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+++ b/src/HOL/Datatype_Examples/Milner_Tofte.thy Tue Nov 24 10:54:21 2015 +0100
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+(* Title: HOL/Datatype_Examples/Milner_Tofte.thy
+ Author: Dmitriy Traytel, ETH Zürich
+ Copyright 2015
+
+Modernized version of HOL/ex/MT.thy by Jacob Frost
+
+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).
+*)
+
+section \<open>Milner-Tofte: Co-induction in Relational Semantics\<close>
+
+theory Milner_Tofte
+imports Main
+begin
+
+typedecl Const
+typedecl ExVar
+typedecl TyConst
+
+datatype Ex =
+ e_const (e_const_fst: Const)
+| e_var ExVar
+| e_fn ExVar Ex ("fn _ => _" [0,51] 1000)
+| e_fix ExVar ExVar Ex ("fix _ ( _ ) = _" [0,51,51] 1000)
+| e_app Ex Ex ("_ @@ _" [51,51] 1000)
+
+datatype Ty =
+ t_const TyConst
+| t_fun Ty Ty ("_ -> _" [51,51] 1000)
+
+datatype 'a genClos =
+ clos_mk ExVar Ex "ExVar \<rightharpoonup> 'a" ("\<langle>_ , _ , _\<rangle>" [0,0,0] 1000)
+
+codatatype Val =
+ v_const Const
+| v_clos "Val genClos"
+
+type_synonym Clos = "Val genClos"
+type_synonym ValEnv = "ExVar \<rightharpoonup> Val"
+type_synonym TyEnv = "ExVar \<rightharpoonup> Ty"
+
+axiomatization
+ c_app :: "[Const, Const] => Const" and
+ isof :: "[Const, Ty] => bool" ("_ isof _" [36,36] 50) where
+ isof_app: "\<lbrakk>c1 isof t1 -> t2; c2 isof t1\<rbrakk> \<Longrightarrow> c_app c1 c2 isof t2"
+
+text \<open>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.\<close>
+
+inductive eval :: "[ValEnv, Ex, Val] => bool" ("_ \<turnstile> _ ---> _" [36,0,36] 50) where
+ eval_const: "ve \<turnstile> e_const c ---> v_const c"
+| eval_var2: "ev \<in> dom ve \<Longrightarrow> ve \<turnstile> e_var ev ---> the (ve ev)"
+| eval_fn: "ve \<turnstile> fn ev => e ---> v_clos \<langle>ev, e, ve\<rangle>"
+| eval_fix: "cl = \<langle>ev1, e, ve(ev2 \<mapsto> v_clos cl)\<rangle> \<Longrightarrow> ve \<turnstile> fix ev2(ev1) = e ---> v_clos(cl)"
+| eval_app1: "\<lbrakk>ve \<turnstile> e1 ---> v_const c1; ve \<turnstile> e2 ---> v_const c2\<rbrakk> \<Longrightarrow>
+ ve \<turnstile> e1 @@ e2 ---> v_const (c_app c1 c2)"
+| eval_app2: "\<lbrakk>ve \<turnstile> e1 ---> v_clos \<langle>xm, em, vem\<rangle>; ve \<turnstile> e2 ---> v2; vem(xm \<mapsto> v2) \<turnstile> em ---> v\<rbrakk> \<Longrightarrow>
+ ve \<turnstile> e1 @@ e2 ---> v"
+
+declare eval.intros[intro]
+
+text \<open>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.\<close>
+
+inductive elab :: "[TyEnv, Ex, Ty] => bool" ("_ \<turnstile> _ ===> _" [36,0,36] 50) where
+ elab_const: "c isof ty \<Longrightarrow> te \<turnstile> e_const c ===> ty"
+| elab_var: "x \<in> dom te \<Longrightarrow> te \<turnstile> e_var x ===> the (te x)"
+| elab_fn: "te(x \<mapsto> ty1) \<turnstile> e ===> ty2 \<Longrightarrow> te \<turnstile> fn x => e ===> ty1 -> ty2"
+| elab_fix: "te(f \<mapsto> ty1 -> ty2, x \<mapsto> ty1) \<turnstile> e ===> ty2 \<Longrightarrow> te \<turnstile> fix f x = e ===> ty1 -> ty2"
+| elab_app: "\<lbrakk>te \<turnstile> e1 ===> ty1 -> ty2; te \<turnstile> e2 ===> ty1\<rbrakk> \<Longrightarrow> te \<turnstile> e1 @@ e2 ===> ty2"
+
+declare elab.intros[intro]
+inductive_cases elabE[elim!]:
+ "te \<turnstile> e_const(c) ===> ty"
+ "te \<turnstile> e_var(x) ===> ty"
+ "te \<turnstile> fn x => e ===> ty"
+ "te \<turnstile> fix f(x) = e ===> ty"
+ "te \<turnstile> e1 @@ e2 ===> ty"
+
+(* The original correspondence relation *)
+
+abbreviation isof_env :: "[ValEnv,TyEnv] => bool" ("_ isofenv _") where
+ "ve isofenv te \<equiv> (dom(ve) = dom(te) \<and>
+ (\<forall>x. x \<in> dom ve \<longrightarrow> (\<exists>c. the (ve x) = v_const(c) \<and> c isof the (te x))))"
+
+coinductive hasty :: "[Val, Ty] => bool" ("_ hasty _" [36,36] 50) where
+ hasty_const: "c isof t \<Longrightarrow> v_const c hasty t"
+| hasty_clos: "\<lbrakk>te \<turnstile> fn ev => e ===> t; dom(ve) = dom(te) \<and>
+ (\<forall>x. x \<in> dom ve --> the (ve x) hasty the (te x)); cl = \<langle>ev,e,ve\<rangle>\<rbrakk> \<Longrightarrow> v_clos cl hasty t"
+
+declare hasty.intros[intro]
+inductive_cases hastyE[elim!]:
+ "v_const c hasty t"
+ "v_clos \<langle>xm , em , evm\<rangle> hasty u -> t"
+
+definition hasty_env :: "[ValEnv, TyEnv] => bool" ("_ hastyenv _ " [36,36] 35) where
+ [simp]: "ve hastyenv te \<equiv> (dom(ve) = dom(te) \<and>
+ (\<forall>x. x \<in> dom ve --> the (ve x) hasty the (te x)))"
+
+theorem consistency: "\<lbrakk>ve \<turnstile> e ---> v; ve hastyenv te; te \<turnstile> e ===> t\<rbrakk> \<Longrightarrow> v hasty t"
+proof (induct ve e v arbitrary: t te rule: eval.induct)
+ case (eval_fix cl x e ve f)
+ then show ?case
+ by coinduction
+ (auto 0 11 intro: exI[of _ "te(f \<mapsto> _)"] exI[of _ x] exI[of _ e] exI[of _ "ve(f \<mapsto> v_clos cl)"])
+next
+ case (eval_app2 ve e1 xm em evm e2 v2 v)
+ then obtain u where "te \<turnstile> e1 ===> u -> t" "te \<turnstile> e2 ===> u" by auto
+ with eval_app2(2)[of te "u -> t"] eval_app2(4)[of te u] eval_app2(1,3,5,7) show ?case
+ by (auto 0 4 elim!: eval_app2(6)[rotated])
+qed (auto 8 0 intro!: isof_app)
+
+lemma basic_consistency_aux:
+ "ve isofenv te \<Longrightarrow> ve hastyenv te"
+ using hasty_const hasty_env_def by force
+
+theorem basic_consistency:
+ "\<lbrakk>ve isofenv te; ve \<turnstile> e ---> v_const c; te \<turnstile> e ===> t\<rbrakk> \<Longrightarrow> c isof t"
+ by (auto dest: consistency intro!: basic_consistency_aux)
+
+end