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theory MatrixGen
use int.Int
type mat 'a
function rows (mat 'a) : int
function cols (mat 'a) : int
axiom rows_and_cols_nonnegative:
forall m: mat 'a. 0 <= rows m /\ 0 <= cols m
function get (mat 'a) int int : 'a
function set (mat 'a) int int 'a : mat 'a
axiom set_def1:
forall m: mat 'a, i j: int, v: 'a. 0 <= i < rows m -> 0 <= j < cols m ->
rows (set m i j v) = rows m
axiom set_def2:
forall m: mat 'a, i j: int, v: 'a. 0 <= i < rows m -> 0 <= j < cols m ->
cols (set m i j v) = cols m
axiom set_def3:
forall m: mat 'a, i j: int, v: 'a. 0 <= i < rows m -> 0 <= j < cols m ->
get (set m i j v) i j = v
axiom set_def4:
forall m: mat 'a, i j: int, v: 'a. 0 <= i < rows m -> 0 <= j < cols m ->
forall i' j': int. 0 <= i' < rows m /\ 0 <= j' < cols m /\
(i <> i' \/ j <> j') ->
get (set m i j v) i' j' = get m i' j'
predicate (==) (m1 m2: mat 'a) =
rows m1 = rows m2 && cols m1 = cols m2 &&
forall i j: int. 0 <= i < rows m1 -> 0 <= j < cols m1 ->
get m1 i j = get m2 i j
axiom extensionality:
forall m1 m2: mat 'a. m1 == m2 -> m1 = m2
predicate (===) (a b: mat 'a) =
rows a = rows b /\ cols a = cols b
end
theory Matrix
use int.Int
type mat 'a
function rows (mat 'a) : int
function cols (mat 'a) : int
function get (mat 'a) int int : 'a
function create (r c: int) (f: int -> int -> 'a) : mat 'a
axiom create_rows:
forall r c: int, f: int -> int -> 'a.
0 <= r -> rows (create r c f) = r
axiom create_cols:
forall r c: int, f: int -> int -> 'a.
0 <= c -> cols (create r c f) = c
axiom create_get:
forall r c: int, f: int -> int -> 'a, i j: int.
0 <= i < r -> 0 <= j < c -> get (create r c f) i j = f i j
function set (m:mat 'a) (x y:int) (z:'a) : mat 'a =
create m.rows m.cols (fun x1 y1 -> if x1 = x && y1 = y then z else get m x1 y1)
clone export MatrixGen with type mat 'a = mat 'a,
function rows = rows,
function cols = cols,
function get = get,
function set = set,
lemma set_def1,
lemma set_def2,
lemma set_def3,
lemma set_def4,
axiom rows_and_cols_nonnegative,
axiom extensionality
use int.Int
end
module MatrixArithmetic
use int.Int
use int.Sum
use sum_extended.Sum_extended
use Matrix
(* Zero matrix *)
constant zerof : int -> int -> int = fun _ _ -> 0
function zero (r c: int) : mat int = create r c zerof
(* Matrix addition *)
function add2f (a b: mat int) : int -> int -> int =
fun x y -> get a x y + get b x y
function add (a b: mat int) : mat int =
create (rows a) (cols a) (add2f a b)
(* Matrix additive inverse *)
function opp2f (a: mat int) : int -> int -> int =
fun x y -> - get a x y
function opp (a: mat int) : mat int =
create (rows a) (cols a) (opp2f a)
function sub (a b: mat int) : mat int =
add a (opp b)
(* Matrix multiplication *)
function mul_atom (a b: mat int) (i j:int) : int -> int =
fun k -> get a i k * get b k j
function mul_cell (a b: mat int): int -> int -> int =
fun i j -> sum (mul_atom a b i j) 0 (cols a)
function mul (a b: mat int) : mat int =
create (rows a) (cols b) (mul_cell a b)
lemma zero_neutral:
forall a. add a (zero a.rows a.cols) = a
by add a (zero a.rows a.cols) == a
lemma add_commutative:
forall a b. a === b ->
add a b = add b a
by add a b == add b a
lemma add_associative:
forall a b c. a === b === c ->
add a (add b c) = add (add a b) c
by add a (add b c) == add (add a b) c
lemma add_opposite:
forall a. add a (opp a) = zero a.rows a.cols
by add a (opp a) == zero a.rows a.cols
lemma opp_involutive:
forall m. opp (opp m) = m
lemma opposite_add: forall m1 m2.
m1 === m2 -> opp (add m1 m2) = add (opp m1) (opp m2)
function ft1 (a b c: mat int) (i j: int) : int -> int -> int =
fun k -> smulf (mul_atom a b i k) (get c k j)
function ft2 (a b c: mat int) (i j: int) : int -> int -> int =
fun k -> smulf (mul_atom b c k j) (get a i k)
let lemma mul_assoc_get (a b c: mat int) (i j: int)
requires { cols a = rows b }
requires { cols b = rows c }
requires { 0 <= i < (rows a) /\ 0 <= j < (cols c) }
ensures { get (mul (mul a b) c) i j = get (mul a (mul b c)) i j }
= let ft1 = ft1 a b c i j in
let ft2 = ft2 a b c i j in
fubini ft1 ft2 0 (cols b) 0 (cols a);
sum_ext (mul_atom (mul a b) c i j) (sumf ft1 0 (cols a)) 0 (cols b);
assert { get (mul (mul a b) c) i j = sum (sumf ft1 0 (cols a)) 0 (cols b) };
sum_ext (mul_atom a (mul b c) i j) (sumf ft2 0 (cols b)) 0 (cols a);
assert { get (mul a (mul b c)) i j = sum (sumf ft2 0 (cols b)) 0 (cols a) }
lemma mul_assoc:
forall a b c. cols a = rows b -> cols b = rows c ->
let ab = mul a b in
let bc = mul b c in
let a_bc = mul a bc in
let ab_c = mul ab c in
a_bc = ab_c
by a_bc == ab_c
let lemma mul_distr_right_get (a b c: mat int) (i j: int)
requires { 0 <= i < rows a /\ 0 <= j < cols c }
requires { cols b = rows c}
requires { a === b }
ensures { get (mul (add a b) c) i j = get (add (mul a c) (mul b c)) i j }
= let mac = mul_atom a c i j in
let mbc = mul_atom b c i j in
assert { get (add (mul a c) (mul b c)) i j =
get (mul a c) i j + get (mul b c) i j =
sum (addf mac mbc) 0 (cols b) };
sum_ext (addf mac mbc) (mul_atom (add a b) c i j) 0 (cols b)
lemma mul_distr_right:
forall a b c. a === b -> cols b = rows c ->
mul (add a b) c = add (mul a c) (mul b c)
by mul (add a b) c == add (mul a c) (mul b c)
let lemma mul_distr_left_get (a b c: mat int) (i j : int)
requires { 0 <= i < rows a /\ 0 <= j < cols c }
requires { cols a = rows b }
requires { b === c }
ensures { get (mul a (add b c)) i j = get (add (mul a b) (mul a c)) i j }
= let mab = mul_atom a b i j in
let mac = mul_atom a c i j in
assert { get (add (mul a b) (mul a c)) i j =
get (mul a b) i j + get (mul a c) i j =
sum (addf mab mac) 0 (cols a) };
sum_ext (addf mab mac) (mul_atom a (add b c) i j) 0 (cols a)
lemma mul_distr_left:
forall a b c. b === c -> cols a = rows b ->
mul a (add b c) = add (mul a b) (mul a c)
by mul a (add b c) == add (mul a b) (mul a c)
lemma mul_zero_right:
forall a c. 0 <= c -> mul a (zero a.cols c) = zero a.rows c
lemma mul_zero_left:
forall a r. 0 <= r -> mul (zero r a.rows) a = zero r a.cols
lemma mul_opp:
forall a b. a.cols = b.rows ->
let oa = opp a in
let ob = opp b in
let ab = mul a b in
let oab = opp ab in
mul oa b = oab = mul a ob
by add (mul oa b) (add ab oab) = oab = add (mul a ob) (add ab oab)
end
module BlockMul
use int.Int
use int.Sum
use sum_extended.Sum_extended
use Matrix
use MatrixArithmetic
function ofs2 (a: mat int) (ai aj: int) : int -> int -> int
= fun i j -> get a (ai + i) (aj + j)
function block (a: mat int) (r dr c dc: int) : mat int =
create dr dc (ofs2 a r c)
predicate c_blocks (a a1 a2: mat int) =
0 <= a1.cols <= a.cols /\ a1 = block a 0 a.rows 0 a1.cols /\
a2 = block a 0 a.rows a1.cols (a.cols - a1.cols)
predicate r_blocks (a a1 a2: mat int) =
0 <= a1.rows <= a.rows /\ a1 = block a 0 a1.rows 0 a.cols /\
a2 = block a a1.rows (a.rows - a1.rows) 0 a.cols
let rec ghost block_mul_ij (a a1 a2 b b1 b2: mat int) (k: int) : unit
requires { a.cols = b.rows /\ a1.cols = b1.rows}
requires { 0 <= k <= a.cols }
requires { c_blocks a a1 a2 /\ r_blocks b b1 b2 }
ensures { forall i j. 0 <= i < a.rows -> 0 <= j < b.cols ->
0 <= k <= a1.cols ->
sum (mul_atom a b i j) 0 k = sum (mul_atom a1 b1 i j) 0 k }
ensures { forall i j. 0 <= i < a.rows -> 0 <= j < b.cols ->
a1.cols <= k <= a.cols ->
sum (mul_atom a b i j) 0 k =
sum (mul_atom a1 b1 i j) 0 a1.cols +
sum (mul_atom a2 b2 i j) 0 (k - a1.cols) }
variant { k }
= if 0 < k then begin
let k = k - 1 in
assert { forall i j. 0 <= i < a.rows -> 0 <= j < b.cols ->
mul_atom a b i j k = if k < a1.cols then mul_atom a1 b1 i j k
else mul_atom a2 b2 i j (k - a1.cols) };
block_mul_ij a a1 a2 b b1 b2 k
end
let lemma mul_split (a a1 a2 b b1 b2: mat int) : unit
requires { a.cols = b.rows /\ a1.cols = b1.rows}
requires { c_blocks a a1 a2 /\ r_blocks b b1 b2 }
ensures {add (mul a1 b1) (mul a2 b2) = mul a b }
= block_mul_ij a a1 a2 b b1 b2 a.cols;
assert { add (mul a1 b1) (mul a2 b2) == mul a b }
let lemma mul_block_cell (a b: mat int) (r dr c dc i j: int) : unit
requires { a.cols = b.rows }
requires { 0 <= r /\ r + dr <= a.rows }
requires { 0 <= c /\ c + dc <= b.cols }
requires { 0 <= i < dr /\ 0 <= j < dc }
ensures { ofs2 (mul a b) r c i j =
get (mul (block a r dr 0 a.cols) (block b 0 b.rows c dc)) i j }
= let a' = block a r dr 0 a.cols in
let b' = block b 0 b.rows c dc in
sum_ext (mul_atom a b (i + r) (j + c)) (mul_atom a' b' i j) 0 a.cols
lemma mul_block:
forall a b: mat int, r dr c dc: int.
a.cols = b.rows -> 0 <= r <= r + dr <= a.rows ->
0 <= c <= c + dc <= b.cols ->
let a' = block a r dr 0 a.cols in
let b' = block b 0 b.rows c dc in
let m' = block (mul a b) r dr c dc in
m' = mul a' b'
by m' == mul a' b'
end
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