till Basics/Proof by Simplification

basics/main.lean

@@ -152,5 +152,86 @@ end TuplePlayground
 --- Numbers
 namespace NatPlayground
 
+#eval Nat.zero
+#eval Nat.zero.succ
+#eval Nat.zero.succ.succ
+example : Nat.zero.succ.succ = 2 := by rfl
+
 end NatPlayground
 
+open Nat in
+#check succ (succ (succ 0))
+
+def minustwo (n : Nat) : Nat :=
+  match n with
+  | 0 => 0
+  | .succ .zero => 0
+  | .succ (.succ n') => n'
+
+#eval minustwo 4
+
+def even (n : Nat) : Bool :=
+  match n with
+  | .zero => true
+  | .succ .zero => false
+  | .succ (.succ n') => even n'
+
+example : even 4 = true := by
+  rfl
+example : even 51 = false := by
+  rfl
+
+namespace NatPlayground2
+
+def plus (n : Nat) (m : Nat) : Nat :=
+  match n with
+  | .zero => m
+  | .succ n' => .succ (plus n' m)
+
+#eval plus 3 4
+
+def mult (n m : Nat) : Nat :=
+  match n with
+  | .zero => .zero
+  | .succ n' => plus m (mult n' m)
+
+end NatPlayground2
+
+namespace Exercise1
+
+def factorial (n : Nat) : Nat :=
+  match n with
+  | 0 => 1
+  | .succ n' => (n' + 1) * factorial n'
+
+#eval factorial 3
+
+example : factorial 3 = 6 := by
+  rfl
+
+example : factorial 5 = .mul 10 12 := by
+  rfl
+
+end Exercise1
+
+#check LT.lt 1 2
+#check LT Nat
+#check (LT Nat)
+#check Nat.lt 1 2
+
+namespace Exercise1
+
+def ltb (n m : Nat) : Bool :=
+  match n, m with
+  | 0, _ => true
+  | _, 0 => false
+  | .succ n', .succ m' => ltb n' m'
+
+#eval ltb 1 2
+#eval ltb 10242 1055090
+example : (ltb 1 2) = true := by rfl
+
+end Exercise1
+
+--- Proof by Simplification
+