(* AUTOMATICALLY GENERATED, DO NOT EDIT! *)
theory HOL4Vec imports HOL4Base begin
;setup_theory res_quan
lemma RES_FORALL_CONJ_DIST: "ALL (P::'a::type => bool) (Q::'a::type => bool) R::'a::type => bool.
RES_FORALL P (%i::'a::type. Q i & R i) =
(RES_FORALL P Q & RES_FORALL P R)"
by (import res_quan RES_FORALL_CONJ_DIST)
lemma RES_FORALL_DISJ_DIST: "ALL (P::'a::type => bool) (Q::'a::type => bool) R::'a::type => bool.
RES_FORALL (%j::'a::type. P j | Q j) R =
(RES_FORALL P R & RES_FORALL Q R)"
by (import res_quan RES_FORALL_DISJ_DIST)
lemma RES_FORALL_UNIQUE: "ALL (x::'a::type => bool) xa::'a::type. RES_FORALL (op = xa) x = x xa"
by (import res_quan RES_FORALL_UNIQUE)
lemma RES_FORALL_FORALL: "ALL (P::'a::type => bool) (R::'a::type => 'b::type => bool) x::'b::type.
(ALL x::'b::type. RES_FORALL P (%i::'a::type. R i x)) =
RES_FORALL P (%i::'a::type. All (R i))"
by (import res_quan RES_FORALL_FORALL)
lemma RES_FORALL_REORDER: "ALL (P::'a::type => bool) (Q::'b::type => bool)
R::'a::type => 'b::type => bool.
RES_FORALL P (%i::'a::type. RES_FORALL Q (R i)) =
RES_FORALL Q (%j::'b::type. RES_FORALL P (%i::'a::type. R i j))"
by (import res_quan RES_FORALL_REORDER)
lemma RES_FORALL_EMPTY: "All (RES_FORALL EMPTY)"
by (import res_quan RES_FORALL_EMPTY)
lemma RES_FORALL_UNIV: "ALL p::'a::type => bool. RES_FORALL pred_set.UNIV p = All p"
by (import res_quan RES_FORALL_UNIV)
lemma RES_FORALL_NULL: "ALL (p::'a::type => bool) m::bool.
RES_FORALL p (%x::'a::type. m) = (p = EMPTY | m)"
by (import res_quan RES_FORALL_NULL)
lemma RES_EXISTS_DISJ_DIST: "ALL (P::'a::type => bool) (Q::'a::type => bool) R::'a::type => bool.
RES_EXISTS P (%i::'a::type. Q i | R i) =
(RES_EXISTS P Q | RES_EXISTS P R)"
by (import res_quan RES_EXISTS_DISJ_DIST)
lemma RES_DISJ_EXISTS_DIST: "ALL (P::'a::type => bool) (Q::'a::type => bool) R::'a::type => bool.
RES_EXISTS (%i::'a::type. P i | Q i) R =
(RES_EXISTS P R | RES_EXISTS Q R)"
by (import res_quan RES_DISJ_EXISTS_DIST)
lemma RES_EXISTS_EQUAL: "ALL (x::'a::type => bool) xa::'a::type. RES_EXISTS (op = xa) x = x xa"
by (import res_quan RES_EXISTS_EQUAL)
lemma RES_EXISTS_REORDER: "ALL (P::'a::type => bool) (Q::'b::type => bool)
R::'a::type => 'b::type => bool.
RES_EXISTS P (%i::'a::type. RES_EXISTS Q (R i)) =
RES_EXISTS Q (%j::'b::type. RES_EXISTS P (%i::'a::type. R i j))"
by (import res_quan RES_EXISTS_REORDER)
lemma RES_EXISTS_EMPTY: "ALL p::'a::type => bool. ~ RES_EXISTS EMPTY p"
by (import res_quan RES_EXISTS_EMPTY)
lemma RES_EXISTS_UNIV: "ALL p::'a::type => bool. RES_EXISTS pred_set.UNIV p = Ex p"
by (import res_quan RES_EXISTS_UNIV)
lemma RES_EXISTS_NULL: "ALL (p::'a::type => bool) m::bool.
RES_EXISTS p (%x::'a::type. m) = (p ~= EMPTY & m)"
by (import res_quan RES_EXISTS_NULL)
lemma RES_EXISTS_ALT: "ALL (p::'a::type => bool) m::'a::type => bool.
RES_EXISTS p m = (IN (RES_SELECT p m) p & m (RES_SELECT p m))"
by (import res_quan RES_EXISTS_ALT)
lemma RES_EXISTS_UNIQUE_EMPTY: "ALL p::'a::type => bool. ~ RES_EXISTS_UNIQUE EMPTY p"
by (import res_quan RES_EXISTS_UNIQUE_EMPTY)
lemma RES_EXISTS_UNIQUE_UNIV: "ALL p::'a::type => bool. RES_EXISTS_UNIQUE pred_set.UNIV p = Ex1 p"
by (import res_quan RES_EXISTS_UNIQUE_UNIV)
lemma RES_EXISTS_UNIQUE_NULL: "ALL (p::'a::type => bool) m::bool.
RES_EXISTS_UNIQUE p (%x::'a::type. m) =
((EX x::'a::type. p = INSERT x EMPTY) & m)"
by (import res_quan RES_EXISTS_UNIQUE_NULL)
lemma RES_EXISTS_UNIQUE_ALT: "ALL (p::'a::type => bool) m::'a::type => bool.
RES_EXISTS_UNIQUE p m =
RES_EXISTS p
(%x::'a::type. m x & RES_FORALL p (%y::'a::type. m y --> y = x))"
by (import res_quan RES_EXISTS_UNIQUE_ALT)
lemma RES_SELECT_EMPTY: "ALL p::'a::type => bool. RES_SELECT EMPTY p = (SOME x::'a::type. False)"
by (import res_quan RES_SELECT_EMPTY)
lemma RES_SELECT_UNIV: "ALL p::'a::type => bool. RES_SELECT pred_set.UNIV p = Eps p"
by (import res_quan RES_SELECT_UNIV)
lemma RES_ABSTRACT: "ALL (p::'a::type => bool) (m::'a::type => 'b::type) x::'a::type.
IN x p --> RES_ABSTRACT p m x = m x"
by (import res_quan RES_ABSTRACT)
lemma RES_ABSTRACT_EQUAL: "ALL (p::'a::type => bool) (m1::'a::type => 'b::type)
m2::'a::type => 'b::type.
(ALL x::'a::type. IN x p --> m1 x = m2 x) -->
RES_ABSTRACT p m1 = RES_ABSTRACT p m2"
by (import res_quan RES_ABSTRACT_EQUAL)
lemma RES_ABSTRACT_IDEMPOT: "ALL (p::'a::type => bool) m::'a::type => 'b::type.
RES_ABSTRACT p (RES_ABSTRACT p m) = RES_ABSTRACT p m"
by (import res_quan RES_ABSTRACT_IDEMPOT)
lemma RES_ABSTRACT_EQUAL_EQ: "ALL (p::'a::type => bool) (m1::'a::type => 'b::type)
m2::'a::type => 'b::type.
(RES_ABSTRACT p m1 = RES_ABSTRACT p m2) =
(ALL x::'a::type. IN x p --> m1 x = m2 x)"
by (import res_quan RES_ABSTRACT_EQUAL_EQ)
;end_setup
;setup_theory word_base
typedef (open) ('a) word = "(Collect::('a::type list recspace => bool) => 'a::type list recspace set)
(%x::'a::type list recspace.
(All::(('a::type list recspace => bool) => bool) => bool)
(%word::'a::type list recspace => bool.
(op -->::bool => bool => bool)
((All::('a::type list recspace => bool) => bool)
(%a0::'a::type list recspace.
(op -->::bool => bool => bool)
((Ex::('a::type list => bool) => bool)
(%a::'a::type list.
(op =::'a::type list recspace
=> 'a::type list recspace => bool)
a0 ((CONSTR::nat
=> 'a::type list
=> (nat => 'a::type list recspace) => 'a::type list recspace)
(0::nat) a
(%n::nat. BOTTOM::'a::type list recspace))))
(word a0)))
(word x)))"
by (rule typedef_helper,import word_base word_TY_DEF)
lemmas word_TY_DEF = typedef_hol2hol4 [OF type_definition_word]
consts
mk_word :: "'a::type list recspace => 'a::type word"
dest_word :: "'a::type word => 'a::type list recspace"
specification (dest_word mk_word) word_repfns: "(ALL a::'a::type word. mk_word (dest_word a) = a) &
(ALL r::'a::type list recspace.
(ALL word::'a::type list recspace => bool.
(ALL a0::'a::type list recspace.
(EX a::'a::type list.
a0 = CONSTR (0::nat) a (%n::nat. BOTTOM)) -->
word a0) -->
word r) =
(dest_word (mk_word r) = r))"
by (import word_base word_repfns)
consts
word_base0 :: "'a::type list => 'a::type word"
defs
word_base0_primdef: "word_base0 ==
%a::'a::type list. mk_word (CONSTR (0::nat) a (%n::nat. BOTTOM))"
lemma word_base0_def: "word_base0 =
(%a::'a::type list. mk_word (CONSTR (0::nat) a (%n::nat. BOTTOM)))"
by (import word_base word_base0_def)
constdefs
WORD :: "'a::type list => 'a::type word"
"WORD == word_base0"
lemma WORD: "WORD = word_base0"
by (import word_base WORD)
consts
word_case :: "('a::type list => 'b::type) => 'a::type word => 'b::type"
specification (word_case_primdef: word_case) word_case_def: "ALL (f::'a::type list => 'b::type) a::'a::type list.
word_case f (WORD a) = f a"
by (import word_base word_case_def)
consts
word_size :: "('a::type => nat) => 'a::type word => nat"
specification (word_size_primdef: word_size) word_size_def: "ALL (f::'a::type => nat) a::'a::type list.
word_size f (WORD a) = (1::nat) + list_size f a"
by (import word_base word_size_def)
lemma word_11: "ALL (a::'a::type list) a'::'a::type list. (WORD a = WORD a') = (a = a')"
by (import word_base word_11)
lemma word_case_cong: "ALL (M::'a::type word) (M'::'a::type word) f::'a::type list => 'b::type.
M = M' &
(ALL a::'a::type list.
M' = WORD a --> f a = (f'::'a::type list => 'b::type) a) -->
word_case f M = word_case f' M'"
by (import word_base word_case_cong)
lemma word_nchotomy: "ALL x::'a::type word. EX l::'a::type list. x = WORD l"
by (import word_base word_nchotomy)
lemma word_Axiom: "ALL f::'a::type list => 'b::type.
EX fn::'a::type word => 'b::type. ALL a::'a::type list. fn (WORD a) = f a"
by (import word_base word_Axiom)
lemma word_induction: "ALL P::'a::type word => bool. (ALL a::'a::type list. P (WORD a)) --> All P"
by (import word_base word_induction)
lemma word_Ax: "ALL f::'a::type list => 'b::type.
EX fn::'a::type word => 'b::type. ALL a::'a::type list. fn (WORD a) = f a"
by (import word_base word_Ax)
lemma WORD_11: "ALL (x::'a::type list) xa::'a::type list. (WORD x = WORD xa) = (x = xa)"
by (import word_base WORD_11)
lemma word_induct: "ALL x::'a::type word => bool. (ALL l::'a::type list. x (WORD l)) --> All x"
by (import word_base word_induct)
lemma word_cases: "ALL x::'a::type word. EX l::'a::type list. x = WORD l"
by (import word_base word_cases)
consts
WORDLEN :: "'a::type word => nat"
specification (WORDLEN) WORDLEN_DEF: "ALL l::'a::type list. WORDLEN (WORD l) = length l"
by (import word_base WORDLEN_DEF)
consts
PWORDLEN :: "nat => 'a::type word => bool"
defs
PWORDLEN_primdef: "PWORDLEN == %n::nat. GSPEC (%w::'a::type word. (w, WORDLEN w = n))"
lemma PWORDLEN_def: "ALL n::nat. PWORDLEN n = GSPEC (%w::'a::type word. (w, WORDLEN w = n))"
by (import word_base PWORDLEN_def)
lemma IN_PWORDLEN: "ALL (n::nat) l::'a::type list. IN (WORD l) (PWORDLEN n) = (length l = n)"
by (import word_base IN_PWORDLEN)
lemma PWORDLEN: "ALL (n::nat) w::'a::type word. IN w (PWORDLEN n) = (WORDLEN w = n)"
by (import word_base PWORDLEN)
lemma PWORDLEN0: "ALL w::'a::type word. IN w (PWORDLEN (0::nat)) --> w = WORD []"
by (import word_base PWORDLEN0)
lemma PWORDLEN1: "ALL x::'a::type. IN (WORD [x]) (PWORDLEN (1::nat))"
by (import word_base PWORDLEN1)
consts
WSEG :: "nat => nat => 'a::type word => 'a::type word"
specification (WSEG) WSEG_DEF: "ALL (m::nat) (k::nat) l::'a::type list.
WSEG m k (WORD l) = WORD (LASTN m (BUTLASTN k l))"
by (import word_base WSEG_DEF)
lemma WSEG0: "ALL (k::nat) w::'a::type word. WSEG (0::nat) k w = WORD []"
by (import word_base WSEG0)
lemma WSEG_PWORDLEN: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m::nat) k::nat. m + k <= n --> IN (WSEG m k w) (PWORDLEN m))"
by (import word_base WSEG_PWORDLEN)
lemma WSEG_WORDLEN: "ALL x::nat.
RES_FORALL (PWORDLEN x)
(%xa::'a::type word.
ALL (xb::nat) xc::nat.
xb + xc <= x --> WORDLEN (WSEG xb xc xa) = xb)"
by (import word_base WSEG_WORDLEN)
lemma WSEG_WORD_LENGTH: "ALL n::nat.
RES_FORALL (PWORDLEN n) (%w::'a::type word. WSEG n (0::nat) w = w)"
by (import word_base WSEG_WORD_LENGTH)
consts
bit :: "nat => 'a::type word => 'a::type"
specification (bit) BIT_DEF: "ALL (k::nat) l::'a::type list. bit k (WORD l) = ELL k l"
by (import word_base BIT_DEF)
lemma BIT0: "ALL x::'a::type. bit (0::nat) (WORD [x]) = x"
by (import word_base BIT0)
lemma WSEG_BIT: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w::'a::type word.
(All::(nat => bool) => bool)
(%k::nat.
(op -->::bool => bool => bool)
((op <::nat => nat => bool) k n)
((op =::'a::type word => 'a::type word => bool)
((WSEG::nat => nat => 'a::type word => 'a::type word)
(1::nat) k w)
((WORD::'a::type list => 'a::type word)
((op #::'a::type => 'a::type list => 'a::type list)
((bit::nat => 'a::type word => 'a::type) k w)
([]::'a::type list)))))))"
by (import word_base WSEG_BIT)
lemma BIT_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m::nat) (k::nat) j::nat.
m + k <= n --> j < m --> bit j (WSEG m k w) = bit (j + k) w)"
by (import word_base BIT_WSEG)
consts
MSB :: "'a::type word => 'a::type"
specification (MSB) MSB_DEF: "ALL l::'a::type list. MSB (WORD l) = hd l"
by (import word_base MSB_DEF)
lemma MSB: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word. (0::nat) < n --> MSB w = bit (PRE n) w)"
by (import word_base MSB)
consts
LSB :: "'a::type word => 'a::type"
specification (LSB) LSB_DEF: "ALL l::'a::type list. LSB (WORD l) = last l"
by (import word_base LSB_DEF)
lemma LSB: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word. (0::nat) < n --> LSB w = bit (0::nat) w)"
by (import word_base LSB)
consts
WSPLIT :: "nat => 'a::type word => 'a::type word * 'a::type word"
specification (WSPLIT) WSPLIT_DEF: "ALL (m::nat) l::'a::type list.
WSPLIT m (WORD l) = (WORD (BUTLASTN m l), WORD (LASTN m l))"
by (import word_base WSPLIT_DEF)
consts
WCAT :: "'a::type word * 'a::type word => 'a::type word"
specification (WCAT) WCAT_DEF: "ALL (l1::'a::type list) l2::'a::type list.
WCAT (WORD l1, WORD l2) = WORD (l1 @ l2)"
by (import word_base WCAT_DEF)
lemma WORD_PARTITION: "(op &::bool => bool => bool)
((All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w::'a::type word.
(All::(nat => bool) => bool)
(%m::nat.
(op -->::bool => bool => bool)
((op <=::nat => nat => bool) m n)
((op =::'a::type word => 'a::type word => bool)
((WCAT::'a::type word * 'a::type word => 'a::type word)
((WSPLIT::nat
=> 'a::type word
=> 'a::type word * 'a::type word)
m w))
w)))))
((All::(nat => bool) => bool)
(%n::nat.
(All::(nat => bool) => bool)
(%m::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w1::'a::type word.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) m)
(%w2::'a::type word.
(op =::'a::type word * 'a::type word
=> 'a::type word * 'a::type word => bool)
((WSPLIT::nat
=> 'a::type word
=> 'a::type word * 'a::type word)
m ((WCAT::'a::type word * 'a::type word
=> 'a::type word)
((Pair::'a::type word
=> 'a::type word
=> 'a::type word * 'a::type word)
w1 w2)))
((Pair::'a::type word
=> 'a::type word
=> 'a::type word * 'a::type word)
w1 w2))))))"
by (import word_base WORD_PARTITION)
lemma WCAT_ASSOC: "ALL (w1::'a::type word) (w2::'a::type word) w3::'a::type word.
WCAT (w1, WCAT (w2, w3)) = WCAT (WCAT (w1, w2), w3)"
by (import word_base WCAT_ASSOC)
lemma WCAT0: "ALL w::'a::type word. WCAT (WORD [], w) = w & WCAT (w, WORD []) = w"
by (import word_base WCAT0)
lemma WCAT_11: "ALL (m::nat) n::nat.
RES_FORALL (PWORDLEN m)
(%wm1::'a::type word.
RES_FORALL (PWORDLEN m)
(%wm2::'a::type word.
RES_FORALL (PWORDLEN n)
(%wn1::'a::type word.
RES_FORALL (PWORDLEN n)
(%wn2::'a::type word.
(WCAT (wm1, wn1) = WCAT (wm2, wn2)) =
(wm1 = wm2 & wn1 = wn2)))))"
by (import word_base WCAT_11)
lemma WSPLIT_PWORDLEN: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w::'a::type word.
(All::(nat => bool) => bool)
(%m::nat.
(op -->::bool => bool => bool)
((op <=::nat => nat => bool) m n)
((op &::bool => bool => bool)
((IN::'a::type word => ('a::type word => bool) => bool)
((fst::'a::type word * 'a::type word => 'a::type word)
((WSPLIT::nat
=> 'a::type word
=> 'a::type word * 'a::type word)
m w))
((PWORDLEN::nat => 'a::type word => bool)
((op -::nat => nat => nat) n m)))
((IN::'a::type word => ('a::type word => bool) => bool)
((snd::'a::type word * 'a::type word => 'a::type word)
((WSPLIT::nat
=> 'a::type word
=> 'a::type word * 'a::type word)
m w))
((PWORDLEN::nat => 'a::type word => bool) m))))))"
by (import word_base WSPLIT_PWORDLEN)
lemma WCAT_PWORDLEN: "ALL n1::nat.
RES_FORALL (PWORDLEN n1)
(%w1::'a::type word.
ALL n2::nat.
RES_FORALL (PWORDLEN n2)
(%w2::'a::type word. IN (WCAT (w1, w2)) (PWORDLEN (n1 + n2))))"
by (import word_base WCAT_PWORDLEN)
lemma WORDLEN_SUC_WCAT: "ALL (n::nat) w::'a::type word.
IN w (PWORDLEN (Suc n)) -->
RES_EXISTS (PWORDLEN (1::nat))
(%b::'a::type word.
RES_EXISTS (PWORDLEN n) (%w'::'a::type word. w = WCAT (b, w')))"
by (import word_base WORDLEN_SUC_WCAT)
lemma WSEG_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m1::nat) (k1::nat) (m2::nat) k2::nat.
m1 + k1 <= n & m2 + k2 <= m1 -->
WSEG m2 k2 (WSEG m1 k1 w) = WSEG m2 (k1 + k2) w)"
by (import word_base WSEG_WSEG)
lemma WSPLIT_WSEG: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w::'a::type word.
(All::(nat => bool) => bool)
(%k::nat.
(op -->::bool => bool => bool)
((op <=::nat => nat => bool) k n)
((op =::'a::type word * 'a::type word
=> 'a::type word * 'a::type word => bool)
((WSPLIT::nat
=> 'a::type word
=> 'a::type word * 'a::type word)
k w)
((Pair::'a::type word
=> 'a::type word => 'a::type word * 'a::type word)
((WSEG::nat => nat => 'a::type word => 'a::type word)
((op -::nat => nat => nat) n k) k w)
((WSEG::nat => nat => 'a::type word => 'a::type word) k
(0::nat) w))))))"
by (import word_base WSPLIT_WSEG)
lemma WSPLIT_WSEG1: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w::'a::type word.
(All::(nat => bool) => bool)
(%k::nat.
(op -->::bool => bool => bool)
((op <=::nat => nat => bool) k n)
((op =::'a::type word => 'a::type word => bool)
((fst::'a::type word * 'a::type word => 'a::type word)
((WSPLIT::nat
=> 'a::type word
=> 'a::type word * 'a::type word)
k w))
((WSEG::nat => nat => 'a::type word => 'a::type word)
((op -::nat => nat => nat) n k) k w)))))"
by (import word_base WSPLIT_WSEG1)
lemma WSPLIT_WSEG2: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w::'a::type word.
(All::(nat => bool) => bool)
(%k::nat.
(op -->::bool => bool => bool)
((op <=::nat => nat => bool) k n)
((op =::'a::type word => 'a::type word => bool)
((snd::'a::type word * 'a::type word => 'a::type word)
((WSPLIT::nat
=> 'a::type word
=> 'a::type word * 'a::type word)
k w))
((WSEG::nat => nat => 'a::type word => 'a::type word) k
(0::nat) w)))))"
by (import word_base WSPLIT_WSEG2)
lemma WCAT_WSEG_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m1::nat) (m2::nat) k::nat.
m1 + (m2 + k) <= n -->
WCAT (WSEG m2 (m1 + k) w, WSEG m1 k w) = WSEG (m1 + m2) k w)"
by (import word_base WCAT_WSEG_WSEG)
lemma WORD_SPLIT: "ALL (x::nat) xa::nat.
RES_FORALL (PWORDLEN (x + xa))
(%w::'a::type word. w = WCAT (WSEG x xa w, WSEG xa (0::nat) w))"
by (import word_base WORD_SPLIT)
lemma WORDLEN_SUC_WCAT_WSEG_WSEG: "RES_FORALL (PWORDLEN (Suc (n::nat)))
(%w::'a::type word. w = WCAT (WSEG (1::nat) n w, WSEG n (0::nat) w))"
by (import word_base WORDLEN_SUC_WCAT_WSEG_WSEG)
lemma WORDLEN_SUC_WCAT_WSEG_WSEG_RIGHT: "RES_FORALL (PWORDLEN (Suc (n::nat)))
(%w::'a::type word. w = WCAT (WSEG n (1::nat) w, WSEG (1::nat) (0::nat) w))"
by (import word_base WORDLEN_SUC_WCAT_WSEG_WSEG_RIGHT)
lemma WORDLEN_SUC_WCAT_BIT_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
(%w::'a::type word. w = WCAT (WORD [bit n w], WSEG n (0::nat) w))"
by (import word_base WORDLEN_SUC_WCAT_BIT_WSEG)
lemma WORDLEN_SUC_WCAT_BIT_WSEG_RIGHT: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
(%w::'a::type word. w = WCAT (WSEG n (1::nat) w, WORD [bit (0::nat) w]))"
by (import word_base WORDLEN_SUC_WCAT_BIT_WSEG_RIGHT)
lemma WSEG_WCAT1: "ALL (n1::nat) n2::nat.
RES_FORALL (PWORDLEN n1)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n2)
(%w2::'a::type word. WSEG n1 n2 (WCAT (w1, w2)) = w1))"
by (import word_base WSEG_WCAT1)
lemma WSEG_WCAT2: "ALL (n1::nat) n2::nat.
RES_FORALL (PWORDLEN n1)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n2)
(%w2::'a::type word. WSEG n2 (0::nat) (WCAT (w1, w2)) = w2))"
by (import word_base WSEG_WCAT2)
lemma WSEG_SUC: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (k::nat) m1::nat.
k + Suc m1 < n -->
WSEG (Suc m1) k w = WCAT (WSEG (1::nat) (k + m1) w, WSEG m1 k w))"
by (import word_base WSEG_SUC)
lemma WORD_CONS_WCAT: "ALL (x::'a::type) l::'a::type list. WORD (x # l) = WCAT (WORD [x], WORD l)"
by (import word_base WORD_CONS_WCAT)
lemma WORD_SNOC_WCAT: "ALL (l::'a::type list) x::'a::type.
WORD (SNOC x l) = WCAT (WORD l, WORD [x])"
by (import word_base WORD_SNOC_WCAT)
lemma BIT_WCAT_FST: "ALL (n1::nat) n2::nat.
RES_FORALL (PWORDLEN n1)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n2)
(%w2::'a::type word.
ALL k::nat.
n2 <= k & k < n1 + n2 -->
bit k (WCAT (w1, w2)) = bit (k - n2) w1))"
by (import word_base BIT_WCAT_FST)
lemma BIT_WCAT_SND: "(All::(nat => bool) => bool)
(%n1::nat.
(All::(nat => bool) => bool)
(%n2::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n1)
(%w1::'a::type word.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n2)
(%w2::'a::type word.
(All::(nat => bool) => bool)
(%k::nat.
(op -->::bool => bool => bool)
((op <::nat => nat => bool) k n2)
((op =::'a::type => 'a::type => bool)
((bit::nat => 'a::type word => 'a::type) k
((WCAT::'a::type word * 'a::type word
=> 'a::type word)
((Pair::'a::type word
=> 'a::type word => 'a::type word * 'a::type word)
w1 w2)))
((bit::nat => 'a::type word => 'a::type) k
w2)))))))"
by (import word_base BIT_WCAT_SND)
lemma BIT_WCAT1: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word. ALL b::'a::type. bit n (WCAT (WORD [b], w)) = b)"
by (import word_base BIT_WCAT1)
lemma WSEG_WCAT_WSEG1: "ALL (n1::nat) n2::nat.
RES_FORALL (PWORDLEN n1)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n2)
(%w2::'a::type word.
ALL (m::nat) k::nat.
m <= n1 & n2 <= k -->
WSEG m k (WCAT (w1, w2)) = WSEG m (k - n2) w1))"
by (import word_base WSEG_WCAT_WSEG1)
lemma WSEG_WCAT_WSEG2: "ALL (n1::nat) n2::nat.
RES_FORALL (PWORDLEN n1)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n2)
(%w2::'a::type word.
ALL (m::nat) k::nat.
m + k <= n2 --> WSEG m k (WCAT (w1, w2)) = WSEG m k w2))"
by (import word_base WSEG_WCAT_WSEG2)
lemma WSEG_WCAT_WSEG: "ALL (n1::nat) n2::nat.
RES_FORALL (PWORDLEN n1)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n2)
(%w2::'a::type word.
ALL (m::nat) k::nat.
m + k <= n1 + n2 & k < n2 & n2 <= m + k -->
WSEG m k (WCAT (w1, w2)) =
WCAT (WSEG (m + k - n2) (0::nat) w1, WSEG (n2 - k) k w2)))"
by (import word_base WSEG_WCAT_WSEG)
lemma BIT_EQ_IMP_WORD_EQ: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w1::'a::type word.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w2::'a::type word.
(op -->::bool => bool => bool)
((All::(nat => bool) => bool)
(%k::nat.
(op -->::bool => bool => bool)
((op <::nat => nat => bool) k n)
((op =::'a::type => 'a::type => bool)
((bit::nat => 'a::type word => 'a::type) k w1)
((bit::nat => 'a::type word => 'a::type) k w2))))
((op =::'a::type word => 'a::type word => bool) w1 w2))))"
by (import word_base BIT_EQ_IMP_WORD_EQ)
;end_setup
;setup_theory word_num
constdefs
LVAL :: "('a::type => nat) => nat => 'a::type list => nat"
"LVAL ==
%(f::'a::type => nat) b::nat.
foldl (%(e::nat) x::'a::type. b * e + f x) (0::nat)"
lemma LVAL_DEF: "ALL (f::'a::type => nat) (b::nat) l::'a::type list.
LVAL f b l = foldl (%(e::nat) x::'a::type. b * e + f x) (0::nat) l"
by (import word_num LVAL_DEF)
consts
NVAL :: "('a::type => nat) => nat => 'a::type word => nat"
specification (NVAL) NVAL_DEF: "ALL (f::'a::type => nat) (b::nat) l::'a::type list.
NVAL f b (WORD l) = LVAL f b l"
by (import word_num NVAL_DEF)
lemma LVAL: "(ALL (x::'a::type => nat) xa::nat. LVAL x xa [] = (0::nat)) &
(ALL (x::'a::type list) (xa::'a::type => nat) (xb::nat) xc::'a::type.
LVAL xa xb (xc # x) = xa xc * xb ^ length x + LVAL xa xb x)"
by (import word_num LVAL)
lemma LVAL_SNOC: "ALL (l::'a::type list) (h::'a::type) (f::'a::type => nat) b::nat.
LVAL f b (SNOC h l) = LVAL f b l * b + f h"
by (import word_num LVAL_SNOC)
lemma LVAL_MAX: "ALL (l::'a::type list) (f::'a::type => nat) b::nat.
(ALL x::'a::type. f x < b) --> LVAL f b l < b ^ length l"
by (import word_num LVAL_MAX)
lemma NVAL_MAX: "ALL (f::'a::type => nat) b::nat.
(ALL x::'a::type. f x < b) -->
(ALL n::nat.
RES_FORALL (PWORDLEN n) (%w::'a::type word. NVAL f b w < b ^ n))"
by (import word_num NVAL_MAX)
lemma NVAL0: "ALL (x::'a::type => nat) xa::nat. NVAL x xa (WORD []) = (0::nat)"
by (import word_num NVAL0)
lemma NVAL1: "ALL (x::'a::type => nat) (xa::nat) xb::'a::type.
NVAL x xa (WORD [xb]) = x xb"
by (import word_num NVAL1)
lemma NVAL_WORDLEN_0: "RES_FORALL (PWORDLEN (0::nat))
(%w::'a::type word.
ALL (fv::'a::type => nat) r::nat. NVAL fv r w = (0::nat))"
by (import word_num NVAL_WORDLEN_0)
lemma NVAL_WCAT1: "ALL (w::'a::type word) (f::'a::type => nat) (b::nat) x::'a::type.
NVAL f b (WCAT (w, WORD [x])) = NVAL f b w * b + f x"
by (import word_num NVAL_WCAT1)
lemma NVAL_WCAT2: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (f::'a::type => nat) (b::nat) x::'a::type.
NVAL f b (WCAT (WORD [x], w)) = f x * b ^ n + NVAL f b w)"
by (import word_num NVAL_WCAT2)
lemma NVAL_WCAT: "ALL (n::nat) m::nat.
RES_FORALL (PWORDLEN n)
(%w1::'a::type word.
RES_FORALL (PWORDLEN m)
(%w2::'a::type word.
ALL (f::'a::type => nat) b::nat.
NVAL f b (WCAT (w1, w2)) =
NVAL f b w1 * b ^ m + NVAL f b w2))"
by (import word_num NVAL_WCAT)
consts
NLIST :: "nat => (nat => 'a::type) => nat => nat => 'a::type list"
specification (NLIST) NLIST_DEF: "(ALL (frep::nat => 'a::type) (b::nat) m::nat.
NLIST (0::nat) frep b m = []) &
(ALL (n::nat) (frep::nat => 'a::type) (b::nat) m::nat.
NLIST (Suc n) frep b m =
SNOC (frep (m mod b)) (NLIST n frep b (m div b)))"
by (import word_num NLIST_DEF)
constdefs
NWORD :: "nat => (nat => 'a::type) => nat => nat => 'a::type word"
"NWORD ==
%(n::nat) (frep::nat => 'a::type) (b::nat) m::nat. WORD (NLIST n frep b m)"
lemma NWORD_DEF: "ALL (n::nat) (frep::nat => 'a::type) (b::nat) m::nat.
NWORD n frep b m = WORD (NLIST n frep b m)"
by (import word_num NWORD_DEF)
lemma NWORD_LENGTH: "ALL (x::nat) (xa::nat => 'a::type) (xb::nat) xc::nat.
WORDLEN (NWORD x xa xb xc) = x"
by (import word_num NWORD_LENGTH)
lemma NWORD_PWORDLEN: "ALL (x::nat) (xa::nat => 'a::type) (xb::nat) xc::nat.
IN (NWORD x xa xb xc) (PWORDLEN x)"
by (import word_num NWORD_PWORDLEN)
;end_setup
;setup_theory word_bitop
consts
PBITOP :: "('a::type word => 'b::type word) => bool"
defs
PBITOP_primdef: "PBITOP ==
GSPEC
(%oper::'a::type word => 'b::type word.
(oper,
ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
IN (oper w) (PWORDLEN n) &
(ALL (m::nat) k::nat.
m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w)))))"
lemma PBITOP_def: "PBITOP =
GSPEC
(%oper::'a::type word => 'b::type word.
(oper,
ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
IN (oper w) (PWORDLEN n) &
(ALL (m::nat) k::nat.
m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w)))))"
by (import word_bitop PBITOP_def)
lemma IN_PBITOP: "ALL oper::'a::type word => 'b::type word.
IN oper PBITOP =
(ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
IN (oper w) (PWORDLEN n) &
(ALL (m::nat) k::nat.
m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w))))"
by (import word_bitop IN_PBITOP)
lemma PBITOP_PWORDLEN: "RES_FORALL PBITOP
(%oper::'a::type word => 'b::type word.
ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word. IN (oper w) (PWORDLEN n)))"
by (import word_bitop PBITOP_PWORDLEN)
lemma PBITOP_WSEG: "RES_FORALL PBITOP
(%oper::'a::type word => 'b::type word.
ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k <= n --> oper (WSEG m k w) = WSEG m k (oper w)))"
by (import word_bitop PBITOP_WSEG)
lemma PBITOP_BIT: "(RES_FORALL::(('a::type word => 'b::type word) => bool)
=> (('a::type word => 'b::type word) => bool) => bool)
(PBITOP::('a::type word => 'b::type word) => bool)
(%oper::'a::type word => 'b::type word.
(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w::'a::type word.
(All::(nat => bool) => bool)
(%k::nat.
(op -->::bool => bool => bool)
((op <::nat => nat => bool) k n)
((op =::'b::type word => 'b::type word => bool)
(oper
((WORD::'a::type list => 'a::type word)
((op #::'a::type
=> 'a::type list => 'a::type list)
((bit::nat => 'a::type word => 'a::type) k w)
([]::'a::type list))))
((WORD::'b::type list => 'b::type word)
((op #::'b::type => 'b::type list => 'b::type list)
((bit::nat => 'b::type word => 'b::type) k
(oper w))
([]::'b::type list))))))))"
by (import word_bitop PBITOP_BIT)
consts
PBITBOP :: "('a::type word => 'b::type word => 'c::type word) => bool"
defs
PBITBOP_primdef: "PBITBOP ==
GSPEC
(%oper::'a::type word => 'b::type word => 'c::type word.
(oper,
ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n)
(%w2::'b::type word.
IN (oper w1 w2) (PWORDLEN n) &
(ALL (m::nat) k::nat.
m + k <= n -->
oper (WSEG m k w1) (WSEG m k w2) =
WSEG m k (oper w1 w2))))))"
lemma PBITBOP_def: "PBITBOP =
GSPEC
(%oper::'a::type word => 'b::type word => 'c::type word.
(oper,
ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n)
(%w2::'b::type word.
IN (oper w1 w2) (PWORDLEN n) &
(ALL (m::nat) k::nat.
m + k <= n -->
oper (WSEG m k w1) (WSEG m k w2) =
WSEG m k (oper w1 w2))))))"
by (import word_bitop PBITBOP_def)
lemma IN_PBITBOP: "ALL oper::'a::type word => 'b::type word => 'c::type word.
IN oper PBITBOP =
(ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n)
(%w2::'b::type word.
IN (oper w1 w2) (PWORDLEN n) &
(ALL (m::nat) k::nat.
m + k <= n -->
oper (WSEG m k w1) (WSEG m k w2) =
WSEG m k (oper w1 w2)))))"
by (import word_bitop IN_PBITBOP)
lemma PBITBOP_PWORDLEN: "RES_FORALL PBITBOP
(%oper::'a::type word => 'b::type word => 'c::type word.
ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n)
(%w2::'b::type word. IN (oper w1 w2) (PWORDLEN n))))"
by (import word_bitop PBITBOP_PWORDLEN)
lemma PBITBOP_WSEG: "RES_FORALL PBITBOP
(%oper::'a::type word => 'b::type word => 'c::type word.
ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::'a::type word.
RES_FORALL (PWORDLEN n)
(%w2::'b::type word.
ALL (m::nat) k::nat.
m + k <= n -->
oper (WSEG m k w1) (WSEG m k w2) =
WSEG m k (oper w1 w2))))"
by (import word_bitop PBITBOP_WSEG)
lemma PBITBOP_EXISTS: "ALL f::'a::type => 'b::type => 'c::type.
EX x::'a::type word => 'b::type word => 'c::type word.
ALL (l1::'a::type list) l2::'b::type list.
x (WORD l1) (WORD l2) = WORD (map2 f l1 l2)"
by (import word_bitop PBITBOP_EXISTS)
consts
WMAP :: "('a::type => 'b::type) => 'a::type word => 'b::type word"
specification (WMAP) WMAP_DEF: "ALL (f::'a::type => 'b::type) l::'a::type list.
WMAP f (WORD l) = WORD (map f l)"
by (import word_bitop WMAP_DEF)
lemma WMAP_PWORDLEN: "RES_FORALL (PWORDLEN (n::nat))
(%w::'a::type word.
ALL f::'a::type => 'b::type. IN (WMAP f w) (PWORDLEN n))"
by (import word_bitop WMAP_PWORDLEN)
lemma WMAP_0: "ALL x::'a::type => 'b::type. WMAP x (WORD []) = WORD []"
by (import word_bitop WMAP_0)
lemma WMAP_BIT: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w::'a::type word.
(All::(nat => bool) => bool)
(%k::nat.
(op -->::bool => bool => bool)
((op <::nat => nat => bool) k n)
((All::(('a::type => 'b::type) => bool) => bool)
(%f::'a::type => 'b::type.
(op =::'b::type => 'b::type => bool)
((bit::nat => 'b::type word => 'b::type) k
((WMAP::('a::type => 'b::type)
=> 'a::type word => 'b::type word)
f w))
(f ((bit::nat => 'a::type word => 'a::type) k
w)))))))"
by (import word_bitop WMAP_BIT)
lemma WMAP_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k <= n -->
(ALL f::'a::type => 'b::type.
WMAP f (WSEG m k w) = WSEG m k (WMAP f w)))"
by (import word_bitop WMAP_WSEG)
lemma WMAP_PBITOP: "ALL f::'a::type => 'b::type. IN (WMAP f) PBITOP"
by (import word_bitop WMAP_PBITOP)
lemma WMAP_WCAT: "ALL (w1::'a::type word) (w2::'a::type word) f::'a::type => 'b::type.
WMAP f (WCAT (w1, w2)) = WCAT (WMAP f w1, WMAP f w2)"
by (import word_bitop WMAP_WCAT)
lemma WMAP_o: "ALL (w::'a::type word) (f::'a::type => 'b::type) g::'b::type => 'c::type.
WMAP g (WMAP f w) = WMAP (g o f) w"
by (import word_bitop WMAP_o)
consts
FORALLBITS :: "('a::type => bool) => 'a::type word => bool"
specification (FORALLBITS) FORALLBITS_DEF: "ALL (P::'a::type => bool) l::'a::type list.
FORALLBITS P (WORD l) = list_all P l"
by (import word_bitop FORALLBITS_DEF)
lemma FORALLBITS: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w::'a::type word.
(All::(('a::type => bool) => bool) => bool)
(%P::'a::type => bool.
(op =::bool => bool => bool)
((FORALLBITS::('a::type => bool) => 'a::type word => bool) P
w)
((All::(nat => bool) => bool)
(%k::nat.
(op -->::bool => bool => bool)
((op <::nat => nat => bool) k n)
(P ((bit::nat => 'a::type word => 'a::type) k
w)))))))"
by (import word_bitop FORALLBITS)
lemma FORALLBITS_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL P::'a::type => bool.
FORALLBITS P w -->
(ALL (m::nat) k::nat. m + k <= n --> FORALLBITS P (WSEG m k w)))"
by (import word_bitop FORALLBITS_WSEG)
lemma FORALLBITS_WCAT: "ALL (w1::'a::type word) (w2::'a::type word) P::'a::type => bool.
FORALLBITS P (WCAT (w1, w2)) = (FORALLBITS P w1 & FORALLBITS P w2)"
by (import word_bitop FORALLBITS_WCAT)
consts
EXISTSABIT :: "('a::type => bool) => 'a::type word => bool"
specification (EXISTSABIT) EXISTSABIT_DEF: "ALL (P::'a::type => bool) l::'a::type list.
EXISTSABIT P (WORD l) = list_exists P l"
by (import word_bitop EXISTSABIT_DEF)
lemma NOT_EXISTSABIT: "ALL (P::'a::type => bool) w::'a::type word.
(~ EXISTSABIT P w) = FORALLBITS (Not o P) w"
by (import word_bitop NOT_EXISTSABIT)
lemma NOT_FORALLBITS: "ALL (P::'a::type => bool) w::'a::type word.
(~ FORALLBITS P w) = EXISTSABIT (Not o P) w"
by (import word_bitop NOT_FORALLBITS)
lemma EXISTSABIT: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::('a::type word => bool)
=> ('a::type word => bool) => bool)
((PWORDLEN::nat => 'a::type word => bool) n)
(%w::'a::type word.
(All::(('a::type => bool) => bool) => bool)
(%P::'a::type => bool.
(op =::bool => bool => bool)
((EXISTSABIT::('a::type => bool) => 'a::type word => bool) P
w)
((Ex::(nat => bool) => bool)
(%k::nat.
(op &::bool => bool => bool)
((op <::nat => nat => bool) k n)
(P ((bit::nat => 'a::type word => 'a::type) k
w)))))))"
by (import word_bitop EXISTSABIT)
lemma EXISTSABIT_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k <= n -->
(ALL P::'a::type => bool.
EXISTSABIT P (WSEG m k w) --> EXISTSABIT P w))"
by (import word_bitop EXISTSABIT_WSEG)
lemma EXISTSABIT_WCAT: "ALL (w1::'a::type word) (w2::'a::type word) P::'a::type => bool.
EXISTSABIT P (WCAT (w1, w2)) = (EXISTSABIT P w1 | EXISTSABIT P w2)"
by (import word_bitop EXISTSABIT_WCAT)
constdefs
SHR :: "bool => 'a::type => 'a::type word => 'a::type word * 'a::type"
"SHR ==
%(f::bool) (b::'a::type) w::'a::type word.
(WCAT
(if f then WSEG (1::nat) (PRE (WORDLEN w)) w else WORD [b],
WSEG (PRE (WORDLEN w)) (1::nat) w),
bit (0::nat) w)"
lemma SHR_DEF: "ALL (f::bool) (b::'a::type) w::'a::type word.
SHR f b w =
(WCAT
(if f then WSEG (1::nat) (PRE (WORDLEN w)) w else WORD [b],
WSEG (PRE (WORDLEN w)) (1::nat) w),
bit (0::nat) w)"
by (import word_bitop SHR_DEF)
constdefs
SHL :: "bool => 'a::type word => 'a::type => 'a::type * 'a::type word"
"SHL ==
%(f::bool) (w::'a::type word) b::'a::type.
(bit (PRE (WORDLEN w)) w,
WCAT
(WSEG (PRE (WORDLEN w)) (0::nat) w,
if f then WSEG (1::nat) (0::nat) w else WORD [b]))"
lemma SHL_DEF: "ALL (f::bool) (w::'a::type word) b::'a::type.
SHL f w b =
(bit (PRE (WORDLEN w)) w,
WCAT
(WSEG (PRE (WORDLEN w)) (0::nat) w,
if f then WSEG (1::nat) (0::nat) w else WORD [b]))"
by (import word_bitop SHL_DEF)
lemma SHR_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k <= n -->
(0::nat) < m -->
(ALL (f::bool) b::'a::type.
SHR f b (WSEG m k w) =
(if f
then WCAT
(WSEG (1::nat) (k + (m - (1::nat))) w,
WSEG (m - (1::nat)) (k + (1::nat)) w)
else WCAT (WORD [b], WSEG (m - (1::nat)) (k + (1::nat)) w),
bit k w)))"
by (import word_bitop SHR_WSEG)
lemma SHR_WSEG_1F: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (b::'a::type) (m::nat) k::nat.
m + k <= n -->
(0::nat) < m -->
SHR False b (WSEG m k w) =
(WCAT (WORD [b], WSEG (m - (1::nat)) (k + (1::nat)) w), bit k w))"
by (import word_bitop SHR_WSEG_1F)
lemma SHR_WSEG_NF: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k < n -->
(0::nat) < m -->
SHR False (bit (m + k) w) (WSEG m k w) =
(WSEG m (k + (1::nat)) w, bit k w))"
by (import word_bitop SHR_WSEG_NF)
lemma SHL_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k <= n -->
(0::nat) < m -->
(ALL (f::bool) b::'a::type.
SHL f (WSEG m k w) b =
(bit (k + (m - (1::nat))) w,
if f then WCAT (WSEG (m - (1::nat)) k w, WSEG (1::nat) k w)
else WCAT (WSEG (m - (1::nat)) k w, WORD [b]))))"
by (import word_bitop SHL_WSEG)
lemma SHL_WSEG_1F: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (b::'a::type) (m::nat) k::nat.
m + k <= n -->
(0::nat) < m -->
SHL False (WSEG m k w) b =
(bit (k + (m - (1::nat))) w,
WCAT (WSEG (m - (1::nat)) k w, WORD [b])))"
by (import word_bitop SHL_WSEG_1F)
lemma SHL_WSEG_NF: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::'a::type word.
ALL (m::nat) k::nat.
m + k <= n -->
(0::nat) < m -->
(0::nat) < k -->
SHL False (WSEG m k w) (bit (k - (1::nat)) w) =
(bit (k + (m - (1::nat))) w, WSEG m (k - (1::nat)) w))"
by (import word_bitop SHL_WSEG_NF)
lemma WSEG_SHL: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
(%w::'a::type word.
ALL (m::nat) k::nat.
(0::nat) < k & m + k <= Suc n -->
(ALL b::'a::type.
WSEG m k (snd (SHL (f::bool) w b)) =
WSEG m (k - (1::nat)) w))"
by (import word_bitop WSEG_SHL)
lemma WSEG_SHL_0: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
(%w::'a::type word.
ALL (m::nat) b::'a::type.
(0::nat) < m & m <= Suc n -->
WSEG m (0::nat) (snd (SHL (f::bool) w b)) =
WCAT
(WSEG (m - (1::nat)) (0::nat) w,
if f then WSEG (1::nat) (0::nat) w else WORD [b]))"
by (import word_bitop WSEG_SHL_0)
;end_setup
;setup_theory bword_num
constdefs
BV :: "bool => nat"
"(BV == (%(b::bool). (if b then (Suc (0::nat)) else (0::nat))))"
lemma BV_DEF: "(ALL (b::bool). ((BV b) = (if b then (Suc (0::nat)) else (0::nat))))"
by (import bword_num BV_DEF)
consts
BNVAL :: "bool word => nat"
specification (BNVAL) BNVAL_DEF: "ALL l::bool list. BNVAL (WORD l) = LVAL BV (2::nat) l"
by (import bword_num BNVAL_DEF)
lemma BV_LESS_2: "ALL x::bool. BV x < (2::nat)"
by (import bword_num BV_LESS_2)
lemma BNVAL_NVAL: "ALL w::bool word. BNVAL w = NVAL BV (2::nat) w"
by (import bword_num BNVAL_NVAL)
lemma BNVAL0: "BNVAL (WORD []) = (0::nat)"
by (import bword_num BNVAL0)
lemma BNVAL_11: "ALL (w1::bool word) w2::bool word.
WORDLEN w1 = WORDLEN w2 --> BNVAL w1 = BNVAL w2 --> w1 = w2"
by (import bword_num BNVAL_11)
lemma BNVAL_ONTO: "ALL w::bool word. Ex (op = (BNVAL w))"
by (import bword_num BNVAL_ONTO)
lemma BNVAL_MAX: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::bool word. BNVAL w < (2::nat) ^ n)"
by (import bword_num BNVAL_MAX)
lemma BNVAL_WCAT1: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::bool word.
ALL x::bool. BNVAL (WCAT (w, WORD [x])) = BNVAL w * (2::nat) + BV x)"
by (import bword_num BNVAL_WCAT1)
lemma BNVAL_WCAT2: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::bool word.
ALL x::bool.
BNVAL (WCAT (WORD [x], w)) = BV x * (2::nat) ^ n + BNVAL w)"
by (import bword_num BNVAL_WCAT2)
lemma BNVAL_WCAT: "ALL (n::nat) m::nat.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN m)
(%w2::bool word.
BNVAL (WCAT (w1, w2)) = BNVAL w1 * (2::nat) ^ m + BNVAL w2))"
by (import bword_num BNVAL_WCAT)
constdefs
VB :: "nat => bool"
"VB == %n::nat. n mod (2::nat) ~= (0::nat)"
lemma VB_DEF: "ALL n::nat. VB n = (n mod (2::nat) ~= (0::nat))"
by (import bword_num VB_DEF)
constdefs
NBWORD :: "nat => nat => bool word"
"NBWORD == %(n::nat) m::nat. WORD (NLIST n VB (2::nat) m)"
lemma NBWORD_DEF: "ALL (n::nat) m::nat. NBWORD n m = WORD (NLIST n VB (2::nat) m)"
by (import bword_num NBWORD_DEF)
lemma NBWORD0: "ALL x::nat. NBWORD (0::nat) x = WORD []"
by (import bword_num NBWORD0)
lemma WORDLEN_NBWORD: "ALL (x::nat) xa::nat. WORDLEN (NBWORD x xa) = x"
by (import bword_num WORDLEN_NBWORD)
lemma PWORDLEN_NBWORD: "ALL (x::nat) xa::nat. IN (NBWORD x xa) (PWORDLEN x)"
by (import bword_num PWORDLEN_NBWORD)
lemma NBWORD_SUC: "ALL (n::nat) m::nat.
NBWORD (Suc n) m =
WCAT (NBWORD n (m div (2::nat)), WORD [VB (m mod (2::nat))])"
by (import bword_num NBWORD_SUC)
lemma VB_BV: "ALL x::bool. VB (BV x) = x"
by (import bword_num VB_BV)
lemma BV_VB: "(All::(nat => bool) => bool)
(%x::nat.
(op -->::bool => bool => bool)
((op <::nat => nat => bool) x
((number_of::bin => nat)
((op BIT::bin => bit => bin)
((op BIT::bin => bit => bin) (Numeral.Pls::bin) (bit.B1::bit))
(bit.B0::bit))))
((op =::nat => nat => bool) ((BV::bool => nat) ((VB::nat => bool) x))
x))"
by (import bword_num BV_VB)
lemma NBWORD_BNVAL: "ALL n::nat. RES_FORALL (PWORDLEN n) (%w::bool word. NBWORD n (BNVAL w) = w)"
by (import bword_num NBWORD_BNVAL)
lemma BNVAL_NBWORD: "ALL (n::nat) m::nat. m < (2::nat) ^ n --> BNVAL (NBWORD n m) = m"
by (import bword_num BNVAL_NBWORD)
lemma ZERO_WORD_VAL: "RES_FORALL (PWORDLEN (n::nat))
(%w::bool word. (w = NBWORD n (0::nat)) = (BNVAL w = (0::nat)))"
by (import bword_num ZERO_WORD_VAL)
lemma WCAT_NBWORD_0: "ALL (n1::nat) n2::nat.
WCAT (NBWORD n1 (0::nat), NBWORD n2 (0::nat)) = NBWORD (n1 + n2) (0::nat)"
by (import bword_num WCAT_NBWORD_0)
lemma WSPLIT_NBWORD_0: "ALL (n::nat) m::nat.
m <= n -->
WSPLIT m (NBWORD n (0::nat)) =
(NBWORD (n - m) (0::nat), NBWORD m (0::nat))"
by (import bword_num WSPLIT_NBWORD_0)
lemma EQ_NBWORD0_SPLIT: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::(bool word => bool) => (bool word => bool) => bool)
((PWORDLEN::nat => bool word => bool) n)
(%w::bool word.
(All::(nat => bool) => bool)
(%m::nat.
(op -->::bool => bool => bool)
((op <=::nat => nat => bool) m n)
((op =::bool => bool => bool)
((op =::bool word => bool word => bool) w
((NBWORD::nat => nat => bool word) n (0::nat)))
((op &::bool => bool => bool)
((op =::bool word => bool word => bool)
((WSEG::nat => nat => bool word => bool word)
((op -::nat => nat => nat) n m) m w)
((NBWORD::nat => nat => bool word)
((op -::nat => nat => nat) n m) (0::nat)))
((op =::bool word => bool word => bool)
((WSEG::nat => nat => bool word => bool word) m
(0::nat) w)
((NBWORD::nat => nat => bool word) m (0::nat))))))))"
by (import bword_num EQ_NBWORD0_SPLIT)
lemma NBWORD_MOD: "ALL (n::nat) m::nat. NBWORD n (m mod (2::nat) ^ n) = NBWORD n m"
by (import bword_num NBWORD_MOD)
lemma WSEG_NBWORD_SUC: "ALL (n::nat) m::nat. WSEG n (0::nat) (NBWORD (Suc n) m) = NBWORD n m"
by (import bword_num WSEG_NBWORD_SUC)
lemma NBWORD_SUC_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
(%w::bool word. NBWORD n (BNVAL w) = WSEG n (0::nat) w)"
by (import bword_num NBWORD_SUC_WSEG)
lemma DOUBL_EQ_SHL: "ALL x>0::nat.
RES_FORALL (PWORDLEN x)
(%xa::bool word.
ALL xb::bool.
NBWORD x (BNVAL xa + BNVAL xa + BV xb) = snd (SHL False xa xb))"
by (import bword_num DOUBL_EQ_SHL)
lemma MSB_NBWORD: "ALL (n::nat) m::nat.
bit n (NBWORD (Suc n) m) = VB (m div (2::nat) ^ n mod (2::nat))"
by (import bword_num MSB_NBWORD)
lemma NBWORD_SPLIT: "ALL (n1::nat) (n2::nat) m::nat.
NBWORD (n1 + n2) m = WCAT (NBWORD n1 (m div (2::nat) ^ n2), NBWORD n2 m)"
by (import bword_num NBWORD_SPLIT)
lemma WSEG_NBWORD: "ALL (m::nat) (k::nat) n::nat.
m + k <= n -->
(ALL l::nat. WSEG m k (NBWORD n l) = NBWORD m (l div (2::nat) ^ k))"
by (import bword_num WSEG_NBWORD)
lemma NBWORD_SUC_FST: "ALL (n::nat) x::nat.
NBWORD (Suc n) x =
WCAT (WORD [VB (x div (2::nat) ^ n mod (2::nat))], NBWORD n x)"
by (import bword_num NBWORD_SUC_FST)
lemma BIT_NBWORD0: "ALL (k::nat) n::nat. k < n --> bit k (NBWORD n (0::nat)) = False"
by (import bword_num BIT_NBWORD0)
lemma ADD_BNVAL_LEFT: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
(%w1::bool word.
RES_FORALL (PWORDLEN (Suc n))
(%w2::bool word.
BNVAL w1 + BNVAL w2 =
(BV (bit n w1) + BV (bit n w2)) * (2::nat) ^ n +
(BNVAL (WSEG n (0::nat) w1) + BNVAL (WSEG n (0::nat) w2))))"
by (import bword_num ADD_BNVAL_LEFT)
lemma ADD_BNVAL_RIGHT: "ALL n::nat.
RES_FORALL (PWORDLEN (Suc n))
(%w1::bool word.
RES_FORALL (PWORDLEN (Suc n))
(%w2::bool word.
BNVAL w1 + BNVAL w2 =
(BNVAL (WSEG n (1::nat) w1) + BNVAL (WSEG n (1::nat) w2)) *
(2::nat) +
(BV (bit (0::nat) w1) + BV (bit (0::nat) w2))))"
by (import bword_num ADD_BNVAL_RIGHT)
lemma ADD_BNVAL_SPLIT: "ALL (n1::nat) n2::nat.
RES_FORALL (PWORDLEN (n1 + n2))
(%w1::bool word.
RES_FORALL (PWORDLEN (n1 + n2))
(%w2::bool word.
BNVAL w1 + BNVAL w2 =
(BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2)) *
(2::nat) ^ n2 +
(BNVAL (WSEG n2 (0::nat) w1) + BNVAL (WSEG n2 (0::nat) w2))))"
by (import bword_num ADD_BNVAL_SPLIT)
;end_setup
;setup_theory bword_arith
consts
ACARRY :: "nat => bool word => bool word => bool => bool"
specification (ACARRY) ACARRY_DEF: "(ALL (w1::bool word) (w2::bool word) cin::bool.
ACARRY (0::nat) w1 w2 cin = cin) &
(ALL (n::nat) (w1::bool word) (w2::bool word) cin::bool.
ACARRY (Suc n) w1 w2 cin =
VB ((BV (bit n w1) + BV (bit n w2) + BV (ACARRY n w1 w2 cin)) div
(2::nat)))"
by (import bword_arith ACARRY_DEF)
consts
ICARRY :: "nat => bool word => bool word => bool => bool"
specification (ICARRY) ICARRY_DEF: "(ALL (w1::bool word) (w2::bool word) cin::bool.
ICARRY (0::nat) w1 w2 cin = cin) &
(ALL (n::nat) (w1::bool word) (w2::bool word) cin::bool.
ICARRY (Suc n) w1 w2 cin =
(bit n w1 & bit n w2 | (bit n w1 | bit n w2) & ICARRY n w1 w2 cin))"
by (import bword_arith ICARRY_DEF)
lemma ACARRY_EQ_ICARRY: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
ALL (cin::bool) k::nat.
k <= n --> ACARRY k w1 w2 cin = ICARRY k w1 w2 cin))"
by (import bword_arith ACARRY_EQ_ICARRY)
lemma BNVAL_LESS_EQ: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w::bool word. BNVAL w <= (2::nat) ^ n - (1::nat))"
by (import bword_arith BNVAL_LESS_EQ)
lemma ADD_BNVAL_LESS_EQ1: "ALL (n::nat) cin::bool.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
(BNVAL w1 + (BNVAL w2 + BV cin)) div (2::nat) ^ n
<= Suc (0::nat)))"
by (import bword_arith ADD_BNVAL_LESS_EQ1)
lemma ADD_BV_BNVAL_DIV_LESS_EQ1: "ALL (n::nat) (x1::bool) (x2::bool) cin::bool.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
(BV x1 + BV x2 +
(BNVAL w1 + (BNVAL w2 + BV cin)) div (2::nat) ^ n) div
(2::nat)
<= (1::nat)))"
by (import bword_arith ADD_BV_BNVAL_DIV_LESS_EQ1)
lemma ADD_BV_BNVAL_LESS_EQ: "ALL (n::nat) (x1::bool) (x2::bool) cin::bool.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
BV x1 + BV x2 + (BNVAL w1 + (BNVAL w2 + BV cin))
<= Suc ((2::nat) ^ Suc n)))"
by (import bword_arith ADD_BV_BNVAL_LESS_EQ)
lemma ADD_BV_BNVAL_LESS_EQ1: "ALL (n::nat) (x1::bool) (x2::bool) cin::bool.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
(BV x1 + BV x2 + (BNVAL w1 + (BNVAL w2 + BV cin))) div
(2::nat) ^ Suc n
<= (1::nat)))"
by (import bword_arith ADD_BV_BNVAL_LESS_EQ1)
lemma ACARRY_EQ_ADD_DIV: "(All::(nat => bool) => bool)
(%n::nat.
(RES_FORALL::(bool word => bool) => (bool word => bool) => bool)
((PWORDLEN::nat => bool word => bool) n)
(%w1::bool word.
(RES_FORALL::(bool word => bool) => (bool word => bool) => bool)
((PWORDLEN::nat => bool word => bool) n)
(%w2::bool word.
(All::(nat => bool) => bool)
(%k::nat.
(op -->::bool => bool => bool)
((op <::nat => nat => bool) k n)
((op =::nat => nat => bool)
((BV::bool => nat)
((ACARRY::nat
=> bool word
=> bool word => bool => bool)
k w1 w2 (cin::bool)))
((op div::nat => nat => nat)
((op +::nat => nat => nat)
((op +::nat => nat => nat)
((BNVAL::bool word => nat)
((WSEG::nat => nat => bool word => bool word)
k (0::nat) w1))
((BNVAL::bool word => nat)
((WSEG::nat => nat => bool word => bool word)
k (0::nat) w2)))
((BV::bool => nat) cin))
((op ^::nat => nat => nat)
((number_of::bin => nat)
((op BIT::bin => bit => bin)
((op BIT::bin => bit => bin)
(Numeral.Pls::bin) (bit.B1::bit))
(bit.B0::bit)))
k)))))))"
by (import bword_arith ACARRY_EQ_ADD_DIV)
lemma ADD_WORD_SPLIT: "ALL (n1::nat) n2::nat.
RES_FORALL (PWORDLEN (n1 + n2))
(%w1::bool word.
RES_FORALL (PWORDLEN (n1 + n2))
(%w2::bool word.
ALL cin::bool.
NBWORD (n1 + n2) (BNVAL w1 + BNVAL w2 + BV cin) =
WCAT
(NBWORD n1
(BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2) +
BV (ACARRY n2 w1 w2 cin)),
NBWORD n2
(BNVAL (WSEG n2 (0::nat) w1) +
BNVAL (WSEG n2 (0::nat) w2) +
BV cin))))"
by (import bword_arith ADD_WORD_SPLIT)
lemma WSEG_NBWORD_ADD: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
ALL (m::nat) (k::nat) cin::bool.
m + k <= n -->
WSEG m k (NBWORD n (BNVAL w1 + BNVAL w2 + BV cin)) =
NBWORD m
(BNVAL (WSEG m k w1) + BNVAL (WSEG m k w2) +
BV (ACARRY k w1 w2 cin))))"
by (import bword_arith WSEG_NBWORD_ADD)
lemma ADD_NBWORD_EQ0_SPLIT: "ALL (n1::nat) n2::nat.
RES_FORALL (PWORDLEN (n1 + n2))
(%w1::bool word.
RES_FORALL (PWORDLEN (n1 + n2))
(%w2::bool word.
ALL cin::bool.
(NBWORD (n1 + n2) (BNVAL w1 + BNVAL w2 + BV cin) =
NBWORD (n1 + n2) (0::nat)) =
(NBWORD n1
(BNVAL (WSEG n1 n2 w1) + BNVAL (WSEG n1 n2 w2) +
BV (ACARRY n2 w1 w2 cin)) =
NBWORD n1 (0::nat) &
NBWORD n2
(BNVAL (WSEG n2 (0::nat) w1) +
BNVAL (WSEG n2 (0::nat) w2) +
BV cin) =
NBWORD n2 (0::nat))))"
by (import bword_arith ADD_NBWORD_EQ0_SPLIT)
lemma ACARRY_MSB: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
ALL cin::bool.
ACARRY n w1 w2 cin =
bit n (NBWORD (Suc n) (BNVAL w1 + BNVAL w2 + BV cin))))"
by (import bword_arith ACARRY_MSB)
lemma ACARRY_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
ALL (cin::bool) (k::nat) m::nat.
k < m & m <= n -->
ACARRY k (WSEG m (0::nat) w1) (WSEG m (0::nat) w2) cin =
ACARRY k w1 w2 cin))"
by (import bword_arith ACARRY_WSEG)
lemma ICARRY_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
ALL (cin::bool) (k::nat) m::nat.
k < m & m <= n -->
ICARRY k (WSEG m (0::nat) w1) (WSEG m (0::nat) w2) cin =
ICARRY k w1 w2 cin))"
by (import bword_arith ICARRY_WSEG)
lemma ACARRY_ACARRY_WSEG: "ALL n::nat.
RES_FORALL (PWORDLEN n)
(%w1::bool word.
RES_FORALL (PWORDLEN n)
(%w2::bool word.
ALL (cin::bool) (m::nat) (k1::nat) k2::nat.
k1 < m & k2 < n & m + k2 <= n -->
ACARRY k1 (WSEG m k2 w1) (WSEG m k2 w2)
(ACARRY k2 w1 w2 cin) =
ACARRY (k1 + k2) w1 w2 cin))"
by (import bword_arith ACARRY_ACARRY_WSEG)
;end_setup
;setup_theory bword_bitop
consts
WNOT :: "bool word => bool word"
specification (WNOT) WNOT_DEF: "ALL l::bool list. WNOT (WORD l) = WORD (map Not l)"
by (import bword_bitop WNOT_DEF)
lemma PBITOP_WNOT: "IN WNOT PBITOP"
by (import bword_bitop PBITOP_WNOT)
lemma WNOT_WNOT: "ALL w::bool word. WNOT (WNOT w) = w"
by (import bword_bitop WNOT_WNOT)
lemma WCAT_WNOT: "ALL (n1::nat) n2::nat.
RES_FORALL (PWORDLEN n1)
(%w1::bool word.
RES_FORALL (PWORDLEN n2)
(%w2::bool word. WCAT (WNOT w1, WNOT w2) = WNOT (WCAT (w1, w2))))"
by (import bword_bitop WCAT_WNOT)
consts
WAND :: "bool word => bool word => bool word"
specification (WAND) WAND_DEF: "ALL (l1::bool list) l2::bool list.
WAND (WORD l1) (WORD l2) = WORD (map2 op & l1 l2)"
by (import bword_bitop WAND_DEF)
lemma PBITBOP_WAND: "IN WAND PBITBOP"
by (import bword_bitop PBITBOP_WAND)
consts
WOR :: "bool word => bool word => bool word"
specification (WOR) WOR_DEF: "ALL (l1::bool list) l2::bool list.
WOR (WORD l1) (WORD l2) = WORD (map2 op | l1 l2)"
by (import bword_bitop WOR_DEF)
lemma PBITBOP_WOR: "IN WOR PBITBOP"
by (import bword_bitop PBITBOP_WOR)
consts
WXOR :: "bool word => bool word => bool word"
specification (WXOR) WXOR_DEF: "ALL (l1::bool list) l2::bool list.
WXOR (WORD l1) (WORD l2) = WORD (map2 (%(x::bool) y::bool. x ~= y) l1 l2)"
by (import bword_bitop WXOR_DEF)
lemma PBITBOP_WXOR: "IN WXOR PBITBOP"
by (import bword_bitop PBITBOP_WXOR)
;end_setup
end