theory Classical

imports "../Nominal2"

begin

(* example from Urban's PhD *)

atom_decl name

atom_decl coname

nominal_datatype trm =

  Ax "name" "coname"

| Cut c::"coname" t1::"trm" n::"name" t2::"trm"             bind n in t1, bind c in t2  

     ("Cut <_>._ '(_')._" [100,100,100,100] 100)

| NotR n::"name" t::"trm" "coname"                            bind n in t

     ("NotR '(_')._ _" [100,100,100] 100)

| NotL c::"coname" t::"trm" "name"                            bind c in t   

     ("NotL <_>._ _" [100,100,100] 100)

| AndR c1::"coname" t1::"trm" c2::"coname" t2::"trm" "coname" bind c1 in t1, bind c2 in t2

     ("AndR <_>._ <_>._ _" [100,100,100,100,100] 100)

| AndL1 n::"name" t::"trm" "name"                             bind n in t

     ("AndL1 '(_')._ _" [100,100,100] 100)

| AndL2 n::"name" t::"trm" "name"                             bind n in t

     ("AndL2 '(_')._ _" [100,100,100] 100)

| OrR1 c::"coname" t::"trm" "coname"                          bind c in t             

     ("OrR1 <_>._ _" [100,100,100] 100)

| OrR2 c::"coname" t::"trm" "coname"                          bind c in t     

     ("OrR2 <_>._ _" [100,100,100] 100)

| OrL n1::"name" t1::"trm" n2::"name" t2::"trm" "name"        bind n1 in t1, bind n2 in t2       

     ("OrL '(_')._ '(_')._ _" [100,100,100,100,100] 100)

| ImpL c::"coname" t1::"trm" n::"name" t2::"trm" "name"       bind c in t1, bind n in t2

     ("ImpL <_>._ '(_')._ _" [100,100,100,100,100] 100)

| ImpR c::"coname" n::"name" t::"trm" "coname"                bind n c in t

     ("ImpR '(_').<_>._ _" [100,100,100,100] 100)

thm trm.distinct

thm trm.induct

thm trm.exhaust

thm trm.strong_exhaust

thm trm.strong_exhaust[simplified]

thm trm.fv_defs

thm trm.bn_defs

thm trm.perm_simps

thm trm.eq_iff

thm trm.fv_bn_eqvt

thm trm.size_eqvt

thm trm.supp

thm trm.supp[simplified]

lemma swap_at_base_sort: "sort_of (atom a) \<noteq> sort_of (atom x) \<Longrightarrow> sort_of (atom b) \<noteq> sort_of (atom x) \<Longrightarrow> (atom a \<rightleftharpoons> atom b) \<bullet> x = x"

  by (rule swap_fresh_fresh) (simp_all add: fresh_at_base(1))

nominal_primrec (* (invariant "\<lambda>(_, e, d) y. atom e \<sharp> y \<and> atom d \<sharp> y") *)

  crename :: "trm \<Rightarrow> coname \<Rightarrow> coname \<Rightarrow> trm"  ("_[_\<turnstile>c>_]" [100,100,100] 100) 

where

  "(Ax x a)[d\<turnstile>c>e] = (if a=d then Ax x e else Ax x a)" 

| "atom a \<sharp> (d, e) \<Longrightarrow> (Cut <a>.M (x).N)[d\<turnstile>c>e] = Cut <a>.(M[d\<turnstile>c>e]) (x).(N[d\<turnstile>c>e])" 

| "(NotR (x).M a)[d\<turnstile>c>e] = (if a=d then NotR (x).(M[d\<turnstile>c>e]) e else NotR (x).(M[d\<turnstile>c>e]) a)" 

| "atom a \<sharp> (d, e) \<Longrightarrow> (NotL <a>.M x)[d\<turnstile>c>e] = (NotL <a>.(M[d\<turnstile>c>e]) x)" 

| "\<lbrakk>atom a \<sharp> (d, e); atom b \<sharp> (d, e)\<rbrakk> \<Longrightarrow> (AndR <a>.M <b>.N c)[d\<turnstile>c>e] = 

          (if c=d then AndR <a>.(M[d\<turnstile>c>e]) <b>.(N[d \<turnstile>c>e]) e else AndR <a>.(M[d\<turnstile>c>e]) <b>.(N[d\<turnstile>c>e]) c)" 

| "(AndL1 (x).M y)[d\<turnstile>c>e] = AndL1 (x).(M[d\<turnstile>c>e]) y"

| "(AndL2 (x).M y)[d\<turnstile>c>e] = AndL2 (x).(M[d\<turnstile>c>e]) y"

| "atom a \<sharp> (d, e) \<Longrightarrow> (OrR1 <a>.M b)[d\<turnstile>c>e] = 

          (if b=d then OrR1 <a>.(M[d\<turnstile>c>e]) e else OrR1 <a>.(M[d\<turnstile>c>e]) b)"

| "atom a \<sharp> (d, e) \<Longrightarrow> (OrR2 <a>.M b)[d\<turnstile>c>e] = 

          (if b=d then OrR2 <a>.(M[d\<turnstile>c>e]) e else OrR2 <a>.(M[d\<turnstile>c>e]) b)"

| "(OrL (x).M (y).N z)[d\<turnstile>c>e] = OrL (x).(M[d\<turnstile>c>e]) (y).(N[d\<turnstile>c>e]) z"

| "atom a \<sharp> (d, e) \<Longrightarrow> (ImpR (x).<a>.M b)[d\<turnstile>c>e] = 

          (if b=d then ImpR (x).<a>.(M[d\<turnstile>c>e]) e else ImpR (x).<a>.(M[d\<turnstile>c>e]) b)"

| "atom a \<sharp> (d, e) \<Longrightarrow> (ImpL <a>.M (x).N y)[d\<turnstile>c>e] = ImpL <a>.(M[d\<turnstile>c>e]) (x).(N[d\<turnstile>c>e]) y"

  apply(simp only: eqvt_def)

  apply(simp only: crename_graph_def)

  apply (rule, perm_simp, rule)

  (*apply(erule crename_graph.induct)

  apply(simp add: trm.fresh)*)

  apply(rule TrueI)

  -- "covered all cases"

  apply(case_tac x)

  apply(rule_tac y="a" and c="(b, c)" in trm.strong_exhaust)

  apply (simp_all add: fresh_star_def)[12]

  apply(metis)+

  -- "compatibility"

  apply(simp_all)

  apply(rule conjI)

  apply(elim conjE)

  apply(erule Abs_lst1_fcb)

  apply(simp add: Abs_fresh_iff)

  apply(simp add: Abs_fresh_iff)

  apply(erule fresh_eqvt_at)

  apply(simp add: finite_supp)

  apply(simp add: fresh_Pair fresh_at_base(1))

  apply(simp add: eqvt_at_def swap_at_base_sort)

  apply simp

  oops

end