# Theory SparseMatrix

theory SparseMatrix
imports Matrix
```(*  Title:      HOL/Matrix_LP/SparseMatrix.thy
Author:     Steven Obua
*)

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 ⇒ '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 ⇒ '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"

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

lemmas [code] = sparse_row_vector_empty [symmetric]

lemma foldl_distrstart: "! a x y. (f (g x y) a = g x (f y a)) ⟹ (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)"
apply (induct arr)
apply (simp add: foldl_distrstart [of "λm x. m + singleton_matrix 0 (fst x) (snd x)" "λx m. singleton_matrix 0 (fst x) (snd x) + m"])
done

lemma sparse_row_vector_append[simp]:
"sparse_row_vector (a @ b) = (sparse_row_vector a) + (sparse_row_vector b)"
by (induct a) auto

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

lemma sparse_row_matrix_cons: "sparse_row_matrix (a#arr) = ((move_matrix (sparse_row_vector (snd a)) (int (fst a)) 0)) + sparse_row_matrix arr"
apply (induct arr)
apply (simp add: foldl_distrstart[of "λm x. m + (move_matrix (sparse_row_vector (snd x)) (int (fst x)) 0)"
"% a m. (move_matrix (sparse_row_vector (snd a)) (int (fst a)) 0) + m"])
done

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

primrec sorted_spvec :: "'a spvec ⇒ bool"
where
"sorted_spvec [] = True"
| sorted_spvec_step: "sorted_spvec (a#as) = (case as of [] ⇒ True | b#bs ⇒ ((fst a < fst b) & (sorted_spvec as)))"

primrec sorted_spmat :: "'a spmat ⇒ bool"
where
"sorted_spmat [] = True"
| "sorted_spmat (a#as) = ((sorted_spvec (snd a)) & (sorted_spmat as))"

declare sorted_spvec.simps [simp del]

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

lemma sorted_spvec_cons1: "sorted_spvec (a#as) ⟹ sorted_spvec as"
apply (induct as)
done

lemma sorted_spvec_cons2: "sorted_spvec (a#b#t) ⟹ sorted_spvec (a#t)"
apply (induct t)
done

lemma sorted_spvec_cons3: "sorted_spvec(a#b#t) ⟹ fst a < fst b"
done

lemma sorted_sparse_row_vector_zero[rule_format]: "m <= n ⟹ sorted_spvec ((n,a)#arr) ⟶ Rep_matrix (sparse_row_vector arr) j m = 0"
apply (induct arr)
apply (auto)
apply (frule sorted_spvec_cons2,simp)+
apply (frule sorted_spvec_cons3, simp)
done

lemma sorted_sparse_row_matrix_zero[rule_format]: "m <= n ⟹ sorted_spvec ((n,a)#arr) ⟶ Rep_matrix (sparse_row_matrix arr) m j = 0"
apply (induct arr)
apply (auto)
apply (frule sorted_spvec_cons2, simp)
apply (frule sorted_spvec_cons3, simp)
done

primrec minus_spvec :: "('a::ab_group_add) spvec ⇒ '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 ⇒ 'a spvec"
where
"abs_spvec [] = []"
| "abs_spvec (a#as) = (fst a, ¦snd a¦)#(abs_spvec as)"

lemma sparse_row_vector_minus:
"sparse_row_vector (minus_spvec v) = - (sparse_row_vector v)"
apply (induct v)
apply (rule ext)+
apply simp
done

apply standard
unfolding abs_matrix_def
apply rule
done
(*FIXME move*)

lemma sparse_row_vector_abs:
"sorted_spvec (v :: 'a::lattice_ring spvec) ⟹ sparse_row_vector (abs_spvec v) = ¦sparse_row_vector v¦"
apply (induct v)
apply simp_all
apply (frule_tac sorted_spvec_cons1, simp)
apply (simp only: Rep_matrix_inject[symmetric])
apply (rule ext)+
apply auto
apply (subgoal_tac "Rep_matrix (sparse_row_vector v) 0 a = 0")
apply (simp)
apply (rule sorted_sparse_row_vector_zero)
apply auto
done

lemma sorted_spvec_minus_spvec:
"sorted_spvec v ⟹ sorted_spvec (minus_spvec v)"
apply (induct v)
apply (simp)
apply (frule sorted_spvec_cons1, simp)
done

lemma sorted_spvec_abs_spvec:
"sorted_spvec v ⟹ sorted_spvec (abs_spvec v)"
apply (induct v)
apply (simp)
apply (frule sorted_spvec_cons1, simp)
done

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

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

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

fun addmult_spvec :: "('a::ring) ⇒ 'a spvec ⇒ 'a spvec ⇒ '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))" *)

by (induct a) auto

by (induct a) auto

lemma sparse_row_vector_map: "(! x y. f (x+y) = (f x) + (f y)) ⟹ (f::'a⇒('a::lattice_ring)) 0 = 0 ⟹
sparse_row_vector (map (% x. (fst x, f (snd x))) a) = apply_matrix f (sparse_row_vector a)"
apply (induct a)
done

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

(sparse_row_vector a) + (scalar_mult y (sparse_row_vector b))"
apply (induct y a b rule: addmult_spvec.induct)
done

lemma sorted_smult_spvec: "sorted_spvec a ⟹ sorted_spvec (smult_spvec y a)"
apply (induct a)
apply (auto simp add: sorted_spvec.simps split:list.split_asm)
done

lemma sorted_spvec_addmult_spvec_helper: "⟦sorted_spvec (addmult_spvec y ((a, b) # arr) brr); aa < a; sorted_spvec ((a, b) # arr);
sorted_spvec ((aa, ba) # brr)⟧ ⟹ sorted_spvec ((aa, y * ba) # addmult_spvec y ((a, b) # arr) brr)"
apply (induct brr)
done

"⟦sorted_spvec (addmult_spvec y arr ((aa, ba) # brr)); a < aa; sorted_spvec ((a, b) # arr); sorted_spvec ((aa, ba) # brr)⟧
⟹ sorted_spvec ((a, b) # addmult_spvec y arr ((aa, ba) # brr))"
apply (induct arr)
apply (auto simp add: smult_spvec_def sorted_spvec.simps)
done

"sorted_spvec (addmult_spvec y arr brr) ⟶ sorted_spvec ((aa, b) # arr) ⟶ sorted_spvec ((aa, ba) # brr)
⟶ sorted_spvec ((aa, b + y * ba) # (addmult_spvec y arr brr))"
apply (induct y arr brr rule: addmult_spvec.induct)
apply (simp_all add: sorted_spvec.simps smult_spvec_def split:list.split)
done

lemma sorted_addmult_spvec: "sorted_spvec a ⟹ sorted_spvec b ⟹ sorted_spvec (addmult_spvec y a b)"
apply (induct y a b rule: addmult_spvec.induct)
apply (rule conjI, intro strip)
apply (case_tac "~(i < j)")
apply (simp_all)
apply (frule_tac as=brr in sorted_spvec_cons1)
apply (intro strip | rule conjI)+
apply (frule_tac as=arr in sorted_spvec_cons1)
apply (intro strip)
apply (frule_tac as=arr in sorted_spvec_cons1)
apply (frule_tac as=brr in sorted_spvec_cons1)
apply (simp)
done

fun mult_spvec_spmat :: "('a::lattice_ring) spvec ⇒ 'a spvec ⇒ 'a spmat  ⇒ '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[rule_format]: "sorted_spvec (a::('a::lattice_ring) spvec) ⟶ sorted_spvec B ⟶
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 ⟹ Suc 0 <= nat ((int b)-(int a))" by arith
have not_iff: "!! a b. a = b ⟹ (~ a) = (~ b)" by simp
have max_helper: "!! a b. ~ (a <= max (Suc a) b) ⟹ False"
by arith
{
fix a
fix v
assume a:"a < nrows(sparse_row_vector v)"
have b:"nrows(sparse_row_vector v) <= 1" by simp
note dummy = less_le_trans[of a "nrows (sparse_row_vector v)" 1, OF a b]
then have "a = 0" by simp
}
note nrows_helper = this
show ?thesis
apply (induct c a B rule: mult_spvec_spmat.induct)
apply simp+
apply (rule conjI)
apply (intro strip)
apply (frule_tac as=brr in sorted_spvec_cons1)
apply (simplesubst Rep_matrix_zero_imp_mult_zero)
apply (simp)
apply (rule disjI2)
apply (intro strip)
apply (subst nrows)
apply (rule  order_trans[of _ 1])
apply (subst Rep_matrix_zero_imp_mult_zero)
apply (intro strip)
apply (case_tac "k <= j")
apply (rule_tac m1 = k and n1 = i and a1 = a in ssubst[OF sorted_sparse_row_vector_zero])
apply (simp_all)
apply (rule disjI2)
apply (rule nrows)
apply (rule order_trans[of _ 1])

apply (intro strip | rule conjI)+
apply (frule_tac as=arr in sorted_spvec_cons1)
apply (subst Rep_matrix_zero_imp_mult_zero)
apply (simp)
apply (rule disjI2)
apply (intro strip)
apply (case_tac "i <= j")
apply (erule sorted_sparse_row_matrix_zero)
apply (simp_all)
apply (intro strip)
apply (case_tac "i=j")
apply (simp_all)
apply (frule_tac as=arr in sorted_spvec_cons1)
apply (frule_tac as=brr in sorted_spvec_cons1)
apply (rule_tac B1 = "sparse_row_matrix brr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])
apply (auto)
apply (rule sorted_sparse_row_matrix_zero)
apply (simp_all)
apply (rule_tac A1 = "sparse_row_vector arr" in ssubst[OF Rep_matrix_zero_imp_mult_zero])
apply (auto)
apply (rule_tac m=k and n = j and a = a and arr=arr in sorted_sparse_row_vector_zero)
apply (simp_all)
apply (drule nrows_notzero)
apply (drule nrows_helper)
apply (arith)

apply (subst Rep_matrix_inject[symmetric])
apply (rule ext)+
apply (simp)
apply (subst Rep_matrix_mult)
apply (rule_tac j1=j in ssubst[OF foldseq_almostzero])
apply (simp_all)
apply (intro strip, rule conjI)
apply (intro strip)
apply (drule_tac max_helper)
apply (simp)
apply (auto)
apply (rule zero_imp_mult_zero)
apply (rule disjI2)
apply (rule nrows)
apply (rule order_trans[of _ 1])
apply (simp)
apply (simp)
done
qed

lemma sorted_mult_spvec_spmat[rule_format]:
"sorted_spvec (c::('a::lattice_ring) spvec) ⟶ sorted_spmat B ⟶ sorted_spvec (mult_spvec_spmat c a B)"
apply (induct c a B rule: mult_spvec_spmat.induct)
done

primrec mult_spmat :: "('a::lattice_ring) spmat ⇒ 'a spmat ⇒ '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 ⟹ sorted_spvec B ⟹
sparse_row_matrix (mult_spmat A B) = (sparse_row_matrix A) * (sparse_row_matrix B)"
apply (induct A)
apply (auto simp add: sparse_row_matrix_cons sparse_row_mult_spvec_spmat algebra_simps move_matrix_mult)
done

lemma sorted_spvec_mult_spmat[rule_format]:
"sorted_spvec (A::('a::lattice_ring) spmat) ⟶ sorted_spvec (mult_spmat A B)"
apply (induct A)
apply (auto)
apply (drule sorted_spvec_cons1, simp)
apply (case_tac A)
done

lemma sorted_spmat_mult_spmat:
"sorted_spmat (B::('a::lattice_ring) spmat) ⟹ sorted_spmat (mult_spmat A B)"
apply (induct A)
done

where
(* "measure (% (a, b). length a + (length b))" *)
| "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)"

by (cases a, auto)

lemma sparse_row_vector_add: "sparse_row_vector (add_spvec a b) = (sparse_row_vector a) + (sparse_row_vector b)"
apply (induct a b rule: add_spvec.induct)
done

where
(* "measure (% (A,B). (length A)+(length B))" *)
| "add_spmat as [] = as"
| "add_spmat ((i,a)#as) ((j,b)#bs) = (
if i < j then
else if j < i then
else

by(cases as) auto

lemma sparse_row_add_spmat: "sparse_row_matrix (add_spmat A B) = (sparse_row_matrix A) + (sparse_row_matrix B)"
apply (induct A B rule: add_spmat.induct)
done

lemma sorted_add_spvec_helper1[rule_format]: "add_spvec ((a,b)#arr) brr = (ab, bb) # list ⟶ (ab = a | (brr ≠ [] & ab = fst (hd brr)))"
proof -
have "(! x ab a. x = (a,b)#arr ⟶ add_spvec x brr = (ab, bb) # list ⟶ (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 ⟶ (ab = a | (brr ≠ [] & ab = fst (hd brr)))"
proof -
have "(! x ab a. x = (a,b)#arr ⟶ add_spmat x brr = (ab, bb) # list ⟶ (ab = a | (ab = fst (hd brr))))"
then show ?thesis
by (case_tac brr, auto)
qed

lemma sorted_add_spvec_helper: "add_spvec arr brr = (ab, bb) # list ⟹ ((arr ≠ [] & ab = fst (hd arr)) | (brr ≠ [] & ab = fst (hd brr)))"
apply (induct arr brr rule: add_spvec.induct)
apply (auto split:if_splits)
done

lemma sorted_add_spmat_helper: "add_spmat arr brr = (ab, bb) # list ⟹ ((arr ≠ [] & ab = fst (hd arr)) | (brr ≠ [] & ab = fst (hd brr)))"
apply (induct arr brr rule: add_spmat.induct)
apply (auto split:if_splits)
done

by (induct a b rule: add_spvec.induct) auto

apply (induct a b rule: add_spmat.induct)
done

lemma sorted_add_spvec_helper2: "add_spvec ((a,b)#arr) brr = (ab, bb) # list ⟹ aa < a ⟹ sorted_spvec ((aa, ba) # brr) ⟹ aa < ab"
apply (auto)
apply (case_tac brr)
apply (simp_all)
apply (drule_tac sorted_spvec_cons3)
apply (simp)
done

lemma sorted_add_spmat_helper2: "add_spmat ((a,b)#arr) brr = (ab, bb) # list ⟹ aa < a ⟹ sorted_spvec ((aa, ba) # brr) ⟹ aa < ab"
apply (auto)
apply (case_tac brr)
apply (simp_all)
apply (drule_tac sorted_spvec_cons3)
apply (simp)
done

lemma sorted_spvec_add_spvec[rule_format]: "sorted_spvec a ⟶ sorted_spvec b ⟶ sorted_spvec (add_spvec a b)"
apply (induct a b rule: add_spvec.induct)
apply (simp_all)
apply (rule conjI)
apply (clarsimp)
apply (frule_tac as=brr in sorted_spvec_cons1)
apply (simp)
apply (subst sorted_spvec_step)
apply (clarsimp simp: sorted_add_spvec_helper2 split: list.split)
apply (clarify)
apply (rule conjI)
apply (clarify)
apply (frule_tac as=arr in sorted_spvec_cons1, simp)
apply (subst sorted_spvec_step)
apply (clarify)
apply (frule_tac as=arr in sorted_spvec_cons1)
apply (frule_tac as=brr in sorted_spvec_cons1)
apply (simp)
apply (subst sorted_spvec_step)
apply (simp split: list.split)
apply (clarsimp)
apply (auto simp: neq_Nil_conv)
apply (drule sorted_spvec_cons3)
apply (simp)
apply (drule sorted_spvec_cons3)
apply (simp)
done

lemma sorted_spvec_add_spmat[rule_format]: "sorted_spvec A ⟶ sorted_spvec B ⟶ sorted_spvec (add_spmat A B)"
apply (induct A B rule: add_spmat.induct)
apply (simp_all)
apply (rule conjI)
apply (intro strip)
apply (simp)
apply (frule_tac as=bs in sorted_spvec_cons1)
apply (simp)
apply (subst sorted_spvec_step)
apply (simp split: list.split)
apply (clarify, simp)
apply (clarify)
apply (rule conjI)
apply (clarify)
apply (frule_tac as=as in sorted_spvec_cons1, simp)
apply (subst sorted_spvec_step)
apply (clarsimp)
apply (frule_tac as=as in sorted_spvec_cons1)
apply (frule_tac as=bs in sorted_spvec_cons1)
apply (simp)
apply (subst sorted_spvec_step)
apply (simp split: list.split)
apply (clarify, simp)
apply (auto simp:neq_Nil_conv)
apply (drule sorted_spvec_cons3)
apply (simp)
apply (drule sorted_spvec_cons3)
apply (simp)
done

lemma sorted_spmat_add_spmat[rule_format]: "sorted_spmat A ⟹ sorted_spmat B ⟹ sorted_spmat (add_spmat A B)"
apply (induct A B rule: add_spmat.induct)
done

fun le_spvec :: "('a::lattice_ab_group_add) spvec ⇒ 'a spvec ⇒ bool"
where
(* "measure (% (a,b). (length a) + (length b))" *)
"le_spvec [] [] = True"
| "le_spvec ((_,a)#as) [] = (a <= 0 & le_spvec as [])"
| "le_spvec [] ((_,b)#bs) = (0 <= b & le_spvec [] bs)"
| "le_spvec ((i,a)#as) ((j,b)#bs) = (
if (i < j) then a <= 0 & le_spvec as ((j,b)#bs)
else if (j < i) then 0 <= b & le_spvec ((i,a)#as) bs
else a <= b & le_spvec as bs)"

fun le_spmat :: "('a::lattice_ab_group_add) spmat ⇒ 'a spmat ⇒ 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 ⇒ 'a matrix ⇒ bool" where
"disj_matrices A B ⟷
(! j i. (Rep_matrix A j i ≠ 0) ⟶ (Rep_matrix B j i = 0)) & (! j i. (Rep_matrix B j i ≠ 0) ⟶ (Rep_matrix A j i = 0))"

declare [[simp_depth_limit = 6]]

lemma disj_matrices_contr1: "disj_matrices A B ⟹ Rep_matrix A j i ≠ 0 ⟹ Rep_matrix B j i = 0"

lemma disj_matrices_contr2: "disj_matrices A B ⟹ Rep_matrix B j i ≠ 0 ⟹ Rep_matrix A j i = 0"

lemma disj_matrices_add: "disj_matrices A B ⟹ disj_matrices C D ⟹ disj_matrices A D ⟹ disj_matrices B C ⟹
(A + B <= C + D) = (A <= C & B <= (D::('a::lattice_ab_group_add) matrix))"
apply (auto)
apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)
apply (intro strip)
apply (erule conjE)+
apply (drule_tac j=j and i=i in spec2)+
apply (case_tac "Rep_matrix B j i = 0")
apply (case_tac "Rep_matrix D j i = 0")
apply (simp_all)
apply (simp (no_asm_use) only: le_matrix_def disj_matrices_def)
apply (intro strip)
apply (erule conjE)+
apply (drule_tac j=j and i=i in spec2)+
apply (case_tac "Rep_matrix A j i = 0")
apply (case_tac "Rep_matrix C j i = 0")
apply (simp_all)
apply (assumption)
done

lemma disj_matrices_zero1[simp]: "disj_matrices 0 B"

lemma disj_matrices_zero2[simp]: "disj_matrices A 0"

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

lemma disj_matrices_add_le_zero: "disj_matrices A B ⟹
(A + B <= 0) = (A <= 0 & (B::('a::lattice_ab_group_add) matrix) <= 0)"
by (rule disj_matrices_add[of A B 0 0, simplified])

lemma disj_matrices_add_zero_le: "disj_matrices A B ⟹
(0 <= A + B) = (0 <= A & 0 <= (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 ⟹ disj_matrices B C ⟹
(A <= B + C) = (A <= C & 0 <= (B::('a::lattice_ab_group_add) matrix))"

lemma disj_matrices_add_le_x: "disj_matrices A B ⟹ disj_matrices B C ⟹
(B + A <= C) = (A <= C &  (B::('a::lattice_ab_group_add) matrix) <= 0)"

lemma disj_sparse_row_singleton: "i <= j ⟹ sorted_spvec((j,y)#v) ⟹ disj_matrices (sparse_row_vector v) (singleton_matrix 0 i x)"
apply (rule conjI)
apply (rule neg_imp)
apply (simp)
apply (intro strip)
apply (rule sorted_sparse_row_vector_zero)
apply (simp_all)
apply (intro strip)
apply (rule sorted_sparse_row_vector_zero)
apply (simp_all)
done

lemma disj_matrices_x_add: "disj_matrices A B ⟹ disj_matrices A C ⟹ disj_matrices (A::('a::lattice_ab_group_add) matrix) (B+C)"
apply (auto)
apply (drule_tac j=j and i=i in spec2)+
apply (case_tac "Rep_matrix B j i = 0")
apply (case_tac "Rep_matrix C j i = 0")
apply (simp_all)
done

lemma disj_matrices_add_x: "disj_matrices A B ⟹ disj_matrices A C ⟹ disj_matrices (B+C) (A::('a::lattice_ab_group_add) matrix)"

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

lemma disj_move_sparse_vec_mat[simplified disj_matrices_commute]:
"j <= a ⟹ sorted_spvec((a,c)#as) ⟹ disj_matrices (move_matrix (sparse_row_vector b) (int j) i) (sparse_row_matrix as)"
apply (drule nrows_notzero)
apply (drule less_le_trans[OF _ nrows_spvec])
apply (subgoal_tac "ja = j")
apply (arith)
apply (rule nrows)
apply (rule order_trans[of _ 1 _])
apply (simp)
apply (case_tac "nat (int ja - int j) = 0")
apply (case_tac "ja = j")
apply arith+
done

lemma disj_move_sparse_row_vector_twice:
"j ≠ u ⟹ disj_matrices (move_matrix (sparse_row_vector a) j i) (move_matrix (sparse_row_vector b) u v)"
apply (rule nrows, rule order_trans[of _ 1], simp, drule nrows_notzero, drule less_le_trans[OF _ nrows_spvec], arith)+
done

lemma le_spvec_iff_sparse_row_le[rule_format]: "(sorted_spvec a) ⟶ (sorted_spvec b) ⟶ (le_spvec a b) = (sparse_row_vector a <= sparse_row_vector b)"
apply (induct a b rule: le_spvec.induct)
disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
apply (rule conjI, intro strip)
apply (simp add: disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
apply (simp, blast)
apply (intro strip, rule conjI, intro strip)
apply (blast)
apply (intro strip)
apply (case_tac "a=b", simp_all)
apply (simp_all add: disj_sparse_row_singleton[OF order_refl] disj_matrices_commute)
done

lemma le_spvec_empty2_sparse_row[rule_format]: "sorted_spvec b ⟶ le_spvec b [] = (sparse_row_vector b <= 0)"
apply (induct b)
apply (intro strip)
apply (auto simp add: disj_matrices_commute disj_sparse_row_singleton[OF order_refl] sorted_spvec_cons1)
done

lemma le_spvec_empty1_sparse_row[rule_format]: "(sorted_spvec b) ⟶ (le_spvec [] b = (0 <= sparse_row_vector b))"
apply (induct b)
apply (intro strip)
apply (auto simp add: disj_matrices_commute disj_sparse_row_singleton[OF order_refl] sorted_spvec_cons1)
done

lemma le_spmat_iff_sparse_row_le[rule_format]: "(sorted_spvec A) ⟶ (sorted_spmat A) ⟶ (sorted_spvec B) ⟶ (sorted_spmat B) ⟶
le_spmat A B = (sparse_row_matrix A <= sparse_row_matrix B)"
apply (induct A B rule: le_spmat.induct)
disj_matrices_commute sorted_spvec_cons1 le_spvec_empty2_sparse_row le_spvec_empty1_sparse_row)+
apply (rule conjI, intro strip)
apply (simp add: disj_move_sparse_vec_mat[OF less_imp_le] disj_matrices_commute)
apply (simp add: disj_move_sparse_vec_mat[OF order_refl] disj_matrices_commute)
apply (simp, blast)
apply (intro strip, rule conjI, intro strip)
apply (simp add: disj_move_sparse_vec_mat[OF less_imp_le] disj_matrices_commute)
apply (simp, blast)
apply (intro strip)
apply (case_tac "i=j")
apply (simp_all)
apply (simp_all add: disj_matrices_commute disj_move_sparse_vec_mat[OF order_refl])
done

declare [[simp_depth_limit = 999]]

primrec abs_spmat :: "('a::lattice_ring) spmat ⇒ '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 ⇒ '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)"
apply (induct A)
apply (subst Rep_matrix_inject[symmetric])
apply (rule ext)+
apply simp
done

lemma Rep_sparse_row_vector_zero: "x ≠ 0 ⟹ Rep_matrix (sparse_row_vector v) x y = 0"
proof -
assume x:"x ≠ 0"
have r:"nrows (sparse_row_vector v) <= Suc 0" by (rule nrows_spvec)
show ?thesis
apply (rule nrows)
apply (subgoal_tac "Suc 0 <= x")
apply (insert r)
apply (simp only:)
apply (insert x)
apply arith
done
qed

lemma sparse_row_matrix_abs:
"sorted_spvec A ⟹ sorted_spmat A ⟹ sparse_row_matrix (abs_spmat A) = ¦sparse_row_matrix A¦"
apply (induct A)
apply (frule_tac sorted_spvec_cons1, simp)
apply (simplesubst Rep_matrix_inject[symmetric])
apply (rule ext)+
apply auto
apply (case_tac "x=a")
apply (simp)
apply (simplesubst sorted_sparse_row_matrix_zero)
apply auto
apply (simplesubst Rep_sparse_row_vector_zero)
apply simp_all
done

lemma sorted_spvec_minus_spmat: "sorted_spvec A ⟹ sorted_spvec (minus_spmat A)"
apply (induct A)
apply (simp)
apply (frule sorted_spvec_cons1, simp)
done

lemma sorted_spvec_abs_spmat: "sorted_spvec A ⟹ sorted_spvec (abs_spmat A)"
apply (induct A)
apply (simp)
apply (frule sorted_spvec_cons1, simp)
done

lemma sorted_spmat_minus_spmat: "sorted_spmat A ⟹ sorted_spmat (minus_spmat A)"
apply (induct A)
done

lemma sorted_spmat_abs_spmat: "sorted_spmat A ⟹ sorted_spmat (abs_spmat A)"
apply (induct A)
done

definition diff_spmat :: "('a::lattice_ring) spmat ⇒ 'a spmat ⇒ 'a spmat"
where "diff_spmat A B = add_spmat A (minus_spmat B)"

lemma sorted_spmat_diff_spmat: "sorted_spmat A ⟹ sorted_spmat B ⟹ sorted_spmat (diff_spmat A B)"

lemma sorted_spvec_diff_spmat: "sorted_spvec A ⟹ sorted_spvec B ⟹ sorted_spvec (diff_spmat A B)"

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

definition sorted_sparse_matrix :: "'a spmat ⇒ bool"
where "sorted_sparse_matrix A ⟷ sorted_spvec A & sorted_spmat A"

lemma sorted_sparse_matrix_imp_spvec: "sorted_sparse_matrix A ⟹ sorted_spvec A"

lemma sorted_sparse_matrix_imp_spmat: "sorted_sparse_matrix A ⟹ sorted_spmat A"

lemmas sorted_sp_simps =
sorted_spvec.simps
sorted_spmat.simps
sorted_sparse_matrix_def

lemma bool1: "(¬ True) = False"  by blast
lemma bool2: "(¬ False) = True"  by blast
lemma bool3: "((P::bool) ∧ True) = P" by blast
lemma bool4: "(True ∧ (P::bool)) = P" by blast
lemma bool5: "((P::bool) ∧ False) = False" by blast
lemma bool6: "(False ∧ (P::bool)) = False" by blast
lemma bool7: "((P::bool) ∨ True) = True" by blast
lemma bool8: "(True ∨ (P::bool)) = True" by blast
lemma bool9: "((P::bool) ∨ False) = P" by blast
lemma bool10: "(False ∨ (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 ⇒ '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 ⇒ '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 ⇒ '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 ⇒ '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) ⟹ pprt (A+B) = pprt A + pprt B"
apply (rule ext)+
apply simp
apply (case_tac "Rep_matrix A x xa ≠ 0")
done

lemma nprt_add: "disj_matrices A (B::(_::lattice_ring) matrix) ⟹ nprt (A+B) = nprt A + nprt B"
apply (rule ext)+
apply simp
apply (case_tac "Rep_matrix A x xa ≠ 0")
done

lemma pprt_singleton[simp]: "pprt (singleton_matrix j i (x::_::lattice_ring)) = singleton_matrix j i (pprt x)"
apply (rule ext)+
apply simp
done

lemma nprt_singleton[simp]: "nprt (singleton_matrix j i (x::_::lattice_ring)) = singleton_matrix j i (nprt x)"
apply (rule ext)+
apply simp
done

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

lemma sparse_row_vector_pprt: "sorted_spvec (v :: 'a::lattice_ring spvec) ⟹ sparse_row_vector (pprt_spvec v) = pprt (sparse_row_vector v)"
apply (induct v)
apply (simp_all)
apply (frule sorted_spvec_cons1, auto)
apply (subst disj_matrices_commute)
apply (rule disj_sparse_row_singleton)
apply auto
done

lemma sparse_row_vector_nprt: "sorted_spvec (v :: 'a::lattice_ring spvec) ⟹ sparse_row_vector (nprt_spvec v) = nprt (sparse_row_vector v)"
apply (induct v)
apply (simp_all)
apply (frule sorted_spvec_cons1, auto)
apply (subst disj_matrices_commute)
apply (rule disj_sparse_row_singleton)
apply auto
done

lemma pprt_move_matrix: "pprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (pprt A) j i"
apply (rule ext)+
apply (simp)
done

lemma nprt_move_matrix: "nprt (move_matrix (A::('a::lattice_ring) matrix) j i) = move_matrix (nprt A) j i"
apply (rule ext)+
apply (simp)
done

lemma sparse_row_matrix_pprt: "sorted_spvec (m :: 'a::lattice_ring spmat) ⟹ sorted_spmat m ⟹ sparse_row_matrix (pprt_spmat m) = pprt (sparse_row_matrix m)"
apply (induct m)
apply simp
apply simp
apply (frule sorted_spvec_cons1)
apply (subst disj_matrices_commute)
apply (rule disj_move_sparse_vec_mat)
apply auto
apply (simp split: list.split)
apply auto
done

lemma sparse_row_matrix_nprt: "sorted_spvec (m :: 'a::lattice_ring spmat) ⟹ sorted_spmat m ⟹ sparse_row_matrix (nprt_spmat m) = nprt (sparse_row_matrix m)"
apply (induct m)
apply simp
apply simp
apply (frule sorted_spvec_cons1)
apply (subst disj_matrices_commute)
apply (rule disj_move_sparse_vec_mat)
apply auto
apply (simp split: list.split)
apply auto
done

lemma sorted_pprt_spvec: "sorted_spvec v ⟹ sorted_spvec (pprt_spvec v)"
apply (induct v)
apply (simp)
apply (frule sorted_spvec_cons1)
apply simp
done

lemma sorted_nprt_spvec: "sorted_spvec v ⟹ sorted_spvec (nprt_spvec v)"
apply (induct v)
apply (simp)
apply (frule sorted_spvec_cons1)
apply simp
done

lemma sorted_spvec_pprt_spmat: "sorted_spvec m ⟹ sorted_spvec (pprt_spmat m)"
apply (induct m)
apply (simp)
apply (frule sorted_spvec_cons1)
apply simp
done

lemma sorted_spvec_nprt_spmat: "sorted_spvec m ⟹ sorted_spvec (nprt_spmat m)"
apply (induct m)
apply (simp)
apply (frule sorted_spvec_cons1)
apply simp
done

lemma sorted_spmat_pprt_spmat: "sorted_spmat m ⟹ sorted_spmat (pprt_spmat m)"
apply (induct m)
done

lemma sorted_spmat_nprt_spmat: "sorted_spmat m ⟹ sorted_spmat (nprt_spmat m)"
apply (induct m)
done

definition mult_est_spmat :: "('a::lattice_ring) spmat ⇒ 'a spmat ⇒ 'a spmat ⇒ 'a spmat ⇒ '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_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
smult_spvec_empty smult_spvec_cons
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 ≤ sparse_row_matrix (b::('a::lattice_ring) spmat)"
"sparse_row_matrix A1 <= A"
"A <= sparse_row_matrix A2"
"sparse_row_matrix c1 <= c"
"c <= sparse_row_matrix c2"
"¦x¦ ≤ sparse_row_matrix r"
shows
"c * x ≤ 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
```