# Theory Recdef

Up to index of Isabelle/HOL

theory Recdef
imports Wellfounded_Relations Datatype
uses (../TFL/casesplit.ML) (../TFL/utils.ML) (../TFL/usyntax.ML) (../TFL/dcterm.ML) (../TFL/thms.ML) (../TFL/rules.ML) (../TFL/thry.ML) (../TFL/tfl.ML) (../TFL/post.ML) (Tools/recdef_package.ML)
begin

    (*  Title:      HOL/Recdef.thy
        ID:         $Id: Recdef.thy,v 1.21 2005/08/09 09:44:38 nipkow Exp $
        Author:     Konrad Slind and Markus Wenzel, TU Muenchen
    *)

    header {* TFL: recursive function definitions *}

    theory Recdef
    imports Wellfounded_Relations Datatype
    uses
      ("../TFL/casesplit.ML")
      ("../TFL/utils.ML")
      ("../TFL/usyntax.ML")
      ("../TFL/dcterm.ML")
      ("../TFL/thms.ML")
      ("../TFL/rules.ML")
      ("../TFL/thry.ML")
      ("../TFL/tfl.ML")
      ("../TFL/post.ML")
      ("Tools/recdef_package.ML")
    begin

    lemma tfl_eq_True: "(x = True) --> x"
      by blast

    lemma tfl_rev_eq_mp: "(x = y) --> y --> x";
      by blast

    lemma tfl_simp_thm: "(x --> y) --> (x = x') --> (x' --> y)"
      by blast

    lemma tfl_P_imp_P_iff_True: "P ==> P = True"
      by blast

    lemma tfl_imp_trans: "(A --> B) ==> (B --> C) ==> (A --> C)"
      by blast

    lemma tfl_disj_assoc: "(a ∨ b) ∨ c == a ∨ (b ∨ c)"
      by simp

    lemma tfl_disjE: "P ∨ Q ==> P --> R ==> Q --> R ==> R"
      by blast

    lemma tfl_exE: "∃x. P x ==> ∀x. P x --> Q ==> Q"
      by blast

    use "../TFL/casesplit.ML"
    use "../TFL/utils.ML"
    use "../TFL/usyntax.ML"
    use "../TFL/dcterm.ML"
    use "../TFL/thms.ML"
    use "../TFL/rules.ML"
    use "../TFL/thry.ML"
    use "../TFL/tfl.ML"
    use "../TFL/post.ML"
    use "Tools/recdef_package.ML"
    setup RecdefPackage.setup

    lemmas [recdef_simp] =
      inv_image_def
      measure_def
      lex_prod_def
      same_fst_def
      less_Suc_eq [THEN iffD2]

    lemmas [recdef_cong] = if_cong image_cong

    lemma let_cong [recdef_cong]:
        "M = N ==> (!!x. x = N ==> f x = g x) ==> Let M f = Let N g"
      by (unfold Let_def) blast

    lemmas [recdef_wf] =
      wf_trancl
      wf_less_than
      wf_lex_prod
      wf_inv_image
      wf_measure
      wf_pred_nat
      wf_same_fst
      wf_empty

    (* The following should really go into Datatype or Finite_Set, but
    each one lacks the other theory as a parent . . . *)

    lemma insert_None_conv_UNIV: "insert None (range Some) = UNIV"
    by (rule set_ext, case_tac x, auto)

    instance option :: (finite) finite
    proof
      have "finite (UNIV :: 'a set)" by (rule finite)
      hence "finite (insert None (Some ` (UNIV :: 'a set)))" by simp
      also have "insert None (Some ` (UNIV :: 'a set)) = UNIV"
        by (rule insert_None_conv_UNIV)
      finally show "finite (UNIV :: 'a option set)" .
    qed

    end

lemma tfl_eq_True:

      x = True --> x

lemma tfl_rev_eq_mp:

      x = y --> y --> x

lemma tfl_simp_thm:

      (x --> y) --> x = x' --> x' --> y

lemma tfl_P_imp_P_iff_True:

      P ==> P = True

lemma tfl_imp_trans:

      [| A --> B; B --> C |] ==> A --> C

lemma tfl_disj_assoc:

      (a ∨ b) ∨ c == a ∨ b ∨ c

lemma tfl_disjE:

      [| P ∨ Q; P --> R; Q --> R |] ==> R

lemma tfl_exE:

      [| ∃x. P x; ∀x. P x --> Q |] ==> Q

lemmas

      inv_image r f == {(x, y). (f x, f y) ∈ r}

      measure == inv_image less_than

      ra <*lex*> rb == {((a, b), a', b'). (a, a') ∈ ra ∨ a = a' ∧ (b, b') ∈ rb}

      same_fst P R == {((x', y'), x, y). x' = x ∧ P x ∧ (y', y) ∈ R x}

      m1 < n1 ∨ m1 = n1 ==> m1 < Suc n1

lemmas

      inv_image r f == {(x, y). (f x, f y) ∈ r}

      measure == inv_image less_than

      ra <*lex*> rb == {((a, b), a', b'). (a, a') ∈ ra ∨ a = a' ∧ (b, b') ∈ rb}

      same_fst P R == {((x', y'), x, y). x' = x ∧ P x ∧ (y', y) ∈ R x}

      m1 < n1 ∨ m1 = n1 ==> m1 < Suc n1

lemmas

      [| b = c; c ==> x = u; ¬ c ==> y = v |]
      ==> (if b then x else y) = (if c then u else v)

      [| M = N; !!x. x ∈ N ==> f x = g x |] ==> f ` M = g ` N

lemmas

      [| b = c; c ==> x = u; ¬ c ==> y = v |]
      ==> (if b then x else y) = (if c then u else v)

      [| M = N; !!x. x ∈ N ==> f x = g x |] ==> f ` M = g ` N

lemma let_cong:

      [| M = N; !!x. x = N ==> f x = g x |] ==> Let M f = Let N g

lemmas

      wf r ==> wf (r+)

      wf less_than

      [| wf ra; wf rb |] ==> wf (ra <*lex*> rb)

      wf r ==> wf (inv_image r f)

      wf (measure f)

      wf pred_nat

      (!!x. P x ==> wf (R x)) ==> wf (same_fst P R)

      wf {}

lemmas

      wf r ==> wf (r+)

      wf less_than

      [| wf ra; wf rb |] ==> wf (ra <*lex*> rb)

      wf r ==> wf (inv_image r f)

      wf (measure f)

      wf pred_nat

      (!!x. P x ==> wf (R x)) ==> wf (same_fst P R)

      wf {}

lemma insert_None_conv_UNIV:

      insert None (range Some) = UNIV