src/HOL/IMP/Fold.thy

Added inf_Int_eq to pred_set_conv database as well

header "Constant Folding"

theory Fold imports Sem_Equiv begin

subsection "Simple folding of arithmetic expressions"

type_synonym
  tab = "vname \<Rightarrow> val option"

(* maybe better as the composition of substitution and the existing simp_const(0) *)
fun simp_const :: "aexp \<Rightarrow> tab \<Rightarrow> aexp" where
"simp_const (N n) _ = N n" |
"simp_const (V x) t = (case t x of None \<Rightarrow> V x | Some k \<Rightarrow> N k)" |
"simp_const (Plus e1 e2) t = (case (simp_const e1 t, simp_const e2 t) of
  (N n1, N n2) \<Rightarrow> N(n1+n2) | (e1',e2') \<Rightarrow> Plus e1' e2')" 

definition "approx t s \<longleftrightarrow> (ALL x k. t x = Some k \<longrightarrow> s x = k)"

theorem aval_simp_const[simp]:
assumes "approx t s"
shows "aval (simp_const a t) s = aval a s"
  using assms 
  by (induct a) (auto simp: approx_def split: aexp.split option.split)

theorem aval_simp_const_N:
assumes "approx t s"
shows "simp_const a t = N n \<Longrightarrow> aval a s = n"
  using assms
  by (induct a arbitrary: n)
     (auto simp: approx_def split: aexp.splits option.splits)

definition
  "merge t1 t2 = (\<lambda>m. if t1 m = t2 m then t1 m else None)"

primrec lnames :: "com \<Rightarrow> vname set" where
"lnames SKIP = {}" |
"lnames (x ::= a) = {x}" |
"lnames (c1; c2) = lnames c1 \<union> lnames c2" |
"lnames (IF b THEN c1 ELSE c2) = lnames c1 \<union> lnames c2" |
"lnames (WHILE b DO c) = lnames c"

primrec "defs" :: "com \<Rightarrow> tab \<Rightarrow> tab" where
"defs SKIP t = t" |
"defs (x ::= a) t =
  (case simp_const a t of N k \<Rightarrow> t(x \<mapsto> k) | _ \<Rightarrow> t(x:=None))" |
"defs (c1;c2) t = (defs c2 o defs c1) t" |
"defs (IF b THEN c1 ELSE c2) t = merge (defs c1 t) (defs c2 t)" |
"defs (WHILE b DO c) t = t |` (-lnames c)" 

primrec fold where
"fold SKIP _ = SKIP" |
"fold (x ::= a) t = (x ::= (simp_const a t))" |
"fold (c1;c2) t = (fold c1 t; fold c2 (defs c1 t))" |
"fold (IF b THEN c1 ELSE c2) t = IF b THEN fold c1 t ELSE fold c2 t" |
"fold (WHILE b DO c) t = WHILE b DO fold c (t |` (-lnames c))"

lemma approx_merge:
  "approx t1 s \<or> approx t2 s \<Longrightarrow> approx (merge t1 t2) s"
  by (fastforce simp: merge_def approx_def)

lemma approx_map_le:
  "approx t2 s \<Longrightarrow> t1 \<subseteq>\<^sub>m t2 \<Longrightarrow> approx t1 s"
  by (clarsimp simp: approx_def map_le_def dom_def)

lemma restrict_map_le [intro!, simp]: "t |` S \<subseteq>\<^sub>m t"
  by (clarsimp simp: restrict_map_def map_le_def)

lemma merge_restrict:
  assumes "t1 |` S = t |` S"
  assumes "t2 |` S = t |` S"
  shows "merge t1 t2 |` S = t |` S"
proof -
  from assms
  have "\<forall>x. (t1 |` S) x = (t |` S) x" 
   and "\<forall>x. (t2 |` S) x = (t |` S) x" by auto
  thus ?thesis
    by (auto simp: merge_def restrict_map_def 
             split: if_splits intro: ext)
qed


lemma defs_restrict:
  "defs c t |` (- lnames c) = t |` (- lnames c)"
proof (induction c arbitrary: t)
  case (Semi c1 c2)
  hence "defs c1 t |` (- lnames c1) = t |` (- lnames c1)" 
    by simp
  hence "defs c1 t |` (- lnames c1) |` (-lnames c2) = 
         t |` (- lnames c1) |` (-lnames c2)" by simp
  moreover
  from Semi
  have "defs c2 (defs c1 t) |` (- lnames c2) = 
        defs c1 t |` (- lnames c2)"
    by simp
  hence "defs c2 (defs c1 t) |` (- lnames c2) |` (- lnames c1) =
         defs c1 t |` (- lnames c2) |` (- lnames c1)"
    by simp
  ultimately
  show ?case by (clarsimp simp: Int_commute)
next
  case (If b c1 c2)
  hence "defs c1 t |` (- lnames c1) = t |` (- lnames c1)" by simp
  hence "defs c1 t |` (- lnames c1) |` (-lnames c2) = 
         t |` (- lnames c1) |` (-lnames c2)" by simp
  moreover
  from If
  have "defs c2 t |` (- lnames c2) = t |` (- lnames c2)" by simp
  hence "defs c2 t |` (- lnames c2) |` (-lnames c1) = 
         t |` (- lnames c2) |` (-lnames c1)" by simp
  ultimately
  show ?case by (auto simp: Int_commute intro: merge_restrict)
qed (auto split: aexp.split)


lemma big_step_pres_approx:
  "(c,s) \<Rightarrow> s' \<Longrightarrow> approx t s \<Longrightarrow> approx (defs c t) s'"
proof (induction arbitrary: t rule: big_step_induct)
  case Skip thus ?case by simp
next
  case Assign
  thus ?case
    by (clarsimp simp: aval_simp_const_N approx_def split: aexp.split)
next
  case (Semi c1 s1 s2 c2 s3)
  have "approx (defs c1 t) s2" by (rule Semi.IH(1)[OF Semi.prems])
  hence "approx (defs c2 (defs c1 t)) s3" by (rule Semi.IH(2))
  thus ?case by simp
next
  case (IfTrue b s c1 s')
  hence "approx (defs c1 t) s'" by simp
  thus ?case by (simp add: approx_merge)
next
  case (IfFalse b s c2 s')
  hence "approx (defs c2 t) s'" by simp
  thus ?case by (simp add: approx_merge)
next
  case WhileFalse
  thus ?case by (simp add: approx_def restrict_map_def)
next
  case (WhileTrue b s1 c s2 s3)
  hence "approx (defs c t) s2" by simp
  with WhileTrue
  have "approx (defs c t |` (-lnames c)) s3" by simp
  thus ?case by (simp add: defs_restrict)
qed

corollary approx_defs_inv [simp]:
  "\<Turnstile> {approx t} c {approx (defs c t)}"
  by (simp add: hoare_valid_def big_step_pres_approx)


lemma big_step_pres_approx_restrict:
  "(c,s) \<Rightarrow> s' \<Longrightarrow> approx (t |` (-lnames c)) s \<Longrightarrow> approx (t |` (-lnames c)) s'"
proof (induction arbitrary: t rule: big_step_induct)
  case Assign
  thus ?case by (clarsimp simp: approx_def)
next
  case (Semi c1 s1 s2 c2 s3)
  hence "approx (t |` (-lnames c2) |` (-lnames c1)) s1" 
    by (simp add: Int_commute)
  hence "approx (t |` (-lnames c2) |` (-lnames c1)) s2"
    by (rule Semi)
  hence "approx (t |` (-lnames c1) |` (-lnames c2)) s2"
    by (simp add: Int_commute)
  hence "approx (t |` (-lnames c1) |` (-lnames c2)) s3"
    by (rule Semi)
  thus ?case by simp
next
  case (IfTrue b s c1 s' c2)
  hence "approx (t |` (-lnames c2) |` (-lnames c1)) s" 
    by (simp add: Int_commute)
  hence "approx (t |` (-lnames c2) |` (-lnames c1)) s'" 
    by (rule IfTrue)
  thus ?case by (simp add: Int_commute) 
next
  case (IfFalse b s c2 s' c1)
  hence "approx (t |` (-lnames c1) |` (-lnames c2)) s" 
    by simp
  hence "approx (t |` (-lnames c1) |` (-lnames c2)) s'" 
    by (rule IfFalse)
  thus ?case by simp
qed auto


lemma approx_restrict_inv:
  "\<Turnstile> {approx (t |` (-lnames c))} c {approx (t |` (-lnames c))}"
  by (simp add: hoare_valid_def big_step_pres_approx_restrict)

declare assign_simp [simp]

lemma approx_eq:
  "approx t \<Turnstile> c \<sim> fold c t"
proof (induction c arbitrary: t)
  case SKIP show ?case by simp
next
  case Assign
  show ?case by (simp add: equiv_up_to_def)
next
  case Semi 
  thus ?case by (auto intro!: equiv_up_to_semi)
next
  case If
  thus ?case by (auto intro!: equiv_up_to_if_weak)
next
  case (While b c)
  hence "approx (t |` (- lnames c)) \<Turnstile> 
         WHILE b DO c \<sim> WHILE b DO fold c (t |` (- lnames c))"
    by (auto intro: equiv_up_to_while_weak approx_restrict_inv)
  thus ?case 
    by (auto intro: equiv_up_to_weaken approx_map_le)
qed
  

lemma approx_empty [simp]: 
  "approx empty = (\<lambda>_. True)"
  by (auto intro!: ext simp: approx_def)

lemma equiv_sym:
  "c \<sim> c' \<longleftrightarrow> c' \<sim> c"
  by (auto simp add: equiv_def)

theorem constant_folding_equiv:
  "fold c empty \<sim> c"
  using approx_eq [of empty c]
  by (simp add: equiv_up_to_True equiv_sym)



subsection {* More ambitious folding including boolean expressions *}


fun bsimp_const :: "bexp \<Rightarrow> tab \<Rightarrow> bexp" where
"bsimp_const (Less a1 a2) t = less (simp_const a1 t) (simp_const a2 t)" |
"bsimp_const (And b1 b2) t = and (bsimp_const b1 t) (bsimp_const b2 t)" |
"bsimp_const (Not b) t = not(bsimp_const b t)" |
"bsimp_const (Bc bc) _ = Bc bc"

theorem bvalsimp_const [simp]:
  assumes "approx t s"
  shows "bval (bsimp_const b t) s = bval b s"
  using assms by (induct b) auto

lemma not_Bc [simp]: "not (Bc v) = Bc (\<not>v)"
  by (cases v) auto

lemma not_Bc_eq [simp]: "(not b = Bc v) = (b = Bc (\<not>v))"
  by (cases b) auto

lemma and_Bc_eq: 
  "(and a b = Bc v) =
   (a = Bc False \<and> \<not>v  \<or>  b = Bc False \<and> \<not>v \<or> 
    (\<exists>v1 v2. a = Bc v1 \<and> b = Bc v2 \<and> v = v1 \<and> v2))"
  by (rule and.induct) auto

lemma less_Bc_eq [simp]:
  "(less a b = Bc v) = (\<exists>n1 n2. a = N n1 \<and> b = N n2 \<and> v = (n1 < n2))"
  by (rule less.induct) auto
    
theorem bvalsimp_const_Bc:
assumes "approx t s"
shows "bsimp_const b t = Bc v \<Longrightarrow> bval b s = v"
  using assms
  by (induct b arbitrary: v)
     (auto simp: approx_def and_Bc_eq aval_simp_const_N 
           split: bexp.splits bool.splits)


primrec "bdefs" :: "com \<Rightarrow> tab \<Rightarrow> tab" where
"bdefs SKIP t = t" |
"bdefs (x ::= a) t =
  (case simp_const a t of N k \<Rightarrow> t(x \<mapsto> k) | _ \<Rightarrow> t(x:=None))" |
"bdefs (c1;c2) t = (bdefs c2 o bdefs c1) t" |
"bdefs (IF b THEN c1 ELSE c2) t = (case bsimp_const b t of
    Bc True \<Rightarrow> bdefs c1 t
  | Bc False \<Rightarrow> bdefs c2 t
  | _ \<Rightarrow> merge (bdefs c1 t) (bdefs c2 t))" |
"bdefs (WHILE b DO c) t = t |` (-lnames c)" 


primrec bfold where
"bfold SKIP _ = SKIP" |
"bfold (x ::= a) t = (x ::= (simp_const a t))" |
"bfold (c1;c2) t = (bfold c1 t; bfold c2 (bdefs c1 t))" |
"bfold (IF b THEN c1 ELSE c2) t = (case bsimp_const b t of
    Bc True \<Rightarrow> bfold c1 t
  | Bc False \<Rightarrow> bfold c2 t
  | _ \<Rightarrow> IF bsimp_const b t THEN bfold c1 t ELSE bfold c2 t)" |
"bfold (WHILE b DO c) t = (case bsimp_const b t of
    Bc False \<Rightarrow> SKIP
  | _ \<Rightarrow> WHILE bsimp_const b (t |` (-lnames c)) DO bfold c (t |` (-lnames c)))"


lemma bdefs_restrict:
  "bdefs c t |` (- lnames c) = t |` (- lnames c)"
proof (induction c arbitrary: t)
  case (Semi c1 c2)
  hence "bdefs c1 t |` (- lnames c1) = t |` (- lnames c1)" 
    by simp
  hence "bdefs c1 t |` (- lnames c1) |` (-lnames c2) = 
         t |` (- lnames c1) |` (-lnames c2)" by simp
  moreover
  from Semi
  have "bdefs c2 (bdefs c1 t) |` (- lnames c2) = 
        bdefs c1 t |` (- lnames c2)"
    by simp
  hence "bdefs c2 (bdefs c1 t) |` (- lnames c2) |` (- lnames c1) =
         bdefs c1 t |` (- lnames c2) |` (- lnames c1)"
    by simp
  ultimately
  show ?case by (clarsimp simp: Int_commute)
next
  case (If b c1 c2)
  hence "bdefs c1 t |` (- lnames c1) = t |` (- lnames c1)" by simp
  hence "bdefs c1 t |` (- lnames c1) |` (-lnames c2) = 
         t |` (- lnames c1) |` (-lnames c2)" by simp
  moreover
  from If
  have "bdefs c2 t |` (- lnames c2) = t |` (- lnames c2)" by simp
  hence "bdefs c2 t |` (- lnames c2) |` (-lnames c1) = 
         t |` (- lnames c2) |` (-lnames c1)" by simp
  ultimately
  show ?case 
    by (auto simp: Int_commute intro: merge_restrict 
             split: bexp.splits bool.splits)
qed (auto split: aexp.split bexp.split bool.split)


lemma big_step_pres_approx_b:
  "(c,s) \<Rightarrow> s' \<Longrightarrow> approx t s \<Longrightarrow> approx (bdefs c t) s'" 
proof (induction arbitrary: t rule: big_step_induct)
  case Skip thus ?case by simp
next
  case Assign
  thus ?case
    by (clarsimp simp: aval_simp_const_N approx_def split: aexp.split)
next
  case (Semi c1 s1 s2 c2 s3)
  have "approx (bdefs c1 t) s2" by (rule Semi.IH(1)[OF Semi.prems])
  hence "approx (bdefs c2 (bdefs c1 t)) s3" by (rule Semi.IH(2))
  thus ?case by simp
next
  case (IfTrue b s c1 s')
  hence "approx (bdefs c1 t) s'" by simp
  thus ?case using `bval b s` `approx t s`
    by (clarsimp simp: approx_merge bvalsimp_const_Bc
                 split: bexp.splits bool.splits)
next
  case (IfFalse b s c2 s')
  hence "approx (bdefs c2 t) s'" by simp
  thus ?case using `\<not>bval b s` `approx t s`
    by (clarsimp simp: approx_merge bvalsimp_const_Bc
                 split: bexp.splits bool.splits)
next
  case WhileFalse
  thus ?case 
    by (clarsimp simp: bvalsimp_const_Bc approx_def restrict_map_def
                 split: bexp.splits bool.splits)
next
  case (WhileTrue b s1 c s2 s3)
  hence "approx (bdefs c t) s2" by simp
  with WhileTrue
  have "approx (bdefs c t |` (- lnames c)) s3"
    by simp
  thus ?case 
    by (simp add: bdefs_restrict)
qed

corollary approx_bdefs_inv [simp]:
  "\<Turnstile> {approx t} c {approx (bdefs c t)}"
  by (simp add: hoare_valid_def big_step_pres_approx_b)

lemma bfold_equiv: 
  "approx t \<Turnstile> c \<sim> bfold c t"
proof (induction c arbitrary: t)
  case SKIP show ?case by simp
next
  case Assign
  thus ?case by (simp add: equiv_up_to_def)
next
  case Semi
  thus ?case by (auto intro!: equiv_up_to_semi)           
next
  case (If b c1 c2)
  hence "approx t \<Turnstile> IF b THEN c1 ELSE c2 \<sim> 
                   IF Fold.bsimp_const b t THEN bfold c1 t ELSE bfold c2 t"
    by (auto intro: equiv_up_to_if_weak simp: bequiv_up_to_def) 
  thus ?case using If
    by (fastforce simp: bvalsimp_const_Bc split: bexp.splits bool.splits 
                 intro: equiv_up_to_trans)
  next
  case (While b c)
  hence "approx (t |` (- lnames c)) \<Turnstile> 
                   WHILE b DO c \<sim>
                   WHILE bsimp_const b (t |` (- lnames c)) 
                      DO bfold c (t |` (- lnames c))" (is "_ \<Turnstile> ?W \<sim> ?W'") 
    by (auto intro: equiv_up_to_while_weak approx_restrict_inv 
             simp: bequiv_up_to_def)
  hence "approx t \<Turnstile> ?W \<sim> ?W'"
    by (auto intro: equiv_up_to_weaken approx_map_le)
  thus ?case
    by (auto split: bexp.splits bool.splits 
             intro: equiv_up_to_while_False 
             simp: bvalsimp_const_Bc)
qed


theorem constant_bfolding_equiv:
  "bfold c empty \<sim> c"
  using bfold_equiv [of empty c]
  by (simp add: equiv_up_to_True equiv_sym)


end