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;;c⇩_{2},s) -> (c⇩_{2},s)" |

Seq2: "(c⇩_{1},s) -> (c⇩_{1}',s') ==> (c⇩_{1};;c⇩_{2},s) -> (c⇩_{1}';;c⇩_{2},s')" |

IfTrue: "bval b s ==> (IF b THEN c⇩_{1}ELSE c⇩_{2},s) -> (c⇩_{1},s)" |

IfFalse: "¬bval b s ==> (IF b THEN c⇩_{1}ELSE c⇩_{2},s) -> (c⇩_{2},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