theory fix (isabelle2005: october 2005)
theory fix (isabelle2005: october 2005)
theory fix
up to index of isabelle/ccl
theory fix
imports type
uses [fix.ml]
begin
(* title: ccl/fix.thy
id: $id: fix.thy,v 1.4 2005/09/17 15:35:27 wenzelm exp $
author: martin coen
copyright 1993 university of cambridge
*)
header {* tentative attempt at including fixed point induction; justified by smith *}
theory fix
imports type
begin
consts
idgen :: "[i]=>i"
incl :: "[i=>o]=>o"
axioms
idgen_def:
"idgen(f) == lam t. case(t,true,false,%x y.<f`x, f`y>,%u. lam x. f ` u(x))"
incl_def: "incl(%x. p(x)) == (all f.(all n:nat. p(f^n`bot)) --> p(fix(f)))"
po_incl: "incl(%x. a(x) [= b(x))"
incl_subst: "incl(p) ==> incl(%x. p((g::i=>i)(x)))"
ml {* use_legacy_bindings (the_context ()) *}
end
theorem fix_ind:
[| p(bot); !!x. p(x) ==> p(f(x)); incl(p) |] ==> p(fix(f))
theorem inclxh:
incl(p) <-> (∀f. (all n:nat. p(f ^ n ` bot)) --> p(fix(f)))
theorem incli:
(!!f. all n:nat. p(f ^ n ` bot) ==> p(fix(f))) ==> incl(p)
theorem incld:
[| incl(p); !!n. n : nat ==> p(f ^ n ` bot) |] ==> p(fix(f))
theorem incle:
[| incl(p); (all n:nat. p(f ^ n ` bot)) --> p(fix(f)) ==> r |] ==> r
theorem npo_incl:
incl(%x. ¬ a(x) [= t)
theorem conj_incl:
[| incl(p); incl(q) |] ==> incl(%x. p(x) ∧ q(x))
theorem all_incl:
(!!a. incl(p(a))) ==> incl(%x. ∀a. p(a, x))
theorem ball_incl:
(!!a. a : a ==> incl(p(a))) ==> incl(%x. all a:a. p(a, x))
theorem eq_incl:
incl(%x. a(x) = b(x))
theorem fix_idgenfp:
idgen(fix(idgen)) = fix(idgen)
theorem id_idgenfp:
idgen(lam x. x) = lam x. x
theorem idgenfp_lam:
idgen(d) = d
==> d = lam x. case(x, true, false, %x y. <d ` x,d ` y>, %u. lam x. d ` u(x))
theorem l_lemma:
[| a = b; a ` t = u |] ==> b ` t = u
theorem po_eta:
[| ∀x. t ` x [= u ` x; ∃f. t = lam x. f(x); ∃f. u = lam x. f(x) |] ==> t [= u
theorem po_eta_lemma:
idgen(d) = d
==> d = lam x. case(x, true, false, %x y. <d ` x,d ` y>, %u. lam x. d ` u(x))
theorem lemma1:
idgen(d) = d
==> {p. ∃a b. p = <a,b> ∧ (∃t. a = fix(idgen) ` t ∧ b = d ` t)} <=
pogen({p. ∃a b. p = <a,b> ∧ (∃t. a = fix(idgen) ` t ∧ b = d ` t)})
theorem fix_least_idgen:
idgen(d) = d ==> fix(idgen) [= d
theorem lemma2:
idgen(d) = d
==> {p. ∃a b. p = <a,b> ∧ b = d ` a} <= pogen({p. ∃a b. p = <a,b> ∧ b = d ` a})
theorem id_least_idgen:
idgen(d) = d ==> lam x. x [= d
theorem reachability:
fix(idgen) = lam x. x
theorem id_apply:
f = lam x. x ==> f ` t = t
theorem term_ind:
[| p(bot); p(true); p(false); !!x y. [| p(x); p(y) |] ==> p(<x,y>);
!!u. (!!x. p(u(x))) ==> p(lam x. u(x)); incl(p) |]
==> p(t)
Acceuil
suivante
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