--- a/src/HOL/Import/HOL4/Generated/HOL4Vec.thy Sun Apr 01 14:50:47 2012 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,1105 +0,0 @@
-(* AUTOMATICALLY GENERATED, DO NOT EDIT! *)
-
-theory HOL4Vec imports HOL4Base begin
-
-setup_theory "~~/src/HOL/Import/HOL4/Generated" res_quan
-
-lemma RES_FORALL_CONJ_DIST: "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: "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: "RES_FORALL (op = xa) x = x xa"
- by (import res_quan RES_FORALL_UNIQUE)
-
-lemma RES_FORALL_FORALL: "(ALL x::'b.
- RES_FORALL (P::'a => bool) (%i::'a. (R::'a => 'b => bool) i x)) =
-RES_FORALL P (%i::'a. All (R i))"
- by (import res_quan RES_FORALL_FORALL)
-
-lemma RES_FORALL_REORDER: "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: "RES_FORALL EMPTY x"
- by (import res_quan RES_FORALL_EMPTY)
-
-lemma RES_FORALL_UNIV: "RES_FORALL pred_set.UNIV p = All p"
- by (import res_quan RES_FORALL_UNIV)
-
-lemma RES_FORALL_NULL: "RES_FORALL p (%x. m) = (p = EMPTY | m)"
- by (import res_quan RES_FORALL_NULL)
-
-lemma RES_EXISTS_DISJ_DIST: "RES_EXISTS P (%i. 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: "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: "RES_EXISTS (op = xa) x = x xa"
- by (import res_quan RES_EXISTS_EQUAL)
-
-lemma RES_EXISTS_REORDER: "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: "~ RES_EXISTS EMPTY p"
- by (import res_quan RES_EXISTS_EMPTY)
-
-lemma RES_EXISTS_UNIV: "RES_EXISTS pred_set.UNIV p = Ex p"
- by (import res_quan RES_EXISTS_UNIV)
-
-lemma RES_EXISTS_NULL: "RES_EXISTS p (%x. m) = (p ~= EMPTY & m)"
- by (import res_quan RES_EXISTS_NULL)
-
-lemma RES_EXISTS_ALT: "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: "~ RES_EXISTS_UNIQUE EMPTY p"
- by (import res_quan RES_EXISTS_UNIQUE_EMPTY)
-
-lemma RES_EXISTS_UNIQUE_UNIV: "RES_EXISTS_UNIQUE pred_set.UNIV p = Ex1 p"
- by (import res_quan RES_EXISTS_UNIQUE_UNIV)
-
-lemma RES_EXISTS_UNIQUE_NULL: "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: "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: "RES_SELECT EMPTY p = (SOME x. False)"
- by (import res_quan RES_SELECT_EMPTY)
-
-lemma RES_SELECT_UNIV: "RES_SELECT pred_set.UNIV p = Eps p"
- by (import res_quan RES_SELECT_UNIV)
-
-lemma RES_ABSTRACT: "IN x p ==> RES_ABSTRACT p m x = m x"
- by (import res_quan RES_ABSTRACT)
-
-lemma RES_ABSTRACT_EQUAL: "(!!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: "RES_ABSTRACT p (RES_ABSTRACT p m) = RES_ABSTRACT p m"
- by (import res_quan RES_ABSTRACT_IDEMPOT)
-
-lemma RES_ABSTRACT_EQUAL_EQ: "(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 "~~/src/HOL/Import/HOL4/Generated" word_base
-
-typedef (open) ('a) word = "{x. ALL word.
- (ALL a0. (EX a. a0 = CONSTR 0 a (%n. BOTTOM)) --> word a0) -->
- word x} :: ('a::type list recspace set)"
- 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)
-
-definition
- WORD :: "'a list => 'a word" where
- "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_base.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_base.word_size f (WORD a) = 1 + Compatibility.list_size f a"
- by (import word_base word_size_def)
-
-lemma word_11: "(WORD a = WORD a') = (a = a')"
- by (import word_base word_11)
-
-lemma word_case_cong: "M = M' & (ALL a. M' = WORD a --> f a = f' a)
-==> word_base.word_case f M = word_base.word_case f' M'"
- by (import word_base word_case_cong)
-
-lemma word_nchotomy: "EX l. x = WORD l"
- by (import word_base word_nchotomy)
-
-lemma word_Axiom: "EX fn. ALL a. fn (WORD a) = f a"
- by (import word_base word_Axiom)
-
-lemma word_induction: "(!!a. P (WORD a)) ==> P x"
- by (import word_base word_induction)
-
-lemma word_Ax: "EX fn. ALL a. fn (WORD a) = f a"
- by (import word_base word_Ax)
-
-lemma WORD_11: "(WORD x = WORD xa) = (x = xa)"
- by (import word_base WORD_11)
-
-lemma word_induct: "(!!l. x (WORD l)) ==> x xa"
- by (import word_base word_induct)
-
-lemma word_cases: "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: "PWORDLEN n = GSPEC (%w. (w, WORDLEN w = n))"
- by (import word_base PWORDLEN_def)
-
-lemma IN_PWORDLEN: "IN (WORD l) (PWORDLEN n) = (length l = n)"
- by (import word_base IN_PWORDLEN)
-
-lemma PWORDLEN: "IN w (PWORDLEN n) = (WORDLEN w = n)"
- by (import word_base PWORDLEN)
-
-lemma PWORDLEN0: "IN w (PWORDLEN 0) ==> w = WORD []"
- by (import word_base PWORDLEN0)
-
-lemma PWORDLEN1: "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: "WSEG 0 k w = WORD []"
- by (import word_base WSEG0)
-
-lemma WSEG_PWORDLEN: "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: "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: "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: "bit 0 (WORD [x]) = x"
- by (import word_base BIT0)
-
-lemma WSEG_BIT: "RES_FORALL (PWORDLEN n) (%w. ALL k<n. WSEG 1 k w = WORD [bit k w])"
- by (import word_base WSEG_BIT)
-
-lemma BIT_WSEG: "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: "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: "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: "(ALL n::nat.
- RES_FORALL (PWORDLEN n)
- (%w::'a word. ALL m<=n. WCAT (WSPLIT m w) = w)) &
-(ALL (n::nat) m::nat.
- RES_FORALL (PWORDLEN n)
- (%w1::'a word.
- RES_FORALL (PWORDLEN m)
- (%w2::'a word. WSPLIT m (WCAT (w1, w2)) = (w1, w2))))"
- by (import word_base WORD_PARTITION)
-
-lemma WCAT_ASSOC: "WCAT (w1, WCAT (w2, w3)) = WCAT (WCAT (w1, w2), w3)"
- by (import word_base WCAT_ASSOC)
-
-lemma WCAT0: "WCAT (WORD [], w) = w & WCAT (w, WORD []) = w"
- by (import word_base WCAT0)
-
-lemma WCAT_11: "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: "RES_FORALL (PWORDLEN n)
- (%w. ALL m<=n.
- IN (fst (WSPLIT m w)) (PWORDLEN (n - m)) &
- IN (snd (WSPLIT m w)) (PWORDLEN m))"
- by (import word_base WSPLIT_PWORDLEN)
-
-lemma WCAT_PWORDLEN: "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: "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: "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: "RES_FORALL (PWORDLEN n)
- (%w. ALL k<=n. WSPLIT k w = (WSEG (n - k) k w, WSEG k 0 w))"
- by (import word_base WSPLIT_WSEG)
-
-lemma WSPLIT_WSEG1: "RES_FORALL (PWORDLEN n) (%w. ALL k<=n. fst (WSPLIT k w) = WSEG (n - k) k w)"
- by (import word_base WSPLIT_WSEG1)
-
-lemma WSPLIT_WSEG2: "RES_FORALL (PWORDLEN n) (%w. ALL k<=n. snd (WSPLIT k w) = WSEG k 0 w)"
- by (import word_base WSPLIT_WSEG2)
-
-lemma WCAT_WSEG_WSEG: "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: "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: "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: "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: "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: "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: "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: "WORD (x # l) = WCAT (WORD [x], WORD l)"
- by (import word_base WORD_CONS_WCAT)
-
-lemma WORD_SNOC_WCAT: "WORD (SNOC x l) = WCAT (WORD l, WORD [x])"
- by (import word_base WORD_SNOC_WCAT)
-
-lemma BIT_WCAT_FST: "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: "RES_FORALL (PWORDLEN n1)
- (%w1. RES_FORALL (PWORDLEN n2)
- (%w2. ALL k<n2. bit k (WCAT (w1, w2)) = bit k w2))"
- by (import word_base BIT_WCAT_SND)
-
-lemma BIT_WCAT1: "RES_FORALL (PWORDLEN n) (%w. ALL b. bit n (WCAT (WORD [b], w)) = b)"
- by (import word_base BIT_WCAT1)
-
-lemma WSEG_WCAT_WSEG1: "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: "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: "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: "RES_FORALL (PWORDLEN n)
- (%w1. RES_FORALL (PWORDLEN n)
- (%w2. (ALL k<n. bit k w1 = bit k w2) --> w1 = w2))"
- by (import word_base BIT_EQ_IMP_WORD_EQ)
-
-;end_setup
-
-setup_theory "~~/src/HOL/Import/HOL4/Generated" word_num
-
-definition
- LVAL :: "('a => nat) => nat => 'a list => nat" where
- "LVAL == %f b. foldl (%e x. b * e + f x) 0"
-
-lemma LVAL_DEF: "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: "LVAL f b (SNOC h l) = LVAL f b l * b + f h"
- by (import word_num LVAL_SNOC)
-
-lemma LVAL_MAX: "(!!x. f x < b) ==> LVAL f b l < b ^ length l"
- by (import word_num LVAL_MAX)
-
-lemma NVAL_MAX: "(!!x. f x < b) ==> RES_FORALL (PWORDLEN n) (%w. NVAL f b w < b ^ n)"
- by (import word_num NVAL_MAX)
-
-lemma NVAL0: "NVAL x xa (WORD []) = 0"
- by (import word_num NVAL0)
-
-lemma NVAL1: "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: "NVAL f b (WCAT (w, WORD [x])) = NVAL f b w * b + f x"
- by (import word_num NVAL_WCAT1)
-
-lemma NVAL_WCAT2: "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: "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)
-
-definition
- NWORD :: "nat => (nat => 'a) => nat => nat => 'a word" where
- "NWORD == %n frep b m. WORD (NLIST n frep b m)"
-
-lemma NWORD_DEF: "NWORD n frep b m = WORD (NLIST n frep b m)"
- by (import word_num NWORD_DEF)
-
-lemma NWORD_LENGTH: "WORDLEN (NWORD x xa xb xc) = x"
- by (import word_num NWORD_LENGTH)
-
-lemma NWORD_PWORDLEN: "IN (NWORD x xa xb xc) (PWORDLEN x)"
- by (import word_num NWORD_PWORDLEN)
-
-;end_setup
-
-setup_theory "~~/src/HOL/Import/HOL4/Generated" 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: "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 PBITOP
- (%oper.
- ALL n.
- RES_FORALL (PWORDLEN n)
- (%w. ALL k<n. oper (WORD [bit k w]) = WORD [bit k (oper w)]))"
- 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: "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: "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: "WMAP (x::'a => 'b) (WORD []) = WORD []"
- by (import word_bitop WMAP_0)
-
-lemma WMAP_BIT: "RES_FORALL (PWORDLEN n) (%w. ALL k<n. ALL f. bit k (WMAP f w) = f (bit k w))"
- by (import word_bitop WMAP_BIT)
-
-lemma WMAP_WSEG: "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: "IN (WMAP f) PBITOP"
- by (import word_bitop WMAP_PBITOP)
-
-lemma WMAP_WCAT: "WMAP (f::'a => 'b) (WCAT (w1::'a word, w2::'a word)) =
-WCAT (WMAP f w1, WMAP f w2)"
- by (import word_bitop WMAP_WCAT)
-
-lemma WMAP_o: "WMAP (g::'b => 'c) (WMAP (f::'a => 'b) (w::'a word)) = 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: "RES_FORALL (PWORDLEN n) (%w. ALL P. FORALLBITS P w = (ALL k<n. P (bit k w)))"
- by (import word_bitop FORALLBITS)
-
-lemma FORALLBITS_WSEG: "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: "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_ex P l"
- by (import word_bitop EXISTSABIT_DEF)
-
-lemma NOT_EXISTSABIT: "(~ EXISTSABIT P w) = FORALLBITS (Not o P) w"
- by (import word_bitop NOT_EXISTSABIT)
-
-lemma NOT_FORALLBITS: "(~ FORALLBITS P w) = EXISTSABIT (Not o P) w"
- by (import word_bitop NOT_FORALLBITS)
-
-lemma EXISTSABIT: "RES_FORALL (PWORDLEN n) (%w. ALL P. EXISTSABIT P w = (EX k<n. P (bit k w)))"
- by (import word_bitop EXISTSABIT)
-
-lemma EXISTSABIT_WSEG: "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: "EXISTSABIT P (WCAT (w1, w2)) = (EXISTSABIT P w1 | EXISTSABIT P w2)"
- by (import word_bitop EXISTSABIT_WCAT)
-
-definition
- SHR :: "bool => 'a => 'a word => 'a word * 'a" where
- "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: "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)
-
-definition
- SHL :: "bool => 'a word => 'a => 'a * 'a word" where
- "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: "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: "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: "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: "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: "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: "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: "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: "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: "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 "~~/src/HOL/Import/HOL4/Generated" bword_num
-
-definition
- BV :: "bool => nat" where
- "BV == %b. if b then Suc 0 else 0"
-
-lemma BV_DEF: "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: "BV x < 2"
- by (import bword_num BV_LESS_2)
-
-lemma BNVAL_NVAL: "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: "[| WORDLEN w1 = WORDLEN w2; BNVAL w1 = BNVAL w2 |] ==> w1 = w2"
- by (import bword_num BNVAL_11)
-
-lemma BNVAL_ONTO: "Ex (op = (BNVAL w))"
- by (import bword_num BNVAL_ONTO)
-
-lemma BNVAL_MAX: "RES_FORALL (PWORDLEN n) (%w. BNVAL w < 2 ^ n)"
- by (import bword_num BNVAL_MAX)
-
-lemma BNVAL_WCAT1: "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: "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: "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)
-
-definition
- VB :: "nat => bool" where
- "VB == %n. n mod 2 ~= 0"
-
-lemma VB_DEF: "VB n = (n mod 2 ~= 0)"
- by (import bword_num VB_DEF)
-
-definition
- NBWORD :: "nat => nat => bool word" where
- "NBWORD == %n m. WORD (NLIST n VB 2 m)"
-
-lemma NBWORD_DEF: "NBWORD n m = WORD (NLIST n VB 2 m)"
- by (import bword_num NBWORD_DEF)
-
-lemma NBWORD0: "NBWORD 0 x = WORD []"
- by (import bword_num NBWORD0)
-
-lemma WORDLEN_NBWORD: "WORDLEN (NBWORD x xa) = x"
- by (import bword_num WORDLEN_NBWORD)
-
-lemma PWORDLEN_NBWORD: "IN (NBWORD x xa) (PWORDLEN x)"
- by (import bword_num PWORDLEN_NBWORD)
-
-lemma NBWORD_SUC: "NBWORD (Suc n) m = WCAT (NBWORD n (m div 2), WORD [VB (m mod 2)])"
- by (import bword_num NBWORD_SUC)
-
-lemma VB_BV: "VB (BV x) = x"
- by (import bword_num VB_BV)
-
-lemma BV_VB: "x < 2 ==> BV (VB x) = x"
- by (import bword_num BV_VB)
-
-lemma NBWORD_BNVAL: "RES_FORALL (PWORDLEN n) (%w. NBWORD n (BNVAL w) = w)"
- by (import bword_num NBWORD_BNVAL)
-
-lemma BNVAL_NBWORD: "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: "WCAT (NBWORD n1 0, NBWORD n2 0) = NBWORD (n1 + n2) 0"
- by (import bword_num WCAT_NBWORD_0)
-
-lemma WSPLIT_NBWORD_0: "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: "RES_FORALL (PWORDLEN n)
- (%w. ALL m<=n.
- (w = NBWORD n 0) =
- (WSEG (n - m) m w = NBWORD (n - m) 0 & WSEG m 0 w = NBWORD m 0))"
- by (import bword_num EQ_NBWORD0_SPLIT)
-
-lemma NBWORD_MOD: "NBWORD n (m mod 2 ^ n) = NBWORD n m"
- by (import bword_num NBWORD_MOD)
-
-lemma WSEG_NBWORD_SUC: "WSEG n 0 (NBWORD (Suc n) m) = NBWORD n m"
- by (import bword_num WSEG_NBWORD_SUC)
-
-lemma NBWORD_SUC_WSEG: "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: "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: "bit n (NBWORD (Suc n) m) = VB (m div 2 ^ n mod 2)"
- by (import bword_num MSB_NBWORD)
-
-lemma NBWORD_SPLIT: "NBWORD (n1 + n2) m = WCAT (NBWORD n1 (m div 2 ^ n2), NBWORD n2 m)"
- by (import bword_num NBWORD_SPLIT)
-
-lemma WSEG_NBWORD: "m + k <= n ==> WSEG m k (NBWORD n l) = NBWORD m (l div 2 ^ k)"
- by (import bword_num WSEG_NBWORD)
-
-lemma NBWORD_SUC_FST: "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: "k < n ==> bit k (NBWORD n 0) = False"
- by (import bword_num BIT_NBWORD0)
-
-lemma ADD_BNVAL_LEFT: "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: "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: "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 "~~/src/HOL/Import/HOL4/Generated" 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: "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: "RES_FORALL (PWORDLEN n) (%w. BNVAL w <= 2 ^ n - 1)"
- by (import bword_arith BNVAL_LESS_EQ)
-
-lemma ADD_BNVAL_LESS_EQ1: "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: "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: "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: "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: "RES_FORALL (PWORDLEN n)
- (%w1. RES_FORALL (PWORDLEN n)
- (%w2. ALL k<n.
- BV (ACARRY k w1 w2 cin) =
- (BNVAL (WSEG k 0 w1) + BNVAL (WSEG k 0 w2) + BV cin) div
- 2 ^ k))"
- by (import bword_arith ACARRY_EQ_ADD_DIV)
-
-lemma ADD_WORD_SPLIT: "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: "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: "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: "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: "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: "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: "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 "~~/src/HOL/Import/HOL4/Generated" 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: "WNOT (WNOT w) = w"
- by (import bword_bitop WNOT_WNOT)
-
-lemma WCAT_WNOT: "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 op ~= l1 l2)"
- by (import bword_bitop WXOR_DEF)
-
-lemma PBITBOP_WXOR: "IN WXOR PBITBOP"
- by (import bword_bitop PBITBOP_WXOR)
-
-;end_setup
-
-end
-