Structure integer_wordTheory
signature integer_wordTheory =
sig
type thm = Thm.thm
(* Definitions *)
val INT_MAX_def : thm
val INT_MIN_def : thm
val UINT_MAX_def : thm
val fromString_primitive_def : thm
val i2w_def : thm
val saturate_i2sw_def : thm
val saturate_i2w_def : thm
val saturate_sw2sw_def : thm
val saturate_sw2w_def : thm
val saturate_w2sw_def : thm
val signed_saturate_add_def : thm
val signed_saturate_sub_def : thm
val toString_def : thm
val w2i_def : thm
(* Theorems *)
val INT_BOUND_ORDER : thm
val INT_MAX : thm
val INT_MAX_MONOTONIC : thm
val INT_MIN : thm
val INT_MIN_MONOTONIC : thm
val INT_ZERO_LE_INT_MAX : thm
val INT_ZERO_LT_INT_MAX : thm
val INT_ZERO_LT_INT_MIN : thm
val INT_ZERO_LT_UINT_MAX : thm
val MULT_MINUS_ONE : thm
val ONE_LE_TWOEXP : thm
val UINT_MAX : thm
val WORD_GEi : thm
val WORD_GTi : thm
val WORD_LEi : thm
val WORD_LTi : thm
val different_sign_then_no_overflow : thm
val fromString_def : thm
val fromString_ind : thm
val i2w_0 : thm
val i2w_DIV : thm
val i2w_INT_MAX : thm
val i2w_INT_MIN : thm
val i2w_UINT_MAX : thm
val i2w_minus_1 : thm
val i2w_w2i : thm
val int_word_nchotomy : thm
val overflow : thm
val ranged_int_word_nchotomy : thm
val saturate_i2sw : thm
val saturate_i2sw_0 : thm
val saturate_i2w_0 : thm
val saturate_sw2sw : thm
val saturate_sw2w : thm
val saturate_w2sw : thm
val signed_saturate_add : thm
val signed_saturate_sub : thm
val sw2sw_i2w : thm
val w2i_1 : thm
val w2i_11 : thm
val w2i_11_lift : thm
val w2i_INT_MAXw : thm
val w2i_INT_MINw : thm
val w2i_UINT_MAXw : thm
val w2i_ge : thm
val w2i_i2w : thm
val w2i_i2w_id : thm
val w2i_i2w_neg : thm
val w2i_i2w_pos : thm
val w2i_le : thm
val w2i_lt_0 : thm
val w2i_minus_1 : thm
val w2i_n2w_mod : thm
val w2i_n2w_neg : thm
val w2i_n2w_pos : thm
val w2i_neg : thm
val w2i_sw2sw_bounds : thm
val w2i_w2n_pos : thm
val w2w_i2w : thm
val word_0_w2i : thm
val word_abs_i2w : thm
val word_abs_w2i : thm
val word_add_i2w : thm
val word_add_i2w_w2n : thm
val word_i2w_add : thm
val word_i2w_mul : thm
val word_msb_i2w : thm
val word_mul_i2w : thm
val word_mul_i2w_w2n : thm
val word_sub_i2w : thm
val word_sub_i2w_w2n : thm
val integer_word_grammars : type_grammar.grammar * term_grammar.grammar
(*
[Omega] Parent theory of "integer_word"
[int_arith] Parent theory of "integer_word"
[words] Parent theory of "integer_word"
[INT_MAX_def] Definition
|- INT_MAX (:α) = &INT_MIN (:α) − 1
[INT_MIN_def] Definition
|- INT_MIN (:α) = -INT_MAX (:α) − 1
[UINT_MAX_def] Definition
|- UINT_MAX (:α) = &dimword (:α) − 1
[fromString_primitive_def] Definition
|- 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] Definition
|- ∀i. i2w i = if i < 0 then -n2w (Num (-i)) else n2w (Num i)
[saturate_i2sw_def] Definition
|- ∀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_i2w_def] Definition
|- ∀i.
saturate_i2w i =
if UINT_MAX (:α) ≤ i then UINT_MAXw
else if i < 0 then 0w
else n2w (Num i)
[saturate_sw2sw_def] Definition
|- ∀w. saturate_sw2sw w = saturate_i2sw (w2i w)
[saturate_sw2w_def] Definition
|- ∀w. saturate_sw2w w = saturate_i2w (w2i w)
[saturate_w2sw_def] Definition
|- ∀w. saturate_w2sw w = saturate_i2sw (&w2n w)
[signed_saturate_add_def] Definition
|- ∀a b. signed_saturate_add a b = saturate_i2sw (w2i a + w2i b)
[signed_saturate_sub_def] Definition
|- ∀a b. signed_saturate_sub a b = saturate_i2sw (w2i a − w2i b)
[toString_def] Definition
|- ∀i.
toString i =
if i < 0 then STRCAT "~" (toString (Num (-i)))
else toString (Num i)
[w2i_def] Definition
|- ∀w. w2i w = if word_msb w then -&w2n (-w) else &w2n w
[INT_BOUND_ORDER] Theorem
|- INT_MIN (:α) < INT_MAX (:α) ∧ INT_MAX (:α) < UINT_MAX (:α) ∧
UINT_MAX (:α) < &dimword (:α)
[INT_MAX] Theorem
|- INT_MAX (:α) = &INT_MAX (:α)
[INT_MAX_MONOTONIC] Theorem
|- dimindex (:α) ≤ dimindex (:β) ⇒ INT_MAX (:α) ≤ INT_MAX (:β)
[INT_MIN] Theorem
|- INT_MIN (:α) = -&INT_MIN (:α)
[INT_MIN_MONOTONIC] Theorem
|- dimindex (:α) ≤ dimindex (:β) ⇒ INT_MIN (:β) ≤ INT_MIN (:α)
[INT_ZERO_LE_INT_MAX] Theorem
|- 0 ≤ INT_MAX (:α)
[INT_ZERO_LT_INT_MAX] Theorem
|- 1 < dimindex (:α) ⇒ 0 < INT_MAX (:α)
[INT_ZERO_LT_INT_MIN] Theorem
|- INT_MIN (:α) < 0
[INT_ZERO_LT_UINT_MAX] Theorem
|- 0 < UINT_MAX (:α)
[MULT_MINUS_ONE] Theorem
|- ∀i. -1w * i2w i = i2w (-i)
[ONE_LE_TWOEXP] Theorem
|- ∀n. 1 ≤ 2 ** n
[UINT_MAX] Theorem
|- UINT_MAX (:α) = &UINT_MAX (:α)
[WORD_GEi] Theorem
|- ∀a b. a ≥ b ⇔ w2i a ≥ w2i b
[WORD_GTi] Theorem
|- ∀a b. a > b ⇔ w2i a > w2i b
[WORD_LEi] Theorem
|- ∀a b. a ≤ b ⇔ w2i a ≤ w2i b
[WORD_LTi] Theorem
|- ∀a b. a < b ⇔ w2i a < w2i b
[different_sign_then_no_overflow] Theorem
|- ∀x y. (word_msb x ⇎ word_msb y) ⇒ (w2i (x + y) = w2i x + w2i y)
[fromString_def] Theorem
|- (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))
[fromString_ind] Theorem
|- ∀P.
(∀t. P (STRING #"~" t)) ∧ (∀t. P (STRING #"-" t)) ∧ P "" ∧
(∀v4 v1. P (STRING v4 v1)) ⇒
∀v. P v
[i2w_0] Theorem
|- i2w 0 = 0w
[i2w_DIV] Theorem
|- ∀n i.
n < dimindex (:α) ∧ INT_MIN (:α) ≤ i ∧ i ≤ INT_MAX (:α) ⇒
(i2w (i / 2 ** n) = i2w i ≫ n)
[i2w_INT_MAX] Theorem
|- i2w (INT_MAX (:α)) = INT_MAXw
[i2w_INT_MIN] Theorem
|- i2w (INT_MIN (:α)) = INT_MINw
[i2w_UINT_MAX] Theorem
|- i2w (UINT_MAX (:α)) = UINT_MAXw
[i2w_minus_1] Theorem
|- i2w (-1) = -1w
[i2w_w2i] Theorem
|- ∀w. i2w (w2i w) = w
[int_word_nchotomy] Theorem
|- ∀w. ∃i. w = i2w i
[overflow] Theorem
|- ∀x y.
w2i (x + y) ≠ w2i x + w2i y ⇔
(word_msb x ⇔ word_msb y) ∧ (word_msb x ⇎ word_msb (x + y))
[ranged_int_word_nchotomy] Theorem
|- ∀w. ∃i. (w = i2w i) ∧ INT_MIN (:α) ≤ i ∧ i ≤ INT_MAX (:α)
[saturate_i2sw] Theorem
|- ∀i. saturate_i2w i = if i < 0 then 0w else saturate_n2w (Num i)
[saturate_i2sw_0] Theorem
|- saturate_i2sw 0 = 0w
[saturate_i2w_0] Theorem
|- saturate_i2w 0 = 0w
[saturate_sw2sw] Theorem
|- ∀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
[saturate_sw2w] Theorem
|- ∀w. saturate_sw2w w = if w < 0w then 0w else saturate_w2w w
[saturate_w2sw] Theorem
|- ∀w.
saturate_w2sw w =
if dimindex (:β) ≤ dimindex (:α) ∧ w2w INT_MAXw ≤₊ w then
INT_MAXw
else w2w w
[signed_saturate_add] Theorem
|- ∀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)
[signed_saturate_sub] Theorem
|- ∀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)
[sw2sw_i2w] Theorem
|- ∀j.
INT_MIN (:β) ≤ j ∧ j ≤ INT_MAX (:β) ∧
dimindex (:β) ≤ dimindex (:α) ⇒
(sw2sw (i2w j) = i2w j)
[w2i_1] Theorem
|- w2i 1w = if dimindex (:α) = 1 then -1 else 1
[w2i_11] Theorem
|- ∀v w. (w2i v = w2i w) ⇔ (v = w)
[w2i_11_lift] Theorem
|- ∀a b.
dimindex (:α) ≤ dimindex (:γ) ∧ dimindex (:β) ≤ dimindex (:γ) ⇒
((w2i a = w2i b) ⇔ (sw2sw a = sw2sw b))
[w2i_INT_MAXw] Theorem
|- w2i INT_MAXw = INT_MAX (:α)
[w2i_INT_MINw] Theorem
|- w2i INT_MINw = INT_MIN (:α)
[w2i_UINT_MAXw] Theorem
|- w2i UINT_MAXw = -1
[w2i_ge] Theorem
|- ∀w. INT_MIN (:α) ≤ w2i w
[w2i_i2w] Theorem
|- ∀i. INT_MIN (:α) ≤ i ∧ i ≤ INT_MAX (:α) ⇒ (w2i (i2w i) = i)
[w2i_i2w_id] Theorem
|- ∀i.
INT_MIN (:α) ≤ i ∧ i ≤ INT_MAX (:α) ∧
dimindex (:β) ≤ dimindex (:α) ⇒
((i = w2i (i2w i)) ⇔ (i2w i = sw2sw (i2w i)))
[w2i_i2w_neg] Theorem
|- ∀n. n ≤ INT_MIN (:α) ⇒ (w2i (i2w (-&n)) = -&n)
[w2i_i2w_pos] Theorem
|- ∀n. n ≤ INT_MAX (:α) ⇒ (w2i (i2w (&n)) = &n)
[w2i_le] Theorem
|- ∀w. w2i w ≤ INT_MAX (:α)
[w2i_lt_0] Theorem
|- ∀w. w2i w < 0 ⇔ w < 0w
[w2i_minus_1] Theorem
|- w2i (-1w) = -1
[w2i_n2w_mod] Theorem
|- ∀n m.
n < dimword (:α) ∧ m ≤ dimindex (:α) ⇒
(Num (w2i (n2w n) % 2 ** m) = n MOD 2 ** m)
[w2i_n2w_neg] Theorem
|- ∀n.
INT_MIN (:α) ≤ n ∧ n < dimword (:α) ⇒
(w2i (n2w n) = -&(dimword (:α) − n))
[w2i_n2w_pos] Theorem
|- ∀n. n < INT_MIN (:α) ⇒ (w2i (n2w n) = &n)
[w2i_neg] Theorem
|- ∀w. 1 < dimindex (:α) ∧ w ≠ INT_MINw ⇒ (w2i (-w) = -w2i w)
[w2i_sw2sw_bounds] Theorem
|- ∀w. INT_MIN (:α) ≤ w2i (sw2sw w) ∧ w2i (sw2sw w) ≤ INT_MAX (:α)
[w2i_w2n_pos] Theorem
|- ∀w n. ¬word_msb w ∧ w2i w < &n ⇒ w2n w < n
[w2w_i2w] Theorem
|- ∀i. dimindex (:α) ≤ dimindex (:β) ⇒ (w2w (i2w i) = i2w i)
[word_0_w2i] Theorem
|- w2i 0w = 0
[word_abs_i2w] Theorem
|- ∀i.
INT_MIN (:α) ≤ i ∧ i ≤ INT_MAX (:α) ⇒
(word_abs (i2w i) = n2w (Num (ABS i)))
[word_abs_w2i] Theorem
|- ∀w. word_abs w = n2w (Num (ABS (w2i w)))
[word_add_i2w] Theorem
|- ∀a b. i2w (w2i a + w2i b) = a + b
[word_add_i2w_w2n] Theorem
|- ∀a b. i2w (&w2n a + &w2n b) = a + b
[word_i2w_add] Theorem
|- ∀a b. i2w a + i2w b = i2w (a + b)
[word_i2w_mul] Theorem
|- ∀a b. i2w a * i2w b = i2w (a * b)
[word_msb_i2w] Theorem
|- ∀i. word_msb (i2w i) ⇔ &INT_MIN (:α) ≤ i % &dimword (:α)
[word_mul_i2w] Theorem
|- ∀a b. i2w (w2i a * w2i b) = a * b
[word_mul_i2w_w2n] Theorem
|- ∀a b. i2w (&w2n a * &w2n b) = a * b
[word_sub_i2w] Theorem
|- ∀a b. i2w (w2i a − w2i b) = a − b
[word_sub_i2w_w2n] Theorem
|- ∀a b. i2w (&w2n a − &w2n b) = a − b
*)
end
HOL 4, Kananaskis-10