In a local proof block in Isabelle I can use the very convenient obtain command, which allows me to define a constant with given properties:
proof
...
from `∃x. P x ∧ Q x`
obtain x where "P x" and "Q x" by blast
...
What is the most convenient way to do that on the theory or locale level?
I can do it by hand using SOME, but it seems to be unnecessary complicated:
lemma ex: "∃x. P x ∧ Q x"
sorry
definition x where "x = (SOME x. P x ∧ Q x)"
lemma P_x: "P x" and Q_x: "Q x"
unfolding atomize_conj x_def by (rule someI_ex, rule ex)
Another, more direct, way seems to be specification:
consts x :: nat
specification (x) P_x: "P x" Q_x: "Q x" by (rule ex)
but it requires the somewhat low-level consts command, and worse, it does not work in a local context.
Would it be possible to have something as nice as the obtain command on the theory level?
Related
I'm having some issues trying to do exercise 4.5 of 'Concrete Semantics' in Isar:
inductive S :: "alpha list ⇒ bool" where
Sε : "S []" |
aSb : "S m ⟹ S (a#m # [b])" |
SS : "S l ⟹ S r ⟹ S (l # r)"
inductive T :: "alpha list ⇒ bool" where
Tε : "T []" |
TaTb : "T l ⟹ T r ⟹ T (l # a#(r # [b]))"
lemma TS: "T w ⟹ S w"
proof (induction w rule: T.induct)
case Tε
show ?case by (simp add:Sε)
case (TaTb l r) show ?case using TaTb.IH(1) (* This being S l, which allows us to case-split on l using S.induct *)
proof (cases "l" rule: S.induct)
case Sε
then show ?case by (simp add: TaTb.IH(2) aSb)
next case (aSb m)
I'm getting Illegal schematic variable(s) in case "aSb"⌂
Also I find suspicious that in Sε I cannot refer to ?case, I get Unbound schematic variable: ?case. I'm thinking that maybe the problem is that I have a cases in an induction?
As summarized by the comments, you have two problems:
cases "l" rule: S.induct makes little sense and you should either use a nested induction induction l rule: S.induct or a case distinction cases l rule: S.cases
In cases you should use ?thesis instead of cases as the Isabelle/jEdit outline tells you (you can click on that thing to insert it into the buffer!). That way you would also have given a name to all variable in the case TaTb.
So you probably want something like:
lemma TS: "T w ⟹ S w"
proof (induction w rule: T.induct)
case Tε
show ?case by (simp add:Sε)
next
case (TaTb l r a b) show ?case using TaTb.IH(1)
proof (cases "l" rule: S.cases)
case Sε
then show ?thesis sorry
next
case (aSb m a b)
then show ?thesis sorry
next
case (SS l r)
then show ?thesis sorry
qed
qed
inductive T :: "alpha list ⇒ bool" where
Tε : "T []" |
TaTb : "T l ⟹ T r ⟹ T (l # a#(r # [b]))"
lemma Tapp: "⟦T l; T r⟧ ⟹ T (l#r)"
proof (induction r rule: T.induct)
I get 'Failed to apply initial proof method⌂'
In Isabelle one could use rotate_tac I guess to get induction to work on the desired argument, what's the Isar equivalent? Would it help to reformulate the lemma with 'assumes' & 'shows'?
Rule induction is always on the leftmost premise of the goal. Therefore, the Isabelle/Isar solution consists on inverting the order of the premises:
lemma Tapp: "⟦T r; T l⟧ ⟹ T (l#r)"
proof (induction r rule: T.induct)
...
Or, using assumes and shows:
lemma Tapp: assumes "T r" and "T l" shows "T (l#r)"
using assms proof (induction r rule: T.induct)
...
Isabelle has some automation for quotient reasoning through the quotient package. I would like to see if that automation is of any use for my example. The relevant definitions is:
definition e_proj where "e_proj = e'_aff_bit // gluing"
So I try to write:
typedef e_aff_t = e'_aff_bit
quotient_type e_proj_t = "e'_aff_bit" / "gluing
However, I get the error:
Extra type variables in representing set: "'a"
The error(s) above occurred in typedef "e_aff_t"
Because as Manuel Eberl explains here, we cannot have type definitions that depend on type parameters. In the past, I was suggested to use the type-to-sets approach.
How would that approach work in my example? Would it lead to more automation?
In the past, I was suggested to use the type-to-sets approach ...
The suggestion that was made in my previous answer was to use the standard set-based infrastructure for reasoning about quotients. I only mentioned that there exist other options for completeness.
I still believe that it is best not to use Types-To-Sets, provided that the definition of a quotient type is the only reason why you wish to use Types-To-Sets:
Even with Types-To-Sets, you will only be able to mimic the behavior of a quotient type in a local context with certain additional assumptions. Upon leaving the local context, the theorems that use locally defined quotient types would need to be converted to the set-based theorems that would inevitably rely on the standard set-based infrastructure for reasoning about quotients.
One would need to develop additional Isabelle/ML infrastructure before Local Typedef Rule can be used to define quotient types locally conveniently. It should not be too difficult to develop an infrastructure that is useable, but it would take some time to develop something that is universally applicable. Personally, I do not consider this application to be sufficiently important to invest my time in it.
In my view, it is only viable to use Types-To-Sets for the definition of quotient types locally if you are already using Types-To-Sets for its intended purpose in a given development. Then, the possibility of using the framework for the definition of quotient types locally can be seen as a 'value-added benefit'.
For completeness, I provide an example that I developed for an answer on the mailing list some time ago. Of course, this is merely the demonstration of the concept, not a solution that can be used for work that is meant to be published in some form. To make this useable, one would need to convert this development to an Isabelle/ML command that would take care of all the details automatically.
theory Scratch
imports Main
"HOL-Types_To_Sets.Prerequisites"
"HOL-Types_To_Sets.Types_To_Sets"
begin
locale local_typedef =
fixes R :: "['a, 'a] ⇒ bool"
assumes is_equivalence: "equivp R"
begin
(*The exposition subsumes some of the content of
HOL/Types_To_Sets/Examples/Prerequisites.thy*)
context
fixes S and s :: "'s itself"
defines S: "S ≡ {x. ∃u. x = {v. R u v}}"
assumes Ex_type_definition_S:
"∃(Rep::'s ⇒ 'a set) (Abs::'a set ⇒ 's). type_definition Rep Abs S"
begin
definition "rep = fst (SOME (Rep::'s ⇒ 'a set, Abs). type_definition Rep
Abs S)"
definition "Abs = snd (SOME (Rep::'s ⇒ 'a set, Abs). type_definition Rep
Abs S)"
definition "rep' a = (SOME x. a ∈ S ⟶ x ∈ a)"
definition "Abs' x = (SOME a. a ∈ S ∧ a = {v. R x v})"
definition "rep'' = rep' o rep"
definition "Abs'' = Abs o Abs'"
lemma type_definition_S: "type_definition rep Abs S"
unfolding Abs_def rep_def split_beta'
by (rule someI_ex) (use Ex_type_definition_S in auto)
lemma rep_in_S[simp]: "rep x ∈ S"
and rep_inverse[simp]: "Abs (rep x) = x"
and Abs_inverse[simp]: "y ∈ S ⟹ rep (Abs y) = y"
using type_definition_S
unfolding type_definition_def by auto
definition cr_S where "cr_S ≡ λs b. s = rep b"
lemmas Domainp_cr_S = type_definition_Domainp[OF type_definition_S
cr_S_def, transfer_domain_rule]
lemmas right_total_cr_S = typedef_right_total[OF type_definition_S
cr_S_def, transfer_rule]
and bi_unique_cr_S = typedef_bi_unique[OF type_definition_S cr_S_def,
transfer_rule]
and left_unique_cr_S = typedef_left_unique[OF type_definition_S cr_S_def,
transfer_rule]
and right_unique_cr_S = typedef_right_unique[OF type_definition_S
cr_S_def, transfer_rule]
lemma cr_S_rep[intro, simp]: "cr_S (rep a) a" by (simp add: cr_S_def)
lemma cr_S_Abs[intro, simp]: "a∈S ⟹ cr_S a (Abs a)" by (simp add: cr_S_def)
(* this part was sledgehammered - please do not pay attention to the
(absence of) proof style *)
lemma r1: "∀a. Abs'' (rep'' a) = a"
unfolding Abs''_def rep''_def comp_def
proof-
{
fix s'
note repS = rep_in_S[of s']
then have "∃x. x ∈ rep s'" using S equivp_reflp is_equivalence by force
then have "rep' (rep s') ∈ rep s'"
using repS unfolding rep'_def by (metis verit_sko_ex')
moreover with is_equivalence repS have "rep s' = {v. R (rep' (rep s'))
v}"
by (smt CollectD S equivp_def)
ultimately have arr: "Abs' (rep' (rep s')) = rep s'"
unfolding Abs'_def by (smt repS some_sym_eq_trivial verit_sko_ex')
have "Abs (Abs' (rep' (rep s'))) = s'" unfolding arr by (rule
rep_inverse)
}
then show "∀a. Abs (Abs' (rep' (rep a))) = a" by auto
qed
lemma r2: "∀a. R (rep'' a) (rep'' a)"
unfolding rep''_def rep'_def
using is_equivalence unfolding equivp_def by blast
lemma r3: "∀r s. R r s = (R r r ∧ R s s ∧ Abs'' r = Abs'' s)"
apply(intro allI)
apply standard
subgoal unfolding Abs''_def Abs'_def
using is_equivalence unfolding equivp_def by auto
subgoal unfolding Abs''_def Abs'_def
using is_equivalence unfolding equivp_def
by (smt Abs''_def Abs'_def CollectD S comp_apply local.Abs_inverse
mem_Collect_eq someI_ex)
done
definition cr_Q where "cr_Q = (λx y. R x x ∧ Abs'' x = y)"
lemma quotient_Q: "Quotient R Abs'' rep'' cr_Q"
unfolding Quotient_def
apply(intro conjI)
subgoal by (rule r1)
subgoal by (rule r2)
subgoal by (rule r3)
subgoal by (rule cr_Q_def)
done
(* instantiate the quotient lemmas from the theory Lifting *)
lemmas Q_Quotient_abs_rep = Quotient_abs_rep[OF quotient_Q]
(*...*)
(* prove the statements about the quotient type 's *)
(*...*)
(* transfer the results back to 'a using the capabilities of transfer -
not demonstrated in the example *)
lemma aa: "(a::'a) = (a::'a)"
by auto
end
thm aa[cancel_type_definition]
(* this shows {x. ∃u. x = {v. R u v}} ≠ {} ⟹ ?a = ?a *)
end
Here is a sample theory:
datatype ty = A | B | C
inductive test where
"test A B"
| "test B C"
inductive test2 where
"¬(∃z. test x z) ⟹ test2 x"
code_pred [show_modes] test .
code_pred [show_modes] test2 .
values "{x. test2 A}"
The generated code tries to enumerate over ty. And so it fails.
I'm tring to define an executable version of test predicate:
definition "test_ex x ≡ ∃y. test x y"
definition "test_ex_fun x ≡
Predicate.singleton (λ_. False)
(Predicate.map (λ_. True) (test_i_o x))"
lemma test_ex_code [code_abbrev, simp]:
"test_ex_fun = test_ex"
apply (intro ext)
unfolding test_ex_def test_ex_fun_def Predicate.singleton_def
apply (simp split: if_split)
But I can't prove the lemma. Could you suggest a better approach?
Existential quantifiers over an argument to an inductive predicate can be made executable by introducing another inductive predicate. For example:
inductive test2_aux where "test x z ==> test2_aux x"
inductive test2 where "~ test2_aux x ==> test2 x"
with appropriate code_pred statements. The free variable z in the premise of test2_aux acts like an existential. Since this transformation is canonical, code_pred has a preprocessor to do so:
code_pred [inductify] test2 .
does the job.
Well, values complains about the fact that ty is not of sort enum. So, in this particular case it is easiest to perform this instantiation.
instantiation ty :: enum
begin
definition enum_ty :: "ty list" where
"enum_ty = [A,B,C]"
definition "enum_all_ty f = list_all f [A,B,C]"
definition "enum_ex_ty f = list_ex f [A,B,C]"
instance
proof (intro_classes)
let ?U = "UNIV :: ty set"
show id: "?U = set enum_class.enum"
unfolding enum_ty_def
using ty.exhaust by auto
fix P
show "enum_class.enum_all P = Ball ?U P"
"enum_class.enum_ex P = Bex ?U P"
unfolding id enum_all_ty_def enum_ex_ty_def enum_ty_def by auto
show "distinct (enum_class.enum :: ty list)" unfolding enum_ty_def by auto
qed
Afterwards, your values-command evaluates without problems.
I thought that the lemma is unprovable, and I should find another approach. But it can be proven as follows:
lemma test_ex_code [code_abbrev, simp]:
"Predicate.singleton (λ_. False)
(Predicate.map (λ_. True) (test_i_o x)) = (∃y. test x y)"
apply (intro ext iffI)
unfolding Predicate.singleton_def
apply (simp_all split: if_split)
apply (metis SUP1_E mem_Collect_eq pred.sel test_i_o_def)
apply (intro conjI impI)
apply (smt SUP1_E the_equality)
apply (metis (full_types) SUP1_E SUP1_I mem_Collect_eq pred.sel test_i_o_def)
done
The interesting thing is that the lemma structure and the proof structure seems to be independent of the concrete predicate. I guess there could be a general solution for any predicate.
I’m trying to express that a function f is constant on a set S, with value r My first idea was
f ` S = {r}
but that does not work, as S can be empty. So I am currently working with
f ` S ⊆ {r}
and it works okish, but I have the impression that this is still not ideal for the standard automation. In particular, auto would fail leaving this goal (irrelevant facts erased)
2. ⋀xa. thunks (delete x Γ) ⊆ thunks Γ ⟹
ae ` thunks Γ ⊆ {up⋅0} ⟹
xa ∈ thunks (delete x Γ) ⟹
ae xa = up⋅0
Sledgehammer of course has no problem (metis image_eqI singletonD subsetCE), but there are a few occurrences of this. (In general, ⊆ does not seem to work with auto as good as I’d expect).
There there a better way to express this, i.e. one that can be used by auto more easily when occurring as an assumption?
I didn't try it, since I didn't have any examples handy. But you might try the following setup.
definition "const f S r ≡ ∀x ∈ S. f x = r"
Which is equivalent to your definition:
lemma
"const f S r ⟷ f ` S ⊆ {r}"
by (auto simp: const_def)
Then employ the following simp rule:
lemma [simp]:
"const f S r ⟹ x ∈ S ⟹ f x = r"
by (simp add: const_def)
The Analysis library defines
definition constant_on (infixl "(constant'_on)" 50)
where "f constant_on A \<equiv> \<exists>y. \<forall>x\<in>A. f x = y"