--- a/Tutorial/Tutorial3s.thy	Tue Feb 19 05:38:46 2013 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,162 +0,0 @@
-
-theory Tutorial3s
-imports Lambda
-begin
-
-section {* Formalising Barendregt's Proof of the Substitution Lemma *}
-
-text {*
-  The substitution lemma is another theorem where the variable
-  convention plays a crucial role.
-
-  Barendregt's proof of this lemma needs in the variable case a 
-  case distinction. One way to do this in Isar is to use blocks. 
-  A block consist of some assumptions and reasoning steps 
-  enclosed in curly braces, like
-
-  { \<dots>
-    have "statement"
-    have "last_statement_in_the_block"
-  }
-
-  Such a block may contain local assumptions like
-
-  { assume "A"
-    assume "B"
-    \<dots>
-    have "C" by \<dots>
-  }
-
-  Where "C" is the last have-statement in this block. The behaviour 
-  of such a block to the 'outside' is the implication
-
-   A \<Longrightarrow> B \<Longrightarrow> C 
-
-  Now if we want to prove a property "smth" using the case-distinctions
-  P1, P2 and P3 then we can use the following reasoning:
-
-    { assume "P1"
-      \<dots>
-      have "smth"
-    }
-    moreover
-    { assume "P2"
-      \<dots>
-      have "smth"
-    }
-    moreover
-    { assume "P3"
-      \<dots>
-      have "smth"
-    }
-    ultimately have "smth" by blast
-
-  The blocks establish the implications
-
-    P1 \<Longrightarrow> smth
-    P2 \<Longrightarrow> smth
-    P3 \<Longrightarrow> smth
-
-  If we know that P1, P2 and P3 cover all the cases, that is P1 \<or> P2 \<or> P3 
-  holds, then we have 'ultimately' established the property "smth" 
-  
-*}
-
-subsection {* Two preliminary facts *}
-
-lemma forget:
-  shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"
-by (nominal_induct t avoiding: x s rule: lam.strong_induct)
-   (auto simp add: lam.fresh fresh_at_base)
-
-lemma fresh_fact:
-  assumes a: "atom z \<sharp> s"
-  and b: "z = y \<or> atom z \<sharp> t"
-  shows "atom z \<sharp> t[y ::= s]"
-using a b
-by (nominal_induct t avoiding: z y s rule: lam.strong_induct)
-   (auto simp add: lam.fresh fresh_at_base)
-
-
-
-section {* EXERCISE 10 *}
-
-text {*
-  Fill in the cases 1.2 and 1.3 and the equational reasoning 
-  in the lambda-case.
-*}
-
-lemma 
-  assumes a: "x \<noteq> y"
-  and     b: "atom x \<sharp> L"
-  shows "M[x ::= N][y ::= L] = M[y ::= L][x ::= N[y ::= L]]"
-using a b
-proof (nominal_induct M avoiding: x y N L rule: lam.strong_induct)
-  case (Var z)
-  have a1: "x \<noteq> y" by fact
-  have a2: "atom x \<sharp> L" by fact
-  show "Var z[x::=N][y::=L] = Var z[y::=L][x::=N[y::=L]]" (is "?LHS = ?RHS")
-  proof -
-    { -- {* Case 1.1 *}
-      assume c1: "z = x"
-      have "(1)": "?LHS = N[y::=L]" using c1 by simp
-      have "(2)": "?RHS = N[y::=L]" using c1 a1 by simp
-      have "?LHS = ?RHS" using "(1)" "(2)" by simp
-    }
-    moreover 
-    { -- {* Case 1.2 *}
-      assume c2: "z = y" "z \<noteq> x" 
-      have "(1)": "?LHS = L" using c2 by simp
-      have "(2)": "?RHS = L[x::=N[y::=L]]" using c2 by simp
-      have "(3)": "L[x::=N[y::=L]] = L" using a2 forget by simp
-      have "?LHS = ?RHS" using "(1)" "(2)" "(3)" by simp
-    }
-    moreover 
-    { -- {* Case 1.3 *}
-      assume c3: "z \<noteq> x" "z \<noteq> y"
-      have "(1)": "?LHS = Var z" using c3 by simp
-      have "(2)": "?RHS = Var z" using c3 by simp
-      have "?LHS = ?RHS" using "(1)" "(2)" by simp
-    }
-    ultimately show "?LHS = ?RHS" by blast
-  qed
-next
-  case (Lam z M1) -- {* case 2: lambdas *}
-  have ih: "\<lbrakk>x \<noteq> y; atom x \<sharp> L\<rbrakk> \<Longrightarrow> M1[x ::= N][y ::= L] = M1[y ::= L][x ::= N[y ::= L]]" by fact
-  have a1: "x \<noteq> y" by fact
-  have a2: "atom x \<sharp> L" by fact
-  have fs: "atom z \<sharp> x" "atom z \<sharp> y" "atom z \<sharp> N" "atom z \<sharp> L" by fact+   -- {* !! *}
-  then have b: "atom z \<sharp> N[y::=L]" by (simp add: fresh_fact)
-  show "(Lam [z].M1)[x ::= N][y ::= L] = (Lam [z].M1)[y ::= L][x ::= N[y ::= L]]" (is "?LHS=?RHS") 
-  proof - 
-    have "?LHS = Lam [z].(M1[x ::= N][y ::= L])" using fs by simp
-    also have "\<dots> = Lam [z].(M1[y ::= L][x ::= N[y ::= L]])" using ih a1 a2 by simp
-    also have "\<dots> = (Lam [z].(M1[y ::= L]))[x ::= N[y ::= L]]" using b fs by simp
-    also have "\<dots> = ?RHS" using fs by simp
-    finally show "?LHS = ?RHS" by simp
-  qed
-next
-  case (App M1 M2) -- {* case 3: applications *}
-  then show "(App M1 M2)[x::=N][y::=L] = (App M1 M2)[y::=L][x::=N[y::=L]]" by simp
-qed
-
-text {* 
-  Again the strong induction principle enables Isabelle to find
-  the proof of the substitution lemma completely automatically. 
-*}
-
-lemma substitution_lemma_version:  
-  assumes asm: "x \<noteq> y" "atom x \<sharp> L"
-  shows "M[x::=N][y::=L] = M[y::=L][x::=N[y::=L]]"
-  using asm 
-by (nominal_induct M avoiding: x y N L rule: lam.strong_induct)
-   (auto simp add: fresh_fact forget)
-
-subsection {* MINI EXERCISE *}
-
-text {*
-  Compare and contrast Barendregt's reasoning and the 
-  formalised proofs.
-*}
-
-end