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* *****************************************************************
* NB: This CHANGES file no longer gives a comprehensive list of *
* changes made to the system. In particular, most changes in the *
* Multivariate theories are excluded, simply because there are *
* so many of them that tracking them would be tedious. For more *
* detailed update lists, consult the git logs ("git log" if you *
* have the system downloaded) or the list of commits on the Web *
* page: https://github.com/jrh13/hol-light/commits/master *
* *****************************************************************
Mon 1st Aug 2022 Library/words.ml
Added a straightforward variant of existing word lemmas:
REAL_VAL_WORD_XOR =
|- !x y.
&(val(word_xor x y)) =
(&(val x) + &(val y)) - &2 * &(val(word_and x y))
Wed 27th Jul 2022 tactics.ml, define.ml, meson.ml, Examples/holby.ml, Examples/mizar.ml, 100/e_is_transcendental.ml
Made "prove" explicitly check that the result does not include any
additional assumptions. This seems usually what is expected (if not
TAC_PROOF can be used directly). There were just a few places
where the original behavior was used essentially, and those have
been updated.
Mon 25th Jul 2022 EC/computegroup.ml
Slightly improved the efficiency of ECGROUP_MUL_CONV using a more
refined reduction strategy.
Mon 25th Jul 2022 pair.ml
Added SUBLET_CONV, to apply a conversion to the RHSs of the toplevel
let-term but neither expand it as with let_CONV nor apply the
conversion to the body, e.g.
# SUBLET_CONV NUM_ADD_CONV
`let x = 5 + 2 and y = 8 + 17 and z = 3 + 7 in x + y + z`;;
val it : thm =
|- (let x = 5 + 2 and y = 8 + 17 and z = 3 + 7 in x + y + z) =
(let x = 7 and y = 25 and z = 10 in x + y + z)
Tue 10th May 2022 EC/edwards448.ml [new file], Library/grouptheory.ml
Added a formalization of the "Goldilocks" curve edwards448, based on Mike
Hamburg's paper "Ed448-Goldilocks" (https://eprint.iacr.org/2015/625.pdf).
So far, we just have the Edwards form of it as in that paper. This is all
highly analogous to curve25519.ml, except that the group order computation
is a bit more involved: to avoid using a more refined Hasse-type bound
we need to do a bit of explicit analysis of the low-order points because
the group order is less than half of the very naive bound.
Also made a couple of trivial variable name tweaks to grouptheory.ml to
avoid name clashes with "decode" and "encode" when loaded with some other
theories.
Fri 22nd Apr 2022 GL/*
Added an update from Marco Maggesi and Cosimo Perini Brogi to their GL
provability logic theory. This now includes a decision procedure (GL_RULE
and GL_TAC) for the GL logic that will prove a valid formula or generate
a countermodel for an invalid one (stored in "!the_gl_countermodel").
This example, in the provability interpretation, corresponds to a
formalized Goedel's Second Incompleteness Theorem: if a system is
consistent it does not prove its own consistency. The file GL/decid.ml
has a number of applications of this kind:
# GL_RULE `|-- (Not (Box False) --> Not (Box (Not (Box False))))`;;
val it : thm = |- |-- (Not Box False --> Not Box Not Box False)
Wed 6th Apr 2022 EC/* [new directory], Examples/nist_curves.ml [deleted]
Replaced the increasingly unwieldy and poorly-named "nist_curves.ml"
file with a new directory "EC" (for Elliptic Curves) that develops
elliptic curve theory in a more systematic way, with separate files
for all the specific curves, as well as adding new material,
particularly around Edwards curves. The general short Weierstrass,
Montgomery and Edwards forms are defined and their properties derived
separately to minimize interdependencies, and here the only field
characteristic assumptions made are common or natural ones: char = 2
and char = 3 are excluded for short Weierstrass, just char = 2 is
excluded for Montgomery, and in principle there are no restrictions at
all on the Edwards theory (though "edwards_nonsingular" cannot be
satisfied in a *finite* field of characteristic 2 since then every
element is a square).
Wed 6th Apr 2022 lists.ml
Added a little clausal rewrite for BUTLAST:
BUTLAST_CLAUSES =
|- BUTLAST [] = [] /\
(!a. BUTLAST [a] = []) /\
(!a h t. BUTLAST(CONS a (CONS h t)) = CONS a (BUTLAST(CONS h t)))
Mon 4th Apr 2022 Library/ringtheory.ml
Fixed a prenormalization bug where RING_RULE and RING_TAC were not
handling the general case of "ring_of_num r n".
Fri 1st Apr 2022 Library/ringtheory.ml
Added a few more trivial theorems about terms involving inversion being
zero in a field and a handy collection of the various individual closure
properties (RING_ADD, RING_POW etc.) into a single clausal theorem:
FIELD_DIV_EQ_0 =
|- !f x y.
field f /\ x IN ring_carrier f /\ y IN ring_carrier f
==> (ring_div f x y = ring_0 f <=> x = ring_0 f \/ y = ring_0 f)
FIELD_INV_EQ_0 =
|- !f x.
field f /\ x IN ring_carrier f
==> (ring_inv f x = ring_0 f <=> x = ring_0 f)
RING_1_DIV =
|- !r x. x IN ring_carrier r ==> ring_div r (ring_1 r) x = ring_inv r x
RING_CLAUSES =
|- (!r. ring_0 r IN ring_carrier r) /\
(!r. ring_1 r IN ring_carrier r) /\
(!r n. ring_of_num r n IN ring_carrier r) /\
(!r x. x IN ring_carrier r ==> ring_neg r x IN ring_carrier r) /\
(!r x. x IN ring_carrier r ==> ring_inv r x IN ring_carrier r) /\
(!r x y.
x IN ring_carrier r /\ y IN ring_carrier r
==> ring_add r x y IN ring_carrier r) /\
(!r x y.
x IN ring_carrier r /\ y IN ring_carrier r
==> ring_sub r x y IN ring_carrier r) /\
(!r x y.
x IN ring_carrier r /\ y IN ring_carrier r
==> ring_mul r x y IN ring_carrier r) /\
(!r x y.
x IN ring_carrier r /\ y IN ring_carrier r
==> ring_div r x y IN ring_carrier r) /\
(!r x n. x IN ring_carrier r ==> ring_pow r x n IN ring_carrier r)
Tue 29th Mar 2022 Library/grouptheory.ml
Added a couple more lemmas about orders of powers in groups:
GROUP_ELEMENT_ORDER_EQ_MUL_GEN =
|- !G x k n.
x IN group_carrier G /\ ~(k = 0)
==> (group_element_order G x = k * n <=>
k divides group_element_order G x /\
group_element_order G (group_pow G x k) = n)
GROUP_ELEMENT_ORDER_EQ_MUL =
|- !G x k n.
x IN group_carrier G /\ ~(k = 0) /\ k divides n
==> (group_element_order G x = k * n <=>
group_element_order G (group_pow G x k) = n)
Wed 23rd Mar 2022 Library/ringtheory.ml
Added one more trivial rewrite that's handy to hide explicit case splits
over the trivial case of integers modulo:
IN_INTEGER_MOD_RING_CARRIER =
|- !n a.
a IN ring_carrier(integer_mod_ring n) <=>
&n = &0 \/ &0 <= a /\ a < &n
Wed 16th Mar 2022 Library/ringtheory.ml
Added some basic support for generic field automation to the ring theory,
adapted and extended from material already there in "nist_curves.ml".
This includes an elimination-based tactic FIELD_TAC as well as a more
directed "pull division terms up" tactic RING_PULL_DIV_TAC and its
contextual conversion RING_PULL_DIV_CONV, these based on the following
new clausal theorem:
|- !f. field f
==> (!x y n.
x IN ring_carrier f /\ y IN ring_carrier f
==> ring_pow f (ring_div f x y) n =
ring_div f (ring_pow f x n) (ring_pow f y n)) /\
(!x1 y1 x2 y2.
x1 IN ring_carrier f /\
x2 IN ring_carrier f /\
y1 IN ring_carrier f /\
y2 IN ring_carrier f
==> ring_div f (ring_div f x1 y1) (ring_div f x2 y2) =
ring_div f (ring_mul f x1 y2) (ring_mul f x2 y1)) /\
(!x1 x2 y.
x1 IN ring_carrier f /\
x2 IN ring_carrier f /\
y IN ring_carrier f /\
~(y = ring_0 f)
==> ring_add f (ring_div f x1 y) (ring_div f x2 y) =
ring_div f (ring_add f x1 x2) y /\
ring_sub f (ring_div f x1 y) (ring_div f x2 y) =
ring_div f (ring_sub f x1 x2) y /\
(ring_div f x1 y = ring_div f x2 y <=> x1 = x2) /\
ring_add f (ring_div f x1 y) x2 =
ring_div f (ring_add f x1 (ring_mul f x2 y)) y /\
ring_add f x1 (ring_div f x2 y) =
ring_div f (ring_add f (ring_mul f x1 y) x2) y /\
ring_sub f (ring_div f x1 y) x2 =
ring_div f (ring_sub f x1 (ring_mul f x2 y)) y /\
ring_sub f x1 (ring_div f x2 y) =
ring_div f (ring_sub f (ring_mul f x1 y) x2) y /\
ring_mul f (ring_div f x1 y) x2 =
ring_div f (ring_mul f x1 x2) y /\
ring_mul f x1 (ring_div f x2 y) =
ring_div f (ring_mul f x1 x2) y /\
(ring_div f x1 y = x2 <=> x1 = ring_mul f x2 y) /\
(x1 = ring_div f x2 y <=> ring_mul f x1 y = x2)) /\
(!x y.
x IN ring_carrier f /\
y IN ring_carrier f /\
~(x = ring_0 f) /\
~(y = ring_0 f)
==> ~(ring_mul f x y = ring_0 f)) /\
(!x n.
x IN ring_carrier f /\ ~(x = ring_0 f)
==> ~(ring_pow f x n = ring_0 f))
Wed 16th Mar 2022 Library/pocklington.ml
Removed some functions around primality proving from "nist_curves.ml" and
put them in the main "pocklington.ml" file, as well as filling out the
functions a bit. The new functions are:
general_certify_prime - Like certify_prime with any factorizing function
guided_certify_prime - Taking a set of hereditary sub-factors as a hint
extract_primes_from_certificate - Getting such sub-factor hints from number
GUIDED_PROVE_PRIME, PRIME_RULE - Proving primality with sub-factor hints
Mon 14th Mar 2022 Library/integer.ml
Did a bit of renaming to resolve a name clash with the core "int_coprime"
theorem from int.ml, renaming int_coprime -> INT_COPRIME and the former
INT_COPRIME -> INT_COPRIME_ALT in this file, so now:
int_coprime =
|- !a b. coprime(a,b) <=> (?x y. a * x + b * y = &1)
INT_COPRIME =
|- !a b.
coprime(a,b) <=> (!d. d divides a /\ d divides b ==> d divides &1)
INT_COPRIME_ALT =
|- !a b.
coprime(a,b) <=> (!d. d divides a /\ d divides b <=> d divides &1)
Fri 11th Mar 2022 Library/ringtheory.ml
Added a few more handy facts about "integer_mod_ring", as well
as defining the ring/field of real numbers explicitly as a ring
structure and proving the usual facts about that too:
real_ring =
|- real_ring = ring ((:real),&0,&1,(--),(+),(*))
FIELD_REAL_RING =
|- field real_ring
INTEGER_MOD_RING_CARRIER_REM =
|- !n x. x rem &n IN ring_carrier (integer_mod_ring n)
INTEGER_MOD_RING_CLAUSES =
|- ring_carrier (integer_mod_ring 0) = (:int) /\
(!n. 0 < n
==> ring_carrier (integer_mod_ring n) = {m | &0 <= m /\ m < &n}) /\
(!n. ring_0 (integer_mod_ring n) = &0) /\
(!n. ring_1 (integer_mod_ring n) = &1 rem &n) /\
(!n. ring_neg (integer_mod_ring n) = (\a. --a rem &n)) /\
(!n. ring_add (integer_mod_ring n) = (\a b. (a + b) rem &n)) /\
(!n. ring_sub (integer_mod_ring n) = (\a b. (a - b) rem &n)) /\
(!n. ring_mul (integer_mod_ring n) = (\a b. (a * b) rem &n)) /\
(!n. ring_pow (integer_mod_ring n) = (\a k. a pow k rem &n)) /\
(!n. ring_of_num (integer_mod_ring n) = (\k. &k rem &n)) /\
(!n. ring_of_int (integer_mod_ring n) = (\x. x rem &n))
INTEGER_MOD_RING_SUB =
|- !n. ring_sub (integer_mod_ring n) = (\a b. (a - b) rem &n)
INTEGRAL_DOMAIN_REAL_RING =
|- integral_domain real_ring
REAL_FIELD_CLAUSES =
|- ring_carrier real_ring = (:real) /\
ring_0 real_ring = &0 /\
ring_1 real_ring = &1 /\
ring_neg real_ring = (--) /\
ring_add real_ring = (+) /\
ring_mul real_ring = (*) /\
ring_of_num real_ring = & /\
ring_sub real_ring = (-) /\
ring_inv real_ring = inv /\
ring_div real_ring = (/) /\
ring_pow real_ring = (pow)
REAL_RING_CHAR =
|- ring_char real_ring = 0
REAL_RING_CLAUSES =
|- ring_carrier real_ring = (:real) /\
ring_0 real_ring = &0 /\
ring_1 real_ring = &1 /\
ring_neg real_ring = (--) /\
ring_add real_ring = (+) /\
ring_mul real_ring = (*)
REAL_RING_DIV =
|- ring_div real_ring = (/)
REAL_RING_INV =
|- ring_inv real_ring = inv
REAL_RING_OF_INT =
|- ring_of_int real_ring = real_of_int
REAL_RING_OF_NUM =
|- ring_of_num real_ring = &
REAL_RING_POW =
|- ring_pow real_ring = (pow)
REAL_RING_SUB =
|- ring_sub real_ring = (-)
Fri 4th Mar 2022 Examples/nist_curves.ml
Added a formalization of curve25519, and Montgomery curves in general,
to the increasingly inaccurately-named "nist_curves.ml" file. Mainly
for ease of re-use of the existing short Weierstrass material, the
properties of Montgomery operations are first derived from their
Weierstrass versions via the standard mapping, even defining the
"wei25519" curve as the short Weierstrass variant (nomenclature taken
from Struik's "Alternative Elliptic Curve Representations"). After
that, all the usual things are proved about curve25519, including the
y-free doubling and differential addition operations (following
Bernstein and Lange, "Montgomery cuves and the Montgomery ladder") and
a y-recovery formula (from Okeya and Sakurai). Also filled out the
earlier curve material with some additional facts, most interestingly
a standard easily computable endomorphism of secp256k1.
Thu 24th Feb 2022 Library/grouptheory.ml
Added this somewhat technical lemma which is handy for explicitly
creating a group as an isomorphic image and not then doing a
separate proof of group-ness and the isomorphism with the original:
CREATE_ISOMORPHIC_COPY_OF_GROUP =
|- !f g G s z i m.
z IN s /\
(!x. x IN group_carrier G ==> f x IN s /\ g (f x) = x) /\
(!y. y IN s ==> g y IN group_carrier G /\ f (g y) = y) /\
g z = group_id G /\
(!x. x IN s ==> i x = f (group_inv G (g x))) /\
(!x y. x IN s /\ y IN s ==> m x y = f (group_mul G (g x) (g y)))
==> group_isomorphisms (G,group (s,z,i,m)) (f,g) /\
group_carrier (group (s,z,i,m)) = s /\
group_id (group (s,z,i,m)) = z /\
group_inv (group (s,z,i,m)) = i /\
group_mul (group (s,z,i,m)) = m
Wed 23rd Feb 2022 Library/ringtheory.ml
Added an explicit conversion INTEGER_MOD_RING_RED_CONV for computing
terms in "integer_mod_ring n" (for nonzero n, the case of n = 0 being
just the integers anyway). This also adds the theorem:
RING_INV_INTEGER_MOD_RING =
|- !n a.
ring_inv (integer_mod_ring n) (&a) =
(if (n = 0 \/ ~(n = 1) /\ a < n) /\ coprime(a,n)
then &(inverse_mod n a) else &0)
This introduces a dependency on the Library/pocklington.ml file for
some additional material about "inverse_mod".
Wed 23rd Feb 2022 calc_num.ml, int.ml, Library/pocklington.ml
Moved EXP_MOD_CONV from "Library/pocklington.ml" into the core (in the
process slightly improving the implementation) and added an analogous
integer form INT_POW_REM_CONV.
Wed 23rd Feb 2022 Library/pocklington.ml
Added INVERSE_MOD_CONV, an explicit calculation conversion for modular
inverse terms of the form `inverse_mod m n` for numerals m and n.
Tue 7th Dec 2021 Library/words.ml
Added one more little missing lemma:
BIT_WORD_BITVAL =
|- !b i. bit i (word(bitval b):N word) <=> i = 0 /\ b
Thu 25th Nov 2021 Examples/nist_curves.ml
Added the other SECG curves defined over prime order fields: secp192k1,
secp224k1 and secp256k1 (the Bitcoin curve). This extends the existing
NIST curve material with the same verification of basic data like
number of points on the curve, and adds some possible computational
formulas in projective and Jacobian coordinates for Weierstrass curves
like these with a = 0.
Wed 10th Nov 2021 Library/words.ml
Added a new construct "bits_of_num" analogous to "bits_of_word" but
for actual numbers not words. The suite of theorems about it helps
with some kinds of "effectively bitwise" reasoning on natural numbers
such as pushing division and modulus w.r.t. powers of 2 through sums
based on the intuition that the bit patterns are not overlapping
(DIV_MOD_DISJOINT_BITS, DISJOINT_BITS_CLAUSES)
bits_of_num =
|- !n. bits_of_num n = {i | numbit i n}
BITSUM_BOUND =
|- !s k.
FINITE s
==> (nsum s (\i. 2 EXP i) < 2 EXP k <=> s SUBSET {i | i < k})
BITSUM_DIVIDES =
|- !s k.
FINITE s
==> (2 EXP k divides nsum s (\i. 2 EXP i) <=>
DISJOINT {i | i < k} s)
BITS_OF_NUM_0 =
|- bits_of_num 0 = {}
BITS_OF_NUM_1 =
|- bits_of_num 1 = {0}
BITS_OF_NUM_ADD =
|- !m n.
DISJOINT (bits_of_num m) (bits_of_num n)
==> bits_of_num(m + n) = bits_of_num m UNION bits_of_num n
BITS_OF_NUM_DISJOINT_NUMSEG_EQ =
|- !n k. DISJOINT {i | i < k} (bits_of_num n) <=> 2 EXP k divides n
BITS_OF_NUM_DIV =
|- !n k. bits_of_num(n DIV 2 EXP k) = {i | k + i IN bits_of_num n}
BITS_OF_NUM_EQ =
|- !m n. bits_of_num m = bits_of_num n <=> m = n
BITS_OF_NUM_GALOIS =
|- !n s. bits_of_num n = s <=> FINITE s /\ nsum s (\i. 2 EXP i) = n
BITS_OF_NUM_MOD =
|- !n k. bits_of_num(n MOD 2 EXP k) = {i | i IN bits_of_num n /\ i < k}
BITS_OF_NUM_MUL =
|- (!n k. bits_of_num(2 EXP k * n) = IMAGE (\i. k + i) (bits_of_num n)) /\
(!n k. bits_of_num(n * 2 EXP k) = IMAGE (\i. k + i) (bits_of_num n))
BITS_OF_NUM_MUL_ALT =
|- (!n k.
bits_of_num(2 EXP k * n) =
{i | k <= i /\ i - k IN bits_of_num n}) /\
(!n k.
bits_of_num(n * 2 EXP k) =
{i | k <= i /\ i - k IN bits_of_num n})
BITS_OF_NUM_NSUM =
|- !s. FINITE s ==> bits_of_num(nsum s (\i. 2 EXP i)) = s
BITS_OF_NUM_POW2 =
|- !k. bits_of_num(2 EXP k) = {k}
BITS_OF_NUM_SUBSET_NUMSEG_EQ =
|- !n k. bits_of_num n SUBSET {i | i < k} <=> n < 2 EXP k
BITS_OF_NUM_SUBSET_NUMSEG_LT =
|- !n. bits_of_num n SUBSET {i | i < n}
BITS_OF_NUM_VAL =
|- !x. bits_of_num(val x) = bits_of_word x
BITS_OF_WORD_WORD =
|- !n. bits_of_word (word n) = {i | i < dimindex (:N)} INTER bits_of_num n
DISJOINT_BITS_CLAUSES =
|- (!k h l.
l < 2 EXP k
==> DISJOINT (bits_of_num(2 EXP k * h)) (bits_of_num l)) /\
(!k h l.
l < 2 EXP k
==> DISJOINT (bits_of_num(h * 2 EXP k)) (bits_of_num l)) /\
(!k h l.
l < 2 EXP k
==> DISJOINT (bits_of_num l) (bits_of_num(2 EXP k * h))) /\
(!k h l.
l < 2 EXP k
==> DISJOINT (bits_of_num l) (bits_of_num(h * 2 EXP k))) /\
(!m n k.
DISJOINT (bits_of_num m) (bits_of_num n)
==> DISJOINT (bits_of_num(2 EXP k * m))
(bits_of_num(2 EXP k * n))) /\
(!m n k.
DISJOINT (bits_of_num m) (bits_of_num n)
==> DISJOINT (bits_of_num(m * 2 EXP k))
(bits_of_num(n * 2 EXP k))) /\
(!m n k.
DISJOINT (bits_of_num m) (bits_of_num n)
==> DISJOINT (bits_of_num(m DIV 2 EXP k))
(bits_of_num(n DIV 2 EXP k))) /\
(!m n k.
DISJOINT (bits_of_num m) (bits_of_num n)
==> DISJOINT (bits_of_num(m MOD 2 EXP k))
(bits_of_num(n MOD 2 EXP k)))
DISJOINT_BITS_HILO =
|- !k h l.
l < 2 EXP k ==> DISJOINT (bits_of_num(2 EXP k * h)) (bits_of_num l)
DIV_MOD_DISJOINT_BITS =
|- (!m n.
DISJOINT (bits_of_num m) (bits_of_num n)
==> (m + n) DIV 2 EXP k = m DIV 2 EXP k + n DIV 2 EXP k) /\
(!m n.
DISJOINT (bits_of_num m) (bits_of_num n)
==> (m + n) MOD 2 EXP k = m MOD 2 EXP k + n MOD 2 EXP k)
FINITE_BITS_OF_NUM =
|- !n. FINITE (bits_of_num n)
IN_BITS_OF_NUM =
|- !n i. i IN bits_of_num n <=> ODD (n DIV 2 EXP i)
NSUM_BITS_DIV =
|- !s k.
FINITE s
==> nsum s (\i. 2 EXP i) DIV 2 EXP k =
nsum {i | i IN s /\ k <= i} (\i. 2 EXP (i - k))
NSUM_BITS_EQ =
|- !s t.
FINITE s /\ FINITE t
==> (nsum s (\i. 2 EXP i) = nsum t (\i. 2 EXP i) <=> s = t)
NSUM_BITS_MOD =
|- !s k.
FINITE s
==> nsum s (\i. 2 EXP i) MOD 2 EXP k =
nsum {i | i IN s /\ i < k} (\i. 2 EXP i)
NSUM_BITS_OF_NUM =
|- !n. nsum (bits_of_num n) (\i. 2 EXP i) = n
Tue 9th Nov 2021 int.ml, real.ml, Library/words.ml
Added a few miscelleneous lemmas:
IVAL_WORD_CONG =
|- !n. (ival (word n) == &n) (mod (&2 pow dimindex(:N)))
MULT_DIV =
|- (!m n p. p divides m ==> (m * n) DIV p = m DIV p * n) /\
(!m n p. p divides n ==> (m * n) DIV p = m * n DIV p)
REAL_OF_NUM_DIV =
|- !m n. &(m DIV n) = &m / &n - &(m MOD n) / &n
VAL_WORD_AND_NOT_MASK_WORD =
|- !x k.
val (word_and x (word_not (word(2 EXP k - 1)))) =
2 EXP k * val x DIV 2 EXP k
WORD_AND_MASK_WORDS =
|- !i j.
word_and (word(2 EXP j - 1)) (word(2 EXP k - 1)) =
word(2 EXP MIN j k - 1)
WORD_AND_NOT_MASK_WORD =
|- (!x k.
word_and x (word_not (word(2 EXP k - 1))) =
word(2 EXP k * val x DIV 2 EXP k)) /\
(!x k.
word_and (word_not (word(2 EXP k - 1))) x =
word(2 EXP k * val x DIV 2 EXP k))
WORD_OR_MASK_WORDS =
|- !i j.
word_or (word(2 EXP j - 1)) (word(2 EXP k - 1)) =
word(2 EXP MAX j k - 1)
WORD_USHR_MASK_WORD =
|- !k i.
k <= dimindex(:N)
==> word_ushr (word(2 EXP k - 1)) i = word(2 EXP (k - i) - 1)
Fri 5th Nov 2021 Library/words.ml
Added more word lemmas, mainly connected with selecting trailing bits and
emulating ctz in terms of clz, very much in line with material in
the "Manipulating Rightmost Bits" section of "Hacker's Delight":
BIT_WORD_XOR_ADD_1 =
|- !x i.
bit i (word_xor x (word_add x (word 1))) <=>
i < dimindex(:N) /\ (!j. j < i ==> bit j x)
BIT_WORD_XOR_NEG =
|- !x i.
bit i (word_xor x (word_neg x)) <=>
i < dimindex(:N) /\ (?j. j < i /\ bit j x)
BIT_WORD_XOR_SUB_1 =
|- !x i.
bit i (word_xor x (word_sub x (word 1))) <=>
i < dimindex(:N) /\ (!j. j < i ==> ~bit j x)
WORD_AND_NEG_CTZ =
|- !x. word_and x (word_neg x) = word_of_bits {word_ctz x}
WORD_AND_NOT_SUB_1_CTZ =
|- !x. word_and (word_not x) (word_sub x (word 1)) =
word(2 EXP word_ctz x - 1)
WORD_CLZ_BIT =
|- !k. word_clz (word_of_bits {k}) =
(if k < dimindex(:N) then dimindex(:N) - 1 - k else dimindex(:N))
WORD_CLZ_MASK_WORD =
|- !k. word_clz (word(2 EXP k - 1)) = dimindex(:N) - k
WORD_CTZ_BIT =
|- !k. word_ctz (word_of_bits {k}) = MIN k (dimindex(:N))
WORD_CTZ_EMULATION_AND_NEG =
|- !x. word_ctz x =
(if x = word 0
then dimindex(:N)
else dimindex(:N) - 1 - word_clz (word_and x (word_neg x)))
WORD_CTZ_EMULATION_AND_NEG_REV =
|- !x. word_clz (word_and x (word_neg x)) =
(if x = word 0
then dimindex(:N)
else dimindex(:N) - 1 - word_ctz x)
WORD_CTZ_EMULATION_XOR_SUB_1 =
|- !x. word_ctz x =
(if x = word 0
then dimindex(:N)
else dimindex(:N) -
1 -
word_clz (word_xor x (word_sub x (word 1))))
WORD_CTZ_EMULATION_XOR_SUB_1_REV =
|- !x. word_clz (word_xor x (word_sub x (word 1))) =
dimindex(:N) - 1 - word_ctz x
WORD_CTZ_MASK_WORD =
|- !k. word_ctz (word(2 EXP k - 1)) = (if k = 0 then dimindex(:N) else 0)
WORD_XOR_SUB_1_CTZ =
|- !x. word_xor x (word_sub x (word 1)) = word(2 EXP (word_ctz x + 1) - 1)
Tue 2nd Nov 2021 Library/words.ml
Added missing conversions WORD_JROL_CONV and WORD_JROR_CONV (for
word-word rotate operations) and included them in WORD_RED_CONV
and hence WORD_REDUCE_CONV.
Mon 1st Nov 2021 parser.ml
Fixed a trivial typo in the error message "closing square bracket
of list expected".
Tue 12th Oct 2021 Library/words.ml
Added a couple more word lemmas about pushing logical complements through
other constructs; the side-conditions make them a bit ugly but they can be
useful:
WORD_JOIN_NOT =
|- !v w.
dimindex(:P) <= dimindex(:M) + dimindex(:N)
==> word_join (word_not v) (word_not w) = word_not (word_join v w)
WORD_SUBWORD_NOT =
|- !x pos len.
dimindex(:N) <= len /\ pos + len <= dimindex(:M)
==> word_subword (word_not x) (pos,len) =
word_not (word_subword x (pos,len))
Sat 9th Oct 2021 Library/words.ml
Added a few more elementary word lemmas like a simple upper bound on
a concrete masking of a word:
VAL_WORD_AND_LE =
|- (!x y. val (word_and x y) <= val x) /\
(!x y. val (word_and x y) <= val y)
VAL_WORD_AND_LE_MIN =
|- !x y. val (word_and x y) <= MIN (val x) (val y)
VAL_WORD_AND_WORD_LE =
|- (!x n. val (word_and x (word n)) <= n) /\
(!x n. val (word_and (word n) x) <= n)
VAL_WORD_OR_LE_MAX =
|- !x y. MAX (val x) (val y) <= val (word_or x y)
Fri 8th Oct 2021 int.ml
Added a systematic set of missing analogs of real theorems for ints:
INT_ABS_BOUNDS : thm =
|- !x k. abs x <= k <=> --k <= x /\ x <= k
INT_EQ_LCANCEL_IMP : thm =
|- !x y z. ~(z = &0) /\ z * x = z * y ==> x = y
INT_EQ_RCANCEL_IMP : thm =
|- !x y z. ~(z = &0) /\ x * z = y * z ==> x = y
INT_LE_LCANCEL_IMP : thm =
|- !x y z. &0 < x /\ x * y <= x * z ==> y <= z
INT_LE_POW_2 : thm =
|- !x. &0 <= x pow 2
INT_LE_RCANCEL_IMP : thm =
|- !x y z. &0 < z /\ x * z <= y * z ==> x <= y
INT_LE_RMUL_EQ : thm =
|- !x y z. &0 < z ==> (x * z <= y * z <=> x <= y)
INT_LT_LADD_IMP : thm =
|- !x y z. y < z ==> x + y < x + z
INT_LT_LCANCEL_IMP : thm =
|- !x y z. &0 < x /\ x * y < x * z ==> y < z
INT_LT_LMUL : thm =
|- !x y z. &0 < x /\ y < z ==> x * y < x * z
INT_LT_LNEG : thm =
|- !x y. --x < y <=> &0 < x + y
INT_LT_POW_2 : thm =
|- !x. &0 < x pow 2 <=> ~(x = &0)
INT_LT_RCANCEL_IMP : thm =
|- !x y z. &0 < z /\ x * z < y * z ==> x < y
INT_LT_RMUL : thm =
|- !x y z. x < y /\ &0 < z ==> x * z < y * z
INT_LT_RNEG : thm =
|- !x y. x < --y <=> x + y < &0
INT_LT_SQUARE : thm =
|- !x. &0 < x * x <=> ~(x = &0)
INT_OF_NUM_SUB_CASES : thm =
|- !m n. &m - &n = (if n <= m then &(m - n) else -- &(n - m))
INT_POS_EQ_SQUARE : thm =
|- !x. &0 <= x <=> (?y. y pow 2 = real_of_int x)
INT_POW_EQ_1 : thm =
|- !x n. x pow n = &1 <=> abs x = &1 /\ (x < &0 ==> EVEN n) \/ n = 0
INT_POW_EQ_1_IMP : thm =
|- !x n. ~(n = 0) /\ x pow n = &1 ==> abs x = &1
INT_POW_EQ_EQ : thm =
|- !n x y.
x pow n = y pow n <=>
(if EVEN n then n = 0 \/ abs x = abs y else x = y)
INT_POW_EQ_ODD : thm =
|- !n x y. ODD n /\ x pow n = y pow n ==> x = y
INT_POW_EQ_ODD_EQ : thm =
|- !n x y. ODD n ==> (x pow n = y pow n <=> x = y)
INT_POW_LBOUND : thm =
|- !x n. &0 <= x ==> &1 + &n * x <= (&1 + x) pow n
INT_POW_LE2_ODD_EQ : thm =
|- !n x y. ODD n ==> (x pow n <= y pow n <=> x <= y)
INT_POW_LT2_ODD : thm =
|- !n x y. x < y /\ ODD n ==> x pow n < y pow n
INT_POW_LT2_ODD_EQ : thm =
|- !n x y. ODD n ==> (x pow n < y pow n <=> x < y)
Fri 8th Oct 2021 real.ml, int.ml
Added integer and real forms of this simple lemma:
INT_EVENPOW_ABS =
|- !x n. EVEN n ==> abs x pow n = x pow n
REAL_EVENPOW_ABS =
|- !x n. EVEN n ==> abs x pow n = x pow n
Wed 6th Oct 2021 int.ml
Added a natural integer analog of MOD_UNIQUE:
INT_REM_UNIQUE =
|- !m n p.
m rem n = p <=>
(n = &0 /\ m = p \/ &0 <= p /\ p < abs n) /\ (m == p) (mod n)
Tue 5th Oct 2021 Library/words.ml
Added a few trivial theorems about unsigned word max/min:
WORD_UMAX =
|- !x y. word_umax x y = (if val x <= val y then y else x)
WORD_UMAX_ASSOC =
|- !x y z. word_umax x (word_umax y z) = word_umax (word_umax x y) z
WORD_UMAX_SYM =
|- !x y. word_umax x y = word_umax y x
WORD_UMIN =
|- !x y. word_umin x y = (if val x <= val y then x else y)
WORD_UMIN_ASSOC =
|- !x y z. word_umin x (word_umin y z) = word_umin (word_umin x y) z
WORD_UMIN_SYM =
|- !x y. word_umin x y = word_umin y x
Fri 20th Aug 2021 Library/words.ml
Added three more word lemmas connecting shifts and subwords:
WORD_SHL_SUBWORD =
|- !x d l.
dimindex(:N) <= l + d
==> word_shl (word_subword x (0,l)) d = word_shl x d
WORD_SUBWORD_AS_USHR =
|- !x k l. dimindex(:N) <= k + l ==> word_subword x (k,l) = word_ushr x k
WORD_USHR_AS_SUBWORD =
|- !x k. word_ushr x k = word_subword x (k,dimindex(:N) - k)
Fri 30th Jul 2021 Library/words.ml
Made the trivial generalization to WORD_ARITH / WORD_ARITH_TAC of also
breaking apart conjunctions at the start. Added these lemmas
characterizing exactness of additions or additions-with-carry in terms
of a later word comparison on the result, all of which are just
proved automatically by WORD_ARITH_TAC.
WORD_ADC_LE_EXACT =
|- (!x y.
val(word_add (word_add x y) (word 1)) <= val x <=>
val(word_add (word_add x y) (word 1)) + 2 EXP dimindex(:N) =
val x + val y + 1) /\
(!x y.
val(word_add (word_add x y) (word 1)) <= val y <=>
val(word_add (word_add x y) (word 1)) + 2 EXP dimindex(:N) =
val x + val y + 1)
WORD_ADC_LE_INEXACT =
|- (!x y.
val(word_add (word_add x y) (word 1)) <= val x <=>
~(val(word_add (word_add x y) (word 1)) = val x + val y + 1)) /\
(!x y.
val(word_add (word_add x y) (word 1)) <= val y <=>
~(val(word_add (word_add x y) (word 1)) = val x + val y + 1))
WORD_ADD_LT_EXACT =
|- (!x y.
val(word_add x y) < val x <=>
val(word_add x y) + 2 EXP dimindex(:N) = val x + val y) /\
(!x y.
val(word_add x y) < val y <=>
val(word_add x y) + 2 EXP dimindex(:N) = val x + val y)
WORD_ADD_LT_INEXACT =
|- (!x y.
val(word_add x y) < val x <=>
~(val(word_add x y) = val x + val y)) /\
(!x y.
val(word_add x y) < val y <=>
~(val(word_add x y) = val x + val y))
WORD_LE_ADD_EXACT =
|- (!x y.
val x <= val(word_add x y) <=> val(word_add x y) = val x + val y) /\
(!x y.
val y <= val(word_add x y) <=> val(word_add x y) = val x + val y)
WORD_LT_ADC_EXACT =
|- (!x y.
val x < val(word_add (word_add x y) (word 1)) <=>
val(word_add (word_add x y) (word 1)) = val x + val y + 1) /\
(!x y.
val y < val(word_add (word_add x y) (word 1)) <=>
val(word_add (word_add x y) (word 1)) = val x + val y + 1)
Wed 28th Jul 2021 Library/words.ml
Added two more simple word lemmas (the de Morgan laws in this setting)
WORD_NOT_AND =
|- !x y. word_not (word_and x y) = word_or (word_not x) (word_not y)
WORD_NOT_OR =
|- !x y. word_not (word_or x y) = word_and (word_not x) (word_not y)
Tue 20th Jul 2021 Library/words.ml
Added another couple of elementary lemmas about AND with a word
consisting of a single bit expressed as a power of 2.
VAL_WORD_AND_POW2 =
|- (!x k. val(word_and x (word(2 EXP k))) = 2 EXP k * bitval(bit k x)) /\
(!x k. val(word_and (word(2 EXP k)) x) = 2 EXP k * bitval(bit k x))
WORD_AND_POW2 =
|- (!x k. word_and x (word(2 EXP k)) = word(2 EXP k * bitval(bit k x))) /\
(!x k. word_and (word(2 EXP k)) x = word(2 EXP k * bitval(bit k x)))
Fri 16th Jul 2021 Library/words.ml
Added one more little word lemma that zero-extension of a 1-bit
Boolean value is trivial:
WORD_ZX_BITVAL = |- !b. word_zx(word(bitval b)) = word(bitval b)
Wed 7th Jul 2021 Library/words.ml
Added a few miscellaneous word theorems with an emphasis on commuting
word_zx, mainly as a shortening, through other word operations:
BIT_GUARD =
|- !x i. bit i x <=> i < dimindex(:N) /\ bit i x
VAL_WORD_SUBWORD_DIMINDEX =
|- !pos w.
val (word_subword w (pos,dimindex(:N))) =
(val w DIV 2 EXP pos) MOD 2 EXP dimindex(:N)
VAL_WORD_SUBWORD_SIMPLE =
|- !w. val (word_subword w (0,dimindex(:N))) =
val w MOD 2 EXP dimindex(:N)
WORD_SUBWORD_AND =
|- !x y pos len.
word_subword (word_and x y) (pos,len) =
word_and (word_subword x (pos,len)) (word_subword y (pos,len))
WORD_SUBWORD_OR =
|- !x y pos len.
word_subword (word_or x y) (pos,len) =
word_or (word_subword x (pos,len)) (word_subword y (pos,len))
WORD_SUBWORD_XOR =
|- !x y pos len.
word_subword (word_xor x y) (pos,len) =
word_xor (word_subword x (pos,len)) (word_subword y (pos,len))
WORD_ZX_ADD =
|- !x y.
dimindex(:N) <= dimindex(:M)
==> word_zx (word_add x y) = word_add (word_zx x) (word_zx y)
WORD_ZX_AND =
|- !x y. word_zx (word_and x y) = word_and (word_zx x) (word_zx y)
WORD_ZX_MUL =
|- !x y.
dimindex(:N) <= dimindex(:M)
==> word_zx (word_mul x y) = word_mul (word_zx x) (word_zx y)
WORD_ZX_NEG =
|- !x. dimindex(:N) <= dimindex(:M)
==> word_zx (word_neg x) = word_neg (word_zx x)
WORD_ZX_NOT =
|- !x. dimindex(:N) <= dimindex(:M)
==> word_zx (word_not x) = word_not (word_zx x)
WORD_ZX_OR =
|- !x y. word_zx (word_or x y) = word_or (word_zx x) (word_zx y)
WORD_ZX_SHL =
|- !x n.