Theorem Proving in Lean. Part of exercises 8.

.gitignore

@@ -1,2 +1,4 @@
 /build
 /lake-packages/*
+/_target
+/leanpkg.path

src/theorem-proving-in-lean/exercises-7.lean

@@ -183,11 +183,11 @@ end ex2
 -- assignment of values to the variables.
 namespace ex3
 
-inductive foo : Type _
+inductive Foo : Type _
-  | const : Nat → foo
+  | const : Nat → Foo
-  | var : Nat → foo
+  | var : Nat → Foo
-  | plus : foo → foo → foo
+  | plus : Foo → Foo → Foo
-  | times : foo → foo → foo
+  | times : Foo → Foo → Foo
 
 def value_at : Nat → List Nat → Nat
 | _,       [] => default
@@ -195,10 +195,10 @@ def value_at : Nat → List Nat → Nat
 | (i + 1), vs => value_at i (List.tail! vs)
 
 -- The provided "variables" are supplied in a 0-indexed list.
-def eval_foo : foo → List Nat → Nat
+def eval_foo : Foo → List Nat → Nat
-  | (foo.const n)  , _  => n
+  | (Foo.const n)  , _  => n
-  | (foo.var n)    , vs => value_at n vs
+  | (Foo.var n)    , vs => value_at n vs
-  | (foo.plus m n) , vs => eval_foo m vs + eval_foo n vs
+  | (Foo.plus m n) , vs => eval_foo m vs + eval_foo n vs
-  | (foo.times m n), vs => eval_foo m vs * eval_foo n vs
+  | (Foo.times m n), vs => eval_foo m vs * eval_foo n vs
 
 end ex3

src/theorem-proving-in-lean/exercises-8.lean

@@ -0,0 +1,138 @@
+/- Exercises 8
+ -
+ - Avigad, Jeremy. ‘Theorem Proving in Lean’, n.d.
+-/
+
+-- Exercise 1
+--
+-- Open a namespace `Hidden` to avoid naming conflicts, and use the equation
+-- compiler to define addition, multiplication, and exponentiation on the
+-- natural numbers. Then use the equation compiler to derive some of their basic
+-- properties.
+namespace ex1
+
+def add : Nat → Nat → Nat
+  | m, Nat.zero   => m
+  | m, Nat.succ n => Nat.succ (add m n)
+
+def mul : Nat → Nat → Nat
+  | _, Nat.zero => 0
+  | m, Nat.succ n => add m (mul m n)
+
+def exp : Nat → Nat → Nat
+  | _, Nat.zero => 1
+  | m, Nat.succ n => mul m (exp m n)
+
+end ex1
+
+-- Exercise 2
+--
+-- Similarly, use the equation compiler to define some basic operations on lists
+-- (like the reverse function) and prove theorems about lists by induction (such
+-- as the fact that `reverse (reverse xs) = xs` for any list `xs`).
+namespace ex2
+
+variable {α : Type _}
+
+def reverse : List α → List α
+  | [] => []
+  | (head :: tail) => reverse tail ++ [head]
+
+-- Proof of `reverse (reverse xs) = xs` shown in previous exercise.
+
+end ex2
+
+-- What does it mean to "derecurse" a function?  A recursive function can be written iteratively.
+-- What class of recursive functions can be written iteratively? Primitive recursive functions.
+
+-- Exercise 3
+--
+-- Define your own function to carry out course-of-value recursion on the
+-- natural numbers. Similarly, see if you can figure out how to define
+-- `WellFounded.fix` on your own.
+namespace ex3
+
+-- TODO: Answer this.
+
+end ex3
+
+-- Exercise 4
+--
+-- Following the examples in Section Dependent Pattern Matching, define a
+-- function that will append two vectors. This is tricky; you will have to
+-- define an auxiliary function.
+namespace ex4
+
+inductive Vector (α : Type u) : Nat → Type u
+  | nil  : Vector α 0
+  | cons : α → {n : Nat} → Vector α n → Vector α (n + 1)
+
+namespace Vector
+
+def append (v₁ : Vector α m) (v₂ : Vector α n) : Vector α (m + n) := sorry
+
+-- TODO: Answer this.
+
+end Vector
+
+end ex4
+
+-- Exercise 5
+--
+-- Consider the following type of arithmetic expressions. The idea is that
+-- `var n` is a variable, `vₙ`, and `const n` is the constant whose value is
+-- `n`.
+namespace ex5
+
+inductive Expr where
+  | const : Nat → Expr
+  | var : Nat → Expr
+  | plus : Expr → Expr → Expr
+  | times : Expr → Expr → Expr
+  deriving Repr
+
+open Expr
+
+def sampleExpr : Expr :=
+  plus (times (var 0) (const 7)) (times (const 2) (var 1))
+
+-- Here `sampleExpr` represents `(v₀ * 7) + (2 * v₁)`. Write a function that
+-- evaluates such an expression, evaluating each `var n` to `v n`.
+
+def eval (v : Nat → Nat) : Expr → Nat
+  | const n     => sorry
+  | var n       => v n
+  | plus e₁ e₂  => sorry
+  | times e₁ e₂ => sorry
+
+def sampleVal : Nat → Nat
+  | 0 => 5
+  | 1 => 6
+  | _ => 0
+
+-- Try it out. You should get 47 here.
+-- #eval eval sampleVal sampleExpr
+
+-- Implement "constant fusion," a procedure that simplifies subterms like
+-- `5 + 7` to `12`. Using the auxiliary function `simpConst`, define a function
+-- "fuse": to simplify a plus or a times, first simplify the arguments
+-- recursively, and then apply `simpConst` to try to simplify the result.
+
+def simpConst : Expr → Expr
+  | plus (const n₁) (const n₂)  => const (n₁ + n₂)
+  | times (const n₁) (const n₂) => const (n₁ * n₂)
+  | e                           => e
+
+def fuse : Expr → Expr := sorry
+
+theorem simpConst_eq (v : Nat → Nat)
+        : ∀ e : Expr, eval v (simpConst e) = eval v e :=
+  sorry
+
+theorem fuse_eq (v : Nat → Nat)
+        : ∀ e : Expr, eval v (fuse e) = eval v e :=
+  sorry
+
+-- The last two theorems show that the definitions preserve the value.
+
+end ex5