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(* Title: HOL/ex/Simult.ML
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1993 University of Cambridge
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Primitives for simultaneous recursive type definitions
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includes worked example of trees & forests
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This is essentially the same data structure that on ex/term.ML, which is
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simpler because it uses list as a new type former. The approach in this
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file may be superior for other simultaneous recursions.
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*)
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open Simult;
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(*** Monotonicity and unfolding of the function ***)
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goal Simult.thy "mono(%Z. A <*> Part Z In1 \
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\ <+> ({Numb(0)} <+> Part Z In0 <*> Part Z In1))";
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by (REPEAT (ares_tac [monoI, subset_refl, usum_mono, uprod_mono,
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Part_mono] 1));
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qed "TF_fun_mono";
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val TF_unfold = TF_fun_mono RS (TF_def RS def_lfp_Tarski);
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goalw Simult.thy [TF_def] "!!A B. A<=B ==> TF(A) <= TF(B)";
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by (REPEAT (ares_tac [lfp_mono, subset_refl, usum_mono, uprod_mono] 1));
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qed "TF_mono";
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goalw Simult.thy [TF_def] "TF(sexp) <= sexp";
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by (rtac lfp_lowerbound 1);
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by (fast_tac (univ_cs addIs sexp.intrs@[sexp_In0I, sexp_In1I]
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addSEs [PartE]) 1);
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qed "TF_sexp";
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(* A <= sexp ==> TF(A) <= sexp *)
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val TF_subset_sexp = standard
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(TF_mono RS (TF_sexp RSN (2,subset_trans)));
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(** Elimination -- structural induction on the set TF **)
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val TF_Rep_defs = [TCONS_def,FNIL_def,FCONS_def,NIL_def,CONS_def];
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val major::prems = goalw Simult.thy TF_Rep_defs
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"[| i: TF(A); \
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\ !!M N. [| M: A; N: Part (TF A) In1; R(N) |] ==> R(TCONS M N); \
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\ R(FNIL); \
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\ !!M N. [| M: Part (TF A) In0; N: Part (TF A) In1; R(M); R(N) \
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\ |] ==> R(FCONS M N) \
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\ |] ==> R(i)";
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by (rtac ([TF_def, TF_fun_mono, major] MRS def_induct) 1);
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by (fast_tac (set_cs addIs (prems@[PartI])
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addEs [usumE, uprodE, PartE]) 1);
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qed "TF_induct";
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(*This lemma replaces a use of subgoal_tac to prove tree_forest_induct*)
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val prems = goalw Simult.thy [Part_def]
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"! M: TF(A). (M: Part (TF A) In0 --> P(M)) & (M: Part (TF A) In1 --> Q(M)) \
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\ ==> (! M: Part (TF A) In0. P(M)) & (! M: Part (TF A) In1. Q(M))";
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by (cfast_tac prems 1);
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qed "TF_induct_lemma";
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val uplus_cs = set_cs addSIs [PartI]
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addSDs [In0_inject, In1_inject]
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addSEs [In0_neq_In1, In1_neq_In0, PartE];
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(*Could prove ~ TCONS M N : Part (TF A) In1 etc. *)
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(*Induction on TF with separate predicates P, Q*)
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val prems = goalw Simult.thy TF_Rep_defs
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"[| !!M N. [| M: A; N: Part (TF A) In1; Q(N) |] ==> P(TCONS M N); \
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\ Q(FNIL); \
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\ !!M N. [| M: Part (TF A) In0; N: Part (TF A) In1; P(M); Q(N) \
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\ |] ==> Q(FCONS M N) \
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\ |] ==> (! M: Part (TF A) In0. P(M)) & (! N: Part (TF A) In1. Q(N))";
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by (rtac (ballI RS TF_induct_lemma) 1);
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by (etac TF_induct 1);
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by (rewrite_goals_tac TF_Rep_defs);
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by (ALLGOALS (fast_tac (uplus_cs addIs prems)));
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(*29 secs??*)
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qed "Tree_Forest_induct";
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(*Induction for the abstract types 'a tree, 'a forest*)
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val prems = goalw Simult.thy [Tcons_def,Fnil_def,Fcons_def]
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"[| !!x ts. Q(ts) ==> P(Tcons x ts); \
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\ Q(Fnil); \
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\ !!t ts. [| P(t); Q(ts) |] ==> Q(Fcons t ts) \
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\ |] ==> (! t. P(t)) & (! ts. Q(ts))";
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by (res_inst_tac [("P1","%z.P(Abs_Tree(z))"),
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("Q1","%z.Q(Abs_Forest(z))")]
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(Tree_Forest_induct RS conjE) 1);
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(*Instantiates ?A1 to range(Leaf). *)
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by (fast_tac (set_cs addSEs [Rep_Tree_inverse RS subst,
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Rep_Forest_inverse RS subst]
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addSIs [Rep_Tree,Rep_Forest]) 4);
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(*Cannot use simplifier: the rewrites work in the wrong direction!*)
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by (ALLGOALS (fast_tac (set_cs addSEs [Abs_Tree_inverse RS subst,
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Abs_Forest_inverse RS subst]
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addSIs prems)));
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qed "tree_forest_induct";
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(*** Isomorphisms ***)
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goal Simult.thy "inj(Rep_Tree)";
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by (rtac inj_inverseI 1);
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by (rtac Rep_Tree_inverse 1);
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qed "inj_Rep_Tree";
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goal Simult.thy "inj_onto Abs_Tree (Part (TF(range Leaf)) In0)";
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by (rtac inj_onto_inverseI 1);
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by (etac Abs_Tree_inverse 1);
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qed "inj_onto_Abs_Tree";
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goal Simult.thy "inj(Rep_Forest)";
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by (rtac inj_inverseI 1);
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by (rtac Rep_Forest_inverse 1);
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qed "inj_Rep_Forest";
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goal Simult.thy "inj_onto Abs_Forest (Part (TF(range Leaf)) In1)";
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by (rtac inj_onto_inverseI 1);
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by (etac Abs_Forest_inverse 1);
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qed "inj_onto_Abs_Forest";
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(** Introduction rules for constructors **)
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(* c : A <*> Part (TF A) In1
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<+> {Numb(0)} <+> Part (TF A) In0 <*> Part (TF A) In1 ==> c : TF(A) *)
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val TF_I = TF_unfold RS equalityD2 RS subsetD;
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(*For reasoning about the representation*)
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val TF_Rep_cs = uplus_cs addIs [TF_I, uprodI, usum_In0I, usum_In1I]
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addSEs [Scons_inject];
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val prems = goalw Simult.thy TF_Rep_defs
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"[| a: A; M: Part (TF A) In1 |] ==> TCONS a M : Part (TF A) In0";
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by (fast_tac (TF_Rep_cs addIs prems) 1);
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qed "TCONS_I";
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(* FNIL is a TF(A) -- this also justifies the type definition*)
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goalw Simult.thy TF_Rep_defs "FNIL: Part (TF A) In1";
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by (fast_tac TF_Rep_cs 1);
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qed "FNIL_I";
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val prems = goalw Simult.thy TF_Rep_defs
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"[| M: Part (TF A) In0; N: Part (TF A) In1 |] ==> \
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\ FCONS M N : Part (TF A) In1";
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by (fast_tac (TF_Rep_cs addIs prems) 1);
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qed "FCONS_I";
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(** Injectiveness of TCONS and FCONS **)
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goalw Simult.thy TF_Rep_defs "(TCONS K M=TCONS L N) = (K=L & M=N)";
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by (fast_tac TF_Rep_cs 1);
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qed "TCONS_TCONS_eq";
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bind_thm ("TCONS_inject", (TCONS_TCONS_eq RS iffD1 RS conjE));
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goalw Simult.thy TF_Rep_defs "(FCONS K M=FCONS L N) = (K=L & M=N)";
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by (fast_tac TF_Rep_cs 1);
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qed "FCONS_FCONS_eq";
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bind_thm ("FCONS_inject", (FCONS_FCONS_eq RS iffD1 RS conjE));
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(** Distinctness of TCONS, FNIL and FCONS **)
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goalw Simult.thy TF_Rep_defs "TCONS M N ~= FNIL";
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by (fast_tac TF_Rep_cs 1);
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qed "TCONS_not_FNIL";
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bind_thm ("FNIL_not_TCONS", (TCONS_not_FNIL RS not_sym));
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bind_thm ("TCONS_neq_FNIL", (TCONS_not_FNIL RS notE));
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val FNIL_neq_TCONS = sym RS TCONS_neq_FNIL;
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goalw Simult.thy TF_Rep_defs "FCONS M N ~= FNIL";
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by (fast_tac TF_Rep_cs 1);
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qed "FCONS_not_FNIL";
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bind_thm ("FNIL_not_FCONS", (FCONS_not_FNIL RS not_sym));
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bind_thm ("FCONS_neq_FNIL", (FCONS_not_FNIL RS notE));
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val FNIL_neq_FCONS = sym RS FCONS_neq_FNIL;
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goalw Simult.thy TF_Rep_defs "TCONS M N ~= FCONS K L";
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by (fast_tac TF_Rep_cs 1);
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qed "TCONS_not_FCONS";
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bind_thm ("FCONS_not_TCONS", (TCONS_not_FCONS RS not_sym));
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bind_thm ("TCONS_neq_FCONS", (TCONS_not_FCONS RS notE));
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val FCONS_neq_TCONS = sym RS TCONS_neq_FCONS;
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(*???? Too many derived rules ????
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Automatically generate symmetric forms? Always expand TF_Rep_defs? *)
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(** Injectiveness of Tcons and Fcons **)
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(*For reasoning about abstract constructors*)
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val TF_cs = set_cs addSIs [Rep_Tree, Rep_Forest, TCONS_I, FNIL_I, FCONS_I]
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addSEs [TCONS_inject, FCONS_inject,
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TCONS_neq_FNIL, FNIL_neq_TCONS,
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FCONS_neq_FNIL, FNIL_neq_FCONS,
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TCONS_neq_FCONS, FCONS_neq_TCONS]
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addSDs [inj_onto_Abs_Tree RS inj_ontoD,
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inj_onto_Abs_Forest RS inj_ontoD,
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inj_Rep_Tree RS injD, inj_Rep_Forest RS injD,
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Leaf_inject];
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goalw Simult.thy [Tcons_def] "(Tcons x xs=Tcons y ys) = (x=y & xs=ys)";
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by (fast_tac TF_cs 1);
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qed "Tcons_Tcons_eq";
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bind_thm ("Tcons_inject", (Tcons_Tcons_eq RS iffD1 RS conjE));
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goalw Simult.thy [Fcons_def,Fnil_def] "Fcons x xs ~= Fnil";
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by (fast_tac TF_cs 1);
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qed "Fcons_not_Fnil";
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bind_thm ("Fcons_neq_Fnil", Fcons_not_Fnil RS notE);
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val Fnil_neq_Fcons = sym RS Fcons_neq_Fnil;
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(** Injectiveness of Fcons **)
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goalw Simult.thy [Fcons_def] "(Fcons x xs=Fcons y ys) = (x=y & xs=ys)";
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by (fast_tac TF_cs 1);
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qed "Fcons_Fcons_eq";
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bind_thm ("Fcons_inject", Fcons_Fcons_eq RS iffD1 RS conjE);
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(*** TF_rec -- by wf recursion on pred_sexp ***)
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val TF_rec_unfold =
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wf_pred_sexp RS wf_trancl RS (TF_rec_def RS def_wfrec);
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(** conversion rules for TF_rec **)
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goalw Simult.thy [TCONS_def]
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"!!M N. [| M: sexp; N: sexp |] ==> \
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\ TF_rec (TCONS M N) b c d = b M N (TF_rec N b c d)";
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by (rtac (TF_rec_unfold RS trans) 1);
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by (simp_tac (HOL_ss addsimps [Case_In0, Split]) 1);
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by (asm_simp_tac (pred_sexp_ss addsimps [In0_def]) 1);
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qed "TF_rec_TCONS";
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goalw Simult.thy [FNIL_def] "TF_rec FNIL b c d = c";
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by (rtac (TF_rec_unfold RS trans) 1);
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by (simp_tac (HOL_ss addsimps [Case_In1, List_case_NIL]) 1);
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qed "TF_rec_FNIL";
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goalw Simult.thy [FCONS_def]
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"!!M N. [| M: sexp; N: sexp |] ==> \
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\ TF_rec (FCONS M N) b c d = d M N (TF_rec M b c d) (TF_rec N b c d)";
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by (rtac (TF_rec_unfold RS trans) 1);
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by (simp_tac (HOL_ss addsimps [Case_In1, List_case_CONS]) 1);
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by (asm_simp_tac (pred_sexp_ss addsimps [CONS_def,In1_def]) 1);
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qed "TF_rec_FCONS";
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(*** tree_rec, forest_rec -- by TF_rec ***)
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val Rep_Tree_in_sexp =
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[range_Leaf_subset_sexp RS TF_subset_sexp RS (Part_subset RS subset_trans),
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Rep_Tree] MRS subsetD;
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val Rep_Forest_in_sexp =
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[range_Leaf_subset_sexp RS TF_subset_sexp RS (Part_subset RS subset_trans),
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Rep_Forest] MRS subsetD;
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val tf_rec_simps = [TF_rec_TCONS, TF_rec_FNIL, TF_rec_FCONS,
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TCONS_I, FNIL_I, FCONS_I, Rep_Tree, Rep_Forest,
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Rep_Tree_inverse, Rep_Forest_inverse,
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Abs_Tree_inverse, Abs_Forest_inverse,
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inj_Leaf, Inv_f_f, sexp.LeafI, range_eqI,
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Rep_Tree_in_sexp, Rep_Forest_in_sexp];
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val tf_rec_ss = HOL_ss addsimps tf_rec_simps;
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goalw Simult.thy [tree_rec_def, forest_rec_def, Tcons_def]
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"tree_rec (Tcons a tf) b c d = b a tf (forest_rec tf b c d)";
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by (simp_tac tf_rec_ss 1);
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qed "tree_rec_Tcons";
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goalw Simult.thy [forest_rec_def, Fnil_def] "forest_rec Fnil b c d = c";
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by (simp_tac tf_rec_ss 1);
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qed "forest_rec_Fnil";
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goalw Simult.thy [tree_rec_def, forest_rec_def, Fcons_def]
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"forest_rec (Fcons t tf) b c d = \
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\ d t tf (tree_rec t b c d) (forest_rec tf b c d)";
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by (simp_tac tf_rec_ss 1);
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qed "forest_rec_Cons";
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