Theory "integer_word"

Signature

Definitions

toString_def

|- ∀i.
     toString i =
     if i < 0 then STRCAT "~" (toString (Num (-i))) else toString (Num i)

fromString_primitive_def

|- fromString =
   WFREC (@R. WF R)
     (λfromString a.
        case a of
          "" => I (&toNum "")
        | STRING #"~" t => I (-&toNum t)
        | STRING #"-" t => I (-&toNum t)
        | STRING v4 t => I (&toNum (STRING v4 t)))

i2w_def

|- ∀i. i2w i = if i < 0 then -n2w (Num (-i)) else n2w (Num i)

w2i_def

|- ∀w. w2i w = if word_msb w then -&w2n (-w) else &w2n w

UINT_MAX_def

|- UINT_MAX (:α) = &dimword (:α) − 1

INT_MAX_def

|- INT_MAX (:α) = &INT_MIN (:α) − 1

INT_MIN_def

|- INT_MIN (:α) = -INT_MAX (:α) − 1

saturate_i2w_def

|- ∀i.
     saturate_i2w i =
     if UINT_MAX (:α) ≤ i then UINT_MAXw
     else if i < 0 then 0w
     else n2w (Num i)

saturate_i2sw_def

|- ∀i.
     saturate_i2sw i =
     if INT_MAX (:α) ≤ i then INT_MAXw
     else if i ≤ INT_MIN (:α) then INT_MINw
     else i2w i

saturate_sw2sw_def

|- ∀w. saturate_sw2sw w = saturate_i2sw (w2i w)

saturate_w2sw_def

|- ∀w. saturate_w2sw w = saturate_i2sw (&w2n w)

saturate_sw2w_def

|- ∀w. saturate_sw2w w = saturate_i2w (w2i w)

signed_saturate_add_def

|- ∀a b. signed_saturate_add a b = saturate_i2sw (w2i a + w2i b)

signed_saturate_sub_def

|- ∀a b. signed_saturate_sub a b = saturate_i2sw (w2i a − w2i b)

Theorems

fromString_ind

|- ∀P.
     (∀t. P (STRING #"~" t)) ∧ (∀t. P (STRING #"-" t)) ∧ P "" ∧
     (∀v4 v1. P (STRING v4 v1)) ⇒
     ∀v. P v

fromString_def

|- (fromString (STRING #"~" t) = -&toNum t) ∧
   (fromString (STRING #"-" t) = -&toNum t) ∧ (fromString "" = &toNum "") ∧
   (fromString (STRING v4 v1) =
    if v4 = #"~" then -&toNum v1
    else if v4 = #"-" then -&toNum v1
    else &toNum (STRING v4 v1))

ONE_LE_TWOEXP

|- ∀n. 1 ≤ 2 ** n

w2i_w2n_pos

|- ∀w n. ¬word_msb w ∧ w2i w < &n ⇒ w2n w < n

w2i_n2w_pos

|- ∀n. n < INT_MIN (:α) ⇒ (w2i (n2w n) = &n)

w2i_n2w_neg

|- ∀n.
     INT_MIN (:α) ≤ n ∧ n < dimword (:α) ⇒
     (w2i (n2w n) = -&(dimword (:α) − n))

i2w_w2i

|- ∀w. i2w (w2i w) = w

w2i_i2w

|- ∀i. INT_MIN (:α) ≤ i ∧ i ≤ INT_MAX (:α) ⇒ (w2i (i2w i) = i)

word_msb_i2w

|- ∀i. word_msb (i2w i) ⇔ &INT_MIN (:α) ≤ i % &dimword (:α)

w2i_11

|- ∀v w. (w2i v = w2i w) ⇔ (v = w)

int_word_nchotomy

|- ∀w. ∃i. w = i2w i

w2i_le

|- ∀w. w2i w ≤ INT_MAX (:α)

w2i_ge

|- ∀w. INT_MIN (:α) ≤ w2i w

ranged_int_word_nchotomy

|- ∀w. ∃i. (w = i2w i) ∧ INT_MIN (:α) ≤ i ∧ i ≤ INT_MAX (:α)

sw2sw_i2w

|- ∀j.
     INT_MIN (:β) ≤ j ∧ j ≤ INT_MAX (:β) ∧ dimindex (:β) ≤ dimindex (:α) ⇒
     (sw2sw (i2w j) = i2w j)

w2w_i2w

|- ∀i. dimindex (:α) ≤ dimindex (:β) ⇒ (w2w (i2w i) = i2w i)

WORD_LTi

|- ∀a b. a < b ⇔ w2i a < w2i b

WORD_GTi

|- ∀a b. a > b ⇔ w2i a > w2i b

WORD_LEi

|- ∀a b. a ≤ b ⇔ w2i a ≤ w2i b

WORD_GEi

|- ∀a b. a ≥ b ⇔ w2i a ≥ w2i b

word_add_i2w_w2n

|- ∀a b. i2w (&w2n a + &w2n b) = a + b

word_add_i2w

|- ∀a b. i2w (w2i a + w2i b) = a + b

word_sub_i2w_w2n

|- ∀a b. i2w (&w2n a − &w2n b) = a − b

word_sub_i2w

|- ∀a b. i2w (w2i a − w2i b) = a − b

word_mul_i2w_w2n

|- ∀a b. i2w (&w2n a * &w2n b) = a * b

word_mul_i2w

|- ∀a b. i2w (w2i a * w2i b) = a * b

word_i2w_add

|- ∀a b. i2w a + i2w b = i2w (a + b)

word_i2w_mul

|- ∀a b. i2w a * i2w b = i2w (a * b)

MULT_MINUS_ONE

|- ∀i. -1w * i2w i = i2w (-i)

word_0_w2i

|- w2i 0w = 0

i2w_DIV

|- ∀n i.
     n < dimindex (:α) ∧ INT_MIN (:α) ≤ i ∧ i ≤ INT_MAX (:α) ⇒
     (i2w (i / 2 ** n) = i2w i ≫ n)

INT_MIN_MONOTONIC

|- dimindex (:α) ≤ dimindex (:β) ⇒ INT_MIN (:β) ≤ INT_MIN (:α)

INT_MAX_MONOTONIC

|- dimindex (:α) ≤ dimindex (:β) ⇒ INT_MAX (:α) ≤ INT_MAX (:β)

w2i_sw2sw_bounds

|- ∀w. INT_MIN (:α) ≤ w2i (sw2sw w) ∧ w2i (sw2sw w) ≤ INT_MAX (:α)

w2i_i2w_id

|- ∀i.
     INT_MIN (:α) ≤ i ∧ i ≤ INT_MAX (:α) ∧ dimindex (:β) ≤ dimindex (:α) ⇒
     ((i = w2i (i2w i)) ⇔ (i2w i = sw2sw (i2w i)))

w2i_11_lift

|- ∀a b.
     dimindex (:α) ≤ dimindex (:γ) ∧ dimindex (:β) ≤ dimindex (:γ) ⇒
     ((w2i a = w2i b) ⇔ (sw2sw a = sw2sw b))

w2i_n2w_mod

|- ∀n m.
     n < dimword (:α) ∧ m ≤ dimindex (:α) ⇒
     (Num (w2i (n2w n) % 2 ** m) = n MOD 2 ** m)

word_abs_w2i

|- ∀w. word_abs w = n2w (Num (ABS (w2i w)))

word_abs_i2w

|- ∀i.
     INT_MIN (:α) ≤ i ∧ i ≤ INT_MAX (:α) ⇒
     (word_abs (i2w i) = n2w (Num (ABS i)))

INT_MIN

|- INT_MIN (:α) = -&INT_MIN (:α)

INT_MAX

|- INT_MAX (:α) = &INT_MAX (:α)

UINT_MAX

|- UINT_MAX (:α) = &UINT_MAX (:α)

INT_BOUND_ORDER

|- INT_MIN (:α) < INT_MAX (:α) ∧ INT_MAX (:α) < UINT_MAX (:α) ∧
   UINT_MAX (:α) < &dimword (:α)

INT_ZERO_LT_INT_MIN

|- INT_MIN (:α) < 0

INT_ZERO_LT_INT_MAX

|- 1 < dimindex (:α) ⇒ 0 < INT_MAX (:α)

INT_ZERO_LE_INT_MAX

|- 0 ≤ INT_MAX (:α)

INT_ZERO_LT_UINT_MAX

|- 0 < UINT_MAX (:α)

w2i_1

|- w2i 1w = if dimindex (:α) = 1 then -1 else 1

w2i_INT_MINw

|- w2i INT_MINw = INT_MIN (:α)

w2i_UINT_MAXw

|- w2i UINT_MAXw = -1

w2i_INT_MAXw

|- w2i INT_MAXw = INT_MAX (:α)

w2i_minus_1

|- w2i (-1w) = -1

w2i_lt_0

|- ∀w. w2i w < 0 ⇔ w < 0w

w2i_neg

|- ∀w. 1 < dimindex (:α) ∧ w ≠ INT_MINw ⇒ (w2i (-w) = -w2i w)

i2w_0

|- i2w 0 = 0w

i2w_minus_1

|- i2w (-1) = -1w

i2w_INT_MIN

|- i2w (INT_MIN (:α)) = INT_MINw

i2w_INT_MAX

|- i2w (INT_MAX (:α)) = INT_MAXw

i2w_UINT_MAX

|- i2w (UINT_MAX (:α)) = UINT_MAXw

saturate_i2w_0

|- saturate_i2w 0 = 0w

saturate_i2sw_0

|- saturate_i2sw 0 = 0w

saturate_w2sw

|- ∀w.
     saturate_w2sw w =
     if dimindex (:β) ≤ dimindex (:α) ∧ w2w INT_MAXw ≤₊ w then INT_MAXw
     else w2w w

saturate_i2sw

|- ∀i. saturate_i2w i = if i < 0 then 0w else saturate_n2w (Num i)

saturate_sw2w

|- ∀w. saturate_sw2w w = if w < 0w then 0w else saturate_w2w w

saturate_sw2sw

|- ∀w.
     saturate_sw2sw w =
     if dimindex (:α) ≤ dimindex (:β) then sw2sw w
     else if sw2sw INT_MAXw ≤ w then INT_MAXw
     else if w ≤ sw2sw INT_MINw then INT_MINw
     else w2w w

signed_saturate_sub

|- ∀a b.
     signed_saturate_sub a b =
     if b = INT_MINw then if 0w ≤ a then INT_MAXw else a + INT_MINw
     else if dimindex (:α) = 1 then a && ¬b
     else signed_saturate_add a (-b)

signed_saturate_add

|- ∀a b.
     signed_saturate_add a b =
     (let sum = a + b and msba = word_msb a
      in
        if (msba ⇔ word_msb b) ∧ (msba ⇎ word_msb sum) then
          if msba then INT_MINw else INT_MAXw
        else sum)

different_sign_then_no_overflow

|- ∀x y. (word_msb x ⇎ word_msb y) ⇒ (w2i (x + y) = w2i x + w2i y)

w2i_i2w_pos

|- ∀n. n ≤ INT_MAX (:α) ⇒ (w2i (i2w (&n)) = &n)

w2i_i2w_neg

|- ∀n. n ≤ INT_MIN (:α) ⇒ (w2i (i2w (-&n)) = -&n)

overflow

|- ∀x y.
     w2i (x + y) ≠ w2i x + w2i y ⇔
     (word_msb x ⇔ word_msb y) ∧ (word_msb x ⇎ word_msb (x + y))