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(in-package "PROOF-CHECKER-ARRAY")
(include-book "ternary")
(include-book "literal")
(include-book "unique")
(include-book "sets")
(include-book "assignment")
(include-book "clause")
(include-book "all-literals")
;; ===================================================================
;; ======================= LITERAL EVALUATION ========================
(defun evaluate-literal (literal assignment)
(declare (xargs :guard (and (literalp literal)
(assignmentp assignment))))
(cond
((member literal assignment) (true))
((member (negate literal) assignment) (false))
(t (undef))))
;; ===================================================================
;; ========================= EVALUATE-CLAUSE =========================
(defun evaluate-clause (clause assignment)
(declare (xargs :guard (and (clausep clause)
(assignmentp assignment))))
(if (atom clause)
(false)
(let* ((literal (car clause))
(literal-value (evaluate-literal literal assignment)))
(if (truep literal-value)
(true)
(let* ((remaining-clause (cdr clause))
(remaining-clause-value (evaluate-clause remaining-clause
assignment)))
(cond
((truep remaining-clause-value) (true))
((undefp literal-value) (undef))
(t remaining-clause-value)))))))
(defthm ternaryp-evaluate-clause
(ternaryp (evaluate-clause clause assignment)))
;; ===================================================================
;; ======================== EVALUATE-FORMULA =========================
(defun evaluate-formula (formula assignment)
(declare (xargs :guard (and (formulap formula)
(assignmentp assignment))))
(if (atom formula)
(true)
(let* ((clause (car formula))
(clause-value (evaluate-clause clause assignment)))
(if (falsep clause-value)
(false)
(let* ((remaining-formula (cdr formula))
(remaining-formula-value (evaluate-formula remaining-formula
assignment)))
(cond
((falsep remaining-formula-value) (false))
((undefp clause-value) (undef))
(t remaining-formula-value)))))))
(defthm ternaryp-evaluate-formula
(ternaryp (evaluate-formula formula assignment)))
;; ===================================================================
;; ============================== CONS ===============================
(defthm iff-truep-evaluate-clause-cons
(iff (truep (evaluate-clause clause (cons literal assignment)))
(or (member literal clause)
(truep (evaluate-clause clause assignment)))))
(defthm truep-evaluate-formula-cons
(equal (truep (evaluate-formula (cons clause formula) assignment))
(and (truep (evaluate-clause clause assignment))
(truep (evaluate-formula formula assignment)))))
(defthm not-undefp-evaluate-clause-cons
(implies (not (undefp (evaluate-clause clause assignment)))
(not (undefp (evaluate-clause clause (cons literal assignment))))))
;; ===================================================================
;; =================== SET EQUALITY AND EVALUATION ===================
; We can reduce any of these to truep implies truep. They might be stronger
; than needed.
(defthm set-equalp-implies-equal-evaluate-literal
(implies (and (set-equalp x y)
(literalp literal)
(assignmentp x)
(assignmentp y))
(equal (evaluate-literal literal y)
(evaluate-literal literal x))))
(defthm set-equalp-implies-equal-evaluate-clause
(implies (and (set-equalp x y)
(clausep clause)
(assignmentp x)
(assignmentp y))
(equal (evaluate-clause clause y)
(evaluate-clause clause x))))
;; :hints (("Goal" :in-theory (disable
;; NOT-SET-DIFFERENCE-VARIABLES-IMPLIES-SUBSET-VARIABLESP
;; SUBSETP-AND-SET-EQUAL-VARIABLESP-IMPLIES-SUBSETP
;; SUBSET-VARIABLESP-AND-SUBSETP-IMPLIES-SUBSETP))))
(defthm set-equalp-implies-equal-evaluate-formula
(implies (and (set-equalp x y)
(formulap formula)
(assignmentp x)
(assignmentp y))
(equal (evaluate-formula formula y)
(evaluate-formula formula x)))
:hints (("Goal" :in-theory (disable set-equalp)))) ; want to remove this
;; ===================================================================
;; ===================== SHORTEN-LONG-ASSIGNMENT =====================
(defun shorten-long-assignment (assignment all-literals)
(declare (xargs :guard (and (literal-listp assignment)
(literal-listp all-literals))))
(union-variables assignment all-literals))
;; =============== ASSIGNMENTP-SHORTEN-LONG-ASSIGNMENT ===============
(encapsulate
()
(local
(defthm literal-listp-shorten-long-assignment
(implies (assignmentp assignment)
(literal-listp (shorten-long-assignment assignment all-literals)))))
(local
(defthm unique-literalsp-shorten-long-assignment
(implies (assignmentp x)
(unique-literalsp (shorten-long-assignment x y)))))
(local
(defthm no-conflicting-literalsp-shorten-long-assignment
(implies (assignmentp x)
(no-conflicting-literalsp (shorten-long-assignment x y)))))
(defthm assignmentp-shorten-long-assignment
(implies (assignmentp x)
(assignmentp (shorten-long-assignment x y)))
:hints (("Goal"
:in-theory (enable assignmentp))))
)
;; ============ SUBSET-VARIABLESP-SHORTEN-LONG-ASSIGNMENT ============
(defthm subsetp-shorten-long-assignment
(subsetp (shorten-long-assignment x y) x))
(defthm subset-variablesp-shorten-long-assignment
(subset-variablesp (shorten-long-assignment x y) y))
;; ========= TRUEP-EVALUATE-FORMULA-SHORTEN-LONG-ASSIGNMENT ==========
(defthm truep-evaluate-literal-shorten-long-assignment
(implies (and (truep (evaluate-literal literal assignment))
(member literal all-literals))
(truep (evaluate-literal literal (shorten-long-assignment
assignment
all-literals)))))
(defthm truep-evaluate-clause-shorten-long-assignment
(implies (and (truep (evaluate-clause clause assignment))
(subsetp clause all-literals))
(truep (evaluate-clause clause (shorten-long-assignment
assignment
all-literals)))))
(defthm subsetp-list-truep-evaluate-formula-shorten-long-assignment
(implies (and (truep (evaluate-formula formula assignment))
(subsetp-list formula all-literals))
(truep (evaluate-formula formula
(shorten-long-assignment
assignment
all-literals))))
:hints (("Goal"
:in-theory (disable shorten-long-assignment))))
(defthm truep-evaluate-formula-shorten-long-assignment
(implies (truep (evaluate-formula formula assignment))
(truep (evaluate-formula formula
(shorten-long-assignment
assignment
(all-literals formula)))))
:hints (("Goal"
:in-theory (disable shorten-long-assignment))))
;; ============ STRONG-ASSIGNMENT-SHORTEN-LONG-ASSIGNMENT ============
(defthm strong-assignment-shorten-long-assignment
(implies (and (assignmentp assignment)
(truep (evaluate-formula formula assignment)))
(and (assignmentp (shorten-long-assignment
assignment
(all-literals formula)))
(truep (evaluate-formula formula
(shorten-long-assignment
assignment
(all-literals formula))))
(subset-variablesp (shorten-long-assignment
assignment
(all-literals formula))
(all-literals formula))))
:hints (("Goal"
:in-theory (disable shorten-long-assignment))))
;; ===================================================================
;; ===================== EXTEND-SHORT-ASSIGNMENT =====================
; consider switching append order?
(defun extend-short-assignment (assignment all-literals)
(declare (xargs :guard (and (literal-listp assignment)
(literal-listp all-literals))))
(append (set-difference-variables (remove-conflicting-literals all-literals)
assignment)
assignment))
;; =============== ASSIGNMENTP-EXTEND-SHORT-ASSIGNMENT ===============
(encapsulate
()
(local
(defthm literal-listp-extend-short-assignment
(implies (and (assignmentp assignment)
(literal-listp all-literals))
(literal-listp (extend-short-assignment assignment
all-literals)))))
(local
(defthm unique-literalsp-extend-short-assignment
(implies (and (assignmentp assignment)
(unique-literalsp all-literals))
(unique-literalsp (extend-short-assignment assignment
all-literals)))))
(local
(defthm no-conflicting-literalsp-extend-short-assignment
(implies (assignmentp assignment)
(no-conflicting-literalsp (extend-short-assignment
assignment
all-literals)))))
(defthm assignmentp-extend-short-assignment
(implies (and (assignmentp assignment)
(literal-listp all-literals)
(unique-literalsp all-literals))
(assignmentp (extend-short-assignment assignment all-literals))))
)
;; ============ SUBSET-VARIABLESP-EXTEND-SHORT-ASSIGNMENT ============
(defthm subset-variablesp-extend-short-assignment
(implies (literal-listp all-literals)
(subset-variablesp all-literals
(extend-short-assignment assignment
all-literals))))
(defthm subset-variablesp-extend-short-assignment2
(implies (subset-variablesp assignment all-literals)
(subset-variablesp (extend-short-assignment assignment
all-literals)
all-literals)))
;; ========= TRUEP-EVALUATE-FORMULA-EXTEND-SHORT-ASSIGNMENT ==========
(defthm truep-evaluate-literal-extend-long-assignment
(implies (and (truep (evaluate-literal literal assignment))
(member literal all-literals))
(truep (evaluate-literal literal (extend-short-assignment
assignment
all-literals)))))
(defthm truep-evaluate-clause-extend-short-assignment
(implies (and (truep (evaluate-clause clause assignment))
(subsetp clause all-literals))
(truep (evaluate-clause clause (extend-short-assignment
assignment
all-literals)))))
(defthm subsetp-list-truep-evaluate-formula-extend-short-assignment
(implies (and (truep (evaluate-formula formula assignment))
(subsetp-list formula all-literals))
(truep (evaluate-formula formula (extend-short-assignment
assignment
all-literals))))
:hints (("Goal"
:in-theory (disable extend-short-assignment))))
(defthm truep-evaluate-formula-extend-short-assignment
(implies (and (assignmentp assignment)
(truep (evaluate-formula formula assignment)))
(truep (evaluate-formula formula (extend-short-assignment
assignment
(all-literals formula)))))
:hints (("Goal"
:in-theory (disable extend-short-assignment))))
;; ============ STRONG-ASSIGNMENT-EXTEND-SHORT-ASSIGNMENT ============
(defthm strong-assignment-extend-short-assignment
(implies (and (formulap formula)
(assignmentp assignment)
(truep (evaluate-formula formula assignment)))
(and (assignmentp (extend-short-assignment
assignment
(all-literals formula)))
(truep (evaluate-formula formula (extend-short-assignment
assignment
(all-literals formula))))
(subset-variablesp (all-literals formula)
(extend-short-assignment
assignment
(all-literals formula)))))
:hints (("Goal"
:in-theory (disable extend-short-assignment))))
;; ===================================================================
;; ============== SATISFYING-IMPLIES-STRONG-SATISFYING ===============
(defun-sk exists-strong-satisfying-assignment (formula)
(exists assignment (and (assignmentp assignment)
(truep (evaluate-formula formula assignment))
(set-equal-variablesp (all-literals formula)
assignment))))
(in-theory (disable exists-strong-satisfying-assignment
exists-strong-satisfying-assignment-suff))
(defthm satisfying-assignment-implies-exists-strong-satisfying-assignment
(implies (and (formulap formula)
(assignmentp assignment)
(truep (evaluate-formula formula assignment)))
(exists-strong-satisfying-assignment formula))
:hints (("Goal"
:in-theory (disable extend-short-assignment
shorten-long-assignment
)
:use ((:instance exists-strong-satisfying-assignment-suff
(assignment (extend-short-assignment
(shorten-long-assignment
assignment
(all-literals formula))
(all-literals formula))))))))
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