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|
(in-package "ACL2")
(include-book "bcv-model")
(include-book "bcv-simple-model")
(include-book "generic")
(defthm member-extract-code-implied-by-member-merged-code-and-inst
(implies (and (merged-code-safe merged-code sig-frame)
(member inst merged-code)
(inst? inst))
(member inst (extract-code merged-code))))
(local
(defthm g-pc-collect-witness-pc
(implies (bound? pc (collect-witness-merged-code-safe merged-code
sig-frame))
(equal (g 'pc (cdr (assoc-equal pc
(collect-witness-merged-code-safe
merged-code sig-frame))))
pc))))
(defthm member-inst-extract-code-implies-bound-pc
(implies (and (merged-code-safe merged-code sig-frame)
(member inst (extract-code merged-code)))
(ASSOC-EQUAL (G 'PC INST)
(COLLECT-WITNESS-MERGED-CODE-SAFE MERGED-CODE SIG-FRAME))))
;----------------------------------------------------------------------
(defthm merged-code-safe-implies-current-inst-is-bound
(implies (AND (MERGED-CODE-SAFE MERGED-CODE SIG-FRAME)
(MEMBER INST MERGED-CODE)
(INST? INST))
(equal (g 'pc (cdr (assoc-equal (g 'pc inst)
(collect-witness-merged-code-safe
merged-code sig-frame))))
(g 'pc inst))))
;; (defthm merged-code-safe-implies-PUSH-is-NOT-THE-LAST-INST
;; (implies (AND (EQUAL (CAR (G 'INST INST)) 'PUSH)
;; (MERGED-CODE-SAFE MERGED-CODE SIG-FRAME)
;; (MEMBER INST MERGED-CODE)
;; (INST? INST))
;; (assoc-equal
;; (+ 1 (g 'pc inst))
;; (collect-witness-merged-code-safe
;; merged-code sig-frame)))
;; :hints (("Goal" :do-not '(generalize))))
(defthm car-bcv-simple-execute-step-is-bcv-execute-step
(implies (member (car (g 'inst inst)) '(PUSH POP IFEQ INVOKE))
(equal (car (bcv-simple-execute-step (g 'inst inst) sig-frame))
(bcv-execute-step inst sig-frame))))
;----------------------------------------------------------------------
;; ;;;;
;; ;;;; Shall we prove the following by induction directly? no.
;; ;;;;
;; ;;;; We will first prove when the first instruction is an instruction
;; ;;;;
;; (defthm merge-code-safe-implies-sig-frame-compatible-lemma
;; (implies (and ;;; (case-split (member (bcv-op-code inst) '(PUSH POP IFEQ INVOKE))))))
;; ;;; (not (member (bcv-op-code inst) '(HALT RETURN)))
;; (case-split (member (bcv-op-code inst) '(PUSH)))
;; (merged-code-safe merged-code sig-frame)
;; (wff-code1 (extract-code merged-code) (g 'pc (car merged-code)))
;; (member inst merged-code)
;; (inst? inst)
;; (equal pc (g 'pc inst)))
;; (sig-frame-compatible
;; (bcv-execute-step inst
;; (cdr (assoc-equal pc
;; (collect-witness-merged-code-safe merged-code sig-frame))))
;; (cdr (assoc-equal (+ 1 pc)
;; (collect-witness-merged-code-safe
;; merged-code sig-frame)))))
;; :hints (("Goal" :in-theory (disable inst? stack-map?
;; sig-frame-compatible
;; bcv-simple-execute-step
;; bcv-execute-step
;; bcv-check-step-pre)
;; :do-not '(generalize fertilize))))
(defthm not-equal-g-not-equal
(implies (not (equal (g 'pc X) (g 'pc Y)))
(not (equal X Y)))
:rule-classes nil)
(defthm s-never-AFTERGOTO
(not (equal (s 'pc (+ 1 x) Any2) 'AFTERGOTO))
:hints (("Goal" :use ((:instance not-equal-g-not-equal
(X (s 'pc (+ 1 x) any2))
(Y 'AFTERGOTO))))))
(defthm bcv-execute-step-is-not-aftergoto
(implies (not (member (car (g 'inst inst)) '(HALT RETURN)))
(not (equal (bcv-execute-step inst sig-frame)
'AFTERGOTO))))
;----------------------------------------------------------------------
(defthm not-merge-code-safe-aftr-goto
(implies (inst? (car merged-code))
(not (merged-code-safe merged-code 'aftergoto))))
(defthm inst-stack-map-more-general
(implies (and (merged-code-safe merged-code sig-frame)
(inst? (car merged-code))
(not (member (bcv-op-code (car merged-code)) '(HALT RETURN)))
(stack-map? (cadr merged-code)))
(sig-frame-compatible
(bcv-execute-step (car merged-code) sig-frame)
(cadr merged-code)))
:hints (("Goal" :in-theory (e/d () (bcv-execute-step
inst? stack-map?
sig-frame-compatible))
:do-not '(generalize)
:do-not-induct t)))
(defun stack-map-prefix (merge-code)
(if (endp merge-code) nil
(if (inst? (car merge-code)) nil
(cons (car merge-code)
(stack-map-prefix (cdr merge-code))))))
(defthm merge-code-safe-implies-first-assignable-to-any-in-the-prefix
(implies (and (merged-code-safe merge-code sig-frame)
(stack-map? (car merge-code))
(member any-stack-map (stack-map-prefix merge-code)))
(sig-frame-compatible
(car merge-code)
any-stack-map)))
(defthm merge-code-safe-equal-assoc-equal-is-last
(implies (and (merged-code-safe merge-code sig-frame)
(stack-map? (car merge-code)))
(equal (cdr (assoc-equal (g 'pc (car merge-code))
(collect-witness-merged-code-safe
merge-code sig-frame)))
(car (last (stack-map-prefix merge-code)))))
:hints (("Goal" :in-theory (e/d () (stack-map? inst?)))))
(defthm member-last-stack-map-prefix
(implies (consp l)
(member (car (last l)) l)))
(defthm merge-code-safe-equal-assoc-equal-is-last-general
(implies (and (merged-code-safe merge-code sig-frame)
(stack-map? (car merge-code))
(equal pc (g 'pc (car merge-code))))
(equal (cdr (assoc-equal pc
(collect-witness-merged-code-safe
merge-code sig-frame)))
(car (last (stack-map-prefix merge-code)))))
:hints (("Goal" :in-theory (e/d () (stack-map? inst?)))))
(encapsulate ()
(local (include-book "bcv-find-correct-witness-bcv-check-pre"))
(defthm wff-code1-uniqueness
(IMPLIES
(AND (WFF-CODE1 (EXTRACT-CODE MERGED-CODE4) any)
(MERGED-CODE-SAFE MERGED-CODE4 MERGED-CODE3))
(wff-code1 (extract-code merged-code4)
(g 'pc (car merged-code4))))
:rule-classes :forward-chaining))
(encapsulate ()
(local (include-book "bcv-find-correct-witness-bcv-check-pre"))
(defthm merged-code-safe-implies-extract-code-pc-is-same-strong
(implies (merged-code-safe merged-code sig-frame)
(equal (g 'pc (car (extract-code merged-code)))
(g 'pc (car merged-code))))))
(defthm two-adjacent-instructions-pc-adjacent-2
(implies (and (merged-code-safe merged-code sig-frame)
(wff-code1 (extract-code merged-code)
(g 'pc (car merged-code)))
(inst? (car merged-code))
(stack-map? (cadr merged-code)))
(equal (+ 1 (g 'pc (car merged-code)))
(g 'pc (cadr merged-code))))
:hints (("Goal" :in-theory (e/d () (inst? stack-map?))
:do-not '(generalize fertilize))))
(local
(defthm merged-code-safe-implies-pc-equal
(implies (and (merged-code-safe merged-code sig-frame)
(inst? (car merged-code)))
(equal (g 'pc sig-frame)
(g 'pc (car merged-code))))))
(defthm two-adjacent-instructions-pc-adjacent-2-specific
(implies (and (merged-code-safe merged-code sig-frame)
(wff-code1 (extract-code merged-code)
(g 'pc (car merged-code)))
(inst? (car merged-code))
(stack-map? (cadr merged-code)))
(equal (+ 1 (g 'pc sig-frame))
(g 'pc (cadr merged-code))))
:hints (("Goal" :in-theory (e/d () (inst? wff-code1
merged-code-safe
stack-map?)))))
(local (in-theory (disable merged-code-safe-implies-pc-equal)))
(local
(defthm sig-frame-compatible-transitive
(implies (and (sig-frame-compatible frame1 frame2)
(sig-frame-compatible frame2 frame3))
(sig-frame-compatible frame1 frame3))))
(defthm wff-code-implies-not-equal
(implies (and (merged-code-safe merged-code sig-frame)
(WFF-CODE1 (EXTRACT-CODE MERGED-CODE)
(+ 1 pc))
(stack-map? (car merged-code)))
(not (equal (g 'pc (car merged-code)) pc))))
(local
(defthm sig-frame-compatible-reflexive
(SIG-FRAME-COMPATIBLE stack-map stack-map)))
(defthm member-last-stack-map-prefix
(implies (consp l)
(member (car (last l)) l)))
(defthm consp-stack-map-prefix
(implies (stack-map? (car merged-code))
(consp (stack-map-prefix merged-code))))
(defthm inst-stack-map-more-general-specific
(implies (and (merged-code-safe merged-code sig-frame)
(inst? (car merged-code))
(not (member (bcv-op-code (car merged-code)) '(HALT RETURN)))
(stack-map? (cadr merged-code)))
(sig-frame-compatible
(bcv-execute-step (car merged-code) sig-frame)
(car (last (stack-map-prefix (cdr merged-code))))))
:hints (("Goal" :in-theory (e/d () (bcv-execute-step
bcv-check-step-pre
inst? stack-map?
sig-frame-compatible))
:use ((:instance
sig-frame-compatible-transitive
(frame1 (bcv-execute-step (car merged-code) sig-frame))
(frame2 (cadr merged-code))
(frame3 (car (last (stack-map-prefix (cdr merged-code)))))))
:restrict
((merge-code-safe-implies-first-assignable-to-any-in-the-prefix
((sig-frame (bcv-execute-step (car merged-code) sig-frame)))))
:do-not '(generalize)
:do-not-induct t)))
(defthm execute-inst-compatible-with-stack-1
(implies (and (merged-code-safe merged-code sig-frame)
(inst? (car merged-code))
(wff-code1 (extract-code merged-code)
(g 'pc sig-frame))
(not (member (bcv-op-code (car merged-code)) '(HALT RETURN)))
(stack-map? (cadr merged-code))
(equal pc (+ 1 (g 'pc (car merged-code)))))
(sig-frame-compatible
(bcv-execute-step (car merged-code) sig-frame)
(cdr (assoc-equal pc (collect-witness-merged-code-safe merged-code sig-frame)))))
:hints (("Goal" :in-theory (e/d () (bcv-execute-step
bcv-check-step-pre
inst? stack-map?
;;wff-code1
;;extract-code
sig-frame-compatible))
:restrict ((two-adjacent-instructions-pc-adjacent-2-specific
((merged-code merged-code))))
:do-not '(generalize)
:do-not-induct t)))
(defthm two-adjacent-instructions-pc-adjacent
(implies (and (merged-code-safe merged-code sig-frame)
(inst? (car merged-code))
(inst? (cadr merged-code)))
(equal (+ 1 (g 'pc (car merged-code)))
(g 'pc (cadr merged-code)))))
(defthm bcv-execute-step-produces-a-specific-state
(implies (and (merged-code-safe merged-code sig-frame)
(inst? (car merged-code))
(inst? (cadr merged-code))
(not (member (bcv-op-code (car merged-code)) '(HALT RETURN)))
(equal pc (g 'pc (cadr merged-code))))
(equal (cdr (assoc-equal pc
(collect-witness-merged-code-safe
merged-code sig-frame)))
(bcv-execute-step (car merged-code) sig-frame)))
:hints (("Goal" :in-theory (e/d ()(bcv-execute-step
bcv-check-step-pre
two-adjacent-instructions-pc-adjacent
sig-frame-compatible
inst? stack-map?))
:do-not '(generalize fertilize)
:use ((:instance two-adjacent-instructions-pc-adjacent))
:do-not-induct t)))
(defthm execute-inst-compatible-with-stack-2
(implies (and (merged-code-safe merged-code sig-frame)
(inst? (car merged-code))
(wff-code1 (extract-code merged-code)
(g 'pc sig-frame))
(not (member (bcv-op-code (car merged-code)) '(HALT RETURN)))
(inst? (cadr merged-code))
(equal pc (+ 1 (g 'pc (car merged-code)))))
(sig-frame-compatible
(bcv-execute-step (car merged-code) sig-frame)
(cdr (assoc-equal pc (collect-witness-merged-code-safe merged-code sig-frame)))))
:hints (("Goal" :in-theory (e/d () (bcv-execute-step
bcv-check-step-pre
inst? stack-map?
sig-frame-compatible))
:do-not '(generalize)
:do-not-induct t)))
;----------------------------------------------------------------------
(defthm execute-inst-compatible-with-stack
(implies (and (merged-code-safe merged-code sig-frame)
(inst? (car merged-code))
(wff-code1 (extract-code merged-code)
(g 'pc sig-frame))
(not (member (bcv-op-code (car merged-code)) '(HALT RETURN)))
(equal pc (+ 1 (g 'pc (car merged-code)))))
(sig-frame-compatible
(bcv-execute-step (car merged-code) sig-frame)
(cdr (assoc-equal pc (collect-witness-merged-code-safe merged-code sig-frame)))))
:hints (("Goal" :in-theory (e/d () (bcv-execute-step
bcv-check-step-pre
inst? stack-map?
execute-inst-compatible-with-stack-1
sig-frame-compatible))
:cases ((inst? (cadr merged-code))
(stack-map? (cadr merged-code)))
:do-not '(generalize)
:do-not-induct t)
("Subgoal 1.1" :use ((:instance execute-inst-compatible-with-stack-1
(pc (+ 1 (g 'pc (car merged-code)))))))))
;----------------------------------------------------------------------
(defthm |Subgoal *1/9.3|
(IMPLIES
(AND (CONSP MERGED-CODE2)
(NOT (EQUAL SIG-FRAME 'AFTERGOTO))
(NOT (MEMBER MERGED-CODE1 MERGED-CODE2))
(NOT (EQUAL (CAR (G 'INST MERGED-CODE1))
'HALT))
(NOT (EQUAL (CAR (G 'INST MERGED-CODE1))
'RETURN))
(WFF-CODE1 (EXTRACT-CODE MERGED-CODE2)
(+ 1 (G 'PC MERGED-CODE1)))
(INST? MERGED-CODE1)
(EQUAL (G 'PC SIG-FRAME)
(G 'PC MERGED-CODE1))
(BCV-CHECK-STEP-PRE MERGED-CODE1 SIG-FRAME)
(MERGED-CODE-SAFE MERGED-CODE2
(BCV-EXECUTE-STEP MERGED-CODE1 SIG-FRAME))
(NOT (EQUAL (+ 1 (G 'PC MERGED-CODE1))
(G 'PC MERGED-CODE1))))
(SIG-FRAME-COMPATIBLE
(BCV-EXECUTE-STEP MERGED-CODE1 SIG-FRAME)
(CDR (ASSOC-EQUAL (+ 1 (G 'PC MERGED-CODE1))
(COLLECT-WITNESS-MERGED-CODE-SAFE
MERGED-CODE2
(BCV-EXECUTE-STEP MERGED-CODE1 SIG-FRAME))))))
:hints (("Goal" :in-theory (disable inst? stack-map?
sig-frame-compatible
execute-inst-compatible-with-stack
bcv-simple-execute-step
bcv-execute-step
bcv-check-step-pre)
:use ((:instance execute-inst-compatible-with-stack
(pc (+ 1 (g 'pc (car merged-code))))))
:do-not '(generalize fertilize))))
(defthm collect-witness-merged-code-safe-normalize
(implies (and (merged-code-safe merged-code sig-frame)
(inst? (car merged-code))
(< (g 'pc (car merged-code)) pc))
(equal
(CDR
(ASSOC-EQUAL PC
(COLLECT-WITNESS-MERGED-CODE-SAFE MERGED-CODE SIG-FRAME)))
(cdr (ASSOC-EQUAL PC
(COLLECT-WITNESS-MERGED-CODE-SAFE (CDR MERGED-CODE)
(BCV-EXECUTE-STEP (CAR MERGED-CODE)
SIG-FRAME))))))
:hints (("Goal" :in-theory (disable bcv-execute-step bcv-check-step-pre
sig-frame-compatible
inst?)
:do-not '(generalize))))
;; (defthm collect-witness-merged-member-remainer
;; (implies (and (wff-code1 (extract-code merged-code)
;; (g 'pc (car merged-code)))
;; (inst? (car merged-code))
;; (member inst (cdr merged-code))
;; (inst? inst))
;; (< (g 'pc (car merged-code))
;; (g 'pc inst)))
;; :hints (("Goal" :do-not '(generalize))))
(defthm member-inst-merged-code-implies-not-equal
(implies (and (member inst code)
(wff-code1 code pc))
(<= pc (g 'pc inst)))
:hints (("Goal" :in-theory (e/d () (inst? stack-map?))
:do-not '(generalize))))
(defthm member-inst-merged-code-implies-not-equal-specific
(implies (and (member inst merged-code)
(merged-code-safe merged-code sig-frame)
(inst? inst)
(wff-code1 (extract-code merged-code) (+ 1 pc)))
(< pc (g 'pc inst)))
:hints (("Goal" :in-theory (e/d () (inst? stack-map?))
:use ((:instance member-inst-merged-code-implies-not-equal
(code (extract-code merged-code))
(pc (+ 1 pc))))
:do-not '(generalize)
:do-not-induct t)))
(defthm member-inst-merged-code-implies-not-equal-futher
(implies (and (member inst merged-code)
(merged-code-safe merged-code sig-frame)
(inst? inst)
(wff-code1 (extract-code merged-code) (+ 1 pc)))
(not (equal (+ 1 (g 'pc inst)) pc)))
:hints (("Goal" :in-theory (e/d () (inst? stack-map?))
:use ((:instance member-inst-merged-code-implies-not-equal
(code (extract-code merged-code))
(pc (+ 1 pc))))
:do-not '(generalize)
:do-not-induct t)))
(defthm member-inst-merged-code-implies-not-equal-futher-2
(implies (and (member inst merged-code)
(merged-code-safe merged-code sig-frame)
(inst? inst)
(wff-code1 (extract-code merged-code) (+ 1 pc)))
(not (equal (g 'pc inst) pc)))
:hints (("Goal" :in-theory (e/d () (inst? stack-map?))
:use ((:instance member-inst-merged-code-implies-not-equal
(code (extract-code merged-code))
(pc (+ 1 pc))))
:do-not '(generalize)
:do-not-induct t)))
(defthm |Subgoal *1/9.12.1''|
(IMPLIES
(AND
(CONSP MERGED-CODE2)
(NOT (EQUAL SIG-FRAME 'AFTERGOTO))
(SIG-FRAME-COMPATIBLE
(BCV-EXECUTE-STEP
MERGED-CODE1
(CDR (ASSOC-EQUAL (G 'PC MERGED-CODE1)
(COLLECT-WITNESS-MERGED-CODE-SAFE
MERGED-CODE2
(BCV-EXECUTE-STEP MERGED-CODE1 SIG-FRAME)))))
(CDR (ASSOC-EQUAL (+ 1 (G 'PC MERGED-CODE1))
(COLLECT-WITNESS-MERGED-CODE-SAFE
MERGED-CODE2
(BCV-EXECUTE-STEP MERGED-CODE1 SIG-FRAME)))))
(NOT (EQUAL (CAR (G 'INST MERGED-CODE1))
'HALT))
(NOT (EQUAL (CAR (G 'INST MERGED-CODE1))
'RETURN))
(WFF-CODE1 (EXTRACT-CODE MERGED-CODE2)
(+ 1 (G 'PC MERGED-CODE1)))
(INST? MERGED-CODE1)
(EQUAL (G 'PC SIG-FRAME)
(G 'PC MERGED-CODE1))
(BCV-CHECK-STEP-PRE MERGED-CODE1 SIG-FRAME)
(MERGED-CODE-SAFE MERGED-CODE2
(BCV-EXECUTE-STEP MERGED-CODE1 SIG-FRAME))
(NOT (EQUAL (+ 1 (G 'PC MERGED-CODE1))
(G 'PC MERGED-CODE1))))
(SIG-FRAME-COMPATIBLE
(BCV-EXECUTE-STEP MERGED-CODE1 SIG-FRAME)
(CDR (ASSOC-EQUAL (+ 1 (G 'PC MERGED-CODE1))
(COLLECT-WITNESS-MERGED-CODE-SAFE
MERGED-CODE2
(BCV-EXECUTE-STEP MERGED-CODE1 SIG-FRAME))))))
:hints (("Goal" :in-theory (disable inst? stack-map?
sig-frame-compatible
execute-inst-compatible-with-stack
bcv-simple-execute-step
bcv-execute-step
bcv-check-step-pre)
:use ((:instance execute-inst-compatible-with-stack
(pc (+ 1 (g 'pc (car merged-code))))))
:do-not '(generalize fertilize))))
;;;
;;; not sure what the above says!!
;;; Don't understand why the above need induction!!!
;;; anyway!!
;;;
(defthm merge-code-safe-implies-sig-frame-compatible-lemma-lemma
(IMPLIES
(AND (not (member (bcv-op-code inst) '(HALT RETURN)))
(WFF-CODE1 (extract-code MERGED-CODE) (G 'PC (car MERGED-CODE)))
(inst? inst)
(MERGED-CODE-SAFE MERGED-CODE sig-frame)
(member inst merged-code)
(equal pc (g 'pc inst)))
(SIG-FRAME-COMPATIBLE
(BCV-EXECUTE-STEP
inst
(cdr (assoc-equal pc (collect-witness-merged-code-safe merged-code sig-frame))))
(CDR (ASSOC-EQUAL (+ 1 pc)
(COLLECT-WITNESS-MERGED-CODE-SAFE
merged-code sig-frame)))))
:hints (("Goal" :in-theory (disable inst? stack-map?
sig-frame-compatible
bcv-simple-execute-step
bcv-execute-step
bcv-check-step-pre)
:do-not '(generalize fertilize))))
;----------------------------------------------------------------------
(defthm merge-code-safe-implies-sig-frame-compatible-lemma
(implies (and (case-split (member (bcv-op-code inst) '(PUSH POP IFEQ INVOKE)))
(merged-code-safe merged-code sig-frame)
(wff-code1 (extract-code merged-code)
(g 'pc (car merged-code)))
(member inst merged-code)
(inst? inst)
(equal pc (g 'pc inst)))
(sig-frame-compatible
(car (bcv-simple-execute-step (g 'inst inst)
(cdr (assoc-equal pc
(collect-witness-merged-code-safe merged-code sig-frame)))))
(cdr (assoc-equal (+ 1 pc)
(collect-witness-merged-code-safe
merged-code sig-frame)))))
:hints (("Goal" :in-theory (disable inst?
sig-frame-compatible
bcv-simple-execute-step
bcv-execute-step
bcv-check-step-pre)
:do-not-induct t)))
;----------------------------------------------------------------------
(defthm member-extract-code-member-merged-code-f
(implies (member inst (extract-code merged-code))
(member inst merged-code))
:rule-classes :forward-chaining)
(defthm member-extract-code-inst-f
(implies (member inst (extract-code merged-code))
(inst? inst))
:rule-classes :forward-chaining)
(defthm merge-code-safe-implies-sig-frame-compatible
(implies (and (case-split (member (bcv-op-code inst) '(PUSH POP IFEQ INVOKE)))
(merged-code-safe merged-code sig-frame)
(equal (g 'pc sig-frame) 0)
(wff-code (extract-code merged-code))
(member inst (extract-code merged-code))
(equal pc (g 'pc inst)))
(sig-frame-compatible
(car (bcv-simple-execute-step (g 'inst inst)
(cdr (assoc-equal pc
(collect-witness-merged-code-safe merged-code sig-frame)))))
(cdr (assoc-equal (+ 1 pc)
(collect-witness-merged-code-safe
merged-code sig-frame)))))
:hints (("Goal" :in-theory (disable inst? bcv-simple-execute-step
bcv-execute-step
sig-frame-compatible
inst? member bcv-op-code
bcv-check-step-pre)
:use ((:instance merge-code-safe-implies-sig-frame-compatible-lemma))
:do-not-induct t)))
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