064ecd3e3d
We also define key theorems that will be used to generate the automatically generated a well-founded subterm relation for inductive datatypes. We also prove decidability and wf theorems asap.
36 lines
1.1 KiB
Text
36 lines
1.1 KiB
Text
import logic data.nat
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open eq.ops nat
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inductive tree (A : Type) :=
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leaf : A → tree A,
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node : tree A → tree A → tree A
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namespace tree
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definition height {A : Type} (t : tree A) : nat :=
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rec_on t
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(λ a, zero)
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(λ t₁ t₂ h₁ h₂, succ (max h₁ h₂))
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definition height_lt {A : Type} : tree A → tree A → Prop :=
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inv_image lt (@height A)
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definition height_lt.wf (A : Type) : well_founded (@height_lt A) :=
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inv_image.wf height lt.wf
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theorem height_lt.node_left {A : Type} (t₁ t₂ : tree A) : height_lt t₁ (node t₁ t₂) :=
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le_imp_lt_succ (max.left (height t₁) (height t₂))
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theorem height_lt.node_right {A : Type} (t₁ t₂ : tree A) : height_lt t₂ (node t₁ t₂) :=
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le_imp_lt_succ (max.right (height t₁) (height t₂))
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theorem height_lt.trans {A : Type} : transitive (@height_lt A) :=
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inv_image.trans lt height @lt.trans
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example : height_lt (leaf 2) (node (leaf 1) (leaf 2)) :=
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!height_lt.node_right
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example : height_lt (leaf 2) (node (node (leaf 1) (leaf 2)) (leaf 3)) :=
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height_lt.trans !height_lt.node_right !height_lt.node_left
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end tree
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