Theory Small_Step

theory Small_Step
imports Star Big_Step
header "Small-Step Semantics of Commands"

theory Small_Step imports Star Big_Step begin

subsection "The transition relation"

text_raw{*\snip{SmallStepDef}{0}{2}{% *}
inductive
small_step :: "com * state => com * state => bool" (infix "->" 55)
where
Assign: "(x ::= a, s) -> (SKIP, s(x := aval a s))" |

Seq1: "(SKIP;;c2,s) -> (c2,s)" |
Seq2: "(c1,s) -> (c1',s') ==> (c1;;c2,s) -> (c1';;c2,s')" |

IfTrue: "bval b s ==> (IF b THEN c1 ELSE c2,s) -> (c1,s)" |
IfFalse: "¬bval b s ==> (IF b THEN c1 ELSE c2,s) -> (c2,s)" |

While: "(WHILE b DO c,s) ->
(IF b THEN c;; WHILE b DO c ELSE SKIP,s)"

text_raw{*}%endsnip*}


abbreviation
small_steps :: "com * state => com * state => bool" (infix "->*" 55)
where "x ->* y == star small_step x y"

subsection{* Executability *}

code_pred small_step .

values "{(c',map t [''x'',''y'',''z'']) |c' t.
(''x'' ::= V ''z'';; ''y'' ::= V ''x'',
<''x'' := 3, ''y'' := 7, ''z'' := 5>) ->* (c',t)}"



subsection{* Proof infrastructure *}

subsubsection{* Induction rules *}

text{* The default induction rule @{thm[source] small_step.induct} only works
for lemmas of the form @{text"a -> b ==> …"} where @{text a} and @{text b} are
not already pairs @{text"(DUMMY,DUMMY)"}. We can generate a suitable variant
of @{thm[source] small_step.induct} for pairs by ``splitting'' the arguments
@{text"->"} into pairs: *}

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


subsubsection{* Proof automation *}

declare small_step.intros[simp,intro]

text{* Rule inversion: *}

inductive_cases SkipE[elim!]: "(SKIP,s) -> ct"
thm SkipE
inductive_cases AssignE[elim!]: "(x::=a,s) -> ct"
thm AssignE
inductive_cases SeqE[elim]: "(c1;;c2,s) -> ct"
thm SeqE
inductive_cases IfE[elim!]: "(IF b THEN c1 ELSE c2,s) -> ct"
inductive_cases WhileE[elim]: "(WHILE b DO c, s) -> ct"


text{* A simple property: *}
lemma deterministic:
"cs -> cs' ==> cs -> cs'' ==> cs'' = cs'"
apply(induction arbitrary: cs'' rule: small_step.induct)
apply blast+
done


subsection "Equivalence with big-step semantics"

lemma star_seq2: "(c1,s) ->* (c1',s') ==> (c1;;c2,s) ->* (c1';;c2,s')"
proof(induction rule: star_induct)
case refl thus ?case by simp
next
case step
thus ?case by (metis Seq2 star.step)
qed

lemma seq_comp:
"[| (c1,s1) ->* (SKIP,s2); (c2,s2) ->* (SKIP,s3) |]
==> (c1;;c2, s1) ->* (SKIP,s3)"

by(blast intro: star.step star_seq2 star_trans)

text{* The following proof corresponds to one on the board where one would
show chains of @{text "->"} and @{text "->*"} steps. *}


lemma big_to_small:
"cs => t ==> cs ->* (SKIP,t)"
proof (induction rule: big_step.induct)
fix s show "(SKIP,s) ->* (SKIP,s)" by simp
next
fix x a s show "(x ::= a,s) ->* (SKIP, s(x := aval a s))" by auto
next
fix c1 c2 s1 s2 s3
assume "(c1,s1) ->* (SKIP,s2)" and "(c2,s2) ->* (SKIP,s3)"
thus "(c1;;c2, s1) ->* (SKIP,s3)" by (rule seq_comp)
next
fix s::state and b c0 c1 t
assume "bval b s"
hence "(IF b THEN c0 ELSE c1,s) -> (c0,s)" by simp
moreover assume "(c0,s) ->* (SKIP,t)"
ultimately
show "(IF b THEN c0 ELSE c1,s) ->* (SKIP,t)" by (metis star.simps)
next
fix s::state and b c0 c1 t
assume "¬bval b s"
hence "(IF b THEN c0 ELSE c1,s) -> (c1,s)" by simp
moreover assume "(c1,s) ->* (SKIP,t)"
ultimately
show "(IF b THEN c0 ELSE c1,s) ->* (SKIP,t)" by (metis star.simps)
next
fix b c and s::state
assume b: "¬bval b s"
let ?if = "IF b THEN c;; WHILE b DO c ELSE SKIP"
have "(WHILE b DO c,s) -> (?if, s)" by blast
moreover have "(?if,s) -> (SKIP, s)" by (simp add: b)
ultimately show "(WHILE b DO c,s) ->* (SKIP,s)" by(metis star.refl star.step)
next
fix b c s s' t
let ?w = "WHILE b DO c"
let ?if = "IF b THEN c;; ?w ELSE SKIP"
assume w: "(?w,s') ->* (SKIP,t)"
assume c: "(c,s) ->* (SKIP,s')"
assume b: "bval b s"
have "(?w,s) -> (?if, s)" by blast
moreover have "(?if, s) -> (c;; ?w, s)" by (simp add: b)
moreover have "(c;; ?w,s) ->* (SKIP,t)" by(rule seq_comp[OF c w])
ultimately show "(WHILE b DO c,s) ->* (SKIP,t)" by (metis star.simps)
qed

text{* Each case of the induction can be proved automatically: *}
lemma "cs => t ==> cs ->* (SKIP,t)"
proof (induction rule: big_step.induct)
case Skip show ?case by blast
next
case Assign show ?case by blast
next
case Seq thus ?case by (blast intro: seq_comp)
next
case IfTrue thus ?case by (blast intro: star.step)
next
case IfFalse thus ?case by (blast intro: star.step)
next
case WhileFalse thus ?case
by (metis star.step star_step1 small_step.IfFalse small_step.While)
next
case WhileTrue
thus ?case
by(metis While seq_comp small_step.IfTrue star.step[of small_step])
qed

lemma small1_big_continue:
"cs -> cs' ==> cs' => t ==> cs => t"
apply (induction arbitrary: t rule: small_step.induct)
apply auto
done

lemma small_big_continue:
"cs ->* cs' ==> cs' => t ==> cs => t"
apply (induction rule: star.induct)
apply (auto intro: small1_big_continue)
done

lemma small_to_big: "cs ->* (SKIP,t) ==> cs => t"
by (metis small_big_continue Skip)

text {*
Finally, the equivalence theorem:
*}

theorem big_iff_small:
"cs => t = cs ->* (SKIP,t)"
by(metis big_to_small small_to_big)


subsection "Final configurations and infinite reductions"

definition "final cs <-> ¬(EX cs'. cs -> cs')"

lemma finalD: "final (c,s) ==> c = SKIP"
apply(simp add: final_def)
apply(induction c)
apply blast+
done

lemma final_iff_SKIP: "final (c,s) = (c = SKIP)"
by (metis SkipE finalD final_def)

text{* Now we can show that @{text"=>"} yields a final state iff @{text"->"}
terminates: *}


lemma big_iff_small_termination:
"(EX t. cs => t) <-> (EX cs'. cs ->* cs' ∧ final cs')"
by(simp add: big_iff_small final_iff_SKIP)

text{* This is the same as saying that the absence of a big step result is
equivalent with absence of a terminating small step sequence, i.e.\ with
nontermination. Since @{text"->"} is determininistic, there is no difference
between may and must terminate. *}


end