
diff -r 78d828f43cdf -r 4b4742aa43f2 Tutorial/Tutorial2s.thy
--- a/Tutorial/Tutorial2s.thy	Sat Dec 17 16:58:11 2011 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,354 +0,0 @@
-
-theory Tutorial2s
-imports Lambda
-begin
-
-section {* Height of a Lambda-Term *}
-
-text {*
-  The theory Lambda defines the notions of capture-avoiding
-  substitution and the height of lambda terms. 
-*}
-
-thm height.simps
-thm subst.simps
-
-subsection {* EXERCISE 7 *}
-
-text {* 
-  Lets prove a property about the height of substitutions.
-
-  Assume that the height of a lambda-term is always greater 
-  or equal to 1.
-*}
-
-lemma height_ge_one: 
-  shows "1 \<le> height t"
-by (induct t rule: lam.induct) (auto)
-
-text {* Remove the sorries *}
-
-theorem
-  shows "height (t[x ::= t']) \<le> height t - 1 + height t'"
-proof (nominal_induct t avoiding: x t' rule: lam.strong_induct)
-  case (Var y)
-  have "1 \<le> height t'" using height_ge_one by simp
-  then show "height (Var y[x ::= t']) \<le> height (Var y) - 1 + height t'" by simp
-next    
-  case (App t1 t2)
-  have ih1: "height (t1[x::=t']) \<le> (height t1) - 1 + height t'" 
-  and  ih2: "height (t2[x::=t']) \<le> (height t2) - 1 + height t'" by fact+
-  then show "height ((App t1 t2)[x ::= t']) \<le> height (App t1 t2) - 1 + height t'" by simp  
-next
-  case (Lam y t1)
-  have ih: "height (t1[x::=t']) \<le> height t1 - 1 + height t'" by fact
-  moreover
-  have vc: "atom y \<sharp> x" "atom y \<sharp> t'" by fact+ -- {* usual variable convention *}
-  ultimately 
-  show "height ((Lam [y].t1)[x ::= t']) \<le> height (Lam [y].t1) - 1 + height t'" by simp
-qed
-
-
-section {* Types and the Weakening Lemma *}
-
-nominal_datatype ty =
-  tVar "string"
-| tArr "ty" "ty" (infixr "\<rightarrow>" 100)
-
-
-text {* 
-  Having defined them as nominal datatypes gives us additional
-  definitions and theorems that come with types (see below).
- 
-  We next define the type of typing contexts, a predicate that
-  defines valid contexts (i.e. lists that contain only unique
-  variables) and the typing judgement.
-*}
-
-type_synonym ty_ctx = "(name \<times> ty) list"
-
-inductive
-  valid :: "ty_ctx \<Rightarrow> bool"
-where
-  v1[intro]: "valid []"
-| v2[intro]: "\<lbrakk>valid \<Gamma>; atom x \<sharp> \<Gamma>\<rbrakk>\<Longrightarrow> valid ((x, T) # \<Gamma>)"
-
-
-inductive
-  typing :: "ty_ctx \<Rightarrow> lam \<Rightarrow> ty \<Rightarrow> bool" ("_ \<turnstile> _ : _" [60, 60, 60] 60) 
-where
-  t_Var[intro]:  "\<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Var x : T"
-| t_App[intro]:  "\<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> App t1 t2 : T2"
-| t_Lam[intro]:  "\<lbrakk>atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].t : T1 \<rightarrow> T2"
-
-
-text {*
-  The predicate atom x \<sharp> \<Gamma>, read as x fresh for \<Gamma>, is defined by 
-  Nominal Isabelle. It is defined for lambda-terms, products, lists etc. 
-  For example we have:
-*}
-
-lemma
-  fixes x::"name"
-  shows "atom x \<sharp> Lam [x].t"
-  and   "atom x \<sharp> (t1, t2) \<Longrightarrow> atom x \<sharp> App t1 t2"
-  and   "atom x \<sharp> Var y \<Longrightarrow> atom x \<sharp> y" 
-  and   "\<lbrakk>atom x \<sharp> t1; atom x \<sharp> t2\<rbrakk> \<Longrightarrow> atom x \<sharp> (t1, t2)"
-  and   "\<lbrakk>atom x \<sharp> l1; atom x \<sharp> l2\<rbrakk> \<Longrightarrow> atom x \<sharp> (l1 @ l2)"
-  and   "atom x \<sharp> y \<Longrightarrow> x \<noteq> y"
-  by (simp_all add: lam.fresh fresh_append fresh_at_base) 
-
-text {* 
-  We can also prove that every variable is fresh for a type. 
-*}
-
-lemma ty_fresh:
-  fixes x::"name"
-  and   T::"ty"
-  shows "atom x \<sharp> T"
-by (induct T rule: ty.induct)
-   (simp_all add: ty.fresh pure_fresh)
-
-text {* 
-  In order to state the weakening lemma in a convenient form, we 
-  using the following abbreviation. Abbreviations behave like 
-  definitions, except that they are always automatically folded 
-  and unfolded.
-*}
-
-abbreviation
-  "sub_ty_ctx" :: "ty_ctx \<Rightarrow> ty_ctx \<Rightarrow> bool" ("_ \<sqsubseteq> _" [60, 60] 60) 
-where
-  "\<Gamma>1 \<sqsubseteq> \<Gamma>2 \<equiv> \<forall>x. x \<in> set \<Gamma>1 \<longrightarrow> x \<in> set \<Gamma>2"
-
-
-subsection {* EXERCISE 8 *}
-
-text {*
-  Fill in the details and give a proof of the weakening lemma. 
-*}
-
-lemma 
-  assumes a: "\<Gamma>1 \<turnstile> t : T"
-  and     b: "valid \<Gamma>2" 
-  and     c: "\<Gamma>1 \<sqsubseteq> \<Gamma>2"
-  shows "\<Gamma>2 \<turnstile> t : T"
-using a b c
-proof (induct arbitrary: \<Gamma>2)
-  case (t_Var \<Gamma>1 x T)
-  have a1: "valid \<Gamma>1" by fact
-  have a2: "(x, T) \<in> set \<Gamma>1" by fact
-  have a3: "valid \<Gamma>2" by fact
-  have a4: "\<Gamma>1 \<sqsubseteq> \<Gamma>2" by fact
-
-  show "\<Gamma>2 \<turnstile> Var x : T" sorry
-next
-  case (t_Lam x \<Gamma>1 T1 t T2) 
-  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; (x, T1) # \<Gamma>1 \<sqsubseteq> \<Gamma>3\<rbrakk> \<Longrightarrow> \<Gamma>3 \<turnstile> t : T2" by fact
-  have a0: "atom x \<sharp> \<Gamma>1" by fact
-  have a1: "valid \<Gamma>2" by fact
-  have a2: "\<Gamma>1 \<sqsubseteq> \<Gamma>2" by fact
-
-  show "\<Gamma>2 \<turnstile> Lam [x].t : T1 \<rightarrow> T2" sorry
-qed (auto) -- {* the application case is automatic*}
-
-
-text {* 
-  Despite the frequent claim that the weakening lemma is trivial,
-  routine or obvious, the proof in the lambda-case does not go 
-  through smoothly. Painful variable renamings seem to be necessary. 
-  But the proof using renamings is horribly complicated (see below). 
-*}
-
-equivariance valid
-equivariance typing
-
-lemma weakening_NOT_TO_BE_TRIED_AT_HOME: 
-  assumes a: "\<Gamma>1 \<turnstile> t : T"
-  and     b: "valid \<Gamma>2" 
-  and     c: "\<Gamma>1 \<sqsubseteq> \<Gamma>2"
-  shows "\<Gamma>2 \<turnstile> t : T"
-using a b c
-proof (induct arbitrary: \<Gamma>2)
-  -- {* lambda case *}
-  case (t_Lam x \<Gamma>1 T1 t T2) 
-  have fc0: "atom x \<sharp> \<Gamma>1" by fact
-  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; (x, T1) # \<Gamma>1 \<sqsubseteq> \<Gamma>3\<rbrakk> \<Longrightarrow> \<Gamma>3 \<turnstile> t : T2" by fact
-  -- {* we choose a fresh variable *}
-  obtain c::"name" where fc1: "atom c \<sharp> (x, t, \<Gamma>1, \<Gamma>2)" by (rule obtain_fresh)
-  -- {* we alpha-rename the abstraction *}
-  have "Lam [c].((c \<leftrightarrow> x) \<bullet> t) = Lam [x].t" using fc1
-    by (auto simp add: lam.eq_iff Abs1_eq_iff flip_def)
-  moreover
-  -- {* we can then alpha rename the goal *}
-  have "\<Gamma>2 \<turnstile> Lam [c].((c \<leftrightarrow> x) \<bullet> t) : T1 \<rightarrow> T2" 
-  proof - 
-    -- {* we need to establish *} 
-    -- {* (1)   (x, T1) # \<Gamma>1 \<sqsubseteq> (x, T1) # ((c \<leftrightarrow> x) \<bullet> \<Gamma>2) *}
-    -- {* (2)   valid ((x, T1) # ((c \<leftrightarrow> x) \<bullet> \<Gamma>2)) *}
-    have "(1)": "(x, T1) # \<Gamma>1 \<sqsubseteq> (x, T1) # ((c \<leftrightarrow> x) \<bullet> \<Gamma>2)" 
-    proof -
-      have "\<Gamma>1 \<sqsubseteq> \<Gamma>2" by fact
-      then have "(c \<leftrightarrow> x) \<bullet> ((x, T1) # \<Gamma>1 \<sqsubseteq> (x, T1) # ((c \<leftrightarrow> x) \<bullet> \<Gamma>2))" using fc0 fc1
-        by (perm_simp) (simp add: flip_fresh_fresh)
-      then show "(x, T1) # \<Gamma>1 \<sqsubseteq> (x, T1) # ((c \<leftrightarrow> x) \<bullet> \<Gamma>2)" by (rule permute_boolE)
-    qed
-    moreover 
-    have "(2)": "valid ((x, T1) # ((c \<leftrightarrow> x) \<bullet> \<Gamma>2))" 
-    proof -
-      have "valid \<Gamma>2" by fact
-      then show "valid ((x, T1) # ((c \<leftrightarrow> x) \<bullet> \<Gamma>2))" using fc1
-        by (auto simp add: fresh_permute_left atom_eqvt valid.eqvt)	
-    qed
-    -- {* these two facts give us by induction hypothesis the following *}
-    ultimately have "(x, T1) # ((c \<leftrightarrow> x) \<bullet> \<Gamma>2) \<turnstile> t : T2" using ih by simp 
-    -- {* we now apply renamings to get to our goal *}
-    then have "(c \<leftrightarrow> x) \<bullet> ((x, T1) # ((c \<leftrightarrow> x) \<bullet> \<Gamma>2) \<turnstile> t : T2)" by (rule permute_boolI)
-    then have "(c, T1) # \<Gamma>2 \<turnstile> ((c \<leftrightarrow> x) \<bullet> t) : T2" using fc1
-      by (perm_simp) (simp add: flip_fresh_fresh ty_fresh)
-    then show "\<Gamma>2 \<turnstile> Lam [c].((c \<leftrightarrow> x) \<bullet> t) : T1 \<rightarrow> T2" using fc1 by auto
-  qed
-  ultimately show "\<Gamma>2 \<turnstile> Lam [x].t : T1 \<rightarrow> T2" by simp
-qed (auto) -- {* var and app cases, luckily, are automatic *}
-
-
-text {* 
-  The remedy is to use again a stronger induction principle that
-  has the usual "variable convention" already build in. The
-  following command derives this induction principle for the typing
-  relation. (We shall explain what happens here later.)
-*}
-
-nominal_inductive typing
-  avoids t_Lam: "x"
-  unfolding fresh_star_def
-  by (simp_all add: fresh_Pair lam.fresh ty_fresh)
-
-text {* Compare the two induction principles *}
-
-thm typing.induct
-thm typing.strong_induct -- {* note the additional assumption {atom x} \<sharp> c *}
-
-text {* 
-  We can use the stronger induction principle by replacing
-  the line
-
-      proof (induct arbitrary: \<Gamma>2)
-
-  with 
-  
-      proof (nominal_induct avoiding: \<Gamma>2 rule: typing.strong_induct)
-
-  Try now the proof.
-*}
- 
-subsection {* EXERCISE 9 *} 
-
-lemma weakening: 
-  assumes a: "\<Gamma>1 \<turnstile> t : T"
-  and     b: "valid \<Gamma>2" 
-  and     c: "\<Gamma>1 \<sqsubseteq> \<Gamma>2"
-  shows "\<Gamma>2 \<turnstile> t : T"
-using a b c
-proof (nominal_induct avoiding: \<Gamma>2 rule: typing.strong_induct)
-  case (t_Var \<Gamma>1 x T)  -- {* variable case is as before *}
-  have "\<Gamma>1 \<sqsubseteq> \<Gamma>2" by fact 
-  moreover  
-  have "valid \<Gamma>2" by fact 
-  moreover 
-  have "(x, T)\<in> set \<Gamma>1" by fact
-  ultimately show "\<Gamma>2 \<turnstile> Var x : T" by auto
-next
-  case (t_Lam x \<Gamma>1 T1 t T2) 
-  have vc: "atom x \<sharp> \<Gamma>2" by fact  -- {* additional fact afforded by the strong induction *}
-  have ih: "\<And>\<Gamma>3. \<lbrakk>valid \<Gamma>3; (x, T1) # \<Gamma>1 \<sqsubseteq> \<Gamma>3\<rbrakk> \<Longrightarrow> \<Gamma>3 \<turnstile> t : T2" by fact
-  have a0: "atom x \<sharp> \<Gamma>1" by fact
-  have a1: "valid \<Gamma>2" by fact
-  have a2: "\<Gamma>1 \<sqsubseteq> \<Gamma>2" by fact
-  have "valid ((x, T1) # \<Gamma>2)" using a1 vc by auto
-  moreover
-  have "(x, T1) # \<Gamma>1 \<sqsubseteq> (x, T1) # \<Gamma>2" using a2 by auto
-  ultimately 
-  have "(x, T1) # \<Gamma>2 \<turnstile> t : T2" using ih by simp 
-  then show "\<Gamma>2 \<turnstile> Lam [x].t : T1 \<rightarrow> T2" using vc by auto
-qed (auto) -- {* app case *}
-
-
-
-section {* Unbind and an Inconsistency Example *}
-
-text {*
-  You might wonder why we had to discharge some proof
-  obligations in order to obtain the stronger induction
-  principle for the typing relation. (Remember for
-  lambda terms this stronger induction principle is
-  derived automatically.)
-  
-  This proof obligation is really needed, because if we 
-  assume universally a stronger induction principle, then 
-  in some cases we can derive false. This is "shown" below.
-*}
-
-
-inductive
-  unbind :: "lam \<Rightarrow> lam \<Rightarrow> bool" (infixr "\<mapsto>" 60)
-where
-  u_Var[intro]: "Var x \<mapsto> Var x"
-| u_App[intro]: "App t1 t2 \<mapsto> App t1 t2"
-| u_Lam[intro]: "t \<mapsto> t' \<Longrightarrow> Lam [x].t \<mapsto> t'"
-
-text {* It is clear that the following judgement holds. *}
-
-lemma unbind_simple:
-  shows "Lam [x].Var x \<mapsto> Var x"
-  by auto
-
-text {*
-  Now lets derive the strong induction principle for unbind.
-  The point is that we cannot establish the proof obligations,
-  therefore we force Isabelle to accept them by using "sorry".
-*}
-
-equivariance unbind
-nominal_inductive unbind
-  avoids u_Lam: "x"
-  sorry
-
-text {*
-  Using the stronger induction principle, we can establish
-  th follwoing bogus property.
-*}
-
-lemma unbind_fake:
-  fixes y::"name"
-  assumes a: "t \<mapsto> t'"
-  and     b: "atom y \<sharp> t"
-  shows "atom y \<sharp> t'"
-using a b
-proof (nominal_induct avoiding: y rule: unbind.strong_induct)
-  case (u_Lam t t' x y)
-  have ih: "atom y \<sharp> t \<Longrightarrow> atom y \<sharp> t'" by fact
-  have asm: "atom y \<sharp> Lam [x]. t" by fact
-  have fc: "atom x \<sharp> y" by fact
-  then have in_eq: "x \<noteq> y" by (simp add: fresh_at_base)
-  then have "atom y \<sharp> t" using asm by (simp add: lam.fresh)
-  then show "atom y \<sharp> t'" using ih by simp
-qed
-
-text {*
-  And from this follows the inconsitency:
-*}
-
-lemma
-  shows "False"
-proof -
-  have "atom x \<sharp> Lam [x]. Var x" by (simp add: lam.fresh)
-  moreover 
-  have "Lam [x]. Var x \<mapsto> Var x" using unbind_simple by auto
-  ultimately 
-  have "atom x \<sharp> Var x" using unbind_fake by auto
-  then have "x \<noteq> x" by (simp add: lam.fresh fresh_at_base)
-  then show "False" by simp
-qed
-
-end




