author  blanchet 
Mon, 15 Sep 2014 10:49:07 +0200  
changeset 58335  a5a3b576fcfb 
parent 32153  a0e57fb1b930 
child 58889  5b7a9633cfa8 
permissions  rwrr 
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(* Title: CCL/Gfp.thy 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
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Copyright 1992 University of Cambridge 
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*) 

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header {* Greatest fixed points *} 
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theory Gfp 

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imports Lfp 

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begin 

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definition 
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eb85850d3eb7
more robust syntax for definition/abbreviation/notation;
wenzelm
parents:
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changeset

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gfp :: "['a set=>'a set] => 'a set" where  "greatest fixed point" 
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"gfp(f) == Union({u. u <= f(u)})" 
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(* gfp(f) is the least upper bound of {u. u <= f(u)} *) 
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lemma gfp_upperbound: "[ A <= f(A) ] ==> A <= gfp(f)" 

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unfolding gfp_def by blast 

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lemma gfp_least: "[ !!u. u <= f(u) ==> u<=A ] ==> gfp(f) <= A" 

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unfolding gfp_def by blast 

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lemma gfp_lemma2: "mono(f) ==> gfp(f) <= f(gfp(f))" 

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by (rule gfp_least, rule subset_trans, assumption, erule monoD, 

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rule gfp_upperbound, assumption) 

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lemma gfp_lemma3: "mono(f) ==> f(gfp(f)) <= gfp(f)" 

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by (rule gfp_upperbound, frule monoD, rule gfp_lemma2, assumption+) 

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lemma gfp_Tarski: "mono(f) ==> gfp(f) = f(gfp(f))" 

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by (rule equalityI gfp_lemma2 gfp_lemma3  assumption)+ 

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(*** Coinduction rules for greatest fixed points ***) 

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(*weak version*) 

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lemma coinduct: "[ a: A; A <= f(A) ] ==> a : gfp(f)" 

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by (blast dest: gfp_upperbound) 

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lemma coinduct2_lemma: 

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"[ A <= f(A) Un gfp(f); mono(f) ] ==> 

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A Un gfp(f) <= f(A Un gfp(f))" 

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apply (rule subset_trans) 

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prefer 2 

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apply (erule mono_Un) 

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apply (rule subst, erule gfp_Tarski) 

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apply (erule Un_least) 

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apply (rule Un_upper2) 

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done 

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(*strong version, thanks to Martin Coen*) 

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lemma coinduct2: 

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"[ a: A; A <= f(A) Un gfp(f); mono(f) ] ==> a : gfp(f)" 

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apply (rule coinduct) 

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prefer 2 

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apply (erule coinduct2_lemma, assumption) 

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apply blast 

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done 

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(*** Even Stronger version of coinduct [by Martin Coen] 

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 instead of the condition A <= f(A) 

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consider A <= (f(A) Un f(f(A)) ...) Un gfp(A) ***) 

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lemma coinduct3_mono_lemma: "mono(f) ==> mono(%x. f(x) Un A Un B)" 

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by (rule monoI) (blast dest: monoD) 

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lemma coinduct3_lemma: 

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assumes prem: "A <= f(lfp(%x. f(x) Un A Un gfp(f)))" 

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and mono: "mono(f)" 

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shows "lfp(%x. f(x) Un A Un gfp(f)) <= f(lfp(%x. f(x) Un A Un gfp(f)))" 

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apply (rule subset_trans) 

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apply (rule mono [THEN coinduct3_mono_lemma, THEN lfp_lemma3]) 

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apply (rule Un_least [THEN Un_least]) 

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apply (rule subset_refl) 

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apply (rule prem) 

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apply (rule mono [THEN gfp_Tarski, THEN equalityD1, THEN subset_trans]) 

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apply (rule mono [THEN monoD]) 

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apply (subst mono [THEN coinduct3_mono_lemma, THEN lfp_Tarski]) 

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apply (rule Un_upper2) 

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done 

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lemma coinduct3: 

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assumes 1: "a:A" 

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and 2: "A <= f(lfp(%x. f(x) Un A Un gfp(f)))" 

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and 3: "mono(f)" 

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shows "a : gfp(f)" 

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apply (rule coinduct) 

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prefer 2 

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apply (rule coinduct3_lemma [OF 2 3]) 

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apply (subst lfp_Tarski [OF coinduct3_mono_lemma, OF 3]) 

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using 1 apply blast 

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done 

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subsection {* Definition forms of @{text "gfp_Tarski"}, to control unfolding *} 

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lemma def_gfp_Tarski: "[ h==gfp(f); mono(f) ] ==> h = f(h)" 

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apply unfold 

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apply (erule gfp_Tarski) 

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done 

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lemma def_coinduct: "[ h==gfp(f); a:A; A <= f(A) ] ==> a: h" 

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apply unfold 

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apply (erule coinduct) 

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apply assumption 

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done 

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lemma def_coinduct2: "[ h==gfp(f); a:A; A <= f(A) Un h; mono(f) ] ==> a: h" 

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apply unfold 

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apply (erule coinduct2) 

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apply assumption 

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apply assumption 

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done 

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lemma def_coinduct3: "[ h==gfp(f); a:A; A <= f(lfp(%x. f(x) Un A Un h)); mono(f) ] ==> a: h" 

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apply unfold 

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apply (erule coinduct3) 

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apply assumption 

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apply assumption 

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done 

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(*Monotonicity of gfp!*) 

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lemma gfp_mono: "[ mono(f); !!Z. f(Z)<=g(Z) ] ==> gfp(f) <= gfp(g)" 

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apply (rule gfp_upperbound) 

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apply (rule subset_trans) 

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apply (rule gfp_lemma2) 

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apply assumption 

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apply (erule meta_spec) 

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done 

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end 