Theory Predicate

Up to index of Isabelle/HOL

theory Predicate
imports Relation
begin

(*  Title:      HOL/Predicate.thy
    ID:         $Id: Predicate.thy,v 1.8 2007/08/20 16:07:29 haftmann Exp $
    Author:     Stefan Berghofer, TU Muenchen
*)

header {* Predicates *}

theory Predicate
imports Inductive Relation
begin

subsection {* Equality and Subsets *}

lemma pred_equals_eq [pred_set_conv]: "((λx. x ∈ R) = (λx. x ∈ S)) = (R = S)"
  by (auto simp add: expand_fun_eq)

lemma pred_equals_eq2 [pred_set_conv]: "((λx y. (x, y) ∈ R) = (λx y. (x, y) ∈ S)) = (R = S)"
  by (auto simp add: expand_fun_eq)

lemma pred_subset_eq [pred_set_conv]: "((λx. x ∈ R) <= (λx. x ∈ S)) = (R <= S)"
  by fast

lemma pred_subset_eq2 [pred_set_conv]: "((λx y. (x, y) ∈ R) <= (λx y. (x, y) ∈ S)) = (R <= S)"
  by fast


subsection {* Top and bottom elements *}

lemma top1I [intro!]: "top x"
  by (simp add: top_fun_eq top_bool_eq)

lemma top2I [intro!]: "top x y"
  by (simp add: top_fun_eq top_bool_eq)

lemma bot1E [elim!]: "bot x ==> P"
  by (simp add: bot_fun_eq bot_bool_eq)

lemma bot2E [elim!]: "bot x y ==> P"
  by (simp add: bot_fun_eq bot_bool_eq)


subsection {* The empty set *}

lemma bot_empty_eq: "bot = (λx. x ∈ {})"
  by (auto simp add: expand_fun_eq)

lemma bot_empty_eq2: "bot = (λx y. (x, y) ∈ {})"
  by (auto simp add: expand_fun_eq)


subsection {* Binary union *}

lemma sup1_iff [simp]: "sup A B x <-> A x | B x"
  by (simp add: sup_fun_eq sup_bool_eq)

lemma sup2_iff [simp]: "sup A B x y <-> A x y | B x y"
  by (simp add: sup_fun_eq sup_bool_eq)

lemma sup_Un_eq [pred_set_conv]: "sup (λx. x ∈ R) (λx. x ∈ S) = (λx. x ∈ R ∪ S)"
  by (simp add: expand_fun_eq)

lemma sup_Un_eq2 [pred_set_conv]: "sup (λx y. (x, y) ∈ R) (λx y. (x, y) ∈ S) = (λx y. (x, y) ∈ R ∪ S)"
  by (simp add: expand_fun_eq)

lemma sup1I1 [elim?]: "A x ==> sup A B x"
  by simp

lemma sup2I1 [elim?]: "A x y ==> sup A B x y"
  by simp

lemma sup1I2 [elim?]: "B x ==> sup A B x"
  by simp

lemma sup2I2 [elim?]: "B x y ==> sup A B x y"
  by simp

text {*
  \medskip Classical introduction rule: no commitment to @{text A} vs
  @{text B}.
*}

lemma sup1CI [intro!]: "(~ B x ==> A x) ==> sup A B x"
  by auto

lemma sup2CI [intro!]: "(~ B x y ==> A x y) ==> sup A B x y"
  by auto

lemma sup1E [elim!]: "sup A B x ==> (A x ==> P) ==> (B x ==> P) ==> P"
  by simp iprover

lemma sup2E [elim!]: "sup A B x y ==> (A x y ==> P) ==> (B x y ==> P) ==> P"
  by simp iprover


subsection {* Binary intersection *}

lemma inf1_iff [simp]: "inf A B x <-> A x ∧ B x"
  by (simp add: inf_fun_eq inf_bool_eq)

lemma inf2_iff [simp]: "inf A B x y <-> A x y ∧ B x y"
  by (simp add: inf_fun_eq inf_bool_eq)

lemma inf_Int_eq [pred_set_conv]: "inf (λx. x ∈ R) (λx. x ∈ S) = (λx. x ∈ R ∩ S)"
  by (simp add: expand_fun_eq)

lemma inf_Int_eq2 [pred_set_conv]: "inf (λx y. (x, y) ∈ R) (λx y. (x, y) ∈ S) = (λx y. (x, y) ∈ R ∩ S)"
  by (simp add: expand_fun_eq)

lemma inf1I [intro!]: "A x ==> B x ==> inf A B x"
  by simp

lemma inf2I [intro!]: "A x y ==> B x y ==> inf A B x y"
  by simp

lemma inf1D1: "inf A B x ==> A x"
  by simp

lemma inf2D1: "inf A B x y ==> A x y"
  by simp

lemma inf1D2: "inf A B x ==> B x"
  by simp

lemma inf2D2: "inf A B x y ==> B x y"
  by simp

lemma inf1E [elim!]: "inf A B x ==> (A x ==> B x ==> P) ==> P"
  by simp

lemma inf2E [elim!]: "inf A B x y ==> (A x y ==> B x y ==> P) ==> P"
  by simp


subsection {* Unions of families *}

lemma SUP1_iff [simp]: "(SUP x:A. B x) b = (EX x:A. B x b)"
  by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast

lemma SUP2_iff [simp]: "(SUP x:A. B x) b c = (EX x:A. B x b c)"
  by (simp add: SUPR_def Sup_fun_def Sup_bool_def) blast

lemma SUP1_I [intro]: "a : A ==> B a b ==> (SUP x:A. B x) b"
  by auto

lemma SUP2_I [intro]: "a : A ==> B a b c ==> (SUP x:A. B x) b c"
  by auto

lemma SUP1_E [elim!]: "(SUP x:A. B x) b ==> (!!x. x : A ==> B x b ==> R) ==> R"
  by auto

lemma SUP2_E [elim!]: "(SUP x:A. B x) b c ==> (!!x. x : A ==> B x b c ==> R) ==> R"
  by auto

lemma SUP_UN_eq: "(SUP i. (λx. x ∈ r i)) = (λx. x ∈ (UN i. r i))"
  by (simp add: expand_fun_eq)

lemma SUP_UN_eq2: "(SUP i. (λx y. (x, y) ∈ r i)) = (λx y. (x, y) ∈ (UN i. r i))"
  by (simp add: expand_fun_eq)


subsection {* Intersections of families *}

lemma INF1_iff [simp]: "(INF x:A. B x) b = (ALL x:A. B x b)"
  by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast

lemma INF2_iff [simp]: "(INF x:A. B x) b c = (ALL x:A. B x b c)"
  by (simp add: INFI_def Inf_fun_def Inf_bool_def) blast

lemma INF1_I [intro!]: "(!!x. x : A ==> B x b) ==> (INF x:A. B x) b"
  by auto

lemma INF2_I [intro!]: "(!!x. x : A ==> B x b c) ==> (INF x:A. B x) b c"
  by auto

lemma INF1_D [elim]: "(INF x:A. B x) b ==> a : A ==> B a b"
  by auto

lemma INF2_D [elim]: "(INF x:A. B x) b c ==> a : A ==> B a b c"
  by auto

lemma INF1_E [elim]: "(INF x:A. B x) b ==> (B a b ==> R) ==> (a ~: A ==> R) ==> R"
  by auto

lemma INF2_E [elim]: "(INF x:A. B x) b c ==> (B a b c ==> R) ==> (a ~: A ==> R) ==> R"
  by auto

lemma INF_INT_eq: "(INF i. (λx. x ∈ r i)) = (λx. x ∈ (INT i. r i))"
  by (simp add: expand_fun_eq)

lemma INF_INT_eq2: "(INF i. (λx y. (x, y) ∈ r i)) = (λx y. (x, y) ∈ (INT i. r i))"
  by (simp add: expand_fun_eq)


subsection {* Composition of two relations *}

inductive
  pred_comp  :: "['b => 'c => bool, 'a => 'b => bool] => 'a => 'c => bool"
    (infixr "OO" 75)
  for r :: "'b => 'c => bool" and s :: "'a => 'b => bool"
where
  pred_compI [intro]: "s a b ==> r b c ==> (r OO s) a c"

inductive_cases pred_compE [elim!]: "(r OO s) a c"

lemma pred_comp_rel_comp_eq [pred_set_conv]:
  "((λx y. (x, y) ∈ r) OO (λx y. (x, y) ∈ s)) = (λx y. (x, y) ∈ r O s)"
  by (auto simp add: expand_fun_eq elim: pred_compE)


subsection {* Converse *}

inductive
  conversep :: "('a => 'b => bool) => 'b => 'a => bool"
    ("(_^--1)" [1000] 1000)
  for r :: "'a => 'b => bool"
where
  conversepI: "r a b ==> r^--1 b a"

notation (xsymbols)
  conversep  ("(_¯¯)" [1000] 1000)

lemma conversepD:
  assumes ab: "r^--1 a b"
  shows "r b a" using ab
  by cases simp

lemma conversep_iff [iff]: "r^--1 a b = r b a"
  by (iprover intro: conversepI dest: conversepD)

lemma conversep_converse_eq [pred_set_conv]:
  "(λx y. (x, y) ∈ r)^--1 = (λx y. (x, y) ∈ r^-1)"
  by (auto simp add: expand_fun_eq)

lemma conversep_conversep [simp]: "(r^--1)^--1 = r"
  by (iprover intro: order_antisym conversepI dest: conversepD)

lemma converse_pred_comp: "(r OO s)^--1 = s^--1 OO r^--1"
  by (iprover intro: order_antisym conversepI pred_compI
    elim: pred_compE dest: conversepD)

lemma converse_meet: "(inf r s)^--1 = inf r^--1 s^--1"
  by (simp add: inf_fun_eq inf_bool_eq)
    (iprover intro: conversepI ext dest: conversepD)

lemma converse_join: "(sup r s)^--1 = sup r^--1 s^--1"
  by (simp add: sup_fun_eq sup_bool_eq)
    (iprover intro: conversepI ext dest: conversepD)

lemma conversep_noteq [simp]: "(op ~=)^--1 = op ~="
  by (auto simp add: expand_fun_eq)

lemma conversep_eq [simp]: "(op =)^--1 = op ="
  by (auto simp add: expand_fun_eq)


subsection {* Domain *}

inductive
  DomainP :: "('a => 'b => bool) => 'a => bool"
  for r :: "'a => 'b => bool"
where
  DomainPI [intro]: "r a b ==> DomainP r a"

inductive_cases DomainPE [elim!]: "DomainP r a"

lemma DomainP_Domain_eq [pred_set_conv]: "DomainP (λx y. (x, y) ∈ r) = (λx. x ∈ Domain r)"
  by (blast intro!: Orderings.order_antisym)


subsection {* Range *}

inductive
  RangeP :: "('a => 'b => bool) => 'b => bool"
  for r :: "'a => 'b => bool"
where
  RangePI [intro]: "r a b ==> RangeP r b"

inductive_cases RangePE [elim!]: "RangeP r b"

lemma RangeP_Range_eq [pred_set_conv]: "RangeP (λx y. (x, y) ∈ r) = (λx. x ∈ Range r)"
  by (blast intro!: Orderings.order_antisym)


subsection {* Inverse image *}

definition
  inv_imagep :: "('b => 'b => bool) => ('a => 'b) => 'a => 'a => bool" where
  "inv_imagep r f == %x y. r (f x) (f y)"

lemma [pred_set_conv]: "inv_imagep (λx y. (x, y) ∈ r) f = (λx y. (x, y) ∈ inv_image r f)"
  by (simp add: inv_image_def inv_imagep_def)

lemma in_inv_imagep [simp]: "inv_imagep r f x y = r (f x) (f y)"
  by (simp add: inv_imagep_def)


subsection {* The Powerset operator *}

definition Powp :: "('a => bool) => 'a set => bool" where
  "Powp A == λB. ∀x ∈ B. A x"

lemma Powp_Pow_eq [pred_set_conv]: "Powp (λx. x ∈ A) = (λx. x ∈ Pow A)"
  by (auto simp add: Powp_def expand_fun_eq)


subsection {* Properties of relations - predicate versions *}

abbreviation antisymP :: "('a => 'a => bool) => bool" where
  "antisymP r == antisym {(x, y). r x y}"

abbreviation transP :: "('a => 'a => bool) => bool" where
  "transP r == trans {(x, y). r x y}"

abbreviation single_valuedP :: "('a => 'b => bool) => bool" where
  "single_valuedP r == single_valued {(x, y). r x y}"

end

Equality and Subsets

lemma pred_equals_eq:

  ((λx. xR) = (λx. xS)) = (R = S)

lemma pred_equals_eq2:

  ((λx y. (x, y) ∈ R) = (λx y. (x, y) ∈ S)) = (R = S)

lemma pred_subset_eq:

  ((λx. xR) x. xS)) = (R  S)

lemma pred_subset_eq2:

  ((λx y. (x, y) ∈ R) x y. (x, y) ∈ S)) = (R  S)

Top and bottom elements

lemma top1I:

  top x

lemma top2I:

  top x y

lemma bot1E:

  bot x ==> P

lemma bot2E:

  bot x y ==> P

The empty set

lemma bot_empty_eq:

  bot = (λx. x ∈ {})

lemma bot_empty_eq2:

  bot = (λx y. (x, y) ∈ {})

Binary union

lemma sup1_iff:

  sup A B x = (A xB x)

lemma sup2_iff:

  sup A B x y = (A x yB x y)

lemma sup_Un_eq:

  supx. xR) (λx. xS) = (λx. xRS)

lemma sup_Un_eq2:

  supx y. (x, y) ∈ R) (λx y. (x, y) ∈ S) = (λx y. (x, y) ∈ RS)

lemma sup1I1:

  A x ==> sup A B x

lemma sup2I1:

  A x y ==> sup A B x y

lemma sup1I2:

  B x ==> sup A B x

lemma sup2I2:

  B x y ==> sup A B x y

lemma sup1CI:

  B x ==> A x) ==> sup A B x

lemma sup2CI:

  B x y ==> A x y) ==> sup A B x y

lemma sup1E:

  [| sup A B x; A x ==> P; B x ==> P |] ==> P

lemma sup2E:

  [| sup A B x y; A x y ==> P; B x y ==> P |] ==> P

Binary intersection

lemma inf1_iff:

  inf A B x = (A xB x)

lemma inf2_iff:

  inf A B x y = (A x yB x y)

lemma inf_Int_eq:

  infx. xR) (λx. xS) = (λx. xRS)

lemma inf_Int_eq2:

  infx y. (x, y) ∈ R) (λx y. (x, y) ∈ S) = (λx y. (x, y) ∈ RS)

lemma inf1I:

  [| A x; B x |] ==> inf A B x

lemma inf2I:

  [| A x y; B x y |] ==> inf A B x y

lemma inf1D1:

  inf A B x ==> A x

lemma inf2D1:

  inf A B x y ==> A x y

lemma inf1D2:

  inf A B x ==> B x

lemma inf2D2:

  inf A B x y ==> B x y

lemma inf1E:

  [| inf A B x; [| A x; B x |] ==> P |] ==> P

lemma inf2E:

  [| inf A B x y; [| A x y; B x y |] ==> P |] ==> P

Unions of families

lemma SUP1_iff:

  (SUP x:A. B x) b = (∃xA. B x b)

lemma SUP2_iff:

  (SUP x:A. B x) b c = (∃xA. B x b c)

lemma SUP1_I:

  [| aA; B a b |] ==> (SUP x:A. B x) b

lemma SUP2_I:

  [| aA; B a b c |] ==> (SUP x:A. B x) b c

lemma SUP1_E:

  [| (SUP x:A. B x) b; !!x. [| xA; B x b |] ==> R |] ==> R

lemma SUP2_E:

  [| (SUP x:A. B x) b c; !!x. [| xA; B x b c |] ==> R |] ==> R

lemma SUP_UN_eq:

  (SUP i. (λx. xr i)) = (λx. x ∈ (UN i. r i))

lemma SUP_UN_eq2:

  (SUP i. (λx y. (x, y) ∈ r i)) = (λx y. (x, y) ∈ (UN i. r i))

Intersections of families

lemma INF1_iff:

  (INF x:A. B x) b = (∀xA. B x b)

lemma INF2_iff:

  (INF x:A. B x) b c = (∀xA. B x b c)

lemma INF1_I:

  (!!x. xA ==> B x b) ==> (INF x:A. B x) b

lemma INF2_I:

  (!!x. xA ==> B x b c) ==> (INF x:A. B x) b c

lemma INF1_D:

  [| (INF x:A. B x) b; aA |] ==> B a b

lemma INF2_D:

  [| (INF x:A. B x) b c; aA |] ==> B a b c

lemma INF1_E:

  [| (INF x:A. B x) b; B a b ==> R; a  A ==> R |] ==> R

lemma INF2_E:

  [| (INF x:A. B x) b c; B a b c ==> R; a  A ==> R |] ==> R

lemma INF_INT_eq:

  (INF i. (λx. xr i)) = (λx. x ∈ (INT i. r i))

lemma INF_INT_eq2:

  (INF i. (λx y. (x, y) ∈ r i)) = (λx y. (x, y) ∈ (INT i. r i))

Composition of two relations

lemma pred_comp_rel_comp_eq:

  x y. (x, y) ∈ r) OOx y. (x, y) ∈ s) = (λx y. (x, y) ∈ r O s)

Converse

lemma conversepD:

  r^--1 a b ==> r b a

lemma conversep_iff:

  r^--1 a b = r b a

lemma conversep_converse_eq:

  x y. (x, y) ∈ r)^--1 = (λx y. (x, y) ∈ r^-1)

lemma conversep_conversep:

  r^--1^--1 = r

lemma converse_pred_comp:

  (r OO s)^--1 = s^--1 OO r^--1

lemma converse_meet:

  (inf r s)^--1 = inf r^--1 s^--1

lemma converse_join:

  (sup r s)^--1 = sup r^--1 s^--1

lemma conversep_noteq:

  op ^--1 = op 

lemma conversep_eq:

  op =^--1 = op =

Domain

lemma DomainP_Domain_eq:

  DomainPx y. (x, y) ∈ r) = (λx. xDomain r)

Range

lemma RangeP_Range_eq:

  RangePx y. (x, y) ∈ r) = (λx. xRange r)

Inverse image

lemma

  inv_imagepx y. (x, y) ∈ r) f = (λx y. (x, y) ∈ inv_image r f)

lemma in_inv_imagep:

  inv_imagep r f x y = r (f x) (f y)

The Powerset operator

lemma Powp_Pow_eq:

  Powpx. xA) = (λx. x ∈ Pow A)

Properties of relations - predicate versions