(* Title: HOL/ex/mt.thy
ID: $Id$
Author: Jacob Frost, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Based upon the article
Robin Milner and Mads Tofte,
Co-induction in Relational Semantics,
Theoretical Computer Science 87 (1991), pages 209-220.
Written up as
Jacob Frost, A Case Study of Co_induction in Isabelle/HOL
Report 308, Computer Lab, University of Cambridge (1993).
*)
MT = Inductive +
types
Const
ExVar
Ex
TyConst
Ty
Clos
Val
ValEnv
TyEnv
arities
Const :: type
ExVar :: type
Ex :: type
TyConst :: type
Ty :: type
Clos :: type
Val :: type
ValEnv :: type
TyEnv :: type
consts
c_app :: "[Const, Const] => Const"
e_const :: "Const => Ex"
e_var :: "ExVar => Ex"
e_fn :: "[ExVar, Ex] => Ex" ("fn _ => _" [0,51] 1000)
e_fix :: "[ExVar, ExVar, Ex] => Ex" ("fix _ ( _ ) = _" [0,51,51] 1000)
e_app :: "[Ex, Ex] => Ex" ("_ @ _" [51,51] 1000)
e_const_fst :: "Ex => Const"
t_const :: "TyConst => Ty"
t_fun :: "[Ty, Ty] => Ty" ("_ -> _" [51,51] 1000)
v_const :: "Const => Val"
v_clos :: "Clos => Val"
ve_emp :: ValEnv
ve_owr :: "[ValEnv, ExVar, Val] => ValEnv" ("_ + { _ |-> _ }" [36,0,0] 50)
ve_dom :: "ValEnv => ExVar set"
ve_app :: "[ValEnv, ExVar] => Val"
clos_mk :: "[ExVar, Ex, ValEnv] => Clos" ("<| _ , _ , _ |>" [0,0,0] 1000)
te_emp :: TyEnv
te_owr :: "[TyEnv, ExVar, Ty] => TyEnv" ("_ + { _ |=> _ }" [36,0,0] 50)
te_app :: "[TyEnv, ExVar] => Ty"
te_dom :: "TyEnv => ExVar set"
eval_fun :: "((ValEnv * Ex) * Val) set => ((ValEnv * Ex) * Val) set"
eval_rel :: "((ValEnv * Ex) * Val) set"
eval :: "[ValEnv, Ex, Val] => bool" ("_ |- _ ---> _" [36,0,36] 50)
elab_fun :: "((TyEnv * Ex) * Ty) set => ((TyEnv * Ex) * Ty) set"
elab_rel :: "((TyEnv * Ex) * Ty) set"
elab :: "[TyEnv, Ex, Ty] => bool" ("_ |- _ ===> _" [36,0,36] 50)
isof :: "[Const, Ty] => bool" ("_ isof _" [36,36] 50)
isof_env :: "[ValEnv,TyEnv] => bool" ("_ isofenv _")
hasty_fun :: "(Val * Ty) set => (Val * Ty) set"
hasty_rel :: "(Val * Ty) set"
hasty :: "[Val, Ty] => bool" ("_ hasty _" [36,36] 50)
hasty_env :: "[ValEnv,TyEnv] => bool" ("_ hastyenv _ " [36,36] 35)
rules
(*
Expression constructors must be injective, distinct and it must be possible
to do induction over expressions.
*)
(* All the constructors are injective *)
e_const_inj "e_const(c1) = e_const(c2) ==> c1 = c2"
e_var_inj "e_var(ev1) = e_var(ev2) ==> ev1 = ev2"
e_fn_inj "fn ev1 => e1 = fn ev2 => e2 ==> ev1 = ev2 & e1 = e2"
e_fix_inj
" fix ev11e(v12) = e1 = fix ev21(ev22) = e2 ==>
ev11 = ev21 & ev12 = ev22 & e1 = e2
"
e_app_inj "e11 @ e12 = e21 @ e22 ==> e11 = e21 & e12 = e22"
(* All constructors are distinct *)
e_disj_const_var "~e_const(c) = e_var(ev)"
e_disj_const_fn "~e_const(c) = fn ev => e"
e_disj_const_fix "~e_const(c) = fix ev1(ev2) = e"
e_disj_const_app "~e_const(c) = e1 @ e2"
e_disj_var_fn "~e_var(ev1) = fn ev2 => e"
e_disj_var_fix "~e_var(ev) = fix ev1(ev2) = e"
e_disj_var_app "~e_var(ev) = e1 @ e2"
e_disj_fn_fix "~fn ev1 => e1 = fix ev21(ev22) = e2"
e_disj_fn_app "~fn ev1 => e1 = e21 @ e22"
e_disj_fix_app "~fix ev11(ev12) = e1 = e21 @ e22"
(* Strong elimination, induction on expressions *)
e_ind
" [| !!ev. P(e_var(ev));
!!c. P(e_const(c));
!!ev e. P(e) ==> P(fn ev => e);
!!ev1 ev2 e. P(e) ==> P(fix ev1(ev2) = e);
!!e1 e2. P(e1) ==> P(e2) ==> P(e1 @ e2)
|] ==>
P(e)
"
(* Types - same scheme as for expressions *)
(* All constructors are injective *)
t_const_inj "t_const(c1) = t_const(c2) ==> c1 = c2"
t_fun_inj "t11 -> t12 = t21 -> t22 ==> t11 = t21 & t12 = t22"
(* All constructors are distinct, not needed so far ... *)
(* Strong elimination, induction on types *)
t_ind
"[| !!p. P(t_const p); !!t1 t2. P(t1) ==> P(t2) ==> P(t_fun t1 t2) |]
==> P(t)"
(* Values - same scheme again *)
(* All constructors are injective *)
v_const_inj "v_const(c1) = v_const(c2) ==> c1 = c2"
v_clos_inj
" v_clos(<|ev1,e1,ve1|>) = v_clos(<|ev2,e2,ve2|>) ==>
ev1 = ev2 & e1 = e2 & ve1 = ve2"
(* All constructors are distinct *)
v_disj_const_clos "~v_const(c) = v_clos(cl)"
(* No induction on values: they are a codatatype! ... *)
(*
Value environments bind variables to values. Only the following trivial
properties are needed.
*)
ve_dom_owr "ve_dom(ve + {ev |-> v}) = ve_dom(ve) Un {ev}"
ve_app_owr1 "ve_app (ve + {ev |-> v}) ev=v"
ve_app_owr2 "~ev1=ev2 ==> ve_app (ve+{ev1 |-> v}) ev2=ve_app ve ev2"
(* Type Environments bind variables to types. The following trivial
properties are needed. *)
te_dom_owr "te_dom(te + {ev |=> t}) = te_dom(te) Un {ev}"
te_app_owr1 "te_app (te + {ev |=> t}) ev=t"
te_app_owr2 "~ev1=ev2 ==> te_app (te+{ev1 |=> t}) ev2=te_app te ev2"
(* The dynamic semantics is defined inductively by a set of inference
rules. These inference rules allows one to draw conclusions of the form ve
|- e ---> v, read the expression e evaluates to the value v in the value
environment ve. Therefore the relation _ |- _ ---> _ is defined in Isabelle
as the least fixpoint of the functor eval_fun below. From this definition
introduction rules and a strong elimination (induction) rule can be
derived.
*)
eval_fun_def
" eval_fun(s) ==
{ pp.
(? ve c. pp=((ve,e_const(c)),v_const(c))) |
(? ve x. pp=((ve,e_var(x)),ve_app ve x) & x:ve_dom(ve)) |
(? ve e x. pp=((ve,fn x => e),v_clos(<|x,e,ve|>)))|
( ? ve e x f cl.
pp=((ve,fix f(x) = e),v_clos(cl)) &
cl=<|x, e, ve+{f |-> v_clos(cl)} |>
) |
( ? ve e1 e2 c1 c2.
pp=((ve,e1 @ e2),v_const(c_app c1 c2)) &
((ve,e1),v_const(c1)):s & ((ve,e2),v_const(c2)):s
) |
( ? ve vem e1 e2 em xm v v2.
pp=((ve,e1 @ e2),v) &
((ve,e1),v_clos(<|xm,em,vem|>)):s &
((ve,e2),v2):s &
((vem+{xm |-> v2},em),v):s
)
}"
eval_rel_def "eval_rel == lfp(eval_fun)"
eval_def "ve |- e ---> v == ((ve,e),v):eval_rel"
(* The static semantics is defined in the same way as the dynamic
semantics. The relation te |- e ===> t express the expression e has the
type t in the type environment te.
*)
elab_fun_def
"elab_fun(s) ==
{ pp.
(? te c t. pp=((te,e_const(c)),t) & c isof t) |
(? te x. pp=((te,e_var(x)),te_app te x) & x:te_dom(te)) |
(? te x e t1 t2. pp=((te,fn x => e),t1->t2) & ((te+{x |=> t1},e),t2):s) |
(? te f x e t1 t2.
pp=((te,fix f(x)=e),t1->t2) & ((te+{f |=> t1->t2}+{x |=> t1},e),t2):s
) |
(? te e1 e2 t1 t2.
pp=((te,e1 @ e2),t2) & ((te,e1),t1->t2):s & ((te,e2),t1):s
)
}"
elab_rel_def "elab_rel == lfp(elab_fun)"
elab_def "te |- e ===> t == ((te,e),t):elab_rel"
(* The original correspondence relation *)
isof_env_def
" ve isofenv te ==
ve_dom(ve) = te_dom(te) &
( ! x.
x:ve_dom(ve) -->
(? c. ve_app ve x = v_const(c) & c isof te_app te x)
)
"
isof_app "[| c1 isof t1->t2; c2 isof t1 |] ==> c_app c1 c2 isof t2"
(* The extented correspondence relation *)
hasty_fun_def
" hasty_fun(r) ==
{ p.
( ? c t. p = (v_const(c),t) & c isof t) |
( ? ev e ve t te.
p = (v_clos(<|ev,e,ve|>),t) &
te |- fn ev => e ===> t &
ve_dom(ve) = te_dom(te) &
(! ev1. ev1:ve_dom(ve) --> (ve_app ve ev1,te_app te ev1) : r)
)
}
"
hasty_rel_def "hasty_rel == gfp(hasty_fun)"
hasty_def "v hasty t == (v,t) : hasty_rel"
hasty_env_def
" ve hastyenv te ==
ve_dom(ve) = te_dom(te) &
(! x. x: ve_dom(ve) --> ve_app ve x hasty te_app te x)"
end