(* Title: HOL/ex/mt.ML
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).
*)
open MT;
val prems = goal MT.thy "~a:{b} ==> ~a=b";
by (cut_facts_tac prems 1);
by (rtac notI 1);
by (dtac notE 1);
by (hyp_subst_tac 1);
by (rtac singletonI 1);
by (assume_tac 1);
qed "notsingletonI";
val prems = goalw MT.thy [Un_def]
"[| c : A Un B; c : A & ~c : B ==> P; c : B ==> P |] ==> P";
by (cut_facts_tac prems 1);bd CollectD 1;be disjE 1;
by (cut_facts_tac [excluded_middle] 1);be disjE 1;
by (resolve_tac prems 1);br conjI 1;ba 1;ba 1;
by (eresolve_tac prems 1);
by (eresolve_tac prems 1);
qed "UnSE";
(* ############################################################ *)
(* Inference systems *)
(* ############################################################ *)
val infsys_mono_tac =
(rewtac subset_def) THEN (safe_tac HOL_cs) THEN (rtac ballI 1) THEN
(rtac CollectI 1) THEN (dtac CollectD 1) THEN
REPEAT
( (TRY ((etac disjE 1) THEN (rtac disjI2 2) THEN (rtac disjI1 1))) THEN
(REPEAT (etac exE 1)) THEN (REPEAT (rtac exI 1)) THEN (fast_tac set_cs 1)
);
val prems = goal MT.thy "P a b ==> P (fst <a,b>) (snd <a,b>)";
by (rtac (fst_conv RS ssubst) 1);
by (rtac (snd_conv RS ssubst) 1);
by (resolve_tac prems 1);
qed "infsys_p1";
val prems = goal MT.thy "P (fst <a,b>) (snd <a,b>) ==> P a b";
by (cut_facts_tac prems 1);
by (dtac (fst_conv RS subst) 1);
by (dtac (snd_conv RS subst) 1);
by (assume_tac 1);
qed "infsys_p2";
val prems = goal MT.thy
"P a b c ==> P (fst(fst <<a,b>,c>)) (snd(fst <<a,b>,c>)) (snd <<a,b>,c>)";
by (rtac (fst_conv RS ssubst) 1);
by (rtac (fst_conv RS ssubst) 1);
by (rtac (snd_conv RS ssubst) 1);
by (rtac (snd_conv RS ssubst) 1);
by (resolve_tac prems 1);
qed "infsys_pp1";
val prems = goal MT.thy
"P (fst(fst <<a,b>,c>)) (snd(fst <<a,b>,c>)) (snd <<a,b>,c>) ==> P a b c";
by (cut_facts_tac prems 1);
by (dtac (fst_conv RS subst) 1);
by (dtac (fst_conv RS subst) 1);
by (dtac (snd_conv RS subst) 1);
by (dtac (snd_conv RS subst) 1);
by (assume_tac 1);
qed "infsys_pp2";
(* ############################################################ *)
(* Fixpoints *)
(* ############################################################ *)
(* Least fixpoints *)
val prems = goal MT.thy "[| mono(f); x:f(lfp(f)) |] ==> x:lfp(f)";
by (rtac subsetD 1);
by (rtac lfp_lemma2 1);
by (resolve_tac prems 1);brs prems 1;
qed "lfp_intro2";
val prems = goal MT.thy
" [| x:lfp(f); mono(f); !!y. y:f(lfp(f)) ==> P(y) |] ==> \
\ P(x)";
by (cut_facts_tac prems 1);
by (resolve_tac prems 1);br subsetD 1;
by (rtac lfp_lemma3 1);ba 1;ba 1;
qed "lfp_elim2";
val prems = goal MT.thy
" [| x:lfp(f); mono(f); !!y. y:f(lfp(f) Int {x.P(x)}) ==> P(y) |] ==> \
\ P(x)";
by (cut_facts_tac prems 1);
by (etac induct 1);ba 1;
by (eresolve_tac prems 1);
qed "lfp_ind2";
(* Greatest fixpoints *)
(* Note : "[| x:S; S <= f(S Un gfp(f)); mono(f) |] ==> x:gfp(f)" *)
val [cih,monoh] = goal MT.thy "[| x:f({x} Un gfp(f)); mono(f) |] ==> x:gfp(f)";
by (rtac (cih RSN (2,gfp_upperbound RS subsetD)) 1);
by (rtac (monoh RS monoD) 1);
by (rtac (UnE RS subsetI) 1);ba 1;
by (fast_tac (set_cs addSIs [cih]) 1);
by (rtac (monoh RS monoD RS subsetD) 1);
by (rtac Un_upper2 1);
by (etac (monoh RS gfp_lemma2 RS subsetD) 1);
qed "gfp_coind2";
val [gfph,monoh,caseh] = goal MT.thy
"[| x:gfp(f); mono(f); !! y. y:f(gfp(f)) ==> P(y) |] ==> P(x)";
by (rtac caseh 1);br subsetD 1;br gfp_lemma2 1;br monoh 1;br gfph 1;
qed "gfp_elim2";
(* ############################################################ *)
(* Expressions *)
(* ############################################################ *)
val e_injs = [e_const_inj, e_var_inj, e_fn_inj, e_fix_inj, e_app_inj];
val 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
];
val e_disj_si = e_disjs @ (e_disjs RL [not_sym]);
val e_disj_se = (e_disj_si RL [notE]);
fun e_ext_cs cs = cs addSIs e_disj_si addSEs e_disj_se addSDs e_injs;
(* ############################################################ *)
(* Values *)
(* ############################################################ *)
val v_disjs = [v_disj_const_clos];
val v_disj_si = v_disjs @ (v_disjs RL [not_sym]);
val v_disj_se = (v_disj_si RL [notE]);
val v_injs = [v_const_inj, v_clos_inj];
fun v_ext_cs cs = cs addSIs v_disj_si addSEs v_disj_se addSDs v_injs;
(* ############################################################ *)
(* Evaluations *)
(* ############################################################ *)
(* Monotonicity of eval_fun *)
goalw MT.thy [mono_def, eval_fun_def] "mono(eval_fun)";
by infsys_mono_tac;
qed "eval_fun_mono";
(* Introduction rules *)
goalw MT.thy [eval_def, eval_rel_def] "ve |- e_const(c) ---> v_const(c)";
by (rtac lfp_intro2 1);
by (rtac eval_fun_mono 1);
by (rewtac eval_fun_def);
by (rtac CollectI 1);br disjI1 1;
by (fast_tac HOL_cs 1);
qed "eval_const";
val prems = goalw MT.thy [eval_def, eval_rel_def]
"ev:ve_dom(ve) ==> ve |- e_var(ev) ---> ve_app ve ev";
by (cut_facts_tac prems 1);
by (rtac lfp_intro2 1);
by (rtac eval_fun_mono 1);
by (rewtac eval_fun_def);
by (rtac CollectI 1);br disjI2 1;br disjI1 1;
by (fast_tac HOL_cs 1);
qed "eval_var";
val prems = goalw MT.thy [eval_def, eval_rel_def]
"ve |- fn ev => e ---> v_clos(<|ev,e,ve|>)";
by (cut_facts_tac prems 1);
by (rtac lfp_intro2 1);
by (rtac eval_fun_mono 1);
by (rewtac eval_fun_def);
by (rtac CollectI 1);br disjI2 1;br disjI2 1;br disjI1 1;
by (fast_tac HOL_cs 1);
qed "eval_fn";
val prems = goalw MT.thy [eval_def, eval_rel_def]
" cl = <| ev1, e, ve + {ev2 |-> v_clos(cl)} |> ==> \
\ ve |- fix ev2(ev1) = e ---> v_clos(cl)";
by (cut_facts_tac prems 1);
by (rtac lfp_intro2 1);
by (rtac eval_fun_mono 1);
by (rewtac eval_fun_def);
by (rtac CollectI 1);br disjI2 1;br disjI2 1;br disjI2 1;br disjI1 1;
by (fast_tac HOL_cs 1);
qed "eval_fix";
val prems = goalw MT.thy [eval_def, eval_rel_def]
" [| ve |- e1 ---> v_const(c1); ve |- e2 ---> v_const(c2) |] ==> \
\ ve |- e1 @ e2 ---> v_const(c_app c1 c2)";
by (cut_facts_tac prems 1);
by (rtac lfp_intro2 1);
by (rtac eval_fun_mono 1);
by (rewtac eval_fun_def);
by (rtac CollectI 1);br disjI2 1;br disjI2 1;br disjI2 1;br disjI2 1;br disjI1 1;
by (fast_tac HOL_cs 1);
qed "eval_app1";
val prems = goalw MT.thy [eval_def, eval_rel_def]
" [| ve |- e1 ---> v_clos(<|xm,em,vem|>); \
\ ve |- e2 ---> v2; \
\ vem + {xm |-> v2} |- em ---> v \
\ |] ==> \
\ ve |- e1 @ e2 ---> v";
by (cut_facts_tac prems 1);
by (rtac lfp_intro2 1);
by (rtac eval_fun_mono 1);
by (rewtac eval_fun_def);
by (rtac CollectI 1);br disjI2 1;br disjI2 1;br disjI2 1;br disjI2 1;br disjI2 1;
by (fast_tac HOL_cs 1);
qed "eval_app2";
(* Strong elimination, induction on evaluations *)
val prems = goalw MT.thy [eval_def, eval_rel_def]
" [| 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>)";
by (resolve_tac (prems RL [lfp_ind2]) 1);
by (rtac eval_fun_mono 1);
by (rewtac eval_fun_def);
by (dtac CollectD 1);
by (safe_tac HOL_cs);
by (ALLGOALS (resolve_tac prems));
by (ALLGOALS (fast_tac set_cs));
qed "eval_ind0";
val prems = goal MT.thy
" [| 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";
by (res_inst_tac [("P","P")] infsys_pp2 1);
by (rtac eval_ind0 1);
by (ALLGOALS (rtac infsys_pp1));
by (ALLGOALS (resolve_tac prems));
by (REPEAT ((assume_tac 1) ORELSE (dtac infsys_pp2 1)));
qed "eval_ind";
(* ############################################################ *)
(* Elaborations *)
(* ############################################################ *)
goalw MT.thy [mono_def, elab_fun_def] "mono(elab_fun)";
by infsys_mono_tac;
qed "elab_fun_mono";
(* Introduction rules *)
val prems = goalw MT.thy [elab_def, elab_rel_def]
"c isof ty ==> te |- e_const(c) ===> ty";
by (cut_facts_tac prems 1);
by (rtac lfp_intro2 1);
by (rtac elab_fun_mono 1);
by (rewtac elab_fun_def);
by (rtac CollectI 1);br disjI1 1;
by (fast_tac HOL_cs 1);
qed "elab_const";
val prems = goalw MT.thy [elab_def, elab_rel_def]
"x:te_dom(te) ==> te |- e_var(x) ===> te_app te x";
by (cut_facts_tac prems 1);
by (rtac lfp_intro2 1);
by (rtac elab_fun_mono 1);
by (rewtac elab_fun_def);
by (rtac CollectI 1);br disjI2 1;br disjI1 1;
by (fast_tac HOL_cs 1);
qed "elab_var";
val prems = goalw MT.thy [elab_def, elab_rel_def]
"te + {x |=> ty1} |- e ===> ty2 ==> te |- fn x => e ===> ty1->ty2";
by (cut_facts_tac prems 1);
by (rtac lfp_intro2 1);
by (rtac elab_fun_mono 1);
by (rewtac elab_fun_def);
by (rtac CollectI 1);br disjI2 1;br disjI2 1;br disjI1 1;
by (fast_tac HOL_cs 1);
qed "elab_fn";
val prems = goalw MT.thy [elab_def, elab_rel_def]
" te + {f |=> ty1->ty2} + {x |=> ty1} |- e ===> ty2 ==> \
\ te |- fix f(x) = e ===> ty1->ty2";
by (cut_facts_tac prems 1);
by (rtac lfp_intro2 1);
by (rtac elab_fun_mono 1);
by (rewtac elab_fun_def);
by (rtac CollectI 1);br disjI2 1;br disjI2 1;br disjI2 1;br disjI1 1;
by (fast_tac HOL_cs 1);
qed "elab_fix";
val prems = goalw MT.thy [elab_def, elab_rel_def]
" [| te |- e1 ===> ty1->ty2; te |- e2 ===> ty1 |] ==> \
\ te |- e1 @ e2 ===> ty2";
by (cut_facts_tac prems 1);
by (rtac lfp_intro2 1);
by (rtac elab_fun_mono 1);
by (rewtac elab_fun_def);
by (rtac CollectI 1);br disjI2 1;br disjI2 1;br disjI2 1;br disjI2 1;
by (fast_tac HOL_cs 1);
qed "elab_app";
(* Strong elimination, induction on elaborations *)
val prems = goalw MT.thy [elab_def, elab_rel_def]
" [| 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>)";
by (resolve_tac (prems RL [lfp_ind2]) 1);
by (rtac elab_fun_mono 1);
by (rewtac elab_fun_def);
by (dtac CollectD 1);
by (safe_tac HOL_cs);
by (ALLGOALS (resolve_tac prems));
by (ALLGOALS (fast_tac set_cs));
qed "elab_ind0";
val prems = goal MT.thy
" [| 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";
by (res_inst_tac [("P","P")] infsys_pp2 1);
by (rtac elab_ind0 1);
by (ALLGOALS (rtac infsys_pp1));
by (ALLGOALS (resolve_tac prems));
by (REPEAT ((assume_tac 1) ORELSE (dtac infsys_pp2 1)));
qed "elab_ind";
(* Weak elimination, case analysis on elaborations *)
val prems = goalw MT.thy [elab_def, elab_rel_def]
" [| 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>)";
by (resolve_tac (prems RL [lfp_elim2]) 1);
by (rtac elab_fun_mono 1);
by (rewtac elab_fun_def);
by (dtac CollectD 1);
by (safe_tac HOL_cs);
by (ALLGOALS (resolve_tac prems));
by (ALLGOALS (fast_tac set_cs));
qed "elab_elim0";
val prems = goal MT.thy
" [| 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";
by (res_inst_tac [("P","P")] infsys_pp2 1);
by (rtac elab_elim0 1);
by (ALLGOALS (rtac infsys_pp1));
by (ALLGOALS (resolve_tac prems));
by (REPEAT ((assume_tac 1) ORELSE (dtac infsys_pp2 1)));
qed "elab_elim";
(* Elimination rules for each expression *)
fun elab_e_elim_tac p =
( (rtac elab_elim 1) THEN
(resolve_tac p 1) THEN
(REPEAT (fast_tac (e_ext_cs HOL_cs) 1))
);
val prems = goal MT.thy "te |- e ===> t ==> (e = e_const(c) --> c isof t)";
by (elab_e_elim_tac prems);
qed "elab_const_elim_lem";
val prems = goal MT.thy "te |- e_const(c) ===> t ==> c isof t";
by (cut_facts_tac prems 1);
by (dtac elab_const_elim_lem 1);
by (fast_tac prop_cs 1);
qed "elab_const_elim";
val prems = goal MT.thy
"te |- e ===> t ==> (e = e_var(x) --> t=te_app te x & x:te_dom(te))";
by (elab_e_elim_tac prems);
qed "elab_var_elim_lem";
val prems = goal MT.thy
"te |- e_var(ev) ===> t ==> t=te_app te ev & ev : te_dom(te)";
by (cut_facts_tac prems 1);
by (dtac elab_var_elim_lem 1);
by (fast_tac prop_cs 1);
qed "elab_var_elim";
val prems = goal MT.thy
" te |- e ===> t ==> \
\ ( e = fn x1 => e1 --> \
\ (? t1 t2.t=t_fun t1 t2 & te + {x1 |=> t1} |- e1 ===> t2) \
\ )";
by (elab_e_elim_tac prems);
qed "elab_fn_elim_lem";
val prems = goal MT.thy
" te |- fn x1 => e1 ===> t ==> \
\ (? t1 t2. t=t1->t2 & te + {x1 |=> t1} |- e1 ===> t2)";
by (cut_facts_tac prems 1);
by (dtac elab_fn_elim_lem 1);
by (fast_tac prop_cs 1);
qed "elab_fn_elim";
val prems = goal MT.thy
" te |- e ===> t ==> \
\ (e = fix f(x) = e1 --> \
\ (? t1 t2. t=t1->t2 & te + {f |=> t1->t2} + {x |=> t1} |- e1 ===> t2))";
by (elab_e_elim_tac prems);
qed "elab_fix_elim_lem";
val prems = goal MT.thy
" te |- fix ev1(ev2) = e1 ===> t ==> \
\ (? t1 t2. t=t1->t2 & te + {ev1 |=> t1->t2} + {ev2 |=> t1} |- e1 ===> t2)";
by (cut_facts_tac prems 1);
by (dtac elab_fix_elim_lem 1);
by (fast_tac prop_cs 1);
qed "elab_fix_elim";
val prems = goal MT.thy
" te |- e ===> t2 ==> \
\ (e = e1 @ e2 --> (? t1 . te |- e1 ===> t1->t2 & te |- e2 ===> t1))";
by (elab_e_elim_tac prems);
qed "elab_app_elim_lem";
val prems = goal MT.thy
"te |- e1 @ e2 ===> t2 ==> (? t1 . te |- e1 ===> t1->t2 & te |- e2 ===> t1)";
by (cut_facts_tac prems 1);
by (dtac elab_app_elim_lem 1);
by (fast_tac prop_cs 1);
qed "elab_app_elim";
(* ############################################################ *)
(* The extended correspondence relation *)
(* ############################################################ *)
(* Monotonicity of hasty_fun *)
goalw MT.thy [mono_def,MT.hasty_fun_def] "mono(hasty_fun)";
by infsys_mono_tac;
bind_thm("mono_hasty_fun", result());
(*
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 *)
val prems = goalw MT.thy [hasty_rel_def] "c isof t ==> <v_const(c),t> : hasty_rel";
by (cut_facts_tac prems 1);
by (rtac gfp_coind2 1);
by (rewtac MT.hasty_fun_def);
by (rtac CollectI 1);br disjI1 1;
by (fast_tac HOL_cs 1);
by (rtac mono_hasty_fun 1);
qed "hasty_rel_const_coind";
(* Second strong introduction (co-induction) rule for hasty_rel *)
val prems = goalw MT.thy [hasty_rel_def]
" [| 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";
by (cut_facts_tac prems 1);
by (rtac gfp_coind2 1);
by (rewtac hasty_fun_def);
by (rtac CollectI 1);br disjI2 1;
by (fast_tac HOL_cs 1);
by (rtac mono_hasty_fun 1);
qed "hasty_rel_clos_coind";
(* Elimination rule for hasty_rel *)
val prems = goalw MT.thy [hasty_rel_def]
" [| !! 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>)";
by (cut_facts_tac prems 1);
by (etac gfp_elim2 1);
by (rtac mono_hasty_fun 1);
by (rewtac hasty_fun_def);
by (dtac CollectD 1);
by (fold_goals_tac [hasty_fun_def]);
by (safe_tac HOL_cs);
by (ALLGOALS (resolve_tac prems));
by (ALLGOALS (fast_tac set_cs));
qed "hasty_rel_elim0";
val prems = goal MT.thy
" [| <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";
by (res_inst_tac [("P","P")] infsys_p2 1);
by (rtac hasty_rel_elim0 1);
by (ALLGOALS (rtac infsys_p1));
by (ALLGOALS (resolve_tac prems));
by (REPEAT ((assume_tac 1) ORELSE (dtac infsys_p2 1)));
qed "hasty_rel_elim";
(* Introduction rules for hasty *)
val prems = goalw MT.thy [hasty_def] "c isof t ==> v_const(c) hasty t";
by (resolve_tac (prems RL [hasty_rel_const_coind]) 1);
qed "hasty_const";
val prems = goalw MT.thy [hasty_def,hasty_env_def]
"te |- fn ev => e ===> t & ve hastyenv te ==> v_clos(<|ev,e,ve|>) hasty t";
by (cut_facts_tac prems 1);
by (rtac hasty_rel_clos_coind 1);
by (ALLGOALS (fast_tac set_cs));
qed "hasty_clos";
(* Elimination on constants for hasty *)
val prems = goalw MT.thy [hasty_def]
"v hasty t ==> (!c.(v = v_const(c) --> c isof t))";
by (cut_facts_tac prems 1);
by (rtac hasty_rel_elim 1);
by (ALLGOALS (fast_tac (v_ext_cs HOL_cs)));
qed "hasty_elim_const_lem";
val prems = goal MT.thy "v_const(c) hasty t ==> c isof t";
by (cut_facts_tac (prems RL [hasty_elim_const_lem]) 1);
by (fast_tac HOL_cs 1);
qed "hasty_elim_const";
(* Elimination on closures for hasty *)
val prems = goalw MT.thy [hasty_env_def,hasty_def]
" v hasty t ==> \
\ ! x e ve. \
\ v=v_clos(<|x,e,ve|>) --> (? te.te |- fn x => e ===> t & ve hastyenv te)";
by (cut_facts_tac prems 1);
by (rtac hasty_rel_elim 1);
by (ALLGOALS (fast_tac (v_ext_cs HOL_cs)));
qed "hasty_elim_clos_lem";
val prems = goal MT.thy
"v_clos(<|ev,e,ve|>) hasty t ==> ? te.te |- fn ev => e ===> t & ve hastyenv te ";
by (cut_facts_tac (prems RL [hasty_elim_clos_lem]) 1);
by (fast_tac HOL_cs 1);
qed "hasty_elim_clos";
(* ############################################################ *)
(* The pointwise extension of hasty to environments *)
(* ############################################################ *)
val prems = goal MT.thy
"[| ve hastyenv te; v hasty t |] ==> \
\ ve + {ev |-> v} hastyenv te + {ev |=> t}";
by (cut_facts_tac prems 1);
by (SELECT_GOAL (rewtac hasty_env_def) 1);
by (safe_tac HOL_cs);
by (rtac (ve_dom_owr RS ssubst) 1);
by (rtac (te_dom_owr RS ssubst) 1);
by (etac subst 1);br refl 1;
by (dtac (ve_dom_owr RS subst) 1);back();back();back();
by (etac UnSE 1);be conjE 1;
by (dtac notsingletonI 1);bd not_sym 1;
by (rtac (ve_app_owr2 RS ssubst) 1);ba 1;
by (rtac (te_app_owr2 RS ssubst) 1);ba 1;
by (fast_tac HOL_cs 1);
by (dtac singletonD 1);by (hyp_subst_tac 1);
by (rtac (ve_app_owr1 RS ssubst) 1);
by (rtac (te_app_owr1 RS ssubst) 1);
by (assume_tac 1);
qed "hasty_env1";
(* ############################################################ *)
(* The Consistency theorem *)
(* ############################################################ *)
val prems = goal MT.thy
"[| ve hastyenv te ; te |- e_const(c) ===> t |] ==> v_const(c) hasty t";
by (cut_facts_tac prems 1);
by (dtac elab_const_elim 1);
by (etac hasty_const 1);
qed "consistency_const";
val prems = goalw MT.thy [hasty_env_def]
" [| ev : ve_dom(ve); ve hastyenv te ; te |- e_var(ev) ===> t |] ==> \
\ ve_app ve ev hasty t";
by (cut_facts_tac prems 1);
by (dtac elab_var_elim 1);
by (fast_tac HOL_cs 1);
qed "consistency_var";
val prems = goal MT.thy
" [| ve hastyenv te ; te |- fn ev => e ===> t |] ==> \
\ v_clos(<| ev, e, ve |>) hasty t";
by (cut_facts_tac prems 1);
by (rtac hasty_clos 1);
by (fast_tac prop_cs 1);
qed "consistency_fn";
val prems = goalw MT.thy [hasty_env_def,hasty_def]
" [| cl = <| ev1, e, ve + { ev2 |-> v_clos(cl) } |>; \
\ ve hastyenv te ; \
\ te |- fix ev2 ev1 = e ===> t \
\ |] ==> \
\ v_clos(cl) hasty t";
by (cut_facts_tac prems 1);
by (dtac elab_fix_elim 1);
by (safe_tac HOL_cs);
by ((forward_tac [ssubst] 1) THEN (assume_tac 2) THEN
(rtac hasty_rel_clos_coind 1));
by (etac elab_fn 1);
by (rtac (ve_dom_owr RS ssubst) 1);
by (rtac (te_dom_owr RS ssubst) 1);
by ((etac subst 1) THEN (rtac refl 1));
by (rtac (ve_dom_owr RS ssubst) 1);
by (safe_tac HOL_cs);
by (dtac (Un_commute RS subst) 1);
by (etac UnSE 1);
by (safe_tac HOL_cs);
by (dtac notsingletonI 1);bd not_sym 1;
by (rtac (ve_app_owr2 RS ssubst) 1);ba 1;
by (rtac (te_app_owr2 RS ssubst) 1);ba 1;
by (fast_tac set_cs 1);
by (etac singletonE 1);
by (hyp_subst_tac 1);
by (rtac (ve_app_owr1 RS ssubst) 1);
by (rtac (te_app_owr1 RS ssubst) 1);
by (etac subst 1);
by (fast_tac set_cs 1);
qed "consistency_fix";
val prems = goal MT.thy
" [| ! 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";
by (cut_facts_tac prems 1);
by (dtac elab_app_elim 1);
by (safe_tac HOL_cs);
by (rtac hasty_const 1);
by (rtac isof_app 1);
by (rtac hasty_elim_const 1);
by (fast_tac HOL_cs 1);
by (rtac hasty_elim_const 1);
by (fast_tac HOL_cs 1);
qed "consistency_app1";
val prems = goal MT.thy
" [| ! 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";
by (cut_facts_tac prems 1);
by (dtac elab_app_elim 1);
by (safe_tac HOL_cs);
by ((etac allE 1) THEN (etac allE 1) THEN (etac impE 1));ba 1;be impE 1;ba 1;
by ((etac allE 1) THEN (etac allE 1) THEN (etac impE 1));ba 1;be impE 1;ba 1;
by (dtac hasty_elim_clos 1);
by (safe_tac HOL_cs);
by (dtac elab_fn_elim 1);
by (safe_tac HOL_cs);
by (dtac t_fun_inj 1);
by (safe_tac prop_cs);
by ((dtac hasty_env1 1) THEN (assume_tac 1) THEN (fast_tac HOL_cs 1));
qed "consistency_app2";
val prems = goal MT.thy
"ve |- e ---> v ==> (! t te. ve hastyenv te --> te |- e ===> t --> v hasty t)";
(* Proof by induction on the structure of evaluations *)
by (resolve_tac (prems RL [eval_ind]) 1);
by (safe_tac HOL_cs);
by (rtac consistency_const 1);ba 1;ba 1;
by (rtac consistency_var 1);ba 1;ba 1;ba 1;
by (rtac consistency_fn 1);ba 1;ba 1;
by (rtac consistency_fix 1);ba 1;ba 1;ba 1;
by (rtac consistency_app1 1);ba 1;ba 1;ba 1;ba 1;
by (rtac consistency_app2 1);ba 5;ba 4;ba 3;ba 2;ba 1;
qed "consistency";
(* ############################################################ *)
(* The Basic Consistency theorem *)
(* ############################################################ *)
val prems = goalw MT.thy [isof_env_def,hasty_env_def]
"ve isofenv te ==> ve hastyenv te";
by (cut_facts_tac prems 1);
by (safe_tac HOL_cs);
by (etac allE 1);be impE 1;ba 1;be exE 1;be conjE 1;
by (dtac hasty_const 1);
by ((rtac ssubst 1) THEN (assume_tac 1) THEN (assume_tac 1));
qed "basic_consistency_lem";
val prems = goal MT.thy
"[| ve isofenv te; ve |- e ---> v_const(c); te |- e ===> t |] ==> c isof t";
by (cut_facts_tac prems 1);
by (rtac hasty_elim_const 1);
by (dtac consistency 1);
by (fast_tac (HOL_cs addSIs [basic_consistency_lem]) 1);
qed "basic_consistency";