# Theory Small_Step

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

theory Small_Step imports Star Big_Step begin

subsection "The transition relation"

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)"

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_to_big:
"cs →* (SKIP,t) ⟹ cs ⇒ t"
apply (induction cs "(SKIP,t)" rule: star.induct)
apply (auto intro: small1_big_continue)
done

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(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')"