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; See the top-level arithmetic-3 LICENSE file for authorship,
; copyright, and license information.
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;
;; non-linear.lisp
;;
;;
;; This book contains a minimal theory for use in the non-linear
;; extension to the linear arithmetic decision procedure.
;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(in-package "ACL2")
(local (include-book "basic-arithmetic"))
(local (include-book "expt"))
;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;
(defthm arith-associativity-of-*
(equal (* (* x y) z)
(* x (* y z))))
(defthm arith-commutativity-of-*
(equal (* x y)
(* y x)))
(defthm arith-commutativity-2-of-*
(equal (* x y z)
(* y x z)))
(defthm arith-unicity-of-1
(and (equal (* 1 x)
(fix x))
(equal (* x 1)
(fix x))))
(defthm arith-times-zero
(and (equal (* 0 x)
0)
(equal (* x 0)
0)))
(defthm arith-inverse-of-*-1
(implies (and (rationalp x) (not (equal x 0)))
(equal (* x (/ x))
1)))
(defthm arith-inverse-of-*-2
(implies (and (rationalp x) (not (equal x 0)))
(equal (* x (/ x) y)
(fix y))))
(defthm arith-inverse-of-*-3
(implies (and (rationalp x) (not (equal x 0)))
(equal (* x y (/ x))
(fix y))))
(defthm arith-functional-self-inversion-of-/
(equal (/ (/ x))
(fix x)))
(defthm arith-distributivity-of-/-over-*
(equal (/ (* x y))
(* (/ x) (/ y))))
(defthm arith-functional-commutativity-of-minus-*-right
(equal (* x (- y))
(- (* x y))))
(defthm arith-functional-commutativity-of-minus-*-left
(equal (* (- x) y)
(- (* x y))))
(defthm arith-reciprocal-minusa
(equal (/ (- x))
(- (/ x))))
(defthm arith-distributivity-1
(equal (* x (+ y z)) (+ (* x y) (* x z))))
(defthm arith-distributivity-2
(equal (* (+ y z) x) (+ (* x y) (* x z))))
(defthm arith-fold-consts-in-*
(implies (and (syntaxp (quotep c))
(syntaxp (quotep d)))
(equal (* c d x) (* (* c d) x))))
(defthm arith-expt-0
(and (equal (expt x 0)
1)
(equal (expt 0 i)
(if (zip i)
1
0))))
(defthm arith-expt-1
(and (equal (expt x 1)
(fix x))
(equal (expt 1 i)
1)))
(defthm arith-expt-minus-1
(equal (expt x -1)
(/ x)))
(defthm arith-functional-commutativity-of-expt-/
(equal (/ (expt r i))
(expt (/ r) i)))
(local
(in-theory (disable FUNCTIONAL-COMMUTATIVITY-OF-EXPT-/-BASE)))
(defthm arith-expt-minus-exponent
(equal (expt r (- i))
(expt (/ r) i)))
(defthm arith-expt-negative-constant-exponent
(implies (syntaxp (quotep i))
(equal (expt (/ r) (* i j))
(expt r (* (- i) j)))))
(defthm arith-exponents-multiply
(implies (and (integerp i)
(integerp j))
(equal (expt (expt r i) j)
(expt r (* i j)))))
(defthm arith-distributivity-of-expt-over-*
(equal (expt (* a b) i)
(* (expt a i)
(expt b i))))
(defthm arith-fix-revealed
(implies (acl2-numberp x)
(equal (fix x) x)))
(defthm arith-Rational-implies2
(implies (rationalp x)
(equal (* (numerator x) (/ (denominator x))) x)))
(defthm arith-exponents-add-1
(implies (and (integerp i)
(integerp j)
(not (equal (+ i j) 0)))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j)))))
(defthm arith-exponents-add-for-nonpos-exponentsa
(implies (and (<= i 0)
(<= j 0)
(integerp i)
(integerp j))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j)))))
(defthm arith-exponents-add-for-nonneg-exponentsa
(implies (and (<= 0 i)
(<= 0 j)
(integerp i)
(integerp j))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j)))))
(defthm arith-exponents-add-2
(implies (and (not (equal 0 r))
(acl2-numberp r)
(integerp i)
(integerp j))
(equal (expt r (+ i j))
(* (expt r i)
(expt r j)))))
(in-theory (disable arith-associativity-of-* arith-commutativity-of-*
arith-commutativity-2-of-* arith-unicity-of-1
arith-times-zero arith-inverse-of-*-1
arith-inverse-of-*-2 arith-inverse-of-*-3
arith-functional-self-inversion-of-/
arith-distributivity-of-/-over-*
arith-functional-commutativity-of-minus-*-right
arith-functional-commutativity-of-minus-*-left
arith-reciprocal-minusa arith-distributivity-1
arith-distributivity-2 arith-fold-consts-in-*
arith-expt-0
arith-expt-1
arith-expt-minus-1
arith-functional-commutativity-of-expt-/
arith-expt-minus-exponent
arith-expt-negative-constant-exponent
arith-exponents-multiply
arith-distributivity-of-expt-over-*
arith-fix-revealed arith-Rational-implies2
arith-exponents-add-1
arith-exponents-add-for-nonpos-exponentsa
arith-exponents-add-for-nonneg-exponentsa
arith-exponents-add-2))
(in-arithmetic-theory '((:REWRITE arith-associativity-of-*)
(:REWRITE arith-commutativity-of-*)
(:REWRITE arith-commutativity-2-of-*)
(:REWRITE arith-unicity-of-1)
(:REWRITE arith-times-zero)
(:REWRITE arith-inverse-of-*-1)
(:REWRITE arith-inverse-of-*-2)
(:REWRITE arith-inverse-of-*-3)
(:REWRITE arith-functional-self-inversion-of-/ )
(:REWRITE arith-distributivity-of-/-over-*)
(:REWRITE arith-functional-commutativity-of-minus-*-right)
(:REWRITE arith-functional-commutativity-of-minus-*-left)
(:REWRITE arith-reciprocal-minusa)
(:REWRITE arith-distributivity-1)
(:REWRITE arith-distributivity-2)
(:REWRITE arith-fold-consts-in-*)
(:REWRITE arith-expt-0)
(:REWRITE arith-expt-1)
(:REWRITE arith-expt-minus-1)
(:REWRITE arith-functional-commutativity-of-expt-/)
(:REWRITE arith-expt-minus-exponent)
(:REWRITE arith-expt-negative-constant-exponent)
(:REWRITE arith-exponents-multiply)
(:REWRITE arith-distributivity-of-expt-over-*)
(:REWRITE arith-fix-revealed)
(:REWRITE arith-Rational-implies2)
(:REWRITE arith-exponents-add-1)
(:REWRITE arith-exponents-add-for-nonpos-exponentsa)
(:REWRITE arith-exponents-add-for-nonneg-exponentsa)
(:REWRITE arith-exponents-add-2)))
|
