src/CCL/Type.thy
author wenzelm
Mon Sep 06 19:13:10 2010 +0200 (2010-09-06)
changeset 39159 0dec18004e75
parent 35409 5c5bb83f2bae
child 41526 54b4686704af
permissions -rw-r--r--
more antiquotations;
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(*  Title:      CCL/Type.thy
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    Author:     Martin Coen
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    Copyright   1993  University of Cambridge
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*)
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header {* Types in CCL are defined as sets of terms *}
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theory Type
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imports Term
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begin
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consts
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  Subtype       :: "['a set, 'a => o] => 'a set"
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  Bool          :: "i set"
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  Unit          :: "i set"
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  Plus           :: "[i set, i set] => i set"        (infixr "+" 55)
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  Pi            :: "[i set, i => i set] => i set"
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  Sigma         :: "[i set, i => i set] => i set"
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  Nat           :: "i set"
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  List          :: "i set => i set"
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  Lists         :: "i set => i set"
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  ILists        :: "i set => i set"
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  TAll          :: "(i set => i set) => i set"       (binder "TALL " 55)
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  TEx           :: "(i set => i set) => i set"       (binder "TEX " 55)
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  Lift          :: "i set => i set"                  ("(3[_])")
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  SPLIT         :: "[i, [i, i] => i set] => i set"
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syntax
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  "_Pi"         :: "[idt, i set, i set] => i set"    ("(3PROD _:_./ _)"
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                                [0,0,60] 60)
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  "_Sigma"      :: "[idt, i set, i set] => i set"    ("(3SUM _:_./ _)"
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                                [0,0,60] 60)
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  "_arrow"      :: "[i set, i set] => i set"         ("(_ ->/ _)"  [54, 53] 53)
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  "_star"       :: "[i set, i set] => i set"         ("(_ */ _)" [56, 55] 55)
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  "_Subtype"    :: "[idt, 'a set, o] => 'a set"      ("(1{_: _ ./ _})")
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translations
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  "PROD x:A. B" => "CONST Pi(A, %x. B)"
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  "A -> B"      => "CONST Pi(A, %_. B)"
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  "SUM x:A. B"  => "CONST Sigma(A, %x. B)"
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  "A * B"       => "CONST Sigma(A, %_. B)"
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  "{x: A. B}"   == "CONST Subtype(A, %x. B)"
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print_translation {*
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 [(@{const_syntax Pi}, dependent_tr' (@{syntax_const "_Pi"}, @{syntax_const "_arrow"})),
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  (@{const_syntax Sigma}, dependent_tr' (@{syntax_const "_Sigma"}, @{syntax_const "_star"}))]
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*}
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axioms
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  Subtype_def: "{x:A. P(x)} == {x. x:A & P(x)}"
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  Unit_def:          "Unit == {x. x=one}"
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  Bool_def:          "Bool == {x. x=true | x=false}"
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  Plus_def:           "A+B == {x. (EX a:A. x=inl(a)) | (EX b:B. x=inr(b))}"
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  Pi_def:         "Pi(A,B) == {x. EX b. x=lam x. b(x) & (ALL x:A. b(x):B(x))}"
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  Sigma_def:   "Sigma(A,B) == {x. EX a:A. EX b:B(a).x=<a,b>}"
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  Nat_def:            "Nat == lfp(% X. Unit + X)"
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  List_def:       "List(A) == lfp(% X. Unit + A*X)"
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  Lists_def:     "Lists(A) == gfp(% X. Unit + A*X)"
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  ILists_def:   "ILists(A) == gfp(% X.{} + A*X)"
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  Tall_def:   "TALL X. B(X) == Inter({X. EX Y. X=B(Y)})"
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  Tex_def:     "TEX X. B(X) == Union({X. EX Y. X=B(Y)})"
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  Lift_def:           "[A] == A Un {bot}"
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  SPLIT_def:   "SPLIT(p,B) == Union({A. EX x y. p=<x,y> & A=B(x,y)})"
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lemmas simp_type_defs =
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    Subtype_def Unit_def Bool_def Plus_def Sigma_def Pi_def Lift_def Tall_def Tex_def
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  and ind_type_defs = Nat_def List_def
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  and simp_data_defs = one_def inl_def inr_def
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  and ind_data_defs = zero_def succ_def nil_def cons_def
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lemma subsetXH: "A <= B <-> (ALL x. x:A --> x:B)"
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  by blast
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subsection {* Exhaustion Rules *}
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lemma EmptyXH: "!!a. a : {} <-> False"
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  and SubtypeXH: "!!a A P. a : {x:A. P(x)} <-> (a:A & P(a))"
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  and UnitXH: "!!a. a : Unit          <-> a=one"
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  and BoolXH: "!!a. a : Bool          <-> a=true | a=false"
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  and PlusXH: "!!a A B. a : A+B           <-> (EX x:A. a=inl(x)) | (EX x:B. a=inr(x))"
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  and PiXH: "!!a A B. a : PROD x:A. B(x) <-> (EX b. a=lam x. b(x) & (ALL x:A. b(x):B(x)))"
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  and SgXH: "!!a A B. a : SUM x:A. B(x)  <-> (EX x:A. EX y:B(x).a=<x,y>)"
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  unfolding simp_type_defs by blast+
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lemmas XHs = EmptyXH SubtypeXH UnitXH BoolXH PlusXH PiXH SgXH
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lemma LiftXH: "a : [A] <-> (a=bot | a:A)"
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  and TallXH: "a : TALL X. B(X) <-> (ALL X. a:B(X))"
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  and TexXH: "a : TEX X. B(X) <-> (EX X. a:B(X))"
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  unfolding simp_type_defs by blast+
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ML {*
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bind_thms ("case_rls", XH_to_Es @{thms XHs});
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*}
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subsection {* Canonical Type Rules *}
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lemma oneT: "one : Unit"
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  and trueT: "true : Bool"
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  and falseT: "false : Bool"
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  and lamT: "!!b B. [| !!x. x:A ==> b(x):B(x) |] ==> lam x. b(x) : Pi(A,B)"
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  and pairT: "!!b B. [| a:A; b:B(a) |] ==> <a,b>:Sigma(A,B)"
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  and inlT: "a:A ==> inl(a) : A+B"
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  and inrT: "b:B ==> inr(b) : A+B"
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  by (blast intro: XHs [THEN iffD2])+
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lemmas canTs = oneT trueT falseT pairT lamT inlT inrT
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subsection {* Non-Canonical Type Rules *}
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lemma lem: "[| a:B(u);  u=v |] ==> a : B(v)"
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  by blast
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ML {*
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fun mk_ncanT_tac top_crls crls =
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  SUBPROOF (fn {context = ctxt, prems = major :: prems, ...} =>
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    resolve_tac ([major] RL top_crls) 1 THEN
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    REPEAT_SOME (eresolve_tac (crls @ [@{thm exE}, @{thm bexE}, @{thm conjE}, @{thm disjE}])) THEN
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    ALLGOALS (asm_simp_tac (simpset_of ctxt)) THEN
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    ALLGOALS (ares_tac (prems RL [@{thm lem}]) ORELSE' etac @{thm bspec}) THEN
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    safe_tac (claset_of ctxt addSIs prems))
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*}
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method_setup ncanT = {*
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  Scan.succeed (SIMPLE_METHOD' o mk_ncanT_tac @{thms case_rls} @{thms case_rls})
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*} ""
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lemma ifT:
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  "[| b:Bool; b=true ==> t:A(true); b=false ==> u:A(false) |] ==>
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    if b then t else u : A(b)"
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  by ncanT
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lemma applyT: "[| f : Pi(A,B);  a:A |] ==> f ` a : B(a)"
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  by ncanT
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lemma splitT:
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  "[| p:Sigma(A,B); !!x y. [| x:A;  y:B(x); p=<x,y> |] ==> c(x,y):C(<x,y>) |]
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    ==> split(p,c):C(p)"
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  by ncanT
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lemma whenT:
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  "[| p:A+B; !!x.[| x:A;  p=inl(x) |] ==> a(x):C(inl(x)); !!y.[| y:B;  p=inr(y) |]
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    ==> b(y):C(inr(y)) |] ==> when(p,a,b) : C(p)"
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  by ncanT
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lemmas ncanTs = ifT applyT splitT whenT
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subsection {* Subtypes *}
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lemma SubtypeD1: "a : Subtype(A, P) ==> a : A"
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  and SubtypeD2: "a : Subtype(A, P) ==> P(a)"
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  by (simp_all add: SubtypeXH)
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lemma SubtypeI: "[| a:A;  P(a) |] ==> a : {x:A. P(x)}"
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  by (simp add: SubtypeXH)
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lemma SubtypeE: "[| a : {x:A. P(x)};  [| a:A;  P(a) |] ==> Q |] ==> Q"
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  by (simp add: SubtypeXH)
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subsection {* Monotonicity *}
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lemma idM: "mono (%X. X)"
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  apply (rule monoI)
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  apply assumption
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  done
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lemma constM: "mono(%X. A)"
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  apply (rule monoI)
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  apply (rule subset_refl)
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  done
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lemma "mono(%X. A(X)) ==> mono(%X.[A(X)])"
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  apply (rule subsetI [THEN monoI])
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  apply (drule LiftXH [THEN iffD1])
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  apply (erule disjE)
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   apply (erule disjI1 [THEN LiftXH [THEN iffD2]])
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  apply (rule disjI2 [THEN LiftXH [THEN iffD2]])
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  apply (drule (1) monoD)
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  apply blast
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  done
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lemma SgM:
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  "[| mono(%X. A(X)); !!x X. x:A(X) ==> mono(%X. B(X,x)) |] ==>
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    mono(%X. Sigma(A(X),B(X)))"
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  by (blast intro!: subsetI [THEN monoI] canTs elim!: case_rls
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    dest!: monoD [THEN subsetD])
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lemma PiM:
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  "[| !!x. x:A ==> mono(%X. B(X,x)) |] ==> mono(%X. Pi(A,B(X)))"
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  by (blast intro!: subsetI [THEN monoI] canTs elim!: case_rls
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    dest!: monoD [THEN subsetD])
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lemma PlusM:
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    "[| mono(%X. A(X));  mono(%X. B(X)) |] ==> mono(%X. A(X)+B(X))"
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  by (blast intro!: subsetI [THEN monoI] canTs elim!: case_rls
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    dest!: monoD [THEN subsetD])
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subsection {* Recursive types *}
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subsubsection {* Conversion Rules for Fixed Points via monotonicity and Tarski *}
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lemma NatM: "mono(%X. Unit+X)";
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  apply (rule PlusM constM idM)+
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  done
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lemma def_NatB: "Nat = Unit + Nat"
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  apply (rule def_lfp_Tarski [OF Nat_def])
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  apply (rule NatM)
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  done
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lemma ListM: "mono(%X.(Unit+Sigma(A,%y. X)))"
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  apply (rule PlusM SgM constM idM)+
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  done
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lemma def_ListB: "List(A) = Unit + A * List(A)"
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  apply (rule def_lfp_Tarski [OF List_def])
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  apply (rule ListM)
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  done
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lemma def_ListsB: "Lists(A) = Unit + A * Lists(A)"
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  apply (rule def_gfp_Tarski [OF Lists_def])
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  apply (rule ListM)
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  done
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lemma IListsM: "mono(%X.({} + Sigma(A,%y. X)))"
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  apply (rule PlusM SgM constM idM)+
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  done
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lemma def_IListsB: "ILists(A) = {} + A * ILists(A)"
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  apply (rule def_gfp_Tarski [OF ILists_def])
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  apply (rule IListsM)
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  done
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lemmas ind_type_eqs = def_NatB def_ListB def_ListsB def_IListsB
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subsection {* Exhaustion Rules *}
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lemma NatXH: "a : Nat <-> (a=zero | (EX x:Nat. a=succ(x)))"
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  and ListXH: "a : List(A) <-> (a=[] | (EX x:A. EX xs:List(A).a=x$xs))"
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  and ListsXH: "a : Lists(A) <-> (a=[] | (EX x:A. EX xs:Lists(A).a=x$xs))"
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  and IListsXH: "a : ILists(A) <-> (EX x:A. EX xs:ILists(A).a=x$xs)"
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  unfolding ind_data_defs
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  by (rule ind_type_eqs [THEN XHlemma1], blast intro!: canTs elim!: case_rls)+
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lemmas iXHs = NatXH ListXH
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ML {* bind_thms ("icase_rls", XH_to_Es @{thms iXHs}) *}
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subsection {* Type Rules *}
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lemma zeroT: "zero : Nat"
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  and succT: "n:Nat ==> succ(n) : Nat"
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  and nilT: "[] : List(A)"
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  and consT: "[| h:A;  t:List(A) |] ==> h$t : List(A)"
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  by (blast intro: iXHs [THEN iffD2])+
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lemmas icanTs = zeroT succT nilT consT
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method_setup incanT = {*
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  Scan.succeed (SIMPLE_METHOD' o mk_ncanT_tac @{thms icase_rls} @{thms case_rls})
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*} ""
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lemma ncaseT:
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  "[| n:Nat; n=zero ==> b:C(zero); !!x.[| x:Nat;  n=succ(x) |] ==> c(x):C(succ(x)) |]
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    ==> ncase(n,b,c) : C(n)"
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  by incanT
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lemma lcaseT:
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  "[| l:List(A); l=[] ==> b:C([]); !!h t.[| h:A;  t:List(A); l=h$t |] ==>
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    c(h,t):C(h$t) |] ==> lcase(l,b,c) : C(l)"
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  by incanT
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lemmas incanTs = ncaseT lcaseT
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subsection {* Induction Rules *}
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lemmas ind_Ms = NatM ListM
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lemma Nat_ind: "[| n:Nat; P(zero); !!x.[| x:Nat; P(x) |] ==> P(succ(x)) |] ==> P(n)"
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  apply (unfold ind_data_defs)
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  apply (erule def_induct [OF Nat_def _ NatM])
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  apply (blast intro: canTs elim!: case_rls)
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  done
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lemma List_ind:
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  "[| l:List(A); P([]); !!x xs.[| x:A;  xs:List(A); P(xs) |] ==> P(x$xs) |] ==> P(l)"
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  apply (unfold ind_data_defs)
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  apply (erule def_induct [OF List_def _ ListM])
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  apply (blast intro: canTs elim!: case_rls)
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  done
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lemmas inds = Nat_ind List_ind
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subsection {* Primitive Recursive Rules *}
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lemma nrecT:
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  "[| n:Nat; b:C(zero);
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      !!x g.[| x:Nat; g:C(x) |] ==> c(x,g):C(succ(x)) |] ==>
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      nrec(n,b,c) : C(n)"
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  by (erule Nat_ind) auto
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lemma lrecT:
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  "[| l:List(A); b:C([]);
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      !!x xs g.[| x:A;  xs:List(A); g:C(xs) |] ==> c(x,xs,g):C(x$xs) |] ==>
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      lrec(l,b,c) : C(l)"
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  by (erule List_ind) auto
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lemmas precTs = nrecT lrecT
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subsection {* Theorem proving *}
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lemma SgE2:
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  "[| <a,b> : Sigma(A,B);  [| a:A;  b:B(a) |] ==> P |] ==> P"
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  unfolding SgXH by blast
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(* General theorem proving ignores non-canonical term-formers,             *)
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(*         - intro rules are type rules for canonical terms                *)
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(*         - elim rules are case rules (no non-canonical terms appear)     *)
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ML {* bind_thms ("XHEs", XH_to_Es @{thms XHs}) *}
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lemmas [intro!] = SubtypeI canTs icanTs
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  and [elim!] = SubtypeE XHEs
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subsection {* Infinite Data Types *}
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lemma lfp_subset_gfp: "mono(f) ==> lfp(f) <= gfp(f)"
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  apply (rule lfp_lowerbound [THEN subset_trans])
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   apply (erule gfp_lemma3)
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  apply (rule subset_refl)
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  done
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lemma gfpI:
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  assumes "a:A"
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    and "!!x X.[| x:A;  ALL y:A. t(y):X |] ==> t(x) : B(X)"
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  shows "t(a) : gfp(B)"
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  apply (rule coinduct)
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   apply (rule_tac P = "%x. EX y:A. x=t (y)" in CollectI)
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   apply (blast intro!: prems)+
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  done
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lemma def_gfpI:
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  "[| C==gfp(B);  a:A;  !!x X.[| x:A;  ALL y:A. t(y):X |] ==> t(x) : B(X) |] ==>
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    t(a) : C"
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  apply unfold
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  apply (erule gfpI)
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  apply blast
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  done
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(* EG *)
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lemma "letrec g x be zero$g(x) in g(bot) : Lists(Nat)"
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  apply (rule refl [THEN UnitXH [THEN iffD2], THEN Lists_def [THEN def_gfpI]])
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  apply (subst letrecB)
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  apply (unfold cons_def)
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  apply blast
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  done
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subsection {* Lemmas and tactics for using the rule @{text
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  "coinduct3"} on @{text "[="} and @{text "="} *}
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lemma lfpI: "[| mono(f);  a : f(lfp(f)) |] ==> a : lfp(f)"
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  apply (erule lfp_Tarski [THEN ssubst])
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  apply assumption
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  done
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lemma ssubst_single: "[| a=a';  a' : A |] ==> a : A"
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  by simp
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lemma ssubst_pair: "[| a=a';  b=b';  <a',b'> : A |] ==> <a,b> : A"
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  by simp
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ML {*
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  val coinduct3_tac = SUBPROOF (fn {context = ctxt, prems = mono :: prems, ...} =>
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    (fast_tac (claset_of ctxt addIs
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        (mono RS @{thm coinduct3_mono_lemma} RS @{thm lfpI}) :: prems) 1));
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*}
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method_setup coinduct3 = {*
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  Scan.succeed (SIMPLE_METHOD' o coinduct3_tac)
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*} ""
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lemma ci3_RI: "[| mono(Agen);  a : R |] ==> a : lfp(%x. Agen(x) Un R Un A)"
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  by coinduct3
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lemma ci3_AgenI: "[| mono(Agen);  a : Agen(lfp(%x. Agen(x) Un R Un A)) |] ==>
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    a : lfp(%x. Agen(x) Un R Un A)"
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  by coinduct3
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lemma ci3_AI: "[| mono(Agen);  a : A |] ==> a : lfp(%x. Agen(x) Un R Un A)"
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  by coinduct3
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ML {*
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fun genIs_tac ctxt genXH gen_mono =
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  rtac (genXH RS iffD2) THEN'
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  simp_tac (simpset_of ctxt) THEN'
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  TRY o fast_tac (claset_of ctxt addIs
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        [genXH RS iffD2, gen_mono RS @{thm coinduct3_mono_lemma} RS @{thm lfpI}])
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*}
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method_setup genIs = {*
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  Attrib.thm -- Attrib.thm >> (fn (genXH, gen_mono) => fn ctxt =>
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    SIMPLE_METHOD' (genIs_tac ctxt genXH gen_mono))
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*} ""
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subsection {* POgen *}
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lemma PO_refl: "<a,a> : PO"
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  by (rule po_refl [THEN PO_iff [THEN iffD1]])
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lemma POgenIs:
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  "<true,true> : POgen(R)"
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  "<false,false> : POgen(R)"
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  "[| <a,a'> : R;  <b,b'> : R |] ==> <<a,b>,<a',b'>> : POgen(R)"
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  "!!b b'. [|!!x. <b(x),b'(x)> : R |] ==><lam x. b(x),lam x. b'(x)> : POgen(R)"
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  "<one,one> : POgen(R)"
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  "<a,a'> : lfp(%x. POgen(x) Un R Un PO) ==>
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    <inl(a),inl(a')> : POgen(lfp(%x. POgen(x) Un R Un PO))"
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  "<b,b'> : lfp(%x. POgen(x) Un R Un PO) ==>
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    <inr(b),inr(b')> : POgen(lfp(%x. POgen(x) Un R Un PO))"
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  "<zero,zero> : POgen(lfp(%x. POgen(x) Un R Un PO))"
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  "<n,n'> : lfp(%x. POgen(x) Un R Un PO) ==>
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    <succ(n),succ(n')> : POgen(lfp(%x. POgen(x) Un R Un PO))"
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  "<[],[]> : POgen(lfp(%x. POgen(x) Un R Un PO))"
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  "[| <h,h'> : lfp(%x. POgen(x) Un R Un PO);  <t,t'> : lfp(%x. POgen(x) Un R Un PO) |]
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    ==> <h$t,h'$t'> : POgen(lfp(%x. POgen(x) Un R Un PO))"
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  unfolding data_defs by (genIs POgenXH POgen_mono)+
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ML {*
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fun POgen_tac ctxt (rla, rlb) i =
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  SELECT_GOAL (safe_tac (claset_of ctxt)) i THEN
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  rtac (rlb RS (rla RS @{thm ssubst_pair})) i THEN
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   457
  (REPEAT (resolve_tac
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   458
      (@{thms POgenIs} @ [@{thm PO_refl} RS (@{thm POgen_mono} RS @{thm ci3_AI})] @
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        (@{thms POgenIs} RL [@{thm POgen_mono} RS @{thm ci3_AgenI}]) @
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        [@{thm POgen_mono} RS @{thm ci3_RI}]) i))
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*}
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subsection {* EQgen *}
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lemma EQ_refl: "<a,a> : EQ"
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   467
  by (rule refl [THEN EQ_iff [THEN iffD1]])
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lemma EQgenIs:
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  "<true,true> : EQgen(R)"
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   471
  "<false,false> : EQgen(R)"
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  "[| <a,a'> : R;  <b,b'> : R |] ==> <<a,b>,<a',b'>> : EQgen(R)"
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  "!!b b'. [|!!x. <b(x),b'(x)> : R |] ==> <lam x. b(x),lam x. b'(x)> : EQgen(R)"
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  "<one,one> : EQgen(R)"
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  "<a,a'> : lfp(%x. EQgen(x) Un R Un EQ) ==>
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   476
    <inl(a),inl(a')> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
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   477
  "<b,b'> : lfp(%x. EQgen(x) Un R Un EQ) ==>
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   478
    <inr(b),inr(b')> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
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   479
  "<zero,zero> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
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   480
  "<n,n'> : lfp(%x. EQgen(x) Un R Un EQ) ==>
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   481
    <succ(n),succ(n')> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
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   482
  "<[],[]> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
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   483
  "[| <h,h'> : lfp(%x. EQgen(x) Un R Un EQ); <t,t'> : lfp(%x. EQgen(x) Un R Un EQ) |]
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   484
    ==> <h$t,h'$t'> : EQgen(lfp(%x. EQgen(x) Un R Un EQ))"
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  unfolding data_defs by (genIs EQgenXH EQgen_mono)+
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   486
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   487
ML {*
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   488
fun EQgen_raw_tac i =
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   489
  (REPEAT (resolve_tac (@{thms EQgenIs} @
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   490
        [@{thm EQ_refl} RS (@{thm EQgen_mono} RS @{thm ci3_AI})] @
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   491
        (@{thms EQgenIs} RL [@{thm EQgen_mono} RS @{thm ci3_AgenI}]) @
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   492
        [@{thm EQgen_mono} RS @{thm ci3_RI}]) i))
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   493
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   494
(* Goals of the form R <= EQgen(R) - rewrite elements <a,b> : EQgen(R) using rews and *)
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   495
(* then reduce this to a goal <a',b'> : R (hopefully?)                                *)
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   496
(*      rews are rewrite rules that would cause looping in the simpifier              *)
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   497
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   498
fun EQgen_tac ctxt rews i =
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   499
 SELECT_GOAL
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   500
   (TRY (safe_tac (claset_of ctxt)) THEN
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   501
    resolve_tac ((rews @ [@{thm refl}]) RL ((rews @ [@{thm refl}]) RL [@{thm ssubst_pair}])) i THEN
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   502
    ALLGOALS (simp_tac (simpset_of ctxt)) THEN
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   503
    ALLGOALS EQgen_raw_tac) i
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   504
*}
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   505
clasohm@0
   506
end