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theory Domain(* Title: HOLCF/Domain.thy
Author: Brian Huffman
*)
header {* Domain package *}
theory Domain
imports Ssum Sprod Up One Tr Fixrec Representable
uses
("Tools/cont_consts.ML")
("Tools/cont_proc.ML")
("Tools/Domain/domain_constructors.ML")
("Tools/Domain/domain_library.ML")
("Tools/Domain/domain_axioms.ML")
("Tools/Domain/domain_theorems.ML")
("Tools/Domain/domain_extender.ML")
begin
default_sort pcpo
subsection {* Casedist *}
lemma ex_one_defined_iff:
"(∃x. P x ∧ x ≠ ⊥) = P ONE"
apply safe
apply (rule_tac p=x in oneE)
apply simp
apply simp
apply force
done
lemma ex_up_defined_iff:
"(∃x. P x ∧ x ≠ ⊥) = (∃x. P (up·x))"
apply safe
apply (rule_tac p=x in upE)
apply simp
apply fast
apply (force intro!: up_defined)
done
lemma ex_sprod_defined_iff:
"(∃y. P y ∧ y ≠ ⊥) =
(∃x y. (P (:x, y:) ∧ x ≠ ⊥) ∧ y ≠ ⊥)"
apply safe
apply (rule_tac p=y in sprodE)
apply simp
apply fast
apply (force intro!: spair_defined)
done
lemma ex_sprod_up_defined_iff:
"(∃y. P y ∧ y ≠ ⊥) =
(∃x y. P (:up·x, y:) ∧ y ≠ ⊥)"
apply safe
apply (rule_tac p=y in sprodE)
apply simp
apply (rule_tac p=x in upE)
apply simp
apply fast
apply (force intro!: spair_defined)
done
lemma ex_ssum_defined_iff:
"(∃x. P x ∧ x ≠ ⊥) =
((∃x. P (sinl·x) ∧ x ≠ ⊥) ∨
(∃x. P (sinr·x) ∧ x ≠ ⊥))"
apply (rule iffI)
apply (erule exE)
apply (erule conjE)
apply (rule_tac p=x in ssumE)
apply simp
apply (rule disjI1, fast)
apply (rule disjI2, fast)
apply (erule disjE)
apply force
apply force
done
lemma exh_start: "p = ⊥ ∨ (∃x. p = x ∧ x ≠ ⊥)"
by auto
lemmas ex_defined_iffs =
ex_ssum_defined_iff
ex_sprod_up_defined_iff
ex_sprod_defined_iff
ex_up_defined_iff
ex_one_defined_iff
text {* Rules for turning exh into casedist *}
lemma exh_casedist0: "[|R; R ==> P|] ==> P" (* like make_elim *)
by auto
lemma exh_casedist1: "((P ∨ Q ==> R) ==> S) ≡ ([|P ==> R; Q ==> R|] ==> S)"
by rule auto
lemma exh_casedist2: "(∃x. P x ==> Q) ≡ (!!x. P x ==> Q)"
by rule auto
lemma exh_casedist3: "(P ∧ Q ==> R) ≡ (P ==> Q ==> R)"
by rule auto
lemmas exh_casedists = exh_casedist1 exh_casedist2 exh_casedist3
subsection {* Combinators for building copy functions *}
lemmas domain_map_stricts =
ssum_map_strict sprod_map_strict u_map_strict
lemmas domain_map_simps =
ssum_map_sinl ssum_map_sinr sprod_map_spair u_map_up
subsection {* Installing the domain package *}
lemmas con_strict_rules =
sinl_strict sinr_strict spair_strict1 spair_strict2
lemmas con_defin_rules =
sinl_defined sinr_defined spair_defined up_defined ONE_defined
lemmas con_defined_iff_rules =
sinl_defined_iff sinr_defined_iff spair_strict_iff up_defined ONE_defined
lemmas con_below_iff_rules =
sinl_below sinr_below sinl_below_sinr sinr_below_sinl con_defined_iff_rules
lemmas con_eq_iff_rules =
sinl_eq sinr_eq sinl_eq_sinr sinr_eq_sinl con_defined_iff_rules
lemmas sel_strict_rules =
cfcomp2 sscase1 sfst_strict ssnd_strict fup1
lemma sel_app_extra_rules:
"sscase·ID·⊥·(sinr·x) = ⊥"
"sscase·ID·⊥·(sinl·x) = x"
"sscase·⊥·ID·(sinl·x) = ⊥"
"sscase·⊥·ID·(sinr·x) = x"
"fup·ID·(up·x) = x"
by (cases "x = ⊥", simp, simp)+
lemmas sel_app_rules =
sel_strict_rules sel_app_extra_rules
ssnd_spair sfst_spair up_defined spair_defined
lemmas sel_defined_iff_rules =
cfcomp2 sfst_defined_iff ssnd_defined_iff
lemmas take_con_rules =
ssum_map_sinl' ssum_map_sinr' sprod_map_spair' u_map_up
deflation_strict deflation_ID ID1 cfcomp2
use "Tools/cont_consts.ML"
use "Tools/cont_proc.ML"
use "Tools/Domain/domain_library.ML"
use "Tools/Domain/domain_axioms.ML"
use "Tools/Domain/domain_constructors.ML"
use "Tools/Domain/domain_theorems.ML"
use "Tools/Domain/domain_extender.ML"
end