1(* The definitions for a challenge suggested by Adam Chlipala *)
2
3theory Compile
4imports "HOL-Nominal.Nominal"
5begin
6
7atom_decl name 
8
9nominal_datatype data = 
10    DNat
11  | DProd "data" "data"
12  | DSum "data" "data"
13
14nominal_datatype ty = 
15    Data "data"
16  | Arrow "ty" "ty" ("_\<rightarrow>_" [100,100] 100)
17
18nominal_datatype trm = 
19  Var "name"
20  | Lam "\<guillemotleft>name\<guillemotright>trm" ("Lam [_]._" [100,100] 100)
21  | App "trm" "trm"
22  | Const "nat"
23  | Pr "trm" "trm"
24  | Fst "trm"
25  | Snd "trm"
26  | InL "trm"
27  | InR "trm"
28  | Case "trm" "\<guillemotleft>name\<guillemotright>trm" "\<guillemotleft>name\<guillemotright>trm" 
29          ("Case _ of inl _ \<rightarrow> _ | inr _ \<rightarrow> _" [100,100,100,100,100] 100)
30
31nominal_datatype dataI = OneI | NatI
32
33nominal_datatype tyI = 
34    DataI "dataI"
35  | ArrowI "tyI" "tyI" ("_\<rightarrow>_" [100,100] 100)
36
37nominal_datatype trmI = 
38    IVar "name"
39  | ILam "\<guillemotleft>name\<guillemotright>trmI" ("ILam [_]._" [100,100] 100)
40  | IApp "trmI" "trmI"
41  | IUnit
42  | INat "nat"
43  | ISucc "trmI"
44  | IAss "trmI" "trmI" ("_\<mapsto>_" [100,100] 100)
45  | IRef "trmI" 
46  | ISeq "trmI" "trmI" ("_;;_" [100,100] 100)
47  | Iif "trmI" "trmI" "trmI"
48
49text \<open>valid contexts\<close>
50
51inductive 
52  valid :: "(name\<times>'a::pt_name) list \<Rightarrow> bool"
53where
54  v1[intro]: "valid []"
55| v2[intro]: "\<lbrakk>valid \<Gamma>;a\<sharp>\<Gamma>\<rbrakk>\<Longrightarrow> valid ((a,\<sigma>)#\<Gamma>)" (* maybe dom of \<Gamma> *)
56
57text \<open>typing judgements for trms\<close>
58
59inductive 
60  typing :: "(name\<times>ty) list\<Rightarrow>trm\<Rightarrow>ty\<Rightarrow>bool" (" _ \<turnstile> _ : _ " [80,80,80] 80)
61where
62  t0[intro]: "\<lbrakk>valid \<Gamma>; (x,\<tau>)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> Var x : \<tau>"
63| t1[intro]: "\<lbrakk>\<Gamma> \<turnstile> e1 : \<tau>1\<rightarrow>\<tau>2; \<Gamma> \<turnstile> e2 : \<tau>1\<rbrakk>\<Longrightarrow> \<Gamma> \<turnstile> App e1 e2 : \<tau>2"
64| t2[intro]: "\<lbrakk>x\<sharp>\<Gamma>;((x,\<tau>1)#\<Gamma>) \<turnstile> t : \<tau>2\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Lam [x].t : \<tau>1\<rightarrow>\<tau>2"
65| t3[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> \<turnstile> Const n : Data(DNat)"
66| t4[intro]: "\<lbrakk>\<Gamma> \<turnstile> e1 : Data(\<sigma>1); \<Gamma> \<turnstile> e2 : Data(\<sigma>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Pr e1 e2 : Data (DProd \<sigma>1 \<sigma>2)"
67| t5[intro]: "\<lbrakk>\<Gamma> \<turnstile> e : Data(DProd \<sigma>1 \<sigma>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Fst e : Data(\<sigma>1)"
68| t6[intro]: "\<lbrakk>\<Gamma> \<turnstile> e : Data(DProd \<sigma>1 \<sigma>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> Snd e : Data(\<sigma>2)"
69| t7[intro]: "\<lbrakk>\<Gamma> \<turnstile> e : Data(\<sigma>1)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> InL e : Data(DSum \<sigma>1 \<sigma>2)"
70| t8[intro]: "\<lbrakk>\<Gamma> \<turnstile> e : Data(\<sigma>2)\<rbrakk> \<Longrightarrow> \<Gamma> \<turnstile> InR e : Data(DSum \<sigma>1 \<sigma>2)"
71| t9[intro]: "\<lbrakk>x1\<sharp>\<Gamma>; x2\<sharp>\<Gamma>; \<Gamma> \<turnstile> e: Data(DSum \<sigma>1 \<sigma>2); 
72             ((x1,Data(\<sigma>1))#\<Gamma>) \<turnstile> e1 : \<tau>; ((x2,Data(\<sigma>2))#\<Gamma>) \<turnstile> e2 : \<tau>\<rbrakk> 
73             \<Longrightarrow> \<Gamma> \<turnstile> (Case e of inl x1 \<rightarrow> e1 | inr x2 \<rightarrow> e2) : \<tau>"
74
75text \<open>typing judgements for Itrms\<close>
76
77inductive 
78  Ityping :: "(name\<times>tyI) list\<Rightarrow>trmI\<Rightarrow>tyI\<Rightarrow>bool" (" _ I\<turnstile> _ : _ " [80,80,80] 80)
79where
80  t0[intro]: "\<lbrakk>valid \<Gamma>; (x,\<tau>)\<in>set \<Gamma>\<rbrakk>\<Longrightarrow> \<Gamma> I\<turnstile> IVar x : \<tau>"
81| t1[intro]: "\<lbrakk>\<Gamma> I\<turnstile> e1 : \<tau>1\<rightarrow>\<tau>2; \<Gamma> I\<turnstile> e2 : \<tau>1\<rbrakk>\<Longrightarrow> \<Gamma> I\<turnstile> IApp e1 e2 : \<tau>2"
82| t2[intro]: "\<lbrakk>x\<sharp>\<Gamma>;((x,\<tau>1)#\<Gamma>) I\<turnstile> t : \<tau>2\<rbrakk> \<Longrightarrow> \<Gamma> I\<turnstile> ILam [x].t : \<tau>1\<rightarrow>\<tau>2"
83| t3[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> I\<turnstile> IUnit : DataI(OneI)"
84| t4[intro]: "valid \<Gamma> \<Longrightarrow> \<Gamma> I\<turnstile> INat(n) : DataI(NatI)"
85| t5[intro]: "\<Gamma> I\<turnstile> e : DataI(NatI) \<Longrightarrow> \<Gamma> I\<turnstile> ISucc(e) : DataI(NatI)"
86| t6[intro]: "\<lbrakk>\<Gamma> I\<turnstile> e : DataI(NatI)\<rbrakk> \<Longrightarrow> \<Gamma> I\<turnstile> IRef e : DataI (NatI)"
87| t7[intro]: "\<lbrakk>\<Gamma> I\<turnstile> e1 : DataI(NatI); \<Gamma> I\<turnstile> e2 : DataI(NatI)\<rbrakk> \<Longrightarrow> \<Gamma> I\<turnstile> e1\<mapsto>e2 : DataI(OneI)"
88| t8[intro]: "\<lbrakk>\<Gamma> I\<turnstile> e1 : DataI(NatI); \<Gamma> I\<turnstile> e2 : \<tau>\<rbrakk> \<Longrightarrow> \<Gamma> I\<turnstile> e1;;e2 : \<tau>"
89| t9[intro]: "\<lbrakk>\<Gamma> I\<turnstile> e: DataI(NatI); \<Gamma> I\<turnstile> e1 : \<tau>; \<Gamma> I\<turnstile> e2 : \<tau>\<rbrakk> \<Longrightarrow> \<Gamma> I\<turnstile> Iif e e1 e2 : \<tau>"
90
91text \<open>capture-avoiding substitution\<close>
92
93class subst =
94  fixes subst :: "'a \<Rightarrow> name \<Rightarrow> 'a \<Rightarrow> 'a"  ("_[_::=_]" [100,100,100] 100)
95
96instantiation trm :: subst
97begin
98
99nominal_primrec subst_trm
100where
101  "(Var x)[y::=t'] = (if x=y then t' else (Var x))"
102| "(App t1 t2)[y::=t'] = App (t1[y::=t']) (t2[y::=t'])"
103| "\<lbrakk>x\<sharp>y; x\<sharp>t'\<rbrakk> \<Longrightarrow> (Lam [x].t)[y::=t'] = Lam [x].(t[y::=t'])"
104| "(Const n)[y::=t'] = Const n"
105| "(Pr e1 e2)[y::=t'] = Pr (e1[y::=t']) (e2[y::=t'])"
106| "(Fst e)[y::=t'] = Fst (e[y::=t'])"
107| "(Snd e)[y::=t'] = Snd (e[y::=t'])"
108| "(InL e)[y::=t'] = InL (e[y::=t'])"
109| "(InR e)[y::=t'] = InR (e[y::=t'])"
110| "\<lbrakk>z\<noteq>x; x\<sharp>y; x\<sharp>e; x\<sharp>e2; z\<sharp>y; z\<sharp>e; z\<sharp>e1; x\<sharp>t'; z\<sharp>t'\<rbrakk> \<Longrightarrow>
111     (Case e of inl x \<rightarrow> e1 | inr z \<rightarrow> e2)[y::=t'] =
112       (Case (e[y::=t']) of inl x \<rightarrow> (e1[y::=t']) | inr z \<rightarrow> (e2[y::=t']))"
113  apply(finite_guess)+
114  apply(rule TrueI)+
115  apply(simp add: abs_fresh)+
116  apply(fresh_guess)+
117  done
118
119instance ..
120
121end
122
123instantiation trmI :: subst
124begin
125
126nominal_primrec subst_trmI
127where
128  "(IVar x)[y::=t'] = (if x=y then t' else (IVar x))"
129| "(IApp t1 t2)[y::=t'] = IApp (t1[y::=t']) (t2[y::=t'])"
130| "\<lbrakk>x\<sharp>y; x\<sharp>t'\<rbrakk> \<Longrightarrow> (ILam [x].t)[y::=t'] = ILam [x].(t[y::=t'])"
131| "(INat n)[y::=t'] = INat n"
132| "(IUnit)[y::=t'] = IUnit"
133| "(ISucc e)[y::=t'] = ISucc (e[y::=t'])"
134| "(IAss e1 e2)[y::=t'] = IAss (e1[y::=t']) (e2[y::=t'])"
135| "(IRef e)[y::=t'] = IRef (e[y::=t'])"
136| "(ISeq e1 e2)[y::=t'] = ISeq (e1[y::=t']) (e2[y::=t'])"
137| "(Iif e e1 e2)[y::=t'] = Iif (e[y::=t']) (e1[y::=t']) (e2[y::=t'])"
138  apply(finite_guess)+
139  apply(rule TrueI)+
140  apply(simp add: abs_fresh)+
141  apply(fresh_guess)+
142  done
143
144instance ..
145
146end
147
148lemma Isubst_eqvt[eqvt]:
149  fixes pi::"name prm"
150  and   t1::"trmI"
151  and   t2::"trmI"
152  and   x::"name"
153  shows "pi\<bullet>(t1[x::=t2]) = ((pi\<bullet>t1)[(pi\<bullet>x)::=(pi\<bullet>t2)])"
154  apply (nominal_induct t1 avoiding: x t2 rule: trmI.strong_induct)
155  apply (simp_all add: subst_trmI.simps eqvts fresh_bij)
156  done
157
158lemma Isubst_supp:
159  fixes t1::"trmI"
160  and   t2::"trmI"
161  and   x::"name"
162  shows "((supp (t1[x::=t2]))::name set) \<subseteq> (supp t2)\<union>((supp t1)-{x})"
163  apply (nominal_induct t1 avoiding: x t2 rule: trmI.strong_induct)
164  apply (auto simp add: subst_trmI.simps trmI.supp supp_atm abs_supp supp_nat)
165  apply blast+
166  done
167
168lemma Isubst_fresh:
169  fixes x::"name"
170  and   y::"name"
171  and   t1::"trmI"
172  and   t2::"trmI"
173  assumes a: "x\<sharp>[y].t1" "x\<sharp>t2"
174  shows "x\<sharp>(t1[y::=t2])"
175using a
176apply(auto simp add: fresh_def Isubst_supp)
177apply(drule rev_subsetD)
178apply(rule Isubst_supp)
179apply(simp add: abs_supp)
180done
181
182text \<open>big-step evaluation for trms\<close>
183
184inductive 
185  big :: "trm\<Rightarrow>trm\<Rightarrow>bool" ("_ \<Down> _" [80,80] 80)
186where
187  b0[intro]: "Lam [x].e \<Down> Lam [x].e"
188| b1[intro]: "\<lbrakk>e1\<Down>Lam [x].e; e2\<Down>e2'; e[x::=e2']\<Down>e'\<rbrakk> \<Longrightarrow> App e1 e2 \<Down> e'"
189| b2[intro]: "Const n \<Down> Const n"
190| b3[intro]: "\<lbrakk>e1\<Down>e1'; e2\<Down>e2'\<rbrakk> \<Longrightarrow> Pr e1 e2 \<Down> Pr e1' e2'"
191| b4[intro]: "e\<Down>Pr e1 e2 \<Longrightarrow> Fst e\<Down>e1"
192| b5[intro]: "e\<Down>Pr e1 e2 \<Longrightarrow> Snd e\<Down>e2"
193| b6[intro]: "e\<Down>e' \<Longrightarrow> InL e \<Down> InL e'"
194| b7[intro]: "e\<Down>e' \<Longrightarrow> InR e \<Down> InR e'"
195| b8[intro]: "\<lbrakk>e\<Down>InL e'; e1[x::=e']\<Down>e''\<rbrakk> \<Longrightarrow> Case e of inl x1 \<rightarrow> e1 | inr x2 \<rightarrow> e2 \<Down> e''"
196| b9[intro]: "\<lbrakk>e\<Down>InR e'; e2[x::=e']\<Down>e''\<rbrakk> \<Longrightarrow> Case e of inl x1 \<rightarrow> e1 | inr x2 \<rightarrow> e2 \<Down> e''"
197
198inductive 
199  Ibig :: "((nat\<Rightarrow>nat)\<times>trmI)\<Rightarrow>((nat\<Rightarrow>nat)\<times>trmI)\<Rightarrow>bool" ("_ I\<Down> _" [80,80] 80)
200where
201  m0[intro]: "(m,ILam [x].e) I\<Down> (m,ILam [x].e)"
202| m1[intro]: "\<lbrakk>(m,e1)I\<Down>(m',ILam [x].e); (m',e2)I\<Down>(m'',e3); (m'',e[x::=e3])I\<Down>(m''',e4)\<rbrakk> 
203            \<Longrightarrow> (m,IApp e1 e2) I\<Down> (m''',e4)"
204| m2[intro]: "(m,IUnit) I\<Down> (m,IUnit)"
205| m3[intro]: "(m,INat(n))I\<Down>(m,INat(n))"
206| m4[intro]: "(m,e)I\<Down>(m',INat(n)) \<Longrightarrow> (m,ISucc(e))I\<Down>(m',INat(n+1))"
207| m5[intro]: "(m,e)I\<Down>(m',INat(n)) \<Longrightarrow> (m,IRef(e))I\<Down>(m',INat(m' n))"
208| m6[intro]: "\<lbrakk>(m,e1)I\<Down>(m',INat(n1)); (m',e2)I\<Down>(m'',INat(n2))\<rbrakk> \<Longrightarrow> (m,e1\<mapsto>e2)I\<Down>(m''(n1:=n2),IUnit)"
209| m7[intro]: "\<lbrakk>(m,e1)I\<Down>(m',IUnit); (m',e2)I\<Down>(m'',e)\<rbrakk> \<Longrightarrow> (m,e1;;e2)I\<Down>(m'',e)"
210| m8[intro]: "\<lbrakk>(m,e)I\<Down>(m',INat(n)); n\<noteq>0; (m',e1)I\<Down>(m'',e)\<rbrakk> \<Longrightarrow> (m,Iif e e1 e2)I\<Down>(m'',e)"
211| m9[intro]: "\<lbrakk>(m,e)I\<Down>(m',INat(0)); (m',e2)I\<Down>(m'',e)\<rbrakk> \<Longrightarrow> (m,Iif e e1 e2)I\<Down>(m'',e)"
212
213text \<open>Translation functions\<close>
214
215nominal_primrec
216  trans :: "trm \<Rightarrow> trmI"
217where
218  "trans (Var x) = (IVar x)"
219| "trans (App e1 e2) = IApp (trans e1) (trans e2)"
220| "trans (Lam [x].e) = ILam [x].(trans e)"
221| "trans (Const n) = INat n"
222| "trans (Pr e1 e2) = 
223       (let limit = IRef(INat 0) in 
224        let v1 = (trans e1) in 
225        let v2 = (trans e2) in 
226        (((ISucc limit)\<mapsto>v1);;(ISucc(ISucc limit)\<mapsto>v2));;(INat 0 \<mapsto> ISucc(ISucc(limit))))"
227| "trans (Fst e) = IRef (ISucc (trans e))"
228| "trans (Snd e) = IRef (ISucc (ISucc (trans e)))"
229| "trans (InL e) = 
230        (let limit = IRef(INat 0) in 
231         let v = (trans e) in 
232         (((ISucc limit)\<mapsto>INat(0));;(ISucc(ISucc limit)\<mapsto>v));;(INat 0 \<mapsto> ISucc(ISucc(limit))))"
233| "trans (InR e) = 
234        (let limit = IRef(INat 0) in 
235         let v = (trans e) in 
236         (((ISucc limit)\<mapsto>INat(1));;(ISucc(ISucc limit)\<mapsto>v));;(INat 0 \<mapsto> ISucc(ISucc(limit))))"
237| "\<lbrakk>x2\<noteq>x1; x1\<sharp>e; x1\<sharp>e2; x2\<sharp>e; x2\<sharp>e1\<rbrakk> \<Longrightarrow> 
238   trans (Case e of inl x1 \<rightarrow> e1 | inr x2 \<rightarrow> e2) =
239       (let v = (trans e) in
240        let v1 = (trans e1) in
241        let v2 = (trans e2) in 
242        Iif (IRef (ISucc v)) (v2[x2::=IRef (ISucc (ISucc v))]) (v1[x1::=IRef (ISucc (ISucc v))]))"
243  apply(finite_guess add: Let_def)+
244  apply(rule TrueI)+
245  apply(simp add: abs_fresh Isubst_fresh)+
246  apply(fresh_guess add: Let_def)+
247  done
248
249nominal_primrec
250  trans_type :: "ty \<Rightarrow> tyI"
251where
252  "trans_type (Data \<sigma>) = DataI(NatI)"
253| "trans_type (\<tau>1\<rightarrow>\<tau>2) = (trans_type \<tau>1)\<rightarrow>(trans_type \<tau>2)"
254  by (rule TrueI)+
255
256end
257