_spvec.simps split:list.split_asm)
lemma sorted_spvec_nprt_spmat: "sorted_spvec m \<Longrightarrow> sorted_spvec (nprt_spmat m)"
by (induct m rule: sorted_spvec.induct) (simp_all add: sorted_spvec.simps split:list.split_asm)
lemma sorted_spmat_pprt_spmat: "sorted_spmat m \<Longrightarrow> sorted_spmat (pprt_spmat m)"
by (induct m) (simp_all add: sorted_pprt_spvec)
lemma sorted_spmat_nprt_spmat: "sorted_spmat m \<Longrightarrow> sorted_spmat (nprt_spmat m)"
by (induct m) (simp_all add: sorted_nprt_spvec)
definition mult_est_spmat :: "('a::lattice_ring) spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat \<Rightarrow> 'a spmat" where
"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
lemmas sparse_row_matrix_arith_simps =
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::lattice_ring) spmat)"
"sparse_row_matrix A1 \<le> A"
"A \<le> sparse_row_matrix A2"
"sparse_row_matrix c1 \<le> c"
"c \<le> sparse_row_matrix c2"
"\<bar>x\<bar> \<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