69 lines
2.1 KiB
Text
69 lines
2.1 KiB
Text
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open nat
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inductive tree (A : Type) :=
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leaf : A → tree A,
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node : tree_list A → tree A
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with tree_list :=
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nil : tree_list A,
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cons : tree A → tree_list A → tree_list A
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namespace tree
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open tree_list
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definition size {A : Type} : tree A → nat
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with size_l : tree_list A → nat,
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size (leaf a) := 1,
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size (node l) := size_l l,
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size_l !nil := 0,
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size_l (cons t l) := size t + size_l l
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variables {A : Type}
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theorem size_leaf (a : A) : size (leaf a) = 1 :=
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rfl
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theorem size_node (l : tree_list A) : size (node l) = size_l l :=
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rfl
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theorem size_l_nil : size_l (nil A) = 0 :=
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rfl
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theorem size_l_cons (t : tree A) (l : tree_list A) : size_l (cons t l) = size t + size_l l :=
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rfl
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definition eq_tree {A : Type} : tree A → tree A → Prop
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with eq_tree_list : tree_list A → tree_list A → Prop,
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eq_tree (leaf a₁) (leaf a₂) := a₁ = a₂,
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eq_tree (node l₁) (node l₂) := eq_tree_list l₁ l₂,
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eq_tree _ _ := false,
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eq_tree_list !nil !nil := true,
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eq_tree_list (cons t₁ l₁) (cons t₂ l₂) := eq_tree t₁ t₂ ∧ eq_tree_list l₁ l₂,
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eq_tree_list _ _ := false
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theorem eq_tree_leaf (a₁ a₂ : A) : eq_tree (leaf a₁) (leaf a₂) = (a₁ = a₂) :=
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rfl
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theorem eq_tree_node (l₁ l₂ : tree_list A) : eq_tree (node l₁) (node l₂) = eq_tree_list l₁ l₂ :=
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rfl
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theorem eq_tree_leaf_node (a₁ : A) (l₂ : tree_list A) : eq_tree (leaf a₁) (node l₂) = false :=
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rfl
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theorem eq_tree_node_leaf (l₁ : tree_list A) (a₂ : A) : eq_tree (node l₁) (leaf a₂) = false :=
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rfl
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theorem eq_tree_list_nil : eq_tree_list (nil A) (nil A) = true :=
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rfl
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theorem eq_tree_list_cons (t₁ t₂ : tree A) (l₁ l₂ : tree_list A) :
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eq_tree_list (cons t₁ l₁) (cons t₂ l₂) = (eq_tree t₁ t₂ ∧ eq_tree_list l₁ l₂) :=
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rfl
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theorem eq_tree_list_cons_nil (t : tree A) (l : tree_list A) : eq_tree_list (cons t l) (nil A) = false :=
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rfl
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theorem eq_tree_list_nil_cons (t : tree A) (l : tree_list A) : eq_tree_list (nil A) (cons t l) = false :=
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rfl
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end tree
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