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(* Title: HOL/BCV/Fixpoint.ML
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ID: $Id$
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Author: Tobias Nipkow
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Copyright 1999 TUM
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*)
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goalw_cterm [] (cterm_of (sign_of thy) (HOLogic.mk_Trueprop (hd fix.tcs)));
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by (asm_full_simp_tac (simpset() addsimps [wf_iff_no_infinite_down_chain]
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addsplits [split_split]) 1);
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by (Clarify_tac 1);
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by (rtac ccontr 1);
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by (Asm_full_simp_tac 1);
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by (res_inst_tac[("x","0")] allE 1);
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by (assume_tac 1);
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by (Clarify_tac 1);
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by (rename_tac "next s0" 1);
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by (subgoal_tac "!i. fst(f i) = next" 1);
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by (eres_inst_tac[("x","%i. snd(f i)")] allE 1);
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by (Asm_full_simp_tac 1);
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by (Clarify_tac 1);
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by (eres_inst_tac[("x","i")] allE 1);
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by (eres_inst_tac[("x","i")] allE 1);
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by (Force_tac 1);
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by (rtac allI 1);
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by (induct_tac "i" 1);
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by (Asm_simp_tac 1);
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by (eres_inst_tac[("x","n")] allE 1);
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by (Auto_tac);
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val fix_tc = result();
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Goalw [wf_iff_no_infinite_down_chain RS eq_reflection]
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"[| wf{(t,s). s:A & next s = Some t & t ~= s}; \
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\ !a:A. next a : option A; s:A |] ==> \
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\ fix(next,s) = (case next s of None => False \
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\ | Some t => if t=s then True else fix(next,t))";
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by (stac (fix_tc RS (hd fix.rules)) 1);
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by (Simp_tac 1);
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by (Clarify_tac 1);
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by (subgoal_tac "!i. f i : A" 1);
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by (Blast_tac 1);
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by (rtac allI 1);
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by (induct_tac "i" 1);
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by (Asm_simp_tac 1);
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by (Auto_tac);
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qed "fix_unfold";
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(* Thm: next has fixpoint above s iff fix(next,s) *)
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Goal "[| x = Some y; x : option A |] ==> y : A";
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by (Blast_tac 1);
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val lemma = result();
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Goal
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"[| acc L; \
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\ !t:L. !s:L. next t = Some t & s <= t --> (? u. next s = Some u & u <= t); \
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\ !a:L. next a : option L; !a:L. !b. next a = Some b --> a <= b |] \
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\ ==> s:L --> fix(next,s) = (? t:L. s <= t & next t = Some t)";
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by (subgoal_tac "wf{(y,x::'a::order). x : L & next x = Some y & y ~= x}" 1);
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by (full_simp_tac (simpset() addsimps [acc_def]) 2);
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by (etac wf_subset 2);
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by (simp_tac (simpset() addsimps [order_less_le]) 2);
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by (blast_tac (claset() addDs [lemma]) 2);
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by (res_inst_tac [("a","s")] wf_induct 1);
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by (assume_tac 1);
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by (Clarify_tac 1);
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by (Full_simp_tac 1);
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by (stac fix_unfold 1);
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by (assume_tac 1);
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by (assume_tac 1);
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by (assume_tac 1);
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by (split_tac [option.split] 1);
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by (rtac conjI 1);
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by (thin_tac "!a:L. !b. next a = Some b --> a <= b" 1);
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by (Force_tac 1);
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by (Clarify_tac 1);
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by (split_tac [split_if] 1);
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by (rtac conjI 1);
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by (Blast_tac 1);
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by (Clarify_tac 1);
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by (EVERY[etac allE 1, etac impE 1, etac impE 2, etac trans 3]);
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by (Blast_tac 1);
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by (etac lemma 1);
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by (Blast_tac 1);
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by (rtac iffI 1);
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by (blast_tac (claset() addIs [order_trans]) 1);
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by (Clarify_tac 1);
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by (EVERY[rtac bexI 1, atac 2, rtac conjI 1, atac 2]);
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by (thin_tac "!a:L. !b. next a = Some b --> a <= b" 1);
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by (Force_tac 1);
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qed_spec_mp "fix_iff_has_fixpoint";
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(* This lemma looks more pleasing to the eye, because of the monotonicity
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assumption for next, instead of the strange assumption above.
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However, function next as defined in DFAimpl is NOT monotone because
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None is not required to be detected as early as possible. Thus the following
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does not hold: sos <= tos & next sos = None ==> next tos = None
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Goal
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"[| wf{(y,x::'a::order). x : L & next x = Some y & y ~= x}; \
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\ !t:L. !s:L. s <= t & next s = None --> next t = None; \
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\ !t:L. !s:L. !u. s <= t & next s = Some u --> \
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\ next t = None | (? v. next t = Some v & u <= v); \
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\ !a:L. next a : option L; !a:L. !b. next a = Some b --> a <= b |] \
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\ ==> s:L --> fix(next,s) = (? t:L. s <= t & next t = Some t)";
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by (res_inst_tac [("a","s")] wf_induct 1);
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by (assume_tac 1);
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by (Clarify_tac 1);
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by (Full_simp_tac 1);
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by (stac fix_unfold 1);
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by (assume_tac 1);
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by (assume_tac 1);
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by (assume_tac 1);
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by (split_tac [option.split] 1);
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by (rtac conjI 1);
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by (blast_tac (claset() addDs [sym RS trans]) 1);
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by (Clarify_tac 1);
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by (split_tac [split_if] 1);
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by (rtac conjI 1);
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by (Blast_tac 1);
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by (Clarify_tac 1);
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by (EVERY[etac allE 1, etac impE 1, etac impE 2, etac trans 3]);
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by (Blast_tac 1);
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by (etac lemma 1);
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by (Blast_tac 1);
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by (rtac iffI 1);
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by (subgoal_tac "next a ~= None" 1);
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by (Clarify_tac 1);
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by (EVERY[etac ballE 1, etac ballE 1, etac allE 1, etac impE 1,etac disjE 2]);
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by (rtac conjI 1);
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by (assume_tac 1);
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by (assume_tac 1);
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by (blast_tac (claset() addDs [sym RS trans]) 1);
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by (blast_tac (claset() addIs [order_trans]) 1);
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by (blast_tac (claset() addIs [order_trans]) 1);
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by (Blast_tac 1);
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by (rtac notI 1);
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by (Clarify_tac 1);
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by (blast_tac (claset() addDs [sym RS trans,lemma]) 1);
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by (blast_tac (claset() addDs [sym RS trans]) 1);
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qed_spec_mp "fix_iff_has_fixpoint";
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*)
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