theory CoreHaskell

imports "../Nominal2"

begin

(* Core Haskell *)

atom_decl var

atom_decl cvar

atom_decl tvar

nominal_datatype core_haskell: 

 tkind =

  KStar

| KFun "tkind" "tkind"

and ckind =

  CKEq "ty" "ty"

and ty =

  TVar "tvar"

| TC "string"

| TApp "ty" "ty"

| TFun "string" "ty_lst"

| TAll tv::"tvar" "tkind" T::"ty"  bind (set) tv in T

| TEq "ckind" "ty"

and ty_lst =

  TsNil

| TsCons "ty" "ty_lst"

and co =

  CVar "cvar"

| CConst "string"

| CApp "co" "co"

| CFun "string" "co_lst"

| CAll cv::"cvar" "ckind" C::"co"  bind (set) cv in C

| CEq "ckind" "co"

| CRefl "ty"

| CSym "co"

| CCir "co" "co"

| CAt "co" "ty"

| CLeft "co"

| CRight "co"

| CSim "co" "co"

| CRightc "co"

| CLeftc "co"

| CCoe "co" "co"

and co_lst =

  CsNil

| CsCons "co" "co_lst"

and trm =

  Var "var"

| K "string"

| LAMT tv::"tvar" "tkind" t::"trm" bind (set) tv in t

| LAMC cv::"cvar" "ckind" t::"trm" bind (set) cv in t

| AppT "trm" "ty"

| AppC "trm" "co"

| Lam v::"var" "ty" t::"trm"       bind (set) v in t

| App "trm" "trm"

| Let x::"var" "ty" "trm" t::"trm" bind (set) x in t

| Case "trm" "assoc_lst"

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

and assoc_lst =

  ANil

| ACons p::"pat" t::"trm" "assoc_lst" bind "bv p" in t

and pat =

  Kpat "string" "tvars" "cvars" "vars"

and vars =

  VsNil

| VsCons "var" "ty" "vars"

and tvars =

  TvsNil

| TvsCons "tvar" "tkind" "tvars"

and cvars =

  CvsNil

| CvsCons "cvar" "ckind" "cvars"

binder

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

and bv_vs :: "vars \<Rightarrow> atom list"

and bv_tvs :: "tvars \<Rightarrow> atom list"

and bv_cvs :: "cvars \<Rightarrow> atom list"

where

  "bv (Kpat s tvts tvcs vs) = append (bv_tvs tvts) (append (bv_cvs tvcs) (bv_vs vs))"

| "bv_vs VsNil = []"

| "bv_vs (VsCons v k t) = (atom v) # bv_vs t"

| "bv_tvs TvsNil = []"

| "bv_tvs (TvsCons v k t) = (atom v) # bv_tvs t"

| "bv_cvs CvsNil = []"

| "bv_cvs (CvsCons v k t) = (atom v) # bv_cvs t"

(* generated theorems *)

thm core_haskell.permute_bn

thm core_haskell.perm_bn_alpha

thm core_haskell.perm_bn_simps

thm core_haskell.bn_finite

thm core_haskell.distinct

thm core_haskell.induct

thm core_haskell.exhaust

thm core_haskell.fv_defs

thm core_haskell.bn_defs

thm core_haskell.perm_simps

thm core_haskell.eq_iff

thm core_haskell.fv_bn_eqvt

thm core_haskell.size_eqvt

thm core_haskell.supp

lemma strong_induction_principle:

  assumes a01: "\<And>b. P1 b KStar"

  and a02: "\<And>tkind1 tkind2 b. \<lbrakk>\<And>c. P1 c tkind1; \<And>c. P1 c tkind2\<rbrakk> \<Longrightarrow> P1 b (KFun tkind1 tkind2)"

  and a03: "\<And>ty1 ty2 b. \<lbrakk>\<And>c. P3 c ty1; \<And>c. P3 c ty2\<rbrakk> \<Longrightarrow> P2 b (CKEq ty1 ty2)"

  and a04: "\<And>tvar b. P3 b (TVar tvar)"

  and a05: "\<And>string b. P3 b (TC string)"

  and a06: "\<And>ty1 ty2 b. \<lbrakk>\<And>c. P3 c ty1; \<And>c. P3 c ty2\<rbrakk> \<Longrightarrow> P3 b (TApp ty1 ty2)"

  and a07: "\<And>string ty_lst b. \<lbrakk>\<And>c. P4 c ty_lst\<rbrakk> \<Longrightarrow> P3 b (TFun string ty_lst)"

  and a08: "\<And>tvar tkind ty b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P3 c ty; atom tvar \<sharp> b\<rbrakk>

    \<Longrightarrow> P3 b (TAll tvar tkind ty)"

  and a09: "\<And>ck ty b. \<lbrakk>\<And>c. P2 c ck; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P3 b (TEq ck ty)"

  and a10: "\<And>b. P4 b TsNil"

  and a11: "\<And>ty ty_lst b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P4 c ty_lst\<rbrakk> \<Longrightarrow> P4 b (TsCons ty ty_lst)"

  and a12: "\<And>string b. P5 b (CVar string)"

  and a12a:"\<And>str b. P5 b (CConst str)"

  and a13: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CApp co1 co2)"

  and a14: "\<And>string co_lst b. \<lbrakk>\<And>c. P6 c co_lst\<rbrakk> \<Longrightarrow> P5 b (CFun string co_lst)"

  and a15: "\<And>tvar ckind co b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P5 c co; atom tvar \<sharp> b\<rbrakk>

    \<Longrightarrow> P5 b (CAll tvar ckind co)"

  and a16: "\<And>ck co b. \<lbrakk>\<And>c. P2 c ck; \<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CEq ck co)"

  and a17: "\<And>ty b. \<lbrakk>\<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P5 b (CRefl ty)"

  and a17a: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CSym co)"

  and a18: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CCir co1 co2)"

  and a18a:"\<And>co ty b. \<lbrakk>\<And>c. P5 c co; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P5 b (CAt co ty)"

  and a19: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CLeft co)"

  and a20: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CRight co)"

  and a21: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CSim co1 co2)"

  and a22: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CRightc co)"

  and a23: "\<And>co b. \<lbrakk>\<And>c. P5 c co\<rbrakk> \<Longrightarrow> P5 b (CLeftc co)"

  and a24: "\<And>co1 co2 b. \<lbrakk>\<And>c. P5 c co1; \<And>c. P5 c co2\<rbrakk> \<Longrightarrow> P5 b (CCoe co1 co2)"

  and a25: "\<And>b. P6 b CsNil"

  and a26: "\<And>co co_lst b. \<lbrakk>\<And>c. P5 c co; \<And>c. P6 c co_lst\<rbrakk> \<Longrightarrow> P6 b (CsCons co co_lst)"

  and a27: "\<And>var b. P7 b (Var var)"

  and a28: "\<And>string b. P7 b (K string)"

  and a29: "\<And>tvar tkind trm b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P7 c trm; atom tvar \<sharp> b\<rbrakk>

    \<Longrightarrow> P7 b (LAMT tvar tkind trm)"

  and a30: "\<And>tvar ckind trm b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P7 c trm; atom tvar \<sharp> b\<rbrakk>

    \<Longrightarrow> P7 b (LAMC tvar ckind trm)"

  and a31: "\<And>trm ty b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P7 b (AppT trm ty)"

  and a31a:"\<And>trm co b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P5 c co\<rbrakk> \<Longrightarrow> P7 b (AppC trm co)"

  and a32: "\<And>var ty trm b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P7 c trm; atom var \<sharp> b\<rbrakk> \<Longrightarrow> P7 b (Lam var ty trm)"

  and a33: "\<And>trm1 trm2 b. \<lbrakk>\<And>c. P7 c trm1; \<And>c. P7 c trm2\<rbrakk> \<Longrightarrow> P7 b (App trm1 trm2)"

  and a34: "\<And>var ty trm1 trm2 b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P7 c trm1; \<And>c. P7 c trm2; atom var \<sharp> b\<rbrakk>

    \<Longrightarrow> P7 b (Let var ty trm1 trm2)"

  and a35: "\<And>trm assoc_lst b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P8 c assoc_lst\<rbrakk> \<Longrightarrow> P7 b (Case trm assoc_lst)"

  and a36: "\<And>trm ty b. \<lbrakk>\<And>c. P7 c trm; \<And>c. P3 c ty\<rbrakk> \<Longrightarrow> P7 b (Cast trm ty)"

  and a37: "\<And>b. P8 b ANil"

  and a38: "\<And>pt trm assoc_lst b. \<lbrakk>\<And>c. P9 c pt; \<And>c. P7 c trm; \<And>c. P8 c assoc_lst; set (bv (pt)) \<sharp>* b\<rbrakk>

    \<Longrightarrow> P8 b (ACons pt trm assoc_lst)"

  and a39: "\<And>string tvars cvars vars b. \<lbrakk>\<And>c. P11 c tvars; \<And>c. P12 c cvars; \<And>c. P10 c vars\<rbrakk>

    \<Longrightarrow> P9 b (Kpat string tvars cvars vars)"

  and a40: "\<And>b. P10 b VsNil"

  and a41: "\<And>var ty vars b. \<lbrakk>\<And>c. P3 c ty; \<And>c. P10 c vars\<rbrakk> \<Longrightarrow> P10 b (VsCons var ty vars)"

  and a42: "\<And>b. P11 b TvsNil"

  and a43: "\<And>tvar tkind tvars b. \<lbrakk>\<And>c. P1 c tkind; \<And>c. P11 c tvars\<rbrakk>

    \<Longrightarrow> P11 b (TvsCons tvar tkind tvars)"

  and a44: "\<And>b. P12 b CvsNil"

  and a45: "\<And>tvar ckind cvars b. \<lbrakk>\<And>c. P2 c ckind; \<And>c. P12 c cvars\<rbrakk>

    \<Longrightarrow> P12 b (CvsCons tvar ckind cvars)"

end