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// Test if even number addition is commutative for gven numbers.
// Test if even number addition is commutative for given numbers.
func testCommutative(e1, e2 *EvenNumber) {
func testCommutative(e1, e2 *EvenNumber) {
c1, c2 := countEven(e1), countEven(e2)
c1, c2 := countEven(e1), countEven(e2)

Revision as of 00:23, 10 January 2020

Proof is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.

This task only makes sense for dependently-typed languages and proof assistants, or for languages with a type system strong enough to emulate certain dependent types. It does not ask you to implement a theorem prover yourself.

In the following task the word "define" implies the need to build the system of Peano axioms using the language itself, that is a way to construct natural and even natural numbers in the canonical forms, as well as a definition of the rules of addition and a way to construct all other acceptable terms. The word "prove" means that some form of logical unification is used (i.e., it requires a type checker in the case of languages ​​with dependent types and a verifying algorithm in the case of proof assistants). Thus, the metatheory of a language must be expressive enough to allow embedding of the Peano axioms and the opportunity to carry out constructive proofs. Examples of the trusted mathematical metatheories can be given: SystemF for Haskell, MLTT for Agda, CoC/CoIC for Coq.

Task:

  1. To illustrate the possibility of type formation and type introduction:
    1. Define a countably infinite set of natural numbers {0, 1, 2, 3, ...}.
    2. Define a countably infinite set of even natural numbers {0, 2, 4, 6, ...} within the previously defined set of natural numbers.
    3. Define a countably infinite set of odd natural numbers {1, 3, 5, 7, ...} within the previously defined set of natural numbers.
  2. To illustrate the possibility of type elimination:
    1. Define the addition on natural numbers.
  3. To demonstrate constructive proofs:
    1. Prove that the addition of any two even numbers is even.
    2. Prove that the addition is always associative.
    3. Prove that the addition is always commutative.
    4. Try to prove that the addition of any two even numbers is odd (it should be rejected).
  4. To demonstrate the ability of disproofs:
    1. Prove that the addition of any two even numbers cannot be odd.
    2. Try to prove that the addition of any two even numbers cannot be even (it should be rejected).

3. and 4. can't be done using a simple number enumeration since there is a countable many natural numbers which is quantified in propositions.

ACL2

3.1. using built-in natural numbers:

<lang Lisp>(thm (implies (and (evenp x) (evenp y)) 

             (evenp (+ x y))))</lang>

Agda

<lang agda>module PeanoArithmetic where

-- 1.1. The natural numbers. -- -- ℕ-formation: ℕ is set. -- -- ℕ-introduction: 0 ∈ ℕ, -- a ∈ ℕ | (1 + a) ∈ ℕ. -- data ℕ : Set where 

 zero : ℕ
 1+_  : ℕ → ℕ

{-# BUILTIN NATURAL ℕ #-} {-# BUILTIN ZERO zero #-} {-# BUILTIN SUC 1+_ #-}

-- 2.1. The rule of addition. -- -- via ℕ-elimination. -- infixl 6 _+_ _+_ : ℕ → ℕ → ℕ 0 + n = n 1+ m + n = 1+ (m + n)

-- 1.2. The even natural numbers. -- data 2×ℕ : ℕ → Set where 

 zero : 2×ℕ 0
 2+_  : {m : ℕ} → 2×ℕ m → 2×ℕ (2 + m)

-- 1.3. The odd natural numbers. -- data 2×ℕ+1 : ℕ → Set where 

 one : 2×ℕ+1 1
 2+_ : {m : ℕ} → 2×ℕ+1 m → 2×ℕ+1 (2 + m)

-- 3.1. Sum of any two even numbers is even. -- -- This function takes any two even numbers and returns their sum as an even -- number, this is the type, i.e. logical proposition, algorithm itself is a -- proof which builds a required term of a given (inhabited) type, and the -- typechecker performs that proof (by unification, so that this is a form of -- compile-time verification). -- even+even≡even : {m n : ℕ} → 2×ℕ m → 2×ℕ n → 2×ℕ (m + n) even+even≡even zero n = n even+even≡even (2+ m) n = 2+ (even+even≡even m n)

-- The identity type for ℕ (for propositional equality). -- infix 4 _≡_ data _≡_ (m : ℕ) : ℕ → Set where 

 refl : m ≡ m

sym : {m n : ℕ} → m ≡ n → n ≡ m sym refl = refl

trans : {m n p : ℕ} → m ≡ n → n ≡ p → m ≡ p trans refl n≡p = n≡p

-- refl, sym and trans forms an equivalence relation.

cong : {m n : ℕ} → m ≡ n → 1 + m ≡ 1 + n cong refl = refl

-- 3.2.1. Direct proof of the associativity of addition using propositional -- equality. -- +-associative : (m n p : ℕ) → (m + n) + p ≡ m + (n + p) +-associative 0 _ _ = refl +-associative (1+ m) n p = cong (+-associative m n p)

-- Proof _of_ mathematical induction on the natural numbers. -- -- P 0, ∀ x. P x → P (1 + x) | ∀ x. P x. -- ind : (P : ℕ → Set) → P 0 → ((m : ℕ) → P m → P (1 + m)) → (m : ℕ) → P m ind _ P₀ _ 0 = P₀ ind P P₀ next (1+ n) = next n (ind P P₀ next n)

-- 3.2.2. The associativity of addition by induction (with propositional -- equality, again). -- +-associative′ : (m n p : ℕ) → (m + n) + p ≡ m + (n + p) +-associative′ m n p = ind P P₀ is m 

 where
   P : ℕ → Set
   P m = m + n + p ≡ m + (n + p)
   P₀ : P 0
   P₀ = refl
   is : (m : ℕ) → P m → P (1 + m)
   is _ Pi = cong Pi

-- Syntactic sugar for equational reasoning (we don't use preorders here).

infix 4 _≋_ data _≋_ (m n : ℕ) : Set where 

 refl : m ≡ n → m ≋ n

infix 1 begin_ begin_ : {m n : ℕ} → m ≋ n → m ≡ n begin refl m≡n = m≡n

infixr 2 _~⟨_⟩_ _~⟨_⟩_ : (m : ℕ){n p : ℕ} → m ≡ n → n ≋ p → m ≋ p _ ~⟨ m≡n ⟩ refl n≡p = refl (trans m≡n n≡p)

infix 2 _∎ _∎ : (m : ℕ) → m ≋ m _∎ _ = refl refl

-- Some helper proofs.

m+0≡m : (m : ℕ) → m + 0 ≡ m m+0≡m 0 = refl m+0≡m (1+ m) = cong (m+0≡m m)

m+1+n≡1+m+n : (m n : ℕ) → m + (1 + n) ≡ 1 + (m + n) m+1+n≡1+m+n 0 n = refl m+1+n≡1+m+n (1+ m) n = cong (m+1+n≡1+m+n m n)

-- 3.3. The commutativity of addition using equational reasoning. -- +-commutative : (m n : ℕ) → m + n ≡ n + m +-commutative 0 n = sym (m+0≡m n) +-commutative (1+ m) n = 

 begin
   1+ m + n   ~⟨ refl ⟩
   1+ (m + n) ~⟨ cong (+-commutative m n) ⟩
   1+ (n + m) ~⟨ sym (m+1+n≡1+m+n n m) ⟩
   n + 1+ m
 ∎

-- 3.4. -- even+even≡odd : {m n : ℕ} → 2×ℕ m → 2×ℕ n → 2×ℕ+1 (m + n) even+even≡odd zero zero = {!!} even+even≡odd _ _ = {!!} -- ^ -- That gives -- -- ?0 : 2×ℕ+1 (zero + zero) -- ?1 : 2×ℕ+1 (.m + .n) -- -- but 2×ℕ+1 (zero + zero) = 2×ℕ+1 0 which is uninhabited, so that this proof -- can not be writen. --

-- The absurd (obviously uninhabited) type. -- -- ⊥-introduction is empty. -- data ⊥ : Set where

-- The negation of a proposition. -- infix 6 ¬_ ¬_ : Set → Set ¬ A = A → ⊥

-- 4.1. Disproof or proof by contradiction. -- -- To disprove even+even≡odd we assume that even+even≡odd and derive -- absurdity, i.e. uninhabited type. -- even+even≢odd : {m n : ℕ} → 2×ℕ m → 2×ℕ n → ¬ 2×ℕ+1 (m + n) even+even≢odd zero zero () even+even≢odd zero (2+ n) (2+ m+n) = even+even≢odd zero n m+n even+even≢odd (2+ m) n (2+ m+n) = even+even≢odd m n m+n

-- 4.2. -- -- even+even≢even : {m n : ℕ} → 2×ℕ m → 2×ℕ n → ¬ 2×ℕ (m + n) -- even+even≢even zero zero () -- ^ -- rejected with the following message: -- -- 2×ℕ zero should be empty, but the following constructor patterns -- are valid: -- zero -- when checking that the clause even+even≢even zero zero () has type -- {m n : ℕ} → 2×ℕ m → 2×ℕ n → ¬ 2×ℕ (m + n) -- </lang>

Coq

1.1., 1.2., 2.1. and 3.1.:

<lang coq>Inductive nat : Set := 

 | O : nat
 | S : nat -> nat.

Fixpoint plus (n m:nat) {struct n} : nat := 

 match n with
   | O => m
   | S p => S (p + m)
 end

where "n + m" := (plus n m) : nat_scope.


Inductive even : nat -> Set := 

 | even_O : even O
 | even_SSn : forall n:nat,
               even n -> even (S (S n)).


Theorem even_plus_even : forall n m:nat, 

 even n -> even m -> even (n + m).

Proof. 

 intros n m H H0.
 
 elim H.
 trivial.
 
 intros.
 simpl.
 
 case even_SSn.
 intros.
 apply even_SSn; assumption.
 
 assumption.

Qed. </lang>

Go

Go lacks the infrastructure for theorem proving and so there's a limit to what can be done here. Rather than give up altogether, I've therefore just done what I can.

It's not difficult (though extremely inefficient) to program a Peano representation of the natural numbers (including addition thereon) in a non-rigorous way and I've based this on the code here which was written by the Go authors themselves to test heavy recursion (part 1.1 and 2).

Even numbers of the form (2 * x) and odd numbers of the form (2 * x + 1) can then be easily constructed by creating types with only one field, the natural number 'x' (parts 1.2 and 1.3).

We can then define addition for the even number type by simply adding the fields 'x' of the operands. Given the method of construction it is obvious that 3.1 must be true and 3.4, 4.1 and 4.2 must be false. If it were otherwise the 'addEven' function would simply not compile in a strongly-typed language such as Go.

However, there is no obvious way to show that 3.2 and 3.3 must always be true (though you could argue that addition must be commutative because the 'addEven' function is symmetric in relation to its arguments). So all I've been able to do here is write functions to test these assertions for given even numbers whilst accepting that this doesn't constitute a proof for all such cases. <lang go>package main

import "fmt"

// Represents a natural number. type Number struct { 

   pred *Number

}

// Represents an even natural number: 2 * half. type EvenNumber struct { 

   half *Number

}

// Represents an odd natural number: 2 * half + 1. type OddNumber struct { 

   half *Number

}

// Returns the natural number zero by a nil pointer. func zero() *Number { return nil }

// Tests whether a natural number is zero. func isZero(x *Number) bool { return x == nil }

// Increments a natural number. func add1(x *Number) *Number { 

   y := new(Number)
   y.pred = x
   return y

}

// Decrements a natural number. func sub1(x *Number) *Number { return x.pred }

// Adds two natural numbers. func add(x, y *Number) *Number { 

   if isZero(y) {
       return x
   }
   return add(add1(x), sub1(y))

}

// Adds two even natural numbers. func addEven(x, y *EvenNumber) *EvenNumber { 

   if isZero(y.half) {
       return x
   }
   return genEven(add(x.half, y.half))

}

// Adds two odd natural numbers. func addOdd(x, y *OddNumber) *EvenNumber { 

   z := add(x.half, y.half)
   return genEven(add1(z))

}

// Generate a natural number from a uint. func gen(n uint) *Number { 

   if n > 0 {
       return add1(gen(n - 1))
   }
   return zero()

}

// Generate an even natural number: 2 * half. func genEven(half *Number) *EvenNumber { 

   return &EvenNumber{half}

}

// Generate an odd natural number: 2 * half + 1. func genOdd(half *Number) *OddNumber { 

   return &OddNumber{half}

}

// Counts a natural number i.e returns its integer representation. func count(x *Number) uint { 

   if isZero(x) {
       return 0
   }
   return count(sub1(x)) + 1

}

// Counts an even natural number i.e returns its integer representation. func countEven(x *EvenNumber) uint { 

   return count(x.half) * 2

}

// Counts an odd natural number i.e returns its integer representation. func countOdd(x *OddNumber) uint { 

   return count(x.half)*2 + 1

}

// Test if even number addition is commutative for given numbers. func testCommutative(e1, e2 *EvenNumber) { 

   c1, c2 := countEven(e1), countEven(e2)
   passed := countEven(addEven(e1, e2)) == countEven(addEven(e2, e1))
   symbol := "=="
   if !passed {
       symbol = "!="
   }
   fmt.Printf("\n%d + %d %s %d + %d\n", c1, c2, symbol, c2, c1)

}

// Test if even number arithmetic is associative for given numbers. func testAssociative(e1, e2, e3 *EvenNumber) { 

   c1, c2, c3 := countEven(e1), countEven(e2), countEven(e3)
   passed := countEven(addEven(addEven(e1, e2), e3)) == countEven(addEven(e1, addEven(e2, e3)))
   symbol := "=="
   if !passed {
       symbol = "!="
   }
   fmt.Printf("\n(%d + %d) + %d %s %d + (%d + %d)\n", c1, c2, c3, symbol, c1, c2, c3)

}

func main() { 

   numbers := [10]*Number{}
   fmt.Println("The first 10 natural numbers are:")
   for i := uint(0); i < 10; i++ {
       numbers[i] = gen(i)
       fmt.Printf("%d ", count(numbers[i]))
   }
   fmt.Println("\n")

   fmt.Println("The first 10 even natural numbers are:")
   for i := uint(0); i < 10; i++ {
       e := genEven(numbers[i])
       fmt.Printf("%d ", countEven(e))
   }
   fmt.Println("\n")

   fmt.Println("The first 10 odd natural numbers are:")
   for i := uint(0); i < 10; i++ {
       o := genOdd(numbers[i])
       fmt.Printf("%d ", countOdd(o))
   }
   fmt.Println("\n")

   fmt.Print("The sum of the first 10 natural numbers is: ")
   sum := numbers[0]
   for i := 1; i < 10; i++ {
       sum = add(sum, numbers[i])
   }
   fmt.Println(count(sum))

   fmt.Print("\nThe sum of the first 10 even natural numbers (increasing order) is: ")
   sumEven := genEven(numbers[0])
   for i := 1; i < 10; i++ {
       sumEven = addEven(sumEven, genEven(numbers[i]))
   }
   fmt.Println(countEven(sumEven))

   fmt.Print("\nThe sum of the first 10 even natural numbers (decreasing order) is: ")
   sumEven = genEven(numbers[9])
   for i := 8; i >= 0; i-- {
       sumEven = addEven(sumEven, genEven(numbers[i]))
   }
   fmt.Println(countEven(sumEven))

   testCommutative(genEven(numbers[8]), genEven(numbers[9]))

   testAssociative(genEven(numbers[7]), genEven(numbers[8]), genEven(numbers[9]))

}</lang>

Output:
The first 10 natural numbers are:
0 1 2 3 4 5 6 7 8 9 

The first 10 even natural numbers are:
0 2 4 6 8 10 12 14 16 18 

The first 10 odd natural numbers are:
1 3 5 7 9 11 13 15 17 19 

The sum of the first 10 natural numbers is: 45

The sum of the first 10 even natural numbers (increasing order) is: 90

The sum of the first 10 even natural numbers (decreasing order) is: 90

16 + 18 == 18 + 16

(14 + 16) + 18 == 14 + (16 + 18)

Haskell

Using GADTs and type families it is possible to write a partial adaptation of the Agda version:

<lang haskell>{-# LANGUAGE TypeOperators, TypeFamilies, GADTs #-}

module PeanoArithmetic where

-- 1.1. Natural numbers.

data Z = Z data S m = S m

-- 2.1. Addition.

infixl 6 :+ type family x :+ y type instance Z :+ n = n type instance S m :+ n = S (m :+ n)

-- 1.2. Even natural numbers.

data En :: * -> * where 

 Ez :: En Z
 Es :: En m -> En (S (S m))

-- 1.3. Odd natural numbers.

data On :: * -> * where 

 Oo :: On (S Z)
 Os :: On m -> On (S (S m))

-- 3.1. Sum of any two even numbers is even.

sum_of_even_is_even :: En m -> En n -> En (m :+ n) sum_of_even_is_even Ez n = n sum_of_even_is_even (Es m) n = Es $ sum_of_even_is_even m n

-- The identity type for natural numbers.

infix 4 := data (:=) m :: * -> * where 

 Refl :: m := m

sym :: m := n -> n := m sym Refl = Refl

trans :: m := n -> n := p -> m := p trans Refl np = np

cong :: m := n -> S m := S n cong Refl = Refl

-- 3.2. Associativity of addition (via propositional equality).

class AssocAdd m where 

 proof :: m -> n -> p -> (m :+ n) :+ p := m :+ (n :+ p)

instance AssocAdd Z where 

 proof Z _ _ = Refl

instance AssocAdd m => AssocAdd (S m) where 

 proof (S m) n p = cong $ proof m n p

-- Induction, associativity of addition by induction, equational reasoning and -- commutativity of addition is too tricky.

-- 3.4. Bad proof.

sum_of_even_is_odd :: En m -> En n -> On (m :+ n) -- ^ -- Сan not be written totally: -- sum_of_even_is_odd Ez Ez = undefined -- ^ -- then, in GHCi: -- -- *PeanoArithmetic> :t sum_of_even_is_odd Ez Ez -- sum_of_even_is_odd Ez Ez :: On (Z :+ Z) -- *PeanoArithmetic> :t undefined :: On (Z :+ Z) -- undefined :: On (Z :+ Z) :: On Z -- *PeanoArithmetic> :t sum_of_even_is_odd Ez Ez :: On Z -- sum_of_even_is_odd Ez Ez :: On Z :: On Z -- *PeanoArithmetic> :t Oo -- Oo :: On (S Z) -- *PeanoArithmetic> :t Os Oo -- Os Oo :: On (S (S (S Z))) -- *PeanoArithmetic> :t Os (Os Oo) -- Os (Os Oo) :: On (S (S (S (S (S Z))))) -- -- so that sum_of_even_is_odd Ez Ez :: On Z, but On Z is empty, it is impossible -- to write such a proof. --

-- Uninhabited type.

data Bot

-- Negation.

type Not a = a -> Bot

-- 4.1. Disproof.

sum_of_even_is_not_odd :: En m -> En n -> Not (On (m :+ n)) sum_of_even_is_not_odd Ez (Es n) (Os mn) = sum_of_even_is_not_odd Ez n mn sum_of_even_is_not_odd (Es m) n (Os mn) = sum_of_even_is_not_odd m n mn sum_of_even_is_not_odd Ez Ez _ = 

 error "impossible happened in sum_of_even_is_not_odd!"

-- ^ -- partial, however, we know that Ez :: En Z, Z + Z = Z and On Z is -- uninhabited, so that this clause is unreachable. -- -- Also, GHC complains: -- -- Warning: Pattern match(es) are non-exhaustive -- In an equation for `sum_of_even_is_not_odd': -- Patterns not matched: -- Ez (Es _) Oo -- (Es _) _ Oo -- -- and can't find that this clauses is unreachable too, since this isn't type check: -- -- sum_of_even_is_not_odd Ez (Es _) Oo = undefined -- sum_of_even_is_not_odd (Es _) _ Oo = undefined --

-- 4.2. Bad disproof.

sum_of_even_is_not_even :: En m -> En n -> Not (En (m :+ n)) -- -- Starting from a partial definition: -- sum_of_even_is_not_even Ez Ez _ = undefined -- -- we can show that it can not be rewritten totally: -- -- *PeanoArithmetic> :t sum_of_even_is_not_even Ez Ez -- sum_of_even_is_not_even Ez Ez :: Not (En (Z :+ Z)) -- *PeanoArithmetic> :t sum_of_even_is_not_even Ez Ez :: Not (En Z) -- sum_of_even_is_not_even Ez Ez :: Not (En Z) :: Not (En Z) -- *PeanoArithmetic> :t sum_of_even_is_not_even Ez Ez :: En Z -> Bot -- sum_of_even_is_not_even Ez Ez :: En Z -> Bot :: En Z -> Bot -- *PeanoArithmetic> :t Ez -- Ez :: En Z -- *PeanoArithmetic> :t (sum_of_even_is_not_even Ez Ez :: En Z -> Bot) Ez -- (sum_of_even_is_not_even Ez Ez :: En Z -> Bot) Ez :: Bot -- -- since we have a "citizen" of an uninhabited type here (contradiction!). -- </lang>

See also Proof/Haskell for implementation of a small theorem prover.


Idris

Idris supports two types of proofs: using tactics, like Coq, or just write functions like Agda. These ways can be combined.

<lang Idris> module Proof

-- Enable terminator checker by default %default total

-- 1.1 Natural numbers %elim data MyNat 

 = Z 
 | S MyNat


-- 1.2 Even naturals %elim data EvNat : MyNat -> Type where 

 EvO  : EvNat Z
 EvSS : EvNat x -> EvNat (S (S x))


-- 1.3 Odd naturals %elim data OddNat : MyNat -> Type where 

 Odd1  : OddNat (S Z)
 OddSS : OddNat x -> OddNat (S (S x))


-- 2.1 addition

infixl 4 :+

(:+) : MyNat -> MyNat -> MyNat (:+) Z b = b (:+) (S a) b = S (a :+ b)


-- 3.1, Prove that the addition of any two even numbers is even.


evensPlus1 : {a : MyNat} -> {b : MyNat} -> (EvNat a) -> (EvNat b) -> (EvNat (a :+ b)) evensPlus1 ea eb = ?proof31

congS : {a : MyNat} -> {b : MyNat} -> (a = b) -> (S a = S b) congS refl = refl

evensPlus2 : {a : MyNat} -> {b : MyNat} -> (EvNat a) ->(EvNat b) -> (EvNat (a :+ b)) evensPlus2 EvO eb = eb evensPlus2 {a=(S (S a))} (EvSS ea) eb = EvSS (evensPlus2 ea eb)

-- 3.2 Prove that the addition is always associative. plusAssoc : (a : MyNat) -> (b : MyNat) -> (c : MyNat) -> (a :+ b) :+ c = a :+ (b :+ c) plusAssoc Z b c = refl plusAssoc (S a) b c = congS (plusAssoc a b c)


-- 3.3 Prove that the addition is always commutative.

plus0_r : (a : MyNat) -> a :+ Z = a plus0_r Z = refl plus0_r (S a) = congS (plus0_r a)

plusS_r : (a : MyNat) -> (b : MyNat) -> a :+ S b = S (a :+ b) plusS_r Z b = refl plusS_r (S a) b = congS (plusS_r a b)

plusComm : (a : MyNat) -> (b : MyNat) -> a :+ b = b :+ a plusComm a b = ?proof33


-- 4.1 Prove that the addition of any two even numbers cannot be odd.

evenNotOdd : (ea : EvNat a) -> (oa : OddNat a) -> _|_ evenNotOdd (EvSS e) (OddSS o) = evenNotOdd e o

evensPlusNotOdd : (ea : EvNat a) -> (eb : EvNat b) -> (OddNat (a :+ b)) -> _|_ evensPlusNotOdd EvO (EvSS eb) (OddSS ob) = evenNotOdd eb ob evensPlusNotOdd (EvSS y) EvO oab = ?epno_so evensPlusNotOdd (EvSS y) (EvSS z) oab = ?epno_ss


Proofs ----------


Proof.proof31 = proof 

 intro a
 intro b
 intro ea
 intro eb
 induction ea
 compute
 exact eb
 intro x
 intro ex
 intro exh
 exact (EvSS exh)


Proof.proof33 = proof 

 intros
 induction a
 compute
 rewrite sym (plus0_r b)
 trivial
 intro a'
 intro ha'
 compute
 rewrite sym (plusS_r b a')
 exact (congS ha')


Proof.epno_ss = proof 

 intro x
 intro ex
 intro y
 intro ey
 compute
 rewrite sym ( plusS_r x (S y))
 rewrite sym ( plusS_r x y)
 intro os4xy
 let es4xy = EvSS(EvSS (evensPlus1 ex ey))
 exact evenNotOdd es4xy os4xy


Proof.epno_so = proof 

 intro x
 intro ex
 rewrite sym (plus0_r x)
 compute
 rewrite sym (plus0_r x)
 intro ossx
 exact evenNotOdd (EvSS ex) ossx

</lang>

J

A Peano number needs a zero, a mechanism for distinguishing equality from its absence, a way of getting a successor to a number, and a system of induction.

We know that a computer can never fully implement peano numbers because a computer can only represent a finite number of distinct values (if necessary, Ackerman's function can be used to illustrate the existence of this limitation). So our implementation of Peano Numbers will represent these mechanisms rather than be a complete implementation of these mechanisms.

So, these can be our definitions:

<lang J>context=:3 :0 

 if. 0 = L. y do. context (,: ; ]) y return. end.
   kernel=. > {: y
   symbols=. (#$kernel) {. > {. y
   order=. /: symbols
   (symbols /: order); order |: kernel

) symbols=: >@{. kernel=: >@{:

zero=: context0x

monadic=: (1 :'[:context u@symbols; u&kernel')( :[:) dyadic=: (1 :'[:context ,&symbols; u&kernel')([: :) successor=: +&1x monadic equals=: -:&kernel all=: i. >. %: kernel successor@$: ::] zero is_member_of=: e.&(-. -.&all)&,&kernel ([: :) exists_in=: 1&e.@e.&kernel ([: :) not=: -. :[: induction=:4 :0 

  3 :'(y)=:context ?~#all'&.>;:x
  assert (#all) > #;._1 LF,y
  assert 0 = # y -.&;: LF,defined,x
  assert 0!:3 y

)

addition=: +/ dyadic isN=: ([ assert@is_member_of&(context all)) even=: [: context kernel@addition~"0@,@kernel :[: @isN odd=: successor@even

defined=: '(zero not exists_in odd successor equals is_member_of addition even)'</lang>

Here, even is a function which, given a natural number, produces a corresponding even natural number. odd is a similar function which gives us odd numbers.

And, induction is a verb where the left argument lists values which represent any natural number and the right argument represents an expression to be considered. If induction succeeds, the expression is true for all natural numbers.

Meanwhile -. is J's implementation of "logical negation" -- a function which has been grafted onto boolean algebra as a convenience for people working with logical systems. ("Grafted on" here means: [a] logical negation was not originally a part of the definition of boolean algebra, and [b] logical negation cannot be a valid part of an infinite set of systems that were valid boolean algebras before someone decided that logical negation should be a part of boolean algebra.)

Also is_member_of represents universal set membership, exists_in represents the existential quantifier.

Note that we do not have to prove that our definitions are correct -- they are our axioms that we use in our proof.

Thus, this is proof that

  1. all natural numbers are even, and
  2. addition is associative, and
  3. addition is commutative, and
  4. the sum of two even numbers can never be odd.

<lang J>'A B C' induction 0 :0 

 ((even A) addition (even B)) is_member_of (even C)
 ((A addition B) addition C) equals (A addition (B addition C))
 (A addition B) equals (B addition A)
 not ((even A) addition (even B)) exists_in (odd C)

)</lang>

Meanwhile, here is how the invalid proofs fail:

<lang J> 'A B C' induction '((even A) addition (even B)) is_member_of (odd C)' |assertion failure: assert</lang>

and

<lang J> 'A B C' induction 'not ((even A) addition (even B)) is_member_of (even C)' |assertion failure: assert</lang>


As an aside, note that peano numbers show us that numbers can represent recursive processes.

OCaml

Using GADT, we can port the Coq version to OCaml.

<lang OCaml> type zero = Zero type 'a succ = Succ

(* 1.1 *) type _ nat = 

 | Zero : zero nat
 | Succ : 'a nat -> 'a succ nat

(* 1.2 *) type _ even = 

 | Even_zero : zero even
 | Even_ss : 'a even -> 'a succ succ even

(* 2.1: define the addition relation *) type (_, _, _) plus = 

 | Plus_zero : ('a, zero, 'a) plus
 | Plus_succ : ('a, 'b, 'c) plus -> ('a, 'b succ, 'c succ) plus

(* 3.1 *)

(* Define the property: there exists a number 'sum that is the sum of 'a and 'b and is even. *) type ('a, 'b) exists_plus_even = Exists_plus_even : ('a, 'b, 'sum) plus * 'sum even -> ('a, 'b) exists_plus_even

(* The proof that if a and b are even, there exists sum that is the sum of a and b that is even. 

  This is a valid proof since it terminated, and it is total (no assert false, and the exhaustiveness
  test of pattern matching doesn't generate a warning) *)

let rec even_plus_even : type a b. a even -> b even -> (a, b) exists_plus_even = 

 fun a_even b_even ->
   match b_even with
   | Even_zero ->
     Exists_plus_even (Plus_zero, a_even)
   | Even_ss b_even' ->
     let Exists_plus_even (plus, even) = even_plus_even a_even b_even' in
     Exists_plus_even (Plus_succ (Plus_succ plus), Even_ss even)

(* 3.3 *)

(* 0 + n = n *) let rec plus_zero : type a. a nat -> (zero, a, a) plus = function 

 | Zero -> Plus_zero
 | Succ a -> Plus_succ (plus_zero a)

(* a + b = n => (a + 1) + b = (n + 1) *) let rec plus_succ_left : type a b sum. (a, b, sum) plus -> (a succ, b, sum succ) plus = 

 function
 | Plus_zero -> Plus_zero
 | Plus_succ plus -> Plus_succ (plus_succ_left plus)

(* a + b = n => b + a = n *) let rec plus_commutative : type a b sum. a nat -> (a, b, sum) plus -> (b, a, sum) plus = 

 fun a plus ->
   match plus with
   | Plus_zero -> plus_zero a
   | Plus_succ plus' ->
     plus_succ_left (plus_commutative a plus')

</lang>

Omega

<lang omega>data Even :: Nat ~> *0 where 

  EZ:: Even Z
  ES:: Even n -> Even (S (S n))

plus:: Nat ~> Nat ~> Nat {plus Z m} = m {plus (S n) m} = S {plus n m}

even_plus:: Even m -> Even n -> Even {plus m n} even_plus EZ en = en even_plus (ES em) en = ES (even_plus em en) </lang>

Phix

Silly text-based version, just for fun.
Obviously 1.1/1.2/1.3 are (loosely) eumlated by the strings "int"/"even"/"odd", and addition by axioms[4].
Clearly 3.2 and 3.3 are not attempted, and I'm not sure which of 3.4/4.1/4.2 the last test is closest to, or for that matter what the difference between them is supposed to be. <lang Phix>constant axioms = {{"even+1","odd"}, 

                  {"even","2*int"},
                  {"2*int+2*int","2*(int+int)"},
                  {"(int+int)","int"},
                  {"2*int","even"},
                  {"odd","2*int+1"}}

procedure proof(string a, b) -- try to convert a into b using only the above axioms 

   string w = a
   sequence seen = {}          -- (avoid infinite loops)
   while not find(w,seen) do
       seen = append(seen,w)
       ?w
       if w=b then exit end if
       integer hit = 0
       for i=1 to length(axioms) do
           string {r,s} = axioms[i]
           integer k = match(r,w)
           if k then
               w[k..k+length(r)-1] = s
               hit = i
               exit
           end if
       end for
       if hit=0 then exit end if
       puts(1,"== ")
   end while
   ?{a,b,iff(w=b?"true","false")}

end procedure

proof("even+even","even") proof("even+1","odd") --bad proofs: proof("int","even") proof("even+even","odd")</lang>

Output:
"even+even"
== "2*int+even"
== "2*int+2*int"
== "2*(int+int)"
== "2*int"
== "even"
{"even+even","even","true"}
"even+1"
== "odd"
{"even+1","odd","true"}
"int"
{"int","even","false"}
"even+even"
== "2*int+even"
== "2*int+2*int"
== "2*(int+int)"
== "2*int"
== "even"
== {"even+even","odd","false"}

Racket

Via #lang cur. Currently (August 29th, 2019), I am using a development version, which could be installed by the following commands: 

raco pkg install https://github.com/stchang/macrotypes.git?path=macrotypes-lib#cur
raco pkg install https://github.com/stchang/macrotypes.git?path=turnstile-lib#cur
raco pkg install https://github.com/wilbowma/cur.git?path=cur-lib#turnstile-core

<lang racket>#lang cur

(require rackunit 

        cur/stdlib/equality
        cur/stdlib/sugar
        cur/ntac/base
        cur/ntac/standard
        cur/ntac/rewrite)
Task 1.1

(data nat : 0 Type 

     [O : nat]
     [S : (-> nat nat)])

(define-syntax #%datum 

 (syntax-parser
   [(_ . n:exact-nonnegative-integer)
    #:when (zero? (syntax-e #'n))
    #'O]
   [(_ . n:exact-nonnegative-integer)
    #`(S (#%datum . #,(- (syntax-e #'n) 1)))]))

(check-equal? (S (S (S (S O)))) 4)

Task 1.2

(data even : 0 (-> nat Type) 

     [even-O : (even 0)]
     [even-SS : (forall [n : nat] (-> (even n) (even (S (S n)))))])
Task 1.3

(data odd : 0 (-> nat Type) 

     [odd-O : (odd 1)]
     [odd-SS : (forall [n : nat] (-> (odd n) (odd (S (S n)))))])
Task 2.1

(define/rec/match + : nat [m : nat] -> nat 

 [O => m]
 [(S n-1) => (S (+ n-1 m))])

(check-equal? (+ 2 3) 5)

Task 3.1

(define-theorem even-plus-even-is-even 

 (∀ [n : nat] [m : nat] (-> (even n) (even m) (even (+ n m))))
 (by-intros n m Hn Hm)
 (by-induction Hn #:as [() (n* IHn)])

 ;; subgoal 1
 simpl
 by-assumption

 ;; subgoal 2
 (by-apply even-SS)
 by-assumption)
Task 3.2

(define-theorem addition-assoc 

 (∀ [a : nat] [b : nat] [c : nat] (== nat (+ (+ a b) c) (+ a (+ b c))))

 (by-intros a b c)
 (by-induction a #:as [() (a-1 IHa)])

 ;; subgoal 1
 reflexivity

 ;; subgoal 2
 display-focus ; show how the context and goal are before rewrite
 (by-rewrite IHa)
 display-focus ; show how the context and goal are after rewrite
 reflexivity)</lang>
Output:
b : nat
c : nat
a-1 : nat
IHa : (== nat (+ (+ a-1 b) c) (+ a-1 (+ b c)))
--------------------------------
(== nat (S (+ (+ a-1 b) c)) (S (+ a-1 (+ b c))))

b : nat
c : nat
a-1 : nat
IHa : (== nat (+ (+ a-1 b) c) (+ a-1 (+ b c)))
--------------------------------
(== nat (S (+ a-1 (+ b c))) (S (+ a-1 (+ b c))))

The fact that there's no error indicates that all proofs are verified.

Salmon

Note that the only current implementation of Salmon is an interpreter that ignores proofs and doesn't try to check them, but in the future when there is an implementation that checks proofs, it should be able to check the proof in this Salmon code.

<lang Salmon>pure function even(x) returns boolean ((x in [0...+oo)) && ((x % 2) == 0)); theorem(forall(x : even, y : even) ((x + y) in even)) proof 

 {
   forall (x : even, y : even)
     {
     L1:
       x in even;
     L2:
       ((x in [0...+oo)) && ((x % 2) == 0)) because type_definition(L1);
     L3:
       ((x % 2) == 0) because simplification(L2);
     L4:
       y in even;
     L5:
       ((y in [0...+oo)) && ((y % 2) == 0)) because type_definition(L4);
     L6:
       ((y % 2) == 0) because simplification(L5);
     L7:
       (((x + y) % 2) == (x % 2) + (y % 2)); // axiom of % and +
     L8:
       (((x + y) % 2) == 0 + (y % 2)) because substitution(L3, L7);
     L9:
       (((x + y) % 2) == 0 + 0) because substitution(L6, L8);
     L10:
       (((x + y) % 2) == 0) because simplification(L9);
       (x + y) in even because type_definition(even, L10);
     };
 };</lang>

Tcl

Using the datatype package from the Pattern Matching task...

Works with: Tcl version 8.5

<lang tcl>package require datatype datatype define Int = Zero | Succ val datatype define EO = Even | Odd

proc evenOdd val { 

   global environment
   datatype match $val {

case Zero -> { Even } case [Succ [Succ x]] -> { evenOdd $x } case t -> { set term [list evenOdd $t] if {[info exists environment($term)]} { return $environment($term) } elseif {[info exists environment($t)]} { return [evenOdd $environment($t)] } else { return $term } } 

   }

}

proc add {a b} { 

   global environment
   datatype match $a {

case Zero -> { return $b } case [Succ x] -> { Succ [add $x $b] } case t -> { datatype match $b { case Zero -> { return $t } case [Succ x] -> { Succ [add $t $x] } case t2 -> { set term [list add $t $t2] if {[info exists environment($term)]} { return $environment($term) } elseif {[info exists environment($t)]} { return [add $environment($t) $t2] } elseif {[info exists environment($t2)]} { return [add $t $environment($t2)] } else { return $term } } } } 

   }

}

puts "BASE CASE" puts "evenOdd Zero = [evenOdd Zero]" puts "evenOdd \[add Zero Zero\] = [evenOdd [add Zero Zero]]"

puts "\nITERATIVE CASE" set environment([list evenOdd p]) Even puts "if evenOdd p = Even..." puts "\tevenOdd \[Succ \[Succ p\]\] = [evenOdd [Succ [Succ p]]]" unset environment puts "if evenOdd \[add p q\] = Even..." set environment([list evenOdd [add p q]]) Even foreach {a b} { 

   p q
   {Succ {Succ p}} q
   p {Succ {Succ q}}
   {Succ {Succ p}} {Succ {Succ q}}

} { 

   puts "\tevenOdd \[[list add $a $b]\] = [evenOdd [add $a $b]]"

}</lang> Output: 

BASE CASE
evenOdd Zero = Even
evenOdd [add Zero Zero] = Even

ITERATIVE CASE
if evenOdd p = Even...
	evenOdd [Succ [Succ p]] = Even
if evenOdd [add p q] = Even...
	evenOdd [add p q] = Even
	evenOdd [add {Succ {Succ p}} q] = Even
	evenOdd [add p {Succ {Succ q}}] = Even
	evenOdd [add {Succ {Succ p}} {Succ {Succ q}}] = Even

It is up to the caller to take the output of this program and interpret it as a proof.

Twelf

<lang twelf>nat : type. z  : nat. s  : nat -> nat.


plus  : nat -> nat -> nat -> type. plus-z : plus z N2 N2. plus-s : plus (s N1) N2 (s N3) 

         <- plus N1 N2 N3.


%% declare totality assertion %mode plus +N1 +N2 -N3. %worlds () (plus _ _ _).

%% check totality assertion %total N1 (plus N1 _ _).


even  : nat -> type. even-z : even z. even-s : even (s (s N)) 

         <- even N.


sum-evens : even N1 -> even N2 -> plus N1 N2 N3 -> even N3 -> type. %mode sum-evens +D1 +D2 +Dplus -D3.

sez : sum-evens 

      even-z 
      (DevenN2 : even N2)
      (plus-z : plus z N2 N2)
      DevenN2.

ses : sum-evens 

      ( (even-s DevenN1') : even (s (s N1')))
      (DevenN2 : even N2)
      ( (plus-s (plus-s Dplus)) : plus (s (s N1')) N2 (s (s N3')))
      (even-s DevenN3')
      <- sum-evens DevenN1' DevenN2 Dplus DevenN3'.

%worlds () (sum-evens _ _ _ _). %total D (sum-evens D _ _ _). </lang>

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