# Theory Word

```(*  Author:     Sebastian Skalberg, TU Muenchen
*)

section ‹Binary Words›

theory Word
imports Main
begin

subsection ‹Auxilary Lemmas›

lemma max_le [intro!]: "[| x ≤ z; y ≤ z |] ==> max x y ≤ z"

lemma max_mono:
fixes x :: "'a::linorder"
assumes mf: "mono f"
shows       "max (f x) (f y) ≤ f (max x y)"
proof -
from mf and max.cobounded1 [of x y]
have fx: "f x ≤ f (max x y)" by (rule monoD)
from mf and max.cobounded2 [of y x]
have fy: "f y ≤ f (max x y)" by (rule monoD)
from fx and fy
show "max (f x) (f y) ≤ f (max x y)" by auto
qed

declare zero_le_power [intro]
and zero_less_power [intro]

lemma int_nat_two_exp: "2 ^ k = int (2 ^ k)"
by simp

subsection ‹Bits›

datatype bit =
Zero ("𝟬")
| One ("𝟭")

primrec bitval :: "bit => nat" where
"bitval 𝟬 = 0"
| "bitval 𝟭 = 1"

primrec bitnot :: "bit => bit"  ("¬⇩b _" [40] 40) where
bitnot_zero: "(¬⇩b 𝟬) = 𝟭"
| bitnot_one : "(¬⇩b 𝟭) = 𝟬"

primrec bitand :: "bit => bit => bit"  (infixr "∧⇩b" 35) where
bitand_zero: "(𝟬 ∧⇩b y) = 𝟬"
| bitand_one:  "(𝟭 ∧⇩b y) = y"

primrec bitor :: "bit => bit => bit"  (infixr "∨⇩b" 30) where
bitor_zero: "(𝟬 ∨⇩b y) = y"
| bitor_one:  "(𝟭 ∨⇩b y) = 𝟭"

primrec bitxor :: "bit => bit => bit"  (infixr "⊕⇩b" 30) where
bitxor_zero: "(𝟬 ⊕⇩b y) = y"
| bitxor_one:  "(𝟭 ⊕⇩b y) = (bitnot y)"

lemma bitnot_bitnot [simp]: "(bitnot (bitnot b)) = b"
by (cases b) simp_all

lemma bitand_cancel [simp]: "(b ∧⇩b b) = b"
by (cases b) simp_all

lemma bitor_cancel [simp]: "(b ∨⇩b b) = b"
by (cases b) simp_all

lemma bitxor_cancel [simp]: "(b ⊕⇩b b) = 𝟬"
by (cases b) simp_all

subsection ‹Bit Vectors›

text ‹First, a couple of theorems expressing case analysis and
induction principles for bit vectors.›

lemma bit_list_cases:
assumes empty: "w = [] ==> P w"
and     zero:  "!!bs. w = 𝟬 # bs ==> P w"
and     one:   "!!bs. w = 𝟭 # bs ==> P w"
shows   "P w"
proof (cases w)
assume "w = []"
thus ?thesis by (rule empty)
next
fix b bs
assume [simp]: "w = b # bs"
show "P w"
proof (cases b)
assume "b = 𝟬"
hence "w = 𝟬 # bs" by simp
thus ?thesis by (rule zero)
next
assume "b = 𝟭"
hence "w = 𝟭 # bs" by simp
thus ?thesis by (rule one)
qed
qed

lemma bit_list_induct:
assumes empty: "P []"
and     zero:  "!!bs. P bs ==> P (𝟬#bs)"
and     one:   "!!bs. P bs ==> P (𝟭#bs)"
shows   "P w"
proof (induct w, simp_all add: empty)
fix b bs
assume "P bs"
then show "P (b#bs)"
by (cases b) (auto intro!: zero one)
qed

definition
bv_msb :: "bit list => bit" where
"bv_msb w = (if w = [] then 𝟬 else hd w)"

definition
bv_extend :: "[nat,bit,bit list]=>bit list" where
"bv_extend i b w = (replicate (i - length w) b) @ w"

definition
bv_not :: "bit list => bit list" where
"bv_not w = map bitnot w"

lemma bv_length_extend [simp]: "length w ≤ i ==> length (bv_extend i b w) = i"

lemma bv_not_Nil [simp]: "bv_not [] = []"

lemma bv_not_Cons [simp]: "bv_not (b#bs) = (bitnot b) # bv_not bs"

lemma bv_not_bv_not [simp]: "bv_not (bv_not w) = w"
by (rule bit_list_induct [of _ w]) simp_all

lemma bv_msb_Nil [simp]: "bv_msb [] = 𝟬"

lemma bv_msb_Cons [simp]: "bv_msb (b#bs) = b"

lemma bv_msb_bv_not [simp]: "0 < length w ==> bv_msb (bv_not w) = (bitnot (bv_msb w))"
by (cases w) simp_all

lemma bv_msb_one_length [simp,intro]: "bv_msb w = 𝟭 ==> 0 < length w"
by (cases w) simp_all

lemma length_bv_not [simp]: "length (bv_not w) = length w"
by (induct w) simp_all

definition
bv_to_nat :: "bit list => nat" where
"bv_to_nat = foldl (%bn b. 2 * bn + bitval b) 0"

lemma bv_to_nat_Nil [simp]: "bv_to_nat [] = 0"

lemma bv_to_nat_helper [simp]: "bv_to_nat (b # bs) = bitval b * 2 ^ length bs + bv_to_nat bs"
proof -
let ?bv_to_nat' = "foldl (λbn b. 2 * bn + bitval b)"
have helper: "⋀base. ?bv_to_nat' base bs = base * 2 ^ length bs + ?bv_to_nat' 0 bs"
proof (induct bs)
case Nil
show ?case by simp
next
case (Cons x xs base)
show ?case
apply (simp only: foldl_Cons)
apply (subst Cons [of "2 * base + bitval x"])
apply simp
apply (subst Cons [of "bitval x"])
done
qed
show ?thesis by (simp add: bv_to_nat_def) (rule helper)
qed

lemma bv_to_nat0 [simp]: "bv_to_nat (𝟬#bs) = bv_to_nat bs"
by simp

lemma bv_to_nat1 [simp]: "bv_to_nat (𝟭#bs) = 2 ^ length bs + bv_to_nat bs"
by simp

lemma bv_to_nat_upper_range: "bv_to_nat w < 2 ^ length w"
proof (induct w, simp_all)
fix b bs
assume "bv_to_nat bs < 2 ^ length bs"
show "bitval b * 2 ^ length bs + bv_to_nat bs < 2 * 2 ^ length bs"
proof (cases b, simp_all)
have "bv_to_nat bs < 2 ^ length bs" by fact
also have "... < 2 * 2 ^ length bs" by auto
finally show "bv_to_nat bs < 2 * 2 ^ length bs" by simp
next
have "bv_to_nat bs < 2 ^ length bs" by fact
hence "2 ^ length bs + bv_to_nat bs < 2 ^ length bs + 2 ^ length bs" by arith
also have "... = 2 * (2 ^ length bs)" by simp
finally show "bv_to_nat bs < 2 ^ length bs" by simp
qed
qed

lemma bv_extend_longer [simp]:
assumes wn: "n ≤ length w"
shows       "bv_extend n b w = w"

lemma bv_extend_shorter [simp]:
assumes wn: "length w < n"
shows       "bv_extend n b w = bv_extend n b (b#w)"
proof -
from wn
have s: "n - Suc (length w) + 1 = n - length w"
by arith
have "bv_extend n b w = replicate (n - length w) b @ w"
also have "... = replicate (n - Suc (length w) + 1) b @ w"
by (subst s) rule
also have "... = (replicate (n - Suc (length w)) b @ replicate 1 b) @ w"
also have "... = replicate (n - Suc (length w)) b @ b # w"
by simp
also have "... = bv_extend n b (b#w)"
finally show "bv_extend n b w = bv_extend n b (b#w)" .
qed

primrec rem_initial :: "bit => bit list => bit list" where
"rem_initial b [] = []"
| "rem_initial b (x#xs) = (if b = x then rem_initial b xs else x#xs)"

lemma rem_initial_length: "length (rem_initial b w) ≤ length w"
by (rule bit_list_induct [of _ w],simp_all (no_asm),safe,simp_all)

lemma rem_initial_equal:
assumes p: "length (rem_initial b w) = length w"
shows      "rem_initial b w = w"
proof -
have "length (rem_initial b w) = length w --> rem_initial b w = w"
proof (induct w, simp_all, clarify)
fix xs
assume "length (rem_initial b xs) = length xs --> rem_initial b xs = xs"
assume f: "length (rem_initial b xs) = Suc (length xs)"
with rem_initial_length [of b xs]
show "rem_initial b xs = b#xs"
by auto
qed
from this and p show ?thesis ..
qed

lemma bv_extend_rem_initial: "bv_extend (length w) b (rem_initial b w) = w"
proof (induct w, simp_all, safe)
fix xs
assume ind: "bv_extend (length xs) b (rem_initial b xs) = xs"
from rem_initial_length [of b xs]
have [simp]: "Suc (length xs) - length (rem_initial b xs) =
1 + (length xs - length (rem_initial b xs))"
by arith
have "bv_extend (Suc (length xs)) b (rem_initial b xs) =
replicate (Suc (length xs) - length (rem_initial b xs)) b @ (rem_initial b xs)"
also have "... =
replicate (1 + (length xs - length (rem_initial b xs))) b @ rem_initial b xs"
by simp
also have "... =
(replicate 1 b @ replicate (length xs - length (rem_initial b xs)) b) @ rem_initial b xs"
also have "... = b # bv_extend (length xs) b (rem_initial b xs)"
by (auto simp add: bv_extend_def [symmetric])
also have "... = b # xs"
finally show "bv_extend (Suc (length xs)) b (rem_initial b xs) = b # xs" .
qed

lemma rem_initial_append1:
assumes "rem_initial b xs ~= []"
shows   "rem_initial b (xs @ ys) = rem_initial b xs @ ys"
using assms by (induct xs) auto

lemma rem_initial_append2:
assumes "rem_initial b xs = []"
shows   "rem_initial b (xs @ ys) = rem_initial b ys"
using assms by (induct xs) auto

definition
norm_unsigned :: "bit list => bit list" where
"norm_unsigned = rem_initial 𝟬"

lemma norm_unsigned_Nil [simp]: "norm_unsigned [] = []"

lemma norm_unsigned_Cons0 [simp]: "norm_unsigned (𝟬#bs) = norm_unsigned bs"

lemma norm_unsigned_Cons1 [simp]: "norm_unsigned (𝟭#bs) = 𝟭#bs"

lemma norm_unsigned_idem [simp]: "norm_unsigned (norm_unsigned w) = norm_unsigned w"
by (rule bit_list_induct [of _ w],simp_all)

fun
nat_to_bv_helper :: "nat => bit list => bit list"
where
"nat_to_bv_helper n bs = (if n = 0 then bs
else nat_to_bv_helper (n div 2) ((if n mod 2 = 0 then 𝟬 else 𝟭)#bs))"

definition
nat_to_bv :: "nat => bit list" where
"nat_to_bv n = nat_to_bv_helper n []"

lemma nat_to_bv0 [simp]: "nat_to_bv 0 = []"

lemmas [simp del] = nat_to_bv_helper.simps

lemma n_div_2_cases:
assumes zero: "(n::nat) = 0 ==> R"
and     div : "[| n div 2 < n ; 0 < n |] ==> R"
shows         "R"
proof (cases "n = 0")
assume "n = 0"
thus R by (rule zero)
next
assume "n ~= 0"
hence "0 < n" by simp
hence "n div 2 < n" by arith
from this and ‹0 < n› show R by (rule div)
qed

lemma int_wf_ge_induct:
assumes ind :  "!!i::int. (!!j. [| k ≤ j ; j < i |] ==> P j) ==> P i"
shows          "P i"
proof (rule wf_induct_rule [OF wf_int_ge_less_than])
fix x
assume ih: "(⋀y::int. (y, x) ∈ int_ge_less_than k ⟹ P y)"
thus "P x"
by (rule ind) (simp add: int_ge_less_than_def)
qed

lemma unfold_nat_to_bv_helper:
"nat_to_bv_helper b l = nat_to_bv_helper b [] @ l"
proof -
have "∀l. nat_to_bv_helper b l = nat_to_bv_helper b [] @ l"
proof (induct b rule: less_induct)
fix n
assume ind: "!!j. j < n ⟹ ∀ l. nat_to_bv_helper j l = nat_to_bv_helper j [] @ l"
show "∀l. nat_to_bv_helper n l = nat_to_bv_helper n [] @ l"
proof
fix l
show "nat_to_bv_helper n l = nat_to_bv_helper n [] @ l"
proof (cases "n < 0")
assume "n < 0"
thus ?thesis
next
assume "~n < 0"
show ?thesis
proof (rule n_div_2_cases [of n])
assume [simp]: "n = 0"
show ?thesis
apply (simp only: nat_to_bv_helper.simps [of n])
apply simp
done
next
assume n2n: "n div 2 < n"
assume [simp]: "0 < n"
hence n20: "0 ≤ n div 2"
by arith
from ind [of "n div 2"] and n2n n20
have ind': "∀l. nat_to_bv_helper (n div 2) l = nat_to_bv_helper (n div 2) [] @ l"
by blast
show ?thesis
apply (simp only: nat_to_bv_helper.simps [of n])
apply (cases "n=0")
apply simp
apply (simp only: if_False)
apply simp
apply (subst spec [OF ind',of "𝟬#l"])
apply (subst spec [OF ind',of "𝟭#l"])
apply (subst spec [OF ind',of "[𝟭]"])
apply (subst spec [OF ind',of "[𝟬]"])
apply simp
done
qed
qed
qed
qed
thus ?thesis ..
qed

lemma nat_to_bv_non0 [simp]: "n≠0 ==> nat_to_bv n = nat_to_bv (n div 2) @ [if n mod 2 = 0 then 𝟬 else 𝟭]"
proof -
assume n: "n≠0"
show ?thesis
apply (subst nat_to_bv_def [of n])
apply (simp only: nat_to_bv_helper.simps [of n])
apply (subst unfold_nat_to_bv_helper)
apply (subst nat_to_bv_def [of "n div 2"])
apply auto
done
qed

lemma bv_to_nat_dist_append:
"bv_to_nat (l1 @ l2) = bv_to_nat l1 * 2 ^ length l2 + bv_to_nat l2"
proof -
have "∀l2. bv_to_nat (l1 @ l2) = bv_to_nat l1 * 2 ^ length l2 + bv_to_nat l2"
proof (induct l1, simp_all)
fix x xs
assume ind: "∀l2. bv_to_nat (xs @ l2) = bv_to_nat xs * 2 ^ length l2 + bv_to_nat l2"
show "∀l2::bit list. bitval x * 2 ^ (length xs + length l2) + bv_to_nat xs * 2 ^ length l2 = (bitval x * 2 ^ length xs + bv_to_nat xs) * 2 ^ length l2"
proof
fix l2
show "bitval x * 2 ^ (length xs + length l2) + bv_to_nat xs * 2 ^ length l2 = (bitval x * 2 ^ length xs + bv_to_nat xs) * 2 ^ length l2"
proof -
have "(2::nat) ^ (length xs + length l2) = 2 ^ length xs * 2 ^ length l2"
by (induct ("length xs")) simp_all
hence "bitval x * 2 ^ (length xs + length l2) + bv_to_nat xs * 2 ^ length l2 =
bitval x * 2 ^ length xs * 2 ^ length l2 + bv_to_nat xs * 2 ^ length l2"
by simp
also have "... = (bitval x * 2 ^ length xs + bv_to_nat xs) * 2 ^ length l2"
finally show ?thesis by simp
qed
qed
qed
thus ?thesis ..
qed

lemma bv_nat_bv [simp]: "bv_to_nat (nat_to_bv n) = n"
proof (induct n rule: less_induct)
fix n
assume ind: "!!j. j < n ⟹ bv_to_nat (nat_to_bv j) = j"
show "bv_to_nat (nat_to_bv n) = n"
proof (rule n_div_2_cases [of n])
assume "n = 0" then show ?thesis by simp
next
assume nn: "n div 2 < n"
assume n0: "0 < n"
from ind and nn
have ind': "bv_to_nat (nat_to_bv (n div 2)) = n div 2" by blast
from n0 have n0': "n ≠ 0" by simp
show ?thesis
apply (subst nat_to_bv_def)
apply (simp only: nat_to_bv_helper.simps [of n])
apply (simp only: n0' if_False)
apply (subst unfold_nat_to_bv_helper)
apply (subst bv_to_nat_dist_append)
apply (fold nat_to_bv_def)
apply (simp add: ind' split del: if_split)
apply (cases "n mod 2 = 0")
proof (simp_all)
assume "n mod 2 = 0"
with div_mult_mod_eq [of n 2]
show "n div 2 * 2 = n" by simp
next
assume "n mod 2 = Suc 0"
with div_mult_mod_eq [of n 2]
show "Suc (n div 2 * 2) = n" by arith
qed
qed
qed

lemma bv_to_nat_type [simp]: "bv_to_nat (norm_unsigned w) = bv_to_nat w"
by (rule bit_list_induct) simp_all

lemma length_norm_unsigned_le [simp]: "length (norm_unsigned w) ≤ length w"
by (rule bit_list_induct) simp_all

lemma bv_to_nat_rew_msb: "bv_msb w = 𝟭 ==> bv_to_nat w = 2 ^ (length w - 1) + bv_to_nat (tl w)"
by (rule bit_list_cases [of w]) simp_all

lemma norm_unsigned_result: "norm_unsigned xs = [] ∨ bv_msb (norm_unsigned xs) = 𝟭"
proof (rule length_induct [of _ xs])
fix xs :: "bit list"
assume ind: "∀ys. length ys < length xs --> norm_unsigned ys = [] ∨ bv_msb (norm_unsigned ys) = 𝟭"
show "norm_unsigned xs = [] ∨ bv_msb (norm_unsigned xs) = 𝟭"
proof (rule bit_list_cases [of xs],simp_all)
fix bs
assume [simp]: "xs = 𝟬#bs"
from ind
have "length bs < length xs --> norm_unsigned bs = [] ∨ bv_msb (norm_unsigned bs) = 𝟭" ..
thus "norm_unsigned bs = [] ∨ bv_msb (norm_unsigned bs) = 𝟭" by simp
qed
qed

lemma norm_empty_bv_to_nat_zero:
assumes nw: "norm_unsigned w = []"
shows       "bv_to_nat w = 0"
proof -
have "bv_to_nat w = bv_to_nat (norm_unsigned w)" by simp
also have "... = bv_to_nat []" by (subst nw) (rule refl)
also have "... = 0" by simp
finally show ?thesis .
qed

lemma bv_to_nat_lower_limit:
assumes w0: "0 < bv_to_nat w"
shows "2 ^ (length (norm_unsigned w) - 1) ≤ bv_to_nat w"
proof -
from w0 and norm_unsigned_result [of w]
have msbw: "bv_msb (norm_unsigned w) = 𝟭"
have "2 ^ (length (norm_unsigned w) - 1) ≤ bv_to_nat (norm_unsigned w)"
by (subst bv_to_nat_rew_msb [OF msbw],simp)
thus ?thesis by simp
qed

lemmas [simp del] = nat_to_bv_non0

lemma norm_unsigned_length [intro!]: "length (norm_unsigned w) ≤ length w"
by (subst norm_unsigned_def,rule rem_initial_length)

lemma norm_unsigned_equal:
"length (norm_unsigned w) = length w ==> norm_unsigned w = w"

lemma bv_extend_norm_unsigned: "bv_extend (length w) 𝟬 (norm_unsigned w) = w"

lemma norm_unsigned_append1 [simp]:
"norm_unsigned xs ≠ [] ==> norm_unsigned (xs @ ys) = norm_unsigned xs @ ys"

lemma norm_unsigned_append2 [simp]:
"norm_unsigned xs = [] ==> norm_unsigned (xs @ ys) = norm_unsigned ys"

lemma bv_to_nat_zero_imp_empty:
"bv_to_nat w = 0 ⟹ norm_unsigned w = []"
by (atomize (full), induct w rule: bit_list_induct) simp_all

lemma bv_to_nat_nzero_imp_nempty:
"bv_to_nat w ≠ 0 ⟹ norm_unsigned w ≠ []"
by (induct w rule: bit_list_induct) simp_all

lemma nat_helper1:
assumes ass: "nat_to_bv (bv_to_nat w) = norm_unsigned w"
shows        "nat_to_bv (2 * bv_to_nat w + bitval x) = norm_unsigned (w @ [x])"
proof (cases x)
assume [simp]: "x = 𝟭"
have "nat_to_bv (Suc (2 * bv_to_nat w) div 2) @ [𝟭] =
nat_to_bv ((1 + 2 * bv_to_nat w) div 2) @ [𝟭]"
also have "... = nat_to_bv (bv_to_nat w) @ [𝟭]"
also have "... = norm_unsigned w @ [𝟭]"
by (subst ass) (rule refl)
also have "... = norm_unsigned (w @ [𝟭])"
by (cases "norm_unsigned w") simp_all
finally have "nat_to_bv (Suc (2 * bv_to_nat w) div 2) @ [𝟭] = norm_unsigned (w @ [𝟭])" .
then show ?thesis by (simp add: nat_to_bv_non0)
next
assume [simp]: "x = 𝟬"
show ?thesis
proof (cases "bv_to_nat w = 0")
assume "bv_to_nat w = 0"
thus ?thesis
next
assume "bv_to_nat w ≠ 0"
thus ?thesis
apply simp
apply (subst nat_to_bv_non0)
apply simp
apply auto
apply (subst ass)
apply (cases "norm_unsigned w")
done
qed
qed

lemma nat_helper2: "nat_to_bv (2 ^ length xs + bv_to_nat xs) = 𝟭 # xs"
proof -
have "∀xs. nat_to_bv (2 ^ length (rev xs) + bv_to_nat (rev xs)) = 𝟭 # (rev xs)" (is "∀xs. ?P xs")
proof
fix xs
show "?P xs"
proof (rule length_induct [of _ xs])
fix xs :: "bit list"
assume ind: "∀ys. length ys < length xs --> ?P ys"
show "?P xs"
proof (cases xs)
assume "xs = []"
then show ?thesis by (simp add: nat_to_bv_non0)
next
fix y ys
assume [simp]: "xs = y # ys"
show ?thesis
apply simp
apply (subst bv_to_nat_dist_append)
apply simp
proof -
have "nat_to_bv (2 * 2 ^ length ys + (bv_to_nat (rev ys) * 2 + bitval y)) =
nat_to_bv (2 * (2 ^ length ys + bv_to_nat (rev ys)) + bitval y)"
also have "... = nat_to_bv (2 * (bv_to_nat (𝟭#rev ys)) + bitval y)"
by simp
also have "... = norm_unsigned (𝟭#rev ys) @ [y]"
proof -
from ind
have "nat_to_bv (2 ^ length (rev ys) + bv_to_nat (rev ys)) = 𝟭 # rev ys"
by auto
hence [simp]: "nat_to_bv (2 ^ length ys + bv_to_nat (rev ys)) = 𝟭 # rev ys"
by simp
show ?thesis
apply (subst nat_helper1)
apply simp_all
done
qed
also have "... = (𝟭#rev ys) @ [y]"
by simp
also have "... = 𝟭 # rev ys @ [y]"
by simp
finally show "nat_to_bv (2 * 2 ^ length ys + (bv_to_nat (rev ys) * 2 + bitval y)) =
𝟭 # rev ys @ [y]" .
qed
qed
qed
qed
hence "nat_to_bv (2 ^ length (rev (rev xs)) + bv_to_nat (rev (rev xs))) =
𝟭 # rev (rev xs)" ..
thus ?thesis by simp
qed

lemma nat_bv_nat [simp]: "nat_to_bv (bv_to_nat w) = norm_unsigned w"
proof (rule bit_list_induct [of _ w],simp_all)
fix xs
assume "nat_to_bv (bv_to_nat xs) = norm_unsigned xs"
have "bv_to_nat xs = bv_to_nat (norm_unsigned xs)" by simp
have "bv_to_nat xs < 2 ^ length xs"
by (rule bv_to_nat_upper_range)
show "nat_to_bv (2 ^ length xs + bv_to_nat xs) = 𝟭 # xs"
by (rule nat_helper2)
qed

lemma bv_to_nat_qinj:
assumes one: "bv_to_nat xs = bv_to_nat ys"
and     len: "length xs = length ys"
shows        "xs = ys"
proof -
from one
have "nat_to_bv (bv_to_nat xs) = nat_to_bv (bv_to_nat ys)"
by simp
hence xsys: "norm_unsigned xs = norm_unsigned ys"
by simp
have "xs = bv_extend (length xs) 𝟬 (norm_unsigned xs)"
also have "... = bv_extend (length ys) 𝟬 (norm_unsigned ys)"
also have "... = ys"
finally show ?thesis .
qed

lemma norm_unsigned_nat_to_bv [simp]:
"norm_unsigned (nat_to_bv n) = nat_to_bv n"
proof -
have "norm_unsigned (nat_to_bv n) = nat_to_bv (bv_to_nat (norm_unsigned (nat_to_bv n)))"
by (subst nat_bv_nat) simp
also have "... = nat_to_bv n" by simp
finally show ?thesis .
qed

lemma length_nat_to_bv_upper_limit:
assumes nk: "n ≤ 2 ^ k - 1"
shows       "length (nat_to_bv n) ≤ k"
proof (cases "n = 0")
case True
thus ?thesis
next
case False
hence n0: "0 < n" by simp
show ?thesis
proof (rule ccontr)
assume "~ length (nat_to_bv n) ≤ k"
hence "k < length (nat_to_bv n)" by simp
hence "k ≤ length (nat_to_bv n) - 1" by arith
hence "(2::nat) ^ k ≤ 2 ^ (length (nat_to_bv n) - 1)" by simp
also have "... = 2 ^ (length (norm_unsigned (nat_to_bv n)) - 1)" by simp
also have "... ≤ bv_to_nat (nat_to_bv n)"
by (rule bv_to_nat_lower_limit) (simp add: n0)
also have "... = n" by simp
finally have "2 ^ k ≤ n" .
with n0 have "2 ^ k - 1 < n" by arith
with nk show False by simp
qed
qed

lemma length_nat_to_bv_lower_limit:
assumes nk: "2 ^ k ≤ n"
shows       "k < length (nat_to_bv n)"
proof (rule ccontr)
assume "~ k < length (nat_to_bv n)"
hence lnk: "length (nat_to_bv n) ≤ k" by simp
have "n = bv_to_nat (nat_to_bv n)" by simp
also have "... < 2 ^ length (nat_to_bv n)"
by (rule bv_to_nat_upper_range)
also from lnk have "... ≤ 2 ^ k" by simp
finally have "n < 2 ^ k" .
with nk show False by simp
qed

subsection ‹Unsigned Arithmetic Operations›

definition
bv_add :: "[bit list, bit list ] => bit list" where
"bv_add w1 w2 = nat_to_bv (bv_to_nat w1 + bv_to_nat w2)"

lemma bv_add_length: "length (bv_add w1 w2) ≤ Suc (max (length w1) (length w2))"
from bv_to_nat_upper_range [of w1] and bv_to_nat_upper_range [of w2]
have "bv_to_nat w1 + bv_to_nat w2 ≤ (2 ^ length w1 - 1) + (2 ^ length w2 - 1)"
by arith
also have "... ≤
max (2 ^ length w1 - 1) (2 ^ length w2 - 1) + max (2 ^ length w1 - 1) (2 ^ length w2 - 1)"
by (rule add_mono,safe intro!: max.cobounded1 max.cobounded2)
also have "... = 2 * max (2 ^ length w1 - 1) (2 ^ length w2 - 1)" by simp
also have "... ≤ 2 ^ Suc (max (length w1) (length w2)) - 2"
proof (cases "length w1 ≤ length w2")
assume w1w2: "length w1 ≤ length w2"
hence "(2::nat) ^ length w1 ≤ 2 ^ length w2" by simp
hence "(2::nat) ^ length w1 - 1 ≤ 2 ^ length w2 - 1" by arith
with w1w2 show ?thesis
by (simp add: diff_mult_distrib2 split: split_max)
next
assume [simp]: "~ (length w1 ≤ length w2)"
have "~ ((2::nat) ^ length w1 - 1 ≤ 2 ^ length w2 - 1)"
proof
assume "(2::nat) ^ length w1 - 1 ≤ 2 ^ length w2 - 1"
hence "((2::nat) ^ length w1 - 1) + 1 ≤ (2 ^ length w2 - 1) + 1"
hence "(2::nat) ^ length w1 ≤ 2 ^ length w2" by simp
hence "length w1 ≤ length w2" by simp
thus False by simp
qed
thus ?thesis
by (simp add: diff_mult_distrib2 split: split_max)
qed
finally show "bv_to_nat w1 + bv_to_nat w2 ≤ 2 ^ Suc (max (length w1) (length w2)) - 1"
by arith
qed

definition
bv_mult :: "[bit list, bit list ] => bit list" where
"bv_mult w1 w2 = nat_to_bv (bv_to_nat w1 * bv_to_nat w2)"

lemma bv_mult_type1 [simp]: "bv_mult (norm_unsigned w1) w2 = bv_mult w1 w2"

lemma bv_mult_type2 [simp]: "bv_mult w1 (norm_unsigned w2) = bv_mult w1 w2"

lemma bv_mult_returntype [simp]: "norm_unsigned (bv_mult w1 w2) = bv_mult w1 w2"

lemma bv_mult_length: "length (bv_mult w1 w2) ≤ length w1 + length w2"
proof (unfold bv_mult_def,rule length_nat_to_bv_upper_limit)
from bv_to_nat_upper_range [of w1] and bv_to_nat_upper_range [of w2]
have h: "bv_to_nat w1 ≤ 2 ^ length w1 - 1 ∧ bv_to_nat w2 ≤ 2 ^ length w2 - 1"
by arith
have "bv_to_nat w1 * bv_to_nat w2 ≤ (2 ^ length w1 - 1) * (2 ^ length w2 - 1)"
apply (cut_tac h)
apply (rule mult_mono)
apply auto
done
also have "... < 2 ^ length w1 * 2 ^ length w2"
by (rule mult_strict_mono,auto)
also have "... = 2 ^ (length w1 + length w2)"
finally show "bv_to_nat w1 * bv_to_nat w2 ≤ 2 ^ (length w1 + length w2) - 1"
by arith
qed

subsection ‹Signed Vectors›

primrec norm_signed :: "bit list => bit list" where
norm_signed_Nil: "norm_signed [] = []"
| norm_signed_Cons: "norm_signed (b#bs) =
(case b of
𝟬 => if norm_unsigned bs = [] then [] else b#norm_unsigned bs
| 𝟭 => b#rem_initial b bs)"

lemma norm_signed0 [simp]: "norm_signed [𝟬] = []"
by simp

lemma norm_signed1 [simp]: "norm_signed [𝟭] = [𝟭]"
by simp

lemma norm_signed01 [simp]: "norm_signed (𝟬#𝟭#xs) = 𝟬#𝟭#xs"
by simp

lemma norm_signed00 [simp]: "norm_signed (𝟬#𝟬#xs) = norm_signed (𝟬#xs)"
by simp

lemma norm_signed10 [simp]: "norm_signed (𝟭#𝟬#xs) = 𝟭#𝟬#xs"
by simp

lemma norm_signed11 [simp]: "norm_signed (𝟭#𝟭#xs) = norm_signed (𝟭#xs)"
by simp

lemmas [simp del] = norm_signed_Cons

definition
int_to_bv :: "int => bit list" where
"int_to_bv n = (if 0 ≤ n
then norm_signed (𝟬#nat_to_bv (nat n))
else norm_signed (bv_not (𝟬#nat_to_bv (nat (-n- 1)))))"

lemma int_to_bv_ge0 [simp]: "0 ≤ n ==> int_to_bv n = norm_signed (𝟬 # nat_to_bv (nat n))"

lemma int_to_bv_lt0 [simp]:
"n < 0 ==> int_to_bv n = norm_signed (bv_not (𝟬#nat_to_bv (nat (-n- 1))))"

lemma norm_signed_idem [simp]: "norm_signed (norm_signed w) = norm_signed w"
proof (rule bit_list_induct [of _ w], simp_all)
fix xs
assume eq: "norm_signed (norm_signed xs) = norm_signed xs"
show "norm_signed (norm_signed (𝟬#xs)) = norm_signed (𝟬#xs)"
proof (rule bit_list_cases [of xs],simp_all)
fix ys
assume "xs = 𝟬#ys"
from this [symmetric] and eq
show "norm_signed (norm_signed (𝟬#ys)) = norm_signed (𝟬#ys)"
by simp
qed
next
fix xs
assume eq: "norm_signed (norm_signed xs) = norm_signed xs"
show "norm_signed (norm_signed (𝟭#xs)) = norm_signed (𝟭#xs)"
proof (rule bit_list_cases [of xs],simp_all)
fix ys
assume "xs = 𝟭#ys"
from this [symmetric] and eq
show "norm_signed (norm_signed (𝟭#ys)) = norm_signed (𝟭#ys)"
by simp
qed
qed

definition
bv_to_int :: "bit list => int" where
"bv_to_int w =
(case bv_msb w of 𝟬 => int (bv_to_nat w)
| 𝟭 => - int (bv_to_nat (bv_not w) + 1))"

lemma bv_to_int_Nil [simp]: "bv_to_int [] = 0"

lemma bv_to_int_Cons0 [simp]: "bv_to_int (𝟬#bs) = int (bv_to_nat bs)"

lemma bv_to_int_Cons1 [simp]: "bv_to_int (𝟭#bs) = - int (bv_to_nat (bv_not bs) + 1)"

lemma bv_to_int_type [simp]: "bv_to_int (norm_signed w) = bv_to_int w"
proof (rule bit_list_induct [of _ w], simp_all)
fix xs
assume ind: "bv_to_int (norm_signed xs) = bv_to_int xs"
show "bv_to_int (norm_signed (𝟬#xs)) = int (bv_to_nat xs)"
proof (rule bit_list_cases [of xs], simp_all)
fix ys
assume [simp]: "xs = 𝟬#ys"
from ind
show "bv_to_int (norm_signed (𝟬#ys)) = int (bv_to_nat ys)"
by simp
qed
next
fix xs
assume ind: "bv_to_int (norm_signed xs) = bv_to_int xs"
show "bv_to_int (norm_signed (𝟭#xs)) = -1 - int (bv_to_nat (bv_not xs))"
proof (rule bit_list_cases [of xs], simp_all)
fix ys
assume [simp]: "xs = 𝟭#ys"
from ind
show "bv_to_int (norm_signed (𝟭#ys)) = -1 - int (bv_to_nat (bv_not ys))"
by simp
qed
qed

lemma bv_to_int_upper_range: "bv_to_int w < 2 ^ (length w - 1)"
proof (rule bit_list_cases [of w],simp_all add: bv_to_nat_upper_range)
fix bs
have "-1 - int (bv_to_nat (bv_not bs)) ≤ 0" by simp
also have "... < 2 ^ length bs" by (induct bs) simp_all
finally show "-1 - int (bv_to_nat (bv_not bs)) < 2 ^ length bs" .
qed

lemma bv_to_int_lower_range: "- (2 ^ (length w - 1)) ≤ bv_to_int w"
proof (rule bit_list_cases [of w],simp_all)
fix bs :: "bit list"
have "- (2 ^ length bs) ≤ (0::int)" by (induct bs) simp_all
also have "... ≤ int (bv_to_nat bs)" by simp
finally show "- (2 ^ length bs) ≤ int (bv_to_nat bs)" .
next
fix bs
from bv_to_nat_upper_range [of "bv_not bs"]
show "- (2 ^ length bs) ≤ -1 - int (bv_to_nat (bv_not bs))"
qed

lemma int_bv_int [simp]: "int_to_bv (bv_to_int w) = norm_signed w"
proof (rule bit_list_cases [of w],simp)
fix xs
assume [simp]: "w = 𝟬#xs"
show ?thesis
apply simp
apply (subst norm_signed_Cons [of "𝟬" "xs"])
apply simp
using norm_unsigned_result [of xs]
apply safe
apply (rule bit_list_cases [of "norm_unsigned xs"])
apply simp_all
done
next
fix xs
assume [simp]: "w = 𝟭#xs"
show ?thesis
apply (simp del: int_to_bv_lt0)
apply (rule bit_list_induct [of _ xs], simp)
apply (subst int_to_bv_lt0)
apply linarith
apply simp
apply (metis add.commute bitnot_zero bv_not_Cons bv_not_bv_not int_nat_two_exp length_bv_not nat_helper2 nat_int norm_signed10 of_nat_add)
apply simp
done
qed

lemma bv_int_bv [simp]: "bv_to_int (int_to_bv i) = i"
by (cases "0 ≤ i") simp_all

lemma bv_msb_norm [simp]: "bv_msb (norm_signed w) = bv_msb w"
by (rule bit_list_cases [of w]) (simp_all add: norm_signed_Cons)

lemma norm_signed_length: "length (norm_signed w) ≤ length w"
apply (cases w, simp_all)
apply (subst norm_signed_Cons)
apply (case_tac a, simp_all)
apply (rule rem_initial_length)
done

lemma norm_signed_equal: "length (norm_signed w) = length w ==> norm_signed w = w"
proof (rule bit_list_cases [of w], simp_all)
fix xs
assume "length (norm_signed (𝟬#xs)) = Suc (length xs)"
thus "norm_signed (𝟬#xs) = 𝟬#xs"
by (simp add: norm_signed_Cons norm_unsigned_equal [THEN eqTrueI]
split: if_split_asm)
next
fix xs
assume "length (norm_signed (𝟭#xs)) = Suc (length xs)"
thus "norm_signed (𝟭#xs) = 𝟭#xs"
apply (rule rem_initial_equal)
apply assumption
done
qed

lemma bv_extend_norm_signed: "bv_msb w = b ==> bv_extend (length w) b (norm_signed w) = w"
proof (rule bit_list_cases [of w],simp_all)
fix xs
show "bv_extend (Suc (length xs)) 𝟬 (norm_signed (𝟬#xs)) = 𝟬#xs"
assume "norm_unsigned xs = []"
hence xx: "rem_initial 𝟬 xs = []"
have "bv_extend (Suc (length xs)) 𝟬 (𝟬#rem_initial 𝟬 xs) = 𝟬#xs"
apply (fold bv_extend_def)
apply (rule bv_extend_rem_initial)
done
thus "bv_extend (Suc (length xs)) 𝟬 [𝟬] = 𝟬#xs"
next
show "bv_extend (Suc (length xs)) 𝟬 (𝟬#norm_unsigned xs) = 𝟬#xs"
apply (fold bv_extend_def)
apply (rule bv_extend_rem_initial)
done
qed
next
fix xs
show "bv_extend (Suc (length xs)) 𝟭 (norm_signed (𝟭#xs)) = 𝟭#xs"
apply (fold bv_extend_def)
apply (rule bv_extend_rem_initial)
done
qed

lemma bv_to_int_qinj:
assumes one: "bv_to_int xs = bv_to_int ys"
and     len: "length xs = length ys"
shows        "xs = ys"
proof -
from one
have "int_to_bv (bv_to_int xs) = int_to_bv (bv_to_int ys)" by simp
hence xsys: "norm_signed xs = norm_signed ys" by simp
hence xsys': "bv_msb xs = bv_msb ys"
proof -
have "bv_msb xs = bv_msb (norm_signed xs)" by simp
also have "... = bv_msb (norm_signed ys)" by (simp add: xsys)
also have "... = bv_msb ys" by simp
finally show ?thesis .
qed
have "xs = bv_extend (length xs) (bv_msb xs) (norm_signed xs)"
also have "... = bv_extend (length ys) (bv_msb ys) (norm_signed ys)"
by (simp add: xsys xsys' len)
also have "... = ys"
finally show ?thesis .
qed

lemma int_to_bv_returntype [simp]: "norm_signed (int_to_bv w) = int_to_bv w"

lemma bv_to_int_msb0: "0 ≤ bv_to_int w1 ==> bv_msb w1 = 𝟬"
by (rule bit_list_cases,simp_all)

lemma bv_to_int_msb1: "bv_to_int w1 < 0 ==> bv_msb w1 = 𝟭"
by (rule bit_list_cases,simp_all)

lemma bv_to_int_lower_limit_gt0:
assumes w0: "0 < bv_to_int w"
shows       "2 ^ (length (norm_signed w) - 2) ≤ bv_to_int w"
proof -
from w0
have "0 ≤ bv_to_int w" by simp
hence [simp]: "bv_msb w = 𝟬" by (rule bv_to_int_msb0)
have "2 ^ (length (norm_signed w) - 2) ≤ bv_to_int (norm_signed w)"
proof (rule bit_list_cases [of w])
assume "w = []"
with w0 show ?thesis by simp
next
fix w'
assume weq: "w = 𝟬 # w'"
thus ?thesis
assume "norm_unsigned w' = []"
with weq and w0 show False
next
assume w'0: "norm_unsigned w' ≠ []"
have "0 < bv_to_nat w'"
proof (rule ccontr)
assume "~ (0 < bv_to_nat w')"
hence "bv_to_nat w' = 0"
by arith
hence "norm_unsigned w' = []"
with w'0
show False by simp
qed
with bv_to_nat_lower_limit [of w']
show "2 ^ (length (norm_unsigned w') - Suc 0) ≤ bv_to_nat w'"
using One_nat_def int_nat_two_exp by presburger
qed
next
fix w'
assume weq: "w = 𝟭 # w'"
from w0 have "bv_msb w = 𝟬" by simp
with weq show ?thesis by simp
qed
also have "...  = bv_to_int w" by simp
finally show ?thesis .
qed

lemma norm_signed_result: "norm_signed w = [] ∨ norm_signed w = [𝟭] ∨ bv_msb (norm_signed w) ≠ bv_msb (tl (norm_signed w))"
apply (rule bit_list_cases [of w],simp_all)
apply (case_tac "bs",simp_all)
apply (case_tac "a",simp_all)
apply safe
apply simp
proof -
fix l
assume msb: "𝟬 = bv_msb (norm_unsigned l)"
assume "norm_unsigned l ≠ []"
with norm_unsigned_result [of l]
have "bv_msb (norm_unsigned l) = 𝟭" by simp
with msb show False by simp
next
fix xs
assume p: "𝟭 = bv_msb (tl (norm_signed (𝟭 # xs)))"
have "𝟭 ≠ bv_msb (tl (norm_signed (𝟭 # xs)))"
by (rule bit_list_induct [of _ xs],simp_all)
with p show False by simp
qed

lemma bv_to_int_upper_limit_lem1:
assumes w0: "bv_to_int w < -1"
shows       "bv_to_int w < - (2 ^ (length (norm_signed w) - 2))"
proof -
from w0
have "bv_to_int w < 0" by simp
hence msbw [simp]: "bv_msb w = 𝟭"
by (rule bv_to_int_msb1)
have "bv_to_int w = bv_to_int (norm_signed w)" by simp
also from norm_signed_result [of w]
have "... < - (2 ^ (length (norm_signed w) - 2))"
proof safe
assume "norm_signed w = []"
hence "bv_to_int (norm_signed w) = 0" by simp
with w0 show ?thesis by simp
next
assume "norm_signed w = [𝟭]"
hence "bv_to_int (norm_signed w) = -1" by simp
with w0 show ?thesis by simp
next
assume "bv_msb (norm_signed w) ≠ bv_msb (tl (norm_signed w))"
hence msb_tl: "𝟭 ≠ bv_msb (tl (norm_signed w))" by simp
show "bv_to_int (norm_signed w) < - (2 ^ (length (norm_signed w) - 2))"
proof (rule bit_list_cases [of "norm_signed w"])
assume "norm_signed w = []"
hence "bv_to_int (norm_signed w) = 0" by simp
with w0 show ?thesis by simp
next
fix w'
assume nw: "norm_signed w = 𝟬 # w'"
from msbw have "bv_msb (norm_signed w) = 𝟭" by simp
with nw show ?thesis by simp
next
fix w'
assume weq: "norm_signed w = 𝟭 # w'"
show ?thesis
proof (rule bit_list_cases [of w'])
assume w'eq: "w' = []"
from w0 have "bv_to_int (norm_signed w) < -1" by simp
with w'eq and weq show ?thesis by simp
next
fix w''
assume w'eq: "w' = 𝟬 # w''"
show ?thesis
next
fix w''
assume w'eq: "w' = 𝟭 # w''"
with weq and msb_tl show ?thesis by simp
qed
qed
qed
finally show ?thesis .
qed

lemma length_int_to_bv_upper_limit_gt0:
assumes w0: "0 < i"
and     wk: "i ≤ 2 ^ (k - 1) - 1"
shows       "length (int_to_bv i) ≤ k"
proof (rule ccontr)
from w0 wk
have k1: "1 < k"
by (cases "k - 1",simp_all)
assume "~ length (int_to_bv i) ≤ k"
hence "k < length (int_to_bv i)" by simp
hence "k ≤ length (int_to_bv i) - 1" by arith
hence a: "k - 1 ≤ length (int_to_bv i) - 2" by arith
hence "(2::int) ^ (k - 1) ≤ 2 ^ (length (int_to_bv i) - 2)" by simp
also have "... ≤ i"
proof -
have "2 ^ (length (norm_signed (int_to_bv i)) - 2) ≤ bv_to_int (int_to_bv i)"
proof (rule bv_to_int_lower_limit_gt0)
from w0 show "0 < bv_to_int (int_to_bv i)" by simp
qed
thus ?thesis by simp
qed
finally have "2 ^ (k - 1) ≤ i" .
with wk show False by simp
qed

lemma pos_length_pos:
assumes i0: "0 < bv_to_int w"
shows       "0 < length w"
proof -
from norm_signed_result [of w]
have "0 < length (norm_signed w)"
proof (auto)
assume ii: "norm_signed w = []"
have "bv_to_int (norm_signed w) = 0" by (subst ii) simp
hence "bv_to_int w = 0" by simp
with i0 show False by simp
next
assume ii: "norm_signed w = []"
assume jj: "bv_msb w ≠ 𝟬"
have "𝟬 = bv_msb (norm_signed w)"
by (subst ii) simp
also have "... ≠ 𝟬"
finally show False by simp
qed
also have "... ≤ length w"
by (rule norm_signed_length)
finally show ?thesis .
qed

lemma neg_length_pos:
assumes i0: "bv_to_int w < -1"
shows       "0 < length w"
proof -
from norm_signed_result [of w]
have "0 < length (norm_signed w)"
proof (auto)
assume ii: "norm_signed w = []"
have "bv_to_int (norm_signed w) = 0"
by (subst ii) simp
hence "bv_to_int w = 0" by simp
with i0 show False by simp
next
assume ii: "norm_signed w = []"
assume jj: "bv_msb w ≠ 𝟬"
have "𝟬 = bv_msb (norm_signed w)" by (subst ii) simp
also have "... ≠ 𝟬" by (simp add: jj)
finally show False by simp
qed
also have "... ≤ length w"
by (rule norm_signed_length)
finally show ?thesis .
qed

lemma length_int_to_bv_lower_limit_gt0:
assumes wk: "2 ^ (k - 1) ≤ i"
shows       "k < length (int_to_bv i)"
proof (rule ccontr)
have "0 < (2::int) ^ (k - 1)"
by (rule zero_less_power) simp
also have "... ≤ i" by (rule wk)
finally have i0: "0 < i" .
have lii0: "0 < length (int_to_bv i)"
apply (rule pos_length_pos)
apply (simp,rule i0)
done
assume "~ k < length (int_to_bv i)"
hence "length (int_to_bv i) ≤ k" by simp
with lii0
have a: "length (int_to_bv i) - 1 ≤ k - 1"
by arith
have "i < 2 ^ (length (int_to_bv i) - 1)"
proof -
have "i = bv_to_int (int_to_bv i)"
by simp
also have "... < 2 ^ (length (int_to_bv i) - 1)"
by (rule bv_to_int_upper_range)
finally show ?thesis .
qed
also have "(2::int) ^ (length (int_to_bv i) - 1) ≤ 2 ^ (k - 1)" using a
by simp
finally have "i < 2 ^ (k - 1)" .
with wk show False by simp
qed

lemma length_int_to_bv_upper_limit_lem1:
assumes w1: "i < -1"
and     wk: "- (2 ^ (k - 1)) ≤ i"
shows       "length (int_to_bv i) ≤ k"
proof (rule ccontr)
from w1 wk
have k1: "1 < k" by (cases "k - 1") simp_all
assume "~ length (int_to_bv i) ≤ k"
hence "k < length (int_to_bv i)" by simp
hence "k ≤ length (int_to_bv i) - 1" by arith
hence a: "k - 1 ≤ length (int_to_bv i) - 2" by arith
have "i < - (2 ^ (length (int_to_bv i) - 2))"
proof -
have "i = bv_to_int (int_to_bv i)"
by simp
also have "... < - (2 ^ (length (norm_signed (int_to_bv i)) - 2))"
by (rule bv_to_int_upper_limit_lem1,simp,rule w1)
finally show ?thesis by simp
qed
also have "... ≤ -(2 ^ (k - 1))"
proof -
have "(2::int) ^ (k - 1) ≤ 2 ^ (length (int_to_bv i) - 2)" using a by simp
thus ?thesis by simp
qed
finally have "i < -(2 ^ (k - 1))" .
with wk show False by simp
qed

lemma length_int_to_bv_lower_limit_lem1:
assumes wk: "i < -(2 ^ (k - 1))"
shows       "k < length (int_to_bv i)"
proof (rule ccontr)
from wk have "i ≤ -(2 ^ (k - 1)) - 1" by simp
also have "... < -1"
proof -
have "0 < (2::int) ^ (k - 1)"
by (rule zero_less_power) simp
hence "-((2::int) ^ (k - 1)) < 0" by simp
thus ?thesis by simp
qed
finally have i1: "i < -1" .
have lii0: "0 < length (int_to_bv i)"
apply (rule neg_length_pos)
apply (simp, rule i1)
done
assume "~ k < length (int_to_bv i)"
hence "length (int_to_bv i) ≤ k"
by simp
with lii0 have a: "length (int_to_bv i) - 1 ≤ k - 1" by arith
hence "(2::int) ^ (length (int_to_bv i) - 1) ≤ 2 ^ (k - 1)" by simp
hence "-((2::int) ^ (k - 1)) ≤ - (2 ^ (length (int_to_bv i) - 1))" by simp
also have "... ≤ i"
proof -
have "- (2 ^ (length (int_to_bv i) - 1)) ≤ bv_to_int (int_to_bv i)"
by (rule bv_to_int_lower_range)
also have "... = i"
by simp
finally show ?thesis .
qed
finally have "-(2 ^ (k - 1)) ≤ i" .
with wk show False by simp
qed

subsection ‹Signed Arithmetic Operations›

subsubsection ‹Conversion from unsigned to signed›

definition
utos :: "bit list => bit list" where
"utos w = norm_signed (𝟬 # w)"

lemma utos_type [simp]: "utos (norm_unsigned w) = utos w"

lemma utos_returntype [simp]: "norm_signed (utos w) = utos w"

lemma utos_length: "length (utos w) ≤ Suc (length w)"

lemma bv_to_int_utos: "bv_to_int (utos w) = int (bv_to_nat w)"
proof (simp add: utos_def norm_signed_Cons, safe)
assume "norm_unsigned w = []"
hence "bv_to_nat (norm_unsigned w) = 0" by simp
thus "bv_to_nat w = 0" by simp
qed

subsubsection ‹Unary minus›

definition
bv_uminus :: "bit list => bit list" where
"bv_uminus w = int_to_bv (- bv_to_int w)"

lemma bv_uminus_type [simp]: "bv_uminus (norm_signed w) = bv_uminus w"

lemma bv_uminus_returntype [simp]: "norm_signed (bv_uminus w) = bv_uminus w"

lemma bv_uminus_length: "length (bv_uminus w) ≤ Suc (length w)"
proof -
have "1 < -bv_to_int w ∨ -bv_to_int w = 1 ∨ -bv_to_int w = 0 ∨ -bv_to_int w = -1 ∨ -bv_to_int w < -1"
by arith
thus ?thesis
proof safe
assume p: "1 < - bv_to_int w"
have lw: "0 < length w"
apply (rule neg_length_pos)
using p
apply simp
done
show ?thesis
from p show "bv_to_int w < 0" by simp
next
have "-(2^(length w - 1)) ≤ bv_to_int w"
by (rule bv_to_int_lower_range)
hence "- bv_to_int w ≤ 2^(length w - 1)" by simp
also from lw have "... < 2 ^ length w" by simp
finally show "- bv_to_int w < 2 ^ length w" by simp
qed
next
assume p: "- bv_to_int w = 1"
hence lw: "0 < length w" by (cases w) simp_all
from p
show ?thesis
using lw
done
next
assume "- bv_to_int w = 0"
thus ?thesis by (simp add: bv_uminus_def)
next
assume p: "- bv_to_int w = -1"
thus ?thesis by (simp add: bv_uminus_def)
next
assume p: "- bv_to_int w < -1"
show ?thesis
apply (rule length_int_to_bv_upper_limit_lem1)
apply (rule p)
apply simp
proof -
have "bv_to_int w < 2 ^ (length w - 1)"
by (rule bv_to_int_upper_range)
also have "... ≤ 2 ^ length w" by simp
finally show "bv_to_int w ≤ 2 ^ length w" by simp
qed
qed
qed

lemma bv_uminus_length_utos: "length (bv_uminus (utos w)) ≤ Suc (length w)"
proof -
have "-bv_to_int (utos w) = 0 ∨ -bv_to_int (utos w) = -1 ∨ -bv_to_int (utos w) < -1"
thus ?thesis
proof safe
assume "-bv_to_int (utos w) = 0"
thus ?thesis by (simp add: bv_uminus_def)
next
assume "-bv_to_int (utos w) = -1"
thus ?thesis by (simp add: bv_uminus_def)
next
assume p: "-bv_to_int (utos w) < -1"
show ?thesis
apply (rule length_int_to_bv_upper_limit_lem1)
apply (rule p)
using bv_to_nat_upper_range [of w] int_nat_two_exp apply presburger
done
qed
qed

definition
bv_sadd :: "[bit list, bit list ] => bit list" where
"bv_sadd w1 w2 = int_to_bv (bv_to_int w1 + bv_to_int w2)"

assumes lw: "0 < max (length w1) (length w2)"
shows   "((2::int) ^ (length w1 - 1)) + (2 ^ (length w2 - 1)) ≤ 2 ^ max (length w1) (length w2)"
proof -
have "((2::int) ^ (length w1 - 1)) + (2 ^ (length w2 - 1)) ≤
2 ^ (max (length w1) (length w2) - 1) + 2 ^ (max (length w1) (length w2) - 1)"
by (auto simp:max_def)
also have "... = 2 ^ max (length w1) (length w2)"
proof -
from lw
show ?thesis
apply simp
apply (subst power_Suc [symmetric])
apply simp
done
qed
finally show ?thesis .
qed

lemma bv_sadd_length: "length (bv_sadd w1 w2) ≤ Suc (max (length w1) (length w2))"
proof -
let ?Q = "bv_to_int w1 + bv_to_int w2"

have helper: "?Q ≠ 0 ==> 0 < max (length w1) (length w2)"
proof -
assume p: "?Q ≠ 0"
show "0 < max (length w1) (length w2)"
assume [simp]: "w1 = []"
show "w2 ≠ []"
proof (rule ccontr,simp)
assume [simp]: "w2 = []"
from p show False by simp
qed
qed
qed

have "0 < ?Q ∨ ?Q = 0 ∨ ?Q = -1 ∨ ?Q < -1" by arith
thus ?thesis
proof safe
assume "?Q = 0"
thus ?thesis
next
assume "?Q = -1"
thus ?thesis
next
assume p: "0 < ?Q"
show ?thesis
apply (rule length_int_to_bv_upper_limit_gt0)
apply (rule p)
proof simp
from bv_to_int_upper_range [of w2]
have "bv_to_int w2 ≤ 2 ^ (length w2 - 1)"
by simp
with bv_to_int_upper_range [of w1]
have "bv_to_int w1 + bv_to_int w2 < (2 ^ (length w1 - 1)) + (2 ^ (length w2 - 1))"
also have "... ≤ 2 ^ max (length w1) (length w2)"
apply (rule helper)
using p
apply simp
done
finally show "?Q < 2 ^ max (length w1) (length w2)" .
qed
next
assume p: "?Q < -1"
show ?thesis
apply (rule length_int_to_bv_upper_limit_lem1,simp_all)
apply (rule p)
proof -
have "(2 ^ (length w1 - 1)) + 2 ^ (length w2 - 1) ≤ (2::int) ^ max (length w1) (length w2)"
apply (rule helper)
using p
apply simp
done
hence "-((2::int) ^ max (length w1) (length w2)) ≤ - (2 ^ (length w1 - 1)) + -(2 ^ (length w2 - 1))"
by simp
also have "- (2 ^ (length w1 - 1)) + -(2 ^ (length w2 - 1)) ≤ ?Q"
apply (rule bv_to_int_lower_range [of w1])
apply (rule bv_to_int_lower_range [of w2])
done
finally show "- (2^max (length w1) (length w2)) ≤ ?Q" .
qed
qed
qed

definition
bv_sub :: "[bit list, bit list] => bit list" where
"bv_sub w1 w2 = bv_sadd w1 (bv_uminus w2)"

lemma bv_sub_type1 [simp]: "bv_sub (norm_signed w1) w2 = bv_sub w1 w2"

lemma bv_sub_type2 [simp]: "bv_sub w1 (norm_signed w2) = bv_sub w1 w2"

lemma bv_sub_returntype [simp]: "norm_signed (bv_sub w1 w2) = bv_sub w1 w2"

lemma bv_sub_length: "length (bv_sub w1 w2) ≤ Suc (max (length w1) (length w2))"
proof (cases "bv_to_int w2 = 0")
assume p: "bv_to_int w2 = 0"
show ?thesis
have "length (norm_signed w1) ≤ length w1"
by (rule norm_signed_length)
also have "... ≤ max (length w1) (length w2)"
by (rule max.cobounded1)
also have "... ≤ Suc (max (length w1) (length w2))"
by arith
finally show "length (norm_signed w1) ≤ Suc (max (length w1) (length w2))" .
qed
next
assume "bv_to_int w2 ≠ 0"
hence "0 < length w2" by (cases w2,simp_all)
hence lmw: "0 < max (length w1) (length w2)" by arith

let ?Q = "bv_to_int w1 - bv_to_int w2"

have "0 < ?Q ∨ ?Q = 0 ∨ ?Q = -1 ∨ ?Q < -1" by arith
thus ?thesis
proof safe
assume "?Q = 0"
thus ?thesis
next
assume "?Q = -1"
thus ?thesis
next
assume p: "0 < ?Q"
show ?thesis
apply (rule length_int_to_bv_upper_limit_gt0)
apply (rule p)
proof simp
from bv_to_int_lower_range [of w2]
have v2: "- bv_to_int w2 ≤ 2 ^ (length w2 - 1)" by simp
have "bv_to_int w1 + - bv_to_int w2 < (2 ^ (length w1 - 1)) + (2 ^ (length w2 - 1))"
apply (rule bv_to_int_upper_range [of w1])
apply (rule v2)
done
also have "... ≤ 2 ^ max (length w1) (length w2)"
apply (rule lmw)
done
finally show "?Q < 2 ^ max (length w1) (length w2)" by simp
qed
next
assume p: "?Q < -1"
show ?thesis