Theory Typing_Framework_err

theory Typing_Framework_err
imports SemilatAlg
```(*  Title:      HOL/MicroJava/DFA/Typing_Framework_err.thy
Author:     Gerwin Klein
*)

section ‹Lifting the Typing Framework to err, app, and eff›

theory Typing_Framework_err
imports Typing_Framework SemilatAlg
begin

definition wt_err_step :: "'s ord ⇒ 's err step_type ⇒ 's err list ⇒ bool" where
"wt_err_step r step ts ≡ wt_step (Err.le r) Err step ts"

definition wt_app_eff :: "'s ord ⇒ (nat ⇒ 's ⇒ bool) ⇒ 's step_type ⇒ 's list ⇒ bool" where
"wt_app_eff r app step ts ≡
∀p < size ts. app p (ts!p) ∧ (∀(q,t) ∈ set (step p (ts!p)). t <=_r ts!q)"

definition map_snd :: "('b ⇒ 'c) ⇒ ('a × 'b) list ⇒ ('a × 'c) list" where
"map_snd f ≡ map (λ(x,y). (x, f y))"

definition error :: "nat ⇒ (nat × 'a err) list" where
"error n ≡ map (λx. (x,Err)) [0..<n]"

definition err_step :: "nat ⇒ (nat ⇒ 's ⇒ bool) ⇒ 's step_type ⇒ 's err step_type" where
"err_step n app step p t ≡
case t of
Err   ⇒ error n
| OK t' ⇒ if app p t' then map_snd OK (step p t') else error n"

definition app_mono :: "'s ord ⇒ (nat ⇒ 's ⇒ bool) ⇒ nat ⇒ 's set ⇒ bool" where
"app_mono r app n A ≡
∀s p t. s ∈ A ∧ p < n ∧ s <=_r t ⟶ app p t ⟶ app p s"

lemmas err_step_defs = err_step_def map_snd_def error_def

lemma bounded_err_stepD:
"bounded (err_step n app step) n ⟹
p < n ⟹  app p a ⟹ (q,b) ∈ set (step p a) ⟹
q < n"
apply (erule allE, erule impE, assumption)
apply (erule_tac x = "OK a" in allE, drule bspec)
apply fast
apply simp
done

lemma in_map_sndD: "(a,b) ∈ set (map_snd f xs) ⟹ ∃b'. (a,b') ∈ set xs"
apply (induct xs)
done

lemma bounded_err_stepI:
"∀p. p < n ⟶ (∀s. ap p s ⟶ (∀(q,s') ∈ set (step p s). q < n))
⟹ bounded (err_step n ap step) n"
apply (clarsimp simp: bounded_def err_step_def split: err.splits)
apply (blast dest: in_map_sndD)
done

lemma bounded_lift:
"bounded step n ⟹ bounded (err_step n app step) n"
apply (unfold bounded_def err_step_def error_def)
apply clarify
apply (erule allE, erule impE, assumption)
apply (case_tac s)
apply (auto simp add: map_snd_def split: if_split_asm)
done

lemma le_list_map_OK [simp]:
"⋀b. map OK a <=[Err.le r] map OK b = (a <=[r] b)"
apply (induct a)
apply simp
apply simp
apply (case_tac b)
apply simp
apply simp
done

lemma map_snd_lessI:
"x ≤|r| y ⟹ map_snd OK x ≤|Err.le r| map_snd OK y"
apply (induct x)
apply (unfold lesubstep_type_def map_snd_def)
apply auto
done

lemma mono_lift:
"order r ⟹ app_mono r app n A ⟹ bounded (err_step n app step) n ⟹
∀s p t. s ∈ A ∧ p < n ∧ s <=_r t ⟶ app p t ⟶ step p s ≤|r| step p t ⟹
mono (Err.le r) (err_step n app step) n (err A)"
apply (simp only: app_mono_def mono_def err_step_def)
apply clarify
apply (case_tac s)
apply simp
apply simp
apply (case_tac t)
apply simp
apply clarify
apply clarify
apply (drule in_map_sndD)
apply clarify
apply (drule bounded_err_stepD, assumption+)
apply (rule exI [of _ Err])
apply simp
apply simp
apply (erule allE, erule allE, erule allE, erule impE)
apply (rule conjI, assumption)
apply (rule conjI, assumption)
apply assumption
apply (rule conjI)
apply clarify
apply (erule allE, erule allE, erule allE, erule impE)
apply (rule conjI, assumption)
apply (rule conjI, assumption)
apply assumption
apply (erule impE, assumption)
apply (rule map_snd_lessI, assumption)
apply clarify
apply clarify
apply (drule in_map_sndD)
apply clarify
apply (drule bounded_err_stepD, assumption+)
apply (rule exI [of _ Err])
apply simp
done

lemma in_errorD:
"(x,y) ∈ set (error n) ⟹ y = Err"

lemma pres_type_lift:
"∀s∈A. ∀p. p < n ⟶ app p s ⟶ (∀(q, s')∈set (step p s). s' ∈ A)
⟹ pres_type (err_step n app step) n (err A)"
apply (unfold pres_type_def err_step_def)
apply clarify
apply (case_tac b)
apply simp
apply (case_tac s)
apply simp
apply (drule in_errorD)
apply simp
apply (simp add: map_snd_def split: if_split_asm)
apply fast
apply (drule in_errorD)
apply simp
done

text ‹
There used to be a condition here that each instruction must have a
successor. This is not needed any more, because the definition of
@{term error} trivially ensures that there is a successor for
the critical case where @{term app} does not hold.
›
lemma wt_err_imp_wt_app_eff:
assumes wt: "wt_err_step r (err_step (size ts) app step) ts"
assumes b:  "bounded (err_step (size ts) app step) (size ts)"
shows "wt_app_eff r app step (map ok_val ts)"
proof (unfold wt_app_eff_def, intro strip, rule conjI)
fix p assume "p < size (map ok_val ts)"
hence lp: "p < size ts" by simp
hence ts: "0 < size ts" by (cases p) auto
hence err: "(0,Err) ∈ set (error (size ts))" by (simp add: error_def)

from wt lp
have [intro?]: "⋀p. p < size ts ⟹ ts ! p ≠ Err"
by (unfold wt_err_step_def wt_step_def) simp

show app: "app p (map ok_val ts ! p)"
proof (rule ccontr)
from wt lp obtain s where
OKp:  "ts ! p = OK s" and
less: "∀(q,t) ∈ set (err_step (size ts) app step p (ts!p)). t <=_(Err.le r) ts!q"
by (unfold wt_err_step_def wt_step_def stable_def)
(auto iff: not_Err_eq)
assume "¬ app p (map ok_val ts ! p)"
with OKp lp have "¬ app p s" by simp
with OKp have "err_step (size ts) app step p (ts!p) = error (size ts)"
with err ts obtain q where
"(q,Err) ∈ set (err_step (size ts) app step p (ts!p))" and
q: "q < size ts" by auto
with less have "ts!q = Err" by auto
moreover from q have "ts!q ≠ Err" ..
ultimately show False by blast
qed

show "∀(q,t)∈set(step p (map ok_val ts ! p)). t <=_r map ok_val ts ! q"
proof clarify
fix q t assume q: "(q,t) ∈ set (step p (map ok_val ts ! p))"

from wt lp q
obtain s where
OKp:  "ts ! p = OK s" and
less: "∀(q,t) ∈ set (err_step (size ts) app step p (ts!p)). t <=_(Err.le r) ts!q"
by (unfold wt_err_step_def wt_step_def stable_def)
(auto iff: not_Err_eq)

from b lp app q have lq: "q < size ts" by (rule bounded_err_stepD)
hence "ts!q ≠ Err" ..
then obtain s' where OKq: "ts ! q = OK s'" by (auto iff: not_Err_eq)

from lp lq OKp OKq app less q
show "t <=_r map ok_val ts ! q"
by (auto simp add: err_step_def map_snd_def)
qed
qed

lemma wt_app_eff_imp_wt_err:
assumes app_eff: "wt_app_eff r app step ts"
assumes bounded: "bounded (err_step (size ts) app step) (size ts)"
shows "wt_err_step r (err_step (size ts) app step) (map OK ts)"
proof (unfold wt_err_step_def wt_step_def, intro strip, rule conjI)
fix p assume "p < size (map OK ts)"
hence p: "p < size ts" by simp
thus "map OK ts ! p ≠ Err" by simp
{ fix q t
assume q: "(q,t) ∈ set (err_step (size ts) app step p (map OK ts ! p))"
with p app_eff obtain
"app p (ts ! p)" "∀(q,t) ∈ set (step p (ts!p)). t <=_r ts!q"
by (unfold wt_app_eff_def) blast
moreover
from q p bounded have "q < size ts"
by - (rule boundedD)
hence "map OK ts ! q = OK (ts!q)" by simp
moreover
have "p < size ts" by (rule p)
moreover note q
ultimately
have "t <=_(Err.le r) map OK ts ! q"
by (auto simp add: err_step_def map_snd_def)
}
thus "stable (Err.le r) (err_step (size ts) app step) (map OK ts) p"
by (unfold stable_def) blast
qed

end

```