added addD2, addE2, addSD2, and addSE2
added not_Some_eq, also to simpset() and claset()
(* Title: Option.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1996 TU Muenchen
Derived rules
*)
open Option;
qed_goal "not_None_eq" thy "(x ~= None) = (? y. x = Some y)"
(K [induct_tac "x" 1, Auto_tac]);
AddIffs[not_None_eq];
qed_goal "not_Some_eq" thy "(!y. x ~= Some y) = (x = None)"
(K [induct_tac "x" 1, Auto_tac]);
AddIffs[not_Some_eq];
section "case analysis in premises";
val optionE = prove_goal thy
"[| opt = None ==> P; !!x. opt = Some x ==> P |] ==> P" (fn prems => [
case_tac "opt = None" 1,
eresolve_tac prems 1,
dtac (not_None_eq RS iffD1) 1,
etac exE 1,
eresolve_tac prems 1]);
fun optionE_tac s = res_inst_tac [("opt",s)] optionE THEN_ALL_NEW hyp_subst_tac;
qed_goal "option_caseE" thy "[|case x of None => P | Some y => Q y; \
\ [|x = None; P |] ==> R; \
\ !!y. [|x = Some y; Q y|] ==> R|] ==> R" (fn prems => [
cut_facts_tac prems 1,
res_inst_tac [("opt","x")] optionE 1,
forward_tac prems 1,
forward_tac prems 3,
Auto_tac]);
fun option_case_tac i = EVERY [
etac option_caseE i,
hyp_subst_tac (i+1),
hyp_subst_tac i];
section "the";
qed_goalw "the_Some" thy [the_def]
"the (Some x) = x" (K [Simp_tac 1]);
Addsimps [the_Some];
section "option_map";
qed_goalw "option_map_None" thy [option_map_def]
"option_map f None = None" (K [Simp_tac 1]);
qed_goalw "option_map_Some" thy [option_map_def]
"option_map f (Some x) = Some (f x)" (K [Simp_tac 1]);
Addsimps [option_map_None, option_map_Some];
val option_map_eq_Some = prove_goalw thy [option_map_def]
"(option_map f xo = Some y) = (? z. xo = Some z & f z = y)"
(K [asm_full_simp_tac (simpset() addsplits [option.split]) 1]);
AddIffs[option_map_eq_Some];
section "o2s";
qed_goal "ospec" thy
"!!x. [| !x:o2s A. P x; A = Some x |] ==> P x" (K [Auto_tac]);
AddDs[ospec];
claset_ref() := claset() addSD2 ("ospec", ospec);
val elem_o2s = prove_goal thy "!!X. x : o2s xo = (xo = Some x)"
(K [optionE_tac "xo" 1, Auto_tac]);
AddIffs [elem_o2s];
val o2s_empty_eq = prove_goal thy "(o2s xo = {}) = (xo = None)"
(K [optionE_tac "xo" 1, Auto_tac]);
Addsimps [o2s_empty_eq];