--- a/src/HOL/ex/MT.thy Sat Aug 18 13:32:28 2007 +0200
+++ b/src/HOL/ex/MT.thy Sat Aug 18 17:42:38 2007 +0200
@@ -1,4 +1,4 @@
-(* Title: HOL/ex/mt.thy
+(* Title: HOL/ex/MT.thy
ID: $Id$
Author: Jacob Frost, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
@@ -13,6 +13,8 @@
Report 308, Computer Lab, University of Cambridge (1993).
*)
+header {* Milner-Tofte: Co-induction in Relational Semantics *}
+
theory MT
imports Main
begin
@@ -264,6 +266,735 @@
ve_dom(ve) = te_dom(te) &
(! x. x: ve_dom(ve) --> ve_app ve x hasty te_app te x)"
-ML {* use_legacy_bindings (the_context ()) *}
+
+(* ############################################################ *)
+(* Inference systems *)
+(* ############################################################ *)
+
+ML {*
+val infsys_mono_tac = REPEAT (ares_tac (@{thms basic_monos} @ [allI, impI]) 1)
+*}
+
+lemma infsys_p1: "P a b ==> P (fst (a,b)) (snd (a,b))"
+ by simp
+
+lemma infsys_p2: "P (fst (a,b)) (snd (a,b)) ==> P a b"
+ by simp
+
+lemma infsys_pp1: "P a b c ==> P (fst(fst((a,b),c))) (snd(fst ((a,b),c))) (snd ((a,b),c))"
+ by simp
+
+lemma infsys_pp2: "P (fst(fst((a,b),c))) (snd(fst((a,b),c))) (snd((a,b),c)) ==> P a b c"
+ by simp
+
+
+(* ############################################################ *)
+(* Fixpoints *)
+(* ############################################################ *)
+
+(* Least fixpoints *)
+
+lemma lfp_intro2: "[| mono(f); x:f(lfp(f)) |] ==> x:lfp(f)"
+apply (rule subsetD)
+apply (rule lfp_lemma2)
+apply assumption+
+done
+
+lemma lfp_elim2:
+ assumes lfp: "x:lfp(f)"
+ and mono: "mono(f)"
+ and r: "!!y. y:f(lfp(f)) ==> P(y)"
+ shows "P(x)"
+apply (rule r)
+apply (rule subsetD)
+apply (rule lfp_lemma3)
+apply (rule mono)
+apply (rule lfp)
+done
+
+lemma lfp_ind2:
+ assumes lfp: "x:lfp(f)"
+ and mono: "mono(f)"
+ and r: "!!y. y:f(lfp(f) Int {x. P(x)}) ==> P(y)"
+ shows "P(x)"
+apply (rule lfp_induct_set [OF lfp mono])
+apply (erule r)
+done
+
+(* Greatest fixpoints *)
+
+(* Note : "[| x:S; S <= f(S Un gfp(f)); mono(f) |] ==> x:gfp(f)" *)
+
+lemma gfp_coind2:
+ assumes cih: "x:f({x} Un gfp(f))"
+ and monoh: "mono(f)"
+ shows "x:gfp(f)"
+apply (rule cih [THEN [2] gfp_upperbound [THEN subsetD]])
+apply (rule monoh [THEN monoD])
+apply (rule UnE [THEN subsetI])
+apply assumption
+apply (blast intro!: cih)
+apply (rule monoh [THEN monoD [THEN subsetD]])
+apply (rule Un_upper2)
+apply (erule monoh [THEN gfp_lemma2, THEN subsetD])
+done
+
+lemma gfp_elim2:
+ assumes gfph: "x:gfp(f)"
+ and monoh: "mono(f)"
+ and caseh: "!!y. y:f(gfp(f)) ==> P(y)"
+ shows "P(x)"
+apply (rule caseh)
+apply (rule subsetD)
+apply (rule gfp_lemma2)
+apply (rule monoh)
+apply (rule gfph)
+done
+
+(* ############################################################ *)
+(* Expressions *)
+(* ############################################################ *)
+
+lemmas e_injs = e_const_inj e_var_inj e_fn_inj e_fix_inj e_app_inj
+
+lemmas e_disjs =
+ e_disj_const_var
+ e_disj_const_fn
+ e_disj_const_fix
+ e_disj_const_app
+ e_disj_var_fn
+ e_disj_var_fix
+ e_disj_var_app
+ e_disj_fn_fix
+ e_disj_fn_app
+ e_disj_fix_app
+
+lemmas e_disj_si = e_disjs e_disjs [symmetric]
+
+lemmas e_disj_se = e_disj_si [THEN notE]
+
+(* ############################################################ *)
+(* Values *)
+(* ############################################################ *)
+
+lemmas v_disjs = v_disj_const_clos
+lemmas v_disj_si = v_disjs v_disjs [symmetric]
+lemmas v_disj_se = v_disj_si [THEN notE]
+
+lemmas v_injs = v_const_inj v_clos_inj
+
+(* ############################################################ *)
+(* Evaluations *)
+(* ############################################################ *)
+
+(* Monotonicity of eval_fun *)
+
+lemma eval_fun_mono: "mono(eval_fun)"
+unfolding mono_def eval_fun_def
+apply (tactic infsys_mono_tac)
+done
+
+(* Introduction rules *)
+
+lemma eval_const: "ve |- e_const(c) ---> v_const(c)"
+unfolding eval_def eval_rel_def
+apply (rule lfp_intro2)
+apply (rule eval_fun_mono)
+apply (unfold eval_fun_def)
+ (*Naughty! But the quantifiers are nested VERY deeply...*)
+apply (blast intro!: exI)
+done
+
+lemma eval_var2:
+ "ev:ve_dom(ve) ==> ve |- e_var(ev) ---> ve_app ve ev"
+apply (unfold eval_def eval_rel_def)
+apply (rule lfp_intro2)
+apply (rule eval_fun_mono)
+apply (unfold eval_fun_def)
+apply (blast intro!: exI)
+done
+
+lemma eval_fn:
+ "ve |- fn ev => e ---> v_clos(<|ev,e,ve|>)"
+apply (unfold eval_def eval_rel_def)
+apply (rule lfp_intro2)
+apply (rule eval_fun_mono)
+apply (unfold eval_fun_def)
+apply (blast intro!: exI)
+done
+
+lemma eval_fix:
+ " cl = <| ev1, e, ve + {ev2 |-> v_clos(cl)} |> ==>
+ ve |- fix ev2(ev1) = e ---> v_clos(cl)"
+apply (unfold eval_def eval_rel_def)
+apply (rule lfp_intro2)
+apply (rule eval_fun_mono)
+apply (unfold eval_fun_def)
+apply (blast intro!: exI)
+done
+
+lemma eval_app1:
+ " [| ve |- e1 ---> v_const(c1); ve |- e2 ---> v_const(c2) |] ==>
+ ve |- e1 @@ e2 ---> v_const(c_app c1 c2)"
+apply (unfold eval_def eval_rel_def)
+apply (rule lfp_intro2)
+apply (rule eval_fun_mono)
+apply (unfold eval_fun_def)
+apply (blast intro!: exI)
+done
+
+lemma eval_app2:
+ " [| ve |- e1 ---> v_clos(<|xm,em,vem|>);
+ ve |- e2 ---> v2;
+ vem + {xm |-> v2} |- em ---> v
+ |] ==>
+ ve |- e1 @@ e2 ---> v"
+apply (unfold eval_def eval_rel_def)
+apply (rule lfp_intro2)
+apply (rule eval_fun_mono)
+apply (unfold eval_fun_def)
+apply (blast intro!: disjI2)
+done
+
+(* Strong elimination, induction on evaluations *)
+
+lemma eval_ind0:
+ " [| ve |- e ---> v;
+ !!ve c. P(((ve,e_const(c)),v_const(c)));
+ !!ev ve. ev:ve_dom(ve) ==> P(((ve,e_var(ev)),ve_app ve ev));
+ !!ev ve e. P(((ve,fn ev => e),v_clos(<|ev,e,ve|>)));
+ !!ev1 ev2 ve cl e.
+ cl = <| ev1, e, ve + {ev2 |-> v_clos(cl)} |> ==>
+ P(((ve,fix ev2(ev1) = e),v_clos(cl)));
+ !!ve c1 c2 e1 e2.
+ [| P(((ve,e1),v_const(c1))); P(((ve,e2),v_const(c2))) |] ==>
+ P(((ve,e1 @@ e2),v_const(c_app c1 c2)));
+ !!ve vem xm e1 e2 em v v2.
+ [| P(((ve,e1),v_clos(<|xm,em,vem|>)));
+ P(((ve,e2),v2));
+ P(((vem + {xm |-> v2},em),v))
+ |] ==>
+ P(((ve,e1 @@ e2),v))
+ |] ==>
+ P(((ve,e),v))"
+unfolding eval_def eval_rel_def
+apply (erule lfp_ind2)
+apply (rule eval_fun_mono)
+apply (unfold eval_fun_def)
+apply (drule CollectD)
+apply safe
+apply auto
+done
+
+lemma eval_ind:
+ " [| ve |- e ---> v;
+ !!ve c. P ve (e_const c) (v_const c);
+ !!ev ve. ev:ve_dom(ve) ==> P ve (e_var ev) (ve_app ve ev);
+ !!ev ve e. P ve (fn ev => e) (v_clos <|ev,e,ve|>);
+ !!ev1 ev2 ve cl e.
+ cl = <| ev1, e, ve + {ev2 |-> v_clos(cl)} |> ==>
+ P ve (fix ev2(ev1) = e) (v_clos cl);
+ !!ve c1 c2 e1 e2.
+ [| P ve e1 (v_const c1); P ve e2 (v_const c2) |] ==>
+ P ve (e1 @@ e2) (v_const(c_app c1 c2));
+ !!ve vem evm e1 e2 em v v2.
+ [| P ve e1 (v_clos <|evm,em,vem|>);
+ P ve e2 v2;
+ P (vem + {evm |-> v2}) em v
+ |] ==> P ve (e1 @@ e2) v
+ |] ==> P ve e v"
+apply (rule_tac P = "P" in infsys_pp2)
+apply (rule eval_ind0)
+apply (rule infsys_pp1)
+apply auto
+done
+
+(* ############################################################ *)
+(* Elaborations *)
+(* ############################################################ *)
+
+lemma elab_fun_mono: "mono(elab_fun)"
+unfolding mono_def elab_fun_def
+apply (tactic infsys_mono_tac)
+done
+
+(* Introduction rules *)
+
+lemma elab_const:
+ "c isof ty ==> te |- e_const(c) ===> ty"
+apply (unfold elab_def elab_rel_def)
+apply (rule lfp_intro2)
+apply (rule elab_fun_mono)
+apply (unfold elab_fun_def)
+apply (blast intro!: exI)
+done
+
+lemma elab_var:
+ "x:te_dom(te) ==> te |- e_var(x) ===> te_app te x"
+apply (unfold elab_def elab_rel_def)
+apply (rule lfp_intro2)
+apply (rule elab_fun_mono)
+apply (unfold elab_fun_def)
+apply (blast intro!: exI)
+done
+
+lemma elab_fn:
+ "te + {x |=> ty1} |- e ===> ty2 ==> te |- fn x => e ===> ty1->ty2"
+apply (unfold elab_def elab_rel_def)
+apply (rule lfp_intro2)
+apply (rule elab_fun_mono)
+apply (unfold elab_fun_def)
+apply (blast intro!: exI)
+done
+
+lemma elab_fix:
+ "te + {f |=> ty1->ty2} + {x |=> ty1} |- e ===> ty2 ==>
+ te |- fix f(x) = e ===> ty1->ty2"
+apply (unfold elab_def elab_rel_def)
+apply (rule lfp_intro2)
+apply (rule elab_fun_mono)
+apply (unfold elab_fun_def)
+apply (blast intro!: exI)
+done
+
+lemma elab_app:
+ "[| te |- e1 ===> ty1->ty2; te |- e2 ===> ty1 |] ==>
+ te |- e1 @@ e2 ===> ty2"
+apply (unfold elab_def elab_rel_def)
+apply (rule lfp_intro2)
+apply (rule elab_fun_mono)
+apply (unfold elab_fun_def)
+apply (blast intro!: disjI2)
+done
+
+(* Strong elimination, induction on elaborations *)
+
+lemma elab_ind0:
+ assumes 1: "te |- e ===> t"
+ and 2: "!!te c t. c isof t ==> P(((te,e_const(c)),t))"
+ and 3: "!!te x. x:te_dom(te) ==> P(((te,e_var(x)),te_app te x))"
+ and 4: "!!te x e t1 t2.
+ [| te + {x |=> t1} |- e ===> t2; P(((te + {x |=> t1},e),t2)) |] ==>
+ P(((te,fn x => e),t1->t2))"
+ and 5: "!!te f x e t1 t2.
+ [| te + {f |=> t1->t2} + {x |=> t1} |- e ===> t2;
+ P(((te + {f |=> t1->t2} + {x |=> t1},e),t2))
+ |] ==>
+ P(((te,fix f(x) = e),t1->t2))"
+ and 6: "!!te e1 e2 t1 t2.
+ [| te |- e1 ===> t1->t2; P(((te,e1),t1->t2));
+ te |- e2 ===> t1; P(((te,e2),t1))
+ |] ==>
+ P(((te,e1 @@ e2),t2))"
+ shows "P(((te,e),t))"
+apply (rule lfp_ind2 [OF 1 [unfolded elab_def elab_rel_def]])
+apply (rule elab_fun_mono)
+apply (unfold elab_fun_def)
+apply (drule CollectD)
+apply safe
+apply (erule 2)
+apply (erule 3)
+apply (rule 4 [unfolded elab_def elab_rel_def]) apply blast+
+apply (rule 5 [unfolded elab_def elab_rel_def]) apply blast+
+apply (rule 6 [unfolded elab_def elab_rel_def]) apply blast+
+done
+
+lemma elab_ind:
+ " [| te |- e ===> t;
+ !!te c t. c isof t ==> P te (e_const c) t;
+ !!te x. x:te_dom(te) ==> P te (e_var x) (te_app te x);
+ !!te x e t1 t2.
+ [| te + {x |=> t1} |- e ===> t2; P (te + {x |=> t1}) e t2 |] ==>
+ P te (fn x => e) (t1->t2);
+ !!te f x e t1 t2.
+ [| te + {f |=> t1->t2} + {x |=> t1} |- e ===> t2;
+ P (te + {f |=> t1->t2} + {x |=> t1}) e t2
+ |] ==>
+ P te (fix f(x) = e) (t1->t2);
+ !!te e1 e2 t1 t2.
+ [| te |- e1 ===> t1->t2; P te e1 (t1->t2);
+ te |- e2 ===> t1; P te e2 t1
+ |] ==>
+ P te (e1 @@ e2) t2
+ |] ==>
+ P te e t"
+apply (rule_tac P = "P" in infsys_pp2)
+apply (erule elab_ind0)
+apply (rule_tac [!] infsys_pp1)
+apply auto
+done
+
+(* Weak elimination, case analysis on elaborations *)
+
+lemma elab_elim0:
+ assumes 1: "te |- e ===> t"
+ and 2: "!!te c t. c isof t ==> P(((te,e_const(c)),t))"
+ and 3: "!!te x. x:te_dom(te) ==> P(((te,e_var(x)),te_app te x))"
+ and 4: "!!te x e t1 t2.
+ te + {x |=> t1} |- e ===> t2 ==> P(((te,fn x => e),t1->t2))"
+ and 5: "!!te f x e t1 t2.
+ te + {f |=> t1->t2} + {x |=> t1} |- e ===> t2 ==>
+ P(((te,fix f(x) = e),t1->t2))"
+ and 6: "!!te e1 e2 t1 t2.
+ [| te |- e1 ===> t1->t2; te |- e2 ===> t1 |] ==>
+ P(((te,e1 @@ e2),t2))"
+ shows "P(((te,e),t))"
+apply (rule lfp_elim2 [OF 1 [unfolded elab_def elab_rel_def]])
+apply (rule elab_fun_mono)
+apply (unfold elab_fun_def)
+apply (drule CollectD)
+apply safe
+apply (erule 2)
+apply (erule 3)
+apply (rule 4 [unfolded elab_def elab_rel_def]) apply blast+
+apply (rule 5 [unfolded elab_def elab_rel_def]) apply blast+
+apply (rule 6 [unfolded elab_def elab_rel_def]) apply blast+
+done
+
+lemma elab_elim:
+ " [| te |- e ===> t;
+ !!te c t. c isof t ==> P te (e_const c) t;
+ !!te x. x:te_dom(te) ==> P te (e_var x) (te_app te x);
+ !!te x e t1 t2.
+ te + {x |=> t1} |- e ===> t2 ==> P te (fn x => e) (t1->t2);
+ !!te f x e t1 t2.
+ te + {f |=> t1->t2} + {x |=> t1} |- e ===> t2 ==>
+ P te (fix f(x) = e) (t1->t2);
+ !!te e1 e2 t1 t2.
+ [| te |- e1 ===> t1->t2; te |- e2 ===> t1 |] ==>
+ P te (e1 @@ e2) t2
+ |] ==>
+ P te e t"
+apply (rule_tac P = "P" in infsys_pp2)
+apply (rule elab_elim0)
+apply auto
+done
+
+(* Elimination rules for each expression *)
+
+lemma elab_const_elim_lem:
+ "te |- e ===> t ==> (e = e_const(c) --> c isof t)"
+apply (erule elab_elim)
+apply (fast intro!: e_disj_si elim!: e_disj_se dest!: e_injs)+
+done
+
+lemma elab_const_elim: "te |- e_const(c) ===> t ==> c isof t"
+apply (drule elab_const_elim_lem)
+apply blast
+done
+
+lemma elab_var_elim_lem:
+ "te |- e ===> t ==> (e = e_var(x) --> t=te_app te x & x:te_dom(te))"
+apply (erule elab_elim)
+apply (fast intro!: e_disj_si elim!: e_disj_se dest!: e_injs)+
+done
+
+lemma elab_var_elim: "te |- e_var(ev) ===> t ==> t=te_app te ev & ev : te_dom(te)"
+apply (drule elab_var_elim_lem)
+apply blast
+done
+
+lemma elab_fn_elim_lem:
+ " te |- e ===> t ==>
+ ( e = fn x1 => e1 -->
+ (? t1 t2. t=t_fun t1 t2 & te + {x1 |=> t1} |- e1 ===> t2)
+ )"
+apply (erule elab_elim)
+apply (fast intro!: e_disj_si elim!: e_disj_se dest!: e_injs)+
+done
+
+lemma elab_fn_elim: " te |- fn x1 => e1 ===> t ==>
+ (? t1 t2. t=t1->t2 & te + {x1 |=> t1} |- e1 ===> t2)"
+apply (drule elab_fn_elim_lem)
+apply blast
+done
+
+lemma elab_fix_elim_lem:
+ " te |- e ===> t ==>
+ (e = fix f(x) = e1 -->
+ (? t1 t2. t=t1->t2 & te + {f |=> t1->t2} + {x |=> t1} |- e1 ===> t2))"
+apply (erule elab_elim)
+apply (fast intro!: e_disj_si elim!: e_disj_se dest!: e_injs)+
+done
+
+lemma elab_fix_elim: " te |- fix ev1(ev2) = e1 ===> t ==>
+ (? t1 t2. t=t1->t2 & te + {ev1 |=> t1->t2} + {ev2 |=> t1} |- e1 ===> t2)"
+apply (drule elab_fix_elim_lem)
+apply blast
+done
+
+lemma elab_app_elim_lem:
+ " te |- e ===> t2 ==>
+ (e = e1 @@ e2 --> (? t1 . te |- e1 ===> t1->t2 & te |- e2 ===> t1))"
+apply (erule elab_elim)
+apply (fast intro!: e_disj_si elim!: e_disj_se dest!: e_injs)+
+done
+
+lemma elab_app_elim: "te |- e1 @@ e2 ===> t2 ==> (? t1 . te |- e1 ===> t1->t2 & te |- e2 ===> t1)"
+apply (drule elab_app_elim_lem)
+apply blast
+done
+
+(* ############################################################ *)
+(* The extended correspondence relation *)
+(* ############################################################ *)
+
+(* Monotonicity of hasty_fun *)
+
+lemma mono_hasty_fun: "mono(hasty_fun)"
+unfolding mono_def hasty_fun_def
+apply (tactic infsys_mono_tac)
+apply blast
+done
+
+(*
+ Because hasty_rel has been defined as the greatest fixpoint of hasty_fun it
+ enjoys two strong indtroduction (co-induction) rules and an elimination rule.
+*)
+
+(* First strong indtroduction (co-induction) rule for hasty_rel *)
+
+lemma hasty_rel_const_coind: "c isof t ==> (v_const(c),t) : hasty_rel"
+apply (unfold hasty_rel_def)
+apply (rule gfp_coind2)
+apply (unfold hasty_fun_def)
+apply (rule CollectI)
+apply (rule disjI1)
+apply blast
+apply (rule mono_hasty_fun)
+done
+
+(* Second strong introduction (co-induction) rule for hasty_rel *)
+
+lemma hasty_rel_clos_coind:
+ " [| te |- fn ev => e ===> t;
+ ve_dom(ve) = te_dom(te);
+ ! ev1.
+ ev1:ve_dom(ve) -->
+ (ve_app ve ev1,te_app te ev1) : {(v_clos(<|ev,e,ve|>),t)} Un hasty_rel
+ |] ==>
+ (v_clos(<|ev,e,ve|>),t) : hasty_rel"
+apply (unfold hasty_rel_def)
+apply (rule gfp_coind2)
+apply (unfold hasty_fun_def)
+apply (rule CollectI)
+apply (rule disjI2)
+apply blast
+apply (rule mono_hasty_fun)
+done
+
+(* Elimination rule for hasty_rel *)
+
+lemma hasty_rel_elim0:
+ " [| !! c t. c isof t ==> P((v_const(c),t));
+ !! te ev e t ve.
+ [| te |- fn ev => e ===> t;
+ ve_dom(ve) = te_dom(te);
+ !ev1. ev1:ve_dom(ve) --> (ve_app ve ev1,te_app te ev1) : hasty_rel
+ |] ==> P((v_clos(<|ev,e,ve|>),t));
+ (v,t) : hasty_rel
+ |] ==> P(v,t)"
+unfolding hasty_rel_def
+apply (erule gfp_elim2)
+apply (rule mono_hasty_fun)
+apply (unfold hasty_fun_def)
+apply (drule CollectD)
+apply (fold hasty_fun_def)
+apply auto
+done
+
+lemma hasty_rel_elim:
+ " [| (v,t) : hasty_rel;
+ !! c t. c isof t ==> P (v_const c) t;
+ !! te ev e t ve.
+ [| te |- fn ev => e ===> t;
+ ve_dom(ve) = te_dom(te);
+ !ev1. ev1:ve_dom(ve) --> (ve_app ve ev1,te_app te ev1) : hasty_rel
+ |] ==> P (v_clos <|ev,e,ve|>) t
+ |] ==> P v t"
+apply (rule_tac P = "P" in infsys_p2)
+apply (rule hasty_rel_elim0)
+apply auto
+done
+
+(* Introduction rules for hasty *)
+
+lemma hasty_const: "c isof t ==> v_const(c) hasty t"
+apply (unfold hasty_def)
+apply (erule hasty_rel_const_coind)
+done
+
+lemma hasty_clos:
+ "te |- fn ev => e ===> t & ve hastyenv te ==> v_clos(<|ev,e,ve|>) hasty t"
+apply (unfold hasty_def hasty_env_def)
+apply (rule hasty_rel_clos_coind)
+apply (blast del: equalityI)+
+done
+
+(* Elimination on constants for hasty *)
+
+lemma hasty_elim_const_lem:
+ "v hasty t ==> (!c.(v = v_const(c) --> c isof t))"
+apply (unfold hasty_def)
+apply (rule hasty_rel_elim)
+apply (blast intro!: v_disj_si elim!: v_disj_se dest!: v_injs)+
+done
+
+lemma hasty_elim_const: "v_const(c) hasty t ==> c isof t"
+apply (drule hasty_elim_const_lem)
+apply blast
+done
+
+(* Elimination on closures for hasty *)
+
+lemma hasty_elim_clos_lem:
+ " v hasty t ==>
+ ! x e ve.
+ v=v_clos(<|x,e,ve|>) --> (? te. te |- fn x => e ===> t & ve hastyenv te)"
+apply (unfold hasty_env_def hasty_def)
+apply (rule hasty_rel_elim)
+apply (blast intro!: v_disj_si elim!: v_disj_se dest!: v_injs)+
+done
+
+lemma hasty_elim_clos: "v_clos(<|ev,e,ve|>) hasty t ==>
+ ? te. te |- fn ev => e ===> t & ve hastyenv te "
+apply (drule hasty_elim_clos_lem)
+apply blast
+done
+
+(* ############################################################ *)
+(* The pointwise extension of hasty to environments *)
+(* ############################################################ *)
+
+lemma hasty_env1: "[| ve hastyenv te; v hasty t |] ==>
+ ve + {ev |-> v} hastyenv te + {ev |=> t}"
+apply (unfold hasty_env_def)
+apply (simp del: mem_simps add: ve_dom_owr te_dom_owr)
+apply (tactic {* safe_tac HOL_cs *})
+apply (case_tac "ev=x")
+apply (simp (no_asm_simp) add: ve_app_owr1 te_app_owr1)
+apply (simp add: ve_app_owr2 te_app_owr2)
+done
+
+(* ############################################################ *)
+(* The Consistency theorem *)
+(* ############################################################ *)
+
+lemma consistency_const: "[| ve hastyenv te ; te |- e_const(c) ===> t |] ==> v_const(c) hasty t"
+apply (drule elab_const_elim)
+apply (erule hasty_const)
+done
+
+lemma consistency_var:
+ "[| ev : ve_dom(ve); ve hastyenv te ; te |- e_var(ev) ===> t |] ==>
+ ve_app ve ev hasty t"
+apply (unfold hasty_env_def)
+apply (drule elab_var_elim)
+apply blast
+done
+
+lemma consistency_fn: "[| ve hastyenv te ; te |- fn ev => e ===> t |] ==>
+ v_clos(<| ev, e, ve |>) hasty t"
+apply (rule hasty_clos)
+apply blast
+done
+
+lemma consistency_fix:
+ "[| cl = <| ev1, e, ve + { ev2 |-> v_clos(cl) } |>;
+ ve hastyenv te ;
+ te |- fix ev2 ev1 = e ===> t
+ |] ==>
+ v_clos(cl) hasty t"
+apply (unfold hasty_env_def hasty_def)
+apply (drule elab_fix_elim)
+apply (tactic {* safe_tac HOL_cs *})
+(*Do a single unfolding of cl*)
+apply (frule ssubst) prefer 2 apply assumption
+apply (rule hasty_rel_clos_coind)
+apply (erule elab_fn)
+apply (simp (no_asm_simp) add: ve_dom_owr te_dom_owr)
+
+apply (simp (no_asm_simp) del: mem_simps add: ve_dom_owr)
+apply (tactic {* safe_tac HOL_cs *})
+apply (case_tac "ev2=ev1a")
+apply (simp (no_asm_simp) del: mem_simps add: ve_app_owr1 te_app_owr1)
+apply blast
+apply (simp add: ve_app_owr2 te_app_owr2)
+done
+
+lemma consistency_app1: "[| ! t te. ve hastyenv te --> te |- e1 ===> t --> v_const(c1) hasty t;
+ ! t te. ve hastyenv te --> te |- e2 ===> t --> v_const(c2) hasty t;
+ ve hastyenv te ; te |- e1 @@ e2 ===> t
+ |] ==>
+ v_const(c_app c1 c2) hasty t"
+apply (drule elab_app_elim)
+apply safe
+apply (rule hasty_const)
+apply (rule isof_app)
+apply (rule hasty_elim_const)
+apply blast
+apply (rule hasty_elim_const)
+apply blast
+done
+
+lemma consistency_app2: "[| ! t te.
+ ve hastyenv te -->
+ te |- e1 ===> t --> v_clos(<|evm, em, vem|>) hasty t;
+ ! t te. ve hastyenv te --> te |- e2 ===> t --> v2 hasty t;
+ ! t te.
+ vem + { evm |-> v2 } hastyenv te --> te |- em ===> t --> v hasty t;
+ ve hastyenv te ;
+ te |- e1 @@ e2 ===> t
+ |] ==>
+ v hasty t"
+apply (drule elab_app_elim)
+apply safe
+apply (erule allE, erule allE, erule impE)
+apply assumption
+apply (erule impE)
+apply assumption
+apply (erule allE, erule allE, erule impE)
+apply assumption
+apply (erule impE)
+apply assumption
+apply (drule hasty_elim_clos)
+apply safe
+apply (drule elab_fn_elim)
+apply (blast intro: hasty_env1 dest!: t_fun_inj)
+done
+
+lemma consistency: "ve |- e ---> v ==>
+ (! t te. ve hastyenv te --> te |- e ===> t --> v hasty t)"
+
+(* Proof by induction on the structure of evaluations *)
+
+apply (erule eval_ind)
+apply safe
+apply (blast intro: consistency_const consistency_var consistency_fn consistency_fix consistency_app1 consistency_app2)+
+done
+
+(* ############################################################ *)
+(* The Basic Consistency theorem *)
+(* ############################################################ *)
+
+lemma basic_consistency_lem:
+ "ve isofenv te ==> ve hastyenv te"
+apply (unfold isof_env_def hasty_env_def)
+apply safe
+apply (erule allE)
+apply (erule impE)
+apply assumption
+apply (erule exE)
+apply (erule conjE)
+apply (drule hasty_const)
+apply (simp (no_asm_simp))
+done
+
+lemma basic_consistency:
+ "[| ve isofenv te; ve |- e ---> v_const(c); te |- e ===> t |] ==> c isof t"
+apply (rule hasty_elim_const)
+apply (drule consistency)
+apply (blast intro!: basic_consistency_lem)
+done
end