src/HOL/Wellfounded_Recursion.thy

First usable version of the new function definition package (HOL/function_packake/...).
Moved Accessible_Part.thy from Library to Main.

+
−(*  ID:         $Id$
+
−    Author:     Tobias Nipkow
+
−    Copyright   1992  University of Cambridge
+
−*)
+
−
+
−header {*Well-founded Recursion*}
+
−
+
−theory Wellfounded_Recursion
+
−imports Transitive_Closure
+
−begin
+
−
+
−consts
+
−  wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => ('a * 'b) set"
+
−
+
−inductive "wfrec_rel R F"
+
−intros
+
−  wfrecI: "ALL z. (z, x) : R --> (z, g z) : wfrec_rel R F ==>
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−            (x, F g x) : wfrec_rel R F"
+
−
+
−constdefs
+
−  wf         :: "('a * 'a)set => bool"
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−  "wf(r) == (!P. (!x. (!y. (y,x):r --> P(y)) --> P(x)) --> (!x. P(x)))"
+
−
+
−  acyclic :: "('a*'a)set => bool"
+
−  "acyclic r == !x. (x,x) ~: r^+"
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−
+
−  cut        :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b"
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−  "cut f r x == (%y. if (y,x):r then f y else arbitrary)"
+
−
+
−  adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool"
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−  "adm_wf R F == ALL f g x.
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−     (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x"
+
−
+
−  wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b"
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−  "wfrec R F == %x. THE y. (x, y) : wfrec_rel R (%f x. F (cut f R x) x)"
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−
+
−axclass wellorder \<subseteq> linorder
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−  wf: "wf {(x,y::'a::ord). x<y}"
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−
+
−
+
−lemma wfUNIVI: 
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−   "(!!P x. (ALL x. (ALL y. (y,x) : r --> P(y)) --> P(x)) ==> P(x)) ==> wf(r)"
+
−by (unfold wf_def, blast)
+
−
+
−text{*Restriction to domain @{term A}.  
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−  If @{term r} is well-founded over @{term A} then @{term "wf r"}*}
+
−lemma wfI: 
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− "[| r <= A <*> A;   
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−     !!x P. [| ALL x. (ALL y. (y,x) : r --> P y) --> P x;  x:A |] ==> P x |]   
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−  ==>  wf r"
+
−by (unfold wf_def, blast)
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−
+
−lemma wf_induct: 
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−    "[| wf(r);           
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−        !!x.[| ALL y. (y,x): r --> P(y) |] ==> P(x)  
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−     |]  ==>  P(a)"
+
−by (unfold wf_def, blast)
+
−
+
−lemmas wf_induct_rule = wf_induct [rule_format, consumes 1, case_names less, induct set: wf]
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−
+
−lemma wf_not_sym [rule_format]: "wf(r) ==> ALL x. (a,x):r --> (x,a)~:r"
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−by (erule_tac a=a in wf_induct, blast)
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−
+
−(* [| wf r;  ~Z ==> (a,x) : r;  (x,a) ~: r ==> Z |] ==> Z *)
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−lemmas wf_asym = wf_not_sym [elim_format]
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−
+
−lemma wf_not_refl [simp]: "wf(r) ==> (a,a) ~: r"
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−by (blast elim: wf_asym)
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−
+
−(* [| wf r;  (a,a) ~: r ==> PROP W |] ==> PROP W *)
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−lemmas wf_irrefl = wf_not_refl [elim_format]
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−
+
−text{*transitive closure of a well-founded relation is well-founded! *}
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−lemma wf_trancl: "wf(r) ==> wf(r^+)"
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−apply (subst wf_def, clarify)
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−apply (rule allE, assumption)
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−  --{*Retains the universal formula for later use!*}
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−apply (erule mp)
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−apply (erule_tac a = x in wf_induct)
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−apply (blast elim: tranclE)
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−done
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−
+
−lemma wf_converse_trancl: "wf (r^-1) ==> wf ((r^+)^-1)"
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−apply (subst trancl_converse [symmetric])
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−apply (erule wf_trancl)
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−done
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−
+
−
+
−subsubsection{*Minimal-element characterization of well-foundedness*}
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−
+
−lemma lemma1: "wf r ==> x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q)"
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−apply (unfold wf_def)
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−apply (drule spec)
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−apply (erule mp [THEN spec], blast)
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−done
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−
+
−lemma lemma2: "(ALL Q x. x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q)) ==> wf r"
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−apply (unfold wf_def, clarify)
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−apply (drule_tac x = "{x. ~ P x}" in spec, blast)
+
−done
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−
+
−lemma wf_eq_minimal: "wf r = (ALL Q x. x:Q --> (EX z:Q. ALL y. (y,z):r --> y~:Q))"
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−by (blast intro!: lemma1 lemma2)
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−
+
−subsubsection{*Other simple well-foundedness results*}
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−
+
−
+
−text{*Well-foundedness of subsets*}
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−lemma wf_subset: "[| wf(r);  p<=r |] ==> wf(p)"
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−apply (simp (no_asm_use) add: wf_eq_minimal)
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−apply fast
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−done
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−
+
−text{*Well-foundedness of the empty relation*}
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−lemma wf_empty [iff]: "wf({})"
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−by (simp add: wf_def)
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−
+
−text{*Well-foundedness of insert*}
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−lemma wf_insert [iff]: "wf(insert (y,x) r) = (wf(r) & (x,y) ~: r^*)"
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−apply (rule iffI)
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− apply (blast elim: wf_trancl [THEN wf_irrefl]
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−              intro: rtrancl_into_trancl1 wf_subset 
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−                     rtrancl_mono [THEN [2] rev_subsetD])
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−apply (simp add: wf_eq_minimal, safe)
+
−apply (rule allE, assumption, erule impE, blast) 
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−apply (erule bexE)
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−apply (rename_tac "a", case_tac "a = x")
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− prefer 2
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−apply blast 
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−apply (case_tac "y:Q")
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− prefer 2 apply blast
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−apply (rule_tac x = "{z. z:Q & (z,y) : r^*}" in allE)
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− apply assumption
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−apply (erule_tac V = "ALL Q. (EX x. x : Q) --> ?P Q" in thin_rl) 
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−  --{*essential for speed*}
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−txt{*Blast with new substOccur fails*}
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−apply (fast intro: converse_rtrancl_into_rtrancl)
+
−done
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−
+
−text{*Well-foundedness of image*}
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−lemma wf_prod_fun_image: "[| wf r; inj f |] ==> wf(prod_fun f f ` r)"
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−apply (simp only: wf_eq_minimal, clarify)
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−apply (case_tac "EX p. f p : Q")
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−apply (erule_tac x = "{p. f p : Q}" in allE)
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−apply (fast dest: inj_onD, blast)
+
−done
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−
+
−
+
−subsubsection{*Well-Foundedness Results for Unions*}
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−
+
−text{*Well-foundedness of indexed union with disjoint domains and ranges*}
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−
+
−lemma wf_UN: "[| ALL i:I. wf(r i);  
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−         ALL i:I. ALL j:I. r i ~= r j --> Domain(r i) Int Range(r j) = {}  
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−      |] ==> wf(UN i:I. r i)"
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−apply (simp only: wf_eq_minimal, clarify)
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−apply (rename_tac A a, case_tac "EX i:I. EX a:A. EX b:A. (b,a) : r i")
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− prefer 2
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− apply force 
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−apply clarify
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−apply (drule bspec, assumption)  
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−apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r i) }" in allE)
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−apply (blast elim!: allE)  
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−done
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−
+
−lemma wf_Union: 
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− "[| ALL r:R. wf r;  
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−     ALL r:R. ALL s:R. r ~= s --> Domain r Int Range s = {}  
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−  |] ==> wf(Union R)"
+
−apply (simp add: Union_def)
+
−apply (blast intro: wf_UN)
+
−done
+
−
+
−(*Intuition: we find an (R u S)-min element of a nonempty subset A
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−             by case distinction.
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−  1. There is a step a -R-> b with a,b : A.
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−     Pick an R-min element z of the (nonempty) set {a:A | EX b:A. a -R-> b}.
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−     By definition, there is z':A s.t. z -R-> z'. Because z is R-min in the
+
−     subset, z' must be R-min in A. Because z' has an R-predecessor, it cannot
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−     have an S-successor and is thus S-min in A as well.
+
−  2. There is no such step.
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−     Pick an S-min element of A. In this case it must be an R-min
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−     element of A as well.
+
−
+
−*)
+
−lemma wf_Un:
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−     "[| wf r; wf s; Domain r Int Range s = {} |] ==> wf(r Un s)"
+
−apply (simp only: wf_eq_minimal, clarify) 
+
−apply (rename_tac A a)
+
−apply (case_tac "EX a:A. EX b:A. (b,a) : r") 
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− prefer 2
+
− apply simp
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− apply (drule_tac x=A in spec)+
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− apply blast 
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−apply (erule_tac x="{a. a:A & (EX b:A. (b,a) : r) }" in allE)+
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−apply (blast elim!: allE)  
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−done
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−
+
−subsubsection {*acyclic*}
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−
+
−lemma acyclicI: "ALL x. (x, x) ~: r^+ ==> acyclic r"
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−by (simp add: acyclic_def)
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−
+
−lemma wf_acyclic: "wf r ==> acyclic r"
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−apply (simp add: acyclic_def)
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−apply (blast elim: wf_trancl [THEN wf_irrefl])
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−done
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−
+
−lemma acyclic_insert [iff]:
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−     "acyclic(insert (y,x) r) = (acyclic r & (x,y) ~: r^*)"
+
−apply (simp add: acyclic_def trancl_insert)
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−apply (blast intro: rtrancl_trans)
+
−done
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−
+
−lemma acyclic_converse [iff]: "acyclic(r^-1) = acyclic r"
+
−by (simp add: acyclic_def trancl_converse)
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−
+
−lemma acyclic_impl_antisym_rtrancl: "acyclic r ==> antisym(r^*)"
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−apply (simp add: acyclic_def antisym_def)
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−apply (blast elim: rtranclE intro: rtrancl_into_trancl1 rtrancl_trancl_trancl)
+
−done
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−
+
−(* Other direction:
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−acyclic = no loops
+
−antisym = only self loops
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−Goalw [acyclic_def,antisym_def] "antisym( r^* ) ==> acyclic(r - Id)
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−==> antisym( r^* ) = acyclic(r - Id)";
+
−*)
+
−
+
−lemma acyclic_subset: "[| acyclic s; r <= s |] ==> acyclic r"
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−apply (simp add: acyclic_def)
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−apply (blast intro: trancl_mono)
+
−done
+
−
+
−
+
−subsection{*Well-Founded Recursion*}
+
−
+
−text{*cut*}
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−
+
−lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))"
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−by (simp add: expand_fun_eq cut_def)
+
−
+
−lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)"
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−by (simp add: cut_def)
+
−
+
−text{*Inductive characterization of wfrec combinator; for details see:  
+
−John Harrison, "Inductive definitions: automation and application"*}
+
−
+
−lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. (x, y) : wfrec_rel R F"
+
−apply (simp add: adm_wf_def)
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−apply (erule_tac a=x in wf_induct) 
+
−apply (rule ex1I)
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−apply (rule_tac g = "%x. THE y. (x, y) : wfrec_rel R F" in wfrec_rel.wfrecI)
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−apply (fast dest!: theI')
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−apply (erule wfrec_rel.cases, simp)
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−apply (erule allE, erule allE, erule allE, erule mp)
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−apply (fast intro: the_equality [symmetric])
+
−done
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−
+
−lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)"
+
−apply (simp add: adm_wf_def)
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−apply (intro strip)
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−apply (rule cuts_eq [THEN iffD2, THEN subst], assumption)
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−apply (rule refl)
+
−done
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−
+
−lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a"
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−apply (simp add: wfrec_def)
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−apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption)
+
−apply (rule wfrec_rel.wfrecI)
+
−apply (intro strip)
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−apply (erule adm_lemma [THEN wfrec_unique, THEN theI'])
+
−done
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−
+
−
+
−text{** This form avoids giant explosions in proofs.  NOTE USE OF ==*}
+
−lemma def_wfrec: "[| f==wfrec r H;  wf(r) |] ==> f(a) = H (cut f r a) a"
+
−apply auto
+
−apply (blast intro: wfrec)
+
−done
+
−
+
−
+
−subsection {* Code generator setup *}
+
−
+
−consts_code
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−  "wfrec"   ("\<module>wfrec?")
+
−attach {*
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−fun wfrec f x = f (wfrec f) x;
+
−*}
+
−
+
−code_primconst wfrec
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−ml {*
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−fun `_` f x = f (`_` f) x;
+
−*}
+
−haskell {*
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−`_` f x = f (`_` f) x
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−*}
+
−
+
−(* code_syntax_const
+
−  wfrec
+
−    ml ("{*wfrec*}?")
+
−    haskell ("{*wfrec*}?") *)
+
−
+
−subsection{*Variants for TFL: the Recdef Package*}
+
−
+
−lemma tfl_wf_induct: "ALL R. wf R -->  
+
−       (ALL P. (ALL x. (ALL y. (y,x):R --> P y) --> P x) --> (ALL x. P x))"
+
−apply clarify
+
−apply (rule_tac r = R and P = P and a = x in wf_induct, assumption, blast)
+
−done
+
−
+
−lemma tfl_cut_apply: "ALL f R. (x,a):R --> (cut f R a)(x) = f(x)"
+
−apply clarify
+
−apply (rule cut_apply, assumption)
+
−done
+
−
+
−lemma tfl_wfrec:
+
−     "ALL M R f. (f=wfrec R M) --> wf R --> (ALL x. f x = M (cut f R x) x)"
+
−apply clarify
+
−apply (erule wfrec)
+
−done
+
−
+
−subsection {*LEAST and wellorderings*}
+
−
+
−text{* See also @{text wf_linord_ex_has_least} and its consequences in
+
− @{text Wellfounded_Relations.ML}*}
+
−
+
−lemma wellorder_Least_lemma [rule_format]:
+
−     "P (k::'a::wellorder) --> P (LEAST x. P(x)) & (LEAST x. P(x)) <= k"
+
−apply (rule_tac a = k in wf [THEN wf_induct])
+
−apply (rule impI)
+
−apply (rule classical)
+
−apply (rule_tac s = x in Least_equality [THEN ssubst], auto)
+
−apply (auto simp add: linorder_not_less [symmetric])
+
−done
+
−
+
−lemmas LeastI   = wellorder_Least_lemma [THEN conjunct1, standard]
+
−lemmas Least_le = wellorder_Least_lemma [THEN conjunct2, standard]
+
−
+
−-- "The following 3 lemmas are due to Brian Huffman"
+
−lemma LeastI_ex: "EX x::'a::wellorder. P x ==> P (Least P)"
+
−apply (erule exE)
+
−apply (erule LeastI)
+
−done
+
−
+
−lemma LeastI2:
+
−  "[| P (a::'a::wellorder); !!x. P x ==> Q x |] ==> Q (Least P)"
+
−by (blast intro: LeastI)
+
−
+
−lemma LeastI2_ex:
+
−  "[| EX a::'a::wellorder. P a; !!x. P x ==> Q x |] ==> Q (Least P)"
+
−by (blast intro: LeastI_ex)
+
−
+
−lemma not_less_Least: "[| k < (LEAST x. P x) |] ==> ~P (k::'a::wellorder)"
+
−apply (simp (no_asm_use) add: linorder_not_le [symmetric])
+
−apply (erule contrapos_nn)
+
−apply (erule Least_le)
+
−done
+
−
+
−ML
+
−{*
+
−val wf_def = thm "wf_def";
+
−val wfUNIVI = thm "wfUNIVI";
+
−val wfI = thm "wfI";
+
−val wf_induct = thm "wf_induct";
+
−val wf_not_sym = thm "wf_not_sym";
+
−val wf_asym = thm "wf_asym";
+
−val wf_not_refl = thm "wf_not_refl";
+
−val wf_irrefl = thm "wf_irrefl";
+
−val wf_trancl = thm "wf_trancl";
+
−val wf_converse_trancl = thm "wf_converse_trancl";
+
−val wf_eq_minimal = thm "wf_eq_minimal";
+
−val wf_subset = thm "wf_subset";
+
−val wf_empty = thm "wf_empty";
+
−val wf_insert = thm "wf_insert";
+
−val wf_UN = thm "wf_UN";
+
−val wf_Union = thm "wf_Union";
+
−val wf_Un = thm "wf_Un";
+
−val wf_prod_fun_image = thm "wf_prod_fun_image";
+
−val acyclicI = thm "acyclicI";
+
−val wf_acyclic = thm "wf_acyclic";
+
−val acyclic_insert = thm "acyclic_insert";
+
−val acyclic_converse = thm "acyclic_converse";
+
−val acyclic_impl_antisym_rtrancl = thm "acyclic_impl_antisym_rtrancl";
+
−val acyclic_subset = thm "acyclic_subset";
+
−val cuts_eq = thm "cuts_eq";
+
−val cut_apply = thm "cut_apply";
+
−val wfrec_unique = thm "wfrec_unique";
+
−val wfrec = thm "wfrec";
+
−val def_wfrec = thm "def_wfrec";
+
−val tfl_wf_induct = thm "tfl_wf_induct";
+
−val tfl_cut_apply = thm "tfl_cut_apply";
+
−val tfl_wfrec = thm "tfl_wfrec";
+
−val LeastI = thm "LeastI";
+
−val Least_le = thm "Least_le";
+
−val not_less_Least = thm "not_less_Least";
+
−*}
+
−
+
−end