Constructive Mathematics, in Theory and Programming Practice
... implicitly, what we (the constructing intelligence) must do in order to construct an element of the set, and what we must do to show that two elements of the set are equal. ([3], p. 2) There are two points to emphasise in this quotation. First, Bishop does not require that the property characterisin ...
... implicitly, what we (the constructing intelligence) must do in order to construct an element of the set, and what we must do to show that two elements of the set are equal. ([3], p. 2) There are two points to emphasise in this quotation. First, Bishop does not require that the property characterisin ...
Introduction to ML
... if x>0 then x-1 else x+1 case num of Int x -> x div 2 | Real y -> y / 2.0 (expr_1; expr_2; …; expr_n) raise SomeException “Fatal Error” some_expr handle SomeExpection s => print s ...
... if x>0 then x-1 else x+1 case num of Int x -> x div 2 | Real y -> y / 2.0 (expr_1; expr_2; …; expr_n) raise SomeException “Fatal Error” some_expr handle SomeExpection s => print s ...
Bounded Functional Interpretation
... a particular assignment of formulas of the language of first-order arithmetic to quantifier-free formulas of the language of T. Gödel’s so-called functional interpretation (a.k.a. Gödel’s Dialectica interpretation, after the journal where it was published [9]) interprets certain principles that a ...
... a particular assignment of formulas of the language of first-order arithmetic to quantifier-free formulas of the language of T. Gödel’s so-called functional interpretation (a.k.a. Gödel’s Dialectica interpretation, after the journal where it was published [9]) interprets certain principles that a ...
PDF
... There are many examples of using these principles to prove statements that can also be proved by induction. For example, one way properties of the natural numbers are proved is to say that if some property P (x) holds, then there is a least number for which it holds. This is used in some algebra boo ...
... There are many examples of using these principles to prove statements that can also be proved by induction. For example, one way properties of the natural numbers are proved is to say that if some property P (x) holds, then there is a least number for which it holds. This is used in some algebra boo ...
Syntax and Semantics of Dependent Types
... view becomes important if one wants to see type theory as a foundation of constructive mathematics which accordingly is to be justi ed by a philosophical argument rather than via an interpretation in some other system, see (Martin-Lof 1975;(1984)). For us the distinction between canonical and nonca ...
... view becomes important if one wants to see type theory as a foundation of constructive mathematics which accordingly is to be justi ed by a philosophical argument rather than via an interpretation in some other system, see (Martin-Lof 1975;(1984)). For us the distinction between canonical and nonca ...
- Free Documents
... In the next section we give the syntax for an extensible calculus of dependent types which encompasses various namedquot type theories like MartinLofs type theory or the Calculus of Constructions. In xx. amp . we introduce presyntax and syntactic context morphisms. Both are auxiliary syntactic notio ...
... In the next section we give the syntax for an extensible calculus of dependent types which encompasses various namedquot type theories like MartinLofs type theory or the Calculus of Constructions. In xx. amp . we introduce presyntax and syntactic context morphisms. Both are auxiliary syntactic notio ...
Lect_8_9
... (syntactic terms) to yield values (abstract entities that we regard as answers). Every value has an associated type. 5 :: Integer 'a' :: Char inc :: Integer -> Integer [1,2,3] :: [Integer] ('b',4) :: (Char,Integer) ...
... (syntactic terms) to yield values (abstract entities that we regard as answers). Every value has an associated type. 5 :: Integer 'a' :: Char inc :: Integer -> Integer [1,2,3] :: [Integer] ('b',4) :: (Char,Integer) ...
A short introduction to the Lambda Calculus
... Whatever our intuition about N , the result will be the same (namely, 12). 5. The official definition. Function formation and function application are all that there is. They can be mixed freely and used as often as desired or needed, which is another way of saying that λ-terms are constructed accor ...
... Whatever our intuition about N , the result will be the same (namely, 12). 5. The official definition. Function formation and function application are all that there is. They can be mixed freely and used as often as desired or needed, which is another way of saying that λ-terms are constructed accor ...
Weyl`s Predicative Classical Mathematics as a Logic
... Four Colour Theorem [Gonthier 2005], which extended the proof checker Coq with axioms for the real numbers that imply the axiom of excluded middle. However, this approach introduces non-canonical objects into the datatypes. (Further discussion on these points can be found in [Luo 2006].) This proble ...
... Four Colour Theorem [Gonthier 2005], which extended the proof checker Coq with axioms for the real numbers that imply the axiom of excluded middle. However, this approach introduces non-canonical objects into the datatypes. (Further discussion on these points can be found in [Luo 2006].) This proble ...
Functional Languages and Higher
... • Local variables need to be stored on heap if they can escape and be accessed after the defining function returns • It happens only if – the variable is referenced from within some nested function – the nested function is returned or passed to some function that might store it in a data structure ...
... • Local variables need to be stored on heap if they can escape and be accessed after the defining function returns • It happens only if – the variable is referenced from within some nested function – the nested function is returned or passed to some function that might store it in a data structure ...
Functional Programming - II
... f : int*int -> int fun g x y = x + y g : int -> int -> int (g 3) : int -> int ((g 3) 4) : int Function application is left associative; -> is right associative ...
... f : int*int -> int fun g x y = x + y g : int -> int -> int (g 3) : int -> int ((g 3) 4) : int Function application is left associative; -> is right associative ...
Interactive Theorem Proving in Coq and the Curry
... proofs, which reflects the precise rules of proof theory [10]. This will ensure, that every proof can be verified step by step. However, the complete proof of a theorem, in such formal languages, quickly becomes a huge text. It is almost impractical, to verify them manually. It becomes necessary, to ...
... proofs, which reflects the precise rules of proof theory [10]. This will ensure, that every proof can be verified step by step. However, the complete proof of a theorem, in such formal languages, quickly becomes a huge text. It is almost impractical, to verify them manually. It becomes necessary, to ...
Constructive Set Theory and Brouwerian Principles1
... forever incomplete. A generic sequence α is purely extensional, in the sense that at any given moment we can know nothing about α other than a finite number of its terms. It follows that for a given sequence α, our procedure for finding an n ∈ N such that (α, n) ∈ P must be able to calculate n from ...
... forever incomplete. A generic sequence α is purely extensional, in the sense that at any given moment we can know nothing about α other than a finite number of its terms. It follows that for a given sequence α, our procedure for finding an n ∈ N such that (α, n) ∈ P must be able to calculate n from ...
Propositions as Types - Informatics Homepages Server
... off, to match the use of the symbol ∃ for the existential quantification introduced by Peano, Gentzen introduced the symbol ∀ to denote universal quantification. He wrote implication as A ⊃ B (if A holds then B holds), conjunction as A & B (both A and B hold), and disjunction as A ∨ B (at least one ...
... off, to match the use of the symbol ∃ for the existential quantification introduced by Peano, Gentzen introduced the symbol ∀ to denote universal quantification. He wrote implication as A ⊃ B (if A holds then B holds), conjunction as A & B (both A and B hold), and disjunction as A ∨ B (at least one ...
Supplemental Reading 1
... have only one primitive set, some innite set (usually !). Sometimes the empty set, , is primitive as well, although it is denable by separation from !. ...
... have only one primitive set, some innite set (usually !). Sometimes the empty set, , is primitive as well, although it is denable by separation from !. ...
Powerpoint - Princeton University
... • Let-bindings, function parameters, and pattern matches (below) bind variables/names in their respective scope. • Occurrences of variables that are not bound are free. • Note: an expression may contains bound and free occurrences of the same name. ...
... • Let-bindings, function parameters, and pattern matches (below) bind variables/names in their respective scope. • Occurrences of variables that are not bound are free. • Note: an expression may contains bound and free occurrences of the same name. ...
