test: contradiction

tests/lean/match3.lean

@@ -1,5 +1,3 @@
-
-
 def f (x : Nat) : Nat :=
 match x with
 | 30  => 31
@@ -39,14 +37,14 @@ by {
 }
 
 theorem ex7 (a : Bool) (p q : Prop) (h₁ : a = true → p) (h₂ : a = false → q) : p ∨ q :=
-match h:a with
-| true  => Or.inl $ h₁ h
-| false => Or.inr $ h₂ h
+  match h:a with
+  | true  => Or.inl $ h₁ h
+  | false => Or.inr $ h₂ h
 
 def head {α} (xs : List α) (h : xs = [] → False) : α :=
-match he:xs with
-| []   => False.elim $ h he
-| x::_ => x
+  match he:xs with
+  | []   => by contradiction
+  | x::_ => x
 
 variable {α : Type u} {p : α → Prop}
 

tests/lean/run/casesUsing.lean

@@ -59,12 +59,12 @@ theorem ex3b (n : Nat) : Exists (fun m => n = m + m ∨ n = m + m + 1) := by
 
 def ex4 {α} (xs : List α) (h : xs = [] → False) : α := by
   cases he:xs with
-  | nil      => apply False.elim; exact h he; done
+  | nil      => contradiction
   | cons x _ => exact x
 
 def ex5 {α} (xs : List α) (h : xs = [] → False) : α := by
   cases he:xs using List.casesOn with
-  | nil      => apply False.elim; exact h he; done
+  | nil      => contradiction
   | cons x _ => exact x
 
 theorem ex6 {α} (f : List α → Bool) (h₁ : {xs : List α} → f xs = true → xs = []) (xs : List α) (h₂ : xs ≠ []) : f xs = false :=
@@ -82,9 +82,9 @@ theorem ex8 {α} (f : List α → Bool) (h₁ : {xs : List α} → f xs = true
   | true  => exact False.elim (h₂ (h₁ he))
   | false => rfl
 
-theorem ex9 {α} (xs : List α) (h : xs = [] → False) : Nonempty α := by
+theorem ex9 (xs : List α) (h : xs = [] → False) : Nonempty α := by
   cases xs using List.rec with
-  | nil      => apply False.elim; apply h; rfl
+  | nil      => contradiction
   | cons x _ => apply Nonempty.intro; assumption
 
 theorem modLt (x : Nat) {y : Nat} (h : y > 0) : x % y < y := by

tests/lean/run/contra.lean

@@ -39,3 +39,11 @@ theorem ex13 (h : id False) : α := by
 
 theorem ex14 (h : 100000000 ≤ 20) : α := by
   contradiction
+
+theorem ex15 (x : α) (h : x = x → False) : α := by
+  contradiction
+
+theorem ex16 (xs : List α) (h : xs = [] → False) : Nonempty α := by
+  cases xs using List.rec with
+  | nil      => contradiction
+  | cons x _ => apply Nonempty.intro; assumption