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Homework 6_1

This is the task corresponding to homework 6_1.

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Definitions File 

theory Defs
  imports "HOL-IMP.Star"
begin

declare [[syntax_ambiguity_warning=false]]

datatype val = Iv int | Pv val val

type_synonym vname = string
type_synonym state = "vname \<Rightarrow> val"

datatype aexp =  N int | V vname | Plus aexp aexp | Pair aexp aexp | Fst aexp | Snd aexp

datatype bexp = Bc bool | Not bexp | And bexp bexp | Less aexp aexp

datatype
  com = SKIP 
      | Assign vname aexp       ("_ ::= _" [1000, 61] 61)
      | Seq    com  com         ("_;; _"  [60, 61] 60)
      | If     bexp com com     ("IF _ THEN _ ELSE _"  [0, 0, 61] 61)
      | While  bexp com         ("WHILE _ DO _"  [0, 61] 61)
      | SWAP   vname

datatype ty = Ity | Pty ty ty

type_synonym tyenv = "vname \<Rightarrow> ty"

fun type :: "val \<Rightarrow> ty" where
"type (Iv i) = Ity" |
"type (Pv v1 v2) = Pty (type v1) (type v2)"

lemma type_eq_Ity[simp]: "type v = Ity \<longleftrightarrow> (\<exists>i. v = Iv i)"
  by (cases v) simp_all

definition styping :: "tyenv \<Rightarrow> state \<Rightarrow> bool" (infix "\<turnstile>" 50)
where "\<Gamma> \<turnstile> s  \<longleftrightarrow>  (\<forall>x. type (s x) = \<Gamma> x)"


consts small_step :: "(com \<times> state) \<Rightarrow> (com \<times> state) \<Rightarrow> bool"

consts taval :: "aexp \<Rightarrow> state \<Rightarrow> val \<Rightarrow> bool"

consts tbval :: "bexp \<Rightarrow> state \<Rightarrow> bool \<Rightarrow> bool"

consts atyping :: "tyenv \<Rightarrow> aexp \<Rightarrow> ty \<Rightarrow> bool"

consts btyping :: "tyenv \<Rightarrow> bexp \<Rightarrow> bool"

consts ctyping :: "tyenv \<Rightarrow> com \<Rightarrow> bool"

consts small_steps :: "com * state \<Rightarrow> com * state \<Rightarrow> bool"


end

Template File 

theory Submission
  imports Defs
begin

inductive taval :: "aexp \<Rightarrow> state \<Rightarrow> val \<Rightarrow> bool" where
"taval (N i) s (Iv i)" |
"taval (V x) s (s x)" |
"taval a1 s (Iv i1) \<Longrightarrow> taval a2 s (Iv i2)
 \<Longrightarrow> taval (Plus a1 a2) s (Iv (i1 + i2))"

inductive_cases [elim!]:
  "taval (N i) s v"
  "taval (V x) s v"
  "taval (Plus a1 a2) s v"


inductive tbval :: "bexp \<Rightarrow> state \<Rightarrow> bool \<Rightarrow> bool" where
"tbval (Bc v) s v" |
"tbval b s bv \<Longrightarrow> tbval (Not b) s (\<not> bv)" |
"tbval b1 s bv1 \<Longrightarrow> tbval b2 s bv2 \<Longrightarrow> tbval (And b1 b2) s (bv1 \<and> bv2)" |
"taval a1 s (Iv i1) \<Longrightarrow> taval a2 s (Iv i2) \<Longrightarrow> tbval (Less a1 a2) s (i1 < i2)"


inductive
  small_step :: "(com \<times> state) \<Rightarrow> (com \<times> state) \<Rightarrow> bool" (infix "\<rightarrow>" 55) 
where
Assign:  "taval a s v \<Longrightarrow> (x ::= a, s) \<rightarrow> (SKIP, s(x := v))" |
Seq1:   "(SKIP;;c,s) \<rightarrow> (c,s)" |
Seq2:   "(c1,s) \<rightarrow> (c1',s') \<Longrightarrow> (c1;;c2,s) \<rightarrow> (c1';;c2,s')" |
IfTrue:  "tbval b s True \<Longrightarrow> (IF b THEN c1 ELSE c2,s) \<rightarrow> (c1,s)" |
IfFalse: "tbval b s False \<Longrightarrow> (IF b THEN c1 ELSE c2,s) \<rightarrow> (c2,s)" |
While:   "(WHILE b DO c,s) \<rightarrow> (IF b THEN c;; WHILE b DO c ELSE SKIP,s)"

lemmas small_step_induct = small_step.induct[split_format(complete)]

inductive atyping :: "tyenv \<Rightarrow> aexp \<Rightarrow> ty \<Rightarrow> bool"
  ("(1_/ \<turnstile>/ (_ :/ _))" [50,0,50] 50) where
Ic_ty: "\<Gamma> \<turnstile> N i : Ity" |
V_ty: "\<Gamma> \<turnstile> V x : \<Gamma> x"


declare atyping.intros [intro!]
inductive_cases [elim!]:
  "\<Gamma> \<turnstile> V x : \<tau>" "\<Gamma> \<turnstile> N i : \<tau>" "\<Gamma> \<turnstile> Plus a1 a2 : \<tau>"

inductive btyping :: "tyenv \<Rightarrow> bexp \<Rightarrow> bool" (infix "\<turnstile>" 50) where
B_ty: "\<Gamma> \<turnstile> Bc v" |
Not_ty: "\<Gamma> \<turnstile> b \<Longrightarrow> \<Gamma> \<turnstile> Not b" |
And_ty: "\<Gamma> \<turnstile> b1 \<Longrightarrow> \<Gamma> \<turnstile> b2 \<Longrightarrow> \<Gamma> \<turnstile> And b1 b2" |
Less_ty: "\<Gamma> \<turnstile> a1 : Ity \<Longrightarrow> \<Gamma> \<turnstile> a2 : Ity \<Longrightarrow> \<Gamma> \<turnstile> Less a1 a2"


declare btyping.intros [intro!]
inductive_cases [elim!]: "\<Gamma> \<turnstile> Not b" "\<Gamma> \<turnstile> And b1 b2" "\<Gamma> \<turnstile> Less a1 a2"

inductive ctyping :: "tyenv \<Rightarrow> com \<Rightarrow> bool" (infix "\<turnstile>" 50) where
Skip_ty: "\<Gamma> \<turnstile> SKIP" |
Assign_ty: "\<Gamma> \<turnstile> a : \<Gamma>(x) \<Longrightarrow> \<Gamma> \<turnstile> x ::= a" |
Seq_ty: "\<Gamma> \<turnstile> c1 \<Longrightarrow> \<Gamma> \<turnstile> c2 \<Longrightarrow> \<Gamma> \<turnstile> c1;;c2" |
If_ty: "\<Gamma> \<turnstile> b \<Longrightarrow> \<Gamma> \<turnstile> c1 \<Longrightarrow> \<Gamma> \<turnstile> c2 \<Longrightarrow> \<Gamma> \<turnstile> IF b THEN c1 ELSE c2" |
While_ty: "\<Gamma> \<turnstile> b \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> WHILE b DO c"


declare ctyping.intros [intro!]
inductive_cases [elim!]:
  "\<Gamma> \<turnstile> x ::= a" "\<Gamma> \<turnstile> c1;;c2"
  "\<Gamma> \<turnstile> IF b THEN c1 ELSE c2"
  "\<Gamma> \<turnstile> WHILE b DO c"

theorem apreservation:
  "\<Gamma> \<turnstile> a : \<tau> \<Longrightarrow> taval a s v \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> type v = \<tau>"
  sorry

theorem aprogress: "\<Gamma> \<turnstile> a : \<tau> \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> \<exists>v. taval a s v"
  sorry

theorem bprogress: "\<Gamma> \<turnstile> b \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> \<exists>v. tbval b s v"
  sorry

theorem progress:
  "\<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> c \<noteq> SKIP \<Longrightarrow> \<exists>cs'. (c,s) \<rightarrow> cs'"
  sorry

theorem styping_preservation:
  "(c,s) \<rightarrow> (c',s') \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> \<Gamma> \<turnstile> s'"
  sorry

theorem ctyping_preservation:
  "(c,s) \<rightarrow> (c',s') \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> c'"
  sorry

abbreviation small_steps :: "com * state \<Rightarrow> com * state \<Rightarrow> bool" (infix "\<rightarrow>*" 55)
where "x \<rightarrow>* y == star small_step x y"

theorem type_sound:
  "(c,s) \<rightarrow>* (c',s') \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> c' \<noteq> SKIP
   \<Longrightarrow> \<exists>cs''. (c',s') \<rightarrow> cs''"
  sorry

end

Check File 

theory Check
  imports Submission
begin

theorem apreservation: "\<Gamma> \<turnstile> a : \<tau> \<Longrightarrow> taval a s v \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> type v = \<tau>"
  by (rule Submission.apreservation)

theorem aprogress: "\<Gamma> \<turnstile> a : \<tau> \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> \<exists>v. taval a s v"
  by (rule Submission.aprogress)

theorem bprogress: "\<Gamma> \<turnstile> b \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> \<exists>v. tbval b s v"
  by (rule Submission.bprogress)

theorem progress: "\<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> c \<noteq> SKIP \<Longrightarrow> \<exists>cs'. (c,s) \<rightarrow> cs'"
  by (rule Submission.progress)

theorem styping_preservation: "(c,s) \<rightarrow> (c',s') \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> \<Gamma> \<turnstile> s'"
  by (rule Submission.styping_preservation)

theorem ctyping_preservation: "(c,s) \<rightarrow> (c',s') \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> c'"
  by (rule Submission.ctyping_preservation)

theorem type_sound: "(c,s) \<rightarrow>* (c',s') \<Longrightarrow> \<Gamma> \<turnstile> c \<Longrightarrow> \<Gamma> \<turnstile> s \<Longrightarrow> c' \<noteq> SKIP
   \<Longrightarrow> \<exists>cs''. (c',s') \<rightarrow> cs''"
  by (rule Submission.type_sound)

end
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