theory Chap03

imports Main

begin

(* 2.5.6 Case Study: Boolean Expressions *)

datatype boolex = Const bool | Var nat | Neg boolex

| And boolex boolex

primrec "value2" :: "boolex \<Rightarrow> (nat \<Rightarrow> bool) \<Rightarrow> bool" where

"value2 (Const b)  env = b" |

"value2 (Var x)    env = env x" |

"value2 (Neg b)    env = (\<not> value2 b env)" |

"value2 (And b c)  env = (value2 b env \<and> value2 c env)"

value "Const true"

value "Var (Suc(0))"

value "value2 (Const False) (\<lambda>x. False)"

value "value2 (Var 11) (\<lambda>x. if (x = 10 | x = 11) then True else False)"

value "value2 (Var 11) (\<lambda>x. True )"

definition

  "en1 \<equiv>  (\<lambda>x. if x = 10 | x = 11 then True else False)"

abbreviation

  "en2 \<equiv>  (\<lambda>x. if x = 10 | x = 11 then True else False)"

value "value2 (And (Var 10) (Var 11)) en2"

lemma "value2 (And (Var 10) (Var 11)) en2 = True"

apply(simp)

done

datatype ifex = 

  CIF bool 

| VIF nat 

| IF ifex ifex ifex

primrec valif :: "ifex \<Rightarrow> (nat \<Rightarrow> bool) \<Rightarrow> bool" where

"valif (CIF b)    env = b" |

"valif (VIF x)    env = env x" |

"valif (IF b t e) env = (if valif b env then valif t env else valif e env)"

abbreviation "vif1 \<equiv> valif (CIF False) (\<lambda>x. False)"

abbreviation "vif2 \<equiv> valif (VIF 11) (\<lambda>x. False)"

abbreviation "vif3 \<equiv> valif (VIF 13) (\<lambda>x. True)"

value "valif (CIF False) (\<lambda>x. False)"

value "valif (VIF 11) (\<lambda>x. True)"

value "valif (IF (CIF False) (CIF True) (CIF True))"

primrec bool2if :: "boolex \<Rightarrow> ifex" where

"bool2if (Const b)  = CIF b" |

"bool2if (Var x)    = VIF x" |

"bool2if (Neg b)    = IF (bool2if b) (CIF False) (CIF True)" |

"bool2if (And b c)  = IF (bool2if b) (bool2if c) (CIF False)"

lemma "valif (bool2if b) env = value2 b env"

apply(induct_tac b)

apply(auto)

done

primrec normif :: "ifex \<Rightarrow> ifex \<Rightarrow> ifex \<Rightarrow> ifex" where

"normif (CIF b)     t e = IF (CIF b) t e" |

"normif (VIF x)     t e = IF (VIF x) t e" |

"normif (IF b t e)  u f = normif b (normif t u f) (normif e u f)"

primrec norm :: "ifex \<Rightarrow> ifex" where

"norm (CIF b)     = CIF b" |

"norm (VIF x)     = VIF x" |

"norm (IF b t e)  = normif b (norm t) (norm e)"  

(***************   CHAPTER-3   ********************************)

lemma "\<lbrakk> xs @ zs = ys @ xs; [] @ xs = [] @ [] \<rbrakk> \<Longrightarrow> ys = zs"

apply simp

done

lemma "\<forall> x. f x = g (f (g x)) \<Longrightarrow> f [] = f [] @ []"

apply (simp (no_asm))

done

definition xor :: "bool \<Rightarrow> bool \<Rightarrow> bool" where

"xor A B \<equiv> (A \<and> \<not>B) \<or> (\<not>A \<and> B)"

lemma "xor A (\<not>A)"

apply(simp only: xor_def)

apply(simp add: xor_def)

done

lemma "(let xs = [] in xs@ys@xs) = ys"

apply(simp only: Let_def)

apply(simp add: Let_def)

done

(* 3.1.8 Conditioal Simplification Rules  *)

lemma hd_Cons_tl: "xs \<noteq> [] \<Longrightarrow> hd xs # tl xs = xs"

using [[simp_trace=true]]

apply(case_tac xs)

apply(simp)

apply(simp)

done

lemma "xs \<noteq> [] \<Longrightarrow> hd(rev xs) # tl(rev xs) = rev xs"

apply(case_tac xs)

using [[simp_trace=true]]

apply(simp)

apply(simp)

done

(* 3.1.9 Automatic Case Splits  *)

lemma "\<forall> xs. if xs = [] then rev xs = [] else rev xs \<noteq> []"

apply(split split_if)

apply(simp)

done

lemma "(case xs of [] \<Rightarrow> zs | y#ys \<Rightarrow> y#(ys@zs)) = xs@zs"

apply(split list.split)

apply(simp split: list.split)

done

lemma "if xs = [] then ys \<noteq> [] else ys = [] \<Longrightarrow> xs @ ys \<noteq> []"

apply(split split_if_asm)

apply(simp)

apply(auto)

done

(* 3.2 Induction Heuristics  *)

primrec itrev :: "'a list \<Rightarrow> 'a list \<Rightarrow> 'a list" where

"itrev [] ys = ys" |

"itrev (x#xs) ys = itrev xs (x#ys)"

lemma "itrev xs [] = rev xs"

apply(induct_tac xs)

apply(simp)

apply(auto)

oops

lemma "itrev xs ys = rev xs @ ys"

apply(induct_tac xs)

apply(simp_all)

oops

lemma "\<forall> ys. itrev xs ys = rev xs @ ys"

apply(induct_tac xs)

apply(simp)

apply(simp)

done

primrec add1 :: "nat \<Rightarrow> nat \<Rightarrow> nat" where

"add1 m 0 = m" |

"add1 m (Suc n) = add1 (Suc m) n"

primrec add2 :: "nat \<Rightarrow> nat \<Rightarrow> nat" where

"add2 m 0 = m" |

"add2 m (Suc n) = Suc (add2 m n)"

value "add1 1 3"

value "1 + 3"

lemma abc [simp]: "add1 m 0 = m"

apply(induction m)

apply(simp)

apply(simp)

done

lemma abc2: "\<forall>m. add1 m n = m + n"

apply(induction n)

apply(simp)

apply(simp del: add1.simps)

apply(simp (no_asm) only: add1.simps)

apply(simp only: )

apply(simp)

done

thm abc2

lemma abc3: "add1 m n = m + n"

apply(induction n arbitrary: m)

apply(simp)

apply(simp del: add1.simps)

apply(simp (no_asm) only: add1.simps)

apply(simp only: )

done

thm abc3

lemma abc4: "add1 m n = add2 m n"

apply(induction n arbitrary: m)

apply(simp)

apply(simp add: abc3)

done

find_theorems "_ \<and> _ "

(* added anottest *)

lemma abc5: "add2 m n = m + n"

apply(induction n)

apply(simp)

apply(simp)

done

(* 3.3 Case Study: Compiling Expressions  *)

type_synonym 'v binop = "'v \<Rightarrow> 'v \<Rightarrow> 'v"

datatype ('a, 'v)expr = 

  Cex 'v

| Vex 'a

| Bex "'v binop" "('a,'v)expr" "('a,'v)expr"

primrec "value" :: "('a,'v)expr \<Rightarrow> ('a \<Rightarrow> 'v) \<Rightarrow> 'v" where

"value (Cex v) env = v" |

"value (Vex a) env = env a" |

"value (Bex f e1 e2) env = f (value e1 env) (value e2 env)"

value "value (Cex a) (\<lambda>x. True)"

datatype ('a,'v)instr = 

  Const 'v

| Load 'a

| Apply "'v binop"

primrec exec :: "('a,'v)instr list \<Rightarrow> ('a\<Rightarrow>'v) \<Rightarrow> 'v list \<Rightarrow> 'v list" where

"exec [] s vs = vs" |

"exec (i#is) s vs = (case i of

    Const v \<Rightarrow> exec is s (v#vs)

  | Load a \<Rightarrow> exec is s ((s a)#vs)

  | Apply f \<Rightarrow> exec is s ((f (hd vs) (hd(tl vs)))#(tl(tl vs))))"

primrec compile :: "('a,'v)expr \<Rightarrow> ('a,'v)instr list" where

"compile (Cex v) = [Const v]" |

"compile (Vex a) = [Load a]" |

"compile (Bex f e1 e2) = (compile e2) @ (compile e1) @ [Apply f]"

theorem "exec (compile e) s [] = [value e s]"

(*the theorem needs to be generalized*)

oops

(*more generalized theorem*)

theorem "\<forall> vs. exec (compile e) s vs = (value e s) # vs"

apply(induct_tac e)

apply(simp)

apply(simp)

oops

lemma exec_app[simp]: "\<forall> vs. exec (xs@ys) s vs = exec ys s (exec xs s vs)"

apply(induct_tac xs)

apply(simp)

apply(simp)

apply(simp split: instr.split)

done

(* 3.4 Advanced Datatypes  *)

datatype 'a aexp = IF "'a bexp" "'a aexp" "'a aexp" 

                | Sum "'a aexp" "'a aexp"

                | Diff "'a aexp" "'a aexp"

                | Var 'a

                | Num nat

and      'a bexp = Less "'a aexp" "'a aexp"

                | And "'a bexp" "'a bexp"

                | Neg "'a bexp"

(*  Total Recursive Functions: Fun  *)

(*  3.5.1 Definition    *)

fun fib :: "nat \<Rightarrow> nat" where

"fib 0 = 0" |

"fib (Suc 0) = 1" |

"fib (Suc(Suc x)) = fib x + fib (Suc x)"

value "fib (Suc(Suc(Suc(Suc(Suc 0)))))"

fun sep :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where

"sep a [] = []" |

"sep a [x] = [x]" |

"sep a (x#y#zs) = x # a # sep a (y#zs)"

fun last :: "'a list \<Rightarrow> 'a" where

"last [x] = x" |

"last (_#y#zs) = last (y#zs)"

fun sep1 :: "'a \<Rightarrow> 'a list \<Rightarrow> 'a list" where

"sep1 a (x#y#zs) = x # a # sep1 a (y#zs)" |

"sep1 _ xs = xs"

fun swap12:: "'a list \<Rightarrow> 'a list" where

"swap12 (x#y#zs) = y#x#zs" |

"swap12 zs = zs"