theory Test

imports "Parser" "../Attic/Prove"

begin

atom_decl name

ML {* val _ = recursive := false *}

text {* example 1, equivalent to example 2 from Terms *}

nominal_datatype lam =

  VAR "name"

| APP "lam" "lam"

| LAM x::"name" t::"lam" bind x in t

| LET bp::"bp" t::"lam"   bind "bi bp" in t

and bp =

  BP "name" "lam"

binder

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

where

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

thm lam_bp.fv

thm lam_bp.supp

thm lam_bp.eq_iff

thm lam_bp.bn

thm lam_bp.perm

thm lam_bp.induct

thm lam_bp.inducts

thm lam_bp.distinct

ML {* Sign.of_sort @{theory} (@{typ lam}, @{sort fs}) *}

thm lam_bp.fv[simplified lam_bp.supp]

ML {* val _ = recursive := true *}

nominal_datatype lam' =

  VAR' "name"

| APP' "lam'" "lam'"

| LAM' x::"name" t::"lam'"  bind x in t

| LET' bp::"bp'" t::"lam'"   bind "bi' bp" in t

and bp' =

  BP' "name" "lam'"

binder

  bi'::"bp' \<Rightarrow> atom set"

where

  "bi' (BP' x t) = {atom x}"

thm lam'_bp'.fv

thm lam'_bp'.eq_iff[no_vars]

thm lam'_bp'.bn

thm lam'_bp'.perm

thm lam'_bp'.induct

thm lam'_bp'.inducts

thm lam'_bp'.distinct

ML {* Sign.of_sort @{theory} (@{typ lam'}, @{sort fs}) *}

thm lam'_bp'.fv[simplified lam'_bp'.supp]

text {* example 2 *}

ML {* val _ = recursive := false  *}

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> atom set"

where

  "f PN = {}"

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

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

thm trm'_pat'.fv

thm trm'_pat'.eq_iff

thm trm'_pat'.bn

thm trm'_pat'.perm

thm trm'_pat'.induct

thm trm'_pat'.distinct

thm trm'_pat'.fv[simplified trm'_pat'.supp]

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> atom set"

where

  "f0 PN0 = {}"

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

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

thm trm0_pat0.fv

thm trm0_pat0.eq_iff

thm trm0_pat0.bn

thm trm0_pat0.perm

thm trm0_pat0.induct

thm trm0_pat0.distinct

thm trm0_pat0.fv[simplified trm0_pat0.supp,no_vars]

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

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

thm trm1_bp1.fv

thm trm1_bp1.eq_iff

thm trm1_bp1.bn

thm trm1_bp1.perm

thm trm1_bp1.induct

thm trm1_bp1.distinct

thm trm1_bp1.fv[simplified trm1_bp1.supp]

text {* 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)"

thm trm3_rassigns3.fv

thm trm3_rassigns3.eq_iff

thm trm3_rassigns3.bn

thm trm3_rassigns3.perm

thm trm3_rassigns3.induct

thm trm3_rassigns3.distinct

thm trm3_rassigns3.fv[simplified trm3_rassigns3.supp]

(* 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)"

thm trm5_lts.fv

thm trm5_lts.eq_iff

thm trm5_lts.bn

thm trm5_lts.perm

thm trm5_lts.induct

thm trm5_lts.distinct

thm trm5_lts.fv[simplified trm5_lts.supp]

(* example form Leroy 96 about modules; OTT *)

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}"

thm mexp_body_defn_sexp_sbody_spec_tyty_path_trmtrm.fv

thm mexp_body_defn_sexp_sbody_spec_tyty_path_trmtrm.eq_iff

thm mexp_body_defn_sexp_sbody_spec_tyty_path_trmtrm.bn

thm mexp_body_defn_sexp_sbody_spec_tyty_path_trmtrm.perm

thm mexp_body_defn_sexp_sbody_spec_tyty_path_trmtrm.induct

thm mexp_body_defn_sexp_sbody_spec_tyty_path_trmtrm.inducts

thm mexp_body_defn_sexp_sbody_spec_tyty_path_trmtrm.distinct

thm mexp_body_defn_sexp_sbody_spec_tyty_path_trmtrm.supp

thm mexp_body_defn_sexp_sbody_spec_tyty_path_trmtrm.fv[simplified mexp_body_defn_sexp_sbody_spec_tyty_path_trmtrm.supp]

(* example 3 from Peter Sewell's bestiary *)

nominal_datatype exp =

  VarP "name"

| AppP "exp" "exp"

| LamP x::"name" e::"exp" bind x in e

| LetP x::"name" p::"pat3" e1::"exp" e2::"exp" bind x in e2, bind "bp'' p" in e1

and pat3 =

  PVar "name"

| PUnit

| PPair "pat3" "pat3"

binder

  bp'' :: "pat3 \<Rightarrow> atom set"

where

  "bp'' (PVar x) = {atom x}"

| "bp'' (PUnit) = {}"

| "bp'' (PPair p1 p2) = bp'' p1 \<union> bp'' p2"

thm exp_pat3.fv

thm exp_pat3.eq_iff

thm exp_pat3.bn

thm exp_pat3.perm

thm exp_pat3.induct

thm exp_pat3.distinct

thm exp_pat3.fv[simplified exp_pat3.supp]

(* example 6 from Peter Sewell's bestiary *)

nominal_datatype exp6 =

  EVar name

| EPair exp6 exp6

| ELetRec x::name p::pat6 e1::exp6 e2::exp6 bind x in e1, bind x in e2, bind "bp6 p" in e1

and pat6 =

  PVar' name

| PUnit'

| PPair' pat6 pat6

binder

  bp6 :: "pat6 \<Rightarrow> atom set"

where

  "bp6 (PVar' x) = {atom x}"

| "bp6 (PUnit') = {}"

| "bp6 (PPair' p1 p2) = bp6 p1 \<union> bp6 p2"

thm exp6_pat6.fv

thm exp6_pat6.eq_iff

thm exp6_pat6.bn

thm exp6_pat6.perm

thm exp6_pat6.induct

thm exp6_pat6.distinct

(* THE REST ARE NOT SUPPOSED TO WORK YET *)

(* example 7 from Peter Sewell's bestiary *)

(* dest_Const raised

nominal_datatype exp7 =

  EVar' name

| EUnit'

| EPair' exp7 exp7

| ELetRec' l::lrbs e::exp7 bind "b7s l" in e, bind "b7s l" in l

and lrb =

  Assign' name exp7

and lrbs =

  Single' lrb

| More' lrb lrbs

binder

  b7 :: "lrb \<Rightarrow> atom set" and

  b7s :: "lrbs \<Rightarrow> atom set"

where

  "b7 (Assign x e) = {atom x}"

| "b7s (Single a) = b7 a"

| "b7s (More a as) = (b7 a) \<union> (b7s as)"

thm alpha_exp7_raw_alpha_lrb_raw_alpha_lrbs_raw.intros

*)

(* example 8 from Peter Sewell's bestiary *)

(*

*** fv_bn: recursive argument, but wrong datatype.

*** At command "nominal_datatype".

nominal_datatype exp8 =

  EVar' name

| EUnit'

| EPair' exp8 exp8

| ELetRec' l::lrbs8 e::exp8 bind "b_lrbs8 l" in e, bind "b_lrbs8 l" in l

and fnclause =

  K' x::name p::pat8 e::exp8 bind "b_pat p" in e

and fnclauses =

  S' fnclause

| ORs' fnclause fnclauses

and lrb8 =

  Clause' fnclauses

and lrbs8 =

  Single' lrb8

| More' lrb8 lrbs8

and pat8 =

  PVar'' name

| PUnit''

| PPair'' pat8 pat8

binder

  b_lrbs8 :: "lrbs8 \<Rightarrow> atom set" and

  b_pat :: "pat8 \<Rightarrow> atom set" and

  b_fnclauses :: "fnclauses \<Rightarrow> atom set" and

  b_fnclause :: "fnclause \<Rightarrow> atom set" and

  b_lrb8 :: "lrb8 \<Rightarrow> atom set"

where

  "b_lrbs8 (Single' l) = b_lrb8 l"

| "b_lrbs8 (More' l ls) = b_lrb8 l \<union> b_lrbs8 ls"

| "b_pat (PVar'' x) = {atom x}"

| "b_pat (PUnit'') = {}"

| "b_pat (PPair'' p1 p2) = b_pat p1 \<union> b_pat p2"

| "b_fnclauses (S' fc) = (b_fnclause fc)"

| "b_fnclauses (ORs' fc fcs) = (b_fnclause fc) \<union> (b_fnclauses fcs)"

| "b_lrb8 (Clause' fcs) = (b_fnclauses fcs)"

| "b_fnclause (K' x pat exp8) = {atom x}"

thm alpha_exp8_raw_alpha_fnclause_raw_alpha_fnclauses_raw_alpha_lrb8_raw_alpha_lrbs8_raw_alpha_pat8_raw.intros

*)

(* example 4 from Terms.thy *)

(* fv_eqvt does not work, we need to repaire defined permute functions

   defined fv and defined alpha... *)

(* lists-datastructure does not work yet

nominal_datatype trm4 =

  Vr4 "name"

| Ap4 "trm4" "trm4 list"

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

thm alpha_trm4_raw_alpha_trm4_raw_list.intros[no_vars]

thm fv_trm4_raw_fv_trm4_raw_list.simps[no_vars]

*)

(* core haskell *)

atom_decl var

atom_decl tvar

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

(* list-data-structure

nominal_datatype tkind =

  KStar

| KFun "tkind" "tkind"

and ckind =

  CKEq "ty" "ty"

and ty =

  TVar "tvar"

| TC "string"

| TApp "ty" "ty"

| TFun "string" "ty list"

| TAll tv::"tvar" "tkind" T::"ty"  bind tv in T

| TEq "ty" "ty" "ty"

and co =

  CC "string"

| CApp "co" "co"

| CFun "string" "co list"

| CAll tv::"tvar" "ckind" C::"co"  bind tv in C

| CEq "co" "co" "co"

| CSym "co"

| CCir "co" "co"

| CLeft "co"

| CRight "co"

| CSim "co"

| CRightc "co"

| CLeftc "co"

| CCoe "co" "co"

abbreviation

  "atoms A \<equiv> atom ` A"

nominal_datatype trm =

  Var "var"

| C "string"

| LAM tv::"tvar" "kind" t::"trm"   bind tv 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)))"

*)

(* example 8 from Terms.thy *)

(* Binding in a term under a bn, needs to fail *)

(*

nominal_datatype foo8 =

  Foo0 "name"

| Foo1 b::"bar8" f::"foo8" bind "bv8 b" in f --"check fo error if this is called 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}"

*)

(* Type schemes with separate datatypes *)

nominal_datatype T =

  TVar "name"

| TFun "T" "T"

(* PROBLEM: 

*** exception Datatype raised 

*** (line 218 of "/usr/local/src/Isabelle_16-Mar-2010/src/HOL/Tools/Datatype/datatype_aux.ML")

*** At command "nominal_datatype".

nominal_datatype TyS =

  TAll xs::"name list" ty::"T" bind xs in ty

*)

(* example 9 from Peter Sewell's bestiary *)

(* run out of steam at the moment *)

end