cleanup for Fun.thy:
merged Update.{thy|ML} into Fun.{thy|ML}
moved o_def from HOL.thy to Fun.thy
added Id_def to Fun.thy
moved image_compose from Set.ML to Fun.ML
moved o_apply and o_assoc from simpdata.ML to Fun.ML
moved fun_upd_same and fun_upd_other (from Map.ML) to Fun.ML
added fun_upd_twist to Fun.ML
(* Title: HOL/wf.ML
ID: $Id$
Author: Tobias Nipkow, with minor changes by Konrad Slind
Copyright 1992 University of Cambridge/1995 TU Munich
Wellfoundedness, induction, and recursion
*)
open WF;
val H_cong = read_instantiate [("f","H")] (standard(refl RS cong RS cong));
val H_cong1 = refl RS H_cong;
(*Restriction to domain A. If r is well-founded over A then wf(r)*)
val [prem1,prem2] = goalw WF.thy [wf_def]
"[| r <= A Times A; \
\ !!x P. [| ! x. (! y. (y,x) : r --> P(y)) --> P(x); x:A |] ==> P(x) |] \
\ ==> wf(r)";
by (Clarify_tac 1);
by (rtac allE 1);
by (assume_tac 1);
by (best_tac (claset() addSEs [prem1 RS subsetD RS SigmaE2] addIs [prem2]) 1);
qed "wfI";
val major::prems = goalw WF.thy [wf_def]
"[| wf(r); \
\ !!x.[| ! y. (y,x): r --> P(y) |] ==> P(x) \
\ |] ==> P(a)";
by (rtac (major RS spec RS mp RS spec) 1);
by (blast_tac (claset() addIs prems) 1);
qed "wf_induct";
(*Perform induction on i, then prove the wf(r) subgoal using prems. *)
fun wf_ind_tac a prems i =
EVERY [res_inst_tac [("a",a)] wf_induct i,
rename_last_tac a ["1"] (i+1),
ares_tac prems i];
val prems = goal WF.thy "[| wf(r); (a,x):r; (x,a):r |] ==> P";
by (subgoal_tac "! x. (a,x):r --> (x,a):r --> P" 1);
by (blast_tac (claset() addIs prems) 1);
by (wf_ind_tac "a" prems 1);
by (Blast_tac 1);
qed "wf_asym";
val prems = goal WF.thy "[| wf(r); (a,a): r |] ==> P";
by (rtac wf_asym 1);
by (REPEAT (resolve_tac prems 1));
qed "wf_irrefl";
(*transitive closure of a wf relation is wf! *)
val [prem] = goal WF.thy "wf(r) ==> wf(r^+)";
by (rewtac wf_def);
by (Clarify_tac 1);
(*must retain the universal formula for later use!*)
by (rtac allE 1 THEN assume_tac 1);
by (etac mp 1);
by (res_inst_tac [("a","x")] (prem RS wf_induct) 1);
by (rtac (impI RS allI) 1);
by (etac tranclE 1);
by (Blast_tac 1);
by (Blast_tac 1);
qed "wf_trancl";
val wf_converse_trancl = prove_goal thy
"!!X. wf (r^-1) ==> wf ((r^+)^-1)" (K [
stac (trancl_converse RS sym) 1,
etac wf_trancl 1]);
(*----------------------------------------------------------------------------
* Minimal-element characterization of well-foundedness
*---------------------------------------------------------------------------*)
val wfr::_ = goalw WF.thy [wf_def]
"wf r ==> x:Q --> (? z:Q. ! y. (y,z):r --> y~:Q)";
by (rtac (wfr RS spec RS mp RS spec) 1);
by (Blast_tac 1);
val lemma1 = result();
Goalw [wf_def]
"(! Q x. x:Q --> (? z:Q. ! y. (y,z):r --> y~:Q)) ==> wf r";
by (Clarify_tac 1);
by (dres_inst_tac [("x", "{x. ~ P x}")] spec 1);
by (Blast_tac 1);
val lemma2 = result();
Goal "wf r = (! Q x. x:Q --> (? z:Q. ! y. (y,z):r --> y~:Q))";
by (blast_tac (claset() addSIs [lemma1, lemma2]) 1);
qed "wf_eq_minimal";
(*---------------------------------------------------------------------------
* Wellfoundedness of subsets
*---------------------------------------------------------------------------*)
Goal "[| wf(r); p<=r |] ==> wf(p)";
by (full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1);
by (Fast_tac 1);
qed "wf_subset";
(*---------------------------------------------------------------------------
* Wellfoundedness of the empty relation.
*---------------------------------------------------------------------------*)
Goal "wf({})";
by (simp_tac (simpset() addsimps [wf_def]) 1);
qed "wf_empty";
AddIffs [wf_empty];
(*---------------------------------------------------------------------------
* Wellfoundedness of `insert'
*---------------------------------------------------------------------------*)
Goal "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)";
by (rtac iffI 1);
by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl]
addIs [rtrancl_into_trancl1,wf_subset,impOfSubs rtrancl_mono]) 1);
by (asm_full_simp_tac (simpset() addsimps [wf_eq_minimal]) 1);
by Safe_tac;
by (EVERY1[rtac allE, atac, etac impE, Blast_tac]);
by (etac bexE 1);
by (rename_tac "a" 1);
by (case_tac "a = x" 1);
by (res_inst_tac [("x","a")]bexI 2);
by (assume_tac 3);
by (Blast_tac 2);
by (case_tac "y:Q" 1);
by (Blast_tac 2);
by (res_inst_tac [("x","{z. z:Q & (z,y) : r^*}")] allE 1);
by (assume_tac 1);
by (thin_tac "! Q. (? x. x : Q) --> ?P Q" 1); (*essential for speed*)
(*Blast_tac with new substOccur fails*)
by (best_tac (claset() addIs [rtrancl_into_rtrancl2]) 1);
qed "wf_insert";
AddIffs [wf_insert];
(*---------------------------------------------------------------------------
* Wellfoundedness of `disjoint union'
*---------------------------------------------------------------------------*)
Goal
"[| !i:I. wf(r i); \
\ !i:I.!j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {} & \
\ Domain(r j) Int Range(r i) = {} \
\ |] ==> wf(UN i:I. r i)";
by(asm_full_simp_tac (HOL_basic_ss addsimps [wf_eq_minimal]) 1);
by(Clarify_tac 1);
by(rename_tac "A a" 1);
by(case_tac "? i:I. ? a:A. ? b:A. (b,a) : r i" 1);
by(Clarify_tac 1);
by(EVERY1[dtac bspec, atac,
eres_inst_tac[("x","{a. a:A & (? b:A. (b,a) : r i)}")]allE]);
by(EVERY1[etac allE,etac impE]);
by(Blast_tac 1);
by(Clarify_tac 1);
by(rename_tac "z'" 1);
by(res_inst_tac [("x","z'")] bexI 1);
ba 2;
by(Clarify_tac 1);
by(rename_tac "j" 1);
by(case_tac "r j = r i" 1);
by(EVERY1[etac allE,etac impE,atac]);
by(Asm_full_simp_tac 1);
by(Blast_tac 1);
by(blast_tac (claset() addEs [equalityE]) 1);
by(Asm_full_simp_tac 1);
by(case_tac "? i. i:I" 1);
by(Clarify_tac 1);
by(EVERY1[dtac bspec, atac, eres_inst_tac[("x","A")]allE]);
by(Blast_tac 1);
by(Blast_tac 1);
qed "wf_UN";
Goalw [Union_def]
"[| !r:R. wf r; \
\ !r:R.!s:R. r ~= s --> Domain r Int Range s = {} & \
\ Domain s Int Range r = {} \
\ |] ==> wf(Union R)";
br wf_UN 1;
by(Blast_tac 1);
by(Blast_tac 1);
qed "wf_Union";
Goal
"[| wf r; wf s; Domain r Int Range s = {}; Domain s Int Range r = {} \
\ |] ==> wf(r Un s)";
br(simplify (simpset()) (read_instantiate[("R","{r,s}")]wf_Union)) 1;
by(Blast_tac 1);
by(Blast_tac 1);
qed "wf_Un";
(*---------------------------------------------------------------------------
* Wellfoundedness of `image'
*---------------------------------------------------------------------------*)
Goal "[| wf r; inj f |] ==> wf(prod_fun f f `` r)";
by(asm_full_simp_tac (HOL_basic_ss addsimps [wf_eq_minimal]) 1);
by(Clarify_tac 1);
by(case_tac "? p. f p : Q" 1);
by(eres_inst_tac [("x","{p. f p : Q}")]allE 1);
by(fast_tac (claset() addDs [injD]) 1);
by(Blast_tac 1);
qed "wf_prod_fun_image";
(*** acyclic ***)
val acyclicI = prove_goalw WF.thy [acyclic_def]
"!!r. !x. (x, x) ~: r^+ ==> acyclic r" (K [atac 1]);
Goalw [acyclic_def]
"wf r ==> acyclic r";
by (blast_tac (claset() addEs [wf_trancl RS wf_irrefl]) 1);
qed "wf_acyclic";
Goalw [acyclic_def]
"acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)";
by (simp_tac (simpset() addsimps [trancl_insert]) 1);
by (blast_tac (claset() addEs [make_elim rtrancl_trans]) 1);
qed "acyclic_insert";
AddIffs [acyclic_insert];
Goalw [acyclic_def] "acyclic(r^-1) = acyclic r";
by (simp_tac (simpset() addsimps [trancl_converse]) 1);
qed "acyclic_converse";
(** cut **)
(*This rewrite rule works upon formulae; thus it requires explicit use of
H_cong to expose the equality*)
Goalw [cut_def]
"(cut f r x = cut g r x) = (!y. (y,x):r --> f(y)=g(y))";
by (simp_tac (HOL_ss addsimps [expand_fun_eq]) 1);
qed "cuts_eq";
Goalw [cut_def] "(x,a):r ==> (cut f r a)(x) = f(x)";
by (asm_simp_tac HOL_ss 1);
qed "cut_apply";
(*** is_recfun ***)
Goalw [is_recfun_def,cut_def]
"[| is_recfun r H a f; ~(b,a):r |] ==> f(b) = arbitrary";
by (etac ssubst 1);
by (asm_simp_tac HOL_ss 1);
qed "is_recfun_undef";
(*** NOTE! some simplifications need a different finish_tac!! ***)
fun indhyp_tac hyps =
(cut_facts_tac hyps THEN'
DEPTH_SOLVE_1 o (ares_tac [TrueI] ORELSE'
eresolve_tac [transD, mp, allE]));
val wf_super_ss = HOL_ss addSolver indhyp_tac;
val prems = goalw WF.thy [is_recfun_def,cut_def]
"[| wf(r); trans(r); is_recfun r H a f; is_recfun r H b g |] ==> \
\ (x,a):r --> (x,b):r --> f(x)=g(x)";
by (cut_facts_tac prems 1);
by (etac wf_induct 1);
by (REPEAT (rtac impI 1 ORELSE etac ssubst 1));
by (asm_simp_tac (wf_super_ss addcongs [if_cong]) 1);
qed_spec_mp "is_recfun_equal";
val prems as [wfr,transr,recfa,recgb,_] = goalw WF.thy [cut_def]
"[| wf(r); trans(r); \
\ is_recfun r H a f; is_recfun r H b g; (b,a):r |] ==> \
\ cut f r b = g";
val gundef = recgb RS is_recfun_undef
and fisg = recgb RS (recfa RS (transr RS (wfr RS is_recfun_equal)));
by (cut_facts_tac prems 1);
by (rtac ext 1);
by (asm_simp_tac (wf_super_ss addsimps [gundef,fisg]) 1);
qed "is_recfun_cut";
(*** Main Existence Lemma -- Basic Properties of the_recfun ***)
val prems = goalw WF.thy [the_recfun_def]
"is_recfun r H a f ==> is_recfun r H a (the_recfun r H a)";
by (res_inst_tac [("P", "is_recfun r H a")] selectI 1);
by (resolve_tac prems 1);
qed "is_the_recfun";
val prems = goal WF.thy
"!!r. [| wf(r); trans(r) |] ==> is_recfun r H a (the_recfun r H a)";
by (wf_ind_tac "a" prems 1);
by (res_inst_tac [("f","cut (%y. H (the_recfun r H y) y) r a1")]
is_the_recfun 1);
by (rewtac is_recfun_def);
by (stac cuts_eq 1);
by (Clarify_tac 1);
by (rtac (refl RSN (2,H_cong)) 1);
by (subgoal_tac
"the_recfun r H y = cut(%x. H(cut(the_recfun r H y) r x) x) r y" 1);
by (etac allE 2);
by (dtac impE 2);
by (atac 2);
by (atac 3);
by (atac 2);
by (etac ssubst 1);
by (simp_tac (HOL_ss addsimps [cuts_eq]) 1);
by (Clarify_tac 1);
by (stac cut_apply 1);
by (fast_tac (claset() addDs [transD]) 1);
by (rtac (refl RSN (2,H_cong)) 1);
by (fold_tac [is_recfun_def]);
by (asm_simp_tac (wf_super_ss addsimps[is_recfun_cut]) 1);
qed "unfold_the_recfun";
val unwind1_the_recfun = rewrite_rule[is_recfun_def] unfold_the_recfun;
(*--------------Old proof-----------------------------------------------------
val prems = goal WF.thy
"[| wf(r); trans(r) |] ==> is_recfun r H a (the_recfun r H a)";
by (cut_facts_tac prems 1);
by (wf_ind_tac "a" prems 1);
by (res_inst_tac [("f", "cut (%y. wftrec r H y) r a1")] is_the_recfun 1);
by (rewrite_goals_tac [is_recfun_def, wftrec_def]);
by (stac cuts_eq 1);
(*Applying the substitution: must keep the quantified assumption!!*)
by (EVERY1 [Clarify_tac, rtac H_cong1, rtac allE, atac,
etac (mp RS ssubst), atac]);
by (fold_tac [is_recfun_def]);
by (asm_simp_tac (wf_super_ss addsimps[cut_apply,is_recfun_cut,cuts_eq]) 1);
qed "unfold_the_recfun";
---------------------------------------------------------------------------*)
(** Removal of the premise trans(r) **)
val th = rewrite_rule[is_recfun_def]
(trans_trancl RSN (2,(wf_trancl RS unfold_the_recfun)));
Goalw [wfrec_def]
"wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a";
by (rtac H_cong 1);
by (rtac refl 2);
by (simp_tac (HOL_ss addsimps [cuts_eq]) 1);
by (rtac allI 1);
by (rtac impI 1);
by (simp_tac(HOL_ss addsimps [wfrec_def]) 1);
by (res_inst_tac [("a1","a")] (th RS ssubst) 1);
by (atac 1);
by (forward_tac[wf_trancl] 1);
by (forward_tac[r_into_trancl] 1);
by (asm_simp_tac (HOL_ss addsimps [cut_apply]) 1);
by (rtac H_cong 1); (*expose the equality of cuts*)
by (rtac refl 2);
by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1);
by (Clarify_tac 1);
by (res_inst_tac [("r","r^+")] is_recfun_equal 1);
by (atac 1);
by (rtac trans_trancl 1);
by (rtac unfold_the_recfun 1);
by (atac 1);
by (rtac trans_trancl 1);
by (rtac unfold_the_recfun 1);
by (atac 1);
by (rtac trans_trancl 1);
by (rtac transD 1);
by (rtac trans_trancl 1);
by (forw_inst_tac [("p","(ya,y)")] r_into_trancl 1);
by (atac 1);
by (atac 1);
by (forw_inst_tac [("p","(ya,y)")] r_into_trancl 1);
by (atac 1);
qed "wfrec";
(*--------------Old proof-----------------------------------------------------
Goalw [wfrec_def]
"wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a";
by (etac (wf_trancl RS wftrec RS ssubst) 1);
by (rtac trans_trancl 1);
by (rtac (refl RS H_cong) 1); (*expose the equality of cuts*)
by (simp_tac (HOL_ss addsimps [cuts_eq, cut_apply, r_into_trancl]) 1);
qed "wfrec";
---------------------------------------------------------------------------*)
(*---------------------------------------------------------------------------
* This form avoids giant explosions in proofs. NOTE USE OF ==
*---------------------------------------------------------------------------*)
val rew::prems = goal WF.thy
"[| f==wfrec r H; wf(r) |] ==> f(a) = H (cut f r a) a";
by (rewtac rew);
by (REPEAT (resolve_tac (prems@[wfrec]) 1));
qed "def_wfrec";
(**** TFL variants ****)
Goal "!R. wf R --> (!P. (!x. (!y. (y,x):R --> P y) --> P x) --> (!x. P x))";
by (Clarify_tac 1);
by (res_inst_tac [("r","R"),("P","P"), ("a","x")] wf_induct 1);
by (assume_tac 1);
by (Blast_tac 1);
qed"tfl_wf_induct";
Goal "!f R. (x,a):R --> (cut f R a)(x) = f(x)";
by (Clarify_tac 1);
by (rtac cut_apply 1);
by (assume_tac 1);
qed"tfl_cut_apply";
Goal "!M R f. (f=wfrec R M) --> wf R --> (!x. f x = M (cut f R x) x)";
by (Clarify_tac 1);
by (etac wfrec 1);
qed "tfl_wfrec";