I started using Isabelle/Simpl and write the following theory according to the user guide:
theory Scratch
imports Simpl.Simpl
begin
hoarestate newvars =
N :: nat
lemma (in newvars) "Γ ⊢ {} ´N :== ´N + 1 {}"
sorry
end
But Isabelle complains that type unification fails:
Type unification failed
Type error in application: operator not of function type
Operator: N_' :: 'a
Operand: s :: ??'a
Simpl itself (including its user guide) successfully compiles.
How can I make it pass?
As Javier Díaz pointed out, importing only Simpl.Vcg instead of Simpl.Simpl does the trick.
The cause of the error seems a name collision against a record defined in Simpl.SyntaxTest. It contains the following record definition:
record 'g vars = "'g state" +
A_' :: "nat list"
AA_' :: "nat list list"
I_' :: nat
M_' :: nat
N_' :: nat
R_' :: int
S_' :: int
B_' :: bool
Abr_':: string
p_' :: ref
q_' :: ref
Isabelle seems to prefer N_' in the record to N in the hoarestate, although the hoarestate is defined later😥. I don't know why this is the case.
Related
I wrote a simple function which should perform the function of a halfadder.
fun halfadder :: "bool * bool ⇒ bool * bool"
where
"halfadder (a,b) = (
let s = xor a b in
let cout = and a b in
(cout,s))"
However, I get the following error:
Type unification failed: No type arity bool :: semiring_bit_operations
Type error in application: incompatible operand type
Operator: xor :: ??'a ⇒ ??'a ⇒ ??'a
Operand: a :: bool
Why is it not able to perform a XOR operation on a bool? What is going wrong here?
I have tried using the XOR operator with different data types and it still faces the same error
The type bool in Isabelle/HOL is the type for logical formulas (of the object logic HOL) and so not intended to be used as a type for bit datas. For example, a diagnosis
term "P ∧ Q"
gives
"P ∧ Q"
:: "bool"
If you jump to the definition of xor, you find that xor is defined for (or fixed for or being a method of) the class semiring_bit_operations. A possible solution is to define your halfadder inside the context of semiring_bit_operations, e.g.
context semiring_bit_operations
begin
fun halfadder :: "'a * 'a ⇒ 'a * 'a"
where
"halfadder (a, b) = (
let s = xor a b in
let cout = and a b in
(cout,s))"
end
which worked for me.
However, semiring_bit_operations is not a class for one bits but bit sequences. Thus, it's better to define your own boolean datatype, and and xor (or use some built-in type if any (which I don't know)).
(Note: I'm a beginner of Isabelle and so it is probably better to wait for more comprehensive answers from specialists.)
theory Scratch
imports
"HOL.Fields"
"HOL.Groups"
begin
locale Field_is_group=
fixes F :: "'a :: field"
begin
typedef 'a mul_group = "{x :: 'a set. x ≠ 0}"
end
end
I'm new to Isabelle, and I have learned something about proving in first-order logic but not about data structures. I tried to formalize the proof of the fact that a field has a multiplicative group, but I don't know how to define the multiplicative group. It is the set of the whole field except zero, and a field in Isabelle is defined as a type. By looking about the references, I know I need use typedef, but it fails:
Type unification failed: No type arity set :: zero
Failed to meet type constraint:
Term: λx. x ≠ 0 :: ??'b ⇒ bool
Type: 'a set ⇒ ??'a
The error(s) above occurred in typedef "mul_group"
Im really new to isabelle, this is my problem
theory MyTheory3
imports Main
begin
(* 3.1) a) define a type natpair, whose elements are pairs of natural numbers.*)
datatype natpair = Natpair "nat × nat"
(*3.1 b)define a function of type natpair ⇒ nat that returns the sum of the elements of a pair of natural
numbers.*)
definition natpair_sum :: "natpair ⇒ nat" where
"Natpair((a::nat) × (b::nat)) = a + b"
end
i get the error
Type unification failed: Clash of types "nat" and "_ set"
Type error in application: incompatible operand type
Operator: (×) :: ??'a set ⇒ ??'b set ⇒ (??'a × ??'b) set Operand: a
:: nat
in addition, can someone show me an example of natpair with a parameterized
constructor
Thanks in advance
youre nuric
Your code has several issues:
Operator (×) is not a constructor for pairs but the Cartesian product of sets; pairs are constructed using the syntax (a, b) or Pair a b.
The definition natpair_sum does not use its name in the equation, that is, the correct syntax is definition natpair_sum :: "natpair ⇒ nat" where "natpair_sum ... = ...".
Pattern matching is not allowed in the left-hand side of a definition; you can use the case construct in the right-hand side of the definition or you can use functions instead.
There is no need to add type annotations for the components of the pair.
As examples, here are two possible definitions for natpair_sum:
definition natpair_sum :: "natpair ⇒ nat" where
"natpair_sum np = (case np of Natpair (a, b) ⇒ a + b)"
fun natpair_sum :: "natpair ⇒ nat" where
"natpair_sum (Natpair (a, b)) = a + b"
As for your last question, I'm not quite sure what you mean by "an example of natpair with a parameterized constructor". Natpair is already a data constructor which takes a pair of nat's as the only parameter.
I'm trying to define a generic operations for a programming language:
type_synonym vname = "string"
type_synonym 'a env = "vname ⇒ 'a option"
locale language =
fixes big_step :: "'exp × 'val env ⇒ 'val ⇒ bool" (infix "⇒" 55)
fixes typing :: "'type env ⇒ 'exp ⇒ 'type ⇒ bool" ("(1_/ ⊢/ (_ :/ _))" [50,0,50] 50)
For example this is a particular language:
datatype foo_exp =
FooBConst bool |
FooLet vname foo_exp foo_exp |
FooVar vname |
FooAnd foo_exp foo_exp
datatype foo_val = FooBValue bool | FooIValue int
type_synonym foo_env = "foo_val env"
datatype foo_type = FooBType | FooIType
type_synonym foo_tenv = "foo_type env"
inductive foo_big_step :: "foo_exp × foo_env ⇒ foo_val ⇒ bool"
inductive foo_typing :: "foo_tenv ⇒ foo_exp ⇒ foo_type ⇒ bool"
How to make it an instance of language locale?
Is it possible to use same notation (⇒ and _ ⊢ _ : _) for different languages in one theory? Could this notation be polymorphic?
To specialize the parameters of a locale, you need to do an interpretation as in
interpretation foo: language foo_big_step foo_typing .
This will generate an abbreviation foo.f for every definition f in the locale language specialised to foo_big_step and foo_typing and every theorem thm of language becomes specialised to foo.thm. The mixfix syntax annotations of parameters and all constants in the locale will not be inherited.
Type classes cannot be used in this context because your locale depends on multiple type variables and type classes in Isabelle support only exactly one type variable.
If you want to use some kind of polymorphic notation for the big-step semantics and type judgements, Adhoc_Overloading might work, provided that Isabelle's parser can statically resolve the overloading uniquely. Here's how this might work:
theory Language imports Main "~~/src/Tools/Adhoc_Overloading" begin
type_synonym 'a env = "vname ⇒ 'a option"
consts
big_step :: "'exp × 'val env ⇒ 'val ⇒ bool" (infix "⇒" 55)
typing :: "'type env ⇒ 'exp ⇒ 'type ⇒ bool" ("(1_/ ⊢/ (_ :/ _))" [50,0,50] 50)
locale language =
fixes big_step :: "'exp × 'val env ⇒ 'val ⇒ bool"
fixes typing :: "'type env ⇒ 'exp ⇒ 'type ⇒ bool"
begin
adhoc_overloading Language.big_step big_step
adhoc_overloading Language.typing typing
end
After the interpretation, you have to register foo's semantics and type judgement constants foo_big_step and foo_typing for adhoc overloading with the syntactic constants big_step and typing again.
interpretation foo: language foo_big_step foo_typing .
adhoc_overloading Language.big_step foo_big_step
adhoc_overloading Language.typing foo_typing
So when you write
term "(x :: foo_exp, E) ⇒ v"
thereafter, Isabelle's parser will figure out by the types that this refers to foo_big_step, and inside the locale Language, term "(x :: 'exp, E) ⇒ v" is resolved to the locale parameter big_step.
This should also work for multiple interpretations of the locale Language provided that the types are sufficient to uniquely resolve the overloading. If not, you'll get error messages, which are not always easy to understand.
according to the codegen documentation section "7.3 Locales and interpretation", exporting code from locales is a bit tricky but achievable. The following example works fine:
locale localTest =
fixes A :: "string"
begin
fun concatA :: "string ⇒ string" where "concatA x = x#A"
definition concatAA :: "string ⇒ string" where "concatAA x = x#A#A"
end
definition localtest_concatA :: "string ⇒ string " where
[code del]: "localtest_concatA = localTest.concatA ''a''"
definition localtest_concatAA :: "string ⇒ string " where
[code del]: "localtest_concatAA = localTest.concatAA ''a''"
interpretation localTest "''a''"
where "localTest.concatA ''a'' = localtest_concatA"
and "localTest.concatAA ''a'' = localtest_concatAA"
apply unfold_locales
apply(simp_all add: localtest_concatA_def localtest_concatAA_def)
done
export_code localtest_concatA localtest_concatAA in Scala file -
How can I export code for locales with multiple parameters? Given the following locale:
locale localTest =
fixes A :: "string"
fixes B :: "string"
begin
fun concatA :: "string ⇒ string" where "concatA x = x#A"
definition concatB :: "string ⇒ string" where "concatB x = x#B"
end
I can interpret it with
interpretation localTest "''a''" "''b''" .
But I canot use this interpretation in a definition
definition localtest_concatA :: "string ⇒ string " where
[code del]: "localtest_concatA = localTest.concatA ''a'' ''b''"
It fails with
Type unification failed: Clash of types "_ list" and "_ ⇒ _"
Type error in application: incompatible operand type
Operator: op = localtest_concatA :: (char list ⇒ char list) ⇒ bool
Operand: localTest.concatA ''a'' ''b'' :: char list
Look at your introduced constants, e.g., by the term command. We have
term localTest.concatA
with output
"localTest.concatA" :: "char list ⇒ char list ⇒ char list"
You see that in addition to the single parameter that you gave in the original definition (inside the locale), there is an additional one (but only 1 not 2, since the definition does not rely on B).
Now, after your interpretation (since you did not explicitly provide a name, the constants of localTest will be in scope without qualifier) we have
term concatA
with output
"localTest.concatA ''a''" :: "char list ⇒ char list"
That is, localTest.concatA ''a'' is already of type string => string. You additionally add ''b'' and obtain type string, but your type annotation says string => string. So there is really a clash of types and the reason was that you gave too many arguments to localTest.concatA. Try using
definition localtest_concatA :: "string ⇒ string " where
[code del]: "localtest_concatA = concatA
instead.