Fixing function for ipasir$a structures.
(ipasir$a-fix x) → new-x
Function:
(defun ipasir$a-fix$inline (x) (declare (xargs :guard (ipasir$a-p x))) (let ((__function__ 'ipasir$a-fix)) (declare (ignorable __function__)) (mbe :logic (b* ((formula (lit-list-list-fix (cdr (std::da-nth 0 x)))) (assumption (lit-list-fix (cdr (std::da-nth 1 x)))) (new-clause (lit-list-fix (cdr (std::da-nth 2 x)))) (status (ipasir-status-fix (cdr (std::da-nth 3 x)))) (solution (lit-list-fix (cdr (std::da-nth 4 x)))) (solved-assumption (lit-list-fix (cdr (std::da-nth 5 x)))) (callback-count (nfix (cdr (std::da-nth 6 x)))) (history (cdr (std::da-nth 7 x)))) (list (cons 'formula formula) (cons 'assumption assumption) (cons 'new-clause new-clause) (cons 'status status) (cons 'solution solution) (cons 'solved-assumption solved-assumption) (cons 'callback-count callback-count) (cons 'history history))) :exec x)))
Theorem:
(defthm ipasir$a-p-of-ipasir$a-fix (b* ((new-x (ipasir$a-fix$inline x))) (ipasir$a-p new-x)) :rule-classes :rewrite)
Theorem:
(defthm ipasir$a-fix-when-ipasir$a-p (implies (ipasir$a-p x) (equal (ipasir$a-fix x) x)))
Function:
(defun ipasir$a-equiv$inline (acl2::x acl2::y) (declare (xargs :guard (and (ipasir$a-p acl2::x) (ipasir$a-p acl2::y)))) (equal (ipasir$a-fix acl2::x) (ipasir$a-fix acl2::y)))
Theorem:
(defthm ipasir$a-equiv-is-an-equivalence (and (booleanp (ipasir$a-equiv x y)) (ipasir$a-equiv x x) (implies (ipasir$a-equiv x y) (ipasir$a-equiv y x)) (implies (and (ipasir$a-equiv x y) (ipasir$a-equiv y z)) (ipasir$a-equiv x z))) :rule-classes (:equivalence))
Theorem:
(defthm ipasir$a-equiv-implies-equal-ipasir$a-fix-1 (implies (ipasir$a-equiv acl2::x x-equiv) (equal (ipasir$a-fix acl2::x) (ipasir$a-fix x-equiv))) :rule-classes (:congruence))
Theorem:
(defthm ipasir$a-fix-under-ipasir$a-equiv (ipasir$a-equiv (ipasir$a-fix acl2::x) acl2::x) :rule-classes (:rewrite :rewrite-quoted-constant))
Theorem:
(defthm equal-of-ipasir$a-fix-1-forward-to-ipasir$a-equiv (implies (equal (ipasir$a-fix acl2::x) acl2::y) (ipasir$a-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm equal-of-ipasir$a-fix-2-forward-to-ipasir$a-equiv (implies (equal acl2::x (ipasir$a-fix acl2::y)) (ipasir$a-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm ipasir$a-equiv-of-ipasir$a-fix-1-forward (implies (ipasir$a-equiv (ipasir$a-fix acl2::x) acl2::y) (ipasir$a-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)
Theorem:
(defthm ipasir$a-equiv-of-ipasir$a-fix-2-forward (implies (ipasir$a-equiv acl2::x (ipasir$a-fix acl2::y)) (ipasir$a-equiv acl2::x acl2::y)) :rule-classes :forward-chaining)