src/HOLCF/Porder.thy
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(*  Title:      HOLCF/porder.thy
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    ID:         $Id$
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    Author:     Franz Regensburger
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    License:    GPL (GNU GENERAL PUBLIC LICENSE)
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Definition of class porder (partial order).
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Conservative extension of theory Porder0 by constant definitions 
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*)
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header {* Type class of partial orders *}
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theory Porder
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imports Main
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begin
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	(* introduce a (syntactic) class for the constant << *)
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axclass sq_ord < type
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	(* characteristic constant << for po *)
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consts
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  "<<"          :: "['a,'a::sq_ord] => bool"        (infixl 55)
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syntax (xsymbols)
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  "op <<"       :: "['a,'a::sq_ord] => bool"        (infixl "\<sqsubseteq>" 55)
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axclass po < sq_ord
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        (* class axioms: *)
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refl_less [iff]: "x << x"        
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antisym_less:    "[|x << y; y << x |] ==> x = y"    
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trans_less:      "[|x << y; y << z |] ==> x << z"
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text {* minimal fixes least element *}
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lemma minimal2UU[OF allI] : "!x::'a::po. uu<<x ==> uu=(@u.!y. u<<y)"
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apply (blast intro: someI2 antisym_less)
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done
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text {* the reverse law of anti-symmetry of @{term "op <<"} *}
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lemma antisym_less_inverse: "(x::'a::po)=y ==> x << y & y << x"
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apply blast
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done
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lemma box_less: "[| (a::'a::po) << b; c << a; b << d|] ==> c << d"
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apply (erule trans_less)
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apply (erule trans_less)
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apply assumption
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done
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lemma po_eq_conv: "((x::'a::po)=y) = (x << y & y << x)"
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apply (fast elim!: antisym_less_inverse intro!: antisym_less)
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done
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subsection {* Constant definitions *}
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consts  
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        "<|"    ::      "['a set,'a::po] => bool"       (infixl 55)
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        "<<|"   ::      "['a set,'a::po] => bool"       (infixl 55)
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        lub     ::      "'a set => 'a::po"
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        tord ::      "'a::po set => bool"
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        chain ::     "(nat=>'a::po) => bool"
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        max_in_chain :: "[nat,nat=>'a::po]=>bool"
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        finite_chain :: "(nat=>'a::po)=>bool"
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syntax
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  "@LUB"	:: "('b => 'a) => 'a"	(binder "LUB " 10)
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translations
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  "LUB x. t"	== "lub(range(%x. t))"
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syntax (xsymbols)
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  "LUB "	:: "[idts, 'a] => 'a"		("(3\<Squnion>_./ _)"[0,10] 10)
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defs
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(* class definitions *)
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is_ub_def:       "S  <| x == ! y. y:S --> y<<x"
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is_lub_def:      "S <<| x == S <| x & (!u. S <| u  --> x << u)"
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(* Arbitrary chains are total orders    *)                  
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tord_def:     "tord S == !x y. x:S & y:S --> (x<<y | y<<x)"
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(* Here we use countable chains and I prefer to code them as functions! *)
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chain_def:        "chain F == !i. F i << F (Suc i)"
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(* finite chains, needed for monotony of continouous functions *)
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max_in_chain_def: "max_in_chain i C == ! j. i <= j --> C(i) = C(j)" 
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finite_chain_def: "finite_chain C == chain(C) & (? i. max_in_chain i C)"
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lub_def:          "lub S == (@x. S <<| x)"
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text {* lubs are unique *}
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lemma unique_lub: 
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        "[| S <<| x ; S <<| y |] ==> x=y"
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apply (unfold is_lub_def is_ub_def)
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apply (blast intro: antisym_less)
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done
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text {* chains are monotone functions *}
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lemma chain_mono [rule_format]: "chain F ==> x<y --> F x<<F y"
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apply (unfold chain_def)
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apply (induct_tac "y")
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apply auto
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prefer 2 apply (blast intro: trans_less)
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apply (blast elim!: less_SucE)
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done
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lemma chain_mono3: "[| chain F; x <= y |] ==> F x << F y"
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apply (drule le_imp_less_or_eq)
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apply (blast intro: chain_mono)
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done
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text {* The range of a chain is a totally ordered *}
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lemma chain_tord: "chain(F) ==> tord(range(F))"
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apply (unfold tord_def)
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apply safe
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apply (rule nat_less_cases)
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apply (fast intro: chain_mono)+
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done
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text {* technical lemmas about @{term lub} and @{term is_lub} *}
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lemmas lub = lub_def [THEN meta_eq_to_obj_eq, standard]
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lemma lubI[OF exI]: "EX x. M <<| x ==> M <<| lub(M)"
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apply (simp add: lub some_eq_ex)
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done
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lemma thelubI: "M <<| l ==> lub(M) = l"
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apply (rule unique_lub)
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apply (subst lub)
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apply (erule someI)
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apply assumption
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done
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lemma lub_singleton [simp]: "lub{x} = x"
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apply (simp (no_asm) add: thelubI is_lub_def is_ub_def)
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done
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text {* access to some definition as inference rule *}
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lemma is_lubD1: "S <<| x ==> S <| x"
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apply (unfold is_lub_def)
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apply auto
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done
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lemma is_lub_lub: "[| S <<| x; S <| u |] ==> x << u"
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apply (unfold is_lub_def)
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apply auto
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done
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lemma is_lubI:
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        "[| S <| x; !!u. S <| u ==> x << u |] ==> S <<| x"
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apply (unfold is_lub_def)
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apply blast
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done
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lemma chainE: "chain F ==> F(i) << F(Suc(i))"
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apply (unfold chain_def)
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apply auto
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done
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lemma chainI: "(!!i. F i << F(Suc i)) ==> chain F"
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apply (unfold chain_def)
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apply blast
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done
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lemma chain_shift: "chain Y ==> chain (%i. Y (i + j))"
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apply (rule chainI)
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apply clarsimp
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apply (erule chainE)
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done
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text {* technical lemmas about (least) upper bounds of chains *}
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lemma ub_rangeD: "range S <| x  ==> S(i) << x"
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apply (unfold is_ub_def)
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apply blast
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done
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lemma ub_rangeI: "(!!i. S i << x) ==> range S <| x"
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apply (unfold is_ub_def)
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apply blast
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done
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lemmas is_ub_lub = is_lubD1 [THEN ub_rangeD, standard]
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(* range(?S1) <<| ?x1 ==> ?S1(?x) << ?x1                                    *)
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text {* results about finite chains *}
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lemma lub_finch1: 
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        "[| chain C; max_in_chain i C|] ==> range C <<| C i"
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apply (unfold max_in_chain_def)
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apply (rule is_lubI)
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apply (rule ub_rangeI)
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apply (rule_tac m = "i" in nat_less_cases)
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apply (rule antisym_less_inverse [THEN conjunct2])
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apply (erule disjI1 [THEN less_or_eq_imp_le, THEN rev_mp])
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apply (erule spec)
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apply (rule antisym_less_inverse [THEN conjunct2])
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apply (erule disjI2 [THEN less_or_eq_imp_le, THEN rev_mp])
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apply (erule spec)
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apply (erule chain_mono)
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apply assumption
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apply (erule ub_rangeD)
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done
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lemma lub_finch2: 
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        "finite_chain(C) ==> range(C) <<| C(@ i. max_in_chain i C)"
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apply (unfold finite_chain_def)
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apply (rule lub_finch1)
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prefer 2 apply (best intro: someI)
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apply blast
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done
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lemma bin_chain: "x<<y ==> chain (%i. if i=0 then x else y)"
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apply (rule chainI)
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apply (induct_tac "i")
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apply auto
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done
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lemma bin_chainmax: 
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        "x<<y ==> max_in_chain (Suc 0) (%i. if (i=0) then x else y)"
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apply (unfold max_in_chain_def le_def)
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apply (rule allI)
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apply (induct_tac "j")
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apply auto
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done
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lemma lub_bin_chain: "x << y ==> range(%i::nat. if (i=0) then x else y) <<| y"
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apply (rule_tac s = "if (Suc 0) = 0 then x else y" in subst , rule_tac [2] lub_finch1)
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apply (erule_tac [2] bin_chain)
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apply (erule_tac [2] bin_chainmax)
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apply (simp (no_asm))
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done
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text {* the maximal element in a chain is its lub *}
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lemma lub_chain_maxelem: "[| Y i = c;  ALL i. Y i<<c |] ==> lub(range Y) = c"
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apply (blast dest: ub_rangeD intro: thelubI is_lubI ub_rangeI)
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done
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text {* the lub of a constant chain is the constant *}
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lemma lub_const: "range(%x. c) <<| c"
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apply (blast dest: ub_rangeD intro: is_lubI ub_rangeI)
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done
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243
c22b85994e17 Franz Regensburger's Higher-Order Logic of Computable Functions embedding LCF
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end 
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74be52691d62 The curried version of HOLCF is now just called HOLCF. The old
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