theory Test

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

begin

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

atom_decl name

ML {* val cheat_alpha_eqvt = ref false *}

ML {* val cheat_fv_eqvt = ref false *}

(*

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}"

*)

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"

binder

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

where

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

thm lam_bp_fv

thm lam_bp_inject

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}) *}

term "supp (x :: lam)"

(* maybe should be added to Infinite.thy *)

lemma infinite_Un:

  shows "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T"

  by simp

lemma bi_eqvt:

  shows "(p \<bullet> (bi b)) = bi (p \<bullet> b)"

  by (rule eqvts)

term alpha_bi

lemma alpha_bi:

  shows "alpha_bi pi b b' = True"

apply(induct b rule: lam_bp_inducts(2))

sorry

lemma fv_bi:

  shows "fv_bi b = {}"

apply(induct b rule: lam_bp_inducts(2))

apply(auto)[1]

apply(simp_all)

apply(simp add: lam_bp_fv)

done

lemma supp_fv:

  "supp t = fv_lam t" and 

  "supp b = fv_bp b"

apply(induct t and b rule: lam_bp_inducts)

apply(simp_all add: lam_bp_fv)

(* VAR case *)

apply(simp only: supp_def)

apply(simp only: lam_bp_perm)

apply(simp only: lam_bp_inject)

apply(simp only: supp_def[symmetric])

apply(simp only: supp_at_base)

(* APP case *)

apply(simp only: supp_def)

apply(simp only: lam_bp_perm)

apply(simp only: lam_bp_inject)

apply(simp only: de_Morgan_conj)

apply(simp only: Collect_disj_eq)

apply(simp only: infinite_Un)

apply(simp only: Collect_disj_eq)

(* LAM case *)

apply(rule_tac t="supp (LAM name lam_raw)" and s="supp (Abs {atom name} lam_raw)" in subst)

apply(simp (no_asm) only: supp_def)

apply(simp only: lam_bp_perm)

apply(simp only: permute_ABS)

apply(simp only: lam_bp_inject)

apply(simp only: Abs_eq_iff)

apply(simp only: insert_eqvt atom_eqvt empty_eqvt)

apply(simp only: alpha_gen)

apply(simp only: supp_eqvt[symmetric])

apply(simp only: eqvts)

apply(simp only: supp_Abs)

(* LET case *)

defer

(* BP case *)

apply(simp only: supp_def)

apply(simp only: lam_bp_perm)

apply(simp only: lam_bp_inject)

(*

apply(simp only: de_Morgan_conj)

apply(simp only: Collect_disj_eq)

apply(simp only: infinite_Un)

apply(simp only: Collect_disj_eq)

*)

apply(simp only: supp_def[symmetric])

apply(simp only: supp_at_base)

(*apply(simp)*)

(* LET case *)

apply(rule_tac t="supp (LET bp_raw lam_raw)" and s="supp (Abs_tst bi bp_raw lam_raw)" in subst)

apply(simp (no_asm) only: supp_def)

apply(simp only: lam_bp_perm)

apply(simp only: permute_ABS_tst)

apply(simp only: lam_bp_inject)

apply(simp only: eqvts_raw)

apply(simp only: Abs_tst_eq_iff)

apply(simp only: alpha_bi)

apply(simp only: alpha_gen)

apply(simp only: alpha_tst) 

apply(simp only: supp_eqvt[symmetric])

apply(simp only: eqvts)

apply(simp add: supp_Abs_tst)

apply(simp add: fv_bi)

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

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

thm trm'_pat'_fv

thm trm'_pat'_inject

thm trm'_pat'_bn

thm trm'_pat'_perm

thm trm'_pat'_induct

thm trm'_pat'_distinct

(* compat should be

compat (PN) pi (PN) == True

compat (PS x) pi (PS x') == pi o x = x'

compat (PD p1 p2) pi (PD p1' p2') == compat p1 pi p1' & compat p2 pi p2'

*)

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_inject

thm trm0_pat0_bn

thm trm0_pat0_perm

thm trm0_pat0_induct

thm trm0_pat0_distinct

text {* example type schemes *}

nominal_datatype t =

  VarTS "name"

| FunTS "t" "t"

and  tyS =

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

thm t_tyS_fv

thm t_tyS_inject

thm t_tyS_bn

thm t_tyS_perm

thm t_tyS_induct

thm t_tyS_distinct

(* 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_inject

thm trm1_bp1_bn

thm trm1_bp1_perm

thm trm1_bp1_induct

thm trm1_bp1_distinct

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_inject

thm trm3_rassigns3_bn

thm trm3_rassigns3_perm

thm trm3_rassigns3_induct

thm trm3_rassigns3_distinct

(* 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 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_inject

thm trm5_lts_bn

thm trm5_lts_perm

thm trm5_lts_induct

thm trm5_lts_distinct

(* 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 phd_fv

thm phd_inject

thm phd_bn

thm phd_perm

thm phd_induct

thm phd_distinct

(* 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_inject

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

(* 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_inject

thm exp_pat3_bn

thm exp_pat3_perm

thm exp_pat3_induct

thm exp_pat3_distinct

(* 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_inject

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 *)

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 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]

(* core haskell *)

atom_decl var

atom_decl tvar

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