--- a/ex/MT.thy Tue Oct 24 14:59:17 1995 +0100
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,278 +0,0 @@
-(* 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 = Gfp + Sum +
-
-types
- Const
-
- ExVar
- Ex
-
- TyConst
- Ty
-
- Clos
- Val
-
- ValEnv
- TyEnv
-
-arities
- Const :: term
-
- ExVar :: term
- Ex :: term
-
- TyConst :: term
- Ty :: term
-
- Clos :: term
- Val :: term
-
- ValEnv :: term
- TyEnv :: term
-
-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)"
-
-(* Strong elimination, induction on values, not needed yet ... *)
-
-
-(*
- 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