theory Compile
imports "../Nominal"
begin
atom_decl name
nominal_datatype data =
DNat
| DProd "data" "data"
| DSum "data" "data"
nominal_datatype ty =
Data "data"
| Arrow "ty" "ty" ("_->_" [100,100] 100)
nominal_datatype trm =
Var "name"
| Lam "«name»trm" ("Lam [_]._" [100,100] 100)
| App "trm" "trm"
| Const "nat"
| Pr "trm" "trm"
| Fst "trm"
| Snd "trm"
| InL "trm"
| InR "trm"
| Case "trm" "«name»trm" "«name»trm"
("Case _ of inl _ -> _ | inr _ -> _" [100,100,100,100,100] 100)
nominal_datatype dataI = OneI | NatI
nominal_datatype tyI =
DataI "dataI"
| ArrowI "tyI" "tyI" ("_->_" [100,100] 100)
nominal_datatype trmI =
IVar "name"
| ILam "«name»trmI" ("ILam [_]._" [100,100] 100)
| IApp "trmI" "trmI"
| IUnit
| INat "nat"
| ISucc "trmI"
| IAss "trmI" "trmI" ("_\<mapsto>_" [100,100] 100)
| IRef "trmI"
| ISeq "trmI" "trmI" ("_;;_" [100,100] 100)
| Iif "trmI" "trmI" "trmI"
text {* valid contexts *}
inductive
valid :: "(name×'a::pt_name) list => bool"
where
v1[intro]: "valid []"
| v2[intro]: "[|valid Γ;a\<sharp>Γ|]==> valid ((a,σ)#Γ)"
text {* typing judgements for trms *}
inductive
typing :: "(name×ty) list=>trm=>ty=>bool" (" _ \<turnstile> _ : _ " [80,80,80] 80)
where
t0[intro]: "[|valid Γ; (x,τ)∈set Γ|]==> Γ \<turnstile> Var x : τ"
| t1[intro]: "[|Γ \<turnstile> e1 : τ1->τ2; Γ \<turnstile> e2 : τ1|]==> Γ \<turnstile> App e1 e2 : τ2"
| t2[intro]: "[|x\<sharp>Γ;((x,τ1)#Γ) \<turnstile> t : τ2|] ==> Γ \<turnstile> Lam [x].t : τ1->τ2"
| t3[intro]: "valid Γ ==> Γ \<turnstile> Const n : Data(DNat)"
| t4[intro]: "[|Γ \<turnstile> e1 : Data(σ1); Γ \<turnstile> e2 : Data(σ2)|] ==> Γ \<turnstile> Pr e1 e2 : Data (DProd σ1 σ2)"
| t5[intro]: "[|Γ \<turnstile> e : Data(DProd σ1 σ2)|] ==> Γ \<turnstile> Fst e : Data(σ1)"
| t6[intro]: "[|Γ \<turnstile> e : Data(DProd σ1 σ2)|] ==> Γ \<turnstile> Snd e : Data(σ2)"
| t7[intro]: "[|Γ \<turnstile> e : Data(σ1)|] ==> Γ \<turnstile> InL e : Data(DSum σ1 σ2)"
| t8[intro]: "[|Γ \<turnstile> e : Data(σ2)|] ==> Γ \<turnstile> InR e : Data(DSum σ1 σ2)"
| t9[intro]: "[|x1\<sharp>Γ; x2\<sharp>Γ; Γ \<turnstile> e: Data(DSum σ1 σ2);
((x1,Data(σ1))#Γ) \<turnstile> e1 : τ; ((x2,Data(σ2))#Γ) \<turnstile> e2 : τ|]
==> Γ \<turnstile> (Case e of inl x1 -> e1 | inr x2 -> e2) : τ"
text {* typing judgements for Itrms *}
inductive
Ityping :: "(name×tyI) list=>trmI=>tyI=>bool" (" _ I\<turnstile> _ : _ " [80,80,80] 80)
where
t0[intro]: "[|valid Γ; (x,τ)∈set Γ|]==> Γ I\<turnstile> IVar x : τ"
| t1[intro]: "[|Γ I\<turnstile> e1 : τ1->τ2; Γ I\<turnstile> e2 : τ1|]==> Γ I\<turnstile> IApp e1 e2 : τ2"
| t2[intro]: "[|x\<sharp>Γ;((x,τ1)#Γ) I\<turnstile> t : τ2|] ==> Γ I\<turnstile> ILam [x].t : τ1->τ2"
| t3[intro]: "valid Γ ==> Γ I\<turnstile> IUnit : DataI(OneI)"
| t4[intro]: "valid Γ ==> Γ I\<turnstile> INat(n) : DataI(NatI)"
| t5[intro]: "Γ I\<turnstile> e : DataI(NatI) ==> Γ I\<turnstile> ISucc(e) : DataI(NatI)"
| t6[intro]: "[|Γ I\<turnstile> e : DataI(NatI)|] ==> Γ I\<turnstile> IRef e : DataI (NatI)"
| t7[intro]: "[|Γ I\<turnstile> e1 : DataI(NatI); Γ I\<turnstile> e2 : DataI(NatI)|] ==> Γ I\<turnstile> e1\<mapsto>e2 : DataI(OneI)"
| t8[intro]: "[|Γ I\<turnstile> e1 : DataI(NatI); Γ I\<turnstile> e2 : τ|] ==> Γ I\<turnstile> e1;;e2 : τ"
| t9[intro]: "[|Γ I\<turnstile> e: DataI(NatI); Γ I\<turnstile> e1 : τ; Γ I\<turnstile> e2 : τ|] ==> Γ I\<turnstile> Iif e e1 e2 : τ"
text {* capture-avoiding substitution *}
class subst =
fixes subst :: "'a => name => 'a => 'a" ("_[_::=_]" [100,100,100] 100)
instantiation trm :: subst
begin
nominal_primrec subst_trm
where
"(Var x)[y::=t'] = (if x=y then t' else (Var x))"
| "(App t1 t2)[y::=t'] = App (t1[y::=t']) (t2[y::=t'])"
| "[|x\<sharp>y; x\<sharp>t'|] ==> (Lam [x].t)[y::=t'] = Lam [x].(t[y::=t'])"
| "(Const n)[y::=t'] = Const n"
| "(Pr e1 e2)[y::=t'] = Pr (e1[y::=t']) (e2[y::=t'])"
| "(Fst e)[y::=t'] = Fst (e[y::=t'])"
| "(Snd e)[y::=t'] = Snd (e[y::=t'])"
| "(InL e)[y::=t'] = InL (e[y::=t'])"
| "(InR e)[y::=t'] = InR (e[y::=t'])"
| "[|z≠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'|] ==>
(Case e of inl x -> e1 | inr z -> e2)[y::=t'] =
(Case (e[y::=t']) of inl x -> (e1[y::=t']) | inr z -> (e2[y::=t']))"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)+
apply(fresh_guess)+
done
instance ..
end
instantiation trmI :: subst
begin
nominal_primrec subst_trmI
where
"(IVar x)[y::=t'] = (if x=y then t' else (IVar x))"
| "(IApp t1 t2)[y::=t'] = IApp (t1[y::=t']) (t2[y::=t'])"
| "[|x\<sharp>y; x\<sharp>t'|] ==> (ILam [x].t)[y::=t'] = ILam [x].(t[y::=t'])"
| "(INat n)[y::=t'] = INat n"
| "(IUnit)[y::=t'] = IUnit"
| "(ISucc e)[y::=t'] = ISucc (e[y::=t'])"
| "(IAss e1 e2)[y::=t'] = IAss (e1[y::=t']) (e2[y::=t'])"
| "(IRef e)[y::=t'] = IRef (e[y::=t'])"
| "(ISeq e1 e2)[y::=t'] = ISeq (e1[y::=t']) (e2[y::=t'])"
| "(Iif e e1 e2)[y::=t'] = Iif (e[y::=t']) (e1[y::=t']) (e2[y::=t'])"
apply(finite_guess)+
apply(rule TrueI)+
apply(simp add: abs_fresh)+
apply(fresh_guess)+
done
instance ..
end
lemma Isubst_eqvt[eqvt]:
fixes pi::"name prm"
and t1::"trmI"
and t2::"trmI"
and x::"name"
shows "pi•(t1[x::=t2]) = ((pi•t1)[(pi•x)::=(pi•t2)])"
apply (nominal_induct t1 avoiding: x t2 rule: trmI.strong_induct)
apply (simp_all add: subst_trmI.simps eqvts fresh_bij)
done
lemma Isubst_supp:
fixes t1::"trmI"
and t2::"trmI"
and x::"name"
shows "((supp (t1[x::=t2]))::name set) ⊆ (supp t2)∪((supp t1)-{x})"
apply (nominal_induct t1 avoiding: x t2 rule: trmI.strong_induct)
apply (auto simp add: subst_trmI.simps trmI.supp supp_atm abs_supp supp_nat)
apply blast+
done
lemma Isubst_fresh:
fixes x::"name"
and y::"name"
and t1::"trmI"
and t2::"trmI"
assumes a: "x\<sharp>[y].t1" "x\<sharp>t2"
shows "x\<sharp>(t1[y::=t2])"
using a
apply(auto simp add: fresh_def Isubst_supp)
apply(drule rev_subsetD)
apply(rule Isubst_supp)
apply(simp add: abs_supp)
done
text {* big-step evaluation for trms *}
inductive
big :: "trm=>trm=>bool" ("_ \<Down> _" [80,80] 80)
where
b0[intro]: "Lam [x].e \<Down> Lam [x].e"
| b1[intro]: "[|e1\<Down>Lam [x].e; e2\<Down>e2'; e[x::=e2']\<Down>e'|] ==> App e1 e2 \<Down> e'"
| b2[intro]: "Const n \<Down> Const n"
| b3[intro]: "[|e1\<Down>e1'; e2\<Down>e2'|] ==> Pr e1 e2 \<Down> Pr e1' e2'"
| b4[intro]: "e\<Down>Pr e1 e2 ==> Fst e\<Down>e1"
| b5[intro]: "e\<Down>Pr e1 e2 ==> Snd e\<Down>e2"
| b6[intro]: "e\<Down>e' ==> InL e \<Down> InL e'"
| b7[intro]: "e\<Down>e' ==> InR e \<Down> InR e'"
| b8[intro]: "[|e\<Down>InL e'; e1[x::=e']\<Down>e''|] ==> Case e of inl x1 -> e1 | inr x2 -> e2 \<Down> e''"
| b9[intro]: "[|e\<Down>InR e'; e2[x::=e']\<Down>e''|] ==> Case e of inl x1 -> e1 | inr x2 -> e2 \<Down> e''"
inductive
Ibig :: "((nat=>nat)×trmI)=>((nat=>nat)×trmI)=>bool" ("_ I\<Down> _" [80,80] 80)
where
m0[intro]: "(m,ILam [x].e) I\<Down> (m,ILam [x].e)"
| m1[intro]: "[|(m,e1)I\<Down>(m',ILam [x].e); (m',e2)I\<Down>(m'',e3); (m'',e[x::=e3])I\<Down>(m''',e4)|]
==> (m,IApp e1 e2) I\<Down> (m''',e4)"
| m2[intro]: "(m,IUnit) I\<Down> (m,IUnit)"
| m3[intro]: "(m,INat(n))I\<Down>(m,INat(n))"
| m4[intro]: "(m,e)I\<Down>(m',INat(n)) ==> (m,ISucc(e))I\<Down>(m',INat(n+1))"
| m5[intro]: "(m,e)I\<Down>(m',INat(n)) ==> (m,IRef(e))I\<Down>(m',INat(m' n))"
| m6[intro]: "[|(m,e1)I\<Down>(m',INat(n1)); (m',e2)I\<Down>(m'',INat(n2))|] ==> (m,e1\<mapsto>e2)I\<Down>(m''(n1:=n2),IUnit)"
| m7[intro]: "[|(m,e1)I\<Down>(m',IUnit); (m',e2)I\<Down>(m'',e)|] ==> (m,e1;;e2)I\<Down>(m'',e)"
| m8[intro]: "[|(m,e)I\<Down>(m',INat(n)); n≠0; (m',e1)I\<Down>(m'',e)|] ==> (m,Iif e e1 e2)I\<Down>(m'',e)"
| m9[intro]: "[|(m,e)I\<Down>(m',INat(0)); (m',e2)I\<Down>(m'',e)|] ==> (m,Iif e e1 e2)I\<Down>(m'',e)"
text {* Translation functions *}
nominal_primrec
trans :: "trm => trmI"
where
"trans (Var x) = (IVar x)"
| "trans (App e1 e2) = IApp (trans e1) (trans e2)"
| "trans (Lam [x].e) = ILam [x].(trans e)"
| "trans (Const n) = INat n"
| "trans (Pr e1 e2) =
(let limit = IRef(INat 0) in
let v1 = (trans e1) in
let v2 = (trans e2) in
(((ISucc limit)\<mapsto>v1);;(ISucc(ISucc limit)\<mapsto>v2));;(INat 0 \<mapsto> ISucc(ISucc(limit))))"
| "trans (Fst e) = IRef (ISucc (trans e))"
| "trans (Snd e) = IRef (ISucc (ISucc (trans e)))"
| "trans (InL e) =
(let limit = IRef(INat 0) in
let v = (trans e) in
(((ISucc limit)\<mapsto>INat(0));;(ISucc(ISucc limit)\<mapsto>v));;(INat 0 \<mapsto> ISucc(ISucc(limit))))"
| "trans (InR e) =
(let limit = IRef(INat 0) in
let v = (trans e) in
(((ISucc limit)\<mapsto>INat(1));;(ISucc(ISucc limit)\<mapsto>v));;(INat 0 \<mapsto> ISucc(ISucc(limit))))"
| "[|x2≠x1; x1\<sharp>e; x1\<sharp>e2; x2\<sharp>e; x2\<sharp>e1|] ==>
trans (Case e of inl x1 -> e1 | inr x2 -> e2) =
(let v = (trans e) in
let v1 = (trans e1) in
let v2 = (trans e2) in
Iif (IRef (ISucc v)) (v2[x2::=IRef (ISucc (ISucc v))]) (v1[x1::=IRef (ISucc (ISucc v))]))"
apply(finite_guess add: Let_def)+
apply(rule TrueI)+
apply(simp add: abs_fresh Isubst_fresh)+
apply(fresh_guess add: Let_def)+
done
nominal_primrec
trans_type :: "ty => tyI"
where
"trans_type (Data σ) = DataI(NatI)"
| "trans_type (τ1->τ2) = (trans_type τ1)->(trans_type τ2)"
by (rule TrueI)+
end