theory Collecting_ITP

imports Complete_Lattice_ix "ACom_ITP"

begin

subsection "Collecting Semantics of Commands"

subsubsection "Annotated commands as a complete lattice"

(* Orderings could also be lifted generically (thus subsuming the

instantiation for preord and order), but then less_eq_acom would need to

become a definition, eg less_eq_acom = lift2 less_eq, and then proofs would

need to unfold this defn first. *)

instantiation acom :: (order) order

begin

fun less_eq_acom :: "('a::order)acom => 'a acom => bool" where

"(SKIP {S}) ≤ (SKIP {S'}) = (S ≤ S')" |

"(x ::= e {S}) ≤ (x' ::= e' {S'}) = (x=x' ∧ e=e' ∧ S ≤ S')" |

"(c1;;c2) ≤ (c1';;c2') = (c1 ≤ c1' ∧ c2 ≤ c2')" |

"(IF b THEN c1 ELSE c2 {S}) ≤ (IF b' THEN c1' ELSE c2' {S'}) =

(b=b' ∧ c1 ≤ c1' ∧ c2 ≤ c2' ∧ S ≤ S')" |

"({Inv} WHILE b DO c {P}) ≤ ({Inv'} WHILE b' DO c' {P'}) =

(b=b' ∧ c ≤ c' ∧ Inv ≤ Inv' ∧ P ≤ P')" |

"less_eq_acom _ _ = False"

lemma SKIP_le: "SKIP {S} ≤ c <-> (∃S'. c = SKIP {S'} ∧ S ≤ S')"

by (cases c) auto

lemma Assign_le: "x ::= e {S} ≤ c <-> (∃S'. c = x ::= e {S'} ∧ S ≤ S')"

by (cases c) auto

lemma Seq_le: "c1;;c2 ≤ c <-> (∃c1' c2'. c = c1';;c2' ∧ c1 ≤ c1' ∧ c2 ≤ c2')"

by (cases c) auto

lemma If_le: "IF b THEN c1 ELSE c2 {S} ≤ c <->

(∃c1' c2' S'. c= IF b THEN c1' ELSE c2' {S'} ∧ c1 ≤ c1' ∧ c2 ≤ c2' ∧ S ≤ S')"

by (cases c) auto

lemma While_le: "{Inv} WHILE b DO c {P} ≤ w <->

(∃Inv' c' P'. w = {Inv'} WHILE b DO c' {P'} ∧ c ≤ c' ∧ Inv ≤ Inv' ∧ P ≤ P')"

by (cases w) auto

definition less_acom :: "'a acom => 'a acom => bool" where

"less_acom x y = (x ≤ y ∧ ¬ y ≤ x)"

instance

proof

case goal1 show ?case by(simp add: less_acom_def)

next

case goal2 thus ?case by (induct x) auto

next

case goal3 thus ?case

apply(induct x y arbitrary: z rule: less_eq_acom.induct)

apply (auto intro: le_trans simp: SKIP_le Assign_le Seq_le If_le While_le)

done

next

case goal4 thus ?case

apply(induct x y rule: less_eq_acom.induct)

apply (auto intro: le_antisym)

done

qed

end

fun sub⇩_{1}:: "'a acom => 'a acom" where

"sub⇩_{1}(c1;;c2) = c1" |

"sub⇩_{1}(IF b THEN c1 ELSE c2 {S}) = c1" |

"sub⇩_{1}({I} WHILE b DO c {P}) = c"

fun sub⇩_{2}:: "'a acom => 'a acom" where

"sub⇩_{2}(c1;;c2) = c2" |

"sub⇩_{2}(IF b THEN c1 ELSE c2 {S}) = c2"

fun invar :: "'a acom => 'a" where

"invar({I} WHILE b DO c {P}) = I"

fun lift :: "('a set => 'b) => com => 'a acom set => 'b acom"

where

"lift F com.SKIP M = (SKIP {F (post ` M)})" |

"lift F (x ::= a) M = (x ::= a {F (post ` M)})" |

"lift F (c1;;c2) M =

lift F c1 (sub⇩_{1}` M);; lift F c2 (sub⇩_{2}` M)" |

"lift F (IF b THEN c1 ELSE c2) M =

IF b THEN lift F c1 (sub⇩_{1}` M) ELSE lift F c2 (sub⇩_{2}` M)

{F (post ` M)}" |

"lift F (WHILE b DO c) M =

{F (invar ` M)}

WHILE b DO lift F c (sub⇩_{1}` M)

{F (post ` M)}"

interpretation Complete_Lattice_ix "%c. {c'. strip c' = c}" "lift Inter"

proof

case goal1

have "a:A ==> lift Inter (strip a) A ≤ a"

proof(induction a arbitrary: A)

case Seq from Seq.prems show ?case by(force intro!: Seq.IH)

next

case If from If.prems show ?case by(force intro!: If.IH)

next

case While from While.prems show ?case by(force intro!: While.IH)

qed force+

with goal1 show ?case by auto

next

case goal2

thus ?case

proof(induction b arbitrary: i A)

case SKIP thus ?case by (force simp:SKIP_le)

next

case Assign thus ?case by (force simp:Assign_le)

next

case Seq from Seq.prems show ?case

by (force intro!: Seq.IH simp:Seq_le)

next

case If from If.prems show ?case by (force simp: If_le intro!: If.IH)

next

case While from While.prems show ?case

by(fastforce simp: While_le intro: While.IH)

qed

next

case goal3

have "strip(lift Inter i A) = i"

proof(induction i arbitrary: A)

case Seq from Seq.prems show ?case

by (fastforce simp: strip_eq_Seq subset_iff intro!: Seq.IH)

next

case If from If.prems show ?case

by (fastforce intro!: If.IH simp: strip_eq_If)

next

case While from While.prems show ?case

by(fastforce intro: While.IH simp: strip_eq_While)

qed auto

thus ?case by auto

qed

lemma le_post: "c ≤ d ==> post c ≤ post d"

by(induction c d rule: less_eq_acom.induct) auto

subsubsection "Collecting semantics"

fun step :: "state set => state set acom => state set acom" where

"step S (SKIP {P}) = (SKIP {S})" |

"step S (x ::= e {P}) =

(x ::= e {{s'. EX s:S. s' = s(x := aval e s)}})" |

"step S (c1;; c2) = step S c1;; step (post c1) c2" |

"step S (IF b THEN c1 ELSE c2 {P}) =

IF b THEN step {s:S. bval b s} c1 ELSE step {s:S. ¬ bval b s} c2

{post c1 ∪ post c2}" |

"step S ({Inv} WHILE b DO c {P}) =

{S ∪ post c} WHILE b DO (step {s:Inv. bval b s} c) {{s:Inv. ¬ bval b s}}"

definition CS :: "com => state set acom" where

"CS c = lfp (step UNIV) c"

lemma mono2_step: "c1 ≤ c2 ==> S1 ⊆ S2 ==> step S1 c1 ≤ step S2 c2"

proof(induction c1 c2 arbitrary: S1 S2 rule: less_eq_acom.induct)

case 2 thus ?case by fastforce

next

case 3 thus ?case by(simp add: le_post)

next

case 4 thus ?case by(simp add: subset_iff)(metis le_post set_mp)+

next

case 5 thus ?case by(simp add: subset_iff) (metis le_post set_mp)

qed auto

lemma mono_step: "mono (step S)"

by(blast intro: monoI mono2_step)

lemma strip_step: "strip(step S c) = strip c"

by (induction c arbitrary: S) auto

lemma lfp_cs_unfold: "lfp (step S) c = step S (lfp (step S) c)"

apply(rule lfp_unfold[OF _ mono_step])

apply(simp add: strip_step)

done

lemma CS_unfold: "CS c = step UNIV (CS c)"

by (metis CS_def lfp_cs_unfold)

lemma strip_CS[simp]: "strip(CS c) = c"

by(simp add: CS_def index_lfp[simplified])

end