other
(in-package "ACL2")
include-book
(include-book "arithmetic/top" :dir :system)
include-book
(include-book "ihs-init")
include-book
(include-book "std/util/defrule" :dir :system)
other
(defxdoc math-lemmas :parents (ihs) :short "A book of theories about +, -, *, /, and EXPT, built on the arithmetic package of Matt Kaufmann.")
local
(local (set-default-parents math-lemmas))
cancel-equal-+-*theorem
(defrule cancel-equal-+-* :short "Rewrite: @('(equal (+ x y) x)') and @('(equal (* x y) x)'); also commutative forms." (and (equal (equal (+ x y) x) (and (acl2-numberp x) (equal (fix y) 0))) (equal (equal (+ y x) x) (and (acl2-numberp x) (equal (fix y) 0))) (equal (equal (* x y) x) (and (acl2-numberp x) (or (equal x 0) (equal y 1)))) (equal (equal (* x y) y) (and (acl2-numberp y) (or (equal y 0) (equal x 1))))) :enable equal-*-x-y-x)
normalize-equal-0theorem
(defrule normalize-equal-0 :short "Rewrite @('(equal (- x) 0)'), @('(equal (+ x y) 0)'), and @('(equal (* x y) 0)')." (and (equal (equal (- x) 0) (equal (fix x) 0)) (equal (equal (+ x y) 0) (equal (fix x) (- y))) (equal (equal (* x y) 0) (or (equal (fix x) 0) (equal (fix y) 0)))))
other
(defsection acl2-numberp-algebra :short "A basic theory of algebra for all @(see acl2-numberp)s." :long "<p>The ACL2-NUMBERP-ALGEBRA theory is designed to be a simple, compact basis for building other theories. This theory contains a minimal set of rules for basic algebraic manipulation including associativity and commutativity, simplification, cancellation, and normalization. It is extended by the theories RATIONALP-ALGEBRA and INTEGERP-ALGEBRA to include selected linear rules and rules for integers respectively. This theory also contains the DEFUN-THEORY (which see) of all built-in function symbols that would normally occur during reasoning about the ACL2-NUMBERPs.</p> <p>We used keep this theory (and book) separate but roughly equal to the books maintained by Matt K. in order to have a solid, simple, and predictable foundation on which to build the rest of the books in the IHS hierarchy. However it was decided that this was too much trouble and we just select the rules of Matt K. that we want.</p>" (deftheory acl2-numberp-algebra (union-theories (defun-theory '(equal eql = /= iff force binary-+ binary-* unary-- unary-/ acl2-numberp zerop fix zp zip)) '(eqlablep-recog commutativity-of-+ commutativity-of-* inverse-of-+ associativity-of-+ associativity-of-* commutativity-2-of-+ commutativity-2-of-* unicity-of-0 functional-self-inversion-of-minus unicity-of-1 default-*-1 default-*-2 default-<-1 default-<-2 default-+-1 default-+-2 inverse-of-* functional-self-inversion-of-/ minus-cancellation-on-right minus-cancellation-on-left /-cancellation-on-left /-cancellation-on-right equal-*-x-y-y cancel-equal-+-* normalize-equal-0 left-cancellation-for-* left-cancellation-for-+ equal-minus-0 zero-is-only-zero-divisor equal-minus-minus equal-/-/ default-unary-minus equal-/ equal-*-/-2 functional-commutativity-of-minus-*-left functional-commutativity-of-minus-*-right reciprocal-minus equal-minus-minus distributivity-of-/-over-* distributivity distributivity-of-minus-over-+))))
rewrite-linear-equalities-to-ifftheorem
(defrule rewrite-linear-equalities-to-iff :short "Rewrite: @('(EQUAL (< w x) (< y z))') → @('(IFF (< w x) (< y z))')." :long "<p>Some proofs of linear equalities don't work when presented as equalities because they need to be proved by linear arithmetic, but linear arithmetic only works at the literal level. This lemma allows you to state the equality as an equality rewrite rule, but breaks the equality into literals for the proof.</p>" (equal (equal (< w x) (< y z)) (iff (< w x) (< y z))))
normalize-<-minus-/theorem
(defrule normalize-<-minus-/ :short "Rewrite inequalities between 0 and negated or reciprocal terms, and @('(< (- x) (- y))')." (and (equal (< (- x) 0) (< 0 x)) (equal (< 0 (- x)) (< x 0)) (equal (< (- x) (- y)) (> x y)) (implies (real/rationalp x) (and (equal (< 0 (/ x)) (< 0 x)) (equal (< (/ x) 0) (< x 0))))))
other
(defsection rationalp-algebra :short "A basic theory of algebra for all @(see rationalp)s." :long "<p>This theory includes the @(see acl2-numberp-algebra) theory along with additional lemmas about the rationals.</p> <p>This theory extends ACL2-NUMBERP-ALGEBRA to include theorems about NUMERATOR and DENOMINATOR, and simple cancellation and normalization theorems and other simple theorems for inequalities.</p>" (deftheory rationalp-algebra (union-theories (theory 'acl2-numberp-algebra) (union-theories (defun-theory '(numerator denominator < abs plusp minusp min max signum rfix)) '(equal-*-/-1 *-r-denominator-r default-denominator numerator-minus equal-denominator-1 numerator-when-integerp <-y-*-y-x <-*-y-x-y <-*-/-right <-*-/-right-commuted <-*-/-left <-*-/-left-commuted <-*-left-cancel <-0-minus /-preserves-positive /-preserves-negative rewrite-linear-equalities-to-iff normalize-<-minus-/ <-unary-/-negative-left <-unary-/-negative-right <-unary-/-positive-left <-unary-/-positive-right)))))
normalize-<-/-to-*theorem
(defrule normalize-<-/-to-* :parents (math-lemmas prefer-*-to-/) :short "Rewrite: Replace @('x < 1/y') with @('x*y < 1') or @('x*y > 1'), based on the sign of y." (implies (and (real/rationalp x) (real/rationalp y) (not (equal y 0))) (and (equal (< x (/ y)) (if (< y 0) (< 1 (* x y)) (< (* x y) 1))) (equal (< (/ y) x) (if (< y 0) (< (* x y) 1) (< 1 (* x y)))))))
normalize-<-/-to-*-3theorem
(defrule normalize-<-/-to-*-3 :parents (math-lemmas prefer-*-to-/) :short "Rewrite: Replace @('x < y/z') and @('x > y/z') with @('x*z < y') or @('x*z > y'), depending on the sign of z." (implies (and (real/rationalp x) (real/rationalp y) (real/rationalp z) (not (equal z 0))) (and (equal (< x (* y (/ z))) (if (< z 0) (< y (* x z)) (< (* x z) y))) (equal (< x (* (/ z) y)) (if (< z 0) (< y (* x z)) (< (* x z) y))) (equal (< (* y (/ z)) x) (if (< z 0) (< (* x z) y) (< y (* x z)))) (equal (< (* (/ z) y) x) (if (< z 0) (< (* x z) y) (< y (* x z)))))) :disable (<-unary-/-negative-left <-unary-/-negative-right <-unary-/-positive-left <-unary-/-positive-left <-*-right-cancel) :use (:instance <-*-right-cancel (x (* x z)) (y y) (z (/ z))))
normalize-equal-/-to-*theorem
(defrule normalize-equal-/-to-* :parents (math-lemmas prefer-*-to-/) :short "Rewrite: Replace @('x = y/z') with @('x*z = y')." (implies (and (acl2-numberp z) (not (equal z 0))) (and (equal (equal x (* y (/ z))) (and (acl2-numberp x) (equal (* x z) (fix y)))) (equal (equal x (* (/ z) y)) (and (acl2-numberp x) (equal (* x z) (fix y)))))))
other
(defsection prefer-*-to-/ :short "A small theory of lemmas that eliminate / in favor of *." :long "<p>This is a small theory of rules that eliminate / from equalites and inequalities in favor of *, e.g., @('x < y/z') is rewritten to @('x*y < z') for positive z. This theory is compatible with the ALGEBRA theories, i.e., it should not cause looping.</p> <p>These rules are not included in @(see rationalp-algebra) because it is not clear that we should prefer @('x*y < z') to @('x < y/z'), or @('x*y = z') to @('x = y/z'). In the case of the lemma @(see normalize-equal-/-to-*), there is no reason to suspect that `y' is a better term than `x'; in fact, the whole point of the proofs using these libraries may have to do with a representation involving /. So, unless someone provides a convincing reason to the contrary, these rules will remain separate from the @(see rationalp-algebra) theory.</p> <p>Note, however, that in certain cases this theory is just the thing that needs to be ENABLEd to make the proofs work. Keep it in mind.</p>" (deftheory prefer-*-to-/ '(normalize-<-/-to-* normalize-<-/-to-*-3 normalize-equal-/-to-*)))
in-theory
(in-theory (disable prefer-*-to-/))
integerp-+-minus-*theorem
(defrule integerp-+-minus-* :short "Rewrite: -i, i + j, i - j, and i * j are integers, when i and j are integers." :long "<p>The system has powerful enough type reasoning to `get' these facts automatically most of the time. There are cases, however, where we need to bring the full power of the rewriter to bear on the problem. In general one would like to keep lemmas like this to a minimum so as to avoid swamping the rewriter.</p>" (and (implies (integerp i) (integerp (- i))) (implies (and (integerp i) (integerp j)) (and (integerp (+ i j)) (integerp (- i j)) (integerp (* i j))))))
other
(defsection integerp-algebra :parents (math-lemmas ihs-math) :short "A basic theory of algebra for all INTEGERPs." :long "<p>this theory consists of the @(see acl2-numberp-algebra) and @(see rationalp-algebra) theories, along with additional lemmas about the integers.</p>" (deftheory integerp-algebra (union-theories (theory 'rationalp-algebra) (union-theories (defun-theory '(integerp integer-abs)) '(integerp-+-minus-* integerp==>denominator=1 <-minus-zero natp-rw posp-rw)))))
other
(defsection expt-algebra :parents (math-lemmas ihs-math) :short "A theory of EXPT which is compatible with the ALGEBRA theories." :long "<p>This theory contains :TYPE-PRESCRIPTIONS, simpification, normalization and selected :LINEAR rules for @(tsee EXPT). This theory will not be useful unless the @(see integerp-algebra) theory, or something similar is ENABLEd.</p>" (deftheory expt-algebra '((expt) (:type-prescription expt) expt-type-prescription-nonzero expt-type-prescription-positive expt-type-prescription-integerp right-unicity-of-1-for-expt functional-commutativity-of-expt-/-base expt-minus exponents-add exponents-multiply expt->-1 expt-is-increasing-for-base>1 expt-is-decreasing-for-pos-base<1 expt-is-weakly-increasing-for-base>1 expt-is-weakly-decreasing-for-pos-base<1)))
other
(defsection ihs-math :short "The default theory of +, -, *, /, and EXPT for the IHS library." :long "<p>This theory simply consists of the theories @(see INTEGERP-ALGEBRA) and @(see EXPT-ALGEBRA).</p> <p>This theory is the default theory exported by the @('ihs/math-lemmas') book. This theory will normally be ENABLEd by every book in the IHS library.</p>" (deftheory ihs-math (union-theories (theory 'integerp-algebra) (theory 'expt-algebra))))