theory Tutorial2

begin

nominal_datatype ty =

  tVar "string"

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"

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

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 {* 

  has the usual "variable convention" already build in. The

*}

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

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.

*}

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)

  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

qed (auto) -- {* app case *}

inductive

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'"

lemma unbind_simple:

  shows "Lam [x].Var x \<mapsto> Var x"

  by auto

equivariance unbind

nominal_inductive unbind

  avoids u_Lam: "x"

  sorry

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

lemma

  shows "False"

proof -

  have "atom x \<sharp> Lam [x]. Var x" by (simp add: lam.fresh)

qed

end