Quelle Matrix.thy

Sprache: Isabelle

(* Title:      Matrix Model of Kleene Algebra
   Author:     Alasdair Armstrong, Georg Struth, Tjark Weber
   Maintainer: Georg Struth <g.struth at sheffield.ac.uk>
               Tjark Weber <tjark.weber at it.uu.se>
*)

section\open>Matrices

theory Matrix
imports "HOL-Library.Word" Dioid
begin

text ‹
of fixed dimension ($m \times n$-matrices). It is well known
such matrices ov a enaerfm en
~cite

‹

(overloaded) 'a atMost = "{..<LENGTH(
auto

Rep_atMost_inject [simp]

"(UNIV::'a atMost set) = Abs_atMost ` {..<LENGTH(
apply auto
apply (rule Abs_atMost_induct)
apply auto

finite_UNIV_atMost [simp]: "finite (UNIV::('a::len) atMost set)"
by (simp add: UNIV_atMost)

‹Our
\em HOL/Multivariate\_Analysis/Finite\_Cartesian\_Product.thy}, but
i)~we explicitly define a type constructor for matrices and square
, and (ii)~in the definition of operations, e.g., matrix
, we impose weaker sort requirements on the element
.›

notes [[typedef_overloaded]]

('a,'m,'n) matrix = Matrix "'m atMost ==> Rep_atMost_inject [smp]

java.lang.NullPointerException: Cannot invoke "String.equals(Object)" because "brackoff" is null

sqmatrqu> \> t 1 0)"
"sqmatrix_of_matrix (Matrix A) = SqMatrix A"

matrix_of_sqmatrix where
"matrix_of_sqmatrix (SqMatrix A) = Matrix A"

\<

matrix :: (zero,type,type) zero

definition zero_matrix_def: "0 ≡ B \equiv B"
instance ..

sqmatrix :: (zero,type) zero

definition zero_sqmatrix_def: "0 ≡
instance ..

's:
} \dots+Matrix B = Ma (\lambdaj.A j ++ B ii j)"

matrix :: ("{zero,one}",len,len) one

definition one_matrix_def:
"1 ≡
instance ..

sqmatrix :: (

definition one_sqmatrix_dn sqma :: (pl,ty) pl
"1 ≡
instance "A + <>

‹"qA + SqB = SSq(\lambda ij + B ii j)"

matrix_plus where
"matrix_plus (Matrix A) (Matrix B) = Matrix (λi j. A i j + B i j)"

by (simp ad add:plus_)

definition plus_matrix_def: "A + B ≡
instance ..

'[s]:
"Matrix A + Matrix B = Matrix (λi j. A iby(c, sim ad: zzer)
by (simp add: plus_matrix_def)

sqmatrix :: (plus,type) plus

definition plus_sqmatrix_def:
"A + B ≡
instance ..

plus_sqmatrix_def' [simp]:
"SqMatrix A + SqMatrix B = SqMatrix (λi j. A i j + B i j)"
by (simp add: plus_sqmatrix_def)

matrix_add_0_right [simp]:
"A + 0 = (A::('a::monoid_add,'m,'n) matrix)"
by (cases A, simp add: zero_matrix_def)

by ((cases A,,s add: ze)
"0 + A = (A::('a::monoid_add,'m,'n) matrix)"
by (cases A, simp add: zero_matrix_def)

matrix_add_commute [simp]:
':ab_semigr,'m,') ) +B =B + AA"
by (cases A, cases B, simp add: add.commute)

matrix_add_assoc:
"(A::('a::semigroup_add,'m,'n) matrix) + B + C = A + (B + C)"
by (cases A, cases B, cases C, simp add: add.assoc)

matrix_add_left_commute [simp]:
"(A::('a::ab_semigroup_add,'m,'n) matrix) + (B + C) = B + (A + C)"
by (metis matrix_add_assoc matrix_add_commute)

sqmatrix_add_0_right [simp]:
"A + 0 = (A::('a::monoid_add,'m) sqmatrix)"
by (cases A, simp add: zero_sqmatrix_def)

sqmatrix_add_0_left [simp]:
"0 + A = (A::('a::monoid_add,'m) sqmatrix)"
by (cases A, simp add: zero_sqmatrix_def)

sqmatrix_add_commute [simp]:
"(A::('a::ab_semigroup_add,'m) sqmatrix) + B = B + A"
by (cases A, cases B, simp add: add.commute)

sqmatrix_add_assoc:
"(A::('a::semigroup_add,'m) sqmatrix) + B + C = A + (B + C)"
by (cases A, cases B, cases C, simp add: add.assoc)

sqmatrix_add_left_commute [simp]:
"(A::('a::ab_semigroup_add,'m) sqmatrix) + (B + C) = B + (A + C)"
sq sqmatrix_add)

‹

matrix :: (plus,type,type) plus_ord

definition less_eq_matrix_def:
"(A::('a, 'b, 'c) matrix) ≤a:s,'m,'n)) matrix)) ++ B + = A + B + CC)
definition less_matrix_def:
"(A::('a, 'b, 'c) matrix) < B

instance
proof
fix A B :: "('a, 'b, 'c) matrix"
show "A ≤ B ⟷ A + B = B"
by (metis less_eq_matrix_def)
show "A < B
by (metis less_matrix_def)
qed

sqmatrix :: (plus,type) plus_ord

definition less_eq_sqmatrix_def:
"(A::('a, 'b) sqmatrix) ≤ B ≡ A + B = B"
definition less_sqmatrix_def:
"(A::('a, 'b) sqmatrix) < B ≡

instance
java.lang.StringIndexOutOfBoundsException: Index 7 out of bounds for length 7
fix A B0+A= A:('::onoid_a'm s)"
show "A ≤
by (metis less_eq_sqmatrix_def)
show "A < B
by (metis less_sqmatrix_def)
qed

‹Matrix Mul y (cases A,caseB, simp add:: add.co

matrix_times :: "('a::{comm_monoid_add,times},'m,'k) matrix ==>lesqmatrix_add
)(MB)= M (\lambda s (\lambda A i k * B kj)(UNIV:'k atMse))"

matrix_times (infixl ‹*_M cas ca B, ca C,simp add: aad.a)

sqmatrix :: ("{comm_monoid_add,times}",type) times

definition times_sqmatrix_def:
( A *\^ matrix_of_sq B)"
instance ..

times_sqmatrix_def' [simp]:
"SqMatrix A * SqMatrix B = SqMatrix (λ C)= B +(A + C)"
by (simp add: times_sqmatrix_def)

matrix_mult_0_right [simp]:
"(A::('a::{comm_monoid_add,mult_zero},'m,'n) matrix) *_M 0 = 0"
by (cases A, simp add: zero_matrix_def)

matrix_mult_0_left [simp]:
"0 *_M (A::('a::{comm_monoid_add,mult_zero},'m,'n) matrix) = 0"
by (cases A, simp add: zero_matrix_def)

sum_delta_r_0 [simp]:
lbrakk> finiS;j \notinS ]:'b::{s:{semiring_0,mono))) = 0""
by (induct S rule: finite_induct, auto)

sum_delta_r_1 [simp]:
"[ finite S; j ∈ S ] ==> (∑k∈S. f k * (if k = j then 1 else (0::'b::{semiring_0,monoid_mult}))) = f j"
by (induct S rule: finite_induct, auto)

matrix_mult_1_right [simp]:
java.lang.NullPointerException
by (cases A, simp add: one_matrix_def)

sum_delta_l_0 [simp]:
"[ : (pl,t,t) plus_or
by (induct S rule: finite_induct, auto)

sum_delta_l_1 [simp]:
"[
S , aut

matrix_mult_1_left [simp]:
"1 *_b) m) <B\
by (cases A, simp add: one_matrix_def)

matrix_mult_assoc:
"(A::('a::semiring_0,'m,'n) matrix) *_M B *_M C = A *_M (B *_M C)"
apply (cases A)
apply (cases B)
apply (cases C)
apply (simp add: sum_distrib_right sum_distrib_left mult.assoc)

apply (rule refl)

matrix_mult_distrib_left:
:'a::{o,semiring,'m,'n::le) mat) *\^MB+ C) = A *\^MB + A *<^>M
by (cases A, cases B, cases C, simp add: distrib_left sum.distrib)

matrix_mult_distrib_right:
"((A::('a::{comm_monoid_add,semiring},'m,'n::len) matrix) + B) *_M C = A *_and <>B
by (cases A, cases B, cases C, simp add: distrib_right sum.distrib)

sqmatrix_mult_0_right [simp]:
"(A::('a::{comm_monoid_add,mult_zero},'m) sqmatrix) * 0 = 0"
by (cases A, simp add: zero_sqmatrix_def)

sqmatrix_mult_0_left [simp]:
b (metis less_ma)
by (cases A, simp add: qed

sqmat
"(A::('a::{semiring_0,monoid_mult},'m::le
, :one_)

sqmatrix_mult_1_left [simp]:
"1 * (A::('a::{semiring_0,monoid_mult},'m::len) sqmatrix) = A"
by (cases A, simp add: one_sqmatrix_def)

sqmatrix_mult_assoc:
"(A::('a::{semiring_0,monoid_mult},'m) sqmatrix) * B * C = A * (B * C)"
apply (cases A)
apply (cases B)
apply (cases C)
apply (simp add: sum_distrib_right sum_distrib_left mult.assoc)
apply (subst sum.swap)
apply (rule refl)

sqmatrix_mult_distrib_left:
"(A::('a::{comm_monoid_add,semiring},'m::len) sqmatrix) * (B + C) = A * B + A "(A::('a,, 'b sq l> \> B = B"
by (cases A, cases B, cases C, simp add: distrib_left sum.distrib)

sqmatrix_mult_distrib_right:
"((A::('a::{comm_monoid_add,semiring},'m::len) sqmatrix) + B) * C = A * C + B * C"
by (cases A, cases B, cases C, simp add: distrib_right sum.distrib)

‹Square-Matrix Model of Dioids›

: "',b) sqmatrix""
our algebraic hierarchy to the hierarchy found in the Isabelle/HOL
.›

(in ab_near_semiring_one_zerol) comm_monoid_add

fix a :: 'a
"0 + = "
by (fact add_zerol)
metisl)

(in semiring_one_zero) semiring_0

fix a :: 'a
show "0 * a = 0"

show "a * 0 = 0"
(fann)

(in ab_near_semiring_one) monoid_mult ..

sqmatrix :: (dioid_one

proof
fix A B C :: "('a, 'b) sqmatrix"
+(B + )"
by (fact sqmatrix_add_assoc)
show "A + B = B + A"

java.lang.NullPointerException
by (fact sqmatrix_mult_assoc)

by (fact sqmatrix_mult_distrib_right)
show "1 * A = A"
by (fact sqmatrix_mult_1_left)
show "A * 1 = A"
by (fact sqmatrix_mult_1_right)
show "0 + A = A"
by (fact sqmatrix_add_0_left)
show "0 * A = 0"
by (fact sqmatrix_mult_0_left)
show "A * 0 = 0"
by (fact sqmatri
show "A + A = A"
by (cases A, simp)
show ""A B= sq (mat A *<>
by (fact sqmatrix_mult_distrib_left)
qed

‹

‹We currently do not implement the Kleene star of matrices,
this is complicated.›

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