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