Theory AutoCorres2.Vanilla32
theory Vanilla32
imports "Word_Mem_Encoding" CTypes
begin
type_synonym exit_status = "32 sword"
instantiation word :: (len8) c_type
begin
definition "typ_name_itself (w::'a::len8 word itself) =
''word'' @ nat_to_bin_string (len_of TYPE('a))"
definition
typ_info_word: "typ_info_t (w::'a::len8 word itself) ≡ word_tag w"
instance by (intro_classes)
(simp add: typ_name_itself_word_def typ_info_word word_tag_def)
end
lemma align_of_word:
"align_of TYPE('a::len8 word) = len_of TYPE('a) div 8"
by (simp add: align_of_def typ_info_word word_tag_def len8_bytes)
instantiation word :: (len8) mem_type
begin
instance
proof intro_classes
show "⋀bs v w. length bs = size_of TYPE('a word) ⟶
update_ti_t (typ_info_t TYPE('a word)) bs v =
update_ti_t (typ_info_t TYPE('a word)) bs w"
by (simp add: typ_info_word word_tag_def size_of_def update_ti_t_def)
next
show "wf_desc (typ_info_t TYPE('a word))"
by (simp add: typ_info_word word_tag_def)
next
have "8 dvd len_of TYPE('a)" by (rule len8_dv8)
moreover have "0 < len_of TYPE('a)" by simp
ultimately have "0 < len_of TYPE('a) div 8" by - (erule dvdE, simp)
thus "wf_size_desc (typ_info_t TYPE('a word))"
by (simp add: typ_info_word word_tag_def len8_pow)
next
have "8 dvd len_of TYPE('a)" by (rule len8_dv8)
thus "wf_lf (lf_set (typ_info_t TYPE('a word)) [])"
by (auto simp: typ_info_word word_tag_def wf_lf_def
fd_cons_def fd_cons_double_update_def fd_cons_update_access_def
word_rcat_rsplit fd_cons_access_update_def fd_cons_length_def
length_word_rsplit_exp_size' word_size fd_cons_update_normalise_def
fd_cons_desc_def field_norm_def)
next
show "size_of TYPE('a word) < addr_card"
by (simp add: size_of_def typ_info_word word_tag_def)
(rule len8_size)
next
show "align_of TYPE('a word) dvd size_of TYPE('a word)"
by (clarsimp simp: size_of_def align_of_word typ_info_word
word_tag_def)
next
show "align_field (typ_info_t TYPE('a word))"
by (clarsimp simp: typ_info_word word_tag_def align_field_def)
next
show "wf_align (typ_info_t TYPE('a word))"
by (clarsimp simp: typ_info_word word_tag_def)
qed
end
instantiation word :: (len8) simple_mem_type
begin
instance
apply intro_classes
apply(clarsimp simp: typ_info_word word_tag_def typ_uinfo_t_def)
done
end
class len_eq1 = len +
assumes len_eq1: "len_of TYPE('a::len) = 1"
instance num1 :: len_eq1
by (intro_classes, simp)
class len_lg01 = len +
assumes len_lg01: "len_of TYPE('a::len) ∈ {1,2}"
instance bit0 :: (len_eq1) len_lg01
by (intro_classes, simp add: len_eq1)
instance num1 :: len_lg01
by (intro_classes, simp)
class len_lg02 = len +
assumes len_lg02: "len_of TYPE('a::len) ∈ {1,2,4}"
instance bit0 :: (len_lg01) len_lg02
apply intro_classes
apply (insert len_lg01[where 'a = 'a])
apply simp
done
instance num1 :: len_lg02
by (intro_classes, simp)
class len_lg03 = len +
assumes len_lg03: "len_of TYPE('a::len) ∈ {1,2,4,8}"
instance bit0 :: (len_lg02) len_lg03
apply intro_classes
apply (insert len_lg02[where 'a = 'a])
apply simp
done
instance num1 :: len_lg03
by (intro_classes, simp)
class len_lg04 = len +
assumes len_lg04: "len_of TYPE('a::len) ∈ {1,2,4,8,16}"
instance bit0 :: (len_lg03) len_lg04
apply intro_classes
apply (insert len_lg03[where 'a = 'a])
apply simp
done
instance num1 :: len_lg04
by (intro_classes, simp)
class len_lg15 = len +
assumes len_lg15: "len_of TYPE('a::len) ∈ {2,4,8,16,32}"
instance bit0 :: (len_lg04) len_lg15
apply intro_classes
apply (insert len_lg04[where 'a = 'a])
apply simp
done
class len_lg26 = len +
assumes len_lg26: "len_of TYPE('a::len) ∈ {4,8,16,32,64}"
instance bit0 :: (len_lg15) len_lg26
apply intro_classes
apply (insert len_lg15[where 'a = 'a])
apply simp
done
instance bit0 :: (len_lg26) len8
apply intro_classes
apply simp
apply (insert len_lg26[where 'a = 'a])
apply (simp add: len_exp_def)
apply (erule disjE)
apply (simp add: bogus_log2lessthree_def len8_bytes len8_width)
apply (erule disjE)
apply (simp add: bogus_log2lessthree_def)
apply (erule disjE)
apply (simp add: bogus_log2lessthree_def)
apply (erule disjE)
apply (simp add: bogus_log2lessthree_def)
apply (simp add: bogus_log2lessthree_def)
apply simp
apply auto
done
instance signed :: (len_eq1) len_eq1 using len_eq1 by (intro_classes, simp)
instance signed :: (len_lg01) len_lg01 using len_lg01 by (intro_classes, simp)
instance signed :: (len_lg02) len_lg02 using len_lg02 by (intro_classes, simp)
instance signed :: (len_lg03) len_lg03 using len_lg03 by (intro_classes, simp)
instance signed :: (len_lg04) len_lg04 using len_lg04 by (intro_classes, simp)
instance signed :: (len_lg15) len_lg15 using len_lg15 by (intro_classes, simp)
instance signed :: (len_lg26) len_lg26 using len_lg26 by (intro_classes, simp)
lemma
"to_bytes (1*256*256*256 + 2*256*256 + 3*256 + (4::32 word)) bs = [4, 3, 2, 1]"
by (simp add: to_bytes_def typ_info_word word_tag_def Let_def)
lemma size_td_words [simp]:
"size_td (typ_info_t TYPE(8 word)) = 1"
"size_td (typ_info_t TYPE(16 word)) = 2"
"size_td (typ_info_t TYPE(32 word)) = 4"
"size_td (typ_info_t TYPE(64 word)) = 8"
by (auto simp: typ_info_word word_tag_def)
lemma align_td_words [simp]:
"align_td (typ_info_t TYPE(8 word)) = 0"
"align_td (typ_info_t TYPE(16 word)) = 1"
"align_td (typ_info_t TYPE(32 word)) = 2"
"align_td (typ_info_t TYPE(64 word)) = 3"
by (auto simp: typ_info_word word_tag_def len8_bytes)
lemma size_of_words [simp]:
"size_of TYPE(8 word) = 1"
"size_of TYPE(16 word) = 2"
"size_of TYPE(32 word) = 4"
"size_of TYPE(64 word) = 8"
by (auto simp: size_of_def)
lemma align_of_words [simp]:
"align_of TYPE(8 word) = 1"
"align_of TYPE(16 word) = 2"
"align_of TYPE(32 word) = 4"
"align_of TYPE(64 word) = 8"
by (auto simp: align_of_word)
lemma size_td_swords [simp]:
"size_td (typ_info_t TYPE(8 sword)) = 1"
"size_td (typ_info_t TYPE(16 sword)) = 2"
"size_td (typ_info_t TYPE(32 sword)) = 4"
"size_td (typ_info_t TYPE(64 sword)) = 8"
by (auto simp: typ_info_word word_tag_def)
lemma align_td_swords [simp]:
"align_td (typ_info_t TYPE(8 sword)) = 0"
"align_td (typ_info_t TYPE(16 sword)) = 1"
"align_td (typ_info_t TYPE(32 sword)) = 2"
"align_td (typ_info_t TYPE(64 sword)) = 3"
by (auto simp: typ_info_word word_tag_def len8_bytes)
lemma size_of_swords [simp]:
"size_of TYPE(8 sword) = 1"
"size_of TYPE(16 sword) = 2"
"size_of TYPE(32 sword) = 4"
"size_of TYPE(64 sword) = 8"
by (auto simp: size_of_def)
lemma align_of_swords [simp]:
"align_of TYPE(8 sword) = 1"
"align_of TYPE(16 sword) = 2"
"align_of TYPE(32 sword) = 4"
"align_of TYPE(64 sword) = 8"
by (auto simp: align_of_word)
instantiation ptr :: (c_type_name) c_type
begin
definition "typ_name_itself (p::'a::c_type_name ptr itself) = typ_name_itself TYPE('a) @ ''+ptr''"
definition
typ_info_ptr:
"typ_info_t (p::'a::c_type_name ptr itself) ≡ TypDesc addr_align
(TypScalar (addr_bitsize div 8) addr_align ⦇
field_access = λp bs. rev (word_rsplit (ptr_val p)),
field_update = λbs v. Ptr (word_rcat (rev (take (addr_bitsize div 8) bs))::addr),
field_sz = (addr_bitsize div 8) ⦈ )
(typ_name_itself TYPE('a) @ ''+ptr'')"
instance by (intro_classes) (simp add: typ_name_itself_ptr_def typ_info_ptr)
end
lemma align_of_ptr [simp]:
"align_of (p::'a::c_type ptr itself) = 2 ^ addr_align"
by (simp add: align_of_def typ_info_ptr)
instantiation ptr :: (c_type) mem_type
begin
instance
apply (intro_classes)
apply (auto simp: to_bytes_def from_bytes_def
length_word_rsplit_exp_size' word_size
word_rcat_rsplit size_of_def addr_card
typ_info_ptr fd_cons_double_update_def
fd_cons_update_access_def fd_cons_access_update_def
fd_cons_length_def fd_cons_update_normalise_def field_norm_def
super_update_bs_def word_rsplit_rcat_size norm_bytes_def
fd_consistent_def fd_cons_def wf_lf_def
fd_cons_desc_def align_field_def update_ti_t_def)
done
end
lemma div_eq: "8 dvd n ⟹ (7 + n::nat) div 8 = n div 8"
by (simp add: div_plus_div_distrib_dvd_right)
lemma length_word_rsplit_len8: "length ((word_rsplit::('a::len8 word) ⇒ 8 word list) w) = (LENGTH('a) div (8::nat))"
proof -
have "8 dvd len_of TYPE('a)" by (rule len8_dv8)
then show ?thesis
by (simp add: length_word_rsplit_exp_size div_eq word_size)
qed
lemma all_value_byte_word: "i < LENGTH('a::len8) div 8 ⟹
padding_base.is_value_byte (λv bs. rev (word_rsplit v))
(λbs v. word_rcat (rev (take (LENGTH('a) div 8) bs))::'a word)
(LENGTH('a) div 8) i"
by (clarsimp simp add: padding_base.is_value_byte_def len8_bytes word_size word_rsplit_rcat_size)
lemma no_padding_byte_word: "i < LENGTH('a::len8) div 8 ⟹
¬ padding_base.is_padding_byte (λv bs. rev (word_rsplit v))
(λbs v. word_rcat (rev (take (LENGTH('a) div 8) bs))::'a word) (LENGTH('a) div 8)
i"
apply (clarsimp simp add: padding_base.is_padding_byte_def len8_bytes word_size )
by (metis (mono_tags, opaque_lifting) Ex_list_of_length len8 len_gt_0 length_list_update less_irrefl_nat nth_list_update unat_mono word_power_less_1)
instantiation word :: (len8) xmem_contained_type
begin
instance
apply intro_classes
apply (clarsimp simp add: typ_info_word word_tag_def)
apply (clarsimp simp add: typ_info_word word_tag_def)
apply (clarsimp simp add: typ_info_word word_tag_def)
apply (unfold_locales; clarsimp simp add: padding_base.eq_padding_def padding_base.eq_upto_padding_def
length_word_rsplit_exp_size word_size word_rcat_rsplit div_eq length_word_rsplit_len8)
apply (simp add: all_value_byte_word no_padding_byte_word)
apply (simp add: wf_padding typ_info_word word_tag_def)
apply (clarsimp simp add: typ_info_word word_tag_def)
done
end
instantiation ptr :: (c_type) simple_mem_type
begin
instance
apply intro_classes
apply(clarsimp simp: typ_info_ptr typ_uinfo_t_def)
done
end
lemma all_value_byte_ptr: "⋀i. i < (addr_bitsize div 8) ⟹
padding_base.is_value_byte (λp bs. rev (word_rsplit (ptr_val p)))
(λbs v. PTR('a) (word_rcat (rev (take (addr_bitsize div 8) bs)))) (addr_bitsize div 8) i"
by (clarsimp simp add: padding_base.is_value_byte_def len8_bytes word_size word_rsplit_rcat_size)
lemma no_padding_byte_ptr: "⋀i. i < (addr_bitsize div 8) ⟹
¬ padding_base.is_padding_byte (λp bs. rev (word_rsplit (ptr_val p)))
(λbs v. PTR('a::c_type) (word_rcat (rev (take (addr_bitsize div 8) bs)))) (addr_bitsize div 8) i"
apply (clarsimp simp add: padding_base.is_padding_byte_def len8_bytes word_size )
by (metis (mono_tags, opaque_lifting) Ex_list_of_length len8 len_gt_0 length_list_update less_irrefl_nat
nth_list_update unat_mono word_power_less_1)
instantiation ptr :: (c_type) xmem_contained_type
begin
instance
apply intro_classes
apply(clarsimp simp: typ_info_ptr typ_uinfo_t_def)
apply(clarsimp simp: typ_info_ptr typ_uinfo_t_def)
apply(clarsimp simp: typ_info_ptr typ_uinfo_t_def)
apply (unfold_locales; clarsimp simp add: padding_base.eq_padding_def padding_base.eq_upto_padding_def
length_word_rsplit_exp_size word_size word_rcat_rsplit div_eq length_word_rsplit_len8)
apply (simp add: no_padding_byte_ptr [simplified] all_value_byte_ptr [simplified])
apply(clarsimp simp: typ_info_ptr typ_uinfo_t_def)
apply (clarsimp simp add: typ_info_ptr typ_uinfo_t_def)
done
end
lemma size_td_ptr [simp]:
"size_td (typ_info_t TYPE('a::c_type ptr)) = addr_bitsize div 8"
by (simp add: typ_info_ptr)
lemma size_of_ptr [simp]:
"size_of TYPE('a::c_type ptr) = addr_bitsize div 8"
by (simp add: size_of_def)
lemma align_td_ptr [simp]: "align_td (typ_info_t TYPE('a::c_type ptr)) = addr_align"
by (simp add: typ_info_ptr)
lemma ptr_add_word32_signed [simp]:
fixes a :: "32 word ptr"
shows "ptr_val (a +⇩p x) = ptr_val a + 4 * of_int x"
by (cases a) (simp add: CTypesDefs.ptr_add_def scast_id)
lemma ptr_add_word32 [simp]:
fixes a :: "32 word ptr"
shows "ptr_val (a +⇩p uint x) = ptr_val a + 4 * x"
by (cases a) (simp add: ptr_add_def scast_id)
lemma ptr_add_word64_signed [simp]:
fixes a :: "64 word ptr"
shows "ptr_val (a +⇩p x) = ptr_val a + 8 * of_int x"
by (cases a) (simp add: CTypesDefs.ptr_add_def scast_id)
lemma ptr_add_word64 [simp]:
fixes a :: "64 word ptr"
shows "ptr_val (a +⇩p uint x) = ptr_val a + 8 * x"
by (cases a) (simp add: ptr_add_def scast_id)
lemma ptr_add_0_id[simp]:"x +⇩p 0 = x"
by (simp add:CTypesDefs.ptr_add_def)
lemma from_bytes_ptr_to_bytes_ptr:
"from_bytes (to_bytes (v::addr_bitsize word) bs) = (Ptr v :: 'a::c_type ptr)"
by (simp add: from_bytes_def to_bytes_def typ_info_ptr word_tag_def
typ_info_word word_size
length_word_rsplit_exp_size' word_rcat_rsplit update_ti_t_def)
lemma ptr_aligned_coerceI:
"ptr_aligned (ptr_coerce x::addr ptr) ⟹ ptr_aligned (x::'a::mem_type ptr ptr)"
by (simp add: ptr_aligned_def)
lemma lift_ptr_ptr [simp]:
"⋀p::'a::mem_type ptr ptr.
lift h (ptr_coerce p) = ptr_val (lift h p)"
by (simp add: lift_def h_val_def from_bytes_def
typ_info_word word_tag_def typ_info_ptr update_ti_t_def)
thm typ_name.simps
lemma typ_name_itself_ptr:
"typ_name_itself (T::'a::c_type ptr itself) = typ_name_itself TYPE('a) @ ''+ptr''"
by (simp add: typ_info_ptr typ_name_itself_ptr_def )
lemma typ_name_itself_word:
"typ_name_itself (T::'a::len8 word itself) = typ_name (typ_info_t T)"
by (simp add: typ_info_word word_tag_def typ_name_itself_word_def )
lemmas Vanilla32_tags [simp] =
typ_info_word
typ_info_ptr
typ_name_itself_ptr
word_tag_def
typ_name_itself_word
lemma ptr_typ_name [simp]:
"typ_name (typ_info_t TYPE(('a :: c_type) ptr)) = typ_name_itself TYPE('a) @ ''+ptr'' "
by simp
lemma word_typ_name [simp]:
"typ_name (typ_info_t TYPE('a::len8 word)) = ''word'' @ nat_to_bin_string (len_of TYPE('a))"
by simp
lemma typ_name_words [simp]:
"typ_name (typ_uinfo_t TYPE(8 word)) = ''word00010''"
"typ_name (typ_uinfo_t TYPE(16 word)) = ''word000010''"
"typ_name (typ_uinfo_t TYPE(32 word)) = ''word0000010''"
"typ_name (typ_uinfo_t TYPE(64 word)) = ''word00000010''"
"typ_name (typ_info_t TYPE(8 word)) = ''word00010''"
"typ_name (typ_info_t TYPE(16 word)) = ''word000010''"
"typ_name (typ_info_t TYPE(32 word)) = ''word0000010''"
"typ_name (typ_info_t TYPE(64 word)) = ''word00000010''"
by (auto simp: typ_uinfo_t_def nat_to_bin_string.simps export_uinfo_def)
lemma typ_name_swords [simp]:
"typ_name (typ_uinfo_t TYPE(8 sword)) = ''word00010''"
"typ_name (typ_uinfo_t TYPE(16 sword)) = ''word000010''"
"typ_name (typ_uinfo_t TYPE(32 sword)) = ''word0000010''"
"typ_name (typ_uinfo_t TYPE(64 sword)) = ''word00000010''"
"typ_name (typ_info_t TYPE(8 sword)) = ''word00010''"
"typ_name (typ_info_t TYPE(16 sword)) = ''word000010''"
"typ_name (typ_info_t TYPE(32 sword)) = ''word0000010''"
"typ_name (typ_info_t TYPE(64 sword)) = ''word00000010''"
by (auto simp: typ_uinfo_t_def nat_to_bin_string.simps export_uinfo_def)
lemma ptr_arith[simp]:
"(x +⇩p a = y +⇩p a) = ((x::('a::c_type) ptr) = (y::'a ptr))"
by (clarsimp simp:CTypesDefs.ptr_add_def)
lemma ptr_arith'[simp]:
"(ptr_coerce (x +⇩p a) = ptr_coerce (y +⇩p a)) = ((x::('a::c_type) ptr) = (y::'a ptr))"
by (clarsimp simp:CTypesDefs.ptr_add_def)
lemma typ_uinfo_t_signed_word_word_conv: "typ_uinfo_t TYPE(('a::len8) signed word) = typ_uinfo_t TYPE('a word)"
by (simp add: typ_uinfo_t_def field_norm_def
len8_bytes word_rsplit_rcat_size word_size fun_eq_iff)
end