(*  Title:      FOL/ex/LocaleInst.thy

    ID:         $Id$

    Author:     Clemens Ballarin

    Copyright (c) 2004 by Clemens Ballarin

Test of locale instantiation mechanism, also provides a few examples.

*)

header {* Test of Locale instantiation *}

theory LocaleInst = FOL:

ML {* set show_hyps *}

section {* Locale without assumptions *}

locale L1 = notes rev_conjI [intro] = conjI [THEN iffD1 [OF conj_commute]]

lemma "[| A; B |] ==> A & B"

proof -

  instantiate my: L1   txt {* No chained fact required. *}

  assume B and A  txt {* order reversed *}

  then show "A & B" ..

  txt {* Applies @{thm my.rev_conjI}. *}

qed

section {* Simple locale with assumptions *}

typedecl i

arities  i :: "term"

consts bin :: "[i, i] => i" (infixl "#" 60)

axioms i_assoc: "(x # y) # z = x # (y # z)"

  i_comm: "x # y = y # x"

locale L2 =

  fixes OP (infixl "+" 60)

  assumes assoc: "(x + y) + z = x + (y + z)"

    and comm: "x + y = y + x"

lemma (in L2) lcomm: "x + (y + z) = y + (x + z)"

proof -

  have "x + (y + z) = (x + y) + z" by (simp add: assoc)

  also have "... = (y + x) + z" by (simp add: comm)

  also have "... = y + (x + z)" by (simp add: assoc)

  finally show ?thesis .

qed

lemmas (in L2) AC = comm assoc lcomm

lemma "(x::i) # y # z # w = y # x # w # z"

proof -

  have "L2 (op #)" by (rule L2.intro [of "op #", OF i_assoc i_comm])

  then instantiate my: L2

    txt {* Chained fact required to discharge assumptions of @{text L2}

      and instantiate parameters. *}

  show ?thesis by (simp only: my.OP.AC)  (* or simply AC *)

qed

section {* Nested locale with assumptions *}

locale L3 =

  fixes OP (infixl "+" 60)

  assumes assoc: "(x + y) + z = x + (y + z)"

locale L4 = L3 +

  assumes comm: "x + y = y + x"

lemma (in L4) lcomm: "x + (y + z) = y + (x + z)"

proof -

  have "x + (y + z) = (x + y) + z" by (simp add: assoc)

  also have "... = (y + x) + z" by (simp add: comm)

  also have "... = y + (x + z)" by (simp add: assoc)

  finally show ?thesis .

qed

lemmas (in L4) AC = comm assoc lcomm

text {* Conceptually difficult locale:

   2nd context fragment contains facts with differing metahyps. *}

lemma L4_intro:

  fixes OP (infixl "+" 60)

  assumes assoc: "!!x y z. (x + y) + z = x + (y + z)"

    and comm: "!!x y. x + y = y + x"

  shows "L4 (op+)"

    by (blast intro: L4.intro L3.intro assoc L4_axioms.intro comm)

lemma "(x::i) # y # z # w = y # x # w # z"

proof -

  have "L4 (op #)" by (rule L4_intro [of "op #", OF i_assoc i_comm])

  then instantiate my: L4

  show ?thesis by (simp only: my.OP.AC)  (* or simply AC *)

qed

section {* Locale with definition *}

text {* This example is admittedly not very creative :-) *}

locale L5 = L4 + var A +

  defines A_def: "A == True"

lemma (in L5) lem: A

  by (unfold A_def) rule

lemma "L5(op #) ==> True"

proof -

  assume "L5(op #)"

  then instantiate my: L5

  show ?thesis by (rule lem)  (* lem instantiated to True *)

qed

end