theory Test_compat
imports "Parser" "../Attic/Prove"
begin

text {* 
  example 1 
  
  single let binding
*}

nominal_datatype lam =
  VAR "name"
| APP "lam" "lam"
| 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 alpha_lam_raw_alpha_bp_raw.intros[no_vars]
thm fv_lam_raw_fv_bp_raw.simps[no_vars]

abbreviation "VAR \<equiv> VAR_raw"
abbreviation "APP \<equiv> APP_raw"
abbreviation "LET \<equiv> LET_raw"
abbreviation "BP \<equiv> BP_raw"
abbreviation "bi \<equiv> bi_raw"

(* non-recursive case *)
primrec
    fv_lam :: "lam_raw \<Rightarrow> atom set"
and fv_compat :: "bp_raw \<Rightarrow> atom set"
where
  "fv_lam (VAR name) = {atom name}"
| "fv_lam (APP lam1 lam2) = fv_lam lam1 \<union> fv_lam lam2"
| "fv_lam (LET bp lam) = (fv_compat bp) \<union> (fv_lam lam - bi bp)"
| "fv_compat (BP name lam) = fv_lam lam"

primrec
    fv_bp :: "bp_raw \<Rightarrow> atom set"
where
  "fv_bp (BP name lam) = {atom name} \<union> fv_lam lam"


inductive
  alpha_lam :: "lam_raw \<Rightarrow> lam_raw \<Rightarrow> bool" and
  alpha_bp  :: "bp_raw \<Rightarrow> bp_raw \<Rightarrow> bool" and
  compat_bp :: "bp_raw \<Rightarrow> perm \<Rightarrow> bp_raw \<Rightarrow> bool"
where
  "x = y \<Longrightarrow> alpha_lam (VAR x) (VAR y)"
| "alpha_lam l1 s1 \<and> alpha_lam l2 s2 \<Longrightarrow> alpha_lam (APP l1 l2) (APP s1 s2)"
| "\<exists>pi. (bi bp, lam) \<approx>gen alpha_lam fv_lam_raw pi (bi bp', lam') \<and> compat_bp bp pi bp' 
   \<Longrightarrow> alpha_lam (LET bp lam) (LET bp' lam')"
| "alpha_lam lam lam' \<and> name = name' \<Longrightarrow> alpha_bp (BP name lam) (BP name' lam')"
| "alpha_lam t t' \<and> pi \<bullet> x = x' \<Longrightarrow> compat_bp (BP x t) pi (BP x' t')" 

lemma test1:
  shows "alpha_lam (LET (BP x (VAR x)) (VAR x))
                   (LET (BP y (VAR x)) (VAR y))"
apply(rule alpha_lam_alpha_bp_compat_bp.intros)
apply(rule_tac x="(x \<leftrightarrow> y)" in exI)
apply(simp add: alpha_gen fresh_star_def)
apply(simp add: alpha_lam_alpha_bp_compat_bp.intros(1))
apply(rule alpha_lam_alpha_bp_compat_bp.intros)
apply(simp add: alpha_lam_alpha_bp_compat_bp.intros(1))
done

lemma test2:
  assumes asm: "distinct [x, y]"
  shows "\<not> alpha_lam (LET (BP x (VAR x)) (VAR x))
                     (LET (BP y (VAR y)) (VAR y))"
using asm
apply(clarify)
apply(erule alpha_lam.cases)
apply(simp_all)
apply(erule exE)
apply(clarify)
apply(simp add: alpha_gen fresh_star_def)
apply(erule alpha_lam.cases)
apply(simp_all)
apply(clarify)
apply(erule compat_bp.cases)
apply(simp_all)
apply(clarify)
apply(erule alpha_lam.cases)
apply(simp_all)
done

(* recursive case where we have also bind "bi bp" in bp *)

inductive
  Alpha_lam :: "lam_raw \<Rightarrow> lam_raw \<Rightarrow> bool" and
  Alpha_bp  :: "bp_raw \<Rightarrow> bp_raw \<Rightarrow> bool" and
  Compat_bp :: "bp_raw \<Rightarrow> perm \<Rightarrow> bp_raw \<Rightarrow> bool"
where
  "x = y \<Longrightarrow> Alpha_lam (VAR x) (VAR y)"
| "Alpha_lam l1 s1 \<and> Alpha_lam l2 s2 \<Longrightarrow> Alpha_lam (APP l1 l2) (APP s1 s2)"
| "\<exists>pi. (bi bp, lam) \<approx>gen Alpha_lam fv_lam_raw pi (bi bp', lam') \<and> Compat_bp bp pi bp' 
   \<Longrightarrow> Alpha_lam (LET bp lam) (LET bp' lam')"
| "Alpha_lam lam lam' \<and> name = name' \<Longrightarrow> Alpha_bp (BP name lam) (BP name' lam')"
| "Alpha_lam (pi \<bullet> t) t' \<and> pi \<bullet> x = x' \<Longrightarrow> Compat_bp (BP x t) pi (BP x' t')"

lemma Test1:
  assumes "distinct [x, y]"
  shows "Alpha_lam (LET (BP x (VAR x)) (VAR x))
                   (LET (BP y (VAR y)) (VAR y))"
apply(rule Alpha_lam_Alpha_bp_Compat_bp.intros)
apply(rule_tac x="(x \<leftrightarrow> y)" in exI)
apply(simp add: alpha_gen fresh_star_def)
apply(simp add: Alpha_lam_Alpha_bp_Compat_bp.intros(1))
apply(rule Alpha_lam_Alpha_bp_Compat_bp.intros)
apply(simp add: Alpha_lam_Alpha_bp_Compat_bp.intros(1))
done

lemma Test2:
  assumes asm: "distinct [x, y]"
  shows "\<not> Alpha_lam (LET (BP x (VAR x)) (VAR x))
                     (LET (BP y (VAR x)) (VAR y))"
using asm
apply(clarify)
apply(erule Alpha_lam.cases)
apply(simp_all)
apply(erule exE)
apply(clarify)
apply(simp add: alpha_gen fresh_star_def)
apply(erule Alpha_lam.cases)
apply(simp_all)
apply(clarify)
apply(erule Compat_bp.cases)
apply(simp_all)
apply(clarify)
apply(erule Alpha_lam.cases)
apply(simp_all)
done


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> atom set"
where 
  "f PN = {}"
| "f (PS x) = {atom x}"
| "f (PD x y) = {atom x} \<union> {atom y}"

thm alpha_trm'_raw_alpha_pat'_raw.intros[no_vars]

abbreviation "Var \<equiv> Var_raw"
abbreviation "App \<equiv> App_raw"
abbreviation "Lam \<equiv> Lam_raw"
abbreviation "Lett \<equiv> Let_raw"
abbreviation "PN \<equiv> PN_raw"
abbreviation "PS \<equiv> PS_raw"
abbreviation "PD \<equiv> PD_raw"
abbreviation "f \<equiv> f_raw"

(* not_yet_done *)
inductive 
  alpha_trm' :: "trm'_raw \<Rightarrow> trm'_raw \<Rightarrow> bool" and
  alpha_pat'  :: "pat'_raw \<Rightarrow> pat'_raw \<Rightarrow> bool" and
  compat_pat' :: "pat'_raw \<Rightarrow> perm \<Rightarrow> pat'_raw \<Rightarrow> bool"
where
  "name = name' \<Longrightarrow> alpha_trm' (Var name) (Var name')"
| "alpha_trm' t2 t2' \<and> alpha_trm' t1 t1' \<Longrightarrow> alpha_trm' (App t1 t2) (App t1' t2')"
| "\<exists>pi. ({atom x}, t) \<approx>gen alpha_trm' fv_trm'_raw pi ({atom x'}, t') \<Longrightarrow> alpha_trm' (Lam x t) (Lam x' t')"
| "\<exists>pi. (f p, t) \<approx>gen alpha_trm' fv_trm'_raw pi (f p', t') \<and> alpha_trm' s s' \<and>
        compat_pat' p pi p' \<Longrightarrow> alpha_trm' (Lett p s t) (Lett p' s' t')"
| "alpha_pat' PN PN"
| "name = name' \<Longrightarrow> alpha_pat' (PS name) (PS name')"
| "name2 = name2' \<and> name1 = name1' \<Longrightarrow> alpha_pat' (PD name1 name2) (PD name1' name2')"
| "compat_pat' PN pi PN"
| "pi \<bullet> x = x' \<Longrightarrow> compat_pat' (PS x) pi (PS x')"
| "pi \<bullet> p1 = p1' \<and> pi \<bullet> p2 = p2' \<Longrightarrow> compat_pat' (PD p1 p2) pi (PD p1' p2')"

lemma
  assumes a: "distinct [x, y, z, u]"
  shows "alpha_trm' (Lett (PD x u) t (App (Var x) (Var y)))
                    (Lett (PD z u) t (App (Var z) (Var y)))"
using a
apply -
apply(rule alpha_trm'_alpha_pat'_compat_pat'.intros)
apply(auto simp add: alpha_gen)
apply(rule_tac x="(x \<leftrightarrow> z)" in exI)
apply(auto simp add: fresh_star_def)
defer
apply(rule alpha_trm'_alpha_pat'_compat_pat'.intros)
apply(simp add: alpha_trm'_alpha_pat'_compat_pat'.intros)
defer
apply(rule alpha_trm'_alpha_pat'_compat_pat'.intros)
apply(simp)
(* they can be proved *)
oops

lemma
  assumes a: "distinct [x, y, z, u]"
  shows "\<not> alpha_trm' (Lett (PD x u) t (App (Var x) (Var y)))
                      (Lett (PD z z) t (App (Var z) (Var y)))"
using a
apply(clarify)
apply(erule alpha_trm'.cases)
apply(simp_all)
apply(auto simp add: alpha_gen)
apply(erule alpha_trm'.cases)
apply(simp_all)
apply(clarify)
apply(erule compat_pat'.cases)
apply(simp_all)
apply(clarify)
apply(erule alpha_trm'.cases)
apply(simp_all)
done

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 f0_raw.simps
(*thm trm0_pat0_induct
thm trm0_pat0_perm
thm trm0_pat0_fv
thm trm0_pat0_bn*)

text {* example type schemes *}

(* does not work yet
nominal_datatype t =
  Var "name"
| Fun "t" "t"

nominal_datatype tyS =
  All xs::"name list" ty::"t_raw" bind xs in ty
*)


nominal_datatype t = 
  Var "name" 
| Fun "t" "t"
and  tyS = 
  All xs::"name set" 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::"assign" t::"trm2"    bind "bv2 r" in t
and assign = 
  As "name" "trm2"
binder
  bv2
where
  "bv2 (As x t) = {atom x}"

(* compat should be
compat (As x t) pi (As x' t') == pi o x = x' & alpha t t'
*)


thm fv_trm2_raw_fv_assign_raw.simps[no_vars]
thm alpha_trm2_raw_alpha_assign_raw.intros[no_vars]



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)"


(* compat should be
compat (ANil) pi (PNil) \<equiv> TRue
compat (ACons x t ts) pi (ACons x' t' ts') \<equiv> pi o x = x' \<and> alpha t t' \<and> compat ts pi ts'
*)

(* example 4 from Terms.thy *)

(* fv_eqvt does not work, we need to repaire defined permute functions
   defined fv and defined alpha... *)
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]

(* 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 *)

(* BV is not respectful, needs to fail*)
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 *)

(* BV is not respectful, needs to fail*)
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 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}"

(* example 9 from Terms.thy *)

(* BV is not respectful, needs to fail*)
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 from my PHD *)

atom_decl coname

nominal_datatype phd =
   Ax "name" "coname"
|  Cut n::"coname" t1::"phd" c::"coname" t2::"phd"              bind n in t1, bind c in t2
|  AndR c1::"coname" t1::"phd" c2::"coname" t2::"phd" "coname"  bind c1 in t1, bind c2 in t2
|  AndL1 n::"name" t::"phd" "name"                              bind n in t
|  AndL2 n::"name" t::"phd" "name"                              bind n in t
|  ImpL c::"coname" t1::"phd" n::"name" t2::"phd" "name"        bind c in t1, bind n in t2
|  ImpR c::"coname" n::"name" t::"phd" "coname"                 bind n in t, bind c in t

thm alpha_phd_raw.intros[no_vars]
thm fv_phd_raw.simps[no_vars]


(* 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}"  

(* core haskell *)
print_theorems

atom_decl var
atom_decl tvar


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


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

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)))"

(*
compat (K s ts vs) pi (K s' ts' vs') ==
  s = s' & 

*)


(*thm bv_raw.simps*)

(* 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::"pat" e1::"exp" e2::"exp" bind x in e2, bind "bp p" in e1
and pat =
  PVar "name"
| PUnit
| PPair "pat" "pat"
binder
  bp :: "pat \<Rightarrow> atom set"
where
  "bp (PVar x) = {atom x}"
| "bp (PUnit) = {}"
| "bp (PPair p1 p2) = bp p1 \<union> bp p2"
thm alpha_exp_raw_alpha_pat_raw.intros

(* 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 alpha_exp6_raw_alpha_pat6_raw.intros

(* example 7 from Peter Sewell's bestiary *)
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 *)
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 9 from Peter Sewell's bestiary *)
(* run out of steam at the moment *)

end



