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• 4vec-<<=
• Svex-env

Svex-env-<<=

(svex-env-<<= x y) checks whether an entire svex-env conservatively approximates another: i.e., is every variable's value in x an approximation of its value in y?

Definitions and Theorems

Theorem: svex-env-<<=-necc

(defthm svex-env-<<=-necc
  (implies (svex-env-<<= x y)
           (4vec-<<= (svex-env-lookup var x)
                     (svex-env-lookup var y))))

Theorem: svex-env-<<=-witnessing-witness-rule-correct

(defthm svex-env-<<=-witnessing-witness-rule-correct
  (implies (not ((lambda (var y x)
                   (not (4vec-<<= (svex-env-lookup var x)
                                  (svex-env-lookup var y))))
                 (svex-env-<<=-witness x y)
                 y x))
           (svex-env-<<= x y))
  :rule-classes nil)

Theorem: svex-env-<<=-instancing-instance-rule-correct

(defthm svex-env-<<=-instancing-instance-rule-correct
  (implies (not (4vec-<<= (svex-env-lookup var x)
                          (svex-env-lookup var y)))
           (not (svex-env-<<= x y)))
  :rule-classes nil)

Theorem: svex-env-<<=-of-svex-env-fix-x

(defthm svex-env-<<=-of-svex-env-fix-x
  (equal (svex-env-<<= (svex-env-fix x) y)
         (svex-env-<<= x y)))

Theorem: svex-env-<<=-svex-env-equiv-congruence-on-x

(defthm svex-env-<<=-svex-env-equiv-congruence-on-x
  (implies (svex-env-equiv x x-equiv)
           (equal (svex-env-<<= x y)
                  (svex-env-<<= x-equiv y)))
  :rule-classes :congruence)

Theorem: svex-env-<<=-of-svex-env-fix-y

(defthm svex-env-<<=-of-svex-env-fix-y
  (equal (svex-env-<<= x (svex-env-fix y))
         (svex-env-<<= x y)))

Theorem: svex-env-<<=-svex-env-equiv-congruence-on-y

(defthm svex-env-<<=-svex-env-equiv-congruence-on-y
  (implies (svex-env-equiv y y-equiv)
           (equal (svex-env-<<= x y)
                  (svex-env-<<= x y-equiv)))
  :rule-classes :congruence)

Theorem: svex-env-<<=-empty

(defthm svex-env-<<=-empty
  (svex-env-<<= nil x))