--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/Attic/Chap03.thy	Fri Feb 05 10:16:10 2016 +0000
@@ -0,0 +1,295 @@
+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"
+