Theory Recdef

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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