INSTALLATION OF INDUCTIVE DEFINITIONS
HOL/ex/MT.thy: now mentions dependence upon Sum.thy
HOL/ex/Acc: new example, borrowed & adapted from ZF
HOL/ex/Simult, ex/Term: updated refs to Sexp intr rules
HOL/Sexp,List,LList,ex/Term: converted as follows
node *set -> item
Sexp -> sexp
LList_corec -> <self>
LList_ -> llist_
LList\> -> llist
List_case -> <self>
List_rec -> <self>
List_ -> list_
List\> -> list
Term_rec -> <self>
Term_ -> term_
Term\> -> term
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ex/Acc.ML Thu Aug 25 10:47:33 1994 +0200
@@ -0,0 +1,63 @@
+(* Title: HOL/ex/Acc
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 University of Cambridge
+
+Inductive definition of acc(r)
+
+See Ch. Paulin-Mohring, Inductive Definitions in the System Coq.
+Research Report 92-49, LIP, ENS Lyon. Dec 1992.
+*)
+
+open Acc;
+
+(*The intended introduction rule*)
+val prems = goal Acc.thy
+ "[| !!b. <b,a>:r ==> b: acc(r) |] ==> a: acc(r)";
+by (fast_tac (set_cs addIs (prems @
+ map (rewrite_rule [pred_def]) acc.intrs)) 1);
+val accI = result();
+
+goal Acc.thy "!!a b r. [| b: acc(r); <a,b>: r |] ==> a: acc(r)";
+by (etac acc.elim 1);
+by (rewtac pred_def);
+by (fast_tac set_cs 1);
+val acc_downward = result();
+
+val [major,indhyp] = goal Acc.thy
+ "[| a : acc(r); \
+\ !!x. [| x: acc(r); ALL y. <y,x>:r --> P(y) |] ==> P(x) \
+\ |] ==> P(a)";
+by (rtac (major RS acc.induct) 1);
+by (rtac indhyp 1);
+by (resolve_tac acc.intrs 1);
+by (rewtac pred_def);
+by (fast_tac set_cs 2);
+be (Int_lower1 RS Pow_mono RS subsetD) 1;
+val acc_induct = result();
+
+
+val [major] = goal Acc.thy "r <= Sigma(acc(r), %u. acc(r)) ==> wf(r)";
+by (rtac (major RS wfI) 1);
+by (etac acc.induct 1);
+by (rewtac pred_def);
+by (fast_tac set_cs 1);
+val acc_wfI = result();
+
+val [major] = goal Acc.thy "wf(r) ==> ALL x. <x,y>: r | <y,x>:r --> y: acc(r)";
+by (rtac (major RS wf_induct) 1);
+br (impI RS allI) 1;
+br accI 1;
+by (fast_tac set_cs 1);
+val acc_wfD_lemma = result();
+
+val [major] = goal Acc.thy "wf(r) ==> r <= Sigma(acc(r), %u. acc(r))";
+by (rtac subsetI 1);
+by (res_inst_tac [("p", "x")] PairE 1);
+by (fast_tac (set_cs addSIs [SigmaI,
+ major RS acc_wfD_lemma RS spec RS mp]) 1);
+val acc_wfD = result();
+
+goal Acc.thy "wf(r) = (r <= Sigma(acc(r), %u. acc(r)))";
+by (EVERY1 [rtac iffI, etac acc_wfD, etac acc_wfI]);
+val wf_acc_iff = result();
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ex/Acc.thy Thu Aug 25 10:47:33 1994 +0200
@@ -0,0 +1,26 @@
+(* Title: HOL/ex/Acc.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1994 University of Cambridge
+
+Inductive definition of acc(r)
+
+See Ch. Paulin-Mohring, Inductive Definitions in the System Coq.
+Research Report 92-49, LIP, ENS Lyon. Dec 1992.
+*)
+
+Acc = WF +
+
+consts
+ pred :: "['b, ('a * 'b)set] => 'a set" (*Set of predecessors*)
+ acc :: "('a * 'a)set => 'a set" (*Accessible part*)
+
+defs
+ pred_def "pred(x,r) == {y. <y,x>:r}"
+
+inductive "acc(r)"
+ intrs
+ pred "pred(a,r): Pow(acc(r)) ==> a: acc(r)"
+ monos "[Pow_mono]"
+
+end
--- a/ex/PL.ML Wed Aug 24 18:49:29 1994 +0200
+++ b/ex/PL.ML Thu Aug 25 10:47:33 1994 +0200
@@ -101,8 +101,7 @@
\ !!x. P(((x->false)->false)->x); \
\ !!x y. [| H |- x->y; H |- x; P(x->y); P(x) |] ==> P(y) \
\ |] ==> P(a)";
-by (rtac (major RS (thms_def RS def_induct)) 1);
-by (rtac thms_bnd_mono 1);
+by (rtac ([thms_def, thms_bnd_mono, major] MRS def_induct) 1);
by (rewrite_tac rule_defs);
by (fast_tac (set_cs addIs prems) 1);
val conseq_induct = result();
@@ -267,7 +266,7 @@
goal PL.thy "hyps(p,t) : Fin(UN v:{x.True}. {#v, #v->false})";
by (PL.pl.induct_tac "p" 1);
by (ALLGOALS (simp_tac (pl_ss setloop (split_tac [expand_if])) THEN'
- fast_tac (set_cs addSIs [Fin_0I, Fin_insertI, Fin_UnI])));
+ fast_tac (set_cs addSIs Fin.intrs@[Fin_UnI])));
val hyps_finite = result();
val Diff_weaken_left = subset_refl RSN (2, Diff_mono) RS weaken_left;
--- a/ex/ROOT.ML Wed Aug 24 18:49:29 1994 +0200
+++ b/ex/ROOT.ML Thu Aug 25 10:47:33 1994 +0200
@@ -20,6 +20,7 @@
time_use_thy "Puzzle";
time_use_thy "NatSum";
time_use "ex/set.ML";
+time_use_thy "Acc";
time_use_thy "PL";
time_use_thy "Term";
time_use_thy "Simult";
--- a/ex/Simult.ML Wed Aug 24 18:49:29 1994 +0200
+++ b/ex/Simult.ML Thu Aug 25 10:47:33 1994 +0200
@@ -3,13 +3,11 @@
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
-For Simult.thy.
-
Primitives for simultaneous recursive type definitions
includes worked example of trees & forests
This is essentially the same data structure that on ex/term.ML, which is
-simpler because it uses List as a new type former. The approach in this
+simpler because it uses list as a new type former. The approach in this
file may be superior for other simultaneous recursions.
*)
@@ -29,15 +27,15 @@
by (REPEAT (ares_tac [lfp_mono, subset_refl, usum_mono, uprod_mono] 1));
val TF_mono = result();
-goalw Simult.thy [TF_def] "TF(Sexp) <= Sexp";
+goalw Simult.thy [TF_def] "TF(sexp) <= sexp";
by (rtac lfp_lowerbound 1);
-by (fast_tac (univ_cs addIs [Sexp_NumbI,Sexp_In0I,Sexp_In1I,Sexp_SconsI]
+by (fast_tac (univ_cs addIs sexp.intrs@[sexp_In0I, sexp_In1I]
addSEs [PartE]) 1);
-val TF_Sexp = result();
+val TF_sexp = result();
-(* A <= Sexp ==> TF(A) <= Sexp *)
-val TF_subset_Sexp = standard
- (TF_mono RS (TF_Sexp RSN (2,subset_trans)));
+(* A <= sexp ==> TF(A) <= sexp *)
+val TF_subset_sexp = standard
+ (TF_mono RS (TF_sexp RSN (2,subset_trans)));
(** Elimination -- structural induction on the set TF **)
@@ -51,8 +49,7 @@
\ !!M N. [| M: Part(TF(A),In0); N: Part(TF(A),In1); R(M); R(N) \
\ |] ==> R(FCONS(M,N)) \
\ |] ==> R(i)";
-by (rtac (major RS (TF_def RS def_induct)) 1);
-by (rtac TF_fun_mono 1);
+by (rtac ([TF_def, TF_fun_mono, major] MRS def_induct) 1);
by (fast_tac (set_cs addIs (prems@[PartI])
addEs [usumE, uprodE, PartE]) 1);
val TF_induct = result();
@@ -228,19 +225,19 @@
val Fcons_inject = standard (Fcons_Fcons_eq RS iffD1 RS conjE);
-(*** TF_rec -- by wf recursion on pred_Sexp ***)
+(*** TF_rec -- by wf recursion on pred_sexp ***)
val TF_rec_unfold =
- wf_pred_Sexp RS wf_trancl RS (TF_rec_def RS def_wfrec);
+ wf_pred_sexp RS wf_trancl RS (TF_rec_def RS def_wfrec);
(** conversion rules for TF_rec **)
goalw Simult.thy [TCONS_def]
- "!!M N. [| M: Sexp; N: Sexp |] ==> \
+ "!!M N. [| M: sexp; N: sexp |] ==> \
\ TF_rec(TCONS(M,N),b,c,d) = b(M, N, TF_rec(N,b,c,d))";
by (rtac (TF_rec_unfold RS trans) 1);
by (simp_tac (HOL_ss addsimps [Case_In0, Split]) 1);
-by (asm_simp_tac (pred_Sexp_ss addsimps [In0_def]) 1);
+by (asm_simp_tac (pred_sexp_ss addsimps [In0_def]) 1);
val TF_rec_TCONS = result();
goalw Simult.thy [FNIL_def] "TF_rec(FNIL,b,c,d) = c";
@@ -249,29 +246,29 @@
val TF_rec_FNIL = result();
goalw Simult.thy [FCONS_def]
- "!!M N. [| M: Sexp; N: Sexp |] ==> \
+ "!!M N. [| M: sexp; N: sexp |] ==> \
\ TF_rec(FCONS(M,N),b,c,d) = d(M, N, TF_rec(M,b,c,d), TF_rec(N,b,c,d))";
by (rtac (TF_rec_unfold RS trans) 1);
by (simp_tac (HOL_ss addsimps [Case_In1, List_case_CONS]) 1);
-by (asm_simp_tac (pred_Sexp_ss addsimps [CONS_def,In1_def]) 1);
+by (asm_simp_tac (pred_sexp_ss addsimps [CONS_def,In1_def]) 1);
val TF_rec_FCONS = result();
(*** tree_rec, forest_rec -- by TF_rec ***)
-val Rep_Tree_in_Sexp =
- [range_Leaf_subset_Sexp RS TF_subset_Sexp RS (Part_subset RS subset_trans),
+val Rep_Tree_in_sexp =
+ [range_Leaf_subset_sexp RS TF_subset_sexp RS (Part_subset RS subset_trans),
Rep_Tree] MRS subsetD;
-val Rep_Forest_in_Sexp =
- [range_Leaf_subset_Sexp RS TF_subset_Sexp RS (Part_subset RS subset_trans),
+val Rep_Forest_in_sexp =
+ [range_Leaf_subset_sexp RS TF_subset_sexp RS (Part_subset RS subset_trans),
Rep_Forest] MRS subsetD;
val tf_rec_simps = [TF_rec_TCONS, TF_rec_FNIL, TF_rec_FCONS,
TCONS_I, FNIL_I, FCONS_I, Rep_Tree, Rep_Forest,
Rep_Tree_inverse, Rep_Forest_inverse,
Abs_Tree_inverse, Abs_Forest_inverse,
- inj_Leaf, Inv_f_f, Sexp_LeafI, range_eqI,
- Rep_Tree_in_Sexp, Rep_Forest_in_Sexp];
+ inj_Leaf, Inv_f_f, sexp.LeafI, range_eqI,
+ Rep_Tree_in_sexp, Rep_Forest_in_sexp];
val tf_rec_ss = HOL_ss addsimps tf_rec_simps;
goalw Simult.thy [tree_rec_def, forest_rec_def, Tcons_def]
--- a/ex/Simult.thy Wed Aug 24 18:49:29 1994 +0200
+++ b/ex/Simult.thy Thu Aug 25 10:47:33 1994 +0200
@@ -1,4 +1,4 @@
-(* Title: HOL/ex/simult
+(* Title: HOL/ex/Simult
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
@@ -6,8 +6,11 @@
A simultaneous recursive type definition: trees & forests
This is essentially the same data structure that on ex/term.ML, which is
-simpler because it uses List as a new type former. The approach in this
+simpler because it uses list as a new type former. The approach in this
file may be superior for other simultaneous recursions.
+
+The inductive definition package does not help defining this sort of mutually
+recursive data structure because it uses Inl, Inr instead of In0, In1.
*)
Simult = List +
@@ -18,27 +21,38 @@
arities tree,forest :: (term)term
consts
- TF :: "'a node set set => 'a node set set"
- FNIL :: "'a node set"
- TCONS,FCONS :: "['a node set, 'a node set] => 'a node set"
- Rep_Tree :: "'a tree => 'a node set"
- Abs_Tree :: "'a node set => 'a tree"
- Rep_Forest :: "'a forest => 'a node set"
- Abs_Forest :: "'a node set => 'a forest"
+ TF :: "'a item set => 'a item set"
+ FNIL :: "'a item"
+ TCONS,FCONS :: "['a item, 'a item] => 'a item"
+ Rep_Tree :: "'a tree => 'a item"
+ Abs_Tree :: "'a item => 'a tree"
+ Rep_Forest :: "'a forest => 'a item"
+ Abs_Forest :: "'a item => 'a forest"
Tcons :: "['a, 'a forest] => 'a tree"
Fcons :: "['a tree, 'a forest] => 'a forest"
Fnil :: "'a forest"
- TF_rec :: "['a node set, ['a node set , 'a node set, 'b]=>'b, \
-\ 'b, ['a node set , 'a node set, 'b, 'b]=>'b] => 'b"
+ TF_rec :: "['a item, ['a item , 'a item, 'b]=>'b, \
+\ 'b, ['a item , 'a item, 'b, 'b]=>'b] => 'b"
tree_rec :: "['a tree, ['a, 'a forest, 'b]=>'b, \
\ 'b, ['a tree, 'a forest, 'b, 'b]=>'b] => 'b"
forest_rec :: "['a forest, ['a, 'a forest, 'b]=>'b, \
\ 'b, ['a tree, 'a forest, 'b, 'b]=>'b] => 'b"
-rules
+defs
+ (*the concrete constants*)
+ TCONS_def "TCONS(M,N) == In0(M $ N)"
+ FNIL_def "FNIL == In1(NIL)"
+ FCONS_def "FCONS(M,N) == In1(CONS(M,N))"
+ (*the abstract constants*)
+ Tcons_def "Tcons(a,ts) == Abs_Tree(TCONS(Leaf(a), Rep_Forest(ts)))"
+ Fnil_def "Fnil == Abs_Forest(FNIL)"
+ Fcons_def "Fcons(t,ts) == Abs_Forest(FCONS(Rep_Tree(t), Rep_Forest(ts)))"
+
TF_def "TF(A) == lfp(%Z. A <*> Part(Z,In1) \
\ <+> ({Numb(0)} <+> Part(Z,In0) <*> Part(Z,In1)))"
- (*faking a type definition for tree...*)
+
+rules
+ (*faking a type definition for tree...*)
Rep_Tree "Rep_Tree(n): Part(TF(range(Leaf)),In0)"
Rep_Tree_inverse "Abs_Tree(Rep_Tree(t)) = t"
Abs_Tree_inverse "z: Part(TF(range(Leaf)),In0) ==> Rep_Tree(Abs_Tree(z)) = z"
@@ -48,18 +62,11 @@
Abs_Forest_inverse
"z: Part(TF(range(Leaf)),In1) ==> Rep_Forest(Abs_Forest(z)) = z"
- (*the concrete constants*)
- TCONS_def "TCONS(M,N) == In0(M $ N)"
- FNIL_def "FNIL == In1(NIL)"
- FCONS_def "FCONS(M,N) == In1(CONS(M,N))"
- (*the abstract constants*)
- Tcons_def "Tcons(a,ts) == Abs_Tree(TCONS(Leaf(a), Rep_Forest(ts)))"
- Fnil_def "Fnil == Abs_Forest(FNIL)"
- Fcons_def "Fcons(t,ts) == Abs_Forest(FCONS(Rep_Tree(t), Rep_Forest(ts)))"
+defs
(*recursion*)
TF_rec_def
- "TF_rec(M,b,c,d) == wfrec(trancl(pred_Sexp), M, \
+ "TF_rec(M,b,c,d) == wfrec(trancl(pred_sexp), M, \
\ Case(Split(%x y g. b(x,y,g(y))), \
\ List_case(%g.c, %x y g. d(x,y,g(x),g(y)))))"
--- a/ex/Term.ML Wed Aug 24 18:49:29 1994 +0200
+++ b/ex/Term.ML Thu Aug 25 10:47:33 1994 +0200
@@ -1,9 +1,9 @@
-(* Title: HOL/ex/term
+(* Title: HOL/ex/Term
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
-For term.thy. illustrates List functor
+Terms over a given alphabet -- function applications; illustrates list functor
(essentially the same type as in Trees & Forests)
*)
@@ -11,85 +11,83 @@
(*** Monotonicity and unfolding of the function ***)
-goal Term.thy "mono(%Z. A <*> List(Z))";
-by (REPEAT (ares_tac [monoI, subset_refl, List_mono, uprod_mono] 1));
-val Term_fun_mono = result();
+goal Term.thy "term(A) = A <*> list(term(A))";
+by (fast_tac (univ_cs addSIs (equalityI :: term.intrs)
+ addEs [term.elim]) 1);
+val term_unfold = result();
-val Term_unfold = Term_fun_mono RS (Term_def RS def_lfp_Tarski);
-
-(*This justifies using Term in other recursive type definitions*)
-goalw Term.thy [Term_def] "!!A B. A<=B ==> Term(A) <= Term(B)";
-by (REPEAT (ares_tac [lfp_mono, subset_refl, List_mono, uprod_mono] 1));
-val Term_mono = result();
+(*This justifies using term in other recursive type definitions*)
+goalw Term.thy term.defs "!!A B. A<=B ==> term(A) <= term(B)";
+by (REPEAT (ares_tac ([lfp_mono, list_mono] @ basic_monos) 1));
+val term_mono = result();
-(** Type checking rules -- Term creates well-founded sets **)
+(** Type checking -- term creates well-founded sets **)
-val prems = goalw Term.thy [Term_def] "Term(Sexp) <= Sexp";
+goalw Term.thy term.defs "term(sexp) <= sexp";
by (rtac lfp_lowerbound 1);
-by (fast_tac (univ_cs addIs [Sexp_SconsI, List_Sexp RS subsetD]) 1);
-val Term_Sexp = result();
+by (fast_tac (univ_cs addIs [sexp.SconsI, list_sexp RS subsetD]) 1);
+val term_sexp = result();
-(* A <= Sexp ==> Term(A) <= Sexp *)
-val Term_subset_Sexp = standard
- (Term_mono RS (Term_Sexp RSN (2,subset_trans)));
+(* A <= sexp ==> term(A) <= sexp *)
+val term_subset_sexp = standard ([term_mono, term_sexp] MRS subset_trans);
-(** Elimination -- structural induction on the set Term(A) **)
+(** Elimination -- structural induction on the set term(A) **)
-(*Induction for the set Term(A) *)
+(*Induction for the set term(A) *)
val [major,minor] = goal Term.thy
- "[| M: Term(A); \
-\ !!x zs. [| x: A; zs: List(Term(A)); zs: List({x.R(x)}) \
+ "[| M: term(A); \
+\ !!x zs. [| x: A; zs: list(term(A)); zs: list({x.R(x)}) \
\ |] ==> R(x$zs) \
\ |] ==> R(M)";
-by (rtac (major RS (Term_def RS def_induct)) 1);
-by (rtac Term_fun_mono 1);
-by (REPEAT (eresolve_tac ([uprodE, ssubst, minor] @
- ([Int_lower1,Int_lower2] RL [List_mono RS subsetD])) 1));
-val Term_induct = result();
+by (rtac (major RS term.induct) 1);
+by (REPEAT (eresolve_tac ([minor] @
+ ([Int_lower1,Int_lower2] RL [list_mono RS subsetD])) 1));
+(*Proof could also use mono_Int RS subsetD RS IntE *)
+val term_induct = result();
-(*Induction on Term(A) followed by induction on List *)
+(*Induction on term(A) followed by induction on list *)
val major::prems = goal Term.thy
- "[| M: Term(A); \
+ "[| M: term(A); \
\ !!x. [| x: A |] ==> R(x$NIL); \
-\ !!x z zs. [| x: A; z: Term(A); zs: List(Term(A)); R(x$zs) \
+\ !!x z zs. [| x: A; z: term(A); zs: list(term(A)); R(x$zs) \
\ |] ==> R(x $ CONS(z,zs)) \
\ |] ==> R(M)";
-by (rtac (major RS Term_induct) 1);
-by (etac List_induct 1);
+by (rtac (major RS term_induct) 1);
+by (etac list.induct 1);
by (REPEAT (ares_tac prems 1));
-val Term_induct2 = result();
+val term_induct2 = result();
(*** Structural Induction on the abstract type 'a term ***)
val list_all_ss = map_ss addsimps [list_all_Nil, list_all_Cons];
-val Rep_Term_in_Sexp =
- Rep_Term RS (range_Leaf_subset_Sexp RS Term_subset_Sexp RS subsetD);
+val Rep_term_in_sexp =
+ Rep_term RS (range_Leaf_subset_sexp RS term_subset_sexp RS subsetD);
(*Induction for the abstract type 'a term*)
-val prems = goalw Term.thy [App_def,Rep_TList_def,Abs_TList_def]
+val prems = goalw Term.thy [App_def,Rep_Tlist_def,Abs_Tlist_def]
"[| !!x ts. list_all(R,ts) ==> R(App(x,ts)) \
\ |] ==> R(t)";
-by (rtac (Rep_Term_inverse RS subst) 1); (*types force good instantiation*)
-by (res_inst_tac [("P","Rep_Term(t) : Sexp")] conjunct2 1);
-by (rtac (Rep_Term RS Term_induct) 1);
-by (REPEAT (ares_tac [conjI, Sexp_SconsI, Term_subset_Sexp RS
- List_subset_Sexp,range_Leaf_subset_Sexp] 1
+by (rtac (Rep_term_inverse RS subst) 1); (*types force good instantiation*)
+by (res_inst_tac [("P","Rep_term(t) : sexp")] conjunct2 1);
+by (rtac (Rep_term RS term_induct) 1);
+by (REPEAT (ares_tac [conjI, sexp.SconsI, term_subset_sexp RS
+ list_subset_sexp, range_Leaf_subset_sexp] 1
ORELSE etac rev_subsetD 1));
-by (eres_inst_tac [("A1","Term(?u)"), ("f1","Rep_Term"), ("g1","Abs_Term")]
+by (eres_inst_tac [("A1","term(?u)"), ("f1","Rep_term"), ("g1","Abs_term")]
(Abs_map_inverse RS subst) 1);
-by (rtac (range_Leaf_subset_Sexp RS Term_subset_Sexp) 1);
-by (etac Abs_Term_inverse 1);
+by (rtac (range_Leaf_subset_sexp RS term_subset_sexp) 1);
+by (etac Abs_term_inverse 1);
by (etac rangeE 1);
by (hyp_subst_tac 1);
by (resolve_tac prems 1);
-by (etac List_induct 1);
+by (etac list.induct 1);
by (etac CollectE 2);
by (stac Abs_map_CONS 2);
by (etac conjunct1 2);
by (etac rev_subsetD 2);
-by (rtac List_subset_Sexp 2);
+by (rtac list_subset_sexp 2);
by (fast_tac set_cs 2);
by (ALLGOALS (asm_simp_tac list_all_ss));
val term_induct = result();
@@ -111,66 +109,56 @@
rename_last_tac a ["1","s"] (i+1)];
-(** Introduction rule for Term **)
-(* c : A <*> List(Term(A)) ==> c : Term(A) *)
-val TermI = standard (Term_unfold RS equalityD2 RS subsetD);
-
-(*The constant APP is not declared; it is simply . *)
-val prems = goal Term.thy "[| M: A; N : List(Term(A)) |] ==> M$N : Term(A)";
-by (REPEAT (resolve_tac (prems@[TermI, ListI, uprodI]) 1));
-val APP_I = result();
-
-
-(*** Term_rec -- by wf recursion on pred_Sexp ***)
+(*** Term_rec -- by wf recursion on pred_sexp ***)
val Term_rec_unfold =
- wf_pred_Sexp RS wf_trancl RS (Term_rec_def RS def_wfrec);
+ wf_pred_sexp RS wf_trancl RS (Term_rec_def RS def_wfrec);
(** conversion rules **)
val [prem] = goal Term.thy
- "N: List(Term(A)) ==> \
-\ !M. <N,M>: pred_Sexp^+ --> \
-\ Abs_map(cut(h, pred_Sexp^+, M), N) = \
+ "N: list(term(A)) ==> \
+\ !M. <N,M>: pred_sexp^+ --> \
+\ Abs_map(cut(h, pred_sexp^+, M), N) = \
\ Abs_map(h,N)";
-by (rtac (prem RS List_induct) 1);
+by (rtac (prem RS list.induct) 1);
by (simp_tac list_all_ss 1);
by (strip_tac 1);
-by (etac (pred_Sexp_CONS_D RS conjE) 1);
-by (asm_simp_tac (list_all_ss addsimps [trancl_pred_SexpD1, cut_apply]) 1);
+by (etac (pred_sexp_CONS_D RS conjE) 1);
+by (asm_simp_tac (list_all_ss addsimps [trancl_pred_sexpD1, cut_apply]) 1);
val Abs_map_lemma = result();
-val [prem1,prem2,A_subset_Sexp] = goal Term.thy
- "[| M: Sexp; N: List(Term(A)); A<=Sexp |] ==> \
+val [prem1,prem2,A_subset_sexp] = goal Term.thy
+ "[| M: sexp; N: list(term(A)); A<=sexp |] ==> \
\ Term_rec(M$N, d) = d(M, N, Abs_map(%Z. Term_rec(Z,d), N))";
by (rtac (Term_rec_unfold RS trans) 1);
by (simp_tac (HOL_ss addsimps
[Split,
- prem2 RS Abs_map_lemma RS spec RS mp, pred_SexpI2 RS r_into_trancl,
- prem1, prem2 RS rev_subsetD, List_subset_Sexp,
- Term_subset_Sexp, A_subset_Sexp])1);
+ prem2 RS Abs_map_lemma RS spec RS mp, pred_sexpI2 RS r_into_trancl,
+ prem1, prem2 RS rev_subsetD, list_subset_sexp,
+ term_subset_sexp, A_subset_sexp])1);
val Term_rec = result();
(*** term_rec -- by Term_rec ***)
local
val Rep_map_type1 = read_instantiate_sg (sign_of Term.thy)
- [("f","Rep_Term")] Rep_map_type;
- val Rep_TList = Rep_Term RS Rep_map_type1;
- val Rep_Term_rec = range_Leaf_subset_Sexp RSN (2,Rep_TList RSN(2,Term_rec));
+ [("f","Rep_term")] Rep_map_type;
+ val Rep_Tlist = Rep_term RS Rep_map_type1;
+ val Rep_Term_rec = range_Leaf_subset_sexp RSN (2,Rep_Tlist RSN(2,Term_rec));
- (*Now avoids conditional rewriting with the premise N: List(Term(A)),
+ (*Now avoids conditional rewriting with the premise N: list(term(A)),
since A will be uninstantiated and will cause rewriting to fail. *)
val term_rec_ss = HOL_ss
- addsimps [Rep_TList RS (rangeI RS APP_I RS Abs_Term_inverse),
- Rep_Term_in_Sexp, Rep_Term_rec, Rep_Term_inverse,
+ addsimps [Rep_Tlist RS (rangeI RS term.APP_I RS Abs_term_inverse),
+ Rep_term_in_sexp, Rep_Term_rec, Rep_term_inverse,
inj_Leaf, Inv_f_f,
- Abs_Rep_map, map_ident, Sexp_LeafI]
+ Abs_Rep_map, map_ident, sexp.LeafI]
in
val term_rec = prove_goalw Term.thy
- [term_rec_def, App_def, Rep_TList_def, Abs_TList_def]
+ [term_rec_def, App_def, Rep_Tlist_def, Abs_Tlist_def]
"term_rec(App(f,ts), d) = d(f, ts, map (%t. term_rec(t,d), ts))"
(fn _ => [simp_tac term_rec_ss 1])
--- a/ex/Term.thy Wed Aug 24 18:49:29 1994 +0200
+++ b/ex/Term.thy Thu Aug 25 10:47:33 1994 +0200
@@ -1,12 +1,12 @@
-(* Title: HOL/ex/term
+(* Title: HOL/ex/Term
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1992 University of Cambridge
-Terms over a given alphabet -- function applications; illustrates List functor
+Terms over a given alphabet -- function applications; illustrates list functor
(essentially the same type as in Trees & Forests)
-There is no constructor APP because it is simply cons (.)
+There is no constructor APP because it is simply cons ($)
*)
Term = List +
@@ -16,33 +16,40 @@
arities term :: (term)term
consts
- Term :: "'a node set set => 'a node set set"
- Rep_Term :: "'a term => 'a node set"
- Abs_Term :: "'a node set => 'a term"
- Rep_TList :: "'a term list => 'a node set"
- Abs_TList :: "'a node set => 'a term list"
+ term :: "'a item set => 'a item set"
+ Rep_term :: "'a term => 'a item"
+ Abs_term :: "'a item => 'a term"
+ Rep_Tlist :: "'a term list => 'a item"
+ Abs_Tlist :: "'a item => 'a term list"
App :: "['a, ('a term)list] => 'a term"
- Term_rec ::
- "['a node set, ['a node set , 'a node set, 'b list]=>'b] => 'b"
+ Term_rec :: "['a item, ['a item , 'a item, 'b list]=>'b] => 'b"
term_rec :: "['a term, ['a ,'a term list, 'b list]=>'b] => 'b"
-rules
- Term_def "Term(A) == lfp(%Z. A <*> List(Z))"
- (*faking a type definition for term...*)
- Rep_Term "Rep_Term(n): Term(range(Leaf))"
- Rep_Term_inverse "Abs_Term(Rep_Term(t)) = t"
- Abs_Term_inverse "M: Term(range(Leaf)) ==> Rep_Term(Abs_Term(M)) = M"
- (*defining abstraction/representation functions for term list...*)
- Rep_TList_def "Rep_TList == Rep_map(Rep_Term)"
- Abs_TList_def "Abs_TList == Abs_map(Abs_Term)"
- (*defining the abstract constants*)
- App_def "App(a,ts) == Abs_Term(Leaf(a) $ Rep_TList(ts))"
- (*list recursion*)
+inductive "term(A)"
+ intrs
+ APP_I "[| M: A; N : list(term(A)) |] ==> M$N : term(A)"
+ monos "[list_mono]"
+
+defs
+ (*defining abstraction/representation functions for term list...*)
+ Rep_Tlist_def "Rep_Tlist == Rep_map(Rep_term)"
+ Abs_Tlist_def "Abs_Tlist == Abs_map(Abs_term)"
+
+ (*defining the abstract constants*)
+ App_def "App(a,ts) == Abs_term(Leaf(a) $ Rep_Tlist(ts))"
+
+ (*list recursion*)
Term_rec_def
- "Term_rec(M,d) == wfrec(trancl(pred_Sexp), M, \
+ "Term_rec(M,d) == wfrec(trancl(pred_sexp), M, \
\ Split(%x y g. d(x,y, Abs_map(g,y))))"
term_rec_def
"term_rec(t,d) == \
-\ Term_rec(Rep_Term(t), %x y r. d(Inv(Leaf,x), Abs_TList(y), r))"
+\ Term_rec(Rep_term(t), %x y r. d(Inv(Leaf,x), Abs_Tlist(y), r))"
+
+rules
+ (*faking a type definition for term...*)
+ Rep_term "Rep_term(n): term(range(Leaf))"
+ Rep_term_inverse "Abs_term(Rep_term(t)) = t"
+ Abs_term_inverse "M: term(range(Leaf)) ==> Rep_term(Abs_term(M)) = M"
end