theory Test

imports "Parser"

begin

text {* example 1 *}

nominal_datatype lam =

  VAR "name"

| APP "lam" "lam"

| LET bp::"bp" t::"lam"   bind "bi bp" in t ("Let _ in _" [100,100] 100)

and bp = 

  BP "name" "lam"  ("_ ::= _" [100,100] 100)

binder

  bi::"bp \<Rightarrow> name set"

where

  "bi (BP x t) = {x}"

typ lam_raw

term VAR_raw

term Test.BP_raw

thm bi_raw.simps

print_theorems

text {* example 2 *}

nominal_datatype trm' =

  Var "name"

| App "trm'" "trm'"

| Lam x::"name" t::"trm'"          bind x in t 

| Let p::"pat'" "trm'" t::"trm'"   bind "f p" in t

and pat' =

  PN

| PS "name"

| PD "name" "name"

binder

  f::"pat' \<Rightarrow> name set"

where 

  "f PN = {}"

| "f (PS x) = {x}"

| "f (PD x y) = {x,y}"

thm f_raw.simps

nominal_datatype trm0 =

  Var0 "name"

| App0 "trm0" "trm0"

| Lam0 x::"name" t::"trm0"          bind x in t 

| Let0 p::"pat0" "trm0" t::"trm0"   bind "f0 p" in t

and pat0 =

  PN0

| PS0 "name"

| PD0 "pat0" "pat0"

binder

  f0::"pat0 \<Rightarrow> name set"

where 

  "f0 PN0 = {}"

| "f0 (PS0 x) = {x}"

| "f0 (PD0 p1 p2) = (f0 p1) \<union> (f0 p2)"

thm f0_raw.simps

text {* example type schemes *}

datatype t = 

  Var "name" 

| Fun "t" "t"

nominal_datatype tyS = 

  All xs::"name list" ty::"t" bind xs in ty

(* example 1 from Terms.thy *)

nominal_datatype trm1 =

  Vr1 "name"

| Ap1 "trm1" "trm1"

| Lm1 x::"name" t::"trm1"      bind x in t 

| Lt1 p::"bp1" "trm1" t::"trm1" bind "bv1 p" in t 

and bp1 =

  BUnit1

| BV1 "name"

| BP1 "bp1" "bp1"

binder

  bv1

where

  "bv1 (BUnit1) = {}"

| "bv1 (BV1 x) = {atom x}"

| "bv1 (BP1 bp1 bp2) = (bv1 bp1) \<union> (bv1 bp2)"

thm bv1_raw.simps

(* example 2 from Terms.thy *)

nominal_datatype trm2 =

  Vr2 "name"

| Ap2 "trm2" "trm2"

| Lm2 x::"name" t::"trm2"       bind x in t

| Lt2 r::"rassign" t::"trm2"    bind "bv2 r" in t

and rassign = 

  As "name" "trm2"

binder

  bv2

where

  "bv2 (As x t) = {atom x}"

(* example 3 from Terms.thy *)

nominal_datatype trm3 =

  Vr3 "name"

| Ap3 "trm3" "trm3"

| Lm3 x::"name" t::"trm3"        bind x in t

| Lt3 r::"rassigns3" t::"trm3"   bind "bv3 r" in t

and rassigns3 =

  ANil

| ACons "name" "trm3" "rassigns3"

binder

  bv3

where

  "bv3 ANil = {}"

| "bv3 (ACons x t as) = {atom x} \<union> (bv3 as)"

(* example 4 from Terms.thy *)

nominal_datatype trm4 =

  Vr4 "name"

| Ap4 "trm4" "trm4 list"

| Lm4 x::"name" t::"trm4"  bind x in t

(* example 5 from Terms.thy *)

nominal_datatype trm5 =

  Vr5 "name"

| Ap5 "trm5" "trm5"

| Lt5 l::"lts" t::"trm5"  bind "bv5 l" in t

and lts =

  Lnil

| Lcons "name" "trm5" "lts"

binder

  bv5

where

  "bv5 Lnil = {}"

| "bv5 (Lcons n t ltl) = {atom n} \<union> (bv5 ltl)"

(* example 6 from Terms.thy *)

nominal_datatype trm6 =

  Vr6 "name"

| Lm6 x::"name" t::"trm6"         bind x in t

| Lt6 left::"trm6" right::"trm6"  bind "bv6 left" in right

binder

  bv6

where

  "bv6 (Vr6 n) = {}"

| "bv6 (Lm6 n t) = {atom n} \<union> bv6 t"

| "bv6 (Lt6 l r) = bv6 l \<union> bv6 r"

(* example 7 from Terms.thy *)

nominal_datatype trm7 =

  Vr7 "name"

| Lm7 l::"name" r::"trm7"   bind l in r

| Lt7 l::"trm7" r::"trm7"   bind "bv7 l" in r

binder 

  bv7 

where

  "bv7 (Vr7 n) = {atom n}"

| "bv7 (Lm7 n t) = bv7 t - {atom n}"

| "bv7 (Lt7 l r) = bv7 l \<union> bv7 r"

(* example 8 from Terms.thy *)

nominal_datatype foo8 =

  Foo0 "name"

| Foo1 b::"bar8" f::"foo8" bind "bv8 b" in foo

and bar8 =

  Bar0 "name"

| Bar1 "name" s::"name" b::"bar8" bind s in b

binder 

  bv8

where

  "bv8 (Bar0 x) = {}"

| "bv8 (Bar1 v x b) = {atom v}"

(* example 9 from Terms.thy *)

nominal_datatype lam9 =

  Var9 "name"

| Lam9 n::"name" l::"lam9" bind n in l

and bla9 =

  Bla9 f::"lam9" s::"lam9" bind "bv9 f" in s

binder

  bv9

where

  "bv9 (Var9 x) = {}"

| "bv9 (Lam9 x b) = {atom x}"

(* example form Leroy 96 about modules *)

nominal_datatype mexp =

  Acc path

| Stru body

| Funct x::name sexp m::mexp    bind x in m

| FApp mexp path

| Ascr mexp sexp

and body =

  Empty

| Seq c::defn d::body bind "cbinders c" in d

and defn =  

  Type name tyty

| Dty name

| DStru name mexp

| Val name trmtrm

and sexp =

  Sig sbody

| SFunc name sexp sexp

and sbody = 

  SEmpty

| SSeq C::spec D::sbody  bind "Cbinders C" in D

and spec =

  Type1 name 

| Type2 name tyty

| SStru name sexp

| SVal name tyty

and tyty =

  Tyref1 name

| Tyref2 path tyty

| Fun tyty tyty

and path =

  Sref1 name

| Sref2 path name

and trmtrm =

  Tref1 name

| Tref2 path name

| Lam v::name tyty M::trmtrm  bind v in M

| App trmtrm trmtrm

| Let body trmtrm

binder

    cbinders :: "defn \<Rightarrow> atom set"

and Cbinders :: "spec \<Rightarrow> atom set"

where

  "cbinders (Type t T) = {atom t}"

| "cbinders (Dty t) = {atom t}"

| "cbinders (DStru x s) = {atom x}"

| "cbinders (Val v M) = {atom v}"

| "Cbinders (Type1 t) = {atom t}"

| "Cbinders (Type2 t T) = {atom t}"

| "Cbinders (SStru x S) = {atom x}"

| "Cbinders (SVal v T) = {atom v}"  

(* core haskell *)

atom_decl var

atom_decl tvar

atom_decl co

datatype sort = 

  TY tvar

| CO co

datatype kind = 

  KStar

| KFun kind kind

| KEq kind kind

(* there are types, coercion types and regular types *)

nominal_datatype ty =

  TVar tvar

| TFun string "ty list"

| TAll tvar kind ty --"some binding"

| TSym ty

| TCir ty ty

| TApp ty ty

| TLeft ty

| TRight ty

| TEq ty

| TRightc ty

| TLeftc ty

| TCoe ty ty

typedecl ty --"hack since ty is not yet defined"

abbreviation 

  "atoms A \<equiv> atom ` A"

nominal_datatype trm =

  Var var

| LAM tv::tvar kind t::trm   bind v in t 

| APP trm ty

| Lam v::var ty t::trm       bind v in t

| App trm trm

| Let x::var ty trm t::trm   bind x in t

| Case trm "assoc list"

| Cast trm ty   --"ty is supposed to be a coercion type only"

and assoc = 

  A p::pat t::trm bind "bv p" in t 

and pat = 

  K string "(tvar \<times> kind) list" "(var \<times> ty) list"

binder

 bv :: "pat \<Rightarrow> atom set"

where

 "bv (K s ts vs) = (atoms (set (map fst ts))) \<union> (atoms (set (map fst vs)))"

thm bv_raw.simps

end