diff -r f04b33ce250f -r a4dc62a46ee4 ex/MT.thy --- 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=<,v_const(c)>) | - (? ve x. pp=<,ve_app(ve,x)> & x:ve_dom(ve)) | - (? ve e x. pp=< e>,v_clos(<|x,e,ve|>)>)| - ( ? ve e x f cl. - pp=<,v_clos(cl)> & - cl=<|x, e, ve+{f |-> v_clos(cl)} |> - ) | - ( ? ve e1 e2 c1 c2. - pp=<,v_const(c_app(c1,c2))> & - <,v_const(c1)>:s & <,v_const(c2)>:s - ) | - ( ? ve vem e1 e2 em xm v v2. - pp=<,v> & - <,v_clos(<|xm,em,vem|>)>:s & - <,v2>:s & - < v2},em>,v>:s - ) - }" - - eval_rel_def "eval_rel == lfp(eval_fun)" - eval_def "ve |- e ---> v == <,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=<,t> & c isof t) | - (? te x. pp=<,te_app(te,x)> & x:te_dom(te)) | - (? te x e t1 t2. pp=< e>,t1->t2> & < t1},e>,t2>:s) | - (? te f x e t1 t2. - pp=<,t1->t2> & < t1->t2}+{x |=> t1},e>,t2>:s - ) | - (? te e1 e2 t1 t2. - pp=<,t2> & <,t1->t2>:s & <,t1>:s - ) - }" - - elab_rel_def "elab_rel == lfp(elab_fun)" - elab_def "te |- e ===> t == <,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 = & c isof t) | - ( ? ev e ve t te. - p = ),t> & - te |- fn ev => e ===> t & - ve_dom(ve) = te_dom(te) & - (! ev1.ev1:ve_dom(ve) --> : r) - ) - } - " - - hasty_rel_def "hasty_rel == gfp(hasty_fun)" - hasty_def "v hasty 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