theory Test

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

begin

text {* weirdo example from Peter Sewell's bestiary *}

nominal_datatype weird =

  WBind x::"name" y::"name" p1::"weird" p2::"weird" p3::"weird"

    bind x in p1, bind x in p2, bind y in p2, bind y in p3

| WV "name"

| WP "weird" "weird"

thm permute_weird_raw.simps[no_vars]

thm alpha_weird_raw.intros[no_vars]

thm fv_weird_raw.simps[no_vars]

thm eqvts

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding weird_inj}, []), (build_alpha_inj @{thms alpha_weird_raw.intros} @{thms weird_raw.distinct weird_raw.inject} @{thms alpha_weird_raw.cases} ctxt)) ctxt)) *}

thm weird_inj

(*local_setup {*

(fn ctxt => snd (Local_Theory.note ((@{binding alpha_eqvt}, []),

build_alpha_eqvts [@{term alpha_weird_raw}] [@{term "permute :: perm \<Rightarrow> weird_raw \<Rightarrow> weird_raw"}] @{thms permute_weird_raw.simps weird_inj} @{thm alpha_weird_raw.induct} ctxt) ctxt)) *}

(*prove {* (snd o snd) (build_alpha_refl_gl [@{term alpha_weird_raw}] ("x","y","z")) *}

apply (tactic {* transp_tac @{context} @{thm alpha_weird_raw.induct} @{thms weird_inj}  @{thms weird_raw.inject} @{thms weird_raw.distinct} @{thms alpha_weird_raw.cases} @{thms alpha_eqvt} 1 *})

*)

lemma "alpha_weird_raw x y \<longrightarrow> (\<forall>z. alpha_weird_raw y z \<longrightarrow> alpha_weird_raw x z)"

apply (rule impI)

apply (erule alpha_weird_raw.induct)

apply (simp_all add: weird_inj)

defer

apply (rule allI)

apply (rule impI)

apply (erule alpha_weird_raw.cases)

apply (simp_all add: weird_inj)

apply (rule allI)

apply (rule impI)

apply (erule alpha_weird_raw.cases)

apply (simp_all add: weird_inj)

apply (erule conjE)+

apply (erule exE)+

apply (erule conjE)+

apply (erule exE)+

apply (rule conjI)

apply (rule_tac x="pica + pic" in exI)

apply (erule alpha_gen_compose_trans)

apply assumption

apply (simp add: alpha_eqvt)

apply (rule_tac x="pia + pib" in exI)

apply (rule_tac x="piaa + piba" in exI)

apply (rule conjI)

apply (erule alpha_gen_compose_trans)

apply assumption

apply (simp add: alpha_eqvt)

apply (rule conjI)

defer

apply (rule_tac x="pid + pi" in exI)

apply (erule alpha_gen_compose_trans)

apply assumption

apply (simp add: alpha_eqvt)

sorry

lemma "alpha_weird_raw x y \<Longrightarrow> alpha_weird_raw y x"

apply (erule alpha_weird_raw.induct)

apply (simp_all add: weird_inj)

apply (erule conjE)+

apply (erule exE)+

apply (erule conjE)+

apply (erule exE)+

apply (rule conjI)

apply (rule_tac x="- pic" in exI)

apply (erule alpha_gen_compose_sym)

apply (simp_all add: alpha_eqvt)

apply (rule_tac x="- pia" in exI)

apply (rule_tac x="- pib" in exI)

apply (simp add: minus_add[symmetric])

apply (rule conjI)

apply (erule alpha_gen_compose_sym)

apply (simp_all add: alpha_eqvt)

apply (rule conjI)

apply (simp add: supp_minus_perm Int_commute)

apply (rule_tac x="- pi" in exI)

apply (erule alpha_gen_compose_sym)

apply (simp_all add: alpha_eqvt)

done

abbreviation "WBind \<equiv> WBind_raw"

abbreviation "WP \<equiv> WP_raw"

abbreviation "WV \<equiv> WV_raw"

lemma test:

  assumes a: "distinct [x, y, z]"

  shows "alpha_weird_raw (WBind x y (WP (WV x) (WV z)) (WP (WV x) (WV y)) (WP (WV z) (WV y)))  

                         (WBind y x (WP (WV y) (WV z)) (WP (WV y) (WV x)) (WP (WV z) (WV x)))"

apply(rule_tac alpha_weird_raw.intros)

unfolding alpha_gen

using a

apply(auto)

apply(rule_tac x="(x \<leftrightarrow> y)" in exI)

apply(auto)

apply(simp add: fresh_star_def flip_def fresh_def supp_swap)

apply(rule alpha_weird_raw.intros)

apply(simp add: alpha_weird_raw.intros(2))

apply(rule_tac x="(x \<leftrightarrow> y)" in exI)

apply(rule_tac x="0" in exI)

apply(simp add: fresh_star_def)

apply(auto)

apply(rule alpha_weird_raw.intros)

apply(simp add: alpha_weird_raw.intros(2))

apply(simp add: flip_def supp_swap supp_perm)

apply(rule_tac x="(x \<leftrightarrow> y)" in exI)

apply(simp)

apply(auto)

apply(simp add: flip_def fresh_def supp_swap)

apply(rule alpha_weird_raw.intros)

apply(simp add: alpha_weird_raw.intros(2))

done*)

text {* example 1 *}

(* ML {* set show_hyps *} *)

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}"

typ lam_raw

term VAR_raw

term APP_raw

term LET_raw

term Test.BP_raw

thm bi_raw.simps

thm permute_lam_raw_permute_bp_raw.simps

thm alpha_lam_raw_alpha_bp_raw.intros[no_vars]

thm fv_lam_raw_fv_bp_raw.simps[no_vars]

(*thm lam_bp_induct

thm lam_bp_perm

thm lam_bp_fv

thm lam_bp_bn

thm lam_bp_inject

thm lam_bp_distinct*)

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, atom y}"

thm alpha_trm'_raw_alpha_pat'_raw.intros[no_vars]

thm fv_trm'_raw_fv_pat'_raw.simps[no_vars]

thm f_raw.simps

(*thm trm'_pat'_induct

thm trm'_pat'_perm

thm trm'_pat'_fv

thm trm'_pat'_bn

thm trm'_pat'_inject*)

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::"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 *)

(* 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)))"

(*thm bv_raw.simps*)

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