Theory BVExample

theory BVExample
imports JVMListExample BVSpecTypeSafe JVM
(*  Title:      HOL/MicroJava/BV/BVExample.thy
Author: Gerwin Klein
*)


header {* \isaheader{Example Welltypings}\label{sec:BVExample} *}

theory BVExample
imports
"../JVM/JVMListExample"
BVSpecTypeSafe
JVM
begin

text {*
This theory shows type correctness of the example program in section
\ref{sec:JVMListExample} (p. \pageref{sec:JVMListExample}) by
explicitly providing a welltyping. It also shows that the start
state of the program conforms to the welltyping; hence type safe
execution is guaranteed.
*}


section "Setup"

text {* Abbreviations for definitions we will have to use often in the
proofs below: *}

lemmas name_defs = list_name_def test_name_def val_name_def next_name_def
lemmas system_defs = SystemClasses_def ObjectC_def NullPointerC_def
OutOfMemoryC_def ClassCastC_def
lemmas class_defs = list_class_def test_class_def

text {* These auxiliary proofs are for efficiency: class lookup,
subclass relation, method and field lookup are computed only once:
*}

lemma class_Object [simp]:
"class E Object = Some (undefined, [],[])"
by (simp add: class_def system_defs E_def)

lemma class_NullPointer [simp]:
"class E (Xcpt NullPointer) = Some (Object, [], [])"
by (simp add: class_def system_defs E_def)

lemma class_OutOfMemory [simp]:
"class E (Xcpt OutOfMemory) = Some (Object, [], [])"
by (simp add: class_def system_defs E_def)

lemma class_ClassCast [simp]:
"class E (Xcpt ClassCast) = Some (Object, [], [])"
by (simp add: class_def system_defs E_def)

lemma class_list [simp]:
"class E list_name = Some list_class"
by (simp add: class_def system_defs E_def name_defs distinct_classes [symmetric])

lemma class_test [simp]:
"class E test_name = Some test_class"
by (simp add: class_def system_defs E_def name_defs distinct_classes [symmetric])

lemma E_classes [simp]:
"{C. is_class E C} = {list_name, test_name, Xcpt NullPointer,
Xcpt ClassCast, Xcpt OutOfMemory, Object}"

by (auto simp add: is_class_def class_def system_defs E_def name_defs class_defs)

text {* The subclass releation spelled out: *}
lemma subcls1:
"subcls1 E = {(list_name,Object), (test_name,Object), (Xcpt NullPointer, Object),
(Xcpt ClassCast, Object), (Xcpt OutOfMemory, Object)}"

apply (simp add: subcls1_def2)
apply (simp add: name_defs class_defs system_defs E_def class_def)
apply (simp add: Sigma_def)
apply auto
done

text {* The subclass relation is acyclic; hence its converse is well founded: *}
lemma notin_rtrancl:
"(a, b) ∈ r* ==> a ≠ b ==> (!!y. (a, y) ∉ r) ==> False"
by (auto elim: converse_rtranclE)

lemma acyclic_subcls1_E: "acyclic (subcls1 E)"
apply (rule acyclicI)
apply (simp add: subcls1)
apply (auto dest!: tranclD)
apply (auto elim!: notin_rtrancl simp add: name_defs distinct_classes)
done

lemma wf_subcls1_E: "wf ((subcls1 E)¯)"
apply (rule finite_acyclic_wf_converse)
apply (simp add: subcls1 del: insert_iff)
apply (rule acyclic_subcls1_E)
done

text {* Method and field lookup: *}
lemma method_Object [simp]:
"method (E, Object) = Map.empty"
by (simp add: method_rec_lemma [OF class_Object wf_subcls1_E])

lemma method_append [simp]:
"method (E, list_name) (append_name, [Class list_name]) =
Some (list_name, PrimT Void, 3, 0, append_ins, [(1, 2, 8, Xcpt NullPointer)])"

apply (insert class_list)
apply (unfold list_class_def)
apply (drule method_rec_lemma [OF _ wf_subcls1_E])
apply simp
done

lemma method_makelist [simp]:
"method (E, test_name) (makelist_name, []) =
Some (test_name, PrimT Void, 3, 2, make_list_ins, [])"

apply (insert class_test)
apply (unfold test_class_def)
apply (drule method_rec_lemma [OF _ wf_subcls1_E])
apply simp
done

lemma field_val [simp]:
"field (E, list_name) val_name = Some (list_name, PrimT Integer)"
apply (unfold TypeRel.field_def)
apply (insert class_list)
apply (unfold list_class_def)
apply (drule fields_rec_lemma [OF _ wf_subcls1_E])
apply simp
done

lemma field_next [simp]:
"field (E, list_name) next_name = Some (list_name, Class list_name)"
apply (unfold TypeRel.field_def)
apply (insert class_list)
apply (unfold list_class_def)
apply (drule fields_rec_lemma [OF _ wf_subcls1_E])
apply (simp add: name_defs distinct_fields [symmetric])
done

lemma [simp]: "fields (E, Object) = []"
by (simp add: fields_rec_lemma [OF class_Object wf_subcls1_E])

lemma [simp]: "fields (E, Xcpt NullPointer) = []"
by (simp add: fields_rec_lemma [OF class_NullPointer wf_subcls1_E])

lemma [simp]: "fields (E, Xcpt ClassCast) = []"
by (simp add: fields_rec_lemma [OF class_ClassCast wf_subcls1_E])

lemma [simp]: "fields (E, Xcpt OutOfMemory) = []"
by (simp add: fields_rec_lemma [OF class_OutOfMemory wf_subcls1_E])

lemma [simp]: "fields (E, test_name) = []"
apply (insert class_test)
apply (unfold test_class_def)
apply (drule fields_rec_lemma [OF _ wf_subcls1_E])
apply simp
done

lemmas [simp] = is_class_def

text {*
The next definition and three proof rules implement an algorithm to
enumarate natural numbers. The command @{text "apply (elim pc_end pc_next pc_0"}
transforms a goal of the form
@{prop [display] "pc < n ==> P pc"}
into a series of goals
@{prop [display] "P 0"}
@{prop [display] "P (Suc 0)"}

@{text "…"}

@{prop [display] "P n"}
*}

definition intervall :: "nat => nat => nat => bool" ("_ ∈ [_, _')") where
"x ∈ [a, b) ≡ a ≤ x ∧ x < b"

lemma pc_0: "x < n ==> (x ∈ [0, n) ==> P x) ==> P x"
by (simp add: intervall_def)

lemma pc_next: "x ∈ [n0, n) ==> P n0 ==> (x ∈ [Suc n0, n) ==> P x) ==> P x"
apply (cases "x=n0")
apply (auto simp add: intervall_def)
done

lemma pc_end: "x ∈ [n,n) ==> P x"
by (unfold intervall_def) arith


section "Program structure"

text {*
The program is structurally wellformed:
*}


lemma wf_struct:
"wf_prog (λG C mb. True) E" (is "wf_prog ?mb E")
proof -
have "unique E"
by (simp add: system_defs E_def class_defs name_defs distinct_classes)
moreover
have "set SystemClasses ⊆ set E" by (simp add: system_defs E_def)
hence "wf_syscls E" by (rule wf_syscls)
moreover
have "wf_cdecl ?mb E ObjectC" by (simp add: wf_cdecl_def ObjectC_def)
moreover
have "wf_cdecl ?mb E NullPointerC"
by (auto elim: notin_rtrancl
simp add: wf_cdecl_def name_defs NullPointerC_def subcls1)
moreover
have "wf_cdecl ?mb E ClassCastC"
by (auto elim: notin_rtrancl
simp add: wf_cdecl_def name_defs ClassCastC_def subcls1)
moreover
have "wf_cdecl ?mb E OutOfMemoryC"
by (auto elim: notin_rtrancl
simp add: wf_cdecl_def name_defs OutOfMemoryC_def subcls1)
moreover
have "wf_cdecl ?mb E (list_name, list_class)"
apply (auto elim!: notin_rtrancl
simp add: wf_cdecl_def wf_fdecl_def list_class_def
wf_mdecl_def wf_mhead_def subcls1)
apply (auto simp add: name_defs distinct_classes distinct_fields)
done
moreover
have "wf_cdecl ?mb E (test_name, test_class)"
apply (auto elim!: notin_rtrancl
simp add: wf_cdecl_def wf_fdecl_def test_class_def
wf_mdecl_def wf_mhead_def subcls1)
apply (auto simp add: name_defs distinct_classes distinct_fields)
done
ultimately
show ?thesis
by (simp add: wf_prog_def ws_prog_def wf_cdecl_mrT_cdecl_mdecl E_def SystemClasses_def)
qed

section "Welltypings"
text {*
We show welltypings of the methods @{term append_name} in class @{term list_name},
and @{term makelist_name} in class @{term test_name}:
*}

lemmas eff_simps [simp] = eff_def norm_eff_def xcpt_eff_def
declare appInvoke [simp del]

definition phi_append :: method_type (a") where
a ≡ map (λ(x,y). Some (x, map OK y)) [
( [], [Class list_name, Class list_name]),
( [Class list_name], [Class list_name, Class list_name]),
( [Class list_name], [Class list_name, Class list_name]),
( [Class list_name, Class list_name], [Class list_name, Class list_name]),
([NT, Class list_name, Class list_name], [Class list_name, Class list_name]),
( [Class list_name], [Class list_name, Class list_name]),
( [Class list_name, Class list_name], [Class list_name, Class list_name]),
( [PrimT Void], [Class list_name, Class list_name]),
( [Class Object], [Class list_name, Class list_name]),
( [], [Class list_name, Class list_name]),
( [Class list_name], [Class list_name, Class list_name]),
( [Class list_name, Class list_name], [Class list_name, Class list_name]),
( [], [Class list_name, Class list_name]),
( [PrimT Void], [Class list_name, Class list_name])]"



lemma bounded_append [simp]:
"check_bounded append_ins [(Suc 0, 2, 8, Xcpt NullPointer)]"
apply (simp add: check_bounded_def)
apply (simp add: eval_nat_numeral append_ins_def)
apply (rule allI, rule impI)
apply (elim pc_end pc_next pc_0)
apply auto
done

lemma types_append [simp]: "check_types E 3 (Suc (Suc 0)) (map OK φa)"
apply (auto simp add: check_types_def phi_append_def JVM_states_unfold)
apply (unfold list_def)
apply auto
done

lemma wt_append [simp]:
"wt_method E list_name [Class list_name] (PrimT Void) 3 0 append_ins
[(Suc 0, 2, 8, Xcpt NullPointer)] φa"

apply (simp add: wt_method_def wt_start_def wt_instr_def)
apply (simp add: phi_append_def append_ins_def)
apply clarify
apply (elim pc_end pc_next pc_0)
apply simp
apply (fastforce simp add: match_exception_entry_def sup_state_conv subcls1)
apply simp
apply simp
apply (fastforce simp add: sup_state_conv subcls1)
apply simp
apply (simp add: app_def xcpt_app_def)
apply simp
apply simp
apply simp
apply (simp add: match_exception_entry_def)
apply (simp add: match_exception_entry_def)
apply simp
apply simp
done

text {* Some abbreviations for readability *}
abbreviation Clist :: ty
where "Clist == Class list_name"
abbreviation Ctest :: ty
where "Ctest == Class test_name"

definition phi_makelist :: method_type (m") where
m ≡ map (λ(x,y). Some (x, y)) [
( [], [OK Ctest, Err , Err ]),
( [Clist], [OK Ctest, Err , Err ]),
( [Clist, Clist], [OK Ctest, Err , Err ]),
( [Clist], [OK Clist, Err , Err ]),
( [PrimT Integer, Clist], [OK Clist, Err , Err ]),
( [], [OK Clist, Err , Err ]),
( [Clist], [OK Clist, Err , Err ]),
( [Clist, Clist], [OK Clist, Err , Err ]),
( [Clist], [OK Clist, OK Clist, Err ]),
( [PrimT Integer, Clist], [OK Clist, OK Clist, Err ]),
( [], [OK Clist, OK Clist, Err ]),
( [Clist], [OK Clist, OK Clist, Err ]),
( [Clist, Clist], [OK Clist, OK Clist, Err ]),
( [Clist], [OK Clist, OK Clist, OK Clist]),
( [PrimT Integer, Clist], [OK Clist, OK Clist, OK Clist]),
( [], [OK Clist, OK Clist, OK Clist]),
( [Clist], [OK Clist, OK Clist, OK Clist]),
( [Clist, Clist], [OK Clist, OK Clist, OK Clist]),
( [PrimT Void], [OK Clist, OK Clist, OK Clist]),
( [], [OK Clist, OK Clist, OK Clist]),
( [Clist], [OK Clist, OK Clist, OK Clist]),
( [Clist, Clist], [OK Clist, OK Clist, OK Clist]),
( [PrimT Void], [OK Clist, OK Clist, OK Clist])]"


lemma bounded_makelist [simp]: "check_bounded make_list_ins []"
apply (simp add: check_bounded_def)
apply (simp add: eval_nat_numeral make_list_ins_def)
apply (rule allI, rule impI)
apply (elim pc_end pc_next pc_0)
apply auto
done

lemma types_makelist [simp]: "check_types E 3 (Suc (Suc (Suc 0))) (map OK φm)"
apply (auto simp add: check_types_def phi_makelist_def JVM_states_unfold)
apply (unfold list_def)
apply auto
done

lemma wt_makelist [simp]:
"wt_method E test_name [] (PrimT Void) 3 2 make_list_ins [] φm"
apply (simp add: wt_method_def)
apply (simp add: make_list_ins_def phi_makelist_def)
apply (simp add: wt_start_def eval_nat_numeral)
apply (simp add: wt_instr_def)
apply clarify
apply (elim pc_end pc_next pc_0)
apply (simp add: match_exception_entry_def)
apply simp
apply simp
apply simp
apply (simp add: match_exception_entry_def)
apply (simp add: match_exception_entry_def)
apply simp
apply simp
apply simp
apply (simp add: match_exception_entry_def)
apply (simp add: match_exception_entry_def)
apply simp
apply simp
apply simp
apply (simp add: match_exception_entry_def)
apply (simp add: match_exception_entry_def)
apply simp
apply (simp add: app_def xcpt_app_def)
apply simp
apply simp
apply simp
apply (simp add: app_def xcpt_app_def)
apply simp
done

text {* The whole program is welltyped: *}
definition Phi :: prog_type ("Φ") where
"Φ C sg ≡ if C = test_name ∧ sg = (makelist_name, []) then φm else
if C = list_name ∧ sg = (append_name, [Class list_name]) then φa else []"


lemma wf_prog:
"wt_jvm_prog E Φ"
apply (unfold wt_jvm_prog_def)
apply (rule wf_mb'E [OF wf_struct])
apply (simp add: E_def)
apply clarify
apply (fold E_def)
apply (simp add: system_defs class_defs Phi_def)
apply auto
done


section "Conformance"
text {* Execution of the program will be typesafe, because its
start state conforms to the welltyping: *}


lemma "E,Φ \<turnstile>JVM start_state E test_name makelist_name \<surd>"
apply (rule BV_correct_initial)
apply (rule wf_prog)
apply simp
apply simp
done


section "Example for code generation: inferring method types"

definition test_kil :: "jvm_prog => cname => ty list => ty => nat => nat =>
exception_table => instr list => JVMType.state list"
where
"test_kil G C pTs rT mxs mxl et instr =
(let first = Some ([],(OK (Class C))#((map OK pTs))@(replicate mxl Err));
start = OK first#(replicate (size instr - 1) (OK None))
in kiljvm G mxs (1+size pTs+mxl) rT et instr start)"


lemma [code]:
"unstables r step ss =
fold (λp A. if ¬stable r step ss p then insert p A else A) [0..<size ss] {}"

proof -
have "unstables r step ss = (UN p:{..<size ss}. if ¬stable r step ss p then {p} else {})"
apply (unfold unstables_def)
apply (rule equalityI)
apply (rule subsetI)
apply (erule CollectE)
apply (erule conjE)
apply (rule UN_I)
apply simp
apply simp
apply (rule subsetI)
apply (erule UN_E)
apply (case_tac "¬ stable r step ss p")
apply simp_all
done
also have "!!f. (UN p:{..<size ss}. f p) = Union (set (map f [0..<size ss]))" by auto
also note Sup_set_fold also note fold_map
also have "op ∪ o (λp. if ¬ stable r step ss p then {p} else {}) =
(λp A. if ¬stable r step ss p then insert p A else A)"

by(auto simp add: fun_eq_iff)
finally show ?thesis .
qed

definition some_elem :: "'a set => 'a" where [code del]:
"some_elem = (%S. SOME x. x : S)"
code_printing
constant some_elem \<rightharpoonup> (SML) "(case/ _ of/ Set/ xs/ =>/ hd/ xs)"

text {* This code setup is just a demonstration and \emph{not} sound! *}

lemma False
proof -
have "some_elem (set [False, True]) = False"
by eval
moreover have "some_elem (set [True, False]) = True"
by eval
ultimately show False
by (simp add: some_elem_def)
qed

lemma [code]:
"iter f step ss w = while (λ(ss, w). ¬ Set.is_empty w)
(λ(ss, w).
let p = some_elem w in propa f (step p (ss ! p)) ss (w - {p}))
(ss, w)"

unfolding iter_def Set.is_empty_def some_elem_def ..

lemma JVM_sup_unfold [code]:
"JVMType.sup S m n = lift2 (Opt.sup
(Product.sup (Listn.sup (JType.sup S))
(λx y. OK (map2 (lift2 (JType.sup S)) x y))))"

apply (unfold JVMType.sup_def JVMType.sl_def Opt.esl_def Err.sl_def
stk_esl_def reg_sl_def Product.esl_def
Listn.sl_def upto_esl_def JType.esl_def Err.esl_def)
by simp

lemmas [code] = JType.sup_def [unfolded exec_lub_def] JVM_le_unfold
lemmas [code] = lesub_def plussub_def

lemmas [code] =
JType.sup_def [unfolded exec_lub_def]
wf_class_code
widen.equation
match_exception_entry_def

definition test1 where
"test1 = test_kil E list_name [Class list_name] (PrimT Void) 3 0
[(Suc 0, 2, 8, Xcpt NullPointer)] append_ins"

definition test2 where
"test2 = test_kil E test_name [] (PrimT Void) 3 2 [] make_list_ins"

ML_val {*
@{code test1};
@{code test2};
*}


end