111 lines
3.0 KiB
Plaintext
111 lines
3.0 KiB
Plaintext
/-
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Chapter 2
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Dependent Type Theory
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-/
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-- ========================================
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-- Exercise 1
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--
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-- Define the function `Do_Twice`, as described in Section 2.4.
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-- ========================================
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namespace ex1
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def double (x : Nat) := x + x
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def doTwice (f : Nat → Nat) (x : Nat) : Nat := f (f x)
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def doTwiceTwice (f : (Nat → Nat) → (Nat → Nat)) (x : Nat → Nat) := f (f x)
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#reduce doTwiceTwice doTwice double 2
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end ex1
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-- ========================================
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-- Exercise 2
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--
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-- Define the functions `curry` and `uncurry`, as described in Section 2.4.
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-- ========================================
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namespace ex2
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def curry (f : α × β → γ) : (α → β → γ) :=
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fun α β => f (α, β)
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def uncurry (f : α → β → γ) : (α × β → γ) :=
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fun x => f x.1 x.2
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end ex2
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-- ========================================
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-- Exercise 3
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--
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-- Above, we used the example `vec α n` for vectors of elements of type `α` of
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-- length `n`. Declare a constant `vec_add` that could represent a function that
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-- adds two vectors of natural numbers of the same length, and a constant
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-- `vec_reverse` that can represent a function that reverses its argument. Use
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-- implicit arguments for parameters that can be inferred. Declare some
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-- variables and check some expressions involving the constants that you have
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-- declared.
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-- ========================================
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namespace ex3
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universe u
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axiom vec : Type u → Nat → Type u
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namespace vec
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axiom empty : ∀ (α : Type u), vec α 0
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axiom cons : ∀ (α : Type u) (n : Nat), α → vec α n → vec α (n + 1)
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axiom append : ∀ (α : Type u) (n m : Nat), vec α m → vec α n → vec α (n + m)
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axiom add : ∀ {α : Type u} {n : Nat}, vec α n → vec α n → vec α n
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axiom reverse : ∀ {α : Type u} {n : Nat}, vec α n → vec α n
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end vec
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-- Check results.
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variable (a b : vec Prop 1)
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variable (c d : vec Prop 2)
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#check vec.add a b
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-- #check vec.add a c
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#check vec.reverse a
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end ex3
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-- ========================================
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-- Exercise 4
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--
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-- Similarly, declare a constant `matrix` so that `matrix α m n` could represent
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-- the type of `m` by `n` matrices. Declare some constants to represent
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-- functions on this type, such as matrix addition and multiplication, and
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-- (using vec) multiplication of a matrix by a vector. Once again, declare some
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-- variables and check some expressions involving the constants that you have
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-- declared.
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-- ========================================
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namespace ex4
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universe u
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axiom matrix : Type u → Nat → Nat → Type u
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namespace matrix
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axiom add : ∀ {α : Type u} {m n : Nat},
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matrix α m n → matrix α m n → matrix α m n
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axiom mul : ∀ {α : Type u} {m n p : Nat},
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matrix α m n → matrix α n p → matrix α m p
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axiom app : ∀ {α : Type u} {m n : Nat},
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matrix α m n → ex3.vec α n → ex3.vec α m
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end matrix
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variable (a b : matrix Prop 5 7)
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variable (c : matrix Prop 7 3)
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variable (d : ex3.vec Prop 3)
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#check matrix.add a b
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-- #check matrix.add a c
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#check matrix.mul a c
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#check matrix.app c d
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end ex4
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