theory Lambda

begin

atom_decl name

  Var "name"

| App "lam" "lam"

| Lam x::"name" l::"lam"  bind x in l

thm lam.fv

thm lam.supp

lemmas supp_fn' = lam.fv[simplified lam.supp]

declare lam.perm[eqvt]

section {* Strong Induction Principles*}

(*

  Old way of establishing strong induction

  principles by chosing a fresh name.

*)

lemma

  fixes c::"'a::fs"

  assumes a1: "\<And>name c. P c (Var name)"

  and     a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"

  and     a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"

  shows "P c lam"

proof -

  have "\<And>p. P c (p \<bullet> lam)"

    apply(induct lam arbitrary: c rule: lam.induct)

    apply(perm_simp)

    apply(rule a1)

    apply(perm_simp)

    apply(rule a2)

    apply(assumption)

    apply(assumption)

    apply(subgoal_tac "\<exists>new::name. (atom new) \<sharp> (c, Lam (p \<bullet> name) (p \<bullet> lam))")

    defer

    apply(simp add: fresh_def)

    apply(rule_tac X="supp (c, Lam (p \<bullet> name) (p \<bullet> lam))" in obtain_at_base)

    apply(simp add: supp_Pair finite_supp)

    apply(blast)

    apply(erule exE)

    apply(rule_tac t="p \<bullet> Lam name lam" and 

                   s="(((p \<bullet> name) \<leftrightarrow> new) + p) \<bullet> Lam name lam" in subst)

    apply(simp del: lam.perm)

    apply(subst lam.perm)

    apply(subst (2) lam.perm)

    apply(rule flip_fresh_fresh)

    apply(simp add: fresh_def)

    apply(simp only: supp_fn')

    apply(simp)

    apply(simp add: fresh_Pair)

    apply(simp)

    apply(rule a3)

    apply(simp add: fresh_Pair)

    apply(drule_tac x="((p \<bullet> name) \<leftrightarrow> new) + p" in meta_spec)

    apply(simp)

    done

  then have "P c (0 \<bullet> lam)" by blast

  then show "P c lam" by simp

qed

(* 

  New way of establishing strong induction

  principles by using a appropriate permutation.

*)

lemma

  fixes c::"'a::fs"

  assumes a1: "\<And>name c. P c (Var name)"

  and     a2: "\<And>lam1 lam2 c. \<lbrakk>\<And>d. P d lam1; \<And>d. P d lam2\<rbrakk> \<Longrightarrow> P c (App lam1 lam2)"

  and     a3: "\<And>name lam c. \<lbrakk>atom name \<sharp> c; \<And>d. P d lam\<rbrakk> \<Longrightarrow> P c (Lam name lam)"

  shows "P c lam"

proof -

  have "\<And>p. P c (p \<bullet> lam)"

    apply(induct lam arbitrary: c rule: lam.induct)

    apply(perm_simp)

    apply(rule a1)

    apply(perm_simp)

    apply(rule a2)

    apply(assumption)

    apply(assumption)

    apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom name}) \<sharp>* c \<and> supp (p \<bullet> Lam name lam) \<sharp>* q")

    apply(erule exE)

    apply(rule_tac t="p \<bullet> Lam name lam" and 

                   s="q \<bullet> p \<bullet> Lam name lam" in subst)

    defer

    apply(simp)

    apply(rule a3)

    apply(simp add: eqvts fresh_star_def)

    apply(drule_tac x="q + p" in meta_spec)

    apply(simp)

    apply(rule at_set_avoiding2)

    apply(simp add: finite_supp)

    apply(simp add: finite_supp)

    apply(simp add: finite_supp)

    apply(perm_simp)

    apply(simp add: fresh_star_def fresh_def supp_fn')

    apply(rule supp_perm_eq)

    apply(simp)

    done

  then have "P c (0 \<bullet> lam)" by blast

  then show "P c lam" by simp

qed

section {* Typing *}

nominal_datatype ty =

  TVar string

| TFun ty ty

notation

 TFun ("_ \<rightarrow> _") 

declare ty.perm[eqvt]

inductive

  valid :: "(name \<times> ty) list \<Rightarrow> bool"

where

  "valid []"

| "\<lbrakk>atom x \<sharp> Gamma; valid Gamma\<rbrakk> \<Longrightarrow> valid ((x, T)#Gamma)"

inductive

  typing :: "(name\<times>ty) list \<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"

equivariance valid

equivariance typing

thm valid.eqvt

thm typing.eqvt

thm eqvts

thm eqvts_raw

thm typing.induct[of "\<Gamma>" "t" "T", no_vars]

lemma

  fixes c::"'a::fs"

  assumes a: "\<Gamma> \<turnstile> t : T" 

  and a1: "\<And>\<Gamma> x T c. \<lbrakk>valid \<Gamma>; (x, T) \<in> set \<Gamma>\<rbrakk> \<Longrightarrow> P c \<Gamma> (Var x) T"

  and a2: "\<And>\<Gamma> t1 T1 T2 t2 c. \<lbrakk>\<Gamma> \<turnstile> t1 : T1 \<rightarrow> T2; \<And>d. P d \<Gamma> t1 T1 \<rightarrow> T2; \<Gamma> \<turnstile> t2 : T1; \<And>d. P d \<Gamma> t2 T1\<rbrakk> 

           \<Longrightarrow> P c \<Gamma> (App t1 t2) T2"

  and a3: "\<And>x \<Gamma> T1 t T2 c. \<lbrakk>atom x \<sharp> c; atom x \<sharp> \<Gamma>; (x, T1) # \<Gamma> \<turnstile> t : T2; \<And>d. P d ((x, T1) # \<Gamma>) t T2\<rbrakk> 

           \<Longrightarrow> P c \<Gamma> (Lam x t) T1 \<rightarrow> T2"

  shows "P c \<Gamma> t T"

proof -

  from a have "\<And>p c. P c (p \<bullet> \<Gamma>) (p \<bullet> t) (p \<bullet> T)"

  proof (induct)

    case (t_Var \<Gamma> x T p c)

    then show ?case

      apply -

      apply(perm_strict_simp)

      apply(rule a1)

      apply(drule_tac p="p" in permute_boolI)

      apply(perm_strict_simp add: permute_minus_cancel)

      apply(assumption)

      apply(rotate_tac 1)

      apply(drule_tac p="p" in permute_boolI)

      apply(perm_strict_simp add: permute_minus_cancel)

      apply(assumption)

      done

  next

    case (t_App \<Gamma> t1 T1 T2 t2 p c)

    then show ?case

      apply -

      apply(perm_strict_simp)

      apply(rule a2)

      apply(drule_tac p="p" in permute_boolI)

      apply(perm_strict_simp add: permute_minus_cancel)

      apply(assumption)

      apply(assumption)

      apply(rotate_tac 2)

      apply(drule_tac p="p" in permute_boolI)

      apply(perm_strict_simp add: permute_minus_cancel)

      apply(assumption)

      apply(assumption)

      done

  next

    case (t_Lam x \<Gamma> T1 t T2 p c)

    then show ?case

      apply -

      apply(subgoal_tac "\<exists>q. (q \<bullet> {p \<bullet> atom x}) \<sharp>* c \<and> 

        supp (p \<bullet> \<Gamma>, p \<bullet> Lam x t, p \<bullet> (T1 \<rightarrow> T2)) \<sharp>* q")

      apply(erule exE)

      apply(rule_tac t="p \<bullet> \<Gamma>" and  s="(q + p) \<bullet> \<Gamma>" in subst)

      apply(simp only: permute_plus)

      apply(rule supp_perm_eq)

      apply(simp add: supp_Pair fresh_star_union)

      apply(rule_tac t="p \<bullet> Lam x t" and  s="(q + p) \<bullet> Lam x t" in subst)

      apply(simp only: permute_plus)

      apply(rule supp_perm_eq)

      apply(simp add: supp_Pair fresh_star_union)

      apply(rule_tac t="p \<bullet> (T1 \<rightarrow> T2)" and  s="(q + p) \<bullet> (T1 \<rightarrow> T2)" in subst)

      apply(simp only: permute_plus)

      apply(rule supp_perm_eq)

      apply(simp add: supp_Pair fresh_star_union)

      apply(simp (no_asm) only: eqvts)

      apply(rule a3)

      apply(simp only: eqvts permute_plus)

      apply(simp add: fresh_star_def)

      apply(drule_tac p="q + p" in permute_boolI)

      apply(perm_strict_simp add: permute_minus_cancel)

      apply(assumption)

      apply(rotate_tac 1)

      apply(drule_tac p="q + p" in permute_boolI)

      apply(perm_strict_simp add: permute_minus_cancel)

      apply(assumption)

      apply(drule_tac x="d" in meta_spec)

      apply(drule_tac x="q + p" in meta_spec)

      apply(perm_strict_simp add: permute_minus_cancel)

      apply(assumption)

      apply(rule at_set_avoiding2)

      apply(simp add: finite_supp)

      apply(simp add: finite_supp)

      apply(simp add: finite_supp)

      apply(rule_tac p="-p" in permute_boolE)

      apply(perm_strict_simp add: permute_minus_cancel)

	(* supplied by the user *)

      apply(simp add: fresh_star_prod)

      apply(simp add: fresh_star_def)

      sorry

  qed

  then have "P c (0 \<bullet> \<Gamma>) (0 \<bullet> t) (0 \<bullet> T)" .

  then show "P c \<Gamma> t T" by simp

qed

inductive

  tt :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'a \<Rightarrow> bool)"  

  for r :: "('a \<Rightarrow> 'a \<Rightarrow> bool)"

where

    aa: "tt r a a"

  | bb: "tt r a b ==> tt r a c"

(* PROBLEM: derived eqvt-theorem does not conform with [eqvt]

equivariance tt

*)

inductive

 alpha_lam_raw'

where

  a1: "name = namea \<Longrightarrow> alpha_lam_raw' (Var_raw name) (Var_raw namea)"

| a2: "\<lbrakk>alpha_lam_raw' lam_raw1 lam_raw1a; alpha_lam_raw' lam_raw2 lam_raw2a\<rbrakk> \<Longrightarrow>

   alpha_lam_raw' (App_raw lam_raw1 lam_raw2) (App_raw lam_raw1a lam_raw2a)"

| a3: "\<exists>pi. ({atom name}, lam_raw) \<approx>gen alpha_lam_raw' fv_lam_raw pi ({atom namea}, lam_rawa) \<Longrightarrow>

   alpha_lam_raw' (Lam_raw name lam_raw) (Lam_raw namea lam_rawa)"

equivariance alpha_lam_raw'

thm eqvts_raw

section {* size function *}

lemma size_eqvt_raw:

  fixes t::"lam_raw"

  shows "size (pi \<bullet> t)  = size t"

  apply (induct rule: lam_raw.inducts)

  apply simp_all

  done

instantiation lam :: size 

begin

quotient_definition

  "size_lam :: lam \<Rightarrow> nat"

is

  "size :: lam_raw \<Rightarrow> nat"

lemma size_rsp:

  "alpha_lam_raw x y \<Longrightarrow> size x = size y"

  apply (induct rule: alpha_lam_raw.inducts)

  apply (simp_all only: lam_raw.size)

  apply (simp_all only: alphas)

  apply clarify

  apply (simp_all only: size_eqvt_raw)

  done

lemma [quot_respect]:

  "(alpha_lam_raw ===> op =) size size"

  by (simp_all add: size_rsp)

lemma [quot_preserve]:

  "(rep_lam ---> id) size = size"

  by (simp_all add: size_lam_def)

instance

  by default

end

lemmas size_lam[simp] = 

  lam_raw.size(4)[quot_lifted]

  lam_raw.size(5)[quot_lifted]

  lam_raw.size(6)[quot_lifted]

(* is this needed? *)

lemma [measure_function]: 

  "is_measure (size::lam\<Rightarrow>nat)" 

  by (rule is_measure_trivial)

section {* Matching *}

definition

  MATCH :: "('c::pt \<Rightarrow> (bool * 'a::pt * 'b::pt)) \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'b"

where

  "MATCH M d x \<equiv> if (\<exists>!r. \<exists>q. M q = (True, x, r)) then (THE r. \<exists>q. M q = (True, x, r)) else d"

(*

lemma MATCH_eqvt:

  shows "p \<bullet> (MATCH M d x) = MATCH (p \<bullet> M) (p \<bullet> d) (p \<bullet> x)"

unfolding MATCH_def

apply(perm_simp the_eqvt)

apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})

apply(simp)

thm eqvts_raw 

apply(subst if_eqvt)

apply(subst ex1_eqvt)

apply(subst permute_fun_def)

apply(subst ex_eqvt)

apply(subst permute_fun_def)

apply(subst eq_eqvt)

apply(subst permute_fun_app_eq[where f="M"])

apply(simp only: permute_minus_cancel)

apply(subst permute_prod.simps)

apply(subst permute_prod.simps)

apply(simp only: permute_minus_cancel)

apply(simp only: permute_bool_def)

apply(simp)

apply(subst ex1_eqvt)

apply(subst permute_fun_def)

apply(subst ex_eqvt)

apply(subst permute_fun_def)

apply(subst eq_eqvt)

apply(simp only: eqvts)

apply(simp)

apply(subgoal_tac "(p \<bullet> (\<exists>!r. \<exists>q. M q = (True, x, r))) = (\<exists>!r. \<exists>q. (p \<bullet> M) q = (True, p \<bullet> x, r))")

apply(drule sym)

apply(simp)

apply(rule impI)

apply(simp add: perm_bool)

apply(rule trans)

apply(rule pt_the_eqvt[OF pta at])

apply(assumption)

apply(simp add: pt_ex_eqvt[OF pt at])

apply(simp add: pt_eq_eqvt[OF ptb at])

apply(rule cheat)

apply(rule trans)

apply(rule pt_ex1_eqvt)

apply(rule pta)

apply(rule at)

apply(simp add: pt_ex_eqvt[OF pt at])

apply(simp add: pt_eq_eqvt[OF ptb at])

apply(subst pt_pi_rev[OF pta at])

apply(subst pt_fun_app_eq[OF pt at])

apply(subst pt_pi_rev[OF pt at])

apply(simp)

done

lemma MATCH_cng:

  assumes a: "M1 = M2" "d1 = d2"

  shows "MATCH M1 d1 x = MATCH M2 d2 x"

using a by simp

lemma MATCH_eq:

  assumes a: "t = l x" "G x" "\<And>x'. t = l x' \<Longrightarrow> G x' \<Longrightarrow> r x' = r x"

  shows "MATCH (\<lambda>x. (G x, l x, r x)) d t = r x"

using a

unfolding MATCH_def

apply(subst if_P)

apply(rule_tac a="r x" in ex1I)

apply(rule_tac x="x" in exI)

apply(blast)

apply(erule exE)

apply(drule_tac x="q" in meta_spec)

apply(auto)[1]

apply(rule the_equality)